LMFDB - Embedded newform 2952.2.j.c.2377.5

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2952,2,Mod(2377,2952)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://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);

sage: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")

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:traces = [6,0,0,0,4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

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

Self dual:	no
Analytic conductor:	$23.5718386767$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.0.1229312.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} + 10x^{4} + 24x^{2} + 8 $ x^6 + 10x^4 + 24x^2 + 8
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{5} $
Twist minimal:	no (minimal twist has level 328)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			2377.5
Root			$2.54832i$ of defining polynomial
Character	$\chi$	$=$	2952.2377
Dual form			2952.2.j.c.2377.6

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+3.60388 q^{5} -4.31182i q^{7} -2.04360i q^{11} -6.35542i q^{13} +5.65685i q^{17} +0.784829i q^{19} +4.98792 q^{23} +7.98792 q^{25} +0.698564i q^{29} +4.98792 q^{31} -15.5393i q^{35} -3.60388 q^{37} +(0.780167 + 6.35542i) q^{41} -8.98792 q^{43} -4.62271i q^{47} -11.5918 q^{49} -1.81897i q^{53} -7.36487i q^{55} -5.42758 q^{59} +11.4276 q^{61} -22.9041i q^{65} -10.6672i q^{67} -12.7971i q^{71} -2.61596 q^{73} -8.81163 q^{77} +4.62271i q^{79} -0.439665 q^{83} +20.3866i q^{85} +13.3326i q^{89} -27.4034 q^{91} +2.82843i q^{95} +9.57152i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 4 q^{5} - 8 q^{23} + 10 q^{25} - 8 q^{31} - 4 q^{37} + 2 q^{41} - 16 q^{43} - 14 q^{49} + 36 q^{61} - 36 q^{73} - 8 q^{83} - 56 q^{91}+O(q^{100}) $ 6 * q + 4 * q^5 - 8 * q^23 + 10 * q^25 - 8 * q^31 - 4 * q^37 + 2 * q^41 - 16 * q^43 - 14 * q^49 + 36 * q^61 - 36 * q^73 - 8 * q^83 - 56 * q^91

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
$5$	3.60388			1.61170			0.805851	−	0.592118i	$-0.201708\pi$
$5$	3.60388			1.61170			0.805851	+	0.592118i	$0.201708\pi$
$6$		0			0
$7$		−	4.31182i		−	1.62971i	−0.579661	−	0.814857i	$-0.696815\pi$
$7$		−	4.31182i		−	1.62971i	0.579661	−	0.814857i	$-0.303185\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$		−	2.04360i		−	0.616168i	−0.951359	−	0.308084i	$-0.900312\pi$
$11$		−	2.04360i		−	0.616168i	0.951359	−	0.308084i	$-0.0996878\pi$
$12$		0			0
$13$		−	6.35542i		−	1.76268i	−0.472486	−	0.881338i	$-0.656643\pi$
$13$		−	6.35542i		−	1.76268i	0.472486	−	0.881338i	$-0.343357\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$			5.65685i			1.37199i	0.727607	+	0.685994i	$0.240633\pi$
$17$			5.65685i			1.37199i	−0.727607	+	0.685994i	$0.759367\pi$
$18$		0			0
$19$			0.784829i			0.180052i	0.995939	+	0.0900261i	$0.0286950\pi$
$19$			0.784829i			0.180052i	−0.995939	+	0.0900261i	$0.971305\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	4.98792			1.04005			0.520026	−	0.854150i	$-0.325922\pi$
$23$	4.98792			1.04005			0.520026	+	0.854150i	$0.325922\pi$
$24$		0			0
$25$	7.98792			1.59758
$26$		0			0
$27$		0			0
$28$		0			0
$29$			0.698564i			0.129720i	0.997894	+	0.0648600i	$0.0206601\pi$
$29$			0.698564i			0.129720i	−0.997894	+	0.0648600i	$0.979340\pi$
$30$		0			0
$31$	4.98792			0.895857			0.447928	−	0.894069i	$-0.352162\pi$
$31$	4.98792			0.895857			0.447928	+	0.894069i	$0.352162\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		−	15.5393i		−	2.62661i
$36$		0			0
$37$	−3.60388			−0.592473			−0.296237	−	0.955115i	$-0.595732\pi$
$37$	−3.60388			−0.592473			−0.296237	+	0.955115i	$0.595732\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	0.780167	+	6.35542i	0.121842	+	0.992550i
$41$	0.780167	+	6.35542i	0.121842	+	0.992550i
$42$		0			0
$43$	−8.98792			−1.37064			−0.685322	−	0.728240i	$-0.740339\pi$
$43$	−8.98792			−1.37064			−0.685322	+	0.728240i	$0.740339\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		−	4.62271i		−	0.674292i	−0.941452	−	0.337146i	$-0.890538\pi$
$47$		−	4.62271i		−	0.674292i	0.941452	−	0.337146i	$-0.109462\pi$
$48$		0			0
$49$	−11.5918			−1.65597
$50$		0			0
$51$		0			0
$52$		0			0
$53$		−	1.81897i		−	0.249855i	−0.992166	−	0.124928i	$-0.960130\pi$
$53$		−	1.81897i		−	0.249855i	0.992166	−	0.124928i	$-0.0398699\pi$
$54$		0			0
$55$		−	7.36487i		−	0.993079i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−5.42758			−0.706611			−0.353306	−	0.935508i	$-0.614942\pi$
$59$	−5.42758			−0.706611			−0.353306	+	0.935508i	$0.614942\pi$
$60$		0			0
$61$	11.4276			1.46315			0.731576	−	0.681760i	$-0.238785\pi$
$61$	11.4276			1.46315			0.731576	+	0.681760i	$0.238785\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$		−	22.9041i		−	2.84091i
$66$		0			0
$67$		−	10.6672i		−	1.30321i	−0.758558	−	0.651605i	$-0.774096\pi$
$67$		−	10.6672i		−	1.30321i	0.758558	−	0.651605i	$-0.225904\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		−	12.7971i		−	1.51874i	−0.650661	−	0.759368i	$-0.725508\pi$
