LMFDB - Embedded newform 3150.2.d.a.3149.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3150,2,Mod(3149,3150)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3150, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 1, 1]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("3150.3149");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3150 = 2 \cdot 3^{2} \cdot 5^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3150.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:	$25.1528766367$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.7442857984.4
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 26x^{6} + 205x^{4} + 540x^{2} + 324 $ x^8 + 26x^6 + 205x^4 + 540*x^2 + 324
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{4}\cdot 3^{2} $
Twist minimal:	no (minimal twist has level 630)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			3149.2
Root			$-3.73923i$ of defining polynomial
Character	$\chi$	$=$	3150.3149
Dual form			3150.2.d.a.3149.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} +1.00000 q^{4} +(-2.64404 + 0.0951965i) q^{7} -1.00000 q^{8} +O(q^{10})$	$q-1.00000 q^{2} +1.00000 q^{4} +(-2.64404 + 0.0951965i) q^{7} -1.00000 q^{8} +5.28808i q^{11} -2.19039 q^{13} +(2.64404 - 0.0951965i) q^{14} +1.00000 q^{16} -1.04544i q^{17} +6.43303i q^{19} -5.28808i q^{22} +7.47847 q^{23} +2.19039 q^{26} +(-2.64404 + 0.0951965i) q^{28} -7.47847i q^{29} +9.09768i q^{31} -1.00000 q^{32} +1.04544i q^{34} -0.855043i q^{37} -6.43303i q^{38} +2.19039 q^{41} +0.954564i q^{43} +5.28808i q^{44} -7.47847 q^{46} -11.0092i q^{47} +(6.98188 - 0.503406i) q^{49} -2.19039 q^{52} -3.09768 q^{53} +(2.64404 - 0.0951965i) q^{56} +7.47847i q^{58} -13.7734 q^{59} -8.05225i q^{61} -9.09768i q^{62} +1.00000 q^{64} +5.33535i q^{67} -1.04544i q^{68} -6.43303i q^{71} -4.57615 q^{73} +0.855043i q^{74} +6.43303i q^{76} +(-0.503406 - 13.9819i) q^{77} -15.6738 q^{79} -2.19039 q^{82} +4.38079i q^{83} -0.954564i q^{86} -5.28808i q^{88} -4.28126 q^{89} +(5.79148 - 0.208518i) q^{91} +7.47847 q^{92} +11.0092i q^{94} -11.8593 q^{97} +(-6.98188 + 0.503406i) q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 8 q^{2} + 8 q^{4} - 8 q^{8}+O(q^{10}) $ 8 * q - 8 * q^2 + 8 * q^4 - 8 * q^8	$ 8 q - 8 q^{2} + 8 q^{4} - 8 q^{8} - 8 q^{13} + 8 q^{16} + 8 q^{23} + 8 q^{26} - 8 q^{32} + 8 q^{41} - 8 q^{46} - 4 q^{49} - 8 q^{52} + 8 q^{53} + 8 q^{64} + 48 q^{73} + 4 q^{77} - 8 q^{79} - 8 q^{82} - 8 q^{89} - 4 q^{91} + 8 q^{92} - 24 q^{97} + 4 q^{98}+O(q^{100}) $ 8 * q - 8 * q^2 + 8 * q^4 - 8 * q^8 - 8 * q^13 + 8 * q^16 + 8 * q^23 + 8 * q^26 - 8 * q^32 + 8 * q^41 - 8 * q^46 - 4 * q^49 - 8 * q^52 + 8 * q^53 + 8 * q^64 + 48 * q^73 + 4 * q^77 - 8 * q^79 - 8 * q^82 - 8 * q^89 - 4 * q^91 + 8 * q^92 - 24 * q^97 + 4 * q^98

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3150\mathbb{Z}\right)^\times$.

$n$	$127$	$451$	$2801$
$\chi(n)$	$-1$	$-1$	$-1$

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 $

$2$	−1.00000			−0.707107
$2$	−1.00000			−0.707107
$3$		0			0
$3$		0			0
$4$	1.00000			0.500000
$5$		0			0
$5$		0			0
$6$		0			0
$7$	−2.64404	+	0.0951965i	−0.999352	+	0.0359809i
$7$	−2.64404	+	0.0951965i	−0.999352	+	0.0359809i
$8$	−1.00000			−0.353553
$9$		0			0
$10$		0			0
$11$			5.28808i			1.59441i	0.603705	+	0.797207i	$0.293690\pi$
$11$			5.28808i			1.59441i	−0.603705	+	0.797207i	$0.706310\pi$
$12$		0			0
$13$	−2.19039			−0.607506			−0.303753	−	0.952751i	$-0.598240\pi$
$13$	−2.19039			−0.607506			−0.303753	+	0.952751i	$0.598240\pi$
$14$	2.64404	−	0.0951965i	0.706649	−	0.0254423i
$15$		0			0
$16$	1.00000			0.250000
$17$		−	1.04544i		−	0.253555i	−0.991931	−	0.126778i	$-0.959537\pi$
$17$		−	1.04544i		−	0.253555i	0.991931	−	0.126778i	$-0.0404635\pi$
$18$		0			0
$19$			6.43303i			1.47584i	0.674889	+	0.737920i	$0.264192\pi$
$19$			6.43303i			1.47584i	−0.674889	+	0.737920i	$0.735808\pi$
$20$		0			0
$21$		0			0
$22$		−	5.28808i		−	1.12742i
$23$	7.47847			1.55937			0.779684	−	0.626173i	$-0.215380\pi$
$23$	7.47847			1.55937			0.779684	+	0.626173i	$0.215380\pi$
$24$		0			0
$25$		0			0
$26$	2.19039			0.429571
$27$		0			0
$28$	−2.64404	+	0.0951965i	−0.499676	+	0.0179904i
$29$		−	7.47847i		−	1.38872i	−0.719629	−	0.694358i	$-0.755688\pi$
$29$		−	7.47847i		−	1.38872i	0.719629	−	0.694358i	$-0.244312\pi$
$30$		0			0
$31$			9.09768i			1.63399i	0.576643	+	0.816996i	$0.304362\pi$
$31$			9.09768i			1.63399i	−0.576643	+	0.816996i	$0.695638\pi$
$32$	−1.00000			−0.176777
$33$		0			0
$34$			1.04544i			0.179291i
$35$		0			0
$36$		0			0
$37$		−	0.855043i		−	0.140568i	−0.997527	−	0.0702841i	$-0.977609\pi$
$37$		−	0.855043i		−	0.140568i	0.997527	−	0.0702841i	$-0.0223906\pi$
$38$		−	6.43303i		−	1.04358i
$39$		0			0
$40$		0			0
$41$	2.19039			0.342082			0.171041	−	0.985264i	$-0.445287\pi$
$41$	2.19039			0.342082			0.171041	+	0.985264i	$0.445287\pi$
$42$		0			0
$43$			0.954564i			0.145570i	0.997348	+	0.0727849i	$0.0231886\pi$
$43$			0.954564i			0.145570i	−0.997348	+	0.0727849i	$0.976811\pi$
$44$			5.28808i			0.797207i
$45$		0			0
$46$	−7.47847			−1.10264
$47$		−	11.0092i		−	1.60585i	−0.596077	−	0.802927i	$-0.703275\pi$
$47$		−	11.0092i		−	1.60585i	0.596077	−	0.802927i	$-0.296725\pi$
$48$		0			0
$49$	6.98188	−	0.503406i	0.997411	−	0.0719152i
$50$		0			0
$51$		0			0
$52$	−2.19039			−0.303753
$53$	−3.09768			−0.425500			−0.212750	−	0.977107i	$-0.568242\pi$
$53$	−3.09768			−0.425500			−0.212750	+	0.977107i	$0.568242\pi$
$54$		0			0
$55$		0			0
$56$	2.64404	−	0.0951965i	0.353324	−	0.0127212i
$57$		0			0
$58$			7.47847i			0.981971i
$59$	−13.7734			−1.79314			−0.896569	−	0.442904i	$-0.853948\pi$
$59$	−13.7734			−1.79314			−0.896569	+	0.442904i	$0.853948\pi$
$60$		0			0
$61$		−	8.05225i		−	1.03098i	−0.856894	−	0.515492i	$-0.827609\pi$
