LMFDB - Embedded newform 2166.4.a.t.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2166,4,Mod(1,2166)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2166, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("2166.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 2166 = 2 \cdot 3 \cdot 19^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	2166.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:	$127.798137072$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.14457.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 32x - 3 $ x^3 - x^2 - 32*x - 3
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{3}\cdot 3 $
Twist minimal:	no (minimal twist has level 114)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-0.0940524$ of defining polynomial
Character	$\chi$	$=$	2166.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.00000 q^{2} +3.00000 q^{3} +4.00000 q^{4} +4.18810 q^{5} -6.00000 q^{6} +3.18810 q^{7} -8.00000 q^{8} +9.00000 q^{9} +O(q^{10})$	\(q-2.00000 q^{2} +3.00000 q^{3} +4.00000 q^{4} +4.18810 q^{5} -6.00000 q^{6} +3.18810 q^{7} -8.00000 q^{8} +9.00000 q^{9} -8.37621 q^{10} +69.4003 q^{11} +12.0000 q^{12} +8.12863 q^{13} -6.37621 q^{14} +12.5643 q^{15} +16.0000 q^{16} -106.084 q^{17} -18.0000 q^{18} +16.7524 q^{20} +9.56431 q^{21} -138.801 q^{22} -176.494 q^{23} -24.0000 q^{24} -107.460 q^{25} -16.2573 q^{26} +27.0000 q^{27} +12.7524 q^{28} +66.2219 q^{29} -25.1286 q^{30} +140.915 q^{31} -32.0000 q^{32} +208.201 q^{33} +212.167 q^{34} +13.3521 q^{35} +36.0000 q^{36} +156.003 q^{37} +24.3859 q^{39} -33.5048 q^{40} +414.563 q^{41} -19.1286 q^{42} +115.850 q^{43} +277.601 q^{44} +37.6929 q^{45} +352.987 q^{46} +620.283 q^{47} +48.0000 q^{48} -332.836 q^{49} +214.920 q^{50} -318.251 q^{51} +32.5145 q^{52} +371.986 q^{53} -54.0000 q^{54} +290.656 q^{55} -25.5048 q^{56} -132.444 q^{58} -91.6929 q^{59} +50.2573 q^{60} +218.621 q^{61} -281.830 q^{62} +28.6929 q^{63} +64.0000 q^{64} +34.0436 q^{65} -416.402 q^{66} +145.342 q^{67} -424.334 q^{68} -529.481 q^{69} -26.7042 q^{70} -887.829 q^{71} -72.0000 q^{72} +199.016 q^{73} -312.006 q^{74} -322.379 q^{75} +221.255 q^{77} -48.7718 q^{78} +389.558 q^{79} +67.0097 q^{80} +81.0000 q^{81} -829.125 q^{82} +380.039 q^{83} +38.2573 q^{84} -444.289 q^{85} -231.701 q^{86} +198.666 q^{87} -555.202 q^{88} +425.799 q^{89} -75.3859 q^{90} +25.9149 q^{91} -705.974 q^{92} +422.744 q^{93} -1240.57 q^{94} -96.0000 q^{96} -419.846 q^{97} +665.672 q^{98} +624.603 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 6 q^{2} + 9 q^{3} + 12 q^{4} + 10 q^{5} - 18 q^{6} + 7 q^{7} - 24 q^{8} + 27 q^{9}+O(q^{10}) $ 3 * q - 6 * q^2 + 9 * q^3 + 12 * q^4 + 10 * q^5 - 18 * q^6 + 7 * q^7 - 24 * q^8 + 27 * q^9	\( 3 q - 6 q^{2} + 9 q^{3} + 12 q^{4} + 10 q^{5} - 18 q^{6} + 7 q^{7} - 24 q^{8} + 27 q^{9} - 20 q^{10} - 44 q^{11} + 36 q^{12} + 9 q^{13} - 14 q^{14} + 30 q^{15} + 48 q^{16} - 84 q^{17} - 54 q^{18} + 40 q^{20} + 21 q^{21} + 88 q^{22} - 2 q^{23} - 72 q^{24} - 83 q^{25} - 18 q^{26} + 81 q^{27} + 28 q^{28} - 92 q^{29} - 60 q^{30} + 109 q^{31} - 96 q^{32} - 132 q^{33} + 168 q^{34} + 282 q^{35} + 108 q^{36} - 245 q^{37} + 27 q^{39} - 80 q^{40} + 688 q^{41} - 42 q^{42} - 103 q^{43} - 176 q^{44} + 90 q^{45} + 4 q^{46} + 322 q^{47} + 144 q^{48} - 754 q^{49} + 166 q^{50} - 252 q^{51} + 36 q^{52} + 1322 q^{53} - 162 q^{54} - 248 q^{55} - 56 q^{56} + 184 q^{58} - 252 q^{59} + 120 q^{60} - 435 q^{61} - 218 q^{62} + 63 q^{63} + 192 q^{64} + 1582 q^{65} + 264 q^{66} + 719 q^{67} - 336 q^{68} - 6 q^{69} - 564 q^{70} + 62 q^{71} - 216 q^{72} - 581 q^{73} + 490 q^{74} - 249 q^{75} - 204 q^{77} - 54 q^{78} + 489 q^{79} + 160 q^{80} + 243 q^{81} - 1376 q^{82} + 2496 q^{83} + 84 q^{84} + 1632 q^{85} + 206 q^{86} - 276 q^{87} + 352 q^{88} - 1584 q^{89} - 180 q^{90} + 1573 q^{91} - 8 q^{92} + 327 q^{93} - 644 q^{94} - 288 q^{96} - 974 q^{97} + 1508 q^{98} - 396 q^{99}+O(q^{100}) \) 3 * q - 6 * q^2 + 9 * q^3 + 12 * q^4 + 10 * q^5 - 18 * q^6 + 7 * q^7 - 24 * q^8 + 27 * q^9 - 20 * q^10 - 44 * q^11 + 36 * q^12 + 9 * q^13 - 14 * q^14 + 30 * q^15 + 48 * q^16 - 84 * q^17 - 54 * q^18 + 40 * q^20 + 21 * q^21 + 88 * q^22 - 2 * q^23 - 72 * q^24 - 83 * q^25 - 18 * q^26 + 81 * q^27 + 28 * q^28 - 92 * q^29 - 60 * q^30 + 109 * q^31 - 96 * q^32 - 132 * q^33 + 168 * q^34 + 282 * q^35 + 108 * q^36 - 245 * q^37 + 27 * q^39 - 80 * q^40 + 688 * q^41 - 42 * q^42 - 103 * q^43 - 176 * q^44 + 90 * q^45 + 4 * q^46 + 322 * q^47 + 144 * q^48 - 754 * q^49 + 166 * q^50 - 252 * q^51 + 36 * q^52 + 1322 * q^53 - 162 * q^54 - 248 * q^55 - 56 * q^56 + 184 * q^58 - 252 * q^59 + 120 * q^60 - 435 * q^61 - 218 * q^62 + 63 * q^63 + 192 * q^64 + 1582 * q^65 + 264 * q^66 + 719 * q^67 - 336 * q^68 - 6 * q^69 - 564 * q^70 + 62 * q^71 - 216 * q^72 - 581 * q^73 + 490 * q^74 - 249 * q^75 - 204 * q^77 - 54 * q^78 + 489 * q^79 + 160 * q^80 + 243 * q^81 - 1376 * q^82 + 2496 * q^83 + 84 * q^84 + 1632 * q^85 + 206 * q^86 - 276 * q^87 + 352 * q^88 - 1584 * q^89 - 180 * q^90 + 1573 * q^91 - 8 * q^92 + 327 * q^93 - 644 * q^94 - 288 * q^96 - 974 * q^97 + 1508 * q^98 - 396 * q^99

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 $

$2$	−2.00000	−0.707107
$2$	−2.00000	−0.707107
$3$	3.00000	0.577350
$3$	3.00000	0.577350
$4$	4.00000	0.500000
$5$	4.18810	0.374595	0.187298	−	0.982303i	$-0.440027\pi$
$5$	4.18810	0.374595	0.187298	+	0.982303i	$0.440027\pi$
$6$	−6.00000	−0.408248
$7$	3.18810	0.172141	0.0860707	−	0.996289i	$-0.472569\pi$
$7$	3.18810	0.172141	0.0860707	+	0.996289i	$0.472569\pi$
$8$	−8.00000	−0.353553
$9$	9.00000	0.333333
$10$	−8.37621	−0.264879
$11$	69.4003	1.90227	0.951135	−	0.308774i	$-0.0999187\pi$
$11$	69.4003	1.90227	0.951135	+	0.308774i	$0.0999187\pi$
$12$	12.0000	0.288675
$13$	8.12863	0.173421	0.0867106	−	0.996234i	$-0.472364\pi$
$13$	8.12863	0.173421	0.0867106	+	0.996234i	$0.472364\pi$
$14$	−6.37621	−0.121722
$15$	12.5643	0.216273
$16$	16.0000	0.250000
$17$	−106.084	−1.51347	−0.756737	−	0.653720i	$-0.773207\pi$
$17$	−106.084	−1.51347	−0.756737	+	0.653720i	$0.773207\pi$
$18$	−18.0000	−0.235702
$19$	0	0
$19$	0	0
$20$	16.7524	0.187298
$21$	9.56431	0.0993859
$22$	−138.801	−1.34511
$23$	−176.494	−1.60006	−0.800031	−	0.599958i	$-0.795184\pi$
