LMFDB - Embedded newform 675.3.c.r.26.6

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [675,3,Mod(26,675)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("675.26"); S:= CuspForms(chi, 3); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(675, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0])) N = Newforms(chi, 3, names="a")

Level:	$ N $	$=$	$ 675 = 3^{3} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	675.c (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [6,0,0,-14,0,0,-16] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$18.3924178443$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.0.60217600.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} + 16x^{4} + 64x^{2} + 4 $ x^6 + 16x^4 + 64x^2 + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2\cdot 3^{4} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			26.6
Root			$2.69399i$ of defining polynomial
Character	$\chi$	$=$	675.26
Dual form			675.3.c.r.26.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+3.69399i q^{2} -9.64560 q^{4} -9.20921 q^{7} -20.8548i q^{8} -15.5488i q^{11} +20.6004 q^{13} -34.0188i q^{14} +38.4552 q^{16} +22.4668i q^{17} +2.33644 q^{19} +57.4372 q^{22} -19.3092i q^{23} +76.0976i q^{26} +88.8283 q^{28} +2.92117i q^{29} -28.1912 q^{31} +58.6339i q^{32} -82.9923 q^{34} +65.4099 q^{37} +8.63079i q^{38} -7.48875i q^{41} +11.9915 q^{43} +149.978i q^{44} +71.3279 q^{46} -50.5075i q^{47} +35.8096 q^{49} -198.703 q^{52} +80.8323i q^{53} +192.056i q^{56} -10.7908 q^{58} -9.46697i q^{59} +52.7557 q^{61} -104.138i q^{62} -62.7728 q^{64} +65.2195 q^{67} -216.706i q^{68} -0.942951i q^{71} -12.8471 q^{73} +241.624i q^{74} -22.5363 q^{76} +143.192i q^{77} +68.4459 q^{79} +27.6634 q^{82} -38.9905i q^{83} +44.2965i q^{86} -324.267 q^{88} -63.7095i q^{89} -189.713 q^{91} +186.248i q^{92} +186.575 q^{94} +186.330 q^{97} +132.280i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 14 q^{4} - 16 q^{7} + 22 q^{13} + 46 q^{16} - 16 q^{19} + 86 q^{22} + 212 q^{28} - 56 q^{31} - 80 q^{34} + 150 q^{37} - 92 q^{43} + 234 q^{46} + 74 q^{49} - 354 q^{52} - 104 q^{58} - 46 q^{61} - 342 q^{64}+ \cdots + 162 q^{97}+O(q^{100}) $ 6 * q - 14 * q^4 - 16 * q^7 + 22 * q^13 + 46 * q^16 - 16 * q^19 + 86 * q^22 + 212 * q^28 - 56 * q^31 - 80 * q^34 + 150 * q^37 - 92 * q^43 + 234 * q^46 + 74 * q^49 - 354 * q^52 - 104 * q^58 - 46 * q^61 - 342 * q^64 + 8 * q^67 + 424 * q^73 - 156 * q^76 + 316 * q^79 + 196 * q^82 - 858 * q^88 + 44 * q^91 + 526 * q^94 + 162 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/675\mathbb{Z}\right)^\times$.

$n$	$326$	$352$
$\chi(n)$	$-1$	$1$

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 $

$2$	3.69399i	1.84700i	0.383602	+	0.923499i	$0.374684\pi$
$2$	3.69399i	1.84700i	−0.383602	+	0.923499i	$0.625316\pi$
$3$	0	0
$3$	0	0
$4$	−9.64560	−2.41140
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−9.20921	−1.31560	−0.657801	−	0.753192i	$-0.728513\pi$
$7$	−9.20921	−1.31560	−0.657801	+	0.753192i	$0.728513\pi$
$8$	− 20.8548i	− 2.60685i
$9$	0	0
$10$	0	0
$11$	− 15.5488i	− 1.41353i	−0.707450	−	0.706764i	$-0.750154\pi$
$11$	− 15.5488i	− 1.41353i	0.707450	−	0.706764i	$-0.249846\pi$
$12$	0	0
$13$	20.6004	1.58464	0.792321	−	0.610104i	$-0.208872\pi$
$13$	20.6004	1.58464	0.792321	+	0.610104i	$0.208872\pi$
$14$	− 34.0188i	− 2.42991i
$15$	0	0
$16$	38.4552	2.40345
$17$	22.4668i	1.32158i	0.750572	+	0.660789i	$0.229778\pi$
$17$	22.4668i	1.32158i	−0.750572	+	0.660789i	$0.770222\pi$
$18$	0	0
$19$	2.33644	0.122970	0.0614852	−	0.998108i	$-0.480416\pi$
$19$	2.33644	0.122970	0.0614852	+	0.998108i	$0.480416\pi$
$20$	0	0
$21$	0	0
$22$	57.4372	2.61078
$23$	− 19.3092i	− 0.839529i	−0.907633	−	0.419764i	$-0.862113\pi$
$23$	− 19.3092i	− 0.839529i	0.907633	−	0.419764i	$-0.137887\pi$
$24$	0	0
$25$	0	0
$26$	76.0976i	2.92683i
$27$	0	0
$28$	88.8283	3.17244
$29$	2.92117i	0.100730i	0.998731	+	0.0503650i	$0.0160385\pi$
$29$	2.92117i	0.100730i	−0.998731	+	0.0503650i	$0.983962\pi$
$30$	0	0
$31$	−28.1912	−0.909395	−0.454698	−	0.890646i	$-0.650253\pi$
$31$	−28.1912	−0.909395	−0.454698	+	0.890646i	$0.650253\pi$
$32$	58.6339i	1.83231i
$33$	0	0
$34$	−82.9923	−2.44095
$35$	0	0
$36$	0	0
$37$	65.4099	1.76784	0.883918	−	0.467642i	$-0.154896\pi$
$37$	65.4099	1.76784	0.883918	+	0.467642i	$0.154896\pi$
$38$	8.63079i	0.227126i
$39$	0	0
$40$	0	0
$41$	− 7.48875i	− 0.182652i	−0.995821	−	0.0913262i	$-0.970889\pi$
$41$	− 7.48875i	− 0.182652i	0.995821	−	0.0913262i	$-0.0291106\pi$
$42$	0	0
$43$	11.9915	0.278872	0.139436	−	0.990231i	$-0.455471\pi$
$43$	11.9915	0.278872	0.139436	+	0.990231i	$0.455471\pi$
$44$	149.978i	3.40858i
$45$	0	0
$46$	71.3279	1.55061
$47$	− 50.5075i	− 1.07463i	−0.843382	−	0.537314i	$-0.819439\pi$
$47$	− 50.5075i	− 1.07463i	0.843382	−	0.537314i	$-0.180561\pi$
$48$	0	0
$49$	35.8096	0.730807
$50$	0	0
$51$	0	0
$52$	−198.703	−3.82121
$53$	80.8323i	1.52514i	0.646907	+	0.762569i	$0.276062\pi$
$53$	80.8323i	1.52514i	−0.646907	+	0.762569i	$0.723938\pi$
