LMFDB - Embedded newform 2166.4.a.be.1.6

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:	$6$
Coefficient field:	$\mathbb{Q}[x]/(x^{6} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - x^{5} - 626x^{4} + 3719x^{3} + 95418x^{2} - 1061685x + 2892769 $ x^6 - x^5 - 626x^4 + 3719x^3 + 95418x^2 - 1061685x + 2892769
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$-18.9451$ 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)

$f(q)$	$=$	$q+2.00000 q^{2} -3.00000 q^{3} +4.00000 q^{4} +21.1812 q^{5} -6.00000 q^{6} +34.5236 q^{7} +8.00000 q^{8} +9.00000 q^{9} +O(q^{10})$	\(q+2.00000 q^{2} -3.00000 q^{3} +4.00000 q^{4} +21.1812 q^{5} -6.00000 q^{6} +34.5236 q^{7} +8.00000 q^{8} +9.00000 q^{9} +42.3624 q^{10} +34.6833 q^{11} -12.0000 q^{12} -41.8088 q^{13} +69.0473 q^{14} -63.5436 q^{15} +16.0000 q^{16} -36.0789 q^{17} +18.0000 q^{18} +84.7247 q^{20} -103.571 q^{21} +69.3666 q^{22} -129.032 q^{23} -24.0000 q^{24} +323.643 q^{25} -83.6176 q^{26} -27.0000 q^{27} +138.095 q^{28} -193.127 q^{29} -127.087 q^{30} +287.144 q^{31} +32.0000 q^{32} -104.050 q^{33} -72.1578 q^{34} +731.252 q^{35} +36.0000 q^{36} +382.606 q^{37} +125.426 q^{39} +169.449 q^{40} +51.4641 q^{41} -207.142 q^{42} -218.894 q^{43} +138.733 q^{44} +190.631 q^{45} -258.063 q^{46} +150.538 q^{47} -48.0000 q^{48} +848.882 q^{49} +647.285 q^{50} +108.237 q^{51} -167.235 q^{52} +112.130 q^{53} -54.0000 q^{54} +734.634 q^{55} +276.189 q^{56} -386.254 q^{58} -492.668 q^{59} -254.174 q^{60} +258.448 q^{61} +574.287 q^{62} +310.713 q^{63} +64.0000 q^{64} -885.559 q^{65} -208.100 q^{66} +219.050 q^{67} -144.316 q^{68} +387.095 q^{69} +1462.50 q^{70} +24.1481 q^{71} +72.0000 q^{72} -174.347 q^{73} +765.212 q^{74} -970.928 q^{75} +1197.39 q^{77} +250.853 q^{78} -923.646 q^{79} +338.899 q^{80} +81.0000 q^{81} +102.928 q^{82} -863.908 q^{83} -414.284 q^{84} -764.193 q^{85} -437.788 q^{86} +579.381 q^{87} +277.467 q^{88} -344.831 q^{89} +381.261 q^{90} -1443.39 q^{91} -516.127 q^{92} -861.431 q^{93} +301.076 q^{94} -96.0000 q^{96} -270.549 q^{97} +1697.76 q^{98} +312.150 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 12 q^{2} - 18 q^{3} + 24 q^{4} - q^{5} - 36 q^{6} + 28 q^{7} + 48 q^{8} + 54 q^{9}+O(q^{10}) $ 6 * q + 12 * q^2 - 18 * q^3 + 24 * q^4 - q^5 - 36 * q^6 + 28 * q^7 + 48 * q^8 + 54 * q^9	\( 6 q + 12 q^{2} - 18 q^{3} + 24 q^{4} - q^{5} - 36 q^{6} + 28 q^{7} + 48 q^{8} + 54 q^{9} - 2 q^{10} + 127 q^{11} - 72 q^{12} + 12 q^{13} + 56 q^{14} + 3 q^{15} + 96 q^{16} + 61 q^{17} + 108 q^{18} - 4 q^{20} - 84 q^{21} + 254 q^{22} - 130 q^{23} - 144 q^{24} + 543 q^{25} + 24 q^{26} - 162 q^{27} + 112 q^{28} - 570 q^{29} + 6 q^{30} + 679 q^{31} + 192 q^{32} - 381 q^{33} + 122 q^{34} + 478 q^{35} + 216 q^{36} - 37 q^{37} - 36 q^{39} - 8 q^{40} + 346 q^{41} - 168 q^{42} - 20 q^{43} + 508 q^{44} - 9 q^{45} - 260 q^{46} - 215 q^{47} - 288 q^{48} + 2056 q^{49} + 1086 q^{50} - 183 q^{51} + 48 q^{52} - 366 q^{53} - 324 q^{54} - 253 q^{55} + 224 q^{56} - 1140 q^{58} - 263 q^{59} + 12 q^{60} + 413 q^{61} + 1358 q^{62} + 252 q^{63} + 384 q^{64} - 1203 q^{65} - 762 q^{66} - 751 q^{67} + 244 q^{68} + 390 q^{69} + 956 q^{70} - 281 q^{71} + 432 q^{72} - 1120 q^{73} - 74 q^{74} - 1629 q^{75} + 1067 q^{77} - 72 q^{78} + 2045 q^{79} - 16 q^{80} + 486 q^{81} + 692 q^{82} + 43 q^{83} - 336 q^{84} - 627 q^{85} - 40 q^{86} + 1710 q^{87} + 1016 q^{88} - 467 q^{89} - 18 q^{90} - 1101 q^{91} - 520 q^{92} - 2037 q^{93} - 430 q^{94} - 576 q^{96} - 275 q^{97} + 4112 q^{98} + 1143 q^{99}+O(q^{100}) \) 6 * q + 12 * q^2 - 18 * q^3 + 24 * q^4 - q^5 - 36 * q^6 + 28 * q^7 + 48 * q^8 + 54 * q^9 - 2 * q^10 + 127 * q^11 - 72 * q^12 + 12 * q^13 + 56 * q^14 + 3 * q^15 + 96 * q^16 + 61 * q^17 + 108 * q^18 - 4 * q^20 - 84 * q^21 + 254 * q^22 - 130 * q^23 - 144 * q^24 + 543 * q^25 + 24 * q^26 - 162 * q^27 + 112 * q^28 - 570 * q^29 + 6 * q^30 + 679 * q^31 + 192 * q^32 - 381 * q^33 + 122 * q^34 + 478 * q^35 + 216 * q^36 - 37 * q^37 - 36 * q^39 - 8 * q^40 + 346 * q^41 - 168 * q^42 - 20 * q^43 + 508 * q^44 - 9 * q^45 - 260 * q^46 - 215 * q^47 - 288 * q^48 + 2056 * q^49 + 1086 * q^50 - 183 * q^51 + 48 * q^52 - 366 * q^53 - 324 * q^54 - 253 * q^55 + 224 * q^56 - 1140 * q^58 - 263 * q^59 + 12 * q^60 + 413 * q^61 + 1358 * q^62 + 252 * q^63 + 384 * q^64 - 1203 * q^65 - 762 * q^66 - 751 * q^67 + 244 * q^68 + 390 * q^69 + 956 * q^70 - 281 * q^71 + 432 * q^72 - 1120 * q^73 - 74 * q^74 - 1629 * q^75 + 1067 * q^77 - 72 * q^78 + 2045 * q^79 - 16 * q^80 + 486 * q^81 + 692 * q^82 + 43 * q^83 - 336 * q^84 - 627 * q^85 - 40 * q^86 + 1710 * q^87 + 1016 * q^88 - 467 * q^89 - 18 * q^90 - 1101 * q^91 - 520 * q^92 - 2037 * q^93 - 430 * q^94 - 576 * q^96 - 275 * q^97 + 4112 * q^98 + 1143 * 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
$5$	21.1812	1.89450	0.947251	−	0.320492i	$-0.103848\pi$
$5$	21.1812	1.89450	0.947251	+	0.320492i	$0.103848\pi$
$6$	−6.00000	−0.408248
$7$	34.5236	1.86410	0.932051	−	0.362328i	$-0.118018\pi$
$7$	34.5236	1.86410	0.932051	+	0.362328i	$0.118018\pi$
$8$	8.00000	0.353553
$9$	9.00000	0.333333
$10$	42.3624	1.33962
$11$	34.6833	0.950674	0.475337	−	0.879804i	$-0.342326\pi$
$11$	34.6833	0.950674	0.475337	+	0.879804i	$0.342326\pi$
$12$	−12.0000	−0.288675
$13$	−41.8088	−0.891975	−0.445987	−	0.895039i	$-0.647147\pi$
$13$	−41.8088	−0.891975	−0.445987	+	0.895039i	$0.647147\pi$
$14$	69.0473	1.31812
$15$	−63.5436	−1.09379
$16$	16.0000	0.250000
$17$	−36.0789	−0.514730	−0.257365	−	0.966314i	$-0.582854\pi$
$17$	−36.0789	−0.514730	−0.257365	+	0.966314i	$0.582854\pi$
$18$	18.0000	0.235702
$19$	0	0
$19$	0	0
$20$	84.7247	0.947251
$21$	−103.571	−1.07624
$22$	69.3666	0.672228
$23$	−129.032	−1.16978	−0.584890	−	0.811112i	$-0.698862\pi$
$23$	−129.032	−1.16978	−0.584890	+	0.811112i	$0.698862\pi$
$24$	−24.0000	−0.204124
$25$	323.643	2.58914
$26$	−83.6176	−0.630721
$27$	−27.0000	−0.192450
$28$	138.095	0.932051
$29$	−193.127	−1.23665	−0.618324	−	0.785923i	$-0.712188\pi$
$29$	−193.127	−1.23665	−0.618324	+	0.785923i	$0.712188\pi$
$30$	−127.087	−0.773427
$31$	287.144	1.66363	0.831815	−	0.555053i	$-0.187302\pi$
