LMFDB - Embedded newform 966.4.a.n.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [966,4,Mod(1,966)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(966, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("966.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 966 = 2 \cdot 3 \cdot 7 \cdot 23 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	966.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:	$56.9958450655$
Analytic rank:	$0$
Dimension:	$5$
Coefficient field:	$\mathbb{Q}[x]/(x^{5} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 2x^{4} - 267x^{3} + 1502x^{2} + 1857x + 198 $ x^5 - 2x^4 - 267x^3 + 1502x^2 + 1857x + 198
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-0.118152$ of defining polynomial
Character	$\chi$	$=$	966.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.00000 q^{2} +3.00000 q^{3} +4.00000 q^{4} +3.69480 q^{5} -6.00000 q^{6} -7.00000 q^{7} -8.00000 q^{8} +9.00000 q^{9} +O(q^{10})$	\(q-2.00000 q^{2} +3.00000 q^{3} +4.00000 q^{4} +3.69480 q^{5} -6.00000 q^{6} -7.00000 q^{7} -8.00000 q^{8} +9.00000 q^{9} -7.38960 q^{10} -2.09584 q^{11} +12.0000 q^{12} +92.2078 q^{13} +14.0000 q^{14} +11.0844 q^{15} +16.0000 q^{16} +38.7373 q^{17} -18.0000 q^{18} +26.2364 q^{19} +14.7792 q^{20} -21.0000 q^{21} +4.19168 q^{22} +23.0000 q^{23} -24.0000 q^{24} -111.348 q^{25} -184.416 q^{26} +27.0000 q^{27} -28.0000 q^{28} -106.890 q^{29} -22.1688 q^{30} -75.0026 q^{31} -32.0000 q^{32} -6.28752 q^{33} -77.4745 q^{34} -25.8636 q^{35} +36.0000 q^{36} +307.677 q^{37} -52.4729 q^{38} +276.624 q^{39} -29.5584 q^{40} -187.148 q^{41} +42.0000 q^{42} -64.4008 q^{43} -8.38336 q^{44} +33.2532 q^{45} -46.0000 q^{46} +406.429 q^{47} +48.0000 q^{48} +49.0000 q^{49} +222.697 q^{50} +116.212 q^{51} +368.831 q^{52} -127.354 q^{53} -54.0000 q^{54} -7.74371 q^{55} +56.0000 q^{56} +78.7093 q^{57} +213.780 q^{58} +506.493 q^{59} +44.3376 q^{60} +274.652 q^{61} +150.005 q^{62} -63.0000 q^{63} +64.0000 q^{64} +340.690 q^{65} +12.5750 q^{66} -211.711 q^{67} +154.949 q^{68} +69.0000 q^{69} +51.7272 q^{70} -864.233 q^{71} -72.0000 q^{72} +981.054 q^{73} -615.355 q^{74} -334.045 q^{75} +104.946 q^{76} +14.6709 q^{77} -553.247 q^{78} -459.857 q^{79} +59.1168 q^{80} +81.0000 q^{81} +374.296 q^{82} -55.0641 q^{83} -84.0000 q^{84} +143.127 q^{85} +128.802 q^{86} -320.670 q^{87} +16.7667 q^{88} +397.741 q^{89} -66.5064 q^{90} -645.455 q^{91} +92.0000 q^{92} -225.008 q^{93} -812.858 q^{94} +96.9385 q^{95} -96.0000 q^{96} +1517.46 q^{97} -98.0000 q^{98} -18.8625 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - 10 q^{2} + 15 q^{3} + 20 q^{4} - 30 q^{6} - 35 q^{7} - 40 q^{8} + 45 q^{9}+O(q^{10}) $ 5 * q - 10 * q^2 + 15 * q^3 + 20 * q^4 - 30 * q^6 - 35 * q^7 - 40 * q^8 + 45 * q^9	\( 5 q - 10 q^{2} + 15 q^{3} + 20 q^{4} - 30 q^{6} - 35 q^{7} - 40 q^{8} + 45 q^{9} + 18 q^{11} + 60 q^{12} - 22 q^{13} + 70 q^{14} + 80 q^{16} + 44 q^{17} - 90 q^{18} - 14 q^{19} - 105 q^{21} - 36 q^{22} + 115 q^{23} - 120 q^{24} + 43 q^{25} + 44 q^{26} + 135 q^{27} - 140 q^{28} - 39 q^{29} - 100 q^{31} - 160 q^{32} + 54 q^{33} - 88 q^{34} + 180 q^{36} + 255 q^{37} + 28 q^{38} - 66 q^{39} + 69 q^{41} + 210 q^{42} + 912 q^{43} + 72 q^{44} - 230 q^{46} - 319 q^{47} + 240 q^{48} + 245 q^{49} - 86 q^{50} + 132 q^{51} - 88 q^{52} + 745 q^{53} - 270 q^{54} + 2199 q^{55} + 280 q^{56} - 42 q^{57} + 78 q^{58} - 315 q^{59} + 1091 q^{61} + 200 q^{62} - 315 q^{63} + 320 q^{64} + 533 q^{65} - 108 q^{66} + 991 q^{67} + 176 q^{68} + 345 q^{69} + 923 q^{71} - 360 q^{72} + 1144 q^{73} - 510 q^{74} + 129 q^{75} - 56 q^{76} - 126 q^{77} + 132 q^{78} - 110 q^{79} + 405 q^{81} - 138 q^{82} + 218 q^{83} - 420 q^{84} + 2973 q^{85} - 1824 q^{86} - 117 q^{87} - 144 q^{88} - 13 q^{89} + 154 q^{91} + 460 q^{92} - 300 q^{93} + 638 q^{94} - 347 q^{95} - 480 q^{96} + 761 q^{97} - 490 q^{98} + 162 q^{99}+O(q^{100}) \) 5 * q - 10 * q^2 + 15 * q^3 + 20 * q^4 - 30 * q^6 - 35 * q^7 - 40 * q^8 + 45 * q^9 + 18 * q^11 + 60 * q^12 - 22 * q^13 + 70 * q^14 + 80 * q^16 + 44 * q^17 - 90 * q^18 - 14 * q^19 - 105 * q^21 - 36 * q^22 + 115 * q^23 - 120 * q^24 + 43 * q^25 + 44 * q^26 + 135 * q^27 - 140 * q^28 - 39 * q^29 - 100 * q^31 - 160 * q^32 + 54 * q^33 - 88 * q^34 + 180 * q^36 + 255 * q^37 + 28 * q^38 - 66 * q^39 + 69 * q^41 + 210 * q^42 + 912 * q^43 + 72 * q^44 - 230 * q^46 - 319 * q^47 + 240 * q^48 + 245 * q^49 - 86 * q^50 + 132 * q^51 - 88 * q^52 + 745 * q^53 - 270 * q^54 + 2199 * q^55 + 280 * q^56 - 42 * q^57 + 78 * q^58 - 315 * q^59 + 1091 * q^61 + 200 * q^62 - 315 * q^63 + 320 * q^64 + 533 * q^65 - 108 * q^66 + 991 * q^67 + 176 * q^68 + 345 * q^69 + 923 * q^71 - 360 * q^72 + 1144 * q^73 - 510 * q^74 + 129 * q^75 - 56 * q^76 - 126 * q^77 + 132 * q^78 - 110 * q^79 + 405 * q^81 - 138 * q^82 + 218 * q^83 - 420 * q^84 + 2973 * q^85 - 1824 * q^86 - 117 * q^87 - 144 * q^88 - 13 * q^89 + 154 * q^91 + 460 * q^92 - 300 * q^93 + 638 * q^94 - 347 * q^95 - 480 * q^96 + 761 * q^97 - 490 * q^98 + 162 * 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
$5$	3.69480	0.330473	0.165237	−	0.986254i	$-0.447161\pi$
$5$	3.69480	0.330473	0.165237	+	0.986254i	$0.447161\pi$
$6$	−6.00000	−0.408248
$7$	−7.00000	−0.377964
$7$	−7.00000	−0.377964
$8$	−8.00000	−0.353553
$9$	9.00000	0.333333
$10$	−7.38960	−0.233680
$11$	−2.09584	−0.0574472	−0.0287236	−	0.999587i	$-0.509144\pi$
$11$	−2.09584	−0.0574472	−0.0287236	+	0.999587i	$0.509144\pi$
$12$	12.0000	0.288675
$13$	92.2078	1.96722	0.983610	−	0.180311i	$-0.0577103\pi$
$13$	92.2078	1.96722	0.983610	+	0.180311i	$0.0577103\pi$
$14$	14.0000	0.267261
$15$	11.0844	0.190799
$16$	16.0000	0.250000
$17$	38.7373	0.552657	0.276328	−	0.961063i	$-0.410882\pi$
$17$	38.7373	0.552657	0.276328	+	0.961063i	$0.410882\pi$
$18$	−18.0000	−0.235702
$19$	26.2364	0.316792	0.158396	−	0.987376i	$-0.449368\pi$
$19$	26.2364	0.316792	0.158396	+	0.987376i	$0.449368\pi$
$20$	14.7792	0.165237
$21$	−21.0000	−0.218218
$22$	4.19168	0.0406213
$23$	23.0000	0.208514
$23$	23.0000	0.208514
$24$	−24.0000	−0.204124
$25$	−111.348	−0.890787
$26$	−184.416	−1.39103
$27$	27.0000	0.192450
$28$	−28.0000	−0.188982
$29$	−106.890	−0.684448	−0.342224	−	0.939618i	$-0.611180\pi$
$29$	−106.890	−0.684448	−0.342224	+	0.939618i	$0.611180\pi$