$54$	0	0
$55$	0	0
$56$	192.056i	3.42958i
$57$	0	0
$58$	−10.7908	−0.186048
$59$	− 9.46697i	− 0.160457i	−0.996776	−	0.0802286i	$-0.974435\pi$
$59$	− 9.46697i	− 0.160457i	0.996776	−	0.0802286i	$-0.0255650\pi$
$60$	0	0
$61$	52.7557	0.864847	0.432424	−	0.901671i	$-0.357659\pi$
$61$	52.7557	0.864847	0.432424	+	0.901671i	$0.357659\pi$
$62$	− 104.138i	− 1.67965i
$63$	0	0
$64$	−62.7728	−0.980825
$65$	0	0
$66$	0	0
$67$	65.2195	0.973425	0.486713	−	0.873562i	$-0.338196\pi$
$67$	65.2195	0.973425	0.486713	+	0.873562i	$0.338196\pi$
$68$	− 216.706i	− 3.18685i
$69$	0	0
$70$	0	0
$71$	− 0.942951i	− 0.0132810i	−0.999978	−	0.00664050i	$-0.997886\pi$
$71$	− 0.942951i	− 0.0132810i	0.999978	−	0.00664050i	$-0.00211375\pi$
$72$	0	0
$73$	−12.8471	−0.175988	−0.0879940	−	0.996121i	$-0.528046\pi$
$73$	−12.8471	−0.175988	−0.0879940	+	0.996121i	$0.528046\pi$
$74$	241.624i	3.26519i
$75$	0	0
$76$	−22.5363	−0.296531
$77$	143.192i	1.85964i
$78$	0	0
$79$	68.4459	0.866403	0.433202	−	0.901297i	$-0.357384\pi$
$79$	68.4459	0.866403	0.433202	+	0.901297i	$0.357384\pi$
$80$	0	0
$81$	0	0
$82$	27.6634	0.337359
$83$	− 38.9905i	− 0.469765i	−0.972024	−	0.234883i	$-0.924529\pi$
$83$	− 38.9905i	− 0.469765i	0.972024	−	0.234883i	$-0.0754706\pi$
$84$	0	0
$85$	0	0
$86$	44.2965i	0.515076i
$87$	0	0
$88$	−324.267	−3.68486
$89$	− 63.7095i	− 0.715837i	−0.933753	−	0.357918i	$-0.883487\pi$
$89$	− 63.7095i	− 0.715837i	0.933753	−	0.357918i	$-0.116513\pi$
$90$	0	0
$91$	−189.713	−2.08476
$92$	186.248i	2.02444i
$93$	0	0
$94$	186.575	1.98484
$95$	0	0
$96$	0	0
$97$	186.330	1.92093	0.960466	−	0.278398i	$-0.0898034\pi$
$97$	186.330	1.92093	0.960466	+	0.278398i	$0.0898034\pi$
$98$	132.280i	1.34980i
$99$	0	0
$100$	0	0
$101$	− 53.1922i	− 0.526656i	−0.964706	−	0.263328i	$-0.915180\pi$
$101$	− 53.1922i	− 0.526656i	0.964706	−	0.263328i	$-0.0848201\pi$
$102$	0	0
$103$	24.1194	0.234169	0.117084	−	0.993122i	$-0.462645\pi$
$103$	24.1194	0.234169	0.117084	+	0.993122i	$0.462645\pi$
$104$	− 429.616i	− 4.13093i
$105$	0	0
$106$	−298.594	−2.81693
$107$	− 53.4287i	− 0.499334i	−0.968332	−	0.249667i	$-0.919679\pi$
$107$	− 53.4287i	− 0.499334i	0.968332	−	0.249667i	$-0.0803211\pi$
$108$	0	0
$109$	43.8293	0.402104	0.201052	−	0.979581i	$-0.435564\pi$
$109$	43.8293	0.402104	0.201052	+	0.979581i	$0.435564\pi$
$110$	0	0
$111$	0	0
$112$	−354.142	−3.16198
$113$	− 121.798i	− 1.07786i	−0.842351	−	0.538929i	$-0.818829\pi$
$113$	− 121.798i	− 1.07786i	0.842351	−	0.538929i	$-0.181171\pi$
$114$	0	0
$115$	0	0
$116$	− 28.1764i	− 0.242900i
$117$	0	0
$118$	34.9709	0.296364
$119$	− 206.902i	− 1.73867i
$120$	0	0
$121$	−120.765	−0.998060
$122$	194.879i	1.59737i
$123$	0	0
$124$	271.921	2.19291
$125$	0	0
$126$	0	0
$127$	−25.3201	−0.199371	−0.0996854	−	0.995019i	$-0.531784\pi$
$127$	−25.3201	−0.199371	−0.0996854	+	0.995019i	$0.531784\pi$
$128$	2.65328i	0.0207288i
$129$	0	0
$130$	0	0
$131$	− 43.3536i	− 0.330943i	−0.986215	−	0.165472i	$-0.947085\pi$
$131$	− 43.3536i	− 0.330943i	0.986215	−	0.165472i	$-0.0529147\pi$
$132$	0	0
$133$	−21.5167	−0.161780
$134$	240.920i	1.79791i
$135$	0	0
$136$	468.541	3.44516
$137$	− 96.4475i	− 0.703996i	−0.936001	−	0.351998i	$-0.885502\pi$
$137$	− 96.4475i	− 0.703996i	0.936001	−	0.351998i	$-0.114498\pi$
$138$	0	0
$139$	−31.7095	−0.228126	−0.114063	−	0.993474i	$-0.536387\pi$
$139$	−31.7095	−0.228126	−0.114063	+	0.993474i	$0.536387\pi$
$140$	0	0
$141$	0	0
$142$	3.48325	0.0245300
$143$	− 320.311i	− 2.23994i
$144$	0	0
$145$	0	0
$146$	− 47.4572i	− 0.325049i
$147$	0	0
$148$	−630.918	−4.26296
$149$	− 187.489i	− 1.25831i	−0.777278	−	0.629157i	$-0.783400\pi$
$149$	− 187.489i	− 1.25831i	0.777278	−	0.629157i	$-0.216600\pi$
$150$	0	0
$151$	−113.910	−0.754369	−0.377184	−	0.926138i	$-0.623108\pi$
$151$	−113.910	−0.754369	−0.377184	+	0.926138i	$0.623108\pi$
$152$	− 48.7260i	− 0.320566i
$153$	0	0
$154$	−528.951	−3.43475
$155$	0	0
$156$	0	0
$157$	−122.473	−0.780083	−0.390041	−	0.920797i	$-0.627539\pi$
$157$	−122.473	−0.780083	−0.390041	+	0.920797i	$0.627539\pi$
$158$	252.839i	1.60024i
$159$	0	0
$160$	0	0
$161$	177.822i	1.10449i
$162$	0	0
$163$	201.136	1.23396	0.616981	−	0.786978i	$-0.288355\pi$
$163$	201.136	1.23396	0.616981	+	0.786978i	$0.288355\pi$
$164$	72.2335i	0.440448i
$165$	0	0
$166$	144.031	0.867656
$167$	− 32.4447i	− 0.194280i	−0.995271	−	0.0971399i	$-0.969031\pi$
$167$	− 32.4447i	− 0.194280i	0.995271	−	0.0971399i	$-0.0309694\pi$
$168$	0	0
$169$	255.375	1.51109
$170$	0	0
$171$	0	0
$172$	−115.665	−0.672472
$173$	218.485i	1.26292i	0.775409	+	0.631460i	$0.217544\pi$
$173$	218.485i	1.26292i	−0.775409	+	0.631460i	$0.782456\pi$
$174$	0	0
$175$	0	0
$176$	− 597.932i	− 3.39734i
$177$	0	0
$178$	235.342	1.32215
$179$	217.112i	1.21292i	0.795116	+	0.606458i	$0.207410\pi$
$179$	217.112i	1.21292i	−0.795116	+	0.606458i	$0.792590\pi$
$180$	0	0
$181$	1.18977	0.00657333	0.00328667	−	0.999995i	$-0.498954\pi$
$181$	1.18977	0.00657333	0.00328667	+	0.999995i	$0.498954\pi$
$182$	− 700.799i	− 3.85054i
$183$	0	0
