LMFDB - Embedded newform 435.4.a.f.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [435,4,Mod(1,435)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(435, 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("435.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$N$	$=$	$435 = 3 \cdot 5 \cdot 29$
Weight:	$k$	$=$	$4$
Character orbit:	$[\chi]$	$=$	435.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:	$25.6658308525$
Analytic rank:	$0$
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:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-2.12043$ of defining polynomial
Character	$\chi$	$=$	435.1

$q$ -expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.12043 q^{2} -3.00000 q^{3} -3.50376 q^{4} -5.00000 q^{5} +6.36130 q^{6} -1.40136 q^{7} +24.3930 q^{8} +9.00000 q^{9} +10.6022 q^{10} -24.5188 q^{11} +10.5113 q^{12} -32.6137 q^{13} +2.97148 q^{14} +15.0000 q^{15} -23.6935 q^{16} -73.4147 q^{17} -19.0839 q^{18} -63.9410 q^{19} +17.5188 q^{20} +4.20407 q^{21} +51.9905 q^{22} -118.222 q^{23} -73.1789 q^{24} +25.0000 q^{25} +69.1552 q^{26} -27.0000 q^{27} +4.91002 q^{28} -29.0000 q^{29} -31.8065 q^{30} -267.895 q^{31} -144.903 q^{32} +73.5565 q^{33} +155.671 q^{34} +7.00678 q^{35} -31.5339 q^{36} +288.463 q^{37} +135.583 q^{38} +97.8411 q^{39} -121.965 q^{40} +492.803 q^{41} -8.91444 q^{42} -31.1238 q^{43} +85.9082 q^{44} -45.0000 q^{45} +250.681 q^{46} +233.499 q^{47} +71.0806 q^{48} -341.036 q^{49} -53.0108 q^{50} +220.244 q^{51} +114.271 q^{52} +6.75576 q^{53} +57.2517 q^{54} +122.594 q^{55} -34.1832 q^{56} +191.823 q^{57} +61.4926 q^{58} +0.170156 q^{59} -52.5565 q^{60} -253.626 q^{61} +568.054 q^{62} -12.6122 q^{63} +496.806 q^{64} +163.069 q^{65} -155.972 q^{66} -188.557 q^{67} +257.228 q^{68} +354.665 q^{69} -14.8574 q^{70} +671.359 q^{71} +219.537 q^{72} +290.796 q^{73} -611.666 q^{74} -75.0000 q^{75} +224.034 q^{76} +34.3596 q^{77} -207.465 q^{78} -350.592 q^{79} +118.468 q^{80} +81.0000 q^{81} -1044.96 q^{82} +1019.19 q^{83} -14.7301 q^{84} +367.073 q^{85} +65.9959 q^{86} +87.0000 q^{87} -598.087 q^{88} -1008.83 q^{89} +95.4195 q^{90} +45.7034 q^{91} +414.221 q^{92} +803.686 q^{93} -495.119 q^{94} +319.705 q^{95} +434.709 q^{96} -346.423 q^{97} +723.144 q^{98} -220.669 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$5 q + 2 q^{2} - 15 q^{3} + 10 q^{4} - 25 q^{5} - 6 q^{6} - 29 q^{7} + 45 q^{9} - 10 q^{10} + 15 q^{11} - 30 q^{12} - 31 q^{13} - 114 q^{14} + 75 q^{15} - 102 q^{16} + 119 q^{17} + 18 q^{18} + 46 q^{19}+ \cdots + 135 q^{99}+O(q^{100})$ 5 * q + 2 * q^2 - 15 * q^3 + 10 * q^4 - 25 * q^5 - 6 * q^6 - 29 * q^7 + 45 * q^9 - 10 * q^10 + 15 * q^11 - 30 * q^12 - 31 * q^13 - 114 * q^14 + 75 * q^15 - 102 * q^16 + 119 * q^17 + 18 * q^18 + 46 * q^19 - 50 * q^20 + 87 * q^21 + 66 * q^22 + 170 * q^23 + 125 * q^25 + 138 * q^26 - 135 * q^27 + 32 * q^28 - 145 * q^29 + 30 * q^30 - 240 * q^31 + 152 * q^32 - 45 * q^33 + 290 * q^34 + 145 * q^35 + 90 * q^36 + 570 * q^37 + 1380 * q^38 + 93 * q^39 - 180 * q^41 + 342 * q^42 - 110 * q^43 + 1420 * q^44 - 225 * q^45 + 696 * q^46 + 697 * q^47 + 306 * q^48 - 362 * q^49 + 50 * q^50 - 357 * q^51 + 960 * q^52 + 960 * q^53 - 54 * q^54 - 75 * q^55 - 224 * q^56 - 138 * q^57 - 58 * q^58 + 154 * q^59 + 150 * q^60 + 360 * q^61 + 1176 * q^62 - 261 * q^63 - 486 * q^64 + 155 * q^65 - 198 * q^66 + 517 * q^67 + 1812 * q^68 - 510 * q^69 + 570 * q^70 - 172 * q^71 + 1882 * q^73 + 1476 * q^74 - 375 * q^75 + 884 * q^76 + 363 * q^77 - 414 * q^78 - 478 * q^79 + 510 * q^80 + 405 * q^81 - 1412 * q^82 + 2356 * q^83 - 96 * q^84 - 595 * q^85 + 272 * q^86 + 435 * q^87 + 1400 * q^88 + 449 * q^89 - 90 * q^90 - 645 * q^91 + 1072 * q^92 + 720 * q^93 + 466 * q^94 - 230 * q^95 - 456 * q^96 + 88 * q^97 + 1780 * q^98 + 135 * q^99

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	$-0.622303\pi$
$2$	−2.12043	−0.749686	−0.374843	+	0.927088i	$0.622303\pi$
$3$	−3.00000	−0.577350
$3$	−3.00000	−0.577350
$4$	−3.50376	−0.437971
$5$	−5.00000	−0.447214
$5$	−5.00000	−0.447214
$6$	6.36130	0.432832
$7$	−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
$9$	9.00000	0.333333
$10$	10.6022	0.335270
$11$	−24.5188	−0.672064	−0.336032	−	0.941851i	$-0.609085\pi$
$11$	−24.5188	−0.672064	−0.336032	+	0.941851i	$0.609085\pi$
$12$	10.5113	0.252862
$13$	−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
$15$	15.0000	0.258199
$16$	−23.6935	−0.370211
$17$	−73.4147	−1.04739	−0.523696	−	0.851905i	$-0.675447\pi$
$17$	−73.4147	−1.04739	−0.523696	+	0.851905i	$0.675447\pi$
$18$	−19.0839	−0.249895
$19$	−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
$21$	4.20407	0.0436858
$22$	51.9905	0.503837
$23$	−118.222	−1.07178	−0.535890	−	0.844288i	$-0.680024\pi$
$23$	−118.222	−1.07178	−0.535890	+	0.844288i	$0.680024\pi$
