LMFDB - Embedded newform 160.8.d.a.81.10

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)

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);

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")

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			81.10
Character	$\chi$	$=$	160.81
Dual form			160.8.d.a.81.19

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-28.9351i q^{3} -125.000i q^{5} -724.987 q^{7} +1349.76 q^{9} -635.952i q^{11} -4193.99i q^{13} -3616.88 q^{15} +2743.22 q^{17} -40390.6i q^{19} +20977.5i q^{21} -47823.7 q^{23} -15625.0 q^{25} -102336. i q^{27} +121799. i q^{29} -172575. q^{31} -18401.3 q^{33} +90623.4i q^{35} -116482. i q^{37} -121353. q^{39} +653497. q^{41} +277478. i q^{43} -168720. i q^{45} -884726. q^{47} -297937. q^{49} -79375.2i q^{51} +568686. i q^{53} -79494.0 q^{55} -1.16870e6 q^{57} +1.63412e6i q^{59} +2.45255e6i q^{61} -978559. q^{63} -524249. q^{65} -838798. i q^{67} +1.38378e6i q^{69} -4.53571e6 q^{71} -2.64056e6 q^{73} +452110. i q^{75} +461057. i q^{77} +3.79614e6 q^{79} -9183.73 q^{81} +4.04075e6i q^{83} -342902. i q^{85} +3.52426e6 q^{87} -1.18628e7 q^{89} +3.04059e6i q^{91} +4.99348e6i q^{93} -5.04882e6 q^{95} -1.28038e7 q^{97} -858384. i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	− 28.9351i	− 0.618729i	−0.950944	−	0.309364i	$-0.899884\pi$
$3$	− 28.9351i	− 0.618729i	0.950944	−	0.309364i	$-0.100116\pi$
$4$	0	0
$5$	− 125.000i	− 0.447214i
$5$	− 125.000i	− 0.447214i
$6$	0	0
$7$	−724.987	−0.798890	−0.399445	−	0.916757i	$-0.630797\pi$
$7$	−724.987	−0.798890	−0.399445	+	0.916757i	$0.630797\pi$
$8$	0	0
$9$	1349.76	0.617175
$10$	0	0
$11$	− 635.952i	− 0.144062i	−0.997402	−	0.0720311i	$-0.977052\pi$
$11$	− 635.952i	− 0.144062i	0.997402	−	0.0720311i	$-0.0229481\pi$
$12$	0	0
$13$	− 4193.99i	− 0.529451i	−0.964324	−	0.264725i	$-0.914719\pi$
$13$	− 4193.99i	− 0.529451i	0.964324	−	0.264725i	$-0.0852813\pi$
$14$	0	0
$15$	−3616.88	−0.276704
$16$	0	0
$17$	2743.22	0.135422	0.0677110	−	0.997705i	$-0.478430\pi$
$17$	2743.22	0.135422	0.0677110	+	0.997705i	$0.478430\pi$
$18$	0	0
$19$	− 40390.6i	− 1.35096i	−0.737378	−	0.675480i	$-0.763936\pi$
$19$	− 40390.6i	− 1.35096i	0.737378	−	0.675480i	$-0.236064\pi$
$20$	0	0
$21$	20977.5i	0.494296i
$22$	0	0
$23$	−47823.7	−0.819588	−0.409794	−	0.912178i	$-0.634399\pi$
$23$	−47823.7	−0.819588	−0.409794	+	0.912178i	$0.634399\pi$
$24$	0	0
$25$	−15625.0	−0.200000
$26$	0	0
$27$	− 102336.i	− 1.00059i
$28$	0	0
$29$	121799.i	0.927363i	0.886002	+	0.463682i	$0.153472\pi$
$29$	121799.i	0.927363i	−0.886002	+	0.463682i	$0.846528\pi$
$30$	0	0
$31$	−172575.	−1.04043	−0.520215	−	0.854035i	$-0.674148\pi$
$31$	−172575.	−1.04043	−0.520215	+	0.854035i	$0.674148\pi$
$32$	0	0
$33$	−18401.3	−0.0891354
$34$	0	0
$35$	90623.4i	0.357274i
$36$	0	0
$37$	− 116482.i	− 0.378054i	−0.981972	−	0.189027i	$-0.939467\pi$
$37$	− 116482.i	− 0.378054i	0.981972	−	0.189027i	$-0.0605333\pi$
$38$	0	0
$39$	−121353.	−0.327586
$40$	0	0
$41$	653497.	1.48081	0.740407	−	0.672159i	$-0.234633\pi$
$41$	653497.	1.48081	0.740407	+	0.672159i	$0.234633\pi$
$42$	0	0
$43$	277478.i	0.532218i	0.963943	+	0.266109i	$0.0857381\pi$
$43$	277478.i	0.532218i	−0.963943	+	0.266109i	$0.914262\pi$
$44$	0	0
$45$	− 168720.i	− 0.276009i
$46$	0	0
$47$	−884726.	−1.24299	−0.621493	−	0.783420i	$-0.713474\pi$
$47$	−884726.	−1.24299	−0.621493	+	0.783420i	$0.713474\pi$
$48$	0	0
$49$	−297937.	−0.361775
$50$	0	0
$51$	− 79375.2i	− 0.0837894i
$52$	0	0
$53$	568686.i	0.524695i	0.964973	+	0.262347i	$0.0844966\pi$
$53$	568686.i	0.524695i	−0.964973	+	0.262347i	$0.915503\pi$
$54$	0	0
$55$	−79494.0	−0.0644266
$56$	0	0
$57$	−1.16870e6	−0.835877
$58$	0	0
$59$	1.63412e6i	1.03586i	0.855422	+	0.517932i	$0.173298\pi$
$59$	1.63412e6i	1.03586i	−0.855422	+	0.517932i	$0.826702\pi$
$60$	0	0
$61$	2.45255e6i	1.38345i	0.722162	+	0.691724i	$0.243148\pi$
$61$	2.45255e6i	1.38345i	−0.722162	+	0.691724i	$0.756852\pi$
$62$	0	0
$63$	−978559.	−0.493055
$64$	0	0
$65$	−524249.	−0.236778
$66$	0	0
$67$	− 838798.i	− 0.340718i	−0.985382	−	0.170359i	$-0.945507\pi$
$67$	− 838798.i	− 0.340718i	0.985382	−	0.170359i	$-0.0544928\pi$
$68$	0	0
$69$	1.38378e6i	0.507102i
$70$	0	0
$71$	−4.53571e6	−1.50398	−0.751988	−	0.659177i	$-0.770905\pi$
$71$	−4.53571e6	−1.50398	−0.751988	+	0.659177i	$0.770905\pi$
$72$	0	0
$73$	−2.64056e6	−0.794450	−0.397225	−	0.917721i	$-0.630027\pi$
$73$	−2.64056e6	−0.794450	−0.397225	+	0.917721i	$0.630027\pi$
$74$	0	0
$75$	452110.i	0.123746i
$76$	0	0
$77$	461057.i	0.115090i
$78$	0	0
$79$	3.79614e6	0.866258	0.433129	−	0.901332i	$-0.357409\pi$
$79$	3.79614e6	0.866258	0.433129	+	0.901332i	$0.357409\pi$
$80$	0	0
$81$	−9183.73	−0.00192009
$82$	0	0
$83$	4.04075e6i	0.775692i	0.921724	+	0.387846i	$0.126781\pi$
$83$	4.04075e6i	0.775692i	−0.921724	+	0.387846i	$0.873219\pi$
$84$	0	0
$85$	− 342902.i	− 0.0605625i
