LMFDB - Embedded newform 117.3.c.a.53.8

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			53.8
Root			$3.55413i$ of defining polynomial
Character	$\chi$	$=$	117.53
Dual form			117.3.c.a.53.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+3.55413i q^{2} -8.63185 q^{4} -0.118383i q^{5} -12.0700 q^{7} -16.4622i q^{8} +0.420750 q^{10} +6.98988i q^{11} -3.60555 q^{13} -42.8983i q^{14} +23.9815 q^{16} +32.3174i q^{17} +14.8936 q^{19} +1.02187i q^{20} -24.8430 q^{22} -14.8116i q^{23} +24.9860 q^{25} -12.8146i q^{26} +104.186 q^{28} +31.8460i q^{29} -32.3426 q^{31} +19.3844i q^{32} -114.860 q^{34} +1.42889i q^{35} -58.6549 q^{37} +52.9340i q^{38} -1.94885 q^{40} +46.4930i q^{41} -0.929781 q^{43} -60.3356i q^{44} +52.6422 q^{46} -2.46229i q^{47} +96.6845 q^{49} +88.8035i q^{50} +31.1226 q^{52} -27.8759i q^{53} +0.827486 q^{55} +198.699i q^{56} -113.185 q^{58} -61.7728i q^{59} +76.9473 q^{61} -114.950i q^{62} +27.0310 q^{64} +0.426837i q^{65} +17.6531 q^{67} -278.959i q^{68} -5.07845 q^{70} +88.6649i q^{71} +0.904326 q^{73} -208.467i q^{74} -128.560 q^{76} -84.3677i q^{77} +79.2802 q^{79} -2.83901i q^{80} -165.242 q^{82} +15.0213i q^{83} +3.82585 q^{85} -3.30456i q^{86} +115.069 q^{88} -16.9997i q^{89} +43.5189 q^{91} +127.851i q^{92} +8.75130 q^{94} -1.76316i q^{95} -8.87928 q^{97} +343.630i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	3.55413i	1.77707i	0.458813	+	0.888533i	$0.348275\pi$
$2$	3.55413i	1.77707i	−0.458813	+	0.888533i	$0.651725\pi$
$3$	0	0
$3$	0	0
$4$	−8.63185	−2.15796
$5$	− 0.118383i	− 0.0236767i	−0.999930	−	0.0118383i	$-0.996232\pi$
$5$	− 0.118383i	− 0.0236767i	0.999930	−	0.0118383i	$-0.00376835\pi$
$6$	0	0
$7$	−12.0700	−1.72428	−0.862142	−	0.506667i	$-0.830877\pi$
$7$	−12.0700	−1.72428	−0.862142	+	0.506667i	$0.830877\pi$
$8$	− 16.4622i	− 2.05778i
$9$	0	0
$10$	0.420750	0.0420750
$11$	6.98988i	0.635444i	0.948184	+	0.317722i	$0.102918\pi$
$11$	6.98988i	0.635444i	−0.948184	+	0.317722i	$0.897082\pi$
$12$	0	0
$13$	−3.60555	−0.277350
$13$	−3.60555	−0.277350
$14$	− 42.8983i	− 3.06417i
$15$	0	0
$16$	23.9815	1.49884
$17$	32.3174i	1.90103i	0.310686	+	0.950513i	$0.399441\pi$
$17$	32.3174i	1.90103i	−0.310686	+	0.950513i	$0.600559\pi$
$18$	0	0
$19$	14.8936	0.783876	0.391938	−	0.919992i	$-0.371805\pi$
$19$	14.8936	0.783876	0.391938	+	0.919992i	$0.371805\pi$
$20$	1.02187i	0.0510934i
$21$	0	0
$22$	−24.8430	−1.12923
$23$	− 14.8116i	− 0.643981i	−0.946743	−	0.321990i	$-0.895648\pi$
$23$	− 14.8116i	− 0.643981i	0.946743	−	0.321990i	$-0.104352\pi$
$24$	0	0
$25$	24.9860	0.999439
$26$	− 12.8146i	− 0.492869i
$27$	0	0
$28$	104.186	3.72094
$29$	31.8460i	1.09814i	0.835778	+	0.549068i	$0.185017\pi$
$29$	31.8460i	1.09814i	−0.835778	+	0.549068i	$0.814983\pi$
$30$	0	0
$31$	−32.3426	−1.04331	−0.521655	−	0.853157i	$-0.674685\pi$
$31$	−32.3426	−1.04331	−0.521655	+	0.853157i	$0.674685\pi$
$32$	19.3844i	0.605764i
$33$	0	0
$34$	−114.860	−3.37825
$35$	1.42889i	0.0408253i
$36$	0	0
$37$	−58.6549	−1.58527	−0.792633	−	0.609699i	$-0.791290\pi$
$37$	−58.6549	−1.58527	−0.792633	+	0.609699i	$0.791290\pi$
$38$	52.9340i	1.39300i
$39$	0	0
$40$	−1.94885	−0.0487213
$41$	46.4930i	1.13398i	0.823726	+	0.566988i	$0.191891\pi$
$41$	46.4930i	1.13398i	−0.823726	+	0.566988i	$0.808109\pi$
$42$	0	0
$43$	−0.929781	−0.0216228	−0.0108114	−	0.999942i	$-0.503441\pi$
$43$	−0.929781	−0.0216228	−0.0108114	+	0.999942i	$0.503441\pi$
$44$	− 60.3356i	− 1.37126i
$45$	0	0
$46$	52.6422	1.14440
$47$	− 2.46229i	− 0.0523891i	−0.999657	−	0.0261946i	$-0.991661\pi$
$47$	− 2.46229i	− 0.0523891i	0.999657	−	0.0261946i	$-0.00833894\pi$
$48$	0	0
$49$	96.6845	1.97315
$50$	88.8035i	1.77607i
$51$	0	0
$52$	31.1226	0.598511
$53$	− 27.8759i	− 0.525960i	−0.964801	−	0.262980i	$-0.915295\pi$
$53$	− 27.8759i	− 0.525960i	0.964801	−	0.262980i	$-0.0847053\pi$
$54$	0	0
$55$	0.827486	0.0150452
$56$	198.699i	3.54819i
$57$	0	0
$58$	−113.185	−1.95146
$59$	− 61.7728i	− 1.04700i	−0.852027	−	0.523498i	$-0.824627\pi$
$59$	− 61.7728i	− 1.04700i	0.852027	−	0.523498i	$-0.175373\pi$
$60$	0	0
$61$	76.9473	1.26143	0.630716	−	0.776014i	$-0.282761\pi$
$61$	76.9473	1.26143	0.630716	+	0.776014i	$0.282761\pi$
$62$	− 114.950i	− 1.85403i
$63$	0	0
$64$	27.0310	0.422360
$65$	0.426837i	0.00656673i
$66$	0	0
$67$	17.6531	0.263479	0.131740	−	0.991284i	$-0.457944\pi$
$67$	17.6531	0.263479	0.131740	+	0.991284i	$0.457944\pi$
$68$	− 278.959i	− 4.10234i
$69$	0	0
$70$	−5.07845	−0.0725492
$71$	88.6649i	1.24880i	0.781104	+	0.624401i	$0.214657\pi$
$71$	88.6649i	1.24880i	−0.781104	+	0.624401i	$0.785343\pi$
$72$	0	0
$73$	0.904326	0.0123880	0.00619402	−	0.999981i	$-0.498028\pi$
$73$	0.904326	0.0123880	0.00619402	+	0.999981i	$0.498028\pi$
$74$	− 208.467i	− 2.81712i
$75$	0	0
$76$	−128.560	−1.69158
$77$	− 84.3677i	− 1.09569i
$78$	0	0
$79$	79.2802	1.00355	0.501773	−	0.864999i	$-0.332681\pi$
$79$	79.2802	1.00355	0.501773	+	0.864999i	$0.332681\pi$
$80$	− 2.83901i	− 0.0354876i
$81$	0	0
$82$	−165.242	−2.01515
$83$	15.0213i	0.180979i	0.995897	+	0.0904896i	$0.0288432\pi$
