LMFDB - Embedded newform 117.3.c.a.53.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [117,3,Mod(53,117)]

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([1, 0]))

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("117.53");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 117 = 3^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	117.c (of order $2$, degree $1$, minimal)

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:	no
Analytic conductor:	$3.18801909302$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.1574161678336.15
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 20x^{6} + 106x^{4} + 164x^{2} + 9 $ x^8 + 20x^6 + 106x^4 + 164*x^2 + 9
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{4}\cdot 3 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			53.3
Root			$-1.65286i$ of defining polynomial
Character	$\chi$	$=$	117.53
Dual form			117.3.c.a.53.6

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q-1.65286i q^{2} +1.26805 q^{4} +2.99060i q^{5} +9.10296 q^{7} -8.70736i q^{8} +4.94305 q^{10} -0.315116i q^{11} +3.60555 q^{13} -15.0459i q^{14} -9.31982 q^{16} -16.4562i q^{17} -23.4478 q^{19} +3.79225i q^{20} -0.520843 q^{22} +13.6020i q^{23} +16.0563 q^{25} -5.95947i q^{26} +11.5430 q^{28} +42.6918i q^{29} -4.47638 q^{31} -19.4251i q^{32} -27.1998 q^{34} +27.2233i q^{35} -23.2480 q^{37} +38.7559i q^{38} +26.0402 q^{40} +24.6140i q^{41} -57.0229 q^{43} -0.399585i q^{44} +22.4821 q^{46} +44.7433i q^{47} +33.8638 q^{49} -26.5388i q^{50} +4.57204 q^{52} +61.8765i q^{53} +0.942388 q^{55} -79.2627i q^{56} +70.5636 q^{58} -90.7335i q^{59} -89.6900 q^{61} +7.39883i q^{62} -69.3862 q^{64} +10.7828i q^{65} +98.0763 q^{67} -20.8674i q^{68} +44.9964 q^{70} -77.6291i q^{71} -140.161 q^{73} +38.4257i q^{74} -29.7330 q^{76} -2.86849i q^{77} +71.5169 q^{79} -27.8719i q^{80} +40.6835 q^{82} -109.269i q^{83} +49.2140 q^{85} +94.2508i q^{86} -2.74383 q^{88} -119.900i q^{89} +32.8212 q^{91} +17.2480i q^{92} +73.9544 q^{94} -70.1229i q^{95} +39.1083 q^{97} -55.9722i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 8 q^{4} - 16 q^{7} + 24 q^{16} + 64 q^{19} - 80 q^{22} - 24 q^{25} + 184 q^{28} - 40 q^{31} - 272 q^{34} - 104 q^{37} + 32 q^{40} + 128 q^{43} + 232 q^{46} + 136 q^{49} + 104 q^{52} - 224 q^{55} - 88 q^{58}+ \cdots + 328 q^{97}+O(q^{100}) $ 8 * q - 8 * q^4 - 16 * q^7 + 24 * q^16 + 64 * q^19 - 80 * q^22 - 24 * q^25 + 184 * q^28 - 40 * q^31 - 272 * q^34 - 104 * q^37 + 32 * q^40 + 128 * q^43 + 232 * q^46 + 136 * q^49 + 104 * q^52 - 224 * q^55 - 88 * q^58 - 112 * q^61 - 72 * q^64 + 184 * q^67 + 40 * q^70 - 472 * q^73 - 72 * q^76 + 80 * q^79 + 48 * q^82 + 520 * q^85 + 144 * q^88 + 104 * q^91 - 128 * q^94 + 328 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/117\mathbb{Z}\right)^\times$.

$n$	$28$	$92$
$\chi(n)$	$1$	$-1$

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 $

$2$	− 1.65286i	− 0.826430i	−0.910634	−	0.413215i	$-0.864406\pi$
$2$	− 1.65286i	− 0.826430i	0.910634	−	0.413215i	$-0.135594\pi$
$3$	0	0
$3$	0	0
$4$	1.26805	0.317014
$5$	2.99060i	0.598121i	0.954234	+	0.299060i	$0.0966732\pi$
$5$	2.99060i	0.598121i	−0.954234	+	0.299060i	$0.903327\pi$
$6$	0	0
$7$	9.10296	1.30042	0.650211	−	0.759753i	$-0.274680\pi$
$7$	9.10296	1.30042	0.650211	+	0.759753i	$0.274680\pi$
$8$	− 8.70736i	− 1.08842i
$9$	0	0
$10$	4.94305	0.494305
$11$	− 0.315116i	− 0.0286469i	−0.999897	−	0.0143235i	$-0.995441\pi$
$11$	− 0.315116i	− 0.0286469i	0.999897	−	0.0143235i	$-0.00455946\pi$
$12$	0	0
$13$	3.60555	0.277350
$13$	3.60555	0.277350
$14$	− 15.0459i	− 1.07471i
$15$	0	0
$16$	−9.31982	−0.582489
$17$	− 16.4562i	− 0.968012i	−0.875065	−	0.484006i	$-0.839181\pi$
$17$	− 16.4562i	− 0.968012i	0.875065	−	0.484006i	$-0.160819\pi$
$18$	0	0
$19$	−23.4478	−1.23409	−0.617046	−	0.786927i	$-0.711671\pi$
$19$	−23.4478	−1.23409	−0.617046	+	0.786927i	$0.711671\pi$
$20$	3.79225i	0.189612i
$21$	0	0
$22$	−0.520843	−0.0236747
$23$	13.6020i	0.591390i	0.955282	+	0.295695i	$0.0955512\pi$
$23$	13.6020i	0.591390i	−0.955282	+	0.295695i	$0.904449\pi$
$24$	0	0
$25$	16.0563	0.642252
$26$	− 5.95947i	− 0.229210i
$27$	0	0
$28$	11.5430	0.412252
$29$	42.6918i	1.47213i	0.676910	+	0.736066i	$0.263319\pi$
$29$	42.6918i	1.47213i	−0.676910	+	0.736066i	$0.736681\pi$
$30$	0	0
$31$	−4.47638	−0.144399	−0.0721997	−	0.997390i	$-0.523002\pi$
$31$	−4.47638	−0.144399	−0.0721997	+	0.997390i	$0.523002\pi$
$32$	− 19.4251i	− 0.607033i
$33$	0	0
$34$	−27.1998	−0.799994
$35$	27.2233i	0.777810i
$36$	0	0
$37$	−23.2480	−0.628324	−0.314162	−	0.949369i	$-0.601724\pi$
$37$	−23.2480	−0.628324	−0.314162	+	0.949369i	$0.601724\pi$
$38$	38.7559i	1.01989i
$39$	0	0
$40$	26.0402	0.651006
$41$	24.6140i	0.600342i	0.953885	+	0.300171i	$0.0970437\pi$
$41$	24.6140i	0.600342i	−0.953885	+	0.300171i	$0.902956\pi$
$42$	0	0
$43$	−57.0229	−1.32611	−0.663057	−	0.748569i	$-0.730741\pi$
$43$	−57.0229	−1.32611	−0.663057	+	0.748569i	$0.730741\pi$
$44$	− 0.399585i	− 0.00908147i
$45$	0	0
$46$	22.4821	0.488742
$47$	44.7433i	0.951985i	0.879449	+	0.475993i	$0.157911\pi$
$47$	44.7433i	0.951985i	−0.879449	+	0.475993i	$0.842089\pi$
$48$	0	0
$49$	33.8638	0.691099
$50$	− 26.5388i	− 0.530776i
$51$	0	0
$52$	4.57204	0.0879238
$53$	61.8765i	1.16748i	0.811940	+	0.583740i	$0.198411\pi$
