LMFDB - Embedded newform 1110.4.a.m.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1110,4,Mod(1,1110)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1110, 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("1110.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1110 = 2 \cdot 3 \cdot 5 \cdot 37 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1110.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:	$65.4921201064$
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} - 418x^{3} + 1860x^{2} + 42465x - 273730 $ x^5 - 2x^4 - 418x^3 + 1860x^2 + 42465x - 273730
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			1.4
Root			$-13.3914$ of defining polynomial
Character	$\chi$	$=$	1110.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.00000 q^{2} -3.00000 q^{3} +4.00000 q^{4} +5.00000 q^{5} +6.00000 q^{6} +11.7656 q^{7} -8.00000 q^{8} +9.00000 q^{9} +O(q^{10})$	\(q-2.00000 q^{2} -3.00000 q^{3} +4.00000 q^{4} +5.00000 q^{5} +6.00000 q^{6} +11.7656 q^{7} -8.00000 q^{8} +9.00000 q^{9} -10.0000 q^{10} +48.6753 q^{11} -12.0000 q^{12} -52.6102 q^{13} -23.5313 q^{14} -15.0000 q^{15} +16.0000 q^{16} -88.9421 q^{17} -18.0000 q^{18} +99.8826 q^{19} +20.0000 q^{20} -35.2969 q^{21} -97.3506 q^{22} +129.243 q^{23} +24.0000 q^{24} +25.0000 q^{25} +105.220 q^{26} -27.0000 q^{27} +47.0626 q^{28} +85.0845 q^{29} +30.0000 q^{30} +271.143 q^{31} -32.0000 q^{32} -146.026 q^{33} +177.884 q^{34} +58.8282 q^{35} +36.0000 q^{36} +37.0000 q^{37} -199.765 q^{38} +157.831 q^{39} -40.0000 q^{40} -113.642 q^{41} +70.5939 q^{42} -83.0845 q^{43} +194.701 q^{44} +45.0000 q^{45} -258.486 q^{46} +106.573 q^{47} -48.0000 q^{48} -204.570 q^{49} -50.0000 q^{50} +266.826 q^{51} -210.441 q^{52} -255.423 q^{53} +54.0000 q^{54} +243.377 q^{55} -94.1251 q^{56} -299.648 q^{57} -170.169 q^{58} +312.224 q^{59} -60.0000 q^{60} -490.053 q^{61} -542.286 q^{62} +105.891 q^{63} +64.0000 q^{64} -263.051 q^{65} +292.052 q^{66} -279.281 q^{67} -355.769 q^{68} -387.728 q^{69} -117.656 q^{70} -414.025 q^{71} -72.0000 q^{72} +444.535 q^{73} -74.0000 q^{74} -75.0000 q^{75} +399.530 q^{76} +572.696 q^{77} -315.661 q^{78} +346.689 q^{79} +80.0000 q^{80} +81.0000 q^{81} +227.283 q^{82} +142.538 q^{83} -141.188 q^{84} -444.711 q^{85} +166.169 q^{86} -255.254 q^{87} -389.402 q^{88} +600.494 q^{89} -90.0000 q^{90} -618.993 q^{91} +516.971 q^{92} -813.429 q^{93} -213.145 q^{94} +499.413 q^{95} +96.0000 q^{96} +186.740 q^{97} +409.139 q^{98} +438.078 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - 10 q^{2} - 15 q^{3} + 20 q^{4} + 25 q^{5} + 30 q^{6} - 19 q^{7} - 40 q^{8} + 45 q^{9}+O(q^{10}) $ 5 * q - 10 * q^2 - 15 * q^3 + 20 * q^4 + 25 * q^5 + 30 * q^6 - 19 * q^7 - 40 * q^8 + 45 * q^9	\( 5 q - 10 q^{2} - 15 q^{3} + 20 q^{4} + 25 q^{5} + 30 q^{6} - 19 q^{7} - 40 q^{8} + 45 q^{9} - 50 q^{10} + 36 q^{11} - 60 q^{12} - 35 q^{13} + 38 q^{14} - 75 q^{15} + 80 q^{16} - 12 q^{17} - 90 q^{18} + 78 q^{19} + 100 q^{20} + 57 q^{21} - 72 q^{22} - 171 q^{23} + 120 q^{24} + 125 q^{25} + 70 q^{26} - 135 q^{27} - 76 q^{28} + 259 q^{29} + 150 q^{30} + 264 q^{31} - 160 q^{32} - 108 q^{33} + 24 q^{34} - 95 q^{35} + 180 q^{36} + 185 q^{37} - 156 q^{38} + 105 q^{39} - 200 q^{40} + 486 q^{41} - 114 q^{42} - 249 q^{43} + 144 q^{44} + 225 q^{45} + 342 q^{46} - 56 q^{47} - 240 q^{48} + 272 q^{49} - 250 q^{50} + 36 q^{51} - 140 q^{52} - 69 q^{53} + 270 q^{54} + 180 q^{55} + 152 q^{56} - 234 q^{57} - 518 q^{58} + 189 q^{59} - 300 q^{60} + 42 q^{61} - 528 q^{62} - 171 q^{63} + 320 q^{64} - 175 q^{65} + 216 q^{66} - 1215 q^{67} - 48 q^{68} + 513 q^{69} + 190 q^{70} - 17 q^{71} - 360 q^{72} - 668 q^{73} - 370 q^{74} - 375 q^{75} + 312 q^{76} - 390 q^{77} - 210 q^{78} - 575 q^{79} + 400 q^{80} + 405 q^{81} - 972 q^{82} - 328 q^{83} + 228 q^{84} - 60 q^{85} + 498 q^{86} - 777 q^{87} - 288 q^{88} + 909 q^{89} - 450 q^{90} - 533 q^{91} - 684 q^{92} - 792 q^{93} + 112 q^{94} + 390 q^{95} + 480 q^{96} - 814 q^{97} - 544 q^{98} + 324 q^{99}+O(q^{100}) \) 5 * q - 10 * q^2 - 15 * q^3 + 20 * q^4 + 25 * q^5 + 30 * q^6 - 19 * q^7 - 40 * q^8 + 45 * q^9 - 50 * q^10 + 36 * q^11 - 60 * q^12 - 35 * q^13 + 38 * q^14 - 75 * q^15 + 80 * q^16 - 12 * q^17 - 90 * q^18 + 78 * q^19 + 100 * q^20 + 57 * q^21 - 72 * q^22 - 171 * q^23 + 120 * q^24 + 125 * q^25 + 70 * q^26 - 135 * q^27 - 76 * q^28 + 259 * q^29 + 150 * q^30 + 264 * q^31 - 160 * q^32 - 108 * q^33 + 24 * q^34 - 95 * q^35 + 180 * q^36 + 185 * q^37 - 156 * q^38 + 105 * q^39 - 200 * q^40 + 486 * q^41 - 114 * q^42 - 249 * q^43 + 144 * q^44 + 225 * q^45 + 342 * q^46 - 56 * q^47 - 240 * q^48 + 272 * q^49 - 250 * q^50 + 36 * q^51 - 140 * q^52 - 69 * q^53 + 270 * q^54 + 180 * q^55 + 152 * q^56 - 234 * q^57 - 518 * q^58 + 189 * q^59 - 300 * q^60 + 42 * q^61 - 528 * q^62 - 171 * q^63 + 320 * q^64 - 175 * q^65 + 216 * q^66 - 1215 * q^67 - 48 * q^68 + 513 * q^69 + 190 * q^70 - 17 * q^71 - 360 * q^72 - 668 * q^73 - 370 * q^74 - 375 * q^75 + 312 * q^76 - 390 * q^77 - 210 * q^78 - 575 * q^79 + 400 * q^80 + 405 * q^81 - 972 * q^82 - 328 * q^83 + 228 * q^84 - 60 * q^85 + 498 * q^86 - 777 * q^87 - 288 * q^88 + 909 * q^89 - 450 * q^90 - 533 * q^91 - 684 * q^92 - 792 * q^93 + 112 * q^94 + 390 * q^95 + 480 * q^96 - 814 * q^97 - 544 * q^98 + 324 * 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
$5$	5.00000	0.447214
$5$	5.00000	0.447214
$6$	6.00000	0.408248
$7$	11.7656	0.635285	0.317642	−	0.948211i	$-0.397109\pi$
$7$	11.7656	0.635285	0.317642	+	0.948211i	$0.397109\pi$
$8$	−8.00000	−0.353553
$9$	9.00000	0.333333
$10$	−10.0000	−0.316228
$11$	48.6753	1.33420	0.667098	−	0.744970i	$-0.267536\pi$
$11$	48.6753	1.33420	0.667098	+	0.744970i	$0.267536\pi$
$12$	−12.0000	−0.288675
$13$	−52.6102	−1.12242	−0.561209	−	0.827674i	$-0.689664\pi$
$13$	−52.6102	−1.12242	−0.561209	+	0.827674i	$0.689664\pi$
$14$	−23.5313	−0.449214
$15$	−15.0000	−0.258199
$16$	16.0000	0.250000
$17$	−88.9421	−1.26892	−0.634460	−	0.772956i	$-0.718777\pi$
$17$	−88.9421	−1.26892	−0.634460	+	0.772956i	$0.718777\pi$
$18$	−18.0000	−0.235702
$19$	99.8826	1.20603	0.603017	−	0.797728i	$-0.293965\pi$
$19$	99.8826	1.20603	0.603017	+	0.797728i	$0.293965\pi$
$20$	20.0000	0.223607
$21$	−35.2969	−0.366782
$22$	−97.3506	−0.943419
$23$	129.243	1.17169	0.585847	−	0.810421i	$-0.300762\pi$
