LMFDB - Embedded newform 1305.4.a.p.1.3

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:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 4x^{7} - 43x^{6} + 117x^{5} + 586x^{4} - 701x^{3} - 1792x^{2} + 924x + 1424 $ x^8 - 4x^7 - 43x^6 + 117x^5 + 586x^4 - 701x^3 - 1792x^2 + 924*x + 1424
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	no (minimal twist has level 435)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$1.81879$ 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)

$f(q)$	$=$	\(q-0.818792 q^{2} -7.32958 q^{4} +5.00000 q^{5} +13.8632 q^{7} +12.5517 q^{8} -4.09396 q^{10} -71.7303 q^{11} -78.8785 q^{13} -11.3510 q^{14} +48.3594 q^{16} -93.8775 q^{17} -15.4555 q^{19} -36.6479 q^{20} +58.7322 q^{22} +187.363 q^{23} +25.0000 q^{25} +64.5851 q^{26} -101.611 q^{28} -29.0000 q^{29} -73.3710 q^{31} -140.010 q^{32} +76.8662 q^{34} +69.3158 q^{35} -127.123 q^{37} +12.6549 q^{38} +62.7587 q^{40} +124.177 q^{41} -41.9717 q^{43} +525.753 q^{44} -153.411 q^{46} +588.328 q^{47} -150.813 q^{49} -20.4698 q^{50} +578.146 q^{52} +125.128 q^{53} -358.651 q^{55} +174.007 q^{56} +23.7450 q^{58} +55.2122 q^{59} +259.660 q^{61} +60.0755 q^{62} -272.236 q^{64} -394.392 q^{65} -664.277 q^{67} +688.083 q^{68} -56.7552 q^{70} -265.186 q^{71} -5.27903 q^{73} +104.087 q^{74} +113.283 q^{76} -994.408 q^{77} -792.533 q^{79} +241.797 q^{80} -101.675 q^{82} -1424.74 q^{83} -469.388 q^{85} +34.3661 q^{86} -900.339 q^{88} +1499.69 q^{89} -1093.50 q^{91} -1373.29 q^{92} -481.718 q^{94} -77.2777 q^{95} +764.065 q^{97} +123.484 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 4 q^{2} + 38 q^{4} + 40 q^{5} - q^{7} + 9 q^{8} + 20 q^{10} - 11 q^{11} - 97 q^{13} + 69 q^{14} + 242 q^{16} + 119 q^{17} + 220 q^{19} + 190 q^{20} - 129 q^{22} + 262 q^{23} + 200 q^{25} - 43 q^{26}+ \cdots + 7677 q^{98}+O(q^{100}) $ 8 * q + 4 * q^2 + 38 * q^4 + 40 * q^5 - q^7 + 9 * q^8 + 20 * q^10 - 11 * q^11 - 97 * q^13 + 69 * q^14 + 242 * q^16 + 119 * q^17 + 220 * q^19 + 190 * q^20 - 129 * q^22 + 262 * q^23 + 200 * q^25 - 43 * q^26 - 13 * q^28 - 232 * q^29 + 108 * q^31 + 566 * q^32 - 527 * q^34 - 5 * q^35 + 118 * q^37 + 1088 * q^38 + 45 * q^40 + 118 * q^41 - 384 * q^43 + 1605 * q^44 + 154 * q^46 + 1309 * q^47 + 427 * q^49 + 100 * q^50 - 747 * q^52 + 1912 * q^53 - 55 * q^55 + 2022 * q^56 - 116 * q^58 + 1178 * q^59 + 572 * q^61 + 2572 * q^62 - 1771 * q^64 - 485 * q^65 + 1147 * q^67 + 3963 * q^68 + 345 * q^70 - 84 * q^71 - 222 * q^73 + 1758 * q^74 + 1390 * q^76 + 1753 * q^77 + 390 * q^79 + 1210 * q^80 - 2994 * q^82 + 1570 * q^83 + 595 * q^85 - 1668 * q^86 - 1790 * q^88 + 119 * q^89 - 3529 * q^91 + 2934 * q^92 - 2277 * q^94 + 1100 * q^95 - 2100 * q^97 + 7677 * 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 $

$2$	−0.818792	−0.289487	−0.144743	−	0.989469i	$-0.546236\pi$
$2$	−0.818792	−0.289487	−0.144743	+	0.989469i	$0.546236\pi$
$3$	0	0
$3$	0	0
$4$	−7.32958	−0.916198
$5$	5.00000	0.447214
$5$	5.00000	0.447214
$6$	0	0
$7$	13.8632	0.748540	0.374270	−	0.927320i	$-0.377893\pi$
$7$	13.8632	0.748540	0.374270	+	0.927320i	$0.377893\pi$
$8$	12.5517	0.554714
$9$	0	0
$10$	−4.09396	−0.129462
$11$	−71.7303	−1.96614	−0.983068	−	0.183243i	$-0.941340\pi$
$11$	−71.7303	−1.96614	−0.983068	+	0.183243i	$0.941340\pi$
$12$	0	0
$13$	−78.8785	−1.68284	−0.841421	−	0.540379i	$-0.818281\pi$
$13$	−78.8785	−1.68284	−0.841421	+	0.540379i	$0.818281\pi$
$14$	−11.3510	−0.216692
$15$	0	0
$16$	48.3594	0.755615
$17$	−93.8775	−1.33933	−0.669666	−	0.742662i	$-0.733563\pi$
$17$	−93.8775	−1.33933	−0.669666	+	0.742662i	$0.733563\pi$
$18$	0	0
$19$	−15.4555	−0.186618	−0.0933091	−	0.995637i	$-0.529744\pi$
$19$	−15.4555	−0.186618	−0.0933091	+	0.995637i	$0.529744\pi$
$20$	−36.6479	−0.409736
$21$	0	0
$22$	58.7322	0.569170
$23$	187.363	1.69860	0.849301	−	0.527909i	$-0.177024\pi$
$23$	187.363	1.69860	0.849301	+	0.527909i	$0.177024\pi$
$24$	0	0
$25$	25.0000	0.200000
$26$	64.5851	0.487161
$27$	0	0
$28$	−101.611	−0.685811
$29$	−29.0000	−0.185695
$29$	−29.0000	−0.185695
$30$	0	0
$31$	−73.3710	−0.425091	−0.212545	−	0.977151i	$-0.568175\pi$
$31$	−73.3710	−0.425091	−0.212545	+	0.977151i	$0.568175\pi$
$32$	−140.010	−0.773454
$33$	0	0
$34$	76.8662	0.387719
$35$	69.3158	0.334757
$36$	0	0
$37$	−127.123	−0.564836	−0.282418	−	0.959291i	$-0.591137\pi$
$37$	−127.123	−0.564836	−0.282418	+	0.959291i	$0.591137\pi$
$38$	12.6549	0.0540235
$39$	0	0
$40$	62.7587	0.248075
$41$	124.177	0.473004	0.236502	−	0.971631i	$-0.423999\pi$
$41$	124.177	0.473004	0.236502	+	0.971631i	$0.423999\pi$
$42$	0	0
$43$	−41.9717	−0.148852	−0.0744258	−	0.997227i	$-0.523712\pi$
$43$	−41.9717	−0.148852	−0.0744258	+	0.997227i	$0.523712\pi$
$44$	525.753	1.80137
$45$	0	0
$46$	−153.411	−0.491723
$47$	588.328	1.82588	0.912941	−	0.408093i	$-0.133806\pi$
$47$	588.328	1.82588	0.912941	+	0.408093i	$0.133806\pi$
$48$	0	0
$49$	−150.813	−0.439688
$50$	−20.4698	−0.0578973
$51$	0	0
$52$	578.146	1.54182
$53$	125.128	0.324294	0.162147	−	0.986767i	$-0.448158\pi$
$53$	125.128	0.324294	0.162147	+	0.986767i	$0.448158\pi$
$54$	0	0
$55$	−358.651	−0.879282
$56$	174.007	0.415225
$57$	0	0
$58$	23.7450	0.0537563