$61$		−	8.05225i		−	1.03098i	0.856894	−	0.515492i	$-0.172391\pi$
$62$		−	9.09768i		−	1.15541i
$63$		0			0
$64$	1.00000			0.125000
$65$		0			0
$66$		0			0
$67$			5.33535i			0.651817i	0.945401	+	0.325908i	$0.105670\pi$
$67$			5.33535i			0.651817i	−0.945401	+	0.325908i	$0.894330\pi$
$68$		−	1.04544i		−	0.126778i
$69$		0			0
$70$		0			0
$71$		−	6.43303i		−	0.763461i	−0.924274	−	0.381730i	$-0.875328\pi$
$71$		−	6.43303i		−	0.763461i	0.924274	−	0.381730i	$-0.124672\pi$
$72$		0			0
$73$	−4.57615			−0.535598			−0.267799	−	0.963475i	$-0.586296\pi$
$73$	−4.57615			−0.535598			−0.267799	+	0.963475i	$0.586296\pi$
$74$			0.855043i			0.0993967i
$75$		0			0
$76$			6.43303i			0.737920i
$77$	−0.503406	−	13.9819i	−0.0573684	−	1.59338i
$78$		0			0
$79$	−15.6738			−1.76344			−0.881722	−	0.471769i	$-0.843616\pi$
$79$	−15.6738			−1.76344			−0.881722	+	0.471769i	$0.843616\pi$
$80$		0			0
$81$		0			0
$82$	−2.19039			−0.241888
$83$			4.38079i			0.480854i	0.970667	+	0.240427i	$0.0772874\pi$
$83$			4.38079i			0.480854i	−0.970667	+	0.240427i	$0.922713\pi$
$84$		0			0
$85$		0			0
$86$		−	0.954564i		−	0.102933i
$87$		0			0
$88$		−	5.28808i		−	0.563711i
$89$	−4.28126			−0.453813			−0.226907	−	0.973917i	$-0.572861\pi$
$89$	−4.28126			−0.453813			−0.226907	+	0.973917i	$0.572861\pi$
$90$		0			0
$91$	5.79148	−	0.208518i	0.607112	−	0.0218586i
$92$	7.47847			0.779684
$93$		0			0
$94$			11.0092i			1.13551i
$95$		0			0
$96$		0			0
$97$	−11.8593			−1.20412			−0.602062	−	0.798449i	$-0.705654\pi$
$97$	−11.8593			−1.20412			−0.602062	+	0.798449i	$0.705654\pi$
$98$	−6.98188	+	0.503406i	−0.705276	+	0.0508517i
$99$		0			0
$100$		0			0
$101$	11.5830			1.15255			0.576274	−	0.817257i	$-0.304506\pi$
$101$	11.5830			1.15255			0.576274	+	0.817257i	$0.304506\pi$
$102$		0			0
$103$	−16.6757			−1.64310			−0.821552	−	0.570134i	$-0.806891\pi$
$103$	−16.6757			−1.64310			−0.821552	+	0.570134i	$0.806891\pi$
$104$	2.19039			0.214786
$105$		0			0
$106$	3.09768			0.300874
$107$	−7.00681			−0.677374			−0.338687	−	0.940899i	$-0.609983\pi$
$107$	−7.00681			−0.677374			−0.338687	+	0.940899i	$0.609983\pi$
$108$		0			0
$109$	4.28991			0.410899			0.205450	−	0.978668i	$-0.434134\pi$
$109$	4.28991			0.410899			0.205450	+	0.978668i	$0.434134\pi$
$110$		0			0
$111$		0			0
$112$	−2.64404	+	0.0951965i	−0.249838	+	0.00899522i
$113$	−8.09087			−0.761125			−0.380563	−	0.924755i	$-0.624270\pi$
$113$	−8.09087			−0.761125			−0.380563	+	0.924755i	$0.624270\pi$
$114$		0			0
$115$		0			0
$116$		−	7.47847i		−	0.694358i
$117$		0			0
$118$	13.7734			1.26794
$119$	0.0995218	+	2.76417i	0.00912314	+	0.253391i
$120$		0			0
$121$	−16.9638			−1.54216
$122$			8.05225i			0.729016i
$123$		0			0
$124$			9.09768i			0.816996i
$125$		0			0
$126$		0			0
$127$			8.57118i			0.760569i	0.924870	+	0.380285i	$0.124174\pi$
$127$			8.57118i			0.760569i	−0.924870	+	0.380285i	$0.875826\pi$
$128$	−1.00000			−0.0883883
$129$		0			0
$130$		0			0
$131$	5.90048			0.515527			0.257764	−	0.966208i	$-0.417014\pi$
$131$	5.90048			0.515527			0.257764	+	0.966208i	$0.417014\pi$
$132$		0			0
$133$	−0.612402	−	17.0092i	−0.0531020	−	1.47488i
$134$		−	5.33535i		−	0.460904i
$135$		0			0
$136$			1.04544i			0.0896454i
$137$	6.86607			0.586608			0.293304	−	0.956019i	$-0.405245\pi$
$137$	6.86607			0.586608			0.293304	+	0.956019i	$0.405245\pi$
$138$		0			0
$139$			13.9115i			1.17996i	0.807418	+	0.589979i	$0.200864\pi$
$139$			13.9115i			1.17996i	−0.807418	+	0.589979i	$0.799136\pi$
$140$		0			0
$141$		0			0
$142$			6.43303i			0.539848i
$143$		−	11.5830i		−	0.968616i
$144$		0			0
$145$		0			0
$146$	4.57615			0.378725
$147$		0			0
$148$		−	0.855043i		−	0.0702841i
$149$			15.9638i			1.30780i	0.756580	+	0.653901i	$0.226869\pi$
$149$			15.9638i			1.30780i	−0.756580	+	0.653901i	$0.773131\pi$
$150$		0			0
$151$	16.0546			1.30651			0.653253	−	0.757139i	$-0.273404\pi$
$151$	16.0546			1.30651			0.653253	+	0.757139i	$0.273404\pi$
$152$		−	6.43303i		−	0.521788i
$153$		0			0
$154$	0.503406	+	13.9819i	0.0405656	+	1.12669i
$155$		0			0
$156$		0			0
$157$	−24.7665			−1.97659			−0.988293	−	0.152569i	$-0.951245\pi$
$157$	−24.7665			−1.97659			−0.988293	+	0.152569i	$0.951245\pi$
$158$	15.6738			1.24694
$159$		0			0
$160$		0			0
$161$	−19.7734	+	0.711924i	−1.55836	+	0.0561075i
$162$		0			0
$163$		−	7.42622i		−	0.581667i	−0.956774	−	0.290833i	$-0.906068\pi$
$163$		−	7.42622i		−	0.581667i	0.956774	−	0.290833i	$-0.0939325\pi$
$164$	2.19039			0.171041
$165$		0			0
$166$		−	4.38079i		−	0.340015i
$167$			0.573779i			0.0444003i	0.999754	+	0.0222002i	$0.00706711\pi$
$167$			0.573779i			0.0444003i	−0.999754	+	0.0222002i	$0.992933\pi$
$168$		0			0
$169$	−8.20218			−0.630937
$170$		0			0
$171$		0			0
$172$			0.954564i			0.0727849i
$173$			9.96375i			0.757530i	0.925493	+	0.378765i	$0.123651\pi$
$173$			9.96375i			0.757530i	−0.925493	+	0.378765i	$0.876349\pi$
$174$		0			0
$175$		0			0
$176$			5.28808i			0.398604i
$177$		0			0
$178$	4.28126			0.320894
$179$		−	1.77336i		−	0.132547i	−0.997801	−	0.0662735i	$-0.978889\pi$
$179$		−	1.77336i		−	0.132547i	0.997801	−	0.0662735i	$-0.0211110\pi$
$180$		0			0
$181$		−	11.0092i		−	0.818306i	−0.912466	−	0.409153i	$-0.865824\pi$
$181$		−	11.0092i		−	0.818306i	0.912466	−	0.409153i	$-0.134176\pi$
$182$	−5.79148	+	0.208518i	−0.429293	+	0.0154564i
$183$		0			0
$184$	−7.47847			−0.551320
$185$		0			0
$186$		0			0
$187$	5.52834			0.404272
$188$		−	11.0092i		−	0.802927i
$189$		0			0
$190$		0			0
$191$		−	5.56697i		−	0.402812i	−0.979508	−	0.201406i	$-0.935449\pi$
$191$		−	5.56697i		−	0.402812i	0.979508	−	0.201406i	$-0.0645510\pi$
$192$		0			0
$193$			8.47166i			0.609803i	0.952384	+	0.304902i	$0.0986236\pi$
$193$			8.47166i			0.609803i	−0.952384	+	0.304902i	$0.901376\pi$
$194$	11.8593			0.851445
$195$		0			0
