LMFDB - Embedded newform 656.2.d.f.81.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [656,2,Mod(81,656)] 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("656.81"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(656, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 656 = 2^{4} \cdot 41 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	656.d (of order $2$, degree $1$, not 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:	$5.23818637260$
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:	$\Z[a_1, \ldots, a_{7}]$
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			81.3
Root			$-2.54832i$ of defining polynomial
Character	$\chi$	$=$	656.81
Dual form			656.2.d.f.81.4

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-0.784829i q^{3} -3.60388 q^{5} +4.31182i q^{7} +2.38404 q^{9} -2.04360i q^{11} -6.35542i q^{13} +2.82843i q^{15} -5.65685i q^{17} -0.784829i q^{19} +3.38404 q^{21} +4.98792 q^{23} +7.98792 q^{25} -4.22555i q^{27} -0.698564i q^{29} -4.98792 q^{31} -1.60388 q^{33} -15.5393i q^{35} -3.60388 q^{37} -4.98792 q^{39} +(-0.780167 - 6.35542i) q^{41} +8.98792 q^{43} -8.59179 q^{45} -4.62271i q^{47} -11.5918 q^{49} -4.43967 q^{51} +1.81897i q^{53} +7.36487i q^{55} -0.615957 q^{57} -5.42758 q^{59} +11.4276 q^{61} +10.2796i q^{63} +22.9041i q^{65} +10.6672i q^{67} -3.91466i q^{69} -12.7971i q^{71} -2.61596 q^{73} -6.26915i q^{75} +8.81163 q^{77} -4.62271i q^{79} +3.83579 q^{81} -0.439665 q^{83} +20.3866i q^{85} -0.548253 q^{87} -13.3326i q^{89} +27.4034 q^{91} +3.91466i q^{93} +2.82843i q^{95} +9.57152i q^{97} -4.87202i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 4 q^{5} - 6 q^{9} - 8 q^{23} + 10 q^{25} + 8 q^{31} + 8 q^{33} - 4 q^{37} + 8 q^{39} - 2 q^{41} + 16 q^{43} + 4 q^{45} - 14 q^{49} - 32 q^{51} - 24 q^{57} + 36 q^{61} - 36 q^{73} + 46 q^{81} - 8 q^{83}+ \cdots + 56 q^{91}+O(q^{100}) $ 6 * q - 4 * q^5 - 6 * q^9 - 8 * q^23 + 10 * q^25 + 8 * q^31 + 8 * q^33 - 4 * q^37 + 8 * q^39 - 2 * q^41 + 16 * q^43 + 4 * q^45 - 14 * q^49 - 32 * q^51 - 24 * q^57 + 36 * q^61 - 36 * q^73 + 46 * q^81 - 8 * q^83 + 40 * q^87 + 56 * q^91

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/656\mathbb{Z}\right)^\times$.

$n$	$129$	$165$	$575$
$\chi(n)$	$-1$	$1$	$1$

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 $

$2$		0			0
$2$		0			0
$3$		−	0.784829i		−	0.453121i	−0.973997	−	0.226561i	$-0.927252\pi$
$3$		−	0.784829i		−	0.453121i	0.973997	−	0.226561i	$-0.0727482\pi$
$4$		0			0
$5$	−3.60388			−1.61170			−0.805851	−	0.592118i	$-0.798292\pi$
$5$	−3.60388			−1.61170			−0.805851	+	0.592118i	$0.798292\pi$
$6$		0			0
$7$			4.31182i			1.62971i	0.579661	+	0.814857i	$0.303185\pi$
$7$			4.31182i			1.62971i	−0.579661	+	0.814857i	$0.696815\pi$
$8$		0			0
$9$	2.38404			0.794681
$10$		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
$15$			2.82843i			0.730297i
$16$		0			0
$17$		−	5.65685i		−	1.37199i	−0.727607	−	0.685994i	$-0.759367\pi$
$17$		−	5.65685i		−	1.37199i	0.727607	−	0.685994i	$-0.240633\pi$
$18$		0			0
$19$		−	0.784829i		−	0.180052i	−0.995939	−	0.0900261i	$-0.971305\pi$
$19$		−	0.784829i		−	0.180052i	0.995939	−	0.0900261i	$-0.0286950\pi$
$20$		0			0
$21$	3.38404			0.738459
$22$		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
$27$		−	4.22555i		−	0.813208i
$28$		0			0
$29$		−	0.698564i		−	0.129720i	−0.997894	−	0.0648600i	$-0.979340\pi$
$29$		−	0.698564i		−	0.129720i	0.997894	−	0.0648600i	$-0.0206601\pi$
$30$		0			0
$31$	−4.98792			−0.895857			−0.447928	−	0.894069i	$-0.647838\pi$
$31$	−4.98792			−0.895857			−0.447928	+	0.894069i	$0.647838\pi$
$32$		0			0
$33$	−1.60388			−0.279199
$34$		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
$39$	−4.98792			−0.798706
$40$		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	$-0.259661\pi$
$43$	8.98792			1.37064			0.685322	+	0.728240i	$0.259661\pi$
$44$		0			0
$45$	−8.59179			−1.28079
$46$		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
$51$	−4.43967			−0.621677
$52$		0			0
$53$			1.81897i			0.249855i	0.992166	+	0.124928i	$0.0398699\pi$