$59$	55.2122	0.121831	0.0609154	−	0.998143i	$-0.480598\pi$
$59$	55.2122	0.121831	0.0609154	+	0.998143i	$0.480598\pi$
$60$	0	0
$61$	259.660	0.545018	0.272509	−	0.962153i	$-0.412147\pi$
$61$	259.660	0.545018	0.272509	+	0.962153i	$0.412147\pi$
$62$	60.0755	0.123058
$63$	0	0
$64$	−272.236	−0.531711
$65$	−394.392	−0.752590
$66$	0	0
$67$	−664.277	−1.21126	−0.605630	−	0.795747i	$-0.707079\pi$
$67$	−664.277	−1.21126	−0.605630	+	0.795747i	$0.707079\pi$
$68$	688.083	1.22709
$69$	0	0
$70$	−56.7552	−0.0969078
$71$	−265.186	−0.443264	−0.221632	−	0.975130i	$-0.571138\pi$
$71$	−265.186	−0.443264	−0.221632	+	0.975130i	$0.571138\pi$
$72$	0	0
$73$	−5.27903	−0.00846388	−0.00423194	−	0.999991i	$-0.501347\pi$
$73$	−5.27903	−0.00846388	−0.00423194	+	0.999991i	$0.501347\pi$
$74$	104.087	0.163512
$75$	0	0
$76$	113.283	0.170979
$77$	−994.408	−1.47173
$78$	0	0
$79$	−792.533	−1.12870	−0.564348	−	0.825537i	$-0.690872\pi$
$79$	−792.533	−1.12870	−0.564348	+	0.825537i	$0.690872\pi$
$80$	241.797	0.337921
$81$	0	0
$82$	−101.675	−0.136928
$83$	−1424.74	−1.88416	−0.942081	−	0.335385i	$-0.891134\pi$
$83$	−1424.74	−1.88416	−0.942081	+	0.335385i	$0.891134\pi$
$84$	0	0
$85$	−469.388	−0.598968
$86$	34.3661	0.0430906
$87$	0	0
$88$	−900.339	−1.09064
$89$	1499.69	1.78614	0.893071	−	0.449916i	$-0.148546\pi$
$89$	1499.69	1.78614	0.893071	+	0.449916i	$0.148546\pi$
$90$	0	0
$91$	−1093.50	−1.25968
$92$	−1373.29	−1.55626
$93$	0	0
$94$	−481.718	−0.528568
$95$	−77.2777	−0.0834582
$96$	0	0
$97$	764.065	0.799784	0.399892	−	0.916562i	$-0.369048\pi$
$97$	764.065	0.799784	0.399892	+	0.916562i	$0.369048\pi$
$98$	123.484	0.127284
$99$	0	0
$100$	−183.240	−0.183240
$101$	1545.45	1.52256	0.761279	−	0.648425i	$-0.224572\pi$
$101$	1545.45	1.52256	0.761279	+	0.648425i	$0.224572\pi$
$102$	0	0
$103$	204.337	0.195475	0.0977374	−	0.995212i	$-0.468839\pi$
$103$	204.337	0.195475	0.0977374	+	0.995212i	$0.468839\pi$
$104$	−990.062	−0.933496
$105$	0	0
$106$	−102.454	−0.0938789
$107$	342.888	0.309797	0.154898	−	0.987930i	$-0.450495\pi$
$107$	342.888	0.309797	0.154898	+	0.987930i	$0.450495\pi$
$108$	0	0
$109$	1312.12	1.15301	0.576504	−	0.817094i	$-0.304416\pi$
$109$	1312.12	1.15301	0.576504	+	0.817094i	$0.304416\pi$
$110$	293.661	0.254540
$111$	0	0
$112$	670.414	0.565608
$113$	−39.3605	−0.0327675	−0.0163837	−	0.999866i	$-0.505215\pi$
$113$	−39.3605	−0.0327675	−0.0163837	+	0.999866i	$0.505215\pi$
$114$	0	0
$115$	936.814	0.759638
$116$	212.558	0.170134
$117$	0	0
$118$	−45.2073	−0.0352684
$119$	−1301.44	−1.00254
$120$	0	0
$121$	3814.23	2.86569
$122$	−212.608	−0.157775
$123$	0	0
$124$	537.778	0.389467
$125$	125.000	0.0894427
$126$	0	0
$127$	−1843.60	−1.28813	−0.644066	−	0.764970i	$-0.722754\pi$
$127$	−1843.60	−1.28813	−0.644066	+	0.764970i	$0.722754\pi$
$128$	1342.99	0.927377
$129$	0	0
$130$	322.925	0.217865
$131$	2280.13	1.52073	0.760366	−	0.649494i	$-0.225019\pi$
$131$	2280.13	1.52073	0.760366	+	0.649494i	$0.225019\pi$
$132$	0	0
$133$	−214.263	−0.139691
$134$	543.905	0.350643
$135$	0	0
$136$	−1178.33	−0.742946
$137$	−1201.01	−0.748975	−0.374487	−	0.927232i	$-0.622181\pi$
$137$	−1201.01	−0.748975	−0.374487	+	0.927232i	$0.622181\pi$
$138$	0	0
$139$	−9.65588	−0.00589209	−0.00294605	−	0.999996i	$-0.500938\pi$
$139$	−9.65588	−0.00589209	−0.00294605	+	0.999996i	$0.500938\pi$
$140$	−508.056	−0.306704
$141$	0	0
$142$	217.132	0.128319
$143$	5657.98	3.30870
$144$	0	0
$145$	−145.000	−0.0830455
$146$	4.32242	0.00245018
$147$	0	0
$148$	931.760	0.517501
$149$	564.420	0.310329	0.155165	−	0.987889i	$-0.450409\pi$
$149$	564.420	0.310329	0.155165	+	0.987889i	$0.450409\pi$
$150$	0	0
$151$	635.922	0.342719	0.171359	−	0.985209i	$-0.445184\pi$
$151$	635.922	0.342719	0.171359	+	0.985209i	$0.445184\pi$
$152$	−193.994	−0.103520
$153$	0	0
$154$	814.213	0.426046
$155$	−366.855	−0.190106
$156$	0	0
$157$	−573.022	−0.291288	−0.145644	−	0.989337i	$-0.546525\pi$
$157$	−573.022	−0.291288	−0.145644	+	0.989337i	$0.546525\pi$
$158$	648.920	0.326742
$159$	0	0
$160$	−700.051	−0.345899
$161$	2597.44	1.27147
$162$	0	0
$163$	−3278.02	−1.57518	−0.787591	−	0.616199i	$-0.788672\pi$
$163$	−3278.02	−1.57518	−0.787591	+	0.616199i	$0.788672\pi$
$164$	−910.164	−0.433365
$165$	0	0
$166$	1166.57	0.545440
$167$	3924.81	1.81863	0.909314	−	0.416112i	$-0.136607\pi$
$167$	3924.81	1.81863	0.909314	+	0.416112i	$0.136607\pi$
$168$	0	0
$169$	4024.82	1.83196
$170$	384.331	0.173393
$171$	0	0
$172$	307.635	0.136378
$173$	2409.34	1.05884	0.529419	−	0.848361i	$-0.322410\pi$
$173$	2409.34	1.05884	0.529419	+	0.848361i	$0.322410\pi$
$174$	0	0
$175$	346.579	0.149708
$176$	−3468.83	−1.48564
$177$	0	0
$178$	−1227.93	−0.517064
$179$	270.657	0.113016	0.0565081	−	0.998402i	$-0.482003\pi$
$179$	270.657	0.113016	0.0565081	+	0.998402i	$0.482003\pi$
$180$	0	0
$181$	−176.505	−0.0724837	−0.0362418	−	0.999343i	$-0.511539\pi$
$181$	−176.505	−0.0724837	−0.0362418	+	0.999343i	$0.511539\pi$
$182$	895.353	0.364659
$183$	0	0
$184$	2351.73	0.942238
$185$	−635.616	−0.252602
$186$	0	0
$187$	6733.86	2.63331
$188$	−4312.19	−1.67287
$189$	0	0
$190$	63.2744	0.0241600
$191$	−2520.29	−0.954774	−0.477387	−	0.878693i	$-0.658416\pi$
