LMFDB - Embedded newform 2394.2.f.b.2015.11

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2394,2,Mod(2015,2394)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2394, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("2394.2015");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2394 = 2 \cdot 3^{2} \cdot 7 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2394.f (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:	$19.1161862439$
Analytic rank:	$0$
Dimension:	$24$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			2015.11
Character	$\chi$	$=$	2394.2015
Dual form			2394.2.f.b.2015.12

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{2} -1.00000 q^{4} +1.62775 q^{5} +(-2.31472 + 1.28144i) q^{7} +1.00000i q^{8} +O(q^{10})$	\(q-1.00000i q^{2} -1.00000 q^{4} +1.62775 q^{5} +(-2.31472 + 1.28144i) q^{7} +1.00000i q^{8} -1.62775i q^{10} +0.0154711i q^{11} -4.25272i q^{13} +(1.28144 + 2.31472i) q^{14} +1.00000 q^{16} -3.03644 q^{17} -1.00000i q^{19} -1.62775 q^{20} +0.0154711 q^{22} +4.14140i q^{23} -2.35041 q^{25} -4.25272 q^{26} +(2.31472 - 1.28144i) q^{28} -5.72068i q^{29} +3.69445i q^{31} -1.00000i q^{32} +3.03644i q^{34} +(-3.76779 + 2.08587i) q^{35} +4.31429 q^{37} -1.00000 q^{38} +1.62775i q^{40} -4.11129 q^{41} -1.43615 q^{43} -0.0154711i q^{44} +4.14140 q^{46} -5.85351 q^{47} +(3.71583 - 5.93234i) q^{49} +2.35041i q^{50} +4.25272i q^{52} -3.96327i q^{53} +0.0251832i q^{55} +(-1.28144 - 2.31472i) q^{56} -5.72068 q^{58} -9.76418 q^{59} +3.02148i q^{61} +3.69445 q^{62} -1.00000 q^{64} -6.92238i q^{65} -8.17607 q^{67} +3.03644 q^{68} +(2.08587 + 3.76779i) q^{70} -13.5592i q^{71} +10.2103i q^{73} -4.31429i q^{74} +1.00000i q^{76} +(-0.0198253 - 0.0358112i) q^{77} -14.4356 q^{79} +1.62775 q^{80} +4.11129i q^{82} -0.624208 q^{83} -4.94258 q^{85} +1.43615i q^{86} -0.0154711 q^{88} -9.38743 q^{89} +(5.44959 + 9.84383i) q^{91} -4.14140i q^{92} +5.85351i q^{94} -1.62775i q^{95} +1.70163i q^{97} +(-5.93234 - 3.71583i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q - 24 q^{4} + 4 q^{7}+O(q^{10}) $ 24 * q - 24 * q^4 + 4 * q^7	$ 24 q - 24 q^{4} + 4 q^{7} + 4 q^{14} + 24 q^{16} - 32 q^{17} - 8 q^{22} + 16 q^{25} - 4 q^{28} + 24 q^{35} - 24 q^{38} + 8 q^{41} + 16 q^{43} - 8 q^{46} - 16 q^{49} - 4 q^{56} + 16 q^{58} - 16 q^{59} + 16 q^{62} - 24 q^{64} - 24 q^{67} + 32 q^{68} - 16 q^{70} - 8 q^{77} - 40 q^{79} + 64 q^{83} + 40 q^{85} + 8 q^{88} + 64 q^{89} + 8 q^{91} + 16 q^{98}+O(q^{100}) $ 24 * q - 24 * q^4 + 4 * q^7 + 4 * q^14 + 24 * q^16 - 32 * q^17 - 8 * q^22 + 16 * q^25 - 4 * q^28 + 24 * q^35 - 24 * q^38 + 8 * q^41 + 16 * q^43 - 8 * q^46 - 16 * q^49 - 4 * q^56 + 16 * q^58 - 16 * q^59 + 16 * q^62 - 24 * q^64 - 24 * q^67 + 32 * q^68 - 16 * q^70 - 8 * q^77 - 40 * q^79 + 64 * q^83 + 40 * q^85 + 8 * q^88 + 64 * q^89 + 8 * q^91 + 16 * q^98

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2394\mathbb{Z}\right)^\times$.

$n$	$533$	$1009$	$1711$
$\chi(n)$	$-1$	$1$	$-1$

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
$3$		0			0
$3$		0			0
$4$	−1.00000			−0.500000
$5$	1.62775			0.727954			0.363977	−	0.931408i	$-0.381419\pi$
$5$	1.62775			0.727954			0.363977	+	0.931408i	$0.381419\pi$
$6$		0			0
$7$	−2.31472	+	1.28144i	−0.874881	+	0.484338i
$7$	−2.31472	+	1.28144i	−0.874881	+	0.484338i
$8$			1.00000i			0.353553i
$9$		0			0
$10$		−	1.62775i		−	0.514741i
$11$			0.0154711i			0.00466471i	0.999997	+	0.00233236i	$0.000742413\pi$
$11$			0.0154711i			0.00466471i	−0.999997	+	0.00233236i	$0.999258\pi$
$12$		0			0
$13$		−	4.25272i		−	1.17949i	−0.807589	−	0.589746i	$-0.799228\pi$
$13$		−	4.25272i		−	1.17949i	0.807589	−	0.589746i	$-0.200772\pi$
$14$	1.28144	+	2.31472i	0.342479	+	0.618634i
$15$		0			0
$16$	1.00000			0.250000
$17$	−3.03644			−0.736445			−0.368223	−	0.929738i	$-0.620034\pi$
$17$	−3.03644			−0.736445			−0.368223	+	0.929738i	$0.620034\pi$
$18$		0			0
$19$		−	1.00000i		−	0.229416i
$19$		−	1.00000i		−	0.229416i
$20$	−1.62775			−0.363977
$21$		0			0
$22$	0.0154711			0.00329845
$23$			4.14140i			0.863542i	0.901983	+	0.431771i	$0.142111\pi$
$23$			4.14140i			0.863542i	−0.901983	+	0.431771i	$0.857889\pi$
$24$		0			0
$25$	−2.35041			−0.470083
$26$	−4.25272			−0.834026
$27$		0			0
$28$	2.31472	−	1.28144i	0.437440	−	0.242169i
$29$		−	5.72068i		−	1.06230i	−0.847277	−	0.531152i	$-0.821759\pi$
$29$		−	5.72068i		−	1.06230i	0.847277	−	0.531152i	$-0.178241\pi$
$30$		0			0
$31$			3.69445i			0.663543i	0.943360	+	0.331772i	$0.107646\pi$
$31$			3.69445i			0.663543i	−0.943360	+	0.331772i	$0.892354\pi$
$32$		−	1.00000i		−	0.176777i
$33$		0			0
$34$			3.03644i			0.520745i
$35$	−3.76779	+	2.08587i	−0.636873	+	0.352576i
$36$		0			0
$37$	4.31429			0.709265			0.354633	−	0.935006i	$-0.384606\pi$
$37$	4.31429			0.709265			0.354633	+	0.935006i	$0.384606\pi$
$38$	−1.00000			−0.162221
$39$		0			0
$40$			1.62775i			0.257371i
$41$	−4.11129			−0.642076			−0.321038	−	0.947066i	$-0.604032\pi$
$41$	−4.11129			−0.642076			−0.321038	+	0.947066i	$0.604032\pi$
$42$		0			0
$43$	−1.43615			−0.219010			−0.109505	−	0.993986i	$-0.534927\pi$
$43$	−1.43615			−0.219010			−0.109505	+	0.993986i	$0.534927\pi$
$44$		−	0.0154711i		−	0.00233236i
$45$		0			0
$46$	4.14140			0.610617
$47$	−5.85351			−0.853822			−0.426911	−	0.904294i	$-0.640398\pi$
$47$	−5.85351			−0.853822			−0.426911	+	0.904294i	$0.640398\pi$
$48$		0			0
$49$	3.71583	−	5.93234i	0.530833	−	0.847477i
$50$			2.35041i			0.332399i
$51$		0			0
$52$			4.25272i			0.589746i
$53$		−	3.96327i		−	0.544398i	−0.962241	−	0.272199i	$-0.912249\pi$
$53$		−	3.96327i		−	0.544398i	0.962241	−	0.272199i	$-0.0877508\pi$
$54$		0			0
$55$			0.0251832i			0.00339570i
$56$	−1.28144	−	2.31472i	−0.171239	−	0.309317i
$57$		0			0
$58$	−5.72068			−0.751162
$59$	−9.76418			−1.27119			−0.635594	−	0.772024i	$-0.719245\pi$
$59$	−9.76418			−1.27119			−0.635594	+	0.772024i	$0.719245\pi$
$60$		0			0
$61$			3.02148i			0.386861i	0.981114	+	0.193431i	$0.0619614\pi$
$61$			3.02148i			0.386861i	−0.981114	+	0.193431i	$0.938039\pi$
