LMFDB - Embedded newform 1815.4.a.bg.1.7

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} - 3 x^{11} - 69 x^{10} + 157 x^{9} + 1812 x^{8} - 2703 x^{7} - 22379 x^{6} + 16453 x^{5} + \cdots + 196416 $ x^12 - 3x^11 - 69x^10 + 157x^9 + 1812x^8 - 2703x^7 - 22379x^6 + 16453x^5 + 129167x^4 - 22672x^3 - 298084x^2 - 19392*x + 196416
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			1.7
Root			$0.911819$ 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)

$f(q)$	$=$	$q-0.0881808 q^{2} +3.00000 q^{3} -7.99222 q^{4} -5.00000 q^{5} -0.264542 q^{6} -20.0828 q^{7} +1.41021 q^{8} +9.00000 q^{9} +O(q^{10})$	\(q-0.0881808 q^{2} +3.00000 q^{3} -7.99222 q^{4} -5.00000 q^{5} -0.264542 q^{6} -20.0828 q^{7} +1.41021 q^{8} +9.00000 q^{9} +0.440904 q^{10} -23.9767 q^{12} +19.8232 q^{13} +1.77091 q^{14} -15.0000 q^{15} +63.8134 q^{16} +7.50555 q^{17} -0.793627 q^{18} +27.5515 q^{19} +39.9611 q^{20} -60.2483 q^{21} -27.3979 q^{23} +4.23062 q^{24} +25.0000 q^{25} -1.74802 q^{26} +27.0000 q^{27} +160.506 q^{28} -98.3333 q^{29} +1.32271 q^{30} +88.1538 q^{31} -16.9088 q^{32} -0.661845 q^{34} +100.414 q^{35} -71.9300 q^{36} -53.2549 q^{37} -2.42951 q^{38} +59.4696 q^{39} -7.05103 q^{40} +197.630 q^{41} +5.31274 q^{42} -37.1791 q^{43} -45.0000 q^{45} +2.41596 q^{46} +528.855 q^{47} +191.440 q^{48} +60.3179 q^{49} -2.20452 q^{50} +22.5166 q^{51} -158.431 q^{52} -306.951 q^{53} -2.38088 q^{54} -28.3209 q^{56} +82.6546 q^{57} +8.67110 q^{58} -69.0453 q^{59} +119.883 q^{60} +779.999 q^{61} -7.77347 q^{62} -180.745 q^{63} -509.016 q^{64} -99.1160 q^{65} +911.553 q^{67} -59.9860 q^{68} -82.1936 q^{69} -8.85457 q^{70} -100.883 q^{71} +12.6919 q^{72} -173.265 q^{73} +4.69606 q^{74} +75.0000 q^{75} -220.198 q^{76} -5.24407 q^{78} -183.512 q^{79} -319.067 q^{80} +81.0000 q^{81} -17.4271 q^{82} -1141.72 q^{83} +481.518 q^{84} -37.5277 q^{85} +3.27848 q^{86} -295.000 q^{87} +1164.37 q^{89} +3.96813 q^{90} -398.105 q^{91} +218.970 q^{92} +264.461 q^{93} -46.6349 q^{94} -137.758 q^{95} -50.7263 q^{96} -1207.61 q^{97} -5.31888 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 9 q^{2} + 36 q^{3} + 57 q^{4} - 60 q^{5} - 27 q^{6} - 21 q^{7} - 123 q^{8} + 108 q^{9}+O(q^{10}) $ 12 * q - 9 * q^2 + 36 * q^3 + 57 * q^4 - 60 * q^5 - 27 * q^6 - 21 * q^7 - 123 * q^8 + 108 * q^9	\( 12 q - 9 q^{2} + 36 q^{3} + 57 q^{4} - 60 q^{5} - 27 q^{6} - 21 q^{7} - 123 q^{8} + 108 q^{9} + 45 q^{10} + 171 q^{12} - 78 q^{13} + 262 q^{14} - 180 q^{15} + 225 q^{16} - 313 q^{17} - 81 q^{18} - 51 q^{19} - 285 q^{20} - 63 q^{21} - 34 q^{23} - 369 q^{24} + 300 q^{25} + 28 q^{26} + 324 q^{27} - 376 q^{28} - 31 q^{29} + 135 q^{30} + 655 q^{31} - 1578 q^{32} - 10 q^{34} + 105 q^{35} + 513 q^{36} + 84 q^{37} - 1076 q^{38} - 234 q^{39} + 615 q^{40} - 1463 q^{41} + 786 q^{42} + 111 q^{43} - 540 q^{45} + q^{46} + 278 q^{47} + 675 q^{48} - 325 q^{49} - 225 q^{50} - 939 q^{51} - 1957 q^{52} + 517 q^{53} - 243 q^{54} + 1543 q^{56} - 153 q^{57} + 442 q^{58} - 308 q^{59} - 855 q^{60} - 604 q^{61} - 1773 q^{62} - 189 q^{63} + 4323 q^{64} + 390 q^{65} - 357 q^{67} - 2192 q^{68} - 102 q^{69} - 1310 q^{70} - 620 q^{71} - 1107 q^{72} - 1892 q^{73} - 581 q^{74} + 900 q^{75} - 378 q^{76} + 84 q^{78} - 415 q^{79} - 1125 q^{80} + 972 q^{81} - 2802 q^{82} - 3158 q^{83} - 1128 q^{84} + 1565 q^{85} + 747 q^{86} - 93 q^{87} + 1563 q^{89} + 405 q^{90} + 1434 q^{91} - 3466 q^{92} + 1965 q^{93} + 3 q^{94} + 255 q^{95} - 4734 q^{96} + 714 q^{97} - 6586 q^{98}+O(q^{100}) \) 12 * q - 9 * q^2 + 36 * q^3 + 57 * q^4 - 60 * q^5 - 27 * q^6 - 21 * q^7 - 123 * q^8 + 108 * q^9 + 45 * q^10 + 171 * q^12 - 78 * q^13 + 262 * q^14 - 180 * q^15 + 225 * q^16 - 313 * q^17 - 81 * q^18 - 51 * q^19 - 285 * q^20 - 63 * q^21 - 34 * q^23 - 369 * q^24 + 300 * q^25 + 28 * q^26 + 324 * q^27 - 376 * q^28 - 31 * q^29 + 135 * q^30 + 655 * q^31 - 1578 * q^32 - 10 * q^34 + 105 * q^35 + 513 * q^36 + 84 * q^37 - 1076 * q^38 - 234 * q^39 + 615 * q^40 - 1463 * q^41 + 786 * q^42 + 111 * q^43 - 540 * q^45 + q^46 + 278 * q^47 + 675 * q^48 - 325 * q^49 - 225 * q^50 - 939 * q^51 - 1957 * q^52 + 517 * q^53 - 243 * q^54 + 1543 * q^56 - 153 * q^57 + 442 * q^58 - 308 * q^59 - 855 * q^60 - 604 * q^61 - 1773 * q^62 - 189 * q^63 + 4323 * q^64 + 390 * q^65 - 357 * q^67 - 2192 * q^68 - 102 * q^69 - 1310 * q^70 - 620 * q^71 - 1107 * q^72 - 1892 * q^73 - 581 * q^74 + 900 * q^75 - 378 * q^76 + 84 * q^78 - 415 * q^79 - 1125 * q^80 + 972 * q^81 - 2802 * q^82 - 3158 * q^83 - 1128 * q^84 + 1565 * q^85 + 747 * q^86 - 93 * q^87 + 1563 * q^89 + 405 * q^90 + 1434 * q^91 - 3466 * q^92 + 1965 * q^93 + 3 * q^94 + 255 * q^95 - 4734 * q^96 + 714 * q^97 - 6586 * 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 $

$2$	−0.0881808	−0.0311766	−0.0155883	−	0.999878i	$-0.504962\pi$
$2$	−0.0881808	−0.0311766	−0.0155883	+	0.999878i	$0.504962\pi$
$3$	3.00000	0.577350
$3$	3.00000	0.577350
$4$	−7.99222	−0.999028
$5$	−5.00000	−0.447214
$5$	−5.00000	−0.447214
$6$	−0.264542	−0.0179998
$7$	−20.0828	−1.08437	−0.542184	−	0.840260i	$-0.682402\pi$
$7$	−20.0828	−1.08437	−0.542184	+	0.840260i	$0.682402\pi$
$8$	1.41021	0.0623229
$9$	9.00000	0.333333
$10$	0.440904	0.0139426
$11$	0	0
$11$	0	0
$12$	−23.9767	−0.576789
$13$	19.8232	0.422920	0.211460	−	0.977387i	$-0.432178\pi$
$13$	19.8232	0.422920	0.211460	+	0.977387i	$0.432178\pi$
$14$	1.77091	0.0338069
$15$	−15.0000	−0.258199
$16$	63.8134	0.997085
$17$	7.50555	0.107080	0.0535401	−	0.998566i	$-0.482950\pi$
$17$	7.50555	0.107080	0.0535401	+	0.998566i	$0.482950\pi$
$18$	−0.793627	−0.0103922
$19$	27.5515	0.332671	0.166336	−	0.986069i	$-0.446806\pi$
$19$	27.5515	0.332671	0.166336	+	0.986069i	$0.446806\pi$
$20$	39.9611	0.446779
$21$	−60.2483	−0.626060
$22$	0	0
$23$	−27.3979	−0.248385	−0.124192	−	0.992258i	$-0.539634\pi$
$23$	−27.3979	−0.248385	−0.124192	+	0.992258i	$0.539634\pi$
$24$	4.23062	0.0359821
$25$	25.0000	0.200000
$26$	−1.74802	−0.0131852
$27$	27.0000	0.192450
$28$	160.506	1.08331
$29$	−98.3333	−0.629656	−0.314828	−	0.949149i	$-0.601947\pi$
$29$	−98.3333	−0.629656	−0.314828	+	0.949149i	$0.601947\pi$
$30$	1.32271	0.00804976