$191$	−2520.29	−0.954774	−0.477387	+	0.878693i	$0.658416\pi$
$192$	0	0
$193$	1493.85	0.557150	0.278575	−	0.960414i	$-0.410138\pi$
$193$	1493.85	0.557150	0.278575	+	0.960414i	$0.410138\pi$
$194$	−625.610	−0.231527
$195$	0	0
$196$	1105.40	0.402841
$197$	3155.05	1.14106	0.570528	−	0.821278i	$-0.306739\pi$
$197$	3155.05	1.14106	0.570528	+	0.821278i	$0.306739\pi$
$198$	0	0
$199$	5460.52	1.94516	0.972578	−	0.232575i	$-0.0747152\pi$
$199$	5460.52	1.94516	0.972578	+	0.232575i	$0.0747152\pi$
$200$	313.793	0.110943
$201$	0	0
$202$	−1265.40	−0.440760
$203$	−402.032	−0.139000
$204$	0	0
$205$	620.884	0.211534
$206$	−167.309	−0.0565873
$207$	0	0
$208$	−3814.52	−1.27158
$209$	1108.63	0.366917
$210$	0	0
$211$	2596.75	0.847241	0.423621	−	0.905840i	$-0.360759\pi$
$211$	2596.75	0.847241	0.423621	+	0.905840i	$0.360759\pi$
$212$	−917.133	−0.297118
$213$	0	0
$214$	−280.754	−0.0896820
$215$	−209.858	−0.0665685
$216$	0	0
$217$	−1017.15	−0.318197
$218$	−1074.35	−0.333781
$219$	0	0
$220$	2628.76	0.805596
$221$	7404.92	2.25389
$222$	0	0
$223$	−1578.45	−0.473995	−0.236998	−	0.971510i	$-0.576163\pi$
$223$	−1578.45	−0.473995	−0.236998	+	0.971510i	$0.576163\pi$
$224$	−1940.98	−0.578961
$225$	0	0
$226$	32.2281	0.00948575
$227$	2428.63	0.710106	0.355053	−	0.934846i	$-0.384463\pi$
$227$	2428.63	0.710106	0.355053	+	0.934846i	$0.384463\pi$
$228$	0	0
$229$	−4621.51	−1.33362	−0.666808	−	0.745229i	$-0.732340\pi$
$229$	−4621.51	−1.33362	−0.666808	+	0.745229i	$0.732340\pi$
$230$	−767.056	−0.219905
$231$	0	0
$232$	−364.000	−0.103008
$233$	−3337.50	−0.938398	−0.469199	−	0.883092i	$-0.655457\pi$
$233$	−3337.50	−0.938398	−0.469199	+	0.883092i	$0.655457\pi$
$234$	0	0
$235$	2941.64	0.816559
$236$	−404.682	−0.111621
$237$	0	0
$238$	1065.61	0.290223
$239$	757.216	0.204938	0.102469	−	0.994736i	$-0.467326\pi$
$239$	757.216	0.204938	0.102469	+	0.994736i	$0.467326\pi$
$240$	0	0
$241$	−480.856	−0.128525	−0.0642627	−	0.997933i	$-0.520470\pi$
$241$	−480.856	−0.128525	−0.0642627	+	0.997933i	$0.520470\pi$
$242$	−3123.06	−0.829578
$243$	0	0
$244$	−1903.20	−0.499344
$245$	−754.065	−0.196634
$246$	0	0
$247$	1219.11	0.314049
$248$	−920.933	−0.235804
$249$	0	0
$250$	−102.349	−0.0258925
$251$	−2236.26	−0.562356	−0.281178	−	0.959656i	$-0.590725\pi$
$251$	−2236.26	−0.562356	−0.281178	+	0.959656i	$0.590725\pi$
$252$	0	0
$253$	−13439.6	−3.33968
$254$	1509.52	0.372897
$255$	0	0
$256$	1078.26	0.263247
$257$	6940.70	1.68463	0.842314	−	0.538988i	$-0.181193\pi$
$257$	6940.70	1.68463	0.842314	+	0.538988i	$0.181193\pi$
$258$	0	0
$259$	−1762.33	−0.422802
$260$	2890.73	0.689521
$261$	0	0
$262$	−1866.95	−0.440232
$263$	−5980.13	−1.40209	−0.701047	−	0.713115i	$-0.747284\pi$
$263$	−5980.13	−1.40209	−0.701047	+	0.713115i	$0.747284\pi$
$264$	0	0
$265$	625.638	0.145029
$266$	175.436	0.0404387
$267$	0	0
$268$	4868.87	1.10975
$269$	−1822.34	−0.413049	−0.206525	−	0.978441i	$-0.566215\pi$
$269$	−1822.34	−0.413049	−0.206525	+	0.978441i	$0.566215\pi$
$270$	0	0
$271$	3201.60	0.717651	0.358826	−	0.933405i	$-0.383177\pi$
$271$	3201.60	0.717651	0.358826	+	0.933405i	$0.383177\pi$
$272$	−4539.86	−1.01202
$273$	0	0
$274$	983.380	0.216818
$275$	−1793.26	−0.393227
$276$	0	0
$277$	924.254	0.200480	0.100240	−	0.994963i	$-0.468039\pi$
$277$	924.254	0.200480	0.100240	+	0.994963i	$0.468039\pi$
$278$	7.90616	0.00170568
$279$	0	0
$280$	870.033	0.185694
$281$	−3105.33	−0.659247	−0.329623	−	0.944113i	$-0.606922\pi$
$281$	−3105.33	−0.659247	−0.329623	+	0.944113i	$0.606922\pi$
$282$	0	0
$283$	−10.5568	−0.00221745	−0.00110873	−	0.999999i	$-0.500353\pi$
$283$	−10.5568	−0.00221745	−0.00110873	+	0.999999i	$0.500353\pi$
$284$	1943.70	0.406117
$285$	0	0
$286$	−4632.70	−0.957823
$287$	1721.48	0.354063
$288$	0	0
$289$	3899.99	0.793811
$290$	118.725	0.0240406
$291$	0	0
$292$	38.6930	0.00775459
$293$	2892.07	0.576643	0.288322	−	0.957534i	$-0.406903\pi$
$293$	2892.07	0.576643	0.288322	+	0.957534i	$0.406903\pi$
$294$	0	0
$295$	276.061	0.0544844
$296$	−1595.62	−0.313322
$297$	0	0
$298$	−462.142	−0.0898362
$299$	−14778.9	−2.85848
$300$	0	0
$301$	−581.860	−0.111421
$302$	−520.687	−0.0992125
$303$	0	0
$304$	−747.420	−0.141012
$305$	1298.30	0.243739
$306$	0	0
$307$	6869.50	1.27708	0.638539	−	0.769590i	$-0.279539\pi$
$307$	6869.50	1.27708	0.638539	+	0.769590i	$0.279539\pi$
$308$	7288.59	1.34840
$309$	0	0
$310$	300.378	0.0550332
$311$	3754.87	0.684627	0.342313	−	0.939586i	$-0.388790\pi$
$311$	3754.87	0.684627	0.342313	+	0.939586i	$0.388790\pi$
$312$	0	0
$313$	1004.33	0.181368	0.0906839	−	0.995880i	$-0.471095\pi$
$313$	1004.33	0.181368	0.0906839	+	0.995880i	$0.471095\pi$
$314$	469.186	0.0843239
$315$	0	0
$316$	5808.94	1.03411
$317$	4555.66	0.807165	0.403582	−	0.914943i	$-0.367765\pi$
$317$	4555.66	0.807165	0.403582	+	0.914943i	$0.367765\pi$
$318$	0	0
$319$	2080.18	0.365102
$320$	−1361.18	−0.237788
$321$	0	0
$322$	−2126.76	−0.368074
$323$	1450.93	0.249944
$324$	0	0
$325$	−1971.96	−0.336569
$326$	2684.02	0.455994
$327$	0	0
$328$	1558.64	0.262382
$329$	8156.08	1.36675
$330$	0	0
$331$	−4344.37	−0.721414	−0.360707	−	0.932679i	$-0.617465\pi$
$331$	−4344.37	−0.721414	−0.360707	+	0.932679i	$0.617465\pi$
