LMFDB - Embedded newform 675.3.d.e.674.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [675,3,Mod(674,675)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

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")

//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);

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: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$18.3924178443$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{5})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 3x^{2} + 1 $ x^4 + 3*x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 135)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			674.3
Root			$-0.618034i$ of defining polynomial
Character	$\chi$	$=$	675.674
Dual form			675.3.d.e.674.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.381966 q^{2} -3.85410 q^{4} -3.70820i q^{7} +3.00000 q^{8} +O(q^{10})$	$q-0.381966 q^{2} -3.85410 q^{4} -3.70820i q^{7} +3.00000 q^{8} +8.18034i q^{11} +7.70820i q^{13} +1.41641i q^{14} +14.2705 q^{16} -11.9443 q^{17} -5.58359 q^{19} -3.12461i q^{22} +28.4164 q^{23} -2.94427i q^{26} +14.2918i q^{28} -56.0689i q^{29} -31.2492 q^{31} -17.4508 q^{32} +4.56231 q^{34} +53.4164i q^{37} +2.13274 q^{38} -60.7639i q^{41} -71.3738i q^{43} -31.5279i q^{44} -10.8541 q^{46} +46.1378 q^{47} +35.2492 q^{49} -29.7082i q^{52} +73.3607 q^{53} -11.1246i q^{56} +21.4164i q^{58} -90.7639i q^{59} +19.0000 q^{61} +11.9361 q^{62} -50.4164 q^{64} -0.334369i q^{67} +46.0344 q^{68} -81.2624i q^{71} +50.7902i q^{73} -20.4033i q^{74} +21.5197 q^{76} +30.3344 q^{77} +48.1672 q^{79} +23.2098i q^{82} +62.6656 q^{83} +27.2624i q^{86} +24.5410i q^{88} -69.7082i q^{89} +28.5836 q^{91} -109.520 q^{92} -17.6231 q^{94} -159.416i q^{97} -13.4640 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 6 q^{2} - 2 q^{4} + 12 q^{8}+O(q^{10}) $ 4 * q - 6 * q^2 - 2 * q^4 + 12 * q^8	$ 4 q - 6 q^{2} - 2 q^{4} + 12 q^{8} - 10 q^{16} - 12 q^{17} - 76 q^{19} + 60 q^{23} + 36 q^{31} + 42 q^{32} - 22 q^{34} + 174 q^{38} - 30 q^{46} - 48 q^{47} - 20 q^{49} + 204 q^{53} + 76 q^{61} - 234 q^{62} - 148 q^{64} + 126 q^{68} - 142 q^{76} + 336 q^{77} + 300 q^{79} + 36 q^{83} + 168 q^{91} - 210 q^{92} + 332 q^{94} + 210 q^{98}+O(q^{100}) $ 4 * q - 6 * q^2 - 2 * q^4 + 12 * q^8 - 10 * q^16 - 12 * q^17 - 76 * q^19 + 60 * q^23 + 36 * q^31 + 42 * q^32 - 22 * q^34 + 174 * q^38 - 30 * q^46 - 48 * q^47 - 20 * q^49 + 204 * q^53 + 76 * q^61 - 234 * q^62 - 148 * q^64 + 126 * q^68 - 142 * q^76 + 336 * q^77 + 300 * q^79 + 36 * q^83 + 168 * q^91 - 210 * q^92 + 332 * q^94 + 210 * q^98

Display 100 coefficients 10 coefficients

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 $

$2$	−0.381966	−0.190983	−0.0954915	−	0.995430i	$-0.530442\pi$
$2$	−0.381966	−0.190983	−0.0954915	+	0.995430i	$0.530442\pi$
$3$	0	0
$3$	0	0
$4$	−3.85410	−0.963525
$5$	0	0
$5$	0	0
$6$	0	0
$7$	− 3.70820i	− 0.529743i	−0.964284	−	0.264872i	$-0.914670\pi$
$7$	− 3.70820i	− 0.529743i	0.964284	−	0.264872i	$-0.0853296\pi$
$8$	3.00000	0.375000
$9$	0	0
$10$	0	0
$11$	8.18034i	0.743667i	0.928299	+	0.371834i	$0.121271\pi$
$11$	8.18034i	0.743667i	−0.928299	+	0.371834i	$0.878729\pi$
$12$	0	0
$13$	7.70820i	0.592939i	0.955042	+	0.296469i	$0.0958093\pi$
$13$	7.70820i	0.592939i	−0.955042	+	0.296469i	$0.904191\pi$
$14$	1.41641i	0.101172i
$15$	0	0
$16$	14.2705	0.891907
$17$	−11.9443	−0.702604	−0.351302	−	0.936262i	$-0.614261\pi$
$17$	−11.9443	−0.702604	−0.351302	+	0.936262i	$0.614261\pi$
$18$	0	0
$19$	−5.58359	−0.293873	−0.146937	−	0.989146i	$-0.546941\pi$
$19$	−5.58359	−0.293873	−0.146937	+	0.989146i	$0.546941\pi$
$20$	0	0
$21$	0	0
$22$	− 3.12461i	− 0.142028i
$23$	28.4164	1.23550	0.617748	−	0.786376i	$-0.288045\pi$
$23$	28.4164	1.23550	0.617748	+	0.786376i	$0.288045\pi$
$24$	0	0
$25$	0	0
$26$	− 2.94427i	− 0.113241i
$27$	0	0
$28$	14.2918i	0.510421i
$29$	− 56.0689i	− 1.93341i	−0.255894	−	0.966705i	$-0.582370\pi$
$29$	− 56.0689i	− 1.93341i	0.255894	−	0.966705i	$-0.417630\pi$
$30$	0	0
$31$	−31.2492	−1.00804	−0.504020	−	0.863692i	$-0.668146\pi$
$31$	−31.2492	−1.00804	−0.504020	+	0.863692i	$0.668146\pi$
$32$	−17.4508	−0.545339
$33$	0	0
$34$	4.56231	0.134185
$35$	0	0
$36$	0	0
$37$	53.4164i	1.44369i	0.692056	+	0.721843i	$0.256705\pi$
$37$	53.4164i	1.44369i	−0.692056	+	0.721843i	$0.743295\pi$
$38$	2.13274	0.0561248
$39$	0	0
$40$	0	0
$41$	− 60.7639i	− 1.48205i	−0.671479	−	0.741024i	$-0.734341\pi$
$41$	− 60.7639i	− 1.48205i	0.671479	−	0.741024i	$-0.265659\pi$
$42$	0	0
$43$	− 71.3738i	− 1.65986i	−0.557870	−	0.829928i	$-0.688381\pi$
$43$	− 71.3738i	− 1.65986i	0.557870	−	0.829928i	$-0.311619\pi$
$44$	− 31.5279i	− 0.716542i
$45$	0	0
$46$	−10.8541	−0.235959
$47$	46.1378	0.981655	0.490827	−	0.871257i	$-0.336695\pi$
$47$	46.1378	0.981655	0.490827	+	0.871257i	$0.336695\pi$
$48$	0	0
$49$	35.2492	0.719372
$50$	0	0
$51$	0	0
$52$	− 29.7082i	− 0.571312i
$53$	73.3607	1.38416	0.692082	−	0.721819i	$-0.256694\pi$
$53$	73.3607	1.38416	0.692082	+	0.721819i	$0.256694\pi$
$54$	0	0
$55$	0	0
$56$	− 11.1246i	− 0.198654i
$57$	0	0
$58$	21.4164i	0.369248i