$71$		−	12.7971i		−	1.51874i	0.650661	−	0.759368i	$-0.274492\pi$
$72$		0			0
$73$	−2.61596			−0.306175			−0.153087	−	0.988213i	$-0.548922\pi$
$73$	−2.61596			−0.306175			−0.153087	+	0.988213i	$0.548922\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−8.81163			−1.00418
$78$		0			0
$79$			4.62271i			0.520096i	0.965596	+	0.260048i	$0.0837383\pi$
$79$			4.62271i			0.520096i	−0.965596	+	0.260048i	$0.916262\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−0.439665			−0.0482595			−0.0241298	−	0.999709i	$-0.507681\pi$
$83$	−0.439665			−0.0482595			−0.0241298	+	0.999709i	$0.507681\pi$
$84$		0			0
$85$			20.3866i			2.21124i
$86$		0			0
$87$		0			0
$88$		0			0
$89$			13.3326i			1.41325i	0.707586	+	0.706627i	$0.249784\pi$
$89$			13.3326i			1.41325i	−0.707586	+	0.706627i	$0.750216\pi$
$90$		0			0
$91$	−27.4034			−2.87266
$92$		0			0
$93$		0			0
$94$		0			0
$95$			2.82843i			0.290191i
$96$		0			0
$97$			9.57152i			0.971840i	0.874003	+	0.485920i	$0.161515\pi$
$97$			9.57152i			0.971840i	−0.874003	+	0.485920i	$0.838485\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$		−	5.23501i		−	0.520903i	−0.965487	−	0.260451i	$-0.916129\pi$
$101$		−	5.23501i		−	0.520903i	0.965487	−	0.260451i	$-0.0838714\pi$
$102$		0			0
$103$	−6.41550			−0.632138			−0.316069	−	0.948736i	$-0.602363\pi$
$103$	−6.41550			−0.632138			−0.316069	+	0.948736i	$0.602363\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−18.4155			−1.78029			−0.890147	−	0.455674i	$-0.849398\pi$
$107$	−18.4155			−1.78029			−0.890147	+	0.455674i	$0.849398\pi$
$108$		0			0
$109$		−	0.698564i		−	0.0669103i	−0.999440	−	0.0334551i	$-0.989349\pi$
$109$		−	0.698564i		−	0.0669103i	0.999440	−	0.0334551i	$-0.0106511\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−11.7995			−1.11001			−0.555004	−	0.831848i	$-0.687283\pi$
$113$	−11.7995			−1.11001			−0.555004	+	0.831848i	$0.687283\pi$
$114$		0			0
$115$	17.9758			1.67626
$116$		0			0
$117$		0			0
$118$		0			0
$119$	24.3913			2.23595
$120$		0			0
$121$	6.82371			0.620337
$122$		0			0
$123$		0			0
$124$		0			0
$125$	10.7681			0.963127
$126$		0			0
$127$	16.5483			1.46842			0.734210	−	0.678922i	$-0.237553\pi$
$127$	16.5483			1.46842			0.734210	+	0.678922i	$0.237553\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	12.0000			1.04844			0.524222	−	0.851581i	$-0.324356\pi$
$131$	12.0000			1.04844			0.524222	+	0.851581i	$0.324356\pi$
$132$		0			0
$133$	3.38404			0.293434
$134$		0			0
$135$		0			0
$136$		0			0
$137$			2.51754i			0.215088i	0.994200	+	0.107544i	$0.0342986\pi$
$137$			2.51754i			0.215088i	−0.994200	+	0.107544i	$0.965701\pi$
$138$		0			0
$139$	18.9638			1.60848			0.804242	−	0.594301i	$-0.202571\pi$
$139$	18.9638			1.60848			0.804242	+	0.594301i	$0.202571\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−12.9879			−1.08610
$144$		0			0
$145$			2.51754i			0.209070i
$146$		0			0
$147$		0			0
$148$		0			0
$149$			3.83788i			0.314411i	0.987566	+	0.157206i	$0.0502486\pi$
$149$			3.83788i			0.314411i	−0.987566	+	0.157206i	$0.949751\pi$
$150$		0			0
$151$		−	12.6246i		−	1.02737i	−0.857978	−	0.513686i	$-0.828279\pi$
$151$		−	12.6246i		−	1.02737i	0.857978	−	0.513686i	$-0.171721\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	17.9758			1.44385
$156$		0			0
$157$		−	2.44075i		−	0.194793i	−0.995246	−	0.0973967i	$-0.968948\pi$
$157$		−	2.44075i		−	0.194793i	0.995246	−	0.0973967i	$-0.0310515\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		−	21.5070i		−	1.69499i
$162$		0			0
$163$	−10.4155			−0.815805			−0.407903	−	0.913025i	$-0.633740\pi$
$163$	−10.4155			−0.815805			−0.407903	+	0.913025i	$0.633740\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$			15.6255i			1.20914i	0.796552	+	0.604570i	$0.206655\pi$
$167$			15.6255i			1.20914i	−0.796552	+	0.604570i	$0.793345\pi$
$168$		0			0
$169$	−27.3913			−2.10703
$170$		0			0
$171$		0			0
$172$		0			0
$173$	2.00000			0.152057			0.0760286	−	0.997106i	$-0.475776\pi$
$173$	2.00000			0.152057			0.0760286	+	0.997106i	$0.475776\pi$
$174$		0			0
$175$		−	34.4425i		−	2.60361i
$176$		0			0
$177$		0			0
$178$		0			0
$179$			10.6672i			0.797307i	0.917102	+	0.398653i	$0.130522\pi$
$179$			10.6672i			0.797307i	−0.917102	+	0.398653i	$0.869478\pi$
$180$		0			0
$181$			1.81897i			0.135203i	0.997712	+	0.0676016i	$0.0215347\pi$
$181$			1.81897i			0.135203i	−0.997712	+	0.0676016i	$0.978465\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−12.9879			−0.954891
$186$		0			0
$187$	11.5603			0.845375
$188$		0			0
$189$		0			0
$190$		0			0
$191$		−	17.9705i		−	1.30030i	−0.759805	−	0.650151i	$-0.774706\pi$
$191$		−	17.9705i		−	1.30030i	0.759805	−	0.650151i	$-0.225294\pi$
$192$		0			0
$193$			4.53644i			0.326540i	0.986581	+	0.163270i	$0.0522042\pi$
$193$			4.53644i			0.326540i	−0.986581	+	0.163270i	$0.947796\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	9.40342			0.669966			0.334983	−	0.942224i	$-0.391269\pi$
$197$	9.40342			0.669966			0.334983	+	0.942224i	$0.391269\pi$
