LMFDB - Embedded newform 405.4.a.j.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [405,4,Mod(1,405)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(405, 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("405.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 405 = 3^{4} \cdot 5 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	405.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:	$23.8957735523$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.2292.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 13x + 1 $ x^3 - x^2 - 13*x + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 3 $
Twist minimal:	no (minimal twist has level 45)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$0.0765073$ of defining polynomial
Character	$\chi$	$=$	405.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+4.57358 q^{2} +12.9176 q^{4} -5.00000 q^{5} -20.1145 q^{7} +22.4912 q^{8} +O(q^{10})$	\(q+4.57358 q^{2} +12.9176 q^{4} -5.00000 q^{5} -20.1145 q^{7} +22.4912 q^{8} -22.8679 q^{10} -66.3416 q^{11} -46.8006 q^{13} -91.9955 q^{14} -0.475708 q^{16} +47.6233 q^{17} -9.95276 q^{19} -64.5882 q^{20} -303.418 q^{22} -9.59204 q^{23} +25.0000 q^{25} -214.046 q^{26} -259.832 q^{28} +178.735 q^{29} +154.037 q^{31} -182.105 q^{32} +217.809 q^{34} +100.573 q^{35} +248.864 q^{37} -45.5197 q^{38} -112.456 q^{40} -249.664 q^{41} -212.245 q^{43} -856.976 q^{44} -43.8700 q^{46} -475.694 q^{47} +61.5946 q^{49} +114.340 q^{50} -604.553 q^{52} +546.314 q^{53} +331.708 q^{55} -452.400 q^{56} +817.459 q^{58} +419.296 q^{59} -545.210 q^{61} +704.502 q^{62} -829.068 q^{64} +234.003 q^{65} -447.877 q^{67} +615.181 q^{68} +459.977 q^{70} -409.542 q^{71} -358.548 q^{73} +1138.20 q^{74} -128.566 q^{76} +1334.43 q^{77} -651.552 q^{79} +2.37854 q^{80} -1141.86 q^{82} +813.142 q^{83} -238.117 q^{85} -970.719 q^{86} -1492.10 q^{88} +201.000 q^{89} +941.373 q^{91} -123.906 q^{92} -2175.62 q^{94} +49.7638 q^{95} +252.149 q^{97} +281.708 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + q^{2} + 11 q^{4} - 15 q^{5} - 43 q^{7} + 27 q^{8}+O(q^{10}) $ 3 * q + q^2 + 11 * q^4 - 15 * q^5 - 43 * q^7 + 27 * q^8	$ 3 q + q^{2} + 11 q^{4} - 15 q^{5} - 43 q^{7} + 27 q^{8} - 5 q^{10} - 14 q^{11} + 40 q^{13} + 27 q^{14} - 13 q^{16} + 166 q^{17} - 164 q^{19} - 55 q^{20} - 376 q^{22} - 171 q^{23} + 75 q^{25} - 434 q^{26} - 517 q^{28} + 335 q^{29} - 352 q^{31} + 77 q^{32} - 52 q^{34} + 215 q^{35} + 402 q^{37} + 178 q^{38} - 135 q^{40} - 187 q^{41} - 602 q^{43} - 982 q^{44} - 201 q^{46} - 665 q^{47} + 430 q^{49} + 25 q^{50} - 456 q^{52} + 730 q^{53} + 70 q^{55} - 705 q^{56} + 217 q^{58} + 298 q^{59} - 1439 q^{61} + 1614 q^{62} - 1569 q^{64} - 200 q^{65} - 1849 q^{67} + 710 q^{68} - 135 q^{70} - 70 q^{71} - 368 q^{73} + 320 q^{74} + 204 q^{76} + 948 q^{77} - 382 q^{79} + 65 q^{80} - 575 q^{82} + 831 q^{83} - 830 q^{85} - 1580 q^{86} - 1428 q^{88} - 1719 q^{89} - 710 q^{91} + 1623 q^{92} - 2077 q^{94} + 820 q^{95} - 282 q^{97} - 2164 q^{98}+O(q^{100}) $ 3 * q + q^2 + 11 * q^4 - 15 * q^5 - 43 * q^7 + 27 * q^8 - 5 * q^10 - 14 * q^11 + 40 * q^13 + 27 * q^14 - 13 * q^16 + 166 * q^17 - 164 * q^19 - 55 * q^20 - 376 * q^22 - 171 * q^23 + 75 * q^25 - 434 * q^26 - 517 * q^28 + 335 * q^29 - 352 * q^31 + 77 * q^32 - 52 * q^34 + 215 * q^35 + 402 * q^37 + 178 * q^38 - 135 * q^40 - 187 * q^41 - 602 * q^43 - 982 * q^44 - 201 * q^46 - 665 * q^47 + 430 * q^49 + 25 * q^50 - 456 * q^52 + 730 * q^53 + 70 * q^55 - 705 * q^56 + 217 * q^58 + 298 * q^59 - 1439 * q^61 + 1614 * q^62 - 1569 * q^64 - 200 * q^65 - 1849 * q^67 + 710 * q^68 - 135 * q^70 - 70 * q^71 - 368 * q^73 + 320 * q^74 + 204 * q^76 + 948 * q^77 - 382 * q^79 + 65 * q^80 - 575 * q^82 + 831 * q^83 - 830 * q^85 - 1580 * q^86 - 1428 * q^88 - 1719 * q^89 - 710 * q^91 + 1623 * q^92 - 2077 * q^94 + 820 * q^95 - 282 * q^97 - 2164 * q^98

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 $

$2$	4.57358	1.61700	0.808502	−	0.588493i	$-0.200278\pi$
$2$	4.57358	1.61700	0.808502	+	0.588493i	$0.200278\pi$
$3$	0	0
$3$	0	0
$4$	12.9176	1.61470
$5$	−5.00000	−0.447214
$5$	−5.00000	−0.447214
$6$	0	0
$7$	−20.1145	−1.08608	−0.543041	−	0.839706i	$-0.682727\pi$
$7$	−20.1145	−1.08608	−0.543041	+	0.839706i	$0.682727\pi$
$8$	22.4912	0.993981
$9$	0	0
$10$	−22.8679	−0.723147
$11$	−66.3416	−1.81843	−0.909215	−	0.416327i	$-0.863317\pi$
$11$	−66.3416	−1.81843	−0.909215	+	0.416327i	$0.863317\pi$
$12$	0	0
$13$	−46.8006	−0.998473	−0.499237	−	0.866466i	$-0.666386\pi$
$13$	−46.8006	−0.998473	−0.499237	+	0.866466i	$0.666386\pi$
$14$	−91.9955	−1.75620
$15$	0	0
$16$	−0.475708	−0.00743294
$17$	47.6233	0.679432	0.339716	−	0.940528i	$-0.389669\pi$
$17$	47.6233	0.679432	0.339716	+	0.940528i	$0.389669\pi$
$18$	0	0
$19$	−9.95276	−0.120175	−0.0600874	−	0.998193i	$-0.519138\pi$
$19$	−9.95276	−0.120175	−0.0600874	+	0.998193i	$0.519138\pi$
$20$	−64.5882	−0.722118
$21$	0	0
$22$	−303.418	−2.94041
$23$	−9.59204	−0.0869599	−0.0434800	−	0.999054i	$-0.513844\pi$
$23$	−9.59204	−0.0869599	−0.0434800	+	0.999054i	$0.513844\pi$
$24$	0	0
$25$	25.0000	0.200000
$26$	−214.046	−1.61454
$27$	0	0
$28$	−259.832	−1.75370
$29$	178.735	1.14449	0.572246	−	0.820082i	$-0.306072\pi$
$29$	178.735	1.14449	0.572246	+	0.820082i	$0.306072\pi$
$30$	0	0
$31$	154.037	0.892449	0.446224	−	0.894921i	$-0.352768\pi$
$31$	154.037	0.892449	0.446224	+	0.894921i	$0.352768\pi$
$32$	−182.105	−1.00600
$33$	0	0
$34$	217.809	1.09865
$35$	100.573	0.485711
