LMFDB - Embedded newform 160.8.d.a.81.12

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [160,8,Mod(81,160)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(160, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 0])) N = Newforms(chi, 8, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("160.81"); S:= CuspForms(chi, 8); N := Newforms(S);

Level:	$ N $	$=$	$ 160 = 2^{5} \cdot 5 $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	160.d (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

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

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

Self dual:	no
Analytic conductor:	$49.9816040775$
Analytic rank:	$0$
Dimension:	$28$
Twist minimal:	no (minimal twist has level 40)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			81.12
Character	$\chi$	$=$	160.81
Dual form			160.8.d.a.81.17

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-15.3942i q^{3} +125.000i q^{5} +206.895 q^{7} +1950.02 q^{9} -5226.99i q^{11} +180.982i q^{13} +1924.27 q^{15} -34858.6 q^{17} +20572.1i q^{19} -3184.98i q^{21} +89873.4 q^{23} -15625.0 q^{25} -63685.9i q^{27} +82757.7i q^{29} +158725. q^{31} -80465.1 q^{33} +25861.9i q^{35} -125771. i q^{37} +2786.06 q^{39} +382738. q^{41} -855458. i q^{43} +243752. i q^{45} +320160. q^{47} -780737. q^{49} +536619. i q^{51} +63020.0i q^{53} +653374. q^{55} +316691. q^{57} -2.07791e6i q^{59} -2.88950e6i q^{61} +403450. q^{63} -22622.7 q^{65} -2.31917e6i q^{67} -1.38352e6i q^{69} +35994.8 q^{71} +590314. q^{73} +240534. i q^{75} -1.08144e6i q^{77} +3.43960e6 q^{79} +3.28430e6 q^{81} +517777. i q^{83} -4.35733e6i q^{85} +1.27398e6 q^{87} +1.01219e7 q^{89} +37444.2i q^{91} -2.44344e6i q^{93} -2.57152e6 q^{95} +6.76059e6 q^{97} -1.01927e7i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 28 q - 1372 q^{7} - 20412 q^{9} + 13500 q^{15} - 2588 q^{23} - 437500 q^{25} - 268024 q^{31} - 99016 q^{33} + 283944 q^{39} - 601208 q^{41} + 2076460 q^{47} + 4316268 q^{49} - 1331000 q^{55} + 3788536 q^{57}+ \cdots + 15198608 q^{97}+O(q^{100}) $ 28 * q - 1372 * q^7 - 20412 * q^9 + 13500 * q^15 - 2588 * q^23 - 437500 * q^25 - 268024 * q^31 - 99016 * q^33 + 283944 * q^39 - 601208 * q^41 + 2076460 * q^47 + 4316268 * q^49 - 1331000 * q^55 + 3788536 * q^57 + 14839060 * q^63 + 1001656 * q^71 + 2534128 * q^73 + 22040096 * q^79 + 11166300 * q^81 - 34127184 * q^87 + 3807224 * q^89 + 13718000 * q^95 + 15198608 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/160\mathbb{Z}\right)^\times$.

$n$	$31$	$97$	$101$
$\chi(n)$	$1$	$1$	$-1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	− 15.3942i	− 0.329179i	−0.986362	−	0.164589i	$-0.947370\pi$
$3$	− 15.3942i	− 0.329179i	0.986362	−	0.164589i	$-0.0526299\pi$
$4$	0	0
$5$	125.000i	0.447214i
$5$	125.000i	0.447214i
$6$	0	0
$7$	206.895	0.227986	0.113993	−	0.993482i	$-0.463636\pi$
$7$	206.895	0.227986	0.113993	+	0.993482i	$0.463636\pi$
$8$	0	0
$9$	1950.02	0.891642
$10$	0	0
$11$	− 5226.99i	− 1.18407i	−0.805912	−	0.592035i	$-0.798325\pi$
$11$	− 5226.99i	− 1.18407i	0.805912	−	0.592035i	$-0.201675\pi$
$12$	0	0
$13$	180.982i	0.0228472i	0.999935	+	0.0114236i	$0.00363632\pi$
$13$	180.982i	0.0228472i	−0.999935	+	0.0114236i	$0.996364\pi$
$14$	0	0
$15$	1924.27	0.147213
$16$	0	0
$17$	−34858.6	−1.72083	−0.860417	−	0.509591i	$-0.829797\pi$
$17$	−34858.6	−1.72083	−0.860417	+	0.509591i	$0.829797\pi$
$18$	0	0
$19$	20572.1i	0.688084i	0.938954	+	0.344042i	$0.111796\pi$
$19$	20572.1i	0.688084i	−0.938954	+	0.344042i	$0.888204\pi$
$20$	0	0
$21$	− 3184.98i	− 0.0750479i
$22$	0	0
$23$	89873.4	1.54022	0.770112	−	0.637909i	$-0.220200\pi$
$23$	89873.4	1.54022	0.770112	+	0.637909i	$0.220200\pi$
$24$	0	0
$25$	−15625.0	−0.200000
$26$	0	0
$27$	− 63685.9i	− 0.622688i
$28$	0	0
$29$	82757.7i	0.630108i	0.949074	+	0.315054i	$0.102023\pi$
$29$	82757.7i	0.630108i	−0.949074	+	0.315054i	$0.897977\pi$
$30$	0	0
$31$	158725.	0.956929	0.478464	−	0.878107i	$-0.341194\pi$
$31$	158725.	0.956929	0.478464	+	0.878107i	$0.341194\pi$
$32$	0	0
$33$	−80465.1	−0.389770
$34$	0	0
$35$	25861.9i	0.101958i
$36$	0	0
$37$	− 125771.i	− 0.408202i	−0.978950	−	0.204101i	$-0.934573\pi$
$37$	− 125771.i	− 0.408202i	0.978950	−	0.204101i	$-0.0654271\pi$
$38$	0	0
$39$	2786.06	0.00752081
$40$	0	0
$41$	382738.	0.867277	0.433638	−	0.901087i	$-0.357230\pi$
$41$	382738.	0.867277	0.433638	+	0.901087i	$0.357230\pi$
$42$	0	0
$43$	− 855458.i	− 1.64081i	−0.571781	−	0.820406i	$-0.693747\pi$
$43$	− 855458.i	− 1.64081i	0.571781	−	0.820406i	$-0.306253\pi$
$44$	0	0
$45$	243752.i	0.398754i
$46$	0	0
$47$	320160.	0.449806	0.224903	−	0.974381i	$-0.427793\pi$
$47$	320160.	0.449806	0.224903	+	0.974381i	$0.427793\pi$
$48$	0	0
$49$	−780737.	−0.948023
$50$	0	0
$51$	536619.i	0.566462i
$52$	0	0
$53$	63020.0i	0.0581451i	0.999577	+	0.0290725i	$0.00925538\pi$
$53$	63020.0i	0.0581451i	−0.999577	+	0.0290725i	$0.990745\pi$
$54$	0	0
$55$	653374.	0.529532
$56$	0	0
$57$	316691.	0.226503
$58$	0	0
$59$	− 2.07791e6i	− 1.31718i	−0.752504	−	0.658588i	$-0.771154\pi$