$198$		0			0
$199$		−	11.0891i		−	0.786084i	−0.919520	−	0.393042i	$-0.871423\pi$
$199$		−	11.0891i		−	0.786084i	0.919520	−	0.393042i	$-0.128577\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	3.01208			0.211407
$204$		0			0
$205$	2.81163	+	22.9041i	0.196373	+	1.59969i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	1.60388			0.110942
$210$		0			0
$211$			25.5695i			1.76028i	0.474716	+	0.880139i	$0.342551\pi$
$211$			25.5695i			1.76028i	−0.474716	+	0.880139i	$0.657449\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−32.3913			−2.20907
$216$		0			0
$217$		−	21.5070i		−	1.45999i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	35.9517			2.41837
$222$		0			0
$223$	14.4155			0.965333			0.482667	−	0.875804i	$-0.339668\pi$
$223$	14.4155			0.965333			0.482667	+	0.875804i	$0.339668\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$			10.0303i			0.665731i	0.942974	+	0.332866i	$0.108016\pi$
$227$			10.0303i			0.665731i	−0.942974	+	0.332866i	$0.891984\pi$
$228$		0			0
$229$		−	18.4445i		−	1.21885i	−0.792846	−	0.609423i	$-0.791401\pi$
$229$		−	18.4445i		−	1.21885i	0.792846	−	0.609423i	$-0.208599\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		−	16.1269i		−	1.05651i	−0.849087	−	0.528253i	$-0.822847\pi$
$233$		−	16.1269i		−	1.05651i	0.849087	−	0.528253i	$-0.177153\pi$
$234$		0			0
$235$		−	16.6597i		−	1.08676i
$236$		0			0
$237$		0			0
$238$		0			0
$239$			6.19237i			0.400551i	0.979740	+	0.200276i	$0.0641837\pi$
$239$			6.19237i			0.400551i	−0.979740	+	0.200276i	$0.935816\pi$
$240$		0			0
$241$	−0.572417			−0.0368726			−0.0184363	−	0.999830i	$-0.505869\pi$
$241$	−0.572417			−0.0368726			−0.0184363	+	0.999830i	$0.505869\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−41.7754			−2.66893
$246$		0			0
$247$	4.98792			0.317374
$248$		0			0
$249$		0			0
$250$		0			0
$251$	3.12067			0.196975			0.0984875	−	0.995138i	$-0.468600\pi$
$251$	3.12067			0.196975			0.0984875	+	0.995138i	$0.468600\pi$
$252$		0			0
$253$		−	10.1933i		−	0.640847i
$254$		0			0
$255$		0			0
$256$		0			0
$257$		−	10.1933i		−	0.635841i	−0.948117	−	0.317920i	$-0.897016\pi$
$257$		−	10.1933i		−	0.635841i	0.948117	−	0.317920i	$-0.102984\pi$
$258$		0			0
$259$			15.5393i			0.965563i
$260$		0			0
$261$		0			0
$262$		0			0
$263$		−	11.0549i		−	0.681675i	−0.940122	−	0.340838i	$-0.889289\pi$
$263$		−	11.0549i		−	0.681675i	0.940122	−	0.340838i	$-0.110711\pi$
$264$		0			0
$265$		−	6.55535i		−	0.402692i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−2.54825			−0.155370			−0.0776849	−	0.996978i	$-0.524753\pi$
$269$	−2.54825			−0.155370			−0.0776849	+	0.996978i	$0.524753\pi$
$270$		0			0
$271$	−4.98792			−0.302994			−0.151497	−	0.988458i	$-0.548409\pi$
$271$	−4.98792			−0.302994			−0.151497	+	0.988458i	$0.548409\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		−	16.3241i		−	0.984380i
$276$		0			0
$277$	−7.16421			−0.430456			−0.215228	−	0.976564i	$-0.569049\pi$
$277$	−7.16421			−0.430456			−0.215228	+	0.976564i	$0.569049\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$			23.5259i			1.40344i	0.712454	+	0.701719i	$0.247584\pi$
$281$			23.5259i			1.40344i	−0.712454	+	0.701719i	$0.752416\pi$
$282$		0			0
$283$	18.9638			1.12728			0.563639	−	0.826021i	$-0.309401\pi$
$283$	18.9638			1.12728			0.563639	+	0.826021i	$0.309401\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	27.4034	−	3.36394i	1.61757	−	0.198567i
$288$		0			0
$289$	−15.0000			−0.882353
$290$		0			0
$291$		0			0
$292$		0			0
$293$			8.37433i			0.489233i	0.969620	+	0.244617i	$0.0786621\pi$
$293$			8.37433i			0.489233i	−0.969620	+	0.244617i	$0.921338\pi$
$294$		0			0
$295$	−19.5603			−1.13885
$296$		0			0
$297$		0			0
$298$		0			0
$299$		−	31.7003i		−	1.83328i
$300$		0			0
$301$			38.7543i			2.23376i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	41.1836			2.35817
$306$		0			0
$307$	−4.87933			−0.278478			−0.139239	−	0.990259i	$-0.544466\pi$
$307$	−4.87933			−0.278478			−0.139239	+	0.990259i	$0.544466\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$			2.56963i			0.145710i	0.997343	+	0.0728552i	$0.0232111\pi$
$311$			2.56963i			0.145710i	−0.997343	+	0.0728552i	$0.976789\pi$
$312$		0			0
$313$			21.7837i			1.23129i	0.788024	+	0.615644i	$0.211104\pi$
$313$			21.7837i			1.23129i	−0.788024	+	0.615644i	$0.788896\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		−	25.3449i		−	1.42351i	−0.702427	−	0.711755i	$-0.747900\pi$
$317$		−	25.3449i		−	1.42351i	0.702427	−	0.711755i	$-0.252100\pi$
$318$		0			0
$319$	1.42758			0.0799293
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−4.43967			−0.247030
$324$		0			0
$325$		−	50.7666i		−	2.81602i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−19.9323			−1.09890
$330$		0			0
$331$			18.3924i			1.01094i	0.862845	+	0.505468i	$0.168680\pi$
$331$			18.3924i			1.01094i	−0.862845	+	0.505468i	$0.831320\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		−	38.4434i		−	2.10039i
$336$		0			0
