LMFDB - Embedded newform 930.4.a.j.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [930,4,Mod(1,930)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(930, 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("930.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 930 = 2 \cdot 3 \cdot 5 \cdot 31 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	930.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:	$54.8717763053$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\mathbb{Q}[x]/(x^{4} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{3} - 398x^{2} - 420x + 24375 $ x^4 - 2x^3 - 398x^2 - 420*x + 24375
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$7.72870$ of defining polynomial
Character	$\chi$	$=$	930.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.00000 q^{2} -3.00000 q^{3} +4.00000 q^{4} -5.00000 q^{5} +6.00000 q^{6} -16.5903 q^{7} -8.00000 q^{8} +9.00000 q^{9} +O(q^{10})$	\(q-2.00000 q^{2} -3.00000 q^{3} +4.00000 q^{4} -5.00000 q^{5} +6.00000 q^{6} -16.5903 q^{7} -8.00000 q^{8} +9.00000 q^{9} +10.0000 q^{10} +44.4131 q^{11} -12.0000 q^{12} +90.3678 q^{13} +33.1805 q^{14} +15.0000 q^{15} +16.0000 q^{16} -21.7231 q^{17} -18.0000 q^{18} -84.1320 q^{19} -20.0000 q^{20} +49.7708 q^{21} -88.8261 q^{22} +64.4088 q^{23} +24.0000 q^{24} +25.0000 q^{25} -180.736 q^{26} -27.0000 q^{27} -66.3610 q^{28} -241.728 q^{29} -30.0000 q^{30} -31.0000 q^{31} -32.0000 q^{32} -133.239 q^{33} +43.4463 q^{34} +82.9513 q^{35} +36.0000 q^{36} -438.645 q^{37} +168.264 q^{38} -271.103 q^{39} +40.0000 q^{40} +421.928 q^{41} -99.5416 q^{42} +52.0635 q^{43} +177.652 q^{44} -45.0000 q^{45} -128.818 q^{46} -14.8186 q^{47} -48.0000 q^{48} -67.7632 q^{49} -50.0000 q^{50} +65.1694 q^{51} +361.471 q^{52} -359.850 q^{53} +54.0000 q^{54} -222.065 q^{55} +132.722 q^{56} +252.396 q^{57} +483.456 q^{58} +58.9738 q^{59} +60.0000 q^{60} +922.905 q^{61} +62.0000 q^{62} -149.312 q^{63} +64.0000 q^{64} -451.839 q^{65} +266.478 q^{66} -276.466 q^{67} -86.8925 q^{68} -193.226 q^{69} -165.903 q^{70} +486.851 q^{71} -72.0000 q^{72} -96.4576 q^{73} +877.291 q^{74} -75.0000 q^{75} -336.528 q^{76} -736.824 q^{77} +542.207 q^{78} +1111.72 q^{79} -80.0000 q^{80} +81.0000 q^{81} -843.856 q^{82} +509.095 q^{83} +199.083 q^{84} +108.616 q^{85} -104.127 q^{86} +725.183 q^{87} -355.304 q^{88} +331.621 q^{89} +90.0000 q^{90} -1499.23 q^{91} +257.635 q^{92} +93.0000 q^{93} +29.6372 q^{94} +420.660 q^{95} +96.0000 q^{96} -1662.66 q^{97} +135.526 q^{98} +399.718 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 8 q^{2} - 12 q^{3} + 16 q^{4} - 20 q^{5} + 24 q^{6} - 7 q^{7} - 32 q^{8} + 36 q^{9}+O(q^{10}) $ 4 * q - 8 * q^2 - 12 * q^3 + 16 * q^4 - 20 * q^5 + 24 * q^6 - 7 * q^7 - 32 * q^8 + 36 * q^9	\( 4 q - 8 q^{2} - 12 q^{3} + 16 q^{4} - 20 q^{5} + 24 q^{6} - 7 q^{7} - 32 q^{8} + 36 q^{9} + 40 q^{10} + 43 q^{11} - 48 q^{12} - 32 q^{13} + 14 q^{14} + 60 q^{15} + 64 q^{16} - 26 q^{17} - 72 q^{18} - 153 q^{19} - 80 q^{20} + 21 q^{21} - 86 q^{22} + 135 q^{23} + 96 q^{24} + 100 q^{25} + 64 q^{26} - 108 q^{27} - 28 q^{28} + 78 q^{29} - 120 q^{30} - 124 q^{31} - 128 q^{32} - 129 q^{33} + 52 q^{34} + 35 q^{35} + 144 q^{36} - 682 q^{37} + 306 q^{38} + 96 q^{39} + 160 q^{40} + 106 q^{41} - 42 q^{42} - 829 q^{43} + 172 q^{44} - 180 q^{45} - 270 q^{46} - 454 q^{47} - 192 q^{48} - 299 q^{49} - 200 q^{50} + 78 q^{51} - 128 q^{52} - 103 q^{53} + 216 q^{54} - 215 q^{55} + 56 q^{56} + 459 q^{57} - 156 q^{58} + 1042 q^{59} + 240 q^{60} - 100 q^{61} + 248 q^{62} - 63 q^{63} + 256 q^{64} + 160 q^{65} + 258 q^{66} - 742 q^{67} - 104 q^{68} - 405 q^{69} - 70 q^{70} + 969 q^{71} - 288 q^{72} - 845 q^{73} + 1364 q^{74} - 300 q^{75} - 612 q^{76} - 241 q^{77} - 192 q^{78} + 1215 q^{79} - 320 q^{80} + 324 q^{81} - 212 q^{82} - 92 q^{83} + 84 q^{84} + 130 q^{85} + 1658 q^{86} - 234 q^{87} - 344 q^{88} + 2349 q^{89} + 360 q^{90} - 1142 q^{91} + 540 q^{92} + 372 q^{93} + 908 q^{94} + 765 q^{95} + 384 q^{96} - 1228 q^{97} + 598 q^{98} + 387 q^{99}+O(q^{100}) \) 4 * q - 8 * q^2 - 12 * q^3 + 16 * q^4 - 20 * q^5 + 24 * q^6 - 7 * q^7 - 32 * q^8 + 36 * q^9 + 40 * q^10 + 43 * q^11 - 48 * q^12 - 32 * q^13 + 14 * q^14 + 60 * q^15 + 64 * q^16 - 26 * q^17 - 72 * q^18 - 153 * q^19 - 80 * q^20 + 21 * q^21 - 86 * q^22 + 135 * q^23 + 96 * q^24 + 100 * q^25 + 64 * q^26 - 108 * q^27 - 28 * q^28 + 78 * q^29 - 120 * q^30 - 124 * q^31 - 128 * q^32 - 129 * q^33 + 52 * q^34 + 35 * q^35 + 144 * q^36 - 682 * q^37 + 306 * q^38 + 96 * q^39 + 160 * q^40 + 106 * q^41 - 42 * q^42 - 829 * q^43 + 172 * q^44 - 180 * q^45 - 270 * q^46 - 454 * q^47 - 192 * q^48 - 299 * q^49 - 200 * q^50 + 78 * q^51 - 128 * q^52 - 103 * q^53 + 216 * q^54 - 215 * q^55 + 56 * q^56 + 459 * q^57 - 156 * q^58 + 1042 * q^59 + 240 * q^60 - 100 * q^61 + 248 * q^62 - 63 * q^63 + 256 * q^64 + 160 * q^65 + 258 * q^66 - 742 * q^67 - 104 * q^68 - 405 * q^69 - 70 * q^70 + 969 * q^71 - 288 * q^72 - 845 * q^73 + 1364 * q^74 - 300 * q^75 - 612 * q^76 - 241 * q^77 - 192 * q^78 + 1215 * q^79 - 320 * q^80 + 324 * q^81 - 212 * q^82 - 92 * q^83 + 84 * q^84 + 130 * q^85 + 1658 * q^86 - 234 * q^87 - 344 * q^88 + 2349 * q^89 + 360 * q^90 - 1142 * q^91 + 540 * q^92 + 372 * q^93 + 908 * q^94 + 765 * q^95 + 384 * q^96 - 1228 * q^97 + 598 * q^98 + 387 * 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
$7$	−16.5903	−0.895790	−0.447895	−	0.894086i	$-0.647826\pi$
$7$	−16.5903	−0.895790	−0.447895	+	0.894086i	$0.647826\pi$
$8$	−8.00000	−0.353553
$9$	9.00000	0.333333
$10$	10.0000	0.316228
$11$	44.4131	1.21737	0.608684	−	0.793413i	$-0.291698\pi$
$11$	44.4131	1.21737	0.608684	+	0.793413i	$0.291698\pi$
$12$	−12.0000	−0.288675
$13$	90.3678	1.92796	0.963982	−	0.265969i	$-0.0856918\pi$
$13$	90.3678	1.92796	0.963982	+	0.265969i	$0.0856918\pi$
$14$	33.1805	0.633419
$15$	15.0000	0.258199
$16$	16.0000	0.250000
$17$	−21.7231	−0.309920	−0.154960	−	0.987921i	$-0.549525\pi$
$17$	−21.7231	−0.309920	−0.154960	+	0.987921i	$0.549525\pi$
$18$	−18.0000	−0.235702
$19$	−84.1320	−1.01585	−0.507926	−	0.861401i	$-0.669588\pi$
$19$	−84.1320	−1.01585	−0.507926	+	0.861401i	$0.669588\pi$
$20$	−20.0000	−0.223607
$21$	49.7708	0.517185
$22$	−88.8261	−0.860809
$23$	64.4088	0.583920	0.291960	−	0.956430i	$-0.405693\pi$