$23$	129.243	1.17169	0.585847	+	0.810421i	$0.300762\pi$
$24$	24.0000	0.204124
$25$	25.0000	0.200000
$26$	105.220	0.793670
$27$	−27.0000	−0.192450
$28$	47.0626	0.317642
$29$	85.0845	0.544821	0.272410	−	0.962181i	$-0.412179\pi$
$29$	85.0845	0.544821	0.272410	+	0.962181i	$0.412179\pi$
$30$	30.0000	0.182574
$31$	271.143	1.57093	0.785463	−	0.618909i	$-0.212425\pi$
$31$	271.143	1.57093	0.785463	+	0.618909i	$0.212425\pi$
$32$	−32.0000	−0.176777
$33$	−146.026	−0.770299
$34$	177.884	0.897262
$35$	58.8282	0.284108
$36$	36.0000	0.166667
$37$	37.0000	0.164399
$37$	37.0000	0.164399
$38$	−199.765	−0.852795
$39$	157.831	0.648029
$40$	−40.0000	−0.158114
$41$	−113.642	−0.432874	−0.216437	−	0.976297i	$-0.569444\pi$
$41$	−113.642	−0.432874	−0.216437	+	0.976297i	$0.569444\pi$
$42$	70.5939	0.259354
$43$	−83.0845	−0.294657	−0.147329	−	0.989088i	$-0.547068\pi$
$43$	−83.0845	−0.294657	−0.147329	+	0.989088i	$0.547068\pi$
$44$	194.701	0.667098
$45$	45.0000	0.149071
$46$	−258.486	−0.828513
$47$	106.573	0.330749	0.165375	−	0.986231i	$-0.447117\pi$
$47$	106.573	0.330749	0.165375	+	0.986231i	$0.447117\pi$
$48$	−48.0000	−0.144338
$49$	−204.570	−0.596413
$50$	−50.0000	−0.141421
$51$	266.826	0.732611
$52$	−210.441	−0.561209
$53$	−255.423	−0.661981	−0.330990	−	0.943634i	$-0.607383\pi$
$53$	−255.423	−0.661981	−0.330990	+	0.943634i	$0.607383\pi$
$54$	54.0000	0.136083
$55$	243.377	0.596671
$56$	−94.1251	−0.224607
$57$	−299.648	−0.696304
$58$	−170.169	−0.385246
$59$	312.224	0.688950	0.344475	−	0.938795i	$-0.388057\pi$
$59$	312.224	0.688950	0.344475	+	0.938795i	$0.388057\pi$
$60$	−60.0000	−0.129099
$61$	−490.053	−1.02860	−0.514302	−	0.857609i	$-0.671949\pi$
$61$	−490.053	−1.02860	−0.514302	+	0.857609i	$0.671949\pi$
$62$	−542.286	−1.11081
$63$	105.891	0.211762
$64$	64.0000	0.125000
$65$	−263.051	−0.501961
$66$	292.052	0.544683
$67$	−279.281	−0.509248	−0.254624	−	0.967040i	$-0.581952\pi$
$67$	−279.281	−0.509248	−0.254624	+	0.967040i	$0.581952\pi$
$68$	−355.769	−0.634460
$69$	−387.728	−0.676478
$70$	−117.656	−0.200895
$71$	−414.025	−0.692052	−0.346026	−	0.938225i	$-0.612469\pi$
$71$	−414.025	−0.692052	−0.346026	+	0.938225i	$0.612469\pi$
$72$	−72.0000	−0.117851
$73$	444.535	0.712725	0.356362	−	0.934348i	$-0.384017\pi$
$73$	444.535	0.712725	0.356362	+	0.934348i	$0.384017\pi$
$74$	−74.0000	−0.116248
$75$	−75.0000	−0.115470
$76$	399.530	0.603017
$77$	572.696	0.847595
$78$	−315.661	−0.458226
$79$	346.689	0.493741	0.246870	−	0.969049i	$-0.420598\pi$
$79$	346.689	0.493741	0.246870	+	0.969049i	$0.420598\pi$
$80$	80.0000	0.111803
$81$	81.0000	0.111111
$82$	227.283	0.306088
$83$	142.538	0.188501	0.0942507	−	0.995548i	$-0.469954\pi$
$83$	142.538	0.188501	0.0942507	+	0.995548i	$0.469954\pi$
$84$	−141.188	−0.183391
$85$	−444.711	−0.567478
$86$	166.169	0.208354
$87$	−255.254	−0.314552
$88$	−389.402	−0.471710
$89$	600.494	0.715193	0.357597	−	0.933876i	$-0.383596\pi$
$89$	600.494	0.715193	0.357597	+	0.933876i	$0.383596\pi$
$90$	−90.0000	−0.105409
$91$	−618.993	−0.713056
$92$	516.971	0.585847
$93$	−813.429	−0.906975
$94$	−213.145	−0.233875
$95$	499.413	0.539355
$96$	96.0000	0.102062
$97$	186.740	0.195470	0.0977348	−	0.995212i	$-0.468840\pi$
$97$	186.740	0.195470	0.0977348	+	0.995212i	$0.468840\pi$
$98$	409.139	0.421728
$99$	438.078	0.444732
$100$	100.000	0.100000
$101$	1110.10	1.09365	0.546826	−	0.837247i	$-0.315836\pi$
$101$	1110.10	1.09365	0.546826	+	0.837247i	$0.315836\pi$
$102$	−533.653	−0.518034
$103$	−991.711	−0.948701	−0.474351	−	0.880336i	$-0.657317\pi$
$103$	−991.711	−0.948701	−0.474351	+	0.880336i	$0.657317\pi$
$104$	420.882	0.396835
$105$	−176.485	−0.164030
$106$	510.845	0.468091
$107$	2159.58	1.95116	0.975580	−	0.219643i	$-0.0704892\pi$
$107$	2159.58	1.95116	0.975580	+	0.219643i	$0.0704892\pi$
$108$	−108.000	−0.0962250
$109$	−1572.82	−1.38210	−0.691048	−	0.722809i	$-0.742851\pi$
$109$	−1572.82	−1.38210	−0.691048	+	0.722809i	$0.742851\pi$
$110$	−486.753	−0.421910
$111$	−111.000	−0.0949158
$112$	188.250	0.158821
$113$	−2127.17	−1.77086	−0.885431	−	0.464771i	$-0.846137\pi$
$113$	−2127.17	−1.77086	−0.885431	+	0.464771i	$0.846137\pi$
$114$	599.296	0.492361
$115$	646.214	0.523998
$116$	340.338	0.272410
$117$	−473.492	−0.374140
$118$	−624.447	−0.487161
$119$	−1046.46	−0.806126
$120$	120.000	0.0912871
$121$	1038.29	0.780079
$122$	980.105	0.727332
$123$	340.925	0.249920
$124$	1084.57	0.785463
$125$	125.000	0.0894427
$126$	−211.782	−0.149738
$127$	1140.28	0.796718	0.398359	−	0.917230i	$-0.369580\pi$
$127$	1140.28	0.796718	0.398359	+	0.917230i	$0.369580\pi$
$128$	−128.000	−0.0883883
$129$	249.254	0.170121
$130$	526.102	0.354940
$131$	1193.03	0.795688	0.397844	−	0.917453i	$-0.369759\pi$
$131$	1193.03	0.795688	0.397844	+	0.917453i	$0.369759\pi$
$132$	−584.104	−0.385149
$133$	1175.18	0.766175
$134$	558.562	0.360093
$135$	−135.000	−0.0860663
$136$	711.537	0.448631
$137$	699.138	0.435995	0.217998	−	0.975949i	$-0.430047\pi$
$137$	699.138	0.435995	0.217998	+	0.975949i	$0.430047\pi$
$138$	775.457	0.478342
$139$	−2137.12	−1.30409	−0.652044	−	0.758181i	$-0.726088\pi$
$139$	−2137.12	−1.30409	−0.652044	+	0.758181i	$0.726088\pi$
$140$	235.313	0.142054
$141$	−319.718	−0.190958
$142$	828.050	0.489355
$143$	−2560.82	−1.49753
$144$	144.000	0.0833333
$145$	425.423	0.243651
$146$	−889.071	−0.503973
$147$	613.709	0.344339
$148$	148.000	0.0821995
$149$	3085.85	1.69666	0.848331	−	0.529467i	$-0.177608\pi$
$149$	3085.85	1.69666	0.848331	+	0.529467i	$0.177608\pi$
$150$	150.000	0.0816497
$151$	−1135.93	−0.612188	−0.306094	−	0.952001i	$-0.599022\pi$
$151$	−1135.93	−0.612188	−0.306094	+	0.952001i	$0.599022\pi$
$152$	−799.061	−0.426397
$153$	−800.479	−0.422973
$154$	−1145.39	−0.599340
$155$	1355.71	0.702540
$156$	631.322	0.324014
$157$	−2691.98	−1.36843	−0.684215	−	0.729281i	$-0.739855\pi$
$157$	−2691.98	−1.36843	−0.684215	+	0.729281i	$0.739855\pi$
$158$	−693.377	−0.349127
$159$	766.268	0.382195
$160$	−160.000	−0.0790569
$161$	1520.62	0.744360
$162$	−162.000	−0.0785674
$163$	2144.53	1.03050	0.515252	−	0.857039i	$-0.327698\pi$
$163$	2144.53	1.03050	0.515252	+	0.857039i	$0.327698\pi$
$164$	−454.566	−0.216437
$165$	−730.130	−0.344488
$166$	−285.077	−0.133291
$167$	4178.38	1.93612	0.968062	−	0.250710i	$-0.0806639\pi$
$167$	4178.38	1.93612	0.968062	+	0.250710i	$0.0806639\pi$