$83$	15.0213i	0.180979i	−0.995897	+	0.0904896i	$0.971157\pi$
$84$	0	0
$85$	3.82585	0.0450100
$86$	− 3.30456i	− 0.0384252i
$87$	0	0
$88$	115.069	1.30760
$89$	− 16.9997i	− 0.191008i	−0.995429	−	0.0955038i	$-0.969554\pi$
$89$	− 16.9997i	− 0.191008i	0.995429	−	0.0955038i	$-0.0304462\pi$
$90$	0	0
$91$	43.5189	0.478230
$92$	127.851i	1.38969i
$93$	0	0
$94$	8.75130	0.0930989
$95$	− 1.76316i	− 0.0185596i
$96$	0	0
$97$	−8.87928	−0.0915390	−0.0457695	−	0.998952i	$-0.514574\pi$
$97$	−8.87928	−0.0915390	−0.0457695	+	0.998952i	$0.514574\pi$
$98$	343.630i	3.50642i
$99$	0	0
$100$	−215.675	−2.15675
$101$	5.23552i	0.0518368i	0.999664	+	0.0259184i	$0.00825101\pi$
$101$	5.23552i	0.0518368i	−0.999664	+	0.0259184i	$0.991749\pi$
$102$	0	0
$103$	−81.5696	−0.791938	−0.395969	−	0.918264i	$-0.629591\pi$
$103$	−81.5696	−0.791938	−0.395969	+	0.918264i	$0.629591\pi$
$104$	59.3554i	0.570725i
$105$	0	0
$106$	99.0745	0.934665
$107$	16.8385i	0.157369i	0.996900	+	0.0786844i	$0.0250719\pi$
$107$	16.8385i	0.157369i	−0.996900	+	0.0786844i	$0.974928\pi$
$108$	0	0
$109$	−53.7394	−0.493022	−0.246511	−	0.969140i	$-0.579284\pi$
$109$	−53.7394	−0.493022	−0.246511	+	0.969140i	$0.579284\pi$
$110$	2.94099i	0.0267363i
$111$	0	0
$112$	−289.456	−2.58443
$113$	139.740i	1.23664i	0.785928	+	0.618318i	$0.212186\pi$
$113$	139.740i	1.23664i	−0.785928	+	0.618318i	$0.787814\pi$
$114$	0	0
$115$	−1.75344	−0.0152473
$116$	− 274.890i	− 2.36974i
$117$	0	0
$118$	219.549	1.86058
$119$	− 390.071i	− 3.27791i
$120$	0	0
$121$	72.1416	0.596211
$122$	273.481i	2.24165i
$123$	0	0
$124$	279.177	2.25142
$125$	− 5.91751i	− 0.0473401i
$126$	0	0
$127$	−158.558	−1.24848	−0.624242	−	0.781231i	$-0.714592\pi$
$127$	−158.558	−1.24848	−0.624242	+	0.781231i	$0.714592\pi$
$128$	173.610i	1.35632i
$129$	0	0
$130$	−1.51704	−0.0116695
$131$	− 83.3075i	− 0.635935i	−0.948102	−	0.317967i	$-0.897000\pi$
$131$	− 83.3075i	− 0.635935i	0.948102	−	0.317967i	$-0.103000\pi$
$132$	0	0
$133$	−179.766	−1.35162
$134$	62.7414i	0.468220i
$135$	0	0
$136$	532.016	3.91189
$137$	54.2712i	0.396140i	0.980188	+	0.198070i	$0.0634674\pi$
$137$	54.2712i	0.396140i	−0.980188	+	0.198070i	$0.936533\pi$
$138$	0	0
$139$	−201.643	−1.45067	−0.725334	−	0.688397i	$-0.758315\pi$
$139$	−201.643	−1.45067	−0.725334	+	0.688397i	$0.758315\pi$
$140$	− 12.3339i	− 0.0880995i
$141$	0	0
$142$	−315.127	−2.21920
$143$	− 25.2024i	− 0.176240i
$144$	0	0
$145$	3.77003	0.0260002
$146$	3.21409i	0.0220143i
$147$	0	0
$148$	506.300	3.42095
$149$	178.662i	1.19908i	0.800346	+	0.599538i	$0.204649\pi$
$149$	178.662i	1.19908i	−0.800346	+	0.599538i	$0.795351\pi$
$150$	0	0
$151$	165.631	1.09689	0.548447	−	0.836185i	$-0.315219\pi$
$151$	165.631	1.09689	0.548447	+	0.836185i	$0.315219\pi$
$152$	− 245.182i	− 1.61304i
$153$	0	0
$154$	299.854	1.94710
$155$	3.82883i	0.0247021i
$156$	0	0
$157$	144.517	0.920492	0.460246	−	0.887791i	$-0.347761\pi$
$157$	144.517	0.920492	0.460246	+	0.887791i	$0.347761\pi$
$158$	281.772i	1.78337i
$159$	0	0
$160$	2.29480	0.0143425
$161$	178.775i	1.11041i
$162$	0	0
$163$	−92.8520	−0.569644	−0.284822	−	0.958580i	$-0.591935\pi$
$163$	−92.8520	−0.569644	−0.284822	+	0.958580i	$0.591935\pi$
$164$	− 401.321i	− 2.44708i
$165$	0	0
$166$	−53.3876	−0.321612
$167$	− 72.4358i	− 0.433747i	−0.976200	−	0.216874i	$-0.930414\pi$
$167$	− 72.4358i	− 0.433747i	0.976200	−	0.216874i	$-0.0695860\pi$
$168$	0	0
$169$	13.0000	0.0769231
$170$	13.5976i	0.0799857i
$171$	0	0
$172$	8.02573	0.0466612
$173$	− 230.759i	− 1.33387i	−0.745118	−	0.666933i	$-0.767607\pi$
$173$	− 230.759i	− 1.33387i	0.745118	−	0.666933i	$-0.232393\pi$
$174$	0	0
$175$	−301.580	−1.72332
$176$	167.628i	0.952430i
$177$	0	0
$178$	60.4191	0.339433
$179$	306.682i	1.71331i	0.515890	+	0.856655i	$0.327461\pi$
$179$	306.682i	1.71331i	−0.515890	+	0.856655i	$0.672539\pi$
$180$	0	0
$181$	126.649	0.699720	0.349860	−	0.936802i	$-0.386229\pi$
$181$	126.649	0.699720	0.349860	+	0.936802i	$0.386229\pi$
$182$	154.672i	0.849847i
$183$	0	0
$184$	−243.831	−1.32517
$185$	6.94376i	0.0375338i
$186$	0	0
$187$	−225.895	−1.20799
$188$	21.2541i	0.113054i
$189$	0	0
$190$	6.26650	0.0329816
$191$	− 258.683i	− 1.35436i	−0.735817	−	0.677181i	$-0.763201\pi$
$191$	− 258.683i	− 1.35436i	0.735817	−	0.677181i	$-0.236799\pi$
$192$	0	0
$193$	−172.716	−0.894901	−0.447450	−	0.894309i	$-0.647668\pi$
$193$	−172.716	−0.894901	−0.447450	+	0.894309i	$0.647668\pi$
$194$	− 31.5581i	− 0.162671i
$195$	0	0
$196$	−834.567	−4.25799
$197$	349.735i	1.77530i	0.460514	+	0.887652i	$0.347665\pi$
$197$	349.735i	1.77530i	−0.460514	+	0.887652i	$0.652335\pi$
$198$	0	0
$199$	95.0515	0.477646	0.238823	−	0.971063i	$-0.423238\pi$
$199$	95.0515	0.477646	0.238823	+	0.971063i	$0.423238\pi$
$200$	− 411.325i	− 2.05662i
$201$	0	0
$202$	−18.6077	−0.0921175
$203$	− 384.380i	− 1.89350i
$204$	0	0
$205$	5.50400	0.0268488
$206$	− 289.909i	− 1.40733i
$207$	0	0
$208$	−86.4664	−0.415704
$209$	104.105i	0.498109i
$210$	0	0
$211$	−183.562	−0.869962	−0.434981	−	0.900440i	$-0.643245\pi$
$211$	−183.562	−0.869962	−0.434981	+	0.900440i	$0.643245\pi$
