LMFDB - Embedded newform 693.4.a.l.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [693,4,Mod(1,693)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(693, 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("693.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 693 = 3^{2} \cdot 7 \cdot 11 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	693.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:	$40.8883236340$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.522072.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 12x^{2} + 5x + 1 $ x^4 - x^3 - 12x^2 + 5x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 77)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$3.79597$ of defining polynomial
Character	$\chi$	$=$	693.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q-1.53253 q^{2} -5.65135 q^{4} -8.69995 q^{5} -7.00000 q^{7} +20.9211 q^{8} +13.3330 q^{10} +11.0000 q^{11} -76.3572 q^{13} +10.7277 q^{14} +13.1485 q^{16} -39.7278 q^{17} -27.9876 q^{19} +49.1664 q^{20} -16.8579 q^{22} -87.2055 q^{23} -49.3108 q^{25} +117.020 q^{26} +39.5594 q^{28} +38.3019 q^{29} -186.071 q^{31} -187.519 q^{32} +60.8842 q^{34} +60.8997 q^{35} -218.781 q^{37} +42.8919 q^{38} -182.013 q^{40} -80.1687 q^{41} -35.1155 q^{43} -62.1648 q^{44} +133.645 q^{46} +282.620 q^{47} +49.0000 q^{49} +75.5704 q^{50} +431.521 q^{52} -145.296 q^{53} -95.6995 q^{55} -146.448 q^{56} -58.6989 q^{58} -91.0461 q^{59} +808.142 q^{61} +285.160 q^{62} +182.192 q^{64} +664.304 q^{65} +794.222 q^{67} +224.516 q^{68} -93.3307 q^{70} -946.901 q^{71} +801.324 q^{73} +335.288 q^{74} +158.168 q^{76} -77.0000 q^{77} -890.737 q^{79} -114.391 q^{80} +122.861 q^{82} +559.333 q^{83} +345.630 q^{85} +53.8156 q^{86} +230.132 q^{88} +1523.75 q^{89} +534.501 q^{91} +492.829 q^{92} -433.125 q^{94} +243.491 q^{95} +664.651 q^{97} -75.0941 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 2 q^{2} + 26 q^{4} - 10 q^{5} - 28 q^{7} + 18 q^{8} - 2 q^{10} + 44 q^{11} + 58 q^{13} - 14 q^{14} + 2 q^{16} - 4 q^{17} + 258 q^{19} - 182 q^{20} + 22 q^{22} - 8 q^{23} + 80 q^{25} + 482 q^{26} - 182 q^{28}+ \cdots + 98 q^{98}+O(q^{100}) $ 4 * q + 2 * q^2 + 26 * q^4 - 10 * q^5 - 28 * q^7 + 18 * q^8 - 2 * q^10 + 44 * q^11 + 58 * q^13 - 14 * q^14 + 2 * q^16 - 4 * q^17 + 258 * q^19 - 182 * q^20 + 22 * q^22 - 8 * q^23 + 80 * q^25 + 482 * q^26 - 182 * q^28 + 396 * q^29 - 56 * q^31 - 134 * q^32 + 472 * q^34 + 70 * q^35 + 84 * q^37 + 942 * q^38 - 1026 * q^40 - 52 * q^41 + 408 * q^43 + 286 * q^44 + 368 * q^46 - 8 * q^47 + 196 * q^49 + 1642 * q^50 + 2030 * q^52 - 624 * q^53 - 110 * q^55 - 126 * q^56 + 864 * q^58 + 238 * q^59 - 162 * q^61 - 688 * q^62 - 902 * q^64 + 32 * q^65 + 1340 * q^67 + 1384 * q^68 + 14 * q^70 - 1788 * q^71 + 1456 * q^73 - 996 * q^74 + 3042 * q^76 - 308 * q^77 - 1324 * q^79 - 2342 * q^80 + 1984 * q^82 - 450 * q^83 - 1736 * q^85 + 4380 * q^86 + 198 * q^88 + 3072 * q^89 - 406 * q^91 - 544 * q^92 - 1696 * q^94 - 24 * q^95 - 652 * q^97 + 98 * 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 $

$2$	−1.53253	−0.541832	−0.270916	−	0.962603i	$-0.587327\pi$
$2$	−1.53253	−0.541832	−0.270916	+	0.962603i	$0.587327\pi$
$3$	0	0
$3$	0	0
$4$	−5.65135	−0.706418
$5$	−8.69995	−0.778148	−0.389074	−	0.921207i	$-0.627205\pi$
$5$	−8.69995	−0.778148	−0.389074	+	0.921207i	$0.627205\pi$
$6$	0	0
$7$	−7.00000	−0.377964
$7$	−7.00000	−0.377964
$8$	20.9211	0.924592
$9$	0	0
$10$	13.3330	0.421625
$11$	11.0000	0.301511
$11$	11.0000	0.301511
$12$	0	0
$13$	−76.3572	−1.62905	−0.814526	−	0.580127i	$-0.803003\pi$
$13$	−76.3572	−1.62905	−0.814526	+	0.580127i	$0.803003\pi$
$14$	10.7277	0.204793
$15$	0	0
$16$	13.1485	0.205445
$17$	−39.7278	−0.566789	−0.283395	−	0.959003i	$-0.591461\pi$
$17$	−39.7278	−0.566789	−0.283395	+	0.959003i	$0.591461\pi$
$18$	0	0
$19$	−27.9876	−0.337937	−0.168968	−	0.985621i	$-0.554044\pi$
$19$	−27.9876	−0.337937	−0.168968	+	0.985621i	$0.554044\pi$
$20$	49.1664	0.549698
$21$	0	0
$22$	−16.8579	−0.163368
$23$	−87.2055	−0.790592	−0.395296	−	0.918554i	$-0.629358\pi$
$23$	−87.2055	−0.790592	−0.395296	+	0.918554i	$0.629358\pi$
$24$	0	0
$25$	−49.3108	−0.394486
$26$	117.020	0.882673
$27$	0	0
$28$	39.5594	0.267001
$29$	38.3019	0.245258	0.122629	−	0.992453i	$-0.460867\pi$
$29$	38.3019	0.245258	0.122629	+	0.992453i	$0.460867\pi$
$30$	0	0
$31$	−186.071	−1.07804	−0.539021	−	0.842292i	$-0.681206\pi$
$31$	−186.071	−1.07804	−0.539021	+	0.842292i	$0.681206\pi$
$32$	−187.519	−1.03591
$33$	0	0
$34$	60.8842	0.307105
$35$	60.8997	0.294112
$36$	0	0
$37$	−218.781	−0.972090	−0.486045	−	0.873934i	$-0.661561\pi$
$37$	−218.781	−0.972090	−0.486045	+	0.873934i	$0.661561\pi$
$38$	42.8919	0.183105
$39$	0	0
$40$	−182.013	−0.719469
$41$	−80.1687	−0.305372	−0.152686	−	0.988275i	$-0.548792\pi$
$41$	−80.1687	−0.305372	−0.152686	+	0.988275i	$0.548792\pi$
$42$	0	0
$43$	−35.1155	−0.124536	−0.0622681	−	0.998059i	$-0.519833\pi$
$43$	−35.1155	−0.124536	−0.0622681	+	0.998059i	$0.519833\pi$
$44$	−62.1648	−0.212993
$45$	0	0
$46$	133.645	0.428368
$47$	282.620	0.877116	0.438558	−	0.898703i	$-0.355489\pi$
$47$	282.620	0.877116	0.438558	+	0.898703i	$0.355489\pi$
$48$	0	0
$49$	49.0000	0.142857
$50$	75.5704	0.213745
$51$	0	0
