LMFDB - Embedded newform 1815.4.a.bj.1.6

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:	$12$
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} - x^{11} - 76 x^{10} + 86 x^{9} + 2070 x^{8} - 2627 x^{7} - 23872 x^{6} + 33784 x^{5} + \cdots + 9680 $ x^12 - x^11 - 76x^10 + 86x^9 + 2070x^8 - 2627x^7 - 23872x^6 + 33784x^5 + 101637x^4 - 160162x^3 - 65064x^2 + 93280x + 9680
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 11^{2} $
Twist minimal:	no (minimal twist has level 165)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$0.875727$ 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)

$f(q)$	$=$	$q-0.875727 q^{2} -3.00000 q^{3} -7.23310 q^{4} -5.00000 q^{5} +2.62718 q^{6} -27.5517 q^{7} +13.3400 q^{8} +9.00000 q^{9} +O(q^{10})$	\(q-0.875727 q^{2} -3.00000 q^{3} -7.23310 q^{4} -5.00000 q^{5} +2.62718 q^{6} -27.5517 q^{7} +13.3400 q^{8} +9.00000 q^{9} +4.37863 q^{10} +21.6993 q^{12} +55.3696 q^{13} +24.1278 q^{14} +15.0000 q^{15} +46.1826 q^{16} +61.3311 q^{17} -7.88154 q^{18} -79.3314 q^{19} +36.1655 q^{20} +82.6551 q^{21} -16.3145 q^{23} -40.0201 q^{24} +25.0000 q^{25} -48.4887 q^{26} -27.0000 q^{27} +199.284 q^{28} -32.8263 q^{29} -13.1359 q^{30} -7.60276 q^{31} -147.164 q^{32} -53.7093 q^{34} +137.759 q^{35} -65.0979 q^{36} -410.245 q^{37} +69.4726 q^{38} -166.109 q^{39} -66.7002 q^{40} -317.823 q^{41} -72.3833 q^{42} -345.350 q^{43} -45.0000 q^{45} +14.2871 q^{46} -21.0543 q^{47} -138.548 q^{48} +416.096 q^{49} -21.8932 q^{50} -183.993 q^{51} -400.494 q^{52} -156.231 q^{53} +23.6446 q^{54} -367.541 q^{56} +237.994 q^{57} +28.7468 q^{58} +773.871 q^{59} -108.497 q^{60} -838.145 q^{61} +6.65794 q^{62} -247.965 q^{63} -240.586 q^{64} -276.848 q^{65} -514.767 q^{67} -443.614 q^{68} +48.9436 q^{69} -120.639 q^{70} -657.303 q^{71} +120.060 q^{72} +779.375 q^{73} +359.263 q^{74} -75.0000 q^{75} +573.812 q^{76} +145.466 q^{78} +357.059 q^{79} -230.913 q^{80} +81.0000 q^{81} +278.326 q^{82} -1279.38 q^{83} -597.853 q^{84} -306.655 q^{85} +302.432 q^{86} +98.4788 q^{87} +1357.89 q^{89} +39.4077 q^{90} -1525.53 q^{91} +118.005 q^{92} +22.8083 q^{93} +18.4378 q^{94} +396.657 q^{95} +441.491 q^{96} -812.285 q^{97} -364.387 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - q^{2} - 36 q^{3} + 57 q^{4} - 60 q^{5} + 3 q^{6} + 51 q^{7} + 45 q^{8} + 108 q^{9}+O(q^{10}) $ 12 * q - q^2 - 36 * q^3 + 57 * q^4 - 60 * q^5 + 3 * q^6 + 51 * q^7 + 45 * q^8 + 108 * q^9	\( 12 q - q^{2} - 36 q^{3} + 57 q^{4} - 60 q^{5} + 3 q^{6} + 51 q^{7} + 45 q^{8} + 108 q^{9} + 5 q^{10} - 171 q^{12} + 132 q^{13} + 178 q^{14} + 180 q^{15} + 329 q^{16} + 189 q^{17} - 9 q^{18} + 85 q^{19} - 285 q^{20} - 153 q^{21} + 444 q^{23} - 135 q^{24} + 300 q^{25} - 308 q^{26} - 324 q^{27} + 858 q^{28} - 439 q^{29} - 15 q^{30} + 75 q^{31} - 56 q^{32} + 866 q^{34} - 255 q^{35} + 513 q^{36} - 138 q^{37} + 660 q^{38} - 396 q^{39} - 225 q^{40} - 379 q^{41} - 534 q^{42} + 1221 q^{43} - 540 q^{45} - 595 q^{46} - 696 q^{47} - 987 q^{48} + 731 q^{49} - 25 q^{50} - 567 q^{51} + 373 q^{52} - 915 q^{53} + 27 q^{54} + 2181 q^{56} - 255 q^{57} - 1182 q^{58} + 868 q^{59} + 855 q^{60} - 84 q^{61} - 1809 q^{62} + 459 q^{63} - 425 q^{64} - 660 q^{65} + 1569 q^{67} + 1182 q^{68} - 1332 q^{69} - 890 q^{70} - 604 q^{71} + 405 q^{72} + 3156 q^{73} - 2273 q^{74} - 900 q^{75} - 146 q^{76} + 924 q^{78} + 1061 q^{79} - 1645 q^{80} + 972 q^{81} - 3030 q^{82} - 314 q^{83} - 2574 q^{84} - 945 q^{85} - 4975 q^{86} + 1317 q^{87} + 3943 q^{89} + 45 q^{90} + 2726 q^{91} + 1842 q^{92} - 225 q^{93} + 1683 q^{94} - 425 q^{95} + 168 q^{96} + 194 q^{97} + 4008 q^{98}+O(q^{100}) \) 12 * q - q^2 - 36 * q^3 + 57 * q^4 - 60 * q^5 + 3 * q^6 + 51 * q^7 + 45 * q^8 + 108 * q^9 + 5 * q^10 - 171 * q^12 + 132 * q^13 + 178 * q^14 + 180 * q^15 + 329 * q^16 + 189 * q^17 - 9 * q^18 + 85 * q^19 - 285 * q^20 - 153 * q^21 + 444 * q^23 - 135 * q^24 + 300 * q^25 - 308 * q^26 - 324 * q^27 + 858 * q^28 - 439 * q^29 - 15 * q^30 + 75 * q^31 - 56 * q^32 + 866 * q^34 - 255 * q^35 + 513 * q^36 - 138 * q^37 + 660 * q^38 - 396 * q^39 - 225 * q^40 - 379 * q^41 - 534 * q^42 + 1221 * q^43 - 540 * q^45 - 595 * q^46 - 696 * q^47 - 987 * q^48 + 731 * q^49 - 25 * q^50 - 567 * q^51 + 373 * q^52 - 915 * q^53 + 27 * q^54 + 2181 * q^56 - 255 * q^57 - 1182 * q^58 + 868 * q^59 + 855 * q^60 - 84 * q^61 - 1809 * q^62 + 459 * q^63 - 425 * q^64 - 660 * q^65 + 1569 * q^67 + 1182 * q^68 - 1332 * q^69 - 890 * q^70 - 604 * q^71 + 405 * q^72 + 3156 * q^73 - 2273 * q^74 - 900 * q^75 - 146 * q^76 + 924 * q^78 + 1061 * q^79 - 1645 * q^80 + 972 * q^81 - 3030 * q^82 - 314 * q^83 - 2574 * q^84 - 945 * q^85 - 4975 * q^86 + 1317 * q^87 + 3943 * q^89 + 45 * q^90 + 2726 * q^91 + 1842 * q^92 - 225 * q^93 + 1683 * q^94 - 425 * q^95 + 168 * q^96 + 194 * q^97 + 4008 * 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 $

$2$	−0.875727	−0.309616	−0.154808	−	0.987945i	$-0.549476\pi$
$2$	−0.875727	−0.309616	−0.154808	+	0.987945i	$0.549476\pi$
$3$	−3.00000	−0.577350
$3$	−3.00000	−0.577350
$4$	−7.23310	−0.904138
$5$	−5.00000	−0.447214
$5$	−5.00000	−0.447214
$6$	2.62718	0.178757
$7$	−27.5517	−1.48765	−0.743826	−	0.668373i	$-0.766991\pi$
$7$	−27.5517	−1.48765	−0.743826	+	0.668373i	$0.766991\pi$
$8$	13.3400	0.589552
$9$	9.00000	0.333333
$10$	4.37863	0.138465
$11$	0	0
$11$	0	0
$12$	21.6993	0.522004
$13$	55.3696	1.18129	0.590645	−	0.806932i	$-0.298873\pi$
$13$	55.3696	1.18129	0.590645	+	0.806932i	$0.298873\pi$
$14$	24.1278	0.460601
$15$	15.0000	0.258199
$16$	46.1826	0.721603
$17$	61.3311	0.874998	0.437499	−	0.899219i	$-0.355864\pi$
$17$	61.3311	0.874998	0.437499	+	0.899219i	$0.355864\pi$
$18$	−7.88154	−0.103205
$19$	−79.3314	−0.957888	−0.478944	−	0.877845i	$-0.658980\pi$
$19$	−79.3314	−0.957888	−0.478944	+	0.877845i	$0.658980\pi$
$20$	36.1655	0.404343
$21$	82.6551	0.858896
$22$	0	0
$23$	−16.3145	−0.147905	−0.0739525	−	0.997262i	$-0.523561\pi$
$23$	−16.3145	−0.147905	−0.0739525	+	0.997262i	$0.523561\pi$
$24$	−40.0201	−0.340378
$25$	25.0000	0.200000
$26$	−48.4887	−0.365746
$27$	−27.0000	−0.192450
$28$	199.284	1.34504