$53$			1.81897i			0.249855i	−0.992166	+	0.124928i	$0.960130\pi$
$54$		0			0
$55$			7.36487i			0.993079i
$56$		0			0
$57$	−0.615957			−0.0815855
$58$		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
$63$			10.2796i			1.29510i
$64$		0			0
$65$			22.9041i			2.84091i
$66$		0			0
$67$			10.6672i			1.30321i	0.758558	+	0.651605i	$0.225904\pi$
$67$			10.6672i			1.30321i	−0.758558	+	0.651605i	$0.774096\pi$
$68$		0			0
$69$		−	3.91466i		−	0.471270i
$70$		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
$75$		−	6.26915i		−	0.723899i
$76$		0			0
$77$	8.81163			1.00418
$78$		0			0
$79$		−	4.62271i		−	0.520096i	−0.965596	−	0.260048i	$-0.916262\pi$
$79$		−	4.62271i		−	0.520096i	0.965596	−	0.260048i	$-0.0837383\pi$
$80$		0			0
$81$	3.83579			0.426199
$82$		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
$87$	−0.548253			−0.0587789
$88$		0			0
$89$		−	13.3326i		−	1.41325i	−0.707586	−	0.706627i	$-0.750216\pi$
$89$		−	13.3326i		−	1.41325i	0.707586	−	0.706627i	$-0.249784\pi$
$90$		0			0
$91$	27.4034			2.87266
$92$		0			0
$93$			3.91466i			0.405932i
$94$		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
$99$		−	4.87202i		−	0.489657i
$100$		0			0
$101$			5.23501i			0.520903i	0.965487	+	0.260451i	$0.0838714\pi$
$101$			5.23501i			0.520903i	−0.965487	+	0.260451i	$0.916129\pi$
$102$		0			0
$103$	6.41550			0.632138			0.316069	−	0.948736i	$-0.397637\pi$
$103$	6.41550			0.632138			0.316069	+	0.948736i	$0.397637\pi$
$104$		0			0
$105$	−12.1957			−1.19018
$106$		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
$111$			2.82843i			0.268462i
$112$		0			0
$113$	11.7995			1.11001			0.555004	−	0.831848i	$-0.312717\pi$
$113$	11.7995			1.11001			0.555004	+	0.831848i	$0.312717\pi$
$114$		0			0
$115$	−17.9758			−1.67626
$116$		0			0
$117$		−	15.1516i		−	1.40076i
$118$		0			0
$119$	24.3913			2.23595
$120$		0			0
$121$	6.82371			0.620337
$122$		0			0
$123$	−4.98792	+	0.612298i	−0.449745	+	0.0552091i
$124$		0			0
$125$	−10.7681			−0.963127
$126$		0			0
$127$	−16.5483			−1.46842			−0.734210	−	0.678922i	$-0.762447\pi$
$127$	−16.5483			−1.46842			−0.734210	+	0.678922i	$0.762447\pi$
$128$		0			0
$129$		−	7.05398i		−	0.621068i
$130$		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
$135$			15.2284i			1.31065i
$136$		0			0
$137$		−	2.51754i		−	0.215088i	−0.994200	−	0.107544i	$-0.965701\pi$
$137$		−	2.51754i		−	0.215088i	0.994200	−	0.107544i	$-0.0342986\pi$
$138$		0			0
$139$	−18.9638			−1.60848			−0.804242	−	0.594301i	$-0.797429\pi$
$139$	−18.9638			−1.60848			−0.804242	+	0.594301i	$0.797429\pi$
$140$		0			0
$141$	−3.62804			−0.305536
$142$		0			0
$143$	−12.9879			−1.08610
$144$		0			0
$145$			2.51754i			0.209070i
$146$		0			0
$147$			9.09758i			0.750356i
$148$		0			0
$149$		−	3.83788i		−	0.314411i	−0.987566	−	0.157206i	$-0.949751\pi$
$149$		−	3.83788i		−	0.314411i	0.987566	−	0.157206i	$-0.0502486\pi$
$150$		0			0
$151$			12.6246i			1.02737i	0.857978	+	0.513686i	$0.171721\pi$
$151$			12.6246i			1.02737i	−0.857978	+	0.513686i	$0.828279\pi$
$152$		0			0
$153$		−	13.4862i		−	1.09029i
$154$		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
$159$	1.42758			0.113215
$160$		0			0
$161$			21.5070i			1.69499i
$162$		0			0
$163$	10.4155			0.815805			0.407903	−	0.913025i	$-0.366260\pi$
$163$	10.4155			0.815805			0.407903	+	0.913025i	$0.366260\pi$
$164$		0			0
$165$	5.78017			0.449985
$166$		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
$171$		−	1.87107i		−	0.143084i
$172$		0			0
$173$	−2.00000			−0.152057			−0.0760286	−	0.997106i	$-0.524224\pi$
$173$	−2.00000			−0.152057			−0.0760286	+	0.997106i	$0.524224\pi$
$174$		0			0
$175$			34.4425i			2.60361i
$176$		0			0
$177$			4.25973i			0.320181i
$178$		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
$183$		−	8.96870i		−	0.662986i
$184$		0			0
$185$	12.9879			0.954891
$186$		0			0
$187$	−11.5603			−0.845375
$188$		0			0
$189$	18.2198			1.32530
$190$		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
$195$	17.9758			1.28728
$196$		0			0
$197$	−9.40342			−0.669966			−0.334983	−	0.942224i	$-0.608731\pi$
