LMFDB - Embedded newform 441.4.a.s.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [441,4,Mod(1,441)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(441, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("441.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 441 = 3^{2} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	441.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:	$26.0198423125$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.57516.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 24x + 6 $ x^3 - x^2 - 24*x + 6
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 21)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$0.248072$ of defining polynomial
Character	$\chi$	$=$	441.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.248072 q^{2} -7.93846 q^{4} +12.4346 q^{5} +3.95388 q^{8} +O(q^{10})$	$q-0.248072 q^{2} -7.93846 q^{4} +12.4346 q^{5} +3.95388 q^{8} -3.08468 q^{10} -60.3115 q^{11} +36.4269 q^{13} +62.5268 q^{16} +48.7461 q^{17} -50.5500 q^{19} -98.7116 q^{20} +14.9616 q^{22} -138.792 q^{23} +29.6194 q^{25} -9.03649 q^{26} +61.1345 q^{29} -1.16935 q^{31} -47.1422 q^{32} -12.0925 q^{34} +69.5268 q^{37} +12.5400 q^{38} +49.1650 q^{40} -308.115 q^{41} +174.443 q^{43} +478.781 q^{44} +34.4305 q^{46} -389.362 q^{47} -7.34774 q^{50} -289.173 q^{52} -314.935 q^{53} -749.950 q^{55} -15.1657 q^{58} -844.526 q^{59} -338.538 q^{61} +0.290084 q^{62} -488.520 q^{64} +452.954 q^{65} -971.550 q^{67} -386.969 q^{68} +98.4698 q^{71} +710.235 q^{73} -17.2477 q^{74} +401.289 q^{76} -486.884 q^{79} +777.496 q^{80} +76.4348 q^{82} -605.688 q^{83} +606.139 q^{85} -43.2743 q^{86} -238.465 q^{88} -218.069 q^{89} +1101.80 q^{92} +96.5897 q^{94} -628.569 q^{95} -782.288 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - q^{2} + 25 q^{4} - 11 q^{5} - 39 q^{8}+O(q^{10}) $ 3 * q - q^2 + 25 * q^4 - 11 * q^5 - 39 * q^8	$ 3 q - q^{2} + 25 q^{4} - 11 q^{5} - 39 q^{8} - 55 q^{10} - 35 q^{11} + 62 q^{13} + 241 q^{16} - 48 q^{17} - 202 q^{19} - 439 q^{20} - 7 q^{22} - 216 q^{23} + 130 q^{25} - 274 q^{26} - 53 q^{29} - 95 q^{31} - 683 q^{32} - 24 q^{34} + 262 q^{37} + 398 q^{38} + 21 q^{40} - 244 q^{41} + 360 q^{43} + 905 q^{44} - 1056 q^{46} + 210 q^{47} + 1378 q^{50} + 324 q^{52} - 393 q^{53} - 1031 q^{55} - 1249 q^{58} - 1143 q^{59} - 70 q^{61} - 1059 q^{62} - 399 q^{64} + 472 q^{65} - 628 q^{67} - 1944 q^{68} - 318 q^{71} + 988 q^{73} - 1002 q^{74} - 2340 q^{76} + 861 q^{79} - 175 q^{80} + 124 q^{82} - 519 q^{83} + 1800 q^{85} + 3208 q^{86} - 891 q^{88} - 1766 q^{89} + 672 q^{92} - 3294 q^{94} + 736 q^{95} + 19 q^{97}+O(q^{100}) $ 3 * q - q^2 + 25 * q^4 - 11 * q^5 - 39 * q^8 - 55 * q^10 - 35 * q^11 + 62 * q^13 + 241 * q^16 - 48 * q^17 - 202 * q^19 - 439 * q^20 - 7 * q^22 - 216 * q^23 + 130 * q^25 - 274 * q^26 - 53 * q^29 - 95 * q^31 - 683 * q^32 - 24 * q^34 + 262 * q^37 + 398 * q^38 + 21 * q^40 - 244 * q^41 + 360 * q^43 + 905 * q^44 - 1056 * q^46 + 210 * q^47 + 1378 * q^50 + 324 * q^52 - 393 * q^53 - 1031 * q^55 - 1249 * q^58 - 1143 * q^59 - 70 * q^61 - 1059 * q^62 - 399 * q^64 + 472 * q^65 - 628 * q^67 - 1944 * q^68 - 318 * q^71 + 988 * q^73 - 1002 * q^74 - 2340 * q^76 + 861 * q^79 - 175 * q^80 + 124 * q^82 - 519 * q^83 + 1800 * q^85 + 3208 * q^86 - 891 * q^88 - 1766 * q^89 + 672 * q^92 - 3294 * q^94 + 736 * q^95 + 19 * q^97

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 $

$2$	−0.248072	−0.0877067	−0.0438533	−	0.999038i	$-0.513963\pi$
$2$	−0.248072	−0.0877067	−0.0438533	+	0.999038i	$0.513963\pi$
$3$	0	0
$3$	0	0
$4$	−7.93846	−0.992308
$5$	12.4346	1.11218	0.556092	−	0.831120i	$-0.312300\pi$
$5$	12.4346	1.11218	0.556092	+	0.831120i	$0.312300\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	3.95388	0.174739
$9$	0	0
$10$	−3.08468	−0.0975460
$11$	−60.3115	−1.65315	−0.826573	−	0.562829i	$-0.809713\pi$
$11$	−60.3115	−1.65315	−0.826573	+	0.562829i	$0.809713\pi$
$12$	0	0
$13$	36.4269	0.777154	0.388577	−	0.921416i	$-0.372967\pi$
$13$	36.4269	0.777154	0.388577	+	0.921416i	$0.372967\pi$
$14$	0	0
$15$	0	0
$16$	62.5268	0.976982
$17$	48.7461	0.695451	0.347726	−	0.937596i	$-0.386954\pi$
$17$	48.7461	0.695451	0.347726	+	0.937596i	$0.386954\pi$
$18$	0	0
$19$	−50.5500	−0.610366	−0.305183	−	0.952294i	$-0.598718\pi$
$19$	−50.5500	−0.610366	−0.305183	+	0.952294i	$0.598718\pi$
$20$	−98.7116	−1.10363
$21$	0	0
$22$	14.9616	0.144992
$23$	−138.792	−1.25827	−0.629135	−	0.777296i	$-0.716591\pi$
$23$	−138.792	−1.25827	−0.629135	+	0.777296i	$0.716591\pi$
$24$	0	0
$25$	29.6194	0.236955
$26$	−9.03649	−0.0681616
$27$	0	0
$28$	0	0
$29$	61.1345	0.391462	0.195731	−	0.980658i	$-0.437292\pi$
$29$	61.1345	0.391462	0.195731	+	0.980658i	$0.437292\pi$
$30$	0	0
$31$	−1.16935	−0.00677490	−0.00338745	−	0.999994i	$-0.501078\pi$
$31$	−1.16935	−0.00677490	−0.00338745	+	0.999994i	$0.501078\pi$
$32$	−47.1422	−0.260426
$33$	0	0
$34$	−12.0925	−0.0609957
$35$	0	0
$36$	0	0
$37$	69.5268	0.308923	0.154461	−	0.987999i	$-0.450636\pi$
$37$	69.5268	0.308923	0.154461	+	0.987999i	$0.450636\pi$
$38$	12.5400	0.0535332
$39$	0	0
$40$	49.1650	0.194342
$41$	−308.115	−1.17365	−0.586823	−	0.809715i	$-0.699622\pi$
$41$	−308.115	−1.17365	−0.586823	+	0.809715i	$0.699622\pi$
