LMFDB - Embedded newform 1305.4.a.m.1.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1305,4,Mod(1,1305)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1305, 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("1305.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1305 = 3^{2} \cdot 5 \cdot 29 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1305.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:	$76.9974925575$
Analytic rank:	$1$
Dimension:	$7$
Coefficient field:	$\mathbb{Q}[x]/(x^{7} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{7} - x^{6} - 35x^{5} + 18x^{4} + 329x^{3} - 167x^{2} - 767x + 638 $ x^7 - x^6 - 35x^5 + 18x^4 + 329x^3 - 167x^2 - 767*x + 638
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	no (minimal twist has level 435)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$3.07921$ of defining polynomial
Character	$\chi$	$=$	1305.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+3.07921 q^{2} +1.48151 q^{4} +5.00000 q^{5} +23.1532 q^{7} -20.0718 q^{8} +15.3960 q^{10} +43.1165 q^{11} -74.6173 q^{13} +71.2934 q^{14} -73.6572 q^{16} -82.7951 q^{17} -88.2501 q^{19} +7.40754 q^{20} +132.765 q^{22} -175.920 q^{23} +25.0000 q^{25} -229.762 q^{26} +34.3016 q^{28} +29.0000 q^{29} +14.1415 q^{31} -66.2315 q^{32} -254.943 q^{34} +115.766 q^{35} -176.503 q^{37} -271.740 q^{38} -100.359 q^{40} +138.356 q^{41} +64.7784 q^{43} +63.8775 q^{44} -541.695 q^{46} +101.341 q^{47} +193.070 q^{49} +76.9801 q^{50} -110.546 q^{52} -265.178 q^{53} +215.583 q^{55} -464.726 q^{56} +89.2970 q^{58} +645.434 q^{59} -386.696 q^{61} +43.5447 q^{62} +385.317 q^{64} -373.087 q^{65} -737.103 q^{67} -122.662 q^{68} +356.467 q^{70} +1014.53 q^{71} -1158.89 q^{73} -543.489 q^{74} -130.743 q^{76} +998.285 q^{77} +714.039 q^{79} -368.286 q^{80} +426.027 q^{82} -747.521 q^{83} -413.976 q^{85} +199.466 q^{86} -865.426 q^{88} +518.363 q^{89} -1727.63 q^{91} -260.628 q^{92} +312.051 q^{94} -441.250 q^{95} -946.328 q^{97} +594.502 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q + q^{2} + 15 q^{4} + 35 q^{5} - 37 q^{7} + 36 q^{8} + 5 q^{10} + 11 q^{11} - 133 q^{13} + 75 q^{14} - 53 q^{16} - 21 q^{17} - 170 q^{19} + 75 q^{20} - 369 q^{22} + 68 q^{23} + 175 q^{25} - 181 q^{26}+ \cdots - 1068 q^{98}+O(q^{100}) $ 7 * q + q^2 + 15 * q^4 + 35 * q^5 - 37 * q^7 + 36 * q^8 + 5 * q^10 + 11 * q^11 - 133 * q^13 + 75 * q^14 - 53 * q^16 - 21 * q^17 - 170 * q^19 + 75 * q^20 - 369 * q^22 + 68 * q^23 + 175 * q^25 - 181 * q^26 - 637 * q^28 + 203 * q^29 - 480 * q^31 + 779 * q^32 - 897 * q^34 - 185 * q^35 - 1032 * q^37 + 194 * q^38 + 180 * q^40 + 638 * q^41 - 512 * q^43 - 625 * q^44 + 16 * q^46 + 111 * q^47 + 178 * q^49 + 25 * q^50 - 1263 * q^52 - 410 * q^53 + 55 * q^55 - 1174 * q^56 + 29 * q^58 + 426 * q^59 - 1192 * q^61 - 460 * q^62 + 390 * q^64 - 665 * q^65 - 1671 * q^67 - 1509 * q^68 + 375 * q^70 + 1324 * q^71 - 852 * q^73 - 1780 * q^74 - 564 * q^76 + 2107 * q^77 + 366 * q^79 - 265 * q^80 - 318 * q^82 - 470 * q^83 - 105 * q^85 + 2196 * q^86 - 2518 * q^88 - 51 * q^89 - 1297 * q^91 + 684 * q^92 - 1837 * q^94 - 850 * q^95 - 3322 * q^97 - 1068 * q^98

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 $

$2$	3.07921	1.08866	0.544332	−	0.838870i	$-0.316783\pi$
$2$	3.07921	1.08866	0.544332	+	0.838870i	$0.316783\pi$
$3$	0	0
$3$	0	0
$4$	1.48151	0.185189
$5$	5.00000	0.447214
$5$	5.00000	0.447214
$6$	0	0
$7$	23.1532	1.25015	0.625077	−	0.780563i	$-0.285067\pi$
$7$	23.1532	1.25015	0.625077	+	0.780563i	$0.285067\pi$
$8$	−20.0718	−0.887056
$9$	0	0
$10$	15.3960	0.486865
$11$	43.1165	1.18183	0.590915	−	0.806734i	$-0.298767\pi$
$11$	43.1165	1.18183	0.590915	+	0.806734i	$0.298767\pi$
$12$	0	0
$13$	−74.6173	−1.59193	−0.795966	−	0.605341i	$-0.793037\pi$
$13$	−74.6173	−1.59193	−0.795966	+	0.605341i	$0.793037\pi$
$14$	71.2934	1.36100
$15$	0	0
$16$	−73.6572	−1.15089
$17$	−82.7951	−1.18122	−0.590611	−	0.806956i	$-0.701113\pi$
$17$	−82.7951	−1.18122	−0.590611	+	0.806956i	$0.701113\pi$
$18$	0	0
$19$	−88.2501	−1.06558	−0.532788	−	0.846248i	$-0.678856\pi$
$19$	−88.2501	−1.06558	−0.532788	+	0.846248i	$0.678856\pi$
$20$	7.40754	0.0828188
$21$	0	0
$22$	132.765	1.28662
$23$	−175.920	−1.59487	−0.797434	−	0.603407i	$-0.793810\pi$
$23$	−175.920	−1.59487	−0.797434	+	0.603407i	$0.793810\pi$
$24$	0	0
$25$	25.0000	0.200000
$26$	−229.762	−1.73308
$27$	0	0
$28$	34.3016	0.231514
$29$	29.0000	0.185695
$29$	29.0000	0.185695
$30$	0	0
$31$	14.1415	0.0819321	0.0409661	−	0.999161i	$-0.486956\pi$
$31$	14.1415	0.0819321	0.0409661	+	0.999161i	$0.486956\pi$
$32$	−66.2315	−0.365881
$33$	0	0
$34$	−254.943	−1.28595
$35$	115.766	0.559086
$36$	0	0
$37$	−176.503	−0.784241	−0.392120	−	0.919914i	$-0.628258\pi$
$37$	−176.503	−0.784241	−0.392120	+	0.919914i	$0.628258\pi$
$38$	−271.740	−1.16005
$39$	0	0
$40$	−100.359	−0.396703
$41$	138.356	0.527015	0.263508	−	0.964657i	$-0.415121\pi$
$41$	138.356	0.527015	0.263508	+	0.964657i	$0.415121\pi$
$42$	0	0
$43$	64.7784	0.229735	0.114868	−	0.993381i	$-0.463356\pi$
$43$	64.7784	0.229735	0.114868	+	0.993381i	$0.463356\pi$
$44$	63.8775	0.218861
$45$	0	0
$46$	−541.695	−1.73627
$47$	101.341	0.314514	0.157257	−	0.987558i	$-0.449735\pi$
