LMFDB - Embedded newform 693.4.a.t.1.6

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:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 2x^{7} - 45x^{6} + 77x^{5} + 540x^{4} - 915x^{3} - 1452x^{2} + 2660x - 672 $ x^8 - 2x^7 - 45x^6 + 77x^5 + 540x^4 - 915x^3 - 1452x^2 + 2660*x - 672
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$2.21621$ 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)

$f(q)$	$=$	\(q+2.21621 q^{2} -3.08840 q^{4} +22.0394 q^{5} -7.00000 q^{7} -24.5743 q^{8} +48.8440 q^{10} -11.0000 q^{11} +2.76840 q^{13} -15.5135 q^{14} -29.7545 q^{16} +50.6354 q^{17} +56.5022 q^{19} -68.0666 q^{20} -24.3783 q^{22} +172.339 q^{23} +360.735 q^{25} +6.13535 q^{26} +21.6188 q^{28} -282.011 q^{29} +237.587 q^{31} +130.652 q^{32} +112.219 q^{34} -154.276 q^{35} +207.061 q^{37} +125.221 q^{38} -541.602 q^{40} -386.514 q^{41} -18.3909 q^{43} +33.9725 q^{44} +381.939 q^{46} +309.957 q^{47} +49.0000 q^{49} +799.465 q^{50} -8.54993 q^{52} +480.364 q^{53} -242.433 q^{55} +172.020 q^{56} -624.996 q^{58} -114.221 q^{59} +109.928 q^{61} +526.543 q^{62} +527.588 q^{64} +61.0138 q^{65} -567.534 q^{67} -156.383 q^{68} -341.908 q^{70} +780.713 q^{71} -605.836 q^{73} +458.891 q^{74} -174.502 q^{76} +77.0000 q^{77} +686.614 q^{79} -655.772 q^{80} -856.597 q^{82} -316.962 q^{83} +1115.97 q^{85} -40.7582 q^{86} +270.317 q^{88} +1611.41 q^{89} -19.3788 q^{91} -532.252 q^{92} +686.931 q^{94} +1245.27 q^{95} -1231.47 q^{97} +108.594 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 2 q^{2} + 30 q^{4} + 10 q^{5} - 56 q^{7} + 15 q^{8} - 13 q^{10} - 88 q^{11} - 148 q^{13} - 14 q^{14} + 266 q^{16} + 114 q^{17} + 58 q^{19} + 291 q^{20} - 22 q^{22} + 246 q^{23} + 244 q^{25} + 305 q^{26}+ \cdots + 98 q^{98}+O(q^{100}) $ 8 * q + 2 * q^2 + 30 * q^4 + 10 * q^5 - 56 * q^7 + 15 * q^8 - 13 * q^10 - 88 * q^11 - 148 * q^13 - 14 * q^14 + 266 * q^16 + 114 * q^17 + 58 * q^19 + 291 * q^20 - 22 * q^22 + 246 * q^23 + 244 * q^25 + 305 * q^26 - 210 * q^28 - 72 * q^29 + 252 * q^31 + 1272 * q^32 + 630 * q^34 - 70 * q^35 - 80 * q^37 + 1885 * q^38 - 342 * q^40 + 682 * q^41 - 106 * q^43 - 330 * q^44 + 120 * q^46 + 828 * q^47 + 392 * q^49 + 801 * q^50 - 1681 * q^52 + 462 * q^53 - 110 * q^55 - 105 * q^56 - 1087 * q^58 + 626 * q^59 - 854 * q^61 + 1350 * q^62 + 2997 * q^64 + 22 * q^65 + 130 * q^67 + 2202 * q^68 + 91 * q^70 + 326 * q^71 - 390 * q^73 - 359 * q^74 + 2041 * q^76 + 616 * q^77 - 508 * q^79 + 4391 * q^80 + 1528 * q^82 + 1596 * q^83 - 880 * q^85 - 414 * q^86 - 165 * q^88 + 4324 * q^89 + 1036 * q^91 + 2092 * q^92 - 1685 * q^94 + 1076 * q^95 - 964 * 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 $

$2$	2.21621	0.783549	0.391775	−	0.920061i	$-0.371861\pi$
$2$	2.21621	0.783549	0.391775	+	0.920061i	$0.371861\pi$
$3$	0	0
$3$	0	0
$4$	−3.08840	−0.386051
$5$	22.0394	1.97126	0.985632	−	0.168908i	$-0.0540241\pi$
$5$	22.0394	1.97126	0.985632	+	0.168908i	$0.0540241\pi$
$6$	0	0
$7$	−7.00000	−0.377964
$7$	−7.00000	−0.377964
$8$	−24.5743	−1.08604
$9$	0	0
$10$	48.8440	1.54458
$11$	−11.0000	−0.301511
$11$	−11.0000	−0.301511
$12$	0	0
$13$	2.76840	0.0590627	0.0295313	−	0.999564i	$-0.490599\pi$
$13$	2.76840	0.0590627	0.0295313	+	0.999564i	$0.490599\pi$
$14$	−15.5135	−0.296154
$15$	0	0
$16$	−29.7545	−0.464914
$17$	50.6354	0.722405	0.361203	−	0.932487i	$-0.382366\pi$
$17$	50.6354	0.722405	0.361203	+	0.932487i	$0.382366\pi$
$18$	0	0
$19$	56.5022	0.682236	0.341118	−	0.940020i	$-0.389194\pi$
$19$	56.5022	0.682236	0.341118	+	0.940020i	$0.389194\pi$
$20$	−68.0666	−0.761007
$21$	0	0
$22$	−24.3783	−0.236249
$23$	172.339	1.56240	0.781198	−	0.624283i	$-0.214609\pi$
$23$	172.339	1.56240	0.781198	+	0.624283i	$0.214609\pi$
$24$	0	0
$25$	360.735	2.88588
$26$	6.13535	0.0462785
$27$	0	0
$28$	21.6188	0.145913
$29$	−282.011	−1.80580	−0.902899	−	0.429853i	$-0.858566\pi$
$29$	−282.011	−1.80580	−0.902899	+	0.429853i	$0.858566\pi$
$30$	0	0
$31$	237.587	1.37651	0.688256	−	0.725468i	$-0.258377\pi$
$31$	237.587	1.37651	0.688256	+	0.725468i	$0.258377\pi$
$32$	130.652	0.721756
$33$	0	0
$34$	112.219	0.566040
$35$	−154.276	−0.745068
$36$	0	0
$37$	207.061	0.920017	0.460009	−	0.887914i	$-0.347846\pi$
$37$	207.061	0.920017	0.460009	+	0.887914i	$0.347846\pi$
$38$	125.221	0.534566
$39$	0	0
$40$	−541.602	−2.14087
$41$	−386.514	−1.47228	−0.736138	−	0.676831i	$-0.763353\pi$
$41$	−386.514	−1.47228	−0.736138	+	0.676831i	$0.763353\pi$
$42$	0	0
$43$	−18.3909	−0.0652230	−0.0326115	−	0.999468i	$-0.510382\pi$
$43$	−18.3909	−0.0652230	−0.0326115	+	0.999468i	$0.510382\pi$
$44$	33.9725	0.116399
$45$	0	0
$46$	381.939	1.22421
$47$	309.957	0.961955	0.480978	−	0.876733i	$-0.340282\pi$
$47$	309.957	0.961955	0.480978	+	0.876733i	$0.340282\pi$
$48$	0	0
$49$	49.0000	0.142857
$50$	799.465	2.26123
$51$	0	0
$52$	−8.54993	−0.0228012
$53$	480.364	1.24496	0.622482	−	0.782634i	$-0.286124\pi$
$53$	480.364	1.24496	0.622482	+	0.782634i	$0.286124\pi$
$54$	0	0
$55$	−242.433	−0.594358
$56$	172.020	0.410484
$57$	0	0
$58$	−624.996	−1.41493
$59$	−114.221	−0.252039	−0.126020	−	0.992028i	$-0.540220\pi$