$62$	3.69445			0.469196
$63$		0			0
$64$	−1.00000			−0.125000
$65$		−	6.92238i		−	0.858615i
$66$		0			0
$67$	−8.17607			−0.998866			−0.499433	−	0.866353i	$-0.666458\pi$
$67$	−8.17607			−0.998866			−0.499433	+	0.866353i	$0.666458\pi$
$68$	3.03644			0.368223
$69$		0			0
$70$	2.08587	+	3.76779i	0.249309	+	0.450337i
$71$		−	13.5592i		−	1.60918i	−0.593830	−	0.804590i	$-0.702385\pi$
$71$		−	13.5592i		−	1.60918i	0.593830	−	0.804590i	$-0.297615\pi$
$72$		0			0
$73$			10.2103i			1.19503i	0.801858	+	0.597514i	$0.203845\pi$
$73$			10.2103i			1.19503i	−0.801858	+	0.597514i	$0.796155\pi$
$74$		−	4.31429i		−	0.501526i
$75$		0			0
$76$			1.00000i			0.114708i
$77$	−0.0198253	−	0.0358112i	−0.00225930	−	0.00408107i
$78$		0			0
$79$	−14.4356			−1.62413			−0.812065	−	0.583567i	$-0.801657\pi$
$79$	−14.4356			−1.62413			−0.812065	+	0.583567i	$0.801657\pi$
$80$	1.62775			0.181989
$81$		0			0
$82$			4.11129i			0.454016i
$83$	−0.624208			−0.0685157			−0.0342578	−	0.999413i	$-0.510907\pi$
$83$	−0.624208			−0.0685157			−0.0342578	+	0.999413i	$0.510907\pi$
$84$		0			0
$85$	−4.94258			−0.536098
$86$			1.43615i			0.154864i
$87$		0			0
$88$	−0.0154711			−0.00164923
$89$	−9.38743			−0.995065			−0.497533	−	0.867445i	$-0.665761\pi$
$89$	−9.38743			−0.995065			−0.497533	+	0.867445i	$0.665761\pi$
$90$		0			0
$91$	5.44959	+	9.84383i	0.571273	+	1.03191i
$92$		−	4.14140i		−	0.431771i
$93$		0			0
$94$			5.85351i			0.603743i
$95$		−	1.62775i		−	0.167004i
$96$		0			0
$97$			1.70163i			0.172774i	0.996262	+	0.0863871i	$0.0275322\pi$
$97$			1.70163i			0.172774i	−0.996262	+	0.0863871i	$0.972468\pi$
$98$	−5.93234	−	3.71583i	−0.599256	−	0.375355i
$99$		0			0
$100$	2.35041			0.235041
$101$	−14.2767			−1.42058			−0.710291	−	0.703908i	$-0.751437\pi$
$101$	−14.2767			−1.42058			−0.710291	+	0.703908i	$0.751437\pi$
$102$		0			0
$103$		−	16.5212i		−	1.62788i	−0.580948	−	0.813941i	$-0.697318\pi$
$103$		−	16.5212i		−	1.62788i	0.580948	−	0.813941i	$-0.302682\pi$
$104$	4.25272			0.417013
$105$		0			0
$106$	−3.96327			−0.384947
$107$			6.35766i			0.614618i	0.951610	+	0.307309i	$0.0994286\pi$
$107$			6.35766i			0.614618i	−0.951610	+	0.307309i	$0.900571\pi$
$108$		0			0
$109$	−13.9937			−1.34035			−0.670175	−	0.742204i	$-0.733781\pi$
$109$	−13.9937			−1.34035			−0.670175	+	0.742204i	$0.733781\pi$
$110$	0.0251832			0.00240112
$111$		0			0
$112$	−2.31472	+	1.28144i	−0.218720	+	0.121085i
$113$		−	17.9326i		−	1.68695i	−0.537165	−	0.843477i	$-0.680505\pi$
$113$		−	17.9326i		−	1.68695i	0.537165	−	0.843477i	$-0.319495\pi$
$114$		0			0
$115$			6.74119i			0.628619i
$116$			5.72068i			0.531152i
$117$		0			0
$118$			9.76418i			0.898865i
$119$	7.02850	−	3.89101i	0.644302	−	0.356689i
$120$		0			0
$121$	10.9998			0.999978
$122$	3.02148			0.273552
$123$		0			0
$124$		−	3.69445i		−	0.331772i
$125$	−11.9647			−1.07015
$126$		0			0
$127$	7.17364			0.636557			0.318279	−	0.947997i	$-0.396895\pi$
$127$	7.17364			0.636557			0.318279	+	0.947997i	$0.396895\pi$
$128$			1.00000i			0.0883883i
$129$		0			0
$130$	−6.92238			−0.607133
$131$	−5.03917			−0.440274			−0.220137	−	0.975469i	$-0.570650\pi$
$131$	−5.03917			−0.440274			−0.220137	+	0.975469i	$0.570650\pi$
$132$		0			0
$133$	1.28144	+	2.31472i	0.111115	+	0.200711i
$134$			8.17607i			0.706305i
$135$		0			0
$136$		−	3.03644i		−	0.260373i
$137$		−	11.9354i		−	1.01971i	−0.860261	−	0.509854i	$-0.829699\pi$
$137$		−	11.9354i		−	1.01971i	0.860261	−	0.509854i	$-0.170301\pi$
$138$		0			0
$139$			21.0877i			1.78863i	0.447433	+	0.894317i	$0.352338\pi$
$139$			21.0877i			1.78863i	−0.447433	+	0.894317i	$0.647662\pi$
$140$	3.76779	−	2.08587i	0.318436	−	0.176288i
$141$		0			0
$142$	−13.5592			−1.13786
$143$	0.0657942			0.00550199
$144$		0			0
$145$		−	9.31186i		−	0.773308i
$146$	10.2103			0.845013
$147$		0			0
$148$	−4.31429			−0.354633
$149$			2.69714i			0.220959i	0.993878	+	0.110479i	$0.0352386\pi$
$149$			2.69714i			0.220959i	−0.993878	+	0.110479i	$0.964761\pi$
$150$		0			0
$151$	9.78150			0.796008			0.398004	−	0.917384i	$-0.369703\pi$
$151$	9.78150			0.796008			0.398004	+	0.917384i	$0.369703\pi$
$152$	1.00000			0.0811107
$153$		0			0
$154$	−0.0358112	+	0.0198253i	−0.00288575	+	0.00159757i
$155$			6.01366i			0.483029i
$156$		0			0
$157$			11.0994i			0.885828i	0.896564	+	0.442914i	$0.146055\pi$
$157$			11.0994i			0.885828i	−0.896564	+	0.442914i	$0.853945\pi$
$158$			14.4356i			1.14843i
$159$		0			0
$160$		−	1.62775i		−	0.128685i
$161$	−5.30696	−	9.58618i	−0.418247	−	0.755497i
$162$		0			0
$163$	7.07679			0.554297			0.277148	−	0.960827i	$-0.410611\pi$
$163$	7.07679			0.554297			0.277148	+	0.960827i	$0.410611\pi$
$164$	4.11129			0.321038
$165$		0			0
$166$			0.624208i			0.0484479i
$167$	8.40972			0.650764			0.325382	−	0.945583i	$-0.394507\pi$
$167$	8.40972			0.650764			0.325382	+	0.945583i	$0.394507\pi$
$168$		0			0
$169$	−5.08559			−0.391199
$170$			4.94258i			0.379079i
$171$		0			0
$172$	1.43615			0.109505
$173$	−19.9525			−1.51696			−0.758480	−	0.651697i	$-0.774058\pi$
$173$	−19.9525			−1.51696			−0.758480	+	0.651697i	$0.774058\pi$
$174$		0			0
$175$	5.44055	−	3.01191i	0.411267	−	0.227679i
$176$			0.0154711i			0.00116618i
$177$		0			0
$178$			9.38743i			0.703618i
$179$			8.06942i			0.603137i	0.953444	+	0.301569i	$0.0975102\pi$
$179$			8.06942i			0.603137i	−0.953444	+	0.301569i	$0.902490\pi$
$180$		0			0
$181$		−	5.46167i		−	0.405962i	−0.979183	−	0.202981i	$-0.934937\pi$
$181$		−	5.46167i		−	0.405962i	0.979183	−	0.202981i	$-0.0650630\pi$
$182$	9.84383	−	5.44959i	0.729673	−	0.403951i
$183$		0			0
$184$	−4.14140			−0.305308
$185$	7.02261			0.516313
$186$		0			0
$187$		−	0.0469771i		−	0.00343531i
$188$	5.85351			0.426911
$189$		0			0
$190$	−1.62775			−0.118090
$191$			8.65104i			0.625967i	0.949759	+	0.312984i	$0.101329\pi$
$191$			8.65104i			0.625967i	−0.949759	+	0.312984i	$0.898671\pi$
$192$		0			0
$193$	20.0998			1.44681			0.723407	−	0.690422i	$-0.242575\pi$
$193$	20.0998			1.44681			0.723407	+	0.690422i	$0.242575\pi$
$194$	1.70163			0.122170
$195$		0			0
