LMFDB - Embedded newform 1815.4.a.bh.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1815,4,Mod(1,1815)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1815, 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("1815.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1815 = 3 \cdot 5 \cdot 11^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1815.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:	$107.088466660$
Analytic rank:	$1$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 5 x^{11} - 59 x^{10} + 269 x^{9} + 1318 x^{8} - 5253 x^{7} - 13369 x^{6} + 44853 x^{5} + \cdots + 17600 $ x^12 - 5x^11 - 59x^10 + 269x^9 + 1318x^8 - 5253x^7 - 13369x^6 + 44853x^5 + 57849x^4 - 151000x^3 - 76540x^2 + 100800*x + 17600
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 5\cdot 11^{2} $
Twist minimal:	no (minimal twist has level 165)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-3.84968$ of defining polynomial
Character	$\chi$	$=$	1815.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-4.84968 q^{2} +3.00000 q^{3} +15.5194 q^{4} +5.00000 q^{5} -14.5491 q^{6} -26.0919 q^{7} -36.4669 q^{8} +9.00000 q^{9} +O(q^{10})$	\(q-4.84968 q^{2} +3.00000 q^{3} +15.5194 q^{4} +5.00000 q^{5} -14.5491 q^{6} -26.0919 q^{7} -36.4669 q^{8} +9.00000 q^{9} -24.2484 q^{10} +46.5583 q^{12} -14.0897 q^{13} +126.537 q^{14} +15.0000 q^{15} +52.6975 q^{16} +42.1984 q^{17} -43.6472 q^{18} -107.920 q^{19} +77.5972 q^{20} -78.2756 q^{21} +16.3480 q^{23} -109.401 q^{24} +25.0000 q^{25} +68.3304 q^{26} +27.0000 q^{27} -404.931 q^{28} +153.773 q^{29} -72.7453 q^{30} +12.3887 q^{31} +36.1690 q^{32} -204.649 q^{34} -130.459 q^{35} +139.675 q^{36} +82.6216 q^{37} +523.377 q^{38} -42.2690 q^{39} -182.335 q^{40} -309.256 q^{41} +379.612 q^{42} +371.805 q^{43} +45.0000 q^{45} -79.2825 q^{46} +286.111 q^{47} +158.093 q^{48} +337.785 q^{49} -121.242 q^{50} +126.595 q^{51} -218.664 q^{52} -353.229 q^{53} -130.941 q^{54} +951.490 q^{56} -323.759 q^{57} -745.749 q^{58} +106.003 q^{59} +232.792 q^{60} +884.737 q^{61} -60.0813 q^{62} -234.827 q^{63} -596.988 q^{64} -70.4483 q^{65} +856.505 q^{67} +654.895 q^{68} +49.0439 q^{69} +632.686 q^{70} +36.4974 q^{71} -328.202 q^{72} +974.807 q^{73} -400.689 q^{74} +75.0000 q^{75} -1674.85 q^{76} +204.991 q^{78} -1036.71 q^{79} +263.488 q^{80} +81.0000 q^{81} +1499.80 q^{82} -552.661 q^{83} -1214.79 q^{84} +210.992 q^{85} -1803.14 q^{86} +461.318 q^{87} -1187.57 q^{89} -218.236 q^{90} +367.626 q^{91} +253.711 q^{92} +37.1661 q^{93} -1387.55 q^{94} -539.599 q^{95} +108.507 q^{96} -1200.44 q^{97} -1638.15 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 7 q^{2} + 36 q^{3} + 49 q^{4} + 60 q^{5} - 21 q^{6} - 77 q^{7} - 111 q^{8} + 108 q^{9}+O(q^{10}) $ 12 * q - 7 * q^2 + 36 * q^3 + 49 * q^4 + 60 * q^5 - 21 * q^6 - 77 * q^7 - 111 * q^8 + 108 * q^9	\( 12 q - 7 q^{2} + 36 q^{3} + 49 q^{4} + 60 q^{5} - 21 q^{6} - 77 q^{7} - 111 q^{8} + 108 q^{9} - 35 q^{10} + 147 q^{12} - 172 q^{13} - 30 q^{14} + 180 q^{15} + 161 q^{16} - 317 q^{17} - 63 q^{18} - 237 q^{19} + 245 q^{20} - 231 q^{21} + 210 q^{23} - 333 q^{24} + 300 q^{25} + 8 q^{26} + 324 q^{27} - 542 q^{28} - 759 q^{29} - 105 q^{30} - 193 q^{31} - 410 q^{32} - 78 q^{34} - 385 q^{35} + 441 q^{36} + 286 q^{37} - 168 q^{38} - 516 q^{39} - 555 q^{40} - 1189 q^{41} - 90 q^{42} - 775 q^{43} + 540 q^{45} - 529 q^{46} + 382 q^{47} + 483 q^{48} + 1195 q^{49} - 175 q^{50} - 951 q^{51} - 1741 q^{52} + 275 q^{53} - 189 q^{54} - 419 q^{56} - 711 q^{57} - 418 q^{58} + 646 q^{59} + 735 q^{60} - 1340 q^{61} - 983 q^{62} - 693 q^{63} - 1489 q^{64} - 860 q^{65} - 185 q^{67} - 3322 q^{68} + 630 q^{69} - 150 q^{70} - 932 q^{71} - 999 q^{72} - 2860 q^{73} - 4187 q^{74} + 900 q^{75} - 1594 q^{76} + 24 q^{78} - 1429 q^{79} + 805 q^{80} + 972 q^{81} - 30 q^{82} - 2590 q^{83} - 1626 q^{84} - 1585 q^{85} - 3195 q^{86} - 2277 q^{87} - 473 q^{89} - 315 q^{90} - 4302 q^{91} + 5462 q^{92} - 579 q^{93} - 2875 q^{94} - 1185 q^{95} - 1230 q^{96} + 318 q^{97} + 194 q^{98}+O(q^{100}) \) 12 * q - 7 * q^2 + 36 * q^3 + 49 * q^4 + 60 * q^5 - 21 * q^6 - 77 * q^7 - 111 * q^8 + 108 * q^9 - 35 * q^10 + 147 * q^12 - 172 * q^13 - 30 * q^14 + 180 * q^15 + 161 * q^16 - 317 * q^17 - 63 * q^18 - 237 * q^19 + 245 * q^20 - 231 * q^21 + 210 * q^23 - 333 * q^24 + 300 * q^25 + 8 * q^26 + 324 * q^27 - 542 * q^28 - 759 * q^29 - 105 * q^30 - 193 * q^31 - 410 * q^32 - 78 * q^34 - 385 * q^35 + 441 * q^36 + 286 * q^37 - 168 * q^38 - 516 * q^39 - 555 * q^40 - 1189 * q^41 - 90 * q^42 - 775 * q^43 + 540 * q^45 - 529 * q^46 + 382 * q^47 + 483 * q^48 + 1195 * q^49 - 175 * q^50 - 951 * q^51 - 1741 * q^52 + 275 * q^53 - 189 * q^54 - 419 * q^56 - 711 * q^57 - 418 * q^58 + 646 * q^59 + 735 * q^60 - 1340 * q^61 - 983 * q^62 - 693 * q^63 - 1489 * q^64 - 860 * q^65 - 185 * q^67 - 3322 * q^68 + 630 * q^69 - 150 * q^70 - 932 * q^71 - 999 * q^72 - 2860 * q^73 - 4187 * q^74 + 900 * q^75 - 1594 * q^76 + 24 * q^78 - 1429 * q^79 + 805 * q^80 + 972 * q^81 - 30 * q^82 - 2590 * q^83 - 1626 * q^84 - 1585 * q^85 - 3195 * q^86 - 2277 * q^87 - 473 * q^89 - 315 * q^90 - 4302 * q^91 + 5462 * q^92 - 579 * q^93 - 2875 * q^94 - 1185 * q^95 - 1230 * q^96 + 318 * q^97 + 194 * q^98

Display 100 coefficients 10 coefficients

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 $

$2$	−4.84968	−1.71462	−0.857311	−	0.514798i	$-0.827867\pi$
$2$	−4.84968	−1.71462	−0.857311	+	0.514798i	$0.827867\pi$
$3$	3.00000	0.577350
$3$	3.00000	0.577350
$4$	15.5194	1.93993
$5$	5.00000	0.447214
$5$	5.00000	0.447214
$6$	−14.5491	−0.989938
$7$	−26.0919	−1.40883	−0.704414	−	0.709789i	$-0.748790\pi$
$7$	−26.0919	−1.40883	−0.704414	+	0.709789i	$0.748790\pi$
$8$	−36.4669	−1.61163
$9$	9.00000	0.333333
$10$	−24.2484	−0.766802
$11$	0	0
$11$	0	0
$12$	46.5583	1.12002
$13$	−14.0897	−0.300598	−0.150299	−	0.988641i	$-0.548024\pi$
$13$	−14.0897	−0.300598	−0.150299	+	0.988641i	$0.548024\pi$
$14$	126.537	2.41561
$15$	15.0000	0.258199
$16$	52.6975	0.823399
$17$	42.1984	0.602036	0.301018	−	0.953619i	$-0.402674\pi$
$17$	42.1984	0.602036	0.301018	+	0.953619i	$0.402674\pi$
$18$	−43.6472	−0.571541
