LMFDB - Embedded newform 650.4.a.x.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [650,4,Mod(1,650)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("650.1"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(650, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 4, names="a")

Level:	$N$	$=$	$650 = 2 \cdot 5^{2} \cdot 13$
Weight:	$k$	$=$	$4$
Character orbit:	$[\chi]$	$=$	650.a (trivial)

Newform invariants

comment:select newform

sage:traces = [4,8,0,16,0,0,56,32,118] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(9)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$38.3512415037$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\mathbb{Q}[x]/(x^{4} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$x^{4} - 113x^{2} - 48x + 1600$ x^4 - 113x^2 - 48x + 1600
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$1$
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-4.39724$ of defining polynomial
Character	$\chi$	$=$	650.1

$q$ -expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+2.00000 q^{2} +4.39724 q^{3} +4.00000 q^{4} +8.79448 q^{6} -6.72037 q^{7} +8.00000 q^{8} -7.66426 q^{9} +38.0456 q^{11} +17.5890 q^{12} +13.0000 q^{13} -13.4407 q^{14} +16.0000 q^{16} +90.4746 q^{17} -15.3285 q^{18} +7.38883 q^{19} -29.5511 q^{21} +76.0912 q^{22} +20.8794 q^{23} +35.1779 q^{24} +26.0000 q^{26} -152.427 q^{27} -26.8815 q^{28} +128.946 q^{29} +168.038 q^{31} +32.0000 q^{32} +167.296 q^{33} +180.949 q^{34} -30.6570 q^{36} +107.718 q^{37} +14.7777 q^{38} +57.1641 q^{39} +261.857 q^{41} -59.1022 q^{42} +267.651 q^{43} +152.182 q^{44} +41.7588 q^{46} -24.1120 q^{47} +70.3559 q^{48} -297.837 q^{49} +397.839 q^{51} +52.0000 q^{52} -9.17911 q^{53} -304.854 q^{54} -53.7629 q^{56} +32.4905 q^{57} +257.893 q^{58} +162.708 q^{59} +3.89689 q^{61} +336.076 q^{62} +51.5067 q^{63} +64.0000 q^{64} +334.591 q^{66} -880.373 q^{67} +361.899 q^{68} +91.8117 q^{69} +313.477 q^{71} -61.3141 q^{72} +354.042 q^{73} +215.437 q^{74} +29.5553 q^{76} -255.680 q^{77} +114.328 q^{78} -1078.46 q^{79} -463.324 q^{81} +523.714 q^{82} +603.564 q^{83} -118.204 q^{84} +535.302 q^{86} +567.008 q^{87} +304.365 q^{88} -1037.13 q^{89} -87.3648 q^{91} +83.5175 q^{92} +738.904 q^{93} -48.2240 q^{94} +140.712 q^{96} -1151.19 q^{97} -595.673 q^{98} -291.591 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$4 q + 8 q^{2} + 16 q^{4} + 56 q^{7} + 32 q^{8} + 118 q^{9} - 49 q^{11} + 52 q^{13} + 112 q^{14} + 64 q^{16} + 27 q^{17} + 236 q^{18} + 51 q^{19} + 62 q^{21} - 98 q^{22} - 9 q^{23} + 104 q^{26} - 144 q^{27}+ \cdots - 6760 q^{99}+O(q^{100})$ 4 * q + 8 * q^2 + 16 * q^4 + 56 * q^7 + 32 * q^8 + 118 * q^9 - 49 * q^11 + 52 * q^13 + 112 * q^14 + 64 * q^16 + 27 * q^17 + 236 * q^18 + 51 * q^19 + 62 * q^21 - 98 * q^22 - 9 * q^23 + 104 * q^26 - 144 * q^27 + 224 * q^28 - 161 * q^29 + 38 * q^31 + 128 * q^32 + 575 * q^33 + 54 * q^34 + 472 * q^36 + 16 * q^37 + 102 * q^38 + 547 * q^41 + 124 * q^42 + 609 * q^43 - 196 * q^44 - 18 * q^46 - 254 * q^47 + 254 * q^49 - 785 * q^51 + 208 * q^52 + 961 * q^53 - 288 * q^54 + 448 * q^56 - 183 * q^57 - 322 * q^58 - 230 * q^59 + 1393 * q^61 + 76 * q^62 + 3422 * q^63 + 256 * q^64 + 1150 * q^66 + 579 * q^67 + 108 * q^68 + 3 * q^69 + 680 * q^71 + 944 * q^72 + 1321 * q^73 + 32 * q^74 + 204 * q^76 - 2258 * q^77 - 55 * q^79 + 3748 * q^81 + 1094 * q^82 + 2733 * q^83 + 248 * q^84 + 1218 * q^86 - 227 * q^87 - 392 * q^88 - 1233 * q^89 + 728 * q^91 - 36 * q^92 + 3304 * q^93 - 508 * q^94 + 1250 * q^97 + 508 * q^98 - 6760 * 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

)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$\alpha_n$			$\theta_n$
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$\alpha_p$			$\theta_p$

$2$	2.00000	0.707107
$2$	2.00000	0.707107
$3$	4.39724	0.846250	0.423125	−	0.906071i	$-0.360933\pi$
$3$	4.39724	0.846250	0.423125	+	0.906071i	$0.360933\pi$
$4$	4.00000	0.500000
$5$	0	0
$5$	0	0
$6$	8.79448	0.598389
$7$	−6.72037	−0.362866	−0.181433	−	0.983403i	$-0.558074\pi$
$7$	−6.72037	−0.362866	−0.181433	+	0.983403i	$0.558074\pi$
$8$	8.00000	0.353553
$9$	−7.66426	−0.283862
$10$	0	0
$11$	38.0456	1.04283	0.521417	−	0.853302i	$-0.325403\pi$
$11$	38.0456	1.04283	0.521417	+	0.853302i	$0.325403\pi$
$12$	17.5890	0.423125
$13$	13.0000	0.277350
$13$	13.0000	0.277350
$14$	−13.4407	−0.256585
$15$	0	0
$16$	16.0000	0.250000
$17$	90.4746	1.29078	0.645392	−	0.763852i	$-0.276694\pi$
$17$	90.4746	1.29078	0.645392	+	0.763852i	$0.276694\pi$
$18$	−15.3285	−0.200720
$19$	7.38883	0.0892166	0.0446083	−	0.999005i	$-0.485796\pi$
$19$	7.38883	0.0892166	0.0446083	+	0.999005i	$0.485796\pi$
$20$	0	0
$21$	−29.5511	−0.307075
$22$	76.0912	0.737395
$23$	20.8794	0.189289	0.0946446	−	0.995511i	$-0.469829\pi$
$23$	20.8794	0.189289	0.0946446	+	0.995511i	$0.469829\pi$
$24$	35.1779	0.299194
$25$	0	0
$26$	26.0000	0.196116
$27$	−152.427	−1.08647
$28$	−26.8815	−0.181433
$29$	128.946	0.825680	0.412840	−	0.910804i	$-0.364537\pi$
$29$	128.946	0.825680	0.412840	+	0.910804i	$0.364537\pi$
$30$	0	0
$31$	168.038	0.973566	0.486783	−	0.873523i	$-0.338170\pi$
$31$	168.038	0.973566	0.486783	+	0.873523i	$0.338170\pi$
$32$	32.0000	0.176777
$33$	167.296	0.882498
$34$	180.949	0.912722
$35$	0	0
$36$	−30.6570	−0.141931
