LMFDB - Embedded newform 152.8.a.d.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [152,8,Mod(1,152)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(152, 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("152.1");

S:= CuspForms(chi, 8);

N := Newforms(S);

Level:	$ N $	$=$	$ 152 = 2^{3} \cdot 19 $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	152.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:	$47.4825238736$
Analytic rank:	$0$
Dimension:	$9$
Coefficient field:	$\mathbb{Q}[x]/(x^{9} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{9} - 2 x^{8} - 13922 x^{7} - 25112 x^{6} + 57411673 x^{5} + 379057666 x^{4} - 62486804160 x^{3} + \cdots + 69542466153984 $ x^9 - 2x^8 - 13922x^7 - 25112x^6 + 57411673x^5 + 379057666x^4 - 62486804160x^3 - 796122178848x^2 + 6759440997120x + 69542466153984
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{19}\cdot 3\cdot 5 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-18.9403$ of defining polynomial
Character	$\chi$	$=$	152.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-12.9403 q^{3} -448.356 q^{5} -610.684 q^{7} -2019.55 q^{9} +O(q^{10})$	$q-12.9403 q^{3} -448.356 q^{5} -610.684 q^{7} -2019.55 q^{9} -797.626 q^{11} -11152.4 q^{13} +5801.85 q^{15} -23870.4 q^{17} +6859.00 q^{19} +7902.43 q^{21} -88520.0 q^{23} +122898. q^{25} +54433.9 q^{27} -170424. q^{29} +218460. q^{31} +10321.5 q^{33} +273804. q^{35} -28815.2 q^{37} +144315. q^{39} -144799. q^{41} -162752. q^{43} +905477. q^{45} -259026. q^{47} -450607. q^{49} +308889. q^{51} +1.31886e6 q^{53} +357621. q^{55} -88757.4 q^{57} -639322. q^{59} -2.19944e6 q^{61} +1.23331e6 q^{63} +5.00024e6 q^{65} -747422. q^{67} +1.14547e6 q^{69} +792365. q^{71} +6.08000e6 q^{73} -1.59034e6 q^{75} +487098. q^{77} -4.36104e6 q^{79} +3.71236e6 q^{81} +8.78352e6 q^{83} +1.07024e7 q^{85} +2.20533e6 q^{87} -2.13717e6 q^{89} +6.81058e6 q^{91} -2.82693e6 q^{93} -3.07528e6 q^{95} +6.51694e6 q^{97} +1.61085e6 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 9 q + 56 q^{3} + 82 q^{5} + 464 q^{7} + 8513 q^{9}+O(q^{10}) $ 9 * q + 56 * q^3 + 82 * q^5 + 464 * q^7 + 8513 * q^9	$ 9 q + 56 q^{3} + 82 q^{5} + 464 q^{7} + 8513 q^{9} + 3360 q^{11} + 16522 q^{13} - 46608 q^{15} - 12902 q^{17} + 61731 q^{19} + 956 q^{21} - 76956 q^{23} + 188627 q^{25} + 417356 q^{27} + 63186 q^{29} + 314040 q^{31} + 89912 q^{33} + 429444 q^{35} + 389598 q^{37} - 350316 q^{39} + 424802 q^{41} + 1707520 q^{43} + 1515186 q^{45} + 968164 q^{47} + 3215825 q^{49} + 1731172 q^{51} + 1629130 q^{53} + 2745620 q^{55} + 384104 q^{57} + 4118768 q^{59} + 4051834 q^{61} + 6884020 q^{63} + 14144476 q^{65} + 5291736 q^{67} + 17194484 q^{69} + 7132000 q^{71} + 16065346 q^{73} - 2212376 q^{75} + 5883084 q^{77} + 799544 q^{79} + 26315489 q^{81} + 9460268 q^{83} - 2142320 q^{85} + 32521836 q^{87} + 29176746 q^{89} + 16210748 q^{91} + 46195392 q^{93} + 562438 q^{95} + 33616226 q^{97} + 21789816 q^{99}+O(q^{100}) $ 9 * q + 56 * q^3 + 82 * q^5 + 464 * q^7 + 8513 * q^9 + 3360 * q^11 + 16522 * q^13 - 46608 * q^15 - 12902 * q^17 + 61731 * q^19 + 956 * q^21 - 76956 * q^23 + 188627 * q^25 + 417356 * q^27 + 63186 * q^29 + 314040 * q^31 + 89912 * q^33 + 429444 * q^35 + 389598 * q^37 - 350316 * q^39 + 424802 * q^41 + 1707520 * q^43 + 1515186 * q^45 + 968164 * q^47 + 3215825 * q^49 + 1731172 * q^51 + 1629130 * q^53 + 2745620 * q^55 + 384104 * q^57 + 4118768 * q^59 + 4051834 * q^61 + 6884020 * q^63 + 14144476 * q^65 + 5291736 * q^67 + 17194484 * q^69 + 7132000 * q^71 + 16065346 * q^73 - 2212376 * q^75 + 5883084 * q^77 + 799544 * q^79 + 26315489 * q^81 + 9460268 * q^83 - 2142320 * q^85 + 32521836 * q^87 + 29176746 * q^89 + 16210748 * q^91 + 46195392 * q^93 + 562438 * q^95 + 33616226 * q^97 + 21789816 * 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
$3$	−12.9403	−0.276706	−0.138353	−	0.990383i	$-0.544181\pi$
$3$	−12.9403	−0.276706	−0.138353	+	0.990383i	$0.544181\pi$
$4$	0	0
$5$	−448.356	−1.60409	−0.802044	−	0.597265i	$-0.796254\pi$
$5$	−448.356	−1.60409	−0.802044	+	0.597265i	$0.796254\pi$
$6$	0	0
$7$	−610.684	−0.672936	−0.336468	−	0.941695i	$-0.609232\pi$
$7$	−610.684	−0.672936	−0.336468	+	0.941695i	$0.609232\pi$
$8$	0	0
$9$	−2019.55	−0.923434
$10$	0	0
$11$	−797.626	−0.180686	−0.0903431	−	0.995911i	$-0.528796\pi$
$11$	−797.626	−0.180686	−0.0903431	+	0.995911i	$0.528796\pi$
$12$	0	0
$13$	−11152.4	−1.40788	−0.703940	−	0.710259i	$-0.748578\pi$
$13$	−11152.4	−1.40788	−0.703940	+	0.710259i	$0.748578\pi$
$14$	0	0
$15$	5801.85	0.443861
$16$	0	0
$17$	−23870.4	−1.17839	−0.589194	−	0.807992i	$-0.700555\pi$
$17$	−23870.4	−1.17839	−0.589194	+	0.807992i	$0.700555\pi$
$18$	0	0
$19$	6859.00	0.229416
$19$	6859.00	0.229416
$20$	0	0
$21$	7902.43	0.186206
$22$	0	0
$23$	−88520.0	−1.51703	−0.758515	−	0.651656i	$-0.774075\pi$
$23$	−88520.0	−1.51703	−0.758515	+	0.651656i	$0.774075\pi$
$24$	0	0
$25$	122898.	1.57310
$26$	0	0
$27$	54433.9	0.532226
$28$	0	0
$29$	−170424.	−1.29759	−0.648795	−	0.760963i	$-0.724727\pi$
$29$	−170424.	−1.29759	−0.648795	+	0.760963i	$0.724727\pi$
$30$	0	0
$31$	218460.	1.31706	0.658531	−	0.752554i	$-0.271178\pi$
$31$	218460.	1.31706	0.658531	+	0.752554i	$0.271178\pi$
$32$	0	0
