LMFDB - Embedded newform 2394.2.f.b.2015.17

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			2015.17
Character	$\chi$	$=$	2394.2015
Dual form			2394.2.f.b.2015.18

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{2} -1.00000 q^{4} -2.79254 q^{5} +(-1.85020 - 1.89123i) q^{7} +1.00000i q^{8} +O(q^{10})$	\(q-1.00000i q^{2} -1.00000 q^{4} -2.79254 q^{5} +(-1.85020 - 1.89123i) q^{7} +1.00000i q^{8} +2.79254i q^{10} -2.81281i q^{11} -4.25740i q^{13} +(-1.89123 + 1.85020i) q^{14} +1.00000 q^{16} -1.95874 q^{17} -1.00000i q^{19} +2.79254 q^{20} -2.81281 q^{22} -7.85553i q^{23} +2.79827 q^{25} -4.25740 q^{26} +(1.85020 + 1.89123i) q^{28} +5.88457i q^{29} -3.43769i q^{31} -1.00000i q^{32} +1.95874i q^{34} +(5.16675 + 5.28134i) q^{35} +6.46193 q^{37} -1.00000 q^{38} -2.79254i q^{40} -4.41535 q^{41} +2.77474 q^{43} +2.81281i q^{44} -7.85553 q^{46} -12.9034 q^{47} +(-0.153524 + 6.99832i) q^{49} -2.79827i q^{50} +4.25740i q^{52} +11.0632i q^{53} +7.85489i q^{55} +(1.89123 - 1.85020i) q^{56} +5.88457 q^{58} +4.48909 q^{59} -10.2940i q^{61} -3.43769 q^{62} -1.00000 q^{64} +11.8890i q^{65} -9.62535 q^{67} +1.95874 q^{68} +(5.28134 - 5.16675i) q^{70} +7.37038i q^{71} +2.22270i q^{73} -6.46193i q^{74} +1.00000i q^{76} +(-5.31969 + 5.20427i) q^{77} +15.3756 q^{79} -2.79254 q^{80} +4.41535i q^{82} -9.81295 q^{83} +5.46985 q^{85} -2.77474i q^{86} +2.81281 q^{88} -5.78370 q^{89} +(-8.05173 + 7.87704i) q^{91} +7.85553i q^{92} +12.9034i q^{94} +2.79254i q^{95} +5.33976i q^{97} +(6.99832 + 0.153524i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	−2.79254			−1.24886			−0.624431	−	0.781080i	$-0.714669\pi$
$5$	−2.79254			−1.24886			−0.624431	+	0.781080i	$0.714669\pi$
$6$		0			0
$7$	−1.85020	−	1.89123i	−0.699310	−	0.714819i
$7$	−1.85020	−	1.89123i	−0.699310	−	0.714819i
$8$			1.00000i			0.353553i
$9$		0			0
$10$			2.79254i			0.883078i
$11$		−	2.81281i		−	0.848095i	−0.905640	−	0.424048i	$-0.860609\pi$
$11$		−	2.81281i		−	0.848095i	0.905640	−	0.424048i	$-0.139391\pi$
$12$		0			0
$13$		−	4.25740i		−	1.18079i	−0.807114	−	0.590395i	$-0.798972\pi$
$13$		−	4.25740i		−	1.18079i	0.807114	−	0.590395i	$-0.201028\pi$
$14$	−1.89123	+	1.85020i	−0.505453	+	0.494487i
$15$		0			0
$16$	1.00000			0.250000
$17$	−1.95874			−0.475064			−0.237532	−	0.971380i	$-0.576338\pi$
$17$	−1.95874			−0.475064			−0.237532	+	0.971380i	$0.576338\pi$
$18$		0			0
$19$		−	1.00000i		−	0.229416i
$19$		−	1.00000i		−	0.229416i
$20$	2.79254			0.624431
$21$		0			0
$22$	−2.81281			−0.599694
$23$		−	7.85553i		−	1.63799i	−0.573800	−	0.818996i	$-0.694531\pi$
$23$		−	7.85553i		−	1.63799i	0.573800	−	0.818996i	$-0.305469\pi$
$24$		0			0
$25$	2.79827			0.559654
$26$	−4.25740			−0.834945
$27$		0			0
$28$	1.85020	+	1.89123i	0.349655	+	0.357409i
$29$			5.88457i			1.09274i	0.837545	+	0.546369i	$0.183990\pi$
$29$			5.88457i			1.09274i	−0.837545	+	0.546369i	$0.816010\pi$
$30$		0			0
$31$		−	3.43769i		−	0.617427i	−0.951155	−	0.308713i	$-0.900102\pi$
$31$		−	3.43769i		−	0.617427i	0.951155	−	0.308713i	$-0.0998984\pi$
$32$		−	1.00000i		−	0.176777i
$33$		0			0
$34$			1.95874i			0.335921i
$35$	5.16675	+	5.28134i	0.873341	+	0.892710i
$36$		0			0
$37$	6.46193			1.06233			0.531167	−	0.847267i	$-0.321754\pi$
$37$	6.46193			1.06233			0.531167	+	0.847267i	$0.321754\pi$
$38$	−1.00000			−0.162221
$39$		0			0
$40$		−	2.79254i		−	0.441539i
$41$	−4.41535			−0.689562			−0.344781	−	0.938683i	$-0.612047\pi$
$41$	−4.41535			−0.689562			−0.344781	+	0.938683i	$0.612047\pi$
$42$		0			0
$43$	2.77474			0.423144			0.211572	−	0.977362i	$-0.432142\pi$
$43$	2.77474			0.423144			0.211572	+	0.977362i	$0.432142\pi$
$44$			2.81281i			0.424048i
$45$		0			0
$46$	−7.85553			−1.15823
$47$	−12.9034			−1.88215			−0.941076	−	0.338195i	$-0.890184\pi$
$47$	−12.9034			−1.88215			−0.941076	+	0.338195i	$0.890184\pi$
$48$		0			0
$49$	−0.153524	+	6.99832i	−0.0219320	+	0.999759i
$50$		−	2.79827i		−	0.395735i
$51$		0			0
$52$			4.25740i			0.590395i
$53$			11.0632i			1.51965i	0.650127	+	0.759826i	$0.274716\pi$
$53$			11.0632i			1.51965i	−0.650127	+	0.759826i	$0.725284\pi$
$54$		0			0
$55$			7.85489i			1.05915i
$56$	1.89123	−	1.85020i	0.252727	−	0.247243i
$57$		0			0
$58$	5.88457			0.772682
$59$	4.48909			0.584430			0.292215	−	0.956353i	$-0.405608\pi$
$59$	4.48909			0.584430			0.292215	+	0.956353i	$0.405608\pi$
$60$		0			0
$61$		−	10.2940i		−	1.31801i	−0.752139	−	0.659005i	$-0.770978\pi$
$61$		−	10.2940i		−	1.31801i	0.752139	−	0.659005i	$-0.229022\pi$
$62$	−3.43769			−0.436587
$63$		0			0
$64$	−1.00000			−0.125000
$65$			11.8890i			1.47464i
$66$		0			0
$67$	−9.62535			−1.17592			−0.587962	−	0.808889i	$-0.700069\pi$
$67$	−9.62535			−1.17592			−0.587962	+	0.808889i	$0.700069\pi$
$68$	1.95874			0.237532
$69$		0			0
$70$	5.28134	−	5.16675i	0.631241	−	0.617545i
$71$			7.37038i			0.874704i	0.899290	+	0.437352i	$0.144084\pi$
$71$			7.37038i			0.874704i	−0.899290	+	0.437352i	$0.855916\pi$
$72$		0			0
$73$			2.22270i			0.260147i	0.991504	+	0.130074i	$0.0415214\pi$
$73$			2.22270i			0.260147i	−0.991504	+	0.130074i	$0.958479\pi$
$74$		−	6.46193i		−	0.751184i
$75$		0			0
$76$			1.00000i			0.114708i
$77$	−5.31969	+	5.20427i	−0.606234	+	0.593081i
$78$		0			0
$79$	15.3756			1.72989			0.864946	−	0.501866i	$-0.167353\pi$
$79$	15.3756			1.72989			0.864946	+	0.501866i	$0.167353\pi$
$80$	−2.79254			−0.312215
$81$		0			0
$82$			4.41535i			0.487594i
$83$	−9.81295			−1.07711			−0.538556	−	0.842590i	$-0.681030\pi$
$83$	−9.81295			−1.07711			−0.538556	+	0.842590i	$0.681030\pi$
$84$		0			0
$85$	5.46985			0.593288
$86$		−	2.77474i		−	0.299208i
$87$		0			0
$88$	2.81281			0.299847
$89$	−5.78370			−0.613071			−0.306536	−	0.951859i	$-0.599170\pi$
