LMFDB - Embedded newform 2394.2.f.b.2015.5

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			2015.5
Character	$\chi$	$=$	2394.2015
Dual form			2394.2.f.b.2015.6

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{2} -1.00000 q^{4} -3.41122 q^{5} +(0.143354 + 2.64186i) q^{7} +1.00000i q^{8} +O(q^{10})$	\(q-1.00000i q^{2} -1.00000 q^{4} -3.41122 q^{5} +(0.143354 + 2.64186i) q^{7} +1.00000i q^{8} +3.41122i q^{10} -1.10935i q^{11} -6.37897i q^{13} +(2.64186 - 0.143354i) q^{14} +1.00000 q^{16} -7.75444 q^{17} -1.00000i q^{19} +3.41122 q^{20} -1.10935 q^{22} +7.32245i q^{23} +6.63643 q^{25} -6.37897 q^{26} +(-0.143354 - 2.64186i) q^{28} +6.66692i q^{29} -1.05333i q^{31} -1.00000i q^{32} +7.75444i q^{34} +(-0.489012 - 9.01199i) q^{35} +7.54738 q^{37} -1.00000 q^{38} -3.41122i q^{40} -9.61970 q^{41} -2.20410 q^{43} +1.10935i q^{44} +7.32245 q^{46} +11.7695 q^{47} +(-6.95890 + 0.757443i) q^{49} -6.63643i q^{50} +6.37897i q^{52} -12.9351i q^{53} +3.78425i q^{55} +(-2.64186 + 0.143354i) q^{56} +6.66692 q^{58} +7.90580 q^{59} +0.763158i q^{61} -1.05333 q^{62} -1.00000 q^{64} +21.7601i q^{65} +9.80062 q^{67} +7.75444 q^{68} +(-9.01199 + 0.489012i) q^{70} +1.42491i q^{71} -3.02282i q^{73} -7.54738i q^{74} +1.00000i q^{76} +(2.93076 - 0.159030i) q^{77} +6.38653 q^{79} -3.41122 q^{80} +9.61970i q^{82} +12.9476 q^{83} +26.4521 q^{85} +2.20410i q^{86} +1.10935 q^{88} +8.74842 q^{89} +(16.8524 - 0.914451i) q^{91} -7.32245i q^{92} -11.7695i q^{94} +3.41122i q^{95} +7.54136i q^{97} +(0.757443 + 6.95890i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	−3.41122			−1.52554			−0.762772	−	0.646667i	$-0.776162\pi$
$5$	−3.41122			−1.52554			−0.762772	+	0.646667i	$0.776162\pi$
$6$		0			0
$7$	0.143354	+	2.64186i	0.0541827	+	0.998531i
$7$	0.143354	+	2.64186i	0.0541827	+	0.998531i
$8$			1.00000i			0.353553i
$9$		0			0
$10$			3.41122i			1.07872i
$11$		−	1.10935i		−	0.334482i	−0.985916	−	0.167241i	$-0.946514\pi$
$11$		−	1.10935i		−	0.334482i	0.985916	−	0.167241i	$-0.0534858\pi$
$12$		0			0
$13$		−	6.37897i		−	1.76921i	−0.466342	−	0.884604i	$-0.654429\pi$
$13$		−	6.37897i		−	1.76921i	0.466342	−	0.884604i	$-0.345571\pi$
$14$	2.64186	−	0.143354i	0.706068	−	0.0383129i
$15$		0			0
$16$	1.00000			0.250000
$17$	−7.75444			−1.88073			−0.940364	−	0.340170i	$-0.889515\pi$
$17$	−7.75444			−1.88073			−0.940364	+	0.340170i	$0.889515\pi$
$18$		0			0
$19$		−	1.00000i		−	0.229416i
$19$		−	1.00000i		−	0.229416i
$20$	3.41122			0.762772
$21$		0			0
$22$	−1.10935			−0.236515
$23$			7.32245i			1.52684i	0.645905	+	0.763418i	$0.276480\pi$
$23$			7.32245i			1.52684i	−0.645905	+	0.763418i	$0.723520\pi$
$24$		0			0
$25$	6.63643			1.32729
$26$	−6.37897			−1.25102
$27$		0			0
$28$	−0.143354	−	2.64186i	−0.0270913	−	0.499266i
$29$			6.66692i			1.23802i	0.785385	+	0.619008i	$0.212465\pi$
$29$			6.66692i			1.23802i	−0.785385	+	0.619008i	$0.787535\pi$
$30$		0			0
$31$		−	1.05333i		−	0.189183i	−0.995516	−	0.0945916i	$-0.969845\pi$
$31$		−	1.05333i		−	0.189183i	0.995516	−	0.0945916i	$-0.0301545\pi$
$32$		−	1.00000i		−	0.176777i
$33$		0			0
$34$			7.75444i			1.32988i
$35$	−0.489012	−	9.01199i	−0.0826581	−	1.52330i
$36$		0			0
$37$	7.54738			1.24078			0.620391	−	0.784293i	$-0.286974\pi$
$37$	7.54738			1.24078			0.620391	+	0.784293i	$0.286974\pi$
$38$	−1.00000			−0.162221
$39$		0			0
$40$		−	3.41122i		−	0.539362i
$41$	−9.61970			−1.50234			−0.751172	−	0.660106i	$-0.770511\pi$
$41$	−9.61970			−1.50234			−0.751172	+	0.660106i	$0.770511\pi$
$42$		0			0
$43$	−2.20410			−0.336122			−0.168061	−	0.985777i	$-0.553751\pi$
$43$	−2.20410			−0.336122			−0.168061	+	0.985777i	$0.553751\pi$
$44$			1.10935i			0.167241i
$45$		0			0
$46$	7.32245			1.07964
$47$	11.7695			1.71677			0.858383	−	0.513010i	$-0.171470\pi$
$47$	11.7695			1.71677			0.858383	+	0.513010i	$0.171470\pi$
$48$		0			0
$49$	−6.95890	+	0.757443i	−0.994128	+	0.108206i
$50$		−	6.63643i		−	0.938533i
$51$		0			0
$52$			6.37897i			0.884604i
$53$		−	12.9351i		−	1.77678i	−0.459094	−	0.888388i	$-0.651826\pi$
$53$		−	12.9351i		−	1.77678i	0.459094	−	0.888388i	$-0.348174\pi$
$54$		0			0
$55$			3.78425i			0.510268i
$56$	−2.64186	+	0.143354i	−0.353034	+	0.0191565i
$57$		0			0
$58$	6.66692			0.875409
$59$	7.90580			1.02925			0.514624	−	0.857416i	$-0.327932\pi$
$59$	7.90580			1.02925			0.514624	+	0.857416i	$0.327932\pi$
$60$		0			0
$61$			0.763158i			0.0977123i	0.998806	+	0.0488562i	$0.0155576\pi$
$61$			0.763158i			0.0977123i	−0.998806	+	0.0488562i	$0.984442\pi$
$62$	−1.05333			−0.133773
$63$		0			0
$64$	−1.00000			−0.125000
$65$			21.7601i			2.69901i
$66$		0			0
$67$	9.80062			1.19734			0.598668	−	0.800997i	$-0.295697\pi$
$67$	9.80062			1.19734			0.598668	+	0.800997i	$0.295697\pi$
$68$	7.75444			0.940364
$69$		0			0
$70$	−9.01199	+	0.489012i	−1.07714	+	0.0584481i
$71$			1.42491i			0.169106i	0.996419	+	0.0845530i	$0.0269462\pi$
$71$			1.42491i			0.169106i	−0.996419	+	0.0845530i	$0.973054\pi$
$72$		0			0
$73$		−	3.02282i		−	0.353794i	−0.984229	−	0.176897i	$-0.943394\pi$
$73$		−	3.02282i		−	0.353794i	0.984229	−	0.176897i	$-0.0566060\pi$
$74$		−	7.54738i		−	0.877365i
$75$		0			0
$76$			1.00000i			0.114708i
$77$	2.93076	−	0.159030i	0.333991	−	0.0181231i
$78$		0			0
$79$	6.38653			0.718541			0.359270	−	0.933234i	$-0.383026\pi$
$79$	6.38653			0.718541			0.359270	+	0.933234i	$0.383026\pi$
$80$	−3.41122			−0.381386
$81$		0			0
$82$			9.61970i			1.06232i
$83$	12.9476			1.42118			0.710590	−	0.703606i	$-0.248428\pi$
$83$	12.9476			1.42118			0.710590	+	0.703606i	$0.248428\pi$
$84$		0			0
$85$	26.4521			2.86913
$86$			2.20410i			0.237674i
$87$		0			0
$88$	1.10935			0.118257
$89$	8.74842			0.927330			0.463665	−	0.886011i	$-0.346534\pi$