$42$	0	0
$43$	174.443	0.618657	0.309329	−	0.950955i	$-0.399896\pi$
$43$	174.443	0.618657	0.309329	+	0.950955i	$0.399896\pi$
$44$	478.781	1.64043
$45$	0	0
$46$	34.4305	0.110359
$47$	−389.362	−1.20839	−0.604194	−	0.796837i	$-0.706505\pi$
$47$	−389.362	−1.20839	−0.604194	+	0.796837i	$0.706505\pi$
$48$	0	0
$49$	0	0
$50$	−7.34774	−0.0207825
$51$	0	0
$52$	−289.173	−0.771176
$53$	−314.935	−0.816220	−0.408110	−	0.912933i	$-0.633812\pi$
$53$	−314.935	−0.816220	−0.408110	+	0.912933i	$0.633812\pi$
$54$	0	0
$55$	−749.950	−1.83860
$56$	0	0
$57$	0	0
$58$	−15.1657	−0.0343338
$59$	−844.526	−1.86352	−0.931762	−	0.363068i	$-0.881729\pi$
$59$	−844.526	−1.86352	−0.931762	+	0.363068i	$0.881729\pi$
$60$	0	0
$61$	−338.538	−0.710579	−0.355290	−	0.934756i	$-0.615618\pi$
$61$	−338.538	−0.710579	−0.355290	+	0.934756i	$0.615618\pi$
$62$	0.290084	0.000594204 0
$63$	0	0
$64$	−488.520	−0.954141
$65$	452.954	0.864339
$66$	0	0
$67$	−971.550	−1.77155	−0.885774	−	0.464117i	$-0.846372\pi$
$67$	−971.550	−1.77155	−0.885774	+	0.464117i	$0.846372\pi$
$68$	−386.969	−0.690102
$69$	0	0
$70$	0	0
$71$	98.4698	0.164595	0.0822973	−	0.996608i	$-0.473774\pi$
$71$	98.4698	0.164595	0.0822973	+	0.996608i	$0.473774\pi$
$72$	0	0
$73$	710.235	1.13872	0.569361	−	0.822088i	$-0.307191\pi$
$73$	710.235	1.13872	0.569361	+	0.822088i	$0.307191\pi$
$74$	−17.2477	−0.0270946
$75$	0	0
$76$	401.289	0.605671
$77$	0	0
$78$	0	0
$79$	−486.884	−0.693402	−0.346701	−	0.937976i	$-0.612698\pi$
$79$	−486.884	−0.693402	−0.346701	+	0.937976i	$0.612698\pi$
$80$	777.496	1.08658
$81$	0	0
$82$	76.4348	0.102937
$83$	−605.688	−0.800999	−0.400499	−	0.916297i	$-0.631163\pi$
$83$	−605.688	−0.800999	−0.400499	+	0.916297i	$0.631163\pi$
$84$	0	0
$85$	606.139	0.773470
$86$	−43.2743	−0.0542604
$87$	0	0
$88$	−238.465	−0.288869
$89$	−218.069	−0.259722	−0.129861	−	0.991532i	$-0.541453\pi$
$89$	−218.069	−0.259722	−0.129861	+	0.991532i	$0.541453\pi$
$90$	0	0
$91$	0	0
$92$	1101.80	1.24859
$93$	0	0
$94$	96.5897	0.105984
$95$	−628.569	−0.678840
$96$	0	0
$97$	−782.288	−0.818859	−0.409429	−	0.912342i	$-0.634272\pi$
$97$	−782.288	−0.818859	−0.409429	+	0.912342i	$0.634272\pi$
$98$	0	0
$99$	0	0
$100$	−235.132	−0.235132
$101$	311.646	0.307029	0.153514	−	0.988146i	$-0.450941\pi$
$101$	311.646	0.307029	0.153514	+	0.988146i	$0.450941\pi$
$102$	0	0
$103$	−149.258	−0.142784	−0.0713922	−	0.997448i	$-0.522744\pi$
$103$	−149.258	−0.142784	−0.0713922	+	0.997448i	$0.522744\pi$
$104$	144.028	0.135799
$105$	0	0
$106$	78.1265	0.0715879
$107$	−851.519	−0.769341	−0.384670	−	0.923054i	$-0.625685\pi$
$107$	−851.519	−0.769341	−0.384670	+	0.923054i	$0.625685\pi$
$108$	0	0
$109$	1361.88	1.19674	0.598369	−	0.801221i	$-0.295816\pi$
$109$	1361.88	1.19674	0.598369	+	0.801221i	$0.295816\pi$
$110$	186.042	0.161258
$111$	0	0
$112$	0	0
$113$	−1048.55	−0.872917	−0.436459	−	0.899724i	$-0.643767\pi$
$113$	−1048.55	−0.872917	−0.436459	+	0.899724i	$0.643767\pi$
$114$	0	0
$115$	−1725.83	−1.39943
$116$	−485.313	−0.388450
$117$	0	0
$118$	209.503	0.163444
$119$	0	0
$120$	0	0
$121$	2306.48	1.73289
$122$	83.9817	0.0623225
$123$	0	0
$124$	9.28286	0.00672279
$125$	−1186.02	−0.848647
$126$	0	0
$127$	488.408	0.341254	0.170627	−	0.985336i	$-0.445421\pi$
$127$	488.408	0.341254	0.170627	+	0.985336i	$0.445421\pi$
$128$	498.326	0.344111
$129$	0	0
$130$	−112.365	−0.0758083
$131$	1854.23	1.23668	0.618338	−	0.785912i	$-0.287806\pi$
$131$	1854.23	1.23668	0.618338	+	0.785912i	$0.287806\pi$
$132$	0	0
$133$	0	0
$134$	241.014	0.155377
$135$	0	0
$136$	192.737	0.121522
$137$	511.115	0.318741	0.159370	−	0.987219i	$-0.449054\pi$
$137$	511.115	0.318741	0.159370	+	0.987219i	$0.449054\pi$
$138$	0	0
$139$	2266.10	1.38279	0.691397	−	0.722475i	$-0.256995\pi$
$139$	2266.10	1.38279	0.691397	+	0.722475i	$0.256995\pi$
$140$	0	0
$141$	0	0
$142$	−24.4276	−0.0144360
$143$	−2196.96	−1.28475
$144$	0	0
$145$	760.183	0.435378
$146$	−176.189	−0.0998735
$147$	0	0
$148$	−551.936	−0.306546
$149$	−1507.90	−0.829074	−0.414537	−	0.910033i	$-0.636056\pi$
$149$	−1507.90	−0.829074	−0.414537	+	0.910033i	$0.636056\pi$
$150$	0	0
$151$	1591.83	0.857887	0.428943	−	0.903331i	$-0.358886\pi$
$151$	1591.83	0.857887	0.428943	+	0.903331i	$0.358886\pi$
$152$	−199.869	−0.106655
$153$	0	0
$154$	0	0
$155$	−14.5404	−0.00753494
$156$	0	0
$157$	1164.16	0.591784	0.295892	−	0.955221i	$-0.404383\pi$
$157$	1164.16	0.591784	0.295892	+	0.955221i	$0.404383\pi$
$158$	120.782	0.0608160
$159$	0	0
$160$	−586.195	−0.289642
$161$	0	0
$162$	0	0
$163$	−1155.88	−0.555432	−0.277716	−	0.960663i	$-0.589577\pi$
$163$	−1155.88	−0.555432	−0.277716	+	0.960663i	$0.589577\pi$
$164$	2445.96	1.16462
$165$	0	0
$166$	150.254	0.0702529
$167$	2890.61	1.33941	0.669707	−	0.742626i	$-0.266420\pi$
$167$	2890.61	1.33941	0.669707	+	0.742626i	$0.266420\pi$
$168$	0	0
$169$	−870.082	−0.396032
$170$	−150.366	−0.0678385
$171$	0	0
$172$	−1384.81	−0.613898
$173$	−1894.94	−0.832770	−0.416385	−	0.909188i	$-0.636703\pi$
$173$	−1894.94	−0.832770	−0.416385	+	0.909188i	$0.636703\pi$
$174$	0	0
$175$	0	0
$176$	−3771.09	−1.61509
$177$	0	0
$178$	54.0967	0.0227793
$179$	4288.49	1.79071	0.895355	−	0.445354i	$-0.146922\pi$
$179$	4288.49	1.79071	0.895355	+	0.445354i	$0.146922\pi$
$180$	0	0
