LMFDB - Embedded newform 1305.4.a.g.1.4

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:	$5$
Coefficient field:	$\mathbb{Q}[x]/(x^{5} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 2x^{4} - 23x^{3} + 38x^{2} + 90x - 116 $ x^5 - 2x^4 - 23x^3 + 38x^2 + 90x - 116
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 435)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-2.12043$ 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)

$f(q)$	$=$	\(q+2.12043 q^{2} -3.50376 q^{4} +5.00000 q^{5} -1.40136 q^{7} -24.3930 q^{8} +10.6022 q^{10} +24.5188 q^{11} -32.6137 q^{13} -2.97148 q^{14} -23.6935 q^{16} +73.4147 q^{17} -63.9410 q^{19} -17.5188 q^{20} +51.9905 q^{22} +118.222 q^{23} +25.0000 q^{25} -69.1552 q^{26} +4.91002 q^{28} +29.0000 q^{29} -267.895 q^{31} +144.903 q^{32} +155.671 q^{34} -7.00678 q^{35} +288.463 q^{37} -135.583 q^{38} -121.965 q^{40} -492.803 q^{41} -31.1238 q^{43} -85.9082 q^{44} +250.681 q^{46} -233.499 q^{47} -341.036 q^{49} +53.0108 q^{50} +114.271 q^{52} -6.75576 q^{53} +122.594 q^{55} +34.1832 q^{56} +61.4926 q^{58} -0.170156 q^{59} -253.626 q^{61} -568.054 q^{62} +496.806 q^{64} -163.069 q^{65} -188.557 q^{67} -257.228 q^{68} -14.8574 q^{70} -671.359 q^{71} +290.796 q^{73} +611.666 q^{74} +224.034 q^{76} -34.3596 q^{77} -350.592 q^{79} -118.468 q^{80} -1044.96 q^{82} -1019.19 q^{83} +367.073 q^{85} -65.9959 q^{86} -598.087 q^{88} +1008.83 q^{89} +45.7034 q^{91} -414.221 q^{92} -495.119 q^{94} -319.705 q^{95} -346.423 q^{97} -723.144 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - 2 q^{2} + 10 q^{4} + 25 q^{5} - 29 q^{7} - 10 q^{10} - 15 q^{11} - 31 q^{13} + 114 q^{14} - 102 q^{16} - 119 q^{17} + 46 q^{19} + 50 q^{20} + 66 q^{22} - 170 q^{23} + 125 q^{25} - 138 q^{26} + 32 q^{28}+ \cdots - 1780 q^{98}+O(q^{100}) $ 5 * q - 2 * q^2 + 10 * q^4 + 25 * q^5 - 29 * q^7 - 10 * q^10 - 15 * q^11 - 31 * q^13 + 114 * q^14 - 102 * q^16 - 119 * q^17 + 46 * q^19 + 50 * q^20 + 66 * q^22 - 170 * q^23 + 125 * q^25 - 138 * q^26 + 32 * q^28 + 145 * q^29 - 240 * q^31 - 152 * q^32 + 290 * q^34 - 145 * q^35 + 570 * q^37 - 1380 * q^38 + 180 * q^41 - 110 * q^43 - 1420 * q^44 + 696 * q^46 - 697 * q^47 - 362 * q^49 - 50 * q^50 + 960 * q^52 - 960 * q^53 - 75 * q^55 + 224 * q^56 - 58 * q^58 - 154 * q^59 + 360 * q^61 - 1176 * q^62 - 486 * q^64 - 155 * q^65 + 517 * q^67 - 1812 * q^68 + 570 * q^70 + 172 * q^71 + 1882 * q^73 - 1476 * q^74 + 884 * q^76 - 363 * q^77 - 478 * q^79 - 510 * q^80 - 1412 * q^82 - 2356 * q^83 - 595 * q^85 - 272 * q^86 + 1400 * q^88 - 449 * q^89 - 645 * q^91 - 1072 * q^92 + 466 * q^94 + 230 * q^95 + 88 * q^97 - 1780 * 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 $

$2$	2.12043	0.749686	0.374843	−	0.927088i	$-0.377697\pi$
$2$	2.12043	0.749686	0.374843	+	0.927088i	$0.377697\pi$
$3$	0	0
$3$	0	0
$4$	−3.50376	−0.437971
$5$	5.00000	0.447214
$5$	5.00000	0.447214
$6$	0	0
$7$	−1.40136	−0.0756661	−0.0378330	−	0.999284i	$-0.512046\pi$
$7$	−1.40136	−0.0756661	−0.0378330	+	0.999284i	$0.512046\pi$
$8$	−24.3930	−1.07803
$9$	0	0
$10$	10.6022	0.335270
$11$	24.5188	0.672064	0.336032	−	0.941851i	$-0.390915\pi$
$11$	24.5188	0.672064	0.336032	+	0.941851i	$0.390915\pi$
$12$	0	0
$13$	−32.6137	−0.695801	−0.347901	−	0.937531i	$-0.613105\pi$
$13$	−32.6137	−0.695801	−0.347901	+	0.937531i	$0.613105\pi$
$14$	−2.97148	−0.0567258
$15$	0	0
$16$	−23.6935	−0.370211
$17$	73.4147	1.04739	0.523696	−	0.851905i	$-0.324553\pi$
$17$	73.4147	1.04739	0.523696	+	0.851905i	$0.324553\pi$
$18$	0	0
$19$	−63.9410	−0.772056	−0.386028	−	0.922487i	$-0.626153\pi$
$19$	−63.9410	−0.772056	−0.386028	+	0.922487i	$0.626153\pi$
$20$	−17.5188	−0.195866
$21$	0	0
$22$	51.9905	0.503837
$23$	118.222	1.07178	0.535890	−	0.844288i	$-0.319976\pi$
$23$	118.222	1.07178	0.535890	+	0.844288i	$0.319976\pi$
$24$	0	0
$25$	25.0000	0.200000
$26$	−69.1552	−0.521632
$27$	0	0
$28$	4.91002	0.0331395
$29$	29.0000	0.185695
$29$	29.0000	0.185695
$30$	0	0
$31$	−267.895	−1.55211	−0.776055	−	0.630665i	$-0.782782\pi$
$31$	−267.895	−1.55211	−0.776055	+	0.630665i	$0.782782\pi$
$32$	144.903	0.800484
$33$	0	0
$34$	155.671	0.785216
$35$	−7.00678	−0.0338389
$36$	0	0
$37$	288.463	1.28170	0.640851	−	0.767665i	$-0.278582\pi$
$37$	288.463	1.28170	0.640851	+	0.767665i	$0.278582\pi$
$38$	−135.583	−0.578800
$39$	0	0
$40$	−121.965	−0.482108
$41$	−492.803	−1.87714	−0.938572	−	0.345083i	$-0.887851\pi$
$41$	−492.803	−1.87714	−0.938572	+	0.345083i	$0.887851\pi$
$42$	0	0
$43$	−31.1238	−0.110380	−0.0551899	−	0.998476i	$-0.517576\pi$
$43$	−31.1238	−0.110380	−0.0551899	+	0.998476i	$0.517576\pi$
$44$	−85.9082	−0.294344
$45$	0	0
$46$	250.681	0.803498
$47$	−233.499	−0.724667	−0.362333	−	0.932049i	$-0.618020\pi$
$47$	−233.499	−0.724667	−0.362333	+	0.932049i	$0.618020\pi$
$48$	0	0
$49$	−341.036	−0.994275
$50$	53.0108	0.149937
$51$	0	0
$52$	114.271	0.304740
$53$	−6.75576	−0.0175090	−0.00875448	−	0.999962i	$-0.502787\pi$
$53$	−6.75576	−0.0175090	−0.00875448	+	0.999962i	$0.502787\pi$
$54$	0	0
$55$	122.594	0.300556
$56$	34.1832	0.0815700
$57$	0	0
$58$	61.4926	0.139213
$59$	−0.170156	−0.000375465 0	−0.000187732	−	1.00000i	$-0.500060\pi$