$184$	−402.689	−2.18853
$185$	0	0
$186$	0	0
$187$	349.332	1.86809
$188$	487.175i	2.59136i
$189$	0	0
$190$	0	0
$191$	− 343.880i	− 1.80042i	−0.435458	−	0.900209i	$-0.643413\pi$
$191$	− 343.880i	− 1.80042i	0.435458	−	0.900209i	$-0.356587\pi$
$192$	0	0
$193$	124.422	0.644673	0.322337	−	0.946625i	$-0.395532\pi$
$193$	124.422	0.644673	0.322337	+	0.946625i	$0.395532\pi$
$194$	688.303i	3.54796i
$195$	0	0
$196$	−345.405	−1.76227
$197$	311.040i	1.57888i	0.613826	+	0.789442i	$0.289630\pi$
$197$	311.040i	1.57888i	−0.613826	+	0.789442i	$0.710370\pi$
$198$	0	0
$199$	351.943	1.76856	0.884280	−	0.466958i	$-0.154650\pi$
$199$	351.943	1.76856	0.884280	+	0.466958i	$0.154650\pi$
$200$	0	0
$201$	0	0
$202$	196.492	0.972731
$203$	− 26.9017i	− 0.132521i
$204$	0	0
$205$	0	0
$206$	89.0969i	0.432509i
$207$	0	0
$208$	792.190	3.80861
$209$	− 36.3288i	− 0.173822i
$210$	0	0
$211$	301.368	1.42829	0.714143	−	0.700000i	$-0.246817\pi$
$211$	301.368	1.42829	0.714143	+	0.700000i	$0.246817\pi$
$212$	− 779.676i	− 3.67772i
$213$	0	0
$214$	197.365	0.922268
$215$	0	0
$216$	0	0
$217$	259.619	1.19640
$218$	161.905i	0.742685i
$219$	0	0
$220$	0	0
$221$	462.824i	2.09423i
$222$	0	0
$223$	145.658	0.653177	0.326588	−	0.945167i	$-0.394101\pi$
$223$	145.658	0.653177	0.326588	+	0.945167i	$0.394101\pi$
$224$	− 539.972i	− 2.41059i
$225$	0	0
$226$	449.921	1.99080
$227$	309.131i	1.36181i	0.732372	+	0.680904i	$0.238413\pi$
$227$	309.131i	1.36181i	−0.732372	+	0.680904i	$0.761587\pi$
$228$	0	0
$229$	−199.658	−0.871867	−0.435934	−	0.899979i	$-0.643582\pi$
$229$	−199.658	−0.871867	−0.435934	+	0.899979i	$0.643582\pi$
$230$	0	0
$231$	0	0
$232$	60.9205	0.262588
$233$	− 186.675i	− 0.801180i	−0.916257	−	0.400590i	$-0.868805\pi$
$233$	− 186.675i	− 0.801180i	0.916257	−	0.400590i	$-0.131195\pi$
$234$	0	0
$235$	0	0
$236$	91.3146i	0.386926i
$237$	0	0
$238$	764.294	3.21132
$239$	− 37.6186i	− 0.157400i	−0.996898	−	0.0787000i	$-0.974923\pi$
$239$	− 37.6186i	− 0.157400i	0.996898	−	0.0787000i	$-0.0250769\pi$
$240$	0	0
$241$	−84.0894	−0.348919	−0.174459	−	0.984664i	$-0.555818\pi$
$241$	−84.0894	−0.348919	−0.174459	+	0.984664i	$0.555818\pi$
$242$	− 446.106i	− 1.84341i
$243$	0	0
$244$	−508.860	−2.08549
$245$	0	0
$246$	0	0
$247$	48.1314	0.194864
$248$	587.923i	2.37066i
$249$	0	0
$250$	0	0
$251$	− 389.119i	− 1.55028i	−0.631792	−	0.775138i	$-0.717680\pi$
$251$	− 389.119i	− 1.55028i	0.631792	−	0.775138i	$-0.282320\pi$
$252$	0	0
$253$	−300.234	−1.18670
$254$	− 93.5323i	− 0.368237i
$255$	0	0
$256$	−260.893	−1.01911
$257$	168.018i	0.653768i	0.945064	+	0.326884i	$0.105999\pi$
$257$	168.018i	0.653768i	−0.945064	+	0.326884i	$0.894001\pi$
$258$	0	0
$259$	−602.374	−2.32577
$260$	0	0
$261$	0	0
$262$	160.148	0.611252
$263$	345.794i	1.31480i	0.753540	+	0.657402i	$0.228345\pi$
$263$	345.794i	1.31480i	−0.753540	+	0.657402i	$0.771655\pi$
$264$	0	0
$265$	0	0
$266$	− 79.4827i	− 0.298807i
$267$	0	0
$268$	−629.081	−2.34732
$269$	− 502.182i	− 1.86685i	−0.358777	−	0.933423i	$-0.616806\pi$
$269$	− 502.182i	− 1.86685i	0.358777	−	0.933423i	$-0.383194\pi$
$270$	0	0
$271$	253.603	0.935806	0.467903	−	0.883780i	$-0.345010\pi$
$271$	253.603	0.935806	0.467903	+	0.883780i	$0.345010\pi$
$272$	863.965i	3.17634i
$273$	0	0
$274$	356.276	1.30028
$275$	0	0
$276$	0	0
$277$	−415.895	−1.50143	−0.750714	−	0.660628i	$-0.770290\pi$
$277$	−415.895	−1.50143	−0.750714	+	0.660628i	$0.770290\pi$
$278$	− 117.135i	− 0.421347i
$279$	0	0
$280$	0	0
$281$	− 410.907i	− 1.46230i	−0.682215	−	0.731151i	$-0.738983\pi$
$281$	− 410.907i	− 1.46230i	0.682215	−	0.731151i	$-0.261017\pi$
$282$	0	0
$283$	−499.088	−1.76356	−0.881780	−	0.471660i	$-0.843655\pi$
$283$	−499.088	−1.76356	−0.881780	+	0.471660i	$0.843655\pi$
$284$	9.09532i	0.0320258i
$285$	0	0
$286$	1183.23	4.13716
$287$	68.9655i	0.240298i
$288$	0	0
$289$	−215.758	−0.746567
$290$	0	0
$291$	0	0
$292$	123.918	0.424377
$293$	− 466.667i	− 1.59272i	−0.604822	−	0.796361i	$-0.706756\pi$
$293$	− 466.667i	− 1.59272i	0.604822	−	0.796361i	$-0.293244\pi$
$294$	0	0
$295$	0	0
$296$	− 1364.11i	− 4.60848i
$297$	0	0
$298$	692.582	2.32410
$299$	− 397.776i	− 1.33035i
$300$	0	0
$301$	−110.432	−0.366885
$302$	− 420.782i	− 1.39332i
$303$	0	0
$304$	89.8481	0.295553
$305$	0	0
$306$	0	0
$307$	−183.976	−0.599271	−0.299636	−	0.954054i	$-0.596865\pi$
$307$	−183.976	−0.599271	−0.299636	+	0.954054i	$0.596865\pi$
$308$	− 1381.17i	− 4.48433i
$309$	0	0
$310$	0	0
$311$	256.734i	0.825510i	0.910842	+	0.412755i	$0.135433\pi$
$311$	256.734i	0.825510i	−0.910842	+	0.412755i	$0.864567\pi$
$312$	0	0
$313$	257.923	0.824034	0.412017	−	0.911176i	$-0.364824\pi$
$313$	257.923	0.824034	0.412017	+	0.911176i	$0.364824\pi$
$314$	− 452.415i	− 1.44081i
$315$	0	0
$316$	−660.201	−2.08924
$317$	− 173.381i	− 0.546942i	−0.961880	−	0.273471i	$-0.911828\pi$
$317$	− 173.381i	− 0.546942i	0.961880	−	0.273471i	$-0.0881718\pi$
$318$	0	0
$319$	45.4207	0.142385
$320$	0	0
$321$	0	0
$322$	−656.874	−2.03998
$323$	52.4923i	0.162515i
$324$	0	0
$325$	0	0
