LMFDB - Embedded newform 1472.4.a.bi.1.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1472,4,Mod(1,1472)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1472, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("1472.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1472 = 2^{6} \cdot 23 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1472.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:	$86.8508115285$
Analytic rank:	$1$
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} - 4x^{8} - 155x^{7} + 342x^{6} + 7139x^{5} - 8520x^{4} - 113229x^{3} + 12582x^{2} + 530388x + 301320 $ x^9 - 4x^8 - 155x^7 + 342x^6 + 7139x^5 - 8520x^4 - 113229x^3 + 12582x^2 + 530388x + 301320
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{10} $
Twist minimal:	no (minimal twist has level 736)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.8
Root			$5.85314$ of defining polynomial
Character	$\chi$	$=$	1472.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.85314 q^{3} +4.56877 q^{5} -17.3356 q^{7} -12.1533 q^{9} +69.1391 q^{11} -44.7989 q^{13} +17.6041 q^{15} -67.7296 q^{17} +93.8206 q^{19} -66.7965 q^{21} +23.0000 q^{23} -104.126 q^{25} -150.863 q^{27} +75.5707 q^{29} -129.384 q^{31} +266.402 q^{33} -79.2026 q^{35} +97.8395 q^{37} -172.616 q^{39} +293.173 q^{41} -162.107 q^{43} -55.5259 q^{45} -386.762 q^{47} -42.4759 q^{49} -260.971 q^{51} -660.868 q^{53} +315.881 q^{55} +361.504 q^{57} +585.664 q^{59} +32.2709 q^{61} +210.686 q^{63} -204.676 q^{65} +220.160 q^{67} +88.6221 q^{69} -610.582 q^{71} +482.525 q^{73} -401.213 q^{75} -1198.57 q^{77} -774.113 q^{79} -253.156 q^{81} -971.247 q^{83} -309.441 q^{85} +291.184 q^{87} +380.833 q^{89} +776.618 q^{91} -498.532 q^{93} +428.645 q^{95} -1721.54 q^{97} -840.271 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 9 q - 14 q^{3} - 30 q^{5} - 28 q^{7} + 103 q^{9} + 40 q^{11} - 76 q^{13} - 72 q^{15} + 166 q^{17} - 148 q^{19} - 96 q^{21} + 207 q^{23} + 339 q^{25} - 326 q^{27} - 252 q^{29} + 258 q^{31} + 460 q^{33} - 276 q^{35}+ \cdots - 2024 q^{99}+O(q^{100}) $ 9 * q - 14 * q^3 - 30 * q^5 - 28 * q^7 + 103 * q^9 + 40 * q^11 - 76 * q^13 - 72 * q^15 + 166 * q^17 - 148 * q^19 - 96 * q^21 + 207 * q^23 + 339 * q^25 - 326 * q^27 - 252 * q^29 + 258 * q^31 + 460 * q^33 - 276 * q^35 - 30 * q^37 - 22 * q^39 + 476 * q^41 - 200 * q^43 - 554 * q^45 + 290 * q^47 + 1277 * q^49 - 396 * q^51 - 702 * q^53 - 404 * q^55 + 2228 * q^57 - 336 * q^59 - 1518 * q^61 + 208 * q^63 + 1352 * q^65 - 664 * q^67 - 322 * q^69 - 874 * q^71 + 2940 * q^73 - 94 * q^75 - 2924 * q^77 - 1268 * q^79 + 3389 * q^81 - 452 * q^83 - 2424 * q^85 - 2022 * q^87 + 3534 * q^89 - 820 * q^91 - 2270 * q^93 - 2360 * q^95 + 2086 * q^97 - 2024 * q^99

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
$3$	3.85314	0.741536	0.370768	−	0.928725i	$-0.379094\pi$
$3$	3.85314	0.741536	0.370768	+	0.928725i	$0.379094\pi$
$4$	0	0
$5$	4.56877	0.408644	0.204322	−	0.978904i	$-0.434501\pi$
$5$	4.56877	0.408644	0.204322	+	0.978904i	$0.434501\pi$
$6$	0	0
$7$	−17.3356	−0.936036	−0.468018	−	0.883719i	$-0.655032\pi$
$7$	−17.3356	−0.936036	−0.468018	+	0.883719i	$0.655032\pi$
$8$	0	0
$9$	−12.1533	−0.450124
$10$	0	0
$11$	69.1391	1.89511	0.947556	−	0.319591i	$-0.103546\pi$
$11$	69.1391	1.89511	0.947556	+	0.319591i	$0.103546\pi$
$12$	0	0
$13$	−44.7989	−0.955768	−0.477884	−	0.878423i	$-0.658596\pi$
$13$	−44.7989	−0.955768	−0.477884	+	0.878423i	$0.658596\pi$
$14$	0	0
$15$	17.6041	0.303024
$16$	0	0
$17$	−67.7296	−0.966285	−0.483142	−	0.875542i	$-0.660505\pi$
$17$	−67.7296	−0.966285	−0.483142	+	0.875542i	$0.660505\pi$
$18$	0	0
$19$	93.8206	1.13284	0.566419	−	0.824117i	$-0.308328\pi$
$19$	93.8206	1.13284	0.566419	+	0.824117i	$0.308328\pi$
$20$	0	0
$21$	−66.7965	−0.694105
$22$	0	0
$23$	23.0000	0.208514
$23$	23.0000	0.208514
$24$	0	0
$25$	−104.126	−0.833010
$26$	0	0
$27$	−150.863	−1.07532
$28$	0	0
$29$	75.5707	0.483901	0.241951	−	0.970289i	$-0.422213\pi$
$29$	75.5707	0.483901	0.241951	+	0.970289i	$0.422213\pi$
$30$	0	0
$31$	−129.384	−0.749612	−0.374806	−	0.927103i	$-0.622291\pi$
$31$	−129.384	−0.749612	−0.374806	+	0.927103i	$0.622291\pi$
$32$	0	0
$33$	266.402	1.40529
$34$	0	0
$35$	−79.2026	−0.382505
$36$	0	0
$37$	97.8395	0.434722	0.217361	−	0.976091i	$-0.430255\pi$
$37$	97.8395	0.434722	0.217361	+	0.976091i	$0.430255\pi$
$38$	0	0
$39$	−172.616	−0.708737
$40$	0	0
$41$	293.173	1.11673	0.558365	−	0.829595i	$-0.311429\pi$
$41$	293.173	1.11673	0.558365	+	0.829595i	$0.311429\pi$
$42$	0	0
$43$	−162.107	−0.574909	−0.287455	−	0.957794i	$-0.592809\pi$
$43$	−162.107	−0.574909	−0.287455	+	0.957794i	$0.592809\pi$
$44$	0	0
$45$	−55.5259	−0.183940
$46$	0	0
$47$	−386.762	−1.20032	−0.600160	−	0.799880i	$-0.704897\pi$
$47$	−386.762	−1.20032	−0.600160	+	0.799880i	$0.704897\pi$