$59$	−0.170156	−0.000375465 0	−0.000187732		1.00000i	$0.500060\pi$
$60$	0	0
$61$	−253.626	−0.532352	−0.266176	−	0.963925i	$-0.585760\pi$
$61$	−253.626	−0.532352	−0.266176	+	0.963925i	$0.585760\pi$
$62$	−568.054	−1.16360
$63$	0	0
$64$	496.806	0.970323
$65$	−163.069	−0.311172
$66$	0	0
$67$	−188.557	−0.343820	−0.171910	−	0.985113i	$-0.554994\pi$
$67$	−188.557	−0.343820	−0.171910	+	0.985113i	$0.554994\pi$
$68$	−257.228	−0.458727
$69$	0	0
$70$	−14.8574	−0.0253686
$71$	−671.359	−1.12219	−0.561096	−	0.827751i	$-0.689620\pi$
$71$	−671.359	−1.12219	−0.561096	+	0.827751i	$0.689620\pi$
$72$	0	0
$73$	290.796	0.466235	0.233117	−	0.972449i	$-0.425107\pi$
$73$	290.796	0.466235	0.233117	+	0.972449i	$0.425107\pi$
$74$	611.666	0.960874
$75$	0	0
$76$	224.034	0.338138
$77$	−34.3596	−0.0508524
$78$	0	0
$79$	−350.592	−0.499300	−0.249650	−	0.968336i	$-0.580316\pi$
$79$	−350.592	−0.499300	−0.249650	+	0.968336i	$0.580316\pi$
$80$	−118.468	−0.165564
$81$	0	0
$82$	−1044.96	−1.40727
$83$	−1019.19	−1.34784	−0.673920	−	0.738804i	$-0.735391\pi$
$83$	−1019.19	−1.34784	−0.673920	+	0.738804i	$0.735391\pi$
$84$	0	0
$85$	367.073	0.468408
$86$	−65.9959	−0.0827503
$87$	0	0
$88$	−598.087	−0.724503
$89$	1008.83	1.20152	0.600761	−	0.799429i	$-0.294864\pi$
$89$	1008.83	1.20152	0.600761	+	0.799429i	$0.294864\pi$
$90$	0	0
$91$	45.7034	0.0526485
$92$	−414.221	−0.469408
$93$	0	0
$94$	−495.119	−0.543273
$95$	−319.705	−0.345274
$96$	0	0
$97$	−346.423	−0.362617	−0.181309	−	0.983426i	$-0.558033\pi$
$97$	−346.423	−0.362617	−0.181309	+	0.983426i	$0.558033\pi$
$98$	−723.144	−0.745394
$99$	0	0
$100$	−87.5941	−0.0875941
$101$	−1015.12	−1.00008	−0.500039	−	0.866003i	$-0.666681\pi$
$101$	−1015.12	−1.00008	−0.500039	+	0.866003i	$0.666681\pi$
$102$	0	0
$103$	1741.22	1.66571	0.832853	−	0.553494i	$-0.186706\pi$
$103$	1741.22	1.66571	0.832853	+	0.553494i	$0.186706\pi$
$104$	795.545	0.750092
$105$	0	0
$106$	−14.3251	−0.0131262
$107$	−1773.14	−1.60201	−0.801007	−	0.598655i	$-0.795702\pi$
$107$	−1773.14	−1.60201	−0.801007	+	0.598655i	$0.795702\pi$
$108$	0	0
$109$	−900.555	−0.791354	−0.395677	−	0.918390i	$-0.629490\pi$
$109$	−900.555	−0.791354	−0.395677	+	0.918390i	$0.629490\pi$
$110$	259.953	0.225323
$111$	0	0
$112$	33.2030	0.0280124
$113$	−959.629	−0.798887	−0.399444	−	0.916758i	$-0.630797\pi$
$113$	−959.629	−0.798887	−0.399444	+	0.916758i	$0.630797\pi$
$114$	0	0
$115$	591.108	0.479314
$116$	−101.609	−0.0813291
$117$	0	0
$118$	−0.360804	−0.000281481 0
$119$	−102.880	−0.0792521
$120$	0	0
$121$	−729.827	−0.548330
$122$	−537.796	−0.399097
$123$	0	0
$124$	938.642	0.679779
$125$	125.000	0.0894427
$126$	0	0
$127$	90.2767	0.0630769	0.0315384	−	0.999503i	$-0.489959\pi$
$127$	90.2767	0.0630769	0.0315384	+	0.999503i	$0.489959\pi$
$128$	−105.782	−0.0730462
$129$	0	0
$130$	−345.776	−0.233281
$131$	−45.1795	−0.0301324	−0.0150662	−	0.999886i	$-0.504796\pi$
$131$	−45.1795	−0.0301324	−0.0150662	+	0.999886i	$0.504796\pi$
$132$	0	0
$133$	89.6041	0.0584185
$134$	−399.823	−0.257757
$135$	0	0
$136$	−1790.80	−1.12912
$137$	−1963.31	−1.22436	−0.612178	−	0.790720i	$-0.709706\pi$
$137$	−1963.31	−1.22436	−0.612178	+	0.790720i	$0.709706\pi$
$138$	0	0
$139$	210.722	0.128584	0.0642920	−	0.997931i	$-0.479521\pi$
$139$	210.722	0.128584	0.0642920	+	0.997931i	$0.479521\pi$
$140$	24.5501	0.0148204
$141$	0	0
$142$	−1423.57	−0.841292
$143$	−799.649	−0.467623
$144$	0	0
$145$	145.000	0.0830455
$146$	616.614	0.349530
$147$	0	0
$148$	−1010.70	−0.561347
$149$	1384.22	0.761074	0.380537	−	0.924766i	$-0.375739\pi$
$149$	1384.22	0.761074	0.380537	+	0.924766i	$0.375739\pi$
$150$	0	0
$151$	−1229.18	−0.662446	−0.331223	−	0.943553i	$-0.607461\pi$
$151$	−1229.18	−0.662446	−0.331223	+	0.943553i	$0.607461\pi$
$152$	1559.71	0.832298
$153$	0	0
$154$	−72.8572	−0.0381234
$155$	−1339.48	−0.694125
$156$	0	0
$157$	1084.39	0.551236	0.275618	−	0.961267i	$-0.411118\pi$
$157$	1084.39	0.551236	0.275618	+	0.961267i	$0.411118\pi$
$158$	−743.408	−0.374319
$159$	0	0
$160$	724.516	0.357987
$161$	−165.671	−0.0810973
$162$	0	0
$163$	−3602.58	−1.73114	−0.865569	−	0.500790i	$-0.833043\pi$
$163$	−3602.58	−1.73114	−0.865569	+	0.500790i	$0.833043\pi$
$164$	1726.67	0.822134
$165$	0	0
$166$	−2161.13	−1.01046
$167$	2664.11	1.23446	0.617231	−	0.786782i	$-0.288255\pi$
$167$	2664.11	1.23446	0.617231	+	0.786782i	$0.288255\pi$
$168$	0	0
$169$	−1133.35	−0.515861
$170$	778.355	0.351159
$171$	0	0
$172$	109.050	0.0483431
$173$	−2832.16	−1.24465	−0.622326	−	0.782758i	$-0.713812\pi$
$173$	−2832.16	−1.24465	−0.622326	+	0.782758i	$0.713812\pi$
$174$	0	0
$175$	−35.0339	−0.0151332
$176$	−580.937	−0.248806
$177$	0	0
$178$	2139.15	0.900765
$179$	2060.93	0.860565	0.430283	−	0.902694i	$-0.358414\pi$
$179$	2060.93	0.860565	0.430283	+	0.902694i	$0.358414\pi$
$180$	0	0
$181$	4334.37	1.77995	0.889976	−	0.456007i	$-0.150721\pi$
$181$	4334.37	1.77995	0.889976	+	0.456007i	$0.150721\pi$
$182$	96.9110	0.0394699
$183$	0	0
$184$	−2883.78	−1.15541
$185$	1442.31	0.573194
$186$	0	0
$187$	1800.04	0.703915
$188$	818.125	0.317383
$189$	0	0
$190$	−677.913	−0.258847
$191$	−1785.57	−0.676438	−0.338219	−	0.941067i	$-0.609824\pi$