$29$	−32.8263	−0.210196	−0.105098	−	0.994462i	$-0.533516\pi$
$29$	−32.8263	−0.210196	−0.105098	+	0.994462i	$0.533516\pi$
$30$	−13.1359	−0.0799426
$31$	−7.60276	−0.0440483	−0.0220241	−	0.999757i	$-0.507011\pi$
$31$	−7.60276	−0.0440483	−0.0220241	+	0.999757i	$0.507011\pi$
$32$	−147.164	−0.812972
$33$	0	0
$34$	−53.7093	−0.270914
$35$	137.759	0.665298
$36$	−65.0979	−0.301379
$37$	−410.245	−1.82281	−0.911404	−	0.411513i	$-0.865001\pi$
$37$	−410.245	−1.82281	−0.911404	+	0.411513i	$0.865001\pi$
$38$	69.4726	0.296578
$39$	−166.109	−0.682018
$40$	−66.7002	−0.263656
$41$	−317.823	−1.21062	−0.605312	−	0.795988i	$-0.706952\pi$
$41$	−317.823	−1.21062	−0.605312	+	0.795988i	$0.706952\pi$
$42$	−72.3833	−0.265928
$43$	−345.350	−1.22477	−0.612387	−	0.790558i	$-0.709791\pi$
$43$	−345.350	−1.22477	−0.612387	+	0.790558i	$0.709791\pi$
$44$	0	0
$45$	−45.0000	−0.149071
$46$	14.2871	0.0457938
$47$	−21.0543	−0.0653424	−0.0326712	−	0.999466i	$-0.510401\pi$
$47$	−21.0543	−0.0653424	−0.0326712	+	0.999466i	$0.510401\pi$
$48$	−138.548	−0.416618
$49$	416.096	1.21311
$50$	−21.8932	−0.0619232
$51$	−183.993	−0.505181
$52$	−400.494	−1.06805
$53$	−156.231	−0.404905	−0.202452	−	0.979292i	$-0.564891\pi$
$53$	−156.231	−0.404905	−0.202452	+	0.979292i	$0.564891\pi$
$54$	23.6446	0.0595857
$55$	0	0
$56$	−367.541	−0.877048
$57$	237.994	0.553037
$58$	28.7468	0.0650801
$59$	773.871	1.70762	0.853808	−	0.520588i	$-0.174287\pi$
$59$	773.871	1.70762	0.853808	+	0.520588i	$0.174287\pi$
$60$	−108.497	−0.233447
$61$	−838.145	−1.75924	−0.879619	−	0.475679i	$-0.842202\pi$
$61$	−838.145	−1.75924	−0.879619	+	0.475679i	$0.842202\pi$
$62$	6.65794	0.0136381
$63$	−247.965	−0.495884
$64$	−240.586	−0.469894
$65$	−276.848	−0.528289
$66$	0	0
$67$	−514.767	−0.938639	−0.469320	−	0.883028i	$-0.655501\pi$
$67$	−514.767	−0.938639	−0.469320	+	0.883028i	$0.655501\pi$
$68$	−443.614	−0.791119
$69$	48.9436	0.0853930
$70$	−120.639	−0.205987
$71$	−657.303	−1.09870	−0.549349	−	0.835593i	$-0.685124\pi$
$71$	−657.303	−1.09870	−0.549349	+	0.835593i	$0.685124\pi$
$72$	120.060	0.196517
$73$	779.375	1.24957	0.624787	−	0.780795i	$-0.285186\pi$
$73$	779.375	1.24957	0.624787	+	0.780795i	$0.285186\pi$
$74$	359.263	0.564371
$75$	−75.0000	−0.115470
$76$	573.812	0.866063
$77$	0	0
$78$	145.466	0.211164
$79$	357.059	0.508510	0.254255	−	0.967137i	$-0.418170\pi$
$79$	357.059	0.508510	0.254255	+	0.967137i	$0.418170\pi$
$80$	−230.913	−0.322711
$81$	81.0000	0.111111
$82$	278.326	0.374829
$83$	−1279.38	−1.69193	−0.845965	−	0.533238i	$-0.820975\pi$
$83$	−1279.38	−1.69193	−0.845965	+	0.533238i	$0.820975\pi$
$84$	−597.853	−0.776561
$85$	−306.655	−0.391311
$86$	302.432	0.379210
$87$	98.4788	0.121357
$88$	0	0
$89$	1357.89	1.61725	0.808627	−	0.588321i	$-0.200211\pi$
$89$	1357.89	1.61725	0.808627	+	0.588321i	$0.200211\pi$
$90$	39.4077	0.0461549
$91$	−1525.53	−1.75735
$92$	118.005	0.133727
$93$	22.8083	0.0254313
$94$	18.4378	0.0202311
$95$	396.657	0.428381
$96$	441.491	0.469370
$97$	−812.285	−0.850258	−0.425129	−	0.905133i	$-0.639771\pi$
$97$	−812.285	−0.850258	−0.425129	+	0.905133i	$0.639771\pi$
$98$	−364.387	−0.375598
$99$	0	0
$100$	−180.828	−0.180828
$101$	76.0916	0.0749643	0.0374822	−	0.999297i	$-0.488066\pi$
$101$	76.0916	0.0749643	0.0374822	+	0.999297i	$0.488066\pi$
$102$	161.128	0.156412
$103$	−523.744	−0.501030	−0.250515	−	0.968113i	$-0.580600\pi$
$103$	−523.744	−0.501030	−0.250515	+	0.968113i	$0.580600\pi$
$104$	738.633	0.696432
$105$	−413.276	−0.384110
$106$	136.816	0.125365
$107$	1386.53	1.25271	0.626357	−	0.779536i	$-0.284545\pi$
$107$	1386.53	1.25271	0.626357	+	0.779536i	$0.284545\pi$
$108$	195.294	0.174001
$109$	−1162.47	−1.02151	−0.510754	−	0.859727i	$-0.670634\pi$
$109$	−1162.47	−1.02151	−0.510754	+	0.859727i	$0.670634\pi$
$110$	0	0
$111$	1230.74	1.05240
$112$	−1272.41	−1.07349
$113$	1208.93	1.00643	0.503216	−	0.864161i	$-0.332150\pi$
$113$	1208.93	1.00643	0.503216	+	0.864161i	$0.332150\pi$
$114$	−208.418	−0.171229
$115$	81.5727	0.0661452
$116$	237.436	0.190046
$117$	498.327	0.393763
$118$	−677.699	−0.528706
$119$	−1689.78	−1.30169
$120$	200.101	0.152222
$121$	0	0
$122$	733.986	0.544689
$123$	953.469	0.698955
$124$	54.9916	0.0398257
$125$	−125.000	−0.0894427
$126$	217.150	0.153534
$127$	1824.93	1.27509	0.637543	−	0.770414i	$-0.279951\pi$
$127$	1824.93	1.27509	0.637543	+	0.770414i	$0.279951\pi$
$128$	1388.00	0.958459
$129$	1036.05	0.707124
$130$	242.443	0.163567
$131$	901.335	0.601145	0.300572	−	0.953759i	$-0.402822\pi$
$131$	901.335	0.601145	0.300572	+	0.953759i	$0.402822\pi$
$132$	0	0
$133$	2185.72	1.42500
$134$	450.795	0.290618
$135$	135.000	0.0860663
$136$	818.159	0.515857
$137$	−95.1040	−0.0593086	−0.0296543	−	0.999560i	$-0.509441\pi$
$137$	−95.1040	−0.0593086	−0.0296543	+	0.999560i	$0.509441\pi$
$138$	−42.8612	−0.0264391
$139$	240.640	0.146840	0.0734201	−	0.997301i	$-0.476609\pi$
$139$	240.640	0.146840	0.0734201	+	0.997301i	$0.476609\pi$
$140$	−996.421	−0.601521
$141$	63.1630	0.0377254
$142$	575.618	0.340174
$143$	0	0
$144$	415.643	0.240534
$145$	164.131	0.0940025
$146$	−682.519	−0.386888
$147$	−1248.29	−0.700389
$148$	2967.35	1.64807
$149$	−294.075	−0.161688	−0.0808442	−	0.996727i	$-0.525762\pi$
$149$	−294.075	−0.161688	−0.0808442	+	0.996727i	$0.525762\pi$
$150$	65.6795	0.0357514
$151$	−472.689	−0.254748	−0.127374	−	0.991855i	$-0.540655\pi$
$151$	−472.689	−0.254748	−0.127374	+	0.991855i	$0.540655\pi$
$152$	−1058.28	−0.564725
$153$	551.980	0.291666
$154$	0	0
$155$	38.0138	0.0196990
$156$	1201.48	0.616638
$157$	−1807.39	−0.918759	−0.459380	−	0.888240i	$-0.651928\pi$
$157$	−1807.39	−0.918759	−0.459380	+	0.888240i	$0.651928\pi$
$158$	−312.687	−0.157443
$159$	468.692	0.233772
$160$	735.818	0.363572
$161$	449.493	0.220031
$162$	−70.9339	−0.0344018
$163$	−2677.56	−1.28664	−0.643321	−	0.765597i	$-0.722444\pi$
$163$	−2677.56	−1.28664	−0.643321	+	0.765597i	$0.722444\pi$
$164$	2298.85	1.09457
$165$	0	0
$166$	1120.39	0.523849
$167$	90.5716	0.0419679	0.0209840	−	0.999780i	$-0.493320\pi$
$167$	90.5716	0.0419679	0.0209840	+	0.999780i	$0.493320\pi$
$168$	1102.62	0.506364
$169$	868.794	0.395446