$86$	0	0
$87$	3.52426e6	0.573786
$88$	0	0
$89$	−1.18628e7	−1.78370	−0.891848	−	0.452334i	$-0.850591\pi$
$89$	−1.18628e7	−1.78370	−0.891848	+	0.452334i	$0.850591\pi$
$90$	0	0
$91$	3.04059e6i	0.422973i
$92$	0	0
$93$	4.99348e6i	0.643744i
$94$	0	0
$95$	−5.04882e6	−0.604168
$96$	0	0
$97$	−1.28038e7	−1.42442	−0.712208	−	0.701968i	$-0.752305\pi$
$97$	−1.28038e7	−1.42442	−0.712208	+	0.701968i	$0.752305\pi$
$98$	0	0
$99$	− 858384.i	− 0.0889116i
$100$	0	0
$101$	− 1.63109e7i	− 1.57526i	−0.616149	−	0.787629i	$-0.711308\pi$
$101$	− 1.63109e7i	− 1.57526i	0.616149	−	0.787629i	$-0.288692\pi$
$102$	0	0
$103$	−1.26057e7	−1.13668	−0.568338	−	0.822795i	$-0.692413\pi$
$103$	−1.26057e7	−1.13668	−0.568338	+	0.822795i	$0.692413\pi$
$104$	0	0
$105$	2.62219e6	0.221056
$106$	0	0
$107$	1.75324e7i	1.38356i	0.722107	+	0.691782i	$0.243174\pi$
$107$	1.75324e7i	1.38356i	−0.722107	+	0.691782i	$0.756826\pi$
$108$	0	0
$109$	7.43018e6i	0.549549i	0.961509	+	0.274775i	$0.0886032\pi$
$109$	7.43018e6i	0.549549i	−0.961509	+	0.274775i	$0.911397\pi$
$110$	0	0
$111$	−3.37042e6	−0.233913
$112$	0	0
$113$	−1.74971e7	−1.14075	−0.570375	−	0.821384i	$-0.693202\pi$
$113$	−1.74971e7	−1.14075	−0.570375	+	0.821384i	$0.693202\pi$
$114$	0	0
$115$	5.97796e6i	0.366531i
$116$	0	0
$117$	− 5.66088e6i	− 0.326764i
$118$	0	0
$119$	−1.98880e6	−0.108187
$120$	0	0
$121$	1.90827e7	0.979246
$122$	0	0
$123$	− 1.89090e7i	− 0.916221i
$124$	0	0
$125$	1.95312e6i	0.0894427i
$126$	0	0
$127$	1.71199e7	0.741632	0.370816	−	0.928706i	$-0.379078\pi$
$127$	1.71199e7	0.741632	0.370816	+	0.928706i	$0.379078\pi$
$128$	0	0
$129$	8.02886e6	0.329298
$130$	0	0
$131$	− 9.54213e6i	− 0.370848i	−0.982659	−	0.185424i	$-0.940634\pi$
$131$	− 9.54213e6i	− 0.370848i	0.982659	−	0.185424i	$-0.0593658\pi$
$132$	0	0
$133$	2.92826e7i	1.07927i
$134$	0	0
$135$	−1.27921e7	−0.447478
$136$	0	0
$137$	3.76038e7	1.24942	0.624712	−	0.780855i	$-0.285216\pi$
$137$	3.76038e7	1.24942	0.624712	+	0.780855i	$0.285216\pi$
$138$	0	0
$139$	− 3.55473e7i	− 1.12268i	−0.827586	−	0.561339i	$-0.810286\pi$
$139$	− 3.55473e7i	− 1.12268i	0.827586	−	0.561339i	$-0.189714\pi$
$140$	0	0
$141$	2.55996e7i	0.769071i
$142$	0	0
$143$	−2.66718e6	−0.0762738
$144$	0	0
$145$	1.52248e7	0.414729
$146$	0	0
$147$	8.62083e6i	0.223840i
$148$	0	0
$149$	2.55870e7i	0.633677i	0.948479	+	0.316839i	$0.102621\pi$
$149$	2.55870e7i	0.633677i	−0.948479	+	0.316839i	$0.897379\pi$
$150$	0	0
$151$	8.75969e6	0.207047	0.103524	−	0.994627i	$-0.466988\pi$
$151$	8.75969e6	0.207047	0.103524	+	0.994627i	$0.466988\pi$
$152$	0	0
$153$	3.70269e6	0.0835790
$154$	0	0
$155$	2.15719e7i	0.465295i
$156$	0	0
$157$	− 4.82895e7i	− 0.995873i	−0.867213	−	0.497937i	$-0.834091\pi$
$157$	− 4.82895e7i	− 0.995873i	0.867213	−	0.497937i	$-0.165909\pi$
$158$	0	0
$159$	1.64550e7	0.324644
$160$	0	0
$161$	3.46715e7	0.654760
$162$	0	0
$163$	1.65945e7i	0.300129i	0.988676	+	0.150064i	$0.0479481\pi$
$163$	1.65945e7i	0.300129i	−0.988676	+	0.150064i	$0.952052\pi$
$164$	0	0
$165$	2.30016e6i	0.0398625i
$166$	0	0
$167$	−3.75161e7	−0.623319	−0.311660	−	0.950194i	$-0.600885\pi$
$167$	−3.75161e7	−0.623319	−0.311660	+	0.950194i	$0.600885\pi$
$168$	0	0
$169$	4.51590e7	0.719682
$170$	0	0
$171$	− 5.45177e7i	− 0.833779i
$172$	0	0
$173$	− 1.02982e8i	− 1.51217i	−0.654476	−	0.756083i	$-0.727111\pi$
$173$	− 1.02982e8i	− 1.51217i	0.654476	−	0.756083i	$-0.272889\pi$
$174$	0	0
$175$	1.13279e7	0.159778
$176$	0	0
$177$	4.72834e7	0.640918
$178$	0	0
$179$	− 5.26588e7i	− 0.686255i	−0.939289	−	0.343127i	$-0.888514\pi$
$179$	− 5.26588e7i	− 0.686255i	0.939289	−	0.343127i	$-0.111486\pi$
$180$	0	0
$181$	− 6.39330e7i	− 0.801402i	−0.916209	−	0.400701i	$-0.868767\pi$
$181$	− 6.39330e7i	− 0.801402i	0.916209	−	0.400701i	$-0.131233\pi$
$182$	0	0
$183$	7.09646e7	0.855978
$184$	0	0
$185$	−1.45603e7	−0.169071
$186$	0	0
$187$	− 1.74455e6i	− 0.0195092i
$188$	0	0
$189$	7.41926e7i	0.799363i
$190$	0	0
$191$	1.29743e8	1.34730	0.673652	−	0.739049i	$-0.264725\pi$
$191$	1.29743e8	1.34730	0.673652	+	0.739049i	$0.264725\pi$
$192$	0	0
$193$	−1.11020e8	−1.11160	−0.555802	−	0.831315i	$-0.687589\pi$
$193$	−1.11020e8	−1.11160	−0.555802	+	0.831315i	$0.687589\pi$
$194$	0	0
$195$	1.51692e7i	0.146501i
$196$	0	0
$197$	− 1.82291e8i	− 1.69877i	−0.527774	−	0.849385i	$-0.676973\pi$
$197$	− 1.82291e8i	− 1.69877i	0.527774	−	0.849385i	$-0.323027\pi$
$198$	0	0
$199$	1.49332e8	1.34328	0.671640	−	0.740878i	$-0.265590\pi$
$199$	1.49332e8	1.34328	0.671640	+	0.740878i	$0.265590\pi$
$200$	0	0
$201$	−2.42707e7	−0.210812
$202$	0	0
$203$	− 8.83025e7i	− 0.740861i
$204$	0	0
$205$	− 8.16871e7i	− 0.662240i
$206$	0	0
$207$	−6.45505e7	−0.505829
$208$	0	0
$209$	−2.56865e7	−0.194622
$210$	0	0
$211$	− 3.06835e7i	− 0.224862i	−0.993660	−	0.112431i	$-0.964136\pi$
$211$	− 3.06835e7i	− 0.224862i	0.993660	−	0.112431i	$-0.0358637\pi$
$212$	0	0