$30$	−22.1688	−0.134915
$31$	−75.0026	−0.434544	−0.217272	−	0.976111i	$-0.569716\pi$
$31$	−75.0026	−0.434544	−0.217272	+	0.976111i	$0.569716\pi$
$32$	−32.0000	−0.176777
$33$	−6.28752	−0.0331672
$34$	−77.4745	−0.390787
$35$	−25.8636	−0.124907
$36$	36.0000	0.166667
$37$	307.677	1.36708	0.683538	−	0.729915i	$-0.260440\pi$
$37$	307.677	1.36708	0.683538	+	0.729915i	$0.260440\pi$
$38$	−52.4729	−0.224006
$39$	276.624	1.13577
$40$	−29.5584	−0.116840
$41$	−187.148	−0.712868	−0.356434	−	0.934320i	$-0.616008\pi$
$41$	−187.148	−0.712868	−0.356434	+	0.934320i	$0.616008\pi$
$42$	42.0000	0.154303
$43$	−64.4008	−0.228396	−0.114198	−	0.993458i	$-0.536430\pi$
$43$	−64.4008	−0.228396	−0.114198	+	0.993458i	$0.536430\pi$
$44$	−8.38336	−0.0287236
$45$	33.2532	0.110158
$46$	−46.0000	−0.147442
$47$	406.429	1.26136	0.630679	−	0.776044i	$-0.282777\pi$
$47$	406.429	1.26136	0.630679	+	0.776044i	$0.282777\pi$
$48$	48.0000	0.144338
$49$	49.0000	0.142857
$50$	222.697	0.629882
$51$	116.212	0.319077
$52$	368.831	0.983610
$53$	−127.354	−0.330064	−0.165032	−	0.986288i	$-0.552773\pi$
$53$	−127.354	−0.330064	−0.165032	+	0.986288i	$0.552773\pi$
$54$	−54.0000	−0.136083
$55$	−7.74371	−0.0189848
$56$	56.0000	0.133631
$57$	78.7093	0.182900
$58$	213.780	0.483978
$59$	506.493	1.11762	0.558812	−	0.829294i	$-0.311257\pi$
$59$	506.493	1.11762	0.558812	+	0.829294i	$0.311257\pi$
$60$	44.3376	0.0953994
$61$	274.652	0.576484	0.288242	−	0.957558i	$-0.406929\pi$
$61$	274.652	0.576484	0.288242	+	0.957558i	$0.406929\pi$
$62$	150.005	0.307269
$63$	−63.0000	−0.125988
$64$	64.0000	0.125000
$65$	340.690	0.650113
$66$	12.5750	0.0234527
$67$	−211.711	−0.386038	−0.193019	−	0.981195i	$-0.561828\pi$
$67$	−211.711	−0.386038	−0.193019	+	0.981195i	$0.561828\pi$
$68$	154.949	0.276328
$69$	69.0000	0.120386
$70$	51.7272	0.0883227
$71$	−864.233	−1.44459	−0.722293	−	0.691587i	$-0.756912\pi$
$71$	−864.233	−1.44459	−0.722293	+	0.691587i	$0.756912\pi$
$72$	−72.0000	−0.117851
$73$	981.054	1.57293	0.786463	−	0.617637i	$-0.211910\pi$
$73$	981.054	1.57293	0.786463	+	0.617637i	$0.211910\pi$
$74$	−615.355	−0.966669
$75$	−334.045	−0.514296
$76$	104.946	0.158396
$77$	14.6709	0.0217130
$78$	−553.247	−0.803114
$79$	−459.857	−0.654911	−0.327455	−	0.944867i	$-0.606191\pi$
$79$	−459.857	−0.654911	−0.327455	+	0.944867i	$0.606191\pi$
$80$	59.1168	0.0826183
$81$	81.0000	0.111111
$82$	374.296	0.504074
$83$	−55.0641	−0.0728201	−0.0364100	−	0.999337i	$-0.511592\pi$
$83$	−55.0641	−0.0728201	−0.0364100	+	0.999337i	$0.511592\pi$
$84$	−84.0000	−0.109109
$85$	143.127	0.182638
$86$	128.802	0.161500
$87$	−320.670	−0.395166
$88$	16.7667	0.0203107
$89$	397.741	0.473713	0.236856	−	0.971545i	$-0.423883\pi$
$89$	397.741	0.473713	0.236856	+	0.971545i	$0.423883\pi$
$90$	−66.5064	−0.0778933
$91$	−645.455	−0.743539
$92$	92.0000	0.104257
$93$	−225.008	−0.250884
$94$	−812.858	−0.891914
$95$	96.9385	0.104691
$96$	−96.0000	−0.102062
$97$	1517.46	1.58840	0.794202	−	0.607654i	$-0.207889\pi$
$97$	1517.46	1.58840	0.794202	+	0.607654i	$0.207889\pi$
$98$	−98.0000	−0.101015
$99$	−18.8625	−0.0191491
$100$	−445.394	−0.445394
$101$	484.447	0.477270	0.238635	−	0.971109i	$-0.423300\pi$
$101$	484.447	0.477270	0.238635	+	0.971109i	$0.423300\pi$
$102$	−232.424	−0.225621
$103$	−442.436	−0.423248	−0.211624	−	0.977351i	$-0.567875\pi$
$103$	−442.436	−0.423248	−0.211624	+	0.977351i	$0.567875\pi$
$104$	−737.663	−0.695517
$105$	−77.5909	−0.0721152
$106$	254.707	0.233390
$107$	−20.6936	−0.0186965	−0.00934825	−	0.999956i	$-0.502976\pi$
$107$	−20.6936	−0.0186965	−0.00934825	+	0.999956i	$0.502976\pi$
$108$	108.000	0.0962250
$109$	612.027	0.537813	0.268906	−	0.963166i	$-0.413338\pi$
$109$	612.027	0.537813	0.268906	+	0.963166i	$0.413338\pi$
$110$	15.4874	0.0134243
$111$	923.032	0.789282
$112$	−112.000	−0.0944911
$113$	313.566	0.261043	0.130521	−	0.991445i	$-0.458335\pi$
$113$	313.566	0.261043	0.130521	+	0.991445i	$0.458335\pi$
$114$	−157.419	−0.129330
$115$	84.9805	0.0689084
$116$	−427.560	−0.342224
$117$	829.871	0.655740
$118$	−1012.99	−0.790279
$119$	−271.161	−0.208885
$120$	−88.6753	−0.0674576
$121$	−1326.61	−0.996700
$122$	−549.303	−0.407636
$123$	−561.444	−0.411575
$124$	−300.010	−0.217272
$125$	−873.261	−0.624855
$126$	126.000	0.0890871
$127$	762.765	0.532948	0.266474	−	0.963842i	$-0.414141\pi$
$127$	762.765	0.532948	0.266474	+	0.963842i	$0.414141\pi$
$128$	−128.000	−0.0883883
$129$	−193.202	−0.131865
$130$	−681.379	−0.459699
$131$	1228.44	0.819305	0.409652	−	0.912242i	$-0.365650\pi$
$131$	1228.44	0.819305	0.409652	+	0.912242i	$0.365650\pi$
$132$	−25.1501	−0.0165836
$133$	−183.655	−0.119736
$134$	423.421	0.272970
$135$	99.7597	0.0635996
$136$	−309.898	−0.195394
$137$	1904.83	1.18789	0.593944	−	0.804506i	$-0.297570\pi$
$137$	1904.83	1.18789	0.593944	+	0.804506i	$0.297570\pi$
$138$	−138.000	−0.0851257
$139$	1202.46	0.733753	0.366876	−	0.930270i	$-0.380427\pi$
$139$	1202.46	0.733753	0.366876	+	0.930270i	$0.380427\pi$
$140$	−103.454	−0.0624536
$141$	1219.29	0.728245
$142$	1728.47	1.02148
$143$	−193.253	−0.113011
$144$	144.000	0.0833333
$145$	−394.938	−0.226192
$146$	−1962.11	−1.11223
$147$	147.000	0.0824786
$148$	1230.71	0.683538
$149$	−2594.02	−1.42625	−0.713123	−	0.701039i	$-0.752720\pi$
$149$	−2594.02	−1.42625	−0.713123	+	0.701039i	$0.752720\pi$
$150$	668.091	0.363662
$151$	1270.19	0.684547	0.342273	−	0.939600i	$-0.388803\pi$
$151$	1270.19	0.684547	0.342273	+	0.939600i	$0.388803\pi$
$152$	−209.892	−0.112003
$153$	348.635	0.184219
$154$	−29.3417	−0.0153534
$155$	−277.120	−0.143605
$156$	1106.49	0.567887
$157$	1323.20	0.672630	0.336315	−	0.941749i	$-0.390819\pi$
$157$	1323.20	0.672630	0.336315	+	0.941749i	$0.390819\pi$
$158$	919.714	0.463092
$159$	−382.061	−0.190562
$160$	−118.234	−0.0584200
$161$	−161.000	−0.0788110
$162$	−162.000	−0.0785674
$163$	1196.44	0.574921	0.287461	−	0.957792i	$-0.407189\pi$
$163$	1196.44	0.574921	0.287461	+	0.957792i	$0.407189\pi$
$164$	−748.592	−0.356434
$165$	−23.2311	−0.0109609
$166$	110.128	0.0514916
$167$	248.822	0.115296	0.0576479	−	0.998337i	$-0.481640\pi$
$167$	248.822	0.115296	0.0576479	+	0.998337i	$0.481640\pi$
$168$	168.000	0.0771517
$169$	6305.28	2.86995
$170$	−286.253	−0.129145
$171$	236.128	0.105597
$172$	−257.603	−0.114198
