LMFDB - Embedded newform 522.4.a.k.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [522,4,Mod(1,522)] 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("522.1"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(522, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 522 = 2 \cdot 3^{2} \cdot 29 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	522.a (trivial)

Newform invariants

comment:select newform

sage:traces = [3,-6,0,12,-20] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

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

Self dual:	yes
Analytic conductor:	$30.7989970230$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.19816.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 42x - 54 $ x^3 - x^2 - 42*x - 54
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 58)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-5.13291$ of defining polynomial
Character	$\chi$	$=$	522.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-2.00000 q^{2} +4.00000 q^{4} -2.36031 q^{5} -18.5045 q^{7} -8.00000 q^{8} +4.72062 q^{10} +15.3852 q^{11} +27.8241 q^{13} +37.0090 q^{14} +16.0000 q^{16} +62.4231 q^{17} +55.3245 q^{19} -9.44123 q^{20} -30.7703 q^{22} -44.3109 q^{23} -119.429 q^{25} -55.6482 q^{26} -74.0180 q^{28} -29.0000 q^{29} -207.078 q^{31} -32.0000 q^{32} -124.846 q^{34} +43.6763 q^{35} +303.388 q^{37} -110.649 q^{38} +18.8825 q^{40} -125.712 q^{41} +101.246 q^{43} +61.5406 q^{44} +88.6218 q^{46} -50.8663 q^{47} -0.583766 q^{49} +238.858 q^{50} +111.296 q^{52} -692.512 q^{53} -36.3137 q^{55} +148.036 q^{56} +58.0000 q^{58} -557.935 q^{59} +809.678 q^{61} +414.157 q^{62} +64.0000 q^{64} -65.6734 q^{65} -749.866 q^{67} +249.692 q^{68} -87.3526 q^{70} +54.7703 q^{71} -184.533 q^{73} -606.775 q^{74} +221.298 q^{76} -284.694 q^{77} -752.875 q^{79} -37.7649 q^{80} +251.423 q^{82} +902.018 q^{83} -147.338 q^{85} -202.492 q^{86} -123.081 q^{88} -953.385 q^{89} -514.871 q^{91} -177.244 q^{92} +101.733 q^{94} -130.583 q^{95} -1114.68 q^{97} +1.16753 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 6 q^{2} + 12 q^{4} - 20 q^{5} + 24 q^{7} - 24 q^{8} + 40 q^{10} - 10 q^{11} - 4 q^{13} - 48 q^{14} + 48 q^{16} + 66 q^{17} - 164 q^{19} - 80 q^{20} + 20 q^{22} + 204 q^{23} + 79 q^{25} + 8 q^{26}+ \cdots - 1078 q^{98}+O(q^{100}) $ 3 * q - 6 * q^2 + 12 * q^4 - 20 * q^5 + 24 * q^7 - 24 * q^8 + 40 * q^10 - 10 * q^11 - 4 * q^13 - 48 * q^14 + 48 * q^16 + 66 * q^17 - 164 * q^19 - 80 * q^20 + 20 * q^22 + 204 * q^23 + 79 * q^25 + 8 * q^26 + 96 * q^28 - 87 * q^29 - 86 * q^31 - 96 * q^32 - 132 * q^34 - 24 * q^35 - 42 * q^37 + 328 * q^38 + 160 * q^40 - 562 * q^41 + 18 * q^43 - 40 * q^44 - 408 * q^46 - 654 * q^47 + 539 * q^49 - 158 * q^50 - 16 * q^52 - 712 * q^53 + 142 * q^55 - 192 * q^56 + 174 * q^58 - 184 * q^59 + 322 * q^61 + 172 * q^62 + 192 * q^64 - 1494 * q^65 - 228 * q^67 + 264 * q^68 + 48 * q^70 + 52 * q^71 - 494 * q^73 + 84 * q^74 - 656 * q^76 - 872 * q^77 - 2110 * q^79 - 320 * q^80 + 1124 * q^82 + 288 * q^83 - 2704 * q^85 - 36 * q^86 + 80 * q^88 - 914 * q^89 - 2984 * q^91 + 816 * q^92 + 1308 * q^94 + 1900 * q^95 + 218 * q^97 - 1078 * 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 $

$2$	−2.00000	−0.707107
$2$	−2.00000	−0.707107
$3$	0	0
$3$	0	0
$4$	4.00000	0.500000
$5$	−2.36031	−0.211112	−0.105556	−	0.994413i	$-0.533662\pi$
$5$	−2.36031	−0.211112	−0.105556	+	0.994413i	$0.533662\pi$
$6$	0	0
$7$	−18.5045	−0.999149	−0.499574	−	0.866271i	$-0.666510\pi$
$7$	−18.5045	−0.999149	−0.499574	+	0.866271i	$0.666510\pi$
$8$	−8.00000	−0.353553
$9$	0	0
$10$	4.72062	0.149279
$11$	15.3852	0.421709	0.210854	−	0.977517i	$-0.432375\pi$
$11$	15.3852	0.421709	0.210854	+	0.977517i	$0.432375\pi$
$12$	0	0
$13$	27.8241	0.593617	0.296808	−	0.954937i	$-0.404078\pi$
$13$	27.8241	0.593617	0.296808	+	0.954937i	$0.404078\pi$
$14$	37.0090	0.706505
$15$	0	0
$16$	16.0000	0.250000
$17$	62.4231	0.890578	0.445289	−	0.895387i	$-0.353101\pi$
$17$	62.4231	0.890578	0.445289	+	0.895387i	$0.353101\pi$
$18$	0	0
$19$	55.3245	0.668016	0.334008	−	0.942570i	$-0.391599\pi$
$19$	55.3245	0.668016	0.334008	+	0.942570i	$0.391599\pi$
$20$	−9.44123	−0.105556
$21$	0	0
$22$	−30.7703	−0.298193
$23$	−44.3109	−0.401716	−0.200858	−	0.979620i	$-0.564373\pi$
$23$	−44.3109	−0.401716	−0.200858	+	0.979620i	$0.564373\pi$
$24$	0	0
$25$	−119.429	−0.955432
$26$	−55.6482	−0.419750
$27$	0	0
$28$	−74.0180	−0.499574
$29$	−29.0000	−0.185695
$29$	−29.0000	−0.185695
$30$	0	0
$31$	−207.078	−1.19975	−0.599877	−	0.800092i	$-0.704784\pi$
$31$	−207.078	−1.19975	−0.599877	+	0.800092i	$0.704784\pi$
$32$	−32.0000	−0.176777
$33$	0	0
$34$	−124.846	−0.629734
$35$	43.6763	0.210933
$36$	0	0
$37$	303.388	1.34802	0.674008	−	0.738724i	$-0.264571\pi$
$37$	303.388	1.34802	0.674008	+	0.738724i	$0.264571\pi$
$38$	−110.649	−0.472359
$39$	0	0
$40$	18.8825	0.0746395
$41$	−125.712	−0.478850	−0.239425	−	0.970915i	$-0.576959\pi$
$41$	−125.712	−0.478850	−0.239425	+	0.970915i	$0.576959\pi$
$42$	0	0
$43$	101.246	0.359066	0.179533	−	0.983752i	$-0.442541\pi$
$43$	101.246	0.359066	0.179533	+	0.983752i	$0.442541\pi$
$44$	61.5406	0.210854
$45$	0	0
$46$	88.6218	0.284056
$47$	−50.8663	−0.157864	−0.0789320	−	0.996880i	$-0.525151\pi$