$191$	−1785.57	−0.676438	−0.338219	+	0.941067i	$0.609824\pi$
$192$	0	0
$193$	1099.28	0.409990	0.204995	−	0.978763i	$-0.434282\pi$
$193$	1099.28	0.409990	0.204995	+	0.978763i	$0.434282\pi$
$194$	−734.566	−0.271849
$195$	0	0
$196$	1194.91	0.435463
$197$	−3189.78	−1.15362	−0.576808	−	0.816880i	$-0.695702\pi$
$197$	−3189.78	−1.15362	−0.576808	+	0.816880i	$0.695702\pi$
$198$	0	0
$199$	−1103.24	−0.392997	−0.196499	−	0.980504i	$-0.562957\pi$
$199$	−1103.24	−0.392997	−0.196499	+	0.980504i	$0.562957\pi$
$200$	−609.824	−0.215605
$201$	0	0
$202$	−2152.49	−0.749745
$203$	−40.6393	−0.0140508
$204$	0	0
$205$	−2464.02	−0.839484
$206$	3692.15	1.24876
$207$	0	0
$208$	772.734	0.257593
$209$	−1567.76	−0.518871
$210$	0	0
$211$	−4741.90	−1.54714	−0.773569	−	0.633712i	$-0.781530\pi$
$211$	−4741.90	−1.54714	−0.773569	+	0.633712i	$0.781530\pi$
$212$	23.6706	0.00766841
$213$	0	0
$214$	−3759.82	−1.20101
$215$	−155.619	−0.0493634
$216$	0	0
$217$	375.417	0.117442
$218$	−1909.57	−0.593267
$219$	0	0
$220$	−429.541	−0.131635
$221$	−2394.32	−0.728777
$222$	0	0
$223$	−2999.28	−0.900658	−0.450329	−	0.892863i	$-0.648693\pi$
$223$	−2999.28	−0.900658	−0.450329	+	0.892863i	$0.648693\pi$
$224$	−203.061	−0.0605695
$225$	0	0
$226$	−2034.83	−0.598915
$227$	1943.66	0.568305	0.284152	−	0.958779i	$-0.408288\pi$
$227$	1943.66	0.568305	0.284152	+	0.958779i	$0.408288\pi$
$228$	0	0
$229$	−3785.56	−1.09239	−0.546194	−	0.837658i	$-0.683924\pi$
$229$	−3785.56	−1.09239	−0.546194	+	0.837658i	$0.683924\pi$
$230$	1253.41	0.359335
$231$	0	0
$232$	−707.396	−0.200185
$233$	2799.88	0.787237	0.393619	−	0.919274i	$-0.371223\pi$
$233$	2799.88	0.787237	0.393619	+	0.919274i	$0.371223\pi$
$234$	0	0
$235$	−1167.49	−0.324081
$236$	0.596187	0.000164443 0
$237$	0	0
$238$	−218.150	−0.0594142
$239$	−1280.17	−0.346473	−0.173236	−	0.984880i	$-0.555422\pi$
$239$	−1280.17	−0.346473	−0.173236	+	0.984880i	$0.555422\pi$
$240$	0	0
$241$	4966.48	1.32746	0.663732	−	0.747970i	$-0.268971\pi$
$241$	4966.48	1.32746	0.663732	+	0.747970i	$0.268971\pi$
$242$	−1547.55	−0.411076
$243$	0	0
$244$	888.645	0.233154
$245$	−1705.18	−0.444653
$246$	0	0
$247$	2085.35	0.537198
$248$	6534.76	1.67322
$249$	0	0
$250$	265.054	0.0670540
$251$	−6716.64	−1.68904	−0.844522	−	0.535520i	$-0.820116\pi$
$251$	−6716.64	−1.68904	−0.844522	+	0.535520i	$0.820116\pi$
$252$	0	0
$253$	2898.66	0.720304
$254$	191.426	0.0472879
$255$	0	0
$256$	−4198.75	−1.02509
$257$	−6324.65	−1.53510	−0.767550	−	0.640989i	$-0.778525\pi$
$257$	−6324.65	−1.53510	−0.767550	+	0.640989i	$0.778525\pi$
$258$	0	0
$259$	−404.239	−0.0969813
$260$	571.354	0.136284
$261$	0	0
$262$	−95.8000	−0.0225899
$263$	615.504	0.144310	0.0721551	−	0.997393i	$-0.477012\pi$
$263$	615.504	0.144310	0.0721551	+	0.997393i	$0.477012\pi$
$264$	0	0
$265$	−33.7788	−0.00783025
$266$	189.999	0.0437955
$267$	0	0
$268$	660.660	0.150583
$269$	5700.55	1.29208	0.646038	−	0.763305i	$-0.276425\pi$
$269$	5700.55	1.29208	0.646038	+	0.763305i	$0.276425\pi$
$270$	0	0
$271$	4219.97	0.945923	0.472961	−	0.881083i	$-0.343185\pi$
$271$	4219.97	0.945923	0.472961	+	0.881083i	$0.343185\pi$
$272$	−1739.45	−0.387757
$273$	0	0
$274$	−4163.07	−0.917883
$275$	612.971	0.134413
$276$	0	0
$277$	1981.60	0.429829	0.214915	−	0.976633i	$-0.431053\pi$
$277$	1981.60	0.429829	0.214915	+	0.976633i	$0.431053\pi$
$278$	446.821	0.0963977
$279$	0	0
$280$	170.916	0.0364792
$281$	1504.70	0.319442	0.159721	−	0.987162i	$-0.448941\pi$
$281$	1504.70	0.319442	0.159721	+	0.987162i	$0.448941\pi$
$282$	0	0
$283$	5801.55	1.21861	0.609305	−	0.792936i	$-0.291449\pi$
$283$	5801.55	1.21861	0.609305	+	0.792936i	$0.291449\pi$
$284$	2352.28	0.491487
$285$	0	0
$286$	−1695.60	−0.350570
$287$	690.592	0.142036
$288$	0	0
$289$	476.715	0.0970313
$290$	307.463	0.0622581
$291$	0	0
$292$	−1018.88	−0.204197
$293$	726.374	0.144830	0.0724151	−	0.997375i	$-0.476929\pi$
$293$	726.374	0.144830	0.0724151	+	0.997375i	$0.476929\pi$
$294$	0	0
$295$	−0.850780	−0.000167913 0
$296$	−7036.46	−1.38171
$297$	0	0
$298$	2935.15	0.570567
$299$	−3855.65	−0.745745
$300$	0	0
$301$	43.6155	0.00835201
$302$	−2606.40	−0.496627
$303$	0	0
$304$	1514.99	0.285824
$305$	−1268.13	−0.238075
$306$	0	0
$307$	−405.295	−0.0753467	−0.0376733	−	0.999290i	$-0.511995\pi$
$307$	−405.295	−0.0753467	−0.0376733	+	0.999290i	$0.511995\pi$
$308$	120.388	0.0222719
$309$	0	0
$310$	−2840.27	−0.520376
$311$	1618.01	0.295013	0.147507	−	0.989061i	$-0.452875\pi$
$311$	1618.01	0.295013	0.147507	+	0.989061i	$0.452875\pi$
$312$	0	0
$313$	5620.47	1.01498	0.507488	−	0.861659i	$-0.330574\pi$
$313$	5620.47	1.01498	0.507488	+	0.861659i	$0.330574\pi$
$314$	2299.38	0.413254
$315$	0	0
$316$	1228.39	0.218679
$317$	−2250.61	−0.398759	−0.199380	−	0.979922i	$-0.563893\pi$
$317$	−2250.61	−0.398759	−0.199380	+	0.979922i	$0.563893\pi$
$318$	0	0
$319$	711.046	0.124799
$320$	2484.03	0.433942
$321$	0	0
$322$	−351.293	−0.0607975
$323$	−4694.21	−0.808646
$324$	0	0
$325$	−815.343	−0.139160
$326$	−7639.02	−1.29781
$327$	0	0
$328$	12020.9	2.02361
$329$	327.215	0.0548327
$330$	0	0
$331$	5451.78	0.905307	0.452654	−	0.891686i	$-0.350477\pi$
