LMFDB - Embedded newform 192.8.a.u.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [192,8,Mod(1,192)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(192, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 8, names="a")

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

chi := DirichletCharacter("192.1");

S:= CuspForms(chi, 8);

N := Newforms(S);

Level:	$ N $	$=$	$ 192 = 2^{6} \cdot 3 $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	192.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:	$59.9779248930$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{235}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 235 $ x^2 - 235
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	no (minimal twist has level 96)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$15.3297$ of defining polynomial
Character	$\chi$	$=$	192.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+27.0000 q^{3} +155.275 q^{5} -1742.38 q^{7} +729.000 q^{9} +O(q^{10})$	$q+27.0000 q^{3} +155.275 q^{5} -1742.38 q^{7} +729.000 q^{9} -51.6521 q^{11} +10471.6 q^{13} +4192.43 q^{15} +24021.5 q^{17} -1149.27 q^{19} -47044.2 q^{21} -86508.9 q^{23} -54014.6 q^{25} +19683.0 q^{27} -153332. q^{29} +132760. q^{31} -1394.61 q^{33} -270548. q^{35} -504967. q^{37} +282732. q^{39} +99416.4 q^{41} +84318.7 q^{43} +113196. q^{45} -998165. q^{47} +2.21233e6 q^{49} +648580. q^{51} -1.24958e6 q^{53} -8020.30 q^{55} -31030.4 q^{57} -1.67870e6 q^{59} +405326. q^{61} -1.27019e6 q^{63} +1.62598e6 q^{65} -2.16414e6 q^{67} -2.33574e6 q^{69} -1.57024e6 q^{71} -2.88524e6 q^{73} -1.45839e6 q^{75} +89997.5 q^{77} -3.87170e6 q^{79} +531441. q^{81} +796558. q^{83} +3.72994e6 q^{85} -4.13997e6 q^{87} +1.41571e6 q^{89} -1.82454e7 q^{91} +3.58453e6 q^{93} -178454. q^{95} -238942. q^{97} -37654.4 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 54 q^{3} - 180 q^{5} - 1032 q^{7} + 1458 q^{9}+O(q^{10}) $ 2 * q + 54 * q^3 - 180 * q^5 - 1032 * q^7 + 1458 * q^9	$ 2 q + 54 q^{3} - 180 q^{5} - 1032 q^{7} + 1458 q^{9} + 2840 q^{11} + 340 q^{13} - 4860 q^{15} + 9780 q^{17} + 32040 q^{19} - 27864 q^{21} - 11136 q^{23} - 19730 q^{25} + 39366 q^{27} - 304212 q^{29} + 77640 q^{31} + 76680 q^{33} - 508720 q^{35} - 1015820 q^{37} + 9180 q^{39} + 704100 q^{41} - 395496 q^{43} - 131220 q^{45} - 1157488 q^{47} + 1893426 q^{49} + 264060 q^{51} - 1568580 q^{53} - 977520 q^{55} + 865080 q^{57} + 139240 q^{59} - 2603580 q^{61} - 752328 q^{63} + 5022840 q^{65} - 5289768 q^{67} - 300672 q^{69} - 5721760 q^{71} - 1190700 q^{73} - 532710 q^{75} + 2144160 q^{77} + 398280 q^{79} + 1062882 q^{81} - 6986616 q^{83} + 8504760 q^{85} - 8213724 q^{87} - 8166732 q^{89} - 25442640 q^{91} + 2096280 q^{93} - 11306000 q^{95} - 10361500 q^{97} + 2070360 q^{99}+O(q^{100}) $ 2 * q + 54 * q^3 - 180 * q^5 - 1032 * q^7 + 1458 * q^9 + 2840 * q^11 + 340 * q^13 - 4860 * q^15 + 9780 * q^17 + 32040 * q^19 - 27864 * q^21 - 11136 * q^23 - 19730 * q^25 + 39366 * q^27 - 304212 * q^29 + 77640 * q^31 + 76680 * q^33 - 508720 * q^35 - 1015820 * q^37 + 9180 * q^39 + 704100 * q^41 - 395496 * q^43 - 131220 * q^45 - 1157488 * q^47 + 1893426 * q^49 + 264060 * q^51 - 1568580 * q^53 - 977520 * q^55 + 865080 * q^57 + 139240 * q^59 - 2603580 * q^61 - 752328 * q^63 + 5022840 * q^65 - 5289768 * q^67 - 300672 * q^69 - 5721760 * q^71 - 1190700 * q^73 - 532710 * q^75 + 2144160 * q^77 + 398280 * q^79 + 1062882 * q^81 - 6986616 * q^83 + 8504760 * q^85 - 8213724 * q^87 - 8166732 * q^89 - 25442640 * q^91 + 2096280 * q^93 - 11306000 * q^95 - 10361500 * q^97 + 2070360 * 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
$3$	27.0000	0.577350
$3$	27.0000	0.577350
$4$	0	0
$5$	155.275	0.555530	0.277765	−	0.960649i	$-0.410406\pi$
$5$	155.275	0.555530	0.277765	+	0.960649i	$0.410406\pi$
$6$	0	0
$7$	−1742.38	−1.91999	−0.959995	−	0.280017i	$-0.909660\pi$
$7$	−1742.38	−1.91999	−0.959995	+	0.280017i	$0.909660\pi$
$8$	0	0
$9$	729.000	0.333333
$10$	0	0
$11$	−51.6521	−0.0117008	−0.00585038	−	0.999983i	$-0.501862\pi$
$11$	−51.6521	−0.0117008	−0.00585038	+	0.999983i	$0.501862\pi$
$12$	0	0
$13$	10471.6	1.32193	0.660967	−	0.750415i	$-0.270146\pi$
$13$	10471.6	1.32193	0.660967	+	0.750415i	$0.270146\pi$
$14$	0	0
$15$	4192.43	0.320735
$16$	0	0
$17$	24021.5	1.18585	0.592923	−	0.805259i	$-0.297974\pi$
$17$	24021.5	1.18585	0.592923	+	0.805259i	$0.297974\pi$
$18$	0	0
$19$	−1149.27	−0.0384403	−0.0192201	−	0.999815i	$-0.506118\pi$
$19$	−1149.27	−0.0384403	−0.0192201	+	0.999815i	$0.506118\pi$
$20$	0	0
$21$	−47044.2	−1.10851
$22$	0	0
$23$	−86508.9	−1.48256	−0.741282	−	0.671194i	$-0.765782\pi$
$23$	−86508.9	−1.48256	−0.741282	+	0.671194i	$0.765782\pi$
$24$	0	0
$25$	−54014.6	−0.691386
$26$	0	0
$27$	19683.0	0.192450
$28$	0	0
$29$	−153332.	−1.16746	−0.583728	−	0.811949i	$-0.698407\pi$
$29$	−153332.	−1.16746	−0.583728	+	0.811949i	$0.698407\pi$
$30$	0	0
$31$	132760.	0.800392	0.400196	−	0.916430i	$-0.368942\pi$
$31$	132760.	0.800392	0.400196	+	0.916430i	$0.368942\pi$
$32$	0	0
$33$	−1394.61	−0.00675543
$34$	0	0
$35$	−270548.	−1.06661
$36$	0	0
$37$	−504967.	−1.63892	−0.819458	−	0.573139i	$-0.805725\pi$
$37$	−504967.	−1.63892	−0.819458	+	0.573139i	$0.805725\pi$
$38$	0	0
$39$	282732.	0.763219
