LMFDB - Embedded newform 117.4.a.e.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [117,4,Mod(1,117)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(117, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("117.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 117 = 3^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	117.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:	$6.90322347067$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{7}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 7 $ x^2 - 7
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$2.64575$ of defining polynomial
Character	$\chi$	$=$	117.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.64575 q^{2} -1.00000 q^{4} -10.5830 q^{5} -22.0000 q^{7} -23.8118 q^{8} +O(q^{10})$	\(q+2.64575 q^{2} -1.00000 q^{4} -10.5830 q^{5} -22.0000 q^{7} -23.8118 q^{8} -28.0000 q^{10} -5.29150 q^{11} +13.0000 q^{13} -58.2065 q^{14} -55.0000 q^{16} +116.413 q^{17} -126.000 q^{19} +10.5830 q^{20} -14.0000 q^{22} +31.7490 q^{23} -13.0000 q^{25} +34.3948 q^{26} +22.0000 q^{28} +52.9150 q^{29} -182.000 q^{31} +44.9778 q^{32} +308.000 q^{34} +232.826 q^{35} -86.0000 q^{37} -333.365 q^{38} +252.000 q^{40} -444.486 q^{41} +96.0000 q^{43} +5.29150 q^{44} +84.0000 q^{46} +365.114 q^{47} +141.000 q^{49} -34.3948 q^{50} -13.0000 q^{52} -190.494 q^{53} +56.0000 q^{55} +523.859 q^{56} +140.000 q^{58} -587.357 q^{59} +574.000 q^{61} -481.527 q^{62} +559.000 q^{64} -137.579 q^{65} -530.000 q^{67} -116.413 q^{68} +616.000 q^{70} +809.600 q^{71} -154.000 q^{73} -227.535 q^{74} +126.000 q^{76} +116.413 q^{77} -460.000 q^{79} +582.065 q^{80} -1176.00 q^{82} -322.782 q^{83} -1232.00 q^{85} +253.992 q^{86} +126.000 q^{88} +1439.29 q^{89} -286.000 q^{91} -31.7490 q^{92} +966.000 q^{94} +1333.46 q^{95} +70.0000 q^{97} +373.051 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{4} - 44 q^{7}+O(q^{10}) $ 2 * q - 2 * q^4 - 44 * q^7	$ 2 q - 2 q^{4} - 44 q^{7} - 56 q^{10} + 26 q^{13} - 110 q^{16} - 252 q^{19} - 28 q^{22} - 26 q^{25} + 44 q^{28} - 364 q^{31} + 616 q^{34} - 172 q^{37} + 504 q^{40} + 192 q^{43} + 168 q^{46} + 282 q^{49} - 26 q^{52} + 112 q^{55} + 280 q^{58} + 1148 q^{61} + 1118 q^{64} - 1060 q^{67} + 1232 q^{70} - 308 q^{73} + 252 q^{76} - 920 q^{79} - 2352 q^{82} - 2464 q^{85} + 252 q^{88} - 572 q^{91} + 1932 q^{94} + 140 q^{97}+O(q^{100}) $ 2 * q - 2 * q^4 - 44 * q^7 - 56 * q^10 + 26 * q^13 - 110 * q^16 - 252 * q^19 - 28 * q^22 - 26 * q^25 + 44 * q^28 - 364 * q^31 + 616 * q^34 - 172 * q^37 + 504 * q^40 + 192 * q^43 + 168 * q^46 + 282 * q^49 - 26 * q^52 + 112 * q^55 + 280 * q^58 + 1148 * q^61 + 1118 * q^64 - 1060 * q^67 + 1232 * q^70 - 308 * q^73 + 252 * q^76 - 920 * q^79 - 2352 * q^82 - 2464 * q^85 + 252 * q^88 - 572 * q^91 + 1932 * q^94 + 140 * q^97

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 $

$2$	2.64575	0.935414	0.467707	−	0.883883i	$-0.345080\pi$
$2$	2.64575	0.935414	0.467707	+	0.883883i	$0.345080\pi$
$3$	0	0
$3$	0	0
$4$	−1.00000	−0.125000
$5$	−10.5830	−0.946573	−0.473286	−	0.880909i	$-0.656932\pi$
$5$	−10.5830	−0.946573	−0.473286	+	0.880909i	$0.656932\pi$
$6$	0	0
$7$	−22.0000	−1.18789	−0.593944	−	0.804506i	$-0.702430\pi$
$7$	−22.0000	−1.18789	−0.593944	+	0.804506i	$0.702430\pi$
$8$	−23.8118	−1.05234
$9$	0	0
$10$	−28.0000	−0.885438
$11$	−5.29150	−0.145041	−0.0725204	−	0.997367i	$-0.523104\pi$
$11$	−5.29150	−0.145041	−0.0725204	+	0.997367i	$0.523104\pi$
$12$	0	0
$13$	13.0000	0.277350
$13$	13.0000	0.277350
$14$	−58.2065	−1.11117
$15$	0	0
$16$	−55.0000	−0.859375
$17$	116.413	1.66084	0.830421	−	0.557136i	$-0.188100\pi$
$17$	116.413	1.66084	0.830421	+	0.557136i	$0.188100\pi$
$18$	0	0
$19$	−126.000	−1.52139	−0.760694	−	0.649110i	$-0.775141\pi$
$19$	−126.000	−1.52139	−0.760694	+	0.649110i	$0.775141\pi$
$20$	10.5830	0.118322
$21$	0	0
$22$	−14.0000	−0.135673
$23$	31.7490	0.287832	0.143916	−	0.989590i	$-0.454031\pi$
$23$	31.7490	0.287832	0.143916	+	0.989590i	$0.454031\pi$
$24$	0	0
$25$	−13.0000	−0.104000
$26$	34.3948	0.259437
$27$	0	0
$28$	22.0000	0.148486
$29$	52.9150	0.338830	0.169415	−	0.985545i	$-0.445812\pi$
$29$	52.9150	0.338830	0.169415	+	0.985545i	$0.445812\pi$
$30$	0	0
$31$	−182.000	−1.05446	−0.527228	−	0.849724i	$-0.676769\pi$
$31$	−182.000	−1.05446	−0.527228	+	0.849724i	$0.676769\pi$
$32$	44.9778	0.248469
$33$	0	0
$34$	308.000	1.55358
$35$	232.826	1.12442
$36$	0	0
$37$	−86.0000	−0.382117	−0.191058	−	0.981579i	$-0.561192\pi$
$37$	−86.0000	−0.382117	−0.191058	+	0.981579i	$0.561192\pi$
$38$	−333.365	−1.42313
$39$	0	0
$40$	252.000	0.996117
$41$	−444.486	−1.69310	−0.846550	−	0.532310i	$-0.821324\pi$
$41$	−444.486	−1.69310	−0.846550	+	0.532310i	$0.821324\pi$
$42$	0	0
$43$	96.0000	0.340462	0.170231	−	0.985404i	$-0.445549\pi$
$43$	96.0000	0.340462	0.170231	+	0.985404i	$0.445549\pi$
$44$	5.29150	0.0181301
$45$	0	0
$46$	84.0000	0.269242
$47$	365.114	1.13313	0.566567	−	0.824016i	$-0.308271\pi$
$47$	365.114	1.13313	0.566567	+	0.824016i	$0.308271\pi$
$48$	0	0
$49$	141.000	0.411079
$50$	−34.3948	−0.0972831
$51$	0	0
$52$	−13.0000	−0.0346688
$53$	−190.494	−0.493705	−0.246853	−	0.969053i	$-0.579396\pi$
