LMFDB - Embedded newform 325.4.a.d.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [325,4,Mod(1,325)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(325, 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("325.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 325 = 5^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	325.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:	$19.1756207519$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 13)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	325.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+5.00000 q^{2} +7.00000 q^{3} +17.0000 q^{4} +35.0000 q^{6} +13.0000 q^{7} +45.0000 q^{8} +22.0000 q^{9} +O(q^{10})$

\(q+5.00000 q^{2} +7.00000 q^{3} +17.0000 q^{4} +35.0000 q^{6} +13.0000 q^{7} +45.0000 q^{8} +22.0000 q^{9} -26.0000 q^{11} +119.000 q^{12} -13.0000 q^{13} +65.0000 q^{14} +89.0000 q^{16} -77.0000 q^{17} +110.000 q^{18} -126.000 q^{19} +91.0000 q^{21} -130.000 q^{22} +96.0000 q^{23} +315.000 q^{24} -65.0000 q^{26} -35.0000 q^{27} +221.000 q^{28} -82.0000 q^{29} +196.000 q^{31} +85.0000 q^{32} -182.000 q^{33} -385.000 q^{34} +374.000 q^{36} +131.000 q^{37} -630.000 q^{38} -91.0000 q^{39} +336.000 q^{41} +455.000 q^{42} +201.000 q^{43} -442.000 q^{44} +480.000 q^{46} +105.000 q^{47} +623.000 q^{48} -174.000 q^{49} -539.000 q^{51} -221.000 q^{52} +432.000 q^{53} -175.000 q^{54} +585.000 q^{56} -882.000 q^{57} -410.000 q^{58} -294.000 q^{59} -56.0000 q^{61} +980.000 q^{62} +286.000 q^{63} -287.000 q^{64} -910.000 q^{66} -478.000 q^{67} -1309.00 q^{68} +672.000 q^{69} +9.00000 q^{71} +990.000 q^{72} -98.0000 q^{73} +655.000 q^{74} -2142.00 q^{76} -338.000 q^{77} -455.000 q^{78} +1304.00 q^{79} -839.000 q^{81} +1680.00 q^{82} +308.000 q^{83} +1547.00 q^{84} +1005.00 q^{86} -574.000 q^{87} -1170.00 q^{88} -1190.00 q^{89} -169.000 q^{91} +1632.00 q^{92} +1372.00 q^{93} +525.000 q^{94} +595.000 q^{96} -70.0000 q^{97} -870.000 q^{98} -572.000 q^{99} +O(q^{100})\)

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 $

$2$	5.00000	1.76777	0.883883	−	0.467707i	$-0.154920\pi$
$2$	5.00000	1.76777	0.883883	+	0.467707i	$0.154920\pi$
$3$	7.00000	1.34715	0.673575	−	0.739119i	$-0.264758\pi$
$3$	7.00000	1.34715	0.673575	+	0.739119i	$0.264758\pi$
$4$	17.0000	2.12500
$5$	0	0
$5$	0	0
$6$	35.0000	2.38145
$7$	13.0000	0.701934	0.350967	−	0.936388i	$-0.385853\pi$
$7$	13.0000	0.701934	0.350967	+	0.936388i	$0.385853\pi$
$8$	45.0000	1.98874
$9$	22.0000	0.814815
$10$	0	0
$11$	−26.0000	−0.712663	−0.356332	−	0.934360i	$-0.615973\pi$
$11$	−26.0000	−0.712663	−0.356332	+	0.934360i	$0.615973\pi$
$12$	119.000	2.86270
$13$	−13.0000	−0.277350
$13$	−13.0000	−0.277350
$14$	65.0000	1.24086
$15$	0	0
$16$	89.0000	1.39062
$17$	−77.0000	−1.09854	−0.549272	−	0.835644i	$-0.685095\pi$
$17$	−77.0000	−1.09854	−0.549272	+	0.835644i	$0.685095\pi$
$18$	110.000	1.44040
$19$	−126.000	−1.52139	−0.760694	−	0.649110i	$-0.775141\pi$
$19$	−126.000	−1.52139	−0.760694	+	0.649110i	$0.775141\pi$
$20$	0	0
$21$	91.0000	0.945611
$22$	−130.000	−1.25982
$23$	96.0000	0.870321	0.435161	−	0.900353i	$-0.356692\pi$
$23$	96.0000	0.870321	0.435161	+	0.900353i	$0.356692\pi$
$24$	315.000	2.67913
$25$	0	0
$26$	−65.0000	−0.490290
$27$	−35.0000	−0.249472
$28$	221.000	1.49161
$29$	−82.0000	−0.525070	−0.262535	−	0.964923i	$-0.584558\pi$
$29$	−82.0000	−0.525070	−0.262535	+	0.964923i	$0.584558\pi$
$30$	0	0
$31$	196.000	1.13557	0.567785	−	0.823177i	$-0.307801\pi$
$31$	196.000	1.13557	0.567785	+	0.823177i	$0.307801\pi$
$32$	85.0000	0.469563
$33$	−182.000	−0.960065
$34$	−385.000	−1.94197
$35$	0	0
$36$	374.000	1.73148
$37$	131.000	0.582061	0.291031	−	0.956714i	$-0.406002\pi$
$37$	131.000	0.582061	0.291031	+	0.956714i	$0.406002\pi$
$38$	−630.000	−2.68946
$39$	−91.0000	−0.373632
$40$	0	0
$41$	336.000	1.27986	0.639932	−	0.768432i	$-0.278963\pi$
$41$	336.000	1.27986	0.639932	+	0.768432i	$0.278963\pi$
$42$	455.000	1.67162
$43$	201.000	0.712842	0.356421	−	0.934325i	$-0.383997\pi$
$43$	201.000	0.712842	0.356421	+	0.934325i	$0.383997\pi$
$44$	−442.000	−1.51441
$45$	0	0
$46$	480.000	1.53852
$47$	105.000	0.325869	0.162934	−	0.986637i	$-0.447904\pi$
$47$	105.000	0.325869	0.162934	+	0.986637i	$0.447904\pi$
$48$	623.000	1.87338
$49$	−174.000	−0.507289
$50$	0	0
$51$	−539.000	−1.47990
$52$	−221.000	−0.589369
$53$	432.000	1.11962	0.559809	−	0.828622i	$-0.310874\pi$
$53$	432.000	1.11962	0.559809	+	0.828622i	$0.310874\pi$
$54$	−175.000	−0.441009
$55$	0	0
$56$	585.000	1.39596
$57$	−882.000	−2.04954
$58$	−410.000	−0.928201
$59$	−294.000	−0.648738	−0.324369	−	0.945931i	$-0.605152\pi$
$59$	−294.000	−0.648738	−0.324369	+	0.945931i	$0.605152\pi$
$60$	0	0
$61$	−56.0000	−0.117542	−0.0587710	−	0.998271i	$-0.518718\pi$
$61$	−56.0000	−0.117542	−0.0587710	+	0.998271i	$0.518718\pi$
$62$	980.000	2.00742
$63$	286.000	0.571946
$64$	−287.000	−0.560547
$65$	0	0
$66$	−910.000	−1.69717
$67$	−478.000	−0.871597	−0.435798	−	0.900044i	$-0.643534\pi$
$67$	−478.000	−0.871597	−0.435798	+	0.900044i	$0.643534\pi$
$68$	−1309.00	−2.33441
$69$	672.000	1.17245
$70$	0	0
$71$	9.00000	0.0150437	0.00752186	−	0.999972i	$-0.497606\pi$