$40$	0	0
$41$	99416.4	0.225276	0.112638	−	0.993636i	$-0.464070\pi$
$41$	99416.4	0.225276	0.112638	+	0.993636i	$0.464070\pi$
$42$	0	0
$43$	84318.7	0.161728	0.0808638	−	0.996725i	$-0.474232\pi$
$43$	84318.7	0.161728	0.0808638	+	0.996725i	$0.474232\pi$
$44$	0	0
$45$	113196.	0.185177
$46$	0	0
$47$	−998165.	−1.40236	−0.701180	−	0.712984i	$-0.747343\pi$
$47$	−998165.	−1.40236	−0.701180	+	0.712984i	$0.747343\pi$
$48$	0	0
$49$	2.21233e6	2.68636
$50$	0	0
$51$	648580.	0.684649
$52$	0	0
$53$	−1.24958e6	−1.15292	−0.576458	−	0.817127i	$-0.695566\pi$
$53$	−1.24958e6	−1.15292	−0.576458	+	0.817127i	$0.695566\pi$
$54$	0	0
$55$	−8020.30	−0.00650012
$56$	0	0
$57$	−31030.4	−0.0221935
$58$	0	0
$59$	−1.67870e6	−1.06412	−0.532061	−	0.846706i	$-0.678582\pi$
$59$	−1.67870e6	−1.06412	−0.532061	+	0.846706i	$0.678582\pi$
$60$	0	0
$61$	405326.	0.228639	0.114320	−	0.993444i	$-0.463531\pi$
$61$	405326.	0.228639	0.114320	+	0.993444i	$0.463531\pi$
$62$	0	0
$63$	−1.27019e6	−0.639997
$64$	0	0
$65$	1.62598e6	0.734374
$66$	0	0
$67$	−2.16414e6	−0.879072	−0.439536	−	0.898225i	$-0.644857\pi$
$67$	−2.16414e6	−0.879072	−0.439536	+	0.898225i	$0.644857\pi$
$68$	0	0
$69$	−2.33574e6	−0.855958
$70$	0	0
$71$	−1.57024e6	−0.520669	−0.260335	−	0.965518i	$-0.583833\pi$
$71$	−1.57024e6	−0.520669	−0.260335	+	0.965518i	$0.583833\pi$
$72$	0	0
$73$	−2.88524e6	−0.868065	−0.434032	−	0.900897i	$-0.642910\pi$
$73$	−2.88524e6	−0.868065	−0.434032	+	0.900897i	$0.642910\pi$
$74$	0	0
$75$	−1.45839e6	−0.399172
$76$	0	0
$77$	89997.5	0.0224653
$78$	0	0
$79$	−3.87170e6	−0.883500	−0.441750	−	0.897138i	$-0.645642\pi$
$79$	−3.87170e6	−0.883500	−0.441750	+	0.897138i	$0.645642\pi$
$80$	0	0
$81$	531441.	0.111111
$82$	0	0
$83$	796558.	0.152913	0.0764564	−	0.997073i	$-0.475639\pi$
$83$	796558.	0.152913	0.0764564	+	0.997073i	$0.475639\pi$
$84$	0	0
$85$	3.72994e6	0.658773
$86$	0	0
$87$	−4.13997e6	−0.674032
$88$	0	0
$89$	1.41571e6	0.212867	0.106434	−	0.994320i	$-0.466057\pi$
$89$	1.41571e6	0.212867	0.106434	+	0.994320i	$0.466057\pi$
$90$	0	0
$91$	−1.82454e7	−2.53810
$92$	0	0
$93$	3.58453e6	0.462107
$94$	0	0
$95$	−178454.	−0.0213547
$96$	0	0
$97$	−238942.	−0.0265822	−0.0132911	−	0.999912i	$-0.504231\pi$
$97$	−238942.	−0.0265822	−0.0132911	+	0.999912i	$0.504231\pi$
$98$	0	0
$99$	−37654.4	−0.00390025
$100$	0	0
$101$	1.87275e7	1.80865	0.904324	−	0.426847i	$-0.140376\pi$
$101$	1.87275e7	1.80865	0.904324	+	0.426847i	$0.140376\pi$
$102$	0	0
$103$	4.67587e6	0.421630	0.210815	−	0.977526i	$-0.432388\pi$
$103$	4.67587e6	0.421630	0.210815	+	0.977526i	$0.432388\pi$
$104$	0	0
$105$	−7.30480e6	−0.615809
$106$	0	0
$107$	1.38526e7	1.09317	0.546585	−	0.837404i	$-0.315928\pi$
$107$	1.38526e7	1.09317	0.546585	+	0.837404i	$0.315928\pi$
$108$	0	0
$109$	−5.05778e6	−0.374083	−0.187041	−	0.982352i	$-0.559890\pi$
$109$	−5.05778e6	−0.374083	−0.187041	+	0.982352i	$0.559890\pi$
$110$	0	0
$111$	−1.36341e7	−0.946229
$112$	0	0
$113$	−2.79817e7	−1.82431	−0.912157	−	0.409841i	$-0.865584\pi$
$113$	−2.79817e7	−1.82431	−0.912157	+	0.409841i	$0.865584\pi$
$114$	0	0
$115$	−1.34327e7	−0.823608
$116$	0	0
$117$	7.63377e6	0.440645
$118$	0	0
$119$	−4.18545e7	−2.27681
$120$	0	0
$121$	−1.94845e7	−0.999863
$122$	0	0
$123$	2.68424e6	0.130063
$124$	0	0
$125$	−2.05180e7	−0.939616
$126$	0	0
$127$	4.22490e6	0.183022	0.0915109	−	0.995804i	$-0.470830\pi$
$127$	4.22490e6	0.183022	0.0915109	+	0.995804i	$0.470830\pi$
$128$	0	0
$129$	2.27660e6	0.0933735
$130$	0	0
$131$	−4.16738e7	−1.61962	−0.809812	−	0.586690i	$-0.800431\pi$
$131$	−4.16738e7	−1.61962	−0.809812	+	0.586690i	$0.800431\pi$
$132$	0	0
$133$	2.00247e6	0.0738049
$134$	0	0
$135$	3.05628e6	0.106912
$136$	0	0
$137$	2.58796e7	0.859876	0.429938	−	0.902858i	$-0.358535\pi$
$137$	2.58796e7	0.859876	0.429938	+	0.902858i	$0.358535\pi$
$138$	0	0
$139$	−5.07048e7	−1.60139	−0.800696	−	0.599071i	$-0.795537\pi$
$139$	−5.07048e7	−1.60139	−0.800696	+	0.599071i	$0.795537\pi$
$140$	0	0
$141$	−2.69505e7	−0.809653
$142$	0	0
$143$	−540879.	−0.0154676
$144$	0	0
$145$	−2.38087e7	−0.648557
$146$	0	0
$147$	5.97330e7	1.55097
$148$	0	0
$149$	4.36811e7	1.08179	0.540893	−	0.841091i	$-0.318086\pi$
$149$	4.36811e7	1.08179	0.540893	+	0.841091i	$0.318086\pi$
$150$	0	0
$151$	6.81915e7	1.61180	0.805900	−	0.592052i	$-0.201682\pi$
$151$	6.81915e7	1.61180	0.805900	+	0.592052i	$0.201682\pi$
$152$	0	0
$153$	1.75117e7	0.395282
$154$	0	0
$155$	2.06144e7	0.444642
$156$	0	0
$157$	−9.99359e6	−0.206098	−0.103049	−	0.994676i	$-0.532860\pi$
$157$	−9.99359e6	−0.206098	−0.103049	+	0.994676i	$0.532860\pi$
$158$	0	0
$159$	−3.37386e7	−0.665636
$160$	0	0
$161$	1.50731e8	2.84651
$162$	0	0
$163$	4.77375e6	0.0863382	0.0431691	−	0.999068i	$-0.486255\pi$
$163$	4.77375e6	0.0863382	0.0431691	+	0.999068i	$0.486255\pi$
$164$	0	0
$165$	−216548.	−0.00375285
$166$	0	0
$167$	−8.90961e7	−1.48030	−0.740152	−	0.672440i	$-0.765246\pi$
$167$	−8.90961e7	−1.48030	−0.740152	+	0.672440i	$0.765246\pi$
$168$	0	0
$169$	4.69052e7	0.747510
$170$	0	0
$171$	−837821.	−0.0128134
$172$	0	0
$173$	2.58721e7	0.379901	0.189950	−	0.981794i	$-0.439167\pi$
$173$	2.58721e7	0.379901	0.189950	+	0.981794i	$0.439167\pi$
$174$	0	0
$175$	9.41137e7	1.32745
$176$	0	0
$177$	−4.53250e7	−0.614372
$178$	0	0