$168$	282.375	0.129677
$169$	570.833	0.259824
$170$	889.421	0.401268
$171$	898.943	0.402011
$172$	−332.338	−0.147329
$173$	1802.57	0.792179	0.396089	−	0.918212i	$-0.370367\pi$
$173$	1802.57	0.792179	0.396089	+	0.918212i	$0.370367\pi$
$174$	510.507	0.222422
$175$	294.141	0.127057
$176$	778.805	0.333549
$177$	−936.671	−0.397766
$178$	−1200.99	−0.505718
$179$	3042.92	1.27061	0.635303	−	0.772263i	$-0.280875\pi$
$179$	3042.92	1.27061	0.635303	+	0.772263i	$0.280875\pi$
$180$	180.000	0.0745356
$181$	−577.859	−0.237303	−0.118652	−	0.992936i	$-0.537857\pi$
$181$	−577.859	−0.237303	−0.118652	+	0.992936i	$0.537857\pi$
$182$	1237.99	0.504207
$183$	1470.16	0.593864
$184$	−1033.94	−0.414257
$185$	185.000	0.0735215
$186$	1626.86	0.641328
$187$	−4329.29	−1.69299
$188$	426.291	0.165375
$189$	−317.672	−0.122261
$190$	−998.826	−0.381381
$191$	−2503.56	−0.948437	−0.474219	−	0.880407i	$-0.657269\pi$
$191$	−2503.56	−0.948437	−0.474219	+	0.880407i	$0.657269\pi$
$192$	−192.000	−0.0721688
$193$	4251.11	1.58550	0.792750	−	0.609546i	$-0.208648\pi$
$193$	4251.11	1.58550	0.792750	+	0.609546i	$0.208648\pi$
$194$	−373.480	−0.138218
$195$	789.153	0.289807
$196$	−818.279	−0.298207
$197$	972.098	0.351569	0.175784	−	0.984429i	$-0.443754\pi$
$197$	972.098	0.351569	0.175784	+	0.984429i	$0.443754\pi$
$198$	−876.156	−0.314473
$199$	5070.89	1.80636	0.903181	−	0.429260i	$-0.141225\pi$
$199$	5070.89	1.80636	0.903181	+	0.429260i	$0.141225\pi$
$200$	−200.000	−0.0707107
$201$	837.843	0.294014
$202$	−2220.19	−0.773328
$203$	1001.07	0.346116
$204$	1067.31	0.366306
$205$	−568.208	−0.193587
$206$	1983.42	0.670833
$207$	1163.19	0.390565
$208$	−841.763	−0.280605
$209$	4861.82	1.60909
$210$	352.969	0.115987
$211$	−18.8560	−0.00615213	−0.00307607	−	0.999995i	$-0.500979\pi$
$211$	−18.8560	−0.00615213	−0.00307607	+	0.999995i	$0.500979\pi$
$212$	−1021.69	−0.330990
$213$	1242.07	0.399557
$214$	−4319.15	−1.37968
$215$	−415.423	−0.131775
$216$	216.000	0.0680414
$217$	3190.17	0.997986
$218$	3145.63	0.977289
$219$	−1333.61	−0.411492
$220$	973.506	0.298335
$221$	4679.26	1.42426
$222$	222.000	0.0671156
$223$	−22.2903	−0.00669360	−0.00334680	−	0.999994i	$-0.501065\pi$
$223$	−22.2903	−0.00669360	−0.00334680	+	0.999994i	$0.501065\pi$
$224$	−376.501	−0.112304
$225$	225.000	0.0666667
$226$	4254.34	1.25219
$227$	900.592	0.263323	0.131662	−	0.991295i	$-0.457969\pi$
$227$	900.592	0.263323	0.131662	+	0.991295i	$0.457969\pi$
$228$	−1198.59	−0.348152
$229$	5839.29	1.68503	0.842514	−	0.538675i	$-0.181075\pi$
$229$	5839.29	1.68503	0.842514	+	0.538675i	$0.181075\pi$
$230$	−1292.43	−0.370522
$231$	−1718.09	−0.489359
$232$	−680.676	−0.192623
$233$	6071.67	1.70716	0.853580	−	0.520962i	$-0.174427\pi$
$233$	6071.67	1.70716	0.853580	+	0.520962i	$0.174427\pi$
$234$	946.984	0.264557
$235$	532.863	0.147916
$236$	1248.89	0.344475
$237$	−1040.07	−0.285061
$238$	2092.92	0.570017
$239$	−3439.94	−0.931009	−0.465505	−	0.885045i	$-0.654127\pi$
$239$	−3439.94	−0.931009	−0.465505	+	0.885045i	$0.654127\pi$
$240$	−240.000	−0.0645497
$241$	1628.19	0.435189	0.217595	−	0.976039i	$-0.430179\pi$
$241$	1628.19	0.435189	0.217595	+	0.976039i	$0.430179\pi$
$242$	−2076.57	−0.551599
$243$	−243.000	−0.0641500
$244$	−1960.21	−0.514302
$245$	−1022.85	−0.266724
$246$	−681.849	−0.176720
$247$	−5254.84	−1.35367
$248$	−2169.14	−0.555406
$249$	−427.615	−0.108831
$250$	−250.000	−0.0632456
$251$	−438.774	−0.110339	−0.0551696	−	0.998477i	$-0.517570\pi$
$251$	−438.774	−0.110339	−0.0551696	+	0.998477i	$0.517570\pi$
$252$	423.563	0.105881
$253$	6290.93	1.56327
$254$	−2280.55	−0.563364
$255$	1334.13	0.327634
$256$	256.000	0.0625000
$257$	5850.25	1.41996	0.709978	−	0.704224i	$-0.248705\pi$
$257$	5850.25	1.41996	0.709978	+	0.704224i	$0.248705\pi$
$258$	−498.507	−0.120293
$259$	435.329	0.104440
$260$	−1052.20	−0.250980
$261$	765.761	0.181607
$262$	−2386.05	−0.562636
$263$	−1704.64	−0.399667	−0.199833	−	0.979830i	$-0.564040\pi$
$263$	−1704.64	−0.399667	−0.199833	+	0.979830i	$0.564040\pi$
$264$	1168.21	0.272342
$265$	−1277.11	−0.296047
$266$	−2350.37	−0.541768
$267$	−1801.48	−0.412917
$268$	−1117.12	−0.254624
$269$	1223.26	0.277262	0.138631	−	0.990344i	$-0.455730\pi$
$269$	1223.26	0.277262	0.138631	+	0.990344i	$0.455730\pi$
$270$	270.000	0.0608581
$271$	5616.72	1.25901	0.629505	−	0.776997i	$-0.283258\pi$
$271$	5616.72	1.25901	0.629505	+	0.776997i	$0.283258\pi$
$272$	−1423.07	−0.317230
$273$	1856.98	0.411683
$274$	−1398.28	−0.308295
$275$	1216.88	0.266839
$276$	−1550.91	−0.338239
$277$	664.191	0.144070	0.0720350	−	0.997402i	$-0.477051\pi$
$277$	664.191	0.144070	0.0720350	+	0.997402i	$0.477051\pi$
$278$	4274.24	0.922130
$279$	2440.29	0.523642
$280$	−470.626	−0.100447
$281$	−7804.60	−1.65688	−0.828440	−	0.560078i	$-0.810771\pi$
$281$	−7804.60	−1.65688	−0.828440	+	0.560078i	$0.810771\pi$
$282$	639.436	0.135028
$283$	−3738.35	−0.785237	−0.392618	−	0.919701i	$-0.628431\pi$
$283$	−3738.35	−0.785237	−0.392618	+	0.919701i	$0.628431\pi$
$284$	−1656.10	−0.346026
$285$	−1498.24	−0.311397
$286$	5121.64	1.05891
$287$	−1337.07	−0.274998
$288$	−288.000	−0.0589256
$289$	2997.70	0.610158
$290$	−850.845	−0.172287
$291$	−560.220	−0.112854
$292$	1778.14	0.356362
$293$	−6746.96	−1.34526	−0.672631	−	0.739978i	$-0.734836\pi$
$293$	−6746.96	−1.34526	−0.672631	+	0.739978i	$0.734836\pi$
$294$	−1227.42	−0.243485
$295$	1561.12	0.308108
$296$	−296.000	−0.0581238
$297$	−1314.23	−0.256766
$298$	−6171.70	−1.19972
$299$	−6799.49	−1.31513
$300$	−300.000	−0.0577350
$301$	−977.543	−0.187191
$302$	2271.85	0.432882
$303$	−3330.29	−0.631420
$304$	1598.12	0.301508
$305$	−2450.26	−0.460005
$306$	1600.96	0.299087
$307$	−1947.32	−0.362017	−0.181009	−	0.983482i	$-0.557936\pi$
$307$	−1947.32	−0.362017	−0.181009	+	0.983482i	$0.557936\pi$
$308$	2290.78	0.423797
$309$	2975.13	0.547733
$310$	−2711.43	−0.496770
$311$	10720.8	1.95472	0.977360	−	0.211582i	$-0.0678615\pi$
$311$	10720.8	1.95472	0.977360	+	0.211582i	$0.0678615\pi$
$312$	−1262.64	−0.229113
$313$	3250.23	0.586946	0.293473	−	0.955967i	$-0.405189\pi$
$313$	3250.23	0.586946	0.293473	+	0.955967i	$0.405189\pi$
$314$	5383.96	0.967625
$315$	529.454	0.0947027
$316$	1386.75	0.246870
$317$	−5093.19	−0.902404	−0.451202	−	0.892422i	$-0.649005\pi$
$317$	−5093.19	−0.902404	−0.451202	+	0.892422i	$0.649005\pi$