$196$	−3.71583	+	5.93234i	−0.265416	+	0.423738i
$197$		−	12.1825i		−	0.867970i	−0.900920	−	0.433985i	$-0.857107\pi$
$197$		−	12.1825i		−	0.867970i	0.900920	−	0.433985i	$-0.142893\pi$
$198$		0			0
$199$			22.6389i			1.60483i	0.596769	+	0.802413i	$0.296451\pi$
$199$			22.6389i			1.60483i	−0.596769	+	0.802413i	$0.703549\pi$
$200$		−	2.35041i		−	0.166199i
$201$		0			0
$202$			14.2767i			1.00450i
$203$	7.33070	+	13.2418i	0.514514	+	0.929389i
$204$		0			0
$205$	−6.69217			−0.467402
$206$	−16.5212			−1.15109
$207$		0			0
$208$		−	4.25272i		−	0.294873i
$209$	0.0154711			0.00107016
$210$		0			0
$211$	14.3718			0.989396			0.494698	−	0.869065i	$-0.335279\pi$
$211$	14.3718			0.989396			0.494698	+	0.869065i	$0.335279\pi$
$212$			3.96327i			0.272199i
$213$		0			0
$214$	6.35766			0.434601
$215$	−2.33770			−0.159429
$216$		0			0
$217$	−4.73421	−	8.55161i	−0.321379	−	0.580521i
$218$			13.9937i			0.947770i
$219$		0			0
$220$		−	0.0251832i		−	0.00169785i
$221$			12.9131i			0.868631i
$222$		0			0
$223$		−	7.96006i		−	0.533045i	−0.963829	−	0.266522i	$-0.914125\pi$
$223$		−	7.96006i		−	0.533045i	0.963829	−	0.266522i	$-0.0858746\pi$
$224$	1.28144	+	2.31472i	0.0856197	+	0.154659i
$225$		0			0
$226$	−17.9326			−1.19286
$227$	−11.4937			−0.762860			−0.381430	−	0.924398i	$-0.624568\pi$
$227$	−11.4937			−0.762860			−0.381430	+	0.924398i	$0.624568\pi$
$228$		0			0
$229$		−	13.6660i		−	0.903075i	−0.892252	−	0.451537i	$-0.850876\pi$
$229$		−	13.6660i		−	0.903075i	0.892252	−	0.451537i	$-0.149124\pi$
$230$	6.74119			0.444501
$231$		0			0
$232$	5.72068			0.375581
$233$		−	5.61522i		−	0.367865i	−0.982939	−	0.183933i	$-0.941117\pi$
$233$		−	5.61522i		−	0.367865i	0.982939	−	0.183933i	$-0.0588828\pi$
$234$		0			0
$235$	−9.52807			−0.621543
$236$	9.76418			0.635594
$237$		0			0
$238$	−3.89101	−	7.02850i	−0.252217	−	0.455590i
$239$			12.7170i			0.822593i	0.911502	+	0.411296i	$0.134924\pi$
$239$			12.7170i			0.822593i	−0.911502	+	0.411296i	$0.865076\pi$
$240$		0			0
$241$		−	3.25681i		−	0.209790i	−0.994483	−	0.104895i	$-0.966549\pi$
$241$		−	3.25681i		−	0.209790i	0.994483	−	0.104895i	$-0.0334506\pi$
$242$		−	10.9998i		−	0.707091i
$243$		0			0
$244$		−	3.02148i		−	0.193431i
$245$	6.04846	−	9.65639i	0.386422	−	0.616924i
$246$		0			0
$247$	−4.25272			−0.270594
$248$	−3.69445			−0.234598
$249$		0			0
$250$			11.9647i			0.756712i
$251$	−5.85069			−0.369293			−0.184646	−	0.982805i	$-0.559114\pi$
$251$	−5.85069			−0.369293			−0.184646	+	0.982805i	$0.559114\pi$
$252$		0			0
$253$	−0.0640721			−0.00402818
$254$		−	7.17364i		−	0.450114i
$255$		0			0
$256$	1.00000			0.0625000
$257$	−6.44102			−0.401779			−0.200890	−	0.979614i	$-0.564383\pi$
$257$	−6.44102			−0.401779			−0.200890	+	0.979614i	$0.564383\pi$
$258$		0			0
$259$	−9.98637	+	5.52850i	−0.620523	+	0.343524i
$260$			6.92238i			0.429308i
$261$		0			0
$262$			5.03917i			0.311321i
$263$		−	6.73912i		−	0.415552i	−0.978176	−	0.207776i	$-0.933378\pi$
$263$		−	6.73912i		−	0.415552i	0.978176	−	0.207776i	$-0.0666225\pi$
$264$		0			0
$265$		−	6.45124i		−	0.396296i
$266$	2.31472	−	1.28144i	0.141924	−	0.0785700i
$267$		0			0
$268$	8.17607			0.499433
$269$	−12.2085			−0.744367			−0.372184	−	0.928159i	$-0.621391\pi$
$269$	−12.2085			−0.744367			−0.372184	+	0.928159i	$0.621391\pi$
$270$		0			0
$271$			11.7754i			0.715305i	0.933855	+	0.357652i	$0.116423\pi$
$271$			11.7754i			0.715305i	−0.933855	+	0.357652i	$0.883577\pi$
$272$	−3.03644			−0.184111
$273$		0			0
$274$	−11.9354			−0.721043
$275$		−	0.0363635i		−	0.00219280i
$276$		0			0
$277$	−8.42662			−0.506306			−0.253153	−	0.967426i	$-0.581468\pi$
$277$	−8.42662			−0.506306			−0.253153	+	0.967426i	$0.581468\pi$
$278$	21.0877			1.26476
$279$		0			0
$280$	−2.08587	−	3.76779i	−0.124654	−	0.225169i
$281$			21.8260i			1.30203i	0.759065	+	0.651015i	$0.225657\pi$
$281$			21.8260i			1.30203i	−0.759065	+	0.651015i	$0.774343\pi$
$282$		0			0
$283$		−	19.7923i		−	1.17653i	−0.808667	−	0.588266i	$-0.799811\pi$
$283$		−	19.7923i		−	1.17653i	0.808667	−	0.588266i	$-0.200189\pi$
$284$			13.5592i			0.804590i
$285$		0			0
$286$		−	0.0657942i		−	0.00389049i
$287$	9.51648	−	5.26837i	0.561740	−	0.310982i
$288$		0			0
$289$	−7.78002			−0.457648
$290$	−9.31186			−0.546811
$291$		0			0
$292$		−	10.2103i		−	0.597514i
$293$	23.7126			1.38531			0.692654	−	0.721270i	$-0.256441\pi$
$293$	23.7126			1.38531			0.692654	+	0.721270i	$0.256441\pi$
$294$		0			0
$295$	−15.8937			−0.925366
$296$			4.31429i			0.250763i
$297$		0			0
$298$	2.69714			0.156241
$299$	17.6122			1.01854
$300$		0			0
$301$	3.32427	−	1.84033i	0.191608	−	0.106075i
$302$		−	9.78150i		−	0.562862i
$303$		0			0
$304$		−	1.00000i		−	0.0573539i
$305$			4.91823i			0.281617i
$306$		0			0
$307$			5.10919i			0.291597i	0.989314	+	0.145798i	$0.0465751\pi$
$307$			5.10919i			0.291597i	−0.989314	+	0.145798i	$0.953425\pi$
$308$	0.0198253	+	0.0358112i	0.00112965	+	0.00204053i
$309$		0			0
$310$	6.01366			0.341553
$311$	18.1351			1.02835			0.514174	−	0.857686i	$-0.328099\pi$
$311$	18.1351			1.02835			0.514174	+	0.857686i	$0.328099\pi$
$312$		0			0
$313$		−	26.0830i		−	1.47430i	−0.675731	−	0.737149i	$-0.736172\pi$
$313$		−	26.0830i		−	1.47430i	0.675731	−	0.737149i	$-0.263828\pi$
$314$	11.0994			0.626375
$315$		0			0
$316$	14.4356			0.812065
$317$			8.44411i			0.474268i	0.971477	+	0.237134i	$0.0762081\pi$
$317$			8.44411i			0.474268i	−0.971477	+	0.237134i	$0.923792\pi$
$318$		0			0
$319$	0.0885052			0.00495534
$320$	−1.62775			−0.0909943
$321$		0			0
$322$	−9.58618	+	5.30696i	−0.534217	+	0.295745i
$323$			3.03644i			0.168952i
$324$		0			0
$325$			9.99565i			0.554459i
$326$		−	7.07679i		−	0.391947i
$327$		0			0
$328$		−	4.11129i		−	0.227008i
$329$	13.5492	−	7.50091i	0.746992	−	0.413539i
$330$		0			0
$331$	−4.98437			−0.273966			−0.136983	−	0.990573i	$-0.543741\pi$
$331$	−4.98437			−0.273966			−0.136983	+	0.990573i	$0.543741\pi$
$332$	0.624208			0.0342578
$333$		0			0
$334$		−	8.40972i		−	0.460159i
$335$	−13.3086			−0.727128
$336$		0			0
$337$	−7.37345			−0.401657			−0.200829	−	0.979626i	$-0.564363\pi$