$71$	9.00000	0.0150437	0.00752186	+	0.999972i	$0.497606\pi$
$72$	990.000	1.62045
$73$	−98.0000	−0.157124	−0.0785619	−	0.996909i	$-0.525033\pi$
$73$	−98.0000	−0.157124	−0.0785619	+	0.996909i	$0.525033\pi$
$74$	655.000	1.02895
$75$	0	0
$76$	−2142.00	−3.23295
$77$	−338.000	−0.500243
$78$	−455.000	−0.660495
$79$	1304.00	1.85711	0.928554	−	0.371198i	$-0.121053\pi$
$79$	1304.00	1.85711	0.928554	+	0.371198i	$0.121053\pi$
$80$	0	0
$81$	−839.000	−1.15089
$82$	1680.00	2.26250
$83$	308.000	0.407318	0.203659	−	0.979042i	$-0.434717\pi$
$83$	308.000	0.407318	0.203659	+	0.979042i	$0.434717\pi$
$84$	1547.00	2.00942
$85$	0	0
$86$	1005.00	1.26014
$87$	−574.000	−0.707348
$88$	−1170.00	−1.41730
$89$	−1190.00	−1.41730	−0.708650	−	0.705560i	$-0.750696\pi$
$89$	−1190.00	−1.41730	−0.708650	+	0.705560i	$0.750696\pi$
$90$	0	0
$91$	−169.000	−0.194681
$92$	1632.00	1.84943
$93$	1372.00	1.52978
$94$	525.000	0.576060
$95$	0	0
$96$	595.000	0.632572
$97$	−70.0000	−0.0732724	−0.0366362	−	0.999329i	$-0.511664\pi$
$97$	−70.0000	−0.0732724	−0.0366362	+	0.999329i	$0.511664\pi$
$98$	−870.000	−0.896768
$99$	−572.000	−0.580689
$100$	0	0
$101$	420.000	0.413778	0.206889	−	0.978364i	$-0.433666\pi$
$101$	420.000	0.413778	0.206889	+	0.978364i	$0.433666\pi$
$102$	−2695.00	−2.61613
$103$	−588.000	−0.562499	−0.281249	−	0.959635i	$-0.590749\pi$
$103$	−588.000	−0.562499	−0.281249	+	0.959635i	$0.590749\pi$
$104$	−585.000	−0.551577
$105$	0	0
$106$	2160.00	1.97922
$107$	684.000	0.617989	0.308994	−	0.951064i	$-0.400008\pi$
$107$	684.000	0.617989	0.308994	+	0.951064i	$0.400008\pi$
$108$	−595.000	−0.530129
$109$	373.000	0.327770	0.163885	−	0.986479i	$-0.447597\pi$
$109$	373.000	0.327770	0.163885	+	0.986479i	$0.447597\pi$
$110$	0	0
$111$	917.000	0.784124
$112$	1157.00	0.976127
$113$	1734.00	1.44355	0.721774	−	0.692128i	$-0.243327\pi$
$113$	1734.00	1.44355	0.721774	+	0.692128i	$0.243327\pi$
$114$	−4410.00	−3.62311
$115$	0	0
$116$	−1394.00	−1.11577
$117$	−286.000	−0.225989
$118$	−1470.00	−1.14682
$119$	−1001.00	−0.771105
$120$	0	0
$121$	−655.000	−0.492111
$122$	−280.000	−0.207787
$123$	2352.00	1.72417
$124$	3332.00	2.41308
$125$	0	0
$126$	1430.00	1.01107
$127$	−1892.00	−1.32195	−0.660976	−	0.750407i	$-0.729857\pi$
$127$	−1892.00	−1.32195	−0.660976	+	0.750407i	$0.729857\pi$
$128$	−2115.00	−1.46048
$129$	1407.00	0.960306
$130$	0	0
$131$	1435.00	0.957073	0.478536	−	0.878068i	$-0.341167\pi$
$131$	1435.00	0.957073	0.478536	+	0.878068i	$0.341167\pi$
$132$	−3094.00	−2.04014
$133$	−1638.00	−1.06791
$134$	−2390.00	−1.54078
$135$	0	0
$136$	−3465.00	−2.18472
$137$	1776.00	1.10755	0.553773	−	0.832667i	$-0.313187\pi$
$137$	1776.00	1.10755	0.553773	+	0.832667i	$0.313187\pi$
$138$	3360.00	2.07262
$139$	−1869.00	−1.14048	−0.570239	−	0.821479i	$-0.693150\pi$
$139$	−1869.00	−1.14048	−0.570239	+	0.821479i	$0.693150\pi$
$140$	0	0
$141$	735.000	0.438994
$142$	45.0000	0.0265938
$143$	338.000	0.197657
$144$	1958.00	1.13310
$145$	0	0
$146$	−490.000	−0.277758
$147$	−1218.00	−0.683394
$148$	2227.00	1.23688
$149$	2466.00	1.35586	0.677928	−	0.735128i	$-0.262878\pi$
$149$	2466.00	1.35586	0.677928	+	0.735128i	$0.262878\pi$
$150$	0	0
$151$	−3323.00	−1.79087	−0.895437	−	0.445189i	$-0.853137\pi$
$151$	−3323.00	−1.79087	−0.895437	+	0.445189i	$0.853137\pi$
$152$	−5670.00	−3.02564
$153$	−1694.00	−0.895110
$154$	−1690.00	−0.884312
$155$	0	0
$156$	−1547.00	−0.793969
$157$	2730.00	1.38776	0.693878	−	0.720092i	$-0.255901\pi$
$157$	2730.00	1.38776	0.693878	+	0.720092i	$0.255901\pi$
$158$	6520.00	3.28293
$159$	3024.00	1.50829
$160$	0	0
$161$	1248.00	0.610908
$162$	−4195.00	−2.03451
$163$	544.000	0.261407	0.130704	−	0.991421i	$-0.458276\pi$
$163$	544.000	0.261407	0.130704	+	0.991421i	$0.458276\pi$
$164$	5712.00	2.71971
$165$	0	0
$166$	1540.00	0.720043
$167$	−1624.00	−0.752508	−0.376254	−	0.926516i	$-0.622788\pi$
$167$	−1624.00	−0.752508	−0.376254	+	0.926516i	$0.622788\pi$
$168$	4095.00	1.88057
$169$	169.000	0.0769231
$170$	0	0
$171$	−2772.00	−1.23965
$172$	3417.00	1.51479
$173$	336.000	0.147662	0.0738312	−	0.997271i	$-0.476477\pi$
$173$	336.000	0.147662	0.0738312	+	0.997271i	$0.476477\pi$
$174$	−2870.00	−1.25043
$175$	0	0
$176$	−2314.00	−0.991047
$177$	−2058.00	−0.873948
$178$	−5950.00	−2.50546
$179$	−3029.00	−1.26479	−0.632397	−	0.774645i	$-0.717929\pi$
$179$	−3029.00	−1.26479	−0.632397	+	0.774645i	$0.717929\pi$
$180$	0	0
$181$	−28.0000	−0.0114985	−0.00574924	−	0.999983i	$-0.501830\pi$
$181$	−28.0000	−0.0114985	−0.00574924	+	0.999983i	$0.501830\pi$
$182$	−845.000	−0.344151
$183$	−392.000	−0.158347
$184$	4320.00	1.73084
$185$	0	0
$186$	6860.00	2.70430
$187$	2002.00	0.782892
$188$	1785.00	0.692471
$189$	−455.000	−0.175113
$190$	0	0
$191$	422.000	0.159868	0.0799342	−	0.996800i	$-0.474529\pi$
$191$	422.000	0.159868	0.0799342	+	0.996800i	$0.474529\pi$
$192$	−2009.00	−0.755141
$193$	−492.000	−0.183497	−0.0917485	−	0.995782i	$-0.529246\pi$
$193$	−492.000	−0.183497	−0.0917485	+	0.995782i	$0.529246\pi$
$194$	−350.000	−0.129529
$195$	0	0
$196$	−2958.00	−1.07799
$197$	−2991.00	−1.08173	−0.540863	−	0.841111i	$-0.681902\pi$