$31$	88.1538	0.510738	0.255369	−	0.966844i	$-0.417803\pi$
$31$	88.1538	0.510738	0.255369	+	0.966844i	$0.417803\pi$
$32$	−16.9088	−0.0934086
$33$	0	0
$34$	−0.661845	−0.00333840
$35$	100.414	0.484944
$36$	−71.9300	−0.333009
$37$	−53.2549	−0.236623	−0.118312	−	0.992977i	$-0.537748\pi$
$37$	−53.2549	−0.236623	−0.118312	+	0.992977i	$0.537748\pi$
$38$	−2.42951	−0.0103716
$39$	59.4696	0.244173
$40$	−7.05103	−0.0278717
$41$	197.630	0.752795	0.376397	−	0.926458i	$-0.377163\pi$
$41$	197.630	0.752795	0.376397	+	0.926458i	$0.377163\pi$
$42$	5.31274	0.0195184
$43$	−37.1791	−0.131855	−0.0659274	−	0.997824i	$-0.521001\pi$
$43$	−37.1791	−0.131855	−0.0659274	+	0.997824i	$0.521001\pi$
$44$	0	0
$45$	−45.0000	−0.149071
$46$	2.41596	0.00774379
$47$	528.855	1.64131	0.820654	−	0.571426i	$-0.193609\pi$
$47$	528.855	1.64131	0.820654	+	0.571426i	$0.193609\pi$
$48$	191.440	0.575667
$49$	60.3179	0.175854
$50$	−2.20452	−0.00623532
$51$	22.5166	0.0618228
$52$	−158.431	−0.422509
$53$	−306.951	−0.795528	−0.397764	−	0.917488i	$-0.630214\pi$
$53$	−306.951	−0.795528	−0.397764	+	0.917488i	$0.630214\pi$
$54$	−2.38088	−0.00599994
$55$	0	0
$56$	−28.3209	−0.0675810
$57$	82.6546	0.192068
$58$	8.67110	0.0196305
$59$	−69.0453	−0.152355	−0.0761774	−	0.997094i	$-0.524272\pi$
$59$	−69.0453	−0.152355	−0.0761774	+	0.997094i	$0.524272\pi$
$60$	119.883	0.257948
$61$	779.999	1.63719	0.818595	−	0.574371i	$-0.194753\pi$
$61$	779.999	1.63719	0.818595	+	0.574371i	$0.194753\pi$
$62$	−7.77347	−0.0159231
$63$	−180.745	−0.361456
$64$	−509.016	−0.994173
$65$	−99.1160	−0.189136
$66$	0	0
$67$	911.553	1.66215	0.831074	−	0.556162i	$-0.187727\pi$
$67$	911.553	1.66215	0.831074	+	0.556162i	$0.187727\pi$
$68$	−59.9860	−0.106976
$69$	−82.1936	−0.143405
$70$	−8.85457	−0.0151189
$71$	−100.883	−0.168629	−0.0843145	−	0.996439i	$-0.526870\pi$
$71$	−100.883	−0.168629	−0.0843145	+	0.996439i	$0.526870\pi$
$72$	12.6919	0.0207743
$73$	−173.265	−0.277797	−0.138898	−	0.990307i	$-0.544356\pi$
$73$	−173.265	−0.277797	−0.138898	+	0.990307i	$0.544356\pi$
$74$	4.69606	0.00737710
$75$	75.0000	0.115470
$76$	−220.198	−0.332348
$77$	0	0
$78$	−5.24407	−0.00761249
$79$	−183.512	−0.261350	−0.130675	−	0.991425i	$-0.541715\pi$
$79$	−183.512	−0.261350	−0.130675	+	0.991425i	$0.541715\pi$
$80$	−319.067	−0.445910
$81$	81.0000	0.111111
$82$	−17.4271	−0.0234696
$83$	−1141.72	−1.50988	−0.754938	−	0.655797i	$-0.772333\pi$
$83$	−1141.72	−1.50988	−0.754938	+	0.655797i	$0.772333\pi$
$84$	481.518	0.625452
$85$	−37.5277	−0.0478877
$86$	3.27848	0.00411079
$87$	−295.000	−0.363532
$88$	0	0
$89$	1164.37	1.38677	0.693387	−	0.720565i	$-0.256118\pi$
$89$	1164.37	1.38677	0.693387	+	0.720565i	$0.256118\pi$
$90$	3.96813	0.00464753
$91$	−398.105	−0.458601
$92$	218.970	0.248143
$93$	264.461	0.294875
$94$	−46.6349	−0.0511704
$95$	−137.758	−0.148775
$96$	−50.7263	−0.0539295
$97$	−1207.61	−1.26406	−0.632031	−	0.774943i	$-0.717779\pi$
$97$	−1207.61	−1.26406	−0.632031	+	0.774943i	$0.717779\pi$
$98$	−5.31888	−0.00548253
$99$	0	0
$100$	−199.806	−0.199806
$101$	710.017	0.699499	0.349749	−	0.936843i	$-0.386267\pi$
$101$	710.017	0.699499	0.349749	+	0.936843i	$0.386267\pi$
$102$	−1.98553	−0.00192742
$103$	−1826.73	−1.74750	−0.873751	−	0.486373i	$-0.838320\pi$
$103$	−1826.73	−1.74750	−0.873751	+	0.486373i	$0.838320\pi$
$104$	27.9548	0.0263576
$105$	301.242	0.279983
$106$	27.0672	0.0248019
$107$	512.730	0.463248	0.231624	−	0.972805i	$-0.425596\pi$
$107$	512.730	0.463248	0.231624	+	0.972805i	$0.425596\pi$
$108$	−215.790	−0.192263
$109$	−1439.24	−1.26472	−0.632358	−	0.774676i	$-0.717913\pi$
$109$	−1439.24	−1.26472	−0.632358	+	0.774676i	$0.717913\pi$
$110$	0	0
$111$	−159.765	−0.136614
$112$	−1281.55	−1.08121
$113$	170.866	0.142245	0.0711227	−	0.997468i	$-0.477342\pi$
$113$	170.866	0.142245	0.0711227	+	0.997468i	$0.477342\pi$
$114$	−7.28854	−0.00598802
$115$	136.989	0.111081
$116$	785.902	0.629044
$117$	178.409	0.140973
$118$	6.08847	0.00474991
$119$	−150.732	−0.116114
$120$	−21.1531	−0.0160917
$121$	0	0
$122$	−68.7809	−0.0510420
$123$	592.889	0.434626
$124$	−704.545	−0.510242
$125$	−125.000	−0.0894427
$126$	15.9382	0.0112690
$127$	−345.562	−0.241446	−0.120723	−	0.992686i	$-0.538521\pi$
$127$	−345.562	−0.241446	−0.120723	+	0.992686i	$0.538521\pi$
$128$	180.156	0.124404
$129$	−111.537	−0.0761265
$130$	8.74012	0.00589661
$131$	−2130.64	−1.42103	−0.710514	−	0.703683i	$-0.751537\pi$
$131$	−2130.64	−1.42103	−0.710514	+	0.703683i	$0.751537\pi$
$132$	0	0
$133$	−553.311	−0.360738
$134$	−80.3814	−0.0518201
$135$	−135.000	−0.0860663
$136$	10.5844	0.00667355
$137$	−653.207	−0.407352	−0.203676	−	0.979038i	$-0.565289\pi$
$137$	−653.207	−0.407352	−0.203676	+	0.979038i	$0.565289\pi$
$138$	7.24789	0.00447088
$139$	−2328.17	−1.42067	−0.710334	−	0.703865i	$-0.751456\pi$
$139$	−2328.17	−1.42067	−0.710334	+	0.703865i	$0.751456\pi$
$140$	−802.530	−0.484473
$141$	1586.57	0.947609
$142$	8.89598	0.00525728
$143$	0	0
$144$	574.321	0.332362
$145$	491.666	0.281591
$146$	15.2787	0.00866076
$147$	180.954	0.101529
$148$	425.625	0.236393
$149$	−806.342	−0.443343	−0.221671	−	0.975121i	$-0.571151\pi$
$149$	−806.342	−0.443343	−0.221671	+	0.975121i	$0.571151\pi$
$150$	−6.61356	−0.00359996
$151$	−1089.49	−0.587161	−0.293580	−	0.955934i	$-0.594847\pi$
$151$	−1089.49	−0.587161	−0.293580	+	0.955934i	$0.594847\pi$
$152$	38.8533	0.0207330
$153$	67.5499	0.0356934
$154$	0	0
$155$	−440.769	−0.228409
$156$	−475.294	−0.243936
$157$	1356.15	0.689379	0.344689	−	0.938717i	$-0.387984\pi$
$157$	1356.15	0.689379	0.344689	+	0.938717i	$0.387984\pi$
$158$	16.1822	0.00814802
$159$	−920.853	−0.459298
$160$	84.5438	0.0417736
$161$	550.225	0.269340
$162$	−7.14264	−0.00346407
$163$	−1532.98	−0.736641	−0.368320	−	0.929699i	$-0.620067\pi$
$163$	−1532.98	−0.736641	−0.368320	+	0.929699i	$0.620067\pi$
$164$	−1579.50	−0.752063
$165$	0	0
$166$	100.677	0.0470728
$167$	−1194.36	−0.553429	−0.276714	−	0.960952i	$-0.589246\pi$
$167$	−1194.36	−0.553429	−0.276714	+	0.960952i	$0.589246\pi$
$168$	−84.9626	−0.0390179
$169$	−1804.04	−0.821138
$170$	3.30922	0.00149298
$171$	247.964	0.110890
$172$	297.144	0.131727