$36$	0	0
$37$	248.864	1.10576	0.552878	−	0.833262i	$-0.313529\pi$
$37$	248.864	1.10576	0.552878	+	0.833262i	$0.313529\pi$
$38$	−45.5197	−0.194323
$39$	0	0
$40$	−112.456	−0.444522
$41$	−249.664	−0.951000	−0.475500	−	0.879716i	$-0.657733\pi$
$41$	−249.664	−0.951000	−0.475500	+	0.879716i	$0.657733\pi$
$42$	0	0
$43$	−212.245	−0.752722	−0.376361	−	0.926473i	$-0.622825\pi$
$43$	−212.245	−0.752722	−0.376361	+	0.926473i	$0.622825\pi$
$44$	−856.976	−2.93623
$45$	0	0
$46$	−43.8700	−0.140615
$47$	−475.694	−1.47632	−0.738160	−	0.674626i	$-0.764305\pi$
$47$	−475.694	−1.47632	−0.738160	+	0.674626i	$0.764305\pi$
$48$	0	0
$49$	61.5946	0.179576
$50$	114.340	0.323401
$51$	0	0
$52$	−604.553	−1.61224
$53$	546.314	1.41589	0.707944	−	0.706269i	$-0.249623\pi$
$53$	546.314	1.41589	0.707944	+	0.706269i	$0.249623\pi$
$54$	0	0
$55$	331.708	0.813227
$56$	−452.400	−1.07955
$57$	0	0
$58$	817.459	1.85065
$59$	419.296	0.925216	0.462608	−	0.886563i	$-0.346914\pi$
$59$	419.296	0.925216	0.462608	+	0.886563i	$0.346914\pi$
$60$	0	0
$61$	−545.210	−1.14438	−0.572188	−	0.820122i	$-0.693905\pi$
$61$	−545.210	−1.14438	−0.572188	+	0.820122i	$0.693905\pi$
$62$	704.502	1.44309
$63$	0	0
$64$	−829.068	−1.61927
$65$	234.003	0.446531
$66$	0	0
$67$	−447.877	−0.816669	−0.408335	−	0.912832i	$-0.633890\pi$
$67$	−447.877	−0.816669	−0.408335	+	0.912832i	$0.633890\pi$
$68$	615.181	1.09708
$69$	0	0
$70$	459.977	0.785397
$71$	−409.542	−0.684559	−0.342279	−	0.939598i	$-0.611199\pi$
$71$	−409.542	−0.684559	−0.342279	+	0.939598i	$0.611199\pi$
$72$	0	0
$73$	−358.548	−0.574861	−0.287431	−	0.957801i	$-0.592801\pi$
$73$	−358.548	−0.574861	−0.287431	+	0.957801i	$0.592801\pi$
$74$	1138.20	1.78801
$75$	0	0
$76$	−128.566	−0.194047
$77$	1334.43	1.97497
$78$	0	0
$79$	−651.552	−0.927915	−0.463958	−	0.885857i	$-0.653571\pi$
$79$	−651.552	−0.927915	−0.463958	+	0.885857i	$0.653571\pi$
$80$	2.37854	0.00332411
$81$	0	0
$82$	−1141.86	−1.53777
$83$	813.142	1.07535	0.537675	−	0.843152i	$-0.319303\pi$
$83$	813.142	1.07535	0.537675	+	0.843152i	$0.319303\pi$
$84$	0	0
$85$	−238.117	−0.303851
$86$	−970.719	−1.21715
$87$	0	0
$88$	−1492.10	−1.80749
$89$	201.000	0.239393	0.119696	−	0.992811i	$-0.461808\pi$
$89$	201.000	0.239393	0.119696	+	0.992811i	$0.461808\pi$
$90$	0	0
$91$	941.373	1.08442
$92$	−123.906	−0.140415
$93$	0	0
$94$	−2175.62	−2.38722
$95$	49.7638	0.0537438
$96$	0	0
$97$	252.149	0.263936	0.131968	−	0.991254i	$-0.457870\pi$
$97$	252.149	0.263936	0.131968	+	0.991254i	$0.457870\pi$
$98$	281.708	0.290375
$99$	0	0
$100$	322.941	0.322941
$101$	−43.6026	−0.0429567	−0.0214783	−	0.999769i	$-0.506837\pi$
$101$	−43.6026	−0.0429567	−0.0214783	+	0.999769i	$0.506837\pi$
$102$	0	0
$103$	−1440.35	−1.37788	−0.688942	−	0.724816i	$-0.741925\pi$
$103$	−1440.35	−1.37788	−0.688942	+	0.724816i	$0.741925\pi$
$104$	−1052.60	−0.992463
$105$	0	0
$106$	2498.61	2.28950
$107$	−355.755	−0.321422	−0.160711	−	0.987002i	$-0.551379\pi$
$107$	−355.755	−0.321422	−0.160711	+	0.987002i	$0.551379\pi$
$108$	0	0
$109$	−1522.51	−1.33789	−0.668946	−	0.743311i	$-0.733254\pi$
$109$	−1522.51	−1.33789	−0.668946	+	0.743311i	$0.733254\pi$
$110$	1517.09	1.31499
$111$	0	0
$112$	9.56865	0.00807279
$113$	812.998	0.676818	0.338409	−	0.940999i	$-0.390111\pi$
$113$	812.998	0.676818	0.338409	+	0.940999i	$0.390111\pi$
$114$	0	0
$115$	47.9602	0.0388897
$116$	2308.83	1.84802
$117$	0	0
$118$	1917.69	1.49608
$119$	−957.921	−0.737920
$120$	0	0
$121$	3070.20	2.30669
$122$	−2493.56	−1.85046
$123$	0	0
$124$	1989.80	1.44104
$125$	−125.000	−0.0894427
$126$	0	0
$127$	−864.662	−0.604144	−0.302072	−	0.953285i	$-0.597678\pi$
$127$	−864.662	−0.604144	−0.302072	+	0.953285i	$0.597678\pi$
$128$	−2334.97	−1.61237
$129$	0	0
$130$	1070.23	0.722043
$131$	2178.52	1.45296	0.726482	−	0.687186i	$-0.241154\pi$
$131$	2178.52	1.45296	0.726482	+	0.687186i	$0.241154\pi$
$132$	0	0
$133$	200.195	0.130520
$134$	−2048.40	−1.32056
$135$	0	0
$136$	1071.11	0.675343
$137$	2299.16	1.43380	0.716900	−	0.697176i	$-0.245560\pi$
$137$	2299.16	1.43380	0.716900	+	0.697176i	$0.245560\pi$
$138$	0	0
$139$	−2133.95	−1.30215	−0.651077	−	0.759012i	$-0.725682\pi$
$139$	−2133.95	−1.30215	−0.651077	+	0.759012i	$0.725682\pi$
$140$	1299.16	0.784280
$141$	0	0
$142$	−1873.07	−1.10693
$143$	3104.83	1.81565
$144$	0	0
$145$	−893.675	−0.511832
$146$	−1639.85	−0.929554
$147$	0	0
$148$	3214.74	1.78547
$149$	−1750.62	−0.962525	−0.481263	−	0.876576i	$-0.659822\pi$
$149$	−1750.62	−0.962525	−0.481263	+	0.876576i	$0.659822\pi$
$150$	0	0
$151$	875.954	0.472080	0.236040	−	0.971743i	$-0.424150\pi$
$151$	875.954	0.472080	0.236040	+	0.971743i	$0.424150\pi$
$152$	−223.850	−0.119451
$153$	0	0
$154$	6103.12	3.19353
$155$	−770.186	−0.399115
$156$	0	0
$157$	−259.395	−0.131860	−0.0659298	−	0.997824i	$-0.521001\pi$
$157$	−259.395	−0.131860	−0.0659298	+	0.997824i	$0.521001\pi$
$158$	−2979.92	−1.50044
$159$	0	0
$160$	910.527	0.449897
$161$	192.939	0.0944457
$162$	0	0
$163$	−1201.80	−0.577498	−0.288749	−	0.957405i	$-0.593239\pi$
$163$	−1201.80	−0.577498	−0.288749	+	0.957405i	$0.593239\pi$
$164$	−3225.07	−1.53558
$165$	0	0
$166$	3718.97	1.73884
$167$	1678.90	0.777949	0.388975	−	0.921248i	$-0.372829\pi$
$167$	1678.90	0.777949	0.388975	+	0.921248i	$0.372829\pi$
$168$	0	0
$169$	−6.70324	−0.00305109
$170$	−1089.05	−0.491329
$171$	0	0
$172$	−2741.70	−1.21542
$173$	−931.798	−0.409499	−0.204749	−	0.978814i	$-0.565638\pi$
$173$	−931.798	−0.409499	−0.204749	+	0.978814i	$0.565638\pi$
$174$	0	0