$332$	10442.7	1.72626
$333$	0	0
$334$	−3213.60	−0.526468
$335$	−3321.39	−0.541692
$336$	0	0
$337$	8112.36	1.31130	0.655650	−	0.755065i	$-0.272394\pi$
$337$	8112.36	1.31130	0.655650	+	0.755065i	$0.272394\pi$
$338$	−3295.49	−0.530328
$339$	0	0
$340$	3440.41	0.548773
$341$	5262.92	0.835786
$342$	0	0
$343$	−6845.81	−1.07766
$344$	−526.817	−0.0825700
$345$	0	0
$346$	−1972.75	−0.306519
$347$	10739.1	1.66139	0.830696	−	0.556726i	$-0.187943\pi$
$347$	10739.1	1.66139	0.830696	+	0.556726i	$0.187943\pi$
$348$	0	0
$349$	−7695.17	−1.18027	−0.590134	−	0.807306i	$-0.700925\pi$
$349$	−7695.17	−1.18027	−0.590134	+	0.807306i	$0.700925\pi$
$350$	−283.776	−0.0433385
$351$	0	0
$352$	10043.0	1.52072
$353$	6621.09	0.998315	0.499157	−	0.866511i	$-0.333643\pi$
$353$	6621.09	0.998315	0.499157	+	0.866511i	$0.333643\pi$
$354$	0	0
$355$	−1325.93	−0.198234
$356$	−10992.1	−1.63646
$357$	0	0
$358$	−221.612	−0.0327167
$359$	−2419.99	−0.355772	−0.177886	−	0.984051i	$-0.556926\pi$
$359$	−2419.99	−0.355772	−0.177886	+	0.984051i	$0.556926\pi$
$360$	0	0
$361$	−6620.13	−0.965174
$362$	144.521	0.0209831
$363$	0	0
$364$	8014.93	1.15411
$365$	−26.3951	−0.00378516
$366$	0	0
$367$	5254.54	0.747369	0.373685	−	0.927556i	$-0.378094\pi$
$367$	5254.54	0.747369	0.373685	+	0.927556i	$0.378094\pi$
$368$	9060.75	1.28349
$369$	0	0
$370$	520.437	0.0731250
$371$	1734.66	0.242747
$372$	0	0
$373$	−6000.18	−0.832916	−0.416458	−	0.909155i	$-0.636729\pi$
$373$	−6000.18	−0.832916	−0.416458	+	0.909155i	$0.636729\pi$
$374$	−5513.63	−0.762308
$375$	0	0
$376$	7384.53	1.01284
$377$	2287.48	0.312496
$378$	0	0
$379$	9449.12	1.28066	0.640328	−	0.768101i	$-0.278798\pi$
$379$	9449.12	1.28066	0.640328	+	0.768101i	$0.278798\pi$
$380$	566.413	0.0764642
$381$	0	0
$382$	2063.59	0.276394
$383$	856.748	0.114302	0.0571511	−	0.998366i	$-0.481798\pi$
$383$	856.748	0.114302	0.0571511	+	0.998366i	$0.481798\pi$
$384$	0	0
$385$	−4972.04	−0.658178
$386$	−1223.16	−0.161288
$387$	0	0
$388$	−5600.28	−0.732760
$389$	172.900	0.0225356	0.0112678	−	0.999937i	$-0.496413\pi$
$389$	172.900	0.0225356	0.0112678	+	0.999937i	$0.496413\pi$
$390$	0	0
$391$	−17589.2	−2.27499
$392$	−1892.96	−0.243901
$393$	0	0
$394$	−2583.33	−0.330320
$395$	−3962.67	−0.504768
$396$	0	0
$397$	−9250.58	−1.16945	−0.584727	−	0.811230i	$-0.698798\pi$
$397$	−9250.58	−1.16945	−0.584727	+	0.811230i	$0.698798\pi$
$398$	−4471.03	−0.563097
$399$	0	0
$400$	1208.98	0.151123
$401$	−9906.64	−1.23370	−0.616851	−	0.787080i	$-0.711592\pi$
$401$	−9906.64	−1.23370	−0.616851	+	0.787080i	$0.711592\pi$
$402$	0	0
$403$	5787.39	0.715361
$404$	−11327.5	−1.39496
$405$	0	0
$406$	329.180	0.0402388
$407$	9118.58	1.11054
$408$	0	0
$409$	−9457.67	−1.14340	−0.571702	−	0.820462i	$-0.693716\pi$
$409$	−9457.67	−1.14340	−0.571702	+	0.820462i	$0.693716\pi$
$410$	−508.375	−0.0612362
$411$	0	0
$412$	−1497.70	−0.179094
$413$	765.415	0.0911952
$414$	0	0
$415$	−7123.70	−0.842623
$416$	11043.8	1.30160
$417$	0	0
$418$	−907.737	−0.106217
$419$	−4924.06	−0.574119	−0.287060	−	0.957913i	$-0.592678\pi$
$419$	−4924.06	−0.574119	−0.287060	+	0.957913i	$0.592678\pi$
$420$	0	0
$421$	−13100.9	−1.51663	−0.758314	−	0.651889i	$-0.773977\pi$
$421$	−13100.9	−1.51663	−0.758314	+	0.651889i	$0.773977\pi$
$422$	−2126.20	−0.245265
$423$	0	0
$424$	1570.57	0.179891
$425$	−2346.94	−0.267866
$426$	0	0
$427$	3599.71	0.407968
$428$	−2513.23	−0.283835
$429$	0	0
$430$	171.830	0.0192707
$431$	12506.6	1.39774	0.698868	−	0.715251i	$-0.253688\pi$
$431$	12506.6	1.39774	0.698868	+	0.715251i	$0.253688\pi$
$432$	0	0
$433$	3714.42	0.412249	0.206124	−	0.978526i	$-0.433915\pi$
$433$	3714.42	0.412249	0.206124	+	0.978526i	$0.433915\pi$
$434$	832.837	0.0921139
$435$	0	0
$436$	−9617.26	−1.05638
$437$	−2895.79	−0.316990
$438$	0	0
$439$	9298.90	1.01096	0.505481	−	0.862838i	$-0.331315\pi$
$439$	9298.90	1.01096	0.505481	+	0.862838i	$0.331315\pi$
$440$	−4501.70	−0.487750
$441$	0	0
$442$	−6063.09	−0.652470
$443$	10321.3	1.10696	0.553478	−	0.832864i	$-0.313300\pi$
$443$	10321.3	1.10696	0.553478	+	0.832864i	$0.313300\pi$
$444$	0	0
$445$	7498.44	0.798787
$446$	1292.42	0.137215
$447$	0	0
$448$	−3774.05	−0.398007
$449$	−7340.29	−0.771514	−0.385757	−	0.922600i	$-0.626060\pi$
$449$	−7340.29	−0.771514	−0.385757	+	0.922600i	$0.626060\pi$
$450$	0	0
$451$	−8907.24	−0.929990
$452$	288.496	0.0300215
$453$	0	0
$454$	−1988.54	−0.205566
$455$	−5467.52	−0.563344
$456$	0	0
$457$	1423.35	0.145692	0.0728462	−	0.997343i	$-0.476792\pi$
$457$	1423.35	0.145692	0.0728462	+	0.997343i	$0.476792\pi$
$458$	3784.06	0.386064
$459$	0	0
$460$	−6866.45	−0.695978
$461$	7865.08	0.794606	0.397303	−	0.917687i	$-0.369946\pi$
$461$	7865.08	0.794606	0.397303	+	0.917687i	$0.369946\pi$
$462$	0	0
$463$	6096.09	0.611899	0.305950	−	0.952048i	$-0.401026\pi$
$463$	6096.09	0.611899	0.305950	+	0.952048i	$0.401026\pi$
$464$	−1402.42	−0.140314
$465$	0	0
$466$	2732.72	0.271654
$467$	7535.45	0.746679	0.373339	−	0.927695i	$-0.378213\pi$
$467$	7535.45	0.746679	0.373339	+	0.927695i	$0.378213\pi$
$468$	0	0
$469$	−9208.98	−0.906676
$470$	−2408.59	−0.236383
$471$	0	0
$472$	693.009	0.0675812