$170$	268.546	0.121156
$171$	−713.983	−0.319296
$172$	2497.95	1.10737
$173$	−2514.71	−1.10514	−0.552571	−	0.833466i	$-0.686353\pi$
$173$	−2514.71	−1.10514	−0.552571	+	0.833466i	$0.686353\pi$
$174$	−86.2405	−0.0375740
$175$	−688.793	−0.297530
$176$	0	0
$177$	−2321.61	−0.985893
$178$	−1189.14	−0.500728
$179$	−3043.71	−1.27094	−0.635468	−	0.772127i	$-0.719193\pi$
$179$	−3043.71	−1.27094	−0.635468	+	0.772127i	$0.719193\pi$
$180$	325.490	0.134781
$181$	−3527.96	−1.44879	−0.724396	−	0.689384i	$-0.757881\pi$
$181$	−3527.96	−1.44879	−0.724396	+	0.689384i	$0.757881\pi$
$182$	1335.95	0.544103
$183$	2514.44	1.01570
$184$	−217.637	−0.0871977
$185$	2051.23	0.815185
$186$	−19.9738	−0.00787394
$187$	0	0
$188$	152.288	0.0590785
$189$	743.896	0.286299
$190$	−347.363	−0.132634
$191$	−2623.45	−0.993853	−0.496926	−	0.867793i	$-0.665538\pi$
$191$	−2623.45	−0.993853	−0.496926	+	0.867793i	$0.665538\pi$
$192$	721.757	0.271293
$193$	2198.03	0.819782	0.409891	−	0.912134i	$-0.365567\pi$
$193$	2198.03	0.819782	0.409891	+	0.912134i	$0.365567\pi$
$194$	711.340	0.263254
$195$	830.544	0.305008
$196$	−3009.67	−1.09682
$197$	1867.64	0.675451	0.337726	−	0.941245i	$-0.390342\pi$
$197$	1867.64	0.675451	0.337726	+	0.941245i	$0.390342\pi$
$198$	0	0
$199$	−3817.35	−1.35982	−0.679911	−	0.733295i	$-0.737981\pi$
$199$	−3817.35	−1.35982	−0.679911	+	0.733295i	$0.737981\pi$
$200$	333.501	0.117910
$201$	1544.30	0.541924
$202$	−66.6355	−0.0232102
$203$	904.419	0.312698
$204$	1330.84	0.456753
$205$	1589.12	0.541408
$206$	458.657	0.155127
$207$	−146.831	−0.0493017
$208$	2557.11	0.852422
$209$	0	0
$210$	361.916	0.118927
$211$	3447.17	1.12471	0.562353	−	0.826897i	$-0.309896\pi$
$211$	3447.17	1.12471	0.562353	+	0.826897i	$0.309896\pi$
$212$	1130.03	0.366090
$213$	1971.91	0.634333
$214$	−1214.22	−0.387861
$215$	1726.75	0.547736
$216$	−360.181	−0.113459
$217$	209.469	0.0655285
$218$	1018.01	0.316275
$219$	−2338.12	−0.721442
$220$	0	0
$221$	3395.88	1.03363
$222$	−1077.79	−0.325840
$223$	4946.20	1.48530	0.742650	−	0.669680i	$-0.233569\pi$
$223$	4946.20	1.48530	0.742650	+	0.669680i	$0.233569\pi$
$224$	4054.61	1.20942
$225$	225.000	0.0666667
$226$	−1058.69	−0.311607
$227$	700.342	0.204772	0.102386	−	0.994745i	$-0.467352\pi$
$227$	700.342	0.204772	0.102386	+	0.994745i	$0.467352\pi$
$228$	−1721.44	−0.500022
$229$	−2446.51	−0.705982	−0.352991	−	0.935627i	$-0.614835\pi$
$229$	−2446.51	−0.705982	−0.352991	+	0.935627i	$0.614835\pi$
$230$	−71.4354	−0.0204796
$231$	0	0
$232$	−437.903	−0.123921
$233$	5815.09	1.63502	0.817509	−	0.575916i	$-0.195355\pi$
$233$	5815.09	1.63502	0.817509	+	0.575916i	$0.195355\pi$
$234$	−436.398	−0.121915
$235$	105.272	0.0292220
$236$	−5597.48	−1.54392
$237$	−1071.18	−0.293589
$238$	1479.78	0.403025
$239$	−196.886	−0.0532867	−0.0266433	−	0.999645i	$-0.508482\pi$
$239$	−196.886	−0.0532867	−0.0266433	+	0.999645i	$0.508482\pi$
$240$	692.739	0.186317
$241$	−300.533	−0.0803278	−0.0401639	−	0.999193i	$-0.512788\pi$
$241$	−300.533	−0.0803278	−0.0401639	+	0.999193i	$0.512788\pi$
$242$	0	0
$243$	−243.000	−0.0641500
$244$	6062.39	1.59059
$245$	−2080.48	−0.542519
$246$	−834.978	−0.216408
$247$	−4392.55	−1.13154
$248$	−101.421	−0.0259687
$249$	3838.14	0.976837
$250$	109.466	0.0276929
$251$	−7484.83	−1.88222	−0.941112	−	0.338095i	$-0.890217\pi$
$251$	−7484.83	−1.88222	−0.941112	+	0.338095i	$0.890217\pi$
$252$	1793.56	0.448347
$253$	0	0
$254$	−1598.14	−0.394788
$255$	919.966	0.225924
$256$	709.179	0.173139
$257$	−1570.13	−0.381098	−0.190549	−	0.981678i	$-0.561027\pi$
$257$	−1570.13	−0.381098	−0.190549	+	0.981678i	$0.561027\pi$
$258$	−907.296	−0.218937
$259$	11303.0	2.71170
$260$	2002.47	0.477646
$261$	−295.436	−0.0700653
$262$	−789.323	−0.186124
$263$	−4934.79	−1.15700	−0.578502	−	0.815681i	$-0.696363\pi$
$263$	−4934.79	−1.15700	−0.578502	+	0.815681i	$0.696363\pi$
$264$	0	0
$265$	781.154	0.181079
$266$	−1914.09	−0.441204
$267$	−4073.66	−0.933722
$268$	3723.36	0.848659
$269$	2455.47	0.556553	0.278277	−	0.960501i	$-0.410237\pi$
$269$	2455.47	0.556553	0.278277	+	0.960501i	$0.410237\pi$
$270$	−118.223	−0.0266475
$271$	931.456	0.208789	0.104395	−	0.994536i	$-0.466709\pi$
$271$	931.456	0.208789	0.104395	+	0.994536i	$0.466709\pi$
$272$	2832.43	0.631402
$273$	4576.58	1.01461
$274$	83.2851	0.0183629
$275$	0	0
$276$	−354.014	−0.0772071
$277$	4060.90	0.880851	0.440425	−	0.897789i	$-0.354828\pi$
$277$	4060.90	0.880851	0.440425	+	0.897789i	$0.354828\pi$
$278$	−210.735	−0.0454641
$279$	−68.4249	−0.0146828
$280$	1837.70	0.392228
$281$	7744.35	1.64409	0.822044	−	0.569424i	$-0.192834\pi$
$281$	7744.35	1.64409	0.822044	+	0.569424i	$0.192834\pi$
$282$	−55.3135	−0.0116804
$283$	−5650.48	−1.18688	−0.593439	−	0.804879i	$-0.702230\pi$
$283$	−5650.48	−1.18688	−0.593439	+	0.804879i	$0.702230\pi$
$284$	4754.34	0.993374
$285$	−1189.97	−0.247326
$286$	0	0
$287$	8756.56	1.80099
$288$	−1324.47	−0.270991
$289$	−1151.50	−0.234378
$290$	−143.734	−0.0291047
$291$	2436.85	0.490897
$292$	−5637.30	−1.12979
$293$	861.772	0.171827	0.0859134	−	0.996303i	$-0.472619\pi$
$293$	861.772	0.171827	0.0859134	+	0.996303i	$0.472619\pi$
$294$	1093.16	0.216852
$295$	−3869.35	−0.763669
$296$	−5472.69	−1.07464
$297$	0	0
$298$	257.529	0.0500613
$299$	−903.330	−0.174719
$300$	542.483	0.104401
$301$	9514.97	1.82204
$302$	413.947	0.0788740
$303$	−228.275	−0.0432807
$304$	−3663.73	−0.691215
$305$	4190.73	0.786755
$306$	−483.384	−0.0903046
$307$	−1960.72	−0.364509	−0.182254	−	0.983251i	$-0.558339\pi$
$307$	−1960.72	−0.364509	−0.182254	+	0.983251i	$0.558339\pi$
$308$	0	0
$309$	1571.23	0.289270
$310$	−33.2897	−0.00609913
$311$	−8913.67	−1.62523	−0.812617	−	0.582797i	$-0.801958\pi$
$311$	−8913.67	−1.62523	−0.812617	+	0.582797i	$0.801958\pi$
$312$	−2215.90	−0.402085
$313$	9336.36	1.68601	0.843007	−	0.537902i	$-0.180783\pi$
$313$	9336.36	1.68601	0.843007	+	0.537902i	$0.180783\pi$
$314$	1582.78	0.284463
$315$	1239.83	0.221766
$316$	−2582.65	−0.459764
$317$	−617.423	−0.109394	−0.0546971	−	0.998503i	$-0.517419\pi$
$317$	−617.423	−0.109394	−0.0546971	+	0.998503i	$0.517419\pi$
$318$	−410.447	−0.0723796