$24$	−73.1789	−0.622399
$25$	25.0000	0.200000
$26$	69.1552	0.521632
$27$	−27.0000	−0.192450
$28$	4.91002	0.0331395
$29$	−29.0000	−0.185695
$29$	−29.0000	−0.185695
$30$	−31.8065	−0.193568
$31$	−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
$33$	73.5565	0.388016
$34$	155.671	0.785216
$35$	7.00678	0.0338389
$36$	−31.5339	−0.145990
$37$	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
$39$	97.8411	0.401721
$40$	−121.965	−0.482108
$41$	492.803	1.87714	0.938572	−	0.345083i	$-0.112149\pi$
$41$	492.803	1.87714	0.938572	+	0.345083i	$0.112149\pi$
$42$	−8.91444	−0.0327507
$43$	−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
$45$	−45.0000	−0.149071
$46$	250.681	0.803498
$47$	233.499	0.724667	0.362333	−	0.932049i	$-0.381980\pi$
$47$	233.499	0.724667	0.362333	+	0.932049i	$0.381980\pi$
$48$	71.0806	0.213742
$49$	−341.036	−0.994275
$50$	−53.0108	−0.149937
$51$	220.244	0.604712
$52$	114.271	0.304740
$53$	6.75576	0.0175090	0.00875448	−	0.999962i	$-0.497213\pi$
$53$	6.75576	0.0175090	0.00875448	+	0.999962i	$0.497213\pi$
$54$	57.2517	0.144277
$55$	122.594	0.300556
$56$	−34.1832	−0.0815700
$57$	191.823	0.445747
$58$	61.4926	0.139213
$59$	0.170156	0.000375465 0	0.000187732	−	1.00000i	$-0.499940\pi$
$59$	0.170156	0.000375465 0	0.000187732		1.00000i	$0.499940\pi$
$60$	−52.5565	−0.113083
$61$	−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
$63$	−12.6122	−0.0252220
$64$	496.806	0.970323
$65$	163.069	0.311172
$66$	−155.972	−0.290890
$67$	−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
$69$	354.665	0.618792
$70$	−14.8574	−0.0253686
$71$	671.359	1.12219	0.561096	−	0.827751i	$-0.310380\pi$
$71$	671.359	1.12219	0.561096	+	0.827751i	$0.310380\pi$
$72$	219.537	0.359342
$73$	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
$75$	−75.0000	−0.115470
$76$	224.034	0.338138
$77$	34.3596	0.0508524
$78$	−207.465	−0.301165
$79$	−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
$81$	81.0000	0.111111
$82$	−1044.96	−1.40727
$83$	1019.19	1.34784	0.673920	−	0.738804i	$-0.264609\pi$
$83$	1019.19	1.34784	0.673920	+	0.738804i	$0.264609\pi$
$84$	−14.7301	−0.0191331
$85$	367.073	0.468408
$86$	65.9959	0.0827503
$87$	87.0000	0.107211
$88$	−598.087	−0.724503
$89$	−1008.83	−1.20152	−0.600761	−	0.799429i	$-0.705136\pi$
$89$	−1008.83	−1.20152	−0.600761	+	0.799429i	$0.705136\pi$
$90$	95.4195	0.111757
$91$	45.7034	0.0526485
$92$	414.221	0.469408
$93$	803.686	0.896112
$94$	−495.119	−0.543273
$95$	319.705	0.345274
$96$	434.709	0.462160
$97$	−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
$99$	−220.669	−0.224021
$100$	−87.5941	−0.0875941
$101$	1015.12	1.00008	0.500039	−	0.866003i	$-0.333319\pi$
$101$	1015.12	1.00008	0.500039	+	0.866003i	$0.333319\pi$
$102$	−467.013	−0.453345
$103$	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
$105$	−21.0203	−0.0195369
$106$	−14.3251	−0.0131262
$107$	1773.14	1.60201	0.801007	−	0.598655i	$-0.204298\pi$
$107$	1773.14	1.60201	0.801007	+	0.598655i	$0.204298\pi$
$108$	94.6016	0.0842875
$109$	−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
$111$	−865.388	−0.739991
$112$	33.2030	0.0280124
$113$	959.629	0.798887	0.399444	−	0.916758i	$-0.369203\pi$
$113$	959.629	0.798887	0.399444	+	0.916758i	$0.369203\pi$
$114$	−406.748	−0.334170
$115$	591.108	0.479314
$116$	101.609	0.0813291
$117$	−293.523	−0.231934
$118$	−0.360804	−0.000281481 0
$119$	102.880	0.0792521
$120$	365.894	0.278345
$121$	−729.827	−0.548330
$122$	537.796	0.399097
$123$	−1478.41	−1.08377
$124$	938.642	0.679779
$125$	−125.000	−0.0894427
$126$	26.7433	0.0189086
$127$	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
$129$	93.3714	0.0637278
$130$	−345.776	−0.233281
$131$	45.1795	0.0301324	0.0150662	−	0.999886i	$-0.495204\pi$
$131$	45.1795	0.0301324	0.0150662	+	0.999886i	$0.495204\pi$
$132$	−257.724	−0.169940
$133$	89.6041	0.0584185
$134$	399.823	0.257757
$135$	135.000	0.0860663
$136$	−1790.80	−1.12912
$137$	1963.31	1.22436	0.612178	−	0.790720i	$-0.290294\pi$
$137$	1963.31	1.22436	0.612178	+	0.790720i	$0.290294\pi$
$138$	−752.043	−0.463900
$139$	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
$141$	−700.497	−0.418386
$142$	−1423.57	−0.841292
$143$	799.649	0.467623
$144$	−213.242	−0.123404
$145$	145.000	0.0830455
$146$	−616.614	−0.349530
$147$	1023.11	0.574045
$148$	−1010.70	−0.561347
$149$	−1384.22	−0.761074	−0.380537	−	0.924766i	$-0.624261\pi$
$149$	−1384.22	−0.761074	−0.380537	+	0.924766i	$0.624261\pi$
$150$	159.032	0.0865663
$151$	−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
$153$	−660.732	−0.349131
$154$	−72.8572	−0.0381234
$155$	1339.48	0.694125
$156$	−342.812	−0.175942
$157$	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
$159$	−20.2673	−0.0101088
$160$	724.516	0.357987
$161$	165.671	0.0810973
$162$	−171.755	−0.0832985
