LMFDB - Embedded newform 1815.4.a.bm.1.4

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} - 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			1.4
Root			$-2.95717$ 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)

$f(q)$	$=$	$q-2.95717 q^{2} -3.00000 q^{3} +0.744851 q^{4} -5.00000 q^{5} +8.87151 q^{6} +13.4337 q^{7} +21.4547 q^{8} +9.00000 q^{9} +O(q^{10})$	\(q-2.95717 q^{2} -3.00000 q^{3} +0.744851 q^{4} -5.00000 q^{5} +8.87151 q^{6} +13.4337 q^{7} +21.4547 q^{8} +9.00000 q^{9} +14.7858 q^{10} -2.23455 q^{12} +13.8516 q^{13} -39.7257 q^{14} +15.0000 q^{15} -69.4040 q^{16} +25.6449 q^{17} -26.6145 q^{18} +38.1498 q^{19} -3.72425 q^{20} -40.3011 q^{21} -62.5445 q^{23} -64.3641 q^{24} +25.0000 q^{25} -40.9616 q^{26} -27.0000 q^{27} +10.0061 q^{28} +53.6490 q^{29} -44.3575 q^{30} -220.673 q^{31} +33.6018 q^{32} -75.8363 q^{34} -67.1685 q^{35} +6.70366 q^{36} -162.917 q^{37} -112.815 q^{38} -41.5549 q^{39} -107.274 q^{40} +17.8559 q^{41} +119.177 q^{42} +259.519 q^{43} -45.0000 q^{45} +184.955 q^{46} +20.5710 q^{47} +208.212 q^{48} -162.535 q^{49} -73.9292 q^{50} -76.9347 q^{51} +10.3174 q^{52} +123.088 q^{53} +79.8436 q^{54} +288.216 q^{56} -114.449 q^{57} -158.649 q^{58} -697.175 q^{59} +11.1728 q^{60} -829.186 q^{61} +652.566 q^{62} +120.903 q^{63} +455.866 q^{64} -69.2581 q^{65} +445.977 q^{67} +19.1016 q^{68} +187.633 q^{69} +198.629 q^{70} +718.238 q^{71} +193.092 q^{72} -268.461 q^{73} +481.773 q^{74} -75.0000 q^{75} +28.4159 q^{76} +122.885 q^{78} +368.090 q^{79} +347.020 q^{80} +81.0000 q^{81} -52.8030 q^{82} -525.631 q^{83} -30.0183 q^{84} -128.224 q^{85} -767.443 q^{86} -160.947 q^{87} +1188.65 q^{89} +133.073 q^{90} +186.079 q^{91} -46.5863 q^{92} +662.018 q^{93} -60.8318 q^{94} -190.749 q^{95} -100.805 q^{96} +296.908 q^{97} +480.645 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−2.95717	−1.04552	−0.522759	−	0.852481i	$-0.675097\pi$
$2$	−2.95717	−1.04552	−0.522759	+	0.852481i	$0.675097\pi$
$3$	−3.00000	−0.577350
$3$	−3.00000	−0.577350
$4$	0.744851	0.0931063
$5$	−5.00000	−0.447214
$5$	−5.00000	−0.447214
$6$	8.87151	0.603630
$7$	13.4337	0.725352	0.362676	−	0.931915i	$-0.381863\pi$
$7$	13.4337	0.725352	0.362676	+	0.931915i	$0.381863\pi$
$8$	21.4547	0.948173
$9$	9.00000	0.333333
$10$	14.7858	0.467570
$11$	0	0
$11$	0	0
$12$	−2.23455	−0.0537550
$13$	13.8516	0.295519	0.147760	−	0.989023i	$-0.452794\pi$
$13$	13.8516	0.295519	0.147760	+	0.989023i	$0.452794\pi$
$14$	−39.7257	−0.758368
$15$	15.0000	0.258199
$16$	−69.4040	−1.08444
$17$	25.6449	0.365871	0.182935	−	0.983125i	$-0.441440\pi$
$17$	25.6449	0.365871	0.182935	+	0.983125i	$0.441440\pi$
$18$	−26.6145	−0.348506
$19$	38.1498	0.460640	0.230320	−	0.973115i	$-0.426023\pi$
$19$	38.1498	0.460640	0.230320	+	0.973115i	$0.426023\pi$
$20$	−3.72425	−0.0416384
$21$	−40.3011	−0.418782
$22$	0	0
$23$	−62.5445	−0.567019	−0.283509	−	0.958969i	$-0.591499\pi$
$23$	−62.5445	−0.567019	−0.283509	+	0.958969i	$0.591499\pi$
$24$	−64.3641	−0.547428
$25$	25.0000	0.200000
$26$	−40.9616	−0.308970
$27$	−27.0000	−0.192450
$28$	10.0061	0.0675349
$29$	53.6490	0.343530	0.171765	−	0.985138i	$-0.445053\pi$
$29$	53.6490	0.343530	0.171765	+	0.985138i	$0.445053\pi$
$30$	−44.3575	−0.269951
$31$	−220.673	−1.27852	−0.639258	−	0.768993i	$-0.720758\pi$
$31$	−220.673	−1.27852	−0.639258	+	0.768993i	$0.720758\pi$
$32$	33.6018	0.185625
$33$	0	0
$34$	−75.8363	−0.382524
$35$	−67.1685	−0.324387
$36$	6.70366	0.0310354
$37$	−162.917	−0.723876	−0.361938	−	0.932202i	$-0.617885\pi$
$37$	−162.917	−0.723876	−0.361938	+	0.932202i	$0.617885\pi$
$38$	−112.815	−0.481607
$39$	−41.5549	−0.170618
$40$	−107.274	−0.424036
$41$	17.8559	0.0680152	0.0340076	−	0.999422i	$-0.489173\pi$
$41$	17.8559	0.0680152	0.0340076	+	0.999422i	$0.489173\pi$
$42$	119.177	0.437844
$43$	259.519	0.920380	0.460190	−	0.887821i	$-0.347781\pi$
$43$	259.519	0.920380	0.460190	+	0.887821i	$0.347781\pi$
$44$	0	0
$45$	−45.0000	−0.149071
$46$	184.955	0.592828
$47$	20.5710	0.0638422	0.0319211	−	0.999490i	$-0.489837\pi$
$47$	20.5710	0.0638422	0.0319211	+	0.999490i	$0.489837\pi$
$48$	208.212	0.626100
$49$	−162.535	−0.473864
$50$	−73.9292	−0.209103
$51$	−76.9347	−0.211236
$52$	10.3174	0.0275147
$53$	123.088	0.319008	0.159504	−	0.987197i	$-0.449010\pi$
$53$	123.088	0.319008	0.159504	+	0.987197i	$0.449010\pi$
$54$	79.8436	0.201210
$55$	0	0
$56$	288.216	0.687759
$57$	−114.449	−0.265951
$58$	−158.649	−0.359167
$59$	−697.175	−1.53838	−0.769191	−	0.639019i	$-0.779340\pi$
$59$	−697.175	−1.53838	−0.769191	+	0.639019i	$0.779340\pi$
$60$	11.1728	0.0240400
$61$	−829.186	−1.74043	−0.870216	−	0.492670i	$-0.836021\pi$
$61$	−829.186	−1.74043	−0.870216	+	0.492670i	$0.836021\pi$
$62$	652.566	1.33671
$63$	120.903	0.241784
$64$	455.866	0.890363
$65$	−69.2581	−0.132160
$66$	0	0
$67$	445.977	0.813206	0.406603	−	0.913605i	$-0.366713\pi$
$67$	445.977	0.813206	0.406603	+	0.913605i	$0.366713\pi$
$68$	19.1016	0.0340649
$69$	187.633	0.327368
$70$	198.629	0.339153
$71$	718.238	1.20055	0.600276	−	0.799793i	$-0.295057\pi$
$71$	718.238	1.20055	0.600276	+	0.799793i	$0.295057\pi$
$72$	193.092	0.316058
$73$	−268.461	−0.430424	−0.215212	−	0.976567i	$-0.569044\pi$
$73$	−268.461	−0.430424	−0.215212	+	0.976567i	$0.569044\pi$
$74$	481.773	0.756825
$75$	−75.0000	−0.115470
$76$	28.4159	0.0428885
$77$	0	0
$78$	122.885	0.178384
$79$	368.090	0.524219	0.262110	−	0.965038i	$-0.415582\pi$