$197$	−2991.00	−1.08173	−0.540863	+	0.841111i	$0.681902\pi$
$198$	−2860.00	−1.02652
$199$	−70.0000	−0.0249355	−0.0124678	−	0.999922i	$-0.503969\pi$
$199$	−70.0000	−0.0249355	−0.0124678	+	0.999922i	$0.503969\pi$
$200$	0	0
$201$	−3346.00	−1.17417
$202$	2100.00	0.731463
$203$	−1066.00	−0.368564
$204$	−9163.00	−3.14480
$205$	0	0
$206$	−2940.00	−0.994367
$207$	2112.00	0.709150
$208$	−1157.00	−0.385690
$209$	3276.00	1.08424
$210$	0	0
$211$	2851.00	0.930194	0.465097	−	0.885260i	$-0.346019\pi$
$211$	2851.00	0.930194	0.465097	+	0.885260i	$0.346019\pi$
$212$	7344.00	2.37919
$213$	63.0000	0.0202661
$214$	3420.00	1.09246
$215$	0	0
$216$	−1575.00	−0.496135
$217$	2548.00	0.797095
$218$	1865.00	0.579421
$219$	−686.000	−0.211669
$220$	0	0
$221$	1001.00	0.304681
$222$	4585.00	1.38615
$223$	−217.000	−0.0651632	−0.0325816	−	0.999469i	$-0.510373\pi$
$223$	−217.000	−0.0651632	−0.0325816	+	0.999469i	$0.510373\pi$
$224$	1105.00	0.329602
$225$	0	0
$226$	8670.00	2.55186
$227$	2576.00	0.753194	0.376597	−	0.926377i	$-0.377094\pi$
$227$	2576.00	0.753194	0.376597	+	0.926377i	$0.377094\pi$
$228$	−14994.0	−4.35527
$229$	455.000	0.131298	0.0656490	−	0.997843i	$-0.479088\pi$
$229$	455.000	0.131298	0.0656490	+	0.997843i	$0.479088\pi$
$230$	0	0
$231$	−2366.00	−0.673902
$232$	−3690.00	−1.04423
$233$	−3061.00	−0.860656	−0.430328	−	0.902673i	$-0.641602\pi$
$233$	−3061.00	−0.860656	−0.430328	+	0.902673i	$0.641602\pi$
$234$	−1430.00	−0.399496
$235$	0	0
$236$	−4998.00	−1.37857
$237$	9128.00	2.50180
$238$	−5005.00	−1.36313
$239$	−3477.00	−0.941039	−0.470520	−	0.882389i	$-0.655934\pi$
$239$	−3477.00	−0.941039	−0.470520	+	0.882389i	$0.655934\pi$
$240$	0	0
$241$	−1610.00	−0.430329	−0.215164	−	0.976578i	$-0.569029\pi$
$241$	−1610.00	−0.430329	−0.215164	+	0.976578i	$0.569029\pi$
$242$	−3275.00	−0.869938
$243$	−4928.00	−1.30095
$244$	−952.000	−0.249777
$245$	0	0
$246$	11760.0	3.04793
$247$	1638.00	0.421957
$248$	8820.00	2.25835
$249$	2156.00	0.548719
$250$	0	0
$251$	1008.00	0.253484	0.126742	−	0.991936i	$-0.459548\pi$
$251$	1008.00	0.253484	0.126742	+	0.991936i	$0.459548\pi$
$252$	4862.00	1.21539
$253$	−2496.00	−0.620246
$254$	−9460.00	−2.33690
$255$	0	0
$256$	−8279.00	−2.02124
$257$	−6041.00	−1.46625	−0.733127	−	0.680092i	$-0.761940\pi$
$257$	−6041.00	−1.46625	−0.733127	+	0.680092i	$0.761940\pi$
$258$	7035.00	1.69760
$259$	1703.00	0.408569
$260$	0	0
$261$	−1804.00	−0.427834
$262$	7175.00	1.69188
$263$	3708.00	0.869373	0.434686	−	0.900582i	$-0.356859\pi$
$263$	3708.00	0.869373	0.434686	+	0.900582i	$0.356859\pi$
$264$	−8190.00	−1.90932
$265$	0	0
$266$	−8190.00	−1.88782
$267$	−8330.00	−1.90932
$268$	−8126.00	−1.85214
$269$	8344.00	1.89124	0.945618	−	0.325278i	$-0.105458\pi$
$269$	8344.00	1.89124	0.945618	+	0.325278i	$0.105458\pi$
$270$	0	0
$271$	−1617.00	−0.362457	−0.181228	−	0.983441i	$-0.558007\pi$
$271$	−1617.00	−0.362457	−0.181228	+	0.983441i	$0.558007\pi$
$272$	−6853.00	−1.52766
$273$	−1183.00	−0.262265
$274$	8880.00	1.95788
$275$	0	0
$276$	11424.0	2.49146
$277$	3820.00	0.828598	0.414299	−	0.910141i	$-0.364027\pi$
$277$	3820.00	0.828598	0.414299	+	0.910141i	$0.364027\pi$
$278$	−9345.00	−2.01610
$279$	4312.00	0.925278
$280$	0	0
$281$	−6214.00	−1.31920	−0.659602	−	0.751615i	$-0.729275\pi$
$281$	−6214.00	−1.31920	−0.659602	+	0.751615i	$0.729275\pi$
$282$	3675.00	0.776039
$283$	5292.00	1.11158	0.555789	−	0.831323i	$-0.312416\pi$
$283$	5292.00	1.11158	0.555789	+	0.831323i	$0.312416\pi$
$284$	153.000	0.0319679
$285$	0	0
$286$	1690.00	0.349412
$287$	4368.00	0.898379
$288$	1870.00	0.382607
$289$	1016.00	0.206798
$290$	0	0
$291$	−490.000	−0.0987090
$292$	−1666.00	−0.333888
$293$	903.000	0.180047	0.0900236	−	0.995940i	$-0.471306\pi$
$293$	903.000	0.180047	0.0900236	+	0.995940i	$0.471306\pi$
$294$	−6090.00	−1.20808
$295$	0	0
$296$	5895.00	1.15757
$297$	910.000	0.177790
$298$	12330.0	2.39684
$299$	−1248.00	−0.241384
$300$	0	0
$301$	2613.00	0.500368
$302$	−16615.0	−3.16585
$303$	2940.00	0.557421
$304$	−11214.0	−2.11568
$305$	0	0
$306$	−8470.00	−1.58235
$307$	−2114.00	−0.393004	−0.196502	−	0.980503i	$-0.562958\pi$
$307$	−2114.00	−0.393004	−0.196502	+	0.980503i	$0.562958\pi$
$308$	−5746.00	−1.06302
$309$	−4116.00	−0.757770
$310$	0	0
$311$	3402.00	0.620288	0.310144	−	0.950690i	$-0.399623\pi$
$311$	3402.00	0.620288	0.310144	+	0.950690i	$0.399623\pi$
$312$	−4095.00	−0.743057
$313$	10689.0	1.93028	0.965141	−	0.261732i	$-0.0842937\pi$
$313$	10689.0	1.93028	0.965141	+	0.261732i	$0.0842937\pi$
$314$	13650.0	2.45323
$315$	0	0
$316$	22168.0	3.94635
$317$	7054.00	1.24982	0.624909	−	0.780698i	$-0.285136\pi$
$317$	7054.00	1.24982	0.624909	+	0.780698i	$0.285136\pi$
$318$	15120.0	2.66631
$319$	2132.00	0.374198
$320$	0	0
$321$	4788.00	0.832524
$322$	6240.00	1.07994
$323$	9702.00	1.67131
$324$	−14263.0	−2.44564
$325$	0	0
$326$	2720.00	0.462107
$327$	2611.00	0.441555
$328$	15120.0	2.54531
$329$	1365.00	0.228738
$330$	0	0
$331$	9704.00	1.61142	0.805710	−	0.592310i	$-0.201784\pi$
$331$	9704.00	1.61142	0.805710	+	0.592310i	$0.201784\pi$
$332$	5236.00	0.865551
$333$	2882.00	0.474272
$334$	−8120.00	−1.33026
$335$	0	0
$336$	8099.00	1.31499
