LMFDB - Embedded newform 1040.2.k.e.961.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1040,2,Mod(961,1040)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1040, 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("1040.961");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1040 = 2^{4} \cdot 5 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1040.k (of order $2$, degree $1$, not 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:	$8.30444181021$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 2x^{7} + 2x^{6} + 6x^{5} + 36x^{4} - 52x^{3} + 50x^{2} + 140x + 196 $ x^8 - 2x^7 + 2x^6 + 6x^5 + 36x^4 - 52x^3 + 50x^2 + 140*x + 196
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	no (minimal twist has level 520)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			961.8
Root			$-0.772270 + 0.772270i$ of defining polynomial
Character	$\chi$	$=$	1040.961
Dual form			1040.2.k.e.961.7

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.80720 q^{3} +1.00000i q^{5} +3.54454i q^{7} +4.88037 q^{9} +O(q^{10})$	$q+2.80720 q^{3} +1.00000i q^{5} +3.54454i q^{7} +4.88037 q^{9} -1.26266i q^{11} +(2.80720 + 2.26266i) q^{13} +2.80720i q^{15} -5.42491 q^{17} -0.926831i q^{19} +9.95023i q^{21} +7.33252 q^{23} -1.00000 q^{25} +5.27857 q^{27} -7.49477 q^{29} -8.16225i q^{31} -3.54454i q^{33} -3.54454 q^{35} +7.15894i q^{37} +(7.88037 + 6.35174i) q^{39} +2.18949i q^{41} +7.14303 q^{43} +4.88037i q^{45} +7.30528i q^{47} -5.56376 q^{49} -15.2288 q^{51} -2.18949 q^{53} +1.26266 q^{55} -2.60180i q^{57} -4.68757i q^{59} +10.2594 q^{61} +17.2987i q^{63} +(-2.26266 + 2.80720i) q^{65} -9.83060i q^{67} +20.5838 q^{69} -14.1125i q^{71} -0.980781i q^{73} -2.80720 q^{75} +4.47555 q^{77} +12.1397 q^{79} +0.176891 q^{81} +1.15894i q^{83} -5.42491i q^{85} -21.0393 q^{87} -11.0891i q^{89} +(-8.02009 + 9.95023i) q^{91} -22.9131i q^{93} +0.926831 q^{95} -14.1397i q^{97} -6.16225i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 4 q^{3} + 16 q^{9}+O(q^{10}) $ 8 * q - 4 * q^3 + 16 * q^9	$ 8 q - 4 q^{3} + 16 q^{9} - 4 q^{13} - 4 q^{17} + 12 q^{23} - 8 q^{25} - 4 q^{27} + 16 q^{29} - 12 q^{35} + 40 q^{39} + 24 q^{43} - 32 q^{49} - 16 q^{51} - 4 q^{53} + 32 q^{61} - 8 q^{65} + 56 q^{69} + 4 q^{75} - 44 q^{77} + 24 q^{79} + 64 q^{81} - 76 q^{87} + 32 q^{91} + 4 q^{95}+O(q^{100}) $ 8 * q - 4 * q^3 + 16 * q^9 - 4 * q^13 - 4 * q^17 + 12 * q^23 - 8 * q^25 - 4 * q^27 + 16 * q^29 - 12 * q^35 + 40 * q^39 + 24 * q^43 - 32 * q^49 - 16 * q^51 - 4 * q^53 + 32 * q^61 - 8 * q^65 + 56 * q^69 + 4 * q^75 - 44 * q^77 + 24 * q^79 + 64 * q^81 - 76 * q^87 + 32 * q^91 + 4 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1040\mathbb{Z}\right)^\times$.

$n$	$261$	$417$	$561$	$911$
$\chi(n)$	$1$	$1$	$-1$	$1$

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
$3$	2.80720			1.62074			0.810369	−	0.585920i	$-0.199267\pi$
$3$	2.80720			1.62074			0.810369	+	0.585920i	$0.199267\pi$
$4$		0			0
$5$			1.00000i			0.447214i
$5$			1.00000i			0.447214i
$6$		0			0
$7$			3.54454i			1.33971i	0.742492	+	0.669855i	$0.233644\pi$
$7$			3.54454i			1.33971i	−0.742492	+	0.669855i	$0.766356\pi$
$8$		0			0
$9$	4.88037			1.62679
$10$		0			0
$11$		−	1.26266i		−	0.380706i	−0.981716	−	0.190353i	$-0.939037\pi$
$11$		−	1.26266i		−	0.380706i	0.981716	−	0.190353i	$-0.0609633\pi$
$12$		0			0
$13$	2.80720	+	2.26266i	0.778577	+	0.627549i
$13$	2.80720	+	2.26266i	0.778577	+	0.627549i
$14$		0			0
$15$			2.80720i			0.724816i
$16$		0			0
$17$	−5.42491			−1.31573			−0.657867	−	0.753134i	$-0.728541\pi$
$17$	−5.42491			−1.31573			−0.657867	+	0.753134i	$0.728541\pi$
$18$		0			0
$19$		−	0.926831i		−	0.212630i	−0.994333	−	0.106315i	$-0.966095\pi$
$19$		−	0.926831i		−	0.212630i	0.994333	−	0.106315i	$-0.0339051\pi$
$20$		0			0
$21$			9.95023i			2.17132i
$22$		0			0
$23$	7.33252			1.52894			0.764468	−	0.644662i	$-0.223002\pi$
$23$	7.33252			1.52894			0.764468	+	0.644662i	$0.223002\pi$
$24$		0			0
$25$	−1.00000			−0.200000
$26$		0			0
$27$	5.27857			1.01586
$28$		0			0
$29$	−7.49477			−1.39174			−0.695872	−	0.718166i	$-0.744982\pi$
$29$	−7.49477			−1.39174			−0.695872	+	0.718166i	$0.744982\pi$
$30$		0			0
$31$		−	8.16225i		−	1.46598i	−0.680238	−	0.732991i	$-0.738124\pi$
$31$		−	8.16225i		−	1.46598i	0.680238	−	0.732991i	$-0.261876\pi$
$32$		0			0
$33$		−	3.54454i		−	0.617025i
$34$		0			0
$35$	−3.54454			−0.599136
$36$		0			0
$37$			7.15894i			1.17692i	0.808526	+	0.588461i	$0.200266\pi$
$37$			7.15894i			1.17692i	−0.808526	+	0.588461i	$0.799734\pi$
$38$		0			0
$39$	7.88037	+	6.35174i	1.26187	+	1.01709i
$40$		0			0
$41$			2.18949i			0.341941i	0.985276	+	0.170971i	$0.0546903\pi$
$41$			2.18949i			0.341941i	−0.985276	+	0.170971i	$0.945310\pi$
$42$		0			0
$43$	7.14303			1.08930			0.544651	−	0.838663i	$-0.316662\pi$
$43$	7.14303			1.08930			0.544651	+	0.838663i	$0.316662\pi$
$44$		0			0
$45$			4.88037i			0.727522i
$46$		0			0
$47$			7.30528i			1.06558i	0.846246	+	0.532792i	$0.178857\pi$
$47$			7.30528i			1.06558i	−0.846246	+	0.532792i	$0.821143\pi$
$48$		0			0
$49$	−5.56376			−0.794823
$50$		0			0
$51$	−15.2288			−2.13246
$52$		0			0
$53$	−2.18949			−0.300750			−0.150375	−	0.988629i	$-0.548048\pi$
$53$	−2.18949			−0.300750			−0.150375	+	0.988629i	$0.548048\pi$
$54$		0			0
$55$	1.26266			0.170257
$56$		0			0
$57$		−	2.60180i		−	0.344617i
$58$		0			0
$59$		−	4.68757i		−	0.610269i	−0.952309	−	0.305135i	$-0.901299\pi$
$59$		−	4.68757i		−	0.610269i	0.952309	−	0.305135i	$-0.0987015\pi$
$60$		0			0
$61$	10.2594			1.31358			0.656788	−	0.754076i	$-0.271915\pi$