$31$	287.144	1.66363	0.831815	+	0.555053i	$0.187302\pi$
$32$	32.0000	0.176777
$33$	−104.050	−0.548872
$34$	−72.1578	−0.363969
$35$	731.252	3.53155
$36$	36.0000	0.166667
$37$	382.606	1.70000	0.850000	−	0.526783i	$-0.176602\pi$
$37$	382.606	1.70000	0.850000	+	0.526783i	$0.176602\pi$
$38$	0	0
$39$	125.426	0.514982
$40$	169.449	0.669808
$41$	51.4641	0.196033	0.0980165	−	0.995185i	$-0.468750\pi$
$41$	51.4641	0.196033	0.0980165	+	0.995185i	$0.468750\pi$
$42$	−207.142	−0.761016
$43$	−218.894	−0.776302	−0.388151	−	0.921596i	$-0.626886\pi$
$43$	−218.894	−0.776302	−0.388151	+	0.921596i	$0.626886\pi$
$44$	138.733	0.475337
$45$	190.631	0.631501
$46$	−258.063	−0.827160
$47$	150.538	0.467196	0.233598	−	0.972333i	$-0.424950\pi$
$47$	150.538	0.467196	0.233598	+	0.972333i	$0.424950\pi$
$48$	−48.0000	−0.144338
$49$	848.882	2.47487
$50$	647.285	1.83080
$51$	108.237	0.297180
$52$	−167.235	−0.445987
$53$	112.130	0.290607	0.145304	−	0.989387i	$-0.453584\pi$
$53$	112.130	0.290607	0.145304	+	0.989387i	$0.453584\pi$
$54$	−54.0000	−0.136083
$55$	734.634	1.80105
$56$	276.189	0.659059
$57$	0	0
$58$	−386.254	−0.874442
$59$	−492.668	−1.08712	−0.543558	−	0.839371i	$-0.682923\pi$
$59$	−492.668	−1.08712	−0.543558	+	0.839371i	$0.682923\pi$
$60$	−254.174	−0.546896
$61$	258.448	0.542474	0.271237	−	0.962513i	$-0.412567\pi$
$61$	258.448	0.542474	0.271237	+	0.962513i	$0.412567\pi$
$62$	574.287	1.17636
$63$	310.713	0.621367
$64$	64.0000	0.125000
$65$	−885.559	−1.68985
$66$	−208.100	−0.388111
$67$	219.050	0.399420	0.199710	−	0.979855i	$-0.436000\pi$
$67$	219.050	0.399420	0.199710	+	0.979855i	$0.436000\pi$
$68$	−144.316	−0.257365
$69$	387.095	0.675373
$70$	1462.50	2.49718
$71$	24.1481	0.0403641	0.0201820	−	0.999796i	$-0.493575\pi$
$71$	24.1481	0.0403641	0.0201820	+	0.999796i	$0.493575\pi$
$72$	72.0000	0.117851
$73$	−174.347	−0.279531	−0.139766	−	0.990185i	$-0.544635\pi$
$73$	−174.347	−0.279531	−0.139766	+	0.990185i	$0.544635\pi$
$74$	765.212	1.20208
$75$	−970.928	−1.49484
$76$	0	0
$77$	1197.39	1.77215
$78$	250.853	0.364147
$79$	−923.646	−1.31542	−0.657711	−	0.753271i	$-0.728475\pi$
$79$	−923.646	−1.31542	−0.657711	+	0.753271i	$0.728475\pi$
$80$	338.899	0.473626
$81$	81.0000	0.111111
$82$	102.928	0.138616
$83$	−863.908	−1.14249	−0.571243	−	0.820781i	$-0.693538\pi$
$83$	−863.908	−1.14249	−0.571243	+	0.820781i	$0.693538\pi$
$84$	−414.284	−0.538120
$85$	−764.193	−0.975158
$86$	−437.788	−0.548929
$87$	579.381	0.713979
$88$	277.467	0.336114
$89$	−344.831	−0.410697	−0.205348	−	0.978689i	$-0.565833\pi$
$89$	−344.831	−0.410697	−0.205348	+	0.978689i	$0.565833\pi$
$90$	381.261	0.446539
$91$	−1443.39	−1.66273
$92$	−516.127	−0.584890
$93$	−861.431	−0.960497
$94$	301.076	0.330357
$95$	0	0
$96$	−96.0000	−0.102062
$97$	−270.549	−0.283197	−0.141598	−	0.989924i	$-0.545224\pi$
$97$	−270.549	−0.283197	−0.141598	+	0.989924i	$0.545224\pi$
$98$	1697.76	1.75000
$99$	312.150	0.316891
$100$	1294.57	1.29457
$101$	1257.53	1.23890	0.619448	−	0.785037i	$-0.287356\pi$
$101$	1257.53	1.23890	0.619448	+	0.785037i	$0.287356\pi$
$102$	216.473	0.210138
$103$	1028.31	0.983717	0.491858	−	0.870675i	$-0.336318\pi$
$103$	1028.31	0.983717	0.491858	+	0.870675i	$0.336318\pi$
$104$	−334.470	−0.315361
$105$	−2193.75	−2.03894
$106$	224.259	0.205490
$107$	−641.954	−0.580000	−0.290000	−	0.957027i	$-0.593655\pi$
$107$	−641.954	−0.580000	−0.290000	+	0.957027i	$0.593655\pi$
$108$	−108.000	−0.0962250
$109$	−1271.31	−1.11715	−0.558574	−	0.829455i	$-0.688651\pi$
$109$	−1271.31	−1.11715	−0.558574	+	0.829455i	$0.688651\pi$
$110$	1469.27	1.27354
$111$	−1147.82	−0.981495
$112$	552.378	0.466025
$113$	1206.48	1.00439	0.502196	−	0.864754i	$-0.332525\pi$
$113$	1206.48	1.00439	0.502196	+	0.864754i	$0.332525\pi$
$114$	0	0
$115$	−2733.04	−2.21615
$116$	−772.508	−0.618324
$117$	−376.279	−0.297325
$118$	−985.335	−0.768707
$119$	−1245.57	−0.959509
$120$	−508.348	−0.386714
$121$	−128.067	−0.0962186
$122$	516.897	0.383587
$123$	−154.392	−0.113180
$124$	1148.57	0.831815
$125$	4207.48	3.01063
$126$	621.426	0.439373
$127$	552.630	0.386126	0.193063	−	0.981186i	$-0.438158\pi$
$127$	552.630	0.386126	0.193063	+	0.981186i	$0.438158\pi$
$128$	128.000	0.0883883
$129$	656.682	0.448198
$130$	−1771.12	−1.19490
$131$	−1111.72	−0.741460	−0.370730	−	0.928741i	$-0.620892\pi$
$131$	−1111.72	−0.741460	−0.370730	+	0.928741i	$0.620892\pi$
$132$	−416.200	−0.274436
$133$	0	0
$134$	438.099	0.282433
$135$	−571.892	−0.364597
$136$	−288.631	−0.181985
$137$	−774.157	−0.482779	−0.241389	−	0.970428i	$-0.577603\pi$
$137$	−774.157	−0.482779	−0.241389	+	0.970428i	$0.577603\pi$
$138$	774.190	0.477561
$139$	2013.65	1.22874	0.614372	−	0.789016i	$-0.289409\pi$
$139$	2013.65	1.22874	0.614372	+	0.789016i	$0.289409\pi$
$140$	2925.01	1.76577
$141$	−451.614	−0.269736
$142$	48.2962	0.0285417
$143$	−1450.07	−0.847977
$144$	144.000	0.0833333
$145$	−4090.66	−2.34283
$146$	−348.694	−0.197659
$147$	−2546.65	−1.42887
$148$	1530.42	0.850000
$149$	187.063	0.102851	0.0514255	−	0.998677i	$-0.483624\pi$
$149$	187.063	0.102851	0.0514255	+	0.998677i	$0.483624\pi$
$150$	−1941.86	−1.05701
$151$	950.566	0.512291	0.256146	−	0.966638i	$-0.417547\pi$
$151$	950.566	0.512291	0.256146	+	0.966638i	$0.417547\pi$
$152$	0	0
$153$	−324.710	−0.171577
$154$	2394.79	1.25310
$155$	6082.04	3.15175
$156$	501.705	0.257491
$157$	414.205	0.210555	0.105278	−	0.994443i	$-0.466427\pi$
$157$	414.205	0.210555	0.105278	+	0.994443i	$0.466427\pi$
$158$	−1847.29	−0.930144
$159$	−336.389	−0.167782
$160$	677.798	0.334904
$161$	−4454.64	−2.18059
$162$	162.000	0.0785674
$163$	−2660.87	−1.27862	−0.639312	−	0.768948i	$-0.720781\pi$
$163$	−2660.87	−1.27862	−0.639312	+	0.768948i	$0.720781\pi$
$164$	205.857	0.0980165
$165$	−2203.90	−1.03984
$166$	−1727.82	−0.807859
$167$	585.527	0.271314	0.135657	−	0.990756i	$-0.456685\pi$
$167$	585.527	0.271314	0.135657	+	0.990756i	$0.456685\pi$
$168$	−828.567	−0.380508
$169$	−449.026	−0.204381
$170$	−1528.39	−0.689541
$171$	0	0
$172$	−875.575	−0.388151
$173$	−354.185	−0.155654	−0.0778271	−	0.996967i	$-0.524798\pi$
$173$	−354.185	−0.155654	−0.0778271	+	0.996967i	$0.524798\pi$