$37$	107.718	0.478617	0.239308	−	0.970944i	$-0.423079\pi$
$37$	107.718	0.478617	0.239308	+	0.970944i	$0.423079\pi$
$38$	14.7777	0.0630856
$39$	57.1641	0.234707
$40$	0	0
$41$	261.857	0.997443	0.498722	−	0.866762i	$-0.333803\pi$
$41$	261.857	0.997443	0.498722	+	0.866762i	$0.333803\pi$
$42$	−59.1022	−0.217135
$43$	267.651	0.949218	0.474609	−	0.880197i	$-0.342589\pi$
$43$	267.651	0.949218	0.474609	+	0.880197i	$0.342589\pi$
$44$	152.182	0.521417
$45$	0	0
$46$	41.7588	0.133848
$47$	−24.1120	−0.0748318	−0.0374159	−	0.999300i	$-0.511913\pi$
$47$	−24.1120	−0.0748318	−0.0374159	+	0.999300i	$0.511913\pi$
$48$	70.3559	0.211562
$49$	−297.837	−0.868328
$50$	0	0
$51$	397.839	1.09233
$52$	52.0000	0.138675
$53$	−9.17911	−0.0237896	−0.0118948	−	0.999929i	$-0.503786\pi$
$53$	−9.17911	−0.0237896	−0.0118948	+	0.999929i	$0.503786\pi$
$54$	−304.854	−0.768248
$55$	0	0
$56$	−53.7629	−0.128292
$57$	32.4905	0.0754995
$58$	257.893	0.583844
$59$	162.708	0.359031	0.179515	−	0.983755i	$-0.442547\pi$
$59$	162.708	0.359031	0.179515	+	0.983755i	$0.442547\pi$
$60$	0	0
$61$	3.89689	0.00817943	0.00408971	−	0.999992i	$-0.498698\pi$
$61$	3.89689	0.00817943	0.00408971	+	0.999992i	$0.498698\pi$
$62$	336.076	0.688415
$63$	51.5067	0.103004
$64$	64.0000	0.125000
$65$	0	0
$66$	334.591	0.624021
$67$	−880.373	−1.60529	−0.802646	−	0.596455i	$-0.796575\pi$
$67$	−880.373	−1.60529	−0.802646	+	0.596455i	$0.796575\pi$
$68$	361.899	0.645392
$69$	91.8117	0.160186
$70$	0	0
$71$	313.477	0.523984	0.261992	−	0.965070i	$-0.415621\pi$
$71$	313.477	0.523984	0.261992	+	0.965070i	$0.415621\pi$
$72$	−61.3141	−0.100360
$73$	354.042	0.567636	0.283818	−	0.958878i	$-0.408399\pi$
$73$	354.042	0.567636	0.283818	+	0.958878i	$0.408399\pi$
$74$	215.437	0.338433
$75$	0	0
$76$	29.5553	0.0446083
$77$	−255.680	−0.378409
$78$	114.328	0.165963
$79$	−1078.46	−1.53590	−0.767951	−	0.640509i	$-0.778724\pi$
$79$	−1078.46	−1.53590	−0.767951	+	0.640509i	$0.778724\pi$
$80$	0	0
$81$	−463.324	−0.635561
$82$	523.714	0.705299
$83$	603.564	0.798190	0.399095	−	0.916909i	$-0.369324\pi$
$83$	603.564	0.798190	0.399095	+	0.916909i	$0.369324\pi$
$84$	−118.204	−0.153538
$85$	0	0
$86$	535.302	0.671199
$87$	567.008	0.698731
$88$	304.365	0.368698
$89$	−1037.13	−1.23524	−0.617618	−	0.786478i	$-0.711902\pi$
$89$	−1037.13	−1.23524	−0.617618	+	0.786478i	$0.711902\pi$
$90$	0	0
$91$	−87.3648	−0.100641
$92$	83.5175	0.0946446
$93$	738.904	0.823880
$94$	−48.2240	−0.0529141
$95$	0	0
$96$	140.712	0.149597
$97$	−1151.19	−1.20500	−0.602502	−	0.798117i	$-0.705829\pi$
$97$	−1151.19	−1.20500	−0.602502	+	0.798117i	$0.705829\pi$
$98$	−595.673	−0.614001
$99$	−291.591	−0.296021
$100$	0	0
$101$	−647.241	−0.637652	−0.318826	−	0.947813i	$-0.603289\pi$
$101$	−647.241	−0.637652	−0.318826	+	0.947813i	$0.603289\pi$
$102$	795.678	0.772391
$103$	1560.60	1.49292	0.746459	−	0.665431i	$-0.231752\pi$
$103$	1560.60	1.49292	0.746459	+	0.665431i	$0.231752\pi$
$104$	104.000	0.0980581
$105$	0	0
$106$	−18.3582	−0.0168218
$107$	624.707	0.564418	0.282209	−	0.959353i	$-0.408933\pi$
$107$	624.707	0.564418	0.282209	+	0.959353i	$0.408933\pi$
$108$	−609.709	−0.543234
$109$	−1586.72	−1.39432	−0.697159	−	0.716917i	$-0.745553\pi$
$109$	−1586.72	−1.39432	−0.697159	+	0.716917i	$0.745553\pi$
$110$	0	0
$111$	473.664	0.405029
$112$	−107.526	−0.0907165
$113$	1369.14	1.13980	0.569902	−	0.821712i	$-0.306981\pi$
$113$	1369.14	1.13980	0.569902	+	0.821712i	$0.306981\pi$
$114$	64.9810	0.0533862
$115$	0	0
$116$	515.785	0.412840
$117$	−99.6354	−0.0787290
$118$	325.416	0.253873
$119$	−608.023	−0.468381
$120$	0	0
$121$	116.468	0.0875042
$122$	7.79377	0.00578373
$123$	1151.45	0.844086
$124$	672.153	0.486783
$125$	0	0
$126$	103.013	0.0728346
$127$	1021.67	0.713844	0.356922	−	0.934134i	$-0.383826\pi$
$127$	1021.67	0.713844	0.356922	+	0.934134i	$0.383826\pi$
$128$	128.000	0.0883883
$129$	1176.93	0.803276
$130$	0	0
$131$	−1640.33	−1.09402	−0.547009	−	0.837127i	$-0.684234\pi$
$131$	−1640.33	−1.09402	−0.547009	+	0.837127i	$0.684234\pi$
$132$	669.183	0.441249
$133$	−49.6557	−0.0323736
$134$	−1760.75	−1.13511
$135$	0	0
$136$	723.797	0.456361
$137$	759.298	0.473513	0.236756	−	0.971569i	$-0.423916\pi$
$137$	759.298	0.473513	0.236756	+	0.971569i	$0.423916\pi$
$138$	183.623	0.113269
$139$	−1804.88	−1.10135	−0.550676	−	0.834719i	$-0.685630\pi$
$139$	−1804.88	−1.10135	−0.550676	+	0.834719i	$0.685630\pi$
$140$	0	0
$141$	−106.026	−0.0633264
$142$	626.953	0.370512
$143$	494.593	0.289230
$144$	−122.628	−0.0709654
$145$	0	0
$146$	708.083	0.401379
$147$	−1309.66	−0.734823
$148$	430.874	0.239308
$149$	587.018	0.322754	0.161377	−	0.986893i	$-0.448406\pi$
$149$	587.018	0.322754	0.161377	+	0.986893i	$0.448406\pi$
$150$	0	0
$151$	−1290.83	−0.695668	−0.347834	−	0.937556i	$-0.613083\pi$
$151$	−1290.83	−0.695668	−0.347834	+	0.937556i	$0.613083\pi$
$152$	59.1107	0.0315428
$153$	−693.421	−0.366404
$154$	−511.361	−0.267576
$155$	0	0
$156$	228.657	0.117354
$157$	1597.77	0.812203	0.406102	−	0.913828i	$-0.366888\pi$
$157$	1597.77	0.812203	0.406102	+	0.913828i	$0.366888\pi$
$158$	−2156.92	−1.08605
$159$	−40.3627	−0.0201319
$160$	0	0
$161$	−140.317	−0.0686866
$162$	−926.648	−0.449410
$163$	−1156.21	−0.555589	−0.277795	−	0.960641i	$-0.589603\pi$