$79$	368.090	0.524219	0.262110	+	0.965038i	$0.415582\pi$
$80$	347.020	0.484975
$81$	81.0000	0.111111
$82$	−52.8030	−0.0711111
$83$	−525.631	−0.695126	−0.347563	−	0.937657i	$-0.612991\pi$
$83$	−525.631	−0.695126	−0.347563	+	0.937657i	$0.612991\pi$
$84$	−30.0183	−0.0389913
$85$	−128.224	−0.163622
$86$	−767.443	−0.962273
$87$	−160.947	−0.198337
$88$	0	0
$89$	1188.65	1.41569	0.707845	−	0.706367i	$-0.249667\pi$
$89$	1188.65	1.41569	0.707845	+	0.706367i	$0.249667\pi$
$90$	133.073	0.155857
$91$	186.079	0.214355
$92$	−46.5863	−0.0527930
$93$	662.018	0.738151
$94$	−60.8318	−0.0667481
$95$	−190.749	−0.206005
$96$	−100.805	−0.107171
$97$	296.908	0.310789	0.155394	−	0.987853i	$-0.450335\pi$
$97$	296.908	0.310789	0.155394	+	0.987853i	$0.450335\pi$
$98$	480.645	0.495433
$99$	0	0
$100$	18.6213	0.0186213
$101$	246.716	0.243061	0.121531	−	0.992588i	$-0.461220\pi$
$101$	246.716	0.243061	0.121531	+	0.992588i	$0.461220\pi$
$102$	227.509	0.220850
$103$	−213.327	−0.204075	−0.102038	−	0.994781i	$-0.532536\pi$
$103$	−213.327	−0.204075	−0.102038	+	0.994781i	$0.532536\pi$
$104$	297.182	0.280203
$105$	201.506	0.187285
$106$	−363.992	−0.333529
$107$	−79.8372	−0.0721322	−0.0360661	−	0.999349i	$-0.511483\pi$
$107$	−79.8372	−0.0721322	−0.0360661	+	0.999349i	$0.511483\pi$
$108$	−20.1110	−0.0179183
$109$	814.199	0.715469	0.357735	−	0.933823i	$-0.383549\pi$
$109$	814.199	0.715469	0.357735	+	0.933823i	$0.383549\pi$
$110$	0	0
$111$	488.751	0.417930
$112$	−932.353	−0.786599
$113$	1918.61	1.59723	0.798616	−	0.601840i	$-0.205566\pi$
$113$	1918.61	1.59723	0.798616	+	0.601840i	$0.205566\pi$
$114$	338.446	0.278056
$115$	312.722	0.253578
$116$	39.9605	0.0319848
$117$	124.665	0.0985064
$118$	2061.67	1.60840
$119$	344.506	0.265385
$120$	321.821	0.244817
$121$	0	0
$122$	2452.04	1.81965
$123$	−53.5677	−0.0392686
$124$	−164.368	−0.119038
$125$	−125.000	−0.0894427
$126$	−357.532	−0.252789
$127$	569.156	0.397673	0.198837	−	0.980033i	$-0.436284\pi$
$127$	569.156	0.397673	0.198837	+	0.980033i	$0.436284\pi$
$128$	−1616.89	−1.11652
$129$	−778.558	−0.531381
$130$	204.808	0.138176
$131$	2730.18	1.82089	0.910445	−	0.413629i	$-0.135739\pi$
$131$	2730.18	1.82089	0.910445	+	0.413629i	$0.135739\pi$
$132$	0	0
$133$	512.493	0.334126
$134$	−1318.83	−0.850221
$135$	135.000	0.0860663
$136$	550.204	0.346909
$137$	−1193.11	−0.744048	−0.372024	−	0.928223i	$-0.621336\pi$
$137$	−1193.11	−0.744048	−0.372024	+	0.928223i	$0.621336\pi$
$138$	−554.864	−0.342269
$139$	−1266.67	−0.772930	−0.386465	−	0.922304i	$-0.626304\pi$
$139$	−1266.67	−0.772930	−0.386465	+	0.922304i	$0.626304\pi$
$140$	−50.0305	−0.0302025
$141$	−61.7129	−0.0368593
$142$	−2123.95	−1.25520
$143$	0	0
$144$	−624.636	−0.361479
$145$	−268.245	−0.153631
$146$	793.885	0.450016
$147$	487.606	0.273586
$148$	−121.349	−0.0673974
$149$	−849.490	−0.467067	−0.233533	−	0.972349i	$-0.575029\pi$
$149$	−849.490	−0.467067	−0.233533	+	0.972349i	$0.575029\pi$
$150$	221.788	0.120726
$151$	−2887.11	−1.55596	−0.777979	−	0.628290i	$-0.783755\pi$
$151$	−2887.11	−1.55596	−0.777979	+	0.628290i	$0.783755\pi$
$152$	818.492	0.436766
$153$	230.804	0.121957
$154$	0	0
$155$	1103.36	0.571769
$156$	−30.9522	−0.0158856
$157$	753.269	0.382913	0.191457	−	0.981501i	$-0.438679\pi$
$157$	753.269	0.382913	0.191457	+	0.981501i	$0.438679\pi$
$158$	−1088.50	−0.548080
$159$	−369.264	−0.184180
$160$	−168.009	−0.0830141
$161$	−840.204	−0.411288
$162$	−239.531	−0.116169
$163$	1263.95	0.607361	0.303681	−	0.952774i	$-0.401784\pi$
$163$	1263.95	0.607361	0.303681	+	0.952774i	$0.401784\pi$
$164$	13.3000	0.00633265
$165$	0	0
$166$	1554.38	0.726766
$167$	−3454.82	−1.60085	−0.800426	−	0.599432i	$-0.795393\pi$
$167$	−3454.82	−1.60085	−0.800426	+	0.599432i	$0.795393\pi$
$168$	−864.649	−0.397078
$169$	−2005.13	−0.912668
$170$	379.181	0.171070
$171$	343.348	0.153547
$172$	193.303	0.0856932
$173$	−277.970	−0.122160	−0.0610799	−	0.998133i	$-0.519454\pi$
$173$	−277.970	−0.122160	−0.0610799	+	0.998133i	$0.519454\pi$
$174$	475.948	0.207365
$175$	335.843	0.145070
$176$	0	0
$177$	2091.53	0.888185
$178$	−3515.03	−1.48013
$179$	−3518.02	−1.46899	−0.734495	−	0.678614i	$-0.762581\pi$
$179$	−3518.02	−1.46899	−0.734495	+	0.678614i	$0.762581\pi$
$180$	−33.5183	−0.0138795
$181$	183.046	0.0751697	0.0375849	−	0.999293i	$-0.488034\pi$
$181$	183.046	0.0751697	0.0375849	+	0.999293i	$0.488034\pi$
$182$	−550.266	−0.224112
$183$	2487.56	1.00484
$184$	−1341.87	−0.537632
$185$	814.585	0.323727
$186$	−1957.70	−0.771750
$187$	0	0
$188$	15.3223	0.00594411
$189$	−362.710	−0.139594
$190$	564.077	0.215381
$191$	3628.53	1.37462	0.687308	−	0.726367i	$-0.258792\pi$
$191$	3628.53	1.37462	0.687308	+	0.726367i	$0.258792\pi$
$192$	−1367.60	−0.514051
$193$	−3896.63	−1.45329	−0.726647	−	0.687011i	$-0.758922\pi$
$193$	−3896.63	−1.45329	−0.726647	+	0.687011i	$0.758922\pi$
$194$	−878.009	−0.324935
$195$	207.774	0.0763027
$196$	−121.065	−0.0441198
$197$	4224.29	1.52776	0.763878	−	0.645361i	$-0.223293\pi$
$197$	4224.29	1.52776	0.763878	+	0.645361i	$0.223293\pi$
$198$	0	0
$199$	2242.23	0.798732	0.399366	−	0.916791i	$-0.369230\pi$
$199$	2242.23	0.798732	0.399366	+	0.916791i	$0.369230\pi$
$200$	536.368	0.189635
$201$	−1337.93	−0.469505
$202$	−729.582	−0.254125
$203$	720.705	0.249180
$204$	−57.3049	−0.0196674
$205$	−89.2796	−0.0304173
$206$	630.845	0.213364
$207$	−562.900	−0.189006
$208$	−961.358	−0.320472
$209$	0	0
$210$	−595.886	−0.195810