$174$	1158.76	0.504860
$175$	11173.3	4.82642
$176$	554.933	0.237669
$177$	1478.00	0.627647
$178$	−689.662	−0.290406
$179$	1499.66	0.626202	0.313101	−	0.949720i	$-0.398632\pi$
$179$	1499.66	0.626202	0.313101	+	0.949720i	$0.398632\pi$
$180$	762.523	0.315750
$181$	−1532.43	−0.629308	−0.314654	−	0.949206i	$-0.601889\pi$
$181$	−1532.43	−0.629308	−0.314654	+	0.949206i	$0.601889\pi$
$182$	−2886.78	−1.17573
$183$	−775.345	−0.313198
$184$	−1032.25	−0.413580
$185$	8104.04	3.22065
$186$	−1722.86	−0.679174
$187$	−1251.34	−0.489341
$188$	602.151	0.233598
$189$	−932.138	−0.358746
$190$	0	0
$191$	−1891.45	−0.716548	−0.358274	−	0.933617i	$-0.616635\pi$
$191$	−1891.45	−0.716548	−0.358274	+	0.933617i	$0.616635\pi$
$192$	−192.000	−0.0721688
$193$	3056.33	1.13989	0.569947	−	0.821682i	$-0.306964\pi$
$193$	3056.33	1.13989	0.569947	+	0.821682i	$0.306964\pi$
$194$	−541.098	−0.200250
$195$	2656.68	0.975634
$196$	3395.53	1.23744
$197$	−2841.55	−1.02767	−0.513837	−	0.857888i	$-0.671776\pi$
$197$	−2841.55	−1.02767	−0.513837	+	0.857888i	$0.671776\pi$
$198$	624.300	0.224076
$199$	981.999	0.349809	0.174905	−	0.984585i	$-0.444038\pi$
$199$	981.999	0.349809	0.174905	+	0.984585i	$0.444038\pi$
$200$	2589.14	0.915399
$201$	−657.149	−0.230605
$202$	2515.05	0.876032
$203$	−6667.45	−2.30524
$204$	432.947	0.148590
$205$	1090.07	0.371385
$206$	2056.63	0.695593
$207$	−1161.28	−0.389927
$208$	−668.940	−0.222994
$209$	0	0
$210$	−4387.51	−1.44175
$211$	1134.18	0.370049	0.185024	−	0.982734i	$-0.440764\pi$
$211$	1134.18	0.370049	0.185024	+	0.982734i	$0.440764\pi$
$212$	448.518	0.145304
$213$	−72.4442	−0.0233042
$214$	−1283.91	−0.410122
$215$	−4636.43	−1.47071
$216$	−216.000	−0.0680414
$217$	9913.25	3.10118
$218$	−2542.62	−0.789943
$219$	523.042	0.161388
$220$	2938.54	0.900527
$221$	1508.41	0.459126
$222$	−2295.63	−0.694022
$223$	1653.58	0.496556	0.248278	−	0.968689i	$-0.420135\pi$
$223$	1653.58	0.496556	0.248278	+	0.968689i	$0.420135\pi$
$224$	1104.76	0.329530
$225$	2912.78	0.863047
$226$	2412.97	0.710213
$227$	440.641	0.128839	0.0644193	−	0.997923i	$-0.479480\pi$
$227$	440.641	0.128839	0.0644193	+	0.997923i	$0.479480\pi$
$228$	0	0
$229$	−5613.51	−1.61987	−0.809937	−	0.586516i	$-0.800499\pi$
$229$	−5613.51	−1.61987	−0.809937	+	0.586516i	$0.800499\pi$
$230$	−5466.09	−1.56706
$231$	−3592.18	−1.02315
$232$	−1545.02	−0.437221
$233$	−46.3696	−0.0130376	−0.00651882	−	0.999979i	$-0.502075\pi$
$233$	−46.3696	−0.0130376	−0.00651882	+	0.999979i	$0.502075\pi$
$234$	−752.558	−0.210240
$235$	3188.57	0.885104
$236$	−1970.67	−0.543558
$237$	2770.94	0.759459
$238$	−2491.15	−0.678476
$239$	4374.02	1.18381	0.591907	−	0.806006i	$-0.298375\pi$
$239$	4374.02	1.18381	0.591907	+	0.806006i	$0.298375\pi$
$240$	−1016.70	−0.273448
$241$	1895.70	0.506692	0.253346	−	0.967376i	$-0.418469\pi$
$241$	1895.70	0.506692	0.253346	+	0.967376i	$0.418469\pi$
$242$	−256.134	−0.0680369
$243$	−243.000	−0.0641500
$244$	1033.79	0.271237
$245$	17980.3	4.68866
$246$	−308.785	−0.0800301
$247$	0	0
$248$	2297.15	0.588182
$249$	2591.73	0.659614
$250$	8414.97	2.12884
$251$	654.298	0.164537	0.0822687	−	0.996610i	$-0.473783\pi$
$251$	654.298	0.164537	0.0822687	+	0.996610i	$0.473783\pi$
$252$	1242.85	0.310684
$253$	−4475.25	−1.11208
$254$	1105.26	0.273032
$255$	2292.58	0.563008
$256$	256.000	0.0625000
$257$	−5768.71	−1.40016	−0.700082	−	0.714063i	$-0.746853\pi$
$257$	−5768.71	−1.40016	−0.700082	+	0.714063i	$0.746853\pi$
$258$	1313.36	0.316924
$259$	13208.9	3.16897
$260$	−3542.24	−0.844924
$261$	−1738.14	−0.412216
$262$	−2223.44	−0.524291
$263$	−7977.73	−1.87045	−0.935225	−	0.354055i	$-0.884802\pi$
$263$	−7977.73	−1.87045	−0.935225	+	0.354055i	$0.884802\pi$
$264$	−832.400	−0.194056
$265$	2375.04	0.550556
$266$	0	0
$267$	1034.49	0.237116
$268$	876.198	0.199710
$269$	2442.85	0.553692	0.276846	−	0.960914i	$-0.410711\pi$
$269$	2442.85	0.553692	0.276846	+	0.960914i	$0.410711\pi$
$270$	−1143.78	−0.257809
$271$	6446.56	1.44502	0.722511	−	0.691360i	$-0.242988\pi$
$271$	6446.56	1.44502	0.722511	+	0.691360i	$0.242988\pi$
$272$	−577.262	−0.128683
$273$	4330.17	0.959978
$274$	−1548.31	−0.341376
$275$	11225.0	2.46143
$276$	1548.38	0.337687
$277$	−4104.48	−0.890305	−0.445152	−	0.895455i	$-0.646850\pi$
$277$	−4104.48	−0.890305	−0.445152	+	0.895455i	$0.646850\pi$
$278$	4027.30	0.868854
$279$	2584.29	0.554543
$280$	5850.01	1.24859
$281$	−7860.15	−1.66867	−0.834337	−	0.551254i	$-0.814149\pi$
$281$	−7860.15	−1.66867	−0.834337	+	0.551254i	$0.814149\pi$
$282$	−903.227	−0.190732
$283$	−6598.64	−1.38604	−0.693019	−	0.720920i	$-0.743720\pi$
$283$	−6598.64	−1.38604	−0.693019	+	0.720920i	$0.743720\pi$
$284$	96.5923	0.0201820
$285$	0	0
$286$	−2900.13	−0.599610
$287$	1776.73	0.365425
$288$	288.000	0.0589256
$289$	−3611.31	−0.735053
$290$	−8181.32	−1.65663
$291$	811.646	0.163504
$292$	−697.389	−0.139766
$293$	−6722.35	−1.34035	−0.670177	−	0.742201i	$-0.733782\pi$
$293$	−6722.35	−1.34035	−0.670177	+	0.742201i	$0.733782\pi$
$294$	−5093.29	−1.01036
$295$	−10435.3	−2.05955
$296$	3060.85	0.601041
$297$	−936.450	−0.182957
$298$	374.126	0.0727266
$299$	5394.65	1.04341
$300$	−3883.71	−0.747420
$301$	−7557.01	−1.44711
$302$	1901.13	0.362245
$303$	−3772.58	−0.715277
$304$	0	0
$305$	5474.24	1.02772
$306$	−649.420	−0.121323
$307$	−6383.41	−1.18671	−0.593355	−	0.804941i	$-0.702197\pi$
$307$	−6383.41	−1.18671	−0.593355	+	0.804941i	$0.702197\pi$
$308$	4789.58	0.886077
$309$	−3084.94	−0.567949
$310$	12164.1	2.22862
$311$	4181.24	0.762368	0.381184	−	0.924499i	$-0.375516\pi$
$311$	4181.24	0.762368	0.381184	+	0.924499i	$0.375516\pi$
$312$	1003.41	0.182074
$313$	−6383.70	−1.15280	−0.576402	−	0.817166i	$-0.695544\pi$
$313$	−6383.70	−1.15280	−0.576402	+	0.817166i	$0.695544\pi$
$314$	828.410	0.148885
$315$	6581.26	1.17718
$316$	−3694.58	−0.657711
$317$	2761.88	0.489347	0.244673	−	0.969606i	$-0.421319\pi$
$317$	2761.88	0.489347	0.244673	+	0.969606i	$0.421319\pi$
$318$	−672.777	−0.118640
$319$	−6698.29	−1.17565
$320$	1355.60	0.236813
$321$	1925.86	0.334863
$322$	−8909.28	−1.54191
$323$	0	0
$324$	324.000	0.0555556
$325$	−13531.1	−2.30945
$326$	−5321.75	−0.904124