$163$	−1156.21	−0.555589	−0.277795	+	0.960641i	$0.589603\pi$
$164$	1047.43	0.498722
$165$	0	0
$166$	1207.13	0.564406
$167$	287.282	0.133117	0.0665585	−	0.997783i	$-0.478798\pi$
$167$	287.282	0.133117	0.0665585	+	0.997783i	$0.478798\pi$
$168$	−236.409	−0.108567
$169$	169.000	0.0769231
$170$	0	0
$171$	−56.6300	−0.0253252
$172$	1070.60	0.474609
$173$	−1757.66	−0.772440	−0.386220	−	0.922407i	$-0.626220\pi$
$173$	−1757.66	−0.772440	−0.386220	+	0.922407i	$0.626220\pi$
$174$	1134.02	0.494078
$175$	0	0
$176$	608.730	0.260709
$177$	715.468	0.303830
$178$	−2074.27	−0.873443
$179$	394.009	0.164523	0.0822615	−	0.996611i	$-0.473786\pi$
$179$	394.009	0.164523	0.0822615	+	0.996611i	$0.473786\pi$
$180$	0	0
$181$	−388.067	−0.159364	−0.0796818	−	0.996820i	$-0.525390\pi$
$181$	−388.067	−0.159364	−0.0796818	+	0.996820i	$0.525390\pi$
$182$	−174.730	−0.0711638
$183$	17.1355	0.00692184
$184$	167.035	0.0669238
$185$	0	0
$186$	1477.81	0.582571
$187$	3442.16	1.34607
$188$	−96.4480	−0.0374159
$189$	1024.37	0.394242
$190$	0	0
$191$	2703.44	1.02416	0.512078	−	0.858939i	$-0.328876\pi$
$191$	2703.44	1.02416	0.512078	+	0.858939i	$0.328876\pi$
$192$	281.423	0.105781
$193$	−593.948	−0.221520	−0.110760	−	0.993847i	$-0.535328\pi$
$193$	−593.948	−0.221520	−0.110760	+	0.993847i	$0.535328\pi$
$194$	−2302.37	−0.852066
$195$	0	0
$196$	−1191.35	−0.434164
$197$	−1826.01	−0.660396	−0.330198	−	0.943912i	$-0.607116\pi$
$197$	−1826.01	−0.660396	−0.330198	+	0.943912i	$0.607116\pi$
$198$	−583.183	−0.209318
$199$	−4087.46	−1.45604	−0.728021	−	0.685555i	$-0.759560\pi$
$199$	−4087.46	−1.45604	−0.728021	+	0.685555i	$0.759560\pi$
$200$	0	0
$201$	−3871.21	−1.35848
$202$	−1294.48	−0.450888
$203$	−866.566	−0.299611
$204$	1591.36	0.546163
$205$	0	0
$206$	3121.20	1.05565
$207$	−160.025	−0.0537319
$208$	208.000	0.0693375
$209$	281.113	0.0930381
$210$	0	0
$211$	−1393.75	−0.454739	−0.227369	−	0.973809i	$-0.573012\pi$
$211$	−1393.75	−0.454739	−0.227369	+	0.973809i	$0.573012\pi$
$212$	−36.7164	−0.0118948
$213$	1378.43	0.443421
$214$	1249.41	0.399104
$215$	0	0
$216$	−1219.42	−0.384124
$217$	−1129.28	−0.353274
$218$	−3173.45	−0.985931
$219$	1556.81	0.480362
$220$	0	0
$221$	1176.17	0.357999
$222$	947.329	0.286399
$223$	3929.04	1.17986	0.589929	−	0.807455i	$-0.299156\pi$
$223$	3929.04	1.17986	0.589929	+	0.807455i	$0.299156\pi$
$224$	−215.052	−0.0641462
$225$	0	0
$226$	2738.28	0.805963
$227$	759.062	0.221941	0.110971	−	0.993824i	$-0.464604\pi$
$227$	759.062	0.221941	0.110971	+	0.993824i	$0.464604\pi$
$228$	129.962	0.0377497
$229$	4736.20	1.36671	0.683356	−	0.730085i	$-0.260520\pi$
$229$	4736.20	1.36671	0.683356	+	0.730085i	$0.260520\pi$
$230$	0	0
$231$	−1124.29	−0.320229
$232$	1031.57	0.291922
$233$	−1269.96	−0.357074	−0.178537	−	0.983933i	$-0.557136\pi$
$233$	−1269.96	−0.357074	−0.178537	+	0.983933i	$0.557136\pi$
$234$	−199.271	−0.0556698
$235$	0	0
$236$	650.833	0.179515
$237$	−4742.25	−1.29976
$238$	−1216.05	−0.331196
$239$	−3394.38	−0.918678	−0.459339	−	0.888261i	$-0.651914\pi$
$239$	−3394.38	−0.918678	−0.459339	+	0.888261i	$0.651914\pi$
$240$	0	0
$241$	−878.126	−0.234710	−0.117355	−	0.993090i	$-0.537442\pi$
$241$	−878.126	−0.234710	−0.117355	+	0.993090i	$0.537442\pi$
$242$	232.936	0.0618748
$243$	2078.19	0.548624
$244$	15.5875	0.00408971
$245$	0	0
$246$	2302.90	0.596859
$247$	96.0548	0.0247442
$248$	1344.31	0.344207
$249$	2654.02	0.675468
$250$	0	0
$251$	205.197	0.0516013	0.0258006	−	0.999667i	$-0.491786\pi$
$251$	205.197	0.0516013	0.0258006	+	0.999667i	$0.491786\pi$
$252$	206.027	0.0515018
$253$	794.369	0.197397
$254$	2043.33	0.504764
$255$	0	0
$256$	256.000	0.0625000
$257$	−6172.50	−1.49817	−0.749086	−	0.662473i	$-0.769507\pi$
$257$	−6172.50	−1.49817	−0.749086	+	0.662473i	$0.769507\pi$
$258$	2353.85	0.568002
$259$	−723.908	−0.173674
$260$	0	0
$261$	−988.278	−0.234379
$262$	−3280.66	−0.773587
$263$	−1575.03	−0.369280	−0.184640	−	0.982806i	$-0.559112\pi$
$263$	−1575.03	−0.369280	−0.184640	+	0.982806i	$0.559112\pi$
$264$	1338.37	0.312010
$265$	0	0
$266$	−99.3114	−0.0228916
$267$	−4560.53	−1.04532
$268$	−3521.49	−0.802646
$269$	4430.34	1.00417	0.502087	−	0.864817i	$-0.332566\pi$
$269$	4430.34	1.00417	0.502087	+	0.864817i	$0.332566\pi$
$270$	0	0
$271$	−5850.80	−1.31148	−0.655740	−	0.754987i	$-0.727643\pi$
$271$	−5850.80	−1.31148	−0.655740	+	0.754987i	$0.727643\pi$
$272$	1447.59	0.322696
$273$	−384.164	−0.0851673
$274$	1518.60	0.334824
$275$	0	0
$276$	367.247	0.0800930
$277$	1058.99	0.229705	0.114853	−	0.993383i	$-0.463360\pi$
$277$	1058.99	0.229705	0.114853	+	0.993383i	$0.463360\pi$
$278$	−3609.76	−0.778773
$279$	−1287.89	−0.276358
$280$	0	0
$281$	−8935.89	−1.89705	−0.948524	−	0.316707i	$-0.897423\pi$
$281$	−8935.89	−1.89705	−0.948524	+	0.316707i	$0.897423\pi$
$282$	−212.053	−0.0447785
$283$	−6974.55	−1.46500	−0.732498	−	0.680769i	$-0.761646\pi$
$283$	−6974.55	−1.46500	−0.732498	+	0.680769i	$0.761646\pi$
$284$	1253.91	0.261992
$285$	0	0
$286$	989.186	0.204517
$287$	−1759.77	−0.361938
$288$	−245.256	−0.0501801
$289$	3272.66	0.666122
$290$	0	0
$291$	−5062.05	−1.01973
$292$	1416.17	0.283818
$293$	−7752.72	−1.54580	−0.772899	−	0.634529i	$-0.781194\pi$
$293$	−7752.72	−1.54580	−0.772899	+	0.634529i	$0.781194\pi$
$294$	−2619.32	−0.519598
$295$	0	0
$296$	861.748	0.169216