$33$	10321.5	0.0499970
$34$	0	0
$35$	273804.	1.07945
$36$	0	0
$37$	−28815.2	−0.0935223	−0.0467611	−	0.998906i	$-0.514890\pi$
$37$	−28815.2	−0.0935223	−0.0467611	+	0.998906i	$0.514890\pi$
$38$	0	0
$39$	144315.	0.389569
$40$	0	0
$41$	−144799.	−0.328112	−0.164056	−	0.986451i	$-0.552458\pi$
$41$	−144799.	−0.328112	−0.164056	+	0.986451i	$0.552458\pi$
$42$	0	0
$43$	−162752.	−0.312167	−0.156084	−	0.987744i	$-0.549887\pi$
$43$	−162752.	−0.312167	−0.156084	+	0.987744i	$0.549887\pi$
$44$	0	0
$45$	905477.	1.48127
$46$	0	0
$47$	−259026.	−0.363916	−0.181958	−	0.983306i	$-0.558244\pi$
$47$	−259026.	−0.363916	−0.181958	+	0.983306i	$0.558244\pi$
$48$	0	0
$49$	−450607.	−0.547157
$50$	0	0
$51$	308889.	0.326067
$52$	0	0
$53$	1.31886e6	1.21684	0.608422	−	0.793614i	$-0.291803\pi$
$53$	1.31886e6	1.21684	0.608422	+	0.793614i	$0.291803\pi$
$54$	0	0
$55$	357621.	0.289837
$56$	0	0
$57$	−88757.4	−0.0634808
$58$	0	0
$59$	−639322.	−0.405264	−0.202632	−	0.979255i	$-0.564950\pi$
$59$	−639322.	−0.405264	−0.202632	+	0.979255i	$0.564950\pi$
$60$	0	0
$61$	−2.19944e6	−1.24067	−0.620336	−	0.784336i	$-0.713004\pi$
$61$	−2.19944e6	−1.24067	−0.620336	+	0.784336i	$0.713004\pi$
$62$	0	0
$63$	1.23331e6	0.621412
$64$	0	0
$65$	5.00024e6	2.25836
$66$	0	0
$67$	−747422.	−0.303602	−0.151801	−	0.988411i	$-0.548507\pi$
$67$	−747422.	−0.303602	−0.151801	+	0.988411i	$0.548507\pi$
$68$	0	0
$69$	1.14547e6	0.419772
$70$	0	0
$71$	792365.	0.262737	0.131369	−	0.991334i	$-0.458063\pi$
$71$	792365.	0.262737	0.131369	+	0.991334i	$0.458063\pi$
$72$	0	0
$73$	6.08000e6	1.82925	0.914626	−	0.404302i	$-0.132486\pi$
$73$	6.08000e6	1.82925	0.914626	+	0.404302i	$0.132486\pi$
$74$	0	0
$75$	−1.59034e6	−0.435286
$76$	0	0
$77$	487098.	0.121590
$78$	0	0
$79$	−4.36104e6	−0.995165	−0.497583	−	0.867417i	$-0.665779\pi$
$79$	−4.36104e6	−0.995165	−0.497583	+	0.867417i	$0.665779\pi$
$80$	0	0
$81$	3.71236e6	0.776163
$82$	0	0
$83$	8.78352e6	1.68615	0.843073	−	0.537799i	$-0.180744\pi$
$83$	8.78352e6	1.68615	0.843073	+	0.537799i	$0.180744\pi$
$84$	0	0
$85$	1.07024e7	1.89024
$86$	0	0
$87$	2.20533e6	0.359051
$88$	0	0
$89$	−2.13717e6	−0.321346	−0.160673	−	0.987008i	$-0.551366\pi$
$89$	−2.13717e6	−0.321346	−0.160673	+	0.987008i	$0.551366\pi$
$90$	0	0
$91$	6.81058e6	0.947413
$92$	0	0
$93$	−2.82693e6	−0.364439
$94$	0	0
$95$	−3.07528e6	−0.368003
$96$	0	0
$97$	6.51694e6	0.725008	0.362504	−	0.931982i	$-0.381922\pi$
$97$	6.51694e6	0.725008	0.362504	+	0.931982i	$0.381922\pi$
$98$	0	0
$99$	1.61085e6	0.166852
$100$	0	0
$101$	−1.26633e7	−1.22299	−0.611495	−	0.791248i	$-0.709432\pi$
$101$	−1.26633e7	−1.22299	−0.611495	+	0.791248i	$0.709432\pi$
$102$	0	0
$103$	−9.77257e6	−0.881208	−0.440604	−	0.897702i	$-0.645236\pi$
$103$	−9.77257e6	−0.881208	−0.440604	+	0.897702i	$0.645236\pi$
$104$	0	0
$105$	−3.54310e6	−0.298690
$106$	0	0
$107$	−2.26471e7	−1.78718	−0.893592	−	0.448881i	$-0.851823\pi$
$107$	−2.26471e7	−1.78718	−0.893592	+	0.448881i	$0.851823\pi$
$108$	0	0
$109$	−2.97914e6	−0.220342	−0.110171	−	0.993913i	$-0.535140\pi$
$109$	−2.97914e6	−0.220342	−0.110171	+	0.993913i	$0.535140\pi$
$110$	0	0
$111$	372876.	0.0258782
$112$	0	0
$113$	1.03860e7	0.677130	0.338565	−	0.940943i	$-0.390059\pi$
$113$	1.03860e7	0.677130	0.338565	+	0.940943i	$0.390059\pi$
$114$	0	0
$115$	3.96885e7	2.43345
$116$	0	0
$117$	2.25228e7	1.30008
$118$	0	0
$119$	1.45773e7	0.792979
$120$	0	0
$121$	−1.88510e7	−0.967353
$122$	0	0
$123$	1.87374e6	0.0907907
$124$	0	0
$125$	−2.00744e7	−0.919300
$126$	0	0
$127$	5.85859e6	0.253793	0.126897	−	0.991916i	$-0.459498\pi$
$127$	5.85859e6	0.253793	0.126897	+	0.991916i	$0.459498\pi$
$128$	0	0
$129$	2.10606e6	0.0863787
$130$	0	0
$131$	−2.75157e7	−1.06938	−0.534689	−	0.845049i	$-0.679571\pi$
$131$	−2.75157e7	−1.06938	−0.534689	+	0.845049i	$0.679571\pi$
$132$	0	0
$133$	−4.18868e6	−0.154382
$134$	0	0
$135$	−2.44058e7	−0.853738
$136$	0	0
$137$	2.27466e7	0.755780	0.377890	−	0.925851i	$-0.376650\pi$
$137$	2.27466e7	0.755780	0.377890	+	0.925851i	$0.376650\pi$
$138$	0	0
$139$	−4.07689e7	−1.28759	−0.643794	−	0.765199i	$-0.722641\pi$
$139$	−4.07689e7	−1.28759	−0.643794	+	0.765199i	$0.722641\pi$
$140$	0	0
$141$	3.35187e6	0.100698
$142$	0	0
$143$	8.89543e6	0.254384
$144$	0	0
$145$	7.64106e7	2.08145
$146$	0	0
$147$	5.83099e6	0.151402
$148$	0	0
$149$	−1.58867e7	−0.393443	−0.196722	−	0.980459i	$-0.563030\pi$
$149$	−1.58867e7	−0.393443	−0.196722	+	0.980459i	$0.563030\pi$
$150$	0	0
$151$	−7.78860e6	−0.184094	−0.0920471	−	0.995755i	$-0.529341\pi$
$151$	−7.78860e6	−0.184094	−0.0920471	+	0.995755i	$0.529341\pi$
$152$	0	0
$153$	4.82074e7	1.08816
$154$	0	0
$155$	−9.79479e7	−2.11268
$156$	0	0
$157$	−9.22820e6	−0.190313	−0.0951565	−	0.995462i	$-0.530335\pi$
$157$	−9.22820e6	−0.190313	−0.0951565	+	0.995462i	$0.530335\pi$
$158$	0	0
$159$	−1.70665e7	−0.336708
$160$	0	0
$161$	5.40578e7	1.02086
$162$	0	0
$163$	−5.34216e7	−0.966184	−0.483092	−	0.875569i	$-0.660486\pi$
$163$	−5.34216e7	−0.966184	−0.483092	+	0.875569i	$0.660486\pi$
$164$	0	0
$165$	−4.62771e6	−0.0801996
$166$	0	0
$167$	−9.80959e7	−1.62983	−0.814917	−	0.579578i	$-0.803217\pi$
$167$	−9.80959e7	−1.62983	−0.814917	+	0.579578i	$0.803217\pi$
$168$	0	0
$169$	6.16270e7	0.982126
$170$	0	0
$171$	−1.38521e7	−0.211850