$179$	4.29009e7	0.559089	0.279545	−	0.960133i	$-0.409817\pi$
$179$	4.29009e7	0.559089	0.279545	+	0.960133i	$0.409817\pi$
$180$	0	0
$181$	−1.28747e8	−1.61384	−0.806920	−	0.590661i	$-0.798867\pi$
$181$	−1.28747e8	−1.61384	−0.806920	+	0.590661i	$0.798867\pi$
$182$	0	0
$183$	1.09438e7	0.132005
$184$	0	0
$185$	−7.84089e7	−0.910467
$186$	0	0
$187$	−1.24076e6	−0.0138753
$188$	0	0
$189$	−3.42952e7	−0.369502
$190$	0	0
$191$	1.60958e8	1.67146	0.835730	−	0.549140i	$-0.185045\pi$
$191$	1.60958e8	1.67146	0.835730	+	0.549140i	$0.185045\pi$
$192$	0	0
$193$	1.61920e8	1.62125	0.810625	−	0.585566i	$-0.199128\pi$
$193$	1.61920e8	1.62125	0.810625	+	0.585566i	$0.199128\pi$
$194$	0	0
$195$	4.39014e7	0.423991
$196$	0	0
$197$	−1.27020e8	−1.18370	−0.591848	−	0.806049i	$-0.701602\pi$
$197$	−1.27020e8	−1.18370	−0.591848	+	0.806049i	$0.701602\pi$
$198$	0	0
$199$	1.74928e7	0.157353	0.0786763	−	0.996900i	$-0.474931\pi$
$199$	1.74928e7	0.157353	0.0786763	+	0.996900i	$0.474931\pi$
$200$	0	0
$201$	−5.84319e7	−0.507532
$202$	0	0
$203$	2.67163e8	2.24151
$204$	0	0
$205$	1.54369e7	0.125147
$206$	0	0
$207$	−6.30650e7	−0.494188
$208$	0	0
$209$	59362.5	0.000449780 0
$210$	0	0
$211$	−4.40594e7	−0.322887	−0.161443	−	0.986882i	$-0.551615\pi$
$211$	−4.40594e7	−0.322887	−0.161443	+	0.986882i	$0.551615\pi$
$212$	0	0
$213$	−4.23965e7	−0.300609
$214$	0	0
$215$	1.30926e7	0.0898445
$216$	0	0
$217$	−2.31319e8	−1.53674
$218$	0	0
$219$	−7.79015e7	−0.501177
$220$	0	0
$221$	2.51542e8	1.56761
$222$	0	0
$223$	1.25971e8	0.760685	0.380343	−	0.924846i	$-0.375806\pi$
$223$	1.25971e8	0.760685	0.380343	+	0.924846i	$0.375806\pi$
$224$	0	0
$225$	−3.93766e7	−0.230462
$226$	0	0
$227$	1.13624e8	0.644733	0.322366	−	0.946615i	$-0.395522\pi$
$227$	1.13624e8	0.644733	0.322366	+	0.946615i	$0.395522\pi$
$228$	0	0
$229$	1.65608e8	0.911289	0.455645	−	0.890162i	$-0.349409\pi$
$229$	1.65608e8	0.911289	0.455645	+	0.890162i	$0.349409\pi$
$230$	0	0
$231$	2.42993e6	0.0129704
$232$	0	0
$233$	2.77584e8	1.43764	0.718818	−	0.695199i	$-0.244684\pi$
$233$	2.77584e8	1.43764	0.718818	+	0.695199i	$0.244684\pi$
$234$	0	0
$235$	−1.54990e8	−0.779054
$236$	0	0
$237$	−1.04536e8	−0.510089
$238$	0	0
$239$	3.00082e8	1.42183	0.710915	−	0.703278i	$-0.248281\pi$
$239$	3.00082e8	1.42183	0.710915	+	0.703278i	$0.248281\pi$
$240$	0	0
$241$	−3.72418e8	−1.71384	−0.856922	−	0.515446i	$-0.827626\pi$
$241$	−3.72418e8	−1.71384	−0.856922	+	0.515446i	$0.827626\pi$
$242$	0	0
$243$	1.43489e7	0.0641500
$244$	0	0
$245$	3.43521e8	1.49235
$246$	0	0
$247$	−1.20347e7	−0.0508155
$248$	0	0
$249$	2.15071e7	0.0882843
$250$	0	0
$251$	−2.44996e8	−0.977916	−0.488958	−	0.872307i	$-0.662623\pi$
$251$	−2.44996e8	−0.977916	−0.488958	+	0.872307i	$0.662623\pi$
$252$	0	0
$253$	4.46837e6	0.0173471
$254$	0	0
$255$	1.00708e8	0.380343
$256$	0	0
$257$	−5.71767e7	−0.210113	−0.105057	−	0.994466i	$-0.533502\pi$
$257$	−5.71767e7	−0.210113	−0.105057	+	0.994466i	$0.533502\pi$
$258$	0	0
$259$	8.79842e8	3.14670
$260$	0	0
$261$	−1.11779e8	−0.389152
$262$	0	0
$263$	−8.63071e7	−0.292551	−0.146275	−	0.989244i	$-0.546729\pi$
$263$	−8.63071e7	−0.292551	−0.146275	+	0.989244i	$0.546729\pi$
$264$	0	0
$265$	−1.94029e8	−0.640479
$266$	0	0
$267$	3.82241e7	0.122899
$268$	0	0
$269$	1.72851e8	0.541425	0.270712	−	0.962660i	$-0.412741\pi$
$269$	1.72851e8	0.541425	0.270712	+	0.962660i	$0.412741\pi$
$270$	0	0
$271$	−2.93066e8	−0.894484	−0.447242	−	0.894413i	$-0.647594\pi$
$271$	−2.93066e8	−0.894484	−0.447242	+	0.894413i	$0.647594\pi$
$272$	0	0
$273$	−4.92626e8	−1.46537
$274$	0	0
$275$	2.78997e6	0.00808974
$276$	0	0
$277$	−4.02094e8	−1.13671	−0.568353	−	0.822785i	$-0.692419\pi$
$277$	−4.02094e8	−1.13671	−0.568353	+	0.822785i	$0.692419\pi$
$278$	0	0
$279$	9.67824e7	0.266797
$280$	0	0
$281$	2.56183e8	0.688777	0.344388	−	0.938827i	$-0.388086\pi$
$281$	2.56183e8	0.688777	0.344388	+	0.938827i	$0.388086\pi$
$282$	0	0
$283$	−7.55378e7	−0.198112	−0.0990562	−	0.995082i	$-0.531582\pi$
$283$	−7.55378e7	−0.198112	−0.0990562	+	0.995082i	$0.531582\pi$
$284$	0	0
$285$	−4.81826e6	−0.0123292
$286$	0	0
$287$	−1.73221e8	−0.432527
$288$	0	0
$289$	1.66693e8	0.406232
$290$	0	0
$291$	−6.45144e6	−0.0153473
$292$	0	0
$293$	6.87808e8	1.59746	0.798731	−	0.601688i	$-0.205505\pi$
$293$	6.87808e8	1.59746	0.798731	+	0.601688i	$0.205505\pi$
$294$	0	0
$295$	−2.60661e8	−0.591152
$296$	0	0
$297$	−1.01667e6	−0.00225181
$298$	0	0
$299$	−9.05883e8	−1.95985
$300$	0	0
$301$	−1.46915e8	−0.310515
$302$	0	0
$303$	5.05641e8	1.04422
$304$	0	0
$305$	6.29372e7	0.127016
$306$	0	0
$307$	−2.58303e8	−0.509501	−0.254751	−	0.967007i	$-0.581993\pi$
$307$	−2.58303e8	−0.509501	−0.254751	+	0.967007i	$0.581993\pi$
$308$	0	0
$309$	1.26248e8	0.243428
$310$	0	0
$311$	−2.14264e8	−0.403913	−0.201957	−	0.979394i	$-0.564730\pi$
$311$	−2.14264e8	−0.403913	−0.201957	+	0.979394i	$0.564730\pi$
$312$	0	0
$313$	4.31251e8	0.794923	0.397461	−	0.917619i	$-0.369891\pi$
$313$	4.31251e8	0.794923	0.397461	+	0.917619i	$0.369891\pi$
$314$	0	0
$315$	−1.97230e8	−0.355537
$316$	0	0
$317$	−1.13425e8	−0.199986	−0.0999932	−	0.994988i	$-0.531882\pi$
$317$	−1.13425e8	−0.199986	−0.0999932	+	0.994988i	$0.531882\pi$
$318$	0	0
$319$	7.91994e6	0.0136601
$320$	0	0
$321$	3.74020e8	0.631142
$322$	0	0
$323$	−2.76073e7	−0.0455842
$324$	0	0
