LMFDB - Embedded newform 400.10.a.y.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [400,10,Mod(1,400)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(400, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 10, names="a")

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

chi := DirichletCharacter("400.1");

S:= CuspForms(chi, 10);

N := Newforms(S);

Level:	$ N $	$=$	$ 400 = 2^{4} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 10 $
Character orbit:	$[\chi]$	$=$	400.a (trivial)

Newform invariants

comment: select newform

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

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

Self dual:	yes
Analytic conductor:	$206.014334466$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	$\mathbb{Q}[x]/(x^{3} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 652x + 4000 $ x^3 - x^2 - 652*x + 4000
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{4}\cdot 5 $
Twist minimal:	no (minimal twist has level 25)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$22.2334$ of defining polynomial
Character	$\chi$	$=$	400.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-210.171 q^{3} +9905.49 q^{7} +24489.0 q^{9} +O(q^{10})$	$q-210.171 q^{3} +9905.49 q^{7} +24489.0 q^{9} -36453.6 q^{11} -164867. q^{13} -82357.1 q^{17} +609617. q^{19} -2.08185e6 q^{21} +1.88578e6 q^{23} -1.01008e6 q^{27} +339235. q^{29} -547314. q^{31} +7.66150e6 q^{33} -5.25687e6 q^{37} +3.46503e7 q^{39} +2.05812e6 q^{41} -6.76158e6 q^{43} +3.15241e7 q^{47} +5.77651e7 q^{49} +1.73091e7 q^{51} +4.89593e7 q^{53} -1.28124e8 q^{57} -8.77960e7 q^{59} +3.84654e7 q^{61} +2.42575e8 q^{63} -1.36116e8 q^{67} -3.96337e8 q^{69} -3.49218e8 q^{71} -1.61345e8 q^{73} -3.61091e8 q^{77} +1.26975e8 q^{79} -2.69727e8 q^{81} -2.87494e8 q^{83} -7.12976e7 q^{87} -5.63133e8 q^{89} -1.63309e9 q^{91} +1.15030e8 q^{93} +4.71704e8 q^{97} -8.92711e8 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 89 q^{3} + 5258 q^{7} + 58234 q^{9}+O(q^{10}) $ 3 * q + 89 * q^3 + 5258 * q^7 + 58234 * q^9	$ 3 q + 89 q^{3} + 5258 q^{7} + 58234 q^{9} + 54699 q^{11} - 215884 q^{13} - 334983 q^{17} - 818845 q^{19} - 2375394 q^{21} + 3526854 q^{23} + 6633395 q^{27} + 2175480 q^{29} - 4274066 q^{31} + 22122137 q^{33} + 10305042 q^{37} + 50414092 q^{39} + 5926311 q^{41} - 24429956 q^{43} + 66858708 q^{47} - 6453929 q^{49} + 25634699 q^{51} - 132620514 q^{53} - 252946415 q^{57} - 5670960 q^{59} + 125306926 q^{61} + 284323404 q^{63} + 88829483 q^{67} - 314274942 q^{69} - 297550596 q^{71} - 181321729 q^{73} - 561214086 q^{77} + 310025170 q^{79} + 1398847363 q^{81} - 731088801 q^{83} + 1046385560 q^{87} - 1103860035 q^{89} - 1183187656 q^{91} - 107386758 q^{93} + 332236842 q^{97} + 892234522 q^{99}+O(q^{100}) $ 3 * q + 89 * q^3 + 5258 * q^7 + 58234 * q^9 + 54699 * q^11 - 215884 * q^13 - 334983 * q^17 - 818845 * q^19 - 2375394 * q^21 + 3526854 * q^23 + 6633395 * q^27 + 2175480 * q^29 - 4274066 * q^31 + 22122137 * q^33 + 10305042 * q^37 + 50414092 * q^39 + 5926311 * q^41 - 24429956 * q^43 + 66858708 * q^47 - 6453929 * q^49 + 25634699 * q^51 - 132620514 * q^53 - 252946415 * q^57 - 5670960 * q^59 + 125306926 * q^61 + 284323404 * q^63 + 88829483 * q^67 - 314274942 * q^69 - 297550596 * q^71 - 181321729 * q^73 - 561214086 * q^77 + 310025170 * q^79 + 1398847363 * q^81 - 731088801 * q^83 + 1046385560 * q^87 - 1103860035 * q^89 - 1183187656 * q^91 - 107386758 * q^93 + 332236842 * q^97 + 892234522 * q^99