$172$	0	0
$173$	−1.01529e8	−1.49083	−0.745413	−	0.666603i	$-0.767748\pi$
$173$	−1.01529e8	−1.49083	−0.745413	+	0.666603i	$0.767748\pi$
$174$	0	0
$175$	−7.50521e7	−1.05859
$176$	0	0
$177$	8.27301e6	0.112139
$178$	0	0
$179$	−1.13078e8	−1.47364	−0.736820	−	0.676089i	$-0.763674\pi$
$179$	−1.13078e8	−1.47364	−0.736820	+	0.676089i	$0.763674\pi$
$180$	0	0
$181$	−1.03026e8	−1.29144	−0.645719	−	0.763575i	$-0.723442\pi$
$181$	−1.03026e8	−1.29144	−0.645719	+	0.763575i	$0.723442\pi$
$182$	0	0
$183$	2.84613e7	0.343302
$184$	0	0
$185$	1.29195e7	0.150018
$186$	0	0
$187$	1.90396e7	0.212918
$188$	0	0
$189$	−3.32419e7	−0.358154
$190$	0	0
$191$	−9.21955e7	−0.957399	−0.478699	−	0.877979i	$-0.658892\pi$
$191$	−9.21955e7	−0.957399	−0.478699	+	0.877979i	$0.658892\pi$
$192$	0	0
$193$	−9.62052e7	−0.963270	−0.481635	−	0.876372i	$-0.659957\pi$
$193$	−9.62052e7	−0.963270	−0.481635	+	0.876372i	$0.659957\pi$
$194$	0	0
$195$	−6.47045e7	−0.624904
$196$	0	0
$197$	2.92172e7	0.272274	0.136137	−	0.990690i	$-0.456531\pi$
$197$	2.92172e7	0.272274	0.136137	+	0.990690i	$0.456531\pi$
$198$	0	0
$199$	3.87443e6	0.0348516	0.0174258	−	0.999848i	$-0.494453\pi$
$199$	3.87443e6	0.0348516	0.0174258	+	0.999848i	$0.494453\pi$
$200$	0	0
$201$	9.67185e6	0.0840085
$202$	0	0
$203$	1.04075e8	0.873195
$204$	0	0
$205$	6.49216e7	0.526321
$206$	0	0
$207$	1.78771e8	1.40088
$208$	0	0
$209$	−5.47092e6	−0.0414523
$210$	0	0
$211$	2.54992e8	1.86869	0.934347	−	0.356364i	$-0.115984\pi$
$211$	2.54992e8	1.86869	0.934347	+	0.356364i	$0.115984\pi$
$212$	0	0
$213$	−1.02534e7	−0.0727010
$214$	0	0
$215$	7.29710e7	0.500744
$216$	0	0
$217$	−1.33410e8	−0.886298
$218$	0	0
$219$	−7.86768e7	−0.506165
$220$	0	0
$221$	2.66211e8	1.65903
$222$	0	0
$223$	7.04562e7	0.425454	0.212727	−	0.977112i	$-0.431766\pi$
$223$	7.04562e7	0.425454	0.212727	+	0.977112i	$0.431766\pi$
$224$	0	0
$225$	−2.48199e8	−1.45265
$226$	0	0
$227$	−3.33992e8	−1.89516	−0.947580	−	0.319519i	$-0.896479\pi$
$227$	−3.33992e8	−1.89516	−0.947580	+	0.319519i	$0.896479\pi$
$228$	0	0
$229$	1.87397e8	1.03119	0.515593	−	0.856833i	$-0.327571\pi$
$229$	1.87397e8	1.03119	0.515593	+	0.856833i	$0.327571\pi$
$230$	0	0
$231$	−6.30318e6	−0.0336448
$232$	0	0
$233$	2.62037e8	1.35712	0.678558	−	0.734547i	$-0.262605\pi$
$233$	2.62037e8	1.35712	0.678558	+	0.734547i	$0.262605\pi$
$234$	0	0
$235$	1.16136e8	0.583754
$236$	0	0
$237$	5.64331e7	0.275369
$238$	0	0
$239$	2.25478e8	1.06834	0.534172	−	0.845376i	$-0.320623\pi$
$239$	2.25478e8	1.06834	0.534172	+	0.845376i	$0.320623\pi$
$240$	0	0
$241$	3.68196e7	0.169441	0.0847207	−	0.996405i	$-0.473000\pi$
$241$	3.68196e7	0.169441	0.0847207	+	0.996405i	$0.473000\pi$
$242$	0	0
$243$	−1.67086e8	−0.746996
$244$	0	0
$245$	2.02033e8	0.877688
$246$	0	0
$247$	−7.64941e7	−0.322990
$248$	0	0
$249$	−1.13661e8	−0.466567
$250$	0	0
$251$	−1.87349e7	−0.0747815	−0.0373907	−	0.999301i	$-0.511905\pi$
$251$	−1.87349e7	−0.0747815	−0.0373907	+	0.999301i	$0.511905\pi$
$252$	0	0
$253$	7.06059e7	0.274106
$254$	0	0
$255$	−1.38492e8	−0.523041
$256$	0	0
$257$	−1.07261e8	−0.394162	−0.197081	−	0.980387i	$-0.563146\pi$
$257$	−1.07261e8	−0.394162	−0.197081	+	0.980387i	$0.563146\pi$
$258$	0	0
$259$	1.75970e7	0.0629345
$260$	0	0
$261$	3.44179e8	1.19824
$262$	0	0
$263$	−4.14555e7	−0.140520	−0.0702599	−	0.997529i	$-0.522383\pi$
$263$	−4.14555e7	−0.140520	−0.0702599	+	0.997529i	$0.522383\pi$
$264$	0	0
$265$	−5.91321e8	−1.95192
$266$	0	0
$267$	2.76555e7	0.0889185
$268$	0	0
$269$	−2.68602e8	−0.841348	−0.420674	−	0.907212i	$-0.638206\pi$
$269$	−2.68602e8	−0.841348	−0.420674	+	0.907212i	$0.638206\pi$
$270$	0	0
$271$	5.05747e8	1.54362	0.771810	−	0.635853i	$-0.219352\pi$
$271$	5.05747e8	1.54362	0.771810	+	0.635853i	$0.219352\pi$
$272$	0	0
$273$	−8.81308e7	−0.262155
$274$	0	0
$275$	−9.80269e7	−0.284237
$276$	0	0
$277$	2.22715e8	0.629609	0.314804	−	0.949157i	$-0.398061\pi$
$277$	2.22715e8	0.629609	0.314804	+	0.949157i	$0.398061\pi$
$278$	0	0
$279$	−4.41191e8	−1.21622
$280$	0	0
$281$	2.75476e8	0.740649	0.370324	−	0.928902i	$-0.379247\pi$
$281$	2.75476e8	0.740649	0.370324	+	0.928902i	$0.379247\pi$
$282$	0	0
$283$	7.02053e8	1.84127	0.920635	−	0.390424i	$-0.127671\pi$
$283$	7.02053e8	1.84127	0.920635	+	0.390424i	$0.127671\pi$
$284$	0	0
$285$	3.97949e7	0.101829
$286$	0	0
$287$	8.84265e7	0.220798
$288$	0	0
$289$	1.59456e8	0.388597
$290$	0	0
$291$	−8.43310e7	−0.200614
$292$	0	0
$293$	−9.86639e7	−0.229151	−0.114575	−	0.993415i	$-0.536551\pi$
$293$	−9.86639e7	−0.229151	−0.114575	+	0.993415i	$0.536551\pi$
$294$	0	0
$295$	2.86644e8	0.650079
$296$	0	0
$297$	−4.34179e7	−0.0961660
$298$	0	0
$299$	9.87209e8	2.13580
$300$	0	0
$301$	9.93903e7	0.210069
$302$	0	0
$303$	1.63867e8	0.338409
$304$	0	0
$305$	9.86132e8	1.99015
$306$	0	0
$307$	9.66453e8	1.90632	0.953161	−	0.302462i	$-0.0978087\pi$
$307$	9.66453e8	1.90632	0.953161	+	0.302462i	$0.0978087\pi$
$308$	0	0
$309$	1.26460e8	0.243836
$310$	0	0
$311$	−2.68112e8	−0.505423	−0.252711	−	0.967542i	$-0.581322\pi$
$311$	−2.68112e8	−0.505423	−0.252711	+	0.967542i	$0.581322\pi$
$312$	0	0
$313$	−5.50471e8	−1.01468	−0.507341	−	0.861746i	$-0.669371\pi$
$313$	−5.50471e8	−1.01468	−0.507341	+	0.861746i	$0.669371\pi$
$314$	0	0
$315$	−5.52961e8	−0.996799
$316$	0	0