$89$	−5.78370			−0.613071			−0.306536	+	0.951859i	$0.599170\pi$
$90$		0			0
$91$	−8.05173	+	7.87704i	−0.844051	+	0.825738i
$92$			7.85553i			0.818996i
$93$		0			0
$94$			12.9034i			1.33088i
$95$			2.79254i			0.286508i
$96$		0			0
$97$			5.33976i			0.542170i	0.962555	+	0.271085i	$0.0873825\pi$
$97$			5.33976i			0.542170i	−0.962555	+	0.271085i	$0.912618\pi$
$98$	6.99832	+	0.153524i	0.706937	+	0.0155083i
$99$		0			0
$100$	−2.79827			−0.279827
$101$	−3.01320			−0.299825			−0.149912	−	0.988699i	$-0.547899\pi$
$101$	−3.01320			−0.299825			−0.149912	+	0.988699i	$0.547899\pi$
$102$		0			0
$103$			5.65232i			0.556940i	0.960445	+	0.278470i	$0.0898272\pi$
$103$			5.65232i			0.556940i	−0.960445	+	0.278470i	$0.910173\pi$
$104$	4.25740			0.417472
$105$		0			0
$106$	11.0632			1.07456
$107$			8.39026i			0.811117i	0.914069	+	0.405559i	$0.132923\pi$
$107$			8.39026i			0.811117i	−0.914069	+	0.405559i	$0.867077\pi$
$108$		0			0
$109$	−1.70095			−0.162921			−0.0814607	−	0.996677i	$-0.525959\pi$
$109$	−1.70095			−0.162921			−0.0814607	+	0.996677i	$0.525959\pi$
$110$	7.85489			0.748934
$111$		0			0
$112$	−1.85020	−	1.89123i	−0.174827	−	0.178705i
$113$			8.46783i			0.796586i	0.917258	+	0.398293i	$0.130397\pi$
$113$			8.46783i			0.796586i	−0.917258	+	0.398293i	$0.869603\pi$
$114$		0			0
$115$			21.9369i			2.04562i
$116$		−	5.88457i		−	0.546369i
$117$		0			0
$118$		−	4.48909i		−	0.413254i
$119$	3.62405	+	3.70443i	0.332217	+	0.339584i
$120$		0			0
$121$	3.08808			0.280735
$122$	−10.2940			−0.931974
$123$		0			0
$124$			3.43769i			0.308713i
$125$	6.14841			0.549931
$126$		0			0
$127$	−2.22190			−0.197162			−0.0985808	−	0.995129i	$-0.531430\pi$
$127$	−2.22190			−0.197162			−0.0985808	+	0.995129i	$0.531430\pi$
$128$			1.00000i			0.0883883i
$129$		0			0
$130$	11.8890			1.04273
$131$	6.90741			0.603503			0.301752	−	0.953387i	$-0.402429\pi$
$131$	6.90741			0.603503			0.301752	+	0.953387i	$0.402429\pi$
$132$		0			0
$133$	−1.89123	+	1.85020i	−0.163991	+	0.160433i
$134$			9.62535i			0.831503i
$135$		0			0
$136$		−	1.95874i		−	0.167960i
$137$			16.2217i			1.38591i	0.720980	+	0.692955i	$0.243692\pi$
$137$			16.2217i			1.38591i	−0.720980	+	0.692955i	$0.756308\pi$
$138$		0			0
$139$		−	17.1404i		−	1.45383i	−0.686728	−	0.726914i	$-0.740954\pi$
$139$		−	17.1404i		−	1.45383i	0.686728	−	0.726914i	$-0.259046\pi$
$140$	−5.16675	−	5.28134i	−0.436670	−	0.446355i
$141$		0			0
$142$	7.37038			0.618509
$143$	−11.9753			−1.00142
$144$		0			0
$145$		−	16.4329i		−	1.36468i
$146$	2.22270			0.183952
$147$		0			0
$148$	−6.46193			−0.531167
$149$		−	9.28882i		−	0.760970i	−0.924787	−	0.380485i	$-0.875757\pi$
$149$		−	9.28882i		−	0.760970i	0.924787	−	0.380485i	$-0.124243\pi$
$150$		0			0
$151$	17.8911			1.45596			0.727978	−	0.685601i	$-0.240460\pi$
$151$	17.8911			1.45596			0.727978	+	0.685601i	$0.240460\pi$
$152$	1.00000			0.0811107
$153$		0			0
$154$	5.20427	+	5.31969i	0.419372	+	0.428672i
$155$			9.59987i			0.771080i
$156$		0			0
$157$		−	2.32257i		−	0.185361i	−0.995696	−	0.0926806i	$-0.970456\pi$
$157$		−	2.32257i		−	0.185361i	0.995696	−	0.0926806i	$-0.0295435\pi$
$158$		−	15.3756i		−	1.22322i
$159$		0			0
$160$			2.79254i			0.220770i
$161$	−14.8566	+	14.5343i	−1.17087	+	1.14546i
$162$		0			0
$163$	−20.3356			−1.59281			−0.796405	−	0.604763i	$-0.793268\pi$
$163$	−20.3356			−1.59281			−0.796405	+	0.604763i	$0.793268\pi$
$164$	4.41535			0.344781
$165$		0			0
$166$			9.81295i			0.761633i
$167$	−4.01538			−0.310720			−0.155360	−	0.987858i	$-0.549654\pi$
$167$	−4.01538			−0.310720			−0.155360	+	0.987858i	$0.549654\pi$
$168$		0			0
$169$	−5.12545			−0.394265
$170$		−	5.46985i		−	0.419518i
$171$		0			0
$172$	−2.77474			−0.211572
$173$	13.9071			1.05734			0.528670	−	0.848827i	$-0.322691\pi$
$173$	13.9071			1.05734			0.528670	+	0.848827i	$0.322691\pi$
$174$		0			0
$175$	−5.17736	−	5.29218i	−0.391372	−	0.400051i
$176$		−	2.81281i		−	0.212024i
$177$		0			0
$178$			5.78370i			0.433507i
$179$		−	15.7067i		−	1.17397i	−0.809597	−	0.586986i	$-0.800314\pi$
$179$		−	15.7067i		−	1.17397i	0.809597	−	0.586986i	$-0.199686\pi$
$180$		0			0
$181$			11.7653i			0.874507i	0.899338	+	0.437253i	$0.144049\pi$
$181$			11.7653i			0.874507i	−0.899338	+	0.437253i	$0.855951\pi$
$182$	7.87704	+	8.05173i	0.583885	+	0.596834i
$183$		0			0
$184$	7.85553			0.579117
$185$	−18.0452			−1.32671
$186$		0			0
$187$			5.50956i			0.402899i
$188$	12.9034			0.941076
$189$		0			0
$190$	2.79254			0.202592
$191$		−	7.67578i		−	0.555400i	−0.960668	−	0.277700i	$-0.910428\pi$
$191$		−	7.67578i		−	0.555400i	0.960668	−	0.277700i	$-0.0895721\pi$
$192$		0			0
$193$	−12.6193			−0.908360			−0.454180	−	0.890910i	$-0.650068\pi$
$193$	−12.6193			−0.908360			−0.454180	+	0.890910i	$0.650068\pi$
$194$	5.33976			0.383372
$195$		0			0
$196$	0.153524	−	6.99832i	0.0109660	−	0.499880i
$197$		−	2.30302i		−	0.164084i	−0.996629	−	0.0820418i	$-0.973856\pi$
$197$		−	2.30302i		−	0.164084i	0.996629	−	0.0820418i	$-0.0261441\pi$
$198$		0			0
$199$			22.4534i			1.59168i	0.605509	+	0.795839i	$0.292970\pi$
$199$			22.4534i			1.59168i	−0.605509	+	0.795839i	$0.707030\pi$
$200$			2.79827i			0.197868i
$201$		0			0
$202$			3.01320i			0.212008i
$203$	11.1291	−	10.8876i	0.781109	−	0.764162i
$204$		0			0
$205$	12.3300			0.861167
$206$	5.65232			0.393816
$207$		0			0
$208$		−	4.25740i		−	0.295198i
$209$	−2.81281			−0.194566
$210$		0			0
$211$	12.8223			0.882725			0.441363	−	0.897329i	$-0.354495\pi$
$211$	12.8223			0.882725			0.441363	+	0.897329i	$0.354495\pi$
$212$		−	11.0632i		−	0.759826i
$213$		0			0
$214$	8.39026			0.573546
$215$	−7.74858			−0.528449
$216$		0			0
$217$	−6.50146	+	6.36040i	−0.441348	+	0.431772i
$218$			1.70095i			0.115203i
$219$		0			0