$325$	−5.65617e8	−0.913967
$326$	0	0
$327$	−1.36560e8	−0.215977
$328$	0	0
$329$	1.73918e9	2.69252
$330$	0	0
$331$	7.46930e8	1.13209	0.566046	−	0.824374i	$-0.308472\pi$
$331$	7.46930e8	1.13209	0.566046	+	0.824374i	$0.308472\pi$
$332$	0	0
$333$	−3.68121e8	−0.546305
$334$	0	0
$335$	−3.36038e8	−0.488351
$336$	0	0
$337$	−5.55138e8	−0.790126	−0.395063	−	0.918654i	$-0.629277\pi$
$337$	−5.55138e8	−0.790126	−0.395063	+	0.918654i	$0.629277\pi$
$338$	0	0
$339$	−7.55506e8	−1.05327
$340$	0	0
$341$	−6.85736e6	−0.00936519
$342$	0	0
$343$	−2.41980e9	−3.23780
$344$	0	0
$345$	−3.62683e8	−0.475510
$346$	0	0
$347$	−1.45452e9	−1.86881	−0.934407	−	0.356206i	$-0.884070\pi$
$347$	−1.45452e9	−1.86881	−0.934407	+	0.356206i	$0.884070\pi$
$348$	0	0
$349$	7.79394e7	0.0981450	0.0490725	−	0.998795i	$-0.484373\pi$
$349$	7.79394e7	0.0981450	0.0490725	+	0.998795i	$0.484373\pi$
$350$	0	0
$351$	2.06112e8	0.254406
$352$	0	0
$353$	8.54961e8	1.03451	0.517255	−	0.855831i	$-0.326954\pi$
$353$	8.54961e8	1.03451	0.517255	+	0.855831i	$0.326954\pi$
$354$	0	0
$355$	−2.43820e8	−0.289248
$356$	0	0
$357$	−1.13007e9	−1.31452
$358$	0	0
$359$	4.81451e8	0.549189	0.274595	−	0.961560i	$-0.411456\pi$
$359$	4.81451e8	0.549189	0.274595	+	0.961560i	$0.411456\pi$
$360$	0	0
$361$	−8.92551e8	−0.998522
$362$	0	0
$363$	−5.26082e8	−0.577271
$364$	0	0
$365$	−4.48007e8	−0.482236
$366$	0	0
$367$	1.01203e9	1.06872	0.534359	−	0.845257i	$-0.320553\pi$
$367$	1.01203e9	1.06872	0.534359	+	0.845257i	$0.320553\pi$
$368$	0	0
$369$	7.24745e7	0.0750919
$370$	0	0
$371$	2.17723e9	2.21359
$372$	0	0
$373$	−1.31785e9	−1.31487	−0.657437	−	0.753510i	$-0.728359\pi$
$373$	−1.31785e9	−1.31487	−0.657437	+	0.753510i	$0.728359\pi$
$374$	0	0
$375$	−5.53986e8	−0.542487
$376$	0	0
$377$	−1.60563e9	−1.54330
$378$	0	0
$379$	−1.26791e9	−1.19633	−0.598167	−	0.801372i	$-0.704104\pi$
$379$	−1.26791e9	−1.19633	−0.598167	+	0.801372i	$0.704104\pi$
$380$	0	0
$381$	1.14072e8	0.105668
$382$	0	0
$383$	7.44181e8	0.676835	0.338417	−	0.940996i	$-0.390108\pi$
$383$	7.44181e8	0.676835	0.338417	+	0.940996i	$0.390108\pi$
$384$	0	0
$385$	1.39744e7	0.0124802
$386$	0	0
$387$	6.14683e7	0.0539092
$388$	0	0
$389$	4.14243e8	0.356806	0.178403	−	0.983958i	$-0.442907\pi$
$389$	4.14243e8	0.356806	0.178403	+	0.983958i	$0.442907\pi$
$390$	0	0
$391$	−2.07807e9	−1.75809
$392$	0	0
$393$	−1.12519e9	−0.935090
$394$	0	0
$395$	−6.01179e8	−0.490810
$396$	0	0
$397$	2.05816e8	0.165087	0.0825436	−	0.996587i	$-0.473696\pi$
$397$	2.05816e8	0.165087	0.0825436	+	0.996587i	$0.473696\pi$
$398$	0	0
$399$	5.40667e7	0.0426113
$400$	0	0
$401$	1.86415e9	1.44369	0.721846	−	0.692054i	$-0.243294\pi$
$401$	1.86415e9	1.44369	0.721846	+	0.692054i	$0.243294\pi$
$402$	0	0
$403$	1.39021e9	1.05807
$404$	0	0
$405$	8.25197e7	0.0617256
$406$	0	0
$407$	2.60826e7	0.0191766
$408$	0	0
$409$	9.07532e8	0.655889	0.327944	−	0.944697i	$-0.393644\pi$
$409$	9.07532e8	0.655889	0.327944	+	0.944697i	$0.393644\pi$
$410$	0	0
$411$	6.98750e8	0.496450
$412$	0	0
$413$	2.92493e9	2.04311
$414$	0	0
$415$	1.23686e8	0.0849477
$416$	0	0
$417$	−1.36903e9	−0.924564
$418$	0	0
$419$	−4.53137e8	−0.300941	−0.150470	−	0.988615i	$-0.548079\pi$
$419$	−4.53137e8	−0.300941	−0.150470	+	0.988615i	$0.548079\pi$
$420$	0	0
$421$	1.62727e9	1.06285	0.531425	−	0.847106i	$-0.321657\pi$
$421$	1.62727e9	1.06285	0.531425	+	0.847106i	$0.321657\pi$
$422$	0	0
$423$	−7.27662e8	−0.467454
$424$	0	0
$425$	−1.29751e9	−0.819878
$426$	0	0
$427$	−7.06231e8	−0.438985
$428$	0	0
$429$	−1.46037e7	−0.00893024
$430$	0	0
$431$	9.84387e8	0.592237	0.296118	−	0.955151i	$-0.404308\pi$
$431$	9.84387e8	0.592237	0.296118	+	0.955151i	$0.404308\pi$
$432$	0	0
$433$	1.67333e9	0.990545	0.495273	−	0.868738i	$-0.335068\pi$
$433$	1.67333e9	0.990545	0.495273	+	0.868738i	$0.335068\pi$
$434$	0	0
$435$	−6.42836e8	−0.374445
$436$	0	0
$437$	9.94225e7	0.0569901
$438$	0	0
$439$	2.78714e9	1.57229	0.786147	−	0.618040i	$-0.212073\pi$
$439$	2.78714e9	1.57229	0.786147	+	0.618040i	$0.212073\pi$
$440$	0	0
$441$	1.61279e9	0.895454
$442$	0	0
$443$	−1.72313e9	−0.941683	−0.470841	−	0.882218i	$-0.656050\pi$
$443$	−1.72313e9	−0.941683	−0.470841	+	0.882218i	$0.656050\pi$
$444$	0	0
$445$	2.19824e8	0.118254
$446$	0	0
$447$	1.17939e9	0.624570
$448$	0	0
$449$	−2.13732e9	−1.11431	−0.557157	−	0.830407i	$-0.688108\pi$
$449$	−2.13732e9	−1.11431	−0.557157	+	0.830407i	$0.688108\pi$
$450$	0	0
$451$	−5.13507e6	−0.00263590
$452$	0	0
$453$	1.84117e9	0.930573
$454$	0	0
$455$	−2.83306e9	−1.40999
$456$	0	0
$457$	−2.56149e9	−1.25541	−0.627706	−	0.778451i	$-0.716006\pi$
$457$	−2.56149e9	−1.25541	−0.627706	+	0.778451i	$0.716006\pi$
$458$	0	0
$459$	4.72815e8	0.228216
$460$	0	0
$461$	−9.96595e8	−0.473768	−0.236884	−	0.971538i	$-0.576126\pi$
$461$	−9.96595e8	−0.473768	−0.236884	+	0.971538i	$0.576126\pi$
$462$	0	0
$463$	2.61394e9	1.22395	0.611973	−	0.790879i	$-0.290376\pi$
$463$	2.61394e9	1.22395	0.611973	+	0.790879i	$0.290376\pi$
$464$	0	0
$465$	5.56590e8	0.256714
$466$	0	0
$467$	2.57019e9	1.16777	0.583883	−	0.811838i	$-0.301533\pi$
$467$	2.57019e9	1.16777	0.583883	+	0.811838i	$0.301533\pi$
$468$	0	0
$469$	3.77075e9	1.68781
$470$	0	0
$471$	−2.69827e8	−0.118990
$472$	0	0
$473$	−4.35524e6	−0.00189233
$474$	0	0
$475$	6.20776e7	0.0265771
