LMFDB - Embedded newform 1815.4.a.n.1.1

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:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{17}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 4 $ x^2 - x - 4
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 165)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.56155$ 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)

$f(q)$	$=$	$q-1.56155 q^{2} +3.00000 q^{3} -5.56155 q^{4} -5.00000 q^{5} -4.68466 q^{6} +10.2462 q^{7} +21.1771 q^{8} +9.00000 q^{9} +O(q^{10})$	\(q-1.56155 q^{2} +3.00000 q^{3} -5.56155 q^{4} -5.00000 q^{5} -4.68466 q^{6} +10.2462 q^{7} +21.1771 q^{8} +9.00000 q^{9} +7.80776 q^{10} -16.6847 q^{12} +40.8769 q^{13} -16.0000 q^{14} -15.0000 q^{15} +11.4233 q^{16} +98.7083 q^{17} -14.0540 q^{18} +39.6458 q^{19} +27.8078 q^{20} +30.7386 q^{21} +61.6932 q^{23} +63.5312 q^{24} +25.0000 q^{25} -63.8314 q^{26} +27.0000 q^{27} -56.9848 q^{28} +149.093 q^{29} +23.4233 q^{30} +54.7386 q^{31} -187.255 q^{32} -154.138 q^{34} -51.2311 q^{35} -50.0540 q^{36} +44.8939 q^{37} -61.9091 q^{38} +122.631 q^{39} -105.885 q^{40} -336.479 q^{41} -48.0000 q^{42} +2.36745 q^{43} -45.0000 q^{45} -96.3371 q^{46} -333.295 q^{47} +34.2699 q^{48} -238.015 q^{49} -39.0388 q^{50} +296.125 q^{51} -227.339 q^{52} +640.064 q^{53} -42.1619 q^{54} +216.985 q^{56} +118.938 q^{57} -232.816 q^{58} -370.773 q^{59} +83.4233 q^{60} +714.405 q^{61} -85.4773 q^{62} +92.2159 q^{63} +201.022 q^{64} -204.384 q^{65} -404.985 q^{67} -548.972 q^{68} +185.080 q^{69} +80.0000 q^{70} +939.292 q^{71} +190.594 q^{72} +362.570 q^{73} -70.1042 q^{74} +75.0000 q^{75} -220.492 q^{76} -191.494 q^{78} -951.835 q^{79} -57.1165 q^{80} +81.0000 q^{81} +525.430 q^{82} -735.221 q^{83} -170.955 q^{84} -493.542 q^{85} -3.69690 q^{86} +447.278 q^{87} +385.879 q^{89} +70.2699 q^{90} +418.833 q^{91} -343.110 q^{92} +164.216 q^{93} +520.458 q^{94} -198.229 q^{95} -561.764 q^{96} -966.345 q^{97} +371.673 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + q^{2} + 6 q^{3} - 7 q^{4} - 10 q^{5} + 3 q^{6} + 4 q^{7} - 3 q^{8} + 18 q^{9}+O(q^{10}) $ 2 * q + q^2 + 6 * q^3 - 7 * q^4 - 10 * q^5 + 3 * q^6 + 4 * q^7 - 3 * q^8 + 18 * q^9	\( 2 q + q^{2} + 6 q^{3} - 7 q^{4} - 10 q^{5} + 3 q^{6} + 4 q^{7} - 3 q^{8} + 18 q^{9} - 5 q^{10} - 21 q^{12} + 90 q^{13} - 32 q^{14} - 30 q^{15} - 39 q^{16} + 16 q^{17} + 9 q^{18} + 170 q^{19} + 35 q^{20} + 12 q^{21} - 124 q^{23} - 9 q^{24} + 50 q^{25} + 62 q^{26} + 54 q^{27} - 48 q^{28} + 158 q^{29} - 15 q^{30} + 60 q^{31} - 123 q^{32} - 366 q^{34} - 20 q^{35} - 63 q^{36} - 372 q^{37} + 272 q^{38} + 270 q^{39} + 15 q^{40} - 38 q^{41} - 96 q^{42} + 516 q^{43} - 90 q^{45} - 572 q^{46} + 224 q^{47} - 117 q^{48} - 542 q^{49} + 25 q^{50} + 48 q^{51} - 298 q^{52} + 472 q^{53} + 27 q^{54} + 368 q^{56} + 510 q^{57} - 210 q^{58} + 248 q^{59} + 105 q^{60} - 72 q^{61} - 72 q^{62} + 36 q^{63} + 769 q^{64} - 450 q^{65} - 744 q^{67} - 430 q^{68} - 372 q^{69} + 160 q^{70} + 2060 q^{71} - 27 q^{72} + 486 q^{73} - 1138 q^{74} + 150 q^{75} - 408 q^{76} + 186 q^{78} - 642 q^{79} + 195 q^{80} + 162 q^{81} + 1290 q^{82} + 286 q^{83} - 144 q^{84} - 80 q^{85} + 1312 q^{86} + 474 q^{87} + 244 q^{89} - 45 q^{90} + 112 q^{91} - 76 q^{92} + 180 q^{93} + 1948 q^{94} - 850 q^{95} - 369 q^{96} - 168 q^{97} - 407 q^{98}+O(q^{100}) \) 2 * q + q^2 + 6 * q^3 - 7 * q^4 - 10 * q^5 + 3 * q^6 + 4 * q^7 - 3 * q^8 + 18 * q^9 - 5 * q^10 - 21 * q^12 + 90 * q^13 - 32 * q^14 - 30 * q^15 - 39 * q^16 + 16 * q^17 + 9 * q^18 + 170 * q^19 + 35 * q^20 + 12 * q^21 - 124 * q^23 - 9 * q^24 + 50 * q^25 + 62 * q^26 + 54 * q^27 - 48 * q^28 + 158 * q^29 - 15 * q^30 + 60 * q^31 - 123 * q^32 - 366 * q^34 - 20 * q^35 - 63 * q^36 - 372 * q^37 + 272 * q^38 + 270 * q^39 + 15 * q^40 - 38 * q^41 - 96 * q^42 + 516 * q^43 - 90 * q^45 - 572 * q^46 + 224 * q^47 - 117 * q^48 - 542 * q^49 + 25 * q^50 + 48 * q^51 - 298 * q^52 + 472 * q^53 + 27 * q^54 + 368 * q^56 + 510 * q^57 - 210 * q^58 + 248 * q^59 + 105 * q^60 - 72 * q^61 - 72 * q^62 + 36 * q^63 + 769 * q^64 - 450 * q^65 - 744 * q^67 - 430 * q^68 - 372 * q^69 + 160 * q^70 + 2060 * q^71 - 27 * q^72 + 486 * q^73 - 1138 * q^74 + 150 * q^75 - 408 * q^76 + 186 * q^78 - 642 * q^79 + 195 * q^80 + 162 * q^81 + 1290 * q^82 + 286 * q^83 - 144 * q^84 - 80 * q^85 + 1312 * q^86 + 474 * q^87 + 244 * q^89 - 45 * q^90 + 112 * q^91 - 76 * q^92 + 180 * q^93 + 1948 * q^94 - 850 * q^95 - 369 * q^96 - 168 * q^97 - 407 * 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 $

$2$	−1.56155	−0.552092	−0.276046	−	0.961144i	$-0.589024\pi$
$2$	−1.56155	−0.552092	−0.276046	+	0.961144i	$0.589024\pi$
$3$	3.00000	0.577350
$3$	3.00000	0.577350
$4$	−5.56155	−0.695194
$5$	−5.00000	−0.447214
$5$	−5.00000	−0.447214
$6$	−4.68466	−0.318751
$7$	10.2462	0.553243	0.276622	−	0.960979i	$-0.410785\pi$
$7$	10.2462	0.553243	0.276622	+	0.960979i	$0.410785\pi$
$8$	21.1771	0.935904
$9$	9.00000	0.333333
$10$	7.80776	0.246903
$11$	0	0
$11$	0	0
$12$	−16.6847	−0.401371
$13$	40.8769	0.872093	0.436047	−	0.899924i	$-0.356378\pi$
$13$	40.8769	0.872093	0.436047	+	0.899924i	$0.356378\pi$
$14$	−16.0000	−0.305441
$15$	−15.0000	−0.258199
$16$	11.4233	0.178489
$17$	98.7083	1.40825	0.704126	−	0.710075i	$-0.251339\pi$
$17$	98.7083	1.40825	0.704126	+	0.710075i	$0.251339\pi$
$18$	−14.0540	−0.184031
$19$	39.6458	0.478704	0.239352	−	0.970933i	$-0.423065\pi$
$19$	39.6458	0.478704	0.239352	+	0.970933i	$0.423065\pi$
$20$	27.8078	0.310900
$21$	30.7386	0.319415
$22$	0	0
$23$	61.6932	0.559301	0.279650	−	0.960102i	$-0.409781\pi$
$23$	61.6932	0.559301	0.279650	+	0.960102i	$0.409781\pi$
$24$	63.5312	0.540344
$25$	25.0000	0.200000
$26$	−63.8314	−0.481476
$27$	27.0000	0.192450
$28$	−56.9848	−0.384612
$29$	149.093	0.954684	0.477342	−	0.878718i	$-0.341600\pi$
$29$	149.093	0.954684	0.477342	+	0.878718i	$0.341600\pi$
$30$	23.4233	0.142550
$31$	54.7386	0.317140	0.158570	−	0.987348i	$-0.449312\pi$
$31$	54.7386	0.317140	0.158570	+	0.987348i	$0.449312\pi$
$32$	−187.255	−1.03445
$33$	0	0
$34$	−154.138	−0.777485