$212$	240.620i	1.13500i
$213$	0	0
$214$	−59.8461	−0.279655
$215$	0.110071i	0 0.000511956i
$216$	0	0
$217$	390.375	1.79896
$218$	− 190.997i	− 0.876133i
$219$	0	0
$220$	−7.14273	−0.0324670
$221$	− 116.522i	− 0.527250i
$222$	0	0
$223$	−244.838	−1.09793	−0.548964	−	0.835846i	$-0.684978\pi$
$223$	−244.838	−1.09793	−0.548964	+	0.835846i	$0.684978\pi$
$224$	− 233.970i	− 1.04451i
$225$	0	0
$226$	−496.654	−2.19758
$227$	66.2739i	0.291956i	0.989288	+	0.145978i	$0.0466328\pi$
$227$	66.2739i	0.291956i	−0.989288	+	0.145978i	$0.953367\pi$
$228$	0	0
$229$	113.691	0.496469	0.248235	−	0.968700i	$-0.420150\pi$
$229$	113.691	0.496469	0.248235	+	0.968700i	$0.420150\pi$
$230$	− 6.23196i	− 0.0270955i
$231$	0	0
$232$	524.255	2.25972
$233$	97.0699i	0.416609i	0.978064	+	0.208305i	$0.0667945\pi$
$233$	97.0699i	0.416609i	−0.978064	+	0.208305i	$0.933205\pi$
$234$	0	0
$235$	−0.291494	−0.00124040
$236$	533.214i	2.25938i
$237$	0	0
$238$	1386.36	5.82506
$239$	− 169.360i	− 0.708617i	−0.935129	−	0.354309i	$-0.884716\pi$
$239$	− 169.360i	− 0.708617i	0.935129	−	0.354309i	$-0.115284\pi$
$240$	0	0
$241$	95.6559	0.396912	0.198456	−	0.980110i	$-0.436407\pi$
$241$	95.6559	0.396912	0.198456	+	0.980110i	$0.436407\pi$
$242$	256.401i	1.05951i
$243$	0	0
$244$	−664.198	−2.72212
$245$	− 11.4458i	− 0.0467177i
$246$	0	0
$247$	−53.6998	−0.217408
$248$	532.431i	2.14690i
$249$	0	0
$250$	21.0316	0.0841264
$251$	− 160.564i	− 0.639696i	−0.947469	−	0.319848i	$-0.896368\pi$
$251$	− 160.564i	− 0.639696i	0.947469	−	0.319848i	$-0.103632\pi$
$252$	0	0
$253$	103.531	0.409213
$254$	− 563.535i	− 2.21864i
$255$	0	0
$256$	−508.907	−1.98792
$257$	− 179.119i	− 0.696963i	−0.937316	−	0.348481i	$-0.886697\pi$
$257$	− 179.119i	− 0.696963i	0.937316	−	0.348481i	$-0.113303\pi$
$258$	0	0
$259$	707.963	2.73345
$260$	− 3.68440i	− 0.0141708i
$261$	0	0
$262$	296.086	1.13010
$263$	249.989i	0.950530i	0.879843	+	0.475265i	$0.157648\pi$
$263$	249.989i	0.950530i	−0.879843	+	0.475265i	$0.842352\pi$
$264$	0	0
$265$	−3.30004	−0.0124530
$266$	− 638.912i	− 2.40193i
$267$	0	0
$268$	−152.379	−0.568578
$269$	138.899i	0.516354i	0.966098	+	0.258177i	$0.0831218\pi$
$269$	138.899i	0.516354i	−0.966098	+	0.258177i	$0.916878\pi$
$270$	0	0
$271$	185.832	0.685725	0.342863	−	0.939386i	$-0.388603\pi$
$271$	185.832	0.685725	0.342863	+	0.939386i	$0.388603\pi$
$272$	775.019i	2.84934i
$273$	0	0
$274$	−192.887	−0.703968
$275$	174.649i	0.635087i
$276$	0	0
$277$	259.845	0.938068	0.469034	−	0.883180i	$-0.344602\pi$
$277$	259.845	0.938068	0.469034	+	0.883180i	$0.344602\pi$
$278$	− 716.666i	− 2.57793i
$279$	0	0
$280$	23.5226	0.0840093
$281$	− 261.217i	− 0.929600i	−0.885416	−	0.464800i	$-0.846126\pi$
$281$	− 261.217i	− 0.929600i	0.885416	−	0.464800i	$-0.153874\pi$
$282$	0	0
$283$	442.586	1.56391	0.781953	−	0.623337i	$-0.214223\pi$
$283$	442.586	1.56391	0.781953	+	0.623337i	$0.214223\pi$
$284$	− 765.343i	− 2.69487i
$285$	0	0
$286$	89.5725	0.313191
$287$	− 561.170i	− 1.95530i
$288$	0	0
$289$	−755.416	−2.61390
$290$	13.3992i	0.0462041i
$291$	0	0
$292$	−7.80601	−0.0267329
$293$	114.871i	0.392053i	0.980599	+	0.196026i	$0.0628038\pi$
$293$	114.871i	0.392053i	−0.980599	+	0.196026i	$0.937196\pi$
$294$	0	0
$295$	−7.31287	−0.0247894
$296$	965.589i	3.26212i
$297$	0	0
$298$	−634.989	−2.13084
$299$	53.4038i	0.178608i
$300$	0	0
$301$	11.2224	0.0372839
$302$	588.675i	1.94925i
$303$	0	0
$304$	357.172	1.17491
$305$	− 9.10928i	− 0.0298665i
$306$	0	0
$307$	225.948	0.735988	0.367994	−	0.929828i	$-0.380045\pi$
$307$	225.948	0.735988	0.367994	+	0.929828i	$0.380045\pi$
$308$	728.250i	2.36445i
$309$	0	0
$310$	−13.6082	−0.0438973
$311$	556.462i	1.78927i	0.446801	+	0.894633i	$0.352563\pi$
$311$	556.462i	1.78927i	−0.446801	+	0.894633i	$0.647437\pi$
$312$	0	0
$313$	212.533	0.679018	0.339509	−	0.940603i	$-0.389739\pi$
$313$	212.533	0.679018	0.339509	+	0.940603i	$0.389739\pi$
$314$	513.633i	1.63577i
$315$	0	0
$316$	−684.335	−2.16562
$317$	555.595i	1.75267i	0.481706	+	0.876333i	$0.340017\pi$
$317$	555.595i	1.75267i	−0.481706	+	0.876333i	$0.659983\pi$
$318$	0	0
$319$	−222.599	−0.697804
$320$	− 3.20002i	− 0.0100001i
$321$	0	0
$322$	−635.391	−1.97326
$323$	481.324i	1.49017i
$324$	0	0
$325$	−90.0883	−0.277195
$326$	− 330.008i	− 1.01230i
$327$	0	0
$328$	765.378	2.33347
$329$	29.7198i	0.0903337i
$330$	0	0
$331$	545.601	1.64834	0.824171	−	0.566341i	$-0.191642\pi$
$331$	545.601	1.64834	0.824171	+	0.566341i	$0.191642\pi$
$332$	− 129.661i	− 0.390546i
$333$	0	0
$334$	257.446	0.770798
$335$	− 2.08983i	− 0.00623831i
$336$	0	0
$337$	198.163	0.588022	0.294011	−	0.955802i	$-0.405010\pi$
$337$	198.163	0.588022	0.294011	+	0.955802i	$0.405010\pi$
$338$	46.2037i	0.136697i
$339$	0	0
$340$	−33.0241	−0.0971298
$341$	− 226.071i	− 0.662965i
$342$	0	0
$343$	−575.552	−1.67799
$344$	15.3063i	0.0444949i
$345$	0	0
$346$	820.147	2.37037
$347$	− 225.416i	− 0.649614i	−0.945780	−	0.324807i	$-0.894701\pi$
$347$	− 225.416i	− 0.649614i	0.945780	−	0.324807i	$-0.105299\pi$
$348$	0	0
$349$	49.5201	0.141891	0.0709457	−	0.997480i	$-0.477398\pi$