$48$	0	0
$49$	−42.4759	−0.123837
$50$	0	0
$51$	−260.971	−0.716535
$52$	0	0
$53$	−660.868	−1.71278	−0.856389	−	0.516331i	$-0.827298\pi$
$53$	−660.868	−1.71278	−0.856389	+	0.516331i	$0.827298\pi$
$54$	0	0
$55$	315.881	0.774425
$56$	0	0
$57$	361.504	0.840041
$58$	0	0
$59$	585.664	1.29232	0.646161	−	0.763201i	$-0.276373\pi$
$59$	585.664	1.29232	0.646161	+	0.763201i	$0.276373\pi$
$60$	0	0
$61$	32.2709	0.0677354	0.0338677	−	0.999426i	$-0.489218\pi$
$61$	32.2709	0.0677354	0.0338677	+	0.999426i	$0.489218\pi$
$62$	0	0
$63$	210.686	0.421332
$64$	0	0
$65$	−204.676	−0.390569
$66$	0	0
$67$	220.160	0.401445	0.200723	−	0.979648i	$-0.435671\pi$
$67$	220.160	0.401445	0.200723	+	0.979648i	$0.435671\pi$
$68$	0	0
$69$	88.6221	0.154621
$70$	0	0
$71$	−610.582	−1.02060	−0.510301	−	0.859996i	$-0.670466\pi$
$71$	−610.582	−1.02060	−0.510301	+	0.859996i	$0.670466\pi$
$72$	0	0
$73$	482.525	0.773634	0.386817	−	0.922157i	$-0.373575\pi$
$73$	482.525	0.773634	0.386817	+	0.922157i	$0.373575\pi$
$74$	0	0
$75$	−401.213	−0.617707
$76$	0	0
$77$	−1198.57	−1.77389
$78$	0	0
$79$	−774.113	−1.10246	−0.551231	−	0.834353i	$-0.685842\pi$
$79$	−774.113	−1.10246	−0.551231	+	0.834353i	$0.685842\pi$
$80$	0	0
$81$	−253.156	−0.347265
$82$	0	0
$83$	−971.247	−1.28444	−0.642218	−	0.766522i	$-0.721986\pi$
$83$	−971.247	−1.28444	−0.642218	+	0.766522i	$0.721986\pi$
$84$	0	0
$85$	−309.441	−0.394866
$86$	0	0
$87$	291.184	0.358830
$88$	0	0
$89$	380.833	0.453576	0.226788	−	0.973944i	$-0.427178\pi$
$89$	380.833	0.453576	0.226788	+	0.973944i	$0.427178\pi$
$90$	0	0
$91$	776.618	0.894633
$92$	0	0
$93$	−498.532	−0.555864
$94$	0	0
$95$	428.645	0.462927
$96$	0	0
$97$	−1721.54	−1.80202	−0.901008	−	0.433802i	$-0.857172\pi$
$97$	−1721.54	−1.80202	−0.901008	+	0.433802i	$0.857172\pi$
$98$	0	0
$99$	−840.271	−0.853035
$100$	0	0
$101$	−1337.08	−1.31727	−0.658634	−	0.752464i	$-0.728865\pi$
$101$	−1337.08	−1.31727	−0.658634	+	0.752464i	$0.728865\pi$
$102$	0	0
$103$	−11.5831	−0.0110807	−0.00554036	−	0.999985i	$-0.501764\pi$
$103$	−11.5831	−0.0110807	−0.00554036	+	0.999985i	$0.501764\pi$
$104$	0	0
$105$	−305.178	−0.283641
$106$	0	0
$107$	−2050.49	−1.85260	−0.926299	−	0.376790i	$-0.877028\pi$
$107$	−2050.49	−1.85260	−0.926299	+	0.376790i	$0.877028\pi$
$108$	0	0
$109$	−483.687	−0.425035	−0.212517	−	0.977157i	$-0.568166\pi$
$109$	−483.687	−0.425035	−0.212517	+	0.977157i	$0.568166\pi$
$110$	0	0
$111$	376.989	0.322362
$112$	0	0
$113$	1413.48	1.17671	0.588357	−	0.808602i	$-0.299775\pi$
$113$	1413.48	1.17671	0.588357	+	0.808602i	$0.299775\pi$
$114$	0	0
$115$	105.082	0.0852081
$116$	0	0
$117$	544.457	0.430214
$118$	0	0
$119$	1174.14	0.904477
$120$	0	0
$121$	3449.21	2.59145
$122$	0	0
$123$	1129.64	0.828096
$124$	0	0
$125$	−1046.83	−0.749048
$126$	0	0
$127$	−231.190	−0.161534	−0.0807669	−	0.996733i	$-0.525737\pi$
$127$	−231.190	−0.161534	−0.0807669	+	0.996733i	$0.525737\pi$
$128$	0	0
$129$	−624.620	−0.426316
$130$	0	0
$131$	164.235	0.109536	0.0547682	−	0.998499i	$-0.482558\pi$
$131$	164.235	0.109536	0.0547682	+	0.998499i	$0.482558\pi$
$132$	0	0
$133$	−1626.44	−1.06038
$134$	0	0
$135$	−689.260	−0.439422
$136$	0	0
$137$	−2151.89	−1.34196	−0.670978	−	0.741477i	$-0.734126\pi$
$137$	−2151.89	−1.34196	−0.670978	+	0.741477i	$0.734126\pi$
$138$	0	0
$139$	−601.833	−0.367243	−0.183622	−	0.982997i	$-0.558782\pi$
$139$	−601.833	−0.367243	−0.183622	+	0.982997i	$0.558782\pi$
$140$	0	0
$141$	−1490.25	−0.890082
$142$	0	0
$143$	−3097.36	−1.81129
$144$	0	0
$145$	345.266	0.197743
$146$	0	0
$147$	−163.666	−0.0918293
$148$	0	0
$149$	−1153.20	−0.634053	−0.317027	−	0.948417i	$-0.602684\pi$
$149$	−1153.20	−0.634053	−0.317027	+	0.948417i	$0.602684\pi$
$150$	0	0
$151$	−104.145	−0.0561270	−0.0280635	−	0.999606i	$-0.508934\pi$
$151$	−104.145	−0.0561270	−0.0280635	+	0.999606i	$0.508934\pi$
$152$	0	0
$153$	823.141	0.434948
$154$	0	0
$155$	−591.124	−0.306324
$156$	0	0
$157$	2177.29	1.10679	0.553396	−	0.832918i	$-0.313331\pi$
$157$	2177.29	1.10679	0.553396	+	0.832918i	$0.313331\pi$
$158$	0	0
$159$	−2546.42	−1.27009
$160$	0	0
$161$	−398.719	−0.195177
$162$	0	0
$163$	−1646.44	−0.791158	−0.395579	−	0.918432i	$-0.629456\pi$
$163$	−1646.44	−0.791158	−0.395579	+	0.918432i	$0.629456\pi$
$164$	0	0
$165$	1217.13	0.574264
$166$	0	0
$167$	3040.31	1.40878	0.704389	−	0.709814i	$-0.251221\pi$
$167$	3040.31	1.40878	0.704389	+	0.709814i	$0.251221\pi$
$168$	0	0
$169$	−190.056	−0.0865072
$170$	0	0
$171$	−1140.23	−0.509918
$172$	0	0
$173$	3578.59	1.57269	0.786343	−	0.617790i	$-0.211972\pi$
$173$	3578.59	1.57269	0.786343	+	0.617790i	$0.211972\pi$
$174$	0	0
$175$	1805.09	0.779728
$176$	0	0
$177$	2256.64	0.958304
$178$	0	0
$179$	1864.08	0.778367	0.389183	−	0.921160i	$-0.372757\pi$
$179$	1864.08	0.778367	0.389183	+	0.921160i	$0.372757\pi$
$180$	0	0