$35$	−51.2311	−0.247418
$36$	−50.0540	−0.231731
$37$	44.8939	0.199473	0.0997367	−	0.995014i	$-0.468200\pi$
$37$	44.8939	0.199473	0.0997367	+	0.995014i	$0.468200\pi$
$38$	−61.9091	−0.264289
$39$	122.631	0.503503
$40$	−105.885	−0.418549
$41$	−336.479	−1.28169	−0.640844	−	0.767671i	$-0.721415\pi$
$41$	−336.479	−1.28169	−0.640844	+	0.767671i	$0.721415\pi$
$42$	−48.0000	−0.176347
$43$	2.36745	0.00839611	0.00419806	−	0.999991i	$-0.498664\pi$
$43$	2.36745	0.00839611	0.00419806	+	0.999991i	$0.498664\pi$
$44$	0	0
$45$	−45.0000	−0.149071
$46$	−96.3371	−0.308786
$47$	−333.295	−1.03439	−0.517193	−	0.855869i	$-0.673023\pi$
$47$	−333.295	−1.03439	−0.517193	+	0.855869i	$0.673023\pi$
$48$	34.2699	0.103051
$49$	−238.015	−0.693922
$50$	−39.0388	−0.110418
$51$	296.125	0.813055
$52$	−227.339	−0.606274
$53$	640.064	1.65886	0.829430	−	0.558610i	$-0.188665\pi$
$53$	640.064	1.65886	0.829430	+	0.558610i	$0.188665\pi$
$54$	−42.1619	−0.106250
$55$	0	0
$56$	216.985	0.517782
$57$	118.938	0.276380
$58$	−232.816	−0.527074
$59$	−370.773	−0.818144	−0.409072	−	0.912502i	$-0.634147\pi$
$59$	−370.773	−0.818144	−0.409072	+	0.912502i	$0.634147\pi$
$60$	83.4233	0.179498
$61$	714.405	1.49951	0.749756	−	0.661715i	$-0.230171\pi$
$61$	714.405	1.49951	0.749756	+	0.661715i	$0.230171\pi$
$62$	−85.4773	−0.175091
$63$	92.2159	0.184414
$64$	201.022	0.392621
$65$	−204.384	−0.390012
$66$	0	0
$67$	−404.985	−0.738459	−0.369230	−	0.929338i	$-0.620378\pi$
$67$	−404.985	−0.738459	−0.369230	+	0.929338i	$0.620378\pi$
$68$	−548.972	−0.979009
$69$	185.080	0.322912
$70$	80.0000	0.136598
$71$	939.292	1.57005	0.785024	−	0.619465i	$-0.212651\pi$
$71$	939.292	1.57005	0.785024	+	0.619465i	$0.212651\pi$
$72$	190.594	0.311968
$73$	362.570	0.581310	0.290655	−	0.956828i	$-0.406127\pi$
$73$	362.570	0.581310	0.290655	+	0.956828i	$0.406127\pi$
$74$	−70.1042	−0.110128
$75$	75.0000	0.115470
$76$	−220.492	−0.332792
$77$	0	0
$78$	−191.494	−0.277980
$79$	−951.835	−1.35557	−0.677784	−	0.735261i	$-0.737059\pi$
$79$	−951.835	−1.35557	−0.677784	+	0.735261i	$0.737059\pi$
$80$	−57.1165	−0.0798227
$81$	81.0000	0.111111
$82$	525.430	0.707610
$83$	−735.221	−0.972302	−0.486151	−	0.873875i	$-0.661599\pi$
$83$	−735.221	−0.972302	−0.486151	+	0.873875i	$0.661599\pi$
$84$	−170.955	−0.222056
$85$	−493.542	−0.629789
$86$	−3.69690	−0.00463543
$87$	447.278	0.551187
$88$	0	0
$89$	385.879	0.459585	0.229793	−	0.973240i	$-0.426195\pi$
$89$	385.879	0.459585	0.229793	+	0.973240i	$0.426195\pi$
$90$	70.2699	0.0823011
$91$	418.833	0.482480
$92$	−343.110	−0.388823
$93$	164.216	0.183101
$94$	520.458	0.571076
$95$	−198.229	−0.214083
$96$	−561.764	−0.597238
$97$	−966.345	−1.01152	−0.505760	−	0.862674i	$-0.668788\pi$
$97$	−966.345	−1.01152	−0.505760	+	0.862674i	$0.668788\pi$
$98$	371.673	0.383109
$99$	0	0
$100$	−139.039	−0.139039
$101$	−348.600	−0.343436	−0.171718	−	0.985146i	$-0.554932\pi$
$101$	−348.600	−0.343436	−0.171718	+	0.985146i	$0.554932\pi$
$102$	−462.415	−0.448881
$103$	−1536.38	−1.46975	−0.734873	−	0.678204i	$-0.762758\pi$
$103$	−1536.38	−1.46975	−0.734873	+	0.678204i	$0.762758\pi$
$104$	865.653	0.816195
$105$	−153.693	−0.142847
$106$	−999.494	−0.915844
$107$	779.180	0.703983	0.351991	−	0.936003i	$-0.385505\pi$
$107$	779.180	0.703983	0.351991	+	0.936003i	$0.385505\pi$
$108$	−150.162	−0.133790
$109$	1501.79	1.31968	0.659842	−	0.751404i	$-0.270623\pi$
$109$	1501.79	1.31968	0.659842	+	0.751404i	$0.270623\pi$
$110$	0	0
$111$	134.682	0.115166
$112$	117.045	0.0987478
$113$	170.000	0.141524	0.0707622	−	0.997493i	$-0.477457\pi$
$113$	170.000	0.141524	0.0707622	+	0.997493i	$0.477457\pi$
$114$	−185.727	−0.152587
$115$	−308.466	−0.250127
$116$	−829.187	−0.663691
$117$	367.892	0.290698
$118$	578.981	0.451691
$119$	1011.39	0.779106
$120$	−317.656	−0.241649
$121$	0	0
$122$	−1115.58	−0.827869
$123$	−1009.44	−0.739983
$124$	−304.432	−0.220474
$125$	−125.000	−0.0894427
$126$	−144.000	−0.101814
$127$	1739.82	1.21562	0.607811	−	0.794082i	$-0.292048\pi$
$127$	1739.82	1.21562	0.607811	+	0.794082i	$0.292048\pi$
$128$	1184.13	0.817683
$129$	7.10235	0.00484750
$130$	319.157	0.215323
$131$	−312.837	−0.208647	−0.104323	−	0.994543i	$-0.533268\pi$
$131$	−312.837	−0.208647	−0.104323	+	0.994543i	$0.533268\pi$
$132$	0	0
$133$	406.220	0.264840
$134$	632.405	0.407698
$135$	−135.000	−0.0860663
$136$	2090.35	1.31799
$137$	−716.928	−0.447090	−0.223545	−	0.974694i	$-0.571763\pi$
$137$	−716.928	−0.447090	−0.223545	+	0.974694i	$0.571763\pi$
$138$	−289.011	−0.178277
$139$	876.483	0.534837	0.267418	−	0.963581i	$-0.413829\pi$
$139$	876.483	0.534837	0.267418	+	0.963581i	$0.413829\pi$
$140$	284.924	0.172004
$141$	−999.886	−0.597203
$142$	−1466.75	−0.866811
$143$	0	0
$144$	102.810	0.0594963
$145$	−745.464	−0.426948
$146$	−566.172	−0.320937
$147$	−714.045	−0.400636
$148$	−249.680	−0.138673
$149$	2376.36	1.30657	0.653285	−	0.757112i	$-0.273390\pi$
$149$	2376.36	1.30657	0.653285	+	0.757112i	$0.273390\pi$
$150$	−117.116	−0.0637501
$151$	92.8466	0.0500381	0.0250190	−	0.999687i	$-0.492035\pi$
$151$	92.8466	0.0500381	0.0250190	+	0.999687i	$0.492035\pi$
$152$	839.583	0.448021
$153$	888.375	0.469417
$154$	0	0
$155$	−273.693	−0.141829
$156$	−682.017	−0.350032
$157$	−1881.24	−0.956301	−0.478150	−	0.878278i	$-0.658693\pi$
$157$	−1881.24	−0.956301	−0.478150	+	0.878278i	$0.658693\pi$
$158$	1486.34	0.748398
$159$	1920.19	0.957743
$160$	936.274	0.462618
$161$	632.121	0.309429
$162$	−126.486	−0.0613436
$163$	−2465.49	−1.18474	−0.592369	−	0.805667i	$-0.701807\pi$
$163$	−2465.49	−1.18474	−0.592369	+	0.805667i	$0.701807\pi$
$164$	1871.35	0.891022
$165$	0	0
$166$	1148.09	0.536800
$167$	1254.30	0.581200	0.290600	−	0.956845i	$-0.406145\pi$
$167$	1254.30	0.581200	0.290600	+	0.956845i	$0.406145\pi$
$168$	650.955	0.298942
$169$	−526.080	−0.239454
$170$	770.691	0.347702
$171$	356.813	0.159568
$172$	−13.1667	−0.00583693
$173$	1206.71	0.530314	0.265157	−	0.964205i	$-0.414576\pi$
$173$	1206.71	0.530314	0.265157	+	0.964205i	$0.414576\pi$
$174$	−698.449	−0.304306
$175$	256.155	0.110649
$176$	0	0
$177$	−1112.32	−0.472356