Display 100 coefficients 10 coefficients

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
$3$	−210.171	−1.49805	−0.749027	−	0.662539i	$-0.769479\pi$
$3$	−210.171	−1.49805	−0.749027	+	0.662539i	$0.769479\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	9905.49	1.55932	0.779659	−	0.626204i	$-0.215392\pi$
$7$	9905.49	1.55932	0.779659	+	0.626204i	$0.215392\pi$
$8$	0	0
$9$	24489.0	1.24417
$10$	0	0
$11$	−36453.6	−0.750712	−0.375356	−	0.926881i	$-0.622480\pi$
$11$	−36453.6	−0.750712	−0.375356	+	0.926881i	$0.622480\pi$
$12$	0	0
$13$	−164867.	−1.60099	−0.800494	−	0.599341i	$-0.795429\pi$
$13$	−164867.	−1.60099	−0.800494	+	0.599341i	$0.795429\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−82357.1	−0.239156	−0.119578	−	0.992825i	$-0.538154\pi$
$17$	−82357.1	−0.239156	−0.119578	+	0.992825i	$0.538154\pi$
$18$	0	0
$19$	609617.	1.07316	0.536582	−	0.843848i	$-0.319715\pi$
$19$	609617.	1.07316	0.536582	+	0.843848i	$0.319715\pi$
$20$	0	0
$21$	−2.08185e6	−2.33594
$22$	0	0
$23$	1.88578e6	1.40513	0.702564	−	0.711620i	$-0.252038\pi$
$23$	1.88578e6	1.40513	0.702564	+	0.711620i	$0.252038\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	−1.01008e6	−0.365778
$28$	0	0
$29$	339235.	0.0890657	0.0445328	−	0.999008i	$-0.485820\pi$
$29$	339235.	0.0890657	0.0445328	+	0.999008i	$0.485820\pi$
$30$	0	0
$31$	−547314.	−0.106441	−0.0532205	−	0.998583i	$-0.516949\pi$
$31$	−547314.	−0.106441	−0.0532205	+	0.998583i	$0.516949\pi$
$32$	0	0
$33$	7.66150e6	1.12461
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−5.25687e6	−0.461126	−0.230563	−	0.973057i	$-0.574057\pi$
$37$	−5.25687e6	−0.461126	−0.230563	+	0.973057i	$0.574057\pi$
$38$	0	0
$39$	3.46503e7	2.39837
$40$	0	0
$41$	2.05812e6	0.113748	0.0568739	−	0.998381i	$-0.481887\pi$
$41$	2.05812e6	0.113748	0.0568739	+	0.998381i	$0.481887\pi$
$42$	0	0
$43$	−6.76158e6	−0.301606	−0.150803	−	0.988564i	$-0.548186\pi$
$43$	−6.76158e6	−0.301606	−0.150803	+	0.988564i	$0.548186\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	3.15241e7	0.942330	0.471165	−	0.882045i	$-0.343834\pi$
$47$	3.15241e7	0.942330	0.471165	+	0.882045i	$0.343834\pi$
$48$	0	0
$49$	5.77651e7	1.43147
$50$	0	0
$51$	1.73091e7	0.358268
$52$	0	0
$53$	4.89593e7	0.852303	0.426151	−	0.904652i	$-0.359869\pi$
$53$	4.89593e7	0.852303	0.426151	+	0.904652i	$0.359869\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−1.28124e8	−1.60766
$58$	0	0
$59$	−8.77960e7	−0.943281	−0.471640	−	0.881791i	$-0.656338\pi$
$59$	−8.77960e7	−0.943281	−0.471640	+	0.881791i	$0.656338\pi$
$60$	0	0
$61$	3.84654e7	0.355701	0.177851	−	0.984057i	$-0.443086\pi$
$61$	3.84654e7	0.355701	0.177851	+	0.984057i	$0.443086\pi$
$62$	0	0
$63$	2.42575e8	1.94005
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−1.36116e8	−0.825227	−0.412613	−	0.910906i	$-0.635384\pi$
$67$	−1.36116e8	−0.825227	−0.412613	+	0.910906i	$0.635384\pi$
$68$	0	0
$69$	−3.96337e8	−2.10496
$70$	0	0
$71$	−3.49218e8	−1.63092	−0.815462	−	0.578810i	$-0.803517\pi$
$71$	−3.49218e8	−1.63092	−0.815462	+	0.578810i	$0.803517\pi$
$72$	0	0
$73$	−1.61345e8	−0.664971	−0.332486	−	0.943108i	$-0.607887\pi$
$73$	−1.61345e8	−0.664971	−0.332486	+	0.943108i	$0.607887\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−3.61091e8	−1.17060
$78$	0	0
$79$	1.26975e8	0.366772	0.183386	−	0.983041i	$-0.441294\pi$
$79$	1.26975e8	0.366772	0.183386	+	0.983041i	$0.441294\pi$
$80$	0	0
$81$	−2.69727e8	−0.696213
$82$	0	0
$83$	−2.87494e8	−0.664931	−0.332466	−	0.943115i	$-0.607881\pi$
$83$	−2.87494e8	−0.664931	−0.332466	+	0.943115i	$0.607881\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	−7.12976e7	−0.133425
$88$	0	0
$89$	−5.63133e8	−0.951385	−0.475692	−	0.879612i	$-0.657802\pi$
$89$	−5.63133e8	−0.951385	−0.475692	+	0.879612i	$0.657802\pi$
$90$	0	0
$91$	−1.63309e9	−2.49645
$92$	0	0
$93$	1.15030e8	0.159454
$94$	0	0
$95$	0	0
$96$	0	0
$97$	4.71704e8	0.541000	0.270500	−	0.962720i	$-0.412811\pi$
$97$	4.71704e8	0.541000	0.270500	+	0.962720i	$0.412811\pi$
$98$	0	0
$99$	−8.92711e8	−0.934012
$100$	0	0
$101$	−1.25673e9	−1.20170	−0.600851	−	0.799361i	$-0.705172\pi$
$101$	−1.25673e9	−1.20170	−0.600851	+	0.799361i	$0.705172\pi$
$102$	0	0
$103$	1.95398e9	1.71062	0.855309	−	0.518119i	$-0.173367\pi$
$103$	1.95398e9	1.71062	0.855309	+	0.518119i	$0.173367\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	2.12306e9	1.56580	0.782899	−	0.622148i	$-0.213740\pi$
$107$	2.12306e9	1.56580	0.782899	+	0.622148i	$0.213740\pi$
$108$	0	0
$109$	−1.18863e8	−0.0806544	−0.0403272	−	0.999187i	$-0.512840\pi$
$109$	−1.18863e8	−0.0806544	−0.0403272	+	0.999187i	$0.512840\pi$
$110$	0	0
$111$	1.10484e9	0.690791
$112$	0	0
$113$	1.85455e9	1.07001	0.535003	−	0.844850i	$-0.320311\pi$
$113$	1.85455e9	1.07001	0.535003	+	0.844850i	$0.320311\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	−4.03742e9	−1.99190
$118$	0	0
$119$	−8.15787e8	−0.372920
$120$	0	0
$121$	−1.02908e9	−0.436432
$122$	0	0
$123$	−4.32557e8	−0.170400
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−5.39208e9	−1.83924	−0.919622	−	0.392805i	$-0.871505\pi$
$127$	−5.39208e9	−1.83924	−0.919622	+	0.392805i	$0.871505\pi$
$128$	0	0
$129$	1.42109e9	0.451823
$130$	0	0
$131$	−3.83690e9	−1.13831	−0.569154	−	0.822231i	$-0.692729\pi$
$131$	−3.83690e9	−1.13831	−0.569154	+	0.822231i	$0.692729\pi$
$132$	0	0
$133$	6.03856e9	1.67340
$134$	0	0
$135$	0	0
$136$	0	0
$137$	2.55408e9	0.619430	0.309715	−	0.950829i	$-0.399766\pi$
$137$	2.55408e9	0.619430	0.309715	+	0.950829i	$0.399766\pi$
$138$	0	0
$139$	4.09121e9	0.929577	0.464789	−	0.885422i	$-0.346130\pi$
$139$	4.09121e9	0.929577	0.464789	+	0.885422i	$0.346130\pi$
$140$	0	0
$141$	−6.62547e9	−1.41166
$142$	0	0
$143$	6.00999e9	1.20188
$144$	0	0
$145$	0	0
$146$	0	0
$147$	−1.21406e10	−2.14443
$148$	0	0
$149$	−6.13643e9	−1.01995	−0.509974	−	0.860190i	$-0.670345\pi$
$149$	−6.13643e9	−1.01995	−0.509974	+	0.860190i	$0.670345\pi$
$150$	0	0
$151$	5.83754e9	0.913763	0.456882	−	0.889528i	$-0.348966\pi$
$151$	5.83754e9	0.913763	0.456882	+	0.889528i	$0.348966\pi$
$152$	0	0
$153$	−2.01684e9	−0.297550
$154$	0	0
$155$	0	0
$156$	0	0
$157$	1.35031e9	0.177371	0.0886857	−	0.996060i	$-0.471733\pi$
$157$	1.35031e9	0.177371	0.0886857	+	0.996060i	$0.471733\pi$
$158$	0	0
$159$	−1.02898e10	−1.27680
$160$	0	0
$161$	1.86796e10	2.19104
$162$	0	0
$163$	−4.50806e9	−0.500202	−0.250101	−	0.968220i	$-0.580464\pi$
$163$	−4.50806e9	−0.500202	−0.250101	+	0.968220i	$0.580464\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	1.81836e10	1.80908	0.904538	−	0.426393i	$-0.140216\pi$
$167$	1.81836e10	1.80908	0.904538	+	0.426393i	$0.140216\pi$
$168$	0	0
$169$	1.65766e10	1.56316
$170$	0	0
$171$	1.49289e10	1.33520
$172$	0	0
$173$	−9.77143e9	−0.829375	−0.414687	−	0.909964i	$-0.636109\pi$
$173$	−9.77143e9	−0.829375	−0.414687	+	0.909964i	$0.636109\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	1.84522e10	1.41309
$178$	0	0
