LMFDB - Embedded newform 960.2.b.b.671.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [960,2,Mod(671,960)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(960, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("960.671");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 960 = 2^{6} \cdot 3 \cdot 5 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	960.b (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$7.66563859404$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{8})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 1 $ x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			671.2
Root			$0.707107 - 0.707107i$ of defining polynomial
Character	$\chi$	$=$	960.671
Dual form			960.2.b.b.671.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.41421 + 1.00000i) q^{3} +1.00000 q^{5} -2.00000i q^{7} +(1.00000 - 2.82843i) q^{9} +O(q^{10})$	$q+(-1.41421 + 1.00000i) q^{3} +1.00000 q^{5} -2.00000i q^{7} +(1.00000 - 2.82843i) q^{9} -2.00000i q^{11} +2.82843i q^{13} +(-1.41421 + 1.00000i) q^{15} -2.82843i q^{17} -5.65685 q^{19} +(2.00000 + 2.82843i) q^{21} -8.48528 q^{23} +1.00000 q^{25} +(1.41421 + 5.00000i) q^{27} -2.00000 q^{29} -6.00000i q^{31} +(2.00000 + 2.82843i) q^{33} -2.00000i q^{35} -2.82843i q^{37} +(-2.82843 - 4.00000i) q^{39} -11.3137i q^{41} +8.48528 q^{43} +(1.00000 - 2.82843i) q^{45} -2.82843 q^{47} +3.00000 q^{49} +(2.82843 + 4.00000i) q^{51} -10.0000 q^{53} -2.00000i q^{55} +(8.00000 - 5.65685i) q^{57} -10.0000i q^{59} +(-5.65685 - 2.00000i) q^{63} +2.82843i q^{65} +2.82843 q^{67} +(12.0000 - 8.48528i) q^{69} +5.65685 q^{71} +10.0000 q^{73} +(-1.41421 + 1.00000i) q^{75} -4.00000 q^{77} +2.00000i q^{79} +(-7.00000 - 5.65685i) q^{81} +2.00000i q^{83} -2.82843i q^{85} +(2.82843 - 2.00000i) q^{87} -5.65685i q^{89} +5.65685 q^{91} +(6.00000 + 8.48528i) q^{93} -5.65685 q^{95} -14.0000 q^{97} +(-5.65685 - 2.00000i) q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{5} + 4 q^{9}+O(q^{10}) $ 4 * q + 4 * q^5 + 4 * q^9	$ 4 q + 4 q^{5} + 4 q^{9} + 8 q^{21} + 4 q^{25} - 8 q^{29} + 8 q^{33} + 4 q^{45} + 12 q^{49} - 40 q^{53} + 32 q^{57} + 48 q^{69} + 40 q^{73} - 16 q^{77} - 28 q^{81} + 24 q^{93} - 56 q^{97}+O(q^{100}) $ 4 * q + 4 * q^5 + 4 * q^9 + 8 * q^21 + 4 * q^25 - 8 * q^29 + 8 * q^33 + 4 * q^45 + 12 * q^49 - 40 * q^53 + 32 * q^57 + 48 * q^69 + 40 * q^73 - 16 * q^77 - 28 * q^81 + 24 * q^93 - 56 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/960\mathbb{Z}\right)^\times$.

$n$	$511$	$577$	$641$	$901$
$\chi(n)$	$-1$	$1$	$-1$	$-1$

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
$3$	−1.41421	+	1.00000i	−0.816497	+	0.577350i
$3$	−1.41421	+	1.00000i	−0.816497	+	0.577350i
$4$		0			0
$5$	1.00000			0.447214
$5$	1.00000			0.447214
$6$		0			0
$7$		−	2.00000i		−	0.755929i	−0.925820	−	0.377964i	$-0.876624\pi$
$7$		−	2.00000i		−	0.755929i	0.925820	−	0.377964i	$-0.123376\pi$
$8$		0			0
$9$	1.00000	−	2.82843i	0.333333	−	0.942809i
$10$		0			0
$11$		−	2.00000i		−	0.603023i	−0.953463	−	0.301511i	$-0.902509\pi$
$11$		−	2.00000i		−	0.603023i	0.953463	−	0.301511i	$-0.0974911\pi$
$12$		0			0
$13$			2.82843i			0.784465i	0.919866	+	0.392232i	$0.128297\pi$
$13$			2.82843i			0.784465i	−0.919866	+	0.392232i	$0.871703\pi$
$14$		0			0
$15$	−1.41421	+	1.00000i	−0.365148	+	0.258199i
$16$		0			0
$17$		−	2.82843i		−	0.685994i	−0.939336	−	0.342997i	$-0.888558\pi$
$17$		−	2.82843i		−	0.685994i	0.939336	−	0.342997i	$-0.111442\pi$
$18$		0			0
$19$	−5.65685			−1.29777			−0.648886	−	0.760886i	$-0.724765\pi$
$19$	−5.65685			−1.29777			−0.648886	+	0.760886i	$0.724765\pi$
$20$		0			0
$21$	2.00000	+	2.82843i	0.436436	+	0.617213i
$22$		0			0
$23$	−8.48528			−1.76930			−0.884652	−	0.466252i	$-0.845604\pi$
$23$	−8.48528			−1.76930			−0.884652	+	0.466252i	$0.845604\pi$
$24$		0			0
$25$	1.00000			0.200000
$26$		0			0
$27$	1.41421	+	5.00000i	0.272166	+	0.962250i
$28$		0			0
$29$	−2.00000			−0.371391			−0.185695	−	0.982607i	$-0.559454\pi$
$29$	−2.00000			−0.371391			−0.185695	+	0.982607i	$0.559454\pi$
$30$		0			0
$31$		−	6.00000i		−	1.07763i	−0.842424	−	0.538816i	$-0.818872\pi$
$31$		−	6.00000i		−	1.07763i	0.842424	−	0.538816i	$-0.181128\pi$
$32$		0			0
$33$	2.00000	+	2.82843i	0.348155	+	0.492366i
$34$		0			0
$35$		−	2.00000i		−	0.338062i
$36$		0			0
$37$		−	2.82843i		−	0.464991i	−0.972598	−	0.232495i	$-0.925311\pi$
$37$		−	2.82843i		−	0.464991i	0.972598	−	0.232495i	$-0.0746890\pi$
$38$		0			0
$39$	−2.82843	−	4.00000i	−0.452911	−	0.640513i
$40$		0			0
$41$		−	11.3137i		−	1.76690i	−0.468521	−	0.883452i	$-0.655213\pi$
$41$		−	11.3137i		−	1.76690i	0.468521	−	0.883452i	$-0.344787\pi$
$42$		0			0
$43$	8.48528			1.29399			0.646997	−	0.762493i	$-0.276025\pi$
$43$	8.48528			1.29399			0.646997	+	0.762493i	$0.276025\pi$
$44$		0			0
$45$	1.00000	−	2.82843i	0.149071	−	0.421637i
$46$		0			0
$47$	−2.82843			−0.412568			−0.206284	−	0.978492i	$-0.566137\pi$
$47$	−2.82843			−0.412568			−0.206284	+	0.978492i	$0.566137\pi$
$48$		0			0
$49$	3.00000			0.428571
$50$		0			0
$51$	2.82843	+	4.00000i	0.396059	+	0.560112i
$52$		0			0
$53$	−10.0000			−1.37361			−0.686803	−	0.726844i	$-0.740986\pi$
$53$	−10.0000			−1.37361			−0.686803	+	0.726844i	$0.740986\pi$
$54$		0			0
$55$		−	2.00000i		−	0.269680i
$56$		0			0
$57$	8.00000	−	5.65685i	1.05963	−	0.749269i
$58$		0			0
$59$		−	10.0000i		−	1.30189i	−0.759125	−	0.650945i	$-0.774373\pi$
$59$		−	10.0000i		−	1.30189i	0.759125	−	0.650945i	$-0.225627\pi$
$60$		0			0
$61$		0			0		1.00000			$0$
$61$		0			0		−1.00000			$\pi$
$62$		0			0
$63$	−5.65685	−	2.00000i	−0.712697	−	0.251976i
$64$		0			0
$65$			2.82843i			0.350823i
$66$		0			0
$67$	2.82843			0.345547			0.172774	−	0.984962i	$-0.444727\pi$