$19$	−107.920	−1.30308	−0.651539	−	0.758615i	$-0.725876\pi$
$19$	−107.920	−1.30308	−0.651539	+	0.758615i	$0.725876\pi$
$20$	77.5972	0.867563
$21$	−78.2756	−0.813387
$22$	0	0
$23$	16.3480	0.148208	0.0741041	−	0.997251i	$-0.476390\pi$
$23$	16.3480	0.148208	0.0741041	+	0.997251i	$0.476390\pi$
$24$	−109.401	−0.930472
$25$	25.0000	0.200000
$26$	68.3304	0.515412
$27$	27.0000	0.192450
$28$	−404.931	−2.73303
$29$	153.773	0.984651	0.492326	−	0.870411i	$-0.336147\pi$
$29$	153.773	0.984651	0.492326	+	0.870411i	$0.336147\pi$
$30$	−72.7453	−0.442714
$31$	12.3887	0.0717767	0.0358883	−	0.999356i	$-0.488574\pi$
$31$	12.3887	0.0717767	0.0358883	+	0.999356i	$0.488574\pi$
$32$	36.1690	0.199807
$33$	0	0
$34$	−204.649	−1.03226
$35$	−130.459	−0.630047
$36$	139.675	0.646643
$37$	82.6216	0.367105	0.183553	−	0.983010i	$-0.441240\pi$
$37$	82.6216	0.367105	0.183553	+	0.983010i	$0.441240\pi$
$38$	523.377	2.23429
$39$	−42.2690	−0.173550
$40$	−182.335	−0.720741
$41$	−309.256	−1.17799	−0.588997	−	0.808135i	$-0.700477\pi$
$41$	−309.256	−1.17799	−0.588997	+	0.808135i	$0.700477\pi$
$42$	379.612	1.39465
$43$	371.805	1.31860	0.659299	−	0.751881i	$-0.270853\pi$
$43$	371.805	1.31860	0.659299	+	0.751881i	$0.270853\pi$
$44$	0	0
$45$	45.0000	0.149071
$46$	−79.2825	−0.254121
$47$	286.111	0.887950	0.443975	−	0.896039i	$-0.353568\pi$
$47$	286.111	0.887950	0.443975	+	0.896039i	$0.353568\pi$
$48$	158.093	0.475390
$49$	337.785	0.984797
$50$	−121.242	−0.342924
$51$	126.595	0.347585
$52$	−218.664	−0.583139
$53$	−353.229	−0.915467	−0.457734	−	0.889089i	$-0.651339\pi$
$53$	−353.229	−0.915467	−0.457734	+	0.889089i	$0.651339\pi$
$54$	−130.941	−0.329979
$55$	0	0
$56$	951.490	2.27050
$57$	−323.759	−0.752332
$58$	−745.749	−1.68830
$59$	106.003	0.233906	0.116953	−	0.993137i	$-0.462687\pi$
$59$	106.003	0.233906	0.116953	+	0.993137i	$0.462687\pi$
$60$	232.792	0.500888
$61$	884.737	1.85703	0.928516	−	0.371291i	$-0.121085\pi$
$61$	884.737	1.85703	0.928516	+	0.371291i	$0.121085\pi$
$62$	−60.0813	−0.123070
$63$	−234.827	−0.469609
$64$	−596.988	−1.16599
$65$	−70.4483	−0.134431
$66$	0	0
$67$	856.505	1.56177	0.780886	−	0.624673i	$-0.214768\pi$
$67$	856.505	1.56177	0.780886	+	0.624673i	$0.214768\pi$
$68$	654.895	1.16791
$69$	49.0439	0.0855680
$70$	632.686	1.08029
$71$	36.4974	0.0610063	0.0305031	−	0.999535i	$-0.490289\pi$
$71$	36.4974	0.0610063	0.0305031	+	0.999535i	$0.490289\pi$
$72$	−328.202	−0.537208
$73$	974.807	1.56291	0.781456	−	0.623961i	$-0.214477\pi$
$73$	974.807	1.56291	0.781456	+	0.623961i	$0.214477\pi$
$74$	−400.689	−0.629447
$75$	75.0000	0.115470
$76$	−1674.85	−2.52788
$77$	0	0
$78$	204.991	0.297573
$79$	−1036.71	−1.47644	−0.738218	−	0.674562i	$-0.764332\pi$
$79$	−1036.71	−1.47644	−0.738218	+	0.674562i	$0.764332\pi$
$80$	263.488	0.368235
$81$	81.0000	0.111111
$82$	1499.80	2.01981
$83$	−552.661	−0.730872	−0.365436	−	0.930836i	$-0.619080\pi$
$83$	−552.661	−0.730872	−0.365436	+	0.930836i	$0.619080\pi$
$84$	−1214.79	−1.57791
$85$	210.992	0.269238
$86$	−1803.14	−2.26090
$87$	461.318	0.568489
$88$	0	0
$89$	−1187.57	−1.41441	−0.707204	−	0.707009i	$-0.750044\pi$
$89$	−1187.57	−1.41441	−0.707204	+	0.707009i	$0.750044\pi$
$90$	−218.236	−0.255601
$91$	367.626	0.423490
$92$	253.711	0.287513
$93$	37.1661	0.0414403
$94$	−1387.55	−1.52250
$95$	−539.599	−0.582754
$96$	108.507	0.115359
$97$	−1200.44	−1.25655	−0.628277	−	0.777990i	$-0.716240\pi$
$97$	−1200.44	−1.25655	−0.628277	+	0.777990i	$0.716240\pi$
$98$	−1638.15	−1.68855
$99$	0	0
$100$	387.986	0.387986
$101$	−1695.89	−1.67077	−0.835385	−	0.549665i	$-0.814755\pi$
$101$	−1695.89	−1.67077	−0.835385	+	0.549665i	$0.814755\pi$
$102$	−613.946	−0.595978
$103$	−1561.14	−1.49343	−0.746717	−	0.665142i	$-0.768371\pi$
$103$	−1561.14	−1.49343	−0.746717	+	0.665142i	$0.768371\pi$
$104$	513.807	0.484451
$105$	−391.378	−0.363758
$106$	1713.05	1.56968
$107$	−511.468	−0.462107	−0.231054	−	0.972941i	$-0.574217\pi$
$107$	−511.468	−0.462107	−0.231054	+	0.972941i	$0.574217\pi$
$108$	419.025	0.373340
$109$	−1456.73	−1.28009	−0.640045	−	0.768337i	$-0.721084\pi$
$109$	−1456.73	−1.28009	−0.640045	+	0.768337i	$0.721084\pi$
$110$	0	0
$111$	247.865	0.211948
$112$	−1374.98	−1.16003
$113$	−643.005	−0.535299	−0.267650	−	0.963516i	$-0.586247\pi$
$113$	−643.005	−0.535299	−0.267650	+	0.963516i	$0.586247\pi$
$114$	1570.13	1.28997
$115$	81.7398	0.0662807
$116$	2386.47	1.91015
$117$	−126.807	−0.100199
$118$	−514.082	−0.401060
$119$	−1101.03	−0.848165
$120$	−547.004	−0.416120
$121$	0	0
$122$	−4290.70	−3.18411
$123$	−927.769	−0.680115
$124$	192.266	0.139242
$125$	125.000	0.0894427
$126$	1138.84	0.805203
$127$	2754.75	1.92476	0.962379	−	0.271712i	$-0.0875899\pi$
$127$	2754.75	1.92476	0.962379	+	0.271712i	$0.0875899\pi$
$128$	2605.85	1.79943
$129$	1115.41	0.761293
$130$	341.652	0.230499
$131$	−1793.93	−1.19646	−0.598230	−	0.801324i	$-0.704129\pi$
$131$	−1793.93	−1.19646	−0.598230	+	0.801324i	$0.704129\pi$
$132$	0	0
$133$	2815.83	1.83581
$134$	−4153.78	−2.67785
$135$	135.000	0.0860663
$136$	−1538.84	−0.970256
$137$	2485.63	1.55008	0.775042	−	0.631910i	$-0.217729\pi$
$137$	2485.63	1.55008	0.775042	+	0.631910i	$0.217729\pi$
$138$	−237.847	−0.146717
$139$	−295.244	−0.180160	−0.0900800	−	0.995935i	$-0.528712\pi$
$139$	−295.244	−0.180160	−0.0900800	+	0.995935i	$0.528712\pi$
$140$	−2024.66	−1.22225
$141$	858.334	0.512658
$142$	−177.001	−0.104603
$143$	0	0
$144$	474.278	0.274466
$145$	768.864	0.440349
$146$	−4727.51	−2.67980
$147$	1013.36	0.568573
$148$	1282.24	0.712159
$149$	67.8012	0.0372785	0.0186392	−	0.999826i	$-0.494067\pi$
$149$	67.8012	0.0372785	0.0186392	+	0.999826i	$0.494067\pi$
$150$	−363.726	−0.197988
$151$	2351.74	1.26743	0.633715	−	0.773567i	$-0.281529\pi$
$151$	2351.74	1.26743	0.633715	+	0.773567i	$0.281529\pi$
$152$	3935.50	2.10007
$153$	379.785	0.200679
$154$	0	0
$155$	61.9435	0.0320995
$156$	−655.991	−0.336675
$157$	−2521.08	−1.28155	−0.640776	−	0.767728i	$-0.721387\pi$
$157$	−2521.08	−1.28155	−0.640776	+	0.767728i	$0.721387\pi$
$158$	5027.69	2.53153
$159$	−1059.69	−0.528545
$160$	180.845	0.0893565
$161$	−426.549	−0.208800
$162$	−392.824	−0.190514