$59$	− 2.07791e6i	− 1.31718i	0.752504	−	0.658588i	$-0.228846\pi$
$60$	0	0
$61$	− 2.88950e6i	− 1.62993i	−0.579513	−	0.814963i	$-0.696757\pi$
$61$	− 2.88950e6i	− 1.62993i	0.579513	−	0.814963i	$-0.303243\pi$
$62$	0	0
$63$	403450.	0.203281
$64$	0	0
$65$	−22622.7	−0.0102176
$66$	0	0
$67$	− 2.31917e6i	− 0.942041i	−0.882122	−	0.471021i	$-0.843886\pi$
$67$	− 2.31917e6i	− 0.942041i	0.882122	−	0.471021i	$-0.156114\pi$
$68$	0	0
$69$	− 1.38352e6i	− 0.507008i
$70$	0	0
$71$	35994.8	0.0119354	0.00596768	−	0.999982i	$-0.498100\pi$
$71$	35994.8	0.0119354	0.00596768	+	0.999982i	$0.498100\pi$
$72$	0	0
$73$	590314.	0.177604	0.0888020	−	0.996049i	$-0.471696\pi$
$73$	590314.	0.177604	0.0888020	+	0.996049i	$0.471696\pi$
$74$	0	0
$75$	240534.i	0.0658357i
$76$	0	0
$77$	− 1.08144e6i	− 0.269951i
$78$	0	0
$79$	3.43960e6	0.784898	0.392449	−	0.919774i	$-0.371628\pi$
$79$	3.43960e6	0.784898	0.392449	+	0.919774i	$0.371628\pi$
$80$	0	0
$81$	3.28430e6	0.686666
$82$	0	0
$83$	517777.i	0.0993961i	0.998764	+	0.0496981i	$0.0158259\pi$
$83$	517777.i	0.0993961i	−0.998764	+	0.0496981i	$0.984174\pi$
$84$	0	0
$85$	− 4.35733e6i	− 0.769580i
$86$	0	0
$87$	1.27398e6	0.207418
$88$	0	0
$89$	1.01219e7	1.52193	0.760966	−	0.648792i	$-0.224725\pi$
$89$	1.01219e7	1.52193	0.760966	+	0.648792i	$0.224725\pi$
$90$	0	0
$91$	37444.2i	0.00520883i
$92$	0	0
$93$	− 2.44344e6i	− 0.315000i
$94$	0	0
$95$	−2.57152e6	−0.307721
$96$	0	0
$97$	6.76059e6	0.752113	0.376057	−	0.926597i	$-0.377280\pi$
$97$	6.76059e6	0.752113	0.376057	+	0.926597i	$0.377280\pi$
$98$	0	0
$99$	− 1.01927e7i	− 1.05577i
$100$	0	0
$101$	− 1.47774e7i	− 1.42716i	−0.700573	−	0.713581i	$-0.747072\pi$
$101$	− 1.47774e7i	− 1.42716i	0.700573	−	0.713581i	$-0.252928\pi$
$102$	0	0
$103$	−6.70183e6	−0.604315	−0.302157	−	0.953258i	$-0.597707\pi$
$103$	−6.70183e6	−0.604315	−0.302157	+	0.953258i	$0.597707\pi$
$104$	0	0
$105$	398122.	0.0335625
$106$	0	0
$107$	− 7.61603e6i	− 0.601015i	−0.953779	−	0.300508i	$-0.902844\pi$
$107$	− 7.61603e6i	− 0.601015i	0.953779	−	0.300508i	$-0.0971561\pi$
$108$	0	0
$109$	6.38375e6i	0.472153i	0.971734	+	0.236077i	$0.0758616\pi$
$109$	6.38375e6i	0.472153i	−0.971734	+	0.236077i	$0.924138\pi$
$110$	0	0
$111$	−1.93614e6	−0.134371
$112$	0	0
$113$	1.34939e7	0.879760	0.439880	−	0.898056i	$-0.355021\pi$
$113$	1.34939e7	0.879760	0.439880	+	0.898056i	$0.355021\pi$
$114$	0	0
$115$	1.12342e7i	0.688809i
$116$	0	0
$117$	352918.i	0.0203715i
$118$	0	0
$119$	−7.21208e6	−0.392325
$120$	0	0
$121$	−7.83426e6	−0.402021
$122$	0	0
$123$	− 5.89192e6i	− 0.285489i
$124$	0	0
$125$	− 1.95312e6i	− 0.0894427i
$126$	0	0
$127$	3.13882e7	1.35973	0.679866	−	0.733336i	$-0.262038\pi$
$127$	3.13882e7	1.35973	0.679866	+	0.733336i	$0.262038\pi$
$128$	0	0
$129$	−1.31690e7	−0.540120
$130$	0	0
$131$	− 5.71983e6i	− 0.222297i	−0.993804	−	0.111149i	$-0.964547\pi$
$131$	− 5.71983e6i	− 0.222297i	0.993804	−	0.111149i	$-0.0354529\pi$
$132$	0	0
$133$	4.25628e6i	0.156873i
$134$	0	0
$135$	7.96074e6	0.278474
$136$	0	0
$137$	−4.18692e7	−1.39115	−0.695573	−	0.718456i	$-0.744849\pi$
$137$	−4.18692e7	−1.39115	−0.695573	+	0.718456i	$0.744849\pi$
$138$	0	0
$139$	− 5.36605e6i	− 0.169474i	−0.996403	−	0.0847370i	$-0.972995\pi$
$139$	− 5.36605e6i	− 0.169474i	0.996403	−	0.0847370i	$-0.0270050\pi$
$140$	0	0
$141$	− 4.92860e6i	− 0.148066i
$142$	0	0
$143$	945990.	0.0270527
$144$	0	0
$145$	−1.03447e7	−0.281793
$146$	0	0
$147$	1.20188e7i	0.312069i
$148$	0	0
$149$	7.03967e7i	1.74341i	0.490027	+	0.871707i	$0.336987\pi$
$149$	7.03967e7i	1.74341i	−0.490027	+	0.871707i	$0.663013\pi$
$150$	0	0
$151$	3.90363e7	0.922675	0.461338	−	0.887225i	$-0.347370\pi$
$151$	3.90363e7	0.922675	0.461338	+	0.887225i	$0.347370\pi$
$152$	0	0
$153$	−6.79750e7	−1.53437
$154$	0	0
$155$	1.98406e7i	0.427952i
$156$	0	0
$157$	− 2.60014e7i	− 0.536227i	−0.963387	−	0.268113i	$-0.913600\pi$
$157$	− 2.60014e7i	− 0.536227i	0.963387	−	0.268113i	$-0.0864002\pi$
$158$	0	0
$159$	970140.	0.0191401
$160$	0	0
$161$	1.85944e7	0.351149
$162$	0	0
$163$	− 8.25776e6i	− 0.149350i	−0.997208	−	0.0746751i	$-0.976208\pi$
$163$	− 8.25776e6i	− 0.149350i	0.997208	−	0.0746751i	$-0.0237920\pi$
$164$	0	0
$165$	− 1.00581e7i	− 0.174311i
$166$	0	0
$167$	−7.79727e7	−1.29549	−0.647746	−	0.761856i	$-0.724288\pi$
$167$	−7.79727e7	−1.29549	−0.647746	+	0.761856i	$0.724288\pi$
$168$	0	0
$169$	6.27158e7	0.999478
$170$	0	0
$171$	4.01161e7i	0.613525i
$172$	0	0
$173$	1.20250e8i	1.76572i	0.469636	+	0.882860i	$0.344385\pi$
$173$	1.20250e8i	1.76572i	−0.469636	+	0.882860i	$0.655615\pi$
$174$	0	0
$175$	−3.23274e6	−0.0455971
$176$	0	0
$177$	−3.19876e7	−0.433586
$178$	0	0
$179$	− 1.01450e7i	− 0.132210i	−0.997813	−	0.0661052i	$-0.978943\pi$
$179$	− 1.01450e7i	− 0.132210i	0.997813	−	0.0661052i	$-0.0210573\pi$
$180$	0	0
$181$	8.28437e7i	1.03845i	0.854638	+	0.519224i	$0.173779\pi$
$181$	8.28437e7i	1.03845i	−0.854638	+	0.519224i	$0.826221\pi$
$182$	0	0
$183$	−4.44814e7	−0.536537
$184$	0	0
$185$	1.57214e7	0.182553
$186$	0	0
$187$	1.82206e8i	2.03759i
$188$	0	0
$189$	− 1.31763e7i	− 0.141964i
$190$	0	0
$191$	−1.47539e8	−1.53211	−0.766055	−	0.642774i	$-0.777783\pi$