$67$	2.82843			0.345547			0.172774	+	0.984962i	$0.444727\pi$
$68$		0			0
$69$	12.0000	−	8.48528i	1.44463	−	1.02151i
$70$		0			0
$71$	5.65685			0.671345			0.335673	−	0.941979i	$-0.391036\pi$
$71$	5.65685			0.671345			0.335673	+	0.941979i	$0.391036\pi$
$72$		0			0
$73$	10.0000			1.17041			0.585206	−	0.810885i	$-0.301014\pi$
$73$	10.0000			1.17041			0.585206	+	0.810885i	$0.301014\pi$
$74$		0			0
$75$	−1.41421	+	1.00000i	−0.163299	+	0.115470i
$76$		0			0
$77$	−4.00000			−0.455842
$78$		0			0
$79$			2.00000i			0.225018i	0.993651	+	0.112509i	$0.0358886\pi$
$79$			2.00000i			0.225018i	−0.993651	+	0.112509i	$0.964111\pi$
$80$		0			0
$81$	−7.00000	−	5.65685i	−0.777778	−	0.628539i
$82$		0			0
$83$			2.00000i			0.219529i	0.993958	+	0.109764i	$0.0350096\pi$
$83$			2.00000i			0.219529i	−0.993958	+	0.109764i	$0.964990\pi$
$84$		0			0
$85$		−	2.82843i		−	0.306786i
$86$		0			0
$87$	2.82843	−	2.00000i	0.303239	−	0.214423i
$88$		0			0
$89$		−	5.65685i		−	0.599625i	−0.953998	−	0.299813i	$-0.903076\pi$
$89$		−	5.65685i		−	0.599625i	0.953998	−	0.299813i	$-0.0969242\pi$
$90$		0			0
$91$	5.65685			0.592999
$92$		0			0
$93$	6.00000	+	8.48528i	0.622171	+	0.879883i
$94$		0			0
$95$	−5.65685			−0.580381
$96$		0			0
$97$	−14.0000			−1.42148			−0.710742	−	0.703452i	$-0.751641\pi$
$97$	−14.0000			−1.42148			−0.710742	+	0.703452i	$0.751641\pi$
$98$		0			0
$99$	−5.65685	−	2.00000i	−0.568535	−	0.201008i
$100$		0			0
$101$	−18.0000			−1.79107			−0.895533	−	0.444994i	$-0.853206\pi$
$101$	−18.0000			−1.79107			−0.895533	+	0.444994i	$0.853206\pi$
$102$		0			0
$103$			6.00000i			0.591198i	0.955312	+	0.295599i	$0.0955191\pi$
$103$			6.00000i			0.591198i	−0.955312	+	0.295599i	$0.904481\pi$
$104$		0			0
$105$	2.00000	+	2.82843i	0.195180	+	0.276026i
$106$		0			0
$107$		−	14.0000i		−	1.35343i	−0.736245	−	0.676716i	$-0.763403\pi$
$107$		−	14.0000i		−	1.35343i	0.736245	−	0.676716i	$-0.236597\pi$
$108$		0			0
$109$		−	16.9706i		−	1.62549i	−0.582623	−	0.812743i	$-0.697974\pi$
$109$		−	16.9706i		−	1.62549i	0.582623	−	0.812743i	$-0.302026\pi$
$110$		0			0
$111$	2.82843	+	4.00000i	0.268462	+	0.379663i
$112$		0			0
$113$			14.1421i			1.33038i	0.746674	+	0.665190i	$0.231650\pi$
$113$			14.1421i			1.33038i	−0.746674	+	0.665190i	$0.768350\pi$
$114$		0			0
$115$	−8.48528			−0.791257
$116$		0			0
$117$	8.00000	+	2.82843i	0.739600	+	0.261488i
$118$		0			0
$119$	−5.65685			−0.518563
$120$		0			0
$121$	7.00000			0.636364
$122$		0			0
$123$	11.3137	+	16.0000i	1.02012	+	1.44267i
$124$		0			0
$125$	1.00000			0.0894427
$126$		0			0
$127$		−	2.00000i		−	0.177471i	−0.996055	−	0.0887357i	$-0.971717\pi$
$127$		−	2.00000i		−	0.177471i	0.996055	−	0.0887357i	$-0.0282826\pi$
$128$		0			0
$129$	−12.0000	+	8.48528i	−1.05654	+	0.747087i
$130$		0			0
$131$			6.00000i			0.524222i	0.965038	+	0.262111i	$0.0844187\pi$
$131$			6.00000i			0.524222i	−0.965038	+	0.262111i	$0.915581\pi$
$132$		0			0
$133$			11.3137i			0.981023i
$134$		0			0
$135$	1.41421	+	5.00000i	0.121716	+	0.430331i
$136$		0			0
$137$		−	2.82843i		−	0.241649i	−0.992674	−	0.120824i	$-0.961446\pi$
$137$		−	2.82843i		−	0.241649i	0.992674	−	0.120824i	$-0.0385538\pi$
$138$		0			0
$139$	−16.9706			−1.43942			−0.719712	−	0.694273i	$-0.755726\pi$
$139$	−16.9706			−1.43942			−0.719712	+	0.694273i	$0.755726\pi$
$140$		0			0
$141$	4.00000	−	2.82843i	0.336861	−	0.238197i
$142$		0			0
$143$	5.65685			0.473050
$144$		0			0
$145$	−2.00000			−0.166091
$146$		0			0
$147$	−4.24264	+	3.00000i	−0.349927	+	0.247436i
$148$		0			0
$149$	14.0000			1.14692			0.573462	−	0.819232i	$-0.305600\pi$
$149$	14.0000			1.14692			0.573462	+	0.819232i	$0.305600\pi$
$150$		0			0
$151$			2.00000i			0.162758i	0.996683	+	0.0813788i	$0.0259324\pi$
$151$			2.00000i			0.162758i	−0.996683	+	0.0813788i	$0.974068\pi$
$152$		0			0
$153$	−8.00000	−	2.82843i	−0.646762	−	0.228665i
$154$		0			0
$155$		−	6.00000i		−	0.481932i
$156$		0			0
$157$			19.7990i			1.58013i	0.613022	+	0.790066i	$0.289954\pi$
$157$			19.7990i			1.58013i	−0.613022	+	0.790066i	$0.710046\pi$
$158$		0			0
$159$	14.1421	−	10.0000i	1.12154	−	0.793052i
$160$		0			0
$161$			16.9706i			1.33747i
$162$		0			0
$163$	8.48528			0.664619			0.332309	−	0.943170i	$-0.392172\pi$
$163$	8.48528			0.664619			0.332309	+	0.943170i	$0.392172\pi$
$164$		0			0
$165$	2.00000	+	2.82843i	0.155700	+	0.220193i
$166$		0			0
$167$	19.7990			1.53209			0.766046	−	0.642786i	$-0.222221\pi$
$167$	19.7990			1.53209			0.766046	+	0.642786i	$0.222221\pi$
$168$		0			0
$169$	5.00000			0.384615
$170$		0			0
$171$	−5.65685	+	16.0000i	−0.432590	+	1.22355i
$172$		0			0
$173$	−18.0000			−1.36851			−0.684257	−	0.729241i	$-0.739873\pi$
$173$	−18.0000			−1.36851			−0.684257	+	0.729241i	$0.739873\pi$
$174$		0			0
$175$		−	2.00000i		−	0.151186i
$176$		0			0
$177$	10.0000	+	14.1421i	0.751646	+	1.06299i
$178$		0			0
$179$		−	18.0000i		−	1.34538i	−0.739923	−	0.672692i	$-0.765138\pi$
$179$		−	18.0000i		−	1.34538i	0.739923	−	0.672692i	$-0.234862\pi$
$180$		0			0
$181$			22.6274i			1.68188i	0.541126	+	0.840941i	$0.317998\pi$
$181$			22.6274i			1.68188i	−0.541126	+	0.840941i	$0.682002\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		−	2.82843i		−	0.207950i
$186$		0			0
$187$	−5.65685			−0.413670
$188$		0			0
$189$	10.0000	−	2.82843i	0.727393	−	0.205738i
$190$		0			0
$191$	5.65685			0.409316			0.204658	−	0.978834i	$-0.434392\pi$
$191$	5.65685			0.409316			0.204658	+	0.978834i	$0.434392\pi$
$192$		0			0
$193$	−14.0000			−1.00774			−0.503871	−	0.863779i	$-0.668091\pi$
$193$	−14.0000			−1.00774			−0.503871	+	0.863779i	$0.668091\pi$
$194$		0			0
$195$	−2.82843	−	4.00000i	−0.202548	−	0.286446i
$196$		0			0
$197$	−10.0000			−0.712470			−0.356235	−	0.934396i	$-0.615940\pi$
