LMFDB - Embedded newform 847.4.a.d.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [847,4,Mod(1,847)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(847, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("847.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 847 = 7 \cdot 11^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	847.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:	$49.9746177749$
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:	$\Z[a_1, a_2, a_3]$
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	$\chi$	$=$	847.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q-1.53253 q^{2} +10.1459 q^{3} -5.65135 q^{4} +8.69995 q^{5} -15.5490 q^{6} +7.00000 q^{7} +20.9211 q^{8} +75.9402 q^{9} -13.3330 q^{10} -57.3382 q^{12} +76.3572 q^{13} -10.7277 q^{14} +88.2693 q^{15} +13.1485 q^{16} -39.7278 q^{17} -116.381 q^{18} +27.9876 q^{19} -49.1664 q^{20} +71.0216 q^{21} +87.2055 q^{23} +212.265 q^{24} -49.3108 q^{25} -117.020 q^{26} +496.545 q^{27} -39.5594 q^{28} +38.3019 q^{29} -135.275 q^{30} -186.071 q^{31} -187.519 q^{32} +60.8842 q^{34} +60.8997 q^{35} -429.164 q^{36} -218.781 q^{37} -42.8919 q^{38} +774.716 q^{39} +182.013 q^{40} -80.1687 q^{41} -108.843 q^{42} +35.1155 q^{43} +660.676 q^{45} -133.645 q^{46} -282.620 q^{47} +133.404 q^{48} +49.0000 q^{49} +75.5704 q^{50} -403.077 q^{51} -431.521 q^{52} +145.296 q^{53} -760.971 q^{54} +146.448 q^{56} +283.961 q^{57} -58.6989 q^{58} +91.0461 q^{59} -498.840 q^{60} -808.142 q^{61} +285.160 q^{62} +531.582 q^{63} +182.192 q^{64} +664.304 q^{65} +794.222 q^{67} +224.516 q^{68} +884.783 q^{69} -93.3307 q^{70} +946.901 q^{71} +1588.75 q^{72} -801.324 q^{73} +335.288 q^{74} -500.305 q^{75} -158.168 q^{76} -1187.28 q^{78} +890.737 q^{79} +114.391 q^{80} +2987.53 q^{81} +122.861 q^{82} +559.333 q^{83} -401.368 q^{84} -345.630 q^{85} -53.8156 q^{86} +388.609 q^{87} -1523.75 q^{89} -1012.51 q^{90} +534.501 q^{91} -492.829 q^{92} -1887.87 q^{93} +433.125 q^{94} +243.491 q^{95} -1902.56 q^{96} +664.651 q^{97} -75.0941 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 2 q^{2} + 14 q^{3} + 26 q^{4} + 10 q^{5} - 14 q^{6} + 28 q^{7} + 18 q^{8} + 76 q^{9} + 2 q^{10} + 70 q^{12} - 58 q^{13} + 14 q^{14} + 284 q^{15} + 2 q^{16} - 4 q^{17} + 62 q^{18} - 258 q^{19} + 182 q^{20}+ \cdots + 98 q^{98}+O(q^{100}) $ 4 * q + 2 * q^2 + 14 * q^3 + 26 * q^4 + 10 * q^5 - 14 * q^6 + 28 * q^7 + 18 * q^8 + 76 * q^9 + 2 * q^10 + 70 * q^12 - 58 * q^13 + 14 * q^14 + 284 * q^15 + 2 * q^16 - 4 * q^17 + 62 * q^18 - 258 * q^19 + 182 * q^20 + 98 * q^21 + 8 * q^23 + 498 * q^24 + 80 * q^25 - 482 * q^26 + 428 * q^27 + 182 * q^28 + 396 * q^29 + 628 * q^30 - 56 * q^31 - 134 * q^32 + 472 * q^34 + 70 * q^35 - 418 * q^36 + 84 * q^37 - 942 * q^38 + 412 * q^39 + 1026 * q^40 - 52 * q^41 - 98 * q^42 - 408 * q^43 + 826 * q^45 - 368 * q^46 + 8 * q^47 + 982 * q^48 + 196 * q^49 + 1642 * q^50 + 388 * q^51 - 2030 * q^52 + 624 * q^53 - 92 * q^54 + 126 * q^56 - 48 * q^57 + 864 * q^58 - 238 * q^59 + 1420 * q^60 + 162 * q^61 - 688 * q^62 + 532 * q^63 - 902 * q^64 + 32 * q^65 + 1340 * q^67 + 1384 * q^68 + 2416 * q^69 + 14 * q^70 + 1788 * q^71 + 2622 * q^72 - 1456 * q^73 - 996 * q^74 - 806 * q^75 - 3042 * q^76 - 2632 * q^78 + 1324 * q^79 + 2342 * q^80 + 1444 * q^81 + 1984 * q^82 - 450 * q^83 + 490 * q^84 + 1736 * q^85 - 4380 * q^86 - 588 * q^87 - 3072 * q^89 + 218 * q^90 - 406 * q^91 + 544 * q^92 - 1264 * q^93 + 1696 * q^94 - 24 * q^95 - 862 * q^96 - 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	$0.587327\pi$
$3$	10.1459	1.95259	0.976294	−	0.216448i	$-0.0694472\pi$
$3$	10.1459	1.95259	0.976294	+	0.216448i	$0.0694472\pi$
$4$	−5.65135	−0.706418
$5$	8.69995	0.778148	0.389074	−	0.921207i	$-0.372795\pi$
$5$	8.69995	0.778148	0.389074	+	0.921207i	$0.372795\pi$
$6$	−15.5490	−1.05797
$7$	7.00000	0.377964
$7$	7.00000	0.377964
$8$	20.9211	0.924592
$9$	75.9402	2.81260
$10$	−13.3330	−0.421625
$11$	0	0
$11$	0	0
$12$	−57.3382	−1.37934
$13$	76.3572	1.62905	0.814526	−	0.580127i	$-0.196997\pi$
$13$	76.3572	1.62905	0.814526	+	0.580127i	$0.196997\pi$
$14$	−10.7277	−0.204793
$15$	88.2693	1.51940
$16$	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	$0.591461\pi$
$18$	−116.381	−1.52396
$19$	27.9876	0.337937	0.168968	−	0.985621i	$-0.445956\pi$
$19$	27.9876	0.337937	0.168968	+	0.985621i	$0.445956\pi$
$20$	−49.1664	−0.549698
$21$	71.0216	0.738009
$22$	0	0
$23$	87.2055	0.790592	0.395296	−	0.918554i	$-0.370642\pi$
$23$	87.2055	0.790592	0.395296	+	0.918554i	$0.370642\pi$
$24$	212.265	1.80535
$25$	−49.3108	−0.394486
$26$	−117.020	−0.882673
$27$	496.545	3.53926
$28$	−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	$0.460867\pi$
$30$	−135.275	−0.823260
$31$	−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
$36$	−429.164	−1.98687