$47$	101.341	0.314514	0.157257	+	0.987558i	$0.449735\pi$
$48$	0	0
$49$	193.070	0.562886
$50$	76.9801	0.217733
$51$	0	0
$52$	−110.546	−0.294808
$53$	−265.178	−0.687265	−0.343633	−	0.939104i	$-0.611657\pi$
$53$	−265.178	−0.687265	−0.343633	+	0.939104i	$0.611657\pi$
$54$	0	0
$55$	215.583	0.528530
$56$	−464.726	−1.10896
$57$	0	0
$58$	89.2970	0.202160
$59$	645.434	1.42421	0.712105	−	0.702073i	$-0.247742\pi$
$59$	645.434	1.42421	0.712105	+	0.702073i	$0.247742\pi$
$60$	0	0
$61$	−386.696	−0.811662	−0.405831	−	0.913948i	$-0.633018\pi$
$61$	−386.696	−0.811662	−0.405831	+	0.913948i	$0.633018\pi$
$62$	43.5447	0.0891966
$63$	0	0
$64$	385.317	0.752573
$65$	−373.087	−0.711934
$66$	0	0
$67$	−737.103	−1.34405	−0.672026	−	0.740528i	$-0.734576\pi$
$67$	−737.103	−1.34405	−0.672026	+	0.740528i	$0.734576\pi$
$68$	−122.662	−0.218749
$69$	0	0
$70$	356.467	0.608657
$71$	1014.53	1.69580	0.847902	−	0.530152i	$-0.177865\pi$
$71$	1014.53	1.69580	0.847902	+	0.530152i	$0.177865\pi$
$72$	0	0
$73$	−1158.89	−1.85805	−0.929025	−	0.370016i	$-0.879352\pi$
$73$	−1158.89	−1.85805	−0.929025	+	0.370016i	$0.879352\pi$
$74$	−543.489	−0.853775
$75$	0	0
$76$	−130.743	−0.197333
$77$	998.285	1.47747
$78$	0	0
$79$	714.039	1.01691	0.508453	−	0.861089i	$-0.330217\pi$
$79$	714.039	1.01691	0.508453	+	0.861089i	$0.330217\pi$
$80$	−368.286	−0.514695
$81$	0	0
$82$	426.027	0.573742
$83$	−747.521	−0.988567	−0.494284	−	0.869301i	$-0.664570\pi$
$83$	−747.521	−0.988567	−0.494284	+	0.869301i	$0.664570\pi$
$84$	0	0
$85$	−413.976	−0.528258
$86$	199.466	0.250104
$87$	0	0
$88$	−865.426	−1.04835
$89$	518.363	0.617375	0.308688	−	0.951163i	$-0.400110\pi$
$89$	518.363	0.617375	0.308688	+	0.951163i	$0.400110\pi$
$90$	0	0
$91$	−1727.63	−1.99016
$92$	−260.628	−0.295351
$93$	0	0
$94$	312.051	0.342400
$95$	−441.250	−0.476540
$96$	0	0
$97$	−946.328	−0.990568	−0.495284	−	0.868731i	$-0.664936\pi$
$97$	−946.328	−0.990568	−0.495284	+	0.868731i	$0.664936\pi$
$98$	594.502	0.612794
$99$	0	0
$100$	37.0377	0.0370377
$101$	−864.251	−0.851448	−0.425724	−	0.904853i	$-0.639981\pi$
$101$	−864.251	−0.851448	−0.425724	+	0.904853i	$0.639981\pi$
$102$	0	0
$103$	−1385.74	−1.32564	−0.662819	−	0.748780i	$-0.730640\pi$
$103$	−1385.74	−1.32564	−0.662819	+	0.748780i	$0.730640\pi$
$104$	1497.70	1.41213
$105$	0	0
$106$	−816.539	−0.748201
$107$	335.496	0.303118	0.151559	−	0.988448i	$-0.451571\pi$
$107$	335.496	0.303118	0.151559	+	0.988448i	$0.451571\pi$
$108$	0	0
$109$	−769.498	−0.676189	−0.338094	−	0.941112i	$-0.609782\pi$
$109$	−769.498	−0.676189	−0.338094	+	0.941112i	$0.609782\pi$
$110$	663.824	0.575392
$111$	0	0
$112$	−1705.40	−1.43879
$113$	−1066.98	−0.888261	−0.444131	−	0.895962i	$-0.646487\pi$
$113$	−1066.98	−0.888261	−0.444131	+	0.895962i	$0.646487\pi$
$114$	0	0
$115$	−879.602	−0.713246
$116$	42.9637	0.0343887
$117$	0	0
$118$	1987.43	1.55049
$119$	−1916.97	−1.47671
$120$	0	0
$121$	528.036	0.396722
$122$	−1190.72	−0.883627
$123$	0	0
$124$	20.9508	0.0151729
$125$	125.000	0.0894427
$126$	0	0
$127$	−1071.24	−0.748482	−0.374241	−	0.927332i	$-0.622097\pi$
$127$	−1071.24	−0.748482	−0.374241	+	0.927332i	$0.622097\pi$
$128$	1716.32	1.18518
$129$	0	0
$130$	−1148.81	−0.775057
$131$	2192.32	1.46217	0.731083	−	0.682288i	$-0.239015\pi$
$131$	2192.32	1.46217	0.731083	+	0.682288i	$0.239015\pi$
$132$	0	0
$133$	−2043.27	−1.33214
$134$	−2269.69	−1.46322
$135$	0	0
$136$	1661.85	1.04781
$137$	−592.283	−0.369359	−0.184679	−	0.982799i	$-0.559125\pi$
$137$	−592.283	−0.369359	−0.184679	+	0.982799i	$0.559125\pi$
$138$	0	0
$139$	1980.19	1.20833	0.604164	−	0.796860i	$-0.293507\pi$
$139$	1980.19	1.20833	0.604164	+	0.796860i	$0.293507\pi$
$140$	171.508	0.103536
$141$	0	0
$142$	3123.94	1.84616
$143$	−3217.24	−1.88139
$144$	0	0
$145$	145.000	0.0830455
$146$	−3568.46	−2.02279
$147$	0	0
$148$	−261.491	−0.145232
$149$	−1366.49	−0.751326	−0.375663	−	0.926756i	$-0.622585\pi$
$149$	−1366.49	−0.751326	−0.375663	+	0.926756i	$0.622585\pi$
$150$	0	0
$151$	−2204.75	−1.18821	−0.594105	−	0.804387i	$-0.702494\pi$
$151$	−2204.75	−1.18821	−0.594105	+	0.804387i	$0.702494\pi$
$152$	1771.34	0.945226
$153$	0	0
$154$	3073.93	1.60847
$155$	70.7077	0.0366412
$156$	0	0
$157$	−1798.31	−0.914144	−0.457072	−	0.889430i	$-0.651102\pi$
$157$	−1798.31	−0.914144	−0.457072	+	0.889430i	$0.651102\pi$
$158$	2198.67	1.10707
$159$	0	0
$160$	−331.157	−0.163627
$161$	−4073.12	−1.99383
$162$	0	0
$163$	358.581	0.172308	0.0861540	−	0.996282i	$-0.472542\pi$
$163$	358.581	0.172308	0.0861540	+	0.996282i	$0.472542\pi$
$164$	204.976	0.0975972
$165$	0	0
$166$	−2301.77	−1.07622
$167$	3993.78	1.85059	0.925293	−	0.379253i	$-0.123819\pi$
$167$	3993.78	1.85059	0.925293	+	0.379253i	$0.123819\pi$
$168$	0	0
$169$	3370.75	1.53425
$170$	−1274.72	−0.575096
$171$	0	0
$172$	95.9698	0.0425443
$173$	−902.727	−0.396723	−0.198362	−	0.980129i	$-0.563562\pi$
$173$	−902.727	−0.396723	−0.198362	+	0.980129i	$0.563562\pi$
$174$	0	0
$175$	578.830	0.250031
$176$	−3175.84	−1.36016
$177$	0	0
$178$	1596.15	0.672114
$179$	−438.007	−0.182895	−0.0914474	−	0.995810i	$-0.529149\pi$
$179$	−438.007	−0.182895	−0.0914474	+	0.995810i	$0.529149\pi$
$180$	0	0
$181$	4385.45	1.80093	0.900464	−	0.434930i	$-0.143227\pi$