$53$	−190.494	−0.493705	−0.246853	+	0.969053i	$0.579396\pi$
$54$	0	0
$55$	56.0000	0.137292
$56$	523.859	1.25006
$57$	0	0
$58$	140.000	0.316947
$59$	−587.357	−1.29606	−0.648028	−	0.761616i	$-0.724406\pi$
$59$	−587.357	−1.29606	−0.648028	+	0.761616i	$0.724406\pi$
$60$	0	0
$61$	574.000	1.20481	0.602403	−	0.798192i	$-0.294210\pi$
$61$	574.000	1.20481	0.602403	+	0.798192i	$0.294210\pi$
$62$	−481.527	−0.986354
$63$	0	0
$64$	559.000	1.09180
$65$	−137.579	−0.262532
$66$	0	0
$67$	−530.000	−0.966415	−0.483208	−	0.875506i	$-0.660528\pi$
$67$	−530.000	−0.966415	−0.483208	+	0.875506i	$0.660528\pi$
$68$	−116.413	−0.207605
$69$	0	0
$70$	616.000	1.05180
$71$	809.600	1.35327	0.676633	−	0.736321i	$-0.263439\pi$
$71$	809.600	1.35327	0.676633	+	0.736321i	$0.263439\pi$
$72$	0	0
$73$	−154.000	−0.246909	−0.123454	−	0.992350i	$-0.539397\pi$
$73$	−154.000	−0.246909	−0.123454	+	0.992350i	$0.539397\pi$
$74$	−227.535	−0.357437
$75$	0	0
$76$	126.000	0.190174
$77$	116.413	0.172292
$78$	0	0
$79$	−460.000	−0.655114	−0.327557	−	0.944831i	$-0.606225\pi$
$79$	−460.000	−0.655114	−0.327557	+	0.944831i	$0.606225\pi$
$80$	582.065	0.813461
$81$	0	0
$82$	−1176.00	−1.58375
$83$	−322.782	−0.426866	−0.213433	−	0.976958i	$-0.568465\pi$
$83$	−322.782	−0.426866	−0.213433	+	0.976958i	$0.568465\pi$
$84$	0	0
$85$	−1232.00	−1.57211
$86$	253.992	0.318473
$87$	0	0
$88$	126.000	0.152632
$89$	1439.29	1.71421	0.857103	−	0.515145i	$-0.172262\pi$
$89$	1439.29	1.71421	0.857103	+	0.515145i	$0.172262\pi$
$90$	0	0
$91$	−286.000	−0.329461
$92$	−31.7490	−0.0359790
$93$	0	0
$94$	966.000	1.05995
$95$	1333.46	1.44010
$96$	0	0
$97$	70.0000	0.0732724	0.0366362	−	0.999329i	$-0.488336\pi$
$97$	70.0000	0.0732724	0.0366362	+	0.999329i	$0.488336\pi$
$98$	373.051	0.384529
$99$	0	0
$100$	13.0000	0.0130000
$101$	−1460.45	−1.43882	−0.719409	−	0.694586i	$-0.755587\pi$
$101$	−1460.45	−1.43882	−0.719409	+	0.694586i	$0.755587\pi$
$102$	0	0
$103$	−1428.00	−1.36607	−0.683034	−	0.730387i	$-0.739340\pi$
$103$	−1428.00	−1.36607	−0.683034	+	0.730387i	$0.739340\pi$
$104$	−309.553	−0.291867
$105$	0	0
$106$	−504.000	−0.461819
$107$	−1619.20	−1.46293	−0.731467	−	0.681877i	$-0.761164\pi$
$107$	−1619.20	−1.46293	−0.731467	+	0.681877i	$0.761164\pi$
$108$	0	0
$109$	−338.000	−0.297014	−0.148507	−	0.988911i	$-0.547447\pi$
$109$	−338.000	−0.297014	−0.148507	+	0.988911i	$0.547447\pi$
$110$	148.162	0.128425
$111$	0	0
$112$	1210.00	1.02084
$113$	−1682.70	−1.40084	−0.700420	−	0.713731i	$-0.747004\pi$
$113$	−1682.70	−1.40084	−0.700420	+	0.713731i	$0.747004\pi$
$114$	0	0
$115$	−336.000	−0.272454
$116$	−52.9150	−0.0423538
$117$	0	0
$118$	−1554.00	−1.21235
$119$	−2561.09	−1.97289
$120$	0	0
$121$	−1303.00	−0.978963
$122$	1518.66	1.12699
$123$	0	0
$124$	182.000	0.131807
$125$	1460.45	1.04502
$126$	0	0
$127$	−376.000	−0.262713	−0.131357	−	0.991335i	$-0.541933\pi$
$127$	−376.000	−0.262713	−0.131357	+	0.991335i	$0.541933\pi$
$128$	1119.15	0.772813
$129$	0	0
$130$	−364.000	−0.245576
$131$	−687.895	−0.458792	−0.229396	−	0.973333i	$-0.573675\pi$
$131$	−687.895	−0.458792	−0.229396	+	0.973333i	$0.573675\pi$
$132$	0	0
$133$	2772.00	1.80724
$134$	−1402.25	−0.903998
$135$	0	0
$136$	−2772.00	−1.74777
$137$	1396.96	0.871168	0.435584	−	0.900148i	$-0.356542\pi$
$137$	1396.96	0.871168	0.435584	+	0.900148i	$0.356542\pi$
$138$	0	0
$139$	2100.00	1.28144	0.640718	−	0.767776i	$-0.278637\pi$
$139$	2100.00	1.28144	0.640718	+	0.767776i	$0.278637\pi$
$140$	−232.826	−0.140553
$141$	0	0
$142$	2142.00	1.26586
$143$	−68.7895	−0.0402271
$144$	0	0
$145$	−560.000	−0.320727
$146$	−407.446	−0.230962
$147$	0	0
$148$	86.0000	0.0477646
$149$	−2000.19	−1.09974	−0.549872	−	0.835249i	$-0.685323\pi$
$149$	−2000.19	−1.09974	−0.549872	+	0.835249i	$0.685323\pi$
$150$	0	0
$151$	3526.00	1.90028	0.950138	−	0.311828i	$-0.100941\pi$
$151$	3526.00	1.90028	0.950138	+	0.311828i	$0.100941\pi$
$152$	3000.28	1.60102
$153$	0	0
$154$	308.000	0.161165
$155$	1926.11	0.998120
$156$	0	0
$157$	3066.00	1.55856	0.779278	−	0.626678i	$-0.215586\pi$
$157$	3066.00	1.55856	0.779278	+	0.626678i	$0.215586\pi$
$158$	−1217.05	−0.612803
$159$	0	0
$160$	−476.000	−0.235194
$161$	−698.478	−0.341912
$162$	0	0
$163$	−3442.00	−1.65398	−0.826988	−	0.562219i	$-0.809948\pi$
$163$	−3442.00	−1.65398	−0.826988	+	0.562219i	$0.809948\pi$
$164$	444.486	0.211637
$165$	0	0
$166$	−854.000	−0.399297
$167$	2693.37	1.24802	0.624011	−	0.781416i	$-0.285502\pi$
$167$	2693.37	1.24802	0.624011	+	0.781416i	$0.285502\pi$
$168$	0	0
$169$	169.000	0.0769231
$170$	−3259.57	−1.47057
$171$	0	0
$172$	−96.0000	−0.0425577
$173$	3492.39	1.53481	0.767404	−	0.641164i	$-0.221548\pi$
$173$	3492.39	1.53481	0.767404	+	0.641164i	$0.221548\pi$
$174$	0	0
$175$	286.000	0.123540
$176$	291.033	0.124644
$177$	0	0
$178$	3808.00	1.60349
$179$	169.328	0.0707049	0.0353524	−	0.999375i	$-0.488745\pi$
$179$	169.328	0.0707049	0.0353524	+	0.999375i	$0.488745\pi$
$180$	0	0
$181$	3374.00	1.38557	0.692783	−	0.721146i	$-0.256384\pi$
$181$	3374.00	1.38557	0.692783	+	0.721146i	$0.256384\pi$
$182$	−756.685	−0.308182
$183$	0	0
$184$	−756.000	−0.302897
$185$	910.138	0.361701
$186$	0	0
$187$	−616.000	−0.240890
$188$	−365.114	−0.141642
$189$	0	0
