LMFDB - Embedded newform 2952.2.j.d.2377.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2952,2,Mod(2377,2952)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2952, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("2952.2377");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2952 = 2^{3} \cdot 3^{2} \cdot 41 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2952.j (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:	$23.5718386767$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.265727878144.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 15x^{6} + 67x^{4} + 77x^{2} + 4 $ x^8 + 15x^6 + 67x^4 + 77*x^2 + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 984)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			2377.1
Root			$2.54814i$ of defining polynomial
Character	$\chi$	$=$	2952.2377
Dual form			2952.2.j.d.2377.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.54814 q^{5} -5.04115i q^{7} +O(q^{10})$	$q-3.54814 q^{5} -5.04115i q^{7} -1.25627i q^{11} -4.25627i q^{13} +0.236747i q^{17} -0.840013i q^{19} +7.74928 q^{23} +7.58929 q^{25} -10.3525i q^{29} -4.69415 q^{31} +17.8867i q^{35} +5.58929 q^{37} +(-6.33303 - 0.944874i) q^{41} +7.82392 q^{43} -2.07464i q^{47} -18.4132 q^{49} -1.29187i q^{53} +4.45740i q^{55} -7.40767 q^{59} -10.7709 q^{61} +15.1018i q^{65} +5.18717i q^{67} +6.74928i q^{71} +8.41625 q^{73} -6.33303 q^{77} -11.1786i q^{79} -10.0607 q^{83} -0.840013i q^{85} +3.21511i q^{89} -21.4565 q^{91} +2.98048i q^{95} -3.94487i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 10 q^{5}+O(q^{10}) $ 8 * q - 10 * q^5	$ 8 q - 10 q^{5} + 20 q^{23} + 2 q^{25} + 8 q^{31} - 14 q^{37} - 12 q^{41} + 6 q^{43} - 32 q^{49} - 6 q^{59} - 22 q^{61} + 64 q^{73} - 12 q^{77} - 22 q^{83} - 12 q^{91}+O(q^{100}) $ 8 * q - 10 * q^5 + 20 * q^23 + 2 * q^25 + 8 * q^31 - 14 * q^37 - 12 * q^41 + 6 * q^43 - 32 * q^49 - 6 * q^59 - 22 * q^61 + 64 * q^73 - 12 * q^77 - 22 * q^83 - 12 * q^91

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2952\mathbb{Z}\right)^\times$.

$n$	$1441$	$1477$	$2215$	$2297$
$\chi(n)$	$-1$	$1$	$1$	$1$

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 $

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$	−3.54814			−1.58678			−0.793388	−	0.608716i	$-0.791685\pi$
$5$	−3.54814			−1.58678			−0.793388	+	0.608716i	$0.791685\pi$
$6$		0			0
$7$		−	5.04115i		−	1.90538i	−0.303949	−	0.952688i	$-0.598305\pi$
$7$		−	5.04115i		−	1.90538i	0.303949	−	0.952688i	$-0.401695\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$		−	1.25627i		−	0.378778i	−0.981902	−	0.189389i	$-0.939349\pi$
$11$		−	1.25627i		−	0.378778i	0.981902	−	0.189389i	$-0.0606508\pi$
$12$		0			0
$13$		−	4.25627i		−	1.18048i	−0.807229	−	0.590238i	$-0.799034\pi$
$13$		−	4.25627i		−	1.18048i	0.807229	−	0.590238i	$-0.200966\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$			0.236747i			0.0574197i	0.999588	+	0.0287098i	$0.00913988\pi$
$17$			0.236747i			0.0574197i	−0.999588	+	0.0287098i	$0.990860\pi$
$18$		0			0
$19$		−	0.840013i		−	0.192712i	−0.995347	−	0.0963561i	$-0.969281\pi$
$19$		−	0.840013i		−	0.192712i	0.995347	−	0.0963561i	$-0.0307187\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	7.74928			1.61584			0.807918	−	0.589295i	$-0.200594\pi$
$23$	7.74928			1.61584			0.807918	+	0.589295i	$0.200594\pi$
$24$		0			0
$25$	7.58929			1.51786
$26$		0			0
$27$		0			0
$28$		0			0
$29$		−	10.3525i		−	1.92242i	−0.275819	−	0.961210i	$-0.588949\pi$
$29$		−	10.3525i		−	1.92242i	0.275819	−	0.961210i	$-0.411051\pi$
$30$		0			0
$31$	−4.69415			−0.843095			−0.421547	−	0.906806i	$-0.638513\pi$
$31$	−4.69415			−0.843095			−0.421547	+	0.906806i	$0.638513\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$			17.8867i			3.02341i
$36$		0			0
$37$	5.58929			0.918874			0.459437	−	0.888210i	$-0.348051\pi$
$37$	5.58929			0.918874			0.459437	+	0.888210i	$0.348051\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−6.33303	−	0.944874i	−0.989052	−	0.147564i
$41$	−6.33303	−	0.944874i	−0.989052	−	0.147564i
$42$		0			0
$43$	7.82392			1.19314			0.596569	−	0.802562i	$-0.296530\pi$
$43$	7.82392			1.19314			0.596569	+	0.802562i	$0.296530\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		−	2.07464i		−	0.302618i	−0.988487	−	0.151309i	$-0.951651\pi$
$47$		−	2.07464i		−	0.302618i	0.988487	−	0.151309i	$-0.0483488\pi$
$48$		0			0
$49$	−18.4132			−2.63046
$50$		0			0
$51$		0			0
$52$		0			0
$53$		−	1.29187i		−	0.177452i	−0.996056	−	0.0887262i	$-0.971720\pi$
$53$		−	1.29187i		−	0.177452i	0.996056	−	0.0887262i	$-0.0282796\pi$
$54$		0			0
$55$			4.45740i			0.601036i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−7.40767			−0.964396			−0.482198	−	0.876062i	$-0.660162\pi$
$59$	−7.40767			−0.964396			−0.482198	+	0.876062i	$0.660162\pi$
$60$		0			0
$61$	−10.7709			−1.37907			−0.689537	−	0.724250i	$-0.742186\pi$
$61$	−10.7709			−1.37907			−0.689537	+	0.724250i	$0.742186\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$			15.1018i			1.87315i
$66$		0			0