$23$	−176.494	−1.60006	−0.800031	+	0.599958i	$0.795184\pi$
$24$	−24.0000	−0.204124
$25$	−107.460	−0.859678
$26$	−16.2573	−0.122627
$27$	27.0000	0.192450
$28$	12.7524	0.0860707
$29$	66.2219	0.424038	0.212019	−	0.977266i	$-0.431996\pi$
$29$	66.2219	0.424038	0.212019	+	0.977266i	$0.431996\pi$
$30$	−25.1286	−0.152928
$31$	140.915	0.816421	0.408210	−	0.912888i	$-0.366153\pi$
$31$	140.915	0.816421	0.408210	+	0.912888i	$0.366153\pi$
$32$	−32.0000	−0.176777
$33$	208.201	1.09828
$34$	212.167	1.07019
$35$	13.3521	0.0644834
$36$	36.0000	0.166667
$37$	156.003	0.693156	0.346578	−	0.938021i	$-0.387344\pi$
$37$	156.003	0.693156	0.346578	+	0.938021i	$0.387344\pi$
$38$	0	0
$39$	24.3859	0.100125
$40$	−33.5048	−0.132440
$41$	414.563	1.57912	0.789559	−	0.613675i	$-0.210309\pi$
$41$	414.563	1.57912	0.789559	+	0.613675i	$0.210309\pi$
$42$	−19.1286	−0.0702765
$43$	115.850	0.410861	0.205430	−	0.978672i	$-0.434141\pi$
$43$	115.850	0.410861	0.205430	+	0.978672i	$0.434141\pi$
$44$	277.601	0.951135
$45$	37.6929	0.124865
$46$	352.987	1.13142
$47$	620.283	1.92505	0.962527	−	0.271185i	$-0.0874156\pi$
$47$	620.283	1.92505	0.962527	+	0.271185i	$0.0874156\pi$
$48$	48.0000	0.144338
$49$	−332.836	−0.970367
$50$	214.920	0.607884
$51$	−318.251	−0.873804
$52$	32.5145	0.0867106
$53$	371.986	0.964078	0.482039	−	0.876150i	$-0.339896\pi$
$53$	371.986	0.964078	0.482039	+	0.876150i	$0.339896\pi$
$54$	−54.0000	−0.136083
$55$	290.656	0.712582
$56$	−25.5048	−0.0608612
$57$	0	0
$58$	−132.444	−0.299840
$59$	−91.6929	−0.202329	−0.101164	−	0.994870i	$-0.532257\pi$
$59$	−91.6929	−0.202329	−0.101164	+	0.994870i	$0.532257\pi$
$60$	50.2573	0.108136
$61$	218.621	0.458877	0.229438	−	0.973323i	$-0.426311\pi$
$61$	218.621	0.458877	0.229438	+	0.973323i	$0.426311\pi$
$62$	−281.830	−0.577297
$63$	28.6929	0.0573805
$64$	64.0000	0.125000
$65$	34.0436	0.0649628
$66$	−416.402	−0.776599
$67$	145.342	0.265021	0.132510	−	0.991182i	$-0.457696\pi$
$67$	145.342	0.265021	0.132510	+	0.991182i	$0.457696\pi$
$68$	−424.334	−0.756737
$69$	−529.481	−0.923797
$70$	−26.7042	−0.0455967
$71$	−887.829	−1.48403	−0.742014	−	0.670385i	$-0.766129\pi$
$71$	−887.829	−1.48403	−0.742014	+	0.670385i	$0.766129\pi$
$72$	−72.0000	−0.117851
$73$	199.016	0.319083	0.159542	−	0.987191i	$-0.448998\pi$
$73$	199.016	0.319083	0.159542	+	0.987191i	$0.448998\pi$
$74$	−312.006	−0.490135
$75$	−322.379	−0.496335
$76$	0	0
$77$	221.255	0.327460
$78$	−48.7718	−0.0707989
$79$	389.558	0.554793	0.277397	−	0.960755i	$-0.410528\pi$
$79$	389.558	0.554793	0.277397	+	0.960755i	$0.410528\pi$
$80$	67.0097	0.0936489
$81$	81.0000	0.111111
$82$	−829.125	−1.11660
$83$	380.039	0.502586	0.251293	−	0.967911i	$-0.419144\pi$
$83$	380.039	0.502586	0.251293	+	0.967911i	$0.419144\pi$
$84$	38.2573	0.0496930
$85$	−444.289	−0.566940
$86$	−231.701	−0.290523
$87$	198.666	0.244818
$88$	−555.202	−0.672554
$89$	425.799	0.507130	0.253565	−	0.967318i	$-0.418397\pi$
$89$	425.799	0.507130	0.253565	+	0.967318i	$0.418397\pi$
$90$	−75.3859	−0.0882930
$91$	25.9149	0.0298530
$92$	−705.974	−0.800031
$93$	422.744	0.471361
$94$	−1240.57	−1.36122
$95$	0	0
$96$	−96.0000	−0.102062
$97$	−419.846	−0.439473	−0.219736	−	0.975559i	$-0.570520\pi$
$97$	−419.846	−0.439473	−0.219736	+	0.975559i	$0.570520\pi$
$98$	665.672	0.686153
$99$	624.603	0.634090
$100$	−429.839	−0.429839
$101$	−1241.71	−1.22331	−0.611657	−	0.791123i	$-0.709497\pi$
$101$	−1241.71	−1.22331	−0.611657	+	0.791123i	$0.709497\pi$
$102$	636.501	0.617873
$103$	593.606	0.567862	0.283931	−	0.958845i	$-0.408361\pi$
$103$	593.606	0.567862	0.283931	+	0.958845i	$0.408361\pi$
$104$	−65.0290	−0.0613137
$105$	40.0564	0.0372295
$106$	−743.971	−0.681706
$107$	1778.56	1.60691	0.803457	−	0.595363i	$-0.202992\pi$
$107$	1778.56	1.60691	0.803457	+	0.595363i	$0.202992\pi$
$108$	108.000	0.0962250
$109$	1069.99	0.940243	0.470121	−	0.882602i	$-0.344210\pi$
$109$	1069.99	0.940243	0.470121	+	0.882602i	$0.344210\pi$
$110$	−581.311	−0.503872
$111$	468.009	0.400194
$112$	51.0097	0.0430354
$113$	−583.197	−0.485510	−0.242755	−	0.970088i	$-0.578051\pi$
$113$	−583.197	−0.485510	−0.242755	+	0.970088i	$0.578051\pi$
$114$	0	0
$115$	−739.173	−0.599376
$116$	264.887	0.212019
$117$	73.1577	0.0578071
$118$	183.386	0.143068
$119$	−338.206	−0.260532
$120$	−100.515	−0.0764640
$121$	3485.40	2.61863
$122$	−437.241	−0.324475
$123$	1243.69	0.911704
$124$	563.659	0.408210
$125$	−973.566	−0.696627
$126$	−57.3859	−0.0405741
$127$	−1659.35	−1.15940	−0.579700	−	0.814830i	$-0.696830\pi$
$127$	−1659.35	−1.15940	−0.579700	+	0.814830i	$0.696830\pi$
$128$	−128.000	−0.0883883
$129$	347.551	0.237211
$130$	−68.0871	−0.0459356
$131$	597.546	0.398533	0.199267	−	0.979945i	$-0.436144\pi$
$131$	597.546	0.398533	0.199267	+	0.979945i	$0.436144\pi$
$132$	832.804	0.549138
$133$	0	0
$134$	−290.685	−0.187398
$135$	113.079	0.0720909
$136$	848.669	0.535094
$137$	−1777.31	−1.10836	−0.554182	−	0.832396i	$-0.686969\pi$
$137$	−1777.31	−1.10836	−0.554182	+	0.832396i	$0.686969\pi$
$138$	1058.96	0.653223
$139$	163.567	0.0998099	0.0499050	−	0.998754i	$-0.484108\pi$
$139$	163.567	0.0998099	0.0499050	+	0.998754i	$0.484108\pi$
$140$	53.4085	0.0322417
$141$	1860.85	1.11143
$142$	1775.66	1.04937
$143$	564.129	0.329894
$144$	144.000	0.0833333
$145$	277.344	0.158843
$146$	−398.032	−0.225626
$147$	−998.508	−0.560242
$148$	624.012	0.346578
$149$	−1119.80	−0.615688	−0.307844	−	0.951437i	$-0.599607\pi$
$149$	−1119.80	−0.615688	−0.307844	+	0.951437i	$0.599607\pi$
$150$	644.759	0.350962
$151$	2807.15	1.51286	0.756432	−	0.654073i	$-0.226941\pi$
$151$	2807.15	1.51286	0.756432	+	0.654073i	$0.226941\pi$
$152$	0	0
$153$	−954.752	−0.504491
$154$	−442.511	−0.231549
$155$	590.166	0.305828
$156$	97.5436	0.0500624
$157$	1003.94	0.510339	0.255170	−	0.966896i	$-0.417869\pi$
$157$	1003.94	0.510339	0.255170	+	0.966896i	$0.417869\pi$
$158$	−779.116	−0.392298
$159$	1115.96	0.556611
$160$	−134.019	−0.0662198
$161$	−562.680	−0.275437
$162$	−162.000	−0.0785674
$163$	1271.99	0.611225	0.305612	−	0.952156i	$-0.401139\pi$
$163$	1271.99	0.611225	0.305612	+	0.952156i	$0.401139\pi$
$164$	1658.25	0.789559
$165$	871.967	0.411409
$166$	−760.077	−0.355382
$167$	2984.98	1.38314	0.691572	−	0.722307i	$-0.256918\pi$
$167$	2984.98	1.38314	0.691572	+	0.722307i	$0.256918\pi$