$89$	8.74842			0.927330			0.463665	+	0.886011i	$0.346534\pi$
$90$		0			0
$91$	16.8524	−	0.914451i	1.76661	−	0.0958605i
$92$		−	7.32245i		−	0.763418i
$93$		0			0
$94$		−	11.7695i		−	1.21394i
$95$			3.41122i			0.349984i
$96$		0			0
$97$			7.54136i			0.765709i	0.923809	+	0.382854i	$0.125059\pi$
$97$			7.54136i			0.765709i	−0.923809	+	0.382854i	$0.874941\pi$
$98$	0.757443	+	6.95890i	0.0765133	+	0.702955i
$99$		0			0
$100$	−6.63643			−0.663643
$101$	−4.13372			−0.411321			−0.205660	−	0.978623i	$-0.565934\pi$
$101$	−4.13372			−0.411321			−0.205660	+	0.978623i	$0.565934\pi$
$102$		0			0
$103$			9.90178i			0.975652i	0.872941	+	0.487826i	$0.162210\pi$
$103$			9.90178i			0.975652i	−0.872941	+	0.487826i	$0.837790\pi$
$104$	6.37897			0.625510
$105$		0			0
$106$	−12.9351			−1.25637
$107$			8.19252i			0.792001i	0.918250	+	0.396001i	$0.129602\pi$
$107$			8.19252i			0.792001i	−0.918250	+	0.396001i	$0.870398\pi$
$108$		0			0
$109$	1.20641			0.115553			0.0577766	−	0.998330i	$-0.481599\pi$
$109$	1.20641			0.115553			0.0577766	+	0.998330i	$0.481599\pi$
$110$	3.78425			0.360814
$111$		0			0
$112$	0.143354	+	2.64186i	0.0135457	+	0.249633i
$113$		−	7.75674i		−	0.729693i	−0.931068	−	0.364846i	$-0.881122\pi$
$113$		−	7.75674i		−	0.729693i	0.931068	−	0.364846i	$-0.118878\pi$
$114$		0			0
$115$		−	24.9785i		−	2.32926i
$116$		−	6.66692i		−	0.619008i
$117$		0			0
$118$		−	7.90580i		−	0.727788i
$119$	−1.11163	−	20.4862i	−0.101903	−	1.87797i
$120$		0			0
$121$	9.76934			0.888122
$122$	0.763158			0.0690931
$123$		0			0
$124$			1.05333i			0.0945916i
$125$	−5.58224			−0.499290
$126$		0			0
$127$	1.65345			0.146720			0.0733601	−	0.997306i	$-0.476628\pi$
$127$	1.65345			0.146720			0.0733601	+	0.997306i	$0.476628\pi$
$128$			1.00000i			0.0883883i
$129$		0			0
$130$	21.7601			1.90849
$131$	14.4366			1.26134			0.630668	−	0.776053i	$-0.282781\pi$
$131$	14.4366			1.26134			0.630668	+	0.776053i	$0.282781\pi$
$132$		0			0
$133$	2.64186	−	0.143354i	0.229079	−	0.0124304i
$134$		−	9.80062i		−	0.846645i
$135$		0			0
$136$		−	7.75444i		−	0.664938i
$137$		−	2.02858i		−	0.173314i	−0.996238	−	0.0866568i	$-0.972382\pi$
$137$		−	2.02858i		−	0.173314i	0.996238	−	0.0866568i	$-0.0276184\pi$
$138$		0			0
$139$		−	0.461013i		−	0.0391026i	−0.999809	−	0.0195513i	$-0.993776\pi$
$139$		−	0.461013i		−	0.0391026i	0.999809	−	0.0195513i	$-0.00622377\pi$
$140$	0.489012	+	9.01199i	0.0413291	+	0.761652i
$141$		0			0
$142$	1.42491			0.119576
$143$	−7.07653			−0.591769
$144$		0			0
$145$		−	22.7423i		−	1.88865i
$146$	−3.02282			−0.250170
$147$		0			0
$148$	−7.54738			−0.620391
$149$		−	15.0557i		−	1.23341i	−0.787195	−	0.616704i	$-0.788468\pi$
$149$		−	15.0557i		−	1.23341i	0.787195	−	0.616704i	$-0.211532\pi$
$150$		0			0
$151$	9.82278			0.799366			0.399683	−	0.916653i	$-0.369120\pi$
$151$	9.82278			0.799366			0.399683	+	0.916653i	$0.369120\pi$
$152$	1.00000			0.0811107
$153$		0			0
$154$	−0.159030	−	2.93076i	−0.0128150	−	0.236167i
$155$			3.59313i			0.288607i
$156$		0			0
$157$			12.2486i			0.977541i	0.872412	+	0.488771i	$0.162555\pi$
$157$			12.2486i			0.977541i	−0.872412	+	0.488771i	$0.837445\pi$
$158$		−	6.38653i		−	0.508085i
$159$		0			0
$160$			3.41122i			0.269681i
$161$	−19.3449	+	1.04970i	−1.52459	+	0.0827281i
$162$		0			0
$163$	−4.89771			−0.383618			−0.191809	−	0.981432i	$-0.561435\pi$
$163$	−4.89771			−0.383618			−0.191809	+	0.981432i	$0.561435\pi$
$164$	9.61970			0.751172
$165$		0			0
$166$		−	12.9476i		−	1.00493i
$167$	5.85191			0.452834			0.226417	−	0.974030i	$-0.427299\pi$
$167$	5.85191			0.452834			0.226417	+	0.974030i	$0.427299\pi$
$168$		0			0
$169$	−27.6913			−2.13010
$170$		−	26.4521i		−	2.02878i
$171$		0			0
$172$	2.20410			0.168061
$173$	−8.82975			−0.671313			−0.335657	−	0.941984i	$-0.608958\pi$
$173$	−8.82975			−0.671313			−0.335657	+	0.941984i	$0.608958\pi$
$174$		0			0
$175$	0.951359	+	17.5326i	0.0719160	+	1.32534i
$176$		−	1.10935i		−	0.0836206i
$177$		0			0
$178$		−	8.74842i		−	0.655722i
$179$			19.4923i			1.45692i	0.685088	+	0.728460i	$0.259764\pi$
$179$			19.4923i			1.45692i	−0.685088	+	0.728460i	$0.740236\pi$
$180$		0			0
$181$		−	5.77116i		−	0.428967i	−0.976728	−	0.214483i	$-0.931193\pi$
$181$		−	5.77116i		−	0.428967i	0.976728	−	0.214483i	$-0.0688068\pi$
$182$	−0.914451	−	16.8524i	−0.0677836	−	1.24918i
$183$		0			0
$184$	−7.32245			−0.539818
$185$	−25.7458			−1.89287
$186$		0			0
$187$			8.60241i			0.629070i
$188$	−11.7695			−0.858383
$189$		0			0
$190$	3.41122			0.247476
$191$		−	15.5161i		−	1.12270i	−0.827577	−	0.561352i	$-0.810281\pi$
$191$		−	15.5161i		−	1.12270i	0.827577	−	0.561352i	$-0.189719\pi$
$192$		0			0
$193$	−1.89748			−0.136584			−0.0682919	−	0.997665i	$-0.521755\pi$
$193$	−1.89748			−0.136584			−0.0682919	+	0.997665i	$0.521755\pi$
$194$	7.54136			0.541438
$195$		0			0
$196$	6.95890	−	0.757443i	0.497064	−	0.0541031i
$197$		−	21.2526i		−	1.51418i	−0.653309	−	0.757091i	$-0.726620\pi$
$197$		−	21.2526i		−	1.51418i	0.653309	−	0.757091i	$-0.273380\pi$
$198$		0			0
$199$		−	3.73822i		−	0.264995i	−0.991183	−	0.132498i	$-0.957700\pi$
$199$		−	3.73822i		−	0.264995i	0.991183	−	0.132498i	$-0.0422997\pi$
$200$			6.63643i			0.469267i
$201$		0			0
$202$			4.13372i			0.290848i
$203$	−17.6131	+	0.955729i	−1.23620	+	0.0670790i
$204$		0			0
$205$	32.8149			2.29189
$206$	9.90178			0.689890
$207$		0			0
$208$		−	6.37897i		−	0.442302i
$209$	−1.10935			−0.0767355
$210$		0			0
$211$	13.0726			0.899952			0.449976	−	0.893041i	$-0.351433\pi$
$211$	13.0726			0.899952			0.449976	+	0.893041i	$0.351433\pi$
$212$			12.9351i			0.888388i
$213$		0			0
$214$	8.19252			0.560029
$215$	7.51867			0.512769
$216$		0			0
$217$	2.78275	−	0.150999i	0.188905	−	0.0102505i
$218$		−	1.20641i		−	0.0817084i