$163$	1069.50	0.513923	0.256962	−	0.966422i	$-0.417279\pi$
$163$	1069.50	0.513923	0.256962	+	0.966422i	$0.417279\pi$
$164$	−4799.49	−2.28523
$165$	0	0
$166$	2680.23	1.25317
$167$	833.926	0.386414	0.193207	−	0.981158i	$-0.438111\pi$
$167$	833.926	0.386414	0.193207	+	0.981158i	$0.438111\pi$
$168$	2854.47	1.31088
$169$	−1998.48	−0.909641
$170$	−1023.24	−0.461642
$171$	−971.277	−0.434359
$172$	5770.20	2.55799
$173$	3495.80	1.53631	0.768153	−	0.640266i	$-0.221176\pi$
$173$	3495.80	1.53631	0.768153	+	0.640266i	$0.221176\pi$
$174$	−2237.25	−0.974743
$175$	−652.297	−0.281766
$176$	0	0
$177$	318.010	0.135046
$178$	5759.35	2.42518
$179$	2284.45	0.953899	0.476950	−	0.878931i	$-0.341742\pi$
$179$	2284.45	0.953899	0.476950	+	0.878931i	$0.341742\pi$
$180$	698.375	0.289188
$181$	−2885.50	−1.18496	−0.592480	−	0.805585i	$-0.701851\pi$
$181$	−2885.50	−1.18496	−0.592480	+	0.805585i	$0.701851\pi$
$182$	−1782.87	−0.726126
$183$	2654.21	1.07216
$184$	−596.160	−0.238856
$185$	413.108	0.164175
$186$	−180.244	−0.0710544
$187$	0	0
$188$	4440.29	1.72256
$189$	−704.480	−0.271129
$190$	2616.88	0.999203
$191$	1340.38	0.507782	0.253891	−	0.967233i	$-0.418290\pi$
$191$	1340.38	0.507782	0.253891	+	0.967233i	$0.418290\pi$
$192$	−1790.96	−0.673186
$193$	−3541.53	−1.32086	−0.660428	−	0.750890i	$-0.729625\pi$
$193$	−3541.53	−1.32086	−0.660428	+	0.750890i	$0.729625\pi$
$194$	5821.73	2.15452
$195$	−211.345	−0.0776140
$196$	5242.24	1.91044
$197$	−2965.93	−1.07266	−0.536329	−	0.844009i	$-0.680190\pi$
$197$	−2965.93	−1.07266	−0.536329	+	0.844009i	$0.680190\pi$
$198$	0	0
$199$	1424.47	0.507428	0.253714	−	0.967279i	$-0.418348\pi$
$199$	1424.47	0.507428	0.253714	+	0.967279i	$0.418348\pi$
$200$	−911.673	−0.322325
$201$	2569.52	0.901690
$202$	8224.55	2.86474
$203$	−4012.22	−1.38720
$204$	1964.68	0.674291
$205$	−1546.28	−0.526815
$206$	7571.03	2.56067
$207$	147.132	0.0494027
$208$	−742.490	−0.247512
$209$	0	0
$210$	1898.06	0.623707
$211$	3692.11	1.20462	0.602311	−	0.798261i	$-0.294247\pi$
$211$	3692.11	1.20462	0.602311	+	0.798261i	$0.294247\pi$
$212$	−5481.92	−1.77594
$213$	109.492	0.0352220
$214$	2480.46	0.792340
$215$	1859.02	0.589695
$216$	−984.607	−0.310157
$217$	−323.244	−0.101121
$218$	7064.70	2.19487
$219$	2924.42	0.902348
$220$	0	0
$221$	−594.561	−0.180970
$222$	−1202.07	−0.363412
$223$	622.367	0.186891	0.0934457	−	0.995624i	$-0.470212\pi$
$223$	622.367	0.186891	0.0934457	+	0.995624i	$0.470212\pi$
$224$	−943.716	−0.281494
$225$	225.000	0.0666667
$226$	3118.37	0.917836
$227$	−6162.40	−1.80182	−0.900909	−	0.434008i	$-0.857099\pi$
$227$	−6162.40	−1.80182	−0.900909	+	0.434008i	$0.857099\pi$
$228$	−5024.56	−1.45947
$229$	−2675.21	−0.771978	−0.385989	−	0.922503i	$-0.626140\pi$
$229$	−2675.21	−0.771978	−0.385989	+	0.922503i	$0.626140\pi$
$230$	−396.412	−0.113646
$231$	0	0
$232$	−5607.62	−1.58689
$233$	−1247.85	−0.350855	−0.175427	−	0.984492i	$-0.556131\pi$
$233$	−1247.85	−0.350855	−0.175427	+	0.984492i	$0.556131\pi$
$234$	614.974	0.171804
$235$	1430.56	0.397103
$236$	1645.11	0.453761
$237$	−3110.12	−0.852421
$238$	5339.67	1.45428
$239$	−4049.92	−1.09610	−0.548050	−	0.836446i	$-0.684629\pi$
$239$	−4049.92	−1.09610	−0.548050	+	0.836446i	$0.684629\pi$
$240$	790.463	0.212601
$241$	3339.20	0.892517	0.446259	−	0.894904i	$-0.352756\pi$
$241$	3339.20	0.892517	0.446259	+	0.894904i	$0.352756\pi$
$242$	0	0
$243$	243.000	0.0641500
$244$	13730.6	3.60251
$245$	1688.93	0.440414
$246$	4499.39	1.16614
$247$	1520.55	0.391702
$248$	−451.778	−0.115677
$249$	−1657.98	−0.421969
$250$	−606.211	−0.153360
$251$	121.987	0.0306763	0.0153382	−	0.999882i	$-0.495118\pi$
$251$	121.987	0.0306763	0.0153382	+	0.999882i	$0.495118\pi$
$252$	−3644.38	−0.911009
$253$	0	0
$254$	−13359.6	−3.30023
$255$	632.975	0.155445
$256$	−7861.66	−1.91935
$257$	−3487.97	−0.846589	−0.423295	−	0.905992i	$-0.639126\pi$
$257$	−3487.97	−0.846589	−0.423295	+	0.905992i	$0.639126\pi$
$258$	−5409.41	−1.30533
$259$	−2155.75	−0.517189
$260$	−1093.32	−0.260787
$261$	1383.95	0.328217
$262$	8699.99	2.05148
$263$	−1135.38	−0.266200	−0.133100	−	0.991103i	$-0.542493\pi$
$263$	−1135.38	−0.266200	−0.133100	+	0.991103i	$0.542493\pi$
$264$	0	0
$265$	−1766.15	−0.409409
$266$	−13655.9	−3.14773
$267$	−3562.71	−0.816609
$268$	13292.5	3.02973
$269$	4573.09	1.03653	0.518264	−	0.855221i	$-0.326578\pi$
$269$	4573.09	1.03653	0.518264	+	0.855221i	$0.326578\pi$
$270$	−654.707	−0.147571
$271$	249.779	0.0559890	0.0279945	−	0.999608i	$-0.491088\pi$
$271$	249.779	0.0559890	0.0279945	+	0.999608i	$0.491088\pi$
$272$	2223.75	0.495715
$273$	1102.88	0.244502
$274$	−12054.5	−2.65781
$275$	0	0
$276$	761.134	0.165996
$277$	−5042.52	−1.09377	−0.546887	−	0.837206i	$-0.684187\pi$
$277$	−5042.52	−1.09377	−0.546887	+	0.837206i	$0.684187\pi$
$278$	1431.84	0.308906
$279$	111.498	0.0239256
$280$	4757.45	1.01540
$281$	−3444.35	−0.731220	−0.365610	−	0.930768i	$-0.619140\pi$
$281$	−3444.35	−0.731220	−0.365610	+	0.930768i	$0.619140\pi$
$282$	−4162.65	−0.879015
$283$	−925.416	−0.194383	−0.0971913	−	0.995266i	$-0.530986\pi$
$283$	−925.416	−0.194383	−0.0971913	+	0.995266i	$0.530986\pi$
$284$	566.420	0.118348
$285$	−1618.80	−0.336453
$286$	0	0
$287$	8069.08	1.65959
$288$	325.521	0.0666024
$289$	−3132.30	−0.637553
$290$	−3728.75	−0.755033
$291$	−3601.31	−0.725472
$292$	15128.5	3.03194
$293$	4338.81	0.865106	0.432553	−	0.901609i	$-0.357613\pi$
$293$	4338.81	0.865106	0.432553	+	0.901609i	$0.357613\pi$
$294$	−4914.46	−0.974887
$295$	530.016	0.104606
$296$	−3012.95	−0.591636
$297$	0	0
$298$	−328.814	−0.0639185
$299$	−230.337	−0.0445510
$300$	1163.96	0.224004
$301$	−9701.08	−1.85768
$302$	−11405.2	−2.17316
$303$	−5087.68	−0.964620
$304$	−5687.10	−1.07295
$305$	4423.69	0.830490
$306$	−1841.84	−0.344088
$307$	−2226.65	−0.413947	−0.206974	−	0.978347i	$-0.566361\pi$
$307$	−2226.65	−0.413947	−0.206974	+	0.978347i	$0.566361\pi$
$308$	0	0
$309$	−4683.42	−0.862234
$310$	−300.407	−0.0550385
$311$	−3262.55	−0.594862	−0.297431	−	0.954743i	$-0.596130\pi$
$311$	−3262.55	−0.594862	−0.297431	+	0.954743i	$0.596130\pi$
$312$	1541.42	0.279698
$313$	−2641.00	−0.476927	−0.238464	−	0.971151i	$-0.576644\pi$