$181$	4385.45	1.80093	0.900464	+	0.434930i	$0.143227\pi$
$182$	−5319.72	−2.16662
$183$	0	0
$184$	3531.04	1.41474
$185$	−882.515	−0.350723
$186$	0	0
$187$	−3569.84	−1.39600
$188$	150.138	0.0582444
$189$	0	0
$190$	−1358.70	−0.518792
$191$	−1188.68	−0.450313	−0.225156	−	0.974323i	$-0.572289\pi$
$191$	−1188.68	−0.450313	−0.225156	+	0.974323i	$0.572289\pi$
$192$	0	0
$193$	2256.44	0.841565	0.420782	−	0.907162i	$-0.361756\pi$
$193$	2256.44	0.841565	0.420782	+	0.907162i	$0.361756\pi$
$194$	−2913.94	−1.07840
$195$	0	0
$196$	286.035	0.104240
$197$	944.551	0.341606	0.170803	−	0.985305i	$-0.445364\pi$
$197$	944.551	0.341606	0.170803	+	0.985305i	$0.445364\pi$
$198$	0	0
$199$	−172.934	−0.0616030	−0.0308015	−	0.999526i	$-0.509806\pi$
$199$	−172.934	−0.0616030	−0.0308015	+	0.999526i	$0.509806\pi$
$200$	−501.794	−0.177411
$201$	0	0
$202$	−2661.21	−0.926940
$203$	671.442	0.232148
$204$	0	0
$205$	691.781	0.235688
$206$	−4266.97	−1.44317
$207$	0	0
$208$	5496.10	1.83215
$209$	−3805.04	−1.25933
$210$	0	0
$211$	−3980.90	−1.29885	−0.649423	−	0.760427i	$-0.724990\pi$
$211$	−3980.90	−1.29885	−0.649423	+	0.760427i	$0.724990\pi$
$212$	−392.864	−0.127274
$213$	0	0
$214$	1033.06	0.329994
$215$	323.892	0.102741
$216$	0	0
$217$	327.422	0.102428
$218$	−2369.44	−0.736142
$219$	0	0
$220$	319.388	0.0978778
$221$	6177.95	1.88043
$222$	0	0
$223$	1939.43	0.582392	0.291196	−	0.956663i	$-0.405947\pi$
$223$	1939.43	0.582392	0.291196	+	0.956663i	$0.405947\pi$
$224$	−1533.47	−0.457407
$225$	0	0
$226$	−3285.47	−0.967017
$227$	6457.84	1.88820	0.944101	−	0.329656i	$-0.106933\pi$
$227$	6457.84	1.88820	0.944101	+	0.329656i	$0.106933\pi$
$228$	0	0
$229$	−1271.99	−0.367054	−0.183527	−	0.983015i	$-0.558751\pi$
$229$	−1271.99	−0.367054	−0.183527	+	0.983015i	$0.558751\pi$
$230$	−2708.48	−0.776485
$231$	0	0
$232$	−582.082	−0.164722
$233$	−1029.14	−0.289361	−0.144681	−	0.989478i	$-0.546215\pi$
$233$	−1029.14	−0.289361	−0.144681	+	0.989478i	$0.546215\pi$
$234$	0	0
$235$	506.706	0.140655
$236$	956.216	0.263747
$237$	0	0
$238$	−5902.75	−1.60764
$239$	−2592.45	−0.701639	−0.350819	−	0.936443i	$-0.614097\pi$
$239$	−2592.45	−0.701639	−0.350819	+	0.936443i	$0.614097\pi$
$240$	0	0
$241$	−5132.31	−1.37179	−0.685894	−	0.727701i	$-0.740589\pi$
$241$	−5132.31	−1.37179	−0.685894	+	0.727701i	$0.740589\pi$
$242$	1625.93	0.431896
$243$	0	0
$244$	−572.894	−0.150311
$245$	965.350	0.251730
$246$	0	0
$247$	6584.99	1.69633
$248$	−283.846	−0.0726784
$249$	0	0
$250$	384.901	0.0973730
$251$	−96.4840	−0.0242630	−0.0121315	−	0.999926i	$-0.503862\pi$
$251$	−96.4840	−0.0242630	−0.0121315	+	0.999926i	$0.503862\pi$
$252$	0	0
$253$	−7585.08	−1.88486
$254$	−3298.57	−0.814845
$255$	0	0
$256$	2202.37	0.537689
$257$	−1022.96	−0.248290	−0.124145	−	0.992264i	$-0.539619\pi$
$257$	−1022.96	−0.248290	−0.124145	+	0.992264i	$0.539619\pi$
$258$	0	0
$259$	−4086.61	−0.980422
$260$	−552.731	−0.131842
$261$	0	0
$262$	6750.60	1.59181
$263$	−2925.05	−0.685805	−0.342902	−	0.939371i	$-0.611410\pi$
$263$	−2925.05	−0.685805	−0.342902	+	0.939371i	$0.611410\pi$
$264$	0	0
$265$	−1325.89	−0.307354
$266$	−6291.65	−1.45025
$267$	0	0
$268$	−1092.02	−0.248903
$269$	−3935.68	−0.892055	−0.446028	−	0.895019i	$-0.647162\pi$
$269$	−3935.68	−0.892055	−0.446028	+	0.895019i	$0.647162\pi$
$270$	0	0
$271$	7099.87	1.59146	0.795731	−	0.605650i	$-0.207087\pi$
$271$	7099.87	1.59146	0.795731	+	0.605650i	$0.207087\pi$
$272$	6098.46	1.35946
$273$	0	0
$274$	−1823.76	−0.402107
$275$	1077.91	0.236366
$276$	0	0
$277$	5100.47	1.10634	0.553172	−	0.833067i	$-0.313417\pi$
$277$	5100.47	1.10634	0.553172	+	0.833067i	$0.313417\pi$
$278$	6097.41	1.31546
$279$	0	0
$280$	−2323.63	−0.495940
$281$	−7835.19	−1.66337	−0.831687	−	0.555245i	$-0.812625\pi$
$281$	−7835.19	−1.66337	−0.831687	+	0.555245i	$0.812625\pi$
$282$	0	0
$283$	7148.56	1.50155	0.750773	−	0.660560i	$-0.229681\pi$
$283$	7148.56	1.50155	0.750773	+	0.660560i	$0.229681\pi$
$284$	1503.03	0.314044
$285$	0	0
$286$	−9906.55	−2.04820
$287$	3203.39	0.658850
$288$	0	0
$289$	1942.04	0.395285
$290$	446.485	0.0904086
$291$	0	0
$292$	−1716.90	−0.344090
$293$	6053.67	1.20703	0.603514	−	0.797352i	$-0.293767\pi$
$293$	6053.67	1.20703	0.603514	+	0.797352i	$0.293767\pi$
$294$	0	0
$295$	3227.17	0.636926
$296$	3542.73	0.695665
$297$	0	0
$298$	−4207.72	−0.817941
$299$	13126.7	2.53892
$300$	0	0
$301$	1499.83	0.287204
$302$	−6788.87	−1.29356
$303$	0	0
$304$	6500.25	1.22637
$305$	−1933.48	−0.362986
$306$	0	0
$307$	5697.09	1.05912	0.529560	−	0.848272i	$-0.322357\pi$
$307$	5697.09	1.05912	0.529560	+	0.848272i	$0.322357\pi$
$308$	1478.97	0.273611
$309$	0	0
$310$	217.724	0.0398899
$311$	10242.2	1.86747	0.933737	−	0.357960i	$-0.116528\pi$
$311$	10242.2	1.86747	0.933737	+	0.357960i	$0.116528\pi$
$312$	0	0
$313$	3018.38	0.545077	0.272539	−	0.962145i	$-0.412137\pi$
$313$	3018.38	0.545077	0.272539	+	0.962145i	$0.412137\pi$
$314$	−5537.36	−0.995195
$315$	0	0
$316$	1057.85	0.188320
$317$	−3757.10	−0.665678	−0.332839	−	0.942984i	$-0.608007\pi$
$317$	−3757.10	−0.665678	−0.332839	+	0.942984i	$0.608007\pi$
$318$	0	0
$319$	1250.38	0.219460
$320$	1926.59	0.336561
$321$	0	0
$322$	−12542.0	−2.17061