$196$	6.98188	−	0.503406i	0.498705	−	0.0359576i
$197$	12.8661			0.916669			0.458335	−	0.888780i	$-0.348446\pi$
$197$	12.8661			0.916669			0.458335	+	0.888780i	$0.348446\pi$
$198$		0			0
$199$		−	17.5830i		−	1.24642i	−0.782053	−	0.623212i	$-0.785828\pi$
$199$		−	17.5830i		−	1.24642i	0.782053	−	0.623212i	$-0.214172\pi$
$200$		0			0
$201$		0			0
$202$	−11.5830			−0.814975
$203$	0.711924	+	19.7734i	0.0499673	+	1.38782i
$204$		0			0
$205$		0			0
$206$	16.6757			1.16185
$207$		0			0
$208$	−2.19039			−0.151876
$209$	−34.0184			−2.35310
$210$		0			0
$211$	4.00000			0.275371			0.137686	−	0.990476i	$-0.456034\pi$
$211$	4.00000			0.275371			0.137686	+	0.990476i	$0.456034\pi$
$212$	−3.09768			−0.212750
$213$		0			0
$214$	7.00681			0.478976
$215$		0			0
$216$		0			0
$217$	−0.866067	−	24.0546i	−0.0587925	−	1.63293i
$218$	−4.28991			−0.290550
$219$		0			0
$220$		0			0
$221$			2.28991i			0.154036i
$222$		0			0
$223$	−10.8710			−0.727979			−0.363989	−	0.931403i	$-0.618586\pi$
$223$	−10.8710			−0.727979			−0.363989	+	0.931403i	$0.618586\pi$
$224$	2.64404	−	0.0951965i	0.176662	−	0.00636058i
$225$		0			0
$226$	8.09087			0.538197
$227$		−	14.0909i		−	0.935244i	−0.883929	−	0.467622i	$-0.845111\pi$
$227$		−	14.0909i		−	0.935244i	0.883929	−	0.467622i	$-0.154889\pi$
$228$		0			0
$229$			14.5239i			0.959767i	0.877332	+	0.479883i	$0.159321\pi$
$229$			14.5239i			0.959767i	−0.877332	+	0.479883i	$0.840679\pi$
$230$		0			0
$231$		0			0
$232$			7.47847i			0.490986i
$233$	−20.6806			−1.35483			−0.677417	−	0.735599i	$-0.736901\pi$
$233$	−20.6806			−1.35483			−0.677417	+	0.735599i	$0.736901\pi$
$234$		0			0
$235$		0			0
$236$	−13.7734			−0.896569
$237$		0			0
$238$	−0.0995218	−	2.76417i	−0.00645104	−	0.179175i
$239$		−	5.56697i		−	0.360097i	−0.983658	−	0.180049i	$-0.942375\pi$
$239$		−	5.56697i		−	0.360097i	0.983658	−	0.180049i	$-0.0576255\pi$
$240$		0			0
$241$		−	11.3839i		−	0.733303i	−0.930358	−	0.366651i	$-0.880504\pi$
$241$		−	11.3839i		−	0.733303i	0.930358	−	0.366651i	$-0.119496\pi$
$242$	16.9638			1.09047
$243$		0			0
$244$		−	8.05225i		−	0.515492i
$245$		0			0
$246$		0			0
$247$		−	14.0909i		−	0.896581i
$248$		−	9.09768i		−	0.577703i
$249$		0			0
$250$		0			0
$251$	1.71874			0.108486			0.0542428	−	0.998528i	$-0.482725\pi$
$251$	1.71874			0.108486			0.0542428	+	0.998528i	$0.482725\pi$
$252$		0			0
$253$			39.5467i			2.48628i
$254$		−	8.57118i		−	0.537804i
$255$		0			0
$256$	1.00000			0.0625000
$257$			2.18779i			0.136471i	0.997669	+	0.0682354i	$0.0217369\pi$
$257$			2.18779i			0.136471i	−0.997669	+	0.0682354i	$0.978263\pi$
$258$		0			0
$259$	0.0813970	+	2.26077i	0.00505777	+	0.140477i
$260$		0			0
$261$		0			0
$262$	−5.90048			−0.364533
$263$	17.3876			1.07217			0.536083	−	0.844166i	$-0.319904\pi$
$263$	17.3876			1.07217			0.536083	+	0.844166i	$0.319904\pi$
$264$		0			0
$265$		0			0
$266$	0.612402	+	17.0092i	0.0375488	+	1.04290i
$267$		0			0
$268$			5.33535i			0.325908i
$269$	−20.3445			−1.24043			−0.620214	−	0.784433i	$-0.712954\pi$
$269$	−20.3445			−1.24043			−0.620214	+	0.784433i	$0.712954\pi$
$270$		0			0
$271$		−	1.47847i		−	0.0898106i	−0.998991	−	0.0449053i	$-0.985701\pi$
$271$		−	1.47847i		−	0.0898106i	0.998991	−	0.0449053i	$-0.0142986\pi$
$272$		−	1.04544i		−	0.0633888i
$273$		0			0
$274$	−6.86607			−0.414794
$275$		0			0
$276$		0			0
$277$			10.7642i			0.646756i	0.946270	+	0.323378i	$0.104819\pi$
$277$			10.7642i			0.646756i	−0.946270	+	0.323378i	$0.895181\pi$
$278$		−	13.9115i		−	0.834356i
$279$		0			0
$280$		0			0
$281$			2.42806i			0.144846i	0.997374	+	0.0724230i	$0.0230731\pi$
$281$			2.42806i			0.144846i	−0.997374	+	0.0724230i	$0.976927\pi$
$282$		0			0
$283$	24.5898			1.46171			0.730855	−	0.682532i	$-0.239121\pi$
$283$	24.5898			1.46171			0.730855	+	0.682532i	$0.239121\pi$
$284$		−	6.43303i		−	0.381730i
$285$		0			0
$286$			11.5830i			0.684915i
$287$	−5.79148	+	0.208518i	−0.341860	+	0.0123084i
$288$		0			0
$289$	15.9071			0.935710
$290$		0			0
$291$		0			0
$292$	−4.57615			−0.267799
$293$		−	12.6707i		−	0.740230i	−0.928986	−	0.370115i	$-0.879318\pi$
$293$		−	12.6707i		−	0.740230i	0.928986	−	0.370115i	$-0.120682\pi$
$294$		0			0
$295$		0			0
$296$			0.855043i			0.0496983i
$297$		0			0
$298$		−	15.9638i		−	0.924755i
$299$	−16.3808			−0.947325
$300$		0			0
$301$	−0.0908711	−	2.52390i	−0.00523773	−	0.145475i
$302$	−16.0546			−0.923840
$303$		0			0
$304$			6.43303i			0.368960i
$305$		0			0
$306$		0			0
$307$	0.866067			0.0494291			0.0247145	−	0.999695i	$-0.492132\pi$
$307$	0.866067			0.0494291			0.0247145	+	0.999695i	$0.492132\pi$
$308$	−0.503406	−	13.9819i	−0.0286842	−	0.796691i
$309$		0			0
$310$		0			0
$311$	−9.71009			−0.550608			−0.275304	−	0.961357i	$-0.588779\pi$
$311$	−9.71009			−0.550608			−0.275304	+	0.961357i	$0.588779\pi$
$312$		0			0
$313$	−5.80096			−0.327889			−0.163945	−	0.986470i	$-0.552422\pi$
$313$	−5.80096			−0.327889			−0.163945	+	0.986470i	$0.552422\pi$
$314$	24.7665			1.39766
$315$		0			0
$316$	−15.6738			−0.881722
$317$	−28.1591			−1.58157			−0.790787	−	0.612092i	$-0.790328\pi$
$317$	−28.1591			−1.58157			−0.790787	+	0.612092i	$0.790328\pi$
$318$		0			0
$319$	39.5467			2.21419
$320$		0			0
$321$		0			0
$322$	19.7734	−	0.711924i	1.10193	−	0.0396740i
$323$	6.72532			0.374207
$324$		0			0
$325$		0			0
$326$			7.42622i			0.411300i
$327$		0			0
$328$	−2.19039			−0.120944
$329$	1.04804	+	29.1087i	0.0577801	+	1.60482i
$330$		0			0
$331$	−19.5830			−1.07638			−0.538189	−	0.842824i	$-0.680891\pi$
$331$	−19.5830			−1.07638			−0.538189	+	0.842824i	$0.680891\pi$
$332$			4.38079i			0.240427i
$333$		0			0
$334$		−	0.573779i		−	0.0313958i
$335$		0			0
$336$		0			0
$337$		−	28.3992i		−	1.54700i	−0.633796	−	0.773500i	$-0.718504\pi$
$337$		−	28.3992i		−	1.54700i	0.633796	−	0.773500i	$-0.281496\pi$