$37$	−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
$39$	774.716	3.18087
$40$	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	$0.548792\pi$
$42$	−108.843	−0.399877
$43$	35.1155	0.124536	0.0622681	−	0.998059i	$-0.480167\pi$
$43$	35.1155	0.124536	0.0622681	+	0.998059i	$0.480167\pi$
$44$	0	0
$45$	660.676	2.18862
$46$	−133.645	−0.428368
$47$	−282.620	−0.877116	−0.438558	−	0.898703i	$-0.644511\pi$
$47$	−282.620	−0.877116	−0.438558	+	0.898703i	$0.644511\pi$
$48$	133.404	0.401149
$49$	49.0000	0.142857
$50$	75.5704	0.213745
$51$	−403.077	−1.10671
$52$	−431.521	−1.15079
$53$	145.296	0.376566	0.188283	−	0.982115i	$-0.439708\pi$
$53$	145.296	0.376566	0.188283	+	0.982115i	$0.439708\pi$
$54$	−760.971	−1.91769
$55$	0	0
$56$	146.448	0.349463
$57$	283.961	0.659851
$58$	−58.6989	−0.132889
$59$	91.0461	0.200901	0.100451	−	0.994942i	$-0.467972\pi$
$59$	91.0461	0.200901	0.100451	+	0.994942i	$0.467972\pi$
$60$	−498.840	−1.07333
$61$	−808.142	−1.69626	−0.848131	−	0.529787i	$-0.822272\pi$
$61$	−808.142	−1.69626	−0.848131	+	0.529787i	$0.822272\pi$
$62$	285.160	0.584118
$63$	531.582	1.06306
$64$	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
$69$	884.783	1.54370
$70$	−93.3307	−0.159359
$71$	946.901	1.58277	0.791384	−	0.611319i	$-0.209361\pi$
$71$	946.901	1.58277	0.791384	+	0.611319i	$0.209361\pi$
$72$	1588.75	2.60051
$73$	−801.324	−1.28477	−0.642383	−	0.766384i	$-0.722054\pi$
$73$	−801.324	−1.28477	−0.642383	+	0.766384i	$0.722054\pi$
$74$	335.288	0.526709
$75$	−500.305	−0.770270
$76$	−158.168	−0.238725
$77$	0	0
$78$	−1187.28	−1.72350
$79$	890.737	1.26855	0.634277	−	0.773106i	$-0.281298\pi$
$79$	890.737	1.26855	0.634277	+	0.773106i	$0.281298\pi$
$80$	114.391	0.159866
$81$	2987.53	4.09812
$82$	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	$0.379410\pi$
$84$	−401.368	−0.521343
$85$	−345.630	−0.441046
$86$	−53.8156	−0.0674777
$87$	388.609	0.478888
$88$	0	0
$89$	−1523.75	−1.81480	−0.907401	−	0.420265i	$-0.861937\pi$
$89$	−1523.75	−1.81480	−0.907401	+	0.420265i	$0.861937\pi$
$90$	−1012.51	−1.18586
$91$	534.501	0.615724
$92$	−492.829	−0.558488
$93$	−1887.87	−2.10497
$94$	433.125	0.475249
$95$	243.491	0.262965
$96$	−1902.56	−2.02270
$97$	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	$0.763646\pi$
$102$	617.728	0.599649
$103$	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
$105$	617.885	0.574280
$106$	−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	$0.615214\pi$
$108$	−2806.15	−2.50020
$109$	1351.08	1.18725	0.593623	−	0.804743i	$-0.297697\pi$
$109$	1351.08	1.18725	0.593623	+	0.804743i	$0.297697\pi$
$110$	0	0
$111$	−2219.74	−1.89809
$112$	92.0393	0.0776508
$113$	−188.362	−0.156811	−0.0784055	−	0.996922i	$-0.524983\pi$
$113$	−188.362	−0.156811	−0.0784055	+	0.996922i	$0.524983\pi$
$114$	−435.179	−0.357529
$115$	758.684	0.615197
$116$	−216.457	−0.173255
$117$	5798.58	4.58187
$118$	−139.531	−0.108855
$119$	−278.095	−0.214226
$120$	1846.69	1.40483
$121$	0	0
$122$	1238.50	0.919089
$123$	−813.388	−0.596266
$124$	1051.55	0.761549
$125$	−1516.50	−1.08512
$126$	−814.666	−0.576002
$127$	−586.957	−0.410110	−0.205055	−	0.978750i	$-0.565737\pi$
$127$	−586.957	−0.410110	−0.205055	+	0.978750i	$0.565737\pi$
$128$	1220.94	0.843101
$129$	356.280	0.243168
$130$	−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
$135$	4319.92	2.75407
$136$	−831.151	−0.524049
$137$	−954.859	−0.595468	−0.297734	−	0.954649i	$-0.596231\pi$
$137$	−954.859	−0.595468	−0.297734	+	0.954649i	$0.596231\pi$
$138$	−1355.96	−0.836426
$139$	−1590.62	−0.970610	−0.485305	−	0.874345i	$-0.661291\pi$
$139$	−1590.62	−0.970610	−0.485305	+	0.874345i	$0.661291\pi$
$140$	−344.165	−0.207766
$141$	−2867.45	−1.71265
$142$	−1451.16	−0.857594
$143$	0	0
$144$	998.498	0.577834
$145$	333.225	0.190847
$146$	1228.05	0.696127
$147$	497.151	0.278941
$148$	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	$0.372741\pi$
$150$	766.733	0.417357
$151$	−411.564	−0.221806	−0.110903	−	0.993831i	$-0.535374\pi$
$151$	−411.564	−0.221806	−0.110903	+	0.993831i	$0.535374\pi$
$152$	585.532	0.312454
$153$	−3016.94	−1.59415
$154$	0	0
$155$	−1618.81	−0.838876
$156$	−4378.19	−2.24702
$157$	−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
$159$	1474.17	0.735278
$160$	−1631.41	−0.806090
$161$	610.439	0.298816
$162$	−4578.49	−2.22049
$163$	−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	$0.388092\pi$
$168$	1485.85	0.682357
$169$	3633.43	1.65381
$170$	529.690	0.238973
$171$	2125.39	0.950481
$172$	−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	$0.274834\pi$
$174$	−595.555	−0.259477
$175$	−345.176	−0.149102
$176$	0	0
$177$	923.748	0.392278
$178$	2335.20	0.983318
$179$	−24.4424	−0.0102062	−0.00510310	−	0.999987i	$-0.501624\pi$
$179$	−24.4424	−0.0102062	−0.00510310	+	0.999987i	$0.501624\pi$
