LMFDB - Embedded newform 117.3.c.a.53.7

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			53.7
Root			$2.13992i$ of defining polynomial
Character	$\chi$	$=$	117.53
Dual form			117.3.c.a.53.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+2.13992i q^{2} -0.579250 q^{4} +3.56642i q^{5} +0.858882 q^{7} +7.32013i q^{8} -7.63185 q^{10} +7.84626i q^{11} -3.60555 q^{13} +1.83794i q^{14} -17.9815 q^{16} -5.04921i q^{17} +8.31746 q^{19} -2.06585i q^{20} -16.7904 q^{22} -19.3527i q^{23} +12.2806 q^{25} -7.71558i q^{26} -0.497507 q^{28} -2.17368i q^{29} +7.92040 q^{31} -9.19837i q^{32} +10.8049 q^{34} +3.06314i q^{35} +68.7104 q^{37} +17.7987i q^{38} -26.1067 q^{40} -72.7172i q^{41} +76.1964 q^{43} -4.54494i q^{44} +41.4133 q^{46} +62.8508i q^{47} -48.2623 q^{49} +26.2795i q^{50} +2.08852 q^{52} -53.9288i q^{53} -27.9831 q^{55} +6.28712i q^{56} +4.65149 q^{58} -53.4683i q^{59} -32.8363 q^{61} +16.9490i q^{62} -52.2421 q^{64} -12.8589i q^{65} -87.0307 q^{67} +2.92475i q^{68} -6.55486 q^{70} -9.99222i q^{71} -39.5822 q^{73} +147.035i q^{74} -4.81789 q^{76} +6.73901i q^{77} -88.1246 q^{79} -64.1295i q^{80} +155.609 q^{82} +24.3629i q^{83} +18.0076 q^{85} +163.054i q^{86} -57.4356 q^{88} +40.5038i q^{89} -3.09674 q^{91} +11.2101i q^{92} -134.496 q^{94} +29.6636i q^{95} +170.201 q^{97} -103.277i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	2.13992i	1.06996i	0.844865	+	0.534980i	$0.179681\pi$
$2$	2.13992i	1.06996i	−0.844865	+	0.534980i	$0.820319\pi$
$3$	0	0
$3$	0	0
$4$	−0.579250	−0.144812
$5$	3.56642i	0.713285i	0.934241	+	0.356642i	$0.116078\pi$
$5$	3.56642i	0.713285i	−0.934241	+	0.356642i	$0.883922\pi$
$6$	0	0
$7$	0.858882	0.122697	0.0613487	−	0.998116i	$-0.480460\pi$
$7$	0.858882	0.122697	0.0613487	+	0.998116i	$0.480460\pi$
$8$	7.32013i	0.915016i
$9$	0	0
$10$	−7.63185	−0.763185
$11$	7.84626i	0.713296i	0.934239	+	0.356648i	$0.116080\pi$
$11$	7.84626i	0.713296i	−0.934239	+	0.356648i	$0.883920\pi$
$12$	0	0
$13$	−3.60555	−0.277350
$13$	−3.60555	−0.277350
$14$	1.83794i	0.131281i
$15$	0	0
$16$	−17.9815	−1.12384
$17$	− 5.04921i	− 0.297012i	−0.988911	−	0.148506i	$-0.952554\pi$
$17$	− 5.04921i	− 0.297012i	0.988911	−	0.148506i	$-0.0474465\pi$
$18$	0	0
$19$	8.31746	0.437761	0.218880	−	0.975752i	$-0.429760\pi$
$19$	8.31746	0.437761	0.218880	+	0.975752i	$0.429760\pi$
$20$	− 2.06585i	− 0.103293i
$21$	0	0
$22$	−16.7904	−0.763198
$23$	− 19.3527i	− 0.841424i	−0.907194	−	0.420712i	$-0.861780\pi$
$23$	− 19.3527i	− 0.841424i	0.907194	−	0.420712i	$-0.138220\pi$
$24$	0	0
$25$	12.2806	0.491225
$26$	− 7.71558i	− 0.296753i
$27$	0	0
$28$	−0.497507	−0.0177681
$29$	− 2.17368i	− 0.0749544i	−0.999297	−	0.0374772i	$-0.988068\pi$
$29$	− 2.17368i	− 0.0749544i	0.999297	−	0.0374772i	$-0.0119322\pi$
$30$	0	0
$31$	7.92040	0.255497	0.127748	−	0.991807i	$-0.459225\pi$
$31$	7.92040	0.255497	0.127748	+	0.991807i	$0.459225\pi$
$32$	− 9.19837i	− 0.287449i
$33$	0	0
$34$	10.8049	0.317791
$35$	3.06314i	0.0875182i
$36$	0	0
$37$	68.7104	1.85704	0.928519	−	0.371286i	$-0.121083\pi$
$37$	68.7104	1.85704	0.928519	+	0.371286i	$0.121083\pi$
$38$	17.7987i	0.468386i
$39$	0	0
$40$	−26.1067	−0.652667
$41$	− 72.7172i	− 1.77359i	−0.462162	−	0.886796i	$-0.652926\pi$
$41$	− 72.7172i	− 1.77359i	0.462162	−	0.886796i	$-0.347074\pi$
$42$	0	0
$43$	76.1964	1.77201	0.886005	−	0.463676i	$-0.153470\pi$
$43$	76.1964	1.77201	0.886005	+	0.463676i	$0.153470\pi$
$44$	− 4.54494i	− 0.103294i
$45$	0	0
$46$	41.4133	0.900289
$47$	62.8508i	1.33725i	0.743599	+	0.668626i	$0.233117\pi$
$47$	62.8508i	1.33725i	−0.743599	+	0.668626i	$0.766883\pi$
$48$	0	0
$49$	−48.2623	−0.984945
$50$	26.2795i	0.525591i
$51$	0	0
$52$	2.08852	0.0401638
$53$	− 53.9288i	− 1.01752i	−0.860907	−	0.508762i	$-0.830103\pi$
$53$	− 53.9288i	− 1.01752i	0.860907	−	0.508762i	$-0.169897\pi$
$54$	0	0
$55$	−27.9831	−0.508783
$56$	6.28712i	0.112270i
$57$	0	0
$58$	4.65149	0.0801981
$59$	− 53.4683i	− 0.906242i	−0.891449	−	0.453121i	$-0.850311\pi$
$59$	− 53.4683i	− 0.906242i	0.891449	−	0.453121i	$-0.149689\pi$
$60$	0	0
$61$	−32.8363	−0.538300	−0.269150	−	0.963098i	$-0.586743\pi$
$61$	−32.8363	−0.538300	−0.269150	+	0.963098i	$0.586743\pi$
$62$	16.9490i	0.273371i
$63$	0	0
$64$	−52.2421	−0.816283
$65$	− 12.8589i	− 0.197830i
$66$	0	0
$67$	−87.0307	−1.29897	−0.649483	−	0.760376i	$-0.725015\pi$
$67$	−87.0307	−1.29897	−0.649483	+	0.760376i	$0.725015\pi$
$68$	2.92475i	0.0430111i
$69$	0	0
$70$	−6.55486	−0.0936409
$71$	− 9.99222i	− 0.140735i	−0.997521	−	0.0703677i	$-0.977583\pi$
$71$	− 9.99222i	− 0.140735i	0.997521	−	0.0703677i	$-0.0224173\pi$
$72$	0	0
$73$	−39.5822	−0.542222	−0.271111	−	0.962548i	$-0.587391\pi$
$73$	−39.5822	−0.542222	−0.271111	+	0.962548i	$0.587391\pi$
$74$	147.035i	1.98695i
$75$	0	0
$76$	−4.81789	−0.0633932
$77$	6.73901i	0.0875196i
$78$	0	0
$79$	−88.1246	−1.11550	−0.557750	−	0.830009i	$-0.688335\pi$
$79$	−88.1246	−1.11550	−0.557750	+	0.830009i	$0.688335\pi$
$80$	− 64.1295i	− 0.801619i
$81$	0	0
$82$	155.609	1.89767
$83$	24.3629i	0.293529i	0.989171	+	0.146765i	$0.0468860\pi$