$213$	1.31241e8i	0.930552i
$214$	0	0
$215$	3.46848e7	0.238015
$216$	0	0
$217$	1.25115e8	0.831189
$218$	0	0
$219$	7.64049e7i	0.491549i
$220$	0	0
$221$	− 1.15050e7i	− 0.0716992i
$222$	0	0
$223$	−7.89432e7	−0.476703	−0.238351	−	0.971179i	$-0.576607\pi$
$223$	−7.89432e7	−0.476703	−0.238351	+	0.971179i	$0.576607\pi$
$224$	0	0
$225$	−2.10900e7	−0.123435
$226$	0	0
$227$	2.00872e8i	1.13980i	0.821714	+	0.569900i	$0.193018\pi$
$227$	2.00872e8i	1.13980i	−0.821714	+	0.569900i	$0.806982\pi$
$228$	0	0
$229$	2.38925e8i	1.31473i	0.753572	+	0.657365i	$0.228329\pi$
$229$	2.38925e8i	1.31473i	−0.753572	+	0.657365i	$0.771671\pi$
$230$	0	0
$231$	1.33407e7	0.0712094
$232$	0	0
$233$	1.13633e8	0.588519	0.294259	−	0.955726i	$-0.404927\pi$
$233$	1.13633e8	0.588519	0.294259	+	0.955726i	$0.404927\pi$
$234$	0	0
$235$	1.10591e8i	0.555880i
$236$	0	0
$237$	− 1.09842e8i	− 0.535979i
$238$	0	0
$239$	1.48286e8	0.702599	0.351299	−	0.936263i	$-0.385740\pi$
$239$	1.48286e8	0.702599	0.351299	+	0.936263i	$0.385740\pi$
$240$	0	0
$241$	−3.54688e8	−1.63225	−0.816126	−	0.577875i	$-0.803882\pi$
$241$	−3.54688e8	−1.63225	−0.816126	+	0.577875i	$0.803882\pi$
$242$	0	0
$243$	− 2.23544e8i	− 0.999404i
$244$	0	0
$245$	3.72421e7i	0.161791i
$246$	0	0
$247$	−1.69398e8	−0.715267
$248$	0	0
$249$	1.16920e8	0.479943
$250$	0	0
$251$	− 1.33392e8i	− 0.532443i	−0.963912	−	0.266221i	$-0.914225\pi$
$251$	− 1.33392e8i	− 0.532443i	0.963912	−	0.266221i	$-0.0857753\pi$
$252$	0	0
$253$	3.04136e7i	0.118072i
$254$	0	0
$255$	−9.92190e6	−0.0374718
$256$	0	0
$257$	5.09013e7	0.187052	0.0935261	−	0.995617i	$-0.470186\pi$
$257$	5.09013e7	0.187052	0.0935261	+	0.995617i	$0.470186\pi$
$258$	0	0
$259$	8.44481e7i	0.302023i
$260$	0	0
$261$	1.64399e8i	0.572345i
$262$	0	0
$263$	−1.65767e8	−0.561892	−0.280946	−	0.959724i	$-0.590648\pi$
$263$	−1.65767e8	−0.561892	−0.280946	+	0.959724i	$0.590648\pi$
$264$	0	0
$265$	7.10857e7	0.234651
$266$	0	0
$267$	3.43250e8i	1.10362i
$268$	0	0
$269$	3.34898e8i	1.04901i	0.851407	+	0.524505i	$0.175750\pi$
$269$	3.34898e8i	1.04901i	−0.851407	+	0.524505i	$0.824250\pi$
$270$	0	0
$271$	−1.88305e8	−0.574736	−0.287368	−	0.957820i	$-0.592780\pi$
$271$	−1.88305e8	−0.574736	−0.287368	+	0.957820i	$0.592780\pi$
$272$	0	0
$273$	8.79796e7	0.261705
$274$	0	0
$275$	9.93675e6i	0.0288124i
$276$	0	0
$277$	− 7.27368e7i	− 0.205625i	−0.994701	−	0.102812i	$-0.967216\pi$
$277$	− 7.27368e7i	− 0.205625i	0.994701	−	0.102812i	$-0.0327841\pi$
$278$	0	0
$279$	−2.32936e8	−0.642128
$280$	0	0
$281$	−2.87068e8	−0.771814	−0.385907	−	0.922538i	$-0.626112\pi$
$281$	−2.87068e8	−0.771814	−0.385907	+	0.922538i	$0.626112\pi$
$282$	0	0
$283$	− 4.41650e8i	− 1.15831i	−0.815216	−	0.579157i	$-0.803382\pi$
$283$	− 4.41650e8i	− 1.15831i	0.815216	−	0.579157i	$-0.196618\pi$
$284$	0	0
$285$	1.46088e8i	0.373816i
$286$	0	0
$287$	−4.73777e8	−1.18301
$288$	0	0
$289$	−4.02813e8	−0.981661
$290$	0	0
$291$	3.70478e8i	0.881327i
$292$	0	0
$293$	− 5.90905e8i	− 1.37240i	−0.727412	−	0.686201i	$-0.759277\pi$
$293$	− 5.90905e8i	− 1.37240i	0.727412	−	0.686201i	$-0.240723\pi$
$294$	0	0
$295$	2.04265e8	0.463252
$296$	0	0
$297$	−6.50811e7	−0.144147
$298$	0	0
$299$	2.00572e8i	0.433931i
$300$	0	0
$301$	− 2.01168e8i	− 0.425184i
$302$	0	0
$303$	−4.71956e8	−0.974658
$304$	0	0
$305$	3.06568e8	0.618696
$306$	0	0
$307$	1.33591e8i	0.263507i	0.991282	+	0.131754i	$0.0420608\pi$
$307$	1.33591e8i	0.263507i	−0.991282	+	0.131754i	$0.957939\pi$
$308$	0	0
$309$	3.64747e8i	0.703294i
$310$	0	0
$311$	−7.26451e8	−1.36945	−0.684723	−	0.728804i	$-0.740077\pi$
$311$	−7.26451e8	−1.36945	−0.684723	+	0.728804i	$0.740077\pi$
$312$	0	0
$313$	−5.86492e8	−1.08108	−0.540539	−	0.841319i	$-0.681780\pi$
$313$	−5.86492e8	−1.08108	−0.540539	+	0.841319i	$0.681780\pi$
$314$	0	0
$315$	1.22320e8i	0.220501i
$316$	0	0
$317$	5.45883e8i	0.962482i	0.876588	+	0.481241i	$0.159814\pi$
$317$	5.45883e8i	0.962482i	−0.876588	+	0.481241i	$0.840186\pi$
$318$	0	0
$319$	7.74581e7	0.133598
$320$	0	0
$321$	5.07302e8	0.856050
$322$	0	0
$323$	− 1.10800e8i	− 0.182950i
$324$	0	0
$325$	6.55311e7i	0.105890i
$326$	0	0
$327$	2.14993e8	0.340022
$328$	0	0
$329$	6.41414e8	0.993009
$330$	0	0
$331$	− 2.19781e8i	− 0.333114i	−0.986032	−	0.166557i	$-0.946735\pi$
$331$	− 2.19781e8i	− 0.333114i	0.986032	−	0.166557i	$-0.0532650\pi$
$332$	0	0
$333$	− 1.57223e8i	− 0.233325i
$334$	0	0
$335$	−1.04850e8	−0.152374
$336$	0	0
$337$	−1.17733e9	−1.67568	−0.837842	−	0.545913i	$-0.816183\pi$
$337$	−1.17733e9	−1.67568	−0.837842	+	0.545913i	$0.816183\pi$
$338$	0	0
$339$	5.06279e8i	0.705815i
$340$	0	0
$341$	1.09750e8i	0.149887i
$342$	0	0
$343$	8.13058e8	1.08791
$344$	0	0
$345$	1.72973e8	0.226783
$346$	0	0
$347$	− 1.26996e9i	− 1.63169i	−0.578270	−	0.815846i	$-0.696272\pi$
$347$	− 1.26996e9i	− 1.63169i	0.578270	−	0.815846i	$-0.303728\pi$
$348$	0	0
$349$	4.38281e8i	0.551904i	0.961171	+	0.275952i	$0.0889931\pi$