$175$	−502.863	−0.217217
$176$	31.5592	0.0135163
$177$	0	0
$178$	919.289	0.387099
$179$	−1023.40	−0.427333	−0.213667	−	0.976907i	$-0.568541\pi$
$179$	−1023.40	−0.427333	−0.213667	+	0.976907i	$0.568541\pi$
$180$	0	0
$181$	2639.93	1.08411	0.542056	−	0.840342i	$-0.317646\pi$
$181$	2639.93	1.08411	0.542056	+	0.840342i	$0.317646\pi$
$182$	4305.44	1.75352
$183$	0	0
$184$	−215.737	−0.0864365
$185$	−1244.32	−0.494509
$186$	0	0
$187$	−3159.41	−1.23550
$188$	−6144.84	−2.38382
$189$	0	0
$190$	227.599	0.0869039
$191$	813.281	0.308099	0.154050	−	0.988063i	$-0.450768\pi$
$191$	813.281	0.308099	0.154050	+	0.988063i	$0.450768\pi$
$192$	0	0
$193$	−814.242	−0.303681	−0.151840	−	0.988405i	$-0.548520\pi$
$193$	−814.242	−0.303681	−0.151840	+	0.988405i	$0.548520\pi$
$194$	1153.22	0.426787
$195$	0	0
$196$	795.657	0.289962
$197$	−4078.41	−1.47500	−0.737499	−	0.675348i	$-0.763994\pi$
$197$	−4078.41	−1.47500	−0.737499	+	0.675348i	$0.763994\pi$
$198$	0	0
$199$	−1342.49	−0.478224	−0.239112	−	0.970992i	$-0.576856\pi$
$199$	−1342.49	−0.478224	−0.239112	+	0.970992i	$0.576856\pi$
$200$	562.280	0.198796
$201$	0	0
$202$	−199.420	−0.0694611
$203$	−3595.17	−1.24301
$204$	0	0
$205$	1248.32	0.425300
$206$	−6587.56	−2.22805
$207$	0	0
$208$	22.2634	0.00742159
$209$	660.282	0.218529
$210$	0	0
$211$	−2954.97	−0.964118	−0.482059	−	0.876139i	$-0.660111\pi$
$211$	−2954.97	−0.964118	−0.482059	+	0.876139i	$0.660111\pi$
$212$	7057.09	2.28624
$213$	0	0
$214$	−1627.07	−0.519740
$215$	1061.22	0.336627
$216$	0	0
$217$	−3098.39	−0.969273
$218$	−6963.33	−2.16338
$219$	0	0
$220$	4284.88	1.31312
$221$	−2228.80	−0.678395
$222$	0	0
$223$	3506.86	1.05308	0.526539	−	0.850151i	$-0.323489\pi$
$223$	3506.86	1.05308	0.526539	+	0.850151i	$0.323489\pi$
$224$	3662.97	1.09260
$225$	0	0
$226$	3718.31	1.09442
$227$	−652.826	−0.190879	−0.0954396	−	0.995435i	$-0.530426\pi$
$227$	−652.826	−0.190879	−0.0954396	+	0.995435i	$0.530426\pi$
$228$	0	0
$229$	4583.55	1.32266	0.661331	−	0.750094i	$-0.269992\pi$
$229$	4583.55	1.32266	0.661331	+	0.750094i	$0.269992\pi$
$230$	219.350	0.0628848
$231$	0	0
$232$	4019.97	1.13760
$233$	−317.527	−0.0892785	−0.0446392	−	0.999003i	$-0.514214\pi$
$233$	−317.527	−0.0892785	−0.0446392	+	0.999003i	$0.514214\pi$
$234$	0	0
$235$	2378.47	0.660230
$236$	5416.32	1.49395
$237$	0	0
$238$	−4381.13	−1.19322
$239$	−1857.67	−0.502773	−0.251386	−	0.967887i	$-0.580887\pi$
$239$	−1857.67	−0.502773	−0.251386	+	0.967887i	$0.580887\pi$
$240$	0	0
$241$	−3266.94	−0.873204	−0.436602	−	0.899655i	$-0.643818\pi$
$241$	−3266.94	−0.873204	−0.436602	+	0.899655i	$0.643818\pi$
$242$	14041.8	3.72993
$243$	0	0
$244$	−7042.82	−1.84783
$245$	−307.973	−0.0803089
$246$	0	0
$247$	465.795	0.119991
$248$	3464.49	0.887077
$249$	0	0
$250$	−571.698	−0.144629
$251$	5641.37	1.41865	0.709323	−	0.704884i	$-0.249001\pi$
$251$	5641.37	1.41865	0.709323	+	0.704884i	$0.249001\pi$
$252$	0	0
$253$	636.351	0.158131
$254$	−3954.60	−0.976904
$255$	0	0
$256$	−4046.61	−0.987943
$257$	1173.46	0.284819	0.142410	−	0.989808i	$-0.454515\pi$
$257$	1173.46	0.284819	0.142410	+	0.989808i	$0.454515\pi$
$258$	0	0
$259$	−5005.79	−1.20094
$260$	3022.77	0.721016
$261$	0	0
$262$	9963.64	2.34945
$263$	2897.96	0.679452	0.339726	−	0.940524i	$-0.389666\pi$
$263$	2897.96	0.679452	0.339726	+	0.940524i	$0.389666\pi$
$264$	0	0
$265$	−2731.57	−0.633204
$266$	915.609	0.211051
$267$	0	0
$268$	−5785.51	−1.31868
$269$	−2930.13	−0.664138	−0.332069	−	0.943255i	$-0.607747\pi$
$269$	−2930.13	−0.664138	−0.332069	+	0.943255i	$0.607747\pi$
$270$	0	0
$271$	−668.881	−0.149932	−0.0749661	−	0.997186i	$-0.523885\pi$
$271$	−668.881	−0.149932	−0.0749661	+	0.997186i	$0.523885\pi$
$272$	−22.6548	−0.00505018
$273$	0	0
$274$	10515.4	2.31846
$275$	−1658.54	−0.363686
$276$	0	0
$277$	−632.630	−0.137224	−0.0686121	−	0.997643i	$-0.521857\pi$
$277$	−632.630	−0.137224	−0.0686121	+	0.997643i	$0.521857\pi$
$278$	−9759.80	−2.10559
$279$	0	0
$280$	2262.00	0.482787
$281$	3595.30	0.763265	0.381633	−	0.924314i	$-0.375362\pi$
$281$	3595.30	0.763265	0.381633	+	0.924314i	$0.375362\pi$
$282$	0	0
$283$	504.368	0.105942	0.0529710	−	0.998596i	$-0.483131\pi$
$283$	504.368	0.105942	0.0529710	+	0.998596i	$0.483131\pi$
$284$	−5290.31	−1.10536
$285$	0	0
$286$	14200.2	2.93592
$287$	5021.88	1.03286
$288$	0	0
$289$	−2645.02	−0.538372
$290$	−4087.29	−0.827635
$291$	0	0
$292$	−4631.60	−0.928232
$293$	−6198.50	−1.23591	−0.617953	−	0.786215i	$-0.712038\pi$
$293$	−6198.50	−1.23591	−0.617953	+	0.786215i	$0.712038\pi$
$294$	0	0
$295$	−2096.48	−0.413769
$296$	5597.26	1.09910
$297$	0	0
$298$	−8006.60	−1.55641
$299$	448.913	0.0868271
$300$	0	0
$301$	4269.21	0.817518
$302$	4006.25	0.763356
$303$	0	0
$304$	4.73461	0.000893251 0
$305$	2726.05	0.511781
$306$	0	0
$307$	−1966.79	−0.365636	−0.182818	−	0.983147i	$-0.558522\pi$
$307$	−1966.79	−0.365636	−0.182818	+	0.983147i	$0.558522\pi$
$308$	17237.7	3.18899
$309$	0	0
$310$	−3522.51	−0.645371
$311$	2306.11	0.420474	0.210237	−	0.977650i	$-0.432576\pi$
$311$	2306.11	0.420474	0.210237	+	0.977650i	$0.432576\pi$
$312$	0	0
$313$	10302.4	1.86047	0.930234	−	0.366968i	$-0.119604\pi$
$313$	10302.4	1.86047	0.930234	+	0.366968i	$0.119604\pi$
$314$	−1186.36	−0.213218
$315$	0	0
$316$	−8416.51	−1.49831
$317$	1701.83	0.301528	0.150764	−	0.988570i	$-0.451827\pi$
$317$	1701.83	0.301528	0.150764	+	0.988570i	$0.451827\pi$
$318$	0	0
$319$	−11857.6	−2.08118
$320$	4145.34	0.724161
$321$	0	0
$322$	882.424	0.152719
$323$	−473.983	−0.0816506
$324$	0	0