$476$	0	0
$477$	−9.10942e8	−0.384305
$478$	0	0
$479$	−7.93225e8	−0.329778	−0.164889	−	0.986312i	$-0.552727\pi$
$479$	−7.93225e8	−0.329778	−0.164889	+	0.986312i	$0.552727\pi$
$480$	0	0
$481$	−5.28779e9	−2.16654
$482$	0	0
$483$	4.06974e9	1.64343
$484$	0	0
$485$	−3.71018e7	−0.0147672
$486$	0	0
$487$	1.26410e9	0.495940	0.247970	−	0.968768i	$-0.420237\pi$
$487$	1.26410e9	0.495940	0.247970	+	0.968768i	$0.420237\pi$
$488$	0	0
$489$	1.28891e8	0.0498474
$490$	0	0
$491$	3.37678e9	1.28741	0.643706	−	0.765273i	$-0.277396\pi$
$491$	3.37678e9	1.28741	0.643706	+	0.765273i	$0.277396\pi$
$492$	0	0
$493$	−3.68327e9	−1.38442
$494$	0	0
$495$	−5.84680e6	−0.00216671
$496$	0	0
$497$	2.73595e9	0.999680
$498$	0	0
$499$	9.76331e8	0.351759	0.175879	−	0.984412i	$-0.443723\pi$
$499$	9.76331e8	0.351759	0.175879	+	0.984412i	$0.443723\pi$
$500$	0	0
$501$	−2.40559e9	−0.854654
$502$	0	0
$503$	−1.35477e9	−0.474654	−0.237327	−	0.971430i	$-0.576271\pi$
$503$	−1.35477e9	−0.474654	−0.237327	+	0.971430i	$0.576271\pi$
$504$	0	0
$505$	2.90791e9	1.00476
$506$	0	0
$507$	1.26644e9	0.431575
$508$	0	0
$509$	−2.02597e9	−0.680959	−0.340479	−	0.940252i	$-0.610589\pi$
$509$	−2.02597e9	−0.680959	−0.340479	+	0.940252i	$0.610589\pi$
$510$	0	0
$511$	5.02718e9	1.66668
$512$	0	0
$513$	−2.26212e7	−0.00739783
$514$	0	0
$515$	7.26047e8	0.234228
$516$	0	0
$517$	5.15573e7	0.0164087
$518$	0	0
$519$	6.98546e8	0.219336
$520$	0	0
$521$	−1.99361e8	−0.0617601	−0.0308800	−	0.999523i	$-0.509831\pi$
$521$	−1.99361e8	−0.0617601	−0.0308800	+	0.999523i	$0.509831\pi$
$522$	0	0
$523$	−4.02783e9	−1.23116	−0.615581	−	0.788074i	$-0.711079\pi$
$523$	−4.02783e9	−1.23116	−0.615581	+	0.788074i	$0.711079\pi$
$524$	0	0
$525$	2.54107e9	0.766406
$526$	0	0
$527$	3.18910e9	0.949142
$528$	0	0
$529$	4.07896e9	1.19799
$530$	0	0
$531$	−1.22377e9	−0.354708
$532$	0	0
$533$	1.04105e9	0.297800
$534$	0	0
$535$	2.15096e9	0.607288
$536$	0	0
$537$	1.15833e9	0.322790
$538$	0	0
$539$	−1.14272e8	−0.0314325
$540$	0	0
$541$	−1.55841e9	−0.423148	−0.211574	−	0.977362i	$-0.567859\pi$
$541$	−1.55841e9	−0.423148	−0.211574	+	0.977362i	$0.567859\pi$
$542$	0	0
$543$	−3.47616e9	−0.931751
$544$	0	0
$545$	−7.85349e8	−0.207814
$546$	0	0
$547$	3.01460e9	0.787542	0.393771	−	0.919209i	$-0.371170\pi$
$547$	3.01460e9	0.787542	0.393771	+	0.919209i	$0.371170\pi$
$548$	0	0
$549$	2.95483e8	0.0762130
$550$	0	0
$551$	1.76221e8	0.0448773
$552$	0	0
$553$	6.74595e9	1.69631
$554$	0	0
$555$	−2.11704e9	−0.525658
$556$	0	0
$557$	−6.97397e9	−1.70996	−0.854981	−	0.518659i	$-0.826431\pi$
$557$	−6.97397e9	−1.70996	−0.854981	+	0.518659i	$0.826431\pi$
$558$	0	0
$559$	8.82948e8	0.213793
$560$	0	0
$561$	−3.35005e7	−0.00801091
$562$	0	0
$563$	4.91575e9	1.16094	0.580471	−	0.814281i	$-0.302869\pi$
$563$	4.91575e9	1.16094	0.580471	+	0.814281i	$0.302869\pi$
$564$	0	0
$565$	−4.34487e9	−1.01346
$566$	0	0
$567$	−9.25970e8	−0.213332
$568$	0	0
$569$	−2.91661e9	−0.663720	−0.331860	−	0.943329i	$-0.607676\pi$
$569$	−2.91661e9	−0.663720	−0.331860	+	0.943329i	$0.607676\pi$
$570$	0	0
$571$	−2.78071e9	−0.625071	−0.312536	−	0.949906i	$-0.601178\pi$
$571$	−2.78071e9	−0.625071	−0.312536	+	0.949906i	$0.601178\pi$
$572$	0	0
$573$	4.34587e9	0.965018
$574$	0	0
$575$	4.67274e9	1.02502
$576$	0	0
$577$	−1.95173e9	−0.422965	−0.211482	−	0.977382i	$-0.567829\pi$
$577$	−1.95173e9	−0.422965	−0.211482	+	0.977382i	$0.567829\pi$
$578$	0	0
$579$	4.37184e9	0.936029
$580$	0	0
$581$	−1.38790e9	−0.293591
$582$	0	0
$583$	6.45433e7	0.0134900
$584$	0	0
$585$	1.18534e9	0.244791
$586$	0	0
$587$	5.31559e9	1.08472	0.542361	−	0.840146i	$-0.317531\pi$
$587$	5.31559e9	1.08472	0.542361	+	0.840146i	$0.317531\pi$
$588$	0	0
$589$	−1.52578e8	−0.0307673
$590$	0	0
$591$	−3.42954e9	−0.683407
$592$	0	0
$593$	−1.17508e9	−0.231407	−0.115703	−	0.993284i	$-0.536912\pi$
$593$	−1.17508e9	−0.231407	−0.115703	+	0.993284i	$0.536912\pi$
$594$	0	0
$595$	−6.49897e9	−1.26484
$596$	0	0
$597$	4.72306e8	0.0908475
$598$	0	0
$599$	−4.88790e8	−0.0929242	−0.0464621	−	0.998920i	$-0.514795\pi$
$599$	−4.88790e8	−0.0929242	−0.0464621	+	0.998920i	$0.514795\pi$
$600$	0	0
$601$	−1.92744e9	−0.362176	−0.181088	−	0.983467i	$-0.557962\pi$
$601$	−1.92744e9	−0.362176	−0.181088	+	0.983467i	$0.557962\pi$
$602$	0	0
$603$	−1.57766e9	−0.293024
$604$	0	0
$605$	−3.02546e9	−0.555454
$606$	0	0
$607$	−6.50300e9	−1.18019	−0.590097	−	0.807332i	$-0.700911\pi$
$607$	−6.50300e9	−1.18019	−0.590097	+	0.807332i	$0.700911\pi$
$608$	0	0
$609$	7.21339e9	1.29413
$610$	0	0
$611$	−1.04523e10	−1.85383
$612$	0	0
$613$	−2.40924e9	−0.422444	−0.211222	−	0.977438i	$-0.567744\pi$
$613$	−2.40924e9	−0.422444	−0.211222	+	0.977438i	$0.567744\pi$
$614$	0	0
$615$	4.16797e8	0.0722539
$616$	0	0
$617$	−8.02270e9	−1.37506	−0.687531	−	0.726155i	$-0.741306\pi$
$617$	−8.02270e9	−1.37506	−0.687531	+	0.726155i	$0.741306\pi$
$618$	0	0
$619$	−7.38409e9	−1.25135	−0.625677	−	0.780083i	$-0.715177\pi$
$619$	−7.38409e9	−1.25135	−0.625677	+	0.780083i	$0.715177\pi$
$620$	0	0
$621$	−1.70275e9	−0.285319
$622$	0	0
$623$	−2.46670e9	−0.408703
$624$	0	0
$625$	1.03395e9	0.169402
$626$	0	0
$627$	1.60279e6	0.000259681 0
$628$	0	0
$629$	−1.21300e10	−1.94350
$630$	0	0
$631$	−7.86728e8	−0.124659	−0.0623293	−	0.998056i	$-0.519853\pi$
