LMFDB - Embedded newform 3234.2.e.d.2155.12

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3234,2,Mod(2155,3234)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3234, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1, 1]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("3234.2155");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3234 = 2 \cdot 3 \cdot 7^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3234.e (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$25.8236200137$
Analytic rank:	$0$
Dimension:	$24$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			2155.12
Character	$\chi$	$=$	3234.2155
Dual form			3234.2.e.d.2155.13

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{2} +1.00000i q^{3} -1.00000 q^{4} +3.51377i q^{5} +1.00000 q^{6} +1.00000i q^{8} -1.00000 q^{9} +O(q^{10})$	\(q-1.00000i q^{2} +1.00000i q^{3} -1.00000 q^{4} +3.51377i q^{5} +1.00000 q^{6} +1.00000i q^{8} -1.00000 q^{9} +3.51377 q^{10} +(-2.26644 - 2.42142i) q^{11} -1.00000i q^{12} +5.40909 q^{13} -3.51377 q^{15} +1.00000 q^{16} +2.00058 q^{17} +1.00000i q^{18} +4.64637 q^{19} -3.51377i q^{20} +(-2.42142 + 2.26644i) q^{22} +7.74100 q^{23} -1.00000 q^{24} -7.34659 q^{25} -5.40909i q^{26} -1.00000i q^{27} -8.62977i q^{29} +3.51377i q^{30} +5.01372i q^{31} -1.00000i q^{32} +(2.42142 - 2.26644i) q^{33} -2.00058i q^{34} +1.00000 q^{36} +5.41510 q^{37} -4.64637i q^{38} +5.40909i q^{39} -3.51377 q^{40} +0.994819 q^{41} +9.82762i q^{43} +(2.26644 + 2.42142i) q^{44} -3.51377i q^{45} -7.74100i q^{46} -10.6987i q^{47} +1.00000i q^{48} +7.34659i q^{50} +2.00058i q^{51} -5.40909 q^{52} +2.47223 q^{53} -1.00000 q^{54} +(8.50830 - 7.96375i) q^{55} +4.64637i q^{57} -8.62977 q^{58} -0.0847471i q^{59} +3.51377 q^{60} -1.37655 q^{61} +5.01372 q^{62} -1.00000 q^{64} +19.0063i q^{65} +(-2.26644 - 2.42142i) q^{66} -11.8945 q^{67} -2.00058 q^{68} +7.74100i q^{69} +12.6946 q^{71} -1.00000i q^{72} -7.10779 q^{73} -5.41510i q^{74} -7.34659i q^{75} -4.64637 q^{76} +5.40909 q^{78} -8.52598i q^{79} +3.51377i q^{80} +1.00000 q^{81} -0.994819i q^{82} +3.88469 q^{83} +7.02957i q^{85} +9.82762 q^{86} +8.62977 q^{87} +(2.42142 - 2.26644i) q^{88} +3.47826i q^{89} -3.51377 q^{90} -7.74100 q^{92} -5.01372 q^{93} -10.6987 q^{94} +16.3263i q^{95} +1.00000 q^{96} +8.90188i q^{97} +(2.26644 + 2.42142i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q - 24 q^{4} + 24 q^{6} - 24 q^{9}+O(q^{10}) $ 24 * q - 24 * q^4 + 24 * q^6 - 24 * q^9	$ 24 q - 24 q^{4} + 24 q^{6} - 24 q^{9} + 24 q^{16} - 16 q^{17} + 32 q^{19} - 8 q^{22} - 24 q^{24} - 8 q^{25} + 8 q^{33} + 24 q^{36} + 16 q^{37} + 16 q^{41} - 24 q^{54} - 16 q^{55} + 16 q^{62} - 24 q^{64} - 64 q^{67} + 16 q^{68} + 64 q^{71} - 32 q^{76} + 24 q^{81} + 16 q^{83} + 8 q^{88} - 16 q^{93} - 64 q^{94} + 24 q^{96}+O(q^{100}) $ 24 * q - 24 * q^4 + 24 * q^6 - 24 * q^9 + 24 * q^16 - 16 * q^17 + 32 * q^19 - 8 * q^22 - 24 * q^24 - 8 * q^25 + 8 * q^33 + 24 * q^36 + 16 * q^37 + 16 * q^41 - 24 * q^54 - 16 * q^55 + 16 * q^62 - 24 * q^64 - 64 * q^67 + 16 * q^68 + 64 * q^71 - 32 * q^76 + 24 * q^81 + 16 * q^83 + 8 * q^88 - 16 * q^93 - 64 * q^94 + 24 * q^96

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3234\mathbb{Z}\right)^\times$.

$n$	$199$	$1079$	$2059$
$\chi(n)$	$-1$	$1$	$-1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		−	1.00000i		−	0.707107i
$2$		−	1.00000i		−	0.707107i
$3$			1.00000i			0.577350i
$3$			1.00000i			0.577350i
$4$	−1.00000			−0.500000
$5$			3.51377i			1.57141i	0.618604	+	0.785703i	$0.287699\pi$
$5$			3.51377i			1.57141i	−0.618604	+	0.785703i	$0.712301\pi$
$6$	1.00000			0.408248
$7$		0			0
$7$		0			0
$8$			1.00000i			0.353553i
$9$	−1.00000			−0.333333
$10$	3.51377			1.11115
$11$	−2.26644	−	2.42142i	−0.683357	−	0.730084i
$11$	−2.26644	−	2.42142i	−0.683357	−	0.730084i
$12$		−	1.00000i		−	0.288675i
$13$	5.40909			1.50021			0.750105	−	0.661318i	$-0.230003\pi$
$13$	5.40909			1.50021			0.750105	+	0.661318i	$0.230003\pi$
$14$		0			0
$15$	−3.51377			−0.907252
$16$	1.00000			0.250000
$17$	2.00058			0.485211			0.242606	−	0.970125i	$-0.421998\pi$
$17$	2.00058			0.485211			0.242606	+	0.970125i	$0.421998\pi$
$18$			1.00000i			0.235702i
$19$	4.64637			1.06595			0.532976	−	0.846131i	$-0.321074\pi$
$19$	4.64637			1.06595			0.532976	+	0.846131i	$0.321074\pi$
$20$		−	3.51377i		−	0.785703i
$21$		0			0
$22$	−2.42142	+	2.26644i	−0.516247	+	0.483207i
$23$	7.74100			1.61411			0.807055	−	0.590476i	$-0.201060\pi$
$23$	7.74100			1.61411			0.807055	+	0.590476i	$0.201060\pi$
$24$	−1.00000			−0.204124
$25$	−7.34659			−1.46932
$26$		−	5.40909i		−	1.06081i
$27$		−	1.00000i		−	0.192450i
$28$		0			0
$29$		−	8.62977i		−	1.60251i	−0.598324	−	0.801254i	$-0.704167\pi$
$29$		−	8.62977i		−	1.60251i	0.598324	−	0.801254i	$-0.295833\pi$
$30$			3.51377i			0.641524i
$31$			5.01372i			0.900492i	0.892905	+	0.450246i	$0.148664\pi$
$31$			5.01372i			0.900492i	−0.892905	+	0.450246i	$0.851336\pi$
$32$		−	1.00000i		−	0.176777i
$33$	2.42142	−	2.26644i	0.421514	−	0.394537i
$34$		−	2.00058i		−	0.343096i
$35$		0			0
$36$	1.00000			0.166667
$37$	5.41510			0.890238			0.445119	−	0.895472i	$-0.353161\pi$
$37$	5.41510			0.890238			0.445119	+	0.895472i	$0.353161\pi$
$38$		−	4.64637i		−	0.753741i
$39$			5.40909i			0.866147i
$40$	−3.51377			−0.555576
$41$	0.994819			0.155365			0.0776823	−	0.996978i	$-0.475248\pi$
$41$	0.994819			0.155365			0.0776823	+	0.996978i	$0.475248\pi$
$42$		0			0
$43$			9.82762i			1.49870i	0.662175	+	0.749349i	$0.269633\pi$
$43$			9.82762i			1.49870i	−0.662175	+	0.749349i	$0.730367\pi$
$44$	2.26644	+	2.42142i	0.341679	+	0.365042i
$45$		−	3.51377i		−	0.523802i
$46$		−	7.74100i		−	1.14135i
$47$		−	10.6987i		−	1.56057i	−0.625422	−	0.780286i	$-0.715073\pi$
$47$		−	10.6987i		−	1.56057i	0.625422	−	0.780286i	$-0.284927\pi$
$48$			1.00000i			0.144338i
$49$		0			0
$50$			7.34659i			1.03896i
$51$			2.00058i			0.280137i
$52$	−5.40909			−0.750105
$53$	2.47223			0.339588			0.169794	−	0.985480i	$-0.445690\pi$
$53$	2.47223			0.339588			0.169794	+	0.985480i	$0.445690\pi$
$54$	−1.00000			−0.136083
$55$	8.50830	−	7.96375i	1.14726	−	1.07383i
$56$		0			0
$57$			4.64637i			0.615427i
$58$	−8.62977			−1.13314
$59$		−	0.0847471i		−	0.0110331i	−0.999985	−	0.00551657i	$-0.998244\pi$
$59$		−	0.0847471i		−	0.0110331i	0.999985	−	0.00551657i	$-0.00175599\pi$
$60$	3.51377			0.453626
$61$	−1.37655			−0.176249			−0.0881243	−	0.996109i	$-0.528087\pi$