$337$	−7.37345			−0.401657			−0.200829	+	0.979626i	$0.564363\pi$
$338$			5.08559i			0.276620i
$339$		0			0
$340$	4.94258			0.268049
$341$	−0.0571572			−0.00309524
$342$		0			0
$343$	−0.999169	+	18.4933i	−0.0539501	+	0.998544i
$344$		−	1.43615i		−	0.0774319i
$345$		0			0
$346$			19.9525i			1.07265i
$347$			1.29322i			0.0694235i	0.999397	+	0.0347117i	$0.0110513\pi$
$347$			1.29322i			0.0694235i	−0.999397	+	0.0347117i	$0.988949\pi$
$348$		0			0
$349$			5.77557i			0.309159i	0.987980	+	0.154580i	$0.0494023\pi$
$349$			5.77557i			0.309159i	−0.987980	+	0.154580i	$0.950598\pi$
$350$	−3.01191	−	5.44055i	−0.160993	−	0.290809i
$351$		0			0
$352$	0.0154711			0.000824613
$353$	−30.2054			−1.60767			−0.803835	−	0.594852i	$-0.797211\pi$
$353$	−30.2054			−1.60767			−0.803835	+	0.594852i	$0.797211\pi$
$354$		0			0
$355$		−	22.0710i		−	1.17141i
$356$	9.38743			0.497533
$357$		0			0
$358$	8.06942			0.426482
$359$		−	17.7226i		−	0.935361i	−0.883898	−	0.467680i	$-0.845090\pi$
$359$		−	17.7226i		−	0.935361i	0.883898	−	0.467680i	$-0.154910\pi$
$360$		0			0
$361$	−1.00000			−0.0526316
$362$	−5.46167			−0.287059
$363$		0			0
$364$	−5.44959	−	9.84383i	−0.285636	−	0.515957i
$365$			16.6199i			0.869926i
$366$		0			0
$367$			8.75590i			0.457054i	0.973538	+	0.228527i	$0.0733909\pi$
$367$			8.75590i			0.457054i	−0.973538	+	0.228527i	$0.926609\pi$
$368$			4.14140i			0.215886i
$369$		0			0
$370$		−	7.02261i		−	0.365088i
$371$	5.07869	+	9.17386i	0.263673	+	0.476283i
$372$		0			0
$373$	11.7997			0.610964			0.305482	−	0.952198i	$-0.401182\pi$
$373$	11.7997			0.610964			0.305482	+	0.952198i	$0.401182\pi$
$374$	−0.0469771			−0.00242913
$375$		0			0
$376$		−	5.85351i		−	0.301872i
$377$	−24.3284			−1.25298
$378$		0			0
$379$	29.1086			1.49521			0.747604	−	0.664145i	$-0.231204\pi$
$379$	29.1086			1.49521			0.747604	+	0.664145i	$0.231204\pi$
$380$			1.62775i			0.0835021i
$381$		0			0
$382$	8.65104			0.442626
$383$	27.6391			1.41229			0.706146	−	0.708066i	$-0.250432\pi$
$383$	27.6391			1.41229			0.706146	+	0.708066i	$0.250432\pi$
$384$		0			0
$385$	−0.0322707	−	0.0582919i	−0.00164467	−	0.00297083i
$386$		−	20.0998i		−	1.02305i
$387$		0			0
$388$		−	1.70163i		−	0.0863871i
$389$		−	0.477596i		−	0.0242151i	−0.999927	−	0.0121076i	$-0.996146\pi$
$389$		−	0.477596i		−	0.0242151i	0.999927	−	0.0121076i	$-0.00385405\pi$
$390$		0			0
$391$		−	12.5751i		−	0.635952i
$392$	5.93234	+	3.71583i	0.299628	+	0.187678i
$393$		0			0
$394$	−12.1825			−0.613747
$395$	−23.4976			−1.18229
$396$		0			0
$397$		−	30.9515i		−	1.55341i	−0.629863	−	0.776707i	$-0.716889\pi$
$397$		−	30.9515i		−	1.55341i	0.629863	−	0.776707i	$-0.283111\pi$
$398$	22.6389			1.13478
$399$		0			0
$400$	−2.35041			−0.117521
$401$		−	0.116312i		−	0.00580836i	−0.999996	−	0.00290418i	$-0.999076\pi$
$401$		−	0.116312i		−	0.00580836i	0.999996	−	0.00290418i	$-0.000924431\pi$
$402$		0			0
$403$	15.7115			0.782643
$404$	14.2767			0.710291
$405$		0			0
$406$	13.2418	−	7.33070i	0.657177	−	0.363817i
$407$			0.0667469i			0.00330852i
$408$		0			0
$409$			3.22965i			0.159696i	0.996807	+	0.0798479i	$0.0254435\pi$
$409$			3.22965i			0.159696i	−0.996807	+	0.0798479i	$0.974557\pi$
$410$			6.69217i			0.330503i
$411$		0			0
$412$			16.5212i			0.813941i
$413$	22.6013	−	12.5122i	1.11214	−	0.615685i
$414$		0			0
$415$	−1.01606			−0.0498763
$416$	−4.25272			−0.208507
$417$		0			0
$418$		−	0.0154711i		−	0.000756716i
$419$	12.5742			0.614291			0.307146	−	0.951663i	$-0.400626\pi$
$419$	12.5742			0.614291			0.307146	+	0.951663i	$0.400626\pi$
$420$		0			0
$421$	8.06413			0.393022			0.196511	−	0.980502i	$-0.437039\pi$
$421$	8.06413			0.393022			0.196511	+	0.980502i	$0.437039\pi$
$422$		−	14.3718i		−	0.699609i
$423$		0			0
$424$	3.96327			0.192474
$425$	7.13690			0.346190
$426$		0			0
$427$	−3.87185	−	6.99388i	−0.187372	−	0.338458i
$428$		−	6.35766i		−	0.307309i
$429$		0			0
$430$			2.33770i			0.112734i
$431$		−	16.3403i		−	0.787086i	−0.919306	−	0.393543i	$-0.871249\pi$
$431$		−	16.3403i		−	0.787086i	0.919306	−	0.393543i	$-0.128751\pi$
$432$		0			0
$433$		−	29.0783i		−	1.39742i	−0.715407	−	0.698708i	$-0.753759\pi$
$433$		−	29.0783i		−	1.39742i	0.715407	−	0.698708i	$-0.246241\pi$
$434$	−8.55161	+	4.73421i	−0.410490	+	0.227250i
$435$		0			0
$436$	13.9937			0.670175
$437$	4.14140			0.198110
$438$		0			0
$439$			20.6815i			0.987072i	0.869725	+	0.493536i	$0.164296\pi$
$439$			20.6815i			0.987072i	−0.869725	+	0.493536i	$0.835704\pi$
$440$	−0.0251832			−0.00120056
$441$		0			0
$442$	12.9131			0.614215
$443$			8.39857i			0.399028i	0.979895	+	0.199514i	$0.0639364\pi$
$443$			8.39857i			0.399028i	−0.979895	+	0.199514i	$0.936064\pi$
$444$		0			0
$445$	−15.2804			−0.724362
$446$	−7.96006			−0.376920
$447$		0			0
$448$	2.31472	−	1.28144i	0.109360	−	0.0605423i
$449$		−	11.7462i		−	0.554339i	−0.960821	−	0.277169i	$-0.910604\pi$
$449$		−	11.7462i		−	0.554339i	0.960821	−	0.277169i	$-0.0893963\pi$
$450$		0			0
$451$		−	0.0636062i		−	0.00299510i
$452$			17.9326i			0.843477i
$453$		0			0
$454$			11.4937i			0.539424i
$455$	8.87060	+	16.0233i	0.415860	+	0.751186i
$456$		0			0
$457$	34.7101			1.62367			0.811836	−	0.583886i	$-0.198468\pi$
$457$	34.7101			1.62367			0.811836	+	0.583886i	$0.198468\pi$
$458$	−13.6660			−0.638570
$459$		0			0
$460$		−	6.74119i		−	0.314310i
$461$	−2.82437			−0.131544			−0.0657720	−	0.997835i	$-0.520951\pi$
$461$	−2.82437			−0.131544			−0.0657720	+	0.997835i	$0.520951\pi$
$462$		0			0
$463$	30.6440			1.42415			0.712073	−	0.702105i	$-0.247756\pi$
$463$	30.6440			1.42415			0.712073	+	0.702105i	$0.247756\pi$
$464$		−	5.72068i		−	0.265576i
$465$		0			0
$466$	−5.61522			−0.260120
$467$	−12.6806			−0.586789			−0.293395	−	0.955991i	$-0.594785\pi$
$467$	−12.6806			−0.586789			−0.293395	+	0.955991i	$0.594785\pi$
$468$		0			0
$469$	18.9253	−	10.4771i	0.873888	−	0.483789i
$470$			9.52807i			0.439497i
$471$		0			0
$472$		−	9.76418i		−	0.449433i
$473$		−	0.0222188i		−	0.00102162i
$474$		0			0
$475$			2.35041i			0.107844i