$83$	24.3629i	0.293529i	−0.989171	+	0.146765i	$0.953114\pi$
$84$	0	0
$85$	18.0076	0.211854
$86$	163.054i	1.89598i
$87$	0	0
$88$	−57.4356	−0.652677
$89$	40.5038i	0.455098i	0.973767	+	0.227549i	$0.0730713\pi$
$89$	40.5038i	0.455098i	−0.973767	+	0.227549i	$0.926929\pi$
$90$	0	0
$91$	−3.09674	−0.0340301
$92$	11.2101i	0.121849i
$93$	0	0
$94$	−134.496	−1.43080
$95$	29.6636i	0.312248i
$96$	0	0
$97$	170.201	1.75465	0.877327	−	0.479893i	$-0.159325\pi$
$97$	170.201	1.75465	0.877327	+	0.479893i	$0.159325\pi$
$98$	− 103.277i	− 1.05385i
$99$	0	0
$100$	−7.11355	−0.0711355
$101$	− 102.920i	− 1.01901i	−0.860467	−	0.509506i	$-0.829828\pi$
$101$	− 102.920i	− 1.01901i	0.860467	−	0.509506i	$-0.170172\pi$
$102$	0	0
$103$	−101.075	−0.981307	−0.490654	−	0.871355i	$-0.663242\pi$
$103$	−101.075	−0.981307	−0.490654	+	0.871355i	$0.663242\pi$
$104$	− 26.3931i	− 0.253780i
$105$	0	0
$106$	115.403	1.08871
$107$	200.168i	1.87073i	0.353690	+	0.935363i	$0.384927\pi$
$107$	200.168i	1.87073i	−0.353690	+	0.935363i	$0.615073\pi$
$108$	0	0
$109$	29.7394	0.272839	0.136419	−	0.990651i	$-0.456441\pi$
$109$	29.7394	0.272839	0.136419	+	0.990651i	$0.456441\pi$
$110$	− 59.8815i	− 0.544377i
$111$	0	0
$112$	−15.4440	−0.137893
$113$	35.8897i	0.317608i	0.987310	+	0.158804i	$0.0507638\pi$
$113$	35.8897i	0.317608i	−0.987310	+	0.158804i	$0.949236\pi$
$114$	0	0
$115$	69.0201	0.600174
$116$	1.25910i	0.0108543i
$117$	0	0
$118$	114.418	0.969642
$119$	− 4.33668i	− 0.0364426i
$120$	0	0
$121$	59.4362	0.491208
$122$	− 70.2670i	− 0.575959i
$123$	0	0
$124$	−4.58789	−0.0369991
$125$	132.958i	1.06367i
$126$	0	0
$127$	22.4466	0.176745	0.0883723	−	0.996088i	$-0.471833\pi$
$127$	22.4466	0.176745	0.0883723	+	0.996088i	$0.471833\pi$
$128$	− 148.587i	− 1.16084i
$129$	0	0
$130$	27.5170	0.211670
$131$	− 42.5938i	− 0.325144i	−0.986697	−	0.162572i	$-0.948021\pi$
$131$	− 42.5938i	− 0.325144i	0.986697	−	0.162572i	$-0.0519789\pi$
$132$	0	0
$133$	7.14371	0.0537121
$134$	− 186.239i	− 1.38984i
$135$	0	0
$136$	36.9608	0.271771
$137$	− 169.512i	− 1.23732i	−0.785660	−	0.618658i	$-0.787677\pi$
$137$	− 169.512i	− 1.23732i	0.785660	−	0.618658i	$-0.212323\pi$
$138$	0	0
$139$	−198.690	−1.42943	−0.714713	−	0.699418i	$-0.753443\pi$
$139$	−198.690	−1.42943	−0.714713	+	0.699418i	$0.753443\pi$
$140$	− 1.77432i	− 0.0126737i
$141$	0	0
$142$	21.3825	0.150581
$143$	− 28.2901i	− 0.197833i
$144$	0	0
$145$	7.75225	0.0534638
$146$	− 84.7027i	− 0.580155i
$147$	0	0
$148$	−39.8005	−0.268922
$149$	− 42.7329i	− 0.286798i	−0.989665	−	0.143399i	$-0.954197\pi$
$149$	− 42.7329i	− 0.286798i	0.989665	−	0.143399i	$-0.0458033\pi$
$150$	0	0
$151$	−173.831	−1.15120	−0.575600	−	0.817731i	$-0.695231\pi$
$151$	−173.831	−1.15120	−0.575600	+	0.817731i	$0.695231\pi$
$152$	60.8848i	0.400558i
$153$	0	0
$154$	−14.4209	−0.0936424
$155$	28.2475i	0.182242i
$156$	0	0
$157$	64.4382	0.410434	0.205217	−	0.978716i	$-0.434210\pi$
$157$	64.4382	0.410434	0.205217	+	0.978716i	$0.434210\pi$
$158$	− 188.579i	− 1.19354i
$159$	0	0
$160$	32.8053	0.205033
$161$	− 16.6217i	− 0.103241i
$162$	0	0
$163$	152.796	0.937402	0.468701	−	0.883357i	$-0.344722\pi$
$163$	152.796	0.937402	0.468701	+	0.883357i	$0.344722\pi$
$164$	42.1215i	0.256838i
$165$	0	0
$166$	−52.1347	−0.314064
$167$	158.953i	0.951814i	0.879496	+	0.475907i	$0.157880\pi$
$167$	158.953i	0.951814i	−0.879496	+	0.475907i	$0.842120\pi$
$168$	0	0
$169$	13.0000	0.0769231
$170$	38.5348i	0.226675i
$171$	0	0
$172$	−44.1368	−0.256609
$173$	− 259.562i	− 1.50036i	−0.661236	−	0.750178i	$-0.729968\pi$
$173$	− 259.562i	− 1.50036i	0.661236	−	0.750178i	$-0.270032\pi$
$174$	0	0
$175$	10.5476	0.0602721
$176$	− 141.087i	− 0.801632i
$177$	0	0
$178$	−86.6747	−0.486937
$179$	290.623i	1.62359i	0.583941	+	0.811796i	$0.301510\pi$
$179$	290.623i	1.62359i	−0.583941	+	0.811796i	$0.698490\pi$
$180$	0	0
$181$	50.4172	0.278548	0.139274	−	0.990254i	$-0.455523\pi$
$181$	50.4172	0.278548	0.139274	+	0.990254i	$0.455523\pi$
$182$	− 6.62678i	− 0.0364109i
$183$	0	0
$184$	141.664	0.769916
$185$	245.050i	1.32460i
$186$	0	0
$187$	39.6174	0.211858
$188$	− 36.4063i	− 0.193651i
$189$	0	0
$190$	−63.4776	−0.334093
$191$	193.582i	1.01352i	0.862087	+	0.506760i	$0.169157\pi$
$191$	193.582i	1.01352i	−0.862087	+	0.506760i	$0.830843\pi$
$192$	0	0
$193$	−123.506	−0.639928	−0.319964	−	0.947430i	$-0.603671\pi$
$193$	−123.506	−0.639928	−0.319964	+	0.947430i	$0.603671\pi$
$194$	364.217i	1.87741i
$195$	0	0
$196$	27.9559	0.142632
$197$	132.645i	0.673327i	0.941625	+	0.336664i	$0.109299\pi$
$197$	132.645i	0.673327i	−0.941625	+	0.336664i	$0.890701\pi$
$198$	0	0
$199$	−294.873	−1.48178	−0.740888	−	0.671629i	$-0.765595\pi$
$199$	−294.873	−1.48178	−0.740888	+	0.671629i	$0.765595\pi$
$200$	89.8958i	0.449479i
$201$	0	0
$202$	220.241	1.09030
$203$	− 1.86693i	− 0.00919671i
$204$	0	0
$205$	259.340	1.26508
$206$	− 216.291i	− 1.04996i
$207$	0	0
$208$	64.8331	0.311698
$209$	65.2609i	0.312253i
$210$	0	0
$211$	138.095	0.654479	0.327240	−	0.944941i	$-0.393882\pi$
$211$	138.095	0.654479	0.327240	+	0.944941i	$0.393882\pi$