$181$	2320.20	0.952812	0.476406	−	0.879225i	$-0.341939\pi$
$181$	2320.20	0.952812	0.476406	+	0.879225i	$0.341939\pi$
$182$	0	0
$183$	124.344	0.0502283
$184$	0	0
$185$	447.007	0.177646
$186$	0	0
$187$	−4682.76	−1.83122
$188$	0	0
$189$	2615.31	1.00654
$190$	0	0
$191$	−3430.77	−1.29970	−0.649848	−	0.760064i	$-0.725167\pi$
$191$	−3430.77	−1.29970	−0.649848	+	0.760064i	$0.725167\pi$
$192$	0	0
$193$	3154.29	1.17643	0.588214	−	0.808705i	$-0.299831\pi$
$193$	3154.29	1.17643	0.588214	+	0.808705i	$0.299831\pi$
$194$	0	0
$195$	−788.645	−0.289621
$196$	0	0
$197$	−5061.75	−1.83063	−0.915316	−	0.402735i	$-0.868060\pi$
$197$	−5061.75	−1.83063	−0.915316	+	0.402735i	$0.868060\pi$
$198$	0	0
$199$	−4327.35	−1.54149	−0.770747	−	0.637141i	$-0.780117\pi$
$199$	−4327.35	−1.54149	−0.770747	+	0.637141i	$0.780117\pi$
$200$	0	0
$201$	848.307	0.297686
$202$	0	0
$203$	−1310.07	−0.452949
$204$	0	0
$205$	1339.44	0.456344
$206$	0	0
$207$	−279.527	−0.0938573
$208$	0	0
$209$	6486.67	2.14685
$210$	0	0
$211$	57.9032	0.0188920	0.00944602	−	0.999955i	$-0.496993\pi$
$211$	57.9032	0.0188920	0.00944602	+	0.999955i	$0.496993\pi$
$212$	0	0
$213$	−2352.65	−0.756813
$214$	0	0
$215$	−740.631	−0.234933
$216$	0	0
$217$	2242.94	0.701664
$218$	0	0
$219$	1859.23	0.573678
$220$	0	0
$221$	3034.21	0.923544
$222$	0	0
$223$	1399.57	0.420279	0.210140	−	0.977671i	$-0.432608\pi$
$223$	1399.57	0.420279	0.210140	+	0.977671i	$0.432608\pi$
$224$	0	0
$225$	1265.48	0.374958
$226$	0	0
$227$	3998.15	1.16902	0.584508	−	0.811388i	$-0.301287\pi$
$227$	3998.15	1.16902	0.584508	+	0.811388i	$0.301287\pi$
$228$	0	0
$229$	4795.02	1.38368	0.691842	−	0.722049i	$-0.256799\pi$
$229$	4795.02	1.38368	0.691842	+	0.722049i	$0.256799\pi$
$230$	0	0
$231$	−4618.25	−1.31541
$232$	0	0
$233$	−6055.21	−1.70253	−0.851266	−	0.524735i	$-0.824164\pi$
$233$	−6055.21	−1.70253	−0.851266	+	0.524735i	$0.824164\pi$
$234$	0	0
$235$	−1767.03	−0.490504
$236$	0	0
$237$	−2982.76	−0.817515
$238$	0	0
$239$	−3833.06	−1.03741	−0.518703	−	0.854954i	$-0.673585\pi$
$239$	−3833.06	−1.03741	−0.518703	+	0.854954i	$0.673585\pi$
$240$	0	0
$241$	935.932	0.250160	0.125080	−	0.992147i	$-0.460081\pi$
$241$	935.932	0.250160	0.125080	+	0.992147i	$0.460081\pi$
$242$	0	0
$243$	3097.86	0.817810
$244$	0	0
$245$	−194.063	−0.0506050
$246$	0	0
$247$	−4203.06	−1.08273
$248$	0	0
$249$	−3742.35	−0.952456
$250$	0	0
$251$	−2398.48	−0.603149	−0.301575	−	0.953443i	$-0.597512\pi$
$251$	−2398.48	−0.603149	−0.301575	+	0.953443i	$0.597512\pi$
$252$	0	0
$253$	1590.20	0.395158
$254$	0	0
$255$	−1192.32	−0.292808
$256$	0	0
$257$	−5905.80	−1.43344	−0.716719	−	0.697362i	$-0.754357\pi$
$257$	−5905.80	−1.43344	−0.716719	+	0.697362i	$0.754357\pi$
$258$	0	0
$259$	−1696.11	−0.406916
$260$	0	0
$261$	−918.437	−0.217815
$262$	0	0
$263$	−4247.46	−0.995853	−0.497927	−	0.867219i	$-0.665905\pi$
$263$	−4247.46	−0.995853	−0.497927	+	0.867219i	$0.665905\pi$
$264$	0	0
$265$	−3019.36	−0.699916
$266$	0	0
$267$	1467.40	0.336343
$268$	0	0
$269$	4464.06	1.01182	0.505908	−	0.862587i	$-0.331158\pi$
$269$	4464.06	1.01182	0.505908	+	0.862587i	$0.331158\pi$
$270$	0	0
$271$	5218.04	1.16964	0.584822	−	0.811162i	$-0.301164\pi$
$271$	5218.04	1.16964	0.584822	+	0.811162i	$0.301164\pi$
$272$	0	0
$273$	2992.41	0.663403
$274$	0	0
$275$	−7199.20	−1.57865
$276$	0	0
$277$	−501.732	−0.108831	−0.0544155	−	0.998518i	$-0.517330\pi$
$277$	−501.732	−0.108831	−0.0544155	+	0.998518i	$0.517330\pi$
$278$	0	0
$279$	1572.44	0.337418
$280$	0	0
$281$	−4957.48	−1.05245	−0.526225	−	0.850345i	$-0.676393\pi$
$281$	−4957.48	−1.05245	−0.526225	+	0.850345i	$0.676393\pi$
$282$	0	0
$283$	2390.56	0.502135	0.251067	−	0.967970i	$-0.419218\pi$
$283$	2390.56	0.502135	0.251067	+	0.967970i	$0.419218\pi$
$284$	0	0
$285$	1651.63	0.343277
$286$	0	0
$287$	−5082.34	−1.04530
$288$	0	0
$289$	−325.701	−0.0662938
$290$	0	0
$291$	−6633.32	−1.33626
$292$	0	0
$293$	−6814.51	−1.35873	−0.679365	−	0.733801i	$-0.737745\pi$
$293$	−6814.51	−1.35873	−0.679365	+	0.733801i	$0.737745\pi$
$294$	0	0
$295$	2675.77	0.528099
$296$	0	0
$297$	−10430.5	−2.03785
$298$	0	0
$299$	−1030.38	−0.199291
$300$	0	0
$301$	2810.23	0.538136
$302$	0	0
$303$	−5151.94	−0.976802
$304$	0	0
$305$	147.438	0.0276796
$306$	0	0
$307$	−6029.97	−1.12101	−0.560503	−	0.828153i	$-0.689392\pi$
$307$	−6029.97	−1.12101	−0.560503	+	0.828153i	$0.689392\pi$
$308$	0	0
$309$	−44.6312	−0.00821676
$310$	0	0
$311$	6221.73	1.13441	0.567206	−	0.823576i	$-0.308024\pi$
$311$	6221.73	1.13441	0.567206	+	0.823576i	$0.308024\pi$
$312$	0	0
$313$	−6078.62	−1.09771	−0.548856	−	0.835917i	$-0.684936\pi$
$313$	−6078.62	−1.09771	−0.548856	+	0.835917i	$0.684936\pi$
$314$	0	0
$315$	962.576	0.172175
$316$	0	0
$317$	2804.08	0.496822	0.248411	−	0.968655i	$-0.420092\pi$
$317$	2804.08	0.496822	0.248411	+	0.968655i	$0.420092\pi$