$476$	−7.02850	+	3.89101i	−0.322151	+	0.178344i
$477$		0			0
$478$	12.7170			0.581661
$479$	17.4486			0.797246			0.398623	−	0.917115i	$-0.369488\pi$
$479$	17.4486			0.797246			0.398623	+	0.917115i	$0.369488\pi$
$480$		0			0
$481$		−	18.3475i		−	0.836572i
$482$	−3.25681			−0.148344
$483$		0			0
$484$	−10.9998			−0.499989
$485$			2.76983i			0.125772i
$486$		0			0
$487$	−4.73684			−0.214647			−0.107323	−	0.994224i	$-0.534228\pi$
$487$	−4.73684			−0.214647			−0.107323	+	0.994224i	$0.534228\pi$
$488$	−3.02148			−0.136776
$489$		0			0
$490$	−9.65639	−	6.04846i	−0.436231	−	0.273242i
$491$			1.91335i			0.0863482i	0.999068	+	0.0431741i	$0.0137470\pi$
$491$			1.91335i			0.0863482i	−0.999068	+	0.0431741i	$0.986253\pi$
$492$		0			0
$493$			17.3705i			0.782328i
$494$			4.25272i			0.191339i
$495$		0			0
$496$			3.69445i			0.165886i
$497$	17.3753	+	31.3857i	0.779388	+	1.40784i
$498$		0			0
$499$	−5.32052			−0.238179			−0.119090	−	0.992884i	$-0.537998\pi$
$499$	−5.32052			−0.238179			−0.119090	+	0.992884i	$0.537998\pi$
$500$	11.9647			0.535076
$501$		0			0
$502$			5.85069i			0.261129i
$503$	43.9132			1.95799			0.978996	−	0.203878i	$-0.0653547\pi$
$503$	43.9132			1.95799			0.978996	+	0.203878i	$0.0653547\pi$
$504$		0			0
$505$	−23.2389			−1.03412
$506$			0.0640721i			0.00284835i
$507$		0			0
$508$	−7.17364			−0.318279
$509$	17.6684			0.783139			0.391569	−	0.920149i	$-0.371932\pi$
$509$	17.6684			0.783139			0.391569	+	0.920149i	$0.371932\pi$
$510$		0			0
$511$	−13.0839	−	23.6340i	−0.578798	−	1.04551i
$512$		−	1.00000i		−	0.0441942i
$513$		0			0
$514$			6.44102i			0.284101i
$515$		−	26.8924i		−	1.18502i
$516$		0			0
$517$		−	0.0905602i		−	0.00398283i
$518$	5.52850	+	9.98637i	0.242908	+	0.438776i
$519$		0			0
$520$	6.92238			0.303566
$521$	5.29541			0.231996			0.115998	−	0.993249i	$-0.462993\pi$
$521$	5.29541			0.231996			0.115998	+	0.993249i	$0.462993\pi$
$522$		0			0
$523$		−	32.3844i		−	1.41607i	−0.706178	−	0.708035i	$-0.749582\pi$
$523$		−	32.3844i		−	1.41607i	0.706178	−	0.708035i	$-0.250418\pi$
$524$	5.03917			0.220137
$525$		0			0
$526$	−6.73912			−0.293840
$527$		−	11.2180i		−	0.488663i
$528$		0			0
$529$	5.84877			0.254295
$530$	−6.45124			−0.280224
$531$		0			0
$532$	−1.28144	−	2.31472i	−0.0555574	−	0.100356i
$533$			17.4842i			0.757323i
$534$		0			0
$535$			10.3487i			0.447414i
$536$		−	8.17607i		−	0.353152i
$537$		0			0
$538$			12.2085i			0.526347i
$539$	0.0917798	+	0.0574880i	0.00395324	+	0.00247618i
$540$		0			0
$541$	−2.74432			−0.117987			−0.0589937	−	0.998258i	$-0.518789\pi$
$541$	−2.74432			−0.117987			−0.0589937	+	0.998258i	$0.518789\pi$
$542$	11.7754			0.505797
$543$		0			0
$544$			3.03644i			0.130186i
$545$	−22.7782			−0.975712
$546$		0			0
$547$	3.02491			0.129336			0.0646679	−	0.997907i	$-0.479401\pi$
$547$	3.02491			0.129336			0.0646679	+	0.997907i	$0.479401\pi$
$548$			11.9354i			0.509854i
$549$		0			0
$550$	−0.0363635			−0.00155055
$551$	−5.72068			−0.243709
$552$		0			0
$553$	33.4143	−	18.4983i	1.42092	−	0.786628i
$554$			8.42662i			0.358013i
$555$		0			0
$556$		−	21.0877i		−	0.894317i
$557$			34.3692i			1.45627i	0.685434	+	0.728134i	$0.259612\pi$
$557$			34.3692i			1.45627i	−0.685434	+	0.728134i	$0.740388\pi$
$558$		0			0
$559$			6.10753i			0.258321i
$560$	−3.76779	+	2.08587i	−0.159218	+	0.0881440i
$561$		0			0
$562$	21.8260			0.920675
$563$	4.35244			0.183433			0.0917166	−	0.995785i	$-0.470765\pi$
$563$	4.35244			0.183433			0.0917166	+	0.995785i	$0.470765\pi$
$564$		0			0
$565$		−	29.1898i		−	1.22803i
$566$	−19.7923			−0.831934
$567$		0			0
$568$	13.5592			0.568931
$569$		−	27.8879i		−	1.16912i	−0.811350	−	0.584561i	$-0.801267\pi$
$569$		−	27.8879i		−	1.16912i	0.811350	−	0.584561i	$-0.198733\pi$
$570$		0			0
$571$	−5.13600			−0.214935			−0.107468	−	0.994209i	$-0.534274\pi$
$571$	−5.13600			−0.214935			−0.107468	+	0.994209i	$0.534274\pi$
$572$	−0.0657942			−0.00275099
$573$		0			0
$574$	−5.26837	−	9.51648i	−0.219897	−	0.397210i
$575$		−	9.73402i		−	0.405937i
$576$		0			0
$577$			6.66630i			0.277522i	0.990326	+	0.138761i	$0.0443120\pi$
$577$			6.66630i			0.277522i	−0.990326	+	0.138761i	$0.955688\pi$
$578$			7.78002i			0.323606i
$579$		0			0
$580$			9.31186i			0.386654i
$581$	1.44486	−	0.799884i	0.0599431	−	0.0331848i
$582$		0			0
$583$	0.0613162			0.00253946
$584$	−10.2103			−0.422507
$585$		0			0
$586$		−	23.7126i		−	0.979560i
$587$	−27.7139			−1.14388			−0.571938	−	0.820297i	$-0.693808\pi$
$587$	−27.7139			−1.14388			−0.571938	+	0.820297i	$0.693808\pi$
$588$		0			0
$589$	3.69445			0.152227
$590$			15.8937i			0.654333i
$591$		0			0
$592$	4.31429			0.177316
$593$	−37.2610			−1.53013			−0.765064	−	0.643954i	$-0.777293\pi$
$593$	−37.2610			−1.53013			−0.765064	+	0.643954i	$0.777293\pi$
$594$		0			0
$595$	11.4407	−	6.33362i	0.469022	−	0.259653i
$596$		−	2.69714i		−	0.110479i
$597$		0			0
$598$		−	17.6122i		−	0.720217i
$599$		−	19.9389i		−	0.814683i	−0.913276	−	0.407341i	$-0.866456\pi$
$599$		−	19.9389i		−	0.814683i	0.913276	−	0.407341i	$-0.133544\pi$
$600$		0			0
$601$			27.1853i			1.10891i	0.832213	+	0.554456i	$0.187074\pi$
$601$			27.1853i			1.10891i	−0.832213	+	0.554456i	$0.812926\pi$
$602$	−1.84033	−	3.32427i	−0.0750064	−	0.135487i
$603$		0			0
$604$	−9.78150			−0.398004
$605$	17.9049			0.727938
$606$		0			0
$607$		−	21.8608i		−	0.887302i	−0.896200	−	0.443651i	$-0.853683\pi$
$607$		−	21.8608i		−	0.887302i	0.896200	−	0.443651i	$-0.146317\pi$
$608$	−1.00000			−0.0405554
$609$		0			0
$610$	4.91823			0.199133
$611$			24.8933i			1.00708i
$612$		0			0
$613$	30.6744			1.23893			0.619464	−	0.785025i	$-0.287350\pi$
$613$	30.6744			1.23893			0.619464	+	0.785025i	$0.287350\pi$
$614$	5.10919			0.206190
$615$		0			0
$616$	0.0358112	−	0.0198253i	0.00144288	−	0.000798783i
$617$		−	28.3980i		−	1.14326i	−0.820511	−	0.571631i	$-0.806311\pi$
$617$		−	28.3980i		−	1.14326i	0.820511	−	0.571631i	$-0.193689\pi$
$618$		0			0
$619$		−	11.5569i		−	0.464510i	−0.972655	−	0.232255i	$-0.925390\pi$
$619$		−	11.5569i		−	0.464510i	0.972655	−	0.232255i	$-0.0746103\pi$