$212$	31.2383i	0.147350i
$213$	0	0
$214$	−428.342	−2.00160
$215$	271.749i	1.26395i
$216$	0	0
$217$	6.80269	0.0313488
$218$	63.6399i	0.291926i
$219$	0	0
$220$	16.2092	0.0736782
$221$	18.2052i	0.0823764i
$222$	0	0
$223$	198.949	0.892148	0.446074	−	0.894996i	$-0.352822\pi$
$223$	198.949	0.892148	0.446074	+	0.894996i	$0.352822\pi$
$224$	− 7.90032i	− 0.0352693i
$225$	0	0
$226$	−76.8011	−0.339828
$227$	− 307.416i	− 1.35426i	−0.735865	−	0.677128i	$-0.763224\pi$
$227$	− 307.416i	− 1.35426i	0.735865	−	0.677128i	$-0.236776\pi$
$228$	0	0
$229$	−421.180	−1.83921	−0.919607	−	0.392839i	$-0.871493\pi$
$229$	−421.180	−1.83921	−0.919607	+	0.392839i	$0.871493\pi$
$230$	147.697i	0.642162i
$231$	0	0
$232$	15.9116	0.0685844
$233$	− 207.599i	− 0.890981i	−0.895287	−	0.445491i	$-0.853029\pi$
$233$	− 207.599i	− 0.890981i	0.895287	−	0.445491i	$-0.146971\pi$
$234$	0	0
$235$	−224.153	−0.953841
$236$	30.9715i	0.131235i
$237$	0	0
$238$	9.28013	0.0389921
$239$	117.639i	0.492213i	0.969243	+	0.246107i	$0.0791514\pi$
$239$	117.639i	0.492213i	−0.969243	+	0.246107i	$0.920849\pi$
$240$	0	0
$241$	−38.9561	−0.161644	−0.0808218	−	0.996729i	$-0.525754\pi$
$241$	−38.9561	−0.161644	−0.0808218	+	0.996729i	$0.525754\pi$
$242$	127.189i	0.525573i
$243$	0	0
$244$	19.0204	0.0779525
$245$	− 172.124i	− 0.702546i
$246$	0	0
$247$	−29.9890	−0.121413
$248$	57.9784i	0.233784i
$249$	0	0
$250$	−284.520	−1.13808
$251$	− 352.849i	− 1.40577i	−0.711302	−	0.702887i	$-0.751894\pi$
$251$	− 352.849i	− 1.40577i	0.711302	−	0.702887i	$-0.248106\pi$
$252$	0	0
$253$	151.847	0.600184
$254$	48.0338i	0.189109i
$255$	0	0
$256$	108.996	0.425767
$257$	− 281.021i	− 1.09347i	−0.837306	−	0.546734i	$-0.815871\pi$
$257$	− 281.021i	− 1.09347i	0.837306	−	0.546734i	$-0.184129\pi$
$258$	0	0
$259$	59.0141	0.227854
$260$	7.44853i	0.0286482i
$261$	0	0
$262$	91.1473	0.347890
$263$	234.575i	0.891919i	0.895053	+	0.445959i	$0.147137\pi$
$263$	234.575i	0.891919i	−0.895053	+	0.445959i	$0.852863\pi$
$264$	0	0
$265$	192.333	0.725784
$266$	15.2870i	0.0574698i
$267$	0	0
$268$	50.4125	0.188107
$269$	− 249.428i	− 0.927241i	−0.886034	−	0.463621i	$-0.846550\pi$
$269$	− 249.428i	− 0.927241i	0.886034	−	0.463621i	$-0.153450\pi$
$270$	0	0
$271$	−410.732	−1.51561	−0.757807	−	0.652478i	$-0.773729\pi$
$271$	−410.732	−1.51561	−0.757807	+	0.652478i	$0.773729\pi$
$272$	90.7922i	0.333795i
$273$	0	0
$274$	362.742	1.32388
$275$	96.3570i	0.350389i
$276$	0	0
$277$	−339.001	−1.22383	−0.611914	−	0.790924i	$-0.709600\pi$
$277$	−339.001	−1.22383	−0.611914	+	0.790924i	$0.709600\pi$
$278$	− 425.181i	− 1.52943i
$279$	0	0
$280$	−22.4225	−0.0800805
$281$	− 308.915i	− 1.09934i	−0.835381	−	0.549672i	$-0.814753\pi$
$281$	− 308.915i	− 1.09934i	0.835381	−	0.549672i	$-0.185247\pi$
$282$	0	0
$283$	303.991	1.07417	0.537087	−	0.843527i	$-0.319525\pi$
$283$	303.991	1.07417	0.537087	+	0.843527i	$0.319525\pi$
$284$	5.78799i	0.0203803i
$285$	0	0
$286$	60.5385	0.211673
$287$	− 62.4555i	− 0.217615i
$288$	0	0
$289$	263.505	0.911784
$290$	16.5892i	0.0572041i
$291$	0	0
$292$	22.9280	0.0785205
$293$	− 188.736i	− 0.644150i	−0.946714	−	0.322075i	$-0.895620\pi$
$293$	− 188.736i	− 0.644150i	0.946714	−	0.322075i	$-0.104380\pi$
$294$	0	0
$295$	190.691	0.646409
$296$	502.969i	1.69922i
$297$	0	0
$298$	91.4450	0.306862
$299$	69.7773i	0.233369i
$300$	0	0
$301$	65.4437	0.217421
$302$	− 371.985i	− 1.23174i
$303$	0	0
$304$	−149.560	−0.491974
$305$	− 117.108i	− 0.383961i
$306$	0	0
$307$	−71.1039	−0.231609	−0.115804	−	0.993272i	$-0.536945\pi$
$307$	−71.1039	−0.231609	−0.115804	+	0.993272i	$0.536945\pi$
$308$	− 3.90357i	− 0.0126739i
$309$	0	0
$310$	−60.4474	−0.194991
$311$	− 404.873i	− 1.30184i	−0.759146	−	0.650921i	$-0.774383\pi$
$311$	− 404.873i	− 1.30184i	0.759146	−	0.650921i	$-0.225617\pi$
$312$	0	0
$313$	161.711	0.516649	0.258325	−	0.966058i	$-0.416830\pi$
$313$	161.711	0.516649	0.258325	+	0.966058i	$0.416830\pi$
$314$	137.893i	0.439148i
$315$	0	0
$316$	51.0462	0.161538
$317$	380.854i	1.20143i	0.799463	+	0.600715i	$0.205117\pi$
$317$	380.854i	1.20143i	−0.799463	+	0.600715i	$0.794883\pi$
$318$	0	0
$319$	17.0552	0.0534647
$320$	− 186.317i	− 0.582242i
$321$	0	0
$322$	35.5691	0.110463
$323$	− 41.9966i	− 0.130020i
$324$	0	0
$325$	−44.2784	−0.136241
$326$	326.972i	1.00298i
$327$	0	0
$328$	532.299	1.62286
$329$	53.9814i	0.164077i
$330$	0	0
$331$	412.554	1.24639	0.623193	−	0.782068i	$-0.285835\pi$
$331$	412.554	1.24639	0.623193	+	0.782068i	$0.285835\pi$
$332$	− 14.1122i	− 0.0425067i
$333$	0	0
$334$	−340.146	−1.01840
$335$	− 310.388i	− 0.926533i
$336$	0	0
$337$	42.5700	0.126320	0.0631602	−	0.998003i	$-0.479882\pi$
$337$	42.5700	0.126320	0.0631602	+	0.998003i	$0.479882\pi$
$338$	27.8189i	0.0823045i
$339$	0	0
$340$	−10.4309	−0.0306791
$341$	62.1455i	0.182245i
$342$	0	0
$343$	−83.5369	−0.243548
$344$	557.767i	1.62142i
$345$	0	0
$346$	555.441	1.60532
$347$	− 257.692i	− 0.742629i	−0.928507	−	0.371314i	$-0.878907\pi$
$347$	− 257.692i	− 0.742629i	0.928507	−	0.371314i	$-0.121093\pi$
$348$	0	0
$349$	198.002	0.567341	0.283671	−	0.958922i	$-0.408448\pi$