$326$	742.995i	2.27913i
$327$	0	0
$328$	−156.176	−0.476148
$329$	465.134i	1.41378i
$330$	0	0
$331$	−25.2825	−0.0763823	−0.0381912	−	0.999270i	$-0.512160\pi$
$331$	−25.2825	−0.0763823	−0.0381912	+	0.999270i	$0.512160\pi$
$332$	376.087i	1.13279i
$333$	0	0
$334$	119.851	0.358834
$335$	0	0
$336$	0	0
$337$	−384.796	−1.14183	−0.570914	−	0.821010i	$-0.693411\pi$
$337$	−384.796	−1.14183	−0.570914	+	0.821010i	$0.693411\pi$
$338$	943.353i	2.79098i
$339$	0	0
$340$	0	0
$341$	438.340i	1.28545i
$342$	0	0
$343$	121.473	0.354150
$344$	− 250.080i	− 0.726978i
$345$	0	0
$346$	−807.083	−2.33261
$347$	543.336i	1.56581i	0.622142	+	0.782904i	$0.286263\pi$
$347$	543.336i	1.56581i	−0.622142	+	0.782904i	$0.713737\pi$
$348$	0	0
$349$	102.381	0.293356	0.146678	−	0.989184i	$-0.453142\pi$
$349$	102.381	0.293356	0.146678	+	0.989184i	$0.453142\pi$
$350$	0	0
$351$	0	0
$352$	911.688	2.59002
$353$	− 609.853i	− 1.72763i	−0.503810	−	0.863814i	$-0.668069\pi$
$353$	− 609.853i	− 1.72763i	0.503810	−	0.863814i	$-0.331931\pi$
$354$	0	0
$355$	0	0
$356$	614.516i	1.72617i
$357$	0	0
$358$	−802.010	−2.24025
$359$	− 322.621i	− 0.898665i	−0.893364	−	0.449333i	$-0.851662\pi$
$359$	− 322.621i	− 0.898665i	0.893364	−	0.449333i	$-0.148338\pi$
$360$	0	0
$361$	−355.541	−0.984878
$362$	4.39502i	0.0121409i
$363$	0	0
$364$	1829.90	5.02719
$365$	0	0
$366$	0	0
$367$	−509.283	−1.38769	−0.693846	−	0.720123i	$-0.744085\pi$
$367$	−509.283	−1.38769	−0.693846	+	0.720123i	$0.744085\pi$
$368$	− 742.537i	− 2.01776i
$369$	0	0
$370$	0	0
$371$	− 744.402i	− 2.00647i
$372$	0	0
$373$	291.668	0.781952	0.390976	−	0.920401i	$-0.372138\pi$
$373$	291.668	0.781952	0.390976	+	0.920401i	$0.372138\pi$
$374$	1290.43i	3.45035i
$375$	0	0
$376$	−1053.32	−2.80140
$377$	60.1772i	0.159621i
$378$	0	0
$379$	132.116	0.348591	0.174295	−	0.984693i	$-0.444235\pi$
$379$	132.116	0.348591	0.174295	+	0.984693i	$0.444235\pi$
$380$	0	0
$381$	0	0
$382$	1270.29	3.32537
$383$	− 1.23436i	− 0.00322286i	−0.999999	−	0.00161143i	$-0.999487\pi$
$383$	− 1.23436i	− 0.00322286i	0.999999	−	0.00161143i	$-0.000512934\pi$
$384$	0	0
$385$	0	0
$386$	459.614i	1.19071i
$387$	0	0
$388$	−1797.27	−4.63213
$389$	− 97.6209i	− 0.250953i	−0.992097	−	0.125477i	$-0.959954\pi$
$389$	− 97.6209i	− 0.250953i	0.992097	−	0.125477i	$-0.0400460\pi$
$390$	0	0
$391$	433.815	1.10950
$392$	− 746.802i	− 1.90511i
$393$	0	0
$394$	−1148.98	−2.91619
$395$	0	0
$396$	0	0
$397$	259.982	0.654866	0.327433	−	0.944874i	$-0.393816\pi$
$397$	259.982	0.654866	0.327433	+	0.944874i	$0.393816\pi$
$398$	1300.08i	3.26652i
$399$	0	0
$400$	0	0
$401$	− 158.010i	− 0.394040i	−0.980400	−	0.197020i	$-0.936874\pi$
$401$	− 158.010i	− 0.394040i	0.980400	−	0.197020i	$-0.0631263\pi$
$402$	0	0
$403$	−580.750	−1.44107
$404$	513.071i	1.26998i
$405$	0	0
$406$	99.3747	0.244765
$407$	− 1017.05i	− 2.49888i
$408$	0	0
$409$	−300.817	−0.735495	−0.367747	−	0.929926i	$-0.619871\pi$
$409$	−300.817	−0.735495	−0.367747	+	0.929926i	$0.619871\pi$
$410$	0	0
$411$	0	0
$412$	−232.646	−0.564675
$413$	87.1833i	0.211098i
$414$	0	0
$415$	0	0
$416$	1207.88i	2.90356i
$417$	0	0
$418$	134.198	0.321049
$419$	162.498i	0.387823i	0.981019	+	0.193911i	$0.0621175\pi$
$419$	162.498i	0.387823i	−0.981019	+	0.193911i	$0.937883\pi$
$420$	0	0
$421$	201.966	0.479730	0.239865	−	0.970806i	$-0.422897\pi$
$421$	201.966	0.479730	0.239865	+	0.970806i	$0.422897\pi$
$422$	1113.25i	2.63804i
$423$	0	0
$424$	1685.74	3.97581
$425$	0	0
$426$	0	0
$427$	−485.838	−1.13779
$428$	515.352i	1.20409i
$429$	0	0
$430$	0	0
$431$	335.503i	0.778429i	0.921147	+	0.389215i	$0.127254\pi$
$431$	335.503i	0.778429i	−0.921147	+	0.389215i	$0.872746\pi$
$432$	0	0
$433$	−104.346	−0.240985	−0.120492	−	0.992714i	$-0.538447\pi$
$433$	−104.346	−0.240985	−0.120492	+	0.992714i	$0.538447\pi$
$434$	959.032i	2.20975i
$435$	0	0
$436$	−422.760	−0.969633
$437$	− 45.1146i	− 0.103237i
$438$	0	0
$439$	256.928	0.585256	0.292628	−	0.956226i	$-0.405470\pi$
$439$	256.928	0.585256	0.292628	+	0.956226i	$0.405470\pi$
$440$	0	0
$441$	0	0
$442$	−1709.67	−3.86803
$443$	764.139i	1.72492i	0.506127	+	0.862459i	$0.331077\pi$
$443$	764.139i	1.72492i	−0.506127	+	0.862459i	$0.668923\pi$
$444$	0	0
$445$	0	0
$446$	538.062i	1.20642i
$447$	0	0
$448$	578.088	1.29038
$449$	543.053i	1.20947i	0.796426	+	0.604736i	$0.206722\pi$
$449$	543.053i	1.20947i	−0.796426	+	0.604736i	$0.793278\pi$
$450$	0	0
$451$	−116.441	−0.258184
$452$	1174.82i	2.59915i
$453$	0	0
$454$	−1141.93	−2.51526
$455$	0	0
$456$	0	0
$457$	−447.094	−0.978323	−0.489161	−	0.872193i	$-0.662697\pi$
$457$	−447.094	−0.978323	−0.489161	+	0.872193i	$0.662697\pi$
$458$	− 737.534i	− 1.61034i
$459$	0	0
$460$	0	0
$461$	− 221.719i	− 0.480953i	−0.970655	−	0.240477i	$-0.922696\pi$
$461$	− 221.719i	− 0.480953i	0.970655	−	0.240477i	$-0.0773037\pi$
$462$	0	0
$463$	−612.697	−1.32332	−0.661659	−	0.749805i	$-0.730147\pi$
$463$	−612.697	−1.32332	−0.661659	+	0.749805i	$0.730147\pi$
$464$	112.334i	0.242099i
$465$	0	0
$466$	689.576	1.47978