$173$	1117.46	0.491091	0.245545	−	0.969385i	$-0.421033\pi$
$173$	1117.46	0.491091	0.245545	+	0.969385i	$0.421033\pi$
$174$	641.341	0.279425
$175$	779.439	0.336686
$176$	−33.5334	−0.0143618
$177$	1519.48	0.645260
$178$	−795.482	−0.334966
$179$	−451.436	−0.188502	−0.0942512	−	0.995548i	$-0.530046\pi$
$179$	−451.436	−0.188502	−0.0942512	+	0.995548i	$0.530046\pi$
$180$	133.013	0.0550789
$181$	1076.00	0.441871	0.220936	−	0.975288i	$-0.429089\pi$
$181$	1076.00	0.441871	0.220936	+	0.975288i	$0.429089\pi$
$182$	1290.91	0.525761
$183$	823.955	0.332833
$184$	−184.000	−0.0737210
$185$	1136.81	0.451782
$186$	450.015	0.177402
$187$	−81.1871	−0.0317486
$188$	1625.72	0.630679
$189$	−189.000	−0.0727393
$190$	−193.877	−0.0740279
$191$	−3628.80	−1.37472	−0.687358	−	0.726319i	$-0.741230\pi$
$191$	−3628.80	−1.37472	−0.687358	+	0.726319i	$0.741230\pi$
$192$	192.000	0.0721688
$193$	1435.64	0.535438	0.267719	−	0.963497i	$-0.413730\pi$
$193$	1435.64	0.535438	0.267719	+	0.963497i	$0.413730\pi$
$194$	−3034.93	−1.12317
$195$	1022.07	0.375343
$196$	196.000	0.0714286
$197$	−1185.07	−0.428594	−0.214297	−	0.976769i	$-0.568746\pi$
$197$	−1185.07	−0.428594	−0.214297	+	0.976769i	$0.568746\pi$
$198$	37.7251	0.0135404
$199$	1038.87	0.370068	0.185034	−	0.982732i	$-0.440760\pi$
$199$	1038.87	0.370068	0.185034	+	0.982732i	$0.440760\pi$
$200$	890.787	0.314941
$201$	−635.132	−0.222879
$202$	−968.894	−0.337481
$203$	748.231	0.258697
$204$	464.847	0.159538
$205$	−691.475	−0.235584
$206$	884.873	0.299282
$207$	207.000	0.0695048
$208$	1475.33	0.491805
$209$	−54.9873	−0.0181988
$210$	155.182	0.0509931
$211$	1446.04	0.471799	0.235899	−	0.971778i	$-0.424196\pi$
$211$	1446.04	0.471799	0.235899	+	0.971778i	$0.424196\pi$
$212$	−509.415	−0.165032
$213$	−2592.70	−0.834032
$214$	41.3872	0.0132204
$215$	−237.948	−0.0754788
$216$	−216.000	−0.0680414
$217$	525.018	0.164242
$218$	−1224.05	−0.380291
$219$	2943.16	0.908130
$220$	−30.9748	−0.00949238
$221$	3571.88	1.08720
$222$	−1846.06	−0.558107
$223$	5631.48	1.69108	0.845542	−	0.533908i	$-0.179277\pi$
$223$	5631.48	1.69108	0.845542	+	0.533908i	$0.179277\pi$
$224$	224.000	0.0668153
$225$	−1002.14	−0.296929
$226$	−627.133	−0.184585
$227$	−1631.70	−0.477091	−0.238545	−	0.971131i	$-0.576671\pi$
$227$	−1631.70	−0.477091	−0.238545	+	0.971131i	$0.576671\pi$
$228$	314.837	0.0914500
$229$	1882.19	0.543138	0.271569	−	0.962419i	$-0.412457\pi$
$229$	1882.19	0.543138	0.271569	+	0.962419i	$0.412457\pi$
$230$	−169.961	−0.0487256
$231$	44.0126	0.0125360
$232$	855.121	0.241989
$233$	−3239.52	−0.910851	−0.455426	−	0.890274i	$-0.650513\pi$
$233$	−3239.52	−0.910851	−0.455426	+	0.890274i	$0.650513\pi$
$234$	−1659.74	−0.463678
$235$	1501.68	0.416845
$236$	2025.97	0.558812
$237$	−1379.57	−0.378113
$238$	542.322	0.147704
$239$	−5412.08	−1.46476	−0.732381	−	0.680895i	$-0.761591\pi$
$239$	−5412.08	−1.46476	−0.732381	+	0.680895i	$0.761591\pi$
$240$	177.351	0.0476997
$241$	6778.56	1.81181	0.905903	−	0.423486i	$-0.139194\pi$
$241$	6778.56	1.81181	0.905903	+	0.423486i	$0.139194\pi$
$242$	2653.21	0.704773
$243$	243.000	0.0641500
$244$	1098.61	0.288242
$245$	181.045	0.0472105
$246$	1122.89	0.291027
$247$	2419.21	0.623200
$248$	600.021	0.153634
$249$	−165.192	−0.0420427
$250$	1746.52	0.441839
$251$	229.077	0.0576065	0.0288033	−	0.999585i	$-0.490830\pi$
$251$	229.077	0.0576065	0.0288033	+	0.999585i	$0.490830\pi$
$252$	−252.000	−0.0629941
$253$	−48.2043	−0.0119786
$254$	−1525.53	−0.376851
$255$	429.380	0.105446
$256$	256.000	0.0625000
$257$	−155.416	−0.0377221	−0.0188610	−	0.999822i	$-0.506004\pi$
$257$	−155.416	−0.0377221	−0.0188610	+	0.999822i	$0.506004\pi$
$258$	386.405	0.0932423
$259$	−2153.74	−0.516706
$260$	1362.76	0.325057
$261$	−962.011	−0.228149
$262$	−2456.87	−0.579336
$263$	6378.22	1.49543	0.747715	−	0.664020i	$-0.231151\pi$
$263$	6378.22	1.49543	0.747715	+	0.664020i	$0.231151\pi$
$264$	50.3001	0.0117264
$265$	−470.547	−0.109077
$266$	367.310	0.0846663
$267$	1193.22	0.273498
$268$	−846.842	−0.193019
$269$	−2444.99	−0.554176	−0.277088	−	0.960844i	$-0.589369\pi$
$269$	−2444.99	−0.554176	−0.277088	+	0.960844i	$0.589369\pi$
$270$	−199.519	−0.0449717
$271$	−224.644	−0.0503548	−0.0251774	−	0.999683i	$-0.508015\pi$
$271$	−224.644	−0.0503548	−0.0251774	+	0.999683i	$0.508015\pi$
$272$	619.796	0.138164
$273$	−1936.36	−0.429282
$274$	−3809.66	−0.839964
$275$	233.368	0.0511732
$276$	276.000	0.0601929
$277$	2652.71	0.575400	0.287700	−	0.957721i	$-0.407109\pi$
$277$	2652.71	0.575400	0.287700	+	0.957721i	$0.407109\pi$
$278$	−2404.93	−0.518842
$279$	−675.023	−0.144848
$280$	206.909	0.0441613
$281$	4772.05	1.01308	0.506542	−	0.862215i	$-0.330924\pi$
$281$	4772.05	1.01308	0.506542	+	0.862215i	$0.330924\pi$
$282$	−2438.57	−0.514947
$283$	−766.164	−0.160932	−0.0804659	−	0.996757i	$-0.525641\pi$
$283$	−766.164	−0.160932	−0.0804659	+	0.996757i	$0.525641\pi$
$284$	−3456.93	−0.722293
$285$	290.815	0.0604436
$286$	386.506	0.0799110
$287$	1310.04	0.269439
$288$	−288.000	−0.0589256
$289$	−3412.42	−0.694570
$290$	789.875	0.159942
$291$	4552.39	0.917065
$292$	3924.21	0.786463
$293$	−2782.53	−0.554802	−0.277401	−	0.960754i	$-0.589473\pi$
$293$	−2782.53	−0.554802	−0.277401	+	0.960754i	$0.589473\pi$
$294$	−294.000	−0.0583212
$295$	1871.39	0.369345
$296$	−2461.42	−0.483335
$297$	−56.5876	−0.0110557
$298$	5188.05	1.00851
$299$	2120.78	0.410194
$300$	−1336.18	−0.257148
$301$	450.806	0.0863256
$302$	−2540.38	−0.484048
$303$	1453.34	0.275552
$304$	419.783	0.0791981
$305$	1014.78	0.190513
$306$	−697.271	−0.130262
$307$	192.160	0.0357236	0.0178618	−	0.999840i	$-0.494314\pi$
$307$	192.160	0.0357236	0.0178618	+	0.999840i	$0.494314\pi$
$308$	58.6835	0.0108565
$309$	−1327.31	−0.244362
$310$	554.239	0.101544
$311$	6133.90	1.11840	0.559198	−	0.829034i	$-0.311109\pi$
$311$	6133.90	1.11840	0.559198	+	0.829034i	$0.311109\pi$
$312$	−2212.99	−0.401557
$313$	3169.51	0.572368	0.286184	−	0.958175i	$-0.407613\pi$
$313$	3169.51	0.572368	0.286184	+	0.958175i	$0.407613\pi$
$314$	−2646.40	−0.475622
$315$	−232.773	−0.0416357
$316$	−1839.43	−0.327455
$317$	−6348.33	−1.12479	−0.562394	−	0.826870i	$-0.690119\pi$
$317$	−6348.33	−1.12479	−0.562394	+	0.826870i	$0.690119\pi$
$318$	764.122	0.134748
$319$	224.024	0.0393196
$320$	236.467	0.0413091
$321$	−62.0808	−0.0107944