$337$	16.2392			0.884606			0.442303	−	0.896866i	$-0.354162\pi$
$337$	16.2392			0.884606			0.442303	+	0.896866i	$0.354162\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		−	10.1933i		−	0.551998i
$342$		0			0
$343$			19.7990i			1.06904i
$344$		0			0
$345$		0			0
$346$		0			0
$347$			18.7033i			1.00404i	0.864855	+	0.502022i	$0.167410\pi$
$347$			18.7033i			1.00404i	−0.864855	+	0.502022i	$0.832590\pi$
$348$		0			0
$349$	23.2513			1.24461			0.622307	−	0.782774i	$-0.286196\pi$
$349$	23.2513			1.24461			0.622307	+	0.782774i	$0.286196\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−19.4082			−1.03299			−0.516497	−	0.856289i	$-0.672764\pi$
$353$	−19.4082			−1.03299			−0.516497	+	0.856289i	$0.672764\pi$
$354$		0			0
$355$		−	46.1192i		−	2.44775i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	11.5603			0.610131			0.305066	−	0.952331i	$-0.401322\pi$
$359$	11.5603			0.610131			0.305066	+	0.952331i	$0.401322\pi$
$360$		0			0
$361$	18.3840			0.967581
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−9.42758			−0.493462
$366$		0			0
$367$	−32.9396			−1.71943			−0.859716	−	0.510772i	$-0.829360\pi$
$367$	−32.9396			−1.71943			−0.859716	+	0.510772i	$0.829360\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−7.84309			−0.407193
$372$		0			0
$373$	6.00000			0.310668			0.155334	−	0.987862i	$-0.450355\pi$
$373$	6.00000			0.310668			0.155334	+	0.987862i	$0.450355\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	4.43967			0.228654
$378$		0			0
$379$	2.57242			0.132136			0.0660681	−	0.997815i	$-0.478955\pi$
$379$	2.57242			0.132136			0.0660681	+	0.997815i	$0.478955\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$			17.0227i			0.869817i	0.900475	+	0.434909i	$0.143219\pi$
$383$			17.0227i			0.869817i	−0.900475	+	0.434909i	$0.856781\pi$
$384$		0			0
$385$	−31.7560			−1.61844
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−13.1207			−0.665244			−0.332622	−	0.943060i	$-0.607933\pi$
$389$	−13.1207			−0.665244			−0.332622	+	0.943060i	$0.607933\pi$
$390$		0			0
$391$			28.2159i			1.42694i
$392$		0			0
$393$		0			0
$394$		0			0
$395$			16.6597i			0.838239i
$396$		0			0
$397$			11.7356i			0.588991i	0.955653	+	0.294495i	$0.0951515\pi$
$397$			11.7356i			0.588991i	−0.955653	+	0.294495i	$0.904848\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	24.1521			1.20610			0.603050	−	0.797703i	$-0.293952\pi$
$401$	24.1521			1.20610			0.603050	+	0.797703i	$0.293952\pi$
$402$		0			0
$403$		−	31.7003i		−	1.57910i
$404$		0			0
$405$		0			0
$406$		0			0
$407$			7.36487i			0.365063i
$408$		0			0
$409$	24.9444			1.23342			0.616710	−	0.787190i	$-0.288465\pi$
$409$	24.9444			1.23342			0.616710	+	0.787190i	$0.288465\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$			23.4028i			1.15157i
$414$		0			0
$415$	−1.58450			−0.0777800
$416$		0			0
$417$		0			0
$418$		0			0
$419$	29.4276			1.43763			0.718816	−	0.695201i	$-0.244684\pi$
$419$	29.4276			1.43763			0.718816	+	0.695201i	$0.244684\pi$
$420$		0			0
$421$		−	28.1391i		−	1.37142i	−0.727876	−	0.685709i	$-0.759492\pi$
$421$		−	28.1391i		−	1.37142i	0.727876	−	0.685709i	$-0.240508\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$			45.1865i			2.19187i
$426$		0			0
$427$		−	49.2737i		−	2.38452i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−6.57242			−0.316582			−0.158291	−	0.987392i	$-0.550598\pi$
$431$	−6.57242			−0.316582			−0.158291	+	0.987392i	$0.550598\pi$
$432$		0			0
$433$	26.3913			1.26829			0.634143	−	0.773215i	$-0.281353\pi$
$433$	26.3913			1.26829			0.634143	+	0.773215i	$0.281353\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$			3.91466i			0.187264i
$438$		0			0
$439$			9.79614i			0.467544i	0.972291	+	0.233772i	$0.0751070\pi$
$439$			9.79614i			0.467544i	−0.972291	+	0.233772i	$0.924893\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−17.5362			−0.833169			−0.416584	−	0.909097i	$-0.636773\pi$
$443$	−17.5362			−0.833169			−0.416584	+	0.909097i	$0.636773\pi$
$444$		0			0
$445$			48.0491i			2.27775i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	17.8431			0.842067			0.421034	−	0.907045i	$-0.361667\pi$
$449$	17.8431			0.842067			0.421034	+	0.907045i	$0.361667\pi$
$450$		0			0
$451$	12.9879	−	1.59435i	0.611577	−	0.0750749i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−98.7585			−4.62987
$456$		0			0
$457$			17.8691i			0.835879i	0.908475	+	0.417940i	$0.137248\pi$
$457$			17.8691i			0.835879i	−0.908475	+	0.417940i	$0.862752\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	18.0194			0.839246			0.419623	−	0.907698i	$-0.362162\pi$
$461$	18.0194			0.839246			0.419623	+	0.907698i	$0.362162\pi$
$462$		0			0
$463$			36.8216i			1.71125i	0.517599	+	0.855624i	$0.326826\pi$
$463$			36.8216i			1.71125i	−0.517599	+	0.855624i	$0.673174\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	3.84309			0.177837			0.0889184	−	0.996039i	$-0.471659\pi$
$467$	3.84309			0.177837			0.0889184	+	0.996039i	$0.471659\pi$
$468$		0			0
$469$	−45.9952			−2.12386
$470$		0			0
$471$		0			0