$168$	−76.5145	−0.0351382
$169$	−2130.93	−0.969925
$170$	888.578	0.400887
$171$	0	0
$172$	463.402	0.205430
$173$	−56.7747	−0.0249509	−0.0124754	−	0.999922i	$-0.503971\pi$
$173$	−56.7747	−0.0249509	−0.0124754	+	0.999922i	$0.503971\pi$
$174$	−397.331	−0.173113
$175$	−342.593	−0.147986
$176$	1110.40	0.475568
$177$	−275.079	−0.116815
$178$	−851.598	−0.358595
$179$	4640.98	1.93790	0.968948	−	0.247266i	$-0.0795323\pi$
$179$	4640.98	1.93790	0.968948	+	0.247266i	$0.0795323\pi$
$180$	150.772	0.0624326
$181$	−3775.41	−1.55041	−0.775204	−	0.631711i	$-0.782353\pi$
$181$	−3775.41	−1.55041	−0.775204	+	0.631711i	$0.782353\pi$
$182$	−51.8298	−0.0211093
$183$	655.862	0.264933
$184$	1411.95	0.565708
$185$	653.357	0.259653
$186$	−845.489	−0.333302
$187$	−7362.23	−2.87904
$188$	2481.13	0.962527
$189$	86.0788	0.0331286
$190$	0	0
$191$	−2762.53	−1.04654	−0.523271	−	0.852166i	$-0.675288\pi$
$191$	−2762.53	−1.04654	−0.523271	+	0.852166i	$0.675288\pi$
$192$	192.000	0.0721688
$193$	−2061.51	−0.768866	−0.384433	−	0.923153i	$-0.625603\pi$
$193$	−2061.51	−0.768866	−0.384433	+	0.923153i	$0.625603\pi$
$194$	839.691	0.310754
$195$	102.131	0.0375063
$196$	−1331.34	−0.485184
$197$	2094.82	0.757614	0.378807	−	0.925476i	$-0.376334\pi$
$197$	2094.82	0.757614	0.378807	+	0.925476i	$0.376334\pi$
$198$	−1249.21	−0.448370
$199$	−1717.36	−0.611762	−0.305881	−	0.952070i	$-0.598951\pi$
$199$	−1717.36	−0.611762	−0.305881	+	0.952070i	$0.598951\pi$
$200$	859.678	0.303942
$201$	436.027	0.153010
$202$	2483.42	0.865014
$203$	211.122	0.0729945
$204$	−1273.00	−0.436902
$205$	1736.23	0.591530
$206$	−1187.21	−0.401539
$207$	−1588.44	−0.533354
$208$	130.058	0.0433553
$209$	0	0
$210$	−80.1127	−0.0263252
$211$	2291.05	0.747500	0.373750	−	0.927529i	$-0.378072\pi$
$211$	2291.05	0.747500	0.373750	+	0.927529i	$0.378072\pi$
$212$	1487.94	0.482039
$213$	−2663.49	−0.856804
$214$	−3557.12	−1.13626
$215$	485.194	0.153907
$216$	−216.000	−0.0680414
$217$	449.251	0.140540
$218$	−2139.98	−0.664852
$219$	597.048	0.184223
$220$	1162.62	0.356291
$221$	−862.314	−0.262468
$222$	−936.019	−0.282980
$223$	−3256.19	−0.977805	−0.488902	−	0.872338i	$-0.662603\pi$
$223$	−3256.19	−0.977805	−0.488902	+	0.872338i	$0.662603\pi$
$224$	−102.019	−0.0304306
$225$	−967.138	−0.286559
$226$	1166.39	0.343307
$227$	998.044	0.291817	0.145909	−	0.989298i	$-0.453389\pi$
$227$	998.044	0.291817	0.145909	+	0.989298i	$0.453389\pi$
$228$	0	0
$229$	−1028.59	−0.296816	−0.148408	−	0.988926i	$-0.547415\pi$
$229$	−1028.59	−0.296816	−0.148408	+	0.988926i	$0.547415\pi$
$230$	1478.35	0.423823
$231$	663.766	0.189059
$232$	−529.775	−0.149920
$233$	−125.486	−0.0352827	−0.0176414	−	0.999844i	$-0.505616\pi$
$233$	−125.486	−0.0352827	−0.0176414	+	0.999844i	$0.505616\pi$
$234$	−146.315	−0.0408758
$235$	2597.81	0.721117
$236$	−366.772	−0.101164
$237$	1168.67	0.320310
$238$	676.411	0.184224
$239$	3591.03	0.971901	0.485950	−	0.873986i	$-0.338474\pi$
$239$	3591.03	0.971901	0.485950	+	0.873986i	$0.338474\pi$
$240$	201.029	0.0540682
$241$	3691.84	0.986773	0.493386	−	0.869810i	$-0.335759\pi$
$241$	3691.84	0.986773	0.493386	+	0.869810i	$0.335759\pi$
$242$	−6970.80	−1.85165
$243$	243.000	0.0641500
$244$	874.482	0.229438
$245$	−1393.95	−0.363495
$246$	−2487.38	−0.644672
$247$	0	0
$248$	−1127.32	−0.288648
$249$	1140.12	0.290168
$250$	1947.13	0.492590
$251$	7294.88	1.83446	0.917228	−	0.398362i	$-0.130421\pi$
$251$	7294.88	1.83446	0.917228	+	0.398362i	$0.130421\pi$
$252$	114.772	0.0286902
$253$	−12248.7	−3.04375
$254$	3318.71	0.819820
$255$	−1332.87	−0.327323
$256$	256.000	0.0625000
$257$	−2303.20	−0.559026	−0.279513	−	0.960142i	$-0.590173\pi$
$257$	−2303.20	−0.559026	−0.279513	+	0.960142i	$0.590173\pi$
$258$	−695.102	−0.167733
$259$	497.354	0.119321
$260$	136.174	0.0324814
$261$	595.997	0.141346
$262$	−1195.09	−0.281806
$263$	7138.38	1.67365	0.836827	−	0.547467i	$-0.184408\pi$
$263$	7138.38	1.67365	0.836827	+	0.547467i	$0.184408\pi$
$264$	−1665.61	−0.388299
$265$	1557.91	0.361139
$266$	0	0
$267$	1277.40	0.292792
$268$	581.370	0.132510
$269$	5446.66	1.23453	0.617265	−	0.786755i	$-0.288241\pi$
$269$	5446.66	1.23453	0.617265	+	0.786755i	$0.288241\pi$
$270$	−226.158	−0.0509760
$271$	−3403.68	−0.762947	−0.381474	−	0.924380i	$-0.624583\pi$
$271$	−3403.68	−0.762947	−0.381474	+	0.924380i	$0.624583\pi$
$272$	−1697.34	−0.378368
$273$	77.7448	0.0172356
$274$	3554.62	0.783731
$275$	−7457.74	−1.63534
$276$	−2117.92	−0.461898
$277$	−5131.93	−1.11317	−0.556584	−	0.830791i	$-0.687888\pi$
$277$	−5131.93	−1.11317	−0.556584	+	0.830791i	$0.687888\pi$
$278$	−327.134	−0.0705763
$279$	1268.23	0.272140
$280$	−106.817	−0.0227983
$281$	−3366.68	−0.714731	−0.357365	−	0.933965i	$-0.616325\pi$
$281$	−3366.68	−0.714731	−0.357365	+	0.933965i	$0.616325\pi$
$282$	−3721.70	−0.785900
$283$	6685.76	1.40434	0.702168	−	0.712011i	$-0.252215\pi$
$283$	6685.76	1.40434	0.702168	+	0.712011i	$0.252215\pi$
$284$	−3551.32	−0.742014
$285$	0	0
$286$	−1128.26	−0.233270
$287$	1321.67	0.271832
$288$	−288.000	−0.0589256
$289$	6340.72	1.29060
$290$	−554.688	−0.112319
$291$	−1259.54	−0.253730
$292$	796.065	0.159542
$293$	5625.93	1.12174	0.560871	−	0.827903i	$-0.310466\pi$
$293$	5625.93	1.12174	0.560871	+	0.827903i	$0.310466\pi$
$294$	1997.02	0.396151
$295$	−384.020	−0.0757915
$296$	−1248.02	−0.245067
$297$	1873.81	0.366092
$298$	2239.60	0.435357
$299$	−1434.65	−0.277485
$300$	−1289.52	−0.248168
$301$	369.343	0.0707262
$302$	−5614.29	−1.06976
$303$	−3725.13	−0.706281
$304$	0	0
$305$	915.606	0.171893
$306$	1909.50	0.356729
$307$	−5603.54	−1.04173	−0.520865	−	0.853639i	$-0.674390\pi$
$307$	−5603.54	−1.04173	−0.520865	+	0.853639i	$0.674390\pi$
$308$	885.022	0.163730
$309$	1780.82	0.327855
$310$	−1180.33	−0.216253
$311$	6668.20	1.21582	0.607908	−	0.794008i	$-0.292009\pi$
$311$	6668.20	1.21582	0.607908	+	0.794008i	$0.292009\pi$
$312$	−195.087	−0.0353995
$313$	4491.40	0.811083	0.405541	−	0.914077i	$-0.367083\pi$
$313$	4491.40	0.811083	0.405541	+	0.914077i	$0.367083\pi$
$314$	−2007.88	−0.360865
$315$	120.169	0.0214945
$316$	1558.23	0.277397
$317$	1266.05	0.224316	0.112158	−	0.993690i	$-0.464224\pi$
$317$	1266.05	0.224316	0.112158	+	0.993690i	$0.464224\pi$
$318$	−2231.91	−0.393583
$319$	4595.82	0.806635