$190$	3528.00	1.34709
$191$	−1185.30	−0.449032	−0.224516	−	0.974470i	$-0.572080\pi$
$191$	−1185.30	−0.449032	−0.224516	+	0.974470i	$0.572080\pi$
$192$	0	0
$193$	−1542.00	−0.575107	−0.287553	−	0.957765i	$-0.592842\pi$
$193$	−1542.00	−0.575107	−0.287553	+	0.957765i	$0.592842\pi$
$194$	185.203	0.0685401
$195$	0	0
$196$	−141.000	−0.0513848
$197$	2127.18	0.769318	0.384659	−	0.923059i	$-0.374319\pi$
$197$	2127.18	0.769318	0.384659	+	0.923059i	$0.374319\pi$
$198$	0	0
$199$	−952.000	−0.339123	−0.169562	−	0.985520i	$-0.554235\pi$
$199$	−952.000	−0.339123	−0.169562	+	0.985520i	$0.554235\pi$
$200$	309.553	0.109443
$201$	0	0
$202$	−3864.00	−1.34589
$203$	−1164.13	−0.402492
$204$	0	0
$205$	4704.00	1.60264
$206$	−3778.13	−1.27784
$207$	0	0
$208$	−715.000	−0.238348
$209$	666.729	0.220663
$210$	0	0
$211$	−1640.00	−0.535082	−0.267541	−	0.963547i	$-0.586211\pi$
$211$	−1640.00	−0.535082	−0.267541	+	0.963547i	$0.586211\pi$
$212$	190.494	0.0617132
$213$	0	0
$214$	−4284.00	−1.36845
$215$	−1015.97	−0.322272
$216$	0	0
$217$	4004.00	1.25258
$218$	−894.264	−0.277831
$219$	0	0
$220$	−56.0000	−0.0171615
$221$	1513.37	0.460635
$222$	0	0
$223$	−4886.00	−1.46722	−0.733612	−	0.679569i	$-0.762167\pi$
$223$	−4886.00	−1.46722	−0.733612	+	0.679569i	$0.762167\pi$
$224$	−989.511	−0.295154
$225$	0	0
$226$	−4452.00	−1.31037
$227$	1867.90	0.546154	0.273077	−	0.961992i	$-0.411959\pi$
$227$	1867.90	0.546154	0.273077	+	0.961992i	$0.411959\pi$
$228$	0	0
$229$	5558.00	1.60386	0.801928	−	0.597421i	$-0.203808\pi$
$229$	5558.00	1.60386	0.801928	+	0.597421i	$0.203808\pi$
$230$	−888.972	−0.254857
$231$	0	0
$232$	−1260.00	−0.356565
$233$	−3577.06	−1.00575	−0.502877	−	0.864358i	$-0.667725\pi$
$233$	−3577.06	−1.00575	−0.502877	+	0.864358i	$0.667725\pi$
$234$	0	0
$235$	−3864.00	−1.07259
$236$	587.357	0.162007
$237$	0	0
$238$	−6776.00	−1.84547
$239$	−2936.78	−0.794832	−0.397416	−	0.917639i	$-0.630093\pi$
$239$	−2936.78	−0.794832	−0.397416	+	0.917639i	$0.630093\pi$
$240$	0	0
$241$	−602.000	−0.160906	−0.0804528	−	0.996758i	$-0.525637\pi$
$241$	−602.000	−0.160906	−0.0804528	+	0.996758i	$0.525637\pi$
$242$	−3447.41	−0.915736
$243$	0	0
$244$	−574.000	−0.150601
$245$	−1492.20	−0.389116
$246$	0	0
$247$	−1638.00	−0.421957
$248$	4333.74	1.10965
$249$	0	0
$250$	3864.00	0.977523
$251$	−3524.14	−0.886222	−0.443111	−	0.896467i	$-0.646125\pi$
$251$	−3524.14	−0.886222	−0.443111	+	0.896467i	$0.646125\pi$
$252$	0	0
$253$	−168.000	−0.0417473
$254$	−994.802	−0.245746
$255$	0	0
$256$	−1511.00	−0.368896
$257$	2942.08	0.714092	0.357046	−	0.934087i	$-0.383784\pi$
$257$	2942.08	0.714092	0.357046	+	0.934087i	$0.383784\pi$
$258$	0	0
$259$	1892.00	0.453912
$260$	137.579	0.0328165
$261$	0	0
$262$	−1820.00	−0.429160
$263$	857.223	0.200984	0.100492	−	0.994938i	$-0.467958\pi$
$263$	857.223	0.200984	0.100492	+	0.994938i	$0.467958\pi$
$264$	0	0
$265$	2016.00	0.467328
$266$	7334.02	1.69052
$267$	0	0
$268$	530.000	0.120802
$269$	328.073	0.0743605	0.0371802	−	0.999309i	$-0.488162\pi$
$269$	328.073	0.0743605	0.0371802	+	0.999309i	$0.488162\pi$
$270$	0	0
$271$	−2814.00	−0.630769	−0.315384	−	0.948964i	$-0.602133\pi$
$271$	−2814.00	−0.630769	−0.315384	+	0.948964i	$0.602133\pi$
$272$	−6402.72	−1.42729
$273$	0	0
$274$	3696.00	0.814903
$275$	68.7895	0.0150842
$276$	0	0
$277$	−3190.00	−0.691944	−0.345972	−	0.938245i	$-0.612451\pi$
$277$	−3190.00	−0.691944	−0.345972	+	0.938245i	$0.612451\pi$
$278$	5556.08	1.19867
$279$	0	0
$280$	−5544.00	−1.18328
$281$	6116.98	1.29861	0.649303	−	0.760530i	$-0.275061\pi$
$281$	6116.98	1.29861	0.649303	+	0.760530i	$0.275061\pi$
$282$	0	0
$283$	−4788.00	−1.00571	−0.502857	−	0.864370i	$-0.667718\pi$
$283$	−4788.00	−1.00571	−0.502857	+	0.864370i	$0.667718\pi$
$284$	−809.600	−0.169158
$285$	0	0
$286$	−182.000	−0.0376290
$287$	9778.70	2.01121
$288$	0	0
$289$	8639.00	1.75840
$290$	−1481.62	−0.300013
$291$	0	0
$292$	154.000	0.0308636
$293$	−6699.04	−1.33571	−0.667854	−	0.744293i	$-0.732787\pi$
$293$	−6699.04	−1.33571	−0.667854	+	0.744293i	$0.732787\pi$
$294$	0	0
$295$	6216.00	1.22681
$296$	2047.81	0.402117
$297$	0	0
$298$	−5292.00	−1.02872
$299$	412.737	0.0798301
$300$	0	0
$301$	−2112.00	−0.404431
$302$	9328.92	1.77755
$303$	0	0
$304$	6930.00	1.30744
$305$	−6074.65	−1.14044
$306$	0	0
$307$	−406.000	−0.0754777	−0.0377388	−	0.999288i	$-0.512015\pi$
$307$	−406.000	−0.0754777	−0.0377388	+	0.999288i	$0.512015\pi$
$308$	−116.413	−0.0215365
$309$	0	0
$310$	5096.00	0.933656
$311$	8286.49	1.51088	0.755440	−	0.655217i	$-0.227423\pi$
$311$	8286.49	1.51088	0.755440	+	0.655217i	$0.227423\pi$
$312$	0	0
$313$	−5586.00	−1.00875	−0.504376	−	0.863484i	$-0.668277\pi$
$313$	−5586.00	−1.00875	−0.504376	+	0.863484i	$0.668277\pi$
$314$	8111.87	1.45790
$315$	0	0
$316$	460.000	0.0818893
$317$	−8392.32	−1.48694	−0.743470	−	0.668770i	$-0.766821\pi$
$317$	−8392.32	−1.48694	−0.743470	+	0.668770i	$0.766821\pi$
$318$	0	0
$319$	−280.000	−0.0491442
$320$	−5915.90	−1.03347
$321$	0	0
$322$	−1848.00	−0.319829
$323$	−14668.0	−2.52679
$324$	0	0
$325$	−169.000	−0.0288444
$326$	−9106.68	−1.54715
$327$	0	0
$328$	10584.0	1.78172
$329$	−8032.50	−1.34604
$330$	0	0
$331$	−4426.00	−0.734970	−0.367485	−	0.930030i	$-0.619781\pi$
