LMFDB - Embedded newform 2166.4.a.bk.1.9

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:	$1$
Dimension:	$9$
Coefficient field:	$\mathbb{Q}[x]/(x^{9} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{9} - 3 x^{8} - 753 x^{7} + 41 x^{6} + 190713 x^{5} + 502293 x^{4} - 15827924 x^{3} - 81474654 x^{2} + \cdots + 33478707 $ x^9 - 3x^8 - 753x^7 + 41x^6 + 190713x^5 + 502293x^4 - 15827924x^3 - 81474654x^2 - 84330765x + 33478707
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 19 $
Twist minimal:	no (minimal twist has level 114)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.9
Root			$-14.7729$ 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)

$f(q)$	$=$	$q-2.00000 q^{2} +3.00000 q^{3} +4.00000 q^{4} +16.3050 q^{5} -6.00000 q^{6} -22.6136 q^{7} -8.00000 q^{8} +9.00000 q^{9} +O(q^{10})$	\(q-2.00000 q^{2} +3.00000 q^{3} +4.00000 q^{4} +16.3050 q^{5} -6.00000 q^{6} -22.6136 q^{7} -8.00000 q^{8} +9.00000 q^{9} -32.6099 q^{10} +3.51051 q^{11} +12.0000 q^{12} -40.6523 q^{13} +45.2272 q^{14} +48.9149 q^{15} +16.0000 q^{16} -39.8840 q^{17} -18.0000 q^{18} +65.2198 q^{20} -67.8407 q^{21} -7.02102 q^{22} +11.0950 q^{23} -24.0000 q^{24} +140.852 q^{25} +81.3045 q^{26} +27.0000 q^{27} -90.4543 q^{28} +240.905 q^{29} -97.8297 q^{30} -247.934 q^{31} -32.0000 q^{32} +10.5315 q^{33} +79.7680 q^{34} -368.713 q^{35} +36.0000 q^{36} -391.300 q^{37} -121.957 q^{39} -130.440 q^{40} +290.524 q^{41} +135.681 q^{42} +506.224 q^{43} +14.0420 q^{44} +146.745 q^{45} -22.1901 q^{46} +356.819 q^{47} +48.0000 q^{48} +168.374 q^{49} -281.703 q^{50} -119.652 q^{51} -162.609 q^{52} +274.523 q^{53} -54.0000 q^{54} +57.2387 q^{55} +180.909 q^{56} -481.809 q^{58} +179.575 q^{59} +195.659 q^{60} -527.699 q^{61} +495.869 q^{62} -203.522 q^{63} +64.0000 q^{64} -662.833 q^{65} -21.0631 q^{66} -162.272 q^{67} -159.536 q^{68} +33.2851 q^{69} +737.427 q^{70} -514.015 q^{71} -72.0000 q^{72} -404.127 q^{73} +782.600 q^{74} +422.555 q^{75} -79.3852 q^{77} +243.914 q^{78} -730.298 q^{79} +260.879 q^{80} +81.0000 q^{81} -581.049 q^{82} +37.0267 q^{83} -271.363 q^{84} -650.307 q^{85} -1012.45 q^{86} +722.714 q^{87} -28.0841 q^{88} -1260.76 q^{89} -293.489 q^{90} +919.293 q^{91} +44.3802 q^{92} -743.803 q^{93} -713.638 q^{94} -96.0000 q^{96} -1641.89 q^{97} -336.748 q^{98} +31.5946 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 9 q - 18 q^{2} + 27 q^{3} + 36 q^{4} - 3 q^{5} - 54 q^{6} + 30 q^{7} - 72 q^{8} + 81 q^{9}+O(q^{10}) $ 9 * q - 18 * q^2 + 27 * q^3 + 36 * q^4 - 3 * q^5 - 54 * q^6 + 30 * q^7 - 72 * q^8 + 81 * q^9	\( 9 q - 18 q^{2} + 27 q^{3} + 36 q^{4} - 3 q^{5} - 54 q^{6} + 30 q^{7} - 72 q^{8} + 81 q^{9} + 6 q^{10} - 57 q^{11} + 108 q^{12} - 93 q^{13} - 60 q^{14} - 9 q^{15} + 144 q^{16} - 147 q^{17} - 162 q^{18} - 12 q^{20} + 90 q^{21} + 114 q^{22} - 24 q^{23} - 216 q^{24} + 390 q^{25} + 186 q^{26} + 243 q^{27} + 120 q^{28} - 225 q^{29} + 18 q^{30} - 429 q^{31} - 288 q^{32} - 171 q^{33} + 294 q^{34} - 438 q^{35} + 324 q^{36} - 159 q^{37} - 279 q^{39} + 24 q^{40} - 567 q^{41} - 180 q^{42} + 1230 q^{43} - 228 q^{44} - 27 q^{45} + 48 q^{46} - 342 q^{47} + 432 q^{48} + 249 q^{49} - 780 q^{50} - 441 q^{51} - 372 q^{52} + 285 q^{53} - 486 q^{54} + 174 q^{55} - 240 q^{56} + 450 q^{58} + 171 q^{59} - 36 q^{60} - 816 q^{61} + 858 q^{62} + 270 q^{63} + 576 q^{64} - 1209 q^{65} + 342 q^{66} - 2388 q^{67} - 588 q^{68} - 72 q^{69} + 876 q^{70} - 2022 q^{71} - 648 q^{72} + 645 q^{73} + 318 q^{74} + 1170 q^{75} - 3843 q^{77} + 558 q^{78} - 3516 q^{79} - 48 q^{80} + 729 q^{81} + 1134 q^{82} - 156 q^{83} + 360 q^{84} - 291 q^{85} - 2460 q^{86} - 675 q^{87} + 456 q^{88} - 1959 q^{89} + 54 q^{90} - 501 q^{91} - 96 q^{92} - 1287 q^{93} + 684 q^{94} - 864 q^{96} + 2457 q^{97} - 498 q^{98} - 513 q^{99}+O(q^{100}) \) 9 * q - 18 * q^2 + 27 * q^3 + 36 * q^4 - 3 * q^5 - 54 * q^6 + 30 * q^7 - 72 * q^8 + 81 * q^9 + 6 * q^10 - 57 * q^11 + 108 * q^12 - 93 * q^13 - 60 * q^14 - 9 * q^15 + 144 * q^16 - 147 * q^17 - 162 * q^18 - 12 * q^20 + 90 * q^21 + 114 * q^22 - 24 * q^23 - 216 * q^24 + 390 * q^25 + 186 * q^26 + 243 * q^27 + 120 * q^28 - 225 * q^29 + 18 * q^30 - 429 * q^31 - 288 * q^32 - 171 * q^33 + 294 * q^34 - 438 * q^35 + 324 * q^36 - 159 * q^37 - 279 * q^39 + 24 * q^40 - 567 * q^41 - 180 * q^42 + 1230 * q^43 - 228 * q^44 - 27 * q^45 + 48 * q^46 - 342 * q^47 + 432 * q^48 + 249 * q^49 - 780 * q^50 - 441 * q^51 - 372 * q^52 + 285 * q^53 - 486 * q^54 + 174 * q^55 - 240 * q^56 + 450 * q^58 + 171 * q^59 - 36 * q^60 - 816 * q^61 + 858 * q^62 + 270 * q^63 + 576 * q^64 - 1209 * q^65 + 342 * q^66 - 2388 * q^67 - 588 * q^68 - 72 * q^69 + 876 * q^70 - 2022 * q^71 - 648 * q^72 + 645 * q^73 + 318 * q^74 + 1170 * q^75 - 3843 * q^77 + 558 * q^78 - 3516 * q^79 - 48 * q^80 + 729 * q^81 + 1134 * q^82 - 156 * q^83 + 360 * q^84 - 291 * q^85 - 2460 * q^86 - 675 * q^87 + 456 * q^88 - 1959 * q^89 + 54 * q^90 - 501 * q^91 - 96 * q^92 - 1287 * q^93 + 684 * q^94 - 864 * q^96 + 2457 * q^97 - 498 * q^98 - 513 * 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
$5$	16.3050	1.45836	0.729180	−	0.684322i	$-0.239902\pi$
$5$	16.3050	1.45836	0.729180	+	0.684322i	$0.239902\pi$
$6$	−6.00000	−0.408248
$7$	−22.6136	−1.22102	−0.610509	−	0.792009i	$-0.709035\pi$
$7$	−22.6136	−1.22102	−0.610509	+	0.792009i	$0.709035\pi$
$8$	−8.00000	−0.353553
$9$	9.00000	0.333333
$10$	−32.6099	−1.03122
$11$	3.51051	0.0962235	0.0481117	−	0.998842i	$-0.484680\pi$
$11$	3.51051	0.0962235	0.0481117	+	0.998842i	$0.484680\pi$
$12$	12.0000	0.288675
$13$	−40.6523	−0.867301	−0.433650	−	0.901081i	$-0.642775\pi$
$13$	−40.6523	−0.867301	−0.433650	+	0.901081i	$0.642775\pi$
$14$	45.2272	0.863391
$15$	48.9149	0.841984
$16$	16.0000	0.250000
$17$	−39.8840	−0.569017	−0.284509	−	0.958673i	$-0.591830\pi$
$17$	−39.8840	−0.569017	−0.284509	+	0.958673i	$0.591830\pi$
$18$	−18.0000	−0.235702
$19$	0	0
$19$	0	0
$20$	65.2198	0.729180
$21$	−67.8407	−0.704955
$22$	−7.02102	−0.0680403
$23$	11.0950	0.100586	0.0502929	−	0.998735i	$-0.483985\pi$
$23$	11.0950	0.100586	0.0502929	+	0.998735i	$0.483985\pi$
$24$	−24.0000	−0.204124
$25$	140.852	1.12681
$26$	81.3045	0.613274
$27$	27.0000	0.192450