$180$	−3733.71	−1.54608
$181$	−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
$183$	−8199.37	−3.31210
$184$	1824.44	0.730975
$185$	−1903.38	−0.756429
$186$	2893.22	1.14054
$187$	0	0
$188$	1597.19	0.619610
$189$	3475.81	1.33772
$190$	−373.158	−0.142483
$191$	−2059.27	−0.780123	−0.390061	−	0.920789i	$-0.627546\pi$
$191$	−2059.27	−0.780123	−0.390061	+	0.920789i	$0.627546\pi$
$192$	1848.51	0.694816
$193$	4371.31	1.63033	0.815165	−	0.579229i	$-0.196646\pi$
$193$	4371.31	1.63033	0.815165	+	0.579229i	$0.196646\pi$
$194$	−1018.60	−0.376965
$195$	6740.00	2.47519
$196$	−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
$201$	8058.13	2.82774
$202$	2292.18	0.798401
$203$	268.113	0.0926988
$204$	2277.93	0.781797
$205$	−697.464	−0.237624
$206$	−1340.01	−0.453219
$207$	6622.41	2.22362
$208$	1003.98	0.334680
$209$	0	0
$210$	−946.928	−0.311163
$211$	3888.39	1.26866	0.634331	−	0.773062i	$-0.281276\pi$
$211$	3888.39	1.26866	0.634331	+	0.773062i	$0.281276\pi$
$212$	−821.120	−0.266013
$213$	9607.21	3.09049
$214$	1201.28	0.383728
$215$	305.503	0.0969076
$216$	10388.3	3.27237
$217$	−1302.50	−0.407462
$218$	−2070.57	−0.643288
$219$	−8130.19	−2.50862
$220$	0	0
$221$	−3033.51	−0.923330
$222$	3401.82	1.02845
$223$	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
$225$	−3744.67	−1.10953
$226$	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	$0.677468\pi$
$228$	−1604.76	−0.466131
$229$	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	$0.299620\pi$
$234$	−8886.52	−2.48261
$235$	−2458.79	−0.682525
$236$	−514.533	−0.141920
$237$	9037.37	2.47696
$238$	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	$0.613645\pi$
$240$	1160.61	0.312153
$241$	−1522.09	−0.406832	−0.203416	−	0.979092i	$-0.565204\pi$
$241$	−1522.09	−0.406832	−0.203416	+	0.979092i	$0.565204\pi$
$242$	0	0
$243$	16904.6	4.46268
$244$	4567.09	1.19827
$245$	426.298	0.111164
$246$	1246.54	0.323076
$247$	2137.06	0.550517
$248$	−3892.81	−0.996750
$249$	5674.96	1.44432
$250$	2324.08	0.587951
$251$	1463.19	0.367952	0.183976	−	0.982931i	$-0.441103\pi$
$251$	1463.19	0.367952	0.183976	+	0.982931i	$0.441103\pi$
$252$	−3004.15	−0.750967
$253$	0	0
$254$	899.531	0.222211
$255$	−3506.75	−0.861181
$256$	−3328.67	−0.812662
$257$	−1183.89	−0.287350	−0.143675	−	0.989625i	$-0.545892\pi$
$257$	−1183.89	−0.287350	−0.143675	+	0.989625i	$0.545892\pi$
$258$	−546.010	−0.131756
$259$	−1531.46	−0.367415
$260$	−3754.21	−0.895486
$261$	2908.65	0.689813
$262$	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
$267$	−15459.9	−3.54356
$268$	−4488.42	−1.02304
$269$	−255.543	−0.0579209	−0.0289604	−	0.999581i	$-0.509220\pi$
$269$	−255.543	−0.0579209	−0.0289604	+	0.999581i	$0.509220\pi$
$270$	−6620.41	−1.49224
$271$	−2589.73	−0.580497	−0.290248	−	0.956951i	$-0.593738\pi$
$271$	−2589.73	−0.580497	−0.290248	+	0.956951i	$0.593738\pi$
$272$	−522.360	−0.116444
$273$	5423.01	1.20226
$274$	1463.35	0.322644
$275$	0	0
$276$	−5000.21	−1.09050
$277$	4441.49	0.963404	0.481702	−	0.876335i	$-0.340019\pi$
$277$	4441.49	0.963404	0.481702	+	0.876335i	$0.340019\pi$
$278$	2437.68	0.525907
$279$	−14130.3	−3.03210
$280$	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	$0.553547\pi$
$282$	4394.46	0.927966
$283$	−3429.29	−0.720317	−0.360159	−	0.932891i	$-0.617277\pi$
$283$	−3429.29	−0.720317	−0.360159	+	0.932891i	$0.617277\pi$
$284$	−5351.27	−1.11810
$285$	2470.45	0.513462
$286$	0	0
$287$	−561.181	−0.115420
$288$	−14240.3	−2.91360
$289$	−3334.70	−0.678750
$290$	−510.677	−0.103407
$291$	6743.51	1.35846
$292$	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	$0.651695\pi$
$294$	−761.900	−0.151139
$295$	792.096	0.156331
$296$	−4577.14	−0.898786
$297$	0	0
$298$	−2169.82	−0.421794
$299$	6658.77	1.28792
$300$	2827.40	0.544132
$301$	245.808	0.0470703
$302$	630.736	0.120181
$303$	−15175.1	−2.87718
$304$	367.994	0.0694274
$305$	−7030.80	−1.31994
$306$	4623.56	0.863762
$307$	−4990.18	−0.927702	−0.463851	−	0.885913i	$-0.653533\pi$
$307$	−4990.18	−0.927702	−0.463851	+	0.885913i	$0.653533\pi$
$308$	0	0
$309$	8871.40	1.63326
$310$	2480.88	0.454530
$311$	3139.78	0.572478	0.286239	−	0.958158i	$-0.407595\pi$
$311$	3139.78	0.572478	0.286239	+	0.958158i	$0.407595\pi$
$312$	16207.9	2.94101
$313$	−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
$315$	4624.74	0.827220
$316$	−5033.86	−0.896130
$317$	5935.83	1.05170	0.525851	−	0.850577i	$-0.323747\pi$
$317$	5935.83	1.05170	0.525851	+	0.850577i	$0.323747\pi$
$318$	−2259.21	−0.398397
$319$	0	0
$320$	1585.06	0.276899
$321$	−7952.94	−1.38283
$322$	−935.517	−0.161908
$323$	−1111.89	−0.191539
$324$	−16883.6	−2.89499
$325$	−3765.24	−0.642639
$326$	676.461	0.114925
$327$	13708.0	2.31820
$328$	−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
$333$	−16614.3	−2.73410
$334$	−2277.93	−0.373183