$47$	−50.8663	−0.157864	−0.0789320	+	0.996880i	$0.525151\pi$
$48$	0	0
$49$	−0.583766	−0.00170194
$50$	238.858	0.675592
$51$	0	0
$52$	111.296	0.296808
$53$	−692.512	−1.79479	−0.897395	−	0.441229i	$-0.854543\pi$
$53$	−692.512	−1.79479	−0.897395	+	0.441229i	$0.854543\pi$
$54$	0	0
$55$	−36.3137	−0.0890280
$56$	148.036	0.353252
$57$	0	0
$58$	58.0000	0.131306
$59$	−557.935	−1.23113	−0.615567	−	0.788085i	$-0.711073\pi$
$59$	−557.935	−1.23113	−0.615567	+	0.788085i	$0.711073\pi$
$60$	0	0
$61$	809.678	1.69948	0.849742	−	0.527198i	$-0.176757\pi$
$61$	809.678	1.69948	0.849742	+	0.527198i	$0.176757\pi$
$62$	414.157	0.848354
$63$	0	0
$64$	64.0000	0.125000
$65$	−65.6734	−0.125320
$66$	0	0
$67$	−749.866	−1.36732	−0.683662	−	0.729799i	$-0.739614\pi$
$67$	−749.866	−1.36732	−0.683662	+	0.729799i	$0.739614\pi$
$68$	249.692	0.445289
$69$	0	0
$70$	−87.3526	−0.149152
$71$	54.7703	0.0915499	0.0457749	−	0.998952i	$-0.485424\pi$
$71$	54.7703	0.0915499	0.0457749	+	0.998952i	$0.485424\pi$
$72$	0	0
$73$	−184.533	−0.295862	−0.147931	−	0.988998i	$-0.547261\pi$
$73$	−184.533	−0.295862	−0.147931	+	0.988998i	$0.547261\pi$
$74$	−606.775	−0.953192
$75$	0	0
$76$	221.298	0.334008
$77$	−284.694	−0.421350
$78$	0	0
$79$	−752.875	−1.07222	−0.536108	−	0.844149i	$-0.680106\pi$
$79$	−752.875	−1.07222	−0.536108	+	0.844149i	$0.680106\pi$
$80$	−37.7649	−0.0527781
$81$	0	0
$82$	251.423	0.338598
$83$	902.018	1.19288	0.596441	−	0.802657i	$-0.296581\pi$
$83$	902.018	1.19288	0.596441	+	0.802657i	$0.296581\pi$
$84$	0	0
$85$	−147.338	−0.188012
$86$	−202.492	−0.253898
$87$	0	0
$88$	−123.081	−0.149097
$89$	−953.385	−1.13549	−0.567745	−	0.823204i	$-0.692184\pi$
$89$	−953.385	−1.13549	−0.567745	+	0.823204i	$0.692184\pi$
$90$	0	0
$91$	−514.871	−0.593111
$92$	−177.244	−0.200858
$93$	0	0
$94$	101.733	0.111627
$95$	−130.583	−0.141026
$96$	0	0
$97$	−1114.68	−1.16679	−0.583395	−	0.812189i	$-0.698276\pi$
$97$	−1114.68	−1.16679	−0.583395	+	0.812189i	$0.698276\pi$
$98$	1.16753	0.00120345
$99$	0	0
$100$	−477.716	−0.477716
$101$	−742.802	−0.731798	−0.365899	−	0.930655i	$-0.619238\pi$
$101$	−742.802	−0.731798	−0.365899	+	0.930655i	$0.619238\pi$
$102$	0	0
$103$	−1782.16	−1.70487	−0.852434	−	0.522835i	$-0.824874\pi$
$103$	−1782.16	−1.70487	−0.852434	+	0.522835i	$0.824874\pi$
$104$	−222.593	−0.209875
$105$	0	0
$106$	1385.02	1.26911
$107$	−1758.89	−1.58914	−0.794571	−	0.607172i	$-0.792304\pi$
$107$	−1758.89	−1.58914	−0.794571	+	0.607172i	$0.792304\pi$
$108$	0	0
$109$	1596.84	1.40321	0.701604	−	0.712567i	$-0.252468\pi$
$109$	1596.84	1.40321	0.701604	+	0.712567i	$0.252468\pi$
$110$	72.6274	0.0629523
$111$	0	0
$112$	−296.072	−0.249787
$113$	−991.893	−0.825747	−0.412874	−	0.910788i	$-0.635475\pi$
$113$	−991.893	−0.825747	−0.412874	+	0.910788i	$0.635475\pi$
$114$	0	0
$115$	104.587	0.0848072
$116$	−116.000	−0.0928477
$117$	0	0
$118$	1115.87	0.870543
$119$	−1155.11	−0.889819
$120$	0	0
$121$	−1094.30	−0.822162
$122$	−1619.36	−1.20172
$123$	0	0
$124$	−828.314	−0.599877
$125$	576.928	0.412816
$126$	0	0
$127$	363.708	0.254125	0.127063	−	0.991895i	$-0.459445\pi$
$127$	363.708	0.254125	0.127063	+	0.991895i	$0.459445\pi$
$128$	−128.000	−0.0883883
$129$	0	0
$130$	131.347	0.0886145
$131$	665.920	0.444135	0.222068	−	0.975031i	$-0.428719\pi$
$131$	665.920	0.444135	0.222068	+	0.975031i	$0.428719\pi$
$132$	0	0
$133$	−1023.75	−0.667448
$134$	1499.73	0.966844
$135$	0	0
$136$	−499.385	−0.314867
$137$	−343.259	−0.214063	−0.107031	−	0.994256i	$-0.534135\pi$
$137$	−343.259	−0.214063	−0.107031	+	0.994256i	$0.534135\pi$
$138$	0	0
$139$	−755.103	−0.460770	−0.230385	−	0.973100i	$-0.573999\pi$
$139$	−755.103	−0.460770	−0.230385	+	0.973100i	$0.573999\pi$
$140$	174.705	0.105466
$141$	0	0
$142$	−109.541	−0.0647355
$143$	428.078	0.250333
$144$	0	0
$145$	68.4489	0.0392026
$146$	369.065	0.209206
$147$	0	0
$148$	1213.55	0.674008
$149$	3033.96	1.66813	0.834066	−	0.551664i	$-0.186007\pi$
$149$	3033.96	1.66813	0.834066	+	0.551664i	$0.186007\pi$
$150$	0	0
$151$	−723.924	−0.390146	−0.195073	−	0.980789i	$-0.562494\pi$
$151$	−723.924	−0.390146	−0.195073	+	0.980789i	$0.562494\pi$
$152$	−442.596	−0.236179
$153$	0	0
$154$	569.389	0.297939
$155$	488.769	0.253283
$156$	0	0
$157$	647.348	0.329070	0.164535	−	0.986371i	$-0.447388\pi$
$157$	647.348	0.329070	0.164535	+	0.986371i	$0.447388\pi$
$158$	1505.75	0.758171
$159$	0	0
$160$	75.5299	0.0373197
$161$	819.951	0.401374
$162$	0	0
$163$	−1514.06	−0.727546	−0.363773	−	0.931488i	$-0.618512\pi$
$163$	−1514.06	−0.727546	−0.363773	+	0.931488i	$0.618512\pi$
$164$	−502.847	−0.239425
$165$	0	0
$166$	−1804.04	−0.843496
$167$	−1841.05	−0.853084	−0.426542	−	0.904468i	$-0.640268\pi$
$167$	−1841.05	−0.853084	−0.426542	+	0.904468i	$0.640268\pi$
$168$	0	0
$169$	−1422.82	−0.647619
$170$	294.675	0.132945
$171$	0	0
$172$	404.983	0.179533
$173$	303.426	0.133347	0.0666736	−	0.997775i	$-0.478761\pi$
$173$	303.426	0.133347	0.0666736	+	0.997775i	$0.478761\pi$
$174$	0	0
$175$	2209.97	0.954618
$176$	246.162	0.105427
$177$	0	0
$178$	1906.77	0.802913
$179$	3036.60	1.26797	0.633983	−	0.773347i	$-0.281419\pi$
$179$	3036.60	1.26797	0.633983	+	0.773347i	$0.281419\pi$
$180$	0	0
$181$	−2339.77	−0.960851	−0.480425	−	0.877036i	$-0.659518\pi$