$173$	−4229.23	−1.85863	−0.929314	−	0.369291i	$-0.879601\pi$
$173$	−4229.23	−1.85863	−0.929314	+	0.369291i	$0.879601\pi$
$174$	26.0133	0.0113337
$175$	−502.069	−0.216874
$176$	0	0
$177$	−207.136	−0.0879621
$178$	−102.675	−0.0432349
$179$	2095.07	0.874821	0.437410	−	0.899262i	$-0.355896\pi$
$179$	2095.07	0.874821	0.437410	+	0.899262i	$0.355896\pi$
$180$	359.650	0.148926
$181$	3480.04	1.42911	0.714557	−	0.699577i	$-0.246628\pi$
$181$	3480.04	1.42911	0.714557	+	0.699577i	$0.246628\pi$
$182$	35.1052	0.0142976
$183$	2340.00	0.945232
$184$	−38.6366	−0.0154801
$185$	266.275	0.105821
$186$	−23.3204	−0.00919320
$187$	0	0
$188$	−4226.73	−1.63971
$189$	−542.235	−0.208687
$190$	12.1476	0.00463830
$191$	600.211	0.227381	0.113690	−	0.993516i	$-0.463733\pi$
$191$	600.211	0.227381	0.113690	+	0.993516i	$0.463733\pi$
$192$	−1527.05	−0.573986
$193$	−4193.60	−1.56405	−0.782027	−	0.623245i	$-0.785814\pi$
$193$	−4193.60	−1.56405	−0.782027	+	0.623245i	$0.785814\pi$
$194$	106.488	0.0394092
$195$	−297.348	−0.109198
$196$	−482.074	−0.175683
$197$	−1352.45	−0.489126	−0.244563	−	0.969633i	$-0.578645\pi$
$197$	−1352.45	−0.489126	−0.244563	+	0.969633i	$0.578645\pi$
$198$	0	0
$199$	2489.09	0.886669	0.443334	−	0.896356i	$-0.353795\pi$
$199$	2489.09	0.886669	0.443334	+	0.896356i	$0.353795\pi$
$200$	35.2552	0.0124646
$201$	2734.66	0.959642
$202$	−62.6099	−0.0218080
$203$	1974.80	0.682779
$204$	−179.958	−0.0617627
$205$	−988.149	−0.336660
$206$	161.082	0.0544812
$207$	−246.581	−0.0827949
$208$	1264.99	0.421688
$209$	0	0
$210$	−26.5637	−0.00872891
$211$	−5650.38	−1.84355	−0.921773	−	0.387729i	$-0.873260\pi$
$211$	−5650.38	−1.84355	−0.921773	+	0.387729i	$0.873260\pi$
$212$	2453.22	0.794755
$213$	−302.650	−0.0973580
$214$	−45.2130	−0.0144425
$215$	185.896	0.0589673
$216$	38.0756	0.0119940
$217$	−1770.37	−0.553828
$218$	126.913	0.0394296
$219$	−519.796	−0.160386
$220$	0	0
$221$	148.784	0.0452864
$222$	14.0882	0.00425917
$223$	3556.08	1.06786	0.533930	−	0.845529i	$-0.320715\pi$
$223$	3556.08	1.06786	0.533930	+	0.845529i	$0.320715\pi$
$224$	339.575	0.101289
$225$	225.000	0.0666667
$226$	−15.0671	−0.00443473
$227$	2100.75	0.614237	0.307118	−	0.951671i	$-0.400635\pi$
$227$	2100.75	0.614237	0.307118	+	0.951671i	$0.400635\pi$
$228$	−660.594	−0.191881
$229$	−1520.85	−0.438868	−0.219434	−	0.975627i	$-0.570421\pi$
$229$	−1520.85	−0.438868	−0.219434	+	0.975627i	$0.570421\pi$
$230$	−12.0798	−0.00346313
$231$	0	0
$232$	−138.670	−0.0392420
$233$	−5200.06	−1.46209	−0.731045	−	0.682329i	$-0.760967\pi$
$233$	−5200.06	−1.46209	−0.731045	+	0.682329i	$0.760967\pi$
$234$	−15.7322	−0.00439507
$235$	−2644.28	−0.734015
$236$	551.826	0.152207
$237$	−550.535	−0.150891
$238$	13.2917	0.00362005
$239$	−5727.41	−1.55011	−0.775053	−	0.631896i	$-0.782277\pi$
$239$	−5727.41	−1.55011	−0.775053	+	0.631896i	$0.782277\pi$
$240$	−957.202	−0.257446
$241$	−2985.87	−0.798077	−0.399038	−	0.916934i	$-0.630656\pi$
$241$	−2985.87	−0.798077	−0.399038	+	0.916934i	$0.630656\pi$
$242$	0	0
$243$	243.000	0.0641500
$244$	−6233.92	−1.63560
$245$	−301.589	−0.0786442
$246$	−52.2814	−0.0135502
$247$	546.159	0.140693
$248$	124.315	0.0318307
$249$	−3425.15	−0.871727
$250$	11.0226	0.00278852
$251$	−5521.86	−1.38859	−0.694296	−	0.719689i	$-0.744284\pi$
$251$	−5521.86	−1.38859	−0.694296	+	0.719689i	$0.744284\pi$
$252$	1444.55	0.361105
$253$	0	0
$254$	30.4719	0.00752747
$255$	−112.583	−0.0276480
$256$	4056.25	0.990294
$257$	3195.44	0.775587	0.387794	−	0.921746i	$-0.373237\pi$
$257$	3195.44	0.775587	0.387794	+	0.921746i	$0.373237\pi$
$258$	9.83544	0.00237336
$259$	1069.51	0.256586
$260$	792.157	0.188952
$261$	−884.999	−0.209885
$262$	187.881	0.0443028
$263$	2201.29	0.516112	0.258056	−	0.966130i	$-0.416918\pi$
$263$	2201.29	0.516112	0.258056	+	0.966130i	$0.416918\pi$
$264$	0	0
$265$	1534.76	0.355771
$266$	48.7914	0.0112466
$267$	3493.11	0.800654
$268$	−7285.34	−1.66053
$269$	1865.27	0.422780	0.211390	−	0.977402i	$-0.432201\pi$
$269$	1865.27	0.422780	0.211390	+	0.977402i	$0.432201\pi$
$270$	11.9044	0.00268325
$271$	6317.26	1.41604	0.708018	−	0.706194i	$-0.249589\pi$
$271$	6317.26	1.41604	0.708018	+	0.706194i	$0.249589\pi$
$272$	478.955	0.106768
$273$	−1194.31	−0.264774
$274$	57.6003	0.0126999
$275$	0	0
$276$	656.910	0.143266
$277$	3826.01	0.829901	0.414951	−	0.909844i	$-0.363799\pi$
$277$	3826.01	0.829901	0.414951	+	0.909844i	$0.363799\pi$
$278$	205.300	0.0442916
$279$	793.384	0.170246
$280$	141.604	0.0302231
$281$	1108.60	0.235351	0.117675	−	0.993052i	$-0.462456\pi$
$281$	1108.60	0.235351	0.117675	+	0.993052i	$0.462456\pi$
$282$	−139.905	−0.0295432
$283$	6171.64	1.29635	0.648173	−	0.761493i	$-0.275533\pi$
$283$	6171.64	1.29635	0.648173	+	0.761493i	$0.275533\pi$
$284$	806.283	0.168465
$285$	−413.273	−0.0858953
$286$	0	0
$287$	−3968.96	−0.816307
$288$	−152.179	−0.0311362
$289$	−4856.67	−0.988534
$290$	−43.3555	−0.00877904
$291$	−3622.83	−0.729807
$292$	1384.77	0.277527
$293$	6105.23	1.21731	0.608655	−	0.793435i	$-0.291710\pi$
$293$	6105.23	1.21731	0.608655	+	0.793435i	$0.291710\pi$
$294$	−15.9566	−0.00316534
$295$	345.227	0.0681352
$296$	−75.1004	−0.0147470
$297$	0	0
$298$	71.1038	0.0138219
$299$	−543.113	−0.105047
$300$	−599.417	−0.115358
$301$	746.660	0.142979
$302$	96.0719	0.0183057
$303$	2130.05	0.403856
$304$	1758.16	0.331702
$305$	−3899.99	−0.732174
$306$	−5.95660	−0.00111280
$307$	−9352.89	−1.73875	−0.869377	−	0.494149i	$-0.835480\pi$
$307$	−9352.89	−1.73875	−0.869377	+	0.494149i	$0.835480\pi$
$308$	0	0
$309$	−5480.18	−1.00892
$310$	38.8673	0.00712102
$311$	8888.35	1.62062	0.810309	−	0.586003i	$-0.199299\pi$
$311$	8888.35	1.62062	0.810309	+	0.586003i	$0.199299\pi$
$312$	83.8644	0.0152176
$313$	7425.51	1.34094	0.670471	−	0.741936i	$-0.266092\pi$
$313$	7425.51	1.34094	0.670471	+	0.741936i	$0.266092\pi$
$314$	−119.586	−0.0214925
$315$	903.725	0.161648
$316$	1466.67	0.261096
$317$	−818.391	−0.145001	−0.0725007	−	0.997368i	$-0.523098\pi$
$317$	−818.391	−0.145001	−0.0725007	+	0.997368i	$0.523098\pi$
$318$	81.2015	0.0143194
$319$	0	0
$320$	2545.08	0.444608
$321$	1538.19	0.267456