$61$	−1.37655			−0.176249			−0.0881243	+	0.996109i	$0.528087\pi$
$62$	5.01372			0.636744
$63$		0			0
$64$	−1.00000			−0.125000
$65$			19.0063i			2.35744i
$66$	−2.26644	−	2.42142i	−0.278979	−	0.298056i
$67$	−11.8945			−1.45315			−0.726574	−	0.687088i	$-0.758889\pi$
$67$	−11.8945			−1.45315			−0.726574	+	0.687088i	$0.758889\pi$
$68$	−2.00058			−0.242606
$69$			7.74100i			0.931907i
$70$		0			0
$71$	12.6946			1.50657			0.753287	−	0.657692i	$-0.228467\pi$
$71$	12.6946			1.50657			0.753287	+	0.657692i	$0.228467\pi$
$72$		−	1.00000i		−	0.117851i
$73$	−7.10779			−0.831903			−0.415952	−	0.909387i	$-0.636551\pi$
$73$	−7.10779			−0.831903			−0.415952	+	0.909387i	$0.636551\pi$
$74$		−	5.41510i		−	0.629493i
$75$		−	7.34659i		−	0.848311i
$76$	−4.64637			−0.532976
$77$		0			0
$78$	5.40909			0.612458
$79$		−	8.52598i		−	0.959248i	−0.877474	−	0.479624i	$-0.840773\pi$
$79$		−	8.52598i		−	0.959248i	0.877474	−	0.479624i	$-0.159227\pi$
$80$			3.51377i			0.392852i
$81$	1.00000			0.111111
$82$		−	0.994819i		−	0.109859i
$83$	3.88469			0.426400			0.213200	−	0.977009i	$-0.431611\pi$
$83$	3.88469			0.426400			0.213200	+	0.977009i	$0.431611\pi$
$84$		0			0
$85$			7.02957i			0.762464i
$86$	9.82762			1.05974
$87$	8.62977			0.925208
$88$	2.42142	−	2.26644i	0.258124	−	0.241603i
$89$			3.47826i			0.368695i	0.982861	+	0.184347i	$0.0590172\pi$
$89$			3.47826i			0.368695i	−0.982861	+	0.184347i	$0.940983\pi$
$90$	−3.51377			−0.370384
$91$		0			0
$92$	−7.74100			−0.807055
$93$	−5.01372			−0.519899
$94$	−10.6987			−1.10349
$95$			16.3263i			1.67504i
$96$	1.00000			0.102062
$97$			8.90188i			0.903849i	0.892056	+	0.451924i	$0.149262\pi$
$97$			8.90188i			0.903849i	−0.892056	+	0.451924i	$0.850738\pi$
$98$		0			0
$99$	2.26644	+	2.42142i	0.227786	+	0.243361i
$100$	7.34659			0.734659
$101$	11.2857			1.12297			0.561483	−	0.827488i	$-0.310231\pi$
$101$	11.2857			1.12297			0.561483	+	0.827488i	$0.310231\pi$
$102$	2.00058			0.198087
$103$			0.391882i			0.0386133i	0.999814	+	0.0193066i	$0.00614588\pi$
$103$			0.391882i			0.0386133i	−0.999814	+	0.0193066i	$0.993854\pi$
$104$			5.40909i			0.530405i
$105$		0			0
$106$		−	2.47223i		−	0.240125i
$107$		−	7.74721i		−	0.748951i	−0.927237	−	0.374476i	$-0.877823\pi$
$107$		−	7.74721i		−	0.748951i	0.927237	−	0.374476i	$-0.122177\pi$
$108$			1.00000i			0.0962250i
$109$			0.244483i			0.0234172i	0.999931	+	0.0117086i	$0.00372705\pi$
$109$			0.244483i			0.0234172i	−0.999931	+	0.0117086i	$0.996273\pi$
$110$	−7.96375	−	8.50830i	−0.759314	−	0.811235i
$111$			5.41510i			0.513979i
$112$		0			0
$113$	5.79811			0.545440			0.272720	−	0.962093i	$-0.412077\pi$
$113$	5.79811			0.545440			0.272720	+	0.962093i	$0.412077\pi$
$114$	4.64637			0.435173
$115$			27.2001i			2.53642i
$116$			8.62977i			0.801254i
$117$	−5.40909			−0.500070
$118$	−0.0847471			−0.00780161
$119$		0			0
$120$		−	3.51377i		−	0.320762i
$121$	−0.726502	+	10.9760i	−0.0660456	+	0.997817i
$122$			1.37655i			0.124627i
$123$			0.994819i			0.0896998i
$124$		−	5.01372i		−	0.450246i
$125$		−	8.24539i		−	0.737490i
$126$		0			0
$127$			16.9452i			1.50364i	0.659366	+	0.751822i	$0.270825\pi$
$127$			16.9452i			1.50364i	−0.659366	+	0.751822i	$0.729175\pi$
$128$			1.00000i			0.0883883i
$129$	−9.82762			−0.865274
$130$	19.0063			1.66696
$131$	17.1337			1.49697			0.748487	−	0.663149i	$-0.230780\pi$
$131$	17.1337			1.49697			0.748487	+	0.663149i	$0.230780\pi$
$132$	−2.42142	+	2.26644i	−0.210757	+	0.197268i
$133$		0			0
$134$			11.8945i			1.02753i
$135$	3.51377			0.302417
$136$			2.00058i			0.171548i
$137$	−17.1721			−1.46711			−0.733554	−	0.679632i	$-0.762140\pi$
$137$	−17.1721			−1.46711			−0.733554	+	0.679632i	$0.762140\pi$
$138$	7.74100			0.658958
$139$	−11.2370			−0.953107			−0.476554	−	0.879145i	$-0.658114\pi$
$139$	−11.2370			−0.953107			−0.476554	+	0.879145i	$0.658114\pi$
$140$		0			0
$141$	10.6987			0.900997
$142$		−	12.6946i		−	1.06531i
$143$	−12.2594	−	13.0976i	−1.02518	−	1.09528i
$144$	−1.00000			−0.0833333
$145$	30.3230			2.51819
$146$			7.10779i			0.588245i
$147$		0			0
$148$	−5.41510			−0.445119
$149$			21.5631i			1.76652i	0.468883	+	0.883260i	$0.344657\pi$
$149$			21.5631i			1.76652i	−0.468883	+	0.883260i	$0.655343\pi$
$150$	−7.34659			−0.599847
$151$		−	10.6870i		−	0.869696i	−0.900504	−	0.434848i	$-0.856802\pi$
$151$		−	10.6870i		−	0.869696i	0.900504	−	0.434848i	$-0.143198\pi$
$152$			4.64637i			0.376871i
$153$	−2.00058			−0.161737
$154$		0			0
$155$	−17.6171			−1.41504
$156$		−	5.40909i		−	0.433074i
$157$			20.9034i			1.66827i	0.551558	+	0.834137i	$0.314034\pi$
$157$			20.9034i			1.66827i	−0.551558	+	0.834137i	$0.685966\pi$
$158$	−8.52598			−0.678291
$159$			2.47223i			0.196061i
$160$	3.51377			0.277788
$161$		0			0
$162$		−	1.00000i		−	0.0785674i
$163$	−15.8867			−1.24434			−0.622171	−	0.782881i	$-0.713749\pi$
$163$	−15.8867			−1.24434			−0.622171	+	0.782881i	$0.713749\pi$
$164$	−0.994819			−0.0776823
$165$	7.96375	+	8.50830i	0.619977	+	0.662370i
$166$		−	3.88469i		−	0.301510i
$167$	−4.57564			−0.354074			−0.177037	−	0.984204i	$-0.556651\pi$
$167$	−4.57564			−0.354074			−0.177037	+	0.984204i	$0.556651\pi$
$168$		0			0
$169$	16.2582			1.25063
$170$	7.02957			0.539143
$171$	−4.64637			−0.355317
$172$		−	9.82762i		−	0.749349i
$173$	−6.74941			−0.513148			−0.256574	−	0.966525i	$-0.582594\pi$
$173$	−6.74941			−0.513148			−0.256574	+	0.966525i	$0.582594\pi$
$174$		−	8.62977i		−	0.654221i
$175$		0			0
$176$	−2.26644	−	2.42142i	−0.170839	−	0.182521i
$177$	0.0847471			0.00636999
$178$	3.47826			0.260707
$179$	2.52905			0.189030			0.0945152	−	0.995523i	$-0.469870\pi$
$179$	2.52905			0.189030			0.0945152	+	0.995523i	$0.469870\pi$
$180$			3.51377i			0.261901i
$181$			12.9953i			0.965931i	0.875639	+	0.482966i	$0.160440\pi$
$181$			12.9953i			0.965931i	−0.875639	+	0.482966i	$0.839560\pi$
$182$		0			0
$183$		−	1.37655i		−	0.101757i
$184$			7.74100i			0.570674i
$185$			19.0274i			1.39893i
$186$			5.01372i			0.367624i
$187$	−4.53419	−	4.84423i	−0.331572	−	0.354245i
$188$			10.6987i			0.780286i
$189$		0			0
$190$	16.3263			1.18443
$191$	16.7559			1.21242			0.606208	−	0.795306i	$-0.292690\pi$
$191$	16.7559			1.21242			0.606208	+	0.795306i	$0.292690\pi$
$192$		−	1.00000i		−	0.0721688i
$193$			16.2642i			1.17072i	0.810773	+	0.585361i	$0.199047\pi$
$193$			16.2642i			1.17072i	−0.810773	+	0.585361i	$0.800953\pi$
$194$	8.90188			0.639118