$319$	0	0
$320$	1202.93	0.210143
$321$	−4159.58	−0.723255
$322$	−393.633	−0.0681253
$323$	−4865.48	−0.838151
$324$	−585.881	−0.100460
$325$	1384.24	0.236258
$326$	2344.81	0.398365
$327$	3487.41	0.589768
$328$	−4239.77	−0.713726
$329$	580.083	0.0972067
$330$	0	0
$331$	2716.68	0.451125	0.225562	−	0.974229i	$-0.427578\pi$
$331$	2716.68	0.451125	0.225562	+	0.974229i	$0.427578\pi$
$332$	9253.89	1.52974
$333$	−3692.21	−0.607603
$334$	−79.3160	−0.0129939
$335$	2573.84	0.419772
$336$	3817.23	0.619782
$337$	2680.86	0.433340	0.216670	−	0.976245i	$-0.430480\pi$
$337$	2680.86	0.433340	0.216670	+	0.976245i	$0.430480\pi$
$338$	−760.827	−0.122436
$339$	−3626.79	−0.581063
$340$	2218.07	0.353799
$341$	0	0
$342$	625.254	0.0988592
$343$	−2013.93	−0.317031
$344$	−4606.98	−0.722068
$345$	−244.718	−0.0381889
$346$	2202.20	0.342170
$347$	1513.54	0.234153	0.117076	−	0.993123i	$-0.462648\pi$
$347$	1513.54	0.234153	0.117076	+	0.993123i	$0.462648\pi$
$348$	−712.307	−0.109723
$349$	−574.974	−0.0881882	−0.0440941	−	0.999027i	$-0.514040\pi$
$349$	−574.974	−0.0881882	−0.0440941	+	0.999027i	$0.514040\pi$
$350$	603.194	0.0921202
$351$	−1494.98	−0.227339
$352$	0	0
$353$	6541.80	0.986360	0.493180	−	0.869927i	$-0.335834\pi$
$353$	6541.80	0.986360	0.493180	+	0.869927i	$0.335834\pi$
$354$	2033.10	0.305248
$355$	3286.51	0.491352
$356$	−9821.73	−1.46222
$357$	5069.33	0.751533
$358$	2665.46	0.393502
$359$	−6680.74	−0.982161	−0.491080	−	0.871114i	$-0.663398\pi$
$359$	−6680.74	−0.982161	−0.491080	+	0.871114i	$0.663398\pi$
$360$	−600.302	−0.0878852
$361$	−565.528	−0.0824506
$362$	3089.53	0.448569
$363$	0	0
$364$	11034.3	1.58889
$365$	−3896.87	−0.558826
$366$	−2201.96	−0.314476
$367$	5932.89	0.843853	0.421927	−	0.906630i	$-0.361354\pi$
$367$	5932.89	0.843853	0.421927	+	0.906630i	$0.361354\pi$
$368$	−753.448	−0.106729
$369$	−2860.41	−0.403542
$370$	−1796.31	−0.252394
$371$	4304.42	0.602357
$372$	−164.975	−0.0229934
$373$	7005.39	0.972454	0.486227	−	0.873833i	$-0.338373\pi$
$373$	7005.39	0.972454	0.486227	+	0.873833i	$0.338373\pi$
$374$	0	0
$375$	375.000	0.0516398
$376$	−280.866	−0.0385227
$377$	−1817.58	−0.248302
$378$	−651.450	−0.0886427
$379$	−822.960	−0.111537	−0.0557686	−	0.998444i	$-0.517761\pi$
$379$	−822.960	−0.111537	−0.0557686	+	0.998444i	$0.517761\pi$
$380$	−2869.06	−0.387315
$381$	−5474.78	−0.736172
$382$	2297.42	0.307713
$383$	−12547.2	−1.67398	−0.836990	−	0.547219i	$-0.815687\pi$
$383$	−12547.2	−1.67398	−0.836990	+	0.547219i	$0.815687\pi$
$384$	−4163.99	−0.553366
$385$	0	0
$386$	−1924.88	−0.253818
$387$	−3108.15	−0.408258
$388$	5875.34	0.768751
$389$	13378.6	1.74376	0.871881	−	0.489718i	$-0.162900\pi$
$389$	13378.6	1.74376	0.871881	+	0.489718i	$0.162900\pi$
$390$	−727.330	−0.0944353
$391$	−1000.59	−0.129417
$392$	5550.74	0.715190
$393$	−2704.00	−0.347071
$394$	−1635.54	−0.209131
$395$	−1785.30	−0.227413
$396$	0	0
$397$	10867.6	1.37387	0.686937	−	0.726717i	$-0.258955\pi$
$397$	10867.6	1.37387	0.686937	+	0.726717i	$0.258955\pi$
$398$	3342.95	0.421023
$399$	−6557.15	−0.822726
$400$	1154.56	0.144321
$401$	8629.51	1.07466	0.537328	−	0.843373i	$-0.319434\pi$
$401$	8629.51	1.07466	0.537328	+	0.843373i	$0.319434\pi$
$402$	−1352.39	−0.167788
$403$	−420.962	−0.0520338
$404$	−550.378	−0.0677781
$405$	−405.000	−0.0496904
$406$	−792.024	−0.0968165
$407$	0	0
$408$	−2454.48	−0.297830
$409$	15218.8	1.83990	0.919951	−	0.392033i	$-0.128228\pi$
$409$	15218.8	1.83990	0.919951	+	0.392033i	$0.128228\pi$
$410$	−1391.63	−0.167629
$411$	285.312	0.0342419
$412$	3788.30	0.453000
$413$	−21321.4	−2.54034
$414$	128.584	0.0152646
$415$	6396.90	0.756654
$416$	−8148.39	−0.960355
$417$	−721.919	−0.0847783
$418$	0	0
$419$	−14287.6	−1.66585	−0.832927	−	0.553382i	$-0.813337\pi$
$419$	−14287.6	−1.66585	−0.832927	+	0.553382i	$0.813337\pi$
$420$	2989.26	0.347288
$421$	8808.02	1.01966	0.509830	−	0.860275i	$-0.329708\pi$
$421$	8808.02	1.01966	0.509830	+	0.860275i	$0.329708\pi$
$422$	−3018.78	−0.348227
$423$	−189.489	−0.0217808
$424$	−2084.12	−0.238712
$425$	1533.28	0.175000
$426$	−1726.85	−0.196400
$427$	23092.3	2.61713
$428$	−10028.9	−1.13263
$429$	0	0
$430$	−1512.16	−0.169588
$431$	−11977.6	−1.33861	−0.669304	−	0.742988i	$-0.733408\pi$
$431$	−11977.6	−1.33861	−0.669304	+	0.742988i	$0.733408\pi$
$432$	−1246.93	−0.138873
$433$	−8202.31	−0.910341	−0.455171	−	0.890404i	$-0.650422\pi$
$433$	−8202.31	−0.910341	−0.455171	+	0.890404i	$0.650422\pi$
$434$	−183.438	−0.0202887
$435$	−492.394	−0.0542724
$436$	8408.26	0.923584
$437$	1294.26	0.141677
$438$	2047.56	0.223370
$439$	−1276.34	−0.138762	−0.0693810	−	0.997590i	$-0.522102\pi$
$439$	−1276.34	−0.138762	−0.0693810	+	0.997590i	$0.522102\pi$
$440$	0	0
$441$	3744.87	0.404370
$442$	−2973.86	−0.320028
$443$	−10307.2	−1.10544	−0.552720	−	0.833367i	$-0.686410\pi$
$443$	−10307.2	−1.10544	−0.552720	+	0.833367i	$0.686410\pi$
$444$	−8902.04	−0.951514
$445$	−6789.43	−0.723258
$446$	−4331.52	−0.459873
$447$	882.225	0.0933508
$448$	6628.54	0.699038
$449$	9236.54	0.970822	0.485411	−	0.874286i	$-0.338670\pi$
$449$	9236.54	0.970822	0.485411	+	0.874286i	$0.338670\pi$
$450$	−197.039	−0.0206411
$451$	0	0
$452$	−8744.33	−0.909952
$453$	1418.07	0.147079
$454$	−613.308	−0.0634009
$455$	7627.64	0.785910
$456$	3174.85	0.326044
$457$	9168.55	0.938483	0.469241	−	0.883070i	$-0.344527\pi$
$457$	9168.55	0.938483	0.469241	+	0.883070i	$0.344527\pi$
$458$	2142.48	0.218584
$459$	−1655.94	−0.168394
$460$	−590.024	−0.0598043
$461$	−12052.1	−1.21762	−0.608811	−	0.793315i	$-0.708353\pi$
$461$	−12052.1	−1.21762	−0.608811	+	0.793315i	$0.708353\pi$
$462$	0	0
$463$	7021.75	0.704813	0.352406	−	0.935847i	$-0.385363\pi$
$463$	7021.75	0.704813	0.352406	+	0.935847i	$0.385363\pi$
$464$	−1516.00	−0.151678
$465$	−114.041	−0.0113732
$466$	−5092.43	−0.506228
$467$	2702.55	0.267792	0.133896	−	0.990995i	$-0.457251\pi$
$467$	2702.55	0.267792	0.133896	+	0.990995i	$0.457251\pi$
$468$	−3604.45	−0.356016
$469$	14182.7	1.39637
$470$	−92.1892	−0.00904760
$471$	5422.16	0.530446
$472$	10323.5	1.00673
$473$	0	0
$474$	938.060	0.0908998
$475$	−1983.29	−0.191578