$322$	−48.5193	−0.00839712
$323$	206.789	0.0356225
$324$	−647.370	−0.111003
$325$	495.580	0.0845841
$326$	135.180	0.0229660
$327$	−4317.72	−0.730184
$328$	278.699	0.0469164
$329$	−10620.9	−1.77978
$330$	0	0
$331$	4244.26	0.704791	0.352395	−	0.935851i	$-0.385367\pi$
$331$	4244.26	0.704791	0.352395	+	0.935851i	$0.385367\pi$
$332$	9124.85	1.50841
$333$	−479.294	−0.0788744
$334$	105.320	0.0172540
$335$	−4557.77	−0.743335
$336$	−3844.65	−0.624235
$337$	3267.90	0.528230	0.264115	−	0.964491i	$-0.414920\pi$
$337$	3267.90	0.528230	0.264115	+	0.964491i	$0.414920\pi$
$338$	159.082	0.0256003
$339$	512.598	0.0821254
$340$	299.930	0.0478412
$341$	0	0
$342$	−21.8656	−0.00345719
$343$	5677.04	0.893678
$344$	−52.4302	−0.00821758
$345$	410.968	0.0641327
$346$	372.937	0.0579457
$347$	−1343.08	−0.207782	−0.103891	−	0.994589i	$-0.533129\pi$
$347$	−1343.08	−0.207782	−0.103891	+	0.994589i	$0.533129\pi$
$348$	2357.70	0.363179
$349$	2176.71	0.333859	0.166929	−	0.985969i	$-0.446615\pi$
$349$	2176.71	0.333859	0.166929	+	0.985969i	$0.446615\pi$
$350$	44.2729	0.00676138
$351$	535.226	0.0813911
$352$	0	0
$353$	−5747.94	−0.866663	−0.433332	−	0.901235i	$-0.642662\pi$
$353$	−5747.94	−0.866663	−0.433332	+	0.901235i	$0.642662\pi$
$354$	18.2654	0.00274236
$355$	504.417	0.0754132
$356$	−9305.90	−1.38543
$357$	−452.197	−0.0670386
$358$	−184.745	−0.0272739
$359$	−9826.30	−1.44460	−0.722301	−	0.691579i	$-0.756916\pi$
$359$	−9826.30	−1.44460	−0.722301	+	0.691579i	$0.756916\pi$
$360$	−63.4593	−0.00929055
$361$	−6099.91	−0.889330
$362$	−306.873	−0.0445549
$363$	0	0
$364$	3181.74	0.458156
$365$	866.326	0.124234
$366$	−206.343	−0.0294691
$367$	4005.71	0.569744	0.284872	−	0.958566i	$-0.408049\pi$
$367$	4005.71	0.569744	0.284872	+	0.958566i	$0.408049\pi$
$368$	−1748.35	−0.247661
$369$	1778.67	0.250932
$370$	−23.4803	−0.00329914
$371$	6164.43	0.862645
$372$	−2113.63	−0.294588
$373$	−3573.29	−0.496027	−0.248013	−	0.968757i	$-0.579778\pi$
$373$	−3573.29	−0.496027	−0.248013	+	0.968757i	$0.579778\pi$
$374$	0	0
$375$	−375.000	−0.0516398
$376$	745.795	0.102291
$377$	−1949.28	−0.266294
$378$	47.8147	0.00650614
$379$	7111.36	0.963816	0.481908	−	0.876222i	$-0.339944\pi$
$379$	7111.36	0.963816	0.481908	+	0.876222i	$0.339944\pi$
$380$	1100.99	0.148630
$381$	−1036.69	−0.139399
$382$	−52.9271	−0.00708897
$383$	−10596.4	−1.41371	−0.706853	−	0.707360i	$-0.749886\pi$
$383$	−10596.4	−1.41371	−0.706853	+	0.707360i	$0.749886\pi$
$384$	540.467	0.0718244
$385$	0	0
$386$	369.795	0.0487619
$387$	−334.612	−0.0439516
$388$	9651.48	1.26283
$389$	−8851.41	−1.15369	−0.576844	−	0.816855i	$-0.695716\pi$
$389$	−8851.41	−1.15369	−0.576844	+	0.816855i	$0.695716\pi$
$390$	26.2204	0.00340441
$391$	−205.636	−0.0265971
$392$	85.0607	0.0109597
$393$	−6391.91	−0.820431
$394$	119.260	0.0152493
$395$	917.559	0.116879
$396$	0	0
$397$	−7546.49	−0.954024	−0.477012	−	0.878897i	$-0.658280\pi$
$397$	−7546.49	−0.954024	−0.477012	+	0.878897i	$0.658280\pi$
$398$	−219.490	−0.0276433
$399$	−1659.93	−0.208272
$400$	1595.34	0.199417
$401$	−1850.65	−0.230466	−0.115233	−	0.993338i	$-0.536762\pi$
$401$	−1850.65	−0.230466	−0.115233	+	0.993338i	$0.536762\pi$
$402$	−241.144	−0.0299184
$403$	1747.49	0.216002
$404$	−5674.62	−0.698819
$405$	−405.000	−0.0496904
$406$	−174.140	−0.0212867
$407$	0	0
$408$	31.7531	0.00385297
$409$	5802.46	0.701499	0.350750	−	0.936469i	$-0.385927\pi$
$409$	5802.46	0.701499	0.350750	+	0.936469i	$0.385927\pi$
$410$	87.1357	0.0104959
$411$	−1959.62	−0.235185
$412$	14599.6	1.74580
$413$	1386.62	0.165209
$414$	21.7437	0.00258126
$415$	5708.58	0.675237
$416$	−335.186	−0.0395044
$417$	−6984.51	−0.820223
$418$	0	0
$419$	16642.8	1.94046	0.970230	−	0.242185i	$-0.0778642\pi$
$419$	16642.8	1.94046	0.970230	+	0.242185i	$0.0778642\pi$
$420$	−2407.59	−0.279710
$421$	−13385.0	−1.54951	−0.774755	−	0.632262i	$-0.782127\pi$
$421$	−13385.0	−1.54951	−0.774755	+	0.632262i	$0.782127\pi$
$422$	498.255	0.0574755
$423$	4759.70	0.547103
$424$	−432.864	−0.0495796
$425$	187.639	0.0214160
$426$	26.6879	0.00303529
$427$	−15664.5	−1.77532
$428$	−4097.86	−0.462798
$429$	0	0
$430$	−16.3924	−0.00183840
$431$	−14706.6	−1.64360	−0.821800	−	0.569775i	$-0.807030\pi$
$431$	−14706.6	−1.64360	−0.821800	+	0.569775i	$0.807030\pi$
$432$	1722.96	0.191889
$433$	−2910.63	−0.323039	−0.161520	−	0.986869i	$-0.551640\pi$
$433$	−2910.63	−0.323039	−0.161520	+	0.986869i	$0.551640\pi$
$434$	156.113	0.0172665
$435$	1475.00	0.162577
$436$	11502.7	1.26349
$437$	−754.853	−0.0826305
$438$	45.8360	0.00500029
$439$	−14928.0	−1.62294	−0.811472	−	0.584392i	$-0.801333\pi$
$439$	−14928.0	−1.62294	−0.811472	+	0.584392i	$0.801333\pi$
$440$	0	0
$441$	542.861	0.0586180
$442$	−13.1199	−0.00141188
$443$	−3688.74	−0.395614	−0.197807	−	0.980241i	$-0.563382\pi$
$443$	−3688.74	−0.395614	−0.197807	+	0.980241i	$0.563382\pi$
$444$	1276.88	0.136482
$445$	−5821.85	−0.620184
$446$	−313.578	−0.0332922
$447$	−2419.03	−0.255964
$448$	10222.5	1.07805
$449$	4403.63	0.462852	0.231426	−	0.972853i	$-0.425661\pi$
$449$	4403.63	0.462852	0.231426	+	0.972853i	$0.425661\pi$
$450$	−19.8407	−0.00207844
$451$	0	0
$452$	−1365.60	−0.142107
$453$	−3268.46	−0.338997
$454$	−185.246	−0.0191498
$455$	1990.52	0.205093
$456$	116.560	0.0119702
$457$	5738.40	0.587376	0.293688	−	0.955901i	$-0.405117\pi$
$457$	5738.40	0.587376	0.293688	+	0.955901i	$0.405117\pi$
$458$	134.110	0.0136824
$459$	202.650	0.0206076
$460$	−1094.85	−0.110973
$461$	18259.4	1.84474	0.922372	−	0.386303i	$-0.126248\pi$
$461$	18259.4	1.84474	0.922372	+	0.386303i	$0.126248\pi$
$462$	0	0
$463$	13374.3	1.34246	0.671229	−	0.741250i	$-0.265767\pi$
$463$	13374.3	1.34246	0.671229	+	0.741250i	$0.265767\pi$
$464$	−6274.98	−0.627821
$465$	−1322.31	−0.131872
$466$	458.545	0.0455830
$467$	−8172.00	−0.809753	−0.404877	−	0.914371i	$-0.632686\pi$
$467$	−8172.00	−0.809753	−0.404877	+	0.914371i	$0.632686\pi$
$468$	−1425.88	−0.140836
$469$	−18306.5	−1.80238
$470$	233.174	0.0228841
$471$	4068.45	0.398013
$472$	−97.3682	−0.00949520
$473$	0	0
$474$	48.5466	0.00470426
$475$	688.788	0.0665342
$476$	1204.69	0.116001
$477$	−2762.56	−0.265176
$478$	505.047	0.0483270