$23$	64.4088	0.583920	0.291960	+	0.956430i	$0.405693\pi$
$24$	24.0000	0.204124
$25$	25.0000	0.200000
$26$	−180.736	−1.36328
$27$	−27.0000	−0.192450
$28$	−66.3610	−0.447895
$29$	−241.728	−1.54785	−0.773926	−	0.633276i	$-0.781710\pi$
$29$	−241.728	−1.54785	−0.773926	+	0.633276i	$0.781710\pi$
$30$	−30.0000	−0.182574
$31$	−31.0000	−0.179605
$31$	−31.0000	−0.179605
$32$	−32.0000	−0.176777
$33$	−133.239	−0.702847
$34$	43.4463	0.219146
$35$	82.9513	0.400609
$36$	36.0000	0.166667
$37$	−438.645	−1.94900	−0.974498	−	0.224397i	$-0.927959\pi$
$37$	−438.645	−1.94900	−0.974498	+	0.224397i	$0.927959\pi$
$38$	168.264	0.718316
$39$	−271.103	−1.11311
$40$	40.0000	0.158114
$41$	421.928	1.60717	0.803586	−	0.595188i	$-0.202923\pi$
$41$	421.928	1.60717	0.803586	+	0.595188i	$0.202923\pi$
$42$	−99.5416	−0.365705
$43$	52.0635	0.184642	0.0923210	−	0.995729i	$-0.470571\pi$
$43$	52.0635	0.184642	0.0923210	+	0.995729i	$0.470571\pi$
$44$	177.652	0.608684
$45$	−45.0000	−0.149071
$46$	−128.818	−0.412894
$47$	−14.8186	−0.0459897	−0.0229948	−	0.999736i	$-0.507320\pi$
$47$	−14.8186	−0.0459897	−0.0229948	+	0.999736i	$0.507320\pi$
$48$	−48.0000	−0.144338
$49$	−67.7632	−0.197560
$50$	−50.0000	−0.141421
$51$	65.1694	0.178932
$52$	361.471	0.963982
$53$	−359.850	−0.932626	−0.466313	−	0.884620i	$-0.654418\pi$
$53$	−359.850	−0.932626	−0.466313	+	0.884620i	$0.654418\pi$
$54$	54.0000	0.136083
$55$	−222.065	−0.544423
$56$	132.722	0.316710
$57$	252.396	0.586503
$58$	483.456	1.09450
$59$	58.9738	0.130131	0.0650656	−	0.997881i	$-0.479274\pi$
$59$	58.9738	0.130131	0.0650656	+	0.997881i	$0.479274\pi$
$60$	60.0000	0.129099
$61$	922.905	1.93714	0.968572	−	0.248732i	$-0.0800138\pi$
$61$	922.905	1.93714	0.968572	+	0.248732i	$0.0800138\pi$
$62$	62.0000	0.127000
$63$	−149.312	−0.298597
$64$	64.0000	0.125000
$65$	−451.839	−0.862211
$66$	266.478	0.496988
$67$	−276.466	−0.504116	−0.252058	−	0.967712i	$-0.581107\pi$
$67$	−276.466	−0.504116	−0.252058	+	0.967712i	$0.581107\pi$
$68$	−86.8925	−0.154960
$69$	−193.226	−0.337127
$70$	−165.903	−0.283274
$71$	486.851	0.813783	0.406892	−	0.913476i	$-0.366613\pi$
$71$	486.851	0.813783	0.406892	+	0.913476i	$0.366613\pi$
$72$	−72.0000	−0.117851
$73$	−96.4576	−0.154651	−0.0773254	−	0.997006i	$-0.524638\pi$
$73$	−96.4576	−0.154651	−0.0773254	+	0.997006i	$0.524638\pi$
$74$	877.291	1.37815
$75$	−75.0000	−0.115470
$76$	−336.528	−0.507926
$77$	−736.824	−1.09051
$78$	542.207	0.787088
$79$	1111.72	1.58326	0.791632	−	0.610998i	$-0.209232\pi$
$79$	1111.72	1.58326	0.791632	+	0.610998i	$0.209232\pi$
$80$	−80.0000	−0.111803
$81$	81.0000	0.111111
$82$	−843.856	−1.13644
$83$	509.095	0.673259	0.336629	−	0.941637i	$-0.390713\pi$
$83$	509.095	0.673259	0.336629	+	0.941637i	$0.390713\pi$
$84$	199.083	0.258592
$85$	108.616	0.138600
$86$	−104.127	−0.130562
$87$	725.183	0.893653
$88$	−355.304	−0.430404
$89$	331.621	0.394963	0.197482	−	0.980307i	$-0.436724\pi$
$89$	331.621	0.394963	0.197482	+	0.980307i	$0.436724\pi$
$90$	90.0000	0.105409
$91$	−1499.23	−1.72705
$92$	257.635	0.291960
$93$	93.0000	0.103695
$94$	29.6372	0.0325196
$95$	420.660	0.454303
$96$	96.0000	0.102062
$97$	−1662.66	−1.74038	−0.870192	−	0.492713i	$-0.836005\pi$
$97$	−1662.66	−1.74038	−0.870192	+	0.492713i	$0.836005\pi$
$98$	135.526	0.139696
$99$	399.718	0.405789
$100$	100.000	0.100000
$101$	1479.04	1.45713	0.728565	−	0.684976i	$-0.240188\pi$
$101$	1479.04	1.45713	0.728565	+	0.684976i	$0.240188\pi$
$102$	−130.339	−0.126524
$103$	−1102.32	−1.05452	−0.527258	−	0.849705i	$-0.676780\pi$
$103$	−1102.32	−1.05452	−0.527258	+	0.849705i	$0.676780\pi$
$104$	−722.943	−0.681638
$105$	−248.854	−0.231292
$106$	719.699	0.659466
$107$	593.968	0.536645	0.268323	−	0.963329i	$-0.413531\pi$
$107$	593.968	0.536645	0.268323	+	0.963329i	$0.413531\pi$
$108$	−108.000	−0.0962250
$109$	−1188.25	−1.04416	−0.522082	−	0.852895i	$-0.674845\pi$
$109$	−1188.25	−1.04416	−0.522082	+	0.852895i	$0.674845\pi$
$110$	444.131	0.384965
$111$	1315.94	1.12525
$112$	−265.444	−0.223947
$113$	−2349.96	−1.95633	−0.978165	−	0.207829i	$-0.933360\pi$
$113$	−2349.96	−1.95633	−0.978165	+	0.207829i	$0.933360\pi$
$114$	−504.792	−0.414720
$115$	−322.044	−0.261137
$116$	−966.911	−0.773926
$117$	813.310	0.642654
$118$	−117.948	−0.0920166
$119$	360.392	0.277623
$120$	−120.000	−0.0912871
$121$	641.520	0.481983
$122$	−1845.81	−1.36977
$123$	−1265.78	−0.927901
$124$	−124.000	−0.0898027
$125$	−125.000	−0.0894427
$126$	298.625	0.211140
$127$	787.614	0.550311	0.275155	−	0.961400i	$-0.411271\pi$
$127$	787.614	0.550311	0.275155	+	0.961400i	$0.411271\pi$
$128$	−128.000	−0.0883883
$129$	−156.190	−0.106603
$130$	903.678	0.609676
$131$	1990.14	1.32733	0.663663	−	0.748032i	$-0.269001\pi$
$131$	1990.14	1.32733	0.663663	+	0.748032i	$0.269001\pi$
$132$	−532.957	−0.351424
$133$	1395.77	0.909990
$134$	552.933	0.356464
$135$	135.000	0.0860663
$136$	173.785	0.109573
$137$	1904.97	1.18797	0.593986	−	0.804475i	$-0.297553\pi$
$137$	1904.97	1.18797	0.593986	+	0.804475i	$0.297553\pi$
$138$	386.453	0.238385
$139$	1889.14	1.15277	0.576385	−	0.817178i	$-0.304463\pi$
$139$	1889.14	1.15277	0.576385	+	0.817178i	$0.304463\pi$
$140$	331.805	0.200305
$141$	44.4558	0.0265522
$142$	−973.702	−0.575432
$143$	4013.51	2.34704
$144$	144.000	0.0833333
$145$	1208.64	0.692221
$146$	192.915	0.109355
$147$	203.290	0.114062
$148$	−1754.58	−0.974498
$149$	1705.78	0.937872	0.468936	−	0.883232i	$-0.344637\pi$
$149$	1705.78	0.937872	0.468936	+	0.883232i	$0.344637\pi$
$150$	150.000	0.0816497
$151$	744.906	0.401454	0.200727	−	0.979647i	$-0.435670\pi$
$151$	744.906	0.401454	0.200727	+	0.979647i	$0.435670\pi$
$152$	673.056	0.359158
$153$	−195.508	−0.103307
$154$	1473.65	0.771104
$155$	155.000	0.0803219
$156$	−1084.41	−0.556555
$157$	−935.766	−0.475683	−0.237842	−	0.971304i	$-0.576440\pi$
$157$	−935.766	−0.475683	−0.237842	+	0.971304i	$0.576440\pi$
$158$	−2223.43	−1.11954
$159$	1079.55	0.538452
$160$	160.000	0.0790569
$161$	−1068.56	−0.523070
$162$	−162.000	−0.0785674
$163$	2004.75	0.963340	0.481670	−	0.876353i	$-0.340030\pi$
$163$	2004.75	0.963340	0.481670	+	0.876353i	$0.340030\pi$
$164$	1687.71	0.803586
$165$	666.196	0.314323
$166$	−1018.19	−0.476066
$167$	3117.58	1.44459	0.722293	−	0.691587i	$-0.243088\pi$
$167$	3117.58	1.44459	0.722293	+	0.691587i	$0.243088\pi$
$168$	−398.166	−0.182852