$338$	8.20218			0.446140
$339$		0			0
$340$		0			0
$341$	−48.1092			−2.60526
$342$		0			0
$343$	−18.4124	+	1.99567i	−0.994177	+	0.107756i
$344$		−	0.954564i		−	0.0514667i
$345$		0			0
$346$		−	9.96375i		−	0.535655i
$347$	12.5898			0.675855			0.337927	−	0.941172i	$-0.390274\pi$
$347$	12.5898			0.675855			0.337927	+	0.941172i	$0.390274\pi$
$348$		0			0
$349$		−	20.9956i		−	1.12387i	−0.827183	−	0.561933i	$-0.810058\pi$
$349$		−	20.9956i		−	1.12387i	0.827183	−	0.561933i	$-0.189942\pi$
$350$		0			0
$351$		0			0
$352$		−	5.28808i		−	0.281855i
$353$			15.1363i			0.805624i	0.915283	+	0.402812i	$0.131967\pi$
$353$			15.1363i			0.805624i	−0.915283	+	0.402812i	$0.868033\pi$
$354$		0			0
$355$		0			0
$356$	−4.28126			−0.226907
$357$		0			0
$358$			1.77336i			0.0937249i
$359$		−	33.1909i		−	1.75175i	−0.482538	−	0.875875i	$-0.660285\pi$
$359$		−	33.1909i		−	1.75175i	0.482538	−	0.875875i	$-0.339715\pi$
$360$		0			0
$361$	−22.3839			−1.17810
$362$			11.0092i			0.578630i
$363$		0			0
$364$	5.79148	−	0.208518i	0.303556	−	0.0109293i
$365$		0			0
$366$		0			0
$367$	−11.6825			−0.609821			−0.304910	−	0.952381i	$-0.598627\pi$
$367$	−11.6825			−0.609821			−0.304910	+	0.952381i	$0.598627\pi$
$368$	7.47847			0.389842
$369$		0			0
$370$		0			0
$371$	8.19039	−	0.294888i	0.425224	−	0.0153098i
$372$		0			0
$373$		−	12.8550i		−	0.665609i	−0.942996	−	0.332804i	$-0.892005\pi$
$373$		−	12.8550i		−	0.665609i	0.942996	−	0.332804i	$-0.107995\pi$
$374$	−5.52834			−0.285864
$375$		0			0
$376$			11.0092i			0.567755i
$377$			16.3808i			0.843653i
$378$		0			0
$379$	−25.0751			−1.28802			−0.644010	−	0.765017i	$-0.722730\pi$
$379$	−25.0751			−1.28802			−0.644010	+	0.765017i	$0.722730\pi$
$380$		0			0
$381$		0			0
$382$			5.56697i			0.284831i
$383$			23.0092i			1.17571i	0.808965	+	0.587857i	$0.200028\pi$
$383$			23.0092i			1.17571i	−0.808965	+	0.587857i	$0.799972\pi$
$384$		0			0
$385$		0			0
$386$		−	8.47166i		−	0.431196i
$387$		0			0
$388$	−11.8593			−0.602062
$389$		−	10.1591i		−	0.515088i	−0.966267	−	0.257544i	$-0.917087\pi$
$389$		−	10.1591i		−	0.515088i	0.966267	−	0.257544i	$-0.0829132\pi$
$390$		0			0
$391$		−	7.81826i		−	0.395386i
$392$	−6.98188	+	0.503406i	−0.352638	+	0.0254258i
$393$		0			0
$394$	−12.8661			−0.648183
$395$		0			0
$396$		0			0
$397$	−13.9141			−0.698329			−0.349164	−	0.937061i	$-0.613535\pi$
$397$	−13.9141			−0.698329			−0.349164	+	0.937061i	$0.613535\pi$
$398$			17.5830i			0.881354i
$399$		0			0
$400$		0			0
$401$		−	14.6197i		−	0.730075i	−0.930993	−	0.365038i	$-0.881056\pi$
$401$		−	14.6197i		−	0.730075i	0.930993	−	0.365038i	$-0.118944\pi$
$402$		0			0
$403$		−	19.9275i		−	0.992660i
$404$	11.5830			0.576274
$405$		0			0
$406$	−0.711924	−	19.7734i	−0.0353322	−	0.981335i
$407$	4.52153			0.224124
$408$		0			0
$409$		−	12.0000i		−	0.593362i	−0.954977	−	0.296681i	$-0.904120\pi$
$409$		−	12.0000i		−	0.593362i	0.954977	−	0.296681i	$-0.0958798\pi$
$410$		0			0
$411$		0			0
$412$	−16.6757			−0.821552
$413$	36.4173	−	1.31117i	1.79198	−	0.0645187i
$414$		0			0
$415$		0			0
$416$	2.19039			0.107393
$417$		0			0
$418$	34.0184			1.66389
$419$	−2.60743			−0.127381			−0.0636906	−	0.997970i	$-0.520287\pi$
$419$	−2.60743			−0.127381			−0.0636906	+	0.997970i	$0.520287\pi$
$420$		0			0
$421$	0.852443			0.0415455			0.0207728	−	0.999784i	$-0.493387\pi$
$421$	0.852443			0.0415455			0.0207728	+	0.999784i	$0.493387\pi$
$422$	−4.00000			−0.194717
$423$		0			0
$424$	3.09768			0.150437
$425$		0			0
$426$		0			0
$427$	0.766545	+	21.2905i	0.0370957	+	1.03032i
$428$	−7.00681			−0.338687
$429$		0			0
$430$		0			0
$431$			20.2476i			0.975293i	0.873041	+	0.487647i	$0.162145\pi$
$431$			20.2476i			0.975293i	−0.873041	+	0.487647i	$0.837855\pi$
$432$		0			0
$433$	−30.6444			−1.47268			−0.736338	−	0.676614i	$-0.763447\pi$
$433$	−30.6444			−1.47268			−0.736338	+	0.676614i	$0.763447\pi$
$434$	0.866067	+	24.0546i	0.0415726	+	1.15466i
$435$		0			0
$436$	4.28991			0.205450
$437$			48.1092i			2.30138i
$438$		0			0
$439$		−	22.6308i		−	1.08011i	−0.841630	−	0.540054i	$-0.818404\pi$
$439$		−	22.6308i		−	1.08011i	0.841630	−	0.540054i	$-0.181596\pi$
$440$		0			0
$441$		0			0
$442$		−	2.28991i		−	0.108920i
$443$	−36.3083			−1.72506			−0.862529	−	0.506007i	$-0.831121\pi$
$443$	−36.3083			−1.72506			−0.862529	+	0.506007i	$0.831121\pi$
$444$		0			0
$445$		0			0
$446$	10.8710			0.514759
$447$		0			0
$448$	−2.64404	+	0.0951965i	−0.124919	+	0.00449761i
$449$		−	25.6764i		−	1.21175i	−0.795561	−	0.605873i	$-0.792824\pi$
$449$		−	25.6764i		−	1.21175i	0.795561	−	0.605873i	$-0.207176\pi$
$450$		0			0
$451$			11.5830i			0.545421i
$452$	−8.09087			−0.380563
$453$		0			0
$454$			14.0909i			0.661317i
$455$		0			0
$456$		0			0
$457$			25.3477i			1.18571i	0.805308	+	0.592857i	$0.202000\pi$
$457$			25.3477i			1.18571i	−0.805308	+	0.592857i	$0.798000\pi$
$458$		−	14.5239i		−	0.678657i
$459$		0			0
$460$		0			0
$461$	−12.7253			−0.592677			−0.296339	−	0.955083i	$-0.595766\pi$
$461$	−12.7253			−0.592677			−0.296339	+	0.955083i	$0.595766\pi$
$462$		0			0
$463$			21.2655i			0.988289i	0.869380	+	0.494145i	$0.164519\pi$
$463$			21.2655i			0.988289i	−0.869380	+	0.494145i	$0.835481\pi$
$464$		−	7.47847i		−	0.347179i
$465$		0			0
$466$	20.6806			0.958013
$467$		−	13.1424i		−	0.608156i	−0.952647	−	0.304078i	$-0.901652\pi$
$467$		−	13.1424i		−	0.608156i	0.952647	−	0.304078i	$-0.0983483\pi$
$468$		0			0
$469$	−0.507906	−	14.1069i	−0.0234529	−	0.651395i
$470$		0			0
$471$		0			0
$472$	13.7734			0.633970
$473$	−5.04781			−0.232099
$474$		0			0
$475$		0			0
$476$	0.0995218	+	2.76417i	0.00456157	+	0.126696i
$477$		0			0
$478$			5.56697i			0.254627i
$479$	26.2899			1.20122			0.600608	−	0.799543i	$-0.294925\pi$
$479$	26.2899			1.20122			0.600608	+	0.799543i	$0.294925\pi$
$480$		0			0