$473$	3010.64	0.292663
$474$	0	0
$475$	−386.389	−0.0373236
$476$	9539.00	0.918528
$477$	0	0
$478$	−620.002	−0.0593269
$479$	−16971.5	−1.61889	−0.809444	−	0.587197i	$-0.800232\pi$
$479$	−16971.5	−1.61889	−0.809444	+	0.587197i	$0.800232\pi$
$480$	0	0
$481$	10027.3	0.950530
$482$	393.721	0.0372064
$483$	0	0
$484$	−27956.7	−2.62554
$485$	3820.33	0.357674
$486$	0	0
$487$	17333.1	1.61281	0.806405	−	0.591364i	$-0.201410\pi$
$487$	17333.1	1.61281	0.806405	+	0.591364i	$0.201410\pi$
$488$	3259.19	0.302329
$489$	0	0
$490$	617.422	0.0569230
$491$	−2468.47	−0.226885	−0.113442	−	0.993545i	$-0.536188\pi$
$491$	−2468.47	−0.226885	−0.113442	+	0.993545i	$0.536188\pi$
$492$	0	0
$493$	2722.45	0.248708
$494$	−998.197	−0.0909130
$495$	0	0
$496$	−3548.17	−0.321205
$497$	−3676.31	−0.331801
$498$	0	0
$499$	9085.35	0.815062	0.407531	−	0.913191i	$-0.366390\pi$
$499$	9085.35	0.815062	0.407531	+	0.913191i	$0.366390\pi$
$500$	−916.198	−0.0819472
$501$	0	0
$502$	1831.03	0.162795
$503$	−4534.24	−0.401932	−0.200966	−	0.979598i	$-0.564408\pi$
$503$	−4534.24	−0.401932	−0.200966	+	0.979598i	$0.564408\pi$
$504$	0	0
$505$	7727.26	0.680908
$506$	11004.2	0.966793
$507$	0	0
$508$	13512.8	1.18018
$509$	13065.3	1.13774	0.568871	−	0.822426i	$-0.307380\pi$
$509$	13065.3	1.13774	0.568871	+	0.822426i	$0.307380\pi$
$510$	0	0
$511$	−73.1840	−0.00633555
$512$	−11626.8	−1.00358
$513$	0	0
$514$	−5682.99	−0.487677
$515$	1021.68	0.0874190
$516$	0	0
$517$	−42200.9	−3.58993
$518$	1442.98	0.122396
$519$	0	0
$520$	−4950.31	−0.417472
$521$	−11144.0	−0.937101	−0.468550	−	0.883437i	$-0.655224\pi$
$521$	−11144.0	−0.937101	−0.468550	+	0.883437i	$0.655224\pi$
$522$	0	0
$523$	−4525.71	−0.378385	−0.189193	−	0.981940i	$-0.560587\pi$
$523$	−4525.71	−0.378385	−0.189193	+	0.981940i	$0.560587\pi$
$524$	−16712.4	−1.39329
$525$	0	0
$526$	4896.49	0.405888
$527$	6887.89	0.569338
$528$	0	0
$529$	22937.8	1.88525
$530$	−512.268	−0.0419839
$531$	0	0
$532$	1570.45	0.127985
$533$	−9794.88	−0.795992
$534$	0	0
$535$	1714.44	0.138545
$536$	−8337.83	−0.671902
$537$	0	0
$538$	1492.12	0.119572
$539$	10817.9	0.864486
$540$	0	0
$541$	12028.0	0.955864	0.477932	−	0.878397i	$-0.341387\pi$
$541$	12028.0	0.955864	0.477932	+	0.878397i	$0.341387\pi$
$542$	−2621.45	−0.207750
$543$	0	0
$544$	13143.8	1.03591
$545$	6560.58	0.515641
$546$	0	0
$547$	861.602	0.0673482	0.0336741	−	0.999433i	$-0.489279\pi$
$547$	861.602	0.0673482	0.0336741	+	0.999433i	$0.489279\pi$
$548$	8802.93	0.686209
$549$	0	0
$550$	1468.30	0.113834
$551$	448.211	0.0346541
$552$	0	0
$553$	−10987.0	−0.844874
$554$	−756.772	−0.0580364
$555$	0	0
$556$	70.7736	0.00539832
$557$	−26268.4	−1.99826	−0.999129	−	0.0417358i	$-0.986711\pi$
$557$	−26268.4	−1.99826	−0.999129	+	0.0417358i	$0.986711\pi$
$558$	0	0
$559$	3310.66	0.250494
$560$	3352.07	0.252948
$561$	0	0
$562$	2542.62	0.190843
$563$	−24246.0	−1.81501	−0.907503	−	0.420045i	$-0.862014\pi$
$563$	−24246.0	−1.81501	−0.907503	+	0.420045i	$0.862014\pi$
$564$	0	0
$565$	−196.803	−0.0146541
$566$	8.64386	0.000641923 0
$567$	0	0
$568$	−3328.54	−0.245885
$569$	16401.5	1.20841	0.604206	−	0.796828i	$-0.293490\pi$
$569$	16401.5	1.20841	0.604206	+	0.796828i	$0.293490\pi$
$570$	0	0
$571$	13490.3	0.988704	0.494352	−	0.869262i	$-0.335405\pi$
$571$	13490.3	0.988704	0.494352	+	0.869262i	$0.335405\pi$
$572$	−41470.6	−3.03142
$573$	0	0
$574$	−1409.54	−0.102496
$575$	4684.07	0.339720
$576$	0	0
$577$	25614.9	1.84811	0.924057	−	0.382254i	$-0.124852\pi$
$577$	25614.9	1.84811	0.924057	+	0.382254i	$0.124852\pi$
$578$	−3193.28	−0.229798
$579$	0	0
$580$	1062.79	0.0760861
$581$	−19751.4	−1.41037
$582$	0	0
$583$	−8975.44	−0.637607
$584$	−66.2609	−0.00469503
$585$	0	0
$586$	−2368.00	−0.166930
$587$	389.674	0.0273996	0.0136998	−	0.999906i	$-0.495639\pi$
$587$	389.674	0.0273996	0.0136998	+	0.999906i	$0.495639\pi$
$588$	0	0
$589$	1133.99	0.0793296
$590$	−226.036	−0.0157725
$591$	0	0
$592$	−6147.60	−0.426799
$593$	−3592.08	−0.248750	−0.124375	−	0.992235i	$-0.539693\pi$
$593$	−3592.08	−0.248750	−0.124375	+	0.992235i	$0.539693\pi$
$594$	0	0
$595$	−6507.19	−0.448351
$596$	−4136.96	−0.284323
$597$	0	0
$598$	12100.8	0.827492
$599$	−6355.87	−0.433546	−0.216773	−	0.976222i	$-0.569553\pi$
$599$	−6355.87	−0.433546	−0.216773	+	0.976222i	$0.569553\pi$
$600$	0	0
$601$	−24779.9	−1.68185	−0.840927	−	0.541149i	$-0.817990\pi$
$601$	−24779.9	−1.68185	−0.840927	+	0.541149i	$0.817990\pi$
$602$	476.422	0.0322550
$603$	0	0
$604$	−4661.04	−0.313998
$605$	19071.2	1.28157
$606$	0	0
$607$	14087.1	0.941971	0.470986	−	0.882141i	$-0.343898\pi$
$607$	14087.1	0.941971	0.470986	+	0.882141i	$0.343898\pi$
$608$	2163.93	0.144341
$609$	0	0
$610$	−1063.04	−0.0705593
$611$	−46406.4	−3.07267
$612$	0	0
$613$	−18561.9	−1.22302	−0.611508	−	0.791238i	$-0.709437\pi$
$613$	−18561.9	−1.22302	−0.611508	+	0.791238i	$0.709437\pi$
$614$	−5624.69	−0.369697
$615$	0	0
$616$	−12481.5	−0.816389
$617$	21599.2	1.40932	0.704661	−	0.709544i	$-0.251099\pi$
$617$	21599.2	1.40932	0.704661	+	0.709544i	$0.251099\pi$
$618$	0	0
$619$	−14290.1	−0.927893	−0.463947	−	0.885863i	$-0.653567\pi$
$619$	−14290.1	−0.927893	−0.463947	+	0.885863i	$0.653567\pi$