$479$	−13543.8	−1.29192	−0.645961	−	0.763371i	$-0.723543\pi$
$479$	−13543.8	−1.29192	−0.645961	+	0.763371i	$0.723543\pi$
$480$	253.632	0.0241180
$481$	−1055.68	−0.100073
$482$	263.296	0.0248813
$483$	1650.68	0.155504
$484$	0	0
$485$	6038.04	0.565306
$486$	−21.4279	−0.00199998
$487$	5849.84	0.544315	0.272157	−	0.962253i	$-0.412263\pi$
$487$	5849.84	0.544315	0.272157	+	0.962253i	$0.412263\pi$
$488$	1099.96	0.102034
$489$	−4598.95	−0.425300
$490$	26.5944	0.00245186
$491$	−5006.46	−0.460159	−0.230080	−	0.973172i	$-0.573899\pi$
$491$	−5006.46	−0.460159	−0.230080	+	0.973172i	$0.573899\pi$
$492$	−4738.51	−0.434204
$493$	−738.045	−0.0674237
$494$	−48.1607	−0.00438634
$495$	0	0
$496$	5625.40	0.509250
$497$	2026.02	0.182856
$498$	302.032	0.0271775
$499$	5115.41	0.458912	0.229456	−	0.973319i	$-0.426305\pi$
$499$	5115.41	0.458912	0.229456	+	0.973319i	$0.426305\pi$
$500$	999.028	0.0893558
$501$	−3583.09	−0.319522
$502$	486.922	0.0432916
$503$	3234.06	0.286679	0.143340	−	0.989674i	$-0.454216\pi$
$503$	3234.06	0.286679	0.143340	+	0.989674i	$0.454216\pi$
$504$	−254.888	−0.0225270
$505$	−3550.09	−0.312825
$506$	0	0
$507$	−5412.12	−0.474084
$508$	2761.81	0.241212
$509$	6199.30	0.539841	0.269921	−	0.962883i	$-0.413003\pi$
$509$	6199.30	0.539841	0.269921	+	0.962883i	$0.413003\pi$
$510$	9.92767	0.000861970 0
$511$	3479.65	0.301234
$512$	−1798.93	−0.155278
$513$	743.891	0.0640226
$514$	−281.776	−0.0241802
$515$	9133.64	0.781507
$516$	891.431	0.0760525
$517$	0	0
$518$	−94.3099	−0.00799950
$519$	−12687.7	−1.07308
$520$	−139.774	−0.0117875
$521$	−7540.13	−0.634048	−0.317024	−	0.948417i	$-0.602684\pi$
$521$	−7540.13	−0.634048	−0.317024	+	0.948417i	$0.602684\pi$
$522$	78.0399	0.00654351
$523$	3051.98	0.255170	0.127585	−	0.991828i	$-0.459277\pi$
$523$	3051.98	0.255170	0.127585	+	0.991828i	$0.459277\pi$
$524$	17028.5	1.41965
$525$	−1506.21	−0.125212
$526$	−194.111	−0.0160906
$527$	661.642	0.0546899
$528$	0	0
$529$	−11416.4	−0.938305
$530$	−135.336	−0.0110917
$531$	−621.408	−0.0507850
$532$	4422.19	0.360387
$533$	3917.65	0.318372
$534$	−308.025	−0.0249617
$535$	−2563.65	−0.207171
$536$	1285.48	0.103590
$537$	6285.21	0.505078
$538$	−164.481	−0.0131808
$539$	0	0
$540$	1078.95	0.0859826
$541$	16403.8	1.30361	0.651805	−	0.758387i	$-0.274012\pi$
$541$	16403.8	1.30361	0.651805	+	0.758387i	$0.274012\pi$
$542$	−557.060	−0.0441472
$543$	10440.1	0.825100
$544$	−126.910	−0.0100022
$545$	7196.20	0.565598
$546$	105.316	0.00825474
$547$	−2461.16	−0.192379	−0.0961896	−	0.995363i	$-0.530666\pi$
$547$	−2461.16	−0.192379	−0.0961896	+	0.995363i	$0.530666\pi$
$548$	5220.58	0.406956
$549$	7019.99	0.545730
$550$	0	0
$551$	−2709.23	−0.209469
$552$	−115.910	−0.00893742
$553$	3685.43	0.283400
$554$	−337.380	−0.0258735
$555$	798.824	0.0610958
$556$	18607.3	1.41929
$557$	−12014.8	−0.913974	−0.456987	−	0.889473i	$-0.651071\pi$
$557$	−12014.8	−0.913974	−0.456987	+	0.889473i	$0.651071\pi$
$558$	−69.9612	−0.00530770
$559$	−737.009	−0.0557641
$560$	6407.76	0.483530
$561$	0	0
$562$	−97.7571	−0.00733743
$563$	−23892.2	−1.78852	−0.894258	−	0.447552i	$-0.852296\pi$
$563$	−23892.2	−1.78852	−0.894258	+	0.447552i	$0.852296\pi$
$564$	−12680.2	−0.946688
$565$	−854.331	−0.0636141
$566$	−544.220	−0.0404157
$567$	−1626.70	−0.120485
$568$	−142.266	−0.0105095
$569$	−18934.8	−1.39506	−0.697529	−	0.716556i	$-0.745717\pi$
$569$	−18934.8	−1.39506	−0.697529	+	0.716556i	$0.745717\pi$
$570$	36.4427	0.00267793
$571$	16650.5	1.22032	0.610158	−	0.792280i	$-0.291106\pi$
$571$	16650.5	1.22032	0.610158	+	0.792280i	$0.291106\pi$
$572$	0	0
$573$	1800.63	0.131278
$574$	349.985	0.0254497
$575$	−684.947	−0.0496769
$576$	−4581.15	−0.331391
$577$	−17564.5	−1.26728	−0.633638	−	0.773630i	$-0.718439\pi$
$577$	−17564.5	−1.26728	−0.633638	+	0.773630i	$0.718439\pi$
$578$	428.265	0.0308191
$579$	−12580.8	−0.903006
$580$	−3929.51	−0.281317
$581$	22928.8	1.63726
$582$	319.463	0.0227529
$583$	0	0
$584$	−244.340	−0.0173131
$585$	−892.044	−0.0630452
$586$	−538.364	−0.0379516
$587$	6037.02	0.424488	0.212244	−	0.977217i	$-0.431923\pi$
$587$	6037.02	0.424488	0.212244	+	0.977217i	$0.431923\pi$
$588$	−1446.22	−0.101431
$589$	2428.77	0.169908
$590$	−30.4423	−0.00212422
$591$	−4057.34	−0.282397
$592$	−3398.38	−0.235933
$593$	−10694.3	−0.740579	−0.370289	−	0.928916i	$-0.620741\pi$
$593$	−10694.3	−0.740579	−0.370289	+	0.928916i	$0.620741\pi$
$594$	0	0
$595$	753.661	0.0519279
$596$	6444.47	0.442912
$597$	7467.28	0.511919
$598$	47.8921	0.00327501
$599$	−2073.12	−0.141411	−0.0707057	−	0.997497i	$-0.522525\pi$
$599$	−2073.12	−0.141411	−0.0707057	+	0.997497i	$0.522525\pi$
$600$	105.765	0.00719643
$601$	−3402.66	−0.230944	−0.115472	−	0.993311i	$-0.536838\pi$
$601$	−3402.66	−0.230944	−0.115472	+	0.993311i	$0.536838\pi$
$602$	−65.8410	−0.00445761
$603$	8203.98	0.554049
$604$	8707.43	0.586590
$605$	0	0
$606$	−187.830	−0.0125909
$607$	9287.61	0.621042	0.310521	−	0.950567i	$-0.399497\pi$
$607$	9287.61	0.621042	0.310521	+	0.950567i	$0.399497\pi$
$608$	−465.862	−0.0310744
$609$	5924.41	0.394203
$610$	343.904	0.0228267
$611$	10483.6	0.694142
$612$	−539.874	−0.0356587
$613$	10648.9	0.701642	0.350821	−	0.936443i	$-0.385903\pi$
$613$	10648.9	0.701642	0.350821	+	0.936443i	$0.385903\pi$
$614$	824.745	0.0542085
$615$	−2964.45	−0.194371
$616$	0	0
$617$	−17340.5	−1.13145	−0.565724	−	0.824595i	$-0.691403\pi$
$617$	−17340.5	−1.13145	−0.565724	+	0.824595i	$0.691403\pi$
$618$	483.247	0.0314547
$619$	−14103.7	−0.915794	−0.457897	−	0.889005i	$-0.651397\pi$
$619$	−14103.7	−0.915794	−0.457897	+	0.889005i	$0.651397\pi$
$620$	3522.72	0.228187
$621$	−739.742	−0.0478017
$622$	−783.782	−0.0505254
$623$	−23383.8	−1.50377
$624$	3794.96	0.243461
$625$	625.000	0.0400000
$626$	−654.787	−0.0418060
$627$	0	0
$628$	−10838.6	−0.688708
$629$	−399.707	−0.0253376
$630$	−79.6911	−0.00503964
$631$	−18456.1	−1.16438	−0.582192	−	0.813051i	$-0.697805\pi$
$631$	−18456.1	−1.16438	−0.582192	+	0.813051i	$0.697805\pi$
$632$	−258.789	−0.0162881
$633$	−16951.1	−1.06437
$634$	72.1663	0.00452065
$635$	1727.81	0.107978
$636$	7359.67	0.458852
$637$	1195.69	0.0743722