$467$	132.891i	0.284564i	0.989826	+	0.142282i	$0.0454440\pi$
$467$	132.891i	0.284564i	−0.989826	+	0.142282i	$0.954556\pi$
$468$	0	0
$469$	−600.620	−1.28064
$470$	0	0
$471$	0	0
$472$	−197.432	−0.418288
$473$	− 186.453i	− 0.394193i
$474$	0	0
$475$	0	0
$476$	1995.69i	4.19263i
$477$	0	0
$478$	138.963	0.290717
$479$	− 525.127i	− 1.09630i	−0.836381	−	0.548149i	$-0.815333\pi$
$479$	− 525.127i	− 1.09630i	0.836381	−	0.548149i	$-0.184667\pi$
$480$	0	0
$481$	1347.47	2.80139
$482$	− 310.626i	− 0.644452i
$483$	0	0
$484$	1164.85	2.40672
$485$	0	0
$486$	0	0
$487$	−100.812	−0.207007	−0.103503	−	0.994629i	$-0.533005\pi$
$487$	−100.812	−0.207007	−0.103503	+	0.994629i	$0.533005\pi$
$488$	− 1100.21i	− 2.25453i
$489$	0	0
$490$	0	0
$491$	321.463i	0.654711i	0.944901	+	0.327356i	$0.106158\pi$
$491$	321.463i	0.654711i	−0.944901	+	0.327356i	$0.893842\pi$
$492$	0	0
$493$	−65.6294	−0.133123
$494$	177.797i	0.359914i
$495$	0	0
$496$	−1084.10	−2.18568
$497$	8.68383i	0.0174725i
$498$	0	0
$499$	−320.785	−0.642856	−0.321428	−	0.946934i	$-0.604163\pi$
$499$	−320.785	−0.642856	−0.321428	+	0.946934i	$0.604163\pi$
$500$	0	0
$501$	0	0
$502$	1437.40	2.86336
$503$	31.7672i	0.0631555i	0.999501	+	0.0315778i	$0.0100532\pi$
$503$	31.7672i	0.0631555i	−0.999501	+	0.0315778i	$0.989947\pi$
$504$	0	0
$505$	0	0
$506$	− 1109.06i	− 2.19183i
$507$	0	0
$508$	244.227	0.480763
$509$	− 107.552i	− 0.211300i	−0.994403	−	0.105650i	$-0.966308\pi$
$509$	− 107.552i	− 0.211300i	0.994403	−	0.105650i	$-0.0336924\pi$
$510$	0	0
$511$	118.312	0.231530
$512$	− 953.122i	− 1.86157i
$513$	0	0
$514$	−620.659	−1.20751
$515$	0	0
$516$	0	0
$517$	−785.332	−1.51902
$518$	− 2225.17i	− 4.29569i
$519$	0	0
$520$	0	0
$521$	509.938i	0.978767i	0.872069	+	0.489384i	$0.162778\pi$
$521$	509.938i	0.978767i	−0.872069	+	0.489384i	$0.837222\pi$
$522$	0	0
$523$	10.6268	0.0203188	0.0101594	−	0.999948i	$-0.496766\pi$
$523$	10.6268	0.0203188	0.0101594	+	0.999948i	$0.496766\pi$
$524$	418.171i	0.798037i
$525$	0	0
$526$	−1277.36	−2.42844
$527$	− 633.368i	− 1.20184i
$528$	0	0
$529$	156.156	0.295192
$530$	0	0
$531$	0	0
$532$	207.542	0.390116
$533$	− 154.271i	− 0.289439i
$534$	0	0
$535$	0	0
$536$	− 1360.14i	− 2.53757i
$537$	0	0
$538$	1855.06	3.44806
$539$	− 556.796i	− 1.03302i
$540$	0	0
$541$	407.714	0.753630	0.376815	−	0.926289i	$-0.377019\pi$
$541$	407.714	0.753630	0.376815	+	0.926289i	$0.377019\pi$
$542$	936.810i	1.72843i
$543$	0	0
$544$	−1317.32	−2.42154
$545$	0	0
$546$	0	0
$547$	690.828	1.26294	0.631470	−	0.775400i	$-0.282452\pi$
$547$	690.828	1.26294	0.631470	+	0.775400i	$0.282452\pi$
$548$	930.294i	1.69762i
$549$	0	0
$550$	0	0
$551$	6.82513i	0.0123868i
$552$	0	0
$553$	−630.332	−1.13984
$554$	− 1536.32i	− 2.77313i
$555$	0	0
$556$	305.857	0.550102
$557$	751.961i	1.35002i	0.737809	+	0.675010i	$0.235861\pi$
$557$	751.961i	1.35002i	−0.737809	+	0.675010i	$0.764139\pi$
$558$	0	0
$559$	247.029	0.441913
$560$	0	0
$561$	0	0
$562$	1517.89	2.70087
$563$	262.222i	0.465759i	0.972506	+	0.232879i	$0.0748148\pi$
$563$	262.222i	0.465759i	−0.972506	+	0.232879i	$0.925185\pi$
$564$	0	0
$565$	0	0
$566$	− 1843.63i	− 3.25729i
$567$	0	0
$568$	−19.6651	−0.0346216
$569$	− 131.216i	− 0.230607i	−0.993330	−	0.115304i	$-0.963216\pi$
$569$	− 131.216i	− 0.230607i	0.993330	−	0.115304i	$-0.0367841\pi$
$570$	0	0
$571$	−372.767	−0.652831	−0.326416	−	0.945226i	$-0.605841\pi$
$571$	−372.767	−0.652831	−0.326416	+	0.945226i	$0.605841\pi$
$572$	3089.59i	5.40138i
$573$	0	0
$574$	−254.758	−0.443830
$575$	0	0
$576$	0	0
$577$	125.906	0.218208	0.109104	−	0.994030i	$-0.465202\pi$
$577$	125.906	0.218208	0.109104	+	0.994030i	$0.465202\pi$
$578$	− 797.009i	− 1.37891i
$579$	0	0
$580$	0	0
$581$	359.072i	0.618024i
$582$	0	0
$583$	1256.85	2.15582
$584$	267.924i	0.458774i
$585$	0	0
$586$	1723.87	2.94175
$587$	− 569.861i	− 0.970802i	−0.874292	−	0.485401i	$-0.838674\pi$
$587$	− 569.861i	− 0.970802i	0.874292	−	0.485401i	$-0.161326\pi$
$588$	0	0
$589$	−65.8671	−0.111829
$590$	0	0
$591$	0	0
$592$	2515.35	4.24890
$593$	− 584.546i	− 0.985744i	−0.870102	−	0.492872i	$-0.835947\pi$
$593$	− 584.546i	− 0.985744i	0.870102	−	0.492872i	$-0.164053\pi$
$594$	0	0
$595$	0	0
$596$	1808.44i	3.03430i
$597$	0	0
$598$	1469.38	2.45716
$599$	− 986.412i	− 1.64677i	−0.567487	−	0.823383i	$-0.692084\pi$
$599$	− 986.412i	− 1.64677i	0.567487	−	0.823383i	$-0.307916\pi$
$600$	0	0
$601$	−677.603	−1.12746	−0.563729	−	0.825960i	$-0.690634\pi$
$601$	−677.603	−1.12746	−0.563729	+	0.825960i	$0.690634\pi$
$602$	− 407.936i	− 0.677635i
$603$	0	0
$604$	1098.73	1.81908
$605$	0	0
$606$	0	0
$607$	−897.241	−1.47816	−0.739078	−	0.673620i	$-0.764739\pi$
$607$	−897.241	−1.47816	−0.739078	+	0.673620i	$0.764739\pi$
$608$	136.995i	0.225320i
$609$	0	0
$610$	0	0
$611$	− 1040.47i	− 1.70290i
$612$	0	0
$613$	301.192	0.491341	0.245671	−	0.969353i	$-0.420992\pi$
$613$	301.192	0.491341	0.245671	+	0.969353i	$0.420992\pi$
$614$	− 679.607i	− 1.10685i
$615$	0	0
$616$	2986.25	4.84780
$617$	832.739i	1.34966i	0.737974	+	0.674829i	$0.235783\pi$