$481$			1.87288i			0.0853959i
$482$			11.3839i			0.518523i
$483$		0			0
$484$	−16.9638			−0.771080
$485$		0			0
$486$		0			0
$487$			39.5381i			1.79164i	0.444416	+	0.895820i	$0.353411\pi$
$487$			39.5381i			1.79164i	−0.444416	+	0.895820i	$0.646589\pi$
$488$			8.05225i			0.364508i
$489$		0			0
$490$		0			0
$491$		−	4.62105i		−	0.208545i	−0.994549	−	0.104273i	$-0.966749\pi$
$491$		−	4.62105i		−	0.208545i	0.994549	−	0.104273i	$-0.0332514\pi$
$492$		0			0
$493$	−7.81826			−0.352117
$494$			14.0909i			0.633978i
$495$		0			0
$496$			9.09768i			0.408498i
$497$	0.612402	+	17.0092i	0.0274700	+	0.762966i
$498$		0			0
$499$	−31.0152			−1.38843			−0.694216	−	0.719766i	$-0.744249\pi$
$499$	−31.0152			−1.38843			−0.694216	+	0.719766i	$0.744249\pi$
$500$		0			0
$501$		0			0
$502$	−1.71874			−0.0767109
$503$		−	28.3385i		−	1.26355i	−0.775152	−	0.631775i	$-0.782327\pi$
$503$		−	28.3385i		−	1.26355i	0.775152	−	0.631775i	$-0.217673\pi$
$504$		0			0
$505$		0			0
$506$		−	39.5467i		−	1.75807i
$507$		0			0
$508$			8.57118i			0.380285i
$509$	22.3808			0.992011			0.496005	−	0.868319i	$-0.334800\pi$
$509$	22.3808			0.992011			0.496005	+	0.868319i	$0.334800\pi$
$510$		0			0
$511$	12.0995	−	0.435633i	0.535251	−	0.0192713i
$512$	−1.00000			−0.0441942
$513$		0			0
$514$		−	2.18779i		−	0.0964994i
$515$		0			0
$516$		0			0
$517$	58.2174			2.56040
$518$	−0.0813970	−	2.26077i	−0.00357638	−	0.0993323i
$519$		0			0
$520$		0			0
$521$	−28.5896			−1.25253			−0.626265	−	0.779610i	$-0.715417\pi$
$521$	−28.5896			−1.25253			−0.626265	+	0.779610i	$0.715417\pi$
$522$		0			0
$523$	3.51472			0.153688			0.0768440	−	0.997043i	$-0.475516\pi$
$523$	3.51472			0.153688			0.0768440	+	0.997043i	$0.475516\pi$
$524$	5.90048			0.257764
$525$		0			0
$526$	−17.3876			−0.758135
$527$	9.51104			0.414307
$528$		0			0
$529$	32.9275			1.43163
$530$		0			0
$531$		0			0
$532$	−0.612402	−	17.0092i	−0.0265510	−	0.737442i
$533$	−4.79782			−0.207817
$534$		0			0
$535$		0			0
$536$		−	5.33535i		−	0.230452i
$537$		0			0
$538$	20.3445			0.877115
$539$	2.66205	+	36.9207i	0.114663	+	1.59029i
$540$		0			0
$541$	30.4900			1.31087			0.655434	−	0.755252i	$-0.272486\pi$
$541$	30.4900			1.31087			0.655434	+	0.755252i	$0.272486\pi$
$542$			1.47847i			0.0635057i
$543$		0			0
$544$			1.04544i			0.0448227i
$545$		0			0
$546$		0			0
$547$		−	6.86369i		−	0.293470i	−0.989176	−	0.146735i	$-0.953123\pi$
$547$		−	6.86369i		−	0.293470i	0.989176	−	0.146735i	$-0.0468765\pi$
$548$	6.86607			0.293304
$549$		0			0
$550$		0			0
$551$	48.1092			2.04952
$552$		0			0
$553$	41.4422	−	1.49209i	1.76230	−	0.0634503i
$554$		−	10.7642i		−	0.457326i
$555$		0			0
$556$			13.9115i			0.589979i
$557$	1.81826			0.0770421			0.0385210	−	0.999258i	$-0.487735\pi$
$557$	1.81826			0.0770421			0.0385210	+	0.999258i	$0.487735\pi$
$558$		0			0
$559$		−	2.09087i		−	0.0884344i
$560$		0			0
$561$		0			0
$562$		−	2.42806i		−	0.102422i
$563$			34.0184i			1.43370i	0.697226	+	0.716852i	$0.254418\pi$
$563$			34.0184i			1.43370i	−0.697226	+	0.716852i	$0.745582\pi$
$564$		0			0
$565$		0			0
$566$	−24.5898			−1.03359
$567$		0			0
$568$			6.43303i			0.269924i
$569$			5.10871i			0.214168i	0.994250	+	0.107084i	$0.0341514\pi$
$569$			5.10871i			0.214168i	−0.994250	+	0.107084i	$0.965849\pi$
$570$		0			0
$571$	10.8214			0.452861			0.226431	−	0.974027i	$-0.427294\pi$
$571$	10.8214			0.452861			0.226431	+	0.974027i	$0.427294\pi$
$572$		−	11.5830i		−	0.484308i
$573$		0			0
$574$	5.79148	−	0.208518i	0.241732	−	0.00870336i
$575$		0			0
$576$		0			0
$577$	39.1523			1.62993			0.814966	−	0.579509i	$-0.196756\pi$
$577$	39.1523			1.62993			0.814966	+	0.579509i	$0.196756\pi$
$578$	−15.9071			−0.661647
$579$		0			0
$580$		0			0
$581$	−0.417035	−	11.5830i	−0.0173015	−	0.480542i
$582$		0			0
$583$		−	16.3808i		−	0.678423i
$584$	4.57615			0.189363
$585$		0			0
$586$			12.6707i			0.523422i
$587$		−	5.52834i		−	0.228179i	−0.993470	−	0.114090i	$-0.963605\pi$
$587$		−	5.52834i		−	0.228179i	0.993470	−	0.114090i	$-0.0363951\pi$
$588$		0			0
$589$	−58.5257			−2.41151
$590$		0			0
$591$		0			0
$592$		−	0.855043i		−	0.0351420i
$593$		−	18.6830i		−	0.767220i	−0.923495	−	0.383610i	$-0.874681\pi$
$593$		−	18.6830i		−	0.767220i	0.923495	−	0.383610i	$-0.125319\pi$
$594$		0			0
$595$		0			0
$596$			15.9638i			0.653901i
$597$		0			0
$598$	16.3808			0.669860
$599$		−	4.06587i		−	0.166127i	−0.996544	−	0.0830635i	$-0.973530\pi$
$599$		−	4.06587i		−	0.166127i	0.996544	−	0.0830635i	$-0.0264704\pi$
$600$		0			0
$601$		−	25.1161i		−	1.02451i	−0.858835	−	0.512253i	$-0.828811\pi$
$601$		−	25.1161i		−	1.02451i	0.858835	−	0.512253i	$-0.171189\pi$
$602$	0.0908711	+	2.52390i	0.00370363	+	0.102867i
$603$		0			0
$604$	16.0546			0.653253
$605$		0			0
$606$		0			0
$607$	31.4336			1.27585			0.637925	−	0.770099i	$-0.279793\pi$
$607$	31.4336			1.27585			0.637925	+	0.770099i	$0.279793\pi$
$608$		−	6.43303i		−	0.260894i
$609$		0			0
$610$		0			0
$611$			24.1144i			0.975566i
$612$		0			0
$613$		−	39.4532i		−	1.59350i	−0.604308	−	0.796751i	$-0.706550\pi$
$613$		−	39.4532i		−	1.59350i	0.604308	−	0.796751i	$-0.293450\pi$
$614$	−0.866067			−0.0349516
$615$		0			0
$616$	0.503406	+	13.9819i	0.0202828	+	0.563346i
$617$	39.7421			1.59996			0.799978	−	0.600029i	$-0.204844\pi$
$617$	39.7421			1.59996			0.799978	+	0.600029i	$0.204844\pi$
$618$		0			0
$619$		−	14.7776i		−	0.593961i	−0.954884	−	0.296980i	$-0.904020\pi$
$619$		−	14.7776i		−	0.593961i	0.954884	−	0.296980i	$-0.0959796\pi$
$620$		0			0
$621$		0			0
$622$	9.71009			0.389339
$623$	11.3198	−	0.407561i	0.453519	−	0.0163286i
$624$		0			0
$625$		0			0
$626$	5.80096			0.231853
$627$		0			0
$628$	−24.7665			−0.988293
$629$	−0.893892			−0.0356418
$630$		0			0
$631$	−16.3083			−0.649223			−0.324611	−	0.945847i	$-0.605233\pi$