$219$		0			0
$220$		−	3.78425i		−	0.255134i
$221$			49.4654i			3.32740i
$222$		0			0
$223$			24.1584i			1.61776i	0.587972	+	0.808881i	$0.299927\pi$
$223$			24.1584i			1.61776i	−0.587972	+	0.808881i	$0.700073\pi$
$224$	2.64186	−	0.143354i	0.176517	−	0.00957824i
$225$		0			0
$226$	−7.75674			−0.515971
$227$	−14.0002			−0.929227			−0.464614	−	0.885514i	$-0.653807\pi$
$227$	−14.0002			−0.929227			−0.464614	+	0.885514i	$0.653807\pi$
$228$		0			0
$229$			19.2950i			1.27505i	0.770429	+	0.637526i	$0.220042\pi$
$229$			19.2950i			1.27505i	−0.770429	+	0.637526i	$0.779958\pi$
$230$	−24.9785			−1.64703
$231$		0			0
$232$	−6.66692			−0.437705
$233$			1.83418i			0.120161i	0.998194	+	0.0600806i	$0.0191358\pi$
$233$			1.83418i			0.120161i	−0.998194	+	0.0600806i	$0.980864\pi$
$234$		0			0
$235$	−40.1485			−2.61900
$236$	−7.90580			−0.514624
$237$		0			0
$238$	−20.4862	+	1.11163i	−1.32792	+	0.0720562i
$239$		−	3.92223i		−	0.253708i	−0.991921	−	0.126854i	$-0.959512\pi$
$239$		−	3.92223i		−	0.253708i	0.991921	−	0.126854i	$-0.0404880\pi$
$240$		0			0
$241$		−	12.2645i		−	0.790027i	−0.918675	−	0.395013i	$-0.870740\pi$
$241$		−	12.2645i		−	0.790027i	0.918675	−	0.395013i	$-0.129260\pi$
$242$		−	9.76934i		−	0.627997i
$243$		0			0
$244$		−	0.763158i		−	0.0488562i
$245$	23.7383	−	2.58381i	1.51659	−	0.165073i
$246$		0			0
$247$	−6.37897			−0.405884
$248$	1.05333			0.0668864
$249$		0			0
$250$			5.58224i			0.353052i
$251$	0.363288			0.0229305			0.0114653	−	0.999934i	$-0.496350\pi$
$251$	0.363288			0.0229305			0.0114653	+	0.999934i	$0.496350\pi$
$252$		0			0
$253$	8.12318			0.510700
$254$		−	1.65345i		−	0.103747i
$255$		0			0
$256$	1.00000			0.0625000
$257$	19.6511			1.22580			0.612902	−	0.790159i	$-0.290002\pi$
$257$	19.6511			1.22580			0.612902	+	0.790159i	$0.290002\pi$
$258$		0			0
$259$	1.08195	+	19.9392i	0.0672289	+	1.23896i
$260$		−	21.7601i		−	1.34950i
$261$		0			0
$262$		−	14.4366i		−	0.891899i
$263$		−	2.40409i		−	0.148243i	−0.997249	−	0.0741213i	$-0.976385\pi$
$263$		−	2.40409i		−	0.148243i	0.997249	−	0.0741213i	$-0.0236152\pi$
$264$		0			0
$265$			44.1246i			2.71055i
$266$	−0.143354	−	2.64186i	−0.00878959	−	0.161983i
$267$		0			0
$268$	−9.80062			−0.598668
$269$	19.8712			1.21157			0.605785	−	0.795628i	$-0.292859\pi$
$269$	19.8712			1.21157			0.605785	+	0.795628i	$0.292859\pi$
$270$		0			0
$271$			8.91962i			0.541828i	0.962604	+	0.270914i	$0.0873259\pi$
$271$			8.91962i			0.541828i	−0.962604	+	0.270914i	$0.912674\pi$
$272$	−7.75444			−0.470182
$273$		0			0
$274$	−2.02858			−0.122551
$275$		−	7.36214i		−	0.443954i
$276$		0			0
$277$	−31.3719			−1.88496			−0.942478	−	0.334267i	$-0.891511\pi$
$277$	−31.3719			−1.88496			−0.942478	+	0.334267i	$0.891511\pi$
$278$	−0.461013			−0.0276497
$279$		0			0
$280$	9.01199	−	0.489012i	0.538569	−	0.0292241i
$281$		−	4.19449i		−	0.250222i	−0.992143	−	0.125111i	$-0.960071\pi$
$281$		−	4.19449i		−	0.250222i	0.992143	−	0.125111i	$-0.0399287\pi$
$282$		0			0
$283$			31.9196i			1.89743i	0.316138	+	0.948713i	$0.397614\pi$
$283$			31.9196i			1.89743i	−0.316138	+	0.948713i	$0.602386\pi$
$284$		−	1.42491i		−	0.0845530i
$285$		0			0
$286$			7.07653i			0.418444i
$287$	−1.37902	−	25.4139i	−0.0814011	−	1.50014i
$288$		0			0
$289$	43.1313			2.53714
$290$	−22.7423			−1.33548
$291$		0			0
$292$			3.02282i			0.176897i
$293$	−4.23835			−0.247607			−0.123804	−	0.992307i	$-0.539509\pi$
$293$	−4.23835			−0.247607			−0.123804	+	0.992307i	$0.539509\pi$
$294$		0			0
$295$	−26.9684			−1.57016
$296$			7.54738i			0.438682i
$297$		0			0
$298$	−15.0557			−0.872151
$299$	46.7097			2.70129
$300$		0			0
$301$	−0.315966	−	5.82293i	−0.0182120	−	0.335628i
$302$		−	9.82278i		−	0.565237i
$303$		0			0
$304$		−	1.00000i		−	0.0573539i
$305$		−	2.60330i		−	0.149065i
$306$		0			0
$307$		−	25.8057i		−	1.47281i	−0.676541	−	0.736405i	$-0.736522\pi$
$307$		−	25.8057i		−	1.47281i	0.676541	−	0.736405i	$-0.263478\pi$
$308$	−2.93076	+	0.159030i	−0.166995	+	0.00906157i
$309$		0			0
$310$	3.59313			0.204076
$311$	−18.3786			−1.04215			−0.521077	−	0.853510i	$-0.674470\pi$
$311$	−18.3786			−1.04215			−0.521077	+	0.853510i	$0.674470\pi$
$312$		0			0
$313$			10.7582i			0.608090i	0.952658	+	0.304045i	$0.0983374\pi$
$313$			10.7582i			0.608090i	−0.952658	+	0.304045i	$0.901663\pi$
$314$	12.2486			0.691226
$315$		0			0
$316$	−6.38653			−0.359270
$317$			27.0813i			1.52104i	0.649315	+	0.760520i	$0.275056\pi$
$317$			27.0813i			1.52104i	−0.649315	+	0.760520i	$0.724944\pi$
$318$		0			0
$319$	7.39596			0.414094
$320$	3.41122			0.190693
$321$		0			0
$322$	1.04970	+	19.3449i	0.0584976	+	1.07805i
$323$			7.75444i			0.431469i
$324$		0			0
$325$		−	42.3336i		−	2.34825i
$326$			4.89771i			0.271259i
$327$		0			0
$328$		−	9.61970i		−	0.531159i
$329$	1.68721	+	31.0936i	0.0930189	+	1.71424i
$330$		0			0
$331$	3.59739			0.197730			0.0988651	−	0.995101i	$-0.468479\pi$
$331$	3.59739			0.197730			0.0988651	+	0.995101i	$0.468479\pi$
$332$	−12.9476			−0.710590
$333$		0			0
$334$		−	5.85191i		−	0.320202i
$335$	−33.4321			−1.82659
$336$		0			0
$337$	−8.61773			−0.469438			−0.234719	−	0.972063i	$-0.575417\pi$
$337$	−8.61773			−0.469438			−0.234719	+	0.972063i	$0.575417\pi$
$338$			27.6913i			1.50621i
$339$		0			0
$340$	−26.4521			−1.43457
$341$	−1.16851			−0.0632784
$342$		0			0
$343$	−2.99865	−	18.2759i	−0.161912	−	0.986805i
$344$		−	2.20410i		−	0.118837i
$345$		0			0
$346$			8.82975i			0.474690i
$347$			33.4662i			1.79656i	0.439425	+	0.898279i	$0.355182\pi$
$347$			33.4662i			1.79656i	−0.439425	+	0.898279i	$0.644818\pi$
$348$		0			0
$349$		−	27.7640i		−	1.48617i	−0.669195	−	0.743087i	$-0.733361\pi$
$349$		−	27.7640i		−	1.48617i	0.669195	−	0.743087i	$-0.266639\pi$