$349$	4.38281e8i	0.551904i	−0.961171	+	0.275952i	$0.911007\pi$
$350$	0	0
$351$	−4.29198e8	−0.529764
$352$	0	0
$353$	−7.38394e8	−0.893463	−0.446731	−	0.894668i	$-0.647412\pi$
$353$	−7.38394e8	−0.893463	−0.446731	+	0.894668i	$0.647412\pi$
$354$	0	0
$355$	5.66963e8i	0.672598i
$356$	0	0
$357$	5.75460e7i	0.0669385i
$358$	0	0
$359$	1.04520e9	1.19225	0.596127	−	0.802890i	$-0.296705\pi$
$359$	1.04520e9	1.19225	0.596127	+	0.802890i	$0.296705\pi$
$360$	0	0
$361$	−7.37527e8	−0.825093
$362$	0	0
$363$	− 5.52160e8i	− 0.605888i
$364$	0	0
$365$	3.30070e8i	0.355289i
$366$	0	0
$367$	1.52124e9	1.60644	0.803222	−	0.595680i	$-0.203118\pi$
$367$	1.52124e9	1.60644	0.803222	+	0.595680i	$0.203118\pi$
$368$	0	0
$369$	8.82065e8	0.913921
$370$	0	0
$371$	− 4.12290e8i	− 0.419173i
$372$	0	0
$373$	− 1.71513e9i	− 1.71126i	−0.517587	−	0.855631i	$-0.673170\pi$
$373$	− 1.71513e9i	− 1.71126i	0.517587	−	0.855631i	$-0.326830\pi$
$374$	0	0
$375$	5.65138e7	0.0553408
$376$	0	0
$377$	5.10823e8	0.490993
$378$	0	0
$379$	1.49965e9i	1.41498i	0.706721	+	0.707492i	$0.250174\pi$
$379$	1.49965e9i	1.41498i	−0.706721	+	0.707492i	$0.749826\pi$
$380$	0	0
$381$	− 4.95366e8i	− 0.458869i
$382$	0	0
$383$	−125718.	−0.000114341 0	−5.71705e−5	−	1.00000i	$-0.500018\pi$
$383$	−125718.	−0.000114341 0	−5.71705e−5		1.00000i	$0.500018\pi$
$384$	0	0
$385$	5.76321e7	0.0514697
$386$	0	0
$387$	3.74530e8i	0.328472i
$388$	0	0
$389$	8.21407e8i	0.707513i	0.935338	+	0.353757i	$0.115096\pi$
$389$	8.21407e8i	0.707513i	−0.935338	+	0.353757i	$0.884904\pi$
$390$	0	0
$391$	−1.31191e8	−0.110990
$392$	0	0
$393$	−2.76102e8	−0.229454
$394$	0	0
$395$	− 4.74517e8i	− 0.387402i
$396$	0	0
$397$	− 9.97785e8i	− 0.800332i	−0.916443	−	0.400166i	$-0.868952\pi$
$397$	− 9.97785e8i	− 0.800332i	0.916443	−	0.400166i	$-0.131048\pi$
$398$	0	0
$399$	8.47295e8	0.667774
$400$	0	0
$401$	1.81089e9	1.40245	0.701225	−	0.712940i	$-0.252637\pi$
$401$	1.81089e9	1.40245	0.701225	+	0.712940i	$0.252637\pi$
$402$	0	0
$403$	7.23779e8i	0.550857i
$404$	0	0
$405$	1.14797e6i	0 0.000858690i
$406$	0	0
$407$	−7.40771e7	−0.0544632
$408$	0	0
$409$	7.95530e8	0.574943	0.287472	−	0.957789i	$-0.407185\pi$
$409$	7.95530e8	0.574943	0.287472	+	0.957789i	$0.407185\pi$
$410$	0	0
$411$	− 1.08807e9i	− 0.773055i
$412$	0	0
$413$	− 1.18472e9i	− 0.827541i
$414$	0	0
$415$	5.05094e8	0.346900
$416$	0	0
$417$	−1.02856e9	−0.694633
$418$	0	0
$419$	2.70680e9i	1.79765i	0.438303	+	0.898827i	$0.355580\pi$
$419$	2.70680e9i	1.79765i	−0.438303	+	0.898827i	$0.644420\pi$
$420$	0	0
$421$	− 1.90994e9i	− 1.24748i	−0.781633	−	0.623739i	$-0.785613\pi$
$421$	− 1.90994e9i	− 1.24748i	0.781633	−	0.623739i	$-0.214387\pi$
$422$	0	0
$423$	−1.19417e9	−0.767140
$424$	0	0
$425$	−4.28628e7	−0.0270844
$426$	0	0
$427$	− 1.77806e9i	− 1.10522i
$428$	0	0
$429$	7.71749e7i	0.0471928i
$430$	0	0
$431$	1.23043e9	0.740266	0.370133	−	0.928979i	$-0.379312\pi$
$431$	1.23043e9	0.740266	0.370133	+	0.928979i	$0.379312\pi$
$432$	0	0
$433$	1.30549e9	0.772799	0.386399	−	0.922332i	$-0.373719\pi$
$433$	1.30549e9	0.772799	0.386399	+	0.922332i	$0.373719\pi$
$434$	0	0
$435$	− 4.40532e8i	− 0.256605i
$436$	0	0
$437$	1.93163e9i	1.10723i
$438$	0	0
$439$	1.03574e9	0.584283	0.292142	−	0.956375i	$-0.405632\pi$
$439$	1.03574e9	0.584283	0.292142	+	0.956375i	$0.405632\pi$
$440$	0	0
$441$	−4.02144e8	−0.223278
$442$	0	0
$443$	− 3.39339e9i	− 1.85448i	−0.374473	−	0.927238i	$-0.622176\pi$
$443$	− 3.39339e9i	− 1.85448i	0.374473	−	0.927238i	$-0.377824\pi$
$444$	0	0
$445$	1.48285e9i	0.797693i
$446$	0	0
$447$	7.40363e8	0.392074
$448$	0	0
$449$	3.03655e9	1.58314	0.791569	−	0.611080i	$-0.209265\pi$
$449$	3.03655e9	1.58314	0.791569	+	0.611080i	$0.209265\pi$
$450$	0	0
$451$	− 4.15593e8i	− 0.213329i
$452$	0	0
$453$	− 2.53462e8i	− 0.128106i
$454$	0	0
$455$	3.80073e8	0.189159
$456$	0	0
$457$	1.73227e8	0.0849001	0.0424500	−	0.999099i	$-0.486484\pi$
$457$	1.73227e8	0.0849001	0.0424500	+	0.999099i	$0.486484\pi$
$458$	0	0
$459$	− 2.80731e8i	− 0.135502i
$460$	0	0
$461$	− 1.70634e9i	− 0.811171i	−0.914057	−	0.405586i	$-0.867068\pi$
$461$	− 1.70634e9i	− 0.811171i	0.914057	−	0.405586i	$-0.132932\pi$
$462$	0	0
$463$	−3.50776e9	−1.64247	−0.821233	−	0.570593i	$-0.806714\pi$
$463$	−3.50776e9	−1.64247	−0.821233	+	0.570593i	$0.806714\pi$
$464$	0	0
$465$	6.24185e8	0.287891
$466$	0	0
$467$	− 3.55199e9i	− 1.61385i	−0.590655	−	0.806924i	$-0.701130\pi$
$467$	− 3.55199e9i	− 1.61385i	0.590655	−	0.806924i	$-0.298870\pi$
$468$	0	0
$469$	6.08117e8i	0.272196i
$470$	0	0
$471$	−1.39726e9	−0.616175
$472$	0	0
$473$	1.76463e8	0.0766725
$474$	0	0
$475$	6.31103e8i	0.270192i
$476$	0	0
$477$	7.67590e8i	0.323829i
$478$	0	0
$479$	9.68267e8	0.402551	0.201276	−	0.979535i	$-0.435491\pi$
$479$	9.68267e8	0.402551	0.201276	+	0.979535i	$0.435491\pi$
$480$	0	0
$481$	−4.88525e8	−0.200161
$482$	0	0
$483$	− 1.00322e9i	− 0.405119i