$631$	−7.86728e8	−0.124659	−0.0623293	+	0.998056i	$0.519853\pi$
$632$	0	0
$633$	−1.18960e9	−0.186419
$634$	0	0
$635$	6.56022e8	0.101674
$636$	0	0
$637$	2.31666e10	3.55119
$638$	0	0
$639$	−1.14471e9	−0.173556
$640$	0	0
$641$	5.81312e8	0.0871779	0.0435890	−	0.999050i	$-0.486121\pi$
$641$	5.81312e8	0.0871779	0.0435890	+	0.999050i	$0.486121\pi$
$642$	0	0
$643$	8.39677e9	1.24559	0.622793	−	0.782386i	$-0.285998\pi$
$643$	8.39677e9	1.24559	0.622793	+	0.782386i	$0.285998\pi$
$644$	0	0
$645$	3.53500e8	0.0518718
$646$	0	0
$647$	−8.27887e9	−1.20173	−0.600864	−	0.799351i	$-0.705177\pi$
$647$	−8.27887e9	−1.20173	−0.600864	+	0.799351i	$0.705177\pi$
$648$	0	0
$649$	8.67086e7	0.0124510
$650$	0	0
$651$	−6.24561e9	−0.887240
$652$	0	0
$653$	7.05982e9	0.992195	0.496098	−	0.868267i	$-0.334766\pi$
$653$	7.05982e9	0.992195	0.496098	+	0.868267i	$0.334766\pi$
$654$	0	0
$655$	−6.47092e9	−0.899749
$656$	0	0
$657$	−2.10334e9	−0.289355
$658$	0	0
$659$	2.61653e9	0.356145	0.178072	−	0.984017i	$-0.443014\pi$
$659$	2.61653e9	0.356145	0.178072	+	0.984017i	$0.443014\pi$
$660$	0	0
$661$	8.77438e9	1.18171	0.590855	−	0.806778i	$-0.298790\pi$
$661$	8.77438e9	1.18171	0.590855	+	0.806778i	$0.298790\pi$
$662$	0	0
$663$	6.79165e9	0.905061
$664$	0	0
$665$	3.10934e8	0.0410008
$666$	0	0
$667$	1.32646e10	1.73083
$668$	0	0
$669$	3.40123e9	0.439182
$670$	0	0
$671$	−2.09360e7	−0.00267525
$672$	0	0
$673$	4.93417e9	0.623967	0.311983	−	0.950088i	$-0.399007\pi$
$673$	4.93417e9	0.623967	0.311983	+	0.950088i	$0.399007\pi$
$674$	0	0
$675$	−1.06317e9	−0.133057
$676$	0	0
$677$	1.15194e10	1.42682	0.713411	−	0.700746i	$-0.247149\pi$
$677$	1.15194e10	1.42682	0.713411	+	0.700746i	$0.247149\pi$
$678$	0	0
$679$	4.16327e8	0.0510376
$680$	0	0
$681$	3.06785e9	0.372237
$682$	0	0
$683$	−5.83610e9	−0.700891	−0.350445	−	0.936583i	$-0.613970\pi$
$683$	−5.83610e9	−0.700891	−0.350445	+	0.936583i	$0.613970\pi$
$684$	0	0
$685$	4.01847e9	0.477687
$686$	0	0
$687$	4.47141e9	0.526133
$688$	0	0
$689$	−1.30850e10	−1.52408
$690$	0	0
$691$	−1.34502e10	−1.55080	−0.775402	−	0.631468i	$-0.782453\pi$
$691$	−1.34502e10	−1.55080	−0.775402	+	0.631468i	$0.782453\pi$
$692$	0	0
$693$	6.56082e7	0.00748844
$694$	0	0
$695$	−7.87321e9	−0.889621
$696$	0	0
$697$	2.38813e9	0.267142
$698$	0	0
$699$	7.49477e9	0.830019
$700$	0	0
$701$	−9.35211e9	−1.02541	−0.512704	−	0.858565i	$-0.671356\pi$
$701$	−9.35211e9	−1.02541	−0.512704	+	0.858565i	$0.671356\pi$
$702$	0	0
$703$	5.80346e8	0.0630004
$704$	0	0
$705$	−4.18474e9	−0.449787
$706$	0	0
$707$	−3.26303e10	−3.47259
$708$	0	0
$709$	−1.07639e10	−1.13425	−0.567126	−	0.823631i	$-0.691945\pi$
$709$	−1.07639e10	−1.13425	−0.567126	+	0.823631i	$0.691945\pi$
$710$	0	0
$711$	−2.82247e9	−0.294500
$712$	0	0
$713$	−1.14850e10	−1.18663
$714$	0	0
$715$	−8.39851e7	−0.00859273
$716$	0	0
$717$	8.10222e9	0.820894
$718$	0	0
$719$	−2.95823e8	−0.0296811	−0.0148406	−	0.999890i	$-0.504724\pi$
$719$	−2.95823e8	−0.0296811	−0.0148406	+	0.999890i	$0.504724\pi$
$720$	0	0
$721$	−8.14712e9	−0.809526
$722$	0	0
$723$	−1.00553e10	−0.989488
$724$	0	0
$725$	8.28218e9	0.807164
$726$	0	0
$727$	−7.92934e9	−0.765362	−0.382681	−	0.923881i	$-0.624999\pi$
$727$	−7.92934e9	−0.765362	−0.382681	+	0.923881i	$0.624999\pi$
$728$	0	0
$729$	3.87420e8	0.0370370
$730$	0	0
$731$	2.02546e9	0.191784
$732$	0	0
$733$	−1.06379e10	−0.997681	−0.498841	−	0.866694i	$-0.666241\pi$
$733$	−1.06379e10	−0.997681	−0.498841	+	0.866694i	$0.666241\pi$
$734$	0	0
$735$	9.27506e9	0.861611
$736$	0	0
$737$	1.11783e8	0.0102858
$738$	0	0
$739$	1.21560e10	1.10799	0.553993	−	0.832522i	$-0.313104\pi$
$739$	1.21560e10	1.10799	0.553993	+	0.832522i	$0.313104\pi$
$740$	0	0
$741$	−3.24937e8	−0.0293383
$742$	0	0
$743$	5.04486e9	0.451220	0.225610	−	0.974218i	$-0.427562\pi$
$743$	5.04486e9	0.451220	0.225610	+	0.974218i	$0.427562\pi$
$744$	0	0
$745$	6.78260e9	0.600965
$746$	0	0
$747$	5.80691e8	0.0509710
$748$	0	0
$749$	−2.41364e10	−2.09887
$750$	0	0
$751$	−1.58691e10	−1.36714	−0.683568	−	0.729887i	$-0.739573\pi$
$751$	−1.58691e10	−1.36714	−0.683568	+	0.729887i	$0.739573\pi$
$752$	0	0
$753$	−6.61490e9	−0.564600
$754$	0	0
$755$	1.05885e10	0.895403
$756$	0	0
$757$	−3.76028e9	−0.315054	−0.157527	−	0.987515i	$-0.550352\pi$
$757$	−3.76028e9	−0.315054	−0.157527	+	0.987515i	$0.550352\pi$
$758$	0	0
$759$	1.20646e8	0.0100154
$760$	0	0
$761$	−2.38635e9	−0.196285	−0.0981427	−	0.995172i	$-0.531290\pi$
$761$	−2.38635e9	−0.196285	−0.0981427	+	0.995172i	$0.531290\pi$
$762$	0	0
$763$	8.81256e9	0.718235
$764$	0	0
$765$	2.71913e9	0.219591
$766$	0	0
$767$	−1.75786e10	−1.40670
$768$	0	0
$769$	−1.63901e9	−0.129969	−0.0649845	−	0.997886i	$-0.520700\pi$
$769$	−1.63901e9	−0.129969	−0.0649845	+	0.997886i	$0.520700\pi$
$770$	0	0
$771$	−1.54377e9	−0.121309
$772$	0	0
$773$	−8.01355e9	−0.624017	−0.312009	−	0.950079i	$-0.601002\pi$
$773$	−8.01355e9	−0.624017	−0.312009	+	0.950079i	$0.601002\pi$
$774$	0	0
$775$	−7.17100e9	−0.553380
$776$	0	0
$777$	2.37557e10	1.81675
$778$	0	0
$779$	−1.14257e8	−0.00865966
$780$	0	0
$781$	8.11063e7	0.00609223
$782$	0	0
$783$	−3.01804e9	−0.224677
$784$	0	0
$785$	−1.55176e9	−0.114493
$786$	0	0
$787$	−1.21205e9	−0.0886361	−0.0443181	−	0.999017i	$-0.514111\pi$
$787$	−1.21205e9	−0.0886361	−0.0443181	+	0.999017i	$0.514111\pi$