$67$			5.18717i			0.633713i	0.948473	+	0.316857i	$0.102627\pi$
$67$			5.18717i			0.633713i	−0.948473	+	0.316857i	$0.897373\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$			6.74928i			0.800992i	0.916299	+	0.400496i	$0.131162\pi$
$71$			6.74928i			0.800992i	−0.916299	+	0.400496i	$0.868838\pi$
$72$		0			0
$73$	8.41625			0.985048			0.492524	−	0.870299i	$-0.336074\pi$
$73$	8.41625			0.985048			0.492524	+	0.870299i	$0.336074\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−6.33303			−0.721715
$78$		0			0
$79$		−	11.1786i		−	1.25769i	−0.777531	−	0.628844i	$-0.783528\pi$
$79$		−	11.1786i		−	1.25769i	0.777531	−	0.628844i	$-0.216472\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−10.0607			−1.10430			−0.552151	−	0.833744i	$-0.686193\pi$
$83$	−10.0607			−1.10430			−0.552151	+	0.833744i	$0.686193\pi$
$84$		0			0
$85$		−	0.840013i		−	0.0911122i
$86$		0			0
$87$		0			0
$88$		0			0
$89$			3.21511i			0.340801i	0.985375	+	0.170401i	$0.0545062\pi$
$89$			3.21511i			0.340801i	−0.985375	+	0.170401i	$0.945494\pi$
$90$		0			0
$91$	−21.4565			−2.24925
$92$		0			0
$93$		0			0
$94$		0			0
$95$			2.98048i			0.305791i
$96$		0			0
$97$		−	3.94487i		−	0.400541i	−0.979741	−	0.200271i	$-0.935818\pi$
$97$		−	3.94487i		−	0.400541i	0.979741	−	0.200271i	$-0.0641821\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$			14.6304i			1.45578i	0.685692	+	0.727892i	$0.259500\pi$
$101$			14.6304i			1.45578i	−0.685692	+	0.727892i	$0.740500\pi$
$102$		0			0
$103$	16.5809			1.63376			0.816880	−	0.576807i	$-0.195702\pi$
$103$	16.5809			1.63376			0.816880	+	0.576807i	$0.195702\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−2.46491			−0.238292			−0.119146	−	0.992877i	$-0.538016\pi$
$107$	−2.46491			−0.238292			−0.119146	+	0.992877i	$0.538016\pi$
$108$		0			0
$109$			5.15140i			0.493415i	0.969090	+	0.246708i	$0.0793487\pi$
$109$			5.15140i			0.493415i	−0.969090	+	0.246708i	$0.920651\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	6.37206			0.599433			0.299717	−	0.954028i	$-0.403108\pi$
$113$	6.37206			0.599433			0.299717	+	0.954028i	$0.403108\pi$
$114$		0			0
$115$	−27.4955			−2.56397
$116$		0			0
$117$		0			0
$118$		0			0
$119$	1.19348			0.109406
$120$		0			0
$121$	9.42180			0.856527
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−9.18717			−0.821725
$126$		0			0
$127$	−8.47904			−0.752393			−0.376197	−	0.926540i	$-0.622768\pi$
$127$	−8.47904			−0.752393			−0.376197	+	0.926540i	$0.622768\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−17.9028			−1.56417			−0.782087	−	0.623169i	$-0.785845\pi$
$131$	−17.9028			−1.56417			−0.782087	+	0.623169i	$0.785845\pi$
$132$		0			0
$133$	−4.23463			−0.367189
$134$		0			0
$135$		0			0
$136$		0			0
$137$		−	10.5061i		−	0.897594i	−0.893634	−	0.448797i	$-0.851853\pi$
$137$		−	10.5061i		−	0.897594i	0.893634	−	0.448797i	$-0.148147\pi$
$138$		0			0
$139$	−5.29187			−0.448851			−0.224425	−	0.974491i	$-0.572051\pi$
$139$	−5.29187			−0.448851			−0.224425	+	0.974491i	$0.572051\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−5.34700			−0.447139
$144$		0			0
$145$			36.7323i			3.05045i
$146$		0			0
$147$		0			0
$148$		0			0
$149$			0.986026i			0.0807784i	0.999184	+	0.0403892i	$0.0128598\pi$
$149$			0.986026i			0.0807784i	−0.999184	+	0.0403892i	$0.987140\pi$
$150$		0			0
$151$			14.7763i			1.20248i	0.799069	+	0.601239i	$0.205326\pi$
$151$			14.7763i			1.20248i	−0.799069	+	0.601239i	$0.794674\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	16.6555			1.33780
$156$		0			0
$157$		−	17.6088i		−	1.40534i	−0.711518	−	0.702668i	$-0.751992\pi$
$157$		−	17.6088i		−	1.40534i	0.711518	−	0.702668i	$-0.248008\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		−	39.0653i		−	3.07878i
$162$		0			0
$163$	−14.5612			−1.14052			−0.570260	−	0.821464i	$-0.693158\pi$
$163$	−14.5612			−1.14052			−0.570260	+	0.821464i	$0.693158\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$			0.909266i			0.0703611i	0.999381	+	0.0351805i	$0.0112006\pi$
$167$			0.909266i			0.0703611i	−0.999381	+	0.0351805i	$0.988799\pi$
$168$		0			0
$169$	−5.11580			−0.393523
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−12.3721			−0.940630			−0.470315	−	0.882498i	$-0.655860\pi$
$173$	−12.3721			−0.940630			−0.470315	+	0.882498i	$0.655860\pi$
$174$		0			0
$175$		−	38.2588i		−	2.89209i
$176$		0			0
$177$		0			0
$178$		0			0
$179$		−	18.8509i		−	1.40899i	−0.709711	−	0.704493i	$-0.751174\pi$
$179$		−	18.8509i		−	1.40899i	0.709711	−	0.704493i	$-0.248826\pi$
$180$		0			0
$181$		−	9.92020i		−	0.737363i	−0.929556	−	0.368681i	$-0.879809\pi$
$181$		−	9.92020i		−	0.737363i	0.929556	−	0.368681i	$-0.120191\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−19.8316			−1.45805
$186$		0			0
$187$	0.297418			0.0217493
$188$		0			0
$189$		0			0
$190$		0			0
$191$		−	3.60881i		−	0.261124i	−0.991440	−	0.130562i	$-0.958322\pi$
$191$		−	3.60881i		−	0.261124i	0.991440	−	0.130562i	$-0.0416782\pi$
$192$		0			0
$193$			23.3872i			1.68345i	0.539907	+	0.841725i	$0.318459\pi$