$620$		−	6.01366i		−	0.241514i
$621$		0			0
$622$		−	18.1351i		−	0.727152i
$623$	21.7292	−	12.0294i	0.870564	−	0.481948i
$624$		0			0
$625$	−7.72348			−0.308939
$626$	−26.0830			−1.04249
$627$		0			0
$628$		−	11.0994i		−	0.442914i
$629$	−13.1001			−0.522335
$630$		0			0
$631$	−37.2610			−1.48334			−0.741669	−	0.670766i	$-0.765966\pi$
$631$	−37.2610			−1.48334			−0.741669	+	0.670766i	$0.765966\pi$
$632$		−	14.4356i		−	0.574217i
$633$		0			0
$634$	8.44411			0.335358
$635$	11.6769			0.463384
$636$		0			0
$637$	−25.2285	−	15.8024i	−0.999591	−	0.626113i
$638$		−	0.0885052i		−	0.00350396i
$639$		0			0
$640$			1.62775i			0.0643427i
$641$		−	39.2760i		−	1.55131i	−0.631158	−	0.775655i	$-0.717420\pi$
$641$		−	39.2760i		−	1.55131i	0.631158	−	0.775655i	$-0.282580\pi$
$642$		0			0
$643$		−	15.2183i		−	0.600152i	−0.953915	−	0.300076i	$-0.902988\pi$
$643$		−	15.2183i		−	0.600152i	0.953915	−	0.300076i	$-0.0970120\pi$
$644$	5.30696	+	9.58618i	0.209123	+	0.377748i
$645$		0			0
$646$	3.03644			0.119467
$647$	−41.0628			−1.61434			−0.807172	−	0.590316i	$-0.799003\pi$
$647$	−41.0628			−1.61434			−0.807172	+	0.590316i	$0.799003\pi$
$648$		0			0
$649$		−	0.151063i		−	0.00592973i
$650$	9.99565			0.392061
$651$		0			0
$652$	−7.07679			−0.277148
$653$		−	10.4599i		−	0.409328i	−0.978832	−	0.204664i	$-0.934390\pi$
$653$		−	10.4599i		−	0.409328i	0.978832	−	0.204664i	$-0.0656102\pi$
$654$		0			0
$655$	−8.20253			−0.320499
$656$	−4.11129			−0.160519
$657$		0			0
$658$	−7.50091	−	13.5492i	−0.292416	−	0.528203i
$659$			25.6992i			1.00110i	0.865707	+	0.500550i	$0.166869\pi$
$659$			25.6992i			1.00110i	−0.865707	+	0.500550i	$0.833131\pi$
$660$		0			0
$661$			31.9219i			1.24162i	0.783962	+	0.620808i	$0.213195\pi$
$661$			31.9219i			1.24162i	−0.783962	+	0.620808i	$0.786805\pi$
$662$			4.98437i			0.193723i
$663$		0			0
$664$		−	0.624208i		−	0.0242240i
$665$	2.08587	+	3.76779i	0.0808865	+	0.146109i
$666$		0			0
$667$	23.6916			0.917344
$668$	−8.40972			−0.325382
$669$		0			0
$670$			13.3086i			0.514157i
$671$	−0.0467457			−0.00180460
$672$		0			0
$673$	−5.14720			−0.198410			−0.0992049	−	0.995067i	$-0.531630\pi$
$673$	−5.14720			−0.198410			−0.0992049	+	0.995067i	$0.531630\pi$
$674$			7.37345i			0.284014i
$675$		0			0
$676$	5.08559			0.195600
$677$	−38.5104			−1.48008			−0.740038	−	0.672565i	$-0.765193\pi$
$677$	−38.5104			−1.48008			−0.740038	+	0.672565i	$0.765193\pi$
$678$		0			0
$679$	−2.18053	−	3.93879i	−0.0836811	−	0.151157i
$680$		−	4.94258i		−	0.189539i
$681$		0			0
$682$			0.0571572i			0.00218866i
$683$		−	34.9110i		−	1.33583i	−0.744237	−	0.667916i	$-0.767187\pi$
$683$		−	34.9110i		−	1.33583i	0.744237	−	0.667916i	$-0.232813\pi$
$684$		0			0
$685$		−	19.4279i		−	0.742301i
$686$	18.4933	+	0.999169i	0.706077	+	0.0381485i
$687$		0			0
$688$	−1.43615			−0.0547526
$689$	−16.8547			−0.642112
$690$		0			0
$691$			0.867963i			0.0330189i	0.999864	+	0.0165094i	$0.00525536\pi$
$691$			0.867963i			0.0330189i	−0.999864	+	0.0165094i	$0.994745\pi$
$692$	19.9525			0.758480
$693$		0			0
$694$	1.29322			0.0490898
$695$			34.3256i			1.30204i
$696$		0			0
$697$	12.4837			0.472854
$698$	5.77557			0.218609
$699$		0			0
$700$	−5.44055	+	3.01191i	−0.205633	+	0.113840i
$701$		−	18.4238i		−	0.695858i	−0.937521	−	0.347929i	$-0.886885\pi$
$701$		−	18.4238i		−	0.695858i	0.937521	−	0.347929i	$-0.113115\pi$
$702$		0			0
$703$		−	4.31429i		−	0.162717i
$704$		−	0.0154711i		−	0.000583089i
$705$		0			0
$706$			30.2054i			1.13679i
$707$	33.0465	−	18.2947i	1.24284	−	0.688042i
$708$		0			0
$709$	30.5094			1.14581			0.572903	−	0.819623i	$-0.305817\pi$
$709$	30.5094			1.14581			0.572903	+	0.819623i	$0.305817\pi$
$710$	−22.0710			−0.828312
$711$		0			0
$712$		−	9.38743i		−	0.351809i
$713$	−15.3002			−0.572998
$714$		0			0
$715$	0.107097			0.00400519
$716$		−	8.06942i		−	0.301569i
$717$		0			0
$718$	−17.7226			−0.661400
$719$	31.8999			1.18966			0.594832	−	0.803850i	$-0.297218\pi$
$719$	31.8999			1.18966			0.594832	+	0.803850i	$0.297218\pi$
$720$		0			0
$721$	21.1709	+	38.2419i	0.788445	+	1.42420i
$722$			1.00000i			0.0372161i
$723$		0			0
$724$			5.46167i			0.202981i
$725$			13.4460i			0.499371i
$726$		0			0
$727$			2.48700i			0.0922376i	0.998936	+	0.0461188i	$0.0146853\pi$
$727$			2.48700i			0.0922376i	−0.998936	+	0.0461188i	$0.985315\pi$
$728$	−9.84383	+	5.44959i	−0.364837	+	0.201975i
$729$		0			0
$730$	16.6199			0.615131
$731$	4.36078			0.161289
$732$		0			0
$733$		−	7.74602i		−	0.286106i	−0.989715	−	0.143053i	$-0.954308\pi$
$733$		−	7.74602i		−	0.286106i	0.989715	−	0.143053i	$-0.0456919\pi$
$734$	8.75590			0.323186
$735$		0			0
$736$	4.14140			0.152654
$737$		−	0.126493i		−	0.00465942i
$738$		0			0
$739$	−45.5042			−1.67390			−0.836949	−	0.547281i	$-0.815663\pi$
$739$	−45.5042			−1.67390			−0.836949	+	0.547281i	$0.815663\pi$
$740$	−7.02261			−0.258156
$741$		0			0
$742$	9.17386	−	5.07869i	0.336783	−	0.186445i
$743$			33.7214i			1.23712i	0.785738	+	0.618559i	$0.212283\pi$
$743$			33.7214i			1.23712i	−0.785738	+	0.618559i	$0.787717\pi$
$744$		0			0
$745$			4.39029i			0.160848i
$746$		−	11.7997i		−	0.432016i
$747$		0			0
$748$			0.0469771i			0.00171765i
$749$	−8.14695	−	14.7162i	−0.297683	−	0.537718i
$750$		0			0
$751$	−42.5587			−1.55299			−0.776495	−	0.630124i	$-0.783004\pi$
$751$	−42.5587			−1.55299			−0.776495	+	0.630124i	$0.783004\pi$
$752$	−5.85351			−0.213455
$753$		0			0
$754$			24.3284i			0.885989i
$755$	15.9219			0.579457
$756$		0			0
$757$	−11.0401			−0.401261			−0.200630	−	0.979667i	$-0.564299\pi$
$757$	−11.0401			−0.401261			−0.200630	+	0.979667i	$0.564299\pi$
$758$		−	29.1086i		−	1.05727i
$759$		0			0
$760$	1.62775			0.0590449
$761$	24.0493			0.871786			0.435893	−	0.899998i	$-0.356433\pi$
$761$	24.0493			0.871786			0.435893	+	0.899998i	$0.356433\pi$
$762$		0			0
$763$	32.3914	−	17.9320i	1.17265	−	0.649182i
$764$		−	8.65104i		−	0.312984i
$765$		0			0
$766$		−	27.6391i		−	0.998642i
$767$			41.5243i			1.49935i
$768$		0			0
$769$		−	35.8822i		−	1.29394i	−0.762514	−	0.646972i	$-0.776035\pi$