$28$	−90.4543	−0.610509
$29$	240.905	1.54258	0.771291	−	0.636483i	$-0.219611\pi$
$29$	240.905	1.54258	0.771291	+	0.636483i	$0.219611\pi$
$30$	−97.8297	−0.595373
$31$	−247.934	−1.43646	−0.718231	−	0.695805i	$-0.755048\pi$
$31$	−247.934	−1.43646	−0.718231	+	0.695805i	$0.755048\pi$
$32$	−32.0000	−0.176777
$33$	10.5315	0.0555546
$34$	79.7680	0.402356
$35$	−368.713	−1.78068
$36$	36.0000	0.166667
$37$	−391.300	−1.73863	−0.869315	−	0.494259i	$-0.835439\pi$
$37$	−391.300	−1.73863	−0.869315	+	0.494259i	$0.835439\pi$
$38$	0	0
$39$	−121.957	−0.500736
$40$	−130.440	−0.515608
$41$	290.524	1.10664	0.553320	−	0.832968i	$-0.313360\pi$
$41$	290.524	1.10664	0.553320	+	0.832968i	$0.313360\pi$
$42$	135.681	0.498479
$43$	506.224	1.79531	0.897656	−	0.440697i	$-0.145269\pi$
$43$	506.224	1.79531	0.897656	+	0.440697i	$0.145269\pi$
$44$	14.0420	0.0481117
$45$	146.745	0.486120
$46$	−22.1901	−0.0711250
$47$	356.819	1.10739	0.553696	−	0.832719i	$-0.313217\pi$
$47$	356.819	1.10739	0.553696	+	0.832719i	$0.313217\pi$
$48$	48.0000	0.144338
$49$	168.374	0.490887
$50$	−281.703	−0.796777
$51$	−119.652	−0.328522
$52$	−162.609	−0.433650
$53$	274.523	0.711483	0.355742	−	0.934584i	$-0.384228\pi$
$53$	274.523	0.711483	0.355742	+	0.934584i	$0.384228\pi$
$54$	−54.0000	−0.136083
$55$	57.2387	0.140328
$56$	180.909	0.431695
$57$	0	0
$58$	−481.809	−1.09077
$59$	179.575	0.396248	0.198124	−	0.980177i	$-0.436515\pi$
$59$	179.575	0.396248	0.198124	+	0.980177i	$0.436515\pi$
$60$	195.659	0.420992
$61$	−527.699	−1.10762	−0.553811	−	0.832643i	$-0.686827\pi$
$61$	−527.699	−1.10762	−0.553811	+	0.832643i	$0.686827\pi$
$62$	495.869	1.01573
$63$	−203.522	−0.407006
$64$	64.0000	0.125000
$65$	−662.833	−1.26484
$66$	−21.0631	−0.0392831
$67$	−162.272	−0.295890	−0.147945	−	0.988996i	$-0.547266\pi$
$67$	−162.272	−0.295890	−0.147945	+	0.988996i	$0.547266\pi$
$68$	−159.536	−0.284509
$69$	33.2851	0.0580733
$70$	737.427	1.25913
$71$	−514.015	−0.859189	−0.429594	−	0.903022i	$-0.641343\pi$
$71$	−514.015	−0.859189	−0.429594	+	0.903022i	$0.641343\pi$
$72$	−72.0000	−0.117851
$73$	−404.127	−0.647939	−0.323969	−	0.946068i	$-0.605017\pi$
$73$	−404.127	−0.647939	−0.323969	+	0.946068i	$0.605017\pi$
$74$	782.600	1.22940
$75$	422.555	0.650565
$76$	0	0
$77$	−79.3852	−0.117491
$78$	243.914	0.354074
$79$	−730.298	−1.04006	−0.520031	−	0.854147i	$-0.674080\pi$
$79$	−730.298	−1.04006	−0.520031	+	0.854147i	$0.674080\pi$
$80$	260.879	0.364590
$81$	81.0000	0.111111
$82$	−581.049	−0.782513
$83$	37.0267	0.0489664	0.0244832	−	0.999700i	$-0.492206\pi$
$83$	37.0267	0.0489664	0.0244832	+	0.999700i	$0.492206\pi$
$84$	−271.363	−0.352478
$85$	−650.307	−0.829832
$86$	−1012.45	−1.26948
$87$	722.714	0.890610
$88$	−28.0841	−0.0340201
$89$	−1260.76	−1.50157	−0.750786	−	0.660545i	$-0.770325\pi$
$89$	−1260.76	−1.50157	−0.750786	+	0.660545i	$0.770325\pi$
$90$	−293.489	−0.343739
$91$	919.293	1.05899
$92$	44.3802	0.0502929
$93$	−743.803	−0.829342
$94$	−713.638	−0.783044
$95$	0	0
$96$	−96.0000	−0.102062
$97$	−1641.89	−1.71864	−0.859321	−	0.511436i	$-0.829114\pi$
$97$	−1641.89	−1.71864	−0.859321	+	0.511436i	$0.829114\pi$
$98$	−336.748	−0.347109
$99$	31.5946	0.0320745
$100$	563.406	0.563406
$101$	−58.3107	−0.0574469	−0.0287234	−	0.999587i	$-0.509144\pi$
$101$	−58.3107	−0.0574469	−0.0287234	+	0.999587i	$0.509144\pi$
$102$	239.304	0.232300
$103$	45.5629	0.0435869	0.0217934	−	0.999762i	$-0.493062\pi$
$103$	45.5629	0.0435869	0.0217934	+	0.999762i	$0.493062\pi$
$104$	325.218	0.306637
$105$	−1106.14	−1.02808
$106$	−549.045	−0.503094
$107$	−957.132	−0.864761	−0.432380	−	0.901691i	$-0.642326\pi$
$107$	−957.132	−0.864761	−0.432380	+	0.901691i	$0.642326\pi$
$108$	108.000	0.0962250
$109$	−970.474	−0.852794	−0.426397	−	0.904536i	$-0.640217\pi$
$109$	−970.474	−0.852794	−0.426397	+	0.904536i	$0.640217\pi$
$110$	−114.477	−0.0992272
$111$	−1173.90	−1.00380
$112$	−361.817	−0.305255
$113$	1052.17	0.875929	0.437964	−	0.898992i	$-0.355700\pi$
$113$	1052.17	0.875929	0.437964	+	0.898992i	$0.355700\pi$
$114$	0	0
$115$	180.904	0.146690
$116$	963.618	0.771291
$117$	−365.870	−0.289100
$118$	−359.150	−0.280190
$119$	901.920	0.694781
$120$	−391.319	−0.297686
$121$	−1318.68	−0.990741
$122$	1055.40	0.783206
$123$	871.573	0.638919
$124$	−991.737	−0.718231
$125$	258.458	0.184938
$126$	407.044	0.287797
$127$	−1159.36	−0.810055	−0.405027	−	0.914305i	$-0.632738\pi$
$127$	−1159.36	−0.810055	−0.405027	+	0.914305i	$0.632738\pi$
$128$	−128.000	−0.0883883
$129$	1518.67	1.03652
$130$	1325.67	0.894374
$131$	−592.895	−0.395431	−0.197715	−	0.980259i	$-0.563352\pi$
$131$	−592.895	−0.395431	−0.197715	+	0.980259i	$0.563352\pi$
$132$	42.1261	0.0277773
$133$	0	0
$134$	324.543	0.209226
$135$	440.234	0.280661
$136$	319.072	0.201178
$137$	297.378	0.185451	0.0927253	−	0.995692i	$-0.470442\pi$
$137$	297.378	0.185451	0.0927253	+	0.995692i	$0.470442\pi$
$138$	−66.5702	−0.0410640
$139$	203.209	0.124000	0.0619998	−	0.998076i	$-0.480252\pi$
$139$	203.209	0.124000	0.0619998	+	0.998076i	$0.480252\pi$
$140$	−1474.85	−0.890342
$141$	1070.46	0.639353
$142$	1028.03	0.607538
$143$	−142.710	−0.0834547
$144$	144.000	0.0833333
$145$	3927.94	2.24964
$146$	808.255	0.458162
$147$	505.122	0.283414
$148$	−1565.20	−0.869315
$149$	−249.492	−0.137176	−0.0685879	−	0.997645i	$-0.521849\pi$
$149$	−249.492	−0.137176	−0.0685879	+	0.997645i	$0.521849\pi$
$150$	−845.109	−0.460019
$151$	−1838.05	−0.990585	−0.495293	−	0.868726i	$-0.664939\pi$
$151$	−1838.05	−0.990585	−0.495293	+	0.868726i	$0.664939\pi$
$152$	0	0
$153$	−358.956	−0.189672
$154$	158.770	0.0830784
$155$	−4042.56	−2.09488
$156$	−487.827	−0.250368
$157$	−3226.99	−1.64039	−0.820197	−	0.572081i	$-0.806136\pi$
$157$	−3226.99	−1.64039	−0.820197	+	0.572081i	$0.806136\pi$
$158$	1460.60	0.735436
$159$	823.568	0.410775
$160$	−521.759	−0.257804
$161$	−250.899	−0.122817
$162$	−162.000	−0.0785674
$163$	1901.48	0.913713	0.456856	−	0.889541i	$-0.348975\pi$
$163$	1901.48	0.913713	0.456856	+	0.889541i	$0.348975\pi$
$164$	1162.10	0.553320
$165$	171.716	0.0810186
$166$	−74.0535	−0.0346245
$167$	−3979.10	−1.84378	−0.921891	−	0.387448i	$-0.873357\pi$
$167$	−3979.10	−1.84378	−0.921891	+	0.387448i	$0.873357\pi$
$168$	542.726	0.249239