$317$	2.60829e7	0.0459885	0.0229942	−	0.999736i	$-0.492680\pi$
$317$	2.60829e7	0.0459885	0.0229942	+	0.999736i	$0.492680\pi$
$318$	0	0
$319$	1.35934e8	0.234457
$320$	0	0
$321$	2.93060e8	0.494525
$322$	0	0
$323$	−1.63727e8	−0.270341
$324$	0	0
$325$	−1.37061e9	−2.21473
$326$	0	0
$327$	3.85509e7	0.0609702
$328$	0	0
$329$	1.58183e8	0.244892
$330$	0	0
$331$	−1.18133e9	−1.79049	−0.895246	−	0.445573i	$-0.853000\pi$
$331$	−1.18133e9	−1.79049	−0.895246	+	0.445573i	$0.853000\pi$
$332$	0	0
$333$	5.81936e7	0.0863616
$334$	0	0
$335$	3.35111e8	0.487004
$336$	0	0
$337$	−3.15915e8	−0.449641	−0.224820	−	0.974400i	$-0.572180\pi$
$337$	−3.15915e8	−0.449641	−0.224820	+	0.974400i	$0.572180\pi$
$338$	0	0
$339$	−1.34397e8	−0.187366
$340$	0	0
$341$	−1.74249e8	−0.237975
$342$	0	0
$343$	7.78104e8	1.04114
$344$	0	0
$345$	−5.13580e8	−0.673351
$346$	0	0
$347$	1.44435e9	1.85575	0.927874	−	0.372894i	$-0.121634\pi$
$347$	1.44435e9	1.85575	0.927874	+	0.372894i	$0.121634\pi$
$348$	0	0
$349$	−9.95757e8	−1.25390	−0.626952	−	0.779058i	$-0.715698\pi$
$349$	−9.95757e8	−1.25390	−0.626952	+	0.779058i	$0.715698\pi$
$350$	0	0
$351$	−6.07067e8	−0.749311
$352$	0	0
$353$	−6.13500e8	−0.742341	−0.371170	−	0.928565i	$-0.621043\pi$
$353$	−6.13500e8	−0.742341	−0.371170	+	0.928565i	$0.621043\pi$
$354$	0	0
$355$	−3.55262e8	−0.421453
$356$	0	0
$357$	−1.88634e8	−0.219422
$358$	0	0
$359$	4.27255e8	0.487368	0.243684	−	0.969855i	$-0.421644\pi$
$359$	4.27255e8	0.487368	0.243684	+	0.969855i	$0.421644\pi$
$360$	0	0
$361$	4.70459e7	0.0526316
$362$	0	0
$363$	2.43937e8	0.267673
$364$	0	0
$365$	−2.72600e9	−2.93428
$366$	0	0
$367$	−6.24024e8	−0.658977	−0.329488	−	0.944160i	$-0.606876\pi$
$367$	−6.24024e8	−0.658977	−0.329488	+	0.944160i	$0.606876\pi$
$368$	0	0
$369$	2.92429e8	0.302990
$370$	0	0
$371$	−8.05410e8	−0.818858
$372$	0	0
$373$	1.22124e8	0.121848	0.0609241	−	0.998142i	$-0.480595\pi$
$373$	1.22124e8	0.121848	0.0609241	+	0.998142i	$0.480595\pi$
$374$	0	0
$375$	2.59768e8	0.254376
$376$	0	0
$377$	1.90063e9	1.82685
$378$	0	0
$379$	8.08681e8	0.763027	0.381513	−	0.924363i	$-0.375403\pi$
$379$	8.08681e8	0.763027	0.381513	+	0.924363i	$0.375403\pi$
$380$	0	0
$381$	−7.58117e7	−0.0702262
$382$	0	0
$383$	−1.80693e9	−1.64341	−0.821703	−	0.569916i	$-0.806976\pi$
$383$	−1.80693e9	−1.64341	−0.821703	+	0.569916i	$0.806976\pi$
$384$	0	0
$385$	−2.18393e8	−0.195041
$386$	0	0
$387$	3.28686e8	0.288266
$388$	0	0
$389$	−1.01333e9	−0.872826	−0.436413	−	0.899746i	$-0.643751\pi$
$389$	−1.01333e9	−0.872826	−0.436413	+	0.899746i	$0.643751\pi$
$390$	0	0
$391$	2.11301e9	1.78765
$392$	0	0
$393$	3.56061e8	0.295904
$394$	0	0
$395$	1.95530e9	1.59633
$396$	0	0
$397$	2.41712e8	0.193879	0.0969395	−	0.995290i	$-0.469095\pi$
$397$	2.41712e8	0.193879	0.0969395	+	0.995290i	$0.469095\pi$
$398$	0	0
$399$	5.42027e7	0.0427185
$400$	0	0
$401$	1.62242e9	1.25649	0.628244	−	0.778017i	$-0.283774\pi$
$401$	1.62242e9	1.25649	0.628244	+	0.778017i	$0.283774\pi$
$402$	0	0
$403$	−2.43635e9	−1.85426
$404$	0	0
$405$	−1.66446e9	−1.24503
$406$	0	0
$407$	2.29837e7	0.0168982
$408$	0	0
$409$	−1.92933e9	−1.39436	−0.697181	−	0.716895i	$-0.745563\pi$
$409$	−1.92933e9	−1.39436	−0.697181	+	0.716895i	$0.745563\pi$
$410$	0	0
$411$	−2.94348e8	−0.209129
$412$	0	0
$413$	3.90424e8	0.272717
$414$	0	0
$415$	−3.93814e9	−2.70473
$416$	0	0
$417$	5.27560e8	0.356284
$418$	0	0
$419$	−1.95566e9	−1.29880	−0.649402	−	0.760445i	$-0.724981\pi$
$419$	−1.95566e9	−1.29880	−0.649402	+	0.760445i	$0.724981\pi$
$420$	0	0
$421$	8.97950e8	0.586496	0.293248	−	0.956036i	$-0.405264\pi$
$421$	8.97950e8	0.586496	0.293248	+	0.956036i	$0.405264\pi$
$422$	0	0
$423$	5.23117e8	0.336053
$424$	0	0
$425$	−2.93363e9	−1.85372
$426$	0	0
$427$	1.34316e9	0.834893
$428$	0	0
$429$	−1.15109e8	−0.0703898
$430$	0	0
$431$	−8.16396e7	−0.0491169	−0.0245584	−	0.999698i	$-0.507818\pi$
$431$	−8.16396e7	−0.0491169	−0.0245584	+	0.999698i	$0.507818\pi$
$432$	0	0
$433$	1.80911e9	1.07092	0.535462	−	0.844560i	$-0.320138\pi$
$433$	1.80911e9	1.07092	0.535462	+	0.844560i	$0.320138\pi$
$434$	0	0
$435$	−9.88774e8	−0.575950
$436$	0	0
$437$	−6.07159e8	−0.348030
$438$	0	0
$439$	1.97358e9	1.11334	0.556670	−	0.830734i	$-0.312079\pi$
$439$	1.97358e9	1.11334	0.556670	+	0.830734i	$0.312079\pi$
$440$	0	0
$441$	9.10024e8	0.505263
$442$	0	0
$443$	1.59486e9	0.871583	0.435791	−	0.900048i	$-0.356469\pi$
$443$	1.59486e9	0.871583	0.435791	+	0.900048i	$0.356469\pi$
$444$	0	0
$445$	9.58212e8	0.515468
$446$	0	0
$447$	2.05579e8	0.108868
$448$	0	0
$449$	−9.62423e8	−0.501769	−0.250884	−	0.968017i	$-0.580721\pi$
$449$	−9.62423e8	−0.501769	−0.250884	+	0.968017i	$0.580721\pi$
$450$	0	0
$451$	1.15496e8	0.0592853
$452$	0	0
$453$	1.00787e8	0.0509400
$454$	0	0
$455$	−3.05357e9	−1.51973
$456$	0	0
$457$	3.67165e9	1.79951	0.899755	−	0.436394i	$-0.143745\pi$
$457$	3.67165e9	1.79951	0.899755	+	0.436394i	$0.143745\pi$
$458$	0	0
$459$	−1.29936e9	−0.627169
$460$	0	0
$461$	−3.61303e9	−1.71758	−0.858792	−	0.512324i	$-0.828785\pi$
$461$	−3.61303e9	−1.71758	−0.858792	+	0.512324i	$0.828785\pi$
$462$	0	0
$463$	5.76619e8	0.269995	0.134997	−	0.990846i	$-0.456897\pi$
$463$	5.76619e8	0.269995	0.134997	+	0.990846i	$0.456897\pi$
$464$	0	0
$465$	1.26747e9	0.584593
$466$	0	0