$179$	−6.65866e9	−0.484784	−0.242392	−	0.970178i	$-0.577932\pi$
$179$	−6.65866e9	−0.484784	−0.242392	+	0.970178i	$0.577932\pi$
$180$	0	0
$181$	−7.32873e9	−0.507546	−0.253773	−	0.967264i	$-0.581672\pi$
$181$	−7.32873e9	−0.507546	−0.253773	+	0.967264i	$0.581672\pi$
$182$	0	0
$183$	−8.08432e9	−0.532860
$184$	0	0
$185$	0	0
$186$	0	0
$187$	3.00221e9	0.179537
$188$	0	0
$189$	−1.00053e10	−0.570364
$190$	0	0
$191$	−8.06956e9	−0.438732	−0.219366	−	0.975643i	$-0.570399\pi$
$191$	−8.06956e9	−0.438732	−0.219366	+	0.975643i	$0.570399\pi$
$192$	0	0
$193$	−3.12603e9	−0.162176	−0.0810878	−	0.996707i	$-0.525839\pi$
$193$	−3.12603e9	−0.162176	−0.0810878	+	0.996707i	$0.525839\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	2.40133e10	1.13594	0.567968	−	0.823051i	$-0.307730\pi$
$197$	2.40133e10	1.13594	0.567968	+	0.823051i	$0.307730\pi$
$198$	0	0
$199$	−9.73605e9	−0.440093	−0.220046	−	0.975489i	$-0.570621\pi$
$199$	−9.73605e9	−0.440093	−0.220046	+	0.975489i	$0.570621\pi$
$200$	0	0
$201$	2.86077e10	1.23623
$202$	0	0
$203$	3.36029e9	0.138882
$204$	0	0
$205$	0	0
$206$	0	0
$207$	4.61808e10	1.74822
$208$	0	0
$209$	−2.22227e10	−0.805637
$210$	0	0
$211$	2.39305e10	0.831151	0.415576	−	0.909559i	$-0.363580\pi$
$211$	2.39305e10	0.831151	0.415576	+	0.909559i	$0.363580\pi$
$212$	0	0
$213$	7.33956e10	2.44321
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−5.42141e9	−0.165975
$218$	0	0
$219$	3.39101e10	0.996163
$220$	0	0
$221$	1.35779e10	0.382885
$222$	0	0
$223$	2.02289e10	0.547773	0.273886	−	0.961762i	$-0.411691\pi$
$223$	2.02289e10	0.547773	0.273886	+	0.961762i	$0.411691\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	4.42152e10	1.10524	0.552618	−	0.833434i	$-0.313629\pi$
$227$	4.42152e10	1.10524	0.552618	+	0.833434i	$0.313629\pi$
$228$	0	0
$229$	−4.89483e10	−1.17619	−0.588096	−	0.808791i	$-0.700122\pi$
$229$	−4.89483e10	−1.17619	−0.588096	+	0.808791i	$0.700122\pi$
$230$	0	0
$231$	7.58909e10	1.75362
$232$	0	0
$233$	−6.68132e10	−1.48512	−0.742559	−	0.669781i	$-0.766388\pi$
$233$	−6.68132e10	−1.48512	−0.742559	+	0.669781i	$0.766388\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	−2.66865e10	−0.549444
$238$	0	0
$239$	2.21153e10	0.438431	0.219216	−	0.975676i	$-0.429650\pi$
$239$	2.21153e10	0.438431	0.219216	+	0.975676i	$0.429650\pi$
$240$	0	0
$241$	9.97637e10	1.90500	0.952502	−	0.304532i	$-0.0985002\pi$
$241$	9.97637e10	1.90500	0.952502	+	0.304532i	$0.0985002\pi$
$242$	0	0
$243$	7.65703e10	1.40874
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−1.00506e11	−1.71812
$248$	0	0
$249$	6.04229e10	0.996104
$250$	0	0
$251$	4.97276e10	0.790799	0.395399	−	0.918509i	$-0.370606\pi$
$251$	4.97276e10	0.790799	0.395399	+	0.918509i	$0.370606\pi$
$252$	0	0
$253$	−6.87435e10	−1.05485
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−5.81308e10	−0.831204	−0.415602	−	0.909547i	$-0.636429\pi$
$257$	−5.81308e10	−0.831204	−0.415602	+	0.909547i	$0.636429\pi$
$258$	0	0
$259$	−5.20718e10	−0.719041
$260$	0	0
$261$	8.30753e9	0.110813
$262$	0	0
$263$	−3.23476e10	−0.416909	−0.208454	−	0.978032i	$-0.566843\pi$
$263$	−3.23476e10	−0.416909	−0.208454	+	0.978032i	$0.566843\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	1.18354e11	1.42523
$268$	0	0
$269$	6.24220e9	0.0726863	0.0363431	−	0.999339i	$-0.488429\pi$
$269$	6.24220e9	0.0726863	0.0363431	+	0.999339i	$0.488429\pi$
$270$	0	0
$271$	−8.05816e10	−0.907557	−0.453779	−	0.891114i	$-0.649924\pi$
$271$	−8.05816e10	−0.907557	−0.453779	+	0.891114i	$0.649924\pi$
$272$	0	0
$273$	3.43228e11	3.73982
$274$	0	0
$275$	0	0
$276$	0	0
$277$	7.07976e10	0.722536	0.361268	−	0.932462i	$-0.382344\pi$
$277$	7.07976e10	0.722536	0.361268	+	0.932462i	$0.382344\pi$
$278$	0	0
$279$	−1.34031e10	−0.132430
$280$	0	0
$281$	8.15619e10	0.780385	0.390193	−	0.920733i	$-0.372408\pi$
$281$	8.15619e10	0.780385	0.390193	+	0.920733i	$0.372408\pi$
$282$	0	0
$283$	1.89301e11	1.75434	0.877168	−	0.480184i	$-0.159430\pi$
$283$	1.89301e11	1.75434	0.877168	+	0.480184i	$0.159430\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	2.03867e10	0.177369
$288$	0	0
$289$	−1.11805e11	−0.942805
$290$	0	0
$291$	−9.91387e10	−0.810447
$292$	0	0
$293$	5.39489e10	0.427640	0.213820	−	0.976873i	$-0.431409\pi$
$293$	5.39489e10	0.427640	0.213820	+	0.976873i	$0.431409\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	3.68209e10	0.274594
$298$	0	0
$299$	−3.10903e11	−2.24959
$300$	0	0
$301$	−6.69768e10	−0.470300
$302$	0	0
$303$	2.64129e11	1.80022
$304$	0	0
$305$	0	0
$306$	0	0
$307$	−1.30037e11	−0.835496	−0.417748	−	0.908563i	$-0.637181\pi$
$307$	−1.30037e11	−0.835496	−0.417748	+	0.908563i	$0.637181\pi$
$308$	0	0
$309$	−4.10671e11	−2.56260
$310$	0	0
$311$	−2.09580e9	−0.0127036	−0.00635182	−	0.999980i	$-0.502022\pi$
$311$	−2.09580e9	−0.0127036	−0.00635182	+	0.999980i	$0.502022\pi$
$312$	0	0
$313$	2.43788e11	1.43570	0.717848	−	0.696200i	$-0.245127\pi$
$313$	2.43788e11	1.43570	0.717848	+	0.696200i	$0.245127\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	3.46483e11	1.92715	0.963573	−	0.267445i	$-0.0861794\pi$
$317$	3.46483e11	1.92715	0.963573	+	0.267445i	$0.0861794\pi$
$318$	0	0
$319$	−1.23664e10	−0.0668626
$320$	0	0
$321$	−4.46207e11	−2.34565
$322$	0	0
$323$	−5.02063e10	−0.256653
$324$	0	0
$325$	0	0
$326$	0	0
$327$	2.49816e10	0.120825
$328$	0	0
$329$	3.12262e11	1.46939
$330$	0	0
$331$	1.43303e11	0.656189	0.328094	−	0.944645i	$-0.393594\pi$
$331$	1.43303e11	0.656189	0.328094	+	0.944645i	$0.393594\pi$
$332$	0	0
$333$	−1.28735e11	−0.573718
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−3.34820e11	−1.41409	−0.707044	−	0.707170i	$-0.749972\pi$
$337$	−3.34820e11	−1.41409	−0.707044	+	0.707170i	$0.749972\pi$
$338$	0	0
$339$	−3.89773e11	−1.60293
$340$	0	0
$341$	1.99515e10	0.0799064
$342$	0	0
$343$	1.72469e11	0.672804
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−9.22101e10	−0.341425	−0.170713	−	0.985321i	$-0.554607\pi$
$347$	−9.22101e10	−0.341425	−0.170713	+	0.985321i	$0.554607\pi$
$348$	0	0
$349$	−4.71738e11	−1.70211	−0.851053	−	0.525080i	$-0.824035\pi$
$349$	−4.71738e11	−1.70211	−0.851053	+	0.525080i	$0.824035\pi$
$350$	0	0
$351$	1.66528e11	0.585606
$352$	0	0
$353$	4.01029e11	1.37464	0.687322	−	0.726353i	$-0.258786\pi$
$353$	4.01029e11	1.37464	0.687322	+	0.726353i	$0.258786\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	1.71455e11	0.558654
$358$	0	0
$359$	2.36110e10	0.0750220	0.0375110	−	0.999296i	$-0.488057\pi$
$359$	2.36110e10	0.0750220	0.0375110	+	0.999296i	$0.488057\pi$
$360$	0	0
$361$	4.89457e10	0.151681
$362$	0	0
$363$	2.16284e11	0.653799
$364$	0	0
$365$	0	0
$366$	0	0
$367$	1.23035e11	0.354022	0.177011	−	0.984209i	$-0.443357\pi$
$367$	1.23035e11	0.354022	0.177011	+	0.984209i	$0.443357\pi$
$368$	0	0
$369$	5.04012e10	0.141521
$370$	0	0
$371$	4.84966e11	1.32901
$372$	0	0
$373$	2.39248e11	0.639968	0.319984	−	0.947423i	$-0.396322\pi$