$484$	0	0
$485$	1.60047e9i	0.637019i
$486$	0	0
$487$	−8.60336e8	−0.337533	−0.168767	−	0.985656i	$-0.553978\pi$
$487$	−8.60336e8	−0.337533	−0.168767	+	0.985656i	$0.553978\pi$
$488$	0	0
$489$	4.80163e8	0.185698
$490$	0	0
$491$	− 1.00221e9i	− 0.382097i	−0.981581	−	0.191048i	$-0.938811\pi$
$491$	− 1.00221e9i	− 0.382097i	0.981581	−	0.191048i	$-0.0611888\pi$
$492$	0	0
$493$	3.34120e8i	0.125585i
$494$	0	0
$495$	−1.07298e8	−0.0397625
$496$	0	0
$497$	3.28833e9	1.20151
$498$	0	0
$499$	5.40650e9i	1.94789i	0.226788	+	0.973944i	$0.427178\pi$
$499$	5.40650e9i	1.94789i	−0.226788	+	0.973944i	$0.572822\pi$
$500$	0	0
$501$	1.08553e9i	0.385665i
$502$	0	0
$503$	−2.06071e9	−0.721986	−0.360993	−	0.932569i	$-0.617562\pi$
$503$	−2.06071e9	−0.721986	−0.360993	+	0.932569i	$0.617562\pi$
$504$	0	0
$505$	−2.03886e9	−0.704477
$506$	0	0
$507$	− 1.30668e9i	− 0.445288i
$508$	0	0
$509$	− 2.77559e9i	− 0.932917i	−0.884543	−	0.466459i	$-0.845530\pi$
$509$	− 2.77559e9i	− 0.932917i	0.884543	−	0.466459i	$-0.154470\pi$
$510$	0	0
$511$	1.91437e9	0.634678
$512$	0	0
$513$	−4.13343e9	−1.35176
$514$	0	0
$515$	1.57571e9i	0.508337i
$516$	0	0
$517$	5.62643e8i	0.179067i
$518$	0	0
$519$	−2.97979e9	−0.935621
$520$	0	0
$521$	−2.62973e9	−0.814666	−0.407333	−	0.913280i	$-0.633541\pi$
$521$	−2.62973e9	−0.814666	−0.407333	+	0.913280i	$0.633541\pi$
$522$	0	0
$523$	2.10802e9i	0.644347i	0.946681	+	0.322173i	$0.104413\pi$
$523$	2.10802e9i	0.644347i	−0.946681	+	0.322173i	$0.895587\pi$
$524$	0	0
$525$	− 3.27774e8i	− 0.0988592i
$526$	0	0
$527$	−4.73412e8	−0.140897
$528$	0	0
$529$	−1.11772e9	−0.328276
$530$	0	0
$531$	2.20568e9i	0.639309i
$532$	0	0
$533$	− 2.74076e9i	− 0.784017i
$534$	0	0
$535$	2.19155e9	0.618748
$536$	0	0
$537$	−1.52369e9	−0.424605
$538$	0	0
$539$	1.89474e8i	0.0521180i
$540$	0	0
$541$	1.57237e9i	0.426936i	0.976950	+	0.213468i	$0.0684760\pi$
$541$	1.57237e9i	0.426936i	−0.976950	+	0.213468i	$0.931524\pi$
$542$	0	0
$543$	−1.84991e9	−0.495850
$544$	0	0
$545$	9.28772e8	0.245766
$546$	0	0
$547$	7.12156e9i	1.86046i	0.366980	+	0.930229i	$0.380392\pi$
$547$	7.12156e9i	1.86046i	−0.366980	+	0.930229i	$0.619608\pi$
$548$	0	0
$549$	3.31035e9i	0.853829i
$550$	0	0
$551$	4.91952e9	1.25283
$552$	0	0
$553$	−2.75215e9	−0.692045
$554$	0	0
$555$	4.21303e8i	0.104609i
$556$	0	0
$557$	2.13776e9i	0.524161i	0.965046	+	0.262081i	$0.0844087\pi$
$557$	2.13776e9i	0.524161i	−0.965046	+	0.262081i	$0.915591\pi$
$558$	0	0
$559$	1.16374e9	0.281783
$560$	0	0
$561$	−5.04788e7	−0.0120709
$562$	0	0
$563$	1.62576e9i	0.383953i	0.981400	+	0.191977i	$0.0614897\pi$
$563$	1.62576e9i	0.383953i	−0.981400	+	0.191977i	$0.938510\pi$
$564$	0	0
$565$	2.18713e9i	0.510159i
$566$	0	0
$567$	6.65808e6	0.00153394
$568$	0	0
$569$	2.12335e9	0.483202	0.241601	−	0.970376i	$-0.422327\pi$
$569$	2.12335e9	0.483202	0.241601	+	0.970376i	$0.422327\pi$
$570$	0	0
$571$	− 4.99851e9i	− 1.12361i	−0.827271	−	0.561803i	$-0.810108\pi$
$571$	− 4.99851e9i	− 1.12361i	0.827271	−	0.561803i	$-0.189892\pi$
$572$	0	0
$573$	− 3.75411e9i	− 0.833616i
$574$	0	0
$575$	7.47245e8	0.163918
$576$	0	0
$577$	−4.67621e9	−1.01339	−0.506697	−	0.862124i	$-0.669134\pi$
$577$	−4.67621e9	−1.01339	−0.506697	+	0.862124i	$0.669134\pi$
$578$	0	0
$579$	3.21237e9i	0.687781i
$580$	0	0
$581$	− 2.92949e9i	− 0.619692i
$582$	0	0
$583$	3.61657e8	0.0755887
$584$	0	0
$585$	−7.07611e8	−0.146133
$586$	0	0
$587$	− 5.82916e8i	− 0.118952i	−0.998230	−	0.0594761i	$-0.981057\pi$
$587$	− 5.82916e8i	− 0.118952i	0.998230	−	0.0594761i	$-0.0189430\pi$
$588$	0	0
$589$	6.97042e9i	1.40558i
$590$	0	0
$591$	−5.27462e9	−1.05108
$592$	0	0
$593$	−3.60198e9	−0.709333	−0.354666	−	0.934993i	$-0.615406\pi$
$593$	−3.60198e9	−0.709333	−0.354666	+	0.934993i	$0.615406\pi$
$594$	0	0
$595$	2.48600e8i	0.0483828i
$596$	0	0
$597$	− 4.32092e9i	− 0.831125i
$598$	0	0
$599$	7.85978e8	0.149423	0.0747114	−	0.997205i	$-0.476196\pi$
$599$	7.85978e8	0.149423	0.0747114	+	0.997205i	$0.476196\pi$
$600$	0	0
$601$	−6.82136e9	−1.28177	−0.640885	−	0.767637i	$-0.721433\pi$
$601$	−6.82136e9	−1.28177	−0.640885	+	0.767637i	$0.721433\pi$
$602$	0	0
$603$	− 1.13218e9i	− 0.210283i
$604$	0	0
$605$	− 2.38534e9i	− 0.437932i
$606$	0	0
$607$	−7.46800e9	−1.35533	−0.677663	−	0.735373i	$-0.737007\pi$
$607$	−7.46800e9	−1.35533	−0.677663	+	0.735373i	$0.737007\pi$
$608$	0	0
$609$	−2.55504e9	−0.458392
$610$	0	0
$611$	3.71053e9i	0.658100i
$612$	0	0
$613$	3.65909e9i	0.641596i	0.947148	+	0.320798i	$0.103951\pi$
$613$	3.65909e9i	0.641596i	−0.947148	+	0.320798i	$0.896049\pi$
$614$	0	0
$615$	−2.36362e9	−0.409747
$616$	0	0
$617$	5.11426e9	0.876567	0.438284	−	0.898837i	$-0.355587\pi$
$617$	5.11426e9	0.876567	0.438284	+	0.898837i	$0.355587\pi$
$618$	0	0
$619$	− 9.24516e9i	− 1.56674i	−0.621555	−	0.783371i	$-0.713499\pi$
$619$	− 9.24516e9i	− 1.56674i	0.621555	−	0.783371i	$-0.286501\pi$
$620$	0	0
$621$	4.89410e9i	0.820073i