$335$	6909.69	1.12692
$336$	933.826	0.151620
$337$	−8916.41	−1.44127	−0.720635	−	0.693315i	$-0.756149\pi$
$337$	−8916.41	−1.44127	−0.720635	+	0.693315i	$0.756149\pi$
$338$	−5568.34	−0.896088
$339$	−1911.12	−0.306187
$340$	1953.28	0.311563
$341$	0	0
$342$	−3257.22	−0.515001
$343$	343.000	0.0539949
$344$	734.655	0.115145
$345$	7697.57	1.20123
$346$	−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	$0.534552\pi$
$348$	−2196.16	−0.338295
$349$	8985.39	1.37816	0.689079	−	0.724686i	$-0.258015\pi$
$349$	8985.39	1.37816	0.689079	+	0.724686i	$0.258015\pi$
$350$	528.993	0.0807881
$351$	37914.8	5.76565
$352$	0	0
$353$	1767.91	0.266562	0.133281	−	0.991078i	$-0.457449\pi$
$353$	1767.91	0.266562	0.133281	+	0.991078i	$0.457449\pi$
$354$	−1415.67	−0.212549
$355$	8238.00	1.23163
$356$	8611.25	1.28201
$357$	−2821.54	−0.418296
$358$	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	$0.473988\pi$
$360$	13822.1	2.02358
$361$	−6075.69	−0.885799
$362$	184.096	0.0267290
$363$	0	0
$364$	−3020.65	−0.434959
$365$	−6971.48	−0.999737
$366$	12565.8	1.79460
$367$	−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
$369$	−6088.03	−0.858890
$370$	2916.99	0.409858
$371$	1017.07	0.142329
$372$	10669.0	1.48699
$373$	−6829.95	−0.948101	−0.474050	−	0.880498i	$-0.657209\pi$
$373$	−6829.95	−0.948101	−0.474050	+	0.880498i	$0.657209\pi$
$374$	0	0
$375$	−15386.3	−2.11879
$376$	−5912.74	−0.810974
$377$	2924.63	0.399538
$378$	−5326.80	−0.724817
$379$	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
$381$	−5955.24	−0.800777
$382$	3155.90	0.422695
$383$	−7534.69	−1.00523	−0.502617	−	0.864509i	$-0.667629\pi$
$383$	−7534.69	−1.00523	−0.502617	+	0.864509i	$0.667629\pi$
$384$	12387.6	1.64623
$385$	0	0
$386$	−6699.17	−0.883364
$387$	2666.68	0.350271
$388$	−3756.17	−0.491471
$389$	−13071.8	−1.70376	−0.851882	−	0.523734i	$-0.824539\pi$
$389$	−13071.8	−1.70376	−0.851882	+	0.523734i	$0.824539\pi$
$390$	−10329.3	−1.34113
$391$	−3464.49	−0.448099
$392$	1025.14	0.132085
$393$	−6321.47	−0.811390
$394$	−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
$399$	1987.73	0.249400
$400$	−648.362	−0.0810452
$401$	8223.85	1.02414	0.512069	−	0.858944i	$-0.328879\pi$
$401$	8223.85	1.02414	0.512069	+	0.858944i	$0.328879\pi$
$402$	−12349.3	−1.53216
$403$	−14207.9	−1.75619
$404$	8452.61	1.04092
$405$	25991.4	3.18894
$406$	−410.892	−0.0502272
$407$	0	0
$408$	−8432.82	−1.02325
$409$	−3689.61	−0.446062	−0.223031	−	0.974811i	$-0.571595\pi$
$409$	−3689.61	−0.446062	−0.223031	+	0.974811i	$0.571595\pi$
$410$	1068.89	0.128753
$411$	−9687.95	−1.16270
$412$	−4941.42	−0.590889
$413$	637.322	0.0759336
$414$	−10149.1	−1.20483
$415$	4866.17	0.575593
$416$	−14318.5	−1.68755
$417$	−16138.4	−1.89520
$418$	0	0
$419$	15657.0	1.82552	0.912762	−	0.408492i	$-0.133945\pi$
$419$	15657.0	1.82552	0.912762	+	0.408492i	$0.133945\pi$
$420$	−3491.88	−0.405682
$421$	−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
$423$	−21462.3	−2.46698
$424$	3039.76	0.348170
$425$	1959.01	0.223591
$426$	−14723.4	−1.67453
$427$	−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	$0.762444\pi$
$432$	6528.81	0.727123
$433$	−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
$435$	3380.88	0.372645
$436$	−7635.41	−0.838692
$437$	2440.67	0.267170
$438$	12459.8	1.35925
$439$	−12676.2	−1.37813	−0.689066	−	0.724699i	$-0.741979\pi$
$439$	−12676.2	−1.37813	−0.689066	+	0.724699i	$0.741979\pi$
$440$	0	0
$441$	3721.07	0.401800
$442$	4648.95	0.500289
$443$	−17489.2	−1.87571	−0.937855	−	0.347028i	$-0.887191\pi$
$443$	−17489.2	−1.87571	−0.937855	+	0.347028i	$0.887191\pi$
$444$	12544.5	1.34085
$445$	−13256.6	−1.41218
$446$	−983.069	−0.104371
$447$	14365.1	1.52001
$448$	1275.34	0.134496
$449$	−15491.8	−1.62830	−0.814148	−	0.580658i	$-0.802796\pi$
$449$	−15491.8	−1.62830	−0.814148	+	0.580658i	$0.802796\pi$
$450$	5738.83	0.601180
$451$	0	0
$452$	1064.50	0.110774
$453$	−4175.71	−0.433095
$454$	5546.40	0.573360
$455$	4650.13	0.479124
$456$	5940.78	0.610093
$457$	7789.99	0.797375	0.398688	−	0.917087i	$-0.369466\pi$
$457$	7789.99	0.797375	0.398688	+	0.917087i	$0.369466\pi$
$458$	−3859.13	−0.393723
$459$	−19726.7	−2.00602
$460$	−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
$465$	−16424.3	−1.63798
$466$	−6418.08	−0.638008
$467$	17652.5	1.74917	0.874584	−	0.484874i	$-0.161134\pi$
$467$	17652.5	1.74917	0.874584	+	0.484874i	$0.161134\pi$
$468$	−32769.8	−3.23672
$469$	5559.55	0.547369
$470$	3768.17	0.369814
$471$	−14379.8	−1.40676
$472$	1904.79	0.185752
$473$	0	0
$474$	−13850.1	−1.34210
$475$	−1380.09	−0.133311
$476$	1571.61	0.151333
$477$	11033.8	1.05913
$478$	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	$0.547693\pi$
$480$	−16552.2	−1.57396
$481$	−16705.5	−1.58359
$482$	2332.66	0.220435
$483$	6193.48	0.583464
$484$	0	0
$485$	5782.43	0.541375
