LMFDB - Embedded newform 3120.2.l.p.1249.9

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [3120,2,Mod(1249,3120)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3120, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 1, 0])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("3120.1249"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 3120 = 2^{4} \cdot 3 \cdot 5 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3120.l (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [10,0,0,0,-2,0,0,0,-10,0,-10,0,0,0,0,0,0,0,16] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(19)] == traces)

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

Self dual:	no
Analytic conductor:	$24.9133254306$
Analytic rank:	$0$
Dimension:	$10$
Coefficient field:	10.0.13266844647424.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} + 16x^{8} + 90x^{6} + 208x^{4} + 169x^{2} + 16 $ x^10 + 16x^8 + 90x^6 + 208x^4 + 169x^2 + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{6} $
Twist minimal:	no (minimal twist has level 195)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1249.9
Root			$-1.77159i$ of defining polynomial
Character	$\chi$	$=$	3120.1249
Dual form			3120.2.l.p.1249.4

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{3} +(1.51036 - 1.64888i) q^{5} +0.437634i q^{7} -1.00000 q^{9} -5.73540 q^{11} -1.00000i q^{13} +(1.64888 + 1.51036i) q^{15} +3.98081i q^{17} +4.77531 q^{19} -0.437634 q^{21} +0.337673i q^{23} +(-0.437634 - 4.98081i) q^{25} -1.00000i q^{27} -1.72295 q^{29} +7.86166 q^{31} -5.73540i q^{33} +(0.721608 + 0.660985i) q^{35} +1.66233i q^{37} +1.00000 q^{39} +2.68304 q^{41} +2.27705i q^{43} +(-1.51036 + 1.64888i) q^{45} +11.7162i q^{47} +6.80848 q^{49} -3.98081 q^{51} +14.2993i q^{53} +(-8.66251 + 9.45701i) q^{55} +4.77531i q^{57} +11.4392 q^{59} -2.25786 q^{61} -0.437634i q^{63} +(-1.64888 - 1.51036i) q^{65} -2.17977i q^{67} -0.337673 q^{69} +13.6970 q^{71} +2.55410i q^{73} +(4.98081 - 0.437634i) q^{75} -2.51001i q^{77} +2.55144 q^{79} +1.00000 q^{81} -5.41158i q^{83} +(6.56389 + 6.01245i) q^{85} -1.72295i q^{87} +4.72448 q^{89} +0.437634 q^{91} +7.86166i q^{93} +(7.21243 - 7.87393i) q^{95} -14.7901i q^{97} +5.73540 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q - 2 q^{5} - 10 q^{9} - 10 q^{11} + 16 q^{19} + 10 q^{21} + 10 q^{25} - 16 q^{29} - 24 q^{31} - 12 q^{35} + 10 q^{39} + 10 q^{41} + 2 q^{45} - 44 q^{49} + 10 q^{51} - 2 q^{55} + 16 q^{59} + 26 q^{61}+ \cdots + 10 q^{99}+O(q^{100}) $ 10 * q - 2 * q^5 - 10 * q^9 - 10 * q^11 + 16 * q^19 + 10 * q^21 + 10 * q^25 - 16 * q^29 - 24 * q^31 - 12 * q^35 + 10 * q^39 + 10 * q^41 + 2 * q^45 - 44 * q^49 + 10 * q^51 - 2 * q^55 + 16 * q^59 + 26 * q^61 + 14 * q^69 - 10 * q^71 + 2 * q^79 + 10 * q^81 - 4 * q^85 - 38 * q^89 - 10 * q^91 + 48 * q^95 + 10 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3120\mathbb{Z}\right)^\times$.

$n$	$1951$	$2081$	$2341$	$2497$	$2641$
$\chi(n)$	$1$	$1$	$1$	$-1$	$1$

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
$3$			1.00000i			0.577350i
$3$			1.00000i			0.577350i
$4$		0			0
$5$	1.51036	−	1.64888i	0.675453	−	0.737403i
$5$	1.51036	−	1.64888i	0.675453	−	0.737403i
$6$		0			0
$7$			0.437634i			0.165410i	0.996574	+	0.0827051i	$0.0263560\pi$
$7$			0.437634i			0.165410i	−0.996574	+	0.0827051i	$0.973644\pi$
$8$		0			0
$9$	−1.00000			−0.333333
$10$		0			0
$11$	−5.73540			−1.72929			−0.864644	−	0.502385i	$-0.832456\pi$
$11$	−5.73540			−1.72929			−0.864644	+	0.502385i	$0.832456\pi$
$12$		0			0
$13$		−	1.00000i		−	0.277350i
$13$		−	1.00000i		−	0.277350i
$14$		0			0
$15$	1.64888	+	1.51036i	0.425740	+	0.389973i
$16$		0			0
$17$			3.98081i			0.965488i	0.875761	+	0.482744i	$0.160360\pi$
$17$			3.98081i			0.965488i	−0.875761	+	0.482744i	$0.839640\pi$
$18$		0			0
$19$	4.77531			1.09553			0.547765	−	0.836632i	$-0.315479\pi$
$19$	4.77531			1.09553			0.547765	+	0.836632i	$0.315479\pi$
$20$		0			0
$21$	−0.437634			−0.0954996
$22$		0			0
$23$			0.337673i			0.0704098i	0.999380	+	0.0352049i	$0.0112084\pi$
$23$			0.337673i			0.0704098i	−0.999380	+	0.0352049i	$0.988792\pi$
$24$		0			0
$25$	−0.437634	−	4.98081i	−0.0875268	−	0.996162i
$26$		0			0
$27$		−	1.00000i		−	0.192450i
$28$		0			0
$29$	−1.72295			−0.319944			−0.159972	−	0.987122i	$-0.551140\pi$
$29$	−1.72295			−0.319944			−0.159972	+	0.987122i	$0.551140\pi$
$30$		0			0
$31$	7.86166			1.41200			0.705998	−	0.708214i	$-0.250499\pi$
$31$	7.86166			1.41200			0.705998	+	0.708214i	$0.250499\pi$
$32$		0			0
$33$		−	5.73540i		−	0.998405i
$34$		0			0
$35$	0.721608	+	0.660985i	0.121974	+	0.111727i
$36$		0			0
$37$			1.66233i			0.273285i	0.990620	+	0.136642i	$0.0436311\pi$
$37$			1.66233i			0.273285i	−0.990620	+	0.136642i	$0.956369\pi$
$38$		0			0
$39$	1.00000			0.160128
$40$		0			0
$41$	2.68304			0.419021			0.209511	−	0.977806i	$-0.432813\pi$
$41$	2.68304			0.419021			0.209511	+	0.977806i	$0.432813\pi$
$42$		0			0
$43$			2.27705i			0.347247i	0.984812	+	0.173623i	$0.0555476\pi$
$43$			2.27705i			0.347247i	−0.984812	+	0.173623i	$0.944452\pi$
$44$		0			0
$45$	−1.51036	+	1.64888i	−0.225151	+	0.245801i
$46$		0			0
$47$			11.7162i			1.70899i	0.519464	+	0.854493i	$0.326132\pi$
$47$			11.7162i			1.70899i	−0.519464	+	0.854493i	$0.673868\pi$
$48$		0			0
$49$	6.80848			0.972639
$50$		0			0
$51$	−3.98081			−0.557425
$52$		0			0
$53$			14.2993i			1.96416i	0.188467	+	0.982080i	$0.439648\pi$
$53$			14.2993i			1.96416i	−0.188467	+	0.982080i	$0.560352\pi$
$54$		0			0
$55$	−8.66251	+	9.45701i	−1.16805	+	1.27518i
$56$		0			0
$57$			4.77531i			0.632505i
$58$		0			0
$59$	11.4392			1.48925			0.744626	−	0.667482i	$-0.232628\pi$
$59$	11.4392			1.48925			0.744626	+	0.667482i	$0.232628\pi$