$163$	−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
$165$	−367.782	−0.173526
$166$	−2161.13	−1.01046
$167$	−2664.11	−1.23446	−0.617231	−	0.786782i	$-0.711745\pi$
$167$	−2664.11	−1.23446	−0.617231	+	0.786782i	$0.711745\pi$
$168$	102.550	0.0470945
$169$	−1133.35	−0.515861
$170$	−778.355	−0.351159
$171$	−575.469	−0.257352
$172$	109.050	0.0483431
$173$	2832.16	1.24465	0.622326	−	0.782758i	$-0.286188\pi$
$173$	2832.16	1.24465	0.622326	+	0.782758i	$0.286188\pi$
$174$	−184.478	−0.0803748
$175$	−35.0339	−0.0151332
$176$	580.937	0.248806
$177$	−0.510468	−0.000216775 0
$178$	2139.15	0.900765
$179$	−2060.93	−0.860565	−0.430283	−	0.902694i	$-0.641586\pi$
$179$	−2060.93	−0.860565	−0.430283	+	0.902694i	$0.641586\pi$
$180$	157.669	0.0652888
$181$	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
$183$	760.877	0.307353
$184$	−2883.78	−1.15541
$185$	−1442.31	−0.573194
$186$	−1704.16	−0.671803
$187$	1800.04	0.703915
$188$	−818.125	−0.317383
$189$	37.8366	0.0145619
$190$	−677.913	−0.258847
$191$	1785.57	0.676438	0.338219	−	0.941067i	$-0.390176\pi$
$191$	1785.57	0.676438	0.338219	+	0.941067i	$0.390176\pi$
$192$	−1490.42	−0.560217
$193$	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
$195$	−489.206	−0.179655
$196$	1194.91	0.435463
$197$	3189.78	1.15362	0.576808	−	0.816880i	$-0.304298\pi$
$197$	3189.78	1.15362	0.576808	+	0.816880i	$0.304298\pi$
$198$	467.915	0.167946
$199$	−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
$201$	565.672	0.198505
$202$	−2152.49	−0.749745
$203$	40.6393	0.0140508
$204$	−771.683	−0.264846
$205$	−2464.02	−0.839484
$206$	−3692.15	−1.24876
$207$	−1063.99	−0.357260
$208$	772.734	0.257593
$209$	1567.76	0.518871
$210$	44.5722	0.0146465
$211$	−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
$213$	−2014.08	−0.647898
$214$	−3759.82	−1.20101
$215$	155.619	0.0493634
$216$	−658.610	−0.207466
$217$	375.417	0.117442
$218$	1909.57	0.593267
$219$	−872.389	−0.269181
$220$	−429.541	−0.131635
$221$	2394.32	0.728777
$222$	1835.00	0.554761
$223$	−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
$225$	225.000	0.0666667
$226$	−2034.83	−0.598915
$227$	−1943.66	−0.568305	−0.284152	−	0.958779i	$-0.591712\pi$
$227$	−1943.66	−0.568305	−0.284152	+	0.958779i	$0.591712\pi$
$228$	−672.103	−0.195224
$229$	−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
$231$	−103.079	−0.0293597
$232$	−707.396	−0.200185
$233$	−2799.88	−0.787237	−0.393619	−	0.919274i	$-0.628777\pi$
$233$	−2799.88	−0.787237	−0.393619	+	0.919274i	$0.628777\pi$
$234$	622.396	0.173877
$235$	−1167.49	−0.324081
$236$	−0.596187	−0.000164443 0
$237$	1051.78	0.288271
$238$	−218.150	−0.0594142
$239$	1280.17	0.346473	0.173236	−	0.984880i	$-0.444578\pi$
$239$	1280.17	0.346473	0.173236	+	0.984880i	$0.444578\pi$
$240$	−355.403	−0.0955882
$241$	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
$243$	−243.000	−0.0641500
$244$	888.645	0.233154
$245$	1705.18	0.444653
$246$	3134.87	0.812487
$247$	2085.35	0.537198
$248$	−6534.76	−1.67322
$249$	−3057.57	−0.778176
$250$	265.054	0.0670540
$251$	6716.64	1.68904	0.844522	−	0.535520i	$-0.179884\pi$
$251$	6716.64	1.68904	0.844522	+	0.535520i	$0.179884\pi$
$252$	44.1902	0.0110465
$253$	2898.66	0.720304
$254$	−191.426	−0.0472879
$255$	−1101.22	−0.270436
$256$	−4198.75	−1.02509
$257$	6324.65	1.53510	0.767550	−	0.640989i	$-0.221475\pi$
$257$	6324.65	1.53510	0.767550	+	0.640989i	$0.221475\pi$
$258$	−197.988	−0.0477759
$259$	−404.239	−0.0969813
$260$	−571.354	−0.136284
$261$	−261.000	−0.0618984
$262$	−95.8000	−0.0225899
$263$	−615.504	−0.144310	−0.0721551	−	0.997393i	$-0.522988\pi$
$263$	−615.504	−0.144310	−0.0721551	+	0.997393i	$0.522988\pi$
$264$	1794.26	0.418292
$265$	−33.7788	−0.00783025
$266$	−189.999	−0.0437955
$267$	3026.48	0.693699
$268$	660.660	0.150583
$269$	−5700.55	−1.29208	−0.646038	−	0.763305i	$-0.723575\pi$
$269$	−5700.55	−1.29208	−0.646038	+	0.763305i	$0.723575\pi$
$270$	−286.258	−0.0645227
$271$	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
$273$	−137.110	−0.0303966
$274$	−4163.07	−0.917883
$275$	−612.971	−0.134413
$276$	−1242.66	−0.271013
$277$	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
$279$	−2411.06	−0.517370
$280$	170.916	0.0364792
$281$	−1504.70	−0.319442	−0.159721	−	0.987162i	$-0.551059\pi$
$281$	−1504.70	−0.319442	−0.159721	+	0.987162i	$0.551059\pi$
$282$	1485.36	0.313659
$283$	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
$285$	−959.115	−0.199344
$286$	−1695.60	−0.350570
$287$	−690.592	−0.142036
$288$	−1304.13	−0.266828
$289$	476.715	0.0970313
$290$	−307.463	−0.0622581
$291$	1039.27	0.209357
$292$	−1018.88	−0.204197
$293$	−726.374	−0.144830	−0.0724151	−	0.997375i	$-0.523071\pi$
$293$	−726.374	−0.144830	−0.0724151	+	0.997375i	$0.523071\pi$
$294$	−2169.43	−0.430353
