LMFDB - Embedded newform 693.4.a.n.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [693,4,Mod(1,693)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(693, 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("693.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 693 = 3^{2} \cdot 7 \cdot 11 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	693.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:	$40.8883236340$
Analytic rank:	$1$
Dimension:	$5$
Coefficient field:	$\mathbb{Q}[x]/(x^{5} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 28x^{3} - 11x^{2} + 108x - 64 $ x^5 - 28x^3 - 11x^2 + 108*x - 64
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 231)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-2.85551$ of defining polynomial
Character	$\chi$	$=$	693.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q-3.85551 q^{2} +6.86494 q^{4} -18.0422 q^{5} -7.00000 q^{7} +4.37623 q^{8} +69.5618 q^{10} -11.0000 q^{11} +16.8358 q^{13} +26.9886 q^{14} -71.7921 q^{16} -76.0041 q^{17} +11.8937 q^{19} -123.858 q^{20} +42.4106 q^{22} +175.501 q^{23} +200.520 q^{25} -64.9104 q^{26} -48.0546 q^{28} +219.409 q^{29} -214.355 q^{31} +241.785 q^{32} +293.035 q^{34} +126.295 q^{35} +272.754 q^{37} -45.8562 q^{38} -78.9567 q^{40} -204.925 q^{41} +406.384 q^{43} -75.5144 q^{44} -676.647 q^{46} -178.653 q^{47} +49.0000 q^{49} -773.107 q^{50} +115.577 q^{52} +238.316 q^{53} +198.464 q^{55} -30.6336 q^{56} -845.931 q^{58} +584.798 q^{59} -910.607 q^{61} +826.449 q^{62} -357.868 q^{64} -303.754 q^{65} -387.749 q^{67} -521.764 q^{68} -486.932 q^{70} +668.091 q^{71} +390.298 q^{73} -1051.60 q^{74} +81.6495 q^{76} +77.0000 q^{77} -250.647 q^{79} +1295.29 q^{80} +790.090 q^{82} +54.2027 q^{83} +1371.28 q^{85} -1566.82 q^{86} -48.1385 q^{88} +906.680 q^{89} -117.850 q^{91} +1204.81 q^{92} +688.796 q^{94} -214.588 q^{95} +618.333 q^{97} -188.920 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - 5 q^{2} + 21 q^{4} - 7 q^{5} - 35 q^{7} - 60 q^{8} + 55 q^{10} - 55 q^{11} + 111 q^{13} + 35 q^{14} + 201 q^{16} - 136 q^{17} + 111 q^{19} - 219 q^{20} + 55 q^{22} + 28 q^{23} + 190 q^{25} + q^{26}+ \cdots - 245 q^{98}+O(q^{100}) $ 5 * q - 5 * q^2 + 21 * q^4 - 7 * q^5 - 35 * q^7 - 60 * q^8 + 55 * q^10 - 55 * q^11 + 111 * q^13 + 35 * q^14 + 201 * q^16 - 136 * q^17 + 111 * q^19 - 219 * q^20 + 55 * q^22 + 28 * q^23 + 190 * q^25 + q^26 - 147 * q^28 - 61 * q^29 - 280 * q^31 - 535 * q^32 - 572 * q^34 + 49 * q^35 - 41 * q^37 - 267 * q^38 - 336 * q^40 - 426 * q^41 + 424 * q^43 - 231 * q^44 + 140 * q^46 - 75 * q^47 + 245 * q^49 - 490 * q^50 - 269 * q^52 - 1500 * q^53 + 77 * q^55 + 420 * q^56 - 1767 * q^58 - 757 * q^59 + 658 * q^61 + 568 * q^62 - 748 * q^64 - 537 * q^65 - 583 * q^67 + 1650 * q^68 - 385 * q^70 + 764 * q^71 + 875 * q^73 + 825 * q^74 + 213 * q^76 + 385 * q^77 - 244 * q^79 + 2577 * q^80 - 2006 * q^82 - 924 * q^83 - 1402 * q^85 - 1272 * q^86 + 660 * q^88 + 1110 * q^89 - 777 * q^91 + 2046 * q^92 - 3349 * q^94 - 1923 * q^95 - 852 * q^97 - 245 * q^98

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 $

$2$	−3.85551	−1.36313	−0.681564	−	0.731759i	$-0.738700\pi$
$2$	−3.85551	−1.36313	−0.681564	+	0.731759i	$0.738700\pi$
$3$	0	0
$3$	0	0
$4$	6.86494	0.858118
$5$	−18.0422	−1.61374	−0.806871	−	0.590728i	$-0.798841\pi$
$5$	−18.0422	−1.61374	−0.806871	+	0.590728i	$0.798841\pi$
$6$	0	0
$7$	−7.00000	−0.377964
$7$	−7.00000	−0.377964
$8$	4.37623	0.193404
$9$	0	0
$10$	69.5618	2.19974
$11$	−11.0000	−0.301511
$11$	−11.0000	−0.301511
$12$	0	0
$13$	16.8358	0.359185	0.179592	−	0.983741i	$-0.442522\pi$
$13$	16.8358	0.359185	0.179592	+	0.983741i	$0.442522\pi$
$14$	26.9886	0.515214
$15$	0	0
$16$	−71.7921	−1.12175
$17$	−76.0041	−1.08434	−0.542168	−	0.840270i	$-0.682396\pi$
$17$	−76.0041	−1.08434	−0.542168	+	0.840270i	$0.682396\pi$
$18$	0	0
$19$	11.8937	0.143611	0.0718053	−	0.997419i	$-0.477124\pi$
$19$	11.8937	0.143611	0.0718053	+	0.997419i	$0.477124\pi$
$20$	−123.858	−1.38478
$21$	0	0
$22$	42.4106	0.410999
$23$	175.501	1.59107	0.795534	−	0.605909i	$-0.207190\pi$
$23$	175.501	1.59107	0.795534	+	0.605909i	$0.207190\pi$
$24$	0	0
$25$	200.520	1.60416
$26$	−64.9104	−0.489615
$27$	0	0
$28$	−48.0546	−0.324338
$29$	219.409	1.40494	0.702468	−	0.711715i	$-0.252081\pi$
$29$	219.409	1.40494	0.702468	+	0.711715i	$0.252081\pi$
$30$	0	0
$31$	−214.355	−1.24192	−0.620958	−	0.783844i	$-0.713256\pi$
$31$	−214.355	−1.24192	−0.620958	+	0.783844i	$0.713256\pi$
$32$	241.785	1.33569
$33$	0	0
$34$	293.035	1.47809
$35$	126.295	0.609937
$36$	0	0
$37$	272.754	1.21190	0.605952	−	0.795501i	$-0.292792\pi$
$37$	272.754	1.21190	0.605952	+	0.795501i	$0.292792\pi$
$38$	−45.8562	−0.195760
$39$	0	0
$40$	−78.9567	−0.312104
$41$	−204.925	−0.780583	−0.390292	−	0.920691i	$-0.627626\pi$
$41$	−204.925	−0.780583	−0.390292	+	0.920691i	$0.627626\pi$
$42$	0	0
$43$	406.384	1.44123	0.720616	−	0.693334i	$-0.243859\pi$
$43$	406.384	1.44123	0.720616	+	0.693334i	$0.243859\pi$
$44$	−75.5144	−0.258732
$45$	0	0
$46$	−676.647	−2.16883
$47$	−178.653	−0.554450	−0.277225	−	0.960805i	$-0.589415\pi$
$47$	−178.653	−0.554450	−0.277225	+	0.960805i	$0.589415\pi$
$48$	0	0