$486$	−25906.9	−2.41802
$487$	−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
$489$	−4478.43	−0.414154
$490$	−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	$0.266329\pi$
$492$	4596.73	0.421213
$493$	−1521.65	−0.139010
$494$	−3275.11	−0.298288
$495$	0	0
$496$	−2446.55	−0.221478
$497$	6628.31	0.598230
$498$	−8697.06	−0.782580
$499$	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
$501$	15080.8	1.34483
$502$	−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	$0.412125\pi$
$504$	11121.3	0.982900
$505$	−13012.3	−1.14662
$506$	0	0
$507$	36864.5	3.22921
$508$	3317.10	0.289709
$509$	−16132.0	−1.40479	−0.702396	−	0.711787i	$-0.747886\pi$
$509$	−16132.0	−1.40479	−0.702396	+	0.711787i	$0.747886\pi$
$510$	5374.20	0.466615
$511$	−5609.27	−0.485596
$512$	−4666.24	−0.402775
$513$	13897.1	1.19605
$514$	1814.35	0.155695
$515$	7607.06	0.650887
$516$	−2013.46	−0.171778
$517$	0	0
$518$	2347.02	0.199077
$519$	30005.5	2.53775
$520$	13898.0	1.17205
$521$	7654.31	0.643650	0.321825	−	0.946799i	$-0.395704\pi$
$521$	7654.31	0.643650	0.321825	+	0.946799i	$0.395704\pi$
$522$	−4457.60	−0.373763
$523$	−21493.7	−1.79705	−0.898523	−	0.438926i	$-0.855359\pi$
$523$	−21493.7	−1.79705	−0.898523	+	0.438926i	$0.855359\pi$
$524$	3521.09	0.293549
$525$	−3502.13	−0.291135
$526$	232.904	0.0193063
$527$	7392.20	0.611023
$528$	0	0
$529$	−4562.20	−0.374965
$530$	−1937.23	−0.158770
$531$	6914.06	0.565056
$532$	−1107.17	−0.0902294
$533$	−6121.46	−0.497467
$534$	23692.8	1.92001
$535$	−6819.49	−0.551088
$536$	16616.0	1.33900
$537$	−247.991	−0.0199285
$538$	391.628	0.0313834
$539$	0	0
$540$	−24413.3	−1.94552
$541$	8661.03	0.688293	0.344147	−	0.938916i	$-0.388168\pi$
$541$	8661.03	0.688293	0.344147	+	0.938916i	$0.388168\pi$
$542$	3968.84	0.314532
$543$	−1218.79	−0.0963227
$544$	7449.74	0.587142
$545$	11754.3	0.923853
$546$	−8310.94	−0.651420
$547$	−21372.9	−1.67064	−0.835321	−	0.549763i	$-0.814718\pi$
$547$	−21372.9	−1.67064	−0.835321	+	0.549763i	$0.814718\pi$
$548$	5396.24	0.420650
$549$	−61370.5	−4.77091
$550$	0	0
$551$	1071.98	0.0828817
$552$	18510.6	1.42729
$553$	6235.16	0.479468
$554$	−6806.72	−0.522003
$555$	−19311.6	−1.47700
$556$	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	$0.741071\pi$
$558$	21655.1	1.64289
$559$	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	$0.511467\pi$
$564$	16205.0	1.20984
$565$	−1638.74	−0.122022
$566$	5255.49	0.390291
$567$	20912.7	1.54894
$568$	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	$0.114448\pi$
$570$	−3786.04	−0.278210
$571$	5211.13	0.381925	0.190962	−	0.981597i	$-0.438839\pi$
$571$	5211.13	0.381925	0.190962	+	0.981597i	$0.438839\pi$
$572$	0	0
$573$	−20893.2	−1.52326
$574$	860.028	0.0625381
$575$	−4300.17	−0.311878
$576$	13835.7	1.00085
$577$	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
$579$	44351.0	3.18336
$580$	−1883.17	−0.134818
$581$	3915.33	0.279579
$582$	−10334.7	−0.736057
$583$	0	0
$584$	−16764.6	−1.18788
$585$	50447.4	3.56537
$586$	7051.75	0.497108
$587$	−11756.7	−0.826661	−0.413331	−	0.910581i	$-0.635635\pi$
$587$	−11756.7	−0.826661	−0.413331	+	0.910581i	$0.635635\pi$
$588$	−2809.57	−0.197049
$589$	−5207.68	−0.364310
$590$	−1213.91	−0.0847051
$591$	22176.0	1.54348
$592$	−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
$597$	−24561.7	−1.68383
$598$	−10204.8	−0.697834
$599$	22486.4	1.53384	0.766918	−	0.641745i	$-0.221789\pi$
$599$	22486.4	1.53384	0.766918	+	0.641745i	$0.221789\pi$
$600$	−10466.9	−0.712185
$601$	−25019.6	−1.69812	−0.849061	−	0.528295i	$-0.822832\pi$
$601$	−25019.6	−1.69812	−0.849061	+	0.528295i	$0.822832\pi$
$602$	−376.709	−0.0255042
$603$	60313.4	4.07322
$604$	2325.89	0.156687
$605$	0	0
$606$	23256.3	1.55895
$607$	−7074.22	−0.473037	−0.236519	−	0.971627i	$-0.576006\pi$
$607$	−7074.22	−0.473037	−0.236519	+	0.971627i	$0.576006\pi$
$608$	−5248.22	−0.350072
$609$	2720.26	0.181003
$610$	10774.9	0.715187
$611$	−21580.1	−1.42887
$612$	17049.8	1.12614
$613$	−4934.94	−0.325155	−0.162578	−	0.986696i	$-0.551981\pi$
$613$	−4934.94	−0.325155	−0.162578	+	0.986696i	$0.551981\pi$
$614$	7647.61	0.502658
$615$	−7076.43	−0.463983
$616$	0	0
$617$	2125.51	0.138687	0.0693434	−	0.997593i	$-0.477910\pi$
$617$	2125.51	0.138687	0.0693434	+	0.997593i	$0.477910\pi$
$618$	−13595.7	−0.884951
$619$	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
$621$	43301.5	2.79811
$622$	−4811.81	−0.310187
$623$	−10666.3	−0.685931
$624$	10186.3	0.653493
$625$	−7029.59	−0.449894
$626$	7238.34	0.462144
$627$	0	0
$628$	8009.60	0.508946
$629$	8691.69	0.550970
$630$	−7087.55	−0.448214
$631$	−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
$633$	39451.4	2.47717
$634$	−9096.85	−0.569846
$635$	−5106.50	−0.319126
$636$	−8331.04	−0.519414
$637$	3741.50	0.232722
$638$	0	0
$639$	71907.9	4.45169