$52$	431.521	1.15079
$53$	−145.296	−0.376566	−0.188283	−	0.982115i	$-0.560292\pi$
$53$	−145.296	−0.376566	−0.188283	+	0.982115i	$0.560292\pi$
$54$	0	0
$55$	−95.6995	−0.234620
$56$	−146.448	−0.349463
$57$	0	0
$58$	−58.6989	−0.132889
$59$	−91.0461	−0.200901	−0.100451	−	0.994942i	$-0.532028\pi$
$59$	−91.0461	−0.200901	−0.100451	+	0.994942i	$0.532028\pi$
$60$	0	0
$61$	808.142	1.69626	0.848131	−	0.529787i	$-0.177728\pi$
$61$	808.142	1.69626	0.848131	+	0.529787i	$0.177728\pi$
$62$	285.160	0.584118
$63$	0	0
$64$	182.192	0.355843
$65$	664.304	1.26764
$66$	0	0
$67$	794.222	1.44820	0.724102	−	0.689693i	$-0.242255\pi$
$67$	794.222	1.44820	0.724102	+	0.689693i	$0.242255\pi$
$68$	224.516	0.400390
$69$	0	0
$70$	−93.3307	−0.159359
$71$	−946.901	−1.58277	−0.791384	−	0.611319i	$-0.790639\pi$
$71$	−946.901	−1.58277	−0.791384	+	0.611319i	$0.790639\pi$
$72$	0	0
$73$	801.324	1.28477	0.642383	−	0.766384i	$-0.277946\pi$
$73$	801.324	1.28477	0.642383	+	0.766384i	$0.277946\pi$
$74$	335.288	0.526709
$75$	0	0
$76$	158.168	0.238725
$77$	−77.0000	−0.113961
$78$	0	0
$79$	−890.737	−1.26855	−0.634277	−	0.773106i	$-0.718702\pi$
$79$	−890.737	−1.26855	−0.634277	+	0.773106i	$0.718702\pi$
$80$	−114.391	−0.159866
$81$	0	0
$82$	122.861	0.165460
$83$	559.333	0.739696	0.369848	−	0.929092i	$-0.379410\pi$
$83$	559.333	0.739696	0.369848	+	0.929092i	$0.379410\pi$
$84$	0	0
$85$	345.630	0.441046
$86$	53.8156	0.0674777
$87$	0	0
$88$	230.132	0.278775
$89$	1523.75	1.81480	0.907401	−	0.420265i	$-0.138063\pi$
$89$	1523.75	1.81480	0.907401	+	0.420265i	$0.138063\pi$
$90$	0	0
$91$	534.501	0.615724
$92$	492.829	0.558488
$93$	0	0
$94$	−433.125	−0.475249
$95$	243.491	0.262965
$96$	0	0
$97$	664.651	0.695723	0.347861	−	0.937546i	$-0.386908\pi$
$97$	664.651	0.695723	0.347861	+	0.937546i	$0.386908\pi$
$98$	−75.0941	−0.0774046
$99$	0	0
$100$	278.672	0.278672
$101$	−1495.68	−1.47352	−0.736761	−	0.676153i	$-0.763646\pi$
$101$	−1495.68	−1.47352	−0.736761	+	0.676153i	$0.763646\pi$
$102$	0	0
$103$	874.379	0.836457	0.418229	−	0.908342i	$-0.362651\pi$
$103$	874.379	0.836457	0.418229	+	0.908342i	$0.362651\pi$
$104$	−1597.48	−1.50621
$105$	0	0
$106$	222.671	0.204035
$107$	−783.854	−0.708205	−0.354103	−	0.935207i	$-0.615214\pi$
$107$	−783.854	−0.708205	−0.354103	+	0.935207i	$0.615214\pi$
$108$	0	0
$109$	−1351.08	−1.18725	−0.593623	−	0.804743i	$-0.702303\pi$
$109$	−1351.08	−1.18725	−0.593623	+	0.804743i	$0.702303\pi$
$110$	146.663	0.127125
$111$	0	0
$112$	−92.0393	−0.0776508
$113$	188.362	0.156811	0.0784055	−	0.996922i	$-0.475017\pi$
$113$	188.362	0.156811	0.0784055	+	0.996922i	$0.475017\pi$
$114$	0	0
$115$	758.684	0.615197
$116$	−216.457	−0.173255
$117$	0	0
$118$	139.531	0.108855
$119$	278.095	0.214226
$120$	0	0
$121$	121.000	0.0909091
$122$	−1238.50	−0.919089
$123$	0	0
$124$	1051.55	0.761549
$125$	1516.50	1.08512
$126$	0	0
$127$	586.957	0.410110	0.205055	−	0.978750i	$-0.434263\pi$
$127$	586.957	0.410110	0.205055	+	0.978750i	$0.434263\pi$
$128$	1220.94	0.843101
$129$	0	0
$130$	−1018.07	−0.686850
$131$	−623.054	−0.415546	−0.207773	−	0.978177i	$-0.566621\pi$
$131$	−623.054	−0.415546	−0.207773	+	0.978177i	$0.566621\pi$
$132$	0	0
$133$	195.913	0.127728
$134$	−1217.17	−0.784683
$135$	0	0
$136$	−831.151	−0.524049
$137$	954.859	0.595468	0.297734	−	0.954649i	$-0.403769\pi$
$137$	954.859	0.595468	0.297734	+	0.954649i	$0.403769\pi$
$138$	0	0
$139$	1590.62	0.970610	0.485305	−	0.874345i	$-0.338709\pi$
$139$	1590.62	0.970610	0.485305	+	0.874345i	$0.338709\pi$
$140$	−344.165	−0.207766
$141$	0	0
$142$	1451.16	0.857594
$143$	−839.929	−0.491178
$144$	0	0
$145$	−333.225	−0.190847
$146$	−1228.05	−0.696127
$147$	0	0
$148$	1236.41	0.686702
$149$	1415.84	0.778459	0.389229	−	0.921141i	$-0.372741\pi$
$149$	1415.84	0.778459	0.389229	+	0.921141i	$0.372741\pi$
$150$	0	0
$151$	411.564	0.221806	0.110903	−	0.993831i	$-0.464626\pi$
$151$	411.564	0.221806	0.110903	+	0.993831i	$0.464626\pi$
$152$	−585.532	−0.312454
$153$	0	0
$154$	118.005	0.0617475
$155$	1618.81	0.838876
$156$	0	0
$157$	−1417.29	−0.720460	−0.360230	−	0.932864i	$-0.617302\pi$
$157$	−1417.29	−0.720460	−0.360230	+	0.932864i	$0.617302\pi$
$158$	1365.08	0.687343
$159$	0	0
$160$	1631.41	0.806090
$161$	610.439	0.298816
$162$	0	0
$163$	−441.401	−0.212105	−0.106053	−	0.994361i	$-0.533821\pi$
$163$	−441.401	−0.212105	−0.106053	+	0.994361i	$0.533821\pi$
$164$	453.061	0.215720
$165$	0	0
$166$	−857.196	−0.400791
$167$	1486.39	0.688742	0.344371	−	0.938834i	$-0.388092\pi$
$167$	1486.39	0.688742	0.344371	+	0.938834i	$0.388092\pi$
$168$	0	0
$169$	3633.43	1.65381
$170$	−529.690	−0.238973
$171$	0	0
$172$	198.450	0.0879747
$173$	2957.39	1.29969	0.649844	−	0.760068i	$-0.274834\pi$
$173$	2957.39	1.29969	0.649844	+	0.760068i	$0.274834\pi$
$174$	0	0
$175$	345.176	0.149102
$176$	144.633	0.0619439
$177$	0	0
$178$	−2335.20	−0.983318
$179$	24.4424	0.0102062	0.00510310	−	0.999987i	$-0.498376\pi$
$179$	24.4424	0.0102062	0.00510310	+	0.999987i	$0.498376\pi$
$180$	0	0
$181$	−120.126	−0.0493308	−0.0246654	−	0.999696i	$-0.507852\pi$
$181$	−120.126	−0.0493308	−0.0246654	+	0.999696i	$0.507852\pi$
$182$	−819.139	−0.333619
$183$	0	0
$184$	−1824.44	−0.730975
$185$	1903.38	0.756429
$186$	0	0
$187$	−437.006	−0.170893