$638$	0	0
$639$	−907.951	−0.0562097
$640$	−900.778	−0.0556350
$641$	−31761.6	−1.95711	−0.978555	−	0.205985i	$-0.933960\pi$
$641$	−31761.6	−1.95711	−0.978555	+	0.205985i	$0.933960\pi$
$642$	−135.639	−0.00833838
$643$	−17274.1	−1.05944	−0.529722	−	0.848172i	$-0.677704\pi$
$643$	−17274.1	−1.05944	−0.529722	+	0.848172i	$0.677704\pi$
$644$	−4397.52	−0.269079
$645$	557.687	0.0340448
$646$	−18.2348	−0.00111059
$647$	27736.8	1.68539	0.842695	−	0.538391i	$-0.180967\pi$
$647$	27736.8	1.68539	0.842695	+	0.538391i	$0.180967\pi$
$648$	114.227	0.00692477
$649$	0	0
$650$	−43.7006	−0.00263704
$651$	−5311.12	−0.319753
$652$	12251.9	0.735925
$653$	21586.8	1.29366	0.646828	−	0.762636i	$-0.276095\pi$
$653$	21586.8	1.29366	0.646828	+	0.762636i	$0.276095\pi$
$654$	380.740	0.0227647
$655$	10653.2	0.635503
$656$	12611.4	0.750601
$657$	−1559.39	−0.0925989
$658$	936.557	0.0554875
$659$	−4944.01	−0.292248	−0.146124	−	0.989266i	$-0.546680\pi$
$659$	−4944.01	−0.292248	−0.146124	+	0.989266i	$0.546680\pi$
$660$	0	0
$661$	11172.9	0.657452	0.328726	−	0.944425i	$-0.393381\pi$
$661$	11172.9	0.657452	0.328726	+	0.944425i	$0.393381\pi$
$662$	−374.262	−0.0219730
$663$	446.352	0.0261461
$664$	−1610.06	−0.0940998
$665$	2766.56	0.161327
$666$	42.2645	0.00245903
$667$	2694.12	0.156397
$668$	9545.62	0.552891
$669$	10668.2	0.616529
$670$	401.907	0.0231747
$671$	0	0
$672$	1018.72	0.0584794
$673$	25068.3	1.43583	0.717914	−	0.696132i	$-0.245097\pi$
$673$	25068.3	1.43583	0.717914	+	0.696132i	$0.245097\pi$
$674$	−288.166	−0.0164684
$675$	675.000	0.0384900
$676$	14418.3	0.820340
$677$	−24068.6	−1.36637	−0.683183	−	0.730247i	$-0.739405\pi$
$677$	−24068.6	−1.36637	−0.683183	+	0.730247i	$0.739405\pi$
$678$	−45.2013	−0.00256039
$679$	24252.1	1.37071
$680$	−52.9219	−0.00298450
$681$	6302.26	0.354630
$682$	0	0
$683$	5099.82	0.285709	0.142855	−	0.989744i	$-0.454372\pi$
$683$	5099.82	0.285709	0.142855	+	0.989744i	$0.454372\pi$
$684$	−1981.78	−0.110783
$685$	3266.04	0.182173
$686$	−500.606	−0.0278618
$687$	−4562.55	−0.253380
$688$	−2372.53	−0.131471
$689$	−6084.75	−0.336445
$690$	−36.2395	−0.00199944
$691$	10323.0	0.568313	0.284157	−	0.958778i	$-0.408286\pi$
$691$	10323.0	0.568313	0.284157	+	0.958778i	$0.408286\pi$
$692$	33801.0	1.85682
$693$	0	0
$694$	118.434	0.00647794
$695$	11640.8	0.635342
$696$	−416.011	−0.0226564
$697$	1483.32	0.0806094
$698$	−191.944	−0.0104086
$699$	−15600.2	−0.844139
$700$	4012.65	0.216663
$701$	−25771.4	−1.38855	−0.694274	−	0.719711i	$-0.744274\pi$
$701$	−25771.4	−1.38855	−0.694274	+	0.719711i	$0.744274\pi$
$702$	−47.1967	−0.00253750
$703$	−1467.25	−0.0787177
$704$	0	0
$705$	−7932.83	−0.423784
$706$	506.858	0.0270196
$707$	−14259.1	−0.758514
$708$	1655.48	0.0878766
$709$	−5995.56	−0.317586	−0.158793	−	0.987312i	$-0.550760\pi$
$709$	−5995.56	−0.317586	−0.158793	+	0.987312i	$0.550760\pi$
$710$	−44.4799	−0.00235113
$711$	−1651.61	−0.0871168
$712$	1642.00	0.0864278
$713$	−2415.23	−0.126860
$714$	39.8750	0.00209004
$715$	0	0
$716$	−16744.3	−0.873970
$717$	−17182.2	−0.894954
$718$	866.490	0.0450378
$719$	−807.103	−0.0418635	−0.0209318	−	0.999781i	$-0.506663\pi$
$719$	−807.103	−0.0418635	−0.0209318	+	0.999781i	$0.506663\pi$
$720$	−2871.60	−0.148637
$721$	36685.8	1.89494
$722$	537.895	0.0277263
$723$	−8957.60	−0.460770
$724$	−27813.3	−1.42773
$725$	−2458.33	−0.125931
$726$	0	0
$727$	17369.5	0.886108	0.443054	−	0.896495i	$-0.353895\pi$
$727$	17369.5	0.886108	0.443054	+	0.896495i	$0.353895\pi$
$728$	−561.410	−0.0285814
$729$	729.000	0.0370370
$730$	−76.3933	−0.00387321
$731$	−279.050	−0.0141190
$732$	−18701.8	−0.944313
$733$	−4366.09	−0.220007	−0.110004	−	0.993931i	$-0.535086\pi$
$733$	−4366.09	−0.220007	−0.110004	+	0.993931i	$0.535086\pi$
$734$	−353.226	−0.0177627
$735$	−904.768	−0.0454053
$736$	463.264	0.0232013
$737$	0	0
$738$	−156.844	−0.00782320
$739$	20634.7	1.02714	0.513572	−	0.858046i	$-0.328322\pi$
$739$	20634.7	1.02714	0.513572	+	0.858046i	$0.328322\pi$
$740$	−2128.13	−0.105718
$741$	1638.48	0.0812294
$742$	−543.584	−0.0268943
$743$	6525.27	0.322192	0.161096	−	0.986939i	$-0.448497\pi$
$743$	6525.27	0.322192	0.161096	+	0.986939i	$0.448497\pi$
$744$	372.945	0.0183775
$745$	4031.71	0.198269
$746$	315.096	0.0154644
$747$	−10275.4	−0.503292
$748$	0	0
$749$	−10297.1	−0.502331
$750$	33.0678	0.00160995
$751$	−17472.1	−0.848958	−0.424479	−	0.905438i	$-0.639543\pi$
$751$	−17472.1	−0.848958	−0.424479	+	0.905438i	$0.639543\pi$
$752$	33748.1	1.63652
$753$	−16565.6	−0.801704
$754$	171.889	0.00830216
$755$	5447.44	0.262586
$756$	4333.66	0.208484
$757$	−22721.3	−1.09091	−0.545456	−	0.838140i	$-0.683643\pi$
$757$	−22721.3	−1.09091	−0.545456	+	0.838140i	$0.683643\pi$
$758$	−627.086	−0.0300485
$759$	0	0
$760$	−194.267	−0.00927210
$761$	−10151.1	−0.483544	−0.241772	−	0.970333i	$-0.577729\pi$
$761$	−10151.1	−0.483544	−0.241772	+	0.970333i	$0.577729\pi$
$762$	91.4157	0.00434599
$763$	28903.9	1.37142
$764$	−4797.02	−0.227160
$765$	−337.750	−0.0159626
$766$	934.396	0.0440746
$767$	−1368.70	−0.0644340
$768$	12168.7	0.571747
$769$	41155.8	1.92993	0.964965	−	0.262378i	$-0.0845068\pi$
$769$	41155.8	1.92993	0.964965	+	0.262378i	$0.0845068\pi$
$770$	0	0
$771$	9586.32	0.447786
$772$	33516.2	1.56253
$773$	−19318.3	−0.898875	−0.449437	−	0.893312i	$-0.648375\pi$
$773$	−19318.3	−0.898875	−0.449437	+	0.893312i	$0.648375\pi$
$774$	29.5063	0.00137026
$775$	2203.84	0.102148
$776$	−1702.98	−0.0787800
$777$	3208.52	0.148140
$778$	780.524	0.0359681
$779$	5445.00	0.250433
$780$	2376.47	0.109091
$781$	0	0
$782$	18.1331	0.000829207 0
$783$	−2655.00	−0.121177
$784$	3849.09	0.175341
$785$	−6780.74	−0.308299
$786$	563.644	0.0255782
$787$	−17984.4	−0.814579	−0.407290	−	0.913299i	$-0.633526\pi$
$787$	−17984.4	−0.814579	−0.407290	+	0.913299i	$0.633526\pi$
$788$	10809.1	0.488651
$789$	6603.87	0.297977
$790$	−80.9110	−0.00364391
$791$	−3431.47	−0.154246
$792$	0	0
$793$	15462.1	0.692401
$794$	665.455	0.0297432
$795$	4604.27	0.205404
$796$	−19893.4	−0.885807
$797$	−23632.6	−1.05033	−0.525164	−	0.851001i	$-0.675996\pi$
$797$	−23632.6	−1.05033	−0.525164	+	0.851001i	$0.675996\pi$