$53$	61.8765i	1.16748i	−0.811940	+	0.583740i	$0.801589\pi$
$54$	0	0
$55$	0.942388	0.0171343
$56$	− 79.2627i	− 1.41541i
$57$	0	0
$58$	70.5636	1.21661
$59$	− 90.7335i	− 1.53786i	−0.639335	−	0.768928i	$-0.720790\pi$
$59$	− 90.7335i	− 1.53786i	0.639335	−	0.768928i	$-0.279210\pi$
$60$	0	0
$61$	−89.6900	−1.47033	−0.735164	−	0.677890i	$-0.762895\pi$
$61$	−89.6900	−1.47033	−0.735164	+	0.677890i	$0.762895\pi$
$62$	7.39883i	0.119336i
$63$	0	0
$64$	−69.3862	−1.08416
$65$	10.7828i	0.165889i
$66$	0	0
$67$	98.0763	1.46383	0.731913	−	0.681399i	$-0.238628\pi$
$67$	98.0763	1.46383	0.731913	+	0.681399i	$0.238628\pi$
$68$	− 20.8674i	− 0.306873i
$69$	0	0
$70$	44.9964	0.642805
$71$	− 77.6291i	− 1.09337i	−0.837339	−	0.546684i	$-0.815890\pi$
$71$	− 77.6291i	− 1.09337i	0.837339	−	0.546684i	$-0.184110\pi$
$72$	0	0
$73$	−140.161	−1.92001	−0.960004	−	0.279986i	$-0.909670\pi$
$73$	−140.161	−1.92001	−0.960004	+	0.279986i	$0.909670\pi$
$74$	38.4257i	0.519266i
$75$	0	0
$76$	−29.7330	−0.391224
$77$	− 2.86849i	− 0.0372531i
$78$	0	0
$79$	71.5169	0.905278	0.452639	−	0.891694i	$-0.350483\pi$
$79$	71.5169	0.905278	0.452639	+	0.891694i	$0.350483\pi$
$80$	− 27.8719i	− 0.348399i
$81$	0	0
$82$	40.6835	0.496140
$83$	− 109.269i	− 1.31649i	−0.752804	−	0.658244i	$-0.771299\pi$
$83$	− 109.269i	− 1.31649i	0.752804	−	0.658244i	$-0.228701\pi$
$84$	0	0
$85$	49.2140	0.578988
$86$	94.2508i	1.09594i
$87$	0	0
$88$	−2.74383	−0.0311799
$89$	− 119.900i	− 1.34720i	−0.739098	−	0.673598i	$-0.764748\pi$
$89$	− 119.900i	− 1.34720i	0.739098	−	0.673598i	$-0.235252\pi$
$90$	0	0
$91$	32.8212	0.360672
$92$	17.2480i	0.187479i
$93$	0	0
$94$	73.9544	0.786749
$95$	− 70.1229i	− 0.738136i
$96$	0	0
$97$	39.1083	0.403179	0.201589	−	0.979470i	$-0.435389\pi$
$97$	39.1083	0.403179	0.201589	+	0.979470i	$0.435389\pi$
$98$	− 55.9722i	− 0.571145i
$99$	0	0
$100$	20.3603	0.203603
$101$	88.0927i	0.872205i	0.899897	+	0.436102i	$0.143641\pi$
$101$	88.0927i	0.872205i	−0.899897	+	0.436102i	$0.856359\pi$
$102$	0	0
$103$	31.9825	0.310510	0.155255	−	0.987874i	$-0.450380\pi$
$103$	31.9825	0.310510	0.155255	+	0.987874i	$0.450380\pi$
$104$	− 31.3948i	− 0.301873i
$105$	0	0
$106$	102.273	0.964841
$107$	161.425i	1.50865i	0.656503	+	0.754324i	$0.272035\pi$
$107$	161.425i	1.50865i	−0.656503	+	0.754324i	$0.727965\pi$
$108$	0	0
$109$	−83.7344	−0.768205	−0.384103	−	0.923290i	$-0.625489\pi$
$109$	−83.7344	−0.768205	−0.384103	+	0.923290i	$0.625489\pi$
$110$	− 1.55763i	− 0.0141603i
$111$	0	0
$112$	−84.8379	−0.757481
$113$	187.622i	1.66037i	0.557487	+	0.830185i	$0.311766\pi$
$113$	187.622i	1.66037i	−0.557487	+	0.830185i	$0.688234\pi$
$114$	0	0
$115$	−40.6781	−0.353722
$116$	54.1355i	0.466686i
$117$	0	0
$118$	−149.970	−1.27093
$119$	− 149.800i	− 1.25882i
$120$	0	0
$121$	120.901	0.999179
$122$	148.245i	1.21512i
$123$	0	0
$124$	−5.67630	−0.0457766
$125$	122.783i	0.982265i
$126$	0	0
$127$	109.019	0.858419	0.429209	−	0.903205i	$-0.358792\pi$
$127$	109.019	0.858419	0.429209	+	0.903205i	$0.358792\pi$
$128$	36.9854i	0.288948i
$129$	0	0
$130$	17.8224	0.137095
$131$	− 84.9320i	− 0.648336i	−0.946000	−	0.324168i	$-0.894916\pi$
$131$	− 84.9320i	− 0.648336i	0.946000	−	0.324168i	$-0.105084\pi$
$132$	0	0
$133$	−213.444	−1.60484
$134$	− 162.106i	− 1.20975i
$135$	0	0
$136$	−143.290	−1.05360
$137$	− 104.797i	− 0.764943i	−0.923967	−	0.382471i	$-0.875073\pi$
$137$	− 104.797i	− 0.764943i	0.923967	−	0.382471i	$-0.124927\pi$
$138$	0	0
$139$	−42.1929	−0.303546	−0.151773	−	0.988415i	$-0.548498\pi$
$139$	−42.1929	−0.303546	−0.151773	+	0.988415i	$0.548498\pi$
$140$	34.5207i	0.246576i
$141$	0	0
$142$	−128.310	−0.903592
$143$	− 1.13617i	− 0.00794523i
$144$	0	0
$145$	−127.674	−0.880512
$146$	231.666i	1.58675i
$147$	0	0
$148$	−29.4797	−0.199187
$149$	− 90.5070i	− 0.607429i	−0.952763	−	0.303715i	$-0.901773\pi$
$149$	− 90.5070i	− 0.607429i	0.952763	−	0.303715i	$-0.0982269\pi$
$150$	0	0
$151$	−0.358353	−0.00237320	−0.00118660	−	0.999999i	$-0.500378\pi$
$151$	−0.358353	−0.00237320	−0.00118660	+	0.999999i	$0.500378\pi$
$152$	204.168i	1.34321i
$153$	0	0
$154$	−4.74121	−0.0307871
$155$	− 13.3871i	− 0.0863683i
$156$	0	0
$157$	113.271	0.721473	0.360736	−	0.932668i	$-0.382525\pi$
$157$	113.271	0.721473	0.360736	+	0.932668i	$0.382525\pi$
$158$	− 118.207i	− 0.748149i
$159$	0	0
$160$	58.0927	0.363079
$161$	123.818i	0.769057i
$162$	0	0
$163$	208.478	1.27901	0.639505	−	0.768787i	$-0.279140\pi$
$163$	208.478	1.27901	0.639505	+	0.768787i	$0.279140\pi$
$164$	31.2119i	0.190316i
$165$	0	0
$166$	−180.606	−1.08799
$167$	141.160i	0.845268i	0.906300	+	0.422634i	$0.138894\pi$
$167$	141.160i	0.845268i	−0.906300	+	0.422634i	$0.861106\pi$
$168$	0	0
$169$	13.0000	0.0769231
$170$	− 81.3438i	− 0.478493i
$171$	0	0
$172$	−72.3081	−0.420396
$173$	− 99.9542i	− 0.577770i	−0.957364	−	0.288885i	$-0.906715\pi$
$173$	− 99.9542i	− 0.577770i	0.957364	−	0.288885i	$-0.0932845\pi$
$174$	0	0
$175$	146.160	0.835199
$176$	2.93683i	0.0166865i
$177$	0	0
$178$	−198.179	−1.11336
$179$	− 208.837i	− 1.16669i	−0.812225	−	0.583345i	$-0.801744\pi$
$179$	− 208.837i	− 1.16669i	0.812225	−	0.583345i	$-0.198256\pi$
$180$	0	0
$181$	159.604	0.881792	0.440896	−	0.897558i	$-0.354661\pi$
$181$	159.604	0.881792	0.440896	+	0.897558i	$0.354661\pi$
$182$	− 54.2488i	− 0.298070i
$183$	0	0
$184$	118.437	0.643680