$197$	−10.0000			−0.712470			−0.356235	+	0.934396i	$0.615940\pi$
$198$		0			0
$199$			26.0000i			1.84309i	0.388270	+	0.921546i	$0.373073\pi$
$199$			26.0000i			1.84309i	−0.388270	+	0.921546i	$0.626927\pi$
$200$		0			0
$201$	−4.00000	+	2.82843i	−0.282138	+	0.199502i
$202$		0			0
$203$			4.00000i			0.280745i
$204$		0			0
$205$		−	11.3137i		−	0.790184i
$206$		0			0
$207$	−8.48528	+	24.0000i	−0.589768	+	1.66812i
$208$		0			0
$209$			11.3137i			0.782586i
$210$		0			0
$211$	11.3137			0.778868			0.389434	−	0.921054i	$-0.372671\pi$
$211$	11.3137			0.778868			0.389434	+	0.921054i	$0.372671\pi$
$212$		0			0
$213$	−8.00000	+	5.65685i	−0.548151	+	0.387601i
$214$		0			0
$215$	8.48528			0.578691
$216$		0			0
$217$	−12.0000			−0.814613
$218$		0			0
$219$	−14.1421	+	10.0000i	−0.955637	+	0.675737i
$220$		0			0
$221$	8.00000			0.538138
$222$		0			0
$223$			14.0000i			0.937509i	0.883328	+	0.468755i	$0.155297\pi$
$223$			14.0000i			0.937509i	−0.883328	+	0.468755i	$0.844703\pi$
$224$		0			0
$225$	1.00000	−	2.82843i	0.0666667	−	0.188562i
$226$		0			0
$227$			10.0000i			0.663723i	0.943328	+	0.331862i	$0.107677\pi$
$227$			10.0000i			0.663723i	−0.943328	+	0.331862i	$0.892323\pi$
$228$		0			0
$229$		−	28.2843i		−	1.86908i	−0.355862	−	0.934539i	$-0.615813\pi$
$229$		−	28.2843i		−	1.86908i	0.355862	−	0.934539i	$-0.384187\pi$
$230$		0			0
$231$	5.65685	−	4.00000i	0.372194	−	0.263181i
$232$		0			0
$233$		−	8.48528i		−	0.555889i	−0.960597	−	0.277945i	$-0.910347\pi$
$233$		−	8.48528i		−	0.555889i	0.960597	−	0.277945i	$-0.0896532\pi$
$234$		0			0
$235$	−2.82843			−0.184506
$236$		0			0
$237$	−2.00000	−	2.82843i	−0.129914	−	0.183726i
$238$		0			0
$239$	22.6274			1.46365			0.731823	−	0.681495i	$-0.238670\pi$
$239$	22.6274			1.46365			0.731823	+	0.681495i	$0.238670\pi$
$240$		0			0
$241$	−14.0000			−0.901819			−0.450910	−	0.892570i	$-0.648900\pi$
$241$	−14.0000			−0.901819			−0.450910	+	0.892570i	$0.648900\pi$
$242$		0			0
$243$	15.5563	+	1.00000i	0.997940	+	0.0641500i
$244$		0			0
$245$	3.00000			0.191663
$246$		0			0
$247$		−	16.0000i		−	1.01806i
$248$		0			0
$249$	−2.00000	−	2.82843i	−0.126745	−	0.179244i
$250$		0			0
$251$		−	18.0000i		−	1.13615i	−0.822977	−	0.568075i	$-0.807688\pi$
$251$		−	18.0000i		−	1.13615i	0.822977	−	0.568075i	$-0.192312\pi$
$252$		0			0
$253$			16.9706i			1.06693i
$254$		0			0
$255$	2.82843	+	4.00000i	0.177123	+	0.250490i
$256$		0			0
$257$			19.7990i			1.23503i	0.786560	+	0.617514i	$0.211860\pi$
$257$			19.7990i			1.23503i	−0.786560	+	0.617514i	$0.788140\pi$
$258$		0			0
$259$	−5.65685			−0.351500
$260$		0			0
$261$	−2.00000	+	5.65685i	−0.123797	+	0.350150i
$262$		0			0
$263$	14.1421			0.872041			0.436021	−	0.899937i	$-0.356387\pi$
$263$	14.1421			0.872041			0.436021	+	0.899937i	$0.356387\pi$
$264$		0			0
$265$	−10.0000			−0.614295
$266$		0			0
$267$	5.65685	+	8.00000i	0.346194	+	0.489592i
$268$		0			0
$269$	6.00000			0.365826			0.182913	−	0.983129i	$-0.441447\pi$
$269$	6.00000			0.365826			0.182913	+	0.983129i	$0.441447\pi$
$270$		0			0
$271$			10.0000i			0.607457i	0.952759	+	0.303728i	$0.0982315\pi$
$271$			10.0000i			0.607457i	−0.952759	+	0.303728i	$0.901768\pi$
$272$		0			0
$273$	−8.00000	+	5.65685i	−0.484182	+	0.342368i
$274$		0			0
$275$		−	2.00000i		−	0.120605i
$276$		0			0
$277$		−	25.4558i		−	1.52949i	−0.644331	−	0.764747i	$-0.722864\pi$
$277$		−	25.4558i		−	1.52949i	0.644331	−	0.764747i	$-0.277136\pi$
$278$		0			0
$279$	−16.9706	−	6.00000i	−1.01600	−	0.359211i
$280$		0			0
$281$			11.3137i			0.674919i	0.941340	+	0.337460i	$0.109568\pi$
$281$			11.3137i			0.674919i	−0.941340	+	0.337460i	$0.890432\pi$
$282$		0			0
$283$	8.48528			0.504398			0.252199	−	0.967675i	$-0.418846\pi$
$283$	8.48528			0.504398			0.252199	+	0.967675i	$0.418846\pi$
$284$		0			0
$285$	8.00000	−	5.65685i	0.473879	−	0.335083i
$286$		0			0
$287$	−22.6274			−1.33565
$288$		0			0
$289$	9.00000			0.529412
$290$		0			0
$291$	19.7990	−	14.0000i	1.16064	−	0.820695i
$292$		0			0
$293$	6.00000			0.350524			0.175262	−	0.984522i	$-0.443923\pi$
$293$	6.00000			0.350524			0.175262	+	0.984522i	$0.443923\pi$
$294$		0			0
$295$		−	10.0000i		−	0.582223i
$296$		0			0
$297$	10.0000	−	2.82843i	0.580259	−	0.164122i
$298$		0			0
$299$		−	24.0000i		−	1.38796i
$300$		0			0
$301$		−	16.9706i		−	0.978167i
$302$		0			0
$303$	25.4558	−	18.0000i	1.46240	−	1.03407i
$304$		0			0
$305$		0			0
$306$		0			0
$307$	14.1421			0.807134			0.403567	−	0.914950i	$-0.367770\pi$
$307$	14.1421			0.807134			0.403567	+	0.914950i	$0.367770\pi$
$308$		0			0
$309$	−6.00000	−	8.48528i	−0.341328	−	0.482711i
$310$		0			0
$311$	−28.2843			−1.60385			−0.801927	−	0.597422i	$-0.796192\pi$
$311$	−28.2843			−1.60385			−0.801927	+	0.597422i	$0.796192\pi$
$312$		0			0
$313$	10.0000			0.565233			0.282617	−	0.959233i	$-0.408798\pi$
$313$	10.0000			0.565233			0.282617	+	0.959233i	$0.408798\pi$
$314$		0			0
$315$	−5.65685	−	2.00000i	−0.318728	−	0.112687i
$316$		0			0
$317$	−18.0000			−1.01098			−0.505490	−	0.862832i	$-0.668688\pi$
$317$	−18.0000			−1.01098			−0.505490	+	0.862832i	$0.668688\pi$
$318$		0			0
$319$			4.00000i			0.223957i
$320$		0			0
$321$	14.0000	+	19.7990i	0.781404	+	1.10507i
$322$		0			0
$323$			16.0000i			0.890264i
$324$		0			0
$325$			2.82843i			0.156893i
$326$		0			0
$327$	16.9706	+	24.0000i	0.938474	+	1.32720i
$328$		0			0
$329$			5.65685i			0.311872i
$330$		0			0
$331$		0			0			−	1.00000i	$-0.5\pi$
$331$		0			0				1.00000i	$0.5\pi$
$332$		0			0
$333$	−8.00000	−	2.82843i	−0.438397	−	0.154997i
$334$		0			0
$335$	2.82843			0.154533
$336$		0			0
$337$	−14.0000			−0.762629			−0.381314	−	0.924445i	$-0.624528\pi$
$337$	−14.0000			−0.762629			−0.381314	+	0.924445i	$0.624528\pi$
$338$		0			0
$339$	−14.1421	−	20.0000i	−0.768095	−	1.08625i