$349$	198.002	0.567341	0.283671	+	0.958922i	$0.408448\pi$
$350$	22.5710i	0.0644886i
$351$	0	0
$352$	72.1728	0.205036
$353$	226.372i	0.641279i	0.947201	+	0.320640i	$0.103898\pi$
$353$	226.372i	0.641279i	−0.947201	+	0.320640i	$0.896102\pi$
$354$	0	0
$355$	35.6365	0.100384
$356$	− 23.4618i	− 0.0659039i
$357$	0	0
$358$	−621.909	−1.73718
$359$	112.701i	0.313929i	0.987604	+	0.156965i	$0.0501709\pi$
$359$	112.701i	0.313929i	−0.987604	+	0.156965i	$0.949829\pi$
$360$	0	0
$361$	−291.820	−0.808365
$362$	107.889i	0.298035i
$363$	0	0
$364$	1.79379	0.00492799
$365$	− 141.167i	− 0.386758i
$366$	0	0
$367$	−107.425	−0.292710	−0.146355	−	0.989232i	$-0.546754\pi$
$367$	−107.425	−0.292710	−0.146355	+	0.989232i	$0.546754\pi$
$368$	347.991i	0.945627i
$369$	0	0
$370$	−524.387	−1.41726
$371$	− 46.3185i	− 0.124848i
$372$	0	0
$373$	325.706	0.873206	0.436603	−	0.899654i	$-0.356182\pi$
$373$	325.706	0.873206	0.436603	+	0.899654i	$0.356182\pi$
$374$	84.7780i	0.226679i
$375$	0	0
$376$	−460.076	−1.22361
$377$	7.83730i	0.0207886i
$378$	0	0
$379$	−245.105	−0.646716	−0.323358	−	0.946277i	$-0.604812\pi$
$379$	−245.105	−0.646716	−0.323358	+	0.946277i	$0.604812\pi$
$380$	− 17.1826i	− 0.0452174i
$381$	0	0
$382$	−414.250	−1.08442
$383$	306.030i	0.799034i	0.916726	+	0.399517i	$0.130822\pi$
$383$	306.030i	0.799034i	−0.916726	+	0.399517i	$0.869178\pi$
$384$	0	0
$385$	−24.0342	−0.0624264
$386$	− 264.293i	− 0.684697i
$387$	0	0
$388$	−98.5892	−0.254096
$389$	269.837i	0.693668i	0.937926	+	0.346834i	$0.112743\pi$
$389$	269.837i	0.693668i	−0.937926	+	0.346834i	$0.887257\pi$
$390$	0	0
$391$	−97.7161	−0.249913
$392$	− 353.286i	− 0.901240i
$393$	0	0
$394$	−283.850	−0.720433
$395$	− 314.289i	− 0.795670i
$396$	0	0
$397$	−333.218	−0.839339	−0.419670	−	0.907677i	$-0.637854\pi$
$397$	−333.218	−0.839339	−0.419670	+	0.907677i	$0.637854\pi$
$398$	− 631.005i	− 1.58544i
$399$	0	0
$400$	−220.824	−0.552059
$401$	64.1145i	0.159886i	0.996799	+	0.0799432i	$0.0254739\pi$
$401$	64.1145i	0.159886i	−0.996799	+	0.0799432i	$0.974526\pi$
$402$	0	0
$403$	−28.5574	−0.0708621
$404$	59.6166i	0.147566i
$405$	0	0
$406$	3.99508	0.00984010
$407$	539.119i	1.32462i
$408$	0	0
$409$	−43.8559	−0.107227	−0.0536135	−	0.998562i	$-0.517074\pi$
$409$	−43.8559	−0.107227	−0.0536135	+	0.998562i	$0.517074\pi$
$410$	554.967i	1.35358i
$411$	0	0
$412$	58.5475	0.142106
$413$	− 45.9229i	− 0.111194i
$414$	0	0
$415$	−86.8885	−0.209370
$416$	33.1652i	0.0797240i
$417$	0	0
$418$	−139.653	−0.334098
$419$	− 92.1735i	− 0.219984i	−0.993932	−	0.109992i	$-0.964917\pi$
$419$	− 92.1735i	− 0.219984i	0.993932	−	0.109992i	$-0.0350826\pi$
$420$	0	0
$421$	582.913	1.38459	0.692296	−	0.721614i	$-0.256599\pi$
$421$	582.913	1.38459	0.692296	+	0.721614i	$0.256599\pi$
$422$	295.512i	0.700266i
$423$	0	0
$424$	394.766	0.931051
$425$	− 62.0075i	− 0.145900i
$426$	0	0
$427$	−28.2025	−0.0660480
$428$	− 115.947i	− 0.270904i
$429$	0	0
$430$	−581.520	−1.35237
$431$	624.264i	1.44841i	0.689585	+	0.724205i	$0.257793\pi$
$431$	624.264i	1.44841i	−0.689585	+	0.724205i	$0.742207\pi$
$432$	0	0
$433$	211.885	0.489341	0.244670	−	0.969606i	$-0.421320\pi$
$433$	211.885	0.489341	0.244670	+	0.969606i	$0.421320\pi$
$434$	14.5572i	0.0335419i
$435$	0	0
$436$	−17.2266	−0.0395105
$437$	− 160.966i	− 0.368342i
$438$	0	0
$439$	263.984	0.601331	0.300665	−	0.953730i	$-0.402791\pi$
$439$	263.984	0.601331	0.300665	+	0.953730i	$0.402791\pi$
$440$	− 204.840i	− 0.465545i
$441$	0	0
$442$	−38.9576	−0.0881394
$443$	− 189.371i	− 0.427474i	−0.976891	−	0.213737i	$-0.931437\pi$
$443$	− 189.371i	− 0.427474i	0.976891	−	0.213737i	$-0.0685635\pi$
$444$	0	0
$445$	−144.454	−0.324615
$446$	425.735i	0.954562i
$447$	0	0
$448$	−44.8698	−0.100156
$449$	802.379i	1.78704i	0.449027	+	0.893518i	$0.351771\pi$
$449$	802.379i	1.78704i	−0.449027	+	0.893518i	$0.648229\pi$
$450$	0	0
$451$	570.558	1.26510
$452$	− 20.7891i	− 0.0459936i
$453$	0	0
$454$	657.846	1.44900
$455$	− 11.0443i	− 0.0242732i
$456$	0	0
$457$	−709.974	−1.55355	−0.776777	−	0.629776i	$-0.783147\pi$
$457$	−709.974	−1.55355	−0.776777	+	0.629776i	$0.783147\pi$
$458$	− 901.291i	− 1.96788i
$459$	0	0
$460$	−39.9799	−0.0869127
$461$	− 118.909i	− 0.257937i	−0.991649	−	0.128968i	$-0.958833\pi$
$461$	− 118.909i	− 0.257937i	0.991649	−	0.128968i	$-0.0411666\pi$
$462$	0	0
$463$	407.849	0.880884	0.440442	−	0.897781i	$-0.354822\pi$
$463$	407.849	0.880884	0.440442	+	0.897781i	$0.354822\pi$
$464$	39.0859i	0.0842369i
$465$	0	0
$466$	444.244	0.953313
$467$	− 856.028i	− 1.83304i	−0.399994	−	0.916518i	$-0.630988\pi$
$467$	− 856.028i	− 1.83304i	0.399994	−	0.916518i	$-0.369012\pi$
$468$	0	0
$469$	−74.7491	−0.159380
$470$	− 479.668i	− 1.02057i
$471$	0	0
$472$	391.395	0.829226
$473$	597.857i	1.26397i
$474$	0	0
$475$	102.144	0.215039
$476$	2.51202i	0.00527735i
$477$	0	0
$478$	−251.738	−0.526648
$479$	845.092i	1.76428i	0.470983	+	0.882142i	$0.343899\pi$
$479$	845.092i	1.76428i	−0.470983	+	0.882142i	$0.656101\pi$
$480$	0	0
$481$	−247.739	−0.515049
$482$	− 83.3628i	− 0.172952i
$483$	0	0