$60$		0			0
$61$	−2.25786			−0.289089			−0.144545	−	0.989498i	$-0.546172\pi$
$61$	−2.25786			−0.289089			−0.144545	+	0.989498i	$0.546172\pi$
$62$		0			0
$63$		−	0.437634i		−	0.0551367i
$64$		0			0
$65$	−1.64888	−	1.51036i	−0.204519	−	0.187337i
$66$		0			0
$67$		−	2.17977i		−	0.266302i	−0.991096	−	0.133151i	$-0.957491\pi$
$67$		−	2.17977i		−	0.266302i	0.991096	−	0.133151i	$-0.0425095\pi$
$68$		0			0
$69$	−0.337673			−0.0406511
$70$		0			0
$71$	13.6970			1.62554			0.812769	−	0.582586i	$-0.197959\pi$
$71$	13.6970			1.62554			0.812769	+	0.582586i	$0.197959\pi$
$72$		0			0
$73$			2.55410i			0.298935i	0.988767	+	0.149467i	$0.0477559\pi$
$73$			2.55410i			0.298935i	−0.988767	+	0.149467i	$0.952244\pi$
$74$		0			0
$75$	4.98081	−	0.437634i	0.575134	−	0.0505336i
$76$		0			0
$77$		−	2.51001i		−	0.286042i
$78$		0			0
$79$	2.55144			0.287060			0.143530	−	0.989646i	$-0.454155\pi$
$79$	2.55144			0.287060			0.143530	+	0.989646i	$0.454155\pi$
$80$		0			0
$81$	1.00000			0.111111
$82$		0			0
$83$		−	5.41158i		−	0.593998i	−0.954878	−	0.296999i	$-0.904014\pi$
$83$		−	5.41158i		−	0.593998i	0.954878	−	0.296999i	$-0.0959857\pi$
$84$		0			0
$85$	6.56389	+	6.01245i	0.711954	+	0.652142i
$86$		0			0
$87$		−	1.72295i		−	0.184720i
$88$		0			0
$89$	4.72448			0.500794			0.250397	−	0.968143i	$-0.419439\pi$
$89$	4.72448			0.500794			0.250397	+	0.968143i	$0.419439\pi$
$90$		0			0
$91$	0.437634			0.0458765
$92$		0			0
$93$			7.86166i			0.815216i
$94$		0			0
$95$	7.21243	−	7.87393i	0.739979	−	0.807848i
$96$		0			0
$97$		−	14.7901i		−	1.50171i	−0.660468	−	0.750854i	$-0.729642\pi$
$97$		−	14.7901i		−	1.50171i	0.660468	−	0.750854i	$-0.270358\pi$
$98$		0			0
$99$	5.73540			0.576430
$100$		0			0
$101$	12.5955			1.25330			0.626651	−	0.779300i	$-0.284425\pi$
$101$	12.5955			1.25330			0.626651	+	0.779300i	$0.284425\pi$
$102$		0			0
$103$		−	6.23867i		−	0.614715i	−0.951594	−	0.307357i	$-0.900555\pi$
$103$		−	6.23867i		−	0.614715i	0.951594	−	0.307357i	$-0.0994447\pi$
$104$		0			0
$105$	−0.660985	+	0.721608i	−0.0645055	+	0.0704217i
$106$		0			0
$107$			1.10554i			0.106877i	0.998571	+	0.0534384i	$0.0170181\pi$
$107$			1.10554i			0.106877i	−0.998571	+	0.0534384i	$0.982982\pi$
$108$		0			0
$109$	−7.92019			−0.758616			−0.379308	−	0.925270i	$-0.623838\pi$
$109$	−7.92019			−0.758616			−0.379308	+	0.925270i	$0.623838\pi$
$110$		0			0
$111$	−1.66233			−0.157781
$112$		0			0
$113$			11.4432i			1.07649i	0.842789	+	0.538244i	$0.180912\pi$
$113$			11.4432i			1.07649i	−0.842789	+	0.538244i	$0.819088\pi$
$114$		0			0
$115$	0.556784	+	0.510008i	0.0519204	+	0.0475585i
$116$		0			0
$117$			1.00000i			0.0924500i
$118$		0			0
$119$	−1.74214			−0.159702
$120$		0			0
$121$	21.8948			1.99044
$122$		0			0
$123$			2.68304i			0.241922i
$124$		0			0
$125$	−8.87376	−	6.80120i	−0.793693	−	0.608318i
$126$		0			0
$127$		−	1.42937i		−	0.126836i	−0.997987	−	0.0634180i	$-0.979800\pi$
$127$		−	1.42937i		−	0.126836i	0.997987	−	0.0634180i	$-0.0202001\pi$
$128$		0			0
$129$	−2.27705			−0.200483
$130$		0			0
$131$	−17.6819			−1.54487			−0.772437	−	0.635092i	$-0.780962\pi$
$131$	−17.6819			−1.54487			−0.772437	+	0.635092i	$0.780962\pi$
$132$		0			0
$133$			2.08984i			0.181212i
$134$		0			0
$135$	−1.64888	−	1.51036i	−0.141913	−	0.129991i
$136$		0			0
$137$			10.7686i			0.920021i	0.887913	+	0.460011i	$0.152154\pi$
$137$			10.7686i			0.920021i	−0.887913	+	0.460011i	$0.847846\pi$
$138$		0			0
$139$	3.80848			0.323031			0.161515	−	0.986870i	$-0.448362\pi$
$139$	3.80848			0.323031			0.161515	+	0.986870i	$0.448362\pi$
$140$		0			0
$141$	−11.7162			−0.986683
$142$		0			0
$143$			5.73540i			0.479618i
$144$		0			0
$145$	−2.60227	+	2.84094i	−0.216107	+	0.235928i
$146$		0			0
$147$			6.80848i			0.561554i
$148$		0			0
$149$	16.3649			1.34067			0.670334	−	0.742060i	$-0.266151\pi$
$149$	16.3649			1.34067			0.670334	+	0.742060i	$0.266151\pi$
$150$		0			0
$151$	−20.3774			−1.65829			−0.829144	−	0.559035i	$-0.811172\pi$
$151$	−20.3774			−1.65829			−0.829144	+	0.559035i	$0.811172\pi$
$152$		0			0
$153$		−	3.98081i		−	0.321829i
$154$		0			0
$155$	11.8739	−	12.9630i	0.953737	−	1.04121i
$156$		0			0
$157$		−	13.1911i		−	1.05276i	−0.850249	−	0.526381i	$-0.823549\pi$
$157$		−	13.1911i		−	1.05276i	0.850249	−	0.526381i	$-0.176451\pi$
$158$		0			0
$159$	−14.2993			−1.13401
$160$		0			0
$161$	−0.147777			−0.0116465
$162$		0			0
$163$			10.3578i			0.811287i	0.914031	+	0.405644i	$0.132953\pi$
$163$			10.3578i			0.811287i	−0.914031	+	0.405644i	$0.867047\pi$
$164$		0			0
$165$	−9.45701	−	8.66251i	−0.736227	−	0.674376i
$166$		0			0
$167$			10.6713i			0.825769i	0.910783	+	0.412885i	$0.135479\pi$
$167$			10.6713i			0.825769i	−0.910783	+	0.412885i	$0.864521\pi$
$168$		0			0
$169$	−1.00000			−0.0769231
$170$		0			0
$171$	−4.77531			−0.365177
$172$		0			0
$173$		−	5.64045i		−	0.428836i	−0.976742	−	0.214418i	$-0.931215\pi$
$173$		−	5.64045i		−	0.428836i	0.976742	−	0.214418i	$-0.0687854\pi$
$174$		0			0
$175$	2.17977	−	0.191524i	0.164775	−	0.0144778i
$176$		0			0
$177$			11.4392i			0.859820i
$178$		0			0
$179$	−11.9616			−0.894054			−0.447027	−	0.894521i	$-0.647517\pi$
$179$	−11.9616			−0.894054			−0.447027	+	0.894521i	$0.647517\pi$
$180$		0			0
$181$	11.3442			0.843209			0.421604	−	0.906780i	$-0.361467\pi$
$181$	11.3442			0.843209			0.421604	+	0.906780i	$0.361467\pi$
$182$		0			0
$183$		−	2.25786i		−	0.166906i
$184$		0			0
$185$	2.74098	+	2.51071i	0.201521	+	0.184591i
$186$		0			0
$187$		−	22.8315i		−	1.66961i
$188$		0			0
$189$	0.437634			0.0318332
$190$		0			0
$191$	19.8691			1.43768			0.718839	−	0.695177i	$-0.244674\pi$
$191$	19.8691			1.43768			0.718839	+	0.695177i	$0.244674\pi$
$192$		0			0