$640$	10622.1	0.656057
$641$	27238.9	1.67843	0.839213	−	0.543803i	$-0.183016\pi$
$641$	27238.9	1.67843	0.839213	+	0.543803i	$0.183016\pi$
$642$	12188.1	0.749263
$643$	−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
$645$	3099.62	0.189221
$646$	1704.00	0.103782
$647$	−9788.76	−0.594801	−0.297400	−	0.954753i	$-0.596120\pi$
$647$	−9788.76	−0.594801	−0.297400	+	0.954753i	$0.596120\pi$
$648$	62502.5	3.78909
$649$	0	0
$650$	5770.35	0.348202
$651$	−13215.1	−0.795605
$652$	2494.51	0.149835
$653$	5539.90	0.331996	0.165998	−	0.986126i	$-0.446916\pi$
$653$	5539.90	0.331996	0.165998	+	0.986126i	$0.446916\pi$
$654$	−21007.9	−1.25608
$655$	−5420.54	−0.323356
$656$	−1054.10	−0.0627371
$657$	−60852.7	−3.61353
$658$	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
$663$	−30777.8	−1.80288
$664$	11701.9	0.683917
$665$	1704.44	0.0993913
$666$	25461.9	1.48142
$667$	3340.14	0.193899
$668$	−8400.08	−0.486540
$669$	6508.29	0.376121
$670$	−10589.3	−0.610599
$671$	0	0
$672$	−13317.9	−0.764510
$673$	9532.72	0.546002	0.273001	−	0.962014i	$-0.411984\pi$
$673$	9532.72	0.546002	0.273001	+	0.962014i	$0.411984\pi$
$674$	13664.7	0.780926
$675$	−24485.0	−1.39619
$676$	−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	$0.440520\pi$
$678$	2928.85	0.165902
$679$	4652.56	0.262958
$680$	−7230.98	−0.407787
$681$	−36719.3	−2.06621
$682$	0	0
$683$	8438.16	0.472734	0.236367	−	0.971664i	$-0.424043\pi$
$683$	8438.16	0.472734	0.236367	+	0.971664i	$0.424043\pi$
$684$	−12011.3	−0.671437
$685$	−8307.23	−0.463362
$686$	−525.658	−0.0292562
$687$	25548.9	1.41885
$688$	461.715	0.0255853
$689$	11094.4	0.613446
$690$	−11796.8	−0.650863
$691$	−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
$696$	8130.13	0.442776
$697$	3184.93	0.173082
$698$	−13770.4	−0.746730
$699$	42490.1	2.29918
$700$	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	$0.438109\pi$
$702$	−58105.6	−3.12401
$703$	−6123.15	−0.328505
$704$	0	0
$705$	−24946.7	−1.33269
$706$	−2709.38	−0.144432
$707$	−10469.8	−0.556939
$708$	−5220.42	−0.277112
$709$	−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
$711$	67642.8	3.56794
$712$	−31878.6	−1.67795
$713$	−16226.4	−0.852292
$714$	4324.09	0.226646
$715$	0	0
$716$	138.132	0.00720984
$717$	−26203.2	−1.36482
$718$	−1701.84	−0.0884571
$719$	−5923.31	−0.307235	−0.153618	−	0.988130i	$-0.549092\pi$
$719$	−5923.31	−0.307235	−0.153618	+	0.988130i	$0.549092\pi$
$720$	8686.88	0.449640
$721$	6120.65	0.316151
$722$	9311.20	0.479954
$723$	−15443.1	−0.794376
$724$	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
$729$	90850.1	4.61566
$730$	10684.0	0.541689
$731$	−1395.06	−0.0705858
$732$	46337.4	2.33973
$733$	−24556.4	−1.23740	−0.618698	−	0.785629i	$-0.712340\pi$
$733$	−24556.4	−1.23740	−0.618698	+	0.785629i	$0.712340\pi$
$734$	8187.72	0.411736
$735$	4325.19	0.217057
$736$	−16352.7	−0.818981
$737$	0	0
$738$	9330.10	0.465374
$739$	−27603.8	−1.37405	−0.687025	−	0.726634i	$-0.741084\pi$
$739$	−27603.8	−1.37405	−0.687025	+	0.726634i	$0.741084\pi$
$740$	10756.7	0.534355
$741$	21682.5	1.07493
$742$	−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	$0.662653\pi$
$744$	−39496.3	−1.94624
$745$	12317.8	0.605756
$746$	10467.1	0.513711
$747$	42475.9	2.08047
$748$	0	0
$749$	−5486.98	−0.267677
$750$	23580.0	1.14803
$751$	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
$753$	14845.5	0.718459
$754$	−4482.08	−0.216482
$755$	−3580.59	−0.172597
$756$	−19643.0	−0.944987
$757$	−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	$0.399524\pi$
$762$	9126.59	0.433886
$763$	9457.55	0.448737
$764$	11637.6	0.551093
$765$	−26247.3	−1.24049
$766$	11547.2	0.544668
$767$	6952.02	0.327279
$768$	−33772.5	−1.58680
$769$	38530.6	1.80683	0.903413	−	0.428770i	$-0.141053\pi$
$769$	38530.6	1.80683	0.903413	+	0.428770i	$0.141053\pi$
$770$	0	0
$771$	−12011.7	−0.561076
$772$	−24703.8	−1.15169
$773$	13835.1	0.643745	0.321873	−	0.946783i	$-0.395688\pi$
$773$	13835.1	0.643745	0.321873	+	0.946783i	$0.395688\pi$
$774$	−4086.77	−0.189788
$775$	9175.31	0.425273
$776$	13905.2	0.643259
$777$	−15538.2	−0.717411
$778$	20032.9	0.923154
$779$	−2243.73	−0.103196
$780$	−38090.0	−1.74852
$781$	0	0
$782$	5309.44	0.242794
$783$	19018.6	0.868032
$784$	644.275	0.0293493
$785$	−12330.4	−0.560624
$786$	9687.86	0.439637
$787$	−11704.2	−0.530125	−0.265062	−	0.964231i	$-0.585393\pi$
$787$	−11704.2	−0.530125	−0.265062	+	0.964231i	$0.585393\pi$
$788$	−12352.1	−0.558409
$789$	−1541.91	−0.0695737
$790$	−11876.2	−0.534854
$791$	−1318.54	−0.0592690
$792$	0	0
$793$	−61707.5	−2.76330
$794$	5658.89	0.252930
$795$	12825.2	0.572155
$796$	13681.0	0.609184
$797$	−3367.27	−0.149655	−0.0748274	−	0.997196i	$-0.523841\pi$
$797$	−3367.27	−0.149655	−0.0748274	+	0.997196i	$0.523841\pi$
$798$	−3046.25	−0.135133
$799$	11227.9	0.497140
$800$	9246.73	0.408652