$472$		0			0
$473$			18.3677i			0.844547i
$474$		0			0
$475$			6.26915i			0.287648i
$476$		0			0
$477$		0			0
$478$		0			0
$479$			12.4862i			0.570510i	0.958452	+	0.285255i	$0.0920782\pi$
$479$			12.4862i			0.570510i	−0.958452	+	0.285255i	$0.907922\pi$
$480$		0			0
$481$			22.9041i			1.04434i
$482$		0			0
$483$		0			0
$484$		0			0
$485$			34.4946i			1.56632i
$486$		0			0
$487$	−17.2707			−0.782609			−0.391304	−	0.920261i	$-0.627976\pi$
$487$	−17.2707			−0.782609			−0.391304	+	0.920261i	$0.627976\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	10.9638			0.494787			0.247394	−	0.968915i	$-0.420426\pi$
$491$	10.9638			0.494787			0.247394	+	0.968915i	$0.420426\pi$
$492$		0			0
$493$	−3.95167			−0.177974
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−55.1788			−2.47511
$498$		0			0
$499$		−	13.8218i		−	0.618747i	−0.950941	−	0.309374i	$-0.899881\pi$
$499$		−	13.8218i		−	0.618747i	0.950941	−	0.309374i	$-0.100119\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		−	35.9780i		−	1.60418i	−0.597205	−	0.802089i	$-0.703722\pi$
$503$		−	35.9780i		−	1.60418i	0.597205	−	0.802089i	$-0.296278\pi$
$504$		0			0
$505$		−	18.8663i		−	0.839540i
$506$		0			0
$507$		0			0
$508$		0			0
$509$			26.7420i			1.18532i	0.805453	+	0.592660i	$0.201922\pi$
$509$			26.7420i			1.18532i	−0.805453	+	0.592660i	$0.798078\pi$
$510$		0			0
$511$			11.2795i			0.498977i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−23.1207			−1.01882
$516$		0			0
$517$	−9.44696			−0.415477
$518$		0			0
$519$		0			0
$520$		0			0
$521$		−	14.9517i		−	0.655044i	−0.944844	−	0.327522i	$-0.893786\pi$
$521$		−	14.9517i		−	0.655044i	0.944844	−	0.327522i	$-0.106214\pi$
$522$		0			0
$523$	−39.2465			−1.71613			−0.858065	−	0.513541i	$-0.828333\pi$
$523$	−39.2465			−1.71613			−0.858065	+	0.513541i	$0.828333\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$			28.2159i			1.22911i
$528$		0			0
$529$	1.87933			0.0817100
$530$		0			0
$531$		0			0
$532$		0			0
$533$	40.3913	−	4.95829i	1.74954	−	0.214767i
$534$		0			0
$535$	−66.3672			−2.86930
$536$		0			0
$537$		0			0
$538$		0			0
$539$			23.6890i			1.02036i
$540$		0			0
$541$	3.69096			0.158687			0.0793433	−	0.996847i	$-0.474718\pi$
$541$	3.69096			0.158687			0.0793433	+	0.996847i	$0.474718\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		−	2.51754i		−	0.107839i
$546$		0			0
$547$		−	3.75162i		−	0.160407i	−0.996778	−	0.0802037i	$-0.974443\pi$
$547$		−	3.75162i		−	0.160407i	0.996778	−	0.0802037i	$-0.0255571\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−0.548253			−0.0233564
$552$		0			0
$553$	19.9323			0.847607
$554$		0			0
$555$		0			0
$556$		0			0
$557$			20.4634i			0.867061i	0.901139	+	0.433531i	$0.142732\pi$
$557$			20.4634i			0.867061i	−0.901139	+	0.433531i	$0.857268\pi$
$558$		0			0
$559$			57.1220i			2.41600i
$560$		0			0
$561$		0			0
$562$		0			0
$563$			26.0188i			1.09656i	0.836295	+	0.548280i	$0.184717\pi$
$563$			26.0188i			1.09656i	−0.836295	+	0.548280i	$0.815283\pi$
$564$		0			0
$565$	−42.5241			−1.78900
$566$		0			0
$567$		0			0
$568$		0			0
$569$	2.61596			0.109667			0.0548333	−	0.998496i	$-0.482537\pi$
$569$	2.61596			0.109667			0.0548333	+	0.998496i	$0.482537\pi$
$570$		0			0
$571$		−	36.9874i		−	1.54787i	−0.633262	−	0.773937i	$-0.718284\pi$
$571$		−	36.9874i		−	1.54787i	0.633262	−	0.773937i	$-0.281716\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	39.8431			1.66157
$576$		0			0
$577$		−	10.1933i		−	0.424353i	−0.977231	−	0.212176i	$-0.931945\pi$
$577$		−	10.1933i		−	0.424353i	0.977231	−	0.212176i	$-0.0680551\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$			1.89576i			0.0786492i
$582$		0			0
$583$	−3.71725			−0.153953
$584$		0			0
$585$		0			0
$586$		0			0
$587$			27.9145i			1.15216i	0.817395	+	0.576078i	$0.195417\pi$
$587$			27.9145i			1.15216i	−0.817395	+	0.576078i	$0.804583\pi$
$588$		0			0
$589$			3.91466i			0.161301i
$590$		0			0
$591$		0			0
$592$		0			0
$593$			36.8585i			1.51360i	0.653647	+	0.756799i	$0.273238\pi$
$593$			36.8585i			1.51360i	−0.653647	+	0.756799i	$0.726762\pi$
$594$		0			0
$595$	87.9033			3.60369
$596$		0			0
$597$		0			0
$598$		0			0
$599$	10.3069			0.421129			0.210565	−	0.977580i	$-0.432470\pi$
$599$	10.3069			0.421129			0.210565	+	0.977580i	$0.432470\pi$
$600$		0			0
$601$			35.9600i			1.46684i	0.679776	+	0.733420i	$0.262077\pi$
$601$			35.9600i			1.46684i	−0.679776	+	0.733420i	$0.737923\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	24.5918			0.999799
$606$		0			0
$607$	33.4276			1.35678			0.678392	−	0.734700i	$-0.262677\pi$
$607$	33.4276			1.35678			0.678392	+	0.734700i	$0.262677\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−29.3793			−1.18856
$612$		0			0
$613$	32.3478			1.30652			0.653258	−	0.757136i	$-0.273402\pi$
$613$	32.3478			1.30652			0.653258	+	0.757136i	$0.273402\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−22.8310			−0.919142			−0.459571	−	0.888141i	$-0.651997\pi$
$617$	−22.8310			−0.919142			−0.459571	+	0.888141i	$0.651997\pi$