$320$	268.039	0.0468244
$321$	5335.68	0.927752
$322$	1125.36	0.194764
$323$	0	0
$324$	324.000	0.0555556
$325$	−873.501	−0.149086
$326$	−2543.97	−0.432201
$327$	3209.97	0.542849
$328$	−3316.50	−0.558302
$329$	1977.53	0.331382
$330$	−1743.93	−0.290910
$331$	5068.09	0.841594	0.420797	−	0.907155i	$-0.361751\pi$
$331$	5068.09	0.841594	0.420797	+	0.907155i	$0.361751\pi$
$332$	1520.15	0.251293
$333$	1404.03	0.231052
$334$	−5969.97	−0.978031
$335$	608.709	0.0992757
$336$	153.029	0.0248465
$337$	−10744.1	−1.73670	−0.868352	−	0.495948i	$-0.834821\pi$
$337$	−10744.1	−1.73670	−0.868352	+	0.495948i	$0.834821\pi$
$338$	4261.85	0.685841
$339$	−1749.59	−0.280309
$340$	−1777.16	−0.283470
$341$	9779.53	1.55305
$342$	0	0
$343$	−2154.64	−0.339182
$344$	−926.803	−0.145261
$345$	−2217.52	−0.346050
$346$	113.549	0.0176429
$347$	−4851.34	−0.750528	−0.375264	−	0.926918i	$-0.622448\pi$
$347$	−4851.34	−0.750528	−0.375264	+	0.926918i	$0.622448\pi$
$348$	794.662	0.122409
$349$	−7611.35	−1.16741	−0.583705	−	0.811966i	$-0.698398\pi$
$349$	−7611.35	−1.16741	−0.583705	+	0.811966i	$0.698398\pi$
$350$	685.186	0.104642
$351$	219.473	0.0333749
$352$	−2220.81	−0.336277
$353$	5567.64	0.839477	0.419739	−	0.907645i	$-0.362122\pi$
$353$	5567.64	0.839477	0.419739	+	0.907645i	$0.362122\pi$
$354$	550.158	0.0826004
$355$	−3718.32	−0.555910
$356$	1703.20	0.253565
$357$	−1014.62	−0.150418
$358$	−9281.96	−1.37030
$359$	4341.31	0.638233	0.319116	−	0.947716i	$-0.396614\pi$
$359$	4341.31	0.638233	0.319116	+	0.947716i	$0.396614\pi$
$360$	−301.544	−0.0441465
$361$	0	0
$362$	7550.82	1.09630
$363$	10456.2	1.51187
$364$	103.660	0.0149265
$365$	833.500	0.119527
$366$	−1311.72	−0.187336
$367$	−8964.48	−1.27505	−0.637524	−	0.770431i	$-0.720041\pi$
$367$	−8964.48	−1.27505	−0.637524	+	0.770431i	$0.720041\pi$
$368$	−2823.90	−0.400016
$369$	3731.06	0.526372
$370$	−1306.71	−0.183602
$371$	1185.93	0.165958
$372$	1690.98	0.235680
$373$	−5048.11	−0.700755	−0.350377	−	0.936609i	$-0.613947\pi$
$373$	−5048.11	−0.700755	−0.350377	+	0.936609i	$0.613947\pi$
$374$	14724.5	2.03579
$375$	−2920.70	−0.402198
$376$	−4962.26	−0.680609
$377$	538.293	0.0735371
$378$	−172.158	−0.0234255
$379$	9290.41	1.25915	0.629573	−	0.776941i	$-0.283230\pi$
$379$	9290.41	1.25915	0.629573	+	0.776941i	$0.283230\pi$
$380$	0	0
$381$	−4978.06	−0.669380
$382$	5525.06	0.740017
$383$	−4162.26	−0.555304	−0.277652	−	0.960682i	$-0.589556\pi$
$383$	−4162.26	−0.555304	−0.277652	+	0.960682i	$0.589556\pi$
$384$	−384.000	−0.0510310
$385$	926.641	0.122665
$386$	4123.03	0.543670
$387$	1042.65	0.136954
$388$	−1679.38	−0.219736
$389$	4185.23	0.545500	0.272750	−	0.962085i	$-0.412067\pi$
$389$	4185.23	0.545500	0.272750	+	0.962085i	$0.412067\pi$
$390$	−204.261	−0.0265210
$391$	18723.1	2.42165
$392$	2662.69	0.343077
$393$	1792.64	0.230093
$394$	−4189.64	−0.535714
$395$	1631.51	0.207823
$396$	2498.41	0.317045
$397$	−4529.76	−0.572650	−0.286325	−	0.958133i	$-0.592434\pi$
$397$	−4529.76	−0.572650	−0.286325	+	0.958133i	$0.592434\pi$
$398$	3434.72	0.432581
$399$	0	0
$400$	−1719.36	−0.214920
$401$	9497.05	1.18269	0.591347	−	0.806417i	$-0.298596\pi$
$401$	9497.05	1.18269	0.591347	+	0.806417i	$0.298596\pi$
$402$	−872.055	−0.108194
$403$	1145.44	0.141585
$404$	−4966.84	−0.611657
$405$	339.236	0.0416217
$406$	−422.245	−0.0516149
$407$	10826.7	1.31857
$408$	2546.01	0.308936
$409$	5354.44	0.647335	0.323667	−	0.946171i	$-0.395084\pi$
$409$	5354.44	0.647335	0.323667	+	0.946171i	$0.395084\pi$
$410$	−3472.46	−0.418275
$411$	−5331.93	−0.639914
$412$	2374.43	0.283931
$413$	−292.327	−0.0348292
$414$	3176.88	0.377138
$415$	1591.64	0.188267
$416$	−260.116	−0.0306568
$417$	490.701	0.0576253
$418$	0	0
$419$	14506.1	1.69134	0.845668	−	0.533709i	$-0.179202\pi$
$419$	14506.1	1.69134	0.845668	+	0.533709i	$0.179202\pi$
$420$	160.225	0.0186148
$421$	1263.45	0.146263	0.0731315	−	0.997322i	$-0.476701\pi$
$421$	1263.45	0.146263	0.0731315	+	0.997322i	$0.476701\pi$
$422$	−4582.10	−0.528562
$423$	5582.55	0.641685
$424$	−2975.88	−0.340853
$425$	11399.7	1.30110
$426$	5326.98	0.605852
$427$	696.985	0.0789918
$428$	7114.24	0.803457
$429$	1692.39	0.190464
$430$	−970.387	−0.108828
$431$	8187.62	0.915043	0.457522	−	0.889198i	$-0.348737\pi$
$431$	8187.62	0.915043	0.457522	+	0.889198i	$0.348737\pi$
$432$	432.000	0.0481125
$433$	9242.94	1.02584	0.512919	−	0.858437i	$-0.328564\pi$
$433$	9242.94	1.02584	0.512919	+	0.858437i	$0.328564\pi$
$434$	−898.502	−0.0993767
$435$	832.032	0.0917078
$436$	4279.96	0.470121
$437$	0	0
$438$	−1194.10	−0.130265
$439$	2043.05	0.222117	0.111059	−	0.993814i	$-0.464576\pi$
$439$	2043.05	0.222117	0.111059	+	0.993814i	$0.464576\pi$
$440$	−2325.25	−0.251936
$441$	−2995.52	−0.323456
$442$	1724.63	0.185593
$443$	−9669.13	−1.03701	−0.518504	−	0.855075i	$-0.673511\pi$
$443$	−9669.13	−1.03701	−0.518504	+	0.855075i	$0.673511\pi$
$444$	1872.04	0.200097
$445$	1783.29	0.189969
$446$	6512.38	0.691412
$447$	−3359.40	−0.355467
$448$	204.039	0.0215177
$449$	13227.2	1.39027	0.695134	−	0.718881i	$-0.255345\pi$
$449$	13227.2	1.39027	0.695134	+	0.718881i	$0.255345\pi$
$450$	1934.28	0.202628
$451$	28770.8	3.00391
$452$	−2332.79	−0.242755
$453$	8421.44	0.873452
$454$	−1996.09	−0.206346
$455$	108.534	0.0111828
$456$	0	0
$457$	−5380.98	−0.550791	−0.275396	−	0.961331i	$-0.588809\pi$
$457$	−5380.98	−0.550791	−0.275396	+	0.961331i	$0.588809\pi$
$458$	2057.17	0.209880
$459$	−2864.26	−0.291268
$460$	−2956.69	−0.299688
$461$	−2844.67	−0.287396	−0.143698	−	0.989622i	$-0.545899\pi$
$461$	−2844.67	−0.287396	−0.143698	+	0.989622i	$0.545899\pi$
$462$	−1327.53	−0.133685
$463$	6625.39	0.665028	0.332514	−	0.943098i	$-0.392103\pi$
$463$	6625.39	0.665028	0.332514	+	0.943098i	$0.392103\pi$
$464$	1059.55	0.106009
$465$	1770.50	0.176570
$466$	250.972	0.0249487
$467$	−18635.1	−1.84653	−0.923264	−	0.384167i	$-0.874489\pi$
$467$	−18635.1	−1.84653	−0.923264	+	0.384167i	$0.874489\pi$
$468$	292.631	0.0289035
$469$	463.367	0.0456211
$470$	−5195.62	−0.509906
$471$	3011.83	0.294645
$472$	733.544	0.0715341
$473$	8040.05	0.781569
$474$	−2337.35	−0.226493
$475$	0	0
$476$	−1352.82	−0.130266
$477$	3347.87	0.321359
$478$	−7182.05	−0.687237
$479$	1556.62	0.148484	0.0742420	−	0.997240i	$-0.476346\pi$
$479$	1556.62	0.148484	0.0742420	+	0.997240i	$0.476346\pi$