$297$	−5799.18	−1.13301
$298$	1174.04	0.228222
$299$	271.432	0.0524994
$300$	0	0
$301$	−1798.71	−0.344439
$302$	−2581.65	−0.491912
$303$	−2846.07	−0.539613
$304$	118.221	0.0223041
$305$	0	0
$306$	−1386.84	−0.259087
$307$	946.266	0.175916	0.0879581	−	0.996124i	$-0.471966\pi$
$307$	946.266	0.175916	0.0879581	+	0.996124i	$0.471966\pi$
$308$	−1022.72	−0.189205
$309$	6862.34	1.26338
$310$	0	0
$311$	4976.16	0.907306	0.453653	−	0.891178i	$-0.350121\pi$
$311$	4976.16	0.907306	0.453653	+	0.891178i	$0.350121\pi$
$312$	457.313	0.0829816
$313$	8053.10	1.45428	0.727138	−	0.686492i	$-0.240850\pi$
$313$	8053.10	1.45428	0.727138	+	0.686492i	$0.240850\pi$
$314$	3195.54	0.574315
$315$	0	0
$316$	−4313.84	−0.767951
$317$	5169.13	0.915859	0.457929	−	0.888989i	$-0.348591\pi$
$317$	5169.13	0.915859	0.457929	+	0.888989i	$0.348591\pi$
$318$	−80.7255	−0.0142354
$319$	4905.84	0.861048
$320$	0	0
$321$	2746.99	0.477638
$322$	−280.634	−0.0485687
$323$	668.502	0.115159
$324$	−1853.30	−0.317781
$325$	0	0
$326$	−2312.41	−0.392861
$327$	−6977.21	−1.17994
$328$	2094.86	0.352649
$329$	162.042	0.0271539
$330$	0	0
$331$	−5221.40	−0.867052	−0.433526	−	0.901141i	$-0.642731\pi$
$331$	−5221.40	−0.867052	−0.433526	+	0.901141i	$0.642731\pi$
$332$	2414.26	0.399095
$333$	−825.583	−0.135861
$334$	574.564	0.0941280
$335$	0	0
$336$	−472.817	−0.0767688
$337$	−7493.10	−1.21120	−0.605602	−	0.795768i	$-0.707067\pi$
$337$	−7493.10	−1.21120	−0.605602	+	0.795768i	$0.707067\pi$
$338$	338.000	0.0543928
$339$	6020.44	0.964559
$340$	0	0
$341$	6393.11	1.01527
$342$	−113.260	−0.0179076
$343$	4306.66	0.677952
$344$	2141.21	0.335599
$345$	0	0
$346$	−3515.31	−0.546198
$347$	3169.63	0.490359	0.245179	−	0.969478i	$-0.421153\pi$
$347$	3169.63	0.490359	0.245179	+	0.969478i	$0.421153\pi$
$348$	2268.03	0.349366
$349$	485.549	0.0744723	0.0372362	−	0.999306i	$-0.488145\pi$
$349$	485.549	0.0744723	0.0372362	+	0.999306i	$0.488145\pi$
$350$	0	0
$351$	−1981.55	−0.301332
$352$	1217.46	0.184349
$353$	−9984.58	−1.50545	−0.752727	−	0.658333i	$-0.771262\pi$
$353$	−9984.58	−1.50545	−0.752727	+	0.658333i	$0.771262\pi$
$354$	1430.94	0.214840
$355$	0	0
$356$	−4148.53	−0.617618
$357$	−2673.62	−0.396367
$358$	788.018	0.116335
$359$	−3284.46	−0.482861	−0.241431	−	0.970418i	$-0.577617\pi$
$359$	−3284.46	−0.482861	−0.241431	+	0.970418i	$0.577617\pi$
$360$	0	0
$361$	−6804.41	−0.992040
$362$	−776.134	−0.112687
$363$	512.138	0.0740504
$364$	−349.459	−0.0503204
$365$	0	0
$366$	34.2711	0.00489448
$367$	5866.77	0.834449	0.417224	−	0.908803i	$-0.363003\pi$
$367$	5866.77	0.834449	0.417224	+	0.908803i	$0.363003\pi$
$368$	334.070	0.0473223
$369$	−2006.94	−0.283136
$370$	0	0
$371$	61.6870	0.00863242
$372$	2955.62	0.411940
$373$	8887.99	1.23379	0.616894	−	0.787046i	$-0.288391\pi$
$373$	8887.99	1.23379	0.616894	+	0.787046i	$0.288391\pi$
$374$	6884.32	0.951818
$375$	0	0
$376$	−192.896	−0.0264570
$377$	1676.30	0.229002
$378$	2048.73	0.278771
$379$	−7715.51	−1.04570	−0.522849	−	0.852426i	$-0.675131\pi$
$379$	−7715.51	−1.04570	−0.522849	+	0.852426i	$0.675131\pi$
$380$	0	0
$381$	4492.51	0.604090
$382$	5406.88	0.724188
$383$	−5610.84	−0.748566	−0.374283	−	0.927315i	$-0.622111\pi$
$383$	−5610.84	−0.748566	−0.374283	+	0.927315i	$0.622111\pi$
$384$	562.847	0.0747986
$385$	0	0
$386$	−1187.90	−0.156638
$387$	−2051.35	−0.269447
$388$	−4604.75	−0.602502
$389$	−14287.7	−1.86225	−0.931123	−	0.364704i	$-0.881170\pi$
$389$	−14287.7	−1.86225	−0.931123	+	0.364704i	$0.881170\pi$
$390$	0	0
$391$	1889.05	0.244331
$392$	−2382.69	−0.307000
$393$	−7212.93	−0.925812
$394$	−3652.03	−0.466971
$395$	0	0
$396$	−1166.37	−0.148010
$397$	12153.5	1.53643	0.768217	−	0.640189i	$-0.221144\pi$
$397$	12153.5	1.53643	0.768217	+	0.640189i	$0.221144\pi$
$398$	−8174.92	−1.02958
$399$	−218.348	−0.0273962
$400$	0	0
$401$	3184.20	0.396537	0.198268	−	0.980148i	$-0.436468\pi$
$401$	3184.20	0.396537	0.198268	+	0.980148i	$0.436468\pi$
$402$	−7742.42	−0.960589
$403$	2184.50	0.270019
$404$	−2588.96	−0.318826
$405$	0	0
$406$	−1733.13	−0.211857
$407$	4098.22	0.499118
$408$	3182.71	0.386195
$409$	−2538.13	−0.306851	−0.153426	−	0.988160i	$-0.549031\pi$
$409$	−2538.13	−0.306851	−0.153426	+	0.988160i	$0.549031\pi$
$410$	0	0
$411$	3338.82	0.400710
$412$	6242.40	0.746459
$413$	−1093.46	−0.130280
$414$	−320.050	−0.0379942
$415$	0	0
$416$	416.000	0.0490290
$417$	−7936.49	−0.932018
$418$	562.225	0.0657879
$419$	−5348.26	−0.623579	−0.311789	−	0.950151i	$-0.600928\pi$
$419$	−5348.26	−0.623579	−0.311789	+	0.950151i	$0.600928\pi$
$420$	0	0
$421$	3021.69	0.349806	0.174903	−	0.984586i	$-0.444039\pi$
$421$	3021.69	0.349806	0.174903	+	0.984586i	$0.444039\pi$
$422$	−2787.51	−0.321549
$423$	184.801	0.0212419
$424$	−73.4328	−0.00841088
$425$	0	0
$426$	2756.87	0.313546
$427$	−26.1885	−0.00296803
$428$	2498.83	0.282209
$429$	2174.84	0.244761
$430$	0	0
$431$	−1505.69	−0.168275	−0.0841376	−	0.996454i	$-0.526814\pi$
$431$	−1505.69	−0.168275	−0.0841376	+	0.996454i	$0.526814\pi$
$432$	−2438.83	−0.271617
$433$	−9174.06	−1.01819	−0.509096	−	0.860710i	$-0.670020\pi$
$433$	−9174.06	−1.01819	−0.509096	+	0.860710i	$0.670020\pi$
$434$	−2258.56	−0.249802
$435$	0	0
$436$	−6346.90	−0.697159
$437$	154.274	0.0168877
$438$	3113.61	0.339667
$439$	11316.2	1.23028	0.615142	−	0.788416i	$-0.289099\pi$