$337$	10449.0	1.68900	0.844500	−	0.535555i	$-0.179897\pi$
$337$	10449.0	1.68900	0.844500	+	0.535555i	$0.179897\pi$
$338$	845.000	0.135982
$339$	12138.0	1.94468
$340$	0	0
$341$	−5096.00	−0.809278
$342$	−13860.0	−2.19141
$343$	−6721.00	−1.05802
$344$	9045.00	1.41766
$345$	0	0
$346$	1680.00	0.261033
$347$	621.000	0.0960721	0.0480361	−	0.998846i	$-0.484704\pi$
$347$	621.000	0.0960721	0.0480361	+	0.998846i	$0.484704\pi$
$348$	−9758.00	−1.50311
$349$	12481.0	1.91431	0.957153	−	0.289584i	$-0.0935168\pi$
$349$	12481.0	1.91431	0.957153	+	0.289584i	$0.0935168\pi$
$350$	0	0
$351$	455.000	0.0691912
$352$	−2210.00	−0.334640
$353$	1400.00	0.211089	0.105545	−	0.994415i	$-0.466341\pi$
$353$	1400.00	0.211089	0.105545	+	0.994415i	$0.466341\pi$
$354$	−10290.0	−1.54494
$355$	0	0
$356$	−20230.0	−3.01176
$357$	−7007.00	−1.03879
$358$	−15145.0	−2.23586
$359$	−4968.00	−0.730365	−0.365182	−	0.930936i	$-0.618993\pi$
$359$	−4968.00	−0.730365	−0.365182	+	0.930936i	$0.618993\pi$
$360$	0	0
$361$	9017.00	1.31462
$362$	−140.000	−0.0203266
$363$	−4585.00	−0.662948
$364$	−2873.00	−0.413698
$365$	0	0
$366$	−1960.00	−0.279920
$367$	−8722.00	−1.24056	−0.620279	−	0.784381i	$-0.712981\pi$
$367$	−8722.00	−1.24056	−0.620279	+	0.784381i	$0.712981\pi$
$368$	8544.00	1.21029
$369$	7392.00	1.04285
$370$	0	0
$371$	5616.00	0.785898
$372$	23324.0	3.25079
$373$	−10012.0	−1.38982	−0.694908	−	0.719098i	$-0.744555\pi$
$373$	−10012.0	−1.38982	−0.694908	+	0.719098i	$0.744555\pi$
$374$	10010.0	1.38397
$375$	0	0
$376$	4725.00	0.648067
$377$	1066.00	0.145628
$378$	−2275.00	−0.309559
$379$	−3372.00	−0.457013	−0.228507	−	0.973542i	$-0.573384\pi$
$379$	−3372.00	−0.457013	−0.228507	+	0.973542i	$0.573384\pi$
$380$	0	0
$381$	−13244.0	−1.78087
$382$	2110.00	0.282610
$383$	847.000	0.113002	0.0565009	−	0.998403i	$-0.482006\pi$
$383$	847.000	0.113002	0.0565009	+	0.998403i	$0.482006\pi$
$384$	−14805.0	−1.96749
$385$	0	0
$386$	−2460.00	−0.324380
$387$	4422.00	0.580834
$388$	−1190.00	−0.155704
$389$	11314.0	1.47466	0.737330	−	0.675533i	$-0.236086\pi$
$389$	11314.0	1.47466	0.737330	+	0.675533i	$0.236086\pi$
$390$	0	0
$391$	−7392.00	−0.956086
$392$	−7830.00	−1.00886
$393$	10045.0	1.28932
$394$	−14955.0	−1.91224
$395$	0	0
$396$	−9724.00	−1.23396
$397$	−1862.00	−0.235393	−0.117697	−	0.993050i	$-0.537551\pi$
$397$	−1862.00	−0.235393	−0.117697	+	0.993050i	$0.537551\pi$
$398$	−350.000	−0.0440802
$399$	−11466.0	−1.43864
$400$	0	0
$401$	6820.00	0.849313	0.424657	−	0.905355i	$-0.360395\pi$
$401$	6820.00	0.849313	0.424657	+	0.905355i	$0.360395\pi$
$402$	−16730.0	−2.07566
$403$	−2548.00	−0.314950
$404$	7140.00	0.879278
$405$	0	0
$406$	−5330.00	−0.651536
$407$	−3406.00	−0.414814
$408$	−24255.0	−2.94314
$409$	−12992.0	−1.57069	−0.785346	−	0.619057i	$-0.787515\pi$
$409$	−12992.0	−1.57069	−0.785346	+	0.619057i	$0.787515\pi$
$410$	0	0
$411$	12432.0	1.49203
$412$	−9996.00	−1.19531
$413$	−3822.00	−0.455371
$414$	10560.0	1.25361
$415$	0	0
$416$	−1105.00	−0.130233
$417$	−13083.0	−1.53640
$418$	16380.0	1.91668
$419$	−7343.00	−0.856155	−0.428078	−	0.903742i	$-0.640809\pi$
$419$	−7343.00	−0.856155	−0.428078	+	0.903742i	$0.640809\pi$
$420$	0	0
$421$	−5059.00	−0.585655	−0.292827	−	0.956165i	$-0.594596\pi$
$421$	−5059.00	−0.585655	−0.292827	+	0.956165i	$0.594596\pi$
$422$	14255.0	1.64437
$423$	2310.00	0.265523
$424$	19440.0	2.22663
$425$	0	0
$426$	315.000	0.0358258
$427$	−728.000	−0.0825068
$428$	11628.0	1.31323
$429$	2366.00	0.266274
$430$	0	0
$431$	3243.00	0.362436	0.181218	−	0.983443i	$-0.441996\pi$
$431$	3243.00	0.362436	0.181218	+	0.983443i	$0.441996\pi$
$432$	−3115.00	−0.346922
$433$	−11599.0	−1.28733	−0.643663	−	0.765309i	$-0.722586\pi$
$433$	−11599.0	−1.28733	−0.643663	+	0.765309i	$0.722586\pi$
$434$	12740.0	1.40908
$435$	0	0
$436$	6341.00	0.696511
$437$	−12096.0	−1.32410
$438$	−3430.00	−0.374182
$439$	−17374.0	−1.88887	−0.944437	−	0.328692i	$-0.893392\pi$
$439$	−17374.0	−1.88887	−0.944437	+	0.328692i	$0.893392\pi$
$440$	0	0
$441$	−3828.00	−0.413346
$442$	5005.00	0.538605
$443$	−989.000	−0.106070	−0.0530348	−	0.998593i	$-0.516889\pi$
$443$	−989.000	−0.106070	−0.0530348	+	0.998593i	$0.516889\pi$
$444$	15589.0	1.66626
$445$	0	0
$446$	−1085.00	−0.115193
$447$	17262.0	1.82654
$448$	−3731.00	−0.393467
$449$	−14474.0	−1.52131	−0.760657	−	0.649154i	$-0.775123\pi$
$449$	−14474.0	−1.52131	−0.760657	+	0.649154i	$0.775123\pi$
$450$	0	0
$451$	−8736.00	−0.912111
$452$	29478.0	3.06754
$453$	−23261.0	−2.41258
$454$	12880.0	1.33147
$455$	0	0
$456$	−39690.0	−4.07600
$457$	1594.00	0.163160	0.0815801	−	0.996667i	$-0.474003\pi$
$457$	1594.00	0.163160	0.0815801	+	0.996667i	$0.474003\pi$
$458$	2275.00	0.232104
$459$	2695.00	0.274056
$460$	0	0
$461$	−5915.00	−0.597590	−0.298795	−	0.954317i	$-0.596585\pi$
$461$	−5915.00	−0.597590	−0.298795	+	0.954317i	$0.596585\pi$
$462$	−11830.0	−1.19130
$463$	11072.0	1.11136	0.555680	−	0.831396i	$-0.312458\pi$
$463$	11072.0	1.11136	0.555680	+	0.831396i	$0.312458\pi$
$464$	−7298.00	−0.730175
$465$	0	0
$466$	−15305.0	−1.52144
$467$	−1260.00	−0.124852	−0.0624260	−	0.998050i	$-0.519884\pi$
$467$	−1260.00	−0.124852	−0.0624260	+	0.998050i	$0.519884\pi$