$484$	−34.4284	−0.0711331
$485$	607.010i	1.25157i
$486$	0	0
$487$	247.650	0.508523	0.254261	−	0.967136i	$-0.418168\pi$
$487$	247.650	0.508523	0.254261	+	0.967136i	$0.418168\pi$
$488$	− 240.366i	− 0.492553i
$489$	0	0
$490$	368.331	0.751696
$491$	− 835.638i	− 1.70191i	−0.525238	−	0.850955i	$-0.676024\pi$
$491$	− 835.638i	− 1.70191i	0.525238	−	0.850955i	$-0.323976\pi$
$492$	0	0
$493$	−10.9754	−0.0222624
$494$	− 64.1740i	− 0.129907i
$495$	0	0
$496$	−142.421	−0.287138
$497$	− 8.58214i	− 0.0172679i
$498$	0	0
$499$	45.5119	0.0912061	0.0456031	−	0.998960i	$-0.485479\pi$
$499$	45.5119	0.0912061	0.0456031	+	0.998960i	$0.485479\pi$
$500$	− 77.0162i	− 0.154032i
$501$	0	0
$502$	755.068	1.50412
$503$	− 565.557i	− 1.12437i	−0.827012	−	0.562184i	$-0.809961\pi$
$503$	− 565.557i	− 1.12437i	0.827012	−	0.562184i	$-0.190039\pi$
$504$	0	0
$505$	367.057	0.726846
$506$	324.939i	0.642173i
$507$	0	0
$508$	−13.0022	−0.0255948
$509$	414.808i	0.814946i	0.913217	+	0.407473i	$0.133590\pi$
$509$	414.808i	0.814946i	−0.913217	+	0.407473i	$0.866410\pi$
$510$	0	0
$511$	−33.9964	−0.0665292
$512$	− 361.106i	− 0.705286i
$513$	0	0
$514$	601.363	1.16997
$515$	− 360.475i	− 0.699951i
$516$	0	0
$517$	−493.144	−0.953857
$518$	126.285i	0.243794i
$519$	0	0
$520$	94.1289	0.181017
$521$	− 760.000i	− 1.45873i	−0.684123	−	0.729367i	$-0.739815\pi$
$521$	− 760.000i	− 1.45873i	0.684123	−	0.729367i	$-0.260185\pi$
$522$	0	0
$523$	−115.493	−0.220827	−0.110414	−	0.993886i	$-0.535218\pi$
$523$	−115.493	−0.220827	−0.110414	+	0.993886i	$0.535218\pi$
$524$	24.6725i	0.0470849i
$525$	0	0
$526$	−501.971	−0.954317
$527$	− 39.9918i	− 0.0758857i
$528$	0	0
$529$	154.471	0.292006
$530$	411.577i	0.776560i
$531$	0	0
$532$	−4.13800	−0.00777819
$533$	262.186i	0.491906i
$534$	0	0
$535$	−713.882	−1.33436
$536$	− 637.076i	− 1.18857i
$537$	0	0
$538$	533.755	0.992110
$539$	− 378.679i	− 0.702558i
$540$	0	0
$541$	34.3309	0.0634582	0.0317291	−	0.999497i	$-0.489899\pi$
$541$	34.3309	0.0634582	0.0317291	+	0.999497i	$0.489899\pi$
$542$	− 878.932i	− 1.62165i
$543$	0	0
$544$	−46.4445	−0.0853759
$545$	106.063i	0.194612i
$546$	0	0
$547$	−565.016	−1.03294	−0.516468	−	0.856307i	$-0.672753\pi$
$547$	−565.016	−1.03294	−0.516468	+	0.856307i	$0.672753\pi$
$548$	98.1900i	0.179179i
$549$	0	0
$550$	−206.196	−0.374902
$551$	− 18.0795i	− 0.0328121i
$552$	0	0
$553$	−75.6886	−0.136869
$554$	− 725.433i	− 1.30945i
$555$	0	0
$556$	115.091	0.206999
$557$	371.688i	0.667303i	0.942696	+	0.333652i	$0.108281\pi$
$557$	371.688i	0.667303i	−0.942696	+	0.333652i	$0.891719\pi$
$558$	0	0
$559$	−274.730	−0.491467
$560$	− 55.0797i	− 0.0983566i
$561$	0	0
$562$	661.054	1.17625
$563$	331.462i	0.588742i	0.955691	+	0.294371i	$0.0951101\pi$
$563$	331.462i	0.588742i	−0.955691	+	0.294371i	$0.904890\pi$
$564$	0	0
$565$	−127.998	−0.226545
$566$	650.517i	1.14932i
$567$	0	0
$568$	73.1443	0.128775
$569$	− 915.855i	− 1.60959i	−0.593555	−	0.804794i	$-0.702276\pi$
$569$	− 915.855i	− 1.60959i	0.593555	−	0.804794i	$-0.297724\pi$
$570$	0	0
$571$	291.517	0.510537	0.255269	−	0.966870i	$-0.417836\pi$
$571$	291.517	0.510537	0.255269	+	0.966870i	$0.417836\pi$
$572$	16.3870i	0.0286487i
$573$	0	0
$574$	133.650	0.232839
$575$	− 237.664i	− 0.413328i
$576$	0	0
$577$	−736.436	−1.27632	−0.638159	−	0.769904i	$-0.720304\pi$
$577$	−736.436	−1.27632	−0.638159	+	0.769904i	$0.720304\pi$
$578$	563.880i	0.975571i
$579$	0	0
$580$	−4.49049	−0.00774223
$581$	20.9249i	0.0360153i
$582$	0	0
$583$	423.139	0.725796
$584$	− 289.747i	− 0.496142i
$585$	0	0
$586$	403.880	0.689215
$587$	− 446.404i	− 0.760484i	−0.924887	−	0.380242i	$-0.875841\pi$
$587$	− 446.404i	− 0.760484i	0.924887	−	0.380242i	$-0.124159\pi$
$588$	0	0
$589$	65.8776	0.111847
$590$	408.062i	0.691631i
$591$	0	0
$592$	−1235.51	−2.08702
$593$	1116.11i	1.88214i	0.338217	+	0.941068i	$0.390176\pi$
$593$	1116.11i	1.88214i	−0.338217	+	0.941068i	$0.609824\pi$
$594$	0	0
$595$	15.4664	0.0259940
$596$	24.7531i	0.0415320i
$597$	0	0
$598$	−149.318	−0.249695
$599$	702.968i	1.17357i	0.809743	+	0.586785i	$0.199607\pi$
$599$	702.968i	1.17357i	−0.809743	+	0.586785i	$0.800393\pi$
$600$	0	0
$601$	−968.678	−1.61178	−0.805888	−	0.592067i	$-0.798312\pi$
$601$	−968.678	−1.61178	−0.805888	+	0.592067i	$0.798312\pi$
$602$	140.044i	0.232632i
$603$	0	0
$604$	100.692	0.166708
$605$	211.975i	0.350371i
$606$	0	0
$607$	−943.964	−1.55513	−0.777565	−	0.628803i	$-0.783545\pi$
$607$	−943.964	−1.55513	−0.777565	+	0.628803i	$0.783545\pi$
$608$	− 76.5071i	− 0.125834i
$609$	0	0
$610$	250.602	0.410823
$611$	− 226.612i	− 0.370887i
$612$	0	0
$613$	−415.139	−0.677226	−0.338613	−	0.940926i	$-0.609958\pi$
$613$	−415.139	−0.677226	−0.338613	+	0.940926i	$0.609958\pi$
$614$	− 152.157i	− 0.247812i
$615$	0	0
$616$	−49.3304	−0.0800818
$617$	318.708i	0.516544i	0.966072	+	0.258272i	$0.0831531\pi$
$617$	318.708i	0.516544i	−0.966072	+	0.258272i	$0.916847\pi$
$618$	0	0
$619$	931.448	1.50476	0.752381	−	0.658728i	$-0.228905\pi$
$619$	931.448	1.50476	0.752381	+	0.658728i	$0.228905\pi$
$620$	− 16.3624i	− 0.0263909i
$621$	0	0