$193$		−	1.10823i		−	0.0797719i	−0.999204	−	0.0398859i	$-0.987301\pi$
$193$		−	1.10823i		−	0.0797719i	0.999204	−	0.0398859i	$-0.0126995\pi$
$194$		0			0
$195$	1.51036	−	1.64888i	0.108159	−	0.118079i
$196$		0			0
$197$			5.37758i			0.383137i	0.981479	+	0.191568i	$0.0613574\pi$
$197$			5.37758i			0.383137i	−0.981479	+	0.191568i	$0.938643\pi$
$198$		0			0
$199$	−11.2736			−0.799162			−0.399581	−	0.916698i	$-0.630844\pi$
$199$	−11.2736			−0.799162			−0.399581	+	0.916698i	$0.630844\pi$
$200$		0			0
$201$	2.17977			0.153749
$202$		0			0
$203$		−	0.754022i		−	0.0529220i
$204$		0			0
$205$	4.05236	−	4.42403i	0.283029	−	0.308987i
$206$		0			0
$207$		−	0.337673i		−	0.0234699i
$208$		0			0
$209$	−27.3883			−1.89449
$210$		0			0
$211$	16.4498			1.13245			0.566224	−	0.824251i	$-0.308404\pi$
$211$	16.4498			1.13245			0.566224	+	0.824251i	$0.308404\pi$
$212$		0			0
$213$			13.6970i			0.938505i
$214$		0			0
$215$	3.75459	+	3.43916i	0.256061	+	0.234549i
$216$		0			0
$217$			3.44053i			0.233559i
$218$		0			0
$219$	−2.55410			−0.172590
$220$		0			0
$221$	3.98081			0.267778
$222$		0			0
$223$			25.0527i			1.67765i	0.544397	+	0.838827i	$0.316758\pi$
$223$			25.0527i			1.67765i	−0.544397	+	0.838827i	$0.683242\pi$
$224$		0			0
$225$	0.437634	+	4.98081i	0.0291756	+	0.332054i
$226$		0			0
$227$			8.11414i			0.538554i	0.963063	+	0.269277i	$0.0867847\pi$
$227$			8.11414i			0.538554i	−0.963063	+	0.269277i	$0.913215\pi$
$228$		0			0
$229$	−9.57717			−0.632877			−0.316439	−	0.948613i	$-0.602487\pi$
$229$	−9.57717			−0.632877			−0.316439	+	0.948613i	$0.602487\pi$
$230$		0			0
$231$	2.51001			0.165146
$232$		0			0
$233$			16.3791i			1.07303i	0.843890	+	0.536516i	$0.180260\pi$
$233$			16.3791i			1.07303i	−0.843890	+	0.536516i	$0.819740\pi$
$234$		0			0
$235$	19.3187	+	17.6957i	1.26021	+	1.15434i
$236$		0			0
$237$			2.55144i			0.165734i
$238$		0			0
$239$	−22.2511			−1.43931			−0.719653	−	0.694334i	$-0.755699\pi$
$239$	−22.2511			−1.43931			−0.719653	+	0.694334i	$0.755699\pi$
$240$		0			0
$241$	23.8031			1.53329			0.766647	−	0.642068i	$-0.221924\pi$
$241$	23.8031			1.53329			0.766647	+	0.642068i	$0.221924\pi$
$242$		0			0
$243$			1.00000i			0.0641500i
$244$		0			0
$245$	10.2832	−	11.2264i	0.656972	−	0.717227i
$246$		0			0
$247$		−	4.77531i		−	0.303846i
$248$		0			0
$249$	5.41158			0.342945
$250$		0			0
$251$	4.35992			0.275196			0.137598	−	0.990488i	$-0.456062\pi$
$251$	4.35992			0.275196			0.137598	+	0.990488i	$0.456062\pi$
$252$		0			0
$253$		−	1.93669i		−	0.121759i
$254$		0			0
$255$	−6.01245	+	6.56389i	−0.376514	+	0.411047i
$256$		0			0
$257$		−	14.9965i		−	0.935457i	−0.883872	−	0.467728i	$-0.845073\pi$
$257$		−	14.9965i		−	0.935457i	0.883872	−	0.467728i	$-0.154927\pi$
$258$		0			0
$259$	−0.727491			−0.0452041
$260$		0			0
$261$	1.72295			0.106648
$262$		0			0
$263$		−	10.7029i		−	0.659971i	−0.943986	−	0.329986i	$-0.892956\pi$
$263$		−	10.7029i		−	0.659971i	0.943986	−	0.329986i	$-0.107044\pi$
$264$		0			0
$265$	23.5779	+	21.5971i	1.44838	+	1.32670i
$266$		0			0
$267$			4.72448i			0.289133i
$268$		0			0
$269$	−15.0076			−0.915028			−0.457514	−	0.889202i	$-0.651260\pi$
$269$	−15.0076			−0.915028			−0.457514	+	0.889202i	$0.651260\pi$
$270$		0			0
$271$	1.71386			0.104109			0.0520547	−	0.998644i	$-0.483423\pi$
$271$	1.71386			0.104109			0.0520547	+	0.998644i	$0.483423\pi$
$272$		0			0
$273$			0.437634i			0.0264868i
$274$		0			0
$275$	2.51001	+	28.5669i	0.151359	+	1.72265i
$276$		0			0
$277$			21.1527i			1.27094i	0.772125	+	0.635471i	$0.219194\pi$
$277$			21.1527i			1.27094i	−0.772125	+	0.635471i	$0.780806\pi$
$278$		0			0
$279$	−7.86166			−0.470665
$280$		0			0
$281$	18.7440			1.11818			0.559088	−	0.829108i	$-0.311151\pi$
$281$	18.7440			1.11818			0.559088	+	0.829108i	$0.311151\pi$
$282$		0			0
$283$		−	9.29744i		−	0.552675i	−0.961061	−	0.276338i	$-0.910879\pi$
$283$		−	9.29744i		−	0.552675i	0.961061	−	0.276338i	$-0.0891208\pi$
$284$		0			0
$285$	7.87393	+	7.21243i	0.466411	+	0.427227i
$286$		0			0
$287$			1.17419i			0.0693103i
$288$		0			0
$289$	1.15315			0.0678321
$290$		0			0
$291$	14.7901			0.867012
$292$		0			0
$293$		−	17.9762i		−	1.05018i	−0.851047	−	0.525090i	$-0.824032\pi$
$293$		−	17.9762i		−	1.05018i	0.851047	−	0.525090i	$-0.175968\pi$
$294$		0			0
$295$	17.2772	−	18.8618i	1.00592	−	1.09818i
$296$		0			0
$297$			5.73540i			0.332802i
$298$		0			0
$299$	0.337673			0.0195282
$300$		0			0
$301$	−0.996515			−0.0574382
$302$		0			0
$303$			12.5955i			0.723595i
$304$		0			0
$305$	−3.41018	+	3.72295i	−0.195266	+	0.213175i
$306$		0			0
$307$		−	13.3129i		−	0.759807i	−0.925026	−	0.379904i	$-0.875957\pi$
$307$		−	13.3129i		−	0.759807i	0.925026	−	0.379904i	$-0.124043\pi$
$308$		0			0
$309$	6.23867			0.354906
$310$		0			0
$311$	−2.64776			−0.150141			−0.0750703	−	0.997178i	$-0.523918\pi$
$311$	−2.64776			−0.150141			−0.0750703	+	0.997178i	$0.523918\pi$
$312$		0			0
$313$			15.8718i			0.897126i	0.893751	+	0.448563i	$0.148064\pi$
$313$			15.8718i			0.897126i	−0.893751	+	0.448563i	$0.851936\pi$
$314$		0			0
$315$	−0.721608	−	0.660985i	−0.0406580	−	0.0372423i
$316$		0			0
$317$		−	3.85187i		−	0.216342i	−0.994132	−	0.108171i	$-0.965501\pi$
$317$		−	3.85187i		−	0.216342i	0.994132	−	0.108171i	$-0.0344995\pi$
$318$		0			0
$319$	9.88181			0.553275
$320$		0			0
$321$	−1.10554			−0.0617054
$322$		0			0
$323$			19.0096i			1.05772i
$324$		0			0
$325$	−4.98081	+	0.437634i	−0.276286	+	0.0242756i
$326$		0			0
$327$		−	7.92019i		−	0.437987i
$328$		0			0
$329$	−5.12742			−0.282684
$330$		0			0
$331$	8.15256			0.448105			0.224053	−	0.974577i	$-0.428071\pi$
$331$	8.15256			0.448105			0.224053	+	0.974577i	$0.428071\pi$
$332$		0			0