$618$		0			0
$619$	0.596580			0.0239786			0.0119893	−	0.999928i	$-0.496184\pi$
$619$	0.596580			0.0239786			0.0119893	+	0.999928i	$0.496184\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	57.4878			2.30320
$624$		0			0
$625$	−1.13275			−0.0453101
$626$		0			0
$627$		0			0
$628$		0			0
$629$		−	20.3866i		−	0.812867i
$630$		0			0
$631$	2.85517			0.113662			0.0568312	−	0.998384i	$-0.481900\pi$
$631$	2.85517			0.113662			0.0568312	+	0.998384i	$0.481900\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	59.6378			2.36666
$636$		0			0
$637$			73.6707i			2.91894i
$638$		0			0
$639$		0			0
$640$		0			0
$641$			0.498629i			0.0196947i	0.999952	+	0.00984733i	$0.00313455\pi$
$641$			0.498629i			0.0196947i	−0.999952	+	0.00984733i	$0.996865\pi$
$642$		0			0
$643$		−	35.0368i		−	1.38172i	−0.722989	−	0.690859i	$-0.757232\pi$
$643$		−	35.0368i		−	1.38172i	0.722989	−	0.690859i	$-0.242768\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−35.9517			−1.41341			−0.706703	−	0.707510i	$-0.749818\pi$
$647$	−35.9517			−1.41341			−0.706703	+	0.707510i	$0.749818\pi$
$648$		0			0
$649$			11.0918i			0.435391i
$650$		0			0
$651$		0			0
$652$		0			0
$653$		−	43.8661i		−	1.71661i	−0.513136	−	0.858307i	$-0.671516\pi$
$653$		−	43.8661i		−	1.71661i	0.513136	−	0.858307i	$-0.328484\pi$
$654$		0			0
$655$	43.2465			1.68978
$656$		0			0
$657$		0			0
$658$		0			0
$659$			21.8426i			0.850866i	0.904990	+	0.425433i	$0.139878\pi$
$659$			21.8426i			0.850866i	−0.904990	+	0.425433i	$0.860122\pi$
$660$		0			0
$661$	16.3478			0.635856			0.317928	−	0.948115i	$-0.397013\pi$
$661$	16.3478			0.635856			0.317928	+	0.948115i	$0.397013\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	12.1957			0.472928
$666$		0			0
$667$			3.48438i			0.134916i
$668$		0			0
$669$		0			0
$670$		0			0
$671$		−	23.3534i		−	0.901547i
$672$		0			0
$673$			33.8728i			1.30570i	0.757487	+	0.652850i	$0.226427\pi$
$673$			33.8728i			1.30570i	−0.757487	+	0.652850i	$0.773573\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	0.0435405			0.00167340			0.000836699	−	1.00000i	$-0.499734\pi$
$677$	0.0435405			0.00167340			0.000836699		1.00000i	$0.499734\pi$
$678$		0			0
$679$	41.2707			1.58382
$680$		0			0
$681$		0			0
$682$		0			0
$683$		−	8.46059i		−	0.323736i	−0.986812	−	0.161868i	$-0.948248\pi$
$683$		−	8.46059i		−	0.323736i	0.986812	−	0.161868i	$-0.0517518\pi$
$684$		0			0
$685$			9.07289i			0.346657i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−11.5603			−0.440414
$690$		0			0
$691$			8.11553i			0.308729i	0.988014	+	0.154365i	$0.0493331\pi$
$691$			8.11553i			0.308729i	−0.988014	+	0.154365i	$0.950667\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	68.3430			2.59240
$696$		0			0
$697$	−35.9517	+	4.41329i	−1.36177	+	0.167165i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−27.2030			−1.02744			−0.513721	−	0.857957i	$-0.671733\pi$
$701$	−27.2030			−1.02744			−0.513721	+	0.857957i	$0.671733\pi$
$702$		0			0
$703$		−	2.82843i		−	0.106676i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−22.5724			−0.848923
$708$		0			0
$709$			3.99145i			0.149902i	0.997187	+	0.0749510i	$0.0238800\pi$
$709$			3.99145i			0.149902i	−0.997187	+	0.0749510i	$0.976120\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	24.8793			0.931738
$714$		0			0
$715$	−46.8068			−1.75048
$716$		0			0
$717$		0			0
$718$		0			0
$719$			34.4083i			1.28321i	0.767034	+	0.641606i	$0.221732\pi$
$719$			34.4083i			1.28321i	−0.767034	+	0.641606i	$0.778268\pi$
$720$		0			0
$721$			27.6625i			1.03020i
$722$		0			0
$723$		0			0
$724$		0			0
$725$			5.58007i			0.207239i
$726$		0			0
$727$			38.7174i			1.43595i	0.696069	+	0.717974i	$0.254931\pi$
$727$			38.7174i			1.43595i	−0.696069	+	0.717974i	$0.745069\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$		−	50.8433i		−	1.88051i
$732$		0			0
$733$	−49.4034			−1.82476			−0.912378	−	0.409348i	$-0.865756\pi$
$733$	−49.4034			−1.82476			−0.912378	+	0.409348i	$0.865756\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−21.7995			−0.802997
$738$		0			0
$739$	13.9758			0.514109			0.257055	−	0.966397i	$-0.417248\pi$
$739$	13.9758			0.514109			0.257055	+	0.966397i	$0.417248\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	6.02416			0.221005			0.110503	−	0.993876i	$-0.464754\pi$
$743$	6.02416			0.221005			0.110503	+	0.993876i	$0.464754\pi$
$744$		0			0
$745$			13.8312i			0.506738i
$746$		0			0
$747$		0			0
$748$		0			0
$749$			79.4043i			2.90137i
$750$		0			0
$751$		−	13.0396i		−	0.475824i	−0.971287	−	0.237912i	$-0.923537\pi$
$751$		−	13.0396i		−	0.475824i	0.971287	−	0.237912i	$-0.0764629\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		−	45.4974i		−	1.65582i
$756$		0			0
$757$		−	18.4445i		−	0.670376i	−0.942151	−	0.335188i	$-0.891200\pi$
$757$		−	18.4445i		−	0.670376i	0.942151	−	0.335188i	$-0.108800\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	26.1280			0.947138			0.473569	−	0.880757i	$-0.342965\pi$