$622$	866.395	1.39292
$623$	34.7879i	0.0558394i
$624$	0	0
$625$	−167.170	−0.267473
$626$	346.049i	0.552794i
$627$	0	0
$628$	−37.3258	−0.0594360
$629$	− 346.933i	− 0.551563i
$630$	0	0
$631$	−363.382	−0.575883	−0.287942	−	0.957648i	$-0.592971\pi$
$631$	−363.382	−0.575883	−0.287942	+	0.957648i	$0.592971\pi$
$632$	− 645.083i	− 1.02070i
$633$	0	0
$634$	−814.995	−1.28548
$635$	80.0539i	0.126069i
$636$	0	0
$637$	174.012	0.273175
$638$	36.4968i	0.0572050i
$639$	0	0
$640$	529.925	0.828008
$641$	509.363i	0.794638i	0.917680	+	0.397319i	$0.130059\pi$
$641$	509.363i	0.794638i	−0.917680	+	0.397319i	$0.869941\pi$
$642$	0	0
$643$	442.556	0.688268	0.344134	−	0.938921i	$-0.388173\pi$
$643$	442.556	0.688268	0.344134	+	0.938921i	$0.388173\pi$
$644$	9.62813i	0.0149505i
$645$	0	0
$646$	89.8693	0.139116
$647$	1044.65i	1.61461i	0.590136	+	0.807304i	$0.299074\pi$
$647$	1044.65i	1.61461i	−0.590136	+	0.807304i	$0.700926\pi$
$648$	0	0
$649$	419.526	0.646419
$650$	− 94.7522i	− 0.145773i
$651$	0	0
$652$	−88.5073	−0.135747
$653$	417.370i	0.639158i	0.947560	+	0.319579i	$0.103541\pi$
$653$	417.370i	0.639158i	−0.947560	+	0.319579i	$0.896459\pi$
$654$	0	0
$655$	151.908	0.231920
$656$	1307.56i	1.99324i
$657$	0	0
$658$	−115.516	−0.175556
$659$	− 164.313i	− 0.249337i	−0.992198	−	0.124669i	$-0.960213\pi$
$659$	− 164.313i	− 0.249337i	0.992198	−	0.124669i	$-0.0397868\pi$
$660$	0	0
$661$	−797.369	−1.20631	−0.603153	−	0.797625i	$-0.706089\pi$
$661$	−797.369	−1.20631	−0.603153	+	0.797625i	$0.706089\pi$
$662$	882.831i	1.33358i
$663$	0	0
$664$	−178.340	−0.268584
$665$	25.4775i	0.0383120i
$666$	0	0
$667$	−42.0666	−0.0630684
$668$	− 92.0734i	− 0.137834i
$669$	0	0
$670$	664.206	0.991352
$671$	− 257.642i	− 0.383967i
$672$	0	0
$673$	566.237	0.841363	0.420682	−	0.907208i	$-0.361791\pi$
$673$	566.237	0.841363	0.420682	+	0.907208i	$0.361791\pi$
$674$	91.0963i	0.135158i
$675$	0	0
$676$	−7.53025	−0.0111394
$677$	793.541i	1.17214i	0.810259	+	0.586072i	$0.199326\pi$
$677$	793.541i	1.17214i	−0.810259	+	0.586072i	$0.800674\pi$
$678$	0	0
$679$	146.183	0.215291
$680$	131.818i	0.193850i
$681$	0	0
$682$	−132.986	−0.194995
$683$	358.976i	0.525586i	0.964852	+	0.262793i	$0.0846437\pi$
$683$	358.976i	0.525586i	−0.964852	+	0.262793i	$0.915356\pi$
$684$	0	0
$685$	604.553	0.882558
$686$	− 178.762i	− 0.260586i
$687$	0	0
$688$	−1370.12	−1.99146
$689$	194.443i	0.282211i
$690$	0	0
$691$	1048.54	1.51742	0.758712	−	0.651426i	$-0.225829\pi$
$691$	1048.54	1.51742	0.758712	+	0.651426i	$0.225829\pi$
$692$	150.351i	0.217270i
$693$	0	0
$694$	551.440	0.794582
$695$	− 708.613i	− 1.01959i
$696$	0	0
$697$	−367.165	−0.526778
$698$	423.708i	0.607032i
$699$	0	0
$700$	−6.10970	−0.00872815
$701$	− 954.340i	− 1.36140i	−0.732563	−	0.680699i	$-0.761676\pi$
$701$	− 954.340i	− 1.36140i	0.732563	−	0.680699i	$-0.238324\pi$
$702$	0	0
$703$	571.496	0.812938
$704$	− 409.905i	− 0.582252i
$705$	0	0
$706$	−484.417	−0.686142
$707$	− 88.3964i	− 0.125030i
$708$	0	0
$709$	−675.395	−0.952603	−0.476301	−	0.879282i	$-0.658023\pi$
$709$	−675.395	−0.952603	−0.476301	+	0.879282i	$0.658023\pi$
$710$	76.2591i	0.107407i
$711$	0	0
$712$	−296.493	−0.416422
$713$	− 153.282i	− 0.214981i
$714$	0	0
$715$	100.894	0.141111
$716$	− 168.343i	− 0.235116i
$717$	0	0
$718$	−241.170	−0.335892
$719$	− 624.734i	− 0.868893i	−0.900698	−	0.434447i	$-0.856944\pi$
$719$	− 624.734i	− 0.868893i	0.900698	−	0.434447i	$-0.143056\pi$
$720$	0	0
$721$	−86.8112	−0.120404
$722$	− 624.471i	− 0.864918i
$723$	0	0
$724$	−29.2041	−0.0403372
$725$	− 26.6941i	− 0.0368195i
$726$	0	0
$727$	474.492	0.652672	0.326336	−	0.945254i	$-0.394186\pi$
$727$	474.492	0.652672	0.326336	+	0.945254i	$0.394186\pi$
$728$	− 22.6685i	− 0.0311381i
$729$	0	0
$730$	302.086	0.413816
$731$	− 384.732i	− 0.526309i
$732$	0	0
$733$	−924.471	−1.26122	−0.630608	−	0.776102i	$-0.717194\pi$
$733$	−924.471	−1.26122	−0.630608	+	0.776102i	$0.717194\pi$
$734$	− 229.880i	− 0.313188i
$735$	0	0
$736$	−178.014	−0.241866
$737$	− 682.866i	− 0.926548i
$738$	0	0
$739$	−911.308	−1.23316	−0.616582	−	0.787291i	$-0.711483\pi$
$739$	−911.308	−1.23316	−0.616582	+	0.787291i	$0.711483\pi$
$740$	− 141.945i	− 0.191818i
$741$	0	0
$742$	99.1177	0.133582
$743$	577.651i	0.777458i	0.921352	+	0.388729i	$0.127086\pi$
$743$	577.651i	0.777458i	−0.921352	+	0.388729i	$0.872914\pi$
$744$	0	0
$745$	152.404	0.204569
$746$	696.983i	0.934294i
$747$	0	0
$748$	−22.9484	−0.0306797
$749$	171.920i	0.229533i
$750$	0	0
$751$	−6.67335	−0.00888595	−0.00444297	−	0.999990i	$-0.501414\pi$
$751$	−6.67335	−0.00888595	−0.00444297	+	0.999990i	$0.501414\pi$
$752$	− 1130.15i	− 1.50286i
$753$	0	0
$754$	−16.7712	−0.0222430
$755$	− 619.955i	− 0.821133i
$756$	0	0
$757$	746.101	0.985602	0.492801	−	0.870142i	$-0.335973\pi$
$757$	746.101	0.985602	0.492801	+	0.870142i	$0.335973\pi$
$758$	− 524.506i	− 0.691960i
$759$	0	0
$760$	−217.141	−0.285712
$761$	− 631.409i	− 0.829710i	−0.909888	−	0.414855i	$-0.863832\pi$
$761$	− 631.409i	− 0.829710i	0.909888	−	0.414855i	$-0.136168\pi$
$762$	0	0
$763$	25.5427	0.0334766