$480$	−402.058	−0.0382320
$481$	1268.09	0.120208
$482$	−7383.68	−0.697754
$483$	−1688.04	−0.159024
$484$	13941.6	1.30932
$485$	−1758.36	−0.164625
$486$	−486.000	−0.0453609
$487$	−19339.5	−1.79950	−0.899751	−	0.436405i	$-0.856252\pi$
$487$	−19339.5	−1.79950	−0.899751	+	0.436405i	$0.856252\pi$
$488$	−1748.96	−0.162238
$489$	3815.96	0.352891
$490$	2787.90	0.257030
$491$	−6838.25	−0.628525	−0.314262	−	0.949336i	$-0.601757\pi$
$491$	−6838.25	−0.628525	−0.314262	+	0.949336i	$0.601757\pi$
$492$	4974.75	0.455852
$493$	−7025.05	−0.641770
$494$	0	0
$495$	2615.90	0.237527
$496$	2254.64	0.204105
$497$	−2830.49	−0.255463
$498$	−2280.23	−0.205180
$499$	−18115.0	−1.62513	−0.812563	−	0.582874i	$-0.801928\pi$
$499$	−18115.0	−1.62513	−0.812563	+	0.582874i	$0.801928\pi$
$500$	−3894.26	−0.348314
$501$	8954.95	0.798559
$502$	−14589.8	−1.29716
$503$	−5831.85	−0.516957	−0.258478	−	0.966017i	$-0.583221\pi$
$503$	−5831.85	−0.516957	−0.258478	+	0.966017i	$0.583221\pi$
$504$	−229.544	−0.0202871
$505$	−5200.41	−0.458248
$506$	24497.4	2.15226
$507$	−6392.78	−0.559987
$508$	−6637.41	−0.579700
$509$	−9914.65	−0.863377	−0.431689	−	0.902023i	$-0.642082\pi$
$509$	−9914.65	−0.863377	−0.431689	+	0.902023i	$0.642082\pi$
$510$	2665.73	0.231452
$511$	634.484	0.0549275
$512$	−512.000	−0.0441942
$513$	0	0
$514$	4606.40	0.395291
$515$	2486.09	0.212718
$516$	1390.20	0.118605
$517$	43047.8	3.66197
$518$	−994.709	−0.0843726
$519$	−170.324	−0.0144054
$520$	−272.348	−0.0229678
$521$	5851.91	0.492086	0.246043	−	0.969259i	$-0.420869\pi$
$521$	5851.91	0.492086	0.246043	+	0.969259i	$0.420869\pi$
$522$	−1191.99	−0.0999466
$523$	−6894.09	−0.576401	−0.288200	−	0.957570i	$-0.593057\pi$
$523$	−6894.09	−0.576401	−0.288200	+	0.957570i	$0.593057\pi$
$524$	2390.19	0.199267
$525$	−1027.78	−0.0854399
$526$	−14276.8	−1.18345
$527$	−14948.7	−1.23563
$528$	3331.21	0.274569
$529$	18983.0	1.56020
$530$	−3115.83	−0.255364
$531$	−825.236	−0.0674430
$532$	0	0
$533$	3369.83	0.273853
$534$	−2554.79	−0.207035
$535$	7448.79	0.601943
$536$	−1162.74	−0.0936991
$537$	13922.9	1.11884
$538$	−10893.3	−0.872945
$539$	−23098.9	−1.84590
$540$	452.315	0.0360455
$541$	−5624.99	−0.447019	−0.223509	−	0.974702i	$-0.571751\pi$
$541$	−5624.99	−0.447019	−0.223509	+	0.974702i	$0.571751\pi$
$542$	6807.36	0.539485
$543$	−11326.2	−0.895129
$544$	3394.67	0.267547
$545$	4481.23	0.352211
$546$	−155.490	−0.0121874
$547$	−3596.14	−0.281096	−0.140548	−	0.990074i	$-0.544886\pi$
$547$	−3596.14	−0.281096	−0.140548	+	0.990074i	$0.544886\pi$
$548$	−7109.24	−0.554182
$549$	1967.59	0.152959
$550$	14915.5	1.15636
$551$	0	0
$552$	4235.85	0.326611
$553$	1241.95	0.0955029
$554$	10263.9	0.787129
$555$	1960.07	0.149911
$556$	654.269	0.0499050
$557$	18381.5	1.39829	0.699145	−	0.714980i	$-0.253564\pi$
$557$	18381.5	1.39829	0.699145	+	0.714980i	$0.253564\pi$
$558$	−2536.47	−0.192432
$559$	941.705	0.0712520
$560$	213.634	0.0161209
$561$	−22086.7	−1.66221
$562$	6733.36	0.505391
$563$	−5578.33	−0.417582	−0.208791	−	0.977960i	$-0.566953\pi$
$563$	−5578.33	−0.417582	−0.208791	+	0.977960i	$0.566953\pi$
$564$	7443.39	0.555715
$565$	−2442.49	−0.181870
$566$	−13371.5	−0.993016
$567$	258.236	0.0191268
$568$	7102.63	0.524683
$569$	−16981.3	−1.25113	−0.625564	−	0.780173i	$-0.715131\pi$
$569$	−16981.3	−1.25113	−0.625564	+	0.780173i	$0.715131\pi$
$570$	0	0
$571$	−19520.5	−1.43066	−0.715332	−	0.698785i	$-0.753724\pi$
$571$	−19520.5	−1.43066	−0.715332	+	0.698785i	$0.753724\pi$
$572$	2256.52	0.164947
$573$	−8287.59	−0.604221
$574$	−2643.34	−0.192214
$575$	18966.0	1.37554
$576$	576.000	0.0416667
$577$	6371.35	0.459693	0.229846	−	0.973227i	$-0.426178\pi$
$577$	6371.35	0.459693	0.229846	+	0.973227i	$0.426178\pi$
$578$	−12681.4	−0.912593
$579$	−6184.54	−0.443905
$580$	1109.38	0.0794213
$581$	1211.60	0.0865159
$582$	2519.07	0.179414
$583$	25815.9	1.83394
$584$	−1592.13	−0.112813
$585$	306.392	0.0216543
$586$	−11251.9	−0.793191
$587$	10759.0	0.756513	0.378257	−	0.925701i	$-0.376524\pi$
$587$	10759.0	0.756513	0.378257	+	0.925701i	$0.376524\pi$
$588$	−3994.03	−0.280121
$589$	0	0
$590$	768.039	0.0535927
$591$	6284.47	0.437408
$592$	2496.05	0.173289
$593$	3611.76	0.250113	0.125057	−	0.992150i	$-0.460089\pi$
$593$	3611.76	0.250113	0.125057	+	0.992150i	$0.460089\pi$
$594$	−3747.62	−0.258866
$595$	−1416.44	−0.0975939
$596$	−4479.19	−0.307844
$597$	−5152.09	−0.353201
$598$	2869.30	0.196211
$599$	9180.23	0.626201	0.313100	−	0.949720i	$-0.398632\pi$
$599$	9180.23	0.626201	0.313100	+	0.949720i	$0.398632\pi$
$600$	2579.03	0.175481
$601$	−12700.4	−0.862000	−0.431000	−	0.902352i	$-0.641839\pi$
$601$	−12700.4	−0.862000	−0.431000	+	0.902352i	$0.641839\pi$
$602$	−738.686	−0.0500110
$603$	1308.08	0.0883403
$604$	11228.6	0.756432
$605$	14597.2	0.980928
$606$	7450.26	0.499416
$607$	24227.5	1.62004	0.810019	−	0.586404i	$-0.199457\pi$
$607$	24227.5	1.62004	0.810019	+	0.586404i	$0.199457\pi$
$608$	0	0
$609$	633.367	0.0421434
$610$	−1831.21	−0.121547
$611$	5042.05	0.333845
$612$	−3819.01	−0.252246
$613$	−16213.5	−1.06828	−0.534141	−	0.845396i	$-0.679365\pi$
$613$	−16213.5	−1.06828	−0.534141	+	0.845396i	$0.679365\pi$
$614$	11207.1	0.736614
$615$	5208.70	0.341520
$616$	−1770.04	−0.115774
$617$	−21515.3	−1.40385	−0.701924	−	0.712252i	$-0.747675\pi$
$617$	−21515.3	−1.40385	−0.701924	+	0.712252i	$0.747675\pi$
$618$	−3561.64	−0.231829
$619$	−15878.5	−1.03103	−0.515516	−	0.856880i	$-0.672400\pi$
$619$	−15878.5	−1.03103	−0.515516	+	0.856880i	$0.672400\pi$
$620$	2360.66	0.152914
$621$	−4765.33	−0.307932
$622$	−13336.4	−0.859712
$623$	1357.49	0.0872982
$624$	390.174	0.0250312
$625$	9355.08	0.598725
$626$	−8982.80	−0.573522
$627$	0	0
$628$	4015.77	0.255170
$629$	−16549.4	−1.04907
$630$	−240.338	−0.0151989
$631$	5559.75	0.350761	0.175380	−	0.984501i	$-0.443884\pi$
$631$	5559.75	0.350761	0.175380	+	0.984501i	$0.443884\pi$
$632$	−3116.46	−0.196149
$633$	6873.16	0.431569
$634$	−2532.09	−0.158616
$635$	−6949.55	−0.434306
$636$	4463.83	0.278305
$637$	−2705.50	−0.168282
$638$	−9191.64	−0.570377
$639$	−7990.46	−0.494676
$640$	−536.077	−0.0331099
$641$	20486.9	1.26238	0.631190	−	0.775628i	$-0.282567\pi$
$641$	20486.9	1.26238	0.631190	+	0.775628i	$0.282567\pi$
$642$	−10671.4	−0.656020