$220$		−	7.85489i		−	0.529577i
$221$			8.33913i			0.560950i
$222$		0			0
$223$			16.6824i			1.11713i	0.829459	+	0.558567i	$0.188649\pi$
$223$			16.6824i			1.11713i	−0.829459	+	0.558567i	$0.811351\pi$
$224$	−1.89123	+	1.85020i	−0.126363	+	0.123622i
$225$		0			0
$226$	8.46783			0.563271
$227$	−23.8762			−1.58472			−0.792360	−	0.610054i	$-0.791148\pi$
$227$	−23.8762			−1.58472			−0.792360	+	0.610054i	$0.791148\pi$
$228$		0			0
$229$		−	17.0991i		−	1.12994i	−0.825110	−	0.564972i	$-0.808887\pi$
$229$		−	17.0991i		−	1.12994i	0.825110	−	0.564972i	$-0.191113\pi$
$230$	21.9369			1.44647
$231$		0			0
$232$	−5.88457			−0.386341
$233$		−	29.8614i		−	1.95629i	−0.207929	−	0.978144i	$-0.566672\pi$
$233$		−	29.8614i		−	1.95629i	0.207929	−	0.978144i	$-0.433328\pi$
$234$		0			0
$235$	36.0332			2.35055
$236$	−4.48909			−0.292215
$237$		0			0
$238$	3.70443	−	3.62405i	0.240122	−	0.234913i
$239$		−	16.5510i		−	1.07060i	−0.844663	−	0.535298i	$-0.820199\pi$
$239$		−	16.5510i		−	1.07060i	0.844663	−	0.535298i	$-0.179801\pi$
$240$		0			0
$241$			20.8786i			1.34491i	0.740140	+	0.672453i	$0.234759\pi$
$241$			20.8786i			1.34491i	−0.740140	+	0.672453i	$0.765241\pi$
$242$		−	3.08808i		−	0.198509i
$243$		0			0
$244$			10.2940i			0.659005i
$245$	0.428722	−	19.5431i	0.0273901	−	1.24856i
$246$		0			0
$247$	−4.25740			−0.270892
$248$	3.43769			0.218293
$249$		0			0
$250$		−	6.14841i		−	0.388860i
$251$	−6.19829			−0.391233			−0.195616	−	0.980681i	$-0.562671\pi$
$251$	−6.19829			−0.391233			−0.195616	+	0.980681i	$0.562671\pi$
$252$		0			0
$253$	−22.0961			−1.38917
$254$			2.22190i			0.139414i
$255$		0			0
$256$	1.00000			0.0625000
$257$	10.8375			0.676024			0.338012	−	0.941142i	$-0.390246\pi$
$257$	10.8375			0.676024			0.338012	+	0.941142i	$0.390246\pi$
$258$		0			0
$259$	−11.9559	−	12.2210i	−0.742900	−	0.759376i
$260$		−	11.8890i		−	0.737321i
$261$		0			0
$262$		−	6.90741i		−	0.426741i
$263$		−	22.7232i		−	1.40117i	−0.713568	−	0.700586i	$-0.752922\pi$
$263$		−	22.7232i		−	1.40117i	0.713568	−	0.700586i	$-0.247078\pi$
$264$		0			0
$265$		−	30.8945i		−	1.89783i
$266$	1.85020	+	1.89123i	0.113443	+	0.115959i
$267$		0			0
$268$	9.62535			0.587962
$269$	−9.13996			−0.557273			−0.278636	−	0.960397i	$-0.589882\pi$
$269$	−9.13996			−0.557273			−0.278636	+	0.960397i	$0.589882\pi$
$270$		0			0
$271$		−	21.8969i		−	1.33014i	−0.746780	−	0.665072i	$-0.768401\pi$
$271$		−	21.8969i		−	1.33014i	0.746780	−	0.665072i	$-0.231599\pi$
$272$	−1.95874			−0.118766
$273$		0			0
$274$	16.2217			0.979987
$275$		−	7.87102i		−	0.474640i
$276$		0			0
$277$	12.4160			0.746008			0.373004	−	0.927830i	$-0.378328\pi$
$277$	12.4160			0.746008			0.373004	+	0.927830i	$0.378328\pi$
$278$	−17.1404			−1.02801
$279$		0			0
$280$	−5.28134	+	5.16675i	−0.315620	+	0.308773i
$281$			11.0752i			0.660690i	0.943860	+	0.330345i	$0.107165\pi$
$281$			11.0752i			0.660690i	−0.943860	+	0.330345i	$0.892835\pi$
$282$		0			0
$283$			20.5934i			1.22415i	0.790799	+	0.612075i	$0.209665\pi$
$283$			20.5934i			1.22415i	−0.790799	+	0.612075i	$0.790335\pi$
$284$		−	7.37038i		−	0.437352i
$285$		0			0
$286$			11.9753i			0.708112i
$287$	8.16928	+	8.35046i	0.482217	+	0.492912i
$288$		0			0
$289$	−13.1633			−0.774315
$290$	−16.4329			−0.964973
$291$		0			0
$292$		−	2.22270i		−	0.130074i
$293$	−7.50953			−0.438711			−0.219356	−	0.975645i	$-0.570396\pi$
$293$	−7.50953			−0.438711			−0.219356	+	0.975645i	$0.570396\pi$
$294$		0			0
$295$	−12.5360			−0.729872
$296$			6.46193i			0.375592i
$297$		0			0
$298$	−9.28882			−0.538087
$299$	−33.4441			−1.93412
$300$		0			0
$301$	−5.13383	−	5.24769i	−0.295909	−	0.302472i
$302$		−	17.8911i		−	1.02952i
$303$		0			0
$304$		−	1.00000i		−	0.0573539i
$305$			28.7464i			1.64601i
$306$		0			0
$307$			16.1047i			0.919142i	0.888141	+	0.459571i	$0.151997\pi$
$307$			16.1047i			0.919142i	−0.888141	+	0.459571i	$0.848003\pi$
$308$	5.31969	−	5.20427i	0.303117	−	0.296541i
$309$		0			0
$310$	9.59987			0.545236
$311$	21.7770			1.23486			0.617431	−	0.786625i	$-0.288173\pi$
$311$	21.7770			1.23486			0.617431	+	0.786625i	$0.288173\pi$
$312$		0			0
$313$		−	27.9439i		−	1.57948i	−0.613439	−	0.789742i	$-0.710214\pi$
$313$		−	27.9439i		−	1.57948i	0.613439	−	0.789742i	$-0.289786\pi$
$314$	−2.32257			−0.131070
$315$		0			0
$316$	−15.3756			−0.864946
$317$			14.8126i			0.831960i	0.909374	+	0.415980i	$0.136561\pi$
$317$			14.8126i			0.831960i	−0.909374	+	0.415980i	$0.863439\pi$
$318$		0			0
$319$	16.5522			0.926745
$320$	2.79254			0.156108
$321$		0			0
$322$	14.5343	+	14.8566i	0.809965	+	0.827928i
$323$			1.95874i			0.108987i
$324$		0			0
$325$		−	11.9134i		−	0.660834i
$326$			20.3356i			1.12629i
$327$		0			0
$328$		−	4.41535i		−	0.243797i
$329$	23.8738	+	24.4033i	1.31621	+	1.34540i
$330$		0			0
$331$	−20.0031			−1.09947			−0.549735	−	0.835339i	$-0.685271\pi$
$331$	−20.0031			−1.09947			−0.549735	+	0.835339i	$0.685271\pi$
$332$	9.81295			0.538556
$333$		0			0
$334$			4.01538i			0.219712i
$335$	26.8791			1.46856
$336$		0			0
$337$	−6.97964			−0.380205			−0.190103	−	0.981764i	$-0.560882\pi$
$337$	−6.97964			−0.380205			−0.190103	+	0.981764i	$0.560882\pi$
$338$			5.12545i			0.278788i
$339$		0			0
$340$	−5.46985			−0.296644
$341$	−9.66957			−0.523636
$342$		0			0
$343$	13.5195	−	12.6579i	0.729984	−	0.683464i
$344$			2.77474i			0.149604i
$345$		0			0
$346$		−	13.9071i		−	0.747653i
$347$			11.7560i			0.631097i	0.948910	+	0.315548i	$0.102188\pi$
$347$			11.7560i			0.631097i	−0.948910	+	0.315548i	$0.897812\pi$
$348$		0			0
$349$			1.26201i			0.0675539i	0.999429	+	0.0337770i	$0.0107536\pi$
$349$			1.26201i			0.0675539i	−0.999429	+	0.0337770i	$0.989246\pi$
$350$	−5.29218	+	5.17736i	−0.282879	+	0.276742i
$351$		0			0