$333$		−	1.66233i		−	0.0910949i
$334$		0			0
$335$	−3.59419	−	3.29224i	−0.196372	−	0.179874i
$336$		0			0
$337$		−	0.343016i		−	0.0186852i	−0.999956	−	0.00934262i	$-0.997026\pi$
$337$		−	0.343016i		−	0.0186852i	0.999956	−	0.00934262i	$-0.00297389\pi$
$338$		0			0
$339$	−11.4432			−0.621510
$340$		0			0
$341$	−45.0898			−2.44175
$342$		0			0
$343$			6.04306i			0.326295i
$344$		0			0
$345$	−0.510008	+	0.556784i	−0.0274579	+	0.0299763i
$346$		0			0
$347$			32.2306i			1.73023i	0.501572	+	0.865116i	$0.332755\pi$
$347$			32.2306i			1.73023i	−0.501572	+	0.865116i	$0.667245\pi$
$348$		0			0
$349$	−23.6186			−1.26427			−0.632137	−	0.774856i	$-0.717822\pi$
$349$	−23.6186			−1.26427			−0.632137	+	0.774856i	$0.717822\pi$
$350$		0			0
$351$	−1.00000			−0.0533761
$352$		0			0
$353$			30.5552i			1.62629i	0.582063	+	0.813144i	$0.302246\pi$
$353$			30.5552i			1.62629i	−0.582063	+	0.813144i	$0.697754\pi$
$354$		0			0
$355$	20.6874	−	22.5848i	1.09797	−	1.19868i
$356$		0			0
$357$		−	1.74214i		−	0.0922038i
$358$		0			0
$359$	−30.2293			−1.59544			−0.797719	−	0.603029i	$-0.793960\pi$
$359$	−30.2293			−1.59544			−0.797719	+	0.603029i	$0.793960\pi$
$360$		0			0
$361$	3.80356			0.200188
$362$		0			0
$363$			21.8948i			1.14918i
$364$		0			0
$365$	4.21141	+	3.85761i	0.220435	+	0.201916i
$366$		0			0
$367$		−	10.0441i		−	0.524299i	−0.965027	−	0.262149i	$-0.915569\pi$
$367$		−	10.0441i		−	0.524299i	0.965027	−	0.262149i	$-0.0844313\pi$
$368$		0			0
$369$	−2.68304			−0.139674
$370$		0			0
$371$	−6.25786			−0.324892
$372$		0			0
$373$		−	28.8097i		−	1.49171i	−0.666109	−	0.745854i	$-0.732042\pi$
$373$		−	28.8097i		−	1.49171i	0.666109	−	0.745854i	$-0.267958\pi$
$374$		0			0
$375$	6.80120	−	8.87376i	0.351213	−	0.458239i
$376$		0			0
$377$			1.72295i			0.0887364i
$378$		0			0
$379$	17.0581			0.876216			0.438108	−	0.898922i	$-0.355649\pi$
$379$	17.0581			0.876216			0.438108	+	0.898922i	$0.355649\pi$
$380$		0			0
$381$	1.42937			0.0732288
$382$		0			0
$383$		−	6.58842i		−	0.336653i	−0.985731	−	0.168326i	$-0.946164\pi$
$383$		−	6.58842i		−	0.336653i	0.985731	−	0.168326i	$-0.0538363\pi$
$384$		0			0
$385$	−4.13871	−	3.79101i	−0.210928	−	0.193208i
$386$		0			0
$387$		−	2.27705i		−	0.115749i
$388$		0			0
$389$	−30.5741			−1.55017			−0.775083	−	0.631859i	$-0.782292\pi$
$389$	−30.5741			−1.55017			−0.775083	+	0.631859i	$0.782292\pi$
$390$		0			0
$391$	−1.34421			−0.0679798
$392$		0			0
$393$		−	17.6819i		−	0.891933i
$394$		0			0
$395$	3.85359	−	4.20703i	0.193895	−	0.211679i
$396$		0			0
$397$		−	21.5069i		−	1.07940i	−0.841857	−	0.539700i	$-0.818538\pi$
$397$		−	21.5069i		−	1.07940i	0.841857	−	0.539700i	$-0.181462\pi$
$398$		0			0
$399$	−2.08984			−0.104623
$400$		0			0
$401$	−17.8990			−0.893835			−0.446918	−	0.894575i	$-0.647478\pi$
$401$	−17.8990			−0.893835			−0.446918	+	0.894575i	$0.647478\pi$
$402$		0			0
$403$		−	7.86166i		−	0.391617i
$404$		0			0
$405$	1.51036	−	1.64888i	0.0750503	−	0.0819337i
$406$		0			0
$407$		−	9.53411i		−	0.472588i
$408$		0			0
$409$	−22.3844			−1.10684			−0.553420	−	0.832902i	$-0.686678\pi$
$409$	−22.3844			−1.10684			−0.553420	+	0.832902i	$0.686678\pi$
$410$		0			0
$411$	−10.7686			−0.531174
$412$		0			0
$413$			5.00617i			0.246337i
$414$		0			0
$415$	−8.92306	−	8.17342i	−0.438016	−	0.401217i
$416$		0			0
$417$			3.80848i			0.186502i
$418$		0			0
$419$	−27.5153			−1.34421			−0.672105	−	0.740456i	$-0.734610\pi$
$419$	−27.5153			−1.34421			−0.672105	+	0.740456i	$0.734610\pi$
$420$		0			0
$421$	6.34607			0.309289			0.154644	−	0.987970i	$-0.450577\pi$
$421$	6.34607			0.309289			0.154644	+	0.987970i	$0.450577\pi$
$422$		0			0
$423$		−	11.7162i		−	0.569662i
$424$		0			0
$425$	19.8277	−	1.74214i	0.961783	−	0.0845062i
$426$		0			0
$427$		−	0.988117i		−	0.0478183i
$428$		0			0
$429$	−5.73540			−0.276908
$430$		0			0
$431$	3.78383			0.182261			0.0911304	−	0.995839i	$-0.470952\pi$
$431$	3.78383			0.182261			0.0911304	+	0.995839i	$0.470952\pi$
$432$		0			0
$433$			2.55410i			0.122742i	0.998115	+	0.0613711i	$0.0195473\pi$
$433$			2.55410i			0.122742i	−0.998115	+	0.0613711i	$0.980453\pi$
$434$		0			0
$435$	−2.84094	−	2.60227i	−0.136213	−	0.124769i
$436$		0			0
$437$			1.61249i			0.0771361i
$438$		0			0
$439$	2.34073			0.111717			0.0558585	−	0.998439i	$-0.482210\pi$
$439$	2.34073			0.111717			0.0558585	+	0.998439i	$0.482210\pi$
$440$		0			0
$441$	−6.80848			−0.324213
$442$		0			0
$443$		−	0.468574i		−	0.0222626i	−0.999938	−	0.0111313i	$-0.996457\pi$
$443$		−	0.468574i		−	0.0222626i	0.999938	−	0.0111313i	$-0.00354328\pi$
$444$		0			0
$445$	7.13566	−	7.79011i	0.338262	−	0.369287i
$446$		0			0
$447$			16.3649i			0.774035i
$448$		0			0
$449$	17.5863			0.829947			0.414974	−	0.909833i	$-0.363791\pi$
$449$	17.5863			0.829947			0.414974	+	0.909833i	$0.363791\pi$
$450$		0			0
$451$	−15.3883			−0.724608
$452$		0			0
$453$		−	20.3774i		−	0.957413i
$454$		0			0
$455$	0.660985	−	0.721608i	0.0309874	−	0.0338295i
$456$		0			0
$457$		−	11.9132i		−	0.557276i	−0.960396	−	0.278638i	$-0.910117\pi$
$457$		−	11.9132i		−	0.557276i	0.960396	−	0.278638i	$-0.0898829\pi$
$458$		0			0
$459$	3.98081			0.185808
$460$		0			0
$461$	−17.7540			−0.826886			−0.413443	−	0.910530i	$-0.635674\pi$
$461$	−17.7540			−0.826886			−0.413443	+	0.910530i	$0.635674\pi$
$462$		0			0
$463$			39.4856i			1.83505i	0.397676	+	0.917526i	$0.369817\pi$
$463$			39.4856i			1.83505i	−0.397676	+	0.917526i	$0.630183\pi$
$464$		0			0
$465$	12.9630	+	11.8739i	0.601143	+	0.550640i
$466$		0			0
$467$		−	19.2793i		−	0.892139i	−0.894998	−	0.446069i	$-0.852823\pi$
$467$		−	19.2793i		−	0.892139i	0.894998	−	0.446069i	$-0.147177\pi$
$468$		0			0
$469$	0.953943			0.0440490
$470$		0			0