$331$	−4426.00	−0.734970	−0.367485	+	0.930030i	$0.619781\pi$
$332$	322.782	0.0533583
$333$	0	0
$334$	7126.00	1.16742
$335$	5608.99	0.914782
$336$	0	0
$337$	8370.00	1.35295	0.676473	−	0.736467i	$-0.263507\pi$
$337$	8370.00	1.35295	0.676473	+	0.736467i	$0.263507\pi$
$338$	447.132	0.0719549
$339$	0	0
$340$	1232.00	0.196513
$341$	963.053	0.152939
$342$	0	0
$343$	4444.00	0.699573
$344$	−2285.93	−0.358282
$345$	0	0
$346$	9240.00	1.43568
$347$	6095.81	0.943056	0.471528	−	0.881851i	$-0.343703\pi$
$347$	6095.81	0.943056	0.471528	+	0.881851i	$0.343703\pi$
$348$	0	0
$349$	4354.00	0.667806	0.333903	−	0.942607i	$-0.391634\pi$
$349$	4354.00	0.667806	0.333903	+	0.942607i	$0.391634\pi$
$350$	756.685	0.115561
$351$	0	0
$352$	−238.000	−0.0360382
$353$	3407.73	0.513810	0.256905	−	0.966437i	$-0.417297\pi$
$353$	3407.73	0.513810	0.256905	+	0.966437i	$0.417297\pi$
$354$	0	0
$355$	−8568.00	−1.28096
$356$	−1439.29	−0.214276
$357$	0	0
$358$	448.000	0.0661384
$359$	−7762.63	−1.14121	−0.570607	−	0.821223i	$-0.693292\pi$
$359$	−7762.63	−1.14121	−0.570607	+	0.821223i	$0.693292\pi$
$360$	0	0
$361$	9017.00	1.31462
$362$	8926.76	1.29608
$363$	0	0
$364$	286.000	0.0411826
$365$	1629.78	0.233717
$366$	0	0
$367$	−7784.00	−1.10714	−0.553572	−	0.832802i	$-0.686735\pi$
$367$	−7784.00	−1.10714	−0.553572	+	0.832802i	$0.686735\pi$
$368$	−1746.20	−0.247355
$369$	0	0
$370$	2408.00	0.338340
$371$	4190.87	0.586467
$372$	0	0
$373$	−8510.00	−1.18132	−0.590658	−	0.806922i	$-0.701132\pi$
$373$	−8510.00	−1.18132	−0.590658	+	0.806922i	$0.701132\pi$
$374$	−1629.78	−0.225332
$375$	0	0
$376$	−8694.00	−1.19244
$377$	687.895	0.0939746
$378$	0	0
$379$	1650.00	0.223627	0.111814	−	0.993729i	$-0.464334\pi$
$379$	1650.00	0.223627	0.111814	+	0.993729i	$0.464334\pi$
$380$	−1333.46	−0.180013
$381$	0	0
$382$	−3136.00	−0.420031
$383$	−8662.19	−1.15566	−0.577829	−	0.816158i	$-0.696100\pi$
$383$	−8662.19	−1.15566	−0.577829	+	0.816158i	$0.696100\pi$
$384$	0	0
$385$	−1232.00	−0.163087
$386$	−4079.75	−0.537963
$387$	0	0
$388$	−70.0000	−0.00915905
$389$	2423.51	0.315879	0.157939	−	0.987449i	$-0.449515\pi$
$389$	2423.51	0.315879	0.157939	+	0.987449i	$0.449515\pi$
$390$	0	0
$391$	3696.00	0.478043
$392$	−3357.46	−0.432595
$393$	0	0
$394$	5628.00	0.719631
$395$	4868.18	0.620114
$396$	0	0
$397$	−1414.00	−0.178757	−0.0893786	−	0.995998i	$-0.528488\pi$
$397$	−1414.00	−0.178757	−0.0893786	+	0.995998i	$0.528488\pi$
$398$	−2518.76	−0.317221
$399$	0	0
$400$	715.000	0.0893750
$401$	−5228.00	−0.651058	−0.325529	−	0.945532i	$-0.605542\pi$
$401$	−5228.00	−0.651058	−0.325529	+	0.945532i	$0.605542\pi$
$402$	0	0
$403$	−2366.00	−0.292454
$404$	1460.45	0.179852
$405$	0	0
$406$	−3080.00	−0.376497
$407$	455.069	0.0554225
$408$	0	0
$409$	5782.00	0.699026	0.349513	−	0.936932i	$-0.386347\pi$
$409$	5782.00	0.699026	0.349513	+	0.936932i	$0.386347\pi$
$410$	12445.6	1.49913
$411$	0	0
$412$	1428.00	0.170759
$413$	12921.8	1.53957
$414$	0	0
$415$	3416.00	0.404060
$416$	584.711	0.0689130
$417$	0	0
$418$	1764.00	0.206412
$419$	−11482.6	−1.33881	−0.669403	−	0.742899i	$-0.733450\pi$
$419$	−11482.6	−1.33881	−0.669403	+	0.742899i	$0.733450\pi$
$420$	0	0
$421$	−14194.0	−1.64317	−0.821583	−	0.570088i	$-0.806909\pi$
$421$	−14194.0	−1.64317	−0.821583	+	0.570088i	$0.806909\pi$
$422$	−4339.03	−0.500523
$423$	0	0
$424$	4536.00	0.519546
$425$	−1513.37	−0.172728
$426$	0	0
$427$	−12628.0	−1.43118
$428$	1619.20	0.182867
$429$	0	0
$430$	−2688.00	−0.301458
$431$	5222.71	0.583687	0.291844	−	0.956466i	$-0.405731\pi$
$431$	5222.71	0.583687	0.291844	+	0.956466i	$0.405731\pi$
$432$	0	0
$433$	−686.000	−0.0761364	−0.0380682	−	0.999275i	$-0.512120\pi$
$433$	−686.000	−0.0761364	−0.0380682	+	0.999275i	$0.512120\pi$
$434$	10593.6	1.17168
$435$	0	0
$436$	338.000	0.0371268
$437$	−4000.38	−0.437904
$438$	0	0
$439$	−1372.00	−0.149162	−0.0745809	−	0.997215i	$-0.523762\pi$
$439$	−1372.00	−0.149162	−0.0745809	+	0.997215i	$0.523762\pi$
$440$	−1333.46	−0.144478
$441$	0	0
$442$	4004.00	0.430884
$443$	2338.84	0.250839	0.125420	−	0.992104i	$-0.459972\pi$
$443$	2338.84	0.250839	0.125420	+	0.992104i	$0.459972\pi$
$444$	0	0
$445$	−15232.0	−1.62262
$446$	−12927.1	−1.37246
$447$	0	0
$448$	−12298.0	−1.29693
$449$	17250.3	1.81312	0.906561	−	0.422074i	$-0.138698\pi$
$449$	17250.3	1.81312	0.906561	+	0.422074i	$0.138698\pi$
$450$	0	0
$451$	2352.00	0.245568
$452$	1682.70	0.175105
$453$	0	0
$454$	4942.00	0.510880
$455$	3026.74	0.311859
$456$	0	0
$457$	−17866.0	−1.82874	−0.914372	−	0.404875i	$-0.867315\pi$
$457$	−17866.0	−1.82874	−0.914372	+	0.404875i	$0.867315\pi$
$458$	14705.1	1.50027
$459$	0	0
$460$	336.000	0.0340567
$461$	2106.02	0.212770	0.106385	−	0.994325i	$-0.466072\pi$
$461$	2106.02	0.212770	0.106385	+	0.994325i	$0.466072\pi$
$462$	0	0
$463$	−13718.0	−1.37695	−0.688477	−	0.725258i	$-0.741720\pi$
$463$	−13718.0	−1.37695	−0.688477	+	0.725258i	$0.741720\pi$
$464$	−2910.33	−0.291182
$465$	0	0
$466$	−9464.00	−0.940797
$467$	4095.62	0.405830	0.202915	−	0.979196i	$-0.434958\pi$
$467$	4095.62	0.405830	0.202915	+	0.979196i	$0.434958\pi$
$468$	0	0
$469$	11660.0	1.14799
$470$	−10223.2	−1.00332
$471$	0	0
$472$	13986.0	1.36389