$181$	383.732	0.157583	0.0787917	−	0.996891i	$-0.474894\pi$
$181$	383.732	0.157583	0.0787917	+	0.996891i	$0.474894\pi$
$182$	0	0
$183$	0	0
$184$	−548.769	−0.219868
$185$	864.539	0.343579
$186$	0	0
$187$	−2939.95	−1.14968
$188$	3090.93	1.19909
$189$	0	0
$190$	155.930	0.0595388
$191$	−385.311	−0.145969	−0.0729845	−	0.997333i	$-0.523252\pi$
$191$	−385.311	−0.145969	−0.0729845	+	0.997333i	$0.523252\pi$
$192$	0	0
$193$	630.224	0.235049	0.117525	−	0.993070i	$-0.462504\pi$
$193$	630.224	0.235049	0.117525	+	0.993070i	$0.462504\pi$
$194$	194.064	0.0718194
$195$	0	0
$196$	0	0
$197$	1250.23	0.452158	0.226079	−	0.974109i	$-0.427409\pi$
$197$	1250.23	0.452158	0.226079	+	0.974109i	$0.427409\pi$
$198$	0	0
$199$	−1092.24	−0.389081	−0.194541	−	0.980894i	$-0.562322\pi$
$199$	−1092.24	−0.389081	−0.194541	+	0.980894i	$0.562322\pi$
$200$	117.112	0.0414052
$201$	0	0
$202$	−77.3105	−0.0269285
$203$	0	0
$204$	0	0
$205$	−3831.29	−1.30531
$206$	37.0267	0.0125232
$207$	0	0
$208$	2277.66	0.759265
$209$	3048.75	1.00902
$210$	0	0
$211$	−3620.05	−1.18111	−0.590556	−	0.806997i	$-0.701091\pi$
$211$	−3620.05	−1.18111	−0.590556	+	0.806997i	$0.701091\pi$
$212$	2500.10	0.809941
$213$	0	0
$214$	211.238	0.0674763
$215$	2169.13	0.688061
$216$	0	0
$217$	0	0
$218$	−337.844	−0.104962
$219$	0	0
$220$	5953.45	1.82446
$221$	1775.67	0.540473
$222$	0	0
$223$	−183.844	−0.0552069	−0.0276034	−	0.999619i	$-0.508788\pi$
$223$	−183.844	−0.0552069	−0.0276034	+	0.999619i	$0.508788\pi$
$224$	0	0
$225$	0	0
$226$	260.117	0.0765607
$227$	−2279.52	−0.666506	−0.333253	−	0.942837i	$-0.608146\pi$
$227$	−2279.52	−0.666506	−0.333253	+	0.942837i	$0.608146\pi$
$228$	0	0
$229$	5412.67	1.56192	0.780960	−	0.624582i	$-0.214730\pi$
$229$	5412.67	1.56192	0.780960	+	0.624582i	$0.214730\pi$
$230$	428.130	0.122739
$231$	0	0
$232$	241.719	0.0684035
$233$	−1138.37	−0.320073	−0.160036	−	0.987111i	$-0.551161\pi$
$233$	−1138.37	−0.320073	−0.160036	+	0.987111i	$0.551161\pi$
$234$	0	0
$235$	−4841.56	−1.34395
$236$	6704.24	1.84919
$237$	0	0
$238$	0	0
$239$	6226.36	1.68515	0.842573	−	0.538583i	$-0.181040\pi$
$239$	6226.36	1.68515	0.842573	+	0.538583i	$0.181040\pi$
$240$	0	0
$241$	−3196.20	−0.854295	−0.427147	−	0.904182i	$-0.640481\pi$
$241$	−3196.20	−0.854295	−0.427147	+	0.904182i	$0.640481\pi$
$242$	−572.173	−0.151986
$243$	0	0
$244$	2687.47	0.705113
$245$	0	0
$246$	0	0
$247$	−1841.38	−0.474349
$248$	−4.62349	−0.00118384
$249$	0	0
$250$	294.218	0.0744320
$251$	−239.608	−0.0602546	−0.0301273	−	0.999546i	$-0.509591\pi$
$251$	−239.608	−0.0602546	−0.0301273	+	0.999546i	$0.509591\pi$
$252$	0	0
$253$	8370.78	2.08010
$254$	−121.160	−0.0299302
$255$	0	0
$256$	3784.54	0.923960
$257$	−699.117	−0.169688	−0.0848439	−	0.996394i	$-0.527039\pi$
$257$	−699.117	−0.169688	−0.0848439	+	0.996394i	$0.527039\pi$
$258$	0	0
$259$	0	0
$260$	−3595.76	−0.857690
$261$	0	0
$262$	−459.982	−0.108465
$263$	919.040	0.215477	0.107738	−	0.994179i	$-0.465639\pi$
$263$	919.040	0.215477	0.107738	+	0.994179i	$0.465639\pi$
$264$	0	0
$265$	−3916.09	−0.907787
$266$	0	0
$267$	0	0
$268$	7712.61	1.75792
$269$	2779.17	0.629923	0.314961	−	0.949104i	$-0.398008\pi$
$269$	2779.17	0.629923	0.314961	+	0.949104i	$0.398008\pi$
$270$	0	0
$271$	2226.98	0.499186	0.249593	−	0.968351i	$-0.419703\pi$
$271$	2226.98	0.499186	0.249593	+	0.968351i	$0.419703\pi$
$272$	3047.94	0.679443
$273$	0	0
$274$	−126.793	−0.0279557
$275$	−1786.39	−0.391721
$276$	0	0
$277$	7307.69	1.58511	0.792557	−	0.609797i	$-0.208749\pi$
$277$	7307.69	1.58511	0.792557	+	0.609797i	$0.208749\pi$
$278$	−562.157	−0.121280
$279$	0	0
$280$	0	0
$281$	−2730.61	−0.579696	−0.289848	−	0.957073i	$-0.593605\pi$
$281$	−2730.61	−0.579696	−0.289848	+	0.957073i	$0.593605\pi$
$282$	0	0
$283$	−1769.85	−0.371755	−0.185878	−	0.982573i	$-0.559513\pi$
$283$	−1769.85	−0.371755	−0.185878	+	0.982573i	$0.559513\pi$
$284$	−781.698	−0.163328
$285$	0	0
$286$	545.004	0.112681
$287$	0	0
$288$	0	0
$289$	−2536.81	−0.516347
$290$	−188.580	−0.0381855
$291$	0	0
$292$	−5638.17	−1.12996
$293$	−8228.81	−1.64072	−0.820362	−	0.571844i	$-0.806228\pi$
$293$	−8228.81	−1.64072	−0.820362	+	0.571844i	$0.806228\pi$
$294$	0	0
$295$	−10501.4	−2.07258
$296$	274.901	0.0539807
$297$	0	0
$298$	374.068	0.0727153
$299$	−5055.78	−0.977870
$300$	0	0
$301$	0	0
$302$	−394.887	−0.0752424
$303$	0	0
$304$	−3160.73	−0.596317
$305$	−4209.58	−0.790295
$306$	0	0
$307$	6019.62	1.11908	0.559541	−	0.828803i	$-0.310977\pi$
$307$	6019.62	1.11908	0.559541	+	0.828803i	$0.310977\pi$
$308$	0	0
$309$	0	0
$310$	3.60707	0.000660865 0
$311$	−1193.71	−0.217650	−0.108825	−	0.994061i	$-0.534709\pi$
$311$	−1193.71	−0.217650	−0.108825	+	0.994061i	$0.534709\pi$
$312$	0	0
$313$	−8846.04	−1.59747	−0.798734	−	0.601684i	$-0.794497\pi$
$313$	−8846.04	−1.59747	−0.798734	+	0.601684i	$0.794497\pi$
$314$	−288.795	−0.0519034
$315$	0	0
$316$	3865.11	0.688068
$317$	−6081.43	−1.07750	−0.538750	−	0.842466i	$-0.681103\pi$
$317$	−6081.43	−1.07750	−0.538750	+	0.842466i	$0.681103\pi$
$318$	0	0
$319$	−3687.11	−0.647143
$320$	−6074.55	−1.06118
$321$	0	0
$322$	0	0
$323$	−2464.12	−0.424480
$324$	0	0
$325$	1078.94	0.184151
$326$	286.741	0.0487151
$327$	0	0
$328$	−1218.25	−0.205082
$329$	0	0
$330$	0	0