$323$	7306.68	1.25868
$324$	0	0
$325$	−1865.43	−0.318386
$326$	1104.14	0.187585
$327$	0	0
$328$	−2777.06	−0.467492
$329$	2346.37	0.393191
$330$	0	0
$331$	2883.87	0.478888	0.239444	−	0.970910i	$-0.423035\pi$
$331$	2883.87	0.478888	0.239444	+	0.970910i	$0.423035\pi$
$332$	−1107.46	−0.183071
$333$	0	0
$334$	12297.7	2.01467
$335$	−3685.52	−0.601078
$336$	0	0
$337$	−1692.21	−0.273532	−0.136766	−	0.990603i	$-0.543671\pi$
$337$	−1692.21	−0.273532	−0.136766	+	0.990603i	$0.543671\pi$
$338$	10379.2	1.67028
$339$	0	0
$340$	−613.309	−0.0978274
$341$	609.735	0.0968298
$342$	0	0
$343$	−3471.36	−0.546460
$344$	−1300.22	−0.203788
$345$	0	0
$346$	−2779.68	−0.431898
$347$	−4744.08	−0.733935	−0.366967	−	0.930234i	$-0.619604\pi$
$347$	−4744.08	−0.733935	−0.366967	+	0.930234i	$0.619604\pi$
$348$	0	0
$349$	−6690.03	−1.02610	−0.513050	−	0.858359i	$-0.671485\pi$
$349$	−6690.03	−1.02610	−0.513050	+	0.858359i	$0.671485\pi$
$350$	1782.34	0.272200
$351$	0	0
$352$	−2855.67	−0.432409
$353$	−2218.76	−0.334541	−0.167270	−	0.985911i	$-0.553495\pi$
$353$	−2218.76	−0.334541	−0.167270	+	0.985911i	$0.553495\pi$
$354$	0	0
$355$	5072.63	0.758387
$356$	767.959	0.114331
$357$	0	0
$358$	−1348.71	−0.199111
$359$	2749.37	0.404195	0.202098	−	0.979365i	$-0.435224\pi$
$359$	2749.37	0.404195	0.202098	+	0.979365i	$0.435224\pi$
$360$	0	0
$361$	929.079	0.135454
$362$	13503.7	1.96061
$363$	0	0
$364$	−2559.50	−0.368555
$365$	−5794.44	−0.830945
$366$	0	0
$367$	8213.16	1.16818	0.584092	−	0.811687i	$-0.301451\pi$
$367$	8213.16	1.16818	0.584092	+	0.811687i	$0.301451\pi$
$368$	12957.8	1.83552
$369$	0	0
$370$	−2717.45	−0.381820
$371$	−6139.73	−0.859188
$372$	0	0
$373$	−8062.31	−1.11917	−0.559585	−	0.828773i	$-0.689039\pi$
$373$	−8062.31	−1.11917	−0.559585	+	0.828773i	$0.689039\pi$
$374$	−10992.3	−1.51978
$375$	0	0
$376$	−2034.10	−0.278991
$377$	−2163.90	−0.295614
$378$	0	0
$379$	−5523.37	−0.748593	−0.374296	−	0.927309i	$-0.622116\pi$
$379$	−5523.37	−0.748593	−0.374296	+	0.927309i	$0.622116\pi$
$380$	−653.716	−0.0882498
$381$	0	0
$382$	−3660.18	−0.490239
$383$	−10435.1	−1.39219	−0.696095	−	0.717950i	$-0.745081\pi$
$383$	−10435.1	−1.39219	−0.696095	+	0.717950i	$0.745081\pi$
$384$	0	0
$385$	4991.43	0.660745
$386$	6948.04	0.916181
$387$	0	0
$388$	−1401.99	−0.183442
$389$	4033.70	0.525749	0.262875	−	0.964830i	$-0.415329\pi$
$389$	4033.70	0.525749	0.262875	+	0.964830i	$0.415329\pi$
$390$	0	0
$391$	14565.4	1.88389
$392$	−3875.26	−0.499311
$393$	0	0
$394$	2908.47	0.371894
$395$	3570.19	0.454775
$396$	0	0
$397$	−14474.2	−1.82982	−0.914911	−	0.403655i	$-0.867740\pi$
$397$	−14474.2	−1.82982	−0.914911	+	0.403655i	$0.867740\pi$
$398$	−532.500	−0.0670649
$399$	0	0
$400$	−1841.43	−0.230179
$401$	3695.34	0.460191	0.230095	−	0.973168i	$-0.426096\pi$
$401$	3695.34	0.460191	0.230095	+	0.973168i	$0.426096\pi$
$402$	0	0
$403$	−1055.20	−0.130430
$404$	−1280.40	−0.157678
$405$	0	0
$406$	2067.51	0.252731
$407$	−7610.20	−0.926839
$408$	0	0
$409$	5880.15	0.710892	0.355446	−	0.934697i	$-0.384329\pi$
$409$	5880.15	0.710892	0.355446	+	0.934697i	$0.384329\pi$
$410$	2130.14	0.256585
$411$	0	0
$412$	−2052.98	−0.245493
$413$	14943.9	1.78048
$414$	0	0
$415$	−3737.60	−0.442101
$416$	4942.01	0.582457
$417$	0	0
$418$	−11716.5	−1.37099
$419$	10799.8	1.25921	0.629603	−	0.776917i	$-0.283218\pi$
$419$	10799.8	1.25921	0.629603	+	0.776917i	$0.283218\pi$
$420$	0	0
$421$	5520.52	0.639083	0.319541	−	0.947572i	$-0.396471\pi$
$421$	5520.52	0.639083	0.319541	+	0.947572i	$0.396471\pi$
$422$	−12258.0	−1.41401
$423$	0	0
$424$	5322.60	0.609642
$425$	−2069.88	−0.236244
$426$	0	0
$427$	−8953.25	−1.01470
$428$	497.040	0.0561340
$429$	0	0
$430$	997.330	0.111850
$431$	3360.04	0.375516	0.187758	−	0.982215i	$-0.439878\pi$
$431$	3360.04	0.375516	0.187758	+	0.982215i	$0.439878\pi$
$432$	0	0
$433$	−5326.35	−0.591150	−0.295575	−	0.955320i	$-0.595511\pi$
$433$	−5326.35	−0.591150	−0.295575	+	0.955320i	$0.595511\pi$
$434$	1008.20	0.111509
$435$	0	0
$436$	−1140.02	−0.125222
$437$	15525.0	1.69945
$438$	0	0
$439$	−2837.77	−0.308518	−0.154259	−	0.988030i	$-0.549299\pi$
$439$	−2837.77	−0.308518	−0.154259	+	0.988030i	$0.549299\pi$
$440$	−4327.13	−0.468836
$441$	0	0
$442$	19023.2	2.04715
$443$	−10702.1	−1.14779	−0.573896	−	0.818928i	$-0.694569\pi$
$443$	−10702.1	−1.14779	−0.573896	+	0.818928i	$0.694569\pi$
$444$	0	0
$445$	2591.82	0.276099
$446$	5971.89	0.634029
$447$	0	0
$448$	8921.32	0.940832
$449$	−7956.43	−0.836275	−0.418137	−	0.908384i	$-0.637317\pi$
$449$	−7956.43	−0.836275	−0.418137	+	0.908384i	$0.637317\pi$
$450$	0	0
$451$	5965.44	0.622842
$452$	−1580.75	−0.164496
$453$	0	0
$454$	19885.0	2.05562
$455$	−8638.14	−0.890027
$456$	0	0
$457$	7365.10	0.753884	0.376942	−	0.926237i	$-0.376976\pi$
$457$	7365.10	0.753884	0.376942	+	0.926237i	$0.376976\pi$
$458$	−3916.71	−0.399598
$459$	0	0
$460$	−1303.14	−0.132085
$461$	−15891.5	−1.60551	−0.802754	−	0.596310i	$-0.796633\pi$
$461$	−15891.5	−1.60551	−0.802754	+	0.596310i	$0.796633\pi$
$462$	0	0
$463$	14456.8	1.45111	0.725555	−	0.688164i	$-0.241583\pi$
$463$	14456.8	1.45111	0.725555	+	0.688164i	$0.241583\pi$
$464$	−2136.06	−0.213716
$465$	0	0
$466$	−3168.93	−0.315017