$643$	−4102.30	−0.251600	−0.125800	−	0.992056i	$-0.540150\pi$
$643$	−4102.30	−0.251600	−0.125800	+	0.992056i	$0.540150\pi$
$644$	−2250.72	−0.137719
$645$	1455.58	0.0888580
$646$	0	0
$647$	22860.4	1.38908	0.694539	−	0.719455i	$-0.255608\pi$
$647$	22860.4	1.38908	0.694539	+	0.719455i	$0.255608\pi$
$648$	−648.000	−0.0392837
$649$	−6363.52	−0.384884
$650$	1747.00	0.105420
$651$	1347.75	0.0811408
$652$	5087.94	0.305612
$653$	26388.5	1.58141	0.790706	−	0.612197i	$-0.209714\pi$
$653$	26388.5	1.58141	0.790706	+	0.612197i	$0.209714\pi$
$654$	−6419.94	−0.383853
$655$	2502.59	0.149289
$656$	6633.00	0.394779
$657$	1791.15	0.106361
$658$	−3955.05	−0.234322
$659$	−3223.29	−0.190534	−0.0952668	−	0.995452i	$-0.530370\pi$
$659$	−3223.29	−0.190534	−0.0952668	+	0.995452i	$0.530370\pi$
$660$	3487.87	0.205705
$661$	27480.2	1.61703	0.808513	−	0.588478i	$-0.200273\pi$
$661$	27480.2	1.61703	0.808513	+	0.588478i	$0.200273\pi$
$662$	−10136.2	−0.595097
$663$	−2586.94	−0.151536
$664$	−3040.31	−0.177691
$665$	0	0
$666$	−2808.06	−0.163378
$667$	−11687.7	−0.678487
$668$	11939.9	0.691572
$669$	−9768.56	−0.564536
$670$	−1217.42	−0.0701985
$671$	15172.3	0.872908
$672$	−306.058	−0.0175691
$673$	2165.00	0.124004	0.0620020	−	0.998076i	$-0.480251\pi$
$673$	2165.00	0.124004	0.0620020	+	0.998076i	$0.480251\pi$
$674$	21488.2	1.22804
$675$	−2901.41	−0.165445
$676$	−8523.70	−0.484963
$677$	−11999.0	−0.681179	−0.340589	−	0.940212i	$-0.610627\pi$
$677$	−11999.0	−0.681179	−0.340589	+	0.940212i	$0.610627\pi$
$678$	3499.18	0.198209
$679$	−1338.51	−0.0756515
$680$	3554.31	0.200444
$681$	2994.13	0.168481
$682$	−19559.1	−1.09817
$683$	−5234.35	−0.293246	−0.146623	−	0.989192i	$-0.546840\pi$
$683$	−5234.35	−0.293246	−0.146623	+	0.989192i	$0.546840\pi$
$684$	0	0
$685$	−7443.56	−0.415188
$686$	4309.27	0.239838
$687$	−3085.76	−0.171367
$688$	1853.61	0.102715
$689$	3023.73	0.167192
$690$	4435.04	0.244694
$691$	−5160.63	−0.284109	−0.142055	−	0.989859i	$-0.545371\pi$
$691$	−5160.63	−0.284109	−0.142055	+	0.989859i	$0.545371\pi$
$692$	−227.099	−0.0124754
$693$	1991.30	0.109153
$694$	9702.67	0.530704
$695$	685.036	0.0373884
$696$	−1589.32	−0.0865563
$697$	−43978.3	−2.38995
$698$	15222.7	0.825484
$699$	−376.459	−0.0203705
$700$	−1370.37	−0.0739931
$701$	−3315.03	−0.178612	−0.0893059	−	0.996004i	$-0.528465\pi$
$701$	−3315.03	−0.178612	−0.0893059	+	0.996004i	$0.528465\pi$
$702$	−438.946	−0.0235996
$703$	0	0
$704$	4441.62	0.237784
$705$	7793.43	0.416337
$706$	−11135.3	−0.593600
$707$	−3958.70	−0.210583
$708$	−1100.32	−0.0584073
$709$	−18333.2	−0.971111	−0.485555	−	0.874206i	$-0.661383\pi$
$709$	−18333.2	−0.971111	−0.485555	+	0.874206i	$0.661383\pi$
$710$	7436.64	0.393088
$711$	3506.02	0.184931
$712$	−3406.39	−0.179298
$713$	−24870.6	−1.30632
$714$	2029.23	0.106362
$715$	2362.63	0.123577
$716$	18563.9	0.968948
$717$	10773.1	0.561127
$718$	−8682.62	−0.451299
$719$	9747.00	0.505566	0.252783	−	0.967523i	$-0.418654\pi$
$719$	9747.00	0.505566	0.252783	+	0.967523i	$0.418654\pi$
$720$	603.087	0.0312163
$721$	1892.48	0.0977526
$722$	0	0
$723$	11075.5	0.569713
$724$	−15101.6	−0.775204
$725$	−7116.19	−0.364536
$726$	−20912.4	−1.06905
$727$	32723.6	1.66939	0.834697	−	0.550709i	$-0.185643\pi$
$727$	32723.6	1.66939	0.834697	+	0.550709i	$0.185643\pi$
$728$	−207.319	−0.0105546
$729$	729.000	0.0370370
$730$	−1667.00	−0.0845185
$731$	−12289.8	−0.621827
$732$	2623.45	0.132466
$733$	36938.2	1.86131	0.930657	−	0.365894i	$-0.119237\pi$
$733$	36938.2	1.86131	0.930657	+	0.365894i	$0.119237\pi$
$734$	17929.0	0.901594
$735$	−4181.86	−0.209864
$736$	5647.79	0.282854
$737$	10086.8	0.504142
$738$	−7462.13	−0.372202
$739$	−9987.19	−0.497138	−0.248569	−	0.968614i	$-0.579960\pi$
$739$	−9987.19	−0.497138	−0.248569	+	0.968614i	$0.579960\pi$
$740$	2613.43	0.129826
$741$	0	0
$742$	−2371.86	−0.117350
$743$	−21739.8	−1.07343	−0.536713	−	0.843765i	$-0.680334\pi$
$743$	−21739.8	−1.07343	−0.536713	+	0.843765i	$0.680334\pi$
$744$	−3381.96	−0.166651
$745$	−4689.83	−0.230634
$746$	10096.2	0.495508
$747$	3420.35	0.167529
$748$	−29448.9	−1.43952
$749$	5670.23	0.276617
$750$	5841.40	0.284397
$751$	38067.4	1.84967	0.924833	−	0.380374i	$-0.124205\pi$
$751$	38067.4	1.84967	0.924833	+	0.380374i	$0.124205\pi$
$752$	9924.53	0.481264
$753$	21884.6	1.05912
$754$	−1076.59	−0.0519986
$755$	11756.6	0.566712
$756$	344.315	0.0165643
$757$	−11365.2	−0.545675	−0.272838	−	0.962060i	$-0.587962\pi$
$757$	−11365.2	−0.545675	−0.272838	+	0.962060i	$0.587962\pi$
$758$	−18580.8	−0.890351
$759$	−36746.1	−1.75731
$760$	0	0
$761$	24549.3	1.16940	0.584699	−	0.811250i	$-0.301213\pi$
$761$	24549.3	1.16940	0.584699	+	0.811250i	$0.301213\pi$
$762$	9956.12	0.473323
$763$	3411.24	0.161855
$764$	−11050.1	−0.523271
$765$	−3998.60	−0.188980
$766$	8324.51	0.392659
$767$	−745.338	−0.0350881
$768$	768.000	0.0360844
$769$	14222.5	0.666941	0.333471	−	0.942760i	$-0.391780\pi$
$769$	14222.5	0.666941	0.333471	+	0.942760i	$0.391780\pi$
$770$	−1853.28	−0.0867372
$771$	−6909.60	−0.322754
$772$	−8246.06	−0.384433
$773$	−16082.6	−0.748319	−0.374159	−	0.927364i	$-0.622069\pi$
$773$	−16082.6	−0.748319	−0.374159	+	0.927364i	$0.622069\pi$
$774$	−2085.31	−0.0968409
$775$	−15142.7	−0.701859
$776$	3358.77	0.155377
$777$	1492.06	0.0688899
$778$	−8370.46	−0.385727
$779$	0	0
$780$	408.523	0.0187531
$781$	−61615.6	−2.82302
$782$	−37446.1	−1.71237
$783$	1787.99	0.0816061
$784$	−5325.38	−0.242592
$785$	4204.61	0.191171
$786$	−3585.28	−0.162701
$787$	−30058.0	−1.36144	−0.680720	−	0.732544i	$-0.738333\pi$
$787$	−30058.0	−1.36144	−0.680720	+	0.732544i	$0.738333\pi$
$788$	8379.29	0.378807
$789$	21415.1	0.966285
$790$	−3263.02	−0.146953
$791$	−1859.29	−0.0835764
$792$	−4996.82	−0.224185
$793$	1777.09	0.0795790
$794$	9059.52	0.404925
$795$	4673.74	0.208504
$796$	−6869.45	−0.305881
$797$	20008.2	0.889243	0.444621	−	0.895719i	$-0.353338\pi$
$797$	20008.2	0.889243	0.444621	+	0.895719i	$0.353338\pi$
$798$	0	0
$799$	−65801.8	−2.91352
$800$	3438.71	0.151971
$801$	3832.19	0.169043
$802$	−18994.1	−0.836291
$803$	13811.8	0.606983
$804$	1744.11	0.0765050
$805$	−2356.56	−0.103178
$806$	−2290.89	−0.100116
$807$	16340.0	0.712757
$808$	9933.68	0.432507
$809$	35302.3	1.53419	0.767097	−	0.641532i	$-0.221701\pi$