$331$	5451.78	0.905307	0.452654	+	0.891686i	$0.350477\pi$
$332$	3571.00	0.590314
$333$	0	0
$334$	5649.07	0.925459
$335$	−942.786	−0.153761
$336$	0	0
$337$	1521.29	0.245904	0.122952	−	0.992413i	$-0.460764\pi$
$337$	1521.29	0.245904	0.122952	+	0.992413i	$0.460764\pi$
$338$	−2403.19	−0.386734
$339$	0	0
$340$	−1286.14	−0.205149
$341$	−6568.48	−1.04312
$342$	0	0
$343$	958.578	0.150899
$344$	759.201	0.118992
$345$	0	0
$346$	−6005.40	−0.933099
$347$	11163.3	1.72703	0.863515	−	0.504323i	$-0.168258\pi$
$347$	11163.3	1.72703	0.863515	+	0.504323i	$0.168258\pi$
$348$	0	0
$349$	−3430.69	−0.526191	−0.263095	−	0.964770i	$-0.584743\pi$
$349$	−3430.69	−0.526191	−0.263095	+	0.964770i	$0.584743\pi$
$350$	−74.2870	−0.0113452
$351$	0	0
$352$	3552.85	0.537977
$353$	−10103.1	−1.52333	−0.761663	−	0.647973i	$-0.775617\pi$
$353$	−10103.1	−1.52333	−0.761663	+	0.647973i	$0.775617\pi$
$354$	0	0
$355$	−3356.79	−0.501860
$356$	−3534.69	−0.526231
$357$	0	0
$358$	4370.06	0.645154
$359$	8241.29	1.21158	0.605792	−	0.795623i	$-0.292856\pi$
$359$	8241.29	1.21158	0.605792	+	0.795623i	$0.292856\pi$
$360$	0	0
$361$	−2770.55	−0.403929
$362$	9190.75	1.33441
$363$	0	0
$364$	−160.134	−0.0230585
$365$	1453.98	0.208507
$366$	0	0
$367$	1511.51	0.214987	0.107493	−	0.994206i	$-0.465718\pi$
$367$	1511.51	0.214987	0.107493	+	0.994206i	$0.465718\pi$
$368$	−2801.09	−0.396785
$369$	0	0
$370$	3058.33	0.429716
$371$	9.46722	0.00132483
$372$	0	0
$373$	273.737	0.0379988	0.0189994	−	0.999819i	$-0.493952\pi$
$373$	273.737	0.0379988	0.0189994	+	0.999819i	$0.493952\pi$
$374$	3816.87	0.527715
$375$	0	0
$376$	5695.73	0.781210
$377$	−945.797	−0.129207
$378$	0	0
$379$	13732.5	1.86118	0.930592	−	0.366057i	$-0.119293\pi$
$379$	13732.5	1.86118	0.930592	+	0.366057i	$0.119293\pi$
$380$	1120.17	0.151220
$381$	0	0
$382$	−3786.19	−0.507116
$383$	8317.66	1.10969	0.554846	−	0.831953i	$-0.312777\pi$
$383$	8317.66	1.10969	0.554846	+	0.831953i	$0.312777\pi$
$384$	0	0
$385$	−171.798	−0.0227419
$386$	2330.95	0.307364
$387$	0	0
$388$	1213.78	0.158816
$389$	−12184.5	−1.58813	−0.794063	−	0.607835i	$-0.792038\pi$
$389$	−12184.5	−1.58813	−0.794063	+	0.607835i	$0.792038\pi$
$390$	0	0
$391$	8679.20	1.12257
$392$	8318.88	1.07185
$393$	0	0
$394$	−6763.71	−0.864850
$395$	−1752.96	−0.223294
$396$	0	0
$397$	10293.0	1.30123	0.650617	−	0.759406i	$-0.274510\pi$
$397$	10293.0	1.30123	0.650617	+	0.759406i	$0.274510\pi$
$398$	−2339.34	−0.294625
$399$	0	0
$400$	−592.338	−0.0740423
$401$	12177.3	1.51647	0.758236	−	0.651980i	$-0.226062\pi$
$401$	12177.3	1.51647	0.758236	+	0.651980i	$0.226062\pi$
$402$	0	0
$403$	8737.06	1.07996
$404$	3556.73	0.438005
$405$	0	0
$406$	−86.1729	−0.0105337
$407$	7072.76	0.861385
$408$	0	0
$409$	10320.0	1.24765	0.623826	−	0.781563i	$-0.285577\pi$
$409$	10320.0	1.24765	0.623826	+	0.781563i	$0.285577\pi$
$410$	−5224.78	−0.629350
$411$	0	0
$412$	−6100.84	−0.729531
$413$	0.238449	2.84100e−5 0
$414$	0	0
$415$	−5095.95	−0.602772
$416$	−4725.83	−0.556978
$417$	0	0
$418$	−3324.33	−0.388991
$419$	1416.16	0.165117	0.0825583	−	0.996586i	$-0.473691\pi$
$419$	1416.16	0.165117	0.0825583	+	0.996586i	$0.473691\pi$
$420$	0	0
$421$	1770.07	0.204912	0.102456	−	0.994738i	$-0.467330\pi$
$421$	1770.07	0.204912	0.102456	+	0.994738i	$0.467330\pi$
$422$	−10054.9	−1.15987
$423$	0	0
$424$	164.793	0.0188751
$425$	1835.37	0.209479
$426$	0	0
$427$	355.420	0.0402809
$428$	6212.65	0.701635
$429$	0	0
$430$	−329.980	−0.0370070
$431$	−8676.34	−0.969662	−0.484831	−	0.874608i	$-0.661119\pi$
$431$	−8676.34	−0.969662	−0.484831	+	0.874608i	$0.661119\pi$
$432$	0	0
$433$	−10365.6	−1.15043	−0.575217	−	0.818001i	$-0.695082\pi$
$433$	−10365.6	−1.15043	−0.575217	+	0.818001i	$0.695082\pi$
$434$	796.046	0.0880448
$435$	0	0
$436$	3155.33	0.346589
$437$	−7559.21	−0.827474
$438$	0	0
$439$	−6186.08	−0.672541	−0.336270	−	0.941765i	$-0.609166\pi$
$439$	−6186.08	−0.672541	−0.336270	+	0.941765i	$0.609166\pi$
$440$	−2990.43	−0.324008
$441$	0	0
$442$	−5077.00	−0.546354
$443$	−7340.11	−0.787221	−0.393611	−	0.919277i	$-0.628774\pi$
$443$	−7340.11	−0.787221	−0.393611	+	0.919277i	$0.628774\pi$
$444$	0	0
$445$	5044.14	0.537337
$446$	−6359.78	−0.675211
$447$	0	0
$448$	−696.201	−0.0734206
$449$	4998.65	0.525392	0.262696	−	0.964879i	$-0.415388\pi$
$449$	4998.65	0.525392	0.262696	+	0.964879i	$0.415388\pi$
$450$	0	0
$451$	−12083.0	−1.26156
$452$	3362.31	0.349889
$453$	0	0
$454$	4121.40	0.426050
$455$	228.517	0.0235451
$456$	0	0
$457$	−2458.87	−0.251687	−0.125844	−	0.992050i	$-0.540164\pi$
$457$	−2458.87	−0.251687	−0.125844	+	0.992050i	$0.540164\pi$
$458$	−8027.03	−0.818949
$459$	0	0
$460$	−2071.10	−0.209925
$461$	−8913.54	−0.900532	−0.450266	−	0.892895i	$-0.648671\pi$
$461$	−8913.54	−0.900532	−0.450266	+	0.892895i	$0.648671\pi$
$462$	0	0
$463$	−4530.84	−0.454787	−0.227393	−	0.973803i	$-0.573020\pi$
$463$	−4530.84	−0.454787	−0.227393	+	0.973803i	$0.573020\pi$
$464$	−687.112	−0.0687465
$465$	0	0
$466$	5936.96	0.590181
$467$	−7900.81	−0.782882	−0.391441	−	0.920203i	$-0.628023\pi$
$467$	−7900.81	−0.782882	−0.391441	+	0.920203i	$0.628023\pi$
$468$	0	0
$469$	264.236	0.0260155
$470$	−2475.59	−0.242959