$769$		−	35.8822i		−	1.29394i	0.762514	−	0.646972i	$-0.223965\pi$
$770$	−0.0582919	+	0.0322707i	−0.00210069	+	0.00116295i
$771$		0			0
$772$	−20.0998			−0.723407
$773$	−50.1543			−1.80393			−0.901963	−	0.431813i	$-0.857874\pi$
$773$	−50.1543			−1.80393			−0.901963	+	0.431813i	$0.857874\pi$
$774$		0			0
$775$		−	8.68349i		−	0.311920i
$776$	−1.70163			−0.0610849
$777$		0			0
$778$	−0.477596			−0.0171227
$779$			4.11129i			0.147302i
$780$		0			0
$781$	0.209776			0.00750637
$782$	−12.5751			−0.449686
$783$		0			0
$784$	3.71583	−	5.93234i	0.132708	−	0.211869i
$785$			18.0671i			0.644842i
$786$		0			0
$787$			52.7697i			1.88104i	0.339741	+	0.940519i	$0.389661\pi$
$787$			52.7697i			1.88104i	−0.339741	+	0.940519i	$0.610339\pi$
$788$			12.1825i			0.433985i
$789$		0			0
$790$			23.4976i			0.836007i
$791$	22.9795	+	41.5088i	0.817057	+	1.47588i
$792$		0			0
$793$	12.8495			0.456299
$794$	−30.9515			−1.09843
$795$		0			0
$796$		−	22.6389i		−	0.802413i
$797$	17.2744			0.611889			0.305945	−	0.952049i	$-0.401028\pi$
$797$	17.2744			0.611889			0.305945	+	0.952049i	$0.401028\pi$
$798$		0			0
$799$	17.7738			0.628793
$800$			2.35041i			0.0830997i
$801$		0			0
$802$	−0.116312			−0.00410713
$803$	−0.157965			−0.00557447
$804$		0			0
$805$	−8.63842	−	15.6039i	−0.304464	−	0.549967i
$806$		−	15.7115i		−	0.553412i
$807$		0			0
$808$		−	14.2767i		−	0.502252i
$809$			39.0471i			1.37282i	0.727213	+	0.686411i	$0.240815\pi$
$809$			39.0471i			1.37282i	−0.727213	+	0.686411i	$0.759185\pi$
$810$		0			0
$811$			21.8993i			0.768988i	0.923128	+	0.384494i	$0.125624\pi$
$811$			21.8993i			0.768988i	−0.923128	+	0.384494i	$0.874376\pi$
$812$	−7.33070	−	13.2418i	−0.257257	−	0.464694i
$813$		0			0
$814$	0.0667469			0.00233948
$815$	11.5193			0.403503
$816$		0			0
$817$			1.43615i			0.0502444i
$818$	3.22965			0.112922
$819$		0			0
$820$	6.69217			0.233701
$821$			10.4501i			0.364709i	0.983233	+	0.182355i	$0.0583719\pi$
$821$			10.4501i			0.364709i	−0.983233	+	0.182355i	$0.941628\pi$
$822$		0			0
$823$	−18.1785			−0.633663			−0.316832	−	0.948482i	$-0.602619\pi$
$823$	−18.1785			−0.633663			−0.316832	+	0.948482i	$0.602619\pi$
$824$	16.5212			0.575543
$825$		0			0
$826$	−12.5122	−	22.6013i	−0.435355	−	0.786400i
$827$		−	10.7526i		−	0.373904i	−0.982369	−	0.186952i	$-0.940139\pi$
$827$		−	10.7526i		−	0.373904i	0.982369	−	0.186952i	$-0.0598608\pi$
$828$		0			0
$829$			4.73146i			0.164330i	0.996619	+	0.0821652i	$0.0261835\pi$
$829$			4.73146i			0.164330i	−0.996619	+	0.0821652i	$0.973816\pi$
$830$			1.01606i			0.0352678i
$831$		0			0
$832$			4.25272i			0.147436i
$833$	−11.2829	+	18.0132i	−0.390929	+	0.624120i
$834$		0			0
$835$	13.6890			0.473726
$836$	−0.0154711			−0.000535079
$837$		0			0
$838$		−	12.5742i		−	0.434370i
$839$	−20.2176			−0.697991			−0.348995	−	0.937124i	$-0.613477\pi$
$839$	−20.2176			−0.697991			−0.348995	+	0.937124i	$0.613477\pi$
$840$		0			0
$841$	−3.72617			−0.128489
$842$		−	8.06413i		−	0.277908i
$843$		0			0
$844$	−14.3718			−0.494698
$845$	−8.27809			−0.284775
$846$		0			0
$847$	−25.4613	+	14.0955i	−0.874862	+	0.484328i
$848$		−	3.96327i		−	0.136099i
$849$		0			0
$850$		−	7.13690i		−	0.244794i
$851$			17.8672i			0.612481i
$852$		0			0
$853$			1.89316i			0.0648207i	0.999475	+	0.0324104i	$0.0103183\pi$
$853$			1.89316i			0.0648207i	−0.999475	+	0.0324104i	$0.989682\pi$
$854$	−6.99388	+	3.87185i	−0.239326	+	0.132492i
$855$		0			0
$856$	−6.35766			−0.217300
$857$	34.2350			1.16945			0.584723	−	0.811233i	$-0.301203\pi$
$857$	34.2350			1.16945			0.584723	+	0.811233i	$0.301203\pi$
$858$		0			0
$859$		−	25.9859i		−	0.886629i	−0.896366	−	0.443314i	$-0.853803\pi$
$859$		−	25.9859i		−	0.886629i	0.896366	−	0.443314i	$-0.146197\pi$
$860$	2.33770			0.0797147
$861$		0			0
$862$	−16.3403			−0.556554
$863$		−	24.9692i		−	0.849961i	−0.905202	−	0.424981i	$-0.860281\pi$
$863$		−	24.9692i		−	0.849961i	0.905202	−	0.424981i	$-0.139719\pi$
$864$		0			0
$865$	−32.4777			−1.10428
$866$	−29.0783			−0.988123
$867$		0			0
$868$	4.73421	+	8.55161i	0.160690	+	0.290261i
$869$		−	0.223334i		−	0.00757610i
$870$		0			0
$871$			34.7705i			1.17815i
$872$		−	13.9937i		−	0.473885i
$873$		0			0
$874$		−	4.14140i		−	0.140085i
$875$	27.6948	−	15.3320i	0.936256	−	0.518316i
$876$		0			0
$877$	13.6662			0.461475			0.230738	−	0.973016i	$-0.425886\pi$
$877$	13.6662			0.461475			0.230738	+	0.973016i	$0.425886\pi$
$878$	20.6815			0.697965
$879$		0			0
$880$			0.0251832i			0.000848924i
$881$	−44.8789			−1.51201			−0.756004	−	0.654567i	$-0.772851\pi$
$881$	−44.8789			−1.51201			−0.756004	+	0.654567i	$0.772851\pi$
$882$		0			0
$883$	−39.7647			−1.33819			−0.669095	−	0.743177i	$-0.733318\pi$
$883$	−39.7647			−1.33819			−0.669095	+	0.743177i	$0.733318\pi$
$884$		−	12.9131i		−	0.434315i
$885$		0			0
$886$	8.39857			0.282156
$887$	22.8160			0.766086			0.383043	−	0.923731i	$-0.374876\pi$
$887$	22.8160			0.766086			0.383043	+	0.923731i	$0.374876\pi$
$888$		0			0
$889$	−16.6049	+	9.19258i	−0.556912	+	0.308309i
$890$			15.2804i			0.512201i
$891$		0			0
$892$			7.96006i			0.266522i
$893$			5.85351i			0.195880i
$894$		0			0
$895$			13.1350i			0.439056i
$896$	−1.28144	−	2.31472i	−0.0428099	−	0.0773293i
$897$		0			0
$898$	−11.7462			−0.391977
$899$	21.1348			0.704884
$900$		0			0
$901$			12.0342i			0.400919i
$902$	−0.0636062			−0.00211786
$903$		0			0
$904$	17.9326			0.596428
$905$		−	8.89025i		−	0.295522i
$906$		0			0
$907$	45.9101			1.52442			0.762211	−	0.647329i	$-0.224114\pi$
$907$	45.9101			1.52442			0.762211	+	0.647329i	$0.224114\pi$
$908$	11.4937			0.381430
$909$		0			0
$910$	16.0233	−	8.87060i	0.531169	−	0.294058i
$911$			3.44757i			0.114223i	0.998368	+	0.0571116i	$0.0181891\pi$
$911$			3.44757i			0.114223i	−0.998368	+	0.0571116i	$0.981811\pi$
$912$		0			0
$913$		−	0.00965718i		−	0.000319606i
$914$		−	34.7101i		−	1.14811i
$915$		0			0
$916$			13.6660i			0.451537i
$917$	11.6642	−	6.45738i	0.385187	−	0.213242i
$918$		0			0
$919$	2.37032			0.0781895			0.0390947	−	0.999236i	$-0.487553\pi$