$468$	−4862.00	−0.480227
$469$	−6214.00	−0.611804
$470$	0	0
$471$	19110.0	1.86952
$472$	−13230.0	−1.29017
$473$	−5226.00	−0.508016
$474$	45640.0	4.42260
$475$	0	0
$476$	−17017.0	−1.63860
$477$	9504.00	0.912281
$478$	−17385.0	−1.66354
$479$	−12033.0	−1.14781	−0.573906	−	0.818921i	$-0.694572\pi$
$479$	−12033.0	−1.14781	−0.573906	+	0.818921i	$0.694572\pi$
$480$	0	0
$481$	−1703.00	−0.161435
$482$	−8050.00	−0.760721
$483$	8736.00	0.822985
$484$	−11135.0	−1.04574
$485$	0	0
$486$	−24640.0	−2.29978
$487$	2280.00	0.212149	0.106075	−	0.994358i	$-0.466172\pi$
$487$	2280.00	0.212149	0.106075	+	0.994358i	$0.466172\pi$
$488$	−2520.00	−0.233760
$489$	3808.00	0.352155
$490$	0	0
$491$	16767.0	1.54111	0.770554	−	0.637375i	$-0.219980\pi$
$491$	16767.0	1.54111	0.770554	+	0.637375i	$0.219980\pi$
$492$	39984.0	3.66386
$493$	6314.00	0.576812
$494$	8190.00	0.745922
$495$	0	0
$496$	17444.0	1.57915
$497$	117.000	0.0105597
$498$	10780.0	0.970007
$499$	12840.0	1.15190	0.575949	−	0.817485i	$-0.304633\pi$
$499$	12840.0	1.15190	0.575949	+	0.817485i	$0.304633\pi$
$500$	0	0
$501$	−11368.0	−1.01374
$502$	5040.00	0.448100
$503$	2198.00	0.194839	0.0974195	−	0.995243i	$-0.468941\pi$
$503$	2198.00	0.194839	0.0974195	+	0.995243i	$0.468941\pi$
$504$	12870.0	1.13745
$505$	0	0
$506$	−12480.0	−1.09645
$507$	1183.00	0.103627
$508$	−32164.0	−2.80915
$509$	−17066.0	−1.48612	−0.743062	−	0.669223i	$-0.766627\pi$
$509$	−17066.0	−1.48612	−0.743062	+	0.669223i	$0.766627\pi$
$510$	0	0
$511$	−1274.00	−0.110290
$512$	−24475.0	−2.11260
$513$	4410.00	0.379544
$514$	−30205.0	−2.59200
$515$	0	0
$516$	23919.0	2.04065
$517$	−2730.00	−0.232235
$518$	8515.00	0.722254
$519$	2352.00	0.198924
$520$	0	0
$521$	2583.00	0.217204	0.108602	−	0.994085i	$-0.465363\pi$
$521$	2583.00	0.217204	0.108602	+	0.994085i	$0.465363\pi$
$522$	−9020.00	−0.756312
$523$	−18620.0	−1.55678	−0.778390	−	0.627781i	$-0.783963\pi$
$523$	−18620.0	−1.55678	−0.778390	+	0.627781i	$0.783963\pi$
$524$	24395.0	2.03378
$525$	0	0
$526$	18540.0	1.53685
$527$	−15092.0	−1.24747
$528$	−16198.0	−1.33509
$529$	−2951.00	−0.242541
$530$	0	0
$531$	−6468.00	−0.528601
$532$	−27846.0	−2.26932
$533$	−4368.00	−0.354970
$534$	−41650.0	−3.37523
$535$	0	0
$536$	−21510.0	−1.73338
$537$	−21203.0	−1.70387
$538$	41720.0	3.34327
$539$	4524.00	0.361526
$540$	0	0
$541$	−16833.0	−1.33772	−0.668861	−	0.743388i	$-0.733218\pi$
$541$	−16833.0	−1.33772	−0.668861	+	0.743388i	$0.733218\pi$
$542$	−8085.00	−0.640739
$543$	−196.000	−0.0154902
$544$	−6545.00	−0.515836
$545$	0	0
$546$	−5915.00	−0.463624
$547$	8615.00	0.673402	0.336701	−	0.941612i	$-0.390689\pi$
$547$	8615.00	0.673402	0.336701	+	0.941612i	$0.390689\pi$
$548$	30192.0	2.35354
$549$	−1232.00	−0.0957750
$550$	0	0
$551$	10332.0	0.798835
$552$	30240.0	2.33170
$553$	16952.0	1.30357
$554$	19100.0	1.46477
$555$	0	0
$556$	−31773.0	−2.42352
$557$	−8535.00	−0.649263	−0.324632	−	0.945841i	$-0.605240\pi$
$557$	−8535.00	−0.649263	−0.324632	+	0.945841i	$0.605240\pi$
$558$	21560.0	1.63568
$559$	−2613.00	−0.197707
$560$	0	0
$561$	14014.0	1.05467
$562$	−31070.0	−2.33204
$563$	4641.00	0.347415	0.173708	−	0.984797i	$-0.444425\pi$
$563$	4641.00	0.347415	0.173708	+	0.984797i	$0.444425\pi$
$564$	12495.0	0.932862
$565$	0	0
$566$	26460.0	1.96501
$567$	−10907.0	−0.807850
$568$	405.000	0.0299180
$569$	−4793.00	−0.353134	−0.176567	−	0.984289i	$-0.556499\pi$
$569$	−4793.00	−0.353134	−0.176567	+	0.984289i	$0.556499\pi$
$570$	0	0
$571$	−5563.00	−0.407713	−0.203857	−	0.979001i	$-0.565348\pi$
$571$	−5563.00	−0.407713	−0.203857	+	0.979001i	$0.565348\pi$
$572$	5746.00	0.420022
$573$	2954.00	0.215367
$574$	21840.0	1.58813
$575$	0	0
$576$	−6314.00	−0.456742
$577$	−24038.0	−1.73434	−0.867171	−	0.498011i	$-0.834064\pi$
$577$	−24038.0	−1.73434	−0.867171	+	0.498011i	$0.834064\pi$
$578$	5080.00	0.365571
$579$	−3444.00	−0.247198
$580$	0	0
$581$	4004.00	0.285910
$582$	−2450.00	−0.174494
$583$	−11232.0	−0.797911
$584$	−4410.00	−0.312478
$585$	0	0
$586$	4515.00	0.318281
$587$	21224.0	1.49235	0.746174	−	0.665751i	$-0.231889\pi$
$587$	21224.0	1.49235	0.746174	+	0.665751i	$0.231889\pi$
$588$	−20706.0	−1.45221
$589$	−24696.0	−1.72764
$590$	0	0
$591$	−20937.0	−1.45725
$592$	11659.0	0.809429
$593$	−4354.00	−0.301513	−0.150757	−	0.988571i	$-0.548171\pi$
$593$	−4354.00	−0.301513	−0.150757	+	0.988571i	$0.548171\pi$
$594$	4550.00	0.314291
$595$	0	0
$596$	41922.0	2.88119
$597$	−490.000	−0.0335919
$598$	−6240.00	−0.426710
$599$	7310.00	0.498629	0.249314	−	0.968423i	$-0.419795\pi$
$599$	7310.00	0.498629	0.249314	+	0.968423i	$0.419795\pi$
$600$	0	0
$601$	−7595.00	−0.515485	−0.257743	−	0.966214i	$-0.582979\pi$
$601$	−7595.00	−0.515485	−0.257743	+	0.966214i	$0.582979\pi$
$602$	13065.0	0.884534
$603$	−10516.0	−0.710190
$604$	−56491.0	−3.80561
$605$	0	0
$606$	14700.0	0.985391
$607$	826.000	0.0552328	0.0276164	−	0.999619i	$-0.491208\pi$
$607$	826.000	0.0552328	0.0276164	+	0.999619i	$0.491208\pi$
$608$	−10710.0	−0.714388
$609$	−7462.00	−0.496511
$610$	0	0
$611$	−1365.00	−0.0903797
$612$	−28798.0	−1.90211
$613$	−14590.0	−0.961312	−0.480656	−	0.876909i	$-0.659602\pi$
$613$	−14590.0	−0.961312	−0.480656	+	0.876909i	$0.659602\pi$
