LMFDB - Embedded newform 675.3.d.j.674.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [675,3,Mod(674,675)] 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("675.674"); 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, 1])) N = Newforms(chi, 3, names="a")

Level:	$ N $	$=$	$ 675 = 3^{3} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	675.d (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [6,-6,0,14,0,0,0,-42,0,0,0,0,0,0,0,46,-84,0,16] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(19)] == 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:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2\cdot 5^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			674.2
Root			$2.69399i$ of defining polynomial
Character	$\chi$	$=$	675.674
Dual form			675.3.d.j.674.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-3.69399 q^{2} +9.64560 q^{4} +9.20921i q^{7} -20.8548 q^{8} +15.5488i q^{11} +20.6004i q^{13} -34.0188i q^{14} +38.4552 q^{16} -22.4668 q^{17} -2.33644 q^{19} -57.4372i q^{22} -19.3092 q^{23} -76.0976i q^{26} +88.8283i q^{28} +2.92117i q^{29} -28.1912 q^{31} -58.6339 q^{32} +82.9923 q^{34} -65.4099i q^{37} +8.63079 q^{38} +7.48875i q^{41} +11.9915i q^{43} +149.978i q^{44} +71.3279 q^{46} +50.5075 q^{47} -35.8096 q^{49} +198.703i q^{52} +80.8323 q^{53} -192.056i q^{56} -10.7908i q^{58} -9.46697i q^{59} +52.7557 q^{61} +104.138 q^{62} +62.7728 q^{64} -65.2195i q^{67} -216.706 q^{68} +0.942951i q^{71} -12.8471i q^{73} +241.624i q^{74} -22.5363 q^{76} -143.192 q^{77} -68.4459 q^{79} -27.6634i q^{82} -38.9905 q^{83} -44.2965i q^{86} -324.267i q^{88} -63.7095i q^{89} -189.713 q^{91} -186.248 q^{92} -186.575 q^{94} -186.330i q^{97} +132.280 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 6 q^{2} + 14 q^{4} - 42 q^{8} + 46 q^{16} - 84 q^{17} + 16 q^{19} - 102 q^{23} - 56 q^{31} - 174 q^{32} + 80 q^{34} - 96 q^{38} + 234 q^{46} - 138 q^{47} - 74 q^{49} - 120 q^{53} - 46 q^{61} + 36 q^{62}+ \cdots + 318 q^{98}+O(q^{100}) $ 6 * q - 6 * q^2 + 14 * q^4 - 42 * q^8 + 46 * q^16 - 84 * q^17 + 16 * q^19 - 102 * q^23 - 56 * q^31 - 174 * q^32 + 80 * q^34 - 96 * q^38 + 234 * q^46 - 138 * q^47 - 74 * q^49 - 120 * q^53 - 46 * q^61 + 36 * q^62 + 342 * q^64 - 432 * q^68 - 156 * q^76 - 492 * q^77 - 316 * q^79 - 324 * q^83 + 44 * q^91 - 210 * q^92 - 526 * q^94 + 318 * q^98

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 $

$2$	−3.69399	−1.84700	−0.923499	−	0.383602i	$-0.874684\pi$
$2$	−3.69399	−1.84700	−0.923499	+	0.383602i	$0.874684\pi$
$3$	0	0
$3$	0	0
$4$	9.64560	2.41140
$5$	0	0
$5$	0	0
$6$	0	0
$7$	9.20921i	1.31560i	0.753192	+	0.657801i	$0.228513\pi$
$7$	9.20921i	1.31560i	−0.753192	+	0.657801i	$0.771487\pi$
$8$	−20.8548	−2.60685
$9$	0	0
$10$	0	0
$11$	15.5488i	1.41353i	0.707450	+	0.706764i	$0.249846\pi$
$11$	15.5488i	1.41353i	−0.707450	+	0.706764i	$0.750154\pi$
$12$	0	0
$13$	20.6004i	1.58464i	0.610104	+	0.792321i	$0.291128\pi$
$13$	20.6004i	1.58464i	−0.610104	+	0.792321i	$0.708872\pi$
$14$	− 34.0188i	− 2.42991i
$15$	0	0
$16$	38.4552	2.40345
$17$	−22.4668	−1.32158	−0.660789	−	0.750572i	$-0.729778\pi$
$17$	−22.4668	−1.32158	−0.660789	+	0.750572i	$0.729778\pi$
$18$	0	0
$19$	−2.33644	−0.122970	−0.0614852	−	0.998108i	$-0.519584\pi$
$19$	−2.33644	−0.122970	−0.0614852	+	0.998108i	$0.519584\pi$
$20$	0	0
$21$	0	0
$22$	− 57.4372i	− 2.61078i
$23$	−19.3092	−0.839529	−0.419764	−	0.907633i	$-0.637887\pi$
$23$	−19.3092	−0.839529	−0.419764	+	0.907633i	$0.637887\pi$
$24$	0	0
$25$	0	0
$26$	− 76.0976i	− 2.92683i
$27$	0	0
$28$	88.8283i	3.17244i
$29$	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	$0.650253\pi$
$32$	−58.6339	−1.83231
$33$	0	0
$34$	82.9923	2.44095
$35$	0	0
$36$	0	0
$37$	− 65.4099i	− 1.76784i	−0.467642	−	0.883918i	$-0.654896\pi$
$37$	− 65.4099i	− 1.76784i	0.467642	−	0.883918i	$-0.345104\pi$
$38$	8.63079	0.227126
$39$	0	0
$40$	0	0
$41$	7.48875i	0.182652i	0.995821	+	0.0913262i	$0.0291106\pi$
$41$	7.48875i	0.182652i	−0.995821	+	0.0913262i	$0.970889\pi$
$42$	0	0
$43$	11.9915i	0.278872i	0.990231	+	0.139436i	$0.0445290\pi$
$43$	11.9915i	0.278872i	−0.990231	+	0.139436i	$0.955471\pi$
$44$	149.978i	3.40858i
$45$	0	0
$46$	71.3279	1.55061
$47$	50.5075	1.07463	0.537314	−	0.843382i	$-0.319439\pi$
$47$	50.5075	1.07463	0.537314	+	0.843382i	$0.319439\pi$
$48$	0	0
$49$	−35.8096	−0.730807
$50$	0	0
$51$	0	0
$52$	198.703i	3.82121i
$53$	80.8323	1.52514	0.762569	−	0.646907i	$-0.223938\pi$