$185$	− 69.5255i	− 0.375814i
$186$	0	0
$187$	−5.18562	−0.0277306
$188$	56.7370i	0.301792i
$189$	0	0
$190$	−115.903	−0.610018
$191$	− 233.056i	− 1.22019i	−0.792329	−	0.610094i	$-0.791132\pi$
$191$	− 233.056i	− 1.22019i	0.792329	−	0.610094i	$-0.208868\pi$
$192$	0	0
$193$	180.834	0.936964	0.468482	−	0.883473i	$-0.344801\pi$
$193$	180.834	0.936964	0.468482	+	0.883473i	$0.344801\pi$
$194$	− 64.6406i	− 0.333199i
$195$	0	0
$196$	42.9412	0.219088
$197$	51.4340i	0.261086i	0.991443	+	0.130543i	$0.0416721\pi$
$197$	51.4340i	0.261086i	−0.991443	+	0.130543i	$0.958328\pi$
$198$	0	0
$199$	330.096	1.65877	0.829387	−	0.558674i	$-0.188690\pi$
$199$	330.096	1.65877	0.829387	+	0.558674i	$0.188690\pi$
$200$	− 139.808i	− 0.699039i
$201$	0	0
$202$	145.605	0.720816
$203$	388.622i	1.91439i
$204$	0	0
$205$	−73.6107	−0.359077
$206$	− 52.8626i	− 0.256615i
$207$	0	0
$208$	−33.6031	−0.161553
$209$	7.38877i	0.0353530i
$210$	0	0
$211$	−259.140	−1.22815	−0.614077	−	0.789246i	$-0.710472\pi$
$211$	−259.140	−1.22815	−0.614077	+	0.789246i	$0.710472\pi$
$212$	78.4628i	0.370107i
$213$	0	0
$214$	266.813	1.24679
$215$	− 170.533i	− 0.793176i
$216$	0	0
$217$	−40.7483	−0.187780
$218$	138.401i	0.634868i
$219$	0	0
$220$	1.19500	0.00543181
$221$	− 59.3337i	− 0.268478i
$222$	0	0
$223$	24.1302	0.108207	0.0541036	−	0.998535i	$-0.482770\pi$
$223$	24.1302	0.108207	0.0541036	+	0.998535i	$0.482770\pi$
$224$	− 176.826i	− 0.789400i
$225$	0	0
$226$	310.113	1.37218
$227$	− 295.237i	− 1.30060i	−0.759675	−	0.650302i	$-0.774642\pi$
$227$	− 295.237i	− 1.30060i	0.759675	−	0.650302i	$-0.225358\pi$
$228$	0	0
$229$	84.0045	0.366832	0.183416	−	0.983035i	$-0.441284\pi$
$229$	84.0045	0.366832	0.183416	+	0.983035i	$0.441284\pi$
$230$	67.2352i	0.292327i
$231$	0	0
$232$	371.733	1.60230
$233$	312.060i	1.33931i	0.742671	+	0.669657i	$0.233559\pi$
$233$	312.060i	1.33931i	−0.742671	+	0.669657i	$0.766441\pi$
$234$	0	0
$235$	−133.809	−0.569402
$236$	− 115.055i	− 0.487521i
$237$	0	0
$238$	−247.599	−1.04033
$239$	− 261.249i	− 1.09309i	−0.837429	−	0.546546i	$-0.815942\pi$
$239$	− 261.249i	− 1.09309i	0.837429	−	0.546546i	$-0.184058\pi$
$240$	0	0
$241$	−346.533	−1.43790	−0.718948	−	0.695064i	$-0.755376\pi$
$241$	−346.533	−1.43790	−0.718948	+	0.695064i	$0.755376\pi$
$242$	− 199.832i	− 0.825752i
$243$	0	0
$244$	−113.732	−0.466114
$245$	101.273i	0.413360i
$246$	0	0
$247$	−84.5421	−0.342276
$248$	38.9774i	0.157167i
$249$	0	0
$250$	202.943	0.811773
$251$	121.070i	0.482351i	0.970481	+	0.241176i	$0.0775330\pi$
$251$	121.070i	0.482351i	−0.970481	+	0.241176i	$0.922467\pi$
$252$	0	0
$253$	4.28620	0.0169415
$254$	− 180.193i	− 0.709423i
$255$	0	0
$256$	−216.413	−0.845364
$257$	− 428.208i	− 1.66618i	−0.553138	−	0.833089i	$-0.686570\pi$
$257$	− 428.208i	− 1.66618i	0.553138	−	0.833089i	$-0.313430\pi$
$258$	0	0
$259$	−211.626	−0.817087
$260$	13.6731i	0.0525890i
$261$	0	0
$262$	−140.381	−0.535804
$263$	− 119.167i	− 0.453108i	−0.973999	−	0.226554i	$-0.927254\pi$
$263$	− 119.167i	− 0.453108i	0.973999	−	0.226554i	$-0.0727459\pi$
$264$	0	0
$265$	−185.048	−0.698294
$266$	352.793i	1.32629i
$267$	0	0
$268$	124.366	0.464053
$269$	229.439i	0.852932i	0.904504	+	0.426466i	$0.140242\pi$
$269$	229.439i	0.852932i	−0.904504	+	0.426466i	$0.859758\pi$
$270$	0	0
$271$	−93.1363	−0.343676	−0.171838	−	0.985125i	$-0.554971\pi$
$271$	−93.1363	−0.343676	−0.171838	+	0.985125i	$0.554971\pi$
$272$	153.369i	0.563856i
$273$	0	0
$274$	−173.215	−0.632171
$275$	− 5.05960i	− 0.0183985i
$276$	0	0
$277$	88.9120	0.320982	0.160491	−	0.987037i	$-0.448692\pi$
$277$	88.9120	0.320982	0.160491	+	0.987037i	$0.448692\pi$
$278$	69.7389i	0.250859i
$279$	0	0
$280$	237.043	0.846583
$281$	460.348i	1.63825i	0.573615	+	0.819125i	$0.305541\pi$
$281$	460.348i	1.63825i	−0.573615	+	0.819125i	$0.694459\pi$
$282$	0	0
$283$	−257.558	−0.910100	−0.455050	−	0.890466i	$-0.650379\pi$
$283$	−257.558	−0.910100	−0.455050	+	0.890466i	$0.650379\pi$
$284$	− 98.4379i	− 0.346612i
$285$	0	0
$286$	−1.87793	−0.00656617
$287$	224.060i	0.780698i
$288$	0	0
$289$	18.1935	0.0629532
$290$	211.028i	0.727682i
$291$	0	0
$292$	−177.731	−0.608669
$293$	338.179i	1.15419i	0.816675	+	0.577097i	$0.195815\pi$
$293$	338.179i	1.15419i	−0.816675	+	0.577097i	$0.804185\pi$
$294$	0	0
$295$	271.348	0.919823
$296$	202.429i	0.683880i
$297$	0	0
$298$	−149.595	−0.501998
$299$	49.0426i	0.164022i
$300$	0	0
$301$	−519.077	−1.72451
$302$	0.592307i	0.00196128i
$303$	0	0
$304$	218.529	0.718845
$305$	− 268.227i	− 0.879433i
$306$	0	0
$307$	290.813	0.947274	0.473637	−	0.880720i	$-0.342941\pi$
$307$	290.813	0.947274	0.473637	+	0.880720i	$0.342941\pi$
$308$	− 3.63740i	− 0.0118097i
$309$	0	0
$310$	−22.1270	−0.0713773
$311$	− 203.983i	− 0.655894i	−0.944696	−	0.327947i	$-0.893643\pi$
$311$	− 203.983i	− 0.655894i	0.944696	−	0.327947i	$-0.106357\pi$
$312$	0	0
$313$	−68.3636	−0.218414	−0.109207	−	0.994019i	$-0.534831\pi$
$313$	−68.3636	−0.218414	−0.109207	+	0.994019i	$0.534831\pi$
$314$	− 187.222i	− 0.596247i
$315$	0	0
$316$	90.6874	0.286985
$317$	225.498i	0.711349i	0.934610	+	0.355675i	$0.115749\pi$
$317$	225.498i	0.711349i	−0.934610	+	0.355675i	$0.884251\pi$
$318$	0	0
$319$	13.4529	0.0421720
$320$	− 207.507i	− 0.648458i
$321$	0	0
$322$	204.654	0.635571
$323$	385.861i	1.19462i
$324$	0	0
$325$	57.8918	0.178129
$326$	− 344.586i	− 1.05701i