$614$	−10570.0	−0.694740
$615$	0	0
$616$	−15210.0	−0.994851
$617$	−4888.00	−0.318936	−0.159468	−	0.987203i	$-0.550978\pi$
$617$	−4888.00	−0.318936	−0.159468	+	0.987203i	$0.550978\pi$
$618$	−20580.0	−1.33956
$619$	−11004.0	−0.714520	−0.357260	−	0.934005i	$-0.616289\pi$
$619$	−11004.0	−0.714520	−0.357260	+	0.934005i	$0.616289\pi$
$620$	0	0
$621$	−3360.00	−0.217121
$622$	17010.0	1.09653
$623$	−15470.0	−0.994851
$624$	−8099.00	−0.519582
$625$	0	0
$626$	53445.0	3.41229
$627$	22932.0	1.46063
$628$	46410.0	2.94898
$629$	−10087.0	−0.639420
$630$	0	0
$631$	−4975.00	−0.313869	−0.156935	−	0.987609i	$-0.550161\pi$
$631$	−4975.00	−0.313869	−0.156935	+	0.987609i	$0.550161\pi$
$632$	58680.0	3.69330
$633$	19957.0	1.25311
$634$	35270.0	2.20939
$635$	0	0
$636$	51408.0	3.20513
$637$	2262.00	0.140697
$638$	10660.0	0.661494
$639$	198.000	0.0122578
$640$	0	0
$641$	3950.00	0.243394	0.121697	−	0.992567i	$-0.461166\pi$
$641$	3950.00	0.243394	0.121697	+	0.992567i	$0.461166\pi$
$642$	23940.0	1.47171
$643$	3682.00	0.225823	0.112911	−	0.993605i	$-0.463982\pi$
$643$	3682.00	0.225823	0.112911	+	0.993605i	$0.463982\pi$
$644$	21216.0	1.29818
$645$	0	0
$646$	48510.0	2.95449
$647$	−10402.0	−0.632063	−0.316032	−	0.948749i	$-0.602351\pi$
$647$	−10402.0	−0.632063	−0.316032	+	0.948749i	$0.602351\pi$
$648$	−37755.0	−2.28882
$649$	7644.00	0.462332
$650$	0	0
$651$	17836.0	1.07381
$652$	9248.00	0.555490
$653$	31680.0	1.89852	0.949260	−	0.314491i	$-0.101834\pi$
$653$	31680.0	1.89852	0.949260	+	0.314491i	$0.101834\pi$
$654$	13055.0	0.780567
$655$	0	0
$656$	29904.0	1.77981
$657$	−2156.00	−0.128027
$658$	6825.00	0.404356
$659$	21940.0	1.29691	0.648453	−	0.761255i	$-0.275416\pi$
$659$	21940.0	1.29691	0.648453	+	0.761255i	$0.275416\pi$
$660$	0	0
$661$	−31374.0	−1.84615	−0.923077	−	0.384616i	$-0.874334\pi$
$661$	−31374.0	−1.84615	−0.923077	+	0.384616i	$0.874334\pi$
$662$	48520.0	2.84862
$663$	7007.00	0.410451
$664$	13860.0	0.810049
$665$	0	0
$666$	14410.0	0.838403
$667$	−7872.00	−0.456979
$668$	−27608.0	−1.59908
$669$	−1519.00	−0.0877847
$670$	0	0
$671$	1456.00	0.0837679
$672$	7735.00	0.444024
$673$	−18013.0	−1.03172	−0.515862	−	0.856672i	$-0.672528\pi$
$673$	−18013.0	−1.03172	−0.515862	+	0.856672i	$0.672528\pi$
$674$	52245.0	2.98576
$675$	0	0
$676$	2873.00	0.163462
$677$	10640.0	0.604030	0.302015	−	0.953303i	$-0.402341\pi$
$677$	10640.0	0.604030	0.302015	+	0.953303i	$0.402341\pi$
$678$	60690.0	3.43774
$679$	−910.000	−0.0514324
$680$	0	0
$681$	18032.0	1.01467
$682$	−25480.0	−1.43062
$683$	9336.00	0.523034	0.261517	−	0.965199i	$-0.415777\pi$
$683$	9336.00	0.523034	0.261517	+	0.965199i	$0.415777\pi$
$684$	−47124.0	−2.63426
$685$	0	0
$686$	−33605.0	−1.87033
$687$	3185.00	0.176878
$688$	17889.0	0.991296
$689$	−5616.00	−0.310526
$690$	0	0
$691$	4200.00	0.231224	0.115612	−	0.993294i	$-0.463117\pi$
$691$	4200.00	0.231224	0.115612	+	0.993294i	$0.463117\pi$
$692$	5712.00	0.313783
$693$	−7436.00	−0.407605
$694$	3105.00	0.169833
$695$	0	0
$696$	−25830.0	−1.40673
$697$	−25872.0	−1.40599
$698$	62405.0	3.38405
$699$	−21427.0	−1.15943
$700$	0	0
$701$	9872.00	0.531898	0.265949	−	0.963987i	$-0.414315\pi$
$701$	9872.00	0.531898	0.265949	+	0.963987i	$0.414315\pi$
$702$	2275.00	0.122314
$703$	−16506.0	−0.885541
$704$	7462.00	0.399481
$705$	0	0
$706$	7000.00	0.373156
$707$	5460.00	0.290445
$708$	−34986.0	−1.85714
$709$	28450.0	1.50700	0.753499	−	0.657449i	$-0.228364\pi$
$709$	28450.0	1.50700	0.753499	+	0.657449i	$0.228364\pi$
$710$	0	0
$711$	28688.0	1.51320
$712$	−53550.0	−2.81864
$713$	18816.0	0.988310
$714$	−35035.0	−1.83635
$715$	0	0
$716$	−51493.0	−2.68769
$717$	−24339.0	−1.26772
$718$	−24840.0	−1.29111
$719$	32718.0	1.69705	0.848523	−	0.529159i	$-0.177493\pi$
$719$	32718.0	1.69705	0.848523	+	0.529159i	$0.177493\pi$
$720$	0	0
$721$	−7644.00	−0.394837
$722$	45085.0	2.32395
$723$	−11270.0	−0.579718
$724$	−476.000	−0.0244343
$725$	0	0
$726$	−22925.0	−1.17194
$727$	22834.0	1.16488	0.582439	−	0.812874i	$-0.302099\pi$
$727$	22834.0	1.16488	0.582439	+	0.812874i	$0.302099\pi$
$728$	−7605.00	−0.387170
$729$	−11843.0	−0.601687
$730$	0	0
$731$	−15477.0	−0.783088
$732$	−6664.00	−0.336487
$733$	−7875.00	−0.396821	−0.198410	−	0.980119i	$-0.563578\pi$
$733$	−7875.00	−0.396821	−0.198410	+	0.980119i	$0.563578\pi$
$734$	−43610.0	−2.19302
$735$	0	0
$736$	8160.00	0.408671
$737$	12428.0	0.621155
$738$	36960.0	1.84352
$739$	−2140.00	−0.106524	−0.0532620	−	0.998581i	$-0.516962\pi$
$739$	−2140.00	−0.106524	−0.0532620	+	0.998581i	$0.516962\pi$
$740$	0	0
$741$	11466.0	0.568440
$742$	28080.0	1.38928
$743$	−31971.0	−1.57860	−0.789302	−	0.614006i	$-0.789557\pi$
$743$	−31971.0	−1.57860	−0.789302	+	0.614006i	$0.789557\pi$
$744$	61740.0	3.04234
$745$	0	0
$746$	−50060.0	−2.45687
$747$	6776.00	0.331889
$748$	34034.0	1.66364
$749$	8892.00	0.433787
$750$	0	0
$751$	−7432.00	−0.361115	−0.180558	−	0.983564i	$-0.557790\pi$
$751$	−7432.00	−0.361115	−0.180558	+	0.983564i	$0.557790\pi$
$752$	9345.00	0.453161
$753$	7056.00	0.341481
$754$	5330.00	0.257437
$755$	0	0
$756$	−7735.00	−0.372115
$757$	−20176.0	−0.968704	−0.484352	−	0.874873i	$-0.660945\pi$