$59$	−114.221	−0.252039	−0.126020	+	0.992028i	$0.540220\pi$
$60$	0	0
$61$	109.928	0.230734	0.115367	−	0.993323i	$-0.463196\pi$
$61$	109.928	0.230734	0.115367	+	0.993323i	$0.463196\pi$
$62$	526.543	1.07856
$63$	0	0
$64$	527.588	1.03045
$65$	61.0138	0.116428
$66$	0	0
$67$	−567.534	−1.03485	−0.517427	−	0.855727i	$-0.673110\pi$
$67$	−567.534	−1.03485	−0.517427	+	0.855727i	$0.673110\pi$
$68$	−156.383	−0.278885
$69$	0	0
$70$	−341.908	−0.583797
$71$	780.713	1.30498	0.652490	−	0.757797i	$-0.273724\pi$
$71$	780.713	1.30498	0.652490	+	0.757797i	$0.273724\pi$
$72$	0	0
$73$	−605.836	−0.971340	−0.485670	−	0.874142i	$-0.661424\pi$
$73$	−605.836	−0.971340	−0.485670	+	0.874142i	$0.661424\pi$
$74$	458.891	0.720879
$75$	0	0
$76$	−174.502	−0.263378
$77$	77.0000	0.113961
$78$	0	0
$79$	686.614	0.977850	0.488925	−	0.872326i	$-0.337389\pi$
$79$	686.614	0.977850	0.488925	+	0.872326i	$0.337389\pi$
$80$	−655.772	−0.916469
$81$	0	0
$82$	−856.597	−1.15360
$83$	−316.962	−0.419170	−0.209585	−	0.977790i	$-0.567211\pi$
$83$	−316.962	−0.419170	−0.209585	+	0.977790i	$0.567211\pi$
$84$	0	0
$85$	1115.97	1.42405
$86$	−40.7582	−0.0511054
$87$	0	0
$88$	270.317	0.327453
$89$	1611.41	1.91920	0.959601	−	0.281365i	$-0.0907870\pi$
$89$	1611.41	1.91920	0.959601	+	0.281365i	$0.0907870\pi$
$90$	0	0
$91$	−19.3788	−0.0223236
$92$	−532.252	−0.603164
$93$	0	0
$94$	686.931	0.753739
$95$	1245.27	1.34487
$96$	0	0
$97$	−1231.47	−1.28904	−0.644520	−	0.764587i	$-0.722943\pi$
$97$	−1231.47	−1.28904	−0.644520	+	0.764587i	$0.722943\pi$
$98$	108.594	0.111936
$99$	0	0
$100$	−1114.10	−1.11410
$101$	−1184.54	−1.16699	−0.583494	−	0.812117i	$-0.698315\pi$
$101$	−1184.54	−1.16699	−0.583494	+	0.812117i	$0.698315\pi$
$102$	0	0
$103$	−654.836	−0.626436	−0.313218	−	0.949681i	$-0.601407\pi$
$103$	−654.836	−0.626436	−0.313218	+	0.949681i	$0.601407\pi$
$104$	−68.0313	−0.0641444
$105$	0	0
$106$	1064.59	0.975490
$107$	654.327	0.591179	0.295589	−	0.955315i	$-0.404484\pi$
$107$	654.327	0.591179	0.295589	+	0.955315i	$0.404484\pi$
$108$	0	0
$109$	1787.78	1.57099	0.785497	−	0.618866i	$-0.212408\pi$
$109$	1787.78	1.57099	0.785497	+	0.618866i	$0.212408\pi$
$110$	−537.284	−0.465709
$111$	0	0
$112$	208.282	0.175721
$113$	−824.970	−0.686784	−0.343392	−	0.939192i	$-0.611576\pi$
$113$	−824.970	−0.686784	−0.343392	+	0.939192i	$0.611576\pi$
$114$	0	0
$115$	3798.24	3.07989
$116$	870.964	0.697129
$117$	0	0
$118$	−253.138	−0.197485
$119$	−354.448	−0.273043
$120$	0	0
$121$	121.000	0.0909091
$122$	243.623	0.180792
$123$	0	0
$124$	−733.764	−0.531403
$125$	5195.46	3.71757
$126$	0	0
$127$	1754.34	1.22577	0.612885	−	0.790172i	$-0.290009\pi$
$127$	1754.34	1.22577	0.612885	+	0.790172i	$0.290009\pi$
$128$	124.033	0.0856491
$129$	0	0
$130$	135.219	0.0912272
$131$	2065.18	1.37737	0.688687	−	0.725059i	$-0.258188\pi$
$131$	2065.18	1.37737	0.688687	+	0.725059i	$0.258188\pi$
$132$	0	0
$133$	−395.515	−0.257861
$134$	−1257.77	−0.810859
$135$	0	0
$136$	−1244.33	−0.784560
$137$	−658.269	−0.410509	−0.205255	−	0.978709i	$-0.565802\pi$
$137$	−658.269	−0.410509	−0.205255	+	0.978709i	$0.565802\pi$
$138$	0	0
$139$	−704.321	−0.429782	−0.214891	−	0.976638i	$-0.568940\pi$
$139$	−704.321	−0.429782	−0.214891	+	0.976638i	$0.568940\pi$
$140$	476.466	0.287634
$141$	0	0
$142$	1730.22	1.02252
$143$	−30.4524	−0.0178081
$144$	0	0
$145$	−6215.35	−3.55970
$146$	−1342.66	−0.761092
$147$	0	0
$148$	−639.489	−0.355173
$149$	−1246.54	−0.685375	−0.342687	−	0.939450i	$-0.611337\pi$
$149$	−1246.54	−0.685375	−0.342687	+	0.939450i	$0.611337\pi$
$150$	0	0
$151$	1137.10	0.612818	0.306409	−	0.951900i	$-0.400872\pi$
$151$	1137.10	0.612818	0.306409	+	0.951900i	$0.400872\pi$
$152$	−1388.50	−0.740935
$153$	0	0
$154$	170.648	0.0892937
$155$	5236.27	2.71347
$156$	0	0
$157$	−130.308	−0.0662400	−0.0331200	−	0.999451i	$-0.510544\pi$
$157$	−130.308	−0.0662400	−0.0331200	+	0.999451i	$0.510544\pi$
$158$	1521.68	0.766193
$159$	0	0
$160$	2879.49	1.42277
$161$	−1206.37	−0.590530
$162$	0	0
$163$	−907.833	−0.436239	−0.218120	−	0.975922i	$-0.569992\pi$
$163$	−907.833	−0.436239	−0.218120	+	0.975922i	$0.569992\pi$
$164$	1193.71	0.568373
$165$	0	0
$166$	−702.456	−0.328441
$167$	−3709.74	−1.71897	−0.859486	−	0.511159i	$-0.829216\pi$
$167$	−3709.74	−1.71897	−0.859486	+	0.511159i	$0.829216\pi$
$168$	0	0
$169$	−2189.34	−0.996512
$170$	2473.23	1.11581
$171$	0	0
$172$	56.7986	0.0251794
$173$	2172.32	0.954673	0.477336	−	0.878721i	$-0.341602\pi$
$173$	2172.32	0.954673	0.477336	+	0.878721i	$0.341602\pi$
$174$	0	0
$175$	−2525.15	−1.09076
$176$	327.300	0.140177
$177$	0	0
$178$	3571.22	1.50379
$179$	−2149.14	−0.897398	−0.448699	−	0.893683i	$-0.648112\pi$
$179$	−2149.14	−0.897398	−0.448699	+	0.893683i	$0.648112\pi$
$180$	0	0
$181$	−1083.16	−0.444810	−0.222405	−	0.974954i	$-0.571391\pi$
$181$	−1083.16	−0.444810	−0.222405	+	0.974954i	$0.571391\pi$
$182$	−42.9475	−0.0174916
$183$	0	0
$184$	−4235.10	−1.69682
$185$	4563.50	1.81360
$186$	0	0
$187$	−556.989	−0.217813
$188$	−957.273	−0.371363
$189$	0	0
$190$	2759.79	1.05377
$191$	−2847.74	−1.07882	−0.539412	−	0.842042i	$-0.681353\pi$
$191$	−2847.74	−1.07882	−0.539412	+	0.842042i	$0.681353\pi$