$349$	49.5201	0.141891	0.0709457	+	0.997480i	$0.477398\pi$
$350$	− 1071.86i	− 3.06245i
$351$	0	0
$352$	−135.495	−0.384929
$353$	− 483.901i	− 1.37083i	−0.728155	−	0.685413i	$-0.759622\pi$
$353$	− 483.901i	− 1.37083i	0.728155	−	0.685413i	$-0.240378\pi$
$354$	0	0
$355$	10.4965	0.0295675
$356$	146.739i	0.412187i
$357$	0	0
$358$	−1089.99	−3.04466
$359$	60.3045i	0.167979i	0.996467	+	0.0839896i	$0.0267662\pi$
$359$	60.3045i	0.167979i	−0.996467	+	0.0839896i	$0.973234\pi$
$360$	0	0
$361$	−139.179	−0.385538
$362$	450.128i	1.24345i
$363$	0	0
$364$	−375.649	−1.03200
$365$	− 0.107057i	0 0.000293307i
$366$	0	0
$367$	−341.309	−0.929997	−0.464998	−	0.885311i	$-0.653945\pi$
$367$	−341.309	−0.929997	−0.464998	+	0.885311i	$0.653945\pi$
$368$	− 355.203i	− 0.965225i
$369$	0	0
$370$	−24.6790	−0.0667001
$371$	336.461i	0.906904i
$372$	0	0
$373$	−26.5501	−0.0711798	−0.0355899	−	0.999366i	$-0.511331\pi$
$373$	−26.5501	−0.0711798	−0.0355899	+	0.999366i	$0.511331\pi$
$374$	− 802.860i	− 2.14669i
$375$	0	0
$376$	−40.5347	−0.107805
$377$	− 114.822i	− 0.304568i
$378$	0	0
$379$	−272.528	−0.719071	−0.359535	−	0.933131i	$-0.617065\pi$
$379$	−272.528	−0.719071	−0.359535	+	0.933131i	$0.617065\pi$
$380$	15.2193i	0.0400509i
$381$	0	0
$382$	919.394	2.40679
$383$	391.268i	1.02159i	0.859703	+	0.510794i	$0.170649\pi$
$383$	391.268i	1.02159i	−0.859703	+	0.510794i	$0.829351\pi$
$384$	0	0
$385$	−9.98774	−0.0259422
$386$	− 613.855i	− 1.59030i
$387$	0	0
$388$	76.6447	0.197538
$389$	− 402.950i	− 1.03586i	−0.855422	−	0.517931i	$-0.826702\pi$
$389$	− 402.950i	− 1.03586i	0.855422	−	0.517931i	$-0.173298\pi$
$390$	0	0
$391$	478.671	1.22422
$392$	− 1591.64i	− 4.06031i
$393$	0	0
$394$	−1243.00	−3.15483
$395$	− 9.38545i	− 0.0237606i
$396$	0	0
$397$	504.118	1.26982	0.634909	−	0.772587i	$-0.281038\pi$
$397$	504.118	1.26982	0.634909	+	0.772587i	$0.281038\pi$
$398$	337.826i	0.848808i
$399$	0	0
$400$	599.201	1.49800
$401$	− 256.668i	− 0.640070i	−0.947406	−	0.320035i	$-0.896305\pi$
$401$	− 256.668i	− 0.640070i	0.947406	−	0.320035i	$-0.103695\pi$
$402$	0	0
$403$	116.613	0.289362
$404$	− 45.1922i	− 0.111862i
$405$	0	0
$406$	1366.14	3.36487
$407$	− 409.990i	− 1.00735i
$408$	0	0
$409$	−661.177	−1.61657	−0.808285	−	0.588792i	$-0.799604\pi$
$409$	−661.177	−1.61657	−0.808285	+	0.588792i	$0.799604\pi$
$410$	19.5619i	0.0477120i
$411$	0	0
$412$	704.097	1.70897
$413$	745.597i	1.80532i
$414$	0	0
$415$	1.77827	0.00428499
$416$	− 69.8916i	− 0.168009i
$417$	0	0
$418$	−370.002	−0.885173
$419$	451.277i	1.07703i	0.842615	+	0.538517i	$0.181015\pi$
$419$	451.277i	1.07703i	−0.842615	+	0.538517i	$0.818985\pi$
$420$	0	0
$421$	−98.2467	−0.233365	−0.116683	−	0.993169i	$-0.537226\pi$
$421$	−98.2467	−0.233365	−0.116683	+	0.993169i	$0.537226\pi$
$422$	− 652.403i	− 1.54598i
$423$	0	0
$424$	−458.898	−1.08231
$425$	807.483i	1.89996i
$426$	0	0
$427$	−928.753	−2.17507
$428$	− 145.347i	− 0.339596i
$429$	0	0
$430$	−0.391205	−0.000909780 0
$431$	− 555.715i	− 1.28936i	−0.764451	−	0.644681i	$-0.776990\pi$
$431$	− 555.715i	− 1.28936i	0.764451	−	0.644681i	$-0.223010\pi$
$432$	0	0
$433$	−85.9736	−0.198553	−0.0992767	−	0.995060i	$-0.531653\pi$
$433$	−85.9736	−0.198553	−0.0992767	+	0.995060i	$0.531653\pi$
$434$	1387.44i	3.19687i
$435$	0	0
$436$	463.871	1.06392
$437$	− 220.598i	− 0.504801i
$438$	0	0
$439$	466.105	1.06174	0.530871	−	0.847452i	$-0.321865\pi$
$439$	466.105	1.06174	0.530871	+	0.847452i	$0.321865\pi$
$440$	− 13.6222i	− 0.0309596i
$441$	0	0
$442$	414.135	0.936957
$443$	752.196i	1.69796i	0.528426	+	0.848979i	$0.322782\pi$
$443$	752.196i	1.69796i	−0.528426	+	0.848979i	$0.677218\pi$
$444$	0	0
$445$	−2.01248	−0.00452242
$446$	− 870.187i	− 1.95109i
$447$	0	0
$448$	−326.264	−0.728268
$449$	21.8356i	0.0486316i	0.999704	+	0.0243158i	$0.00774073\pi$
$449$	21.8356i	0.0486316i	−0.999704	+	0.0243158i	$0.992259\pi$
$450$	0	0
$451$	−324.981	−0.720578
$452$	− 1206.21i	− 2.66862i
$453$	0	0
$454$	−235.546	−0.518824
$455$	− 5.15192i	− 0.0113229i
$456$	0	0
$457$	209.052	0.457445	0.228722	−	0.973492i	$-0.426545\pi$
$457$	209.052	0.457445	0.228722	+	0.973492i	$0.426545\pi$
$458$	404.075i	0.882259i
$459$	0	0
$460$	15.1355	0.0329032
$461$	657.282i	1.42577i	0.701279	+	0.712887i	$0.252613\pi$
$461$	657.282i	1.42577i	−0.701279	+	0.712887i	$0.747387\pi$
$462$	0	0
$463$	−592.238	−1.27913	−0.639566	−	0.768736i	$-0.720886\pi$
$463$	−592.238	−1.27913	−0.639566	+	0.768736i	$0.720886\pi$
$464$	763.713i	1.64593i
$465$	0	0
$466$	−344.999	−0.740342
$467$	44.2448i	0.0947427i	0.998877	+	0.0473714i	$0.0150844\pi$
$467$	44.2448i	0.0947427i	−0.998877	+	0.0473714i	$0.984916\pi$
$468$	0	0
$469$	−213.073	−0.454313
$470$	− 1.03601i	− 0.00220427i
$471$	0	0
$472$	−1016.92	−2.15448
$473$	− 6.49906i	− 0.0137401i
$474$	0	0
$475$	372.132	0.783437
$476$	3367.03i	7.07360i
$477$	0	0
$478$	601.926	1.25926
$479$	− 553.714i	− 1.15598i	−0.816044	−	0.577989i	$-0.803837\pi$
$479$	− 553.714i	− 1.15598i	0.816044	−	0.577989i	$-0.196163\pi$
$480$	0	0
$481$	211.483	0.439674