$169$	−544.394	−0.247790
$170$	1300.61	0.586780
$171$	0	0
$172$	2024.90	0.897656
$173$	3389.41	1.48955	0.744775	−	0.667316i	$-0.232557\pi$
$173$	3389.41	1.48955	0.744775	+	0.667316i	$0.232557\pi$
$174$	−1445.43	−0.629756
$175$	−3185.16	−1.37586
$176$	56.1681	0.0240559
$177$	538.724	0.228774
$178$	2521.51	1.06177
$179$	38.3967	0.0160330	0.00801649	−	0.999968i	$-0.497448\pi$
$179$	38.3967	0.0160330	0.00801649	+	0.999968i	$0.497448\pi$
$180$	586.978	0.243060
$181$	−4513.62	−1.85356	−0.926782	−	0.375601i	$-0.877436\pi$
$181$	−4513.62	−1.85356	−0.926782	+	0.375601i	$0.877436\pi$
$182$	−1838.59	−0.748819
$183$	−1583.10	−0.639485
$184$	−88.7603	−0.0355625
$185$	−6380.12	−2.53555
$186$	1487.61	0.586433
$187$	−140.013	−0.0547528
$188$	1427.28	0.553696
$189$	−610.567	−0.234985
$190$	0	0
$191$	3291.59	1.24697	0.623484	−	0.781836i	$-0.285717\pi$
$191$	3291.59	1.24697	0.623484	+	0.781836i	$0.285717\pi$
$192$	192.000	0.0721688
$193$	−517.137	−0.192872	−0.0964360	−	0.995339i	$-0.530744\pi$
$193$	−517.137	−0.192872	−0.0964360	+	0.995339i	$0.530744\pi$
$194$	3283.77	1.21526
$195$	−1988.50	−0.730253
$196$	673.496	0.245443
$197$	1797.35	0.650030	0.325015	−	0.945709i	$-0.394631\pi$
$197$	1797.35	0.650030	0.325015	+	0.945709i	$0.394631\pi$
$198$	−63.1892	−0.0226801
$199$	−2168.66	−0.772525	−0.386262	−	0.922389i	$-0.626234\pi$
$199$	−2168.66	−0.772525	−0.386262	+	0.922389i	$0.626234\pi$
$200$	−1126.81	−0.398388
$201$	−486.815	−0.170832
$202$	116.621	0.0406211
$203$	−5447.72	−1.88352
$204$	−478.608	−0.164261
$205$	4736.99	1.61388
$206$	−91.1258	−0.0308206
$207$	99.8554	0.0335286
$208$	−650.436	−0.216825
$209$	0	0
$210$	2212.28	0.726961
$211$	−135.969	−0.0443626	−0.0221813	−	0.999754i	$-0.507061\pi$
$211$	−135.969	−0.0443626	−0.0221813	+	0.999754i	$0.507061\pi$
$212$	1098.09	0.355742
$213$	−1542.05	−0.496053
$214$	1914.26	0.611478
$215$	8253.96	2.61821
$216$	−216.000	−0.0680414
$217$	5606.68	1.75395
$218$	1940.95	0.603017
$219$	−1212.38	−0.374088
$220$	228.955	0.0701642
$221$	1621.37	0.493509
$222$	2347.80	0.709792
$223$	1167.77	0.350670	0.175335	−	0.984509i	$-0.443899\pi$
$223$	1167.77	0.350670	0.175335	+	0.984509i	$0.443899\pi$
$224$	723.635	0.215848
$225$	1267.66	0.375604
$226$	−2104.34	−0.619375
$227$	6684.51	1.95448	0.977238	−	0.212144i	$-0.0680446\pi$
$227$	6684.51	1.95448	0.977238	+	0.212144i	$0.0680446\pi$
$228$	0	0
$229$	3123.34	0.901293	0.450647	−	0.892702i	$-0.351193\pi$
$229$	3123.34	0.901293	0.450647	+	0.892702i	$0.351193\pi$
$230$	−361.808	−0.103726
$231$	−238.156	−0.0678333
$232$	−1927.24	−0.545385
$233$	3846.47	1.08150	0.540752	−	0.841182i	$-0.318140\pi$
$233$	3846.47	1.08150	0.540752	+	0.841182i	$0.318140\pi$
$234$	731.741	0.204425
$235$	5817.92	1.61498
$236$	718.299	0.198124
$237$	−2190.89	−0.600481
$238$	−1803.84	−0.491284
$239$	−6734.17	−1.82258	−0.911291	−	0.411763i	$-0.864913\pi$
$239$	−6734.17	−1.82258	−0.911291	+	0.411763i	$0.864913\pi$
$240$	782.638	0.210496
$241$	1627.05	0.434886	0.217443	−	0.976073i	$-0.430228\pi$
$241$	1627.05	0.434886	0.217443	+	0.976073i	$0.430228\pi$
$242$	2637.35	0.700560
$243$	243.000	0.0641500
$244$	−2110.79	−0.553811
$245$	2745.33	0.715889
$246$	−1743.15	−0.451784
$247$	0	0
$248$	1983.47	0.507866
$249$	111.080	0.0282708
$250$	−516.917	−0.130771
$251$	−4465.46	−1.12294	−0.561469	−	0.827498i	$-0.689764\pi$
$251$	−4465.46	−1.12294	−0.561469	+	0.827498i	$0.689764\pi$
$252$	−814.089	−0.203503
$253$	38.9492	0.00967872
$254$	2318.73	0.572795
$255$	−1950.92	−0.479104
$256$	256.000	0.0625000
$257$	−3204.78	−0.777854	−0.388927	−	0.921269i	$-0.627154\pi$
$257$	−3204.78	−0.777854	−0.388927	+	0.921269i	$0.627154\pi$
$258$	−3037.34	−0.732933
$259$	8848.69	2.12290
$260$	−2651.33	−0.632418
$261$	2168.14	0.514194
$262$	1185.79	0.279612
$263$	6338.95	1.48622	0.743111	−	0.669168i	$-0.233349\pi$
$263$	6338.95	1.48622	0.743111	+	0.669168i	$0.233349\pi$
$264$	−84.2522	−0.0196415
$265$	4476.08	1.03760
$266$	0	0
$267$	−3782.27	−0.866933
$268$	−649.087	−0.147945
$269$	−8074.25	−1.83009	−0.915047	−	0.403346i	$-0.867847\pi$
$269$	−8074.25	−1.83009	−0.915047	+	0.403346i	$0.867847\pi$
$270$	−880.468	−0.198458
$271$	7272.68	1.63020	0.815099	−	0.579321i	$-0.196682\pi$
$271$	7272.68	1.63020	0.815099	+	0.579321i	$0.196682\pi$
$272$	−638.144	−0.142254
$273$	2757.88	0.611408
$274$	−594.756	−0.131133
$275$	494.461	0.108426
$276$	133.140	0.0290366
$277$	−5207.79	−1.12962	−0.564812	−	0.825220i	$-0.691051\pi$
$277$	−5207.79	−1.12962	−0.564812	+	0.825220i	$0.691051\pi$
$278$	−406.418	−0.0876810
$279$	−2231.41	−0.478821
$280$	2949.71	0.629567
$281$	1396.48	0.296466	0.148233	−	0.988952i	$-0.452641\pi$
$281$	1396.48	0.296466	0.148233	+	0.988952i	$0.452641\pi$
$282$	−2140.92	−0.452091
$283$	8280.39	1.73929	0.869643	−	0.493680i	$-0.164349\pi$
$283$	8280.39	1.73929	0.869643	+	0.493680i	$0.164349\pi$
$284$	−2056.06	−0.429594
$285$	0	0
$286$	285.420	0.0590114
$287$	−6569.80	−1.35123
$288$	−288.000	−0.0589256
$289$	−3322.27	−0.676219
$290$	−7855.88	−1.59073
$291$	−4925.66	−0.992259
$292$	−1616.51	−0.323969
$293$	−4910.51	−0.979096	−0.489548	−	0.871976i	$-0.662838\pi$
$293$	−4910.51	−0.979096	−0.489548	+	0.871976i	$0.662838\pi$
$294$	−1010.24	−0.200404
$295$	2927.96	0.577872
$296$	3130.40	0.614698
$297$	94.7837	0.0185182
$298$	498.985	0.0969980
$299$	−451.038	−0.0872382
$300$	1690.22	0.325283
$301$	−11447.5	−2.19211
$302$	3676.10	0.700450
$303$	−174.932	−0.0331670
$304$	0	0
$305$	−8604.10	−1.61531
$306$	717.912	0.134119
$307$	−6149.02	−1.14314	−0.571569	−	0.820554i	$-0.693665\pi$
$307$	−6149.02	−1.14314	−0.571569	+	0.820554i	$0.693665\pi$
$308$	−317.541	−0.0587453
$309$	136.689	0.0251649
$310$	8085.12	1.48130
$311$	−7609.71	−1.38748	−0.693741	−	0.720225i	$-0.744039\pi$
$311$	−7609.71	−1.38748	−0.693741	+	0.720225i	$0.744039\pi$
$312$	975.654	0.177037
$313$	6931.43	1.25172	0.625859	−	0.779937i	$-0.284749\pi$
$313$	6931.43	1.25172	0.625859	+	0.779937i	$0.284749\pi$
$314$	6453.98	1.15993
$315$	−3318.42	−0.593561
$316$	−2921.19	−0.520031
$317$	658.133	0.116607	0.0583035	−	0.998299i	$-0.481431\pi$
$317$	658.133	0.116607	0.0583035	+	0.998299i	$0.481431\pi$
$318$	−1647.14	−0.290462