$350$	17.5326	−	0.951359i	0.937155	−	0.0508523i
$351$		0			0
$352$	−1.10935			−0.0591287
$353$	11.1618			0.594083			0.297042	−	0.954865i	$-0.404000\pi$
$353$	11.1618			0.594083			0.297042	+	0.954865i	$0.404000\pi$
$354$		0			0
$355$		−	4.86069i		−	0.257979i
$356$	−8.74842			−0.463665
$357$		0			0
$358$	19.4923			1.03020
$359$		−	21.3200i		−	1.12522i	−0.826721	−	0.562612i	$-0.809796\pi$
$359$		−	21.3200i		−	1.12522i	0.826721	−	0.562612i	$-0.190204\pi$
$360$		0			0
$361$	−1.00000			−0.0526316
$362$	−5.77116			−0.303325
$363$		0			0
$364$	−16.8524	+	0.914451i	−0.883305	+	0.0479302i
$365$			10.3115i			0.539729i
$366$		0			0
$367$		−	17.2337i		−	0.899591i	−0.893132	−	0.449795i	$-0.851497\pi$
$367$		−	17.2337i		−	0.899591i	0.893132	−	0.449795i	$-0.148503\pi$
$368$			7.32245i			0.381709i
$369$		0			0
$370$			25.7458i			1.33846i
$371$	34.1728	−	1.85430i	1.77417	−	0.0962704i
$372$		0			0
$373$	10.9441			0.566665			0.283332	−	0.959022i	$-0.408560\pi$
$373$	10.9441			0.566665			0.283332	+	0.959022i	$0.408560\pi$
$374$	8.60241			0.444820
$375$		0			0
$376$			11.7695i			0.606968i
$377$	42.5281			2.19031
$378$		0			0
$379$	−20.2540			−1.04038			−0.520190	−	0.854051i	$-0.674139\pi$
$379$	−20.2540			−1.04038			−0.520190	+	0.854051i	$0.674139\pi$
$380$		−	3.41122i		−	0.174992i
$381$		0			0
$382$	−15.5161			−0.793871
$383$	−25.7193			−1.31420			−0.657098	−	0.753805i	$-0.728216\pi$
$383$	−25.7193			−1.31420			−0.657098	+	0.753805i	$0.728216\pi$
$384$		0			0
$385$	−9.99747	+	0.542487i	−0.509518	+	0.0276477i
$386$			1.89748i			0.0965794i
$387$		0			0
$388$		−	7.54136i		−	0.382854i
$389$		−	12.9820i		−	0.658216i	−0.944292	−	0.329108i	$-0.893252\pi$
$389$		−	12.9820i		−	0.658216i	0.944292	−	0.329108i	$-0.106748\pi$
$390$		0			0
$391$		−	56.7815i		−	2.87156i
$392$	−0.757443	−	6.95890i	−0.0382567	−	0.351477i
$393$		0			0
$394$	−21.2526			−1.07069
$395$	−21.7859			−1.09617
$396$		0			0
$397$		−	15.9008i		−	0.798041i	−0.916942	−	0.399020i	$-0.869350\pi$
$397$		−	15.9008i		−	0.798041i	0.916942	−	0.399020i	$-0.130650\pi$
$398$	−3.73822			−0.187380
$399$		0			0
$400$	6.63643			0.331822
$401$			10.9593i			0.547282i	0.961832	+	0.273641i	$0.0882280\pi$
$401$			10.9593i			0.547282i	−0.961832	+	0.273641i	$0.911772\pi$
$402$		0			0
$403$	−6.71915			−0.334704
$404$	4.13372			0.205660
$405$		0			0
$406$	0.955729	+	17.6131i	0.0474320	+	0.874123i
$407$		−	8.37270i		−	0.415019i
$408$		0			0
$409$			4.69585i			0.232195i	0.993238	+	0.116097i	$0.0370385\pi$
$409$			4.69585i			0.232195i	−0.993238	+	0.116097i	$0.962962\pi$
$410$		−	32.8149i		−	1.62061i
$411$		0			0
$412$		−	9.90178i		−	0.487826i
$413$	1.13333	+	20.8861i	0.0557674	+	1.02774i
$414$		0			0
$415$	−44.1670			−2.16807
$416$	−6.37897			−0.312755
$417$		0			0
$418$			1.10935i			0.0542602i
$419$	−8.80429			−0.430118			−0.215059	−	0.976601i	$-0.568994\pi$
$419$	−8.80429			−0.430118			−0.215059	+	0.976601i	$0.568994\pi$
$420$		0			0
$421$	−1.31840			−0.0642549			−0.0321275	−	0.999484i	$-0.510228\pi$
$421$	−1.31840			−0.0642549			−0.0321275	+	0.999484i	$0.510228\pi$
$422$		−	13.0726i		−	0.636362i
$423$		0			0
$424$	12.9351			0.628185
$425$	−51.4618			−2.49627
$426$		0			0
$427$	−2.01616	+	0.109402i	−0.0975688	+	0.00529432i
$428$		−	8.19252i		−	0.396001i
$429$		0			0
$430$		−	7.51867i		−	0.362582i
$431$		−	12.3039i		−	0.592657i	−0.955086	−	0.296328i	$-0.904238\pi$
$431$		−	12.3039i		−	0.592657i	0.955086	−	0.296328i	$-0.0957622\pi$
$432$		0			0
$433$			10.9893i			0.528111i	0.964507	+	0.264056i	$0.0850603\pi$
$433$			10.9893i			0.528111i	−0.964507	+	0.264056i	$0.914940\pi$
$434$	−0.150999	−	2.78275i	−0.00724816	−	0.133576i
$435$		0			0
$436$	−1.20641			−0.0577766
$437$	7.32245			0.350280
$438$		0			0
$439$			27.9495i			1.33396i	0.745077	+	0.666978i	$0.232412\pi$
$439$			27.9495i			1.33396i	−0.745077	+	0.666978i	$0.767588\pi$
$440$	−3.78425			−0.180407
$441$		0			0
$442$	49.4654			2.35283
$443$			27.9924i			1.32996i	0.746862	+	0.664979i	$0.231560\pi$
$443$			27.9924i			1.32996i	−0.746862	+	0.664979i	$0.768440\pi$
$444$		0			0
$445$	−29.8428			−1.41468
$446$	24.1584			1.14393
$447$		0			0
$448$	−0.143354	−	2.64186i	−0.00677284	−	0.124816i
$449$			30.6831i			1.44803i	0.689787	+	0.724013i	$0.257704\pi$
$449$			30.6831i			1.44803i	−0.689787	+	0.724013i	$0.742296\pi$
$450$		0			0
$451$			10.6716i			0.502508i
$452$			7.75674i			0.364846i
$453$		0			0
$454$			14.0002i			0.657063i
$455$	−57.4872	+	3.11939i	−2.69504	+	0.146239i
$456$		0			0
$457$	21.0046			0.982554			0.491277	−	0.871003i	$-0.336530\pi$
$457$	21.0046			0.982554			0.491277	+	0.871003i	$0.336530\pi$
$458$	19.2950			0.901597
$459$		0			0
$460$			24.9785i			1.16463i
$461$	12.1422			0.565519			0.282759	−	0.959191i	$-0.408750\pi$
$461$	12.1422			0.565519			0.282759	+	0.959191i	$0.408750\pi$
$462$		0			0
$463$	−16.0862			−0.747588			−0.373794	−	0.927512i	$-0.621943\pi$
$463$	−16.0862			−0.747588			−0.373794	+	0.927512i	$0.621943\pi$
$464$			6.66692i			0.309504i
$465$		0			0
$466$	1.83418			0.0849669
$467$	25.1312			1.16293			0.581466	−	0.813571i	$-0.302479\pi$
$467$	25.1312			1.16293			0.581466	+	0.813571i	$0.302479\pi$
$468$		0			0
$469$	1.40496	+	25.8919i	0.0648749	+	1.19558i
$470$			40.1485i			1.85191i
$471$		0			0
$472$			7.90580i			0.363894i
$473$			2.44512i			0.112427i
$474$		0			0
$475$		−	6.63643i		−	0.304500i
$476$	1.11163	+	20.4862i	0.0509514	+	0.938983i
$477$		0			0
$478$	−3.92223			−0.179399
$479$	18.8006			0.859019			0.429510	−	0.903062i	$-0.358686\pi$
$479$	18.8006			0.859019			0.429510	+	0.903062i	$0.358686\pi$
$480$		0			0
$481$		−	48.1445i		−	2.19520i
$482$	−12.2645			−0.558633
$483$		0			0
$484$	−9.76934			−0.444061
$485$		−	25.7252i		−	1.16812i