$49$	49.0000	0.142857
$50$	−773.107	−2.18668
$51$	0	0
$52$	115.577	0.308223
$53$	238.316	0.617646	0.308823	−	0.951120i	$-0.400065\pi$
$53$	238.316	0.617646	0.308823	+	0.951120i	$0.400065\pi$
$54$	0	0
$55$	198.464	0.486561
$56$	−30.6336	−0.0730998
$57$	0	0
$58$	−845.931	−1.91511
$59$	584.798	1.29041	0.645205	−	0.764009i	$-0.276772\pi$
$59$	584.798	1.29041	0.645205	+	0.764009i	$0.276772\pi$
$60$	0	0
$61$	−910.607	−1.91133	−0.955666	−	0.294452i	$-0.904863\pi$
$61$	−910.607	−1.91133	−0.955666	+	0.294452i	$0.904863\pi$
$62$	826.449	1.69289
$63$	0	0
$64$	−357.868	−0.698961
$65$	−303.754	−0.579631
$66$	0	0
$67$	−387.749	−0.707030	−0.353515	−	0.935429i	$-0.615014\pi$
$67$	−387.749	−0.707030	−0.353515	+	0.935429i	$0.615014\pi$
$68$	−521.764	−0.930488
$69$	0	0
$70$	−486.932	−0.831422
$71$	668.091	1.11673	0.558365	−	0.829595i	$-0.311429\pi$
$71$	668.091	1.11673	0.558365	+	0.829595i	$0.311429\pi$
$72$	0	0
$73$	390.298	0.625766	0.312883	−	0.949792i	$-0.398705\pi$
$73$	390.298	0.625766	0.312883	+	0.949792i	$0.398705\pi$
$74$	−1051.60	−1.65198
$75$	0	0
$76$	81.6495	0.123235
$77$	77.0000	0.113961
$78$	0	0
$79$	−250.647	−0.356962	−0.178481	−	0.983943i	$-0.557118\pi$
$79$	−250.647	−0.356962	−0.178481	+	0.983943i	$0.557118\pi$
$80$	1295.29	1.81022
$81$	0	0
$82$	790.090	1.06403
$83$	54.2027	0.0716809	0.0358404	−	0.999358i	$-0.488589\pi$
$83$	54.2027	0.0716809	0.0358404	+	0.999358i	$0.488589\pi$
$84$	0	0
$85$	1371.28	1.74984
$86$	−1566.82	−1.96458
$87$	0	0
$88$	−48.1385	−0.0583134
$89$	906.680	1.07986	0.539932	−	0.841709i	$-0.318450\pi$
$89$	906.680	1.07986	0.539932	+	0.841709i	$0.318450\pi$
$90$	0	0
$91$	−117.850	−0.135759
$92$	1204.81	1.36532
$93$	0	0
$94$	688.796	0.755786
$95$	−214.588	−0.231750
$96$	0	0
$97$	618.333	0.647239	0.323620	−	0.946187i	$-0.395100\pi$
$97$	618.333	0.647239	0.323620	+	0.946187i	$0.395100\pi$
$98$	−188.920	−0.194733
$99$	0	0
$100$	1376.56	1.37656
$101$	−443.731	−0.437157	−0.218578	−	0.975819i	$-0.570142\pi$
$101$	−443.731	−0.437157	−0.218578	+	0.975819i	$0.570142\pi$
$102$	0	0
$103$	−1893.39	−1.81128	−0.905638	−	0.424051i	$-0.860608\pi$
$103$	−1893.39	−1.81128	−0.905638	+	0.424051i	$0.860608\pi$
$104$	73.6771	0.0694677
$105$	0	0
$106$	−918.830	−0.841930
$107$	−966.336	−0.873077	−0.436538	−	0.899686i	$-0.643796\pi$
$107$	−966.336	−0.873077	−0.436538	+	0.899686i	$0.643796\pi$
$108$	0	0
$109$	86.9483	0.0764049	0.0382025	−	0.999270i	$-0.487837\pi$
$109$	86.9483	0.0764049	0.0382025	+	0.999270i	$0.487837\pi$
$110$	−765.179	−0.663245
$111$	0	0
$112$	502.545	0.423982
$113$	−1645.50	−1.36987	−0.684937	−	0.728602i	$-0.740170\pi$
$113$	−1645.50	−1.36987	−0.684937	+	0.728602i	$0.740170\pi$
$114$	0	0
$115$	−3166.43	−2.56757
$116$	1506.23	1.20560
$117$	0	0
$118$	−2254.69	−1.75899
$119$	532.029	0.409840
$120$	0	0
$121$	121.000	0.0909091
$122$	3510.85	2.60539
$123$	0	0
$124$	−1471.54	−1.06571
$125$	−1362.55	−0.974961
$126$	0	0
$127$	−978.550	−0.683719	−0.341859	−	0.939751i	$-0.611057\pi$
$127$	−978.550	−0.683719	−0.341859	+	0.939751i	$0.611057\pi$
$128$	−554.519	−0.382914
$129$	0	0
$130$	1171.13	0.790112
$131$	−1494.86	−0.996993	−0.498497	−	0.866892i	$-0.666114\pi$
$131$	−1494.86	−0.996993	−0.498497	+	0.866892i	$0.666114\pi$
$132$	0	0
$133$	−83.2559	−0.0542797
$134$	1494.97	0.963773
$135$	0	0
$136$	−332.611	−0.209715
$137$	−2397.79	−1.49530	−0.747652	−	0.664091i	$-0.768819\pi$
$137$	−2397.79	−1.49530	−0.747652	+	0.664091i	$0.768819\pi$
$138$	0	0
$139$	2672.65	1.63087	0.815436	−	0.578847i	$-0.196497\pi$
$139$	2672.65	1.63087	0.815436	+	0.578847i	$0.196497\pi$
$140$	867.009	0.523398
$141$	0	0
$142$	−2575.83	−1.52225
$143$	−185.193	−0.108298
$144$	0	0
$145$	−3958.61	−2.26720
$146$	−1504.80	−0.853000
$147$	0	0
$148$	1872.44	1.03996
$149$	−2052.12	−1.12830	−0.564149	−	0.825673i	$-0.690796\pi$
$149$	−2052.12	−1.12830	−0.564149	+	0.825673i	$0.690796\pi$
$150$	0	0
$151$	2366.66	1.27547	0.637734	−	0.770256i	$-0.279872\pi$
$151$	2366.66	1.27547	0.637734	+	0.770256i	$0.279872\pi$
$152$	52.0495	0.0277748
$153$	0	0
$154$	−296.874	−0.155343
$155$	3867.44	2.00413
$156$	0	0
$157$	−3763.06	−1.91290	−0.956450	−	0.291897i	$-0.905714\pi$
$157$	−3763.06	−1.91290	−0.956450	+	0.291897i	$0.905714\pi$
$158$	966.371	0.486584
$159$	0	0
$160$	−4362.33	−2.15545
$161$	−1228.51	−0.601367
$162$	0	0
$163$	1394.64	0.670166	0.335083	−	0.942189i	$-0.391236\pi$
$163$	1394.64	0.670166	0.335083	+	0.942189i	$0.391236\pi$
$164$	−1406.80	−0.669832
$165$	0	0
$166$	−208.979	−0.0977102
$167$	384.184	0.178018	0.0890091	−	0.996031i	$-0.471630\pi$
$167$	384.184	0.178018	0.0890091	+	0.996031i	$0.471630\pi$
$168$	0	0
$169$	−1913.56	−0.870986
$170$	−5286.98	−2.38525
$171$	0	0
$172$	2789.80	1.23675
$173$	−1156.21	−0.508121	−0.254061	−	0.967188i	$-0.581766\pi$
$173$	−1156.21	−0.508121	−0.254061	+	0.967188i	$0.581766\pi$
$174$	0	0
$175$	−1403.64	−0.606316
$176$	789.713	0.338221
$177$	0	0
$178$	−3495.71	−1.47199
$179$	−141.765	−0.0591956	−0.0295978	−	0.999562i	$-0.509423\pi$
$179$	−141.765	−0.0591956	−0.0295978	+	0.999562i	$0.509423\pi$
$180$	0	0
$181$	1359.83	0.558428	0.279214	−	0.960229i	$-0.409926\pi$
$181$	1359.83	0.558428	0.279214	+	0.960229i	$0.409926\pi$