$313$	−2641.00	−0.476927	−0.238464	+	0.971151i	$0.576644\pi$
$314$	12226.4	2.19738
$315$	−1174.13	−0.210016
$316$	−16089.1	−2.86418
$317$	−11059.0	−1.95941	−0.979707	−	0.200433i	$-0.935765\pi$
$317$	−11059.0	−1.95941	−0.979707	+	0.200433i	$0.935765\pi$
$318$	5139.15	0.906256
$319$	0	0
$320$	−2984.94	−0.521448
$321$	−1534.40	−0.266798
$322$	2068.63	0.358013
$323$	−4554.03	−0.784499
$324$	1257.07	0.215548
$325$	−352.242	−0.0601195
$326$	−5186.73	−0.881185
$327$	−4370.20	−0.739060
$328$	11277.6	1.89848
$329$	−7465.18	−1.25097
$330$	0	0
$331$	2233.47	0.370884	0.185442	−	0.982655i	$-0.440628\pi$
$331$	2233.47	0.370884	0.185442	+	0.982655i	$0.440628\pi$
$332$	−8576.99	−1.41784
$333$	743.594	0.122368
$334$	−4044.28	−0.662554
$335$	4282.53	0.698446
$336$	−4124.93	−0.669742
$337$	10844.7	1.75297	0.876483	−	0.481432i	$-0.159883\pi$
$337$	10844.7	1.75297	0.876483	+	0.481432i	$0.159883\pi$
$338$	9692.00	1.55969
$339$	−1929.01	−0.309055
$340$	3274.47	0.522304
$341$	0	0
$342$	4710.39	0.744762
$343$	136.063	0.0214190
$344$	−13558.6	−2.12509
$345$	245.220	0.0382672
$346$	−16953.5	−2.63419
$347$	−1338.41	−0.207059	−0.103529	−	0.994626i	$-0.533014\pi$
$347$	−1338.41	−0.207059	−0.103529	+	0.994626i	$0.533014\pi$
$348$	7159.40	1.10283
$349$	−6188.43	−0.949167	−0.474583	−	0.880211i	$-0.657401\pi$
$349$	−6188.43	−0.949167	−0.474583	+	0.880211i	$0.657401\pi$
$350$	3163.43	0.483122
$351$	−380.421	−0.0578501
$352$	0	0
$353$	−7454.73	−1.12401	−0.562005	−	0.827134i	$-0.689970\pi$
$353$	−7454.73	−1.12401	−0.562005	+	0.827134i	$0.689970\pi$
$354$	−1542.25	−0.231552
$355$	182.487	0.0272828
$356$	−18430.4	−2.74385
$357$	−3303.10	−0.489688
$358$	−11078.9	−1.63558
$359$	−10297.1	−1.51381	−0.756906	−	0.653524i	$-0.773290\pi$
$359$	−10297.1	−1.51381	−0.756906	+	0.653524i	$0.773290\pi$
$360$	−1641.01	−0.240247
$361$	4787.66	0.698012
$362$	13993.8	2.03176
$363$	0	0
$364$	5705.34	0.821542
$365$	4874.04	0.698955
$366$	−12872.1	−1.83835
$367$	4748.92	0.675454	0.337727	−	0.941244i	$-0.390342\pi$
$367$	4748.92	0.675454	0.337727	+	0.941244i	$0.390342\pi$
$368$	861.497	0.122034
$369$	−2783.31	−0.392665
$370$	−2003.44	−0.281497
$371$	9216.41	1.28974
$372$	576.797	0.0803913
$373$	12802.5	1.77719	0.888593	−	0.458696i	$-0.151683\pi$
$373$	12802.5	1.77719	0.888593	+	0.458696i	$0.151683\pi$
$374$	0	0
$375$	375.000	0.0516398
$376$	−10433.6	−1.43104
$377$	−2166.61	−0.295984
$378$	3416.51	0.464884
$379$	1104.20	0.149655	0.0748273	−	0.997197i	$-0.476159\pi$
$379$	1104.20	0.149655	0.0748273	+	0.997197i	$0.476159\pi$
$380$	−8374.27	−1.13050
$381$	8264.24	1.11126
$382$	−6500.41	−0.870655
$383$	198.104	0.0264298	0.0132149	−	0.999913i	$-0.495793\pi$
$383$	198.104	0.0264298	0.0132149	+	0.999913i	$0.495793\pi$
$384$	7817.56	1.03890
$385$	0	0
$386$	17175.3	2.26477
$387$	3346.24	0.439533
$388$	−18630.1	−2.43763
$389$	−7267.48	−0.947238	−0.473619	−	0.880730i	$-0.657053\pi$
$389$	−7267.48	−0.947238	−0.473619	+	0.880730i	$0.657053\pi$
$390$	1024.96	0.133079
$391$	689.857	0.0892266
$392$	−12318.0	−1.58712
$393$	−5381.79	−0.690777
$394$	14383.8	1.83921
$395$	−5183.53	−0.660282
$396$	0	0
$397$	−5921.05	−0.748537	−0.374268	−	0.927320i	$-0.622106\pi$
$397$	−5921.05	−0.748537	−0.374268	+	0.927320i	$0.622106\pi$
$398$	−6908.24	−0.870047
$399$	8447.48	1.05991
$400$	1317.44	0.164680
$401$	65.5674	0.00816528	0.00408264	−	0.999992i	$-0.498700\pi$
$401$	65.5674	0.00816528	0.00408264	+	0.999992i	$0.498700\pi$
$402$	−12461.3	−1.54606
$403$	−174.553	−0.0215759
$404$	−26319.3	−3.24118
$405$	405.000	0.0496904
$406$	19458.0	2.37853
$407$	0	0
$408$	−4616.53	−0.560177
$409$	−4079.76	−0.493230	−0.246615	−	0.969113i	$-0.579318\pi$
$409$	−4079.76	−0.493230	−0.246615	+	0.969113i	$0.579318\pi$
$410$	7498.98	0.903289
$411$	7456.88	0.894942
$412$	−24228.0	−2.89716
$413$	−2765.82	−0.329533
$414$	−713.542	−0.0847070
$415$	−2763.30	−0.326856
$416$	−509.609	−0.0600616
$417$	−885.731	−0.104015
$418$	0	0
$419$	−4907.09	−0.572141	−0.286071	−	0.958209i	$-0.592349\pi$
$419$	−4907.09	−0.572141	−0.286071	+	0.958209i	$0.592349\pi$
$420$	−6073.97	−0.705665
$421$	−4399.03	−0.509253	−0.254626	−	0.967039i	$-0.581952\pi$
$421$	−4399.03	−0.509253	−0.254626	+	0.967039i	$0.581952\pi$
$422$	−17905.6	−2.06547
$423$	2575.00	0.295983
$424$	12881.2	1.47539
$425$	1054.96	0.120407
$426$	−531.003	−0.0603924
$427$	−23084.4	−2.61624
$428$	−7937.70	−0.896456
$429$	0	0
$430$	−9015.68	−1.01110
$431$	1432.19	0.160061	0.0800305	−	0.996792i	$-0.474498\pi$
$431$	1432.19	0.160061	0.0800305	+	0.996792i	$0.474498\pi$
$432$	1422.83	0.158463
$433$	1306.35	0.144987	0.0724933	−	0.997369i	$-0.476904\pi$
$433$	1306.35	0.144987	0.0724933	+	0.997369i	$0.476904\pi$
$434$	1567.63	0.173384
$435$	2306.59	0.254236
$436$	−22607.7	−2.48329
$437$	−1764.27	−0.193127
$438$	−14182.5	−1.54719
$439$	4884.01	0.530982	0.265491	−	0.964113i	$-0.414466\pi$
$439$	4884.01	0.530982	0.265491	+	0.964113i	$0.414466\pi$
$440$	0	0
$441$	3040.07	0.328266
$442$	2883.43	0.310296
$443$	14179.1	1.52070	0.760348	−	0.649515i	$-0.225028\pi$
$443$	14179.1	1.52070	0.760348	+	0.649515i	$0.225028\pi$
$444$	3846.72	0.411165
$445$	−5937.86	−0.632543
$446$	−3018.28	−0.320448
$447$	203.404	0.0215227
$448$	15576.5	1.64268
$449$	−13345.6	−1.40272	−0.701358	−	0.712809i	$-0.747423\pi$
$449$	−13345.6	−1.40272	−0.701358	+	0.712809i	$0.747423\pi$
$450$	−1091.18	−0.114308
$451$	0	0
$452$	−9979.08	−1.03844
$453$	7055.22	0.731751
$454$	29885.7	3.08944
$455$	1838.13	0.189391
$456$	11806.5	1.21248
$457$	−14787.7	−1.51365	−0.756824	−	0.653619i	$-0.773250\pi$
$457$	−14787.7	−1.51365	−0.756824	+	0.653619i	$0.773250\pi$
$458$	12973.9	1.32365
$459$	1139.36	0.115862
$460$	1268.56	0.128580
$461$	−9586.13	−0.968483	−0.484242	−	0.874934i	$-0.660904\pi$
$461$	−9586.13	−0.968483	−0.484242	+	0.874934i	$0.660904\pi$
$462$	0	0
$463$	2930.22	0.294122	0.147061	−	0.989127i	$-0.453019\pi$
$463$	2930.22	0.294122	0.147061	+	0.989127i	$0.453019\pi$
$464$	8103.44	0.810761
$465$	185.831	0.0185327
$466$	6051.66	0.601583
$467$	−11934.3	−1.18256	−0.591280	−	0.806466i	$-0.701377\pi$
$467$	−11934.3	−1.18256	−0.591280	+	0.806466i	$0.701377\pi$
$468$	−1967.97	−0.194380