$631$	−16.3083			−0.649223			−0.324611	+	0.945847i	$0.605233\pi$
$632$	15.6738			0.623472
$633$		0			0
$634$	28.1591			1.11834
$635$		0			0
$636$		0			0
$637$	−15.2931	+	1.10266i	−0.605933	+	0.0436889i
$638$	−39.5467			−1.56567
$639$		0			0
$640$		0			0
$641$		−	12.5289i		−	0.494861i	−0.968906	−	0.247430i	$-0.920414\pi$
$641$		−	12.5289i		−	0.494861i	0.968906	−	0.247430i	$-0.0795862\pi$
$642$		0			0
$643$	14.3992			0.567847			0.283924	−	0.958847i	$-0.408364\pi$
$643$	14.3992			0.567847			0.283924	+	0.958847i	$0.408364\pi$
$644$	−19.7734	+	0.711924i	−0.779179	+	0.0280537i
$645$		0			0
$646$	−6.72532			−0.264604
$647$		−	31.7708i		−	1.24904i	−0.781010	−	0.624519i	$-0.785295\pi$
$647$		−	31.7708i		−	1.24904i	0.781010	−	0.624519i	$-0.214705\pi$
$648$		0			0
$649$		−	72.8346i		−	2.85901i
$650$		0			0
$651$		0			0
$652$		−	7.42622i		−	0.290833i
$653$	−20.7389			−0.811578			−0.405789	−	0.913967i	$-0.633003\pi$
$653$	−20.7389			−0.811578			−0.405789	+	0.913967i	$0.633003\pi$
$654$		0			0
$655$		0			0
$656$	2.19039			0.0855205
$657$		0			0
$658$	−1.04804	−	29.1087i	−0.0408567	−	1.13478i
$659$			2.44038i			0.0950638i	0.998870	+	0.0475319i	$0.0151356\pi$
$659$			2.44038i			0.0950638i	−0.998870	+	0.0475319i	$0.984864\pi$
$660$		0			0
$661$			33.3863i			1.29858i	0.760542	+	0.649288i	$0.224933\pi$
$661$			33.3863i			1.29858i	−0.760542	+	0.649288i	$0.775067\pi$
$662$	19.5830			0.761114
$663$		0			0
$664$		−	4.38079i		−	0.170007i
$665$		0			0
$666$		0			0
$667$		−	55.9275i		−	2.16552i
$668$			0.573779i			0.0222002i
$669$		0			0
$670$		0			0
$671$	42.5809			1.64382
$672$		0			0
$673$			18.6943i			0.720611i	0.932834	+	0.360306i	$0.117328\pi$
$673$			18.6943i			0.720611i	−0.932834	+	0.360306i	$0.882672\pi$
$674$			28.3992i			1.09389i
$675$		0			0
$676$	−8.20218			−0.315468
$677$			36.5861i			1.40612i	0.711131	+	0.703059i	$0.248183\pi$
$677$			36.5861i			1.40612i	−0.711131	+	0.703059i	$0.751817\pi$
$678$		0			0
$679$	31.3563	−	1.12896i	1.20335	−	0.0433255i
$680$		0			0
$681$		0			0
$682$	48.1092			1.84220
$683$	36.9207			1.41273			0.706365	−	0.707847i	$-0.250334\pi$
$683$	36.9207			1.41273			0.706365	+	0.707847i	$0.250334\pi$
$684$		0			0
$685$		0			0
$686$	18.4124	−	1.99567i	0.702990	−	0.0761952i
$687$		0			0
$688$			0.954564i			0.0363924i
$689$	6.78514			0.258493
$690$		0			0
$691$		−	5.20823i		−	0.198130i	−0.995081	−	0.0990652i	$-0.968415\pi$
$691$		−	5.20823i		−	0.198130i	0.995081	−	0.0990652i	$-0.0315852\pi$
$692$			9.96375i			0.378765i
$693$		0			0
$694$	−12.5898			−0.477901
$695$		0			0
$696$		0			0
$697$		−	2.28991i		−	0.0867367i
$698$			20.9956i			0.794694i
$699$		0			0
$700$		0			0
$701$		−	13.2831i		−	0.501696i	−0.968027	−	0.250848i	$-0.919291\pi$
$701$		−	13.2831i		−	0.501696i	0.968027	−	0.250848i	$-0.0807094\pi$
$702$		0			0
$703$	5.50052			0.207456
$704$			5.28808i			0.199302i
$705$		0			0
$706$		−	15.1363i		−	0.569662i
$707$	−30.6258	+	1.10266i	−1.15180	+	0.0414697i
$708$		0			0
$709$	4.09087			0.153636			0.0768179	−	0.997045i	$-0.475524\pi$
$709$	4.09087			0.153636			0.0768179	+	0.997045i	$0.475524\pi$
$710$		0			0
$711$		0			0
$712$	4.28126			0.160447
$713$			68.0367i			2.54800i
$714$		0			0
$715$		0			0
$716$		−	1.77336i		−	0.0662735i
$717$		0			0
$718$			33.1909i			1.23867i
$719$	16.1817			0.603477			0.301739	−	0.953391i	$-0.402433\pi$
$719$	16.1817			0.603477			0.301739	+	0.953391i	$0.402433\pi$
$720$		0			0
$721$	44.0911	−	1.58747i	1.64204	−	0.0591203i
$722$	22.3839			0.833043
$723$		0			0
$724$		−	11.0092i		−	0.409153i
$725$		0			0
$726$		0			0
$727$	40.7849			1.51263			0.756314	−	0.654208i	$-0.226998\pi$
$727$	40.7849			1.51263			0.756314	+	0.654208i	$0.226998\pi$
$728$	−5.79148	+	0.208518i	−0.214647	+	0.00772818i
$729$		0			0
$730$		0			0
$731$	0.997936			0.0369100
$732$		0			0
$733$	39.2481			1.44966			0.724832	−	0.688926i	$-0.241917\pi$
$733$	39.2481			1.44966			0.724832	+	0.688926i	$0.241917\pi$
$734$	11.6825			0.431208
$735$		0			0
$736$	−7.47847			−0.275660
$737$	−28.2137			−1.03927
$738$		0			0
$739$	16.7305			0.615442			0.307721	−	0.951477i	$-0.400434\pi$
$739$	16.7305			0.615442			0.307721	+	0.951477i	$0.400434\pi$
$740$		0			0
$741$		0			0
$742$	−8.19039	+	0.294888i	−0.300679	+	0.0108257i
$743$	−35.7185			−1.31039			−0.655193	−	0.755462i	$-0.727412\pi$
$743$	−35.7185			−1.31039			−0.655193	+	0.755462i	$0.727412\pi$
$744$		0			0
$745$		0			0
$746$			12.8550i			0.470657i
$747$		0			0
$748$	5.52834			0.202136
$749$	18.5263	−	0.667024i	0.676935	−	0.0243725i
$750$		0			0
$751$	−9.14236			−0.333609			−0.166805	−	0.985990i	$-0.553345\pi$
$751$	−9.14236			−0.333609			−0.166805	+	0.985990i	$0.553345\pi$
$752$		−	11.0092i		−	0.401464i
$753$		0			0
$754$		−	16.3808i		−	0.596553i
$755$		0			0
$756$		0			0
$757$			48.2027i			1.75196i	0.482350	+	0.875979i	$0.339783\pi$
$757$			48.2027i			1.75196i	−0.482350	+	0.875979i	$0.660217\pi$
$758$	25.0751			0.910767
$759$		0			0
$760$		0			0
$761$	−9.61056			−0.348383			−0.174191	−	0.984712i	$-0.555731\pi$
$761$	−9.61056			−0.348383			−0.174191	+	0.984712i	$0.555731\pi$
$762$		0			0
$763$	−11.3427	+	0.408385i	−0.410633	+	0.0147845i
$764$		−	5.56697i		−	0.201406i
$765$		0			0
$766$		−	23.0092i		−	0.831356i
$767$	30.1691			1.08934
$768$		0			0
$769$			33.2931i			1.20058i	0.799783	+	0.600289i	$0.204948\pi$
$769$			33.2931i			1.20058i	−0.799783	+	0.600289i	$0.795052\pi$
$770$		0			0
$771$		0			0
$772$			8.47166i			0.304902i
$773$		−	12.2726i		−	0.441415i	−0.975340	−	0.220708i	$-0.929163\pi$
$773$		−	12.2726i		−	0.441415i	0.975340	−	0.220708i	$-0.0708367\pi$
$774$		0			0
$775$		0			0
$776$	11.8593			0.425722
$777$		0			0
$778$			10.1591i			0.364222i
$779$			14.0909i			0.504858i
$780$		0			0
$781$	34.0184			1.21727
$782$			7.81826i			0.279580i
$783$		0			0