$471$	13.1911			0.607812
$472$		0			0
$473$		−	13.0598i		−	0.600490i
$474$		0			0
$475$	−2.08984	−	23.7849i	−0.0958883	−	1.09133i
$476$		0			0
$477$		−	14.2993i		−	0.654720i
$478$		0			0
$479$	38.8194			1.77371			0.886853	−	0.462052i	$-0.152887\pi$
$479$	38.8194			1.77371			0.886853	+	0.462052i	$0.152887\pi$
$480$		0			0
$481$	1.66233			0.0757956
$482$		0			0
$483$		−	0.147777i		−	0.00672411i
$484$		0			0
$485$	−24.3872	−	22.3384i	−1.10736	−	1.01433i
$486$		0			0
$487$			29.8866i			1.35429i	0.735849	+	0.677145i	$0.236783\pi$
$487$			29.8866i			1.35429i	−0.735849	+	0.677145i	$0.763217\pi$
$488$		0			0
$489$	−10.3578			−0.468397
$490$		0			0
$491$	−26.7474			−1.20709			−0.603547	−	0.797327i	$-0.706246\pi$
$491$	−26.7474			−1.20709			−0.603547	+	0.797327i	$0.706246\pi$
$492$		0			0
$493$		−	6.85874i		−	0.308902i
$494$		0			0
$495$	8.66251	−	9.45701i	0.389351	−	0.425061i
$496$		0			0
$497$			5.99429i			0.268880i
$498$		0			0
$499$	31.9694			1.43115			0.715574	−	0.698537i	$-0.246165\pi$
$499$	31.9694			1.43115			0.715574	+	0.698537i	$0.246165\pi$
$500$		0			0
$501$	−10.6713			−0.476758
$502$		0			0
$503$		−	9.37993i		−	0.418231i	−0.977891	−	0.209115i	$-0.932942\pi$
$503$		−	9.37993i		−	0.418231i	0.977891	−	0.209115i	$-0.0670584\pi$
$504$		0			0
$505$	19.0238	−	20.7686i	0.846547	−	0.924189i
$506$		0			0
$507$		−	1.00000i		−	0.0444116i
$508$		0			0
$509$	−11.0042			−0.487753			−0.243877	−	0.969806i	$-0.578419\pi$
$509$	−11.0042			−0.487753			−0.243877	+	0.969806i	$0.578419\pi$
$510$		0			0
$511$	−1.11776			−0.0494469
$512$		0			0
$513$		−	4.77531i		−	0.210835i
$514$		0			0
$515$	−10.2868	−	9.42263i	−0.453292	−	0.415211i
$516$		0			0
$517$		−	67.1972i		−	2.95533i
$518$		0			0
$519$	5.64045			0.247588
$520$		0			0
$521$	29.1305			1.27623			0.638115	−	0.769941i	$-0.279715\pi$
$521$	29.1305			1.27623			0.638115	+	0.769941i	$0.279715\pi$
$522$		0			0
$523$		−	13.5675i		−	0.593266i	−0.954992	−	0.296633i	$-0.904136\pi$
$523$		−	13.5675i		−	0.593266i	0.954992	−	0.296633i	$-0.0958638\pi$
$524$		0			0
$525$	0.191524	+	2.17977i	0.00835878	+	0.0951331i
$526$		0			0
$527$			31.2958i			1.36327i
$528$		0			0
$529$	22.8860			0.995042
$530$		0			0
$531$	−11.4392			−0.496417
$532$		0			0
$533$		−	2.68304i		−	0.116216i
$534$		0			0
$535$	1.82291	+	1.66977i	0.0788113	+	0.0721902i
$536$		0			0
$537$		−	11.9616i		−	0.516182i
$538$		0			0
$539$	−39.0493			−1.68197
$540$		0			0
$541$	−22.6672			−0.974541			−0.487270	−	0.873251i	$-0.662007\pi$
$541$	−22.6672			−0.974541			−0.487270	+	0.873251i	$0.662007\pi$
$542$		0			0
$543$			11.3442i			0.486827i
$544$		0			0
$545$	−11.9623	+	13.0595i	−0.512410	+	0.559406i
$546$		0			0
$547$		−	33.5014i		−	1.43242i	−0.697886	−	0.716209i	$-0.745876\pi$
$547$		−	33.5014i		−	1.43242i	0.697886	−	0.716209i	$-0.254124\pi$
$548$		0			0
$549$	2.25786			0.0963632
$550$		0			0
$551$	−8.22762			−0.350508
$552$		0			0
$553$			1.11660i			0.0474826i
$554$		0			0
$555$	−2.51071	+	2.74098i	−0.106574	+	0.116348i
$556$		0			0
$557$			42.6185i			1.80580i	0.429846	+	0.902902i	$0.358568\pi$
$557$			42.6185i			1.80580i	−0.429846	+	0.902902i	$0.641432\pi$
$558$		0			0
$559$	2.27705			0.0963090
$560$		0			0
$561$	22.8315			0.963949
$562$		0			0
$563$		−	17.1688i		−	0.723580i	−0.932260	−	0.361790i	$-0.882166\pi$
$563$		−	17.1688i		−	0.723580i	0.932260	−	0.361790i	$-0.117834\pi$
$564$		0			0
$565$	18.8685	+	17.2834i	0.793805	+	0.727116i
$566$		0			0
$567$			0.437634i			0.0183789i
$568$		0			0
$569$	−16.9018			−0.708561			−0.354281	−	0.935139i	$-0.615274\pi$
$569$	−16.9018			−0.708561			−0.354281	+	0.935139i	$0.615274\pi$
$570$		0			0
$571$	29.9408			1.25298			0.626491	−	0.779428i	$-0.284490\pi$
$571$	29.9408			1.25298			0.626491	+	0.779428i	$0.284490\pi$
$572$		0			0
$573$			19.8691i			0.830044i
$574$		0			0
$575$	1.68189	−	0.147777i	0.0701396	−	0.00616275i
$576$		0			0
$577$			44.8430i			1.86684i	0.358789	+	0.933419i	$0.383190\pi$
$577$			44.8430i			1.86684i	−0.358789	+	0.933419i	$0.616810\pi$
$578$		0			0
$579$	1.10823			0.0460563
$580$		0			0
$581$	2.36829			0.0982532
$582$		0			0
$583$		−	82.0122i		−	3.39660i
$584$		0			0
$585$	1.64888	+	1.51036i	0.0681729	+	0.0624456i
$586$		0			0
$587$		−	24.2703i		−	1.00174i	−0.865522	−	0.500872i	$-0.833013\pi$
$587$		−	24.2703i		−	1.00174i	0.865522	−	0.500872i	$-0.166987\pi$
$588$		0			0
$589$	37.5418			1.54688
$590$		0			0
$591$	−5.37758			−0.221204
$592$		0			0
$593$		−	44.2659i		−	1.81778i	−0.417032	−	0.908892i	$-0.636930\pi$
$593$		−	44.2659i		−	1.81778i	0.417032	−	0.908892i	$-0.363070\pi$
$594$		0			0
$595$	−2.63125	+	2.87258i	−0.107871	+	0.117764i
$596$		0			0
$597$		−	11.2736i		−	0.461396i
$598$		0			0
$599$	−18.9996			−0.776301			−0.388151	−	0.921596i	$-0.626886\pi$
$599$	−18.9996			−0.776301			−0.388151	+	0.921596i	$0.626886\pi$
$600$		0			0
$601$	0.722123			0.0294560			0.0147280	−	0.999892i	$-0.495312\pi$
$601$	0.722123			0.0294560			0.0147280	+	0.999892i	$0.495312\pi$
$602$		0			0
$603$			2.17977i			0.0887672i
$604$		0			0
$605$	33.0690	−	36.1020i	1.34445	−	1.46776i
$606$		0			0
$607$		−	29.9236i		−	1.21456i	−0.794487	−	0.607281i	$-0.792260\pi$
$607$		−	29.9236i		−	1.21456i	0.794487	−	0.607281i	$-0.207740\pi$
$608$		0			0
$609$	0.754022			0.0305545
$610$		0			0
$611$	11.7162			0.473987
$612$		0			0
$613$		−	34.6980i		−	1.40144i	−0.713438	−	0.700719i	$-0.752863\pi$
$613$		−	34.6980i		−	1.40144i	0.713438	−	0.700719i	$-0.247137\pi$
$614$		0			0
$615$	4.42403	+	4.05236i	0.178394	+	0.163407i
$616$		0			0
$617$			25.8247i			1.03966i	0.854269	+	0.519831i	$0.174005\pi$
$617$			25.8247i			1.03966i	−0.854269	+	0.519831i	$0.825995\pi$
$618$		0			0