$318$	0	0
$319$	5224.89	0.917046
$320$	0	0
$321$	−7900.80	−1.37377
$322$	0	0
$323$	−6354.43	−1.09464
$324$	0	0
$325$	4664.75	0.796165
$326$	0	0
$327$	−1863.71	−0.315179
$328$	0	0
$329$	6704.77	1.12354
$330$	0	0
$331$	−6477.58	−1.07565	−0.537825	−	0.843056i	$-0.680754\pi$
$331$	−6477.58	−1.07565	−0.537825	+	0.843056i	$0.680754\pi$
$332$	0	0
$333$	−1189.08	−0.195679
$334$	0	0
$335$	1005.86	0.164048
$336$	0	0
$337$	3007.54	0.486146	0.243073	−	0.970008i	$-0.421845\pi$
$337$	3007.54	0.486146	0.243073	+	0.970008i	$0.421845\pi$
$338$	0	0
$339$	5446.31	0.872576
$340$	0	0
$341$	−8945.46	−1.42060
$342$	0	0
$343$	6682.47	1.05195
$344$	0	0
$345$	404.895	0.0631849
$346$	0	0
$347$	−7350.65	−1.13719	−0.568593	−	0.822619i	$-0.692512\pi$
$347$	−7350.65	−1.13719	−0.568593	+	0.822619i	$0.692512\pi$
$348$	0	0
$349$	7719.57	1.18401	0.592004	−	0.805935i	$-0.298337\pi$
$349$	7719.57	1.18401	0.592004	+	0.805935i	$0.298337\pi$
$350$	0	0
$351$	6758.51	1.02776
$352$	0	0
$353$	1123.52	0.169402	0.0847009	−	0.996406i	$-0.473007\pi$
$353$	1123.52	0.169402	0.0847009	+	0.996406i	$0.473007\pi$
$354$	0	0
$355$	−2789.61	−0.417062
$356$	0	0
$357$	4524.10	0.670703
$358$	0	0
$359$	1813.36	0.266590	0.133295	−	0.991076i	$-0.457444\pi$
$359$	1813.36	0.266590	0.133295	+	0.991076i	$0.457444\pi$
$360$	0	0
$361$	1943.31	0.283323
$362$	0	0
$363$	13290.3	1.92165
$364$	0	0
$365$	2204.55	0.316141
$366$	0	0
$367$	−7864.31	−1.11857	−0.559283	−	0.828977i	$-0.688923\pi$
$367$	−7864.31	−1.11857	−0.559283	+	0.828977i	$0.688923\pi$
$368$	0	0
$369$	−3563.03	−0.502667
$370$	0	0
$371$	11456.6	1.60322
$372$	0	0
$373$	8820.93	1.22448	0.612239	−	0.790673i	$-0.290269\pi$
$373$	8820.93	1.22448	0.612239	+	0.790673i	$0.290269\pi$
$374$	0	0
$375$	−4033.56	−0.555446
$376$	0	0
$377$	−3385.49	−0.462497
$378$	0	0
$379$	−765.994	−0.103817	−0.0519083	−	0.998652i	$-0.516530\pi$
$379$	−765.994	−0.103817	−0.0519083	+	0.998652i	$0.516530\pi$
$380$	0	0
$381$	−890.806	−0.119783
$382$	0	0
$383$	−4943.43	−0.659524	−0.329762	−	0.944064i	$-0.606968\pi$
$383$	−4943.43	−0.659524	−0.329762	+	0.944064i	$0.606968\pi$
$384$	0	0
$385$	−5476.00	−0.724890
$386$	0	0
$387$	1970.14	0.258780
$388$	0	0
$389$	11427.9	1.48951	0.744753	−	0.667341i	$-0.232567\pi$
$389$	11427.9	1.48951	0.744753	+	0.667341i	$0.232567\pi$
$390$	0	0
$391$	−1557.78	−0.201484
$392$	0	0
$393$	632.819	0.0812252
$394$	0	0
$395$	−3536.75	−0.450514
$396$	0	0
$397$	−5848.52	−0.739368	−0.369684	−	0.929158i	$-0.620534\pi$
$397$	−5848.52	−0.739368	−0.369684	+	0.929158i	$0.620534\pi$
$398$	0	0
$399$	−6266.89	−0.786309
$400$	0	0
$401$	1397.62	0.174049	0.0870247	−	0.996206i	$-0.472264\pi$
$401$	1397.62	0.174049	0.0870247	+	0.996206i	$0.472264\pi$
$402$	0	0
$403$	5796.24	0.716455
$404$	0	0
$405$	−1156.61	−0.141907
$406$	0	0
$407$	6764.54	0.823847
$408$	0	0
$409$	107.129	0.0129515	0.00647577	−	0.999979i	$-0.497939\pi$
$409$	107.129	0.0129515	0.00647577	+	0.999979i	$0.497939\pi$
$410$	0	0
$411$	−8291.51	−0.995109
$412$	0	0
$413$	−10152.9	−1.20966
$414$	0	0
$415$	−4437.41	−0.524877
$416$	0	0
$417$	−2318.94	−0.272324
$418$	0	0
$419$	−6219.42	−0.725152	−0.362576	−	0.931954i	$-0.618103\pi$
$419$	−6219.42	−0.725152	−0.362576	+	0.931954i	$0.618103\pi$
$420$	0	0
$421$	15866.6	1.83679	0.918395	−	0.395665i	$-0.129486\pi$
$421$	15866.6	1.83679	0.918395	+	0.395665i	$0.129486\pi$
$422$	0	0
$423$	4700.46	0.540293
$424$	0	0
$425$	7052.43	0.804925
$426$	0	0
$427$	−559.436	−0.0634028
$428$	0	0
$429$	−11934.5	−1.34313
$430$	0	0
$431$	9482.26	1.05973	0.529866	−	0.848081i	$-0.322242\pi$
$431$	9482.26	1.05973	0.529866	+	0.848081i	$0.322242\pi$
$432$	0	0
$433$	−547.564	−0.0607719	−0.0303860	−	0.999538i	$-0.509674\pi$
$433$	−547.564	−0.0607719	−0.0303860	+	0.999538i	$0.509674\pi$
$434$	0	0
$435$	1330.36	0.146634
$436$	0	0
$437$	2157.87	0.236213
$438$	0	0
$439$	−3952.22	−0.429679	−0.214840	−	0.976649i	$-0.568923\pi$
$439$	−3952.22	−0.429679	−0.214840	+	0.976649i	$0.568923\pi$
$440$	0	0
$441$	516.225	0.0557418
$442$	0	0
$443$	2920.17	0.313186	0.156593	−	0.987663i	$-0.449949\pi$
$443$	2920.17	0.313186	0.156593	+	0.987663i	$0.449949\pi$
$444$	0	0
$445$	1739.94	0.185351
$446$	0	0
$447$	−4443.44	−0.470174
$448$	0	0
$449$	13672.2	1.43704	0.718521	−	0.695506i	$-0.244820\pi$
$449$	13672.2	1.43704	0.718521	+	0.695506i	$0.244820\pi$
$450$	0	0
$451$	20269.7	2.11633
$452$	0	0
$453$	−401.284	−0.0416202
$454$	0	0
$455$	3548.19	0.365586
$456$	0	0
$457$	−2770.01	−0.283535	−0.141767	−	0.989900i	$-0.545279\pi$
$457$	−2770.01	−0.283535	−0.141767	+	0.989900i	$0.545279\pi$
$458$	0	0
$459$	10217.9	1.03906
$460$	0	0
$461$	−9689.50	−0.978927	−0.489463	−	0.872024i	$-0.662807\pi$
$461$	−9689.50	−0.978927	−0.489463	+	0.872024i	$0.662807\pi$
$462$	0	0
$463$	−9354.03	−0.938917	−0.469458	−	0.882955i	$-0.655551\pi$