$195$	−19.0063			−1.36107
$196$		0			0
$197$			23.8308i			1.69788i	0.528493	+	0.848938i	$0.322757\pi$
$197$			23.8308i			1.69788i	−0.528493	+	0.848938i	$0.677243\pi$
$198$	2.42142	−	2.26644i	0.172082	−	0.161069i
$199$			0.608638i			0.0431452i	0.999767	+	0.0215726i	$0.00686731\pi$
$199$			0.608638i			0.0431452i	−0.999767	+	0.0215726i	$0.993133\pi$
$200$		−	7.34659i		−	0.519482i
$201$		−	11.8945i		−	0.838976i
$202$		−	11.2857i		−	0.794057i
$203$		0			0
$204$		−	2.00058i		−	0.140068i
$205$			3.49557i			0.244141i
$206$	0.391882			0.0273037
$207$	−7.74100			−0.538037
$208$	5.40909			0.375053
$209$	−10.5307	−	11.2508i	−0.728425	−	0.778234i
$210$		0			0
$211$			0.499969i			0.0344193i	0.999852	+	0.0172096i	$0.00547827\pi$
$211$			0.499969i			0.0344193i	−0.999852	+	0.0172096i	$0.994522\pi$
$212$	−2.47223			−0.169794
$213$			12.6946i			0.869821i
$214$	−7.74721			−0.529588
$215$	−34.5320			−2.35506
$216$	1.00000			0.0680414
$217$		0			0
$218$	0.244483			0.0165585
$219$		−	7.10779i		−	0.480300i
$220$	−8.50830	+	7.96375i	−0.573629	+	0.536916i
$221$	10.8213			0.727919
$222$	5.41510			0.363438
$223$		−	19.0402i		−	1.27503i	−0.770440	−	0.637513i	$-0.779963\pi$
$223$		−	19.0402i		−	1.27503i	0.770440	−	0.637513i	$-0.220037\pi$
$224$		0			0
$225$	7.34659			0.489773
$226$		−	5.79811i		−	0.385685i
$227$	−18.3944			−1.22088			−0.610439	−	0.792063i	$-0.709007\pi$
$227$	−18.3944			−1.22088			−0.610439	+	0.792063i	$0.709007\pi$
$228$		−	4.64637i		−	0.307714i
$229$		−	9.03881i		−	0.597302i	−0.954362	−	0.298651i	$-0.903463\pi$
$229$		−	9.03881i		−	0.597302i	0.954362	−	0.298651i	$-0.0965366\pi$
$230$	27.2001			1.79352
$231$		0			0
$232$	8.62977			0.566572
$233$			24.8457i			1.62770i	0.581076	+	0.813849i	$0.302632\pi$
$233$			24.8457i			1.62770i	−0.581076	+	0.813849i	$0.697368\pi$
$234$			5.40909i			0.353603i
$235$	37.5930			2.45229
$236$			0.0847471i			0.00551657i
$237$	8.52598			0.553822
$238$		0			0
$239$		−	23.6327i		−	1.52867i	−0.644818	−	0.764336i	$-0.723067\pi$
$239$		−	23.6327i		−	1.52867i	0.644818	−	0.764336i	$-0.276933\pi$
$240$	−3.51377			−0.226813
$241$	24.7570			1.59474			0.797370	−	0.603491i	$-0.206224\pi$
$241$	24.7570			1.59474			0.797370	+	0.603491i	$0.206224\pi$
$242$	10.9760	+	0.726502i	0.705563	+	0.0467013i
$243$			1.00000i			0.0641500i
$244$	1.37655			0.0881243
$245$		0			0
$246$	0.994819			0.0634274
$247$	25.1326			1.59915
$248$	−5.01372			−0.318372
$249$			3.88469i			0.246182i
$250$	−8.24539			−0.521484
$251$		−	14.3590i		−	0.906335i	−0.891425	−	0.453168i	$-0.850294\pi$
$251$		−	14.3590i		−	0.906335i	0.891425	−	0.453168i	$-0.149706\pi$
$252$		0			0
$253$	−17.5445	−	18.7442i	−1.10301	−	1.17844i
$254$	16.9452			1.06324
$255$	−7.02957			−0.440209
$256$	1.00000			0.0625000
$257$			9.31913i			0.581311i	0.956828	+	0.290656i	$0.0938734\pi$
$257$			9.31913i			0.581311i	−0.956828	+	0.290656i	$0.906127\pi$
$258$			9.82762i			0.611841i
$259$		0			0
$260$		−	19.0063i		−	1.17872i
$261$			8.62977i			0.534169i
$262$		−	17.1337i		−	1.05852i
$263$		−	19.8128i		−	1.22171i	−0.791742	−	0.610856i	$-0.790825\pi$
$263$		−	19.8128i		−	1.22171i	0.791742	−	0.610856i	$-0.209175\pi$
$264$	2.26644	+	2.42142i	0.139490	+	0.149028i
$265$			8.68687i			0.533630i
$266$		0			0
$267$	−3.47826			−0.212866
$268$	11.8945			0.726574
$269$		−	18.1073i		−	1.10402i	−0.833837	−	0.552010i	$-0.813861\pi$
$269$		−	18.1073i		−	1.10402i	0.833837	−	0.552010i	$-0.186139\pi$
$270$		−	3.51377i		−	0.213841i
$271$	−1.57400			−0.0956136			−0.0478068	−	0.998857i	$-0.515223\pi$
$271$	−1.57400			−0.0956136			−0.0478068	+	0.998857i	$0.515223\pi$
$272$	2.00058			0.121303
$273$		0			0
$274$			17.1721i			1.03740i
$275$	16.6506	+	17.7891i	1.00407	+	1.07273i
$276$		−	7.74100i		−	0.465954i
$277$		−	0.0551403i		−	0.00331306i	−0.999999	−	0.00165653i	$-0.999473\pi$
$277$		−	0.0551403i		−	0.00331306i	0.999999	−	0.00165653i	$-0.000527290\pi$
$278$			11.2370i			0.673949i
$279$		−	5.01372i		−	0.300164i
$280$		0			0
$281$		−	21.1973i		−	1.26452i	−0.774755	−	0.632261i	$-0.782127\pi$
$281$		−	21.1973i		−	1.26452i	0.774755	−	0.632261i	$-0.217873\pi$
$282$		−	10.6987i		−	0.637101i
$283$	28.2314			1.67818			0.839090	−	0.543993i	$-0.183088\pi$
$283$	28.2314			1.67818			0.839090	+	0.543993i	$0.183088\pi$
$284$	−12.6946			−0.753287
$285$	−16.3263			−0.967086
$286$	−13.0976	+	12.2594i	−0.774480	+	0.724912i
$287$		0			0
$288$			1.00000i			0.0589256i
$289$	−12.9977			−0.764570
$290$		−	30.3230i		−	1.78063i
$291$	−8.90188			−0.521837
$292$	7.10779			0.415952
$293$	21.8815			1.27833			0.639166	−	0.769069i	$-0.279280\pi$
$293$	21.8815			1.27833			0.639166	+	0.769069i	$0.279280\pi$
$294$		0			0
$295$	0.297782			0.0173375
$296$			5.41510i			0.314746i
$297$	−2.42142	+	2.26644i	−0.140505	+	0.131512i
$298$	21.5631			1.24912
$299$	41.8718			2.42151
$300$			7.34659i			0.424156i
$301$		0			0
$302$	−10.6870			−0.614968
$303$			11.2857i			0.648345i
$304$	4.64637			0.266488
$305$		−	4.83687i		−	0.276958i
$306$			2.00058i			0.114365i
$307$	−18.8721			−1.07709			−0.538543	−	0.842598i	$-0.681025\pi$
$307$	−18.8721			−1.07709			−0.538543	+	0.842598i	$0.681025\pi$
$308$		0			0
$309$	−0.391882			−0.0222934
$310$			17.6171i			1.00058i
$311$			4.23812i			0.240322i	0.992754	+	0.120161i	$0.0383410\pi$
$311$			4.23812i			0.240322i	−0.992754	+	0.120161i	$0.961659\pi$
$312$	−5.40909			−0.306229
$313$		−	6.89858i		−	0.389931i	−0.980810	−	0.194965i	$-0.937541\pi$
$313$		−	6.89858i		−	0.389931i	0.980810	−	0.194965i	$-0.0624595\pi$
$314$	20.9034			1.17965
$315$		0			0
$316$			8.52598i			0.479624i
$317$	−15.3735			−0.863460			−0.431730	−	0.902003i	$-0.642097\pi$
$317$	−15.3735			−0.863460			−0.431730	+	0.902003i	$0.642097\pi$
$318$	2.47223			0.138636
$319$	−20.8963	+	19.5589i	−1.16997	+	1.09509i
$320$		−	3.51377i		−	0.196426i
$321$	7.74721			0.432407
$322$		0			0
$323$	9.29542			0.517211
$324$	−1.00000			−0.0555556
$325$	−39.7383			−2.20429
$326$			15.8867i			0.879883i
$327$	−0.244483			−0.0135199
$328$			0.994819i			0.0549297i
$329$		0			0
$330$	8.50830	−	7.96375i	0.468366	−	0.438390i
$331$	7.01665			0.385670			0.192835	−	0.981231i	$-0.438232\pi$
$331$	7.01665			0.385670			0.192835	+	0.981231i	$0.438232\pi$
$332$	−3.88469			−0.213200
$333$	−5.41510			−0.296746
$334$			4.57564i			0.250368i
$335$		−	41.7947i		−	2.28349i
$336$		0			0
$337$		−	13.2060i		−	0.719375i	−0.933073	−	0.359688i	$-0.882883\pi$