$318$	−1532.54	−0.270253
$319$	4141.52	0.726898
$320$	320.000	0.0559017
$321$	−6478.73	−1.12650
$322$	−3041.25	−0.526342
$323$	−8883.77	−1.53036
$324$	324.000	0.0555556
$325$	−1315.25	−0.224484
$326$	−4289.05	−0.728677
$327$	4718.45	0.797953
$328$	909.133	0.153044
$329$	1253.90	0.210120
$330$	1460.26	0.243590
$331$	−323.912	−0.0537879	−0.0268940	−	0.999638i	$-0.508562\pi$
$331$	−323.912	−0.0537879	−0.0268940	+	0.999638i	$0.508562\pi$
$332$	570.153	0.0942507
$333$	333.000	0.0547997
$334$	−8356.76	−1.36905
$335$	−1396.40	−0.227743
$336$	−564.751	−0.0916955
$337$	−9445.05	−1.52672	−0.763360	−	0.645973i	$-0.776452\pi$
$337$	−9445.05	−1.52672	−0.763360	+	0.645973i	$0.776452\pi$
$338$	−1141.67	−0.183723
$339$	6381.51	1.02241
$340$	−1778.84	−0.283739
$341$	13198.0	2.09592
$342$	−1797.89	−0.284265
$343$	−6442.51	−1.01418
$344$	664.676	0.104177
$345$	−1938.64	−0.302530
$346$	−3605.14	−0.560155
$347$	821.406	0.127076	0.0635380	−	0.997979i	$-0.479762\pi$
$347$	821.406	0.127076	0.0635380	+	0.997979i	$0.479762\pi$
$348$	−1021.01	−0.157276
$349$	5887.28	0.902977	0.451488	−	0.892277i	$-0.350893\pi$
$349$	5887.28	0.902977	0.451488	+	0.892277i	$0.350893\pi$
$350$	−588.282	−0.0898429
$351$	1420.48	0.216010
$352$	−1557.61	−0.235855
$353$	7959.04	1.20005	0.600024	−	0.799982i	$-0.295158\pi$
$353$	7959.04	1.20005	0.600024	+	0.799982i	$0.295158\pi$
$354$	1873.34	0.281263
$355$	−2070.12	−0.309495
$356$	2401.98	0.357597
$357$	3139.38	0.465417
$358$	−6085.84	−0.898454
$359$	5387.23	0.791998	0.395999	−	0.918251i	$-0.370398\pi$
$359$	5387.23	0.791998	0.395999	+	0.918251i	$0.370398\pi$
$360$	−360.000	−0.0527046
$361$	3117.53	0.454517
$362$	1155.72	0.167799
$363$	−3114.86	−0.450379
$364$	−2475.97	−0.356528
$365$	2222.68	0.318740
$366$	−2940.32	−0.419926
$367$	−11495.0	−1.63497	−0.817485	−	0.575950i	$-0.804632\pi$
$367$	−11495.0	−1.63497	−0.817485	+	0.575950i	$0.804632\pi$
$368$	2067.88	0.292924
$369$	−1022.77	−0.144291
$370$	−370.000	−0.0519875
$371$	−3005.21	−0.420546
$372$	−3253.72	−0.453487
$373$	−9637.43	−1.33782	−0.668910	−	0.743343i	$-0.733239\pi$
$373$	−9637.43	−1.33782	−0.668910	+	0.743343i	$0.733239\pi$
$374$	8658.57	1.19712
$375$	−375.000	−0.0516398
$376$	−852.581	−0.116938
$377$	−4476.31	−0.611517
$378$	635.345	0.0864513
$379$	7429.18	1.00689	0.503445	−	0.864027i	$-0.332066\pi$
$379$	7429.18	1.00689	0.503445	+	0.864027i	$0.332066\pi$
$380$	1997.65	0.269677
$381$	−3420.83	−0.459985
$382$	5007.13	0.670646
$383$	6446.41	0.860042	0.430021	−	0.902819i	$-0.358506\pi$
$383$	6446.41	0.860042	0.430021	+	0.902819i	$0.358506\pi$
$384$	384.000	0.0510310
$385$	2863.48	0.379056
$386$	−8502.22	−1.12112
$387$	−747.761	−0.0982192
$388$	746.959	0.0977348
$389$	13326.7	1.73699	0.868494	−	0.495699i	$-0.165088\pi$
$389$	13326.7	1.73699	0.868494	+	0.495699i	$0.165088\pi$
$390$	−1578.31	−0.204925
$391$	−11495.1	−1.48679
$392$	1636.56	0.210864
$393$	−3579.08	−0.459391
$394$	−1944.20	−0.248597
$395$	1733.44	0.220808
$396$	1752.31	0.222366
$397$	6757.02	0.854220	0.427110	−	0.904200i	$-0.359532\pi$
$397$	6757.02	0.854220	0.427110	+	0.904200i	$0.359532\pi$
$398$	−10141.8	−1.27729
$399$	−3525.55	−0.442351
$400$	400.000	0.0500000
$401$	14279.5	1.77826	0.889132	−	0.457651i	$-0.151309\pi$
$401$	14279.5	1.77826	0.889132	+	0.457651i	$0.151309\pi$
$402$	−1675.69	−0.207900
$403$	−14264.9	−1.76324
$404$	4440.39	0.546826
$405$	405.000	0.0496904
$406$	−2002.15	−0.244741
$407$	1800.99	0.219340
$408$	−2134.61	−0.259017
$409$	−5313.36	−0.642368	−0.321184	−	0.947017i	$-0.604081\pi$
$409$	−5313.36	−0.642368	−0.321184	+	0.947017i	$0.604081\pi$
$410$	1136.42	0.136887
$411$	−2097.41	−0.251722
$412$	−3966.84	−0.474351
$413$	3673.51	0.437680
$414$	−2326.37	−0.276171
$415$	712.692	0.0843004
$416$	1683.53	0.198417
$417$	6411.36	0.752916
$418$	−9723.63	−1.13780
$419$	11052.3	1.28864	0.644320	−	0.764756i	$-0.277141\pi$
$419$	11052.3	1.28864	0.644320	+	0.764756i	$0.277141\pi$
$420$	−705.939	−0.0820149
$421$	−5391.17	−0.624108	−0.312054	−	0.950064i	$-0.601017\pi$
$421$	−5391.17	−0.624108	−0.312054	+	0.950064i	$0.601017\pi$
$422$	37.7120	0.00435022
$423$	959.154	0.110250
$424$	2043.38	0.234046
$425$	−2223.55	−0.253784
$426$	−2484.15	−0.282529
$427$	−5765.78	−0.653456
$428$	8638.31	0.975580
$429$	7682.45	0.864597
$430$	830.845	0.0931789
$431$	−6672.93	−0.745762	−0.372881	−	0.927879i	$-0.621630\pi$
$431$	−6672.93	−0.745762	−0.372881	+	0.927879i	$0.621630\pi$
$432$	−432.000	−0.0481125
$433$	−8448.98	−0.937718	−0.468859	−	0.883273i	$-0.655335\pi$
$433$	−8448.98	−0.937718	−0.468859	+	0.883273i	$0.655335\pi$
$434$	−6380.34	−0.705682
$435$	−1276.27	−0.140672
$436$	−6291.26	−0.691048
$437$	12909.1	1.41310
$438$	2667.21	0.290969
$439$	8458.82	0.919629	0.459815	−	0.888015i	$-0.347916\pi$
$439$	8458.82	0.919629	0.459815	+	0.888015i	$0.347916\pi$
$440$	−1947.01	−0.210955
$441$	−1841.13	−0.198804
$442$	−9358.53	−1.00710
$443$	965.534	0.103553	0.0517764	−	0.998659i	$-0.483512\pi$
$443$	965.534	0.103553	0.0517764	+	0.998659i	$0.483512\pi$
$444$	−444.000	−0.0474579
$445$	3002.47	0.319844
$446$	44.5807	0.00473309
$447$	−9257.54	−0.979568
$448$	753.001	0.0794106
$449$	−8400.13	−0.882910	−0.441455	−	0.897283i	$-0.645538\pi$
$449$	−8400.13	−0.882910	−0.441455	+	0.897283i	$0.645538\pi$
$450$	−450.000	−0.0471405
$451$	−5531.54	−0.577539
$452$	−8508.68	−0.885431
$453$	3407.78	0.353447
$454$	−1801.18	−0.186198
$455$	−3094.96	−0.318888
$456$	2397.18	0.246181
$457$	2292.25	0.234632	0.117316	−	0.993095i	$-0.462571\pi$
$457$	2292.25	0.234632	0.117316	+	0.993095i	$0.462571\pi$
$458$	−11678.6	−1.19149
$459$	2401.44	0.244204
$460$	2584.86	0.261999
$461$	−11081.3	−1.11954	−0.559768	−	0.828650i	$-0.689110\pi$
$461$	−11081.3	−1.11954	−0.559768	+	0.828650i	$0.689110\pi$
$462$	3436.18	0.346029
$463$	−2974.07	−0.298524	−0.149262	−	0.988798i	$-0.547690\pi$
$463$	−2974.07	−0.298524	−0.149262	+	0.988798i	$0.547690\pi$
$464$	1361.35	0.136205
$465$	−4067.14	−0.405611
$466$	−12143.3	−1.20714
$467$	8268.92	0.819357	0.409679	−	0.912230i	$-0.365641\pi$
$467$	8268.92	0.819357	0.409679	+	0.912230i	$0.365641\pi$
$468$	−1893.97	−0.187070
$469$	−3285.92	−0.323517
$470$	−1065.73	−0.104592
$471$	8075.94	0.790063
$472$	−2497.79	−0.243581
$473$	−4044.17	−0.393131
$474$	2080.13	0.201569