$463$	−9354.03	−0.938917	−0.469458	+	0.882955i	$0.655551\pi$
$464$	0	0
$465$	−2277.68	−0.227150
$466$	0	0
$467$	10489.7	1.03941	0.519707	−	0.854345i	$-0.326041\pi$
$467$	10489.7	1.03941	0.519707	+	0.854345i	$0.326041\pi$
$468$	0	0
$469$	−3816.62	−0.375767
$470$	0	0
$471$	8389.39	0.820727
$472$	0	0
$473$	−11207.9	−1.08952
$474$	0	0
$475$	−9769.20	−0.943666
$476$	0	0
$477$	8031.76	0.770962
$478$	0	0
$479$	2252.33	0.214847	0.107424	−	0.994213i	$-0.465740\pi$
$479$	2252.33	0.214847	0.107424	+	0.994213i	$0.465740\pi$
$480$	0	0
$481$	−4383.11	−0.415494
$482$	0	0
$483$	−1536.32	−0.144731
$484$	0	0
$485$	−7865.31	−0.736383
$486$	0	0
$487$	17154.6	1.59620	0.798100	−	0.602525i	$-0.205839\pi$
$487$	17154.6	1.59620	0.798100	+	0.602525i	$0.205839\pi$
$488$	0	0
$489$	−6343.94	−0.586672
$490$	0	0
$491$	−17525.9	−1.61086	−0.805429	−	0.592692i	$-0.798065\pi$
$491$	−17525.9	−1.61086	−0.805429	+	0.592692i	$0.798065\pi$
$492$	0	0
$493$	−5118.38	−0.467586
$494$	0	0
$495$	−3839.01	−0.348587
$496$	0	0
$497$	10584.8	0.955320
$498$	0	0
$499$	2981.27	0.267455	0.133727	−	0.991018i	$-0.457305\pi$
$499$	2981.27	0.267455	0.133727	+	0.991018i	$0.457305\pi$
$500$	0	0
$501$	11714.7	1.04466
$502$	0	0
$503$	−2766.37	−0.245222	−0.122611	−	0.992455i	$-0.539127\pi$
$503$	−2766.37	−0.245222	−0.122611	+	0.992455i	$0.539127\pi$
$504$	0	0
$505$	−6108.80	−0.538293
$506$	0	0
$507$	−732.313	−0.0641482
$508$	0	0
$509$	7139.63	0.621726	0.310863	−	0.950455i	$-0.399382\pi$
$509$	7139.63	0.621726	0.310863	+	0.950455i	$0.399382\pi$
$510$	0	0
$511$	−8364.87	−0.724149
$512$	0	0
$513$	−14154.1	−1.21816
$514$	0	0
$515$	−52.9205	−0.00452807
$516$	0	0
$517$	−26740.4	−2.27474
$518$	0	0
$519$	13788.8	1.16620
$520$	0	0
$521$	5449.24	0.458226	0.229113	−	0.973400i	$-0.426418\pi$
$521$	5449.24	0.458226	0.229113	+	0.973400i	$0.426418\pi$
$522$	0	0
$523$	−379.720	−0.0317476	−0.0158738	−	0.999874i	$-0.505053\pi$
$523$	−379.720	−0.0317476	−0.0158738	+	0.999874i	$0.505053\pi$
$524$	0	0
$525$	6955.28	0.578196
$526$	0	0
$527$	8763.09	0.724338
$528$	0	0
$529$	529.000	0.0434783
$530$	0	0
$531$	−7117.78	−0.581705
$532$	0	0
$533$	−13133.8	−1.06733
$534$	0	0
$535$	−9368.21	−0.757052
$536$	0	0
$537$	7182.54	0.577187
$538$	0	0
$539$	−2936.75	−0.234684
$540$	0	0
$541$	−18938.7	−1.50506	−0.752529	−	0.658559i	$-0.771166\pi$
$541$	−18938.7	−1.50506	−0.752529	+	0.658559i	$0.771166\pi$
$542$	0	0
$543$	8940.04	0.706545
$544$	0	0
$545$	−2209.86	−0.173688
$546$	0	0
$547$	18261.2	1.42741	0.713706	−	0.700446i	$-0.247015\pi$
$547$	18261.2	1.42741	0.713706	+	0.700446i	$0.247015\pi$
$548$	0	0
$549$	−392.199	−0.0304893
$550$	0	0
$551$	7090.09	0.548182
$552$	0	0
$553$	13419.7	1.03194
$554$	0	0
$555$	1722.38	0.131731
$556$	0	0
$557$	4344.87	0.330517	0.165259	−	0.986250i	$-0.447154\pi$
$557$	4344.87	0.330517	0.165259	+	0.986250i	$0.447154\pi$
$558$	0	0
$559$	7262.22	0.549480
$560$	0	0
$561$	−18043.3	−1.35791
$562$	0	0
$563$	420.144	0.0314511	0.0157255	−	0.999876i	$-0.494994\pi$
$563$	420.144	0.0314511	0.0157255	+	0.999876i	$0.494994\pi$
$564$	0	0
$565$	6457.85	0.480856
$566$	0	0
$567$	4388.62	0.325052
$568$	0	0
$569$	21741.5	1.60185	0.800924	−	0.598766i	$-0.204342\pi$
$569$	21741.5	1.60185	0.800924	+	0.598766i	$0.204342\pi$
$570$	0	0
$571$	20511.9	1.50332	0.751659	−	0.659552i	$-0.229254\pi$
$571$	20511.9	1.50332	0.751659	+	0.659552i	$0.229254\pi$
$572$	0	0
$573$	−13219.2	−0.963772
$574$	0	0
$575$	−2394.90	−0.173695
$576$	0	0
$577$	−12408.5	−0.895273	−0.447636	−	0.894216i	$-0.647734\pi$
$577$	−12408.5	−0.895273	−0.447636	+	0.894216i	$0.647734\pi$
$578$	0	0
$579$	12153.9	0.872365
$580$	0	0
$581$	16837.2	1.20228
$582$	0	0
$583$	−45691.8	−3.24591
$584$	0	0
$585$	2487.50	0.175804
$586$	0	0
$587$	972.118	0.0683537	0.0341768	−	0.999416i	$-0.489119\pi$
$587$	972.118	0.0683537	0.0341768	+	0.999416i	$0.489119\pi$
$588$	0	0
$589$	−12138.8	−0.849189
$590$	0	0
$591$	−19503.6	−1.35748
$592$	0	0
$593$	27008.9	1.87036	0.935178	−	0.354177i	$-0.115239\pi$
$593$	27008.9	1.87036	0.935178	+	0.354177i	$0.115239\pi$
$594$	0	0
$595$	5364.36	0.369609
$596$	0	0
$597$	−16673.9	−1.14307
$598$	0	0
$599$	18928.2	1.29113	0.645563	−	0.763707i	$-0.276623\pi$
$599$	18928.2	1.29113	0.645563	+	0.763707i	$0.276623\pi$
$600$	0	0
$601$	−12364.7	−0.839212	−0.419606	−	0.907706i	$-0.637832\pi$
$601$	−12364.7	−0.839212	−0.419606	+	0.907706i	$0.637832\pi$
$602$	0	0
$603$	−2675.68	−0.180700
$604$	0	0
$605$	15758.7	1.05898
$606$	0	0
$607$	−220.686	−0.0147568	−0.00737838	−	0.999973i	$-0.502349\pi$
$607$	−220.686	−0.0147568	−0.00737838	+	0.999973i	$0.502349\pi$
$608$	0	0
$609$	−5047.86	−0.335878
$610$	0	0
$611$	17326.5	1.14723
$612$	0	0
$613$	16105.1	1.06114	0.530569	−	0.847642i	$-0.321978\pi$
$613$	16105.1	1.06114	0.530569	+	0.847642i	$0.321978\pi$
$614$	0	0