$476$	12222.3	1.17691
$477$	−1406.08	−0.134968
$478$	172.419	0.0164984
$479$	12401.7	1.18298	0.591490	−	0.806313i	$-0.298540\pi$
$479$	12401.7	1.18298	0.591490	+	0.806313i	$0.298540\pi$
$480$	−2207.45	−0.209908
$481$	−22715.1	−2.15326
$482$	263.184	0.0248708
$483$	−1348.48	−0.127035
$484$	0	0
$485$	4061.42	0.380247
$486$	212.802	0.0198619
$487$	15149.8	1.40965	0.704826	−	0.709380i	$-0.251025\pi$
$487$	15149.8	1.40965	0.704826	+	0.709380i	$0.251025\pi$
$488$	−11180.9	−1.03716
$489$	8032.67	0.742843
$490$	1821.93	0.167973
$491$	−368.368	−0.0338579	−0.0169289	−	0.999857i	$-0.505389\pi$
$491$	−368.368	−0.0338579	−0.0169289	+	0.999857i	$0.505389\pi$
$492$	−6896.54	−0.631951
$493$	−2013.27	−0.183921
$494$	3846.67	0.350344
$495$	0	0
$496$	−351.115	−0.0317854
$497$	18109.8	1.63448
$498$	−3361.16	−0.302444
$499$	8709.84	0.781375	0.390687	−	0.920523i	$-0.372237\pi$
$499$	8709.84	0.781375	0.390687	+	0.920523i	$0.372237\pi$
$500$	904.138	0.0808685
$501$	−271.715	−0.0242302
$502$	6554.67	0.582767
$503$	11260.2	0.998145	0.499072	−	0.866560i	$-0.333674\pi$
$503$	11260.2	0.998145	0.499072	+	0.866560i	$0.333674\pi$
$504$	−3307.87	−0.292349
$505$	−380.458	−0.0335251
$506$	0	0
$507$	−2606.38	−0.228311
$508$	−13199.9	−1.15285
$509$	5264.42	0.458431	0.229215	−	0.973376i	$-0.426384\pi$
$509$	5264.42	0.458431	0.229215	+	0.973376i	$0.426384\pi$
$510$	−805.639	−0.0699496
$511$	−21473.1	−1.85893
$512$	−11725.0	−1.01207
$513$	2141.95	0.184346
$514$	1375.01	0.117994
$515$	2618.72	0.224067
$516$	−7493.85	−0.639338
$517$	0	0
$518$	−9898.30	−0.839588
$519$	7544.12	0.638054
$520$	−3693.16	−0.311454
$521$	6206.39	0.521894	0.260947	−	0.965353i	$-0.415965\pi$
$521$	6206.39	0.521894	0.260947	+	0.965353i	$0.415965\pi$
$522$	258.722	0.0216934
$523$	15111.5	1.26345	0.631723	−	0.775195i	$-0.282348\pi$
$523$	15111.5	1.26345	0.631723	+	0.775195i	$0.282348\pi$
$524$	−6519.45	−0.543518
$525$	2066.38	0.171779
$526$	4321.53	0.358227
$527$	−466.286	−0.0385422
$528$	0	0
$529$	−11900.8	−0.978124
$530$	−684.078	−0.0560650
$531$	6964.83	0.569205
$532$	−15809.5	−1.28840
$533$	−17597.7	−1.43010
$534$	3567.41	0.289096
$535$	−6932.63	−0.560231
$536$	−6867.01	−0.553376
$537$	9131.13	0.733775
$538$	−2150.32	−0.172318
$539$	0	0
$540$	−976.469	−0.0778158
$541$	−13166.5	−1.04635	−0.523174	−	0.852226i	$-0.675252\pi$
$541$	−13166.5	−1.04635	−0.523174	+	0.852226i	$0.675252\pi$
$542$	−815.701	−0.0646446
$543$	10583.9	0.836460
$544$	−9025.71	−0.711349
$545$	5812.35	0.456832
$546$	−4007.84	−0.314138
$547$	−894.476	−0.0699178	−0.0349589	−	0.999389i	$-0.511130\pi$
$547$	−894.476	−0.0699178	−0.0349589	+	0.999389i	$0.511130\pi$
$548$	687.897	0.0536232
$549$	−7543.31	−0.586413
$550$	0	0
$551$	2604.15	0.201344
$552$	652.910	0.0503436
$553$	−9837.59	−0.756487
$554$	−3556.24	−0.272726
$555$	−6153.68	−0.470647
$556$	−1740.57	−0.132764
$557$	−5200.49	−0.395605	−0.197802	−	0.980242i	$-0.563380\pi$
$557$	−5200.49	−0.395605	−0.197802	+	0.980242i	$0.563380\pi$
$558$	59.9215	0.00454602
$559$	−19121.9	−1.44681
$560$	6362.04	0.480081
$561$	0	0
$562$	−6781.93	−0.509036
$563$	3377.56	0.252837	0.126419	−	0.991977i	$-0.459652\pi$
$563$	3377.56	0.252837	0.126419	+	0.991977i	$0.459652\pi$
$564$	−456.865	−0.0341090
$565$	−6044.66	−0.450090
$566$	4948.28	0.367476
$567$	−2231.69	−0.165295
$568$	−8768.45	−0.647739
$569$	−4859.26	−0.358015	−0.179008	−	0.983848i	$-0.557289\pi$
$569$	−4859.26	−0.358015	−0.179008	+	0.983848i	$0.557289\pi$
$570$	1042.09	0.0765760
$571$	13321.4	0.976328	0.488164	−	0.872752i	$-0.337667\pi$
$571$	13321.4	0.976328	0.488164	+	0.872752i	$0.337667\pi$
$572$	0	0
$573$	7870.34	0.573801
$574$	−7668.36	−0.557615
$575$	−407.864	−0.0295810
$576$	−2165.27	−0.156631
$577$	−14598.5	−1.05328	−0.526642	−	0.850087i	$-0.676549\pi$
$577$	−14598.5	−1.05328	−0.526642	+	0.850087i	$0.676549\pi$
$578$	1008.40	0.0725671
$579$	−6594.10	−0.473301
$580$	−1187.18	−0.0849912
$581$	35249.1	2.51700
$582$	−2134.02	−0.151990
$583$	0	0
$584$	10396.9	0.736689
$585$	−2491.63	−0.176096
$586$	−754.677	−0.0532004
$587$	−7052.72	−0.495906	−0.247953	−	0.968772i	$-0.579758\pi$
$587$	−7052.72	−0.495906	−0.247953	+	0.968772i	$0.579758\pi$
$588$	9029.00	0.633248
$589$	603.138	0.0421933
$590$	3388.50	0.236444
$591$	−5602.92	−0.389972
$592$	−18946.2	−1.31534
$593$	10386.8	0.719280	0.359640	−	0.933091i	$-0.382900\pi$
$593$	10386.8	0.719280	0.359640	+	0.933091i	$0.382900\pi$
$594$	0	0
$595$	8448.88	0.582135
$596$	2127.07	0.146189
$597$	11452.0	0.785093
$598$	791.070	0.0540958
$599$	23289.6	1.58862	0.794312	−	0.607510i	$-0.207832\pi$
$599$	23289.6	1.58862	0.794312	+	0.607510i	$0.207832\pi$
$600$	−1000.50	−0.0680756
$601$	22970.1	1.55902	0.779509	−	0.626392i	$-0.215469\pi$
$601$	22970.1	1.55902	0.779509	+	0.626392i	$0.215469\pi$
$602$	−8332.51	−0.564133
$603$	−4632.90	−0.312880
$604$	3419.01	0.230327
$605$	0	0
$606$	199.906	0.0134004
$607$	14919.5	0.997631	0.498815	−	0.866708i	$-0.333769\pi$
$607$	14919.5	0.997631	0.498815	+	0.866708i	$0.333769\pi$
$608$	11674.7	0.778736
$609$	−2713.26	−0.180537
$610$	−3669.93	−0.243592
$611$	−1165.77	−0.0771883
$612$	−3992.53	−0.263706
$613$	15942.9	1.05045	0.525226	−	0.850963i	$-0.323981\pi$
$613$	15942.9	1.05045	0.525226	+	0.850963i	$0.323981\pi$
$614$	1717.06	0.112858
$615$	−4767.35	−0.312582
$616$	0	0
$617$	−13657.7	−0.891147	−0.445573	−	0.895245i	$-0.647000\pi$
$617$	−13657.7	−0.891147	−0.445573	+	0.895245i	$0.647000\pi$
$618$	−1375.97	−0.0895626
$619$	11469.7	0.744762	0.372381	−	0.928080i	$-0.378542\pi$
$619$	11469.7	0.744762	0.372381	+	0.928080i	$0.378542\pi$
$620$	−274.958	−0.0178106
$621$	440.493	0.0284643
$622$	7805.94	0.503199
$623$	−37412.1	−2.40591
$624$	−7671.34	−0.492146
$625$	625.000	0.0400000
$626$	−8176.10	−0.522017
$627$	0	0
$628$	13073.0	0.830685
$629$	−25160.8	−1.59495
$630$	−1085.75	−0.0686624
$631$	268.561	0.0169433	0.00847167	−	0.999964i	$-0.497303\pi$
$631$	268.561	0.0169433	0.00847167	+	0.999964i	$0.497303\pi$
$632$	4763.19	0.299793
$633$	−10341.5	−0.649349
$634$	540.694	0.0338702
$635$	−9124.63	−0.570236
$636$	−3390.10	−0.211362
$637$	23039.1	1.43303