$188$	−1597.19	−0.619610
$189$	0	0
$190$	−373.158	−0.142483
$191$	2059.27	0.780123	0.390061	−	0.920789i	$-0.372454\pi$
$191$	2059.27	0.780123	0.390061	+	0.920789i	$0.372454\pi$
$192$	0	0
$193$	−4371.31	−1.63033	−0.815165	−	0.579229i	$-0.803354\pi$
$193$	−4371.31	−1.63033	−0.815165	+	0.579229i	$0.803354\pi$
$194$	−1018.60	−0.376965
$195$	0	0
$196$	−276.916	−0.100917
$197$	2185.70	0.790479	0.395240	−	0.918578i	$-0.370662\pi$
$197$	2185.70	0.790479	0.395240	+	0.918578i	$0.370662\pi$
$198$	0	0
$199$	−2420.84	−0.862356	−0.431178	−	0.902267i	$-0.641902\pi$
$199$	−2420.84	−0.862356	−0.431178	+	0.902267i	$0.641902\pi$
$200$	−1031.64	−0.364739
$201$	0	0
$202$	2292.18	0.798401
$203$	−268.113	−0.0926988
$204$	0	0
$205$	697.464	0.237624
$206$	−1340.01	−0.453219
$207$	0	0
$208$	−1003.98	−0.334680
$209$	−307.864	−0.101892
$210$	0	0
$211$	−3888.39	−1.26866	−0.634331	−	0.773062i	$-0.718724\pi$
$211$	−3888.39	−1.26866	−0.634331	+	0.773062i	$0.718724\pi$
$212$	821.120	0.266013
$213$	0	0
$214$	1201.28	0.383728
$215$	305.503	0.0969076
$216$	0	0
$217$	1302.50	0.407462
$218$	2070.57	0.643288
$219$	0	0
$220$	540.831	0.165740
$221$	3033.51	0.923330
$222$	0	0
$223$	641.467	0.192627	0.0963135	−	0.995351i	$-0.469295\pi$
$223$	641.467	0.192627	0.0963135	+	0.995351i	$0.469295\pi$
$224$	1312.64	0.391537
$225$	0	0
$226$	−288.672	−0.0849652
$227$	−3619.11	−1.05819	−0.529094	−	0.848563i	$-0.677468\pi$
$227$	−3619.11	−1.05819	−0.529094	+	0.848563i	$0.677468\pi$
$228$	0	0
$229$	2518.14	0.726652	0.363326	−	0.931662i	$-0.381641\pi$
$229$	2518.14	0.726652	0.363326	+	0.931662i	$0.381641\pi$
$230$	−1162.71	−0.333333
$231$	0	0
$232$	801.318	0.226763
$233$	4187.89	1.17750	0.588751	−	0.808315i	$-0.299620\pi$
$233$	4187.89	1.17750	0.588751	+	0.808315i	$0.299620\pi$
$234$	0	0
$235$	−2458.79	−0.682525
$236$	514.533	0.141920
$237$	0	0
$238$	−426.189	−0.116075
$239$	−2582.63	−0.698981	−0.349490	−	0.936940i	$-0.613645\pi$
$239$	−2582.63	−0.698981	−0.349490	+	0.936940i	$0.613645\pi$
$240$	0	0
$241$	1522.09	0.406832	0.203416	−	0.979092i	$-0.434796\pi$
$241$	1522.09	0.406832	0.203416	+	0.979092i	$0.434796\pi$
$242$	−185.436	−0.0492574
$243$	0	0
$244$	−4567.09	−1.19827
$245$	−426.298	−0.111164
$246$	0	0
$247$	2137.06	0.550517
$248$	−3892.81	−0.996750
$249$	0	0
$250$	−2324.08	−0.587951
$251$	−1463.19	−0.367952	−0.183976	−	0.982931i	$-0.558897\pi$
$251$	−1463.19	−0.367952	−0.183976	+	0.982931i	$0.558897\pi$
$252$	0	0
$253$	−959.261	−0.238372
$254$	−899.531	−0.222211
$255$	0	0
$256$	−3328.67	−0.812662
$257$	1183.89	0.287350	0.143675	−	0.989625i	$-0.454108\pi$
$257$	1183.89	0.287350	0.143675	+	0.989625i	$0.454108\pi$
$258$	0	0
$259$	1531.46	0.367415
$260$	−3754.21	−0.895486
$261$	0	0
$262$	954.850	0.225156
$263$	−151.973	−0.0356315	−0.0178158	−	0.999841i	$-0.505671\pi$
$263$	−151.973	−0.0356315	−0.0178158	+	0.999841i	$0.505671\pi$
$264$	0	0
$265$	1264.07	0.293024
$266$	−300.243	−0.0692072
$267$	0	0
$268$	−4488.42	−1.02304
$269$	255.543	0.0579209	0.0289604	−	0.999581i	$-0.490780\pi$
$269$	255.543	0.0579209	0.0289604	+	0.999581i	$0.490780\pi$
$270$	0	0
$271$	2589.73	0.580497	0.290248	−	0.956951i	$-0.406262\pi$
$271$	2589.73	0.580497	0.290248	+	0.956951i	$0.406262\pi$
$272$	−522.360	−0.116444
$273$	0	0
$274$	−1463.35	−0.322644
$275$	−542.419	−0.118942
$276$	0	0
$277$	−4441.49	−0.963404	−0.481702	−	0.876335i	$-0.659981\pi$
$277$	−4441.49	−0.963404	−0.481702	+	0.876335i	$0.659981\pi$
$278$	−2437.68	−0.525907
$279$	0	0
$280$	1274.09	0.271934
$281$	−1577.34	−0.334861	−0.167431	−	0.985884i	$-0.553547\pi$
$281$	−1577.34	−0.334861	−0.167431	+	0.985884i	$0.553547\pi$
$282$	0	0
$283$	3429.29	0.720317	0.360159	−	0.932891i	$-0.382723\pi$
$283$	3429.29	0.720317	0.360159	+	0.932891i	$0.382723\pi$
$284$	5351.27	1.11810
$285$	0	0
$286$	1287.22	0.266136
$287$	561.181	0.115420
$288$	0	0
$289$	−3334.70	−0.678750
$290$	510.677	0.103407
$291$	0	0
$292$	−4528.56	−0.907581
$293$	−4601.37	−0.917458	−0.458729	−	0.888576i	$-0.651695\pi$
$293$	−4601.37	−0.917458	−0.458729	+	0.888576i	$0.651695\pi$
$294$	0	0
$295$	792.096	0.156331
$296$	−4577.14	−0.898786
$297$	0	0
$298$	−2169.82	−0.421794
$299$	6658.77	1.28792
$300$	0	0
$301$	245.808	0.0470703
$302$	−630.736	−0.120181
$303$	0	0
$304$	−367.994	−0.0694274
$305$	−7030.80	−1.31994
$306$	0	0
$307$	4990.18	0.927702	0.463851	−	0.885913i	$-0.346467\pi$
$307$	4990.18	0.927702	0.463851	+	0.885913i	$0.346467\pi$
$308$	435.154	0.0805038
$309$	0	0
$310$	−2480.88	−0.454530
$311$	−3139.78	−0.572478	−0.286239	−	0.958158i	$-0.592405\pi$
$311$	−3139.78	−0.572478	−0.286239	+	0.958158i	$0.592405\pi$
$312$	0	0
$313$	−4723.12	−0.852929	−0.426464	−	0.904504i	$-0.640241\pi$
$313$	−4723.12	−0.852929	−0.426464	+	0.904504i	$0.640241\pi$
$314$	2172.04	0.390368
$315$	0	0
$316$	5033.86	0.896130
$317$	−5935.83	−1.05170	−0.525851	−	0.850577i	$-0.676253\pi$
$317$	−5935.83	−1.05170	−0.525851	+	0.850577i	$0.676253\pi$
$318$	0	0
$319$	421.321	0.0739481
$320$	−1585.06	−0.276899
$321$	0	0
$322$	−935.517	−0.161908
$323$	1111.89	0.191539
$324$	0	0
$325$	3765.24	0.642639
$326$	676.461	0.114925
$327$	0	0
$328$	−1677.22	−0.282344
$329$	−1978.34	−0.331519
$330$	0	0
$331$	−6390.75	−1.06123	−0.530615	−	0.847613i	$-0.678039\pi$