$211$	4156.66	1.35619	0.678096	−	0.734974i	$-0.262805\pi$
$211$	4156.66	1.35619	0.678096	+	0.734974i	$0.262805\pi$
$212$	91.6823	0.0297017
$213$	−2154.71	−0.693139
$214$	236.092	0.0754155
$215$	−1297.60	−0.411606
$216$	−579.277	−0.182476
$217$	−2964.45	−0.927374
$218$	−2407.73	−0.748036
$219$	805.383	0.248506
$220$	0	0
$221$	355.223	0.108122
$222$	−1445.32	−0.436953
$223$	−3686.16	−1.10692	−0.553461	−	0.832875i	$-0.686693\pi$
$223$	−3686.16	−1.10692	−0.553461	+	0.832875i	$0.686693\pi$
$224$	451.396	0.134644
$225$	225.000	0.0666667
$226$	−5673.64	−1.66993
$227$	−4852.44	−1.41880	−0.709400	−	0.704806i	$-0.751034\pi$
$227$	−4852.44	−1.41880	−0.709400	+	0.704806i	$0.751034\pi$
$228$	−85.2477	−0.0247617
$229$	−6525.07	−1.88292	−0.941460	−	0.337124i	$-0.890546\pi$
$229$	−6525.07	−1.88292	−0.941460	+	0.337124i	$0.890546\pi$
$230$	−924.773	−0.265121
$231$	0	0
$232$	1151.02	0.325726
$233$	−949.548	−0.266983	−0.133491	−	0.991050i	$-0.542619\pi$
$233$	−949.548	−0.266983	−0.133491	+	0.991050i	$0.542619\pi$
$234$	−368.654	−0.102990
$235$	−102.855	−0.0285511
$236$	−519.291	−0.143233
$237$	−1104.27	−0.302658
$238$	−1018.76	−0.277465
$239$	5268.44	1.42589	0.712943	−	0.701222i	$-0.247362\pi$
$239$	5268.44	1.42589	0.712943	+	0.701222i	$0.247362\pi$
$240$	−1041.06	−0.280001
$241$	−1128.47	−0.301622	−0.150811	−	0.988563i	$-0.548189\pi$
$241$	−1128.47	−0.301622	−0.150811	+	0.988563i	$0.548189\pi$
$242$	0	0
$243$	−243.000	−0.0641500
$244$	−617.620	−0.162045
$245$	812.677	0.211919
$246$	158.409	0.0410560
$247$	528.436	0.136128
$248$	−4734.47	−1.21225
$249$	1576.89	0.401331
$250$	369.646	0.0935139
$251$	2186.48	0.549837	0.274919	−	0.961467i	$-0.411349\pi$
$251$	2186.48	0.549837	0.274919	+	0.961467i	$0.411349\pi$
$252$	90.0550	0.0225116
$253$	0	0
$254$	−1683.09	−0.415774
$255$	384.673	0.0944674
$256$	1134.48	0.276973
$257$	338.497	0.0821591	0.0410796	−	0.999156i	$-0.486920\pi$
$257$	338.497	0.0821591	0.0410796	+	0.999156i	$0.486920\pi$
$258$	2302.33	0.555568
$259$	−2188.58	−0.525065
$260$	−51.5870	−0.0123050
$261$	482.841	0.114510
$262$	−8073.60	−1.90377
$263$	3560.56	0.834803	0.417402	−	0.908722i	$-0.362941\pi$
$263$	3560.56	0.834803	0.417402	+	0.908722i	$0.362941\pi$
$264$	0	0
$265$	−615.440	−0.142665
$266$	−1515.53	−0.349335
$267$	−3565.95	−0.817349
$268$	332.186	0.0757146
$269$	7224.32	1.63745	0.818726	−	0.574184i	$-0.194680\pi$
$269$	7224.32	1.63745	0.818726	+	0.574184i	$0.194680\pi$
$270$	−399.218	−0.0899838
$271$	844.801	0.189365	0.0946826	−	0.995508i	$-0.469816\pi$
$271$	844.801	0.189365	0.0946826	+	0.995508i	$0.469816\pi$
$272$	−1779.86	−0.396764
$273$	−558.236	−0.123758
$274$	3528.24	0.777915
$275$	0	0
$276$	139.759	0.0304801
$277$	−3237.43	−0.702231	−0.351116	−	0.936332i	$-0.614198\pi$
$277$	−3237.43	−0.702231	−0.351116	+	0.936332i	$0.614198\pi$
$278$	3745.75	0.808111
$279$	−1986.05	−0.426172
$280$	−1441.08	−0.307575
$281$	5492.06	1.16594	0.582970	−	0.812494i	$-0.301891\pi$
$281$	5492.06	1.16594	0.582970	+	0.812494i	$0.301891\pi$
$282$	182.495	0.0385370
$283$	3736.57	0.784862	0.392431	−	0.919781i	$-0.371634\pi$
$283$	3736.57	0.784862	0.392431	+	0.919781i	$0.371634\pi$
$284$	534.980	0.111779
$285$	572.247	0.118937
$286$	0	0
$287$	239.871	0.0493350
$288$	302.416	0.0618751
$289$	−4255.34	−0.866139
$290$	793.246	0.160624
$291$	−890.725	−0.179434
$292$	−199.963	−0.0400752
$293$	−9509.23	−1.89602	−0.948012	−	0.318234i	$-0.896910\pi$
$293$	−9509.23	−1.89602	−0.948012	+	0.318234i	$0.896910\pi$
$294$	−1441.93	−0.286039
$295$	3485.88	0.687985
$296$	−3495.34	−0.686359
$297$	0	0
$298$	2512.09	0.488326
$299$	−866.343	−0.167565
$300$	−55.8638	−0.0107510
$301$	3486.31	0.667599
$302$	8537.68	1.62678
$303$	−740.149	−0.140332
$304$	−2647.75	−0.499535
$305$	4145.93	0.778345
$306$	−682.527	−0.127508
$307$	−6485.38	−1.20567	−0.602834	−	0.797867i	$-0.705962\pi$
$307$	−6485.38	−1.20567	−0.602834	+	0.797867i	$0.705962\pi$
$308$	0	0
$309$	639.982	0.117823
$310$	−3262.83	−0.597795
$311$	−10727.9	−1.95602	−0.978008	−	0.208569i	$-0.933119\pi$
$311$	−10727.9	−1.95602	−0.978008	+	0.208569i	$0.933119\pi$
$312$	−891.547	−0.161775
$313$	−8221.69	−1.48472	−0.742360	−	0.670001i	$-0.766294\pi$
$313$	−8221.69	−1.48472	−0.742360	+	0.670001i	$0.766294\pi$
$314$	−2227.54	−0.400343
$315$	−604.517	−0.108129
$316$	274.172	0.0488081
$317$	3456.12	0.612351	0.306176	−	0.951975i	$-0.400951\pi$
$317$	3456.12	0.612351	0.306176	+	0.951975i	$0.400951\pi$
$318$	1091.98	0.192563
$319$	0	0
$320$	−2279.33	−0.398183
$321$	239.511	0.0416456
$322$	2484.63	0.430009
$323$	978.347	0.168535
$324$	60.3329	0.0103451
$325$	346.291	0.0591038
$326$	−3737.70	−0.635006
$327$	−2442.60	−0.413076
$328$	383.093	0.0644902
$329$	276.344	0.0463081
$330$	0	0
$331$	9266.82	1.53882	0.769411	−	0.638753i	$-0.220550\pi$
$331$	9266.82	1.53882	0.769411	+	0.638753i	$0.220550\pi$
$332$	−391.516	−0.0647207
$333$	−1466.25	−0.241292
$334$	10216.5	1.67372
$335$	−2229.89	−0.363677
$336$	2797.06	0.454143
$337$	−7306.41	−1.18102	−0.590512	−	0.807029i	$-0.701074\pi$
$337$	−7306.41	−1.18102	−0.590512	+	0.807029i	$0.701074\pi$
$338$	5929.52	0.954211
$339$	−5755.82	−0.922163
$340$	−95.5081	−0.0152343
$341$	0	0
$342$	−1015.34	−0.160536
$343$	−6791.22	−1.06907
$344$	5567.91	0.872679
$345$	−938.167	−0.146404
$346$	822.003	0.127720
$347$	684.673	0.105923	0.0529614	−	0.998597i	$-0.483134\pi$
$347$	684.673	0.105923	0.0529614	+	0.998597i	$0.483134\pi$
$348$	−119.882	−0.0184665