$191$	−1.47539e8	−1.53211	−0.766055	+	0.642774i	$0.777783\pi$
$192$	0	0
$193$	−4.97985e6	−0.0498615	−0.0249307	−	0.999689i	$-0.507937\pi$
$193$	−4.97985e6	−0.0498615	−0.0249307	+	0.999689i	$0.507937\pi$
$194$	0	0
$195$	348258.i	0.00336341i
$196$	0	0
$197$	− 1.72498e8i	− 1.60751i	−0.594961	−	0.803755i	$-0.702832\pi$
$197$	− 1.72498e8i	− 1.60751i	0.594961	−	0.803755i	$-0.297168\pi$
$198$	0	0
$199$	−8.94277e7	−0.804427	−0.402213	−	0.915546i	$-0.631759\pi$
$199$	−8.94277e7	−0.804427	−0.402213	+	0.915546i	$0.631759\pi$
$200$	0	0
$201$	−3.57016e7	−0.310100
$202$	0	0
$203$	1.71222e7i	0.143656i
$204$	0	0
$205$	4.78422e7i	0.387858i
$206$	0	0
$207$	1.75255e8	1.37333
$208$	0	0
$209$	1.07530e8	0.814740
$210$	0	0
$211$	− 8.01536e7i	− 0.587401i	−0.955898	−	0.293700i	$-0.905113\pi$
$211$	− 8.01536e7i	− 0.587401i	0.955898	−	0.293700i	$-0.0948868\pi$
$212$	0	0
$213$	− 554110.i	− 0.00392886i
$214$	0	0
$215$	1.06932e8	0.733794
$216$	0	0
$217$	3.28395e7	0.218166
$218$	0	0
$219$	− 9.08738e6i	− 0.0584634i
$220$	0	0
$221$	− 6.30877e6i	− 0.0393162i
$222$	0	0
$223$	−2.67224e8	−1.61365	−0.806824	−	0.590791i	$-0.798816\pi$
$223$	−2.67224e8	−1.61365	−0.806824	+	0.590791i	$0.798816\pi$
$224$	0	0
$225$	−3.04691e7	−0.178328
$226$	0	0
$227$	2.95568e8i	1.67713i	0.544799	+	0.838567i	$0.316606\pi$
$227$	2.95568e8i	1.67713i	−0.544799	+	0.838567i	$0.683394\pi$
$228$	0	0
$229$	2.49496e8i	1.37290i	0.727177	+	0.686450i	$0.240832\pi$
$229$	2.49496e8i	1.37290i	−0.727177	+	0.686450i	$0.759168\pi$
$230$	0	0
$231$	−1.66478e7	−0.0888620
$232$	0	0
$233$	4.03293e7	0.208870	0.104435	−	0.994532i	$-0.466697\pi$
$233$	4.03293e7	0.208870	0.104435	+	0.994532i	$0.466697\pi$
$234$	0	0
$235$	4.00201e7i	0.201159i
$236$	0	0
$237$	− 5.29498e7i	− 0.258372i
$238$	0	0
$239$	1.88275e8	0.892072	0.446036	−	0.895015i	$-0.352835\pi$
$239$	1.88275e8	0.892072	0.446036	+	0.895015i	$0.352835\pi$
$240$	0	0
$241$	−8.07295e7	−0.371512	−0.185756	−	0.982596i	$-0.559473\pi$
$241$	−8.07295e7	−0.371512	−0.185756	+	0.982596i	$0.559473\pi$
$242$	0	0
$243$	− 1.89840e8i	− 0.848723i
$244$	0	0
$245$	− 9.75922e7i	− 0.423969i
$246$	0	0
$247$	−3.72318e6	−0.0157208
$248$	0	0
$249$	7.97074e6	0.0327191
$250$	0	0
$251$	− 4.09776e8i	− 1.63564i	−0.575472	−	0.817821i	$-0.695182\pi$
$251$	− 4.09776e8i	− 1.63564i	0.575472	−	0.817821i	$-0.304818\pi$
$252$	0	0
$253$	− 4.69767e8i	− 1.82373i
$254$	0	0
$255$	−6.70774e7	−0.253329
$256$	0	0
$257$	−1.56457e8	−0.574950	−0.287475	−	0.957788i	$-0.592816\pi$
$257$	−1.56457e8	−0.574950	−0.287475	+	0.957788i	$0.592816\pi$
$258$	0	0
$259$	− 2.60214e7i	− 0.0930641i
$260$	0	0
$261$	1.61379e8i	0.561831i
$262$	0	0
$263$	1.72821e8	0.585801	0.292901	−	0.956143i	$-0.405379\pi$
$263$	1.72821e8	0.585801	0.292901	+	0.956143i	$0.405379\pi$
$264$	0	0
$265$	−7.87750e6	−0.0260033
$266$	0	0
$267$	− 1.55817e8i	− 0.500987i
$268$	0	0
$269$	2.07890e8i	0.651179i	0.945511	+	0.325589i	$0.105563\pi$
$269$	2.07890e8i	0.651179i	−0.945511	+	0.325589i	$0.894437\pi$
$270$	0	0
$271$	1.26178e8	0.385115	0.192558	−	0.981286i	$-0.438322\pi$
$271$	1.26178e8	0.385115	0.192558	+	0.981286i	$0.438322\pi$
$272$	0	0
$273$	576423.	0.00171463
$274$	0	0
$275$	8.16717e7i	0.236814i
$276$	0	0
$277$	− 6.18270e8i	− 1.74783i	−0.486079	−	0.873915i	$-0.661573\pi$
$277$	− 6.18270e8i	− 1.74783i	0.486079	−	0.873915i	$-0.338427\pi$
$278$	0	0
$279$	3.09517e8	0.853238
$280$	0	0
$281$	−1.01028e8	−0.271625	−0.135812	−	0.990735i	$-0.543364\pi$
$281$	−1.01028e8	−0.271625	−0.135812	+	0.990735i	$0.543364\pi$
$282$	0	0
$283$	− 2.99096e8i	− 0.784437i	−0.919872	−	0.392218i	$-0.871708\pi$
$283$	− 2.99096e8i	− 0.784437i	0.919872	−	0.392218i	$-0.128292\pi$
$284$	0	0
$285$	3.95863e7i	0.101295i
$286$	0	0
$287$	7.91866e7	0.197727
$288$	0	0
$289$	8.04785e8	1.96127
$290$	0	0
$291$	− 1.04074e8i	− 0.247580i
$292$	0	0
$293$	3.84747e7i	0.0893590i	0.999001	+	0.0446795i	$0.0142267\pi$
$293$	3.84747e7i	0.0893590i	−0.999001	+	0.0446795i	$0.985773\pi$
$294$	0	0
$295$	2.59738e8	0.589059
$296$	0	0
$297$	−3.32886e8	−0.737306
$298$	0	0
$299$	1.62654e7i	0.0351898i
$300$	0	0
$301$	− 1.76990e8i	− 0.374082i
$302$	0	0
$303$	−2.27486e8	−0.469791
$304$	0	0
$305$	3.61187e8	0.728925
$306$	0	0
$307$	− 5.27312e8i	− 1.04012i	−0.854130	−	0.520060i	$-0.825910\pi$
$307$	− 5.27312e8i	− 1.04012i	0.854130	−	0.520060i	$-0.174090\pi$
$308$	0	0
$309$	1.03169e8i	0.198927i
$310$	0	0
$311$	−1.06674e8	−0.201093	−0.100546	−	0.994932i	$-0.532059\pi$
$311$	−1.06674e8	−0.201093	−0.100546	+	0.994932i	$0.532059\pi$
$312$	0	0
$313$	−2.65874e8	−0.490085	−0.245042	−	0.969512i	$-0.578802\pi$
$313$	−2.65874e8	−0.490085	−0.245042	+	0.969512i	$0.578802\pi$
$314$	0	0
$315$	5.04312e7i	0.0909102i
$316$	0	0
$317$	6.09184e8i	1.07409i	0.843553	+	0.537046i	$0.180460\pi$
$317$	6.09184e8i	1.07409i	−0.843553	+	0.537046i	$0.819540\pi$
$318$	0	0
$319$	4.32574e8	0.746092
$320$	0	0
$321$	−1.17242e8	−0.197841
$322$	0	0
$323$	− 7.17116e8i	− 1.18408i
$324$	0	0
$325$	− 2.82784e6i	− 0.00456944i
$326$	0	0
$327$	9.82724e7	0.155423
$328$	0	0
$329$	6.62397e7	0.102549
$330$	0	0