$340$		0			0
$341$	−12.0000			−0.649836
$342$		0			0
$343$		−	20.0000i		−	1.07990i
$344$		0			0
$345$	12.0000	−	8.48528i	0.646058	−	0.456832i
$346$		0			0
$347$			18.0000i			0.966291i	0.875540	+	0.483145i	$0.160506\pi$
$347$			18.0000i			0.966291i	−0.875540	+	0.483145i	$0.839494\pi$
$348$		0			0
$349$			5.65685i			0.302804i	0.988472	+	0.151402i	$0.0483789\pi$
$349$			5.65685i			0.302804i	−0.988472	+	0.151402i	$0.951621\pi$
$350$		0			0
$351$	−14.1421	+	4.00000i	−0.754851	+	0.213504i
$352$		0			0
$353$		−	31.1127i		−	1.65596i	−0.560756	−	0.827981i	$-0.689490\pi$
$353$		−	31.1127i		−	1.65596i	0.560756	−	0.827981i	$-0.310510\pi$
$354$		0			0
$355$	5.65685			0.300235
$356$		0			0
$357$	8.00000	−	5.65685i	0.423405	−	0.299392i
$358$		0			0
$359$	−11.3137			−0.597115			−0.298557	−	0.954392i	$-0.596505\pi$
$359$	−11.3137			−0.597115			−0.298557	+	0.954392i	$0.596505\pi$
$360$		0			0
$361$	13.0000			0.684211
$362$		0			0
$363$	−9.89949	+	7.00000i	−0.519589	+	0.367405i
$364$		0			0
$365$	10.0000			0.523424
$366$		0			0
$367$		−	26.0000i		−	1.35719i	−0.734513	−	0.678594i	$-0.762589\pi$
$367$		−	26.0000i		−	1.35719i	0.734513	−	0.678594i	$-0.237411\pi$
$368$		0			0
$369$	−32.0000	−	11.3137i	−1.66585	−	0.588968i
$370$		0			0
$371$			20.0000i			1.03835i
$372$		0			0
$373$			14.1421i			0.732252i	0.930565	+	0.366126i	$0.119316\pi$
$373$			14.1421i			0.732252i	−0.930565	+	0.366126i	$0.880684\pi$
$374$		0			0
$375$	−1.41421	+	1.00000i	−0.0730297	+	0.0516398i
$376$		0			0
$377$		−	5.65685i		−	0.291343i
$378$		0			0
$379$	−16.9706			−0.871719			−0.435860	−	0.900015i	$-0.643556\pi$
$379$	−16.9706			−0.871719			−0.435860	+	0.900015i	$0.643556\pi$
$380$		0			0
$381$	2.00000	+	2.82843i	0.102463	+	0.144905i
$382$		0			0
$383$	14.1421			0.722629			0.361315	−	0.932444i	$-0.382328\pi$
$383$	14.1421			0.722629			0.361315	+	0.932444i	$0.382328\pi$
$384$		0			0
$385$	−4.00000			−0.203859
$386$		0			0
$387$	8.48528	−	24.0000i	0.431331	−	1.21999i
$388$		0			0
$389$	6.00000			0.304212			0.152106	−	0.988364i	$-0.451394\pi$
$389$	6.00000			0.304212			0.152106	+	0.988364i	$0.451394\pi$
$390$		0			0
$391$			24.0000i			1.21373i
$392$		0			0
$393$	−6.00000	−	8.48528i	−0.302660	−	0.428026i
$394$		0			0
$395$			2.00000i			0.100631i
$396$		0			0
$397$			8.48528i			0.425864i	0.977067	+	0.212932i	$0.0683013\pi$
$397$			8.48528i			0.425864i	−0.977067	+	0.212932i	$0.931699\pi$
$398$		0			0
$399$	−11.3137	−	16.0000i	−0.566394	−	0.801002i
$400$		0			0
$401$		−	11.3137i		−	0.564980i	−0.959270	−	0.282490i	$-0.908840\pi$
$401$		−	11.3137i		−	0.564980i	0.959270	−	0.282490i	$-0.0911603\pi$
$402$		0			0
$403$	16.9706			0.845364
$404$		0			0
$405$	−7.00000	−	5.65685i	−0.347833	−	0.281091i
$406$		0			0
$407$	−5.65685			−0.280400
$408$		0			0
$409$	26.0000			1.28562			0.642809	−	0.766027i	$-0.277769\pi$
$409$	26.0000			1.28562			0.642809	+	0.766027i	$0.277769\pi$
$410$		0			0
$411$	2.82843	+	4.00000i	0.139516	+	0.197305i
$412$		0			0
$413$	−20.0000			−0.984136
$414$		0			0
$415$			2.00000i			0.0981761i
$416$		0			0
$417$	24.0000	−	16.9706i	1.17529	−	0.831052i
$418$		0			0
$419$			30.0000i			1.46560i	0.680446	+	0.732798i	$0.261786\pi$
$419$			30.0000i			1.46560i	−0.680446	+	0.732798i	$0.738214\pi$
$420$		0			0
$421$			22.6274i			1.10279i	0.834243	+	0.551396i	$0.185905\pi$
$421$			22.6274i			1.10279i	−0.834243	+	0.551396i	$0.814095\pi$
$422$		0			0
$423$	−2.82843	+	8.00000i	−0.137523	+	0.388973i
$424$		0			0
$425$		−	2.82843i		−	0.137199i
$426$		0			0
$427$		0			0
$428$		0			0
$429$	−8.00000	+	5.65685i	−0.386244	+	0.273115i
$430$		0			0
$431$	−5.65685			−0.272481			−0.136241	−	0.990676i	$-0.543502\pi$
$431$	−5.65685			−0.272481			−0.136241	+	0.990676i	$0.543502\pi$
$432$		0			0
$433$	18.0000			0.865025			0.432512	−	0.901628i	$-0.357627\pi$
$433$	18.0000			0.865025			0.432512	+	0.901628i	$0.357627\pi$
$434$		0			0
$435$	2.82843	−	2.00000i	0.135613	−	0.0958927i
$436$		0			0
$437$	48.0000			2.29615
$438$		0			0
$439$		−	6.00000i		−	0.286364i	−0.989696	−	0.143182i	$-0.954267\pi$
$439$		−	6.00000i		−	0.286364i	0.989696	−	0.143182i	$-0.0457335\pi$
$440$		0			0
$441$	3.00000	−	8.48528i	0.142857	−	0.404061i
$442$		0			0
$443$		−	14.0000i		−	0.665160i	−0.943075	−	0.332580i	$-0.892081\pi$
$443$		−	14.0000i		−	0.665160i	0.943075	−	0.332580i	$-0.107919\pi$
$444$		0			0
$445$		−	5.65685i		−	0.268161i
$446$		0			0
$447$	−19.7990	+	14.0000i	−0.936460	+	0.662177i
$448$		0			0
$449$		−	5.65685i		−	0.266963i	−0.991051	−	0.133482i	$-0.957384\pi$
$449$		−	5.65685i		−	0.266963i	0.991051	−	0.133482i	$-0.0426157\pi$
$450$		0			0
$451$	−22.6274			−1.06548
$452$		0			0
$453$	−2.00000	−	2.82843i	−0.0939682	−	0.132891i
$454$		0			0
$455$	5.65685			0.265197
$456$		0			0
$457$	−6.00000			−0.280668			−0.140334	−	0.990104i	$-0.544818\pi$
$457$	−6.00000			−0.280668			−0.140334	+	0.990104i	$0.544818\pi$
$458$		0			0
$459$	14.1421	−	4.00000i	0.660098	−	0.186704i
$460$		0			0
$461$	14.0000			0.652045			0.326023	−	0.945362i	$-0.394291\pi$
$461$	14.0000			0.652045			0.326023	+	0.945362i	$0.394291\pi$
$462$		0			0
$463$			6.00000i			0.278844i	0.990233	+	0.139422i	$0.0445244\pi$
$463$			6.00000i			0.278844i	−0.990233	+	0.139422i	$0.955476\pi$
$464$		0			0
$465$	6.00000	+	8.48528i	0.278243	+	0.393496i
$466$		0			0
$467$			2.00000i			0.0925490i	0.998929	+	0.0462745i	$0.0147349\pi$
$467$			2.00000i			0.0925490i	−0.998929	+	0.0462745i	$0.985265\pi$
$468$		0			0
$469$		−	5.65685i		−	0.261209i
$470$		0			0
$471$	−19.7990	−	28.0000i	−0.912289	−	1.29017i
$472$		0			0
$473$		−	16.9706i		−	0.780307i
$474$		0			0
$475$	−5.65685			−0.259554
$476$		0			0
$477$	−10.0000	+	28.2843i	−0.457869	+	1.29505i
$478$		0			0