$349$	3392.79	0.520378	0.260189	−	0.965558i	$-0.416215\pi$
$349$	3392.79	0.520378	0.260189	+	0.965558i	$0.416215\pi$
$350$	−993.144	−0.151674
$351$	−373.994	−0.0568727
$352$	0	0
$353$	−10934.4	−1.64866	−0.824331	−	0.566108i	$-0.808448\pi$
$353$	−10934.4	−1.64866	−0.824331	+	0.566108i	$0.808448\pi$
$354$	−6185.00	−0.928612
$355$	−3591.19	−0.536903
$356$	885.366	0.131810
$357$	−1033.52	−0.153220
$358$	10403.4	1.53585
$359$	−2575.42	−0.378622	−0.189311	−	0.981917i	$-0.560625\pi$
$359$	−2575.42	−0.378622	−0.189311	+	0.981917i	$0.560625\pi$
$360$	−965.462	−0.141345
$361$	−5403.59	−0.787811
$362$	−541.299	−0.0785912
$363$	0	0
$364$	138.601	0.0199578
$365$	1342.30	0.192492
$366$	−7356.13	−1.05058
$367$	−1398.59	−0.198925	−0.0994626	−	0.995041i	$-0.531712\pi$
$367$	−1398.59	−0.198925	−0.0994626	+	0.995041i	$0.531712\pi$
$368$	4340.84	0.614896
$369$	160.703	0.0226717
$370$	−2408.87	−0.338462
$371$	1653.53	0.231393
$372$	493.105	0.0687266
$373$	−623.943	−0.0866127	−0.0433063	−	0.999062i	$-0.513789\pi$
$373$	−623.943	−0.0866127	−0.0433063	+	0.999062i	$0.513789\pi$
$374$	0	0
$375$	375.000	0.0516398
$376$	441.344	0.0605334
$377$	743.126	0.101520
$378$	1072.60	0.145948
$379$	−5504.16	−0.745988	−0.372994	−	0.927834i	$-0.621669\pi$
$379$	−5504.16	−0.745988	−0.372994	+	0.927834i	$0.621669\pi$
$380$	−142.079	−0.0191803
$381$	−1707.47	−0.229597
$382$	−10730.2	−1.43718
$383$	−7440.83	−0.992711	−0.496356	−	0.868119i	$-0.665329\pi$
$383$	−7440.83	−0.992711	−0.496356	+	0.868119i	$0.665329\pi$
$384$	4850.66	0.644620
$385$	0	0
$386$	11523.0	1.51944
$387$	2335.67	0.306793
$388$	221.153	0.0289364
$389$	6043.79	0.787744	0.393872	−	0.919165i	$-0.371135\pi$
$389$	6043.79	0.787744	0.393872	+	0.919165i	$0.371135\pi$
$390$	−614.424	−0.0797758
$391$	−1603.95	−0.207455
$392$	−3487.15	−0.449305
$393$	−8190.53	−1.05129
$394$	−12491.9	−1.59730
$395$	−1840.45	−0.234438
$396$	0	0
$397$	−5697.21	−0.720239	−0.360119	−	0.932906i	$-0.617264\pi$
$397$	−5697.21	−0.720239	−0.360119	+	0.932906i	$0.617264\pi$
$398$	−6630.67	−0.835089
$399$	−1537.48	−0.192908
$400$	−1735.10	−0.216888
$401$	15823.4	1.97053	0.985266	−	0.171027i	$-0.0547086\pi$
$401$	15823.4	1.97053	0.985266	+	0.171027i	$0.0547086\pi$
$402$	3956.49	0.490875
$403$	−3056.67	−0.377826
$404$	183.767	0.0226306
$405$	−405.000	−0.0496904
$406$	−2131.25	−0.260522
$407$	0	0
$408$	−1650.61	−0.200288
$409$	3371.10	0.407555	0.203778	−	0.979017i	$-0.434678\pi$
$409$	3371.10	0.407555	0.203778	+	0.979017i	$0.434678\pi$
$410$	264.015	0.0318019
$411$	3579.34	0.429576
$412$	−158.897	−0.0190007
$413$	−9365.65	−1.11587
$414$	1664.59	0.197609
$415$	2628.15	0.310870
$416$	465.439	0.0548558
$417$	3800.00	0.446251
$418$	0	0
$419$	−713.843	−0.0832304	−0.0416152	−	0.999134i	$-0.513250\pi$
$419$	−713.843	−0.0832304	−0.0416152	+	0.999134i	$0.513250\pi$
$420$	150.092	0.0174374
$421$	−1067.26	−0.123551	−0.0617754	−	0.998090i	$-0.519676\pi$
$421$	−1067.26	−0.123551	−0.0617754	+	0.998090i	$0.519676\pi$
$422$	−12292.0	−1.41792
$423$	185.139	0.0212807
$424$	2640.82	0.302475
$425$	641.122	0.0731741
$426$	6371.85	0.724689
$427$	−11139.0	−1.26243
$428$	−59.4668	−0.00671597
$429$	0	0
$430$	3837.21	0.430342
$431$	−4779.89	−0.534197	−0.267099	−	0.963669i	$-0.586065\pi$
$431$	−4779.89	−0.534197	−0.267099	+	0.963669i	$0.586065\pi$
$432$	1873.91	0.208700
$433$	9747.37	1.08182	0.540911	−	0.841080i	$-0.318080\pi$
$433$	9747.37	1.08182	0.540911	+	0.841080i	$0.318080\pi$
$434$	8766.39	0.969585
$435$	804.735	0.0886991
$436$	606.457	0.0666147
$437$	−2386.06	−0.261191
$438$	−2381.65	−0.259817
$439$	−16880.1	−1.83517	−0.917587	−	0.397536i	$-0.869866\pi$
$439$	−16880.1	−1.83517	−0.917587	+	0.397536i	$0.869866\pi$
$440$	0	0
$441$	−1462.82	−0.157955
$442$	−1050.46	−0.113043
$443$	−3952.86	−0.423941	−0.211971	−	0.977276i	$-0.567988\pi$
$443$	−3952.86	−0.423941	−0.211971	+	0.977276i	$0.567988\pi$
$444$	364.047	0.0389119
$445$	−5943.24	−0.633116
$446$	10900.6	1.15731
$447$	2548.47	0.269661
$448$	6123.97	0.645827
$449$	−2612.11	−0.274550	−0.137275	−	0.990533i	$-0.543834\pi$
$449$	−2612.11	−0.274550	−0.137275	+	0.990533i	$0.543834\pi$
$450$	−665.363	−0.0697012
$451$	0	0
$452$	1429.08	0.148713
$453$	8661.33	0.898333
$454$	14349.5	1.48338
$455$	−930.393	−0.0958626
$456$	−2455.48	−0.252167
$457$	−8039.48	−0.822913	−0.411456	−	0.911429i	$-0.634980\pi$
$457$	−8039.48	−0.822913	−0.411456	+	0.911429i	$0.634980\pi$
$458$	19295.7	1.96863
$459$	−692.412	−0.0704118
$460$	232.932	0.0236098
$461$	2749.59	0.277790	0.138895	−	0.990307i	$-0.455645\pi$
$461$	2749.59	0.277790	0.138895	+	0.990307i	$0.455645\pi$
$462$	0	0
$463$	−17487.3	−1.75530	−0.877650	−	0.479303i	$-0.840890\pi$
$463$	−17487.3	−1.75530	−0.877650	+	0.479303i	$0.840890\pi$
$464$	−3723.46	−0.372537
$465$	−3310.09	−0.330111
$466$	2807.98	0.279135
$467$	4642.11	0.459982	0.229991	−	0.973193i	$-0.426130\pi$
$467$	4642.11	0.459982	0.229991	+	0.973193i	$0.426130\pi$
$468$	92.8565	0.00917157
$469$	5991.13	0.589861
$470$	304.159	0.0298507
$471$	−2259.81	−0.221075
$472$	−14957.7	−1.45865
$473$	0	0
$474$	3265.51	0.316434
$475$	953.745	0.0921280
$476$	256.606	0.0247090
$477$	1107.79	0.106336
$478$	−15579.7	−1.49079
$479$	8868.05	0.845911	0.422956	−	0.906150i	$-0.360993\pi$
$479$	8868.05	0.845911	0.422956	+	0.906150i	$0.360993\pi$
$480$	504.026	0.0479282
$481$	−2256.67	−0.213919
$482$	3337.07	0.315351
$483$	2520.61	0.237457
$484$	0	0