$193$			23.3872i			1.68345i	−0.539907	+	0.841725i	$0.681541\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−5.63044			−0.401152			−0.200576	−	0.979678i	$-0.564281\pi$
$197$	−5.63044			−0.401152			−0.200576	+	0.979678i	$0.564281\pi$
$198$		0			0
$199$			16.5397i			1.17247i	0.810141	+	0.586234i	$0.199390\pi$
$199$			16.5397i			1.17247i	−0.810141	+	0.586234i	$0.800610\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−52.1887			−3.66293
$204$		0			0
$205$	22.4705	+	3.35254i	1.56940	+	0.234152i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−1.05528			−0.0729952
$210$		0			0
$211$			5.14813i			0.354412i	0.984174	+	0.177206i	$0.0567059\pi$
$211$			5.14813i			0.354412i	−0.984174	+	0.177206i	$0.943294\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−27.7604			−1.89324
$216$		0			0
$217$			23.6639i			1.60641i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	1.00766			0.0677825
$222$		0			0
$223$	15.1872			1.01701			0.508504	−	0.861060i	$-0.330199\pi$
$223$	15.1872			1.01701			0.508504	+	0.861060i	$0.330199\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$			13.2283i			0.877994i	0.898488	+	0.438997i	$0.144666\pi$
$227$			13.2283i			0.877994i	−0.898488	+	0.438997i	$0.855334\pi$
$228$		0			0
$229$			18.8425i			1.24515i	0.782561	+	0.622574i	$0.213913\pi$
$229$			18.8425i			1.24515i	−0.782561	+	0.622574i	$0.786087\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		−	28.5039i		−	1.86736i	−0.358114	−	0.933678i	$-0.616580\pi$
$233$		−	28.5039i		−	1.86736i	0.358114	−	0.933678i	$-0.383420\pi$
$234$		0			0
$235$			7.36113i			0.480187i
$236$		0			0
$237$		0			0
$238$		0			0
$239$		−	6.52111i		−	0.421816i	−0.977506	−	0.210908i	$-0.932358\pi$
$239$		−	6.52111i		−	0.421816i	0.977506	−	0.210908i	$-0.0676420\pi$
$240$		0			0
$241$	18.4565			1.18889			0.594443	−	0.804138i	$-0.297372\pi$
$241$	18.4565			1.18889			0.594443	+	0.804138i	$0.297372\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	65.3326			4.17395
$246$		0			0
$247$	−3.57532			−0.227492
$248$		0			0
$249$		0			0
$250$		0			0
$251$	16.7051			1.05442			0.527208	−	0.849736i	$-0.323239\pi$
$251$	16.7051			1.05442			0.527208	+	0.849736i	$0.323239\pi$
$252$		0			0
$253$		−	9.73515i		−	0.612044i
$254$		0			0
$255$		0			0
$256$		0			0
$257$			25.6555i			1.60035i	0.599769	+	0.800173i	$0.295259\pi$
$257$			25.6555i			1.60035i	−0.599769	+	0.800173i	$0.704741\pi$
$258$		0			0
$259$		−	28.1765i		−	1.75080i
$260$		0			0
$261$		0			0
$262$		0			0
$263$		−	22.8175i		−	1.40698i	−0.710703	−	0.703492i	$-0.751623\pi$
$263$		−	22.8175i		−	1.40698i	0.710703	−	0.703492i	$-0.248377\pi$
$264$		0			0
$265$			4.58375i			0.281577i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	22.8783			1.39491			0.697457	−	0.716627i	$-0.254315\pi$
$269$	22.8783			1.39491			0.697457	+	0.716627i	$0.254315\pi$
$270$		0			0
$271$	−24.6856			−1.49954			−0.749771	−	0.661698i	$-0.769836\pi$
$271$	−24.6856			−1.49954			−0.749771	+	0.661698i	$0.769836\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		−	9.53416i		−	0.574932i
$276$		0			0
$277$	8.09073			0.486125			0.243063	−	0.970011i	$-0.421848\pi$
$277$	8.09073			0.486125			0.243063	+	0.970011i	$0.421848\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$			14.0691i			0.839292i	0.907688	+	0.419646i	$0.137846\pi$
$281$			14.0691i			0.839292i	−0.907688	+	0.419646i	$0.862154\pi$
$282$		0			0
$283$	−4.17304			−0.248061			−0.124031	−	0.992278i	$-0.539582\pi$
$283$	−4.17304			−0.248061			−0.124031	+	0.992278i	$0.539582\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−4.76325	+	31.9257i	−0.281166	+	1.88452i
$288$		0			0
$289$	16.9440			0.996703
$290$		0			0
$291$		0			0
$292$		0			0
$293$		−	8.77973i		−	0.512917i	−0.966555	−	0.256459i	$-0.917444\pi$
$293$		−	8.77973i		−	0.512917i	0.966555	−	0.256459i	$-0.0825558\pi$
$294$		0			0
$295$	26.2834			1.53028
$296$		0			0
$297$		0			0
$298$		0			0
$299$		−	32.9830i		−	1.90746i
$300$		0			0
$301$		−	39.4416i		−	2.27338i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	38.2167			2.18828
$306$		0			0
$307$	−20.3656			−1.16233			−0.581163	−	0.813787i	$-0.697402\pi$
$307$	−20.3656			−1.16233			−0.581163	+	0.813787i	$0.697402\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		−	7.95238i		−	0.450938i	−0.974250	−	0.225469i	$-0.927609\pi$
$311$		−	7.95238i		−	0.450938i	0.974250	−	0.225469i	$-0.0723915\pi$
$312$		0			0
$313$		−	12.9699i		−	0.733104i	−0.930398	−	0.366552i	$-0.880538\pi$
$313$		−	12.9699i		−	0.733104i	0.930398	−	0.366552i	$-0.119462\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$			15.2391i			0.855913i	0.903799	+	0.427957i	$0.140766\pi$
$317$			15.2391i			0.855913i	−0.903799	+	0.427957i	$0.859234\pi$
$318$		0			0
$319$	−13.0055			−0.728171
$320$		0			0
$321$		0			0
$322$		0			0
$323$	0.198871			0.0110655
$324$		0			0
$325$		−	32.3020i		−	1.79179i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−10.4586			−0.576601
$330$		0			0
$331$			27.9560i			1.53660i	0.640091	+	0.768299i	$0.278897\pi$