$331$	−6390.75	−1.06123	−0.530615	+	0.847613i	$0.678039\pi$
$332$	−3160.98	−0.522535
$333$	0	0
$334$	−2277.93	−0.373183
$335$	−6909.69	−1.12692
$336$	0	0
$337$	8916.41	1.44127	0.720635	−	0.693315i	$-0.243851\pi$
$337$	8916.41	1.44127	0.720635	+	0.693315i	$0.243851\pi$
$338$	−5568.34	−0.896088
$339$	0	0
$340$	−1953.28	−0.311563
$341$	−2046.78	−0.325042
$342$	0	0
$343$	−343.000	−0.0539949
$344$	−734.655	−0.115145
$345$	0	0
$346$	−4532.29	−0.704212
$347$	−1400.52	−0.216668	−0.108334	−	0.994115i	$-0.534552\pi$
$347$	−1400.52	−0.216668	−0.108334	+	0.994115i	$0.534552\pi$
$348$	0	0
$349$	−8985.39	−1.37816	−0.689079	−	0.724686i	$-0.741985\pi$
$349$	−8985.39	−1.37816	−0.689079	+	0.724686i	$0.741985\pi$
$350$	−528.993	−0.0807881
$351$	0	0
$352$	−2062.71	−0.312338
$353$	−1767.91	−0.266562	−0.133281	−	0.991078i	$-0.542551\pi$
$353$	−1767.91	−0.266562	−0.133281	+	0.991078i	$0.542551\pi$
$354$	0	0
$355$	8238.00	1.23163
$356$	−8611.25	−1.28201
$357$	0	0
$358$	−37.4587	−0.00553004
$359$	1110.48	0.163256	0.0816278	−	0.996663i	$-0.473988\pi$
$359$	1110.48	0.163256	0.0816278	+	0.996663i	$0.473988\pi$
$360$	0	0
$361$	−6075.69	−0.885799
$362$	184.096	0.0267290
$363$	0	0
$364$	−3020.65	−0.434959
$365$	−6971.48	−0.999737
$366$	0	0
$367$	−5342.61	−0.759896	−0.379948	−	0.925008i	$-0.624058\pi$
$367$	−5342.61	−0.759896	−0.379948	+	0.925008i	$0.624058\pi$
$368$	−1146.62	−0.162423
$369$	0	0
$370$	−2916.99	−0.409858
$371$	1017.07	0.142329
$372$	0	0
$373$	6829.95	0.948101	0.474050	−	0.880498i	$-0.342791\pi$
$373$	6829.95	0.948101	0.474050	+	0.880498i	$0.342791\pi$
$374$	669.726	0.0925955
$375$	0	0
$376$	5912.74	0.810974
$377$	−2924.63	−0.399538
$378$	0	0
$379$	7826.62	1.06076	0.530378	−	0.847761i	$-0.322050\pi$
$379$	7826.62	1.06076	0.530378	+	0.847761i	$0.322050\pi$
$380$	−1376.05	−0.185763
$381$	0	0
$382$	−3155.90	−0.422695
$383$	7534.69	1.00523	0.502617	−	0.864509i	$-0.332371\pi$
$383$	7534.69	1.00523	0.502617	+	0.864509i	$0.332371\pi$
$384$	0	0
$385$	669.896	0.0886781
$386$	6699.17	0.883364
$387$	0	0
$388$	−3756.17	−0.491471
$389$	13071.8	1.70376	0.851882	−	0.523734i	$-0.175461\pi$
$389$	13071.8	1.70376	0.851882	+	0.523734i	$0.175461\pi$
$390$	0	0
$391$	3464.49	0.448099
$392$	1025.14	0.132085
$393$	0	0
$394$	−3349.65	−0.428307
$395$	7749.37	0.987122
$396$	0	0
$397$	−3692.51	−0.466806	−0.233403	−	0.972380i	$-0.574986\pi$
$397$	−3692.51	−0.466806	−0.233403	+	0.972380i	$0.574986\pi$
$398$	3710.02	0.467252
$399$	0	0
$400$	−648.362	−0.0810452
$401$	−8223.85	−1.02414	−0.512069	−	0.858944i	$-0.671121\pi$
$401$	−8223.85	−1.02414	−0.512069	+	0.858944i	$0.671121\pi$
$402$	0	0
$403$	14207.9	1.75619
$404$	8452.61	1.04092
$405$	0	0
$406$	410.892	0.0502272
$407$	−2406.59	−0.293096
$408$	0	0
$409$	3689.61	0.446062	0.223031	−	0.974811i	$-0.428405\pi$
$409$	3689.61	0.446062	0.223031	+	0.974811i	$0.428405\pi$
$410$	−1068.89	−0.128753
$411$	0	0
$412$	−4941.42	−0.590889
$413$	637.322	0.0759336
$414$	0	0
$415$	−4866.17	−0.575593
$416$	14318.5	1.68755
$417$	0	0
$418$	471.811	0.0552082
$419$	−15657.0	−1.82552	−0.912762	−	0.408492i	$-0.866055\pi$
$419$	−15657.0	−1.82552	−0.912762	+	0.408492i	$0.866055\pi$
$420$	0	0
$421$	−6007.30	−0.695435	−0.347717	−	0.937599i	$-0.613043\pi$
$421$	−6007.30	−0.695435	−0.347717	+	0.937599i	$0.613043\pi$
$422$	5959.08	0.687402
$423$	0	0
$424$	−3039.76	−0.348170
$425$	1959.01	0.223591
$426$	0	0
$427$	−5656.99	−0.641127
$428$	4429.83	0.500289
$429$	0	0
$430$	−468.193	−0.0525076
$431$	−13139.0	−1.46841	−0.734203	−	0.678931i	$-0.762444\pi$
$431$	−13139.0	−1.46841	−0.734203	+	0.678931i	$0.762444\pi$
$432$	0	0
$433$	−4392.47	−0.487502	−0.243751	−	0.969838i	$-0.578378\pi$
$433$	−4392.47	−0.487502	−0.243751	+	0.969838i	$0.578378\pi$
$434$	−1996.12	−0.220776
$435$	0	0
$436$	7635.41	0.838692
$437$	2440.67	0.267170
$438$	0	0
$439$	12676.2	1.37813	0.689066	−	0.724699i	$-0.258021\pi$
$439$	12676.2	1.37813	0.689066	+	0.724699i	$0.258021\pi$
$440$	−2002.14	−0.216928
$441$	0	0
$442$	−4648.95	−0.500289
$443$	17489.2	1.87571	0.937855	−	0.347028i	$-0.112809\pi$
$443$	17489.2	1.87571	0.937855	+	0.347028i	$0.112809\pi$
$444$	0	0
$445$	−13256.6	−1.41218
$446$	−983.069	−0.104371
$447$	0	0
$448$	−1275.34	−0.134496
$449$	15491.8	1.62830	0.814148	−	0.580658i	$-0.197204\pi$
$449$	15491.8	1.62830	0.814148	+	0.580658i	$0.197204\pi$
$450$	0	0
$451$	−881.856	−0.0920731
$452$	−1064.50	−0.110774
$453$	0	0
$454$	5546.40	0.573360
$455$	−4650.13	−0.479124
$456$	0	0
$457$	−7789.99	−0.797375	−0.398688	−	0.917087i	$-0.630534\pi$
$457$	−7789.99	−0.797375	−0.398688	+	0.917087i	$0.630534\pi$
$458$	−3859.13	−0.393723
$459$	0	0
$460$	−4287.59	−0.434586
$461$	8497.44	0.858493	0.429247	−	0.903187i	$-0.358779\pi$
$461$	8497.44	0.858493	0.429247	+	0.903187i	$0.358779\pi$
$462$	0	0
$463$	875.113	0.0878401	0.0439200	−	0.999035i	$-0.486015\pi$
$463$	875.113	0.0878401	0.0439200	+	0.999035i	$0.486015\pi$
$464$	503.611	0.0503870
$465$	0	0
$466$	−6418.08	−0.638008
$467$	−17652.5	−1.74917	−0.874584	−	0.484874i	$-0.838866\pi$
$467$	−17652.5	−1.74917	−0.874584	+	0.484874i	$0.838866\pi$
$468$	0	0
$469$	−5559.55	−0.547369
$470$	3768.17	0.369814
$471$	0	0
$472$	−1904.79	−0.185752
$473$	−386.270	−0.0375491