$295$	−0.850780	−0.000167913 0
$296$	7036.46	1.38171
$297$	662.008	0.129339
$298$	2935.15	0.570567
$299$	3855.65	0.745745
$300$	262.782	0.0505725
$301$	43.6155	0.00835201
$302$	2606.40	0.496627
$303$	−3045.35	−0.577395
$304$	1514.99	0.285824
$305$	1268.13	0.238075
$306$	1401.04	0.261739
$307$	−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
$309$	−5223.67	−0.961696
$310$	−2840.27	−0.520376
$311$	−1618.01	−0.295013	−0.147507	−	0.989061i	$-0.547125\pi$
$311$	−1618.01	−0.295013	−0.147507	+	0.989061i	$0.547125\pi$
$312$	2386.63	0.433066
$313$	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
$315$	63.0610	0.0112796
$316$	1228.39	0.218679
$317$	2250.61	0.398759	0.199380	−	0.979922i	$-0.436107\pi$
$317$	2250.61	0.398759	0.199380	+	0.979922i	$0.436107\pi$
$318$	42.9754	0.00757843
$319$	711.046	0.124799
$320$	−2484.03	−0.433942
$321$	−5319.41	−0.924924
$322$	−351.293	−0.0607975
$323$	4694.21	0.808646
$324$	−283.805	−0.0486634
$325$	−815.343	−0.139160
$326$	7639.02	1.29781
$327$	2701.67	0.456888
$328$	12020.9	2.02361
$329$	−327.215	−0.0548327
$330$	779.858	0.130090
$331$	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
$333$	2596.16	0.427234
$334$	5649.07	0.925459
$335$	942.786	0.153761
$336$	−99.6091	−0.0161730
$337$	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
$339$	−2878.89	−0.461238
$340$	−1286.14	−0.205149
$341$	6568.48	1.04312
$342$	1220.24	0.192933
$343$	958.578	0.150899
$344$	−759.201	−0.118992
$345$	−1773.32	−0.276732
$346$	−6005.40	−0.933099
$347$	−11163.3	−1.72703	−0.863515	−	0.504323i	$-0.831742\pi$
$347$	−11163.3	−1.72703	−0.863515	+	0.504323i	$0.831742\pi$
$348$	−304.827	−0.0469554
$349$	−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
$351$	880.570	0.133907
$352$	3552.85	0.537977
$353$	10103.1	1.52333	0.761663	−	0.647973i	$-0.224383\pi$
$353$	10103.1	1.52333	0.761663	+	0.647973i	$0.224383\pi$
$354$	1.08241	0.000162513 0
$355$	−3356.79	−0.501860
$356$	3534.69	0.526231
$357$	−308.640	−0.0457562
$358$	4370.06	0.645154
$359$	−8241.29	−1.21158	−0.605792	−	0.795623i	$-0.707144\pi$
$359$	−8241.29	−1.21158	−0.605792	+	0.795623i	$0.707144\pi$
$360$	−1097.68	−0.160703
$361$	−2770.55	−0.403929
$362$	−9190.75	−1.33441
$363$	2189.48	0.316579
$364$	−160.134	−0.0230585
$365$	−1453.98	−0.208507
$366$	−1613.39	−0.230419
$367$	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
$369$	4435.23	0.625715
$370$	3058.33	0.429716
$371$	−9.46722	−0.00132483
$372$	−2815.93	−0.392470
$373$	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
$375$	375.000	0.0516398
$376$	5695.73	0.781210
$377$	945.797	0.129207
$378$	−80.2300	−0.0109169
$379$	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
$381$	−270.830	−0.0364174
$382$	−3786.19	−0.507116
$383$	−8317.66	−1.10969	−0.554846	−	0.831953i	$-0.687223\pi$
$383$	−8317.66	−1.10969	−0.554846	+	0.831953i	$0.687223\pi$
$384$	−317.347	−0.0421732
$385$	−171.798	−0.0227419
$386$	−2330.95	−0.307364
$387$	−280.114	−0.0367933
$388$	1213.78	0.158816
$389$	12184.5	1.58813	0.794063	−	0.607835i	$-0.207962\pi$
$389$	12184.5	1.58813	0.794063	+	0.607835i	$0.207962\pi$
$390$	1037.33	0.134685
$391$	8679.20	1.12257
$392$	−8318.88	−1.07185
$393$	−135.538	−0.0173970
$394$	−6763.71	−0.864850
$395$	1752.96	0.223294
$396$	773.173	0.0981147
$397$	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
$399$	−268.812	−0.0337279
$400$	−592.338	−0.0740423
$401$	−12177.3	−1.51647	−0.758236	−	0.651980i	$-0.773938\pi$
$401$	−12177.3	−1.51647	−0.758236	+	0.651980i	$0.773938\pi$
$402$	−1199.47	−0.148816
$403$	8737.06	1.07996
$404$	−3556.73	−0.438005
$405$	−405.000	−0.0496904
$406$	−86.1729	−0.0105337
$407$	−7072.76	−0.861385
$408$	5372.40	0.651896
$409$	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
$411$	−5889.93	−0.706882
$412$	−6100.84	−0.729531
$413$	−0.238449	−2.84100e−5 0
$414$	2256.13	0.267833
$415$	−5095.95	−0.602772
$416$	4725.83	0.556978
$417$	−632.165	−0.0742380
$418$	−3324.33	−0.388991
$419$	−1416.16	−0.165117	−0.0825583	−	0.996586i	$-0.526309\pi$
$419$	−1416.16	−0.165117	−0.0825583	+	0.996586i	$0.526309\pi$
$420$	73.6503	0.00855658
$421$	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
$423$	2101.49	0.241556
$424$	164.793	0.0188751
$425$	−1835.37	−0.209479
$426$	4270.71	0.485720
$427$	355.420	0.0402809
$428$	−6212.65	−0.701635
$429$	−2398.95	−0.269982
$430$	−329.980	−0.0370070
$431$	8676.34	0.969662	0.484831	−	0.874608i	$-0.338881\pi$
$431$	8676.34	0.969662	0.484831	+	0.874608i	$0.338881\pi$
$432$	639.725	0.0712472
$433$	−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
$435$	−435.000	−0.0479463
$436$	3155.33	0.346589
$437$	7559.21	0.827474
$438$	1849.84	0.201801
$439$	−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
$441$	−3069.33	−0.331425
$442$	−5077.00	−0.546354