$192$	0	0
$193$	−4265.96	−1.59104	−0.795519	−	0.605929i	$-0.792802\pi$
$193$	−4265.96	−1.59104	−0.795519	+	0.605929i	$0.792802\pi$
$194$	−2729.20	−1.01003
$195$	0	0
$196$	−151.332	−0.0551501
$197$	−3081.31	−1.11439	−0.557194	−	0.830382i	$-0.688122\pi$
$197$	−3081.31	−1.11439	−0.557194	+	0.830382i	$0.688122\pi$
$198$	0	0
$199$	−1880.41	−0.669843	−0.334921	−	0.942246i	$-0.608710\pi$
$199$	−1880.41	−0.669843	−0.334921	+	0.942246i	$0.608710\pi$
$200$	−8864.79	−3.13418
$201$	0	0
$202$	−2625.18	−0.914392
$203$	1974.08	0.682528
$204$	0	0
$205$	−8518.54	−2.90225
$206$	−1451.26	−0.490844
$207$	0	0
$208$	−82.3723	−0.0274591
$209$	−621.524	−0.205702
$210$	0	0
$211$	−1383.75	−0.451475	−0.225738	−	0.974188i	$-0.572479\pi$
$211$	−1383.75	−0.451475	−0.225738	+	0.974188i	$0.572479\pi$
$212$	−1483.56	−0.480619
$213$	0	0
$214$	1450.13	0.463218
$215$	−405.325	−0.128572
$216$	0	0
$217$	−1663.11	−0.520272
$218$	3962.10	1.23095
$219$	0	0
$220$	748.732	0.229452
$221$	140.179	0.0426672
$222$	0	0
$223$	1565.56	0.470123	0.235062	−	0.971980i	$-0.424471\pi$
$223$	1565.56	0.470123	0.235062	+	0.971980i	$0.424471\pi$
$224$	−914.562	−0.272798
$225$	0	0
$226$	−1828.31	−0.538129
$227$	2467.42	0.721448	0.360724	−	0.932673i	$-0.382530\pi$
$227$	2467.42	0.721448	0.360724	+	0.932673i	$0.382530\pi$
$228$	0	0
$229$	−1294.61	−0.373581	−0.186791	−	0.982400i	$-0.559809\pi$
$229$	−1294.61	−0.373581	−0.186791	+	0.982400i	$0.559809\pi$
$230$	8417.71	2.41325
$231$	0	0
$232$	6930.21	1.96117
$233$	−5470.03	−1.53800	−0.768999	−	0.639249i	$-0.779245\pi$
$233$	−5470.03	−1.53800	−0.768999	+	0.639249i	$0.779245\pi$
$234$	0	0
$235$	6831.27	1.89627
$236$	352.761	0.0972999
$237$	0	0
$238$	−785.531	−0.213943
$239$	370.090	0.100164	0.0500819	−	0.998745i	$-0.484052\pi$
$239$	370.090	0.100164	0.0500819	+	0.998745i	$0.484052\pi$
$240$	0	0
$241$	−1364.30	−0.364658	−0.182329	−	0.983238i	$-0.558364\pi$
$241$	−1364.30	−0.364658	−0.182329	+	0.983238i	$0.558364\pi$
$242$	268.162	0.0712317
$243$	0	0
$244$	−339.501	−0.0890751
$245$	1079.93	0.281609
$246$	0	0
$247$	156.420	0.0402947
$248$	−5838.52	−1.49495
$249$	0	0
$250$	11514.2	2.91290
$251$	3715.37	0.934311	0.467155	−	0.884175i	$-0.345279\pi$
$251$	3715.37	0.934311	0.467155	+	0.884175i	$0.345279\pi$
$252$	0	0
$253$	−1895.73	−0.471080
$254$	3888.00	0.960451
$255$	0	0
$256$	−3945.82	−0.963335
$257$	−450.250	−0.109283	−0.0546417	−	0.998506i	$-0.517402\pi$
$257$	−450.250	−0.109283	−0.0546417	+	0.998506i	$0.517402\pi$
$258$	0	0
$259$	−1449.43	−0.347734
$260$	−188.435	−0.0449471
$261$	0	0
$262$	4576.88	1.07924
$263$	−1842.16	−0.431910	−0.215955	−	0.976403i	$-0.569286\pi$
$263$	−1842.16	−0.431910	−0.215955	+	0.976403i	$0.569286\pi$
$264$	0	0
$265$	10586.9	2.45415
$266$	−876.546	−0.202047
$267$	0	0
$268$	1752.77	0.399506
$269$	−2039.63	−0.462298	−0.231149	−	0.972918i	$-0.574249\pi$
$269$	−2039.63	−0.462298	−0.231149	+	0.972918i	$0.574249\pi$
$270$	0	0
$271$	7810.34	1.75072	0.875359	−	0.483473i	$-0.160625\pi$
$271$	7810.34	1.75072	0.875359	+	0.483473i	$0.160625\pi$
$272$	−1506.63	−0.335857
$273$	0	0
$274$	−1458.86	−0.321654
$275$	−3968.09	−0.870126
$276$	0	0
$277$	6066.15	1.31581	0.657905	−	0.753101i	$-0.271443\pi$
$277$	6066.15	1.31581	0.657905	+	0.753101i	$0.271443\pi$
$278$	−1560.93	−0.336756
$279$	0	0
$280$	3791.21	0.809172
$281$	4527.28	0.961121	0.480560	−	0.876962i	$-0.340433\pi$
$281$	4527.28	0.961121	0.480560	+	0.876962i	$0.340433\pi$
$282$	0	0
$283$	−3694.07	−0.775935	−0.387968	−	0.921673i	$-0.626823\pi$
$283$	−3694.07	−0.775935	−0.387968	+	0.921673i	$0.626823\pi$
$284$	−2411.16	−0.503788
$285$	0	0
$286$	−67.4889	−0.0139535
$287$	2705.60	0.556468
$288$	0	0
$289$	−2349.06	−0.478131
$290$	−13774.5	−2.78920
$291$	0	0
$292$	1871.07	0.374986
$293$	902.927	0.180033	0.0900163	−	0.995940i	$-0.471308\pi$
$293$	902.927	0.180033	0.0900163	+	0.995940i	$0.471308\pi$
$294$	0	0
$295$	−2517.36	−0.496836
$296$	−5088.37	−0.999175
$297$	0	0
$298$	−2762.60	−0.537025
$299$	477.102	0.0922793
$300$	0	0
$301$	128.736	0.0246520
$302$	2520.05	0.480173
$303$	0	0
$304$	−1681.20	−0.317181
$305$	2422.74	0.454838
$306$	0	0
$307$	−3119.31	−0.579898	−0.289949	−	0.957042i	$-0.593638\pi$
$307$	−3119.31	−0.579898	−0.289949	+	0.957042i	$0.593638\pi$
$308$	−237.807	−0.0439945
$309$	0	0
$310$	11604.7	2.12614
$311$	−3520.52	−0.641898	−0.320949	−	0.947097i	$-0.604002\pi$
$311$	−3520.52	−0.641898	−0.320949	+	0.947097i	$0.604002\pi$
$312$	0	0
$313$	−10174.8	−1.83742	−0.918709	−	0.394935i	$-0.870767\pi$
$313$	−10174.8	−1.83742	−0.918709	+	0.394935i	$0.870767\pi$
$314$	−288.789	−0.0519023
$315$	0	0
$316$	−2120.54	−0.377499
$317$	−3907.40	−0.692307	−0.346153	−	0.938178i	$-0.612512\pi$
$317$	−3907.40	−0.692307	−0.346153	+	0.938178i	$0.612512\pi$
$318$	0	0
$319$	3102.12	0.544469
$320$	11627.7	2.03128
$321$	0	0
$322$	−2673.57	−0.462710
$323$	2861.01	0.492851
$324$	0	0
$325$	998.657	0.170448
$326$	−2011.95	−0.341815
$327$	0	0
$328$	9498.29	1.59895
$329$	−2169.70	−0.363585
$330$	0	0
$331$	9015.12	1.49703	0.748513	−	0.663120i	$-0.230768\pi$
$331$	9015.12	1.49703	0.748513	+	0.663120i	$0.230768\pi$