$638$	0	0
$639$	−5915.73	−0.366232
$640$	−6939.98	−0.428636
$641$	−4944.80	−0.304693	−0.152346	−	0.988327i	$-0.548683\pi$
$641$	−4944.80	−0.304693	−0.152346	+	0.988327i	$0.548683\pi$
$642$	3642.65	0.223932
$643$	−24275.7	−1.48886	−0.744432	−	0.667698i	$-0.767280\pi$
$643$	−24275.7	−1.48886	−0.744432	+	0.667698i	$0.767280\pi$
$644$	−3251.23	−0.198939
$645$	−5180.24	−0.316236
$646$	4260.83	0.259505
$647$	−16690.0	−1.01415	−0.507074	−	0.861903i	$-0.669273\pi$
$647$	−16690.0	−1.01415	−0.507074	+	0.861903i	$0.669273\pi$
$648$	1080.54	0.0655058
$649$	0	0
$650$	−1212.22	−0.0731493
$651$	−628.407	−0.0378329
$652$	19367.0	1.16330
$653$	6888.09	0.412790	0.206395	−	0.978469i	$-0.433827\pi$
$653$	6888.09	0.412790	0.206395	+	0.978469i	$0.433827\pi$
$654$	−3054.02	−0.182602
$655$	−4506.67	−0.268840
$656$	−14677.9	−0.873590
$657$	7014.37	0.416525
$658$	−507.994	−0.0300968
$659$	11965.8	0.707315	0.353657	−	0.935375i	$-0.384938\pi$
$659$	11965.8	0.707315	0.353657	+	0.935375i	$0.384938\pi$
$660$	0	0
$661$	−18799.4	−1.10622	−0.553111	−	0.833108i	$-0.686559\pi$
$661$	−18799.4	−1.10622	−0.553111	+	0.833108i	$0.686559\pi$
$662$	−2379.07	−0.139675
$663$	−10187.6	−0.596765
$664$	−17067.0	−0.997481
$665$	−10928.6	−0.637281
$666$	3233.36	0.188124
$667$	535.545	0.0310891
$668$	−655.114	−0.0379448
$669$	−14838.6	−0.857538
$670$	−2253.98	−0.129968
$671$	0	0
$672$	−12163.8	−0.698259
$673$	−21026.0	−1.20430	−0.602148	−	0.798384i	$-0.705688\pi$
$673$	−21026.0	−1.20430	−0.602148	+	0.798384i	$0.705688\pi$
$674$	−2347.70	−0.134169
$675$	−675.000	−0.0384900
$676$	−6284.08	−0.357537
$677$	9575.29	0.543587	0.271793	−	0.962356i	$-0.412383\pi$
$677$	9575.29	0.543587	0.271793	+	0.962356i	$0.412383\pi$
$678$	3176.08	0.179907
$679$	22379.8	1.26489
$680$	−4090.79	−0.230698
$681$	−2101.03	−0.118225
$682$	0	0
$683$	18439.6	1.03305	0.516524	−	0.856273i	$-0.327226\pi$
$683$	18439.6	1.03305	0.516524	+	0.856273i	$0.327226\pi$
$684$	5164.31	0.288688
$685$	475.520	0.0265236
$686$	1763.65	0.0981580
$687$	7339.53	0.407599
$688$	−15949.1	−0.883801
$689$	−8650.44	−0.478310
$690$	214.306	0.0118239
$691$	−4577.48	−0.252005	−0.126002	−	0.992030i	$-0.540215\pi$
$691$	−4577.48	−0.252005	−0.126002	+	0.992030i	$0.540215\pi$
$692$	18189.1	0.999200
$693$	0	0
$694$	−1325.45	−0.0724975
$695$	−1203.20	−0.0656690
$696$	1313.71	0.0715461
$697$	−19492.4	−1.05929
$698$	503.520	0.0273045
$699$	−17445.3	−0.943978
$700$	4982.11	0.269008
$701$	−22172.1	−1.19462	−0.597309	−	0.802011i	$-0.703763\pi$
$701$	−22172.1	−1.19462	−0.597309	+	0.802011i	$0.703763\pi$
$702$	1309.19	0.0703879
$703$	32545.3	1.74605
$704$	0	0
$705$	−315.815	−0.0168713
$706$	−5728.83	−0.305393
$707$	−2096.45	−0.111521
$708$	16792.5	0.891383
$709$	7382.08	0.391029	0.195515	−	0.980701i	$-0.437362\pi$
$709$	7382.08	0.391029	0.195515	+	0.980701i	$0.437362\pi$
$710$	−2878.09	−0.152131
$711$	3213.53	0.169503
$712$	18114.3	0.953455
$713$	124.036	0.00651497
$714$	−4439.35	−0.232687
$715$	0	0
$716$	22015.5	1.14910
$717$	590.659	0.0307651
$718$	5850.50	0.304093
$719$	22377.8	1.16071	0.580356	−	0.814363i	$-0.302913\pi$
$719$	22377.8	1.16071	0.580356	+	0.814363i	$0.302913\pi$
$720$	−2078.22	−0.107570
$721$	14430.0	0.745358
$722$	495.248	0.0255280
$723$	901.598	0.0463773
$724$	25518.1	1.30991
$725$	−820.656	−0.0420392
$726$	0	0
$727$	−36356.1	−1.85471	−0.927353	−	0.374187i	$-0.877922\pi$
$727$	−36356.1	−1.85471	−0.927353	+	0.374187i	$0.877922\pi$
$728$	−20350.6	−1.03605
$729$	729.000	0.0370370
$730$	3412.60	0.173022
$731$	−21180.7	−1.07168
$732$	−18187.2	−0.918330
$733$	27151.9	1.36818	0.684091	−	0.729397i	$-0.260199\pi$
$733$	27151.9	1.36818	0.684091	+	0.729397i	$0.260199\pi$
$734$	−5195.59	−0.261271
$735$	6241.44	0.313223
$736$	2400.91	0.120243
$737$	0	0
$738$	2504.94	0.124943
$739$	−13023.5	−0.648277	−0.324138	−	0.946010i	$-0.605074\pi$
$739$	−13023.5	−0.648277	−0.324138	+	0.946010i	$0.605074\pi$
$740$	−14836.7	−0.737039
$741$	13177.6	0.653297
$742$	−3769.50	−0.186500
$743$	−20228.2	−0.998791	−0.499396	−	0.866374i	$-0.666445\pi$
$743$	−20228.2	−0.998791	−0.499396	+	0.866374i	$0.666445\pi$
$744$	304.263	0.0149931
$745$	1470.38	0.0723092
$746$	−6134.81	−0.301088
$747$	−11514.4	−0.563977
$748$	0	0
$749$	−38201.1	−1.86360
$750$	−328.398	−0.0159885
$751$	−16475.6	−0.800538	−0.400269	−	0.916398i	$-0.631083\pi$
$751$	−16475.6	−0.800538	−0.400269	+	0.916398i	$0.631083\pi$
$752$	−972.344	−0.0471512
$753$	22454.5	1.08670
$754$	1591.70	0.0768784
$755$	2363.45	0.113927
$756$	−5380.68	−0.258854
$757$	31518.3	1.51328	0.756641	−	0.653831i	$-0.226839\pi$
$757$	31518.3	1.51328	0.756641	+	0.653831i	$0.226839\pi$
$758$	720.688	0.0345337
$759$	0	0
$760$	5291.42	0.252553
$761$	25376.7	1.20881	0.604405	−	0.796677i	$-0.293411\pi$
$761$	25376.7	1.20881	0.604405	+	0.796677i	$0.293411\pi$
$762$	4794.41	0.227931
$763$	32028.0	1.51965
$764$	18975.6	0.898580
$765$	−2759.90	−0.130437
$766$	10988.0	0.518291
$767$	42848.9	2.01719
$768$	−2127.54	−0.0999621
$769$	16916.2	0.793255	0.396628	−	0.917980i	$-0.370180\pi$
$769$	16916.2	0.793255	0.396628	+	0.917980i	$0.370180\pi$
$770$	0	0
$771$	4710.39	0.220027
$772$	−15898.6	−0.741196
$773$	10118.2	0.470797	0.235398	−	0.971899i	$-0.424361\pi$
$773$	10118.2	0.470797	0.235398	+	0.971899i	$0.424361\pi$
$774$	2721.89	0.126403
$775$	−190.069	−0.00880966
$776$	−10835.9	−0.501271
$777$	−33908.9	−1.56560
$778$	−11716.0	−0.539897
$779$	25213.3	1.15964
$780$	−6007.41	−0.275769
$781$	0	0
$782$	876.242	0.0400695
$783$	886.309	0.0404522
$784$	19216.4	0.875383
$785$	9036.94	0.410882
$786$	2367.97	0.107459
$787$	15357.0	0.695573	0.347787	−	0.937574i	$-0.386933\pi$
$787$	15357.0	0.695573	0.347787	+	0.937574i	$0.386933\pi$
$788$	−13508.8	−0.610701
$789$	14804.4	0.667997
$790$	1563.43	0.0704107
$791$	−33308.1	−1.49722
$792$	0	0
$793$	−46407.8	−2.07817
$794$	−9517.02	−0.425373
$795$	−2343.46	−0.104546
$796$	27611.3	1.22947
$797$	37362.6	1.66054	0.830271	−	0.557359i	$-0.188185\pi$
$797$	37362.6	1.66054	0.830271	+	0.557359i	$0.188185\pi$
$798$	5742.27	0.254729
$799$	−1291.29	−0.0571745
$800$	−3679.09	−0.162594