$59$	− 90.7639i	− 1.53837i	−0.639025	−	0.769186i	$-0.720662\pi$
$59$	− 90.7639i	− 1.53837i	0.639025	−	0.769186i	$-0.279338\pi$
$60$	0	0
$61$	19.0000	0.311475	0.155738	−	0.987798i	$-0.450225\pi$
$61$	19.0000	0.311475	0.155738	+	0.987798i	$0.450225\pi$
$62$	11.9361	0.192518
$63$	0	0
$64$	−50.4164	−0.787756
$65$	0	0
$66$	0	0
$67$	− 0.334369i	− 0.00499058i	−0.999997	−	0.00249529i	$-0.999206\pi$
$67$	− 0.334369i	− 0.00499058i	0.999997	−	0.00249529i	$-0.000794276\pi$
$68$	46.0344	0.676977
$69$	0	0
$70$	0	0
$71$	− 81.2624i	− 1.14454i	−0.820065	−	0.572270i	$-0.806063\pi$
$71$	− 81.2624i	− 1.14454i	0.820065	−	0.572270i	$-0.193937\pi$
$72$	0	0
$73$	50.7902i	0.695757i	0.937540	+	0.347878i	$0.113098\pi$
$73$	50.7902i	0.695757i	−0.937540	+	0.347878i	$0.886902\pi$
$74$	− 20.4033i	− 0.275720i
$75$	0	0
$76$	21.5197	0.283154
$77$	30.3344	0.393953
$78$	0	0
$79$	48.1672	0.609711	0.304856	−	0.952399i	$-0.401392\pi$
$79$	48.1672	0.609711	0.304856	+	0.952399i	$0.401392\pi$
$80$	0	0
$81$	0	0
$82$	23.2098i	0.283046i
$83$	62.6656	0.755008	0.377504	−	0.926008i	$-0.376783\pi$
$83$	62.6656	0.755008	0.377504	+	0.926008i	$0.376783\pi$
$84$	0	0
$85$	0	0
$86$	27.2624i	0.317004i
$87$	0	0
$88$	24.5410i	0.278875i
$89$	− 69.7082i	− 0.783238i	−0.920127	−	0.391619i	$-0.871915\pi$
$89$	− 69.7082i	− 0.783238i	0.920127	−	0.391619i	$-0.128085\pi$
$90$	0	0
$91$	28.5836	0.314105
$92$	−109.520	−1.19043
$93$	0	0
$94$	−17.6231	−0.187479
$95$	0	0
$96$	0	0
$97$	− 159.416i	− 1.64347i	−0.569871	−	0.821734i	$-0.693007\pi$
$97$	− 159.416i	− 1.64347i	0.569871	−	0.821734i	$-0.306993\pi$
$98$	−13.4640	−0.137388
$99$	0	0
$100$	0	0
$101$	− 24.0000i	− 0.237624i	−0.992917	−	0.118812i	$-0.962091\pi$
$101$	− 24.0000i	− 0.237624i	0.992917	−	0.118812i	$-0.0379085\pi$
$102$	0	0
$103$	− 25.1672i	− 0.244342i	−0.992509	−	0.122171i	$-0.961014\pi$
$103$	− 25.1672i	− 0.244342i	0.992509	−	0.122171i	$-0.0389856\pi$
$104$	23.1246i	0.222352i
$105$	0	0
$106$	−28.0213	−0.264352
$107$	−47.6656	−0.445473	−0.222737	−	0.974879i	$-0.571499\pi$
$107$	−47.6656	−0.445473	−0.222737	+	0.974879i	$0.571499\pi$
$108$	0	0
$109$	−164.915	−1.51298	−0.756490	−	0.654005i	$-0.773087\pi$
$109$	−164.915	−1.51298	−0.756490	+	0.654005i	$0.773087\pi$
$110$	0	0
$111$	0	0
$112$	− 52.9180i	− 0.472482i
$113$	−68.3870	−0.605195	−0.302597	−	0.953118i	$-0.597854\pi$
$113$	−68.3870	−0.605195	−0.302597	+	0.953118i	$0.597854\pi$
$114$	0	0
$115$	0	0
$116$	216.095i	1.86289i
$117$	0	0
$118$	34.6687i	0.293803i
$119$	44.2918i	0.372200i
$120$	0	0
$121$	54.0820	0.446959
$122$	−7.25735	−0.0594865
$123$	0	0
$124$	120.438	0.971272
$125$	0	0
$126$	0	0
$127$	200.748i	1.58069i	0.612662	+	0.790345i	$0.290099\pi$
$127$	200.748i	1.58069i	−0.612662	+	0.790345i	$0.709901\pi$
$128$	89.0608	0.695787
$129$	0	0
$130$	0	0
$131$	8.16408i	0.0623212i	0.999514	+	0.0311606i	$0.00992033\pi$
$131$	8.16408i	0.0623212i	−0.999514	+	0.0311606i	$0.990080\pi$
$132$	0	0
$133$	20.7051i	0.155677i
$134$	0.127717i	0 0.000953115i
$135$	0	0
$136$	−35.8328	−0.263477
$137$	−50.3313	−0.367381	−0.183691	−	0.982984i	$-0.558804\pi$
$137$	−50.3313	−0.367381	−0.183691	+	0.982984i	$0.558804\pi$
$138$	0	0
$139$	126.498	0.910061	0.455030	−	0.890476i	$-0.349628\pi$
$139$	126.498	0.910061	0.455030	+	0.890476i	$0.349628\pi$
$140$	0	0
$141$	0	0
$142$	31.0395i	0.218588i
$143$	−63.0557	−0.440949
$144$	0	0
$145$	0	0
$146$	− 19.4001i	− 0.132878i
$147$	0	0
$148$	− 205.872i	− 1.39103i
$149$	− 15.8197i	− 0.106172i	−0.998590	−	0.0530861i	$-0.983094\pi$
$149$	− 15.8197i	− 0.106172i	0.998590	−	0.0530861i	$-0.0169058\pi$
$150$	0	0
$151$	−138.997	−0.920509	−0.460255	−	0.887787i	$-0.652242\pi$
$151$	−138.997	−0.920509	−0.460255	+	0.887787i	$0.652242\pi$
$152$	−16.7508	−0.110202
$153$	0	0
$154$	−11.5867	−0.0752383
$155$	0	0
$156$	0	0
$157$	− 190.705i	− 1.21468i	−0.794441	−	0.607341i	$-0.792236\pi$
$157$	− 190.705i	− 1.21468i	0.794441	−	0.607341i	$-0.207764\pi$
$158$	−18.3982	−0.116444
$159$	0	0
$160$	0	0
$161$	− 105.374i	− 0.654496i
$162$	0	0
$163$	− 36.8328i	− 0.225968i	−0.993597	−	0.112984i	$-0.963959\pi$
$163$	− 36.8328i	− 0.225968i	0.993597	−	0.112984i	$-0.0360409\pi$
$164$	234.190i	1.42799i
$165$	0	0
$166$	−23.9361	−0.144194
$167$	258.580	1.54839	0.774193	−	0.632950i	$-0.218156\pi$
$167$	258.580	1.54839	0.774193	+	0.632950i	$0.218156\pi$
$168$	0	0
$169$	109.584	0.648424
$170$	0	0
$171$	0	0
$172$	275.082i	1.59931i
$173$	−182.331	−1.05394	−0.526969	−	0.849885i	$-0.676672\pi$
$173$	−182.331	−1.05394	−0.526969	+	0.849885i	$0.676672\pi$
$174$	0	0
$175$	0	0
$176$	116.738i	0.663282i
$177$	0	0
$178$	26.6262i	0.149585i
$179$	256.774i	1.43449i	0.696820	+	0.717246i	$0.254598\pi$
$179$	256.774i	1.43449i	−0.696820	+	0.717246i	$0.745402\pi$
$180$	0	0
$181$	180.082	0.994928	0.497464	−	0.867485i	$-0.334265\pi$
$181$	180.082	0.994928	0.497464	+	0.867485i	$0.334265\pi$
$182$	−10.9180	−0.0599888
$183$	0	0
$184$	85.2492	0.463311
$185$	0	0
$186$	0	0
$187$	− 97.7082i	− 0.522504i
$188$	−177.820	−0.945849
$189$	0	0
$190$	0	0
$191$	− 128.387i	− 0.672183i	−0.941829	−	0.336092i	$-0.890895\pi$