$325$	−1170.02	−0.199695
$326$	−5496.52	−0.933817
$327$	0	0
$328$	−5615.25	−0.945276
$329$	9568.36	1.60341
$330$	0	0
$331$	−8351.49	−1.38683	−0.693413	−	0.720540i	$-0.743894\pi$
$331$	−8351.49	−1.38683	−0.693413	+	0.720540i	$0.743894\pi$
$332$	10503.9	1.73637
$333$	0	0
$334$	7678.61	1.25795
$335$	2239.38	0.365226
$336$	0	0
$337$	−7857.42	−1.27009	−0.635046	−	0.772474i	$-0.719019\pi$
$337$	−7857.42	−1.27009	−0.635046	+	0.772474i	$0.719019\pi$
$338$	−30.6578	−0.00493362
$339$	0	0
$340$	−3075.90	−0.490630
$341$	−10219.1	−1.62286
$342$	0	0
$343$	5660.34	0.891048
$344$	−4773.64	−0.748191
$345$	0	0
$346$	−4261.65	−0.662162
$347$	−1212.18	−0.187530	−0.0937652	−	0.995594i	$-0.529890\pi$
$347$	−1212.18	−0.187530	−0.0937652	+	0.995594i	$0.529890\pi$
$348$	0	0
$349$	1398.66	0.214524	0.107262	−	0.994231i	$-0.465792\pi$
$349$	1398.66	0.214524	0.107262	+	0.994231i	$0.465792\pi$
$350$	−2299.89	−0.351240
$351$	0	0
$352$	12081.2	1.82934
$353$	−2628.43	−0.396309	−0.198154	−	0.980171i	$-0.563495\pi$
$353$	−2628.43	−0.396309	−0.198154	+	0.980171i	$0.563495\pi$
$354$	0	0
$355$	2047.71	0.306144
$356$	2596.44	0.386548
$357$	0	0
$358$	−4680.61	−0.691000
$359$	−3677.48	−0.540640	−0.270320	−	0.962770i	$-0.587130\pi$
$359$	−3677.48	−0.540640	−0.270320	+	0.962770i	$0.587130\pi$
$360$	0	0
$361$	−6759.94	−0.985558
$362$	12073.9	1.75301
$363$	0	0
$364$	12160.3	1.75103
$365$	1792.74	0.257086
$366$	0	0
$367$	11429.8	1.62569	0.812846	−	0.582479i	$-0.197917\pi$
$367$	11429.8	1.62569	0.812846	+	0.582479i	$0.197917\pi$
$368$	4.56301	0.000646368 0
$369$	0	0
$370$	−5691.00	−0.799624
$371$	−10988.9	−1.53777
$372$	0	0
$373$	2259.90	0.313708	0.156854	−	0.987622i	$-0.449865\pi$
$373$	2259.90	0.313708	0.156854	+	0.987622i	$0.449865\pi$
$374$	−14449.8	−1.99781
$375$	0	0
$376$	−10698.9	−1.46743
$377$	−8364.90	−1.14274
$378$	0	0
$379$	−11815.8	−1.60142	−0.800709	−	0.599053i	$-0.795544\pi$
$379$	−11815.8	−1.60142	−0.800709	+	0.599053i	$0.795544\pi$
$380$	642.831	0.0867803
$381$	0	0
$382$	3719.60	0.498198
$383$	8080.22	1.07802	0.539008	−	0.842301i	$-0.318799\pi$
$383$	8080.22	1.07802	0.539008	+	0.842301i	$0.318799\pi$
$384$	0	0
$385$	−6672.15	−0.883232
$386$	−3724.00	−0.491054
$387$	0	0
$388$	3257.17	0.426180
$389$	−3100.43	−0.404108	−0.202054	−	0.979374i	$-0.564762\pi$
$389$	−3100.43	−0.404108	−0.202054	+	0.979374i	$0.564762\pi$
$390$	0	0
$391$	−456.805	−0.0590834
$392$	1385.34	0.178495
$393$	0	0
$394$	−18652.9	−2.38508
$395$	3257.76	0.414976
$396$	0	0
$397$	−11990.1	−1.51578	−0.757890	−	0.652382i	$-0.773770\pi$
$397$	−11990.1	−1.51578	−0.757890	+	0.652382i	$0.773770\pi$
$398$	−6139.98	−0.773291
$399$	0	0
$400$	−11.8927	−0.00148659
$401$	−12853.3	−1.60065	−0.800326	−	0.599565i	$-0.795340\pi$
$401$	−12853.3	−1.60065	−0.800326	+	0.599565i	$0.795340\pi$
$402$	0	0
$403$	−7209.04	−0.891086
$404$	−563.243	−0.0693623
$405$	0	0
$406$	−16442.8	−2.00996
$407$	−16510.0	−2.01074
$408$	0	0
$409$	2225.09	0.269006	0.134503	−	0.990913i	$-0.457056\pi$
$409$	2225.09	0.269006	0.134503	+	0.990913i	$0.457056\pi$
$410$	5709.30	0.687712
$411$	0	0
$412$	−18605.9	−2.22488
$413$	−8433.95	−1.00486
$414$	0	0
$415$	−4065.71	−0.480911
$416$	8522.65	1.00446
$417$	0	0
$418$	3019.85	0.353363
$419$	−9676.65	−1.12825	−0.564123	−	0.825691i	$-0.690786\pi$
$419$	−9676.65	−1.12825	−0.564123	+	0.825691i	$0.690786\pi$
$420$	0	0
$421$	−9962.60	−1.15332	−0.576660	−	0.816984i	$-0.695644\pi$
$421$	−9962.60	−1.15332	−0.576660	+	0.816984i	$0.695644\pi$
$422$	−13514.8	−1.55898
$423$	0	0
$424$	12287.3	1.40737
$425$	1190.58	0.135886
$426$	0	0
$427$	10966.6	1.24289
$428$	−4595.51	−0.519001
$429$	0	0
$430$	4853.59	0.544328
$431$	2461.47	0.275092	0.137546	−	0.990495i	$-0.456078\pi$
$431$	2461.47	0.275092	0.137546	+	0.990495i	$0.456078\pi$
$432$	0	0
$433$	7818.49	0.867743	0.433871	−	0.900975i	$-0.357147\pi$
$433$	7818.49	0.867743	0.433871	+	0.900975i	$0.357147\pi$
$434$	−14170.7	−1.56732
$435$	0	0
$436$	−19667.3	−2.16030
$437$	95.4672	0.0104504
$438$	0	0
$439$	6211.83	0.675340	0.337670	−	0.941264i	$-0.390361\pi$
$439$	6211.83	0.675340	0.337670	+	0.941264i	$0.390361\pi$
$440$	7460.51	0.808332
$441$	0	0
$442$	−10193.6	−1.09697
$443$	−2984.35	−0.320069	−0.160034	−	0.987111i	$-0.551161\pi$
$443$	−2984.35	−0.320069	−0.160034	+	0.987111i	$0.551161\pi$
$444$	0	0
$445$	−1005.00	−0.107060
$446$	16038.9	1.70283
$447$	0	0
$448$	16676.3	1.75867
$449$	810.476	0.0851865	0.0425932	−	0.999092i	$-0.486438\pi$
$449$	810.476	0.0851865	0.0425932	+	0.999092i	$0.486438\pi$
$450$	0	0
$451$	16563.1	1.72933
$452$	10502.0	1.09286
$453$	0	0
$454$	−2985.75	−0.308653
$455$	−4706.86	−0.484970
$456$	0	0
$457$	1571.77	0.160885	0.0804425	−	0.996759i	$-0.474367\pi$
$457$	1571.77	0.160885	0.0804425	+	0.996759i	$0.474367\pi$
$458$	20963.2	2.13875
$459$	0	0
$460$	619.532	0.0627953
$461$	−2062.69	−0.208393	−0.104196	−	0.994557i	$-0.533227\pi$
$461$	−2062.69	−0.208393	−0.104196	+	0.994557i	$0.533227\pi$
$462$	0	0
$463$	2783.23	0.279369	0.139684	−	0.990196i	$-0.455391\pi$
$463$	2783.23	0.279369	0.139684	+	0.990196i	$0.455391\pi$
$464$	−85.0256	−0.00850693
$465$	0	0
$466$	−1452.24	−0.144364
$467$	−10939.7	−1.08400	−0.541999	−	0.840379i	$-0.682332\pi$
$467$	−10939.7	−1.08400	−0.541999	+	0.840379i	$0.682332\pi$
$468$	0	0
$469$	9008.83	0.886970
$470$	10878.1	1.06760
$471$	0	0
$472$	9430.49	0.919647
$473$	14080.7	1.36877
$474$	0	0
$475$	−248.819	−0.0240349
$476$	−12374.1	−1.19152
$477$	0	0
$478$	−8496.21	−0.812986