$169$	5969.34	2.71704
$170$	−217.231	−0.0980052
$171$	−757.188	−0.338617
$172$	208.254	0.0923210
$173$	2296.77	1.00936	0.504682	−	0.863305i	$-0.331610\pi$
$173$	2296.77	1.00936	0.504682	+	0.863305i	$0.331610\pi$
$174$	−1450.37	−0.631908
$175$	−414.757	−0.179158
$176$	710.609	0.304342
$177$	−176.921	−0.0751312
$178$	−663.242	−0.279281
$179$	−1284.79	−0.536480	−0.268240	−	0.963352i	$-0.586442\pi$
$179$	−1284.79	−0.536480	−0.268240	+	0.963352i	$0.586442\pi$
$180$	−180.000	−0.0745356
$181$	380.721	0.156347	0.0781733	−	0.996940i	$-0.475091\pi$
$181$	380.721	0.156347	0.0781733	+	0.996940i	$0.475091\pi$
$182$	2998.45	1.22121
$183$	−2768.71	−1.11841
$184$	−515.271	−0.206447
$185$	2193.23	0.871617
$186$	−186.000	−0.0733236
$187$	−964.791	−0.377286
$188$	−59.2744	−0.0229948
$189$	447.937	0.172395
$190$	−841.320	−0.321241
$191$	3067.12	1.16193	0.580967	−	0.813927i	$-0.302675\pi$
$191$	3067.12	1.16193	0.580967	+	0.813927i	$0.302675\pi$
$192$	−192.000	−0.0721688
$193$	−1217.93	−0.454241	−0.227120	−	0.973867i	$-0.572931\pi$
$193$	−1217.93	−0.454241	−0.227120	+	0.973867i	$0.572931\pi$
$194$	3325.31	1.23064
$195$	1355.52	0.497798
$196$	−271.053	−0.0987802
$197$	4674.67	1.69064	0.845321	−	0.534259i	$-0.179409\pi$
$197$	4674.67	1.69064	0.845321	+	0.534259i	$0.179409\pi$
$198$	−799.435	−0.286936
$199$	−5221.83	−1.86013	−0.930065	−	0.367396i	$-0.880250\pi$
$199$	−5221.83	−1.86013	−0.930065	+	0.367396i	$0.880250\pi$
$200$	−200.000	−0.0707107
$201$	829.399	0.291051
$202$	−2958.08	−1.03035
$203$	4010.33	1.38655
$204$	260.678	0.0894661
$205$	−2109.64	−0.718749
$206$	2204.65	0.745655
$207$	579.679	0.194640
$208$	1445.89	0.481991
$209$	−3736.56	−1.23667
$210$	497.708	0.163548
$211$	−4628.78	−1.51023	−0.755115	−	0.655592i	$-0.772419\pi$
$211$	−4628.78	−1.51023	−0.755115	+	0.655592i	$0.772419\pi$
$212$	−1439.40	−0.466313
$213$	−1460.55	−0.469838
$214$	−1187.94	−0.379466
$215$	−260.317	−0.0825744
$216$	216.000	0.0680414
$217$	514.298	0.160889
$218$	2376.51	0.738336
$219$	289.373	0.0892877
$220$	−888.261	−0.272212
$221$	−1963.07	−0.597514
$222$	−2631.87	−0.795674
$223$	−3994.45	−1.19950	−0.599749	−	0.800188i	$-0.704733\pi$
$223$	−3994.45	−1.19950	−0.599749	+	0.800188i	$0.704733\pi$
$224$	530.888	0.158355
$225$	225.000	0.0666667
$226$	4699.91	1.38333
$227$	1889.42	0.552447	0.276223	−	0.961093i	$-0.410917\pi$
$227$	1889.42	0.552447	0.276223	+	0.961093i	$0.410917\pi$
$228$	1009.58	0.293251
$229$	6609.38	1.90725	0.953624	−	0.300999i	$-0.0973201\pi$
$229$	6609.38	1.90725	0.953624	+	0.300999i	$0.0973201\pi$
$230$	644.088	0.184652
$231$	2210.47	0.629604
$232$	1933.82	0.547249
$233$	−3331.37	−0.936676	−0.468338	−	0.883549i	$-0.655147\pi$
$233$	−3331.37	−0.936676	−0.468338	+	0.883549i	$0.655147\pi$
$234$	−1626.62	−0.454425
$235$	74.0930	0.0205672
$236$	235.895	0.0650656
$237$	−3335.15	−0.914098
$238$	−720.785	−0.196309
$239$	1704.50	0.461319	0.230659	−	0.973035i	$-0.425912\pi$
$239$	1704.50	0.461319	0.230659	+	0.973035i	$0.425912\pi$
$240$	240.000	0.0645497
$241$	5969.23	1.59549	0.797743	−	0.602997i	$-0.206027\pi$
$241$	5969.23	1.59549	0.797743	+	0.602997i	$0.206027\pi$
$242$	−1283.04	−0.340814
$243$	−243.000	−0.0641500
$244$	3691.62	0.968572
$245$	338.816	0.0883517
$246$	2531.57	0.656125
$247$	−7602.82	−1.95853
$248$	248.000	0.0635001
$249$	−1527.29	−0.388706
$250$	250.000	0.0632456
$251$	2304.30	0.579467	0.289733	−	0.957107i	$-0.406433\pi$
$251$	2304.30	0.579467	0.289733	+	0.957107i	$0.406433\pi$
$252$	−597.249	−0.149298
$253$	2860.59	0.710846
$254$	−1575.23	−0.389128
$255$	−325.847	−0.0800209
$256$	256.000	0.0625000
$257$	4519.34	1.09692	0.548460	−	0.836177i	$-0.315214\pi$
$257$	4519.34	1.09692	0.548460	+	0.836177i	$0.315214\pi$
$258$	312.381	0.0753798
$259$	7277.24	1.74589
$260$	−1807.36	−0.431106
$261$	−2175.55	−0.515951
$262$	−3980.29	−0.938562
$263$	4268.37	1.00076	0.500379	−	0.865807i	$-0.333194\pi$
$263$	4268.37	1.00076	0.500379	+	0.865807i	$0.333194\pi$
$264$	1065.91	0.248494
$265$	1799.25	0.417083
$266$	−2791.54	−0.643460
$267$	−994.862	−0.228032
$268$	−1105.87	−0.252058
$269$	−5750.00	−1.30328	−0.651642	−	0.758527i	$-0.725920\pi$
$269$	−5750.00	−1.30328	−0.651642	+	0.758527i	$0.725920\pi$
$270$	−270.000	−0.0608581
$271$	4783.32	1.07220	0.536100	−	0.844155i	$-0.319897\pi$
$271$	4783.32	1.07220	0.536100	+	0.844155i	$0.319897\pi$
$272$	−347.570	−0.0774799
$273$	4497.68	0.997113
$274$	−3809.93	−0.840024
$275$	1110.33	0.243473
$276$	−772.906	−0.168563
$277$	−4495.26	−0.975069	−0.487535	−	0.873104i	$-0.662104\pi$
$277$	−4495.26	−0.975069	−0.487535	+	0.873104i	$0.662104\pi$
$278$	−3778.29	−0.815131
$279$	−279.000	−0.0598684
$280$	−663.610	−0.141637
$281$	−7712.68	−1.63737	−0.818683	−	0.574245i	$-0.805296\pi$
$281$	−7712.68	−1.63737	−0.818683	+	0.574245i	$0.805296\pi$
$282$	−88.9116	−0.0187752
$283$	6379.42	1.33999	0.669995	−	0.742365i	$-0.266296\pi$
$283$	6379.42	1.33999	0.669995	+	0.742365i	$0.266296\pi$
$284$	1947.40	0.406892
$285$	−1261.98	−0.262292
$286$	−8027.02	−1.65961
$287$	−6999.89	−1.43969
$288$	−288.000	−0.0589256
$289$	−4441.11	−0.903950
$290$	−2417.28	−0.489474
$291$	4987.97	1.00481
$292$	−385.830	−0.0773254
$293$	−1240.26	−0.247293	−0.123647	−	0.992326i	$-0.539459\pi$
$293$	−1240.26	−0.247293	−0.123647	+	0.992326i	$0.539459\pi$
$294$	−406.579	−0.0806537
$295$	−294.869	−0.0581964
$296$	3509.16	0.689074
$297$	−1199.15	−0.234282
$298$	−3411.56	−0.663176
$299$	5820.49	1.12578
$300$	−300.000	−0.0577350
$301$	−863.747	−0.165400
$302$	−1489.81	−0.283871
$303$	−4437.13	−0.841275
$304$	−1346.11	−0.253963
$305$	−4614.52	−0.866317
$306$	391.016	0.0730487
$307$	−5979.32	−1.11159	−0.555794	−	0.831320i	$-0.687586\pi$
$307$	−5979.32	−1.11159	−0.555794	+	0.831320i	$0.687586\pi$
$308$	−2947.30	−0.545253
$309$	3306.97	0.608825
$310$	−310.000	−0.0567962
$311$	−2373.03	−0.432677	−0.216338	−	0.976318i	$-0.569411\pi$
$311$	−2373.03	−0.432677	−0.216338	+	0.976318i	$0.569411\pi$
$312$	2168.83	0.393544
$313$	8024.12	1.44904	0.724521	−	0.689252i	$-0.242061\pi$
$313$	8024.12	1.44904	0.724521	+	0.689252i	$0.242061\pi$
$314$	1871.53	0.336359
$315$	746.562	0.133536
$316$	4446.87	0.791632
$317$	1670.68	0.296009	0.148004	−	0.988987i	$-0.452715\pi$
$317$	1670.68	0.296009	0.148004	+	0.988987i	$0.452715\pi$
$318$	−2159.10	−0.380743