$485$	−1484.54	−0.138989
$486$	718.592	0.0670700
$487$	−1960.53	−0.182423	−0.0912117	−	0.995832i	$-0.529074\pi$
$487$	−1960.53	−0.182423	−0.0912117	+	0.995832i	$0.529074\pi$
$488$	−17789.9	−1.65023
$489$	−3791.84	−0.350660
$490$	−2403.22	−0.221565
$491$	−13106.9	−1.20470	−0.602348	−	0.798234i	$-0.705768\pi$
$491$	−13106.9	−1.20470	−0.602348	+	0.798234i	$0.705768\pi$
$492$	−39.9000	−0.00365616
$493$	1375.82	0.125688
$494$	−1562.68	−0.142324
$495$	0	0
$496$	15315.6	1.38647
$497$	9648.60	0.870823
$498$	−4663.14	−0.419599
$499$	−8815.60	−0.790863	−0.395431	−	0.918496i	$-0.629405\pi$
$499$	−8815.60	−0.790863	−0.395431	+	0.918496i	$0.629405\pi$
$500$	−93.1063	−0.00832768
$501$	10364.5	0.924252
$502$	−6465.78	−0.574864
$503$	14750.9	1.30757	0.653787	−	0.756679i	$-0.273179\pi$
$503$	14750.9	1.30757	0.653787	+	0.756679i	$0.273179\pi$
$504$	2593.95	0.229253
$505$	−1233.58	−0.108700
$506$	0	0
$507$	6015.40	0.526929
$508$	423.937	0.0370259
$509$	6003.97	0.522832	0.261416	−	0.965226i	$-0.415811\pi$
$509$	6003.97	0.522832	0.261416	+	0.965226i	$0.415811\pi$
$510$	−1137.54	−0.0987673
$511$	−3606.43	−0.312209
$512$	9580.24	0.826935
$513$	−1030.04	−0.0886502
$514$	−1000.99	−0.0858988
$515$	1066.64	0.0912653
$516$	−579.909	−0.0494750
$517$	0	0
$518$	6472.00	0.548964
$519$	833.909	0.0705289
$520$	−1485.91	−0.125311
$521$	−6769.02	−0.569206	−0.284603	−	0.958645i	$-0.591862\pi$
$521$	−6769.02	−0.569206	−0.284603	+	0.958645i	$0.591862\pi$
$522$	−1427.84	−0.119722
$523$	−18106.8	−1.51387	−0.756937	−	0.653488i	$-0.773305\pi$
$523$	−18106.8	−1.51387	−0.756937	+	0.653488i	$0.773305\pi$
$524$	2033.57	0.169536
$525$	−1007.53	−0.0837564
$526$	−10529.2	−0.872801
$527$	−5659.13	−0.467771
$528$	0	0
$529$	−8255.19	−0.678490
$530$	1819.96	0.149159
$531$	−6274.58	−0.512794
$532$	381.731	0.0311093
$533$	247.333	0.0200998
$534$	10545.1	0.854553
$535$	399.186	0.0322585
$536$	9568.31	0.771060
$537$	10554.1	0.848122
$538$	−21363.6	−1.71199
$539$	0	0
$540$	100.555	0.00801332
$541$	14549.7	1.15627	0.578134	−	0.815941i	$-0.303781\pi$
$541$	14549.7	1.15627	0.578134	+	0.815941i	$0.303781\pi$
$542$	−2498.22	−0.197985
$543$	−549.139	−0.0433992
$544$	861.713	0.0679148
$545$	−4071.00	−0.319968
$546$	1650.80	0.129391
$547$	−7927.94	−0.619697	−0.309849	−	0.950786i	$-0.600278\pi$
$547$	−7927.94	−0.619697	−0.309849	+	0.950786i	$0.600278\pi$
$548$	−888.691	−0.0692756
$549$	−7462.67	−0.580144
$550$	0	0
$551$	2046.70	0.158244
$552$	4025.62	0.310402
$553$	4944.81	0.380244
$554$	9573.62	0.734195
$555$	−2443.76	−0.186904
$556$	−943.477	−0.0719647
$557$	−12856.6	−0.978013	−0.489006	−	0.872280i	$-0.662640\pi$
$557$	−12856.6	−0.978013	−0.489006	+	0.872280i	$0.662640\pi$
$558$	5873.10	0.445570
$559$	3594.76	0.271990
$560$	4661.77	0.351778
$561$	0	0
$562$	−16241.0	−1.21901
$563$	−9751.32	−0.729963	−0.364981	−	0.931015i	$-0.618925\pi$
$563$	−9751.32	−0.729963	−0.364981	+	0.931015i	$0.618925\pi$
$564$	−45.9669	−0.00343183
$565$	−9593.03	−0.714304
$566$	−11049.7	−0.820587
$567$	1088.13	0.0805947
$568$	15409.6	1.13833
$569$	−3962.75	−0.291963	−0.145982	−	0.989287i	$-0.546634\pi$
$569$	−3962.75	−0.291963	−0.145982	+	0.989287i	$0.546634\pi$
$570$	−1692.23	−0.124350
$571$	19004.4	1.39283	0.696416	−	0.717638i	$-0.254777\pi$
$571$	19004.4	1.39283	0.696416	+	0.717638i	$0.254777\pi$
$572$	0	0
$573$	−10885.6	−0.793634
$574$	−709.340	−0.0515806
$575$	−1563.61	−0.113404
$576$	4102.79	0.296788
$577$	10410.7	0.751134	0.375567	−	0.926795i	$-0.377448\pi$
$577$	10410.7	0.751134	0.375567	+	0.926795i	$0.377448\pi$
$578$	12583.8	0.905563
$579$	11689.9	0.839060
$580$	−199.803	−0.0143041
$581$	−7061.17	−0.504211
$582$	2634.03	0.187601
$583$	0	0
$584$	−5759.75	−0.408117
$585$	−623.323	−0.0440534
$586$	28120.4	1.98233
$587$	−1636.78	−0.115089	−0.0575445	−	0.998343i	$-0.518327\pi$
$587$	−1636.78	−0.115089	−0.0575445	+	0.998343i	$0.518327\pi$
$588$	363.194	0.0254726
$589$	−8418.61	−0.588935
$590$	−10308.3	−0.719300
$591$	−12672.9	−0.882050
$592$	11307.1	0.784998
$593$	11586.5	0.802361	0.401181	−	0.915999i	$-0.368600\pi$
$593$	11586.5	0.802361	0.401181	+	0.915999i	$0.368600\pi$
$594$	0	0
$595$	−1722.53	−0.118684
$596$	−632.743	−0.0434869
$597$	−6726.70	−0.461148
$598$	2561.92	0.175192
$599$	7828.53	0.533998	0.266999	−	0.963697i	$-0.413968\pi$
$599$	7828.53	0.533998	0.266999	+	0.963697i	$0.413968\pi$
$600$	−1609.10	−0.109486
$601$	−15473.7	−1.05023	−0.525113	−	0.851032i	$-0.675977\pi$
$601$	−15473.7	−1.05023	−0.525113	+	0.851032i	$0.675977\pi$
$602$	−10309.6	−0.697987
$603$	4013.80	0.271069
$604$	−2150.47	−0.144870
$605$	0	0
$606$	2188.75	0.146719
$607$	20781.3	1.38960	0.694799	−	0.719204i	$-0.255493\pi$
$607$	20781.3	1.38960	0.694799	+	0.719204i	$0.255493\pi$
$608$	1281.90	0.0855064
$609$	−2162.12	−0.143864
$610$	−12260.2	−0.813773
$611$	284.941	0.0188666
$612$	171.915	0.0113550
$613$	15758.2	1.03829	0.519143	−	0.854688i	$-0.326251\pi$
$613$	15758.2	1.03829	0.519143	+	0.854688i	$0.326251\pi$
$614$	19178.4	1.26055
$615$	267.839	0.0175615
$616$	0	0
$617$	−14137.0	−0.922421	−0.461211	−	0.887291i	$-0.652585\pi$
$617$	−14137.0	−0.922421	−0.461211	+	0.887291i	$0.652585\pi$
$618$	−1892.53	−0.123186
$619$	−22204.7	−1.44181	−0.720907	−	0.693032i	$-0.756275\pi$
$619$	−22204.7	−1.44181	−0.720907	+	0.693032i	$0.756275\pi$
$620$	821.841	0.0532354
$621$	1688.70	0.109123
$622$	31724.1	2.04505
$623$	15968.0	1.02687
$624$	2884.07	0.185025
$625$	625.000	0.0400000