$473$	−507.984	−0.0493808
$474$	0	0
$475$	1638.00	0.158224
$476$	2561.09	0.246612
$477$	0	0
$478$	−7770.00	−0.743497
$479$	8715.10	0.831322	0.415661	−	0.909520i	$-0.363550\pi$
$479$	8715.10	0.831322	0.415661	+	0.909520i	$0.363550\pi$
$480$	0	0
$481$	−1118.00	−0.105980
$482$	−1592.74	−0.150513
$483$	0	0
$484$	1303.00	0.122370
$485$	−740.810	−0.0693577
$486$	0	0
$487$	4506.00	0.419274	0.209637	−	0.977779i	$-0.432772\pi$
$487$	4506.00	0.419274	0.209637	+	0.977779i	$0.432772\pi$
$488$	−13668.0	−1.26787
$489$	0	0
$490$	−3948.00	−0.363985
$491$	−21621.1	−1.98726	−0.993631	−	0.112683i	$-0.964056\pi$
$491$	−21621.1	−1.98726	−0.993631	+	0.112683i	$0.964056\pi$
$492$	0	0
$493$	6160.00	0.562743
$494$	−4333.74	−0.394705
$495$	0	0
$496$	10010.0	0.906174
$497$	−17811.2	−1.60753
$498$	0	0
$499$	−786.000	−0.0705134	−0.0352567	−	0.999378i	$-0.511225\pi$
$499$	−786.000	−0.0705134	−0.0352567	+	0.999378i	$0.511225\pi$
$500$	−1460.45	−0.130627
$501$	0	0
$502$	−9324.00	−0.828985
$503$	2106.02	0.186685	0.0933426	−	0.995634i	$-0.470245\pi$
$503$	2106.02	0.186685	0.0933426	+	0.995634i	$0.470245\pi$
$504$	0	0
$505$	15456.0	1.36195
$506$	−444.486	−0.0390510
$507$	0	0
$508$	376.000	0.0328392
$509$	8392.32	0.730812	0.365406	−	0.930848i	$-0.380930\pi$
$509$	8392.32	0.730812	0.365406	+	0.930848i	$0.380930\pi$
$510$	0	0
$511$	3388.00	0.293300
$512$	−12951.0	−1.11788
$513$	0	0
$514$	7784.00	0.667972
$515$	15112.5	1.29308
$516$	0	0
$517$	−1932.00	−0.164351
$518$	5005.76	0.424596
$519$	0	0
$520$	3276.00	0.276273
$521$	3905.13	0.328382	0.164191	−	0.986429i	$-0.447499\pi$
$521$	3905.13	0.328382	0.164191	+	0.986429i	$0.447499\pi$
$522$	0	0
$523$	−17668.0	−1.47718	−0.738592	−	0.674152i	$-0.764509\pi$
$523$	−17668.0	−1.47718	−0.738592	+	0.674152i	$0.764509\pi$
$524$	687.895	0.0573489
$525$	0	0
$526$	2268.00	0.188003
$527$	−21187.2	−1.75129
$528$	0	0
$529$	−11159.0	−0.917153
$530$	5333.83	0.437145
$531$	0	0
$532$	−2772.00	−0.225905
$533$	−5778.32	−0.469581
$534$	0	0
$535$	17136.0	1.38477
$536$	12620.2	1.01700
$537$	0	0
$538$	868.000	0.0695579
$539$	−746.102	−0.0596232
$540$	0	0
$541$	−1650.00	−0.131126	−0.0655629	−	0.997848i	$-0.520884\pi$
$541$	−1650.00	−0.131126	−0.0655629	+	0.997848i	$0.520884\pi$
$542$	−7445.14	−0.590030
$543$	0	0
$544$	5236.00	0.412668
$545$	3577.06	0.281145
$546$	0	0
$547$	3796.00	0.296719	0.148359	−	0.988934i	$-0.452601\pi$
$547$	3796.00	0.296719	0.148359	+	0.988934i	$0.452601\pi$
$548$	−1396.96	−0.108896
$549$	0	0
$550$	182.000	0.0141100
$551$	−6667.29	−0.515492
$552$	0	0
$553$	10120.0	0.778203
$554$	−8439.95	−0.647254
$555$	0	0
$556$	−2100.00	−0.160180
$557$	−2063.69	−0.156986	−0.0784930	−	0.996915i	$-0.525011\pi$
$557$	−2063.69	−0.156986	−0.0784930	+	0.996915i	$0.525011\pi$
$558$	0	0
$559$	1248.00	0.0944271
$560$	−12805.4	−0.966301
$561$	0	0
$562$	16184.0	1.21473
$563$	26034.2	1.94886	0.974432	−	0.224683i	$-0.0721346\pi$
$563$	26034.2	1.94886	0.974432	+	0.224683i	$0.0721346\pi$
$564$	0	0
$565$	17808.0	1.32600
$566$	−12667.9	−0.940759
$567$	0	0
$568$	−19278.0	−1.42410
$569$	−3640.55	−0.268225	−0.134112	−	0.990966i	$-0.542818\pi$
$569$	−3640.55	−0.268225	−0.134112	+	0.990966i	$0.542818\pi$
$570$	0	0
$571$	−19612.0	−1.43737	−0.718684	−	0.695337i	$-0.755255\pi$
$571$	−19612.0	−1.43737	−0.718684	+	0.695337i	$0.755255\pi$
$572$	68.7895	0.00502838
$573$	0	0
$574$	25872.0	1.88132
$575$	−412.737	−0.0299345
$576$	0	0
$577$	15722.0	1.13434	0.567171	−	0.823600i	$-0.308038\pi$
$577$	15722.0	1.13434	0.567171	+	0.823600i	$0.308038\pi$
$578$	22856.6	1.64483
$579$	0	0
$580$	560.000	0.0400909
$581$	7101.20	0.507069
$582$	0	0
$583$	1008.00	0.0716074
$584$	3667.01	0.259832
$585$	0	0
$586$	−17724.0	−1.24944
$587$	2725.12	0.191615	0.0958074	−	0.995400i	$-0.469457\pi$
$587$	2725.12	0.191615	0.0958074	+	0.995400i	$0.469457\pi$
$588$	0	0
$589$	22932.0	1.60424
$590$	16446.0	1.14758
$591$	0	0
$592$	4730.00	0.328381
$593$	18012.3	1.24734	0.623672	−	0.781686i	$-0.285640\pi$
$593$	18012.3	1.24734	0.623672	+	0.781686i	$0.285640\pi$
$594$	0	0
$595$	27104.0	1.86749
$596$	2000.19	0.137468
$597$	0	0
$598$	1092.00	0.0746742
$599$	12181.0	0.830891	0.415446	−	0.909618i	$-0.363626\pi$
$599$	12181.0	0.830891	0.415446	+	0.909618i	$0.363626\pi$
$600$	0	0
$601$	5950.00	0.403836	0.201918	−	0.979402i	$-0.435283\pi$
$601$	5950.00	0.403836	0.201918	+	0.979402i	$0.435283\pi$
$602$	−5587.83	−0.378310
$603$	0	0
$604$	−3526.00	−0.237535
$605$	13789.7	0.926660
$606$	0	0
$607$	14168.0	0.947383	0.473691	−	0.880691i	$-0.342921\pi$
$607$	14168.0	0.947383	0.473691	+	0.880691i	$0.342921\pi$
$608$	−5667.20	−0.378019
$609$	0	0
$610$	−16072.0	−1.06678
$611$	4746.48	0.314275
$612$	0	0
$613$	−6326.00	−0.416810	−0.208405	−	0.978043i	$-0.566827\pi$
$613$	−6326.00	−0.416810	−0.208405	+	0.978043i	$0.566827\pi$
$614$	−1074.18	−0.0706029
$615$	0	0
$616$	−2772.00	−0.181310
$617$	8805.06	0.574519	0.287260	−	0.957853i	$-0.407256\pi$
$617$	8805.06	0.574519	0.287260	+	0.957853i	$0.407256\pi$
$618$	0	0
$619$	−24486.0	−1.58994	−0.794972	−	0.606646i	$-0.792515\pi$
$619$	−24486.0	−1.58994	−0.794972	+	0.606646i	$0.792515\pi$
$620$	−1926.11	−0.124765
$621$	0	0
$622$	21924.0	1.41330