$53$	80.8323	1.52514	0.762569	+	0.646907i	$0.223938\pi$
$54$	0	0
$55$	0	0
$56$	− 192.056i	− 3.42958i
$57$	0	0
$58$	− 10.7908i	− 0.186048i
$59$	− 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	$0.357659\pi$
$62$	104.138	1.67965
$63$	0	0
$64$	62.7728	0.980825
$65$	0	0
$66$	0	0
$67$	− 65.2195i	− 0.973425i	−0.873562	−	0.486713i	$-0.838196\pi$
$67$	− 65.2195i	− 0.973425i	0.873562	−	0.486713i	$-0.161804\pi$
$68$	−216.706	−3.18685
$69$	0	0
$70$	0	0
$71$	0.942951i	0.0132810i	0.999978	+	0.00664050i	$0.00211375\pi$
$71$	0.942951i	0.0132810i	−0.999978	+	0.00664050i	$0.997886\pi$
$72$	0	0
$73$	− 12.8471i	− 0.175988i	−0.996121	−	0.0879940i	$-0.971954\pi$
$73$	− 12.8471i	− 0.175988i	0.996121	−	0.0879940i	$-0.0280456\pi$
$74$	241.624i	3.26519i
$75$	0	0
$76$	−22.5363	−0.296531
$77$	−143.192	−1.85964
$78$	0	0
$79$	−68.4459	−0.866403	−0.433202	−	0.901297i	$-0.642616\pi$
$79$	−68.4459	−0.866403	−0.433202	+	0.901297i	$0.642616\pi$
$80$	0	0
$81$	0	0
$82$	− 27.6634i	− 0.337359i
$83$	−38.9905	−0.469765	−0.234883	−	0.972024i	$-0.575471\pi$
$83$	−38.9905	−0.469765	−0.234883	+	0.972024i	$0.575471\pi$
$84$	0	0
$85$	0	0
$86$	− 44.2965i	− 0.515076i
$87$	0	0
$88$	− 324.267i	− 3.68486i
$89$	− 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
$92$	−186.248	−2.02444
$93$	0	0
$94$	−186.575	−1.98484
$95$	0	0
$96$	0	0
$97$	− 186.330i	− 1.92093i	−0.278398	−	0.960466i	$-0.589803\pi$
$97$	− 186.330i	− 1.92093i	0.278398	−	0.960466i	$-0.410197\pi$
$98$	132.280	1.34980
$99$	0	0
$100$	0	0
$101$	53.1922i	0.526656i	0.964706	+	0.263328i	$0.0848201\pi$
$101$	53.1922i	0.526656i	−0.964706	+	0.263328i	$0.915180\pi$
$102$	0	0
$103$	24.1194i	0.234169i	0.993122	+	0.117084i	$0.0373548\pi$
$103$	24.1194i	0.234169i	−0.993122	+	0.117084i	$0.962645\pi$
$104$	− 429.616i	− 4.13093i
$105$	0	0
$106$	−298.594	−2.81693
$107$	53.4287	0.499334	0.249667	−	0.968332i	$-0.419679\pi$
$107$	53.4287	0.499334	0.249667	+	0.968332i	$0.419679\pi$
$108$	0	0
$109$	−43.8293	−0.402104	−0.201052	−	0.979581i	$-0.564436\pi$
$109$	−43.8293	−0.402104	−0.201052	+	0.979581i	$0.564436\pi$
$110$	0	0
$111$	0	0
$112$	354.142i	3.16198i
$113$	−121.798	−1.07786	−0.538929	−	0.842351i	$-0.681171\pi$
$113$	−121.798	−1.07786	−0.538929	+	0.842351i	$0.681171\pi$
$114$	0	0
$115$	0	0
$116$	28.1764i	0.242900i
$117$	0	0
$118$	34.9709i	0.296364i
$119$	− 206.902i	− 1.73867i
$120$	0	0
$121$	−120.765	−0.998060
$122$	−194.879	−1.59737
$123$	0	0
$124$	−271.921	−2.19291
$125$	0	0
$126$	0	0
$127$	25.3201i	0.199371i	0.995019	+	0.0996854i	$0.0317836\pi$
$127$	25.3201i	0.199371i	−0.995019	+	0.0996854i	$0.968216\pi$
$128$	2.65328	0.0207288
$129$	0	0
$130$	0	0
$131$	43.3536i	0.330943i	0.986215	+	0.165472i	$0.0529147\pi$
$131$	43.3536i	0.330943i	−0.986215	+	0.165472i	$0.947085\pi$
$132$	0	0
$133$	− 21.5167i	− 0.161780i
$134$	240.920i	1.79791i
$135$	0	0
$136$	468.541	3.44516
$137$	96.4475	0.703996	0.351998	−	0.936001i	$-0.385502\pi$
$137$	96.4475	0.703996	0.351998	+	0.936001i	$0.385502\pi$
$138$	0	0
$139$	31.7095	0.228126	0.114063	−	0.993474i	$-0.463613\pi$
$139$	31.7095	0.228126	0.114063	+	0.993474i	$0.463613\pi$
$140$	0	0
$141$	0	0
$142$	− 3.48325i	− 0.0245300i
$143$	−320.311	−2.23994
$144$	0	0
$145$	0	0
$146$	47.4572i	0.325049i
$147$	0	0
$148$	− 630.918i	− 4.26296i
$149$	− 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	$0.623108\pi$
$152$	48.7260	0.320566
$153$	0	0
$154$	528.951	3.43475
$155$	0	0
$156$	0	0
$157$	122.473i	0.780083i	0.920797	+	0.390041i	$0.127539\pi$
$157$	122.473i	0.780083i	−0.920797	+	0.390041i	$0.872461\pi$
$158$	252.839	1.60024
$159$	0	0
$160$	0	0
$161$	− 177.822i	− 1.10449i
$162$	0	0
$163$	201.136i	1.23396i	0.786978	+	0.616981i	$0.211645\pi$
$163$	201.136i	1.23396i	−0.786978	+	0.616981i	$0.788355\pi$
$164$	72.2335i	0.440448i
$165$	0	0
$166$	144.031	0.867656
$167$	32.4447	0.194280	0.0971399	−	0.995271i	$-0.469031\pi$
$167$	32.4447	0.194280	0.0971399	+	0.995271i	$0.469031\pi$
$168$	0	0
$169$	−255.375	−1.51109
$170$	0	0
$171$	0	0
$172$	115.665i	0.672472i
$173$	218.485	1.26292	0.631460	−	0.775409i	$-0.282456\pi$
$173$	218.485	1.26292	0.631460	+	0.775409i	$0.282456\pi$
$174$	0	0
$175$	0	0
$176$	597.932i	3.39734i
$177$	0	0
$178$	235.342i	1.32215i
$179$	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	$0.498954\pi$
$182$	700.799	3.85054
$183$	0	0
$184$	402.689	2.18853
$185$	0	0
$186$	0	0
$187$	− 349.332i	− 1.86809i
$188$	487.175	2.59136