$622$	0	0
$623$	8.60035e9	1.42498
$624$	0	0
$625$	2.44141e8	0.0400000
$626$	0	0
$627$	7.43240e8i	0.120418i
$628$	0	0
$629$	− 3.19536e8i	− 0.0511968i
$630$	0	0
$631$	−9.02091e9	−1.42938	−0.714690	−	0.699442i	$-0.753432\pi$
$631$	−9.02091e9	−1.42938	−0.714690	+	0.699442i	$0.753432\pi$
$632$	0	0
$633$	−8.87829e8	−0.139129
$634$	0	0
$635$	− 2.13999e9i	− 0.331668i
$636$	0	0
$637$	1.24954e9i	0.191542i
$638$	0	0
$639$	−6.12212e9	−0.928216
$640$	0	0
$641$	2.21484e9	0.332154	0.166077	−	0.986113i	$-0.446890\pi$
$641$	2.21484e9	0.332154	0.166077	+	0.986113i	$0.446890\pi$
$642$	0	0
$643$	3.60818e8i	0.0535242i	0.999642	+	0.0267621i	$0.00851966\pi$
$643$	3.60818e8i	0.0535242i	−0.999642	+	0.0267621i	$0.991480\pi$
$644$	0	0
$645$	− 1.00361e9i	− 0.147267i
$646$	0	0
$647$	2.93899e9	0.426612	0.213306	−	0.976985i	$-0.431577\pi$
$647$	2.93899e9	0.426612	0.213306	+	0.976985i	$0.431577\pi$
$648$	0	0
$649$	1.03922e9	0.149229
$650$	0	0
$651$	− 3.62021e9i	− 0.514281i
$652$	0	0
$653$	2.01978e8i	0.0283863i	0.999899	+	0.0141932i	$0.00451797\pi$
$653$	2.01978e8i	0.0283863i	−0.999899	+	0.0141932i	$0.995482\pi$
$654$	0	0
$655$	−1.19277e9	−0.165848
$656$	0	0
$657$	−3.56413e9	−0.490315
$658$	0	0
$659$	5.73039e9i	0.779983i	0.920818	+	0.389992i	$0.127522\pi$
$659$	5.73039e9i	0.779983i	−0.920818	+	0.389992i	$0.872478\pi$
$660$	0	0
$661$	− 2.92528e9i	− 0.393969i	−0.980407	−	0.196985i	$-0.936885\pi$
$661$	− 2.92528e9i	− 0.393969i	0.980407	−	0.196985i	$-0.0631149\pi$
$662$	0	0
$663$	−3.32899e8	−0.0443624
$664$	0	0
$665$	3.66033e9	0.482664
$666$	0	0
$667$	− 5.82486e9i	− 0.760055i
$668$	0	0
$669$	2.28423e9i	0.294949i
$670$	0	0
$671$	1.55970e9	0.199302
$672$	0	0
$673$	1.38147e10	1.74699	0.873493	−	0.486837i	$-0.161849\pi$
$673$	1.38147e10	1.74699	0.873493	+	0.486837i	$0.161849\pi$
$674$	0	0
$675$	1.59901e9i	0.200118i
$676$	0	0
$677$	− 7.08067e9i	− 0.877029i	−0.898724	−	0.438514i	$-0.855505\pi$
$677$	− 7.08067e9i	− 0.877029i	0.898724	−	0.438514i	$-0.144495\pi$
$678$	0	0
$679$	9.28257e9	1.13795
$680$	0	0
$681$	5.81224e9	0.705227
$682$	0	0
$683$	− 1.48335e10i	− 1.78144i	−0.454554	−	0.890719i	$-0.650201\pi$
$683$	− 1.48335e10i	− 1.78144i	0.454554	−	0.890719i	$-0.349799\pi$
$684$	0	0
$685$	− 4.70048e9i	− 0.558760i
$686$	0	0
$687$	6.91330e9	0.813461
$688$	0	0
$689$	2.38506e9	0.277800
$690$	0	0
$691$	8.98381e9i	1.03583i	0.855433	+	0.517914i	$0.173291\pi$
$691$	8.98381e9i	1.03583i	−0.855433	+	0.517914i	$0.826709\pi$
$692$	0	0
$693$	6.22317e8i	0.0710306i
$694$	0	0
$695$	−4.44342e9	−0.502077
$696$	0	0
$697$	1.79269e9	0.200535
$698$	0	0
$699$	− 3.28799e9i	− 0.364133i
$700$	0	0
$701$	− 1.43806e9i	− 0.157676i	−0.996887	−	0.0788379i	$-0.974879\pi$
$701$	− 1.43806e9i	− 0.157676i	0.996887	−	0.0788379i	$-0.0251209\pi$
$702$	0	0
$703$	−4.70478e9	−0.510736
$704$	0	0
$705$	3.19995e9	0.343939
$706$	0	0
$707$	1.18252e10i	1.25846i
$708$	0	0
$709$	1.96630e9i	0.207199i	0.994619	+	0.103600i	$0.0330360\pi$
$709$	1.96630e9i	0.207199i	−0.994619	+	0.103600i	$0.966964\pi$
$710$	0	0
$711$	5.12388e9	0.534633
$712$	0	0
$713$	8.25319e9	0.852724
$714$	0	0
$715$	3.33397e8i	0.0341107i
$716$	0	0
$717$	− 4.29066e9i	− 0.434718i
$718$	0	0
$719$	6.89945e9	0.692251	0.346125	−	0.938188i	$-0.387497\pi$
$719$	6.89945e9	0.692251	0.346125	+	0.938188i	$0.387497\pi$
$720$	0	0
$721$	9.13896e9	0.908079
$722$	0	0
$723$	1.02629e10i	1.00992i
$724$	0	0
$725$	− 1.90311e9i	− 0.185473i
$726$	0	0
$727$	−7.37008e9	−0.711380	−0.355690	−	0.934604i	$-0.615754\pi$
$727$	−7.37008e9	−0.711380	−0.355690	+	0.934604i	$0.615754\pi$
$728$	0	0
$729$	−6.48835e9	−0.620280
$730$	0	0
$731$	7.61183e8i	0.0720740i
$732$	0	0
$733$	1.50977e10i	1.41594i	0.706241	+	0.707971i	$0.250389\pi$
$733$	1.50977e10i	1.41594i	−0.706241	+	0.707971i	$0.749611\pi$
$734$	0	0
$735$	1.07760e9	0.100104
$736$	0	0
$737$	−5.33435e8	−0.0490846
$738$	0	0
$739$	− 1.89400e10i	− 1.72633i	−0.504922	−	0.863165i	$-0.668479\pi$
$739$	− 1.89400e10i	− 1.72633i	0.504922	−	0.863165i	$-0.331521\pi$
$740$	0	0
$741$	4.90153e9i	0.442556i
$742$	0	0
$743$	1.59965e10	1.43075	0.715375	−	0.698741i	$-0.246256\pi$
$743$	1.59965e10	1.43075	0.715375	+	0.698741i	$0.246256\pi$
$744$	0	0
$745$	3.19838e9	0.283389
$746$	0	0
$747$	5.45406e9i	0.478737i
$748$	0	0
$749$	− 1.27108e10i	− 1.10531i
$750$	0	0
$751$	−7.15896e9	−0.616752	−0.308376	−	0.951265i	$-0.599785\pi$
$751$	−7.15896e9	−0.616752	−0.308376	+	0.951265i	$0.599785\pi$
$752$	0	0
$753$	−3.85972e9	−0.329438
$754$	0	0
$755$	− 1.09496e9i	− 0.0925943i
$756$	0	0
$757$	− 1.93055e9i	− 0.161751i	−0.996724	−	0.0808754i	$-0.974228\pi$
$757$	− 1.93055e9i	− 0.161751i	0.996724	−	0.0808754i	$-0.0257716\pi$
$758$	0	0
$759$	8.80018e8	0.0730543
$760$	0	0
$761$	1.10433e10	0.908348	0.454174	−	0.890913i	$-0.349934\pi$
$761$	1.10433e10	0.908348	0.454174	+	0.890913i	$0.349934\pi$
$762$	0	0
$763$	− 5.38678e9i	− 0.439029i