$319$	−10735.9	−1.88431
$320$	−320.000	−0.0559017
$321$	−1781.90	−0.309832
$322$	2137.12	0.369866
$323$	1827.61	0.314833
$324$	324.000	0.0555556
$325$	2259.20	0.385593
$326$	−4009.51	−0.681184
$327$	3564.76	0.602849
$328$	−3375.42	−0.568221
$329$	245.845	0.0411971
$330$	−1332.39	−0.222260
$331$	−3615.94	−0.600453	−0.300227	−	0.953868i	$-0.597062\pi$
$331$	−3615.94	−0.600453	−0.300227	+	0.953868i	$0.597062\pi$
$332$	2036.38	0.336629
$333$	−3947.81	−0.649665
$334$	−6235.17	−1.02148
$335$	1382.33	0.225447
$336$	796.333	0.129296
$337$	8324.10	1.34553	0.672764	−	0.739858i	$-0.265107\pi$
$337$	8324.10	1.34553	0.672764	+	0.739858i	$0.265107\pi$
$338$	−11938.7	−1.92124
$339$	7049.87	1.12949
$340$	434.463	0.0693001
$341$	−1376.80	−0.218646
$342$	1514.38	0.239439
$343$	6814.67	1.07276
$344$	−416.508	−0.0652808
$345$	966.132	0.150768
$346$	−4593.54	−0.713728
$347$	9740.02	1.50683	0.753417	−	0.657543i	$-0.228404\pi$
$347$	9740.02	1.50683	0.753417	+	0.657543i	$0.228404\pi$
$348$	2900.73	0.446827
$349$	−12928.8	−1.98299	−0.991494	−	0.130150i	$-0.958454\pi$
$349$	−12928.8	−1.98299	−0.991494	+	0.130150i	$0.958454\pi$
$350$	829.513	0.126684
$351$	−2439.93	−0.371037
$352$	−1421.22	−0.215202
$353$	−2763.36	−0.416653	−0.208327	−	0.978059i	$-0.566802\pi$
$353$	−2763.36	−0.416653	−0.208327	+	0.978059i	$0.566802\pi$
$354$	353.843	0.0531258
$355$	−2434.26	−0.363935
$356$	1326.48	0.197482
$357$	−1081.18	−0.160286
$358$	2569.59	0.379349
$359$	7287.86	1.07142	0.535708	−	0.844403i	$-0.320045\pi$
$359$	7287.86	1.07142	0.535708	+	0.844403i	$0.320045\pi$
$360$	360.000	0.0527046
$361$	219.187	0.0319561
$362$	−761.441	−0.110554
$363$	−1924.56	−0.278273
$364$	−5996.90	−0.863525
$365$	482.288	0.0691619
$366$	5537.43	0.790836
$367$	−4610.74	−0.655800	−0.327900	−	0.944712i	$-0.606341\pi$
$367$	−4610.74	−0.655800	−0.327900	+	0.944712i	$0.606341\pi$
$368$	1030.54	0.145980
$369$	3797.35	0.535724
$370$	−4386.45	−0.616327
$371$	5970.00	0.835437
$372$	372.000	0.0518476
$373$	−4860.53	−0.674715	−0.337358	−	0.941377i	$-0.609533\pi$
$373$	−4860.53	−0.674715	−0.337358	+	0.941377i	$0.609533\pi$
$374$	1929.58	0.266781
$375$	375.000	0.0516398
$376$	118.549	0.0162598
$377$	−21844.4	−2.98420
$378$	−895.874	−0.121902
$379$	6527.48	0.884681	0.442341	−	0.896847i	$-0.354148\pi$
$379$	6527.48	0.884681	0.442341	+	0.896847i	$0.354148\pi$
$380$	1682.64	0.227152
$381$	−2362.84	−0.317722
$382$	−6134.25	−0.821611
$383$	5004.07	0.667614	0.333807	−	0.942642i	$-0.391667\pi$
$383$	5004.07	0.667614	0.333807	+	0.942642i	$0.391667\pi$
$384$	384.000	0.0510310
$385$	3684.12	0.487689
$386$	2435.86	0.321197
$387$	468.571	0.0615473
$388$	−6650.63	−0.870192
$389$	11204.8	1.46043	0.730214	−	0.683218i	$-0.239420\pi$
$389$	11204.8	1.46043	0.730214	+	0.683218i	$0.239420\pi$
$390$	−2711.03	−0.351996
$391$	−1399.16	−0.180968
$392$	542.106	0.0698482
$393$	−5970.43	−0.766332
$394$	−9349.34	−1.19546
$395$	−5558.58	−0.708057
$396$	1598.87	0.202895
$397$	9946.06	1.25738	0.628688	−	0.777657i	$-0.283592\pi$
$397$	9946.06	1.25738	0.628688	+	0.777657i	$0.283592\pi$
$398$	10443.7	1.31531
$399$	−4187.31	−0.525383
$400$	400.000	0.0500000
$401$	6104.27	0.760182	0.380091	−	0.924949i	$-0.375893\pi$
$401$	6104.27	0.760182	0.380091	+	0.924949i	$0.375893\pi$
$402$	−1658.80	−0.205804
$403$	−2801.40	−0.346272
$404$	5916.17	0.728565
$405$	−405.000	−0.0496904
$406$	−8020.66	−0.980440
$407$	−19481.6	−2.37264
$408$	−521.355	−0.0632621
$409$	−5584.29	−0.675123	−0.337562	−	0.941303i	$-0.609602\pi$
$409$	−5584.29	−0.675123	−0.337562	+	0.941303i	$0.609602\pi$
$410$	4219.28	0.508233
$411$	−5714.90	−0.685877
$412$	−4409.29	−0.527258
$413$	−978.391	−0.116570
$414$	−1159.36	−0.137631
$415$	−2545.48	−0.301091
$416$	−2891.77	−0.340819
$417$	−5667.43	−0.665552
$418$	7473.12	0.874455
$419$	−6066.05	−0.707270	−0.353635	−	0.935384i	$-0.615054\pi$
$419$	−6066.05	−0.707270	−0.353635	+	0.935384i	$0.615054\pi$
$420$	−995.416	−0.115646
$421$	1576.39	0.182490	0.0912451	−	0.995828i	$-0.470915\pi$
$421$	1576.39	0.182490	0.0912451	+	0.995828i	$0.470915\pi$
$422$	9257.56	1.06789
$423$	−133.367	−0.0153299
$424$	2878.80	0.329733
$425$	−543.078	−0.0619839
$426$	2921.11	0.332226
$427$	−15311.2	−1.73527
$428$	2375.87	0.268323
$429$	−12040.5	−1.35506
$430$	520.635	0.0583889
$431$	4355.85	0.486808	0.243404	−	0.969925i	$-0.421736\pi$
$431$	4355.85	0.486808	0.243404	+	0.969925i	$0.421736\pi$
$432$	−432.000	−0.0481125
$433$	−1018.70	−0.113061	−0.0565306	−	0.998401i	$-0.518004\pi$
$433$	−1018.70	−0.113061	−0.0565306	+	0.998401i	$0.518004\pi$
$434$	−1028.60	−0.113765
$435$	−3625.92	−0.399654
$436$	−4753.01	−0.522082
$437$	−5418.84	−0.593177
$438$	−578.745	−0.0631359
$439$	6241.18	0.678531	0.339266	−	0.940691i	$-0.389821\pi$
$439$	6241.18	0.678531	0.339266	+	0.940691i	$0.389821\pi$
$440$	1776.52	0.192483
$441$	−609.869	−0.0658535
$442$	3926.14	0.422506
$443$	11141.6	1.19492	0.597462	−	0.801897i	$-0.296176\pi$
$443$	11141.6	1.19492	0.597462	+	0.801897i	$0.296176\pi$
$444$	5263.74	0.562627
$445$	−1658.10	−0.176633
$446$	7988.90	0.848173
$447$	−5117.34	−0.541481
$448$	−1061.78	−0.111974
$449$	8796.22	0.924542	0.462271	−	0.886739i	$-0.347035\pi$
$449$	8796.22	0.924542	0.462271	+	0.886739i	$0.347035\pi$
$450$	−450.000	−0.0471405
$451$	18739.1	1.95652
$452$	−9399.83	−0.978165
$453$	−2234.72	−0.231780
$454$	−3778.85	−0.390639
$455$	7496.13	0.772360
$456$	−2019.17	−0.207360
$457$	1827.45	0.187055	0.0935277	−	0.995617i	$-0.470186\pi$
$457$	1827.45	0.187055	0.0935277	+	0.995617i	$0.470186\pi$
$458$	−13218.8	−1.34863
$459$	586.524	0.0596440
$460$	−1288.18	−0.130569
$461$	−3458.72	−0.349433	−0.174717	−	0.984619i	$-0.555901\pi$
$461$	−3458.72	−0.349433	−0.174717	+	0.984619i	$0.555901\pi$
$462$	−4420.95	−0.445197
$463$	8469.80	0.850162	0.425081	−	0.905155i	$-0.360246\pi$
$463$	8469.80	0.850162	0.425081	+	0.905155i	$0.360246\pi$
$464$	−3867.65	−0.386963
$465$	−465.000	−0.0463739
$466$	6662.75	0.662330
$467$	−14125.1	−1.39964	−0.699820	−	0.714320i	$-0.746736\pi$
$467$	−14125.1	−1.39964	−0.699820	+	0.714320i	$0.746736\pi$
$468$	3253.24	0.321327
$469$	4586.65	0.451582
$470$	−148.186	−0.0145432
$471$	2807.30	0.274636
$472$	−471.790	−0.0460083
$473$	2312.30	0.224777
$474$	6670.30	0.646365
$475$	−2103.30	−0.203170
$476$	1441.57	0.138811