$479$	−14623.7	−1.39493	−0.697467	−	0.716617i	$-0.745690\pi$
$479$	−14623.7	−1.39493	−0.697467	+	0.716617i	$0.745690\pi$
$480$	0	0
$481$	−11647.0	−1.10407
$482$	−14941.6	−1.41197
$483$	0	0
$484$	39659.8	3.72462
$485$	−1260.74	−0.118036
$486$	0	0
$487$	16473.6	1.53284	0.766419	−	0.642341i	$-0.222037\pi$
$487$	16473.6	1.53284	0.766419	+	0.642341i	$0.222037\pi$
$488$	−12262.4	−1.13749
$489$	0	0
$490$	−1408.54	−0.129860
$491$	−20529.2	−1.88690	−0.943450	−	0.331514i	$-0.892441\pi$
$491$	−20529.2	−1.88690	−0.943450	+	0.331514i	$0.892441\pi$
$492$	0	0
$493$	8511.95	0.777604
$494$	2130.35	0.194026
$495$	0	0
$496$	−73.2768	−0.00663352
$497$	8237.74	0.743488
$498$	0	0
$499$	−14404.6	−1.29226	−0.646131	−	0.763226i	$-0.723614\pi$
$499$	−14404.6	−1.29226	−0.646131	+	0.763226i	$0.723614\pi$
$500$	−1614.70	−0.144424
$501$	0	0
$502$	25801.3	2.29396
$503$	2953.63	0.261821	0.130910	−	0.991394i	$-0.458210\pi$
$503$	2953.63	0.261821	0.130910	+	0.991394i	$0.458210\pi$
$504$	0	0
$505$	218.013	0.0192108
$506$	2910.40	0.255698
$507$	0	0
$508$	−11169.4	−0.975515
$509$	17368.9	1.51250	0.756250	−	0.654283i	$-0.227029\pi$
$509$	17368.9	1.51250	0.756250	+	0.654283i	$0.227029\pi$
$510$	0	0
$511$	7212.03	0.624347
$512$	172.223	0.0148657
$513$	0	0
$514$	5366.92	0.460554
$515$	7201.76	0.616209
$516$	0	0
$517$	31558.3	2.68459
$518$	−22894.4	−1.94193
$519$	0	0
$520$	5263.01	0.443843
$521$	6146.30	0.516841	0.258421	−	0.966033i	$-0.416798\pi$
$521$	6146.30	0.516841	0.258421	+	0.966033i	$0.416798\pi$
$522$	0	0
$523$	−4554.68	−0.380807	−0.190404	−	0.981706i	$-0.560980\pi$
$523$	−4554.68	−0.380807	−0.190404	+	0.981706i	$0.560980\pi$
$524$	28141.3	2.34611
$525$	0	0
$526$	13254.1	1.09868
$527$	7335.77	0.606359
$528$	0	0
$529$	−12075.0	−0.992438
$530$	−12493.1	−1.02389
$531$	0	0
$532$	2586.05	0.210751
$533$	11684.4	0.949548
$534$	0	0
$535$	1778.77	0.143744
$536$	−10073.3	−0.811753
$537$	0	0
$538$	−13401.2	−1.07391
$539$	−4086.28	−0.326547
$540$	0	0
$541$	18091.8	1.43776	0.718879	−	0.695135i	$-0.244655\pi$
$541$	18091.8	1.43776	0.718879	+	0.695135i	$0.244655\pi$
$542$	−3059.18	−0.242441
$543$	0	0
$544$	−8672.47	−0.683509
$545$	7612.56	0.598323
$546$	0	0
$547$	15781.4	1.23357	0.616786	−	0.787131i	$-0.288435\pi$
$547$	15781.4	1.23357	0.616786	+	0.787131i	$0.288435\pi$
$548$	29699.7	2.31516
$549$	0	0
$550$	−7585.46	−0.588082
$551$	−1778.91	−0.137539
$552$	0	0
$553$	13105.7	1.00779
$554$	−2893.39	−0.221892
$555$	0	0
$556$	−27565.6	−2.10259
$557$	13954.5	1.06153	0.530766	−	0.847519i	$-0.321904\pi$
$557$	13954.5	1.06153	0.530766	+	0.847519i	$0.321904\pi$
$558$	0	0
$559$	9933.18	0.751572
$560$	−47.8432	−0.00361026
$561$	0	0
$562$	16443.4	1.23420
$563$	13603.9	1.01836	0.509181	−	0.860659i	$-0.329948\pi$
$563$	13603.9	1.01836	0.509181	+	0.860659i	$0.329948\pi$
$564$	0	0
$565$	−4064.99	−0.302682
$566$	2306.77	0.171309
$567$	0	0
$568$	−9211.10	−0.680438
$569$	12458.6	0.917912	0.458956	−	0.888459i	$-0.348224\pi$
$569$	12458.6	0.917912	0.458956	+	0.888459i	$0.348224\pi$
$570$	0	0
$571$	−13457.1	−0.986275	−0.493137	−	0.869951i	$-0.664150\pi$
$571$	−13457.1	−0.986275	−0.493137	+	0.869951i	$0.664150\pi$
$572$	40107.0	2.93175
$573$	0	0
$574$	22968.0	1.67015
$575$	−239.801	−0.0173920
$576$	0	0
$577$	3722.70	0.268592	0.134296	−	0.990941i	$-0.457123\pi$
$577$	3722.70	0.268592	0.134296	+	0.990941i	$0.457123\pi$
$578$	−12097.2	−0.870550
$579$	0	0
$580$	−11544.2	−0.826458
$581$	−16356.0	−1.16792
$582$	0	0
$583$	−36243.4	−2.57469
$584$	−8064.19	−0.571401
$585$	0	0
$586$	−28349.3	−1.99847
$587$	17540.3	1.23333	0.616667	−	0.787224i	$-0.288483\pi$
$587$	17540.3	1.23333	0.616667	+	0.787224i	$0.288483\pi$
$588$	0	0
$589$	−1533.10	−0.107250
$590$	−9588.43	−0.669067
$591$	0	0
$592$	−118.387	−0.00821902
$593$	22350.6	1.54777	0.773886	−	0.633325i	$-0.218310\pi$
$593$	22350.6	1.54777	0.773886	+	0.633325i	$0.218310\pi$
$594$	0	0
$595$	4789.60	0.330008
$596$	−22613.9	−1.55419
$597$	0	0
$598$	2053.14	0.140400
$599$	2561.62	0.174733	0.0873665	−	0.996176i	$-0.472155\pi$
$599$	2561.62	0.174733	0.0873665	+	0.996176i	$0.472155\pi$
$600$	0	0
$601$	13385.0	0.908463	0.454232	−	0.890884i	$-0.349914\pi$
$601$	13385.0	0.908463	0.454232	+	0.890884i	$0.349914\pi$
$602$	19525.6	1.32193
$603$	0	0
$604$	11315.3	0.762270
$605$	−15351.0	−1.03158
$606$	0	0
$607$	−28628.6	−1.91433	−0.957166	−	0.289539i	$-0.906498\pi$
$607$	−28628.6	−1.91433	−0.957166	+	0.289539i	$0.906498\pi$
$608$	1812.45	0.120896
$609$	0	0
$610$	12467.8	0.827552
$611$	22262.8	1.47407
$612$	0	0
$613$	7188.12	0.473614	0.236807	−	0.971557i	$-0.423899\pi$
$613$	7188.12	0.473614	0.236807	+	0.971557i	$0.423899\pi$
$614$	−8995.25	−0.591236
$615$	0	0
$616$	30013.0	1.96308
$617$	−15533.8	−1.01356	−0.506779	−	0.862076i	$-0.669164\pi$
$617$	−15533.8	−1.01356	−0.506779	+	0.862076i	$0.669164\pi$
$618$	0	0
$619$	−22159.8	−1.43890	−0.719450	−	0.694545i	$-0.755606\pi$
$619$	−22159.8	−1.43890	−0.719450	+	0.694545i	$0.755606\pi$
$620$	−9948.99	−0.644453
$621$	0	0
$622$	10547.2	0.679908
$623$	−4043.02	−0.260000
$624$	0	0
$625$	625.000	0.0400000
$626$	47118.9	3.00839
$627$	0	0
$628$	−3350.77	−0.212914
$629$	11851.7	0.751287
$630$	0	0
$631$	−25582.8	−1.61400	−0.807002	−	0.590549i	$-0.798911\pi$
$631$	−25582.8	−1.61400	−0.807002	+	0.590549i	$0.798911\pi$
$632$	−14654.2	−0.922330
$633$	0	0
$634$	7783.46	0.487572
$635$	4323.31	0.270182
$636$	0	0
$637$	−2882.66	−0.179302
$638$	−54231.5	−3.36527
$639$	0	0
$640$	11674.8	0.721076