$331$	3053.30	0.507022	0.253511	−	0.967333i	$-0.418415\pi$
$331$	3053.30	0.507022	0.253511	+	0.967333i	$0.418415\pi$
$332$	4808.23	0.794837
$333$	0	0
$334$	−717.079	−0.117475
$335$	−12080.8	−1.97029
$336$	0	0
$337$	3865.80	0.624877	0.312438	−	0.949938i	$-0.398854\pi$
$337$	3865.80	0.624877	0.312438	+	0.949938i	$0.398854\pi$
$338$	215.843	0.0347346
$339$	0	0
$340$	−4811.81	−0.767521
$341$	70.5255	0.0111999
$342$	0	0
$343$	0	0
$344$	689.726	0.108103
$345$	0	0
$346$	470.080	0.0730395
$347$	99.5931	0.0154076	0.00770380	−	0.999970i	$-0.497548\pi$
$347$	99.5931	0.0154076	0.00770380	+	0.999970i	$0.497548\pi$
$348$	0	0
$349$	−3607.34	−0.553285	−0.276643	−	0.960973i	$-0.589222\pi$
$349$	−3607.34	−0.553285	−0.276643	+	0.960973i	$0.589222\pi$
$350$	0	0
$351$	0	0
$352$	2843.22	0.430523
$353$	−7130.73	−1.07516	−0.537579	−	0.843214i	$-0.680661\pi$
$353$	−7130.73	−1.07516	−0.537579	+	0.843214i	$0.680661\pi$
$354$	0	0
$355$	1224.43	0.183060
$356$	1731.13	0.257724
$357$	0	0
$358$	−1063.85	−0.157057
$359$	6500.29	0.955632	0.477816	−	0.878460i	$-0.341428\pi$
$359$	6500.29	0.955632	0.477816	+	0.878460i	$0.341428\pi$
$360$	0	0
$361$	−4303.70	−0.627453
$362$	−95.1932	−0.0138211
$363$	0	0
$364$	0	0
$365$	8831.49	1.26647
$366$	0	0
$367$	824.886	0.117326	0.0586631	−	0.998278i	$-0.481316\pi$
$367$	824.886	0.117326	0.0586631	+	0.998278i	$0.481316\pi$
$368$	−8678.25	−1.22931
$369$	0	0
$370$	−214.468	−0.0301342
$371$	0	0
$372$	0	0
$373$	1333.85	0.185159	0.0925793	−	0.995705i	$-0.470489\pi$
$373$	1333.85	0.185159	0.0925793	+	0.995705i	$0.470489\pi$
$374$	729.320	0.100835
$375$	0	0
$376$	−1539.49	−0.211152
$377$	2226.94	0.304226
$378$	0	0
$379$	−1338.29	−0.181380	−0.0906902	−	0.995879i	$-0.528907\pi$
$379$	−1338.29	−0.181380	−0.0906902	+	0.995879i	$0.528907\pi$
$380$	4989.87	0.673618
$381$	0	0
$382$	95.5847	0.0128025
$383$	−353.376	−0.0471453	−0.0235727	−	0.999722i	$-0.507504\pi$
$383$	−353.376	−0.0471453	−0.0235727	+	0.999722i	$0.507504\pi$
$384$	0	0
$385$	0	0
$386$	−156.341	−0.0206154
$387$	0	0
$388$	6210.16	0.812560
$389$	−11737.2	−1.52982	−0.764908	−	0.644139i	$-0.777216\pi$
$389$	−11737.2	−1.52982	−0.764908	+	0.644139i	$0.777216\pi$
$390$	0	0
$391$	−6765.59	−0.875066
$392$	0	0
$393$	0	0
$394$	−310.147	−0.0396573
$395$	−6054.21	−0.771191
$396$	0	0
$397$	13281.4	1.67903	0.839516	−	0.543335i	$-0.182839\pi$
$397$	13281.4	1.67903	0.839516	+	0.543335i	$0.182839\pi$
$398$	270.955	0.0341250
$399$	0	0
$400$	1852.01	0.231501
$401$	7482.36	0.931798	0.465899	−	0.884838i	$-0.345731\pi$
$401$	7482.36	0.931798	0.465899	+	0.884838i	$0.345731\pi$
$402$	0	0
$403$	−42.5959	−0.00526514
$404$	−2473.99	−0.304667
$405$	0	0
$406$	0	0
$407$	−4193.27	−0.510694
$408$	0	0
$409$	−13796.6	−1.66797	−0.833983	−	0.551791i	$-0.813945\pi$
$409$	−13796.6	−1.66797	−0.833983	+	0.551791i	$0.813945\pi$
$410$	950.436	0.114485
$411$	0	0
$412$	1184.88	0.141686
$413$	0	0
$414$	0	0
$415$	−7531.49	−0.890859
$416$	−1717.24	−0.202391
$417$	0	0
$418$	−756.308	−0.0884982
$419$	−9497.56	−1.10737	−0.553683	−	0.832728i	$-0.686778\pi$
$419$	−9497.56	−1.10737	−0.553683	+	0.832728i	$0.686778\pi$
$420$	0	0
$421$	624.367	0.0722797	0.0361399	−	0.999347i	$-0.488494\pi$
$421$	624.367	0.0722797	0.0361399	+	0.999347i	$0.488494\pi$
$422$	898.032	0.103591
$423$	0	0
$424$	−1245.22	−0.142625
$425$	1443.83	0.164791
$426$	0	0
$427$	0	0
$428$	6759.75	0.763423
$429$	0	0
$430$	−538.099	−0.0603475
$431$	13397.3	1.49727	0.748636	−	0.662981i	$-0.230709\pi$
$431$	13397.3	1.49727	0.748636	+	0.662981i	$0.230709\pi$
$432$	0	0
$433$	−14057.3	−1.56016	−0.780079	−	0.625681i	$-0.784821\pi$
$433$	−14057.3	−1.56016	−0.780079	+	0.625681i	$0.784821\pi$
$434$	0	0
$435$	0	0
$436$	−10811.2	−1.18753
$437$	7015.95	0.768006
$438$	0	0
$439$	−16368.8	−1.77960	−0.889798	−	0.456356i	$-0.849155\pi$
$439$	−16368.8	−1.77960	−0.889798	+	0.456356i	$0.849155\pi$
$440$	−2965.22	−0.321275
$441$	0	0
$442$	−440.494	−0.0474031
$443$	1178.71	0.126416	0.0632078	−	0.998000i	$-0.479867\pi$
$443$	1178.71	0.126416	0.0632078	+	0.998000i	$0.479867\pi$
$444$	0	0
$445$	−2711.60	−0.288858
$446$	45.6067	0.00484201
$447$	0	0
$448$	0	0
$449$	12400.9	1.30342	0.651709	−	0.758469i	$-0.274052\pi$
$449$	12400.9	1.30342	0.651709	+	0.758469i	$0.274052\pi$
$450$	0	0
$451$	18582.9	1.94021
$452$	8323.90	0.866202
$453$	0	0
$454$	565.484	0.0584570
$455$	0	0
$456$	0	0
$457$	9925.58	1.01597	0.507986	−	0.861365i	$-0.330390\pi$
$457$	9925.58	1.01597	0.507986	+	0.861365i	$0.330390\pi$
$458$	−1342.73	−0.136991
$459$	0	0
$460$	13700.4	1.38866
$461$	16010.3	1.61751	0.808755	−	0.588146i	$-0.200142\pi$
$461$	16010.3	1.61751	0.808755	+	0.588146i	$0.200142\pi$
$462$	0	0
$463$	17372.4	1.74377	0.871883	−	0.489714i	$-0.162899\pi$
$463$	17372.4	1.74377	0.871883	+	0.489714i	$0.162899\pi$
$464$	3822.54	0.382451
$465$	0	0
$466$	282.397	0.0280725
$467$	−2108.06	−0.208886	−0.104443	−	0.994531i	$-0.533306\pi$
$467$	−2108.06	−0.208886	−0.104443	+	0.994531i	$0.533306\pi$
$468$	0	0
$469$	0	0
$470$	1201.05	0.117873
$471$	0	0
$472$	−3339.16	−0.325630
$473$	−10520.9	−1.02273
$474$	0	0
$475$	−1497.26	−0.144629
$476$	0	0
$477$	0	0
$478$	−1544.59	−0.147798
$479$	2450.04	0.233706	0.116853	−	0.993149i	$-0.462719\pi$
$479$	2450.04	0.233706	0.116853	+	0.993149i	$0.462719\pi$