$197$	−9.40342			−0.669966			−0.334983	+	0.942224i	$0.608731\pi$
$198$		0			0
$199$			11.0891i			0.786084i	0.919520	+	0.393042i	$0.128577\pi$
$199$			11.0891i			0.786084i	−0.919520	+	0.393042i	$0.871423\pi$
$200$		0			0
$201$	8.37196			0.590513
$202$		0			0
$203$	3.01208			0.211407
$204$		0			0
$205$	2.81163	+	22.9041i	0.196373	+	1.59969i
$206$		0			0
$207$	11.8914			0.826510
$208$		0			0
$209$	−1.60388			−0.110942
$210$		0			0
$211$		−	25.5695i		−	1.76028i	−0.474716	−	0.880139i	$-0.657449\pi$
$211$		−	25.5695i		−	1.76028i	0.474716	−	0.880139i	$-0.342551\pi$
$212$		0			0
$213$	−10.0435			−0.688172
$214$		0			0
$215$	−32.3913			−2.20907
$216$		0			0
$217$		−	21.5070i		−	1.45999i
$218$		0			0
$219$			2.05308i			0.138734i
$220$		0			0
$221$	−35.9517			−2.41837
$222$		0			0
$223$	−14.4155			−0.965333			−0.482667	−	0.875804i	$-0.660332\pi$
$223$	−14.4155			−0.965333			−0.482667	+	0.875804i	$0.660332\pi$
$224$		0			0
$225$	19.0435			1.26957
$226$		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
$231$		−	6.91562i		−	0.455015i
$232$		0			0
$233$			16.1269i			1.05651i	0.849087	+	0.528253i	$0.177153\pi$
$233$			16.1269i			1.05651i	−0.849087	+	0.528253i	$0.822847\pi$
$234$		0			0
$235$			16.6597i			1.08676i
$236$		0			0
$237$	−3.62804			−0.235666
$238$		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
$243$		−	15.6871i		−	1.00633i
$244$		0			0
$245$	41.7754			2.66893
$246$		0			0
$247$	−4.98792			−0.317374
$248$		0			0
$249$			0.345062i			0.0218674i
$250$		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
$255$	16.0000			1.00196
$256$		0			0
$257$			10.1933i			0.635841i	0.948117	+	0.317920i	$0.102984\pi$
$257$			10.1933i			0.635841i	−0.948117	+	0.317920i	$0.897016\pi$
$258$		0			0
$259$		−	15.5393i		−	0.965563i
$260$		0			0
$261$		−	1.66541i		−	0.103086i
$262$		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
$267$	−10.4638			−0.640376
$268$		0			0
$269$	2.54825			0.155370			0.0776849	−	0.996978i	$-0.475247\pi$
$269$	2.54825			0.155370			0.0776849	+	0.996978i	$0.475247\pi$
$270$		0			0
$271$	4.98792			0.302994			0.151497	−	0.988458i	$-0.451591\pi$
$271$	4.98792			0.302994			0.151497	+	0.988458i	$0.451591\pi$
$272$		0			0
$273$		−	21.5070i		−	1.30166i
$274$		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
$279$	−11.8914			−0.711920
$280$		0			0
$281$		−	23.5259i		−	1.40344i	−0.712454	−	0.701719i	$-0.752416\pi$
$281$		−	23.5259i		−	1.40344i	0.712454	−	0.701719i	$-0.247584\pi$
$282$		0			0
$283$	−18.9638			−1.12728			−0.563639	−	0.826021i	$-0.690599\pi$
$283$	−18.9638			−1.12728			−0.563639	+	0.826021i	$0.690599\pi$
$284$		0			0
$285$	2.21983			0.131492
$286$		0			0
$287$	27.4034	−	3.36394i	1.61757	−	0.198567i
$288$		0			0
$289$	−15.0000			−0.882353
$290$		0			0
$291$	7.51201			0.440362
$292$		0			0
$293$		−	8.37433i		−	0.489233i	−0.969620	−	0.244617i	$-0.921338\pi$
$293$		−	8.37433i		−	0.489233i	0.969620	−	0.244617i	$-0.0786621\pi$
$294$		0			0
$295$	19.5603			1.13885
$296$		0			0
$297$	−8.63533			−0.501073
$298$		0			0
$299$		−	31.7003i		−	1.83328i
$300$		0			0
$301$			38.7543i			2.23376i
$302$		0			0
$303$	4.10859			0.236032
$304$		0			0
$305$	−41.1836			−2.35817
$306$		0			0
$307$	4.87933			0.278478			0.139239	−	0.990259i	$-0.455534\pi$
$307$	4.87933			0.278478			0.139239	+	0.990259i	$0.455534\pi$
$308$		0			0
$309$		−	5.03507i		−	0.286435i
$310$		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
$315$		−	37.0463i		−	2.08732i
$316$		0			0
$317$			25.3449i			1.42351i	0.702427	+	0.711755i	$0.252100\pi$
$317$			25.3449i			1.42351i	−0.702427	+	0.711755i	$0.747900\pi$
$318$		0			0
$319$	−1.42758			−0.0799293
$320$		0			0
$321$			14.4530i			0.806689i
$322$		0			0
$323$	−4.43967			−0.247030
$324$		0			0
$325$		−	50.7666i		−	2.81602i
$326$		0			0
$327$	−0.548253			−0.0303185
$328$		0			0
$329$	19.9323			1.09890
$330$		0			0
$331$		−	18.3924i		−	1.01094i	−0.862845	−	0.505468i	$-0.831320\pi$
$331$		−	18.3924i		−	1.01094i	0.862845	−	0.505468i	$-0.168680\pi$
$332$		0			0
$333$	−8.59179			−0.470827