$178$	−602.570	−0.253733
$179$	−1442.29	−0.602244	−0.301122	−	0.953586i	$-0.597361\pi$
$179$	−1442.29	−0.602244	−0.301122	+	0.953586i	$0.597361\pi$
$180$	250.270	0.103633
$181$	4261.81	1.75015	0.875076	−	0.483985i	$-0.160811\pi$
$181$	4261.81	1.75015	0.875076	+	0.483985i	$0.160811\pi$
$182$	−654.030	−0.266373
$183$	2143.22	0.865744
$184$	1306.48	0.523451
$185$	−224.470	−0.0892072
$186$	−256.432	−0.101089
$187$	0	0
$188$	1853.64	0.719099
$189$	276.648	0.106472
$190$	309.545	0.118194
$191$	−852.223	−0.322852	−0.161426	−	0.986885i	$-0.551609\pi$
$191$	−852.223	−0.322852	−0.161426	+	0.986885i	$0.551609\pi$
$192$	603.065	0.226680
$193$	2459.95	0.917468	0.458734	−	0.888574i	$-0.348303\pi$
$193$	2459.95	0.917468	0.458734	+	0.888574i	$0.348303\pi$
$194$	1509.00	0.558452
$195$	−613.153	−0.225173
$196$	1323.73	0.482410
$197$	−3477.06	−1.25751	−0.628756	−	0.777602i	$-0.716436\pi$
$197$	−3477.06	−1.25751	−0.628756	+	0.777602i	$0.716436\pi$
$198$	0	0
$199$	3995.04	1.42312	0.711560	−	0.702626i	$-0.247989\pi$
$199$	3995.04	1.42312	0.711560	+	0.702626i	$0.247989\pi$
$200$	529.427	0.187181
$201$	−1214.95	−0.426350
$202$	544.358	0.189608
$203$	1527.64	0.528173
$204$	−1646.91	−0.565231
$205$	1682.40	0.573188
$206$	2399.14	0.811436
$207$	555.239	0.186434
$208$	466.949	0.155659
$209$	0	0
$210$	240.000	0.0788646
$211$	−1046.13	−0.341319	−0.170660	−	0.985330i	$-0.554590\pi$
$211$	−1046.13	−0.341319	−0.170660	+	0.985330i	$0.554590\pi$
$212$	−3559.75	−1.15323
$213$	2817.88	0.906468
$214$	−1216.73	−0.388664
$215$	−11.8373	−0.00375486
$216$	571.781	0.180115
$217$	560.864	0.175456
$218$	−2345.13	−0.728587
$219$	1087.71	0.335619
$220$	0	0
$221$	4034.89	1.22813
$222$	−210.313	−0.0635823
$223$	−506.265	−0.152027	−0.0760135	−	0.997107i	$-0.524219\pi$
$223$	−506.265	−0.152027	−0.0760135	+	0.997107i	$0.524219\pi$
$224$	−1918.65	−0.572300
$225$	225.000	0.0666667
$226$	−265.464	−0.0781345
$227$	4286.29	1.25326	0.626632	−	0.779315i	$-0.284433\pi$
$227$	4286.29	1.25326	0.626632	+	0.779315i	$0.284433\pi$
$228$	−661.477	−0.192138
$229$	5709.37	1.64754	0.823769	−	0.566926i	$-0.191867\pi$
$229$	5709.37	1.64754	0.823769	+	0.566926i	$0.191867\pi$
$230$	481.686	0.138093
$231$	0	0
$232$	3157.35	0.893492
$233$	−2946.09	−0.828348	−0.414174	−	0.910198i	$-0.635929\pi$
$233$	−2946.09	−0.828348	−0.414174	+	0.910198i	$0.635929\pi$
$234$	−574.483	−0.160492
$235$	1666.48	0.462591
$236$	2062.07	0.568769
$237$	−2855.51	−0.782637
$238$	−1579.33	−0.430139
$239$	2078.89	0.562646	0.281323	−	0.959613i	$-0.409227\pi$
$239$	2078.89	0.562646	0.281323	+	0.959613i	$0.409227\pi$
$240$	−171.349	−0.0460856
$241$	−1853.37	−0.495378	−0.247689	−	0.968840i	$-0.579671\pi$
$241$	−1853.37	−0.495378	−0.247689	+	0.968840i	$0.579671\pi$
$242$	0	0
$243$	243.000	0.0641500
$244$	−3973.20	−1.04245
$245$	1190.08	0.310331
$246$	1576.29	0.408539
$247$	1620.60	0.417475
$248$	1159.20	0.296813
$249$	−2205.66	−0.561359
$250$	195.194	0.0493806
$251$	−2358.39	−0.593068	−0.296534	−	0.955022i	$-0.595831\pi$
$251$	−2358.39	−0.593068	−0.296534	+	0.955022i	$0.595831\pi$
$252$	−512.864	−0.128204
$253$	0	0
$254$	−2716.82	−0.671135
$255$	−1480.62	−0.363609
$256$	−3457.26	−0.844057
$257$	5519.25	1.33962	0.669809	−	0.742534i	$-0.266376\pi$
$257$	5519.25	1.33962	0.669809	+	0.742534i	$0.266376\pi$
$258$	−11.0907	−0.00267627
$259$	459.993	0.110357
$260$	1136.70	0.271134
$261$	1341.84	0.318228
$262$	488.512	0.115192
$263$	−2259.65	−0.529795	−0.264898	−	0.964277i	$-0.585338\pi$
$263$	−2259.65	−0.529795	−0.264898	+	0.964277i	$0.585338\pi$
$264$	0	0
$265$	−3200.32	−0.741865
$266$	−634.333	−0.146216
$267$	1157.64	0.265342
$268$	2252.34	0.513373
$269$	7039.53	1.59557	0.797783	−	0.602944i	$-0.206006\pi$
$269$	7039.53	1.59557	0.797783	+	0.602944i	$0.206006\pi$
$270$	210.810	0.0475165
$271$	−5155.08	−1.15553	−0.577765	−	0.816203i	$-0.696075\pi$
$271$	−5155.08	−1.15553	−0.577765	+	0.816203i	$0.696075\pi$
$272$	1127.57	0.251357
$273$	1256.50	0.278560
$274$	1119.52	0.246835
$275$	0	0
$276$	−1029.33	−0.224487
$277$	−9074.52	−1.96836	−0.984179	−	0.177175i	$-0.943304\pi$
$277$	−9074.52	−1.96836	−0.984179	+	0.177175i	$0.943304\pi$
$278$	−1368.67	−0.295279
$279$	492.648	0.105713
$280$	−1084.92	−0.231559
$281$	3407.79	0.723459	0.361729	−	0.932283i	$-0.382186\pi$
$281$	3407.79	0.723459	0.361729	+	0.932283i	$0.382186\pi$
$282$	1561.38	0.329711
$283$	8827.73	1.85425	0.927127	−	0.374746i	$-0.122270\pi$
$283$	8827.73	1.85425	0.927127	+	0.374746i	$0.122270\pi$
$284$	−5223.92	−1.09149
$285$	−594.688	−0.123601
$286$	0	0
$287$	−3447.64	−0.709085
$288$	−1685.29	−0.344815
$289$	4830.33	0.983174
$290$	1164.08	0.235715
$291$	−2899.03	−0.584001
$292$	−2016.45	−0.404123
$293$	4528.29	0.902886	0.451443	−	0.892300i	$-0.350909\pi$
$293$	4528.29	0.902886	0.451443	+	0.892300i	$0.350909\pi$
$294$	1115.02	0.221188
$295$	1853.86	0.365885
$296$	950.722	0.186688
$297$	0	0
$298$	−3710.81	−0.721347
$299$	2521.83	0.487762
$300$	−417.116	−0.0802741
$301$	24.2574	0.00464509
$302$	−144.985	−0.0276256
$303$	−1045.80	−0.198283
$304$	452.886	0.0854434
$305$	−3572.03	−0.670602
$306$	−1387.24	−0.259162
$307$	−568.106	−0.105614	−0.0528071	−	0.998605i	$-0.516817\pi$
$307$	−568.106	−0.105614	−0.0528071	+	0.998605i	$0.516817\pi$
$308$	0	0
$309$	−4609.14	−0.848559
$310$	427.386	0.0783029
$311$	−6853.59	−1.24962	−0.624809	−	0.780778i	$-0.714823\pi$
$311$	−6853.59	−1.24962	−0.624809	+	0.780778i	$0.714823\pi$
$312$	2596.96	0.471230
$313$	−1138.92	−0.205673	−0.102837	−	0.994698i	$-0.532792\pi$
$313$	−1138.92	−0.205673	−0.102837	+	0.994698i	$0.532792\pi$
$314$	2937.65	0.527966
$315$	−461.080	−0.0824727
$316$	5293.68	0.942382
$317$	3207.48	0.568297	0.284148	−	0.958780i	$-0.408289\pi$
$317$	3207.48	0.568297	0.284148	+	0.958780i	$0.408289\pi$
$318$	−2998.48	−0.528763
$319$	0	0
$320$	−1005.11	−0.175585
$321$	2337.54	0.406445
$322$	−987.091	−0.170834
$323$	3913.37	0.674136
$324$	−450.486	−0.0772438
$325$	1021.92	0.174419
$326$	3850.00	0.654085
$327$	4505.37	0.761920
$328$	−7125.65	−1.19954
$329$	−3415.02	−0.572267