$486$		0			0
$487$	32.1207			1.45553			0.727765	−	0.685827i	$-0.240559\pi$
$487$	32.1207			1.45553			0.727765	+	0.685827i	$0.240559\pi$
$488$	−0.763158			−0.0345465
$489$		0			0
$490$	−2.58381	−	23.7383i	−0.116724	−	1.07239i
$491$		−	39.5682i		−	1.78569i	−0.450366	−	0.892844i	$-0.648707\pi$
$491$		−	39.5682i		−	1.78569i	0.450366	−	0.892844i	$-0.351293\pi$
$492$		0			0
$493$		−	51.6982i		−	2.32837i
$494$			6.37897i			0.287004i
$495$		0			0
$496$		−	1.05333i		−	0.0472958i
$497$	−3.76443	+	0.204267i	−0.168858	+	0.00916262i
$498$		0			0
$499$	5.95956			0.266787			0.133393	−	0.991063i	$-0.457413\pi$
$499$	5.95956			0.266787			0.133393	+	0.991063i	$0.457413\pi$
$500$	5.58224			0.249645
$501$		0			0
$502$		−	0.363288i		−	0.0162143i
$503$	14.1591			0.631324			0.315662	−	0.948872i	$-0.397773\pi$
$503$	14.1591			0.631324			0.315662	+	0.948872i	$0.397773\pi$
$504$		0			0
$505$	14.1010			0.627488
$506$		−	8.12318i		−	0.361119i
$507$		0			0
$508$	−1.65345			−0.0733601
$509$	36.3191			1.60982			0.804908	−	0.593400i	$-0.202215\pi$
$509$	36.3191			1.60982			0.804908	+	0.593400i	$0.202215\pi$
$510$		0			0
$511$	7.98589	−	0.433333i	0.353275	−	0.0191695i
$512$		−	1.00000i		−	0.0441942i
$513$		0			0
$514$		−	19.6511i		−	0.866775i
$515$		−	33.7772i		−	1.48840i
$516$		0			0
$517$		−	13.0566i		−	0.574228i
$518$	19.9392	−	1.08195i	0.876076	−	0.0475380i
$519$		0			0
$520$	−21.7601			−0.954243
$521$	−3.03373			−0.132910			−0.0664552	−	0.997789i	$-0.521169\pi$
$521$	−3.03373			−0.132910			−0.0664552	+	0.997789i	$0.521169\pi$
$522$		0			0
$523$			12.6447i			0.552914i	0.961026	+	0.276457i	$0.0891603\pi$
$523$			12.6447i			0.552914i	−0.961026	+	0.276457i	$0.910840\pi$
$524$	−14.4366			−0.630668
$525$		0			0
$526$	−2.40409			−0.104823
$527$			8.16796i			0.355802i
$528$		0			0
$529$	−30.6183			−1.33123
$530$	44.1246			1.91665
$531$		0			0
$532$	−2.64186	+	0.143354i	−0.114539	+	0.00621518i
$533$			61.3638i			2.65796i
$534$		0			0
$535$		−	27.9465i		−	1.20823i
$536$			9.80062i			0.423322i
$537$		0			0
$538$		−	19.8712i		−	0.856709i
$539$	0.840271	+	7.71987i	0.0361931	+	0.332518i
$540$		0			0
$541$	−43.5673			−1.87311			−0.936553	−	0.350526i	$-0.886003\pi$
$541$	−43.5673			−1.87311			−0.936553	+	0.350526i	$0.886003\pi$
$542$	8.91962			0.383130
$543$		0			0
$544$			7.75444i			0.332469i
$545$	−4.11533			−0.176282
$546$		0			0
$547$	30.5858			1.30776			0.653878	−	0.756600i	$-0.273141\pi$
$547$	30.5858			1.30776			0.653878	+	0.756600i	$0.273141\pi$
$548$			2.02858i			0.0866568i
$549$		0			0
$550$	−7.36214			−0.313923
$551$	6.66692			0.284020
$552$		0			0
$553$	0.915534	+	16.8723i	0.0389325	+	0.717485i
$554$			31.3719i			1.33287i
$555$		0			0
$556$			0.461013i			0.0195513i
$557$		−	4.65166i		−	0.197097i	−0.995132	−	0.0985486i	$-0.968580\pi$
$557$		−	4.65166i		−	0.197097i	0.995132	−	0.0985486i	$-0.0314200\pi$
$558$		0			0
$559$			14.0599i			0.594670i
$560$	−0.489012	−	9.01199i	−0.0206645	−	0.380826i
$561$		0			0
$562$	−4.19449			−0.176934
$563$	15.2694			0.643527			0.321764	−	0.946820i	$-0.395724\pi$
$563$	15.2694			0.643527			0.321764	+	0.946820i	$0.395724\pi$
$564$		0			0
$565$			26.4600i			1.11318i
$566$	31.9196			1.34168
$567$		0			0
$568$	−1.42491			−0.0597880
$569$		−	14.3776i		−	0.602740i	−0.953507	−	0.301370i	$-0.902556\pi$
$569$		−	14.3776i		−	0.602740i	0.953507	−	0.301370i	$-0.0974439\pi$
$570$		0			0
$571$	28.7884			1.20476			0.602379	−	0.798210i	$-0.294220\pi$
$571$	28.7884			1.20476			0.602379	+	0.798210i	$0.294220\pi$
$572$	7.07653			0.295884
$573$		0			0
$574$	−25.4139	+	1.37902i	−1.06076	+	0.0575592i
$575$			48.5950i			2.02655i
$576$		0			0
$577$		−	8.69950i		−	0.362165i	−0.983468	−	0.181083i	$-0.942040\pi$
$577$		−	8.69950i		−	0.362165i	0.983468	−	0.181083i	$-0.0579601\pi$
$578$		−	43.1313i		−	1.79403i
$579$		0			0
$580$			22.7423i			0.944324i
$581$	1.85608	+	34.2057i	0.0770034	+	1.41909i
$582$		0			0
$583$	−14.3496			−0.594300
$584$	3.02282			0.125085
$585$		0			0
$586$			4.23835i			0.175085i
$587$	−2.88208			−0.118956			−0.0594780	−	0.998230i	$-0.518944\pi$
$587$	−2.88208			−0.118956			−0.0594780	+	0.998230i	$0.518944\pi$
$588$		0			0
$589$	−1.05333			−0.0434016
$590$			26.9684i			1.11027i
$591$		0			0
$592$	7.54738			0.310195
$593$	20.5670			0.844586			0.422293	−	0.906459i	$-0.361225\pi$
$593$	20.5670			0.844586			0.422293	+	0.906459i	$0.361225\pi$
$594$		0			0
$595$	3.79201	+	69.8829i	0.155457	+	2.86492i
$596$			15.0557i			0.616704i
$597$		0			0
$598$		−	46.7097i		−	1.91010i
$599$			15.1132i			0.617510i	0.951142	+	0.308755i	$0.0999123\pi$
$599$			15.1132i			0.617510i	−0.951142	+	0.308755i	$0.900088\pi$
$600$		0			0
$601$			7.24882i			0.295686i	0.989011	+	0.147843i	$0.0472330\pi$
$601$			7.24882i			0.295686i	−0.989011	+	0.147843i	$0.952767\pi$
$602$	−5.82293	+	0.315966i	−0.237325	+	0.0128778i
$603$		0			0
$604$	−9.82278			−0.399683
$605$	−33.3254			−1.35487
$606$		0			0
$607$		−	21.3609i		−	0.867011i	−0.901151	−	0.433506i	$-0.857276\pi$
$607$		−	21.3609i		−	0.867011i	0.901151	−	0.433506i	$-0.142724\pi$
$608$	−1.00000			−0.0405554
$609$		0			0
$610$	−2.60330			−0.105405
$611$		−	75.0776i		−	3.03732i
$612$		0			0
$613$	−8.14541			−0.328990			−0.164495	−	0.986378i	$-0.552599\pi$
$613$	−8.14541			−0.328990			−0.164495	+	0.986378i	$0.552599\pi$
$614$	−25.8057			−1.04143
$615$		0			0
$616$	0.159030	+	2.93076i	0.00640750	+	0.118084i
$617$			41.7967i			1.68267i	0.540514	+	0.841335i	$0.318230\pi$
$617$			41.7967i			1.68267i	−0.540514	+	0.841335i	$0.681770\pi$
$618$		0			0
$619$		−	6.75046i		−	0.271324i	−0.990755	−	0.135662i	$-0.956684\pi$
$619$		−	6.75046i		−	0.271324i	0.990755	−	0.135662i	$-0.0433161\pi$
$620$		−	3.59313i		−	0.144304i