$189$	0	0
$190$	0	0
$191$	343.880i	1.80042i	0.435458	+	0.900209i	$0.356587\pi$
$191$	343.880i	1.80042i	−0.435458	+	0.900209i	$0.643413\pi$
$192$	0	0
$193$	124.422i	0.644673i	0.946625	+	0.322337i	$0.104468\pi$
$193$	124.422i	0.644673i	−0.946625	+	0.322337i	$0.895532\pi$
$194$	688.303i	3.54796i
$195$	0	0
$196$	−345.405	−1.76227
$197$	−311.040	−1.57888	−0.789442	−	0.613826i	$-0.789630\pi$
$197$	−311.040	−1.57888	−0.789442	+	0.613826i	$0.789630\pi$
$198$	0	0
$199$	−351.943	−1.76856	−0.884280	−	0.466958i	$-0.845350\pi$
$199$	−351.943	−1.76856	−0.884280	+	0.466958i	$0.845350\pi$
$200$	0	0
$201$	0	0
$202$	− 196.492i	− 0.972731i
$203$	−26.9017	−0.132521
$204$	0	0
$205$	0	0
$206$	− 89.0969i	− 0.432509i
$207$	0	0
$208$	792.190i	3.80861i
$209$	− 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	$0.246817\pi$
$212$	779.676	3.67772
$213$	0	0
$214$	−197.365	−0.922268
$215$	0	0
$216$	0	0
$217$	− 259.619i	− 1.19640i
$218$	161.905	0.742685
$219$	0	0
$220$	0	0
$221$	− 462.824i	− 2.09423i
$222$	0	0
$223$	145.658i	0.653177i	0.945167	+	0.326588i	$0.105899\pi$
$223$	145.658i	0.653177i	−0.945167	+	0.326588i	$0.894101\pi$
$224$	− 539.972i	− 2.41059i
$225$	0	0
$226$	449.921	1.99080
$227$	−309.131	−1.36181	−0.680904	−	0.732372i	$-0.738413\pi$
$227$	−309.131	−1.36181	−0.680904	+	0.732372i	$0.738413\pi$
$228$	0	0
$229$	199.658	0.871867	0.435934	−	0.899979i	$-0.356418\pi$
$229$	199.658	0.871867	0.435934	+	0.899979i	$0.356418\pi$
$230$	0	0
$231$	0	0
$232$	− 60.9205i	− 0.262588i
$233$	−186.675	−0.801180	−0.400590	−	0.916257i	$-0.631195\pi$
$233$	−186.675	−0.801180	−0.400590	+	0.916257i	$0.631195\pi$
$234$	0	0
$235$	0	0
$236$	− 91.3146i	− 0.386926i
$237$	0	0
$238$	764.294i	3.21132i
$239$	− 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	$0.555818\pi$
$242$	446.106	1.84341
$243$	0	0
$244$	508.860	2.08549
$245$	0	0
$246$	0	0
$247$	− 48.1314i	− 0.194864i
$248$	587.923	2.37066
$249$	0	0
$250$	0	0
$251$	389.119i	1.55028i	0.631792	+	0.775138i	$0.282320\pi$
$251$	389.119i	1.55028i	−0.631792	+	0.775138i	$0.717680\pi$
$252$	0	0
$253$	− 300.234i	− 1.18670i
$254$	− 93.5323i	− 0.368237i
$255$	0	0
$256$	−260.893	−1.01911
$257$	−168.018	−0.653768	−0.326884	−	0.945064i	$-0.605999\pi$
$257$	−168.018	−0.653768	−0.326884	+	0.945064i	$0.605999\pi$
$258$	0	0
$259$	602.374	2.32577
$260$	0	0
$261$	0	0
$262$	− 160.148i	− 0.611252i
$263$	345.794	1.31480	0.657402	−	0.753540i	$-0.271655\pi$
$263$	345.794	1.31480	0.657402	+	0.753540i	$0.271655\pi$
$264$	0	0
$265$	0	0
$266$	79.4827i	0.298807i
$267$	0	0
$268$	− 629.081i	− 2.34732i
$269$	− 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	$0.345010\pi$
$272$	−863.965	−3.17634
$273$	0	0
$274$	−356.276	−1.30028
$275$	0	0
$276$	0	0
$277$	415.895i	1.50143i	0.660628	+	0.750714i	$0.270290\pi$
$277$	415.895i	1.50143i	−0.660628	+	0.750714i	$0.729710\pi$
$278$	−117.135	−0.421347
$279$	0	0
$280$	0	0
$281$	410.907i	1.46230i	0.682215	+	0.731151i	$0.261017\pi$
$281$	410.907i	1.46230i	−0.682215	+	0.731151i	$0.738983\pi$
$282$	0	0
$283$	− 499.088i	− 1.76356i	−0.471660	−	0.881780i	$-0.656345\pi$
$283$	− 499.088i	− 1.76356i	0.471660	−	0.881780i	$-0.343655\pi$
$284$	9.09532i	0.0320258i
$285$	0	0
$286$	1183.23	4.13716
$287$	−68.9655	−0.240298
$288$	0	0
$289$	215.758	0.746567
$290$	0	0
$291$	0	0
$292$	− 123.918i	− 0.424377i
$293$	−466.667	−1.59272	−0.796361	−	0.604822i	$-0.793244\pi$
$293$	−466.667	−1.59272	−0.796361	+	0.604822i	$0.793244\pi$
$294$	0	0
$295$	0	0
$296$	1364.11i	4.60848i
$297$	0	0
$298$	692.582i	2.32410i
$299$	− 397.776i	− 1.33035i
$300$	0	0
$301$	−110.432	−0.366885
$302$	420.782	1.39332
$303$	0	0
$304$	−89.8481	−0.295553
$305$	0	0
$306$	0	0
$307$	183.976i	0.599271i	0.954054	+	0.299636i	$0.0968651\pi$
$307$	183.976i	0.599271i	−0.954054	+	0.299636i	$0.903135\pi$
$308$	−1381.17	−4.48433
$309$	0	0
$310$	0	0
$311$	− 256.734i	− 0.825510i	−0.910842	−	0.412755i	$-0.864567\pi$
$311$	− 256.734i	− 0.825510i	0.910842	−	0.412755i	$-0.135433\pi$
$312$	0	0
$313$	257.923i	0.824034i	0.911176	+	0.412017i	$0.135176\pi$
$313$	257.923i	0.824034i	−0.911176	+	0.412017i	$0.864824\pi$
$314$	− 452.415i	− 1.44081i
$315$	0	0
$316$	−660.201	−2.08924
$317$	173.381	0.546942	0.273471	−	0.961880i	$-0.411828\pi$
$317$	173.381	0.546942	0.273471	+	0.961880i	$0.411828\pi$
$318$	0	0
$319$	−45.4207	−0.142385
$320$	0	0
$321$	0	0
$322$	656.874i	2.03998i
$323$	52.4923	0.162515
$324$	0	0
$325$	0	0