$352$	−2.81281			−0.149923
$353$	21.4175			1.13994			0.569970	−	0.821666i	$-0.306955\pi$
$353$	21.4175			1.13994			0.569970	+	0.821666i	$0.306955\pi$
$354$		0			0
$355$		−	20.5821i		−	1.09238i
$356$	5.78370			0.306536
$357$		0			0
$358$	−15.7067			−0.830124
$359$		−	15.0187i		−	0.792656i	−0.918109	−	0.396328i	$-0.870284\pi$
$359$		−	15.0187i		−	0.792656i	0.918109	−	0.396328i	$-0.129716\pi$
$360$		0			0
$361$	−1.00000			−0.0526316
$362$	11.7653			0.618370
$363$		0			0
$364$	8.05173	−	7.87704i	0.422026	−	0.412869i
$365$		−	6.20698i		−	0.324888i
$366$		0			0
$367$		−	24.1756i		−	1.26196i	−0.775801	−	0.630978i	$-0.782654\pi$
$367$		−	24.1756i		−	1.26196i	0.775801	−	0.630978i	$-0.217346\pi$
$368$		−	7.85553i		−	0.409498i
$369$		0			0
$370$			18.0452i			0.938124i
$371$	20.9231	−	20.4692i	1.08628	−	1.06271i
$372$		0			0
$373$	−33.9540			−1.75807			−0.879035	−	0.476758i	$-0.841812\pi$
$373$	−33.9540			−1.75807			−0.879035	+	0.476758i	$0.841812\pi$
$374$	5.50956			0.284893
$375$		0			0
$376$		−	12.9034i		−	0.665441i
$377$	25.0530			1.29029
$378$		0			0
$379$	8.08528			0.415313			0.207656	−	0.978202i	$-0.433416\pi$
$379$	8.08528			0.415313			0.207656	+	0.978202i	$0.433416\pi$
$380$		−	2.79254i		−	0.143254i
$381$		0			0
$382$	−7.67578			−0.392727
$383$	13.5617			0.692969			0.346485	−	0.938056i	$-0.387375\pi$
$383$	13.5617			0.692969			0.346485	+	0.938056i	$0.387375\pi$
$384$		0			0
$385$	14.8554	−	14.5331i	0.757103	−	0.740676i
$386$			12.6193i			0.642307i
$387$		0			0
$388$		−	5.33976i		−	0.271085i
$389$		−	0.213733i		−	0.0108367i	−0.999985	−	0.00541834i	$-0.998275\pi$
$389$		−	0.213733i		−	0.0108367i	0.999985	−	0.00541834i	$-0.00172472\pi$
$390$		0			0
$391$			15.3869i			0.778150i
$392$	−6.99832	−	0.153524i	−0.353468	−	0.00775415i
$393$		0			0
$394$	−2.30302			−0.116025
$395$	−42.9370			−2.16039
$396$		0			0
$397$			20.6217i			1.03497i	0.855691	+	0.517487i	$0.173133\pi$
$397$			20.6217i			1.03497i	−0.855691	+	0.517487i	$0.826867\pi$
$398$	22.4534			1.12549
$399$		0			0
$400$	2.79827			0.139914
$401$		−	7.95277i		−	0.397142i	−0.980086	−	0.198571i	$-0.936370\pi$
$401$		−	7.95277i		−	0.397142i	0.980086	−	0.198571i	$-0.0636301\pi$
$402$		0			0
$403$	−14.6356			−0.729051
$404$	3.01320			0.149912
$405$		0			0
$406$	−10.8876	−	11.1291i	−0.540344	−	0.552328i
$407$		−	18.1762i		−	0.900960i
$408$		0			0
$409$			26.3164i			1.30126i	0.759394	+	0.650631i	$0.225496\pi$
$409$			26.3164i			1.30126i	−0.759394	+	0.650631i	$0.774504\pi$
$410$		−	12.3300i		−	0.608937i
$411$		0			0
$412$		−	5.65232i		−	0.278470i
$413$	−8.30571	−	8.48991i	−0.408697	−	0.417761i
$414$		0			0
$415$	27.4031			1.34516
$416$	−4.25740			−0.208736
$417$		0			0
$418$			2.81281i			0.137579i
$419$	25.9792			1.26917			0.634583	−	0.772855i	$-0.281172\pi$
$419$	25.9792			1.26917			0.634583	+	0.772855i	$0.281172\pi$
$420$		0			0
$421$	17.6164			0.858572			0.429286	−	0.903169i	$-0.358765\pi$
$421$	17.6164			0.858572			0.429286	+	0.903169i	$0.358765\pi$
$422$		−	12.8223i		−	0.624181i
$423$		0			0
$424$	−11.0632			−0.537278
$425$	−5.48108			−0.265871
$426$		0			0
$427$	−19.4683	+	19.0459i	−0.942138	+	0.921697i
$428$		−	8.39026i		−	0.405559i
$429$		0			0
$430$			7.74858i			0.373670i
$431$			36.7215i			1.76881i	0.466716	+	0.884407i	$0.345437\pi$
$431$			36.7215i			1.76881i	−0.466716	+	0.884407i	$0.654563\pi$
$432$		0			0
$433$		−	26.6661i		−	1.28149i	−0.767753	−	0.640746i	$-0.778625\pi$
$433$		−	26.6661i		−	1.28149i	0.767753	−	0.640746i	$-0.221375\pi$
$434$	6.36040	+	6.50146i	0.305309	+	0.312080i
$435$		0			0
$436$	1.70095			0.0814607
$437$	−7.85553			−0.375781
$438$		0			0
$439$		−	20.6892i		−	0.987440i	−0.869621	−	0.493720i	$-0.835637\pi$
$439$		−	20.6892i		−	0.987440i	0.869621	−	0.493720i	$-0.164363\pi$
$440$	−7.85489			−0.374467
$441$		0			0
$442$	8.33913			0.396652
$443$			16.3627i			0.777415i	0.921361	+	0.388707i	$0.127078\pi$
$443$			16.3627i			0.777415i	−0.921361	+	0.388707i	$0.872922\pi$
$444$		0			0
$445$	16.1512			0.765641
$446$	16.6824			0.789934
$447$		0			0
$448$	1.85020	+	1.89123i	0.0874137	+	0.0893524i
$449$			37.7578i			1.78190i	0.454099	+	0.890951i	$0.349961\pi$
$449$			37.7578i			1.78190i	−0.454099	+	0.890951i	$0.650039\pi$
$450$		0			0
$451$			12.4196i			0.584814i
$452$		−	8.46783i		−	0.398293i
$453$		0			0
$454$			23.8762i			1.12057i
$455$	22.4848	−	21.9969i	1.05410	−	1.03123i
$456$		0			0
$457$	−2.62175			−0.122640			−0.0613200	−	0.998118i	$-0.519531\pi$
$457$	−2.62175			−0.122640			−0.0613200	+	0.998118i	$0.519531\pi$
$458$	−17.0991			−0.798991
$459$		0			0
$460$		−	21.9369i		−	1.02281i
$461$	−35.6151			−1.65876			−0.829380	−	0.558686i	$-0.811306\pi$
$461$	−35.6151			−1.65876			−0.829380	+	0.558686i	$0.811306\pi$
$462$		0			0
$463$	−17.3748			−0.807478			−0.403739	−	0.914874i	$-0.632290\pi$
$463$	−17.3748			−0.807478			−0.403739	+	0.914874i	$0.632290\pi$
$464$			5.88457i			0.273184i
$465$		0			0
$466$	−29.8614			−1.38330
$467$	38.7528			1.79327			0.896633	−	0.442774i	$-0.146006\pi$
$467$	38.7528			1.79327			0.896633	+	0.442774i	$0.146006\pi$
$468$		0			0
$469$	17.8088	+	18.2038i	0.822334	+	0.840572i
$470$		−	36.0332i		−	1.66209i
$471$		0			0
$472$			4.48909i			0.206627i
$473$		−	7.80483i		−	0.358867i
$474$		0			0
$475$		−	2.79827i		−	0.128394i
$476$	−3.62405	−	3.70443i	−0.166108	−	0.169792i
$477$		0			0
$478$	−16.5510			−0.757025
$479$	10.0355			0.458534			0.229267	−	0.973364i	$-0.426367\pi$
$479$	10.0355			0.458534			0.229267	+	0.973364i	$0.426367\pi$
$480$		0			0
$481$		−	27.5110i		−	1.25439i
$482$	20.8786			0.950992
$483$		0			0
$484$	−3.08808			−0.140367
$485$		−	14.9115i		−	0.677095i
$486$		0			0
$487$	−1.42589			−0.0646133			−0.0323066	−	0.999478i	$-0.510285\pi$