$477$	−3238.65	−0.310875
$478$	−3409.01	−0.326202
$479$	−9989.80	−0.952913	−0.476457	−	0.879198i	$-0.658079\pi$
$479$	−9989.80	−0.952913	−0.476457	+	0.879198i	$0.658079\pi$
$480$	−480.000	−0.0456435
$481$	−39639.4	−3.75759
$482$	−11938.5	−1.12818
$483$	3205.68	0.301995
$484$	2566.08	0.240992
$485$	8313.29	0.778323
$486$	486.000	0.0453609
$487$	7792.15	0.725043	0.362522	−	0.931975i	$-0.381916\pi$
$487$	7792.15	0.725043	0.362522	+	0.931975i	$0.381916\pi$
$488$	−7383.24	−0.684884
$489$	−6014.26	−0.556185
$490$	−677.632	−0.0624741
$491$	19281.9	1.77226	0.886130	−	0.463436i	$-0.153384\pi$
$491$	19281.9	1.77226	0.886130	+	0.463436i	$0.153384\pi$
$492$	−5063.13	−0.463951
$493$	5251.08	0.479710
$494$	15205.6	1.38489
$495$	−1998.59	−0.181474
$496$	−496.000	−0.0449013
$497$	−8076.99	−0.728979
$498$	3054.57	0.274857
$499$	19007.9	1.70523	0.852615	−	0.522540i	$-0.175016\pi$
$499$	19007.9	1.70523	0.852615	+	0.522540i	$0.175016\pi$
$500$	−500.000	−0.0447214
$501$	−9352.75	−0.834032
$502$	−4608.60	−0.409745
$503$	−18982.5	−1.68268	−0.841339	−	0.540507i	$-0.818232\pi$
$503$	−18982.5	−1.68268	−0.841339	+	0.540507i	$0.818232\pi$
$504$	1194.50	0.105570
$505$	−7395.21	−0.651649
$506$	−5721.19	−0.502644
$507$	−17908.0	−1.56869
$508$	3150.46	0.275155
$509$	5319.70	0.463245	0.231622	−	0.972806i	$-0.425597\pi$
$509$	5319.70	0.463245	0.231622	+	0.972806i	$0.425597\pi$
$510$	651.694	0.0565833
$511$	1600.26	0.138535
$512$	−512.000	−0.0441942
$513$	2271.56	0.195501
$514$	−9038.67	−0.775640
$515$	5511.62	0.471594
$516$	−624.762	−0.0533016
$517$	−658.140	−0.0559864
$518$	−14554.5	−1.23453
$519$	−6890.30	−0.582757
$520$	3614.71	0.304838
$521$	7052.40	0.593035	0.296518	−	0.955027i	$-0.404175\pi$
$521$	7052.40	0.593035	0.296518	+	0.955027i	$0.404175\pi$
$522$	4351.10	0.364832
$523$	−3859.16	−0.322656	−0.161328	−	0.986901i	$-0.551578\pi$
$523$	−3859.16	−0.322656	−0.161328	+	0.986901i	$0.551578\pi$
$524$	7960.58	0.663663
$525$	1244.27	0.103437
$526$	−8536.75	−0.707642
$527$	673.417	0.0556632
$528$	−2131.83	−0.175712
$529$	−8018.50	−0.659037
$530$	−3598.50	−0.294922
$531$	530.764	0.0433770
$532$	5583.08	0.454995
$533$	38128.7	3.09857
$534$	1989.72	0.161243
$535$	−2969.84	−0.239995
$536$	2211.73	0.178232
$537$	3854.38	0.309737
$538$	11500.0	0.921561
$539$	−3009.57	−0.240504
$540$	540.000	0.0430331
$541$	9474.08	0.752907	0.376453	−	0.926436i	$-0.377144\pi$
$541$	9474.08	0.752907	0.376453	+	0.926436i	$0.377144\pi$
$542$	−9566.64	−0.758160
$543$	−1142.16	−0.0902668
$544$	695.140	0.0547866
$545$	5941.26	0.466965
$546$	−8995.35	−0.705065
$547$	−9387.26	−0.733767	−0.366883	−	0.930267i	$-0.619575\pi$
$547$	−9387.26	−0.733767	−0.366883	+	0.930267i	$0.619575\pi$
$548$	7619.87	0.593986
$549$	8306.14	0.645715
$550$	−2220.65	−0.172162
$551$	20337.0	1.57239
$552$	1545.81	0.119192
$553$	−18443.7	−1.41827
$554$	8990.53	0.689478
$555$	−6579.68	−0.503229
$556$	7556.57	0.576385
$557$	−8798.95	−0.669342	−0.334671	−	0.942335i	$-0.608625\pi$
$557$	−8798.95	−0.669342	−0.334671	+	0.942335i	$0.608625\pi$
$558$	558.000	0.0423334
$559$	4704.86	0.355983
$560$	1327.22	0.100152
$561$	2894.37	0.217826
$562$	15425.4	1.15779
$563$	7481.75	0.560068	0.280034	−	0.959990i	$-0.409654\pi$
$563$	7481.75	0.560068	0.280034	+	0.959990i	$0.409654\pi$
$564$	177.823	0.0132761
$565$	11749.8	0.874897
$566$	−12758.8	−0.947517
$567$	−1343.81	−0.0995322
$568$	−3894.81	−0.287716
$569$	−1072.54	−0.0790215	−0.0395108	−	0.999219i	$-0.512580\pi$
$569$	−1072.54	−0.0790215	−0.0395108	+	0.999219i	$0.512580\pi$
$570$	2523.96	0.185468
$571$	−5257.46	−0.385320	−0.192660	−	0.981266i	$-0.561712\pi$
$571$	−5257.46	−0.385320	−0.192660	+	0.981266i	$0.561712\pi$
$572$	16054.0	1.17352
$573$	−9201.37	−0.670843
$574$	13999.8	1.01801
$575$	1610.22	0.116784
$576$	576.000	0.0416667
$577$	4941.22	0.356509	0.178254	−	0.983984i	$-0.442955\pi$
$577$	4941.22	0.356509	0.178254	+	0.983984i	$0.442955\pi$
$578$	8882.21	0.639189
$579$	3653.79	0.262256
$580$	4834.56	0.346110
$581$	−8446.03	−0.603099
$582$	−9975.94	−0.710509
$583$	−15982.0	−1.13535
$584$	771.661	0.0546773
$585$	−4066.55	−0.287404
$586$	2480.52	0.174863
$587$	−16543.2	−1.16322	−0.581610	−	0.813468i	$-0.697577\pi$
$587$	−16543.2	−1.16322	−0.581610	+	0.813468i	$0.697577\pi$
$588$	813.159	0.0570308
$589$	2608.09	0.182452
$590$	589.738	0.0411511
$591$	−14024.0	−0.976092
$592$	−7018.32	−0.487249
$593$	−1838.07	−0.127286	−0.0636428	−	0.997973i	$-0.520272\pi$
$593$	−1838.07	−0.127286	−0.0636428	+	0.997973i	$0.520272\pi$
$594$	2398.31	0.165663
$595$	−1801.96	−0.124157
$596$	6823.12	0.468936
$597$	15665.5	1.07395
$598$	−11641.0	−0.796045
$599$	9919.70	0.676641	0.338320	−	0.941031i	$-0.390141\pi$
$599$	9919.70	0.676641	0.338320	+	0.941031i	$0.390141\pi$
$600$	600.000	0.0408248
$601$	−5386.31	−0.365578	−0.182789	−	0.983152i	$-0.558513\pi$
$601$	−5386.31	−0.365578	−0.182789	+	0.983152i	$0.558513\pi$
$602$	1727.49	0.116956
$603$	−2488.20	−0.168039
$604$	2979.62	0.200727
$605$	−3207.60	−0.215550
$606$	8874.25	0.594871
$607$	5068.26	0.338903	0.169452	−	0.985539i	$-0.445800\pi$
$607$	5068.26	0.338903	0.169452	+	0.985539i	$0.445800\pi$
$608$	2692.22	0.179579
$609$	−12031.0	−0.800526
$610$	9229.05	0.612579
$611$	−1339.13	−0.0886665
$612$	−782.033	−0.0516533
$613$	17060.8	1.12411	0.562056	−	0.827099i	$-0.310010\pi$
$613$	17060.8	1.12411	0.562056	+	0.827099i	$0.310010\pi$
$614$	11958.6	0.786012
$615$	6328.92	0.414970
$616$	5894.59	0.385552
$617$	−12384.3	−0.808063	−0.404032	−	0.914745i	$-0.632391\pi$
$617$	−12384.3	−0.808063	−0.404032	+	0.914745i	$0.632391\pi$
$618$	−6613.94	−0.430504
$619$	−5915.30	−0.384097	−0.192049	−	0.981385i	$-0.561513\pi$
$619$	−5915.30	−0.384097	−0.192049	+	0.981385i	$0.561513\pi$
$620$	620.000	0.0401610
$621$	−1739.04	−0.112376
$622$	4746.07	0.305949
$623$	−5501.67	−0.353804
$624$	−4337.66	−0.278278
$625$	625.000	0.0400000
$626$	−16048.2	−1.02463
$627$	11209.7	0.713989
$628$	−3743.06	−0.237842
$629$	9528.75	0.604032
$630$	−1493.12	−0.0944245
$631$	1452.67	0.0916481	0.0458241	−	0.998950i	$-0.485409\pi$
$631$	1452.67	0.0916481	0.0458241	+	0.998950i	$0.485409\pi$
$632$	−8893.73	−0.559768
$633$	13886.3	0.871932
$634$	−3341.36	−0.209310
$635$	−3938.07	−0.246106
$636$	4318.20	0.269226
$637$	−6123.62	−0.380889
$638$	21471.7	1.33241