$443$	7340.11	0.787221	0.393611	−	0.919277i	$-0.371226\pi$
$443$	7340.11	0.787221	0.393611	+	0.919277i	$0.371226\pi$
$444$	3032.11	0.324094
$445$	5044.14	0.537337
$446$	6359.78	0.675211
$447$	4152.67	0.439406
$448$	−696.201	−0.0734206
$449$	−4998.65	−0.525392	−0.262696	−	0.964879i	$-0.584612\pi$
$449$	−4998.65	−0.525392	−0.262696	+	0.964879i	$0.584612\pi$
$450$	−477.097	−0.0499791
$451$	−12083.0	−1.26156
$452$	−3362.31	−0.349889
$453$	3687.54	0.382463
$454$	4121.40	0.426050
$455$	−228.517	−0.0235451
$456$	4679.13	0.480527
$457$	−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
$459$	1982.20	0.201571
$460$	−2071.10	−0.209925
$461$	8913.54	0.900532	0.450266	−	0.892895i	$-0.351329\pi$
$461$	8913.54	0.900532	0.450266	+	0.892895i	$0.351329\pi$
$462$	218.572	0.0220105
$463$	−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
$465$	−4018.43	−0.400753
$466$	5936.96	0.590181
$467$	7900.81	0.782882	0.391441	−	0.920203i	$-0.371977\pi$
$467$	7900.81	0.782882	0.391441	+	0.920203i	$0.371977\pi$
$468$	1028.44	0.101580
$469$	264.236	0.0260155
$470$	2475.59	0.242959
$471$	−3253.18	−0.318256
$472$	4.15061	0.000404761 0
$473$	763.119	0.0741823
$474$	−2230.22	−0.216113
$475$	−1598.53	−0.154411
$476$	−360.467	−0.0347101
$477$	60.8018	0.00583632
$478$	−2714.50	−0.259746
$479$	−20336.4	−1.93986	−0.969931	−	0.243381i	$-0.921744\pi$
$479$	−20336.4	−1.93986	−0.969931	+	0.243381i	$0.921744\pi$
$480$	−2173.55	−0.206684
$481$	−9407.83	−0.891809
$482$	−10531.1	−0.995182
$483$	−497.012	−0.0468216
$484$	2557.14	0.240152
$485$	1732.11	0.162167
$486$	515.265	0.0480924
$487$	−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
$489$	10807.7	0.999473
$490$	−3615.72	−0.333350
$491$	12022.9	1.10506	0.552529	−	0.833493i	$-0.313663\pi$
$491$	12022.9	1.10506	0.552529	+	0.833493i	$0.313663\pi$
$492$	5180.00	0.474659
$493$	2129.03	0.194496
$494$	−4421.85	−0.402730
$495$	1103.35	0.100185
$496$	6347.39	0.574609
$497$	−940.812	−0.0849119
$498$	6483.38	0.583388
$499$	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
$501$	7992.33	0.712717
$502$	−14242.2	−1.26625
$503$	−4957.94	−0.439491	−0.219745	−	0.975557i	$-0.570523\pi$
$503$	−4957.94	−0.439491	−0.219745	+	0.975557i	$0.570523\pi$
$504$	−307.649	−0.0271900
$505$	−5075.58	−0.447249
$506$	−6146.40	−0.540002
$507$	3400.04	0.297832
$508$	−316.308	−0.0276258
$509$	7023.72	0.611633	0.305816	−	0.952091i	$-0.401071\pi$
$509$	7023.72	0.611633	0.305816	+	0.952091i	$0.401071\pi$
$510$	2335.06	0.202742
$511$	−407.509	−0.0352781
$512$	8056.91	0.695446
$513$	1726.41	0.148582
$514$	−13411.0	−1.15084
$515$	−8706.12	−0.744927
$516$	−327.151	−0.0279109
$517$	−5725.12	−0.487022
$518$	857.161	0.0727056
$519$	−8496.47	−0.718600
$520$	3977.72	0.335451
$521$	−17727.1	−1.49067	−0.745333	−	0.666692i	$-0.767710\pi$
$521$	−17727.1	−1.49067	−0.745333	+	0.666692i	$0.767710\pi$
$522$	553.433	0.0464044
$523$	−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
$525$	105.102	0.00873716
$526$	1305.13	0.108187
$527$	19667.5	1.62567
$528$	−1742.81	−0.143648
$529$	1809.36	0.148710
$530$	71.6257	0.00587023
$531$	1.53140	0.000125155 0
$532$	−313.951	−0.0255856
$533$	−16072.1	−1.30612
$534$	−6417.45	−0.520057
$535$	−8865.68	−0.716443
$536$	−4599.47	−0.370647
$537$	6182.79	0.496848
$538$	12087.6	0.968652
$539$	8361.81	0.668216
$540$	−473.008	−0.0376945
$541$	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
$543$	−13003.1	−1.02766
$544$	10638.0	0.838421
$545$	4502.78	0.353904
$546$	290.733	0.0227879
$547$	−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
$549$	−2282.63	−0.177451
$550$	1299.76	0.100767
$551$	1854.29	0.143367
$552$	8651.33	0.667074
$553$	491.304	0.0377801
$554$	−4201.85	−0.322237
$555$	4326.94	0.330934
$556$	−738.319	−0.0563160
$557$	−23665.3	−1.80024	−0.900119	−	0.435644i	$-0.856521\pi$
$557$	−23665.3	−1.80024	−0.900119	+	0.435644i	$0.856521\pi$
$558$	5112.49	0.387865
$559$	1015.06	0.0768024
$560$	−166.015	−0.0125275
$561$	−5400.12	−0.406405
$562$	3190.62	0.239481
$563$	5688.95	0.425863	0.212931	−	0.977067i	$-0.431699\pi$
$563$	5688.95	0.425863	0.212931	+	0.977067i	$0.431699\pi$
$564$	2454.38	0.183241
$565$	−4798.14	−0.357273
$566$	−12301.8	−0.913574
$567$	−113.510	−0.00840734
$568$	16376.4	1.20975
$569$	−5181.22	−0.381736	−0.190868	−	0.981616i	$-0.561130\pi$
$569$	−5181.22	−0.381736	−0.190868	+	0.981616i	$0.561130\pi$
$570$	2033.74	0.149446
$571$	−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
$573$	−5356.72	−0.390542
$574$	1464.35	0.106483
$575$	−2955.54	−0.214356
$576$	4471.25	0.323441
$577$	−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
$579$	−3297.84	−0.236708
$580$	−508.046	−0.0363715
$581$	−1428.25	−0.101986
$582$	−2203.70	−0.156952
$583$	−165.643	−0.0117671
$584$	7093.38	0.502613
$585$	1467.62	0.103724