$471$	0	0
$472$	4.15061	0.000404761 0
$473$	−763.119	−0.0741823
$474$	0	0
$475$	−1598.53	−0.154411
$476$	360.467	0.0347101
$477$	0	0
$478$	−2714.50	−0.259746
$479$	20336.4	1.93986	0.969931	−	0.243381i	$-0.0782565\pi$
$479$	20336.4	1.93986	0.969931	+	0.243381i	$0.0782565\pi$
$480$	0	0
$481$	−9407.83	−0.891809
$482$	10531.1	0.995182
$483$	0	0
$484$	2557.14	0.240152
$485$	−1732.11	−0.162167
$486$	0	0
$487$	−11847.3	−1.10236	−0.551181	−	0.834385i	$-0.685823\pi$
$487$	−11847.3	−1.10236	−0.551181	+	0.834385i	$0.685823\pi$
$488$	6186.68	0.573889
$489$	0	0
$490$	−3615.72	−0.333350
$491$	−12022.9	−1.10506	−0.552529	−	0.833493i	$-0.686337\pi$
$491$	−12022.9	−1.10506	−0.552529	+	0.833493i	$0.686337\pi$
$492$	0	0
$493$	2129.03	0.194496
$494$	4421.85	0.402730
$495$	0	0
$496$	6347.39	0.574609
$497$	940.812	0.0849119
$498$	0	0
$499$	293.738	0.0263518	0.0131759	−	0.999913i	$-0.495806\pi$
$499$	293.738	0.0263518	0.0131759	+	0.999913i	$0.495806\pi$
$500$	−437.971	−0.0391733
$501$	0	0
$502$	−14242.2	−1.26625
$503$	4957.94	0.439491	0.219745	−	0.975557i	$-0.429477\pi$
$503$	4957.94	0.439491	0.219745	+	0.975557i	$0.429477\pi$
$504$	0	0
$505$	−5075.58	−0.447249
$506$	6146.40	0.540002
$507$	0	0
$508$	−316.308	−0.0276258
$509$	−7023.72	−0.611633	−0.305816	−	0.952091i	$-0.598929\pi$
$509$	−7023.72	−0.611633	−0.305816	+	0.952091i	$0.598929\pi$
$510$	0	0
$511$	−407.509	−0.0352781
$512$	−8056.91	−0.695446
$513$	0	0
$514$	−13411.0	−1.15084
$515$	8706.12	0.744927
$516$	0	0
$517$	−5725.12	−0.487022
$518$	−857.161	−0.0727056
$519$	0	0
$520$	3977.72	0.335451
$521$	17727.1	1.49067	0.745333	−	0.666692i	$-0.232290\pi$
$521$	17727.1	1.49067	0.745333	+	0.666692i	$0.232290\pi$
$522$	0	0
$523$	−5487.13	−0.458768	−0.229384	−	0.973336i	$-0.573671\pi$
$523$	−5487.13	−0.458768	−0.229384	+	0.973336i	$0.573671\pi$
$524$	158.298	0.0131971
$525$	0	0
$526$	1305.13	0.108187
$527$	−19667.5	−1.62567
$528$	0	0
$529$	1809.36	0.148710
$530$	−71.6257	−0.00587023
$531$	0	0
$532$	−313.951	−0.0255856
$533$	16072.1	1.30612
$534$	0	0
$535$	−8865.68	−0.716443
$536$	4599.47	0.370647
$537$	0	0
$538$	12087.6	0.968652
$539$	−8361.81	−0.668216
$540$	0	0
$541$	4243.99	0.337270	0.168635	−	0.985679i	$-0.446064\pi$
$541$	4243.99	0.337270	0.168635	+	0.985679i	$0.446064\pi$
$542$	8948.17	0.709145
$543$	0	0
$544$	10638.0	0.838421
$545$	−4502.78	−0.353904
$546$	0	0
$547$	−22519.2	−1.76024	−0.880119	−	0.474753i	$-0.842537\pi$
$547$	−22519.2	−1.76024	−0.880119	+	0.474753i	$0.842537\pi$
$548$	6878.97	0.536232
$549$	0	0
$550$	1299.76	0.100767
$551$	−1854.29	−0.143367
$552$	0	0
$553$	491.304	0.0377801
$554$	4201.85	0.322237
$555$	0	0
$556$	−738.319	−0.0563160
$557$	23665.3	1.80024	0.900119	−	0.435644i	$-0.143479\pi$
$557$	23665.3	1.80024	0.900119	+	0.435644i	$0.143479\pi$
$558$	0	0
$559$	1015.06	0.0768024
$560$	166.015	0.0125275
$561$	0	0
$562$	3190.62	0.239481
$563$	−5688.95	−0.425863	−0.212931	−	0.977067i	$-0.568301\pi$
$563$	−5688.95	−0.425863	−0.212931	+	0.977067i	$0.568301\pi$
$564$	0	0
$565$	−4798.14	−0.357273
$566$	12301.8	0.913574
$567$	0	0
$568$	16376.4	1.20975
$569$	5181.22	0.381736	0.190868	−	0.981616i	$-0.438870\pi$
$569$	5181.22	0.381736	0.190868	+	0.981616i	$0.438870\pi$
$570$	0	0
$571$	−10124.7	−0.742040	−0.371020	−	0.928625i	$-0.620992\pi$
$571$	−10124.7	−0.742040	−0.371020	+	0.928625i	$0.620992\pi$
$572$	2801.78	0.204805
$573$	0	0
$574$	1464.35	0.106483
$575$	2955.54	0.214356
$576$	0	0
$577$	−1502.64	−0.108416	−0.0542079	−	0.998530i	$-0.517263\pi$
$577$	−1502.64	−0.108416	−0.0542079	+	0.998530i	$0.517263\pi$
$578$	1010.84	0.0727431
$579$	0	0
$580$	−508.046	−0.0363715
$581$	1428.25	0.101986
$582$	0	0
$583$	−165.643	−0.0117671
$584$	−7093.38	−0.502613
$585$	0	0
$586$	1540.23	0.108577
$587$	10571.9	0.743351	0.371675	−	0.928363i	$-0.378783\pi$
$587$	10571.9	0.743351	0.371675	+	0.928363i	$0.378783\pi$
$588$	0	0
$589$	17129.5	1.19832
$590$	−1.80402	−0.000125882 0
$591$	0	0
$592$	−6834.70	−0.474500
$593$	6095.11	0.422085	0.211042	−	0.977477i	$-0.432314\pi$
$593$	6095.11	0.422085	0.211042	+	0.977477i	$0.432314\pi$
$594$	0	0
$595$	−514.400	−0.0354426
$596$	−4849.99	−0.333328
$597$	0	0
$598$	−8175.64	−0.559075
$599$	6143.32	0.419047	0.209524	−	0.977804i	$-0.432809\pi$
$599$	6143.32	0.419047	0.209524	+	0.977804i	$0.432809\pi$
$600$	0	0
$601$	17150.2	1.16401	0.582007	−	0.813184i	$-0.302268\pi$
$601$	17150.2	1.16401	0.582007	+	0.813184i	$0.302268\pi$
$602$	92.4837	0.00626139
$603$	0	0
$604$	4306.76	0.290132
$605$	−3649.14	−0.245221
$606$	0	0
$607$	−6196.83	−0.414368	−0.207184	−	0.978302i	$-0.566430\pi$
$607$	−6196.83	−0.414368	−0.207184	+	0.978302i	$0.566430\pi$
$608$	−9265.25	−0.618019
$609$	0	0
$610$	−2688.98	−0.178481
$611$	7615.27	0.504224
$612$	0	0
$613$	15766.1	1.03881	0.519403	−	0.854530i	$-0.326154\pi$
$613$	15766.1	1.03881	0.519403	+	0.854530i	$0.326154\pi$
$614$	−859.401	−0.0564864
$615$	0	0
$616$	838.132	0.0548203
$617$	−18677.5	−1.21868	−0.609340	−	0.792909i	$-0.708566\pi$
$617$	−18677.5	−1.21868	−0.609340	+	0.792909i	$0.708566\pi$
$618$	0	0
$619$	14341.9	0.931263	0.465631	−	0.884979i	$-0.345827\pi$