$469$	−22347.8	−2.20027
$470$	−6937.75	−0.680882
$471$	−7563.23	−0.739905
$472$	−3865.61	−0.376968
$473$	0	0
$474$	15083.1	1.46158
$475$	−2697.99	−0.260616
$476$	−17087.4	−1.64538
$477$	−3179.06	−0.305156
$478$	19640.8	1.87940
$479$	5926.33	0.565304	0.282652	−	0.959222i	$-0.408786\pi$
$479$	5926.33	0.565304	0.282652	+	0.959222i	$0.408786\pi$
$480$	542.535	0.0515900
$481$	−1164.11	−0.110351
$482$	−16194.1	−1.53033
$483$	−1279.65	−0.120551
$484$	0	0
$485$	−6002.18	−0.561948
$486$	−1178.47	−0.109993
$487$	2619.97	0.243782	0.121891	−	0.992543i	$-0.461104\pi$
$487$	2619.97	0.243782	0.121891	+	0.992543i	$0.461104\pi$
$488$	−32263.6	−2.99284
$489$	3208.49	0.296714
$490$	−8190.76	−0.755144
$491$	14231.1	1.30802	0.654012	−	0.756484i	$-0.273084\pi$
$491$	14231.1	1.30802	0.654012	+	0.756484i	$0.273084\pi$
$492$	−14398.5	−1.31938
$493$	6488.96	0.592795
$494$	−7374.20	−0.671621
$495$	0	0
$496$	652.854	0.0591008
$497$	−952.286	−0.0859474
$498$	8040.69	0.723518
$499$	731.919	0.0656617	0.0328308	−	0.999461i	$-0.489548\pi$
$499$	731.919	0.0656617	0.0328308	+	0.999461i	$0.489548\pi$
$500$	1939.93	0.173513
$501$	2501.78	0.223096
$502$	−591.599	−0.0525983
$503$	−16518.2	−1.46423	−0.732117	−	0.681179i	$-0.761467\pi$
$503$	−16518.2	−1.46423	−0.732117	+	0.681179i	$0.761467\pi$
$504$	8563.41	0.756834
$505$	−8479.47	−0.747191
$506$	0	0
$507$	−5995.44	−0.525181
$508$	42752.1	3.73389
$509$	−15437.7	−1.34433	−0.672163	−	0.740403i	$-0.734635\pi$
$509$	−15437.7	−1.34433	−0.672163	+	0.740403i	$0.734635\pi$
$510$	−3069.73	−0.266529
$511$	−25434.5	−2.20187
$512$	17279.7	1.49153
$513$	−2913.83	−0.250777
$514$	16915.5	1.45158
$515$	−7805.70	−0.667884
$516$	17310.6	1.47686
$517$	0	0
$518$	10454.7	0.886783
$519$	10487.4	0.886987
$520$	2569.03	0.216653
$521$	4206.74	0.353744	0.176872	−	0.984234i	$-0.443402\pi$
$521$	4206.74	0.353744	0.176872	+	0.984234i	$0.443402\pi$
$522$	−6711.74	−0.562768
$523$	−8255.69	−0.690241	−0.345120	−	0.938558i	$-0.612162\pi$
$523$	−8255.69	−0.690241	−0.345120	+	0.938558i	$0.612162\pi$
$524$	−27840.8	−2.32105
$525$	−1956.89	−0.162677
$526$	5506.25	0.456433
$527$	522.783	0.0432121
$528$	0	0
$529$	−11899.7	−0.978034
$530$	8565.25	0.701983
$531$	954.029	0.0779686
$532$	43700.0	3.56135
$533$	4357.32	0.354102
$534$	17278.0	1.40018
$535$	−2557.34	−0.206661
$536$	−31234.1	−2.51699
$537$	6853.36	0.550734
$538$	−22178.0	−1.77725
$539$	0	0
$540$	2095.12	0.166963
$541$	9795.46	0.778447	0.389223	−	0.921143i	$-0.372743\pi$
$541$	9795.46	0.778447	0.389223	+	0.921143i	$0.372743\pi$
$542$	−1211.35	−0.0960000
$543$	−8656.51	−0.684137
$544$	1526.27	0.120291
$545$	−7283.67	−0.572474
$546$	−5348.60	−0.419229
$547$	7746.31	0.605500	0.302750	−	0.953070i	$-0.402095\pi$
$547$	7746.31	0.605500	0.302750	+	0.953070i	$0.402095\pi$
$548$	38575.6	3.00706
$549$	7962.64	0.619011
$550$	0	0
$551$	−16595.1	−1.28308
$552$	−1788.48	−0.137904
$553$	27049.6	2.08004
$554$	24454.6	1.87541
$555$	1239.32	0.0947862
$556$	−4582.01	−0.349498
$557$	16592.8	1.26223	0.631113	−	0.775691i	$-0.282598\pi$
$557$	16592.8	1.26223	0.631113	+	0.775691i	$0.282598\pi$
$558$	−540.732	−0.0410233
$559$	−5238.61	−0.396368
$560$	−6874.88	−0.518780
$561$	0	0
$562$	16704.0	1.25377
$563$	17975.8	1.34563	0.672816	−	0.739810i	$-0.265084\pi$
$563$	17975.8	1.34563	0.672816	+	0.739810i	$0.265084\pi$
$564$	13320.9	0.994521
$565$	−3215.02	−0.239393
$566$	4487.98	0.333293
$567$	−2113.44	−0.156536
$568$	−1330.95	−0.0983193
$569$	8001.08	0.589495	0.294747	−	0.955575i	$-0.404764\pi$
$569$	8001.08	0.589495	0.294747	+	0.955575i	$0.404764\pi$
$570$	7850.65	0.576890
$571$	9210.05	0.675006	0.337503	−	0.941324i	$-0.390418\pi$
$571$	9210.05	0.675006	0.337503	+	0.941324i	$0.390418\pi$
$572$	0	0
$573$	4021.14	0.293168
$574$	−39132.5	−2.84557
$575$	408.699	0.0296416
$576$	−5372.89	−0.388664
$577$	18587.4	1.34108	0.670539	−	0.741874i	$-0.266063\pi$
$577$	18587.4	1.34108	0.670539	+	0.741874i	$0.266063\pi$
$578$	15190.7	1.09316
$579$	−10624.6	−0.762596
$580$	11932.3	0.854247
$581$	14419.9	1.02967
$582$	17465.2	1.24391
$583$	0	0
$584$	−35548.2	−2.51883
$585$	−634.035	−0.0448105
$586$	−21041.9	−1.48333
$587$	−14706.1	−1.03405	−0.517023	−	0.855972i	$-0.672960\pi$
$587$	−14706.1	−1.03405	−0.517023	+	0.855972i	$0.672960\pi$
$588$	15726.7	1.10299
$589$	−1336.99	−0.0935306
$590$	−2570.41	−0.179360
$591$	−8897.79	−0.619300
$592$	4353.95	0.302274
$593$	−21065.6	−1.45879	−0.729394	−	0.684093i	$-0.760198\pi$
$593$	−21065.6	−1.45879	−0.729394	+	0.684093i	$0.760198\pi$
$594$	0	0
$595$	−5505.17	−0.379311
$596$	1052.24	0.0723176
$597$	4273.41	0.292963
$598$	1117.06	0.0763882
$599$	−8733.98	−0.595761	−0.297880	−	0.954603i	$-0.596280\pi$
$599$	−8733.98	−0.595761	−0.297880	+	0.954603i	$0.596280\pi$
$600$	−2735.02	−0.186094
$601$	−4024.41	−0.273143	−0.136572	−	0.990630i	$-0.543608\pi$
$601$	−4024.41	−0.273143	−0.136572	+	0.990630i	$0.543608\pi$
$602$	47047.2	3.18522
$603$	7708.55	0.520591
$604$	36497.7	2.45873
$605$	0	0
$606$	24673.7	1.65396
$607$	−24816.9	−1.65945	−0.829726	−	0.558171i	$-0.811503\pi$
$607$	−24816.9	−1.65945	−0.829726	+	0.558171i	$0.811503\pi$
$608$	−3903.35	−0.260364
$609$	−12036.7	−0.800903
$610$	−21453.5	−1.42398
$611$	−4031.21	−0.266916
$612$	5894.05	0.389302
$613$	−145.666	−0.00959770	−0.00479885	−	0.999988i	$-0.501528\pi$
$613$	−145.666	−0.00959770	−0.00479885	+	0.999988i	$0.501528\pi$
$614$	10798.6	0.709764
$615$	−4638.85	−0.304157
$616$	0	0
$617$	−4935.61	−0.322042	−0.161021	−	0.986951i	$-0.551479\pi$
$617$	−4935.61	−0.322042	−0.161021	+	0.986951i	$0.551479\pi$
$618$	22713.1	1.47841
$619$	12155.8	0.789307	0.394654	−	0.918830i	$-0.370865\pi$
$619$	12155.8	0.789307	0.394654	+	0.918830i	$0.370865\pi$
$620$	961.329	0.0622708
$621$	441.395	0.0285227
$622$	15822.3	1.01996
$623$	30986.0	1.99266
$624$	−2227.47	−0.142901
$625$	625.000	0.0400000
$626$	12808.0	0.817750
$627$	0	0
$628$	−39125.7	−2.48612
$629$	3486.49	0.221011
$630$	5694.18	0.360098
$631$	−2503.84	−0.157966	−0.0789829	−	0.996876i	$-0.525167\pi$
$631$	−2503.84	−0.157966	−0.0789829	+	0.996876i	$0.525167\pi$
$632$	37805.4	2.37946
$633$	11076.3	0.695489
$634$	53632.6	3.35966