$482$	339.974i	0.705339i
$483$	0	0
$484$	−622.715	−1.28660
$485$	1.05116i	0.00216734i
$486$	0	0
$487$	55.1939	0.113335	0.0566673	−	0.998393i	$-0.481953\pi$
$487$	55.1939	0.113335	0.0566673	+	0.998393i	$0.481953\pi$
$488$	− 1266.72i	− 2.59574i
$489$	0	0
$490$	40.6800	0.0830205
$491$	84.4643i	0.172025i	0.996294	+	0.0860125i	$0.0274125\pi$
$491$	84.4643i	0.172025i	−0.996294	+	0.0860125i	$0.972587\pi$
$492$	0	0
$493$	−1029.18	−2.08759
$494$	− 190.856i	− 0.386349i
$495$	0	0
$496$	−775.623	−1.56376
$497$	− 1070.18i	− 2.15329i
$498$	0	0
$499$	548.532	1.09926	0.549631	−	0.835407i	$-0.314768\pi$
$499$	548.532	1.09926	0.549631	+	0.835407i	$0.314768\pi$
$500$	51.0791i	0.102158i
$501$	0	0
$502$	570.664	1.13678
$503$	− 417.804i	− 0.830624i	−0.909679	−	0.415312i	$-0.863672\pi$
$503$	− 417.804i	− 0.830624i	0.909679	−	0.415312i	$-0.136328\pi$
$504$	0	0
$505$	0.619798	0.00122732
$506$	367.963i	0.727199i
$507$	0	0
$508$	1368.65	2.69418
$509$	− 499.112i	− 0.980574i	−0.871561	−	0.490287i	$-0.836892\pi$
$509$	− 499.112i	− 0.980574i	0.871561	−	0.490287i	$-0.163108\pi$
$510$	0	0
$511$	−10.9152	−0.0213605
$512$	− 1114.28i	− 2.17634i
$513$	0	0
$514$	636.614	1.23855
$515$	9.65648i	0.0187505i
$516$	0	0
$517$	17.2111	0.0332903
$518$	2516.19i	4.85752i
$519$	0	0
$520$	7.02669	0.0135129
$521$	− 397.286i	− 0.762544i	−0.924463	−	0.381272i	$-0.875486\pi$
$521$	− 397.286i	− 0.762544i	0.924463	−	0.381272i	$-0.124514\pi$
$522$	0	0
$523$	130.826	0.250146	0.125073	−	0.992148i	$-0.460083\pi$
$523$	130.826	0.250146	0.125073	+	0.992148i	$0.460083\pi$
$524$	719.098i	1.37232i
$525$	0	0
$526$	−888.495	−1.68915
$527$	− 1045.23i	− 1.98336i
$528$	0	0
$529$	309.618	0.585289
$530$	− 11.7288i	− 0.0221298i
$531$	0	0
$532$	1551.71	2.91676
$533$	− 167.633i	− 0.314508i
$534$	0	0
$535$	1.99339	0.00372597
$536$	− 290.609i	− 0.542181i
$537$	0	0
$538$	−493.666	−0.917595
$539$	675.813i	1.25383i
$540$	0	0
$541$	389.269	0.719536	0.359768	−	0.933042i	$-0.382856\pi$
$541$	389.269	0.719536	0.359768	+	0.933042i	$0.382856\pi$
$542$	660.470i	1.21858i
$543$	0	0
$544$	−626.455	−1.15157
$545$	6.36185i	0.0116731i
$546$	0	0
$547$	1.34870	0.00246563	0.00123281	−	0.999999i	$-0.499608\pi$
$547$	1.34870	0.00246563	0.00123281	+	0.999999i	$0.499608\pi$
$548$	− 468.461i	− 0.854856i
$549$	0	0
$550$	−620.726	−1.12859
$551$	474.302i	0.860803i
$552$	0	0
$553$	−956.910	−1.73040
$554$	923.523i	1.66701i
$555$	0	0
$556$	1740.55	3.13049
$557$	− 302.631i	− 0.543324i	−0.962393	−	0.271662i	$-0.912427\pi$
$557$	− 302.631i	− 0.543324i	0.962393	−	0.271662i	$-0.0875733\pi$
$558$	0	0
$559$	3.35237	0.00599709
$560$	34.2668i	0.0611907i
$561$	0	0
$562$	928.401	1.65196
$563$	760.848i	1.35142i	0.737169	+	0.675708i	$0.236162\pi$
$563$	760.848i	1.35142i	−0.737169	+	0.675708i	$0.763838\pi$
$564$	0	0
$565$	16.5429	0.0292794
$566$	1573.01i	2.77917i
$567$	0	0
$568$	1459.62	2.56976
$569$	− 345.629i	− 0.607431i	−0.952763	−	0.303716i	$-0.901773\pi$
$569$	− 345.629i	− 0.607431i	0.952763	−	0.303716i	$-0.0982273\pi$
$570$	0	0
$571$	599.127	1.04926	0.524630	−	0.851330i	$-0.324204\pi$
$571$	599.127	1.04926	0.524630	+	0.851330i	$0.324204\pi$
$572$	217.543i	0.380320i
$573$	0	0
$574$	1994.47	3.47469
$575$	− 370.081i	− 0.643620i
$576$	0	0
$577$	−215.164	−0.372901	−0.186450	−	0.982464i	$-0.559698\pi$
$577$	−215.164	−0.372901	−0.186450	+	0.982464i	$0.559698\pi$
$578$	− 2684.85i	− 4.64507i
$579$	0	0
$580$	−32.5424	−0.0561075
$581$	− 181.307i	− 0.312059i
$582$	0	0
$583$	194.849	0.334218
$584$	− 14.8872i	− 0.0254918i
$585$	0	0
$586$	−408.268	−0.696704
$587$	− 324.539i	− 0.552877i	−0.961032	−	0.276439i	$-0.910846\pi$
$587$	− 324.539i	− 0.552877i	0.961032	−	0.276439i	$-0.0891542\pi$
$588$	0	0
$589$	−481.699	−0.817826
$590$	− 25.9909i	− 0.0440524i
$591$	0	0
$592$	−1406.63	−2.37606
$593$	− 849.593i	− 1.43270i	−0.697740	−	0.716352i	$-0.745811\pi$
$593$	− 849.593i	− 1.43270i	0.697740	−	0.716352i	$-0.254189\pi$
$594$	0	0
$595$	−46.1779	−0.0776099
$596$	− 1542.19i	− 2.58756i
$597$	0	0
$598$	−189.804	−0.317398
$599$	− 312.421i	− 0.521570i	−0.965397	−	0.260785i	$-0.916019\pi$
$599$	− 312.421i	− 0.521570i	0.965397	−	0.260785i	$-0.0839814\pi$
$600$	0	0
$601$	589.233	0.980421	0.490211	−	0.871604i	$-0.336920\pi$
$601$	589.233	0.980421	0.490211	+	0.871604i	$0.336920\pi$
$602$	39.8860i	0.0662559i
$603$	0	0
$604$	−1429.70	−2.36706
$605$	− 8.54036i	− 0.0141163i
$606$	0	0
$607$	382.298	0.629816	0.314908	−	0.949122i	$-0.398026\pi$
$607$	382.298	0.629816	0.314908	+	0.949122i	$0.398026\pi$
$608$	288.705i	0.474844i
$609$	0	0
$610$	32.3756	0.0530747
$611$	8.87790i	0.0145301i
$612$	0	0
$613$	142.773	0.232908	0.116454	−	0.993196i	$-0.462847\pi$
$613$	142.773	0.232908	0.116454	+	0.993196i	$0.462847\pi$
$614$	803.050i	1.30790i
$615$	0	0
$616$	−1388.88	−2.25468
$617$	507.404i	0.822372i	0.911551	+	0.411186i	$0.134885\pi$
$617$	507.404i	0.822372i	−0.911551	+	0.411186i	$0.865115\pi$
$618$	0	0
$619$	307.551	0.496851	0.248426	−	0.968651i	$-0.420087\pi$
$619$	307.551	0.496851	0.248426	+	0.968651i	$0.420087\pi$