$191$	− 128.387i	− 0.672183i	0.941829	−	0.336092i	$-0.109105\pi$
$192$	0	0
$193$	64.2918i	0.333118i	0.986031	+	0.166559i	$0.0532656\pi$
$193$	64.2918i	0.333118i	−0.986031	+	0.166559i	$0.946734\pi$
$194$	60.8916i	0.313874i
$195$	0	0
$196$	−135.854	−0.693133
$197$	217.997	1.10658	0.553292	−	0.832988i	$-0.313372\pi$
$197$	217.997	1.10658	0.553292	+	0.832988i	$0.313372\pi$
$198$	0	0
$199$	−320.249	−1.60929	−0.804646	−	0.593754i	$-0.797645\pi$
$199$	−320.249	−1.60929	−0.804646	+	0.593754i	$0.797645\pi$
$200$	0	0
$201$	0	0
$202$	9.16718i	0.0453821i
$203$	−207.915	−1.02421
$204$	0	0
$205$	0	0
$206$	9.61301i	0.0466651i
$207$	0	0
$208$	110.000i	0.528846i
$209$	− 45.6757i	− 0.218544i
$210$	0	0
$211$	−63.8328	−0.302525	−0.151263	−	0.988494i	$-0.548334\pi$
$211$	−63.8328	−0.302525	−0.151263	+	0.988494i	$0.548334\pi$
$212$	−282.740	−1.33368
$213$	0	0
$214$	18.2067	0.0850778
$215$	0	0
$216$	0	0
$217$	115.878i	0.534002i
$218$	62.9919	0.288954
$219$	0	0
$220$	0	0
$221$	− 92.0689i	− 0.416601i
$222$	0	0
$223$	− 8.24922i	− 0.0369920i	−0.999829	−	0.0184960i	$-0.994112\pi$
$223$	− 8.24922i	− 0.0369920i	0.999829	−	0.0184960i	$-0.00588780\pi$
$224$	64.7113i	0.288890i
$225$	0	0
$226$	26.1215	0.115582
$227$	339.525	1.49570	0.747852	−	0.663866i	$-0.231085\pi$
$227$	339.525	1.49570	0.747852	+	0.663866i	$0.231085\pi$
$228$	0	0
$229$	−202.495	−0.884259	−0.442130	−	0.896951i	$-0.645777\pi$
$229$	−202.495	−0.884259	−0.442130	+	0.896951i	$0.645777\pi$
$230$	0	0
$231$	0	0
$232$	− 168.207i	− 0.725029i
$233$	19.0820	0.0818972	0.0409486	−	0.999161i	$-0.486962\pi$
$233$	19.0820	0.0818972	0.0409486	+	0.999161i	$0.486962\pi$
$234$	0	0
$235$	0	0
$236$	349.813i	1.48226i
$237$	0	0
$238$	− 16.9180i	− 0.0710839i
$239$	272.610i	1.14063i	0.821427	+	0.570314i	$0.193178\pi$
$239$	272.610i	1.14063i	−0.821427	+	0.570314i	$0.806822\pi$
$240$	0	0
$241$	232.580	0.965064	0.482532	−	0.875878i	$-0.339717\pi$
$241$	232.580	0.965064	0.482532	+	0.875878i	$0.339717\pi$
$242$	−20.6575	−0.0853616
$243$	0	0
$244$	−73.2279	−0.300114
$245$	0	0
$246$	0	0
$247$	− 43.0395i	− 0.174249i
$248$	−93.7477	−0.378015
$249$	0	0
$250$	0	0
$251$	− 14.4195i	− 0.0574483i	−0.999587	−	0.0287241i	$-0.990856\pi$
$251$	− 14.4195i	− 0.0574483i	0.999587	−	0.0287241i	$-0.00914443\pi$
$252$	0	0
$253$	232.456i	0.918798i
$254$	− 76.6788i	− 0.301885i
$255$	0	0
$256$	167.647	0.654873
$257$	−165.748	−0.644933	−0.322466	−	0.946581i	$-0.604512\pi$
$257$	−165.748	−0.644933	−0.322466	+	0.946581i	$0.604512\pi$
$258$	0	0
$259$	198.079	0.764784
$260$	0	0
$261$	0	0
$262$	− 3.11840i	− 0.0119023i
$263$	−520.026	−1.97729	−0.988643	−	0.150282i	$-0.951982\pi$
$263$	−520.026	−1.97729	−0.988643	+	0.150282i	$0.951982\pi$
$264$	0	0
$265$	0	0
$266$	− 7.90864i	− 0.0297317i
$267$	0	0
$268$	1.28869i	0.00480855i
$269$	87.3576i	0.324749i	0.986729	+	0.162375i	$0.0519153\pi$
$269$	87.3576i	0.324749i	−0.986729	+	0.162375i	$0.948085\pi$
$270$	0	0
$271$	35.5836	0.131305	0.0656524	−	0.997843i	$-0.479087\pi$
$271$	35.5836	0.131305	0.0656524	+	0.997843i	$0.479087\pi$
$272$	−170.451	−0.626658
$273$	0	0
$274$	19.2248	0.0701636
$275$	0	0
$276$	0	0
$277$	− 147.374i	− 0.532036i	−0.963968	−	0.266018i	$-0.914292\pi$
$277$	− 147.374i	− 0.532036i	0.963968	−	0.266018i	$-0.0857080\pi$
$278$	−48.3181	−0.173806
$279$	0	0
$280$	0	0
$281$	190.869i	0.679250i	0.940561	+	0.339625i	$0.110300\pi$
$281$	190.869i	0.679250i	−0.940561	+	0.339625i	$0.889700\pi$
$282$	0	0
$283$	117.708i	0.415930i	0.978136	+	0.207965i	$0.0666840\pi$
$283$	117.708i	0.415930i	−0.978136	+	0.207965i	$0.933316\pi$
$284$	313.193i	1.10279i
$285$	0	0
$286$	24.0851	0.0842138
$287$	−225.325	−0.785105
$288$	0	0
$289$	−146.334	−0.506347
$290$	0	0
$291$	0	0
$292$	− 195.751i	− 0.670379i
$293$	413.243	1.41039	0.705193	−	0.709016i	$-0.250860\pi$
$293$	413.243	1.41039	0.705193	+	0.709016i	$0.250860\pi$
$294$	0	0
$295$	0	0
$296$	160.249i	0.541383i
$297$	0	0
$298$	6.04257i	0.0202771i
$299$	219.039i	0.732573i
$300$	0	0
$301$	−264.669	−0.879298
$302$	53.0921	0.175802
$303$	0	0
$304$	−79.6807	−0.262108
$305$	0	0
$306$	0	0
$307$	458.158i	1.49237i	0.665738	+	0.746185i	$0.268117\pi$
$307$	458.158i	1.49237i	−0.665738	+	0.746185i	$0.731883\pi$
$308$	−116.912	−0.379584
$309$	0	0
$310$	0	0
$311$	118.043i	0.379558i	0.981827	+	0.189779i	$0.0607772\pi$
$311$	118.043i	0.379558i	−0.981827	+	0.189779i	$0.939223\pi$
$312$	0	0
$313$	− 412.827i	− 1.31893i	−0.751733	−	0.659467i	$-0.770782\pi$
$313$	− 412.827i	− 1.31893i	0.751733	−	0.659467i	$-0.229218\pi$
$314$	72.8429i	0.231984i
$315$	0	0
$316$	−185.641	−0.587472
$317$	269.944	0.851559	0.425780	−	0.904827i	$-0.360000\pi$
$317$	269.944	0.851559	0.425780	+	0.904827i	$0.360000\pi$
$318$	0	0
$319$	458.663	1.43781
$320$	0	0
$321$	0	0
$322$	40.2492i	0.124998i
$323$	66.6919	0.206477
$324$	0	0
$325$	0	0
$326$	14.0689i	0.0431561i
$327$	0	0
$328$	− 182.292i	− 0.555768i
$329$	− 171.088i	− 0.520025i
$330$	0	0
$331$	9.91486	0.0299542	0.0149771	−	0.999888i	$-0.495232\pi$
$331$	9.91486	0.0299542	0.0149771	+	0.999888i	$0.495232\pi$
$332$	−241.520	−0.727469
$333$	0	0