$479$	−22.6274			−1.03387			−0.516937	−	0.856024i	$-0.672928\pi$
$479$	−22.6274			−1.03387			−0.516937	+	0.856024i	$0.672928\pi$
$480$		0			0
$481$	8.00000			0.364769
$482$		0			0
$483$	−16.9706	−	24.0000i	−0.772187	−	1.09204i
$484$		0			0
$485$	−14.0000			−0.635707
$486$		0			0
$487$		−	26.0000i		−	1.17817i	−0.808070	−	0.589086i	$-0.799488\pi$
$487$		−	26.0000i		−	1.17817i	0.808070	−	0.589086i	$-0.200512\pi$
$488$		0			0
$489$	−12.0000	+	8.48528i	−0.542659	+	0.383718i
$490$		0			0
$491$			30.0000i			1.35388i	0.736038	+	0.676941i	$0.236695\pi$
$491$			30.0000i			1.35388i	−0.736038	+	0.676941i	$0.763305\pi$
$492$		0			0
$493$			5.65685i			0.254772i
$494$		0			0
$495$	−5.65685	−	2.00000i	−0.254257	−	0.0898933i
$496$		0			0
$497$		−	11.3137i		−	0.507489i
$498$		0			0
$499$	−16.9706			−0.759707			−0.379853	−	0.925047i	$-0.624026\pi$
$499$	−16.9706			−0.759707			−0.379853	+	0.925047i	$0.624026\pi$
$500$		0			0
$501$	−28.0000	+	19.7990i	−1.25095	+	0.884554i
$502$		0			0
$503$	−19.7990			−0.882793			−0.441397	−	0.897312i	$-0.645517\pi$
$503$	−19.7990			−0.882793			−0.441397	+	0.897312i	$0.645517\pi$
$504$		0			0
$505$	−18.0000			−0.800989
$506$		0			0
$507$	−7.07107	+	5.00000i	−0.314037	+	0.222058i
$508$		0			0
$509$	22.0000			0.975133			0.487566	−	0.873086i	$-0.337885\pi$
$509$	22.0000			0.975133			0.487566	+	0.873086i	$0.337885\pi$
$510$		0			0
$511$		−	20.0000i		−	0.884748i
$512$		0			0
$513$	−8.00000	−	28.2843i	−0.353209	−	1.24878i
$514$		0			0
$515$			6.00000i			0.264392i
$516$		0			0
$517$			5.65685i			0.248788i
$518$		0			0
$519$	25.4558	−	18.0000i	1.11739	−	0.790112i
$520$		0			0
$521$		0			0		1.00000			$0$
$521$		0			0		−1.00000			$\pi$
$522$		0			0
$523$	−2.82843			−0.123678			−0.0618392	−	0.998086i	$-0.519697\pi$
$523$	−2.82843			−0.123678			−0.0618392	+	0.998086i	$0.519697\pi$
$524$		0			0
$525$	2.00000	+	2.82843i	0.0872872	+	0.123443i
$526$		0			0
$527$	−16.9706			−0.739249
$528$		0			0
$529$	49.0000			2.13043
$530$		0			0
$531$	−28.2843	−	10.0000i	−1.22743	−	0.433963i
$532$		0			0
$533$	32.0000			1.38607
$534$		0			0
$535$		−	14.0000i		−	0.605273i
$536$		0			0
$537$	18.0000	+	25.4558i	0.776757	+	1.09850i
$538$		0			0
$539$		−	6.00000i		−	0.258438i
$540$		0			0
$541$		−	33.9411i		−	1.45924i	−0.683851	−	0.729621i	$-0.739696\pi$
$541$		−	33.9411i		−	1.45924i	0.683851	−	0.729621i	$-0.260304\pi$
$542$		0			0
$543$	−22.6274	−	32.0000i	−0.971035	−	1.37325i
$544$		0			0
$545$		−	16.9706i		−	0.726939i
$546$		0			0
$547$	−19.7990			−0.846544			−0.423272	−	0.906003i	$-0.639118\pi$
$547$	−19.7990			−0.846544			−0.423272	+	0.906003i	$0.639118\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	11.3137			0.481980
$552$		0			0
$553$	4.00000			0.170097
$554$		0			0
$555$	2.82843	+	4.00000i	0.120060	+	0.169791i
$556$		0			0
$557$	−2.00000			−0.0847427			−0.0423714	−	0.999102i	$-0.513491\pi$
$557$	−2.00000			−0.0847427			−0.0423714	+	0.999102i	$0.513491\pi$
$558$		0			0
$559$			24.0000i			1.01509i
$560$		0			0
$561$	8.00000	−	5.65685i	0.337760	−	0.238833i
$562$		0			0
$563$			42.0000i			1.77009i	0.465506	+	0.885044i	$0.345872\pi$
$563$			42.0000i			1.77009i	−0.465506	+	0.885044i	$0.654128\pi$
$564$		0			0
$565$			14.1421i			0.594964i
$566$		0			0
$567$	−11.3137	+	14.0000i	−0.475131	+	0.587945i
$568$		0			0
$569$		−	16.9706i		−	0.711443i	−0.934592	−	0.355722i	$-0.884235\pi$
$569$		−	16.9706i		−	0.711443i	0.934592	−	0.355722i	$-0.115765\pi$
$570$		0			0
$571$	22.6274			0.946928			0.473464	−	0.880813i	$-0.343003\pi$
$571$	22.6274			0.946928			0.473464	+	0.880813i	$0.343003\pi$
$572$		0			0
$573$	−8.00000	+	5.65685i	−0.334205	+	0.236318i
$574$		0			0
$575$	−8.48528			−0.353861
$576$		0			0
$577$	2.00000			0.0832611			0.0416305	−	0.999133i	$-0.486745\pi$
$577$	2.00000			0.0832611			0.0416305	+	0.999133i	$0.486745\pi$
$578$		0			0
$579$	19.7990	−	14.0000i	0.822818	−	0.581820i
$580$		0			0
$581$	4.00000			0.165948
$582$		0			0
$583$			20.0000i			0.828315i
$584$		0			0
$585$	8.00000	+	2.82843i	0.330759	+	0.116941i
$586$		0			0
$587$		−	30.0000i		−	1.23823i	−0.785299	−	0.619116i	$-0.787491\pi$
$587$		−	30.0000i		−	1.23823i	0.785299	−	0.619116i	$-0.212509\pi$
$588$		0			0
$589$			33.9411i			1.39852i
$590$		0			0
$591$	14.1421	−	10.0000i	0.581730	−	0.411345i
$592$		0			0
$593$			25.4558i			1.04535i	0.852533	+	0.522673i	$0.175065\pi$
$593$			25.4558i			1.04535i	−0.852533	+	0.522673i	$0.824935\pi$
$594$		0			0
$595$	−5.65685			−0.231908
$596$		0			0
$597$	−26.0000	−	36.7696i	−1.06411	−	1.50488i
$598$		0			0
$599$	−22.6274			−0.924531			−0.462266	−	0.886742i	$-0.652963\pi$
$599$	−22.6274			−0.924531			−0.462266	+	0.886742i	$0.652963\pi$
$600$		0			0
$601$	−14.0000			−0.571072			−0.285536	−	0.958368i	$-0.592172\pi$
$601$	−14.0000			−0.571072			−0.285536	+	0.958368i	$0.592172\pi$
$602$		0			0
$603$	2.82843	−	8.00000i	0.115182	−	0.325785i
$604$		0			0
$605$	7.00000			0.284590
$606$		0			0
$607$		−	26.0000i		−	1.05531i	−0.849460	−	0.527654i	$-0.823072\pi$
$607$		−	26.0000i		−	1.05531i	0.849460	−	0.527654i	$-0.176928\pi$
$608$		0			0
$609$	−4.00000	−	5.65685i	−0.162088	−	0.229227i
$610$		0			0
$611$		−	8.00000i		−	0.323645i
$612$		0			0
$613$		−	42.4264i		−	1.71359i	−0.515660	−	0.856793i	$-0.672453\pi$
$613$		−	42.4264i		−	1.71359i	0.515660	−	0.856793i	$-0.327547\pi$
$614$		0			0
$615$	11.3137	+	16.0000i	0.456213	+	0.645182i
$616$		0			0
$617$		−	2.82843i		−	0.113868i	−0.998378	−	0.0569341i	$-0.981868\pi$
$617$		−	2.82843i		−	0.113868i	0.998378	−	0.0569341i	$-0.0181325\pi$
$618$		0			0
$619$	39.5980			1.59158			0.795789	−	0.605575i	$-0.207057\pi$
$619$	39.5980			1.59158			0.795789	+	0.605575i	$0.207057\pi$
$620$		0			0
$621$	−12.0000	−	42.4264i	−0.481543	−	1.70251i