$331$			27.9560i			1.53660i	−0.640091	+	0.768299i	$0.721103\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		−	18.4048i		−	1.00556i
$336$		0			0
$337$	−5.27790			−0.287506			−0.143753	−	0.989614i	$-0.545917\pi$
$337$	−5.27790			−0.287506			−0.143753	+	0.989614i	$0.545917\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$			5.89710i			0.319346i
$342$		0			0
$343$			57.5357i			3.10664i
$344$		0			0
$345$		0			0
$346$		0			0
$347$			8.26720i			0.443807i	0.975069	+	0.221903i	$0.0712269\pi$
$347$			8.26720i			0.443807i	−0.975069	+	0.221903i	$0.928773\pi$
$348$		0			0
$349$	−0.104861			−0.00561309			−0.00280654	−	0.999996i	$-0.500893\pi$
$349$	−0.104861			−0.00561309			−0.00280654	+	0.999996i	$0.500893\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	20.8866			1.11168			0.555840	−	0.831290i	$-0.312397\pi$
$353$	20.8866			1.11168			0.555840	+	0.831290i	$0.312397\pi$
$354$		0			0
$355$		−	23.9474i		−	1.27099i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	29.0814			1.53486			0.767428	−	0.641135i	$-0.221536\pi$
$359$	29.0814			1.53486			0.767428	+	0.641135i	$0.221536\pi$
$360$		0			0
$361$	18.2944			0.962862
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−29.8620			−1.56305
$366$		0			0
$367$	2.84213			0.148358			0.0741789	−	0.997245i	$-0.476366\pi$
$367$	2.84213			0.148358			0.0741789	+	0.997245i	$0.476366\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−6.51253			−0.338114
$372$		0			0
$373$	4.88116			0.252737			0.126369	−	0.991983i	$-0.459668\pi$
$373$	4.88116			0.252737			0.126369	+	0.991983i	$0.459668\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−44.0632			−2.26937
$378$		0			0
$379$	−14.1286			−0.725738			−0.362869	−	0.931840i	$-0.618203\pi$
$379$	−14.1286			−0.725738			−0.362869	+	0.931840i	$0.618203\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$			25.8846i			1.32264i	0.750103	+	0.661320i	$0.230004\pi$
$383$			25.8846i			1.32264i	−0.750103	+	0.661320i	$0.769996\pi$
$384$		0			0
$385$	22.4705			1.14520
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−15.7797			−0.800064			−0.400032	−	0.916501i	$-0.631001\pi$
$389$	−15.7797			−0.800064			−0.400032	+	0.916501i	$0.631001\pi$
$390$		0			0
$391$			1.83462i			0.0927808i
$392$		0			0
$393$		0			0
$394$		0			0
$395$			39.6632i			1.99567i
$396$		0			0
$397$		−	29.4459i		−	1.47785i	−0.673788	−	0.738925i	$-0.735334\pi$
$397$		−	29.4459i		−	1.47785i	0.673788	−	0.738925i	$-0.264666\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	3.93302			0.196405			0.0982027	−	0.995166i	$-0.468691\pi$
$401$	3.93302			0.196405			0.0982027	+	0.995166i	$0.468691\pi$
$402$		0			0
$403$			19.9796i			0.995253i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		−	7.02163i		−	0.348049i
$408$		0			0
$409$	−1.29203			−0.0638866			−0.0319433	−	0.999490i	$-0.510170\pi$
$409$	−1.29203			−0.0638866			−0.0319433	+	0.999490i	$0.510170\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$			37.3432i			1.83754i
$414$		0			0
$415$	35.6967			1.75228
$416$		0			0
$417$		0			0
$418$		0			0
$419$	3.34950			0.163634			0.0818170	−	0.996647i	$-0.473928\pi$
$419$	3.34950			0.163634			0.0818170	+	0.996647i	$0.473928\pi$
$420$		0			0
$421$		−	17.2193i		−	0.839220i	−0.907705	−	0.419610i	$-0.862167\pi$
$421$		−	17.2193i		−	0.839220i	0.907705	−	0.419610i	$-0.137833\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$			1.79674i			0.0871549i
$426$		0			0
$427$			54.2978i			2.62766i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	4.48839			0.216198			0.108099	−	0.994140i	$-0.465524\pi$
$431$	4.48839			0.216198			0.108099	+	0.994140i	$0.465524\pi$
$432$		0			0
$433$	−27.5918			−1.32598			−0.662989	−	0.748630i	$-0.730712\pi$
$433$	−27.5918			−1.32598			−0.662989	+	0.748630i	$0.730712\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		−	6.50949i		−	0.311391i
$438$		0			0
$439$			21.0079i			1.00265i	0.865258	+	0.501326i	$0.167154\pi$
$439$			21.0079i			1.00265i	−0.865258	+	0.501326i	$0.832846\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	3.39689			0.161391			0.0806955	−	0.996739i	$-0.474286\pi$
$443$	3.39689			0.161391			0.0806955	+	0.996739i	$0.474286\pi$
$444$		0			0
$445$		−	11.4077i		−	0.540775i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	6.33529			0.298981			0.149491	−	0.988763i	$-0.452237\pi$
$449$	6.33529			0.298981			0.149491	+	0.988763i	$0.452237\pi$
$450$		0			0
$451$	−1.18701	+	7.95596i	−0.0558942	+	0.374632i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	76.1306			3.56906
$456$		0			0
$457$			22.3962i			1.04765i	0.851826	+	0.523825i	$0.175495\pi$
$457$			22.3962i			1.04765i	−0.851826	+	0.523825i	$0.824505\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	30.0934			1.40159			0.700795	−	0.713363i	$-0.252829\pi$
$461$	30.0934			1.40159			0.700795	+	0.713363i	$0.252829\pi$
$462$		0			0
$463$			0.0830744i			0.00386080i	0.999998	+	0.00193040i	$0.000614465\pi$
$463$			0.0830744i			0.00386080i	−0.999998	+	0.00193040i	$0.999386\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−10.1449			−0.469452			−0.234726	−	0.972062i	$-0.575419\pi$