$327$	3813.92	0.644986
$328$	411.713	0.0693081
$329$	5197.12	0.870900
$330$	−4407.80	−0.735278
$331$	4283.66	0.711333	0.355667	−	0.934613i	$-0.384254\pi$
$331$	4283.66	0.711333	0.355667	+	0.934613i	$0.384254\pi$
$332$	−3455.63	−0.571243
$333$	3443.45	0.566667
$334$	1171.05	0.191848
$335$	4639.73	0.756703
$336$	−1657.13	−0.269060
$337$	6104.46	0.986739	0.493369	−	0.869820i	$-0.335765\pi$
$337$	6104.46	0.986739	0.493369	+	0.869820i	$0.335765\pi$
$338$	−898.052	−0.144519
$339$	−3619.45	−0.579886
$340$	−3056.77	−0.487579
$341$	9959.10	1.58157
$342$	0	0
$343$	17464.9	2.74931
$344$	−1751.15	−0.274464
$345$	8199.13	1.27950
$346$	−708.370	−0.110064
$347$	9287.61	1.43684	0.718422	−	0.695608i	$-0.244865\pi$
$347$	9287.61	1.43684	0.718422	+	0.695608i	$0.244865\pi$
$348$	2317.53	0.356990
$349$	−679.740	−0.104257	−0.0521284	−	0.998640i	$-0.516601\pi$
$349$	−679.740	−0.104257	−0.0521284	+	0.998640i	$0.516601\pi$
$350$	22346.6	3.41279
$351$	1128.84	0.171661
$352$	1109.87	0.168057
$353$	12267.8	1.84971	0.924854	−	0.380323i	$-0.124187\pi$
$353$	12267.8	1.84971	0.924854	+	0.380323i	$0.124187\pi$
$354$	2956.01	0.443813
$355$	511.485	0.0764699
$356$	−1379.32	−0.205348
$357$	3736.72	0.553973
$358$	2999.33	0.442791
$359$	3492.78	0.513487	0.256744	−	0.966480i	$-0.417350\pi$
$359$	3492.78	0.513487	0.256744	+	0.966480i	$0.417350\pi$
$360$	1525.05	0.223269
$361$	0	0
$362$	−3064.86	−0.444988
$363$	384.201	0.0555519
$364$	−5773.57	−0.831366
$365$	−3692.88	−0.529573
$366$	−1550.69	−0.221464
$367$	−6514.48	−0.926576	−0.463288	−	0.886208i	$-0.653330\pi$
$367$	−6514.48	−0.926576	−0.463288	+	0.886208i	$0.653330\pi$
$368$	−2064.51	−0.292445
$369$	463.177	0.0653443
$370$	16208.1	2.27735
$371$	3871.12	0.541721
$372$	−3445.72	−0.480249
$373$	−2859.21	−0.396901	−0.198451	−	0.980111i	$-0.563591\pi$
$373$	−2859.21	−0.396901	−0.198451	+	0.980111i	$0.563591\pi$
$374$	−2502.67	−0.346016
$375$	−12622.5	−1.73819
$376$	1204.30	0.165179
$377$	8074.41	1.10306
$378$	−1864.28	−0.253672
$379$	−6359.22	−0.861877	−0.430938	−	0.902381i	$-0.641817\pi$
$379$	−6359.22	−0.861877	−0.430938	+	0.902381i	$0.641817\pi$
$380$	0	0
$381$	−1657.89	−0.222930
$382$	−3782.90	−0.506676
$383$	−4171.56	−0.556544	−0.278272	−	0.960502i	$-0.589762\pi$
$383$	−4171.56	−0.556544	−0.278272	+	0.960502i	$0.589762\pi$
$384$	−384.000	−0.0510310
$385$	25362.2	3.35735
$386$	6112.66	0.806026
$387$	−1970.04	−0.258767
$388$	−1082.20	−0.141598
$389$	−10065.7	−1.31196	−0.655978	−	0.754780i	$-0.727744\pi$
$389$	−10065.7	−1.31196	−0.655978	+	0.754780i	$0.727744\pi$
$390$	5313.36	0.689878
$391$	4655.32	0.602121
$392$	6791.05	0.875000
$393$	3335.15	0.428082
$394$	−5683.09	−0.726675
$395$	−19563.9	−2.49207
$396$	1248.60	0.158446
$397$	5758.38	0.727972	0.363986	−	0.931404i	$-0.381416\pi$
$397$	5758.38	0.727972	0.363986	+	0.931404i	$0.381416\pi$
$398$	1964.00	0.247352
$399$	0	0
$400$	5178.28	0.647285
$401$	11947.1	1.48780	0.743900	−	0.668291i	$-0.232974\pi$
$401$	11947.1	1.48780	0.743900	+	0.668291i	$0.232974\pi$
$402$	−1314.30	−0.163063
$403$	−12005.1	−1.48392
$404$	5030.11	0.619448
$405$	1715.68	0.210500
$406$	−13334.9	−1.63005
$407$	13270.0	1.61615
$408$	865.893	0.105069
$409$	−2827.19	−0.341798	−0.170899	−	0.985289i	$-0.554667\pi$
$409$	−2827.19	−0.341798	−0.170899	+	0.985289i	$0.554667\pi$
$410$	2180.14	0.262609
$411$	2322.47	0.278732
$412$	4113.26	0.491858
$413$	−17008.7	−2.02650
$414$	−2322.57	−0.275720
$415$	−18298.6	−2.16444
$416$	−1337.88	−0.157680
$417$	−6040.95	−0.709416
$418$	0	0
$419$	2678.13	0.312256	0.156128	−	0.987737i	$-0.450099\pi$
$419$	2678.13	0.312256	0.156128	+	0.987737i	$0.450099\pi$
$420$	−8775.02	−1.01947
$421$	15953.5	1.84686	0.923429	−	0.383769i	$-0.125374\pi$
$421$	15953.5	1.84686	0.923429	+	0.383769i	$0.125374\pi$
$422$	2268.36	0.261664
$423$	1354.84	0.155732
$424$	897.036	0.102745
$425$	−11676.7	−1.33271
$426$	−144.888	−0.0164786
$427$	8922.58	1.01123
$428$	−2567.82	−0.290000
$429$	4350.20	0.489580
$430$	−9272.86	−1.03995
$431$	5342.63	0.597090	0.298545	−	0.954396i	$-0.403499\pi$
$431$	5342.63	0.597090	0.298545	+	0.954396i	$0.403499\pi$
$432$	−432.000	−0.0481125
$433$	541.683	0.0601192	0.0300596	−	0.999548i	$-0.490430\pi$
$433$	541.683	0.0601192	0.0300596	+	0.999548i	$0.490430\pi$
$434$	19826.5	2.19286
$435$	12272.0	1.35264
$436$	−5085.23	−0.558574
$437$	0	0
$438$	1046.08	0.114118
$439$	−1746.85	−0.189915	−0.0949574	−	0.995481i	$-0.530271\pi$
$439$	−1746.85	−0.189915	−0.0949574	+	0.995481i	$0.530271\pi$
$440$	5877.07	0.636769
$441$	7639.94	0.824958
$442$	3016.83	0.324651
$443$	16140.4	1.73105	0.865526	−	0.500865i	$-0.166985\pi$
$443$	16140.4	1.73105	0.865526	+	0.500865i	$0.166985\pi$
$444$	−4591.27	−0.490748
$445$	−7303.93	−0.778066
$446$	3307.16	0.351118
$447$	−561.189	−0.0593810
$448$	2209.51	0.233013
$449$	−1982.14	−0.208336	−0.104168	−	0.994560i	$-0.533218\pi$
$449$	−1982.14	−0.208336	−0.104168	+	0.994560i	$0.533218\pi$
$450$	5825.57	0.610266
$451$	1784.95	0.186363
$452$	4825.93	0.502196
$453$	−2851.70	−0.295772
$454$	881.282	0.0911026
$455$	−30572.7	−3.15005
$456$	0	0
$457$	−4784.58	−0.489744	−0.244872	−	0.969555i	$-0.578746\pi$
$457$	−4784.58	−0.489744	−0.244872	+	0.969555i	$0.578746\pi$
$458$	−11227.0	−1.14542
$459$	974.130	0.0990599
$460$	−10932.2	−1.10808
$461$	842.481	0.0851155	0.0425578	−	0.999094i	$-0.486449\pi$
$461$	842.481	0.0851155	0.0425578	+	0.999094i	$0.486449\pi$
$462$	−7184.37	−0.723478
$463$	−9249.77	−0.928452	−0.464226	−	0.885717i	$-0.653667\pi$
$463$	−9249.77	−0.928452	−0.464226	+	0.885717i	$0.653667\pi$
$464$	−3090.03	−0.309162
$465$	−18246.1	−1.81966
$466$	−92.7391	−0.00921901
$467$	−18451.1	−1.82830	−0.914151	−	0.405375i	$-0.867141\pi$
$467$	−18451.1	−1.82830	−0.914151	+	0.405375i	$0.867141\pi$
$468$	−1505.12	−0.148662
$469$	7562.39	0.744560
$470$	6377.14	0.625863
$471$	−1242.62	−0.121564
$472$	−3941.34	−0.384354
$473$	−7591.97	−0.738011
$474$	5541.88	0.537019
$475$	0	0
$476$	−4982.30	−0.479755
$477$	1009.17	0.0968690
$478$	8748.04	0.837084
$479$	4883.57	0.465837	0.232919	−	0.972496i	$-0.425172\pi$
$479$	4883.57	0.465837	0.232919	+	0.972496i	$0.425172\pi$