$622$		0			0
$623$	−11.3137			−0.453274
$624$		0			0
$625$	1.00000			0.0400000
$626$		0			0
$627$	−11.3137	−	16.0000i	−0.451826	−	0.638978i
$628$		0			0
$629$	−8.00000			−0.318981
$630$		0			0
$631$			2.00000i			0.0796187i	0.999207	+	0.0398094i	$0.0126751\pi$
$631$			2.00000i			0.0796187i	−0.999207	+	0.0398094i	$0.987325\pi$
$632$		0			0
$633$	−16.0000	+	11.3137i	−0.635943	+	0.449680i
$634$		0			0
$635$		−	2.00000i		−	0.0793676i
$636$		0			0
$637$			8.48528i			0.336199i
$638$		0			0
$639$	5.65685	−	16.0000i	0.223782	−	0.632950i
$640$		0			0
$641$		−	22.6274i		−	0.893729i	−0.894602	−	0.446865i	$-0.852541\pi$
$641$		−	22.6274i		−	0.893729i	0.894602	−	0.446865i	$-0.147459\pi$
$642$		0			0
$643$	−14.1421			−0.557711			−0.278856	−	0.960333i	$-0.589955\pi$
$643$	−14.1421			−0.557711			−0.278856	+	0.960333i	$0.589955\pi$
$644$		0			0
$645$	−12.0000	+	8.48528i	−0.472500	+	0.334108i
$646$		0			0
$647$	19.7990			0.778379			0.389189	−	0.921158i	$-0.372755\pi$
$647$	19.7990			0.778379			0.389189	+	0.921158i	$0.372755\pi$
$648$		0			0
$649$	−20.0000			−0.785069
$650$		0			0
$651$	16.9706	−	12.0000i	0.665129	−	0.470317i
$652$		0			0
$653$	46.0000			1.80012			0.900060	−	0.435767i	$-0.143523\pi$
$653$	46.0000			1.80012			0.900060	+	0.435767i	$0.143523\pi$
$654$		0			0
$655$			6.00000i			0.234439i
$656$		0			0
$657$	10.0000	−	28.2843i	0.390137	−	1.10347i
$658$		0			0
$659$			30.0000i			1.16863i	0.811525	+	0.584317i	$0.198638\pi$
$659$			30.0000i			1.16863i	−0.811525	+	0.584317i	$0.801362\pi$
$660$		0			0
$661$			11.3137i			0.440052i	0.975494	+	0.220026i	$0.0706143\pi$
$661$			11.3137i			0.440052i	−0.975494	+	0.220026i	$0.929386\pi$
$662$		0			0
$663$	−11.3137	+	8.00000i	−0.439388	+	0.310694i
$664$		0			0
$665$			11.3137i			0.438727i
$666$		0			0
$667$	16.9706			0.657103
$668$		0			0
$669$	−14.0000	−	19.7990i	−0.541271	−	0.765473i
$670$		0			0
$671$		0			0
$672$		0			0
$673$	18.0000			0.693849			0.346925	−	0.937893i	$-0.387226\pi$
$673$	18.0000			0.693849			0.346925	+	0.937893i	$0.387226\pi$
$674$		0			0
$675$	1.41421	+	5.00000i	0.0544331	+	0.192450i
$676$		0			0
$677$	6.00000			0.230599			0.115299	−	0.993331i	$-0.463217\pi$
$677$	6.00000			0.230599			0.115299	+	0.993331i	$0.463217\pi$
$678$		0			0
$679$			28.0000i			1.07454i
$680$		0			0
$681$	−10.0000	−	14.1421i	−0.383201	−	0.541928i
$682$		0			0
$683$		−	22.0000i		−	0.841807i	−0.907106	−	0.420903i	$-0.861713\pi$
$683$		−	22.0000i		−	0.841807i	0.907106	−	0.420903i	$-0.138287\pi$
$684$		0			0
$685$		−	2.82843i		−	0.108069i
$686$		0			0
$687$	28.2843	+	40.0000i	1.07911	+	1.52610i
$688$		0			0
$689$		−	28.2843i		−	1.07754i
$690$		0			0
$691$	11.3137			0.430394			0.215197	−	0.976571i	$-0.430961\pi$
$691$	11.3137			0.430394			0.215197	+	0.976571i	$0.430961\pi$
$692$		0			0
$693$	−4.00000	+	11.3137i	−0.151947	+	0.429772i
$694$		0			0
$695$	−16.9706			−0.643730
$696$		0			0
$697$	−32.0000			−1.21209
$698$		0			0
$699$	8.48528	+	12.0000i	0.320943	+	0.453882i
$700$		0			0
$701$	30.0000			1.13308			0.566542	−	0.824033i	$-0.308281\pi$
$701$	30.0000			1.13308			0.566542	+	0.824033i	$0.308281\pi$
$702$		0			0
$703$			16.0000i			0.603451i
$704$		0			0
$705$	4.00000	−	2.82843i	0.150649	−	0.106525i
$706$		0			0
$707$			36.0000i			1.35392i
$708$		0			0
$709$		−	5.65685i		−	0.212448i	−0.994342	−	0.106224i	$-0.966124\pi$
$709$		−	5.65685i		−	0.212448i	0.994342	−	0.106224i	$-0.0338760\pi$
$710$		0			0
$711$	5.65685	+	2.00000i	0.212149	+	0.0750059i
$712$		0			0
$713$			50.9117i			1.90666i
$714$		0			0
$715$	5.65685			0.211554
$716$		0			0
$717$	−32.0000	+	22.6274i	−1.19506	+	0.845036i
$718$		0			0
$719$	33.9411			1.26579			0.632895	−	0.774237i	$-0.281866\pi$
$719$	33.9411			1.26579			0.632895	+	0.774237i	$0.281866\pi$
$720$		0			0
$721$	12.0000			0.446903
$722$		0			0
$723$	19.7990	−	14.0000i	0.736332	−	0.520666i
$724$		0			0
$725$	−2.00000			−0.0742781
$726$		0			0
$727$		−	26.0000i		−	0.964287i	−0.876092	−	0.482143i	$-0.839858\pi$
$727$		−	26.0000i		−	0.964287i	0.876092	−	0.482143i	$-0.160142\pi$
$728$		0			0
$729$	−23.0000	+	14.1421i	−0.851852	+	0.523783i
$730$		0			0
$731$		−	24.0000i		−	0.887672i
$732$		0			0
$733$		−	42.4264i		−	1.56706i	−0.621357	−	0.783528i	$-0.713418\pi$
$733$		−	42.4264i		−	1.56706i	0.621357	−	0.783528i	$-0.286582\pi$
$734$		0			0
$735$	−4.24264	+	3.00000i	−0.156492	+	0.110657i
$736$		0			0
$737$		−	5.65685i		−	0.208373i
$738$		0			0
$739$	39.5980			1.45664			0.728318	−	0.685240i	$-0.240303\pi$
$739$	39.5980			1.45664			0.728318	+	0.685240i	$0.240303\pi$
$740$		0			0
$741$	16.0000	+	22.6274i	0.587775	+	0.831239i
$742$		0			0
$743$	2.82843			0.103765			0.0518825	−	0.998653i	$-0.483478\pi$
$743$	2.82843			0.103765			0.0518825	+	0.998653i	$0.483478\pi$
$744$		0			0
$745$	14.0000			0.512920
$746$		0			0
$747$	5.65685	+	2.00000i	0.206973	+	0.0731762i
$748$		0			0
$749$	−28.0000			−1.02310
$750$		0			0
$751$			10.0000i			0.364905i	0.983215	+	0.182453i	$0.0584036\pi$
$751$			10.0000i			0.364905i	−0.983215	+	0.182453i	$0.941596\pi$
$752$		0			0
$753$	18.0000	+	25.4558i	0.655956	+	0.927663i
$754$		0			0
$755$			2.00000i			0.0727875i
$756$		0			0
$757$		−	14.1421i		−	0.514005i	−0.966411	−	0.257002i	$-0.917265\pi$
$757$		−	14.1421i		−	0.514005i	0.966411	−	0.257002i	$-0.0827348\pi$
$758$		0			0
$759$	−16.9706	−	24.0000i	−0.615992	−	0.871145i
$760$		0			0
$761$			33.9411i			1.23036i	0.788385	+	0.615182i	$0.210918\pi$
$761$			33.9411i			1.23036i	−0.788385	+	0.615182i	$0.789082\pi$
$762$		0			0
$763$	−33.9411			−1.22875
$764$		0			0
$765$	−8.00000	−	2.82843i	−0.289241	−	0.102262i
$766$		0			0
$767$	28.2843			1.02129
$768$		0			0
$769$	18.0000			0.649097			0.324548	−	0.945869i	$-0.394788\pi$