$467$	−10.1449			−0.469452			−0.234726	+	0.972062i	$0.575419\pi$
$468$		0			0
$469$	26.1493			1.20746
$470$		0			0
$471$		0			0
$472$		0			0
$473$		−	9.82892i		−	0.451934i
$474$		0			0
$475$		−	6.37510i		−	0.292510i
$476$		0			0
$477$		0			0
$478$		0			0
$479$		−	4.90161i		−	0.223960i	−0.993710	−	0.111980i	$-0.964281\pi$
$479$		−	4.90161i		−	0.223960i	0.993710	−	0.111980i	$-0.0357193\pi$
$480$		0			0
$481$		−	23.7895i		−	1.08471i
$482$		0			0
$483$		0			0
$484$		0			0
$485$			13.9970i			0.635569i
$486$		0			0
$487$	11.1505			0.505277			0.252638	−	0.967561i	$-0.418702\pi$
$487$	11.1505			0.505277			0.252638	+	0.967561i	$0.418702\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−9.08573			−0.410033			−0.205017	−	0.978758i	$-0.565725\pi$
$491$	−9.08573			−0.410033			−0.205017	+	0.978758i	$0.565725\pi$
$492$		0			0
$493$	2.45094			0.110385
$494$		0			0
$495$		0			0
$496$		0			0
$497$	34.0241			1.52619
$498$		0			0
$499$		−	32.4048i		−	1.45064i	−0.688413	−	0.725319i	$-0.741692\pi$
$499$		−	32.4048i		−	1.45064i	0.688413	−	0.725319i	$-0.258308\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		−	13.3192i		−	0.593874i	−0.954897	−	0.296937i	$-0.904035\pi$
$503$		−	13.3192i		−	0.593874i	0.954897	−	0.296937i	$-0.0959651\pi$
$504$		0			0
$505$		−	51.9108i		−	2.31000i
$506$		0			0
$507$		0			0
$508$		0			0
$509$			11.7949i			0.522800i	0.965231	+	0.261400i	$0.0841841\pi$
$509$			11.7949i			0.522800i	−0.965231	+	0.261400i	$0.915816\pi$
$510$		0			0
$511$		−	42.4276i		−	1.87689i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−58.8312			−2.59241
$516$		0			0
$517$	−2.60630			−0.114625
$518$		0			0
$519$		0			0
$520$		0			0
$521$			12.2770i			0.537864i	0.963159	+	0.268932i	$0.0866707\pi$
$521$			12.2770i			0.537864i	−0.963159	+	0.268932i	$0.913329\pi$
$522$		0			0
$523$	−29.7313			−1.30006			−0.650030	−	0.759908i	$-0.725244\pi$
$523$	−29.7313			−1.30006			−0.650030	+	0.759908i	$0.725244\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		−	1.11133i		−	0.0484102i
$528$		0			0
$529$	37.0513			1.61093
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−4.02163	+	26.9550i	−0.174196	+	1.16755i
$534$		0			0
$535$	8.74585			0.378116
$536$		0			0
$537$		0			0
$538$		0			0
$539$			23.1319i			0.996361i
$540$		0			0
$541$	−28.0153			−1.20447			−0.602236	−	0.798318i	$-0.705724\pi$
$541$	−28.0153			−1.20447			−0.602236	+	0.798318i	$0.705724\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		−	18.2779i		−	0.782939i
$546$		0			0
$547$		−	8.12307i		−	0.347317i	−0.984806	−	0.173659i	$-0.944441\pi$
$547$		−	8.12307i		−	0.347317i	0.984806	−	0.173659i	$-0.0555589\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−8.69627			−0.370473
$552$		0			0
$553$	−56.3529			−2.39637
$554$		0			0
$555$		0			0
$556$		0			0
$557$		−	12.7269i		−	0.539255i	−0.962965	−	0.269627i	$-0.913099\pi$
$557$		−	12.7269i		−	0.539255i	0.962965	−	0.269627i	$-0.0869006\pi$
$558$		0			0
$559$		−	33.3007i		−	1.40847i
$560$		0			0
$561$		0			0
$562$		0			0
$563$		−	27.3788i		−	1.15388i	−0.816787	−	0.576939i	$-0.804247\pi$
$563$		−	27.3788i		−	1.15388i	0.816787	−	0.576939i	$-0.195753\pi$
$564$		0			0
$565$	−22.6090			−0.951166
$566$		0			0
$567$		0			0
$568$		0			0
$569$	5.47257			0.229422			0.114711	−	0.993399i	$-0.463406\pi$
$569$	5.47257			0.229422			0.114711	+	0.993399i	$0.463406\pi$
$570$		0			0
$571$			14.7935i			0.619087i	0.950885	+	0.309544i	$0.100176\pi$
$571$			14.7935i			0.619087i	−0.950885	+	0.309544i	$0.899824\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	58.8115			2.45261
$576$		0			0
$577$			35.0515i			1.45921i	0.683868	+	0.729606i	$0.260296\pi$
$577$			35.0515i			1.45921i	−0.683868	+	0.729606i	$0.739704\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$			50.7174i			2.10411i
$582$		0			0
$583$	−1.62294			−0.0672152
$584$		0			0
$585$		0			0
$586$		0			0
$587$			41.5062i			1.71314i	0.516027	+	0.856572i	$0.327410\pi$
$587$			41.5062i			1.71314i	−0.516027	+	0.856572i	$0.672590\pi$
$588$		0			0
$589$			3.94315i			0.162475i
$590$		0			0
$591$		0			0
$592$		0			0
$593$		−	44.5227i		−	1.82833i	−0.405345	−	0.914164i	$-0.632848\pi$
$593$		−	44.5227i		−	1.82833i	0.405345	−	0.914164i	$-0.367152\pi$
$594$		0			0
$595$	−4.23463			−0.173603
$596$		0			0
$597$		0			0
$598$		0			0
$599$	1.07029			0.0437310			0.0218655	−	0.999761i	$-0.493039\pi$
$599$	1.07029			0.0437310			0.0218655	+	0.999761i	$0.493039\pi$
$600$		0			0
$601$		−	27.6803i		−	1.12910i	−0.825397	−	0.564552i	$-0.809049\pi$
$601$		−	27.6803i		−	1.12910i	0.825397	−	0.564552i	$-0.190951\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−33.4298			−1.35912
$606$		0			0
$607$	20.9118			0.848783			0.424391	−	0.905479i	$-0.360488\pi$
$607$	20.9118			0.848783			0.424391	+	0.905479i	$0.360488\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−8.83024			−0.357233
$612$		0			0
$613$	−44.2006			−1.78525			−0.892623	−	0.450804i	$-0.851137\pi$