$182$	454.373	0.185057
$183$	0	0
$184$	768.034	0.307719
$185$	−4921.07	−1.95570
$186$	0	0
$187$	836.045	0.326940
$188$	−1226.44	−0.475783
$189$	0	0
$190$	827.347	0.315905
$191$	−3678.06	−1.39338	−0.696689	−	0.717373i	$-0.745344\pi$
$191$	−3678.06	−1.39338	−0.696689	+	0.717373i	$0.745344\pi$
$192$	0	0
$193$	1107.52	0.413064	0.206532	−	0.978440i	$-0.433782\pi$
$193$	1107.52	0.413064	0.206532	+	0.978440i	$0.433782\pi$
$194$	−2383.99	−0.882270
$195$	0	0
$196$	336.382	0.122588
$197$	693.069	0.250655	0.125328	−	0.992115i	$-0.460002\pi$
$197$	693.069	0.250655	0.125328	+	0.992115i	$0.460002\pi$
$198$	0	0
$199$	145.818	0.0519437	0.0259718	−	0.999663i	$-0.491732\pi$
$199$	145.818	0.0519437	0.0259718	+	0.999663i	$0.491732\pi$
$200$	877.522	0.310251
$201$	0	0
$202$	1710.81	0.595901
$203$	−1535.86	−0.531016
$204$	0	0
$205$	3697.29	1.25966
$206$	7299.99	2.46900
$207$	0	0
$208$	−1208.68	−0.402916
$209$	−130.831	−0.0433002
$210$	0	0
$211$	−3518.11	−1.14785	−0.573927	−	0.818907i	$-0.694580\pi$
$211$	−3518.11	−1.14785	−0.573927	+	0.818907i	$0.694580\pi$
$212$	1636.03	0.530013
$213$	0	0
$214$	3725.72	1.19012
$215$	−7332.05	−2.32578
$216$	0	0
$217$	1500.49	0.469400
$218$	−335.230	−0.104150
$219$	0	0
$220$	1362.44	0.417527
$221$	−1279.59	−0.389477
$222$	0	0
$223$	1630.18	0.489530	0.244765	−	0.969582i	$-0.421289\pi$
$223$	1630.18	0.489530	0.244765	+	0.969582i	$0.421289\pi$
$224$	−1692.50	−0.504842
$225$	0	0
$226$	6344.24	1.86731
$227$	−1841.48	−0.538428	−0.269214	−	0.963080i	$-0.586764\pi$
$227$	−1841.48	−0.538428	−0.269214	+	0.963080i	$0.586764\pi$
$228$	0	0
$229$	4468.48	1.28945	0.644727	−	0.764413i	$-0.276971\pi$
$229$	4468.48	1.28945	0.644727	+	0.764413i	$0.276971\pi$
$230$	12208.2	3.49993
$231$	0	0
$232$	960.182	0.271720
$233$	1914.60	0.538323	0.269162	−	0.963095i	$-0.413253\pi$
$233$	1914.60	0.538323	0.269162	+	0.963095i	$0.413253\pi$
$234$	0	0
$235$	3223.28	0.894739
$236$	4014.60	1.10732
$237$	0	0
$238$	−2051.24	−0.558665
$239$	−6454.45	−1.74688	−0.873438	−	0.486935i	$-0.838115\pi$
$239$	−6454.45	−1.74688	−0.873438	+	0.486935i	$0.838115\pi$
$240$	0	0
$241$	−3461.21	−0.925129	−0.462564	−	0.886586i	$-0.653071\pi$
$241$	−3461.21	−0.925129	−0.462564	+	0.886586i	$0.653071\pi$
$242$	−466.516	−0.123921
$243$	0	0
$244$	−6251.26	−1.64015
$245$	−884.067	−0.230535
$246$	0	0
$247$	200.240	0.0515827
$248$	−938.068	−0.240191
$249$	0	0
$250$	5253.32	1.32900
$251$	3579.75	0.900207	0.450104	−	0.892976i	$-0.351387\pi$
$251$	3579.75	0.900207	0.450104	+	0.892976i	$0.351387\pi$
$252$	0	0
$253$	−1930.51	−0.479725
$254$	3772.81	0.931996
$255$	0	0
$256$	5000.90	1.22092
$257$	−3780.31	−0.917545	−0.458773	−	0.888554i	$-0.651711\pi$
$257$	−3780.31	−0.917545	−0.458773	+	0.888554i	$0.651711\pi$
$258$	0	0
$259$	−1909.28	−0.458057
$260$	−2085.25	−0.497392
$261$	0	0
$262$	5763.43	1.35903
$263$	5063.37	1.18715	0.593575	−	0.804778i	$-0.297716\pi$
$263$	5063.37	1.18715	0.593575	+	0.804778i	$0.297716\pi$
$264$	0	0
$265$	−4299.74	−0.996721
$266$	320.994	0.0739902
$267$	0	0
$268$	−2661.87	−0.606715
$269$	−2910.06	−0.659589	−0.329795	−	0.944053i	$-0.606979\pi$
$269$	−2910.06	−0.659589	−0.329795	+	0.944053i	$0.606979\pi$
$270$	0	0
$271$	5442.93	1.22005	0.610027	−	0.792381i	$-0.291159\pi$
$271$	5442.93	1.22005	0.610027	+	0.792381i	$0.291159\pi$
$272$	5456.50	1.21636
$273$	0	0
$274$	9244.68	2.03829
$275$	−2205.72	−0.483673
$276$	0	0
$277$	4044.19	0.877226	0.438613	−	0.898676i	$-0.355470\pi$
$277$	4044.19	0.877226	0.438613	+	0.898676i	$0.355470\pi$
$278$	−10304.4	−2.22309
$279$	0	0
$280$	552.697	0.117964
$281$	8172.77	1.73504	0.867520	−	0.497402i	$-0.165713\pi$
$281$	8172.77	1.73504	0.867520	+	0.497402i	$0.165713\pi$
$282$	0	0
$283$	2182.31	0.458391	0.229196	−	0.973380i	$-0.426390\pi$
$283$	2182.31	0.458391	0.229196	+	0.973380i	$0.426390\pi$
$284$	4586.41	0.958286
$285$	0	0
$286$	714.015	0.147624
$287$	1434.48	0.295033
$288$	0	0
$289$	863.627	0.175784
$290$	15262.4	3.09049
$291$	0	0
$292$	2679.37	0.536981
$293$	−1709.43	−0.340839	−0.170420	−	0.985372i	$-0.554512\pi$
$293$	−1709.43	−0.340839	−0.170420	+	0.985372i	$0.554512\pi$
$294$	0	0
$295$	−10551.0	−2.08239
$296$	1193.63	0.234387
$297$	0	0
$298$	7911.98	1.53801
$299$	2954.70	0.571487
$300$	0	0
$301$	−2844.69	−0.544735
$302$	−9124.66	−1.73863
$303$	0	0
$304$	−853.874	−0.161095
$305$	16429.3	3.08440
$306$	0	0
$307$	5485.85	1.01985	0.509925	−	0.860219i	$-0.329673\pi$
$307$	5485.85	1.01985	0.509925	+	0.860219i	$0.329673\pi$
$308$	528.600	0.0977916
$309$	0	0
$310$	−14910.9	−2.73189
$311$	−476.818	−0.0869385	−0.0434693	−	0.999055i	$-0.513841\pi$
$311$	−476.818	−0.0869385	−0.0434693	+	0.999055i	$0.513841\pi$
$312$	0	0
$313$	779.948	0.140848	0.0704238	−	0.997517i	$-0.477565\pi$
$313$	779.948	0.140848	0.0704238	+	0.997517i	$0.477565\pi$
$314$	14508.5	2.60753
$315$	0	0
$316$	−1720.68	−0.306315
$317$	−6398.10	−1.13361	−0.566803	−	0.823853i	$-0.691820\pi$
$317$	−6398.10	−1.13361	−0.566803	+	0.823853i	$0.691820\pi$
$318$	0	0
$319$	−2413.49	−0.423604
$320$	6456.72	1.12794
$321$	0	0
$322$	4736.53	0.819740
$323$	−903.970	−0.155722
$324$	0	0
$325$	3375.91	0.576190
$326$	−5377.06	−0.913522
$327$	0	0