$61$	10.2594			1.31358			0.656788	+	0.754076i	$0.271915\pi$
$62$		0			0
$63$			17.2987i			2.17943i
$64$		0			0
$65$	−2.26266	+	2.80720i	−0.280648	+	0.348190i
$66$		0			0
$67$		−	9.83060i		−	1.20100i	−0.799625	−	0.600499i	$-0.794969\pi$
$67$		−	9.83060i		−	1.20100i	0.799625	−	0.600499i	$-0.205031\pi$
$68$		0			0
$69$	20.5838			2.47800
$70$		0			0
$71$		−	14.1125i		−	1.67484i	−0.546558	−	0.837421i	$-0.684062\pi$
$71$		−	14.1125i		−	1.67484i	0.546558	−	0.837421i	$-0.315938\pi$
$72$		0			0
$73$		−	0.980781i		−	0.114792i	−0.998351	−	0.0573959i	$-0.981720\pi$
$73$		−	0.980781i		−	0.114792i	0.998351	−	0.0573959i	$-0.0182797\pi$
$74$		0			0
$75$	−2.80720			−0.324147
$76$		0			0
$77$	4.47555			0.510036
$78$		0			0
$79$	12.1397			1.36583			0.682913	−	0.730500i	$-0.260713\pi$
$79$	12.1397			1.36583			0.682913	+	0.730500i	$0.260713\pi$
$80$		0			0
$81$	0.176891			0.0196546
$82$		0			0
$83$			1.15894i			0.127210i	0.997975	+	0.0636050i	$0.0202598\pi$
$83$			1.15894i			0.127210i	−0.997975	+	0.0636050i	$0.979740\pi$
$84$		0			0
$85$		−	5.42491i		−	0.588414i
$86$		0			0
$87$	−21.0393			−2.25565
$88$		0			0
$89$		−	11.0891i		−	1.17544i	−0.809064	−	0.587720i	$-0.800026\pi$
$89$		−	11.0891i		−	1.17544i	0.809064	−	0.587720i	$-0.199974\pi$
$90$		0			0
$91$	−8.02009	+	9.95023i	−0.840734	+	1.04307i
$92$		0			0
$93$		−	22.9131i		−	2.37597i
$94$		0			0
$95$	0.926831			0.0950908
$96$		0			0
$97$		−	14.1397i		−	1.43567i	−0.696213	−	0.717835i	$-0.745133\pi$
$97$		−	14.1397i		−	1.43567i	0.696213	−	0.717835i	$-0.254867\pi$
$98$		0			0
$99$		−	6.16225i		−	0.619329i
$100$		0			0
$101$	−11.6144			−1.15568			−0.577838	−	0.816152i	$-0.696103\pi$
$101$	−11.6144			−1.15568			−0.577838	+	0.816152i	$0.696103\pi$
$102$		0			0
$103$	−14.4103			−1.41989			−0.709943	−	0.704259i	$-0.751279\pi$
$103$	−14.4103			−1.41989			−0.709943	+	0.704259i	$0.751279\pi$
$104$		0			0
$105$	−9.95023			−0.971043
$106$		0			0
$107$	3.84651			0.371856			0.185928	−	0.982563i	$-0.440471\pi$
$107$	3.84651			0.371856			0.185928	+	0.982563i	$0.440471\pi$
$108$		0			0
$109$		−	2.91092i		−	0.278816i	−0.990235	−	0.139408i	$-0.955480\pi$
$109$		−	2.91092i		−	0.278816i	0.990235	−	0.139408i	$-0.0445199\pi$
$110$		0			0
$111$			20.0966i			1.90748i
$112$		0			0
$113$	−2.33583			−0.219736			−0.109868	−	0.993946i	$-0.535043\pi$
$113$	−2.33583			−0.219736			−0.109868	+	0.993946i	$0.535043\pi$
$114$		0			0
$115$			7.33252i			0.683761i
$116$		0			0
$117$	13.7002	+	11.0426i	1.26658	+	1.02089i
$118$		0			0
$119$		−	19.2288i		−	1.76270i
$120$		0			0
$121$	9.40569			0.855063
$122$		0			0
$123$			6.14634i			0.554197i
$124$		0			0
$125$		−	1.00000i		−	0.0894427i
$126$		0			0
$127$	6.05395			0.537201			0.268601	−	0.963252i	$-0.413439\pi$
$127$	6.05395			0.537201			0.268601	+	0.963252i	$0.413439\pi$
$128$		0			0
$129$	20.0519			1.76547
$130$		0			0
$131$	2.71010			0.236782			0.118391	−	0.992967i	$-0.462226\pi$
$131$	2.71010			0.236782			0.118391	+	0.992967i	$0.462226\pi$
$132$		0			0
$133$	3.28519			0.284862
$134$		0			0
$135$			5.27857i			0.454307i
$136$		0			0
$137$			11.2354i			0.959906i	0.877294	+	0.479953i	$0.159346\pi$
$137$			11.2354i			0.959906i	−0.877294	+	0.479953i	$0.840654\pi$
$138$		0			0
$139$	−1.28990			−0.109408			−0.0547041	−	0.998503i	$-0.517422\pi$
$139$	−1.28990			−0.109408			−0.0547041	+	0.998503i	$0.517422\pi$
$140$		0			0
$141$			20.5074i			1.72703i
$142$		0			0
$143$	2.85697	−	3.54454i	0.238912	−	0.296409i
$144$		0			0
$145$		−	7.49477i		−	0.622407i
$146$		0			0
$147$	−15.6186			−1.28820
$148$		0			0
$149$		−	16.9895i		−	1.39184i	−0.718121	−	0.695918i	$-0.754997\pi$
$149$		−	16.9895i		−	1.39184i	0.718121	−	0.695918i	$-0.245003\pi$
$150$		0			0
$151$			17.0168i			1.38481i	0.721511	+	0.692403i	$0.243448\pi$
$151$			17.0168i			1.38481i	−0.721511	+	0.692403i	$0.756552\pi$
$152$		0			0
$153$	−26.4755			−2.14042
$154$		0			0
$155$	8.16225			0.655607
$156$		0			0
$157$	−17.2354			−1.37554			−0.687768	−	0.725931i	$-0.741409\pi$
$157$	−17.2354			−1.37554			−0.687768	+	0.725931i	$0.741409\pi$
$158$		0			0
$159$	−6.14634			−0.487436
$160$		0			0
$161$			25.9904i			2.04833i
$162$		0			0
$163$		−	14.3944i		−	1.12745i	−0.825962	−	0.563726i	$-0.809367\pi$
$163$		−	14.3944i		−	1.12745i	0.825962	−	0.563726i	$-0.190633\pi$
$164$		0			0
$165$	3.54454			0.275942
$166$		0			0
$167$		−	10.7415i		−	0.831204i	−0.909547	−	0.415602i	$-0.863571\pi$
$167$		−	10.7415i		−	0.831204i	0.909547	−	0.415602i	$-0.136429\pi$
$168$		0			0
$169$	2.76074	+	12.7035i	0.212364	+	0.977191i
$170$		0			0
$171$		−	4.52328i		−	0.345904i
$172$		0			0
$173$	12.8996			0.980737			0.490369	−	0.871515i	$-0.336862\pi$
$173$	12.8996			0.980737			0.490369	+	0.871515i	$0.336862\pi$
$174$		0			0
$175$		−	3.54454i		−	0.267942i
$176$		0			0
$177$		−	13.1589i		−	0.989086i
$178$		0			0
$179$	−2.28606			−0.170868			−0.0854340	−	0.996344i	$-0.527228\pi$
$179$	−2.28606			−0.170868			−0.0854340	+	0.996344i	$0.527228\pi$
$180$		0			0
$181$	−18.5838			−1.38133			−0.690663	−	0.723177i	$-0.742681\pi$
$181$	−18.5838			−1.38133			−0.690663	+	0.723177i	$0.742681\pi$
$182$		0			0
$183$	28.8000			2.12896
$184$		0			0
$185$	−7.15894			−0.526336
$186$		0			0
$187$			6.84982i			0.500908i
$188$		0			0
$189$			18.7101i			1.36096i
$190$		0			0
$191$	−8.00000			−0.578860			−0.289430	−	0.957199i	$-0.593466\pi$
$191$	−8.00000			−0.578860			−0.289430	+	0.957199i	$0.593466\pi$
$192$		0			0
$193$			19.6144i			1.41188i	0.708274	+	0.705938i	$0.249474\pi$