$635$	13773.7	0.860778
$636$	−16445.8	−1.02534
$637$	−4759.28	−0.296028
$638$	0	0
$639$	328.477	0.0203354
$640$	13029.3	0.804730
$641$	−18022.3	−1.11051	−0.555255	−	0.831680i	$-0.687379\pi$
$641$	−18022.3	−1.11051	−0.555255	+	0.831680i	$0.687379\pi$
$642$	7441.38	0.457457
$643$	−9234.89	−0.566390	−0.283195	−	0.959062i	$-0.591394\pi$
$643$	−9234.89	−0.566390	−0.283195	+	0.959062i	$0.591394\pi$
$644$	−6619.80	−0.405057
$645$	5577.07	0.340461
$646$	22085.6	1.34512
$647$	−9504.52	−0.577529	−0.288764	−	0.957400i	$-0.593244\pi$
$647$	−9504.52	−0.577529	−0.288764	+	0.957400i	$0.593244\pi$
$648$	−2953.82	−0.179069
$649$	0	0
$650$	1708.26	0.103082
$651$	−969.733	−0.0583822
$652$	16598.0	0.996976
$653$	−22017.9	−1.31949	−0.659744	−	0.751491i	$-0.729335\pi$
$653$	−22017.9	−1.31949	−0.659744	+	0.751491i	$0.729335\pi$
$654$	21194.1	1.26721
$655$	−8969.65	−0.535073
$656$	−16297.0	−0.969959
$657$	8773.27	0.520971
$658$	36203.8	2.14494
$659$	−29116.2	−1.72110	−0.860552	−	0.509362i	$-0.829881\pi$
$659$	−29116.2	−1.72110	−0.860552	+	0.509362i	$0.829881\pi$
$660$	0	0
$661$	−11403.4	−0.671014	−0.335507	−	0.942038i	$-0.608908\pi$
$661$	−11403.4	−0.671014	−0.335507	+	0.942038i	$0.608908\pi$
$662$	−10831.6	−0.635925
$663$	−1783.68	−0.104483
$664$	20153.8	1.17789
$665$	14079.1	0.821000
$666$	−3606.20	−0.209816
$667$	2513.87	0.145933
$668$	12942.1	0.749616
$669$	1867.10	0.107902
$670$	−20768.9	−1.19757
$671$	0	0
$672$	−2831.15	−0.162521
$673$	−4300.54	−0.246320	−0.123160	−	0.992387i	$-0.539303\pi$
$673$	−4300.54	−0.246320	−0.123160	+	0.992387i	$0.539303\pi$
$674$	−52593.5	−3.00568
$675$	675.000	0.0384900
$676$	−31015.3	−1.76464
$677$	11671.0	0.662560	0.331280	−	0.943533i	$-0.392520\pi$
$677$	11671.0	0.662560	0.331280	+	0.943533i	$0.392520\pi$
$678$	9355.11	0.529913
$679$	31321.6	1.77027
$680$	−7694.22	−0.433912
$681$	−18487.2	−1.04028
$682$	0	0
$683$	14048.2	0.787028	0.393514	−	0.919319i	$-0.371259\pi$
$683$	14048.2	0.787028	0.393514	+	0.919319i	$0.371259\pi$
$684$	−15073.7	−0.842627
$685$	12428.1	0.693219
$686$	−659.864	−0.0367255
$687$	−8025.63	−0.445702
$688$	19593.2	1.08573
$689$	4976.88	0.275187
$690$	−1189.24	−0.0656138
$691$	−20556.7	−1.13171	−0.565856	−	0.824504i	$-0.691454\pi$
$691$	−20556.7	−1.13171	−0.565856	+	0.824504i	$0.691454\pi$
$692$	54252.9	2.98033
$693$	0	0
$694$	6490.85	0.355028
$695$	−1476.22	−0.0805700
$696$	−16822.9	−0.916191
$697$	−13050.1	−0.709194
$698$	30011.9	1.62746
$699$	−3743.54	−0.202566
$700$	−10123.3	−0.546606
$701$	13883.5	0.748037	0.374019	−	0.927421i	$-0.377980\pi$
$701$	13883.5	0.748037	0.374019	+	0.927421i	$0.377980\pi$
$702$	1844.92	0.0991910
$703$	−8916.50	−0.478367
$704$	0	0
$705$	4291.67	0.229268
$706$	36153.1	1.92725
$707$	44249.1	2.35383
$708$	4935.33	0.261979
$709$	−32015.8	−1.69588	−0.847939	−	0.530093i	$-0.822157\pi$
$709$	−32015.8	−1.69588	−0.847939	+	0.530093i	$0.822157\pi$
$710$	−885.005	−0.0467798
$711$	−9330.35	−0.492145
$712$	43307.1	2.27950
$713$	202.530	0.0106379
$714$	16019.0	0.839630
$715$	0	0
$716$	35453.4	1.85050
$717$	−12149.8	−0.632833
$718$	49937.5	2.59561
$719$	8931.92	0.463288	0.231644	−	0.972801i	$-0.425590\pi$
$719$	8931.92	0.463288	0.231644	+	0.972801i	$0.425590\pi$
$720$	2371.39	0.122745
$721$	40733.0	2.10399
$722$	−23218.7	−1.19683
$723$	10017.6	0.515295
$724$	−44781.4	−2.29874
$725$	3844.32	0.196930
$726$	0	0
$727$	−199.112	−0.0101577	−0.00507885	−	0.999987i	$-0.501617\pi$
$727$	−199.112	−0.0101577	−0.00507885	+	0.999987i	$0.501617\pi$
$728$	−13406.2	−0.682508
$729$	729.000	0.0370370
$730$	−23637.5	−1.19844
$731$	15689.6	0.793843
$732$	41191.9	2.07991
$733$	−26469.8	−1.33381	−0.666905	−	0.745143i	$-0.732381\pi$
$733$	−26469.8	−1.33381	−0.666905	+	0.745143i	$0.732381\pi$
$734$	−23030.8	−1.15815
$735$	5066.78	0.254273
$736$	591.289	0.0296131
$737$	0	0
$738$	13498.2	0.673272
$739$	9867.99	0.491204	0.245602	−	0.969371i	$-0.421014\pi$
$739$	9867.99	0.491204	0.245602	+	0.969371i	$0.421014\pi$
$740$	6411.20	0.318487
$741$	4561.66	0.226149
$742$	−44696.7	−2.21141
$743$	−23577.2	−1.16415	−0.582076	−	0.813135i	$-0.697759\pi$
$743$	−23577.2	−1.16415	−0.582076	+	0.813135i	$0.697759\pi$
$744$	−1355.33	−0.0667862
$745$	339.006	0.0166714
$746$	−62088.3	−3.04720
$747$	−4973.95	−0.243624
$748$	0	0
$749$	13345.2	0.651030
$750$	−1818.63	−0.0885427
$751$	13143.1	0.638611	0.319306	−	0.947652i	$-0.396550\pi$
$751$	13143.1	0.638611	0.319306	+	0.947652i	$0.396550\pi$
$752$	15077.4	0.731137
$753$	365.961	0.0177110
$754$	10507.4	0.507501
$755$	11758.7	0.566812
$756$	−10933.1	−0.525971
$757$	22027.4	1.05759	0.528797	−	0.848749i	$-0.322643\pi$
$757$	22027.4	1.05759	0.528797	+	0.848749i	$0.322643\pi$
$758$	−5355.04	−0.256601
$759$	0	0
$760$	19677.5	0.939181
$761$	−25514.6	−1.21538	−0.607689	−	0.794175i	$-0.707903\pi$
$761$	−25514.6	−1.21538	−0.607689	+	0.794175i	$0.707903\pi$
$762$	−40078.9	−1.90539
$763$	38008.9	1.80343
$764$	20801.9	0.985062
$765$	1898.93	0.0897462
$766$	−960.741	−0.0453172
$767$	−1493.55	−0.0703115
$768$	−23585.0	−1.10814
$769$	−36528.0	−1.71292	−0.856459	−	0.516214i	$-0.827341\pi$
$769$	−36528.0	−1.71292	−0.856459	+	0.516214i	$0.827341\pi$
$770$	0	0
$771$	−10463.9	−0.488778
$772$	−54962.6	−2.56237
$773$	−1945.99	−0.0905464	−0.0452732	−	0.998975i	$-0.514416\pi$
$773$	−1945.99	−0.0905464	−0.0452732	+	0.998975i	$0.514416\pi$
$774$	−16228.2	−0.753633
$775$	309.718	0.0143553
$776$	43776.2	2.02509
$777$	−6467.25	−0.298599
$778$	35245.0	1.62416
$779$	33374.9	1.53502
$780$	−3279.96	−0.150566
$781$	0	0
$782$	−3345.59	−0.152990
$783$	4151.86	0.189496
$784$	17800.4	0.810880
$785$	−12605.4	−0.573128
$786$	26100.0	1.18442
$787$	−16138.8	−0.730985	−0.365493	−	0.930814i	$-0.619099\pi$
$787$	−16138.8	−0.730985	−0.365493	+	0.930814i	$0.619099\pi$
$788$	−46029.6	−2.08088
$789$	−3406.15	−0.153691
$790$	25138.5	1.13213
$791$	16777.2	0.754145
$792$	0	0
$793$	−12465.7	−0.558220
$794$	28715.2	1.28346
$795$	−5298.44	−0.236373
$796$	22107.0	0.984374
$797$	32003.2	1.42235	0.711174	−	0.703016i	$-0.248164\pi$
$797$	32003.2	1.42235	0.711174	+	0.703016i	$0.248164\pi$
$798$	−40967.6	−1.81734
$799$	12073.4	0.534577