$480$	0	0
$481$	2532.65	0.240081
$482$	792.887	0.0749274
$483$	0	0
$484$	−18309.9	−1.71956
$485$	−9727.44	−0.910722
$486$	0	0
$487$	645.236	0.0600379	0.0300189	−	0.999549i	$-0.490443\pi$
$487$	645.236	0.0600379	0.0300189	+	0.999549i	$0.490443\pi$
$488$	−1338.54	−0.124166
$489$	0	0
$490$	0	0
$491$	−11766.1	−1.08146	−0.540731	−	0.841196i	$-0.681852\pi$
$491$	−11766.1	−1.08146	−0.540731	+	0.841196i	$0.681852\pi$
$492$	0	0
$493$	2980.07	0.272242
$494$	456.794	0.0416035
$495$	0	0
$496$	−73.1159	−0.00661896
$497$	0	0
$498$	0	0
$499$	−44.0209	−0.00394919	−0.00197459	−	0.999998i	$-0.500629\pi$
$499$	−44.0209	−0.00394919	−0.00197459	+	0.999998i	$0.500629\pi$
$500$	9415.17	0.842119
$501$	0	0
$502$	59.4399	0.00528473
$503$	−8290.27	−0.734880	−0.367440	−	0.930047i	$-0.619766\pi$
$503$	−8290.27	−0.734880	−0.367440	+	0.930047i	$0.619766\pi$
$504$	0	0
$505$	3875.19	0.341473
$506$	−2076.56	−0.182439
$507$	0	0
$508$	−3877.21	−0.338629
$509$	−6915.04	−0.602168	−0.301084	−	0.953598i	$-0.597349\pi$
$509$	−6915.04	−0.602168	−0.301084	+	0.953598i	$0.597349\pi$
$510$	0	0
$511$	0	0
$512$	−4925.45	−0.425148
$513$	0	0
$514$	173.431	0.0148827
$515$	−1855.96	−0.158803
$516$	0	0
$517$	23483.0	1.99764
$518$	0	0
$519$	0	0
$520$	1790.93	0.151033
$521$	−13399.3	−1.12674	−0.563371	−	0.826204i	$-0.690496\pi$
$521$	−13399.3	−1.12674	−0.563371	+	0.826204i	$0.690496\pi$
$522$	0	0
$523$	−9936.99	−0.830811	−0.415406	−	0.909636i	$-0.636360\pi$
$523$	−9936.99	−0.830811	−0.415406	+	0.909636i	$0.636360\pi$
$524$	−14719.7	−1.22716
$525$	0	0
$526$	−227.988	−0.0188988
$527$	−57.0014	−0.00471161
$528$	0	0
$529$	7096.33	0.583244
$530$	971.472	0.0796190
$531$	0	0
$532$	0	0
$533$	−11223.7	−0.912104
$534$	0	0
$535$	−10588.3	−0.855649
$536$	−3841.40	−0.309558
$537$	0	0
$538$	−689.435	−0.0552484
$539$	0	0
$540$	0	0
$541$	9286.17	0.737973	0.368987	−	0.929435i	$-0.379705\pi$
$541$	9286.17	0.737973	0.368987	+	0.929435i	$0.379705\pi$
$542$	−552.452	−0.0437820
$543$	0	0
$544$	−2298.00	−0.181114
$545$	16934.4	1.33099
$546$	0	0
$547$	−16821.6	−1.31488	−0.657438	−	0.753508i	$-0.728360\pi$
$547$	−16821.6	−1.31488	−0.657438	+	0.753508i	$0.728360\pi$
$548$	−4057.47	−0.316289
$549$	0	0
$550$	443.153	0.0343566
$551$	−3090.35	−0.238935
$552$	0	0
$553$	0	0
$554$	−1812.83	−0.139025
$555$	0	0
$556$	−17989.4	−1.37216
$557$	−1805.94	−0.137379	−0.0686897	−	0.997638i	$-0.521882\pi$
$557$	−1805.94	−0.137379	−0.0686897	+	0.997638i	$0.521882\pi$
$558$	0	0
$559$	6354.40	0.480792
$560$	0	0
$561$	0	0
$562$	677.388	0.0508432
$563$	−12214.9	−0.914381	−0.457190	−	0.889369i	$-0.651144\pi$
$563$	−12214.9	−0.914381	−0.457190	+	0.889369i	$0.651144\pi$
$564$	0	0
$565$	−13038.3	−0.970845
$566$	439.050	0.0326054
$567$	0	0
$568$	389.338	0.0287610
$569$	−4283.77	−0.315615	−0.157808	−	0.987470i	$-0.550443\pi$
$569$	−4283.77	−0.315615	−0.157808	+	0.987470i	$0.550443\pi$
$570$	0	0
$571$	6359.94	0.466121	0.233060	−	0.972462i	$-0.425126\pi$
$571$	6359.94	0.466121	0.233060	+	0.972462i	$0.425126\pi$
$572$	17440.5	1.27487
$573$	0	0
$574$	0	0
$575$	−4110.95	−0.298153
$576$	0	0
$577$	14468.7	1.04392	0.521959	−	0.852971i	$-0.325201\pi$
$577$	14468.7	1.04392	0.521959	+	0.852971i	$0.325201\pi$
$578$	629.313	0.0452871
$579$	0	0
$580$	−6034.68	−0.432028
$581$	0	0
$582$	0	0
$583$	18994.2	1.34933
$584$	2808.19	0.198979
$585$	0	0
$586$	2041.34	0.143903
$587$	11132.6	0.782777	0.391388	−	0.920226i	$-0.371995\pi$
$587$	11132.6	0.782777	0.391388	+	0.920226i	$0.371995\pi$
$588$	0	0
$589$	59.1108	0.00413517
$590$	2605.09	0.181779
$591$	0	0
$592$	4347.29	0.301812
$593$	19775.6	1.36946	0.684728	−	0.728799i	$-0.259921\pi$
$593$	19775.6	1.36946	0.684728	+	0.728799i	$0.259921\pi$
$594$	0	0
$595$	0	0
$596$	11970.4	0.822696
$597$	0	0
$598$	1254.20	0.0857657
$599$	23891.0	1.62965	0.814825	−	0.579707i	$-0.196833\pi$
$599$	23891.0	1.62965	0.814825	+	0.579707i	$0.196833\pi$
$600$	0	0
$601$	19395.5	1.31641	0.658204	−	0.752840i	$-0.271317\pi$
$601$	19395.5	1.31641	0.658204	+	0.752840i	$0.271317\pi$
$602$	0	0
$603$	0	0
$604$	−12636.6	−0.851288
$605$	28680.2	1.92730
$606$	0	0
$607$	14596.7	0.976051	0.488025	−	0.872829i	$-0.337717\pi$
$607$	14596.7	0.976051	0.488025	+	0.872829i	$0.337717\pi$
$608$	2383.04	0.158956
$609$	0	0
$610$	1044.28	0.0693142
$611$	−14183.2	−0.939104
$612$	0	0
$613$	1979.80	0.130446	0.0652229	−	0.997871i	$-0.479224\pi$
$613$	1979.80	0.130446	0.0652229	+	0.997871i	$0.479224\pi$
$614$	−1493.30	−0.0981509
$615$	0	0
$616$	0	0
$617$	−16262.4	−1.06110	−0.530551	−	0.847653i	$-0.678015\pi$
$617$	−16262.4	−1.06110	−0.530551	+	0.847653i	$0.678015\pi$
$618$	0	0
$619$	−12021.0	−0.780555	−0.390278	−	0.920697i	$-0.627621\pi$
$619$	−12021.0	−0.780555	−0.390278	+	0.920697i	$0.627621\pi$
$620$	115.429	0.00747698
$621$	0	0
$622$	296.127	0.0190894
$623$	0	0
$624$	0	0
$625$	−18450.1	−1.18081
$626$	2194.45	0.140109
$627$	0	0
$628$	−9241.64	−0.587232
$629$	3389.16	0.214841
$630$	0	0
$631$	25347.6	1.59916	0.799582	−	0.600557i	$-0.205055\pi$
$631$	25347.6	1.59916	0.799582	+	0.600557i	$0.205055\pi$
$632$	−1925.08	−0.121164
$633$	0	0
$634$	1508.63	0.0945039
$635$	6073.16	0.379537
$636$	0	0
$637$	0	0
$638$	914.669	0.0567588
$639$	0	0