$641$	−3810.68	−0.234809	−0.117405	−	0.993084i	$-0.537457\pi$
$641$	−3810.68	−0.234809	−0.117405	+	0.993084i	$0.537457\pi$
$642$	0	0
$643$	26720.1	1.63878	0.819391	−	0.573235i	$-0.194312\pi$
$643$	26720.1	1.63878	0.819391	+	0.573235i	$0.194312\pi$
$644$	2492.32	0.152502
$645$	0	0
$646$	−2167.80	−0.132029
$647$	−5114.23	−0.310759	−0.155380	−	0.987855i	$-0.549660\pi$
$647$	−5114.23	−0.310759	−0.155380	+	0.987855i	$0.549660\pi$
$648$	0	0
$649$	−27816.8	−1.68244
$650$	−5351.16	−0.322907
$651$	0	0
$652$	−15524.4	−0.932489
$653$	8871.05	0.531625	0.265813	−	0.964025i	$-0.414360\pi$
$653$	8871.05	0.531625	0.265813	+	0.964025i	$0.414360\pi$
$654$	0	0
$655$	−10892.6	−0.649785
$656$	118.767	0.00706872
$657$	0	0
$658$	43761.7	2.59272
$659$	24204.5	1.43076	0.715382	−	0.698734i	$-0.246253\pi$
$659$	24204.5	1.43076	0.715382	+	0.698734i	$0.246253\pi$
$660$	0	0
$661$	21379.5	1.25805	0.629023	−	0.777387i	$-0.283455\pi$
$661$	21379.5	1.25805	0.629023	+	0.777387i	$0.283455\pi$
$662$	−38196.2	−2.24250
$663$	0	0
$664$	18288.6	1.06888
$665$	−1000.98	−0.0583702
$666$	0	0
$667$	−1714.43	−0.0995248
$668$	21687.5	1.25616
$669$	0	0
$670$	10242.0	0.590571
$671$	36170.1	2.08097
$672$	0	0
$673$	29701.4	1.70119	0.850597	−	0.525818i	$-0.176241\pi$
$673$	29701.4	1.70119	0.850597	+	0.525818i	$0.176241\pi$
$674$	−35936.6	−2.05375
$675$	0	0
$676$	−86.5900	−0.00492661
$677$	−2661.48	−0.151092	−0.0755459	−	0.997142i	$-0.524070\pi$
$677$	−2661.48	−0.151092	−0.0755459	+	0.997142i	$0.524070\pi$
$678$	0	0
$679$	−5071.86	−0.286657
$680$	−5355.53	−0.302022
$681$	0	0
$682$	−46737.8	−2.62417
$683$	−28698.1	−1.60776	−0.803882	−	0.594789i	$-0.797235\pi$
$683$	−28698.1	−1.60776	−0.803882	+	0.594789i	$0.797235\pi$
$684$	0	0
$685$	−11495.8	−0.641215
$686$	25888.0	1.44083
$687$	0	0
$688$	100.967	0.00559493
$689$	−25567.8	−1.41373
$690$	0	0
$691$	16825.2	0.926284	0.463142	−	0.886284i	$-0.346722\pi$
$691$	16825.2	0.926284	0.463142	+	0.886284i	$0.346722\pi$
$692$	−12036.6	−0.661220
$693$	0	0
$694$	−5543.99	−0.303238
$695$	10669.8	0.582341
$696$	0	0
$697$	−11889.8	−0.646140
$698$	6396.90	0.346886
$699$	0	0
$700$	−6495.81	−0.350741
$701$	998.795	0.0538145	0.0269073	−	0.999638i	$-0.491434\pi$
$701$	998.795	0.0538145	0.0269073	+	0.999638i	$0.491434\pi$
$702$	0	0
$703$	−2476.88	−0.132884
$704$	55001.7	2.94454
$705$	0	0
$706$	−12021.3	−0.640834
$707$	877.047	0.0466545
$708$	0	0
$709$	33253.8	1.76145	0.880727	−	0.473624i	$-0.157054\pi$
$709$	33253.8	1.76145	0.880727	+	0.473624i	$0.157054\pi$
$710$	9365.36	0.495036
$711$	0	0
$712$	4520.73	0.237952
$713$	−1477.53	−0.0776073
$714$	0	0
$715$	−15524.1	−0.811985
$716$	−13219.9	−0.690017
$717$	0	0
$718$	−16819.2	−0.874218
$719$	−1178.94	−0.0611503	−0.0305752	−	0.999532i	$-0.509734\pi$
$719$	−1178.94	−0.0611503	−0.0305752	+	0.999532i	$0.509734\pi$
$720$	0	0
$721$	28972.0	1.49650
$722$	−30917.1	−1.59365
$723$	0	0
$724$	34101.6	1.75052
$725$	4468.37	0.228898
$726$	0	0
$727$	12676.7	0.646703	0.323351	−	0.946279i	$-0.395190\pi$
$727$	12676.7	0.646703	0.323351	+	0.946279i	$0.395190\pi$
$728$	21172.6	1.07790
$729$	0	0
$730$	8199.24	0.415709
$731$	−10107.8	−0.511423
$732$	0	0
$733$	9806.85	0.494167	0.247083	−	0.968994i	$-0.420528\pi$
$733$	9806.85	0.494167	0.247083	+	0.968994i	$0.420528\pi$
$734$	52275.0	2.62875
$735$	0	0
$736$	1746.76	0.0874817
$737$	29712.8	1.48506
$738$	0	0
$739$	−29970.4	−1.49185	−0.745927	−	0.666028i	$-0.767993\pi$
$739$	−29970.4	−1.49185	−0.745927	+	0.666028i	$0.767993\pi$
$740$	−16073.7	−0.798487
$741$	0	0
$742$	−50258.4	−2.48658
$743$	−21697.6	−1.07134	−0.535670	−	0.844427i	$-0.679941\pi$
$743$	−21697.6	−1.07134	−0.535670	+	0.844427i	$0.679941\pi$
$744$	0	0
$745$	8753.09	0.430454
$746$	10335.8	0.507267
$747$	0	0
$748$	−40812.1	−1.99497
$749$	7155.84	0.349090
$750$	0	0
$751$	17024.2	0.827193	0.413596	−	0.910460i	$-0.364272\pi$
$751$	17024.2	0.827193	0.413596	+	0.910460i	$0.364272\pi$
$752$	226.291	0.0109734
$753$	0	0
$754$	−38257.6	−1.84782
$755$	−4379.77	−0.211121
$756$	0	0
$757$	30745.2	1.47616	0.738080	−	0.674714i	$-0.235733\pi$
$757$	30745.2	1.47616	0.738080	+	0.674714i	$0.235733\pi$
$758$	−54040.6	−2.58950
$759$	0	0
$760$	1119.25	0.0534203
$761$	−21496.6	−1.02398	−0.511992	−	0.858990i	$-0.671092\pi$
$761$	−21496.6	−1.02398	−0.511992	+	0.858990i	$0.671092\pi$
$762$	0	0
$763$	30624.6	1.45306
$764$	10505.7	0.497489
$765$	0	0
$766$	36955.5	1.74316
$767$	−19623.3	−0.923803
$768$	0	0
$769$	22057.5	1.03435	0.517174	−	0.855880i	$-0.326984\pi$
$769$	22057.5	1.03435	0.517174	+	0.855880i	$0.326984\pi$
$770$	−30515.6	−1.42819
$771$	0	0
$772$	−10518.1	−0.490355
$773$	−30155.8	−1.40314	−0.701570	−	0.712601i	$-0.747517\pi$
$773$	−30155.8	−1.40314	−0.701570	+	0.712601i	$0.747517\pi$
$774$	0	0
$775$	3850.93	0.178490
$776$	5671.14	0.262348
$777$	0	0
$778$	−14180.1	−0.653445
$779$	2484.85	0.114286
$780$	0	0
$781$	27169.6	1.24482
$782$	−2089.23	−0.0955381
$783$	0	0
$784$	−29.3010	−0.00133478
$785$	1296.97	0.0589694
$786$	0	0
$787$	−3248.72	−0.147147	−0.0735733	−	0.997290i	$-0.523440\pi$
$787$	−3248.72	−0.147147	−0.0735733	+	0.997290i	$0.523440\pi$
$788$	−52683.5	−2.38169
$789$	0	0
$790$	14899.6	0.671019
$791$	−16353.1	−0.735081
$792$	0	0
$793$	25516.1	1.14263
$794$	−54837.6	−2.45103
$795$	0	0
$796$	−17341.8	−0.772191
$797$	27710.2	1.23155	0.615775	−	0.787922i	$-0.288843\pi$
$797$	27710.2	1.23155	0.615775	+	0.787922i	$0.288843\pi$
$798$	0	0
$799$	−22654.1	−1.00306
$800$	−4552.64	−0.201200