$809$	35302.3	1.53419	0.767097	+	0.641532i	$0.221701\pi$
$810$	−678.473	−0.0294310
$811$	−10980.8	−0.475446	−0.237723	−	0.971333i	$-0.576401\pi$
$811$	−10980.8	−0.475446	−0.237723	+	0.971333i	$0.576401\pi$
$812$	844.489	0.0364972
$813$	−10211.0	−0.440488
$814$	−21653.3	−0.932369
$815$	5327.21	0.228962
$816$	−5092.01	−0.218451
$817$	0	0
$818$	−10708.9	−0.457735
$819$	233.234	0.00995100
$820$	6944.93	0.295765
$821$	−5626.68	−0.239187	−0.119594	−	0.992823i	$-0.538159\pi$
$821$	−5626.68	−0.239187	−0.119594	+	0.992823i	$0.538159\pi$
$822$	10663.9	0.452487
$823$	28766.3	1.21838	0.609191	−	0.793023i	$-0.291494\pi$
$823$	28766.3	1.21838	0.609191	+	0.793023i	$0.291494\pi$
$824$	−4748.85	−0.200769
$825$	−22373.2	−0.944164
$826$	584.653	0.0246280
$827$	−44482.4	−1.87038	−0.935191	−	0.354145i	$-0.884772\pi$
$827$	−44482.4	−1.87038	−0.935191	+	0.354145i	$0.884772\pi$
$828$	−6353.77	−0.266677
$829$	−23302.5	−0.976271	−0.488136	−	0.872768i	$-0.662323\pi$
$829$	−23302.5	−0.976271	−0.488136	+	0.872768i	$0.662323\pi$
$830$	−3183.28	−0.133125
$831$	−15395.8	−0.642688
$832$	520.232	0.0216777
$833$	35308.4	1.46862
$834$	−981.403	−0.0407472
$835$	12501.4	0.518120
$836$	0	0
$837$	3804.70	0.157120
$838$	−29012.2	−1.19596
$839$	27683.7	1.13915	0.569575	−	0.821939i	$-0.307108\pi$
$839$	27683.7	1.13915	0.569575	+	0.821939i	$0.307108\pi$
$840$	−320.451	−0.0131626
$841$	−20003.7	−0.820192
$842$	−2526.90	−0.103424
$843$	−10100.0	−0.412650
$844$	9164.21	0.373750
$845$	−8924.54	−0.363330
$846$	−11165.1	−0.453740
$847$	11111.8	0.450776
$848$	5951.77	0.241020
$849$	20057.3	0.810794
$850$	−22799.4	−0.920017
$851$	−27533.5	−1.10909
$852$	−10654.0	−0.428402
$853$	−30609.5	−1.22866	−0.614331	−	0.789049i	$-0.710574\pi$
$853$	−30609.5	−1.22866	−0.614331	+	0.789049i	$0.710574\pi$
$854$	−1393.97	−0.0558556
$855$	0	0
$856$	−14228.5	−0.568130
$857$	−10918.3	−0.435194	−0.217597	−	0.976039i	$-0.569822\pi$
$857$	−10918.3	−0.435194	−0.217597	+	0.976039i	$0.569822\pi$
$858$	−3384.78	−0.134679
$859$	14639.5	0.581482	0.290741	−	0.956802i	$-0.406098\pi$
$859$	14639.5	0.581482	0.290741	+	0.956802i	$0.406098\pi$
$860$	1940.77	0.0769533
$861$	3965.01	0.156942
$862$	−16375.2	−0.647033
$863$	22775.2	0.898353	0.449176	−	0.893443i	$-0.351717\pi$
$863$	22775.2	0.898353	0.449176	+	0.893443i	$0.351717\pi$
$864$	−864.000	−0.0340207
$865$	−237.778	−0.00934648
$866$	−18485.9	−0.725377
$867$	19022.2	0.745129
$868$	1797.00	0.0702700
$869$	27035.4	1.05537
$870$	−1664.06	−0.0648472
$871$	1181.43	0.0459603
$872$	−8559.92	−0.332426
$873$	−3778.61	−0.146491
$874$	0	0
$875$	−3103.83	−0.119918
$876$	2388.19	0.0921114
$877$	−805.908	−0.0310303	−0.0155152	−	0.999880i	$-0.504939\pi$
$877$	−805.908	−0.0310303	−0.0155152	+	0.999880i	$0.504939\pi$
$878$	−4086.10	−0.157061
$879$	16877.8	0.647638
$880$	4650.49	0.178146
$881$	−901.690	−0.0344821	−0.0172410	−	0.999851i	$-0.505488\pi$
$881$	−901.690	−0.0344821	−0.0172410	+	0.999851i	$0.505488\pi$
$882$	5991.05	0.228718
$883$	−25406.3	−0.968279	−0.484140	−	0.874991i	$-0.660867\pi$
$883$	−25406.3	−0.968279	−0.484140	+	0.874991i	$0.660867\pi$
$884$	−3449.26	−0.131234
$885$	−1152.06	−0.0437582
$886$	19338.3	0.733275
$887$	−28560.5	−1.08114	−0.540568	−	0.841300i	$-0.681791\pi$
$887$	−28560.5	−1.08114	−0.540568	+	0.841300i	$0.681791\pi$
$888$	−3744.07	−0.141490
$889$	−5290.19	−0.199581
$890$	−3566.58	−0.134328
$891$	5621.42	0.211363
$892$	−13024.8	−0.488902
$893$	0	0
$894$	6718.79	0.251353
$895$	19436.9	0.725927
$896$	−408.077	−0.0152153
$897$	−4303.95	−0.160206
$898$	−26454.4	−0.983067
$899$	9331.64	0.346193
$900$	−3868.55	−0.143280
$901$	−39461.6	−1.45911
$902$	−57541.6	−2.12408
$903$	1108.03	0.0408338
$904$	4665.58	0.171654
$905$	−15811.8	−0.580776
$906$	−16842.9	−0.617624
$907$	40905.4	1.49751	0.748755	−	0.662847i	$-0.230652\pi$
$907$	40905.4	1.49751	0.748755	+	0.662847i	$0.230652\pi$
$908$	3992.18	0.145909
$909$	−11175.4	−0.407772
$910$	−217.069	−0.00790743
$911$	−30047.9	−1.09279	−0.546395	−	0.837527i	$-0.684000\pi$
$911$	−30047.9	−1.09279	−0.546395	+	0.837527i	$0.684000\pi$
$912$	0	0
$913$	26374.8	0.956055
$914$	10762.0	0.389468
$915$	2746.82	0.0992426
$916$	−4114.34	−0.148408
$917$	1905.04	0.0686041
$918$	5728.51	0.205958
$919$	−49600.1	−1.78036	−0.890182	−	0.455605i	$-0.849423\pi$
$919$	−49600.1	−1.78036	−0.890182	+	0.455605i	$0.849423\pi$
$920$	5913.39	0.211912
$921$	−16810.6	−0.601443
$922$	5689.33	0.203219
$923$	−7216.84	−0.257362
$924$	2655.07	0.0945295
$925$	−16764.1	−0.595891
$926$	−13250.8	−0.470246
$927$	5342.46	0.189287
$928$	−2119.10	−0.0749600
$929$	27556.8	0.973205	0.486603	−	0.873623i	$-0.338236\pi$
$929$	27556.8	0.973205	0.486603	+	0.873623i	$0.338236\pi$
$930$	−3541.00	−0.124854
$931$	0	0
$932$	−501.945	−0.0176414
$933$	20004.6	0.701952
$934$	37270.1	1.30569
$935$	−30833.8	−1.07847
$936$	−585.261	−0.0204379
$937$	−26880.8	−0.937202	−0.468601	−	0.883410i	$-0.655242\pi$
$937$	−26880.8	−0.937202	−0.468601	+	0.883410i	$0.655242\pi$
$938$	−926.734	−0.0322590
$939$	13474.2	0.468279
$940$	10391.2	0.360558
$941$	6411.34	0.222108	0.111054	−	0.993814i	$-0.464577\pi$
$941$	6411.34	0.222108	0.111054	+	0.993814i	$0.464577\pi$
$942$	−6023.65	−0.208345
$943$	−73167.6	−2.52669
$944$	−1467.09	−0.0505822
$945$	360.507	0.0124098
$946$	−16080.1	−0.552653
$947$	−1989.83	−0.0682796	−0.0341398	−	0.999417i	$-0.510869\pi$
$947$	−1989.83	−0.0682796	−0.0341398	+	0.999417i	$0.510869\pi$
$948$	4674.69	0.160155
$949$	1617.73	0.0553358
$950$	0	0
$951$	3798.14	0.129509
$952$	2705.64	0.0921118
$953$	38266.4	1.30070	0.650351	−	0.759633i	$-0.274622\pi$
$953$	38266.4	1.30070	0.650351	+	0.759633i	$0.274622\pi$
$954$	−6695.74	−0.227235
$955$	−11569.8	−0.392030
$956$	14364.1	0.485950
$957$	13787.5	0.465711
$958$	−3113.24	−0.104994
$959$	−5666.25	−0.190795
$960$	804.116	0.0270341
$961$	−9934.01	−0.333457
$962$	−2536.18	−0.0849998
$963$	16007.0	0.535638
$964$	14767.4	0.493386
$965$	−8633.84	−0.288014
$966$	3376.08	0.112447
$967$	220.116	0.00732002	0.00366001	−	0.999993i	$-0.498835\pi$
$967$	220.116	0.00732002	0.00366001	+	0.999993i	$0.498835\pi$
$968$	−27883.2	−0.925827
$969$	0	0
$970$	3516.72	0.116407
$971$	−2175.36	−0.0718955	−0.0359478	−	0.999354i	$-0.511445\pi$
$971$	−2175.36	−0.0718955	−0.0359478	+	0.999354i	$0.511445\pi$