$619$	−29.3106			−1.17809			−0.589047	−	0.808099i	$-0.700497\pi$
$619$	−29.3106			−1.17809			−0.589047	+	0.808099i	$0.700497\pi$
$620$		0			0
$621$	0.337673			0.0135504
$622$		0			0
$623$			2.06759i			0.0828364i
$624$		0			0
$625$	−24.6170	+	4.35955i	−0.984678	+	0.174382i
$626$		0			0
$627$		−	27.3883i		−	1.09378i
$628$		0			0
$629$	−6.61741			−0.263853
$630$		0			0
$631$	22.9480			0.913546			0.456773	−	0.889583i	$-0.349005\pi$
$631$	22.9480			0.913546			0.456773	+	0.889583i	$0.349005\pi$
$632$		0			0
$633$			16.4498i			0.653819i
$634$		0			0
$635$	−2.35686	−	2.15886i	−0.0935292	−	0.0856717i
$636$		0			0
$637$		−	6.80848i		−	0.269762i
$638$		0			0
$639$	−13.6970			−0.541846
$640$		0			0
$641$	24.3185			0.960522			0.480261	−	0.877126i	$-0.340542\pi$
$641$	24.3185			0.960522			0.480261	+	0.877126i	$0.340542\pi$
$642$		0			0
$643$			11.2911i			0.445276i	0.974901	+	0.222638i	$0.0714668\pi$
$643$			11.2911i			0.445276i	−0.974901	+	0.222638i	$0.928533\pi$
$644$		0			0
$645$	−3.43916	+	3.75459i	−0.135417	+	0.147837i
$646$		0			0
$647$		−	7.55633i		−	0.297070i	−0.988907	−	0.148535i	$-0.952544\pi$
$647$		−	7.55633i		−	0.297070i	0.988907	−	0.148535i	$-0.0474558\pi$
$648$		0			0
$649$	−65.6082			−2.57535
$650$		0			0
$651$	−3.44053			−0.134845
$652$		0			0
$653$		−	33.6711i		−	1.31765i	−0.752295	−	0.658826i	$-0.771053\pi$
$653$		−	33.6711i		−	1.31765i	0.752295	−	0.658826i	$-0.228947\pi$
$654$		0			0
$655$	−26.7060	+	29.1554i	−1.04349	+	1.13919i
$656$		0			0
$657$		−	2.55410i		−	0.0996449i
$658$		0			0
$659$	−17.1039			−0.666274			−0.333137	−	0.942878i	$-0.608107\pi$
$659$	−17.1039			−0.666274			−0.333137	+	0.942878i	$0.608107\pi$
$660$		0			0
$661$	19.1964			0.746655			0.373327	−	0.927700i	$-0.378217\pi$
$661$	19.1964			0.746655			0.373327	+	0.927700i	$0.378217\pi$
$662$		0			0
$663$			3.98081i			0.154602i
$664$		0			0
$665$	3.44590	+	3.15640i	0.133626	+	0.122400i
$666$		0			0
$667$		−	0.581794i		−	0.0225272i
$668$		0			0
$669$	−25.0527			−0.968594
$670$		0			0
$671$	12.9497			0.499919
$672$		0			0
$673$			27.9616i			1.07784i	0.842357	+	0.538921i	$0.181168\pi$
$673$			27.9616i			1.07784i	−0.842357	+	0.538921i	$0.818832\pi$
$674$		0			0
$675$	−4.98081	+	0.437634i	−0.191711	+	0.0168445i
$676$		0			0
$677$		−	6.98533i		−	0.268468i	−0.990950	−	0.134234i	$-0.957143\pi$
$677$		−	6.98533i		−	0.268468i	0.990950	−	0.134234i	$-0.0428574\pi$
$678$		0			0
$679$	6.47266			0.248398
$680$		0			0
$681$	−8.11414			−0.310934
$682$		0			0
$683$		−	18.0969i		−	0.692460i	−0.938150	−	0.346230i	$-0.887462\pi$
$683$		−	18.0969i		−	0.692460i	0.938150	−	0.346230i	$-0.112538\pi$
$684$		0			0
$685$	17.7561	+	16.2644i	0.678426	+	0.621431i
$686$		0			0
$687$		−	9.57717i		−	0.365392i
$688$		0			0
$689$	14.2993			0.544760
$690$		0			0
$691$	−13.9818			−0.531892			−0.265946	−	0.963988i	$-0.585684\pi$
$691$	−13.9818			−0.531892			−0.265946	+	0.963988i	$0.585684\pi$
$692$		0			0
$693$			2.51001i			0.0953473i
$694$		0			0
$695$	5.75216	−	6.27973i	0.218192	−	0.238204i
$696$		0			0
$697$			10.6807i			0.404560i
$698$		0			0
$699$	−16.3791			−0.619515
$700$		0			0
$701$	−2.64470			−0.0998891			−0.0499445	−	0.998752i	$-0.515904\pi$
$701$	−2.64470			−0.0998891			−0.0499445	+	0.998752i	$0.515904\pi$
$702$		0			0
$703$			7.93812i			0.299392i
$704$		0			0
$705$	−17.6957	+	19.3187i	−0.666458	+	0.727583i
$706$		0			0
$707$			5.51224i			0.207309i
$708$		0			0
$709$	−9.89834			−0.371740			−0.185870	−	0.982574i	$-0.559510\pi$
$709$	−9.89834			−0.371740			−0.185870	+	0.982574i	$0.559510\pi$
$710$		0			0
$711$	−2.55144			−0.0956866
$712$		0			0
$713$			2.65467i			0.0994183i
$714$		0			0
$715$	9.45701	+	8.66251i	0.353672	+	0.323960i
$716$		0			0
$717$		−	22.2511i		−	0.830983i
$718$		0			0
$719$	21.1288			0.787972			0.393986	−	0.919116i	$-0.371096\pi$
$719$	21.1288			0.787972			0.393986	+	0.919116i	$0.371096\pi$
$720$		0			0
$721$	2.73026			0.101680
$722$		0			0
$723$			23.8031i			0.885248i
$724$		0			0
$725$	0.754022	+	8.58169i	0.0280037	+	0.318716i
$726$		0			0
$727$			18.7471i			0.695290i	0.937626	+	0.347645i	$0.113019\pi$
$727$			18.7471i			0.695290i	−0.937626	+	0.347645i	$0.886981\pi$
$728$		0			0
$729$	−1.00000			−0.0370370
$730$		0			0
$731$	−9.06451			−0.335263
$732$		0			0
$733$		−	47.2675i		−	1.74586i	−0.487842	−	0.872932i	$-0.662216\pi$
$733$		−	47.2675i		−	1.74586i	0.487842	−	0.872932i	$-0.337784\pi$
$734$		0			0
$735$	11.2264	+	10.2832i	0.414091	+	0.379303i
$736$		0			0
$737$			12.5019i			0.460512i
$738$		0			0
$739$	−40.6501			−1.49534			−0.747670	−	0.664071i	$-0.768827\pi$
$739$	−40.6501			−1.49534			−0.747670	+	0.664071i	$0.768827\pi$
$740$		0			0
$741$	4.77531			0.175425
$742$		0			0
$743$		−	1.06271i		−	0.0389871i	−0.999810	−	0.0194936i	$-0.993795\pi$
$743$		−	1.06271i		−	0.0389871i	0.999810	−	0.0194936i	$-0.00620539\pi$
$744$		0			0
$745$	24.7169	−	26.9839i	0.905558	−	0.988612i
$746$		0			0
$747$			5.41158i			0.197999i
$748$		0			0
$749$	−0.483823			−0.0176785
$750$		0			0
$751$	10.8780			0.396945			0.198473	−	0.980106i	$-0.436402\pi$
$751$	10.8780			0.396945			0.198473	+	0.980106i	$0.436402\pi$
$752$		0			0
$753$			4.35992i			0.158884i
$754$		0			0
$755$	−30.7772	+	33.5999i	−1.12010	+	1.22283i
$756$		0			0
$757$		−	16.2279i		−	0.589812i	−0.955526	−	0.294906i	$-0.904712\pi$
$757$		−	16.2279i		−	0.589812i	0.955526	−	0.294906i	$-0.0952884\pi$
$758$		0			0
$759$	1.93669			0.0702975
$760$		0			0
$761$	37.3880			1.35531			0.677657	−	0.735379i	$-0.262996\pi$
$761$	37.3880			1.35531			0.677657	+	0.735379i	$0.262996\pi$
$762$		0			0
$763$		−	3.46615i		−	0.125483i
$764$		0			0
$765$	−6.56389	−	6.01245i	−0.237318	−	0.217381i
$766$		0			0
$767$		−	11.4392i		−	0.413044i