$801$	0	0
$802$	−58785.4	−2.58826
$803$	23786.6	1.04535
$804$	0	0
$805$	−964.697	−0.0422374
$806$	−32971.1	−1.44089
$807$	0	0
$808$	−980.676	−0.0426981
$809$	2244.10	0.0975259	0.0487630	−	0.998810i	$-0.484472\pi$
$809$	2244.10	0.0975259	0.0487630	+	0.998810i	$0.484472\pi$
$810$	0	0
$811$	−2739.73	−0.118625	−0.0593126	−	0.998239i	$-0.518891\pi$
$811$	−2739.73	−0.118625	−0.0593126	+	0.998239i	$0.518891\pi$
$812$	−46441.1	−2.00710
$813$	0	0
$814$	−75510.0	−3.25138
$815$	6008.99	0.258265
$816$	0	0
$817$	2112.42	0.0904581
$818$	10176.6	0.434984
$819$	0	0
$820$	16125.4	0.686734
$821$	17232.1	0.732528	0.366264	−	0.930511i	$-0.380637\pi$
$821$	17232.1	0.732528	0.366264	+	0.930511i	$0.380637\pi$
$822$	0	0
$823$	19285.9	0.816845	0.408422	−	0.912793i	$-0.366079\pi$
$823$	19285.9	0.816845	0.408422	+	0.912793i	$0.366079\pi$
$824$	−32395.3	−1.36959
$825$	0	0
$826$	−38573.4	−1.62487
$827$	−26379.4	−1.10919	−0.554595	−	0.832120i	$-0.687127\pi$
$827$	−26379.4	−1.10919	−0.554595	+	0.832120i	$0.687127\pi$
$828$	0	0
$829$	−8718.15	−0.365252	−0.182626	−	0.983182i	$-0.558460\pi$
$829$	−8718.15	−0.365252	−0.182626	+	0.983182i	$0.558460\pi$
$830$	−18594.9	−0.777635
$831$	0	0
$832$	38800.9	1.61680
$833$	2933.34	0.122010
$834$	0	0
$835$	−8394.52	−0.347910
$836$	8529.28	0.352860
$837$	0	0
$838$	−44256.9	−1.82438
$839$	13476.2	0.554529	0.277265	−	0.960794i	$-0.410572\pi$
$839$	13476.2	0.554529	0.277265	+	0.960794i	$0.410572\pi$
$840$	0	0
$841$	7557.17	0.309860
$842$	−45564.8	−1.86492
$843$	0	0
$844$	−38171.3	−1.55677
$845$	33.5162	0.00136449
$846$	0	0
$847$	−61755.7	−2.50526
$848$	−259.886	−0.0105242
$849$	0	0
$850$	5445.23	0.219729
$851$	−2387.11	−0.0961565
$852$	0	0
$853$	−28236.6	−1.13342	−0.566708	−	0.823919i	$-0.691783\pi$
$853$	−28236.6	−1.13342	−0.566708	+	0.823919i	$0.691783\pi$
$854$	50156.8	2.00975
$855$	0	0
$856$	−8001.36	−0.319487
$857$	−8458.16	−0.337136	−0.168568	−	0.985690i	$-0.553914\pi$
$857$	−8458.16	−0.337136	−0.168568	+	0.985690i	$0.553914\pi$
$858$	0	0
$859$	−23054.3	−0.915720	−0.457860	−	0.889024i	$-0.651384\pi$
$859$	−23054.3	−0.915720	−0.457860	+	0.889024i	$0.651384\pi$
$860$	13708.5	0.543554
$861$	0	0
$862$	11257.7	0.444826
$863$	42314.4	1.66906	0.834529	−	0.550963i	$-0.185740\pi$
$863$	42314.4	1.66906	0.834529	+	0.550963i	$0.185740\pi$
$864$	0	0
$865$	4658.99	0.183133
$866$	35758.5	1.40314
$867$	0	0
$868$	−40023.9	−1.56509
$869$	43225.0	1.68735
$870$	0	0
$871$	20960.9	0.815422
$872$	−34243.1	−1.32984
$873$	0	0
$874$	436.627	0.0168983
$875$	2514.32	0.0971422
$876$	0	0
$877$	−10180.0	−0.391966	−0.195983	−	0.980607i	$-0.562790\pi$
$877$	−10180.0	−0.391966	−0.195983	+	0.980607i	$0.562790\pi$
$878$	28410.3	1.09203
$879$	0	0
$880$	−157.796	−0.00604466
$881$	−6030.64	−0.230621	−0.115311	−	0.993329i	$-0.536786\pi$
$881$	−6030.64	−0.230621	−0.115311	+	0.993329i	$0.536786\pi$
$882$	0	0
$883$	−30238.0	−1.15242	−0.576211	−	0.817301i	$-0.695469\pi$
$883$	−30238.0	−1.15242	−0.576211	+	0.817301i	$0.695469\pi$
$884$	−28790.8	−1.09541
$885$	0	0
$886$	−13649.1	−0.517553
$887$	−18776.2	−0.710758	−0.355379	−	0.934722i	$-0.615648\pi$
$887$	−18776.2	−0.710758	−0.355379	+	0.934722i	$0.615648\pi$
$888$	0	0
$889$	17392.3	0.656151
$890$	−4596.44	−0.173116
$891$	0	0
$892$	45300.3	1.70041
$893$	4734.46	0.177416
$894$	0	0
$895$	5117.01	0.191109
$896$	46966.8	1.75117
$897$	0	0
$898$	3706.78	0.137747
$899$	27531.8	1.02140
$900$	0	0
$901$	26017.3	0.962000
$902$	75752.7	2.79633
$903$	0	0
$904$	18285.3	0.672744
$905$	−13199.6	−0.484829
$906$	0	0
$907$	−37652.5	−1.37843	−0.689213	−	0.724559i	$-0.742044\pi$
$907$	−37652.5	−1.37843	−0.689213	+	0.724559i	$0.742044\pi$
$908$	−8432.97	−0.308214
$909$	0	0
$910$	−21527.2	−0.784198
$911$	47586.1	1.73062	0.865312	−	0.501233i	$-0.167120\pi$
$911$	47586.1	1.73062	0.865312	+	0.501233i	$0.167120\pi$
$912$	0	0
$913$	−53945.1	−1.95545
$914$	7188.63	0.260152
$915$	0	0
$916$	59208.7	2.13571
$917$	−43819.9	−1.57804
$918$	0	0
$919$	32177.3	1.15499	0.577493	−	0.816396i	$-0.304031\pi$
$919$	32177.3	1.15499	0.577493	+	0.816396i	$0.304031\pi$
$920$	1078.68	0.0386556
$921$	0	0
$922$	−9433.89	−0.336972
$923$	19166.8	0.683514
$924$	0	0
$925$	6221.60	0.221151
$926$	12729.3	0.451741
$927$	0	0
$928$	−32548.6	−1.15136
$929$	−40643.7	−1.43539	−0.717694	−	0.696359i	$-0.754802\pi$
$929$	−40643.7	−1.43539	−0.717694	+	0.696359i	$0.754802\pi$
$930$	0	0
$931$	−613.036	−0.0215805
$932$	−4101.70	−0.144158
$933$	0	0
$934$	−50033.4	−1.75283
$935$	15797.0	0.552533
$936$	0	0
$937$	6077.87	0.211905	0.105953	−	0.994371i	$-0.466211\pi$
$937$	6077.87	0.211905	0.105953	+	0.994371i	$0.466211\pi$
$938$	41202.6	1.43424
$939$	0	0
$940$	30724.2	1.06608
$941$	−1464.59	−0.0507377	−0.0253688	−	0.999678i	$-0.508076\pi$
$941$	−1464.59	−0.0507377	−0.0253688	+	0.999678i	$0.508076\pi$
$942$	0	0
$943$	2394.79	0.0826989
$944$	−199.463	−0.00687707
$945$	0	0
$946$	64399.0	2.21331
$947$	10608.7	0.364031	0.182015	−	0.983296i	$-0.441738\pi$
$947$	10608.7	0.364031	0.182015	+	0.983296i	$0.441738\pi$
$948$	0	0
$949$	16780.3	0.573984
$950$	−1137.99	−0.0388646
$951$	0	0
$952$	−21544.8	−0.733478
$953$	43623.4	1.48279	0.741396	−	0.671068i	$-0.234164\pi$
$953$	43623.4	1.48279	0.741396	+	0.671068i	$0.234164\pi$
$954$	0	0
$955$	−4066.40	−0.137786
$956$	−23996.7	−0.811830
$957$	0	0
$958$	−66882.6	−2.25562
$959$	−46246.6	−1.55723
$960$	0	0
$961$	−6063.52	−0.203535
$962$	−53268.4	−1.78528
$963$	0	0
$964$	−42201.2	−1.40997
$965$	4071.21	0.135810