$181$	−2339.77	−0.960851	−0.480425	+	0.877036i	$0.659518\pi$
$182$	1029.74	0.419393
$183$	0	0
$184$	354.487	0.142028
$185$	−716.088	−0.284583
$186$	0	0
$187$	960.389	0.375565
$188$	−203.465	−0.0789320
$189$	0	0
$190$	261.166	0.0997208
$191$	4217.47	1.59772	0.798862	−	0.601514i	$-0.205436\pi$
$191$	4217.47	1.59772	0.798862	+	0.601514i	$0.205436\pi$
$192$	0	0
$193$	−1962.59	−0.731969	−0.365985	−	0.930621i	$-0.619268\pi$
$193$	−1962.59	−0.731969	−0.365985	+	0.930621i	$0.619268\pi$
$194$	2229.36	0.825045
$195$	0	0
$196$	−2.33506	−0.000850971 0
$197$	−2040.68	−0.738032	−0.369016	−	0.929423i	$-0.620305\pi$
$197$	−2040.68	−0.738032	−0.369016	+	0.929423i	$0.620305\pi$
$198$	0	0
$199$	3235.68	1.15262	0.576310	−	0.817231i	$-0.304492\pi$
$199$	3235.68	1.15262	0.576310	+	0.817231i	$0.304492\pi$
$200$	955.432	0.337796
$201$	0	0
$202$	1485.60	0.517459
$203$	536.630	0.185537
$204$	0	0
$205$	296.718	0.101091
$206$	3564.32	1.20552
$207$	0	0
$208$	445.186	0.148404
$209$	851.176	0.281708
$210$	0	0
$211$	−3943.15	−1.28653	−0.643264	−	0.765645i	$-0.722420\pi$
$211$	−3943.15	−1.28653	−0.643264	+	0.765645i	$0.722420\pi$
$212$	−2770.05	−0.897395
$213$	0	0
$214$	3517.78	1.12369
$215$	−238.971	−0.0758033
$216$	0	0
$217$	3831.88	1.19873
$218$	−3193.68	−0.992218
$219$	0	0
$220$	−145.255	−0.0445140
$221$	1736.87	0.528662
$222$	0	0
$223$	−2530.83	−0.759987	−0.379993	−	0.924989i	$-0.624074\pi$
$223$	−2530.83	−0.759987	−0.379993	+	0.924989i	$0.624074\pi$
$224$	592.144	0.176626
$225$	0	0
$226$	1983.79	0.583892
$227$	−5429.77	−1.58761	−0.793803	−	0.608175i	$-0.791902\pi$
$227$	−5429.77	−1.58761	−0.793803	+	0.608175i	$0.791902\pi$
$228$	0	0
$229$	5597.57	1.61527	0.807637	−	0.589680i	$-0.200746\pi$
$229$	5597.57	1.61527	0.807637	+	0.589680i	$0.200746\pi$
$230$	−209.175	−0.0599677
$231$	0	0
$232$	232.000	0.0656532
$233$	1736.74	0.488316	0.244158	−	0.969735i	$-0.421488\pi$
$233$	1736.74	0.488316	0.244158	+	0.969735i	$0.421488\pi$
$234$	0	0
$235$	120.060	0.0333271
$236$	−2231.74	−0.615567
$237$	0	0
$238$	2310.21	0.629197
$239$	−4205.18	−1.13812	−0.569059	−	0.822296i	$-0.692693\pi$
$239$	−4205.18	−1.13812	−0.569059	+	0.822296i	$0.692693\pi$
$240$	0	0
$241$	6050.05	1.61709	0.808543	−	0.588436i	$-0.200256\pi$
$241$	6050.05	1.61709	0.808543	+	0.588436i	$0.200256\pi$
$242$	2188.59	0.581356
$243$	0	0
$244$	3238.71	0.849742
$245$	1.37787	0.000359301 0
$246$	0	0
$247$	1539.35	0.396545
$248$	1656.63	0.424177
$249$	0	0
$250$	−1153.86	−0.291905
$251$	6533.19	1.64291	0.821457	−	0.570271i	$-0.193162\pi$
$251$	6533.19	1.64291	0.821457	+	0.570271i	$0.193162\pi$
$252$	0	0
$253$	−681.730	−0.169407
$254$	−727.417	−0.179694
$255$	0	0
$256$	256.000	0.0625000
$257$	−2423.49	−0.588222	−0.294111	−	0.955771i	$-0.595024\pi$
$257$	−2423.49	−0.588222	−0.294111	+	0.955771i	$0.595024\pi$
$258$	0	0
$259$	−5614.03	−1.34687
$260$	−262.694	−0.0626599
$261$	0	0
$262$	−1331.84	−0.314051
$263$	2692.44	0.631266	0.315633	−	0.948881i	$-0.397783\pi$
$263$	2692.44	0.631266	0.315633	+	0.948881i	$0.397783\pi$
$264$	0	0
$265$	1634.54	0.378902
$266$	2047.50	0.471957
$267$	0	0
$268$	−2999.46	−0.683662
$269$	6688.58	1.51602	0.758011	−	0.652241i	$-0.226171\pi$
$269$	6688.58	1.51602	0.758011	+	0.652241i	$0.226171\pi$
$270$	0	0
$271$	1273.30	0.285415	0.142707	−	0.989765i	$-0.454419\pi$
$271$	1273.30	0.285415	0.142707	+	0.989765i	$0.454419\pi$
$272$	998.769	0.222644
$273$	0	0
$274$	686.517	0.151365
$275$	−1837.43	−0.402914
$276$	0	0
$277$	−1660.50	−0.360179	−0.180090	−	0.983650i	$-0.557639\pi$
$277$	−1660.50	−0.360179	−0.180090	+	0.983650i	$0.557639\pi$
$278$	1510.21	0.325814
$279$	0	0
$280$	−349.410	−0.0745760
$281$	1247.71	0.264883	0.132442	−	0.991191i	$-0.457718\pi$
$281$	1247.71	0.264883	0.132442	+	0.991191i	$0.457718\pi$
$282$	0	0
$283$	6592.72	1.38479	0.692396	−	0.721517i	$-0.256555\pi$
$283$	6592.72	1.38479	0.692396	+	0.721517i	$0.256555\pi$
$284$	219.081	0.0457749
$285$	0	0
$286$	−856.156	−0.177012
$287$	2326.23	0.478443
$288$	0	0
$289$	−1016.36	−0.206871
$290$	−136.898	−0.0277204
$291$	0	0
$292$	−738.130	−0.147931
$293$	−837.525	−0.166992	−0.0834962	−	0.996508i	$-0.526609\pi$
$293$	−837.525	−0.166992	−0.0834962	+	0.996508i	$0.526609\pi$
$294$	0	0
$295$	1316.90	0.259908
$296$	−2427.10	−0.476596
$297$	0	0
$298$	−6067.92	−1.17955
$299$	−1232.91	−0.238465
$300$	0	0
$301$	−1873.50	−0.358761
$302$	1447.85	0.275875
$303$	0	0
$304$	885.192	0.167004
$305$	−1911.09	−0.358782
$306$	0	0
$307$	500.495	0.0930448	0.0465224	−	0.998917i	$-0.485186\pi$
$307$	500.495	0.0930448	0.0465224	+	0.998917i	$0.485186\pi$
$308$	−1138.78	−0.210675
$309$	0	0
$310$	−977.538	−0.179098
$311$	−2016.48	−0.367667	−0.183833	−	0.982957i	$-0.558851\pi$
$311$	−2016.48	−0.367667	−0.183833	+	0.982957i	$0.558851\pi$
$312$	0	0
$313$	8739.06	1.57815	0.789075	−	0.614296i	$-0.210560\pi$
$313$	8739.06	1.57815	0.789075	+	0.614296i	$0.210560\pi$
$314$	−1294.70	−0.232688
$315$	0	0
$316$	−3011.50	−0.536108
$317$	1690.73	0.299560	0.149780	−	0.988719i	$-0.452143\pi$
$317$	1690.73	0.299560	0.149780	+	0.988719i	$0.452143\pi$
$318$	0	0
$319$	−446.169	−0.0783094
$320$	−151.060	−0.0263890
$321$	0	0
$322$	−1639.90	−0.283814
$323$	3453.53	0.594920
$324$	0	0
$325$	−3323.00	−0.567160