$334$		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
$339$		−	9.26063i		−	0.502968i
$340$		0			0
$341$			10.1933i			0.551998i
$342$		0			0
$343$		−	19.7990i		−	1.06904i
$344$		0			0
$345$			14.1080i			0.759547i
$346$		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
$351$	−26.8552			−1.43342
$352$		0			0
$353$	19.4082			1.03299			0.516497	−	0.856289i	$-0.327236\pi$
$353$	19.4082			1.03299			0.516497	+	0.856289i	$0.327236\pi$
$354$		0			0
$355$			46.1192i			2.44775i
$356$		0			0
$357$		−	19.1430i		−	1.01316i
$358$		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
$363$		−	5.35545i		−	0.281088i
$364$		0			0
$365$	9.42758			0.493462
$366$		0			0
$367$	32.9396			1.71943			0.859716	−	0.510772i	$-0.170640\pi$
$367$	32.9396			1.71943			0.859716	+	0.510772i	$0.170640\pi$
$368$		0			0
$369$	−1.85995	−	15.1516i	−0.0968253	−	0.788760i
$370$		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
$375$			8.45111i			0.436413i
$376$		0			0
$377$	−4.43967			−0.228654
$378$		0			0
$379$	−2.57242			−0.132136			−0.0660681	−	0.997815i	$-0.521045\pi$
$379$	−2.57242			−0.132136			−0.0660681	+	0.997815i	$0.521045\pi$
$380$		0			0
$381$			12.9876i			0.665373i
$382$		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
$387$	21.4276			1.08923
$388$		0			0
$389$	13.1207			0.665244			0.332622	−	0.943060i	$-0.392067\pi$
$389$	13.1207			0.665244			0.332622	+	0.943060i	$0.392067\pi$
$390$		0			0
$391$		−	28.2159i		−	1.42694i
$392$		0			0
$393$		−	9.41795i		−	0.475073i
$394$		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
$399$		−	2.65590i		−	0.132961i
$400$		0			0
$401$	−24.1521			−1.20610			−0.603050	−	0.797703i	$-0.706048\pi$
$401$	−24.1521			−1.20610			−0.603050	+	0.797703i	$0.706048\pi$
$402$		0			0
$403$			31.7003i			1.57910i
$404$		0			0
$405$	−13.8237			−0.686906
$406$		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
$411$	−1.97584			−0.0974608
$412$		0			0
$413$		−	23.4028i		−	1.15157i
$414$		0			0
$415$	1.58450			0.0777800
$416$		0			0
$417$			14.8833i			0.728839i
$418$		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
$423$		−	11.0207i		−	0.535847i
$424$		0			0
$425$		−	45.1865i		−	2.19187i
$426$		0			0
$427$			49.2737i			2.38452i
$428$		0			0
$429$			10.1933i			0.492137i
$430$		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
$435$	1.97584			0.0947341
$436$		0			0
$437$		−	3.91466i		−	0.187264i
$438$		0			0
$439$		−	9.79614i		−	0.467544i	−0.972291	−	0.233772i	$-0.924893\pi$
$439$		−	9.79614i		−	0.467544i	0.972291	−	0.233772i	$-0.0751070\pi$
$440$		0			0
$441$	−27.6353			−1.31597
$442$		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
$447$	−3.01208			−0.142467
$448$		0			0
$449$	−17.8431			−0.842067			−0.421034	−	0.907045i	$-0.638333\pi$
$449$	−17.8431			−0.842067			−0.421034	+	0.907045i	$0.638333\pi$
$450$		0			0
$451$	−12.9879	+	1.59435i	−0.611577	+	0.0750749i
$452$		0			0
$453$	9.90813			0.465525
$454$		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
$459$	−23.9033			−1.11571
$460$		0			0
$461$	−18.0194			−0.839246			−0.419623	−	0.907698i	$-0.637838\pi$
$461$	−18.0194			−0.839246			−0.419623	+	0.907698i	$0.637838\pi$
$462$		0			0
$463$		−	36.8216i		−	1.71125i	−0.517599	−	0.855624i	$-0.673174\pi$
$463$		−	36.8216i		−	1.71125i	0.517599	−	0.855624i	$-0.326826\pi$
$464$		0			0
$465$		−	14.1080i		−	0.654241i
$466$		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
$471$	−1.91557			−0.0882650
$472$		0			0
$473$		−	18.3677i		−	0.844547i
$474$		0			0
$475$		−	6.26915i		−	0.287648i
$476$		0			0
$477$			4.33651i			0.198555i
$478$		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
$483$	16.8793			0.768036
$484$		0			0
$485$		−	34.4946i		−	1.56632i
$486$		0			0
$487$	17.2707			0.782609			0.391304	−	0.920261i	$-0.372024\pi$
$487$	17.2707			0.782609			0.391304	+	0.920261i	$0.372024\pi$
$488$		0			0
$489$		−	8.17439i		−	0.369659i
$490$		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
$495$			17.5582i			0.789181i
$496$		0			0
$497$	55.1788			2.47511
$498$		0			0
$499$			13.8218i			0.618747i	0.950941	+	0.309374i	$0.100119\pi$