$331$	9.96510e8i	1.51037i	0.655511	+	0.755186i	$0.272453\pi$
$331$	9.96510e8i	1.51037i	−0.655511	+	0.755186i	$0.727547\pi$
$332$	0	0
$333$	− 2.45256e8i	− 0.363970i
$334$	0	0
$335$	2.89896e8	0.421294
$336$	0	0
$337$	5.29532e8	0.753681	0.376840	−	0.926278i	$-0.377011\pi$
$337$	5.29532e8	0.753681	0.376840	+	0.926278i	$0.377011\pi$
$338$	0	0
$339$	− 2.07728e8i	− 0.289598i
$340$	0	0
$341$	− 8.29655e8i	− 1.13307i
$342$	0	0
$343$	−3.31918e8	−0.444121
$344$	0	0
$345$	1.72941e8	0.226741
$346$	0	0
$347$	5.37311e8i	0.690356i	0.938537	+	0.345178i	$0.112181\pi$
$347$	5.37311e8i	0.690356i	−0.938537	+	0.345178i	$0.887819\pi$
$348$	0	0
$349$	− 5.21999e8i	− 0.657325i	−0.944447	−	0.328663i	$-0.893402\pi$
$349$	− 5.21999e8i	− 0.657325i	0.944447	−	0.328663i	$-0.106598\pi$
$350$	0	0
$351$	1.15260e7	0.0142267
$352$	0	0
$353$	−1.43718e9	−1.73900	−0.869501	−	0.493932i	$-0.835559\pi$
$353$	−1.43718e9	−1.73900	−0.869501	+	0.493932i	$0.835559\pi$
$354$	0	0
$355$	4.49935e6i	0.00533766i
$356$	0	0
$357$	1.11024e8i	0.129145i
$358$	0	0
$359$	9.26349e8	1.05668	0.528341	−	0.849032i	$-0.322814\pi$
$359$	9.26349e8	1.05668	0.528341	+	0.849032i	$0.322814\pi$
$360$	0	0
$361$	4.70659e8	0.526540
$362$	0	0
$363$	1.20602e8i	0.132337i
$364$	0	0
$365$	7.37892e7i	0.0794269i
$366$	0	0
$367$	−1.07900e9	−1.13943	−0.569717	−	0.821841i	$-0.692947\pi$
$367$	−1.07900e9	−1.13943	−0.569717	+	0.821841i	$0.692947\pi$
$368$	0	0
$369$	7.46346e8	0.773300
$370$	0	0
$371$	1.30385e7i	0.0132562i
$372$	0	0
$373$	− 3.73680e8i	− 0.372837i	−0.982470	−	0.186419i	$-0.940312\pi$
$373$	− 3.73680e8i	− 0.372837i	0.982470	−	0.186419i	$-0.0596881\pi$
$374$	0	0
$375$	−3.00667e7	−0.0294426
$376$	0	0
$377$	−1.49776e7	−0.0143962
$378$	0	0
$379$	1.90537e9i	1.79780i	0.438156	+	0.898899i	$0.355632\pi$
$379$	1.90537e9i	1.79780i	−0.438156	+	0.898899i	$0.644368\pi$
$380$	0	0
$381$	− 4.83195e8i	− 0.447595i
$382$	0	0
$383$	−1.51577e9	−1.37860	−0.689298	−	0.724478i	$-0.742081\pi$
$383$	−1.51577e9	−1.37860	−0.689298	+	0.724478i	$0.742081\pi$
$384$	0	0
$385$	1.35180e8	0.120726
$386$	0	0
$387$	− 1.66816e9i	− 1.46302i
$388$	0	0
$389$	2.14501e9i	1.84759i	0.382887	+	0.923795i	$0.374930\pi$
$389$	2.14501e9i	1.84759i	−0.382887	+	0.923795i	$0.625070\pi$
$390$	0	0
$391$	−3.13286e9	−2.65047
$392$	0	0
$393$	−8.80520e7	−0.0731754
$394$	0	0
$395$	4.29950e8i	0.351017i
$396$	0	0
$397$	1.65883e9i	1.33056i	0.746594	+	0.665280i	$0.231688\pi$
$397$	1.65883e9i	1.33056i	−0.746594	+	0.665280i	$0.768312\pi$
$398$	0	0
$399$	6.55218e7	0.0516393
$400$	0	0
$401$	−1.68334e9	−1.30367	−0.651833	−	0.758363i	$-0.726000\pi$
$401$	−1.68334e9	−1.30367	−0.651833	+	0.758363i	$0.726000\pi$
$402$	0	0
$403$	2.87263e7i	0.0218631i
$404$	0	0
$405$	4.10538e8i	0.307086i
$406$	0	0
$407$	−6.57404e8	−0.483339
$408$	0	0
$409$	2.00857e9	1.45163	0.725815	−	0.687890i	$-0.241463\pi$
$409$	2.00857e9	1.45163	0.725815	+	0.687890i	$0.241463\pi$
$410$	0	0
$411$	6.44541e8i	0.457935i
$412$	0	0
$413$	− 4.29909e8i	− 0.300297i
$414$	0	0
$415$	−6.47221e7	−0.0444513
$416$	0	0
$417$	−8.26058e7	−0.0557872
$418$	0	0
$419$	2.31648e9i	1.53843i	0.638988	+	0.769217i	$0.279353\pi$
$419$	2.31648e9i	1.53843i	−0.638988	+	0.769217i	$0.720647\pi$
$420$	0	0
$421$	− 8.89530e8i	− 0.580996i	−0.956876	−	0.290498i	$-0.906179\pi$
$421$	− 8.89530e8i	− 0.580996i	0.956876	−	0.290498i	$-0.0938210\pi$
$422$	0	0
$423$	6.24319e8	0.401066
$424$	0	0
$425$	5.44666e8	0.344167
$426$	0	0
$427$	− 5.97823e8i	− 0.371600i
$428$	0	0
$429$	− 1.45627e7i	− 0.00890516i
$430$	0	0
$431$	2.58409e9	1.55467	0.777334	−	0.629088i	$-0.216572\pi$
$431$	2.58409e9	1.55467	0.777334	+	0.629088i	$0.216572\pi$
$432$	0	0
$433$	−1.00522e9	−0.595049	−0.297525	−	0.954714i	$-0.596161\pi$
$433$	−1.00522e9	−0.595049	−0.297525	+	0.954714i	$0.596161\pi$
$434$	0	0
$435$	1.59248e8i	0.0927602i
$436$	0	0
$437$	1.84889e9i	1.05980i
$438$	0	0
$439$	2.09435e9	1.18147	0.590735	−	0.806866i	$-0.298838\pi$
$439$	2.09435e9	1.18147	0.590735	+	0.806866i	$0.298838\pi$
$440$	0	0
$441$	−1.52245e9	−0.845296
$442$	0	0
$443$	6.18371e8i	0.337937i	0.985621	+	0.168969i	$0.0540436\pi$
$443$	6.18371e8i	0.337937i	−0.985621	+	0.168969i	$0.945956\pi$
$444$	0	0
$445$	1.26523e9i	0.680629i
$446$	0	0
$447$	1.08370e9	0.573894
$448$	0	0
$449$	−2.19183e9	−1.14274	−0.571368	−	0.820694i	$-0.693587\pi$
$449$	−2.19183e9	−1.14274	−0.571368	+	0.820694i	$0.693587\pi$
$450$	0	0
$451$	− 2.00057e9i	− 1.02692i
$452$	0	0
$453$	− 6.00930e8i	− 0.303725i
$454$	0	0
$455$	−4.68053e6	−0.00232946
$456$	0	0
$457$	5.87698e8	0.288037	0.144018	−	0.989575i	$-0.453998\pi$
$457$	5.87698e8	0.288037	0.144018	+	0.989575i	$0.453998\pi$
$458$	0	0
$459$	2.22000e9i	1.07154i
$460$	0	0
$461$	2.08054e9i	0.989060i	0.869161	+	0.494530i	$0.164660\pi$
$461$	2.08054e9i	0.989060i	−0.869161	+	0.494530i	$0.835340\pi$
$462$	0	0
$463$	−3.20279e9	−1.49967	−0.749833	−	0.661627i	$-0.769866\pi$
$463$	−3.20279e9	−1.49967	−0.749833	+	0.661627i	$0.769866\pi$
$464$	0	0
$465$	3.05430e8	0.140872
$466$	0	0
$467$	2.17564e8i	0.0988504i	0.998778	+	0.0494252i	$0.0157389\pi$
$467$	2.17564e8i	0.0988504i	−0.998778	+	0.0494252i	$0.984261\pi$