$639$	4381.66	0.271261
$640$	640.000	0.0395285
$641$	−10978.4	−0.676476	−0.338238	−	0.941061i	$-0.609831\pi$
$641$	−10978.4	−0.676476	−0.338238	+	0.941061i	$0.609831\pi$
$642$	3563.81	0.219085
$643$	16184.1	0.992598	0.496299	−	0.868152i	$-0.334692\pi$
$643$	16184.1	0.992598	0.496299	+	0.868152i	$0.334692\pi$
$644$	−4274.24	−0.261535
$645$	780.952	0.0476744
$646$	−3655.22	−0.222620
$647$	27711.7	1.68386	0.841931	−	0.539585i	$-0.181419\pi$
$647$	27711.7	1.68386	0.841931	+	0.539585i	$0.181419\pi$
$648$	−648.000	−0.0392837
$649$	2619.21	0.158417
$650$	−4518.39	−0.272655
$651$	−1542.89	−0.0928891
$652$	8019.01	0.481670
$653$	9069.18	0.543498	0.271749	−	0.962368i	$-0.412398\pi$
$653$	9069.18	0.543498	0.271749	+	0.962368i	$0.412398\pi$
$654$	−7129.52	−0.426278
$655$	−9950.72	−0.593598
$656$	6750.85	0.401793
$657$	−868.118	−0.0515503
$658$	−491.689	−0.0291308
$659$	−10262.7	−0.606642	−0.303321	−	0.952888i	$-0.598096\pi$
$659$	−10262.7	−0.606642	−0.303321	+	0.952888i	$0.598096\pi$
$660$	2664.78	0.157161
$661$	−16128.2	−0.949039	−0.474519	−	0.880245i	$-0.657378\pi$
$661$	−16128.2	−0.949039	−0.474519	+	0.880245i	$0.657378\pi$
$662$	7231.88	0.424585
$663$	5889.22	0.344975
$664$	−4072.76	−0.238033
$665$	−6978.86	−0.406960
$666$	7895.62	0.459383
$667$	−15569.4	−0.903823
$668$	12470.3	0.722293
$669$	11983.3	0.692530
$670$	−2764.66	−0.159415
$671$	40989.0	2.35822
$672$	−1592.67	−0.0914262
$673$	−3008.49	−0.172316	−0.0861580	−	0.996281i	$-0.527459\pi$
$673$	−3008.49	−0.172316	−0.0861580	+	0.996281i	$0.527459\pi$
$674$	−16648.2	−0.951431
$675$	−675.000	−0.0384900
$676$	23877.4	1.35852
$677$	−18181.2	−1.03214	−0.516072	−	0.856545i	$-0.672606\pi$
$677$	−18181.2	−1.03214	−0.516072	+	0.856545i	$0.672606\pi$
$678$	−14099.7	−0.798668
$679$	27583.9	1.55902
$680$	−868.925	−0.0490026
$681$	−5668.27	−0.318955
$682$	2753.61	0.154606
$683$	24170.7	1.35412	0.677060	−	0.735927i	$-0.263254\pi$
$683$	24170.7	1.35412	0.677060	+	0.735927i	$0.263254\pi$
$684$	−3028.75	−0.169309
$685$	−9524.83	−0.531278
$686$	−13629.3	−0.758558
$687$	−19828.1	−1.10115
$688$	833.016	0.0461605
$689$	−32518.8	−1.79807
$690$	−1932.26	−0.106609
$691$	12965.6	0.713796	0.356898	−	0.934143i	$-0.383834\pi$
$691$	12965.6	0.713796	0.356898	+	0.934143i	$0.383834\pi$
$692$	9187.07	0.504682
$693$	−6631.42	−0.363502
$694$	−19480.0	−1.06549
$695$	−9445.71	−0.515534
$696$	−5801.47	−0.315954
$697$	−9165.59	−0.498094
$698$	25857.6	1.40218
$699$	9994.12	0.540790
$700$	−1659.03	−0.0895790
$701$	3401.68	0.183280	0.0916402	−	0.995792i	$-0.470789\pi$
$701$	3401.68	0.183280	0.0916402	+	0.995792i	$0.470789\pi$
$702$	4879.86	0.262363
$703$	36904.1	1.97989
$704$	2842.44	0.152171
$705$	−222.279	−0.0118745
$706$	5526.71	0.294618
$707$	−24537.7	−1.30528
$708$	−707.686	−0.0375656
$709$	−22607.8	−1.19753	−0.598767	−	0.800923i	$-0.704343\pi$
$709$	−22607.8	−1.19753	−0.598767	+	0.800923i	$0.704343\pi$
$710$	4868.51	0.257341
$711$	10005.4	0.527755
$712$	−2652.97	−0.139641
$713$	−1996.67	−0.104875
$714$	2162.35	0.113339
$715$	−20067.6	−1.04963
$716$	−5139.17	−0.268240
$717$	−5113.51	−0.266343
$718$	−14575.7	−0.757606
$719$	2183.72	0.113267	0.0566336	−	0.998395i	$-0.481963\pi$
$719$	2183.72	0.113267	0.0566336	+	0.998395i	$0.481963\pi$
$720$	−720.000	−0.0372678
$721$	18287.8	0.944625
$722$	−438.374	−0.0225964
$723$	−17907.7	−0.921154
$724$	1522.88	0.0781733
$725$	−6043.20	−0.309571
$726$	3849.12	0.196769
$727$	122.447	0.00624667	0.00312333	−	0.999995i	$-0.499006\pi$
$727$	122.447	0.00624667	0.00312333	+	0.999995i	$0.499006\pi$
$728$	11993.8	0.610604
$729$	729.000	0.0370370
$730$	−964.576	−0.0489049
$731$	−1130.98	−0.0572242
$732$	−11074.9	−0.559206
$733$	−26109.0	−1.31563	−0.657816	−	0.753179i	$-0.728520\pi$
$733$	−26109.0	−1.31563	−0.657816	+	0.753179i	$0.728520\pi$
$734$	9221.48	0.463721
$735$	−1016.45	−0.0510099
$736$	−2061.08	−0.103224
$737$	−12278.7	−0.613694
$738$	−7594.70	−0.378814
$739$	−14352.9	−0.714451	−0.357226	−	0.934018i	$-0.616277\pi$
$739$	−14352.9	−0.714451	−0.357226	+	0.934018i	$0.616277\pi$
$740$	8772.91	0.435809
$741$	22808.5	1.13076
$742$	−11940.0	−0.590743
$743$	32086.0	1.58428	0.792140	−	0.610340i	$-0.208967\pi$
$743$	32086.0	1.58428	0.792140	+	0.610340i	$0.208967\pi$
$744$	−744.000	−0.0366618
$745$	−8528.90	−0.419429
$746$	9721.06	0.477096
$747$	4581.86	0.224420
$748$	−3859.16	−0.188643
$749$	−9854.09	−0.480722
$750$	−750.000	−0.0365148
$751$	30874.4	1.50016	0.750081	−	0.661346i	$-0.230015\pi$
$751$	30874.4	1.50016	0.750081	+	0.661346i	$0.230015\pi$
$752$	−237.098	−0.0114974
$753$	−6912.90	−0.334555
$754$	43688.8	2.11015
$755$	−3724.53	−0.179536
$756$	1791.75	0.0861974
$757$	19232.8	0.923417	0.461709	−	0.887032i	$-0.347237\pi$
$757$	19232.8	0.923417	0.461709	+	0.887032i	$0.347237\pi$
$758$	−13055.0	−0.625564
$759$	−8581.78	−0.410407
$760$	−3365.28	−0.160620
$761$	−21724.5	−1.03484	−0.517420	−	0.855732i	$-0.673107\pi$
$761$	−21724.5	−1.03484	−0.517420	+	0.855732i	$0.673107\pi$
$762$	4725.69	0.224663
$763$	19713.4	0.935352
$764$	12268.5	0.580967
$765$	977.541	0.0462001
$766$	−10008.1	−0.472074
$767$	5329.33	0.250888
$768$	−768.000	−0.0360844
$769$	9632.91	0.451719	0.225859	−	0.974160i	$-0.427481\pi$
$769$	9632.91	0.451719	0.225859	+	0.974160i	$0.427481\pi$
$770$	−7368.24	−0.344848
$771$	−13558.0	−0.633307
$772$	−4871.72	−0.227120
$773$	−14118.3	−0.656920	−0.328460	−	0.944518i	$-0.606530\pi$
$773$	−14118.3	−0.656920	−0.328460	+	0.944518i	$0.606530\pi$
$774$	−937.143	−0.0435205
$775$	−775.000	−0.0359211
$776$	13301.3	0.615319
$777$	−21831.7	−1.00799
$778$	−22409.6	−1.03268
$779$	−35497.6	−1.63265
$780$	5422.07	0.248899
$781$	21622.6	0.990673
$782$	2798.32	0.127964
$783$	6526.65	0.297884
$784$	−1084.21	−0.0493901
$785$	4678.83	0.212732
$786$	11940.9	0.541879
$787$	34498.8	1.56258	0.781290	−	0.624168i	$-0.214562\pi$
$787$	34498.8	1.56258	0.781290	+	0.624168i	$0.214562\pi$
$788$	18698.7	0.845321
$789$	−12805.1	−0.577787
$790$	11117.2	0.500672
$791$	38986.4	1.75246
$792$	−3197.74	−0.143468
$793$	83400.9	3.73474
$794$	−19892.1	−0.889100
$795$	−5397.74	−0.240803
$796$	−20887.3	−0.930065
$797$	81.4061	0.00361801	0.00180900	−	0.999998i	$-0.499424\pi$
$797$	81.4061	0.00361801	0.00180900	+	0.999998i	$0.499424\pi$
$798$	8374.63	0.371502
$799$	321.906	0.0142531