$499$			13.8218i			0.618747i	−0.950941	+	0.309374i	$0.899881\pi$
$500$		0			0
$501$	12.2634			0.547887
$502$		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
$507$			21.4975i			0.954739i
$508$		0			0
$509$		−	26.7420i		−	1.18532i	−0.805453	−	0.592660i	$-0.798078\pi$
$509$		−	26.7420i		−	1.18532i	0.805453	−	0.592660i	$-0.201922\pi$
$510$		0			0
$511$		−	11.2795i		−	0.498977i
$512$		0			0
$513$	−3.31634			−0.146420
$514$		0			0
$515$	−23.1207			−1.01882
$516$		0			0
$517$	−9.44696			−0.415477
$518$		0			0
$519$			1.56966i			0.0689004i
$520$		0			0
$521$			14.9517i			0.655044i	0.944844	+	0.327522i	$0.106214\pi$
$521$			14.9517i			0.655044i	−0.944844	+	0.327522i	$0.893786\pi$
$522$		0			0
$523$	39.2465			1.71613			0.858065	−	0.513541i	$-0.171667\pi$
$523$	39.2465			1.71613			0.858065	+	0.513541i	$0.171667\pi$
$524$		0			0
$525$	27.0315			1.17975
$526$		0			0
$527$			28.2159i			1.22911i
$528$		0			0
$529$	1.87933			0.0817100
$530$		0			0
$531$	−12.9396			−0.561530
$532$		0			0
$533$	−40.3913	+	4.95829i	−1.74954	+	0.214767i
$534$		0			0
$535$	66.3672			2.86930
$536$		0			0
$537$	8.37196			0.361277
$538$		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
$543$	1.42758			0.0612635
$544$		0			0
$545$			2.51754i			0.107839i
$546$		0			0
$547$			3.75162i			0.160407i	0.996778	+	0.0802037i	$0.0255571\pi$
$547$			3.75162i			0.160407i	−0.996778	+	0.0802037i	$0.974443\pi$
$548$		0			0
$549$	27.2438			1.16274
$550$		0			0
$551$	−0.548253			−0.0233564
$552$		0			0
$553$	19.9323			0.847607
$554$		0			0
$555$		−	10.1933i		−	0.432681i
$556$		0			0
$557$		−	20.4634i		−	0.867061i	−0.901139	−	0.433531i	$-0.857268\pi$
$557$		−	20.4634i		−	0.867061i	0.901139	−	0.433531i	$-0.142732\pi$
$558$		0			0
$559$		−	57.1220i		−	2.41600i
$560$		0			0
$561$			9.07289i			0.383058i
$562$		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
$567$			16.5392i			0.694583i
$568$		0			0
$569$	−2.61596			−0.109667			−0.0548333	−	0.998496i	$-0.517463\pi$
$569$	−2.61596			−0.109667			−0.0548333	+	0.998496i	$0.517463\pi$
$570$		0			0
$571$			36.9874i			1.54787i	0.633262	+	0.773937i	$0.281716\pi$
$571$			36.9874i			1.54787i	−0.633262	+	0.773937i	$0.718284\pi$
$572$		0			0
$573$	−14.1038			−0.589195
$574$		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
$579$	3.56033			0.147962
$580$		0			0
$581$		−	1.89576i		−	0.0786492i
$582$		0			0
$583$	3.71725			0.153953
$584$		0			0
$585$			54.6044i			2.25762i
$586$		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
$591$			7.38008i			0.303576i
$592$		0			0
$593$		−	36.8585i		−	1.51360i	−0.653647	−	0.756799i	$-0.726762\pi$
$593$		−	36.8585i		−	1.51360i	0.653647	−	0.756799i	$-0.273238\pi$
$594$		0			0
$595$	−87.9033			−3.60369
$596$		0			0
$597$	8.70304			0.356192
$598$		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
$603$			25.4312i			1.03564i
$604$		0			0
$605$	−24.5918			−0.999799
$606$		0			0
$607$	−33.4276			−1.35678			−0.678392	−	0.734700i	$-0.737323\pi$
$607$	−33.4276			−1.35678			−0.678392	+	0.734700i	$0.737323\pi$
$608$		0			0
$609$		−	2.36397i		−	0.0957929i
$610$		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
$615$	17.9758	−	2.20665i	0.724856	−	0.0889806i
$616$		0			0
$617$	22.8310			0.919142			0.459571	−	0.888141i	$-0.348003\pi$
$617$	22.8310			0.919142			0.459571	+	0.888141i	$0.348003\pi$
$618$		0			0
$619$	−0.596580			−0.0239786			−0.0119893	−	0.999928i	$-0.503816\pi$
$619$	−0.596580			−0.0239786			−0.0119893	+	0.999928i	$0.503816\pi$
$620$		0			0
$621$		−	21.0767i		−	0.845780i
$622$		0			0
$623$	57.4878			2.30320
$624$		0			0
$625$	−1.13275			−0.0453101
$626$		0			0
$627$			1.25877i			0.0502704i
$628$		0			0
$629$			20.3866i			0.812867i
$630$		0			0
$631$	−2.85517			−0.113662			−0.0568312	−	0.998384i	$-0.518100\pi$
$631$	−2.85517			−0.113662			−0.0568312	+	0.998384i	$0.518100\pi$
$632$		0			0
$633$	−20.0677			−0.797620
$634$		0			0
$635$	59.6378			2.36666
$636$		0			0
$637$			73.6707i			2.91894i
$638$		0			0
$639$		−	30.5088i		−	1.20691i
$640$		0			0
$641$		−	0.498629i		−	0.0196947i	−0.999952	−	0.00984733i	$-0.996865\pi$