$761$	26.1280			0.947138			0.473569	+	0.880757i	$0.342965\pi$
$762$		0			0
$763$	−3.01208			−0.109045
$764$		0			0
$765$		0			0
$766$		0			0
$767$			34.4946i			1.24553i
$768$		0			0
$769$	−17.9517			−0.647354			−0.323677	−	0.946168i	$-0.604919\pi$
$769$	−17.9517			−0.647354			−0.323677	+	0.946168i	$0.604919\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$		−	26.2434i		−	0.943909i	−0.881623	−	0.471955i	$-0.843549\pi$
$773$		−	26.2434i		−	0.943909i	0.881623	−	0.471955i	$-0.156451\pi$
$774$		0			0
$775$	39.8431			1.43121
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−4.98792	+	0.612298i	−0.178711	+	0.0219379i
$780$		0			0
$781$	−26.1521			−0.935797
$782$		0			0
$783$		0			0
$784$		0			0
$785$		−	8.79617i		−	0.313949i
$786$		0			0
$787$	−37.9758			−1.35369			−0.676846	−	0.736124i	$-0.736654\pi$
$787$	−37.9758			−1.35369			−0.676846	+	0.736124i	$0.736654\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$			50.8775i			1.80900i
$792$		0			0
$793$		−	72.6271i		−	2.57906i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	11.6931			0.414190			0.207095	−	0.978321i	$-0.433599\pi$
$797$	11.6931			0.414190			0.207095	+	0.978321i	$0.433599\pi$
$798$		0			0
$799$	26.1500			0.925120
$800$		0			0
$801$		0			0
$802$		0			0
$803$			5.34596i			0.188655i
$804$		0			0
$805$		−	77.5086i		−	2.73182i
$806$		0			0
$807$		0			0
$808$		0			0
$809$			40.4965i			1.42378i	0.702291	+	0.711890i	$0.252161\pi$
$809$			40.4965i			1.42378i	−0.702291	+	0.711890i	$0.747839\pi$
$810$		0			0
$811$	−24.4397			−0.858193			−0.429096	−	0.903259i	$-0.641168\pi$
$811$	−24.4397			−0.858193			−0.429096	+	0.903259i	$0.641168\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−37.5362			−1.31483
$816$		0			0
$817$		−	7.05398i		−	0.246788i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−41.2271			−1.43884			−0.719418	−	0.694577i	$-0.755592\pi$
$821$	−41.2271			−1.43884			−0.719418	+	0.694577i	$0.755592\pi$
$822$		0			0
$823$			34.2358i			1.19338i	0.802470	+	0.596692i	$0.203519\pi$
$823$			34.2358i			1.19338i	−0.802470	+	0.596692i	$0.796481\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		−	24.6710i		−	0.857895i	−0.903329	−	0.428948i	$-0.858884\pi$
$827$		−	24.6710i		−	0.857895i	0.903329	−	0.428948i	$-0.141116\pi$
$828$		0			0
$829$	37.8840			1.31576			0.657882	−	0.753121i	$-0.271452\pi$
$829$	37.8840			1.31576			0.657882	+	0.753121i	$0.271452\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		−	65.5731i		−	2.27197i
$834$		0			0
$835$			56.3125i			1.94877i
$836$		0			0
$837$		0			0
$838$		0			0
$839$			4.31182i			0.148861i	0.997226	+	0.0744303i	$0.0237138\pi$
$839$			4.31182i			0.148861i	−0.997226	+	0.0744303i	$0.976286\pi$
$840$		0			0
$841$	28.5120			0.983173
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−98.7150			−3.39590
$846$		0			0
$847$		−	29.4226i		−	1.01097i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−17.9758			−0.616204
$852$		0			0
$853$	41.9517			1.43640			0.718199	−	0.695838i	$-0.244967\pi$
$853$	41.9517			1.43640			0.718199	+	0.695838i	$0.244967\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	23.4470			0.800933			0.400466	−	0.916311i	$-0.368848\pi$
$857$	23.4470			0.800933			0.400466	+	0.916311i	$0.368848\pi$
$858$		0			0
$859$	20.3913			0.695743			0.347872	−	0.937542i	$-0.386904\pi$
$859$	20.3913			0.695743			0.347872	+	0.937542i	$0.386904\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	4.59658			0.156469			0.0782347	−	0.996935i	$-0.475072\pi$
$863$	4.59658			0.156469			0.0782347	+	0.996935i	$0.475072\pi$
$864$		0			0
$865$	7.20775			0.245071
$866$		0			0
$867$		0			0
$868$		0			0
$869$	9.44696			0.320466
$870$		0			0
$871$	−67.7948			−2.29714
$872$		0			0
$873$		0			0
$874$		0			0
$875$		−	46.4301i		−	1.56962i
$876$		0			0
$877$	−40.5241			−1.36840			−0.684201	−	0.729294i	$-0.739849\pi$
$877$	−40.5241			−1.36840			−0.684201	+	0.729294i	$0.739849\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	15.2707			0.514482			0.257241	−	0.966347i	$-0.417187\pi$
$881$	15.2707			0.514482			0.257241	+	0.966347i	$0.417187\pi$
$882$		0			0
$883$		−	15.5640i		−	0.523769i	−0.965099	−	0.261884i	$-0.915656\pi$
$883$		−	15.5640i		−	0.523769i	0.965099	−	0.261884i	$-0.0843439\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$			45.5836i			1.53055i	0.643704	+	0.765275i	$0.277397\pi$
$887$			45.5836i			1.53055i	−0.643704	+	0.765275i	$0.722603\pi$
$888$		0			0
$889$		−	71.3531i		−	2.39311i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	3.62804			0.121408
$894$		0			0
$895$			38.4434i			1.28502i
$896$		0			0
$897$		0			0
$898$		0			0
$899$			3.48438i			0.116211i
$900$		0			0
$901$	10.2897			0.342798
$902$		0			0
$903$		0			0
$904$		0			0
$905$			6.55535i			0.217907i
$906$		0			0
$907$	−28.3913			−0.942719			−0.471359	−	0.881941i	$-0.656237\pi$
$907$	−28.3913			−0.942719			−0.471359	+	0.881941i	$0.656237\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−30.2586			−1.00251			−0.501256	−	0.865299i	$-0.667128\pi$