$467$	−3.54781e9	−1.61195	−0.805974	−	0.591951i	$-0.798358\pi$
$467$	−3.54781e9	−1.61195	−0.805974	+	0.591951i	$0.798358\pi$
$468$	0	0
$469$	4.56439e8	0.204305
$470$	0	0
$471$	1.19415e8	0.0526608
$472$	0	0
$473$	1.29816e8	0.0564043
$474$	0	0
$475$	8.42959e8	0.360893
$476$	0	0
$477$	−2.66351e9	−1.12367
$478$	0	0
$479$	3.58532e9	1.49057	0.745287	−	0.666743i	$-0.232312\pi$
$479$	3.58532e9	1.49057	0.745287	+	0.666743i	$0.232312\pi$
$480$	0	0
$481$	3.21357e8	0.131668
$482$	0	0
$483$	−6.99523e8	−0.282480
$484$	0	0
$485$	−2.92191e9	−1.16298
$486$	0	0
$487$	2.10687e9	0.826582	0.413291	−	0.910599i	$-0.364379\pi$
$487$	2.10687e9	0.826582	0.413291	+	0.910599i	$0.364379\pi$
$488$	0	0
$489$	6.91290e8	0.267349
$490$	0	0
$491$	−1.42095e9	−0.541744	−0.270872	−	0.962615i	$-0.587312\pi$
$491$	−1.42095e9	−0.541744	−0.270872	+	0.962615i	$0.587312\pi$
$492$	0	0
$493$	4.06808e9	1.52906
$494$	0	0
$495$	−7.22232e8	−0.267645
$496$	0	0
$497$	−4.83885e8	−0.176805
$498$	0	0
$499$	1.35904e9	0.489645	0.244822	−	0.969568i	$-0.421270\pi$
$499$	1.35904e9	0.489645	0.244822	+	0.969568i	$0.421270\pi$
$500$	0	0
$501$	1.26939e9	0.450985
$502$	0	0
$503$	−1.38824e9	−0.486382	−0.243191	−	0.969978i	$-0.578194\pi$
$503$	−1.38824e9	−0.486382	−0.243191	+	0.969978i	$0.578194\pi$
$504$	0	0
$505$	5.67768e9	1.96178
$506$	0	0
$507$	−7.97470e8	−0.271761
$508$	0	0
$509$	−3.49832e9	−1.17584	−0.587918	−	0.808921i	$-0.700052\pi$
$509$	−3.49832e9	−1.17584	−0.587918	+	0.808921i	$0.700052\pi$
$510$	0	0
$511$	−3.71296e9	−1.23097
$512$	0	0
$513$	3.73362e8	0.122101
$514$	0	0
$515$	4.38159e9	1.41354
$516$	0	0
$517$	2.06606e8	0.0657547
$518$	0	0
$519$	1.31381e9	0.412521
$520$	0	0
$521$	2.87079e9	0.889345	0.444672	−	0.895693i	$-0.353320\pi$
$521$	2.87079e9	0.889345	0.444672	+	0.895693i	$0.353320\pi$
$522$	0	0
$523$	5.34638e9	1.63420	0.817098	−	0.576499i	$-0.195582\pi$
$523$	5.34638e9	1.63420	0.817098	+	0.576499i	$0.195582\pi$
$524$	0	0
$525$	9.71195e8	0.292920
$526$	0	0
$527$	−5.21472e9	−1.55201
$528$	0	0
$529$	4.43097e9	1.30138
$530$	0	0
$531$	1.29114e9	0.374234
$532$	0	0
$533$	1.61485e9	0.461942
$534$	0	0
$535$	1.01540e10	2.86680
$536$	0	0
$537$	1.46326e9	0.407766
$538$	0	0
$539$	3.59416e8	0.0988637
$540$	0	0
$541$	−5.46213e9	−1.48310	−0.741551	−	0.670896i	$-0.765910\pi$
$541$	−5.46213e9	−1.48310	−0.741551	+	0.670896i	$0.765910\pi$
$542$	0	0
$543$	1.33319e9	0.357349
$544$	0	0
$545$	1.33572e9	0.353449
$546$	0	0
$547$	−1.99084e9	−0.520092	−0.260046	−	0.965596i	$-0.583738\pi$
$547$	−1.99084e9	−0.520092	−0.260046	+	0.965596i	$0.583738\pi$
$548$	0	0
$549$	4.44187e9	1.14568
$550$	0	0
$551$	−1.16894e9	−0.297687
$552$	0	0
$553$	2.66322e9	0.669683
$554$	0	0
$555$	−1.67181e8	−0.0415109
$556$	0	0
$557$	4.05741e9	0.994844	0.497422	−	0.867509i	$-0.334280\pi$
$557$	4.05741e9	0.994844	0.497422	+	0.867509i	$0.334280\pi$
$558$	0	0
$559$	1.81508e9	0.439494
$560$	0	0
$561$	−2.46378e8	−0.0589159
$562$	0	0
$563$	4.89883e8	0.115695	0.0578473	−	0.998325i	$-0.481576\pi$
$563$	4.89883e8	0.115695	0.0578473	+	0.998325i	$0.481576\pi$
$564$	0	0
$565$	−4.65661e9	−1.08618
$566$	0	0
$567$	−2.26708e9	−0.522308
$568$	0	0
$569$	2.34018e9	0.532545	0.266272	−	0.963898i	$-0.414208\pi$
$569$	2.34018e9	0.532545	0.266272	+	0.963898i	$0.414208\pi$
$570$	0	0
$571$	−1.73889e8	−0.0390882	−0.0195441	−	0.999809i	$-0.506221\pi$
$571$	−1.73889e8	−0.0390882	−0.0195441	+	0.999809i	$0.506221\pi$
$572$	0	0
$573$	1.19303e9	0.264918
$574$	0	0
$575$	−1.08790e10	−2.38644
$576$	0	0
$577$	7.68023e9	1.66440	0.832202	−	0.554472i	$-0.187080\pi$
$577$	7.68023e9	1.66440	0.832202	+	0.554472i	$0.187080\pi$
$578$	0	0
$579$	1.24492e9	0.266543
$580$	0	0
$581$	−5.36396e9	−1.13467
$582$	0	0
$583$	−1.05196e9	−0.219867
$584$	0	0
$585$	−1.00982e10	−2.08545
$586$	0	0
$587$	−5.64323e9	−1.15158	−0.575791	−	0.817597i	$-0.695306\pi$
$587$	−5.64323e9	−1.15158	−0.575791	+	0.817597i	$0.695306\pi$
$588$	0	0
$589$	1.49842e9	0.302155
$590$	0	0
$591$	−3.78078e8	−0.0753400
$592$	0	0
$593$	9.38008e9	1.84720	0.923602	−	0.383352i	$-0.125230\pi$
$593$	9.38008e9	1.84720	0.923602	+	0.383352i	$0.125230\pi$
$594$	0	0
$595$	−6.53581e9	−1.27201
$596$	0	0
$597$	−5.01362e7	−0.00964365
$598$	0	0
$599$	7.13757e9	1.35693	0.678463	−	0.734634i	$-0.262646\pi$
$599$	7.13757e9	1.35693	0.678463	+	0.734634i	$0.262646\pi$
$600$	0	0
$601$	1.96406e9	0.369058	0.184529	−	0.982827i	$-0.440924\pi$
$601$	1.96406e9	0.369058	0.184529	+	0.982827i	$0.440924\pi$
$602$	0	0
$603$	1.50946e9	0.280356
$604$	0	0
$605$	8.45195e9	1.55172
$606$	0	0
$607$	−8.03346e9	−1.45795	−0.728974	−	0.684541i	$-0.760003\pi$
$607$	−8.03346e9	−1.45795	−0.728974	+	0.684541i	$0.760003\pi$
$608$	0	0
$609$	−1.34676e9	−0.241619
$610$	0	0
$611$	2.88876e9	0.512351
$612$	0	0
$613$	−2.90624e9	−0.509589	−0.254795	−	0.966995i	$-0.582008\pi$
$613$	−2.90624e9	−0.509589	−0.254795	+	0.966995i	$0.582008\pi$
$614$	0	0
$615$	−8.40103e8	−0.145636
$616$	0	0
$617$	1.29659e9	0.222230	0.111115	−	0.993808i	$-0.464558\pi$
$617$	1.29659e9	0.222230	0.111115	+	0.993808i	$0.464558\pi$
$618$	0	0
$619$	−5.35707e9	−0.907841	−0.453921	−	0.891042i	$-0.649975\pi$
$619$	−5.35707e9	−0.907841	−0.453921	+	0.891042i	$0.649975\pi$
$620$	0	0
$621$	−4.81849e9	−0.807403
$622$	0	0