$764$	− 112.132i	− 0.146770i
$765$	0	0
$766$	−654.880	−0.854934
$767$	192.783i	0.251346i
$768$	0	0
$769$	−464.923	−0.604581	−0.302290	−	0.953216i	$-0.597751\pi$
$769$	−464.923	−0.604581	−0.302290	+	0.953216i	$0.597751\pi$
$770$	− 51.4311i	− 0.0667937i
$771$	0	0
$772$	71.5410	0.0926696
$773$	− 384.272i	− 0.497118i	−0.968617	−	0.248559i	$-0.920043\pi$
$773$	− 384.272i	− 0.497118i	0.968617	−	0.248559i	$-0.0799570\pi$
$774$	0	0
$775$	97.2676	0.125507
$776$	1245.90i	1.60554i
$777$	0	0
$778$	−577.429	−0.742197
$779$	− 604.823i	− 0.776409i
$780$	0	0
$781$	78.4015	0.100386
$782$	− 209.104i	− 0.267397i
$783$	0	0
$784$	867.827	1.10692
$785$	229.814i	0.292757i
$786$	0	0
$787$	1040.08	1.32158	0.660788	−	0.750572i	$-0.270222\pi$
$787$	1040.08	1.32158	0.660788	+	0.750572i	$0.270222\pi$
$788$	− 76.8349i	− 0.0975062i
$789$	0	0
$790$	672.554	0.851334
$791$	30.8250i	0.0389697i
$792$	0	0
$793$	118.393	0.149298
$794$	− 713.059i	− 0.898059i
$795$	0	0
$796$	170.805	0.214580
$797$	− 773.072i	− 0.969977i	−0.874520	−	0.484989i	$-0.838824\pi$
$797$	− 773.072i	− 0.969977i	0.874520	−	0.484989i	$-0.161176\pi$
$798$	0	0
$799$	317.347	0.397180
$800$	− 112.962i	− 0.141202i
$801$	0	0
$802$	−137.200	−0.171072
$803$	− 310.572i	− 0.386765i
$804$	0	0
$805$	59.2801	0.0736399
$806$	− 61.1106i	− 0.0758195i
$807$	0	0
$808$	753.390	0.932413
$809$	527.404i	0.651921i	0.945383	+	0.325961i	$0.105688\pi$
$809$	527.404i	0.651921i	−0.945383	+	0.325961i	$0.894312\pi$
$810$	0	0
$811$	147.398	0.181749	0.0908744	−	0.995862i	$-0.471034\pi$
$811$	147.398	0.181749	0.0908744	+	0.995862i	$0.471034\pi$
$812$	1.08142i	0.00133180i
$813$	0	0
$814$	−1153.67	−1.41729
$815$	544.937i	0.668634i
$816$	0	0
$817$	633.760	0.775716
$818$	− 93.8480i	− 0.114729i
$819$	0	0
$820$	−150.223	−0.183199
$821$	267.199i	0.325455i	0.986671	+	0.162728i	$0.0520292\pi$
$821$	267.199i	0.325455i	−0.986671	+	0.162728i	$0.947971\pi$
$822$	0	0
$823$	124.455	0.151221	0.0756104	−	0.997137i	$-0.475909\pi$
$823$	124.455	0.151221	0.0756104	+	0.997137i	$0.475909\pi$
$824$	− 739.879i	− 0.897912i
$825$	0	0
$826$	98.2714	0.118973
$827$	16.6224i	0.0200996i	0.999949	+	0.0100498i	$0.00319901\pi$
$827$	16.6224i	0.0200996i	−0.999949	+	0.0100498i	$0.996801\pi$
$828$	0	0
$829$	1497.36	1.80622	0.903111	−	0.429407i	$-0.141278\pi$
$829$	1497.36	1.80622	0.903111	+	0.429407i	$0.141278\pi$
$830$	− 185.934i	− 0.224017i
$831$	0	0
$832$	188.362	0.226396
$833$	243.687i	0.292541i
$834$	0	0
$835$	−566.893	−0.678914
$836$	− 37.8024i	− 0.0452182i
$837$	0	0
$838$	197.244	0.235374
$839$	− 729.144i	− 0.869063i	−0.900657	−	0.434531i	$-0.856914\pi$
$839$	− 729.144i	− 0.869063i	0.900657	−	0.434531i	$-0.143086\pi$
$840$	0	0
$841$	836.275	0.994382
$842$	1247.39i	1.48146i
$843$	0	0
$844$	−79.9916	−0.0947768
$845$	46.3635i	0.0548680i
$846$	0	0
$847$	51.0487	0.0602700
$848$	969.719i	1.14354i
$849$	0	0
$850$	132.691	0.156107
$851$	− 1329.73i	− 1.56255i
$852$	0	0
$853$	−1268.25	−1.48681	−0.743404	−	0.668843i	$-0.766790\pi$
$853$	−1268.25	−1.48681	−0.743404	+	0.668843i	$0.766790\pi$
$854$	− 60.3510i	− 0.0706687i
$855$	0	0
$856$	−1465.25	−1.71174
$857$	1423.69i	1.66125i	0.556832	+	0.830625i	$0.312017\pi$
$857$	1423.69i	1.66125i	−0.556832	+	0.830625i	$0.687983\pi$
$858$	0	0
$859$	1167.83	1.35953	0.679764	−	0.733431i	$-0.262082\pi$
$859$	1167.83	1.35953	0.679764	+	0.733431i	$0.262082\pi$
$860$	− 157.410i	− 0.183035i
$861$	0	0
$862$	−1335.87	−1.54974
$863$	− 1234.26i	− 1.43020i	−0.699023	−	0.715100i	$-0.746381\pi$
$863$	− 1234.26i	− 1.43020i	0.699023	−	0.715100i	$-0.253619\pi$
$864$	0	0
$865$	925.707	1.07018
$866$	453.416i	0.523574i
$867$	0	0
$868$	−3.94046	−0.00453970
$869$	− 691.448i	− 0.795683i
$870$	0	0
$871$	313.794	0.360268
$872$	217.696i	0.249652i
$873$	0	0
$874$	344.453	0.394111
$875$	114.196i	0.130509i
$876$	0	0
$877$	−147.934	−0.168682	−0.0843409	−	0.996437i	$-0.526878\pi$
$877$	−147.934	−0.168682	−0.0843409	+	0.996437i	$0.526878\pi$
$878$	564.904i	0.643399i
$879$	0	0
$880$	503.177	0.571792
$881$	274.630i	0.311726i	0.987779	+	0.155863i	$0.0498158\pi$
$881$	274.630i	0.311726i	−0.987779	+	0.155863i	$0.950184\pi$
$882$	0	0
$883$	976.907	1.10635	0.553175	−	0.833065i	$-0.313416\pi$
$883$	976.907	1.10635	0.553175	+	0.833065i	$0.313416\pi$
$884$	− 10.5454i	− 0.0119291i
$885$	0	0
$886$	405.238	0.457379
$887$	1288.92i	1.45312i	0.687103	+	0.726560i	$0.258882\pi$
$887$	1288.92i	1.45312i	−0.687103	+	0.726560i	$0.741118\pi$
$888$	0	0
$889$	19.2789	0.0216861
$890$	− 309.119i	− 0.347324i
$891$	0	0
$892$	−115.241	−0.129194
$893$	522.759i	0.585396i
$894$	0	0
$895$	−1036.48	−1.15808
$896$	− 127.619i	− 0.142432i
$897$	0	0
$898$	−1717.03	−1.91206
$899$	− 17.2164i	− 0.0191506i
$900$	0	0
$901$	−272.298	−0.302217
$902$	1220.95i	1.35360i
$903$	0	0
$904$	−262.717	−0.290616
$905$	179.809i	0.198684i
$906$	0	0
$907$	1174.03	1.29441	0.647204	−	0.762317i	$-0.275938\pi$
$907$	1174.03	1.29441	0.647204	+	0.762317i	$0.275938\pi$
$908$	178.071i	0.196113i
$909$	0	0
$910$	23.6339	0.0259713
$911$	1592.11i	1.74765i	0.486241	+	0.873825i	$0.338368\pi$