$801$	12221.0	0.539085
$802$	−7557.09	−0.332731
$803$	0	0
$804$	−11170.1	−0.489974
$805$	−2247.47	−0.0984010
$806$	368.648	0.0161105
$807$	−7366.42	−0.321326
$808$	1015.06	0.0441954
$809$	−35234.5	−1.53125	−0.765624	−	0.643289i	$-0.777569\pi$
$809$	−35234.5	−1.53125	−0.765624	+	0.643289i	$0.777569\pi$
$810$	354.669	0.0153850
$811$	−15144.4	−0.655724	−0.327862	−	0.944726i	$-0.606328\pi$
$811$	−15144.4	−0.655724	−0.327862	+	0.944726i	$0.606328\pi$
$812$	−6541.76	−0.282723
$813$	−2794.37	−0.120545
$814$	0	0
$815$	13387.8	0.575403
$816$	−8497.29	−0.364540
$817$	27397.1	1.17320
$818$	−13327.5	−0.569664
$819$	−13729.7	−0.585783
$820$	−11494.2	−0.489507
$821$	−30089.4	−1.27909	−0.639543	−	0.768756i	$-0.720876\pi$
$821$	−30089.4	−1.27909	−0.639543	+	0.768756i	$0.720876\pi$
$822$	−249.855	−0.0106018
$823$	1616.31	0.0684580	0.0342290	−	0.999414i	$-0.489102\pi$
$823$	1616.31	0.0684580	0.0342290	+	0.999414i	$0.489102\pi$
$824$	−6986.77	−0.295383
$825$	0	0
$826$	18671.8	0.786530
$827$	12137.5	0.510355	0.255178	−	0.966894i	$-0.417866\pi$
$827$	12137.5	0.510355	0.255178	+	0.966894i	$0.417866\pi$
$828$	1062.04	0.0445755
$829$	−7604.96	−0.318614	−0.159307	−	0.987229i	$-0.550926\pi$
$829$	−7604.96	−0.318614	−0.159307	+	0.987229i	$0.550926\pi$
$830$	−5601.94	−0.234272
$831$	−12182.7	−0.508559
$832$	−13321.1	−0.555081
$833$	25519.6	1.06147
$834$	632.204	0.0262487
$835$	−452.858	−0.0187686
$836$	0	0
$837$	205.275	0.00847710
$838$	12512.0	0.515776
$839$	−20369.0	−0.838158	−0.419079	−	0.907950i	$-0.637647\pi$
$839$	−20369.0	−0.838158	−0.419079	+	0.907950i	$0.637647\pi$
$840$	−5513.11	−0.226453
$841$	−23311.4	−0.955818
$842$	−7713.42	−0.315703
$843$	−23233.0	−0.949215
$844$	−24933.7	−1.01689
$845$	−4343.97	−0.176849
$846$	165.941	0.00674368
$847$	0	0
$848$	−7215.14	−0.292181
$849$	16951.4	0.685244
$850$	−1342.73	−0.0541827
$851$	6692.96	0.269603
$852$	−14263.0	−0.573525
$853$	43982.6	1.76546	0.882730	−	0.469881i	$-0.155703\pi$
$853$	43982.6	1.76546	0.882730	+	0.469881i	$0.155703\pi$
$854$	−20222.6	−0.810307
$855$	3569.91	0.142794
$856$	18496.3	0.738540
$857$	−28723.2	−1.14488	−0.572442	−	0.819946i	$-0.694004\pi$
$857$	−28723.2	−1.14488	−0.572442	+	0.819946i	$0.694004\pi$
$858$	0	0
$859$	25072.7	0.995891	0.497945	−	0.867208i	$-0.334088\pi$
$859$	25072.7	0.995891	0.497945	+	0.867208i	$0.334088\pi$
$860$	−12489.7	−0.495229
$861$	−26269.7	−1.03980
$862$	10489.1	0.414455
$863$	14149.2	0.558106	0.279053	−	0.960276i	$-0.409980\pi$
$863$	14149.2	0.558106	0.279053	+	0.960276i	$0.409980\pi$
$864$	3973.42	0.156457
$865$	12573.5	0.494234
$866$	7182.98	0.281856
$867$	3454.49	0.135318
$868$	−1515.11	−0.0592468
$869$	0	0
$870$	431.203	0.0168036
$871$	−28502.5	−1.10880
$872$	−15507.4	−0.602232
$873$	−7310.56	−0.283419
$874$	−1133.41	−0.0438653
$875$	3443.96	0.133060
$876$	16911.9	0.652283
$877$	25334.2	0.975455	0.487728	−	0.872996i	$-0.337826\pi$
$877$	25334.2	0.975455	0.487728	+	0.872996i	$0.337826\pi$
$878$	1117.73	0.0429630
$879$	−2585.32	−0.0992042
$880$	0	0
$881$	−25153.1	−0.961897	−0.480948	−	0.876749i	$-0.659708\pi$
$881$	−25153.1	−0.961897	−0.480948	+	0.876749i	$0.659708\pi$
$882$	−3279.48	−0.125199
$883$	−21749.1	−0.828896	−0.414448	−	0.910073i	$-0.636025\pi$
$883$	−21749.1	−0.828896	−0.414448	+	0.910073i	$0.636025\pi$
$884$	−24562.7	−0.934541
$885$	11608.1	0.440905
$886$	9026.29	0.342262
$887$	22802.7	0.863179	0.431590	−	0.902070i	$-0.357953\pi$
$887$	22802.7	0.863179	0.431590	+	0.902070i	$0.357953\pi$
$888$	16418.1	0.620444
$889$	−50279.8	−1.89689
$890$	5945.69	0.223932
$891$	0	0
$892$	−35776.3	−1.34292
$893$	1670.27	0.0625907
$894$	−772.588	−0.0289029
$895$	15218.5	0.568380
$896$	−38241.7	−1.42585
$897$	2709.99	0.100874
$898$	−8088.68	−0.300582
$899$	249.570	0.00925877
$900$	−1627.45	−0.0602759
$901$	−9581.81	−0.354291
$902$	0	0
$903$	−28544.9	−1.05195
$904$	16127.2	0.593343
$905$	17639.8	0.647919
$906$	−1241.84	−0.0455379
$907$	32587.7	1.19301	0.596503	−	0.802611i	$-0.296556\pi$
$907$	32587.7	1.19301	0.596503	+	0.802611i	$0.296556\pi$
$908$	−5065.65	−0.185143
$909$	684.825	0.0249881
$910$	−6679.73	−0.243330
$911$	−1832.04	−0.0666281	−0.0333141	−	0.999445i	$-0.510606\pi$
$911$	−1832.04	−0.0666281	−0.0333141	+	0.999445i	$0.510606\pi$
$912$	10991.2	0.399073
$913$	0	0
$914$	−8029.15	−0.290569
$915$	−12572.2	−0.454233
$916$	17695.9	0.638305
$917$	−24833.3	−0.894294
$918$	1450.15	0.0521374
$919$	−8877.26	−0.318644	−0.159322	−	0.987227i	$-0.550931\pi$
$919$	−8877.26	−0.318644	−0.159322	+	0.987227i	$0.550931\pi$
$920$	1088.18	0.0389960
$921$	5882.16	0.210449
$922$	10554.4	0.376996
$923$	−36394.6	−1.29788
$924$	0	0
$925$	−10256.1	−0.364562
$926$	−6149.13	−0.218221
$927$	−4713.70	−0.167010
$928$	4830.83	0.170883
$929$	−26142.9	−0.923274	−0.461637	−	0.887069i	$-0.652738\pi$
$929$	−26142.9	−0.923274	−0.461637	+	0.887069i	$0.652738\pi$
$930$	99.8692	0.00352133
$931$	−33009.5	−1.16202
$932$	−42061.1	−1.47828
$933$	26741.0	0.938330
$934$	−2366.69	−0.0829128
$935$	0	0
$936$	6647.69	0.232144
$937$	−609.632	−0.0212549	−0.0106274	−	0.999944i	$-0.503383\pi$
$937$	−609.632	−0.0212549	−0.0106274	+	0.999944i	$0.503383\pi$
$938$	−12420.2	−0.432338
$939$	−28009.1	−0.973421
$940$	−761.441	−0.0264207
$941$	−1504.10	−0.0521067	−0.0260533	−	0.999661i	$-0.508294\pi$
$941$	−1504.10	−0.0521067	−0.0260533	+	0.999661i	$0.508294\pi$
$942$	−4748.33	−0.164235
$943$	5185.14	0.179058
$944$	35739.3	1.23222
$945$	−3719.48	−0.128037
$946$	0	0
$947$	−12897.7	−0.442577	−0.221289	−	0.975208i	$-0.571026\pi$
$947$	−12897.7	−0.442577	−0.221289	+	0.975208i	$0.571026\pi$
$948$	7747.94	0.265445
$949$	43153.7	1.47611
$950$	1736.82	0.0593155
$951$	1852.27	0.0631588
$952$	−22541.7	−0.767416
$953$	23191.2	0.788286	0.394143	−	0.919049i	$-0.371041\pi$
$953$	23191.2	0.788286	0.394143	+	0.919049i	$0.371041\pi$
$954$	1231.34	0.0417884
$955$	13117.2	0.444464
$956$	1424.10	0.0481785
$957$	0	0
$958$	−10860.5	−0.366270
$959$	2620.28	0.0882306
$960$	−3608.78	−0.121326
$961$	−29733.2	−0.998060
$962$	19892.2	0.666686
$963$	12478.7	0.417572
$964$	2173.78	0.0726274