$617$	832.739i	1.34966i	−0.737974	+	0.674829i	$0.764217\pi$
$618$	0	0
$619$	−702.912	−1.13556	−0.567781	−	0.823180i	$-0.692198\pi$
$619$	−702.912	−1.13556	−0.567781	+	0.823180i	$0.692198\pi$
$620$	0	0
$621$	0	0
$622$	−948.373	−1.52471
$623$	586.714i	0.941756i
$624$	0	0
$625$	0	0
$626$	952.765i	1.52199i
$627$	0	0
$628$	1181.33	1.88109
$629$	1469.55i	2.33633i
$630$	0	0
$631$	93.9508	0.148892	0.0744459	−	0.997225i	$-0.476281\pi$
$631$	93.9508	0.148892	0.0744459	+	0.997225i	$0.476281\pi$
$632$	− 1427.43i	− 2.25858i
$633$	0	0
$634$	640.467	1.01020
$635$	0	0
$636$	0	0
$637$	737.690	1.15807
$638$	167.784i	0.262984i
$639$	0	0
$640$	0	0
$641$	400.819i	0.625303i	0.949868	+	0.312651i	$0.101217\pi$
$641$	400.819i	0.625303i	−0.949868	+	0.312651i	$0.898783\pi$
$642$	0	0
$643$	529.930	0.824152	0.412076	−	0.911149i	$-0.364804\pi$
$643$	529.930	0.824152	0.412076	+	0.911149i	$0.364804\pi$
$644$	− 1715.20i	− 2.66336i
$645$	0	0
$646$	−193.906	−0.300165
$647$	− 803.762i	− 1.24229i	−0.783695	−	0.621145i	$-0.786668\pi$
$647$	− 803.762i	− 1.24229i	0.783695	−	0.621145i	$-0.213332\pi$
$648$	0	0
$649$	−147.200	−0.226811
$650$	0	0
$651$	0	0
$652$	−1940.08	−2.97558
$653$	− 273.937i	− 0.419505i	−0.977755	−	0.209752i	$-0.932734\pi$
$653$	− 273.937i	− 0.419505i	0.977755	−	0.209752i	$-0.0672658\pi$
$654$	0	0
$655$	0	0
$656$	− 287.981i	− 0.438996i
$657$	0	0
$658$	−1718.20	−2.61125
$659$	988.698i	1.50030i	0.661268	+	0.750150i	$0.270019\pi$
$659$	988.698i	1.50030i	−0.661268	+	0.750150i	$0.729981\pi$
$660$	0	0
$661$	7.79904	0.0117988	0.00589942	−	0.999983i	$-0.498122\pi$
$661$	7.79904	0.0117988	0.00589942	+	0.999983i	$0.498122\pi$
$662$	− 93.3936i	− 0.141078i
$663$	0	0
$664$	−813.140	−1.22461
$665$	0	0
$666$	0	0
$667$	56.4054	0.0845657
$668$	312.949i	0.468486i
$669$	0	0
$670$	0	0
$671$	− 820.288i	− 1.22249i
$672$	0	0
$673$	−802.306	−1.19213	−0.596067	−	0.802935i	$-0.703271\pi$
$673$	−802.306	−1.19213	−0.596067	+	0.802935i	$0.703271\pi$
$674$	− 1421.43i	− 2.10895i
$675$	0	0
$676$	−2463.24	−3.64385
$677$	853.169i	1.26022i	0.776506	+	0.630110i	$0.216990\pi$
$677$	853.169i	1.26022i	−0.776506	+	0.630110i	$0.783010\pi$
$678$	0	0
$679$	−1715.96	−2.52718
$680$	0	0
$681$	0	0
$682$	−1619.23	−2.37423
$683$	964.796i	1.41259i	0.707920	+	0.706293i	$0.249634\pi$
$683$	964.796i	1.41259i	−0.707920	+	0.706293i	$0.750366\pi$
$684$	0	0
$685$	0	0
$686$	448.722i	0.654114i
$687$	0	0
$688$	461.135	0.670255
$689$	1665.17i	2.41680i
$690$	0	0
$691$	−386.152	−0.558831	−0.279415	−	0.960170i	$-0.590141\pi$
$691$	−386.152	−0.558831	−0.279415	+	0.960170i	$0.590141\pi$
$692$	− 2107.42i	− 3.04540i
$693$	0	0
$694$	−2007.08	−2.89204
$695$	0	0
$696$	0	0
$697$	168.248	0.241389
$698$	378.196i	0.541827i
$699$	0	0
$700$	0	0
$701$	− 501.171i	− 0.714938i	−0.933925	−	0.357469i	$-0.883640\pi$
$701$	− 501.171i	− 0.714938i	0.933925	−	0.357469i	$-0.116360\pi$
$702$	0	0
$703$	152.826	0.217391
$704$	976.042i	1.38642i
$705$	0	0
$706$	2252.79	3.19093
$707$	489.858i	0.692869i
$708$	0	0
$709$	711.983	1.00421	0.502104	−	0.864807i	$-0.332560\pi$
$709$	711.983	1.00421	0.502104	+	0.864807i	$0.332560\pi$
$710$	0	0
$711$	0	0
$712$	−1328.65	−1.86608
$713$	544.349i	0.763463i
$714$	0	0
$715$	0	0
$716$	− 2094.17i	− 2.92482i
$717$	0	0
$718$	1191.76	1.65983
$719$	653.723i	0.909212i	0.890693	+	0.454606i	$0.150220\pi$
$719$	653.723i	0.909212i	−0.890693	+	0.454606i	$0.849780\pi$
$720$	0	0
$721$	−222.121	−0.308073
$722$	− 1313.37i	− 1.81907i
$723$	0	0
$724$	−11.4761	−0.0158509
$725$	0	0
$726$	0	0
$727$	649.704	0.893678	0.446839	−	0.894614i	$-0.352550\pi$
$727$	649.704	0.893678	0.446839	+	0.894614i	$0.352550\pi$
$728$	3956.43i	5.43465i
$729$	0	0
$730$	0	0
$731$	269.411i	0.368551i
$732$	0	0
$733$	−54.0661	−0.0737600	−0.0368800	−	0.999320i	$-0.511742\pi$
$733$	−54.0661	−0.0737600	−0.0368800	+	0.999320i	$0.511742\pi$
$734$	− 1881.29i	− 2.56306i
$735$	0	0
$736$	1132.17	1.53828
$737$	− 1014.09i	− 1.37596i
$738$	0	0
$739$	1161.91	1.57227	0.786134	−	0.618057i	$-0.212080\pi$
$739$	1161.91	1.57227	0.786134	+	0.618057i	$0.212080\pi$
$740$	0	0
$741$	0	0
$742$	2749.82	3.70595
$743$	176.127i	0.237048i	0.992951	+	0.118524i	$0.0378163\pi$
$743$	176.127i	0.237048i	−0.992951	+	0.118524i	$0.962184\pi$
$744$	0	0
$745$	0	0
$746$	1077.42i	1.44426i
$747$	0	0
$748$	−3369.52	−4.50470
$749$	492.036i	0.656924i
$750$	0	0
$751$	−1116.91	−1.48724	−0.743618	−	0.668605i	$-0.766892\pi$
$751$	−1116.91	−1.48724	−0.743618	+	0.668605i	$0.766892\pi$
$752$	− 1942.28i	− 2.58281i
$753$	0	0
$754$	−222.294	−0.294820
$755$	0	0
$756$	0	0
$757$	1040.37	1.37433	0.687167	−	0.726500i	$-0.258854\pi$
$757$	1040.37	1.37433	0.687167	+	0.726500i	$0.258854\pi$
$758$	488.035i	0.643846i
$759$	0	0
$760$	0	0
$761$	1350.15i	1.77418i	0.461592	+	0.887092i	$0.347278\pi$
$761$	1350.15i	1.77418i	−0.461592	+	0.887092i	$0.652722\pi$
$762$	0	0
$763$	−403.633	−0.529008
$764$	3316.93i	4.34153i
$765$	0	0
$766$	4.55970	0.00595261
$767$	− 195.023i	− 0.254267i
$768$	0	0