$764$	0	0
$765$	− 4.62836e8i	− 0.0373777i
$766$	0	0
$767$	6.85349e9	0.548438
$768$	0	0
$769$	−1.93885e10	−1.53746	−0.768728	−	0.639575i	$-0.779110\pi$
$769$	−1.93885e10	−1.53746	−0.768728	+	0.639575i	$0.779110\pi$
$770$	0	0
$771$	− 1.47283e9i	− 0.115735i
$772$	0	0
$773$	− 2.19721e9i	− 0.171097i	−0.996334	−	0.0855487i	$-0.972736\pi$
$773$	− 2.19721e9i	− 0.171097i	0.996334	−	0.0855487i	$-0.0272643\pi$
$774$	0	0
$775$	2.69649e9	0.208086
$776$	0	0
$777$	2.44351e9	0.186871
$778$	0	0
$779$	− 2.63951e10i	− 2.00052i
$780$	0	0
$781$	2.88449e9i	0.216666i
$782$	0	0
$783$	1.24645e10	0.927912
$784$	0	0
$785$	−6.03619e9	−0.445368
$786$	0	0
$787$	1.04999e10i	0.767848i	0.923365	+	0.383924i	$0.125428\pi$
$787$	1.04999e10i	0.767848i	−0.923365	+	0.383924i	$0.874572\pi$
$788$	0	0
$789$	4.79648e9i	0.347659i
$790$	0	0
$791$	1.26851e10	0.911334
$792$	0	0
$793$	1.02859e10	0.732467
$794$	0	0
$795$	− 2.05687e9i	− 0.145185i
$796$	0	0
$797$	− 2.39080e10i	− 1.67278i	−0.548134	−	0.836390i	$-0.684662\pi$
$797$	− 2.39080e10i	− 1.67278i	0.548134	−	0.836390i	$-0.315338\pi$
$798$	0	0
$799$	−2.42700e9	−0.168328
$800$	0	0
$801$	−1.60119e10	−1.10085
$802$	0	0
$803$	1.67927e9i	0.114450i
$804$	0	0
$805$	− 4.33394e9i	− 0.292818i
$806$	0	0
$807$	9.69030e9	0.649053
$808$	0	0
$809$	−1.43285e9	−0.0951436	−0.0475718	−	0.998868i	$-0.515148\pi$
$809$	−1.43285e9	−0.0951436	−0.0475718	+	0.998868i	$0.515148\pi$
$810$	0	0
$811$	− 2.02071e9i	− 0.133024i	−0.997786	−	0.0665120i	$-0.978813\pi$
$811$	− 2.02071e9i	− 0.133024i	0.997786	−	0.0665120i	$-0.0211871\pi$
$812$	0	0
$813$	5.44860e9i	0.355605i
$814$	0	0
$815$	2.07431e9	0.134222
$816$	0	0
$817$	1.12075e10	0.719005
$818$	0	0
$819$	4.10407e9i	0.261048i
$820$	0	0
$821$	3.97118e9i	0.250448i	0.992128	+	0.125224i	$0.0399650\pi$
$821$	3.97118e9i	0.250448i	−0.992128	+	0.125224i	$0.960035\pi$
$822$	0	0
$823$	−1.59915e10	−0.999979	−0.499989	−	0.866032i	$-0.666663\pi$
$823$	−1.59915e10	−0.999979	−0.499989	+	0.866032i	$0.666663\pi$
$824$	0	0
$825$	2.87521e8	0.0178271
$826$	0	0
$827$	− 3.60022e9i	− 0.221340i	−0.993857	−	0.110670i	$-0.964700\pi$
$827$	− 3.60022e9i	− 0.221340i	0.993857	−	0.110670i	$-0.0352997\pi$
$828$	0	0
$829$	− 2.42858e10i	− 1.48051i	−0.672327	−	0.740254i	$-0.734705\pi$
$829$	− 2.42858e10i	− 1.48051i	0.672327	−	0.740254i	$-0.265295\pi$
$830$	0	0
$831$	−2.10465e9	−0.127226
$832$	0	0
$833$	−8.17306e8	−0.0489922
$834$	0	0
$835$	4.68952e9i	0.278757i
$836$	0	0
$837$	1.76608e10i	1.04105i
$838$	0	0
$839$	−9.51333e9	−0.556117	−0.278058	−	0.960564i	$-0.589691\pi$
$839$	−9.51333e9	−0.556117	−0.278058	+	0.960564i	$0.589691\pi$
$840$	0	0
$841$	2.41494e9	0.139998
$842$	0	0
$843$	8.30634e9i	0.477544i
$844$	0	0
$845$	− 5.64487e9i	− 0.321852i
$846$	0	0
$847$	−1.38347e10	−0.782310
$848$	0	0
$849$	−1.27792e10	−0.716682
$850$	0	0
$851$	5.57061e9i	0.309848i
$852$	0	0
$853$	− 1.30130e10i	− 0.717887i	−0.933359	−	0.358944i	$-0.883137\pi$
$853$	− 1.30130e10i	− 0.717887i	0.933359	−	0.358944i	$-0.116863\pi$
$854$	0	0
$855$	−6.81471e9	−0.372877
$856$	0	0
$857$	−3.57734e8	−0.0194146	−0.00970728	−	0.999953i	$-0.503090\pi$
$857$	−3.57734e8	−0.0194146	−0.00970728	+	0.999953i	$0.503090\pi$
$858$	0	0
$859$	4.46756e8i	0.0240489i	0.999928	+	0.0120244i	$0.00382759\pi$
$859$	4.46756e8i	0.0240489i	−0.999928	+	0.0120244i	$0.996172\pi$
$860$	0	0
$861$	1.37088e10i	0.731960i
$862$	0	0
$863$	2.63664e10	1.39641	0.698206	−	0.715897i	$-0.253982\pi$
$863$	2.63664e10	1.39641	0.698206	+	0.715897i	$0.253982\pi$
$864$	0	0
$865$	−1.28727e10	−0.676261
$866$	0	0
$867$	1.16554e10i	0.607382i
$868$	0	0
$869$	− 2.41416e9i	− 0.124795i
$870$	0	0
$871$	−3.51791e9	−0.180394
$872$	0	0
$873$	−1.72820e10	−0.879114
$874$	0	0
$875$	− 1.41599e9i	− 0.0714549i
$876$	0	0
$877$	2.75765e10i	1.38051i	0.723564	+	0.690257i	$0.242503\pi$
$877$	2.75765e10i	1.38051i	−0.723564	+	0.690257i	$0.757497\pi$
$878$	0	0
$879$	−1.70979e10	−0.849144
$880$	0	0
$881$	−8.48952e9	−0.418280	−0.209140	−	0.977886i	$-0.567067\pi$
$881$	−8.48952e9	−0.418280	−0.209140	+	0.977886i	$0.567067\pi$
$882$	0	0
$883$	3.84331e10i	1.87864i	0.343043	+	0.939320i	$0.388542\pi$
$883$	3.84331e10i	1.87864i	−0.343043	+	0.939320i	$0.611458\pi$
$884$	0	0
$885$	− 5.91043e9i	− 0.286627i
$886$	0	0
$887$	−3.67955e9	−0.177036	−0.0885181	−	0.996075i	$-0.528213\pi$
$887$	−3.67955e9	−0.177036	−0.0885181	+	0.996075i	$0.528213\pi$
$888$	0	0
$889$	−1.24117e10	−0.592483
$890$	0	0
$891$	5.84041e6i	0 0.000276612i
$892$	0	0
$893$	3.57346e10i	1.67922i
$894$	0	0
$895$	−6.58235e9	−0.306902
$896$	0	0
$897$	5.80356e9	0.268486
$898$	0	0
$899$	− 2.10195e10i	− 0.964857i
$900$	0	0
$901$	1.56003e9i	0.0710552i
$902$	0	0
$903$	−5.82081e9	−0.263073
$904$	0	0
$905$	−7.99163e9	−0.358398
$906$	0	0
$907$	− 1.43630e10i	− 0.639177i	−0.947556	−	0.319588i	$-0.896455\pi$
$907$	− 1.43630e10i	− 0.639177i	0.947556	−	0.319588i	$-0.103545\pi$