$468$	0	0
$469$	− 4.79824e8i	− 0.214772i
$470$	0	0
$471$	−4.00270e8	−0.176514
$472$	0	0
$473$	−4.47147e9	−1.94284
$474$	0	0
$475$	− 3.21440e8i	− 0.137617i
$476$	0	0
$477$	1.22890e8i	0.0518446i
$478$	0	0
$479$	4.24977e9	1.76681	0.883407	−	0.468607i	$-0.155244\pi$
$479$	4.24977e9	1.76681	0.883407	+	0.468607i	$0.155244\pi$
$480$	0	0
$481$	2.27623e7	0.00932627
$482$	0	0
$483$	− 2.86245e8i	− 0.115591i
$484$	0	0
$485$	8.45073e8i	0.336355i
$486$	0	0
$487$	−3.58617e8	−0.140695	−0.0703476	−	0.997523i	$-0.522411\pi$
$487$	−3.58617e8	−0.140695	−0.0703476	+	0.997523i	$0.522411\pi$
$488$	0	0
$489$	−1.27121e8	−0.0491629
$490$	0	0
$491$	− 9.53136e8i	− 0.363387i	−0.983355	−	0.181694i	$-0.941842\pi$
$491$	− 9.53136e8i	− 0.363387i	0.983355	−	0.181694i	$-0.0581579\pi$
$492$	0	0
$493$	− 2.88482e9i	− 1.08431i
$494$	0	0
$495$	1.27409e9	0.472153
$496$	0	0
$497$	7.44715e6	0.00272109
$498$	0	0
$499$	− 2.10234e9i	− 0.757446i	−0.925510	−	0.378723i	$-0.876363\pi$
$499$	− 2.10234e9i	− 0.757446i	0.925510	−	0.378723i	$-0.123637\pi$
$500$	0	0
$501$	1.20032e9i	0.426448i
$502$	0	0
$503$	−2.06991e9	−0.725210	−0.362605	−	0.931943i	$-0.618113\pi$
$503$	−2.06991e9	−0.725210	−0.362605	+	0.931943i	$0.618113\pi$
$504$	0	0
$505$	1.84718e9	0.638246
$506$	0	0
$507$	− 9.65456e8i	− 0.329007i
$508$	0	0
$509$	1.29240e8i	0.0434394i	0.999764	+	0.0217197i	$0.00691414\pi$
$509$	1.29240e8i	0.0434394i	−0.999764	+	0.0217197i	$0.993086\pi$
$510$	0	0
$511$	1.22133e8	0.0404911
$512$	0	0
$513$	1.31016e9	0.428462
$514$	0	0
$515$	− 8.37729e8i	− 0.270258i
$516$	0	0
$517$	− 1.67348e9i	− 0.532602i
$518$	0	0
$519$	1.85114e9	0.581237
$520$	0	0
$521$	9.14488e8	0.283300	0.141650	−	0.989917i	$-0.454759\pi$
$521$	9.14488e8	0.283300	0.141650	+	0.989917i	$0.454759\pi$
$522$	0	0
$523$	3.51620e9i	1.07478i	0.843335	+	0.537388i	$0.180589\pi$
$523$	3.51620e9i	1.07478i	−0.843335	+	0.537388i	$0.819411\pi$
$524$	0	0
$525$	4.97653e7i	0.0150096i
$526$	0	0
$527$	−5.53294e9	−1.64672
$528$	0	0
$529$	4.67240e9	1.37229
$530$	0	0
$531$	− 4.05196e9i	− 1.17445i
$532$	0	0
$533$	6.92685e7i	0.0198148i
$534$	0	0
$535$	9.52004e8	0.268782
$536$	0	0
$537$	−1.56173e8	−0.0435208
$538$	0	0
$539$	4.08091e9i	1.12252i
$540$	0	0
$541$	− 2.84793e9i	− 0.773285i	−0.922230	−	0.386642i	$-0.873635\pi$
$541$	− 2.84793e9i	− 0.773285i	0.922230	−	0.386642i	$-0.126365\pi$
$542$	0	0
$543$	1.27531e9	0.341835
$544$	0	0
$545$	−7.97968e8	−0.211153
$546$	0	0
$547$	− 4.35740e9i	− 1.13834i	−0.822220	−	0.569169i	$-0.807265\pi$
$547$	− 4.35740e9i	− 1.13834i	0.822220	−	0.569169i	$-0.192735\pi$
$548$	0	0
$549$	− 5.63458e9i	− 1.45331i
$550$	0	0
$551$	−1.70250e9	−0.433568
$552$	0	0
$553$	7.11637e8	0.178945
$554$	0	0
$555$	− 2.42018e8i	− 0.0600927i
$556$	0	0
$557$	− 6.77954e9i	− 1.66229i	−0.556055	−	0.831145i	$-0.687686\pi$
$557$	− 6.77954e9i	− 1.66229i	0.556055	−	0.831145i	$-0.312314\pi$
$558$	0	0
$559$	1.54822e8	0.0374880
$560$	0	0
$561$	2.80490e9	0.670730
$562$	0	0
$563$	− 6.65687e9i	− 1.57214i	−0.618139	−	0.786069i	$-0.712113\pi$
$563$	− 6.65687e9i	− 1.57214i	0.618139	−	0.786069i	$-0.287887\pi$
$564$	0	0
$565$	1.68674e9i	0.393441i
$566$	0	0
$567$	6.79506e8	0.156550
$568$	0	0
$569$	7.16957e9	1.63155	0.815774	−	0.578370i	$-0.196311\pi$
$569$	7.16957e9	1.63155	0.815774	+	0.578370i	$0.196311\pi$
$570$	0	0
$571$	1.62406e9i	0.365071i	0.983199	+	0.182535i	$0.0584304\pi$
$571$	1.62406e9i	0.365071i	−0.983199	+	0.182535i	$0.941570\pi$
$572$	0	0
$573$	2.27124e9i	0.504338i
$574$	0	0
$575$	−1.40427e9	−0.308045
$576$	0	0
$577$	6.98294e8	0.151329	0.0756646	−	0.997133i	$-0.475892\pi$
$577$	6.98294e8	0.151329	0.0756646	+	0.997133i	$0.475892\pi$
$578$	0	0
$579$	7.66605e7i	0.0164133i
$580$	0	0
$581$	1.07126e8i	0.0226609i
$582$	0	0
$583$	3.29405e8	0.0688478
$584$	0	0
$585$	−4.41147e7	−0.00911042
$586$	0	0
$587$	6.97861e9i	1.42408i	0.702137	+	0.712042i	$0.252229\pi$
$587$	6.97861e9i	1.42408i	−0.702137	+	0.712042i	$0.747771\pi$
$588$	0	0
$589$	3.26531e9i	0.658448i
$590$	0	0
$591$	−2.65547e9	−0.529158
$592$	0	0
$593$	−1.94954e9	−0.383920	−0.191960	−	0.981403i	$-0.561484\pi$
$593$	−1.94954e9	−0.383920	−0.191960	+	0.981403i	$0.561484\pi$
$594$	0	0
$595$	− 9.01510e8i	− 0.175453i
$596$	0	0
$597$	1.37666e9i	0.264800i
$598$	0	0
$599$	−3.06381e9	−0.582463	−0.291232	−	0.956653i	$-0.594065\pi$
$599$	−3.06381e9	−0.582463	−0.291232	+	0.956653i	$0.594065\pi$
$600$	0	0
$601$	−5.08283e8	−0.0955091	−0.0477545	−	0.998859i	$-0.515207\pi$
$601$	−5.08283e8	−0.0955091	−0.0477545	+	0.998859i	$0.515207\pi$
$602$	0	0
$603$	− 4.52242e9i	− 0.839963i
$604$	0	0
$605$	− 9.79282e8i	− 0.179789i
$606$	0	0
$607$	8.96433e9	1.62689	0.813444	−	0.581644i	$-0.197590\pi$
$607$	8.96433e9	1.62689	0.813444	+	0.581644i	$0.197590\pi$
$608$	0	0
$609$	2.63581e8	0.0472883
$610$	0	0
$611$	5.79432e7i	0.0102768i
$612$	0	0
$613$	− 9.47512e9i	− 1.66140i	−0.556724	−	0.830698i	$-0.687942\pi$
$613$	− 9.47512e9i	− 1.66140i	0.556724	−	0.830698i	$-0.312058\pi$
$614$	0	0
$615$	7.36490e8	0.127675
$616$	0	0
$617$	3.17374e9	0.543968	0.271984	−	0.962302i	$-0.412320\pi$
$617$	3.17374e9	0.543968	0.271984	+	0.962302i	$0.412320\pi$