$327$	0	0
$328$	214.323	0.653423
$329$	407.296i	1.23798i
$330$	0	0
$331$	16.0487	0.0484854	0.0242427	−	0.999706i	$-0.492283\pi$
$331$	16.0487	0.0484854	0.0242427	+	0.999706i	$0.492283\pi$
$332$	− 138.558i	− 0.417345i
$333$	0	0
$334$	233.317	0.698555
$335$	293.307i	0.875544i
$336$	0	0
$337$	−22.7096	−0.0673875	−0.0336937	−	0.999432i	$-0.510727\pi$
$337$	−22.7096	−0.0673875	−0.0336937	+	0.999432i	$0.510727\pi$
$338$	− 21.4872i	− 0.0635715i
$339$	0	0
$340$	62.4060	0.183547
$341$	1.41058i	0.00413660i
$342$	0	0
$343$	−137.784	−0.401702
$344$	496.518i	1.44337i
$345$	0	0
$346$	−165.210	−0.477486
$347$	− 337.189i	− 0.971726i	−0.874035	−	0.485863i	$-0.838505\pi$
$347$	− 337.189i	− 0.971726i	0.874035	−	0.485863i	$-0.161495\pi$
$348$	0	0
$349$	−130.123	−0.372846	−0.186423	−	0.982470i	$-0.559689\pi$
$349$	−130.123	−0.372846	−0.186423	+	0.982470i	$0.559689\pi$
$350$	− 241.582i	− 0.690233i
$351$	0	0
$352$	−6.12115	−0.0173896
$353$	573.635i	1.62503i	0.582941	+	0.812515i	$0.301902\pi$
$353$	573.635i	1.62503i	−0.582941	+	0.812515i	$0.698098\pi$
$354$	0	0
$355$	232.158	0.653966
$356$	− 152.040i	− 0.427079i
$357$	0	0
$358$	−345.179	−0.964187
$359$	− 681.535i	− 1.89843i	−0.314635	−	0.949213i	$-0.601882\pi$
$359$	− 681.535i	− 1.89843i	0.314635	−	0.949213i	$-0.398118\pi$
$360$	0	0
$361$	188.797	0.522984
$362$	− 263.804i	− 0.728739i
$363$	0	0
$364$	41.6190	0.114338
$365$	− 419.165i	− 1.14840i
$366$	0	0
$367$	−576.535	−1.57094	−0.785471	−	0.618899i	$-0.787579\pi$
$367$	−576.535	−1.57094	−0.785471	+	0.618899i	$0.787579\pi$
$368$	− 126.768i	− 0.344478i
$369$	0	0
$370$	−114.916	−0.310584
$371$	563.259i	1.51822i
$372$	0	0
$373$	370.680	0.993780	0.496890	−	0.867814i	$-0.334475\pi$
$373$	370.680	0.993780	0.496890	+	0.867814i	$0.334475\pi$
$374$	8.57109i	0.0229174i
$375$	0	0
$376$	389.596	1.03616
$377$	153.927i	0.408296i
$378$	0	0
$379$	115.121	0.303748	0.151874	−	0.988400i	$-0.451469\pi$
$379$	115.121	0.303748	0.151874	+	0.988400i	$0.451469\pi$
$380$	− 88.9197i	− 0.233999i
$381$	0	0
$382$	−385.209	−1.00840
$383$	− 138.743i	− 0.362252i	−0.983460	−	0.181126i	$-0.942026\pi$
$383$	− 138.743i	− 0.362252i	0.983460	−	0.181126i	$-0.0579742\pi$
$384$	0	0
$385$	8.57851	0.0222819
$386$	− 298.893i	− 0.774335i
$387$	0	0
$388$	49.5915	0.127813
$389$	− 716.370i	− 1.84157i	−0.390074	−	0.920783i	$-0.627551\pi$
$389$	− 716.370i	− 1.84157i	0.390074	−	0.920783i	$-0.372449\pi$
$390$	0	0
$391$	223.837	0.572472
$392$	− 294.864i	− 0.752205i
$393$	0	0
$394$	85.0132	0.215770
$395$	213.879i	0.541465i
$396$	0	0
$397$	−158.652	−0.399627	−0.199814	−	0.979834i	$-0.564034\pi$
$397$	−158.652	−0.399627	−0.199814	+	0.979834i	$0.564034\pi$
$398$	− 545.603i	− 1.37086i
$399$	0	0
$400$	−149.642	−0.374104
$401$	− 37.1460i	− 0.0926335i	−0.998927	−	0.0463167i	$-0.985252\pi$
$401$	− 37.1460i	− 0.0926335i	0.998927	−	0.0463167i	$-0.0147484\pi$
$402$	0	0
$403$	−16.1398	−0.0400492
$404$	111.706i	0.276501i
$405$	0	0
$406$	642.337	1.58211
$407$	7.32582i	0.0179996i
$408$	0	0
$409$	62.0756	0.151774	0.0758870	−	0.997116i	$-0.475821\pi$
$409$	62.0756	0.151774	0.0758870	+	0.997116i	$0.475821\pi$
$410$	121.668i	0.296752i
$411$	0	0
$412$	40.5556	0.0984359
$413$	− 825.943i	− 1.99986i
$414$	0	0
$415$	326.779	0.787419
$416$	− 70.0381i	− 0.168361i
$417$	0	0
$418$	12.2126	0.0292167
$419$	250.937i	0.598894i	0.954113	+	0.299447i	$0.0968022\pi$
$419$	250.937i	0.598894i	−0.954113	+	0.299447i	$0.903198\pi$
$420$	0	0
$421$	24.4538	0.0580851	0.0290426	−	0.999578i	$-0.490754\pi$
$421$	24.4538	0.0580851	0.0290426	+	0.999578i	$0.490754\pi$
$422$	428.323i	1.01498i
$423$	0	0
$424$	538.781	1.27071
$425$	− 264.226i	− 0.621707i
$426$	0	0
$427$	−816.444	−1.91205
$428$	204.696i	0.478262i
$429$	0	0
$430$	−281.867	−0.655504
$431$	346.662i	0.804320i	0.915569	+	0.402160i	$0.131740\pi$
$431$	346.662i	0.804320i	−0.915569	+	0.402160i	$0.868260\pi$
$432$	0	0
$433$	−554.932	−1.28160	−0.640799	−	0.767709i	$-0.721397\pi$
$433$	−554.932	−1.28160	−0.640799	+	0.767709i	$0.721397\pi$
$434$	67.3512i	0.155187i
$435$	0	0
$436$	−106.180	−0.243532
$437$	− 318.936i	− 0.729830i
$438$	0	0
$439$	774.134	1.76340	0.881701	−	0.471808i	$-0.156399\pi$
$439$	774.134	1.76340	0.881701	+	0.471808i	$0.156399\pi$
$440$	− 8.20570i	− 0.0186493i
$441$	0	0
$442$	−98.0702	−0.221878
$443$	397.919i	0.898237i	0.893472	+	0.449119i	$0.148262\pi$
$443$	397.919i	0.898237i	−0.893472	+	0.449119i	$0.851738\pi$
$444$	0	0
$445$	358.575	0.805785
$446$	− 39.8838i	− 0.0894256i
$447$	0	0
$448$	−631.620	−1.40987
$449$	− 170.800i	− 0.380401i	−0.981745	−	0.190201i	$-0.939086\pi$
$449$	− 170.800i	− 0.380401i	0.981745	−	0.190201i	$-0.0609138\pi$
$450$	0	0
$451$	7.75627	0.0171979
$452$	237.915i	0.526360i
$453$	0	0
$454$	−487.986	−1.07486
$455$	98.1551i	0.215726i
$456$	0	0
$457$	366.489	0.801946	0.400973	−	0.916090i	$-0.368672\pi$
$457$	366.489	0.801946	0.400973	+	0.916090i	$0.368672\pi$
$458$	− 138.848i	− 0.303161i
$459$	0	0
$460$	−51.5820	−0.112135
$461$	57.6898i	0.125141i	0.998041	+	0.0625703i	$0.0199298\pi$
$461$	57.6898i	0.125141i	−0.998041	+	0.0625703i	$0.980070\pi$
$462$	0	0
$463$	331.654	0.716315	0.358157	−	0.933661i	$-0.383405\pi$
$463$	331.654	0.716315	0.358157	+	0.933661i	$0.383405\pi$
$464$	− 397.880i	− 0.857500i
$465$	0	0
$466$	515.792	1.10685