$620$	2688.89	0.174175
$621$	0	0
$622$	−3074.45	−0.198190
$623$	20790.4	1.33700
$624$	0	0
$625$	625.000	0.0400000
$626$	−822.338	−0.0525036
$627$	0	0
$628$	4200.01	0.266877
$629$	11934.0	0.756503
$630$	0	0
$631$	21344.9	1.34664	0.673318	−	0.739353i	$-0.264869\pi$
$631$	21344.9	1.34664	0.673318	+	0.739353i	$0.264869\pi$
$632$	−9947.67	−0.626103
$633$	0	0
$634$	−3730.14	−0.233663
$635$	−9217.98	−0.576070
$636$	0	0
$637$	11895.9	0.739926
$638$	−1703.23	−0.105692
$639$	0	0
$640$	6714.93	0.414736
$641$	−24178.6	−1.48985	−0.744927	−	0.667146i	$-0.767516\pi$
$641$	−24178.6	−1.48985	−0.744927	+	0.667146i	$0.767516\pi$
$642$	0	0
$643$	11994.8	0.735658	0.367829	−	0.929893i	$-0.380101\pi$
$643$	11994.8	0.735658	0.367829	+	0.929893i	$0.380101\pi$
$644$	−19038.1	−1.16492
$645$	0	0
$646$	−1188.01	−0.0723553
$647$	−5641.51	−0.342798	−0.171399	−	0.985202i	$-0.554829\pi$
$647$	−5641.51	−0.342798	−0.171399	+	0.985202i	$0.554829\pi$
$648$	0	0
$649$	−3960.39	−0.239536
$650$	1614.63	0.0974321
$651$	0	0
$652$	24026.5	1.44318
$653$	−21090.4	−1.26390	−0.631952	−	0.775008i	$-0.717746\pi$
$653$	−21090.4	−1.26390	−0.631952	+	0.775008i	$0.717746\pi$
$654$	0	0
$655$	11400.7	0.680092
$656$	6005.12	0.357409
$657$	0	0
$658$	−6678.13	−0.395654
$659$	13160.3	0.777926	0.388963	−	0.921253i	$-0.372833\pi$
$659$	13160.3	0.777926	0.388963	+	0.921253i	$0.372833\pi$
$660$	0	0
$661$	32282.2	1.89959	0.949796	−	0.312869i	$-0.101290\pi$
$661$	32282.2	1.89959	0.949796	+	0.312869i	$0.101290\pi$
$662$	3557.13	0.208840
$663$	0	0
$664$	−17883.0	−1.04517
$665$	−1071.31	−0.0624718
$666$	0	0
$667$	−5433.52	−0.315423
$668$	−28767.2	−1.66622
$669$	0	0
$670$	2719.52	0.156813
$671$	−18625.5	−1.07158
$672$	0	0
$673$	−2556.97	−0.146455	−0.0732273	−	0.997315i	$-0.523330\pi$
$673$	−2556.97	−0.146455	−0.0732273	+	0.997315i	$0.523330\pi$
$674$	−6642.33	−0.379604
$675$	0	0
$676$	−29500.2	−1.67844
$677$	−24565.3	−1.39457	−0.697284	−	0.716795i	$-0.745608\pi$
$677$	−24565.3	−1.39457	−0.697284	+	0.716795i	$0.745608\pi$
$678$	0	0
$679$	10592.4	0.598670
$680$	−5891.63	−0.332255
$681$	0	0
$682$	−4309.23	−0.241949
$683$	2925.25	0.163882	0.0819411	−	0.996637i	$-0.473888\pi$
$683$	2925.25	0.163882	0.0819411	+	0.996637i	$0.473888\pi$
$684$	0	0
$685$	−6005.07	−0.334952
$686$	5605.29	0.311969
$687$	0	0
$688$	−2029.72	−0.112475
$689$	−9869.88	−0.545737
$690$	0	0
$691$	−3667.88	−0.201929	−0.100964	−	0.994890i	$-0.532193\pi$
$691$	−3667.88	−0.201929	−0.100964	+	0.994890i	$0.532193\pi$
$692$	−17659.5	−0.970104
$693$	0	0
$694$	−8793.06	−0.480951
$695$	−48.2794	−0.00263502
$696$	0	0
$697$	−11657.4	−0.633510
$698$	6300.74	0.341672
$699$	0	0
$700$	−2540.28	−0.137162
$701$	−13843.7	−0.745891	−0.372946	−	0.927853i	$-0.621652\pi$
$701$	−13843.7	−0.745891	−0.372946	+	0.927853i	$0.621652\pi$
$702$	0	0
$703$	1964.76	0.105409
$704$	19527.6	1.04542
$705$	0	0
$706$	−5421.30	−0.288999
$707$	21424.9	1.13970
$708$	0	0
$709$	23902.3	1.26610	0.633052	−	0.774109i	$-0.281802\pi$
$709$	23902.3	1.26610	0.633052	+	0.774109i	$0.281802\pi$
$710$	1085.66	0.0573860
$711$	0	0
$712$	18823.7	0.990797
$713$	−13747.0	−0.722060
$714$	0	0
$715$	28289.9	1.47969
$716$	−1983.81	−0.103545
$717$	0	0
$718$	1981.47	0.102991
$719$	−16629.2	−0.862537	−0.431269	−	0.902224i	$-0.641934\pi$
$719$	−16629.2	−0.862537	−0.431269	+	0.902224i	$0.641934\pi$
$720$	0	0
$721$	2832.75	0.146321
$722$	5420.51	0.279405
$723$	0	0
$724$	1293.71	0.0664094
$725$	−725.000	−0.0371391
$726$	0	0
$727$	−8889.10	−0.453478	−0.226739	−	0.973956i	$-0.572806\pi$
$727$	−8889.10	−0.453478	−0.226739	+	0.973956i	$0.572806\pi$
$728$	−13725.4	−0.698759
$729$	0	0
$730$	21.6121	0.00109575
$731$	3940.20	0.199362
$732$	0	0
$733$	−17622.0	−0.887973	−0.443987	−	0.896033i	$-0.646436\pi$
$733$	−17622.0	−0.887973	−0.443987	+	0.896033i	$0.646436\pi$
$734$	−4302.37	−0.216353
$735$	0	0
$736$	−26232.7	−1.31379
$737$	47648.8	2.38150
$738$	0	0
$739$	−25697.3	−1.27915	−0.639573	−	0.768730i	$-0.720889\pi$
$739$	−25697.3	−1.27915	−0.639573	+	0.768730i	$0.720889\pi$
$740$	4658.80	0.231434
$741$	0	0
$742$	−1420.33	−0.0702721
$743$	35603.8	1.75798	0.878989	−	0.476843i	$-0.158219\pi$
$743$	35603.8	1.75798	0.878989	+	0.476843i	$0.158219\pi$
$744$	0	0
$745$	2822.10	0.138784
$746$	4912.90	0.241118
$747$	0	0
$748$	−49356.4	−2.41263
$749$	4753.51	0.231895
$750$	0	0
$751$	−890.657	−0.0432763	−0.0216382	−	0.999766i	$-0.506888\pi$
$751$	−890.657	−0.0432763	−0.0216382	+	0.999766i	$0.506888\pi$
$752$	28451.2	1.37966
$753$	0	0
$754$	−1872.97	−0.0904634
$755$	3179.61	0.153269
$756$	0	0
$757$	−27316.1	−1.31152	−0.655761	−	0.754969i	$-0.727652\pi$
$757$	−27316.1	−1.31152	−0.655761	+	0.754969i	$0.727652\pi$
$758$	−7736.86	−0.370733
$759$	0	0
$760$	−969.969	−0.0462954
$761$	19155.6	0.912472	0.456236	−	0.889859i	$-0.349197\pi$
$761$	19155.6	0.912472	0.456236	+	0.889859i	$0.349197\pi$
$762$	0	0
$763$	18190.1	0.863073
$764$	18472.7	0.874762
$765$	0	0
$766$	−701.498	−0.0330890
$767$	−4355.06	−0.205022
$768$	0	0
$769$	−6601.93	−0.309586	−0.154793	−	0.987947i	$-0.549471\pi$
$769$	−6601.93	−0.309586	−0.154793	+	0.987947i	$0.549471\pi$
$770$	4071.06	0.190534