$193$			19.6144i			1.41188i	−0.708274	+	0.705938i	$0.750526\pi$
$194$		0			0
$195$	−6.35174	+	7.88037i	−0.454857	+	0.564325i
$196$		0			0
$197$			18.7035i			1.33257i	0.745698	+	0.666284i	$0.232116\pi$
$197$			18.7035i			1.33257i	−0.745698	+	0.666284i	$0.767884\pi$
$198$		0			0
$199$	2.28606			0.162054			0.0810272	−	0.996712i	$-0.474180\pi$
$199$	2.28606			0.162054			0.0810272	+	0.996712i	$0.474180\pi$
$200$		0			0
$201$		−	27.5964i		−	1.94650i
$202$		0			0
$203$		−	26.5655i		−	1.86453i
$204$		0			0
$205$	−2.18949			−0.152921
$206$		0			0
$207$	35.7854			2.48726
$208$		0			0
$209$	−1.17027			−0.0809494
$210$		0			0
$211$	−13.6144			−0.937254			−0.468627	−	0.883396i	$-0.655251\pi$
$211$	−13.6144			−0.937254			−0.468627	+	0.883396i	$0.655251\pi$
$212$		0			0
$213$		−	39.6165i		−	2.71448i
$214$		0			0
$215$			7.14303i			0.487151i
$216$		0			0
$217$	28.9314			1.96399
$218$		0			0
$219$		−	2.75325i		−	0.186047i
$220$		0			0
$221$	−15.2288	−	12.2747i	−1.02440	−	0.825687i
$222$		0			0
$223$			25.8624i			1.73188i	0.500152	+	0.865938i	$0.333278\pi$
$223$			25.8624i			1.73188i	−0.500152	+	0.865938i	$0.666722\pi$
$224$		0			0
$225$	−4.88037			−0.325358
$226$		0			0
$227$		−	25.6776i		−	1.70428i	−0.523310	−	0.852142i	$-0.675303\pi$
$227$		−	25.6776i		−	1.70428i	0.523310	−	0.852142i	$-0.324697\pi$
$228$		0			0
$229$		−	14.8996i		−	0.984592i	−0.870428	−	0.492296i	$-0.836158\pi$
$229$		−	14.8996i		−	0.984592i	0.870428	−	0.492296i	$-0.163842\pi$
$230$		0			0
$231$	12.5638			0.826635
$232$		0			0
$233$	−10.8114			−0.708277			−0.354138	−	0.935193i	$-0.615226\pi$
$233$	−10.8114			−0.708277			−0.354138	+	0.935193i	$0.615226\pi$
$234$		0			0
$235$	−7.30528			−0.476544
$236$		0			0
$237$	34.0786			2.21364
$238$		0			0
$239$			22.1011i			1.42960i	0.699327	+	0.714802i	$0.253483\pi$
$239$			22.1011i			1.42960i	−0.699327	+	0.714802i	$0.746517\pi$
$240$		0			0
$241$			2.42875i			0.156450i	0.996936	+	0.0782249i	$0.0249252\pi$
$241$			2.42875i			0.156450i	−0.996936	+	0.0782249i	$0.975075\pi$
$242$		0			0
$243$	−15.3391			−0.984006
$244$		0			0
$245$		−	5.56376i		−	0.355455i
$246$		0			0
$247$	2.09710	−	2.60180i	0.133435	−	0.165549i
$248$		0			0
$249$			3.25337i			0.206174i
$250$		0			0
$251$	0.703477			0.0444031			0.0222015	−	0.999754i	$-0.492932\pi$
$251$	0.703477			0.0444031			0.0222015	+	0.999754i	$0.492932\pi$
$252$		0			0
$253$		−	9.25848i		−	0.582076i
$254$		0			0
$255$		−	15.2288i		−	0.953664i
$256$		0			0
$257$	17.5215			1.09296			0.546480	−	0.837472i	$-0.315968\pi$
$257$	17.5215			1.09296			0.546480	+	0.837472i	$0.315968\pi$
$258$		0			0
$259$	−25.3751			−1.57673
$260$		0			0
$261$	−36.5772			−2.26407
$262$		0			0
$263$	−1.24257			−0.0766203			−0.0383101	−	0.999266i	$-0.512197\pi$
$263$	−1.24257			−0.0766203			−0.0383101	+	0.999266i	$0.512197\pi$
$264$		0			0
$265$		−	2.18949i		−	0.134499i
$266$		0			0
$267$		−	31.1293i		−	1.90508i
$268$		0			0
$269$	−13.7926			−0.840947			−0.420473	−	0.907305i	$-0.638136\pi$
$269$	−13.7926			−0.840947			−0.420473	+	0.907305i	$0.638136\pi$
$270$		0			0
$271$			12.8771i			0.782226i	0.920343	+	0.391113i	$0.127910\pi$
$271$			12.8771i			0.782226i	−0.920343	+	0.391113i	$0.872090\pi$
$272$		0			0
$273$	−22.5140	+	27.9323i	−1.36261	+	1.69054i
$274$		0			0
$275$			1.26266i			0.0761413i
$276$		0			0
$277$	−28.2681			−1.69847			−0.849233	−	0.528018i	$-0.822935\pi$
$277$	−28.2681			−1.69847			−0.849233	+	0.528018i	$0.822935\pi$
$278$		0			0
$279$		−	39.8348i		−	2.38485i
$280$		0			0
$281$		−	23.3572i		−	1.39337i	−0.717376	−	0.696686i	$-0.754657\pi$
$281$		−	23.3572i		−	1.39337i	0.717376	−	0.696686i	$-0.245343\pi$
$282$		0			0
$283$	5.42073			0.322229			0.161114	−	0.986936i	$-0.448491\pi$
$283$	5.42073			0.322229			0.161114	+	0.986936i	$0.448491\pi$
$284$		0			0
$285$	2.60180			0.154117
$286$		0			0
$287$	−7.76074			−0.458102
$288$		0			0
$289$	12.4296			0.731154
$290$		0			0
$291$		−	39.6930i		−	2.32685i
$292$		0			0
$293$		−	14.0088i		−	0.818400i	−0.912445	−	0.409200i	$-0.865808\pi$
$293$		−	14.0088i		−	0.818400i	0.912445	−	0.409200i	$-0.134192\pi$
$294$		0			0
$295$	4.68757			0.272921
$296$		0			0
$297$		−	6.66504i		−	0.386745i
$298$		0			0
$299$	20.5838	+	16.5910i	1.19039	+	0.959482i
$300$		0			0
$301$			25.3187i			1.45935i
$302$		0			0
$303$	−32.6039			−1.87305
$304$		0			0
$305$			10.2594i			0.587449i
$306$		0			0
$307$		−	5.69088i		−	0.324796i	−0.986725	−	0.162398i	$-0.948077\pi$
$307$		−	5.69088i		−	0.324796i	0.986725	−	0.162398i	$-0.0519228\pi$
$308$		0			0
$309$	−40.4525			−2.30126
$310$		0			0
$311$	−16.4642			−0.933600			−0.466800	−	0.884363i	$-0.654593\pi$
$311$	−16.4642			−0.933600			−0.466800	+	0.884363i	$0.654593\pi$
$312$		0			0
$313$	24.7934			1.40141			0.700704	−	0.713452i	$-0.252870\pi$
$313$	24.7934			1.40141			0.700704	+	0.713452i	$0.252870\pi$
$314$		0			0
$315$	−17.2987			−0.974669
$316$		0			0
$317$		−	15.0192i		−	0.843563i	−0.906697	−	0.421782i	$-0.861405\pi$
$317$		−	15.0192i		−	0.843563i	0.906697	−	0.421782i	$-0.138595\pi$
$318$		0			0
$319$			9.46334i			0.529846i
$320$		0			0
$321$	10.7979			0.602681
$322$		0			0
$323$			5.02797i			0.279764i
$324$		0			0
$325$	−2.80720	−	2.26266i	−0.155715	−	0.125510i
$326$		0			0
$327$		−	8.17154i		−	0.451887i
$328$		0			0
$329$	−25.8938			−1.42757
$330$		0			0
$331$			5.96614i			0.327929i	0.986466	+	0.163964i	$0.0524282\pi$
$331$			5.96614i			0.327929i	−0.986466	+	0.163964i	$0.947572\pi$
$332$		0			0
$333$			34.9383i			1.91460i
$334$		0			0
$335$	9.83060			0.537103
$336$		0			0