$972$	972.000	0.0320750
$973$	521.469	0.0171814
$974$	38679.0	1.27244
$975$	−2620.50	−0.0860751
$976$	3497.93	0.114719
$977$	−13707.7	−0.448872	−0.224436	−	0.974489i	$-0.572054\pi$
$977$	−13707.7	−0.448872	−0.224436	+	0.974489i	$0.572054\pi$
$978$	−7631.91	−0.249531
$979$	29550.6	0.964699
$980$	−5575.81	−0.181748
$981$	9629.91	0.313414
$982$	13676.5	0.444434
$983$	18885.7	0.612779	0.306389	−	0.951906i	$-0.400879\pi$
$983$	18885.7	0.612779	0.306389	+	0.951906i	$0.400879\pi$
$984$	−9949.50	−0.322336
$985$	8773.33	0.283799
$986$	14050.1	0.453800
$987$	5932.58	0.191323
$988$	0	0
$989$	−20446.8	−0.657403
$990$	−5231.80	−0.167957
$991$	40655.0	1.30318	0.651589	−	0.758572i	$-0.274103\pi$
$991$	40655.0	1.30318	0.651589	+	0.758572i	$0.274103\pi$
$992$	−4509.27	−0.144324
$993$	15204.3	0.485894
$994$	5660.99	0.180639
$995$	−7192.49	−0.229163
$996$	4560.46	0.145084
$997$	−16433.3	−0.522014	−0.261007	−	0.965337i	$-0.584055\pi$
$997$	−16433.3	−0.522014	−0.261007	+	0.965337i	$0.584055\pi$
$998$	36229.9	1.14914
$999$	4212.08	0.133398

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2166.4.a.t.1.2		3
19.8	odd	6		114.4.e.d.7.2	✓	6
19.12	odd	6		114.4.e.d.49.2	yes	6
19.18	odd	2		2166.4.a.u.1.2		3
57.8	even	6		342.4.g.h.235.2		6
57.50	even	6		342.4.g.h.163.2		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
114.4.e.d.7.2	✓	6	19.8	odd	6
114.4.e.d.49.2	yes	6	19.12	odd	6
342.4.g.h.163.2		6	57.50	even	6
342.4.g.h.235.2		6	57.8	even	6
2166.4.a.t.1.2		3	1.1	even	1	trivial
2166.4.a.u.1.2		3	19.18	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:	\( N \)	\(=\)	\( 2166 = 2 \cdot 3 \cdot 19^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2166.a (trivial)

\(f(q)\)	\(=\)	\(q-2.00000 q^{2} +3.00000 q^{3} +4.00000 q^{4} +4.18810 q^{5} -6.00000 q^{6} +3.18810 q^{7} -8.00000 q^{8} +9.00000 q^{9} +O(q^{10})\)	\(q-2.00000 q^{2} +3.00000 q^{3} +4.00000 q^{4} +4.18810 q^{5} -6.00000 q^{6} +3.18810 q^{7} -8.00000 q^{8} +9.00000 q^{9} -8.37621 q^{10} +69.4003 q^{11} +12.0000 q^{12} +8.12863 q^{13} -6.37621 q^{14} +12.5643 q^{15} +16.0000 q^{16} -106.084 q^{17} -18.0000 q^{18} +16.7524 q^{20} +9.56431 q^{21} -138.801 q^{22} -176.494 q^{23} -24.0000 q^{24} -107.460 q^{25} -16.2573 q^{26} +27.0000 q^{27} +12.7524 q^{28} +66.2219 q^{29} -25.1286 q^{30} +140.915 q^{31} -32.0000 q^{32} +208.201 q^{33} +212.167 q^{34} +13.3521 q^{35} +36.0000 q^{36} +156.003 q^{37} +24.3859 q^{39} -33.5048 q^{40} +414.563 q^{41} -19.1286 q^{42} +115.850 q^{43} +277.601 q^{44} +37.6929 q^{45} +352.987 q^{46} +620.283 q^{47} +48.0000 q^{48} -332.836 q^{49} +214.920 q^{50} -318.251 q^{51} +32.5145 q^{52} +371.986 q^{53} -54.0000 q^{54} +290.656 q^{55} -25.5048 q^{56} -132.444 q^{58} -91.6929 q^{59} +50.2573 q^{60} +218.621 q^{61} -281.830 q^{62} +28.6929 q^{63} +64.0000 q^{64} +34.0436 q^{65} -416.402 q^{66} +145.342 q^{67} -424.334 q^{68} -529.481 q^{69} -26.7042 q^{70} -887.829 q^{71} -72.0000 q^{72} +199.016 q^{73} -312.006 q^{74} -322.379 q^{75} +221.255 q^{77} -48.7718 q^{78} +389.558 q^{79} +67.0097 q^{80} +81.0000 q^{81} -829.125 q^{82} +380.039 q^{83} +38.2573 q^{84} -444.289 q^{85} -231.701 q^{86} +198.666 q^{87} -555.202 q^{88} +425.799 q^{89} -75.3859 q^{90} +25.9149 q^{91} -705.974 q^{92} +422.744 q^{93} -1240.57 q^{94} -96.0000 q^{96} -419.846 q^{97} +665.672 q^{98} +624.603 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 6 q^{2} + 9 q^{3} + 12 q^{4} + 10 q^{5} - 18 q^{6} + 7 q^{7} - 24 q^{8} + 27 q^{9}+O(q^{10}) \) 3 * q - 6 * q^2 + 9 * q^3 + 12 * q^4 + 10 * q^5 - 18 * q^6 + 7 * q^7 - 24 * q^8 + 27 * q^9	\( 3 q - 6 q^{2} + 9 q^{3} + 12 q^{4} + 10 q^{5} - 18 q^{6} + 7 q^{7} - 24 q^{8} + 27 q^{9} - 20 q^{10} - 44 q^{11} + 36 q^{12} + 9 q^{13} - 14 q^{14} + 30 q^{15} + 48 q^{16} - 84 q^{17} - 54 q^{18} + 40 q^{20} + 21 q^{21} + 88 q^{22} - 2 q^{23} - 72 q^{24} - 83 q^{25} - 18 q^{26} + 81 q^{27} + 28 q^{28} - 92 q^{29} - 60 q^{30} + 109 q^{31} - 96 q^{32} - 132 q^{33} + 168 q^{34} + 282 q^{35} + 108 q^{36} - 245 q^{37} + 27 q^{39} - 80 q^{40} + 688 q^{41} - 42 q^{42} - 103 q^{43} - 176 q^{44} + 90 q^{45} + 4 q^{46} + 322 q^{47} + 144 q^{48} - 754 q^{49} + 166 q^{50} - 252 q^{51} + 36 q^{52} + 1322 q^{53} - 162 q^{54} - 248 q^{55} - 56 q^{56} + 184 q^{58} - 252 q^{59} + 120 q^{60} - 435 q^{61} - 218 q^{62} + 63 q^{63} + 192 q^{64} + 1582 q^{65} + 264 q^{66} + 719 q^{67} - 336 q^{68} - 6 q^{69} - 564 q^{70} + 62 q^{71} - 216 q^{72} - 581 q^{73} + 490 q^{74} - 249 q^{75} - 204 q^{77} - 54 q^{78} + 489 q^{79} + 160 q^{80} + 243 q^{81} - 1376 q^{82} + 2496 q^{83} + 84 q^{84} + 1632 q^{85} + 206 q^{86} - 276 q^{87} + 352 q^{88} - 1584 q^{89} - 180 q^{90} + 1573 q^{91} - 8 q^{92} + 327 q^{93} - 644 q^{94} - 288 q^{96} - 974 q^{97} + 1508 q^{98} - 396 q^{99}+O(q^{100}) \) 3 * q - 6 * q^2 + 9 * q^3 + 12 * q^4 + 10 * q^5 - 18 * q^6 + 7 * q^7 - 24 * q^8 + 27 * q^9 - 20 * q^10 - 44 * q^11 + 36 * q^12 + 9 * q^13 - 14 * q^14 + 30 * q^15 + 48 * q^16 - 84 * q^17 - 54 * q^18 + 40 * q^20 + 21 * q^21 + 88 * q^22 - 2 * q^23 - 72 * q^24 - 83 * q^25 - 18 * q^26 + 81 * q^27 + 28 * q^28 - 92 * q^29 - 60 * q^30 + 109 * q^31 - 96 * q^32 - 132 * q^33 + 168 * q^34 + 282 * q^35 + 108 * q^36 - 245 * q^37 + 27 * q^39 - 80 * q^40 + 688 * q^41 - 42 * q^42 - 103 * q^43 - 176 * q^44 + 90 * q^45 + 4 * q^46 + 322 * q^47 + 144 * q^48 - 754 * q^49 + 166 * q^50 - 252 * q^51 + 36 * q^52 + 1322 * q^53 - 162 * q^54 - 248 * q^55 - 56 * q^56 + 184 * q^58 - 252 * q^59 + 120 * q^60 - 435 * q^61 - 218 * q^62 + 63 * q^63 + 192 * q^64 + 1582 * q^65 + 264 * q^66 + 719 * q^67 - 336 * q^68 - 6 * q^69 - 564 * q^70 + 62 * q^71 - 216 * q^72 - 581 * q^73 + 490 * q^74 - 249 * q^75 - 204 * q^77 - 54 * q^78 + 489 * q^79 + 160 * q^80 + 243 * q^81 - 1376 * q^82 + 2496 * q^83 + 84 * q^84 + 1632 * q^85 + 206 * q^86 - 276 * q^87 + 352 * q^88 - 1584 * q^89 - 180 * q^90 + 1573 * q^91 - 8 * q^92 + 327 * q^93 - 644 * q^94 - 288 * q^96 - 974 * q^97 + 1508 * q^98 - 396 * q^99

Introduction

L-functions

Modular forms

Varieties

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Representations

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Database

Properties

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