$319$	845.698	0.148433
$320$	1043.52	0.182295
$321$	−2871.40	−0.499270
$322$	501.797	0.0868449
$323$	0	0
$324$	324.000	0.0555556
$325$	−5725.93	−0.977285
$326$	−3802.95	−0.646092
$327$	−2911.42	−0.492361
$328$	−2324.19	−0.391257
$329$	−8068.96	−1.35215
$330$	−343.432	−0.0572888
$331$	−7812.34	−1.29730	−0.648649	−	0.761088i	$-0.724665\pi$
$331$	−7812.34	−1.29730	−0.648649	+	0.761088i	$0.724665\pi$
$332$	148.107	0.0244832
$333$	−3521.70	−0.579543
$334$	7958.19	1.30375
$335$	−2645.83	−0.431514
$336$	−1085.45	−0.176239
$337$	3222.72	0.520927	0.260464	−	0.965484i	$-0.416125\pi$
$337$	3222.72	0.520927	0.260464	+	0.965484i	$0.416125\pi$
$338$	1088.79	0.175214
$339$	3156.51	0.505718
$340$	−2601.23	−0.414916
$341$	−870.376	−0.138221
$342$	0	0
$343$	3948.92	0.621637
$344$	−4049.79	−0.634739
$345$	542.712	0.0846917
$346$	−6778.82	−1.05327
$347$	1182.67	0.182966	0.0914829	−	0.995807i	$-0.470839\pi$
$347$	1182.67	0.182966	0.0914829	+	0.995807i	$0.470839\pi$
$348$	2890.86	0.445305
$349$	1224.20	0.187765	0.0938825	−	0.995583i	$-0.470072\pi$
$349$	1224.20	0.187765	0.0938825	+	0.995583i	$0.470072\pi$
$350$	6370.32	0.972879
$351$	−1097.61	−0.166912
$352$	−112.336	−0.0170101
$353$	−2601.03	−0.392179	−0.196089	−	0.980586i	$-0.562824\pi$
$353$	−2601.03	−0.392179	−0.196089	+	0.980586i	$0.562824\pi$
$354$	−1077.45	−0.161768
$355$	−8381.00	−1.25301
$356$	−5043.03	−0.750786
$357$	2705.76	0.401132
$358$	−76.7934	−0.0113370
$359$	−2605.15	−0.382993	−0.191496	−	0.981493i	$-0.561334\pi$
$359$	−2605.15	−0.382993	−0.191496	+	0.981493i	$0.561334\pi$
$360$	−1173.96	−0.171869
$361$	0	0
$362$	9027.25	1.31067
$363$	−3956.03	−0.572005
$364$	3677.17	0.529495
$365$	−6589.28	−0.944928
$366$	3166.19	0.452184
$367$	4566.24	0.649471	0.324735	−	0.945805i	$-0.394725\pi$
$367$	4566.24	0.649471	0.324735	+	0.945805i	$0.394725\pi$
$368$	177.521	0.0251465
$369$	2614.72	0.368880
$370$	12760.2	1.79290
$371$	−6207.94	−0.868734
$372$	−2975.21	−0.414671
$373$	4831.11	0.670631	0.335316	−	0.942106i	$-0.391157\pi$
$373$	4831.11	0.670631	0.335316	+	0.942106i	$0.391157\pi$
$374$	280.026	0.0387161
$375$	775.375	0.106774
$376$	−2854.55	−0.391522
$377$	−9793.32	−1.33788
$378$	1221.13	0.166160
$379$	4412.77	0.598071	0.299036	−	0.954242i	$-0.403335\pi$
$379$	4412.77	0.598071	0.299036	+	0.954242i	$0.403335\pi$
$380$	0	0
$381$	−3478.09	−0.467685
$382$	−6583.17	−0.881739
$383$	−5882.94	−0.784867	−0.392434	−	0.919780i	$-0.628367\pi$
$383$	−5882.94	−0.784867	−0.392434	+	0.919780i	$0.628367\pi$
$384$	−384.000	−0.0510310
$385$	−1294.37	−0.171344
$386$	1034.27	0.136381
$387$	4556.01	0.598437
$388$	−6567.55	−0.859321
$389$	9090.27	1.18482	0.592410	−	0.805636i	$-0.298176\pi$
$389$	9090.27	1.18482	0.592410	+	0.805636i	$0.298176\pi$
$390$	3977.00	0.516367
$391$	−442.515	−0.0572351
$392$	−1346.99	−0.173555
$393$	−1778.68	−0.228302
$394$	−3594.70	−0.459641
$395$	−11907.5	−1.51679
$396$	126.378	0.0160372
$397$	2838.24	0.358809	0.179404	−	0.983775i	$-0.442583\pi$
$397$	2838.24	0.358809	0.179404	+	0.983775i	$0.442583\pi$
$398$	4337.33	0.546257
$399$	0	0
$400$	2253.62	0.281703
$401$	−11665.4	−1.45273	−0.726364	−	0.687311i	$-0.758791\pi$
$401$	−11665.4	−1.45273	−0.726364	+	0.687311i	$0.758791\pi$
$402$	973.630	0.120797
$403$	10079.1	1.24584
$404$	−233.243	−0.0287234
$405$	1320.70	0.162040
$406$	10895.4	1.33185
$407$	−1373.66	−0.167297
$408$	957.216	0.116150
$409$	−5600.05	−0.677029	−0.338514	−	0.940961i	$-0.609924\pi$
$409$	−5600.05	−0.677029	−0.338514	+	0.940961i	$0.609924\pi$
$410$	−9473.97	−1.14119
$411$	892.135	0.107070
$412$	182.252	0.0217934
$413$	−4060.83	−0.483827
$414$	−199.711	−0.0237083
$415$	603.719	0.0714106
$416$	1300.87	0.153319
$417$	609.627	0.0715913
$418$	0	0
$419$	3772.52	0.439856	0.219928	−	0.975516i	$-0.429418\pi$
$419$	3772.52	0.439856	0.219928	+	0.975516i	$0.429418\pi$
$420$	−4424.56	−0.514039
$421$	−10869.5	−1.25831	−0.629153	−	0.777281i	$-0.716598\pi$
$421$	−10869.5	−1.25831	−0.629153	+	0.777281i	$0.716598\pi$
$422$	271.939	0.0313691
$423$	3211.37	0.369131
$424$	−2196.18	−0.251547
$425$	−5617.72	−0.641176
$426$	3084.09	0.350762
$427$	11933.2	1.35243
$428$	−3828.53	−0.432380
$429$	−428.130	−0.0481826
$430$	−16507.9	−1.85135
$431$	11218.6	1.25378	0.626891	−	0.779107i	$-0.284327\pi$
$431$	11218.6	1.25378	0.626891	+	0.779107i	$0.284327\pi$
$432$	432.000	0.0481125
$433$	4158.80	0.461569	0.230784	−	0.973005i	$-0.425871\pi$
$433$	4158.80	0.461569	0.230784	+	0.973005i	$0.425871\pi$
$434$	−11213.4	−1.24023
$435$	11783.8	1.29883
$436$	−3881.90	−0.426397
$437$	0	0
$438$	2424.76	0.264520
$439$	−8545.05	−0.929005	−0.464503	−	0.885572i	$-0.653767\pi$
$439$	−8545.05	−0.929005	−0.464503	+	0.885572i	$0.653767\pi$
$440$	−457.909	−0.0496136
$441$	1515.37	0.163629
$442$	−3242.75	−0.348964
$443$	−16686.0	−1.78956	−0.894780	−	0.446508i	$-0.852667\pi$
$443$	−16686.0	−1.78956	−0.894780	+	0.446508i	$0.852667\pi$
$444$	−4695.60	−0.501899
$445$	−20556.6	−2.18983
$446$	−2335.54	−0.247961
$447$	−748.477	−0.0791985
$448$	−1447.27	−0.152627
$449$	−4915.45	−0.516647	−0.258324	−	0.966058i	$-0.583170\pi$
$449$	−4915.45	−0.516647	−0.258324	+	0.966058i	$0.583170\pi$
$450$	−2535.33	−0.265592
$451$	1019.89	0.106485
$452$	4208.68	0.437964
$453$	−5514.15	−0.571915
$454$	−13369.0	−1.38202
$455$	14989.0	1.54439
$456$	0	0
$457$	−14255.0	−1.45913	−0.729565	−	0.683911i	$-0.760278\pi$
$457$	−14255.0	−1.45913	−0.729565	+	0.683911i	$0.760278\pi$
$458$	−6246.68	−0.637311
$459$	−1076.87	−0.109507
$460$	723.616	0.0733452
$461$	−10213.2	−1.03183	−0.515916	−	0.856639i	$-0.672548\pi$
$461$	−10213.2	−1.03183	−0.515916	+	0.856639i	$0.672548\pi$
$462$	476.311	0.0479654
$463$	4735.33	0.475312	0.237656	−	0.971349i	$-0.423621\pi$
$463$	4735.33	0.475312	0.237656	+	0.971349i	$0.423621\pi$
$464$	3854.47	0.385645
$465$	−12127.7	−1.20948
$466$	−7692.94	−0.764739
$467$	−327.926	−0.0324938	−0.0162469	−	0.999868i	$-0.505172\pi$
$467$	−327.926	−0.0324938	−0.0162469	+	0.999868i	$0.505172\pi$
$468$	−1463.48	−0.144550
$469$	3669.54	0.361287
$470$	−11635.8	−1.14196
$471$	−9680.98	−0.947082
$472$	−1436.60	−0.140095
$473$	1777.10	0.172751
$474$	4381.79	0.424604
$475$	0	0
$476$	3607.68	0.347390