$337$	6.09657			0.332101			0.166051	−	0.986117i	$-0.446899\pi$
$337$	6.09657			0.332101			0.166051	+	0.986117i	$0.446899\pi$
$338$		0			0
$339$	−6.55714			−0.356135
$340$		0			0
$341$	−10.3061			−0.558109
$342$		0			0
$343$			5.09082i			0.274878i
$344$		0			0
$345$			20.5838i			1.10820i
$346$		0			0
$347$	−7.04646			−0.378274			−0.189137	−	0.981951i	$-0.560569\pi$
$347$	−7.04646			−0.378274			−0.189137	+	0.981951i	$0.560569\pi$
$348$		0			0
$349$			6.26810i			0.335524i	0.985828	+	0.167762i	$0.0536540\pi$
$349$			6.26810i			0.335524i	−0.985828	+	0.167762i	$0.946346\pi$
$350$		0			0
$351$	14.8180	+	11.9436i	0.790926	+	0.637503i
$352$		0			0
$353$		−	8.58856i		−	0.457123i	−0.973529	−	0.228561i	$-0.926598\pi$
$353$		−	8.58856i		−	0.457123i	0.973529	−	0.228561i	$-0.0734022\pi$
$354$		0			0
$355$	14.1125			0.749012
$356$		0			0
$357$		−	53.9791i		−	2.85688i
$358$		0			0
$359$			2.11248i			0.111492i	0.998445	+	0.0557461i	$0.0177537\pi$
$359$			2.11248i			0.111492i	−0.998445	+	0.0557461i	$0.982246\pi$
$360$		0			0
$361$	18.1410			0.954789
$362$		0			0
$363$	26.4036			1.38583
$364$		0			0
$365$	0.980781			0.0513364
$366$		0			0
$367$	28.3718			1.48100			0.740499	−	0.672058i	$-0.234589\pi$
$367$	28.3718			1.48100			0.740499	+	0.672058i	$0.234589\pi$
$368$		0			0
$369$			10.6855i			0.556266i
$370$		0			0
$371$		−	7.76074i		−	0.402917i
$372$		0			0
$373$	8.10790			0.419811			0.209906	−	0.977722i	$-0.432684\pi$
$373$	8.10790			0.419811			0.209906	+	0.977722i	$0.432684\pi$
$374$		0			0
$375$		−	2.80720i		−	0.144963i
$376$		0			0
$377$	−21.0393	−	16.9581i	−1.08358	−	0.873387i
$378$		0			0
$379$			9.93432i			0.510292i	0.966903	+	0.255146i	$0.0821235\pi$
$379$			9.93432i			0.510292i	−0.966903	+	0.255146i	$0.917877\pi$
$380$		0			0
$381$	16.9946			0.870662
$382$		0			0
$383$		−	38.0874i		−	1.94617i	−0.230439	−	0.973087i	$-0.574016\pi$
$383$		−	38.0874i		−	1.94617i	0.230439	−	0.973087i	$-0.425984\pi$
$384$		0			0
$385$			4.47555i			0.228095i
$386$		0			0
$387$	34.8606			1.77206
$388$		0			0
$389$	−22.2794			−1.12961			−0.564806	−	0.825224i	$-0.691049\pi$
$389$	−22.2794			−1.12961			−0.564806	+	0.825224i	$0.691049\pi$
$390$		0			0
$391$	−39.7782			−2.01167
$392$		0			0
$393$	7.60778			0.383762
$394$		0			0
$395$			12.1397i			0.610816i
$396$		0			0
$397$		−	11.2987i		−	0.567063i	−0.958963	−	0.283532i	$-0.908494\pi$
$397$		−	11.2987i		−	0.567063i	0.958963	−	0.283532i	$-0.0915061\pi$
$398$		0			0
$399$	9.22218			0.461686
$400$		0			0
$401$			7.62102i			0.380575i	0.981728	+	0.190288i	$0.0609421\pi$
$401$			7.62102i			0.380575i	−0.981728	+	0.190288i	$0.939058\pi$
$402$		0			0
$403$	18.4684	−	22.9131i	0.919976	−	1.14138i
$404$		0			0
$405$			0.176891i			0.00878981i
$406$		0			0
$407$	9.03931			0.448062
$408$		0			0
$409$			15.9887i			0.790589i	0.918555	+	0.395294i	$0.129357\pi$
$409$			15.9887i			0.790589i	−0.918555	+	0.395294i	$0.870643\pi$
$410$		0			0
$411$			31.5401i			1.55576i
$412$		0			0
$413$	16.6153			0.817584
$414$		0			0
$415$	−1.15894			−0.0568900
$416$		0			0
$417$	−3.62102			−0.177322
$418$		0			0
$419$	−29.3751			−1.43507			−0.717535	−	0.696523i	$-0.754729\pi$
$419$	−29.3751			−1.43507			−0.717535	+	0.696523i	$0.754729\pi$
$420$		0			0
$421$			24.3179i			1.18518i	0.805504	+	0.592590i	$0.201895\pi$
$421$			24.3179i			1.18518i	−0.805504	+	0.592590i	$0.798105\pi$
$422$		0			0
$423$			35.6524i			1.73348i
$424$		0			0
$425$	5.42491			0.263147
$426$		0			0
$427$			36.3647i			1.75981i
$428$		0			0
$429$	8.02009	−	9.95023i	0.387213	−	0.480402i
$430$		0			0
$431$			3.83775i			0.184858i	0.995719	+	0.0924290i	$0.0294631\pi$
$431$			3.83775i			0.184858i	−0.995719	+	0.0924290i	$0.970537\pi$
$432$		0			0
$433$	4.87248			0.234157			0.117078	−	0.993123i	$-0.462647\pi$
$433$	4.87248			0.234157			0.117078	+	0.993123i	$0.462647\pi$
$434$		0			0
$435$		−	21.0393i		−	1.00876i
$436$		0			0
$437$		−	6.79601i		−	0.325097i
$438$		0			0
$439$	9.18200			0.438233			0.219117	−	0.975699i	$-0.429683\pi$
$439$	9.18200			0.438233			0.219117	+	0.975699i	$0.429683\pi$
$440$		0			0
$441$	−27.1532			−1.29301
$442$		0			0
$443$	−27.0322			−1.28434			−0.642168	−	0.766564i	$-0.721965\pi$
$443$	−27.0322			−1.28434			−0.642168	+	0.766564i	$0.721965\pi$
$444$		0			0
$445$	11.0891			0.525673
$446$		0			0
$447$		−	47.6930i		−	2.25580i
$448$		0			0
$449$			17.6576i			0.833311i	0.909064	+	0.416656i	$0.136798\pi$
$449$			17.6576i			0.833311i	−0.909064	+	0.416656i	$0.863202\pi$
$450$		0			0
$451$	2.76458			0.130179
$452$		0			0
$453$			47.7695i			2.24441i
$454$		0			0
$455$	−9.95023	−	8.02009i	−0.466474	−	0.375988i
$456$		0			0
$457$		−	28.3179i		−	1.32465i	−0.749215	−	0.662327i	$-0.769569\pi$
$457$		−	28.3179i		−	1.32465i	0.749215	−	0.662327i	$-0.230431\pi$
$458$		0			0
$459$	−28.6358			−1.33660
$460$		0			0
$461$		−	2.15105i		−	0.100185i	−0.998745	−	0.0500923i	$-0.984048\pi$
$461$		−	2.15105i		−	0.100185i	0.998745	−	0.0500923i	$-0.0159515\pi$
$462$		0			0
$463$		−	15.1523i		−	0.704188i	−0.935965	−	0.352094i	$-0.885470\pi$
$463$		−	15.1523i		−	0.704188i	0.935965	−	0.352094i	$-0.114530\pi$
$464$		0			0
$465$	22.9131			1.06257
$466$		0			0
$467$	24.5613			1.13656			0.568281	−	0.822834i	$-0.307609\pi$
$467$	24.5613			1.13656			0.568281	+	0.822834i	$0.307609\pi$
$468$		0			0
$469$	34.8449			1.60899
$470$		0			0
$471$	−48.3833			−2.22938
$472$		0			0
$473$		−	9.01922i		−	0.414704i
$474$		0			0
$475$			0.926831i			0.0425259i
$476$		0			0
$477$	−10.6855			−0.489257
$478$		0			0
$479$			23.6416i			1.08021i	0.841596	+	0.540107i	$0.181616\pi$