$965$	−10990.2	−0.366618
$966$	1180.90	0.0393321
$967$	50632.2	1.68379	0.841893	−	0.539644i	$-0.181441\pi$
$967$	50632.2	1.68379	0.841893	+	0.539644i	$0.181441\pi$
$968$	0	0
$969$	14596.4	0.483906
$970$	−3556.70	−0.117731
$971$	−42622.7	−1.40868	−0.704340	−	0.709863i	$-0.748757\pi$
$971$	−42622.7	−1.40868	−0.704340	+	0.709863i	$0.748757\pi$
$972$	1757.64	0.0580005
$973$	−6630.03	−0.218447
$974$	−13267.0	−0.436451
$975$	−4152.72	−0.136404
$976$	−38707.7	−1.26947
$977$	18699.7	0.612341	0.306171	−	0.951977i	$-0.400952\pi$
$977$	18699.7	0.612341	0.306171	+	0.951977i	$0.400952\pi$
$978$	−7034.43	−0.229996
$979$	0	0
$980$	15048.3	0.490512
$981$	−10462.2	−0.340503
$982$	322.590	0.0104829
$983$	12533.3	0.406665	0.203333	−	0.979110i	$-0.434823\pi$
$983$	12533.3	0.406665	0.203333	+	0.979110i	$0.434823\pi$
$984$	12719.3	0.412070
$985$	−9338.20	−0.302071
$986$	1763.07	0.0569450
$987$	−1740.25	−0.0561223
$988$	31771.8	1.02307
$989$	5634.22	0.181150
$990$	0	0
$991$	23975.1	0.768512	0.384256	−	0.923227i	$-0.374458\pi$
$991$	23975.1	0.768512	0.384256	+	0.923227i	$0.374458\pi$
$992$	1118.85	0.0358100
$993$	−8150.04	−0.260457
$994$	−15859.3	−0.506061
$995$	19086.7	0.608131
$996$	−27761.7	−0.883195
$997$	−55397.7	−1.75974	−0.879871	−	0.475213i	$-0.842371\pi$
$997$	−55397.7	−1.75974	−0.879871	+	0.475213i	$0.842371\pi$
$998$	−7627.44	−0.241926
$999$	11076.6	0.350800

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1815.4.a.bj.1.6		12
11.3	even	5		165.4.m.b.31.3	yes	24
11.4	even	5		165.4.m.b.16.3	✓	24
11.10	odd	2		1815.4.a.bm.1.7		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
165.4.m.b.16.3	✓	24	11.4	even	5
165.4.m.b.31.3	yes	24	11.3	even	5
1815.4.a.bj.1.6		12	1.1	even	1	trivial
1815.4.a.bm.1.7		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)

\(f(q)\)	\(=\)	\(q-0.875727 q^{2} -3.00000 q^{3} -7.23310 q^{4} -5.00000 q^{5} +2.62718 q^{6} -27.5517 q^{7} +13.3400 q^{8} +9.00000 q^{9} +O(q^{10})\)	\(q-0.875727 q^{2} -3.00000 q^{3} -7.23310 q^{4} -5.00000 q^{5} +2.62718 q^{6} -27.5517 q^{7} +13.3400 q^{8} +9.00000 q^{9} +4.37863 q^{10} +21.6993 q^{12} +55.3696 q^{13} +24.1278 q^{14} +15.0000 q^{15} +46.1826 q^{16} +61.3311 q^{17} -7.88154 q^{18} -79.3314 q^{19} +36.1655 q^{20} +82.6551 q^{21} -16.3145 q^{23} -40.0201 q^{24} +25.0000 q^{25} -48.4887 q^{26} -27.0000 q^{27} +199.284 q^{28} -32.8263 q^{29} -13.1359 q^{30} -7.60276 q^{31} -147.164 q^{32} -53.7093 q^{34} +137.759 q^{35} -65.0979 q^{36} -410.245 q^{37} +69.4726 q^{38} -166.109 q^{39} -66.7002 q^{40} -317.823 q^{41} -72.3833 q^{42} -345.350 q^{43} -45.0000 q^{45} +14.2871 q^{46} -21.0543 q^{47} -138.548 q^{48} +416.096 q^{49} -21.8932 q^{50} -183.993 q^{51} -400.494 q^{52} -156.231 q^{53} +23.6446 q^{54} -367.541 q^{56} +237.994 q^{57} +28.7468 q^{58} +773.871 q^{59} -108.497 q^{60} -838.145 q^{61} +6.65794 q^{62} -247.965 q^{63} -240.586 q^{64} -276.848 q^{65} -514.767 q^{67} -443.614 q^{68} +48.9436 q^{69} -120.639 q^{70} -657.303 q^{71} +120.060 q^{72} +779.375 q^{73} +359.263 q^{74} -75.0000 q^{75} +573.812 q^{76} +145.466 q^{78} +357.059 q^{79} -230.913 q^{80} +81.0000 q^{81} +278.326 q^{82} -1279.38 q^{83} -597.853 q^{84} -306.655 q^{85} +302.432 q^{86} +98.4788 q^{87} +1357.89 q^{89} +39.4077 q^{90} -1525.53 q^{91} +118.005 q^{92} +22.8083 q^{93} +18.4378 q^{94} +396.657 q^{95} +441.491 q^{96} -812.285 q^{97} -364.387 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - q^{2} - 36 q^{3} + 57 q^{4} - 60 q^{5} + 3 q^{6} + 51 q^{7} + 45 q^{8} + 108 q^{9}+O(q^{10}) \) 12 * q - q^2 - 36 * q^3 + 57 * q^4 - 60 * q^5 + 3 * q^6 + 51 * q^7 + 45 * q^8 + 108 * q^9	\( 12 q - q^{2} - 36 q^{3} + 57 q^{4} - 60 q^{5} + 3 q^{6} + 51 q^{7} + 45 q^{8} + 108 q^{9} + 5 q^{10} - 171 q^{12} + 132 q^{13} + 178 q^{14} + 180 q^{15} + 329 q^{16} + 189 q^{17} - 9 q^{18} + 85 q^{19} - 285 q^{20} - 153 q^{21} + 444 q^{23} - 135 q^{24} + 300 q^{25} - 308 q^{26} - 324 q^{27} + 858 q^{28} - 439 q^{29} - 15 q^{30} + 75 q^{31} - 56 q^{32} + 866 q^{34} - 255 q^{35} + 513 q^{36} - 138 q^{37} + 660 q^{38} - 396 q^{39} - 225 q^{40} - 379 q^{41} - 534 q^{42} + 1221 q^{43} - 540 q^{45} - 595 q^{46} - 696 q^{47} - 987 q^{48} + 731 q^{49} - 25 q^{50} - 567 q^{51} + 373 q^{52} - 915 q^{53} + 27 q^{54} + 2181 q^{56} - 255 q^{57} - 1182 q^{58} + 868 q^{59} + 855 q^{60} - 84 q^{61} - 1809 q^{62} + 459 q^{63} - 425 q^{64} - 660 q^{65} + 1569 q^{67} + 1182 q^{68} - 1332 q^{69} - 890 q^{70} - 604 q^{71} + 405 q^{72} + 3156 q^{73} - 2273 q^{74} - 900 q^{75} - 146 q^{76} + 924 q^{78} + 1061 q^{79} - 1645 q^{80} + 972 q^{81} - 3030 q^{82} - 314 q^{83} - 2574 q^{84} - 945 q^{85} - 4975 q^{86} + 1317 q^{87} + 3943 q^{89} + 45 q^{90} + 2726 q^{91} + 1842 q^{92} - 225 q^{93} + 1683 q^{94} - 425 q^{95} + 168 q^{96} + 194 q^{97} + 4008 q^{98}+O(q^{100}) \) 12 * q - q^2 - 36 * q^3 + 57 * q^4 - 60 * q^5 + 3 * q^6 + 51 * q^7 + 45 * q^8 + 108 * q^9 + 5 * q^10 - 171 * q^12 + 132 * q^13 + 178 * q^14 + 180 * q^15 + 329 * q^16 + 189 * q^17 - 9 * q^18 + 85 * q^19 - 285 * q^20 - 153 * q^21 + 444 * q^23 - 135 * q^24 + 300 * q^25 - 308 * q^26 - 324 * q^27 + 858 * q^28 - 439 * q^29 - 15 * q^30 + 75 * q^31 - 56 * q^32 + 866 * q^34 - 255 * q^35 + 513 * q^36 - 138 * q^37 + 660 * q^38 - 396 * q^39 - 225 * q^40 - 379 * q^41 - 534 * q^42 + 1221 * q^43 - 540 * q^45 - 595 * q^46 - 696 * q^47 - 987 * q^48 + 731 * q^49 - 25 * q^50 - 567 * q^51 + 373 * q^52 - 915 * q^53 + 27 * q^54 + 2181 * q^56 - 255 * q^57 - 1182 * q^58 + 868 * q^59 + 855 * q^60 - 84 * q^61 - 1809 * q^62 + 459 * q^63 - 425 * q^64 - 660 * q^65 + 1569 * q^67 + 1182 * q^68 - 1332 * q^69 - 890 * q^70 - 604 * q^71 + 405 * q^72 + 3156 * q^73 - 2273 * q^74 - 900 * q^75 - 146 * q^76 + 924 * q^78 + 1061 * q^79 - 1645 * q^80 + 972 * q^81 - 3030 * q^82 - 314 * q^83 - 2574 * q^84 - 945 * q^85 - 4975 * q^86 + 1317 * q^87 + 3943 * q^89 + 45 * q^90 + 2726 * q^91 + 1842 * q^92 - 225 * q^93 + 1683 * q^94 - 425 * q^95 + 168 * q^96 + 194 * q^97 + 4008 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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