$640$	6196.49	0.382715
$641$	5111.60	0.314971	0.157485	−	0.987521i	$-0.449661\pi$
$641$	5111.60	0.314971	0.157485	+	0.987521i	$0.449661\pi$
$642$	0	0
$643$	−10931.3	−0.670435	−0.335217	−	0.942141i	$-0.608810\pi$
$643$	−10931.3	−0.670435	−0.335217	+	0.942141i	$0.608810\pi$
$644$	0	0
$645$	0	0
$646$	611.278	0.0372297
$647$	18406.1	1.11842	0.559211	−	0.829025i	$-0.311104\pi$
$647$	18406.1	1.11842	0.559211	+	0.829025i	$0.311104\pi$
$648$	0	0
$649$	50934.7	3.08068
$650$	−267.655	−0.0161512
$651$	0	0
$652$	9175.91	0.551160
$653$	−19921.4	−1.19385	−0.596926	−	0.802296i	$-0.703611\pi$
$653$	−19921.4	−1.19385	−0.596926	+	0.802296i	$0.703611\pi$
$654$	0	0
$655$	23056.6	1.37541
$656$	−19265.5	−1.14663
$657$	0	0
$658$	0	0
$659$	18858.8	1.11477	0.557385	−	0.830254i	$-0.311805\pi$
$659$	18858.8	1.11477	0.557385	+	0.830254i	$0.311805\pi$
$660$	0	0
$661$	25832.1	1.52005	0.760023	−	0.649896i	$-0.225188\pi$
$661$	25832.1	1.52005	0.760023	+	0.649896i	$0.225188\pi$
$662$	−757.437	−0.0444692
$663$	0	0
$664$	−2394.82	−0.139965
$665$	0	0
$666$	0	0
$667$	−8485.00	−0.492564
$668$	−22947.0	−1.32911
$669$	0	0
$670$	2996.92	0.172807
$671$	20417.7	1.17469
$672$	0	0
$673$	−16275.0	−0.932178	−0.466089	−	0.884738i	$-0.654337\pi$
$673$	−16275.0	−0.932178	−0.466089	+	0.884738i	$0.654337\pi$
$674$	−958.996	−0.0548059
$675$	0	0
$676$	6907.11	0.392985
$677$	26271.8	1.49144	0.745720	−	0.666259i	$-0.232105\pi$
$677$	26271.8	1.49144	0.745720	+	0.666259i	$0.232105\pi$
$678$	0	0
$679$	0	0
$680$	2396.60	0.135155
$681$	0	0
$682$	−17.4954	−0.000982306 0
$683$	8072.29	0.452237	0.226118	−	0.974100i	$-0.427396\pi$
$683$	8072.29	0.452237	0.226118	+	0.974100i	$0.427396\pi$
$684$	0	0
$685$	6355.51	0.354499
$686$	0	0
$687$	0	0
$688$	10907.3	0.604417
$689$	−11472.1	−0.634328
$690$	0	0
$691$	−24485.3	−1.34799	−0.673997	−	0.738734i	$-0.735424\pi$
$691$	−24485.3	−1.34799	−0.673997	+	0.738734i	$0.735424\pi$
$692$	15042.9	0.826364
$693$	0	0
$694$	−24.7062	−0.00135135
$695$	28178.1	1.53792
$696$	0	0
$697$	−15019.4	−0.816214
$698$	894.880	0.0485268
$699$	0	0
$700$	0	0
$701$	−778.448	−0.0419423	−0.0209712	−	0.999780i	$-0.506676\pi$
$701$	−778.448	−0.0419423	−0.0209712	+	0.999780i	$0.506676\pi$
$702$	0	0
$703$	−3514.58	−0.188556
$704$	29463.4	1.57733
$705$	0	0
$706$	1768.93	0.0942985
$707$	0	0
$708$	0	0
$709$	−24172.0	−1.28039	−0.640197	−	0.768211i	$-0.721147\pi$
$709$	−24172.0	−1.28039	−0.640197	+	0.768211i	$0.721147\pi$
$710$	−303.747	−0.0160555
$711$	0	0
$712$	−862.218	−0.0453834
$713$	162.297	0.00852466
$714$	0	0
$715$	−27318.3	−1.42888
$716$	−34044.0	−1.77693
$717$	0	0
$718$	−1612.54	−0.0838153
$719$	81.8835	0.00424720	0.00212360	−	0.999998i	$-0.499324\pi$
$719$	81.8835	0.00424720	0.00212360	+	0.999998i	$0.499324\pi$
$720$	0	0
$721$	0	0
$722$	1067.63	0.0550318
$723$	0	0
$724$	−3046.24	−0.156371
$725$	1810.76	0.0927588
$726$	0	0
$727$	−32542.9	−1.66018	−0.830088	−	0.557632i	$-0.811710\pi$
$727$	−32542.9	−1.66018	−0.830088	+	0.557632i	$0.811710\pi$
$728$	0	0
$729$	0	0
$730$	−2190.85	−0.111078
$731$	8503.40	0.430246
$732$	0	0
$733$	−5068.94	−0.255424	−0.127712	−	0.991811i	$-0.540763\pi$
$733$	−5068.94	−0.255424	−0.127712	+	0.991811i	$0.540763\pi$
$734$	−204.631	−0.0102903
$735$	0	0
$736$	6542.98	0.327687
$737$	58595.6	2.92863
$738$	0	0
$739$	−38428.5	−1.91287	−0.956437	−	0.291939i	$-0.905700\pi$
$739$	−38428.5	−1.91287	−0.956437	+	0.291939i	$0.905700\pi$
$740$	−6863.11	−0.340936
$741$	0	0
$742$	0	0
$743$	−21592.9	−1.06617	−0.533086	−	0.846061i	$-0.678968\pi$
$743$	−21592.9	−1.06617	−0.533086	+	0.846061i	$0.678968\pi$
$744$	0	0
$745$	−18750.1	−0.922083
$746$	−330.891	−0.0162396
$747$	0	0
$748$	23338.7	1.14084
$749$	0	0
$750$	0	0
$751$	8112.60	0.394185	0.197093	−	0.980385i	$-0.436850\pi$
$751$	8112.60	0.394185	0.197093	+	0.980385i	$0.436850\pi$
$752$	−24345.6	−1.18057
$753$	0	0
$754$	−552.441	−0.0266826
$755$	19793.7	0.954129
$756$	0	0
$757$	3108.01	0.149224	0.0746120	−	0.997213i	$-0.476228\pi$
$757$	3108.01	0.149224	0.0746120	+	0.997213i	$0.476228\pi$
$758$	331.992	0.0159083
$759$	0	0
$760$	−2485.29	−0.118620
$761$	7211.93	0.343538	0.171769	−	0.985137i	$-0.445052\pi$
$761$	7211.93	0.343538	0.171769	+	0.985137i	$0.445052\pi$
$762$	0	0
$763$	0	0
$764$	3058.77	0.144846
$765$	0	0
$766$	87.6626	0.00413496
$767$	−30763.5	−1.44825
$768$	0	0
$769$	−7533.07	−0.353250	−0.176625	−	0.984278i	$-0.556518\pi$
$769$	−7533.07	−0.353250	−0.176625	+	0.984278i	$0.556518\pi$
$770$	0	0
$771$	0	0
$772$	−5003.00	−0.233241
$773$	24832.6	1.15546	0.577728	−	0.816229i	$-0.303940\pi$
$773$	24832.6	1.15546	0.577728	+	0.816229i	$0.303940\pi$
$774$	0	0
$775$	−34.6355	−0.00160535
$776$	−3093.08	−0.143086
$777$	0	0
$778$	2911.66	0.134175
$779$	15575.2	0.716355
$780$	0	0
$781$	−5938.86	−0.272099
$782$	1678.35	0.0767491
$783$	0	0
$784$	0	0
$785$	14475.9	0.658173
$786$	0	0
$787$	36313.1	1.64476	0.822378	−	0.568941i	$-0.192647\pi$
$787$	36313.1	1.64476	0.822378	+	0.568941i	$0.192647\pi$
$788$	−9924.89	−0.448680
$789$	0	0
$790$	1501.88	0.0676386
$791$	0	0
$792$	0	0
$793$	−12331.9	−0.552229
$794$	−3294.75	−0.147262
$795$	0	0
$796$	8670.74	0.386088
$797$	−31665.7	−1.40735	−0.703675	−	0.710522i	$-0.748459\pi$
$797$	−31665.7	−1.40735	−0.703675	+	0.710522i	$0.748459\pi$