$480$	−2033.39	−0.193357
$481$	−15996.3	−1.51636
$482$	3791.40	0.358285
$483$	13363.9	1.25896
$484$	−512.268	−0.0481093
$485$	−5730.54	−0.536517
$486$	−486.000	−0.0453609
$487$	1306.77	0.121592	0.0607961	−	0.998150i	$-0.480636\pi$
$487$	1306.77	0.121592	0.0607961	+	0.998150i	$0.480636\pi$
$488$	2067.59	0.191794
$489$	7982.62	0.738214
$490$	35960.6	3.31538
$491$	−8381.44	−0.770365	−0.385182	−	0.922840i	$-0.625861\pi$
$491$	−8381.44	−0.770365	−0.385182	+	0.922840i	$0.625861\pi$
$492$	−617.570	−0.0565898
$493$	6967.81	0.636540
$494$	0	0
$495$	6611.70	0.600352
$496$	4594.30	0.415908
$497$	833.680	0.0752427
$498$	5183.45	0.466418
$499$	12658.2	1.13559	0.567794	−	0.823171i	$-0.307797\pi$
$499$	12658.2	1.13559	0.567794	+	0.823171i	$0.307797\pi$
$500$	16829.9	1.50532
$501$	−1756.58	−0.156643
$502$	1308.60	0.116346
$503$	11010.6	0.976018	0.488009	−	0.872839i	$-0.337723\pi$
$503$	11010.6	0.976018	0.488009	+	0.872839i	$0.337723\pi$
$504$	2485.70	0.219686
$505$	26635.9	2.34709
$506$	−8950.49	−0.786359
$507$	1347.08	0.118000
$508$	2210.52	0.193063
$509$	−11385.1	−0.991427	−0.495714	−	0.868486i	$-0.665093\pi$
$509$	−11385.1	−0.991427	−0.495714	+	0.868486i	$0.665093\pi$
$510$	4585.16	0.398106
$511$	−6019.10	−0.521075
$512$	512.000	0.0441942
$513$	0	0
$514$	−11537.4	−0.990065
$515$	21780.9	1.86365
$516$	2626.73	0.224099
$517$	5221.15	0.444151
$518$	26417.9	2.24080
$519$	1062.55	0.0898670
$520$	−7084.48	−0.597452
$521$	−16050.6	−1.34970	−0.674848	−	0.737957i	$-0.735791\pi$
$521$	−16050.6	−1.34970	−0.674848	+	0.737957i	$0.735791\pi$
$522$	−3476.29	−0.291481
$523$	13911.4	1.16310	0.581550	−	0.813511i	$-0.302446\pi$
$523$	13911.4	1.16310	0.581550	+	0.813511i	$0.302446\pi$
$524$	−4446.87	−0.370730
$525$	−33520.0	−2.78653
$526$	−15955.5	−1.32261
$527$	−10359.8	−0.856321
$528$	−1664.80	−0.137218
$529$	4482.16	0.368387
$530$	4750.07	0.389302
$531$	−4434.01	−0.362372
$532$	0	0
$533$	−2151.65	−0.174856
$534$	2068.99	0.167666
$535$	−13597.3	−1.09881
$536$	1752.40	0.141216
$537$	−4498.99	−0.361538
$538$	4885.70	0.391519
$539$	29442.0	2.35280
$540$	−2287.57	−0.182299
$541$	3548.96	0.282037	0.141018	−	0.990007i	$-0.454962\pi$
$541$	3548.96	0.282037	0.141018	+	0.990007i	$0.454962\pi$
$542$	12893.1	1.02178
$543$	4597.30	0.363331
$544$	−1154.52	−0.0909923
$545$	−26927.8	−2.11644
$546$	8660.35	0.678807
$547$	10609.9	0.829333	0.414667	−	0.909973i	$-0.363898\pi$
$547$	10609.9	0.829333	0.414667	+	0.909973i	$0.363898\pi$
$548$	−3096.63	−0.241389
$549$	2326.04	0.180825
$550$	22450.0	1.74049
$551$	0	0
$552$	3096.76	0.238780
$553$	−31887.6	−2.45208
$554$	−8208.96	−0.629540
$555$	−24312.1	−1.85945
$556$	8054.60	0.614372
$557$	634.039	0.0482318	0.0241159	−	0.999709i	$-0.492323\pi$
$557$	634.039	0.0482318	0.0241159	+	0.999709i	$0.492323\pi$
$558$	5168.59	0.392121
$559$	9151.69	0.692442
$560$	11700.0	0.882886
$561$	3754.01	0.282521
$562$	−15720.3	−1.17993
$563$	881.704	0.0660025	0.0330012	−	0.999455i	$-0.489493\pi$
$563$	881.704	0.0660025	0.0330012	+	0.999455i	$0.489493\pi$
$564$	−1806.45	−0.134868
$565$	25554.7	1.90282
$566$	−13197.3	−0.980077
$567$	2796.41	0.207122
$568$	193.185	0.0142709
$569$	−18995.4	−1.39952	−0.699761	−	0.714377i	$-0.746710\pi$
$569$	−18995.4	−1.39952	−0.699761	+	0.714377i	$0.746710\pi$
$570$	0	0
$571$	1360.33	0.0996989	0.0498495	−	0.998757i	$-0.484126\pi$
$571$	1360.33	0.0996989	0.0498495	+	0.998757i	$0.484126\pi$
$572$	−5800.27	−0.423989
$573$	5674.35	0.413699
$574$	3553.46	0.258395
$575$	−41760.1	−3.02873
$576$	576.000	0.0416667
$577$	27155.9	1.95929	0.979647	−	0.200727i	$-0.0643304\pi$
$577$	27155.9	1.95929	0.979647	+	0.200727i	$0.0643304\pi$
$578$	−7222.63	−0.519761
$579$	−9168.99	−0.658118
$580$	−16362.6	−1.17142
$581$	−29825.3	−2.12971
$582$	1623.29	0.115615
$583$	3889.02	0.276273
$584$	−1394.78	−0.0988293
$585$	−7970.03	−0.563283
$586$	−13444.7	−0.947773
$587$	21756.5	1.52979	0.764894	−	0.644156i	$-0.222791\pi$
$587$	21756.5	1.52979	0.764894	+	0.644156i	$0.222791\pi$
$588$	−10186.6	−0.714435
$589$	0	0
$590$	−20870.6	−1.45632
$591$	8524.64	0.593328
$592$	6121.69	0.425000
$593$	−15723.8	−1.08887	−0.544433	−	0.838805i	$-0.683255\pi$
$593$	−15723.8	−1.08887	−0.544433	+	0.838805i	$0.683255\pi$
$594$	−1872.90	−0.129370
$595$	−26382.7	−1.81779
$596$	748.252	0.0514255
$597$	−2946.00	−0.201962
$598$	10789.3	0.737805
$599$	−15741.9	−1.07378	−0.536891	−	0.843651i	$-0.680401\pi$
$599$	−15741.9	−1.07378	−0.536891	+	0.843651i	$0.680401\pi$
$600$	−7767.42	−0.528506
$601$	−19912.1	−1.35147	−0.675733	−	0.737147i	$-0.736173\pi$
$601$	−19912.1	−1.35147	−0.675733	+	0.737147i	$0.736173\pi$
$602$	−15114.0	−1.02326
$603$	1971.45	0.133140
$604$	3802.27	0.256146
$605$	−2712.61	−0.182286
$606$	−7545.16	−0.505777
$607$	9686.54	0.647718	0.323859	−	0.946105i	$-0.395020\pi$
$607$	9686.54	0.647718	0.323859	+	0.946105i	$0.395020\pi$
$608$	0	0
$609$	20002.4	1.33093
$610$	10948.5	0.726707
$611$	−6293.80	−0.416727
$612$	−1298.84	−0.0857884
$613$	−19025.8	−1.25358	−0.626790	−	0.779188i	$-0.715632\pi$
$613$	−19025.8	−1.25358	−0.626790	+	0.779188i	$0.715632\pi$
$614$	−12766.8	−0.839131
$615$	−3270.21	−0.214419
$616$	9579.16	0.626551
$617$	−15885.5	−1.03651	−0.518256	−	0.855226i	$-0.673419\pi$
$617$	−15885.5	−1.03651	−0.518256	+	0.855226i	$0.673419\pi$
$618$	−6169.89	−0.401601
$619$	−19478.7	−1.26480	−0.632402	−	0.774640i	$-0.717931\pi$
$619$	−19478.7	−1.26480	−0.632402	+	0.774640i	$0.717931\pi$
$620$	24328.2	1.57588
$621$	3483.85	0.225124
$622$	8362.48	0.539075
$623$	−11904.8	−0.765580
$624$	2006.82	0.128745
$625$	48664.2	3.11451
$626$	−12767.4	−0.815156
$627$	0	0
$628$	1656.82	0.105278
$629$	−13804.0	−0.875041
$630$	13162.5	0.832393
$631$	25255.5	1.59335	0.796676	−	0.604407i	$-0.206590\pi$
$631$	25255.5	1.59335	0.796676	+	0.604407i	$0.206590\pi$
$632$	−7389.17	−0.465072
$633$	−3402.55	−0.213648
$634$	5523.77	0.346020
$635$	11705.4	0.731517
$636$	−1345.55	−0.0838910
$637$	−35490.7	−2.20752
$638$	−13396.6	−0.831310
$639$	217.333	0.0134547
$640$	2711.19	0.167452
$641$	14775.8	0.910467	0.455233	−	0.890372i	$-0.349556\pi$