$911$	1592.11i	1.74765i	−0.486241	+	0.873825i	$0.661632\pi$
$912$	0	0
$913$	−191.158	−0.209373
$914$	− 1519.29i	− 1.66224i
$915$	0	0
$916$	243.969	0.266341
$917$	− 36.5831i	− 0.0398943i
$918$	0	0
$919$	−1715.09	−1.86625	−0.933127	−	0.359546i	$-0.882931\pi$
$919$	−1715.09	−1.86625	−0.933127	+	0.359546i	$0.882931\pi$
$920$	505.235i	0.549169i
$921$	0	0
$922$	254.455	0.275982
$923$	36.0275i	0.0390330i
$924$	0	0
$925$	843.807	0.912223
$926$	872.764i	0.942510i
$927$	0	0
$928$	−19.9943	−0.0215456
$929$	− 385.239i	− 0.414681i	−0.978269	−	0.207340i	$-0.933519\pi$
$929$	− 385.239i	− 0.414681i	0.978269	−	0.207340i	$-0.0664808\pi$
$930$	0	0
$931$	−401.420	−0.431171
$932$	120.251i	0.129025i
$933$	0	0
$934$	1831.83	1.96127
$935$	141.292i	0.151115i
$936$	0	0
$937$	1212.65	1.29419	0.647093	−	0.762411i	$-0.275985\pi$
$937$	1212.65	1.29419	0.647093	+	0.762411i	$0.275985\pi$
$938$	− 159.957i	− 0.170530i
$939$	0	0
$940$	129.840	0.138128
$941$	− 219.219i	− 0.232964i	−0.993193	−	0.116482i	$-0.962838\pi$
$941$	− 219.219i	− 0.232964i	0.993193	−	0.116482i	$-0.0371617\pi$
$942$	0	0
$943$	−1407.28	−1.49234
$944$	961.438i	1.01847i
$945$	0	0
$946$	−1279.36	−1.35239
$947$	− 55.5968i	− 0.0587084i	−0.999569	−	0.0293542i	$-0.990655\pi$
$947$	− 55.5968i	− 0.0587084i	0.999569	−	0.0293542i	$-0.00934507\pi$
$948$	0	0
$949$	142.716	0.150385
$950$	218.579i	0.230083i
$951$	0	0
$952$	31.7450	0.0333456
$953$	− 599.958i	− 0.629546i	−0.949167	−	0.314773i	$-0.898072\pi$
$953$	− 599.958i	− 0.629546i	0.949167	−	0.314773i	$-0.101928\pi$
$954$	0	0
$955$	−690.396	−0.722928
$956$	− 68.1424i	− 0.0712786i
$957$	0	0
$958$	−1808.43	−1.88771
$959$	− 145.591i	− 0.151816i
$960$	0	0
$961$	−898.267	−0.934721
$962$	− 530.141i	− 0.551082i
$963$	0	0
$964$	22.5653	0.0234080
$965$	− 440.475i	− 0.456451i
$966$	0	0
$967$	−988.565	−1.02230	−0.511150	−	0.859491i	$-0.670781\pi$
$967$	−988.565	−1.02230	−0.511150	+	0.859491i	$0.670781\pi$
$968$	435.081i	0.449463i
$969$	0	0
$970$	−1298.95	−1.33913
$971$	940.431i	0.968518i	0.874925	+	0.484259i	$0.160911\pi$
$971$	940.431i	0.968518i	−0.874925	+	0.484259i	$0.839089\pi$
$972$	0	0
$973$	−170.651	−0.175387
$974$	529.952i	0.544098i
$975$	0	0
$976$	590.445	0.604964
$977$	− 317.445i	− 0.324918i	−0.986715	−	0.162459i	$-0.948057\pi$
$977$	− 317.445i	− 0.324918i	0.986715	−	0.162459i	$-0.0519426\pi$
$978$	0	0
$979$	−317.803	−0.324620
$980$	99.7027i	0.101737i
$981$	0	0
$982$	1788.20	1.82097
$983$	461.726i	0.469711i	0.972030	+	0.234856i	$0.0754617\pi$
$983$	461.726i	0.469711i	−0.972030	+	0.234856i	$0.924538\pi$
$984$	0	0
$985$	−473.070	−0.480274
$986$	− 23.4864i	− 0.0238198i
$987$	0	0
$988$	17.3711	0.0175821
$989$	− 1474.61i	− 1.49101i
$990$	0	0
$991$	−858.657	−0.866455	−0.433228	−	0.901285i	$-0.642625\pi$
$991$	−858.657	−0.866455	−0.433228	+	0.901285i	$0.642625\pi$
$992$	− 72.8548i	− 0.0734424i
$993$	0	0
$994$	18.3651	0.0184759
$995$	− 1051.64i	− 1.05693i
$996$	0	0
$997$	−933.420	−0.936229	−0.468114	−	0.883668i	$-0.655066\pi$
$997$	−933.420	−0.936229	−0.468114	+	0.883668i	$0.655066\pi$
$998$	97.3917i	0.0975868i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	117.3.c.a.53.7	yes	8
3.2	odd	2	inner	117.3.c.a.53.2	✓	8
4.3	odd	2		1872.3.f.d.1457.7		8
12.11	even	2		1872.3.f.d.1457.2		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
117.3.c.a.53.2	✓	8	3.2	odd	2	inner
117.3.c.a.53.7	yes	8	1.1	even	1	trivial
1872.3.f.d.1457.2		8	12.11	even	2
1872.3.f.d.1457.7		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)

\(f(q)\)	\(=\)	\(q+2.13992i q^{2} -0.579250 q^{4} +3.56642i q^{5} +0.858882 q^{7} +7.32013i q^{8} -7.63185 q^{10} +7.84626i q^{11} -3.60555 q^{13} +1.83794i q^{14} -17.9815 q^{16} -5.04921i q^{17} +8.31746 q^{19} -2.06585i q^{20} -16.7904 q^{22} -19.3527i q^{23} +12.2806 q^{25} -7.71558i q^{26} -0.497507 q^{28} -2.17368i q^{29} +7.92040 q^{31} -9.19837i q^{32} +10.8049 q^{34} +3.06314i q^{35} +68.7104 q^{37} +17.7987i q^{38} -26.1067 q^{40} -72.7172i q^{41} +76.1964 q^{43} -4.54494i q^{44} +41.4133 q^{46} +62.8508i q^{47} -48.2623 q^{49} +26.2795i q^{50} +2.08852 q^{52} -53.9288i q^{53} -27.9831 q^{55} +6.28712i q^{56} +4.65149 q^{58} -53.4683i q^{59} -32.8363 q^{61} +16.9490i q^{62} -52.2421 q^{64} -12.8589i q^{65} -87.0307 q^{67} +2.92475i q^{68} -6.55486 q^{70} -9.99222i q^{71} -39.5822 q^{73} +147.035i q^{74} -4.81789 q^{76} +6.73901i q^{77} -88.1246 q^{79} -64.1295i q^{80} +155.609 q^{82} +24.3629i q^{83} +18.0076 q^{85} +163.054i q^{86} -57.4356 q^{88} +40.5038i q^{89} -3.09674 q^{91} +11.2101i q^{92} -134.496 q^{94} +29.6636i q^{95} +170.201 q^{97} -103.277i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 8 q^{4} - 16 q^{7} + 24 q^{16} + 64 q^{19} - 80 q^{22} - 24 q^{25} + 184 q^{28} - 40 q^{31} - 272 q^{34} - 104 q^{37} + 32 q^{40} + 128 q^{43} + 232 q^{46} + 136 q^{49} + 104 q^{52} - 224 q^{55} - 88 q^{58}+ \cdots + 328 q^{97}+O(q^{100}) \) 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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