$623$	−31664.4	−2.03628
$624$	0	0
$625$	−13831.0	−0.885184
$626$	−14779.2	−0.943601
$627$	0	0
$628$	−3066.00	−0.194820
$629$	−10011.5	−0.634635
$630$	0	0
$631$	22430.0	1.41509	0.707547	−	0.706666i	$-0.249802\pi$
$631$	22430.0	1.41509	0.707547	+	0.706666i	$0.249802\pi$
$632$	10953.4	0.689404
$633$	0	0
$634$	−22204.0	−1.39090
$635$	3979.21	0.248677
$636$	0	0
$637$	1833.00	0.114013
$638$	−740.810	−0.0459702
$639$	0	0
$640$	−11844.0	−0.731524
$641$	27484.1	1.69353	0.846767	−	0.531964i	$-0.178546\pi$
$641$	27484.1	1.69353	0.846767	+	0.531964i	$0.178546\pi$
$642$	0	0
$643$	16478.0	1.01062	0.505310	−	0.862938i	$-0.331378\pi$
$643$	16478.0	1.01062	0.505310	+	0.862938i	$0.331378\pi$
$644$	698.478	0.0427390
$645$	0	0
$646$	−38808.0	−2.36359
$647$	−26563.3	−1.61408	−0.807042	−	0.590494i	$-0.798933\pi$
$647$	−26563.3	−1.61408	−0.807042	+	0.590494i	$0.798933\pi$
$648$	0	0
$649$	3108.00	0.187981
$650$	−447.132	−0.0269815
$651$	0	0
$652$	3442.00	0.206747
$653$	−20287.6	−1.21580	−0.607899	−	0.794014i	$-0.707987\pi$
$653$	−20287.6	−1.21580	−0.607899	+	0.794014i	$0.707987\pi$
$654$	0	0
$655$	7280.00	0.434280
$656$	24446.7	1.45501
$657$	0	0
$658$	−21252.0	−1.25910
$659$	−656.146	−0.0387858	−0.0193929	−	0.999812i	$-0.506173\pi$
$659$	−656.146	−0.0387858	−0.0193929	+	0.999812i	$0.506173\pi$
$660$	0	0
$661$	−14238.0	−0.837812	−0.418906	−	0.908030i	$-0.637586\pi$
$661$	−14238.0	−0.837812	−0.418906	+	0.908030i	$0.637586\pi$
$662$	−11710.1	−0.687501
$663$	0	0
$664$	7686.00	0.449209
$665$	−29336.1	−1.71068
$666$	0	0
$667$	1680.00	0.0975260
$668$	−2693.37	−0.156003
$669$	0	0
$670$	14840.0	0.855700
$671$	−3037.32	−0.174746
$672$	0	0
$673$	−4874.00	−0.279166	−0.139583	−	0.990210i	$-0.544576\pi$
$673$	−4874.00	−0.279166	−0.139583	+	0.990210i	$0.544576\pi$
$674$	22144.9	1.26557
$675$	0	0
$676$	−169.000	−0.00961538
$677$	−21801.0	−1.23764	−0.618818	−	0.785534i	$-0.712388\pi$
$677$	−21801.0	−1.23764	−0.618818	+	0.785534i	$0.712388\pi$
$678$	0	0
$679$	−1540.00	−0.0870394
$680$	29336.1	1.65439
$681$	0	0
$682$	2548.00	0.143062
$683$	−8746.85	−0.490028	−0.245014	−	0.969520i	$-0.578793\pi$
$683$	−8746.85	−0.490028	−0.245014	+	0.969520i	$0.578793\pi$
$684$	0	0
$685$	−14784.0	−0.824624
$686$	11757.7	0.654390
$687$	0	0
$688$	−5280.00	−0.292584
$689$	−2476.42	−0.136929
$690$	0	0
$691$	294.000	0.0161857	0.00809283	−	0.999967i	$-0.497424\pi$
$691$	294.000	0.0161857	0.00809283	+	0.999967i	$0.497424\pi$
$692$	−3492.39	−0.191851
$693$	0	0
$694$	16128.0	0.882148
$695$	−22224.3	−1.21297
$696$	0	0
$697$	−51744.0	−2.81197
$698$	11519.6	0.624675
$699$	0	0
$700$	−286.000	−0.0154425
$701$	−15758.1	−0.849037	−0.424519	−	0.905419i	$-0.639557\pi$
$701$	−15758.1	−0.849037	−0.424519	+	0.905419i	$0.639557\pi$
$702$	0	0
$703$	10836.0	0.581348
$704$	−2957.95	−0.158355
$705$	0	0
$706$	9016.00	0.480626
$707$	32130.0	1.70916
$708$	0	0
$709$	−6722.00	−0.356065	−0.178032	−	0.984025i	$-0.556973\pi$
$709$	−6722.00	−0.356065	−0.178032	+	0.984025i	$0.556973\pi$
$710$	−22668.8	−1.19823
$711$	0	0
$712$	−34272.0	−1.80393
$713$	−5778.32	−0.303506
$714$	0	0
$715$	728.000	0.0380778
$716$	−169.328	−0.00883811
$717$	0	0
$718$	−20538.0	−1.06751
$719$	31.7490	0.00164679	0.000823393	−	1.00000i	$-0.499738\pi$
$719$	31.7490	0.00164679	0.000823393		1.00000i	$0.499738\pi$
$720$	0	0
$721$	31416.0	1.62274
$722$	23856.7	1.22972
$723$	0	0
$724$	−3374.00	−0.173196
$725$	−687.895	−0.0352383
$726$	0	0
$727$	12824.0	0.654217	0.327109	−	0.944987i	$-0.393926\pi$
$727$	12824.0	0.654217	0.327109	+	0.944987i	$0.393926\pi$
$728$	6810.16	0.346705
$729$	0	0
$730$	4312.00	0.218622
$731$	11175.7	0.565453
$732$	0	0
$733$	−29610.0	−1.49205	−0.746023	−	0.665920i	$-0.768039\pi$
$733$	−29610.0	−1.49205	−0.746023	+	0.665920i	$0.768039\pi$
$734$	−20594.5	−1.03564
$735$	0	0
$736$	1428.00	0.0715174
$737$	2804.50	0.140170
$738$	0	0
$739$	−15622.0	−0.777625	−0.388812	−	0.921317i	$-0.627115\pi$
$739$	−15622.0	−0.777625	−0.388812	+	0.921317i	$0.627115\pi$
$740$	−910.138	−0.0452126
$741$	0	0
$742$	11088.0	0.548589
$743$	−8588.11	−0.424047	−0.212024	−	0.977265i	$-0.568005\pi$
$743$	−8588.11	−0.424047	−0.212024	+	0.977265i	$0.568005\pi$
$744$	0	0
$745$	21168.0	1.04099
$746$	−22515.3	−1.10502
$747$	0	0
$748$	616.000	0.0301112
$749$	35622.4	1.73780
$750$	0	0
$751$	29468.0	1.43183	0.715914	−	0.698189i	$-0.246010\pi$
$751$	29468.0	1.43183	0.715914	+	0.698189i	$0.246010\pi$
$752$	−20081.3	−0.973787
$753$	0	0
$754$	1820.00	0.0879052
$755$	−37315.7	−1.79875
$756$	0	0
$757$	−35030.0	−1.68189	−0.840943	−	0.541124i	$-0.817999\pi$
$757$	−35030.0	−1.68189	−0.840943	+	0.541124i	$0.817999\pi$
$758$	4365.49	0.209184
$759$	0	0
$760$	−31752.0	−1.51548
$761$	−22330.1	−1.06369	−0.531844	−	0.846842i	$-0.678501\pi$
$761$	−22330.1	−1.06369	−0.531844	+	0.846842i	$0.678501\pi$
$762$	0	0
$763$	7436.00	0.352819
$764$	1185.30	0.0561290
$765$	0	0
$766$	−22918.0	−1.08102
$767$	−7635.64	−0.359461
$768$	0	0
$769$	−27342.0	−1.28216	−0.641078	−	0.767476i	$-0.721512\pi$
$769$	−27342.0	−1.28216	−0.641078	+	0.767476i	$0.721512\pi$
$770$	−3259.57	−0.152554
$771$	0	0
$772$	1542.00	0.0718883
$773$	11884.7	0.552993	0.276496	−	0.961015i	$-0.410827\pi$