$326$	− 742.995i	− 2.27913i
$327$	0	0
$328$	− 156.176i	− 0.476148i
$329$	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	$0.512160\pi$
$332$	−376.087	−1.13279
$333$	0	0
$334$	−119.851	−0.358834
$335$	0	0
$336$	0	0
$337$	384.796i	1.14183i	0.821010	+	0.570914i	$0.193411\pi$
$337$	384.796i	1.14183i	−0.821010	+	0.570914i	$0.806589\pi$
$338$	943.353	2.79098
$339$	0	0
$340$	0	0
$341$	− 438.340i	− 1.28545i
$342$	0	0
$343$	121.473i	0.354150i
$344$	− 250.080i	− 0.726978i
$345$	0	0
$346$	−807.083	−2.33261
$347$	−543.336	−1.56581	−0.782904	−	0.622142i	$-0.786263\pi$
$347$	−543.336	−1.56581	−0.782904	+	0.622142i	$0.786263\pi$
$348$	0	0
$349$	−102.381	−0.293356	−0.146678	−	0.989184i	$-0.546858\pi$
$349$	−102.381	−0.293356	−0.146678	+	0.989184i	$0.546858\pi$
$350$	0	0
$351$	0	0
$352$	− 911.688i	− 2.59002i
$353$	−609.853	−1.72763	−0.863814	−	0.503810i	$-0.831931\pi$
$353$	−609.853	−1.72763	−0.863814	+	0.503810i	$0.831931\pi$
$354$	0	0
$355$	0	0
$356$	− 614.516i	− 1.72617i
$357$	0	0
$358$	− 802.010i	− 2.24025i
$359$	− 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
$362$	−4.39502	−0.0121409
$363$	0	0
$364$	−1829.90	−5.02719
$365$	0	0
$366$	0	0
$367$	509.283i	1.38769i	0.720123	+	0.693846i	$0.244085\pi$
$367$	509.283i	1.38769i	−0.720123	+	0.693846i	$0.755915\pi$
$368$	−742.537	−2.01776
$369$	0	0
$370$	0	0
$371$	744.402i	2.00647i
$372$	0	0
$373$	291.668i	0.781952i	0.920401	+	0.390976i	$0.127862\pi$
$373$	291.668i	0.781952i	−0.920401	+	0.390976i	$0.872138\pi$
$374$	1290.43i	3.45035i
$375$	0	0
$376$	−1053.32	−2.80140
$377$	−60.1772	−0.159621
$378$	0	0
$379$	−132.116	−0.348591	−0.174295	−	0.984693i	$-0.555765\pi$
$379$	−132.116	−0.348591	−0.174295	+	0.984693i	$0.555765\pi$
$380$	0	0
$381$	0	0
$382$	− 1270.29i	− 3.32537i
$383$	−1.23436	−0.00322286	−0.00161143	−	0.999999i	$-0.500513\pi$
$383$	−1.23436	−0.00322286	−0.00161143	+	0.999999i	$0.500513\pi$
$384$	0	0
$385$	0	0
$386$	− 459.614i	− 1.19071i
$387$	0	0
$388$	− 1797.27i	− 4.63213i
$389$	− 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
$392$	746.802	1.90511
$393$	0	0
$394$	1148.98	2.91619
$395$	0	0
$396$	0	0
$397$	− 259.982i	− 0.654866i	−0.944874	−	0.327433i	$-0.893816\pi$
$397$	− 259.982i	− 0.654866i	0.944874	−	0.327433i	$-0.106184\pi$
$398$	1300.08	3.26652
$399$	0	0
$400$	0	0
$401$	158.010i	0.394040i	0.980400	+	0.197020i	$0.0631263\pi$
$401$	158.010i	0.394040i	−0.980400	+	0.197020i	$0.936874\pi$
$402$	0	0
$403$	− 580.750i	− 1.44107i
$404$	513.071i	1.26998i
$405$	0	0
$406$	99.3747	0.244765
$407$	1017.05	2.49888
$408$	0	0
$409$	300.817	0.735495	0.367747	−	0.929926i	$-0.380129\pi$
$409$	300.817	0.735495	0.367747	+	0.929926i	$0.380129\pi$
$410$	0	0
$411$	0	0
$412$	232.646i	0.564675i
$413$	87.1833	0.211098
$414$	0	0
$415$	0	0
$416$	− 1207.88i	− 2.90356i
$417$	0	0
$418$	134.198i	0.321049i
$419$	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	$0.422897\pi$
$422$	−1113.25	−2.63804
$423$	0	0
$424$	−1685.74	−3.97581
$425$	0	0
$426$	0	0
$427$	485.838i	1.13779i
$428$	515.352	1.20409
$429$	0	0
$430$	0	0
$431$	− 335.503i	− 0.778429i	−0.921147	−	0.389215i	$-0.872746\pi$
$431$	− 335.503i	− 0.778429i	0.921147	−	0.389215i	$-0.127254\pi$
$432$	0	0
$433$	− 104.346i	− 0.240985i	−0.992714	−	0.120492i	$-0.961553\pi$
$433$	− 104.346i	− 0.240985i	0.992714	−	0.120492i	$-0.0384473\pi$
$434$	959.032i	2.20975i
$435$	0	0
$436$	−422.760	−0.969633
$437$	45.1146	0.103237
$438$	0	0
$439$	−256.928	−0.585256	−0.292628	−	0.956226i	$-0.594530\pi$
$439$	−256.928	−0.585256	−0.292628	+	0.956226i	$0.594530\pi$
$440$	0	0
$441$	0	0
$442$	1709.67i	3.86803i
$443$	764.139	1.72492	0.862459	−	0.506127i	$-0.168923\pi$
$443$	764.139	1.72492	0.862459	+	0.506127i	$0.168923\pi$
$444$	0	0
$445$	0	0
$446$	− 538.062i	− 1.20642i
$447$	0	0
$448$	578.088i	1.29038i
$449$	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
$452$	−1174.82	−2.59915
$453$	0	0
$454$	1141.93	2.51526
$455$	0	0
$456$	0	0
$457$	447.094i	0.978323i	0.872193	+	0.489161i	$0.162697\pi$
$457$	447.094i	0.978323i	−0.872193	+	0.489161i	$0.837303\pi$
$458$	−737.534	−1.61034
$459$	0	0
$460$	0	0
$461$	221.719i	0.480953i	0.970655	+	0.240477i	$0.0773037\pi$
$461$	221.719i	0.480953i	−0.970655	+	0.240477i	$0.922696\pi$
$462$	0	0
$463$	− 612.697i	− 1.32332i	−0.749805	−	0.661659i	$-0.769853\pi$
$463$	− 612.697i	− 1.32332i	0.749805	−	0.661659i	$-0.230147\pi$
$464$	112.334i	0.242099i
$465$	0	0
$466$	689.576	1.47978