$788$	0	0
$789$	−2.33029e9	−0.168904
$790$	0	0
$791$	4.87547e10	3.50267
$792$	0	0
$793$	4.24440e9	0.302246
$794$	0	0
$795$	−5.23877e9	−0.369781
$796$	0	0
$797$	−9.31165e8	−0.0651512	−0.0325756	−	0.999469i	$-0.510371\pi$
$797$	−9.31165e8	−0.0651512	−0.0325756	+	0.999469i	$0.510371\pi$
$798$	0	0
$799$	−2.39774e10	−1.66298
$800$	0	0
$801$	1.03205e9	0.0709557
$802$	0	0
$803$	1.49029e8	0.0101570
$804$	0	0
$805$	2.34048e10	1.58132
$806$	0	0
$807$	4.66697e9	0.312592
$808$	0	0
$809$	7.48310e9	0.496891	0.248446	−	0.968646i	$-0.420080\pi$
$809$	7.48310e9	0.496891	0.248446	+	0.968646i	$0.420080\pi$
$810$	0	0
$811$	2.60208e10	1.71296	0.856479	−	0.516181i	$-0.172647\pi$
$811$	2.60208e10	1.71296	0.856479	+	0.516181i	$0.172647\pi$
$812$	0	0
$813$	−7.91277e9	−0.516430
$814$	0	0
$815$	7.41246e8	0.0479635
$816$	0	0
$817$	−9.69053e7	−0.00621685
$818$	0	0
$819$	−1.33009e10	−0.846033
$820$	0	0
$821$	−2.24111e9	−0.141339	−0.0706695	−	0.997500i	$-0.522514\pi$
$821$	−2.24111e9	−0.141339	−0.0706695	+	0.997500i	$0.522514\pi$
$822$	0	0
$823$	2.16414e10	1.35327	0.676636	−	0.736317i	$-0.263437\pi$
$823$	2.16414e10	1.35327	0.676636	+	0.736317i	$0.263437\pi$
$824$	0	0
$825$	7.53291e7	0.00467062
$826$	0	0
$827$	−1.18759e10	−0.730125	−0.365062	−	0.930983i	$-0.618952\pi$
$827$	−1.18759e10	−0.730125	−0.365062	+	0.930983i	$0.618952\pi$
$828$	0	0
$829$	−9.14265e9	−0.557354	−0.278677	−	0.960385i	$-0.589896\pi$
$829$	−9.14265e9	−0.557354	−0.278677	+	0.960385i	$0.589896\pi$
$830$	0	0
$831$	−1.08565e10	−0.656277
$832$	0	0
$833$	5.31435e10	3.18561
$834$	0	0
$835$	−1.38344e10	−0.822353
$836$	0	0
$837$	2.61312e9	0.154036
$838$	0	0
$839$	1.27491e10	0.745267	0.372633	−	0.927979i	$-0.378455\pi$
$839$	1.27491e10	0.745267	0.372633	+	0.927979i	$0.378455\pi$
$840$	0	0
$841$	6.26094e9	0.362956
$842$	0	0
$843$	6.91694e9	0.397665
$844$	0	0
$845$	7.28321e9	0.415264
$846$	0	0
$847$	3.39493e10	1.91973
$848$	0	0
$849$	−2.03952e9	−0.114380
$850$	0	0
$851$	4.36841e10	2.42980
$852$	0	0
$853$	−1.72779e10	−0.953167	−0.476583	−	0.879129i	$-0.658125\pi$
$853$	−1.72779e10	−0.953167	−0.476583	+	0.879129i	$0.658125\pi$
$854$	0	0
$855$	−1.30093e8	−0.00711824
$856$	0	0
$857$	−1.14152e10	−0.619514	−0.309757	−	0.950816i	$-0.600248\pi$
$857$	−1.14152e10	−0.619514	−0.309757	+	0.950816i	$0.600248\pi$
$858$	0	0
$859$	1.27618e10	0.686965	0.343483	−	0.939159i	$-0.388393\pi$
$859$	1.27618e10	0.686965	0.343483	+	0.939159i	$0.388393\pi$
$860$	0	0
$861$	−4.67696e9	−0.249720
$862$	0	0
$863$	−3.19302e8	−0.0169108	−0.00845540	−	0.999964i	$-0.502691\pi$
$863$	−3.19302e8	−0.0169108	−0.00845540	+	0.999964i	$0.502691\pi$
$864$	0	0
$865$	4.01730e9	0.211046
$866$	0	0
$867$	4.50070e9	0.234538
$868$	0	0
$869$	1.99981e8	0.0103376
$870$	0	0
$871$	−2.26620e10	−1.16208
$872$	0	0
$873$	−1.74189e8	−0.00886075
$874$	0	0
$875$	3.57501e10	1.80405
$876$	0	0
$877$	−8.51426e9	−0.426234	−0.213117	−	0.977027i	$-0.568362\pi$
$877$	−8.51426e9	−0.426234	−0.213117	+	0.977027i	$0.568362\pi$
$878$	0	0
$879$	1.85708e10	0.922295
$880$	0	0
$881$	−8.20262e9	−0.404145	−0.202072	−	0.979371i	$-0.564768\pi$
$881$	−8.20262e9	−0.404145	−0.202072	+	0.979371i	$0.564768\pi$
$882$	0	0
$883$	−1.57333e10	−0.769054	−0.384527	−	0.923114i	$-0.625635\pi$
$883$	−1.57333e10	−0.769054	−0.384527	+	0.923114i	$0.625635\pi$
$884$	0	0
$885$	−7.03785e9	−0.341302
$886$	0	0
$887$	1.84173e10	0.886124	0.443062	−	0.896491i	$-0.353892\pi$
$887$	1.84173e10	0.886124	0.443062	+	0.896491i	$0.353892\pi$
$888$	0	0
$889$	−7.36136e9	−0.351400
$890$	0	0
$891$	−2.74501e7	−0.00130008
$892$	0	0
$893$	1.14717e9	0.0539071
$894$	0	0
$895$	6.66146e9	0.310591
$896$	0	0
$897$	−2.44588e10	−1.13152
$898$	0	0
$899$	−2.03565e10	−0.934423
$900$	0	0
$901$	−3.00167e10	−1.36718
$902$	0	0
$903$	−3.96670e9	−0.179276
$904$	0	0
$905$	−1.99912e10	−0.896537
$906$	0	0
$907$	4.44837e9	0.197959	0.0989796	−	0.995089i	$-0.468442\pi$
$907$	4.44837e9	0.197959	0.0989796	+	0.995089i	$0.468442\pi$
$908$	0	0
$909$	1.36523e10	0.602883
$910$	0	0
$911$	−1.83330e10	−0.803377	−0.401688	−	0.915776i	$-0.631576\pi$
$911$	−1.83330e10	−0.803377	−0.401688	+	0.915776i	$0.631576\pi$
$912$	0	0
$913$	−4.11439e7	−0.00178920
$914$	0	0
$915$	1.69930e9	0.0733327
$916$	0	0
$917$	7.26115e10	3.10966
$918$	0	0
$919$	3.38356e10	1.43803	0.719017	−	0.694993i	$-0.244592\pi$
$919$	3.38356e10	1.43803	0.719017	+	0.694993i	$0.244592\pi$
$920$	0	0
$921$	−6.97418e9	−0.294161
$922$	0	0
$923$	−1.64429e10	−0.688291
$924$	0	0
$925$	2.72756e10	1.13312
$926$	0	0
$927$	3.40871e9	0.140543
$928$	0	0
$929$	3.27055e10	1.33834	0.669169	−	0.743110i	$-0.266650\pi$
$929$	3.27055e10	1.33834	0.669169	+	0.743110i	$0.266650\pi$
$930$	0	0
$931$	−2.54258e9	−0.103264
$932$	0	0
$933$	−5.78513e9	−0.233199
$934$	0	0
$935$	−1.92660e8	−0.00770814
$936$	0	0
$937$	−2.13238e10	−0.846791	−0.423395	−	0.905945i	$-0.639162\pi$
$937$	−2.13238e10	−0.846791	−0.423395	+	0.905945i	$0.639162\pi$
$938$	0	0
$939$	1.16438e10	0.458949
$940$	0	0
$941$	−2.37346e10	−0.928578	−0.464289	−	0.885684i	$-0.653690\pi$
$941$	−2.37346e10	−0.928578	−0.464289	+	0.885684i	$0.653690\pi$
$942$	0	0
$943$	−8.60040e9	−0.333986
$944$	0	0
$945$	−5.32520e9	−0.205270
$946$	0	0
$947$	−1.23219e9	−0.0471470	−0.0235735	−	0.999722i	$-0.507504\pi$