$615$	5161.05	0.338396
$616$	0	0
$617$	2563.86	0.167288	0.0836442	−	0.996496i	$-0.473344\pi$
$617$	2563.86	0.167288	0.0836442	+	0.996496i	$0.473344\pi$
$618$	0	0
$619$	−25219.9	−1.63760	−0.818798	−	0.574081i	$-0.805359\pi$
$619$	−25219.9	−1.63760	−0.818798	+	0.574081i	$0.805359\pi$
$620$	0	0
$621$	−3469.85	−0.224220
$622$	0	0
$623$	−6601.98	−0.424563
$624$	0	0
$625$	8233.07	0.526917
$626$	0	0
$627$	24994.0	1.59197
$628$	0	0
$629$	−6626.63	−0.420065
$630$	0	0
$631$	16090.7	1.01515	0.507575	−	0.861607i	$-0.330542\pi$
$631$	16090.7	1.01515	0.507575	+	0.861607i	$0.330542\pi$
$632$	0	0
$633$	223.109	0.0140091
$634$	0	0
$635$	−1056.25	−0.0660098
$636$	0	0
$637$	1902.88	0.118359
$638$	0	0
$639$	7420.61	0.459397
$640$	0	0
$641$	22241.7	1.37051	0.685254	−	0.728304i	$-0.259691\pi$
$641$	22241.7	1.37051	0.685254	+	0.728304i	$0.259691\pi$
$642$	0	0
$643$	−19169.5	−1.17569	−0.587847	−	0.808972i	$-0.700024\pi$
$643$	−19169.5	−1.17569	−0.587847	+	0.808972i	$0.700024\pi$
$644$	0	0
$645$	−2853.75	−0.174211
$646$	0	0
$647$	7574.09	0.460229	0.230114	−	0.973164i	$-0.426090\pi$
$647$	7574.09	0.460229	0.230114	+	0.973164i	$0.426090\pi$
$648$	0	0
$649$	40492.3	2.44909
$650$	0	0
$651$	8642.37	0.520309
$652$	0	0
$653$	−3658.54	−0.219249	−0.109625	−	0.993973i	$-0.534965\pi$
$653$	−3658.54	−0.219249	−0.109625	+	0.993973i	$0.534965\pi$
$654$	0	0
$655$	750.352	0.0447613
$656$	0	0
$657$	−5864.29	−0.348231
$658$	0	0
$659$	21866.2	1.29254	0.646272	−	0.763107i	$-0.276327\pi$
$659$	21866.2	1.29254	0.646272	+	0.763107i	$0.276327\pi$
$660$	0	0
$661$	−20092.0	−1.18228	−0.591141	−	0.806568i	$-0.701322\pi$
$661$	−20092.0	−1.18228	−0.591141	+	0.806568i	$0.701322\pi$
$662$	0	0
$663$	11691.2	0.684842
$664$	0	0
$665$	−7430.84	−0.433317
$666$	0	0
$667$	1738.13	0.100900
$668$	0	0
$669$	5392.74	0.311652
$670$	0	0
$671$	2231.18	0.128366
$672$	0	0
$673$	5414.18	0.310106	0.155053	−	0.987906i	$-0.450445\pi$
$673$	5414.18	0.310106	0.155053	+	0.987906i	$0.450445\pi$
$674$	0	0
$675$	15708.8	0.895752
$676$	0	0
$677$	−17530.0	−0.995173	−0.497586	−	0.867415i	$-0.665780\pi$
$677$	−17530.0	−0.995173	−0.497586	+	0.867415i	$0.665780\pi$
$678$	0	0
$679$	29843.9	1.68675
$680$	0	0
$681$	15405.4	0.866868
$682$	0	0
$683$	−22600.3	−1.26614	−0.633072	−	0.774093i	$-0.718206\pi$
$683$	−22600.3	−1.26614	−0.633072	+	0.774093i	$0.718206\pi$
$684$	0	0
$685$	−9831.48	−0.548382
$686$	0	0
$687$	18475.9	1.02605
$688$	0	0
$689$	29606.2	1.63702
$690$	0	0
$691$	−8476.98	−0.466685	−0.233343	−	0.972395i	$-0.574966\pi$
$691$	−8476.98	−0.466685	−0.233343	+	0.972395i	$0.574966\pi$
$692$	0	0
$693$	14566.6	0.798471
$694$	0	0
$695$	−2749.64	−0.150072
$696$	0	0
$697$	−19856.5	−1.07908
$698$	0	0
$699$	−23331.5	−1.26249
$700$	0	0
$701$	14816.7	0.798316	0.399158	−	0.916882i	$-0.369302\pi$
$701$	14816.7	0.798316	0.399158	+	0.916882i	$0.369302\pi$
$702$	0	0
$703$	9179.37	0.492470
$704$	0	0
$705$	−6808.61	−0.363726
$706$	0	0
$707$	23179.1	1.23301
$708$	0	0
$709$	5860.84	0.310449	0.155225	−	0.987879i	$-0.450390\pi$
$709$	5860.84	0.310449	0.155225	+	0.987879i	$0.450390\pi$
$710$	0	0
$711$	9408.06	0.496244
$712$	0	0
$713$	−2975.82	−0.156305
$714$	0	0
$715$	−14151.1	−0.740171
$716$	0	0
$717$	−14769.3	−0.769274
$718$	0	0
$719$	24195.2	1.25498	0.627489	−	0.778625i	$-0.284083\pi$
$719$	24195.2	1.25498	0.627489	+	0.778625i	$0.284083\pi$
$720$	0	0
$721$	200.800	0.0103720
$722$	0	0
$723$	3606.27	0.185503
$724$	0	0
$725$	−7868.90	−0.403095
$726$	0	0
$727$	−11113.9	−0.566974	−0.283487	−	0.958976i	$-0.591491\pi$
$727$	−11113.9	−0.566974	−0.283487	+	0.958976i	$0.591491\pi$
$728$	0	0
$729$	18771.7	0.953701
$730$	0	0
$731$	10979.4	0.555526
$732$	0	0
$733$	14272.4	0.719184	0.359592	−	0.933110i	$-0.382916\pi$
$733$	14272.4	0.719184	0.359592	+	0.933110i	$0.382916\pi$
$734$	0	0
$735$	−747.751	−0.0375255
$736$	0	0
$737$	15221.7	0.760784
$738$	0	0
$739$	−19737.9	−0.982505	−0.491253	−	0.871017i	$-0.663461\pi$
$739$	−19737.9	−0.982505	−0.491253	+	0.871017i	$0.663461\pi$
$740$	0	0
$741$	−16195.0	−0.802884
$742$	0	0
$743$	8473.54	0.418390	0.209195	−	0.977874i	$-0.432916\pi$
$743$	8473.54	0.418390	0.209195	+	0.977874i	$0.432916\pi$
$744$	0	0
$745$	−5268.72	−0.259102
$746$	0	0
$747$	11803.9	0.578155
$748$	0	0
$749$	35546.5	1.73410
$750$	0	0
$751$	4631.81	0.225056	0.112528	−	0.993649i	$-0.464105\pi$
$751$	4631.81	0.225056	0.112528	+	0.993649i	$0.464105\pi$
$752$	0	0
$753$	−9241.66	−0.447257
$754$	0	0
$755$	−475.814	−0.0229359
$756$	0	0
$757$	26389.6	1.26703	0.633517	−	0.773729i	$-0.281611\pi$
$757$	26389.6	1.26703	0.633517	+	0.773729i	$0.281611\pi$
$758$	0	0
$759$	6127.25	0.293024
$760$	0	0
$761$	12842.0	0.611726	0.305863	−	0.952076i	$-0.401055\pi$
$761$	12842.0	0.611726	0.305863	+	0.952076i	$0.401055\pi$
$762$	0	0
$763$	8385.02	0.397848