$641$	14775.8	0.910467	0.455233	+	0.890372i	$0.349556\pi$
$642$	3851.72	0.236784
$643$	−10786.2	−0.661535	−0.330767	−	0.943712i	$-0.607308\pi$
$643$	−10786.2	−0.661535	−0.330767	+	0.943712i	$0.607308\pi$
$644$	−17818.6	−1.09029
$645$	13909.3	0.849113
$646$	0	0
$647$	−28549.7	−1.73478	−0.867392	−	0.497626i	$-0.834205\pi$
$647$	−28549.7	−1.73478	−0.867392	+	0.497626i	$0.834205\pi$
$648$	648.000	0.0392837
$649$	−17087.4	−1.03349
$650$	−27062.2	−1.63303
$651$	−29739.7	−1.79046
$652$	−10643.5	−0.639312
$653$	28055.8	1.68133	0.840665	−	0.541555i	$-0.182164\pi$
$653$	28055.8	1.68133	0.840665	+	0.541555i	$0.182164\pi$
$654$	7627.85	0.456074
$655$	−23547.5	−1.40470
$656$	823.426	0.0490082
$657$	−1569.12	−0.0931772
$658$	10394.2	0.615820
$659$	14442.2	0.853701	0.426851	−	0.904322i	$-0.359623\pi$
$659$	14442.2	0.853701	0.426851	+	0.904322i	$0.359623\pi$
$660$	−8815.61	−0.519920
$661$	−33733.1	−1.98497	−0.992486	−	0.122362i	$-0.960953\pi$
$661$	−33733.1	−1.98497	−0.992486	+	0.122362i	$0.960953\pi$
$662$	8567.32	0.502989
$663$	−4525.24	−0.265077
$664$	−6911.27	−0.403929
$665$	0	0
$666$	6886.90	0.400694
$667$	24919.5	1.44661
$668$	2342.11	0.135657
$669$	−4960.74	−0.286687
$670$	9279.46	0.535070
$671$	8963.85	0.515716
$672$	−3314.27	−0.190254
$673$	−22248.5	−1.27432	−0.637160	−	0.770732i	$-0.719891\pi$
$673$	−22248.5	−1.27432	−0.637160	+	0.770732i	$0.719891\pi$
$674$	12208.9	0.697730
$675$	−8738.35	−0.498280
$676$	−1796.10	−0.102191
$677$	−15209.6	−0.863443	−0.431722	−	0.902007i	$-0.642094\pi$
$677$	−15209.6	−0.863443	−0.431722	+	0.902007i	$0.642094\pi$
$678$	−7238.90	−0.410042
$679$	−9340.33	−0.527907
$680$	−6113.55	−0.344770
$681$	−1321.92	−0.0743850
$682$	19918.2	1.11834
$683$	−7986.78	−0.447446	−0.223723	−	0.974653i	$-0.571821\pi$
$683$	−7986.78	−0.447446	−0.223723	+	0.974653i	$0.571821\pi$
$684$	0	0
$685$	−16397.6	−0.914625
$686$	34929.8	1.94406
$687$	16840.5	0.935235
$688$	−3502.30	−0.194076
$689$	−4688.00	−0.259214
$690$	16398.3	0.904740
$691$	−19578.9	−1.07788	−0.538940	−	0.842344i	$-0.681175\pi$
$691$	−19578.9	−1.07788	−0.538940	+	0.842344i	$0.681175\pi$
$692$	−1416.74	−0.0778271
$693$	10776.6	0.590718
$694$	18575.2	1.01600
$695$	42651.5	2.32786
$696$	4635.05	0.252430
$697$	−1856.77	−0.100904
$698$	−1359.48	−0.0737207
$699$	139.109	0.00752729
$700$	44693.3	2.41321
$701$	−9665.94	−0.520795	−0.260398	−	0.965501i	$-0.583854\pi$
$701$	−9665.94	−0.520795	−0.260398	+	0.965501i	$0.583854\pi$
$702$	2257.67	0.121382
$703$	0	0
$704$	2219.73	0.118834
$705$	−9565.71	−0.511015
$706$	24535.5	1.30794
$707$	43414.4	2.30943
$708$	5912.01	0.313824
$709$	534.810	0.0283289	0.0141645	−	0.999900i	$-0.495491\pi$
$709$	534.810	0.0283289	0.0141645	+	0.999900i	$0.495491\pi$
$710$	1022.97	0.0540724
$711$	−8312.82	−0.438474
$712$	−2758.65	−0.145203
$713$	−37050.6	−1.94608
$714$	7473.45	0.391718
$715$	−30714.1	−1.60649
$716$	5998.65	0.313101
$717$	−13122.1	−0.683476
$718$	6985.56	0.363090
$719$	26764.9	1.38826	0.694132	−	0.719848i	$-0.255788\pi$
$719$	26764.9	1.38826	0.694132	+	0.719848i	$0.255788\pi$
$720$	3050.09	0.157875
$721$	35501.2	1.83375
$722$	0	0
$723$	−5687.10	−0.292539
$724$	−6129.73	−0.314654
$725$	−62504.1	−3.20186
$726$	768.402	0.0392811
$727$	13054.0	0.665951	0.332975	−	0.942935i	$-0.391947\pi$
$727$	13054.0	0.665951	0.332975	+	0.942935i	$0.391947\pi$
$728$	−11547.1	−0.587864
$729$	729.000	0.0370370
$730$	−7385.76	−0.374465
$731$	7897.45	0.399586
$732$	−3101.38	−0.156599
$733$	−33017.1	−1.66373	−0.831865	−	0.554977i	$-0.812727\pi$
$733$	−33017.1	−1.66373	−0.831865	+	0.554977i	$0.812727\pi$
$734$	−13029.0	−0.655188
$735$	−53941.0	−2.70700
$736$	−4129.01	−0.206790
$737$	7597.37	0.379719
$738$	926.355	0.0462054
$739$	−975.126	−0.0485394	−0.0242697	−	0.999705i	$-0.507726\pi$
$739$	−975.126	−0.0485394	−0.0242697	+	0.999705i	$0.507726\pi$
$740$	32416.2	1.61033
$741$	0	0
$742$	7742.24	0.383055
$743$	−3207.16	−0.158357	−0.0791785	−	0.996860i	$-0.525230\pi$
$743$	−3207.16	−0.158357	−0.0791785	+	0.996860i	$0.525230\pi$
$744$	−6891.45	−0.339587
$745$	3962.21	0.194851
$746$	−5718.42	−0.280652
$747$	−7775.18	−0.380828
$748$	−5005.34	−0.244670
$749$	−22162.6	−1.08118
$750$	−25244.9	−1.22908
$751$	39154.0	1.90246	0.951231	−	0.308480i	$-0.0998205\pi$
$751$	39154.0	1.90246	0.951231	+	0.308480i	$0.0998205\pi$
$752$	2408.61	0.116799
$753$	−1962.89	−0.0949958
$754$	16148.8	0.779980
$755$	20134.1	0.970537
$756$	−3728.55	−0.179373
$757$	−3013.77	−0.144699	−0.0723497	−	0.997379i	$-0.523050\pi$
$757$	−3013.77	−0.144699	−0.0723497	+	0.997379i	$0.523050\pi$
$758$	−12718.4	−0.609439
$759$	13425.7	0.642060
$760$	0	0
$761$	−27726.1	−1.32072	−0.660361	−	0.750948i	$-0.729597\pi$
$761$	−27726.1	−1.32072	−0.660361	+	0.750948i	$0.729597\pi$
$762$	−3315.78	−0.157635
$763$	−43890.2	−2.08248
$764$	−7565.81	−0.358274
$765$	−6877.74	−0.325053
$766$	−8343.11	−0.393536
$767$	20597.8	0.969680
$768$	−768.000	−0.0360844
$769$	32472.2	1.52273	0.761363	−	0.648326i	$-0.224531\pi$
$769$	32472.2	1.52273	0.761363	+	0.648326i	$0.224531\pi$
$770$	50724.5	2.37400
$771$	17306.1	0.808385
$772$	12225.3	0.569947
$773$	4621.29	0.215028	0.107514	−	0.994204i	$-0.465711\pi$
$773$	4621.29	0.215028	0.107514	+	0.994204i	$0.465711\pi$
$774$	−3940.09	−0.182976
$775$	92931.9	4.30737
$776$	−2164.39	−0.100125
$777$	−39626.8	−1.82961
$778$	−20131.4	−0.927693
$779$	0	0
$780$	10626.7	0.487817
$781$	837.536	0.0383731
$782$	9310.63	0.425764
$783$	5214.43	0.237993
$784$	13582.1	0.618719
$785$	8773.35	0.398897
$786$	6670.31	0.302700
$787$	5114.51	0.231655	0.115828	−	0.993269i	$-0.463048\pi$
$787$	5114.51	0.231655	0.115828	+	0.993269i	$0.463048\pi$
$788$	−11366.2	−0.513837
$789$	23933.2	1.07990
$790$	−39127.8	−1.76216
$791$	41652.2	1.87229
$792$	2497.20	0.112038
$793$	−10805.4	−0.483873
$794$	11516.8	0.514754
$795$	−7125.11	−0.317864
$796$	3927.99	0.174905
$797$	856.981	0.0380876	0.0190438	−	0.999819i	$-0.493938\pi$
$797$	856.981	0.0380876	0.0190438	+	0.999819i	$0.493938\pi$
$798$	0	0
$799$	−5431.24	−0.240480
$800$	10356.6	0.457700
$801$	−3103.48	−0.136899
$802$	23894.1	1.05203
$803$	−6046.94	−0.265743
$804$	−2628.59	−0.115303