$439$	11316.2	1.23028	0.615142	+	0.788416i	$0.289099\pi$
$440$	0	0
$441$	2282.70	0.246485
$442$	2352.34	0.253143
$443$	9257.54	0.992865	0.496432	−	0.868075i	$-0.334643\pi$
$443$	9257.54	0.992865	0.496432	+	0.868075i	$0.334643\pi$
$444$	1894.66	0.202515
$445$	0	0
$446$	7858.09	0.834285
$447$	2581.26	0.273131
$448$	−430.104	−0.0453582
$449$	13793.9	1.44983	0.724916	−	0.688837i	$-0.241879\pi$
$449$	13793.9	1.44983	0.724916	+	0.688837i	$0.241879\pi$
$450$	0	0
$451$	9962.50	1.04017
$452$	5476.56	0.569902
$453$	−5676.07	−0.588709
$454$	1518.12	0.156936
$455$	0	0
$456$	259.924	0.0266931
$457$	−8892.09	−0.910184	−0.455092	−	0.890444i	$-0.650394\pi$
$457$	−8892.09	−0.910184	−0.455092	+	0.890444i	$0.650394\pi$
$458$	9472.40	0.966411
$459$	−13790.8	−1.40239
$460$	0	0
$461$	7726.98	0.780654	0.390327	−	0.920676i	$-0.372362\pi$
$461$	7726.98	0.780654	0.390327	+	0.920676i	$0.372362\pi$
$462$	−2248.58	−0.226436
$463$	−17674.5	−1.77409	−0.887046	−	0.461681i	$-0.847246\pi$
$463$	−17674.5	−1.77409	−0.887046	+	0.461681i	$0.847246\pi$
$464$	2063.14	0.206420
$465$	0	0
$466$	−2539.93	−0.252489
$467$	12101.8	1.19915	0.599574	−	0.800319i	$-0.295337\pi$
$467$	12101.8	1.19915	0.599574	+	0.800319i	$0.295337\pi$
$468$	−398.542	−0.0393645
$469$	5916.43	0.582506
$470$	0	0
$471$	7025.78	0.687327
$472$	1301.67	0.126937
$473$	10182.9	0.989878
$474$	−9484.50	−0.919067
$475$	0	0
$476$	−2432.09	−0.234191
$477$	70.3511	0.00675294
$478$	−6788.75	−0.649603
$479$	−20301.3	−1.93651	−0.968255	−	0.249965i	$-0.919581\pi$
$479$	−20301.3	−1.93651	−0.968255	+	0.249965i	$0.919581\pi$
$480$	0	0
$481$	1400.34	0.132744
$482$	−1756.25	−0.165965
$483$	−617.008	−0.0581260
$484$	465.872	0.0437521
$485$	0	0
$486$	4156.37	0.387936
$487$	6287.98	0.585084	0.292542	−	0.956253i	$-0.405499\pi$
$487$	6287.98	0.585084	0.292542	+	0.956253i	$0.405499\pi$
$488$	31.1751	0.00289186
$489$	−5084.12	−0.470167
$490$	0	0
$491$	3476.20	0.319509	0.159754	−	0.987157i	$-0.448930\pi$
$491$	3476.20	0.319509	0.159754	+	0.987157i	$0.448930\pi$
$492$	4605.79	0.422043
$493$	11666.4	1.06577
$494$	192.110	0.0174968
$495$	0	0
$496$	2688.61	0.243391
$497$	−2106.68	−0.190136
$498$	5308.04	0.477628
$499$	8775.84	0.787296	0.393648	−	0.919261i	$-0.371213\pi$
$499$	8775.84	0.787296	0.393648	+	0.919261i	$0.371213\pi$
$500$	0	0
$501$	1263.25	0.112650
$502$	410.394	0.0364876
$503$	−9463.56	−0.838885	−0.419443	−	0.907782i	$-0.637774\pi$
$503$	−9463.56	−0.838885	−0.419443	+	0.907782i	$0.637774\pi$
$504$	412.053	0.0364173
$505$	0	0
$506$	1588.74	0.139581
$507$	743.134	0.0650961
$508$	4086.66	0.356922
$509$	8430.23	0.734113	0.367056	−	0.930199i	$-0.380366\pi$
$509$	8430.23	0.734113	0.367056	+	0.930199i	$0.380366\pi$
$510$	0	0
$511$	−2379.29	−0.205976
$512$	512.000	0.0441942
$513$	−1126.26	−0.0969309
$514$	−12345.0	−1.05937
$515$	0	0
$516$	4707.70	0.401638
$517$	−917.355	−0.0780372
$518$	−1447.82	−0.122806
$519$	−7728.84	−0.653677
$520$	0	0
$521$	13693.3	1.15147	0.575735	−	0.817636i	$-0.304716\pi$
$521$	13693.3	1.15147	0.575735	+	0.817636i	$0.304716\pi$
$522$	−1976.56	−0.165731
$523$	6910.07	0.577737	0.288868	−	0.957369i	$-0.406721\pi$
$523$	6910.07	0.577737	0.288868	+	0.957369i	$0.406721\pi$
$524$	−6561.32	−0.547009
$525$	0	0
$526$	−3150.06	−0.261120
$527$	15203.2	1.25666
$528$	2676.73	0.220625
$529$	−11731.1	−0.964170
$530$	0	0
$531$	−1247.04	−0.101915
$532$	−198.623	−0.0161868
$533$	3404.14	0.276641
$534$	−9121.06	−0.739151
$535$	0	0
$536$	−7042.98	−0.567557
$537$	1732.55	0.139227
$538$	8860.69	0.710058
$539$	−11331.4	−0.905523
$540$	0	0
$541$	−10623.4	−0.844240	−0.422120	−	0.906540i	$-0.638714\pi$
$541$	−10623.4	−0.844240	−0.422120	+	0.906540i	$0.638714\pi$
$542$	−11701.6	−0.927356
$543$	−1706.43	−0.134861
$544$	2895.19	0.228180
$545$	0	0
$546$	−768.328	−0.0602224
$547$	23615.7	1.84595	0.922975	−	0.384859i	$-0.125750\pi$
$547$	23615.7	1.84595	0.922975	+	0.384859i	$0.125750\pi$
$548$	3037.19	0.236756
$549$	−29.8668	−0.00232182
$550$	0	0
$551$	952.762	0.0736643
$552$	734.493	0.0566343
$553$	7247.65	0.557326
$554$	2117.97	0.162426
$555$	0	0
$556$	−7219.52	−0.550676
$557$	19374.0	1.47379	0.736895	−	0.676007i	$-0.236291\pi$
$557$	19374.0	1.47379	0.736895	+	0.676007i	$0.236291\pi$
$558$	−2575.78	−0.195415
$559$	3479.46	0.263266
$560$	0	0
$561$	15136.0	1.13911
$562$	−17871.8	−1.34141
$563$	−8112.11	−0.607255	−0.303628	−	0.952791i	$-0.598198\pi$
$563$	−8112.11	−0.607255	−0.303628	+	0.952791i	$0.598198\pi$
$564$	−424.105	−0.0316632
$565$	0	0
$566$	−13949.1	−1.03591
$567$	3113.71	0.230623
$568$	2507.81	0.185256
$569$	12071.0	0.889358	0.444679	−	0.895690i	$-0.353318\pi$
$569$	12071.0	0.889358	0.444679	+	0.895690i	$0.353318\pi$
$570$	0	0
$571$	15113.6	1.10768	0.553841	−	0.832623i	$-0.313162\pi$
$571$	15113.6	1.10768	0.553841	+	0.832623i	$0.313162\pi$
$572$	1978.37	0.144615
$573$	11887.7	0.866692
$574$	−3519.55	−0.255929
$575$	0	0
$576$	−490.513	−0.0354827
$577$	6562.11	0.473456	0.236728	−	0.971576i	$-0.423925\pi$
$577$	6562.11	0.473456	0.236728	+	0.971576i	$0.423925\pi$
$578$	6545.32	0.471020
$579$	−2611.73	−0.187461
$580$	0	0
$581$	−4056.18	−0.289636
$582$	−10124.1	−0.721061
$583$	−349.225	−0.0248086
$584$	2832.33	0.200690
$585$	0	0
$586$	−15505.4	−1.09304