$621$		0			0
$622$			18.3786i			0.736914i
$623$	1.25412	+	23.1121i	0.0502452	+	0.925968i
$624$		0			0
$625$	−14.1399			−0.565597
$626$	10.7582			0.429985
$627$		0			0
$628$		−	12.2486i		−	0.488771i
$629$	−58.5257			−2.33357
$630$		0			0
$631$	19.4934			0.776022			0.388011	−	0.921655i	$-0.373162\pi$
$631$	19.4934			0.776022			0.388011	+	0.921655i	$0.373162\pi$
$632$			6.38653i			0.254042i
$633$		0			0
$634$	27.0813			1.07554
$635$	−5.64029			−0.223828
$636$		0			0
$637$	4.83171	+	44.3906i	0.191439	+	1.75882i
$638$		−	7.39596i		−	0.292809i
$639$		0			0
$640$		−	3.41122i		−	0.134840i
$641$			16.0639i			0.634484i	0.948345	+	0.317242i	$0.102757\pi$
$641$			16.0639i			0.634484i	−0.948345	+	0.317242i	$0.897243\pi$
$642$		0			0
$643$			9.34808i			0.368652i	0.982865	+	0.184326i	$0.0590103\pi$
$643$			9.34808i			0.368652i	−0.982865	+	0.184326i	$0.940990\pi$
$644$	19.3449	−	1.04970i	0.762297	−	0.0413640i
$645$		0			0
$646$	7.75444			0.305094
$647$	−41.9664			−1.64987			−0.824934	−	0.565229i	$-0.808788\pi$
$647$	−41.9664			−1.64987			−0.824934	+	0.565229i	$0.808788\pi$
$648$		0			0
$649$		−	8.77032i		−	0.344265i
$650$	−42.3336			−1.66046
$651$		0			0
$652$	4.89771			0.191809
$653$			16.1259i			0.631053i	0.948917	+	0.315527i	$0.102181\pi$
$653$			16.1259i			0.631053i	−0.948917	+	0.315527i	$0.897819\pi$
$654$		0			0
$655$	−49.2466			−1.92422
$656$	−9.61970			−0.375586
$657$		0			0
$658$	31.0936	−	1.68721i	1.21215	−	0.0657743i
$659$		−	6.57436i		−	0.256101i	−0.991768	−	0.128050i	$-0.959128\pi$
$659$		−	6.57436i		−	0.256101i	0.991768	−	0.128050i	$-0.0408719\pi$
$660$		0			0
$661$		−	4.23617i		−	0.164768i	−0.996601	−	0.0823839i	$-0.973747\pi$
$661$		−	4.23617i		−	0.164768i	0.996601	−	0.0823839i	$-0.0262534\pi$
$662$		−	3.59739i		−	0.139816i
$663$		0			0
$664$			12.9476i			0.502463i
$665$	−9.01199	+	0.489012i	−0.349470	+	0.0189631i
$666$		0			0
$667$	−48.8182			−1.89025
$668$	−5.85191			−0.226417
$669$		0			0
$670$			33.4321i			1.29159i
$671$	0.846611			0.0326830
$672$		0			0
$673$	14.9886			0.577767			0.288883	−	0.957364i	$-0.406716\pi$
$673$	14.9886			0.577767			0.288883	+	0.957364i	$0.406716\pi$
$674$			8.61773i			0.331943i
$675$		0			0
$676$	27.6913			1.06505
$677$	34.0910			1.31022			0.655111	−	0.755533i	$-0.272622\pi$
$677$	34.0910			1.31022			0.655111	+	0.755533i	$0.272622\pi$
$678$		0			0
$679$	−19.9232	+	1.08108i	−0.764584	+	0.0414882i
$680$			26.4521i			1.01439i
$681$		0			0
$682$			1.16851i			0.0447446i
$683$			14.7289i			0.563585i	0.959475	+	0.281793i	$0.0909290\pi$
$683$			14.7289i			0.563585i	−0.959475	+	0.281793i	$0.909071\pi$
$684$		0			0
$685$			6.91995i			0.264398i
$686$	−18.2759	+	2.99865i	−0.697777	+	0.114489i
$687$		0			0
$688$	−2.20410			−0.0840305
$689$	−82.5128			−3.14349
$690$		0			0
$691$			23.8471i			0.907187i	0.891209	+	0.453593i	$0.149858\pi$
$691$			23.8471i			0.907187i	−0.891209	+	0.453593i	$0.850142\pi$
$692$	8.82975			0.335657
$693$		0			0
$694$	33.4662			1.27036
$695$			1.57262i			0.0596528i
$696$		0			0
$697$	74.5954			2.82550
$698$	−27.7640			−1.05088
$699$		0			0
$700$	−0.951359	−	17.5326i	−0.0359580	−	0.662668i
$701$		−	31.8926i		−	1.20457i	−0.798282	−	0.602284i	$-0.794257\pi$
$701$		−	31.8926i		−	1.20457i	0.798282	−	0.602284i	$-0.205743\pi$
$702$		0			0
$703$		−	7.54738i		−	0.284655i
$704$			1.10935i			0.0418103i
$705$		0			0
$706$		−	11.1618i		−	0.420080i
$707$	−0.592585	−	10.9207i	−0.0222865	−	0.410716i
$708$		0			0
$709$	14.2636			0.535681			0.267841	−	0.963463i	$-0.413690\pi$
$709$	14.2636			0.535681			0.267841	+	0.963463i	$0.413690\pi$
$710$	−4.86069			−0.182419
$711$		0			0
$712$			8.74842i			0.327861i
$713$	7.71294			0.288852
$714$		0			0
$715$	24.1396			0.902770
$716$		−	19.4923i		−	0.728460i
$717$		0			0
$718$	−21.3200			−0.795654
$719$	−40.7527			−1.51982			−0.759910	−	0.650028i	$-0.774757\pi$
$719$	−40.7527			−1.51982			−0.759910	+	0.650028i	$0.774757\pi$
$720$		0			0
$721$	−26.1592	+	1.41946i	−0.974219	+	0.0528634i
$722$			1.00000i			0.0372161i
$723$		0			0
$724$			5.77116i			0.214483i
$725$			44.2446i			1.64320i
$726$		0			0
$727$		−	29.2540i		−	1.08497i	−0.840065	−	0.542485i	$-0.817483\pi$
$727$		−	29.2540i		−	1.08497i	0.840065	−	0.542485i	$-0.182517\pi$
$728$	0.914451	+	16.8524i	0.0338918	+	0.624591i
$729$		0			0
$730$	10.3115			0.381646
$731$	17.0916			0.632154
$732$		0			0
$733$		−	15.8499i		−	0.585431i	−0.956200	−	0.292716i	$-0.905441\pi$
$733$		−	15.8499i		−	0.585431i	0.956200	−	0.292716i	$-0.0945589\pi$
$734$	−17.2337			−0.636107
$735$		0			0
$736$	7.32245			0.269909
$737$		−	10.8723i		−	0.400488i
$738$		0			0
$739$	14.5195			0.534109			0.267055	−	0.963681i	$-0.413950\pi$
$739$	14.5195			0.534109			0.267055	+	0.963681i	$0.413950\pi$
$740$	25.7458			0.946434
$741$		0			0
$742$	−1.85430	−	34.1728i	−0.0680735	−	1.25452i
$743$			2.35090i			0.0862463i	0.999070	+	0.0431232i	$0.0137308\pi$
$743$			2.35090i			0.0862463i	−0.999070	+	0.0431232i	$0.986269\pi$
$744$		0			0
$745$			51.3582i			1.88162i
$746$		−	10.9441i		−	0.400692i
$747$		0			0
$748$		−	8.60241i		−	0.314535i
$749$	−21.6435	+	1.17443i	−0.790838	+	0.0429128i
$750$		0			0
$751$	−33.2500			−1.21331			−0.606656	−	0.794965i	$-0.707489\pi$
$751$	−33.2500			−1.21331			−0.606656	+	0.794965i	$0.707489\pi$
$752$	11.7695			0.429191
$753$		0			0
$754$		−	42.5281i		−	1.54878i
$755$	−33.5077			−1.21947
$756$		0			0
$757$	−18.8896			−0.686556			−0.343278	−	0.939234i	$-0.611537\pi$
$757$	−18.8896			−0.686556			−0.343278	+	0.939234i	$0.611537\pi$
$758$			20.2540i			0.735660i
$759$		0			0
$760$	−3.41122			−0.123738
$761$	35.2880			1.27919			0.639594	−	0.768713i	$-0.279103\pi$
$761$	35.2880			1.27919			0.639594	+	0.768713i	$0.279103\pi$