$332$	978.908	0.161821
$333$	0	0
$334$	−8221.57	−1.34690
$335$	−12508.1	−2.03997
$336$	0	0
$337$	2727.61	0.440898	0.220449	−	0.975399i	$-0.429248\pi$
$337$	2727.61	0.440898	0.220449	+	0.975399i	$0.429248\pi$
$338$	−4852.03	−0.780816
$339$	0	0
$340$	−3446.58	−0.549756
$341$	−2613.46	−0.415034
$342$	0	0
$343$	−343.000	−0.0539949
$344$	451.943	0.0708347
$345$	0	0
$346$	4814.32	0.748033
$347$	919.041	0.142181	0.0710903	−	0.997470i	$-0.477352\pi$
$347$	919.041	0.142181	0.0710903	+	0.997470i	$0.477352\pi$
$348$	0	0
$349$	1634.89	0.250755	0.125378	−	0.992109i	$-0.459986\pi$
$349$	1634.89	0.250755	0.125378	+	0.992109i	$0.459986\pi$
$350$	−5596.26	−0.854664
$351$	0	0
$352$	−1437.17	−0.217618
$353$	−4994.77	−0.753101	−0.376551	−	0.926396i	$-0.622890\pi$
$353$	−4994.77	−0.753101	−0.376551	+	0.926396i	$0.622890\pi$
$354$	0	0
$355$	17206.4	2.57246
$356$	−4976.68	−0.740909
$357$	0	0
$358$	−4762.95	−0.703156
$359$	9765.04	1.43560	0.717798	−	0.696251i	$-0.245150\pi$
$359$	9765.04	1.43560	0.717798	+	0.696251i	$0.245150\pi$
$360$	0	0
$361$	−3666.50	−0.534554
$362$	−2400.51	−0.348531
$363$	0	0
$364$	59.8495	0.00861804
$365$	−13352.3	−1.91477
$366$	0	0
$367$	−9537.33	−1.35652	−0.678262	−	0.734820i	$-0.737267\pi$
$367$	−9537.33	−1.35652	−0.678262	+	0.734820i	$0.737267\pi$
$368$	−5127.86	−0.726380
$369$	0	0
$370$	10113.7	1.42104
$371$	−3362.55	−0.470552
$372$	0	0
$373$	−8576.29	−1.19052	−0.595259	−	0.803534i	$-0.702951\pi$
$373$	−8576.29	−1.19052	−0.595259	+	0.803534i	$0.702951\pi$
$374$	−1234.41	−0.170667
$375$	0	0
$376$	−7616.97	−1.04472
$377$	−780.718	−0.106655
$378$	0	0
$379$	−13710.9	−1.85826	−0.929131	−	0.369752i	$-0.879443\pi$
$379$	−13710.9	−1.85826	−0.929131	+	0.369752i	$0.879443\pi$
$380$	−3845.91	−0.519187
$381$	0	0
$382$	−6311.20	−0.845311
$383$	8428.82	1.12452	0.562262	−	0.826959i	$-0.309931\pi$
$383$	8428.82	1.12452	0.562262	+	0.826959i	$0.309931\pi$
$384$	0	0
$385$	1697.03	0.224646
$386$	−9454.26	−1.24666
$387$	0	0
$388$	3803.28	0.497635
$389$	1757.23	0.229036	0.114518	−	0.993421i	$-0.463468\pi$
$389$	1757.23	0.229036	0.114518	+	0.993421i	$0.463468\pi$
$390$	0	0
$391$	8726.44	1.12868
$392$	−1204.14	−0.155148
$393$	0	0
$394$	−6828.84	−0.873178
$395$	15132.6	1.92760
$396$	0	0
$397$	4018.53	0.508021	0.254010	−	0.967201i	$-0.418250\pi$
$397$	4018.53	0.508021	0.254010	+	0.967201i	$0.418250\pi$
$398$	−4167.39	−0.524855
$399$	0	0
$400$	−10733.5	−1.34169
$401$	−3344.16	−0.416458	−0.208229	−	0.978080i	$-0.566770\pi$
$401$	−3344.16	−0.416458	−0.208229	+	0.978080i	$0.566770\pi$
$402$	0	0
$403$	657.734	0.0813005
$404$	3658.33	0.450516
$405$	0	0
$406$	4374.97	0.534794
$407$	−2277.67	−0.277396
$408$	0	0
$409$	−13540.8	−1.63704	−0.818521	−	0.574477i	$-0.805206\pi$
$409$	−13540.8	−1.63704	−0.818521	+	0.574477i	$0.805206\pi$
$410$	−18878.9	−2.27405
$411$	0	0
$412$	2022.40	0.241836
$413$	799.547	0.0952619
$414$	0	0
$415$	−6985.66	−0.826295
$416$	361.696	0.0426288
$417$	0	0
$418$	−1377.43	−0.161178
$419$	7286.55	0.849573	0.424786	−	0.905294i	$-0.360349\pi$
$419$	7286.55	0.849573	0.424786	+	0.905294i	$0.360349\pi$
$420$	0	0
$421$	−10778.5	−1.24777	−0.623886	−	0.781515i	$-0.714447\pi$
$421$	−10778.5	−1.24777	−0.623886	+	0.781515i	$0.714447\pi$
$422$	−3066.68	−0.353753
$423$	0	0
$424$	−11804.6	−1.35208
$425$	18266.0	2.08477
$426$	0	0
$427$	−769.494	−0.0872094
$428$	−2020.83	−0.228225
$429$	0	0
$430$	−898.285	−0.100742
$431$	8547.58	0.955272	0.477636	−	0.878558i	$-0.341494\pi$
$431$	8547.58	0.955272	0.477636	+	0.878558i	$0.341494\pi$
$432$	0	0
$433$	7140.86	0.792535	0.396267	−	0.918135i	$-0.370305\pi$
$433$	7140.86	0.792535	0.396267	+	0.918135i	$0.370305\pi$
$434$	−3685.80	−0.407659
$435$	0	0
$436$	−5521.39	−0.606483
$437$	9737.52	1.06592
$438$	0	0
$439$	−2411.65	−0.262191	−0.131095	−	0.991370i	$-0.541849\pi$
$439$	−2411.65	−0.262191	−0.131095	+	0.991370i	$0.541849\pi$
$440$	5957.62	0.645496
$441$	0	0
$442$	310.666	0.0334318
$443$	12210.1	1.30953	0.654764	−	0.755834i	$-0.272768\pi$
$443$	12210.1	1.30953	0.654764	+	0.755834i	$0.272768\pi$
$444$	0	0
$445$	35514.5	3.78325
$446$	3469.61	0.368365
$447$	0	0
$448$	−3693.12	−0.389472
$449$	4214.35	0.442956	0.221478	−	0.975165i	$-0.428912\pi$
$449$	4214.35	0.442956	0.221478	+	0.975165i	$0.428912\pi$
$450$	0	0
$451$	4251.65	0.443908
$452$	2547.84	0.265134
$453$	0	0
$454$	5468.33	0.565290
$455$	−427.096	−0.0440057
$456$	0	0
$457$	169.614	0.0173615	0.00868075	−	0.999962i	$-0.497237\pi$
$457$	169.614	0.0173615	0.00868075	+	0.999962i	$0.497237\pi$
$458$	−2869.13	−0.292719
$459$	0	0
$460$	−11730.5	−1.18900
$461$	813.238	0.0821611	0.0410805	−	0.999156i	$-0.486920\pi$
$461$	813.238	0.0821611	0.0410805	+	0.999156i	$0.486920\pi$
$462$	0	0
$463$	−12681.2	−1.27289	−0.636443	−	0.771324i	$-0.719595\pi$
$463$	−12681.2	−1.27289	−0.636443	+	0.771324i	$0.719595\pi$
$464$	8391.10	0.839541
$465$	0	0
$466$	−12122.8	−1.20510
$467$	7297.16	0.723067	0.361534	−	0.932359i	$-0.382253\pi$
$467$	7297.16	0.723067	0.361534	+	0.932359i	$0.382253\pi$
$468$	0	0
$469$	3972.73	0.391138
$470$	15139.5	1.48582
$471$	0	0
$472$	2806.90	0.273724