$769$	18.0000			0.649097			0.324548	+	0.945869i	$0.394788\pi$
$770$		0			0
$771$	−19.7990	−	28.0000i	−0.713043	−	1.00840i
$772$		0			0
$773$	−26.0000			−0.935155			−0.467578	−	0.883952i	$-0.654873\pi$
$773$	−26.0000			−0.935155			−0.467578	+	0.883952i	$0.654873\pi$
$774$		0			0
$775$		−	6.00000i		−	0.215526i
$776$		0			0
$777$	8.00000	−	5.65685i	0.286998	−	0.202939i
$778$		0			0
$779$			64.0000i			2.29304i
$780$		0			0
$781$		−	11.3137i		−	0.404836i
$782$		0			0
$783$	−2.82843	−	10.0000i	−0.101080	−	0.357371i
$784$		0			0
$785$			19.7990i			0.706656i
$786$		0			0
$787$	−19.7990			−0.705758			−0.352879	−	0.935669i	$-0.614797\pi$
$787$	−19.7990			−0.705758			−0.352879	+	0.935669i	$0.614797\pi$
$788$		0			0
$789$	−20.0000	+	14.1421i	−0.712019	+	0.503473i
$790$		0			0
$791$	28.2843			1.00567
$792$		0			0
$793$		0			0
$794$		0			0
$795$	14.1421	−	10.0000i	0.501570	−	0.354663i
$796$		0			0
$797$	−2.00000			−0.0708436			−0.0354218	−	0.999372i	$-0.511277\pi$
$797$	−2.00000			−0.0708436			−0.0354218	+	0.999372i	$0.511277\pi$
$798$		0			0
$799$			8.00000i			0.283020i
$800$		0			0
$801$	−16.0000	−	5.65685i	−0.565332	−	0.199875i
$802$		0			0
$803$		−	20.0000i		−	0.705785i
$804$		0			0
$805$			16.9706i			0.598134i
$806$		0			0
$807$	−8.48528	+	6.00000i	−0.298696	+	0.211210i
$808$		0			0
$809$			28.2843i			0.994422i	0.867630	+	0.497211i	$0.165643\pi$
$809$			28.2843i			0.994422i	−0.867630	+	0.497211i	$0.834357\pi$
$810$		0			0
$811$	−22.6274			−0.794556			−0.397278	−	0.917698i	$-0.630045\pi$
$811$	−22.6274			−0.794556			−0.397278	+	0.917698i	$0.630045\pi$
$812$		0			0
$813$	−10.0000	−	14.1421i	−0.350715	−	0.495986i
$814$		0			0
$815$	8.48528			0.297226
$816$		0			0
$817$	−48.0000			−1.67931
$818$		0			0
$819$	5.65685	−	16.0000i	0.197666	−	0.559085i
$820$		0			0
$821$	−2.00000			−0.0698005			−0.0349002	−	0.999391i	$-0.511111\pi$
$821$	−2.00000			−0.0698005			−0.0349002	+	0.999391i	$0.511111\pi$
$822$		0			0
$823$		−	26.0000i		−	0.906303i	−0.891434	−	0.453152i	$-0.850300\pi$
$823$		−	26.0000i		−	0.906303i	0.891434	−	0.453152i	$-0.149700\pi$
$824$		0			0
$825$	2.00000	+	2.82843i	0.0696311	+	0.0984732i
$826$		0			0
$827$		−	6.00000i		−	0.208640i	−0.994544	−	0.104320i	$-0.966733\pi$
$827$		−	6.00000i		−	0.208640i	0.994544	−	0.104320i	$-0.0332667\pi$
$828$		0			0
$829$			39.5980i			1.37529i	0.726045	+	0.687647i	$0.241356\pi$
$829$			39.5980i			1.37529i	−0.726045	+	0.687647i	$0.758644\pi$
$830$		0			0
$831$	25.4558	+	36.0000i	0.883053	+	1.24883i
$832$		0			0
$833$		−	8.48528i		−	0.293998i
$834$		0			0
$835$	19.7990			0.685172
$836$		0			0
$837$	30.0000	−	8.48528i	1.03695	−	0.293294i
$838$		0			0
$839$	45.2548			1.56237			0.781185	−	0.624299i	$-0.214615\pi$
$839$	45.2548			1.56237			0.781185	+	0.624299i	$0.214615\pi$
$840$		0			0
$841$	−25.0000			−0.862069
$842$		0			0
$843$	−11.3137	−	16.0000i	−0.389665	−	0.551069i
$844$		0			0
$845$	5.00000			0.172005
$846$		0			0
$847$		−	14.0000i		−	0.481046i
$848$		0			0
$849$	−12.0000	+	8.48528i	−0.411839	+	0.291214i
$850$		0			0
$851$			24.0000i			0.822709i
$852$		0			0
$853$		−	19.7990i		−	0.677905i	−0.940804	−	0.338952i	$-0.889927\pi$
$853$		−	19.7990i		−	0.677905i	0.940804	−	0.338952i	$-0.110073\pi$
$854$		0			0
$855$	−5.65685	+	16.0000i	−0.193460	+	0.547188i
$856$		0			0
$857$			19.7990i			0.676321i	0.941089	+	0.338160i	$0.109805\pi$
$857$			19.7990i			0.676321i	−0.941089	+	0.338160i	$0.890195\pi$
$858$		0			0
$859$	16.9706			0.579028			0.289514	−	0.957174i	$-0.406506\pi$
$859$	16.9706			0.579028			0.289514	+	0.957174i	$0.406506\pi$
$860$		0			0
$861$	32.0000	−	22.6274i	1.09056	−	0.771140i
$862$		0			0
$863$	−19.7990			−0.673965			−0.336983	−	0.941511i	$-0.609406\pi$
$863$	−19.7990			−0.673965			−0.336983	+	0.941511i	$0.609406\pi$
$864$		0			0
$865$	−18.0000			−0.612018
$866$		0			0
$867$	−12.7279	+	9.00000i	−0.432263	+	0.305656i
$868$		0			0
$869$	4.00000			0.135691
$870$		0			0
$871$			8.00000i			0.271070i
$872$		0			0
$873$	−14.0000	+	39.5980i	−0.473828	+	1.34019i
$874$		0			0
$875$		−	2.00000i		−	0.0676123i
$876$		0			0
$877$			8.48528i			0.286528i	0.989685	+	0.143264i	$0.0457597\pi$
$877$			8.48528i			0.286528i	−0.989685	+	0.143264i	$0.954240\pi$
$878$		0			0
$879$	−8.48528	+	6.00000i	−0.286201	+	0.202375i
$880$		0			0
$881$		−	45.2548i		−	1.52467i	−0.647180	−	0.762337i	$-0.724052\pi$
$881$		−	45.2548i		−	1.52467i	0.647180	−	0.762337i	$-0.275948\pi$
$882$		0			0
$883$	−25.4558			−0.856657			−0.428329	−	0.903623i	$-0.640897\pi$
$883$	−25.4558			−0.856657			−0.428329	+	0.903623i	$0.640897\pi$
$884$		0			0
$885$	10.0000	+	14.1421i	0.336146	+	0.475383i
$886$		0			0
$887$	−14.1421			−0.474846			−0.237423	−	0.971406i	$-0.576303\pi$
$887$	−14.1421			−0.474846			−0.237423	+	0.971406i	$0.576303\pi$
$888$		0			0
$889$	−4.00000			−0.134156
$890$		0			0
$891$	−11.3137	+	14.0000i	−0.379023	+	0.469018i
$892$		0			0
$893$	16.0000			0.535420
$894$		0			0
$895$		−	18.0000i		−	0.601674i
$896$		0			0
$897$	24.0000	+	33.9411i	0.801337	+	1.13326i
$898$		0			0
$899$			12.0000i			0.400222i
$900$		0			0
$901$			28.2843i			0.942286i
$902$		0			0
$903$	16.9706	+	24.0000i	0.564745	+	0.798670i
$904$		0			0
$905$			22.6274i			0.752161i
$906$		0			0
$907$	14.1421			0.469582			0.234791	−	0.972046i	$-0.424559\pi$
$907$	14.1421			0.469582			0.234791	+	0.972046i	$0.424559\pi$
$908$		0			0
$909$	−18.0000	+	50.9117i	−0.597022	+	1.68863i
$910$		0			0
$911$	5.65685			0.187420			0.0937100	−	0.995600i	$-0.470127\pi$
$911$	5.65685			0.187420			0.0937100	+	0.995600i	$0.470127\pi$
$912$		0			0
$913$	4.00000			0.132381
$914$		0			0
$915$		0			0
$916$		0			0
$917$	12.0000			0.396275
$918$		0			0
$919$			26.0000i			0.857661i	0.903385	+	0.428830i	$0.141074\pi$