$467$	14908.2	1.47724	0.738618	−	0.674124i	$-0.235479\pi$
$467$	14908.2	1.47724	0.738618	+	0.674124i	$0.235479\pi$
$468$	0	0
$469$	−17066.3	−1.68027
$470$	1560.25	0.153126
$471$	0	0
$472$	−12955.0	−1.26335
$473$	2793.02	0.271508
$474$	0	0
$475$	−2206.25	−0.213115
$476$	−2840.01	−0.273470
$477$	0	0
$478$	−7982.68	−0.763848
$479$	−11758.0	−1.12158	−0.560791	−	0.827957i	$-0.689503\pi$
$479$	−11758.0	−1.12158	−0.560791	+	0.827957i	$0.689503\pi$
$480$	0	0
$481$	13170.2	1.24846
$482$	−15803.4	−1.49342
$483$	0	0
$484$	782.290	0.0734683
$485$	−4731.64	−0.442995
$486$	0	0
$487$	−8795.82	−0.818433	−0.409216	−	0.912437i	$-0.634198\pi$
$487$	−8795.82	−0.818433	−0.409216	+	0.912437i	$0.634198\pi$
$488$	7761.68	0.719989
$489$	0	0
$490$	2972.51	0.274050
$491$	14778.8	1.35837	0.679185	−	0.733967i	$-0.262333\pi$
$491$	14778.8	1.35837	0.679185	+	0.733967i	$0.262333\pi$
$492$	0	0
$493$	−2401.06	−0.219347
$494$	20276.5	1.84673
$495$	0	0
$496$	−1041.63	−0.0942952
$497$	23489.5	2.12002
$498$	0	0
$499$	10941.0	0.981538	0.490769	−	0.871290i	$-0.336716\pi$
$499$	10941.0	0.981538	0.490769	+	0.871290i	$0.336716\pi$
$500$	185.189	0.0165638
$501$	0	0
$502$	−297.094	−0.0264143
$503$	7424.36	0.658123	0.329062	−	0.944308i	$-0.393268\pi$
$503$	7424.36	0.658123	0.329062	+	0.944308i	$0.393268\pi$
$504$	0	0
$505$	−4321.26	−0.380779
$506$	−23356.0	−2.05198
$507$	0	0
$508$	−1587.05	−0.138610
$509$	−3745.37	−0.326151	−0.163075	−	0.986614i	$-0.552141\pi$
$509$	−3745.37	−0.326151	−0.163075	+	0.986614i	$0.552141\pi$
$510$	0	0
$511$	−26832.0	−2.32285
$512$	−6949.02	−0.599817
$513$	0	0
$514$	−3149.91	−0.270305
$515$	−6928.68	−0.592843
$516$	0	0
$517$	4369.49	0.371702
$518$	−12583.5	−1.06735
$519$	0	0
$520$	7488.51	0.631525
$521$	−19900.9	−1.67346	−0.836730	−	0.547616i	$-0.815536\pi$
$521$	−19900.9	−1.67346	−0.836730	+	0.547616i	$0.815536\pi$
$522$	0	0
$523$	−4230.73	−0.353722	−0.176861	−	0.984236i	$-0.556594\pi$
$523$	−4230.73	−0.353722	−0.176861	+	0.984236i	$0.556594\pi$
$524$	3247.94	0.270777
$525$	0	0
$526$	−9006.84	−0.746610
$527$	−1170.85	−0.0967800
$528$	0	0
$529$	18781.0	1.54360
$530$	−4082.69	−0.334606
$531$	0	0
$532$	−3027.12	−0.246696
$533$	−10323.8	−0.838972
$534$	0	0
$535$	1677.48	0.135558
$536$	14795.0	1.19225
$537$	0	0
$538$	−12118.8	−0.971148
$539$	8324.51	0.665236
$540$	0	0
$541$	−20053.9	−1.59368	−0.796842	−	0.604188i	$-0.793498\pi$
$541$	−20053.9	−1.59368	−0.796842	+	0.604188i	$0.793498\pi$
$542$	21862.0	1.73257
$543$	0	0
$544$	5483.64	0.432186
$545$	−3847.49	−0.302401
$546$	0	0
$547$	753.829	0.0589240	0.0294620	−	0.999566i	$-0.490621\pi$
$547$	753.829	0.0589240	0.0294620	+	0.999566i	$0.490621\pi$
$548$	−877.472	−0.0684010
$549$	0	0
$550$	3319.12	0.257323
$551$	−2559.25	−0.197873
$552$	0	0
$553$	16532.3	1.27129
$554$	15705.4	1.20444
$555$	0	0
$556$	2933.67	0.223769
$557$	−14038.4	−1.06791	−0.533953	−	0.845514i	$-0.679294\pi$
$557$	−14038.4	−1.06791	−0.533953	+	0.845514i	$0.679294\pi$
$558$	0	0
$559$	−4833.59	−0.365723
$560$	−8526.99	−0.643449
$561$	0	0
$562$	−24126.2	−1.81085
$563$	11290.9	0.845211	0.422606	−	0.906314i	$-0.361116\pi$
$563$	11290.9	0.845211	0.422606	+	0.906314i	$0.361116\pi$
$564$	0	0
$565$	−5334.92	−0.397242
$566$	22011.9	1.63468
$567$	0	0
$568$	−20363.3	−1.50427
$569$	24522.4	1.80674	0.903369	−	0.428864i	$-0.141086\pi$
$569$	24522.4	1.80674	0.903369	+	0.428864i	$0.141086\pi$
$570$	0	0
$571$	9525.67	0.698138	0.349069	−	0.937097i	$-0.386498\pi$
$571$	9525.67	0.698138	0.349069	+	0.937097i	$0.386498\pi$
$572$	−4766.37	−0.348413
$573$	0	0
$574$	9863.89	0.717266
$575$	−4398.01	−0.318973
$576$	0	0
$577$	2493.72	0.179922	0.0899610	−	0.995945i	$-0.471326\pi$
$577$	2493.72	0.179922	0.0899610	+	0.995945i	$0.471326\pi$
$578$	5979.93	0.430332
$579$	0	0
$580$	214.819	0.0153791
$581$	−17307.5	−1.23586
$582$	0	0
$583$	−11433.6	−0.812231
$584$	23261.0	1.64819
$585$	0	0
$586$	18640.5	1.31405
$587$	−17081.6	−1.20108	−0.600539	−	0.799595i	$-0.705047\pi$
$587$	−17081.6	−1.20108	−0.600539	+	0.799595i	$0.705047\pi$
$588$	0	0
$589$	−1247.99	−0.0873050
$590$	9937.13	0.693398
$591$	0	0
$592$	13000.7	0.902578
$593$	9633.41	0.667111	0.333556	−	0.942730i	$-0.391752\pi$
$593$	9633.41	0.667111	0.333556	+	0.942730i	$0.391752\pi$
$594$	0	0
$595$	−9584.86	−0.660405
$596$	−2024.47	−0.139137
$597$	0	0
$598$	40419.8	2.76403
$599$	−11572.5	−0.789379	−0.394690	−	0.918814i	$-0.629148\pi$
$599$	−11572.5	−0.789379	−0.394690	+	0.918814i	$0.629148\pi$
$600$	0	0
$601$	−27804.0	−1.88710	−0.943552	−	0.331225i	$-0.892538\pi$
$601$	−27804.0	−1.88710	−0.943552	+	0.331225i	$0.892538\pi$
$602$	4618.27	0.312669
$603$	0	0
$604$	−3266.35	−0.220043
$605$	2640.18	0.177419
$606$	0	0
$607$	2573.81	0.172105	0.0860526	−	0.996291i	$-0.472575\pi$
$607$	2573.81	0.172105	0.0860526	+	0.996291i	$0.472575\pi$
$608$	5844.93	0.389874
$609$	0	0
$610$	−5953.59	−0.395170
$611$	−7561.82	−0.500685
$612$	0	0
$613$	11274.0	0.742824	0.371412	−	0.928468i	$-0.378874\pi$
$613$	11274.0	0.742824	0.371412	+	0.928468i	$0.378874\pi$
$614$	17542.5	1.15303
$615$	0	0
$616$	−20037.4	−1.31060
$617$	1684.94	0.109940	0.0549701	−	0.998488i	$-0.482494\pi$
$617$	1684.94	0.109940	0.0549701	+	0.998488i	$0.482494\pi$