$586$	1540.23	0.108577
$587$	−10571.9	−0.743351	−0.371675	−	0.928363i	$-0.621217\pi$
$587$	−10571.9	−0.743351	−0.371675	+	0.928363i	$0.621217\pi$
$588$	−3584.73	−0.251415
$589$	17129.5	1.19832
$590$	1.80402	0.000125882 0
$591$	−9569.34	−0.666040
$592$	−6834.70	−0.474500
$593$	−6095.11	−0.422085	−0.211042	−	0.977477i	$-0.567686\pi$
$593$	−6095.11	−0.422085	−0.211042	+	0.977477i	$0.567686\pi$
$594$	−1403.74	−0.0969635
$595$	−514.400	−0.0354426
$596$	4849.99	0.333328
$597$	3309.71	0.226897
$598$	−8175.64	−0.559075
$599$	−6143.32	−0.419047	−0.209524	−	0.977804i	$-0.567191\pi$
$599$	−6143.32	−0.419047	−0.209524	+	0.977804i	$0.567191\pi$
$600$	−1829.47	−0.124480
$601$	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
$603$	−1697.02	−0.114607
$604$	4306.76	0.290132
$605$	3649.14	0.245221
$606$	6457.46	0.432865
$607$	−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
$609$	−121.918	−0.00811225
$610$	−2688.98	−0.178481
$611$	−7615.27	−0.504224
$612$	2315.05	0.152909
$613$	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
$615$	7392.05	0.484677
$616$	838.132	0.0548203
$617$	18677.5	1.21868	0.609340	−	0.792909i	$-0.291434\pi$
$617$	18677.5	1.21868	0.609340	+	0.792909i	$0.291434\pi$
$618$	11076.4	0.720971
$619$	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
$621$	3191.98	0.206264
$622$	3430.89	0.221167
$623$	1413.73	0.0909144
$624$	−2318.20	−0.148722
$625$	625.000	0.0400000
$626$	−11917.8	−0.760914
$627$	−4703.27	−0.299570
$628$	−3799.46	−0.241425
$629$	−21177.4	−1.34244
$630$	−133.717	−0.00845618
$631$	−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
$633$	14225.7	0.893240
$634$	−4772.26	−0.298944
$635$	−451.384	−0.0282088
$636$	71.0118	0.00442736
$637$	11122.5	0.691817
$638$	−1507.72	−0.0935602
$639$	6042.23	0.374064
$640$	−528.911	−0.0326672
$641$	3014.56	0.185753	0.0928767	−	0.995678i	$-0.470394\pi$
$641$	3014.56	0.185753	0.0928767	+	0.995678i	$0.470394\pi$
$642$	11279.4	0.693402
$643$	−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
$645$	−466.857	−0.0285000
$646$	−9953.75	−0.606231
$647$	−24199.3	−1.47044	−0.735218	−	0.677831i	$-0.762920\pi$
$647$	−24199.3	−1.47044	−0.735218	+	0.677831i	$0.762920\pi$
$648$	1975.83	0.119781
$649$	−4.17203	−0.000252336 0
$650$	1728.88	0.104326
$651$	−1126.25	−0.0678052
$652$	12622.6	0.758187
$653$	−8138.48	−0.487723	−0.243862	−	0.969810i	$-0.578414\pi$
$653$	−8138.48	−0.487723	−0.243862	+	0.969810i	$0.578414\pi$
$654$	−5728.70	−0.342523
$655$	−225.897	−0.0134756
$656$	−11676.2	−0.694940
$657$	2617.17	0.155412
$658$	693.838	0.0411073
$659$	−24406.8	−1.44272	−0.721360	−	0.692560i	$-0.756483\pi$
$659$	−24406.8	−1.44272	−0.721360	+	0.692560i	$0.756483\pi$
$660$	1288.62	0.0759993
$661$	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
$663$	−7182.97	−0.420760
$664$	24861.1	1.45301
$665$	−448.020	−0.0261255
$666$	−5504.99	−0.320291
$667$	3428.43	0.199024
$668$	9334.41	0.540658
$669$	8997.85	0.519995
$670$	−1999.12	−0.115272
$671$	6218.60	0.357774
$672$	−609.182	−0.0349698
$673$	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
$675$	−675.000	−0.0384900
$676$	3970.98	0.225932
$677$	24093.6	1.36779	0.683894	−	0.729582i	$-0.260285\pi$
$677$	24093.6	1.36779	0.683894	+	0.729582i	$0.260285\pi$
$678$	6104.48	0.345784
$679$	485.461	0.0274378
$680$	8954.01	0.504957
$681$	5830.98	0.328111
$682$	−13928.0	−0.782011
$683$	−30150.8	−1.68915	−0.844574	−	0.535438i	$-0.820146\pi$
$683$	−30150.8	−1.68915	−0.844574	+	0.535438i	$0.820146\pi$
$684$	2016.31	0.112713
$685$	−9816.55	−0.547549
$686$	−2032.60	−0.113127
$687$	11356.7	0.630691
$688$	737.432	0.0408639
$689$	−220.330	−0.0121828
$690$	3760.22	0.207462
$691$	−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
$693$	309.236	0.0169508
$694$	23671.1	1.29473
$695$	−1053.61	−0.0575045
$696$	2122.19	0.115577
$697$	−36179.0	−1.96611
$698$	7274.54	0.394478
$699$	8399.64	0.454512
$700$	122.750	0.00662790
$701$	35818.0	1.92985	0.964926	−	0.262521i	$-0.0845539\pi$
$701$	35818.0	1.92985	0.964926	+	0.262521i	$0.0845539\pi$
$702$	−1867.19	−0.100388
$703$	−18444.6	−0.989546
$704$	−12181.1	−0.652119
$705$	3502.48	0.187108
$706$	−21423.0	−1.14202
$707$	−1422.54	−0.0756720
$708$	1.78856	9.49409e−5 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
$711$	−3155.33	−0.166433
$712$	−24608.3	−1.29527
$713$	31671.0	1.66352
$714$	654.451	0.0343028
$715$	−3998.25	−0.209127
$716$	7221.01	0.376902
$717$	−3840.50	−0.200036
$718$	17475.1	0.908308
$719$	18782.0	0.974200	0.487100	−	0.873346i	$-0.338055\pi$
$719$	18782.0	0.974200	0.487100	+	0.873346i	$0.338055\pi$
$720$	1066.21	0.0551878
$721$	−2440.07	−0.126037
$722$	5874.76	0.302820
$723$	−14899.4	−0.766412
$724$	−15186.6	−0.779567
$725$	−725.000	−0.0371391
$726$	−4642.65	−0.237335