$373$	2.39248e11	0.639968	0.319984	+	0.947423i	$0.396322\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−5.59287e10	−0.142593
$378$	0	0
$379$	5.90629e11	1.47041	0.735205	−	0.677845i	$-0.237086\pi$
$379$	5.90629e11	1.47041	0.735205	+	0.677845i	$0.237086\pi$
$380$	0	0
$381$	1.13326e12	2.75529
$382$	0	0
$383$	−5.30525e10	−0.125983	−0.0629914	−	0.998014i	$-0.520064\pi$
$383$	−5.30525e10	−0.125983	−0.0629914	+	0.998014i	$0.520064\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−1.65584e11	−0.375249
$388$	0	0
$389$	3.10395e11	0.687293	0.343646	−	0.939099i	$-0.388338\pi$
$389$	3.10395e11	0.687293	0.343646	+	0.939099i	$0.388338\pi$
$390$	0	0
$391$	−1.55307e11	−0.336045
$392$	0	0
$393$	8.06407e11	1.70525
$394$	0	0
$395$	0	0
$396$	0	0
$397$	2.09351e11	0.422979	0.211489	−	0.977380i	$-0.432169\pi$
$397$	2.09351e11	0.422979	0.211489	+	0.977380i	$0.432169\pi$
$398$	0	0
$399$	−1.26913e12	−2.50685
$400$	0	0
$401$	4.43868e11	0.857244	0.428622	−	0.903484i	$-0.358999\pi$
$401$	4.43868e11	0.857244	0.428622	+	0.903484i	$0.358999\pi$
$402$	0	0
$403$	9.02338e10	0.170411
$404$	0	0
$405$	0	0
$406$	0	0
$407$	1.91632e11	0.346172
$408$	0	0
$409$	4.56031e11	0.805823	0.402912	−	0.915239i	$-0.367998\pi$
$409$	4.56031e11	0.805823	0.402912	+	0.915239i	$0.367998\pi$
$410$	0	0
$411$	−5.36794e11	−0.927940
$412$	0	0
$413$	−8.69663e11	−1.47087
$414$	0	0
$415$	0	0
$416$	0	0
$417$	−8.59856e11	−1.39256
$418$	0	0
$419$	1.12392e12	1.78144	0.890720	−	0.454552i	$-0.150201\pi$
$419$	1.12392e12	1.78144	0.890720	+	0.454552i	$0.150201\pi$
$420$	0	0
$421$	−1.02822e12	−1.59521	−0.797605	−	0.603180i	$-0.793900\pi$
$421$	−1.02822e12	−1.59521	−0.797605	+	0.603180i	$0.793900\pi$
$422$	0	0
$423$	7.71994e11	1.17242
$424$	0	0
$425$	0	0
$426$	0	0
$427$	3.81018e11	0.554652
$428$	0	0
$429$	−1.26313e12	−1.80048
$430$	0	0
$431$	1.37714e12	1.92234	0.961171	−	0.275953i	$-0.0889934\pi$
$431$	1.37714e12	1.92234	0.961171	+	0.275953i	$0.0889934\pi$
$432$	0	0
$433$	−1.94790e11	−0.266299	−0.133150	−	0.991096i	$-0.542509\pi$
$433$	−1.94790e11	−0.266299	−0.133150	+	0.991096i	$0.542509\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	1.14961e12	1.50793
$438$	0	0
$439$	5.32533e11	0.684315	0.342157	−	0.939643i	$-0.388842\pi$
$439$	5.32533e11	0.684315	0.342157	+	0.939643i	$0.388842\pi$
$440$	0	0
$441$	1.41461e12	1.78099
$442$	0	0
$443$	7.26633e11	0.896393	0.448196	−	0.893935i	$-0.352067\pi$
$443$	7.26633e11	0.896393	0.448196	+	0.893935i	$0.352067\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	1.28970e12	1.52794
$448$	0	0
$449$	9.34830e11	1.08549	0.542743	−	0.839899i	$-0.317386\pi$
$449$	9.34830e11	1.08549	0.542743	+	0.839899i	$0.317386\pi$
$450$	0	0
$451$	−7.50258e10	−0.0853918
$452$	0	0
$453$	−1.22688e12	−1.36887
$454$	0	0
$455$	0	0
$456$	0	0
$457$	1.09190e12	1.17101	0.585503	−	0.810670i	$-0.300897\pi$
$457$	1.09190e12	1.17101	0.585503	+	0.810670i	$0.300897\pi$
$458$	0	0
$459$	8.31870e10	0.0874779
$460$	0	0
$461$	9.93683e11	1.02469	0.512347	−	0.858779i	$-0.328776\pi$
$461$	9.93683e11	1.02469	0.512347	+	0.858779i	$0.328776\pi$
$462$	0	0
$463$	−6.95734e11	−0.703605	−0.351802	−	0.936074i	$-0.614431\pi$
$463$	−6.95734e11	−0.703605	−0.351802	+	0.936074i	$0.614431\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	1.87957e12	1.82866	0.914328	−	0.404975i	$-0.132720\pi$
$467$	1.87957e12	1.82866	0.914328	+	0.404975i	$0.132720\pi$
$468$	0	0
$469$	−1.34830e12	−1.28679
$470$	0	0
$471$	−2.83795e11	−0.265712
$472$	0	0
$473$	2.46484e11	0.226419
$474$	0	0
$475$	0	0
$476$	0	0
$477$	1.19896e12	1.06041
$478$	0	0
$479$	2.29011e11	0.198768	0.0993838	−	0.995049i	$-0.468313\pi$
$479$	2.29011e11	0.198768	0.0993838	+	0.995049i	$0.468313\pi$
$480$	0	0
$481$	8.66683e11	0.738256
$482$	0	0
$483$	−3.92591e12	−3.28230
$484$	0	0
$485$	0	0
$486$	0	0
$487$	−4.81410e11	−0.387824	−0.193912	−	0.981019i	$-0.562118\pi$
$487$	−4.81410e11	−0.387824	−0.193912	+	0.981019i	$0.562118\pi$
$488$	0	0
$489$	9.47465e11	0.749330
$490$	0	0
$491$	1.70320e12	1.32251	0.661257	−	0.750160i	$-0.270023\pi$
$491$	1.70320e12	1.32251	0.661257	+	0.750160i	$0.270023\pi$
$492$	0	0
$493$	−2.79384e10	−0.0213006
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−3.45917e12	−2.54313
$498$	0	0
$499$	−2.06075e12	−1.48789	−0.743947	−	0.668238i	$-0.767049\pi$
$499$	−2.06075e12	−1.48789	−0.743947	+	0.668238i	$0.767049\pi$
$500$	0	0
$501$	−3.82168e12	−2.71010
$502$	0	0
$503$	−5.97493e11	−0.416176	−0.208088	−	0.978110i	$-0.566724\pi$
$503$	−5.97493e11	−0.416176	−0.208088	+	0.978110i	$0.566724\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	−3.48392e12	−2.34170
$508$	0	0
$509$	−8.46938e11	−0.559270	−0.279635	−	0.960106i	$-0.590213\pi$
$509$	−8.46938e11	−0.559270	−0.279635	+	0.960106i	$0.590213\pi$
$510$	0	0
$511$	−1.59820e12	−1.03690
$512$	0	0
$513$	−6.15760e11	−0.392540
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−1.14917e12	−0.707418
$518$	0	0
$519$	2.05367e12	1.24245
$520$	0	0
$521$	7.31013e11	0.434666	0.217333	−	0.976098i	$-0.430264\pi$
$521$	7.31013e11	0.434666	0.217333	+	0.976098i	$0.430264\pi$
$522$	0	0
$523$	1.04453e12	0.610467	0.305233	−	0.952278i	$-0.401266\pi$
$523$	1.04453e12	0.610467	0.305233	+	0.952278i	$0.401266\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	4.50751e10	0.0254559
$528$	0	0
$529$	1.75502e12	0.974386
$530$	0	0
$531$	−2.15003e12	−1.17360
$532$	0	0
$533$	−3.39315e11	−0.182109
$534$	0	0
$535$	0	0
$536$	0	0
$537$	1.39946e12	0.726233
$538$	0	0
$539$	−2.10575e12	−1.07462
$540$	0	0
$541$	−1.88034e12	−0.943733	−0.471866	−	0.881670i	$-0.656420\pi$
$541$	−1.88034e12	−0.943733	−0.471866	+	0.881670i	$0.656420\pi$
$542$	0	0
$543$	1.54029e12	0.760331
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−1.08624e12	−0.518780	−0.259390	−	0.965773i	$-0.583521\pi$
$547$	−1.08624e12	−0.518780	−0.259390	+	0.965773i	$0.583521\pi$
$548$	0	0
$549$	9.41977e11	0.442553
$550$	0	0
$551$	2.06804e11	0.0955821
$552$	0	0
$553$	1.25775e12	0.571914
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−2.11965e12	−0.933072	−0.466536	−	0.884502i	$-0.654498\pi$
$557$	−2.11965e12	−0.933072	−0.466536	+	0.884502i	$0.654498\pi$
$558$	0	0
$559$	1.11476e12	0.482868
$560$	0	0
$561$	−6.30978e11	−0.268956
$562$	0	0
$563$	3.04489e12	1.27727	0.638636	−	0.769509i	$-0.279499\pi$
$563$	3.04489e12	1.27727	0.638636	+	0.769509i	$0.279499\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	−2.67178e12	−1.08562
$568$	0	0
$569$	3.25275e12	1.30090	0.650452	−	0.759548i	$-0.274580\pi$
$569$	3.25275e12	1.30090	0.650452	+	0.759548i	$0.274580\pi$
$570$	0	0
$571$	1.35916e12	0.535066	0.267533	−	0.963549i	$-0.413792\pi$
$571$	1.35916e12	0.535066	0.267533	+	0.963549i	$0.413792\pi$
$572$	0	0
$573$	1.69599e12	0.657245
$574$	0	0
$575$	0	0
$576$	0	0
$577$	2.35782e12	0.885564	0.442782	−	0.896629i	$-0.353992\pi$
$577$	2.35782e12	0.885564	0.442782	+	0.896629i	$0.353992\pi$
$578$	0	0
$579$	6.57002e11	0.242948
$580$	0	0