$618$	0	0
$619$	14779.1	0.959647	0.479824	−	0.877365i	$-0.340701\pi$
$619$	14779.1	0.959647	0.479824	+	0.877365i	$0.340701\pi$
$620$	104.754	0.00678553
$621$	0	0
$622$	31538.0	2.03305
$623$	12001.8	0.771814
$624$	0	0
$625$	625.000	0.0400000
$626$	9294.22	0.593406
$627$	0	0
$628$	−2664.21	−0.169289
$629$	14613.6	0.926362
$630$	0	0
$631$	6631.39	0.418370	0.209185	−	0.977876i	$-0.432919\pi$
$631$	6631.39	0.418370	0.209185	+	0.977876i	$0.432919\pi$
$632$	−14332.0	−0.902053
$633$	0	0
$634$	−11568.9	−0.724699
$635$	−5356.20	−0.334731
$636$	0	0
$637$	−14406.4	−0.896077
$638$	3850.18	0.238918
$639$	0	0
$640$	8581.61	0.530028
$641$	22946.8	1.41395	0.706977	−	0.707237i	$-0.250059\pi$
$641$	22946.8	1.41395	0.706977	+	0.707237i	$0.250059\pi$
$642$	0	0
$643$	−366.544	−0.0224807	−0.0112404	−	0.999937i	$-0.503578\pi$
$643$	−366.544	−0.0224807	−0.0112404	+	0.999937i	$0.503578\pi$
$644$	−6034.36	−0.369235
$645$	0	0
$646$	22498.8	1.37028
$647$	−16832.6	−1.02281	−0.511404	−	0.859340i	$-0.670874\pi$
$647$	−16832.6	−1.02281	−0.511404	+	0.859340i	$0.670874\pi$
$648$	0	0
$649$	27828.9	1.68317
$650$	−5744.05	−0.346616
$651$	0	0
$652$	531.240	0.0319095
$653$	5342.33	0.320156	0.160078	−	0.987104i	$-0.448825\pi$
$653$	5342.33	0.320156	0.160078	+	0.987104i	$0.448825\pi$
$654$	0	0
$655$	10961.6	0.653901
$656$	−10190.9	−0.606538
$657$	0	0
$658$	7224.97	0.428052
$659$	−27709.9	−1.63798	−0.818988	−	0.573811i	$-0.805464\pi$
$659$	−27709.9	−1.63798	−0.818988	+	0.573811i	$0.805464\pi$
$660$	0	0
$661$	−32828.4	−1.93174	−0.965868	−	0.259036i	$-0.916595\pi$
$661$	−32828.4	−1.93174	−0.965868	+	0.259036i	$0.916595\pi$
$662$	8880.03	0.521348
$663$	0	0
$664$	15004.1	0.876914
$665$	−10216.4	−0.595749
$666$	0	0
$667$	−5101.69	−0.296159
$668$	5916.82	0.342707
$669$	0	0
$670$	−11348.5	−0.654372
$671$	−16673.0	−0.959246
$672$	0	0
$673$	32414.7	1.85660	0.928302	−	0.371827i	$-0.121269\pi$
$673$	32414.7	1.85660	0.928302	+	0.371827i	$0.121269\pi$
$674$	−5210.66	−0.297785
$675$	0	0
$676$	4993.79	0.284125
$677$	24783.8	1.40697	0.703485	−	0.710710i	$-0.251626\pi$
$677$	24783.8	1.40697	0.703485	+	0.710710i	$0.251626\pi$
$678$	0	0
$679$	−21910.5	−1.23836
$680$	8309.23	0.468595
$681$	0	0
$682$	1877.50	0.105415
$683$	−14097.9	−0.789811	−0.394906	−	0.918722i	$-0.629223\pi$
$683$	−14097.9	−0.789811	−0.394906	+	0.918722i	$0.629223\pi$
$684$	0	0
$685$	−2961.41	−0.165182
$686$	−10689.0	−0.594911
$687$	0	0
$688$	−4771.40	−0.264401
$689$	19786.9	1.09408
$690$	0	0
$691$	12672.4	0.697657	0.348829	−	0.937186i	$-0.386579\pi$
$691$	12672.4	0.697657	0.348829	+	0.937186i	$0.386579\pi$
$692$	−1337.40	−0.0734686
$693$	0	0
$694$	−14608.0	−0.799008
$695$	9900.95	0.540381
$696$	0	0
$697$	−11455.2	−0.622522
$698$	−20600.0	−1.11708
$699$	0	0
$700$	857.541	0.0463029
$701$	−28508.3	−1.53601	−0.768005	−	0.640444i	$-0.778750\pi$
$701$	−28508.3	−1.53601	−0.768005	+	0.640444i	$0.778750\pi$
$702$	0	0
$703$	15576.4	0.835669
$704$	16613.5	0.889413
$705$	0	0
$706$	−6832.03	−0.364202
$707$	−20010.2	−1.06444
$708$	0	0
$709$	14243.6	0.754484	0.377242	−	0.926115i	$-0.376873\pi$
$709$	14243.6	0.754484	0.377242	+	0.926115i	$0.376873\pi$
$710$	15619.7	0.825628
$711$	0	0
$712$	−10404.5	−0.547646
$713$	−2487.79	−0.130671
$714$	0	0
$715$	−16086.2	−0.841385
$716$	−648.911	−0.0338700
$717$	0	0
$718$	8465.87	0.440033
$719$	24710.0	1.28168	0.640839	−	0.767675i	$-0.278587\pi$
$719$	24710.0	1.28168	0.640839	+	0.767675i	$0.278587\pi$
$720$	0	0
$721$	−32084.2	−1.65725
$722$	2860.82	0.147464
$723$	0	0
$724$	6497.08	0.333511
$725$	725.000	0.0371391
$726$	0	0
$727$	27341.9	1.39485	0.697425	−	0.716658i	$-0.254329\pi$
$727$	27341.9	1.39485	0.697425	+	0.716658i	$0.254329\pi$
$728$	34676.6	1.76538
$729$	0	0
$730$	−17842.3	−0.904620
$731$	−5363.34	−0.271368
$732$	0	0
$733$	18543.0	0.934379	0.467190	−	0.884157i	$-0.345267\pi$
$733$	18543.0	0.934379	0.467190	+	0.884157i	$0.345267\pi$
$734$	25290.0	1.27176
$735$	0	0
$736$	11651.5	0.583531
$737$	−31781.3	−1.58844
$738$	0	0
$739$	12155.7	0.605082	0.302541	−	0.953136i	$-0.402165\pi$
$739$	12155.7	0.605082	0.302541	+	0.953136i	$0.402165\pi$
$740$	−1307.45	−0.0649499
$741$	0	0
$742$	−18905.5	−0.935366
$743$	−16802.2	−0.829627	−0.414814	−	0.909906i	$-0.636153\pi$
$743$	−16802.2	−0.829627	−0.414814	+	0.909906i	$0.636153\pi$
$744$	0	0
$745$	−6832.47	−0.336003
$746$	−24825.5	−1.21840
$747$	0	0
$748$	−5288.75	−0.258524
$749$	7767.80	0.378944
$750$	0	0
$751$	−19768.5	−0.960538	−0.480269	−	0.877121i	$-0.659461\pi$
$751$	−19768.5	−0.960538	−0.480269	+	0.877121i	$0.659461\pi$
$752$	−7464.52	−0.361972
$753$	0	0
$754$	−6663.10	−0.321825
$755$	−11023.7	−0.531384
$756$	0	0
$757$	−21503.5	−1.03244	−0.516221	−	0.856456i	$-0.672662\pi$
$757$	−21503.5	−1.03244	−0.516221	+	0.856456i	$0.672662\pi$
$758$	−17007.6	−0.814966
$759$	0	0
$760$	8856.68	0.422718
$761$	−26870.1	−1.27995	−0.639974	−	0.768397i	$-0.721055\pi$
$761$	−26870.1	−1.27995	−0.639974	+	0.768397i	$0.721055\pi$
$762$	0	0
$763$	−17816.3	−0.845341
$764$	−1761.04	−0.0833927
$765$	0	0
$766$	−32131.8	−1.51563
$767$	−48160.6	−2.26725
$768$	0	0
$769$	−12864.4	−0.603254	−0.301627	−	0.953426i	$-0.597530\pi$
$769$	−12864.4	−0.603254	−0.301627	+	0.953426i	$0.597530\pi$