$784$	6.98188	−	0.503406i	0.249353	−	0.0179788i
$785$		0			0
$786$		0			0
$787$	21.4286			0.763847			0.381923	−	0.924194i	$-0.375262\pi$
$787$	21.4286			0.763847			0.381923	+	0.924194i	$0.375262\pi$
$788$	12.8661			0.458335
$789$		0			0
$790$		0			0
$791$	21.3926	−	0.770222i	0.760632	−	0.0273859i
$792$		0			0
$793$			17.6376i			0.626329i
$794$	13.9141			0.493793
$795$		0			0
$796$		−	17.5830i		−	0.623212i
$797$			9.96375i			0.352934i	0.984307	+	0.176467i	$0.0564669\pi$
$797$			9.96375i			0.352934i	−0.984307	+	0.176467i	$0.943533\pi$
$798$		0			0
$799$	−11.5094			−0.407173
$800$		0			0
$801$		0			0
$802$			14.6197i			0.516241i
$803$		−	24.1990i		−	0.853966i
$804$		0			0
$805$		0			0
$806$			19.9275i			0.701916i
$807$		0			0
$808$	−11.5830			−0.407487
$809$			32.6234i			1.14698i	0.819213	+	0.573489i	$0.194411\pi$
$809$			32.6234i			1.14698i	−0.819213	+	0.573489i	$0.805589\pi$
$810$		0			0
$811$			56.6914i			1.99071i	0.0962936	+	0.995353i	$0.469301\pi$
$811$			56.6914i			1.99071i	−0.0962936	+	0.995353i	$0.530699\pi$
$812$	0.711924	+	19.7734i	0.0249836	+	0.693909i
$813$		0			0
$814$	−4.52153			−0.158480
$815$		0			0
$816$		0			0
$817$	−6.14075			−0.214837
$818$			12.0000i			0.419570i
$819$		0			0
$820$		0			0
$821$			45.3198i			1.58167i	0.612027	+	0.790837i	$0.290354\pi$
$821$			45.3198i			1.58167i	−0.612027	+	0.790837i	$0.709646\pi$
$822$		0			0
$823$		−	29.6462i		−	1.03340i	−0.856166	−	0.516701i	$-0.827160\pi$
$823$		−	29.6462i		−	1.03340i	0.856166	−	0.516701i	$-0.172840\pi$
$824$	16.6757			0.580925
$825$		0			0
$826$	−36.4173	+	1.31117i	−1.26712	+	0.0456216i
$827$	−42.1129			−1.46441			−0.732205	−	0.681084i	$-0.761509\pi$
$827$	−42.1129			−1.46441			−0.732205	+	0.681084i	$0.761509\pi$
$828$		0			0
$829$			33.3936i			1.15981i	0.814684	+	0.579905i	$0.196910\pi$
$829$			33.3936i			1.15981i	−0.814684	+	0.579905i	$0.803090\pi$
$830$		0			0
$831$		0			0
$832$	−2.19039			−0.0759382
$833$	−0.526279	−	7.29910i	−0.0182345	−	0.252899i
$834$		0			0
$835$		0			0
$836$	−34.0184			−1.17655
$837$		0			0
$838$	2.60743			0.0900721
$839$	41.4385			1.43062			0.715309	−	0.698809i	$-0.246286\pi$
$839$	41.4385			1.43062			0.715309	+	0.698809i	$0.246286\pi$
$840$		0			0
$841$	−26.9275			−0.928535
$842$	−0.852443			−0.0293771
$843$		0			0
$844$	4.00000			0.137686
$845$		0			0
$846$		0			0
$847$	44.8528	−	1.61489i	1.54116	−	0.0554882i
$848$	−3.09768			−0.106375
$849$		0			0
$850$		0			0
$851$		−	6.39441i		−	0.219198i
$852$		0			0
$853$	30.4866			1.04384			0.521920	−	0.852994i	$-0.325216\pi$
$853$	30.4866			1.04384			0.521920	+	0.852994i	$0.325216\pi$
$854$	−0.766545	−	21.2905i	−0.0262306	−	0.728544i
$855$		0			0
$856$	7.00681			0.239488
$857$			5.22718i			0.178557i	0.996007	+	0.0892785i	$0.0284561\pi$
$857$			5.22718i			0.178557i	−0.996007	+	0.0892785i	$0.971544\pi$
$858$		0			0
$859$			43.2728i			1.47645i	0.674555	+	0.738224i	$0.264335\pi$
$859$			43.2728i			1.47645i	−0.674555	+	0.738224i	$0.735665\pi$
$860$		0			0
$861$		0			0
$862$		−	20.2476i		−	0.689636i
$863$	−30.9259			−1.05273			−0.526365	−	0.850259i	$-0.676445\pi$
$863$	−30.9259			−1.05273			−0.526365	+	0.850259i	$0.676445\pi$
$864$		0			0
$865$		0			0
$866$	30.6444			1.04134
$867$		0			0
$868$	−0.866067	−	24.0546i	−0.0293962	−	0.816467i
$869$		−	82.8844i		−	2.81166i
$870$		0			0
$871$		−	11.6865i		−	0.395982i
$872$	−4.28991			−0.145275
$873$		0			0
$874$		−	48.1092i		−	1.62732i
$875$		0			0
$876$		0			0
$877$			1.79836i			0.0607262i	0.999539	+	0.0303631i	$0.00966636\pi$
$877$			1.79836i			0.0607262i	−0.999539	+	0.0303631i	$0.990334\pi$
$878$			22.6308i			0.763752i
$879$		0			0
$880$		0			0
$881$	−19.6289			−0.661316			−0.330658	−	0.943751i	$-0.607271\pi$
$881$	−19.6289			−0.661316			−0.330658	+	0.943751i	$0.607271\pi$
$882$		0			0
$883$			52.5013i			1.76681i	0.468611	+	0.883404i	$0.344754\pi$
$883$			52.5013i			1.76681i	−0.468611	+	0.883404i	$0.655246\pi$
$884$			2.28991i			0.0770182i
$885$		0			0
$886$	36.3083			1.21980
$887$			39.1720i			1.31527i	0.753338	+	0.657633i	$0.228442\pi$
$887$			39.1720i			1.31527i	−0.753338	+	0.657633i	$0.771558\pi$
$888$		0			0
$889$	−0.815946	−	22.6625i	−0.0273659	−	0.760077i
$890$		0			0
$891$		0			0
$892$	−10.8710			−0.363989
$893$	70.8225			2.36998
$894$		0			0
$895$		0			0
$896$	2.64404	−	0.0951965i	0.0883311	−	0.00318029i
$897$		0			0
$898$			25.6764i			0.856834i
$899$	68.0367			2.26915
$900$		0			0
$901$			3.23843i			0.107888i
$902$		−	11.5830i		−	0.385671i
$903$		0			0
$904$	8.09087			0.269098
$905$		0			0
$906$		0			0
$907$		−	47.0822i		−	1.56334i	−0.623693	−	0.781669i	$-0.714369\pi$
$907$		−	47.0822i		−	1.56334i	0.623693	−	0.781669i	$-0.285631\pi$
$908$		−	14.0909i		−	0.467622i
$909$		0			0
$910$		0			0
$911$		−	40.6504i		−	1.34681i	−0.739274	−	0.673405i	$-0.764831\pi$
$911$		−	40.6504i		−	1.34681i	0.739274	−	0.673405i	$-0.235169\pi$
$912$		0			0
$913$	−23.1659			−0.766680
$914$		−	25.3477i		−	0.838426i
$915$		0			0
$916$			14.5239i			0.479883i
$917$	−15.6011	+	0.561705i	−0.515193	+	0.0185491i
$918$		0			0
$919$	−44.4900			−1.46759			−0.733795	−	0.679371i	$-0.762253\pi$
$919$	−44.4900			−1.46759			−0.733795	+	0.679371i	$0.762253\pi$
$920$		0			0
$921$		0			0
$922$	12.7253			0.419086
$923$			14.0909i			0.463807i
$924$		0			0
$925$		0			0
$926$		−	21.2655i		−	0.698826i
$927$		0			0
$928$			7.47847i			0.245493i
$929$	−53.7371			−1.76306			−0.881529	−	0.472131i	$-0.843485\pi$
$929$	−53.7371			−1.76306			−0.881529	+	0.472131i	$0.843485\pi$
$930$		0			0
$931$	3.23843	+	44.9146i	0.106135	+	1.47202i
$932$	−20.6806			−0.677417
$933$		0			0
$934$			13.1424i			0.430031i
$935$		0			0
$936$		0			0
$937$	−44.0682			−1.43965			−0.719823	−	0.694157i	$-0.755777\pi$
$937$	−44.0682			−1.43965			−0.719823	+	0.694157i	$0.755777\pi$