$801$	−115714.	−5.10432
$802$	−12603.3	−0.554911
$803$	0	0
$804$	−45539.3	−1.99757
$805$	5310.79	0.232523
$806$	21774.0	0.951559
$807$	−2592.72	−0.113096
$808$	−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	$0.649818\pi$
$810$	−39832.6	−1.72787
$811$	22445.9	0.971863	0.485931	−	0.873997i	$-0.338480\pi$
$811$	22445.9	0.971863	0.485931	+	0.873997i	$0.338480\pi$
$812$	−1515.20	−0.0654841
$813$	−26275.2	−1.13347
$814$	0	0
$815$	−3840.17	−0.165049
$816$	−5299.84	−0.227367
$817$	982.798	0.0420854
$818$	5654.45	0.241691
$819$	40590.1	1.73179
$820$	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	$0.317537\pi$
$822$	14847.1	0.629990
$823$	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	$0.680265\pi$
$828$	−37425.5	−1.57080
$829$	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
$831$	45063.1	1.88113
$832$	13911.7	0.579688
$833$	−1946.66	−0.0809699
$834$	24732.6	1.02688
$835$	12931.5	0.535943
$836$	0	0
$837$	−92392.6	−3.81548
$838$	−23994.9	−0.989127
$839$	−10538.6	−0.433649	−0.216824	−	0.976211i	$-0.569570\pi$
$839$	−10538.6	−0.433649	−0.216824	+	0.976211i	$0.569570\pi$
$840$	12926.8	0.530974
$841$	−22922.0	−0.939849
$842$	9206.38	0.376809
$843$	−16003.6	−0.653846
$844$	−21974.6	−0.896206
$845$	31610.6	1.28691
$846$	32891.6	1.33669
$847$	0	0
$848$	1910.42	0.0773635
$849$	−34793.3	−1.40648
$850$	−3002.25	−0.121149
$851$	−19078.9	−0.768526
$852$	−54293.7	−2.18318
$853$	−40061.0	−1.60805	−0.804023	−	0.594598i	$-0.797311\pi$
$853$	−40061.0	−1.60805	−0.804023	+	0.594598i	$0.797311\pi$
$854$	8669.52	0.347383
$855$	18490.8	0.739615
$856$	−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
$861$	−5693.71	−0.225367
$862$	20135.9	0.795629
$863$	−1268.51	−0.0500356	−0.0250178	−	0.999687i	$-0.507964\pi$
$863$	−1268.51	−0.0500356	−0.0250178	+	0.999687i	$0.507964\pi$
$864$	−93111.8	−3.66635
$865$	25729.1	1.01135
$866$	6731.60	0.264144
$867$	−33833.7	−1.32532
$868$	7360.86	0.287839
$869$	0	0
$870$	−5181.30	−0.201911
$871$	60644.6	2.35920
$872$	28266.1	1.09772
$873$	50473.8	1.95679
$874$	−3740.41	−0.144761
$875$	−10615.5	−0.410135
$876$	45946.5	1.77213
$877$	−6691.33	−0.257640	−0.128820	−	0.991668i	$-0.541119\pi$
$877$	−6691.33	−0.257640	−0.128820	+	0.991668i	$0.541119\pi$
$878$	19426.6	0.746716
$879$	−46685.3	−1.79142
$880$	0	0
$881$	14514.6	0.555063	0.277531	−	0.960717i	$-0.410484\pi$
$881$	14514.6	0.555063	0.277531	+	0.960717i	$0.410484\pi$
$882$	−5702.66	−0.217708
$883$	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
$885$	8036.57	0.305250
$886$	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	$0.802086\pi$
$888$	−46439.4	−1.75496
$889$	−4108.70	−0.155007
$890$	20316.1	0.765166
$891$	0	0
$892$	−3625.15	−0.136075
$893$	−7909.87	−0.296410
$894$	−22014.9	−0.823590
$895$	−212.648	−0.00794193
$896$	8546.59	0.318662
$897$	67559.5	2.51477
$898$	23741.7	0.882263
$899$	−7126.87	−0.264399
$900$	21162.4	0.783794
$901$	−5772.31	−0.213434
$902$	0	0
$903$	2493.96	0.0919089
$904$	−3940.75	−0.144986
$905$	−1045.09	−0.0383866
$906$	6399.41	0.234665
$907$	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
$909$	−113582.	−4.14443
$910$	−7126.47	−0.259605
$911$	17335.4	0.630460	0.315230	−	0.949015i	$-0.397918\pi$
$911$	17335.4	0.630460	0.315230	+	0.949015i	$0.397918\pi$
$912$	3733.65	0.135563
$913$	0	0
$914$	−11938.4	−0.432043
$915$	−71334.1	−2.57730
$916$	−14230.9	−0.513321
$917$	−4361.38	−0.157061
$918$	30231.7	1.08692
$919$	34995.3	1.25613	0.628067	−	0.778159i	$-0.283846\pi$
$919$	34995.3	1.25613	0.628067	+	0.778159i	$0.283846\pi$
$920$	15872.5	0.568806
$921$	−50630.1	−1.81142
$922$	−13022.6	−0.465159
$923$	72302.8	2.57841
$924$	0	0
$925$	10788.3	0.383476
$926$	−1341.14	−0.0475945
$927$	66400.5	2.35262
$928$	−7182.35	−0.254065
$929$	−10671.6	−0.376882	−0.188441	−	0.982084i	$-0.560343\pi$
$929$	−10671.6	−0.376882	−0.188441	+	0.982084i	$0.560343\pi$
$930$	25170.8	0.887510
$931$	1371.39	0.0482767
$932$	−23667.2	−0.831808
$933$	31856.0	1.11781
$934$	−27053.1	−0.947755
$935$	0	0
$936$	121313.	4.23636
$937$	1855.85	0.0647045	0.0323522	−	0.999477i	$-0.489700\pi$
$937$	1855.85	0.0647045	0.0323522	+	0.999477i	$0.489700\pi$
$938$	−8520.19	−0.296582
$939$	−47920.6	−1.66542
$940$	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	$0.585829\pi$
$942$	22037.4	0.762228
$943$	−6991.16	−0.241425
$944$	1197.12	0.0412742
$945$	30239.4	1.04094
$946$	0	0
$947$	−14600.9	−0.501020	−0.250510	−	0.968114i	$-0.580598\pi$
$947$	−14600.9	−0.501020	−0.250510	+	0.968114i	$0.580598\pi$
$948$	−51073.3	−1.74977
$949$	−61186.9	−2.09295
$950$	2115.03	0.0722324
$951$	60224.6	2.05354
$952$	−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	$0.488560\pi$
$954$	−16909.7	−0.573870
$955$	−17915.5	−0.607051
$956$	14595.3	0.493773
$957$	0	0
$958$	4796.46	0.161761