$581$	−2.84777e12	−1.03684
$582$	0	0
$583$	−1.78474e12	−0.639834
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−2.37065e12	−0.824130	−0.412065	−	0.911154i	$-0.635192\pi$
$587$	−2.37065e12	−0.824130	−0.412065	+	0.911154i	$0.635192\pi$
$588$	0	0
$589$	−3.33652e11	−0.114229
$590$	0	0
$591$	−5.04690e12	−1.70169
$592$	0	0
$593$	−4.56707e12	−1.51667	−0.758336	−	0.651864i	$-0.773987\pi$
$593$	−4.56707e12	−1.51667	−0.758336	+	0.651864i	$0.773987\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	2.04624e12	0.659283
$598$	0	0
$599$	5.30493e12	1.68368	0.841839	−	0.539729i	$-0.181473\pi$
$599$	5.30493e12	1.68368	0.841839	+	0.539729i	$0.181473\pi$
$600$	0	0
$601$	−2.71307e12	−0.848255	−0.424128	−	0.905602i	$-0.639419\pi$
$601$	−2.71307e12	−0.848255	−0.424128	+	0.905602i	$0.639419\pi$
$602$	0	0
$603$	−3.33335e12	−1.02672
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−4.00734e12	−1.19814	−0.599069	−	0.800698i	$-0.704462\pi$
$607$	−4.00734e12	−1.19814	−0.599069	+	0.800698i	$0.704462\pi$
$608$	0	0
$609$	−7.06237e11	−0.208052
$610$	0	0
$611$	−5.19728e12	−1.50866
$612$	0	0
$613$	2.24954e12	0.643459	0.321730	−	0.946832i	$-0.395736\pi$
$613$	2.24954e12	0.643459	0.321730	+	0.946832i	$0.395736\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−1.19664e12	−0.332414	−0.166207	−	0.986091i	$-0.553152\pi$
$617$	−1.19664e12	−0.332414	−0.166207	+	0.986091i	$0.553152\pi$
$618$	0	0
$619$	4.36723e12	1.19563	0.597817	−	0.801633i	$-0.296035\pi$
$619$	4.36723e12	1.19563	0.597817	+	0.801633i	$0.296035\pi$
$620$	0	0
$621$	−1.90478e12	−0.513965
$622$	0	0
$623$	−5.57811e12	−1.48351
$624$	0	0
$625$	0	0
$626$	0	0
$627$	4.67058e12	1.20689
$628$	0	0
$629$	4.32940e11	0.110281
$630$	0	0
$631$	−2.78790e12	−0.700077	−0.350039	−	0.936735i	$-0.613832\pi$
$631$	−2.78790e12	−0.700077	−0.350039	+	0.936735i	$0.613832\pi$
$632$	0	0
$633$	−5.02950e12	−1.24511
$634$	0	0
$635$	0	0
$636$	0	0
$637$	−9.52355e12	−2.29177
$638$	0	0
$639$	−8.55199e12	−2.02915
$640$	0	0
$641$	−2.28295e12	−0.534115	−0.267058	−	0.963681i	$-0.586051\pi$
$641$	−2.28295e12	−0.534115	−0.267058	+	0.963681i	$0.586051\pi$
$642$	0	0
$643$	−2.29995e12	−0.530603	−0.265301	−	0.964166i	$-0.585471\pi$
$643$	−2.29995e12	−0.530603	−0.265301	+	0.964166i	$0.585471\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	1.85647e12	0.416503	0.208251	−	0.978075i	$-0.433223\pi$
$647$	1.85647e12	0.416503	0.208251	+	0.978075i	$0.433223\pi$
$648$	0	0
$649$	3.20048e12	0.708132
$650$	0	0
$651$	1.13942e12	0.248640
$652$	0	0
$653$	−7.75397e12	−1.66884	−0.834420	−	0.551129i	$-0.814197\pi$
$653$	−7.75397e12	−1.66884	−0.834420	+	0.551129i	$0.814197\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	−3.95117e12	−0.827336
$658$	0	0
$659$	−7.42166e12	−1.53291	−0.766455	−	0.642298i	$-0.777981\pi$
$659$	−7.42166e12	−1.53291	−0.766455	+	0.642298i	$0.777981\pi$
$660$	0	0
$661$	5.79783e12	1.18130	0.590648	−	0.806929i	$-0.298872\pi$
$661$	5.79783e12	1.18130	0.590648	+	0.806929i	$0.298872\pi$
$662$	0	0
$663$	−2.85369e12	−0.573583
$664$	0	0
$665$	0	0
$666$	0	0
$667$	6.39724e11	0.125149
$668$	0	0
$669$	−4.25153e12	−0.820593
$670$	0	0
$671$	−1.40220e12	−0.267029
$672$	0	0
$673$	6.32456e12	1.18840	0.594200	−	0.804318i	$-0.297469\pi$
$673$	6.32456e12	1.18840	0.594200	+	0.804318i	$0.297469\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	4.95066e12	0.905762	0.452881	−	0.891571i	$-0.350396\pi$
$677$	4.95066e12	0.905762	0.452881	+	0.891571i	$0.350396\pi$
$678$	0	0
$679$	4.67246e12	0.843591
$680$	0	0
$681$	−9.29277e12	−1.65571
$682$	0	0
$683$	7.32611e12	1.28819	0.644095	−	0.764945i	$-0.277234\pi$
$683$	7.32611e12	1.28819	0.644095	+	0.764945i	$0.277234\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	1.02875e13	1.76200
$688$	0	0
$689$	−8.07176e12	−1.36453
$690$	0	0
$691$	5.66806e12	0.945765	0.472882	−	0.881126i	$-0.343214\pi$
$691$	5.66806e12	0.945765	0.472882	+	0.881126i	$0.343214\pi$
$692$	0	0
$693$	−8.84274e12	−1.45642
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−1.69501e11	−0.0272034
$698$	0	0
$699$	1.40422e13	2.22479
$700$	0	0
$701$	−4.20985e12	−0.658470	−0.329235	−	0.944248i	$-0.606791\pi$
$701$	−4.20985e12	−0.658470	−0.329235	+	0.944248i	$0.606791\pi$
$702$	0	0
$703$	−3.20468e12	−0.494863
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−1.24486e13	−1.87384
$708$	0	0
$709$	1.19654e13	1.77835	0.889176	−	0.457566i	$-0.151279\pi$
$709$	1.19654e13	1.77835	0.889176	+	0.457566i	$0.151279\pi$
$710$	0	0
$711$	3.10949e12	0.456326
$712$	0	0
$713$	−1.03211e12	−0.149563
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−4.64799e12	−0.656794
$718$	0	0
$719$	4.07118e11	0.0568120	0.0284060	−	0.999596i	$-0.490957\pi$
$719$	4.07118e11	0.0568120	0.0284060	+	0.999596i	$0.490957\pi$
$720$	0	0
$721$	1.93551e13	2.66740
$722$	0	0
$723$	−2.09675e13	−2.85380
$724$	0	0
$725$	0	0
$726$	0	0
$727$	6.44424e12	0.855592	0.427796	−	0.903875i	$-0.359290\pi$
$727$	6.44424e12	0.855592	0.427796	+	0.903875i	$0.359290\pi$
$728$	0	0
$729$	−1.07838e13	−1.41416
$730$	0	0
$731$	5.56864e11	0.0721308
$732$	0	0
$733$	−5.46122e12	−0.698749	−0.349375	−	0.936983i	$-0.613606\pi$
$733$	−5.46122e12	−0.698749	−0.349375	+	0.936983i	$0.613606\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	4.96192e12	0.619507
$738$	0	0
$739$	1.88201e12	0.232125	0.116062	−	0.993242i	$-0.462973\pi$
$739$	1.88201e12	0.232125	0.116062	+	0.993242i	$0.462973\pi$
$740$	0	0
$741$	2.11234e13	2.57384
$742$	0	0
$743$	1.36704e13	1.64563	0.822813	−	0.568312i	$-0.192403\pi$
$743$	1.36704e13	1.64563	0.822813	+	0.568312i	$0.192403\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	−7.04042e12	−0.827287
$748$	0	0
$749$	2.10300e13	2.44158
$750$	0	0
$751$	−8.52585e12	−0.978043	−0.489021	−	0.872272i	$-0.662646\pi$
$751$	−8.52585e12	−0.978043	−0.489021	+	0.872272i	$0.662646\pi$
$752$	0	0
$753$	−1.04513e13	−1.18466
$754$	0	0
$755$	0	0
$756$	0	0
$757$	7.50792e12	0.830975	0.415487	−	0.909599i	$-0.363611\pi$
$757$	7.50792e12	0.830975	0.415487	+	0.909599i	$0.363611\pi$
$758$	0	0
$759$	1.44479e13	1.58022
$760$	0	0
$761$	−1.61560e13	−1.74624	−0.873118	−	0.487509i	$-0.837906\pi$
$761$	−1.61560e13	−1.74624	−0.873118	+	0.487509i	$0.837906\pi$
$762$	0	0
$763$	−1.17740e12	−0.125766
$764$	0	0
$765$	0	0
$766$	0	0
$767$	1.44747e13	1.51018
$768$	0	0
$769$	1.11830e12	0.115317	0.0576583	−	0.998336i	$-0.481637\pi$
$769$	1.11830e12	0.115317	0.0576583	+	0.998336i	$0.481637\pi$
$770$	0	0
$771$	1.22174e13	1.24519
$772$	0	0
$773$	1.73165e13	1.74442	0.872212	−	0.489129i	$-0.162685\pi$
$773$	1.73165e13	1.74442	0.872212	+	0.489129i	$0.162685\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	1.09440e13	1.07716
$778$	0	0
$779$	1.25466e12	0.122070
$780$	0	0
$781$	1.27302e13	1.22435
$782$	0	0
$783$	−3.42654e11	−0.0325783
$784$	0	0
$785$	0	0
$786$	0	0
$787$	−3.88515e11	−0.0361012	−0.0180506	−	0.999837i	$-0.505746\pi$