$762$		0			0
$763$	0.172944	+	3.18717i	0.00626098	+	0.115383i
$764$			15.5161i			0.561352i
$765$		0			0
$766$			25.7193i			0.929278i
$767$		−	50.4309i		−	1.82095i
$768$		0			0
$769$			31.9106i			1.15073i	0.817898	+	0.575364i	$0.195139\pi$
$769$			31.9106i			1.15073i	−0.817898	+	0.575364i	$0.804861\pi$
$770$	0.542487	+	9.99747i	0.0195499	+	0.360284i
$771$		0			0
$772$	1.89748			0.0682919
$773$	−40.5722			−1.45928			−0.729639	−	0.683832i	$-0.760312\pi$
$773$	−40.5722			−1.45928			−0.729639	+	0.683832i	$0.760312\pi$
$774$		0			0
$775$		−	6.99034i		−	0.251100i
$776$	−7.54136			−0.270719
$777$		0			0
$778$	−12.9820			−0.465429
$779$			9.61970i			0.344662i
$780$		0			0
$781$	1.58073			0.0565630
$782$	−56.7815			−2.03050
$783$		0			0
$784$	−6.95890	+	0.757443i	−0.248532	+	0.0270515i
$785$		−	41.7825i		−	1.49128i
$786$		0			0
$787$		−	23.6724i		−	0.843830i	−0.906635	−	0.421915i	$-0.861358\pi$
$787$		−	23.6724i		−	0.843830i	0.906635	−	0.421915i	$-0.138642\pi$
$788$			21.2526i			0.757091i
$789$		0			0
$790$			21.7859i			0.775106i
$791$	20.4923	−	1.11196i	0.728621	−	0.0395367i
$792$		0			0
$793$	4.86816			0.172873
$794$	−15.9008			−0.564300
$795$		0			0
$796$			3.73822i			0.132498i
$797$	30.0215			1.06341			0.531707	−	0.846928i	$-0.321551\pi$
$797$	30.0215			1.06341			0.531707	+	0.846928i	$0.321551\pi$
$798$		0			0
$799$	−91.2663			−3.22877
$800$		−	6.63643i		−	0.234633i
$801$		0			0
$802$	10.9593			0.386987
$803$	−3.35337			−0.118338
$804$		0			0
$805$	65.9898	−	3.58077i	2.32584	−	0.126205i
$806$			6.71915i			0.236672i
$807$		0			0
$808$		−	4.13372i		−	0.145424i
$809$		−	40.7356i		−	1.43219i	−0.698004	−	0.716094i	$-0.745928\pi$
$809$		−	40.7356i		−	1.43219i	0.698004	−	0.716094i	$-0.254072\pi$
$810$		0			0
$811$		−	22.6264i		−	0.794519i	−0.917706	−	0.397260i	$-0.869961\pi$
$811$		−	22.6264i		−	0.794519i	0.917706	−	0.397260i	$-0.130039\pi$
$812$	17.6131	−	0.955729i	0.618099	−	0.0335395i
$813$		0			0
$814$	−8.37270			−0.293463
$815$	16.7072			0.585227
$816$		0			0
$817$			2.20410i			0.0771117i
$818$	4.69585			0.164187
$819$		0			0
$820$	−32.8149			−1.14595
$821$		−	32.2184i		−	1.12443i	−0.826991	−	0.562215i	$-0.809949\pi$
$821$		−	32.2184i		−	1.12443i	0.826991	−	0.562215i	$-0.190051\pi$
$822$		0			0
$823$	−49.0894			−1.71115			−0.855574	−	0.517681i	$-0.826796\pi$
$823$	−49.0894			−1.71115			−0.855574	+	0.517681i	$0.826796\pi$
$824$	−9.90178			−0.344945
$825$		0			0
$826$	20.8861	−	1.13333i	0.726719	−	0.0394335i
$827$		−	1.16878i		−	0.0406425i	−0.999794	−	0.0203213i	$-0.993531\pi$
$827$		−	1.16878i		−	0.0406425i	0.999794	−	0.0203213i	$-0.00646890\pi$
$828$		0			0
$829$		−	5.10467i		−	0.177292i	−0.996063	−	0.0886462i	$-0.971746\pi$
$829$		−	5.10467i		−	0.177292i	0.996063	−	0.0886462i	$-0.0282541\pi$
$830$			44.1670i			1.53306i
$831$		0			0
$832$			6.37897i			0.221151i
$833$	53.9624	−	5.87355i	1.86969	−	0.203506i
$834$		0			0
$835$	−19.9621			−0.690819
$836$	1.10935			0.0383678
$837$		0			0
$838$			8.80429i			0.304139i
$839$	31.1931			1.07691			0.538453	−	0.842656i	$-0.319009\pi$
$839$	31.1931			1.07691			0.538453	+	0.842656i	$0.319009\pi$
$840$		0			0
$841$	−15.4478			−0.532683
$842$			1.31840i			0.0454351i
$843$		0			0
$844$	−13.0726			−0.449976
$845$	94.4611			3.24956
$846$		0			0
$847$	1.40047	+	25.8093i	0.0481208	+	0.886817i
$848$		−	12.9351i		−	0.444194i
$849$		0			0
$850$			51.4618i			1.76513i
$851$			55.2653i			1.89447i
$852$		0			0
$853$			3.10371i			0.106269i	0.998587	+	0.0531345i	$0.0169212\pi$
$853$			3.10371i			0.106269i	−0.998587	+	0.0531345i	$0.983079\pi$
$854$	0.109402	+	2.01616i	0.00374365	+	0.0689916i
$855$		0			0
$856$	−8.19252			−0.280015
$857$	−20.8592			−0.712537			−0.356269	−	0.934384i	$-0.615951\pi$
$857$	−20.8592			−0.712537			−0.356269	+	0.934384i	$0.615951\pi$
$858$		0			0
$859$			27.5439i			0.939786i	0.882723	+	0.469893i	$0.155708\pi$
$859$			27.5439i			0.939786i	−0.882723	+	0.469893i	$0.844292\pi$
$860$	−7.51867			−0.256385
$861$		0			0
$862$	−12.3039			−0.419071
$863$		−	3.35840i		−	0.114321i	−0.998365	−	0.0571606i	$-0.981795\pi$
$863$		−	3.35840i		−	0.114321i	0.998365	−	0.0571606i	$-0.0182047\pi$
$864$		0			0
$865$	30.1202			1.02412
$866$	10.9893			0.373431
$867$		0			0
$868$	−2.78275	+	0.150999i	−0.0944526	+	0.00512523i
$869$		−	7.08491i		−	0.240339i
$870$		0			0
$871$		−	62.5179i		−	2.11834i
$872$			1.20641i			0.0408542i
$873$		0			0
$874$		−	7.32245i		−	0.247686i
$875$	−0.800235	−	14.7475i	−0.0270529	−	0.498557i
$876$		0			0
$877$	−18.3628			−0.620068			−0.310034	−	0.950725i	$-0.600340\pi$
$877$	−18.3628			−0.620068			−0.310034	+	0.950725i	$0.600340\pi$
$878$	27.9495			0.943249
$879$		0			0
$880$			3.78425i			0.127567i
$881$	5.96177			0.200857			0.100429	−	0.994944i	$-0.467979\pi$
$881$	5.96177			0.200857			0.100429	+	0.994944i	$0.467979\pi$
$882$		0			0
$883$	−31.4838			−1.05951			−0.529757	−	0.848150i	$-0.677717\pi$
$883$	−31.4838			−1.05951			−0.529757	+	0.848150i	$0.677717\pi$
$884$		−	49.4654i		−	1.66370i
$885$		0			0
$886$	27.9924			0.940422
$887$	36.6144			1.22939			0.614695	−	0.788765i	$-0.289279\pi$
$887$	36.6144			1.22939			0.614695	+	0.788765i	$0.289279\pi$
$888$		0			0
$889$	0.237029	+	4.36820i	0.00794969	+	0.146505i
$890$			29.8428i			1.00033i
$891$		0			0
$892$		−	24.1584i		−	0.808881i
$893$		−	11.7695i		−	0.393853i
$894$		0			0
$895$		−	66.4925i		−	2.22260i
$896$	−2.64186	+	0.143354i	−0.0882585	+	0.00478912i
$897$		0			0
$898$	30.6831			1.02391
$899$	7.02245			0.234212
$900$		0			0
$901$			100.305i			3.34163i
$902$	10.6716			0.355327
$903$		0			0
$904$	7.75674			0.257985
$905$			19.6867i			0.654408i
$906$		0			0
$907$	12.5797			0.417703			0.208851	−	0.977947i	$-0.433027\pi$