$919$	2.37032			0.0781895			0.0390947	+	0.999236i	$0.487553\pi$
$920$	−6.74119			−0.222250
$921$		0			0
$922$			2.82437i			0.0930157i
$923$	−57.6634			−1.89801
$924$		0			0
$925$	−10.1404			−0.333414
$926$		−	30.6440i		−	1.00702i
$927$		0			0
$928$	−5.72068			−0.187791
$929$	46.3022			1.51913			0.759563	−	0.650433i	$-0.225413\pi$
$929$	46.3022			1.51913			0.759563	+	0.650433i	$0.225413\pi$
$930$		0			0
$931$	−5.93234	−	3.71583i	−0.194424	−	0.121781i
$932$			5.61522i			0.183933i
$933$		0			0
$934$			12.6806i			0.414923i
$935$		−	0.0764672i		−	0.00250074i
$936$		0			0
$937$			37.5682i			1.22730i	0.789578	+	0.613650i	$0.210299\pi$
$937$			37.5682i			1.22730i	−0.789578	+	0.613650i	$0.789701\pi$
$938$	−10.4771	−	18.9253i	−0.342090	−	0.617932i
$939$		0			0
$940$	9.52807			0.310772
$941$	32.0436			1.04459			0.522296	−	0.852764i	$-0.325076\pi$
$941$	32.0436			1.04459			0.522296	+	0.852764i	$0.325076\pi$
$942$		0			0
$943$		−	17.0265i		−	0.554460i
$944$	−9.76418			−0.317797
$945$		0			0
$946$	−0.0222188			−0.000722395
$947$		−	17.6805i		−	0.574540i	−0.957850	−	0.287270i	$-0.907252\pi$
$947$		−	17.6805i		−	0.574540i	0.957850	−	0.287270i	$-0.0927477\pi$
$948$		0			0
$949$	43.4216			1.40953
$950$	2.35041			0.0762575
$951$		0			0
$952$	3.89101	+	7.02850i	0.126108	+	0.227795i
$953$			21.6317i			0.700719i	0.936615	+	0.350360i	$0.113941\pi$
$953$			21.6317i			0.700719i	−0.936615	+	0.350360i	$0.886059\pi$
$954$		0			0
$955$			14.0818i			0.455675i
$956$		−	12.7170i		−	0.411296i
$957$		0			0
$958$		−	17.4486i		−	0.563738i
$959$	15.2945	+	27.6270i	0.493884	+	0.892124i
$960$		0			0
$961$	17.3510			0.559710
$962$	−18.3475			−0.591546
$963$		0			0
$964$			3.25681i			0.104895i
$965$	32.7175			1.05321
$966$		0			0
$967$	−59.2349			−1.90487			−0.952433	−	0.304747i	$-0.901428\pi$
$967$	−59.2349			−1.90487			−0.952433	+	0.304747i	$0.901428\pi$
$968$			10.9998i			0.353546i
$969$		0			0
$970$	2.76983			0.0889340
$971$	22.5901			0.724950			0.362475	−	0.931993i	$-0.381932\pi$
$971$	22.5901			0.724950			0.362475	+	0.931993i	$0.381932\pi$
$972$		0			0
$973$	−27.0226	−	48.8120i	−0.866304	−	1.56484i
$974$			4.73684i			0.151778i
$975$		0			0
$976$			3.02148i			0.0967153i
$977$		−	31.9117i		−	1.02095i	−0.859894	−	0.510473i	$-0.829470\pi$
$977$		−	31.9117i		−	1.02095i	0.859894	−	0.510473i	$-0.170530\pi$
$978$		0			0
$979$		−	0.145234i		−	0.00464170i
$980$	−6.04846	+	9.65639i	−0.193211	+	0.308462i
$981$		0			0
$982$	1.91335			0.0610574
$983$	11.8941			0.379362			0.189681	−	0.981846i	$-0.439255\pi$
$983$	11.8941			0.379362			0.189681	+	0.981846i	$0.439255\pi$
$984$		0			0
$985$		−	19.8302i		−	0.631842i
$986$	17.3705			0.553190
$987$		0			0
$988$	4.25272			0.135297
$989$		−	5.94767i		−	0.189125i
$990$		0			0
$991$	27.8517			0.884737			0.442369	−	0.896833i	$-0.354138\pi$
$991$	27.8517			0.884737			0.442369	+	0.896833i	$0.354138\pi$
$992$	3.69445			0.117299
$993$		0			0
$994$	31.3857	−	17.3753i	0.995494	−	0.551110i
$995$			36.8505i			1.16824i
$996$		0			0
$997$			27.1084i			0.858531i	0.903178	+	0.429266i	$0.141228\pi$
$997$			27.1084i			0.858531i	−0.903178	+	0.429266i	$0.858772\pi$
$998$			5.32052i			0.168418i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2394.2.f.b.2015.11	yes	24
3.2	odd	2		2394.2.f.a.2015.12	yes	24
7.6	odd	2		2394.2.f.a.2015.11	✓	24
21.20	even	2	inner	2394.2.f.b.2015.12	yes	24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2394.2.f.a.2015.11	✓	24	7.6	odd	2
2394.2.f.a.2015.12	yes	24	3.2	odd	2
2394.2.f.b.2015.11	yes	24	1.1	even	1	trivial
2394.2.f.b.2015.12	yes	24	21.20	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:	\( N \)	\(=\)	\( 2394 = 2 \cdot 3^{2} \cdot 7 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2394.f (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.00000i q^{2} -1.00000 q^{4} +1.62775 q^{5} +(-2.31472 + 1.28144i) q^{7} +1.00000i q^{8} +O(q^{10})\)	\(q-1.00000i q^{2} -1.00000 q^{4} +1.62775 q^{5} +(-2.31472 + 1.28144i) q^{7} +1.00000i q^{8} -1.62775i q^{10} +0.0154711i q^{11} -4.25272i q^{13} +(1.28144 + 2.31472i) q^{14} +1.00000 q^{16} -3.03644 q^{17} -1.00000i q^{19} -1.62775 q^{20} +0.0154711 q^{22} +4.14140i q^{23} -2.35041 q^{25} -4.25272 q^{26} +(2.31472 - 1.28144i) q^{28} -5.72068i q^{29} +3.69445i q^{31} -1.00000i q^{32} +3.03644i q^{34} +(-3.76779 + 2.08587i) q^{35} +4.31429 q^{37} -1.00000 q^{38} +1.62775i q^{40} -4.11129 q^{41} -1.43615 q^{43} -0.0154711i q^{44} +4.14140 q^{46} -5.85351 q^{47} +(3.71583 - 5.93234i) q^{49} +2.35041i q^{50} +4.25272i q^{52} -3.96327i q^{53} +0.0251832i q^{55} +(-1.28144 - 2.31472i) q^{56} -5.72068 q^{58} -9.76418 q^{59} +3.02148i q^{61} +3.69445 q^{62} -1.00000 q^{64} -6.92238i q^{65} -8.17607 q^{67} +3.03644 q^{68} +(2.08587 + 3.76779i) q^{70} -13.5592i q^{71} +10.2103i q^{73} -4.31429i q^{74} +1.00000i q^{76} +(-0.0198253 - 0.0358112i) q^{77} -14.4356 q^{79} +1.62775 q^{80} +4.11129i q^{82} -0.624208 q^{83} -4.94258 q^{85} +1.43615i q^{86} -0.0154711 q^{88} -9.38743 q^{89} +(5.44959 + 9.84383i) q^{91} -4.14140i q^{92} +5.85351i q^{94} -1.62775i q^{95} +1.70163i q^{97} +(-5.93234 - 3.71583i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q - 24 q^{4} + 4 q^{7}+O(q^{10}) \) 24 * q - 24 * q^4 + 4 * q^7	\( 24 q - 24 q^{4} + 4 q^{7} + 4 q^{14} + 24 q^{16} - 32 q^{17} - 8 q^{22} + 16 q^{25} - 4 q^{28} + 24 q^{35} - 24 q^{38} + 8 q^{41} + 16 q^{43} - 8 q^{46} - 16 q^{49} - 4 q^{56} + 16 q^{58} - 16 q^{59} + 16 q^{62} - 24 q^{64} - 24 q^{67} + 32 q^{68} - 16 q^{70} - 8 q^{77} - 40 q^{79} + 64 q^{83} + 40 q^{85} + 8 q^{88} + 64 q^{89} + 8 q^{91} + 16 q^{98}+O(q^{100}) \) 24 * q - 24 * q^4 + 4 * q^7 + 4 * q^14 + 24 * q^16 - 32 * q^17 - 8 * q^22 + 16 * q^25 - 4 * q^28 + 24 * q^35 - 24 * q^38 + 8 * q^41 + 16 * q^43 - 8 * q^46 - 16 * q^49 - 4 * q^56 + 16 * q^58 - 16 * q^59 + 16 * q^62 - 24 * q^64 - 24 * q^67 + 32 * q^68 - 16 * q^70 - 8 * q^77 - 40 * q^79 + 64 * q^83 + 40 * q^85 + 8 * q^88 + 64 * q^89 + 8 * q^91 + 16 * q^98

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