$771$	0	0
$772$	−10949.3	−0.510460
$773$	−10501.3	−0.488622	−0.244311	−	0.969697i	$-0.578562\pi$
$773$	−10501.3	−0.488622	−0.244311	+	0.969697i	$0.578562\pi$
$774$	0	0
$775$	−1834.27	−0.0850181
$776$	9590.34	0.443651
$777$	0	0
$778$	−141.569	−0.00652376
$779$	−1919.22	−0.0882711
$780$	0	0
$781$	19021.8	0.871517
$782$	14401.9	0.658580
$783$	0	0
$784$	−7293.22	−0.332235
$785$	−2865.11	−0.130268
$786$	0	0
$787$	27169.2	1.23059	0.615297	−	0.788296i	$-0.289036\pi$
$787$	27169.2	1.23059	0.615297	+	0.788296i	$0.289036\pi$
$788$	−23125.2	−1.04543
$789$	0	0
$790$	3244.60	0.146124
$791$	−545.661	−0.0245278
$792$	0	0
$793$	−20481.6	−0.917180
$794$	7574.30	0.338541
$795$	0	0
$796$	−40023.3	−1.78215
$797$	−3072.45	−0.136552	−0.0682759	−	0.997666i	$-0.521750\pi$
$797$	−3072.45	−0.136552	−0.0682759	+	0.997666i	$0.521750\pi$
$798$	0	0
$799$	−55230.8	−2.44546
$800$	−3500.25	−0.154691
$801$	0	0
$802$	8111.48	0.357140
$803$	378.666	0.0166411
$804$	0	0
$805$	12987.2	0.568619
$806$	−4738.67	−0.207087
$807$	0	0
$808$	19398.1	0.844583
$809$	149.629	0.00650269	0.00325134	−	0.999995i	$-0.498965\pi$
$809$	149.629	0.00650269	0.00325134	+	0.999995i	$0.498965\pi$
$810$	0	0
$811$	−10550.4	−0.456813	−0.228406	−	0.973566i	$-0.573351\pi$
$811$	−10550.4	−0.456813	−0.228406	+	0.973566i	$0.573351\pi$
$812$	2946.72	0.127352
$813$	0	0
$814$	−7466.22	−0.321488
$815$	−16390.1	−0.704442
$816$	0	0
$817$	648.695	0.0277784
$818$	7743.86	0.331000
$819$	0	0
$820$	−4550.82	−0.193807
$821$	−29361.9	−1.24816	−0.624078	−	0.781362i	$-0.714525\pi$
$821$	−29361.9	−1.24816	−0.624078	+	0.781362i	$0.714525\pi$
$822$	0	0
$823$	23398.7	0.991042	0.495521	−	0.868596i	$-0.334977\pi$
$823$	23398.7	0.991042	0.495521	+	0.868596i	$0.334977\pi$
$824$	2564.78	0.108433
$825$	0	0
$826$	−626.716	−0.0263998
$827$	−15323.5	−0.644317	−0.322158	−	0.946686i	$-0.604408\pi$
$827$	−15323.5	−0.644317	−0.322158	+	0.946686i	$0.604408\pi$
$828$	0	0
$829$	10782.3	0.451730	0.225865	−	0.974159i	$-0.427479\pi$
$829$	10782.3	0.451730	0.225865	+	0.974159i	$0.427479\pi$
$830$	5832.83	0.243928
$831$	0	0
$832$	21473.6	0.894786
$833$	14157.9	0.588888
$834$	0	0
$835$	19624.0	0.813315
$836$	−8125.79	−0.336168
$837$	0	0
$838$	4031.78	0.166200
$839$	−41081.5	−1.69045	−0.845227	−	0.534408i	$-0.820535\pi$
$839$	−41081.5	−1.69045	−0.845227	+	0.534408i	$0.820535\pi$
$840$	0	0
$841$	841.000	0.0344828
$842$	10726.9	0.439044
$843$	0	0
$844$	−19033.1	−0.776240
$845$	20124.1	0.819278
$846$	0	0
$847$	52877.3	2.14508
$848$	6051.10	0.245042
$849$	0	0
$850$	1921.65	0.0775437
$851$	−23818.2	−0.959432
$852$	0	0
$853$	−23051.2	−0.925274	−0.462637	−	0.886548i	$-0.653097\pi$
$853$	−23051.2	−0.925274	−0.462637	+	0.886548i	$0.653097\pi$
$854$	−2947.41	−0.118101
$855$	0	0
$856$	4303.84	0.171848
$857$	−1000.28	−0.0398704	−0.0199352	−	0.999801i	$-0.506346\pi$
$857$	−1000.28	−0.0398704	−0.0199352	+	0.999801i	$0.506346\pi$
$858$	0	0
$859$	−16370.6	−0.650242	−0.325121	−	0.945672i	$-0.605405\pi$
$859$	−16370.6	−0.650242	−0.325121	+	0.945672i	$0.605405\pi$
$860$	1538.17	0.0609899
$861$	0	0
$862$	−10240.3	−0.404626
$863$	−16312.7	−0.643441	−0.321721	−	0.946835i	$-0.604261\pi$
$863$	−16312.7	−0.643441	−0.321721	+	0.946835i	$0.604261\pi$
$864$	0	0
$865$	12046.7	0.473526
$866$	−3041.34	−0.119340
$867$	0	0
$868$	7455.30	0.291532
$869$	56848.6	2.21917
$870$	0	0
$871$	52397.2	2.03836
$872$	16469.3	0.639589
$873$	0	0
$874$	2371.05	0.0917644
$875$	1732.89	0.0669515
$876$	0	0
$877$	−15674.8	−0.603534	−0.301767	−	0.953382i	$-0.597576\pi$
$877$	−15674.8	−0.603534	−0.301767	+	0.953382i	$0.597576\pi$
$878$	−7613.86	−0.292660
$879$	0	0
$880$	−17344.2	−0.664399
$881$	14789.4	0.565569	0.282784	−	0.959184i	$-0.408742\pi$
$881$	14789.4	0.565569	0.282784	+	0.959184i	$0.408742\pi$
$882$	0	0
$883$	5049.14	0.192432	0.0962158	−	0.995360i	$-0.469326\pi$
$883$	5049.14	0.192432	0.0962158	+	0.995360i	$0.469326\pi$
$884$	−54274.9	−2.06500
$885$	0	0
$886$	−8451.03	−0.320449
$887$	37670.6	1.42599	0.712997	−	0.701167i	$-0.247337\pi$
$887$	37670.6	1.42599	0.712997	+	0.701167i	$0.247337\pi$
$888$	0	0
$889$	−25558.1	−0.964218
$890$	−6139.66	−0.231238
$891$	0	0
$892$	11569.4	0.434273
$893$	−9092.92	−0.340743
$894$	0	0
$895$	1353.29	0.0505423
$896$	18618.0	0.694179
$897$	0	0
$898$	6010.17	0.223343
$899$	2127.76	0.0789374
$900$	0	0
$901$	−11746.7	−0.434338
$902$	7293.17	0.269220
$903$	0	0
$904$	−494.043	−0.0181766
$905$	−882.527	−0.0324157
$906$	0	0
$907$	−29554.5	−1.08196	−0.540981	−	0.841035i	$-0.681947\pi$
$907$	−29554.5	−1.08196	−0.540981	+	0.841035i	$0.681947\pi$
$908$	−17800.9	−0.650597
$909$	0	0
$910$	4476.76	0.163081
$911$	−898.948	−0.0326932	−0.0163466	−	0.999866i	$-0.505204\pi$
$911$	−898.948	−0.0326932	−0.0163466	+	0.999866i	$0.505204\pi$
$912$	0	0
$913$	102197.	3.70452
$914$	−1165.43	−0.0421760
$915$	0	0
$916$	33873.7	1.22186
$917$	31609.8	1.13833
$918$	0	0
$919$	−36963.4	−1.32678	−0.663390	−	0.748274i	$-0.730883\pi$
$919$	−36963.4	−1.32678	−0.663390	+	0.748274i	$0.730883\pi$
$920$	11758.6	0.421381
$921$	0	0
$922$	−6439.87	−0.230028
$923$	20917.4	0.745944
$924$	0	0
$925$	−3178.08	−0.112967