$473$	202.300	0.0196655
$474$	0	0
$475$	20382.3	1.96885
$476$	1094.68	0.105409
$477$	0	0
$478$	820.199	0.0784833
$479$	−9874.50	−0.941915	−0.470958	−	0.882156i	$-0.656092\pi$
$479$	−9874.50	−0.941915	−0.470958	+	0.882156i	$0.656092\pi$
$480$	0	0
$481$	573.227	0.0543387
$482$	−3023.59	−0.285727
$483$	0	0
$484$	−373.697	−0.0350955
$485$	−27140.9	−2.54104
$486$	0	0
$487$	4995.82	0.464851	0.232425	−	0.972614i	$-0.425334\pi$
$487$	4995.82	0.464851	0.232425	+	0.972614i	$0.425334\pi$
$488$	−2701.39	−0.250586
$489$	0	0
$490$	2393.35	0.220655
$491$	−900.456	−0.0827638	−0.0413819	−	0.999143i	$-0.513176\pi$
$491$	−900.456	−0.0827638	−0.0413819	+	0.999143i	$0.513176\pi$
$492$	0	0
$493$	−14279.7	−1.30452
$494$	346.661	0.0315729
$495$	0	0
$496$	−7069.28	−0.639960
$497$	−5464.99	−0.493236
$498$	0	0
$499$	11471.6	1.02914	0.514568	−	0.857450i	$-0.327952\pi$
$499$	11471.6	1.02914	0.514568	+	0.857450i	$0.327952\pi$
$500$	−16045.7	−1.43517
$501$	0	0
$502$	8234.04	0.732078
$503$	4206.65	0.372893	0.186447	−	0.982465i	$-0.440303\pi$
$503$	4206.65	0.372893	0.186447	+	0.982465i	$0.440303\pi$
$504$	0	0
$505$	−26106.5	−2.30044
$506$	−4201.33	−0.369114
$507$	0	0
$508$	−5418.12	−0.473209
$509$	16162.9	1.40748	0.703742	−	0.710456i	$-0.251511\pi$
$509$	16162.9	1.40748	0.703742	+	0.710456i	$0.251511\pi$
$510$	0	0
$511$	4240.85	0.367132
$512$	−9737.04	−0.840470
$513$	0	0
$514$	−997.849	−0.0856289
$515$	−14432.2	−1.23487
$516$	0	0
$517$	−3409.53	−0.290040
$518$	−3212.24	−0.272467
$519$	0	0
$520$	−1499.37	−0.126445
$521$	−11332.7	−0.952964	−0.476482	−	0.879184i	$-0.658088\pi$
$521$	−11332.7	−0.952964	−0.476482	+	0.879184i	$0.658088\pi$
$522$	0	0
$523$	14087.3	1.17781	0.588904	−	0.808203i	$-0.299560\pi$
$523$	14087.3	1.17781	0.588904	+	0.808203i	$0.299560\pi$
$524$	−6378.12	−0.531736
$525$	0	0
$526$	−4082.61	−0.338423
$527$	12030.3	0.994399
$528$	0	0
$529$	17533.6	1.44108
$530$	23462.9	1.92295
$531$	0	0
$532$	1221.51	0.0995474
$533$	−1070.02	−0.0869566
$534$	0	0
$535$	14421.0	1.16537
$536$	13946.7	1.12389
$537$	0	0
$538$	−4520.25	−0.362234
$539$	−539.000	−0.0430730
$540$	0	0
$541$	−6339.91	−0.503833	−0.251917	−	0.967749i	$-0.581061\pi$
$541$	−6339.91	−0.503833	−0.251917	+	0.967749i	$0.581061\pi$
$542$	17309.4	1.37177
$543$	0	0
$544$	6615.60	0.521400
$545$	39401.6	3.09684
$546$	0	0
$547$	959.649	0.0750121	0.0375061	−	0.999296i	$-0.488059\pi$
$547$	959.649	0.0750121	0.0375061	+	0.999296i	$0.488059\pi$
$548$	2033.00	0.158477
$549$	0	0
$550$	−8794.12	−0.681786
$551$	−15934.2	−1.23198
$552$	0	0
$553$	−4806.30	−0.369592
$554$	13443.9	1.03100
$555$	0	0
$556$	2175.23	0.165918
$557$	17184.0	1.30720	0.653598	−	0.756842i	$-0.273259\pi$
$557$	17184.0	1.30720	0.653598	+	0.756842i	$0.273259\pi$
$558$	0	0
$559$	−50.9133	−0.00385224
$560$	4590.40	0.346393
$561$	0	0
$562$	10033.4	0.753086
$563$	−6598.36	−0.493939	−0.246969	−	0.969023i	$-0.579435\pi$
$563$	−6598.36	−0.493939	−0.246969	+	0.969023i	$0.579435\pi$
$564$	0	0
$565$	−18181.8	−1.35383
$566$	−8186.84	−0.607984
$567$	0	0
$568$	−19185.4	−1.41726
$569$	−17565.4	−1.29416	−0.647081	−	0.762422i	$-0.724010\pi$
$569$	−17565.4	−1.29416	−0.647081	+	0.762422i	$0.724010\pi$
$570$	0	0
$571$	−10179.7	−0.746074	−0.373037	−	0.927817i	$-0.621684\pi$
$571$	−10179.7	−0.746074	−0.373037	+	0.927817i	$0.621684\pi$
$572$	94.0492	0.00687482
$573$	0	0
$574$	5996.18	0.436020
$575$	62168.6	4.50889
$576$	0	0
$577$	6287.76	0.453662	0.226831	−	0.973934i	$-0.427164\pi$
$577$	6287.76	0.453662	0.226831	+	0.973934i	$0.427164\pi$
$578$	−5206.01	−0.374639
$579$	0	0
$580$	19195.5	1.37423
$581$	2218.74	0.158432
$582$	0	0
$583$	−5284.01	−0.375371
$584$	14888.0	1.05491
$585$	0	0
$586$	2001.08	0.141064
$587$	−18601.0	−1.30791	−0.653957	−	0.756532i	$-0.726892\pi$
$587$	−18601.0	−1.30791	−0.653957	+	0.756532i	$0.726892\pi$
$588$	0	0
$589$	13424.2	0.939106
$590$	−5579.01	−0.389295
$591$	0	0
$592$	−6161.00	−0.427729
$593$	6362.24	0.440583	0.220292	−	0.975434i	$-0.429299\pi$
$593$	6362.24	0.440583	0.220292	+	0.975434i	$0.429299\pi$
$594$	0	0
$595$	−7811.82	−0.538241
$596$	3849.83	0.264589
$597$	0	0
$598$	1057.36	0.0723054
$599$	−12238.1	−0.834787	−0.417393	−	0.908726i	$-0.637056\pi$
$599$	−12238.1	−0.834787	−0.417393	+	0.908726i	$0.637056\pi$
$600$	0	0
$601$	−25394.1	−1.72354	−0.861770	−	0.507300i	$-0.830644\pi$
$601$	−25394.1	−1.72354	−0.861770	+	0.507300i	$0.830644\pi$
$602$	285.307	0.0193160
$603$	0	0
$604$	−3511.81	−0.236579
$605$	2666.77	0.179206
$606$	0	0
$607$	5680.37	0.379834	0.189917	−	0.981800i	$-0.439178\pi$
$607$	5680.37	0.379834	0.189917	+	0.981800i	$0.439178\pi$
$608$	7382.11	0.492408
$609$	0	0
$610$	5369.30	0.356388
$611$	858.084	0.0568157
$612$	0	0
$613$	682.502	0.0449690	0.0224845	−	0.999747i	$-0.492842\pi$
$613$	682.502	0.0449690	0.0224845	+	0.999747i	$0.492842\pi$
$614$	−6913.06	−0.454378
$615$	0	0
$616$	−1892.22	−0.123766
$617$	4304.44	0.280859	0.140430	−	0.990091i	$-0.455152\pi$
$617$	4304.44	0.280859	0.140430	+	0.990091i	$0.455152\pi$
$618$	0	0
$619$	−13022.7	−0.845602	−0.422801	−	0.906223i	$-0.638953\pi$
$619$	−13022.7	−0.845602	−0.422801	+	0.906223i	$0.638953\pi$