$334$	−98.7690	−0.295715
$335$	0	0
$336$	0	0
$337$	436.450i	1.29510i	0.762022	+	0.647551i	$0.224207\pi$
$337$	436.450i	1.29510i	−0.762022	+	0.647551i	$0.775793\pi$
$338$	−41.8572	−0.123838
$339$	0	0
$340$	0	0
$341$	− 255.629i	− 0.749646i
$342$	0	0
$343$	− 312.413i	− 0.910826i
$344$	− 214.122i	− 0.622446i
$345$	0	0
$346$	69.6443	0.201284
$347$	25.3839	0.0731524	0.0365762	−	0.999331i	$-0.488355\pi$
$347$	25.3839	0.0731524	0.0365762	+	0.999331i	$0.488355\pi$
$348$	0	0
$349$	72.0820	0.206539	0.103269	−	0.994653i	$-0.467070\pi$
$349$	72.0820	0.206539	0.103269	+	0.994653i	$0.467070\pi$
$350$	0	0
$351$	0	0
$352$	− 142.754i	− 0.405551i
$353$	392.164	1.11095	0.555473	−	0.831534i	$-0.312537\pi$
$353$	392.164	1.11095	0.555473	+	0.831534i	$0.312537\pi$
$354$	0	0
$355$	0	0
$356$	268.663i	0.754670i
$357$	0	0
$358$	− 98.0789i	− 0.273964i
$359$	− 399.994i	− 1.11419i	−0.830449	−	0.557094i	$-0.811916\pi$
$359$	− 399.994i	− 1.11419i	0.830449	−	0.557094i	$-0.188084\pi$
$360$	0	0
$361$	−329.823	−0.913639
$362$	−68.7852	−0.190014
$363$	0	0
$364$	−110.164	−0.302649
$365$	0	0
$366$	0	0
$367$	− 206.371i	− 0.562318i	−0.959661	−	0.281159i	$-0.909281\pi$
$367$	− 206.371i	− 0.562318i	0.959661	−	0.281159i	$-0.0907188\pi$
$368$	405.517	1.10195
$369$	0	0
$370$	0	0
$371$	− 272.036i	− 0.733252i
$372$	0	0
$373$	354.833i	0.951294i	0.879636	+	0.475647i	$0.157786\pi$
$373$	354.833i	0.951294i	−0.879636	+	0.475647i	$0.842214\pi$
$374$	37.3212i	0.0997893i
$375$	0	0
$376$	138.413	0.368120
$377$	432.190	1.14639
$378$	0	0
$379$	423.997	1.11873	0.559363	−	0.828923i	$-0.311046\pi$
$379$	423.997	1.11873	0.559363	+	0.828923i	$0.311046\pi$
$380$	0	0
$381$	0	0
$382$	49.0395i	0.128376i
$383$	−207.158	−0.540882	−0.270441	−	0.962737i	$-0.587170\pi$
$383$	−207.158	−0.540882	−0.270441	+	0.962737i	$0.587170\pi$
$384$	0	0
$385$	0	0
$386$	− 24.5573i	− 0.0636199i
$387$	0	0
$388$	614.407i	1.58352i
$389$	− 439.957i	− 1.13100i	−0.824750	−	0.565498i	$-0.808684\pi$
$389$	− 439.957i	− 1.13100i	0.824750	−	0.565498i	$-0.191316\pi$
$390$	0	0
$391$	−339.413	−0.868065
$392$	105.748	0.269764
$393$	0	0
$394$	−83.2674	−0.211339
$395$	0	0
$396$	0	0
$397$	− 453.872i	− 1.14326i	−0.820513	−	0.571628i	$-0.806312\pi$
$397$	− 453.872i	− 1.14326i	0.820513	−	0.571628i	$-0.193688\pi$
$398$	122.324	0.307348
$399$	0	0
$400$	0	0
$401$	− 556.407i	− 1.38755i	−0.720192	−	0.693774i	$-0.755946\pi$
$401$	− 556.407i	− 1.38755i	0.720192	−	0.693774i	$-0.244054\pi$
$402$	0	0
$403$	− 240.875i	− 0.597706i
$404$	92.4984i	0.228957i
$405$	0	0
$406$	79.4164	0.195607
$407$	−436.964	−1.07362
$408$	0	0
$409$	−75.6625	−0.184994	−0.0924970	−	0.995713i	$-0.529485\pi$
$409$	−75.6625	−0.184994	−0.0924970	+	0.995713i	$0.529485\pi$
$410$	0	0
$411$	0	0
$412$	96.9969i	0.235429i
$413$	−336.571	−0.814942
$414$	0	0
$415$	0	0
$416$	− 134.515i	− 0.323353i
$417$	0	0
$418$	17.4466i	0.0417382i
$419$	− 521.060i	− 1.24358i	−0.783184	−	0.621789i	$-0.786406\pi$
$419$	− 521.060i	− 1.24358i	0.783184	−	0.621789i	$-0.213594\pi$
$420$	0	0
$421$	114.745	0.272552	0.136276	−	0.990671i	$-0.456487\pi$
$421$	114.745	0.272552	0.136276	+	0.990671i	$0.456487\pi$
$422$	24.3820	0.0577772
$423$	0	0
$424$	220.082	0.519061
$425$	0	0
$426$	0	0
$427$	− 70.4559i	− 0.165002i
$428$	183.708	0.429225
$429$	0	0
$430$	0	0
$431$	− 233.204i	− 0.541076i	−0.962709	−	0.270538i	$-0.912799\pi$
$431$	− 233.204i	− 0.541076i	0.962709	−	0.270538i	$-0.0872015\pi$
$432$	0	0
$433$	− 501.702i	− 1.15867i	−0.815091	−	0.579333i	$-0.803313\pi$
$433$	− 501.702i	− 1.15867i	0.815091	−	0.579333i	$-0.196687\pi$
$434$	− 44.2616i	− 0.101985i
$435$	0	0
$436$	635.599	1.45780
$437$	−158.666	−0.363079
$438$	0	0
$439$	−365.997	−0.833706	−0.416853	−	0.908974i	$-0.636867\pi$
$439$	−365.997	−0.833706	−0.416853	+	0.908974i	$0.636867\pi$
$440$	0	0
$441$	0	0
$442$	35.1672i	0.0795638i
$443$	632.902	1.42867	0.714337	−	0.699802i	$-0.246728\pi$
$443$	632.902	1.42867	0.714337	+	0.699802i	$0.246728\pi$
$444$	0	0
$445$	0	0
$446$	3.15092i	0.00706485i
$447$	0	0
$448$	186.954i	0.417309i
$449$	− 796.344i	− 1.77360i	−0.462158	−	0.886798i	$-0.652925\pi$
$449$	− 796.344i	− 1.77360i	0.462158	−	0.886798i	$-0.347075\pi$
$450$	0	0
$451$	497.070	1.10215
$452$	263.570	0.583120
$453$	0	0
$454$	−129.687	−0.285654
$455$	0	0
$456$	0	0
$457$	202.577i	0.443277i	0.975129	+	0.221638i	$0.0711404\pi$
$457$	202.577i	0.443277i	−0.975129	+	0.221638i	$0.928860\pi$
$458$	77.3463	0.168878
$459$	0	0
$460$	0	0
$461$	− 727.909i	− 1.57898i	−0.613765	−	0.789489i	$-0.710346\pi$
$461$	− 727.909i	− 1.57898i	0.613765	−	0.789489i	$-0.289654\pi$
$462$	0	0
$463$	493.003i	1.06480i	0.846492	+	0.532401i	$0.178710\pi$
$463$	493.003i	1.06480i	−0.846492	+	0.532401i	$0.821290\pi$
$464$	− 800.132i	− 1.72442i
$465$	0	0
$466$	−7.28869	−0.0156410
$467$	453.190	0.970429	0.485215	−	0.874395i	$-0.338741\pi$
$467$	453.190	0.970429	0.485215	+	0.874395i	$0.338741\pi$
$468$	0	0
$469$	−1.23991	−0.00264372
$470$	0	0
$471$	0	0
$472$	− 272.292i	− 0.576889i
$473$	583.862	1.23438
$474$	0	0
$475$	0	0
$476$	− 170.705i	− 0.358624i
$477$	0	0