$966$	0	0
$967$	5080.37	0.168949	0.0844745	−	0.996426i	$-0.473079\pi$
$967$	5080.37	0.168949	0.0844745	+	0.996426i	$0.473079\pi$
$968$	69052.6	2.29280
$969$	0	0
$970$	−5766.12	−0.190865
$971$	−9875.60	−0.326388	−0.163194	−	0.986594i	$-0.552180\pi$
$971$	−9875.60	−0.326388	−0.163194	+	0.986594i	$0.552180\pi$
$972$	0	0
$973$	42923.4	1.41425
$974$	75343.5	2.47861
$975$	0	0
$976$	259.361	0.00850608
$977$	−24580.8	−0.804923	−0.402461	−	0.915437i	$-0.631845\pi$
$977$	−24580.8	−0.804923	−0.402461	+	0.915437i	$0.631845\pi$
$978$	0	0
$979$	−13334.6	−0.435319
$980$	−3978.28	−0.129675
$981$	0	0
$982$	−93891.8	−3.05113
$983$	31860.6	1.03377	0.516885	−	0.856055i	$-0.327092\pi$
$983$	31860.6	1.03377	0.516885	+	0.856055i	$0.327092\pi$
$984$	0	0
$985$	20392.1	0.659640
$986$	38930.1	1.25739
$987$	0	0
$988$	6016.97	0.193750
$989$	2035.86	0.0654566
$990$	0	0
$991$	−36921.3	−1.18350	−0.591748	−	0.806123i	$-0.701562\pi$
$991$	−36921.3	−1.18350	−0.591748	+	0.806123i	$0.701562\pi$
$992$	−28051.0	−0.897803
$993$	0	0
$994$	37676.0	1.20222
$995$	6712.45	0.213868
$996$	0	0
$997$	−3734.12	−0.118617	−0.0593083	−	0.998240i	$-0.518890\pi$
$997$	−3734.12	−0.118617	−0.0593083	+	0.998240i	$0.518890\pi$
$998$	−65880.7	−2.08959
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	405.4.a.j.1.3		3
3.2	odd	2		405.4.a.h.1.1		3
5.4	even	2		2025.4.a.q.1.1		3
9.2	odd	6		45.4.e.b.31.3	yes	6
9.4	even	3		135.4.e.b.46.1		6
9.5	odd	6		45.4.e.b.16.3	✓	6
9.7	even	3		135.4.e.b.91.1		6
15.14	odd	2		2025.4.a.s.1.3		3
45.2	even	12		225.4.k.c.49.1		12
45.14	odd	6		225.4.e.c.151.1		6
45.23	even	12		225.4.k.c.124.1		12
45.29	odd	6		225.4.e.c.76.1		6
45.32	even	12		225.4.k.c.124.6		12
45.38	even	12		225.4.k.c.49.6		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
45.4.e.b.16.3	✓	6	9.5	odd	6
45.4.e.b.31.3	yes	6	9.2	odd	6
135.4.e.b.46.1		6	9.4	even	3
135.4.e.b.91.1		6	9.7	even	3
225.4.e.c.76.1		6	45.29	odd	6
225.4.e.c.151.1		6	45.14	odd	6
225.4.k.c.49.1		12	45.2	even	12
225.4.k.c.49.6		12	45.38	even	12
225.4.k.c.124.1		12	45.23	even	12
225.4.k.c.124.6		12	45.32	even	12
405.4.a.h.1.1		3	3.2	odd	2
405.4.a.j.1.3		3	1.1	even	1	trivial
2025.4.a.q.1.1		3	5.4	even	2
2025.4.a.s.1.3		3	15.14	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:	\( N \)	\(=\)	\( 405 = 3^{4} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	405.a (trivial)

\(f(q)\)	\(=\)	\(q+4.57358 q^{2} +12.9176 q^{4} -5.00000 q^{5} -20.1145 q^{7} +22.4912 q^{8} +O(q^{10})\)	\(q+4.57358 q^{2} +12.9176 q^{4} -5.00000 q^{5} -20.1145 q^{7} +22.4912 q^{8} -22.8679 q^{10} -66.3416 q^{11} -46.8006 q^{13} -91.9955 q^{14} -0.475708 q^{16} +47.6233 q^{17} -9.95276 q^{19} -64.5882 q^{20} -303.418 q^{22} -9.59204 q^{23} +25.0000 q^{25} -214.046 q^{26} -259.832 q^{28} +178.735 q^{29} +154.037 q^{31} -182.105 q^{32} +217.809 q^{34} +100.573 q^{35} +248.864 q^{37} -45.5197 q^{38} -112.456 q^{40} -249.664 q^{41} -212.245 q^{43} -856.976 q^{44} -43.8700 q^{46} -475.694 q^{47} +61.5946 q^{49} +114.340 q^{50} -604.553 q^{52} +546.314 q^{53} +331.708 q^{55} -452.400 q^{56} +817.459 q^{58} +419.296 q^{59} -545.210 q^{61} +704.502 q^{62} -829.068 q^{64} +234.003 q^{65} -447.877 q^{67} +615.181 q^{68} +459.977 q^{70} -409.542 q^{71} -358.548 q^{73} +1138.20 q^{74} -128.566 q^{76} +1334.43 q^{77} -651.552 q^{79} +2.37854 q^{80} -1141.86 q^{82} +813.142 q^{83} -238.117 q^{85} -970.719 q^{86} -1492.10 q^{88} +201.000 q^{89} +941.373 q^{91} -123.906 q^{92} -2175.62 q^{94} +49.7638 q^{95} +252.149 q^{97} +281.708 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + q^{2} + 11 q^{4} - 15 q^{5} - 43 q^{7} + 27 q^{8}+O(q^{10}) \) 3 * q + q^2 + 11 * q^4 - 15 * q^5 - 43 * q^7 + 27 * q^8	\( 3 q + q^{2} + 11 q^{4} - 15 q^{5} - 43 q^{7} + 27 q^{8} - 5 q^{10} - 14 q^{11} + 40 q^{13} + 27 q^{14} - 13 q^{16} + 166 q^{17} - 164 q^{19} - 55 q^{20} - 376 q^{22} - 171 q^{23} + 75 q^{25} - 434 q^{26} - 517 q^{28} + 335 q^{29} - 352 q^{31} + 77 q^{32} - 52 q^{34} + 215 q^{35} + 402 q^{37} + 178 q^{38} - 135 q^{40} - 187 q^{41} - 602 q^{43} - 982 q^{44} - 201 q^{46} - 665 q^{47} + 430 q^{49} + 25 q^{50} - 456 q^{52} + 730 q^{53} + 70 q^{55} - 705 q^{56} + 217 q^{58} + 298 q^{59} - 1439 q^{61} + 1614 q^{62} - 1569 q^{64} - 200 q^{65} - 1849 q^{67} + 710 q^{68} - 135 q^{70} - 70 q^{71} - 368 q^{73} + 320 q^{74} + 204 q^{76} + 948 q^{77} - 382 q^{79} + 65 q^{80} - 575 q^{82} + 831 q^{83} - 830 q^{85} - 1580 q^{86} - 1428 q^{88} - 1719 q^{89} - 710 q^{91} + 1623 q^{92} - 2077 q^{94} + 820 q^{95} - 282 q^{97} - 2164 q^{98}+O(q^{100}) \) 3 * q + q^2 + 11 * q^4 - 15 * q^5 - 43 * q^7 + 27 * q^8 - 5 * q^10 - 14 * q^11 + 40 * q^13 + 27 * q^14 - 13 * q^16 + 166 * q^17 - 164 * q^19 - 55 * q^20 - 376 * q^22 - 171 * q^23 + 75 * q^25 - 434 * q^26 - 517 * q^28 + 335 * q^29 - 352 * q^31 + 77 * q^32 - 52 * q^34 + 215 * q^35 + 402 * q^37 + 178 * q^38 - 135 * q^40 - 187 * q^41 - 602 * q^43 - 982 * q^44 - 201 * q^46 - 665 * q^47 + 430 * q^49 + 25 * q^50 - 456 * q^52 + 730 * q^53 + 70 * q^55 - 705 * q^56 + 217 * q^58 + 298 * q^59 - 1439 * q^61 + 1614 * q^62 - 1569 * q^64 - 200 * q^65 - 1849 * q^67 + 710 * q^68 - 135 * q^70 - 70 * q^71 - 368 * q^73 + 320 * q^74 + 204 * q^76 + 948 * q^77 - 382 * q^79 + 65 * q^80 - 575 * q^82 + 831 * q^83 - 830 * q^85 - 1580 * q^86 - 1428 * q^88 - 1719 * q^89 - 710 * q^91 + 1623 * q^92 - 2077 * q^94 + 820 * q^95 - 282 * q^97 - 2164 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more