$800$	904.224	0.0399615
$801$	−10688.1	−0.471469
$802$	−317.981	−0.0140004
$803$	0	0
$804$	39877.4	1.74921
$805$	−2132.74	−0.0933781
$806$	846.525	0.0369945
$807$	13719.3	0.598440
$808$	61844.1	2.69266
$809$	−13596.0	−0.590867	−0.295433	−	0.955363i	$-0.595464\pi$
$809$	−13596.0	−0.590867	−0.295433	+	0.955363i	$0.595464\pi$
$810$	−1964.12	−0.0852003
$811$	−29398.6	−1.27290	−0.636451	−	0.771317i	$-0.719598\pi$
$811$	−29398.6	−1.27290	−0.636451	+	0.771317i	$0.719598\pi$
$812$	−62267.4	−2.69108
$813$	749.338	0.0323253
$814$	0	0
$815$	5347.49	0.229834
$816$	6671.25	0.286201
$817$	−40125.1	−1.71824
$818$	19785.6	0.845704
$819$	3308.63	0.141163
$820$	−23997.4	−1.02198
$821$	−36171.5	−1.53763	−0.768815	−	0.639471i	$-0.779153\pi$
$821$	−36171.5	−1.53763	−0.768815	+	0.639471i	$0.779153\pi$
$822$	−36163.5	−1.53449
$823$	−23170.3	−0.981367	−0.490683	−	0.871338i	$-0.663253\pi$
$823$	−23170.3	−0.981367	−0.490683	+	0.871338i	$0.663253\pi$
$824$	56929.9	2.40685
$825$	0	0
$826$	13413.4	0.565025
$827$	23698.2	0.996453	0.498226	−	0.867047i	$-0.333985\pi$
$827$	23698.2	0.996453	0.498226	+	0.867047i	$0.333985\pi$
$828$	2283.40	0.0958378
$829$	−11300.3	−0.473432	−0.236716	−	0.971579i	$-0.576071\pi$
$829$	−11300.3	−0.473432	−0.236716	+	0.971579i	$0.576071\pi$
$830$	13401.2	0.560435
$831$	−15127.6	−0.631491
$832$	8411.37	0.350495
$833$	14254.0	0.592883
$834$	4295.51	0.178347
$835$	4169.63	0.172809
$836$	0	0
$837$	334.495	0.0138134
$838$	23797.9	0.981006
$839$	15487.7	0.637298	0.318649	−	0.947873i	$-0.396771\pi$
$839$	15487.7	0.637298	0.318649	+	0.947873i	$0.396771\pi$
$840$	14272.3	0.586241
$841$	−742.941	−0.0304621
$842$	21333.9	0.873176
$843$	−10333.1	−0.422170
$844$	57299.5	2.33688
$845$	−9992.41	−0.406804
$846$	−12488.0	−0.507500
$847$	0	0
$848$	−18614.3	−0.753795
$849$	−2776.25	−0.112227
$850$	−5116.22	−0.206453
$851$	1350.69	0.0544080
$852$	1699.26	0.0683282
$853$	−11349.9	−0.455584	−0.227792	−	0.973710i	$-0.573151\pi$
$853$	−11349.9	−0.455584	−0.227792	+	0.973710i	$0.573151\pi$
$854$	111952.	4.48586
$855$	−4856.39	−0.194251
$856$	18651.7	0.744744
$857$	16725.5	0.666664	0.333332	−	0.942810i	$-0.391827\pi$
$857$	16725.5	0.666664	0.333332	+	0.942810i	$0.391827\pi$
$858$	0	0
$859$	26206.9	1.04094	0.520470	−	0.853880i	$-0.325757\pi$
$859$	26206.9	1.04094	0.520470	+	0.853880i	$0.325757\pi$
$860$	28851.0	1.14397
$861$	24207.2	0.958165
$862$	−6945.68	−0.274444
$863$	31253.4	1.23277	0.616384	−	0.787446i	$-0.288597\pi$
$863$	31253.4	1.23277	0.616384	+	0.787446i	$0.288597\pi$
$864$	976.562	0.0384529
$865$	17479.0	0.687057
$866$	−6335.39	−0.248597
$867$	−9396.90	−0.368092
$868$	−5016.57	−0.196168
$869$	0	0
$870$	−11186.2	−0.435918
$871$	−12067.9	−0.469465
$872$	53122.6	2.06303
$873$	−10803.9	−0.418851
$874$	8556.14	0.331139
$875$	−3261.48	−0.126009
$876$	45385.4	1.75049
$877$	−24395.5	−0.939315	−0.469657	−	0.882849i	$-0.655623\pi$
$877$	−24395.5	−0.939315	−0.469657	+	0.882849i	$0.655623\pi$
$878$	−23685.9	−0.910434
$879$	13016.4	0.499469
$880$	0	0
$881$	32158.8	1.22980	0.614902	−	0.788604i	$-0.289196\pi$
$881$	32158.8	1.22980	0.614902	+	0.788604i	$0.289196\pi$
$882$	−14743.4	−0.562851
$883$	30667.7	1.16880	0.584400	−	0.811466i	$-0.301330\pi$
$883$	30667.7	1.16880	0.584400	+	0.811466i	$0.301330\pi$
$884$	−9227.25	−0.351070
$885$	1590.05	0.0603942
$886$	−68764.1	−2.60742
$887$	35312.1	1.33671	0.668357	−	0.743841i	$-0.266998\pi$
$887$	35312.1	1.33671	0.668357	+	0.743841i	$0.266998\pi$
$888$	−9038.86	−0.341581
$889$	−71876.4	−2.71165
$890$	28796.7	1.08457
$891$	0	0
$892$	9658.79	0.362556
$893$	−30877.1	−1.15707
$894$	−986.443	−0.0369034
$895$	11422.3	0.426597
$896$	−67991.6	−2.53509
$897$	−691.012	−0.0257215
$898$	64722.2	2.40513
$899$	1905.05	0.0706750
$900$	3491.87	0.129329
$901$	−14905.7	−0.551144
$902$	0	0
$903$	−29103.2	−1.07253
$904$	23448.4	0.862702
$905$	−14427.5	−0.529930
$906$	−34215.6	−1.25468
$907$	26852.5	0.983044	0.491522	−	0.870865i	$-0.336441\pi$
$907$	26852.5	0.983044	0.491522	+	0.870865i	$0.336441\pi$
$908$	−95637.0	−3.49540
$909$	−15263.1	−0.556924
$910$	−8914.34	−0.324734
$911$	−39549.3	−1.43834	−0.719169	−	0.694835i	$-0.755477\pi$
$911$	−39549.3	−1.43834	−0.719169	+	0.694835i	$0.755477\pi$
$912$	−17061.3	−0.619470
$913$	0	0
$914$	71715.5	2.59534
$915$	13271.1	0.479484
$916$	−41517.8	−1.49758
$917$	46807.0	1.68561
$918$	−5525.51	−0.198659
$919$	12746.0	0.457510	0.228755	−	0.973484i	$-0.426535\pi$
$919$	12746.0	0.457510	0.228755	+	0.973484i	$0.426535\pi$
$920$	−2980.80	−0.106820
$921$	−6679.96	−0.238993
$922$	46489.7	1.66058
$923$	−514.236	−0.0183384
$924$	0	0
$925$	2065.54	0.0734211
$926$	−14210.6	−0.504309
$927$	−14050.3	−0.497811
$928$	5561.80	0.196740
$929$	−6953.18	−0.245561	−0.122781	−	0.992434i	$-0.539181\pi$
$929$	−6953.18	−0.245561	−0.122781	+	0.992434i	$0.539181\pi$
$930$	−901.220	−0.0317765
$931$	−36453.7	−1.28327
$932$	−19365.9	−0.680633
$933$	−9787.64	−0.343444
$934$	57877.8	2.02764
$935$	0	0
$936$	4624.26	0.161484
$937$	20418.1	0.711879	0.355939	−	0.934509i	$-0.384161\pi$
$937$	20418.1	0.711879	0.355939	+	0.934509i	$0.384161\pi$
$938$	108380.	3.77263
$939$	−7923.01	−0.275354
$940$	22201.4	0.770352
$941$	8165.00	0.282860	0.141430	−	0.989948i	$-0.454830\pi$
$941$	8165.00	0.282860	0.141430	+	0.989948i	$0.454830\pi$
$942$	36679.3	1.26866
$943$	−5055.71	−0.174588
$944$	5586.11	0.192598
$945$	−3522.40	−0.121253
$946$	0	0
$947$	−94.4852	−0.00324219	−0.00162110	−	0.999999i	$-0.500516\pi$
$947$	−94.4852	−0.00324219	−0.00162110	+	0.999999i	$0.500516\pi$
$948$	−48267.3	−1.65364
$949$	−13734.7	−0.469808
$950$	13084.4	0.446857
$951$	−33177.0	−1.13127
$952$	40151.3	1.36692
$953$	7662.52	0.260455	0.130227	−	0.991484i	$-0.458429\pi$
$953$	7662.52	0.260455	0.130227	+	0.991484i	$0.458429\pi$
$954$	15417.5	0.523227
$955$	6701.89	0.227087
$956$	−62852.5	−2.12636
$957$	0	0
$958$	−28740.8	−0.969284
$959$	−64854.7	−2.18380
$960$	−8954.82	−0.301058
$961$	−29637.5	−0.994848
$962$	5645.57	0.189210
$963$	−4603.21	−0.154036
$964$	51822.5	1.73142
$965$	−17707.7	−0.590705
$966$	6205.88	0.206699
$967$	−25806.8	−0.858213	−0.429107	−	0.903254i	$-0.641172\pi$