$478$	− 104.128i	− 0.217840i
$479$	319.544i	0.667107i	0.942731	+	0.333553i	$0.108248\pi$
$479$	319.544i	0.667107i	−0.942731	+	0.333553i	$0.891752\pi$
$480$	0	0
$481$	−411.745	−0.856018
$482$	−88.8378	−0.184311
$483$	0	0
$484$	−208.438	−0.430656
$485$	0	0
$486$	0	0
$487$	− 792.869i	− 1.62807i	−0.580817	−	0.814034i	$-0.697267\pi$
$487$	− 792.869i	− 1.62807i	0.580817	−	0.814034i	$-0.302733\pi$
$488$	57.0000	0.116803
$489$	0	0
$490$	0	0
$491$	369.437i	0.752417i	0.926535	+	0.376208i	$0.122772\pi$
$491$	369.437i	0.752417i	−0.926535	+	0.376208i	$0.877228\pi$
$492$	0	0
$493$	669.702i	1.35842i
$494$	16.4396i	0.0332786i
$495$	0	0
$496$	−445.942	−0.899077
$497$	−301.337	−0.606313
$498$	0	0
$499$	−609.912	−1.22227	−0.611134	−	0.791527i	$-0.709286\pi$
$499$	−609.912	−1.22227	−0.611134	+	0.791527i	$0.709286\pi$
$500$	0	0
$501$	0	0
$502$	5.50776i	0.0109716i
$503$	305.308	0.606974	0.303487	−	0.952836i	$-0.401849\pi$
$503$	305.308	0.606974	0.303487	+	0.952836i	$0.401849\pi$
$504$	0	0
$505$	0	0
$506$	− 88.7902i	− 0.175475i
$507$	0	0
$508$	− 773.702i	− 1.52304i
$509$	− 204.478i	− 0.401726i	−0.979619	−	0.200863i	$-0.935625\pi$
$509$	− 204.478i	− 0.401726i	0.979619	−	0.200863i	$-0.0643745\pi$
$510$	0	0
$511$	188.341	0.368573
$512$	−420.279	−0.820857
$513$	0	0
$514$	63.3100	0.123171
$515$	0	0
$516$	0	0
$517$	377.423i	0.730024i
$518$	−75.6594	−0.146061
$519$	0	0
$520$	0	0
$521$	590.227i	1.13287i	0.824105	+	0.566436i	$0.191678\pi$
$521$	590.227i	1.13287i	−0.824105	+	0.566436i	$0.808322\pi$
$522$	0	0
$523$	− 19.7933i	− 0.0378458i	−0.999821	−	0.0189229i	$-0.993976\pi$
$523$	− 19.7933i	− 0.0378458i	0.999821	−	0.0189229i	$-0.00602370\pi$
$524$	− 31.4652i	− 0.0600481i
$525$	0	0
$526$	198.632	0.377628
$527$	373.249	0.708253
$528$	0	0
$529$	278.492	0.526450
$530$	0	0
$531$	0	0
$532$	− 79.7996i	− 0.149999i
$533$	468.381	0.878763
$534$	0	0
$535$	0	0
$536$	− 1.00311i	− 0.00187147i
$537$	0	0
$538$	− 33.3676i	− 0.0620216i
$539$	288.351i	0.534973i
$540$	0	0
$541$	428.164	0.791431	0.395715	−	0.918373i	$-0.370497\pi$
$541$	428.164	0.791431	0.395715	+	0.918373i	$0.370497\pi$
$542$	−13.5917	−0.0250770
$543$	0	0
$544$	208.438	0.383158
$545$	0	0
$546$	0	0
$547$	982.705i	1.79654i	0.439448	+	0.898268i	$0.355174\pi$
$547$	982.705i	1.79654i	−0.439448	+	0.898268i	$0.644826\pi$
$548$	193.982	0.353981
$549$	0	0
$550$	0	0
$551$	313.066i	0.568177i
$552$	0	0
$553$	− 178.614i	− 0.322990i
$554$	56.2918i	0.101610i
$555$	0	0
$556$	−487.538	−0.876867
$557$	−310.839	−0.558059	−0.279030	−	0.960282i	$-0.590013\pi$
$557$	−310.839	−0.558059	−0.279030	+	0.960282i	$0.590013\pi$
$558$	0	0
$559$	550.164	0.984193
$560$	0	0
$561$	0	0
$562$	− 72.9055i	− 0.129725i
$563$	−850.997	−1.51154	−0.755770	−	0.654837i	$-0.772737\pi$
$563$	−850.997	−1.51154	−0.755770	+	0.654837i	$0.772737\pi$
$564$	0	0
$565$	0	0
$566$	− 44.9605i	− 0.0794356i
$567$	0	0
$568$	− 243.787i	− 0.429203i
$569$	− 292.377i	− 0.513843i	−0.966432	−	0.256922i	$-0.917292\pi$
$569$	− 292.377i	− 0.513843i	0.966432	−	0.256922i	$-0.0827083\pi$
$570$	0	0
$571$	−357.754	−0.626539	−0.313270	−	0.949664i	$-0.601424\pi$
$571$	−357.754	−0.626539	−0.313270	+	0.949664i	$0.601424\pi$
$572$	243.023	0.424866
$573$	0	0
$574$	86.0665	0.149942
$575$	0	0
$576$	0	0
$577$	− 216.456i	− 0.375140i	−0.982251	−	0.187570i	$-0.939939\pi$
$577$	− 216.456i	− 0.375140i	0.982251	−	0.187570i	$-0.0600612\pi$
$578$	55.8948	0.0967037
$579$	0	0
$580$	0	0
$581$	− 232.377i	− 0.399960i
$582$	0	0
$583$	600.115i	1.02936i
$584$	152.371i	0.260909i
$585$	0	0
$586$	−157.845	−0.269360
$587$	−578.489	−0.985501	−0.492751	−	0.870171i	$-0.664008\pi$
$587$	−578.489	−0.985501	−0.492751	+	0.870171i	$0.664008\pi$
$588$	0	0
$589$	174.483	0.296236
$590$	0	0
$591$	0	0
$592$	762.279i	1.28763i
$593$	159.971	0.269765	0.134882	−	0.990862i	$-0.456934\pi$
$593$	159.971	0.269765	0.134882	+	0.990862i	$0.456934\pi$
$594$	0	0
$595$	0	0
$596$	60.9706i	0.102300i
$597$	0	0
$598$	− 83.6656i	− 0.139909i
$599$	453.755i	0.757520i	0.925495	+	0.378760i	$0.123649\pi$
$599$	453.755i	0.757520i	−0.925495	+	0.378760i	$0.876351\pi$
$600$	0	0
$601$	−653.584	−1.08749	−0.543747	−	0.839249i	$-0.682995\pi$
$601$	−653.584	−1.08749	−0.543747	+	0.839249i	$0.682995\pi$
$602$	101.094	0.167931
$603$	0	0
$604$	535.708	0.886934
$605$	0	0
$606$	0	0
$607$	184.626i	0.304162i	0.988368	+	0.152081i	$0.0485974\pi$
$607$	184.626i	0.304162i	−0.988368	+	0.152081i	$0.951403\pi$
$608$	97.4384	0.160261
$609$	0	0
$610$	0	0
$611$	355.639i	0.582061i
$612$	0	0
$613$	− 903.416i	− 1.47376i	−0.676022	−	0.736881i	$-0.736298\pi$
$613$	− 903.416i	− 1.47376i	0.676022	−	0.736881i	$-0.263702\pi$
$614$	− 175.001i	− 0.285017i
$615$	0	0
$616$	91.0031	0.147732
$617$	−386.489	−0.626401	−0.313200	−	0.949687i	$-0.601401\pi$
$617$	−386.489	−0.626401	−0.313200	+	0.949687i	$0.601401\pi$
$618$	0	0
$619$	−1165.90	−1.88353	−0.941763	−	0.336278i	$-0.890832\pi$
$619$	−1165.90	−1.88353	−0.941763	+	0.336278i	$0.890832\pi$
$620$	0	0
$621$	0	0
$622$	− 45.0883i	− 0.0724891i
$623$	−258.492	−0.414915
$624$	0	0
$625$	0	0
$626$	157.686i	0.251894i
$627$	0	0