$770$	15369.6	0.719329
$771$	0	0
$772$	3342.93	0.155848
$773$	18534.1	0.862388	0.431194	−	0.902259i	$-0.358092\pi$
$773$	18534.1	0.862388	0.431194	+	0.902259i	$0.358092\pi$
$774$	0	0
$775$	353.539	0.0163864
$776$	18994.5	0.878689
$777$	0	0
$778$	12420.6	0.572364
$779$	−12210.0	−0.561575
$780$	0	0
$781$	43742.9	2.00415
$782$	44849.7	2.05092
$783$	0	0
$784$	−14221.0	−0.647822
$785$	−8991.54	−0.408818
$786$	0	0
$787$	−26438.7	−1.19751	−0.598754	−	0.800933i	$-0.704337\pi$
$787$	−26438.7	−1.19751	−0.598754	+	0.800933i	$0.704337\pi$
$788$	1399.36	0.0632616
$789$	0	0
$790$	10993.4	0.495097
$791$	−24704.1	−1.11046
$792$	0	0
$793$	28854.2	1.29211
$794$	−44569.1	−1.99206
$795$	0	0
$796$	−256.204	−0.0114082
$797$	33935.7	1.50823	0.754117	−	0.656740i	$-0.228065\pi$
$797$	33935.7	1.50823	0.754117	+	0.656740i	$0.228065\pi$
$798$	0	0
$799$	−8390.57	−0.371510
$800$	−1655.79	−0.0731761
$801$	0	0
$802$	11378.7	0.500993
$803$	−49967.3	−2.19590
$804$	0	0
$805$	−20365.6	−0.891668
$806$	−3249.19	−0.141995
$807$	0	0
$808$	17347.1	0.755281
$809$	−2918.27	−0.126824	−0.0634121	−	0.997987i	$-0.520198\pi$
$809$	−2918.27	−0.126824	−0.0634121	+	0.997987i	$0.520198\pi$
$810$	0	0
$811$	9351.59	0.404906	0.202453	−	0.979292i	$-0.435109\pi$
$811$	9351.59	0.404906	0.202453	+	0.979292i	$0.435109\pi$
$812$	994.748	0.0429911
$813$	0	0
$814$	−23433.4	−1.00902
$815$	1792.90	0.0770585
$816$	0	0
$817$	−5716.70	−0.244800
$818$	18106.2	0.773922
$819$	0	0
$820$	1024.88	0.0436468
$821$	36272.1	1.54191	0.770954	−	0.636891i	$-0.219780\pi$
$821$	36272.1	1.54191	0.770954	+	0.636891i	$0.219780\pi$
$822$	0	0
$823$	37236.3	1.57713	0.788563	−	0.614954i	$-0.210825\pi$
$823$	37236.3	1.57713	0.788563	+	0.614954i	$0.210825\pi$
$824$	27814.2	1.17591
$825$	0	0
$826$	46015.2	1.93835
$827$	−1886.24	−0.0793118	−0.0396559	−	0.999213i	$-0.512626\pi$
$827$	−1886.24	−0.0793118	−0.0396559	+	0.999213i	$0.512626\pi$
$828$	0	0
$829$	7608.68	0.318770	0.159385	−	0.987216i	$-0.449049\pi$
$829$	7608.68	0.318770	0.159385	+	0.987216i	$0.449049\pi$
$830$	−11508.9	−0.481299
$831$	0	0
$832$	−28751.3	−1.19805
$833$	−15985.3	−0.664893
$834$	0	0
$835$	19968.9	0.827607
$836$	−5637.20	−0.233214
$837$	0	0
$838$	33254.9	1.37085
$839$	−6779.04	−0.278949	−0.139475	−	0.990226i	$-0.544541\pi$
$839$	−6779.04	−0.278949	−0.139475	+	0.990226i	$0.544541\pi$
$840$	0	0
$841$	841.000	0.0344828
$842$	16998.8	0.695746
$843$	0	0
$844$	−5897.74	−0.240531
$845$	16853.7	0.686137
$846$	0	0
$847$	12225.7	0.495963
$848$	19532.3	0.790969
$849$	0	0
$850$	−6373.58	−0.257191
$851$	31050.5	1.25076
$852$	0	0
$853$	8817.97	0.353953	0.176976	−	0.984215i	$-0.443368\pi$
$853$	8817.97	0.353953	0.176976	+	0.984215i	$0.443368\pi$
$854$	−27568.9	−1.10467
$855$	0	0
$856$	−6734.00	−0.268882
$857$	−5429.28	−0.216407	−0.108203	−	0.994129i	$-0.534510\pi$
$857$	−5429.28	−0.216407	−0.108203	+	0.994129i	$0.534510\pi$
$858$	0	0
$859$	−43734.8	−1.73715	−0.868574	−	0.495559i	$-0.834963\pi$
$859$	−43734.8	−1.73715	−0.868574	+	0.495559i	$0.834963\pi$
$860$	479.849	0.0190264
$861$	0	0
$862$	10346.3	0.408811
$863$	20597.8	0.812464	0.406232	−	0.913770i	$-0.366842\pi$
$863$	20597.8	0.812464	0.406232	+	0.913770i	$0.366842\pi$
$864$	0	0
$865$	−4513.64	−0.177420
$866$	−16400.9	−0.643563
$867$	0	0
$868$	485.078	0.0189685
$869$	30786.9	1.20181
$870$	0	0
$871$	55000.7	2.13964
$872$	15445.2	0.599817
$873$	0	0
$874$	47804.6	1.85013
$875$	2894.15	0.111817
$876$	0	0
$877$	22182.9	0.854121	0.427061	−	0.904223i	$-0.359549\pi$
$877$	22182.9	0.854121	0.427061	+	0.904223i	$0.359549\pi$
$878$	−8738.08	−0.335872
$879$	0	0
$880$	−15879.2	−0.608282
$881$	41017.8	1.56859	0.784293	−	0.620391i	$-0.213026\pi$
$881$	41017.8	1.56859	0.784293	+	0.620391i	$0.213026\pi$
$882$	0	0
$883$	18142.8	0.691452	0.345726	−	0.938335i	$-0.387633\pi$
$883$	18142.8	0.691452	0.345726	+	0.938335i	$0.387633\pi$
$884$	9152.69	0.348233
$885$	0	0
$886$	−32954.0	−1.24956
$887$	14723.4	0.557341	0.278671	−	0.960387i	$-0.410106\pi$
$887$	14723.4	0.557341	0.278671	+	0.960387i	$0.410106\pi$
$888$	0	0
$889$	−24802.6	−0.935718
$890$	7980.73	0.300578
$891$	0	0
$892$	2873.27	0.107852
$893$	−8943.38	−0.335139
$894$	0	0
$895$	−2190.03	−0.0817930
$896$	39738.3	1.48166
$897$	0	0
$898$	−24499.5	−0.910422
$899$	410.105	0.0152144
$900$	0	0
$901$	21955.5	0.811813
$902$	18368.8	0.678066
$903$	0	0
$904$	21416.3	0.787937
$905$	21927.3	0.805400
$906$	0	0
$907$	−19250.3	−0.704738	−0.352369	−	0.935861i	$-0.614624\pi$
$907$	−19250.3	−0.704738	−0.352369	+	0.935861i	$0.614624\pi$
$908$	9567.34	0.349673
$909$	0	0
$910$	−26598.6	−0.968940
$911$	−46931.6	−1.70682	−0.853410	−	0.521241i	$-0.825469\pi$
$911$	−46931.6	−1.70682	−0.853410	+	0.521241i	$0.825469\pi$
$912$	0	0
$913$	−32230.5	−1.16832
$914$	22678.7	0.820726
$915$	0	0
$916$	−1884.46	−0.0679742
$917$	50759.2	1.82793
$918$	0	0
$919$	−35021.9	−1.25709	−0.628546	−	0.777773i	$-0.716349\pi$
$919$	−35021.9	−1.25709	−0.628546	+	0.777773i	$0.716349\pi$
$920$	17655.2	0.632689
$921$	0	0
$922$	−48933.1	−1.74786
$923$	−75701.3	−2.69961
$924$	0	0
$925$	−4412.57	−0.156848
$926$	44515.4	1.57977
$927$	0	0
$928$	−1920.71	−0.0679423