$623$	1.30513e9	0.216245
$624$	0	0
$625$	−6.00954e8	−0.0984603
$626$	0	0
$627$	7.07952e7	0.0114701
$628$	0	0
$629$	6.87829e8	0.110205
$630$	0	0
$631$	−3.70368e9	−0.586855	−0.293427	−	0.955981i	$-0.594796\pi$
$631$	−3.70368e9	−0.586855	−0.293427	+	0.955981i	$0.594796\pi$
$632$	0	0
$633$	−3.29967e9	−0.517080
$634$	0	0
$635$	−2.62673e9	−0.407106
$636$	0	0
$637$	5.02534e9	0.770332
$638$	0	0
$639$	−1.60022e9	−0.242620
$640$	0	0
$641$	−1.05174e9	−0.157726	−0.0788632	−	0.996885i	$-0.525129\pi$
$641$	−1.05174e9	−0.157726	−0.0788632	+	0.996885i	$0.525129\pi$
$642$	0	0
$643$	−1.32758e9	−0.196935	−0.0984675	−	0.995140i	$-0.531394\pi$
$643$	−1.32758e9	−0.196935	−0.0984675	+	0.995140i	$0.531394\pi$
$644$	0	0
$645$	−9.44265e8	−0.138559
$646$	0	0
$647$	−1.03277e10	−1.49913	−0.749564	−	0.661932i	$-0.769737\pi$
$647$	−1.03277e10	−1.49913	−0.749564	+	0.661932i	$0.769737\pi$
$648$	0	0
$649$	5.09940e8	0.0732256
$650$	0	0
$651$	1.72636e9	0.245244
$652$	0	0
$653$	5.50814e9	0.774121	0.387060	−	0.922054i	$-0.373491\pi$
$653$	5.50814e9	0.774121	0.387060	+	0.922054i	$0.373491\pi$
$654$	0	0
$655$	1.23368e10	1.71538
$656$	0	0
$657$	−1.22789e10	−1.68919
$658$	0	0
$659$	−8.11699e9	−1.10483	−0.552416	−	0.833569i	$-0.686294\pi$
$659$	−8.11699e9	−1.10483	−0.552416	+	0.833569i	$0.686294\pi$
$660$	0	0
$661$	4.15573e9	0.559684	0.279842	−	0.960046i	$-0.409718\pi$
$661$	4.15573e9	0.559684	0.279842	+	0.960046i	$0.409718\pi$
$662$	0	0
$663$	−3.44485e9	−0.459064
$664$	0	0
$665$	1.87802e9	0.247642
$666$	0	0
$667$	1.50859e10	1.96848
$668$	0	0
$669$	−9.11723e8	−0.117726
$670$	0	0
$671$	1.75433e9	0.224172
$672$	0	0
$673$	1.24619e10	1.57591	0.787954	−	0.615734i	$-0.211141\pi$
$673$	1.24619e10	1.57591	0.787954	+	0.615734i	$0.211141\pi$
$674$	0	0
$675$	6.68984e9	0.837244
$676$	0	0
$677$	−5.44025e9	−0.673843	−0.336921	−	0.941533i	$-0.609386\pi$
$677$	−5.44025e9	−0.673843	−0.336921	+	0.941533i	$0.609386\pi$
$678$	0	0
$679$	−3.97979e9	−0.487884
$680$	0	0
$681$	4.32195e9	0.524403
$682$	0	0
$683$	−5.12121e9	−0.615036	−0.307518	−	0.951542i	$-0.599498\pi$
$683$	−5.12121e9	−0.615036	−0.307518	+	0.951542i	$0.599498\pi$
$684$	0	0
$685$	−1.01986e10	−1.21234
$686$	0	0
$687$	−2.42496e9	−0.285336
$688$	0	0
$689$	−1.47085e10	−1.71317
$690$	0	0
$691$	−1.45730e9	−0.168026	−0.0840129	−	0.996465i	$-0.526774\pi$
$691$	−1.45730e9	−0.168026	−0.0840129	+	0.996465i	$0.526774\pi$
$692$	0	0
$693$	−9.83718e8	−0.112281
$694$	0	0
$695$	1.82790e10	2.06540
$696$	0	0
$697$	3.45641e9	0.386643
$698$	0	0
$699$	−3.39083e9	−0.375523
$700$	0	0
$701$	−9.50026e9	−1.04165	−0.520826	−	0.853663i	$-0.674376\pi$
$701$	−9.50026e9	−1.04165	−0.520826	+	0.853663i	$0.674376\pi$
$702$	0	0
$703$	−1.97643e8	−0.0214555
$704$	0	0
$705$	−1.50283e9	−0.161528
$706$	0	0
$707$	7.73330e9	0.822994
$708$	0	0
$709$	1.31224e9	0.138277	0.0691387	−	0.997607i	$-0.477975\pi$
$709$	1.31224e9	0.138277	0.0691387	+	0.997607i	$0.477975\pi$
$710$	0	0
$711$	8.80733e9	0.918969
$712$	0	0
$713$	−1.93381e10	−1.99802
$714$	0	0
$715$	−3.98832e9	−0.408055
$716$	0	0
$717$	−2.91775e9	−0.295618
$718$	0	0
$719$	−7.33298e8	−0.0735748	−0.0367874	−	0.999323i	$-0.511712\pi$
$719$	−7.33298e8	−0.0735748	−0.0367874	+	0.999323i	$0.511712\pi$
$720$	0	0
$721$	5.96796e9	0.592997
$722$	0	0
$723$	−4.76456e8	−0.0468855
$724$	0	0
$725$	−2.09448e10	−2.04124
$726$	0	0
$727$	−6.87921e9	−0.664000	−0.332000	−	0.943279i	$-0.607723\pi$
$727$	−6.87921e9	−0.664000	−0.332000	+	0.943279i	$0.607723\pi$
$728$	0	0
$729$	−5.95680e9	−0.569465
$730$	0	0
$731$	3.88496e9	0.367854
$732$	0	0
$733$	−1.38323e10	−1.29727	−0.648634	−	0.761101i	$-0.724659\pi$
$733$	−1.38323e10	−1.29727	−0.648634	+	0.761101i	$0.724659\pi$
$734$	0	0
$735$	−2.61436e9	−0.242862
$736$	0	0
$737$	5.96163e8	0.0548566
$738$	0	0
$739$	−7.07576e9	−0.644938	−0.322469	−	0.946580i	$-0.604513\pi$
$739$	−7.07576e9	−0.644938	−0.322469	+	0.946580i	$0.604513\pi$
$740$	0	0
$741$	9.89855e8	0.0893734
$742$	0	0
$743$	−3.32152e9	−0.297081	−0.148541	−	0.988906i	$-0.547458\pi$
$743$	−3.32152e9	−0.297081	−0.148541	+	0.988906i	$0.547458\pi$
$744$	0	0
$745$	7.12291e9	0.631118
$746$	0	0
$747$	−1.77387e10	−1.55704
$748$	0	0
$749$	1.38302e10	1.20266
$750$	0	0
$751$	3.09703e9	0.266812	0.133406	−	0.991061i	$-0.457409\pi$
$751$	3.09703e9	0.266812	0.133406	+	0.991061i	$0.457409\pi$
$752$	0	0
$753$	2.42435e8	0.0206925
$754$	0	0
$755$	3.49207e9	0.295303
$756$	0	0
$757$	−1.66635e10	−1.39614	−0.698072	−	0.716028i	$-0.745958\pi$
$757$	−1.66635e10	−1.39614	−0.698072	+	0.716028i	$0.745958\pi$
$758$	0	0
$759$	−9.13660e8	−0.0758470
$760$	0	0
$761$	1.03368e10	0.850241	0.425120	−	0.905137i	$-0.360232\pi$
$761$	1.03368e10	0.850241	0.425120	+	0.905137i	$0.360232\pi$
$762$	0	0
$763$	1.81931e9	0.148276
$764$	0	0
$765$	−2.16141e10	−1.74551
$766$	0	0
$767$	7.12996e9	0.570563
$768$	0	0
$769$	−5.20185e9	−0.412492	−0.206246	−	0.978500i	$-0.566125\pi$
$769$	−5.20185e9	−0.412492	−0.206246	+	0.978500i	$0.566125\pi$
$770$	0	0
$771$	1.38798e9	0.109067
$772$	0	0
$773$	−1.06405e10	−0.828582	−0.414291	−	0.910144i	$-0.635970\pi$
$773$	−1.06405e10	−0.828582	−0.414291	+	0.910144i	$0.635970\pi$
$774$	0	0
$775$	2.68484e10	2.07187
$776$	0	0
$777$	−2.27710e8	−0.0174144
$778$	0	0
$779$	−9.93177e8	−0.0752741