$967$	−25806.8	−0.858213	−0.429107	+	0.903254i	$0.641172\pi$
$968$	0	0
$969$	−13662.1	−0.452931
$970$	29108.7	0.963529
$971$	−27874.5	−0.921252	−0.460626	−	0.887594i	$-0.652375\pi$
$971$	−27874.5	−0.921252	−0.460626	+	0.887594i	$0.652375\pi$
$972$	3771.22	0.124447
$973$	7703.45	0.253814
$974$	−12706.0	−0.417995
$975$	−1056.72	−0.0347100
$976$	46623.5	1.52908
$977$	−27461.7	−0.899262	−0.449631	−	0.893214i	$-0.648445\pi$
$977$	−27461.7	−0.899262	−0.449631	+	0.893214i	$0.648445\pi$
$978$	−15560.2	−0.508752
$979$	0	0
$980$	26211.2	0.854373
$981$	−13110.6	−0.426697
$982$	−69016.3	−2.24277
$983$	33356.9	1.08232	0.541160	−	0.840920i	$-0.317985\pi$
$983$	33356.9	1.08232	0.541160	+	0.840920i	$0.317985\pi$
$984$	33832.9	1.09609
$985$	−14829.7	−0.479708
$986$	−31469.4	−1.01642
$987$	−22395.5	−0.722247
$988$	23598.1	0.759875
$989$	6078.26	0.195427
$990$	0	0
$991$	13419.0	0.430140	0.215070	−	0.976599i	$-0.431002\pi$
$991$	13419.0	0.430140	0.215070	+	0.976599i	$0.431002\pi$
$992$	448.087	0.0143415
$993$	6700.40	0.214130
$994$	4618.28	0.147367
$995$	7122.36	0.226929
$996$	−25731.0	−0.818591
$997$	−27160.5	−0.862771	−0.431385	−	0.902168i	$-0.641975\pi$
$997$	−27160.5	−0.862771	−0.431385	+	0.902168i	$0.641975\pi$
$998$	−3549.57	−0.112585
$999$	2230.78	0.0706495

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1815.4.a.bh.1.2		12
11.2	odd	10		165.4.m.a.136.1	yes	24
11.6	odd	10		165.4.m.a.91.1	✓	24
11.10	odd	2		1815.4.a.bn.1.11		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
165.4.m.a.91.1	✓	24	11.6	odd	10
165.4.m.a.136.1	yes	24	11.2	odd	10
1815.4.a.bh.1.2		12	1.1	even	1	trivial
1815.4.a.bn.1.11		12	11.10	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:	\( N \)	\(=\)	\( 1815 = 3 \cdot 5 \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1815.a (trivial)

\(f(q)\)	\(=\)	\(q-4.84968 q^{2} +3.00000 q^{3} +15.5194 q^{4} +5.00000 q^{5} -14.5491 q^{6} -26.0919 q^{7} -36.4669 q^{8} +9.00000 q^{9} +O(q^{10})\)	\(q-4.84968 q^{2} +3.00000 q^{3} +15.5194 q^{4} +5.00000 q^{5} -14.5491 q^{6} -26.0919 q^{7} -36.4669 q^{8} +9.00000 q^{9} -24.2484 q^{10} +46.5583 q^{12} -14.0897 q^{13} +126.537 q^{14} +15.0000 q^{15} +52.6975 q^{16} +42.1984 q^{17} -43.6472 q^{18} -107.920 q^{19} +77.5972 q^{20} -78.2756 q^{21} +16.3480 q^{23} -109.401 q^{24} +25.0000 q^{25} +68.3304 q^{26} +27.0000 q^{27} -404.931 q^{28} +153.773 q^{29} -72.7453 q^{30} +12.3887 q^{31} +36.1690 q^{32} -204.649 q^{34} -130.459 q^{35} +139.675 q^{36} +82.6216 q^{37} +523.377 q^{38} -42.2690 q^{39} -182.335 q^{40} -309.256 q^{41} +379.612 q^{42} +371.805 q^{43} +45.0000 q^{45} -79.2825 q^{46} +286.111 q^{47} +158.093 q^{48} +337.785 q^{49} -121.242 q^{50} +126.595 q^{51} -218.664 q^{52} -353.229 q^{53} -130.941 q^{54} +951.490 q^{56} -323.759 q^{57} -745.749 q^{58} +106.003 q^{59} +232.792 q^{60} +884.737 q^{61} -60.0813 q^{62} -234.827 q^{63} -596.988 q^{64} -70.4483 q^{65} +856.505 q^{67} +654.895 q^{68} +49.0439 q^{69} +632.686 q^{70} +36.4974 q^{71} -328.202 q^{72} +974.807 q^{73} -400.689 q^{74} +75.0000 q^{75} -1674.85 q^{76} +204.991 q^{78} -1036.71 q^{79} +263.488 q^{80} +81.0000 q^{81} +1499.80 q^{82} -552.661 q^{83} -1214.79 q^{84} +210.992 q^{85} -1803.14 q^{86} +461.318 q^{87} -1187.57 q^{89} -218.236 q^{90} +367.626 q^{91} +253.711 q^{92} +37.1661 q^{93} -1387.55 q^{94} -539.599 q^{95} +108.507 q^{96} -1200.44 q^{97} -1638.15 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 7 q^{2} + 36 q^{3} + 49 q^{4} + 60 q^{5} - 21 q^{6} - 77 q^{7} - 111 q^{8} + 108 q^{9}+O(q^{10}) \) 12 * q - 7 * q^2 + 36 * q^3 + 49 * q^4 + 60 * q^5 - 21 * q^6 - 77 * q^7 - 111 * q^8 + 108 * q^9	\( 12 q - 7 q^{2} + 36 q^{3} + 49 q^{4} + 60 q^{5} - 21 q^{6} - 77 q^{7} - 111 q^{8} + 108 q^{9} - 35 q^{10} + 147 q^{12} - 172 q^{13} - 30 q^{14} + 180 q^{15} + 161 q^{16} - 317 q^{17} - 63 q^{18} - 237 q^{19} + 245 q^{20} - 231 q^{21} + 210 q^{23} - 333 q^{24} + 300 q^{25} + 8 q^{26} + 324 q^{27} - 542 q^{28} - 759 q^{29} - 105 q^{30} - 193 q^{31} - 410 q^{32} - 78 q^{34} - 385 q^{35} + 441 q^{36} + 286 q^{37} - 168 q^{38} - 516 q^{39} - 555 q^{40} - 1189 q^{41} - 90 q^{42} - 775 q^{43} + 540 q^{45} - 529 q^{46} + 382 q^{47} + 483 q^{48} + 1195 q^{49} - 175 q^{50} - 951 q^{51} - 1741 q^{52} + 275 q^{53} - 189 q^{54} - 419 q^{56} - 711 q^{57} - 418 q^{58} + 646 q^{59} + 735 q^{60} - 1340 q^{61} - 983 q^{62} - 693 q^{63} - 1489 q^{64} - 860 q^{65} - 185 q^{67} - 3322 q^{68} + 630 q^{69} - 150 q^{70} - 932 q^{71} - 999 q^{72} - 2860 q^{73} - 4187 q^{74} + 900 q^{75} - 1594 q^{76} + 24 q^{78} - 1429 q^{79} + 805 q^{80} + 972 q^{81} - 30 q^{82} - 2590 q^{83} - 1626 q^{84} - 1585 q^{85} - 3195 q^{86} - 2277 q^{87} - 473 q^{89} - 315 q^{90} - 4302 q^{91} + 5462 q^{92} - 579 q^{93} - 2875 q^{94} - 1185 q^{95} - 1230 q^{96} + 318 q^{97} + 194 q^{98}+O(q^{100}) \) 12 * q - 7 * q^2 + 36 * q^3 + 49 * q^4 + 60 * q^5 - 21 * q^6 - 77 * q^7 - 111 * q^8 + 108 * q^9 - 35 * q^10 + 147 * q^12 - 172 * q^13 - 30 * q^14 + 180 * q^15 + 161 * q^16 - 317 * q^17 - 63 * q^18 - 237 * q^19 + 245 * q^20 - 231 * q^21 + 210 * q^23 - 333 * q^24 + 300 * q^25 + 8 * q^26 + 324 * q^27 - 542 * q^28 - 759 * q^29 - 105 * q^30 - 193 * q^31 - 410 * q^32 - 78 * q^34 - 385 * q^35 + 441 * q^36 + 286 * q^37 - 168 * q^38 - 516 * q^39 - 555 * q^40 - 1189 * q^41 - 90 * q^42 - 775 * q^43 + 540 * q^45 - 529 * q^46 + 382 * q^47 + 483 * q^48 + 1195 * q^49 - 175 * q^50 - 951 * q^51 - 1741 * q^52 + 275 * q^53 - 189 * q^54 - 419 * q^56 - 711 * q^57 - 418 * q^58 + 646 * q^59 + 735 * q^60 - 1340 * q^61 - 983 * q^62 - 693 * q^63 - 1489 * q^64 - 860 * q^65 - 185 * q^67 - 3322 * q^68 + 630 * q^69 - 150 * q^70 - 932 * q^71 - 999 * q^72 - 2860 * q^73 - 4187 * q^74 + 900 * q^75 - 1594 * q^76 + 24 * q^78 - 1429 * q^79 + 805 * q^80 + 972 * q^81 - 30 * q^82 - 2590 * q^83 - 1626 * q^84 - 1585 * q^85 - 3195 * q^86 - 2277 * q^87 - 473 * q^89 - 315 * q^90 - 4302 * q^91 + 5462 * q^92 - 579 * q^93 - 2875 * q^94 - 1185 * q^95 - 1230 * q^96 + 318 * q^97 + 194 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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