$322$	322.000	0.0557278
$323$	1016.33	0.175077
$324$	324.000	0.0555556
$325$	−10267.2	−1.75237
$326$	−2392.87	−0.406531
$327$	1836.08	0.310506
$328$	1497.18	0.252037
$329$	−2845.00	−0.476748
$330$	46.4623	0.00775050
$331$	−2743.51	−0.455579	−0.227790	−	0.973710i	$-0.573150\pi$
$331$	−2743.51	−0.455579	−0.227790	+	0.973710i	$0.573150\pi$
$332$	−220.256	−0.0364100
$333$	2769.10	0.455692
$334$	−497.644	−0.0815265
$335$	−782.229	−0.127575
$336$	−336.000	−0.0545545
$337$	−8222.46	−1.32910	−0.664549	−	0.747245i	$-0.731376\pi$
$337$	−8222.46	−1.32910	−0.664549	+	0.747245i	$0.731376\pi$
$338$	−12610.6	−2.02936
$339$	940.699	0.150713
$340$	572.506	0.0913191
$341$	157.193	0.0249633
$342$	−472.256	−0.0746686
$343$	−343.000	−0.0539949
$344$	515.206	0.0807502
$345$	254.941	0.0397843
$346$	−2234.92	−0.347254
$347$	−1169.92	−0.180994	−0.0904969	−	0.995897i	$-0.528846\pi$
$347$	−1169.92	−0.180994	−0.0904969	+	0.995897i	$0.528846\pi$
$348$	−1282.68	−0.197583
$349$	4707.48	0.722022	0.361011	−	0.932562i	$-0.382432\pi$
$349$	4707.48	0.722022	0.361011	+	0.932562i	$0.382432\pi$
$350$	−1558.88	−0.238073
$351$	2489.61	0.378592
$352$	67.0668	0.0101553
$353$	−7584.65	−1.14360	−0.571799	−	0.820394i	$-0.693754\pi$
$353$	−7584.65	−1.14360	−0.571799	+	0.820394i	$0.693754\pi$
$354$	−3038.96	−0.456268
$355$	−3193.17	−0.477397
$356$	1590.96	0.236856
$357$	−813.482	−0.120600
$358$	902.872	0.133291
$359$	−5324.41	−0.782763	−0.391381	−	0.920229i	$-0.628003\pi$
$359$	−5324.41	−0.782763	−0.391381	+	0.920229i	$0.628003\pi$
$360$	−266.026	−0.0389466
$361$	−6170.65	−0.899643
$362$	−2152.01	−0.312450
$363$	−3979.82	−0.575445
$364$	−2581.82	−0.371770
$365$	3624.80	0.519810
$366$	−1647.91	−0.235349
$367$	4100.72	0.583259	0.291630	−	0.956531i	$-0.405803\pi$
$367$	4100.72	0.583259	0.291630	+	0.956531i	$0.405803\pi$
$368$	368.000	0.0521286
$369$	−1684.33	−0.237623
$370$	−2273.61	−0.319458
$371$	891.476	0.124752
$372$	−900.031	−0.125442
$373$	−1398.40	−0.194119	−0.0970593	−	0.995279i	$-0.530944\pi$
$373$	−1398.40	−0.194119	−0.0970593	+	0.995279i	$0.530944\pi$
$374$	162.374	0.0224496
$375$	−2619.78	−0.360760
$376$	−3251.43	−0.445957
$377$	−9856.10	−1.34646
$378$	378.000	0.0514344
$379$	−14031.7	−1.90174	−0.950871	−	0.309588i	$-0.899809\pi$
$379$	−14031.7	−1.90174	−0.950871	+	0.309588i	$0.899809\pi$
$380$	387.754	0.0523457
$381$	2288.29	0.307698
$382$	7257.60	0.972071
$383$	745.634	0.0994781	0.0497390	−	0.998762i	$-0.484161\pi$
$383$	745.634	0.0994781	0.0497390	+	0.998762i	$0.484161\pi$
$384$	−384.000	−0.0510310
$385$	54.2060	0.00717556
$386$	−2871.28	−0.378612
$387$	−579.607	−0.0761320
$388$	6069.86	0.794202
$389$	13287.3	1.73186	0.865932	−	0.500161i	$-0.166726\pi$
$389$	13287.3	1.73186	0.865932	+	0.500161i	$0.166726\pi$
$390$	−2044.14	−0.265408
$391$	890.957	0.115237
$392$	−392.000	−0.0505076
$393$	3685.31	0.473026
$394$	2370.15	0.303062
$395$	−1699.08	−0.216430
$396$	−75.4502	−0.00957453
$397$	−2516.30	−0.318109	−0.159055	−	0.987270i	$-0.550845\pi$
$397$	−2516.30	−0.318109	−0.159055	+	0.987270i	$0.550845\pi$
$398$	−2077.74	−0.261678
$399$	−550.965	−0.0691297
$400$	−1781.57	−0.222697
$401$	5382.76	0.670329	0.335165	−	0.942160i	$-0.391208\pi$
$401$	5382.76	0.670329	0.335165	+	0.942160i	$0.391208\pi$
$402$	1270.26	0.157599
$403$	−6915.83	−0.854843
$404$	1937.79	0.238635
$405$	299.279	0.0367192
$406$	−1496.46	−0.182926
$407$	−644.842	−0.0785347
$408$	−929.694	−0.112811
$409$	7472.70	0.903425	0.451713	−	0.892163i	$-0.350813\pi$
$409$	7472.70	0.903425	0.451713	+	0.892163i	$0.350813\pi$
$410$	1382.95	0.166583
$411$	5714.49	0.685827
$412$	−1769.75	−0.211624
$413$	−3545.45	−0.422422
$414$	−414.000	−0.0491473
$415$	−203.451	−0.0240651
$416$	−2950.65	−0.347759
$417$	3607.39	0.423632
$418$	109.975	0.0128685
$419$	−5997.41	−0.699266	−0.349633	−	0.936887i	$-0.613694\pi$
$419$	−5997.41	−0.699266	−0.349633	+	0.936887i	$0.613694\pi$
$420$	−310.363	−0.0360576
$421$	5191.94	0.601045	0.300522	−	0.953775i	$-0.402839\pi$
$421$	5191.94	0.601045	0.300522	+	0.953775i	$0.402839\pi$
$422$	−2892.08	−0.333612
$423$	3657.86	0.420452
$424$	1018.83	0.116695
$425$	−4313.33	−0.492300
$426$	5185.40	0.589750
$427$	−1922.56	−0.217891
$428$	−82.7744	−0.00934825
$429$	−579.758	−0.0652471
$430$	475.896	0.0533715
$431$	−5315.02	−0.594004	−0.297002	−	0.954877i	$-0.595987\pi$
$431$	−5315.02	−0.594004	−0.297002	+	0.954877i	$0.595987\pi$
$432$	432.000	0.0481125
$433$	−7427.63	−0.824363	−0.412182	−	0.911102i	$-0.635233\pi$
$433$	−7427.63	−0.824363	−0.412182	+	0.911102i	$0.635233\pi$
$434$	−1050.04	−0.116137
$435$	−1184.81	−0.130592
$436$	2448.11	0.268906
$437$	603.438	0.0660557
$438$	−5886.32	−0.642145
$439$	4238.03	0.460752	0.230376	−	0.973102i	$-0.426004\pi$
$439$	4238.03	0.460752	0.230376	+	0.973102i	$0.426004\pi$
$440$	61.9497	0.00671213
$441$	441.000	0.0476190
$442$	−7143.76	−0.768764
$443$	−11680.7	−1.25275	−0.626373	−	0.779524i	$-0.715461\pi$
$443$	−11680.7	−1.25275	−0.626373	+	0.779524i	$0.715461\pi$
$444$	3692.13	0.394641
$445$	1469.57	0.156549
$446$	−11263.0	−1.19578
$447$	−7782.07	−0.823443
$448$	−448.000	−0.0472456
$449$	−7220.07	−0.758878	−0.379439	−	0.925217i	$-0.623883\pi$
$449$	−7220.07	−0.758878	−0.379439	+	0.925217i	$0.623883\pi$
$450$	2004.27	0.209961
$451$	392.232	0.0409523
$452$	1254.27	0.130521
$453$	3810.57	0.395223
$454$	3263.40	0.337354
$455$	−2384.83	−0.245720
$456$	−629.675	−0.0646649
$457$	2749.87	0.281474	0.140737	−	0.990047i	$-0.455053\pi$
$457$	2749.87	0.281474	0.140737	+	0.990047i	$0.455053\pi$
$458$	−3764.38	−0.384057
$459$	1045.91	0.106359
$460$	339.922	0.0344542
$461$	−15518.5	−1.56783	−0.783913	−	0.620870i	$-0.786779\pi$
$461$	−15518.5	−1.56783	−0.783913	+	0.620870i	$0.786779\pi$
$462$	−88.0252	−0.00886430
$463$	−15910.7	−1.59704	−0.798522	−	0.601966i	$-0.794384\pi$
$463$	−15910.7	−1.59704	−0.798522	+	0.601966i	$0.794384\pi$
$464$	−1710.24	−0.171112
$465$	−831.359	−0.0829104
$466$	6479.05	0.644069
$467$	7474.09	0.740599	0.370300	−	0.928912i	$-0.379255\pi$
$467$	7474.09	0.740599	0.370300	+	0.928912i	$0.379255\pi$
$468$	3319.48	0.327870
$469$	1481.97	0.145909
$470$	−3003.35	−0.294754
$471$	3969.60	0.388343
$472$	−4051.95	−0.395140
$473$	134.974	0.0131207
$474$	2759.14	0.267366
$475$	−2921.39	−0.282195