$800$	−800.000	−0.0353553
$801$	2984.59	0.131654
$802$	−12208.5	−0.537530
$803$	−4283.98	−0.188267
$804$	3317.60	0.145526
$805$	5342.80	0.233924
$806$	5602.80	0.244852
$807$	17250.0	0.752451
$808$	−11832.3	−0.515173
$809$	−8334.53	−0.362208	−0.181104	−	0.983464i	$-0.557967\pi$
$809$	−8334.53	−0.362208	−0.181104	+	0.983464i	$0.557967\pi$
$810$	810.000	0.0351364
$811$	−44034.8	−1.90662	−0.953310	−	0.301992i	$-0.902348\pi$
$811$	−44034.8	−1.90662	−0.953310	+	0.301992i	$0.902348\pi$
$812$	16041.3	0.693275
$813$	−14350.0	−0.619035
$814$	38963.2	1.67771
$815$	−10023.8	−0.430819
$816$	1042.71	0.0447330
$817$	−4380.20	−0.187569
$818$	11168.6	0.477384
$819$	−13493.0	−0.575683
$820$	−8438.56	−0.359375
$821$	−21382.6	−0.908962	−0.454481	−	0.890756i	$-0.650175\pi$
$821$	−21382.6	−0.908962	−0.454481	+	0.890756i	$0.650175\pi$
$822$	11429.8	0.484988
$823$	−4688.21	−0.198567	−0.0992835	−	0.995059i	$-0.531655\pi$
$823$	−4688.21	−0.198567	−0.0992835	+	0.995059i	$0.531655\pi$
$824$	8818.59	0.372828
$825$	−3330.98	−0.140569
$826$	1956.78	0.0824275
$827$	−19850.2	−0.834655	−0.417328	−	0.908756i	$-0.637033\pi$
$827$	−19850.2	−0.834655	−0.417328	+	0.908756i	$0.637033\pi$
$828$	2318.72	0.0973201
$829$	15021.5	0.629335	0.314667	−	0.949202i	$-0.398107\pi$
$829$	15021.5	0.629335	0.314667	+	0.949202i	$0.398107\pi$
$830$	5090.95	0.212903
$831$	13485.8	0.562956
$832$	5783.54	0.240995
$833$	1472.03	0.0612278
$834$	11334.9	0.470616
$835$	−15587.9	−0.646039
$836$	−14946.2	−0.618333
$837$	837.000	0.0345651
$838$	12132.1	0.500115
$839$	36000.2	1.48136	0.740682	−	0.671856i	$-0.234503\pi$
$839$	36000.2	1.48136	0.740682	+	0.671856i	$0.234503\pi$
$840$	1990.83	0.0817741
$841$	34043.3	1.39585
$842$	−3152.77	−0.129040
$843$	23138.0	0.945334
$844$	−18515.1	−0.755115
$845$	−29846.7	−1.21510
$846$	266.735	0.0108399
$847$	−10643.0	−0.431756
$848$	−5757.59	−0.233156
$849$	−19138.3	−0.773644
$850$	1086.16	0.0438292
$851$	−28252.6	−1.13806
$852$	−5842.21	−0.234919
$853$	−3961.21	−0.159003	−0.0795014	−	0.996835i	$-0.525333\pi$
$853$	−3961.21	−0.159003	−0.0795014	+	0.996835i	$0.525333\pi$
$854$	30622.5	1.22702
$855$	3785.94	0.151434
$856$	−4751.74	−0.189733
$857$	40753.7	1.62441	0.812205	−	0.583372i	$-0.198267\pi$
$857$	40753.7	1.62441	0.812205	+	0.583372i	$0.198267\pi$
$858$	24081.1	0.958175
$859$	39533.0	1.57025	0.785126	−	0.619336i	$-0.212598\pi$
$859$	39533.0	1.57025	0.785126	+	0.619336i	$0.212598\pi$
$860$	−1041.27	−0.0412872
$861$	20999.7	0.831205
$862$	−8711.71	−0.344225
$863$	5901.42	0.232777	0.116388	−	0.993204i	$-0.462868\pi$
$863$	5901.42	0.232777	0.116388	+	0.993204i	$0.462868\pi$
$864$	864.000	0.0340207
$865$	−11483.8	−0.451402
$866$	2037.40	0.0799464
$867$	13323.3	0.521896
$868$	2057.19	0.0804443
$869$	49374.7	1.92741
$870$	7251.83	0.282598
$871$	−24983.7	−0.971917
$872$	9506.02	0.369168
$873$	−14963.9	−0.580128
$874$	10837.7	0.419439
$875$	2073.78	0.0801219
$876$	1157.49	0.0446438
$877$	20170.5	0.776637	0.388319	−	0.921525i	$-0.373056\pi$
$877$	20170.5	0.776637	0.388319	+	0.921525i	$0.373056\pi$
$878$	−12482.4	−0.479794
$879$	3720.79	0.142775
$880$	−3553.04	−0.136106
$881$	−9965.12	−0.381082	−0.190541	−	0.981679i	$-0.561024\pi$
$881$	−9965.12	−0.381082	−0.190541	+	0.981679i	$0.561024\pi$
$882$	1219.74	0.0465654
$883$	26803.4	1.02152	0.510762	−	0.859722i	$-0.329363\pi$
$883$	26803.4	1.02152	0.510762	+	0.859722i	$0.329363\pi$
$884$	−7852.29	−0.298757
$885$	884.607	0.0335997
$886$	−22283.1	−0.844939
$887$	−30659.6	−1.16059	−0.580297	−	0.814405i	$-0.697064\pi$
$887$	−30659.6	−1.16059	−0.580297	+	0.814405i	$0.697064\pi$
$888$	−10527.5	−0.397837
$889$	−13066.7	−0.492963
$890$	3316.21	0.124898
$891$	3597.46	0.135263
$892$	−15977.8	−0.599749
$893$	1246.72	0.0467187
$894$	10234.7	0.382885
$895$	6423.96	0.239921
$896$	2123.55	0.0791774
$897$	−17461.5	−0.649968
$898$	−17592.4	−0.653750
$899$	7493.56	0.278003
$900$	900.000	0.0333333
$901$	7817.06	0.289039
$902$	−37478.2	−1.38347
$903$	2591.24	0.0954940
$904$	18799.7	0.691667
$905$	−1903.60	−0.0699203
$906$	4469.43	0.163893
$907$	−1038.62	−0.0380230	−0.0190115	−	0.999819i	$-0.506052\pi$
$907$	−1038.62	−0.0380230	−0.0190115	+	0.999819i	$0.506052\pi$
$908$	7557.69	0.276223
$909$	13311.4	0.485710
$910$	−14992.3	−0.546141
$911$	2534.63	0.0921801	0.0460901	−	0.998937i	$-0.485324\pi$
$911$	2534.63	0.0921801	0.0460901	+	0.998937i	$0.485324\pi$
$912$	4038.33	0.146626
$913$	22610.5	0.819603
$914$	−3654.89	−0.132268
$915$	13843.6	0.500169
$916$	26437.5	0.953624
$917$	−33017.0	−1.18901
$918$	−1173.05	−0.0421747
$919$	27501.8	0.987159	0.493580	−	0.869701i	$-0.335688\pi$
$919$	27501.8	0.987159	0.493580	+	0.869701i	$0.335688\pi$
$920$	2576.35	0.0923259
$921$	17938.0	0.641776
$922$	6917.45	0.247087
$923$	43995.7	1.56894
$924$	8841.89	0.314802
$925$	−10966.1	−0.389799
$926$	−16939.6	−0.601156
$927$	−9920.91	−0.351505
$928$	7735.29	0.273624
$929$	15131.8	0.534402	0.267201	−	0.963641i	$-0.413901\pi$
$929$	15131.8	0.534402	0.267201	+	0.963641i	$0.413901\pi$
$930$	930.000	0.0327913
$931$	5701.05	0.200692
$932$	−13325.5	−0.468338
$933$	7119.10	0.249806
$934$	28250.2	0.989694
$935$	4823.95	0.168727
$936$	−6506.48	−0.227213
$937$	12916.4	0.450333	0.225166	−	0.974320i	$-0.427707\pi$
$937$	12916.4	0.450333	0.225166	+	0.974320i	$0.427707\pi$
$938$	−9173.30	−0.319317
$939$	−24072.4	−0.836605
$940$	296.372	0.0102836
$941$	−278.273	−0.00964022	−0.00482011	−	0.999988i	$-0.501534\pi$
$941$	−278.273	−0.00964022	−0.00482011	+	0.999988i	$0.501534\pi$
$942$	−5614.60	−0.194197
$943$	27175.9	0.938461
$944$	943.581	0.0325328
$945$	−2239.69	−0.0770973
$946$	−4624.60	−0.158941
$947$	−25547.5	−0.876645	−0.438322	−	0.898818i	$-0.644427\pi$
$947$	−25547.5	−0.876645	−0.438322	+	0.898818i	$0.644427\pi$
$948$	−13340.6	−0.457049
$949$	−8716.66	−0.298161
$950$	4206.60	0.143663
$951$	−5012.05	−0.170901
$952$	−2883.14	−0.0981545
$953$	22161.8	0.753297	0.376649	−	0.926356i	$-0.377076\pi$
$953$	22161.8	0.753297	0.376649	+	0.926356i	$0.377076\pi$
$954$	6477.29	0.219822
$955$	−15335.6	−0.519633
$956$	6818.02	0.230659
$957$	32207.6	1.08790
$958$	19979.6	0.673812
$959$	−31603.9	−1.06417
$960$	960.000	0.0322749
$961$	961.000	0.0322581
$962$	79278.8	2.65702
$963$	5345.71	0.178882
$964$	23876.9	0.797743
$965$	6089.65	0.203143
$966$	−6411.36	−0.213542