$911$	−30.2586			−1.00251			−0.501256	+	0.865299i	$0.667128\pi$
$912$		0			0
$913$			0.898499i			0.0297360i
$914$		0			0
$915$		0			0
$916$		0			0
$917$		−	51.7418i		−	1.70867i
$918$		0			0
$919$			35.7164i			1.17818i	0.808069	+	0.589088i	$0.200513\pi$
$919$			35.7164i			1.17818i	−0.808069	+	0.589088i	$0.799487\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−81.3309			−2.67704
$924$		0			0
$925$	−28.7875			−0.946526
$926$		0			0
$927$		0			0
$928$		0			0
$929$		−	6.90041i		−	0.226395i	−0.993572	−	0.113198i	$-0.963891\pi$
$929$		−	6.90041i		−	0.226395i	0.993572	−	0.113198i	$-0.0361093\pi$
$930$		0			0
$931$		−	9.09758i		−	0.298161i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	41.6620			1.36249
$936$		0			0
$937$		−	35.8369i		−	1.17074i	−0.810766	−	0.585370i	$-0.800949\pi$
$937$		−	35.8369i		−	1.17074i	0.810766	−	0.585370i	$-0.199051\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	23.7103			0.772935			0.386467	−	0.922303i	$-0.373695\pi$
$941$	23.7103			0.772935			0.386467	+	0.922303i	$0.373695\pi$
$942$		0			0
$943$	3.89141	+	31.7003i	0.126722	+	1.03230i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−7.01208			−0.227862			−0.113931	−	0.993489i	$-0.536344\pi$
$947$	−7.01208			−0.227862			−0.113931	+	0.993489i	$0.536344\pi$
$948$		0			0
$949$			16.6255i			0.539687i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−17.8237			−0.577367			−0.288683	−	0.957425i	$-0.593217\pi$
$953$	−17.8237			−0.577367			−0.288683	+	0.957425i	$0.593217\pi$
$954$		0			0
$955$		−	64.7636i		−	2.09570i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	10.8552			0.350532
$960$		0			0
$961$	−6.12067			−0.197441
$962$		0			0
$963$		0			0
$964$		0			0
$965$			16.3488i			0.526286i
$966$		0			0
$967$		−	2.01619i		−	0.0648364i	−0.999474	−	0.0324182i	$-0.989679\pi$
$967$		−	2.01619i		−	0.0648364i	0.999474	−	0.0324182i	$-0.0103208\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$			14.2216i			0.456394i	0.973615	+	0.228197i	$0.0732830\pi$
$971$			14.2216i			0.456394i	−0.973615	+	0.228197i	$0.926717\pi$
$972$		0			0
$973$		−	81.7683i		−	2.62137i
$974$		0			0
$975$		0			0
$976$		0			0
$977$		−	49.2243i		−	1.57483i	−0.616426	−	0.787413i	$-0.711420\pi$
$977$		−	49.2243i		−	1.57483i	0.616426	−	0.787413i	$-0.288580\pi$
$978$		0			0
$979$	27.2465			0.870802
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−50.2103			−1.60146			−0.800729	−	0.599026i	$-0.795554\pi$
$983$	−50.2103			−1.60146			−0.800729	+	0.599026i	$0.795554\pi$
$984$		0			0
$985$	33.8888			1.07979
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−44.8310			−1.42554
$990$		0			0
$991$		−	54.7401i		−	1.73888i	−0.494042	−	0.869438i	$-0.664481\pi$
$991$		−	54.7401i		−	1.73888i	0.494042	−	0.869438i	$-0.335519\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		−	39.9637i		−	1.26693i
$996$		0			0
$997$		−	41.1950i		−	1.30466i	−0.757935	−	0.652330i	$-0.773792\pi$
$997$		−	41.1950i		−	1.30466i	0.757935	−	0.652330i	$-0.226208\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2952.2.j.c.2377.5		6
3.2	odd	2		328.2.d.b.81.4	yes	6
12.11	even	2		656.2.d.f.81.3		6
24.5	odd	2		2624.2.d.o.2049.3		6
24.11	even	2		2624.2.d.n.2049.4		6
41.40	even	2	inner	2952.2.j.c.2377.6		6
123.122	odd	2		328.2.d.b.81.3	✓	6
492.491	even	2		656.2.d.f.81.4		6
984.245	odd	2		2624.2.d.o.2049.4		6
984.491	even	2		2624.2.d.n.2049.3		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
328.2.d.b.81.3	✓	6	123.122	odd	2
328.2.d.b.81.4	yes	6	3.2	odd	2
656.2.d.f.81.3		6	12.11	even	2
656.2.d.f.81.4		6	492.491	even	2
2624.2.d.n.2049.3		6	984.491	even	2
2624.2.d.n.2049.4		6	24.11	even	2
2624.2.d.o.2049.3		6	24.5	odd	2
2624.2.d.o.2049.4		6	984.245	odd	2
2952.2.j.c.2377.5		6	1.1	even	1	trivial
2952.2.j.c.2377.6		6	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 2952 = 2^{3} \cdot 3^{2} \cdot 41 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2952.j (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+3.60388 q^{5} -4.31182i q^{7} -2.04360i q^{11} -6.35542i q^{13} +5.65685i q^{17} +0.784829i q^{19} +4.98792 q^{23} +7.98792 q^{25} +0.698564i q^{29} +4.98792 q^{31} -15.5393i q^{35} -3.60388 q^{37} +(0.780167 + 6.35542i) q^{41} -8.98792 q^{43} -4.62271i q^{47} -11.5918 q^{49} -1.81897i q^{53} -7.36487i q^{55} -5.42758 q^{59} +11.4276 q^{61} -22.9041i q^{65} -10.6672i q^{67} -12.7971i q^{71} -2.61596 q^{73} -8.81163 q^{77} +4.62271i q^{79} -0.439665 q^{83} +20.3866i q^{85} +13.3326i q^{89} -27.4034 q^{91} +2.82843i q^{95} +9.57152i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 4 q^{5} - 8 q^{23} + 10 q^{25} - 8 q^{31} - 4 q^{37} + 2 q^{41} - 16 q^{43} - 14 q^{49} + 36 q^{61} - 36 q^{73} - 8 q^{83} - 56 q^{91}+O(q^{100}) \) 6 * q + 4 * q^5 - 8 * q^23 + 10 * q^25 - 8 * q^31 - 4 * q^37 + 2 * q^41 - 16 * q^43 - 14 * q^49 + 36 * q^61 - 36 * q^73 - 8 * q^83 - 56 * q^91

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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