$328$	−896.799	−0.150968
$329$	1250.57	0.209562
$330$	0	0
$331$	−7792.92	−1.29407	−0.647036	−	0.762460i	$-0.723991\pi$
$331$	−7792.92	−1.29407	−0.647036	+	0.762460i	$0.723991\pi$
$332$	372.098	0.0615106
$333$	0	0
$334$	−1481.22	−0.242662
$335$	6995.83	1.14096
$336$	0	0
$337$	−4106.13	−0.663724	−0.331862	−	0.943328i	$-0.607677\pi$
$337$	−4106.13	−0.663724	−0.331862	+	0.943328i	$0.607677\pi$
$338$	7377.73	1.18727
$339$	0	0
$340$	9413.76	1.50157
$341$	2357.91	0.374452
$342$	0	0
$343$	−343.000	−0.0539949
$344$	1778.43	0.278740
$345$	0	0
$346$	4457.77	0.692634
$347$	3014.89	0.466421	0.233210	−	0.972426i	$-0.425077\pi$
$347$	3014.89	0.466421	0.233210	+	0.972426i	$0.425077\pi$
$348$	0	0
$349$	8717.58	1.33708	0.668541	−	0.743675i	$-0.266919\pi$
$349$	8717.58	1.33708	0.668541	+	0.743675i	$0.266919\pi$
$350$	5411.75	0.826486
$351$	0	0
$352$	−2659.64	−0.402725
$353$	248.227	0.0374272	0.0187136	−	0.999825i	$-0.494043\pi$
$353$	248.227	0.0374272	0.0187136	+	0.999825i	$0.494043\pi$
$354$	0	0
$355$	−12053.8	−1.80211
$356$	6224.31	0.926650
$357$	0	0
$358$	546.576	0.0806912
$359$	9850.33	1.44814	0.724068	−	0.689729i	$-0.242270\pi$
$359$	9850.33	1.44814	0.724068	+	0.689729i	$0.242270\pi$
$360$	0	0
$361$	−6717.54	−0.979376
$362$	−5242.84	−0.761209
$363$	0	0
$364$	−809.036	−0.116497
$365$	−7041.83	−1.00983
$366$	0	0
$367$	−3597.52	−0.511686	−0.255843	−	0.966718i	$-0.582353\pi$
$367$	−3597.52	−0.511686	−0.255843	+	0.966718i	$0.582353\pi$
$368$	−12599.6	−1.78478
$369$	0	0
$370$	18973.2	2.66587
$371$	−1668.21	−0.233448
$372$	0	0
$373$	2275.67	0.315897	0.157949	−	0.987447i	$-0.449512\pi$
$373$	2275.67	0.315897	0.157949	+	0.987447i	$0.449512\pi$
$374$	−3223.38	−0.445660
$375$	0	0
$376$	−781.824	−0.107233
$377$	3693.91	0.504632
$378$	0	0
$379$	3964.58	0.537326	0.268663	−	0.963234i	$-0.413418\pi$
$379$	3964.58	0.537326	0.268663	+	0.963234i	$0.413418\pi$
$380$	−1473.14	−0.198869
$381$	0	0
$382$	14180.8	1.89935
$383$	9289.55	1.23936	0.619679	−	0.784855i	$-0.287263\pi$
$383$	9289.55	1.23936	0.619679	+	0.784855i	$0.287263\pi$
$384$	0	0
$385$	−1389.25	−0.183903
$386$	−4270.07	−0.563059
$387$	0	0
$388$	4244.82	0.555407
$389$	−15086.8	−1.96640	−0.983201	−	0.182528i	$-0.941572\pi$
$389$	−15086.8	−1.96640	−0.983201	+	0.182528i	$0.941572\pi$
$390$	0	0
$391$	−13338.8	−1.72525
$392$	214.435	0.0276291
$393$	0	0
$394$	−2672.13	−0.341675
$395$	4522.21	0.576044
$396$	0	0
$397$	−10215.5	−1.29144	−0.645722	−	0.763572i	$-0.723444\pi$
$397$	−10215.5	−1.29144	−0.645722	+	0.763572i	$0.723444\pi$
$398$	−562.204	−0.0708059
$399$	0	0
$400$	−14395.8	−1.79947
$401$	4143.18	0.515962	0.257981	−	0.966150i	$-0.416943\pi$
$401$	4143.18	0.515962	0.257981	+	0.966150i	$0.416943\pi$
$402$	0	0
$403$	−3608.84	−0.446077
$404$	−3046.18	−0.375132
$405$	0	0
$406$	5921.52	0.723843
$407$	−3000.29	−0.365403
$408$	0	0
$409$	−1958.71	−0.236801	−0.118401	−	0.992966i	$-0.537777\pi$
$409$	−1958.71	−0.236801	−0.118401	+	0.992966i	$0.537777\pi$
$410$	−14254.9	−1.71708
$411$	0	0
$412$	−12998.0	−1.55429
$413$	−4093.59	−0.487729
$414$	0	0
$415$	−977.934	−0.115674
$416$	4070.64	0.479758
$417$	0	0
$418$	504.419	0.0590237
$419$	−616.552	−0.0718868	−0.0359434	−	0.999354i	$-0.511444\pi$
$419$	−616.552	−0.0718868	−0.0359434	+	0.999354i	$0.511444\pi$
$420$	0	0
$421$	−612.327	−0.0708859	−0.0354430	−	0.999372i	$-0.511284\pi$
$421$	−612.327	−0.0708859	−0.0354430	+	0.999372i	$0.511284\pi$
$422$	13564.1	1.56467
$423$	0	0
$424$	1042.93	0.119455
$425$	−15240.4	−1.73945
$426$	0	0
$427$	6374.25	0.722416
$428$	−6633.84	−0.749203
$429$	0	0
$430$	28268.8	3.17033
$431$	2316.95	0.258941	0.129470	−	0.991583i	$-0.458672\pi$
$431$	2316.95	0.258941	0.129470	+	0.991583i	$0.458672\pi$
$432$	0	0
$433$	−9187.01	−1.01963	−0.509815	−	0.860284i	$-0.670286\pi$
$433$	−9187.01	−1.01963	−0.509815	+	0.860284i	$0.670286\pi$
$434$	−5785.14	−0.639852
$435$	0	0
$436$	596.895	0.0655644
$437$	2087.36	0.228494
$438$	0	0
$439$	1907.40	0.207369	0.103685	−	0.994610i	$-0.466937\pi$
$439$	1907.40	0.207369	0.103685	+	0.994610i	$0.466937\pi$
$440$	868.524	0.0941028
$441$	0	0
$442$	4933.46	0.530907
$443$	−7657.14	−0.821223	−0.410611	−	0.911810i	$-0.634685\pi$
$443$	−7657.14	−0.821223	−0.410611	+	0.911810i	$0.634685\pi$
$444$	0	0
$445$	−16358.5	−1.74262
$446$	−6285.19	−0.667292
$447$	0	0
$448$	2505.08	0.264182
$449$	−2156.86	−0.226701	−0.113350	−	0.993555i	$-0.536158\pi$
$449$	−2156.86	−0.226701	−0.113350	+	0.993555i	$0.536158\pi$
$450$	0	0
$451$	2254.18	0.235355
$452$	−11296.3	−1.17551
$453$	0	0
$454$	7099.83	0.733946
$455$	2126.28	0.219080
$456$	0	0
$457$	18073.5	1.84998	0.924990	−	0.379990i	$-0.124073\pi$
$457$	18073.5	1.84998	0.924990	+	0.379990i	$0.124073\pi$
$458$	−17228.2	−1.75769
$459$	0	0
$460$	−21737.3	−2.20328
$461$	−3559.83	−0.359648	−0.179824	−	0.983699i	$-0.557553\pi$
$461$	−3559.83	−0.359648	−0.179824	+	0.983699i	$0.557553\pi$
$462$	0	0
$463$	−11741.3	−1.17855	−0.589273	−	0.807934i	$-0.700586\pi$
$463$	−11741.3	−1.17855	−0.589273	+	0.807934i	$0.700586\pi$
$464$	−15751.8	−1.57599
$465$	0	0
$466$	−7381.74	−0.733804
$467$	−15624.4	−1.54821	−0.774103	−	0.633060i	$-0.781799\pi$
$467$	−15624.4	−1.54821	−0.774103	+	0.633060i	$0.781799\pi$