$768$		0			0
$769$	12.5874			0.453914			0.226957	−	0.973905i	$-0.427122\pi$
$769$	12.5874			0.453914			0.226957	+	0.973905i	$0.427122\pi$
$770$		0			0
$771$	14.9965			0.540086
$772$		0			0
$773$			0.746723i			0.0268577i	0.999910	+	0.0134289i	$0.00427467\pi$
$773$			0.746723i			0.0268577i	−0.999910	+	0.0134289i	$0.995725\pi$
$774$		0			0
$775$	−3.44053	−	39.1574i	−0.123588	−	1.40658i
$776$		0			0
$777$		−	0.727491i		−	0.0260986i
$778$		0			0
$779$	12.8124			0.459050
$780$		0			0
$781$	−78.5579			−2.81102
$782$		0			0
$783$			1.72295i			0.0615732i
$784$		0			0
$785$	−21.7505	−	19.9232i	−0.776310	−	0.711091i
$786$		0			0
$787$		−	33.3585i		−	1.18910i	−0.804058	−	0.594551i	$-0.797330\pi$
$787$		−	33.3585i		−	1.18910i	0.804058	−	0.594551i	$-0.202670\pi$
$788$		0			0
$789$	10.7029			0.381035
$790$		0			0
$791$	−5.00794			−0.178062
$792$		0			0
$793$			2.25786i			0.0801790i
$794$		0			0
$795$	−21.5971	+	23.5779i	−0.765969	+	0.836221i
$796$		0			0
$797$			0.233988i			0.00828829i	0.999991	+	0.00414415i	$0.00131913\pi$
$797$			0.233988i			0.00828829i	−0.999991	+	0.00414415i	$0.998681\pi$
$798$		0			0
$799$	−46.6400			−1.65001
$800$		0			0
$801$	−4.72448			−0.166931
$802$		0			0
$803$		−	14.6488i		−	0.516945i
$804$		0			0
$805$	−0.223197	+	0.243668i	−0.00786666	+	0.00858816i
$806$		0			0
$807$		−	15.0076i		−	0.528292i
$808$		0			0
$809$	22.2003			0.780521			0.390260	−	0.920705i	$-0.372385\pi$
$809$	22.2003			0.780521			0.390260	+	0.920705i	$0.372385\pi$
$810$		0			0
$811$	43.4456			1.52558			0.762791	−	0.646645i	$-0.223829\pi$
$811$	43.4456			1.52558			0.762791	+	0.646645i	$0.223829\pi$
$812$		0			0
$813$			1.71386i			0.0601076i
$814$		0			0
$815$	17.0788	+	15.6440i	0.598246	+	0.547986i
$816$		0			0
$817$			10.8736i			0.380420i
$818$		0			0
$819$	−0.437634			−0.0152922
$820$		0			0
$821$	−6.76172			−0.235986			−0.117993	−	0.993014i	$-0.537646\pi$
$821$	−6.76172			−0.235986			−0.117993	+	0.993014i	$0.537646\pi$
$822$		0			0
$823$		−	28.6208i		−	0.997659i	−0.866700	−	0.498829i	$-0.833763\pi$
$823$		−	28.6208i		−	0.997659i	0.866700	−	0.498829i	$-0.166237\pi$
$824$		0			0
$825$	−28.5669	+	2.51001i	−0.994574	+	0.0873873i
$826$		0			0
$827$		−	8.19215i		−	0.284869i	−0.989804	−	0.142435i	$-0.954507\pi$
$827$		−	8.19215i		−	0.284869i	0.989804	−	0.142435i	$-0.0454930\pi$
$828$		0			0
$829$	3.60208			0.125105			0.0625526	−	0.998042i	$-0.480076\pi$
$829$	3.60208			0.125105			0.0625526	+	0.998042i	$0.480076\pi$
$830$		0			0
$831$	−21.1527			−0.733779
$832$		0			0
$833$			27.1033i			0.939072i
$834$		0			0
$835$	17.5957	+	16.1175i	0.608925	+	0.557768i
$836$		0			0
$837$		−	7.86166i		−	0.271739i
$838$		0			0
$839$	1.50713			0.0520318			0.0260159	−	0.999662i	$-0.491718\pi$
$839$	1.50713			0.0520318			0.0260159	+	0.999662i	$0.491718\pi$
$840$		0			0
$841$	−26.0314			−0.897636
$842$		0			0
$843$			18.7440i			0.645579i
$844$		0			0
$845$	−1.51036	+	1.64888i	−0.0519579	+	0.0567233i
$846$		0			0
$847$			9.58193i			0.329239i
$848$		0			0
$849$	9.29744			0.319087
$850$		0			0
$851$	−0.561324			−0.0192419
$852$		0			0
$853$		−	8.35952i		−	0.286225i	−0.989706	−	0.143112i	$-0.954289\pi$
$853$		−	8.35952i		−	0.286225i	0.989706	−	0.143112i	$-0.0457110\pi$
$854$		0			0
$855$	−7.21243	+	7.87393i	−0.246660	+	0.269283i
$856$		0			0
$857$		−	12.8565i		−	0.439168i	−0.975594	−	0.219584i	$-0.929530\pi$
$857$		−	12.8565i		−	0.439168i	0.975594	−	0.219584i	$-0.0704700\pi$
$858$		0			0
$859$	−16.3734			−0.558652			−0.279326	−	0.960196i	$-0.590111\pi$
$859$	−16.3734			−0.558652			−0.279326	+	0.960196i	$0.590111\pi$
$860$		0			0
$861$	−1.17419			−0.0400163
$862$		0			0
$863$		−	50.0600i		−	1.70406i	−0.523492	−	0.852031i	$-0.675371\pi$
$863$		−	50.0600i		−	1.70406i	0.523492	−	0.852031i	$-0.324629\pi$
$864$		0			0
$865$	−9.30045	−	8.51911i	−0.316225	−	0.289658i
$866$		0			0
$867$			1.15315i			0.0391629i
$868$		0			0
$869$	−14.6335			−0.496409
$870$		0			0
$871$	−2.17977			−0.0738588
$872$		0			0
$873$			14.7901i			0.500570i
$874$		0			0
$875$	2.97644	−	3.88346i	0.100622	−	0.131285i
$876$		0			0
$877$			48.1027i			1.62431i	0.583439	+	0.812157i	$0.301707\pi$
$877$			48.1027i			1.62431i	−0.583439	+	0.812157i	$0.698293\pi$
$878$		0			0
$879$	17.9762			0.606321
$880$		0			0
$881$	−41.2109			−1.38843			−0.694216	−	0.719767i	$-0.744249\pi$
$881$	−41.2109			−1.38843			−0.694216	+	0.719767i	$0.744249\pi$
$882$		0			0
$883$		−	13.9344i		−	0.468930i	−0.972125	−	0.234465i	$-0.924666\pi$
$883$		−	13.9344i		−	0.468930i	0.972125	−	0.234465i	$-0.0753338\pi$
$884$		0			0
$885$	18.8618	+	17.2772i	0.634034	+	0.580768i
$886$		0			0
$887$		−	47.8092i		−	1.60528i	−0.596466	−	0.802638i	$-0.703429\pi$
$887$		−	47.8092i		−	1.60528i	0.596466	−	0.802638i	$-0.296571\pi$
$888$		0			0
$889$	0.625541			0.0209800
$890$		0			0
$891$	−5.73540			−0.192143
$892$		0			0
$893$			55.9485i			1.87225i
$894$		0			0
$895$	−18.0663	+	19.7233i	−0.603891	+	0.659278i
$896$		0			0
$897$			0.337673i			0.0112746i
$898$		0			0
$899$	−13.5452			−0.451759
$900$		0			0
$901$	−56.9228			−1.89637
$902$		0			0
$903$		−	0.996515i		−	0.0331619i
$904$		0			0
$905$	17.1338	−	18.7053i	0.569548	−	0.621785i
$906$		0			0
$907$			8.74334i			0.290318i	0.989408	+	0.145159i	$0.0463693\pi$
$907$			8.74334i			0.290318i	−0.989408	+	0.145159i	$0.953631\pi$
$908$		0			0
$909$	−12.5955			−0.417768
$910$		0			0
$911$	2.67266			0.0885493			0.0442746	−	0.999019i	$-0.485902\pi$
$911$	2.67266			0.0885493			0.0442746	+	0.999019i	$0.485902\pi$
$912$		0			0
$913$			31.0376i			1.02719i
$914$		0			0
$915$	−3.72295	−	3.41018i	−0.123077	−	0.112737i
$916$		0			0
$917$		−	7.73820i		−	0.255538i
$918$		0			0