$727$	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
$729$	729.000	0.0370370
$730$	3083.07	0.156314
$731$	2284.94	0.115611
$732$	−2665.93	−0.134612
$733$	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
$735$	−5115.54	−0.256721
$736$	17130.7	0.857942
$737$	4623.20	0.231069
$738$	−9404.60	−0.469090
$739$	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
$741$	−6256.06	−0.310151
$742$	20.0746	0.000993210 0
$743$	29021.0	1.43295	0.716473	−	0.697615i	$-0.245755\pi$
$743$	29021.0	1.43295	0.716473	+	0.697615i	$0.245755\pi$
$744$	19604.3	0.966032
$745$	6921.12	0.340363
$746$	−580.441	−0.0284872
$747$	9172.72	0.449280
$748$	−6306.92	−0.308294
$749$	−2484.79	−0.121218
$750$	−795.162	−0.0387136
$751$	−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
$753$	−20149.9	−0.975170
$754$	−2005.50	−0.0968647
$755$	6145.90	0.296255
$756$	−132.570	−0.00637770
$757$	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
$759$	−8695.97	−0.415868
$760$	7798.55	0.372215
$761$	−31623.5	−1.50638	−0.753188	−	0.657805i	$-0.771485\pi$
$761$	−31623.5	−1.50638	−0.753188	+	0.657805i	$0.771485\pi$
$762$	574.277	0.0273017
$763$	1262.00	0.0598786
$764$	−6256.23	−0.296260
$765$	3303.66	0.156136
$766$	17637.0	0.831921
$767$	−5.54942	−0.000261249 0
$768$	12596.2	0.591833
$769$	−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
$771$	−18974.0	−0.886291
$772$	−3851.62	−0.179563
$773$	−29837.7	−1.38834	−0.694170	−	0.719811i	$-0.744228\pi$
$773$	−29837.7	−1.38834	−0.694170	+	0.719811i	$0.744228\pi$
$774$	593.963	0.0275834
$775$	−6697.39	−0.310422
$776$	−8450.27	−0.390911
$777$	1212.72	0.0559922
$778$	−25836.5	−1.19060
$779$	−31510.3	−1.44926
$780$	1714.06	0.0786836
$781$	−16460.9	−0.754185
$782$	−18403.7	−0.841578
$783$	783.000	0.0357371
$784$	8080.35	0.368092
$785$	−5421.97	−0.246520
$786$	287.400	0.0130423
$787$	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
$789$	1846.51	0.0833175
$790$	−3717.04	−0.167400
$791$	−1344.78	−0.0604487
$792$	−5382.78	−0.241501
$793$	8271.67	0.370411
$794$	−21825.6	−0.975517
$795$	101.336	0.00452080
$796$	3865.49	0.172121
$797$	25155.7	1.11802	0.559010	−	0.829161i	$-0.311181\pi$
$797$	25155.7	1.11802	0.559010	+	0.829161i	$0.311181\pi$
$798$	569.998	0.0252854
$799$	−17142.3	−0.759010
$800$	−3622.58	−0.160097
$801$	−9079.44	−0.400507
$802$	25821.1	1.13688
$803$	−7129.98	−0.313340
$804$	−1981.98	−0.0869391
$805$	−828.353	−0.0362678
$806$	−18526.4	−0.809631
$807$	17101.6	0.745981
$808$	24761.7	1.07811
$809$	−11402.5	−0.495537	−0.247769	−	0.968819i	$-0.579697\pi$
$809$	−11402.5	−0.495537	−0.247769	+	0.968819i	$0.579697\pi$
$810$	858.775	0.0372522
$811$	−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
$813$	−12659.9	−0.546129
$814$	14997.3	0.645769
$815$	18012.9	0.774189
$816$	−5218.36	−0.223871
$817$	1990.09	0.0852195
$818$	−21882.8	−0.935348
$819$	411.330	0.0175495
$820$	8633.33	0.367669
$821$	32017.0	1.36103	0.680513	−	0.732736i	$-0.261757\pi$
$821$	32017.0	1.36103	0.680513	+	0.732736i	$0.261757\pi$
$822$	12489.2	0.529940
$823$	−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
$825$	1838.91	0.0776032
$826$	0.505615	2.12986e−5 0
$827$	17534.8	0.737295	0.368648	−	0.929569i	$-0.379821\pi$
$827$	17534.8	0.737295	0.368648	+	0.929569i	$0.379821\pi$
$828$	3727.99	0.156469
$829$	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
$831$	−5944.79	−0.248162
$832$	−16202.7	−0.675152
$833$	25037.1	1.04140
$834$	1340.46	0.0556552
$835$	13320.6	0.552068
$836$	−5493.05	−0.227250
$837$	7233.18	0.298704
$838$	3002.87	0.123786
$839$	12398.1	0.510168	0.255084	−	0.966919i	$-0.417897\pi$
$839$	12398.1	0.510168	0.255084	+	0.966919i	$0.417897\pi$
$840$	−512.748	−0.0210613
$841$	841.000	0.0344828
$842$	−3753.31	−0.153620
$843$	4514.11	0.184430
$844$	16614.5	0.677601
$845$	5666.73	0.230700
$846$	−4456.07	−0.181091
$847$	1022.75	0.0414900
$848$	−160.068	−0.00648202
$849$	−17404.7	−0.703564
$850$	3891.77	0.157043
$851$	−34102.5	−1.37370
$852$	7056.85	0.283760
$853$	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
$855$	2877.35	0.115091
$856$	43252.0	1.72701
$857$	−1305.42	−0.0520328	−0.0260164	−	0.999662i	$-0.508282\pi$
$857$	−1305.42	−0.0520328	−0.0260164	+	0.999662i	$0.508282\pi$
$858$	5086.81	0.202402
$859$	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
$861$	2071.78	0.0820046
$862$	−18397.6	−0.726942
$863$	31555.2	1.24467	0.622336	−	0.782750i	$-0.286184\pi$
$863$	31555.2	1.24467	0.622336	+	0.782750i	$0.286184\pi$
$864$	3912.39	0.154053
$865$	−14160.8	−0.556625
$866$	21979.5	0.862465
$867$	−1430.14	−0.0560211
$868$	−1315.37	−0.0514362
$869$	8596.11	0.335562
$870$	922.388	0.0359447
$871$	6149.55	0.239230
$872$	−21967.2	−0.853100
$873$	−3117.80	−0.120872
$874$	−16028.8	−0.620346
$875$	175.169	0.00676778
$876$	3056.64	0.117893