$330$	0	0
$331$	−9135.12	−1.51695	−0.758477	−	0.651700i	$-0.774056\pi$
$331$	−9135.12	−1.51695	−0.758477	+	0.651700i	$0.774056\pi$
$332$	4088.97	0.675938
$333$	404.045	0.0664911
$334$	−1958.65	−0.320876
$335$	2024.92	0.330249
$336$	351.136	0.0570121
$337$	3470.05	0.560907	0.280453	−	0.959868i	$-0.409515\pi$
$337$	3470.05	0.560907	0.280453	+	0.959868i	$0.409515\pi$
$338$	821.501	0.132200
$339$	510.000	0.0817091
$340$	2744.86	0.437826
$341$	0	0
$342$	−557.182	−0.0880963
$343$	−5953.20	−0.937151
$344$	50.1357	0.00785795
$345$	−925.398	−0.144411
$346$	−1884.34	−0.292782
$347$	89.3315	0.0138201	0.00691004	−	0.999976i	$-0.497800\pi$
$347$	89.3315	0.0138201	0.00691004	+	0.999976i	$0.497800\pi$
$348$	−2487.56	−0.383182
$349$	149.375	0.0229107	0.0114554	−	0.999934i	$-0.496354\pi$
$349$	149.375	0.0229107	0.0114554	+	0.999934i	$0.496354\pi$
$350$	−400.000	−0.0610883
$351$	1103.68	0.167834
$352$	0	0
$353$	7867.64	1.18627	0.593133	−	0.805104i	$-0.297891\pi$
$353$	7867.64	1.18627	0.593133	+	0.805104i	$0.297891\pi$
$354$	1736.94	0.260784
$355$	−4696.46	−0.702147
$356$	−2146.09	−0.319501
$357$	3034.16	0.449817
$358$	2252.21	0.332494
$359$	−4974.22	−0.731279	−0.365639	−	0.930757i	$-0.619150\pi$
$359$	−4974.22	−0.731279	−0.365639	+	0.930757i	$0.619150\pi$
$360$	−952.969	−0.139516
$361$	−5287.21	−0.770842
$362$	−6655.04	−0.966246
$363$	0	0
$364$	−2329.36	−0.335417
$365$	−1812.85	−0.259970
$366$	−3346.74	−0.477970
$367$	−13266.7	−1.88696	−0.943479	−	0.331433i	$-0.892468\pi$
$367$	−13266.7	−1.88696	−0.943479	+	0.331433i	$0.892468\pi$
$368$	704.739	0.0998290
$369$	−3028.31	−0.427229
$370$	350.521	0.0492506
$371$	6558.23	0.917754
$372$	−913.295	−0.127291
$373$	4632.77	0.643099	0.321549	−	0.946893i	$-0.395796\pi$
$373$	4632.77	0.643099	0.321549	+	0.946893i	$0.395796\pi$
$374$	0	0
$375$	−375.000	−0.0516398
$376$	−7058.22	−0.968085
$377$	6094.45	0.832573
$378$	−432.000	−0.0587822
$379$	6503.31	0.881406	0.440703	−	0.897653i	$-0.354729\pi$
$379$	6503.31	0.881406	0.440703	+	0.897653i	$0.354729\pi$
$380$	1102.46	0.148829
$381$	5219.45	0.701839
$382$	1330.79	0.178244
$383$	12734.5	1.69897	0.849484	−	0.527614i	$-0.176913\pi$
$383$	12734.5	1.69897	0.849484	+	0.527614i	$0.176913\pi$
$384$	3552.39	0.472090
$385$	0	0
$386$	−3841.35	−0.506527
$387$	21.3071	0.00279870
$388$	5374.38	0.703203
$389$	12024.6	1.56728	0.783639	−	0.621216i	$-0.213361\pi$
$389$	12024.6	1.56728	0.783639	+	0.621216i	$0.213361\pi$
$390$	957.471	0.124317
$391$	6089.63	0.787636
$392$	−5040.47	−0.649444
$393$	−938.511	−0.120462
$394$	5429.61	0.694263
$395$	4759.18	0.606228
$396$	0	0
$397$	−5223.65	−0.660371	−0.330186	−	0.943916i	$-0.607111\pi$
$397$	−5223.65	−0.660371	−0.330186	+	0.943916i	$0.607111\pi$
$398$	−6238.46	−0.785693
$399$	1218.66	0.152905
$400$	285.582	0.0356978
$401$	9648.18	1.20151	0.600757	−	0.799432i	$-0.294866\pi$
$401$	9648.18	1.20151	0.600757	+	0.799432i	$0.294866\pi$
$402$	1897.22	0.235384
$403$	2237.55	0.276576
$404$	1938.76	0.238755
$405$	−405.000	−0.0496904
$406$	−2385.48	−0.291600
$407$	0	0
$408$	6271.06	0.760941
$409$	2010.47	0.243060	0.121530	−	0.992588i	$-0.461220\pi$
$409$	2010.47	0.243060	0.121530	+	0.992588i	$0.461220\pi$
$410$	−2627.15	−0.316453
$411$	−2150.78	−0.258127
$412$	8544.65	1.02176
$413$	−3799.02	−0.452633
$414$	−867.034	−0.102929
$415$	3676.11	0.434827
$416$	−7654.39	−0.902133
$417$	2629.45	0.308788
$418$	0	0
$419$	4435.27	0.517129	0.258565	−	0.965994i	$-0.416750\pi$
$419$	4435.27	0.517129	0.258565	+	0.965994i	$0.416750\pi$
$420$	854.773	0.0993063
$421$	15217.9	1.76170	0.880852	−	0.473392i	$-0.156971\pi$
$421$	15217.9	1.76170	0.880852	+	0.473392i	$0.156971\pi$
$422$	1633.58	0.188440
$423$	−2999.66	−0.344795
$424$	13554.7	1.55253
$425$	2467.71	0.281650
$426$	−4400.26	−0.500454
$427$	7319.95	0.829595
$428$	−4333.45	−0.489405
$429$	0	0
$430$	18.4845	0.00207303
$431$	5622.11	0.628324	0.314162	−	0.949369i	$-0.398277\pi$
$431$	5622.11	0.628324	0.314162	+	0.949369i	$0.398277\pi$
$432$	308.429	0.0343502
$433$	−14306.3	−1.58780	−0.793898	−	0.608051i	$-0.791951\pi$
$433$	−14306.3	−1.58780	−0.793898	+	0.608051i	$0.791951\pi$
$434$	−875.818	−0.0968678
$435$	−2236.39	−0.246498
$436$	−8352.29	−0.917436
$437$	2445.88	0.267740
$438$	−1698.52	−0.185293
$439$	−4384.20	−0.476643	−0.238322	−	0.971186i	$-0.576597\pi$
$439$	−4384.20	−0.476643	−0.238322	+	0.971186i	$0.576597\pi$
$440$	0	0
$441$	−2142.14	−0.231307
$442$	−6300.69	−0.678039
$443$	−10090.0	−1.08214	−0.541071	−	0.840977i	$-0.681981\pi$
$443$	−10090.0	−1.08214	−0.541071	+	0.840977i	$0.681981\pi$
$444$	−749.040	−0.0800627
$445$	−1929.39	−0.205533
$446$	790.560	0.0839330
$447$	7129.07	0.754348
$448$	2059.71	0.217215
$449$	9582.52	1.00719	0.503594	−	0.863941i	$-0.332011\pi$
$449$	9582.52	1.00719	0.503594	+	0.863941i	$0.332011\pi$
$450$	−351.349	−0.0368062
$451$	0	0
$452$	−945.464	−0.0983869
$453$	278.540	0.0288895
$454$	−6693.27	−0.691918
$455$	−2094.17	−0.215772
$456$	2518.75	0.258665
$457$	−9999.34	−1.02352	−0.511761	−	0.859128i	$-0.671007\pi$
$457$	−9999.34	−1.02352	−0.511761	+	0.859128i	$0.671007\pi$
$458$	−8915.49	−0.909593
$459$	2665.12	0.271018
$460$	1715.55	0.173887
$461$	11115.8	1.12302	0.561512	−	0.827468i	$-0.310220\pi$
$461$	11115.8	1.12302	0.561512	+	0.827468i	$0.310220\pi$
$462$	0	0
$463$	−1567.16	−0.157305	−0.0786524	−	0.996902i	$-0.525062\pi$
$463$	−1567.16	−0.157305	−0.0786524	+	0.996902i	$0.525062\pi$
$464$	1703.13	0.170401
$465$	−821.080	−0.0818853
$466$	4600.48	0.457325
$467$	12648.8	1.25335	0.626675	−	0.779281i	$-0.284415\pi$
$467$	12648.8	1.25335	0.626675	+	0.779281i	$0.284415\pi$
$468$	−2046.05	−0.202091
$469$	−4149.56	−0.408548
$470$	−2602.29	−0.255393
$471$	−5643.72	−0.552120
$472$	−7851.88	−0.765704
$473$	0	0
$474$	4459.02	0.432088
$475$	991.146	0.0957408
$476$	−5624.88	−0.541630
$477$	5760.58	0.552953
$478$	−3246.30	−0.310633
$479$	10719.2	1.02249	0.511247	−	0.859434i	$-0.329184\pi$
$479$	10719.2	1.02249	0.511247	+	0.859434i	$0.329184\pi$
$480$	2808.82	0.267093
$481$	1835.12	0.173959
$482$	2894.14	0.273494