$467$	144.500i	0.309422i	0.987960	+	0.154711i	$0.0494447\pi$
$467$	144.500i	0.309422i	−0.987960	+	0.154711i	$0.950555\pi$
$468$	0	0
$469$	892.784	1.90359
$470$	221.168i	0.470571i
$471$	0	0
$472$	−790.049	−1.67383
$473$	17.9688i	0.0379891i
$474$	0	0
$475$	−376.484	−0.792598
$476$	− 189.955i	− 0.399064i
$477$	0	0
$478$	−431.808	−0.903365
$479$	− 92.6210i	− 0.193363i	−0.995315	−	0.0966817i	$-0.969177\pi$
$479$	− 92.6210i	− 0.193363i	0.995315	−	0.0966817i	$-0.0308229\pi$
$480$	0	0
$481$	−83.8219	−0.174266
$482$	572.770i	1.18832i
$483$	0	0
$484$	153.309	0.316753
$485$	116.957i	0.241149i
$486$	0	0
$487$	264.219	0.542543	0.271272	−	0.962503i	$-0.412556\pi$
$487$	264.219	0.542543	0.271272	+	0.962503i	$0.412556\pi$
$488$	780.962i	1.60033i
$489$	0	0
$490$	167.391	0.341613
$491$	812.120i	1.65401i	0.562192	+	0.827007i	$0.309958\pi$
$491$	812.120i	1.65401i	−0.562192	+	0.827007i	$0.690042\pi$
$492$	0	0
$493$	702.545	1.42504
$494$	139.736i	0.282867i
$495$	0	0
$496$	41.7191	0.0841110
$497$	− 706.654i	− 1.42184i
$498$	0	0
$499$	64.0030	0.128263	0.0641313	−	0.997941i	$-0.479572\pi$
$499$	64.0030	0.128263	0.0641313	+	0.997941i	$0.479572\pi$
$500$	155.696i	0.311391i
$501$	0	0
$502$	200.112	0.398629
$503$	− 207.435i	− 0.412396i	−0.978510	−	0.206198i	$-0.933891\pi$
$503$	− 207.435i	− 0.412396i	0.978510	−	0.206198i	$-0.0661091\pi$
$504$	0	0
$505$	−263.450	−0.521684
$506$	− 7.08449i	− 0.0140010i
$507$	0	0
$508$	138.242	0.272131
$509$	673.967i	1.32410i	0.749460	+	0.662050i	$0.230313\pi$
$509$	673.967i	1.32410i	−0.749460	+	0.662050i	$0.769687\pi$
$510$	0	0
$511$	−1275.88	−2.49682
$512$	505.642i	0.987582i
$513$	0	0
$514$	−707.768	−1.37698
$515$	95.6470i	0.185722i
$516$	0	0
$517$	14.0993	0.0272715
$518$	349.787i	0.675265i
$519$	0	0
$520$	93.8894	0.180557
$521$	518.723i	0.995630i	0.867283	+	0.497815i	$0.165864\pi$
$521$	518.723i	0.995630i	−0.867283	+	0.497815i	$0.834136\pi$
$522$	0	0
$523$	153.321	0.293157	0.146579	−	0.989199i	$-0.453174\pi$
$523$	153.321	0.293157	0.146579	+	0.989199i	$0.453174\pi$
$524$	− 107.698i	− 0.205531i
$525$	0	0
$526$	−196.967	−0.374462
$527$	73.6642i	0.139780i
$528$	0	0
$529$	343.987	0.650258
$530$	305.858i	0.577091i
$531$	0	0
$532$	−270.659	−0.508757
$533$	88.7470i	0.166505i
$534$	0	0
$535$	−482.759	−0.902353
$536$	− 853.985i	− 1.59326i
$537$	0	0
$538$	379.230	0.704888
$539$	− 10.6710i	− 0.0197979i
$540$	0	0
$541$	−818.301	−1.51257	−0.756286	−	0.654241i	$-0.772988\pi$
$541$	−818.301	−1.51257	−0.756286	+	0.654241i	$0.772988\pi$
$542$	153.941i	0.284024i
$543$	0	0
$544$	−319.663	−0.587615
$545$	− 250.416i	− 0.459479i
$546$	0	0
$547$	−982.786	−1.79668	−0.898342	−	0.439297i	$-0.855227\pi$
$547$	−982.786	−1.79668	−0.898342	+	0.439297i	$0.855227\pi$
$548$	− 132.888i	− 0.242497i
$549$	0	0
$550$	−8.36281	−0.0152051
$551$	− 1001.03i	− 1.81675i
$552$	0	0
$553$	651.016	1.17724
$554$	− 146.959i	− 0.265269i
$555$	0	0
$556$	−53.5029	−0.0962282
$557$	− 696.090i	− 1.24971i	−0.780740	−	0.624856i	$-0.785158\pi$
$557$	− 696.090i	− 1.24971i	0.780740	−	0.624856i	$-0.214842\pi$
$558$	0	0
$559$	−205.599	−0.367798
$560$	− 253.717i	− 0.453065i
$561$	0	0
$562$	760.891	1.35390
$563$	153.937i	0.273423i	0.990611	+	0.136712i	$0.0436534\pi$
$563$	153.937i	0.273423i	−0.990611	+	0.136712i	$0.956347\pi$
$564$	0	0
$565$	−561.103	−0.993102
$566$	425.708i	0.752134i
$567$	0	0
$568$	−675.944	−1.19004
$569$	− 488.224i	− 0.858039i	−0.903295	−	0.429020i	$-0.858859\pi$
$569$	− 488.224i	− 0.858039i	0.903295	−	0.429020i	$-0.141141\pi$
$570$	0	0
$571$	684.984	1.19962	0.599811	−	0.800142i	$-0.295242\pi$
$571$	684.984	1.19962	0.599811	+	0.800142i	$0.295242\pi$
$572$	− 1.44072i	− 0.00251875i
$573$	0	0
$574$	370.340	0.645192
$575$	218.397i	0.379821i
$576$	0	0
$577$	−13.8816	−0.0240582	−0.0120291	−	0.999928i	$-0.503829\pi$
$577$	−13.8816	−0.0240582	−0.0120291	+	0.999928i	$0.503829\pi$
$578$	− 30.0713i	− 0.0520264i
$579$	0	0
$580$	−161.898	−0.279134
$581$	− 994.667i	− 1.71199i
$582$	0	0
$583$	19.4983	0.0334447
$584$	1220.43i	2.08977i
$585$	0	0
$586$	558.963	0.953861
$587$	782.701i	1.33339i	0.745330	+	0.666696i	$0.232292\pi$
$587$	782.701i	1.33339i	−0.745330	+	0.666696i	$0.767708\pi$
$588$	0	0
$589$	104.961	0.178202
$590$	− 448.500i	− 0.760170i
$591$	0	0
$592$	216.667	0.365992
$593$	− 916.503i	− 1.54554i	−0.634688	−	0.772768i	$-0.718872\pi$
$593$	− 916.503i	− 1.54554i	0.634688	−	0.772768i	$-0.281128\pi$
$594$	0	0
$595$	447.993	0.752929
$596$	− 114.768i	− 0.192563i
$597$	0	0
$598$	81.0605	0.135553
$599$	− 49.3571i	− 0.0823991i	−0.999151	−	0.0411995i	$-0.986882\pi$
$599$	− 49.3571i	− 0.0823991i	0.999151	−	0.0411995i	$-0.0131179\pi$
$600$	0	0
$601$	−60.0406	−0.0999011	−0.0499506	−	0.998752i	$-0.515906\pi$
$601$	−60.0406	−0.0999011	−0.0499506	+	0.998752i	$0.515906\pi$
$602$	857.961i	1.42518i
$603$	0	0
$604$	−0.454411	−0.000752336 0
$605$	361.566i	0.597630i
$606$	0	0
$607$	500.297	0.824213	0.412107	−	0.911136i	$-0.364793\pi$
$607$	500.297	0.824213	0.412107	+	0.911136i	$0.364793\pi$
$608$	455.474i	0.749135i
$609$	0	0
$610$	−443.342	−0.726790
$611$	161.324i	0.264033i
$612$	0	0
$613$	137.391	0.224129	0.112064	−	0.993701i	$-0.464254\pi$
$613$	137.391	0.224129	0.112064	+	0.993701i	$0.464254\pi$
$614$	− 480.673i	− 0.782856i
$615$	0	0
$616$	−24.9770	−0.0405470