$479$			23.6416i			1.08021i	−0.841596	+	0.540107i	$0.818384\pi$
$480$		0			0
$481$	−16.1982	+	20.0966i	−0.738576	+	0.916325i
$482$		0			0
$483$			72.9602i			3.31981i
$484$		0			0
$485$	14.1397			0.642052
$486$		0			0
$487$			37.9026i			1.71753i	0.512369	+	0.858765i	$0.328768\pi$
$487$			37.9026i			1.71753i	−0.512369	+	0.858765i	$0.671232\pi$
$488$		0			0
$489$		−	40.4078i		−	1.82730i
$490$		0			0
$491$	32.6584			1.47385			0.736927	−	0.675972i	$-0.236276\pi$
$491$	32.6584			1.47385			0.736927	+	0.675972i	$0.236276\pi$
$492$		0			0
$493$	40.6584			1.83116
$494$		0			0
$495$	6.16225			0.276972
$496$		0			0
$497$	50.0222			2.24380
$498$		0			0
$499$			6.78049i			0.303537i	0.988416	+	0.151768i	$0.0484967\pi$
$499$			6.78049i			0.303537i	−0.988416	+	0.151768i	$0.951503\pi$
$500$		0			0
$501$		−	30.1536i		−	1.34716i
$502$		0			0
$503$	−24.6542			−1.09928			−0.549639	−	0.835402i	$-0.685235\pi$
$503$	−24.6542			−1.09928			−0.549639	+	0.835402i	$0.685235\pi$
$504$		0			0
$505$		−	11.6144i		−	0.516834i
$506$		0			0
$507$	7.74994	+	35.6612i	0.344187	+	1.58377i
$508$		0			0
$509$			23.6576i			1.04860i	0.851533	+	0.524301i	$0.175673\pi$
$509$			23.6576i			1.04860i	−0.851533	+	0.524301i	$0.824327\pi$
$510$		0			0
$511$	3.47642			0.153788
$512$		0			0
$513$		−	4.89234i		−	0.216002i
$514$		0			0
$515$		−	14.4103i		−	0.634992i
$516$		0			0
$517$	9.22408			0.405675
$518$		0			0
$519$	36.2117			1.58952
$520$		0			0
$521$	41.0414			1.79806			0.899029	−	0.437889i	$-0.144274\pi$
$521$	41.0414			1.79806			0.899029	+	0.437889i	$0.144274\pi$
$522$		0			0
$523$	−17.4722			−0.764008			−0.382004	−	0.924161i	$-0.624766\pi$
$523$	−17.4722			−0.764008			−0.382004	+	0.924161i	$0.624766\pi$
$524$		0			0
$525$		−	9.95023i		−	0.434264i
$526$		0			0
$527$			44.2794i			1.92884i
$528$		0			0
$529$	30.7658			1.33765
$530$		0			0
$531$		−	22.8771i		−	0.992780i
$532$		0			0
$533$	−4.95407	+	6.14634i	−0.214585	+	0.266227i
$534$		0			0
$535$			3.84651i			0.166299i
$536$		0			0
$537$	−6.41742			−0.276932
$538$		0			0
$539$			7.02514i			0.302594i
$540$		0			0
$541$		−	9.24013i		−	0.397264i	−0.980074	−	0.198632i	$-0.936350\pi$
$541$		−	9.24013i		−	0.397264i	0.980074	−	0.198632i	$-0.0636499\pi$
$542$		0			0
$543$	−52.1686			−2.23877
$544$		0			0
$545$	2.91092			0.124690
$546$		0			0
$547$	6.43955			0.275335			0.137668	−	0.990478i	$-0.456039\pi$
$547$	6.43955			0.275335			0.137668	+	0.990478i	$0.456039\pi$
$548$		0			0
$549$	50.0694			2.13691
$550$		0			0
$551$			6.94638i			0.295926i
$552$		0			0
$553$			43.0297i			1.82981i
$554$		0			0
$555$	−20.0966			−0.853052
$556$		0			0
$557$			23.8174i			1.00917i	0.863361	+	0.504587i	$0.168355\pi$
$557$			23.8174i			1.00917i	−0.863361	+	0.504587i	$0.831645\pi$
$558$		0			0
$559$	20.0519	+	16.1622i	0.848105	+	0.683590i
$560$		0			0
$561$			19.2288i			0.811840i
$562$		0			0
$563$	41.2216			1.73729			0.868643	−	0.495439i	$-0.164993\pi$
$563$	41.2216			1.73729			0.868643	+	0.495439i	$0.164993\pi$
$564$		0			0
$565$		−	2.33583i		−	0.0982691i
$566$		0			0
$567$			0.626999i			0.0263315i
$568$		0			0
$569$	−20.8699			−0.874912			−0.437456	−	0.899240i	$-0.644120\pi$
$569$	−20.8699			−0.874912			−0.437456	+	0.899240i	$0.644120\pi$
$570$		0			0
$571$	25.8394			1.08134			0.540672	−	0.841234i	$-0.318170\pi$
$571$	25.8394			1.08134			0.540672	+	0.841234i	$0.318170\pi$
$572$		0			0
$573$	−22.4576			−0.938180
$574$		0			0
$575$	−7.33252			−0.305787
$576$		0			0
$577$			35.0367i			1.45860i	0.684195	+	0.729299i	$0.260154\pi$
$577$			35.0367i			1.45860i	−0.684195	+	0.729299i	$0.739846\pi$
$578$		0			0
$579$			55.0615i			2.28828i
$580$		0			0
$581$	−4.10790			−0.170424
$582$		0			0
$583$			2.76458i			0.114497i
$584$		0			0
$585$	−11.0426	+	13.7002i	−0.456556	+	0.566432i
$586$		0			0
$587$			10.4939i			0.433130i	0.976268	+	0.216565i	$0.0694852\pi$
$587$			10.4939i			0.433130i	−0.976268	+	0.216565i	$0.930515\pi$
$588$		0			0
$589$	−7.56502			−0.311711
$590$		0			0
$591$			52.5044i			2.15974i
$592$		0			0
$593$			42.6892i			1.75303i	0.481371	+	0.876517i	$0.340139\pi$
$593$			42.6892i			1.75303i	−0.481371	+	0.876517i	$0.659861\pi$
$594$		0			0
$595$	19.2288			0.788304
$596$		0			0
$597$	6.41742			0.262647
$598$		0			0
$599$	8.85643			0.361864			0.180932	−	0.983496i	$-0.442089\pi$
$599$	8.85643			0.361864			0.180932	+	0.983496i	$0.442089\pi$
$600$		0			0
$601$	−36.8366			−1.50260			−0.751298	−	0.659963i	$-0.770572\pi$
$601$	−36.8366			−1.50260			−0.751298	+	0.659963i	$0.770572\pi$
$602$		0			0
$603$		−	47.9769i		−	1.95377i
$604$		0			0
$605$			9.40569i			0.382396i
$606$		0			0
$607$	23.1477			0.939538			0.469769	−	0.882789i	$-0.344337\pi$
$607$	23.1477			0.939538			0.469769	+	0.882789i	$0.344337\pi$
$608$		0			0
$609$		−	74.5746i		−	3.02192i
$610$		0			0
$611$	−16.5294	+	20.5074i	−0.668706	+	0.829639i
$612$		0			0
$613$		−	10.6717i		−	0.431024i	−0.976501	−	0.215512i	$-0.930858\pi$
$613$		−	10.6717i		−	0.431024i	0.976501	−	0.215512i	$-0.0691421\pi$
$614$		0			0
$615$	−6.14634			−0.247844
$616$		0			0
$617$			43.0363i			1.73258i	0.499544	+	0.866289i	$0.333501\pi$
$617$			43.0363i			1.73258i	−0.499544	+	0.866289i	$0.666499\pi$
$618$		0			0
$619$		−	10.1941i		−	0.409734i	−0.978790	−	0.204867i	$-0.934324\pi$
$619$		−	10.1941i		−	0.409734i	0.978790	−	0.204867i	$-0.0656762\pi$
$620$		0			0
$621$	38.7052			1.55319
$622$		0			0
$623$	39.3057			1.57475
$624$		0			0
$625$	1.00000			0.0400000
$626$		0			0
$627$	−3.28519			−0.131198
$628$		0			0
$629$		−	38.8366i		−	1.54852i
$630$		0			0