$474$	0	0
$475$	1380.09	0.133311
$476$	−1571.61	−0.151333
$477$	0	0
$478$	3957.96	0.378730
$479$	−3129.76	−0.298544	−0.149272	−	0.988796i	$-0.547693\pi$
$479$	−3129.76	−0.298544	−0.149272	+	0.988796i	$0.547693\pi$
$480$	0	0
$481$	16705.5	1.58359
$482$	−2332.66	−0.220435
$483$	0	0
$484$	−683.813	−0.0642198
$485$	−5782.43	−0.541375
$486$	0	0
$487$	−366.055	−0.0340606	−0.0170303	−	0.999855i	$-0.505421\pi$
$487$	−366.055	−0.0340606	−0.0170303	+	0.999855i	$0.505421\pi$
$488$	16907.2	1.56835
$489$	0	0
$490$	653.315	0.0602322
$491$	14577.2	1.33984	0.669919	−	0.742434i	$-0.266329\pi$
$491$	14577.2	1.33984	0.669919	+	0.742434i	$0.266329\pi$
$492$	0	0
$493$	−1521.65	−0.139010
$494$	−3275.11	−0.298288
$495$	0	0
$496$	−2446.55	−0.221478
$497$	6628.31	0.598230
$498$	0	0
$499$	9504.05	0.852624	0.426312	−	0.904576i	$-0.359813\pi$
$499$	9504.05	0.852624	0.426312	+	0.904576i	$0.359813\pi$
$500$	−8570.24	−0.766546
$501$	0	0
$502$	2242.39	0.199368
$503$	6149.90	0.545150	0.272575	−	0.962134i	$-0.412125\pi$
$503$	6149.90	0.545150	0.272575	+	0.962134i	$0.412125\pi$
$504$	0	0
$505$	13012.3	1.14662
$506$	1470.10	0.129158
$507$	0	0
$508$	−3317.10	−0.289709
$509$	16132.0	1.40479	0.702396	−	0.711787i	$-0.252114\pi$
$509$	16132.0	1.40479	0.702396	+	0.711787i	$0.252114\pi$
$510$	0	0
$511$	−5609.27	−0.485596
$512$	−4666.24	−0.402775
$513$	0	0
$514$	−1814.35	−0.155695
$515$	−7607.06	−0.650887
$516$	0	0
$517$	3108.83	0.264460
$518$	−2347.02	−0.199077
$519$	0	0
$520$	13898.0	1.17205
$521$	−7654.31	−0.643650	−0.321825	−	0.946799i	$-0.604296\pi$
$521$	−7654.31	−0.643650	−0.321825	+	0.946799i	$0.604296\pi$
$522$	0	0
$523$	21493.7	1.79705	0.898523	−	0.438926i	$-0.144641\pi$
$523$	21493.7	1.79705	0.898523	+	0.438926i	$0.144641\pi$
$524$	3521.09	0.293549
$525$	0	0
$526$	232.904	0.0193063
$527$	7392.20	0.611023
$528$	0	0
$529$	−4562.20	−0.374965
$530$	−1937.23	−0.158770
$531$	0	0
$532$	−1107.17	−0.0902294
$533$	6121.46	0.497467
$534$	0	0
$535$	6819.49	0.551088
$536$	16616.0	1.33900
$537$	0	0
$538$	−391.628	−0.0313834
$539$	539.000	0.0430730
$540$	0	0
$541$	−8661.03	−0.688293	−0.344147	−	0.938916i	$-0.611832\pi$
$541$	−8661.03	−0.688293	−0.344147	+	0.938916i	$0.611832\pi$
$542$	−3968.84	−0.314532
$543$	0	0
$544$	7449.74	0.587142
$545$	11754.3	0.923853
$546$	0	0
$547$	21372.9	1.67064	0.835321	−	0.549763i	$-0.185282\pi$
$547$	21372.9	1.67064	0.835321	+	0.549763i	$0.185282\pi$
$548$	−5396.24	−0.420650
$549$	0	0
$550$	831.274	0.0644466
$551$	−1071.98	−0.0828817
$552$	0	0
$553$	6235.16	0.479468
$554$	6806.72	0.522003
$555$	0	0
$556$	−8989.15	−0.685656
$557$	−18062.1	−1.37399	−0.686997	−	0.726660i	$-0.741071\pi$
$557$	−18062.1	−1.37399	−0.686997	+	0.726660i	$0.741071\pi$
$558$	0	0
$559$	2681.32	0.202876
$560$	800.738	0.0604238
$561$	0	0
$562$	2417.32	0.181438
$563$	−962.299	−0.0720356	−0.0360178	−	0.999351i	$-0.511467\pi$
$563$	−962.299	−0.0720356	−0.0360178	+	0.999351i	$0.511467\pi$
$564$	0	0
$565$	−1638.74	−0.122022
$566$	−5255.49	−0.390291
$567$	0	0
$568$	−19810.2	−1.46341
$569$	25409.7	1.87211	0.936055	−	0.351853i	$-0.114448\pi$
$569$	25409.7	1.87211	0.936055	+	0.351853i	$0.114448\pi$
$570$	0	0
$571$	−5211.13	−0.381925	−0.190962	−	0.981597i	$-0.561161\pi$
$571$	−5211.13	−0.381925	−0.190962	+	0.981597i	$0.561161\pi$
$572$	4746.73	0.346977
$573$	0	0
$574$	−860.028	−0.0625381
$575$	4300.17	0.311878
$576$	0	0
$577$	409.463	0.0295428	0.0147714	−	0.999891i	$-0.495298\pi$
$577$	409.463	0.0295428	0.0147714	+	0.999891i	$0.495298\pi$
$578$	5110.53	0.367768
$579$	0	0
$580$	1883.17	0.134818
$581$	−3915.33	−0.279579
$582$	0	0
$583$	−1598.26	−0.113539
$584$	16764.6	1.18788
$585$	0	0
$586$	7051.75	0.497108
$587$	11756.7	0.826661	0.413331	−	0.910581i	$-0.364365\pi$
$587$	11756.7	0.826661	0.413331	+	0.910581i	$0.364365\pi$
$588$	0	0
$589$	5207.68	0.364310
$590$	−1213.91	−0.0847051
$591$	0	0
$592$	−2876.63	−0.199711
$593$	312.172	0.0216178	0.0108089	−	0.999942i	$-0.496559\pi$
$593$	312.172	0.0216178	0.0108089	+	0.999942i	$0.496559\pi$
$594$	0	0
$595$	−2419.41	−0.166700
$596$	−8001.42	−0.549918
$597$	0	0
$598$	−10204.8	−0.697834
$599$	−22486.4	−1.53384	−0.766918	−	0.641745i	$-0.778211\pi$
$599$	−22486.4	−1.53384	−0.766918	+	0.641745i	$0.778211\pi$
$600$	0	0
$601$	25019.6	1.69812	0.849061	−	0.528295i	$-0.177168\pi$
$601$	25019.6	1.69812	0.849061	+	0.528295i	$0.177168\pi$
$602$	−376.709	−0.0255042
$603$	0	0
$604$	−2325.89	−0.156687
$605$	−1052.69	−0.0707407
$606$	0	0
$607$	7074.22	0.473037	0.236519	−	0.971627i	$-0.423994\pi$
$607$	7074.22	0.473037	0.236519	+	0.971627i	$0.423994\pi$
$608$	5248.22	0.350072
$609$	0	0
$610$	10774.9	0.715187
$611$	−21580.1	−1.42887
$612$	0	0
$613$	4934.94	0.325155	0.162578	−	0.986696i	$-0.448019\pi$
$613$	4934.94	0.325155	0.162578	+	0.986696i	$0.448019\pi$
$614$	−7647.61	−0.502658
$615$	0	0
$616$	−1610.93	−0.105367
$617$	−2125.51	−0.138687	−0.0693434	−	0.997593i	$-0.522090\pi$
$617$	−2125.51	−0.138687	−0.0693434	+	0.997593i	$0.522090\pi$
$618$	0	0
$619$	8168.09	0.530377	0.265188	−	0.964197i	$-0.414566\pi$
$619$	8168.09	0.530377	0.265188	+	0.964197i	$0.414566\pi$
$620$	−9148.45	−0.592598
$621$	0	0
$622$	4811.81	0.310187
$623$	−10666.3	−0.685931
$624$	0	0