$757$	−20176.0	−0.968704	−0.484352	+	0.874873i	$0.660945\pi$
$758$	−16860.0	−0.807893
$759$	−17472.0	−0.835564
$760$	0	0
$761$	−9478.00	−0.451481	−0.225741	−	0.974187i	$-0.572480\pi$
$761$	−9478.00	−0.451481	−0.225741	+	0.974187i	$0.572480\pi$
$762$	−66220.0	−3.14816
$763$	4849.00	0.230073
$764$	7174.00	0.339720
$765$	0	0
$766$	4235.00	0.199761
$767$	3822.00	0.179928
$768$	−57953.0	−2.72292
$769$	−12096.0	−0.567221	−0.283610	−	0.958940i	$-0.591532\pi$
$769$	−12096.0	−0.567221	−0.283610	+	0.958940i	$0.591532\pi$
$770$	0	0
$771$	−42287.0	−1.97526
$772$	−8364.00	−0.389931
$773$	−17941.0	−0.834790	−0.417395	−	0.908725i	$-0.637057\pi$
$773$	−17941.0	−0.834790	−0.417395	+	0.908725i	$0.637057\pi$
$774$	22110.0	1.02678
$775$	0	0
$776$	−3150.00	−0.145720
$777$	11921.0	0.550403
$778$	56570.0	2.60685
$779$	−42336.0	−1.94717
$780$	0	0
$781$	−234.000	−0.0107211
$782$	−36960.0	−1.69014
$783$	2870.00	0.130990
$784$	−15486.0	−0.705448
$785$	0	0
$786$	50225.0	2.27922
$787$	−6664.00	−0.301837	−0.150919	−	0.988546i	$-0.548223\pi$
$787$	−6664.00	−0.301837	−0.150919	+	0.988546i	$0.548223\pi$
$788$	−50847.0	−2.29867
$789$	25956.0	1.17118
$790$	0	0
$791$	22542.0	1.01328
$792$	−25740.0	−1.15484
$793$	728.000	0.0326003
$794$	−9310.00	−0.416120
$795$	0	0
$796$	−1190.00	−0.0529880
$797$	1442.00	0.0640882	0.0320441	−	0.999486i	$-0.489798\pi$
$797$	1442.00	0.0640882	0.0320441	+	0.999486i	$0.489798\pi$
$798$	−57330.0	−2.54318
$799$	−8085.00	−0.357981
$800$	0	0
$801$	−26180.0	−1.15484
$802$	34100.0	1.50139
$803$	2548.00	0.111976
$804$	−56882.0	−2.49512
$805$	0	0
$806$	−12740.0	−0.556759
$807$	58408.0	2.54778
$808$	18900.0	0.822896
$809$	30207.0	1.31276	0.656379	−	0.754431i	$-0.272087\pi$
$809$	30207.0	1.31276	0.656379	+	0.754431i	$0.272087\pi$
$810$	0	0
$811$	21140.0	0.915322	0.457661	−	0.889127i	$-0.348687\pi$
$811$	21140.0	0.915322	0.457661	+	0.889127i	$0.348687\pi$
$812$	−18122.0	−0.783199
$813$	−11319.0	−0.488284
$814$	−17030.0	−0.733294
$815$	0	0
$816$	−47971.0	−2.05799
$817$	−25326.0	−1.08451
$818$	−64960.0	−2.77662
$819$	−3718.00	−0.158629
$820$	0	0
$821$	569.000	0.0241879	0.0120939	−	0.999927i	$-0.496150\pi$
$821$	569.000	0.0241879	0.0120939	+	0.999927i	$0.496150\pi$
$822$	62160.0	2.63757
$823$	8538.00	0.361623	0.180812	−	0.983518i	$-0.442128\pi$
$823$	8538.00	0.361623	0.180812	+	0.983518i	$0.442128\pi$
$824$	−26460.0	−1.11866
$825$	0	0
$826$	−19110.0	−0.804990
$827$	32702.0	1.37504	0.687521	−	0.726164i	$-0.258699\pi$
$827$	32702.0	1.37504	0.687521	+	0.726164i	$0.258699\pi$
$828$	35904.0	1.50694
$829$	−21154.0	−0.886259	−0.443130	−	0.896458i	$-0.646132\pi$
$829$	−21154.0	−0.886259	−0.443130	+	0.896458i	$0.646132\pi$
$830$	0	0
$831$	26740.0	1.11625
$832$	3731.00	0.155468
$833$	13398.0	0.557279
$834$	−65415.0	−2.71599
$835$	0	0
$836$	55692.0	2.30400
$837$	−6860.00	−0.283293
$838$	−36715.0	−1.51348
$839$	−2184.00	−0.0898690	−0.0449345	−	0.998990i	$-0.514308\pi$
$839$	−2184.00	−0.0898690	−0.0449345	+	0.998990i	$0.514308\pi$
$840$	0	0
$841$	−17665.0	−0.724302
$842$	−25295.0	−1.03530
$843$	−43498.0	−1.77717
$844$	48467.0	1.97666
$845$	0	0
$846$	11550.0	0.469382
$847$	−8515.00	−0.345430
$848$	38448.0	1.55697
$849$	37044.0	1.49746
$850$	0	0
$851$	12576.0	0.506580
$852$	1071.00	0.0430656
$853$	−36687.0	−1.47261	−0.736307	−	0.676648i	$-0.763432\pi$
$853$	−36687.0	−1.47261	−0.736307	+	0.676648i	$0.763432\pi$
$854$	−3640.00	−0.145853
$855$	0	0
$856$	30780.0	1.22902
$857$	−36806.0	−1.46706	−0.733529	−	0.679658i	$-0.762128\pi$
$857$	−36806.0	−1.46706	−0.733529	+	0.679658i	$0.762128\pi$
$858$	11830.0	0.470710
$859$	4900.00	0.194628	0.0973142	−	0.995254i	$-0.468975\pi$
$859$	4900.00	0.194628	0.0973142	+	0.995254i	$0.468975\pi$
$860$	0	0
$861$	30576.0	1.21025
$862$	16215.0	0.640702
$863$	13697.0	0.540268	0.270134	−	0.962823i	$-0.412932\pi$
$863$	13697.0	0.540268	0.270134	+	0.962823i	$0.412932\pi$
$864$	−2975.00	−0.117143
$865$	0	0
$866$	−57995.0	−2.27569
$867$	7112.00	0.278588
$868$	43316.0	1.69383
$869$	−33904.0	−1.32349
$870$	0	0
$871$	6214.00	0.241737
$872$	16785.0	0.651848
$873$	−1540.00	−0.0597034
$874$	−60480.0	−2.34069
$875$	0	0
$876$	−11662.0	−0.449797
$877$	−6239.00	−0.240224	−0.120112	−	0.992760i	$-0.538325\pi$
$877$	−6239.00	−0.240224	−0.120112	+	0.992760i	$0.538325\pi$
$878$	−86870.0	−3.33909
$879$	6321.00	0.242551
$880$	0	0
$881$	133.000	0.00508613	0.00254307	−	0.999997i	$-0.499191\pi$
$881$	133.000	0.00508613	0.00254307	+	0.999997i	$0.499191\pi$
$882$	−19140.0	−0.730700
$883$	26003.0	0.991020	0.495510	−	0.868602i	$-0.334981\pi$
$883$	26003.0	0.991020	0.495510	+	0.868602i	$0.334981\pi$
$884$	17017.0	0.647448
$885$	0	0
$886$	−4945.00	−0.187506
$887$	31248.0	1.18287	0.591435	−	0.806353i	$-0.298562\pi$
$887$	31248.0	1.18287	0.591435	+	0.806353i	$0.298562\pi$
$888$	41265.0	1.55942
$889$	−24596.0	−0.927923
$890$	0	0
$891$	21814.0	0.820198
$892$	−3689.00	−0.138472
$893$	−13230.0	−0.495773
$894$	86310.0	3.22890
$895$	0	0
$896$	−27495.0	−1.02516
$897$	−8736.00	−0.325180
$898$	−72370.0	−2.68933
$899$	−16072.0	−0.596253
$900$	0	0
$901$	−33264.0	−1.22995
$902$	−43680.0	−1.61240