$631$			3.40536i			0.135565i	0.997700	+	0.0677826i	$0.0215924\pi$
$631$			3.40536i			0.135565i	−0.997700	+	0.0677826i	$0.978408\pi$
$632$		0			0
$633$	−38.2183			−1.51904
$634$		0			0
$635$			6.05395i			0.240244i
$636$		0			0
$637$	−15.6186	−	12.5889i	−0.618831	−	0.498790i
$638$		0			0
$639$		−	68.8741i		−	2.72462i
$640$		0			0
$641$	−6.09954			−0.240917			−0.120459	−	0.992718i	$-0.538437\pi$
$641$	−6.09954			−0.240917			−0.120459	+	0.992718i	$0.538437\pi$
$642$		0			0
$643$		−	42.8135i		−	1.68840i	−0.536028	−	0.844200i	$-0.680076\pi$
$643$		−	42.8135i		−	1.68840i	0.536028	−	0.844200i	$-0.319924\pi$
$644$		0			0
$645$			20.0519i			0.789543i
$646$		0			0
$647$	−39.1766			−1.54019			−0.770095	−	0.637929i	$-0.779791\pi$
$647$	−39.1766			−1.54019			−0.770095	+	0.637929i	$0.779791\pi$
$648$		0			0
$649$	−5.91881			−0.232333
$650$		0			0
$651$	81.2162			3.18311
$652$		0			0
$653$	24.6266			0.963713			0.481857	−	0.876250i	$-0.339963\pi$
$653$	24.6266			0.963713			0.481857	+	0.876250i	$0.339963\pi$
$654$		0			0
$655$			2.71010i			0.105892i
$656$		0			0
$657$		−	4.78657i		−	0.186742i
$658$		0			0
$659$	−3.55329			−0.138417			−0.0692083	−	0.997602i	$-0.522047\pi$
$659$	−3.55329			−0.138417			−0.0692083	+	0.997602i	$0.522047\pi$
$660$		0			0
$661$			29.6078i			1.15161i	0.817587	+	0.575805i	$0.195311\pi$
$661$			29.6078i			1.15161i	−0.817587	+	0.575805i	$0.804689\pi$
$662$		0			0
$663$	−42.7503	−	34.4576i	−1.66028	−	1.33822i
$664$		0			0
$665$			3.28519i			0.127394i
$666$		0			0
$667$	−54.9555			−2.12789
$668$		0			0
$669$			72.6010i			2.80692i
$670$		0			0
$671$		−	12.9541i		−	0.500087i
$672$		0			0
$673$	−27.3638			−1.05480			−0.527399	−	0.849618i	$-0.676833\pi$
$673$	−27.3638			−1.05480			−0.527399	+	0.849618i	$0.676833\pi$
$674$		0			0
$675$	−5.27857			−0.203172
$676$		0			0
$677$	−0.443730			−0.0170539			−0.00852697	−	0.999964i	$-0.502714\pi$
$677$	−0.443730			−0.0170539			−0.00852697	+	0.999964i	$0.502714\pi$
$678$		0			0
$679$	50.1188			1.92338
$680$		0			0
$681$		−	72.0823i		−	2.76220i
$682$		0			0
$683$			4.02199i			0.153897i	0.997035	+	0.0769486i	$0.0245177\pi$
$683$			4.02199i			0.153897i	−0.997035	+	0.0769486i	$0.975482\pi$
$684$		0			0
$685$	−11.2354			−0.429283
$686$		0			0
$687$		−	41.8261i		−	1.59577i
$688$		0			0
$689$	−6.14634	−	4.95407i	−0.234157	−	0.188735i
$690$		0			0
$691$		−	5.69506i		−	0.216650i	−0.994116	−	0.108325i	$-0.965451\pi$
$691$		−	5.69506i		−	0.216650i	0.994116	−	0.108325i	$-0.0345487\pi$
$692$		0			0
$693$	21.8423			0.829721
$694$		0			0
$695$		−	1.28990i		−	0.0489288i
$696$		0			0
$697$		−	11.8778i		−	0.449903i
$698$		0			0
$699$	−30.3497			−1.14793
$700$		0			0
$701$	−16.3856			−0.618876			−0.309438	−	0.950920i	$-0.600141\pi$
$701$	−16.3856			−0.618876			−0.309438	+	0.950920i	$0.600141\pi$
$702$		0			0
$703$	6.63512			0.250248
$704$		0			0
$705$	−20.5074			−0.772352
$706$		0			0
$707$		−	41.1677i		−	1.54827i
$708$		0			0
$709$			10.4672i			0.393104i	0.980493	+	0.196552i	$0.0629744\pi$
$709$			10.4672i			0.393104i	−0.980493	+	0.196552i	$0.937026\pi$
$710$		0			0
$711$	59.2463			2.22191
$712$		0			0
$713$		−	59.8498i		−	2.24139i
$714$		0			0
$715$	3.54454	+	2.85697i	0.132558	+	0.106845i
$716$		0			0
$717$			62.0423i			2.31701i
$718$		0			0
$719$	−5.96156			−0.222329			−0.111164	−	0.993802i	$-0.535458\pi$
$719$	−5.96156			−0.222329			−0.111164	+	0.993802i	$0.535458\pi$
$720$		0			0
$721$		−	51.0777i		−	1.90223i
$722$		0			0
$723$			6.81800i			0.253564i
$724$		0			0
$725$	7.49477			0.278349
$726$		0			0
$727$	−6.38142			−0.236674			−0.118337	−	0.992973i	$-0.537756\pi$
$727$	−6.38142			−0.236674			−0.118337	+	0.992973i	$0.537756\pi$
$728$		0			0
$729$	−43.5907			−1.61447
$730$		0			0
$731$	−38.7503			−1.43323
$732$		0			0
$733$		−	19.7314i		−	0.728798i	−0.931243	−	0.364399i	$-0.881274\pi$
$733$		−	19.7314i		−	0.728798i	0.931243	−	0.364399i	$-0.118726\pi$
$734$		0			0
$735$		−	15.6186i		−	0.576100i
$736$		0			0
$737$	−12.4127			−0.457228
$738$		0			0
$739$		−	20.7164i		−	0.762065i	−0.924562	−	0.381033i	$-0.875569\pi$
$739$		−	20.7164i		−	0.762065i	0.924562	−	0.381033i	$-0.124431\pi$
$740$		0			0
$741$	5.88699	−	7.30377i	0.216264	−	0.268311i
$742$		0			0
$743$			0.649668i			0.0238340i	0.999929	+	0.0119170i	$0.00379339\pi$
$743$			0.649668i			0.0238340i	−0.999929	+	0.0119170i	$0.996207\pi$
$744$		0			0
$745$	16.9895			0.622448
$746$		0			0
$747$			5.65604i			0.206944i
$748$		0			0
$749$			13.6341i			0.498179i
$750$		0			0
$751$	45.5550			1.66233			0.831163	−	0.556028i	$-0.187675\pi$
$751$	45.5550			1.66233			0.831163	+	0.556028i	$0.187675\pi$
$752$		0			0
$753$	1.97480			0.0719657
$754$		0			0
$755$	−17.0168			−0.619304
$756$		0			0
$757$	−35.7494			−1.29933			−0.649667	−	0.760219i	$-0.725092\pi$
$757$	−35.7494			−1.29933			−0.649667	+	0.760219i	$0.725092\pi$
$758$		0			0
$759$		−	25.9904i		−	0.943392i
$760$		0			0
$761$			38.2952i			1.38820i	0.719879	+	0.694100i	$0.244197\pi$
$761$			38.2952i			1.38820i	−0.719879	+	0.694100i	$0.755803\pi$
$762$		0			0
$763$	10.3179			0.373532
$764$		0			0
$765$		−	26.4755i		−	0.957225i
$766$		0			0
$767$	10.6064	−	13.1589i	0.382974	−	0.475142i
$768$		0			0
$769$			39.4548i			1.42278i	0.702799	+	0.711389i	$0.251933\pi$
$769$			39.4548i			1.42278i	−0.702799	+	0.711389i	$0.748067\pi$
$770$		0			0
$771$	49.1863			1.77140
$772$		0			0
$773$		−	36.3091i		−	1.30595i	−0.757380	−	0.652974i	$-0.773521\pi$
$773$		−	36.3091i		−	1.30595i	0.757380	−	0.652974i	$-0.226479\pi$
$774$		0			0
$775$			8.16225i			0.293197i