$483$	1896.36	0.178649
$484$	0	0
$485$	4831.72	0.452365
$486$	−379.457	−0.0354167
$487$	7161.20	0.666335	0.333167	−	0.942868i	$-0.391883\pi$
$487$	7161.20	0.666335	0.333167	+	0.942868i	$0.391883\pi$
$488$	15129.0	1.40340
$489$	−7396.48	−0.684009
$490$	−1858.37	−0.171331
$491$	−14567.3	−1.33893	−0.669463	−	0.742845i	$-0.733476\pi$
$491$	−14567.3	−1.33893	−0.669463	+	0.742845i	$0.733476\pi$
$492$	5614.04	0.514432
$493$	14716.7	1.34444
$494$	−2530.65	−0.230485
$495$	0	0
$496$	625.295	0.0566060
$497$	9624.18	0.868619
$498$	3444.26	0.309922
$499$	−4638.99	−0.416172	−0.208086	−	0.978111i	$-0.566723\pi$
$499$	−4638.99	−0.416172	−0.208086	+	0.978111i	$0.566723\pi$
$500$	695.194	0.0621801
$501$	3762.89	0.335556
$502$	3682.75	0.327428
$503$	12206.3	1.08201	0.541006	−	0.841019i	$-0.318044\pi$
$503$	12206.3	1.08201	0.541006	+	0.841019i	$0.318044\pi$
$504$	1952.86	0.172594
$505$	1743.00	0.153589
$506$	0	0
$507$	−1578.24	−0.138249
$508$	−9676.09	−0.845093
$509$	10018.6	0.872427	0.436214	−	0.899843i	$-0.356319\pi$
$509$	10018.6	0.872427	0.436214	+	0.899843i	$0.356319\pi$
$510$	2312.07	0.200746
$511$	3714.97	0.321606
$512$	−4074.36	−0.351686
$513$	1070.44	0.0921267
$514$	−8618.61	−0.739592
$515$	7681.89	0.657291
$516$	−39.5001	−0.00336995
$517$	0	0
$518$	−718.303	−0.0609274
$519$	3620.12	0.306177
$520$	−4328.27	−0.365014
$521$	1054.72	0.0886916	0.0443458	−	0.999016i	$-0.485880\pi$
$521$	1054.72	0.0886916	0.0443458	+	0.999016i	$0.485880\pi$
$522$	−2095.35	−0.175691
$523$	16234.2	1.35730	0.678652	−	0.734460i	$-0.262564\pi$
$523$	16234.2	1.35730	0.678652	+	0.734460i	$0.262564\pi$
$524$	1739.86	0.145050
$525$	768.466	0.0638830
$526$	3528.57	0.292496
$527$	5403.16	0.446613
$528$	0	0
$529$	−8360.95	−0.687183
$530$	4997.47	0.409578
$531$	−3336.95	−0.272715
$532$	−2259.21	−0.184115
$533$	−13754.2	−1.11775
$534$	−1807.71	−0.146493
$535$	−3895.90	−0.314831
$536$	−8576.40	−0.691127
$537$	−4326.86	−0.347706
$538$	−10992.6	−0.880900
$539$	0	0
$540$	750.810	0.0598328
$541$	−675.936	−0.0537167	−0.0268584	−	0.999639i	$-0.508550\pi$
$541$	−675.936	−0.0537167	−0.0268584	+	0.999639i	$0.508550\pi$
$542$	8049.93	0.637960
$543$	12785.4	1.01045
$544$	−18483.6	−1.45676
$545$	−7508.96	−0.590181
$546$	−1962.09	−0.153791
$547$	−13058.2	−1.02071	−0.510355	−	0.859964i	$-0.670486\pi$
$547$	−13058.2	−1.02071	−0.510355	+	0.859964i	$0.670486\pi$
$548$	3987.23	0.310814
$549$	6429.65	0.499837
$550$	0	0
$551$	5910.91	0.457011
$552$	3919.44	0.302215
$553$	−9752.70	−0.749959
$554$	14170.3	1.08672
$555$	−673.409	−0.0515038
$556$	−4874.61	−0.371815
$557$	6710.48	0.510471	0.255236	−	0.966879i	$-0.417847\pi$
$557$	6710.48	0.510471	0.255236	+	0.966879i	$0.417847\pi$
$558$	−769.295	−0.0583636
$559$	96.7741	0.00732219
$560$	−585.227	−0.0441614
$561$	0	0
$562$	−5321.45	−0.399416
$563$	20820.5	1.55858	0.779288	−	0.626666i	$-0.215581\pi$
$563$	20820.5	1.55858	0.779288	+	0.626666i	$0.215581\pi$
$564$	5560.92	0.415172
$565$	−850.000	−0.0632916
$566$	−13785.0	−1.02372
$567$	829.943	0.0614715
$568$	19891.5	1.46941
$569$	−3251.08	−0.239530	−0.119765	−	0.992802i	$-0.538214\pi$
$569$	−3251.08	−0.239530	−0.119765	+	0.992802i	$0.538214\pi$
$570$	928.636	0.0682391
$571$	4637.50	0.339883	0.169941	−	0.985454i	$-0.445642\pi$
$571$	4637.50	0.339883	0.169941	+	0.985454i	$0.445642\pi$
$572$	0	0
$573$	−2556.67	−0.186399
$574$	5383.67	0.391481
$575$	1542.33	0.111860
$576$	1809.20	0.130874
$577$	14462.4	1.04346	0.521730	−	0.853111i	$-0.325287\pi$
$577$	14462.4	1.04346	0.521730	+	0.853111i	$0.325287\pi$
$578$	−7542.82	−0.542803
$579$	7379.86	0.529700
$580$	4145.94	0.296812
$581$	−7533.23	−0.537920
$582$	4526.99	0.322423
$583$	0	0
$584$	7678.18	0.544050
$585$	−1839.46	−0.130004
$586$	−7071.17	−0.498476
$587$	−22759.7	−1.60033	−0.800166	−	0.599779i	$-0.795255\pi$
$587$	−22759.7	−1.60033	−0.800166	+	0.599779i	$0.795255\pi$
$588$	3971.20	0.278520
$589$	2170.16	0.151816
$590$	−2894.91	−0.202002
$591$	−10431.2	−0.726025
$592$	512.836	0.0356038
$593$	14956.4	1.03573	0.517864	−	0.855463i	$-0.326727\pi$
$593$	14956.4	1.03573	0.517864	+	0.855463i	$0.326727\pi$
$594$	0	0
$595$	−5056.93	−0.348427
$596$	−13216.2	−0.908319
$597$	11985.1	0.821638
$598$	−3937.96	−0.269290
$599$	2150.77	0.146708	0.0733539	−	0.997306i	$-0.476630\pi$
$599$	2150.77	0.146708	0.0733539	+	0.997306i	$0.476630\pi$
$600$	1588.28	0.108069
$601$	−27759.8	−1.88410	−0.942050	−	0.335472i	$-0.891104\pi$
$601$	−27759.8	−1.88410	−0.942050	+	0.335472i	$0.891104\pi$
$602$	−37.8792	−0.00256452
$603$	−3644.86	−0.246153
$604$	−516.371	−0.0347862
$605$	0	0
$606$	1633.07	0.109470
$607$	10991.5	0.734974	0.367487	−	0.930029i	$-0.380218\pi$
$607$	10991.5	0.734974	0.367487	+	0.930029i	$0.380218\pi$
$608$	−7423.87	−0.495194
$609$	4582.91	0.304941
$610$	5577.91	0.370234
$611$	−13624.1	−0.902081
$612$	−4940.74	−0.326336
$613$	−10646.1	−0.701457	−0.350728	−	0.936477i	$-0.614066\pi$
$613$	−10646.1	−0.701457	−0.350728	+	0.936477i	$0.614066\pi$
$614$	887.128	0.0583088
$615$	5047.19	0.330930
$616$	0	0
$617$	7199.92	0.469786	0.234893	−	0.972021i	$-0.424526\pi$
$617$	7199.92	0.469786	0.234893	+	0.972021i	$0.424526\pi$
$618$	7197.41	0.468483
$619$	12186.9	0.791332	0.395666	−	0.918395i	$-0.370514\pi$
$619$	12186.9	0.791332	0.395666	+	0.918395i	$0.370514\pi$
$620$	1522.16	0.0985990
$621$	1665.72	0.107637
$622$	10702.2	0.689905
$623$	3953.80	0.254262
$624$	1400.85	0.0898698
$625$	625.000	0.0400000
$626$	1778.49	0.113551
$627$	0	0
$628$	10462.6	0.664814
$629$	4431.40	0.280909
$630$	720.000	0.0455325
$631$	7370.64	0.465009	0.232505	−	0.972595i	$-0.425308\pi$
$631$	7370.64	0.465009	0.232505	+	0.972595i	$0.425308\pi$
$632$	−20157.1	−1.26868
$633$	−3138.38	−0.197061
$634$	−5008.65	−0.313752
$635$	−8699.09	−0.543642
$636$	−10679.3	−0.665818
$637$	−9729.32	−0.605164
$638$	0	0
$639$	8453.63	0.523349
$640$	−5920.66	−0.365679
$641$	−25014.9	−1.54139	−0.770693	−	0.637207i	$-0.780090\pi$
$641$	−25014.9	−1.54139	−0.770693	+	0.637207i	$0.780090\pi$
$642$	−3650.19	−0.224395
$643$	−21668.2	−1.32894	−0.664472	−	0.747313i	$-0.731343\pi$