$908$	0	0
$909$	− 2.20158e10i	− 0.972210i
$910$	0	0
$911$	1.87441e10	0.821391	0.410696	−	0.911772i	$-0.365286\pi$
$911$	1.87441e10	0.821391	0.410696	+	0.911772i	$0.365286\pi$
$912$	0	0
$913$	2.56973e9	0.111748
$914$	0	0
$915$	− 8.87057e9i	− 0.382805i
$916$	0	0
$917$	6.91792e9i	0.296267i
$918$	0	0
$919$	−3.28361e10	−1.39556	−0.697778	−	0.716314i	$-0.745828\pi$
$919$	−3.28361e10	−1.39556	−0.697778	+	0.716314i	$0.745828\pi$
$920$	0	0
$921$	3.86546e9	0.163040
$922$	0	0
$923$	1.90227e10i	0.796281i
$924$	0	0
$925$	1.82003e9i	0.0756108i
$926$	0	0
$927$	−1.70147e10	−0.701528
$928$	0	0
$929$	3.86364e10	1.58103	0.790517	−	0.612440i	$-0.209812\pi$
$929$	3.86364e10	1.58103	0.790517	+	0.612440i	$0.209812\pi$
$930$	0	0
$931$	1.20338e10i	0.488743i
$932$	0	0
$933$	2.10199e10i	0.847315i
$934$	0	0
$935$	−2.18069e8	−0.00872477
$936$	0	0
$937$	−4.10023e9	−0.162825	−0.0814123	−	0.996681i	$-0.525943\pi$
$937$	−4.10023e9	−0.162825	−0.0814123	+	0.996681i	$0.525943\pi$
$938$	0	0
$939$	1.69702e10i	0.668894i
$940$	0	0
$941$	− 1.80337e10i	− 0.705538i	−0.935710	−	0.352769i	$-0.885240\pi$
$941$	− 1.80337e10i	− 0.705538i	0.935710	−	0.352769i	$-0.114760\pi$
$942$	0	0
$943$	−3.12526e10	−1.21366
$944$	0	0
$945$	9.27407e9	0.357486
$946$	0	0
$947$	− 6.66183e9i	− 0.254899i	−0.991845	−	0.127450i	$-0.959321\pi$
$947$	− 6.66183e9i	− 0.254899i	0.991845	−	0.127450i	$-0.0406791\pi$
$948$	0	0
$949$	1.10745e10i	0.420622i
$950$	0	0
$951$	1.57952e10	0.595515
$952$	0	0
$953$	1.09990e10	0.411651	0.205825	−	0.978589i	$-0.434012\pi$
$953$	1.09990e10	0.411651	0.205825	+	0.978589i	$0.434012\pi$
$954$	0	0
$955$	− 1.62178e10i	− 0.602533i
$956$	0	0
$957$	− 2.24126e9i	− 0.0826608i
$958$	0	0
$959$	−2.72623e10	−0.998153
$960$	0	0
$961$	2.26966e9	0.0824951
$962$	0	0
$963$	2.36646e10i	0.853901i
$964$	0	0
$965$	1.38775e10i	0.497125i
$966$	0	0
$967$	−1.48740e10	−0.528975	−0.264487	−	0.964389i	$-0.585203\pi$
$967$	−1.48740e10	−0.528975	−0.264487	+	0.964389i	$0.585203\pi$
$968$	0	0
$969$	−3.20601e9	−0.113196
$970$	0	0
$971$	− 1.67742e10i	− 0.587995i	−0.955806	−	0.293998i	$-0.905014\pi$
$971$	− 1.67742e10i	− 0.587995i	0.955806	−	0.293998i	$-0.0949858\pi$
$972$	0	0
$973$	2.57713e10i	0.896896i
$974$	0	0
$975$	1.89615e9	0.0655173
$976$	0	0
$977$	1.07958e10	0.370361	0.185181	−	0.982705i	$-0.440713\pi$
$977$	1.07958e10	0.370361	0.185181	+	0.982705i	$0.440713\pi$
$978$	0	0
$979$	7.54415e9i	0.256963i
$980$	0	0
$981$	1.00290e10i	0.339168i
$982$	0	0
$983$	−1.18872e10	−0.399154	−0.199577	−	0.979882i	$-0.563957\pi$
$983$	−1.18872e10	−0.399154	−0.199577	+	0.979882i	$0.563957\pi$
$984$	0	0
$985$	−2.27864e10	−0.759713
$986$	0	0
$987$	− 1.85594e10i	− 0.614403i
$988$	0	0
$989$	− 1.32700e10i	− 0.436199i
$990$	0	0
$991$	1.17102e10	0.382213	0.191106	−	0.981569i	$-0.438792\pi$
$991$	1.17102e10	0.382213	0.191106	+	0.981569i	$0.438792\pi$
$992$	0	0
$993$	−6.35939e9	−0.206107
$994$	0	0
$995$	− 1.86665e10i	− 0.600733i
$996$	0	0
$997$	− 2.29607e9i	− 0.0733757i	−0.999327	−	0.0366879i	$-0.988319\pi$
$997$	− 2.29607e9i	− 0.0733757i	0.999327	−	0.0366879i	$-0.0116807\pi$
$998$	0	0
$999$	−1.19204e10	−0.378278

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	160.8.d.a.81.10		28
4.3	odd	2		40.8.d.a.21.10	yes	28
8.3	odd	2		40.8.d.a.21.9	✓	28
8.5	even	2	inner	160.8.d.a.81.19		28

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
40.8.d.a.21.9	✓	28	8.3	odd	2
40.8.d.a.21.10	yes	28	4.3	odd	2
160.8.d.a.81.10		28	1.1	even	1	trivial
160.8.d.a.81.19		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)

\(f(q)\)	\(=\)	\(q-28.9351i q^{3} -125.000i q^{5} -724.987 q^{7} +1349.76 q^{9} -635.952i q^{11} -4193.99i q^{13} -3616.88 q^{15} +2743.22 q^{17} -40390.6i q^{19} +20977.5i q^{21} -47823.7 q^{23} -15625.0 q^{25} -102336. i q^{27} +121799. i q^{29} -172575. q^{31} -18401.3 q^{33} +90623.4i q^{35} -116482. i q^{37} -121353. q^{39} +653497. q^{41} +277478. i q^{43} -168720. i q^{45} -884726. q^{47} -297937. q^{49} -79375.2i q^{51} +568686. i q^{53} -79494.0 q^{55} -1.16870e6 q^{57} +1.63412e6i q^{59} +2.45255e6i q^{61} -978559. q^{63} -524249. q^{65} -838798. i q^{67} +1.38378e6i q^{69} -4.53571e6 q^{71} -2.64056e6 q^{73} +452110. i q^{75} +461057. i q^{77} +3.79614e6 q^{79} -9183.73 q^{81} +4.04075e6i q^{83} -342902. i q^{85} +3.52426e6 q^{87} -1.18628e7 q^{89} +3.04059e6i q^{91} +4.99348e6i q^{93} -5.04882e6 q^{95} -1.28038e7 q^{97} -858384. i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 28 q - 1372 q^{7} - 20412 q^{9} + 13500 q^{15} - 2588 q^{23} - 437500 q^{25} - 268024 q^{31} - 99016 q^{33} + 283944 q^{39} - 601208 q^{41} + 2076460 q^{47} + 4316268 q^{49} - 1331000 q^{55} + 3788536 q^{57}+ \cdots + 15198608 q^{97}+O(q^{100}) \) 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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