$477$	2470.70	0.237161
$478$	13468.3	1.28876
$479$	−5663.86	−0.540268	−0.270134	−	0.962823i	$-0.587068\pi$
$479$	−5663.86	−0.540268	−0.270134	+	0.962823i	$0.587068\pi$
$480$	−1565.28	−0.148843
$481$	15907.2	1.50791
$482$	−3254.10	−0.307511
$483$	−752.696	−0.0709086
$484$	−5274.71	−0.495371
$485$	−26770.9	−2.50640
$486$	−486.000	−0.0453609
$487$	9218.25	0.857739	0.428869	−	0.903366i	$-0.358912\pi$
$487$	9218.25	0.857739	0.428869	+	0.903366i	$0.358912\pi$
$488$	4221.59	0.391603
$489$	5704.43	0.527532
$490$	−5490.66	−0.506210
$491$	20750.5	1.90724	0.953621	−	0.301009i	$-0.0973235\pi$
$491$	20750.5	1.90724	0.953621	+	0.301009i	$0.0973235\pi$
$492$	3486.29	0.319460
$493$	−9608.24	−0.877755
$494$	0	0
$495$	515.148	0.0467761
$496$	−3966.95	−0.359116
$497$	11623.7	1.04909
$498$	−222.160	−0.0199905
$499$	−5108.14	−0.458260	−0.229130	−	0.973396i	$-0.573588\pi$
$499$	−5108.14	−0.458260	−0.229130	+	0.973396i	$0.573588\pi$
$500$	1033.83	0.0924689
$501$	−11937.3	−1.06451
$502$	8930.92	0.794037
$503$	18484.5	1.63854	0.819268	−	0.573411i	$-0.194380\pi$
$503$	18484.5	1.63854	0.819268	+	0.573411i	$0.194380\pi$
$504$	1628.18	0.143898
$505$	−950.754	−0.0837782
$506$	−77.8985	−0.00684389
$507$	−1633.18	−0.143062
$508$	−4637.46	−0.405027
$509$	−18655.8	−1.62456	−0.812281	−	0.583266i	$-0.801775\pi$
$509$	−18655.8	−1.62456	−0.812281	+	0.583266i	$0.801775\pi$
$510$	3901.84	0.338777
$511$	9138.77	0.791145
$512$	−512.000	−0.0441942
$513$	0	0
$514$	6409.55	0.550026
$515$	742.901	0.0635653
$516$	6074.69	0.518262
$517$	1252.62	0.106557
$518$	−17697.4	−1.50112
$519$	10168.2	0.859992
$520$	5302.67	0.447187
$521$	17624.6	1.48205	0.741026	−	0.671476i	$-0.234339\pi$
$521$	17624.6	1.48205	0.741026	+	0.671476i	$0.234339\pi$
$522$	−4336.28	−0.363590
$523$	21021.9	1.75759	0.878797	−	0.477195i	$-0.158346\pi$
$523$	21021.9	1.75759	0.878797	+	0.477195i	$0.158346\pi$
$524$	−2371.58	−0.197715
$525$	−9555.47	−0.794352
$526$	−12677.9	−1.05092
$527$	9888.62	0.817372
$528$	168.504	0.0138887
$529$	−12043.9	−0.989882
$530$	−8952.16	−0.733693
$531$	1616.17	0.132083
$532$	0	0
$533$	−11810.5	−0.959790
$534$	7564.54	0.613014
$535$	−15606.0	−1.26113
$536$	1298.17	0.104613
$537$	115.190	0.00925664
$538$	16148.5	1.29407
$539$	591.079	0.0472348
$540$	1760.94	0.140331
$541$	2916.75	0.231794	0.115897	−	0.993261i	$-0.463026\pi$
$541$	2916.75	0.231794	0.115897	+	0.993261i	$0.463026\pi$
$542$	−14545.4	−1.15272
$543$	−13540.9	−1.07016
$544$	1276.29	0.100589
$545$	−15823.5	−1.24368
$546$	−5515.76	−0.432331
$547$	3287.25	0.256952	0.128476	−	0.991713i	$-0.458991\pi$
$547$	3287.25	0.256952	0.128476	+	0.991713i	$0.458991\pi$
$548$	1189.51	0.0927253
$549$	−4749.29	−0.369207
$550$	−988.921	−0.0766686
$551$	0	0
$552$	−266.281	−0.0205320
$553$	16514.7	1.26994
$554$	10415.6	0.798764
$555$	−19140.4	−1.46390
$556$	812.836	0.0619998
$557$	−17344.8	−1.31943	−0.659713	−	0.751517i	$-0.729322\pi$
$557$	−17344.8	−1.31943	−0.659713	+	0.751517i	$0.729322\pi$
$558$	4462.82	0.338577
$559$	−20579.1	−1.55707
$560$	−5899.41	−0.445171
$561$	−420.039	−0.0316116
$562$	−2792.96	−0.209633
$563$	14621.7	1.09455	0.547275	−	0.836953i	$-0.315665\pi$
$563$	14621.7	1.09455	0.547275	+	0.836953i	$0.315665\pi$
$564$	4281.83	0.319677
$565$	17155.6	1.27742
$566$	−16560.8	−1.22986
$567$	−1831.70	−0.135669
$568$	4112.12	0.303769
$569$	−15937.5	−1.17423	−0.587113	−	0.809505i	$-0.699735\pi$
$569$	−15937.5	−1.17423	−0.587113	+	0.809505i	$0.699735\pi$
$570$	0	0
$571$	5889.36	0.431632	0.215816	−	0.976434i	$-0.430759\pi$
$571$	5889.36	0.431632	0.215816	+	0.976434i	$0.430759\pi$
$572$	−570.840	−0.0417273
$573$	9874.76	0.719937
$574$	13139.6	0.955463
$575$	1562.75	0.113341
$576$	576.000	0.0416667
$577$	−15343.7	−1.10705	−0.553524	−	0.832833i	$-0.686717\pi$
$577$	−15343.7	−1.10705	−0.553524	+	0.832833i	$0.686717\pi$
$578$	6644.53	0.478159
$579$	−1551.41	−0.111355
$580$	15711.8	1.12482
$581$	−837.307	−0.0597889
$582$	9851.32	0.701633
$583$	963.715	0.0684614
$584$	3233.02	0.229081
$585$	−5965.50	−0.421612
$586$	9821.03	0.692326
$587$	−11695.4	−0.822350	−0.411175	−	0.911556i	$-0.634881\pi$
$587$	−11695.4	−0.822350	−0.411175	+	0.911556i	$0.634881\pi$
$588$	2020.49	0.141707
$589$	0	0
$590$	−5855.92	−0.408618
$591$	5392.05	0.375295
$592$	−6260.80	−0.434657
$593$	−10616.2	−0.735171	−0.367585	−	0.929990i	$-0.619815\pi$
$593$	−10616.2	−0.735171	−0.367585	+	0.929990i	$0.619815\pi$
$594$	−189.567	−0.0130944
$595$	14705.8	1.01324
$596$	−997.969	−0.0685879
$597$	−6505.99	−0.446017
$598$	902.077	0.0616867
$599$	−185.712	−0.0126678	−0.00633388	−	0.999980i	$-0.502016\pi$
$599$	−185.712	−0.0126678	−0.00633388	+	0.999980i	$0.502016\pi$
$600$	−3380.44	−0.230010
$601$	21193.3	1.43843	0.719213	−	0.694790i	$-0.244503\pi$
$601$	21193.3	1.43843	0.719213	+	0.694790i	$0.244503\pi$
$602$	22895.1	1.55006
$603$	−1460.45	−0.0986301
$604$	−7352.20	−0.495293
$605$	−21501.0	−1.44486
$606$	349.864	0.0234526
$607$	−2847.55	−0.190410	−0.0952048	−	0.995458i	$-0.530351\pi$
$607$	−2847.55	−0.190410	−0.0952048	+	0.995458i	$0.530351\pi$
$608$	0	0
$609$	−16343.1	−1.08745
$610$	17208.2	1.14220
$611$	−14505.5	−0.960442
$612$	−1435.82	−0.0948362
$613$	−4658.48	−0.306940	−0.153470	−	0.988153i	$-0.549045\pi$
$613$	−4658.48	−0.306940	−0.153470	+	0.988153i	$0.549045\pi$
$614$	12298.0	0.808320
$615$	14211.0	0.931774
$616$	635.081	0.0415392
$617$	−14742.3	−0.961920	−0.480960	−	0.876743i	$-0.659712\pi$
$617$	−14742.3	−0.961920	−0.480960	+	0.876743i	$0.659712\pi$
$618$	−273.378	−0.0177943
$619$	4822.84	0.313160	0.156580	−	0.987665i	$-0.449953\pi$
$619$	4822.84	0.313160	0.156580	+	0.987665i	$0.449953\pi$
$620$	−16170.2	−1.04744
$621$	299.566	0.0193578
$622$	15219.4	0.981098
$623$	28510.2	1.83345
$624$	−1951.31	−0.125184
$625$	−13392.3	−0.857106
$626$	−13862.9	−0.885098
$627$	0	0
$628$	−12908.0	−0.820197
$629$	15606.6	0.989310
$630$	6636.84	0.419711
$631$	−16204.9	−1.02236	−0.511179	−	0.859474i	$-0.670791\pi$
$631$	−16204.9	−1.02236	−0.511179	+	0.859474i	$0.670791\pi$
$632$	5842.39	0.367718
$633$	−407.908	−0.0256128
$634$	−1316.27	−0.0824536
$635$	−18903.4	−1.18135
$636$	3294.27	0.205387
$637$	−6844.79	−0.425746
$638$	−1691.40	−0.104958