$626$	24312.9	1.55230
$627$	0	0
$628$	561.073	0.0356517
$629$	−4177.99	−0.264845
$630$	1787.66	0.113051
$631$	14169.5	0.893947	0.446973	−	0.894547i	$-0.352502\pi$
$631$	14169.5	0.893947	0.446973	+	0.894547i	$0.352502\pi$
$632$	7897.26	0.497051
$633$	−12470.0	−0.782998
$634$	−10220.3	−0.640224
$635$	−2845.78	−0.177845
$636$	−275.047	−0.0171483
$637$	−2251.38	−0.140036
$638$	0	0
$639$	6464.14	0.400184
$640$	8084.43	0.499321
$641$	4233.79	0.260881	0.130440	−	0.991456i	$-0.458361\pi$
$641$	4233.79	0.260881	0.130440	+	0.991456i	$0.458361\pi$
$642$	−708.276	−0.0435412
$643$	−20028.5	−1.22838	−0.614189	−	0.789159i	$-0.710517\pi$
$643$	−20028.5	−1.22838	−0.614189	+	0.789159i	$0.710517\pi$
$644$	−625.827	−0.0382935
$645$	3892.79	0.237641
$646$	−2893.14	−0.176206
$647$	22677.1	1.37794	0.688971	−	0.724789i	$-0.258063\pi$
$647$	22677.1	1.37794	0.688971	+	0.724789i	$0.258063\pi$
$648$	1737.83	0.105353
$649$	0	0
$650$	−1024.04	−0.0617941
$651$	8893.36	0.535419
$652$	941.451	0.0565492
$653$	−24678.8	−1.47895	−0.739476	−	0.673183i	$-0.764926\pi$
$653$	−24678.8	−1.47895	−0.739476	+	0.673183i	$0.764926\pi$
$654$	7223.18	0.431879
$655$	−13650.9	−0.814327
$656$	−1239.27	−0.0737583
$657$	−2416.15	−0.143475
$658$	−817.197	−0.0484159
$659$	9281.04	0.548616	0.274308	−	0.961642i	$-0.411551\pi$
$659$	9281.04	0.548616	0.274308	+	0.961642i	$0.411551\pi$
$660$	0	0
$661$	13001.3	0.765040	0.382520	−	0.923947i	$-0.375056\pi$
$661$	13001.3	0.765040	0.382520	+	0.923947i	$0.375056\pi$
$662$	−27403.6	−1.60887
$663$	−1065.67	−0.0624241
$664$	−11277.3	−0.659100
$665$	−2562.47	−0.149426
$666$	4335.96	0.252275
$667$	−3355.45	−0.194788
$668$	−2573.33	−0.149049
$669$	11058.5	0.639082
$670$	6594.15	0.380230
$671$	0	0
$672$	−1354.19	−0.0777365
$673$	−5807.93	−0.332659	−0.166329	−	0.986070i	$-0.553191\pi$
$673$	−5807.93	−0.332659	−0.166329	+	0.986070i	$0.553191\pi$
$674$	21606.3	1.23478
$675$	−675.000	−0.0384900
$676$	−1493.52	−0.0849752
$677$	2910.87	0.165249	0.0826245	−	0.996581i	$-0.473670\pi$
$677$	2910.87	0.165249	0.0826245	+	0.996581i	$0.473670\pi$
$678$	17020.9	0.964137
$679$	3988.58	0.225431
$680$	−2751.02	−0.155142
$681$	14557.3	0.819144
$682$	0	0
$683$	−21234.7	−1.18964	−0.594819	−	0.803860i	$-0.702776\pi$
$683$	−21234.7	−1.18964	−0.594819	+	0.803860i	$0.702776\pi$
$684$	255.743	0.0142962
$685$	5965.57	0.332748
$686$	20082.8	1.11773
$687$	19575.2	1.08710
$688$	−18011.7	−0.998094
$689$	1704.97	0.0942731
$690$	2774.32	0.153067
$691$	16554.8	0.911398	0.455699	−	0.890134i	$-0.349389\pi$
$691$	16554.8	0.911398	0.455699	+	0.890134i	$0.349389\pi$
$692$	−207.046	−0.0113738
$693$	0	0
$694$	−2024.70	−0.110744
$695$	6333.33	0.345665
$696$	−3453.07	−0.188058
$697$	457.913	0.0248848
$698$	−10033.1	−0.544065
$699$	2848.64	0.154143
$700$	250.153	0.0135070
$701$	−11287.6	−0.608172	−0.304086	−	0.952645i	$-0.598351\pi$
$701$	−11287.6	−0.608172	−0.304086	+	0.952645i	$0.598351\pi$
$702$	1105.96	0.0594614
$703$	−6215.25	−0.333446
$704$	0	0
$705$	308.564	0.0164840
$706$	32334.8	1.72370
$707$	3314.32	0.176305
$708$	1557.87	0.0826956
$709$	−5742.74	−0.304193	−0.152097	−	0.988366i	$-0.548603\pi$
$709$	−5742.74	−0.304193	−0.152097	+	0.988366i	$0.548603\pi$
$710$	10619.8	0.561341
$711$	3312.81	0.174740
$712$	25502.1	1.34232
$713$	13801.9	0.724942
$714$	3056.29	0.160194
$715$	0	0
$716$	−2620.40	−0.136772
$717$	−15805.3	−0.823236
$718$	7615.95	0.395856
$719$	13803.5	0.715971	0.357986	−	0.933727i	$-0.383464\pi$
$719$	13803.5	0.715971	0.357986	+	0.933727i	$0.383464\pi$
$720$	3123.18	0.161658
$721$	−2865.78	−0.148026
$722$	15979.3	0.823670
$723$	3385.40	0.174142
$724$	136.342	0.00699878
$725$	1341.23	0.0687060
$726$	0	0
$727$	28815.2	1.47001	0.735005	−	0.678062i	$-0.237180\pi$
$727$	28815.2	1.47001	0.735005	+	0.678062i	$0.237180\pi$
$728$	3992.26	0.203246
$729$	729.000	0.0370370
$730$	−3969.42	−0.201253
$731$	6655.35	0.336740
$732$	1852.86	0.0935569
$733$	7588.75	0.382397	0.191198	−	0.981551i	$-0.438763\pi$
$733$	7588.75	0.382397	0.191198	+	0.981551i	$0.438763\pi$
$734$	4135.85	0.207980
$735$	−2438.03	−0.122351
$736$	−2101.60	−0.105253
$737$	0	0
$738$	−475.227	−0.0237037
$739$	−10493.0	−0.522316	−0.261158	−	0.965296i	$-0.584104\pi$
$739$	−10493.0	−0.522316	−0.261158	+	0.965296i	$0.584104\pi$
$740$	606.745	0.0301410
$741$	−1585.31	−0.0785935
$742$	−4889.77	−0.241926
$743$	6109.24	0.301650	0.150825	−	0.988560i	$-0.451807\pi$
$743$	6109.24	0.301650	0.150825	+	0.988560i	$0.451807\pi$
$744$	14203.4	0.699895
$745$	4247.45	0.208879
$746$	1845.10	0.0905551
$747$	−4730.68	−0.231709
$748$	0	0
$749$	−1072.51	−0.0523213
$750$	−1108.94	−0.0539903
$751$	10119.0	0.491673	0.245837	−	0.969311i	$-0.420937\pi$
$751$	10119.0	0.491673	0.245837	+	0.969311i	$0.420937\pi$
$752$	−1427.71	−0.0692329
$753$	−6559.43	−0.317449
$754$	−2197.55	−0.106141
$755$	14435.6	0.695846
$756$	−270.165	−0.0129971
$757$	19631.9	0.942579	0.471290	−	0.881978i	$-0.343789\pi$
$757$	19631.9	0.942579	0.471290	+	0.881978i	$0.343789\pi$
$758$	16276.7	0.779944
$759$	0	0
$760$	−4092.46	−0.195328
$761$	37652.9	1.79358	0.896791	−	0.442454i	$-0.145892\pi$
$761$	37652.9	1.79358	0.896791	+	0.442454i	$0.145892\pi$
$762$	5049.28	0.240047
$763$	10937.7	0.518967
$764$	2702.72	0.127985
$765$	−1154.02	−0.0545408
$766$	22003.8	1.03790
$767$	−9657.01	−0.454621
$768$	−3403.44	−0.159910
$769$	−8180.02	−0.383588	−0.191794	−	0.981435i	$-0.561431\pi$