$620$	− 33.0499i	− 0.0533062i
$621$	0	0
$622$	−1977.74	−3.17964
$623$	205.186i	0.329351i
$624$	0	0
$625$	623.949	0.998319
$626$	755.369i	1.20666i
$627$	0	0
$628$	−1247.45	−1.98639
$629$	− 1895.57i	− 3.01363i
$630$	0	0
$631$	−586.261	−0.929099	−0.464549	−	0.885547i	$-0.653784\pi$
$631$	−586.261	−0.929099	−0.464549	+	0.885547i	$0.653784\pi$
$632$	− 1305.13i	− 2.06507i
$633$	0	0
$634$	−1974.66	−3.11460
$635$	18.7706i	0.0295600i
$636$	0	0
$637$	−348.601	−0.547254
$638$	− 791.148i	− 1.24004i
$639$	0	0
$640$	20.5525	0.0321133
$641$	740.100i	1.15460i	0.816531	+	0.577301i	$0.195894\pi$
$641$	740.100i	1.15460i	−0.816531	+	0.577301i	$0.804106\pi$
$642$	0	0
$643$	1044.44	1.62433	0.812164	−	0.583429i	$-0.198289\pi$
$643$	1044.44	1.62433	0.812164	+	0.583429i	$0.198289\pi$
$644$	− 1543.16i	− 2.39621i
$645$	0	0
$646$	−1710.69	−2.64813
$647$	457.314i	0.706822i	0.935468	+	0.353411i	$0.114978\pi$
$647$	457.314i	0.706822i	−0.935468	+	0.353411i	$0.885022\pi$
$648$	0	0
$649$	431.784	0.665307
$650$	− 320.186i	− 0.492593i
$651$	0	0
$652$	801.485	1.22927
$653$	− 100.342i	− 0.153663i	−0.997044	−	0.0768313i	$-0.975520\pi$
$653$	− 100.342i	− 0.153663i	0.997044	−	0.0768313i	$-0.0244803\pi$
$654$	0	0
$655$	−9.86222	−0.0150568
$656$	1114.97i	1.69965i
$657$	0	0
$658$	−105.628	−0.160529
$659$	679.814i	1.03158i	0.856714	+	0.515792i	$0.172502\pi$
$659$	679.814i	1.03158i	−0.856714	+	0.515792i	$0.827498\pi$
$660$	0	0
$661$	−288.219	−0.436035	−0.218018	−	0.975945i	$-0.569959\pi$
$661$	−288.219	−0.436035	−0.218018	+	0.975945i	$0.569959\pi$
$662$	1939.14i	2.92921i
$663$	0	0
$664$	247.283	0.372415
$665$	21.2813i	0.0320020i
$666$	0	0
$667$	471.688	0.707179
$668$	625.255i	0.936011i
$669$	0	0
$670$	7.42754	0.0110859
$671$	537.853i	0.801569i
$672$	0	0
$673$	−101.770	−0.151218	−0.0756091	−	0.997138i	$-0.524090\pi$
$673$	−101.770	−0.151218	−0.0756091	+	0.997138i	$0.524090\pi$
$674$	704.299i	1.04495i
$675$	0	0
$676$	−112.214	−0.165997
$677$	− 496.502i	− 0.733386i	−0.930342	−	0.366693i	$-0.880490\pi$
$677$	− 496.502i	− 0.733386i	0.930342	−	0.366693i	$-0.119510\pi$
$678$	0	0
$679$	107.173	0.157839
$680$	− 62.9819i	− 0.0926204i
$681$	0	0
$682$	803.486	1.17813
$683$	607.021i	0.888757i	0.895839	+	0.444378i	$0.146575\pi$
$683$	607.021i	0.888757i	−0.895839	+	0.444378i	$0.853425\pi$
$684$	0	0
$685$	6.42481	0.00937929
$686$	− 2045.59i	− 2.98190i
$687$	0	0
$688$	−22.2975	−0.0324092
$689$	100.508i	0.145875i
$690$	0	0
$691$	−311.008	−0.450084	−0.225042	−	0.974349i	$-0.572252\pi$
$691$	−311.008	−0.450084	−0.225042	+	0.974349i	$0.572252\pi$
$692$	1991.88i	2.87843i
$693$	0	0
$694$	801.159	1.15441
$695$	23.8712i	0.0343470i
$696$	0	0
$697$	−1502.53	−2.15572
$698$	176.001i	0.252150i
$699$	0	0
$700$	2603.20	3.71885
$701$	− 407.041i	− 0.580657i	−0.956927	−	0.290329i	$-0.906235\pi$
$701$	− 407.041i	− 0.580657i	0.956927	−	0.290329i	$-0.0937646\pi$
$702$	0	0
$703$	−873.585	−1.24265
$704$	188.944i	0.268386i
$705$	0	0
$706$	1719.85	2.43605
$707$	− 63.1926i	− 0.0893814i
$708$	0	0
$709$	−543.871	−0.767097	−0.383548	−	0.923521i	$-0.625298\pi$
$709$	−543.871	−0.767097	−0.383548	+	0.923521i	$0.625298\pi$
$710$	37.3058i	0.0525433i
$711$	0	0
$712$	−279.852	−0.393051
$713$	479.044i	0.671872i
$714$	0	0
$715$	−2.98354	−0.00417279
$716$	− 2647.24i	− 3.69726i
$717$	0	0
$718$	−214.330	−0.298510
$719$	209.447i	0.291303i	0.989336	+	0.145652i	$0.0465279\pi$
$719$	209.447i	0.291303i	−0.989336	+	0.145652i	$0.953472\pi$
$720$	0	0
$721$	984.544	1.36553
$722$	− 494.662i	− 0.685127i
$723$	0	0
$724$	−1093.22	−1.50997
$725$	795.703i	1.09752i
$726$	0	0
$727$	113.107	0.155581	0.0777905	−	0.996970i	$-0.475213\pi$
$727$	113.107	0.155581	0.0777905	+	0.996970i	$0.475213\pi$
$728$	− 716.418i	− 0.984091i
$729$	0	0
$730$	0.380495	0.000521227 0
$731$	− 30.0481i	− 0.0411055i
$732$	0	0
$733$	−806.994	−1.10095	−0.550473	−	0.834853i	$-0.685553\pi$
$733$	−806.994	−1.10095	−0.550473	+	0.834853i	$0.685553\pi$
$734$	− 1213.06i	− 1.65267i
$735$	0	0
$736$	287.114	0.390100
$737$	123.393i	0.167426i
$738$	0	0
$739$	−105.959	−0.143382	−0.0716909	−	0.997427i	$-0.522840\pi$
$739$	−105.959	−0.143382	−0.0716909	+	0.997427i	$0.522840\pi$
$740$	− 59.9375i	− 0.0809966i
$741$	0	0
$742$	−1195.83	−1.61163
$743$	1136.54i	1.52966i	0.644230	+	0.764832i	$0.277178\pi$
$743$	1136.54i	1.52966i	−0.644230	+	0.764832i	$0.722822\pi$
$744$	0	0
$745$	21.1506	0.0283901
$746$	− 94.3625i	− 0.126491i
$747$	0	0
$748$	1949.89	2.60681
$749$	− 203.240i	− 0.271348i
$750$	0	0
$751$	788.117	1.04942	0.524712	−	0.851280i	$-0.324173\pi$
$751$	788.117	1.04942	0.524712	+	0.851280i	$0.324173\pi$
$752$	− 59.0493i	− 0.0785230i
$753$	0	0
$754$	408.093	0.541238
$755$	− 19.6080i	− 0.0259708i
$756$	0	0
$757$	−118.569	−0.156629	−0.0783147	−	0.996929i	$-0.524954\pi$
$757$	−118.569	−0.156629	−0.0783147	+	0.996929i	$0.524954\pi$
$758$	− 968.600i	− 1.27784i
$759$	0	0
$760$	−29.0255	−0.0381915
$761$	− 984.782i	− 1.29406i	−0.762463	−	0.647031i	$-0.776010\pi$
$761$	− 984.782i	− 1.29406i	0.762463	−	0.647031i	$-0.223990\pi$
$762$	0	0