$919$	−15.4118			−0.508389			−0.254195	−	0.967153i	$-0.581810\pi$
$919$	−15.4118			−0.508389			−0.254195	+	0.967153i	$0.581810\pi$
$920$		0			0
$921$	13.3129			0.438675
$922$		0			0
$923$		−	13.6970i		−	0.450843i
$924$		0			0
$925$	8.27973	−	0.727491i	0.272236	−	0.0239198i
$926$		0			0
$927$			6.23867i			0.204905i
$928$		0			0
$929$	−16.7575			−0.549795			−0.274898	−	0.961473i	$-0.588644\pi$
$929$	−16.7575			−0.549795			−0.274898	+	0.961473i	$0.588644\pi$
$930$		0			0
$931$	32.5126			1.06556
$932$		0			0
$933$		−	2.64776i		−	0.0866838i
$934$		0			0
$935$	−37.6466	−	34.4838i	−1.23117	−	1.12774i
$936$		0			0
$937$		−	34.3945i		−	1.12362i	−0.827267	−	0.561809i	$-0.810105\pi$
$937$		−	34.3945i		−	1.12362i	0.827267	−	0.561809i	$-0.189895\pi$
$938$		0			0
$939$	−15.8718			−0.517956
$940$		0			0
$941$	3.90298			0.127234			0.0636168	−	0.997974i	$-0.479736\pi$
$941$	3.90298			0.127234			0.0636168	+	0.997974i	$0.479736\pi$
$942$		0			0
$943$			0.905993i			0.0295032i
$944$		0			0
$945$	0.660985	−	0.721608i	0.0215018	−	0.0234739i
$946$		0			0
$947$			32.7611i			1.06459i	0.846558	+	0.532297i	$0.178671\pi$
$947$			32.7611i			1.06459i	−0.846558	+	0.532297i	$0.821329\pi$
$948$		0			0
$949$	2.55410			0.0829096
$950$		0			0
$951$	3.85187			0.124905
$952$		0			0
$953$		−	3.05611i		−	0.0989970i	−0.998774	−	0.0494985i	$-0.984238\pi$
$953$		−	3.05611i		−	0.0989970i	0.998774	−	0.0494985i	$-0.0157623\pi$
$954$		0			0
$955$	30.0095	−	32.7618i	0.971083	−	1.06015i
$956$		0			0
$957$			9.88181i			0.319434i
$958$		0			0
$959$	−4.71269			−0.152181
$960$		0			0
$961$	30.8057			0.993733
$962$		0			0
$963$		−	1.10554i		−	0.0356256i
$964$		0			0
$965$	−1.82734	−	1.67382i	−0.0588240	−	0.0538821i
$966$		0			0
$967$			20.5538i			0.660966i	0.943812	+	0.330483i	$0.107212\pi$
$967$			20.5538i			0.660966i	−0.943812	+	0.330483i	$0.892788\pi$
$968$		0			0
$969$	−19.0096			−0.610676
$970$		0			0
$971$	49.3281			1.58301			0.791507	−	0.611160i	$-0.209297\pi$
$971$	49.3281			1.58301			0.791507	+	0.611160i	$0.209297\pi$
$972$		0			0
$973$			1.66672i			0.0534326i
$974$		0			0
$975$	−0.437634	−	4.98081i	−0.0140155	−	0.159514i
$976$		0			0
$977$			19.0164i			0.608390i	0.952610	+	0.304195i	$0.0983874\pi$
$977$			19.0164i			0.608390i	−0.952610	+	0.304195i	$0.901613\pi$
$978$		0			0
$979$	−27.0968			−0.866017
$980$		0			0
$981$	7.92019			0.252872
$982$		0			0
$983$			20.8696i			0.665636i	0.942991	+	0.332818i	$0.107999\pi$
$983$			20.8696i			0.665636i	−0.942991	+	0.332818i	$0.892001\pi$
$984$		0			0
$985$	8.86700	+	8.12207i	0.282526	+	0.258791i
$986$		0			0
$987$		−	5.12742i		−	0.163207i
$988$		0			0
$989$	−0.768899			−0.0244496
$990$		0			0
$991$	−15.6435			−0.496932			−0.248466	−	0.968641i	$-0.579926\pi$
$991$	−15.6435			−0.496932			−0.248466	+	0.968641i	$0.579926\pi$
$992$		0			0
$993$			8.15256i			0.258714i
$994$		0			0
$995$	−17.0271	+	18.5888i	−0.539796	+	0.589304i
$996$		0			0
$997$			6.21108i			0.196707i	0.995152	+	0.0983535i	$0.0313576\pi$
$997$			6.21108i			0.196707i	−0.995152	+	0.0983535i	$0.968642\pi$
$998$		0			0
$999$	1.66233			0.0525937

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3120.2.l.p.1249.9		10
4.3	odd	2		195.2.c.b.79.8	yes	10
5.4	even	2	inner	3120.2.l.p.1249.4		10
12.11	even	2		585.2.c.c.469.3		10
20.3	even	4		975.2.a.s.1.4		5
20.7	even	4		975.2.a.r.1.2		5
20.19	odd	2		195.2.c.b.79.3	✓	10
60.23	odd	4		2925.2.a.bm.1.2		5
60.47	odd	4		2925.2.a.bl.1.4		5
60.59	even	2		585.2.c.c.469.8		10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
195.2.c.b.79.3	✓	10	20.19	odd	2
195.2.c.b.79.8	yes	10	4.3	odd	2
585.2.c.c.469.3		10	12.11	even	2
585.2.c.c.469.8		10	60.59	even	2
975.2.a.r.1.2		5	20.7	even	4
975.2.a.s.1.4		5	20.3	even	4
2925.2.a.bl.1.4		5	60.47	odd	4
2925.2.a.bm.1.2		5	60.23	odd	4
3120.2.l.p.1249.4		10	5.4	even	2	inner
3120.2.l.p.1249.9		10	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 3120 = 2^{4} \cdot 3 \cdot 5 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3120.l (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{3} +(1.51036 - 1.64888i) q^{5} +0.437634i q^{7} -1.00000 q^{9} -5.73540 q^{11} -1.00000i q^{13} +(1.64888 + 1.51036i) q^{15} +3.98081i q^{17} +4.77531 q^{19} -0.437634 q^{21} +0.337673i q^{23} +(-0.437634 - 4.98081i) q^{25} -1.00000i q^{27} -1.72295 q^{29} +7.86166 q^{31} -5.73540i q^{33} +(0.721608 + 0.660985i) q^{35} +1.66233i q^{37} +1.00000 q^{39} +2.68304 q^{41} +2.27705i q^{43} +(-1.51036 + 1.64888i) q^{45} +11.7162i q^{47} +6.80848 q^{49} -3.98081 q^{51} +14.2993i q^{53} +(-8.66251 + 9.45701i) q^{55} +4.77531i q^{57} +11.4392 q^{59} -2.25786 q^{61} -0.437634i q^{63} +(-1.64888 - 1.51036i) q^{65} -2.17977i q^{67} -0.337673 q^{69} +13.6970 q^{71} +2.55410i q^{73} +(4.98081 - 0.437634i) q^{75} -2.51001i q^{77} +2.55144 q^{79} +1.00000 q^{81} -5.41158i q^{83} +(6.56389 + 6.01245i) q^{85} -1.72295i q^{87} +4.72448 q^{89} +0.437634 q^{91} +7.86166i q^{93} +(7.21243 - 7.87393i) q^{95} -14.7901i q^{97} +5.73540 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q - 2 q^{5} - 10 q^{9} - 10 q^{11} + 16 q^{19} + 10 q^{21} + 10 q^{25} - 16 q^{29} - 24 q^{31} - 12 q^{35} + 10 q^{39} + 10 q^{41} + 2 q^{45} - 44 q^{49} + 10 q^{51} - 2 q^{55} + 16 q^{59} + 26 q^{61}+ \cdots + 10 q^{99}+O(q^{100}) \) 10 * q - 2 * q^5 - 10 * q^9 - 10 * q^11 + 16 * q^19 + 10 * q^21 + 10 * q^25 - 16 * q^29 - 24 * q^31 - 12 * q^35 + 10 * q^39 + 10 * q^41 + 2 * q^45 - 44 * q^49 + 10 * q^51 - 2 * q^55 + 16 * q^59 + 26 * q^61 + 14 * q^69 - 10 * q^71 + 2 * q^79 + 10 * q^81 - 4 * q^85 - 38 * q^89 - 10 * q^91 + 48 * q^95 + 10 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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