$780$	0	0
$781$	−6.32011e8	−0.0474729
$782$	0	0
$783$	−9.27684e9	−0.690611
$784$	0	0
$785$	4.13752e9	0.305279
$786$	0	0
$787$	−7.27109e9	−0.531726	−0.265863	−	0.964011i	$-0.585657\pi$
$787$	−7.27109e9	−0.531726	−0.265863	+	0.964011i	$0.585657\pi$
$788$	0	0
$789$	5.36446e8	0.0388827
$790$	0	0
$791$	−6.34254e9	−0.455665
$792$	0	0
$793$	2.45290e10	1.74672
$794$	0	0
$795$	7.65186e9	0.540110
$796$	0	0
$797$	1.74025e10	1.21761	0.608805	−	0.793320i	$-0.291649\pi$
$797$	1.74025e10	1.21761	0.608805	+	0.793320i	$0.291649\pi$
$798$	0	0
$799$	6.18306e9	0.428835
$800$	0	0
$801$	4.31611e9	0.296742
$802$	0	0
$803$	−4.84956e9	−0.330520
$804$	0	0
$805$	−2.42372e10	−1.63756
$806$	0	0
$807$	3.47578e9	0.232806
$808$	0	0
$809$	5.29753e9	0.351766	0.175883	−	0.984411i	$-0.443722\pi$
$809$	5.29753e9	0.351766	0.175883	+	0.984411i	$0.443722\pi$
$810$	0	0
$811$	9.01559e9	0.593501	0.296750	−	0.954955i	$-0.404097\pi$
$811$	9.01559e9	0.593501	0.296750	+	0.954955i	$0.404097\pi$
$812$	0	0
$813$	−6.54450e9	−0.427130
$814$	0	0
$815$	2.39519e10	1.54984
$816$	0	0
$817$	−1.11632e9	−0.0716161
$818$	0	0
$819$	−1.37543e10	−0.874873
$820$	0	0
$821$	1.40553e10	0.886417	0.443208	−	0.896419i	$-0.353840\pi$
$821$	1.40553e10	0.886417	0.443208	+	0.896419i	$0.353840\pi$
$822$	0	0
$823$	−1.04404e10	−0.652859	−0.326429	−	0.945222i	$-0.605846\pi$
$823$	−1.04404e10	−0.652859	−0.326429	+	0.945222i	$0.605846\pi$
$824$	0	0
$825$	1.26850e9	0.0786502
$826$	0	0
$827$	2.00292e10	1.23139	0.615693	−	0.787986i	$-0.288876\pi$
$827$	2.00292e10	1.23139	0.615693	+	0.787986i	$0.288876\pi$
$828$	0	0
$829$	8.53589e9	0.520365	0.260182	−	0.965559i	$-0.416217\pi$
$829$	8.53589e9	0.520365	0.260182	+	0.965559i	$0.416217\pi$
$830$	0	0
$831$	−2.88200e9	−0.174217
$832$	0	0
$833$	1.07562e10	0.644763
$834$	0	0
$835$	4.39819e10	2.61440
$836$	0	0
$837$	1.18916e10	0.700975
$838$	0	0
$839$	−3.02709e10	−1.76953	−0.884766	−	0.466035i	$-0.845682\pi$
$839$	−3.02709e10	−1.76953	−0.884766	+	0.466035i	$0.845682\pi$
$840$	0	0
$841$	1.17944e10	0.683739
$842$	0	0
$843$	−3.56474e9	−0.204942
$844$	0	0
$845$	−2.76308e10	−1.57542
$846$	0	0
$847$	1.15120e10	0.650966
$848$	0	0
$849$	−9.08477e9	−0.509491
$850$	0	0
$851$	2.55072e9	0.141876
$852$	0	0
$853$	1.20968e10	0.667341	0.333671	−	0.942690i	$-0.391713\pi$
$853$	1.20968e10	0.667341	0.333671	+	0.942690i	$0.391713\pi$
$854$	0	0
$855$	6.21067e9	0.339826
$856$	0	0
$857$	−1.15027e9	−0.0624260	−0.0312130	−	0.999513i	$-0.509937\pi$
$857$	−1.15027e9	−0.0624260	−0.0312130	+	0.999513i	$0.509937\pi$
$858$	0	0
$859$	5.83084e9	0.313874	0.156937	−	0.987609i	$-0.449838\pi$
$859$	5.83084e9	0.313874	0.156937	+	0.987609i	$0.449838\pi$
$860$	0	0
$861$	−1.14426e9	−0.0610963
$862$	0	0
$863$	2.61395e9	0.138439	0.0692196	−	0.997601i	$-0.477949\pi$
$863$	2.61395e9	0.138439	0.0692196	+	0.997601i	$0.477949\pi$
$864$	0	0
$865$	4.55210e10	2.39142
$866$	0	0
$867$	−2.06341e9	−0.107527
$868$	0	0
$869$	3.47848e9	0.179813
$870$	0	0
$871$	8.33553e9	0.427435
$872$	0	0
$873$	−1.31613e10	−0.669497
$874$	0	0
$875$	1.22591e10	0.618630
$876$	0	0
$877$	−2.60197e8	−0.0130258	−0.00651290	−	0.999979i	$-0.502073\pi$
$877$	−2.60197e8	−0.0130258	−0.00651290	+	0.999979i	$0.502073\pi$
$878$	0	0
$879$	1.27674e9	0.0634075
$880$	0	0
$881$	−5.20364e6	−0.000256385 0	−0.000128192	−	1.00000i	$-0.500041\pi$
$881$	−5.20364e6	−0.000256385 0	−0.000128192		1.00000i	$0.500041\pi$
$882$	0	0
$883$	−1.74039e9	−0.0850717	−0.0425358	−	0.999095i	$-0.513544\pi$
$883$	−1.74039e9	−0.0850717	−0.0425358	+	0.999095i	$0.513544\pi$
$884$	0	0
$885$	−3.70926e9	−0.179881
$886$	0	0
$887$	−2.55616e10	−1.22986	−0.614929	−	0.788583i	$-0.710815\pi$
$887$	−2.55616e10	−1.22986	−0.614929	+	0.788583i	$0.710815\pi$
$888$	0	0
$889$	−3.57775e9	−0.170787
$890$	0	0
$891$	−2.96108e9	−0.140242
$892$	0	0
$893$	−1.77666e9	−0.0834882
$894$	0	0
$895$	5.06991e10	2.36385
$896$	0	0
$897$	−1.27748e10	−0.590988
$898$	0	0
$899$	−3.72308e10	−1.70900
$900$	0	0
$901$	−3.14818e10	−1.43391
$902$	0	0
$903$	−1.28614e9	−0.0581274
$904$	0	0
$905$	4.61926e10	2.07158
$906$	0	0
$907$	1.09403e10	0.486858	0.243429	−	0.969919i	$-0.421728\pi$
$907$	1.09403e10	0.486858	0.243429	+	0.969919i	$0.421728\pi$
$908$	0	0
$909$	2.55742e10	1.12935
$910$	0	0
$911$	−3.27541e10	−1.43533	−0.717664	−	0.696389i	$-0.754789\pi$
$911$	−3.27541e10	−1.43533	−0.717664	+	0.696389i	$0.754789\pi$
$912$	0	0
$913$	−7.00596e9	−0.304663
$914$	0	0
$915$	−1.27608e10	−0.550687
$916$	0	0
$917$	1.68034e10	0.719623
$918$	0	0
$919$	2.59412e10	1.10252	0.551259	−	0.834334i	$-0.314148\pi$
$919$	2.59412e10	1.10252	0.551259	+	0.834334i	$0.314148\pi$
$920$	0	0
$921$	−1.25062e10	−0.527492
$922$	0	0
$923$	−8.83676e9	−0.369902
$924$	0	0
$925$	−3.54133e9	−0.147120
$926$	0	0
$927$	1.97362e10	0.813737
$928$	0	0
$929$	−1.20517e10	−0.493166	−0.246583	−	0.969122i	$-0.579308\pi$
$929$	−1.20517e10	−0.493166	−0.246583	+	0.969122i	$0.579308\pi$
$930$	0	0
$931$	−3.09072e9	−0.125526
$932$	0	0
$933$	3.46944e9	0.139854
$934$	0	0
$935$	−8.53654e9	−0.341540
$936$	0	0
$937$	−3.36217e9	−0.133515	−0.0667577	−	0.997769i	$-0.521265\pi$
$937$	−3.36217e9	−0.133515	−0.0667577	+	0.997769i	$0.521265\pi$
$938$	0	0
$939$	7.12325e9	0.280769
$940$	0	0