$763$	648.634	0.850110
$764$	2232.91i	2.92266i
$765$	0	0
$766$	−1390.62	−1.81543
$767$	222.725i	0.290385i
$768$	0	0
$769$	−556.366	−0.723493	−0.361746	−	0.932277i	$-0.617819\pi$
$769$	−556.366	−0.723493	−0.361746	+	0.932277i	$0.617819\pi$
$770$	− 35.4977i	− 0.0461010i
$771$	0	0
$772$	1490.86	1.93116
$773$	− 1295.19i	− 1.67554i	−0.546026	−	0.837768i	$-0.683860\pi$
$773$	− 1295.19i	− 1.67554i	0.546026	−	0.837768i	$-0.316140\pi$
$774$	0	0
$775$	−808.112	−1.04273
$776$	146.173i	0.188367i
$777$	0	0
$778$	1432.14	1.84080
$779$	692.450i	0.888896i
$780$	0	0
$781$	−619.757	−0.793543
$782$	1701.26i	2.17553i
$783$	0	0
$784$	2318.64	2.95745
$785$	− 17.1084i	− 0.0217942i
$786$	0	0
$787$	1549.63	1.96903	0.984517	−	0.175289i	$-0.0560859\pi$
$787$	1549.63	1.96903	0.984517	+	0.175289i	$0.0560859\pi$
$788$	− 3018.86i	− 3.83104i
$789$	0	0
$790$	33.3571	0.0422242
$791$	− 1686.66i	− 2.13231i
$792$	0	0
$793$	−277.437	−0.349858
$794$	1791.70i	2.25655i
$795$	0	0
$796$	−820.471	−1.03074
$797$	− 925.776i	− 1.16158i	−0.814055	−	0.580788i	$-0.802745\pi$
$797$	− 925.776i	− 1.16158i	0.814055	−	0.580788i	$-0.197255\pi$
$798$	0	0
$799$	79.5748	0.0995930
$800$	484.339i	0.605424i
$801$	0	0
$802$	912.233	1.13745
$803$	6.32113i	0.00787190i
$804$	0	0
$805$	21.1640	0.0262907
$806$	414.458i	0.514216i
$807$	0	0
$808$	86.1883	0.106669
$809$	811.708i	1.00335i	0.865057	+	0.501674i	$0.167282\pi$
$809$	811.708i	1.00335i	−0.865057	+	0.501674i	$0.832718\pi$
$810$	0	0
$811$	986.434	1.21632	0.608159	−	0.793815i	$-0.291908\pi$
$811$	986.434	1.21632	0.608159	+	0.793815i	$0.291908\pi$
$812$	3317.91i	4.08610i
$813$	0	0
$814$	1457.16	1.79012
$815$	10.9921i	0.0134873i
$816$	0	0
$817$	−13.8478	−0.0169496
$818$	− 2349.91i	− 2.87275i
$819$	0	0
$820$	−47.5097	−0.0579387
$821$	719.802i	0.876738i	0.898795	+	0.438369i	$0.144444\pi$
$821$	719.802i	0.876738i	−0.898795	+	0.438369i	$0.855556\pi$
$822$	0	0
$823$	−1014.90	−1.23317	−0.616584	−	0.787289i	$-0.711484\pi$
$823$	−1014.90	−1.23317	−0.616584	+	0.787289i	$0.711484\pi$
$824$	1342.82i	1.62963i
$825$	0	0
$826$	−2649.95	−3.20817
$827$	1234.39i	1.49261i	0.665601	+	0.746307i	$0.268175\pi$
$827$	1234.39i	1.49261i	−0.665601	+	0.746307i	$0.731825\pi$
$828$	0	0
$829$	1261.86	1.52215	0.761075	−	0.648664i	$-0.224672\pi$
$829$	1261.86	1.52215	0.761075	+	0.648664i	$0.224672\pi$
$830$	6.32020i	0.00761470i
$831$	0	0
$832$	−97.4617	−0.117141
$833$	3124.60i	3.75102i
$834$	0	0
$835$	−8.57519	−0.0102697
$836$	− 898.617i	− 1.07490i
$837$	0	0
$838$	−1603.90	−1.91396
$839$	747.207i	0.890593i	0.895383	+	0.445296i	$0.146902\pi$
$839$	747.207i	0.890593i	−0.895383	+	0.445296i	$0.853098\pi$
$840$	0	0
$841$	−173.165	−0.205904
$842$	− 349.182i	− 0.414705i
$843$	0	0
$844$	1584.48	1.87735
$845$	− 1.53898i	− 0.00182128i
$846$	0	0
$847$	−870.748	−1.02804
$848$	− 668.504i	− 0.788330i
$849$	0	0
$850$	−2869.90	−3.37635
$851$	868.770i	1.02088i
$852$	0	0
$853$	−646.185	−0.757544	−0.378772	−	0.925490i	$-0.623654\pi$
$853$	−646.185	−0.757544	−0.378772	+	0.925490i	$0.623654\pi$
$854$	− 3300.91i	− 3.86523i
$855$	0	0
$856$	277.198	0.323830
$857$	1505.51i	1.75672i	0.477997	+	0.878362i	$0.341363\pi$
$857$	1505.51i	1.75672i	−0.477997	+	0.878362i	$0.658637\pi$
$858$	0	0
$859$	−1669.63	−1.94370	−0.971848	−	0.235610i	$-0.924291\pi$
$859$	−1669.63	−1.94370	−0.971848	+	0.235610i	$0.924291\pi$
$860$	− 0.950113i	− 0.00110478i
$861$	0	0
$862$	1975.09	2.29128
$863$	456.548i	0.529024i	0.964382	+	0.264512i	$0.0852109\pi$
$863$	456.548i	0.529024i	−0.964382	+	0.264512i	$0.914789\pi$
$864$	0	0
$865$	−27.3180	−0.0315815
$866$	− 305.562i	− 0.352843i
$867$	0	0
$868$	−3369.66	−3.88209
$869$	554.159i	0.637697i
$870$	0	0
$871$	−63.6491	−0.0730759
$872$	884.670i	1.01453i
$873$	0	0
$874$	784.035	0.897065
$875$	71.4242i	0.0816277i
$876$	0	0
$877$	1231.17	1.40384	0.701920	−	0.712256i	$-0.252327\pi$
$877$	1231.17	1.40384	0.701920	+	0.712256i	$0.252327\pi$
$878$	1656.60i	1.88679i
$879$	0	0
$880$	19.8443	0.0225504
$881$	− 352.384i	− 0.399982i	−0.979798	−	0.199991i	$-0.935909\pi$
$881$	− 352.384i	− 0.399982i	0.979798	−	0.199991i	$-0.0640912\pi$
$882$	0	0
$883$	−559.440	−0.633568	−0.316784	−	0.948498i	$-0.602603\pi$
$883$	−559.440	−0.633568	−0.316784	+	0.948498i	$0.602603\pi$
$884$	1005.80i	1.13779i
$885$	0	0
$886$	−2673.40	−3.01738
$887$	− 1672.88i	− 1.88600i	−0.332789	−	0.943001i	$-0.607990\pi$
$887$	− 1672.88i	− 1.88600i	0.332789	−	0.943001i	$-0.392010\pi$
$888$	0	0
$889$	1913.79	2.15274
$890$	− 7.15261i	− 0.00803664i
$891$	0	0
$892$	2113.41	2.36929
$893$	− 36.6724i	− 0.0410666i
$894$	0	0
$895$	36.3061	0.0405655
$896$	− 2095.46i	− 2.33869i
$897$	0	0
$898$	−77.6066	−0.0864216
$899$	− 1029.98i	− 1.14570i
$900$	0	0
$901$	900.876	0.999863
$902$	− 1155.02i	− 1.28051i
$903$	0	0
$904$	2300.43	2.54472
$905$	− 14.9932i	− 0.0165670i
$906$	0	0
$907$	−1675.43	−1.84722	−0.923610	−	0.383334i	$-0.874776\pi$
$907$	−1675.43	−1.84722	−0.923610	+	0.383334i	$0.874776\pi$
$908$	− 572.067i	− 0.630029i
$909$	0	0
$910$	18.3106	0.0201215