$776$		0			0
$777$	−71.2331			−2.55547
$778$		0			0
$779$	2.02929			0.0727068
$780$		0			0
$781$	−17.8193			−0.637623
$782$		0			0
$783$	−39.5616			−1.41382
$784$		0			0
$785$		−	17.2354i		−	0.615158i
$786$		0			0
$787$			39.1380i			1.39512i	0.716527	+	0.697560i	$0.245731\pi$
$787$			39.1380i			1.39512i	−0.716527	+	0.697560i	$0.754269\pi$
$788$		0			0
$789$	−3.48815			−0.124181
$790$		0			0
$791$		−	8.27944i		−	0.294383i
$792$		0			0
$793$	28.8000	+	23.2134i	1.02272	+	0.824333i
$794$		0			0
$795$		−	6.14634i		−	0.217988i
$796$		0			0
$797$	−19.4764			−0.689890			−0.344945	−	0.938623i	$-0.612102\pi$
$797$	−19.4764			−0.689890			−0.344945	+	0.938623i	$0.612102\pi$
$798$		0			0
$799$		−	39.6304i		−	1.40202i
$800$		0			0
$801$		−	54.1188i		−	1.91219i
$802$		0			0
$803$	−1.23839			−0.0437020
$804$		0			0
$805$	−25.9904			−0.916041
$806$		0			0
$807$	−38.7185			−1.36295
$808$		0			0
$809$	−21.4431			−0.753899			−0.376949	−	0.926234i	$-0.623027\pi$
$809$	−21.4431			−0.753899			−0.376949	+	0.926234i	$0.623027\pi$
$810$		0			0
$811$		−	5.88091i		−	0.206507i	−0.994655	−	0.103253i	$-0.967075\pi$
$811$		−	5.88091i		−	0.206507i	0.994655	−	0.103253i	$-0.0329252\pi$
$812$		0			0
$813$			36.1485i			1.26778i
$814$		0			0
$815$	14.3944			0.504212
$816$		0			0
$817$		−	6.62038i		−	0.231618i
$818$		0			0
$819$	−39.1410	+	48.5608i	−1.36770	+	1.69685i
$820$		0			0
$821$			12.6969i			0.443123i	0.975146	+	0.221562i	$0.0711154\pi$
$821$			12.6969i			0.443123i	−0.975146	+	0.221562i	$0.928885\pi$
$822$		0			0
$823$	20.7425			0.723036			0.361518	−	0.932365i	$-0.382259\pi$
$823$	20.7425			0.723036			0.361518	+	0.932365i	$0.382259\pi$
$824$		0			0
$825$			3.54454i			0.123405i
$826$		0			0
$827$			24.2480i			0.843186i	0.906785	+	0.421593i	$0.138529\pi$
$827$			24.2480i			0.843186i	−0.906785	+	0.421593i	$0.861471\pi$
$828$		0			0
$829$	42.2901			1.46880			0.734398	−	0.678719i	$-0.237465\pi$
$829$	42.2901			1.46880			0.734398	+	0.678719i	$0.237465\pi$
$830$		0			0
$831$	−79.3542			−2.75277
$832$		0			0
$833$	30.1829			1.04577
$834$		0			0
$835$	10.7415			0.371726
$836$		0			0
$837$		−	43.0850i		−	1.48924i
$838$		0			0
$839$		−	43.2238i		−	1.49225i	−0.665806	−	0.746125i	$-0.731912\pi$
$839$		−	43.2238i		−	1.49225i	0.665806	−	0.746125i	$-0.268088\pi$
$840$		0			0
$841$	27.1715			0.936950
$842$		0			0
$843$		−	65.5683i		−	2.25829i
$844$		0			0
$845$	−12.7035	+	2.76074i	−0.437013	+	0.0949722i
$846$		0			0
$847$			33.3388i			1.14554i
$848$		0			0
$849$	15.2171			0.522248
$850$		0			0
$851$			52.4931i			1.79944i
$852$		0			0
$853$			39.8708i			1.36515i	0.730816	+	0.682575i	$0.239140\pi$
$853$			39.8708i			1.36515i	−0.730816	+	0.682575i	$0.760860\pi$
$854$		0			0
$855$	4.52328			0.154693
$856$		0			0
$857$	14.4960			0.495175			0.247588	−	0.968866i	$-0.420362\pi$
$857$	14.4960			0.495175			0.247588	+	0.968866i	$0.420362\pi$
$858$		0			0
$859$	20.3563			0.694548			0.347274	−	0.937764i	$-0.387107\pi$
$859$	20.3563			0.694548			0.347274	+	0.937764i	$0.387107\pi$
$860$		0			0
$861$	−21.7859			−0.742463
$862$		0			0
$863$		−	16.2162i		−	0.552006i	−0.961157	−	0.276003i	$-0.910990\pi$
$863$		−	16.2162i		−	0.552006i	0.961157	−	0.276003i	$-0.0890099\pi$
$864$		0			0
$865$			12.8996i			0.438599i
$866$		0			0
$867$	34.8924			1.18501
$868$		0			0
$869$		−	15.3283i		−	0.519978i
$870$		0			0
$871$	22.2433	−	27.5964i	0.753685	−	0.935070i
$872$		0			0
$873$		−	69.0070i		−	2.33553i
$874$		0			0
$875$	3.54454			0.119827
$876$		0			0
$877$			25.8009i			0.871235i	0.900132	+	0.435617i	$0.143470\pi$
$877$			25.8009i			0.871235i	−0.900132	+	0.435617i	$0.856530\pi$
$878$		0			0
$879$		−	39.3254i		−	1.32641i
$880$		0			0
$881$	21.7274			0.732015			0.366008	−	0.930612i	$-0.380724\pi$
$881$	21.7274			0.732015			0.366008	+	0.930612i	$0.380724\pi$
$882$		0			0
$883$	−26.0247			−0.875800			−0.437900	−	0.899024i	$-0.644278\pi$
$883$	−26.0247			−0.875800			−0.437900	+	0.899024i	$0.644278\pi$
$884$		0			0
$885$	13.1589			0.442333
$886$		0			0
$887$	−7.38891			−0.248095			−0.124048	−	0.992276i	$-0.539588\pi$
$887$	−7.38891			−0.248095			−0.124048	+	0.992276i	$0.539588\pi$
$888$		0			0
$889$			21.4585i			0.719694i
$890$		0			0
$891$		−	0.223354i		−	0.00748263i
$892$		0			0
$893$	6.77076			0.226575
$894$		0			0
$895$		−	2.28606i		−	0.0764145i
$896$		0			0
$897$	57.7830	+	46.5742i	1.92932	+	1.55507i
$898$		0			0
$899$			61.1741i			2.04027i
$900$		0			0
$901$	11.8778			0.395706
$902$		0			0
$903$			71.0748i			2.36522i
$904$		0			0
$905$		−	18.5838i		−	0.617748i
$906$		0			0
$907$	−26.4684			−0.878869			−0.439434	−	0.898275i	$-0.644821\pi$
$907$	−26.4684			−0.878869			−0.439434	+	0.898275i	$0.644821\pi$
$908$		0			0
$909$	−56.6825			−1.88004
$910$		0			0
$911$	−40.0084			−1.32554			−0.662768	−	0.748825i	$-0.730618\pi$
$911$	−40.0084			−1.32554			−0.662768	+	0.748825i	$0.730618\pi$
$912$		0			0
$913$	1.46334			0.0484296
$914$		0			0
$915$			28.8000i			0.952100i
$916$		0			0
$917$			9.60604i			0.317219i
$918$		0			0
$919$	−45.1226			−1.48846			−0.744229	−	0.667924i	$-0.767183\pi$
$919$	−45.1226			−1.48846			−0.744229	+	0.667924i	$0.767183\pi$
$920$		0			0
$921$		−	15.9754i		−	0.526408i
$922$		0			0
$923$	31.9317	−	39.6165i	1.05105	−	1.30399i
$924$		0			0
$925$		−	7.15894i		−	0.235384i
$926$		0			0
$927$	−70.3274			−2.30985
$928$		0			0
$929$		−	15.1179i		−	0.496003i	−0.968760	−	0.248001i	$-0.920226\pi$
$929$		−	15.1179i		−	0.496003i	0.968760	−	0.248001i	$-0.0797738\pi$
$930$		0			0
$931$			5.15666i			0.169003i