$326$	3028.11	0.514453
$327$	0	0
$328$	1005.69	0.169299
$329$	941.255	0.157730
$330$	0	0
$331$	−7446.19	−1.23649	−0.618247	−	0.785984i	$-0.712157\pi$
$331$	−7446.19	−1.23649	−0.618247	+	0.785984i	$0.712157\pi$
$332$	3608.07	0.596441
$333$	0	0
$334$	3682.11	0.603221
$335$	1769.91	0.288659
$336$	0	0
$337$	−9366.72	−1.51406	−0.757030	−	0.653381i	$-0.773350\pi$
$337$	−9366.72	−1.51406	−0.757030	+	0.653381i	$0.773350\pi$
$338$	2845.64	0.457936
$339$	0	0
$340$	−589.351	−0.0940060
$341$	−3185.93	−0.505947
$342$	0	0
$343$	6357.84	1.00085
$344$	−809.967	−0.126949
$345$	0	0
$346$	−606.852	−0.0942907
$347$	−7951.11	−1.23008	−0.615040	−	0.788496i	$-0.710860\pi$
$347$	−7951.11	−1.23008	−0.615040	+	0.788496i	$0.710860\pi$
$348$	0	0
$349$	−1256.74	−0.192755	−0.0963776	−	0.995345i	$-0.530726\pi$
$349$	−1256.74	−0.192755	−0.0963776	+	0.995345i	$0.530726\pi$
$350$	−4419.94	−0.675017
$351$	0	0
$352$	−492.325	−0.0745483
$353$	−7309.17	−1.10206	−0.551031	−	0.834485i	$-0.685765\pi$
$353$	−7309.17	−1.10206	−0.551031	+	0.834485i	$0.685765\pi$
$354$	0	0
$355$	−129.275	−0.0193273
$356$	−3813.54	−0.567745
$357$	0	0
$358$	−6073.20	−0.896588
$359$	−11555.1	−1.69876	−0.849379	−	0.527783i	$-0.823024\pi$
$359$	−11555.1	−1.69876	−0.849379	+	0.527783i	$0.823024\pi$
$360$	0	0
$361$	−3798.20	−0.553754
$362$	4679.55	0.679424
$363$	0	0
$364$	−2059.48	−0.296556
$365$	435.554	0.0624601
$366$	0	0
$367$	−2097.86	−0.298386	−0.149193	−	0.988808i	$-0.547668\pi$
$367$	−2097.86	−0.298386	−0.149193	+	0.988808i	$0.547668\pi$
$368$	−708.975	−0.100429
$369$	0	0
$370$	1432.18	0.201231
$371$	12814.6	1.79326
$372$	0	0
$373$	13979.4	1.94055	0.970277	−	0.241996i	$-0.0778022\pi$
$373$	13979.4	1.94055	0.970277	+	0.241996i	$0.0778022\pi$
$374$	−1920.78	−0.265564
$375$	0	0
$376$	406.930	0.0558134
$377$	−806.899	−0.110232
$378$	0	0
$379$	−894.071	−0.121175	−0.0605875	−	0.998163i	$-0.519297\pi$
$379$	−894.071	−0.121175	−0.0605875	+	0.998163i	$0.519297\pi$
$380$	−522.331	−0.0705132
$381$	0	0
$382$	−8434.94	−1.12976
$383$	−5312.26	−0.708730	−0.354365	−	0.935107i	$-0.615303\pi$
$383$	−5312.26	−0.708730	−0.354365	+	0.935107i	$0.615303\pi$
$384$	0	0
$385$	671.967	0.0889522
$386$	3925.17	0.517580
$387$	0	0
$388$	−4458.72	−0.583395
$389$	4206.27	0.548242	0.274121	−	0.961695i	$-0.411613\pi$
$389$	4206.27	0.548242	0.274121	+	0.961695i	$0.411613\pi$
$390$	0	0
$391$	−2766.02	−0.357759
$392$	4.67013	0.000601727 0
$393$	0	0
$394$	4081.36	0.521868
$395$	1777.02	0.226358
$396$	0	0
$397$	5702.43	0.720898	0.360449	−	0.932779i	$-0.382623\pi$
$397$	5702.43	0.720898	0.360449	+	0.932779i	$0.382623\pi$
$398$	−6471.36	−0.815025
$399$	0	0
$400$	−1910.86	−0.238858
$401$	−9803.05	−1.22080	−0.610400	−	0.792093i	$-0.708991\pi$
$401$	−9803.05	−1.22080	−0.610400	+	0.792093i	$0.708991\pi$
$402$	0	0
$403$	−5761.77	−0.712194
$404$	−2971.21	−0.365899
$405$	0	0
$406$	−1073.26	−0.131195
$407$	4667.66	0.568471
$408$	0	0
$409$	8384.14	1.01362	0.506808	−	0.862059i	$-0.330825\pi$
$409$	8384.14	1.01362	0.506808	+	0.862059i	$0.330825\pi$
$410$	−593.436	−0.0714823
$411$	0	0
$412$	−7128.64	−0.852434
$413$	10324.3	1.23009
$414$	0	0
$415$	−2129.04	−0.251832
$416$	−890.371	−0.104938
$417$	0	0
$418$	−1702.35	−0.199198
$419$	12350.2	1.43997	0.719986	−	0.693989i	$-0.244148\pi$
$419$	12350.2	1.43997	0.719986	+	0.693989i	$0.244148\pi$
$420$	0	0
$421$	−4776.65	−0.552968	−0.276484	−	0.961018i	$-0.589169\pi$
$421$	−4776.65	−0.552968	−0.276484	+	0.961018i	$0.589169\pi$
$422$	7886.29	0.909712
$423$	0	0
$424$	5540.10	0.634554
$425$	−7455.12	−0.850886
$426$	0	0
$427$	−14982.7	−1.69804
$428$	−7035.55	−0.794571
$429$	0	0
$430$	477.943	0.0536010
$431$	536.087	0.0599128	0.0299564	−	0.999551i	$-0.490463\pi$
$431$	536.087	0.0599128	0.0299564	+	0.999551i	$0.490463\pi$
$432$	0	0
$433$	3530.09	0.391790	0.195895	−	0.980625i	$-0.437239\pi$
$433$	3530.09	0.391790	0.195895	+	0.980625i	$0.437239\pi$
$434$	−7663.76	−0.847632
$435$	0	0
$436$	6387.37	0.701604
$437$	−2451.48	−0.268353
$438$	0	0
$439$	−11536.6	−1.25425	−0.627123	−	0.778920i	$-0.715768\pi$
$439$	−11536.6	−1.25425	−0.627123	+	0.778920i	$0.715768\pi$
$440$	290.510	0.0314761
$441$	0	0
$442$	−3473.73	−0.373820
$443$	−6775.94	−0.726715	−0.363357	−	0.931650i	$-0.618370\pi$
$443$	−6775.94	−0.726715	−0.363357	+	0.931650i	$0.618370\pi$
$444$	0	0
$445$	2250.28	0.239716
$446$	5061.67	0.537392
$447$	0	0
$448$	−1184.29	−0.124894
$449$	6601.58	0.693870	0.346935	−	0.937889i	$-0.387222\pi$
$449$	6601.58	0.693870	0.346935	+	0.937889i	$0.387222\pi$
$450$	0	0
$451$	−1934.09	−0.201935
$452$	−3967.57	−0.412874
$453$	0	0
$454$	10859.5	1.12261
$455$	1215.25	0.125213
$456$	0	0
$457$	−4454.94	−0.456003	−0.228001	−	0.973661i	$-0.573219\pi$
$457$	−4454.94	−0.456003	−0.228001	+	0.973661i	$0.573219\pi$
$458$	−11195.1	−1.14217
$459$	0	0
$460$	418.350	0.0424036
$461$	−1861.62	−0.188078	−0.0940392	−	0.995568i	$-0.529978\pi$
$461$	−1861.62	−0.188078	−0.0940392	+	0.995568i	$0.529978\pi$
$462$	0	0
$463$	−7795.98	−0.782527	−0.391263	−	0.920279i	$-0.627962\pi$
$463$	−7795.98	−0.782527	−0.391263	+	0.920279i	$0.627962\pi$
$464$	−464.000	−0.0464238
$465$	0	0
$466$	−3473.48	−0.345291
$467$	−13072.0	−1.29529	−0.647646	−	0.761941i	$-0.724247\pi$