$798$	0	0
$799$	−18979.9	−0.840375
$800$	−1396.32	−0.0617094
$801$	0	0
$802$	−1856.16	−0.0817249
$803$	−42835.4	−1.88247
$804$	0	0
$805$	0	0
$806$	10.5668	0.000461788 0
$807$	0	0
$808$	1232.21	0.0536498
$809$	−12384.6	−0.538219	−0.269110	−	0.963110i	$-0.586729\pi$
$809$	−12384.6	−0.538219	−0.269110	+	0.963110i	$0.586729\pi$
$810$	0	0
$811$	16742.4	0.724914	0.362457	−	0.932000i	$-0.381938\pi$
$811$	16742.4	0.724914	0.362457	+	0.932000i	$0.381938\pi$
$812$	0	0
$813$	0	0
$814$	1040.23	0.0447913
$815$	−14372.9	−0.617744
$816$	0	0
$817$	−8818.07	−0.377607
$818$	3422.55	0.146292
$819$	0	0
$820$	30414.6	1.29527
$821$	−26456.3	−1.12464	−0.562322	−	0.826918i	$-0.690092\pi$
$821$	−26456.3	−1.12464	−0.562322	+	0.826918i	$0.690092\pi$
$822$	0	0
$823$	23098.5	0.978328	0.489164	−	0.872192i	$-0.337302\pi$
$823$	23098.5	0.978328	0.489164	+	0.872192i	$0.337302\pi$
$824$	−590.148	−0.0249500
$825$	0	0
$826$	0	0
$827$	−20647.6	−0.868183	−0.434092	−	0.900869i	$-0.642931\pi$
$827$	−20647.6	−0.868183	−0.434092	+	0.900869i	$0.642931\pi$
$828$	0	0
$829$	−23368.5	−0.979037	−0.489519	−	0.871993i	$-0.662827\pi$
$829$	−23368.5	−0.979037	−0.489519	+	0.871993i	$0.662827\pi$
$830$	1868.35	0.0781343
$831$	0	0
$832$	−17795.3	−0.741514
$833$	0	0
$834$	0	0
$835$	35943.6	1.48968
$836$	−24202.3	−1.00126
$837$	0	0
$838$	2356.08	0.0971234
$839$	−16735.5	−0.688645	−0.344322	−	0.938851i	$-0.611891\pi$
$839$	−16735.5	−0.688645	−0.344322	+	0.938851i	$0.611891\pi$
$840$	0	0
$841$	−20651.6	−0.846758
$842$	−154.888	−0.00633941
$843$	0	0
$844$	28737.6	1.17203
$845$	−10819.1	−0.440460
$846$	0	0
$847$	0	0
$848$	−19691.9	−0.797432
$849$	0	0
$850$	−358.174	−0.0144532
$851$	−9649.80	−0.388708
$852$	0	0
$853$	−10294.5	−0.413219	−0.206609	−	0.978424i	$-0.566243\pi$
$853$	−10294.5	−0.413219	−0.206609	+	0.978424i	$0.566243\pi$
$854$	0	0
$855$	0	0
$856$	−3366.81	−0.134434
$857$	32788.6	1.30693	0.653463	−	0.756958i	$-0.273315\pi$
$857$	32788.6	1.30693	0.653463	+	0.756958i	$0.273315\pi$
$858$	0	0
$859$	−4909.76	−0.195016	−0.0975081	−	0.995235i	$-0.531087\pi$
$859$	−4909.76	−0.195016	−0.0975081	+	0.995235i	$0.531087\pi$
$860$	−17219.5	−0.682768
$861$	0	0
$862$	−3323.49	−0.131321
$863$	−17795.0	−0.701909	−0.350954	−	0.936393i	$-0.614143\pi$
$863$	−17795.0	−0.701909	−0.350954	+	0.936393i	$0.614143\pi$
$864$	0	0
$865$	−23562.8	−0.926195
$866$	3487.21	0.136836
$867$	0	0
$868$	0	0
$869$	29364.7	1.14629
$870$	0	0
$871$	−35390.5	−1.37677
$872$	5384.71	0.209116
$873$	0	0
$874$	−1740.46	−0.0673592
$875$	0	0
$876$	0	0
$877$	34672.2	1.33500	0.667501	−	0.744609i	$-0.267364\pi$
$877$	34672.2	1.33500	0.667501	+	0.744609i	$0.267364\pi$
$878$	4060.65	0.156082
$879$	0	0
$880$	−46892.0	−1.79628
$881$	−40848.2	−1.56210	−0.781051	−	0.624467i	$-0.785316\pi$
$881$	−40848.2	−1.56210	−0.781051	+	0.624467i	$0.785316\pi$
$882$	0	0
$883$	30035.1	1.14469	0.572345	−	0.820013i	$-0.306034\pi$
$883$	30035.1	1.14469	0.572345	+	0.820013i	$0.306034\pi$
$884$	−14096.1	−0.536315
$885$	0	0
$886$	−292.404	−0.0110875
$887$	−33210.7	−1.25717	−0.628583	−	0.777742i	$-0.716365\pi$
$887$	−33210.7	−1.25717	−0.628583	+	0.777742i	$0.716365\pi$
$888$	0	0
$889$	0	0
$890$	672.671	0.0253348
$891$	0	0
$892$	1459.44	0.0547822
$893$	19682.2	0.737559
$894$	0	0
$895$	53325.7	1.99160
$896$	0	0
$897$	0	0
$898$	−3076.32	−0.114318
$899$	−71.4878	−0.00265211
$900$	0	0
$901$	−15351.9	−0.567641
$902$	−4609.90	−0.170169
$903$	0	0
$904$	−4145.86	−0.152532
$905$	4771.56	0.175262
$906$	0	0
$907$	2497.83	0.0914433	0.0457217	−	0.998954i	$-0.485441\pi$
$907$	2497.83	0.0914433	0.0457217	+	0.998954i	$0.485441\pi$
$908$	18095.9	0.661379
$909$	0	0
$910$	0	0
$911$	1895.00	0.0689180	0.0344590	−	0.999406i	$-0.489029\pi$
$911$	1895.00	0.0689180	0.0344590	+	0.999406i	$0.489029\pi$
$912$	0	0
$913$	36530.0	1.32417
$914$	−2462.26	−0.0891075
$915$	0	0
$916$	−42968.3	−1.54990
$917$	0	0
$918$	0	0
$919$	6270.71	0.225083	0.112542	−	0.993647i	$-0.464101\pi$
$919$	6270.71	0.225083	0.112542	+	0.993647i	$0.464101\pi$
$920$	−6823.73	−0.244534
$921$	0	0
$922$	−3971.69	−0.141866
$923$	3586.95	0.127915
$924$	0	0
$925$	2059.34	0.0732008
$926$	−4309.61	−0.152940
$927$	0	0
$928$	−2882.01	−0.101947
$929$	31552.6	1.11432	0.557161	−	0.830404i	$-0.311890\pi$
$929$	31552.6	1.11432	0.557161	+	0.830404i	$0.311890\pi$
$930$	0	0
$931$	0	0
$932$	9036.89	0.317611
$933$	0	0
$934$	522.951	0.0183207
$935$	−36557.2	−1.27866
$936$	0	0
$937$	−22030.2	−0.768084	−0.384042	−	0.923316i	$-0.625468\pi$
$937$	−22030.2	−0.768084	−0.384042	+	0.923316i	$0.625468\pi$
$938$	0	0
$939$	0	0
$940$	38434.5	1.33361
$941$	32538.6	1.12724	0.563618	−	0.826036i	$-0.309409\pi$
$941$	32538.6	1.12724	0.563618	+	0.826036i	$0.309409\pi$
$942$	0	0
$943$	42764.1	1.47677
$944$	−52805.6	−1.82063
$945$	0	0
$946$	2609.94	0.0897003
$947$	40711.0	1.39697	0.698485	−	0.715625i	$-0.253858\pi$
$947$	40711.0	1.39697	0.698485	+	0.715625i	$0.253858\pi$
$948$	0	0
$949$	25871.7	0.884962
$950$	371.428	0.0126850
$951$	0	0
$952$	0	0
$953$	52516.4	1.78507	0.892536	−	0.450976i	$-0.148924\pi$
$953$	52516.4	1.78507	0.892536	+	0.450976i	$0.148924\pi$
$954$	0	0
$955$	−4791.18	−0.162345
$956$	−49427.7	−1.67218
$957$	0	0
$958$	−607.786	−0.0204976