$628$	734.997i	1.17038i
$629$	− 638.020i	− 1.01434i
$630$	0	0
$631$	50.8297	0.0805542	0.0402771	−	0.999189i	$-0.487176\pi$
$631$	50.8297	0.0805542	0.0402771	+	0.999189i	$0.487176\pi$
$632$	144.502	0.228642
$633$	0	0
$634$	−103.110	−0.162633
$635$	0	0
$636$	0	0
$637$	271.708i	0.426543i
$638$	−175.193	−0.274598
$639$	0	0
$640$	0	0
$641$	548.754i	0.856090i	0.903757	+	0.428045i	$0.140798\pi$
$641$	548.754i	0.856090i	−0.903757	+	0.428045i	$0.859202\pi$
$642$	0	0
$643$	526.711i	0.819147i	0.912277	+	0.409573i	$0.134322\pi$
$643$	526.711i	0.819147i	−0.912277	+	0.409573i	$0.865678\pi$
$644$	406.122i	0.630623i
$645$	0	0
$646$	−25.4741	−0.0394335
$647$	−694.607	−1.07358	−0.536790	−	0.843716i	$-0.680363\pi$
$647$	−694.607	−1.07358	−0.536790	+	0.843716i	$0.680363\pi$
$648$	0	0
$649$	742.480	1.14404
$650$	0	0
$651$	0	0
$652$	141.957i	0.217726i
$653$	1122.29	1.71867	0.859336	−	0.511412i	$-0.170877\pi$
$653$	1122.29	1.71867	0.859336	+	0.511412i	$0.170877\pi$
$654$	0	0
$655$	0	0
$656$	− 867.132i	− 1.32185i
$657$	0	0
$658$	65.3499i	0.0993160i
$659$	670.488i	1.01743i	0.860934	+	0.508717i	$0.169880\pi$
$659$	670.488i	1.01743i	−0.860934	+	0.508717i	$0.830120\pi$
$660$	0	0
$661$	1088.15	1.64622	0.823110	−	0.567882i	$-0.192237\pi$
$661$	1088.15	1.64622	0.823110	+	0.567882i	$0.192237\pi$
$662$	−3.78714	−0.00572075
$663$	0	0
$664$	187.997	0.283128
$665$	0	0
$666$	0	0
$667$	− 1593.28i	− 2.38872i
$668$	−996.596	−1.49191
$669$	0	0
$670$	0	0
$671$	155.426i	0.231634i
$672$	0	0
$673$	− 638.371i	− 0.948545i	−0.880378	−	0.474272i	$-0.842711\pi$
$673$	− 638.371i	− 0.948545i	0.880378	−	0.474272i	$-0.157289\pi$
$674$	− 166.709i	− 0.247343i
$675$	0	0
$676$	−422.346	−0.624773
$677$	798.158	1.17896	0.589481	−	0.807782i	$-0.299332\pi$
$677$	798.158	1.17896	0.589481	+	0.807782i	$0.299332\pi$
$678$	0	0
$679$	−591.149	−0.870616
$680$	0	0
$681$	0	0
$682$	97.6417i	0.143170i
$683$	−757.505	−1.10908	−0.554542	−	0.832156i	$-0.687107\pi$
$683$	−757.505	−1.10908	−0.554542	+	0.832156i	$0.687107\pi$
$684$	0	0
$685$	0	0
$686$	119.331i	0.173952i
$687$	0	0
$688$	− 1018.54i	− 1.48044i
$689$	565.479i	0.820724i
$690$	0	0
$691$	702.331	1.01640	0.508199	−	0.861240i	$-0.330311\pi$
$691$	702.331	1.01640	0.508199	+	0.861240i	$0.330311\pi$
$692$	702.723	1.01550
$693$	0	0
$694$	−9.69578	−0.0139709
$695$	0	0
$696$	0	0
$697$	725.781i	1.04129i
$698$	−27.5329	−0.0394454
$699$	0	0
$700$	0	0
$701$	406.616i	0.580052i	0.957019	+	0.290026i	$0.0936639\pi$
$701$	406.616i	0.580052i	−0.957019	+	0.290026i	$0.906336\pi$
$702$	0	0
$703$	− 298.255i	− 0.424261i
$704$	− 412.423i	− 0.585829i
$705$	0	0
$706$	−149.793	−0.212172
$707$	−88.9969	−0.125880
$708$	0	0
$709$	−564.322	−0.795941	−0.397970	−	0.917398i	$-0.630285\pi$
$709$	−564.322	−0.795941	−0.397970	+	0.917398i	$0.630285\pi$
$710$	0	0
$711$	0	0
$712$	− 209.125i	− 0.293714i
$713$	−887.991	−1.24543
$714$	0	0
$715$	0	0
$716$	− 989.633i	− 1.38217i
$717$	0	0
$718$	152.784i	0.212791i
$719$	1340.07i	1.86379i	0.362725	+	0.931896i	$0.381846\pi$
$719$	1340.07i	1.86379i	−0.362725	+	0.931896i	$0.618154\pi$
$720$	0	0
$721$	−93.3251	−0.129438
$722$	125.981	0.174489
$723$	0	0
$724$	−694.055	−0.958639
$725$	0	0
$726$	0	0
$727$	− 130.991i	− 0.180180i	−0.995934	−	0.0900899i	$-0.971285\pi$
$727$	− 130.991i	− 0.180180i	0.995934	−	0.0900899i	$-0.0287154\pi$
$728$	85.7508	0.117790
$729$	0	0
$730$	0	0
$731$	852.508i	1.16622i
$732$	0	0
$733$	1414.48i	1.92971i	0.262777	+	0.964857i	$0.415362\pi$
$733$	1414.48i	1.92971i	−0.262777	+	0.964857i	$0.584638\pi$
$734$	78.8266i	0.107393i
$735$	0	0
$736$	−495.890	−0.673764
$737$	2.73525	0.00371133
$738$	0	0
$739$	−10.5805	−0.0143173	−0.00715865	−	0.999974i	$-0.502279\pi$
$739$	−10.5805	−0.0143173	−0.00715865	+	0.999974i	$0.502279\pi$
$740$	0	0
$741$	0	0
$742$	103.909i	0.140039i
$743$	−794.053	−1.06871	−0.534356	−	0.845260i	$-0.679446\pi$
$743$	−794.053	−1.06871	−0.534356	+	0.845260i	$0.679446\pi$
$744$	0	0
$745$	0	0
$746$	− 135.534i	− 0.181681i
$747$	0	0
$748$	376.577i	0.503446i
$749$	176.754i	0.235986i
$750$	0	0
$751$	−53.5109	−0.0712528	−0.0356264	−	0.999365i	$-0.511343\pi$
$751$	−53.5109	−0.0712528	−0.0356264	+	0.999365i	$0.511343\pi$
$752$	658.409	0.875544
$753$	0	0
$754$	−165.082	−0.218942
$755$	0	0
$756$	0	0
$757$	− 921.745i	− 1.21763i	−0.793313	−	0.608814i	$-0.791646\pi$
$757$	− 921.745i	− 1.21763i	0.793313	−	0.608814i	$-0.208354\pi$
$758$	−161.952	−0.213658
$759$	0	0
$760$	0	0
$761$	− 578.115i	− 0.759678i	−0.925053	−	0.379839i	$-0.875979\pi$
$761$	− 578.115i	− 0.759678i	0.925053	−	0.379839i	$-0.124021\pi$
$762$	0	0
$763$	611.538i	0.801491i
$764$	494.817i	0.647666i
$765$	0	0
$766$	79.1273	0.103299
$767$	699.627	0.912160
$768$	0	0
$769$	651.656	0.847407	0.423704	−	0.905801i	$-0.360730\pi$
$769$	651.656	0.847407	0.423704	+	0.905801i	$0.360730\pi$
$770$	0	0
$771$	0	0
$772$	− 247.787i	− 0.320968i
$773$	389.754	0.504209	0.252105	−	0.967700i	$-0.418877\pi$
$773$	389.754	0.504209	0.252105	+	0.967700i	$0.418877\pi$
$774$	0	0
$775$	0	0
$776$	− 478.249i	− 0.616301i
$777$	0	0
$778$	168.049i	0.216001i