$805$	−94354.6	−4.13113
$806$	−24010.3	−1.04929
$807$	−7328.55	−0.319674
$808$	10060.2	0.438016
$809$	−6296.58	−0.273642	−0.136821	−	0.990596i	$-0.543688\pi$
$809$	−6296.58	−0.273642	−0.136821	+	0.990596i	$0.543688\pi$
$810$	3431.35	0.148846
$811$	31874.8	1.38012	0.690059	−	0.723753i	$-0.257585\pi$
$811$	31874.8	1.38012	0.690059	+	0.723753i	$0.257585\pi$
$812$	−26669.8	−1.15262
$813$	−19339.7	−0.834284
$814$	26540.1	1.14279
$815$	−56360.4	−2.42236
$816$	1731.79	0.0742949
$817$	0	0
$818$	−5654.37	−0.241688
$819$	−12990.5	−0.554244
$820$	4360.29	0.185692
$821$	−9483.00	−0.403117	−0.201558	−	0.979476i	$-0.564601\pi$
$821$	−9483.00	−0.403117	−0.201558	+	0.979476i	$0.564601\pi$
$822$	4644.94	0.197094
$823$	−34781.1	−1.47314	−0.736570	−	0.676362i	$-0.763556\pi$
$823$	−34781.1	−1.47314	−0.736570	+	0.676362i	$0.763556\pi$
$824$	8226.51	0.347796
$825$	−33675.0	−1.42111
$826$	−34017.4	−1.43295
$827$	−29048.4	−1.22142	−0.610708	−	0.791856i	$-0.709115\pi$
$827$	−29048.4	−1.22142	−0.610708	+	0.791856i	$0.709115\pi$
$828$	−4645.14	−0.194963
$829$	46420.8	1.94483	0.972413	−	0.233266i	$-0.0749413\pi$
$829$	46420.8	1.94483	0.972413	+	0.233266i	$0.0749413\pi$
$830$	−36597.2	−1.53049
$831$	12313.4	0.514018
$832$	−2675.76	−0.111497
$833$	−30626.7	−1.27389
$834$	−12081.9	−0.501633
$835$	12402.2	0.514005
$836$	0	0
$837$	−7752.88	−0.320166
$838$	5356.27	0.220799
$839$	42217.0	1.73718	0.868590	−	0.495532i	$-0.165027\pi$
$839$	42217.0	1.73718	0.868590	+	0.495532i	$0.165027\pi$
$840$	−17550.0	−0.720874
$841$	12909.1	0.529299
$842$	31907.1	1.30593
$843$	23580.5	0.963410
$844$	4536.73	0.185024
$845$	−9510.90	−0.387201
$846$	2709.68	0.110119
$847$	−4421.34	−0.179361
$848$	1794.07	0.0726518
$849$	19795.9	0.800229
$850$	−23353.3	−0.942367
$851$	−49368.2	−1.98863
$852$	−289.777	−0.0116521
$853$	−22406.6	−0.899401	−0.449700	−	0.893179i	$-0.648469\pi$
$853$	−22406.6	−0.899401	−0.449700	+	0.893179i	$0.648469\pi$
$854$	17845.2	0.715045
$855$	0	0
$856$	−5135.63	−0.205061
$857$	−43537.4	−1.73537	−0.867684	−	0.497116i	$-0.834392\pi$
$857$	−43537.4	−1.73537	−0.867684	+	0.497116i	$0.834392\pi$
$858$	8700.40	0.346185
$859$	−17540.7	−0.696719	−0.348359	−	0.937361i	$-0.613261\pi$
$859$	−17540.7	−0.696719	−0.348359	+	0.937361i	$0.613261\pi$
$860$	−18545.7	−0.735353
$861$	−5330.19	−0.210978
$862$	10685.3	0.422206
$863$	−4808.21	−0.189656	−0.0948281	−	0.995494i	$-0.530230\pi$
$863$	−4808.21	−0.189656	−0.0948281	+	0.995494i	$0.530230\pi$
$864$	−864.000	−0.0340207
$865$	−7502.05	−0.294887
$866$	1083.37	0.0425107
$867$	10833.9	0.424383
$868$	39653.0	1.55059
$869$	−32035.1	−1.25054
$870$	24544.0	0.956458
$871$	−9158.19	−0.356273
$872$	−10170.5	−0.394972
$873$	−2434.94	−0.0943989
$874$	0	0
$875$	145258.	5.61212
$876$	2092.17	0.0806938
$877$	435.386	0.0167639	0.00838194	−	0.999965i	$-0.497332\pi$
$877$	435.386	0.0167639	0.00838194	+	0.999965i	$0.497332\pi$
$878$	−3493.70	−0.134290
$879$	20167.0	0.773854
$880$	11754.1	0.450264
$881$	−34927.9	−1.33570	−0.667849	−	0.744296i	$-0.732785\pi$
$881$	−34927.9	−1.33570	−0.667849	+	0.744296i	$0.732785\pi$
$882$	15279.9	0.583333
$883$	39261.4	1.49632	0.748161	−	0.663518i	$-0.230937\pi$
$883$	39261.4	1.49632	0.748161	+	0.663518i	$0.230937\pi$
$884$	6033.66	0.229563
$885$	31305.9	1.18908
$886$	32280.9	1.22404
$887$	47140.5	1.78447	0.892235	−	0.451572i	$-0.149137\pi$
$887$	47140.5	1.78447	0.892235	+	0.451572i	$0.149137\pi$
$888$	−9182.54	−0.347011
$889$	19078.8	0.719778
$890$	−14607.9	−0.550176
$891$	2809.35	0.105630
$892$	6614.33	0.248278
$893$	0	0
$894$	−1122.38	−0.0419887
$895$	31764.6	1.18634
$896$	4419.03	0.164765
$897$	−16184.0	−0.602416
$898$	−3964.28	−0.147316
$899$	−55455.2	−2.05733
$900$	11651.1	0.431523
$901$	−4045.51	−0.149584
$902$	3569.90	0.131779
$903$	22671.0	0.835487
$904$	9651.86	0.355106
$905$	−32458.7	−1.19223
$906$	−5703.40	−0.209142
$907$	24169.7	0.884832	0.442416	−	0.896810i	$-0.354121\pi$
$907$	24169.7	0.884832	0.442416	+	0.896810i	$0.354121\pi$
$908$	1762.56	0.0644193
$909$	11317.7	0.412966
$910$	−61145.5	−2.22742
$911$	23859.2	0.867717	0.433859	−	0.900981i	$-0.357152\pi$
$911$	23859.2	0.867717	0.433859	+	0.900981i	$0.357152\pi$
$912$	0	0
$913$	−29963.2	−1.08613
$914$	−9569.15	−0.346301
$915$	−16422.7	−0.593354
$916$	−22454.0	−0.809937
$917$	−38380.5	−1.38216
$918$	1948.26	0.0700459
$919$	−27129.4	−0.973795	−0.486898	−	0.873459i	$-0.661872\pi$
$919$	−27129.4	−0.973795	−0.486898	+	0.873459i	$0.661872\pi$
$920$	−21864.3	−0.783528
$921$	19150.2	0.685148
$922$	1684.96	0.0601858
$923$	−1009.60	−0.0360037
$924$	−14368.7	−0.511577
$925$	123827.	4.40154
$926$	−18499.5	−0.656515
$927$	9254.83	0.327906
$928$	−6180.07	−0.218611
$929$	−30778.9	−1.08700	−0.543500	−	0.839409i	$-0.682901\pi$
$929$	−30778.9	−1.08700	−0.543500	+	0.839409i	$0.682901\pi$
$930$	−36492.3	−1.28670
$931$	0	0
$932$	−185.478	−0.00651882
$933$	−12543.7	−0.440153
$934$	−36902.3	−1.29280
$935$	−26504.8	−0.927057
$936$	−3010.23	−0.105120
$937$	19866.9	0.692662	0.346331	−	0.938112i	$-0.387427\pi$
$937$	19866.9	0.692662	0.346331	+	0.938112i	$0.387427\pi$
$938$	15124.8	0.526483
$939$	19151.1	0.665572
$940$	12754.3	0.442552
$941$	40634.2	1.40769	0.703845	−	0.710354i	$-0.251465\pi$
$941$	40634.2	1.40769	0.703845	+	0.710354i	$0.251465\pi$
$942$	−2485.23	−0.0859588
$943$	−6640.50	−0.229315
$944$	−7882.68	−0.271779
$945$	−19743.8	−0.679646
$946$	−15183.9	−0.521852
$947$	19679.5	0.675289	0.337645	−	0.941274i	$-0.390370\pi$
$947$	19679.5	0.675289	0.337645	+	0.941274i	$0.390370\pi$
$948$	11083.8	0.379730
$949$	7289.24	0.249335
$950$	0	0
$951$	−8285.65	−0.282524
$952$	−9964.59	−0.339238
$953$	−32558.0	−1.10667	−0.553335	−	0.832959i	$-0.686645\pi$
$953$	−32558.0	−1.10667	−0.553335	+	0.832959i	$0.686645\pi$
$954$	2018.33	0.0684967
$955$	−40063.2	−1.35750
$956$	17496.1	0.591907
$957$	20094.9	0.678762
$958$	9767.14	0.329397
$959$	−26726.7	−0.899948
$960$	−4066.79	−0.136724
$961$	52660.5	1.76767
$962$	−31992.6	−1.07223
$963$	−5777.58	−0.193333
$964$	7582.80	0.253346
$965$	64736.7	2.15953
$966$	26727.8	0.890222
$967$	1646.85	0.0547664	0.0273832	−	0.999625i	$-0.491283\pi$