$787$	−3.88515e11	−0.0361012	−0.0180506	+	0.999837i	$0.505746\pi$
$788$	0	0
$789$	6.79853e12	0.624552
$790$	0	0
$791$	1.83702e13	1.66848
$792$	0	0
$793$	−6.34166e12	−0.569474
$794$	0	0
$795$	0	0
$796$	0	0
$797$	1.08432e12	0.0951910	0.0475955	−	0.998867i	$-0.484844\pi$
$797$	1.08432e12	0.0951910	0.0475955	+	0.998867i	$0.484844\pi$
$798$	0	0
$799$	−2.59624e12	−0.225363
$800$	0	0
$801$	−1.37906e13	−1.18368
$802$	0	0
$803$	5.88161e12	0.499202
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−1.31193e12	−0.108888
$808$	0	0
$809$	4.06803e12	0.333900	0.166950	−	0.985965i	$-0.446608\pi$
$809$	4.06803e12	0.333900	0.166950	+	0.985965i	$0.446608\pi$
$810$	0	0
$811$	−4.87468e12	−0.395688	−0.197844	−	0.980234i	$-0.563394\pi$
$811$	−4.87468e12	−0.395688	−0.197844	+	0.980234i	$0.563394\pi$
$812$	0	0
$813$	1.69359e13	1.35957
$814$	0	0
$815$	0	0
$816$	0	0
$817$	−4.12198e12	−0.323673
$818$	0	0
$819$	−3.99926e13	−3.10600
$820$	0	0
$821$	−1.03829e12	−0.0797582	−0.0398791	−	0.999205i	$-0.512697\pi$
$821$	−1.03829e12	−0.0797582	−0.0398791	+	0.999205i	$0.512697\pi$
$822$	0	0
$823$	1.10923e13	0.842794	0.421397	−	0.906876i	$-0.361540\pi$
$823$	1.10923e13	0.842794	0.421397	+	0.906876i	$0.361540\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−1.45526e13	−1.08184	−0.540922	−	0.841073i	$-0.681925\pi$
$827$	−1.45526e13	−1.08184	−0.540922	+	0.841073i	$0.681925\pi$
$828$	0	0
$829$	1.37985e13	1.01470	0.507348	−	0.861741i	$-0.330626\pi$
$829$	1.37985e13	1.01470	0.507348	+	0.861741i	$0.330626\pi$
$830$	0	0
$831$	−1.48796e13	−1.08240
$832$	0	0
$833$	−4.75736e12	−0.342345
$834$	0	0
$835$	0	0
$836$	0	0
$837$	5.52829e11	0.0389337
$838$	0	0
$839$	−9.20234e11	−0.0641165	−0.0320582	−	0.999486i	$-0.510206\pi$
$839$	−9.20234e11	−0.0641165	−0.0320582	+	0.999486i	$0.510206\pi$
$840$	0	0
$841$	−1.43921e13	−0.992067
$842$	0	0
$843$	−1.71420e13	−1.16906
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−1.01936e13	−0.680536
$848$	0	0
$849$	−3.97855e13	−2.62809
$850$	0	0
$851$	−9.91330e12	−0.647941
$852$	0	0
$853$	7.46964e12	0.483091	0.241545	−	0.970390i	$-0.422346\pi$
$853$	7.46964e12	0.483091	0.241545	+	0.970390i	$0.422346\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−2.05568e13	−1.30179	−0.650897	−	0.759166i	$-0.725607\pi$
$857$	−2.05568e13	−1.30179	−0.650897	+	0.759166i	$0.725607\pi$
$858$	0	0
$859$	1.05923e12	0.0663775	0.0331888	−	0.999449i	$-0.489434\pi$
$859$	1.05923e12	0.0663775	0.0331888	+	0.999449i	$0.489434\pi$
$860$	0	0
$861$	−4.28469e12	−0.265709
$862$	0	0
$863$	1.53023e13	0.939092	0.469546	−	0.882908i	$-0.344418\pi$
$863$	1.53023e13	0.939092	0.469546	+	0.882908i	$0.344418\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	2.34982e13	1.41237
$868$	0	0
$869$	−4.62869e12	−0.275340
$870$	0	0
$871$	2.24410e13	1.32118
$872$	0	0
$873$	1.15516e13	0.673095
$874$	0	0
$875$	0	0
$876$	0	0
$877$	1.74558e11	0.00996416	0.00498208	−	0.999988i	$-0.498414\pi$
$877$	1.74558e11	0.00996416	0.00498208	+	0.999988i	$0.498414\pi$
$878$	0	0
$879$	−1.13385e13	−0.640629
$880$	0	0
$881$	8.59039e12	0.480420	0.240210	−	0.970721i	$-0.422784\pi$
$881$	8.59039e12	0.480420	0.240210	+	0.970721i	$0.422784\pi$
$882$	0	0
$883$	−1.70227e13	−0.942336	−0.471168	−	0.882043i	$-0.656167\pi$
$883$	−1.70227e13	−0.942336	−0.471168	+	0.882043i	$0.656167\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−2.01246e13	−1.09162	−0.545810	−	0.837909i	$-0.683778\pi$
$887$	−2.01246e13	−1.09162	−0.545810	+	0.837909i	$0.683778\pi$
$888$	0	0
$889$	−5.34112e13	−2.86797
$890$	0	0
$891$	9.83253e12	0.522655
$892$	0	0
$893$	1.92177e13	1.01127
$894$	0	0
$895$	0	0
$896$	0	0
$897$	6.53428e13	3.37002
$898$	0	0
$899$	−1.85668e11	−0.00948023
$900$	0	0
$901$	−4.03214e12	−0.203833
$902$	0	0
$903$	1.40766e13	0.704535
$904$	0	0
$905$	0	0
$906$	0	0
$907$	−2.10130e13	−1.03099	−0.515495	−	0.856893i	$-0.672392\pi$
$907$	−2.10130e13	−1.03099	−0.515495	+	0.856893i	$0.672392\pi$
$908$	0	0
$909$	−3.07761e13	−1.49512
$910$	0	0
$911$	−1.95345e12	−0.0939656	−0.0469828	−	0.998896i	$-0.514961\pi$
$911$	−1.95345e12	−0.0939656	−0.0469828	+	0.998896i	$0.514961\pi$
$912$	0	0
$913$	1.04802e13	0.499172
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−3.80064e13	−1.77499
$918$	0	0
$919$	2.47656e12	0.114533	0.0572663	−	0.998359i	$-0.481762\pi$
$919$	2.47656e12	0.114533	0.0572663	+	0.998359i	$0.481762\pi$
$920$	0	0
$921$	2.73301e13	1.25162
$922$	0	0
$923$	5.75744e13	2.61109
$924$	0	0
$925$	0	0
$926$	0	0
$927$	4.78510e13	2.12830
$928$	0	0
$929$	4.40844e13	1.94184	0.970922	−	0.239395i	$-0.0769490\pi$
$929$	4.40844e13	1.94184	0.970922	+	0.239395i	$0.0769490\pi$
$930$	0	0
$931$	3.52146e13	1.53621
$932$	0	0
$933$	4.40477e11	0.0190308
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−4.92250e12	−0.208621	−0.104310	−	0.994545i	$-0.533264\pi$
$937$	−4.92250e12	−0.208621	−0.104310	+	0.994545i	$0.533264\pi$
$938$	0	0
$939$	−5.12372e13	−2.15075
$940$	0	0
$941$	−1.54680e13	−0.643103	−0.321552	−	0.946892i	$-0.604204\pi$
$941$	−1.54680e13	−0.643103	−0.321552	+	0.946892i	$0.604204\pi$
$942$	0	0
$943$	3.88116e12	0.159830
$944$	0	0
$945$	0	0
$946$	0	0
$947$	5.49107e12	0.221862	0.110931	−	0.993828i	$-0.464617\pi$
$947$	5.49107e12	0.221862	0.110931	+	0.993828i	$0.464617\pi$
$948$	0	0
$949$	2.66004e13	1.06461
$950$	0	0
$951$	−7.28207e13	−2.88697
$952$	0	0
$953$	1.76932e13	0.694845	0.347422	−	0.937709i	$-0.387057\pi$
$953$	1.76932e13	0.694845	0.347422	+	0.937709i	$0.387057\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	2.59905e12	0.100164
$958$	0	0
$959$	2.52994e13	0.965888
$960$	0	0
$961$	−2.61401e13	−0.988670
$962$	0	0
$963$	5.19916e13	1.94812
$964$	0	0
$965$	0	0
$966$	0	0
$967$	2.89231e13	1.06372	0.531858	−	0.846834i	$-0.321494\pi$
$967$	2.89231e13	1.06372	0.531858	+	0.846834i	$0.321494\pi$
$968$	0	0
$969$	1.05519e13	0.384481
$970$	0	0
$971$	−2.69482e13	−0.972843	−0.486422	−	0.873724i	$-0.661698\pi$
$971$	−2.69482e13	−0.972843	−0.486422	+	0.873724i	$0.661698\pi$
$972$	0	0
$973$	4.05255e13	1.44951
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−3.90598e13	−1.37153	−0.685764	−	0.727824i	$-0.740532\pi$
$977$	−3.90598e13	−1.37153	−0.685764	+	0.727824i	$0.740532\pi$
$978$	0	0
$979$	2.05282e13	0.714216
$980$	0	0
$981$	−2.91084e12	−0.100348
$982$	0	0
$983$	9.03830e12	0.308742	0.154371	−	0.988013i	$-0.450665\pi$
$983$	9.03830e12	0.308742	0.154371	+	0.988013i	$0.450665\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	−6.56285e13	−2.20123
$988$	0	0
$989$	−1.27509e13	−0.423795
$990$	0	0
$991$	−4.89096e13	−1.61088	−0.805439	−	0.592678i	$-0.798071\pi$
$991$	−4.89096e13	−1.61088	−0.805439	+	0.592678i	$0.798071\pi$
$992$	0	0
$993$	−3.01181e13	−0.983007
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−4.01445e13	−1.28676	−0.643380	−	0.765547i	$-0.722468\pi$
$997$	−4.01445e13	−1.28676	−0.643380	+	0.765547i	$0.722468\pi$
$998$	0	0
$999$	5.30984e12	0.168670