$476$	−1084.64	−0.104442
$477$	−1146.18	−0.110021
$478$	10824.2	1.03574
$479$	3499.73	0.333835	0.166917	−	0.985971i	$-0.446619\pi$
$479$	3499.73	0.333835	0.166917	+	0.985971i	$0.446619\pi$
$480$	−354.701	−0.0337288
$481$	28370.3	2.68934
$482$	−13557.1	−1.28114
$483$	−483.000	−0.0455016
$484$	−5306.43	−0.498350
$485$	5606.73	0.524925
$486$	−486.000	−0.0453609
$487$	3970.07	0.369407	0.184703	−	0.982794i	$-0.440868\pi$
$487$	3970.07	0.369407	0.184703	+	0.982794i	$0.440868\pi$
$488$	−2197.21	−0.203818
$489$	3589.31	0.331931
$490$	−362.091	−0.0333828
$491$	−4030.18	−0.370426	−0.185213	−	0.982698i	$-0.559298\pi$
$491$	−4030.18	−0.370426	−0.185213	+	0.982698i	$0.559298\pi$
$492$	−2245.78	−0.205787
$493$	−4140.63	−0.378265
$494$	−4838.41	−0.440669
$495$	−69.6934	−0.00632825
$496$	−1200.04	−0.108636
$497$	6049.63	0.546002
$498$	330.384	0.0297287
$499$	−14821.5	−1.32966	−0.664830	−	0.746995i	$-0.731496\pi$
$499$	−14821.5	−1.32966	−0.664830	+	0.746995i	$0.731496\pi$
$500$	−3493.04	−0.312427
$501$	746.465	0.0665661
$502$	−458.155	−0.0407340
$503$	−9096.73	−0.806368	−0.403184	−	0.915119i	$-0.632097\pi$
$503$	−9096.73	−0.806368	−0.403184	+	0.915119i	$0.632097\pi$
$504$	504.000	0.0445435
$505$	1789.94	0.157725
$506$	96.4086	0.00847013
$507$	18915.9	1.65697
$508$	3051.06	0.266474
$509$	−2959.71	−0.257734	−0.128867	−	0.991662i	$-0.541134\pi$
$509$	−2959.71	−0.257734	−0.128867	+	0.991662i	$0.541134\pi$
$510$	−858.759	−0.0745618
$511$	−6867.38	−0.594510
$512$	−512.000	−0.0441942
$513$	708.384	0.0609667
$514$	310.832	0.0266735
$515$	−1634.72	−0.139872
$516$	−772.810	−0.0659323
$517$	−851.810	−0.0724614
$518$	4307.48	0.365367
$519$	3352.37	0.283531
$520$	−2725.52	−0.229850
$521$	−3024.94	−0.254366	−0.127183	−	0.991879i	$-0.540594\pi$
$521$	−3024.94	−0.254366	−0.127183	+	0.991879i	$0.540594\pi$
$522$	1924.02	0.161326
$523$	−15013.1	−1.25521	−0.627607	−	0.778530i	$-0.715965\pi$
$523$	−15013.1	−1.25521	−0.627607	+	0.778530i	$0.715965\pi$
$524$	4913.74	0.409652
$525$	2338.32	0.194386
$526$	−12756.4	−1.05743
$527$	−2905.39	−0.240154
$528$	−100.600	−0.00829179
$529$	529.000	0.0434783
$530$	941.094	0.0771292
$531$	4558.44	0.372541
$532$	−734.620	−0.0598681
$533$	−17256.5	−1.40237
$534$	−2386.45	−0.193392
$535$	−76.4588	−0.00617869
$536$	1693.68	0.136485
$537$	−1354.31	−0.108832
$538$	4889.97	0.391862
$539$	−102.696	−0.00820674
$540$	399.039	0.0317998
$541$	−19183.9	−1.52455	−0.762273	−	0.647255i	$-0.775917\pi$
$541$	−19183.9	−1.52455	−0.762273	+	0.647255i	$0.775917\pi$
$542$	449.288	0.0356062
$543$	3228.01	0.255114
$544$	−1239.59	−0.0976968
$545$	2261.32	0.177733
$546$	3872.73	0.303549
$547$	−15117.7	−1.18169	−0.590847	−	0.806784i	$-0.701206\pi$
$547$	−15117.7	−1.18169	−0.590847	+	0.806784i	$0.701206\pi$
$548$	7619.32	0.593944
$549$	2471.86	0.192161
$550$	−466.737	−0.0361849
$551$	−2804.42	−0.216828
$552$	−552.000	−0.0425628
$553$	3219.00	0.247533
$554$	−5305.42	−0.406869
$555$	3410.42	0.260837
$556$	4809.85	0.366876
$557$	3979.52	0.302724	0.151362	−	0.988478i	$-0.451634\pi$
$557$	3979.52	0.302724	0.151362	+	0.988478i	$0.451634\pi$
$558$	1350.05	0.102423
$559$	−5938.26	−0.449305
$560$	−413.818	−0.0312268
$561$	−243.561	−0.0183301
$562$	−9544.09	−0.716358
$563$	16246.8	1.21620	0.608100	−	0.793860i	$-0.291932\pi$
$563$	16246.8	1.21620	0.608100	+	0.793860i	$0.291932\pi$
$564$	4877.15	0.364122
$565$	1158.57	0.0862676
$566$	1532.33	0.113796
$567$	−567.000	−0.0419961
$568$	6913.87	0.510738
$569$	−1504.62	−0.110855	−0.0554277	−	0.998463i	$-0.517652\pi$
$569$	−1504.62	−0.110855	−0.0554277	+	0.998463i	$0.517652\pi$
$570$	−581.631	−0.0427401
$571$	−13023.1	−0.954468	−0.477234	−	0.878776i	$-0.658361\pi$
$571$	−13023.1	−0.954468	−0.477234	+	0.878776i	$0.658361\pi$
$572$	−773.011	−0.0565056
$573$	−10886.4	−0.793693
$574$	−2620.07	−0.190522
$575$	−2561.01	−0.185742
$576$	576.000	0.0416667
$577$	−122.394	−0.00883075	−0.00441537	−	0.999990i	$-0.501405\pi$
$577$	−122.394	−0.00883075	−0.00441537	+	0.999990i	$0.501405\pi$
$578$	6824.85	0.491135
$579$	4306.92	0.309135
$580$	−1579.75	−0.113096
$581$	385.448	0.0275234
$582$	−9104.78	−0.648463
$583$	266.913	0.0189612
$584$	−7848.43	−0.556114
$585$	3066.21	0.216704
$586$	5565.06	0.392304
$587$	−8270.55	−0.581537	−0.290769	−	0.956793i	$-0.593911\pi$
$587$	−8270.55	−0.581537	−0.290769	+	0.956793i	$0.593911\pi$
$588$	588.000	0.0412393
$589$	−1967.80	−0.137660
$590$	−3742.78	−0.261166
$591$	−3555.22	−0.247449
$592$	4922.84	0.341769
$593$	−15426.4	−1.06827	−0.534137	−	0.845398i	$-0.679363\pi$
$593$	−15426.4	−1.06827	−0.534137	+	0.845398i	$0.679363\pi$
$594$	113.175	0.00781757
$595$	−1001.89	−0.0690308
$596$	−10376.1	−0.713123
$597$	3116.61	0.213659
$598$	−4241.56	−0.290051
$599$	−14463.1	−0.986557	−0.493278	−	0.869872i	$-0.664202\pi$
$599$	−14463.1	−0.986557	−0.493278	+	0.869872i	$0.664202\pi$
$600$	2672.36	0.181831
$601$	−10103.8	−0.685761	−0.342881	−	0.939379i	$-0.611403\pi$
$601$	−10103.8	−0.685761	−0.342881	+	0.939379i	$0.611403\pi$
$602$	−901.611	−0.0610414
$603$	−1905.40	−0.128679
$604$	5080.76	0.342273
$605$	−4901.55	−0.329383
$606$	−2906.68	−0.194845
$607$	27805.2	1.85927	0.929636	−	0.368479i	$-0.120121\pi$
$607$	27805.2	1.85927	0.929636	+	0.368479i	$0.120121\pi$
$608$	−839.566	−0.0560015
$609$	2244.69	0.149359
$610$	−2029.57	−0.134713
$611$	37476.0	2.48137
$612$	1394.54	0.0921095
$613$	5946.15	0.391782	0.195891	−	0.980626i	$-0.437240\pi$
$613$	5946.15	0.391782	0.195891	+	0.980626i	$0.437240\pi$
$614$	−384.319	−0.0252604
$615$	−2074.42	−0.136014
$616$	−117.367	−0.00767670
$617$	17646.3	1.15140	0.575699	−	0.817662i	$-0.304730\pi$
$617$	17646.3	1.15140	0.575699	+	0.817662i	$0.304730\pi$
$618$	2654.62	0.172790
$619$	13811.4	0.896810	0.448405	−	0.893830i	$-0.351992\pi$
$619$	13811.4	0.896810	0.448405	+	0.893830i	$0.351992\pi$
$620$	−1108.48	−0.0718026
$621$	621.000	0.0401286
$622$	−12267.8	−0.790826
$623$	−2784.19	−0.179047
$624$	4425.98	0.283944
$625$	10692.0	0.684290
$626$	−6339.02	−0.404726
$627$	−164.962	−0.0105071
$628$	5292.81	0.336315
$629$	11918.6	0.755524
$630$	465.545	0.0294409
$631$	28959.5	1.82703	0.913517	−	0.406801i	$-0.133356\pi$
$631$	28959.5	1.82703	0.913517	+	0.406801i	$0.133356\pi$
$632$	3678.86	0.231546
$633$	4338.12	0.272393