$467$	−13072.0	−1.29529	−0.647646	+	0.761941i	$0.724247\pi$
$468$	0	0
$469$	13875.9	1.36616
$470$	−240.120	−0.0235658
$471$	0	0
$472$	4463.48	0.435272
$473$	1557.68	0.151421
$474$	0	0
$475$	−6607.35	−0.638244
$476$	−4620.43	−0.444910
$477$	0	0
$478$	8410.36	0.804771
$479$	−4847.90	−0.462435	−0.231218	−	0.972902i	$-0.574271\pi$
$479$	−4847.90	−0.462435	−0.231218	+	0.972902i	$0.574271\pi$
$480$	0	0
$481$	8441.48	0.800205
$482$	−12100.1	−1.14345
$483$	0	0
$484$	−4377.19	−0.411081
$485$	2630.99	0.246324
$486$	0	0
$487$	15092.4	1.40432	0.702160	−	0.712020i	$-0.252219\pi$
$487$	15092.4	1.40432	0.702160	+	0.712020i	$0.252219\pi$
$488$	−6477.42	−0.600859
$489$	0	0
$490$	−2.75574	−0.000254064 0
$491$	4438.48	0.407955	0.203978	−	0.978976i	$-0.434613\pi$
$491$	4438.48	0.407955	0.203978	+	0.978976i	$0.434613\pi$
$492$	0	0
$493$	−1810.27	−0.165376
$494$	−3078.71	−0.280400
$495$	0	0
$496$	−3313.26	−0.299939
$497$	−1013.50	−0.0914719
$498$	0	0
$499$	1648.66	0.147904	0.0739519	−	0.997262i	$-0.476439\pi$
$499$	1648.66	0.147904	0.0739519	+	0.997262i	$0.476439\pi$
$500$	2307.71	0.206408
$501$	0	0
$502$	−13066.4	−1.16172
$503$	−16870.8	−1.49549	−0.747746	−	0.663984i	$-0.768864\pi$
$503$	−16870.8	−1.49549	−0.747746	+	0.663984i	$0.768864\pi$
$504$	0	0
$505$	1753.24	0.154492
$506$	1363.46	0.119789
$507$	0	0
$508$	1454.83	0.127063
$509$	−11051.6	−0.962384	−0.481192	−	0.876615i	$-0.659796\pi$
$509$	−11051.6	−0.962384	−0.481192	+	0.876615i	$0.659796\pi$
$510$	0	0
$511$	3414.68	0.295610
$512$	−512.000	−0.0441942
$513$	0	0
$514$	4846.98	0.415936
$515$	4206.45	0.359919
$516$	0	0
$517$	−782.586	−0.0665727
$518$	11228.1	0.952380
$519$	0	0
$520$	525.387	0.0443072
$521$	15369.3	1.29240	0.646201	−	0.763167i	$-0.276357\pi$
$521$	15369.3	1.29240	0.646201	+	0.763167i	$0.276357\pi$
$522$	0	0
$523$	779.256	0.0651520	0.0325760	−	0.999469i	$-0.489629\pi$
$523$	779.256	0.0651520	0.0325760	+	0.999469i	$0.489629\pi$
$524$	2663.68	0.222068
$525$	0	0
$526$	−5384.88	−0.446373
$527$	−12926.5	−1.06847
$528$	0	0
$529$	−10203.5	−0.838624
$530$	−3269.08	−0.267924
$531$	0	0
$532$	−4095.01	−0.333724
$533$	−3497.81	−0.284253
$534$	0	0
$535$	4151.52	0.335487
$536$	5998.92	0.483422
$537$	0	0
$538$	−13377.2	−1.07199
$539$	−8.98133	−0.000717724 0
$540$	0	0
$541$	18847.1	1.49778	0.748889	−	0.662695i	$-0.230587\pi$
$541$	18847.1	1.49778	0.748889	+	0.662695i	$0.230587\pi$
$542$	−2546.60	−0.201819
$543$	0	0
$544$	−1997.54	−0.157433
$545$	−3769.04	−0.296235
$546$	0	0
$547$	−3422.52	−0.267525	−0.133763	−	0.991013i	$-0.542706\pi$
$547$	−3422.52	−0.267525	−0.133763	+	0.991013i	$0.542706\pi$
$548$	−1373.03	−0.107031
$549$	0	0
$550$	3674.87	0.284903
$551$	−1604.41	−0.124048
$552$	0	0
$553$	13931.6	1.07130
$554$	3321.00	0.254685
$555$	0	0
$556$	−3020.41	−0.230385
$557$	24115.0	1.83445	0.917224	−	0.398373i	$-0.130425\pi$
$557$	24115.0	1.83445	0.917224	+	0.398373i	$0.130425\pi$
$558$	0	0
$559$	2817.07	0.213148
$560$	698.821	0.0527332
$561$	0	0
$562$	−2495.42	−0.187301
$563$	−18904.9	−1.41518	−0.707591	−	0.706622i	$-0.750218\pi$
$563$	−18904.9	−1.41518	−0.707591	+	0.706622i	$0.750218\pi$
$564$	0	0
$565$	2341.17	0.174325
$566$	−13185.4	−0.979196
$567$	0	0
$568$	−438.162	−0.0323678
$569$	14072.1	1.03679	0.518394	−	0.855142i	$-0.326530\pi$
$569$	14072.1	1.03679	0.518394	+	0.855142i	$0.326530\pi$
$570$	0	0
$571$	16218.7	1.18867	0.594334	−	0.804218i	$-0.297416\pi$
$571$	16218.7	1.18867	0.594334	+	0.804218i	$0.297416\pi$
$572$	1712.31	0.125167
$573$	0	0
$574$	−4652.46	−0.338310
$575$	5292.01	0.383812
$576$	0	0
$577$	5122.09	0.369559	0.184779	−	0.982780i	$-0.440843\pi$
$577$	5122.09	0.369559	0.184779	+	0.982780i	$0.440843\pi$
$578$	2032.72	0.146280
$579$	0	0
$580$	273.796	0.0196013
$581$	−16691.4	−1.19187
$582$	0	0
$583$	−10654.4	−0.756879
$584$	1476.26	0.104603
$585$	0	0
$586$	1675.05	0.118081
$587$	−22661.7	−1.59344	−0.796721	−	0.604347i	$-0.793434\pi$
$587$	−22661.7	−1.59344	−0.796721	+	0.604347i	$0.793434\pi$
$588$	0	0
$589$	−11456.5	−0.801455
$590$	−2633.79	−0.183782
$591$	0	0
$592$	4854.20	0.337004
$593$	13919.5	0.963921	0.481960	−	0.876193i	$-0.339925\pi$
$593$	13919.5	0.963921	0.481960	+	0.876193i	$0.339925\pi$
$594$	0	0
$595$	2726.41	0.187852
$596$	12135.8	0.834066
$597$	0	0
$598$	2465.82	0.168620
$599$	10239.3	0.698441	0.349221	−	0.937041i	$-0.386446\pi$
$599$	10239.3	0.698441	0.349221	+	0.937041i	$0.386446\pi$
$600$	0	0
$601$	−6948.34	−0.471595	−0.235798	−	0.971802i	$-0.575770\pi$
$601$	−6948.34	−0.471595	−0.235798	+	0.971802i	$0.575770\pi$
$602$	3747.01	0.253682
$603$	0	0
$604$	−2895.70	−0.195073
$605$	2582.88	0.173568
$606$	0	0
$607$	−15449.1	−1.03305	−0.516525	−	0.856272i	$-0.672775\pi$
$607$	−15449.1	−1.03305	−0.516525	+	0.856272i	$0.672775\pi$
$608$	−1770.38	−0.118090
$609$	0	0
$610$	3822.18	0.253697
$611$	−1415.31	−0.0937107
$612$	0	0
$613$	8873.06	0.584632	0.292316	−	0.956322i	$-0.405574\pi$
$613$	8873.06	0.584632	0.292316	+	0.956322i	$0.405574\pi$
$614$	−1000.99	−0.0657926
$615$	0	0
$616$	2277.56	0.148970
$617$	16549.2	1.07981	0.539906	−	0.841725i	$-0.318460\pi$
$617$	16549.2	1.07981	0.539906	+	0.841725i	$0.318460\pi$
$618$	0	0
$619$	−1121.37	−0.0728140	−0.0364070	−	0.999337i	$-0.511591\pi$