$938$	0.507906	+	14.1069i	0.0165837	+	0.460606i
$939$		0			0
$940$		0			0
$941$	20.0362			0.653163			0.326582	−	0.945169i	$-0.394103\pi$
$941$	20.0362			0.653163			0.326582	+	0.945169i	$0.394103\pi$
$942$		0			0
$943$	16.3808			0.533432
$944$	−13.7734			−0.448285
$945$		0			0
$946$	5.04781			0.164118
$947$	−30.9244			−1.00491			−0.502453	−	0.864604i	$-0.667569\pi$
$947$	−30.9244			−1.00491			−0.502453	+	0.864604i	$0.667569\pi$
$948$		0			0
$949$	10.0236			0.325379
$950$		0			0
$951$		0			0
$952$	−0.0995218	−	2.76417i	−0.00322552	−	0.0895873i
$953$	31.8414			1.03144			0.515722	−	0.856756i	$-0.327524\pi$
$953$	31.8414			1.03144			0.515722	+	0.856756i	$0.327524\pi$
$954$		0			0
$955$		0			0
$956$		−	5.56697i		−	0.180049i
$957$		0			0
$958$	−26.2899			−0.849389
$959$	−18.1541	+	0.653625i	−0.586228	+	0.0211067i
$960$		0			0
$961$	−51.7678			−1.66993
$962$		−	1.87288i		−	0.0603841i
$963$		0			0
$964$		−	11.3839i		−	0.366651i
$965$		0			0
$966$		0			0
$967$			32.6804i			1.05093i	0.850815	+	0.525466i	$0.176109\pi$
$967$			32.6804i			1.05093i	−0.850815	+	0.525466i	$0.823891\pi$
$968$	16.9638			0.545236
$969$		0			0
$970$		0			0
$971$	19.2419			0.617501			0.308751	−	0.951143i	$-0.400089\pi$
$971$	19.2419			0.617501			0.308751	+	0.951143i	$0.400089\pi$
$972$		0			0
$973$	−1.32433	−	36.7825i	−0.0424559	−	1.17919i
$974$		−	39.5381i		−	1.26688i
$975$		0			0
$976$		−	8.05225i		−	0.257746i
$977$	−2.48528			−0.0795112			−0.0397556	−	0.999209i	$-0.512658\pi$
$977$	−2.48528			−0.0795112			−0.0397556	+	0.999209i	$0.512658\pi$
$978$		0			0
$979$		−	22.6397i		−	0.723566i
$980$		0			0
$981$		0			0
$982$			4.62105i			0.147464i
$983$			49.4135i			1.57605i	0.615645	+	0.788024i	$0.288896\pi$
$983$			49.4135i			1.57605i	−0.615645	+	0.788024i	$0.711104\pi$
$984$		0			0
$985$		0			0
$986$	7.81826			0.248984
$987$		0			0
$988$		−	14.0909i		−	0.448290i
$989$			7.13868i			0.226997i
$990$		0			0
$991$	−57.0526			−1.81233			−0.906167	−	0.422920i	$-0.861005\pi$
$991$	−57.0526			−1.81233			−0.906167	+	0.422920i	$0.861005\pi$
$992$		−	9.09768i		−	0.288852i
$993$		0			0
$994$	−0.612402	−	17.0092i	−0.0194242	−	0.539499i
$995$		0			0
$996$		0			0
$997$	16.6794			0.528240			0.264120	−	0.964490i	$-0.414918\pi$
$997$	16.6794			0.528240			0.264120	+	0.964490i	$0.414918\pi$
$998$	31.0152			0.981770
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3150.2.d.a.3149.2		8
3.2	odd	2		3150.2.d.d.3149.2		8
5.2	odd	4		3150.2.b.f.251.2		8
5.3	odd	4		630.2.b.b.251.7	yes	8
5.4	even	2		3150.2.d.f.3149.7		8
7.6	odd	2		3150.2.d.c.3149.8		8
15.2	even	4		3150.2.b.e.251.6		8
15.8	even	4		630.2.b.a.251.3	✓	8
15.14	odd	2		3150.2.d.c.3149.7		8
20.3	even	4		5040.2.f.i.881.3		8
21.20	even	2		3150.2.d.f.3149.8		8
35.13	even	4		630.2.b.a.251.7	yes	8
35.27	even	4		3150.2.b.e.251.2		8
35.34	odd	2		3150.2.d.d.3149.1		8
60.23	odd	4		5040.2.f.f.881.3		8
105.62	odd	4		3150.2.b.f.251.6		8
105.83	odd	4		630.2.b.b.251.3	yes	8
105.104	even	2	inner	3150.2.d.a.3149.1		8
140.83	odd	4		5040.2.f.f.881.4		8
420.83	even	4		5040.2.f.i.881.4		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
630.2.b.a.251.3	✓	8	15.8	even	4
630.2.b.a.251.7	yes	8	35.13	even	4
630.2.b.b.251.3	yes	8	105.83	odd	4
630.2.b.b.251.7	yes	8	5.3	odd	4
3150.2.b.e.251.2		8	35.27	even	4
3150.2.b.e.251.6		8	15.2	even	4
3150.2.b.f.251.2		8	5.2	odd	4
3150.2.b.f.251.6		8	105.62	odd	4
3150.2.d.a.3149.1		8	105.104	even	2	inner
3150.2.d.a.3149.2		8	1.1	even	1	trivial
3150.2.d.c.3149.7		8	15.14	odd	2
3150.2.d.c.3149.8		8	7.6	odd	2
3150.2.d.d.3149.1		8	35.34	odd	2
3150.2.d.d.3149.2		8	3.2	odd	2
3150.2.d.f.3149.7		8	5.4	even	2
3150.2.d.f.3149.8		8	21.20	even	2
5040.2.f.f.881.3		8	60.23	odd	4
5040.2.f.f.881.4		8	140.83	odd	4
5040.2.f.i.881.3		8	20.3	even	4
5040.2.f.i.881.4		8	420.83	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:	\( N \)	\(=\)	\( 3150 = 2 \cdot 3^{2} \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3150.d (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000 q^{4} +(-2.64404 + 0.0951965i) q^{7} -1.00000 q^{8} +O(q^{10})\)	\(q-1.00000 q^{2} +1.00000 q^{4} +(-2.64404 + 0.0951965i) q^{7} -1.00000 q^{8} +5.28808i q^{11} -2.19039 q^{13} +(2.64404 - 0.0951965i) q^{14} +1.00000 q^{16} -1.04544i q^{17} +6.43303i q^{19} -5.28808i q^{22} +7.47847 q^{23} +2.19039 q^{26} +(-2.64404 + 0.0951965i) q^{28} -7.47847i q^{29} +9.09768i q^{31} -1.00000 q^{32} +1.04544i q^{34} -0.855043i q^{37} -6.43303i q^{38} +2.19039 q^{41} +0.954564i q^{43} +5.28808i q^{44} -7.47847 q^{46} -11.0092i q^{47} +(6.98188 - 0.503406i) q^{49} -2.19039 q^{52} -3.09768 q^{53} +(2.64404 - 0.0951965i) q^{56} +7.47847i q^{58} -13.7734 q^{59} -8.05225i q^{61} -9.09768i q^{62} +1.00000 q^{64} +5.33535i q^{67} -1.04544i q^{68} -6.43303i q^{71} -4.57615 q^{73} +0.855043i q^{74} +6.43303i q^{76} +(-0.503406 - 13.9819i) q^{77} -15.6738 q^{79} -2.19039 q^{82} +4.38079i q^{83} -0.954564i q^{86} -5.28808i q^{88} -4.28126 q^{89} +(5.79148 - 0.208518i) q^{91} +7.47847 q^{92} +11.0092i q^{94} -11.8593 q^{97} +(-6.98188 + 0.503406i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 8 q^{2} + 8 q^{4} - 8 q^{8}+O(q^{10}) \) 8 * q - 8 * q^2 + 8 * q^4 - 8 * q^8	\( 8 q - 8 q^{2} + 8 q^{4} - 8 q^{8} - 8 q^{13} + 8 q^{16} + 8 q^{23} + 8 q^{26} - 8 q^{32} + 8 q^{41} - 8 q^{46} - 4 q^{49} - 8 q^{52} + 8 q^{53} + 8 q^{64} + 48 q^{73} + 4 q^{77} - 8 q^{79} - 8 q^{82} - 8 q^{89} - 4 q^{91} + 8 q^{92} - 24 q^{97} + 4 q^{98}+O(q^{100}) \) 8 * q - 8 * q^2 + 8 * q^4 - 8 * q^8 - 8 * q^13 + 8 * q^16 + 8 * q^23 + 8 * q^26 - 8 * q^32 + 8 * q^41 - 8 * q^46 - 4 * q^49 - 8 * q^52 + 8 * q^53 + 8 * q^64 + 48 * q^73 + 4 * q^77 - 8 * q^79 - 8 * q^82 - 8 * q^89 - 4 * q^91 + 8 * q^92 - 24 * q^97 + 4 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more