$779$	339.281i	0.435534i
$780$	0	0
$781$	664.754	0.851157
$782$	129.644	0.165786
$783$	0	0
$784$	503.024	0.641613
$785$	0	0
$786$	0	0
$787$	36.4559i	0.0463226i	0.999732	+	0.0231613i	$0.00737313\pi$
$787$	36.4559i	0.0463226i	−0.999732	+	0.0231613i	$0.992627\pi$
$788$	−840.182	−1.06622
$789$	0	0
$790$	0	0
$791$	253.593i	0.320598i
$792$	0	0
$793$	146.456i	0.184686i
$794$	173.364i	0.218342i
$795$	0	0
$796$	1234.27	1.55059
$797$	−731.466	−0.917774	−0.458887	−	0.888495i	$-0.651752\pi$
$797$	−731.466	−0.917774	−0.458887	+	0.888495i	$0.651752\pi$
$798$	0	0
$799$	−551.082	−0.689715
$800$	0	0
$801$	0	0
$802$	212.529i	0.264998i
$803$	−415.481	−0.517412
$804$	0	0
$805$	0	0
$806$	92.0062i	0.114152i
$807$	0	0
$808$	− 72.0000i	− 0.0891089i
$809$	595.767i	0.736424i	0.929742	+	0.368212i	$0.120030\pi$
$809$	595.767i	0.736424i	−0.929742	+	0.368212i	$0.879970\pi$
$810$	0	0
$811$	180.177	0.222166	0.111083	−	0.993811i	$-0.464568\pi$
$811$	180.177	0.222166	0.111083	+	0.993811i	$0.464568\pi$
$812$	801.325	0.986854
$813$	0	0
$814$	166.906	0.205044
$815$	0	0
$816$	0	0
$817$	398.522i	0.487787i
$818$	28.9005	0.0353307
$819$	0	0
$820$	0	0
$821$	− 31.3375i	− 0.0381699i	−0.999818	−	0.0190849i	$-0.993925\pi$
$821$	− 31.3375i	− 0.0381699i	0.999818	−	0.0190849i	$-0.00607529\pi$
$822$	0	0
$823$	− 234.803i	− 0.285301i	−0.989773	−	0.142650i	$-0.954438\pi$
$823$	− 234.803i	− 0.285301i	0.989773	−	0.142650i	$-0.0455625\pi$
$824$	− 75.5016i	− 0.0916281i
$825$	0	0
$826$	128.559	0.155640
$827$	47.8003	0.0577996	0.0288998	−	0.999582i	$-0.490800\pi$
$827$	47.8003	0.0577996	0.0288998	+	0.999582i	$0.490800\pi$
$828$	0	0
$829$	292.760	0.353148	0.176574	−	0.984287i	$-0.443498\pi$
$829$	292.760	0.353148	0.176574	+	0.984287i	$0.443498\pi$
$830$	0	0
$831$	0	0
$832$	− 388.620i	− 0.467091i
$833$	−421.026	−0.505434
$834$	0	0
$835$	0	0
$836$	176.039i	0.210573i
$837$	0	0
$838$	199.027i	0.237502i
$839$	494.387i	0.589257i	0.955612	+	0.294629i	$0.0951960\pi$
$839$	494.387i	0.589257i	−0.955612	+	0.294629i	$0.904804\pi$
$840$	0	0
$841$	−2302.72	−2.73807
$842$	−43.8285	−0.0520529
$843$	0	0
$844$	246.018	0.291491
$845$	0	0
$846$	0	0
$847$	− 200.547i	− 0.236774i
$848$	1046.89	1.23455
$849$	0	0
$850$	0	0
$851$	1517.90i	1.78367i
$852$	0	0
$853$	− 1151.65i	− 1.35011i	−0.737766	−	0.675057i	$-0.764119\pi$
$853$	− 1151.65i	− 1.35011i	0.737766	−	0.675057i	$-0.235881\pi$
$854$	26.9117i	0.0315126i
$855$	0	0
$856$	−142.997	−0.167052
$857$	−1098.02	−1.28124	−0.640620	−	0.767858i	$-0.721323\pi$
$857$	−1098.02	−1.28124	−0.640620	+	0.767858i	$0.721323\pi$
$858$	0	0
$859$	290.757	0.338483	0.169242	−	0.985575i	$-0.445868\pi$
$859$	290.757	0.338483	0.169242	+	0.985575i	$0.445868\pi$
$860$	0	0
$861$	0	0
$862$	89.0758i	0.103336i
$863$	−1069.28	−1.23903	−0.619514	−	0.784985i	$-0.712670\pi$
$863$	−1069.28	−1.23903	−0.619514	+	0.784985i	$0.712670\pi$
$864$	0	0
$865$	0	0
$866$	191.633i	0.221285i
$867$	0	0
$868$	− 446.608i	− 0.514525i
$869$	394.024i	0.453422i
$870$	0	0
$871$	2.57738	0.00295911
$872$	−494.745	−0.567368
$873$	0	0
$874$	60.6049	0.0693420
$875$	0	0
$876$	0	0
$877$	160.043i	0.182489i	0.995829	+	0.0912443i	$0.0290844\pi$
$877$	160.043i	0.182489i	−0.995829	+	0.0912443i	$0.970916\pi$
$878$	139.798	0.159224
$879$	0	0
$880$	0	0
$881$	712.502i	0.808743i	0.914595	+	0.404371i	$0.132510\pi$
$881$	712.502i	0.808743i	−0.914595	+	0.404371i	$0.867490\pi$
$882$	0	0
$883$	− 1244.12i	− 1.40897i	−0.709718	−	0.704486i	$-0.751178\pi$
$883$	− 1244.12i	− 1.40897i	0.709718	−	0.704486i	$-0.248822\pi$
$884$	354.843i	0.401406i
$885$	0	0
$886$	−241.747	−0.272852
$887$	429.827	0.484585	0.242292	−	0.970203i	$-0.422101\pi$
$887$	429.827	0.484585	0.242292	+	0.970203i	$0.422101\pi$
$888$	0	0
$889$	744.413	0.837360
$890$	0	0
$891$	0	0
$892$	31.7933i	0.0356428i
$893$	−257.614	−0.288482
$894$	0	0
$895$	0	0
$896$	− 330.255i	− 0.368589i
$897$	0	0
$898$	304.177i	0.338727i
$899$	1752.11i	1.94895i
$900$	0	0
$901$	−876.240	−0.972519
$902$	−189.864	−0.210492
$903$	0	0
$904$	−205.161	−0.226948
$905$	0	0
$906$	0	0
$907$	553.240i	0.609967i	0.952358	+	0.304983i	$0.0986509\pi$
$907$	553.240i	0.609967i	−0.952358	+	0.304983i	$0.901349\pi$
$908$	−1308.56	−1.44115
$909$	0	0
$910$	0	0
$911$	667.275i	0.732464i	0.930524	+	0.366232i	$0.119352\pi$
$911$	667.275i	0.732464i	−0.930524	+	0.366232i	$0.880648\pi$
$912$	0	0
$913$	512.626i	0.561474i
$914$	− 77.3777i	− 0.0846583i
$915$	0	0
$916$	780.438	0.852006
$917$	30.2741	0.0330143
$918$	0	0
$919$	338.000	0.367791	0.183896	−	0.982946i	$-0.441129\pi$
$919$	338.000	0.367791	0.183896	+	0.982946i	$0.441129\pi$
$920$	0	0
$921$	0	0
$922$	278.036i	0.301558i
$923$	626.387	0.678642
$924$	0	0
$925$	0	0
$926$	− 188.310i	− 0.203359i
$927$	0	0
$928$	978.450i	1.05436i
$929$	− 216.829i	− 0.233400i	−0.993167	−	0.116700i	$-0.962768\pi$
$929$	− 216.829i	− 0.233400i	0.993167	−	0.116700i	$-0.0372317\pi$
$930$	0	0
$931$	−196.817	−0.211404
$932$	−73.5441	−0.0789100
$933$	0	0
$934$	−173.103	−0.185335
$935$	0	0
$936$	0	0
$937$	851.647i	0.908908i	0.890770	+	0.454454i	$0.150166\pi$