$467$	−132.891	−0.284564	−0.142282	−	0.989826i	$-0.545444\pi$
$467$	−132.891	−0.284564	−0.142282	+	0.989826i	$0.545444\pi$
$468$	0	0
$469$	600.620	1.28064
$470$	0	0
$471$	0	0
$472$	197.432i	0.418288i
$473$	−186.453	−0.394193
$474$	0	0
$475$	0	0
$476$	− 1995.69i	− 4.19263i
$477$	0	0
$478$	138.963i	0.290717i
$479$	− 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
$482$	310.626	0.644452
$483$	0	0
$484$	−1164.85	−2.40672
$485$	0	0
$486$	0	0
$487$	100.812i	0.207007i	0.994629	+	0.103503i	$0.0330053\pi$
$487$	100.812i	0.207007i	−0.994629	+	0.103503i	$0.966995\pi$
$488$	−1100.21	−2.25453
$489$	0	0
$490$	0	0
$491$	− 321.463i	− 0.654711i	−0.944901	−	0.327356i	$-0.893842\pi$
$491$	− 321.463i	− 0.654711i	0.944901	−	0.327356i	$-0.106158\pi$
$492$	0	0
$493$	− 65.6294i	− 0.133123i
$494$	177.797i	0.359914i
$495$	0	0
$496$	−1084.10	−2.18568
$497$	−8.68383	−0.0174725
$498$	0	0
$499$	320.785	0.642856	0.321428	−	0.946934i	$-0.395837\pi$
$499$	320.785	0.642856	0.321428	+	0.946934i	$0.395837\pi$
$500$	0	0
$501$	0	0
$502$	− 1437.40i	− 2.86336i
$503$	31.7672	0.0631555	0.0315778	−	0.999501i	$-0.489947\pi$
$503$	31.7672	0.0631555	0.0315778	+	0.999501i	$0.489947\pi$
$504$	0	0
$505$	0	0
$506$	1109.06i	2.19183i
$507$	0	0
$508$	244.227i	0.480763i
$509$	− 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
$512$	953.122	1.86157
$513$	0	0
$514$	620.659	1.20751
$515$	0	0
$516$	0	0
$517$	785.332i	1.51902i
$518$	−2225.17	−4.29569
$519$	0	0
$520$	0	0
$521$	− 509.938i	− 0.978767i	−0.872069	−	0.489384i	$-0.837222\pi$
$521$	− 509.938i	− 0.978767i	0.872069	−	0.489384i	$-0.162778\pi$
$522$	0	0
$523$	10.6268i	0.0203188i	0.999948	+	0.0101594i	$0.00323390\pi$
$523$	10.6268i	0.0203188i	−0.999948	+	0.0101594i	$0.996766\pi$
$524$	418.171i	0.798037i
$525$	0	0
$526$	−1277.36	−2.42844
$527$	633.368	1.20184
$528$	0	0
$529$	−156.156	−0.295192
$530$	0	0
$531$	0	0
$532$	− 207.542i	− 0.390116i
$533$	−154.271	−0.289439
$534$	0	0
$535$	0	0
$536$	1360.14i	2.53757i
$537$	0	0
$538$	1855.06i	3.44806i
$539$	− 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	$0.377019\pi$
$542$	−936.810	−1.72843
$543$	0	0
$544$	1317.32	2.42154
$545$	0	0
$546$	0	0
$547$	− 690.828i	− 1.26294i	−0.775400	−	0.631470i	$-0.782452\pi$
$547$	− 690.828i	− 1.26294i	0.775400	−	0.631470i	$-0.217548\pi$
$548$	930.294	1.69762
$549$	0	0
$550$	0	0
$551$	− 6.82513i	− 0.0123868i
$552$	0	0
$553$	− 630.332i	− 1.13984i
$554$	− 1536.32i	− 2.77313i
$555$	0	0
$556$	305.857	0.550102
$557$	−751.961	−1.35002	−0.675010	−	0.737809i	$-0.735861\pi$
$557$	−751.961	−1.35002	−0.675010	+	0.737809i	$0.735861\pi$
$558$	0	0
$559$	−247.029	−0.441913
$560$	0	0
$561$	0	0
$562$	− 1517.89i	− 2.70087i
$563$	262.222	0.465759	0.232879	−	0.972506i	$-0.425185\pi$
$563$	262.222	0.465759	0.232879	+	0.972506i	$0.425185\pi$
$564$	0	0
$565$	0	0
$566$	1843.63i	3.25729i
$567$	0	0
$568$	− 19.6651i	− 0.0346216i
$569$	− 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	$0.605841\pi$
$572$	−3089.59	−5.40138
$573$	0	0
$574$	254.758	0.443830
$575$	0	0
$576$	0	0
$577$	− 125.906i	− 0.218208i	−0.994030	−	0.109104i	$-0.965202\pi$
$577$	− 125.906i	− 0.218208i	0.994030	−	0.109104i	$-0.0347982\pi$
$578$	−797.009	−1.37891
$579$	0	0
$580$	0	0
$581$	− 359.072i	− 0.618024i
$582$	0	0
$583$	1256.85i	2.15582i
$584$	267.924i	0.458774i
$585$	0	0
$586$	1723.87	2.94175
$587$	569.861	0.970802	0.485401	−	0.874292i	$-0.338674\pi$
$587$	569.861	0.970802	0.485401	+	0.874292i	$0.338674\pi$
$588$	0	0
$589$	65.8671	0.111829
$590$	0	0
$591$	0	0
$592$	− 2515.35i	− 4.24890i
$593$	−584.546	−0.985744	−0.492872	−	0.870102i	$-0.664053\pi$
$593$	−584.546	−0.985744	−0.492872	+	0.870102i	$0.664053\pi$
$594$	0	0
$595$	0	0
$596$	− 1808.44i	− 3.03430i
$597$	0	0
$598$	1469.38i	2.45716i
$599$	− 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	$0.690634\pi$
$602$	407.936	0.677635
$603$	0	0
$604$	−1098.73	−1.81908
$605$	0	0
$606$	0	0
$607$	897.241i	1.47816i	0.673620	+	0.739078i	$0.264739\pi$
$607$	897.241i	1.47816i	−0.673620	+	0.739078i	$0.735261\pi$
$608$	136.995	0.225320
$609$	0	0
$610$	0	0
$611$	1040.47i	1.70290i
$612$	0	0
$613$	301.192i	0.491341i	0.969353	+	0.245671i	$0.0790081\pi$
$613$	301.192i	0.491341i	−0.969353	+	0.245671i	$0.920992\pi$
$614$	− 679.607i	− 1.10685i
$615$	0	0
$616$	2986.25	4.84780
$617$	−832.739	−1.34966	−0.674829	−	0.737974i	$-0.735783\pi$
$617$	−832.739	−1.34966	−0.674829	+	0.737974i	$0.735783\pi$
$618$	0	0