$619$	14341.9	0.931263	0.465631	+	0.884979i	$0.345827\pi$
$620$	4693.21	0.304006
$621$	0	0
$622$	3430.89	0.221167
$623$	−1413.73	−0.0909144
$624$	0	0
$625$	625.000	0.0400000
$626$	11917.8	0.760914
$627$	0	0
$628$	−3799.46	−0.241425
$629$	21177.4	1.34244
$630$	0	0
$631$	−11850.5	−0.747638	−0.373819	−	0.927502i	$-0.621952\pi$
$631$	−11850.5	−0.747638	−0.373819	+	0.927502i	$0.621952\pi$
$632$	8551.99	0.538259
$633$	0	0
$634$	−4772.26	−0.298944
$635$	451.384	0.0282088
$636$	0	0
$637$	11122.5	0.691817
$638$	1507.72	0.0935602
$639$	0	0
$640$	−528.911	−0.0326672
$641$	−3014.56	−0.185753	−0.0928767	−	0.995678i	$-0.529606\pi$
$641$	−3014.56	−0.185753	−0.0928767	+	0.995678i	$0.529606\pi$
$642$	0	0
$643$	−21475.3	−1.31711	−0.658555	−	0.752533i	$-0.728832\pi$
$643$	−21475.3	−1.31711	−0.658555	+	0.752533i	$0.728832\pi$
$644$	580.470	0.0355182
$645$	0	0
$646$	−9953.75	−0.606231
$647$	24199.3	1.47044	0.735218	−	0.677831i	$-0.237080\pi$
$647$	24199.3	1.47044	0.735218	+	0.677831i	$0.237080\pi$
$648$	0	0
$649$	−4.17203	−0.000252336 0
$650$	−1728.88	−0.104326
$651$	0	0
$652$	12622.6	0.758187
$653$	8138.48	0.487723	0.243862	−	0.969810i	$-0.421586\pi$
$653$	8138.48	0.487723	0.243862	+	0.969810i	$0.421586\pi$
$654$	0	0
$655$	−225.897	−0.0134756
$656$	11676.2	0.694940
$657$	0	0
$658$	693.838	0.0411073
$659$	24406.8	1.44272	0.721360	−	0.692560i	$-0.243517\pi$
$659$	24406.8	1.44272	0.721360	+	0.692560i	$0.243517\pi$
$660$	0	0
$661$	11235.2	0.661115	0.330557	−	0.943786i	$-0.392763\pi$
$661$	11235.2	0.661115	0.330557	+	0.943786i	$0.392763\pi$
$662$	11560.1	0.678697
$663$	0	0
$664$	24861.1	1.45301
$665$	448.020	0.0261255
$666$	0	0
$667$	3428.43	0.199024
$668$	−9334.41	−0.540658
$669$	0	0
$670$	−1999.12	−0.115272
$671$	−6218.60	−0.357774
$672$	0	0
$673$	19425.6	1.11263	0.556315	−	0.830971i	$-0.312215\pi$
$673$	19425.6	1.11263	0.556315	+	0.830971i	$0.312215\pi$
$674$	3225.78	0.184351
$675$	0	0
$676$	3970.98	0.225932
$677$	−24093.6	−1.36779	−0.683894	−	0.729582i	$-0.739715\pi$
$677$	−24093.6	−1.36779	−0.683894	+	0.729582i	$0.739715\pi$
$678$	0	0
$679$	485.461	0.0274378
$680$	−8954.01	−0.504957
$681$	0	0
$682$	−13928.0	−0.782011
$683$	30150.8	1.68915	0.844574	−	0.535438i	$-0.179854\pi$
$683$	30150.8	1.68915	0.844574	+	0.535438i	$0.179854\pi$
$684$	0	0
$685$	−9816.55	−0.547549
$686$	2032.60	0.113127
$687$	0	0
$688$	737.432	0.0408639
$689$	220.330	0.0121828
$690$	0	0
$691$	−6867.54	−0.378080	−0.189040	−	0.981969i	$-0.560538\pi$
$691$	−6867.54	−0.378080	−0.189040	+	0.981969i	$0.560538\pi$
$692$	9923.21	0.545121
$693$	0	0
$694$	23671.1	1.29473
$695$	1053.61	0.0575045
$696$	0	0
$697$	−36179.0	−1.96611
$698$	−7274.54	−0.394478
$699$	0	0
$700$	122.750	0.00662790
$701$	−35818.0	−1.92985	−0.964926	−	0.262521i	$-0.915446\pi$
$701$	−35818.0	−1.92985	−0.964926	+	0.262521i	$0.915446\pi$
$702$	0	0
$703$	−18444.6	−0.989546
$704$	12181.1	0.652119
$705$	0	0
$706$	−21423.0	−1.14202
$707$	1422.54	0.0756720
$708$	0	0
$709$	−11409.3	−0.604354	−0.302177	−	0.953252i	$-0.597713\pi$
$709$	−11409.3	−0.604354	−0.302177	+	0.953252i	$0.597713\pi$
$710$	−7117.86	−0.376237
$711$	0	0
$712$	−24608.3	−1.29527
$713$	−31671.0	−1.66352
$714$	0	0
$715$	−3998.25	−0.209127
$716$	−7221.01	−0.376902
$717$	0	0
$718$	17475.1	0.908308
$719$	−18782.0	−0.974200	−0.487100	−	0.873346i	$-0.661945\pi$
$719$	−18782.0	−0.974200	−0.487100	+	0.873346i	$0.661945\pi$
$720$	0	0
$721$	−2440.07	−0.126037
$722$	−5874.76	−0.302820
$723$	0	0
$724$	−15186.6	−0.779567
$725$	725.000	0.0371391
$726$	0	0
$727$	1658.12	0.0845889	0.0422945	−	0.999105i	$-0.486533\pi$
$727$	1658.12	0.0845889	0.0422945	+	0.999105i	$0.486533\pi$
$728$	−1114.84	−0.0567565
$729$	0	0
$730$	3083.07	0.156314
$731$	−2284.94	−0.115611
$732$	0	0
$733$	21347.6	1.07570	0.537852	−	0.843039i	$-0.319236\pi$
$733$	21347.6	1.07570	0.537852	+	0.843039i	$0.319236\pi$
$734$	3205.05	0.161172
$735$	0	0
$736$	17130.7	0.857942
$737$	−4623.20	−0.231069
$738$	0	0
$739$	27818.5	1.38474	0.692369	−	0.721543i	$-0.256567\pi$
$739$	27818.5	1.38474	0.692369	+	0.721543i	$0.256567\pi$
$740$	−5053.52	−0.251042
$741$	0	0
$742$	20.0746	0.000993210 0
$743$	−29021.0	−1.43295	−0.716473	−	0.697615i	$-0.754245\pi$
$743$	−29021.0	−1.43295	−0.716473	+	0.697615i	$0.754245\pi$
$744$	0	0
$745$	6921.12	0.340363
$746$	580.441	0.0284872
$747$	0	0
$748$	−6306.92	−0.308294
$749$	2484.79	0.121218
$750$	0	0
$751$	−27505.6	−1.33648	−0.668239	−	0.743947i	$-0.732952\pi$
$751$	−27505.6	−1.33648	−0.668239	+	0.743947i	$0.732952\pi$
$752$	5532.41	0.268280
$753$	0	0
$754$	−2005.50	−0.0968647
$755$	−6145.90	−0.296255
$756$	0	0
$757$	31401.4	1.50767	0.753833	−	0.657066i	$-0.228203\pi$
$757$	31401.4	1.50767	0.753833	+	0.657066i	$0.228203\pi$
$758$	29118.8	1.39530
$759$	0	0
$760$	7798.55	0.372215
$761$	31623.5	1.50638	0.753188	−	0.657805i	$-0.228515\pi$
$761$	31623.5	1.50638	0.753188	+	0.657805i	$0.228515\pi$
$762$	0	0
$763$	1262.00	0.0598786
$764$	6256.23	0.296260
$765$	0	0
$766$	17637.0	0.831921
$767$	5.54942	0.000261249 0
$768$	0	0
$769$	−24813.8	−1.16360	−0.581801	−	0.813331i	$-0.697652\pi$
$769$	−24813.8	−1.16360	−0.581801	+	0.813331i	$0.697652\pi$