$487$	−1.42589			−0.0646133			−0.0323066	+	0.999478i	$0.510285\pi$
$488$	10.2940			0.465987
$489$		0			0
$490$	−19.5431	−	0.428722i	−0.882866	−	0.0193677i
$491$		−	4.27590i		−	0.192969i	−0.995334	−	0.0964844i	$-0.969240\pi$
$491$		−	4.27590i		−	0.192969i	0.995334	−	0.0964844i	$-0.0307598\pi$
$492$		0			0
$493$		−	11.5263i		−	0.519120i
$494$			4.25740i			0.191549i
$495$		0			0
$496$		−	3.43769i		−	0.154357i
$497$	13.9391	−	13.6367i	0.625255	−	0.611689i
$498$		0			0
$499$	−18.4160			−0.824412			−0.412206	−	0.911091i	$-0.635242\pi$
$499$	−18.4160			−0.824412			−0.412206	+	0.911091i	$0.635242\pi$
$500$	−6.14841			−0.274965
$501$		0			0
$502$			6.19829i			0.276643i
$503$	−4.14287			−0.184721			−0.0923606	−	0.995726i	$-0.529441\pi$
$503$	−4.14287			−0.184721			−0.0923606	+	0.995726i	$0.529441\pi$
$504$		0			0
$505$	8.41448			0.374439
$506$			22.0961i			0.982293i
$507$		0			0
$508$	2.22190			0.0985808
$509$	0.434933			0.0192781			0.00963904	−	0.999954i	$-0.496932\pi$
$509$	0.434933			0.0192781			0.00963904	+	0.999954i	$0.496932\pi$
$510$		0			0
$511$	4.20364	−	4.11244i	0.185958	−	0.181924i
$512$		−	1.00000i		−	0.0441942i
$513$		0			0
$514$		−	10.8375i		−	0.478021i
$515$		−	15.7843i		−	0.695540i
$516$		0			0
$517$			36.2948i			1.59624i
$518$	−12.2210	+	11.9559i	−0.536960	+	0.525310i
$519$		0			0
$520$	−11.8890			−0.521365
$521$	−39.4921			−1.73018			−0.865090	−	0.501617i	$-0.832739\pi$
$521$	−39.4921			−1.73018			−0.865090	+	0.501617i	$0.832739\pi$
$522$		0			0
$523$			4.03447i			0.176415i	0.996102	+	0.0882075i	$0.0281138\pi$
$523$			4.03447i			0.176415i	−0.996102	+	0.0882075i	$0.971886\pi$
$524$	−6.90741			−0.301752
$525$		0			0
$526$	−22.7232			−0.990778
$527$			6.73352i			0.293317i
$528$		0			0
$529$	−38.7093			−1.68302
$530$	−30.8945			−1.34197
$531$		0			0
$532$	1.89123	−	1.85020i	0.0819953	−	0.0802163i
$533$			18.7979i			0.814228i
$534$		0			0
$535$		−	23.4301i		−	1.01297i
$536$		−	9.62535i		−	0.415752i
$537$		0			0
$538$			9.13996i			0.394052i
$539$	19.6850	+	0.431835i	0.847891	+	0.0186005i
$540$		0			0
$541$	−32.9289			−1.41572			−0.707861	−	0.706351i	$-0.750340\pi$
$541$	−32.9289			−1.41572			−0.707861	+	0.706351i	$0.750340\pi$
$542$	−21.8969			−0.940553
$543$		0			0
$544$			1.95874i			0.0839802i
$545$	4.74997			0.203466
$546$		0			0
$547$	−35.9075			−1.53529			−0.767647	−	0.640873i	$-0.778573\pi$
$547$	−35.9075			−1.53529			−0.767647	+	0.640873i	$0.778573\pi$
$548$		−	16.2217i		−	0.692955i
$549$		0			0
$550$	−7.87102			−0.335621
$551$	5.88457			0.250691
$552$		0			0
$553$	−28.4479	−	29.0789i	−1.20973	−	1.23656i
$554$		−	12.4160i		−	0.527507i
$555$		0			0
$556$			17.1404i			0.726914i
$557$		−	10.5834i		−	0.448432i	−0.974539	−	0.224216i	$-0.928018\pi$
$557$		−	10.5834i		−	0.448432i	0.974539	−	0.224216i	$-0.0719821\pi$
$558$		0			0
$559$		−	11.8132i		−	0.499645i
$560$	5.16675	+	5.28134i	0.218335	+	0.223177i
$561$		0			0
$562$	11.0752			0.467178
$563$	−2.56389			−0.108055			−0.0540275	−	0.998539i	$-0.517206\pi$
$563$	−2.56389			−0.108055			−0.0540275	+	0.998539i	$0.517206\pi$
$564$		0			0
$565$		−	23.6467i		−	0.994826i
$566$	20.5934			0.865605
$567$		0			0
$568$	−7.37038			−0.309254
$569$		−	32.7323i		−	1.37221i	−0.727503	−	0.686104i	$-0.759319\pi$
$569$		−	32.7323i		−	1.37221i	0.727503	−	0.686104i	$-0.240681\pi$
$570$		0			0
$571$	−22.8629			−0.956783			−0.478391	−	0.878147i	$-0.658780\pi$
$571$	−22.8629			−0.956783			−0.478391	+	0.878147i	$0.658780\pi$
$572$	11.9753			0.500711
$573$		0			0
$574$	8.35046	−	8.16928i	0.348541	−	0.340979i
$575$		−	21.9819i		−	0.916709i
$576$		0			0
$577$			2.69585i			0.112230i	0.998424	+	0.0561148i	$0.0178713\pi$
$577$			2.69585i			0.112230i	−0.998424	+	0.0561148i	$0.982129\pi$
$578$			13.1633i			0.547523i
$579$		0			0
$580$			16.4329i			0.682339i
$581$	18.1559	+	18.5586i	0.753235	+	0.769940i
$582$		0			0
$583$	31.1188			1.28881
$584$	−2.22270			−0.0919760
$585$		0			0
$586$			7.50953i			0.310216i
$587$	−15.0407			−0.620795			−0.310398	−	0.950607i	$-0.600462\pi$
$587$	−15.0407			−0.620795			−0.310398	+	0.950607i	$0.600462\pi$
$588$		0			0
$589$	−3.43769			−0.141647
$590$			12.5360i			0.516097i
$591$		0			0
$592$	6.46193			0.265583
$593$	−15.3773			−0.631470			−0.315735	−	0.948847i	$-0.602251\pi$
$593$	−15.3773			−0.631470			−0.315735	+	0.948847i	$0.602251\pi$
$594$		0			0
$595$	−10.1203	−	10.3448i	−0.414892	−	0.424094i
$596$			9.28882i			0.380485i
$597$		0			0
$598$			33.4441i			1.36763i
$599$			8.61276i			0.351908i	0.984398	+	0.175954i	$0.0563009\pi$
$599$			8.61276i			0.351908i	−0.984398	+	0.175954i	$0.943699\pi$
$600$		0			0
$601$		−	9.64748i		−	0.393529i	−0.980451	−	0.196765i	$-0.936957\pi$
$601$		−	9.64748i		−	0.393529i	0.980451	−	0.196765i	$-0.0630434\pi$
$602$	−5.24769	+	5.13383i	−0.213880	+	0.209239i
$603$		0			0
$604$	−17.8911			−0.727978
$605$	−8.62359			−0.350599
$606$		0			0
$607$		−	0.150114i		−	0.00609292i	−0.999995	−	0.00304646i	$-0.999030\pi$
$607$		−	0.150114i		−	0.00609292i	0.999995	−	0.00304646i	$-0.000969720\pi$
$608$	−1.00000			−0.0405554
$609$		0			0
$610$	28.7464			1.16391
$611$			54.9349i			2.22243i
$612$		0			0
$613$	−30.1296			−1.21692			−0.608462	−	0.793583i	$-0.708213\pi$
$613$	−30.1296			−1.21692			−0.608462	+	0.793583i	$0.708213\pi$
$614$	16.1047			0.649932
$615$		0			0
$616$	−5.20427	−	5.31969i	−0.209686	−	0.214336i
$617$		−	29.4148i		−	1.18419i	−0.805866	−	0.592097i	$-0.798300\pi$
$617$		−	29.4148i		−	1.18419i	0.805866	−	0.592097i	$-0.201700\pi$
$618$		0			0
$619$			20.8389i			0.837587i	0.908082	+	0.418793i	$0.137547\pi$
$619$			20.8389i			0.837587i	−0.908082	+	0.418793i	$0.862453\pi$
$620$		−	9.59987i		−	0.385540i