$618$	0	0
$619$	8.06198e6i	0.00136623i	1.00000		0.000683116i	$0.000217443\pi$
$619$	8.06198e6i	0.00136623i	−1.00000		0.000683116i	$0.999783\pi$
$620$	0	0
$621$	− 5.72367e9i	− 0.959078i
$622$	0	0
$623$	2.09416e9	0.346978
$624$	0	0
$625$	2.44141e8	0.0400000
$626$	0	0
$627$	− 1.65534e9i	− 0.268195i
$628$	0	0
$629$	4.38421e9i	0.702448i
$630$	0	0
$631$	9.94224e9	1.57537	0.787683	−	0.616080i	$-0.211280\pi$
$631$	9.94224e9	1.57537	0.787683	+	0.616080i	$0.211280\pi$
$632$	0	0
$633$	−1.23390e9	−0.193360
$634$	0	0
$635$	3.92353e9i	0.608091i
$636$	0	0
$637$	− 1.41299e8i	− 0.0216597i
$638$	0	0
$639$	7.01906e7	0.0106421
$640$	0	0
$641$	−7.72262e9	−1.15814	−0.579071	−	0.815277i	$-0.696585\pi$
$641$	−7.72262e9	−1.15814	−0.579071	+	0.815277i	$0.696585\pi$
$642$	0	0
$643$	− 6.30325e9i	− 0.935031i	−0.883985	−	0.467516i	$-0.845149\pi$
$643$	− 6.30325e9i	− 0.935031i	0.883985	−	0.467516i	$-0.154851\pi$
$644$	0	0
$645$	− 1.64613e9i	− 0.241549i
$646$	0	0
$647$	−7.64503e8	−0.110972	−0.0554861	−	0.998459i	$-0.517671\pi$
$647$	−7.64503e8	−0.110972	−0.0554861	+	0.998459i	$0.517671\pi$
$648$	0	0
$649$	−1.08612e10	−1.55963
$650$	0	0
$651$	− 5.05536e8i	− 0.0718155i
$652$	0	0
$653$	1.13012e10i	1.58828i	0.607732	+	0.794142i	$0.292080\pi$
$653$	1.13012e10i	1.58828i	−0.607732	+	0.794142i	$0.707920\pi$
$654$	0	0
$655$	7.14979e8	0.0994143
$656$	0	0
$657$	1.15112e9	0.158359
$658$	0	0
$659$	− 8.18496e9i	− 1.11408i	−0.830484	−	0.557042i	$-0.811936\pi$
$659$	− 8.18496e9i	− 1.11408i	0.830484	−	0.557042i	$-0.188064\pi$
$660$	0	0
$661$	− 2.61103e8i	− 0.0351647i	−0.999845	−	0.0175823i	$-0.994403\pi$
$661$	− 2.61103e8i	− 0.0351647i	0.999845	−	0.0175823i	$-0.00559692\pi$
$662$	0	0
$663$	−9.71182e7	−0.0129421
$664$	0	0
$665$	−5.32034e8	−0.0701559
$666$	0	0
$667$	7.43771e9i	0.970508i
$668$	0	0
$669$	4.11369e9i	0.531178i
$670$	0	0
$671$	−1.51034e10	−1.92995
$672$	0	0
$673$	1.64656e9	0.208221	0.104110	−	0.994566i	$-0.466800\pi$
$673$	1.64656e9	0.208221	0.104110	+	0.994566i	$0.466800\pi$
$674$	0	0
$675$	9.95093e8i	0.124538i
$676$	0	0
$677$	7.17917e9i	0.889230i	0.895722	+	0.444615i	$0.146659\pi$
$677$	7.17917e9i	0.889230i	−0.895722	+	0.444615i	$0.853341\pi$
$678$	0	0
$679$	1.39873e9	0.171471
$680$	0	0
$681$	4.55003e9	0.552076
$682$	0	0
$683$	1.40800e10i	1.69095i	0.534015	+	0.845475i	$0.320682\pi$
$683$	1.40800e10i	1.69095i	−0.534015	+	0.845475i	$0.679318\pi$
$684$	0	0
$685$	− 5.23365e9i	− 0.622139i
$686$	0	0
$687$	3.84078e9	0.451929
$688$	0	0
$689$	−1.14055e7	−0.00132845
$690$	0	0
$691$	− 4.46261e9i	− 0.514536i	−0.966340	−	0.257268i	$-0.917178\pi$
$691$	− 4.46261e9i	− 0.514536i	0.966340	−	0.257268i	$-0.0828224\pi$
$692$	0	0
$693$	− 2.10883e9i	− 0.240699i
$694$	0	0
$695$	6.70757e8	0.0757911
$696$	0	0
$697$	−1.33417e10	−1.49244
$698$	0	0
$699$	− 6.20836e8i	− 0.0687554i
$700$	0	0
$701$	1.39996e10i	1.53497i	0.641064	+	0.767487i	$0.278493\pi$
$701$	1.39996e10i	1.53497i	−0.641064	+	0.767487i	$0.721507\pi$
$702$	0	0
$703$	2.58738e9	0.280877
$704$	0	0
$705$	6.16075e8	0.0662173
$706$	0	0
$707$	− 3.05737e9i	− 0.325372i
$708$	0	0
$709$	8.55580e9i	0.901569i	0.892633	+	0.450785i	$0.148856\pi$
$709$	8.55580e9i	0.901569i	−0.892633	+	0.450785i	$0.851144\pi$
$710$	0	0
$711$	6.70729e9	0.699848
$712$	0	0
$713$	1.42652e10	1.47388
$714$	0	0
$715$	1.18249e8i	0.0120983i
$716$	0	0
$717$	− 2.89833e9i	− 0.293651i
$718$	0	0
$719$	−1.06358e10	−1.06714	−0.533569	−	0.845757i	$-0.679149\pi$
$719$	−1.06358e10	−1.06714	−0.533569	+	0.845757i	$0.679149\pi$
$720$	0	0
$721$	−1.38658e9	−0.137775
$722$	0	0
$723$	1.24276e9i	0.122294i
$724$	0	0
$725$	− 1.29309e9i	− 0.126022i
$726$	0	0
$727$	−5.54600e9	−0.535315	−0.267657	−	0.963514i	$-0.586250\pi$
$727$	−5.54600e9	−0.535315	−0.267657	+	0.963514i	$0.586250\pi$
$728$	0	0
$729$	4.26034e9	0.407285
$730$	0	0
$731$	2.98201e10i	2.82357i
$732$	0	0
$733$	2.09767e9i	0.196731i	0.995150	+	0.0983656i	$0.0313615\pi$
$733$	2.09767e9i	0.196731i	−0.995150	+	0.0983656i	$0.968639\pi$
$734$	0	0
$735$	−1.50235e9	−0.139561
$736$	0	0
$737$	−1.21223e10	−1.11544
$738$	0	0
$739$	− 2.13547e10i	− 1.94643i	−0.229906	−	0.973213i	$-0.573842\pi$
$739$	− 2.13547e10i	− 1.94643i	0.229906	−	0.973213i	$-0.426158\pi$
$740$	0	0
$741$	5.73152e7i	0.00517495i
$742$	0	0
$743$	−1.14077e10	−1.02033	−0.510163	−	0.860078i	$-0.670415\pi$
$743$	−1.14077e10	−1.02033	−0.510163	+	0.860078i	$0.670415\pi$
$744$	0	0
$745$	−8.79959e9	−0.779679
$746$	0	0
$747$	1.00968e9i	0.0886257i
$748$	0	0
$749$	− 1.57572e9i	− 0.137023i
$750$	0	0
$751$	5.67889e9	0.489242	0.244621	−	0.969619i	$-0.421337\pi$
$751$	5.67889e9	0.489242	0.244621	+	0.969619i	$0.421337\pi$
$752$	0	0
$753$	−6.30816e9	−0.538419
$754$	0	0
$755$	4.87953e9i	0.412633i
$756$	0	0
$757$	− 1.85210e9i	− 0.155177i	−0.996985	−	0.0775887i	$-0.975278\pi$
$757$	− 1.85210e9i	− 0.155177i	0.996985	−	0.0775887i	$-0.0247221\pi$
$758$	0	0
$759$	−7.23167e9	−0.600333
$760$	0	0
$761$	−5.34803e9	−0.439893	−0.219947	−	0.975512i	$-0.570588\pi$
$761$	−5.34803e9	−0.439893	−0.219947	+	0.975512i	$0.570588\pi$
$762$	0	0
$763$	1.32077e9i	0.107644i