$769$	−8180.02	−0.383588	−0.191794	+	0.981435i	$0.561431\pi$
$770$	0	0
$771$	−1015.49	−0.0474346
$772$	−2902.41	−0.135311
$773$	−1141.76	−0.0531258	−0.0265629	−	0.999647i	$-0.508456\pi$
$773$	−1141.76	−0.0531258	−0.0265629	+	0.999647i	$0.508456\pi$
$774$	−6906.98	−0.320758
$775$	−5516.82	−0.255703
$776$	6370.08	0.294681
$777$	6565.74	0.303146
$778$	−17872.5	−0.823600
$779$	681.199	0.0313305
$780$	154.761	0.00710427
$781$	0	0
$782$	4743.14	0.216898
$783$	−1448.52	−0.0661124
$784$	11280.6	0.513876
$785$	−3766.34	−0.171244
$786$	24220.8	1.09914
$787$	−39228.0	−1.77678	−0.888390	−	0.459089i	$-0.848176\pi$
$787$	−39228.0	−1.77678	−0.888390	+	0.459089i	$0.848176\pi$
$788$	3146.46	0.142244
$789$	−10681.7	−0.481974
$790$	5442.52	0.245109
$791$	25774.0	1.15856
$792$	0	0
$793$	−11485.6	−0.514331
$794$	16847.6	0.753022
$795$	1846.32	0.0823676
$796$	1670.13	0.0743671
$797$	24547.6	1.09099	0.545495	−	0.838114i	$-0.316342\pi$
$797$	24547.6	1.09099	0.545495	+	0.838114i	$0.316342\pi$
$798$	4546.59	0.201689
$799$	527.540	0.0233580
$800$	840.044	0.0371250
$801$	10697.8	0.471897
$802$	−46792.5	−2.06023
$803$	0	0
$804$	−996.559	−0.0437139
$805$	4201.02	0.183934
$806$	9039.10	0.395023
$807$	−21673.0	−0.945384
$808$	5293.23	0.230464
$809$	−699.945	−0.0304187	−0.0152094	−	0.999884i	$-0.504841\pi$
$809$	−699.945	−0.0304187	−0.0152094	+	0.999884i	$0.504841\pi$
$810$	1197.65	0.0519522
$811$	−33739.5	−1.46086	−0.730428	−	0.682990i	$-0.760679\pi$
$811$	−33739.5	−1.46086	−0.730428	+	0.682990i	$0.760679\pi$
$812$	536.818	0.0232003
$813$	−2534.40	−0.109330
$814$	0	0
$815$	−6319.73	−0.271620
$816$	5339.58	0.229072
$817$	9900.61	0.423964
$818$	−9968.91	−0.426106
$819$	1674.71	0.0714518
$820$	−66.4999	−0.00283205
$821$	32297.4	1.37294	0.686472	−	0.727156i	$-0.259158\pi$
$821$	32297.4	1.37294	0.686472	+	0.727156i	$0.259158\pi$
$822$	−10584.7	−0.449129
$823$	−10124.4	−0.428817	−0.214408	−	0.976744i	$-0.568782\pi$
$823$	−10124.4	−0.428817	−0.214408	+	0.976744i	$0.568782\pi$
$824$	−4576.87	−0.193499
$825$	0	0
$826$	27695.8	1.16666
$827$	−6914.26	−0.290728	−0.145364	−	0.989378i	$-0.546435\pi$
$827$	−6914.26	−0.290728	−0.145364	+	0.989378i	$0.546435\pi$
$828$	−419.277	−0.0175977
$829$	−2321.74	−0.0972707	−0.0486354	−	0.998817i	$-0.515487\pi$
$829$	−2321.74	−0.0972707	−0.0486354	+	0.998817i	$0.515487\pi$
$830$	−7771.90	−0.325020
$831$	9712.28	0.405433
$832$	6314.48	0.263119
$833$	−4168.21	−0.173373
$834$	−11237.2	−0.466563
$835$	17274.1	0.715923
$836$	0	0
$837$	5958.16	0.246050
$838$	2110.96	0.0870188
$839$	−30540.6	−1.25671	−0.628354	−	0.777927i	$-0.716271\pi$
$839$	−30540.6	−1.25671	−0.628354	+	0.777927i	$0.716271\pi$
$840$	4323.24	0.177579
$841$	−21510.8	−0.881987
$842$	3156.06	0.129175
$843$	−16476.2	−0.673155
$844$	3096.09	0.126270
$845$	10025.7	0.408158
$846$	−547.486	−0.0222494
$847$	0	0
$848$	−8542.81	−0.345945
$849$	−11209.7	−0.453140
$850$	−1895.91	−0.0765048
$851$	10189.6	0.410451
$852$	−1604.94	−0.0645356
$853$	−16368.5	−0.657029	−0.328515	−	0.944499i	$-0.606548\pi$
$853$	−16368.5	−0.657029	−0.328515	+	0.944499i	$0.606548\pi$
$854$	32940.0	1.31989
$855$	−1716.74	−0.0686682
$856$	−1712.88	−0.0683938
$857$	−29094.5	−1.15969	−0.579843	−	0.814728i	$-0.696886\pi$
$857$	−29094.5	−1.15969	−0.579843	+	0.814728i	$0.696886\pi$
$858$	0	0
$859$	−34464.5	−1.36893	−0.684467	−	0.729044i	$-0.739965\pi$
$859$	−34464.5	−1.36893	−0.684467	+	0.729044i	$0.739965\pi$
$860$	−966.516	−0.0383232
$861$	−719.613	−0.0284836
$862$	14134.9	0.558513
$863$	−34844.9	−1.37443	−0.687215	−	0.726454i	$-0.741167\pi$
$863$	−34844.9	−1.37443	−0.687215	+	0.726454i	$0.741167\pi$
$864$	−907.247	−0.0357236
$865$	1389.85	0.0546315
$866$	−28824.6	−1.13106
$867$	12766.0	0.500065
$868$	−2208.07	−0.0863444
$869$	0	0
$870$	−2379.74	−0.0927364
$871$	6177.51	0.240318
$872$	17468.4	0.678389
$873$	2672.18	0.103596
$874$	7055.98	0.273080
$875$	−1679.21	−0.0648775
$876$	599.890	0.0231375
$877$	−45487.0	−1.75141	−0.875704	−	0.482848i	$-0.839602\pi$
$877$	−45487.0	−1.75141	−0.875704	+	0.482848i	$0.839602\pi$
$878$	49917.2	1.91871
$879$	28527.7	1.09467
$880$	0	0
$881$	16896.5	0.646149	0.323075	−	0.946373i	$-0.395284\pi$
$881$	16896.5	0.646149	0.323075	+	0.946373i	$0.395284\pi$
$882$	4325.80	0.165144
$883$	47599.3	1.81409	0.907047	−	0.421029i	$-0.138331\pi$
$883$	47599.3	1.81409	0.907047	+	0.421029i	$0.138331\pi$
$884$	264.588	0.0100668
$885$	−10457.6	−0.397208
$886$	11689.3	0.443238
$887$	45471.8	1.72130	0.860650	−	0.509197i	$-0.170058\pi$
$887$	45471.8	1.72130	0.860650	+	0.509197i	$0.170058\pi$
$888$	10486.0	0.396270
$889$	7645.88	0.288453
$890$	17575.2	0.661934
$891$	0	0
$892$	−2745.64	−0.103061
$893$	784.778	0.0294083
$894$	−7536.26	−0.281935
$895$	17590.1	0.656952
$896$	−21720.8	−0.809867
$897$	2599.03	0.0967436
$898$	7724.45	0.287047
$899$	−11838.9	−0.439209
$900$	167.591	0.00620709
$901$	3156.58	0.116716
$902$	0	0
$903$	−10458.9	−0.385439
$904$	41163.1	1.51445
$905$	−915.231	−0.0336169
$906$	−25613.0	−0.939223
$907$	3514.39	0.128659	0.0643293	−	0.997929i	$-0.479509\pi$
$907$	3514.39	0.128659	0.0643293	+	0.997929i	$0.479509\pi$
$908$	−3614.34	−0.132099
$909$	2220.45	0.0810205
$910$	2751.33	0.100226
$911$	−15794.1	−0.574405	−0.287203	−	0.957870i	$-0.592725\pi$
$911$	−15794.1	−0.574405	−0.287203	+	0.957870i	$0.592725\pi$
$912$	7943.24	0.288407
$913$	0	0
$914$	23774.1	0.860369
$915$	−12437.8	−0.449378
$916$	−4860.20	−0.175312