$613$	−44.2006			−1.78525			−0.892623	+	0.450804i	$0.851137\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	18.9669			0.763579			0.381789	−	0.924249i	$-0.375308\pi$
$617$	18.9669			0.763579			0.381789	+	0.924249i	$0.375308\pi$
$618$		0			0
$619$	7.26932			0.292178			0.146089	−	0.989271i	$-0.453331\pi$
$619$	7.26932			0.292178			0.146089	+	0.989271i	$0.453331\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	16.2079			0.649355
$624$		0			0
$625$	−5.34912			−0.213965
$626$		0			0
$627$		0			0
$628$		0			0
$629$			1.32325i			0.0527614i
$630$		0			0
$631$	−41.2328			−1.64145			−0.820726	−	0.571323i	$-0.806431\pi$
$631$	−41.2328			−1.64145			−0.820726	+	0.571323i	$0.806431\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	30.0848			1.19388
$636$		0			0
$637$			78.3715i			3.10519i
$638$		0			0
$639$		0			0
$640$		0			0
$641$		−	25.4771i		−	1.00628i	−0.864204	−	0.503142i	$-0.832177\pi$
$641$		−	25.4771i		−	1.00628i	0.864204	−	0.503142i	$-0.167823\pi$
$642$		0			0
$643$		−	6.78489i		−	0.267570i	−0.991010	−	0.133785i	$-0.957287\pi$
$643$		−	6.78489i		−	0.267570i	0.991010	−	0.133785i	$-0.0427131\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−48.7292			−1.91574			−0.957872	−	0.287196i	$-0.907277\pi$
$647$	−48.7292			−1.91574			−0.957872	+	0.287196i	$0.907277\pi$
$648$		0			0
$649$			9.30600i			0.365292i
$650$		0			0
$651$		0			0
$652$		0			0
$653$		−	1.14601i		−	0.0448470i	−0.999749	−	0.0224235i	$-0.992862\pi$
$653$		−	1.14601i		−	0.0448470i	0.999749	−	0.0224235i	$-0.00713821\pi$
$654$		0			0
$655$	63.5216			2.48200
$656$		0			0
$657$		0			0
$658$		0			0
$659$		−	25.0683i		−	0.976523i	−0.872697	−	0.488262i	$-0.837631\pi$
$659$		−	25.0683i		−	0.976523i	0.872697	−	0.488262i	$-0.162369\pi$
$660$		0			0
$661$	35.2425			1.37078			0.685388	−	0.728178i	$-0.259633\pi$
$661$	35.2425			1.37078			0.685388	+	0.728178i	$0.259633\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	15.0251			0.582647
$666$		0			0
$667$		−	80.2247i		−	3.10631i
$668$		0			0
$669$		0			0
$670$		0			0
$671$			13.5311i			0.522363i
$672$		0			0
$673$		−	24.7441i		−	0.953816i	−0.878953	−	0.476908i	$-0.841758\pi$
$673$		−	24.7441i		−	0.953816i	0.878953	−	0.476908i	$-0.158242\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	21.6025			0.830251			0.415126	−	0.909764i	$-0.363738\pi$
$677$	21.6025			0.830251			0.415126	+	0.909764i	$0.363738\pi$
$678$		0			0
$679$	−19.8867			−0.763182
$680$		0			0
$681$		0			0
$682$		0			0
$683$		−	38.5638i		−	1.47560i	−0.675017	−	0.737802i	$-0.735864\pi$
$683$		−	38.5638i		−	1.47560i	0.675017	−	0.737802i	$-0.264136\pi$
$684$		0			0
$685$			37.2770i			1.42428i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−5.49856			−0.209478
$690$		0			0
$691$			6.72118i			0.255686i	0.991794	+	0.127843i	$0.0408053\pi$
$691$			6.72118i			0.255686i	−0.991794	+	0.127843i	$0.959195\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	18.7763			0.712226
$696$		0			0
$697$	0.223696	−	1.49933i	0.00847311	−	0.0567911i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	4.79347			0.181047			0.0905234	−	0.995894i	$-0.471146\pi$
$701$	4.79347			0.181047			0.0905234	+	0.995894i	$0.471146\pi$
$702$		0			0
$703$		−	4.69508i		−	0.177078i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	73.7543			2.77382
$708$		0			0
$709$		−	10.5583i		−	0.396525i	−0.980149	−	0.198263i	$-0.936470\pi$
$709$		−	10.5583i		−	0.396525i	0.980149	−	0.198263i	$-0.0635299\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−36.3763			−1.36230
$714$		0			0
$715$	18.9719			0.709509
$716$		0			0
$717$		0			0
$718$		0			0
$719$		−	30.6922i		−	1.14463i	−0.820035	−	0.572313i	$-0.806046\pi$
$719$		−	30.6922i		−	1.14463i	0.820035	−	0.572313i	$-0.193954\pi$
$720$		0			0
$721$		−	83.5866i		−	3.11293i
$722$		0			0
$723$		0			0
$724$		0			0
$725$		−	78.5685i		−	2.91796i
$726$		0			0
$727$		−	4.27594i		−	0.158586i	−0.996851	−	0.0792929i	$-0.974734\pi$
$727$		−	4.27594i		−	0.158586i	0.996851	−	0.0792929i	$-0.0252662\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$			1.85229i			0.0685095i
$732$		0			0
$733$	8.65643			0.319732			0.159866	−	0.987139i	$-0.448894\pi$
$733$	8.65643			0.319732			0.159866	+	0.987139i	$0.448894\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	6.51646			0.240037
$738$		0			0
$739$	28.7820			1.05876			0.529382	−	0.848384i	$-0.322424\pi$
$739$	28.7820			1.05876			0.529382	+	0.848384i	$0.322424\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−20.3031			−0.744849			−0.372425	−	0.928062i	$-0.621473\pi$
$743$	−20.3031			−0.744849			−0.372425	+	0.928062i	$0.621473\pi$
$744$		0			0
$745$		−	3.49856i		−	0.128177i
$746$		0			0
$747$		0			0
$748$		0			0
$749$			12.4260i			0.454036i
$750$		0			0
$751$			39.6138i			1.44553i	0.691095	+	0.722764i	$0.257129\pi$
$751$			39.6138i			1.44553i	−0.691095	+	0.722764i	$0.742871\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		−	52.4284i		−	1.90806i
$756$		0			0
$757$		−	5.42626i		−	0.197221i	−0.995126	−	0.0986105i	$-0.968560\pi$