$932$		0			0
$933$	−46.2183			−1.51312
$934$		0			0
$935$	−6.84982			−0.224013
$936$		0			0
$937$	−12.7195			−0.415529			−0.207764	−	0.978179i	$-0.566619\pi$
$937$	−12.7195			−0.415529			−0.207764	+	0.978179i	$0.566619\pi$
$938$		0			0
$939$	69.6001			2.27131
$940$		0			0
$941$		−	58.3309i		−	1.90153i	−0.309906	−	0.950767i	$-0.600297\pi$
$941$		−	58.3309i		−	1.90153i	0.309906	−	0.950767i	$-0.399703\pi$
$942$		0			0
$943$			16.0545i			0.522806i
$944$		0			0
$945$	−18.7101			−0.608639
$946$		0			0
$947$		−	34.9581i		−	1.13599i	−0.823033	−	0.567993i	$-0.807720\pi$
$947$		−	34.9581i		−	1.13599i	0.823033	−	0.567993i	$-0.192280\pi$
$948$		0			0
$949$	2.21918	−	2.75325i	0.0720375	−	0.0893742i
$950$		0			0
$951$		−	42.1619i		−	1.36719i
$952$		0			0
$953$	−3.86202			−0.125103			−0.0625515	−	0.998042i	$-0.519924\pi$
$953$	−3.86202			−0.125103			−0.0625515	+	0.998042i	$0.519924\pi$
$954$		0			0
$955$		−	8.00000i		−	0.258874i
$956$		0			0
$957$			26.5655i			0.858741i
$958$		0			0
$959$	−39.8244			−1.28600
$960$		0			0
$961$	−35.6223			−1.14911
$962$		0			0
$963$	18.7724			0.604931
$964$		0			0
$965$	−19.6144			−0.631410
$966$		0			0
$967$		−	4.56336i		−	0.146748i	−0.997305	−	0.0733739i	$-0.976623\pi$
$967$		−	4.56336i		−	0.146748i	0.997305	−	0.0733739i	$-0.0233766\pi$
$968$		0			0
$969$			14.1145i			0.453424i
$970$		0			0
$971$	24.1079			0.773659			0.386830	−	0.922151i	$-0.373570\pi$
$971$	24.1079			0.773659			0.386830	+	0.922151i	$0.373570\pi$
$972$		0			0
$973$		−	4.57212i		−	0.146575i
$974$		0			0
$975$	−7.88037	−	6.35174i	−0.252374	−	0.203418i
$976$		0			0
$977$		−	5.41318i		−	0.173183i	−0.996244	−	0.0865914i	$-0.972403\pi$
$977$		−	5.41318i		−	0.173183i	0.996244	−	0.0865914i	$-0.0275975\pi$
$978$		0			0
$979$	−14.0017			−0.447498
$980$		0			0
$981$		−	14.2064i		−	0.453575i
$982$		0			0
$983$		−	8.67205i		−	0.276596i	−0.990391	−	0.138298i	$-0.955837\pi$
$983$		−	8.67205i		−	0.276596i	0.990391	−	0.138298i	$-0.0441631\pi$
$984$		0			0
$985$	−18.7035			−0.595942
$986$		0			0
$987$	−72.6892			−2.31372
$988$		0			0
$989$	52.3764			1.66547
$990$		0			0
$991$	46.8816			1.48924			0.744622	−	0.667486i	$-0.232630\pi$
$991$	46.8816			1.48924			0.744622	+	0.667486i	$0.232630\pi$
$992$		0			0
$993$			16.7481i			0.531486i
$994$		0			0
$995$			2.28606i			0.0724729i
$996$		0			0
$997$	−11.4794			−0.363556			−0.181778	−	0.983340i	$-0.558185\pi$
$997$	−11.4794			−0.363556			−0.181778	+	0.983340i	$0.558185\pi$
$998$		0			0
$999$			37.7890i			1.19559i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1040.2.k.e.961.8		8
4.3	odd	2		520.2.k.b.441.2	yes	8
12.11	even	2		4680.2.g.k.2521.2		8
13.12	even	2	inner	1040.2.k.e.961.7		8
20.3	even	4		2600.2.f.e.649.2		8
20.7	even	4		2600.2.f.f.649.7		8
20.19	odd	2		2600.2.k.c.2001.8		8
52.31	even	4		6760.2.a.bd.1.1		4
52.47	even	4		6760.2.a.bc.1.1		4
52.51	odd	2		520.2.k.b.441.1	✓	8
156.155	even	2		4680.2.g.k.2521.7		8
260.103	even	4		2600.2.f.f.649.2		8
260.207	even	4		2600.2.f.e.649.7		8
260.259	odd	2		2600.2.k.c.2001.7		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
520.2.k.b.441.1	✓	8	52.51	odd	2
520.2.k.b.441.2	yes	8	4.3	odd	2
1040.2.k.e.961.7		8	13.12	even	2	inner
1040.2.k.e.961.8		8	1.1	even	1	trivial
2600.2.f.e.649.2		8	20.3	even	4
2600.2.f.e.649.7		8	260.207	even	4
2600.2.f.f.649.2		8	260.103	even	4
2600.2.f.f.649.7		8	20.7	even	4
2600.2.k.c.2001.7		8	260.259	odd	2
2600.2.k.c.2001.8		8	20.19	odd	2
4680.2.g.k.2521.2		8	12.11	even	2
4680.2.g.k.2521.7		8	156.155	even	2
6760.2.a.bc.1.1		4	52.47	even	4
6760.2.a.bd.1.1		4	52.31	even	4

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:	\( N \)	\(=\)	\( 1040 = 2^{4} \cdot 5 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1040.k (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+2.80720 q^{3} +1.00000i q^{5} +3.54454i q^{7} +4.88037 q^{9} +O(q^{10})\)	\(q+2.80720 q^{3} +1.00000i q^{5} +3.54454i q^{7} +4.88037 q^{9} -1.26266i q^{11} +(2.80720 + 2.26266i) q^{13} +2.80720i q^{15} -5.42491 q^{17} -0.926831i q^{19} +9.95023i q^{21} +7.33252 q^{23} -1.00000 q^{25} +5.27857 q^{27} -7.49477 q^{29} -8.16225i q^{31} -3.54454i q^{33} -3.54454 q^{35} +7.15894i q^{37} +(7.88037 + 6.35174i) q^{39} +2.18949i q^{41} +7.14303 q^{43} +4.88037i q^{45} +7.30528i q^{47} -5.56376 q^{49} -15.2288 q^{51} -2.18949 q^{53} +1.26266 q^{55} -2.60180i q^{57} -4.68757i q^{59} +10.2594 q^{61} +17.2987i q^{63} +(-2.26266 + 2.80720i) q^{65} -9.83060i q^{67} +20.5838 q^{69} -14.1125i q^{71} -0.980781i q^{73} -2.80720 q^{75} +4.47555 q^{77} +12.1397 q^{79} +0.176891 q^{81} +1.15894i q^{83} -5.42491i q^{85} -21.0393 q^{87} -11.0891i q^{89} +(-8.02009 + 9.95023i) q^{91} -22.9131i q^{93} +0.926831 q^{95} -14.1397i q^{97} -6.16225i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 4 q^{3} + 16 q^{9}+O(q^{10}) \) 8 * q - 4 * q^3 + 16 * q^9	\( 8 q - 4 q^{3} + 16 q^{9} - 4 q^{13} - 4 q^{17} + 12 q^{23} - 8 q^{25} - 4 q^{27} + 16 q^{29} - 12 q^{35} + 40 q^{39} + 24 q^{43} - 32 q^{49} - 16 q^{51} - 4 q^{53} + 32 q^{61} - 8 q^{65} + 56 q^{69} + 4 q^{75} - 44 q^{77} + 24 q^{79} + 64 q^{81} - 76 q^{87} + 32 q^{91} + 4 q^{95}+O(q^{100}) \) 8 * q - 4 * q^3 + 16 * q^9 - 4 * q^13 - 4 * q^17 + 12 * q^23 - 8 * q^25 - 4 * q^27 + 16 * q^29 - 12 * q^35 + 40 * q^39 + 24 * q^43 - 32 * q^49 - 16 * q^51 - 4 * q^53 + 32 * q^61 - 8 * q^65 + 56 * q^69 + 4 * q^75 - 44 * q^77 + 24 * q^79 + 64 * q^81 - 76 * q^87 + 32 * q^91 + 4 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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