$619$	−1121.37	−0.0728140	−0.0364070	+	0.999337i	$0.511591\pi$
$620$	1955.08	0.126642
$621$	0	0
$622$	4032.97	0.259980
$623$	17641.9	1.13452
$624$	0	0
$625$	13566.9	0.868281
$626$	−17478.1	−1.11592
$627$	0	0
$628$	2589.39	0.164535
$629$	18938.4	1.20051
$630$	0	0
$631$	1944.05	0.122649	0.0613245	−	0.998118i	$-0.480468\pi$
$631$	1944.05	0.122649	0.0613245	+	0.998118i	$0.480468\pi$
$632$	6023.00	0.379086
$633$	0	0
$634$	−3381.45	−0.211821
$635$	−858.464	−0.0536490
$636$	0	0
$637$	−16.2428	−0.00101030
$638$	892.339	0.0553731
$639$	0	0
$640$	302.119	0.0186599
$641$	11807.0	0.727530	0.363765	−	0.931491i	$-0.381491\pi$
$641$	11807.0	0.727530	0.363765	+	0.931491i	$0.381491\pi$
$642$	0	0
$643$	14848.2	0.910665	0.455333	−	0.890321i	$-0.349520\pi$
$643$	14848.2	0.910665	0.455333	+	0.890321i	$0.349520\pi$
$644$	3279.80	0.200687
$645$	0	0
$646$	−6907.05	−0.420672
$647$	4933.68	0.299788	0.149894	−	0.988702i	$-0.452107\pi$
$647$	4933.68	0.299788	0.149894	+	0.988702i	$0.452107\pi$
$648$	0	0
$649$	−8583.91	−0.519180
$650$	6646.00	0.401043
$651$	0	0
$652$	−6056.22	−0.363773
$653$	27029.7	1.61984	0.809919	−	0.586541i	$-0.199511\pi$
$653$	27029.7	1.61984	0.809919	+	0.586541i	$0.199511\pi$
$654$	0	0
$655$	−1571.78	−0.0937625
$656$	−2011.39	−0.119713
$657$	0	0
$658$	−1882.51	−0.111532
$659$	−12211.1	−0.721817	−0.360908	−	0.932601i	$-0.617533\pi$
$659$	−12211.1	−0.721817	−0.360908	+	0.932601i	$0.617533\pi$
$660$	0	0
$661$	−864.848	−0.0508906	−0.0254453	−	0.999676i	$-0.508100\pi$
$661$	−864.848	−0.0508906	−0.0254453	+	0.999676i	$0.508100\pi$
$662$	14892.4	0.874333
$663$	0	0
$664$	−7216.14	−0.421748
$665$	2416.37	0.140906
$666$	0	0
$667$	1285.02	0.0745968
$668$	−7364.21	−0.426542
$669$	0	0
$670$	−3539.83	−0.204113
$671$	12457.0	0.716688
$672$	0	0
$673$	−7448.31	−0.426614	−0.213307	−	0.976985i	$-0.568423\pi$
$673$	−7448.31	−0.426614	−0.213307	+	0.976985i	$0.568423\pi$
$674$	18733.4	1.07060
$675$	0	0
$676$	−5691.28	−0.323810
$677$	−9281.19	−0.526891	−0.263446	−	0.964674i	$-0.584859\pi$
$677$	−9281.19	−0.526891	−0.263446	+	0.964674i	$0.584859\pi$
$678$	0	0
$679$	20626.6	1.16580
$680$	1178.70	0.0664723
$681$	0	0
$682$	6371.87	0.357759
$683$	−32252.9	−1.80691	−0.903456	−	0.428680i	$-0.858979\pi$
$683$	−32252.9	−1.80691	−0.903456	+	0.428680i	$0.858979\pi$
$684$	0	0
$685$	810.196	0.0451913
$686$	−12715.7	−0.707707
$687$	0	0
$688$	1619.93	0.0897666
$689$	−19268.5	−1.06542
$690$	0	0
$691$	4915.05	0.270590	0.135295	−	0.990805i	$-0.456802\pi$
$691$	4915.05	0.270590	0.135295	+	0.990805i	$0.456802\pi$
$692$	1213.70	0.0666736
$693$	0	0
$694$	15902.2	0.869798
$695$	1782.28	0.0972742
$696$	0	0
$697$	−7847.31	−0.426453
$698$	2513.47	0.136298
$699$	0	0
$700$	8839.89	0.477309
$701$	−11610.7	−0.625577	−0.312788	−	0.949823i	$-0.601263\pi$
$701$	−11610.7	−0.625577	−0.312788	+	0.949823i	$0.601263\pi$
$702$	0	0
$703$	16784.8	0.900497
$704$	984.650	0.0527136
$705$	0	0
$706$	14618.3	0.779275
$707$	13745.2	0.731175
$708$	0	0
$709$	−6350.02	−0.336361	−0.168180	−	0.985756i	$-0.553789\pi$
$709$	−6350.02	−0.336361	−0.168180	+	0.985756i	$0.553789\pi$
$710$	258.550	0.0136665
$711$	0	0
$712$	7627.08	0.401456
$713$	9175.84	0.481960
$714$	0	0
$715$	−1010.40	−0.0528485
$716$	12146.4	0.633983
$717$	0	0
$718$	23110.2	1.20120
$719$	22323.6	1.15790	0.578949	−	0.815364i	$-0.303463\pi$
$719$	22323.6	1.15790	0.578949	+	0.815364i	$0.303463\pi$
$720$	0	0
$721$	32978.0	1.70342
$722$	7596.40	0.391563
$723$	0	0
$724$	−9359.10	−0.480425
$725$	3463.44	0.177419
$726$	0	0
$727$	−24589.5	−1.25443	−0.627217	−	0.778844i	$-0.715806\pi$
$727$	−24589.5	−1.25443	−0.627217	+	0.778844i	$0.715806\pi$
$728$	4118.97	0.209696
$729$	0	0
$730$	−871.108	−0.0441659
$731$	6320.08	0.319776
$732$	0	0
$733$	−35064.1	−1.76688	−0.883438	−	0.468548i	$-0.844777\pi$
$733$	−35064.1	−1.76688	−0.883438	+	0.468548i	$0.844777\pi$
$734$	4195.73	0.210991
$735$	0	0
$736$	1417.95	0.0710140
$737$	−11536.8	−0.576612
$738$	0	0
$739$	−33877.0	−1.68631	−0.843157	−	0.537668i	$-0.819305\pi$
$739$	−33877.0	−1.68631	−0.843157	+	0.537668i	$0.819305\pi$
$740$	−2864.35	−0.142291
$741$	0	0
$742$	−25629.2	−1.26803
$743$	−3119.59	−0.154033	−0.0770166	−	0.997030i	$-0.524539\pi$
$743$	−3119.59	−0.154033	−0.0770166	+	0.997030i	$0.524539\pi$
$744$	0	0
$745$	−7161.08	−0.352163
$746$	−27958.8	−1.37218
$747$	0	0
$748$	3841.55	0.187782
$749$	32547.3	1.58779
$750$	0	0
$751$	−26359.7	−1.28080	−0.640398	−	0.768043i	$-0.721231\pi$
$751$	−26359.7	−1.28080	−0.640398	+	0.768043i	$0.721231\pi$
$752$	−813.861	−0.0394660
$753$	0	0
$754$	1613.80	0.0779457
$755$	1708.68	0.0823647
$756$	0	0
$757$	−10385.9	−0.498654	−0.249327	−	0.968419i	$-0.580209\pi$
$757$	−10385.9	−0.498654	−0.249327	+	0.968419i	$0.580209\pi$
$758$	1788.14	0.0856837
$759$	0	0
$760$	1044.66	0.0498604
$761$	−24007.6	−1.14360	−0.571798	−	0.820395i	$-0.693754\pi$
$761$	−24007.6	−1.14360	−0.571798	+	0.820395i	$0.693754\pi$
$762$	0	0
$763$	−29548.7	−1.40201
$764$	16869.9	0.798862
$765$	0	0
$766$	10624.5	0.501148
$767$	−15524.0	−0.730821
$768$	0	0
$769$	12607.2	0.591195	0.295597	−	0.955313i	$-0.404481\pi$
$769$	12607.2	0.591195	0.295597	+	0.955313i	$0.404481\pi$
$770$	−1343.93	−0.0628987
$771$	0	0
$772$	−7850.35	−0.365985