$757$		−	5.42626i		−	0.197221i	0.995126	−	0.0986105i	$-0.0314398\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−0.815340			−0.0295560			−0.0147780	−	0.999891i	$-0.504704\pi$
$761$	−0.815340			−0.0295560			−0.0147780	+	0.999891i	$0.504704\pi$
$762$		0			0
$763$	25.9690			0.940141
$764$		0			0
$765$		0			0
$766$		0			0
$767$			31.5290i			1.13845i
$768$		0			0
$769$	40.8729			1.47391			0.736957	−	0.675940i	$-0.236262\pi$
$769$	40.8729			1.47391			0.736957	+	0.675940i	$0.236262\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$		−	18.9191i		−	0.680474i	−0.940340	−	0.340237i	$-0.889493\pi$
$773$		−	18.9191i		−	0.680474i	0.940340	−	0.340237i	$-0.110507\pi$
$774$		0			0
$775$	−35.6253			−1.27970
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−0.793706	+	5.31982i	−0.0284375	+	0.190602i
$780$		0			0
$781$	8.47889			0.303398
$782$		0			0
$783$		0			0
$784$		0			0
$785$			62.4785i			2.22995i
$786$		0			0
$787$	45.1238			1.60849			0.804246	−	0.594297i	$-0.202570\pi$
$787$	45.1238			1.60849			0.804246	+	0.594297i	$0.202570\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		−	32.1225i		−	1.14215i
$792$		0			0
$793$			45.8439i			1.62796i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	10.6575			0.377507			0.188753	−	0.982025i	$-0.439555\pi$
$797$	10.6575			0.377507			0.188753	+	0.982025i	$0.439555\pi$
$798$		0			0
$799$	0.491167			0.0173762
$800$		0			0
$801$		0			0
$802$		0			0
$803$		−	10.5730i		−	0.373115i
$804$		0			0
$805$			138.609i			4.88533i
$806$		0			0
$807$		0			0
$808$		0			0
$809$		−	20.5211i		−	0.721484i	−0.932666	−	0.360742i	$-0.882523\pi$
$809$		−	20.5211i		−	0.721484i	0.932666	−	0.360742i	$-0.117477\pi$
$810$		0			0
$811$	−15.2973			−0.537160			−0.268580	−	0.963257i	$-0.586554\pi$
$811$	−15.2973			−0.537160			−0.268580	+	0.963257i	$0.586554\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	51.6651			1.80975
$816$		0			0
$817$		−	6.57219i		−	0.229932i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	1.94356			0.0678308			0.0339154	−	0.999425i	$-0.489202\pi$
$821$	1.94356			0.0678308			0.0339154	+	0.999425i	$0.489202\pi$
$822$		0			0
$823$		−	14.4566i		−	0.503927i	−0.967737	−	0.251963i	$-0.918924\pi$
$823$		−	14.4566i		−	0.503927i	0.967737	−	0.251963i	$-0.0810762\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		−	20.9443i		−	0.728306i	−0.931339	−	0.364153i	$-0.881359\pi$
$827$		−	20.9443i		−	0.728306i	0.931339	−	0.364153i	$-0.118641\pi$
$828$		0			0
$829$	36.6048			1.27134			0.635668	−	0.771963i	$-0.280725\pi$
$829$	36.6048			1.27134			0.635668	+	0.771963i	$0.280725\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		−	4.35928i		−	0.151040i
$834$		0			0
$835$		−	3.22620i		−	0.111647i
$836$		0			0
$837$		0			0
$838$		0			0
$839$		−	13.6496i		−	0.471236i	−0.971846	−	0.235618i	$-0.924289\pi$
$839$		−	13.6496i		−	0.471236i	0.971846	−	0.235618i	$-0.0757113\pi$
$840$		0			0
$841$	−78.1752			−2.69570
$842$		0			0
$843$		0			0
$844$		0			0
$845$	18.1516			0.624433
$846$		0			0
$847$		−	47.4967i		−	1.63201i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	43.3130			1.48475
$852$		0			0
$853$	45.4568			1.55641			0.778205	−	0.628010i	$-0.216130\pi$
$853$	45.4568			1.55641			0.778205	+	0.628010i	$0.216130\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	30.5455			1.04341			0.521707	−	0.853125i	$-0.325295\pi$
$857$	30.5455			1.04341			0.521707	+	0.853125i	$0.325295\pi$
$858$		0			0
$859$	13.7305			0.468480			0.234240	−	0.972179i	$-0.424740\pi$
$859$	13.7305			0.468480			0.234240	+	0.972179i	$0.424740\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	27.1595			0.924519			0.462259	−	0.886745i	$-0.347039\pi$
$863$	27.1595			0.924519			0.462259	+	0.886745i	$0.347039\pi$
$864$		0			0
$865$	43.8978			1.49257
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−14.0433			−0.476385
$870$		0			0
$871$	22.0780			0.748083
$872$		0			0
$873$		0			0
$874$		0			0
$875$			46.3139i			1.56570i
$876$		0			0
$877$	−45.5973			−1.53971			−0.769856	−	0.638217i	$-0.779672\pi$
$877$	−45.5973			−1.53971			−0.769856	+	0.638217i	$0.779672\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−11.2219			−0.378074			−0.189037	−	0.981970i	$-0.560537\pi$
$881$	−11.2219			−0.378074			−0.189037	+	0.981970i	$0.560537\pi$
$882$		0			0
$883$			56.0339i			1.88569i	0.333230	+	0.942846i	$0.391862\pi$
$883$			56.0339i			1.88569i	−0.333230	+	0.942846i	$0.608138\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$			36.1204i			1.21280i	0.795158	+	0.606402i	$0.207388\pi$
$887$			36.1204i			1.21280i	−0.795158	+	0.606402i	$0.792612\pi$
$888$		0			0
$889$			42.7441i			1.43359i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−1.74273			−0.0583182
$894$		0			0
$895$			66.8858i			2.23575i
$896$		0			0
$897$		0			0
$898$		0			0
$899$			48.5964i			1.62078i
$900$		0			0
$901$	0.305848			0.0101893
$902$		0			0
$903$		0			0
$904$		0			0
$905$			35.1983i			1.17003i
$906$		0			0
$907$	−18.3452			−0.609142			−0.304571	−	0.952490i	$-0.598513\pi$