$621$		0			0
$622$		−	21.7770i		−	0.873180i
$623$	10.7010	+	10.9383i	0.428727	+	0.438235i
$624$		0			0
$625$	−31.1610			−1.24644
$626$	−27.9439			−1.11686
$627$		0			0
$628$			2.32257i			0.0926806i
$629$	−12.6572			−0.504676
$630$		0			0
$631$	13.0244			0.518494			0.259247	−	0.965811i	$-0.416526\pi$
$631$	13.0244			0.518494			0.259247	+	0.965811i	$0.416526\pi$
$632$			15.3756i			0.611609i
$633$		0			0
$634$	14.8126			0.588285
$635$	6.20474			0.246227
$636$		0			0
$637$	29.7946	+	0.653614i	1.18051	+	0.0258971i
$638$		−	16.5522i		−	0.655308i
$639$		0			0
$640$		−	2.79254i		−	0.110385i
$641$		−	40.5063i		−	1.59990i	−0.600064	−	0.799952i	$-0.704858\pi$
$641$		−	40.5063i		−	1.59990i	0.600064	−	0.799952i	$-0.295142\pi$
$642$		0			0
$643$		−	41.9319i		−	1.65363i	−0.562473	−	0.826816i	$-0.690150\pi$
$643$		−	41.9319i		−	1.65363i	0.562473	−	0.826816i	$-0.309850\pi$
$644$	14.8566	−	14.5343i	0.585433	−	0.572732i
$645$		0			0
$646$	1.95874			0.0770655
$647$	0.420630			0.0165367			0.00826834	−	0.999966i	$-0.497368\pi$
$647$	0.420630			0.0165367			0.00826834	+	0.999966i	$0.497368\pi$
$648$		0			0
$649$		−	12.6270i		−	0.495652i
$650$	−11.9134			−0.467280
$651$		0			0
$652$	20.3356			0.796405
$653$			31.1939i			1.22071i	0.792127	+	0.610356i	$0.208974\pi$
$653$			31.1939i			1.22071i	−0.792127	+	0.610356i	$0.791026\pi$
$654$		0			0
$655$	−19.2892			−0.753692
$656$	−4.41535			−0.172391
$657$		0			0
$658$	24.4033	−	23.8738i	0.951340	−	0.930699i
$659$		−	14.4376i		−	0.562407i	−0.959648	−	0.281204i	$-0.909266\pi$
$659$		−	14.4376i		−	0.562407i	0.959648	−	0.281204i	$-0.0907337\pi$
$660$		0			0
$661$			23.3304i			0.907447i	0.891142	+	0.453724i	$0.149905\pi$
$661$			23.3304i			0.907447i	−0.891142	+	0.453724i	$0.850095\pi$
$662$			20.0031i			0.777443i
$663$		0			0
$664$		−	9.81295i		−	0.380816i
$665$	5.28134	−	5.16675i	0.204802	−	0.200358i
$666$		0			0
$667$	46.2264			1.78989
$668$	4.01538			0.155360
$669$		0			0
$670$		−	26.8791i		−	1.03843i
$671$	−28.9551			−1.11780
$672$		0			0
$673$	−1.76022			−0.0678513			−0.0339257	−	0.999424i	$-0.510801\pi$
$673$	−1.76022			−0.0678513			−0.0339257	+	0.999424i	$0.510801\pi$
$674$			6.97964i			0.268846i
$675$		0			0
$676$	5.12545			0.197133
$677$	−44.9229			−1.72653			−0.863264	−	0.504753i	$-0.831584\pi$
$677$	−44.9229			−1.72653			−0.863264	+	0.504753i	$0.831584\pi$
$678$		0			0
$679$	10.0987	−	9.87961i	0.387553	−	0.379145i
$680$			5.46985i			0.209759i
$681$		0			0
$682$			9.66957i			0.370267i
$683$		−	4.51293i		−	0.172682i	−0.996266	−	0.0863412i	$-0.972482\pi$
$683$		−	4.51293i		−	0.172682i	0.996266	−	0.0863412i	$-0.0275175\pi$
$684$		0			0
$685$		−	45.2996i		−	1.73081i
$686$	−12.6579	−	13.5195i	−0.483282	−	0.516177i
$687$		0			0
$688$	2.77474			0.105786
$689$	47.1006			1.79439
$690$		0			0
$691$			13.0163i			0.495164i	0.968867	+	0.247582i	$0.0796359\pi$
$691$			13.0163i			0.495164i	−0.968867	+	0.247582i	$0.920364\pi$
$692$	−13.9071			−0.528670
$693$		0			0
$694$	11.7560			0.446253
$695$			47.8652i			1.81563i
$696$		0			0
$697$	8.64851			0.327586
$698$	1.26201			0.0477678
$699$		0			0
$700$	5.17736	+	5.29218i	0.195686	+	0.200026i
$701$			2.52708i			0.0954465i	0.998861	+	0.0477232i	$0.0151965\pi$
$701$			2.52708i			0.0954465i	−0.998861	+	0.0477232i	$0.984803\pi$
$702$		0			0
$703$		−	6.46193i		−	0.243716i
$704$			2.81281i			0.106012i
$705$		0			0
$706$		−	21.4175i		−	0.806059i
$707$	5.57502	+	5.69866i	0.209670	+	0.214320i
$708$		0			0
$709$	−8.79153			−0.330173			−0.165086	−	0.986279i	$-0.552790\pi$
$709$	−8.79153			−0.330173			−0.165086	+	0.986279i	$0.552790\pi$
$710$	−20.5821			−0.772432
$711$		0			0
$712$		−	5.78370i		−	0.216753i
$713$	−27.0048			−1.01134
$714$		0			0
$715$	33.4414			1.25064
$716$			15.7067i			0.586986i
$717$		0			0
$718$	−15.0187			−0.560492
$719$	47.6152			1.77575			0.887874	−	0.460087i	$-0.152182\pi$
$719$	47.6152			1.77575			0.887874	+	0.460087i	$0.152182\pi$
$720$		0			0
$721$	10.6899	−	10.4579i	0.398111	−	0.389473i
$722$			1.00000i			0.0372161i
$723$		0			0
$724$		−	11.7653i		−	0.437253i
$725$			16.4666i			0.611555i
$726$		0			0
$727$			2.16700i			0.0803694i	0.999192	+	0.0401847i	$0.0127946\pi$
$727$			2.16700i			0.0803694i	−0.999192	+	0.0401847i	$0.987205\pi$
$728$	−7.87704	−	8.05173i	−0.291942	−	0.298417i
$729$		0			0
$730$	−6.20698			−0.229730
$731$	−5.43499			−0.201020
$732$		0			0
$733$		−	18.1874i		−	0.671767i	−0.941903	−	0.335884i	$-0.890965\pi$
$733$		−	18.1874i		−	0.671767i	0.941903	−	0.335884i	$-0.109035\pi$
$734$	−24.1756			−0.892337
$735$		0			0
$736$	−7.85553			−0.289559
$737$			27.0743i			0.997295i
$738$		0			0
$739$	10.0954			0.371364			0.185682	−	0.982610i	$-0.440551\pi$
$739$	10.0954			0.371364			0.185682	+	0.982610i	$0.440551\pi$
$740$	18.0452			0.663354
$741$		0			0
$742$	−20.4692	−	20.9231i	−0.751447	−	0.768113i
$743$		−	31.0735i		−	1.13998i	−0.821653	−	0.569988i	$-0.806948\pi$
$743$		−	31.0735i		−	1.13998i	0.821653	−	0.569988i	$-0.193052\pi$
$744$		0			0
$745$			25.9394i			0.950346i
$746$			33.9540i			1.24314i
$747$		0			0
$748$		−	5.50956i		−	0.201450i
$749$	15.8679	−	15.5237i	0.579802	−	0.567222i
$750$		0			0
$751$	−20.7729			−0.758013			−0.379006	−	0.925394i	$-0.623734\pi$
$751$	−20.7729			−0.758013			−0.379006	+	0.925394i	$0.623734\pi$
$752$	−12.9034			−0.470538
$753$		0			0
$754$		−	25.0530i		−	0.912375i
$755$	−49.9615			−1.81829
$756$		0			0
$757$	11.0056			0.400004			0.200002	−	0.979796i	$-0.435905\pi$
$757$	11.0056			0.400004			0.200002	+	0.979796i	$0.435905\pi$
$758$		−	8.08528i		−	0.293670i
$759$		0			0
$760$	−2.79254			−0.101296
$761$	−20.4499			−0.741308			−0.370654	−	0.928771i	$-0.620866\pi$
$761$	−20.4499			−0.741308			−0.370654	+	0.928771i	$0.620866\pi$