$773$	−6237.38	−0.290224	−0.145112	−	0.989415i	$-0.546354\pi$
$773$	−6237.38	−0.290224	−0.145112	+	0.989415i	$0.546354\pi$
$774$	0	0
$775$	24731.2	1.14628
$776$	8917.44	0.412522
$777$	0	0
$778$	−8412.54	−0.387666
$779$	−6954.93	−0.319880
$780$	0	0
$781$	842.650	0.0386074
$782$	5532.05	0.252974
$783$	0	0
$784$	−9.34026	−0.000425486 0
$785$	−1527.94	−0.0694707
$786$	0	0
$787$	−6005.74	−0.272022	−0.136011	−	0.990707i	$-0.543428\pi$
$787$	−6005.74	−0.272022	−0.136011	+	0.990707i	$0.543428\pi$
$788$	−8162.72	−0.369016
$789$	0	0
$790$	−3554.03	−0.160059
$791$	18354.5	0.825044
$792$	0	0
$793$	22528.5	1.00884
$794$	−11404.9	−0.509752
$795$	0	0
$796$	12942.7	0.576310
$797$	6169.00	0.274175	0.137087	−	0.990559i	$-0.456226\pi$
$797$	6169.00	0.274175	0.137087	+	0.990559i	$0.456226\pi$
$798$	0	0
$799$	−3175.23	−0.140590
$800$	3821.73	0.168898
$801$	0	0
$802$	19606.1	0.863236
$803$	−2839.06	−0.124768
$804$	0	0
$805$	−1935.34	−0.0847350
$806$	11523.5	0.503597
$807$	0	0
$808$	5942.42	0.258730
$809$	−40910.3	−1.77791	−0.888956	−	0.457993i	$-0.848568\pi$
$809$	−40910.3	−1.77791	−0.888956	+	0.457993i	$0.848568\pi$
$810$	0	0
$811$	−28340.0	−1.22707	−0.613534	−	0.789669i	$-0.710252\pi$
$811$	−28340.0	−1.22707	−0.613534	+	0.789669i	$0.710252\pi$
$812$	2146.52	0.0927686
$813$	0	0
$814$	−9335.33	−0.401969
$815$	3573.64	0.153594
$816$	0	0
$817$	5601.38	0.239862
$818$	−16768.3	−0.716735
$819$	0	0
$820$	1186.87	0.0505456
$821$	17791.4	0.756303	0.378151	−	0.925744i	$-0.376560\pi$
$821$	17791.4	0.756303	0.378151	+	0.925744i	$0.376560\pi$
$822$	0	0
$823$	5375.77	0.227688	0.113844	−	0.993499i	$-0.463684\pi$
$823$	5375.77	0.227688	0.113844	+	0.993499i	$0.463684\pi$
$824$	14257.3	0.602762
$825$	0	0
$826$	−20648.6	−0.869802
$827$	−17894.5	−0.752422	−0.376211	−	0.926534i	$-0.622773\pi$
$827$	−17894.5	−0.752422	−0.376211	+	0.926534i	$0.622773\pi$
$828$	0	0
$829$	14195.1	0.594713	0.297357	−	0.954766i	$-0.403895\pi$
$829$	14195.1	0.594713	0.297357	+	0.954766i	$0.403895\pi$
$830$	4258.08	0.178072
$831$	0	0
$832$	1780.74	0.0742021
$833$	−36.4405	−0.00151571
$834$	0	0
$835$	4345.45	0.180097
$836$	3404.70	0.140854
$837$	0	0
$838$	−24700.5	−1.01821
$839$	18312.2	0.753524	0.376762	−	0.926310i	$-0.377038\pi$
$839$	18312.2	0.753524	0.376762	+	0.926310i	$0.377038\pi$
$840$	0	0
$841$	841.000	0.0344828
$842$	9553.30	0.391008
$843$	0	0
$844$	−15772.6	−0.643264
$845$	3358.29	0.136720
$846$	0	0
$847$	20249.4	0.821462
$848$	−11080.2	−0.448697
$849$	0	0
$850$	14910.2	0.601667
$851$	−13443.4	−0.541520
$852$	0	0
$853$	192.909	0.00774335	0.00387168	−	0.999993i	$-0.498768\pi$
$853$	192.909	0.00774335	0.00387168	+	0.999993i	$0.498768\pi$
$854$	29965.3	1.20069
$855$	0	0
$856$	14071.1	0.561846
$857$	20355.5	0.811355	0.405678	−	0.914016i	$-0.367036\pi$
$857$	20355.5	0.811355	0.405678	+	0.914016i	$0.367036\pi$
$858$	0	0
$859$	35959.3	1.42831	0.714153	−	0.699990i	$-0.246812\pi$
$859$	35959.3	1.42831	0.714153	+	0.699990i	$0.246812\pi$
$860$	−955.886	−0.0379017
$861$	0	0
$862$	−1072.17	−0.0423648
$863$	−7302.77	−0.288052	−0.144026	−	0.989574i	$-0.546005\pi$
$863$	−7302.77	−0.288052	−0.144026	+	0.989574i	$0.546005\pi$
$864$	0	0
$865$	−716.179	−0.0281512
$866$	−7060.17	−0.277038
$867$	0	0
$868$	15327.5	0.599367
$869$	−11583.1	−0.452163
$870$	0	0
$871$	−20864.3	−0.811666
$872$	−12774.7	−0.496109
$873$	0	0
$874$	4902.96	0.189754
$875$	−10675.8	−0.412464
$876$	0	0
$877$	36855.5	1.41907	0.709533	−	0.704673i	$-0.248906\pi$
$877$	36855.5	1.41907	0.709533	+	0.704673i	$0.248906\pi$
$878$	23073.3	0.886886
$879$	0	0
$880$	−581.019	−0.0222570
$881$	−9615.17	−0.367699	−0.183850	−	0.982954i	$-0.558856\pi$
$881$	−9615.17	−0.367699	−0.183850	+	0.982954i	$0.558856\pi$
$882$	0	0
$883$	46143.6	1.75861	0.879306	−	0.476257i	$-0.158007\pi$
$883$	46143.6	1.75861	0.879306	+	0.476257i	$0.158007\pi$
$884$	6947.46	0.264331
$885$	0	0
$886$	13551.9	0.513865
$887$	28495.0	1.07866	0.539328	−	0.842096i	$-0.318678\pi$
$887$	28495.0	1.07866	0.539328	+	0.842096i	$0.318678\pi$
$888$	0	0
$889$	−6730.24	−0.253909
$890$	−4500.57	−0.169505
$891$	0	0
$892$	−10123.3	−0.379993
$893$	−2814.15	−0.105456
$894$	0	0
$895$	−7167.31	−0.267684
$896$	2368.58	0.0883131
$897$	0	0
$898$	−13203.2	−0.490640
$899$	6005.27	0.222789
$900$	0	0
$901$	−43228.7	−1.59840
$902$	3868.19	0.142790
$903$	0	0
$904$	7935.14	0.291946
$905$	5522.59	0.202848
$906$	0	0
$907$	21873.4	0.800764	0.400382	−	0.916348i	$-0.368877\pi$
$907$	21873.4	0.800764	0.400382	+	0.916348i	$0.368877\pi$
$908$	−21719.1	−0.793803
$909$	0	0
$910$	−2430.51	−0.0885390
$911$	−32308.0	−1.17498	−0.587492	−	0.809230i	$-0.699885\pi$
$911$	−32308.0	−1.17498	−0.587492	+	0.809230i	$0.699885\pi$
$912$	0	0
$913$	13877.7	0.503049
$914$	8909.88	0.322443
$915$	0	0
$916$	22390.3	0.807637
$917$	−12322.5	−0.443757
$918$	0	0
$919$	4297.83	0.154268	0.0771340	−	0.997021i	$-0.475423\pi$
$919$	4297.83	0.154268	0.0771340	+	0.997021i	$0.475423\pi$
$920$	−836.699	−0.0299839
$921$	0	0
$922$	3723.23	0.132991
$923$	1523.93	0.0543455
$924$	0	0
$925$	−36233.3	−1.28794
$926$	15592.0	0.553330
$927$	0	0
$928$	928.000	0.0328266
$929$	−36879.6	−1.30245	−0.651227	−	0.758883i	$-0.725746\pi$
$929$	−36879.6	−1.30245	−0.651227	+	0.758883i	$0.725746\pi$
$930$	0	0