$764$	0	0
$765$	3760.75	0.177739
$766$	0	0
$767$	−26237.1	−1.23516
$768$	0	0
$769$	2732.93	0.128156	0.0640779	−	0.997945i	$-0.479589\pi$
$769$	2732.93	0.128156	0.0640779	+	0.997945i	$0.479589\pi$
$770$	0	0
$771$	−22755.9	−1.06295
$772$	0	0
$773$	−27456.8	−1.27756	−0.638780	−	0.769390i	$-0.720560\pi$
$773$	−27456.8	−1.27756	−0.638780	+	0.769390i	$0.720560\pi$
$774$	0	0
$775$	13472.2	0.624434
$776$	0	0
$777$	−6535.34	−0.301743
$778$	0	0
$779$	27505.7	1.26507
$780$	0	0
$781$	−42215.1	−1.93415
$782$	0	0
$783$	−11400.8	−0.520348
$784$	0	0
$785$	9947.54	0.452284
$786$	0	0
$787$	13435.4	0.608538	0.304269	−	0.952586i	$-0.401588\pi$
$787$	13435.4	0.608538	0.304269	+	0.952586i	$0.401588\pi$
$788$	0	0
$789$	−16366.0	−0.738461
$790$	0	0
$791$	−24503.5	−1.10145
$792$	0	0
$793$	−1445.70	−0.0647393
$794$	0	0
$795$	−11634.0	−0.519013
$796$	0	0
$797$	3131.89	0.139194	0.0695968	−	0.997575i	$-0.477829\pi$
$797$	3131.89	0.139194	0.0695968	+	0.997575i	$0.477829\pi$
$798$	0	0
$799$	26195.3	1.15985
$800$	0	0
$801$	−4628.40	−0.204165
$802$	0	0
$803$	33361.3	1.46612
$804$	0	0
$805$	−1821.66	−0.0797578
$806$	0	0
$807$	17200.6	0.750298
$808$	0	0
$809$	20095.9	0.873343	0.436671	−	0.899621i	$-0.356157\pi$
$809$	20095.9	0.873343	0.436671	+	0.899621i	$0.356157\pi$
$810$	0	0
$811$	−6769.96	−0.293126	−0.146563	−	0.989201i	$-0.546821\pi$
$811$	−6769.96	−0.293126	−0.146563	+	0.989201i	$0.546821\pi$
$812$	0	0
$813$	20105.8	0.867333
$814$	0	0
$815$	−7522.19	−0.323302
$816$	0	0
$817$	−15209.0	−0.651279
$818$	0	0
$819$	−9438.50	−0.402696
$820$	0	0
$821$	−4811.60	−0.204538	−0.102269	−	0.994757i	$-0.532610\pi$
$821$	−4811.60	−0.204538	−0.102269	+	0.994757i	$0.532610\pi$
$822$	0	0
$823$	42009.3	1.77928	0.889642	−	0.456659i	$-0.150954\pi$
$823$	42009.3	1.77928	0.889642	+	0.456659i	$0.150954\pi$
$824$	0	0
$825$	−27739.5	−1.17062
$826$	0	0
$827$	17049.0	0.716869	0.358434	−	0.933555i	$-0.383311\pi$
$827$	17049.0	0.716869	0.358434	+	0.933555i	$0.383311\pi$
$828$	0	0
$829$	30850.7	1.29251	0.646254	−	0.763122i	$-0.276335\pi$
$829$	30850.7	1.29251	0.646254	+	0.763122i	$0.276335\pi$
$830$	0	0
$831$	−1933.24	−0.0807021
$832$	0	0
$833$	2876.88	0.119661
$834$	0	0
$835$	13890.5	0.575689
$836$	0	0
$837$	19519.2	0.806072
$838$	0	0
$839$	10339.1	0.425441	0.212721	−	0.977113i	$-0.431768\pi$
$839$	10339.1	0.425441	0.212721	+	0.977113i	$0.431768\pi$
$840$	0	0
$841$	−18678.1	−0.765840
$842$	0	0
$843$	−19101.8	−0.780430
$844$	0	0
$845$	−868.325	−0.0353506
$846$	0	0
$847$	−59794.3	−2.42569
$848$	0	0
$849$	9211.15	0.372351
$850$	0	0
$851$	2250.31	0.0906458
$852$	0	0
$853$	15576.7	0.625248	0.312624	−	0.949877i	$-0.398792\pi$
$853$	15576.7	0.625248	0.312624	+	0.949877i	$0.398792\pi$
$854$	0	0
$855$	−5209.47	−0.208375
$856$	0	0
$857$	4703.84	0.187492	0.0937458	−	0.995596i	$-0.470116\pi$
$857$	4703.84	0.187492	0.0937458	+	0.995596i	$0.470116\pi$
$858$	0	0
$859$	−14624.9	−0.580903	−0.290451	−	0.956890i	$-0.593805\pi$
$859$	−14624.9	−0.580903	−0.290451	+	0.956890i	$0.593805\pi$
$860$	0	0
$861$	−19582.9	−0.775127
$862$	0	0
$863$	4379.13	0.172731	0.0863657	−	0.996264i	$-0.472475\pi$
$863$	4379.13	0.172731	0.0863657	+	0.996264i	$0.472475\pi$
$864$	0	0
$865$	16349.7	0.642668
$866$	0	0
$867$	−1254.97	−0.0491593
$868$	0	0
$869$	−53521.4	−2.08929
$870$	0	0
$871$	−9862.94	−0.383689
$872$	0	0
$873$	20922.4	0.811131
$874$	0	0
$875$	18147.4	0.701136
$876$	0	0
$877$	36538.9	1.40688	0.703439	−	0.710756i	$-0.251647\pi$
$877$	36538.9	1.40688	0.703439	+	0.710756i	$0.251647\pi$
$878$	0	0
$879$	−26257.2	−1.00755
$880$	0	0
$881$	−45420.3	−1.73695	−0.868473	−	0.495736i	$-0.834898\pi$
$881$	−45420.3	−1.73695	−0.868473	+	0.495736i	$0.834898\pi$
$882$	0	0
$883$	182.167	0.00694272	0.00347136	−	0.999994i	$-0.498895\pi$
$883$	182.167	0.00694272	0.00347136	+	0.999994i	$0.498895\pi$
$884$	0	0
$885$	10310.1	0.391605
$886$	0	0
$887$	5859.88	0.221821	0.110911	−	0.993830i	$-0.464623\pi$
$887$	5859.88	0.221821	0.110911	+	0.993830i	$0.464623\pi$
$888$	0	0
$889$	4007.82	0.151201
$890$	0	0
$891$	−17503.0	−0.658105
$892$	0	0
$893$	−36286.3	−1.35977
$894$	0	0
$895$	8516.55	0.318075
$896$	0	0
$897$	−3970.18	−0.147782
$898$	0	0
$899$	−9777.61	−0.362738
$900$	0	0
$901$	44760.4	1.65503
$902$	0	0
$903$	10828.2	0.399047
$904$	0	0
$905$	10600.5	0.389361
$906$	0	0
$907$	22448.3	0.821810	0.410905	−	0.911678i	$-0.365213\pi$
$907$	22448.3	0.821810	0.410905	+	0.911678i	$0.365213\pi$
$908$	0	0
$909$	16249.9	0.592934
$910$	0	0
$911$	−25185.2	−0.915943	−0.457971	−	0.888967i	$-0.651424\pi$
$911$	−25185.2	−0.915943	−0.457971	+	0.888967i	$0.651424\pi$
$912$	0	0
$913$	−67151.1	−2.43415
$914$	0	0
$915$	568.100	0.0205255
$916$	0	0
$917$	−2847.11	−0.102530
$918$	0	0
$919$	−11581.1	−0.415697	−0.207849	−	0.978161i	$-0.566646\pi$