$337$		−	13.2060i		−	0.719375i	0.933073	−	0.359688i	$-0.117117\pi$
$338$		−	16.2582i		−	0.884330i
$339$			5.79811i			0.314910i
$340$		−	7.02957i		−	0.381232i
$341$	12.1403	−	11.3633i	0.657435	−	0.615357i
$342$			4.64637i			0.251247i
$343$		0			0
$344$	−9.82762			−0.529870
$345$	−27.2001			−1.46441
$346$			6.74941i			0.362850i
$347$		−	16.2961i		−	0.874819i	−0.899262	−	0.437410i	$-0.855896\pi$
$347$		−	16.2961i		−	0.874819i	0.899262	−	0.437410i	$-0.144104\pi$
$348$	−8.62977			−0.462604
$349$	−12.8652			−0.688656			−0.344328	−	0.938849i	$-0.611893\pi$
$349$	−12.8652			−0.688656			−0.344328	+	0.938849i	$0.611893\pi$
$350$		0			0
$351$		−	5.40909i		−	0.288716i
$352$	−2.42142	+	2.26644i	−0.129062	+	0.120802i
$353$		−	19.6689i		−	1.04687i	−0.852066	−	0.523434i	$-0.824651\pi$
$353$		−	19.6689i		−	1.04687i	0.852066	−	0.523434i	$-0.175349\pi$
$354$		−	0.0847471i		−	0.00450426i
$355$			44.6060i			2.36744i
$356$		−	3.47826i		−	0.184347i
$357$		0			0
$358$		−	2.52905i		−	0.133665i
$359$		−	3.46810i		−	0.183039i	−0.995803	−	0.0915197i	$-0.970828\pi$
$359$		−	3.46810i		−	0.183039i	0.995803	−	0.0915197i	$-0.0291725\pi$
$360$	3.51377			0.185192
$361$	2.58878			0.136252
$362$	12.9953			0.683016
$363$	−10.9760	−	0.726502i	−0.576090	−	0.0381315i
$364$		0			0
$365$		−	24.9751i		−	1.30726i
$366$	−1.37655			−0.0719532
$367$		−	11.5696i		−	0.603930i	−0.953319	−	0.301965i	$-0.902357\pi$
$367$		−	11.5696i		−	0.603930i	0.953319	−	0.301965i	$-0.0976426\pi$
$368$	7.74100			0.403528
$369$	−0.994819			−0.0517882
$370$	19.0274			0.989189
$371$		0			0
$372$	5.01372			0.259950
$373$		−	29.8566i		−	1.54592i	−0.634457	−	0.772958i	$-0.718776\pi$
$373$		−	29.8566i		−	1.54592i	0.634457	−	0.772958i	$-0.281224\pi$
$374$	−4.84423	+	4.53419i	−0.250489	+	0.234457i
$375$	8.24539			0.425790
$376$	10.6987			0.551746
$377$		−	46.6792i		−	2.40410i
$378$		0			0
$379$	2.32917			0.119642			0.0598208	−	0.998209i	$-0.480947\pi$
$379$	2.32917			0.119642			0.0598208	+	0.998209i	$0.480947\pi$
$380$		−	16.3263i		−	0.837521i
$381$	−16.9452			−0.868130
$382$		−	16.7559i		−	0.857308i
$383$			4.35234i			0.222394i	0.993798	+	0.111197i	$0.0354685\pi$
$383$			4.35234i			0.222394i	−0.993798	+	0.111197i	$0.964532\pi$
$384$	−1.00000			−0.0510310
$385$		0			0
$386$	16.2642			0.827826
$387$		−	9.82762i		−	0.499566i
$388$		−	8.90188i		−	0.451924i
$389$	−37.6920			−1.91106			−0.955531	−	0.294891i	$-0.904717\pi$
$389$	−37.6920			−1.91106			−0.955531	+	0.294891i	$0.904717\pi$
$390$			19.0063i			0.962421i
$391$	15.4865			0.783184
$392$		0			0
$393$			17.1337i			0.864279i
$394$	23.8308			1.20058
$395$	29.9584			1.50737
$396$	−2.26644	−	2.42142i	−0.113893	−	0.121681i
$397$			20.6744i			1.03762i	0.854890	+	0.518809i	$0.173625\pi$
$397$			20.6744i			1.03762i	−0.854890	+	0.518809i	$0.826375\pi$
$398$	0.608638			0.0305083
$399$		0			0
$400$	−7.34659			−0.367330
$401$	−20.5716			−1.02730			−0.513649	−	0.858001i	$-0.671707\pi$
$401$	−20.5716			−1.02730			−0.513649	+	0.858001i	$0.671707\pi$
$402$	−11.8945			−0.593246
$403$			27.1197i			1.35093i
$404$	−11.2857			−0.561483
$405$			3.51377i			0.174601i
$406$		0			0
$407$	−12.2730	−	13.1122i	−0.608350	−	0.649948i
$408$	−2.00058			−0.0990433
$409$	0.669187			0.0330892			0.0165446	−	0.999863i	$-0.494733\pi$
$409$	0.669187			0.0330892			0.0165446	+	0.999863i	$0.494733\pi$
$410$	3.49557			0.172634
$411$		−	17.1721i		−	0.847035i
$412$		−	0.391882i		−	0.0193066i
$413$		0			0
$414$			7.74100i			0.380450i
$415$			13.6499i			0.670047i
$416$		−	5.40909i		−	0.265202i
$417$		−	11.2370i		−	0.550277i
$418$	−11.2508	+	10.5307i	−0.550295	+	0.515075i
$419$		−	18.5889i		−	0.908129i	−0.890969	−	0.454065i	$-0.849974\pi$
$419$		−	18.5889i		−	0.908129i	0.890969	−	0.454065i	$-0.150026\pi$
$420$		0			0
$421$	−13.6636			−0.665922			−0.332961	−	0.942941i	$-0.608048\pi$
$421$	−13.6636			−0.665922			−0.332961	+	0.942941i	$0.608048\pi$
$422$	0.499969			0.0243381
$423$			10.6987i			0.520191i
$424$			2.47223i			0.120062i
$425$	−14.6974			−0.712929
$426$	12.6946			0.615057
$427$		0			0
$428$			7.74721i			0.374476i
$429$	13.0976	−	12.2594i	0.632360	−	0.591888i
$430$			34.5320i			1.66528i
$431$		−	5.99083i		−	0.288568i	−0.989536	−	0.144284i	$-0.953912\pi$
$431$		−	5.99083i		−	0.288568i	0.989536	−	0.144284i	$-0.0460879\pi$
$432$		−	1.00000i		−	0.0481125i
$433$			17.7633i			0.853649i	0.904335	+	0.426824i	$0.140368\pi$
$433$			17.7633i			0.853649i	−0.904335	+	0.426824i	$0.859632\pi$
$434$		0			0
$435$			30.3230i			1.45388i
$436$		−	0.244483i		−	0.0117086i
$437$	35.9676			1.72056
$438$	−7.10779			−0.339623
$439$	−9.34241			−0.445889			−0.222944	−	0.974831i	$-0.571567\pi$
$439$	−9.34241			−0.445889			−0.222944	+	0.974831i	$0.571567\pi$
$440$	7.96375	+	8.50830i	0.379657	+	0.405617i
$441$		0			0
$442$		−	10.8213i		−	0.514716i
$443$	17.1346			0.814091			0.407046	−	0.913408i	$-0.366559\pi$
$443$	17.1346			0.814091			0.407046	+	0.913408i	$0.366559\pi$
$444$		−	5.41510i		−	0.256989i
$445$	−12.2218			−0.579370
$446$	−19.0402			−0.901579
$447$	−21.5631			−1.01990
$448$		0			0
$449$	14.7314			0.695217			0.347608	−	0.937640i	$-0.386994\pi$
$449$	14.7314			0.695217			0.347608	+	0.937640i	$0.386994\pi$
$450$		−	7.34659i		−	0.346322i
$451$	−2.25470	−	2.40887i	−0.106170	−	0.113429i
$452$	−5.79811			−0.272720
$453$	10.6870			0.502119
$454$			18.3944i			0.863291i
$455$		0			0
$456$	−4.64637			−0.217586
$457$			24.8203i			1.16105i	0.814244	+	0.580523i	$0.197152\pi$
$457$			24.8203i			1.16105i	−0.814244	+	0.580523i	$0.802848\pi$
$458$	−9.03881			−0.422356
$459$		−	2.00058i		−	0.0933789i
$460$		−	27.2001i		−	1.26821i
$461$	8.76302			0.408134			0.204067	−	0.978957i	$-0.434584\pi$
$461$	8.76302			0.408134			0.204067	+	0.978957i	$0.434584\pi$
$462$		0			0
$463$	24.6334			1.14481			0.572407	−	0.819970i	$-0.306010\pi$
$463$	24.6334			1.14481			0.572407	+	0.819970i	$0.306010\pi$
$464$		−	8.62977i		−	0.400627i
$465$		−	17.6171i		−	0.816973i
$466$	24.8457			1.15096
$467$			16.5454i			0.765629i	0.923825	+	0.382815i	$0.125045\pi$
$467$			16.5454i			0.765629i	−0.923825	+	0.382815i	$0.874955\pi$
$468$	5.40909			0.250035
$469$		0			0
$470$		−	37.5930i		−	1.73403i
$471$	−20.9034			−0.963178
$472$	0.0847471			0.00390080
$473$	23.7967	−	22.2737i	1.09418	−	1.02415i
$474$		−	8.52598i		−	0.391611i
$475$	−34.1350			−1.56622
$476$		0			0
$477$	−2.47223			−0.113196
$478$	−23.6327			−1.08093