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	400.10.a.y.1.1		3
4.3	odd	2		25.10.a.c.1.2	✓	3
5.2	odd	4		400.10.c.q.49.5		6
5.3	odd	4		400.10.c.q.49.2		6
5.4	even	2		400.10.a.u.1.3		3
12.11	even	2		225.10.a.p.1.2		3
20.3	even	4		25.10.b.c.24.5		6
20.7	even	4		25.10.b.c.24.2		6
20.19	odd	2		25.10.a.d.1.2	yes	3
60.23	odd	4		225.10.b.m.199.2		6
60.47	odd	4		225.10.b.m.199.5		6
60.59	even	2		225.10.a.m.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
25.10.a.c.1.2	✓	3	4.3	odd	2
25.10.a.d.1.2	yes	3	20.19	odd	2
25.10.b.c.24.2		6	20.7	even	4
25.10.b.c.24.5		6	20.3	even	4
225.10.a.m.1.2		3	60.59	even	2
225.10.a.p.1.2		3	12.11	even	2
225.10.b.m.199.2		6	60.23	odd	4
225.10.b.m.199.5		6	60.47	odd	4
400.10.a.u.1.3		3	5.4	even	2
400.10.a.y.1.1		3	1.1	even	1	trivial
400.10.c.q.49.2		6	5.3	odd	4
400.10.c.q.49.5		6	5.2	odd	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 400 = 2^{4} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 10 \)
Character orbit:	\([\chi]\)	\(=\)	400.a (trivial)