$641$		−	0.498629i		−	0.0196947i	0.999952	−	0.00984733i	$-0.00313455\pi$
$642$		0			0
$643$			35.0368i			1.38172i	0.722989	+	0.690859i	$0.242768\pi$
$643$			35.0368i			1.38172i	−0.722989	+	0.690859i	$0.757232\pi$
$644$		0			0
$645$			25.4217i			1.00098i
$646$		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
$651$	−16.8793			−0.661553
$652$		0			0
$653$			43.8661i			1.71661i	0.513136	+	0.858307i	$0.328484\pi$
$653$			43.8661i			1.71661i	−0.513136	+	0.858307i	$0.671516\pi$
$654$		0			0
$655$	−43.2465			−1.68978
$656$		0			0
$657$	−6.23655			−0.243311
$658$		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
$663$			28.2159i			1.09582i
$664$		0			0
$665$	−12.1957			−0.472928
$666$		0			0
$667$		−	3.48438i		−	0.134916i
$668$		0			0
$669$			11.3137i			0.437413i
$670$		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
$675$		−	33.7534i		−	1.29917i
$676$		0			0
$677$	−0.0435405			−0.00167340			−0.000836699	−	1.00000i	$-0.500266\pi$
$677$	−0.0435405			−0.00167340			−0.000836699		1.00000i	$0.500266\pi$
$678$		0			0
$679$	−41.2707			−1.58382
$680$		0			0
$681$	7.87203			0.301657
$682$		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
$687$	−14.4758			−0.552285
$688$		0			0
$689$	11.5603			0.440414
$690$		0			0
$691$		−	8.11553i		−	0.308729i	−0.988014	−	0.154365i	$-0.950667\pi$
$691$		−	8.11553i		−	0.308729i	0.988014	−	0.154365i	$-0.0493331\pi$
$692$		0			0
$693$	21.0073			0.798001
$694$		0			0
$695$	68.3430			2.59240
$696$		0			0
$697$	−35.9517	+	4.41329i	−1.36177	+	0.167165i
$698$		0			0
$699$	12.6568			0.478726
$700$		0			0
$701$	27.2030			1.02744			0.513721	−	0.857957i	$-0.328267\pi$
$701$	27.2030			1.02744			0.513721	+	0.857957i	$0.328267\pi$
$702$		0			0
$703$			2.82843i			0.106676i
$704$		0			0
$705$	13.0750			0.492433
$706$		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
$711$		−	11.0207i		−	0.413310i
$712$		0			0
$713$	−24.8793			−0.931738
$714$		0			0
$715$	46.8068			1.75048
$716$		0			0
$717$	4.85995			0.181498
$718$		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
$723$			0.449249i			0.0167078i
$724$		0			0
$725$		−	5.58007i		−	0.207239i
$726$		0			0
$727$		−	38.7174i		−	1.43595i	−0.696069	−	0.717974i	$-0.745069\pi$
$727$		−	38.7174i		−	1.43595i	0.696069	−	0.717974i	$-0.254931\pi$
$728$		0			0
$729$	−0.804331			−0.0297900
$730$		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
$735$		−	32.7865i		−	1.20935i
$736$		0			0
$737$	21.7995			0.802997
$738$		0			0
$739$	−13.9758			−0.514109			−0.257055	−	0.966397i	$-0.582752\pi$
$739$	−13.9758			−0.514109			−0.257055	+	0.966397i	$0.582752\pi$
$740$		0			0
$741$			3.91466i			0.143809i
$742$		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
$747$	−1.04818			−0.0383509
$748$		0			0
$749$		−	79.4043i		−	2.90137i
$750$		0			0
$751$			13.0396i			0.475824i	0.971287	+	0.237912i	$0.0764629\pi$
$751$			13.0396i			0.475824i	−0.971287	+	0.237912i	$0.923537\pi$
$752$		0			0
$753$		−	2.44919i		−	0.0892536i
$754$		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
$759$	−8.00000			−0.290382
$760$		0			0
$761$	−26.1280			−0.947138			−0.473569	−	0.880757i	$-0.657035\pi$
$761$	−26.1280			−0.947138			−0.473569	+	0.880757i	$0.657035\pi$
$762$		0			0
$763$	3.01208			0.109045
$764$		0			0
$765$			48.6025i			1.75723i
$766$		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
$771$	8.00000			0.288113
$772$		0			0
$773$			26.2434i			0.943909i	0.881623	+	0.471955i	$0.156451\pi$
$773$			26.2434i			0.943909i	−0.881623	+	0.471955i	$0.843549\pi$
$774$		0			0
$775$	−39.8431			−1.43121
$776$		0			0
$777$	−12.1957			−0.437517
$778$		0			0
$779$	−4.98792	+	0.612298i	−0.178711	+	0.0219379i
$780$		0			0
$781$	−26.1521			−0.935797
$782$		0			0
$783$	−2.95182			−0.105489
$784$		0			0
$785$			8.79617i			0.313949i
$786$		0			0
$787$	37.9758			1.35369			0.676846	−	0.736124i	$-0.263346\pi$
$787$	37.9758			1.35369			0.676846	+	0.736124i	$0.263346\pi$
$788$		0			0
$789$	−8.67622			−0.308882
$790$		0			0
$791$			50.8775i			1.80900i
$792$		0			0
$793$		−	72.6271i		−	2.57906i