$762$		0			0
$763$	3.14710	+	3.21689i	0.113933	+	0.116459i
$764$			7.67578i			0.277700i
$765$		0			0
$766$		−	13.5617i		−	0.490003i
$767$		−	19.1118i		−	0.690089i
$768$		0			0
$769$		−	14.3429i		−	0.517220i	−0.965982	−	0.258610i	$-0.916736\pi$
$769$		−	14.3429i		−	0.517220i	0.965982	−	0.258610i	$-0.0832644\pi$
$770$	−14.5331	−	14.8554i	−0.523737	−	0.535352i
$771$		0			0
$772$	12.6193			0.454180
$773$	6.80167			0.244639			0.122320	−	0.992491i	$-0.460967\pi$
$773$	6.80167			0.244639			0.122320	+	0.992491i	$0.460967\pi$
$774$		0			0
$775$		−	9.61958i		−	0.345545i
$776$	−5.33976			−0.191686
$777$		0			0
$778$	−0.213733			−0.00766269
$779$			4.41535i			0.158196i
$780$		0			0
$781$	20.7315			0.741832
$782$	15.3869			0.550235
$783$		0			0
$784$	−0.153524	+	6.99832i	−0.00548301	+	0.249940i
$785$			6.48586i			0.231490i
$786$		0			0
$787$			25.6205i			0.913271i	0.889654	+	0.456636i	$0.150946\pi$
$787$			25.6205i			0.913271i	−0.889654	+	0.456636i	$0.849054\pi$
$788$			2.30302i			0.0820418i
$789$		0			0
$790$			42.9370i			1.52763i
$791$	16.0146	−	15.6672i	0.569415	−	0.557060i
$792$		0			0
$793$	−43.8256			−1.55629
$794$	20.6217			0.731838
$795$		0			0
$796$		−	22.4534i		−	0.795839i
$797$	12.6559			0.448295			0.224147	−	0.974555i	$-0.428040\pi$
$797$	12.6559			0.448295			0.224147	+	0.974555i	$0.428040\pi$
$798$		0			0
$799$	25.2743			0.894142
$800$		−	2.79827i		−	0.0989338i
$801$		0			0
$802$	−7.95277			−0.280822
$803$	6.25204			0.220630
$804$		0			0
$805$	41.4877	−	40.5876i	1.46225	−	1.43052i
$806$			14.6356i			0.515517i
$807$		0			0
$808$		−	3.01320i		−	0.106004i
$809$		−	3.22145i		−	0.113260i	−0.998395	−	0.0566301i	$-0.981964\pi$
$809$		−	3.22145i		−	0.113260i	0.998395	−	0.0566301i	$-0.0180356\pi$
$810$		0			0
$811$			19.6943i			0.691562i	0.938315	+	0.345781i	$0.112386\pi$
$811$			19.6943i			0.691562i	−0.938315	+	0.345781i	$0.887614\pi$
$812$	−11.1291	+	10.8876i	−0.390555	+	0.382081i
$813$		0			0
$814$	−18.1762			−0.637075
$815$	56.7881			1.98920
$816$		0			0
$817$		−	2.77474i		−	0.0970760i
$818$	26.3164			0.920132
$819$		0			0
$820$	−12.3300			−0.430584
$821$		−	23.8292i		−	0.831643i	−0.909446	−	0.415822i	$-0.863494\pi$
$821$		−	23.8292i		−	0.831643i	0.909446	−	0.415822i	$-0.136506\pi$
$822$		0			0
$823$	36.7977			1.28269			0.641343	−	0.767254i	$-0.278378\pi$
$823$	36.7977			1.28269			0.641343	+	0.767254i	$0.278378\pi$
$824$	−5.65232			−0.196908
$825$		0			0
$826$	−8.48991	+	8.30571i	−0.295402	+	0.288993i
$827$			14.3867i			0.500276i	0.968210	+	0.250138i	$0.0804760\pi$
$827$			14.3867i			0.500276i	−0.968210	+	0.250138i	$0.919524\pi$
$828$		0			0
$829$			21.3049i			0.739949i	0.929042	+	0.369975i	$0.120634\pi$
$829$			21.3049i			0.739949i	−0.929042	+	0.369975i	$0.879366\pi$
$830$		−	27.4031i		−	0.951174i
$831$		0			0
$832$			4.25740i			0.147599i
$833$	0.300714	−	13.7079i	0.0104191	−	0.474949i
$834$		0			0
$835$	11.2131			0.388046
$836$	2.81281			0.0972832
$837$		0			0
$838$		−	25.9792i		−	0.897436i
$839$	−1.55642			−0.0537336			−0.0268668	−	0.999639i	$-0.508553\pi$
$839$	−1.55642			−0.0537336			−0.0268668	+	0.999639i	$0.508553\pi$
$840$		0			0
$841$	−5.62818			−0.194075
$842$		−	17.6164i		−	0.607102i
$843$		0			0
$844$	−12.8223			−0.441363
$845$	14.3130			0.492382
$846$		0			0
$847$	−5.71357	−	5.84028i	−0.196320	−	0.200674i
$848$			11.0632i			0.379913i
$849$		0			0
$850$			5.48108i			0.187999i
$851$		−	50.7618i		−	1.74009i
$852$		0			0
$853$			31.7035i			1.08551i	0.839892	+	0.542753i	$0.182618\pi$
$853$			31.7035i			1.08551i	−0.839892	+	0.542753i	$0.817382\pi$
$854$	19.0459	+	19.4683i	0.651738	+	0.666193i
$855$		0			0
$856$	−8.39026			−0.286773
$857$	−38.8043			−1.32553			−0.662765	−	0.748827i	$-0.730617\pi$
$857$	−38.8043			−1.32553			−0.662765	+	0.748827i	$0.730617\pi$
$858$		0			0
$859$			34.5538i			1.17896i	0.807783	+	0.589480i	$0.200667\pi$
$859$			34.5538i			1.17896i	−0.807783	+	0.589480i	$0.799333\pi$
$860$	7.74858			0.264224
$861$		0			0
$862$	36.7215			1.25074
$863$			28.4011i			0.966786i	0.875403	+	0.483393i	$0.160596\pi$
$863$			28.4011i			0.966786i	−0.875403	+	0.483393i	$0.839404\pi$
$864$		0			0
$865$	−38.8362			−1.32047
$866$	−26.6661			−0.906151
$867$		0			0
$868$	6.50146	−	6.36040i	0.220674	−	0.215886i
$869$		−	43.2487i		−	1.46711i
$870$		0			0
$871$			40.9789i			1.38852i
$872$		−	1.70095i		−	0.0576014i
$873$		0			0
$874$			7.85553i			0.265717i
$875$	−11.3758	−	11.6281i	−0.384572	−	0.393101i
$876$		0			0
$877$	−8.10006			−0.273519			−0.136760	−	0.990604i	$-0.543669\pi$
$877$	−8.10006			−0.273519			−0.136760	+	0.990604i	$0.543669\pi$
$878$	−20.6892			−0.698226
$879$		0			0
$880$			7.85489i			0.264788i
$881$	−28.5675			−0.962463			−0.481232	−	0.876594i	$-0.659810\pi$
$881$	−28.5675			−0.962463			−0.481232	+	0.876594i	$0.659810\pi$
$882$		0			0
$883$	−40.7466			−1.37123			−0.685617	−	0.727963i	$-0.740467\pi$
$883$	−40.7466			−1.37123			−0.685617	+	0.727963i	$0.740467\pi$
$884$		−	8.33913i		−	0.280475i
$885$		0			0
$886$	16.3627			0.549715
$887$	31.5744			1.06017			0.530083	−	0.847946i	$-0.322161\pi$
$887$	31.5744			1.06017			0.530083	+	0.847946i	$0.322161\pi$
$888$		0			0
$889$	4.11096	+	4.20213i	0.137877	+	0.140935i
$890$		−	16.1512i		−	0.541390i
$891$		0			0
$892$		−	16.6824i		−	0.558567i
$893$			12.9034i			0.431795i
$894$		0			0
$895$			43.8615i			1.46613i
$896$	1.89123	−	1.85020i	0.0631817	−	0.0618108i
$897$		0			0
$898$	37.7578			1.26000
$899$	20.2293			0.674685
$900$		0			0
$901$		−	21.6700i		−	0.721931i
$902$	12.4196			0.413526
$903$		0			0
$904$	−8.46783			−0.281636
$905$		−	32.8550i		−	1.09214i
$906$		0			0
$907$	−31.7303			−1.05359			−0.526794	−	0.849993i	$-0.676606\pi$