$959$	0	0
$960$	0	0
$961$	−29789.6	−0.999954
$962$	−628.279	−0.0210567
$963$	0	0
$964$	25372.9	0.847723
$965$	7836.58	0.261418
$966$	0	0
$967$	14721.6	0.489570	0.244785	−	0.969577i	$-0.421283\pi$
$967$	14721.6	0.489570	0.244785	+	0.969577i	$0.421283\pi$
$968$	9119.56	0.302803
$969$	0	0
$970$	2413.10	0.0798764
$971$	−13772.5	−0.455181	−0.227590	−	0.973757i	$-0.573085\pi$
$971$	−13772.5	−0.455181	−0.227590	+	0.973757i	$0.573085\pi$
$972$	0	0
$973$	0	0
$974$	−160.065	−0.00526572
$975$	0	0
$976$	−21167.7	−0.694223
$977$	−24782.1	−0.811513	−0.405757	−	0.913981i	$-0.632992\pi$
$977$	−24782.1	−0.811513	−0.405757	+	0.913981i	$0.632992\pi$
$978$	0	0
$979$	13152.0	0.429358
$980$	0	0
$981$	0	0
$982$	2918.84	0.0948514
$983$	−42804.7	−1.38887	−0.694435	−	0.719556i	$-0.744345\pi$
$983$	−42804.7	−1.38887	−0.694435	+	0.719556i	$0.744345\pi$
$984$	0	0
$985$	15546.1	0.502883
$986$	−739.271	−0.0238775
$987$	0	0
$988$	14617.7	0.470700
$989$	−24211.3	−0.778438
$990$	0	0
$991$	449.862	0.0144201	0.00721006	−	0.999974i	$-0.497705\pi$
$991$	449.862	0.0144201	0.00721006	+	0.999974i	$0.497705\pi$
$992$	55.1259	0.00176436
$993$	0	0
$994$	0	0
$995$	−13581.6	−0.432730
$996$	0	0
$997$	−21473.7	−0.682127	−0.341063	−	0.940040i	$-0.610787\pi$
$997$	−21473.7	−0.682127	−0.341063	+	0.940040i	$0.610787\pi$
$998$	10.9203	0.000346370 0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	441.4.a.s.1.2		3
3.2	odd	2		147.4.a.l.1.2		3
7.2	even	3		63.4.e.c.46.2		6
7.3	odd	6		441.4.e.w.226.2		6
7.4	even	3		63.4.e.c.37.2		6
7.5	odd	6		441.4.e.w.361.2		6
7.6	odd	2		441.4.a.t.1.2		3
12.11	even	2		2352.4.a.ci.1.1		3
21.2	odd	6		21.4.e.b.4.2	✓	6
21.5	even	6		147.4.e.n.67.2		6
21.11	odd	6		21.4.e.b.16.2	yes	6
21.17	even	6		147.4.e.n.79.2		6
21.20	even	2		147.4.a.m.1.2		3
84.11	even	6		336.4.q.k.289.3		6
84.23	even	6		336.4.q.k.193.3		6
84.83	odd	2		2352.4.a.cg.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
21.4.e.b.4.2	✓	6	21.2	odd	6
21.4.e.b.16.2	yes	6	21.11	odd	6
63.4.e.c.37.2		6	7.4	even	3
63.4.e.c.46.2		6	7.2	even	3
147.4.a.l.1.2		3	3.2	odd	2
147.4.a.m.1.2		3	21.20	even	2
147.4.e.n.67.2		6	21.5	even	6
147.4.e.n.79.2		6	21.17	even	6
336.4.q.k.193.3		6	84.23	even	6
336.4.q.k.289.3		6	84.11	even	6
441.4.a.s.1.2		3	1.1	even	1	trivial
441.4.a.t.1.2		3	7.6	odd	2
441.4.e.w.226.2		6	7.3	odd	6
441.4.e.w.361.2		6	7.5	odd	6
2352.4.a.cg.1.3		3	84.83	odd	2
2352.4.a.ci.1.1		3	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:	\( N \)	\(=\)	\( 441 = 3^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	441.a (trivial)

\(f(q)\)	\(=\)	\(q-0.248072 q^{2} -7.93846 q^{4} +12.4346 q^{5} +3.95388 q^{8} +O(q^{10})\)	\(q-0.248072 q^{2} -7.93846 q^{4} +12.4346 q^{5} +3.95388 q^{8} -3.08468 q^{10} -60.3115 q^{11} +36.4269 q^{13} +62.5268 q^{16} +48.7461 q^{17} -50.5500 q^{19} -98.7116 q^{20} +14.9616 q^{22} -138.792 q^{23} +29.6194 q^{25} -9.03649 q^{26} +61.1345 q^{29} -1.16935 q^{31} -47.1422 q^{32} -12.0925 q^{34} +69.5268 q^{37} +12.5400 q^{38} +49.1650 q^{40} -308.115 q^{41} +174.443 q^{43} +478.781 q^{44} +34.4305 q^{46} -389.362 q^{47} -7.34774 q^{50} -289.173 q^{52} -314.935 q^{53} -749.950 q^{55} -15.1657 q^{58} -844.526 q^{59} -338.538 q^{61} +0.290084 q^{62} -488.520 q^{64} +452.954 q^{65} -971.550 q^{67} -386.969 q^{68} +98.4698 q^{71} +710.235 q^{73} -17.2477 q^{74} +401.289 q^{76} -486.884 q^{79} +777.496 q^{80} +76.4348 q^{82} -605.688 q^{83} +606.139 q^{85} -43.2743 q^{86} -238.465 q^{88} -218.069 q^{89} +1101.80 q^{92} +96.5897 q^{94} -628.569 q^{95} -782.288 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - q^{2} + 25 q^{4} - 11 q^{5} - 39 q^{8}+O(q^{10}) \) 3 * q - q^2 + 25 * q^4 - 11 * q^5 - 39 * q^8	\( 3 q - q^{2} + 25 q^{4} - 11 q^{5} - 39 q^{8} - 55 q^{10} - 35 q^{11} + 62 q^{13} + 241 q^{16} - 48 q^{17} - 202 q^{19} - 439 q^{20} - 7 q^{22} - 216 q^{23} + 130 q^{25} - 274 q^{26} - 53 q^{29} - 95 q^{31} - 683 q^{32} - 24 q^{34} + 262 q^{37} + 398 q^{38} + 21 q^{40} - 244 q^{41} + 360 q^{43} + 905 q^{44} - 1056 q^{46} + 210 q^{47} + 1378 q^{50} + 324 q^{52} - 393 q^{53} - 1031 q^{55} - 1249 q^{58} - 1143 q^{59} - 70 q^{61} - 1059 q^{62} - 399 q^{64} + 472 q^{65} - 628 q^{67} - 1944 q^{68} - 318 q^{71} + 988 q^{73} - 1002 q^{74} - 2340 q^{76} + 861 q^{79} - 175 q^{80} + 124 q^{82} - 519 q^{83} + 1800 q^{85} + 3208 q^{86} - 891 q^{88} - 1766 q^{89} + 672 q^{92} - 3294 q^{94} + 736 q^{95} + 19 q^{97}+O(q^{100}) \) 3 * q - q^2 + 25 * q^4 - 11 * q^5 - 39 * q^8 - 55 * q^10 - 35 * q^11 + 62 * q^13 + 241 * q^16 - 48 * q^17 - 202 * q^19 - 439 * q^20 - 7 * q^22 - 216 * q^23 + 130 * q^25 - 274 * q^26 - 53 * q^29 - 95 * q^31 - 683 * q^32 - 24 * q^34 + 262 * q^37 + 398 * q^38 + 21 * q^40 - 244 * q^41 + 360 * q^43 + 905 * q^44 - 1056 * q^46 + 210 * q^47 + 1378 * q^50 + 324 * q^52 - 393 * q^53 - 1031 * q^55 - 1249 * q^58 - 1143 * q^59 - 70 * q^61 - 1059 * q^62 - 399 * q^64 + 472 * q^65 - 628 * q^67 - 1944 * q^68 - 318 * q^71 + 988 * q^73 - 1002 * q^74 - 2340 * q^76 + 861 * q^79 - 175 * q^80 + 124 * q^82 - 519 * q^83 + 1800 * q^85 + 3208 * q^86 - 891 * q^88 - 1766 * q^89 + 672 * q^92 - 3294 * q^94 + 736 * q^95 + 19 * q^97

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