$643$	−21668.2	−1.32894	−0.664472	+	0.747313i	$0.731343\pi$
$644$	−3515.58	−0.215113
$645$	−35.5118	−0.00216787
$646$	−6110.94	−0.372185
$647$	−27625.3	−1.67861	−0.839305	−	0.543661i	$-0.817038\pi$
$647$	−27625.3	−1.67861	−0.839305	+	0.543661i	$0.817038\pi$
$648$	1715.34	0.103989
$649$	0	0
$650$	−1595.79	−0.0962952
$651$	1682.59	0.101299
$652$	13712.0	0.823623
$653$	−14314.0	−0.857810	−0.428905	−	0.903350i	$-0.641101\pi$
$653$	−14314.0	−0.857810	−0.428905	+	0.903350i	$0.641101\pi$
$654$	−7035.38	−0.420650
$655$	1564.19	0.0933096
$656$	−3843.70	−0.228767
$657$	3263.13	0.193770
$658$	5332.73	0.315944
$659$	28327.8	1.67450	0.837249	−	0.546822i	$-0.184163\pi$
$659$	28327.8	1.67450	0.837249	+	0.546822i	$0.184163\pi$
$660$	0	0
$661$	−32190.9	−1.89422	−0.947112	−	0.320905i	$-0.896013\pi$
$661$	−32190.9	−1.89422	−0.947112	+	0.320905i	$0.896013\pi$
$662$	14265.0	0.837498
$663$	12104.7	0.709059
$664$	−15569.8	−0.909981
$665$	−2031.10	−0.118440
$666$	−630.938	−0.0367092
$667$	9198.01	0.533955
$668$	−6975.84	−0.404047
$669$	−1518.80	−0.0877729
$670$	−3162.03	−0.182328
$671$	0	0
$672$	−5755.95	−0.330418
$673$	6207.38	0.355538	0.177769	−	0.984072i	$-0.443112\pi$
$673$	6207.38	0.355538	0.177769	+	0.984072i	$0.443112\pi$
$674$	−5418.66	−0.309672
$675$	675.000	0.0384900
$676$	2925.82	0.166467
$677$	−28831.1	−1.63674	−0.818368	−	0.574695i	$-0.805121\pi$
$677$	−28831.1	−1.63674	−0.818368	+	0.574695i	$0.805121\pi$
$678$	−796.392	−0.0451110
$679$	−9901.37	−0.559617
$680$	−10451.8	−0.589422
$681$	12858.9	0.723573
$682$	0	0
$683$	3193.10	0.178888	0.0894441	−	0.995992i	$-0.471491\pi$
$683$	3193.10	0.178888	0.0894441	+	0.995992i	$0.471491\pi$
$684$	−1984.43	−0.110931
$685$	3584.64	0.199945
$686$	9296.24	0.517394
$687$	17128.1	0.951206
$688$	27.0441	0.00149861
$689$	26163.8	1.44668
$690$	1445.06	0.0797281
$691$	7682.49	0.422946	0.211473	−	0.977384i	$-0.432174\pi$
$691$	7682.49	0.422946	0.211473	+	0.977384i	$0.432174\pi$
$692$	−6711.17	−0.368671
$693$	0	0
$694$	−139.496	−0.00762996
$695$	−4382.41	−0.239186
$696$	9472.05	0.515858
$697$	−33213.3	−1.80494
$698$	−233.256	−0.0126488
$699$	−8838.28	−0.478247
$700$	−1424.62	−0.0769223
$701$	−26551.6	−1.43058	−0.715292	−	0.698825i	$-0.753707\pi$
$701$	−26551.6	−1.43058	−0.715292	+	0.698825i	$0.753707\pi$
$702$	−1723.45	−0.0926601
$703$	1779.86	0.0954887
$704$	0	0
$705$	4999.43	0.267077
$706$	−12285.7	−0.654928
$707$	−3571.83	−0.190004
$708$	6186.22	0.328379
$709$	−16304.6	−0.863655	−0.431828	−	0.901956i	$-0.642131\pi$
$709$	−16304.6	−0.863655	−0.431828	+	0.901956i	$0.642131\pi$
$710$	7333.77	0.387650
$711$	−8566.52	−0.451856
$712$	8171.79	0.430127
$713$	3377.00	0.177377
$714$	−4738.00	−0.248341
$715$	0	0
$716$	8021.36	0.418676
$717$	6236.68	0.324844
$718$	7767.50	0.403733
$719$	−3973.62	−0.206107	−0.103053	−	0.994676i	$-0.532861\pi$
$719$	−3973.62	−0.206107	−0.103053	+	0.994676i	$0.532861\pi$
$720$	−514.048	−0.0266076
$721$	−15742.1	−0.813128
$722$	8256.25	0.425576
$723$	−5560.11	−0.286007
$724$	−23702.3	−1.21670
$725$	3727.32	0.190937
$726$	0	0
$727$	−10780.4	−0.549961	−0.274980	−	0.961450i	$-0.588671\pi$
$727$	−10780.4	−0.549961	−0.274980	+	0.961450i	$0.588671\pi$
$728$	8869.67	0.451555
$729$	729.000	0.0370370
$730$	2830.86	0.143527
$731$	233.687	0.0118238
$732$	−11919.6	−0.601860
$733$	9211.46	0.464165	0.232083	−	0.972696i	$-0.425446\pi$
$733$	9211.46	0.464165	0.232083	+	0.972696i	$0.425446\pi$
$734$	20716.6	1.04177
$735$	3570.23	0.179170
$736$	−11552.3	−0.578566
$737$	0	0
$738$	4728.87	0.235870
$739$	−11084.7	−0.551768	−0.275884	−	0.961191i	$-0.588971\pi$
$739$	−11084.7	−0.551768	−0.275884	+	0.961191i	$0.588971\pi$
$740$	1248.40	0.0620163
$741$	4861.80	0.241029
$742$	−10241.0	−0.506685
$743$	27420.4	1.35391	0.676955	−	0.736024i	$-0.263299\pi$
$743$	27420.4	1.35391	0.676955	+	0.736024i	$0.263299\pi$
$744$	3477.61	0.171365
$745$	−11881.8	−0.584316
$746$	−7234.32	−0.355050
$747$	−6616.99	−0.324101
$748$	0	0
$749$	7983.64	0.389474
$750$	585.582	0.0285099
$751$	−11290.8	−0.548614	−0.274307	−	0.961642i	$-0.588448\pi$
$751$	−11290.8	−0.548614	−0.274307	+	0.961642i	$0.588448\pi$
$752$	−3807.33	−0.184626
$753$	−7075.16	−0.342408
$754$	−9516.81	−0.459657
$755$	−464.233	−0.0223777
$756$	−1538.59	−0.0740185
$757$	3739.19	0.179528	0.0897642	−	0.995963i	$-0.471389\pi$
$757$	3739.19	0.179528	0.0897642	+	0.995963i	$0.471389\pi$
$758$	−10155.3	−0.486617
$759$	0	0
$760$	−4197.92	−0.200361
$761$	15621.1	0.744107	0.372053	−	0.928211i	$-0.378654\pi$
$761$	15621.1	0.744107	0.372053	+	0.928211i	$0.378654\pi$
$762$	−8150.45	−0.387480
$763$	15387.7	0.730106
$764$	4739.69	0.224445
$765$	−4441.87	−0.209930
$766$	−19885.7	−0.937987
$767$	−15156.0	−0.713498
$768$	−10371.8	−0.487317
$769$	−40241.7	−1.88706	−0.943531	−	0.331284i	$-0.892518\pi$
$769$	−40241.7	−1.88706	−0.943531	+	0.331284i	$0.892518\pi$
$770$	0	0
$771$	16557.8	0.773428
$772$	−13681.2	−0.637818
$773$	22821.4	1.06187	0.530936	−	0.847412i	$-0.321840\pi$
$773$	22821.4	1.06187	0.530936	+	0.847412i	$0.321840\pi$
$774$	−33.2721	−0.00154514
$775$	1368.47	0.0634281
$776$	−20464.4	−0.946685
$777$	1379.98	0.0637148
$778$	−18777.0	−0.865282
$779$	−13340.0	−0.613549
$780$	3410.09	0.156539
$781$	0	0
$782$	−9509.28	−0.434848
$783$	4025.51	0.183729
$784$	−2718.92	−0.123857
$785$	9406.19	0.427671
$786$	1465.53	0.0665062
$787$	29454.3	1.33410	0.667048	−	0.745015i	$-0.267558\pi$
$787$	29454.3	1.33410	0.667048	+	0.745015i	$0.267558\pi$
$788$	19337.8	0.874216
$789$	−6778.96	−0.305878
$790$	−7431.70	−0.334694
$791$	1741.86	0.0782974
$792$	0	0
$793$	29202.7	1.30771
$794$	8157.00	0.364586
$795$	−9600.97	−0.428316
$796$	−22218.6	−0.989344
$797$	−27440.3	−1.21955	−0.609777	−	0.792573i	$-0.708741\pi$
$797$	−27440.3	−1.21955	−0.609777	+	0.792573i	$0.708741\pi$
$798$	−1903.00	−0.0844179
$799$	−32899.0	−1.45668
$800$	−4681.37	−0.206889
$801$	3472.91	0.153195
$802$	−15066.1	−0.663346
$803$	0	0
$804$	6757.03	0.296396
$805$	−3160.61	−0.138381
$806$	−3494.05	−0.152695
$807$	21118.6	0.921201
$808$	−7382.34	−0.321423