$625$	−7029.59	−0.449894
$626$	7238.34	0.462144
$627$	0	0
$628$	8009.60	0.508946
$629$	8691.69	0.550970
$630$	0	0
$631$	−8419.88	−0.531205	−0.265602	−	0.964083i	$-0.585571\pi$
$631$	−8419.88	−0.531205	−0.265602	+	0.964083i	$0.585571\pi$
$632$	−18635.2	−1.17289
$633$	0	0
$634$	9096.85	0.569846
$635$	−5106.50	−0.319126
$636$	0	0
$637$	−3741.50	−0.232722
$638$	−645.687	−0.0400674
$639$	0	0
$640$	−10622.1	−0.656057
$641$	−27238.9	−1.67843	−0.839213	−	0.543803i	$-0.816984\pi$
$641$	−27238.9	−1.67843	−0.839213	+	0.543803i	$0.816984\pi$
$642$	0	0
$643$	−12438.7	−0.762882	−0.381441	−	0.924393i	$-0.624572\pi$
$643$	−12438.7	−0.762882	−0.381441	+	0.924393i	$0.624572\pi$
$644$	−3449.80	−0.211089
$645$	0	0
$646$	−1704.00	−0.103782
$647$	9788.76	0.594801	0.297400	−	0.954753i	$-0.403880\pi$
$647$	9788.76	0.594801	0.297400	+	0.954753i	$0.403880\pi$
$648$	0	0
$649$	−1001.51	−0.0605741
$650$	−5770.35	−0.348202
$651$	0	0
$652$	2494.51	0.149835
$653$	−5539.90	−0.331996	−0.165998	−	0.986126i	$-0.553084\pi$
$653$	−5539.90	−0.331996	−0.165998	+	0.986126i	$0.553084\pi$
$654$	0	0
$655$	5420.54	0.323356
$656$	−1054.10	−0.0627371
$657$	0	0
$658$	3031.87	0.179627
$659$	18751.7	1.10844	0.554221	−	0.832370i	$-0.313016\pi$
$659$	18751.7	1.10844	0.554221	+	0.832370i	$0.313016\pi$
$660$	0	0
$661$	24849.3	1.46222	0.731108	−	0.682262i	$-0.239004\pi$
$661$	24849.3	1.46222	0.731108	+	0.682262i	$0.239004\pi$
$662$	9794.03	0.575009
$663$	0	0
$664$	11701.9	0.683917
$665$	−1704.44	−0.0993913
$666$	0	0
$667$	−3340.14	−0.193899
$668$	−8400.08	−0.486540
$669$	0	0
$670$	10589.3	0.610599
$671$	8889.56	0.511442
$672$	0	0
$673$	−9532.72	−0.546002	−0.273001	−	0.962014i	$-0.588016\pi$
$673$	−9532.72	−0.546002	−0.273001	+	0.962014i	$0.588016\pi$
$674$	−13664.7	−0.780926
$675$	0	0
$676$	−20533.7	−1.16828
$677$	6544.90	0.371552	0.185776	−	0.982592i	$-0.440520\pi$
$677$	6544.90	0.371552	0.185776	+	0.982592i	$0.440520\pi$
$678$	0	0
$679$	−4652.56	−0.262958
$680$	7230.98	0.407787
$681$	0	0
$682$	3136.76	0.176118
$683$	−8438.16	−0.472734	−0.236367	−	0.971664i	$-0.575957\pi$
$683$	−8438.16	−0.472734	−0.236367	+	0.971664i	$0.575957\pi$
$684$	0	0
$685$	−8307.23	−0.463362
$686$	525.658	0.0292562
$687$	0	0
$688$	−461.715	−0.0255853
$689$	11094.4	0.613446
$690$	0	0
$691$	−8196.16	−0.451225	−0.225613	−	0.974217i	$-0.572438\pi$
$691$	−8196.16	−0.451225	−0.225613	+	0.974217i	$0.572438\pi$
$692$	−16713.2	−0.918123
$693$	0	0
$694$	2146.34	0.117398
$695$	−13838.3	−0.755278
$696$	0	0
$697$	3184.93	0.173082
$698$	13770.4	0.746730
$699$	0	0
$700$	−1950.71	−0.105328
$701$	7172.05	0.386426	0.193213	−	0.981157i	$-0.438109\pi$
$701$	7172.05	0.386426	0.193213	+	0.981157i	$0.438109\pi$
$702$	0	0
$703$	6123.15	0.328505
$704$	2004.11	0.107291
$705$	0	0
$706$	2709.38	0.144432
$707$	10469.8	0.556939
$708$	0	0
$709$	−16766.6	−0.888127	−0.444063	−	0.895995i	$-0.646464\pi$
$709$	−16766.6	−0.888127	−0.444063	+	0.895995i	$0.646464\pi$
$710$	−12625.0	−0.667335
$711$	0	0
$712$	31878.6	1.67795
$713$	16226.4	0.852292
$714$	0	0
$715$	7307.35	0.382209
$716$	−138.132	−0.00720984
$717$	0	0
$718$	−1701.84	−0.0884571
$719$	5923.31	0.307235	0.153618	−	0.988130i	$-0.450908\pi$
$719$	5923.31	0.307235	0.153618	+	0.988130i	$0.450908\pi$
$720$	0	0
$721$	−6120.65	−0.316151
$722$	9311.20	0.479954
$723$	0	0
$724$	678.872	0.0348482
$725$	−1888.70	−0.0967509
$726$	0	0
$727$	−4256.80	−0.217161	−0.108581	−	0.994088i	$-0.534631\pi$
$727$	−4256.80	−0.217161	−0.108581	+	0.994088i	$0.534631\pi$
$728$	11182.4	0.569293
$729$	0	0
$730$	10684.0	0.541689
$731$	1395.06	0.0705858
$732$	0	0
$733$	24556.4	1.23740	0.618698	−	0.785629i	$-0.287660\pi$
$733$	24556.4	1.23740	0.618698	+	0.785629i	$0.287660\pi$
$734$	8187.72	0.411736
$735$	0	0
$736$	16352.7	0.818981
$737$	8736.44	0.436650
$738$	0	0
$739$	27603.8	1.37405	0.687025	−	0.726634i	$-0.258916\pi$
$739$	27603.8	1.37405	0.687025	+	0.726634i	$0.258916\pi$
$740$	−10756.7	−0.534355
$741$	0	0
$742$	−1558.70	−0.0771181
$743$	−19808.8	−0.978082	−0.489041	−	0.872261i	$-0.662653\pi$
$743$	−19808.8	−0.978082	−0.489041	+	0.872261i	$0.662653\pi$
$744$	0	0
$745$	−12317.8	−0.605756
$746$	−10467.1	−0.513711
$747$	0	0
$748$	2469.67	0.120722
$749$	5486.98	0.267677
$750$	0	0
$751$	596.125	0.0289653	0.0144826	−	0.999895i	$-0.495390\pi$
$751$	596.125	0.0289653	0.0144826	+	0.999895i	$0.495390\pi$
$752$	3716.03	0.180199
$753$	0	0
$754$	4482.08	0.216482
$755$	−3580.59	−0.172597
$756$	0	0
$757$	−1845.87	−0.0886253	−0.0443127	−	0.999018i	$-0.514110\pi$
$757$	−1845.87	−0.0886253	−0.0443127	+	0.999018i	$0.514110\pi$
$758$	−11994.6	−0.574752
$759$	0	0
$760$	5094.10	0.243135
$761$	13034.2	0.620877	0.310439	−	0.950593i	$-0.399524\pi$
$761$	13034.2	0.620877	0.310439	+	0.950593i	$0.399524\pi$
$762$	0	0
$763$	9457.55	0.448737
$764$	−11637.6	−0.551093
$765$	0	0
$766$	−11547.2	−0.544668
$767$	6952.02	0.327279
$768$	0	0
$769$	−38530.6	−1.80683	−0.903413	−	0.428770i	$-0.858947\pi$
$769$	−38530.6	−1.80683	−0.903413	+	0.428770i	$0.858947\pi$
$770$	−1026.64	−0.0480486
$771$	0	0
$772$	24703.8	1.15169
$773$	−13835.1	−0.643745	−0.321873	−	0.946783i	$-0.604312\pi$
$773$	−13835.1	−0.643745	−0.321873	+	0.946783i	$0.604312\pi$
$774$	0	0
$775$	9175.31	0.425273