$479$	25.2557			1.15396			0.576981	−	0.816758i	$-0.304231\pi$
$479$	25.2557			1.15396			0.576981	+	0.816758i	$0.304231\pi$
$480$			3.51377i			0.160381i
$481$	29.2908			1.33554
$482$		−	24.7570i		−	1.12765i
$483$		0			0
$484$	0.726502	−	10.9760i	0.0330228	−	0.498908i
$485$	−31.2792			−1.42031
$486$	1.00000			0.0453609
$487$	−33.6204			−1.52349			−0.761743	−	0.647879i	$-0.775656\pi$
$487$	−33.6204			−1.52349			−0.761743	+	0.647879i	$0.775656\pi$
$488$		−	1.37655i		−	0.0623133i
$489$		−	15.8867i		−	0.718421i
$490$		0			0
$491$			5.83403i			0.263286i	0.991297	+	0.131643i	$0.0420253\pi$
$491$			5.83403i			0.263286i	−0.991297	+	0.131643i	$0.957975\pi$
$492$		−	0.994819i		−	0.0448499i
$493$		−	17.2645i		−	0.777555i
$494$		−	25.1326i		−	1.13077i
$495$	−8.50830	+	7.96375i	−0.382420	+	0.357944i
$496$			5.01372i			0.225123i
$497$		0			0
$498$	3.88469			0.174077
$499$	−1.83045			−0.0819422			−0.0409711	−	0.999160i	$-0.513045\pi$
$499$	−1.83045			−0.0819422			−0.0409711	+	0.999160i	$0.513045\pi$
$500$			8.24539i			0.368745i
$501$		−	4.57564i		−	0.204425i
$502$	−14.3590			−0.640876
$503$	−1.65521			−0.0738020			−0.0369010	−	0.999319i	$-0.511749\pi$
$503$	−1.65521			−0.0738020			−0.0369010	+	0.999319i	$0.511749\pi$
$504$		0			0
$505$			39.6553i			1.76464i
$506$	−18.7442	+	17.5445i	−0.833280	+	0.779949i
$507$			16.2582i			0.722053i
$508$		−	16.9452i		−	0.751822i
$509$		−	2.23661i		−	0.0991358i	−0.998771	−	0.0495679i	$-0.984216\pi$
$509$		−	2.23661i		−	0.0991358i	0.998771	−	0.0495679i	$-0.0157844\pi$
$510$			7.02957i			0.311275i
$511$		0			0
$512$		−	1.00000i		−	0.0441942i
$513$		−	4.64637i		−	0.205142i
$514$	9.31913			0.411049
$515$	−1.37698			−0.0606772
$516$	9.82762			0.432637
$517$	−25.9061	+	24.2481i	−1.13935	+	1.06643i
$518$		0			0
$519$		−	6.74941i		−	0.296266i
$520$	−19.0063			−0.833481
$521$			21.4204i			0.938444i	0.883080	+	0.469222i	$0.155466\pi$
$521$			21.4204i			0.938444i	−0.883080	+	0.469222i	$0.844534\pi$
$522$	8.62977			0.377715
$523$	11.3878			0.497953			0.248977	−	0.968509i	$-0.419906\pi$
$523$	11.3878			0.497953			0.248977	+	0.968509i	$0.419906\pi$
$524$	−17.1337			−0.748487
$525$		0			0
$526$	−19.8128			−0.863881
$527$			10.0303i			0.436928i
$528$	2.42142	−	2.26644i	0.105379	−	0.0986341i
$529$	36.9231			1.60535
$530$	8.68687			0.377333
$531$			0.0847471i			0.00367771i
$532$		0			0
$533$	5.38106			0.233080
$534$			3.47826i			0.150519i
$535$	27.2219			1.17691
$536$		−	11.8945i		−	0.513766i
$537$			2.52905i			0.109137i
$538$	−18.1073			−0.780661
$539$		0			0
$540$	−3.51377			−0.151209
$541$		−	34.8629i		−	1.49887i	−0.662076	−	0.749436i	$-0.730325\pi$
$541$		−	34.8629i		−	1.49887i	0.662076	−	0.749436i	$-0.269675\pi$
$542$			1.57400i			0.0676090i
$543$	−12.9953			−0.557681
$544$		−	2.00058i		−	0.0857740i
$545$	−0.859057			−0.0367979
$546$		0			0
$547$		−	13.8893i		−	0.593865i	−0.954898	−	0.296933i	$-0.904036\pi$
$547$		−	13.8893i		−	0.593865i	0.954898	−	0.296933i	$-0.0959637\pi$
$548$	17.1721			0.733554
$549$	1.37655			0.0587496
$550$	17.7891	−	16.6506i	0.758532	−	0.709984i
$551$		−	40.0971i		−	1.70820i
$552$	−7.74100			−0.329479
$553$		0			0
$554$	−0.0551403			−0.00234269
$555$	−19.0274			−0.807670
$556$	11.2370			0.476554
$557$			31.1271i			1.31890i	0.751750	+	0.659449i	$0.229210\pi$
$557$			31.1271i			1.31890i	−0.751750	+	0.659449i	$0.770790\pi$
$558$	−5.01372			−0.212248
$559$			53.1584i			2.24836i
$560$		0			0
$561$	4.84423	−	4.53419i	0.204523	−	0.191433i
$562$	−21.1973			−0.894153
$563$	18.8138			0.792905			0.396453	−	0.918055i	$-0.370241\pi$
$563$	18.8138			0.792905			0.396453	+	0.918055i	$0.370241\pi$
$564$	−10.6987			−0.450499
$565$			20.3732i			0.857109i
$566$		−	28.2314i		−	1.18665i
$567$		0			0
$568$			12.6946i			0.532655i
$569$		−	18.5660i		−	0.778326i	−0.921169	−	0.389163i	$-0.872764\pi$
$569$		−	18.5660i		−	0.778326i	0.921169	−	0.389163i	$-0.127236\pi$
$570$			16.3263i			0.683833i
$571$		−	10.9839i		−	0.459662i	−0.973231	−	0.229831i	$-0.926183\pi$
$571$		−	10.9839i		−	0.459662i	0.973231	−	0.229831i	$-0.0738174\pi$
$572$	12.2594	+	13.0976i	0.512590	+	0.547640i
$573$			16.7559i			0.699989i
$574$		0			0
$575$	−56.8700			−2.37164
$576$	1.00000			0.0416667
$577$		−	45.9493i		−	1.91290i	−0.291902	−	0.956448i	$-0.594288\pi$
$577$		−	45.9493i		−	1.91290i	0.291902	−	0.956448i	$-0.405712\pi$
$578$			12.9977i			0.540633i
$579$	−16.2642			−0.675917
$580$	−30.3230			−1.25910
$581$		0			0
$582$			8.90188i			0.368995i
$583$	−5.60317	−	5.98631i	−0.232060	−	0.247927i
$584$		−	7.10779i		−	0.294122i
$585$		−	19.0063i		−	0.785814i
$586$		−	21.8815i		−	0.903917i
$587$			37.2070i			1.53570i	0.640630	+	0.767849i	$0.278673\pi$
$587$			37.2070i			1.53570i	−0.640630	+	0.767849i	$0.721327\pi$
$588$		0			0
$589$			23.2956i			0.959880i
$590$		−	0.297782i		−	0.0122595i
$591$	−23.8308			−0.980269
$592$	5.41510			0.222559
$593$	−31.0991			−1.27709			−0.638544	−	0.769585i	$-0.720463\pi$
$593$	−31.0991			−1.27709			−0.638544	+	0.769585i	$0.720463\pi$
$594$	2.26644	+	2.42142i	0.0929931	+	0.0993519i
$595$		0			0
$596$		−	21.5631i		−	0.883260i
$597$	−0.608638			−0.0249099
$598$		−	41.8718i		−	1.71226i
$599$	−8.70709			−0.355762			−0.177881	−	0.984052i	$-0.556924\pi$
$599$	−8.70709			−0.355762			−0.177881	+	0.984052i	$0.556924\pi$
$600$	7.34659			0.299923
$601$	9.32795			0.380495			0.190247	−	0.981736i	$-0.439071\pi$
$601$	9.32795			0.380495			0.190247	+	0.981736i	$0.439071\pi$
$602$		0			0
$603$	11.8945			0.484383
$604$			10.6870i			0.434848i
$605$	−38.5671	−	2.55276i	−1.56798	−	0.103785i
$606$	11.2857			0.458449
$607$	7.16638			0.290874			0.145437	−	0.989367i	$-0.453541\pi$
$607$	7.16638			0.290874			0.145437	+	0.989367i	$0.453541\pi$
$608$		−	4.64637i		−	0.188435i
$609$		0			0
$610$	−4.83687			−0.195839
$611$		−	57.8705i		−	2.34119i
$612$	2.00058			0.0808685
$613$			9.58070i			0.386961i	0.981104	+	0.193480i	$0.0619776\pi$
$613$			9.58070i			0.386961i	−0.981104	+	0.193480i	$0.938022\pi$
$614$			18.8721i			0.761614i
$615$	−3.49557			−0.140955
$616$		0			0
$617$	−22.4445			−0.903582			−0.451791	−	0.892124i	$-0.649215\pi$
$617$	−22.4445			−0.903582			−0.451791	+	0.892124i	$0.649215\pi$
$618$			0.391882i			0.0157638i
$619$			34.4743i			1.38564i	0.721111	+	0.692819i	$0.243632\pi$
$619$			34.4743i			1.38564i	−0.721111	+	0.692819i	$0.756368\pi$
$620$	17.6171			0.707519
$621$		−	7.74100i		−	0.310636i
$622$	4.23812			0.169933
$623$		0			0