$794$		0			0
$795$	−5.14483			−0.182468
$796$		0			0
$797$	−11.6931			−0.414190			−0.207095	−	0.978321i	$-0.566401\pi$
$797$	−11.6931			−0.414190			−0.207095	+	0.978321i	$0.566401\pi$
$798$		0			0
$799$	−26.1500			−0.925120
$800$		0			0
$801$		−	31.7855i		−	1.12309i
$802$		0			0
$803$			5.34596i			0.188655i
$804$		0			0
$805$		−	77.5086i		−	2.73182i
$806$		0			0
$807$		−	1.99994i		−	0.0704014i
$808$		0			0
$809$		−	40.4965i		−	1.42378i	−0.702291	−	0.711890i	$-0.747839\pi$
$809$		−	40.4965i		−	1.42378i	0.702291	−	0.711890i	$-0.252161\pi$
$810$		0			0
$811$	24.4397			0.858193			0.429096	−	0.903259i	$-0.358832\pi$
$811$	24.4397			0.858193			0.429096	+	0.903259i	$0.358832\pi$
$812$		0			0
$813$		−	3.91466i		−	0.137293i
$814$		0			0
$815$	−37.5362			−1.31483
$816$		0			0
$817$		−	7.05398i		−	0.246788i
$818$		0			0
$819$	65.3309			2.28285
$820$		0			0
$821$	41.2271			1.43884			0.719418	−	0.694577i	$-0.244408\pi$
$821$	41.2271			1.43884			0.719418	+	0.694577i	$0.244408\pi$
$822$		0			0
$823$		−	34.2358i		−	1.19338i	−0.802470	−	0.596692i	$-0.796481\pi$
$823$		−	34.2358i		−	1.19338i	0.802470	−	0.596692i	$-0.203519\pi$
$824$		0			0
$825$	−12.8116			−0.446044
$826$		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
$831$			5.62268i			0.195049i
$832$		0			0
$833$			65.5731i			2.27197i
$834$		0			0
$835$		−	56.3125i		−	1.94877i
$836$		0			0
$837$			21.0767i			0.728518i
$838$		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
$843$	−18.4638			−0.635928
$844$		0			0
$845$	98.7150			3.39590
$846$		0			0
$847$			29.4226i			1.01097i
$848$		0			0
$849$			14.8833i			0.510794i
$850$		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
$855$			6.74309i			0.230609i
$856$		0			0
$857$	−23.4470			−0.800933			−0.400466	−	0.916311i	$-0.631152\pi$
$857$	−23.4470			−0.800933			−0.400466	+	0.916311i	$0.631152\pi$
$858$		0			0
$859$	−20.3913			−0.695743			−0.347872	−	0.937542i	$-0.613096\pi$
$859$	−20.3913			−0.695743			−0.347872	+	0.937542i	$0.613096\pi$
$860$		0			0
$861$	−2.64012	−	21.5070i	−0.0899751	−	0.732957i
$862$		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
$867$			11.7724i			0.399813i
$868$		0			0
$869$	−9.44696			−0.320466
$870$		0			0
$871$	67.7948			2.29714
$872$		0			0
$873$			22.8189i			0.772303i
$874$		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
$879$	−6.57242			−0.221682
$880$		0			0
$881$	−15.2707			−0.514482			−0.257241	−	0.966347i	$-0.582813\pi$
$881$	−15.2707			−0.514482			−0.257241	+	0.966347i	$0.582813\pi$
$882$		0			0
$883$			15.5640i			0.523769i	0.965099	+	0.261884i	$0.0843439\pi$
$883$			15.5640i			0.523769i	−0.965099	+	0.261884i	$0.915656\pi$
$884$		0			0
$885$		−	15.3515i		−	0.516036i
$886$		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
$891$		−	7.83881i		−	0.262610i
$892$		0			0
$893$	−3.62804			−0.121408
$894$		0			0
$895$		−	38.4434i		−	1.28502i
$896$		0			0
$897$	−24.8793			−0.830697
$898$		0			0
$899$			3.48438i			0.116211i
$900$		0			0
$901$	10.2897			0.342798
$902$		0			0
$903$	30.4155			1.01216
$904$		0			0
$905$		−	6.55535i		−	0.217907i
$906$		0			0
$907$	28.3913			0.942719			0.471359	−	0.881941i	$-0.343763\pi$
$907$	28.3913			0.942719			0.471359	+	0.881941i	$0.343763\pi$
$908$		0			0
$909$			12.4805i			0.413952i
$910$		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
$915$			32.3221i			1.06854i
$916$		0			0
$917$			51.7418i			1.70867i
$918$		0			0
$919$		−	35.7164i		−	1.17818i	−0.808069	−	0.589088i	$-0.799487\pi$
$919$		−	35.7164i		−	1.17818i	0.808069	−	0.589088i	$-0.200513\pi$
$920$		0			0
$921$		−	3.82944i		−	0.126184i
$922$		0			0
$923$	−81.3309			−2.67704
$924$		0			0
$925$	−28.7875			−0.946526
$926$		0			0
$927$	15.2948			0.502348
$928$		0			0
$929$			6.90041i			0.226395i	0.993572	+	0.113198i	$0.0361093\pi$
$929$			6.90041i			0.226395i	−0.993572	+	0.113198i	$0.963891\pi$
$930$		0			0
$931$			9.09758i			0.298161i
$932$		0			0
$933$	2.01672			0.0660245
$934$		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
$939$	17.0965			0.557923
$940$		0			0