$937$	851.647i	0.908908i	−0.890770	+	0.454454i	$0.849834\pi$
$938$	0.473602	0.000504906 0
$939$	0	0
$940$	0	0
$941$	1073.42i	1.14073i	0.821392	+	0.570363i	$0.193198\pi$
$941$	1073.42i	1.14073i	−0.821392	+	0.570363i	$0.806802\pi$
$942$	0	0
$943$	− 1726.69i	− 1.83106i
$944$	− 1295.25i	− 1.37208i
$945$	0	0
$946$	−223.016	−0.235746
$947$	−1645.82	−1.73793	−0.868965	−	0.494873i	$-0.835215\pi$
$947$	−1645.82	−1.73793	−0.868965	+	0.494873i	$0.835215\pi$
$948$	0	0
$949$	−391.502	−0.412541
$950$	0	0
$951$	0	0
$952$	132.875i	0.139575i
$953$	1139.03	1.19520	0.597602	−	0.801793i	$-0.296120\pi$
$953$	1139.03	1.19520	0.597602	+	0.801793i	$0.296120\pi$
$954$	0	0
$955$	0	0
$956$	− 1050.67i	− 1.09902i
$957$	0	0
$958$	− 122.055i	− 0.127406i
$959$	186.639i	0.194618i
$960$	0	0
$961$	15.5140	0.0161436
$962$	157.272	0.163485
$963$	0	0
$964$	−896.389	−0.929864
$965$	0	0
$966$	0	0
$967$	357.489i	0.369689i	0.982768	+	0.184844i	$0.0591781\pi$
$967$	357.489i	0.369689i	−0.982768	+	0.184844i	$0.940822\pi$
$968$	162.246	0.167610
$969$	0	0
$970$	0	0
$971$	46.5836i	0.0479749i	0.999712	+	0.0239874i	$0.00763617\pi$
$971$	46.5836i	0.0479749i	−0.999712	+	0.0239874i	$0.992364\pi$
$972$	0	0
$973$	− 469.082i	− 0.482099i
$974$	302.849i	0.310933i
$975$	0	0
$976$	271.140	0.277807
$977$	−346.616	−0.354776	−0.177388	−	0.984141i	$-0.556765\pi$
$977$	−346.616	−0.354776	−0.177388	+	0.984141i	$0.556765\pi$
$978$	0	0
$979$	570.237	0.582469
$980$	0	0
$981$	0	0
$982$	− 141.112i	− 0.143699i
$983$	−846.975	−0.861623	−0.430811	−	0.902442i	$-0.641773\pi$
$983$	−846.975	−0.861623	−0.430811	+	0.902442i	$0.641773\pi$
$984$	0	0
$985$	0	0
$986$	− 255.803i	− 0.259435i
$987$	0	0
$988$	165.878i	0.167893i
$989$	− 2028.19i	− 2.05075i
$990$	0	0
$991$	1304.33	1.31617	0.658085	−	0.752943i	$-0.271367\pi$
$991$	1304.33	1.31617	0.658085	+	0.752943i	$0.271367\pi$
$992$	545.326	0.549723
$993$	0	0
$994$	115.101	0.115795
$995$	0	0
$996$	0	0
$997$	− 1835.77i	− 1.84130i	−0.390391	−	0.920649i	$-0.627660\pi$
$997$	− 1835.77i	− 1.84130i	0.390391	−	0.920649i	$-0.372340\pi$
$998$	232.966	0.233432
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	675.3.d.e.674.3		4
3.2	odd	2		675.3.d.i.674.1		4
5.2	odd	4		135.3.c.c.26.2	✓	4
5.3	odd	4		675.3.c.p.26.3		4
5.4	even	2		675.3.d.i.674.2		4
15.2	even	4		135.3.c.c.26.3	yes	4
15.8	even	4		675.3.c.p.26.2		4
15.14	odd	2	inner	675.3.d.e.674.4		4
20.7	even	4		2160.3.l.g.161.3		4
45.2	even	12		405.3.i.c.296.3		8
45.7	odd	12		405.3.i.c.296.2		8
45.22	odd	12		405.3.i.c.26.3		8
45.32	even	12		405.3.i.c.26.2		8
60.47	odd	4		2160.3.l.g.161.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
135.3.c.c.26.2	✓	4	5.2	odd	4
135.3.c.c.26.3	yes	4	15.2	even	4
405.3.i.c.26.2		8	45.32	even	12
405.3.i.c.26.3		8	45.22	odd	12
405.3.i.c.296.2		8	45.7	odd	12
405.3.i.c.296.3		8	45.2	even	12
675.3.c.p.26.2		4	15.8	even	4
675.3.c.p.26.3		4	5.3	odd	4
675.3.d.e.674.3		4	1.1	even	1	trivial
675.3.d.e.674.4		4	15.14	odd	2	inner
675.3.d.i.674.1		4	3.2	odd	2
675.3.d.i.674.2		4	5.4	even	2
2160.3.l.g.161.1		4	60.47	odd	4
2160.3.l.g.161.3		4	20.7	even	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 $\F_{q}$

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

\(f(q)\)	\(=\)	\(q-0.381966 q^{2} -3.85410 q^{4} -3.70820i q^{7} +3.00000 q^{8} +O(q^{10})\)	\(q-0.381966 q^{2} -3.85410 q^{4} -3.70820i q^{7} +3.00000 q^{8} +8.18034i q^{11} +7.70820i q^{13} +1.41641i q^{14} +14.2705 q^{16} -11.9443 q^{17} -5.58359 q^{19} -3.12461i q^{22} +28.4164 q^{23} -2.94427i q^{26} +14.2918i q^{28} -56.0689i q^{29} -31.2492 q^{31} -17.4508 q^{32} +4.56231 q^{34} +53.4164i q^{37} +2.13274 q^{38} -60.7639i q^{41} -71.3738i q^{43} -31.5279i q^{44} -10.8541 q^{46} +46.1378 q^{47} +35.2492 q^{49} -29.7082i q^{52} +73.3607 q^{53} -11.1246i q^{56} +21.4164i q^{58} -90.7639i q^{59} +19.0000 q^{61} +11.9361 q^{62} -50.4164 q^{64} -0.334369i q^{67} +46.0344 q^{68} -81.2624i q^{71} +50.7902i q^{73} -20.4033i q^{74} +21.5197 q^{76} +30.3344 q^{77} +48.1672 q^{79} +23.2098i q^{82} +62.6656 q^{83} +27.2624i q^{86} +24.5410i q^{88} -69.7082i q^{89} +28.5836 q^{91} -109.520 q^{92} -17.6231 q^{94} -159.416i q^{97} -13.4640 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 6 q^{2} - 2 q^{4} + 12 q^{8}+O(q^{10}) \) 4 * q - 6 * q^2 - 2 * q^4 + 12 * q^8	\( 4 q - 6 q^{2} - 2 q^{4} + 12 q^{8} - 10 q^{16} - 12 q^{17} - 76 q^{19} + 60 q^{23} + 36 q^{31} + 42 q^{32} - 22 q^{34} + 174 q^{38} - 30 q^{46} - 48 q^{47} - 20 q^{49} + 204 q^{53} + 76 q^{61} - 234 q^{62} - 148 q^{64} + 126 q^{68} - 142 q^{76} + 336 q^{77} + 300 q^{79} + 36 q^{83} + 168 q^{91} - 210 q^{92} + 332 q^{94} + 210 q^{98}+O(q^{100}) \) 4 * q - 6 * q^2 - 2 * q^4 + 12 * q^8 - 10 * q^16 - 12 * q^17 - 76 * q^19 + 60 * q^23 + 36 * q^31 + 42 * q^32 - 22 * q^34 + 174 * q^38 - 30 * q^46 - 48 * q^47 - 20 * q^49 + 204 * q^53 + 76 * q^61 - 234 * q^62 - 148 * q^64 + 126 * q^68 - 142 * q^76 + 336 * q^77 + 300 * q^79 + 36 * q^83 + 168 * q^91 - 210 * q^92 + 332 * q^94 + 210 * q^98

Introduction

L-functions

Modular forms

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