$967$	1646.85	0.0547664	0.0273832	+	0.999625i	$0.491283\pi$
$968$	−1024.54	−0.0340184
$969$	0	0
$970$	−11461.1	−0.379375
$971$	−48397.5	−1.59954	−0.799768	−	0.600309i	$-0.795044\pi$
$971$	−48397.5	−1.59954	−0.799768	+	0.600309i	$0.795044\pi$
$972$	−972.000	−0.0320750
$973$	69518.5	2.29050
$974$	2613.54	0.0859786
$975$	40593.3	1.33336
$976$	4135.17	0.135619
$977$	15848.4	0.518971	0.259486	−	0.965747i	$-0.416447\pi$
$977$	15848.4	0.518971	0.259486	+	0.965747i	$0.416447\pi$
$978$	15965.2	0.521996
$979$	−11959.9	−0.390439
$980$	71921.3	2.34433
$981$	−11441.8	−0.372383
$982$	−16762.9	−0.544730
$983$	−6168.79	−0.200156	−0.100078	−	0.994980i	$-0.531909\pi$
$983$	−6168.79	−0.200156	−0.100078	+	0.994980i	$0.531909\pi$
$984$	−1235.14	−0.0400151
$985$	−60187.3	−1.94693
$986$	13935.6	0.450102
$987$	−15591.3	−0.502815
$988$	0	0
$989$	28244.2	0.908103
$990$	13223.4	0.424513
$991$	−19024.0	−0.609807	−0.304903	−	0.952383i	$-0.598624\pi$
$991$	−19024.0	−0.609807	−0.304903	+	0.952383i	$0.598624\pi$
$992$	9188.60	0.294091
$993$	−12851.0	−0.410689
$994$	1667.36	0.0532047
$995$	20799.9	0.662714
$996$	10366.9	0.329807
$997$	−18687.0	−0.593605	−0.296802	−	0.954939i	$-0.595920\pi$
$997$	−18687.0	−0.593605	−0.296802	+	0.954939i	$0.595920\pi$
$998$	25316.4	0.802982
$999$	−10330.4	−0.327165

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2166.4.a.be.1.6	yes	6
19.18	odd	2		2166.4.a.bc.1.6	✓	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2166.4.a.bc.1.6	✓	6	19.18	odd	2
2166.4.a.be.1.6	yes	6	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:	\( N \)	\(=\)	\( 2166 = 2 \cdot 3 \cdot 19^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2166.a (trivial)

\(f(q)\)	\(=\)	\(q+2.00000 q^{2} -3.00000 q^{3} +4.00000 q^{4} +21.1812 q^{5} -6.00000 q^{6} +34.5236 q^{7} +8.00000 q^{8} +9.00000 q^{9} +O(q^{10})\)	\(q+2.00000 q^{2} -3.00000 q^{3} +4.00000 q^{4} +21.1812 q^{5} -6.00000 q^{6} +34.5236 q^{7} +8.00000 q^{8} +9.00000 q^{9} +42.3624 q^{10} +34.6833 q^{11} -12.0000 q^{12} -41.8088 q^{13} +69.0473 q^{14} -63.5436 q^{15} +16.0000 q^{16} -36.0789 q^{17} +18.0000 q^{18} +84.7247 q^{20} -103.571 q^{21} +69.3666 q^{22} -129.032 q^{23} -24.0000 q^{24} +323.643 q^{25} -83.6176 q^{26} -27.0000 q^{27} +138.095 q^{28} -193.127 q^{29} -127.087 q^{30} +287.144 q^{31} +32.0000 q^{32} -104.050 q^{33} -72.1578 q^{34} +731.252 q^{35} +36.0000 q^{36} +382.606 q^{37} +125.426 q^{39} +169.449 q^{40} +51.4641 q^{41} -207.142 q^{42} -218.894 q^{43} +138.733 q^{44} +190.631 q^{45} -258.063 q^{46} +150.538 q^{47} -48.0000 q^{48} +848.882 q^{49} +647.285 q^{50} +108.237 q^{51} -167.235 q^{52} +112.130 q^{53} -54.0000 q^{54} +734.634 q^{55} +276.189 q^{56} -386.254 q^{58} -492.668 q^{59} -254.174 q^{60} +258.448 q^{61} +574.287 q^{62} +310.713 q^{63} +64.0000 q^{64} -885.559 q^{65} -208.100 q^{66} +219.050 q^{67} -144.316 q^{68} +387.095 q^{69} +1462.50 q^{70} +24.1481 q^{71} +72.0000 q^{72} -174.347 q^{73} +765.212 q^{74} -970.928 q^{75} +1197.39 q^{77} +250.853 q^{78} -923.646 q^{79} +338.899 q^{80} +81.0000 q^{81} +102.928 q^{82} -863.908 q^{83} -414.284 q^{84} -764.193 q^{85} -437.788 q^{86} +579.381 q^{87} +277.467 q^{88} -344.831 q^{89} +381.261 q^{90} -1443.39 q^{91} -516.127 q^{92} -861.431 q^{93} +301.076 q^{94} -96.0000 q^{96} -270.549 q^{97} +1697.76 q^{98} +312.150 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 12 q^{2} - 18 q^{3} + 24 q^{4} - q^{5} - 36 q^{6} + 28 q^{7} + 48 q^{8} + 54 q^{9}+O(q^{10}) \) 6 * q + 12 * q^2 - 18 * q^3 + 24 * q^4 - q^5 - 36 * q^6 + 28 * q^7 + 48 * q^8 + 54 * q^9	\( 6 q + 12 q^{2} - 18 q^{3} + 24 q^{4} - q^{5} - 36 q^{6} + 28 q^{7} + 48 q^{8} + 54 q^{9} - 2 q^{10} + 127 q^{11} - 72 q^{12} + 12 q^{13} + 56 q^{14} + 3 q^{15} + 96 q^{16} + 61 q^{17} + 108 q^{18} - 4 q^{20} - 84 q^{21} + 254 q^{22} - 130 q^{23} - 144 q^{24} + 543 q^{25} + 24 q^{26} - 162 q^{27} + 112 q^{28} - 570 q^{29} + 6 q^{30} + 679 q^{31} + 192 q^{32} - 381 q^{33} + 122 q^{34} + 478 q^{35} + 216 q^{36} - 37 q^{37} - 36 q^{39} - 8 q^{40} + 346 q^{41} - 168 q^{42} - 20 q^{43} + 508 q^{44} - 9 q^{45} - 260 q^{46} - 215 q^{47} - 288 q^{48} + 2056 q^{49} + 1086 q^{50} - 183 q^{51} + 48 q^{52} - 366 q^{53} - 324 q^{54} - 253 q^{55} + 224 q^{56} - 1140 q^{58} - 263 q^{59} + 12 q^{60} + 413 q^{61} + 1358 q^{62} + 252 q^{63} + 384 q^{64} - 1203 q^{65} - 762 q^{66} - 751 q^{67} + 244 q^{68} + 390 q^{69} + 956 q^{70} - 281 q^{71} + 432 q^{72} - 1120 q^{73} - 74 q^{74} - 1629 q^{75} + 1067 q^{77} - 72 q^{78} + 2045 q^{79} - 16 q^{80} + 486 q^{81} + 692 q^{82} + 43 q^{83} - 336 q^{84} - 627 q^{85} - 40 q^{86} + 1710 q^{87} + 1016 q^{88} - 467 q^{89} - 18 q^{90} - 1101 q^{91} - 520 q^{92} - 2037 q^{93} - 430 q^{94} - 576 q^{96} - 275 q^{97} + 4112 q^{98} + 1143 q^{99}+O(q^{100}) \) 6 * q + 12 * q^2 - 18 * q^3 + 24 * q^4 - q^5 - 36 * q^6 + 28 * q^7 + 48 * q^8 + 54 * q^9 - 2 * q^10 + 127 * q^11 - 72 * q^12 + 12 * q^13 + 56 * q^14 + 3 * q^15 + 96 * q^16 + 61 * q^17 + 108 * q^18 - 4 * q^20 - 84 * q^21 + 254 * q^22 - 130 * q^23 - 144 * q^24 + 543 * q^25 + 24 * q^26 - 162 * q^27 + 112 * q^28 - 570 * q^29 + 6 * q^30 + 679 * q^31 + 192 * q^32 - 381 * q^33 + 122 * q^34 + 478 * q^35 + 216 * q^36 - 37 * q^37 - 36 * q^39 - 8 * q^40 + 346 * q^41 - 168 * q^42 - 20 * q^43 + 508 * q^44 - 9 * q^45 - 260 * q^46 - 215 * q^47 - 288 * q^48 + 2056 * q^49 + 1086 * q^50 - 183 * q^51 + 48 * q^52 - 366 * q^53 - 324 * q^54 - 253 * q^55 + 224 * q^56 - 1140 * q^58 - 263 * q^59 + 12 * q^60 + 413 * q^61 + 1358 * q^62 + 252 * q^63 + 384 * q^64 - 1203 * q^65 - 762 * q^66 - 751 * q^67 + 244 * q^68 + 390 * q^69 + 956 * q^70 - 281 * q^71 + 432 * q^72 - 1120 * q^73 - 74 * q^74 - 1629 * q^75 + 1067 * q^77 - 72 * q^78 + 2045 * q^79 - 16 * q^80 + 486 * q^81 + 692 * q^82 + 43 * q^83 - 336 * q^84 - 627 * q^85 - 40 * q^86 + 1710 * q^87 + 1016 * q^88 - 467 * q^89 - 18 * q^90 - 1101 * q^91 - 520 * q^92 - 2037 * q^93 - 430 * q^94 - 576 * q^96 - 275 * q^97 + 4112 * q^98 + 1143 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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