$617$	− 470.896i	− 0.763203i	−0.924327	−	0.381602i	$-0.875373\pi$
$617$	− 470.896i	− 0.763203i	0.924327	−	0.381602i	$-0.124627\pi$
$618$	0	0
$619$	1.90176	0.00307230	0.00153615	−	0.999999i	$-0.499511\pi$
$619$	1.90176	0.00307230	0.00153615	+	0.999999i	$0.499511\pi$
$620$	− 16.9755i	− 0.0273799i
$621$	0	0
$622$	−337.155	−0.542050
$623$	− 1091.45i	− 1.75192i
$624$	0	0
$625$	34.2118	0.0547389
$626$	112.995i	0.180504i
$627$	0	0
$628$	143.634	0.228717
$629$	382.574i	0.608225i
$630$	0	0
$631$	83.6165	0.132514	0.0662571	−	0.997803i	$-0.478894\pi$
$631$	83.6165	0.132514	0.0662571	+	0.997803i	$0.478894\pi$
$632$	− 622.723i	− 0.985322i
$633$	0	0
$634$	372.716	0.587880
$635$	326.033i	0.513438i
$636$	0	0
$637$	122.098	0.191676
$638$	− 22.2357i	− 0.0348522i
$639$	0	0
$640$	−110.609	−0.172826
$641$	− 77.2534i	− 0.120520i	−0.998183	−	0.0602601i	$-0.980807\pi$
$641$	− 77.2534i	− 0.120520i	0.998183	−	0.0602601i	$-0.0191930\pi$
$642$	0	0
$643$	−56.2948	−0.0875502	−0.0437751	−	0.999041i	$-0.513939\pi$
$643$	−56.2948	−0.0875502	−0.0437751	+	0.999041i	$0.513939\pi$
$644$	157.008i	0.243801i
$645$	0	0
$646$	637.774	0.987266
$647$	− 1064.84i	− 1.64581i	−0.568182	−	0.822903i	$-0.692353\pi$
$647$	− 1064.84i	− 1.64581i	0.568182	−	0.822903i	$-0.307647\pi$
$648$	0	0
$649$	−28.5916	−0.0440548
$650$	− 95.6870i	− 0.147211i
$651$	0	0
$652$	264.362	0.405463
$653$	845.231i	1.29438i	0.762328	+	0.647191i	$0.224056\pi$
$653$	845.231i	1.29438i	−0.762328	+	0.647191i	$0.775944\pi$
$654$	0	0
$655$	253.998	0.387783
$656$	− 229.398i	− 0.349692i
$657$	0	0
$658$	673.204	1.02311
$659$	471.835i	0.715987i	0.933724	+	0.357994i	$0.116539\pi$
$659$	471.835i	0.715987i	−0.933724	+	0.357994i	$0.883461\pi$
$660$	0	0
$661$	612.351	0.926400	0.463200	−	0.886254i	$-0.346701\pi$
$661$	612.351	0.926400	0.463200	+	0.886254i	$0.346701\pi$
$662$	− 26.5262i	− 0.0400697i
$663$	0	0
$664$	−951.440	−1.43289
$665$	− 638.326i	− 0.959889i
$666$	0	0
$667$	−580.692	−0.870603
$668$	178.998i	0.267962i
$669$	0	0
$670$	484.796	0.723576
$671$	28.2628i	0.0421204i
$672$	0	0
$673$	1126.87	1.67440	0.837198	−	0.546900i	$-0.184192\pi$
$673$	1126.87	1.67440	0.837198	+	0.546900i	$0.184192\pi$
$674$	37.5358i	0.0556910i
$675$	0	0
$676$	16.4847	0.0243857
$677$	− 851.800i	− 1.25820i	−0.777326	−	0.629099i	$-0.783424\pi$
$677$	− 851.800i	− 1.25820i	0.777326	−	0.629099i	$-0.216576\pi$
$678$	0	0
$679$	356.001	0.524302
$680$	− 428.524i	− 0.630182i
$681$	0	0
$682$	2.33149	0.00341861
$683$	− 473.707i	− 0.693568i	−0.937945	−	0.346784i	$-0.887274\pi$
$683$	− 473.707i	− 0.693568i	0.937945	−	0.346784i	$-0.112726\pi$
$684$	0	0
$685$	313.407	0.457528
$686$	227.737i	0.331979i
$687$	0	0
$688$	531.443	0.772446
$689$	223.099i	0.323801i
$690$	0	0
$691$	−881.925	−1.27630	−0.638151	−	0.769911i	$-0.720301\pi$
$691$	−881.925	−1.27630	−0.638151	+	0.769911i	$0.720301\pi$
$692$	− 126.747i	− 0.183161i
$693$	0	0
$694$	−557.326	−0.803063
$695$	− 126.182i	− 0.181557i
$696$	0	0
$697$	405.053	0.581138
$698$	215.075i	0.308131i
$699$	0	0
$700$	185.339	0.264769
$701$	24.6220i	0.0351240i	0.999846	+	0.0175620i	$0.00559045\pi$
$701$	24.6220i	0.0351240i	−0.999846	+	0.0175620i	$0.994410\pi$
$702$	0	0
$703$	545.113	0.775410
$704$	21.8647i	0.0310578i
$705$	0	0
$706$	948.139	1.34297
$707$	801.904i	1.13423i
$708$	0	0
$709$	−1123.21	−1.58422	−0.792109	−	0.610379i	$-0.791017\pi$
$709$	−1123.21	−1.58422	−0.792109	+	0.610379i	$0.791017\pi$
$710$	− 383.724i	− 0.540457i
$711$	0	0
$712$	−1044.02	−1.46631
$713$	− 60.8876i	− 0.0853963i
$714$	0	0
$715$	3.39783	0.00475221
$716$	− 264.817i	− 0.369856i
$717$	0	0
$718$	−1126.48	−1.56892
$719$	213.798i	0.297354i	0.988886	+	0.148677i	$0.0475015\pi$
$719$	213.798i	0.297354i	−0.988886	+	0.148677i	$0.952499\pi$
$720$	0	0
$721$	291.135	0.403794
$722$	− 312.055i	− 0.432210i
$723$	0	0
$724$	202.387	0.279540
$725$	685.472i	0.945479i
$726$	0	0
$727$	−585.224	−0.804985	−0.402492	−	0.915423i	$-0.631856\pi$
$727$	−585.224	−0.804985	−0.402492	+	0.915423i	$0.631856\pi$
$728$	− 285.786i	− 0.392563i
$729$	0	0
$730$	−692.821	−0.949069
$731$	938.380i	1.28369i
$732$	0	0
$733$	37.3209	0.0509153	0.0254577	−	0.999676i	$-0.491896\pi$
$733$	37.3209	0.0509153	0.0254577	+	0.999676i	$0.491896\pi$
$734$	952.932i	1.29827i
$735$	0	0
$736$	264.219	0.358993
$737$	− 30.9054i	− 0.0419341i
$738$	0	0
$739$	−698.106	−0.944663	−0.472331	−	0.881421i	$-0.656587\pi$
$739$	−698.106	−0.944663	−0.472331	+	0.881421i	$0.656587\pi$
$740$	− 88.1622i	− 0.119138i
$741$	0	0
$742$	930.988	1.25470
$743$	166.716i	0.224382i	0.993687	+	0.112191i	$0.0357869\pi$
$743$	166.716i	0.224382i	−0.993687	+	0.112191i	$0.964213\pi$
$744$	0	0
$745$	270.670	0.363316
$746$	− 612.682i	− 0.821289i
$747$	0	0
$748$	−6.57564	−0.00879097
$749$	1469.45i	1.96188i
$750$	0	0
$751$	−935.532	−1.24571	−0.622857	−	0.782335i	$-0.714028\pi$
$751$	−935.532	−1.24571	−0.622857	+	0.782335i	$0.714028\pi$
$752$	− 417.000i	− 0.554521i
$753$	0	0
$754$	254.421	0.337428
$755$	− 1.07169i	− 0.00141946i
$756$	0	0
$757$	−256.749	−0.339167	−0.169583	−	0.985516i	$-0.554242\pi$
$757$	−256.749	−0.339167	−0.169583	+	0.985516i	$0.554242\pi$
$758$	− 190.278i	− 0.251026i
$759$	0	0
$760$	−610.585	−0.803402
$761$	− 159.934i	− 0.210162i	−0.994464	−	0.105081i	$-0.966490\pi$
$761$	− 159.934i	− 0.210162i	0.994464	−	0.105081i	$-0.0335103\pi$
$762$	0	0