$475$	2497.06	0.241207
$476$	−4185.85	−0.403063
$477$	−2298.80	−0.220660
$478$	6879.88	0.658323
$479$	−2134.00	−0.203559	−0.101780	−	0.994807i	$-0.532454\pi$
$479$	−2134.00	−0.203559	−0.101780	+	0.994807i	$0.532454\pi$
$480$	480.000	0.0456435
$481$	−1946.58	−0.184525
$482$	−3256.37	−0.307725
$483$	−4561.87	−0.429757
$484$	4153.14	0.390040
$485$	933.699	0.0874167
$486$	486.000	0.0453609
$487$	−4042.13	−0.376111	−0.188056	−	0.982158i	$-0.560219\pi$
$487$	−4042.13	−0.376111	−0.188056	+	0.982158i	$0.560219\pi$
$488$	3920.42	0.363666
$489$	−6433.58	−0.594962
$490$	2045.70	0.188602
$491$	−14516.0	−1.33422	−0.667108	−	0.744961i	$-0.732468\pi$
$491$	−14516.0	−1.33422	−0.667108	+	0.744961i	$0.732468\pi$
$492$	1363.70	0.124960
$493$	−7567.60	−0.691334
$494$	10509.7	0.957193
$495$	2190.39	0.198890
$496$	4338.29	0.392732
$497$	−4871.27	−0.439650
$498$	855.230	0.0769554
$499$	−10123.0	−0.908156	−0.454078	−	0.890962i	$-0.650031\pi$
$499$	−10123.0	−0.908156	−0.454078	+	0.890962i	$0.650031\pi$
$500$	500.000	0.0447214
$501$	−12535.1	−1.11782
$502$	877.547	0.0780216
$503$	10934.1	0.969241	0.484620	−	0.874725i	$-0.338958\pi$
$503$	10934.1	0.969241	0.484620	+	0.874725i	$0.338958\pi$
$504$	−847.126	−0.0748690
$505$	5550.48	0.489096
$506$	−12581.9	−1.10540
$507$	−1712.50	−0.150009
$508$	4561.10	0.398359
$509$	5070.34	0.441530	0.220765	−	0.975327i	$-0.429145\pi$
$509$	5070.34	0.441530	0.220765	+	0.975327i	$0.429145\pi$
$510$	−2668.26	−0.231672
$511$	5230.24	0.452783
$512$	−512.000	−0.0441942
$513$	−2696.83	−0.232101
$514$	−11700.5	−1.00406
$515$	−4958.56	−0.424272
$516$	997.014	0.0850603
$517$	5187.46	0.441284
$518$	−870.657	−0.0738504
$519$	−5407.71	−0.457365
$520$	2104.41	0.177470
$521$	15205.7	1.27865	0.639323	−	0.768938i	$-0.279215\pi$
$521$	15205.7	1.27865	0.639323	+	0.768938i	$0.279215\pi$
$522$	−1531.52	−0.128415
$523$	3489.10	0.291717	0.145858	−	0.989305i	$-0.453406\pi$
$523$	3489.10	0.291717	0.145858	+	0.989305i	$0.453406\pi$
$524$	4772.10	0.397844
$525$	−882.423	−0.0733564
$526$	3409.27	0.282607
$527$	−24116.0	−1.99338
$528$	−2336.41	−0.192575
$529$	4536.70	0.372869
$530$	2554.23	0.209337
$531$	2810.01	0.229650
$532$	4700.73	0.383088
$533$	5978.71	0.485866
$534$	3602.96	0.291976
$535$	10797.9	0.872586
$536$	2234.25	0.180046
$537$	−9128.76	−0.733585
$538$	−2446.52	−0.196054
$539$	−9957.49	−0.795732
$540$	−540.000	−0.0430331
$541$	−6810.00	−0.541192	−0.270596	−	0.962693i	$-0.587221\pi$
$541$	−6810.00	−0.541192	−0.270596	+	0.962693i	$0.587221\pi$
$542$	−11233.4	−0.890254
$543$	1733.58	0.137007
$544$	2846.15	0.224315
$545$	−7864.08	−0.618092
$546$	−3713.96	−0.291104
$547$	−9521.48	−0.744258	−0.372129	−	0.928181i	$-0.621372\pi$
$547$	−9521.48	−0.744258	−0.372129	+	0.928181i	$0.621372\pi$
$548$	2796.55	0.217998
$549$	−4410.47	−0.342868
$550$	−2433.77	−0.188684
$551$	8498.46	0.657072
$552$	3101.83	0.239171
$553$	4079.01	0.313666
$554$	−1328.38	−0.101873
$555$	−555.000	−0.0424476
$556$	−8548.49	−0.652044
$557$	−8512.39	−0.647543	−0.323772	−	0.946135i	$-0.604951\pi$
$557$	−8512.39	−0.647543	−0.323772	+	0.946135i	$0.604951\pi$
$558$	−4880.57	−0.370271
$559$	4371.09	0.330729
$560$	941.251	0.0710270
$561$	12987.9	0.977447
$562$	15609.2	1.17159
$563$	−12049.4	−0.901991	−0.450996	−	0.892526i	$-0.648931\pi$
$563$	−12049.4	−0.901991	−0.450996	+	0.892526i	$0.648931\pi$
$564$	−1278.87	−0.0954791
$565$	−10635.9	−0.791954
$566$	7476.71	0.555246
$567$	953.017	0.0705872
$568$	3312.20	0.244677
$569$	5520.25	0.406715	0.203358	−	0.979105i	$-0.434815\pi$
$569$	5520.25	0.406715	0.203358	+	0.979105i	$0.434815\pi$
$570$	2996.48	0.220191
$571$	−7850.99	−0.575400	−0.287700	−	0.957721i	$-0.592891\pi$
$571$	−7850.99	−0.575400	−0.287700	+	0.957721i	$0.592891\pi$
$572$	−10243.3	−0.748763
$573$	7510.69	0.547581
$574$	2674.13	0.194453
$575$	3231.07	0.234339
$576$	576.000	0.0416667
$577$	−16598.2	−1.19756	−0.598780	−	0.800913i	$-0.704348\pi$
$577$	−16598.2	−1.19756	−0.598780	+	0.800913i	$0.704348\pi$
$578$	−5995.41	−0.431447
$579$	−12753.3	−0.915389
$580$	1701.69	0.121826
$581$	1677.06	0.119752
$582$	1120.44	0.0798002
$583$	−12432.8	−0.883212
$584$	−3556.28	−0.251986
$585$	−2367.46	−0.167320
$586$	13493.9	0.951244
$587$	8366.39	0.588276	0.294138	−	0.955763i	$-0.404968\pi$
$587$	8366.39	0.588276	0.294138	+	0.955763i	$0.404968\pi$
$588$	2454.84	0.172170
$589$	27082.5	1.89459
$590$	−3122.24	−0.217865
$591$	−2916.29	−0.202978
$592$	592.000	0.0410997
$593$	−10863.4	−0.752287	−0.376144	−	0.926561i	$-0.622750\pi$
$593$	−10863.4	−0.752287	−0.376144	+	0.926561i	$0.622750\pi$
$594$	2628.47	0.181561
$595$	−5232.31	−0.360510
$596$	12343.4	0.848331
$597$	−15212.7	−1.04290
$598$	13599.0	0.929939
$599$	−1996.44	−0.136181	−0.0680904	−	0.997679i	$-0.521691\pi$
$599$	−1996.44	−0.136181	−0.0680904	+	0.997679i	$0.521691\pi$
$600$	600.000	0.0408248
$601$	452.705	0.0307258	0.0153629	−	0.999882i	$-0.495110\pi$
$601$	452.705	0.0307258	0.0153629	+	0.999882i	$0.495110\pi$
$602$	1955.09	0.132364
$603$	−2513.53	−0.169749
$604$	−4543.70	−0.306094
$605$	5191.43	0.348862
$606$	6660.58	0.446481
$607$	29186.7	1.95165	0.975826	−	0.218549i	$-0.0701325\pi$
$607$	29186.7	1.95165	0.975826	+	0.218549i	$0.0701325\pi$
$608$	−3196.24	−0.213199
$609$	−3003.22	−0.199830
$610$	4900.53	0.325273
$611$	−5606.81	−0.371239
$612$	−3201.92	−0.211487
$613$	13957.3	0.919622	0.459811	−	0.888017i	$-0.347917\pi$
$613$	13957.3	0.919622	0.459811	+	0.888017i	$0.347917\pi$
$614$	3894.63	0.255985
$615$	1704.62	0.111768
$616$	−4581.57	−0.299670
$617$	12102.0	0.789639	0.394819	−	0.918759i	$-0.370807\pi$
$617$	12102.0	0.789639	0.394819	+	0.918759i	$0.370807\pi$
$618$	−5950.27	−0.387306
$619$	1341.17	0.0870859	0.0435429	−	0.999052i	$-0.486135\pi$
$619$	1341.17	0.0870859	0.0435429	+	0.999052i	$0.486135\pi$
$620$	5422.86	0.351270
$621$	−3489.56	−0.225493
$622$	−21441.5	−1.38220
$623$	7065.19	0.454352
$624$	2525.29	0.162007
$625$	625.000	0.0400000
$626$	−6500.47	−0.415033
$627$	−14585.4	−0.929006
$628$	−10767.9	−0.684215
$629$	−3290.86	−0.208609
$630$	−1058.91	−0.0669649
$631$	8725.65	0.550495	0.275248	−	0.961373i	$-0.411240\pi$
$631$	8725.65	0.550495	0.275248	+	0.961373i	$0.411240\pi$
$632$	−2773.51	−0.174564
$633$	56.5680	0.00355194
$634$	10186.4	0.638096
$635$	5701.38	0.356303
$636$	3065.07	0.191097