$587$	−25888.6	−1.82033	−0.910166	−	0.414244i	$-0.864046\pi$
$587$	−25888.6	−1.82033	−0.910166	+	0.414244i	$0.864046\pi$
$588$	−5238.64	−0.367411
$589$	1241.61	0.0868582
$590$	0	0
$591$	−8029.42	−0.558860
$592$	1723.50	0.119654
$593$	11744.8	0.813324	0.406662	−	0.913579i	$-0.366693\pi$
$593$	11744.8	0.813324	0.406662	+	0.913579i	$0.366693\pi$
$594$	−11598.4	−0.801156
$595$	0	0
$596$	2348.07	0.161377
$597$	−17973.6	−1.23218
$598$	542.864	0.0371227
$599$	7436.84	0.507280	0.253640	−	0.967299i	$-0.418372\pi$
$599$	7436.84	0.507280	0.253640	+	0.967299i	$0.418372\pi$
$600$	0	0
$601$	18434.8	1.25120	0.625600	−	0.780144i	$-0.284854\pi$
$601$	18434.8	1.25120	0.625600	+	0.780144i	$0.284854\pi$
$602$	−3597.43	−0.243555
$603$	6747.41	0.455681
$604$	−5163.30	−0.347834
$605$	0	0
$606$	−5692.15	−0.381564
$607$	15636.9	1.04561	0.522804	−	0.852453i	$-0.324886\pi$
$607$	15636.9	1.04561	0.522804	+	0.852453i	$0.324886\pi$
$608$	236.443	0.0157714
$609$	−3810.50	−0.253546
$610$	0	0
$611$	−313.456	−0.0207546
$612$	−2773.69	−0.183202
$613$	14469.5	0.953370	0.476685	−	0.879074i	$-0.341838\pi$
$613$	14469.5	0.953370	0.476685	+	0.879074i	$0.341838\pi$
$614$	1892.53	0.124392
$615$	0	0
$616$	−2045.44	−0.133788
$617$	16273.9	1.06185	0.530925	−	0.847419i	$-0.321845\pi$
$617$	16273.9	1.06185	0.530925	+	0.847419i	$0.321845\pi$
$618$	13724.7	0.893346
$619$	5060.75	0.328608	0.164304	−	0.986410i	$-0.447462\pi$
$619$	5060.75	0.328608	0.164304	+	0.986410i	$0.447462\pi$
$620$	0	0
$621$	−3182.58	−0.205657
$622$	9952.32	0.641562
$623$	6969.92	0.448225
$624$	914.626	0.0586769
$625$	0	0
$626$	16106.2	1.02833
$627$	1236.12	0.0787335
$628$	6391.08	0.406102
$629$	9745.79	0.617790
$630$	0	0
$631$	−7513.63	−0.474030	−0.237015	−	0.971506i	$-0.576169\pi$
$631$	−7513.63	−0.474030	−0.237015	+	0.971506i	$0.576169\pi$
$632$	−8627.68	−0.543023
$633$	−6128.67	−0.384823
$634$	10338.3	0.647610
$635$	0	0
$636$	−161.451	−0.0100660
$637$	−3871.88	−0.240831
$638$	9811.68	0.608853
$639$	−2402.57	−0.148739
$640$	0	0
$641$	22079.0	1.36048	0.680239	−	0.732990i	$-0.261876\pi$
$641$	22079.0	1.36048	0.680239	+	0.732990i	$0.261876\pi$
$642$	5493.97	0.337741
$643$	−3256.85	−0.199748	−0.0998738	−	0.995000i	$-0.531844\pi$
$643$	−3256.85	−0.199748	−0.0998738	+	0.995000i	$0.531844\pi$
$644$	−561.268	−0.0343433
$645$	0	0
$646$	1337.00	0.0814299
$647$	−26658.2	−1.61985	−0.809924	−	0.586534i	$-0.800492\pi$
$647$	−26658.2	−1.61985	−0.809924	+	0.586534i	$0.800492\pi$
$648$	−3706.59	−0.224705
$649$	6190.33	0.374410
$650$	0	0
$651$	−4965.71	−0.298958
$652$	−4624.82	−0.277795
$653$	26493.1	1.58768	0.793839	−	0.608128i	$-0.208079\pi$
$653$	26493.1	1.58768	0.793839	+	0.608128i	$0.208079\pi$
$654$	−13954.4	−0.834344
$655$	0	0
$656$	4189.71	0.249361
$657$	−2713.47	−0.161130
$658$	324.083	0.0192007
$659$	−21341.5	−1.26153	−0.630764	−	0.775975i	$-0.717258\pi$
$659$	−21341.5	−1.26153	−0.630764	+	0.775975i	$0.717258\pi$
$660$	0	0
$661$	−28954.4	−1.70378	−0.851888	−	0.523724i	$-0.824542\pi$
$661$	−28954.4	−1.70378	−0.851888	+	0.523724i	$0.824542\pi$
$662$	−10442.8	−0.613098
$663$	5171.91	0.302956
$664$	4828.52	0.282203
$665$	0	0
$666$	−1651.17	−0.0960681
$667$	2692.32	0.156292
$668$	1149.13	0.0665585
$669$	17277.0	0.998454
$670$	0	0
$671$	148.259	0.00852979
$672$	−945.635	−0.0542837
$673$	29639.0	1.69762	0.848810	−	0.528699i	$-0.177320\pi$
$673$	29639.0	1.69762	0.848810	+	0.528699i	$0.177320\pi$
$674$	−14986.2	−0.856450
$675$	0	0
$676$	676.000	0.0384615
$677$	14469.5	0.821431	0.410716	−	0.911763i	$-0.365279\pi$
$677$	14469.5	0.821431	0.410716	+	0.911763i	$0.365279\pi$
$678$	12040.9	0.682046
$679$	7736.40	0.437255
$680$	0	0
$681$	3337.78	0.187818
$682$	12786.2	0.717903
$683$	−27103.8	−1.51845	−0.759223	−	0.650831i	$-0.774421\pi$
$683$	−27103.8	−1.51845	−0.759223	+	0.650831i	$0.774421\pi$
$684$	−226.520	−0.0126626
$685$	0	0
$686$	8613.32	0.479385
$687$	20826.2	1.15658
$688$	4282.41	0.237305
$689$	−119.328	−0.00659804
$690$	0	0
$691$	−10376.5	−0.571259	−0.285629	−	0.958340i	$-0.592203\pi$
$691$	−10376.5	−0.571259	−0.285629	+	0.958340i	$0.592203\pi$
$692$	−7030.63	−0.386220
$693$	1959.60	0.107416
$694$	6339.25	0.346736
$695$	0	0
$696$	4536.06	0.247039
$697$	23691.4	1.28748
$698$	971.098	0.0526599
$699$	−5584.34	−0.302173
$700$	0	0
$701$	29104.3	1.56812	0.784061	−	0.620684i	$-0.213145\pi$
$701$	29104.3	1.56812	0.784061	+	0.620684i	$0.213145\pi$
$702$	−3963.11	−0.213074
$703$	795.914	0.0427005
$704$	2434.92	0.130354
$705$	0	0
$706$	−19969.2	−1.06452
$707$	4349.69	0.231382
$708$	2861.87	0.151915
$709$	21945.1	1.16243	0.581217	−	0.813749i	$-0.302577\pi$
$709$	21945.1	1.16243	0.581217	+	0.813749i	$0.302577\pi$
$710$	0	0
$711$	8265.60	0.435983
$712$	−8297.07	−0.436722
$713$	3508.53	0.184285
$714$	−5347.25	−0.280274
$715$	0	0
$716$	1576.04	0.0822615
$717$	−14925.9	−0.777431
$718$	−6568.92	−0.341434
$719$	−19029.0	−0.987010	−0.493505	−	0.869743i	$-0.664285\pi$
$719$	−19029.0	−0.987010	−0.493505	+	0.869743i	$0.664285\pi$
$720$	0	0
$721$	−10487.8	−0.541729
$722$	−13608.8	−0.701478
$723$	−3861.33	−0.198623
$724$	−1552.27	−0.0796818
$725$	0	0
$726$	1024.28	0.0523615
$727$	−1944.43	−0.0991950	−0.0495975	−	0.998769i	$-0.515794\pi$
$727$	−1944.43	−0.0991950	−0.0495975	+	0.998769i	$0.515794\pi$