$468$	0	0
$469$	2714.24	0.267232
$470$	−12427.4	−1.21964
$471$	0	0
$472$	2559.21	0.249570
$473$	−4470.22	−0.434548
$474$	0	0
$475$	2384.93	0.230375
$476$	3652.35	0.351691
$477$	0	0
$478$	24885.2	2.38122
$479$	−856.272	−0.0816787	−0.0408393	−	0.999166i	$-0.513003\pi$
$479$	−856.272	−0.0816787	−0.0408393	+	0.999166i	$0.513003\pi$
$480$	0	0
$481$	4592.02	0.435297
$482$	13344.7	1.26107
$483$	0	0
$484$	830.658	0.0780107
$485$	−11156.1	−1.04448
$486$	0	0
$487$	1401.75	0.130430	0.0652148	−	0.997871i	$-0.479227\pi$
$487$	1401.75	0.130430	0.0652148	+	0.997871i	$0.479227\pi$
$488$	−3985.02	−0.369659
$489$	0	0
$490$	3408.53	0.314248
$491$	4413.29	0.405639	0.202820	−	0.979216i	$-0.434990\pi$
$491$	4413.29	0.405639	0.202820	+	0.979216i	$0.434990\pi$
$492$	0	0
$493$	−16676.0	−1.52342
$494$	−772.025	−0.0703139
$495$	0	0
$496$	15389.0	1.39312
$497$	−4676.64	−0.422084
$498$	0	0
$499$	5105.31	0.458006	0.229003	−	0.973426i	$-0.426453\pi$
$499$	5105.31	0.458006	0.229003	+	0.973426i	$0.426453\pi$
$500$	−9353.82	−0.836631
$501$	0	0
$502$	−13801.8	−1.22710
$503$	18573.1	1.64639	0.823195	−	0.567759i	$-0.192190\pi$
$503$	18573.1	1.64639	0.823195	+	0.567759i	$0.192190\pi$
$504$	0	0
$505$	8005.87	0.705458
$506$	7443.11	0.653927
$507$	0	0
$508$	−6717.69	−0.586711
$509$	20788.8	1.81031	0.905153	−	0.425086i	$-0.139756\pi$
$509$	20788.8	1.81031	0.905153	+	0.425086i	$0.139756\pi$
$510$	0	0
$511$	−2732.09	−0.236517
$512$	−14844.8	−1.28136
$513$	0	0
$514$	14575.0	1.25073
$515$	34160.9	2.92293
$516$	0	0
$517$	1965.18	0.167173
$518$	7361.23	0.624390
$519$	0	0
$520$	−1329.30	−0.112103
$521$	12175.6	1.02384	0.511921	−	0.859032i	$-0.328934\pi$
$521$	12175.6	1.02384	0.511921	+	0.859032i	$0.328934\pi$
$522$	0	0
$523$	−104.134	−0.00870642	−0.00435321	−	0.999991i	$-0.501386\pi$
$523$	−104.134	−0.00870642	−0.00435321	+	0.999991i	$0.501386\pi$
$524$	−10262.1	−0.855538
$525$	0	0
$526$	−19521.9	−1.61824
$527$	16291.9	1.34665
$528$	0	0
$529$	18633.7	1.53150
$530$	16577.7	1.35866
$531$	0	0
$532$	−571.547	−0.0465784
$533$	−3450.07	−0.280374
$534$	0	0
$535$	17434.8	1.40892
$536$	−1696.88	−0.136742
$537$	0	0
$538$	11219.8	0.899104
$539$	−539.000	−0.0430730
$540$	0	0
$541$	−2741.20	−0.217844	−0.108922	−	0.994050i	$-0.534740\pi$
$541$	−2741.20	−0.217844	−0.108922	+	0.994050i	$0.534740\pi$
$542$	−20985.3	−1.66309
$543$	0	0
$544$	−18376.7	−1.44833
$545$	−1568.74	−0.123298
$546$	0	0
$547$	−17008.0	−1.32945	−0.664725	−	0.747089i	$-0.731451\pi$
$547$	−17008.0	−1.32945	−0.664725	+	0.747089i	$0.731451\pi$
$548$	−16460.7	−1.28315
$549$	0	0
$550$	8504.18	0.659308
$551$	2609.58	0.201764
$552$	0	0
$553$	1754.53	0.134919
$554$	−15592.4	−1.19577
$555$	0	0
$556$	18347.6	1.39948
$557$	2055.20	0.156340	0.0781702	−	0.996940i	$-0.475092\pi$
$557$	2055.20	0.156340	0.0781702	+	0.996940i	$0.475092\pi$
$558$	0	0
$559$	6841.79	0.517669
$560$	−9067.00	−0.684198
$561$	0	0
$562$	−31510.2	−2.36508
$563$	4425.50	0.331284	0.165642	−	0.986186i	$-0.447030\pi$
$563$	4425.50	0.331284	0.165642	+	0.986186i	$0.447030\pi$
$564$	0	0
$565$	29688.4	2.21062
$566$	−8413.90	−0.624846
$567$	0	0
$568$	2923.72	0.215980
$569$	−6590.24	−0.485549	−0.242775	−	0.970083i	$-0.578058\pi$
$569$	−6590.24	−0.485549	−0.242775	+	0.970083i	$0.578058\pi$
$570$	0	0
$571$	−24640.9	−1.80594	−0.902968	−	0.429707i	$-0.858617\pi$
$571$	−24640.9	−1.80594	−0.902968	+	0.429707i	$0.858617\pi$
$572$	−1271.34	−0.0929327
$573$	0	0
$574$	−5530.63	−0.402167
$575$	35191.6	2.55233
$576$	0	0
$577$	−26063.5	−1.88048	−0.940242	−	0.340507i	$-0.889401\pi$
$577$	−26063.5	−1.88048	−0.940242	+	0.340507i	$0.889401\pi$
$578$	−3329.72	−0.239616
$579$	0	0
$580$	−27175.6	−1.94553
$581$	−379.419	−0.0270928
$582$	0	0
$583$	−2621.48	−0.186227
$584$	1708.03	0.121026
$585$	0	0
$586$	6590.72	0.464607
$587$	−18880.3	−1.32756	−0.663778	−	0.747930i	$-0.731048\pi$
$587$	−18880.3	−1.32756	−0.663778	+	0.747930i	$0.731048\pi$
$588$	0	0
$589$	−2549.48	−0.178352
$590$	40679.6	2.83856
$591$	0	0
$592$	−19581.6	−1.35945
$593$	−20373.4	−1.41085	−0.705426	−	0.708783i	$-0.749244\pi$
$593$	−20373.4	−1.41085	−0.705426	+	0.708783i	$0.749244\pi$
$594$	0	0
$595$	−9598.96	−0.661376
$596$	−14087.7	−0.968213
$597$	0	0
$598$	−11391.9	−0.779010
$599$	3953.48	0.269674	0.134837	−	0.990868i	$-0.456949\pi$
$599$	3953.48	0.269674	0.134837	+	0.990868i	$0.456949\pi$
$600$	0	0
$601$	6736.90	0.457244	0.228622	−	0.973515i	$-0.426578\pi$
$601$	6736.90	0.457244	0.228622	+	0.973515i	$0.426578\pi$
$602$	10967.7	0.742543
$603$	0	0
$604$	16247.0	1.09450
$605$	−2183.10	−0.146704
$606$	0	0
$607$	−28336.7	−1.89481	−0.947406	−	0.320035i	$-0.896305\pi$
$607$	−28336.7	−1.89481	−0.947406	+	0.320035i	$0.896305\pi$
$608$	2875.72	0.191819
$609$	0	0
$610$	−63343.4	−4.20443
$611$	−3007.75	−0.199150
$612$	0	0
$613$	2767.63	0.182355	0.0911774	−	0.995835i	$-0.470937\pi$
$613$	2767.63	0.182355	0.0911774	+	0.995835i	$0.470937\pi$
$614$	−21150.7	−1.39019
$615$	0	0
$616$	336.970	0.0220404
$617$	−25880.0	−1.68864	−0.844318	−	0.535842i	$-0.819994\pi$
$617$	−25880.0	−1.68864	−0.844318	+	0.535842i	$0.819994\pi$
$618$	0	0
$619$	42.9744	0.00279045	0.00139522	−	0.999999i	$-0.499556\pi$