$769$	560.677	0.729099	0.364550	−	0.931184i	$-0.381223\pi$
$769$	560.677	0.729099	0.364550	+	0.931184i	$0.381223\pi$
$770$	0	0
$771$	0	0
$772$	−1200.12	−1.55457
$773$	− 1271.66i	− 1.64510i	−0.568691	−	0.822551i	$-0.692550\pi$
$773$	− 1271.66i	− 1.64510i	0.568691	−	0.822551i	$-0.307450\pi$
$774$	0	0
$775$	0	0
$776$	− 3885.88i	− 5.00758i
$777$	0	0
$778$	360.611	0.463510
$779$	− 17.4970i	− 0.0224608i
$780$	0	0
$781$	−14.6618	−0.0187731
$782$	1602.51i	2.04925i
$783$	0	0
$784$	1377.06	1.75646
$785$	0	0
$786$	0	0
$787$	−673.581	−0.855884	−0.427942	−	0.903806i	$-0.640761\pi$
$787$	−673.581	−0.855884	−0.427942	+	0.903806i	$0.640761\pi$
$788$	− 3000.17i	− 3.80732i
$789$	0	0
$790$	0	0
$791$	1121.66i	1.41803i
$792$	0	0
$793$	1086.79	1.37047
$794$	960.371i	1.20954i
$795$	0	0
$796$	−3394.70	−4.26470
$797$	− 441.048i	− 0.553386i	−0.960958	−	0.276693i	$-0.910762\pi$
$797$	− 441.048i	− 0.553386i	0.960958	−	0.276693i	$-0.0892385\pi$
$798$	0	0
$799$	1134.74	1.42020
$800$	0	0
$801$	0	0
$802$	583.688	0.727790
$803$	199.757i	0.248764i
$804$	0	0
$805$	0	0
$806$	− 2145.29i	− 2.66165i
$807$	0	0
$808$	−1109.31	−1.37291
$809$	− 177.448i	− 0.219342i	−0.993968	−	0.109671i	$-0.965020\pi$
$809$	− 177.448i	− 0.219342i	0.993968	−	0.109671i	$-0.0349797\pi$
$810$	0	0
$811$	−26.5526	−0.0327406	−0.0163703	−	0.999866i	$-0.505211\pi$
$811$	−26.5526	−0.0327406	−0.0163703	+	0.999866i	$0.505211\pi$
$812$	259.483i	0.319560i
$813$	0	0
$814$	3756.96	4.61543
$815$	0	0
$816$	0	0
$817$	28.0174	0.0342930
$818$	− 1111.22i	− 1.35846i
$819$	0	0
$820$	0	0
$821$	561.109i	0.683446i	0.939801	+	0.341723i	$0.111010\pi$
$821$	561.109i	0.683446i	−0.939801	+	0.341723i	$0.888990\pi$
$822$	0	0
$823$	140.819	0.171105	0.0855523	−	0.996334i	$-0.472735\pi$
$823$	140.819	0.171105	0.0855523	+	0.996334i	$0.472735\pi$
$824$	− 503.005i	− 0.610443i
$825$	0	0
$826$	−322.055	−0.389897
$827$	− 445.029i	− 0.538124i	−0.963123	−	0.269062i	$-0.913286\pi$
$827$	− 445.029i	− 0.538124i	0.963123	−	0.269062i	$-0.0867137\pi$
$828$	0	0
$829$	1456.17	1.75653	0.878267	−	0.478170i	$-0.158700\pi$
$829$	1456.17	1.75653	0.878267	+	0.478170i	$0.158700\pi$
$830$	0	0
$831$	0	0
$832$	−1293.14	−1.55426
$833$	804.527i	0.965819i
$834$	0	0
$835$	0	0
$836$	350.413i	0.419154i
$837$	0	0
$838$	−600.266	−0.716308
$839$	205.320i	0.244720i	0.992486	+	0.122360i	$0.0390462\pi$
$839$	205.320i	0.244720i	−0.992486	+	0.122360i	$0.960954\pi$
$840$	0	0
$841$	832.467	0.989853
$842$	746.063i	0.886060i
$843$	0	0
$844$	−2906.88	−3.44417
$845$	0	0
$846$	0	0
$847$	1112.15	1.31305
$848$	3108.42i	3.66559i
$849$	0	0
$850$	0	0
$851$	− 1263.01i	− 1.48415i
$852$	0	0
$853$	−788.618	−0.924523	−0.462261	−	0.886744i	$-0.652962\pi$
$853$	−788.618	−0.924523	−0.462261	+	0.886744i	$0.652962\pi$
$854$	− 1794.68i	− 2.10150i
$855$	0	0
$856$	−1114.25	−1.30169
$857$	− 309.338i	− 0.360954i	−0.983579	−	0.180477i	$-0.942236\pi$
$857$	− 309.338i	− 0.360954i	0.983579	−	0.180477i	$-0.0577641\pi$
$858$	0	0
$859$	1129.78	1.31522	0.657611	−	0.753357i	$-0.271567\pi$
$859$	1129.78	1.31522	0.657611	+	0.753357i	$0.271567\pi$
$860$	0	0
$861$	0	0
$862$	−1239.35	−1.43776
$863$	618.421i	0.716594i	0.933608	+	0.358297i	$0.116642\pi$
$863$	618.421i	0.716594i	−0.933608	+	0.358297i	$0.883358\pi$
$864$	0	0
$865$	0	0
$866$	− 385.455i	− 0.445098i
$867$	0	0
$868$	−2504.18	−2.88500
$869$	− 1064.25i	− 1.22468i
$870$	0	0
$871$	1343.54	1.54253
$872$	− 914.052i	− 1.04822i
$873$	0	0
$874$	166.653	0.190679
$875$	0	0
$876$	0	0
$877$	751.759	0.857194	0.428597	−	0.903496i	$-0.359008\pi$
$877$	751.759	0.857194	0.428597	+	0.903496i	$0.359008\pi$
$878$	949.089i	1.08097i
$879$	0	0
$880$	0	0
$881$	1465.38i	1.66332i	0.555285	+	0.831660i	$0.312609\pi$
$881$	1465.38i	1.66332i	−0.555285	+	0.831660i	$0.687391\pi$
$882$	0	0
$883$	658.936	0.746247	0.373124	−	0.927782i	$-0.378287\pi$
$883$	658.936	0.746247	0.373124	+	0.927782i	$0.378287\pi$
$884$	− 4464.22i	− 5.05002i
$885$	0	0
$886$	−2822.72	−3.18592
$887$	− 962.298i	− 1.08489i	−0.840091	−	0.542445i	$-0.817499\pi$
$887$	− 962.298i	− 1.08489i	0.840091	−	0.542445i	$-0.182501\pi$
$888$	0	0
$889$	233.178	0.262293
$890$	0	0
$891$	0	0
$892$	−1404.96	−1.57507
$893$	− 118.008i	− 0.132147i
$894$	0	0
$895$	0	0
$896$	− 24.4347i	− 0.0272708i
$897$	0	0
$898$	−2006.04	−2.23389
$899$	− 82.3514i	− 0.0916034i
$900$	0	0
$901$	−1816.04	−2.01559
$902$	− 430.133i	− 0.476866i
$903$	0	0
$904$	−2540.08	−2.80982
$905$	0	0
$906$	0	0
$907$	336.048	0.370505	0.185252	−	0.982691i	$-0.440690\pi$
$907$	336.048	0.370505	0.185252	+	0.982691i	$0.440690\pi$
$908$	− 2981.75i	− 3.28386i
$909$	0	0
$910$	0	0
$911$	− 1629.30i	− 1.78848i	−0.447589	−	0.894239i	$-0.647717\pi$
$911$	− 1629.30i	− 1.78848i	0.447589	−	0.894239i	$-0.352283\pi$
$912$	0	0
$913$	−606.256	−0.664026
$914$	− 1651.56i	− 1.80696i
$915$	0	0
$916$	1925.82	2.10242
$917$	399.252i	0.435390i
$918$	0	0
$919$	−396.462	−0.431406	−0.215703	−	0.976459i	$-0.569204\pi$
$919$	−396.462	−0.431406	−0.215703	+	0.976459i	$0.569204\pi$
$920$	0	0
$921$	0	0