$877$	−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
$879$	2179.12	0.0836178
$880$	−2904.69	−0.111269
$881$	−2301.79	−0.0880242	−0.0440121	−	0.999031i	$-0.514014\pi$
$881$	−2301.79	−0.0880242	−0.0440121	+	0.999031i	$0.514014\pi$
$882$	6508.30	0.248465
$883$	−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
$885$	2.55234	9.69446e−5 0
$886$	−15564.2	−0.590169
$887$	17351.4	0.656825	0.328412	−	0.944534i	$-0.393486\pi$
$887$	17351.4	0.656825	0.328412	+	0.944534i	$0.393486\pi$
$888$	−21109.4	−0.797730
$889$	−126.510	−0.00477278
$890$	−10695.8	−0.402834
$891$	−1986.02	−0.0746738
$892$	10508.8	0.394462
$893$	−14930.2	−0.559484
$894$	−8805.46	−0.329417
$895$	10304.7	0.384856
$896$	−148.238	−0.00552712
$897$	−11566.9	−0.430556
$898$	10599.3	0.393879
$899$	7768.97	0.288220
$900$	−788.347	−0.0291980
$901$	−495.972	−0.0183388
$902$	25621.1	0.945775
$903$	−130.846	−0.00482203
$904$	23408.2	0.861222
$905$	−21671.9	−0.796019
$906$	−7819.19	−0.286727
$907$	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
$909$	9136.05	0.333359
$910$	484.555	0.0176515
$911$	32678.5	1.18846	0.594230	−	0.804295i	$-0.297457\pi$
$911$	32678.5	1.18846	0.594230	+	0.804295i	$0.297457\pi$
$912$	−4544.96	−0.165021
$913$	−24989.4	−0.905835
$914$	5213.87	0.188687
$915$	−3804.39	−0.137453
$916$	13263.7	0.478434
$917$	−63.3125	−0.00228000
$918$	−4203.11	−0.151115
$919$	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
$921$	1215.89	0.0435014
$922$	−18900.6	−0.675116
$923$	−21895.5	−0.780822
$924$	361.164	0.0128587
$925$	7211.57	0.256340
$926$	9607.35	0.340947
$927$	15671.0	0.555236
$928$	4202.19	0.148646
$929$	45355.7	1.60180	0.800900	−	0.598798i	$-0.204355\pi$
$929$	45355.7	1.60180	0.800900	+	0.598798i	$0.204355\pi$
$930$	8520.81	0.300439
$931$	21806.2	0.767636
$932$	9810.12	0.344787
$933$	4854.04	0.170326
$934$	−16753.1	−0.586916
$935$	−9000.21	−0.314800
$936$	−7159.90	−0.250031
$937$	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
$939$	−16861.4	−0.585997
$940$	4090.63	0.141938
$941$	−22774.4	−0.788975	−0.394487	−	0.918901i	$-0.629078\pi$
$941$	−22774.4	−0.788975	−0.394487	+	0.918901i	$0.629078\pi$
$942$	6898.15	0.238592
$943$	−58260.0	−2.01188
$944$	−4.03160	−0.000139001 0
$945$	−189.183	−0.00651230
$946$	−1618.14	−0.0556135
$947$	42834.2	1.46983	0.734913	−	0.678162i	$-0.237223\pi$
$947$	42834.2	1.46983	0.734913	+	0.678162i	$0.237223\pi$
$948$	−3685.18	−0.126254
$949$	−9483.94	−0.324407
$950$	3389.57	0.115760
$951$	−6751.82	−0.230224
$952$	2509.55	0.0854359
$953$	−11113.4	−0.377754	−0.188877	−	0.982001i	$-0.560485\pi$
$953$	−11113.4	−0.377754	−0.188877	+	0.982001i	$0.560485\pi$
$954$	−128.926	−0.00437541
$955$	−8927.87	−0.302512
$956$	−4485.40	−0.151745
$957$	−2133.14	−0.0720528
$958$	43122.0	1.45429
$959$	−2751.29	−0.0926422
$960$	7452.08	0.250536
$961$	41977.0	1.40905
$962$	19948.7	0.668577
$963$	15958.2	0.534005
$964$	−17401.4	−0.581390
$965$	−5496.40	−0.183353
$966$	1053.88	0.0351015
$967$	−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
$969$	−14082.6	−0.466872
$970$	−3672.83	−0.121575
$971$	−37840.3	−1.25062	−0.625311	−	0.780375i	$-0.715028\pi$
$971$	−37840.3	−1.25062	−0.625311	+	0.780375i	$0.715028\pi$
$972$	851.415	0.0280958
$973$	−295.296	−0.00972945
$974$	25121.3	0.826426
$975$	2446.03	0.0803442
$976$	6009.29	0.197083
$977$	−40258.4	−1.31830	−0.659150	−	0.752011i	$-0.729084\pi$
$977$	−40258.4	−1.31830	−0.659150	+	0.752011i	$0.729084\pi$
$978$	−22917.1	−0.749291
$979$	24735.2	0.807500
$980$	−5974.55	−0.194745
$981$	−8105.00	−0.263785
$982$	−25493.7	−0.828447
$983$	13109.2	0.425348	0.212674	−	0.977123i	$-0.431783\pi$
$983$	13109.2	0.425348	0.212674	+	0.977123i	$0.431783\pi$
$984$	−36062.8	−1.16833
$985$	−15948.9	−0.515913
$986$	−4514.46	−0.145811
$987$	981.645	0.0316577
$988$	−7306.58	−0.235277
$989$	3679.51	0.118303
$990$	−2339.57	−0.0751076
$991$	−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
$993$	−16355.3	−0.522679
$994$	1994.93	0.0636573
$995$	5516.19	0.175754
$996$	10713.0	0.340818
$997$	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
$999$	−7788.49	−0.246664

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

)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	435.4.a.f.1.2	✓	5
3.2	odd	2		1305.4.a.g.1.4		5
5.4	even	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	1.1	even	1	trivial
1305.4.a.g.1.4		5	3.2	odd	2
2175.4.a.j.1.4		5	5.4	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more

Newspace parameters

Newform invariants

Embedding invariants

$q$ -expansion

Coefficient data

Twists

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	435.4.a.f.1.2

Level	$435$
Weight	$4$
Character	435.1
Self dual	yes
Analytic conductor	$25.666$
Analytic rank	$0$
Dimension	$5$
CM	no
Inner twists	$1$