$763$	−762.230	−0.998991
$764$	− 295.528i	− 0.386816i
$765$	0	0
$766$	−229.322	−0.299376
$767$	− 327.144i	− 0.426524i
$768$	0	0
$769$	−632.916	−0.823038	−0.411519	−	0.911401i	$-0.635002\pi$
$769$	−632.916	−0.823038	−0.411519	+	0.911401i	$0.635002\pi$
$770$	− 14.1791i	− 0.0184144i
$771$	0	0
$772$	229.307	0.297030
$773$	629.049i	0.813776i	0.913478	+	0.406888i	$0.133386\pi$
$773$	629.049i	0.813776i	−0.913478	+	0.406888i	$0.866614\pi$
$774$	0	0
$775$	−71.8741	−0.0927407
$776$	− 340.530i	− 0.438827i
$777$	0	0
$778$	−1184.06	−1.52193
$779$	− 577.143i	− 0.740877i
$780$	0	0
$781$	−24.4622	−0.0313216
$782$	− 369.971i	− 0.473108i
$783$	0	0
$784$	−315.605	−0.402557
$785$	338.749i	0.431528i
$786$	0	0
$787$	565.537	0.718598	0.359299	−	0.933222i	$-0.383016\pi$
$787$	565.537	0.718598	0.359299	+	0.933222i	$0.383016\pi$
$788$	65.2211i	0.0827679i
$789$	0	0
$790$	353.512	0.447483
$791$	1707.91i	2.15918i
$792$	0	0
$793$	−323.382	−0.407795
$794$	262.230i	0.330264i
$795$	0	0
$796$	418.580	0.525854
$797$	− 181.462i	− 0.227681i	−0.993499	−	0.113840i	$-0.963685\pi$
$797$	− 181.462i	− 0.227681i	0.993499	−	0.113840i	$-0.0363153\pi$
$798$	0	0
$799$	736.305	0.921533
$800$	− 311.895i	− 0.389868i
$801$	0	0
$802$	−61.3972	−0.0765551
$803$	44.1669i	0.0550023i
$804$	0	0
$805$	−370.291	−0.459989
$806$	26.6769i	0.0330978i
$807$	0	0
$808$	767.054	0.949325
$809$	680.471i	0.841127i	0.907263	+	0.420563i	$0.138168\pi$
$809$	680.471i	0.841127i	−0.907263	+	0.420563i	$0.861832\pi$
$810$	0	0
$811$	−821.125	−1.01249	−0.506243	−	0.862391i	$-0.668966\pi$
$811$	−821.125	−1.01249	−0.506243	+	0.862391i	$0.668966\pi$
$812$	492.794i	0.606889i
$813$	0	0
$814$	12.1086	0.0148754
$815$	623.476i	0.765002i
$816$	0	0
$817$	1337.06	1.63655
$818$	− 102.602i	− 0.125431i
$819$	0	0
$820$	−93.3424	−0.113832
$821$	− 245.999i	− 0.299633i	−0.988714	−	0.149816i	$-0.952132\pi$
$821$	− 245.999i	− 0.299633i	0.988714	−	0.149816i	$-0.0478683\pi$
$822$	0	0
$823$	1453.51	1.76611	0.883057	−	0.469266i	$-0.155481\pi$
$823$	1453.51	1.76611	0.883057	+	0.469266i	$0.155481\pi$
$824$	− 278.483i	− 0.337965i
$825$	0	0
$826$	−1365.17	−1.65275
$827$	255.461i	0.308900i	0.988001	+	0.154450i	$0.0493606\pi$
$827$	255.461i	0.308900i	−0.988001	+	0.154450i	$0.950639\pi$
$828$	0	0
$829$	−1367.99	−1.65017	−0.825087	−	0.565006i	$-0.808874\pi$
$829$	−1367.99	−1.65017	−0.825087	+	0.565006i	$0.808874\pi$
$830$	− 540.120i	− 0.650747i
$831$	0	0
$832$	−250.175	−0.300692
$833$	− 557.270i	− 0.668992i
$834$	0	0
$835$	−422.153	−0.505572
$836$	9.36936i	0.0112074i
$837$	0	0
$838$	414.763	0.494944
$839$	208.071i	0.247999i	0.992282	+	0.123999i	$0.0395721\pi$
$839$	208.071i	0.247999i	−0.992282	+	0.123999i	$0.960428\pi$
$840$	0	0
$841$	−981.590	−1.16717
$842$	− 40.4188i	− 0.0480033i
$843$	0	0
$844$	−328.604	−0.389342
$845$	38.8778i	0.0460093i
$846$	0	0
$847$	1100.55	1.29936
$848$	− 576.678i	− 0.680044i
$849$	0	0
$850$	−436.728	−0.513797
$851$	− 316.218i	− 0.371585i
$852$	0	0
$853$	−487.480	−0.571488	−0.285744	−	0.958306i	$-0.592241\pi$
$853$	−487.480	−0.571488	−0.285744	+	0.958306i	$0.592241\pi$
$854$	1349.47i	1.58017i
$855$	0	0
$856$	1405.59	1.64204
$857$	1221.14i	1.42490i	0.701721	+	0.712451i	$0.252415\pi$
$857$	1221.14i	1.42490i	−0.701721	+	0.712451i	$0.747585\pi$
$858$	0	0
$859$	−650.258	−0.756995	−0.378497	−	0.925602i	$-0.623559\pi$
$859$	−650.258	−0.756995	−0.378497	+	0.925602i	$0.623559\pi$
$860$	− 216.245i	− 0.251448i
$861$	0	0
$862$	572.984	0.664714
$863$	330.661i	0.383153i	0.981478	+	0.191576i	$0.0613600\pi$
$863$	330.661i	0.383153i	−0.981478	+	0.191576i	$0.938640\pi$
$864$	0	0
$865$	298.923	0.345576
$866$	917.224i	1.05915i
$867$	0	0
$868$	−51.6711	−0.0595289
$869$	− 22.5361i	− 0.0259334i
$870$	0	0
$871$	353.619	0.405992
$872$	729.105i	0.836130i
$873$	0	0
$874$	−527.156	−0.603153
$875$	1117.69i	1.27736i
$876$	0	0
$877$	1424.99	1.62485	0.812425	−	0.583066i	$-0.198147\pi$
$877$	1424.99	1.62485	0.812425	+	0.583066i	$0.198147\pi$
$878$	− 1279.53i	− 1.45733i
$879$	0	0
$880$	−8.78288	−0.00998055
$881$	− 1120.04i	− 1.27133i	−0.771965	−	0.635665i	$-0.780726\pi$
$881$	− 1120.04i	− 1.27133i	0.771965	−	0.635665i	$-0.219274\pi$
$882$	0	0
$883$	20.7974	0.0235532	0.0117766	−	0.999931i	$-0.496251\pi$
$883$	20.7974	0.0235532	0.0117766	+	0.999931i	$0.496251\pi$
$884$	− 75.2383i	− 0.0851112i
$885$	0	0
$886$	657.705	0.742330
$887$	− 495.824i	− 0.558990i	−0.960147	−	0.279495i	$-0.909833\pi$
$887$	− 495.824i	− 0.558990i	0.960147	−	0.279495i	$-0.0901670\pi$
$888$	0	0
$889$	992.397	1.11631
$890$	− 592.673i	− 0.665925i
$891$	0	0
$892$	30.5984	0.0343031
$893$	− 1049.13i	− 1.17484i
$894$	0	0
$895$	624.550	0.697821
$896$	336.676i	0.375755i
$897$	0	0
$898$	−282.309	−0.314375
$899$	− 191.105i	− 0.212575i
$900$	0	0
$901$	1018.25	1.13014
$902$	− 12.8200i	− 0.0142129i
$903$	0	0
$904$	1633.69	1.80718
$905$	477.313i	0.527418i
$906$	0	0
$907$	109.081	0.120266	0.0601331	−	0.998190i	$-0.480847\pi$
$907$	109.081	0.120266	0.0601331	+	0.998190i	$0.480847\pi$
$908$	− 374.377i	− 0.412309i
$909$	0	0
$910$	162.237	0.178282
$911$	263.944i	0.289730i	0.989451	+	0.144865i	$0.0462748\pi$
$911$	263.944i	0.289730i	−0.989451	+	0.144865i	$0.953725\pi$
$912$	0	0
$913$	−34.4323	−0.0377133
$914$	− 605.755i	− 0.662752i
$915$	0	0
$916$	106.522	0.116291