$959$	−6684.02	−0.225066
$960$	16081.9	0.540669
$961$	4831.40	0.162177
$962$	25601.7	0.858037
$963$	−59526.0	−1.99190
$964$	8601.87	0.287394
$965$	38030.2	1.26864
$966$	−9491.70	−0.316139
$967$	−7251.75	−0.241159	−0.120579	−	0.992704i	$-0.538475\pi$
$967$	−7251.75	−0.241159	−0.120579	+	0.992704i	$0.538475\pi$
$968$	0	0
$969$	−11281.2	−0.373997
$970$	−8861.77	−0.293334
$971$	2742.57	0.0906420	0.0453210	−	0.998972i	$-0.485569\pi$
$971$	2742.57	0.0906420	0.0453210	+	0.998972i	$0.485569\pi$
$972$	−95533.9	−3.15252
$973$	−11134.4	−0.366856
$974$	560.991	0.0184551
$975$	−38201.9	−1.25481
$976$	−10625.8	−0.348488
$977$	23770.4	0.778384	0.389192	−	0.921157i	$-0.372754\pi$
$977$	23770.4	0.778384	0.389192	+	0.921157i	$0.372754\pi$
$978$	6863.34	0.224402
$979$	0	0
$980$	−2409.16	−0.0785282
$981$	102601.	3.33925
$982$	−22340.1	−0.725967
$983$	−49265.4	−1.59850	−0.799249	−	0.601000i	$-0.794769\pi$
$983$	−49265.4	−1.59850	−0.799249	+	0.601000i	$0.794769\pi$
$984$	−17017.0	−0.551302
$985$	19015.5	0.615109
$986$	2331.98	0.0753198
$987$	−20072.2	−0.647319
$988$	−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
$993$	−64840.2	−2.07215
$994$	−10158.1	−0.324140
$995$	−21061.2	−0.671040
$996$	−32071.2	−1.02030
$997$	41275.5	1.31114	0.655571	−	0.755134i	$-0.272428\pi$
$997$	41275.5	1.31114	0.655571	+	0.755134i	$0.272428\pi$
$998$	−14565.3	−0.461979
$999$	−108634.	−3.44048

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
77.4.a.d.1.3	✓	4	11.10	odd	2
539.4.a.g.1.3		4	77.76	even	2
693.4.a.l.1.2		4	33.32	even	2
847.4.a.d.1.2		4	1.1	even	1	trivial
1232.4.a.s.1.1		4	44.43	even	2
1925.4.a.p.1.2		4	55.54	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:	\( N \)	\(=\)	\( 847 = 7 \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	847.a (trivial)

\(f(q)\)	\(=\)	\(q-1.53253 q^{2} +10.1459 q^{3} -5.65135 q^{4} +8.69995 q^{5} -15.5490 q^{6} +7.00000 q^{7} +20.9211 q^{8} +75.9402 q^{9} -13.3330 q^{10} -57.3382 q^{12} +76.3572 q^{13} -10.7277 q^{14} +88.2693 q^{15} +13.1485 q^{16} -39.7278 q^{17} -116.381 q^{18} +27.9876 q^{19} -49.1664 q^{20} +71.0216 q^{21} +87.2055 q^{23} +212.265 q^{24} -49.3108 q^{25} -117.020 q^{26} +496.545 q^{27} -39.5594 q^{28} +38.3019 q^{29} -135.275 q^{30} -186.071 q^{31} -187.519 q^{32} +60.8842 q^{34} +60.8997 q^{35} -429.164 q^{36} -218.781 q^{37} -42.8919 q^{38} +774.716 q^{39} +182.013 q^{40} -80.1687 q^{41} -108.843 q^{42} +35.1155 q^{43} +660.676 q^{45} -133.645 q^{46} -282.620 q^{47} +133.404 q^{48} +49.0000 q^{49} +75.5704 q^{50} -403.077 q^{51} -431.521 q^{52} +145.296 q^{53} -760.971 q^{54} +146.448 q^{56} +283.961 q^{57} -58.6989 q^{58} +91.0461 q^{59} -498.840 q^{60} -808.142 q^{61} +285.160 q^{62} +531.582 q^{63} +182.192 q^{64} +664.304 q^{65} +794.222 q^{67} +224.516 q^{68} +884.783 q^{69} -93.3307 q^{70} +946.901 q^{71} +1588.75 q^{72} -801.324 q^{73} +335.288 q^{74} -500.305 q^{75} -158.168 q^{76} -1187.28 q^{78} +890.737 q^{79} +114.391 q^{80} +2987.53 q^{81} +122.861 q^{82} +559.333 q^{83} -401.368 q^{84} -345.630 q^{85} -53.8156 q^{86} +388.609 q^{87} -1523.75 q^{89} -1012.51 q^{90} +534.501 q^{91} -492.829 q^{92} -1887.87 q^{93} +433.125 q^{94} +243.491 q^{95} -1902.56 q^{96} +664.651 q^{97} -75.0941 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 2 q^{2} + 14 q^{3} + 26 q^{4} + 10 q^{5} - 14 q^{6} + 28 q^{7} + 18 q^{8} + 76 q^{9} + 2 q^{10} + 70 q^{12} - 58 q^{13} + 14 q^{14} + 284 q^{15} + 2 q^{16} - 4 q^{17} + 62 q^{18} - 258 q^{19} + 182 q^{20}+ \cdots + 98 q^{98}+O(q^{100}) \) 4 * q + 2 * q^2 + 14 * q^3 + 26 * q^4 + 10 * q^5 - 14 * q^6 + 28 * q^7 + 18 * q^8 + 76 * q^9 + 2 * q^10 + 70 * q^12 - 58 * q^13 + 14 * q^14 + 284 * q^15 + 2 * q^16 - 4 * q^17 + 62 * q^18 - 258 * q^19 + 182 * q^20 + 98 * q^21 + 8 * q^23 + 498 * q^24 + 80 * q^25 - 482 * q^26 + 428 * q^27 + 182 * q^28 + 396 * q^29 + 628 * q^30 - 56 * q^31 - 134 * q^32 + 472 * q^34 + 70 * q^35 - 418 * q^36 + 84 * q^37 - 942 * q^38 + 412 * q^39 + 1026 * q^40 - 52 * q^41 - 98 * q^42 - 408 * q^43 + 826 * q^45 - 368 * q^46 + 8 * q^47 + 982 * q^48 + 196 * q^49 + 1642 * q^50 + 388 * q^51 - 2030 * q^52 + 624 * q^53 - 92 * q^54 + 126 * q^56 - 48 * q^57 + 864 * q^58 - 238 * q^59 + 1420 * q^60 + 162 * q^61 - 688 * q^62 + 532 * q^63 - 902 * q^64 + 32 * q^65 + 1340 * q^67 + 1384 * q^68 + 2416 * q^69 + 14 * q^70 + 1788 * q^71 + 2622 * q^72 - 1456 * q^73 - 996 * q^74 - 806 * q^75 - 3042 * q^76 - 2632 * q^78 + 1324 * q^79 + 2342 * q^80 + 1444 * q^81 + 1984 * q^82 - 450 * q^83 + 490 * q^84 + 1736 * q^85 - 4380 * q^86 - 588 * q^87 - 3072 * q^89 + 218 * q^90 - 406 * q^91 + 544 * q^92 - 1264 * q^93 + 1696 * q^94 - 24 * q^95 - 862 * q^96 - 652 * q^97 + 98 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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