$619$	702.912	1.13556	0.567781	−	0.823180i	$-0.307802\pi$
$619$	702.912	1.13556	0.567781	+	0.823180i	$0.307802\pi$
$620$	0	0
$621$	0	0
$622$	948.373i	1.52471i
$623$	586.714	0.941756
$624$	0	0
$625$	0	0
$626$	− 952.765i	− 1.52199i
$627$	0	0
$628$	1181.33i	1.88109i
$629$	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	$0.476281\pi$
$632$	1427.43	2.25858
$633$	0	0
$634$	−640.467	−1.01020
$635$	0	0
$636$	0	0
$637$	− 737.690i	− 1.15807i
$638$	167.784	0.262984
$639$	0	0
$640$	0	0
$641$	− 400.819i	− 0.625303i	−0.949868	−	0.312651i	$-0.898783\pi$
$641$	− 400.819i	− 0.625303i	0.949868	−	0.312651i	$-0.101217\pi$
$642$	0	0
$643$	529.930i	0.824152i	0.911149	+	0.412076i	$0.135196\pi$
$643$	529.930i	0.824152i	−0.911149	+	0.412076i	$0.864804\pi$
$644$	− 1715.20i	− 2.66336i
$645$	0	0
$646$	−193.906	−0.300165
$647$	803.762	1.24229	0.621145	−	0.783695i	$-0.286668\pi$
$647$	803.762	1.24229	0.621145	+	0.783695i	$0.286668\pi$
$648$	0	0
$649$	147.200	0.226811
$650$	0	0
$651$	0	0
$652$	1940.08i	2.97558i
$653$	−273.937	−0.419505	−0.209752	−	0.977755i	$-0.567266\pi$
$653$	−273.937	−0.419505	−0.209752	+	0.977755i	$0.567266\pi$
$654$	0	0
$655$	0	0
$656$	287.981i	0.438996i
$657$	0	0
$658$	− 1718.20i	− 2.61125i
$659$	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	$0.498122\pi$
$662$	93.3936	0.141078
$663$	0	0
$664$	813.140	1.22461
$665$	0	0
$666$	0	0
$667$	− 56.4054i	− 0.0845657i
$668$	312.949	0.468486
$669$	0	0
$670$	0	0
$671$	820.288i	1.22249i
$672$	0	0
$673$	− 802.306i	− 1.19213i	−0.802935	−	0.596067i	$-0.796729\pi$
$673$	− 802.306i	− 1.19213i	0.802935	−	0.596067i	$-0.203271\pi$
$674$	− 1421.43i	− 2.10895i
$675$	0	0
$676$	−2463.24	−3.64385
$677$	−853.169	−1.26022	−0.630110	−	0.776506i	$-0.716990\pi$
$677$	−853.169	−1.26022	−0.630110	+	0.776506i	$0.716990\pi$
$678$	0	0
$679$	1715.96	2.52718
$680$	0	0
$681$	0	0
$682$	1619.23i	2.37423i
$683$	964.796	1.41259	0.706293	−	0.707920i	$-0.250366\pi$
$683$	964.796	1.41259	0.706293	+	0.707920i	$0.250366\pi$
$684$	0	0
$685$	0	0
$686$	− 448.722i	− 0.654114i
$687$	0	0
$688$	461.135i	0.670255i
$689$	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	$0.590141\pi$
$692$	2107.42	3.04540
$693$	0	0
$694$	2007.08	2.89204
$695$	0	0
$696$	0	0
$697$	− 168.248i	− 0.241389i
$698$	378.196	0.541827
$699$	0	0
$700$	0	0
$701$	501.171i	0.714938i	0.933925	+	0.357469i	$0.116360\pi$
$701$	501.171i	0.714938i	−0.933925	+	0.357469i	$0.883640\pi$
$702$	0	0
$703$	152.826i	0.217391i
$704$	976.042i	1.38642i
$705$	0	0
$706$	2252.79	3.19093
$707$	−489.858	−0.692869
$708$	0	0
$709$	−711.983	−1.00421	−0.502104	−	0.864807i	$-0.667440\pi$
$709$	−711.983	−1.00421	−0.502104	+	0.864807i	$0.667440\pi$
$710$	0	0
$711$	0	0
$712$	1328.65i	1.86608i
$713$	544.349	0.763463
$714$	0	0
$715$	0	0
$716$	2094.17i	2.92482i
$717$	0	0
$718$	1191.76i	1.65983i
$719$	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
$722$	1313.37	1.81907
$723$	0	0
$724$	11.4761	0.0158509
$725$	0	0
$726$	0	0
$727$	− 649.704i	− 0.893678i	−0.894614	−	0.446839i	$-0.852550\pi$
$727$	− 649.704i	− 0.893678i	0.894614	−	0.446839i	$-0.147450\pi$
$728$	3956.43	5.43465
$729$	0	0
$730$	0	0
$731$	− 269.411i	− 0.368551i
$732$	0	0
$733$	− 54.0661i	− 0.0737600i	−0.999320	−	0.0368800i	$-0.988258\pi$
$733$	− 54.0661i	− 0.0737600i	0.999320	−	0.0368800i	$-0.0117419\pi$
$734$	− 1881.29i	− 2.56306i
$735$	0	0
$736$	1132.17	1.53828
$737$	1014.09	1.37596
$738$	0	0
$739$	−1161.91	−1.57227	−0.786134	−	0.618057i	$-0.787920\pi$
$739$	−1161.91	−1.57227	−0.786134	+	0.618057i	$0.787920\pi$
$740$	0	0
$741$	0	0
$742$	− 2749.82i	− 3.70595i
$743$	176.127	0.237048	0.118524	−	0.992951i	$-0.462184\pi$
$743$	176.127	0.237048	0.118524	+	0.992951i	$0.462184\pi$
$744$	0	0
$745$	0	0
$746$	− 1077.42i	− 1.44426i
$747$	0	0
$748$	− 3369.52i	− 4.50470i
$749$	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	$0.766892\pi$
$752$	1942.28	2.58281
$753$	0	0
$754$	222.294	0.294820
$755$	0	0
$756$	0	0
$757$	− 1040.37i	− 1.37433i	−0.726500	−	0.687167i	$-0.758854\pi$
$757$	− 1040.37i	− 1.37433i	0.726500	−	0.687167i	$-0.241146\pi$
$758$	488.035	0.643846
$759$	0	0
$760$	0	0
$761$	− 1350.15i	− 1.77418i	−0.461592	−	0.887092i	$-0.652722\pi$
$761$	− 1350.15i	− 1.77418i	0.461592	−	0.887092i	$-0.347278\pi$
$762$	0	0
$763$	− 403.633i	− 0.529008i
$764$	3316.93i	4.34153i
$765$	0	0
$766$	4.55970	0.00595261
$767$	195.023	0.254267
$768$	0	0
$769$	−560.677	−0.729099	−0.364550	−	0.931184i	$-0.618777\pi$