$931$	−32.2966	−0.00113692
$932$	6946.96	0.244158
$933$	0	0
$934$	26144.1	0.915910
$935$	−2266.81	−0.0792863
$936$	0	0
$937$	−15770.1	−0.549824	−0.274912	−	0.961469i	$-0.588649\pi$
$937$	−15770.1	−0.549824	−0.274912	+	0.961469i	$0.588649\pi$
$938$	−27751.8	−0.966020
$939$	0	0
$940$	480.241	0.0166635
$941$	485.478	0.0168184	0.00840921	−	0.999965i	$-0.497323\pi$
$941$	485.478	0.0168184	0.00840921	+	0.999965i	$0.497323\pi$
$942$	0	0
$943$	5570.40	0.192362
$944$	−8926.95	−0.307783
$945$	0	0
$946$	−3115.37	−0.107071
$947$	−8167.18	−0.280251	−0.140126	−	0.990134i	$-0.544751\pi$
$947$	−8167.18	−0.280251	−0.140126	+	0.990134i	$0.544751\pi$
$948$	0	0
$949$	−5134.45	−0.175628
$950$	13214.7	0.451307
$951$	0	0
$952$	9240.86	0.314599
$953$	29839.4	1.01426	0.507132	−	0.861868i	$-0.330706\pi$
$953$	29839.4	1.01426	0.507132	+	0.861868i	$0.330706\pi$
$954$	0	0
$955$	−9954.53	−0.337299
$956$	−16820.7	−0.569059
$957$	0	0
$958$	9695.81	0.326991
$959$	6351.83	0.213880
$960$	0	0
$961$	13090.5	0.439411
$962$	−16883.0	−0.565830
$963$	0	0
$964$	24200.2	0.808543
$965$	4632.31	0.154528
$966$	0	0
$967$	−8150.60	−0.271050	−0.135525	−	0.990774i	$-0.543272\pi$
$967$	−8150.60	−0.271050	−0.135525	+	0.990774i	$0.543272\pi$
$968$	8754.38	0.290678
$969$	0	0
$970$	−5261.98	−0.174177
$971$	34617.7	1.14411	0.572057	−	0.820214i	$-0.306146\pi$
$971$	34617.7	1.14411	0.572057	+	0.820214i	$0.306146\pi$
$972$	0	0
$973$	13972.8	0.460378
$974$	−30184.9	−0.993004
$975$	0	0
$976$	12954.8	0.424871
$977$	40762.0	1.33479	0.667395	−	0.744704i	$-0.267409\pi$
$977$	40762.0	1.33479	0.667395	+	0.744704i	$0.267409\pi$
$978$	0	0
$979$	−14668.0	−0.478846
$980$	5.51147	0.000179651 0
$981$	0	0
$982$	−8876.97	−0.288468
$983$	15737.6	0.510631	0.255315	−	0.966858i	$-0.417821\pi$
$983$	15737.6	0.510631	0.255315	+	0.966858i	$0.417821\pi$
$984$	0	0
$985$	4816.63	0.155808
$986$	3620.54	0.116939
$987$	0	0
$988$	6157.42	0.198273
$989$	−4486.30	−0.144243
$990$	0	0
$991$	−47989.9	−1.53829	−0.769146	−	0.639073i	$-0.779318\pi$
$991$	−47989.9	−1.53829	−0.769146	+	0.639073i	$0.779318\pi$
$992$	6626.51	0.212089
$993$	0	0
$994$	2026.99	0.0646804
$995$	−7637.20	−0.243332
$996$	0	0
$997$	−51443.4	−1.63413	−0.817066	−	0.576545i	$-0.804401\pi$
$997$	−51443.4	−1.63413	−0.817066	+	0.576545i	$0.804401\pi$
$998$	−3297.31	−0.104584
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	522.4.a.k.1.2		3
3.2	odd	2		58.4.a.d.1.3	✓	3
12.11	even	2		464.4.a.i.1.1		3
15.14	odd	2		1450.4.a.h.1.1		3
24.5	odd	2		1856.4.a.r.1.1		3
24.11	even	2		1856.4.a.s.1.3		3
87.86	odd	2		1682.4.a.d.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
58.4.a.d.1.3	✓	3	3.2	odd	2
464.4.a.i.1.1		3	12.11	even	2
522.4.a.k.1.2		3	1.1	even	1	trivial
1450.4.a.h.1.1		3	15.14	odd	2
1682.4.a.d.1.1		3	87.86	odd	2
1856.4.a.r.1.1		3	24.5	odd	2
1856.4.a.s.1.3		3	24.11	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 522 = 2 \cdot 3^{2} \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	522.a (trivial)

\(f(q)\)	\(=\)	\(q-2.00000 q^{2} +4.00000 q^{4} -2.36031 q^{5} -18.5045 q^{7} -8.00000 q^{8} +4.72062 q^{10} +15.3852 q^{11} +27.8241 q^{13} +37.0090 q^{14} +16.0000 q^{16} +62.4231 q^{17} +55.3245 q^{19} -9.44123 q^{20} -30.7703 q^{22} -44.3109 q^{23} -119.429 q^{25} -55.6482 q^{26} -74.0180 q^{28} -29.0000 q^{29} -207.078 q^{31} -32.0000 q^{32} -124.846 q^{34} +43.6763 q^{35} +303.388 q^{37} -110.649 q^{38} +18.8825 q^{40} -125.712 q^{41} +101.246 q^{43} +61.5406 q^{44} +88.6218 q^{46} -50.8663 q^{47} -0.583766 q^{49} +238.858 q^{50} +111.296 q^{52} -692.512 q^{53} -36.3137 q^{55} +148.036 q^{56} +58.0000 q^{58} -557.935 q^{59} +809.678 q^{61} +414.157 q^{62} +64.0000 q^{64} -65.6734 q^{65} -749.866 q^{67} +249.692 q^{68} -87.3526 q^{70} +54.7703 q^{71} -184.533 q^{73} -606.775 q^{74} +221.298 q^{76} -284.694 q^{77} -752.875 q^{79} -37.7649 q^{80} +251.423 q^{82} +902.018 q^{83} -147.338 q^{85} -202.492 q^{86} -123.081 q^{88} -953.385 q^{89} -514.871 q^{91} -177.244 q^{92} +101.733 q^{94} -130.583 q^{95} -1114.68 q^{97} +1.16753 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 6 q^{2} + 12 q^{4} - 20 q^{5} + 24 q^{7} - 24 q^{8} + 40 q^{10} - 10 q^{11} - 4 q^{13} - 48 q^{14} + 48 q^{16} + 66 q^{17} - 164 q^{19} - 80 q^{20} + 20 q^{22} + 204 q^{23} + 79 q^{25} + 8 q^{26}+ \cdots - 1078 q^{98}+O(q^{100}) \) 3 * q - 6 * q^2 + 12 * q^4 - 20 * q^5 + 24 * q^7 - 24 * q^8 + 40 * q^10 - 10 * q^11 - 4 * q^13 - 48 * q^14 + 48 * q^16 + 66 * q^17 - 164 * q^19 - 80 * q^20 + 20 * q^22 + 204 * q^23 + 79 * q^25 + 8 * q^26 + 96 * q^28 - 87 * q^29 - 86 * q^31 - 96 * q^32 - 132 * q^34 - 24 * q^35 - 42 * q^37 + 328 * q^38 + 160 * q^40 - 562 * q^41 + 18 * q^43 - 40 * q^44 - 408 * q^46 - 654 * q^47 + 539 * q^49 - 158 * q^50 - 16 * q^52 - 712 * q^53 + 142 * q^55 - 192 * q^56 + 174 * q^58 - 184 * q^59 + 322 * q^61 + 172 * q^62 + 192 * q^64 - 1494 * q^65 - 228 * q^67 + 264 * q^68 + 48 * q^70 + 52 * q^71 - 494 * q^73 + 84 * q^74 - 656 * q^76 - 872 * q^77 - 2110 * q^79 - 320 * q^80 + 1124 * q^82 + 288 * q^83 - 2704 * q^85 - 36 * q^86 + 80 * q^88 - 914 * q^89 - 2984 * q^91 + 816 * q^92 + 1308 * q^94 + 1900 * q^95 + 218 * q^97 - 1078 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more