$776$	13905.2	0.643259
$777$	0	0
$778$	−20032.9	−0.923154
$779$	2243.73	0.103196
$780$	0	0
$781$	−10415.9	−0.477222
$782$	−5309.44	−0.242794
$783$	0	0
$784$	644.275	0.0293493
$785$	12330.4	0.560624
$786$	0	0
$787$	11704.2	0.530125	0.265062	−	0.964231i	$-0.414607\pi$
$787$	11704.2	0.530125	0.265062	+	0.964231i	$0.414607\pi$
$788$	−12352.1	−0.558409
$789$	0	0
$790$	−11876.2	−0.534854
$791$	−1318.54	−0.0592690
$792$	0	0
$793$	−61707.5	−2.76330
$794$	5658.89	0.252930
$795$	0	0
$796$	13681.0	0.609184
$797$	3367.27	0.149655	0.0748274	−	0.997196i	$-0.476159\pi$
$797$	3367.27	0.149655	0.0748274	+	0.997196i	$0.476159\pi$
$798$	0	0
$799$	−11227.9	−0.497140
$800$	9246.73	0.408652
$801$	0	0
$802$	12603.3	0.554911
$803$	8814.56	0.387371
$804$	0	0
$805$	−5310.79	−0.232523
$806$	−21774.0	−0.951559
$807$	0	0
$808$	−31291.3	−1.36241
$809$	−20869.5	−0.906961	−0.453481	−	0.891266i	$-0.649818\pi$
$809$	−20869.5	−0.906961	−0.453481	+	0.891266i	$0.649818\pi$
$810$	0	0
$811$	−22445.9	−0.971863	−0.485931	−	0.873997i	$-0.661520\pi$
$811$	−22445.9	−0.971863	−0.485931	+	0.873997i	$0.661520\pi$
$812$	1515.20	0.0654841
$813$	0	0
$814$	3688.17	0.158809
$815$	3840.17	0.165049
$816$	0	0
$817$	982.798	0.0420854
$818$	−5654.45	−0.241691
$819$	0	0
$820$	−3941.61	−0.167862
$821$	25516.4	1.08469	0.542343	−	0.840157i	$-0.317537\pi$
$821$	25516.4	1.08469	0.542343	+	0.840157i	$0.317537\pi$
$822$	0	0
$823$	36376.7	1.54072	0.770360	−	0.637609i	$-0.220076\pi$
$823$	36376.7	1.54072	0.770360	+	0.637609i	$0.220076\pi$
$824$	18293.0	0.773382
$825$	0	0
$826$	−976.717	−0.0411433
$827$	−25520.1	−1.07306	−0.536530	−	0.843881i	$-0.680265\pi$
$827$	−25520.1	−1.07306	−0.536530	+	0.843881i	$0.680265\pi$
$828$	0	0
$829$	23202.5	0.972084	0.486042	−	0.873936i	$-0.338440\pi$
$829$	23202.5	0.972084	0.486042	+	0.873936i	$0.338440\pi$
$830$	7457.56	0.311875
$831$	0	0
$832$	−13911.7	−0.579688
$833$	−1946.66	−0.0809699
$834$	0	0
$835$	−12931.5	−0.535943
$836$	1739.84	0.0719782
$837$	0	0
$838$	23994.9	0.989127
$839$	10538.6	0.433649	0.216824	−	0.976211i	$-0.430430\pi$
$839$	10538.6	0.433649	0.216824	+	0.976211i	$0.430430\pi$
$840$	0	0
$841$	−22922.0	−0.939849
$842$	9206.38	0.376809
$843$	0	0
$844$	21974.6	0.896206
$845$	−31610.6	−1.28691
$846$	0	0
$847$	−847.000	−0.0343604
$848$	−1910.42	−0.0773635
$849$	0	0
$850$	−3002.25	−0.121149
$851$	19078.9	0.768526
$852$	0	0
$853$	40061.0	1.60805	0.804023	−	0.594598i	$-0.202689\pi$
$853$	40061.0	1.60805	0.804023	+	0.594598i	$0.202689\pi$
$854$	8669.52	0.347383
$855$	0	0
$856$	−16399.1	−0.654801
$857$	−775.719	−0.0309195	−0.0154598	−	0.999880i	$-0.504921\pi$
$857$	−775.719	−0.0309195	−0.0154598	+	0.999880i	$0.504921\pi$
$858$	0	0
$859$	−10241.5	−0.406792	−0.203396	−	0.979097i	$-0.565198\pi$
$859$	−10241.5	−0.406792	−0.203396	+	0.979097i	$0.565198\pi$
$860$	−1726.50	−0.0684573
$861$	0	0
$862$	20135.9	0.795629
$863$	1268.51	0.0500356	0.0250178	−	0.999687i	$-0.492036\pi$
$863$	1268.51	0.0500356	0.0250178	+	0.999687i	$0.492036\pi$
$864$	0	0
$865$	−25729.1	−1.01135
$866$	6731.60	0.264144
$867$	0	0
$868$	−7360.86	−0.287839
$869$	−9798.11	−0.382483
$870$	0	0
$871$	−60644.6	−2.35920
$872$	−28266.1	−1.09772
$873$	0	0
$874$	−3740.41	−0.144761
$875$	−10615.5	−0.410135
$876$	0	0
$877$	6691.33	0.257640	0.128820	−	0.991668i	$-0.458881\pi$
$877$	6691.33	0.257640	0.128820	+	0.991668i	$0.458881\pi$
$878$	−19426.6	−0.746716
$879$	0	0
$880$	−1258.30	−0.0482015
$881$	−14514.6	−0.555063	−0.277531	−	0.960717i	$-0.589516\pi$
$881$	−14514.6	−0.555063	−0.277531	+	0.960717i	$0.589516\pi$
$882$	0	0
$883$	10335.2	0.393891	0.196946	−	0.980414i	$-0.436898\pi$
$883$	10335.2	0.393891	0.196946	+	0.980414i	$0.436898\pi$
$884$	−17143.4	−0.652257
$885$	0	0
$886$	−26802.8	−1.01632
$887$	−42946.3	−1.62570	−0.812851	−	0.582472i	$-0.802086\pi$
$887$	−42946.3	−1.62570	−0.812851	+	0.582472i	$0.802086\pi$
$888$	0	0
$889$	−4108.70	−0.155007
$890$	20316.1	0.765166
$891$	0	0
$892$	−3625.15	−0.136075
$893$	−7909.87	−0.296410
$894$	0	0
$895$	−212.648	−0.00794193
$896$	−8546.59	−0.318662
$897$	0	0
$898$	−23741.7	−0.882263
$899$	−7126.87	−0.264399
$900$	0	0
$901$	5772.31	0.213434
$902$	1351.47	0.0498882
$903$	0	0
$904$	3940.75	0.144986
$905$	1045.09	0.0383866
$906$	0	0
$907$	8376.30	0.306649	0.153324	−	0.988176i	$-0.451002\pi$
$907$	8376.30	0.306649	0.153324	+	0.988176i	$0.451002\pi$
$908$	20452.8	0.747524
$909$	0	0
$910$	7126.47	0.259605
$911$	−17335.4	−0.630460	−0.315230	−	0.949015i	$-0.602082\pi$
$911$	−17335.4	−0.630460	−0.315230	+	0.949015i	$0.602082\pi$
$912$	0	0
$913$	6152.66	0.223027
$914$	11938.4	0.432043
$915$	0	0
$916$	−14230.9	−0.513321
$917$	4361.38	0.157061
$918$	0	0
$919$	−34995.3	−1.25613	−0.628067	−	0.778159i	$-0.716154\pi$
$919$	−34995.3	−1.25613	−0.628067	+	0.778159i	$0.716154\pi$
$920$	15872.5	0.568806
$921$	0	0
$922$	−13022.6	−0.465159
$923$	72302.8	2.57841
$924$	0	0
$925$	10788.3	0.383476
$926$	−1341.14	−0.0475945
$927$	0	0
$928$	−7182.35	−0.254065
$929$	10671.6	0.376882	0.188441	−	0.982084i	$-0.439657\pi$
$929$	10671.6	0.376882	0.188441	+	0.982084i	$0.439657\pi$
$930$	0	0
$931$	−1371.39	−0.0482767
$932$	−23667.2	−0.831808
$933$	0	0
$934$	27053.1	0.947755
$935$	3801.93	0.132980
$936$	0	0