$809$	5060.18	0.219909	0.109954	−	0.993937i	$-0.464930\pi$
$809$	5060.18	0.219909	0.109954	+	0.993937i	$0.464930\pi$
$810$	632.429	0.0274337
$811$	−30480.1	−1.31973	−0.659865	−	0.751384i	$-0.729387\pi$
$811$	−30480.1	−1.31973	−0.659865	+	0.751384i	$0.729387\pi$
$812$	−8496.03	−0.367183
$813$	−15465.2	−0.667146
$814$	0	0
$815$	12327.5	0.529831
$816$	3382.72	0.145121
$817$	93.8596	0.00401925
$818$	−3139.46	−0.134192
$819$	3769.50	0.160827
$820$	−9356.73	−0.398477
$821$	−37909.0	−1.61149	−0.805745	−	0.592263i	$-0.798235\pi$
$821$	−37909.0	−1.61149	−0.805745	+	0.592263i	$0.798235\pi$
$822$	3358.56	0.142510
$823$	23636.0	1.00109	0.500546	−	0.865710i	$-0.333133\pi$
$823$	23636.0	1.00109	0.500546	+	0.865710i	$0.333133\pi$
$824$	−32536.0	−1.37554
$825$	0	0
$826$	5932.36	0.249895
$827$	42634.3	1.79267	0.896336	−	0.443376i	$-0.146219\pi$
$827$	42634.3	1.79267	0.896336	+	0.443376i	$0.146219\pi$
$828$	−3087.99	−0.129608
$829$	−45152.5	−1.89169	−0.945845	−	0.324619i	$-0.894764\pi$
$829$	−45152.5	−1.89169	−0.945845	+	0.324619i	$0.894764\pi$
$830$	−5740.44	−0.240064
$831$	−27223.6	−1.13643
$832$	8217.15	0.342402
$833$	−23494.1	−0.977217
$834$	−4106.02	−0.170480
$835$	−6271.49	−0.259921
$836$	0	0
$837$	1477.94	0.0610337
$838$	−6925.91	−0.285503
$839$	30431.5	1.25222	0.626110	−	0.779734i	$-0.284646\pi$
$839$	30431.5	1.25222	0.626110	+	0.779734i	$0.284646\pi$
$840$	−3254.77	−0.133691
$841$	−2160.34	−0.0885784
$842$	−23763.6	−0.972623
$843$	10223.4	0.417689
$844$	5818.09	0.237283
$845$	2630.40	0.107087
$846$	4684.13	0.190359
$847$	0	0
$848$	7311.64	0.296088
$849$	26483.2	1.07055
$850$	−3853.46	−0.155497
$851$	2769.65	0.111566
$852$	−15671.8	−0.630171
$853$	−10367.2	−0.416139	−0.208070	−	0.978114i	$-0.566718\pi$
$853$	−10367.2	−0.416139	−0.208070	+	0.978114i	$0.566718\pi$
$854$	−11430.5	−0.458013
$855$	−1784.06	−0.0713610
$856$	16500.8	0.658860
$857$	−12947.1	−0.516063	−0.258032	−	0.966136i	$-0.583074\pi$
$857$	−12947.1	−0.516063	−0.258032	+	0.966136i	$0.583074\pi$
$858$	0	0
$859$	−20383.5	−0.809636	−0.404818	−	0.914397i	$-0.632665\pi$
$859$	−20383.5	−0.809636	−0.404818	+	0.914397i	$0.632665\pi$
$860$	65.8335	0.00261035
$861$	−10342.9	−0.409391
$862$	−8779.22	−0.346893
$863$	9056.42	0.357224	0.178612	−	0.983920i	$-0.442839\pi$
$863$	9056.42	0.357224	0.178612	+	0.983920i	$0.442839\pi$
$864$	−5055.88	−0.199079
$865$	−6033.54	−0.237164
$866$	22340.0	0.876610
$867$	14491.0	0.567636
$868$	−3119.27	−0.121976
$869$	0	0
$870$	3492.24	0.136090
$871$	−16554.5	−0.644005
$872$	31803.6	1.23510
$873$	−8697.10	−0.337173
$874$	−3819.37	−0.147817
$875$	−1280.78	−0.0494836
$876$	−6049.36	−0.233321
$877$	2867.88	0.110424	0.0552118	−	0.998475i	$-0.482417\pi$
$877$	2867.88	0.110424	0.0552118	+	0.998475i	$0.482417\pi$
$878$	6846.16	0.263151
$879$	13584.9	0.521281
$880$	0	0
$881$	−11862.5	−0.453640	−0.226820	−	0.973937i	$-0.572833\pi$
$881$	−11862.5	−0.453640	−0.226820	+	0.973937i	$0.572833\pi$
$882$	3345.06	0.127703
$883$	33463.8	1.27537	0.637683	−	0.770299i	$-0.279893\pi$
$883$	33463.8	1.27537	0.637683	+	0.770299i	$0.279893\pi$
$884$	−22440.3	−0.853787
$885$	5561.59	0.211244
$886$	15756.0	0.597442
$887$	−2420.75	−0.0916357	−0.0458178	−	0.998950i	$-0.514589\pi$
$887$	−2420.75	−0.0916357	−0.0458178	+	0.998950i	$0.514589\pi$
$888$	2852.17	0.107784
$889$	17826.5	0.672534
$890$	3012.85	0.113473
$891$	0	0
$892$	2815.62	0.105688
$893$	−13213.8	−0.495165
$894$	−11132.4	−0.416470
$895$	7211.44	0.269332
$896$	12132.9	0.452378
$897$	7565.48	0.281610
$898$	−14963.6	−0.556060
$899$	8161.14	0.302769
$900$	−1251.35	−0.0463463
$901$	63179.7	2.33609
$902$	0	0
$903$	72.7722	0.00268185
$904$	3600.10	0.132453
$905$	−21309.0	−0.782692
$906$	−434.955	−0.0159497
$907$	−38154.1	−1.39679	−0.698393	−	0.715714i	$-0.746101\pi$
$907$	−38154.1	−1.39679	−0.698393	+	0.715714i	$0.746101\pi$
$908$	−23838.4	−0.871262
$909$	−3137.40	−0.114479
$910$	3270.15	0.119126
$911$	−35758.0	−1.30045	−0.650227	−	0.759740i	$-0.725326\pi$
$911$	−35758.0	−1.30045	−0.650227	+	0.759740i	$0.725326\pi$
$912$	1358.66	0.0493308
$913$	0	0
$914$	15614.5	0.565079
$915$	−10716.1	−0.387172
$916$	−31753.0	−1.14536
$917$	−3205.39	−0.115432
$918$	−4161.73	−0.149627
$919$	−17387.5	−0.624115	−0.312058	−	0.950063i	$-0.601018\pi$
$919$	−17387.5	−0.624115	−0.312058	+	0.950063i	$0.601018\pi$
$920$	−6532.41	−0.234095
$921$	−1704.32	−0.0609764
$922$	−17357.9	−0.620013
$923$	38395.3	1.36923
$924$	0	0
$925$	1122.35	0.0398947
$926$	2447.20	0.0868467
$927$	−13827.4	−0.489915
$928$	−27918.3	−0.987569
$929$	6955.93	0.245658	0.122829	−	0.992428i	$-0.460803\pi$
$929$	6955.93	0.245658	0.122829	+	0.992428i	$0.460803\pi$
$930$	1282.16	0.0452082
$931$	−9436.31	−0.332183
$932$	16384.9	0.575863
$933$	−20560.8	−0.721467
$934$	−19751.7	−0.691965
$935$	0	0
$936$	7790.88	0.272065
$937$	16074.5	0.560438	0.280219	−	0.959936i	$-0.409593\pi$
$937$	16074.5	0.560438	0.280219	+	0.959936i	$0.409593\pi$
$938$	6479.76	0.225556
$939$	−3416.77	−0.118746
$940$	−9268.20	−0.321591
$941$	687.126	0.0238041	0.0119021	−	0.999929i	$-0.496211\pi$
$941$	687.126	0.0238041	0.0119021	+	0.999929i	$0.496211\pi$
$942$	8812.96	0.304821
$943$	−20758.5	−0.716849
$944$	−4235.44	−0.146030
$945$	−1383.24	−0.0476156
$946$	0	0
$947$	35352.0	1.21308	0.606540	−	0.795053i	$-0.292557\pi$
$947$	35352.0	1.21308	0.606540	+	0.795053i	$0.292557\pi$
$948$	15881.0	0.544085
$949$	14820.7	0.506956
$950$	−1547.73	−0.0528578
$951$	9622.44	0.328106
$952$	21418.2	0.729168
$953$	19390.7	0.659103	0.329552	−	0.944138i	$-0.393102\pi$
$953$	19390.7	0.659103	0.329552	+	0.944138i	$0.393102\pi$
$954$	−8995.45	−0.305281
$955$	4261.12	0.144384
$956$	−11561.9	−0.391148
$957$	0	0
$958$	−16738.7	−0.564511
$959$	−7345.80	−0.247349
$960$	−3015.33	−0.101374
$961$	−26794.7	−0.899422
$962$	−2865.64	−0.0960416
$963$	7012.62	0.234661
$964$	10307.6	0.344384
$965$	−12299.8	−0.410304
$966$	−2961.27	−0.0986308
$967$	−28643.6	−0.952551	−0.476275	−	0.879296i	$-0.658013\pi$
$967$	−28643.6	−0.952551	−0.476275	+	0.879296i	$0.658013\pi$
$968$	0	0
$969$	11740.1	0.389213