$798$	146.374	0.00649322
$799$	3969.35	0.175751
$800$	−422.719	−0.0186817
$801$	10479.3	0.462258
$802$	163.192	0.00718516
$803$	0	0
$804$	−21856.0	−0.958709
$805$	−2751.13	−0.120453
$806$	−154.095	−0.00673420
$807$	5595.82	0.244092
$808$	1001.27	0.0435948
$809$	−27832.2	−1.20955	−0.604777	−	0.796395i	$-0.706738\pi$
$809$	−27832.2	−1.20955	−0.604777	+	0.796395i	$0.706738\pi$
$810$	35.7132	0.00154918
$811$	17855.1	0.773092	0.386546	−	0.922270i	$-0.373668\pi$
$811$	17855.1	0.773092	0.386546	+	0.922270i	$0.373668\pi$
$812$	−15783.1	−0.682115
$813$	18951.8	0.817549
$814$	0	0
$815$	7664.91	0.329436
$816$	1436.86	0.0616425
$817$	−1024.34	−0.0438643
$818$	−511.665	−0.0218704
$819$	−3582.94	−0.152867
$820$	7897.51	0.336333
$821$	32111.6	1.36505	0.682523	−	0.730864i	$-0.260883\pi$
$821$	32111.6	1.36505	0.682523	+	0.730864i	$0.260883\pi$
$822$	172.801	0.00733227
$823$	−10222.5	−0.432968	−0.216484	−	0.976286i	$-0.569459\pi$
$823$	−10222.5	−0.432968	−0.216484	+	0.976286i	$0.569459\pi$
$824$	−2576.06	−0.108909
$825$	0	0
$826$	−122.273	−0.00515065
$827$	12783.0	0.537494	0.268747	−	0.963211i	$-0.413390\pi$
$827$	12783.0	0.537494	0.268747	+	0.963211i	$0.413390\pi$
$828$	1970.73	0.0827144
$829$	16007.9	0.670661	0.335330	−	0.942101i	$-0.391152\pi$
$829$	16007.9	0.670661	0.335330	+	0.942101i	$0.391152\pi$
$830$	−503.387	−0.0210516
$831$	11478.0	0.479144
$832$	−10090.3	−0.420456
$833$	452.719	0.0188305
$834$	615.899	0.0255718
$835$	5971.82	0.247501
$836$	0	0
$837$	2380.15	0.0982916
$838$	−1467.57	−0.0604970
$839$	27812.2	1.14444	0.572219	−	0.820101i	$-0.306083\pi$
$839$	27812.2	1.14444	0.572219	+	0.820101i	$0.306083\pi$
$840$	424.813	0.0174493
$841$	−14719.6	−0.603533
$842$	1180.30	0.0483084
$843$	3325.80	0.135880
$844$	45159.1	1.84175
$845$	9020.20	0.367224
$846$	−419.714	−0.0170568
$847$	0	0
$848$	−19587.6	−0.793209
$849$	18514.9	0.748446
$850$	−16.5461	−0.000667679 0
$851$	1459.07	0.0587736
$852$	2418.85	0.0972634
$853$	12732.9	0.511096	0.255548	−	0.966796i	$-0.417744\pi$
$853$	12732.9	0.511096	0.255548	+	0.966796i	$0.417744\pi$
$854$	1381.31	0.0553483
$855$	−1239.82	−0.0495917
$856$	723.056	0.0288710
$857$	33688.8	1.34281	0.671404	−	0.741092i	$-0.265692\pi$
$857$	33688.8	1.34281	0.671404	+	0.741092i	$0.265692\pi$
$858$	0	0
$859$	9411.51	0.373826	0.186913	−	0.982376i	$-0.440152\pi$
$859$	9411.51	0.373826	0.186913	+	0.982376i	$0.440152\pi$
$860$	−1485.72	−0.0589100
$861$	−11906.9	−0.471295
$862$	1296.84	0.0512419
$863$	−7657.78	−0.302055	−0.151028	−	0.988530i	$-0.548258\pi$
$863$	−7657.78	−0.302055	−0.151028	+	0.988530i	$0.548258\pi$
$864$	−456.537	−0.0179765
$865$	21146.2	0.831204
$866$	256.662	0.0100713
$867$	−14570.0	−0.570730
$868$	14149.2	0.553290
$869$	0	0
$870$	−130.067	−0.00506858
$871$	18069.9	0.702956
$872$	−2029.62	−0.0788208
$873$	−10868.5	−0.421354
$874$	66.5635	0.00257614
$875$	2510.35	0.0969888
$876$	4154.32	0.160230
$877$	41605.1	1.60194	0.800972	−	0.598701i	$-0.204316\pi$
$877$	41605.1	1.60194	0.800972	+	0.598701i	$0.204316\pi$
$878$	1316.36	0.0505979
$879$	18315.7	0.702814
$880$	0	0
$881$	16888.3	0.645835	0.322918	−	0.946427i	$-0.395336\pi$
$881$	16888.3	0.645835	0.322918	+	0.946427i	$0.395336\pi$
$882$	−47.8699	−0.00182751
$883$	−39086.7	−1.48966	−0.744832	−	0.667252i	$-0.767470\pi$
$883$	−39086.7	−1.48966	−0.744832	+	0.667252i	$0.767470\pi$
$884$	−1189.11	−0.0452424
$885$	1035.68	0.0393379
$886$	325.276	0.0123339
$887$	−33161.6	−1.25531	−0.627654	−	0.778492i	$-0.715985\pi$
$887$	−33161.6	−1.25531	−0.627654	+	0.778492i	$0.715985\pi$
$888$	−225.301	−0.00851421
$889$	6939.84	0.261817
$890$	513.375	0.0193352
$891$	0	0
$892$	−28421.0	−1.06682
$893$	14570.8	0.546016
$894$	213.312	0.00798009
$895$	−10475.4	−0.391232
$896$	−3618.02	−0.134899
$897$	−1629.34	−0.0606489
$898$	−388.316	−0.0144301
$899$	−8668.45	−0.321590
$900$	−1798.25	−0.0666019
$901$	−2303.84	−0.0851852
$902$	0	0
$903$	2239.98	0.0825491
$904$	240.956	0.00886515
$905$	−17400.2	−0.639119
$906$	288.216	0.0105688
$907$	17009.5	0.622702	0.311351	−	0.950295i	$-0.399218\pi$
$907$	17009.5	0.622702	0.311351	+	0.950295i	$0.399218\pi$
$908$	−16789.7	−0.613640
$909$	6390.16	0.233166
$910$	−175.526	−0.00639410
$911$	38785.7	1.41057	0.705283	−	0.708926i	$-0.250820\pi$
$911$	38785.7	1.41057	0.705283	+	0.708926i	$0.250820\pi$
$912$	5274.47	0.191508
$913$	0	0
$914$	−506.016	−0.0183124
$915$	−11700.0	−0.422721
$916$	12155.0	0.438441
$917$	42789.1	1.54092
$918$	−17.8698	−0.000642475 0
$919$	28931.7	1.03849	0.519244	−	0.854626i	$-0.326214\pi$
$919$	28931.7	1.03849	0.519244	+	0.854626i	$0.326214\pi$
$920$	193.183	0.00692289
$921$	−28058.7	−1.00387
$922$	−1610.13	−0.0575129
$923$	−1999.83	−0.0713167
$924$	0	0
$925$	−1331.37	−0.0473246
$926$	−1179.36	−0.0418533
$927$	−16440.5	−0.582501
$928$	1662.69	0.0588153
$929$	31733.3	1.12071	0.560353	−	0.828254i	$-0.310666\pi$
$929$	31733.3	1.12071	0.560353	+	0.828254i	$0.310666\pi$
$930$	116.602	0.00411132
$931$	1661.85	0.0585015
$932$	41560.0	1.46067
$933$	26665.1	0.935664
$934$	720.613	0.0252454
$935$	0	0
$936$	251.593	0.00878588
$937$	0.423803	1.47759e−5 0	7.38797e−6	−	1.00000i	$-0.499998\pi$
$937$	0.423803	1.47759e−5 0	7.38797e−6		1.00000i	$0.499998\pi$
$938$	1614.28	0.0561921
$939$	22276.5	0.774193
$940$	21133.6	0.733302
$941$	33841.6	1.17237	0.586187	−	0.810176i	$-0.300628\pi$
$941$	33841.6	1.17237	0.586187	+	0.810176i	$0.300628\pi$
$942$	−358.759	−0.0124087
$943$	−5414.63	−0.186983
$944$	−4406.02	−0.151911
$945$	2711.17	0.0933275
$946$	0	0
$947$	−42055.3	−1.44310	−0.721549	−	0.692363i	$-0.756570\pi$
$947$	−42055.3	−1.44310	−0.721549	+	0.692363i	$0.756570\pi$
$948$	4400.00	0.150744
$949$	−3434.67	−0.117486
$950$	−60.7379	−0.00207431
$951$	−2455.17	−0.0837165
$952$	−212.564	−0.00723658
$953$	−55168.4	−1.87521	−0.937607	−	0.347697i	$-0.886964\pi$
$953$	−55168.4	−1.87521	−0.937607	+	0.347697i	$0.886964\pi$
$954$	243.605	0.00826729
$955$	−3001.06	−0.101688
$956$	45774.7	1.54860
$957$	0	0
$958$	1194.30	0.0402777
$959$	13118.2	0.441720
$960$	7635.25	0.256694
$961$	−22019.9	−0.739146
$962$	93.0909	0.00311993
$963$	4614.57	0.154416