$926$	−4991.43	−0.177137
$927$	0	0
$928$	4060.29	0.143627
$929$	35541.5	1.25520	0.627600	−	0.778536i	$-0.284037\pi$
$929$	35541.5	1.25520	0.627600	+	0.778536i	$0.284037\pi$
$930$	0	0
$931$	2330.90	0.0820537
$932$	24462.5	0.859758
$933$	0	0
$934$	−6169.96	−0.216154
$935$	33669.3	1.17765
$936$	0	0
$937$	17380.4	0.605968	0.302984	−	0.952996i	$-0.402017\pi$
$937$	17380.4	0.605968	0.302984	+	0.952996i	$0.402017\pi$
$938$	7540.24	0.262471
$939$	0	0
$940$	−21561.0	−0.748129
$941$	−12456.3	−0.431525	−0.215763	−	0.976446i	$-0.569224\pi$
$941$	−12456.3	−0.431525	−0.215763	+	0.976446i	$0.569224\pi$
$942$	0	0
$943$	23266.1	0.803446
$944$	2670.03	0.0920572
$945$	0	0
$946$	−2465.09	−0.0847219
$947$	36639.9	1.25727	0.628636	−	0.777700i	$-0.283614\pi$
$947$	36639.9	1.25727	0.628636	+	0.777700i	$0.283614\pi$
$948$	0	0
$949$	416.402	0.0142434
$950$	316.372	0.0108047
$951$	0	0
$952$	−16335.3	−0.556125
$953$	−4554.66	−0.154816	−0.0774082	−	0.996999i	$-0.524664\pi$
$953$	−4554.66	−0.154816	−0.0774082	+	0.996999i	$0.524664\pi$
$954$	0	0
$955$	−12601.5	−0.426988
$956$	−5550.07	−0.187764
$957$	0	0
$958$	13896.1	0.468647
$959$	−16649.8	−0.560638
$960$	0	0
$961$	−24407.7	−0.819298
$962$	−8210.26	−0.275166
$963$	0	0
$964$	3524.47	0.117755
$965$	7469.27	0.249165
$966$	0	0
$967$	13804.0	0.459056	0.229528	−	0.973302i	$-0.426282\pi$
$967$	13804.0	0.459056	0.229528	+	0.973302i	$0.426282\pi$
$968$	47875.2	1.58964
$969$	0	0
$970$	−3128.05	−0.103542
$971$	24931.6	0.823988	0.411994	−	0.911187i	$-0.364832\pi$
$971$	24931.6	0.823988	0.411994	+	0.911187i	$0.364832\pi$
$972$	0	0
$973$	−133.861	−0.00441047
$974$	−14192.2	−0.466887
$975$	0	0
$976$	12557.0	0.411824
$977$	24423.4	0.799770	0.399885	−	0.916565i	$-0.369050\pi$
$977$	24423.4	0.799770	0.399885	+	0.916565i	$0.369050\pi$
$978$	0	0
$979$	−107573.	−3.51180
$980$	5526.98	0.180156
$981$	0	0
$982$	2021.16	0.0656801
$983$	−52340.5	−1.69827	−0.849136	−	0.528174i	$-0.822877\pi$
$983$	−52340.5	−1.69827	−0.849136	+	0.528174i	$0.822877\pi$
$984$	0	0
$985$	15775.2	0.510296
$986$	−2229.12	−0.0719976
$987$	0	0
$988$	−8935.56	−0.287731
$989$	−7863.93	−0.252840
$990$	0	0
$991$	6746.36	0.216251	0.108126	−	0.994137i	$-0.465515\pi$
$991$	6746.36	0.216251	0.108126	+	0.994137i	$0.465515\pi$
$992$	10272.7	0.328788
$993$	0	0
$994$	3010.13	0.0960519
$995$	27302.6	0.869901
$996$	0	0
$997$	15842.8	0.503257	0.251629	−	0.967824i	$-0.419034\pi$
$997$	15842.8	0.503257	0.251629	+	0.967824i	$0.419034\pi$
$998$	−7439.01	−0.235950
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1305.4.a.p.1.3		8
3.2	odd	2		435.4.a.k.1.6	✓	8
15.14	odd	2		2175.4.a.o.1.3		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
435.4.a.k.1.6	✓	8	3.2	odd	2
1305.4.a.p.1.3		8	1.1	even	1	trivial
2175.4.a.o.1.3		8	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)

\(f(q)\)	\(=\)	\(q-0.818792 q^{2} -7.32958 q^{4} +5.00000 q^{5} +13.8632 q^{7} +12.5517 q^{8} -4.09396 q^{10} -71.7303 q^{11} -78.8785 q^{13} -11.3510 q^{14} +48.3594 q^{16} -93.8775 q^{17} -15.4555 q^{19} -36.6479 q^{20} +58.7322 q^{22} +187.363 q^{23} +25.0000 q^{25} +64.5851 q^{26} -101.611 q^{28} -29.0000 q^{29} -73.3710 q^{31} -140.010 q^{32} +76.8662 q^{34} +69.3158 q^{35} -127.123 q^{37} +12.6549 q^{38} +62.7587 q^{40} +124.177 q^{41} -41.9717 q^{43} +525.753 q^{44} -153.411 q^{46} +588.328 q^{47} -150.813 q^{49} -20.4698 q^{50} +578.146 q^{52} +125.128 q^{53} -358.651 q^{55} +174.007 q^{56} +23.7450 q^{58} +55.2122 q^{59} +259.660 q^{61} +60.0755 q^{62} -272.236 q^{64} -394.392 q^{65} -664.277 q^{67} +688.083 q^{68} -56.7552 q^{70} -265.186 q^{71} -5.27903 q^{73} +104.087 q^{74} +113.283 q^{76} -994.408 q^{77} -792.533 q^{79} +241.797 q^{80} -101.675 q^{82} -1424.74 q^{83} -469.388 q^{85} +34.3661 q^{86} -900.339 q^{88} +1499.69 q^{89} -1093.50 q^{91} -1373.29 q^{92} -481.718 q^{94} -77.2777 q^{95} +764.065 q^{97} +123.484 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 4 q^{2} + 38 q^{4} + 40 q^{5} - q^{7} + 9 q^{8} + 20 q^{10} - 11 q^{11} - 97 q^{13} + 69 q^{14} + 242 q^{16} + 119 q^{17} + 220 q^{19} + 190 q^{20} - 129 q^{22} + 262 q^{23} + 200 q^{25} - 43 q^{26}+ \cdots + 7677 q^{98}+O(q^{100}) \) 8 * q + 4 * q^2 + 38 * q^4 + 40 * q^5 - q^7 + 9 * q^8 + 20 * q^10 - 11 * q^11 - 97 * q^13 + 69 * q^14 + 242 * q^16 + 119 * q^17 + 220 * q^19 + 190 * q^20 - 129 * q^22 + 262 * q^23 + 200 * q^25 - 43 * q^26 - 13 * q^28 - 232 * q^29 + 108 * q^31 + 566 * q^32 - 527 * q^34 - 5 * q^35 + 118 * q^37 + 1088 * q^38 + 45 * q^40 + 118 * q^41 - 384 * q^43 + 1605 * q^44 + 154 * q^46 + 1309 * q^47 + 427 * q^49 + 100 * q^50 - 747 * q^52 + 1912 * q^53 - 55 * q^55 + 2022 * q^56 - 116 * q^58 + 1178 * q^59 + 572 * q^61 + 2572 * q^62 - 1771 * q^64 - 485 * q^65 + 1147 * q^67 + 3963 * q^68 + 345 * q^70 - 84 * q^71 - 222 * q^73 + 1758 * q^74 + 1390 * q^76 + 1753 * q^77 + 390 * q^79 + 1210 * q^80 - 2994 * q^82 + 1570 * q^83 + 595 * q^85 - 1668 * q^86 - 1790 * q^88 + 119 * q^89 - 3529 * q^91 + 2934 * q^92 - 2277 * q^94 + 1100 * q^95 - 2100 * q^97 + 7677 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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