$624$			5.40909i			0.216537i
$625$	−7.76055			−0.310422
$626$	−6.89858			−0.275723
$627$	11.2508	−	10.5307i	0.449314	−	0.420557i
$628$		−	20.9034i		−	0.834137i
$629$	10.8333			0.431953
$630$		0			0
$631$	12.1605			0.484103			0.242051	−	0.970263i	$-0.422180\pi$
$631$	12.1605			0.484103			0.242051	+	0.970263i	$0.422180\pi$
$632$	8.52598			0.339145
$633$	−0.499969			−0.0198720
$634$			15.3735i			0.610559i
$635$	−59.5416			−2.36284
$636$		−	2.47223i		−	0.0980305i
$637$		0			0
$638$	19.5589	+	20.8963i	0.774342	+	0.827291i
$639$	−12.6946			−0.502192
$640$	−3.51377			−0.138894
$641$	−33.5514			−1.32520			−0.662600	−	0.748974i	$-0.730547\pi$
$641$	−33.5514			−1.32520			−0.662600	+	0.748974i	$0.730547\pi$
$642$		−	7.74721i		−	0.305758i
$643$		−	6.78101i		−	0.267417i	−0.991021	−	0.133708i	$-0.957311\pi$
$643$		−	6.78101i		−	0.267417i	0.991021	−	0.133708i	$-0.0426886\pi$
$644$		0			0
$645$		−	34.5320i		−	1.35970i
$646$		−	9.29542i		−	0.365724i
$647$		−	25.2651i		−	0.993272i	−0.867959	−	0.496636i	$-0.834568\pi$
$647$		−	25.2651i		−	0.993272i	0.867959	−	0.496636i	$-0.165432\pi$
$648$			1.00000i			0.0392837i
$649$	−0.205208	+	0.192074i	−0.00805512	+	0.00753958i
$650$			39.7383i			1.55867i
$651$		0			0
$652$	15.8867			0.622171
$653$	−4.91960			−0.192519			−0.0962594	−	0.995356i	$-0.530688\pi$
$653$	−4.91960			−0.192519			−0.0962594	+	0.995356i	$0.530688\pi$
$654$			0.244483i			0.00956003i
$655$			60.2038i			2.35236i
$656$	0.994819			0.0388412
$657$	7.10779			0.277301
$658$		0			0
$659$			34.6691i			1.35052i	0.737582	+	0.675258i	$0.235968\pi$
$659$			34.6691i			1.35052i	−0.737582	+	0.675258i	$0.764032\pi$
$660$	−7.96375	−	8.50830i	−0.309989	−	0.331185i
$661$		−	32.7287i		−	1.27300i	−0.771277	−	0.636499i	$-0.780382\pi$
$661$		−	32.7287i		−	1.27300i	0.771277	−	0.636499i	$-0.219618\pi$
$662$		−	7.01665i		−	0.272710i
$663$			10.8213i			0.420264i
$664$			3.88469i			0.150755i
$665$		0			0
$666$			5.41510i			0.209831i
$667$		−	66.8031i		−	2.58663i
$668$	4.57564			0.177037
$669$	19.0402			0.736136
$670$	−41.7947			−1.61467
$671$	3.11986	+	3.33319i	0.120441	+	0.128676i
$672$		0			0
$673$		−	2.75938i		−	0.106366i	−0.998585	−	0.0531832i	$-0.983063\pi$
$673$		−	2.75938i		−	0.106366i	0.998585	−	0.0531832i	$-0.0169367\pi$
$674$	−13.2060			−0.508675
$675$			7.34659i			0.282770i
$676$	−16.2582			−0.625316
$677$	14.6512			0.563093			0.281546	−	0.959548i	$-0.409153\pi$
$677$	14.6512			0.563093			0.281546	+	0.959548i	$0.409153\pi$
$678$	5.79811			0.222675
$679$		0			0
$680$	−7.02957			−0.269572
$681$		−	18.3944i		−	0.704874i
$682$	−11.3633	−	12.1403i	−0.435123	−	0.464876i
$683$	12.3960			0.474321			0.237160	−	0.971471i	$-0.423783\pi$
$683$	12.3960			0.474321			0.237160	+	0.971471i	$0.423783\pi$
$684$	4.64637			0.177659
$685$		−	60.3387i		−	2.30542i
$686$		0			0
$687$	9.03881			0.344852
$688$			9.82762i			0.374674i
$689$	13.3725			0.509453
$690$			27.2001i			1.03549i
$691$			32.8123i			1.24824i	0.781329	+	0.624119i	$0.214542\pi$
$691$			32.8123i			1.24824i	−0.781329	+	0.624119i	$0.785458\pi$
$692$	6.74941			0.256574
$693$		0			0
$694$	−16.2961			−0.618591
$695$		−	39.4841i		−	1.49772i
$696$			8.62977i			0.327111i
$697$	1.99021			0.0753846
$698$			12.8652i			0.486953i
$699$	−24.8457			−0.939752
$700$		0			0
$701$		−	11.1535i		−	0.421263i	−0.977566	−	0.210631i	$-0.932448\pi$
$701$		−	11.1535i		−	0.421263i	0.977566	−	0.210631i	$-0.0675520\pi$
$702$	−5.40909			−0.204153
$703$	25.1606			0.948950
$704$	2.26644	+	2.42142i	0.0854197	+	0.0912605i
$705$			37.5930i			1.41583i
$706$	−19.6689			−0.740247
$707$		0			0
$708$	−0.0847471			−0.00318499
$709$	−13.3277			−0.500534			−0.250267	−	0.968177i	$-0.580518\pi$
$709$	−13.3277			−0.500534			−0.250267	+	0.968177i	$0.580518\pi$
$710$	44.6060			1.67403
$711$			8.52598i			0.319749i
$712$	−3.47826			−0.130353
$713$			38.8113i			1.45349i
$714$		0			0
$715$	46.0221	−	43.0766i	1.72113	−	1.61097i
$716$	−2.52905			−0.0945152
$717$	23.6327			0.882580
$718$	−3.46810			−0.129428
$719$		−	18.8696i		−	0.703716i	−0.936053	−	0.351858i	$-0.885550\pi$
$719$		−	18.8696i		−	0.703716i	0.936053	−	0.351858i	$-0.114450\pi$
$720$		−	3.51377i		−	0.130951i
$721$		0			0
$722$		−	2.58878i		−	0.0963446i
$723$			24.7570i			0.920724i
$724$		−	12.9953i		−	0.482966i
$725$			63.3994i			2.35459i
$726$	−0.726502	+	10.9760i	−0.0269630	+	0.407357i
$727$			42.9624i			1.59339i	0.604384	+	0.796693i	$0.293419\pi$
$727$			42.9624i			1.59339i	−0.604384	+	0.796693i	$0.706581\pi$
$728$		0			0
$729$	−1.00000			−0.0370370
$730$	−24.9751			−0.924371
$731$			19.6609i			0.727185i
$732$			1.37655i			0.0508786i
$733$	8.60588			0.317865			0.158933	−	0.987289i	$-0.449195\pi$
$733$	8.60588			0.317865			0.158933	+	0.987289i	$0.449195\pi$
$734$	−11.5696			−0.427043
$735$		0			0
$736$		−	7.74100i		−	0.285337i
$737$	26.9583	+	28.8016i	0.993020	+	1.06092i
$738$			0.994819i			0.0366198i
$739$			8.08298i			0.297337i	0.988887	+	0.148669i	$0.0474987\pi$
$739$			8.08298i			0.297337i	−0.988887	+	0.148669i	$0.952501\pi$
$740$		−	19.0274i		−	0.699463i
$741$			25.1326i			0.923270i
$742$		0			0
$743$			19.7243i			0.723615i	0.932253	+	0.361808i	$0.117840\pi$
$743$			19.7243i			0.723615i	−0.932253	+	0.361808i	$0.882160\pi$
$744$		−	5.01372i		−	0.183812i
$745$	−75.7679			−2.77592
$746$	−29.8566			−1.09313
$747$	−3.88469			−0.142133
$748$	4.53419	+	4.84423i	0.165786	+	0.177122i
$749$		0			0
$750$		−	8.24539i		−	0.301079i
$751$	3.57361			0.130403			0.0652014	−	0.997872i	$-0.479231\pi$
$751$	3.57361			0.130403			0.0652014	+	0.997872i	$0.479231\pi$
$752$		−	10.6987i		−	0.390143i
$753$	14.3590			0.523273
$754$	−46.6792			−1.69996
$755$	37.5517			1.36665
$756$		0			0
$757$	10.1774			0.369905			0.184953	−	0.982747i	$-0.440787\pi$
$757$	10.1774			0.369905			0.184953	+	0.982747i	$0.440787\pi$
$758$		−	2.32917i		−	0.0845993i
$759$	18.7442	−	17.5445i	0.680371	−	0.636826i
$760$	−16.3263			−0.592217
$761$	24.7282			0.896398			0.448199	−	0.893934i	$-0.352066\pi$
$761$	24.7282			0.896398			0.448199	+	0.893934i	$0.352066\pi$
$762$			16.9452i			0.613860i
$763$		0			0
$764$	−16.7559			−0.606208
$765$		−	7.02957i		−	0.254155i
$766$	4.35234			0.157256
$767$		−	0.458405i		−	0.0165520i
$768$			1.00000i			0.0360844i
$769$	14.2211			0.512824			0.256412	−	0.966568i	$-0.417460\pi$
$769$	14.2211			0.512824			0.256412	+	0.966568i	$0.417460\pi$
$770$		0			0
$771$	−9.31913			−0.335620