$907$	−31.7303			−1.05359			−0.526794	+	0.849993i	$0.676606\pi$
$908$	23.8762			0.792360
$909$		0			0
$910$	−21.9969	−	22.4848i	−0.729191	−	0.745363i
$911$		−	6.03909i		−	0.200084i	−0.994983	−	0.100042i	$-0.968102\pi$
$911$		−	6.03909i		−	0.200084i	0.994983	−	0.100042i	$-0.0318977\pi$
$912$		0			0
$913$			27.6020i			0.913493i
$914$			2.62175i			0.0867196i
$915$		0			0
$916$			17.0991i			0.564972i
$917$	−12.7801	−	13.0635i	−0.422036	−	0.431396i
$918$		0			0
$919$	24.6715			0.813836			0.406918	−	0.913465i	$-0.366603\pi$
$919$	24.6715			0.813836			0.406918	+	0.913465i	$0.366603\pi$
$920$	−21.9369			−0.723237
$921$		0			0
$922$			35.6151i			1.17292i
$923$	31.3787			1.03284
$924$		0			0
$925$	18.0822			0.594540
$926$			17.3748i			0.570973i
$927$		0			0
$928$	5.88457			0.193171
$929$	−46.7617			−1.53420			−0.767100	−	0.641527i	$-0.778301\pi$
$929$	−46.7617			−1.53420			−0.767100	+	0.641527i	$0.778301\pi$
$930$		0			0
$931$	6.99832	+	0.153524i	0.229361	+	0.00503155i
$932$			29.8614i			0.978144i
$933$		0			0
$934$		−	38.7528i		−	1.26803i
$935$		−	15.3857i		−	0.503165i
$936$		0			0
$937$			55.3866i			1.80940i	0.426047	+	0.904701i	$0.359906\pi$
$937$			55.3866i			1.80940i	−0.426047	+	0.904701i	$0.640094\pi$
$938$	18.2038	−	17.8088i	0.594374	−	0.581478i
$939$		0			0
$940$	−36.0332			−1.17527
$941$	29.1503			0.950273			0.475137	−	0.879912i	$-0.342399\pi$
$941$	29.1503			0.950273			0.475137	+	0.879912i	$0.342399\pi$
$942$		0			0
$943$			34.6849i			1.12950i
$944$	4.48909			0.146107
$945$		0			0
$946$	−7.80483			−0.253757
$947$			5.38761i			0.175074i	0.996161	+	0.0875369i	$0.0278996\pi$
$947$			5.38761i			0.175074i	−0.996161	+	0.0875369i	$0.972100\pi$
$948$		0			0
$949$	9.46292			0.307179
$950$	−2.79827			−0.0907879
$951$		0			0
$952$	−3.70443	+	3.62405i	−0.120061	+	0.117456i
$953$			61.3354i			1.98685i	0.114494	+	0.993424i	$0.463475\pi$
$953$			61.3354i			1.98685i	−0.114494	+	0.993424i	$0.536525\pi$
$954$		0			0
$955$			21.4349i			0.693617i
$956$			16.5510i			0.535298i
$957$		0			0
$958$		−	10.0355i		−	0.324233i
$959$	30.6790	−	30.0133i	0.990675	−	0.969181i
$960$		0			0
$961$	19.1823			0.618784
$962$	−27.5110			−0.886990
$963$		0			0
$964$		−	20.8786i		−	0.672453i
$965$	35.2400			1.13442
$966$		0			0
$967$	−13.0236			−0.418809			−0.209405	−	0.977829i	$-0.567153\pi$
$967$	−13.0236			−0.418809			−0.209405	+	0.977829i	$0.567153\pi$
$968$			3.08808i			0.0992547i
$969$		0			0
$970$	−14.9115			−0.478779
$971$	18.7568			0.601935			0.300967	−	0.953634i	$-0.402690\pi$
$971$	18.7568			0.601935			0.300967	+	0.953634i	$0.402690\pi$
$972$		0			0
$973$	−32.4165	+	31.7131i	−1.03922	+	1.01668i
$974$			1.42589i			0.0456885i
$975$		0			0
$976$		−	10.2940i		−	0.329503i
$977$		−	4.55909i		−	0.145858i	−0.997337	−	0.0729292i	$-0.976765\pi$
$977$		−	4.55909i		−	0.145858i	0.997337	−	0.0729292i	$-0.0232347\pi$
$978$		0			0
$979$			16.2685i			0.519943i
$980$	−0.428722	+	19.5431i	−0.0136950	+	0.624280i
$981$		0			0
$982$	−4.27590			−0.136450
$983$	9.62686			0.307049			0.153525	−	0.988145i	$-0.450938\pi$
$983$	9.62686			0.307049			0.153525	+	0.988145i	$0.450938\pi$
$984$		0			0
$985$			6.43128i			0.204918i
$986$	−11.5263			−0.367073
$987$		0			0
$988$	4.25740			0.135446
$989$		−	21.7971i		−	0.693107i
$990$		0			0
$991$	57.4276			1.82425			0.912125	−	0.409913i	$-0.134441\pi$
$991$	57.4276			1.82425			0.912125	+	0.409913i	$0.134441\pi$
$992$	−3.43769			−0.109147
$993$		0			0
$994$	−13.6367	−	13.9391i	−0.432529	−	0.442122i
$995$		−	62.7019i		−	1.98778i
$996$		0			0
$997$		−	8.69843i		−	0.275482i	−0.990468	−	0.137741i	$-0.956016\pi$
$997$		−	8.69843i		−	0.275482i	0.990468	−	0.137741i	$-0.0439842\pi$
$998$			18.4160i			0.582947i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2394.2.f.b.2015.17	yes	24
3.2	odd	2		2394.2.f.a.2015.18	yes	24
7.6	odd	2		2394.2.f.a.2015.17	✓	24
21.20	even	2	inner	2394.2.f.b.2015.18	yes	24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2394.2.f.a.2015.17	✓	24	7.6	odd	2
2394.2.f.a.2015.18	yes	24	3.2	odd	2
2394.2.f.b.2015.17	yes	24	1.1	even	1	trivial
2394.2.f.b.2015.18	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)

\(f(q)\)	\(=\)	\(q-1.00000i q^{2} -1.00000 q^{4} -2.79254 q^{5} +(-1.85020 - 1.89123i) q^{7} +1.00000i q^{8} +O(q^{10})\)	\(q-1.00000i q^{2} -1.00000 q^{4} -2.79254 q^{5} +(-1.85020 - 1.89123i) q^{7} +1.00000i q^{8} +2.79254i q^{10} -2.81281i q^{11} -4.25740i q^{13} +(-1.89123 + 1.85020i) q^{14} +1.00000 q^{16} -1.95874 q^{17} -1.00000i q^{19} +2.79254 q^{20} -2.81281 q^{22} -7.85553i q^{23} +2.79827 q^{25} -4.25740 q^{26} +(1.85020 + 1.89123i) q^{28} +5.88457i q^{29} -3.43769i q^{31} -1.00000i q^{32} +1.95874i q^{34} +(5.16675 + 5.28134i) q^{35} +6.46193 q^{37} -1.00000 q^{38} -2.79254i q^{40} -4.41535 q^{41} +2.77474 q^{43} +2.81281i q^{44} -7.85553 q^{46} -12.9034 q^{47} +(-0.153524 + 6.99832i) q^{49} -2.79827i q^{50} +4.25740i q^{52} +11.0632i q^{53} +7.85489i q^{55} +(1.89123 - 1.85020i) q^{56} +5.88457 q^{58} +4.48909 q^{59} -10.2940i q^{61} -3.43769 q^{62} -1.00000 q^{64} +11.8890i q^{65} -9.62535 q^{67} +1.95874 q^{68} +(5.28134 - 5.16675i) q^{70} +7.37038i q^{71} +2.22270i q^{73} -6.46193i q^{74} +1.00000i q^{76} +(-5.31969 + 5.20427i) q^{77} +15.3756 q^{79} -2.79254 q^{80} +4.41535i q^{82} -9.81295 q^{83} +5.46985 q^{85} -2.77474i q^{86} +2.81281 q^{88} -5.78370 q^{89} +(-8.05173 + 7.87704i) q^{91} +7.85553i q^{92} +12.9034i q^{94} +2.79254i q^{95} +5.33976i q^{97} +(6.99832 + 0.153524i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q - 24 q^{4} + 4 q^{7}+O(q^{10}) \) 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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