$620$	−16171.7	−1.04754
$621$	0	0
$622$	−7802.21	−0.502958
$623$	−11279.9	−0.725390
$624$	0	0
$625$	69412.9	4.44242
$626$	−22549.4	−1.43971
$627$	0	0
$628$	402.443	0.0255720
$629$	10484.6	0.664625
$630$	0	0
$631$	18156.4	1.14548	0.572739	−	0.819738i	$-0.305881\pi$
$631$	18156.4	1.14548	0.572739	+	0.819738i	$0.305881\pi$
$632$	−16873.0	−1.06198
$633$	0	0
$634$	−8659.62	−0.542456
$635$	38664.7	2.41632
$636$	0	0
$637$	135.651	0.00843753
$638$	6874.96	0.426618
$639$	0	0
$640$	2733.62	0.168837
$641$	−10368.7	−0.638905	−0.319452	−	0.947602i	$-0.603499\pi$
$641$	−10368.7	−0.638905	−0.319452	+	0.947602i	$0.603499\pi$
$642$	0	0
$643$	−17679.6	−1.08431	−0.542157	−	0.840277i	$-0.682392\pi$
$643$	−17679.6	−1.08431	−0.542157	+	0.840277i	$0.682392\pi$
$644$	3725.76	0.227975
$645$	0	0
$646$	6340.61	0.386173
$647$	1774.38	0.107817	0.0539087	−	0.998546i	$-0.482832\pi$
$647$	1774.38	0.107817	0.0539087	+	0.998546i	$0.482832\pi$
$648$	0	0
$649$	1256.43	0.0759927
$650$	2213.24	0.133554
$651$	0	0
$652$	2803.76	0.168410
$653$	25514.1	1.52901	0.764506	−	0.644616i	$-0.222983\pi$
$653$	25514.1	1.52901	0.764506	+	0.644616i	$0.222983\pi$
$654$	0	0
$655$	45515.4	2.71517
$656$	11500.5	0.684483
$657$	0	0
$658$	−4808.52	−0.284887
$659$	557.327	0.0329444	0.0164722	−	0.999864i	$-0.494756\pi$
$659$	557.327	0.0329444	0.0164722	+	0.999864i	$0.494756\pi$
$660$	0	0
$661$	11263.2	0.662768	0.331384	−	0.943496i	$-0.392484\pi$
$661$	11263.2	0.662768	0.331384	+	0.943496i	$0.392484\pi$
$662$	19979.4	1.17299
$663$	0	0
$664$	7789.11	0.455235
$665$	−8716.92	−0.508312
$666$	0	0
$667$	−48601.4	−2.82137
$668$	11457.2	0.663610
$669$	0	0
$670$	−27720.6	−1.59842
$671$	−1209.20	−0.0695690
$672$	0	0
$673$	−3615.18	−0.207065	−0.103533	−	0.994626i	$-0.533015\pi$
$673$	−3615.18	−0.207065	−0.103533	+	0.994626i	$0.533015\pi$
$674$	6044.97	0.345465
$675$	0	0
$676$	6761.56	0.384704
$677$	23421.6	1.32964	0.664820	−	0.747004i	$-0.268508\pi$
$677$	23421.6	1.32964	0.664820	+	0.747004i	$0.268508\pi$
$678$	0	0
$679$	8620.30	0.487212
$680$	−27424.2	−1.54657
$681$	0	0
$682$	−5791.97	−0.325199
$683$	−29364.9	−1.64512	−0.822559	−	0.568679i	$-0.807455\pi$
$683$	−29364.9	−1.64512	−0.822559	+	0.568679i	$0.807455\pi$
$684$	0	0
$685$	−14507.9	−0.809222
$686$	−760.161	−0.0423077
$687$	0	0
$688$	547.213	0.0303231
$689$	1329.84	0.0735309
$690$	0	0
$691$	−20430.9	−1.12479	−0.562394	−	0.826869i	$-0.690120\pi$
$691$	−20430.9	−1.12479	−0.562394	+	0.826869i	$0.690120\pi$
$692$	−6709.00	−0.368552
$693$	0	0
$694$	2036.79	0.111406
$695$	−15522.8	−0.847215
$696$	0	0
$697$	−19571.3	−1.06358
$698$	3623.26	0.196479
$699$	0	0
$700$	7798.67	0.421089
$701$	6673.16	0.359546	0.179773	−	0.983708i	$-0.442464\pi$
$701$	6673.16	0.359546	0.179773	+	0.983708i	$0.442464\pi$
$702$	0	0
$703$	11699.4	0.627669
$704$	−5803.47	−0.310691
$705$	0	0
$706$	−11069.5	−0.590092
$707$	8291.75	0.441080
$708$	0	0
$709$	−14706.4	−0.778999	−0.389500	−	0.921027i	$-0.627352\pi$
$709$	−14706.4	−0.778999	−0.389500	+	0.921027i	$0.627352\pi$
$710$	38133.1	2.01565
$711$	0	0
$712$	−39599.2	−2.08433
$713$	40945.4	2.15066
$714$	0	0
$715$	−671.152	−0.0351044
$716$	6637.41	0.346441
$717$	0	0
$718$	21641.4	1.12486
$719$	35481.4	1.84038	0.920189	−	0.391473i	$-0.128035\pi$
$719$	35481.4	1.84038	0.920189	+	0.391473i	$0.128035\pi$
$720$	0	0
$721$	4583.85	0.236771
$722$	−8125.75	−0.418849
$723$	0	0
$724$	3345.24	0.171719
$725$	−101731.	−5.21132
$726$	0	0
$727$	8818.93	0.449898	0.224949	−	0.974371i	$-0.427778\pi$
$727$	8818.93	0.449898	0.224949	+	0.974371i	$0.427778\pi$
$728$	476.219	0.0242443
$729$	0	0
$730$	−29591.5	−1.50031
$731$	−931.231	−0.0471174
$732$	0	0
$733$	−645.865	−0.0325451	−0.0162726	−	0.999868i	$-0.505180\pi$
$733$	−645.865	−0.0325451	−0.0162726	+	0.999868i	$0.505180\pi$
$734$	−21136.7	−1.06290
$735$	0	0
$736$	22516.4	1.12767
$737$	6242.87	0.312020
$738$	0	0
$739$	6264.49	0.311831	0.155915	−	0.987770i	$-0.450167\pi$
$739$	6264.49	0.311831	0.155915	+	0.987770i	$0.450167\pi$
$740$	−14093.9	−0.700140
$741$	0	0
$742$	−7452.12	−0.368701
$743$	−4320.14	−0.213312	−0.106656	−	0.994296i	$-0.534014\pi$
$743$	−4320.14	−0.213312	−0.106656	+	0.994296i	$0.534014\pi$
$744$	0	0
$745$	−27473.1	−1.35105
$746$	−19006.9	−0.932830
$747$	0	0
$748$	1720.21	0.0840870
$749$	−4580.29	−0.223445
$750$	0	0
$751$	32107.5	1.56008	0.780041	−	0.625729i	$-0.215198\pi$
$751$	32107.5	1.56008	0.780041	+	0.625729i	$0.215198\pi$
$752$	−9222.63	−0.447227
$753$	0	0
$754$	−1730.24	−0.0835697
$755$	25060.9	1.20803
$756$	0	0
$757$	25385.0	1.21880	0.609401	−	0.792862i	$-0.291410\pi$
$757$	25385.0	1.21880	0.609401	+	0.792862i	$0.291410\pi$
$758$	−30386.2	−1.45604
$759$	0	0
$760$	−30601.7	−1.46058
$761$	−5092.57	−0.242583	−0.121291	−	0.992617i	$-0.538704\pi$
$761$	−5092.57	−0.242583	−0.121291	+	0.992617i	$0.538704\pi$
$762$	0	0
$763$	−12514.5	−0.593780
$764$	8794.98	0.416480
$765$	0	0
$766$	18680.0	0.881119
$767$	−316.209	−0.0148861
$768$	0	0
$769$	10162.4	0.476547	0.238273	−	0.971198i	$-0.423419\pi$
$769$	10162.4	0.476547	0.238273	+	0.971198i	$0.423419\pi$