$970$	−7544.99	−0.249747
$971$	−19574.8	−0.646946	−0.323473	−	0.946237i	$-0.604851\pi$
$971$	−19574.8	−0.646946	−0.323473	+	0.946237i	$0.604851\pi$
$972$	−1351.46	−0.0445967
$973$	8980.63	0.295895
$974$	−11182.6	−0.367878
$975$	3065.77	0.100701
$976$	8160.86	0.267646
$977$	50095.5	1.64043	0.820213	−	0.572058i	$-0.193855\pi$
$977$	50095.5	1.64043	0.820213	+	0.572058i	$0.193855\pi$
$978$	11550.0	0.377636
$979$	0	0
$980$	−6618.67	−0.215740
$981$	13516.1	0.439895
$982$	22747.6	0.739211
$983$	14445.4	0.468706	0.234353	−	0.972152i	$-0.424703\pi$
$983$	14445.4	0.468706	0.234353	+	0.972152i	$0.424703\pi$
$984$	−21376.9	−0.692553
$985$	17385.3	0.562377
$986$	−22980.9	−0.742253
$987$	−10245.0	−0.330399
$988$	−9013.05	−0.290226
$989$	146.056	0.00469595
$990$	0	0
$991$	29120.1	0.933430	0.466715	−	0.884408i	$-0.345437\pi$
$991$	29120.1	0.933430	0.466715	+	0.884408i	$0.345437\pi$
$992$	−10250.1	−0.328064
$993$	−27405.4	−0.875814
$994$	−15028.7	−0.479558
$995$	−19975.2	−0.636438
$996$	12266.9	0.390253
$997$	9137.45	0.290257	0.145128	−	0.989413i	$-0.453640\pi$
$997$	9137.45	0.290257	0.145128	+	0.989413i	$0.453640\pi$
$998$	7244.03	0.229765
$999$	1212.14	0.0383887

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1815.4.a.n.1.1		2
11.10	odd	2		165.4.a.c.1.2	✓	2
33.32	even	2		495.4.a.d.1.1		2
55.32	even	4		825.4.c.j.199.3		4
55.43	even	4		825.4.c.j.199.2		4
55.54	odd	2		825.4.a.m.1.1		2
165.164	even	2		2475.4.a.n.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
165.4.a.c.1.2	✓	2	11.10	odd	2
495.4.a.d.1.1		2	33.32	even	2
825.4.a.m.1.1		2	55.54	odd	2
825.4.c.j.199.2		4	55.43	even	4
825.4.c.j.199.3		4	55.32	even	4
1815.4.a.n.1.1		2	1.1	even	1	trivial
2475.4.a.n.1.2		2	165.164	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1815 = 3 \cdot 5 \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1815.a (trivial)

\(f(q)\)	\(=\)	\(q-1.56155 q^{2} +3.00000 q^{3} -5.56155 q^{4} -5.00000 q^{5} -4.68466 q^{6} +10.2462 q^{7} +21.1771 q^{8} +9.00000 q^{9} +O(q^{10})\)	\(q-1.56155 q^{2} +3.00000 q^{3} -5.56155 q^{4} -5.00000 q^{5} -4.68466 q^{6} +10.2462 q^{7} +21.1771 q^{8} +9.00000 q^{9} +7.80776 q^{10} -16.6847 q^{12} +40.8769 q^{13} -16.0000 q^{14} -15.0000 q^{15} +11.4233 q^{16} +98.7083 q^{17} -14.0540 q^{18} +39.6458 q^{19} +27.8078 q^{20} +30.7386 q^{21} +61.6932 q^{23} +63.5312 q^{24} +25.0000 q^{25} -63.8314 q^{26} +27.0000 q^{27} -56.9848 q^{28} +149.093 q^{29} +23.4233 q^{30} +54.7386 q^{31} -187.255 q^{32} -154.138 q^{34} -51.2311 q^{35} -50.0540 q^{36} +44.8939 q^{37} -61.9091 q^{38} +122.631 q^{39} -105.885 q^{40} -336.479 q^{41} -48.0000 q^{42} +2.36745 q^{43} -45.0000 q^{45} -96.3371 q^{46} -333.295 q^{47} +34.2699 q^{48} -238.015 q^{49} -39.0388 q^{50} +296.125 q^{51} -227.339 q^{52} +640.064 q^{53} -42.1619 q^{54} +216.985 q^{56} +118.938 q^{57} -232.816 q^{58} -370.773 q^{59} +83.4233 q^{60} +714.405 q^{61} -85.4773 q^{62} +92.2159 q^{63} +201.022 q^{64} -204.384 q^{65} -404.985 q^{67} -548.972 q^{68} +185.080 q^{69} +80.0000 q^{70} +939.292 q^{71} +190.594 q^{72} +362.570 q^{73} -70.1042 q^{74} +75.0000 q^{75} -220.492 q^{76} -191.494 q^{78} -951.835 q^{79} -57.1165 q^{80} +81.0000 q^{81} +525.430 q^{82} -735.221 q^{83} -170.955 q^{84} -493.542 q^{85} -3.69690 q^{86} +447.278 q^{87} +385.879 q^{89} +70.2699 q^{90} +418.833 q^{91} -343.110 q^{92} +164.216 q^{93} +520.458 q^{94} -198.229 q^{95} -561.764 q^{96} -966.345 q^{97} +371.673 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + q^{2} + 6 q^{3} - 7 q^{4} - 10 q^{5} + 3 q^{6} + 4 q^{7} - 3 q^{8} + 18 q^{9}+O(q^{10}) \) 2 * q + q^2 + 6 * q^3 - 7 * q^4 - 10 * q^5 + 3 * q^6 + 4 * q^7 - 3 * q^8 + 18 * q^9	\( 2 q + q^{2} + 6 q^{3} - 7 q^{4} - 10 q^{5} + 3 q^{6} + 4 q^{7} - 3 q^{8} + 18 q^{9} - 5 q^{10} - 21 q^{12} + 90 q^{13} - 32 q^{14} - 30 q^{15} - 39 q^{16} + 16 q^{17} + 9 q^{18} + 170 q^{19} + 35 q^{20} + 12 q^{21} - 124 q^{23} - 9 q^{24} + 50 q^{25} + 62 q^{26} + 54 q^{27} - 48 q^{28} + 158 q^{29} - 15 q^{30} + 60 q^{31} - 123 q^{32} - 366 q^{34} - 20 q^{35} - 63 q^{36} - 372 q^{37} + 272 q^{38} + 270 q^{39} + 15 q^{40} - 38 q^{41} - 96 q^{42} + 516 q^{43} - 90 q^{45} - 572 q^{46} + 224 q^{47} - 117 q^{48} - 542 q^{49} + 25 q^{50} + 48 q^{51} - 298 q^{52} + 472 q^{53} + 27 q^{54} + 368 q^{56} + 510 q^{57} - 210 q^{58} + 248 q^{59} + 105 q^{60} - 72 q^{61} - 72 q^{62} + 36 q^{63} + 769 q^{64} - 450 q^{65} - 744 q^{67} - 430 q^{68} - 372 q^{69} + 160 q^{70} + 2060 q^{71} - 27 q^{72} + 486 q^{73} - 1138 q^{74} + 150 q^{75} - 408 q^{76} + 186 q^{78} - 642 q^{79} + 195 q^{80} + 162 q^{81} + 1290 q^{82} + 286 q^{83} - 144 q^{84} - 80 q^{85} + 1312 q^{86} + 474 q^{87} + 244 q^{89} - 45 q^{90} + 112 q^{91} - 76 q^{92} + 180 q^{93} + 1948 q^{94} - 850 q^{95} - 369 q^{96} - 168 q^{97} - 407 q^{98}+O(q^{100}) \) 2 * q + q^2 + 6 * q^3 - 7 * q^4 - 10 * q^5 + 3 * q^6 + 4 * q^7 - 3 * q^8 + 18 * q^9 - 5 * q^10 - 21 * q^12 + 90 * q^13 - 32 * q^14 - 30 * q^15 - 39 * q^16 + 16 * q^17 + 9 * q^18 + 170 * q^19 + 35 * q^20 + 12 * q^21 - 124 * q^23 - 9 * q^24 + 50 * q^25 + 62 * q^26 + 54 * q^27 - 48 * q^28 + 158 * q^29 - 15 * q^30 + 60 * q^31 - 123 * q^32 - 366 * q^34 - 20 * q^35 - 63 * q^36 - 372 * q^37 + 272 * q^38 + 270 * q^39 + 15 * q^40 - 38 * q^41 - 96 * q^42 + 516 * q^43 - 90 * q^45 - 572 * q^46 + 224 * q^47 - 117 * q^48 - 542 * q^49 + 25 * q^50 + 48 * q^51 - 298 * q^52 + 472 * q^53 + 27 * q^54 + 368 * q^56 + 510 * q^57 - 210 * q^58 + 248 * q^59 + 105 * q^60 - 72 * q^61 - 72 * q^62 + 36 * q^63 + 769 * q^64 - 450 * q^65 - 744 * q^67 - 430 * q^68 - 372 * q^69 + 160 * q^70 + 2060 * q^71 - 27 * q^72 + 486 * q^73 - 1138 * q^74 + 150 * q^75 - 408 * q^76 + 186 * q^78 - 642 * q^79 + 195 * q^80 + 162 * q^81 + 1290 * q^82 + 286 * q^83 - 144 * q^84 - 80 * q^85 + 1312 * q^86 + 474 * q^87 + 244 * q^89 - 45 * q^90 + 112 * q^91 - 76 * q^92 + 180 * q^93 + 1948 * q^94 - 850 * q^95 - 369 * q^96 - 168 * q^97 - 407 * q^98

Introduction

L-functions

Modular forms

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