$947$	−1.23219e9	−0.0471470	−0.0235735	+	0.999722i	$0.507504\pi$
$948$	0	0
$949$	−3.02130e10	−1.14752
$950$	0	0
$951$	−3.06247e9	−0.115462
$952$	0	0
$953$	−2.42965e10	−0.909325	−0.454662	−	0.890664i	$-0.650240\pi$
$953$	−2.42965e10	−0.909325	−0.454662	+	0.890664i	$0.650240\pi$
$954$	0	0
$955$	2.49928e10	0.928547
$956$	0	0
$957$	2.13838e8	0.00788668
$958$	0	0
$959$	−4.50921e10	−1.65095
$960$	0	0
$961$	−9.88727e9	−0.359372
$962$	0	0
$963$	1.00985e10	0.364390
$964$	0	0
$965$	2.51422e10	0.900653
$966$	0	0
$967$	4.54395e10	1.61600	0.807998	−	0.589185i	$-0.200551\pi$
$967$	4.54395e10	1.61600	0.807998	+	0.589185i	$0.200551\pi$
$968$	0	0
$969$	−7.45397e8	−0.0263181
$970$	0	0
$971$	−2.55196e9	−0.0894555	−0.0447277	−	0.998999i	$-0.514242\pi$
$971$	−2.55196e9	−0.0894555	−0.0447277	+	0.998999i	$0.514242\pi$
$972$	0	0
$973$	8.83469e10	3.07466
$974$	0	0
$975$	−1.52717e10	−0.527679
$976$	0	0
$977$	5.12078e9	0.175673	0.0878365	−	0.996135i	$-0.472005\pi$
$977$	5.12078e9	0.175673	0.0878365	+	0.996135i	$0.472005\pi$
$978$	0	0
$979$	−7.31243e7	−0.00249071
$980$	0	0
$981$	−3.68712e9	−0.124694
$982$	0	0
$983$	−2.49695e10	−0.838442	−0.419221	−	0.907884i	$-0.637697\pi$
$983$	−2.49695e10	−0.838442	−0.419221	+	0.907884i	$0.637697\pi$
$984$	0	0
$985$	−1.97231e10	−0.657579
$986$	0	0
$987$	4.69578e10	1.55453
$988$	0	0
$989$	−7.29431e9	−0.239771
$990$	0	0
$991$	2.79851e9	0.0913417	0.0456709	−	0.998957i	$-0.485457\pi$
$991$	2.79851e9	0.0913417	0.0456709	+	0.998957i	$0.485457\pi$
$992$	0	0
$993$	2.01671e10	0.653614
$994$	0	0
$995$	2.71620e9	0.0874141
$996$	0	0
$997$	2.32195e10	0.742026	0.371013	−	0.928628i	$-0.379010\pi$
$997$	2.32195e10	0.742026	0.371013	+	0.928628i	$0.379010\pi$
$998$	0	0
$999$	−9.93926e9	−0.315410

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	192.8.a.u.1.2		2
3.2	odd	2		576.8.a.bl.1.1		2
4.3	odd	2		192.8.a.r.1.2		2
8.3	odd	2		96.8.a.g.1.1	yes	2
8.5	even	2		96.8.a.d.1.1	✓	2
12.11	even	2		576.8.a.bm.1.1		2
24.5	odd	2		288.8.a.i.1.2		2
24.11	even	2		288.8.a.j.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
96.8.a.d.1.1	✓	2	8.5	even	2
96.8.a.g.1.1	yes	2	8.3	odd	2
192.8.a.r.1.2		2	4.3	odd	2
192.8.a.u.1.2		2	1.1	even	1	trivial
288.8.a.i.1.2		2	24.5	odd	2
288.8.a.j.1.2		2	24.11	even	2
576.8.a.bl.1.1		2	3.2	odd	2
576.8.a.bm.1.1		2	12.11	even	2

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:	\( N \)	\(=\)	\( 192 = 2^{6} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	192.a (trivial)

\(f(q)\)	\(=\)	\(q+27.0000 q^{3} +155.275 q^{5} -1742.38 q^{7} +729.000 q^{9} +O(q^{10})\)	\(q+27.0000 q^{3} +155.275 q^{5} -1742.38 q^{7} +729.000 q^{9} -51.6521 q^{11} +10471.6 q^{13} +4192.43 q^{15} +24021.5 q^{17} -1149.27 q^{19} -47044.2 q^{21} -86508.9 q^{23} -54014.6 q^{25} +19683.0 q^{27} -153332. q^{29} +132760. q^{31} -1394.61 q^{33} -270548. q^{35} -504967. q^{37} +282732. q^{39} +99416.4 q^{41} +84318.7 q^{43} +113196. q^{45} -998165. q^{47} +2.21233e6 q^{49} +648580. q^{51} -1.24958e6 q^{53} -8020.30 q^{55} -31030.4 q^{57} -1.67870e6 q^{59} +405326. q^{61} -1.27019e6 q^{63} +1.62598e6 q^{65} -2.16414e6 q^{67} -2.33574e6 q^{69} -1.57024e6 q^{71} -2.88524e6 q^{73} -1.45839e6 q^{75} +89997.5 q^{77} -3.87170e6 q^{79} +531441. q^{81} +796558. q^{83} +3.72994e6 q^{85} -4.13997e6 q^{87} +1.41571e6 q^{89} -1.82454e7 q^{91} +3.58453e6 q^{93} -178454. q^{95} -238942. q^{97} -37654.4 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 54 q^{3} - 180 q^{5} - 1032 q^{7} + 1458 q^{9}+O(q^{10}) \) 2 * q + 54 * q^3 - 180 * q^5 - 1032 * q^7 + 1458 * q^9	\( 2 q + 54 q^{3} - 180 q^{5} - 1032 q^{7} + 1458 q^{9} + 2840 q^{11} + 340 q^{13} - 4860 q^{15} + 9780 q^{17} + 32040 q^{19} - 27864 q^{21} - 11136 q^{23} - 19730 q^{25} + 39366 q^{27} - 304212 q^{29} + 77640 q^{31} + 76680 q^{33} - 508720 q^{35} - 1015820 q^{37} + 9180 q^{39} + 704100 q^{41} - 395496 q^{43} - 131220 q^{45} - 1157488 q^{47} + 1893426 q^{49} + 264060 q^{51} - 1568580 q^{53} - 977520 q^{55} + 865080 q^{57} + 139240 q^{59} - 2603580 q^{61} - 752328 q^{63} + 5022840 q^{65} - 5289768 q^{67} - 300672 q^{69} - 5721760 q^{71} - 1190700 q^{73} - 532710 q^{75} + 2144160 q^{77} + 398280 q^{79} + 1062882 q^{81} - 6986616 q^{83} + 8504760 q^{85} - 8213724 q^{87} - 8166732 q^{89} - 25442640 q^{91} + 2096280 q^{93} - 11306000 q^{95} - 10361500 q^{97} + 2070360 q^{99}+O(q^{100}) \) 2 * q + 54 * q^3 - 180 * q^5 - 1032 * q^7 + 1458 * q^9 + 2840 * q^11 + 340 * q^13 - 4860 * q^15 + 9780 * q^17 + 32040 * q^19 - 27864 * q^21 - 11136 * q^23 - 19730 * q^25 + 39366 * q^27 - 304212 * q^29 + 77640 * q^31 + 76680 * q^33 - 508720 * q^35 - 1015820 * q^37 + 9180 * q^39 + 704100 * q^41 - 395496 * q^43 - 131220 * q^45 - 1157488 * q^47 + 1893426 * q^49 + 264060 * q^51 - 1568580 * q^53 - 977520 * q^55 + 865080 * q^57 + 139240 * q^59 - 2603580 * q^61 - 752328 * q^63 + 5022840 * q^65 - 5289768 * q^67 - 300672 * q^69 - 5721760 * q^71 - 1190700 * q^73 - 532710 * q^75 + 2144160 * q^77 + 398280 * q^79 + 1062882 * q^81 - 6986616 * q^83 + 8504760 * q^85 - 8213724 * q^87 - 8166732 * q^89 - 25442640 * q^91 + 2096280 * q^93 - 11306000 * q^95 - 10361500 * q^97 + 2070360 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more