$907$	12.5797			0.417703			0.208851	+	0.977947i	$0.433027\pi$
$908$	14.0002			0.464614
$909$		0			0
$910$	3.11939	+	57.4872i	0.103407	+	1.90568i
$911$		−	33.8352i		−	1.12101i	−0.828150	−	0.560506i	$-0.810607\pi$
$911$		−	33.8352i		−	1.12101i	0.828150	−	0.560506i	$-0.189393\pi$
$912$		0			0
$913$		−	14.3634i		−	0.475360i
$914$		−	21.0046i		−	0.694771i
$915$		0			0
$916$		−	19.2950i		−	0.637526i
$917$	2.06955	+	38.1397i	0.0683425	+	1.25948i
$918$		0			0
$919$	−21.6226			−0.713262			−0.356631	−	0.934245i	$-0.616075\pi$
$919$	−21.6226			−0.713262			−0.356631	+	0.934245i	$0.616075\pi$
$920$	24.9785			0.823517
$921$		0			0
$922$		−	12.1422i		−	0.399882i
$923$	9.08948			0.299184
$924$		0			0
$925$	50.0877			1.64687
$926$			16.0862i			0.528625i
$927$		0			0
$928$	6.66692			0.218852
$929$	33.6257			1.10322			0.551611	−	0.834101i	$-0.314013\pi$
$929$	33.6257			1.10322			0.551611	+	0.834101i	$0.314013\pi$
$930$		0			0
$931$	0.757443	+	6.95890i	0.0248242	+	0.228069i
$932$		−	1.83418i		−	0.0600806i
$933$		0			0
$934$		−	25.1312i		−	0.822317i
$935$		−	29.3447i		−	0.959675i
$936$		0			0
$937$		−	38.0456i		−	1.24290i	−0.783455	−	0.621448i	$-0.786545\pi$
$937$		−	38.0456i		−	1.24290i	0.783455	−	0.621448i	$-0.213455\pi$
$938$	25.8919	−	1.40496i	0.845401	−	0.0458735i
$939$		0			0
$940$	40.1485			1.30950
$941$	41.1138			1.34027			0.670136	−	0.742239i	$-0.266236\pi$
$941$	41.1138			1.34027			0.670136	+	0.742239i	$0.266236\pi$
$942$		0			0
$943$		−	70.4398i		−	2.29383i
$944$	7.90580			0.257312
$945$		0			0
$946$	2.44512			0.0794978
$947$			23.2029i			0.753992i	0.926215	+	0.376996i	$0.123043\pi$
$947$			23.2029i			0.753992i	−0.926215	+	0.376996i	$0.876957\pi$
$948$		0			0
$949$	−19.2825			−0.625936
$950$	−6.63643			−0.215314
$951$		0			0
$952$	20.4862	−	1.11163i	0.663961	−	0.0360281i
$953$		−	7.41939i		−	0.240338i	−0.992753	−	0.120169i	$-0.961656\pi$
$953$		−	7.41939i		−	0.240338i	0.992753	−	0.120169i	$-0.0383436\pi$
$954$		0			0
$955$			52.9288i			1.71273i
$956$			3.92223i			0.126854i
$957$		0			0
$958$		−	18.8006i		−	0.607418i
$959$	5.35924	−	0.290805i	0.173059	−	0.00939060i
$960$		0			0
$961$	29.8905			0.964210
$962$	−48.1445			−1.55224
$963$		0			0
$964$			12.2645i			0.395013i
$965$	6.47274			0.208365
$966$		0			0
$967$	52.3178			1.68243			0.841214	−	0.540703i	$-0.181842\pi$
$967$	52.3178			1.68243			0.841214	+	0.540703i	$0.181842\pi$
$968$			9.76934i			0.313998i
$969$		0			0
$970$	−25.7252			−0.825988
$971$	−13.9131			−0.446492			−0.223246	−	0.974762i	$-0.571665\pi$
$971$	−13.9131			−0.446492			−0.223246	+	0.974762i	$0.571665\pi$
$972$		0			0
$973$	1.21793	−	0.0660880i	0.0390452	−	0.00211868i
$974$		−	32.1207i		−	1.02921i
$975$		0			0
$976$			0.763158i			0.0244281i
$977$		−	20.2122i		−	0.646645i	−0.946289	−	0.323323i	$-0.895200\pi$
$977$		−	20.2122i		−	0.646645i	0.946289	−	0.323323i	$-0.104800\pi$
$978$		0			0
$979$		−	9.70508i		−	0.310176i
$980$	−23.7383	+	2.58381i	−0.758294	+	0.0825367i
$981$		0			0
$982$	−39.5682			−1.26267
$983$	−50.6272			−1.61476			−0.807378	−	0.590035i	$-0.799114\pi$
$983$	−50.6272			−1.61476			−0.807378	+	0.590035i	$0.799114\pi$
$984$		0			0
$985$			72.4972i			2.30995i
$986$	−51.6982			−1.64641
$987$		0			0
$988$	6.37897			0.202942
$989$		−	16.1394i		−	0.513203i
$990$		0			0
$991$	55.5418			1.76434			0.882172	−	0.470928i	$-0.156081\pi$
$991$	55.5418			1.76434			0.882172	+	0.470928i	$0.156081\pi$
$992$	−1.05333			−0.0334432
$993$		0			0
$994$	0.204267	+	3.76443i	0.00647895	+	0.119400i
$995$			12.7519i			0.404262i
$996$		0			0
$997$			34.0391i			1.07803i	0.842296	+	0.539015i	$0.181203\pi$
$997$			34.0391i			1.07803i	−0.842296	+	0.539015i	$0.818797\pi$
$998$		−	5.95956i		−	0.188647i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2394.2.f.b.2015.5	yes	24
3.2	odd	2		2394.2.f.a.2015.6	yes	24
7.6	odd	2		2394.2.f.a.2015.5	✓	24
21.20	even	2	inner	2394.2.f.b.2015.6	yes	24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2394.2.f.a.2015.5	✓	24	7.6	odd	2
2394.2.f.a.2015.6	yes	24	3.2	odd	2
2394.2.f.b.2015.5	yes	24	1.1	even	1	trivial
2394.2.f.b.2015.6	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)

\(f(q)\)	\(=\)	\(q-1.00000i q^{2} -1.00000 q^{4} -3.41122 q^{5} +(0.143354 + 2.64186i) q^{7} +1.00000i q^{8} +O(q^{10})\)	\(q-1.00000i q^{2} -1.00000 q^{4} -3.41122 q^{5} +(0.143354 + 2.64186i) q^{7} +1.00000i q^{8} +3.41122i q^{10} -1.10935i q^{11} -6.37897i q^{13} +(2.64186 - 0.143354i) q^{14} +1.00000 q^{16} -7.75444 q^{17} -1.00000i q^{19} +3.41122 q^{20} -1.10935 q^{22} +7.32245i q^{23} +6.63643 q^{25} -6.37897 q^{26} +(-0.143354 - 2.64186i) q^{28} +6.66692i q^{29} -1.05333i q^{31} -1.00000i q^{32} +7.75444i q^{34} +(-0.489012 - 9.01199i) q^{35} +7.54738 q^{37} -1.00000 q^{38} -3.41122i q^{40} -9.61970 q^{41} -2.20410 q^{43} +1.10935i q^{44} +7.32245 q^{46} +11.7695 q^{47} +(-6.95890 + 0.757443i) q^{49} -6.63643i q^{50} +6.37897i q^{52} -12.9351i q^{53} +3.78425i q^{55} +(-2.64186 + 0.143354i) q^{56} +6.66692 q^{58} +7.90580 q^{59} +0.763158i q^{61} -1.05333 q^{62} -1.00000 q^{64} +21.7601i q^{65} +9.80062 q^{67} +7.75444 q^{68} +(-9.01199 + 0.489012i) q^{70} +1.42491i q^{71} -3.02282i q^{73} -7.54738i q^{74} +1.00000i q^{76} +(2.93076 - 0.159030i) q^{77} +6.38653 q^{79} -3.41122 q^{80} +9.61970i q^{82} +12.9476 q^{83} +26.4521 q^{85} +2.20410i q^{86} +1.10935 q^{88} +8.74842 q^{89} +(16.8524 - 0.914451i) q^{91} -7.32245i q^{92} -11.7695i q^{94} +3.41122i q^{95} +7.54136i q^{97} +(0.757443 + 6.95890i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q - 24 q^{4} + 4 q^{7}+O(q^{10}) \) 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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