$919$	−11581.1	−0.415697	−0.207849	+	0.978161i	$0.566646\pi$
$920$	0	0
$921$	−23234.3	−0.831266
$922$	0	0
$923$	27353.4	0.975459
$924$	0	0
$925$	−10187.7	−0.362128
$926$	0	0
$927$	140.773	0.00498770
$928$	0	0
$929$	12481.0	0.440784	0.220392	−	0.975411i	$-0.429266\pi$
$929$	12481.0	0.440784	0.220392	+	0.975411i	$0.429266\pi$
$930$	0	0
$931$	−3985.12	−0.140287
$932$	0	0
$933$	23973.2	0.841208
$934$	0	0
$935$	−21394.5	−0.748315
$936$	0	0
$937$	20576.7	0.717408	0.358704	−	0.933451i	$-0.383219\pi$
$937$	20576.7	0.717408	0.358704	+	0.933451i	$0.383219\pi$
$938$	0	0
$939$	−23421.7	−0.813993
$940$	0	0
$941$	−31346.6	−1.08594	−0.542970	−	0.839752i	$-0.682700\pi$
$941$	−31346.6	−1.08594	−0.542970	+	0.839752i	$0.682700\pi$
$942$	0	0
$943$	6742.98	0.232854
$944$	0	0
$945$	11948.8	0.411315
$946$	0	0
$947$	−12127.4	−0.416145	−0.208072	−	0.978113i	$-0.566719\pi$
$947$	−12127.4	−0.416145	−0.208072	+	0.978113i	$0.566719\pi$
$948$	0	0
$949$	−21616.6	−0.739415
$950$	0	0
$951$	10804.5	0.368412
$952$	0	0
$953$	10646.6	0.361886	0.180943	−	0.983494i	$-0.442085\pi$
$953$	10646.6	0.361886	0.180943	+	0.983494i	$0.442085\pi$
$954$	0	0
$955$	−15674.4	−0.531112
$956$	0	0
$957$	20132.2	0.680023
$958$	0	0
$959$	37304.3	1.25612
$960$	0	0
$961$	−13050.9	−0.438082
$962$	0	0
$963$	24920.3	0.833898
$964$	0	0
$965$	14411.2	0.480740
$966$	0	0
$967$	38839.1	1.29160	0.645802	−	0.763505i	$-0.276523\pi$
$967$	38839.1	1.29160	0.645802	+	0.763505i	$0.276523\pi$
$968$	0	0
$969$	−24484.5	−0.811719
$970$	0	0
$971$	51156.6	1.69072	0.845362	−	0.534195i	$-0.179385\pi$
$971$	51156.6	1.69072	0.845362	+	0.534195i	$0.179385\pi$
$972$	0	0
$973$	10433.1	0.343753
$974$	0	0
$975$	17973.9	0.590385
$976$	0	0
$977$	−52839.4	−1.73028	−0.865140	−	0.501531i	$-0.832770\pi$
$977$	−52839.4	−1.73028	−0.865140	+	0.501531i	$0.832770\pi$
$978$	0	0
$979$	26330.5	0.859576
$980$	0	0
$981$	5878.41	0.191318
$982$	0	0
$983$	16728.8	0.542794	0.271397	−	0.962467i	$-0.412514\pi$
$983$	16728.8	0.542794	0.271397	+	0.962467i	$0.412514\pi$
$984$	0	0
$985$	−23126.0	−0.748076
$986$	0	0
$987$	25834.4	0.833148
$988$	0	0
$989$	−3728.46	−0.119877
$990$	0	0
$991$	−41197.2	−1.32056	−0.660279	−	0.751021i	$-0.729562\pi$
$991$	−41197.2	−1.32056	−0.660279	+	0.751021i	$0.729562\pi$
$992$	0	0
$993$	−24959.0	−0.797634
$994$	0	0
$995$	−19770.7	−0.629922
$996$	0	0
$997$	53746.1	1.70728	0.853639	−	0.520866i	$-0.174391\pi$
$997$	53746.1	1.70728	0.853639	+	0.520866i	$0.174391\pi$
$998$	0	0
$999$	−14760.4	−0.467465

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1472.4.a.bi.1.8		9
4.3	odd	2		1472.4.a.bl.1.2		9
8.3	odd	2		736.4.a.g.1.8	✓	9
8.5	even	2		736.4.a.j.1.2	yes	9

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
736.4.a.g.1.8	✓	9	8.3	odd	2
736.4.a.j.1.2	yes	9	8.5	even	2
1472.4.a.bi.1.8		9	1.1	even	1	trivial
1472.4.a.bl.1.2		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:	\( N \)	\(=\)	\( 1472 = 2^{6} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1472.a (trivial)

\(f(q)\)	\(=\)	\(q+3.85314 q^{3} +4.56877 q^{5} -17.3356 q^{7} -12.1533 q^{9} +69.1391 q^{11} -44.7989 q^{13} +17.6041 q^{15} -67.7296 q^{17} +93.8206 q^{19} -66.7965 q^{21} +23.0000 q^{23} -104.126 q^{25} -150.863 q^{27} +75.5707 q^{29} -129.384 q^{31} +266.402 q^{33} -79.2026 q^{35} +97.8395 q^{37} -172.616 q^{39} +293.173 q^{41} -162.107 q^{43} -55.5259 q^{45} -386.762 q^{47} -42.4759 q^{49} -260.971 q^{51} -660.868 q^{53} +315.881 q^{55} +361.504 q^{57} +585.664 q^{59} +32.2709 q^{61} +210.686 q^{63} -204.676 q^{65} +220.160 q^{67} +88.6221 q^{69} -610.582 q^{71} +482.525 q^{73} -401.213 q^{75} -1198.57 q^{77} -774.113 q^{79} -253.156 q^{81} -971.247 q^{83} -309.441 q^{85} +291.184 q^{87} +380.833 q^{89} +776.618 q^{91} -498.532 q^{93} +428.645 q^{95} -1721.54 q^{97} -840.271 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 9 q - 14 q^{3} - 30 q^{5} - 28 q^{7} + 103 q^{9} + 40 q^{11} - 76 q^{13} - 72 q^{15} + 166 q^{17} - 148 q^{19} - 96 q^{21} + 207 q^{23} + 339 q^{25} - 326 q^{27} - 252 q^{29} + 258 q^{31} + 460 q^{33} - 276 q^{35}+ \cdots - 2024 q^{99}+O(q^{100}) \) 9 * q - 14 * q^3 - 30 * q^5 - 28 * q^7 + 103 * q^9 + 40 * q^11 - 76 * q^13 - 72 * q^15 + 166 * q^17 - 148 * q^19 - 96 * q^21 + 207 * q^23 + 339 * q^25 - 326 * q^27 - 252 * q^29 + 258 * q^31 + 460 * q^33 - 276 * q^35 - 30 * q^37 - 22 * q^39 + 476 * q^41 - 200 * q^43 - 554 * q^45 + 290 * q^47 + 1277 * q^49 - 396 * q^51 - 702 * q^53 - 404 * q^55 + 2228 * q^57 - 336 * q^59 - 1518 * q^61 + 208 * q^63 + 1352 * q^65 - 664 * q^67 - 322 * q^69 - 874 * q^71 + 2940 * q^73 - 94 * q^75 - 2924 * q^77 - 1268 * q^79 + 3389 * q^81 - 452 * q^83 - 2424 * q^85 - 2022 * q^87 + 3534 * q^89 - 820 * q^91 - 2270 * q^93 - 2360 * q^95 + 2086 * q^97 - 2024 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more