$637$	10762.5	0.669425
$638$	−8283.03	−0.513994
$639$	−3726.22	−0.230684
$640$	−640.000	−0.0395285
$641$	−31901.1	−1.96570	−0.982852	−	0.184395i	$-0.940967\pi$
$641$	−31901.1	−1.96570	−0.982852	+	0.184395i	$0.940967\pi$
$642$	12957.5	0.796558
$643$	−6606.98	−0.405216	−0.202608	−	0.979260i	$-0.564942\pi$
$643$	−6606.98	−0.405216	−0.202608	+	0.979260i	$0.564942\pi$
$644$	6082.50	0.372180
$645$	1246.27	0.0760802
$646$	17767.5	1.08213
$647$	9754.53	0.592720	0.296360	−	0.955076i	$-0.404227\pi$
$647$	9754.53	0.592720	0.296360	+	0.955076i	$0.404227\pi$
$648$	−648.000	−0.0392837
$649$	15197.6	0.919195
$650$	2630.51	0.158734
$651$	−9570.51	−0.576187
$652$	8578.10	0.515252
$653$	−27632.8	−1.65598	−0.827989	−	0.560744i	$-0.810515\pi$
$653$	−27632.8	−1.65598	−0.827989	+	0.560744i	$0.810515\pi$
$654$	−9436.90	−0.564238
$655$	5965.13	0.355842
$656$	−1818.27	−0.108218
$657$	4000.82	0.237575
$658$	−2507.79	−0.148577
$659$	25232.0	1.49150	0.745750	−	0.666226i	$-0.232091\pi$
$659$	25232.0	1.49150	0.745750	+	0.666226i	$0.232091\pi$
$660$	−2920.52	−0.172244
$661$	−9148.81	−0.538347	−0.269174	−	0.963092i	$-0.586751\pi$
$661$	−9148.81	−0.538347	−0.269174	+	0.963092i	$0.586751\pi$
$662$	647.823	0.0380338
$663$	−14037.8	−0.822297
$664$	−1140.31	−0.0666453
$665$	5875.91	0.342644
$666$	−666.000	−0.0387492
$667$	10996.6	0.638364
$668$	16713.5	0.968062
$669$	66.8710	0.00386455
$670$	2792.81	0.161038
$671$	−23853.5	−1.37236
$672$	1129.50	0.0648385
$673$	−592.398	−0.0339305	−0.0169653	−	0.999856i	$-0.505400\pi$
$673$	−592.398	−0.0339305	−0.0169653	+	0.999856i	$0.505400\pi$
$674$	18890.1	1.07955
$675$	−675.000	−0.0384900
$676$	2283.33	0.129912
$677$	2993.28	0.169928	0.0849639	−	0.996384i	$-0.472922\pi$
$677$	2993.28	0.169928	0.0849639	+	0.996384i	$0.472922\pi$
$678$	−12763.0	−0.722951
$679$	2197.11	0.124179
$680$	3557.69	0.200634
$681$	−2701.78	−0.152030
$682$	−26395.9	−1.48204
$683$	23759.5	1.33109	0.665543	−	0.746359i	$-0.268200\pi$
$683$	23759.5	1.33109	0.665543	+	0.746359i	$0.268200\pi$
$684$	3595.77	0.201006
$685$	3495.69	0.194983
$686$	12885.0	0.717132
$687$	−17517.9	−0.972851
$688$	−1329.35	−0.0736644
$689$	13437.8	0.743020
$690$	3877.28	0.213921
$691$	14635.6	0.805740	0.402870	−	0.915257i	$-0.368013\pi$
$691$	14635.6	0.805740	0.402870	+	0.915257i	$0.368013\pi$
$692$	7210.28	0.396089
$693$	5154.27	0.282532
$694$	−1642.81	−0.0898563
$695$	−10685.6	−0.583206
$696$	2042.03	0.111211
$697$	10107.5	0.549282
$698$	−11774.6	−0.638501
$699$	−18215.0	−0.985629
$700$	1176.56	0.0635285
$701$	−25800.5	−1.39011	−0.695057	−	0.718954i	$-0.744621\pi$
$701$	−25800.5	−1.39011	−0.695057	+	0.718954i	$0.744621\pi$
$702$	−2840.95	−0.152742
$703$	3695.66	0.198271
$704$	3115.22	0.166775
$705$	−1598.59	−0.0853991
$706$	−15918.1	−0.848562
$707$	13061.0	0.694780
$708$	−3746.68	−0.198883
$709$	−5311.39	−0.281345	−0.140672	−	0.990056i	$-0.544926\pi$
$709$	−5311.39	−0.281345	−0.140672	+	0.990056i	$0.544926\pi$
$710$	4140.25	0.218846
$711$	3120.20	0.164580
$712$	−4803.95	−0.252859
$713$	35043.3	1.84065
$714$	−6278.77	−0.329099
$715$	−12804.1	−0.669714
$716$	12171.7	0.635303
$717$	10319.8	0.537518
$718$	−10774.5	−0.560027
$719$	10746.8	0.557426	0.278713	−	0.960374i	$-0.410092\pi$
$719$	10746.8	0.557426	0.278713	+	0.960374i	$0.410092\pi$
$720$	720.000	0.0372678
$721$	−11668.1	−0.602696
$722$	−6235.07	−0.321392
$723$	−4884.56	−0.251257
$724$	−2311.44	−0.118652
$725$	2127.11	0.108964
$726$	6229.71	0.318466
$727$	−29202.3	−1.48976	−0.744879	−	0.667200i	$-0.767493\pi$
$727$	−29202.3	−1.48976	−0.744879	+	0.667200i	$0.767493\pi$
$728$	4951.94	0.252103
$729$	729.000	0.0370370
$730$	−4445.35	−0.225383
$731$	7389.72	0.373897
$732$	5880.63	0.296932
$733$	36069.0	1.81751	0.908757	−	0.417326i	$-0.137033\pi$
$733$	36069.0	1.81751	0.908757	+	0.417326i	$0.137033\pi$
$734$	22990.0	1.15610
$735$	3068.55	0.153993
$736$	−4135.77	−0.207128
$737$	−13594.1	−0.679436
$738$	2045.55	0.102029
$739$	−11683.3	−0.581566	−0.290783	−	0.956789i	$-0.593916\pi$
$739$	−11683.3	−0.581566	−0.290783	+	0.956789i	$0.593916\pi$
$740$	740.000	0.0367607
$741$	15764.5	0.781544
$742$	6010.42	0.297371
$743$	−17716.3	−0.874761	−0.437380	−	0.899277i	$-0.644094\pi$
$743$	−17716.3	−0.874761	−0.437380	+	0.899277i	$0.644094\pi$
$744$	6507.43	0.320664
$745$	15429.2	0.758770
$746$	19274.9	0.945982
$747$	1282.85	0.0628338
$748$	−17317.1	−0.846494
$749$	25408.8	1.23954
$750$	750.000	0.0365148
$751$	−21284.2	−1.03418	−0.517091	−	0.855930i	$-0.672985\pi$
$751$	−21284.2	−1.03418	−0.517091	+	0.855930i	$0.672985\pi$
$752$	1705.16	0.0826873
$753$	1316.32	0.0637044
$754$	8952.63	0.432408
$755$	−5679.63	−0.273779
$756$	−1270.69	−0.0611303
$757$	19930.9	0.956938	0.478469	−	0.878104i	$-0.341192\pi$
$757$	19930.9	0.956938	0.478469	+	0.878104i	$0.341192\pi$
$758$	−14858.4	−0.711978
$759$	−18872.8	−0.902555
$760$	−3995.30	−0.190691
$761$	−19800.6	−0.943194	−0.471597	−	0.881814i	$-0.656322\pi$
$761$	−19800.6	−0.943194	−0.471597	+	0.881814i	$0.656322\pi$
$762$	6841.66	0.325259
$763$	−18505.2	−0.878025
$764$	−10014.3	−0.474219
$765$	−4002.40	−0.189159
$766$	−12892.8	−0.608141
$767$	−16426.1	−0.773291
$768$	−768.000	−0.0360844
$769$	726.705	0.0340776	0.0170388	−	0.999855i	$-0.494576\pi$
$769$	726.705	0.0340776	0.0170388	+	0.999855i	$0.494576\pi$
$770$	−5726.96	−0.268033
$771$	−17550.8	−0.819812
$772$	17004.4	0.792750
$773$	−39162.0	−1.82220	−0.911100	−	0.412185i	$-0.864766\pi$
$773$	−39162.0	−1.82220	−0.911100	+	0.412185i	$0.864766\pi$
$774$	1495.52	0.0694514
$775$	6778.57	0.314185
$776$	−1493.92	−0.0691090
$777$	−1305.99	−0.0602986
$778$	−26653.3	−1.22824
$779$	−11350.8	−0.522061
$780$	3156.61	0.144904
$781$	−20152.8	−0.923333
$782$	22990.3	1.05132
$783$	−2297.28	−0.104851
$784$	−3273.11	−0.149103
$785$	−13459.9	−0.611980
$786$	7158.15	0.324838
$787$	28533.1	1.29237	0.646185	−	0.763181i	$-0.276363\pi$
$787$	28533.1	1.29237	0.646185	+	0.763181i	$0.276363\pi$
$788$	3888.39	0.175784
$789$	5113.91	0.230748
$790$	−3466.89	−0.156135
$791$	−25027.5	−1.12500
$792$	−3504.62	−0.157237
$793$	25781.8	1.15452
$794$	−13514.0	−0.604025
$795$	3831.34	0.170923
$796$	20283.6	0.903181
$797$	26410.8	1.17380	0.586899	−	0.809660i	$-0.300348\pi$
$797$	26410.8	1.17380	0.586899	+	0.809660i	$0.300348\pi$
$798$	7051.10	0.312790