$619$	42.9744	0.00279045	0.00139522	+	0.999999i	$0.499556\pi$
$620$	26549.7	1.71978
$621$	0	0
$622$	1838.38	0.118508
$623$	−6346.76	−0.408150
$624$	0	0
$625$	−481.672	−0.0308270
$626$	−3007.10	−0.191993
$627$	0	0
$628$	−25833.2	−1.64149
$629$	−20730.4	−1.31411
$630$	0	0
$631$	−18766.2	−1.18395	−0.591974	−	0.805957i	$-0.701651\pi$
$631$	−18766.2	−1.18395	−0.591974	+	0.805957i	$0.701651\pi$
$632$	−1096.89	−0.0690377
$633$	0	0
$634$	24667.9	1.54525
$635$	17655.2	1.10335
$636$	0	0
$637$	824.953	0.0513121
$638$	9305.25	0.577427
$639$	0	0
$640$	10004.7	0.617925
$641$	29882.6	1.84133	0.920663	−	0.390358i	$-0.127649\pi$
$641$	29882.6	1.84133	0.920663	+	0.390358i	$0.127649\pi$
$642$	0	0
$643$	−18805.2	−1.15335	−0.576677	−	0.816973i	$-0.695651\pi$
$643$	−18805.2	−1.15335	−0.576677	+	0.816973i	$0.695651\pi$
$644$	−8433.64	−0.516044
$645$	0	0
$646$	3485.26	0.212269
$647$	28514.1	1.73262	0.866309	−	0.499508i	$-0.166486\pi$
$647$	28514.1	1.73262	0.866309	+	0.499508i	$0.166486\pi$
$648$	0	0
$649$	−6432.78	−0.389073
$650$	−13015.9	−0.785421
$651$	0	0
$652$	9574.15	0.575081
$653$	−28136.5	−1.68616	−0.843082	−	0.537785i	$-0.819261\pi$
$653$	−28136.5	−1.68616	−0.843082	+	0.537785i	$0.819261\pi$
$654$	0	0
$655$	26970.4	1.60889
$656$	14712.0	0.875621
$657$	0	0
$658$	−4821.57	−0.285660
$659$	−25412.3	−1.50216	−0.751080	−	0.660211i	$-0.770467\pi$
$659$	−25412.3	−1.50216	−0.751080	+	0.660211i	$0.770467\pi$
$660$	0	0
$661$	−15015.6	−0.883569	−0.441784	−	0.897121i	$-0.645654\pi$
$661$	−15015.6	−0.883569	−0.441784	+	0.897121i	$0.645654\pi$
$662$	30045.7	1.76398
$663$	0	0
$664$	237.203	0.0138634
$665$	1502.12	0.0875934
$666$	0	0
$667$	38506.5	2.23535
$668$	2637.40	0.152761
$669$	0	0
$670$	−26972.5	−1.55528
$671$	10016.7	0.576289
$672$	0	0
$673$	21748.2	1.24566	0.622831	−	0.782357i	$-0.285983\pi$
$673$	21748.2	1.24566	0.622831	+	0.782357i	$0.285983\pi$
$674$	15831.2	0.904741
$675$	0	0
$676$	−13136.5	−0.747409
$677$	−18842.5	−1.06969	−0.534843	−	0.844952i	$-0.679629\pi$
$677$	−18842.5	−1.06969	−0.534843	+	0.844952i	$0.679629\pi$
$678$	0	0
$679$	−4328.33	−0.244633
$680$	6001.03	0.338425
$681$	0	0
$682$	−9090.94	−0.510425
$683$	11932.7	0.668510	0.334255	−	0.942483i	$-0.391515\pi$
$683$	11932.7	0.668510	0.334255	+	0.942483i	$0.391515\pi$
$684$	0	0
$685$	43261.3	2.41303
$686$	1322.44	0.0736020
$687$	0	0
$688$	−29175.2	−1.61670
$689$	4012.23	0.221849
$690$	0	0
$691$	11305.0	0.622376	0.311188	−	0.950348i	$-0.399273\pi$
$691$	11305.0	0.622376	0.311188	+	0.950348i	$0.399273\pi$
$692$	−7937.31	−0.436028
$693$	0	0
$694$	−11623.9	−0.635791
$695$	−48220.4	−2.63181
$696$	0	0
$697$	15575.1	0.846414
$698$	−33610.7	−1.82261
$699$	0	0
$700$	−9635.92	−0.520291
$701$	−9963.42	−0.536823	−0.268412	−	0.963304i	$-0.586499\pi$
$701$	−9963.42	−0.536823	−0.268412	+	0.963304i	$0.586499\pi$
$702$	0	0
$703$	3244.05	0.174042
$704$	3936.55	0.210745
$705$	0	0
$706$	−957.043	−0.0510181
$707$	3106.11	0.165230
$708$	0	0
$709$	11000.9	0.582718	0.291359	−	0.956614i	$-0.405893\pi$
$709$	11000.9	0.582718	0.291359	+	0.956614i	$0.405893\pi$
$710$	46473.6	2.45651
$711$	0	0
$712$	3967.84	0.208850
$713$	−37619.7	−1.97597
$714$	0	0
$715$	3341.29	0.174765
$716$	−973.209	−0.0507968
$717$	0	0
$718$	−37978.0	−1.97399
$719$	16311.5	0.846059	0.423030	−	0.906116i	$-0.360967\pi$
$719$	16311.5	0.846059	0.423030	+	0.906116i	$0.360967\pi$
$720$	0	0
$721$	13253.7	0.684598
$722$	25899.5	1.33501
$723$	0	0
$724$	9335.17	0.479197
$725$	43995.9	2.25374
$726$	0	0
$727$	−17113.7	−0.873056	−0.436528	−	0.899691i	$-0.643792\pi$
$727$	−17113.7	−0.873056	−0.436528	+	0.899691i	$0.643792\pi$
$728$	−515.740	−0.0262563
$729$	0	0
$730$	27149.8	1.37652
$731$	−30886.9	−1.56278
$732$	0	0
$733$	−908.468	−0.0457777	−0.0228888	−	0.999738i	$-0.507286\pi$
$733$	−908.468	−0.0457777	−0.0228888	+	0.999738i	$0.507286\pi$
$734$	13870.3	0.697494
$735$	0	0
$736$	42433.6	2.12517
$737$	4265.24	0.213178
$738$	0	0
$739$	3111.66	0.154891	0.0774453	−	0.996997i	$-0.475324\pi$
$739$	3111.66	0.154891	0.0774453	+	0.996997i	$0.475324\pi$
$740$	−33782.9	−1.67822
$741$	0	0
$742$	6431.81	0.318220
$743$	−34279.0	−1.69257	−0.846283	−	0.532734i	$-0.821165\pi$
$743$	−34279.0	−1.69257	−0.846283	+	0.532734i	$0.821165\pi$
$744$	0	0
$745$	37024.8	1.82078
$746$	−8773.86	−0.430608
$747$	0	0
$748$	5739.40	0.280553
$749$	6764.35	0.329992
$750$	0	0
$751$	18668.4	0.907086	0.453543	−	0.891234i	$-0.350160\pi$
$751$	18668.4	0.907086	0.453543	+	0.891234i	$0.350160\pi$
$752$	12825.8	0.621955
$753$	0	0
$754$	−14241.9	−0.687877
$755$	−42699.6	−2.05828
$756$	0	0
$757$	8879.31	0.426320	0.213160	−	0.977017i	$-0.431624\pi$
$757$	8879.31	0.426320	0.213160	+	0.977017i	$0.431624\pi$
$758$	−15285.5	−0.732444
$759$	0	0
$760$	−939.087	−0.0448214
$761$	20677.5	0.984967	0.492484	−	0.870322i	$-0.336089\pi$
$761$	20677.5	0.984967	0.492484	+	0.870322i	$0.336089\pi$
$762$	0	0
$763$	−608.638	−0.0288783
$764$	−25249.7	−1.19568
$765$	0	0
$766$	−35816.0	−1.68940
$767$	9845.52	0.463496
$768$	0	0
$769$	−27497.6	−1.28945	−0.644725	−	0.764414i	$-0.723028\pi$
$769$	−27497.6	−1.28945	−0.644725	+	0.764414i	$0.723028\pi$
$770$	5356.26	0.250683
$771$	0	0