$941$	−4.15855e10	−1.62697	−0.813483	−	0.581589i	$-0.802431\pi$
$941$	−4.15855e10	−1.62697	−0.813483	+	0.581589i	$0.802431\pi$
$942$	0	0
$943$	1.28176e10	0.497756
$944$	0	0
$945$	1.49042e10	0.574511
$946$	0	0
$947$	−2.96254e10	−1.13355	−0.566773	−	0.823874i	$-0.691808\pi$
$947$	−2.96254e10	−1.13355	−0.566773	+	0.823874i	$0.691808\pi$
$948$	0	0
$949$	−6.78064e10	−2.57537
$950$	0	0
$951$	−3.37520e8	−0.0127253
$952$	0	0
$953$	1.58681e10	0.593882	0.296941	−	0.954896i	$-0.404034\pi$
$953$	1.58681e10	0.593882	0.296941	+	0.954896i	$0.404034\pi$
$954$	0	0
$955$	4.13364e10	1.53575
$956$	0	0
$957$	−1.75903e9	−0.0648756
$958$	0	0
$959$	−1.38910e10	−0.508592
$960$	0	0
$961$	2.02121e10	0.734650
$962$	0	0
$963$	4.57369e10	1.65034
$964$	0	0
$965$	4.31342e10	1.54517
$966$	0	0
$967$	−2.77875e9	−0.0988229	−0.0494114	−	0.998779i	$-0.515735\pi$
$967$	−2.77875e9	−0.0988229	−0.0494114	+	0.998779i	$0.515735\pi$
$968$	0	0
$969$	2.11867e9	0.0748050
$970$	0	0
$971$	−1.74377e10	−0.611254	−0.305627	−	0.952151i	$-0.598866\pi$
$971$	−1.74377e10	−0.611254	−0.305627	+	0.952151i	$0.598866\pi$
$972$	0	0
$973$	2.48969e10	0.866464
$974$	0	0
$975$	1.77360e10	0.612831
$976$	0	0
$977$	−2.69095e10	−0.923154	−0.461577	−	0.887100i	$-0.652716\pi$
$977$	−2.69095e10	−0.923154	−0.461577	+	0.887100i	$0.652716\pi$
$978$	0	0
$979$	1.70466e9	0.0580628
$980$	0	0
$981$	6.01652e9	0.203472
$982$	0	0
$983$	1.77111e10	0.594715	0.297357	−	0.954766i	$-0.403895\pi$
$983$	1.77111e10	0.594715	0.297357	+	0.954766i	$0.403895\pi$
$984$	0	0
$985$	−1.30997e10	−0.436752
$986$	0	0
$987$	−2.04694e9	−0.0677633
$988$	0	0
$989$	1.44068e10	0.473567
$990$	0	0
$991$	2.08537e10	0.680651	0.340326	−	0.940308i	$-0.389463\pi$
$991$	2.08537e10	0.680651	0.340326	+	0.940308i	$0.389463\pi$
$992$	0	0
$993$	1.52867e10	0.495440
$994$	0	0
$995$	−1.73713e9	−0.0559050
$996$	0	0
$997$	−4.13878e10	−1.32263	−0.661317	−	0.750107i	$-0.730002\pi$
$997$	−4.13878e10	−1.32263	−0.661317	+	0.750107i	$0.730002\pi$
$998$	0	0
$999$	−1.56852e9	−0.0497750

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	152.8.a.d.1.4	✓	9
4.3	odd	2		304.8.a.m.1.6		9

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
152.8.a.d.1.4	✓	9	1.1	even	1	trivial
304.8.a.m.1.6		9	4.3	odd	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:	\( N \)	\(=\)	\( 152 = 2^{3} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	152.a (trivial)

\(f(q)\)	\(=\)	\(q-12.9403 q^{3} -448.356 q^{5} -610.684 q^{7} -2019.55 q^{9} +O(q^{10})\)	\(q-12.9403 q^{3} -448.356 q^{5} -610.684 q^{7} -2019.55 q^{9} -797.626 q^{11} -11152.4 q^{13} +5801.85 q^{15} -23870.4 q^{17} +6859.00 q^{19} +7902.43 q^{21} -88520.0 q^{23} +122898. q^{25} +54433.9 q^{27} -170424. q^{29} +218460. q^{31} +10321.5 q^{33} +273804. q^{35} -28815.2 q^{37} +144315. q^{39} -144799. q^{41} -162752. q^{43} +905477. q^{45} -259026. q^{47} -450607. q^{49} +308889. q^{51} +1.31886e6 q^{53} +357621. q^{55} -88757.4 q^{57} -639322. q^{59} -2.19944e6 q^{61} +1.23331e6 q^{63} +5.00024e6 q^{65} -747422. q^{67} +1.14547e6 q^{69} +792365. q^{71} +6.08000e6 q^{73} -1.59034e6 q^{75} +487098. q^{77} -4.36104e6 q^{79} +3.71236e6 q^{81} +8.78352e6 q^{83} +1.07024e7 q^{85} +2.20533e6 q^{87} -2.13717e6 q^{89} +6.81058e6 q^{91} -2.82693e6 q^{93} -3.07528e6 q^{95} +6.51694e6 q^{97} +1.61085e6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 9 q + 56 q^{3} + 82 q^{5} + 464 q^{7} + 8513 q^{9}+O(q^{10}) \) 9 * q + 56 * q^3 + 82 * q^5 + 464 * q^7 + 8513 * q^9	\( 9 q + 56 q^{3} + 82 q^{5} + 464 q^{7} + 8513 q^{9} + 3360 q^{11} + 16522 q^{13} - 46608 q^{15} - 12902 q^{17} + 61731 q^{19} + 956 q^{21} - 76956 q^{23} + 188627 q^{25} + 417356 q^{27} + 63186 q^{29} + 314040 q^{31} + 89912 q^{33} + 429444 q^{35} + 389598 q^{37} - 350316 q^{39} + 424802 q^{41} + 1707520 q^{43} + 1515186 q^{45} + 968164 q^{47} + 3215825 q^{49} + 1731172 q^{51} + 1629130 q^{53} + 2745620 q^{55} + 384104 q^{57} + 4118768 q^{59} + 4051834 q^{61} + 6884020 q^{63} + 14144476 q^{65} + 5291736 q^{67} + 17194484 q^{69} + 7132000 q^{71} + 16065346 q^{73} - 2212376 q^{75} + 5883084 q^{77} + 799544 q^{79} + 26315489 q^{81} + 9460268 q^{83} - 2142320 q^{85} + 32521836 q^{87} + 29176746 q^{89} + 16210748 q^{91} + 46195392 q^{93} + 562438 q^{95} + 33616226 q^{97} + 21789816 q^{99}+O(q^{100}) \) 9 * q + 56 * q^3 + 82 * q^5 + 464 * q^7 + 8513 * q^9 + 3360 * q^11 + 16522 * q^13 - 46608 * q^15 - 12902 * q^17 + 61731 * q^19 + 956 * q^21 - 76956 * q^23 + 188627 * q^25 + 417356 * q^27 + 63186 * q^29 + 314040 * q^31 + 89912 * q^33 + 429444 * q^35 + 389598 * q^37 - 350316 * q^39 + 424802 * q^41 + 1707520 * q^43 + 1515186 * q^45 + 968164 * q^47 + 3215825 * q^49 + 1731172 * q^51 + 1629130 * q^53 + 2745620 * q^55 + 384104 * q^57 + 4118768 * q^59 + 4051834 * q^61 + 6884020 * q^63 + 14144476 * q^65 + 5291736 * q^67 + 17194484 * q^69 + 7132000 * q^71 + 16065346 * q^73 - 2212376 * q^75 + 5883084 * q^77 + 799544 * q^79 + 26315489 * q^81 + 9460268 * q^83 - 2142320 * q^85 + 32521836 * q^87 + 29176746 * q^89 + 16210748 * q^91 + 46195392 * q^93 + 562438 * q^95 + 33616226 * q^97 + 21789816 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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