$919$			26.0000i			0.857661i	−0.903385	+	0.428830i	$0.858926\pi$
$920$		0			0
$921$	−20.0000	+	14.1421i	−0.659022	+	0.465999i
$922$		0			0
$923$			16.0000i			0.526646i
$924$		0			0
$925$		−	2.82843i		−	0.0929981i
$926$		0			0
$927$	16.9706	+	6.00000i	0.557386	+	0.197066i
$928$		0			0
$929$			39.5980i			1.29917i	0.760290	+	0.649584i	$0.225057\pi$
$929$			39.5980i			1.29917i	−0.760290	+	0.649584i	$0.774943\pi$
$930$		0			0
$931$	−16.9706			−0.556188
$932$		0			0
$933$	40.0000	−	28.2843i	1.30954	−	0.925985i
$934$		0			0
$935$	−5.65685			−0.184999
$936$		0			0
$937$	−38.0000			−1.24141			−0.620703	−	0.784046i	$-0.713153\pi$
$937$	−38.0000			−1.24141			−0.620703	+	0.784046i	$0.713153\pi$
$938$		0			0
$939$	−14.1421	+	10.0000i	−0.461511	+	0.326338i
$940$		0			0
$941$	54.0000			1.76035			0.880175	−	0.474650i	$-0.157425\pi$
$941$	54.0000			1.76035			0.880175	+	0.474650i	$0.157425\pi$
$942$		0			0
$943$			96.0000i			3.12619i
$944$		0			0
$945$	10.0000	−	2.82843i	0.325300	−	0.0920087i
$946$		0			0
$947$			2.00000i			0.0649913i	0.999472	+	0.0324956i	$0.0103455\pi$
$947$			2.00000i			0.0649913i	−0.999472	+	0.0324956i	$0.989654\pi$
$948$		0			0
$949$			28.2843i			0.918146i
$950$		0			0
$951$	25.4558	−	18.0000i	0.825462	−	0.583690i
$952$		0			0
$953$			14.1421i			0.458109i	0.973414	+	0.229054i	$0.0735634\pi$
$953$			14.1421i			0.458109i	−0.973414	+	0.229054i	$0.926437\pi$
$954$		0			0
$955$	5.65685			0.183052
$956$		0			0
$957$	−4.00000	−	5.65685i	−0.129302	−	0.182860i
$958$		0			0
$959$	−5.65685			−0.182669
$960$		0			0
$961$	−5.00000			−0.161290
$962$		0			0
$963$	−39.5980	−	14.0000i	−1.27603	−	0.451144i
$964$		0			0
$965$	−14.0000			−0.450676
$966$		0			0
$967$			54.0000i			1.73652i	0.496107	+	0.868261i	$0.334762\pi$
$967$			54.0000i			1.73652i	−0.496107	+	0.868261i	$0.665238\pi$
$968$		0			0
$969$	−16.0000	−	22.6274i	−0.513994	−	0.726897i
$970$		0			0
$971$		−	34.0000i		−	1.09111i	−0.838074	−	0.545556i	$-0.816319\pi$
$971$		−	34.0000i		−	1.09111i	0.838074	−	0.545556i	$-0.183681\pi$
$972$		0			0
$973$			33.9411i			1.08810i
$974$		0			0
$975$	−2.82843	−	4.00000i	−0.0905822	−	0.128103i
$976$		0			0
$977$		−	36.7696i		−	1.17636i	−0.808729	−	0.588181i	$-0.799844\pi$
$977$		−	36.7696i		−	1.17636i	0.808729	−	0.588181i	$-0.200156\pi$
$978$		0			0
$979$	−11.3137			−0.361588
$980$		0			0
$981$	−48.0000	−	16.9706i	−1.53252	−	0.541828i
$982$		0			0
$983$	−8.48528			−0.270638			−0.135319	−	0.990802i	$-0.543206\pi$
$983$	−8.48528			−0.270638			−0.135319	+	0.990802i	$0.543206\pi$
$984$		0			0
$985$	−10.0000			−0.318626
$986$		0			0
$987$	−5.65685	−	8.00000i	−0.180060	−	0.254643i
$988$		0			0
$989$	−72.0000			−2.28947
$990$		0			0
$991$		−	54.0000i		−	1.71537i	−0.514178	−	0.857683i	$-0.671903\pi$
$991$		−	54.0000i		−	1.71537i	0.514178	−	0.857683i	$-0.328097\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$			26.0000i			0.824255i
$996$		0			0
$997$		−	25.4558i		−	0.806195i	−0.915157	−	0.403097i	$-0.867934\pi$
$997$		−	25.4558i		−	0.806195i	0.915157	−	0.403097i	$-0.132066\pi$
$998$		0			0
$999$	14.1421	−	4.00000i	0.447437	−	0.126554i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	960.2.b.b.671.2	yes	4
3.2	odd	2		960.2.b.a.671.1	✓	4
4.3	odd	2	inner	960.2.b.b.671.3	yes	4
8.3	odd	2		960.2.b.a.671.2	yes	4
8.5	even	2		960.2.b.a.671.3	yes	4
12.11	even	2		960.2.b.a.671.4	yes	4
24.5	odd	2	inner	960.2.b.b.671.4	yes	4
24.11	even	2	inner	960.2.b.b.671.1	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
960.2.b.a.671.1	✓	4	3.2	odd	2
960.2.b.a.671.2	yes	4	8.3	odd	2
960.2.b.a.671.3	yes	4	8.5	even	2
960.2.b.a.671.4	yes	4	12.11	even	2
960.2.b.b.671.1	yes	4	24.11	even	2	inner
960.2.b.b.671.2	yes	4	1.1	even	1	trivial
960.2.b.b.671.3	yes	4	4.3	odd	2	inner
960.2.b.b.671.4	yes	4	24.5	odd	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 960 = 2^{6} \cdot 3 \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	960.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(-1.41421 + 1.00000i) q^{3} +1.00000 q^{5} -2.00000i q^{7} +(1.00000 - 2.82843i) q^{9} +O(q^{10})\)	\(q+(-1.41421 + 1.00000i) q^{3} +1.00000 q^{5} -2.00000i q^{7} +(1.00000 - 2.82843i) q^{9} -2.00000i q^{11} +2.82843i q^{13} +(-1.41421 + 1.00000i) q^{15} -2.82843i q^{17} -5.65685 q^{19} +(2.00000 + 2.82843i) q^{21} -8.48528 q^{23} +1.00000 q^{25} +(1.41421 + 5.00000i) q^{27} -2.00000 q^{29} -6.00000i q^{31} +(2.00000 + 2.82843i) q^{33} -2.00000i q^{35} -2.82843i q^{37} +(-2.82843 - 4.00000i) q^{39} -11.3137i q^{41} +8.48528 q^{43} +(1.00000 - 2.82843i) q^{45} -2.82843 q^{47} +3.00000 q^{49} +(2.82843 + 4.00000i) q^{51} -10.0000 q^{53} -2.00000i q^{55} +(8.00000 - 5.65685i) q^{57} -10.0000i q^{59} +(-5.65685 - 2.00000i) q^{63} +2.82843i q^{65} +2.82843 q^{67} +(12.0000 - 8.48528i) q^{69} +5.65685 q^{71} +10.0000 q^{73} +(-1.41421 + 1.00000i) q^{75} -4.00000 q^{77} +2.00000i q^{79} +(-7.00000 - 5.65685i) q^{81} +2.00000i q^{83} -2.82843i q^{85} +(2.82843 - 2.00000i) q^{87} -5.65685i q^{89} +5.65685 q^{91} +(6.00000 + 8.48528i) q^{93} -5.65685 q^{95} -14.0000 q^{97} +(-5.65685 - 2.00000i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{5} + 4 q^{9}+O(q^{10}) \) 4 * q + 4 * q^5 + 4 * q^9	\( 4 q + 4 q^{5} + 4 q^{9} + 8 q^{21} + 4 q^{25} - 8 q^{29} + 8 q^{33} + 4 q^{45} + 12 q^{49} - 40 q^{53} + 32 q^{57} + 48 q^{69} + 40 q^{73} - 16 q^{77} - 28 q^{81} + 24 q^{93} - 56 q^{97}+O(q^{100}) \) 4 * q + 4 * q^5 + 4 * q^9 + 8 * q^21 + 4 * q^25 - 8 * q^29 + 8 * q^33 + 4 * q^45 + 12 * q^49 - 40 * q^53 + 32 * q^57 + 48 * q^69 + 40 * q^73 - 16 * q^77 - 28 * q^81 + 24 * q^93 - 56 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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