$772$	7603.08	0.354457
$773$	−9834.49	−0.457596	−0.228798	−	0.973474i	$-0.573480\pi$
$773$	−9834.49	−0.457596	−0.228798	+	0.973474i	$0.573480\pi$
$774$	0	0
$775$	−42982.6	−1.99223
$776$	2705.97	0.125179
$777$	0	0
$778$	58167.2	2.68046
$779$	−2437.32	−0.112100
$780$	0	0
$781$	−7349.00	−0.336707
$782$	51427.9	2.35174
$783$	0	0
$784$	−3517.81	−0.160250
$785$	67893.9	3.08693
$786$	0	0
$787$	−39329.2	−1.78136	−0.890682	−	0.454627i	$-0.849773\pi$
$787$	−39329.2	−1.78136	−0.890682	+	0.454627i	$0.849773\pi$
$788$	4757.88	0.215092
$789$	0	0
$790$	−17435.4	−0.785221
$791$	11518.5	0.517764
$792$	0	0
$793$	−15330.8	−0.686522
$794$	39386.1	1.76040
$795$	0	0
$796$	1001.03	0.0445738
$797$	22338.1	0.992792	0.496396	−	0.868096i	$-0.334656\pi$
$797$	22338.1	0.992792	0.496396	+	0.868096i	$0.334656\pi$
$798$	0	0
$799$	13578.3	0.601210
$800$	48482.8	2.14266
$801$	0	0
$802$	−15974.1	−0.703322
$803$	−4293.28	−0.188676
$804$	0	0
$805$	22165.0	0.970451
$806$	13913.9	0.608060
$807$	0	0
$808$	−1941.87	−0.0845478
$809$	−37494.8	−1.62948	−0.814739	−	0.579828i	$-0.803120\pi$
$809$	−37494.8	−1.62948	−0.814739	+	0.579828i	$0.803120\pi$
$810$	0	0
$811$	−42506.0	−1.84043	−0.920214	−	0.391417i	$-0.871985\pi$
$811$	−42506.0	−1.84043	−0.920214	+	0.391417i	$0.871985\pi$
$812$	−10543.6	−0.455674
$813$	0	0
$814$	11567.6	0.498091
$815$	−25162.4	−1.08147
$816$	0	0
$817$	4833.41	0.206976
$818$	7551.81	0.322791
$819$	0	0
$820$	25381.7	1.08094
$821$	42443.9	1.80427	0.902133	−	0.431458i	$-0.142001\pi$
$821$	42443.9	1.80427	0.902133	+	0.431458i	$0.142001\pi$
$822$	0	0
$823$	2288.06	0.0969099	0.0484549	−	0.998825i	$-0.484570\pi$
$823$	2288.06	0.0969099	0.0484549	+	0.998825i	$0.484570\pi$
$824$	−8285.92	−0.350308
$825$	0	0
$826$	15782.9	0.664837
$827$	3182.84	0.133831	0.0669155	−	0.997759i	$-0.478684\pi$
$827$	3182.84	0.133831	0.0669155	+	0.997759i	$0.478684\pi$
$828$	0	0
$829$	−1264.12	−0.0529609	−0.0264804	−	0.999649i	$-0.508430\pi$
$829$	−1264.12	−0.0529609	−0.0264804	+	0.999649i	$0.508430\pi$
$830$	3770.43	0.157679
$831$	0	0
$832$	−6024.98	−0.251056
$833$	−3724.20	−0.154905
$834$	0	0
$835$	−6931.52	−0.287275
$836$	−898.145	−0.0371567
$837$	0	0
$838$	2377.12	0.0979908
$839$	2934.36	0.120745	0.0603726	−	0.998176i	$-0.480771\pi$
$839$	2934.36	0.120745	0.0603726	+	0.998176i	$0.480771\pi$
$840$	0	0
$841$	23751.1	0.973845
$842$	2360.83	0.0966266
$843$	0	0
$844$	−24151.6	−0.984993
$845$	34524.7	1.40555
$846$	0	0
$847$	−847.000	−0.0343604
$848$	−17109.2	−0.692845
$849$	0	0
$850$	58759.3	2.37109
$851$	47868.6	1.92822
$852$	0	0
$853$	25305.3	1.01575	0.507877	−	0.861429i	$-0.330430\pi$
$853$	25305.3	1.01575	0.507877	+	0.861429i	$0.330430\pi$
$854$	−24576.0	−0.984745
$855$	0	0
$856$	−4228.91	−0.168856
$857$	34699.9	1.38311	0.691556	−	0.722323i	$-0.256926\pi$
$857$	34699.9	1.38311	0.691556	+	0.722323i	$0.256926\pi$
$858$	0	0
$859$	17543.8	0.696841	0.348421	−	0.937338i	$-0.386718\pi$
$859$	17543.8	0.696841	0.348421	+	0.937338i	$0.386718\pi$
$860$	−50334.1	−1.99579
$861$	0	0
$862$	−8933.01	−0.352969
$863$	−18985.7	−0.748876	−0.374438	−	0.927252i	$-0.622164\pi$
$863$	−18985.7	−0.748876	−0.374438	+	0.927252i	$0.622164\pi$
$864$	0	0
$865$	20860.5	0.819976
$866$	35420.6	1.38989
$867$	0	0
$868$	10300.8	0.402800
$869$	2757.11	0.107628
$870$	0	0
$871$	−6528.05	−0.253955
$872$	380.506	0.0147770
$873$	0	0
$874$	−8047.83	−0.311467
$875$	9537.84	0.368501
$876$	0	0
$877$	4203.00	0.161830	0.0809152	−	0.996721i	$-0.474216\pi$
$877$	4203.00	0.161830	0.0809152	+	0.996721i	$0.474216\pi$
$878$	−7353.98	−0.282671
$879$	0	0
$880$	−14248.1	−0.545801
$881$	−25566.7	−0.977713	−0.488856	−	0.872364i	$-0.662586\pi$
$881$	−25566.7	−0.977713	−0.488856	+	0.872364i	$0.662586\pi$
$882$	0	0
$883$	−39105.9	−1.49039	−0.745197	−	0.666845i	$-0.767645\pi$
$883$	−39105.9	−1.49039	−0.745197	+	0.666845i	$0.767645\pi$
$884$	−8784.29	−0.334217
$885$	0	0
$886$	29522.2	1.11943
$887$	10841.1	0.410382	0.205191	−	0.978722i	$-0.434218\pi$
$887$	10841.1	0.410382	0.205191	+	0.978722i	$0.434218\pi$
$888$	0	0
$889$	6849.85	0.258421
$890$	63070.3	2.37542
$891$	0	0
$892$	11191.1	0.420074
$893$	−2124.84	−0.0796249
$894$	0	0
$895$	2557.75	0.0955264
$896$	3881.63	0.144728
$897$	0	0
$898$	8315.79	0.309022
$899$	−47031.4	−1.74481
$900$	0	0
$901$	−18113.0	−0.669735
$902$	−8690.99	−0.320819
$903$	0	0
$904$	−7201.09	−0.264939
$905$	−24534.3	−0.901159
$906$	0	0
$907$	−21542.0	−0.788632	−0.394316	−	0.918975i	$-0.629018\pi$
$907$	−21542.0	−0.788632	−0.394316	+	0.918975i	$0.629018\pi$
$908$	−12641.6	−0.462035
$909$	0	0
$910$	−8197.88	−0.298634
$911$	−16272.3	−0.591794	−0.295897	−	0.955220i	$-0.595618\pi$
$911$	−16272.3	−0.591794	−0.295897	+	0.955220i	$0.595618\pi$
$912$	0	0
$913$	−596.229	−0.0216126
$914$	−69682.4	−2.52176
$915$	0	0
$916$	30675.8	1.10650
$917$	10464.0	0.376828
$918$	0	0
$919$	46663.1	1.67494	0.837472	−	0.546481i	$-0.184033\pi$
$919$	46663.1	1.67494	0.837472	+	0.546481i	$0.184033\pi$
$920$	−13857.0	−0.496578
$921$	0	0
$922$	13725.0	0.490247
$923$	11247.8	0.401112
$924$	0	0
$925$	54692.6	1.94409
$926$	45268.8	1.60651
$927$	0	0
$928$	53049.7	1.87656
$929$	−37396.2	−1.32070	−0.660349	−	0.750959i	$-0.729592\pi$