$770$	−364.286	−0.0170493
$771$	0	0
$772$	−3851.62	−0.179563
$773$	29837.7	1.38834	0.694170	−	0.719811i	$-0.255772\pi$
$773$	29837.7	1.38834	0.694170	+	0.719811i	$0.255772\pi$
$774$	0	0
$775$	−6697.39	−0.310422
$776$	8450.27	0.390911
$777$	0	0
$778$	−25836.5	−1.19060
$779$	31510.3	1.44926
$780$	0	0
$781$	−16460.9	−0.754185
$782$	18403.7	0.841578
$783$	0	0
$784$	8080.35	0.368092
$785$	5421.97	0.246520
$786$	0	0
$787$	28836.1	1.30609	0.653046	−	0.757318i	$-0.273491\pi$
$787$	28836.1	1.30609	0.653046	+	0.757318i	$0.273491\pi$
$788$	11176.2	0.505250
$789$	0	0
$790$	−3717.04	−0.167400
$791$	1344.78	0.0604487
$792$	0	0
$793$	8271.67	0.370411
$794$	21825.6	0.975517
$795$	0	0
$796$	3865.49	0.172121
$797$	−25155.7	−1.11802	−0.559010	−	0.829161i	$-0.688819\pi$
$797$	−25155.7	−1.11802	−0.559010	+	0.829161i	$0.688819\pi$
$798$	0	0
$799$	−17142.3	−0.759010
$800$	3622.58	0.160097
$801$	0	0
$802$	25821.1	1.13688
$803$	7129.98	0.313340
$804$	0	0
$805$	−828.353	−0.0362678
$806$	18526.4	0.809631
$807$	0	0
$808$	24761.7	1.07811
$809$	11402.5	0.495537	0.247769	−	0.968819i	$-0.420303\pi$
$809$	11402.5	0.495537	0.247769	+	0.968819i	$0.420303\pi$
$810$	0	0
$811$	−36317.0	−1.57246	−0.786229	−	0.617935i	$-0.787969\pi$
$811$	−36317.0	−1.57246	−0.786229	+	0.617935i	$0.787969\pi$
$812$	142.391	0.00615385
$813$	0	0
$814$	14997.3	0.645769
$815$	−18012.9	−0.774189
$816$	0	0
$817$	1990.09	0.0852195
$818$	21882.8	0.935348
$819$	0	0
$820$	8633.33	0.367669
$821$	−32017.0	−1.36103	−0.680513	−	0.732736i	$-0.738243\pi$
$821$	−32017.0	−1.36103	−0.680513	+	0.732736i	$0.738243\pi$
$822$	0	0
$823$	−31912.4	−1.35163	−0.675817	−	0.737069i	$-0.736209\pi$
$823$	−31912.4	−1.35163	−0.675817	+	0.737069i	$0.736209\pi$
$824$	−42473.6	−1.79568
$825$	0	0
$826$	0.505615	2.12986e−5 0
$827$	−17534.8	−0.737295	−0.368648	−	0.929569i	$-0.620179\pi$
$827$	−17534.8	−0.737295	−0.368648	+	0.929569i	$0.620179\pi$
$828$	0	0
$829$	39507.3	1.65518	0.827592	−	0.561331i	$-0.189710\pi$
$829$	39507.3	1.65518	0.827592	+	0.561331i	$0.189710\pi$
$830$	−10805.6	−0.451890
$831$	0	0
$832$	−16202.7	−0.675152
$833$	−25037.1	−1.04140
$834$	0	0
$835$	13320.6	0.552068
$836$	5493.05	0.227250
$837$	0	0
$838$	3002.87	0.123786
$839$	−12398.1	−0.510168	−0.255084	−	0.966919i	$-0.582103\pi$
$839$	−12398.1	−0.510168	−0.255084	+	0.966919i	$0.582103\pi$
$840$	0	0
$841$	841.000	0.0344828
$842$	3753.31	0.153620
$843$	0	0
$844$	16614.5	0.677601
$845$	−5666.73	−0.230700
$846$	0	0
$847$	1022.75	0.0414900
$848$	160.068	0.00648202
$849$	0	0
$850$	3891.77	0.157043
$851$	34102.5	1.37370
$852$	0	0
$853$	9946.94	0.399269	0.199635	−	0.979870i	$-0.436024\pi$
$853$	9946.94	0.399269	0.199635	+	0.979870i	$0.436024\pi$
$854$	753.644	0.0301981
$855$	0	0
$856$	43252.0	1.72701
$857$	1305.42	0.0520328	0.0260164	−	0.999662i	$-0.491718\pi$
$857$	1305.42	0.0520328	0.0260164	+	0.999662i	$0.491718\pi$
$858$	0	0
$859$	38980.3	1.54830	0.774151	−	0.633001i	$-0.218177\pi$
$859$	38980.3	1.54830	0.774151	+	0.633001i	$0.218177\pi$
$860$	545.252	0.0216197
$861$	0	0
$862$	−18397.6	−0.726942
$863$	−31555.2	−1.24467	−0.622336	−	0.782750i	$-0.713816\pi$
$863$	−31555.2	−1.24467	−0.622336	+	0.782750i	$0.713816\pi$
$864$	0	0
$865$	−14160.8	−0.556625
$866$	−21979.5	−0.862465
$867$	0	0
$868$	−1315.37	−0.0514362
$869$	−8596.11	−0.335562
$870$	0	0
$871$	6149.55	0.239230
$872$	21967.2	0.853100
$873$	0	0
$874$	−16028.8	−0.620346
$875$	−175.169	−0.00676778
$876$	0	0
$877$	−35538.1	−1.36834	−0.684171	−	0.729322i	$-0.739836\pi$
$877$	−35538.1	−1.36834	−0.684171	+	0.729322i	$0.739836\pi$
$878$	−13117.2	−0.504195
$879$	0	0
$880$	−2904.69	−0.111269
$881$	2301.79	0.0880242	0.0440121	−	0.999031i	$-0.485986\pi$
$881$	2301.79	0.0880242	0.0440121	+	0.999031i	$0.485986\pi$
$882$	0	0
$883$	−33464.5	−1.27539	−0.637695	−	0.770289i	$-0.720112\pi$
$883$	−33464.5	−1.27539	−0.637695	+	0.770289i	$0.720112\pi$
$884$	8389.15	0.319183
$885$	0	0
$886$	−15564.2	−0.590169
$887$	−17351.4	−0.656825	−0.328412	−	0.944534i	$-0.606514\pi$
$887$	−17351.4	−0.656825	−0.328412	+	0.944534i	$0.606514\pi$
$888$	0	0
$889$	−126.510	−0.00477278
$890$	10695.8	0.402834
$891$	0	0
$892$	10508.8	0.394462
$893$	14930.2	0.559484
$894$	0	0
$895$	10304.7	0.384856
$896$	148.238	0.00552712
$897$	0	0
$898$	10599.3	0.393879
$899$	−7768.97	−0.288220
$900$	0	0
$901$	−495.972	−0.0183388
$902$	−25621.1	−0.945775
$903$	0	0
$904$	23408.2	0.861222
$905$	21671.9	0.796019
$906$	0	0
$907$	13518.7	0.494908	0.247454	−	0.968900i	$-0.420406\pi$
$907$	13518.7	0.494908	0.247454	+	0.968900i	$0.420406\pi$
$908$	−6810.12	−0.248901
$909$	0	0
$910$	484.555	0.0176515
$911$	−32678.5	−1.18846	−0.594230	−	0.804295i	$-0.702543\pi$
$911$	−32678.5	−1.18846	−0.594230	+	0.804295i	$0.702543\pi$
$912$	0	0
$913$	−24989.4	−0.905835
$914$	−5213.87	−0.188687
$915$	0	0
$916$	13263.7	0.478434
$917$	63.3125	0.00228000
$918$	0	0
$919$	30621.0	1.09912	0.549562	−	0.835453i	$-0.314795\pi$
$919$	30621.0	1.09912	0.549562	+	0.835453i	$0.314795\pi$
$920$	−14418.9	−0.516714
$921$	0	0
$922$	−18900.6	−0.675116
$923$	21895.5	0.780822
$924$	0	0
$925$	7211.57	0.256340