$799$	−9478.80	−0.419694
$800$	−800.000	−0.0353553
$801$	5404.44	0.238398
$802$	−28559.0	−1.25742
$803$	21637.9	0.950915
$804$	3351.37	0.147007
$805$	7603.12	0.332888
$806$	28529.8	1.24680
$807$	−3669.77	−0.160077
$808$	−8880.78	−0.386664
$809$	7634.05	0.331766	0.165883	−	0.986145i	$-0.446953\pi$
$809$	7634.05	0.331766	0.165883	+	0.986145i	$0.446953\pi$
$810$	−810.000	−0.0351364
$811$	−37255.2	−1.61308	−0.806539	−	0.591181i	$-0.798662\pi$
$811$	−37255.2	−1.61308	−0.806539	+	0.591181i	$0.798662\pi$
$812$	4004.30	0.173058
$813$	−16850.2	−0.726889
$814$	−3601.97	−0.155097
$815$	10722.6	0.460856
$816$	4269.22	0.183153
$817$	−8298.70	−0.355367
$818$	10626.7	0.454223
$819$	−5570.93	−0.237685
$820$	−2272.83	−0.0967936
$821$	−18854.7	−0.801501	−0.400750	−	0.916187i	$-0.631251\pi$
$821$	−18854.7	−0.801501	−0.400750	+	0.916187i	$0.631251\pi$
$822$	4194.83	0.177994
$823$	24977.3	1.05790	0.528951	−	0.848652i	$-0.322585\pi$
$823$	24977.3	1.05790	0.528951	+	0.848652i	$0.322585\pi$
$824$	7933.69	0.335416
$825$	−3650.65	−0.154060
$826$	−7347.02	−0.309486
$827$	−25271.3	−1.06260	−0.531300	−	0.847184i	$-0.678296\pi$
$827$	−25271.3	−1.06260	−0.531300	+	0.847184i	$0.678296\pi$
$828$	4652.74	0.195282
$829$	5155.44	0.215990	0.107995	−	0.994151i	$-0.465557\pi$
$829$	5155.44	0.215990	0.107995	+	0.994151i	$0.465557\pi$
$830$	−1425.38	−0.0596094
$831$	−1992.57	−0.0831788
$832$	−3367.05	−0.140302
$833$	18194.9	0.756800
$834$	−12822.7	−0.532392
$835$	20891.9	0.865861
$836$	19447.3	0.804543
$837$	−7320.86	−0.302325
$838$	−22104.6	−0.911206
$839$	16618.1	0.683814	0.341907	−	0.939734i	$-0.388927\pi$
$839$	16618.1	0.683814	0.341907	+	0.939734i	$0.388927\pi$
$840$	1411.88	0.0579933
$841$	−17149.6	−0.703170
$842$	10782.3	0.441311
$843$	23413.8	0.956600
$844$	−75.4240	−0.00307607
$845$	2854.16	0.116197
$846$	−1918.31	−0.0779584
$847$	12216.1	0.495573
$848$	−4086.76	−0.165495
$849$	11215.1	0.453357
$850$	4447.11	0.179452
$851$	4781.98	0.192625
$852$	4968.30	0.199778
$853$	6973.47	0.279914	0.139957	−	0.990158i	$-0.455304\pi$
$853$	6973.47	0.279914	0.139957	+	0.990158i	$0.455304\pi$
$854$	11531.6	0.462063
$855$	4494.72	0.179785
$856$	−17276.6	−0.689839
$857$	585.994	0.0233573	0.0116786	−	0.999932i	$-0.496282\pi$
$857$	585.994	0.0233573	0.0116786	+	0.999932i	$0.496282\pi$
$858$	−15364.9	−0.611363
$859$	21793.7	0.865647	0.432823	−	0.901479i	$-0.357517\pi$
$859$	21793.7	0.865647	0.432823	+	0.901479i	$0.357517\pi$
$860$	−1661.69	−0.0658874
$861$	4011.20	0.158770
$862$	13345.9	0.527334
$863$	−44956.1	−1.77326	−0.886630	−	0.462480i	$-0.846960\pi$
$863$	−44956.1	−1.77326	−0.886630	+	0.462480i	$0.846960\pi$
$864$	864.000	0.0340207
$865$	9012.86	0.354273
$866$	16898.0	0.663067
$867$	−8993.11	−0.352275
$868$	12760.7	0.498993
$869$	16875.2	0.658747
$870$	2552.54	0.0994702
$871$	14693.0	0.571589
$872$	12582.5	0.488645
$873$	1680.66	0.0651566
$874$	−25818.2	−0.999215
$875$	1470.71	0.0568216
$876$	−5334.42	−0.205746
$877$	1931.67	0.0743762	0.0371881	−	0.999308i	$-0.488160\pi$
$877$	1931.67	0.0743762	0.0371881	+	0.999308i	$0.488160\pi$
$878$	−16917.6	−0.650276
$879$	20240.9	0.776688
$880$	3894.02	0.149168
$881$	−44869.5	−1.71588	−0.857940	−	0.513750i	$-0.828256\pi$
$881$	−44869.5	−1.71588	−0.857940	+	0.513750i	$0.828256\pi$
$882$	3682.25	0.140576
$883$	34968.6	1.33271	0.666357	−	0.745633i	$-0.267853\pi$
$883$	34968.6	1.33271	0.666357	+	0.745633i	$0.267853\pi$
$884$	18717.1	0.712130
$885$	−4683.35	−0.177886
$886$	−1931.07	−0.0732229
$887$	−15663.5	−0.592930	−0.296465	−	0.955044i	$-0.595808\pi$
$887$	−15663.5	−0.592930	−0.296465	+	0.955044i	$0.595808\pi$
$888$	888.000	0.0335578
$889$	13416.1	0.506143
$890$	−6004.94	−0.226164
$891$	3942.70	0.148244
$892$	−89.1614	−0.00334680
$893$	10644.8	0.398895
$894$	18515.1	0.692659
$895$	15214.6	0.568232
$896$	−1506.00	−0.0561518
$897$	20398.5	0.759292
$898$	16800.3	0.624312
$899$	23070.1	0.855873
$900$	900.000	0.0333333
$901$	22717.8	0.840001
$902$	11063.1	0.408382
$903$	2932.63	0.108075
$904$	17017.4	0.626094
$905$	−2889.29	−0.106125
$906$	−6815.56	−0.249925
$907$	32941.5	1.20596	0.602980	−	0.797757i	$-0.293980\pi$
$907$	32941.5	1.20596	0.602980	+	0.797757i	$0.293980\pi$
$908$	3602.37	0.131662
$909$	9990.87	0.364550
$910$	6189.93	0.225488
$911$	28373.4	1.03189	0.515946	−	0.856621i	$-0.327441\pi$
$911$	28373.4	1.03189	0.515946	+	0.856621i	$0.327441\pi$
$912$	−4794.36	−0.174076
$913$	6938.10	0.251498
$914$	−4584.50	−0.165910
$915$	7350.79	0.265584
$916$	23357.2	0.842514
$917$	14036.7	0.505488
$918$	−4802.88	−0.172678
$919$	2825.60	0.101423	0.0507116	−	0.998713i	$-0.483851\pi$
$919$	2825.60	0.101423	0.0507116	+	0.998713i	$0.483851\pi$
$920$	−5169.71	−0.185261
$921$	5841.95	0.209011
$922$	22162.5	0.791631
$923$	21781.9	0.776772
$924$	−6872.35	−0.244680
$925$	925.000	0.0328798
$926$	5948.14	0.211089
$927$	−8925.40	−0.316234
$928$	−2722.70	−0.0963116
$929$	45741.7	1.61543	0.807717	−	0.589571i	$-0.200703\pi$
$929$	45741.7	1.61543	0.807717	+	0.589571i	$0.200703\pi$
$930$	8134.29	0.286811
$931$	−20433.0	−0.719294
$932$	24286.7	0.853580
$933$	−32162.3	−1.12856
$934$	−16537.8	−0.579373
$935$	−21646.4	−0.757127
$936$	3787.93	0.132278
$937$	6282.97	0.219056	0.109528	−	0.993984i	$-0.465066\pi$
$937$	6282.97	0.219056	0.109528	+	0.993984i	$0.465066\pi$
$938$	6571.84	0.228761
$939$	−9750.70	−0.338873
$940$	2131.45	0.0739578
$941$	−11654.1	−0.403734	−0.201867	−	0.979413i	$-0.564701\pi$
$941$	−11654.1	−0.403734	−0.201867	+	0.979413i	$0.564701\pi$
$942$	−16151.9	−0.558659
$943$	−14687.4	−0.507196
$944$	4995.58	0.172238
$945$	−1588.36	−0.0546766
$946$	8088.33	0.277986
$947$	45596.3	1.56460	0.782302	−	0.622899i	$-0.214045\pi$
$947$	45596.3	1.56460	0.782302	+	0.622899i	$0.214045\pi$
$948$	−4160.26	−0.142531
$949$	−23387.1	−0.799976
$950$	−4994.13	−0.170559
$951$	15279.6	0.521003
$952$	8371.69	0.285008
$953$	5195.55	0.176601	0.0883003	−	0.996094i	$-0.471856\pi$
$953$	5195.55	0.176601	0.0883003	+	0.996094i	$0.471856\pi$
$954$	4597.61	0.156030
$955$	−12517.8	−0.424154
$956$	−13759.8	−0.465505
$957$	−12424.5	−0.419675
$958$	4268.00	0.143938
$959$	8225.80	0.276981
$960$	−960.000	−0.0322749
$961$	43727.5	1.46781
$962$	3893.15	0.130479
$963$	19436.2	0.650387
$964$	6512.74	0.217595
$965$	21255.6	0.709058
$966$	9123.75	0.303884