$929$	−37396.2	−1.32070	−0.660349	+	0.750959i	$0.729592\pi$
$930$	0	0
$931$	582.791	0.0205158
$932$	13143.6	0.461945
$933$	0	0
$934$	60240.1	2.11040
$935$	−15084.1	−0.527596
$936$	0	0
$937$	−27371.8	−0.954321	−0.477161	−	0.878816i	$-0.658334\pi$
$937$	−27371.8	−0.954321	−0.477161	+	0.878816i	$0.658334\pi$
$938$	−10464.8	−0.364272
$939$	0	0
$940$	22127.6	0.767791
$941$	−2345.33	−0.0812491	−0.0406246	−	0.999174i	$-0.512935\pi$
$941$	−2345.33	−0.0812491	−0.0406246	+	0.999174i	$0.512935\pi$
$942$	0	0
$943$	−35964.6	−1.24196
$944$	−41983.9	−1.44752
$945$	0	0
$946$	17235.0	0.592344
$947$	−39562.8	−1.35757	−0.678784	−	0.734338i	$-0.737493\pi$
$947$	−39562.8	−1.35757	−0.678784	+	0.734338i	$0.737493\pi$
$948$	0	0
$949$	6570.97	0.224766
$950$	−9195.10	−0.314030
$951$	0	0
$952$	2328.28	0.0792647
$953$	34016.4	1.15624	0.578121	−	0.815951i	$-0.303786\pi$
$953$	34016.4	1.15624	0.578121	+	0.815951i	$0.303786\pi$
$954$	0	0
$955$	66360.3	2.24855
$956$	−44309.4	−1.49903
$957$	0	0
$958$	3301.36	0.111338
$959$	16784.5	0.565172
$960$	0	0
$961$	16157.2	0.542353
$962$	−17704.6	−0.593366
$963$	0	0
$964$	−23761.0	−0.793869
$965$	−19982.1	−0.666578
$966$	0	0
$967$	−37016.4	−1.23099	−0.615495	−	0.788141i	$-0.711044\pi$
$967$	−37016.4	−1.23099	−0.615495	+	0.788141i	$0.711044\pi$
$968$	529.524	0.0175822
$969$	0	0
$970$	43012.3	1.42376
$971$	−43208.1	−1.42803	−0.714013	−	0.700133i	$-0.753124\pi$
$971$	−43208.1	−1.42803	−0.714013	+	0.700133i	$0.753124\pi$
$972$	0	0
$973$	−18708.6	−0.616412
$974$	−5404.44	−0.177792
$975$	0	0
$976$	65374.4	2.14404
$977$	−12161.9	−0.398253	−0.199126	−	0.979974i	$-0.563810\pi$
$977$	−12161.9	−0.398253	−0.199126	+	0.979974i	$0.563810\pi$
$978$	0	0
$979$	−9973.48	−0.325591
$980$	−6069.07	−0.197826
$981$	0	0
$982$	−17015.5	−0.552938
$983$	−16531.0	−0.536377	−0.268188	−	0.963366i	$-0.586425\pi$
$983$	−16531.0	−0.536377	−0.268188	+	0.963366i	$0.586425\pi$
$984$	0	0
$985$	−12504.5	−0.404493
$986$	64294.3	2.07662
$987$	0	0
$988$	1374.63	0.0442641
$989$	71320.9	2.29310
$990$	0	0
$991$	−16088.6	−0.515712	−0.257856	−	0.966183i	$-0.583016\pi$
$991$	−16088.6	−0.515712	−0.257856	+	0.966183i	$0.583016\pi$
$992$	−51828.0	−1.65881
$993$	0	0
$994$	18030.8	0.575355
$995$	−2630.88	−0.0838237
$996$	0	0
$997$	−31807.7	−1.01039	−0.505196	−	0.863005i	$-0.668580\pi$
$997$	−31807.7	−1.01039	−0.505196	+	0.863005i	$0.668580\pi$
$998$	−19683.6	−0.624321
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	693.4.a.n.1.2		5
3.2	odd	2		231.4.a.l.1.4	✓	5
21.20	even	2		1617.4.a.p.1.4		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
231.4.a.l.1.4	✓	5	3.2	odd	2
693.4.a.n.1.2		5	1.1	even	1	trivial
1617.4.a.p.1.4		5	21.20	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:	\( N \)	\(=\)	\( 693 = 3^{2} \cdot 7 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	693.a (trivial)

\(f(q)\)	\(=\)	\(q-3.85551 q^{2} +6.86494 q^{4} -18.0422 q^{5} -7.00000 q^{7} +4.37623 q^{8} +69.5618 q^{10} -11.0000 q^{11} +16.8358 q^{13} +26.9886 q^{14} -71.7921 q^{16} -76.0041 q^{17} +11.8937 q^{19} -123.858 q^{20} +42.4106 q^{22} +175.501 q^{23} +200.520 q^{25} -64.9104 q^{26} -48.0546 q^{28} +219.409 q^{29} -214.355 q^{31} +241.785 q^{32} +293.035 q^{34} +126.295 q^{35} +272.754 q^{37} -45.8562 q^{38} -78.9567 q^{40} -204.925 q^{41} +406.384 q^{43} -75.5144 q^{44} -676.647 q^{46} -178.653 q^{47} +49.0000 q^{49} -773.107 q^{50} +115.577 q^{52} +238.316 q^{53} +198.464 q^{55} -30.6336 q^{56} -845.931 q^{58} +584.798 q^{59} -910.607 q^{61} +826.449 q^{62} -357.868 q^{64} -303.754 q^{65} -387.749 q^{67} -521.764 q^{68} -486.932 q^{70} +668.091 q^{71} +390.298 q^{73} -1051.60 q^{74} +81.6495 q^{76} +77.0000 q^{77} -250.647 q^{79} +1295.29 q^{80} +790.090 q^{82} +54.2027 q^{83} +1371.28 q^{85} -1566.82 q^{86} -48.1385 q^{88} +906.680 q^{89} -117.850 q^{91} +1204.81 q^{92} +688.796 q^{94} -214.588 q^{95} +618.333 q^{97} -188.920 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - 5 q^{2} + 21 q^{4} - 7 q^{5} - 35 q^{7} - 60 q^{8} + 55 q^{10} - 55 q^{11} + 111 q^{13} + 35 q^{14} + 201 q^{16} - 136 q^{17} + 111 q^{19} - 219 q^{20} + 55 q^{22} + 28 q^{23} + 190 q^{25} + q^{26}+ \cdots - 245 q^{98}+O(q^{100}) \) 5 * q - 5 * q^2 + 21 * q^4 - 7 * q^5 - 35 * q^7 - 60 * q^8 + 55 * q^10 - 55 * q^11 + 111 * q^13 + 35 * q^14 + 201 * q^16 - 136 * q^17 + 111 * q^19 - 219 * q^20 + 55 * q^22 + 28 * q^23 + 190 * q^25 + q^26 - 147 * q^28 - 61 * q^29 - 280 * q^31 - 535 * q^32 - 572 * q^34 + 49 * q^35 - 41 * q^37 - 267 * q^38 - 336 * q^40 - 426 * q^41 + 424 * q^43 - 231 * q^44 + 140 * q^46 - 75 * q^47 + 245 * q^49 - 490 * q^50 - 269 * q^52 - 1500 * q^53 + 77 * q^55 + 420 * q^56 - 1767 * q^58 - 757 * q^59 + 658 * q^61 + 568 * q^62 - 748 * q^64 - 537 * q^65 - 583 * q^67 + 1650 * q^68 - 385 * q^70 + 764 * q^71 + 875 * q^73 + 825 * q^74 + 213 * q^76 + 385 * q^77 - 244 * q^79 + 2577 * q^80 - 2006 * q^82 - 924 * q^83 - 1402 * q^85 - 1272 * q^86 + 660 * q^88 + 1110 * q^89 - 777 * q^91 + 2046 * q^92 - 3349 * q^94 - 1923 * q^95 - 852 * q^97 - 245 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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