$770$	3760.99	0.176021
$771$	0	0
$772$	13175.0	0.614221
$773$	14229.8	0.662108	0.331054	−	0.943612i	$-0.392596\pi$
$773$	14229.8	0.662108	0.331054	+	0.943612i	$0.392596\pi$
$774$	0	0
$775$	85705.9	3.97245
$776$	30262.5	1.39995
$777$	0	0
$778$	3894.40	0.179461
$779$	−21838.9	−1.00444
$780$	0	0
$781$	−8587.84	−0.393466
$782$	19339.6	0.884379
$783$	0	0
$784$	−1457.97	−0.0664163
$785$	−2871.90	−0.130576
$786$	0	0
$787$	13391.2	0.606537	0.303269	−	0.952905i	$-0.401922\pi$
$787$	13391.2	0.606537	0.303269	+	0.952905i	$0.401922\pi$
$788$	9516.34	0.430210
$789$	0	0
$790$	33537.0	1.51037
$791$	5774.79	0.259580
$792$	0	0
$793$	304.323	0.0136278
$794$	8905.91	0.398059
$795$	0	0
$796$	5807.46	0.258593
$797$	−8830.58	−0.392466	−0.196233	−	0.980557i	$-0.562871\pi$
$797$	−8830.58	−0.392466	−0.196233	+	0.980557i	$0.562871\pi$
$798$	0	0
$799$	15694.8	0.694922
$800$	47130.7	2.08290
$801$	0	0
$802$	−7411.38	−0.326315
$803$	6664.20	0.292870
$804$	0	0
$805$	−26587.7	−1.16409
$806$	1457.68	0.0637029
$807$	0	0
$808$	29109.1	1.26739
$809$	44283.6	1.92451	0.962255	−	0.272149i	$-0.0877343\pi$
$809$	44283.6	1.92451	0.962255	+	0.272149i	$0.0877343\pi$
$810$	0	0
$811$	−42502.7	−1.84029	−0.920143	−	0.391582i	$-0.871928\pi$
$811$	−42502.7	−1.84029	−0.920143	+	0.391582i	$0.871928\pi$
$812$	−6096.75	−0.263490
$813$	0	0
$814$	−5047.81	−0.217353
$815$	−20008.1	−0.859942
$816$	0	0
$817$	−1039.13	−0.0444975
$818$	−30009.3	−1.28270
$819$	0	0
$820$	26308.7	1.12041
$821$	−20018.3	−0.850968	−0.425484	−	0.904966i	$-0.639896\pi$
$821$	−20018.3	−0.850968	−0.425484	+	0.904966i	$0.639896\pi$
$822$	0	0
$823$	−31778.1	−1.34595	−0.672973	−	0.739667i	$-0.734983\pi$
$823$	−31778.1	−1.34595	−0.672973	+	0.739667i	$0.734983\pi$
$824$	16092.1	0.680334
$825$	0	0
$826$	1771.97	0.0746424
$827$	9416.88	0.395958	0.197979	−	0.980206i	$-0.436562\pi$
$827$	9416.88	0.395958	0.197979	+	0.980206i	$0.436562\pi$
$828$	0	0
$829$	−30106.3	−1.26132	−0.630661	−	0.776058i	$-0.717216\pi$
$829$	−30106.3	−1.26132	−0.630661	+	0.776058i	$0.717216\pi$
$830$	−15481.7	−0.647443
$831$	0	0
$832$	1460.57	0.0608609
$833$	2481.13	0.103201
$834$	0	0
$835$	−81760.4	−3.38855
$836$	1919.52	0.0794114
$837$	0	0
$838$	16148.5	0.665682
$839$	45186.6	1.85937	0.929687	−	0.368350i	$-0.120077\pi$
$839$	45186.6	1.85937	0.929687	+	0.368350i	$0.120077\pi$
$840$	0	0
$841$	55141.3	2.26091
$842$	−23887.4	−0.977691
$843$	0	0
$844$	4273.58	0.174292
$845$	−48251.6	−1.96439
$846$	0	0
$847$	−847.000	−0.0343604
$848$	−14293.0	−0.578802
$849$	0	0
$850$	40481.2	1.63352
$851$	35684.7	1.43743
$852$	0	0
$853$	−2401.55	−0.0963980	−0.0481990	−	0.998838i	$-0.515348\pi$
$853$	−2401.55	−0.0963980	−0.0481990	+	0.998838i	$0.515348\pi$
$854$	−1705.36	−0.0683328
$855$	0	0
$856$	−16079.6	−0.642043
$857$	4218.83	0.168159	0.0840797	−	0.996459i	$-0.473205\pi$
$857$	4218.83	0.168159	0.0840797	+	0.996459i	$0.473205\pi$
$858$	0	0
$859$	−15508.3	−0.615992	−0.307996	−	0.951388i	$-0.599658\pi$
$859$	−15508.3	−0.615992	−0.307996	+	0.951388i	$0.599658\pi$
$860$	1251.81	0.0496352
$861$	0	0
$862$	18943.2	0.748503
$863$	−18753.2	−0.739706	−0.369853	−	0.929090i	$-0.620592\pi$
$863$	−18753.2	−0.739706	−0.369853	+	0.929090i	$0.620592\pi$
$864$	0	0
$865$	47876.6	1.88191
$866$	15825.6	0.620990
$867$	0	0
$868$	5136.35	0.200851
$869$	−7552.75	−0.294833
$870$	0	0
$871$	−1571.16	−0.0611213
$872$	−43933.4	−1.70616
$873$	0	0
$874$	21580.4	0.835203
$875$	−36368.2	−1.40511
$876$	0	0
$877$	45435.5	1.74943	0.874713	−	0.484641i	$-0.161050\pi$
$877$	45435.5	1.74943	0.874713	+	0.484641i	$0.161050\pi$
$878$	−5344.73	−0.205439
$879$	0	0
$880$	7213.49	0.276326
$881$	−20328.9	−0.777410	−0.388705	−	0.921362i	$-0.627078\pi$
$881$	−20328.9	−0.777410	−0.388705	+	0.921362i	$0.627078\pi$
$882$	0	0
$883$	18438.6	0.702728	0.351364	−	0.936239i	$-0.385718\pi$
$883$	18438.6	0.702728	0.351364	+	0.936239i	$0.385718\pi$
$884$	−432.929	−0.0164717
$885$	0	0
$886$	27060.2	1.02608
$887$	35240.4	1.33400	0.667000	−	0.745058i	$-0.267578\pi$
$887$	35240.4	1.33400	0.667000	+	0.745058i	$0.267578\pi$
$888$	0	0
$889$	−12280.4	−0.463298
$890$	78707.6	2.96436
$891$	0	0
$892$	−4835.08	−0.181491
$893$	17513.3	0.656281
$894$	0	0
$895$	−47365.7	−1.76901
$896$	−868.232	−0.0323723
$897$	0	0
$898$	9339.89	0.347078
$899$	−67002.1	−2.48570
$900$	0	0
$901$	24323.4	0.899368
$902$	9422.57	0.347824
$903$	0	0
$904$	20273.0	0.745875
$905$	−23872.2	−0.876839
$906$	0	0
$907$	−27735.7	−1.01538	−0.507690	−	0.861540i	$-0.669500\pi$
$907$	−27735.7	−1.01538	−0.507690	+	0.861540i	$0.669500\pi$
$908$	−7620.40	−0.278515
$909$	0	0
$910$	−946.536	−0.0344806
$911$	12663.6	0.460554	0.230277	−	0.973125i	$-0.426037\pi$
$911$	12663.6	0.460554	0.230277	+	0.973125i	$0.426037\pi$
$912$	0	0
$913$	3486.59	0.126385
$914$	375.901	0.0136036
$915$	0	0
$916$	3998.28	0.144221
$917$	−14456.3	−0.520598
$918$	0	0
$919$	−16398.4	−0.588612	−0.294306	−	0.955711i	$-0.595088\pi$
$919$	−16398.4	−0.588612	−0.294306	+	0.955711i	$0.595088\pi$
$920$	−93339.0	−3.34489
$921$	0	0
$922$	1802.31	0.0643773
$923$	2161.32	0.0770756
$924$	0	0
$925$	74694.2	2.65506
$926$	−28104.3	−0.997369