$634$	12696.7	0.795345
$635$	2818.27	0.176125
$636$	−1528.24	−0.0952812
$637$	4518.18	0.281031
$638$	−448.049	−0.0278032
$639$	−7778.10	−0.481529
$640$	−472.935	−0.0292100
$641$	15331.1	0.944685	0.472342	−	0.881415i	$-0.343409\pi$
$641$	15331.1	0.944685	0.472342	+	0.881415i	$0.343409\pi$
$642$	124.162	0.00763281
$643$	−18867.8	−1.15719	−0.578595	−	0.815615i	$-0.696399\pi$
$643$	−18867.8	−1.15719	−0.578595	+	0.815615i	$0.696399\pi$
$644$	−644.000	−0.0394055
$645$	−713.845	−0.0435777
$646$	−2032.66	−0.123798
$647$	−31395.1	−1.90768	−0.953839	−	0.300319i	$-0.902907\pi$
$647$	−31395.1	−1.90768	−0.953839	+	0.300319i	$0.902907\pi$
$648$	−648.000	−0.0392837
$649$	−1061.53	−0.0642043
$650$	20534.4	1.23912
$651$	1575.05	0.0948253
$652$	4785.75	0.287461
$653$	4247.89	0.254568	0.127284	−	0.991866i	$-0.459374\pi$
$653$	4247.89	0.254568	0.127284	+	0.991866i	$0.459374\pi$
$654$	−3672.16	−0.219561
$655$	4538.83	0.270758
$656$	−2994.37	−0.178217
$657$	8829.48	0.524309
$658$	5690.01	0.337112
$659$	−3023.55	−0.178727	−0.0893633	−	0.995999i	$-0.528483\pi$
$659$	−3023.55	−0.178727	−0.0893633	+	0.995999i	$0.528483\pi$
$660$	−92.9245	−0.00548043
$661$	−6360.61	−0.374280	−0.187140	−	0.982333i	$-0.559922\pi$
$661$	−6360.61	−0.374280	−0.187140	+	0.982333i	$0.559922\pi$
$662$	5487.01	0.322143
$663$	10715.6	0.627694
$664$	440.513	0.0257458
$665$	−678.569	−0.0395696
$666$	−5538.19	−0.322223
$667$	−2458.47	−0.142717
$668$	995.287	0.0576479
$669$	16894.4	0.976348
$670$	1564.46	0.0902094
$671$	−575.625	−0.0331174
$672$	672.000	0.0385758
$673$	11724.5	0.671542	0.335771	−	0.941944i	$-0.391003\pi$
$673$	11724.5	0.671542	0.335771	+	0.941944i	$0.391003\pi$
$674$	16444.9	0.939814
$675$	−3006.41	−0.171432
$676$	25221.1	1.43498
$677$	16498.9	0.936639	0.468319	−	0.883559i	$-0.344860\pi$
$677$	16498.9	0.936639	0.468319	+	0.883559i	$0.344860\pi$
$678$	−1881.40	−0.106570
$679$	−10622.2	−0.600360
$680$	−1145.01	−0.0645724
$681$	−4895.10	−0.275449
$682$	−314.387	−0.0176517
$683$	12745.1	0.714021	0.357010	−	0.934100i	$-0.383796\pi$
$683$	12745.1	0.714021	0.357010	+	0.934100i	$0.383796\pi$
$684$	944.512	0.0527987
$685$	7037.97	0.392565
$686$	686.000	0.0381802
$687$	5646.57	0.313581
$688$	−1030.41	−0.0570990
$689$	−11743.0	−0.649308
$690$	−509.883	−0.0281317
$691$	18073.3	0.994996	0.497498	−	0.867465i	$-0.334252\pi$
$691$	18073.3	0.994996	0.497498	+	0.867465i	$0.334252\pi$
$692$	4469.83	0.245545
$693$	132.038	0.00723767
$694$	2339.85	0.127982
$695$	4442.87	0.242486
$696$	2565.36	0.139712
$697$	−7249.60	−0.393972
$698$	−9414.96	−0.510547
$699$	−9718.57	−0.525880
$700$	3117.76	0.168343
$701$	24973.3	1.34554	0.672772	−	0.739850i	$-0.265103\pi$
$701$	24973.3	1.34554	0.672772	+	0.739850i	$0.265103\pi$
$702$	−4979.22	−0.267705
$703$	8072.36	0.433079
$704$	−134.134	−0.00718090
$705$	4505.03	0.240665
$706$	15169.3	0.808646
$707$	−3391.13	−0.180391
$708$	6077.92	0.322630
$709$	−24501.6	−1.29785	−0.648927	−	0.760851i	$-0.724782\pi$
$709$	−24501.6	−1.29785	−0.648927	+	0.760851i	$0.724782\pi$
$710$	6386.34	0.337571
$711$	−4138.71	−0.218304
$712$	−3181.93	−0.167483
$713$	−1725.06	−0.0906087
$714$	1626.96	0.0852768
$715$	−714.031	−0.0373472
$716$	−1805.74	−0.0942512
$717$	−16236.2	−0.845681
$718$	10648.8	0.553497
$719$	−18301.1	−0.949257	−0.474628	−	0.880186i	$-0.657418\pi$
$719$	−18301.1	−0.949257	−0.474628	+	0.880186i	$0.657418\pi$
$720$	532.052	0.0275394
$721$	3097.05	0.159973
$722$	12341.3	0.636143
$723$	20335.7	1.04605
$724$	4304.01	0.220936
$725$	11902.0	0.609698
$726$	7959.64	0.406901
$727$	−6701.15	−0.341859	−0.170930	−	0.985283i	$-0.554677\pi$
$727$	−6701.15	−0.341859	−0.170930	+	0.985283i	$0.554677\pi$
$728$	5163.64	0.262881
$729$	729.000	0.0370370
$730$	−7249.60	−0.367561
$731$	−2494.71	−0.126225
$732$	3295.82	0.166417
$733$	−27323.2	−1.37681	−0.688407	−	0.725324i	$-0.741690\pi$
$733$	−27323.2	−1.37681	−0.688407	+	0.725324i	$0.741690\pi$
$734$	−8201.45	−0.412426
$735$	543.136	0.0272570
$736$	−736.000	−0.0368605
$737$	443.711	0.0221768
$738$	3368.66	0.168025
$739$	6401.78	0.318665	0.159332	−	0.987225i	$-0.449066\pi$
$739$	6401.78	0.318665	0.159332	+	0.987225i	$0.449066\pi$
$740$	4547.23	0.225891
$741$	7257.62	0.359805
$742$	−1782.95	−0.0882132
$743$	−17653.2	−0.871645	−0.435822	−	0.900033i	$-0.643542\pi$
$743$	−17653.2	−0.871645	−0.435822	+	0.900033i	$0.643542\pi$
$744$	1800.06	0.0887009
$745$	−9584.40	−0.471336
$746$	2796.79	0.137263
$747$	−495.577	−0.0242734
$748$	−324.748	−0.0158743
$749$	144.855	0.00706661
$750$	5239.56	0.255096
$751$	−22303.7	−1.08372	−0.541861	−	0.840468i	$-0.682280\pi$
$751$	−22303.7	−1.08372	−0.541861	+	0.840468i	$0.682280\pi$
$752$	6502.87	0.315339
$753$	687.232	0.0332591
$754$	19712.2	0.952090
$755$	4693.10	0.226224
$756$	−756.000	−0.0363696
$757$	122.440	0.00587869	0.00293934	−	0.999996i	$-0.499064\pi$
$757$	122.440	0.00587869	0.00293934	+	0.999996i	$0.499064\pi$
$758$	28063.4	1.34473
$759$	−144.613	−0.00691583
$760$	−775.508	−0.0370140
$761$	5300.67	0.252496	0.126248	−	0.991999i	$-0.459707\pi$
$761$	5300.67	0.252496	0.126248	+	0.991999i	$0.459707\pi$
$762$	−4576.59	−0.217575
$763$	−4284.19	−0.203274
$764$	−14515.2	−0.687358
$765$	1288.14	0.0608794
$766$	−1491.27	−0.0703416
$767$	46702.6	2.19861
$768$	768.000	0.0360844
$769$	−3325.45	−0.155941	−0.0779707	−	0.996956i	$-0.524844\pi$
$769$	−3325.45	−0.155941	−0.0779707	+	0.996956i	$0.524844\pi$
$770$	−108.412	−0.00507389
$771$	−466.248	−0.0217789
$772$	5742.56	0.267719
$773$	−37232.9	−1.73244	−0.866219	−	0.499665i	$-0.833456\pi$
$773$	−37232.9	−1.73244	−0.866219	+	0.499665i	$0.833456\pi$
$774$	1159.21	0.0538335
$775$	8351.42	0.387086
$776$	−12139.7	−0.561585
$777$	−6461.22	−0.298321
$778$	−26574.7	−1.22461
$779$	−4910.10	−0.225831
$780$	4088.28	0.187672
$781$	1811.29	0.0829874
$782$	−1781.91	−0.0814848
$783$	−2886.03	−0.131722
$784$	784.000	0.0357143
$785$	4888.97	0.222286
$786$	−7370.61	−0.334480
$787$	−38549.3	−1.74604	−0.873020	−	0.487684i	$-0.837842\pi$
$787$	−38549.3	−1.74604	−0.873020	+	0.487684i	$0.837842\pi$
$788$	−4740.29	−0.214297
$789$	19134.7	0.863387
$790$	3398.16	0.153039
$791$	−2194.96	−0.0986649
$792$	150.900	0.00677022
$793$	25325.0	1.13407
$794$	5032.60	0.224937
$795$	−1411.64	−0.0629758
$796$	4155.48	0.185034