$929$	6090.61	0.215098	0.107549	−	0.994200i	$-0.465700\pi$
$929$	6090.61	0.215098	0.107549	+	0.994200i	$0.465700\pi$
$930$	0	0
$931$	−17038.4	−0.599798
$932$	−1524.68	−0.0535864
$933$	0	0
$934$	45905.4	1.60821
$935$	−17849.2	−0.624312
$936$	0	0
$937$	29684.0	1.03494	0.517468	−	0.855703i	$-0.326875\pi$
$937$	29684.0	1.03494	0.517468	+	0.855703i	$0.326875\pi$
$938$	−52550.6	−1.82925
$939$	0	0
$940$	750.690	0.0260477
$941$	−2691.42	−0.0932390	−0.0466195	−	0.998913i	$-0.514845\pi$
$941$	−2691.42	−0.0932390	−0.0466195	+	0.998913i	$0.514845\pi$
$942$	0	0
$943$	−24339.7	−0.840519
$944$	−47540.9	−1.63911
$945$	0	0
$946$	8600.29	0.295581
$947$	25412.5	0.872012	0.436006	−	0.899944i	$-0.356393\pi$
$947$	25412.5	0.872012	0.436006	+	0.899944i	$0.356393\pi$
$948$	0	0
$949$	86473.2	2.95789
$950$	−6793.51	−0.232011
$951$	0	0
$952$	38477.0	1.30992
$953$	46599.6	1.58395	0.791977	−	0.610551i	$-0.209052\pi$
$953$	46599.6	1.58395	0.791977	+	0.610551i	$0.209052\pi$
$954$	0	0
$955$	−5943.39	−0.201386
$956$	−3840.74	−0.129935
$957$	0	0
$958$	−36205.4	−1.22103
$959$	−13713.2	−0.461755
$960$	0	0
$961$	−29591.0	−0.993287
$962$	40553.7	1.35915
$963$	0	0
$964$	−7603.56	−0.254040
$965$	11282.2	0.376359
$966$	0	0
$967$	−12852.9	−0.427426	−0.213713	−	0.976896i	$-0.568556\pi$
$967$	−12852.9	−0.427426	−0.213713	+	0.976896i	$0.568556\pi$
$968$	−10598.6	−0.351914
$969$	0	0
$970$	−14569.7	−0.482273
$971$	43098.2	1.42439	0.712196	−	0.701980i	$-0.247701\pi$
$971$	43098.2	1.42439	0.712196	+	0.701980i	$0.247701\pi$
$972$	0	0
$973$	45847.7	1.51060
$974$	−27084.1	−0.890998
$975$	0	0
$976$	28483.0	0.934137
$977$	32446.8	1.06250	0.531251	−	0.847215i	$-0.321722\pi$
$977$	32446.8	1.06250	0.531251	+	0.847215i	$0.321722\pi$
$978$	0	0
$979$	22350.0	0.729632
$980$	1430.17	0.0466176
$981$	0	0
$982$	45507.1	1.47881
$983$	16990.2	0.551276	0.275638	−	0.961262i	$-0.411111\pi$
$983$	16990.2	0.551276	0.275638	+	0.961262i	$0.411111\pi$
$984$	0	0
$985$	4722.75	0.152771
$986$	−7393.36	−0.238796
$987$	0	0
$988$	9755.71	0.314140
$989$	−11395.8	−0.366397
$990$	0	0
$991$	−28350.3	−0.908757	−0.454379	−	0.890809i	$-0.650139\pi$
$991$	−28350.3	−0.908757	−0.454379	+	0.890809i	$0.650139\pi$
$992$	−936.615	−0.0299774
$993$	0	0
$994$	72329.1	2.30799
$995$	−864.671	−0.0275497
$996$	0	0
$997$	−41329.0	−1.31284	−0.656420	−	0.754396i	$-0.727930\pi$
$997$	−41329.0	−1.31284	−0.656420	+	0.754396i	$0.727930\pi$
$998$	33689.6	1.06856
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1305.4.a.m.1.6		7
3.2	odd	2		435.4.a.j.1.2	✓	7
15.14	odd	2		2175.4.a.m.1.6		7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
435.4.a.j.1.2	✓	7	3.2	odd	2
1305.4.a.m.1.6		7	1.1	even	1	trivial
2175.4.a.m.1.6		7	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:	\( N \)	\(=\)	\( 1305 = 3^{2} \cdot 5 \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1305.a (trivial)

\(f(q)\)	\(=\)	\(q+3.07921 q^{2} +1.48151 q^{4} +5.00000 q^{5} +23.1532 q^{7} -20.0718 q^{8} +15.3960 q^{10} +43.1165 q^{11} -74.6173 q^{13} +71.2934 q^{14} -73.6572 q^{16} -82.7951 q^{17} -88.2501 q^{19} +7.40754 q^{20} +132.765 q^{22} -175.920 q^{23} +25.0000 q^{25} -229.762 q^{26} +34.3016 q^{28} +29.0000 q^{29} +14.1415 q^{31} -66.2315 q^{32} -254.943 q^{34} +115.766 q^{35} -176.503 q^{37} -271.740 q^{38} -100.359 q^{40} +138.356 q^{41} +64.7784 q^{43} +63.8775 q^{44} -541.695 q^{46} +101.341 q^{47} +193.070 q^{49} +76.9801 q^{50} -110.546 q^{52} -265.178 q^{53} +215.583 q^{55} -464.726 q^{56} +89.2970 q^{58} +645.434 q^{59} -386.696 q^{61} +43.5447 q^{62} +385.317 q^{64} -373.087 q^{65} -737.103 q^{67} -122.662 q^{68} +356.467 q^{70} +1014.53 q^{71} -1158.89 q^{73} -543.489 q^{74} -130.743 q^{76} +998.285 q^{77} +714.039 q^{79} -368.286 q^{80} +426.027 q^{82} -747.521 q^{83} -413.976 q^{85} +199.466 q^{86} -865.426 q^{88} +518.363 q^{89} -1727.63 q^{91} -260.628 q^{92} +312.051 q^{94} -441.250 q^{95} -946.328 q^{97} +594.502 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q + q^{2} + 15 q^{4} + 35 q^{5} - 37 q^{7} + 36 q^{8} + 5 q^{10} + 11 q^{11} - 133 q^{13} + 75 q^{14} - 53 q^{16} - 21 q^{17} - 170 q^{19} + 75 q^{20} - 369 q^{22} + 68 q^{23} + 175 q^{25} - 181 q^{26}+ \cdots - 1068 q^{98}+O(q^{100}) \) 7 * q + q^2 + 15 * q^4 + 35 * q^5 - 37 * q^7 + 36 * q^8 + 5 * q^10 + 11 * q^11 - 133 * q^13 + 75 * q^14 - 53 * q^16 - 21 * q^17 - 170 * q^19 + 75 * q^20 - 369 * q^22 + 68 * q^23 + 175 * q^25 - 181 * q^26 - 637 * q^28 + 203 * q^29 - 480 * q^31 + 779 * q^32 - 897 * q^34 - 185 * q^35 - 1032 * q^37 + 194 * q^38 + 180 * q^40 + 638 * q^41 - 512 * q^43 - 625 * q^44 + 16 * q^46 + 111 * q^47 + 178 * q^49 + 25 * q^50 - 1263 * q^52 - 410 * q^53 + 55 * q^55 - 1174 * q^56 + 29 * q^58 + 426 * q^59 - 1192 * q^61 - 460 * q^62 + 390 * q^64 - 665 * q^65 - 1671 * q^67 - 1509 * q^68 + 375 * q^70 + 1324 * q^71 - 852 * q^73 - 1780 * q^74 - 564 * q^76 + 2107 * q^77 + 366 * q^79 - 265 * q^80 - 318 * q^82 - 470 * q^83 - 105 * q^85 + 2196 * q^86 - 2518 * q^88 - 51 * q^89 - 1297 * q^91 + 684 * q^92 - 1837 * q^94 - 850 * q^95 - 3322 * q^97 - 1068 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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