$922$	819.030	0.888319
$923$	− 19.4251i	− 0.0210456i
$924$	0	0
$925$	0	0
$926$	− 2263.30i	− 2.44417i
$927$	0	0
$928$	−171.280	−0.184569
$929$	− 208.174i	− 0.224084i	−0.993703	−	0.112042i	$-0.964261\pi$
$929$	− 208.174i	− 0.224084i	0.993703	−	0.112042i	$-0.0357390\pi$
$930$	0	0
$931$	83.6668	0.0898677
$932$	1800.59i	1.93197i
$933$	0	0
$934$	−490.900	−0.525589
$935$	0	0
$936$	0	0
$937$	−719.086	−0.767434	−0.383717	−	0.923451i	$-0.625356\pi$
$937$	−719.086	−0.767434	−0.383717	+	0.923451i	$0.625356\pi$
$938$	− 2218.69i	− 2.36534i
$939$	0	0
$940$	0	0
$941$	1150.83i	1.22299i	0.791250	+	0.611493i	$0.209431\pi$
$941$	1150.83i	1.22299i	−0.791250	+	0.611493i	$0.790569\pi$
$942$	0	0
$943$	−144.602	−0.153342
$944$	− 364.054i	− 0.385650i
$945$	0	0
$946$	688.758	0.728074
$947$	480.014i	0.506878i	0.967351	+	0.253439i	$0.0815618\pi$
$947$	480.014i	0.506878i	−0.967351	+	0.253439i	$0.918438\pi$
$948$	0	0
$949$	−264.655	−0.278878
$950$	0	0
$951$	0	0
$952$	−4314.89	−4.53245
$953$	1356.84i	1.42375i	0.702304	+	0.711877i	$0.252155\pi$
$953$	1356.84i	1.42375i	−0.702304	+	0.711877i	$0.747845\pi$
$954$	0	0
$955$	0	0
$956$	362.854i	0.379554i
$957$	0	0
$958$	1939.82	2.02486
$959$	888.205i	0.926178i
$960$	0	0
$961$	−166.254	−0.173001
$962$	4977.54i	5.17416i
$963$	0	0
$964$	811.092	0.841382
$965$	0	0
$966$	0	0
$967$	364.891	0.377343	0.188671	−	0.982040i	$-0.439582\pi$
$967$	364.891	0.377343	0.188671	+	0.982040i	$0.439582\pi$
$968$	2518.54i	2.60179i
$969$	0	0
$970$	0	0
$971$	− 1095.60i	− 1.12832i	−0.825666	−	0.564160i	$-0.809200\pi$
$971$	− 1095.60i	− 1.12832i	0.825666	−	0.564160i	$-0.190800\pi$
$972$	0	0
$973$	292.019	0.300122
$974$	− 372.400i	− 0.382341i
$975$	0	0
$976$	2028.73	2.07861
$977$	207.952i	0.212848i	0.994321	+	0.106424i	$0.0339400\pi$
$977$	207.952i	0.212848i	−0.994321	+	0.106424i	$0.966060\pi$
$978$	0	0
$979$	−990.606	−1.01185
$980$	0	0
$981$	0	0
$982$	−1187.48	−1.20925
$983$	− 1385.34i	− 1.40930i	−0.709556	−	0.704649i	$-0.751104\pi$
$983$	− 1385.34i	− 1.40930i	0.709556	−	0.704649i	$-0.248896\pi$
$984$	0	0
$985$	0	0
$986$	− 242.435i	− 0.245877i
$987$	0	0
$988$	−464.257	−0.469895
$989$	− 231.546i	− 0.234121i
$990$	0	0
$991$	−1605.21	−1.61979	−0.809896	−	0.586573i	$-0.800477\pi$
$991$	−1605.21	−1.61979	−0.809896	+	0.586573i	$0.800477\pi$
$992$	− 1652.96i	− 1.66629i
$993$	0	0
$994$	−32.0780	−0.0322717
$995$	0	0
$996$	0	0
$997$	292.605	0.293485	0.146742	−	0.989175i	$-0.453121\pi$
$997$	292.605	0.293485	0.146742	+	0.989175i	$0.453121\pi$
$998$	− 1184.98i	− 1.18735i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	675.3.c.r.26.6	yes	6
3.2	odd	2	inner	675.3.c.r.26.1	✓	6
5.2	odd	4		675.3.d.j.674.1		6
5.3	odd	4		675.3.d.k.674.6		6
5.4	even	2		675.3.c.s.26.1	yes	6
15.2	even	4		675.3.d.k.674.5		6
15.8	even	4		675.3.d.j.674.2		6
15.14	odd	2		675.3.c.s.26.6	yes	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
675.3.c.r.26.1	✓	6	3.2	odd	2	inner
675.3.c.r.26.6	yes	6	1.1	even	1	trivial
675.3.c.s.26.1	yes	6	5.4	even	2
675.3.c.s.26.6	yes	6	15.14	odd	2
675.3.d.j.674.1		6	5.2	odd	4
675.3.d.j.674.2		6	15.8	even	4
675.3.d.k.674.5		6	15.2	even	4
675.3.d.k.674.6		6	5.3	odd	4

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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 675 = 3^{3} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	675.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+3.69399i q^{2} -9.64560 q^{4} -9.20921 q^{7} -20.8548i q^{8} -15.5488i q^{11} +20.6004 q^{13} -34.0188i q^{14} +38.4552 q^{16} +22.4668i q^{17} +2.33644 q^{19} +57.4372 q^{22} -19.3092i q^{23} +76.0976i q^{26} +88.8283 q^{28} +2.92117i q^{29} -28.1912 q^{31} +58.6339i q^{32} -82.9923 q^{34} +65.4099 q^{37} +8.63079i q^{38} -7.48875i q^{41} +11.9915 q^{43} +149.978i q^{44} +71.3279 q^{46} -50.5075i q^{47} +35.8096 q^{49} -198.703 q^{52} +80.8323i q^{53} +192.056i q^{56} -10.7908 q^{58} -9.46697i q^{59} +52.7557 q^{61} -104.138i q^{62} -62.7728 q^{64} +65.2195 q^{67} -216.706i q^{68} -0.942951i q^{71} -12.8471 q^{73} +241.624i q^{74} -22.5363 q^{76} +143.192i q^{77} +68.4459 q^{79} +27.6634 q^{82} -38.9905i q^{83} +44.2965i q^{86} -324.267 q^{88} -63.7095i q^{89} -189.713 q^{91} +186.248i q^{92} +186.575 q^{94} +186.330 q^{97} +132.280i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 14 q^{4} - 16 q^{7} + 22 q^{13} + 46 q^{16} - 16 q^{19} + 86 q^{22} + 212 q^{28} - 56 q^{31} - 80 q^{34} + 150 q^{37} - 92 q^{43} + 234 q^{46} + 74 q^{49} - 354 q^{52} - 104 q^{58} - 46 q^{61} - 342 q^{64}+ \cdots + 162 q^{97}+O(q^{100}) \) 6 * q - 14 * q^4 - 16 * q^7 + 22 * q^13 + 46 * q^16 - 16 * q^19 + 86 * q^22 + 212 * q^28 - 56 * q^31 - 80 * q^34 + 150 * q^37 - 92 * q^43 + 234 * q^46 + 74 * q^49 - 354 * q^52 - 104 * q^58 - 46 * q^61 - 342 * q^64 + 8 * q^67 + 424 * q^73 - 156 * q^76 + 316 * q^79 + 196 * q^82 - 858 * q^88 + 44 * q^91 + 526 * q^94 + 162 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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