$797$	36158.1	1.60701	0.803504	−	0.595300i	$-0.202967\pi$
$797$	36158.1	1.60701	0.803504	+	0.595300i	$0.202967\pi$
$798$	1101.93	0.0488821
$799$	15744.0	0.697098
$800$	3563.15	0.157470
$801$	3579.67	0.157904
$802$	−10765.5	−0.473995
$803$	−2056.13	−0.0903602
$804$	−2540.53	−0.111440
$805$	−594.863	−0.0260449
$806$	13831.7	0.604465
$807$	−7334.96	−0.319954
$808$	−3875.57	−0.168740
$809$	−15290.6	−0.664512	−0.332256	−	0.943189i	$-0.607810\pi$
$809$	−15290.6	−0.664512	−0.332256	+	0.943189i	$0.607810\pi$
$810$	−598.558	−0.0259644
$811$	31489.7	1.36344	0.681722	−	0.731612i	$-0.261232\pi$
$811$	31489.7	1.36344	0.681722	+	0.731612i	$0.261232\pi$
$812$	2992.92	0.129349
$813$	−673.932	−0.0290724
$814$	1289.68	0.0555324
$815$	4420.60	0.189996
$816$	1859.39	0.0797691
$817$	−1689.65	−0.0723541
$818$	−14945.4	−0.638818
$819$	−5809.09	−0.247846
$820$	−2765.90	−0.117792
$821$	−14187.7	−0.603113	−0.301556	−	0.953448i	$-0.597506\pi$
$821$	−14187.7	−0.603113	−0.301556	+	0.953448i	$0.597506\pi$
$822$	−11429.0	−0.484953
$823$	38513.7	1.63123	0.815615	−	0.578594i	$-0.196399\pi$
$823$	38513.7	1.63123	0.815615	+	0.578594i	$0.196399\pi$
$824$	3539.49	0.149641
$825$	700.105	0.0295449
$826$	7090.90	0.298697
$827$	−29916.0	−1.25790	−0.628950	−	0.777446i	$-0.716515\pi$
$827$	−29916.0	−1.25790	−0.628950	+	0.777446i	$0.716515\pi$
$828$	828.000	0.0347524
$829$	31097.4	1.30284	0.651421	−	0.758716i	$-0.274173\pi$
$829$	31097.4	1.30284	0.651421	+	0.758716i	$0.274173\pi$
$830$	406.902	0.0170166
$831$	7958.12	0.332207
$832$	5901.30	0.245902
$833$	1898.13	0.0789510
$834$	−7214.78	−0.299553
$835$	919.347	0.0381022
$836$	−219.949	−0.00909941
$837$	−2025.07	−0.0836280
$838$	11994.8	0.494456
$839$	−19561.1	−0.804918	−0.402459	−	0.915438i	$-0.631844\pi$
$839$	−19561.1	−0.804918	−0.402459	+	0.915438i	$0.631844\pi$
$840$	620.727	0.0254966
$841$	−12963.5	−0.531531
$842$	−10383.9	−0.425003
$843$	14316.1	0.584904
$844$	5784.16	0.235899
$845$	23296.8	0.948442
$846$	−7315.72	−0.297305
$847$	9286.25	0.376717
$848$	−2037.66	−0.0825159
$849$	−2298.49	−0.0929140
$850$	8626.67	0.348108
$851$	7076.58	0.285055
$852$	−10370.8	−0.417016
$853$	3851.55	0.154601	0.0773004	−	0.997008i	$-0.475370\pi$
$853$	3851.55	0.154601	0.0773004	+	0.997008i	$0.475370\pi$
$854$	3845.12	0.154072
$855$	872.446	0.0348971
$856$	165.549	0.00661021
$857$	−10142.3	−0.404265	−0.202133	−	0.979358i	$-0.564787\pi$
$857$	−10142.3	−0.404265	−0.202133	+	0.979358i	$0.564787\pi$
$858$	1159.52	0.0461366
$859$	−25117.0	−0.997649	−0.498825	−	0.866703i	$-0.666235\pi$
$859$	−25117.0	−0.997649	−0.498825	+	0.866703i	$0.666235\pi$
$860$	−951.793	−0.0377394
$861$	3930.11	0.155561
$862$	10630.0	0.420024
$863$	−30325.2	−1.19615	−0.598077	−	0.801438i	$-0.704068\pi$
$863$	−30325.2	−1.19615	−0.598077	+	0.801438i	$0.704068\pi$
$864$	−864.000	−0.0340207
$865$	4128.79	0.162292
$866$	14855.3	0.582913
$867$	−10237.3	−0.401010
$868$	2100.07	0.0821211
$869$	963.786	0.0376228
$870$	2369.63	0.0923424
$871$	−19521.4	−0.759422
$872$	−4896.22	−0.190146
$873$	13657.2	0.529468
$874$	−1206.88	−0.0467085
$875$	6112.83	0.236173
$876$	11772.6	0.454065
$877$	−1116.76	−0.0429993	−0.0214997	−	0.999769i	$-0.506844\pi$
$877$	−1116.76	−0.0429993	−0.0214997	+	0.999769i	$0.506844\pi$
$878$	−8476.06	−0.325801
$879$	−8347.59	−0.320315
$880$	−123.899	−0.00474619
$881$	37718.0	1.44240	0.721199	−	0.692728i	$-0.243591\pi$
$881$	37718.0	1.44240	0.721199	+	0.692728i	$0.243591\pi$
$882$	−882.000	−0.0336718
$883$	−10456.4	−0.398513	−0.199256	−	0.979947i	$-0.563853\pi$
$883$	−10456.4	−0.398513	−0.199256	+	0.979947i	$0.563853\pi$
$884$	14287.5	0.543599
$885$	5614.18	0.213241
$886$	23361.4	0.885825
$887$	39295.0	1.48748	0.743741	−	0.668467i	$-0.233049\pi$
$887$	39295.0	1.48748	0.743741	+	0.668467i	$0.233049\pi$
$888$	−7384.25	−0.279053
$889$	−5339.35	−0.201436
$890$	−2939.15	−0.110697
$891$	−169.763	−0.00638302
$892$	22525.9	0.845542
$893$	10663.3	0.399588
$894$	15564.1	0.582262
$895$	−1667.97	−0.0622950
$896$	896.000	0.0334077
$897$	6362.34	0.236825
$898$	14440.1	0.536608
$899$	8017.03	0.297423
$900$	−4008.54	−0.148465
$901$	−4933.33	−0.182412
$902$	−784.464	−0.0289576
$903$	1352.42	0.0498401
$904$	−2508.53	−0.0922926
$905$	3975.62	0.146027
$906$	−7621.14	−0.279465
$907$	−27924.6	−1.02229	−0.511147	−	0.859493i	$-0.670779\pi$
$907$	−27924.6	−1.02229	−0.511147	+	0.859493i	$0.670779\pi$
$908$	−6526.79	−0.238545
$909$	4360.02	0.159090
$910$	4769.66	0.173750
$911$	49144.1	1.78728	0.893642	−	0.448780i	$-0.148141\pi$
$911$	49144.1	1.78728	0.893642	+	0.448780i	$0.148141\pi$
$912$	1259.35	0.0457250
$913$	115.405	0.00418331
$914$	−5499.75	−0.199032
$915$	3044.35	0.109992
$916$	7528.76	0.271569
$917$	−8599.05	−0.309668
$918$	−2091.81	−0.0752071
$919$	1924.41	0.0690754	0.0345377	−	0.999403i	$-0.489004\pi$
$919$	1924.41	0.0690754	0.0345377	+	0.999403i	$0.489004\pi$
$920$	−679.844	−0.0243628
$921$	576.479	0.0206250
$922$	31037.0	1.10862
$923$	−79689.1	−2.84182
$924$	176.050	0.00626800
$925$	−34259.4	−1.21777
$926$	31821.3	1.12928
$927$	−3981.93	−0.141083
$928$	3420.48	0.120994
$929$	−940.278	−0.0332072	−0.0166036	−	0.999862i	$-0.505285\pi$
$929$	−940.278	−0.0332072	−0.0166036	+	0.999862i	$0.505285\pi$
$930$	1662.72	0.0586265
$931$	1285.59	0.0452560
$932$	−12958.1	−0.455426
$933$	18401.7	0.645707
$934$	−14948.2	−0.523683
$935$	−299.970	−0.0104921
$936$	−6638.96	−0.231839
$937$	−28146.3	−0.981323	−0.490662	−	0.871350i	$-0.663245\pi$
$937$	−28146.3	−0.981323	−0.490662	+	0.871350i	$0.663245\pi$
$938$	−2963.95	−0.103173
$939$	9508.53	0.330457
$940$	6006.70	0.208422
$941$	11361.7	0.393604	0.196802	−	0.980443i	$-0.436944\pi$
$941$	11361.7	0.393604	0.196802	+	0.980443i	$0.436944\pi$
$942$	−7939.21	−0.274600
$943$	−4304.40	−0.148643
$944$	8103.89	0.279406
$945$	−698.318	−0.0240384
$946$	−269.947	−0.00927775
$947$	8948.74	0.307070	0.153535	−	0.988143i	$-0.450934\pi$
$947$	8948.74	0.307070	0.153535	+	0.988143i	$0.450934\pi$
$948$	−5518.28	−0.189056
$949$	90460.8	3.09429
$950$	5842.77	0.199542
$951$	−19045.0	−0.649396
$952$	2169.29	0.0738519
$953$	20275.0	0.689161	0.344580	−	0.938757i	$-0.388021\pi$
$953$	20275.0	0.689161	0.344580	+	0.938757i	$0.388021\pi$
$954$	2292.37	0.0777968
$955$	−13407.7	−0.454307
$956$	−21648.3	−0.732381