$772$		−	16.2642i		−	0.585361i
$773$		−	42.4877i		−	1.52818i	−0.645112	−	0.764088i	$-0.723189\pi$
$773$		−	42.4877i		−	1.52818i	0.645112	−	0.764088i	$-0.276811\pi$
$774$	−9.82762			−0.353246
$775$		−	36.8338i		−	1.32311i
$776$	−8.90188			−0.319559
$777$		0			0
$778$			37.6920i			1.35132i
$779$	4.62230			0.165611
$780$	19.0063			0.680535
$781$	−28.7716	−	30.7390i	−1.02953	−	1.09993i
$782$		−	15.4865i		−	0.553795i
$783$	−8.62977			−0.308403
$784$		0			0
$785$	−73.4498			−2.62154
$786$	17.1337			0.611137
$787$	33.9890			1.21158			0.605788	−	0.795626i	$-0.292858\pi$
$787$	33.9890			1.21158			0.605788	+	0.795626i	$0.292858\pi$
$788$		−	23.8308i		−	0.848938i
$789$	19.8128			0.705356
$790$		−	29.9584i		−	1.06587i
$791$		0			0
$792$	−2.42142	+	2.26644i	−0.0860412	+	0.0805344i
$793$	−7.44586			−0.264410
$794$	20.6744			0.733707
$795$	−8.68687			−0.308091
$796$		−	0.608638i		−	0.0215726i
$797$			28.6394i			1.01446i	0.861811	+	0.507229i	$0.169330\pi$
$797$			28.6394i			1.01446i	−0.861811	+	0.507229i	$0.830670\pi$
$798$		0			0
$799$		−	21.4037i		−	0.757207i
$800$			7.34659i			0.259741i
$801$		−	3.47826i		−	0.122898i
$802$			20.5716i			0.726409i
$803$	16.1094	+	17.2109i	0.568487	+	0.607359i
$804$			11.8945i			0.419488i
$805$		0			0
$806$	27.1197			0.955250
$807$	18.1073			0.637407
$808$			11.2857i			0.397029i
$809$			48.8805i			1.71855i	0.511515	+	0.859274i	$0.329084\pi$
$809$			48.8805i			1.71855i	−0.511515	+	0.859274i	$0.670916\pi$
$810$	3.51377			0.123461
$811$	3.15799			0.110892			0.0554460	−	0.998462i	$-0.482342\pi$
$811$	3.15799			0.110892			0.0554460	+	0.998462i	$0.482342\pi$
$812$		0			0
$813$		−	1.57400i		−	0.0552025i
$814$	−13.1122	+	12.2730i	−0.459583	+	0.430169i
$815$		−	55.8222i		−	1.95537i
$816$			2.00058i			0.0700342i
$817$			45.6628i			1.59754i
$818$		−	0.669187i		−	0.0233976i
$819$		0			0
$820$		−	3.49557i		−	0.122071i
$821$			41.8653i			1.46111i	0.682854	+	0.730554i	$0.260738\pi$
$821$			41.8653i			1.46111i	−0.682854	+	0.730554i	$0.739262\pi$
$822$	−17.1721			−0.598944
$823$	−27.1268			−0.945582			−0.472791	−	0.881175i	$-0.656753\pi$
$823$	−27.1268			−0.945582			−0.472791	+	0.881175i	$0.656753\pi$
$824$	−0.391882			−0.0136519
$825$	−17.7891	+	16.6506i	−0.619339	+	0.579700i
$826$		0			0
$827$		−	50.7514i		−	1.76480i	−0.470501	−	0.882399i	$-0.655927\pi$
$827$		−	50.7514i		−	1.76480i	0.470501	−	0.882399i	$-0.344073\pi$
$828$	7.74100			0.269018
$829$			40.4033i			1.40327i	0.712539	+	0.701633i	$0.247545\pi$
$829$			40.4033i			1.40327i	−0.712539	+	0.701633i	$0.752455\pi$
$830$	13.6499			0.473795
$831$	0.0551403			0.00191280
$832$	−5.40909			−0.187526
$833$		0			0
$834$	−11.2370			−0.389104
$835$		−	16.0778i		−	0.556394i
$836$	10.5307	+	11.2508i	0.364213	+	0.389117i
$837$	5.01372			0.173300
$838$	−18.5889			−0.642144
$839$		−	11.7816i		−	0.406745i	−0.979101	−	0.203373i	$-0.934810\pi$
$839$		−	11.7816i		−	0.406745i	0.979101	−	0.203373i	$-0.0651903\pi$
$840$		0			0
$841$	−45.4729			−1.56803
$842$			13.6636i			0.470878i
$843$	21.1973			0.730072
$844$		−	0.499969i		−	0.0172096i
$845$			57.1277i			1.96525i
$846$	10.6987			0.367831
$847$		0			0
$848$	2.47223			0.0848969
$849$			28.2314i			0.968897i
$850$			14.6974i			0.504117i
$851$	41.9183			1.43694
$852$		−	12.6946i		−	0.434911i
$853$	55.8734			1.91307			0.956534	−	0.291620i	$-0.0941941\pi$
$853$	55.8734			1.91307			0.956534	+	0.291620i	$0.0941941\pi$
$854$		0			0
$855$		−	16.3263i		−	0.558348i
$856$	7.74721			0.264794
$857$	−16.5483			−0.565280			−0.282640	−	0.959226i	$-0.591210\pi$
$857$	−16.5483			−0.565280			−0.282640	+	0.959226i	$0.591210\pi$
$858$	−12.2594	−	13.0976i	−0.418528	−	0.447146i
$859$			0.983230i			0.0335474i	0.999859	+	0.0167737i	$0.00533948\pi$
$859$			0.983230i			0.0335474i	−0.999859	+	0.0167737i	$0.994661\pi$
$860$	34.5320			1.17753
$861$		0			0
$862$	−5.99083			−0.204048
$863$	−1.50393			−0.0511942			−0.0255971	−	0.999672i	$-0.508149\pi$
$863$	−1.50393			−0.0511942			−0.0255971	+	0.999672i	$0.508149\pi$
$864$	−1.00000			−0.0340207
$865$		−	23.7159i		−	0.806364i
$866$	17.7633			0.603621
$867$		−	12.9977i		−	0.441425i
$868$		0			0
$869$	−20.6449	+	19.3236i	−0.700332	+	0.655509i
$870$	30.3230			1.02805
$871$	−64.3386			−2.18003
$872$	−0.244483			−0.00827923
$873$		−	8.90188i		−	0.301283i
$874$		−	35.9676i		−	1.21662i
$875$		0			0
$876$			7.10779i			0.240150i
$877$		−	2.03193i		−	0.0686134i	−0.999411	−	0.0343067i	$-0.989078\pi$
$877$		−	2.03193i		−	0.0686134i	0.999411	−	0.0343067i	$-0.0109223\pi$
$878$			9.34241i			0.315291i
$879$			21.8815i			0.738045i
$880$	8.50830	−	7.96375i	0.286815	−	0.268458i
$881$		−	21.2271i		−	0.715160i	−0.933883	−	0.357580i	$-0.883602\pi$
$881$		−	21.2271i		−	0.715160i	0.933883	−	0.357580i	$-0.116398\pi$
$882$		0			0
$883$	1.66995			0.0561982			0.0280991	−	0.999605i	$-0.491055\pi$
$883$	1.66995			0.0561982			0.0280991	+	0.999605i	$0.491055\pi$
$884$	−10.8213			−0.363959
$885$			0.297782i			0.0100098i
$886$		−	17.1346i		−	0.575650i
$887$	−51.2155			−1.71965			−0.859824	−	0.510591i	$-0.829427\pi$
$887$	−51.2155			−1.71965			−0.859824	+	0.510591i	$0.829427\pi$
$888$	−5.41510			−0.181719
$889$		0			0
$890$			12.2218i			0.409676i
$891$	−2.26644	−	2.42142i	−0.0759286	−	0.0811205i
$892$			19.0402i			0.637513i
$893$		−	49.7104i		−	1.66349i
$894$			21.5631i			0.721179i
$895$			8.88652i			0.297043i
$896$		0			0
$897$			41.8718i			1.39806i
$898$		−	14.7314i		−	0.491592i
$899$	43.2673			1.44304
$900$	−7.34659			−0.244886
$901$	4.94589			0.164772
$902$	−2.40887	+	2.25470i	−0.0802066	+	0.0750732i
$903$		0			0
$904$			5.79811i			0.192842i
$905$	−45.6624			−1.51787
$906$		−	10.6870i		−	0.355052i
$907$	17.6922			0.587460			0.293730	−	0.955888i	$-0.405103\pi$
$907$	17.6922			0.587460			0.293730	+	0.955888i	$0.405103\pi$
$908$	18.3944			0.610439
$909$	−11.2857			−0.374322
$910$		0			0
$911$	−31.9183			−1.05750			−0.528750	−	0.848778i	$-0.677339\pi$
$911$	−31.9183			−1.05750			−0.528750	+	0.848778i	$0.677339\pi$
$912$			4.64637i			0.153857i
$913$	−8.80441	−	9.40644i	−0.291383	−	0.311308i
$914$	24.8203			0.820984
$915$	4.83687			0.159902
$916$			9.03881i			0.298651i
$917$		0			0
$918$	−2.00058			−0.0660289
$919$			25.5328i			0.842250i	0.907003	+	0.421125i	$0.138365\pi$
$919$			25.5328i			0.842250i	−0.907003	+	0.421125i	$0.861635\pi$
$920$	−27.2001			−0.896761
$921$		−	18.8721i		−	0.621856i
$922$		−	8.76302i		−	0.288595i