$927$	0	0
$928$	−36845.2	−1.30335
$929$	−42073.8	−1.48589	−0.742947	−	0.669350i	$-0.766573\pi$
$929$	−42073.8	−1.48589	−0.742947	+	0.669350i	$0.766573\pi$
$930$	0	0
$931$	2768.61	0.0974623
$932$	16893.7	0.593745
$933$	0	0
$934$	16172.1	0.566559
$935$	−12275.7	−0.429368
$936$	0	0
$937$	11771.2	0.410402	0.205201	−	0.978720i	$-0.434215\pi$
$937$	11771.2	0.410402	0.205201	+	0.978720i	$0.434215\pi$
$938$	8804.42	0.306476
$939$	0	0
$940$	−21097.7	−0.732055
$941$	6056.53	0.209817	0.104908	−	0.994482i	$-0.466545\pi$
$941$	6056.53	0.209817	0.104908	+	0.994482i	$0.466545\pi$
$942$	0	0
$943$	−66611.3	−2.30028
$944$	3398.59	0.117177
$945$	0	0
$946$	448.340	0.0154089
$947$	−2545.73	−0.0873551	−0.0436775	−	0.999046i	$-0.513907\pi$
$947$	−2545.73	−0.0873551	−0.0436775	+	0.999046i	$0.513907\pi$
$948$	0	0
$949$	−1677.19	−0.0573699
$950$	45171.5	1.54269
$951$	0	0
$952$	8710.29	0.296536
$953$	−17699.0	−0.601602	−0.300801	−	0.953687i	$-0.597254\pi$
$953$	−17699.0	−0.601602	−0.300801	+	0.953687i	$0.597254\pi$
$954$	0	0
$955$	−62762.5	−2.12665
$956$	−1142.99	−0.0386683
$957$	0	0
$958$	−21884.0	−0.738037
$959$	4607.89	0.155158
$960$	0	0
$961$	26656.5	0.894784
$962$	1270.39	0.0425770
$963$	0	0
$964$	4213.52	0.140776
$965$	−94019.1	−3.13635
$966$	0	0
$967$	−48183.5	−1.60235	−0.801177	−	0.598427i	$-0.795793\pi$
$967$	−48183.5	−1.60235	−0.801177	+	0.598427i	$0.795793\pi$
$968$	−2973.48	−0.0987308
$969$	0	0
$970$	−60149.9	−1.99103
$971$	9670.55	0.319612	0.159806	−	0.987148i	$-0.448913\pi$
$971$	9670.55	0.319612	0.159806	+	0.987148i	$0.448913\pi$
$972$	0	0
$973$	4930.25	0.162443
$974$	11071.8	0.364234
$975$	0	0
$976$	−3270.84	−0.107272
$977$	14624.6	0.478897	0.239449	−	0.970909i	$-0.423033\pi$
$977$	14624.6	0.478897	0.239449	+	0.970909i	$0.423033\pi$
$978$	0	0
$979$	−17725.5	−0.578661
$980$	−3335.26	−0.108715
$981$	0	0
$982$	−1995.60	−0.0648495
$983$	11183.6	0.362871	0.181436	−	0.983403i	$-0.441926\pi$
$983$	11183.6	0.362871	0.181436	+	0.983403i	$0.441926\pi$
$984$	0	0
$985$	−67910.3	−2.19675
$986$	−31646.9	−1.02215
$987$	0	0
$988$	−483.090	−0.0155558
$989$	−3169.47	−0.101904
$990$	0	0
$991$	18858.3	0.604493	0.302247	−	0.953230i	$-0.402263\pi$
$991$	18858.3	0.604493	0.302247	+	0.953230i	$0.402263\pi$
$992$	31041.1	0.993505
$993$	0	0
$994$	−12111.6	−0.386475
$995$	−41443.1	−1.32044
$996$	0	0
$997$	156.252	0.00496345	0.00248172	−	0.999997i	$-0.499210\pi$
$997$	156.252	0.00496345	0.00248172	+	0.999997i	$0.499210\pi$
$998$	25423.5	0.806379
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	693.4.a.t.1.6	yes	8
3.2	odd	2		693.4.a.s.1.3	✓	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
693.4.a.s.1.3	✓	8	3.2	odd	2
693.4.a.t.1.6	yes	8	1.1	even	1	trivial

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)

\(f(q)\)	\(=\)	\(q+2.21621 q^{2} -3.08840 q^{4} +22.0394 q^{5} -7.00000 q^{7} -24.5743 q^{8} +48.8440 q^{10} -11.0000 q^{11} +2.76840 q^{13} -15.5135 q^{14} -29.7545 q^{16} +50.6354 q^{17} +56.5022 q^{19} -68.0666 q^{20} -24.3783 q^{22} +172.339 q^{23} +360.735 q^{25} +6.13535 q^{26} +21.6188 q^{28} -282.011 q^{29} +237.587 q^{31} +130.652 q^{32} +112.219 q^{34} -154.276 q^{35} +207.061 q^{37} +125.221 q^{38} -541.602 q^{40} -386.514 q^{41} -18.3909 q^{43} +33.9725 q^{44} +381.939 q^{46} +309.957 q^{47} +49.0000 q^{49} +799.465 q^{50} -8.54993 q^{52} +480.364 q^{53} -242.433 q^{55} +172.020 q^{56} -624.996 q^{58} -114.221 q^{59} +109.928 q^{61} +526.543 q^{62} +527.588 q^{64} +61.0138 q^{65} -567.534 q^{67} -156.383 q^{68} -341.908 q^{70} +780.713 q^{71} -605.836 q^{73} +458.891 q^{74} -174.502 q^{76} +77.0000 q^{77} +686.614 q^{79} -655.772 q^{80} -856.597 q^{82} -316.962 q^{83} +1115.97 q^{85} -40.7582 q^{86} +270.317 q^{88} +1611.41 q^{89} -19.3788 q^{91} -532.252 q^{92} +686.931 q^{94} +1245.27 q^{95} -1231.47 q^{97} +108.594 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 2 q^{2} + 30 q^{4} + 10 q^{5} - 56 q^{7} + 15 q^{8} - 13 q^{10} - 88 q^{11} - 148 q^{13} - 14 q^{14} + 266 q^{16} + 114 q^{17} + 58 q^{19} + 291 q^{20} - 22 q^{22} + 246 q^{23} + 244 q^{25} + 305 q^{26}+ \cdots + 98 q^{98}+O(q^{100}) \) 8 * q + 2 * q^2 + 30 * q^4 + 10 * q^5 - 56 * q^7 + 15 * q^8 - 13 * q^10 - 88 * q^11 - 148 * q^13 - 14 * q^14 + 266 * q^16 + 114 * q^17 + 58 * q^19 + 291 * q^20 - 22 * q^22 + 246 * q^23 + 244 * q^25 + 305 * q^26 - 210 * q^28 - 72 * q^29 + 252 * q^31 + 1272 * q^32 + 630 * q^34 - 70 * q^35 - 80 * q^37 + 1885 * q^38 - 342 * q^40 + 682 * q^41 - 106 * q^43 - 330 * q^44 + 120 * q^46 + 828 * q^47 + 392 * q^49 + 801 * q^50 - 1681 * q^52 + 462 * q^53 - 110 * q^55 - 105 * q^56 - 1087 * q^58 + 626 * q^59 - 854 * q^61 + 1350 * q^62 + 2997 * q^64 + 22 * q^65 + 130 * q^67 + 2202 * q^68 + 91 * q^70 + 326 * q^71 - 390 * q^73 - 359 * q^74 + 2041 * q^76 + 616 * q^77 - 508 * q^79 + 4391 * q^80 + 1528 * q^82 + 1596 * q^83 - 880 * q^85 - 414 * q^86 - 165 * q^88 + 4324 * q^89 + 1036 * q^91 + 2092 * q^92 - 1685 * q^94 + 1076 * q^95 - 964 * q^97 + 98 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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