$957$	672.073	0.0227012
$958$	−6999.46	−0.236057
$959$	−13333.8	−0.448979
$960$	709.402	0.0238498
$961$	−24165.6	−0.811172
$962$	−56740.5	−1.90165
$963$	−186.242	−0.00623217
$964$	27114.2	0.905903
$965$	5304.40	0.176948
$966$	966.000	0.0321745
$967$	−15581.1	−0.518155	−0.259077	−	0.965857i	$-0.583418\pi$
$967$	−15581.1	−0.518155	−0.259077	+	0.965857i	$0.583418\pi$
$968$	10612.9	0.352387
$969$	3048.98	0.101081
$970$	−11213.5	−0.371178
$971$	−47008.6	−1.55363	−0.776816	−	0.629728i	$-0.783166\pi$
$971$	−47008.6	−1.55363	−0.776816	+	0.629728i	$0.783166\pi$
$972$	972.000	0.0320750
$973$	−8417.25	−0.277332
$974$	−7940.14	−0.261210
$975$	−30801.6	−1.01173
$976$	4394.43	0.144121
$977$	31738.5	1.03931	0.519654	−	0.854377i	$-0.326061\pi$
$977$	31738.5	1.03931	0.519654	+	0.854377i	$0.326061\pi$
$978$	−7178.62	−0.234711
$979$	−833.601	−0.0272135
$980$	724.181	0.0236052
$981$	5508.25	0.179271
$982$	8060.35	0.261931
$983$	26897.4	0.872731	0.436366	−	0.899769i	$-0.356265\pi$
$983$	26897.4	0.872731	0.436366	+	0.899769i	$0.356265\pi$
$984$	4491.55	0.145514
$985$	−4378.61	−0.141639
$986$	8281.26	0.267474
$987$	−8535.01	−0.275251
$988$	9676.82	0.311600
$989$	−1481.22	−0.0476239
$990$	139.387	0.00447475
$991$	−41787.3	−1.33947	−0.669736	−	0.742599i	$-0.733593\pi$
$991$	−41787.3	−1.33947	−0.669736	+	0.742599i	$0.733593\pi$
$992$	2400.08	0.0768172
$993$	−8230.52	−0.263029
$994$	−12099.3	−0.386082
$995$	3838.42	0.122298
$996$	−660.769	−0.0210213
$997$	−25828.9	−0.820470	−0.410235	−	0.911980i	$-0.634553\pi$
$997$	−25828.9	−0.820470	−0.410235	+	0.911980i	$0.634553\pi$
$998$	29643.0	0.940212
$999$	8307.29	0.263094

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	966.4.a.n.1.3	✓	5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
966.4.a.n.1.3	✓	5	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:	\( N \)	\(=\)	\( 966 = 2 \cdot 3 \cdot 7 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	966.a (trivial)

\(f(q)\)	\(=\)	\(q-2.00000 q^{2} +3.00000 q^{3} +4.00000 q^{4} +3.69480 q^{5} -6.00000 q^{6} -7.00000 q^{7} -8.00000 q^{8} +9.00000 q^{9} +O(q^{10})\)	\(q-2.00000 q^{2} +3.00000 q^{3} +4.00000 q^{4} +3.69480 q^{5} -6.00000 q^{6} -7.00000 q^{7} -8.00000 q^{8} +9.00000 q^{9} -7.38960 q^{10} -2.09584 q^{11} +12.0000 q^{12} +92.2078 q^{13} +14.0000 q^{14} +11.0844 q^{15} +16.0000 q^{16} +38.7373 q^{17} -18.0000 q^{18} +26.2364 q^{19} +14.7792 q^{20} -21.0000 q^{21} +4.19168 q^{22} +23.0000 q^{23} -24.0000 q^{24} -111.348 q^{25} -184.416 q^{26} +27.0000 q^{27} -28.0000 q^{28} -106.890 q^{29} -22.1688 q^{30} -75.0026 q^{31} -32.0000 q^{32} -6.28752 q^{33} -77.4745 q^{34} -25.8636 q^{35} +36.0000 q^{36} +307.677 q^{37} -52.4729 q^{38} +276.624 q^{39} -29.5584 q^{40} -187.148 q^{41} +42.0000 q^{42} -64.4008 q^{43} -8.38336 q^{44} +33.2532 q^{45} -46.0000 q^{46} +406.429 q^{47} +48.0000 q^{48} +49.0000 q^{49} +222.697 q^{50} +116.212 q^{51} +368.831 q^{52} -127.354 q^{53} -54.0000 q^{54} -7.74371 q^{55} +56.0000 q^{56} +78.7093 q^{57} +213.780 q^{58} +506.493 q^{59} +44.3376 q^{60} +274.652 q^{61} +150.005 q^{62} -63.0000 q^{63} +64.0000 q^{64} +340.690 q^{65} +12.5750 q^{66} -211.711 q^{67} +154.949 q^{68} +69.0000 q^{69} +51.7272 q^{70} -864.233 q^{71} -72.0000 q^{72} +981.054 q^{73} -615.355 q^{74} -334.045 q^{75} +104.946 q^{76} +14.6709 q^{77} -553.247 q^{78} -459.857 q^{79} +59.1168 q^{80} +81.0000 q^{81} +374.296 q^{82} -55.0641 q^{83} -84.0000 q^{84} +143.127 q^{85} +128.802 q^{86} -320.670 q^{87} +16.7667 q^{88} +397.741 q^{89} -66.5064 q^{90} -645.455 q^{91} +92.0000 q^{92} -225.008 q^{93} -812.858 q^{94} +96.9385 q^{95} -96.0000 q^{96} +1517.46 q^{97} -98.0000 q^{98} -18.8625 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - 10 q^{2} + 15 q^{3} + 20 q^{4} - 30 q^{6} - 35 q^{7} - 40 q^{8} + 45 q^{9}+O(q^{10}) \) 5 * q - 10 * q^2 + 15 * q^3 + 20 * q^4 - 30 * q^6 - 35 * q^7 - 40 * q^8 + 45 * q^9	\( 5 q - 10 q^{2} + 15 q^{3} + 20 q^{4} - 30 q^{6} - 35 q^{7} - 40 q^{8} + 45 q^{9} + 18 q^{11} + 60 q^{12} - 22 q^{13} + 70 q^{14} + 80 q^{16} + 44 q^{17} - 90 q^{18} - 14 q^{19} - 105 q^{21} - 36 q^{22} + 115 q^{23} - 120 q^{24} + 43 q^{25} + 44 q^{26} + 135 q^{27} - 140 q^{28} - 39 q^{29} - 100 q^{31} - 160 q^{32} + 54 q^{33} - 88 q^{34} + 180 q^{36} + 255 q^{37} + 28 q^{38} - 66 q^{39} + 69 q^{41} + 210 q^{42} + 912 q^{43} + 72 q^{44} - 230 q^{46} - 319 q^{47} + 240 q^{48} + 245 q^{49} - 86 q^{50} + 132 q^{51} - 88 q^{52} + 745 q^{53} - 270 q^{54} + 2199 q^{55} + 280 q^{56} - 42 q^{57} + 78 q^{58} - 315 q^{59} + 1091 q^{61} + 200 q^{62} - 315 q^{63} + 320 q^{64} + 533 q^{65} - 108 q^{66} + 991 q^{67} + 176 q^{68} + 345 q^{69} + 923 q^{71} - 360 q^{72} + 1144 q^{73} - 510 q^{74} + 129 q^{75} - 56 q^{76} - 126 q^{77} + 132 q^{78} - 110 q^{79} + 405 q^{81} - 138 q^{82} + 218 q^{83} - 420 q^{84} + 2973 q^{85} - 1824 q^{86} - 117 q^{87} - 144 q^{88} - 13 q^{89} + 154 q^{91} + 460 q^{92} - 300 q^{93} + 638 q^{94} - 347 q^{95} - 480 q^{96} + 761 q^{97} - 490 q^{98} + 162 q^{99}+O(q^{100}) \) 5 * q - 10 * q^2 + 15 * q^3 + 20 * q^4 - 30 * q^6 - 35 * q^7 - 40 * q^8 + 45 * q^9 + 18 * q^11 + 60 * q^12 - 22 * q^13 + 70 * q^14 + 80 * q^16 + 44 * q^17 - 90 * q^18 - 14 * q^19 - 105 * q^21 - 36 * q^22 + 115 * q^23 - 120 * q^24 + 43 * q^25 + 44 * q^26 + 135 * q^27 - 140 * q^28 - 39 * q^29 - 100 * q^31 - 160 * q^32 + 54 * q^33 - 88 * q^34 + 180 * q^36 + 255 * q^37 + 28 * q^38 - 66 * q^39 + 69 * q^41 + 210 * q^42 + 912 * q^43 + 72 * q^44 - 230 * q^46 - 319 * q^47 + 240 * q^48 + 245 * q^49 - 86 * q^50 + 132 * q^51 - 88 * q^52 + 745 * q^53 - 270 * q^54 + 2199 * q^55 + 280 * q^56 - 42 * q^57 + 78 * q^58 - 315 * q^59 + 1091 * q^61 + 200 * q^62 - 315 * q^63 + 320 * q^64 + 533 * q^65 - 108 * q^66 + 991 * q^67 + 176 * q^68 + 345 * q^69 + 923 * q^71 - 360 * q^72 + 1144 * q^73 - 510 * q^74 + 129 * q^75 - 56 * q^76 - 126 * q^77 + 132 * q^78 - 110 * q^79 + 405 * q^81 - 138 * q^82 + 218 * q^83 - 420 * q^84 + 2973 * q^85 - 1824 * q^86 - 117 * q^87 - 144 * q^88 - 13 * q^89 + 154 * q^91 + 460 * q^92 - 300 * q^93 + 638 * q^94 - 347 * q^95 - 480 * q^96 + 761 * q^97 - 490 * q^98 + 162 * q^99

Introduction

L-functions

Modular forms

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