$773$	11884.7	0.552993	0.276496	+	0.961015i	$0.410827\pi$
$774$	0	0
$775$	2366.00	0.109664
$776$	−1666.82	−0.0771076
$777$	0	0
$778$	6412.00	0.295477
$779$	56005.3	2.57586
$780$	0	0
$781$	−4284.00	−0.196279
$782$	9778.70	0.447168
$783$	0	0
$784$	−7755.00	−0.353271
$785$	−32447.5	−1.47529
$786$	0	0
$787$	22666.0	1.02663	0.513314	−	0.858201i	$-0.328418\pi$
$787$	22666.0	1.02663	0.513314	+	0.858201i	$0.328418\pi$
$788$	−2127.18	−0.0961647
$789$	0	0
$790$	12880.0	0.580063
$791$	37019.4	1.66404
$792$	0	0
$793$	7462.00	0.334153
$794$	−3741.09	−0.167212
$795$	0	0
$796$	952.000	0.0423904
$797$	582.065	0.0258693	0.0129346	−	0.999916i	$-0.495883\pi$
$797$	582.065	0.0258693	0.0129346	+	0.999916i	$0.495883\pi$
$798$	0	0
$799$	42504.0	1.88196
$800$	−584.711	−0.0258408
$801$	0	0
$802$	−13832.0	−0.609009
$803$	814.891	0.0358118
$804$	0	0
$805$	7392.00	0.323644
$806$	−6259.85	−0.273565
$807$	0	0
$808$	34776.0	1.51413
$809$	793.725	0.0344943	0.0172472	−	0.999851i	$-0.494510\pi$
$809$	793.725	0.0344943	0.0172472	+	0.999851i	$0.494510\pi$
$810$	0	0
$811$	−9478.00	−0.410379	−0.205190	−	0.978722i	$-0.565781\pi$
$811$	−9478.00	−0.410379	−0.205190	+	0.978722i	$0.565781\pi$
$812$	1164.13	0.0503115
$813$	0	0
$814$	1204.00	0.0518430
$815$	36426.7	1.56561
$816$	0	0
$817$	−12096.0	−0.517975
$818$	15297.7	0.653879
$819$	0	0
$820$	−4704.00	−0.200330
$821$	3227.82	0.137213	0.0686063	−	0.997644i	$-0.478145\pi$
$821$	3227.82	0.137213	0.0686063	+	0.997644i	$0.478145\pi$
$822$	0	0
$823$	40476.0	1.71434	0.857172	−	0.515031i	$-0.172219\pi$
$823$	40476.0	1.71434	0.857172	+	0.515031i	$0.172219\pi$
$824$	34003.2	1.43757
$825$	0	0
$826$	34188.0	1.44014
$827$	7169.99	0.301481	0.150741	−	0.988573i	$-0.451834\pi$
$827$	7169.99	0.301481	0.150741	+	0.988573i	$0.451834\pi$
$828$	0	0
$829$	27482.0	1.15137	0.575687	−	0.817670i	$-0.304735\pi$
$829$	27482.0	1.15137	0.575687	+	0.817670i	$0.304735\pi$
$830$	9037.89	0.377963
$831$	0	0
$832$	7267.00	0.302810
$833$	16414.2	0.682737
$834$	0	0
$835$	−28504.0	−1.18134
$836$	−666.729	−0.0275829
$837$	0	0
$838$	−30380.0	−1.25234
$839$	19128.8	0.787126	0.393563	−	0.919298i	$-0.371242\pi$
$839$	19128.8	0.787126	0.393563	+	0.919298i	$0.371242\pi$
$840$	0	0
$841$	−21589.0	−0.885194
$842$	−37553.8	−1.53704
$843$	0	0
$844$	1640.00	0.0668852
$845$	−1788.53	−0.0728133
$846$	0	0
$847$	28666.0	1.16290
$848$	10477.2	0.424278
$849$	0	0
$850$	−4004.00	−0.161572
$851$	−2730.42	−0.109985
$852$	0	0
$853$	31962.0	1.28295	0.641476	−	0.767143i	$-0.278322\pi$
$853$	31962.0	1.28295	0.641476	+	0.767143i	$0.278322\pi$
$854$	−33410.5	−1.33874
$855$	0	0
$856$	38556.0	1.53951
$857$	−4931.68	−0.196573	−0.0982865	−	0.995158i	$-0.531336\pi$
$857$	−4931.68	−0.196573	−0.0982865	+	0.995158i	$0.531336\pi$
$858$	0	0
$859$	11704.0	0.464884	0.232442	−	0.972610i	$-0.425328\pi$
$859$	11704.0	0.464884	0.232442	+	0.972610i	$0.425328\pi$
$860$	1015.97	0.0402840
$861$	0	0
$862$	13818.0	0.545989
$863$	2280.64	0.0899581	0.0449790	−	0.998988i	$-0.485678\pi$
$863$	2280.64	0.0899581	0.0449790	+	0.998988i	$0.485678\pi$
$864$	0	0
$865$	−36960.0	−1.45281
$866$	−1814.99	−0.0712191
$867$	0	0
$868$	−4004.00	−0.156572
$869$	2434.09	0.0950183
$870$	0	0
$871$	−6890.00	−0.268035
$872$	8048.38	0.312560
$873$	0	0
$874$	−10584.0	−0.409621
$875$	−32130.0	−1.24136
$876$	0	0
$877$	−1006.00	−0.0387346	−0.0193673	−	0.999812i	$-0.506165\pi$
$877$	−1006.00	−0.0387346	−0.0193673	+	0.999812i	$0.506165\pi$
$878$	−3629.97	−0.139528
$879$	0	0
$880$	−3080.00	−0.117985
$881$	40681.1	1.55571	0.777855	−	0.628444i	$-0.216308\pi$
$881$	40681.1	1.55571	0.777855	+	0.628444i	$0.216308\pi$
$882$	0	0
$883$	124.000	0.00472586	0.00236293	−	0.999997i	$-0.499248\pi$
$883$	124.000	0.00472586	0.00236293	+	0.999997i	$0.499248\pi$
$884$	−1513.37	−0.0575793
$885$	0	0
$886$	6188.00	0.234639
$887$	−16573.0	−0.627358	−0.313679	−	0.949529i	$-0.601562\pi$
$887$	−16573.0	−0.627358	−0.313679	+	0.949529i	$0.601562\pi$
$888$	0	0
$889$	8272.00	0.312074
$890$	−40300.1	−1.51782
$891$	0	0
$892$	4886.00	0.183403
$893$	−46004.3	−1.72394
$894$	0	0
$895$	−1792.00	−0.0669273
$896$	−24621.4	−0.918016
$897$	0	0
$898$	45640.0	1.69602
$899$	−9630.53	−0.357282
$900$	0	0
$901$	−22176.0	−0.819966
$902$	6222.81	0.229708
$903$	0	0
$904$	40068.0	1.47416
$905$	−35707.1	−1.31154
$906$	0	0
$907$	−42996.0	−1.57404	−0.787022	−	0.616924i	$-0.788378\pi$
$907$	−42996.0	−1.57404	−0.787022	+	0.616924i	$0.788378\pi$
$908$	−1867.90	−0.0682692
$909$	0	0
$910$	8008.00	0.291717
$911$	53169.0	1.93366	0.966832	−	0.255413i	$-0.0822113\pi$
$911$	53169.0	1.93366	0.966832	+	0.255413i	$0.0822113\pi$
$912$	0	0
$913$	1708.00	0.0619130
$914$	−47269.0	−1.71063
$915$	0	0
$916$	−5558.00	−0.200482
$917$	15133.7	0.544993
$918$	0	0
$919$	8244.00	0.295913	0.147957	−	0.988994i	$-0.452730\pi$
$919$	8244.00	0.295913	0.147957	+	0.988994i	$0.452730\pi$
$920$	8000.75	0.286714
$921$	0	0
$922$	5572.00	0.199028
$923$	10524.8	0.375328
$924$	0	0
$925$	1118.00	0.0397401
$926$	−36294.4	−1.28802
$927$	0	0
$928$	2380.00	0.0841889
$929$	16192.0	0.571843	0.285922	−	0.958253i	$-0.407700\pi$
$929$	16192.0	0.571843	0.285922	+	0.958253i	$0.407700\pi$