$907$	−18.3452			−0.609142			−0.304571	+	0.952490i	$0.598513\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−50.1842			−1.66268			−0.831339	−	0.555766i	$-0.812425\pi$
$911$	−50.1842			−1.66268			−0.831339	+	0.555766i	$0.812425\pi$
$912$		0			0
$913$			12.6389i			0.418286i
$914$		0			0
$915$		0			0
$916$		0			0
$917$			90.2507i			2.98034i
$918$		0			0
$919$		−	31.1736i		−	1.02832i	−0.857694	−	0.514161i	$-0.828103\pi$
$919$		−	31.1736i		−	1.02832i	0.857694	−	0.514161i	$-0.171897\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	28.7267			0.945552
$924$		0			0
$925$	42.4188			1.39472
$926$		0			0
$927$		0			0
$928$		0			0
$929$		−	33.2699i		−	1.09155i	−0.837932	−	0.545774i	$-0.816236\pi$
$929$		−	33.2699i		−	1.09155i	0.837932	−	0.545774i	$-0.183764\pi$
$930$		0			0
$931$			15.4673i			0.506921i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−1.05528			−0.0345113
$936$		0			0
$937$			1.97205i			0.0644241i	0.999481	+	0.0322121i	$0.0102552\pi$
$937$			1.97205i			0.0644241i	−0.999481	+	0.0322121i	$0.989745\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−8.57940			−0.279680			−0.139840	−	0.990174i	$-0.544659\pi$
$941$	−8.57940			−0.279680			−0.139840	+	0.990174i	$0.544659\pi$
$942$		0			0
$943$	−49.0764	−	7.32209i	−1.59815	−	0.238440i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	2.36729			0.0769265			0.0384633	−	0.999260i	$-0.487754\pi$
$947$	2.36729			0.0769265			0.0384633	+	0.999260i	$0.487754\pi$
$948$		0			0
$949$		−	35.8218i		−	1.16283i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−51.8427			−1.67935			−0.839674	−	0.543090i	$-0.817254\pi$
$953$	−51.8427			−1.67935			−0.839674	+	0.543090i	$0.817254\pi$
$954$		0			0
$955$			12.8046i			0.414346i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−52.9627			−1.71025
$960$		0			0
$961$	−8.96493			−0.289191
$962$		0			0
$963$		0			0
$964$		0			0
$965$		−	82.9811i		−	2.67126i
$966$		0			0
$967$			41.0971i			1.32159i	0.750565	+	0.660796i	$0.229781\pi$
$967$			41.0971i			1.32159i	−0.750565	+	0.660796i	$0.770219\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$			49.1931i			1.57868i	0.613955	+	0.789341i	$0.289578\pi$
$971$			49.1931i			1.57868i	−0.613955	+	0.789341i	$0.710422\pi$
$972$		0			0
$973$			26.6771i			0.855230i
$974$		0			0
$975$		0			0
$976$		0			0
$977$			25.7031i			0.822316i	0.911564	+	0.411158i	$0.134876\pi$
$977$			25.7031i			0.822316i	−0.911564	+	0.411158i	$0.865124\pi$
$978$		0			0
$979$	4.03904			0.129088
$980$		0			0
$981$		0			0
$982$		0			0
$983$	17.0729			0.544542			0.272271	−	0.962221i	$-0.412225\pi$
$983$	17.0729			0.544542			0.272271	+	0.962221i	$0.412225\pi$
$984$		0			0
$985$	19.9776			0.636539
$986$		0			0
$987$		0			0
$988$		0			0
$989$	60.6298			1.92791
$990$		0			0
$991$		−	42.2641i		−	1.34256i	−0.741203	−	0.671281i	$-0.765744\pi$
$991$		−	42.2641i		−	1.34256i	0.741203	−	0.671281i	$-0.234256\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		−	58.6852i		−	1.86045i
$996$		0			0
$997$		−	52.6203i		−	1.66650i	−0.552896	−	0.833251i	$-0.686477\pi$
$997$		−	52.6203i		−	1.66650i	0.552896	−	0.833251i	$-0.313523\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2952.2.j.d.2377.1		8
3.2	odd	2		984.2.j.b.409.4	✓	8
12.11	even	2		1968.2.j.f.1393.8		8
41.40	even	2	inner	2952.2.j.d.2377.2		8
123.122	odd	2		984.2.j.b.409.8	yes	8
492.491	even	2		1968.2.j.f.1393.4		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
984.2.j.b.409.4	✓	8	3.2	odd	2
984.2.j.b.409.8	yes	8	123.122	odd	2
1968.2.j.f.1393.4		8	492.491	even	2
1968.2.j.f.1393.8		8	12.11	even	2
2952.2.j.d.2377.1		8	1.1	even	1	trivial
2952.2.j.d.2377.2		8	41.40	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:	\( N \)	\(=\)	\( 2952 = 2^{3} \cdot 3^{2} \cdot 41 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2952.j (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-3.54814 q^{5} -5.04115i q^{7} +O(q^{10})\)	\(q-3.54814 q^{5} -5.04115i q^{7} -1.25627i q^{11} -4.25627i q^{13} +0.236747i q^{17} -0.840013i q^{19} +7.74928 q^{23} +7.58929 q^{25} -10.3525i q^{29} -4.69415 q^{31} +17.8867i q^{35} +5.58929 q^{37} +(-6.33303 - 0.944874i) q^{41} +7.82392 q^{43} -2.07464i q^{47} -18.4132 q^{49} -1.29187i q^{53} +4.45740i q^{55} -7.40767 q^{59} -10.7709 q^{61} +15.1018i q^{65} +5.18717i q^{67} +6.74928i q^{71} +8.41625 q^{73} -6.33303 q^{77} -11.1786i q^{79} -10.0607 q^{83} -0.840013i q^{85} +3.21511i q^{89} -21.4565 q^{91} +2.98048i q^{95} -3.94487i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 10 q^{5}+O(q^{10}) \) 8 * q - 10 * q^5	\( 8 q - 10 q^{5} + 20 q^{23} + 2 q^{25} + 8 q^{31} - 14 q^{37} - 12 q^{41} + 6 q^{43} - 32 q^{49} - 6 q^{59} - 22 q^{61} + 64 q^{73} - 12 q^{77} - 22 q^{83} - 12 q^{91}+O(q^{100}) \) 8 * q - 10 * q^5 + 20 * q^23 + 2 * q^25 + 8 * q^31 - 14 * q^37 - 12 * q^41 + 6 * q^43 - 32 * q^49 - 6 * q^59 - 22 * q^61 + 64 * q^73 - 12 * q^77 - 22 * q^83 - 12 * q^91

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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