$639$	−4626.14	−0.286396
$640$	−2087.03	−0.128902
$641$	−10332.1	−0.636654	−0.318327	−	0.947981i	$-0.603121\pi$
$641$	−10332.1	−0.636654	−0.318327	+	0.947981i	$0.603121\pi$
$642$	5742.79	0.353037
$643$	7814.75	0.479291	0.239645	−	0.970861i	$-0.422969\pi$
$643$	7814.75	0.479291	0.239645	+	0.970861i	$0.422969\pi$
$644$	−1003.59	−0.0614086
$645$	24761.9	1.51162
$646$	0	0
$647$	−18199.1	−1.10584	−0.552921	−	0.833234i	$-0.686487\pi$
$647$	−18199.1	−1.10584	−0.552921	+	0.833234i	$0.686487\pi$
$648$	−648.000	−0.0392837
$649$	630.399	0.0381284
$650$	11451.9	0.691045
$651$	16820.1	1.01264
$652$	7605.91	0.456856
$653$	12741.3	0.763560	0.381780	−	0.924253i	$-0.375311\pi$
$653$	12741.3	0.763560	0.381780	+	0.924253i	$0.375311\pi$
$654$	5822.85	0.348152
$655$	−9667.12	−0.576680
$656$	4648.39	0.276660
$657$	−3637.15	−0.215980
$658$	16137.9	0.956112
$659$	6569.93	0.388358	0.194179	−	0.980966i	$-0.437796\pi$
$659$	6569.93	0.388358	0.194179	+	0.980966i	$0.437796\pi$
$660$	686.864	0.0405093
$661$	31545.6	1.85625	0.928125	−	0.372268i	$-0.121420\pi$
$661$	31545.6	1.85625	0.928125	+	0.372268i	$0.121420\pi$
$662$	15624.7	0.917328
$663$	4864.12	0.284928
$664$	−296.214	−0.0173122
$665$	0	0
$666$	7043.40	0.409799
$667$	2672.85	0.155162
$668$	−15916.4	−0.921891
$669$	3503.30	0.202460
$670$	5291.67	0.305127
$671$	−1852.49	−0.106579
$672$	2170.90	0.124620
$673$	−9314.62	−0.533510	−0.266755	−	0.963764i	$-0.585951\pi$
$673$	−9314.62	−0.533510	−0.266755	+	0.963764i	$0.585951\pi$
$674$	−6445.43	−0.368351
$675$	3802.99	0.216855
$676$	−2177.58	−0.123895
$677$	30826.5	1.75001	0.875006	−	0.484113i	$-0.160858\pi$
$677$	30826.5	1.75001	0.875006	+	0.484113i	$0.160858\pi$
$678$	−6313.03	−0.357596
$679$	37128.9	2.09849
$680$	5202.46	0.293390
$681$	20053.5	1.12842
$682$	1740.75	0.0977373
$683$	24554.7	1.37564	0.687819	−	0.725882i	$-0.258568\pi$
$683$	24554.7	1.37564	0.687819	+	0.725882i	$0.258568\pi$
$684$	0	0
$685$	4848.74	0.270454
$686$	−7897.83	−0.439564
$687$	9370.02	0.520362
$688$	8099.58	0.448828
$689$	−11160.0	−0.617070
$690$	−1085.42	−0.0598861
$691$	−27481.3	−1.51293	−0.756467	−	0.654032i	$-0.773076\pi$
$691$	−27481.3	−1.51293	−0.756467	+	0.654032i	$0.773076\pi$
$692$	13557.6	0.744775
$693$	−714.467	−0.0391636
$694$	−2365.34	−0.129376
$695$	3313.31	0.180836
$696$	−5781.71	−0.314878
$697$	−11587.3	−0.629698
$698$	−2448.40	−0.132770
$699$	11539.4	0.624407
$700$	−12740.6	−0.687929
$701$	−16486.8	−0.888300	−0.444150	−	0.895953i	$-0.646494\pi$
$701$	−16486.8	−0.888300	−0.444150	+	0.895953i	$0.646494\pi$
$702$	2195.22	0.118025
$703$	0	0
$704$	224.673	0.0120279
$705$	17453.8	0.932407
$706$	5202.07	0.277312
$707$	1318.61	0.0701437
$708$	2154.90	0.114387
$709$	13591.8	0.719960	0.359980	−	0.932960i	$-0.382784\pi$
$709$	13591.8	0.719960	0.359980	+	0.932960i	$0.382784\pi$
$710$	16762.0	0.886009
$711$	−6572.68	−0.346688
$712$	10086.1	0.530886
$713$	−2750.84	−0.144488
$714$	−5411.52	−0.283643
$715$	−2326.88	−0.121707
$716$	153.587	0.00801649
$717$	−20202.5	−1.05227
$718$	5210.29	0.270817
$719$	−16184.5	−0.839474	−0.419737	−	0.907646i	$-0.637878\pi$
$719$	−16184.5	−0.839474	−0.419737	+	0.907646i	$0.637878\pi$
$720$	2347.91	0.121530
$721$	−1030.34	−0.0532204
$722$	0	0
$723$	4881.15	0.251082
$724$	−18054.5	−0.926782
$725$	33931.8	1.73820
$726$	7912.06	0.404468
$727$	−4416.25	−0.225295	−0.112648	−	0.993635i	$-0.535933\pi$
$727$	−4416.25	−0.225295	−0.112648	+	0.993635i	$0.535933\pi$
$728$	−7354.34	−0.374410
$729$	729.000	0.0370370
$730$	13178.6	0.668165
$731$	−20190.2	−1.02156
$732$	−6332.38	−0.319743
$733$	26994.3	1.36024	0.680122	−	0.733099i	$-0.261927\pi$
$733$	26994.3	1.36024	0.680122	+	0.733099i	$0.261927\pi$
$734$	−9132.48	−0.459245
$735$	8236.00	0.413319
$736$	−355.041	−0.0177812
$737$	−569.656	−0.0284716
$738$	−5229.44	−0.260838
$739$	−4293.10	−0.213700	−0.106850	−	0.994275i	$-0.534076\pi$
$739$	−4293.10	−0.213700	−0.106850	+	0.994275i	$0.534076\pi$
$740$	−25520.5	−1.26777
$741$	0	0
$742$	12415.9	0.614288
$743$	11959.9	0.590534	0.295267	−	0.955415i	$-0.404591\pi$
$743$	11959.9	0.590534	0.295267	+	0.955415i	$0.404591\pi$
$744$	5950.42	0.293217
$745$	−4067.96	−0.200052
$746$	−9662.22	−0.474208
$747$	333.241	0.0163221
$748$	−560.053	−0.0273764
$749$	21644.2	1.05589
$750$	−1550.75	−0.0755005
$751$	27023.9	1.31307	0.656536	−	0.754294i	$-0.272021\pi$
$751$	27023.9	1.31307	0.656536	+	0.754294i	$0.272021\pi$
$752$	5709.11	0.276848
$753$	−13396.4	−0.648329
$754$	19586.6	0.946025
$755$	−29969.3	−1.44463
$756$	−2442.27	−0.117493
$757$	−13999.1	−0.672135	−0.336067	−	0.941838i	$-0.609097\pi$
$757$	−13999.1	−0.672135	−0.336067	+	0.941838i	$0.609097\pi$
$758$	−8825.55	−0.422900
$759$	116.848	0.00558801
$760$	0	0
$761$	17713.5	0.843775	0.421888	−	0.906648i	$-0.361368\pi$
$761$	17713.5	0.843775	0.421888	+	0.906648i	$0.361368\pi$
$762$	6956.19	0.330703
$763$	21945.9	1.04128
$764$	13166.3	0.623484
$765$	−5852.76	−0.276611
$766$	11765.9	0.554985
$767$	−7300.12	−0.343666
$768$	768.000	0.0360844
$769$	−3308.66	−0.155154	−0.0775770	−	0.996986i	$-0.524718\pi$
$769$	−3308.66	−0.155154	−0.0775770	+	0.996986i	$0.524718\pi$
$770$	2588.74	0.121158
$771$	−9614.33	−0.449094
$772$	−2068.55	−0.0964360
$773$	8043.72	0.374272	0.187136	−	0.982334i	$-0.440079\pi$
$773$	8043.72	0.374272	0.187136	+	0.982334i	$0.440079\pi$
$774$	−9112.03	−0.423159
$775$	−34921.9	−1.61862
$776$	13135.1	0.607632
$777$	26546.1	1.22566
$778$	−18180.5	−0.837795
$779$	0	0
$780$	−7954.00	−0.365127
$781$	−1804.46	−0.0826741
$782$	885.029	0.0404713
$783$	6504.42	0.296870
$784$	2693.99	0.122722
$785$	−52616.0	−2.39229
$786$	3557.37	0.161434
$787$	29500.7	1.33620	0.668099	−	0.744073i	$-0.267108\pi$
$787$	29500.7	1.33620	0.668099	+	0.744073i	$0.267108\pi$
$788$	7189.40	0.325015
$789$	19016.8	0.858071
$790$	23815.0	1.07253
$791$	−23793.4	−1.06953
$792$	−252.757	−0.0113400
$793$	21452.1	0.960640
$794$	−5676.47	−0.253716
$795$	13428.2	0.599057
$796$	−8674.65	−0.386262
$797$	29002.5	1.28898	0.644492	−	0.764611i	$-0.277069\pi$
$797$	29002.5	1.28898	0.644492	+	0.764611i	$0.277069\pi$
$798$	0	0
$799$	−14231.4	−0.630125
$800$	−4507.25	−0.199194
$801$	−11346.8	−0.500524
$802$	23330.9	1.02723