$911$	− 431.221i	− 0.473349i	−0.971589	−	0.236675i	$-0.923942\pi$
$911$	− 431.221i	− 0.473349i	0.971589	−	0.236675i	$-0.0760575\pi$
$912$	0	0
$913$	−104.997	−0.115002
$914$	742.999i	0.812909i
$915$	0	0
$916$	−981.368	−1.07136
$917$	1005.52i	1.09653i
$918$	0	0
$919$	−510.245	−0.555217	−0.277609	−	0.960694i	$-0.589542\pi$
$919$	−510.245	−0.555217	−0.277609	+	0.960694i	$0.589542\pi$
$920$	28.8655i	0.0313756i
$921$	0	0
$922$	−2336.07	−2.53369
$923$	− 319.686i	− 0.346355i
$924$	0	0
$925$	−1465.55	−1.58438
$926$	− 2104.89i	− 2.27310i
$927$	0	0
$928$	−617.316	−0.665211
$929$	− 1555.07i	− 1.67392i	−0.547267	−	0.836958i	$-0.684332\pi$
$929$	− 1555.07i	− 1.67392i	0.547267	−	0.836958i	$-0.315668\pi$
$930$	0	0
$931$	1439.99	1.54671
$932$	− 837.893i	− 0.899027i
$933$	0	0
$934$	−157.252	−0.168364
$935$	26.7422i	0.0286013i
$936$	0	0
$937$	1006.41	1.07408	0.537040	−	0.843557i	$-0.319542\pi$
$937$	1006.41	1.07408	0.537040	+	0.843557i	$0.319542\pi$
$938$	− 757.288i	− 0.807343i
$939$	0	0
$940$	2.51613	0.00267674
$941$	130.681i	0.138874i	0.997586	+	0.0694371i	$0.0221203\pi$
$941$	130.681i	0.138874i	−0.997586	+	0.0694371i	$0.977880\pi$
$942$	0	0
$943$	688.634	0.730259
$944$	− 1481.40i	− 1.56928i
$945$	0	0
$946$	23.0985	0.0244170
$947$	− 157.837i	− 0.166670i	−0.996522	−	0.0833350i	$-0.973443\pi$
$947$	− 157.837i	− 0.166670i	0.996522	−	0.0833350i	$-0.0265572\pi$
$948$	0	0
$949$	−3.26059	−0.00343582
$950$	1322.61i	1.39222i
$951$	0	0
$952$	−6421.43	−6.74520
$953$	15.3048i	0.0160596i	0.999968	+	0.00802979i	$0.00255599\pi$
$953$	15.3048i	0.0160596i	−0.999968	+	0.00802979i	$0.997444\pi$
$954$	0	0
$955$	−30.6238	−0.0320668
$956$	1461.89i	1.52917i
$957$	0	0
$958$	1967.97	2.05425
$959$	− 655.053i	− 0.683058i
$960$	0	0
$961$	85.0444	0.0884957
$962$	751.639i	0.781329i
$963$	0	0
$964$	−825.687	−0.856522
$965$	20.4467i	0.0211883i
$966$	0	0
$967$	980.476	1.01394	0.506968	−	0.861965i	$-0.330766\pi$
$967$	980.476	1.01394	0.506968	+	0.861965i	$0.330766\pi$
$968$	− 1187.61i	− 1.22687i
$969$	0	0
$970$	−3.73596	−0.00385150
$971$	813.968i	0.838278i	0.907922	+	0.419139i	$0.137668\pi$
$971$	813.968i	0.838278i	−0.907922	+	0.419139i	$0.862332\pi$
$972$	0	0
$973$	2433.83	2.50136
$974$	196.166i	0.201403i
$975$	0	0
$976$	1845.31	1.89069
$977$	− 1812.54i	− 1.85521i	−0.373561	−	0.927606i	$-0.621863\pi$
$977$	− 1812.54i	− 1.85521i	0.373561	−	0.927606i	$-0.378137\pi$
$978$	0	0
$979$	118.826	0.121375
$980$	98.7988i	0.100815i
$981$	0	0
$982$	−300.197	−0.305700
$983$	− 384.892i	− 0.391548i	−0.980649	−	0.195774i	$-0.937278\pi$
$983$	− 384.892i	− 0.391548i	0.980649	−	0.195774i	$-0.0627219\pi$
$984$	0	0
$985$	41.4028	0.0420333
$986$	− 3657.84i	− 3.70978i
$987$	0	0
$988$	463.529	0.469159
$989$	13.7715i	0.0139247i
$990$	0	0
$991$	628.480	0.634188	0.317094	−	0.948394i	$-0.397293\pi$
$991$	628.480	0.634188	0.317094	+	0.948394i	$0.397293\pi$
$992$	− 626.943i	− 0.631999i
$993$	0	0
$994$	3803.58	3.82654
$995$	− 11.2525i	− 0.0113091i
$996$	0	0
$997$	−1573.51	−1.57825	−0.789123	−	0.614236i	$-0.789464\pi$
$997$	−1573.51	−1.57825	−0.789123	+	0.614236i	$0.789464\pi$
$998$	1949.55i	1.95346i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	117.3.c.a.53.8	yes	8
3.2	odd	2	inner	117.3.c.a.53.1	✓	8
4.3	odd	2		1872.3.f.d.1457.4		8
12.11	even	2		1872.3.f.d.1457.5		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
117.3.c.a.53.1	✓	8	3.2	odd	2	inner
117.3.c.a.53.8	yes	8	1.1	even	1	trivial
1872.3.f.d.1457.4		8	4.3	odd	2
1872.3.f.d.1457.5		8	12.11	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q+3.55413i q^{2} -8.63185 q^{4} -0.118383i q^{5} -12.0700 q^{7} -16.4622i q^{8} +0.420750 q^{10} +6.98988i q^{11} -3.60555 q^{13} -42.8983i q^{14} +23.9815 q^{16} +32.3174i q^{17} +14.8936 q^{19} +1.02187i q^{20} -24.8430 q^{22} -14.8116i q^{23} +24.9860 q^{25} -12.8146i q^{26} +104.186 q^{28} +31.8460i q^{29} -32.3426 q^{31} +19.3844i q^{32} -114.860 q^{34} +1.42889i q^{35} -58.6549 q^{37} +52.9340i q^{38} -1.94885 q^{40} +46.4930i q^{41} -0.929781 q^{43} -60.3356i q^{44} +52.6422 q^{46} -2.46229i q^{47} +96.6845 q^{49} +88.8035i q^{50} +31.1226 q^{52} -27.8759i q^{53} +0.827486 q^{55} +198.699i q^{56} -113.185 q^{58} -61.7728i q^{59} +76.9473 q^{61} -114.950i q^{62} +27.0310 q^{64} +0.426837i q^{65} +17.6531 q^{67} -278.959i q^{68} -5.07845 q^{70} +88.6649i q^{71} +0.904326 q^{73} -208.467i q^{74} -128.560 q^{76} -84.3677i q^{77} +79.2802 q^{79} -2.83901i q^{80} -165.242 q^{82} +15.0213i q^{83} +3.82585 q^{85} -3.30456i q^{86} +115.069 q^{88} -16.9997i q^{89} +43.5189 q^{91} +127.851i q^{92} +8.75130 q^{94} -1.76316i q^{95} -8.87928 q^{97} +343.630i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 8 q^{4} - 16 q^{7} + 24 q^{16} + 64 q^{19} - 80 q^{22} - 24 q^{25} + 184 q^{28} - 40 q^{31} - 272 q^{34} - 104 q^{37} + 32 q^{40} + 128 q^{43} + 232 q^{46} + 136 q^{49} + 104 q^{52} - 224 q^{55} - 88 q^{58}+ \cdots + 328 q^{97}+O(q^{100}) \) 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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