$769$	−560.677	−0.729099	−0.364550	+	0.931184i	$0.618777\pi$
$770$	0	0
$771$	0	0
$772$	1200.12i	1.55457i
$773$	−1271.66	−1.64510	−0.822551	−	0.568691i	$-0.807450\pi$
$773$	−1271.66	−1.64510	−0.822551	+	0.568691i	$0.807450\pi$
$774$	0	0
$775$	0	0
$776$	3885.88i	5.00758i
$777$	0	0
$778$	360.611i	0.463510i
$779$	− 17.4970i	− 0.0224608i
$780$	0	0
$781$	−14.6618	−0.0187731
$782$	−1602.51	−2.04925
$783$	0	0
$784$	−1377.06	−1.75646
$785$	0	0
$786$	0	0
$787$	673.581i	0.855884i	0.903806	+	0.427942i	$0.140761\pi$
$787$	673.581i	0.855884i	−0.903806	+	0.427942i	$0.859239\pi$
$788$	−3000.17	−3.80732
$789$	0	0
$790$	0	0
$791$	− 1121.66i	− 1.41803i
$792$	0	0
$793$	1086.79i	1.37047i
$794$	960.371i	1.20954i
$795$	0	0
$796$	−3394.70	−4.26470
$797$	441.048	0.553386	0.276693	−	0.960958i	$-0.410762\pi$
$797$	441.048	0.553386	0.276693	+	0.960958i	$0.410762\pi$
$798$	0	0
$799$	−1134.74	−1.42020
$800$	0	0
$801$	0	0
$802$	− 583.688i	− 0.727790i
$803$	199.757	0.248764
$804$	0	0
$805$	0	0
$806$	2145.29i	2.66165i
$807$	0	0
$808$	− 1109.31i	− 1.37291i
$809$	− 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	$0.505211\pi$
$812$	−259.483	−0.319560
$813$	0	0
$814$	−3756.96	−4.61543
$815$	0	0
$816$	0	0
$817$	− 28.0174i	− 0.0342930i
$818$	−1111.22	−1.35846
$819$	0	0
$820$	0	0
$821$	− 561.109i	− 0.683446i	−0.939801	−	0.341723i	$-0.888990\pi$
$821$	− 561.109i	− 0.683446i	0.939801	−	0.341723i	$-0.111010\pi$
$822$	0	0
$823$	140.819i	0.171105i	0.996334	+	0.0855523i	$0.0272655\pi$
$823$	140.819i	0.171105i	−0.996334	+	0.0855523i	$0.972735\pi$
$824$	− 503.005i	− 0.610443i
$825$	0	0
$826$	−322.055	−0.389897
$827$	445.029	0.538124	0.269062	−	0.963123i	$-0.413286\pi$
$827$	445.029	0.538124	0.269062	+	0.963123i	$0.413286\pi$
$828$	0	0
$829$	−1456.17	−1.75653	−0.878267	−	0.478170i	$-0.841300\pi$
$829$	−1456.17	−1.75653	−0.878267	+	0.478170i	$0.841300\pi$
$830$	0	0
$831$	0	0
$832$	1293.14i	1.55426i
$833$	804.527	0.965819
$834$	0	0
$835$	0	0
$836$	− 350.413i	− 0.419154i
$837$	0	0
$838$	− 600.266i	− 0.716308i
$839$	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
$842$	−746.063	−0.886060
$843$	0	0
$844$	2906.88	3.44417
$845$	0	0
$846$	0	0
$847$	− 1112.15i	− 1.31305i
$848$	3108.42	3.66559
$849$	0	0
$850$	0	0
$851$	1263.01i	1.48415i
$852$	0	0
$853$	− 788.618i	− 0.924523i	−0.886744	−	0.462261i	$-0.847038\pi$
$853$	− 788.618i	− 0.924523i	0.886744	−	0.462261i	$-0.152962\pi$
$854$	− 1794.68i	− 2.10150i
$855$	0	0
$856$	−1114.25	−1.30169
$857$	309.338	0.360954	0.180477	−	0.983579i	$-0.442236\pi$
$857$	309.338	0.360954	0.180477	+	0.983579i	$0.442236\pi$
$858$	0	0
$859$	−1129.78	−1.31522	−0.657611	−	0.753357i	$-0.728433\pi$
$859$	−1129.78	−1.31522	−0.657611	+	0.753357i	$0.728433\pi$
$860$	0	0
$861$	0	0
$862$	1239.35i	1.43776i
$863$	618.421	0.716594	0.358297	−	0.933608i	$-0.383358\pi$
$863$	618.421	0.716594	0.358297	+	0.933608i	$0.383358\pi$
$864$	0	0
$865$	0	0
$866$	385.455i	0.445098i
$867$	0	0
$868$	− 2504.18i	− 2.88500i
$869$	− 1064.25i	− 1.22468i
$870$	0	0
$871$	1343.54	1.54253
$872$	914.052	1.04822
$873$	0	0
$874$	−166.653	−0.190679
$875$	0	0
$876$	0	0
$877$	− 751.759i	− 0.857194i	−0.903496	−	0.428597i	$-0.859008\pi$
$877$	− 751.759i	− 0.857194i	0.903496	−	0.428597i	$-0.140992\pi$
$878$	949.089	1.08097
$879$	0	0
$880$	0	0
$881$	− 1465.38i	− 1.66332i	−0.555285	−	0.831660i	$-0.687391\pi$
$881$	− 1465.38i	− 1.66332i	0.555285	−	0.831660i	$-0.312609\pi$
$882$	0	0
$883$	658.936i	0.746247i	0.927782	+	0.373124i	$0.121713\pi$
$883$	658.936i	0.746247i	−0.927782	+	0.373124i	$0.878287\pi$
$884$	− 4464.22i	− 5.05002i
$885$	0	0
$886$	−2822.72	−3.18592
$887$	962.298	1.08489	0.542445	−	0.840091i	$-0.317499\pi$
$887$	962.298	1.08489	0.542445	+	0.840091i	$0.317499\pi$
$888$	0	0
$889$	−233.178	−0.262293
$890$	0	0
$891$	0	0
$892$	1404.96i	1.57507i
$893$	−118.008	−0.132147
$894$	0	0
$895$	0	0
$896$	24.4347i	0.0272708i
$897$	0	0
$898$	− 2006.04i	− 2.23389i
$899$	− 82.3514i	− 0.0916034i
$900$	0	0
$901$	−1816.04	−2.01559
$902$	430.133	0.476866
$903$	0	0
$904$	2540.08	2.80982
$905$	0	0
$906$	0	0
$907$	− 336.048i	− 0.370505i	−0.982691	−	0.185252i	$-0.940690\pi$
$907$	− 336.048i	− 0.370505i	0.982691	−	0.185252i	$-0.0593102\pi$
$908$	−2981.75	−3.28386
$909$	0	0
$910$	0	0
$911$	1629.30i	1.78848i	0.447589	+	0.894239i	$0.352283\pi$
$911$	1629.30i	1.78848i	−0.447589	+	0.894239i	$0.647717\pi$
$912$	0	0
$913$	− 606.256i	− 0.664026i