$903$	18291.0	0.674071
$904$	78030.0	2.87084
$905$	0	0
$906$	−116305.	−4.26487
$907$	38253.0	1.40041	0.700204	−	0.713943i	$-0.253092\pi$
$907$	38253.0	1.40041	0.700204	+	0.713943i	$0.253092\pi$
$908$	43792.0	1.60054
$909$	9240.00	0.337152
$910$	0	0
$911$	36374.0	1.32286	0.661429	−	0.750007i	$-0.269950\pi$
$911$	36374.0	1.32286	0.661429	+	0.750007i	$0.269950\pi$
$912$	−78498.0	−2.85014
$913$	−8008.00	−0.290281
$914$	7970.00	0.288429
$915$	0	0
$916$	7735.00	0.279008
$917$	18655.0	0.671802
$918$	13475.0	0.484468
$919$	−27648.0	−0.992408	−0.496204	−	0.868206i	$-0.665273\pi$
$919$	−27648.0	−0.992408	−0.496204	+	0.868206i	$0.665273\pi$
$920$	0	0
$921$	−14798.0	−0.529436
$922$	−29575.0	−1.05640
$923$	−117.000	−0.00417237
$924$	−40222.0	−1.43204
$925$	0	0
$926$	55360.0	1.96462
$927$	−12936.0	−0.458332
$928$	−6970.00	−0.246553
$929$	756.000	0.0266992	0.0133496	−	0.999911i	$-0.495751\pi$
$929$	756.000	0.0266992	0.0133496	+	0.999911i	$0.495751\pi$
$930$	0	0
$931$	21924.0	0.771783
$932$	−52037.0	−1.82889
$933$	23814.0	0.835622
$934$	−6300.00	−0.220709
$935$	0	0
$936$	−12870.0	−0.449433
$937$	−20846.0	−0.726797	−0.363399	−	0.931634i	$-0.618384\pi$
$937$	−20846.0	−0.726797	−0.363399	+	0.931634i	$0.618384\pi$
$938$	−31070.0	−1.08153
$939$	74823.0	2.60038
$940$	0	0
$941$	−41321.0	−1.43148	−0.715742	−	0.698365i	$-0.753911\pi$
$941$	−41321.0	−1.43148	−0.715742	+	0.698365i	$0.753911\pi$
$942$	95550.0	3.30487
$943$	32256.0	1.11389
$944$	−26166.0	−0.902151
$945$	0	0
$946$	−26130.0	−0.898055
$947$	−54966.0	−1.88612	−0.943060	−	0.332624i	$-0.892066\pi$
$947$	−54966.0	−1.88612	−0.943060	+	0.332624i	$0.892066\pi$
$948$	155176.	5.31633
$949$	1274.00	0.0435783
$950$	0	0
$951$	49378.0	1.68369
$952$	−45045.0	−1.53353
$953$	44553.0	1.51439	0.757195	−	0.653189i	$-0.226569\pi$
$953$	44553.0	1.51439	0.757195	+	0.653189i	$0.226569\pi$
$954$	47520.0	1.61270
$955$	0	0
$956$	−59109.0	−1.99971
$957$	14924.0	0.504101
$958$	−60165.0	−2.02906
$959$	23088.0	0.777425
$960$	0	0
$961$	8625.00	0.289517
$962$	−8515.00	−0.285379
$963$	15048.0	0.503546
$964$	−27370.0	−0.914448
$965$	0	0
$966$	43680.0	1.45485
$967$	27907.0	0.928054	0.464027	−	0.885821i	$-0.346404\pi$
$967$	27907.0	0.928054	0.464027	+	0.885821i	$0.346404\pi$
$968$	−29475.0	−0.978680
$969$	67914.0	2.25151
$970$	0	0
$971$	−16443.0	−0.543441	−0.271720	−	0.962376i	$-0.587593\pi$
$971$	−16443.0	−0.543441	−0.271720	+	0.962376i	$0.587593\pi$
$972$	−83776.0	−2.76452
$973$	−24297.0	−0.800541
$974$	11400.0	0.375030
$975$	0	0
$976$	−4984.00	−0.163457
$977$	45414.0	1.48713	0.743563	−	0.668666i	$-0.233134\pi$
$977$	45414.0	1.48713	0.743563	+	0.668666i	$0.233134\pi$
$978$	19040.0	0.622528
$979$	30940.0	1.01006
$980$	0	0
$981$	8206.00	0.267072
$982$	83835.0	2.72432
$983$	8981.00	0.291403	0.145702	−	0.989329i	$-0.453456\pi$
$983$	8981.00	0.291403	0.145702	+	0.989329i	$0.453456\pi$
$984$	105840.	3.42892
$985$	0	0
$986$	31570.0	1.01967
$987$	9555.00	0.308145
$988$	27846.0	0.896659
$989$	19296.0	0.620402
$990$	0	0
$991$	−17414.0	−0.558198	−0.279099	−	0.960262i	$-0.590036\pi$
$991$	−17414.0	−0.558198	−0.279099	+	0.960262i	$0.590036\pi$
$992$	16660.0	0.533221
$993$	67928.0	2.17083
$994$	585.000	0.0186671
$995$	0	0
$996$	36652.0	1.16603
$997$	23702.0	0.752909	0.376454	−	0.926435i	$-0.377143\pi$
$997$	23702.0	0.752909	0.376454	+	0.926435i	$0.377143\pi$
$998$	64200.0	2.03629
$999$	−4585.00	−0.145208

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	325.4.a.d.1.1		1
5.2	odd	4		325.4.b.b.274.2		2
5.3	odd	4		325.4.b.b.274.1		2
5.4	even	2		13.4.a.a.1.1	✓	1
15.14	odd	2		117.4.a.b.1.1		1
20.19	odd	2		208.4.a.g.1.1		1
35.34	odd	2		637.4.a.a.1.1		1
40.19	odd	2		832.4.a.a.1.1		1
40.29	even	2		832.4.a.r.1.1		1
55.54	odd	2		1573.4.a.a.1.1		1
60.59	even	2		1872.4.a.k.1.1		1
65.4	even	6		169.4.c.a.146.1		2
65.9	even	6		169.4.c.e.146.1		2
65.19	odd	12		169.4.e.e.23.2		4
65.24	odd	12		169.4.e.e.147.2		4
65.29	even	6		169.4.c.e.22.1		2
65.34	odd	4		169.4.b.a.168.1		2
65.44	odd	4		169.4.b.a.168.2		2
65.49	even	6		169.4.c.a.22.1		2
65.54	odd	12		169.4.e.e.147.1		4
65.59	odd	12		169.4.e.e.23.1		4
65.64	even	2		169.4.a.e.1.1		1
195.194	odd	2		1521.4.a.a.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
13.4.a.a.1.1	✓	1	5.4	even	2
117.4.a.b.1.1		1	15.14	odd	2
169.4.a.e.1.1		1	65.64	even	2
169.4.b.a.168.1		2	65.34	odd	4
169.4.b.a.168.2		2	65.44	odd	4
169.4.c.a.22.1		2	65.49	even	6
169.4.c.a.146.1		2	65.4	even	6
169.4.c.e.22.1		2	65.29	even	6
169.4.c.e.146.1		2	65.9	even	6
169.4.e.e.23.1		4	65.59	odd	12
169.4.e.e.23.2		4	65.19	odd	12
169.4.e.e.147.1		4	65.54	odd	12
169.4.e.e.147.2		4	65.24	odd	12
208.4.a.g.1.1		1	20.19	odd	2
325.4.a.d.1.1		1	1.1	even	1	trivial
325.4.b.b.274.1		2	5.3	odd	4
325.4.b.b.274.2		2	5.2	odd	4
637.4.a.a.1.1		1	35.34	odd	2
832.4.a.a.1.1		1	40.19	odd	2
832.4.a.r.1.1		1	40.29	even	2
1521.4.a.a.1.1		1	195.194	odd	2
1573.4.a.a.1.1		1	55.54	odd	2
1872.4.a.k.1.1		1	60.59	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:	\( N \)	\(=\)	\( 325 = 5^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	325.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more