$803$	−1418.69	−0.0623469
$804$	−1947.26	−0.0854162
$805$	−4090.89	−0.179112
$806$	−20158.2	−0.880945
$807$	−24222.7	−1.05661
$808$	466.486	0.0203105
$809$	−875.472	−0.0380469	−0.0190235	−	0.999819i	$-0.506056\pi$
$809$	−875.472	−0.0380469	−0.0190235	+	0.999819i	$0.506056\pi$
$810$	−2641.40	−0.114580
$811$	6299.10	0.272739	0.136369	−	0.990658i	$-0.456457\pi$
$811$	6299.10	0.272739	0.136369	+	0.990658i	$0.456457\pi$
$812$	−21790.9	−0.941760
$813$	21818.0	0.941196
$814$	2747.32	0.118297
$815$	31003.5	1.33252
$816$	−1914.43	−0.0821306
$817$	0	0
$818$	11200.1	0.478732
$819$	8273.64	0.352997
$820$	18947.9	0.806940
$821$	6795.60	0.288877	0.144438	−	0.989514i	$-0.453862\pi$
$821$	6795.60	0.288877	0.144438	+	0.989514i	$0.453862\pi$
$822$	−1784.27	−0.0757099
$823$	−42134.6	−1.78459	−0.892296	−	0.451450i	$-0.850907\pi$
$823$	−42134.6	−1.78459	−0.892296	+	0.451450i	$0.850907\pi$
$824$	−364.503	−0.0154103
$825$	1483.38	0.0625997
$826$	8121.66	0.342117
$827$	8490.46	0.357004	0.178502	−	0.983940i	$-0.442875\pi$
$827$	8490.46	0.357004	0.178502	+	0.983940i	$0.442875\pi$
$828$	399.421	0.0167643
$829$	−3373.19	−0.141322	−0.0706608	−	0.997500i	$-0.522511\pi$
$829$	−3373.19	−0.141322	−0.0706608	+	0.997500i	$0.522511\pi$
$830$	−1207.44	−0.0504949
$831$	−15623.4	−0.652188
$832$	−2601.74	−0.108413
$833$	−6715.43	−0.279323
$834$	−1219.25	−0.0506227
$835$	−64879.0	−2.68890
$836$	0	0
$837$	−6694.23	−0.276447
$838$	−7545.05	−0.311025
$839$	−12320.7	−0.506982	−0.253491	−	0.967338i	$-0.581579\pi$
$839$	−12320.7	−0.506982	−0.253491	+	0.967338i	$0.581579\pi$
$840$	8849.12	0.363481
$841$	33646.0	1.37956
$842$	21739.0	0.889757
$843$	4189.44	0.171165
$844$	−543.877	−0.0221813
$845$	−8876.32	−0.361367
$846$	−6422.75	−0.261015
$847$	29820.0	1.20971
$848$	4392.36	0.177871
$849$	24841.2	1.00418
$850$	11235.4	0.453380
$851$	−4341.49	−0.174882
$852$	−6168.19	−0.248026
$853$	−1168.04	−0.0468850	−0.0234425	−	0.999725i	$-0.507463\pi$
$853$	−1168.04	−0.0468850	−0.0234425	+	0.999725i	$0.507463\pi$
$854$	−23866.3	−0.956310
$855$	0	0
$856$	7657.05	0.305739
$857$	12186.2	0.485731	0.242866	−	0.970060i	$-0.421913\pi$
$857$	12186.2	0.485731	0.242866	+	0.970060i	$0.421913\pi$
$858$	856.261	0.0340702
$859$	23873.1	0.948241	0.474121	−	0.880460i	$-0.342766\pi$
$859$	23873.1	0.948241	0.474121	+	0.880460i	$0.342766\pi$
$860$	33015.8	1.30910
$861$	−19709.4	−0.780133
$862$	−22437.2	−0.886558
$863$	31328.3	1.23572	0.617860	−	0.786288i	$-0.288000\pi$
$863$	31328.3	1.23572	0.617860	+	0.786288i	$0.288000\pi$
$864$	−864.000	−0.0340207
$865$	55264.2	2.17230
$866$	−8317.60	−0.326378
$867$	−9966.80	−0.390415
$868$	22426.7	0.876974
$869$	−2563.72	−0.100078
$870$	−23567.6	−0.918411
$871$	6596.71	0.256626
$872$	7763.80	0.301508
$873$	−14777.0	−0.572881
$874$	0	0
$875$	−5844.67	−0.225812
$876$	−4849.53	−0.187044
$877$	−31494.1	−1.21263	−0.606316	−	0.795223i	$-0.707354\pi$
$877$	−31494.1	−1.21263	−0.606316	+	0.795223i	$0.707354\pi$
$878$	17090.1	0.656906
$879$	−14731.5	−0.565282
$880$	915.819	0.0350821
$881$	−11684.0	−0.446815	−0.223408	−	0.974725i	$-0.571718\pi$
$881$	−11684.0	−0.446815	−0.223408	+	0.974725i	$0.571718\pi$
$882$	−3030.73	−0.115703
$883$	13339.0	0.508374	0.254187	−	0.967155i	$-0.418192\pi$
$883$	13339.0	0.508374	0.254187	+	0.967155i	$0.418192\pi$
$884$	6485.50	0.246754
$885$	8783.88	0.333635
$886$	33372.0	1.26541
$887$	−8731.75	−0.330534	−0.165267	−	0.986249i	$-0.552849\pi$
$887$	−8731.75	−0.330534	−0.165267	+	0.986249i	$0.552849\pi$
$888$	9391.19	0.354896
$889$	26217.4	0.989092
$890$	41113.2	1.54844
$891$	284.351	0.0106915
$892$	4671.07	0.175335
$893$	0	0
$894$	1496.95	0.0560018
$895$	626.056	0.0233818
$896$	2894.54	0.107924
$897$	−1353.12	−0.0503670
$898$	9830.91	0.365325
$899$	−59728.5	−2.21586
$900$	5070.65	0.187802
$901$	−10949.1	−0.404846
$902$	−2039.78	−0.0752962
$903$	−34342.6	−1.26561
$904$	−8417.37	−0.309688
$905$	−73594.4	−2.70316
$906$	11028.3	0.404405
$907$	−20755.8	−0.759851	−0.379925	−	0.925017i	$-0.624050\pi$
$907$	−20755.8	−0.759851	−0.379925	+	0.925017i	$0.624050\pi$
$908$	26738.0	0.977238
$909$	−524.797	−0.0191490
$910$	−29978.1	−1.09205
$911$	14277.3	0.519239	0.259619	−	0.965711i	$-0.416403\pi$
$911$	14277.3	0.519239	0.259619	+	0.965711i	$0.416403\pi$
$912$	0	0
$913$	129.983	0.00471172
$914$	28510.1	1.03176
$915$	−25812.3	−0.932599
$916$	12493.4	0.450647
$917$	13407.5	0.482829
$918$	2153.74	0.0774334
$919$	−27017.4	−0.969773	−0.484887	−	0.874577i	$-0.661139\pi$
$919$	−27017.4	−0.969773	−0.484887	+	0.874577i	$0.661139\pi$
$920$	−1447.23	−0.0518629
$921$	−18447.1	−0.659991
$922$	20426.3	0.729615
$923$	20895.9	0.745175
$924$	−952.622	−0.0339166
$925$	−55115.2	−1.95911
$926$	−9470.66	−0.336096
$927$	410.066	0.0145290
$928$	−7708.95	−0.272692
$929$	35968.7	1.27029	0.635143	−	0.772395i	$-0.280941\pi$
$929$	35968.7	1.27029	0.635143	+	0.772395i	$0.280941\pi$
$930$	24255.3	0.855230
$931$	0	0
$932$	15385.9	0.540752
$933$	−22829.1	−0.801063
$934$	655.852	0.0229766
$935$	−2282.91	−0.0798493
$936$	2926.96	0.102212
$937$	−26516.3	−0.924493	−0.462247	−	0.886751i	$-0.652956\pi$
$937$	−26516.3	−0.924493	−0.462247	+	0.886751i	$0.652956\pi$
$938$	−7339.09	−0.255469
$939$	20794.3	0.722679
$940$	23271.7	0.807488
$941$	33822.7	1.17172	0.585860	−	0.810412i	$-0.300757\pi$
$941$	33822.7	1.17172	0.585860	+	0.810412i	$0.300757\pi$
$942$	19362.0	0.669688
$943$	3223.38	0.111312
$944$	2873.20	0.0990621
$945$	−9955.26	−0.342693
$946$	−3554.21	−0.122154
$947$	7953.52	0.272919	0.136460	−	0.990646i	$-0.456428\pi$
$947$	7953.52	0.272919	0.136460	+	0.990646i	$0.456428\pi$
$948$	−8763.58	−0.300240
$949$	16428.7	0.561958
$950$	0	0
$951$	1974.40	0.0673231
$952$	−7215.36	−0.245642
$953$	31173.3	1.05960	0.529802	−	0.848121i	$-0.322266\pi$
$953$	31173.3	1.05960	0.529802	+	0.848121i	$0.322266\pi$
$954$	−4941.41	−0.167698
$955$	53669.2	1.81853
$956$	−26936.7	−0.911291
$957$	2537.09	0.0856976
$958$	11327.7	0.382027
$959$	−6724.79	−0.226439
$960$	3130.55	0.105248
$961$	31680.4	1.06342
$962$	−31814.4	−1.06626
$963$	−8614.19	−0.288254
$964$	6508.20	0.217443
$965$	−8431.89	−0.281277
$966$	1505.39	0.0501399
$967$	−46223.9	−1.53719	−0.768594	−	0.639737i	$-0.779043\pi$