\(f(q)\)	\(=\)	\(q-210.171 q^{3} +9905.49 q^{7} +24489.0 q^{9} +O(q^{10})\)	\(q-210.171 q^{3} +9905.49 q^{7} +24489.0 q^{9} -36453.6 q^{11} -164867. q^{13} -82357.1 q^{17} +609617. q^{19} -2.08185e6 q^{21} +1.88578e6 q^{23} -1.01008e6 q^{27} +339235. q^{29} -547314. q^{31} +7.66150e6 q^{33} -5.25687e6 q^{37} +3.46503e7 q^{39} +2.05812e6 q^{41} -6.76158e6 q^{43} +3.15241e7 q^{47} +5.77651e7 q^{49} +1.73091e7 q^{51} +4.89593e7 q^{53} -1.28124e8 q^{57} -8.77960e7 q^{59} +3.84654e7 q^{61} +2.42575e8 q^{63} -1.36116e8 q^{67} -3.96337e8 q^{69} -3.49218e8 q^{71} -1.61345e8 q^{73} -3.61091e8 q^{77} +1.26975e8 q^{79} -2.69727e8 q^{81} -2.87494e8 q^{83} -7.12976e7 q^{87} -5.63133e8 q^{89} -1.63309e9 q^{91} +1.15030e8 q^{93} +4.71704e8 q^{97} -8.92711e8 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 89 q^{3} + 5258 q^{7} + 58234 q^{9}+O(q^{10}) \) 3 * q + 89 * q^3 + 5258 * q^7 + 58234 * q^9	\( 3 q + 89 q^{3} + 5258 q^{7} + 58234 q^{9} + 54699 q^{11} - 215884 q^{13} - 334983 q^{17} - 818845 q^{19} - 2375394 q^{21} + 3526854 q^{23} + 6633395 q^{27} + 2175480 q^{29} - 4274066 q^{31} + 22122137 q^{33} + 10305042 q^{37} + 50414092 q^{39} + 5926311 q^{41} - 24429956 q^{43} + 66858708 q^{47} - 6453929 q^{49} + 25634699 q^{51} - 132620514 q^{53} - 252946415 q^{57} - 5670960 q^{59} + 125306926 q^{61} + 284323404 q^{63} + 88829483 q^{67} - 314274942 q^{69} - 297550596 q^{71} - 181321729 q^{73} - 561214086 q^{77} + 310025170 q^{79} + 1398847363 q^{81} - 731088801 q^{83} + 1046385560 q^{87} - 1103860035 q^{89} - 1183187656 q^{91} - 107386758 q^{93} + 332236842 q^{97} + 892234522 q^{99}+O(q^{100}) \) 3 * q + 89 * q^3 + 5258 * q^7 + 58234 * q^9 + 54699 * q^11 - 215884 * q^13 - 334983 * q^17 - 818845 * q^19 - 2375394 * q^21 + 3526854 * q^23 + 6633395 * q^27 + 2175480 * q^29 - 4274066 * q^31 + 22122137 * q^33 + 10305042 * q^37 + 50414092 * q^39 + 5926311 * q^41 - 24429956 * q^43 + 66858708 * q^47 - 6453929 * q^49 + 25634699 * q^51 - 132620514 * q^53 - 252946415 * q^57 - 5670960 * q^59 + 125306926 * q^61 + 284323404 * q^63 + 88829483 * q^67 - 314274942 * q^69 - 297550596 * q^71 - 181321729 * q^73 - 561214086 * q^77 + 310025170 * q^79 + 1398847363 * q^81 - 731088801 * q^83 + 1046385560 * q^87 - 1103860035 * q^89 - 1183187656 * q^91 - 107386758 * q^93 + 332236842 * q^97 + 892234522 * q^99

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