$917$	36676.4	1.32079
$918$	2047.58	0.0736168
$919$	−21882.5	−0.785459	−0.392730	−	0.919654i	$-0.628469\pi$
$919$	−21882.5	−0.785459	−0.392730	+	0.919654i	$0.628469\pi$
$920$	6709.37	0.240436
$921$	19456.1	0.696093
$922$	−8131.02	−0.290435
$923$	9948.76	0.354786
$924$	0	0
$925$	−4072.93	−0.144775
$926$	51712.9	1.83520
$927$	−1919.95	−0.0680251
$928$	1802.70	0.0637679
$929$	−100.426	−0.00354668	−0.00177334	−	0.999998i	$-0.500564\pi$
$929$	−100.426	−0.00354668	−0.00177334	+	0.999998i	$0.500564\pi$
$930$	9788.50	0.345137
$931$	−6200.69	−0.218281
$932$	−707.272	−0.0248578
$933$	32183.6	1.12931
$934$	−13727.5	−0.480919
$935$	0	0
$936$	2674.64	0.0934011
$937$	−48135.4	−1.67825	−0.839123	−	0.543942i	$-0.816931\pi$
$937$	−48135.4	−1.67825	−0.839123	+	0.543942i	$0.816931\pi$
$938$	−17716.8	−0.616709
$939$	24665.1	0.857204
$940$	−76.6115	−0.00265829
$941$	−10466.4	−0.362588	−0.181294	−	0.983429i	$-0.558029\pi$
$941$	−10466.4	−0.362588	−0.181294	+	0.983429i	$0.558029\pi$
$942$	6682.63	0.231138
$943$	−1116.79	−0.0385659
$944$	48386.8	1.66828
$945$	1813.55	0.0624284
$946$	0	0
$947$	−9541.38	−0.327406	−0.163703	−	0.986510i	$-0.552344\pi$
$947$	−9541.38	−0.327406	−0.163703	+	0.986510i	$0.552344\pi$
$948$	−822.516	−0.0281794
$949$	−3718.62	−0.127199
$950$	−2820.38	−0.0963214
$951$	−10368.4	−0.353541
$952$	7391.28	0.251631
$953$	−17051.3	−0.579586	−0.289793	−	0.957089i	$-0.593586\pi$
$953$	−17051.3	−0.579586	−0.289793	+	0.957089i	$0.593586\pi$
$954$	−3275.93	−0.111176
$955$	−18142.7	−0.614747
$956$	3924.20	0.132759
$957$	0	0
$958$	−26224.3	−0.884415
$959$	−16027.9	−0.539697
$960$	6837.99	0.229891
$961$	18905.4	0.634601
$962$	6673.34	0.223656
$963$	−718.534	−0.0240441
$964$	−840.540	−0.0280830
$965$	19483.2	0.649933
$966$	−7453.88	−0.248266
$967$	−2160.46	−0.0718468	−0.0359234	−	0.999355i	$-0.511437\pi$
$967$	−2160.46	−0.0718468	−0.0359234	+	0.999355i	$0.511437\pi$
$968$	0	0
$969$	−2935.04	−0.0973035
$970$	4390.04	0.145315
$971$	−45614.5	−1.50756	−0.753779	−	0.657128i	$-0.771771\pi$
$971$	−45614.5	−1.50756	−0.753779	+	0.657128i	$0.771771\pi$
$972$	−180.999	−0.00597277
$973$	−17016.0	−0.560646
$974$	5797.62	0.190727
$975$	−1038.87	−0.0341236
$976$	57548.8	1.88739
$977$	19116.0	0.625972	0.312986	−	0.949758i	$-0.398671\pi$
$977$	19116.0	0.625972	0.312986	+	0.949758i	$0.398671\pi$
$978$	11213.1	0.366621
$979$	0	0
$980$	605.323	0.0197310
$981$	7327.80	0.238490
$982$	38759.3	1.25953
$983$	−24497.3	−0.794854	−0.397427	−	0.917634i	$-0.630097\pi$
$983$	−24497.3	−0.794854	−0.397427	+	0.917634i	$0.630097\pi$
$984$	−1149.28	−0.0372334
$985$	−21121.4	−0.683233
$986$	−4068.54	−0.131409
$987$	−829.033	−0.0267360
$988$	393.606	0.0126744
$989$	−16231.5	−0.521872
$990$	0	0
$991$	19625.0	0.629072	0.314536	−	0.949246i	$-0.398151\pi$
$991$	19625.0	0.629072	0.314536	+	0.949246i	$0.398151\pi$
$992$	−7414.99	−0.237325
$993$	−27800.5	−0.888440
$994$	−28532.5	−0.910460
$995$	−11211.2	−0.357204
$996$	1174.55	0.0373665
$997$	−2468.28	−0.0784064	−0.0392032	−	0.999231i	$-0.512482\pi$
$997$	−2468.28	−0.0784064	−0.0392032	+	0.999231i	$0.512482\pi$
$998$	26069.2	0.826861
$999$	4398.76	0.139310

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1815.4.a.bm.1.4		12
11.7	odd	10		165.4.m.b.16.5	✓	24
11.8	odd	10		165.4.m.b.31.5	yes	24
11.10	odd	2		1815.4.a.bj.1.9		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
165.4.m.b.16.5	✓	24	11.7	odd	10
165.4.m.b.31.5	yes	24	11.8	odd	10
1815.4.a.bj.1.9		12	11.10	odd	2
1815.4.a.bm.1.4		12	1.1	even	1	trivial

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)

\(f(q)\)	\(=\)	\(q-2.95717 q^{2} -3.00000 q^{3} +0.744851 q^{4} -5.00000 q^{5} +8.87151 q^{6} +13.4337 q^{7} +21.4547 q^{8} +9.00000 q^{9} +O(q^{10})\)	\(q-2.95717 q^{2} -3.00000 q^{3} +0.744851 q^{4} -5.00000 q^{5} +8.87151 q^{6} +13.4337 q^{7} +21.4547 q^{8} +9.00000 q^{9} +14.7858 q^{10} -2.23455 q^{12} +13.8516 q^{13} -39.7257 q^{14} +15.0000 q^{15} -69.4040 q^{16} +25.6449 q^{17} -26.6145 q^{18} +38.1498 q^{19} -3.72425 q^{20} -40.3011 q^{21} -62.5445 q^{23} -64.3641 q^{24} +25.0000 q^{25} -40.9616 q^{26} -27.0000 q^{27} +10.0061 q^{28} +53.6490 q^{29} -44.3575 q^{30} -220.673 q^{31} +33.6018 q^{32} -75.8363 q^{34} -67.1685 q^{35} +6.70366 q^{36} -162.917 q^{37} -112.815 q^{38} -41.5549 q^{39} -107.274 q^{40} +17.8559 q^{41} +119.177 q^{42} +259.519 q^{43} -45.0000 q^{45} +184.955 q^{46} +20.5710 q^{47} +208.212 q^{48} -162.535 q^{49} -73.9292 q^{50} -76.9347 q^{51} +10.3174 q^{52} +123.088 q^{53} +79.8436 q^{54} +288.216 q^{56} -114.449 q^{57} -158.649 q^{58} -697.175 q^{59} +11.1728 q^{60} -829.186 q^{61} +652.566 q^{62} +120.903 q^{63} +455.866 q^{64} -69.2581 q^{65} +445.977 q^{67} +19.1016 q^{68} +187.633 q^{69} +198.629 q^{70} +718.238 q^{71} +193.092 q^{72} -268.461 q^{73} +481.773 q^{74} -75.0000 q^{75} +28.4159 q^{76} +122.885 q^{78} +368.090 q^{79} +347.020 q^{80} +81.0000 q^{81} -52.8030 q^{82} -525.631 q^{83} -30.0183 q^{84} -128.224 q^{85} -767.443 q^{86} -160.947 q^{87} +1188.65 q^{89} +133.073 q^{90} +186.079 q^{91} -46.5863 q^{92} +662.018 q^{93} -60.8318 q^{94} -190.749 q^{95} -100.805 q^{96} +296.908 q^{97} +480.645 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + q^{2} - 36 q^{3} + 57 q^{4} - 60 q^{5} - 3 q^{6} - 51 q^{7} - 45 q^{8} + 108 q^{9}+O(q^{10}) \) 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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