$926$	−9607.35	−0.340947
$927$	0	0
$928$	4202.19	0.148646
$929$	−45355.7	−1.60180	−0.800900	−	0.598798i	$-0.795645\pi$
$929$	−45355.7	−1.60180	−0.800900	+	0.598798i	$0.795645\pi$
$930$	0	0
$931$	21806.2	0.767636
$932$	−9810.12	−0.344787
$933$	0	0
$934$	−16753.1	−0.586916
$935$	9000.21	0.314800
$936$	0	0
$937$	16518.2	0.575907	0.287953	−	0.957644i	$-0.407025\pi$
$937$	16518.2	0.575907	0.287953	+	0.957644i	$0.407025\pi$
$938$	560.294	0.0195035
$939$	0	0
$940$	4090.63	0.141938
$941$	22774.4	0.788975	0.394487	−	0.918901i	$-0.370922\pi$
$941$	22774.4	0.788975	0.394487	+	0.918901i	$0.370922\pi$
$942$	0	0
$943$	−58260.0	−2.01188
$944$	4.03160	0.000139001 0
$945$	0	0
$946$	−1618.14	−0.0556135
$947$	−42834.2	−1.46983	−0.734913	−	0.678162i	$-0.762777\pi$
$947$	−42834.2	−1.46983	−0.734913	+	0.678162i	$0.762777\pi$
$948$	0	0
$949$	−9483.94	−0.324407
$950$	−3389.57	−0.115760
$951$	0	0
$952$	2509.55	0.0854359
$953$	11113.4	0.377754	0.188877	−	0.982001i	$-0.439515\pi$
$953$	11113.4	0.377754	0.188877	+	0.982001i	$0.439515\pi$
$954$	0	0
$955$	−8927.87	−0.302512
$956$	4485.40	0.151745
$957$	0	0
$958$	43122.0	1.45429
$959$	2751.29	0.0926422
$960$	0	0
$961$	41977.0	1.40905
$962$	−19948.7	−0.668577
$963$	0	0
$964$	−17401.4	−0.581390
$965$	5496.40	0.183353
$966$	0	0
$967$	−27959.0	−0.929784	−0.464892	−	0.885367i	$-0.653907\pi$
$967$	−27959.0	−0.929784	−0.464892	+	0.885367i	$0.653907\pi$
$968$	17802.7	0.591115
$969$	0	0
$970$	−3672.83	−0.121575
$971$	37840.3	1.25062	0.625311	−	0.780375i	$-0.284972\pi$
$971$	37840.3	1.25062	0.625311	+	0.780375i	$0.284972\pi$
$972$	0	0
$973$	−295.296	−0.00972945
$974$	−25121.3	−0.826426
$975$	0	0
$976$	6009.29	0.197083
$977$	40258.4	1.31830	0.659150	−	0.752011i	$-0.270916\pi$
$977$	40258.4	1.31830	0.659150	+	0.752011i	$0.270916\pi$
$978$	0	0
$979$	24735.2	0.807500
$980$	5974.55	0.194745
$981$	0	0
$982$	−25493.7	−0.828447
$983$	−13109.2	−0.425348	−0.212674	−	0.977123i	$-0.568217\pi$
$983$	−13109.2	−0.425348	−0.212674	+	0.977123i	$0.568217\pi$
$984$	0	0
$985$	−15948.9	−0.515913
$986$	4514.46	0.145811
$987$	0	0
$988$	−7306.58	−0.235277
$989$	−3679.51	−0.118303
$990$	0	0
$991$	−14404.6	−0.461732	−0.230866	−	0.972986i	$-0.574156\pi$
$991$	−14404.6	−0.461732	−0.230866	+	0.972986i	$0.574156\pi$
$992$	−38818.9	−1.24244
$993$	0	0
$994$	1994.93	0.0636573
$995$	−5516.19	−0.175754
$996$	0	0
$997$	30046.1	0.954432	0.477216	−	0.878786i	$-0.341646\pi$
$997$	30046.1	0.954432	0.477216	+	0.878786i	$0.341646\pi$
$998$	622.853	0.0197556
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1305.4.a.g.1.4		5
3.2	odd	2		435.4.a.f.1.2	✓	5
15.14	odd	2		2175.4.a.j.1.4		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
435.4.a.f.1.2	✓	5	3.2	odd	2
1305.4.a.g.1.4		5	1.1	even	1	trivial
2175.4.a.j.1.4		5	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)

\(f(q)\)	\(=\)	\(q+2.12043 q^{2} -3.50376 q^{4} +5.00000 q^{5} -1.40136 q^{7} -24.3930 q^{8} +10.6022 q^{10} +24.5188 q^{11} -32.6137 q^{13} -2.97148 q^{14} -23.6935 q^{16} +73.4147 q^{17} -63.9410 q^{19} -17.5188 q^{20} +51.9905 q^{22} +118.222 q^{23} +25.0000 q^{25} -69.1552 q^{26} +4.91002 q^{28} +29.0000 q^{29} -267.895 q^{31} +144.903 q^{32} +155.671 q^{34} -7.00678 q^{35} +288.463 q^{37} -135.583 q^{38} -121.965 q^{40} -492.803 q^{41} -31.1238 q^{43} -85.9082 q^{44} +250.681 q^{46} -233.499 q^{47} -341.036 q^{49} +53.0108 q^{50} +114.271 q^{52} -6.75576 q^{53} +122.594 q^{55} +34.1832 q^{56} +61.4926 q^{58} -0.170156 q^{59} -253.626 q^{61} -568.054 q^{62} +496.806 q^{64} -163.069 q^{65} -188.557 q^{67} -257.228 q^{68} -14.8574 q^{70} -671.359 q^{71} +290.796 q^{73} +611.666 q^{74} +224.034 q^{76} -34.3596 q^{77} -350.592 q^{79} -118.468 q^{80} -1044.96 q^{82} -1019.19 q^{83} +367.073 q^{85} -65.9959 q^{86} -598.087 q^{88} +1008.83 q^{89} +45.7034 q^{91} -414.221 q^{92} -495.119 q^{94} -319.705 q^{95} -346.423 q^{97} -723.144 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - 2 q^{2} + 10 q^{4} + 25 q^{5} - 29 q^{7} - 10 q^{10} - 15 q^{11} - 31 q^{13} + 114 q^{14} - 102 q^{16} - 119 q^{17} + 46 q^{19} + 50 q^{20} + 66 q^{22} - 170 q^{23} + 125 q^{25} - 138 q^{26} + 32 q^{28}+ \cdots - 1780 q^{98}+O(q^{100}) \) 5 * q - 2 * q^2 + 10 * q^4 + 25 * q^5 - 29 * q^7 - 10 * q^10 - 15 * q^11 - 31 * q^13 + 114 * q^14 - 102 * q^16 - 119 * q^17 + 46 * q^19 + 50 * q^20 + 66 * q^22 - 170 * q^23 + 125 * q^25 - 138 * q^26 + 32 * q^28 + 145 * q^29 - 240 * q^31 - 152 * q^32 + 290 * q^34 - 145 * q^35 + 570 * q^37 - 1380 * q^38 + 180 * q^41 - 110 * q^43 - 1420 * q^44 + 696 * q^46 - 697 * q^47 - 362 * q^49 - 50 * q^50 + 960 * q^52 - 960 * q^53 - 75 * q^55 + 224 * q^56 - 58 * q^58 - 154 * q^59 + 360 * q^61 - 1176 * q^62 - 486 * q^64 - 155 * q^65 + 517 * q^67 - 1812 * q^68 + 570 * q^70 + 172 * q^71 + 1882 * q^73 - 1476 * q^74 + 884 * q^76 - 363 * q^77 - 478 * q^79 - 510 * q^80 - 1412 * q^82 - 2356 * q^83 - 595 * q^85 - 272 * q^86 + 1400 * q^88 - 449 * q^89 - 645 * q^91 - 1072 * q^92 + 466 * q^94 + 230 * q^95 + 88 * q^97 - 1780 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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