$967$	−46223.9	−1.53719	−0.768594	+	0.639737i	$0.779043\pi$
$968$	10549.4	0.350280
$969$	0	0
$970$	53541.8	1.77229
$971$	11979.4	0.395920	0.197960	−	0.980210i	$-0.436568\pi$
$971$	11979.4	0.395920	0.197960	+	0.980210i	$0.436568\pi$
$972$	972.000	0.0320750
$973$	−4595.28	−0.151406
$974$	−18436.5	−0.606513
$975$	−17177.8	−0.564236
$976$	−8443.18	−0.276905
$977$	−18380.6	−0.601891	−0.300946	−	0.953641i	$-0.597302\pi$
$977$	−18380.6	−0.601891	−0.300946	+	0.953641i	$0.597302\pi$
$978$	−11408.9	−0.373022
$979$	−4425.90	−0.144486
$980$	10981.3	0.357945
$981$	−8734.27	−0.284265
$982$	−41501.0	−1.34862
$983$	27642.2	0.896898	0.448449	−	0.893808i	$-0.351977\pi$
$983$	27642.2	0.896898	0.448449	+	0.893808i	$0.351977\pi$
$984$	−6972.58	−0.225892
$985$	29305.7	0.947978
$986$	19216.5	0.620667
$987$	−24206.9	−0.780662
$988$	0	0
$989$	5616.57	0.180583
$990$	−1030.30	−0.0330757
$991$	−2836.44	−0.0909209	−0.0454605	−	0.998966i	$-0.514476\pi$
$991$	−2836.44	−0.0909209	−0.0454605	+	0.998966i	$0.514476\pi$
$992$	7933.90	0.253933
$993$	−23437.0	−0.748995
$994$	−23247.5	−0.741816
$995$	−35359.9	−1.12662
$996$	444.321	0.0141354
$997$	29037.6	0.922397	0.461198	−	0.887297i	$-0.347420\pi$
$997$	29037.6	0.922397	0.461198	+	0.887297i	$0.347420\pi$
$998$	10216.3	0.324039
$999$	−10565.1	−0.334599

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2166.4.a.bk.1.9		9
19.9	even	9		114.4.i.c.43.3	✓	18
19.17	even	9		114.4.i.c.61.3	yes	18
19.18	odd	2		2166.4.a.bl.1.9		9

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
114.4.i.c.43.3	✓	18	19.9	even	9
114.4.i.c.61.3	yes	18	19.17	even	9
2166.4.a.bk.1.9		9	1.1	even	1	trivial
2166.4.a.bl.1.9		9	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)

\(f(q)\)	\(=\)	\(q-2.00000 q^{2} +3.00000 q^{3} +4.00000 q^{4} +16.3050 q^{5} -6.00000 q^{6} -22.6136 q^{7} -8.00000 q^{8} +9.00000 q^{9} +O(q^{10})\)	\(q-2.00000 q^{2} +3.00000 q^{3} +4.00000 q^{4} +16.3050 q^{5} -6.00000 q^{6} -22.6136 q^{7} -8.00000 q^{8} +9.00000 q^{9} -32.6099 q^{10} +3.51051 q^{11} +12.0000 q^{12} -40.6523 q^{13} +45.2272 q^{14} +48.9149 q^{15} +16.0000 q^{16} -39.8840 q^{17} -18.0000 q^{18} +65.2198 q^{20} -67.8407 q^{21} -7.02102 q^{22} +11.0950 q^{23} -24.0000 q^{24} +140.852 q^{25} +81.3045 q^{26} +27.0000 q^{27} -90.4543 q^{28} +240.905 q^{29} -97.8297 q^{30} -247.934 q^{31} -32.0000 q^{32} +10.5315 q^{33} +79.7680 q^{34} -368.713 q^{35} +36.0000 q^{36} -391.300 q^{37} -121.957 q^{39} -130.440 q^{40} +290.524 q^{41} +135.681 q^{42} +506.224 q^{43} +14.0420 q^{44} +146.745 q^{45} -22.1901 q^{46} +356.819 q^{47} +48.0000 q^{48} +168.374 q^{49} -281.703 q^{50} -119.652 q^{51} -162.609 q^{52} +274.523 q^{53} -54.0000 q^{54} +57.2387 q^{55} +180.909 q^{56} -481.809 q^{58} +179.575 q^{59} +195.659 q^{60} -527.699 q^{61} +495.869 q^{62} -203.522 q^{63} +64.0000 q^{64} -662.833 q^{65} -21.0631 q^{66} -162.272 q^{67} -159.536 q^{68} +33.2851 q^{69} +737.427 q^{70} -514.015 q^{71} -72.0000 q^{72} -404.127 q^{73} +782.600 q^{74} +422.555 q^{75} -79.3852 q^{77} +243.914 q^{78} -730.298 q^{79} +260.879 q^{80} +81.0000 q^{81} -581.049 q^{82} +37.0267 q^{83} -271.363 q^{84} -650.307 q^{85} -1012.45 q^{86} +722.714 q^{87} -28.0841 q^{88} -1260.76 q^{89} -293.489 q^{90} +919.293 q^{91} +44.3802 q^{92} -743.803 q^{93} -713.638 q^{94} -96.0000 q^{96} -1641.89 q^{97} -336.748 q^{98} +31.5946 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 9 q - 18 q^{2} + 27 q^{3} + 36 q^{4} - 3 q^{5} - 54 q^{6} + 30 q^{7} - 72 q^{8} + 81 q^{9}+O(q^{10}) \) 9 * q - 18 * q^2 + 27 * q^3 + 36 * q^4 - 3 * q^5 - 54 * q^6 + 30 * q^7 - 72 * q^8 + 81 * q^9	\( 9 q - 18 q^{2} + 27 q^{3} + 36 q^{4} - 3 q^{5} - 54 q^{6} + 30 q^{7} - 72 q^{8} + 81 q^{9} + 6 q^{10} - 57 q^{11} + 108 q^{12} - 93 q^{13} - 60 q^{14} - 9 q^{15} + 144 q^{16} - 147 q^{17} - 162 q^{18} - 12 q^{20} + 90 q^{21} + 114 q^{22} - 24 q^{23} - 216 q^{24} + 390 q^{25} + 186 q^{26} + 243 q^{27} + 120 q^{28} - 225 q^{29} + 18 q^{30} - 429 q^{31} - 288 q^{32} - 171 q^{33} + 294 q^{34} - 438 q^{35} + 324 q^{36} - 159 q^{37} - 279 q^{39} + 24 q^{40} - 567 q^{41} - 180 q^{42} + 1230 q^{43} - 228 q^{44} - 27 q^{45} + 48 q^{46} - 342 q^{47} + 432 q^{48} + 249 q^{49} - 780 q^{50} - 441 q^{51} - 372 q^{52} + 285 q^{53} - 486 q^{54} + 174 q^{55} - 240 q^{56} + 450 q^{58} + 171 q^{59} - 36 q^{60} - 816 q^{61} + 858 q^{62} + 270 q^{63} + 576 q^{64} - 1209 q^{65} + 342 q^{66} - 2388 q^{67} - 588 q^{68} - 72 q^{69} + 876 q^{70} - 2022 q^{71} - 648 q^{72} + 645 q^{73} + 318 q^{74} + 1170 q^{75} - 3843 q^{77} + 558 q^{78} - 3516 q^{79} - 48 q^{80} + 729 q^{81} + 1134 q^{82} - 156 q^{83} + 360 q^{84} - 291 q^{85} - 2460 q^{86} - 675 q^{87} + 456 q^{88} - 1959 q^{89} + 54 q^{90} - 501 q^{91} - 96 q^{92} - 1287 q^{93} + 684 q^{94} - 864 q^{96} + 2457 q^{97} - 498 q^{98} - 513 q^{99}+O(q^{100}) \) 9 * q - 18 * q^2 + 27 * q^3 + 36 * q^4 - 3 * q^5 - 54 * q^6 + 30 * q^7 - 72 * q^8 + 81 * q^9 + 6 * q^10 - 57 * q^11 + 108 * q^12 - 93 * q^13 - 60 * q^14 - 9 * q^15 + 144 * q^16 - 147 * q^17 - 162 * q^18 - 12 * q^20 + 90 * q^21 + 114 * q^22 - 24 * q^23 - 216 * q^24 + 390 * q^25 + 186 * q^26 + 243 * q^27 + 120 * q^28 - 225 * q^29 + 18 * q^30 - 429 * q^31 - 288 * q^32 - 171 * q^33 + 294 * q^34 - 438 * q^35 + 324 * q^36 - 159 * q^37 - 279 * q^39 + 24 * q^40 - 567 * q^41 - 180 * q^42 + 1230 * q^43 - 228 * q^44 - 27 * q^45 + 48 * q^46 - 342 * q^47 + 432 * q^48 + 249 * q^49 - 780 * q^50 - 441 * q^51 - 372 * q^52 + 285 * q^53 - 486 * q^54 + 174 * q^55 - 240 * q^56 + 450 * q^58 + 171 * q^59 - 36 * q^60 - 816 * q^61 + 858 * q^62 + 270 * q^63 + 576 * q^64 - 1209 * q^65 + 342 * q^66 - 2388 * q^67 - 588 * q^68 - 72 * q^69 + 876 * q^70 - 2022 * q^71 - 648 * q^72 + 645 * q^73 + 318 * q^74 + 1170 * q^75 - 3843 * q^77 + 558 * q^78 - 3516 * q^79 - 48 * q^80 + 729 * q^81 + 1134 * q^82 - 156 * q^83 + 360 * q^84 - 291 * q^85 - 2460 * q^86 - 675 * q^87 + 456 * q^88 - 1959 * q^89 + 54 * q^90 - 501 * q^91 - 96 * q^92 - 1287 * q^93 + 684 * q^94 - 864 * q^96 + 2457 * q^97 - 498 * q^98 - 513 * q^99

Introduction

L-functions

Modular forms

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