$937$	−1855.85	−0.0647045	−0.0323522	−	0.999477i	$-0.510300\pi$
$937$	−1855.85	−0.0647045	−0.0323522	+	0.999477i	$0.510300\pi$
$938$	8520.19	0.296582
$939$	0	0
$940$	13895.4	0.482148
$941$	−15378.8	−0.532766	−0.266383	−	0.963867i	$-0.585829\pi$
$941$	−15378.8	−0.532766	−0.266383	+	0.963867i	$0.585829\pi$
$942$	0	0
$943$	6991.16	0.241425
$944$	−1197.12	−0.0412742
$945$	0	0
$946$	591.972	0.0203453
$947$	14600.9	0.501020	0.250510	−	0.968114i	$-0.419402\pi$
$947$	14600.9	0.501020	0.250510	+	0.968114i	$0.419402\pi$
$948$	0	0
$949$	−61186.9	−2.09295
$950$	−2115.03	−0.0722324
$951$	0	0
$952$	5818.06	0.198072
$953$	2114.27	0.0718658	0.0359329	−	0.999354i	$-0.488560\pi$
$953$	2114.27	0.0718658	0.0359329	+	0.999354i	$0.488560\pi$
$954$	0	0
$955$	−17915.5	−0.607051
$956$	14595.3	0.493773
$957$	0	0
$958$	4796.46	0.161761
$959$	−6684.02	−0.225066
$960$	0	0
$961$	4831.40	0.162177
$962$	−25601.7	−0.858037
$963$	0	0
$964$	−8601.87	−0.287394
$965$	38030.2	1.26864
$966$	0	0
$967$	7251.75	0.241159	0.120579	−	0.992704i	$-0.461525\pi$
$967$	7251.75	0.241159	0.120579	+	0.992704i	$0.461525\pi$
$968$	2531.46	0.0840538
$969$	0	0
$970$	8861.77	0.293334
$971$	−2742.57	−0.0906420	−0.0453210	−	0.998972i	$-0.514431\pi$
$971$	−2742.57	−0.0906420	−0.0453210	+	0.998972i	$0.514431\pi$
$972$	0	0
$973$	−11134.4	−0.366856
$974$	560.991	0.0184551
$975$	0	0
$976$	10625.8	0.348488
$977$	−23770.4	−0.778384	−0.389192	−	0.921157i	$-0.627246\pi$
$977$	−23770.4	−0.778384	−0.389192	+	0.921157i	$0.627246\pi$
$978$	0	0
$979$	16761.3	0.547184
$980$	2409.16	0.0785282
$981$	0	0
$982$	−22340.1	−0.725967
$983$	49265.4	1.59850	0.799249	−	0.601000i	$-0.205231\pi$
$983$	49265.4	1.59850	0.799249	+	0.601000i	$0.205231\pi$
$984$	0	0
$985$	−19015.5	−0.615109
$986$	2331.98	0.0753198
$987$	0	0
$988$	−12077.2	−0.388895
$989$	3062.26	0.0984574
$990$	0	0
$991$	−17123.5	−0.548884	−0.274442	−	0.961604i	$-0.588493\pi$
$991$	−17123.5	−0.548884	−0.274442	+	0.961604i	$0.588493\pi$
$992$	34891.9	1.11675
$993$	0	0
$994$	−10158.1	−0.324140
$995$	21061.2	0.671040
$996$	0	0
$997$	−41275.5	−1.31114	−0.655571	−	0.755134i	$-0.727572\pi$
$997$	−41275.5	−1.31114	−0.655571	+	0.755134i	$0.727572\pi$
$998$	−14565.3	−0.461979
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	693.4.a.l.1.2		4
3.2	odd	2		77.4.a.d.1.3	✓	4
12.11	even	2		1232.4.a.s.1.1		4
15.14	odd	2		1925.4.a.p.1.2		4
21.20	even	2		539.4.a.g.1.3		4
33.32	even	2		847.4.a.d.1.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
77.4.a.d.1.3	✓	4	3.2	odd	2
539.4.a.g.1.3		4	21.20	even	2
693.4.a.l.1.2		4	1.1	even	1	trivial
847.4.a.d.1.2		4	33.32	even	2
1232.4.a.s.1.1		4	12.11	even	2
1925.4.a.p.1.2		4	15.14	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 693 = 3^{2} \cdot 7 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	693.a (trivial)

\(f(q)\)	\(=\)	\(q-1.53253 q^{2} -5.65135 q^{4} -8.69995 q^{5} -7.00000 q^{7} +20.9211 q^{8} +13.3330 q^{10} +11.0000 q^{11} -76.3572 q^{13} +10.7277 q^{14} +13.1485 q^{16} -39.7278 q^{17} -27.9876 q^{19} +49.1664 q^{20} -16.8579 q^{22} -87.2055 q^{23} -49.3108 q^{25} +117.020 q^{26} +39.5594 q^{28} +38.3019 q^{29} -186.071 q^{31} -187.519 q^{32} +60.8842 q^{34} +60.8997 q^{35} -218.781 q^{37} +42.8919 q^{38} -182.013 q^{40} -80.1687 q^{41} -35.1155 q^{43} -62.1648 q^{44} +133.645 q^{46} +282.620 q^{47} +49.0000 q^{49} +75.5704 q^{50} +431.521 q^{52} -145.296 q^{53} -95.6995 q^{55} -146.448 q^{56} -58.6989 q^{58} -91.0461 q^{59} +808.142 q^{61} +285.160 q^{62} +182.192 q^{64} +664.304 q^{65} +794.222 q^{67} +224.516 q^{68} -93.3307 q^{70} -946.901 q^{71} +801.324 q^{73} +335.288 q^{74} +158.168 q^{76} -77.0000 q^{77} -890.737 q^{79} -114.391 q^{80} +122.861 q^{82} +559.333 q^{83} +345.630 q^{85} +53.8156 q^{86} +230.132 q^{88} +1523.75 q^{89} +534.501 q^{91} +492.829 q^{92} -433.125 q^{94} +243.491 q^{95} +664.651 q^{97} -75.0941 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 2 q^{2} + 26 q^{4} - 10 q^{5} - 28 q^{7} + 18 q^{8} - 2 q^{10} + 44 q^{11} + 58 q^{13} - 14 q^{14} + 2 q^{16} - 4 q^{17} + 258 q^{19} - 182 q^{20} + 22 q^{22} - 8 q^{23} + 80 q^{25} + 482 q^{26} - 182 q^{28}+ \cdots + 98 q^{98}+O(q^{100}) \) 4 * q + 2 * q^2 + 26 * q^4 - 10 * q^5 - 28 * q^7 + 18 * q^8 - 2 * q^10 + 44 * q^11 + 58 * q^13 - 14 * q^14 + 2 * q^16 - 4 * q^17 + 258 * q^19 - 182 * q^20 + 22 * q^22 - 8 * q^23 + 80 * q^25 + 482 * q^26 - 182 * q^28 + 396 * q^29 - 56 * q^31 - 134 * q^32 + 472 * q^34 + 70 * q^35 + 84 * q^37 + 942 * q^38 - 1026 * q^40 - 52 * q^41 + 408 * q^43 + 286 * q^44 + 368 * q^46 - 8 * q^47 + 196 * q^49 + 1642 * q^50 + 2030 * q^52 - 624 * q^53 - 110 * q^55 - 126 * q^56 + 864 * q^58 + 238 * q^59 - 162 * q^61 - 688 * q^62 - 902 * q^64 + 32 * q^65 + 1340 * q^67 + 1384 * q^68 + 14 * q^70 - 1788 * q^71 + 1456 * q^73 - 996 * q^74 + 3042 * q^76 - 308 * q^77 - 1324 * q^79 - 2342 * q^80 + 1984 * q^82 - 450 * q^83 - 1736 * q^85 + 4380 * q^86 + 198 * q^88 + 3072 * q^89 - 406 * q^91 - 544 * q^92 - 1696 * q^94 - 24 * q^95 - 652 * q^97 + 98 * q^98

Introduction

L-functions

Modular forms

Varieties

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Groups

Database

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