$941$	−23.7103			−0.772935			−0.386467	−	0.922303i	$-0.626305\pi$
$941$	−23.7103			−0.772935			−0.386467	+	0.922303i	$0.626305\pi$
$942$		0			0
$943$	−3.89141	−	31.7003i	−0.126722	−	1.03230i
$944$		0			0
$945$	−65.6620			−2.13599
$946$		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
$951$	19.8914			0.645023
$952$		0			0
$953$	17.8237			0.577367			0.288683	−	0.957425i	$-0.406783\pi$
$953$	17.8237			0.577367			0.288683	+	0.957425i	$0.406783\pi$
$954$		0			0
$955$			64.7636i			2.09570i
$956$		0			0
$957$			1.12041i			0.0362177i
$958$		0			0
$959$	10.8552			0.350532
$960$		0			0
$961$	−6.12067			−0.197441
$962$		0			0
$963$	−43.9033			−1.41477
$964$		0			0
$965$		−	16.3488i		−	0.526286i
$966$		0			0
$967$			2.01619i			0.0648364i	0.999474	+	0.0324182i	$0.0103208\pi$
$967$			2.01619i			0.0648364i	−0.999474	+	0.0324182i	$0.989679\pi$
$968$		0			0
$969$			3.48438i			0.111934i
$970$		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
$975$	−39.8431			−1.27600
$976$		0			0
$977$			49.2243i			1.57483i	0.616426	+	0.787413i	$0.288580\pi$
$977$			49.2243i			1.57483i	−0.616426	+	0.787413i	$0.711420\pi$
$978$		0			0
$979$	−27.2465			−0.870802
$980$		0			0
$981$		−	1.66541i		−	0.0531723i
$982$		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
$987$		−	15.6435i		−	0.497937i
$988$		0			0
$989$	44.8310			1.42554
$990$		0			0
$991$			54.7401i			1.73888i	0.494042	+	0.869438i	$0.335519\pi$
$991$			54.7401i			1.73888i	−0.494042	+	0.869438i	$0.664481\pi$
$992$		0			0
$993$	−14.4349			−0.458077
$994$		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
$999$			15.2284i			0.481804i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	656.2.d.f.81.3		6
4.3	odd	2		328.2.d.b.81.4	yes	6
8.3	odd	2		2624.2.d.o.2049.3		6
8.5	even	2		2624.2.d.n.2049.4		6
12.11	even	2		2952.2.j.c.2377.5		6
41.40	even	2	inner	656.2.d.f.81.4		6
164.163	odd	2		328.2.d.b.81.3	✓	6
328.163	odd	2		2624.2.d.o.2049.4		6
328.245	even	2		2624.2.d.n.2049.3		6
492.491	even	2		2952.2.j.c.2377.6		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
328.2.d.b.81.3	✓	6	164.163	odd	2
328.2.d.b.81.4	yes	6	4.3	odd	2
656.2.d.f.81.3		6	1.1	even	1	trivial
656.2.d.f.81.4		6	41.40	even	2	inner
2624.2.d.n.2049.3		6	328.245	even	2
2624.2.d.n.2049.4		6	8.5	even	2
2624.2.d.o.2049.3		6	8.3	odd	2
2624.2.d.o.2049.4		6	328.163	odd	2
2952.2.j.c.2377.5		6	12.11	even	2
2952.2.j.c.2377.6		6	492.491	even	2

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:	\( N \)	\(=\)	\( 656 = 2^{4} \cdot 41 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	656.d (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-0.784829i q^{3} -3.60388 q^{5} +4.31182i q^{7} +2.38404 q^{9} -2.04360i q^{11} -6.35542i q^{13} +2.82843i q^{15} -5.65685i q^{17} -0.784829i q^{19} +3.38404 q^{21} +4.98792 q^{23} +7.98792 q^{25} -4.22555i q^{27} -0.698564i q^{29} -4.98792 q^{31} -1.60388 q^{33} -15.5393i q^{35} -3.60388 q^{37} -4.98792 q^{39} +(-0.780167 - 6.35542i) q^{41} +8.98792 q^{43} -8.59179 q^{45} -4.62271i q^{47} -11.5918 q^{49} -4.43967 q^{51} +1.81897i q^{53} +7.36487i q^{55} -0.615957 q^{57} -5.42758 q^{59} +11.4276 q^{61} +10.2796i q^{63} +22.9041i q^{65} +10.6672i q^{67} -3.91466i q^{69} -12.7971i q^{71} -2.61596 q^{73} -6.26915i q^{75} +8.81163 q^{77} -4.62271i q^{79} +3.83579 q^{81} -0.439665 q^{83} +20.3866i q^{85} -0.548253 q^{87} -13.3326i q^{89} +27.4034 q^{91} +3.91466i q^{93} +2.82843i q^{95} +9.57152i q^{97} -4.87202i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 4 q^{5} - 6 q^{9} - 8 q^{23} + 10 q^{25} + 8 q^{31} + 8 q^{33} - 4 q^{37} + 8 q^{39} - 2 q^{41} + 16 q^{43} + 4 q^{45} - 14 q^{49} - 32 q^{51} - 24 q^{57} + 36 q^{61} - 36 q^{73} + 46 q^{81} - 8 q^{83}+ \cdots + 56 q^{91}+O(q^{100}) \) 6 * q - 4 * q^5 - 6 * q^9 - 8 * q^23 + 10 * q^25 + 8 * q^31 + 8 * q^33 - 4 * q^37 + 8 * q^39 - 2 * q^41 + 16 * q^43 + 4 * q^45 - 14 * q^49 - 32 * q^51 - 24 * q^57 + 36 * q^61 - 36 * q^73 + 46 * q^81 - 8 * q^83 + 40 * q^87 + 56 * q^91

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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