$764$	0	0
$765$	− 8.49688e9i	− 0.686190i
$766$	0	0
$767$	3.76063e8	0.0300938
$768$	0	0
$769$	−8.10347e8	−0.0642583	−0.0321291	−	0.999484i	$-0.510229\pi$
$769$	−8.10347e8	−0.0642583	−0.0321291	+	0.999484i	$0.510229\pi$
$770$	0	0
$771$	2.40853e9i	0.189261i
$772$	0	0
$773$	2.04191e10i	1.59004i	0.606583	+	0.795021i	$0.292540\pi$
$773$	2.04191e10i	1.59004i	−0.606583	+	0.795021i	$0.707460\pi$
$774$	0	0
$775$	−2.48008e9	−0.191386
$776$	0	0
$777$	−4.00578e8	−0.0306347
$778$	0	0
$779$	7.87373e9i	0.596760i
$780$	0	0
$781$	− 1.88144e8i	− 0.0141323i
$782$	0	0
$783$	5.27050e9	0.392361
$784$	0	0
$785$	3.25018e9	0.239808
$786$	0	0
$787$	9.31284e9i	0.681037i	0.940238	+	0.340518i	$0.110603\pi$
$787$	9.31284e9i	0.681037i	−0.940238	+	0.340518i	$0.889397\pi$
$788$	0	0
$789$	− 2.66043e9i	− 0.192833i
$790$	0	0
$791$	2.79183e9	0.200573
$792$	0	0
$793$	5.22946e8	0.0372392
$794$	0	0
$795$	1.21267e8i	0.00855972i
$796$	0	0
$797$	− 3.99689e9i	− 0.279652i	−0.990176	−	0.139826i	$-0.955346\pi$
$797$	− 3.99689e9i	− 0.279652i	0.990176	−	0.139826i	$-0.0446543\pi$
$798$	0	0
$799$	−1.11604e10	−0.774041
$800$	0	0
$801$	1.97378e10	1.35702
$802$	0	0
$803$	− 3.08556e9i	− 0.210296i
$804$	0	0
$805$	2.32430e9i	0.157038i
$806$	0	0
$807$	3.20029e9	0.214354
$808$	0	0
$809$	−2.60637e10	−1.73068	−0.865338	−	0.501189i	$-0.832896\pi$
$809$	−2.60637e10	−1.73068	−0.865338	+	0.501189i	$0.832896\pi$
$810$	0	0
$811$	− 2.92385e9i	− 0.192479i	−0.995358	−	0.0962394i	$-0.969319\pi$
$811$	− 2.92385e9i	− 0.192479i	0.995358	−	0.0962394i	$-0.0306814\pi$
$812$	0	0
$813$	− 1.94240e9i	− 0.126772i
$814$	0	0
$815$	1.03222e9	0.0667914
$816$	0	0
$817$	1.75986e10	1.12902
$818$	0	0
$819$	7.30170e7i	0.00464441i
$820$	0	0
$821$	− 4.25354e9i	− 0.268256i	−0.990964	−	0.134128i	$-0.957177\pi$
$821$	− 4.25354e9i	− 0.268256i	0.990964	−	0.134128i	$-0.0428233\pi$
$822$	0	0
$823$	1.65577e10	1.03538	0.517691	−	0.855568i	$-0.326792\pi$
$823$	1.65577e10	1.03538	0.517691	+	0.855568i	$0.326792\pi$
$824$	0	0
$825$	1.25727e9	0.0779541
$826$	0	0
$827$	2.53453e10i	1.55822i	0.626888	+	0.779110i	$0.284328\pi$
$827$	2.53453e10i	1.55822i	−0.626888	+	0.779110i	$0.715672\pi$
$828$	0	0
$829$	3.78621e9i	0.230815i	0.993318	+	0.115407i	$0.0368173\pi$
$829$	3.78621e9i	0.230815i	−0.993318	+	0.115407i	$0.963183\pi$
$830$	0	0
$831$	−9.51774e9	−0.575348
$832$	0	0
$833$	2.72154e10	1.63139
$834$	0	0
$835$	− 9.74658e9i	− 0.579362i
$836$	0	0
$837$	− 1.01086e10i	− 0.595868i
$838$	0	0
$839$	−2.53943e10	−1.48446	−0.742232	−	0.670143i	$-0.766233\pi$
$839$	−2.53943e10	−1.48446	−0.742232	+	0.670143i	$0.766233\pi$
$840$	0	0
$841$	1.04010e10	0.602963
$842$	0	0
$843$	1.55524e9i	0.0894130i
$844$	0	0
$845$	7.83947e9i	0.446980i
$846$	0	0
$847$	−1.62087e9	−0.0916550
$848$	0	0
$849$	−4.60433e9	−0.258220
$850$	0	0
$851$	− 1.13035e10i	− 0.628722i
$852$	0	0
$853$	− 2.80898e9i	− 0.154963i	−0.996994	−	0.0774815i	$-0.975312\pi$
$853$	− 2.80898e9i	− 0.154963i	0.996994	−	0.0774815i	$-0.0246879\pi$
$854$	0	0
$855$	−5.01451e9	−0.274377
$856$	0	0
$857$	5.95128e9	0.322982	0.161491	−	0.986874i	$-0.448370\pi$
$857$	5.95128e9	0.322982	0.161491	+	0.986874i	$0.448370\pi$
$858$	0	0
$859$	− 2.18106e10i	− 1.17407i	−0.809563	−	0.587033i	$-0.800296\pi$
$859$	− 2.18106e10i	− 1.17407i	0.809563	−	0.587033i	$-0.199704\pi$
$860$	0	0
$861$	− 1.21901e9i	− 0.0650873i
$862$	0	0
$863$	2.04922e10	1.08530	0.542652	−	0.839957i	$-0.317420\pi$
$863$	2.04922e10	1.08530	0.542652	+	0.839957i	$0.317420\pi$
$864$	0	0
$865$	−1.50312e10	−0.789654
$866$	0	0
$867$	− 1.23890e10i	− 0.645608i
$868$	0	0
$869$	− 1.79788e10i	− 0.929374i
$870$	0	0
$871$	4.19726e8	0.0215230
$872$	0	0
$873$	1.31833e10	0.670616
$874$	0	0
$875$	− 4.04092e8i	− 0.0203916i
$876$	0	0
$877$	− 4.67115e9i	− 0.233843i	−0.993141	−	0.116922i	$-0.962697\pi$
$877$	− 4.67115e9i	− 0.233843i	0.993141	−	0.116922i	$-0.0373027\pi$
$878$	0	0
$879$	5.92285e8	0.0294151
$880$	0	0
$881$	3.56670e9	0.175732	0.0878661	−	0.996132i	$-0.471995\pi$
$881$	3.56670e9	0.175732	0.0878661	+	0.996132i	$0.471995\pi$
$882$	0	0
$883$	1.06146e10i	0.518850i	0.965763	+	0.259425i	$0.0835330\pi$
$883$	1.06146e10i	0.518850i	−0.965763	+	0.259425i	$0.916467\pi$
$884$	0	0
$885$	− 3.99845e9i	− 0.193906i
$886$	0	0
$887$	1.86034e10	0.895075	0.447538	−	0.894265i	$-0.352301\pi$
$887$	1.86034e10	0.895075	0.447538	+	0.894265i	$0.352301\pi$
$888$	0	0
$889$	6.49407e9	0.309999
$890$	0	0
$891$	− 1.71670e10i	− 0.813061i
$892$	0	0
$893$	6.58638e9i	0.309504i
$894$	0	0
$895$	1.26812e9	0.0591263
$896$	0	0
$897$	2.50393e8	0.0115837
$898$	0	0
$899$	1.31357e10i	0.602969i
$900$	0	0
$901$	− 2.19679e9i	− 0.100058i
$902$	0	0
$903$	−2.72461e9	−0.123140
$904$	0	0
$905$	−1.03555e10	−0.464408
$906$	0	0
$907$	− 1.30672e10i	− 0.581511i	−0.956797	−	0.290756i	$-0.906093\pi$
$907$	− 1.30672e10i	− 0.581511i	0.956797	−	0.290756i	$-0.0939066\pi$
$908$	0	0
$909$	− 2.88162e10i	− 1.27252i
$910$	0	0
$911$	2.08232e10	0.912499	0.456249	−	0.889852i	$-0.349192\pi$
$911$	2.08232e10	0.912499	0.456249	+	0.889852i	$0.349192\pi$
$912$	0	0
$913$	2.70642e9	0.117692
$914$	0	0