$967$	−8550.65	−0.284354	−0.142177	−	0.989841i	$-0.545410\pi$
$967$	−8550.65	−0.284354	−0.142177	+	0.989841i	$0.545410\pi$
$968$	−8306.29	−0.275800
$969$	26651.3	0.883554
$970$	−1867.40	−0.0618129
$971$	−26602.4	−0.879210	−0.439605	−	0.898191i	$-0.644882\pi$
$971$	−26602.4	−0.879210	−0.439605	+	0.898191i	$0.644882\pi$
$972$	−972.000	−0.0320750
$973$	−25144.6	−0.828468
$974$	8084.25	0.265951
$975$	3945.76	0.129606
$976$	−7840.84	−0.257151
$977$	−34346.8	−1.12472	−0.562360	−	0.826892i	$-0.690107\pi$
$977$	−34346.8	−1.12472	−0.562360	+	0.826892i	$0.690107\pi$
$978$	12867.2	0.420702
$979$	29229.2	0.954208
$980$	−4091.39	−0.133362
$981$	−14155.3	−0.460699
$982$	29032.1	0.943433
$983$	−17387.9	−0.564179	−0.282090	−	0.959388i	$-0.591028\pi$
$983$	−17387.9	−0.564179	−0.282090	+	0.959388i	$0.591028\pi$
$984$	−2727.40	−0.0883600
$985$	4860.49	0.157226
$986$	15135.2	0.488847
$987$	−3761.69	−0.121313
$988$	−21019.4	−0.676837
$989$	−10738.1	−0.345249
$990$	−4380.78	−0.140637
$991$	−50977.0	−1.63404	−0.817022	−	0.576607i	$-0.804376\pi$
$991$	−50977.0	−1.63404	−0.817022	+	0.576607i	$0.804376\pi$
$992$	−8676.57	−0.277703
$993$	971.735	0.0310545
$994$	9742.53	0.310880
$995$	25354.5	0.807830
$996$	−1710.46	−0.0544157
$997$	−27605.8	−0.876915	−0.438458	−	0.898752i	$-0.644475\pi$
$997$	−27605.8	−0.876915	−0.438458	+	0.898752i	$0.644475\pi$
$998$	20246.1	0.642163
$999$	−999.000	−0.0316386

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1110.4.a.m.1.4	✓	5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1110.4.a.m.1.4	✓	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:	\( N \)	\(=\)	\( 1110 = 2 \cdot 3 \cdot 5 \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1110.a (trivial)

\(f(q)\)	\(=\)	\(q-2.00000 q^{2} -3.00000 q^{3} +4.00000 q^{4} +5.00000 q^{5} +6.00000 q^{6} +11.7656 q^{7} -8.00000 q^{8} +9.00000 q^{9} +O(q^{10})\)	\(q-2.00000 q^{2} -3.00000 q^{3} +4.00000 q^{4} +5.00000 q^{5} +6.00000 q^{6} +11.7656 q^{7} -8.00000 q^{8} +9.00000 q^{9} -10.0000 q^{10} +48.6753 q^{11} -12.0000 q^{12} -52.6102 q^{13} -23.5313 q^{14} -15.0000 q^{15} +16.0000 q^{16} -88.9421 q^{17} -18.0000 q^{18} +99.8826 q^{19} +20.0000 q^{20} -35.2969 q^{21} -97.3506 q^{22} +129.243 q^{23} +24.0000 q^{24} +25.0000 q^{25} +105.220 q^{26} -27.0000 q^{27} +47.0626 q^{28} +85.0845 q^{29} +30.0000 q^{30} +271.143 q^{31} -32.0000 q^{32} -146.026 q^{33} +177.884 q^{34} +58.8282 q^{35} +36.0000 q^{36} +37.0000 q^{37} -199.765 q^{38} +157.831 q^{39} -40.0000 q^{40} -113.642 q^{41} +70.5939 q^{42} -83.0845 q^{43} +194.701 q^{44} +45.0000 q^{45} -258.486 q^{46} +106.573 q^{47} -48.0000 q^{48} -204.570 q^{49} -50.0000 q^{50} +266.826 q^{51} -210.441 q^{52} -255.423 q^{53} +54.0000 q^{54} +243.377 q^{55} -94.1251 q^{56} -299.648 q^{57} -170.169 q^{58} +312.224 q^{59} -60.0000 q^{60} -490.053 q^{61} -542.286 q^{62} +105.891 q^{63} +64.0000 q^{64} -263.051 q^{65} +292.052 q^{66} -279.281 q^{67} -355.769 q^{68} -387.728 q^{69} -117.656 q^{70} -414.025 q^{71} -72.0000 q^{72} +444.535 q^{73} -74.0000 q^{74} -75.0000 q^{75} +399.530 q^{76} +572.696 q^{77} -315.661 q^{78} +346.689 q^{79} +80.0000 q^{80} +81.0000 q^{81} +227.283 q^{82} +142.538 q^{83} -141.188 q^{84} -444.711 q^{85} +166.169 q^{86} -255.254 q^{87} -389.402 q^{88} +600.494 q^{89} -90.0000 q^{90} -618.993 q^{91} +516.971 q^{92} -813.429 q^{93} -213.145 q^{94} +499.413 q^{95} +96.0000 q^{96} +186.740 q^{97} +409.139 q^{98} +438.078 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - 10 q^{2} - 15 q^{3} + 20 q^{4} + 25 q^{5} + 30 q^{6} - 19 q^{7} - 40 q^{8} + 45 q^{9}+O(q^{10}) \) 5 * q - 10 * q^2 - 15 * q^3 + 20 * q^4 + 25 * q^5 + 30 * q^6 - 19 * q^7 - 40 * q^8 + 45 * q^9	\( 5 q - 10 q^{2} - 15 q^{3} + 20 q^{4} + 25 q^{5} + 30 q^{6} - 19 q^{7} - 40 q^{8} + 45 q^{9} - 50 q^{10} + 36 q^{11} - 60 q^{12} - 35 q^{13} + 38 q^{14} - 75 q^{15} + 80 q^{16} - 12 q^{17} - 90 q^{18} + 78 q^{19} + 100 q^{20} + 57 q^{21} - 72 q^{22} - 171 q^{23} + 120 q^{24} + 125 q^{25} + 70 q^{26} - 135 q^{27} - 76 q^{28} + 259 q^{29} + 150 q^{30} + 264 q^{31} - 160 q^{32} - 108 q^{33} + 24 q^{34} - 95 q^{35} + 180 q^{36} + 185 q^{37} - 156 q^{38} + 105 q^{39} - 200 q^{40} + 486 q^{41} - 114 q^{42} - 249 q^{43} + 144 q^{44} + 225 q^{45} + 342 q^{46} - 56 q^{47} - 240 q^{48} + 272 q^{49} - 250 q^{50} + 36 q^{51} - 140 q^{52} - 69 q^{53} + 270 q^{54} + 180 q^{55} + 152 q^{56} - 234 q^{57} - 518 q^{58} + 189 q^{59} - 300 q^{60} + 42 q^{61} - 528 q^{62} - 171 q^{63} + 320 q^{64} - 175 q^{65} + 216 q^{66} - 1215 q^{67} - 48 q^{68} + 513 q^{69} + 190 q^{70} - 17 q^{71} - 360 q^{72} - 668 q^{73} - 370 q^{74} - 375 q^{75} + 312 q^{76} - 390 q^{77} - 210 q^{78} - 575 q^{79} + 400 q^{80} + 405 q^{81} - 972 q^{82} - 328 q^{83} + 228 q^{84} - 60 q^{85} + 498 q^{86} - 777 q^{87} - 288 q^{88} + 909 q^{89} - 450 q^{90} - 533 q^{91} - 684 q^{92} - 792 q^{93} + 112 q^{94} + 390 q^{95} + 480 q^{96} - 814 q^{97} - 544 q^{98} + 324 q^{99}+O(q^{100}) \) 5 * q - 10 * q^2 - 15 * q^3 + 20 * q^4 + 25 * q^5 + 30 * q^6 - 19 * q^7 - 40 * q^8 + 45 * q^9 - 50 * q^10 + 36 * q^11 - 60 * q^12 - 35 * q^13 + 38 * q^14 - 75 * q^15 + 80 * q^16 - 12 * q^17 - 90 * q^18 + 78 * q^19 + 100 * q^20 + 57 * q^21 - 72 * q^22 - 171 * q^23 + 120 * q^24 + 125 * q^25 + 70 * q^26 - 135 * q^27 - 76 * q^28 + 259 * q^29 + 150 * q^30 + 264 * q^31 - 160 * q^32 - 108 * q^33 + 24 * q^34 - 95 * q^35 + 180 * q^36 + 185 * q^37 - 156 * q^38 + 105 * q^39 - 200 * q^40 + 486 * q^41 - 114 * q^42 - 249 * q^43 + 144 * q^44 + 225 * q^45 + 342 * q^46 - 56 * q^47 - 240 * q^48 + 272 * q^49 - 250 * q^50 + 36 * q^51 - 140 * q^52 - 69 * q^53 + 270 * q^54 + 180 * q^55 + 152 * q^56 - 234 * q^57 - 518 * q^58 + 189 * q^59 - 300 * q^60 + 42 * q^61 - 528 * q^62 - 171 * q^63 + 320 * q^64 - 175 * q^65 + 216 * q^66 - 1215 * q^67 - 48 * q^68 + 513 * q^69 + 190 * q^70 - 17 * q^71 - 360 * q^72 - 668 * q^73 - 370 * q^74 - 375 * q^75 + 312 * q^76 - 390 * q^77 - 210 * q^78 - 575 * q^79 + 400 * q^80 + 405 * q^81 - 972 * q^82 - 328 * q^83 + 228 * q^84 - 60 * q^85 + 498 * q^86 - 777 * q^87 - 288 * q^88 + 909 * q^89 - 450 * q^90 - 533 * q^91 - 684 * q^92 - 792 * q^93 + 112 * q^94 + 390 * q^95 + 480 * q^96 - 814 * q^97 - 544 * q^98 + 324 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more