$728$	−698.918	−0.0355819
$729$	21648.0	1.09983
$730$	0	0
$731$	24215.6	1.22524
$732$	68.5422	0.00346092
$733$	15466.1	0.779339	0.389669	−	0.920955i	$-0.372589\pi$
$733$	15466.1	0.779339	0.389669	+	0.920955i	$0.372589\pi$
$734$	11733.5	0.590044
$735$	0	0
$736$	668.140	0.0334619
$737$	−33494.3	−1.67406
$738$	−4013.88	−0.200207
$739$	−29589.2	−1.47288	−0.736439	−	0.676504i	$-0.763494\pi$
$739$	−29589.2	−1.47288	−0.736439	+	0.676504i	$0.763494\pi$
$740$	0	0
$741$	422.376	0.0209398
$742$	123.374	0.00610404
$743$	−27269.0	−1.34644	−0.673219	−	0.739443i	$-0.735089\pi$
$743$	−27269.0	−1.34644	−0.673219	+	0.739443i	$0.735089\pi$
$744$	5911.23	0.291285
$745$	0	0
$746$	17776.0	0.872420
$747$	−4625.88	−0.226576
$748$	13768.6	0.673037
$749$	−4198.26	−0.204808
$750$	0	0
$751$	5349.06	0.259907	0.129953	−	0.991520i	$-0.458517\pi$
$751$	5349.06	0.259907	0.129953	+	0.991520i	$0.458517\pi$
$752$	−385.792	−0.0187080
$753$	902.301	0.0436676
$754$	3352.60	0.161929
$755$	0	0
$756$	4097.47	0.197121
$757$	−1692.77	−0.0812744	−0.0406372	−	0.999174i	$-0.512939\pi$
$757$	−1692.77	−0.0812744	−0.0406372	+	0.999174i	$0.512939\pi$
$758$	−15431.0	−0.739420
$759$	3493.03	0.167047
$760$	0	0
$761$	23884.4	1.13773	0.568864	−	0.822432i	$-0.307383\pi$
$761$	23884.4	1.13773	0.568864	+	0.822432i	$0.307383\pi$
$762$	8985.02	0.427156
$763$	10663.4	0.505950
$764$	10813.8	0.512078
$765$	0	0
$766$	−11221.7	−0.529316
$767$	2115.21	0.0995772
$768$	1125.69	0.0528906
$769$	36901.4	1.73043	0.865213	−	0.501404i	$-0.167183\pi$
$769$	36901.4	1.73043	0.865213	+	0.501404i	$0.167183\pi$
$770$	0	0
$771$	−27142.0	−1.26783
$772$	−2375.79	−0.110760
$773$	−10910.3	−0.507654	−0.253827	−	0.967250i	$-0.581689\pi$
$773$	−10910.3	−0.507654	−0.253827	+	0.967250i	$0.581689\pi$
$774$	−4102.69	−0.190528
$775$	0	0
$776$	−9209.50	−0.426033
$777$	−3183.20	−0.146971
$778$	−28575.4	−1.31681
$779$	1934.82	0.0889885
$780$	0	0
$781$	11926.4	0.546428
$782$	3778.11	0.172768
$783$	−19654.9	−0.897074
$784$	−4765.39	−0.217082
$785$	0	0
$786$	−14425.9	−0.654648
$787$	−8182.56	−0.370618	−0.185309	−	0.982680i	$-0.559329\pi$
$787$	−8182.56	−0.370618	−0.185309	+	0.982680i	$0.559329\pi$
$788$	−7304.05	−0.330198
$789$	−6925.80	−0.312503
$790$	0	0
$791$	−9201.13	−0.413596
$792$	−2332.73	−0.104659
$793$	50.6595	0.00226856
$794$	24306.9	1.08642
$795$	0	0
$796$	−16349.8	−0.728021
$797$	3416.06	0.151823	0.0759117	−	0.997115i	$-0.475813\pi$
$797$	3416.06	0.151823	0.0759117	+	0.997115i	$0.475813\pi$
$798$	−436.696	−0.0193720
$799$	−2181.52	−0.0965917
$800$	0	0
$801$	7948.86	0.350636
$802$	6368.39	0.280394
$803$	13469.7	0.591951
$804$	−15484.8	−0.679239
$805$	0	0
$806$	4368.99	0.190932
$807$	19481.3	0.849782
$808$	−5177.92	−0.225444
$809$	4921.79	0.213895	0.106947	−	0.994265i	$-0.465892\pi$
$809$	4921.79	0.213895	0.106947	+	0.994265i	$0.465892\pi$
$810$	0	0
$811$	−31959.9	−1.38380	−0.691902	−	0.721992i	$-0.743227\pi$
$811$	−31959.9	−1.38380	−0.691902	+	0.721992i	$0.743227\pi$
$812$	−3466.27	−0.149806
$813$	−25727.4	−1.10984
$814$	8196.43	0.352930
$815$	0	0
$816$	6365.42	0.273081
$817$	1977.63	0.0846860
$818$	−5076.25	−0.216977
$819$	669.587	0.0285681
$820$	0	0
$821$	20235.9	0.860215	0.430108	−	0.902778i	$-0.358476\pi$
$821$	20235.9	0.860215	0.430108	+	0.902778i	$0.358476\pi$
$822$	6677.64	0.283345
$823$	−31793.7	−1.34661	−0.673304	−	0.739365i	$-0.735126\pi$
$823$	−31793.7	−1.34661	−0.673304	+	0.739365i	$0.735126\pi$
$824$	12484.8	0.527826
$825$	0	0
$826$	−2186.92	−0.0921218
$827$	12551.0	0.527742	0.263871	−	0.964558i	$-0.415001\pi$
$827$	12551.0	0.527742	0.263871	+	0.964558i	$0.415001\pi$
$828$	−640.100	−0.0268660
$829$	43053.5	1.80375	0.901877	−	0.431994i	$-0.142190\pi$
$829$	43053.5	1.80375	0.901877	+	0.431994i	$0.142190\pi$
$830$	0	0
$831$	4656.62	0.194388
$832$	832.000	0.0346688
$833$	−26946.7	−1.12082
$834$	−15873.0	−0.659036
$835$	0	0
$836$	1124.45	0.0465191
$837$	−25613.6	−1.05775
$838$	−10696.5	−0.440937
$839$	−4666.54	−0.192022	−0.0960111	−	0.995380i	$-0.530608\pi$
$839$	−4666.54	−0.192022	−0.0960111	+	0.995380i	$0.530608\pi$
$840$	0	0
$841$	−7761.86	−0.318253
$842$	6043.38	0.247350
$843$	−39293.3	−1.60538
$844$	−5575.01	−0.227369
$845$	0	0
$846$	369.601	0.0150203
$847$	−782.708	−0.0317523
$848$	−146.866	−0.00594739
$849$	−30668.8	−1.23975
$850$	0	0
$851$	2249.10	0.0905969
$852$	5513.73	0.221711
$853$	−229.511	−0.00921256	−0.00460628	−	0.999989i	$-0.501466\pi$
$853$	−229.511	−0.00921256	−0.00460628	+	0.999989i	$0.501466\pi$
$854$	−52.3770	−0.00209872
$855$	0	0
$856$	4997.65	0.199552
$857$	−36924.7	−1.47179	−0.735895	−	0.677095i	$-0.763238\pi$
$857$	−36924.7	−1.47179	−0.735895	+	0.677095i	$0.763238\pi$
$858$	4349.69	0.173072
$859$	28515.5	1.13264	0.566318	−	0.824187i	$-0.308367\pi$
$859$	28515.5	1.13264	0.566318	+	0.824187i	$0.308367\pi$
$860$	0	0
$861$	−7738.16	−0.306290
$862$	−3011.38	−0.118989
$863$	−24308.3	−0.958824	−0.479412	−	0.877590i	$-0.659150\pi$
$863$	−24308.3	−0.958824	−0.479412	+	0.877590i	$0.659150\pi$
$864$	−4877.67	−0.192062
$865$	0	0
$866$	−18348.1	−0.719971
$867$	14390.7	0.563706
$868$	−4517.11	−0.176637
$869$	−41030.7	−1.60169
$870$	0	0
$871$	−11444.8	−0.445228
$872$	−12693.8	−0.492966
$873$	8823.00	0.342054
$874$	308.548	0.0119414
$875$	0	0