$917$	− 773.132i	− 0.843111i
$918$	0	0
$919$	−1474.65	−1.60462	−0.802312	−	0.596905i	$-0.796397\pi$
$919$	−1474.65	−1.60462	−0.802312	+	0.596905i	$0.796397\pi$
$920$	354.199i	0.384998i
$921$	0	0
$922$	95.3532	0.103420
$923$	− 279.896i	− 0.303246i
$924$	0	0
$925$	−373.277	−0.403542
$926$	− 548.177i	− 0.591984i
$927$	0	0
$928$	829.291	0.893633
$929$	− 451.122i	− 0.485600i	−0.970076	−	0.242800i	$-0.921934\pi$
$929$	− 451.122i	− 0.485600i	0.970076	−	0.242800i	$-0.0780659\pi$
$930$	0	0
$931$	−794.031	−0.852880
$932$	395.709i	0.424581i
$933$	0	0
$934$	238.839	0.255716
$935$	− 15.5081i	− 0.0165862i
$936$	0	0
$937$	−1070.93	−1.14293	−0.571465	−	0.820626i	$-0.693625\pi$
$937$	−1070.93	−1.14293	−0.571465	+	0.820626i	$0.693625\pi$
$938$	− 1475.65i	− 1.57318i
$939$	0	0
$940$	−169.678	−0.180508
$941$	− 944.024i	− 1.00321i	−0.865096	−	0.501607i	$-0.832742\pi$
$941$	− 944.024i	− 1.00321i	0.865096	−	0.501607i	$-0.167258\pi$
$942$	0	0
$943$	−334.799	−0.355036
$944$	845.620i	0.895784i
$945$	0	0
$946$	29.7000	0.0313953
$947$	388.622i	0.410372i	0.978723	+	0.205186i	$0.0657799\pi$
$947$	388.622i	0.410372i	−0.978723	+	0.205186i	$0.934220\pi$
$948$	0	0
$949$	−505.356	−0.532514
$950$	622.275i	0.655027i
$951$	0	0
$952$	−1304.36	−1.37013
$953$	− 927.318i	− 0.973052i	−0.873666	−	0.486526i	$-0.838264\pi$
$953$	− 927.318i	− 0.973052i	0.873666	−	0.486526i	$-0.161736\pi$
$954$	0	0
$955$	696.978	0.729819
$956$	− 331.278i	− 0.346525i
$957$	0	0
$958$	−153.090	−0.159801
$959$	− 953.964i	− 0.994748i
$960$	0	0
$961$	−940.962	−0.979149
$962$	138.546i	0.144018i
$963$	0	0
$964$	−439.423	−0.455833
$965$	540.803i	0.560418i
$966$	0	0
$967$	256.120	0.264860	0.132430	−	0.991192i	$-0.457722\pi$
$967$	256.120	0.264860	0.132430	+	0.991192i	$0.457722\pi$
$968$	− 1052.73i	− 1.08753i
$969$	0	0
$970$	193.314	0.199293
$971$	805.661i	0.829722i	0.909885	+	0.414861i	$0.136170\pi$
$971$	805.661i	0.829722i	−0.909885	+	0.414861i	$0.863830\pi$
$972$	0	0
$973$	−384.080	−0.394738
$974$	− 436.716i	− 0.448374i
$975$	0	0
$976$	835.894	0.856449
$977$	− 755.926i	− 0.773722i	−0.922138	−	0.386861i	$-0.873559\pi$
$977$	− 755.926i	− 0.773722i	0.922138	−	0.386861i	$-0.126441\pi$
$978$	0	0
$979$	−37.7826	−0.0385930
$980$	128.420i	0.131041i
$981$	0	0
$982$	1342.32	1.36693
$983$	728.405i	0.741002i	0.928832	+	0.370501i	$0.120814\pi$
$983$	728.405i	0.741002i	−0.928832	+	0.370501i	$0.879186\pi$
$984$	0	0
$985$	−153.819	−0.156161
$986$	− 1161.21i	− 1.17770i
$987$	0	0
$988$	−107.204	−0.108506
$989$	− 775.623i	− 0.784250i
$990$	0	0
$991$	1238.28	1.24953	0.624763	−	0.780814i	$-0.285196\pi$
$991$	1238.28	1.24953	0.624763	+	0.780814i	$0.285196\pi$
$992$	86.9540i	0.0876552i
$993$	0	0
$994$	−1168.00	−1.17505
$995$	987.187i	0.992147i
$996$	0	0
$997$	1564.68	1.56939	0.784694	−	0.619883i	$-0.212820\pi$
$997$	1564.68	1.56939	0.784694	+	0.619883i	$0.212820\pi$
$998$	− 105.788i	− 0.106000i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	117.3.c.a.53.3	✓	8
3.2	odd	2	inner	117.3.c.a.53.6	yes	8
4.3	odd	2		1872.3.f.d.1457.6		8
12.11	even	2		1872.3.f.d.1457.3		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
117.3.c.a.53.3	✓	8	1.1	even	1	trivial
117.3.c.a.53.6	yes	8	3.2	odd	2	inner
1872.3.f.d.1457.3		8	12.11	even	2
1872.3.f.d.1457.6		8	4.3	odd	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:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	117.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.65286i q^{2} +1.26805 q^{4} +2.99060i q^{5} +9.10296 q^{7} -8.70736i q^{8} +4.94305 q^{10} -0.315116i q^{11} +3.60555 q^{13} -15.0459i q^{14} -9.31982 q^{16} -16.4562i q^{17} -23.4478 q^{19} +3.79225i q^{20} -0.520843 q^{22} +13.6020i q^{23} +16.0563 q^{25} -5.95947i q^{26} +11.5430 q^{28} +42.6918i q^{29} -4.47638 q^{31} -19.4251i q^{32} -27.1998 q^{34} +27.2233i q^{35} -23.2480 q^{37} +38.7559i q^{38} +26.0402 q^{40} +24.6140i q^{41} -57.0229 q^{43} -0.399585i q^{44} +22.4821 q^{46} +44.7433i q^{47} +33.8638 q^{49} -26.5388i q^{50} +4.57204 q^{52} +61.8765i q^{53} +0.942388 q^{55} -79.2627i q^{56} +70.5636 q^{58} -90.7335i q^{59} -89.6900 q^{61} +7.39883i q^{62} -69.3862 q^{64} +10.7828i q^{65} +98.0763 q^{67} -20.8674i q^{68} +44.9964 q^{70} -77.6291i q^{71} -140.161 q^{73} +38.4257i q^{74} -29.7330 q^{76} -2.86849i q^{77} +71.5169 q^{79} -27.8719i q^{80} +40.6835 q^{82} -109.269i q^{83} +49.2140 q^{85} +94.2508i q^{86} -2.74383 q^{88} -119.900i q^{89} +32.8212 q^{91} +17.2480i q^{92} +73.9544 q^{94} -70.1229i q^{95} +39.1083 q^{97} -55.9722i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 8 q^{4} - 16 q^{7} + 24 q^{16} + 64 q^{19} - 80 q^{22} - 24 q^{25} + 184 q^{28} - 40 q^{31} - 272 q^{34} - 104 q^{37} + 32 q^{40} + 128 q^{43} + 232 q^{46} + 136 q^{49} + 104 q^{52} - 224 q^{55} - 88 q^{58}+ \cdots + 328 q^{97}+O(q^{100}) \) 8 * q - 8 * q^4 - 16 * q^7 + 24 * q^16 + 64 * q^19 - 80 * q^22 - 24 * q^25 + 184 * q^28 - 40 * q^31 - 272 * q^34 - 104 * q^37 + 32 * q^40 + 128 * q^43 + 232 * q^46 + 136 * q^49 + 104 * q^52 - 224 * q^55 - 88 * q^58 - 112 * q^61 - 72 * q^64 + 184 * q^67 + 40 * q^70 - 472 * q^73 - 72 * q^76 + 80 * q^79 + 48 * q^82 + 520 * q^85 + 144 * q^88 + 104 * q^91 - 128 * q^94 + 328 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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