$915$	− 5.56017e9i	− 0.239946i
$916$	0	0
$917$	− 1.18341e9i	− 0.0506805i
$918$	0	0
$919$	1.75479e10	0.745798	0.372899	−	0.927872i	$-0.378364\pi$
$919$	1.75479e10	0.745798	0.372899	+	0.927872i	$0.378364\pi$
$920$	0	0
$921$	−8.11752e9	−0.342385
$922$	0	0
$923$	6.51440e6i	0 0.000272690i
$924$	0	0
$925$	1.96517e9i	0.0816404i
$926$	0	0
$927$	−1.30687e10	−0.538832
$928$	0	0
$929$	−1.98642e10	−0.812862	−0.406431	−	0.913681i	$-0.633227\pi$
$929$	−1.98642e10	−0.812862	−0.406431	+	0.913681i	$0.633227\pi$
$930$	0	0
$931$	− 1.60614e10i	− 0.652320i
$932$	0	0
$933$	1.64215e9i	0.0661954i
$934$	0	0
$935$	−2.27757e10	−0.911237
$936$	0	0
$937$	1.72786e10	0.686153	0.343076	−	0.939308i	$-0.388531\pi$
$937$	1.72786e10	0.686153	0.343076	+	0.939308i	$0.388531\pi$
$938$	0	0
$939$	4.09291e9i	0.161325i
$940$	0	0
$941$	2.33037e10i	0.911719i	0.890052	+	0.455859i	$0.150668\pi$
$941$	2.33037e10i	0.911719i	−0.890052	+	0.455859i	$0.849332\pi$
$942$	0	0
$943$	3.43979e10	1.33580
$944$	0	0
$945$	1.64704e9	0.0634881
$946$	0	0
$947$	− 1.84941e10i	− 0.707632i	−0.935315	−	0.353816i	$-0.884884\pi$
$947$	− 1.84941e10i	− 0.707632i	0.935315	−	0.353816i	$-0.115116\pi$
$948$	0	0
$949$	1.06836e8i	0.00405775i
$950$	0	0
$951$	9.37788e9	0.353568
$952$	0	0
$953$	−2.84591e10	−1.06511	−0.532556	−	0.846394i	$-0.678769\pi$
$953$	−2.84591e10	−1.06511	−0.532556	+	0.846394i	$0.678769\pi$
$954$	0	0
$955$	− 1.84424e10i	− 0.685181i
$956$	0	0
$957$	− 6.65910e9i	− 0.245598i
$958$	0	0
$959$	−8.66254e9	−0.317161
$960$	0	0
$961$	−2.31896e9	−0.0842870
$962$	0	0
$963$	− 1.48514e10i	− 0.535890i
$964$	0	0
$965$	− 6.22481e8i	− 0.0222987i
$966$	0	0
$967$	1.39464e10	0.495987	0.247994	−	0.968762i	$-0.420229\pi$
$967$	1.39464e10	0.495987	0.247994	+	0.968762i	$0.420229\pi$
$968$	0	0
$969$	−1.10394e10	−0.389773
$970$	0	0
$971$	− 6.58331e9i	− 0.230769i	−0.993321	−	0.115384i	$-0.963190\pi$
$971$	− 6.58331e9i	− 0.230769i	0.993321	−	0.115384i	$-0.0368100\pi$
$972$	0	0
$973$	− 1.11021e9i	− 0.0386376i
$974$	0	0
$975$	−4.35322e7	−0.00150416
$976$	0	0
$977$	−1.75217e10	−0.601098	−0.300549	−	0.953766i	$-0.597170\pi$
$977$	−1.75217e10	−0.601098	−0.300549	+	0.953766i	$0.597170\pi$
$978$	0	0
$979$	− 5.29068e10i	− 1.80207i
$980$	0	0
$981$	1.24484e10i	0.420991i
$982$	0	0
$983$	9.71858e9	0.326336	0.163168	−	0.986598i	$-0.447829\pi$
$983$	9.71858e9	0.326336	0.163168	+	0.986598i	$0.447829\pi$
$984$	0	0
$985$	2.15623e10	0.718900
$986$	0	0
$987$	− 1.01970e9i	− 0.0337570i
$988$	0	0
$989$	− 7.68829e10i	− 2.52722i
$990$	0	0
$991$	4.51199e10	1.47269	0.736343	−	0.676608i	$-0.236551\pi$
$991$	4.51199e10	1.47269	0.736343	+	0.676608i	$0.236551\pi$
$992$	0	0
$993$	1.53404e10	0.497182
$994$	0	0
$995$	− 1.11785e10i	− 0.359750i
$996$	0	0
$997$	− 9.00945e9i	− 0.287916i	−0.989584	−	0.143958i	$-0.954017\pi$
$997$	− 9.00945e9i	− 0.287916i	0.989584	−	0.143958i	$-0.0459830\pi$
$998$	0	0
$999$	−8.00985e9	−0.254182

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	160.8.d.a.81.12		28
4.3	odd	2		40.8.d.a.21.6	yes	28
8.3	odd	2		40.8.d.a.21.5	✓	28
8.5	even	2	inner	160.8.d.a.81.17		28

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
40.8.d.a.21.5	✓	28	8.3	odd	2
40.8.d.a.21.6	yes	28	4.3	odd	2
160.8.d.a.81.12		28	1.1	even	1	trivial
160.8.d.a.81.17		28	8.5	even	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 160 = 2^{5} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	160.d (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-15.3942i q^{3} +125.000i q^{5} +206.895 q^{7} +1950.02 q^{9} -5226.99i q^{11} +180.982i q^{13} +1924.27 q^{15} -34858.6 q^{17} +20572.1i q^{19} -3184.98i q^{21} +89873.4 q^{23} -15625.0 q^{25} -63685.9i q^{27} +82757.7i q^{29} +158725. q^{31} -80465.1 q^{33} +25861.9i q^{35} -125771. i q^{37} +2786.06 q^{39} +382738. q^{41} -855458. i q^{43} +243752. i q^{45} +320160. q^{47} -780737. q^{49} +536619. i q^{51} +63020.0i q^{53} +653374. q^{55} +316691. q^{57} -2.07791e6i q^{59} -2.88950e6i q^{61} +403450. q^{63} -22622.7 q^{65} -2.31917e6i q^{67} -1.38352e6i q^{69} +35994.8 q^{71} +590314. q^{73} +240534. i q^{75} -1.08144e6i q^{77} +3.43960e6 q^{79} +3.28430e6 q^{81} +517777. i q^{83} -4.35733e6i q^{85} +1.27398e6 q^{87} +1.01219e7 q^{89} +37444.2i q^{91} -2.44344e6i q^{93} -2.57152e6 q^{95} +6.76059e6 q^{97} -1.01927e7i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 28 q - 1372 q^{7} - 20412 q^{9} + 13500 q^{15} - 2588 q^{23} - 437500 q^{25} - 268024 q^{31} - 99016 q^{33} + 283944 q^{39} - 601208 q^{41} + 2076460 q^{47} + 4316268 q^{49} - 1331000 q^{55} + 3788536 q^{57}+ \cdots + 15198608 q^{97}+O(q^{100}) \) 28 * q - 1372 * q^7 - 20412 * q^9 + 13500 * q^15 - 2588 * q^23 - 437500 * q^25 - 268024 * q^31 - 99016 * q^33 + 283944 * q^39 - 601208 * q^41 + 2076460 * q^47 + 4316268 * q^49 - 1331000 * q^55 + 3788536 * q^57 + 14839060 * q^63 + 1001656 * q^71 + 2534128 * q^73 + 22040096 * q^79 + 11166300 * q^81 - 34127184 * q^87 + 3807224 * q^89 + 13718000 * q^95 + 15198608 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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