$876$	6227.23	0.240181
$877$	15552.8	0.598837	0.299418	−	0.954122i	$-0.403207\pi$
$877$	15552.8	0.598837	0.299418	+	0.954122i	$0.403207\pi$
$878$	22632.5	0.869943
$879$	−34090.6	−1.30813
$880$	0	0
$881$	9776.72	0.373878	0.186939	−	0.982372i	$-0.440143\pi$
$881$	9776.72	0.373878	0.186939	+	0.982372i	$0.440143\pi$
$882$	4565.40	0.174291
$883$	−39779.8	−1.51608	−0.758038	−	0.652210i	$-0.773842\pi$
$883$	−39779.8	−1.51608	−0.758038	+	0.652210i	$0.773842\pi$
$884$	4704.68	0.178999
$885$	0	0
$886$	18515.1	0.702062
$887$	−34270.8	−1.29729	−0.648647	−	0.761089i	$-0.724665\pi$
$887$	−34270.8	−1.29729	−0.648647	+	0.761089i	$0.724665\pi$
$888$	3789.31	0.143199
$889$	−6865.97	−0.259030
$890$	0	0
$891$	−17627.4	−0.662785
$892$	15716.2	0.589929
$893$	−178.160	−0.00667624
$894$	5162.52	0.193133
$895$	0	0
$896$	−860.207	−0.0320731
$897$	1193.55	0.0444276
$898$	27587.8	1.02519
$899$	21667.9	0.803854
$900$	0	0
$901$	−830.476	−0.0307072
$902$	19925.0	0.735510
$903$	−7909.38	−0.291481
$904$	10953.1	0.402982
$905$	0	0
$906$	−11352.1	−0.416280
$907$	−8926.99	−0.326809	−0.163405	−	0.986559i	$-0.552248\pi$
$907$	−8926.99	−0.326809	−0.163405	+	0.986559i	$0.552248\pi$
$908$	3036.25	0.110971
$909$	4960.62	0.181005
$910$	0	0
$911$	−4788.29	−0.174142	−0.0870709	−	0.996202i	$-0.527751\pi$
$911$	−4788.29	−0.174142	−0.0870709	+	0.996202i	$0.527751\pi$
$912$	519.848	0.0188749
$913$	22963.0	0.832381
$914$	−17784.2	−0.643598
$915$	0	0
$916$	18944.8	0.683356
$917$	11023.6	0.396982
$918$	−27581.6	−0.991643
$919$	40887.6	1.46764	0.733818	−	0.679346i	$-0.237736\pi$
$919$	40887.6	1.46764	0.733818	+	0.679346i	$0.237736\pi$
$920$	0	0
$921$	4160.96	0.148869
$922$	15454.0	0.552005
$923$	4075.20	0.145327
$924$	−4497.16	−0.160114
$925$	0	0
$926$	−35349.0	−1.25447
$927$	−11960.9	−0.423782
$928$	4126.28	0.145961
$929$	−26311.4	−0.929222	−0.464611	−	0.885515i	$-0.653806\pi$
$929$	−26311.4	−0.929222	−0.464611	+	0.885515i	$0.653806\pi$
$930$	0	0
$931$	−2200.67	−0.0774693
$932$	−5079.86	−0.178537
$933$	21881.4	0.767808
$934$	24203.5	0.847926
$935$	0	0
$936$	−797.083	−0.0278349
$937$	−19268.4	−0.671793	−0.335897	−	0.941899i	$-0.609039\pi$
$937$	−19268.4	−0.671793	−0.335897	+	0.941899i	$0.609039\pi$
$938$	11832.9	0.411894
$939$	35411.4	1.23068
$940$	0	0
$941$	−36177.0	−1.25328	−0.626639	−	0.779309i	$-0.715570\pi$
$941$	−36177.0	−1.25328	−0.626639	+	0.779309i	$0.715570\pi$
$942$	14051.6	0.486014
$943$	5467.41	0.188805
$944$	2603.33	0.0897577
$945$	0	0
$946$	20365.9	0.699949
$947$	17.6131	0.000604381 0	0.000302190	−	1.00000i	$-0.499904\pi$
$947$	17.6131	0.000604381 0	0.000302190		1.00000i	$0.499904\pi$
$948$	−18969.0	−0.649878
$949$	4602.54	0.157434
$950$	0	0
$951$	22729.9	0.775045
$952$	−4864.18	−0.165598
$953$	−49640.9	−1.68733	−0.843665	−	0.536869i	$-0.819607\pi$
$953$	−49640.9	−1.68733	−0.843665	+	0.536869i	$0.819607\pi$
$954$	140.702	0.00477505
$955$	0	0
$956$	−13577.5	−0.459339
$957$	21572.2	0.728661
$958$	−40602.5	−1.36932
$959$	−5102.76	−0.171822
$960$	0	0
$961$	−1554.19	−0.0521697
$962$	2800.68	0.0938644
$963$	−4787.92	−0.160216
$964$	−3512.50	−0.117355
$965$	0	0
$966$	−1234.02	−0.0411013
$967$	50123.7	1.66688	0.833438	−	0.552614i	$-0.186369\pi$
$967$	50123.7	1.66688	0.833438	+	0.552614i	$0.186369\pi$
$968$	931.744	0.0309374
$969$	2939.57	0.0974535
$970$	0	0
$971$	−56029.5	−1.85177	−0.925886	−	0.377803i	$-0.876680\pi$
$971$	−56029.5	−1.85177	−0.925886	+	0.377803i	$0.876680\pi$
$972$	8312.74	0.274312
$973$	12129.5	0.399643
$974$	12576.0	0.413717
$975$	0	0
$976$	62.3502	0.00204486
$977$	−53685.2	−1.75797	−0.878987	−	0.476846i	$-0.841780\pi$
$977$	−53685.2	−1.75797	−0.878987	+	0.476846i	$0.841780\pi$
$978$	−10168.2	−0.332458
$979$	−39458.4	−1.28815
$980$	0	0
$981$	12161.1	0.395793
$982$	6952.40	0.225927
$983$	−2358.26	−0.0765177	−0.0382588	−	0.999268i	$-0.512181\pi$
$983$	−2358.26	−0.0765177	−0.0382588	+	0.999268i	$0.512181\pi$
$984$	9211.59	0.298429
$985$	0	0
$986$	23332.7	0.753616
$987$	712.536	0.0229790
$988$	384.219	0.0123721
$989$	5588.39	0.179677
$990$	0	0
$991$	26921.8	0.862965	0.431483	−	0.902121i	$-0.357991\pi$
$991$	26921.8	0.862965	0.431483	+	0.902121i	$0.357991\pi$
$992$	5377.22	0.172104
$993$	−22959.8	−0.733742
$994$	−4213.36	−0.134446
$995$	0	0
$996$	10616.1	0.337734
$997$	6841.04	0.217310	0.108655	−	0.994080i	$-0.465346\pi$
$997$	6841.04	0.217310	0.108655	+	0.994080i	$0.465346\pi$
$998$	17551.7	0.556702
$999$	−16419.2	−0.520001

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Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	650.4.a.x.1.3	yes	4
5.2	odd	4		650.4.b.p.599.6		8
5.3	odd	4		650.4.b.p.599.3		8
5.4	even	2		650.4.a.w.1.2	✓	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
650.4.a.w.1.2	✓	4	5.4	even	2
650.4.a.x.1.3	yes	4	1.1	even	1	trivial
650.4.b.p.599.3		8	5.3	odd	4
650.4.b.p.599.6		8	5.2	odd	4

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

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Embedding invariants

$q$ -expansion

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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	650.4.a.x.1.3

Level	$650$
Weight	$4$
Character	650.1
Self dual	yes
Analytic conductor	$38.351$
Analytic rank	$0$
Dimension	$4$
CM	no
Inner twists	$1$