$914$	− 1651.56i	− 1.80696i
$915$	0	0
$916$	1925.82	2.10242
$917$	−399.252	−0.435390
$918$	0	0
$919$	396.462	0.431406	0.215703	−	0.976459i	$-0.430796\pi$
$919$	396.462	0.431406	0.215703	+	0.976459i	$0.430796\pi$
$920$	0	0
$921$	0	0
$922$	− 819.030i	− 0.888319i
$923$	−19.4251	−0.0210456
$924$	0	0
$925$	0	0
$926$	2263.30i	2.44417i
$927$	0	0
$928$	− 171.280i	− 0.184569i
$929$	− 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
$932$	−1800.59	−1.93197
$933$	0	0
$934$	490.900	0.525589
$935$	0	0
$936$	0	0
$937$	719.086i	0.767434i	0.923451	+	0.383717i	$0.125356\pi$
$937$	719.086i	0.767434i	−0.923451	+	0.383717i	$0.874644\pi$
$938$	−2218.69	−2.36534
$939$	0	0
$940$	0	0
$941$	− 1150.83i	− 1.22299i	−0.791250	−	0.611493i	$-0.790569\pi$
$941$	− 1150.83i	− 1.22299i	0.791250	−	0.611493i	$-0.209431\pi$
$942$	0	0
$943$	− 144.602i	− 0.153342i
$944$	− 364.054i	− 0.385650i
$945$	0	0
$946$	688.758	0.728074
$947$	−480.014	−0.506878	−0.253439	−	0.967351i	$-0.581562\pi$
$947$	−480.014	−0.506878	−0.253439	+	0.967351i	$0.581562\pi$
$948$	0	0
$949$	264.655	0.278878
$950$	0	0
$951$	0	0
$952$	4314.89i	4.53245i
$953$	1356.84	1.42375	0.711877	−	0.702304i	$-0.247845\pi$
$953$	1356.84	1.42375	0.711877	+	0.702304i	$0.247845\pi$
$954$	0	0
$955$	0	0
$956$	− 362.854i	− 0.379554i
$957$	0	0
$958$	1939.82i	2.02486i
$959$	888.205i	0.926178i
$960$	0	0
$961$	−166.254	−0.173001
$962$	−4977.54	−5.17416
$963$	0	0
$964$	−811.092	−0.841382
$965$	0	0
$966$	0	0
$967$	− 364.891i	− 0.377343i	−0.982040	−	0.188671i	$-0.939582\pi$
$967$	− 364.891i	− 0.377343i	0.982040	−	0.188671i	$-0.0604181\pi$
$968$	2518.54	2.60179
$969$	0	0
$970$	0	0
$971$	1095.60i	1.12832i	0.825666	+	0.564160i	$0.190800\pi$
$971$	1095.60i	1.12832i	−0.825666	+	0.564160i	$0.809200\pi$
$972$	0	0
$973$	292.019i	0.300122i
$974$	− 372.400i	− 0.382341i
$975$	0	0
$976$	2028.73	2.07861
$977$	−207.952	−0.212848	−0.106424	−	0.994321i	$-0.533940\pi$
$977$	−207.952	−0.212848	−0.106424	+	0.994321i	$0.533940\pi$
$978$	0	0
$979$	990.606	1.01185
$980$	0	0
$981$	0	0
$982$	1187.48i	1.20925i
$983$	−1385.34	−1.40930	−0.704649	−	0.709556i	$-0.748896\pi$
$983$	−1385.34	−1.40930	−0.704649	+	0.709556i	$0.748896\pi$
$984$	0	0
$985$	0	0
$986$	242.435i	0.245877i
$987$	0	0
$988$	− 464.257i	− 0.469895i
$989$	− 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	$0.800477\pi$
$992$	1652.96	1.66629
$993$	0	0
$994$	32.0780	0.0322717
$995$	0	0
$996$	0	0
$997$	− 292.605i	− 0.293485i	−0.989175	−	0.146742i	$-0.953121\pi$
$997$	− 292.605i	− 0.293485i	0.989175	−	0.146742i	$-0.0468789\pi$
$998$	−1184.98	−1.18735
$999$	0	0

Twists

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

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

Level:	\( N \)	\(=\)	\( 675 = 3^{3} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	675.d (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-3.69399 q^{2} +9.64560 q^{4} +9.20921i q^{7} -20.8548 q^{8} +15.5488i q^{11} +20.6004i q^{13} -34.0188i q^{14} +38.4552 q^{16} -22.4668 q^{17} -2.33644 q^{19} -57.4372i q^{22} -19.3092 q^{23} -76.0976i q^{26} +88.8283i q^{28} +2.92117i q^{29} -28.1912 q^{31} -58.6339 q^{32} +82.9923 q^{34} -65.4099i q^{37} +8.63079 q^{38} +7.48875i q^{41} +11.9915i q^{43} +149.978i q^{44} +71.3279 q^{46} +50.5075 q^{47} -35.8096 q^{49} +198.703i q^{52} +80.8323 q^{53} -192.056i q^{56} -10.7908i q^{58} -9.46697i q^{59} +52.7557 q^{61} +104.138 q^{62} +62.7728 q^{64} -65.2195i q^{67} -216.706 q^{68} +0.942951i q^{71} -12.8471i q^{73} +241.624i q^{74} -22.5363 q^{76} -143.192 q^{77} -68.4459 q^{79} -27.6634i q^{82} -38.9905 q^{83} -44.2965i q^{86} -324.267i q^{88} -63.7095i q^{89} -189.713 q^{91} -186.248 q^{92} -186.575 q^{94} -186.330i q^{97} +132.280 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 6 q^{2} + 14 q^{4} - 42 q^{8} + 46 q^{16} - 84 q^{17} + 16 q^{19} - 102 q^{23} - 56 q^{31} - 174 q^{32} + 80 q^{34} - 96 q^{38} + 234 q^{46} - 138 q^{47} - 74 q^{49} - 120 q^{53} - 46 q^{61} + 36 q^{62}+ \cdots + 318 q^{98}+O(q^{100}) \) 6 * q - 6 * q^2 + 14 * q^4 - 42 * q^8 + 46 * q^16 - 84 * q^17 + 16 * q^19 - 102 * q^23 - 56 * q^31 - 174 * q^32 + 80 * q^34 - 96 * q^38 + 234 * q^46 - 138 * q^47 - 74 * q^49 - 120 * q^53 - 46 * q^61 + 36 * q^62 + 342 * q^64 - 432 * q^68 - 156 * q^76 - 492 * q^77 - 316 * q^79 - 324 * q^83 + 44 * q^91 - 210 * q^92 - 526 * q^94 + 318 * q^98

Introduction

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