$967$	42977.8	1.42924	0.714619	−	0.699514i	$-0.246600\pi$
$967$	42977.8	1.42924	0.714619	+	0.699514i	$0.246600\pi$
$968$	−5132.16	−0.170407
$969$	−5482.83	−0.181769
$970$	−16626.6	−0.550358
$971$	−47904.8	−1.58325	−0.791627	−	0.611005i	$-0.790766\pi$
$971$	−47904.8	−1.58325	−0.791627	+	0.611005i	$0.790766\pi$
$972$	−972.000	−0.0320750
$973$	−31341.4	−1.03264
$974$	−15584.3	−0.512683
$975$	−6777.59	−0.222622
$976$	14766.5	0.484286
$977$	−26813.5	−0.878035	−0.439018	−	0.898478i	$-0.644673\pi$
$977$	−26813.5	−0.878035	−0.439018	+	0.898478i	$0.644673\pi$
$978$	12028.5	0.393282
$979$	14728.3	0.480815
$980$	1355.26	0.0441759
$981$	−10694.3	−0.348055
$982$	−38563.8	−1.25318
$983$	−29353.1	−0.952411	−0.476205	−	0.879334i	$-0.657988\pi$
$983$	−29353.1	−0.952411	−0.476205	+	0.879334i	$0.657988\pi$
$984$	10126.3	0.328063
$985$	−23373.3	−0.756078
$986$	−10502.2	−0.339206
$987$	−737.534	−0.0237852
$988$	−30411.3	−0.979263
$989$	3353.35	0.107816
$990$	3997.18	0.128322
$991$	21234.1	0.680649	0.340325	−	0.940308i	$-0.389463\pi$
$991$	21234.1	0.680649	0.340325	+	0.940308i	$0.389463\pi$
$992$	992.000	0.0317500
$993$	10847.8	0.346672
$994$	16154.0	0.515466
$995$	26109.2	0.831875
$996$	−6109.15	−0.194353
$997$	−38254.9	−1.21519	−0.607595	−	0.794247i	$-0.707866\pi$
$997$	−38254.9	−1.21519	−0.607595	+	0.794247i	$0.707866\pi$
$998$	−38015.7	−1.20578
$999$	11843.4	0.375084

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	930.4.a.j.1.2	✓	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
930.4.a.j.1.2	✓	4	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:	\( N \)	\(=\)	\( 930 = 2 \cdot 3 \cdot 5 \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	930.a (trivial)

\(f(q)\)	\(=\)	\(q-2.00000 q^{2} -3.00000 q^{3} +4.00000 q^{4} -5.00000 q^{5} +6.00000 q^{6} -16.5903 q^{7} -8.00000 q^{8} +9.00000 q^{9} +O(q^{10})\)	\(q-2.00000 q^{2} -3.00000 q^{3} +4.00000 q^{4} -5.00000 q^{5} +6.00000 q^{6} -16.5903 q^{7} -8.00000 q^{8} +9.00000 q^{9} +10.0000 q^{10} +44.4131 q^{11} -12.0000 q^{12} +90.3678 q^{13} +33.1805 q^{14} +15.0000 q^{15} +16.0000 q^{16} -21.7231 q^{17} -18.0000 q^{18} -84.1320 q^{19} -20.0000 q^{20} +49.7708 q^{21} -88.8261 q^{22} +64.4088 q^{23} +24.0000 q^{24} +25.0000 q^{25} -180.736 q^{26} -27.0000 q^{27} -66.3610 q^{28} -241.728 q^{29} -30.0000 q^{30} -31.0000 q^{31} -32.0000 q^{32} -133.239 q^{33} +43.4463 q^{34} +82.9513 q^{35} +36.0000 q^{36} -438.645 q^{37} +168.264 q^{38} -271.103 q^{39} +40.0000 q^{40} +421.928 q^{41} -99.5416 q^{42} +52.0635 q^{43} +177.652 q^{44} -45.0000 q^{45} -128.818 q^{46} -14.8186 q^{47} -48.0000 q^{48} -67.7632 q^{49} -50.0000 q^{50} +65.1694 q^{51} +361.471 q^{52} -359.850 q^{53} +54.0000 q^{54} -222.065 q^{55} +132.722 q^{56} +252.396 q^{57} +483.456 q^{58} +58.9738 q^{59} +60.0000 q^{60} +922.905 q^{61} +62.0000 q^{62} -149.312 q^{63} +64.0000 q^{64} -451.839 q^{65} +266.478 q^{66} -276.466 q^{67} -86.8925 q^{68} -193.226 q^{69} -165.903 q^{70} +486.851 q^{71} -72.0000 q^{72} -96.4576 q^{73} +877.291 q^{74} -75.0000 q^{75} -336.528 q^{76} -736.824 q^{77} +542.207 q^{78} +1111.72 q^{79} -80.0000 q^{80} +81.0000 q^{81} -843.856 q^{82} +509.095 q^{83} +199.083 q^{84} +108.616 q^{85} -104.127 q^{86} +725.183 q^{87} -355.304 q^{88} +331.621 q^{89} +90.0000 q^{90} -1499.23 q^{91} +257.635 q^{92} +93.0000 q^{93} +29.6372 q^{94} +420.660 q^{95} +96.0000 q^{96} -1662.66 q^{97} +135.526 q^{98} +399.718 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 8 q^{2} - 12 q^{3} + 16 q^{4} - 20 q^{5} + 24 q^{6} - 7 q^{7} - 32 q^{8} + 36 q^{9}+O(q^{10}) \) 4 * q - 8 * q^2 - 12 * q^3 + 16 * q^4 - 20 * q^5 + 24 * q^6 - 7 * q^7 - 32 * q^8 + 36 * q^9	\( 4 q - 8 q^{2} - 12 q^{3} + 16 q^{4} - 20 q^{5} + 24 q^{6} - 7 q^{7} - 32 q^{8} + 36 q^{9} + 40 q^{10} + 43 q^{11} - 48 q^{12} - 32 q^{13} + 14 q^{14} + 60 q^{15} + 64 q^{16} - 26 q^{17} - 72 q^{18} - 153 q^{19} - 80 q^{20} + 21 q^{21} - 86 q^{22} + 135 q^{23} + 96 q^{24} + 100 q^{25} + 64 q^{26} - 108 q^{27} - 28 q^{28} + 78 q^{29} - 120 q^{30} - 124 q^{31} - 128 q^{32} - 129 q^{33} + 52 q^{34} + 35 q^{35} + 144 q^{36} - 682 q^{37} + 306 q^{38} + 96 q^{39} + 160 q^{40} + 106 q^{41} - 42 q^{42} - 829 q^{43} + 172 q^{44} - 180 q^{45} - 270 q^{46} - 454 q^{47} - 192 q^{48} - 299 q^{49} - 200 q^{50} + 78 q^{51} - 128 q^{52} - 103 q^{53} + 216 q^{54} - 215 q^{55} + 56 q^{56} + 459 q^{57} - 156 q^{58} + 1042 q^{59} + 240 q^{60} - 100 q^{61} + 248 q^{62} - 63 q^{63} + 256 q^{64} + 160 q^{65} + 258 q^{66} - 742 q^{67} - 104 q^{68} - 405 q^{69} - 70 q^{70} + 969 q^{71} - 288 q^{72} - 845 q^{73} + 1364 q^{74} - 300 q^{75} - 612 q^{76} - 241 q^{77} - 192 q^{78} + 1215 q^{79} - 320 q^{80} + 324 q^{81} - 212 q^{82} - 92 q^{83} + 84 q^{84} + 130 q^{85} + 1658 q^{86} - 234 q^{87} - 344 q^{88} + 2349 q^{89} + 360 q^{90} - 1142 q^{91} + 540 q^{92} + 372 q^{93} + 908 q^{94} + 765 q^{95} + 384 q^{96} - 1228 q^{97} + 598 q^{98} + 387 q^{99}+O(q^{100}) \) 4 * q - 8 * q^2 - 12 * q^3 + 16 * q^4 - 20 * q^5 + 24 * q^6 - 7 * q^7 - 32 * q^8 + 36 * q^9 + 40 * q^10 + 43 * q^11 - 48 * q^12 - 32 * q^13 + 14 * q^14 + 60 * q^15 + 64 * q^16 - 26 * q^17 - 72 * q^18 - 153 * q^19 - 80 * q^20 + 21 * q^21 - 86 * q^22 + 135 * q^23 + 96 * q^24 + 100 * q^25 + 64 * q^26 - 108 * q^27 - 28 * q^28 + 78 * q^29 - 120 * q^30 - 124 * q^31 - 128 * q^32 - 129 * q^33 + 52 * q^34 + 35 * q^35 + 144 * q^36 - 682 * q^37 + 306 * q^38 + 96 * q^39 + 160 * q^40 + 106 * q^41 - 42 * q^42 - 829 * q^43 + 172 * q^44 - 180 * q^45 - 270 * q^46 - 454 * q^47 - 192 * q^48 - 299 * q^49 - 200 * q^50 + 78 * q^51 - 128 * q^52 - 103 * q^53 + 216 * q^54 - 215 * q^55 + 56 * q^56 + 459 * q^57 - 156 * q^58 + 1042 * q^59 + 240 * q^60 - 100 * q^61 + 248 * q^62 - 63 * q^63 + 256 * q^64 + 160 * q^65 + 258 * q^66 - 742 * q^67 - 104 * q^68 - 405 * q^69 - 70 * q^70 + 969 * q^71 - 288 * q^72 - 845 * q^73 + 1364 * q^74 - 300 * q^75 - 612 * q^76 - 241 * q^77 - 192 * q^78 + 1215 * q^79 - 320 * q^80 + 324 * q^81 - 212 * q^82 - 92 * q^83 + 84 * q^84 + 130 * q^85 + 1658 * q^86 - 234 * q^87 - 344 * q^88 + 2349 * q^89 + 360 * q^90 - 1142 * q^91 + 540 * q^92 + 372 * q^93 + 908 * q^94 + 765 * q^95 + 384 * q^96 - 1228 * q^97 + 598 * q^98 + 387 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more