$930$	0	0
$931$	−17766.0	−0.625410
$932$	3577.06	0.125719
$933$	0	0
$934$	10836.0	0.379620
$935$	6519.13	0.228020
$936$	0	0
$937$	−18214.0	−0.635032	−0.317516	−	0.948253i	$-0.602849\pi$
$937$	−18214.0	−0.635032	−0.317516	+	0.948253i	$0.602849\pi$
$938$	30849.5	1.07385
$939$	0	0
$940$	3864.00	0.134074
$941$	13916.7	0.482115	0.241057	−	0.970511i	$-0.422506\pi$
$941$	13916.7	0.482115	0.241057	+	0.970511i	$0.422506\pi$
$942$	0	0
$943$	−14112.0	−0.487328
$944$	32304.6	1.11380
$945$	0	0
$946$	−1344.00	−0.0461916
$947$	−40400.6	−1.38632	−0.693159	−	0.720784i	$-0.743782\pi$
$947$	−40400.6	−1.38632	−0.693159	+	0.720784i	$0.743782\pi$
$948$	0	0
$949$	−2002.00	−0.0684802
$950$	4333.74	0.148005
$951$	0	0
$952$	60984.0	2.07616
$953$	−21271.8	−0.723046	−0.361523	−	0.932363i	$-0.617743\pi$
$953$	−21271.8	−0.723046	−0.361523	+	0.932363i	$0.617743\pi$
$954$	0	0
$955$	12544.0	0.425041
$956$	2936.78	0.0993540
$957$	0	0
$958$	23058.0	0.777631
$959$	−30733.0	−1.03485
$960$	0	0
$961$	3333.00	0.111879
$962$	−2957.95	−0.0991353
$963$	0	0
$964$	602.000	0.0201132
$965$	16319.0	0.544380
$966$	0	0
$967$	9578.00	0.318519	0.159259	−	0.987237i	$-0.449089\pi$
$967$	9578.00	0.318519	0.159259	+	0.987237i	$0.449089\pi$
$968$	31026.7	1.03020
$969$	0	0
$970$	−1960.00	−0.0648782
$971$	−44956.6	−1.48581	−0.742907	−	0.669394i	$-0.766554\pi$
$971$	−44956.6	−1.48581	−0.742907	+	0.669394i	$0.766554\pi$
$972$	0	0
$973$	−46200.0	−1.52220
$974$	11921.8	0.392195
$975$	0	0
$976$	−31570.0	−1.03538
$977$	32765.0	1.07292	0.536461	−	0.843925i	$-0.319761\pi$
$977$	32765.0	1.07292	0.536461	+	0.843925i	$0.319761\pi$
$978$	0	0
$979$	−7616.00	−0.248630
$980$	1492.20	0.0486395
$981$	0	0
$982$	−57204.0	−1.85891
$983$	−47438.3	−1.53921	−0.769607	−	0.638518i	$-0.779548\pi$
$983$	−47438.3	−1.53921	−0.769607	+	0.638518i	$0.779548\pi$
$984$	0	0
$985$	−22512.0	−0.728215
$986$	16297.8	0.526398
$987$	0	0
$988$	1638.00	0.0527447
$989$	3047.91	0.0979957
$990$	0	0
$991$	−6848.00	−0.219509	−0.109755	−	0.993959i	$-0.535007\pi$
$991$	−6848.00	−0.219509	−0.109755	+	0.993959i	$0.535007\pi$
$992$	−8185.95	−0.262000
$993$	0	0
$994$	−47124.0	−1.50370
$995$	10075.0	0.321005
$996$	0	0
$997$	−5810.00	−0.184558	−0.0922791	−	0.995733i	$-0.529415\pi$
$997$	−5810.00	−0.184558	−0.0922791	+	0.995733i	$0.529415\pi$
$998$	−2079.56	−0.0659593
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	117.4.a.e.1.2	yes	2
3.2	odd	2	inner	117.4.a.e.1.1	✓	2
4.3	odd	2		1872.4.a.ba.1.1		2
12.11	even	2		1872.4.a.ba.1.2		2
13.12	even	2		1521.4.a.p.1.1		2
39.38	odd	2		1521.4.a.p.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
117.4.a.e.1.1	✓	2	3.2	odd	2	inner
117.4.a.e.1.2	yes	2	1.1	even	1	trivial
1521.4.a.p.1.1		2	13.12	even	2
1521.4.a.p.1.2		2	39.38	odd	2
1872.4.a.ba.1.1		2	4.3	odd	2
1872.4.a.ba.1.2		2	12.11	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 117 = 3^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	117.a (trivial)

\(f(q)\)	\(=\)	\(q+2.64575 q^{2} -1.00000 q^{4} -10.5830 q^{5} -22.0000 q^{7} -23.8118 q^{8} +O(q^{10})\)	\(q+2.64575 q^{2} -1.00000 q^{4} -10.5830 q^{5} -22.0000 q^{7} -23.8118 q^{8} -28.0000 q^{10} -5.29150 q^{11} +13.0000 q^{13} -58.2065 q^{14} -55.0000 q^{16} +116.413 q^{17} -126.000 q^{19} +10.5830 q^{20} -14.0000 q^{22} +31.7490 q^{23} -13.0000 q^{25} +34.3948 q^{26} +22.0000 q^{28} +52.9150 q^{29} -182.000 q^{31} +44.9778 q^{32} +308.000 q^{34} +232.826 q^{35} -86.0000 q^{37} -333.365 q^{38} +252.000 q^{40} -444.486 q^{41} +96.0000 q^{43} +5.29150 q^{44} +84.0000 q^{46} +365.114 q^{47} +141.000 q^{49} -34.3948 q^{50} -13.0000 q^{52} -190.494 q^{53} +56.0000 q^{55} +523.859 q^{56} +140.000 q^{58} -587.357 q^{59} +574.000 q^{61} -481.527 q^{62} +559.000 q^{64} -137.579 q^{65} -530.000 q^{67} -116.413 q^{68} +616.000 q^{70} +809.600 q^{71} -154.000 q^{73} -227.535 q^{74} +126.000 q^{76} +116.413 q^{77} -460.000 q^{79} +582.065 q^{80} -1176.00 q^{82} -322.782 q^{83} -1232.00 q^{85} +253.992 q^{86} +126.000 q^{88} +1439.29 q^{89} -286.000 q^{91} -31.7490 q^{92} +966.000 q^{94} +1333.46 q^{95} +70.0000 q^{97} +373.051 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{4} - 44 q^{7}+O(q^{10}) \) 2 * q - 2 * q^4 - 44 * q^7	\( 2 q - 2 q^{4} - 44 q^{7} - 56 q^{10} + 26 q^{13} - 110 q^{16} - 252 q^{19} - 28 q^{22} - 26 q^{25} + 44 q^{28} - 364 q^{31} + 616 q^{34} - 172 q^{37} + 504 q^{40} + 192 q^{43} + 168 q^{46} + 282 q^{49} - 26 q^{52} + 112 q^{55} + 280 q^{58} + 1148 q^{61} + 1118 q^{64} - 1060 q^{67} + 1232 q^{70} - 308 q^{73} + 252 q^{76} - 920 q^{79} - 2352 q^{82} - 2464 q^{85} + 252 q^{88} - 572 q^{91} + 1932 q^{94} + 140 q^{97}+O(q^{100}) \) 2 * q - 2 * q^4 - 44 * q^7 - 56 * q^10 + 26 * q^13 - 110 * q^16 - 252 * q^19 - 28 * q^22 - 26 * q^25 + 44 * q^28 - 364 * q^31 + 616 * q^34 - 172 * q^37 + 504 * q^40 + 192 * q^43 + 168 * q^46 + 282 * q^49 - 26 * q^52 + 112 * q^55 + 280 * q^58 + 1148 * q^61 + 1118 * q^64 - 1060 * q^67 + 1232 * q^70 - 308 * q^73 + 252 * q^76 - 920 * q^79 - 2352 * q^82 - 2464 * q^85 + 252 * q^88 - 572 * q^91 + 1932 * q^94 + 140 * q^97

Introduction

L-functions

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