$923$	68.6663			2.26018
$924$		0			0
$925$	−39.7826			−1.30804
$926$		−	24.6334i		−	0.809505i
$927$		−	0.391882i		−	0.0128711i
$928$	−8.62977			−0.283286
$929$		−	35.6897i		−	1.17094i	−0.810694	−	0.585470i	$-0.800910\pi$
$929$		−	35.6897i		−	1.17094i	0.810694	−	0.585470i	$-0.199090\pi$
$930$	−17.6171			−0.577687
$931$		0			0
$932$		−	24.8457i		−	0.813849i
$933$	−4.23812			−0.138750
$934$	16.5454			0.541382
$935$	17.0215	−	15.9321i	0.556663	−	0.521035i
$936$		−	5.40909i		−	0.176802i
$937$	−4.89203			−0.159816			−0.0799079	−	0.996802i	$-0.525463\pi$
$937$	−4.89203			−0.159816			−0.0799079	+	0.996802i	$0.525463\pi$
$938$		0			0
$939$	6.89858			0.225127
$940$	−37.5930			−1.22615
$941$	36.5189			1.19048			0.595241	−	0.803547i	$-0.297057\pi$
$941$	36.5189			1.19048			0.595241	+	0.803547i	$0.297057\pi$
$942$			20.9034i			0.681070i
$943$	7.70090			0.250776
$944$		−	0.0847471i		−	0.00275828i
$945$		0			0
$946$	−22.2737	−	23.7967i	−0.724181	−	0.773699i
$947$	4.55098			0.147887			0.0739434	−	0.997262i	$-0.476442\pi$
$947$	4.55098			0.147887			0.0739434	+	0.997262i	$0.476442\pi$
$948$	−8.52598			−0.276911
$949$	−38.4466			−1.24803
$950$			34.1350i			1.10749i
$951$		−	15.3735i		−	0.498519i
$952$		0			0
$953$			31.0896i			1.00709i	0.863969	+	0.503545i	$0.167971\pi$
$953$			31.0896i			1.00709i	−0.863969	+	0.503545i	$0.832029\pi$
$954$			2.47223i			0.0800415i
$955$			58.8765i			1.90520i
$956$			23.6327i			0.764336i
$957$	−19.5589	−	20.8963i	−0.632248	−	0.675480i
$958$		−	25.2557i		−	0.815974i
$959$		0			0
$960$	3.51377			0.113406
$961$	5.86257			0.189115
$962$		−	29.2908i		−	0.944372i
$963$			7.74721i			0.249650i
$964$	−24.7570			−0.797370
$965$	−57.1487			−1.83968
$966$		0			0
$967$		−	40.8346i		−	1.31315i	−0.754259	−	0.656577i	$-0.772004\pi$
$967$		−	40.8346i		−	1.31315i	0.754259	−	0.656577i	$-0.227996\pi$
$968$	−10.9760	−	0.726502i	−0.352781	−	0.0233507i
$969$			9.29542i			0.298612i
$970$			31.2792i			1.00431i
$971$		−	2.45605i		−	0.0788183i	−0.999223	−	0.0394091i	$-0.987452\pi$
$971$		−	2.45605i		−	0.0788183i	0.999223	−	0.0394091i	$-0.0125476\pi$
$972$		−	1.00000i		−	0.0320750i
$973$		0			0
$974$			33.6204i			1.07727i
$975$		−	39.7383i		−	1.27265i
$976$	−1.37655			−0.0440622
$977$	−26.9513			−0.862249			−0.431125	−	0.902292i	$-0.641883\pi$
$977$	−26.9513			−0.862249			−0.431125	+	0.902292i	$0.641883\pi$
$978$	−15.8867			−0.508001
$979$	8.42231	−	7.88327i	0.269178	−	0.251950i
$980$		0			0
$981$		−	0.244483i		−	0.00780573i
$982$	5.83403			0.186171
$983$			44.2119i			1.41014i	0.709138	+	0.705070i	$0.249084\pi$
$983$			44.2119i			1.41014i	−0.709138	+	0.705070i	$0.750916\pi$
$984$	−0.994819			−0.0317137
$985$	−83.7361			−2.66805
$986$	−17.2645			−0.549814
$987$		0			0
$988$	−25.1326			−0.799576
$989$			76.0756i			2.41906i
$990$	7.96375	+	8.50830i	0.253105	+	0.270412i
$991$	6.19997			0.196948			0.0984742	−	0.995140i	$-0.468604\pi$
$991$	6.19997			0.196948			0.0984742	+	0.995140i	$0.468604\pi$
$992$	5.01372			0.159186
$993$			7.01665i			0.222667i
$994$		0			0
$995$	−2.13862			−0.0677987
$996$		−	3.88469i		−	0.123091i
$997$	58.4393			1.85079			0.925395	−	0.379004i	$-0.123733\pi$
$997$	58.4393			1.85079			0.925395	+	0.379004i	$0.123733\pi$
$998$			1.83045i			0.0579419i
$999$		−	5.41510i		−	0.171326i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3234.2.e.d.2155.12	yes	24
7.6	odd	2		3234.2.e.c.2155.1	✓	24
11.10	odd	2		3234.2.e.c.2155.24	yes	24
77.76	even	2	inner	3234.2.e.d.2155.13	yes	24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3234.2.e.c.2155.1	✓	24	7.6	odd	2
3234.2.e.c.2155.24	yes	24	11.10	odd	2
3234.2.e.d.2155.12	yes	24	1.1	even	1	trivial
3234.2.e.d.2155.13	yes	24	77.76	even	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 3234 = 2 \cdot 3 \cdot 7^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3234.e (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.00000i q^{2} +1.00000i q^{3} -1.00000 q^{4} +3.51377i q^{5} +1.00000 q^{6} +1.00000i q^{8} -1.00000 q^{9} +O(q^{10})\)	\(q-1.00000i q^{2} +1.00000i q^{3} -1.00000 q^{4} +3.51377i q^{5} +1.00000 q^{6} +1.00000i q^{8} -1.00000 q^{9} +3.51377 q^{10} +(-2.26644 - 2.42142i) q^{11} -1.00000i q^{12} +5.40909 q^{13} -3.51377 q^{15} +1.00000 q^{16} +2.00058 q^{17} +1.00000i q^{18} +4.64637 q^{19} -3.51377i q^{20} +(-2.42142 + 2.26644i) q^{22} +7.74100 q^{23} -1.00000 q^{24} -7.34659 q^{25} -5.40909i q^{26} -1.00000i q^{27} -8.62977i q^{29} +3.51377i q^{30} +5.01372i q^{31} -1.00000i q^{32} +(2.42142 - 2.26644i) q^{33} -2.00058i q^{34} +1.00000 q^{36} +5.41510 q^{37} -4.64637i q^{38} +5.40909i q^{39} -3.51377 q^{40} +0.994819 q^{41} +9.82762i q^{43} +(2.26644 + 2.42142i) q^{44} -3.51377i q^{45} -7.74100i q^{46} -10.6987i q^{47} +1.00000i q^{48} +7.34659i q^{50} +2.00058i q^{51} -5.40909 q^{52} +2.47223 q^{53} -1.00000 q^{54} +(8.50830 - 7.96375i) q^{55} +4.64637i q^{57} -8.62977 q^{58} -0.0847471i q^{59} +3.51377 q^{60} -1.37655 q^{61} +5.01372 q^{62} -1.00000 q^{64} +19.0063i q^{65} +(-2.26644 - 2.42142i) q^{66} -11.8945 q^{67} -2.00058 q^{68} +7.74100i q^{69} +12.6946 q^{71} -1.00000i q^{72} -7.10779 q^{73} -5.41510i q^{74} -7.34659i q^{75} -4.64637 q^{76} +5.40909 q^{78} -8.52598i q^{79} +3.51377i q^{80} +1.00000 q^{81} -0.994819i q^{82} +3.88469 q^{83} +7.02957i q^{85} +9.82762 q^{86} +8.62977 q^{87} +(2.42142 - 2.26644i) q^{88} +3.47826i q^{89} -3.51377 q^{90} -7.74100 q^{92} -5.01372 q^{93} -10.6987 q^{94} +16.3263i q^{95} +1.00000 q^{96} +8.90188i q^{97} +(2.26644 + 2.42142i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q - 24 q^{4} + 24 q^{6} - 24 q^{9}+O(q^{10}) \) 24 * q - 24 * q^4 + 24 * q^6 - 24 * q^9	\( 24 q - 24 q^{4} + 24 q^{6} - 24 q^{9} + 24 q^{16} - 16 q^{17} + 32 q^{19} - 8 q^{22} - 24 q^{24} - 8 q^{25} + 8 q^{33} + 24 q^{36} + 16 q^{37} + 16 q^{41} - 24 q^{54} - 16 q^{55} + 16 q^{62} - 24 q^{64} - 64 q^{67} + 16 q^{68} + 64 q^{71} - 32 q^{76} + 24 q^{81} + 16 q^{83} + 8 q^{88} - 16 q^{93} - 64 q^{94} + 24 q^{96}+O(q^{100}) \) 24 * q - 24 * q^4 + 24 * q^6 - 24 * q^9 + 24 * q^16 - 16 * q^17 + 32 * q^19 - 8 * q^22 - 24 * q^24 - 8 * q^25 + 8 * q^33 + 24 * q^36 + 16 * q^37 + 16 * q^41 - 24 * q^54 - 16 * q^55 + 16 * q^62 - 24 * q^64 - 64 * q^67 + 16 * q^68 + 64 * q^71 - 32 * q^76 + 24 * q^81 + 16 * q^83 + 8 * q^88 - 16 * q^93 - 64 * q^94 + 24 * q^96

Introduction

L-functions

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