$964$	23863.7	0.797301
$965$	20968.0	0.699466
$966$	−145.558	−0.00484808
$967$	−44036.1	−1.46443	−0.732217	−	0.681072i	$-0.761514\pi$
$967$	−44036.1	−1.46443	−0.732217	+	0.681072i	$0.761514\pi$
$968$	0	0
$969$	620.368	0.0205667
$970$	−532.439	−0.0176243
$971$	−3016.24	−0.0996867	−0.0498433	−	0.998757i	$-0.515872\pi$
$971$	−3016.24	−0.0996867	−0.0498433	+	0.998757i	$0.515872\pi$
$972$	−1942.11	−0.0640877
$973$	46756.1	1.54053
$974$	−515.843	−0.0169699
$975$	1486.74	0.0488346
$976$	49774.4	1.63242
$977$	14131.0	0.462733	0.231367	−	0.972867i	$-0.425680\pi$
$977$	14131.0	0.462733	0.231367	+	0.972867i	$0.425680\pi$
$978$	405.539	0.0132594
$979$	0	0
$980$	2410.37	0.0785678
$981$	−12953.2	−0.421572
$982$	441.473	0.0143462
$983$	−58080.2	−1.88451	−0.942253	−	0.334902i	$-0.891297\pi$
$983$	−58080.2	−1.88451	−0.942253	+	0.334902i	$0.891297\pi$
$984$	836.096	0.0270872
$985$	6762.24	0.218744
$986$	65.0814	0.00210204
$987$	−31862.6	−1.02756
$988$	−4365.03	−0.140557
$989$	1018.63	0.0327507
$990$	0	0
$991$	56900.8	1.82393	0.911965	−	0.410268i	$-0.134565\pi$
$991$	56900.8	1.82393	0.911965	+	0.410268i	$0.134565\pi$
$992$	−1490.57	−0.0477074
$993$	12732.8	0.406911
$994$	−178.656	−0.00570083
$995$	−12445.5	−0.396530
$996$	27374.6	0.870879
$997$	−37109.5	−1.17880	−0.589402	−	0.807840i	$-0.700637\pi$
$997$	−37109.5	−1.17880	−0.589402	+	0.807840i	$0.700637\pi$
$998$	−451.081	−0.0143073
$999$	−1437.88	−0.0455381

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1815.4.a.bg.1.7		12
11.5	even	5		165.4.m.d.91.3	✓	24
11.9	even	5		165.4.m.d.136.3	yes	24
11.10	odd	2		1815.4.a.bo.1.6		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
165.4.m.d.91.3	✓	24	11.5	even	5
165.4.m.d.136.3	yes	24	11.9	even	5
1815.4.a.bg.1.7		12	1.1	even	1	trivial
1815.4.a.bo.1.6		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)

\(f(q)\)	\(=\)	\(q-0.0881808 q^{2} +3.00000 q^{3} -7.99222 q^{4} -5.00000 q^{5} -0.264542 q^{6} -20.0828 q^{7} +1.41021 q^{8} +9.00000 q^{9} +O(q^{10})\)	\(q-0.0881808 q^{2} +3.00000 q^{3} -7.99222 q^{4} -5.00000 q^{5} -0.264542 q^{6} -20.0828 q^{7} +1.41021 q^{8} +9.00000 q^{9} +0.440904 q^{10} -23.9767 q^{12} +19.8232 q^{13} +1.77091 q^{14} -15.0000 q^{15} +63.8134 q^{16} +7.50555 q^{17} -0.793627 q^{18} +27.5515 q^{19} +39.9611 q^{20} -60.2483 q^{21} -27.3979 q^{23} +4.23062 q^{24} +25.0000 q^{25} -1.74802 q^{26} +27.0000 q^{27} +160.506 q^{28} -98.3333 q^{29} +1.32271 q^{30} +88.1538 q^{31} -16.9088 q^{32} -0.661845 q^{34} +100.414 q^{35} -71.9300 q^{36} -53.2549 q^{37} -2.42951 q^{38} +59.4696 q^{39} -7.05103 q^{40} +197.630 q^{41} +5.31274 q^{42} -37.1791 q^{43} -45.0000 q^{45} +2.41596 q^{46} +528.855 q^{47} +191.440 q^{48} +60.3179 q^{49} -2.20452 q^{50} +22.5166 q^{51} -158.431 q^{52} -306.951 q^{53} -2.38088 q^{54} -28.3209 q^{56} +82.6546 q^{57} +8.67110 q^{58} -69.0453 q^{59} +119.883 q^{60} +779.999 q^{61} -7.77347 q^{62} -180.745 q^{63} -509.016 q^{64} -99.1160 q^{65} +911.553 q^{67} -59.9860 q^{68} -82.1936 q^{69} -8.85457 q^{70} -100.883 q^{71} +12.6919 q^{72} -173.265 q^{73} +4.69606 q^{74} +75.0000 q^{75} -220.198 q^{76} -5.24407 q^{78} -183.512 q^{79} -319.067 q^{80} +81.0000 q^{81} -17.4271 q^{82} -1141.72 q^{83} +481.518 q^{84} -37.5277 q^{85} +3.27848 q^{86} -295.000 q^{87} +1164.37 q^{89} +3.96813 q^{90} -398.105 q^{91} +218.970 q^{92} +264.461 q^{93} -46.6349 q^{94} -137.758 q^{95} -50.7263 q^{96} -1207.61 q^{97} -5.31888 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 9 q^{2} + 36 q^{3} + 57 q^{4} - 60 q^{5} - 27 q^{6} - 21 q^{7} - 123 q^{8} + 108 q^{9}+O(q^{10}) \) 12 * q - 9 * q^2 + 36 * q^3 + 57 * q^4 - 60 * q^5 - 27 * q^6 - 21 * q^7 - 123 * q^8 + 108 * q^9	\( 12 q - 9 q^{2} + 36 q^{3} + 57 q^{4} - 60 q^{5} - 27 q^{6} - 21 q^{7} - 123 q^{8} + 108 q^{9} + 45 q^{10} + 171 q^{12} - 78 q^{13} + 262 q^{14} - 180 q^{15} + 225 q^{16} - 313 q^{17} - 81 q^{18} - 51 q^{19} - 285 q^{20} - 63 q^{21} - 34 q^{23} - 369 q^{24} + 300 q^{25} + 28 q^{26} + 324 q^{27} - 376 q^{28} - 31 q^{29} + 135 q^{30} + 655 q^{31} - 1578 q^{32} - 10 q^{34} + 105 q^{35} + 513 q^{36} + 84 q^{37} - 1076 q^{38} - 234 q^{39} + 615 q^{40} - 1463 q^{41} + 786 q^{42} + 111 q^{43} - 540 q^{45} + q^{46} + 278 q^{47} + 675 q^{48} - 325 q^{49} - 225 q^{50} - 939 q^{51} - 1957 q^{52} + 517 q^{53} - 243 q^{54} + 1543 q^{56} - 153 q^{57} + 442 q^{58} - 308 q^{59} - 855 q^{60} - 604 q^{61} - 1773 q^{62} - 189 q^{63} + 4323 q^{64} + 390 q^{65} - 357 q^{67} - 2192 q^{68} - 102 q^{69} - 1310 q^{70} - 620 q^{71} - 1107 q^{72} - 1892 q^{73} - 581 q^{74} + 900 q^{75} - 378 q^{76} + 84 q^{78} - 415 q^{79} - 1125 q^{80} + 972 q^{81} - 2802 q^{82} - 3158 q^{83} - 1128 q^{84} + 1565 q^{85} + 747 q^{86} - 93 q^{87} + 1563 q^{89} + 405 q^{90} + 1434 q^{91} - 3466 q^{92} + 1965 q^{93} + 3 q^{94} + 255 q^{95} - 4734 q^{96} + 714 q^{97} - 6586 q^{98}+O(q^{100}) \) 12 * q - 9 * q^2 + 36 * q^3 + 57 * q^4 - 60 * q^5 - 27 * q^6 - 21 * q^7 - 123 * q^8 + 108 * q^9 + 45 * q^10 + 171 * q^12 - 78 * q^13 + 262 * q^14 - 180 * q^15 + 225 * q^16 - 313 * q^17 - 81 * q^18 - 51 * q^19 - 285 * q^20 - 63 * q^21 - 34 * q^23 - 369 * q^24 + 300 * q^25 + 28 * q^26 + 324 * q^27 - 376 * q^28 - 31 * q^29 + 135 * q^30 + 655 * q^31 - 1578 * q^32 - 10 * q^34 + 105 * q^35 + 513 * q^36 + 84 * q^37 - 1076 * q^38 - 234 * q^39 + 615 * q^40 - 1463 * q^41 + 786 * q^42 + 111 * q^43 - 540 * q^45 + q^46 + 278 * q^47 + 675 * q^48 - 325 * q^49 - 225 * q^50 - 939 * q^51 - 1957 * q^52 + 517 * q^53 - 243 * q^54 + 1543 * q^56 - 153 * q^57 + 442 * q^58 - 308 * q^59 - 855 * q^60 - 604 * q^61 - 1773 * q^62 - 189 * q^63 + 4323 * q^64 + 390 * q^65 - 357 * q^67 - 2192 * q^68 - 102 * q^69 - 1310 * q^70 - 620 * q^71 - 1107 * q^72 - 1892 * q^73 - 581 * q^74 + 900 * q^75 - 378 * q^76 + 84 * q^78 - 415 * q^79 - 1125 * q^80 + 972 * q^81 - 2802 * q^82 - 3158 * q^83 - 1128 * q^84 + 1565 * q^85 + 747 * q^86 - 93 * q^87 + 1563 * q^89 + 405 * q^90 + 1434 * q^91 - 3466 * q^92 + 1965 * q^93 + 3 * q^94 + 255 * q^95 - 4734 * q^96 + 714 * q^97 - 6586 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more