LMFDB - Embedded newform 3150.2.b.f.251.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3150,2,Mod(251,3150)]

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, 0, 1]))

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

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

chi := DirichletCharacter("3150.251");

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.b (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$25.1528766367$
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^{2}\cdot 3^{2} $
Twist minimal:	no (minimal twist has level 630)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			251.6
Root			$3.73923i$ of defining polynomial
Character	$\chi$	$=$	3150.251
Dual form			3150.2.b.f.251.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{2} -1.00000 q^{4} +(-0.0951965 + 2.64404i) q^{7} -1.00000i q^{8} +O(q^{10})$	$q+1.00000i q^{2} -1.00000 q^{4} +(-0.0951965 + 2.64404i) q^{7} -1.00000i q^{8} -5.28808i q^{11} -2.19039i q^{13} +(-2.64404 - 0.0951965i) q^{14} +1.00000 q^{16} +1.04544 q^{17} +6.43303i q^{19} +5.28808 q^{22} +7.47847i q^{23} +2.19039 q^{26} +(0.0951965 - 2.64404i) q^{28} -7.47847i q^{29} -9.09768i q^{31} +1.00000i q^{32} +1.04544i q^{34} +0.855043 q^{37} -6.43303 q^{38} +2.19039 q^{41} +0.954564 q^{43} +5.28808i q^{44} -7.47847 q^{46} +11.0092 q^{47} +(-6.98188 - 0.503406i) q^{49} +2.19039i q^{52} -3.09768i q^{53} +(2.64404 + 0.0951965i) q^{56} +7.47847 q^{58} +13.7734 q^{59} +8.05225i q^{61} +9.09768 q^{62} -1.00000 q^{64} -5.33535 q^{67} -1.04544 q^{68} +6.43303i q^{71} -4.57615i q^{73} +0.855043i q^{74} -6.43303i q^{76} +(13.9819 + 0.503406i) q^{77} +15.6738 q^{79} +2.19039i q^{82} +4.38079 q^{83} +0.954564i q^{86} -5.28808 q^{88} +4.28126 q^{89} +(5.79148 + 0.208518i) q^{91} -7.47847i q^{92} +11.0092i q^{94} +11.8593i q^{97} +(0.503406 - 6.98188i) q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 8 q^{4} + 4 q^{7}+O(q^{10}) $ 8 * q - 8 * q^4 + 4 * q^7	$ 8 q - 8 q^{4} + 4 q^{7} + 8 q^{16} + 8 q^{26} - 4 q^{28} + 8 q^{37} - 8 q^{38} + 8 q^{41} + 16 q^{43} - 8 q^{46} - 40 q^{47} + 4 q^{49} + 8 q^{58} + 40 q^{62} - 8 q^{64} - 32 q^{67} + 52 q^{77} + 8 q^{79} + 16 q^{83} + 8 q^{89} - 4 q^{91} - 4 q^{98}+O(q^{100}) $ 8 * q - 8 * q^4 + 4 * q^7 + 8 * q^16 + 8 * q^26 - 4 * q^28 + 8 * q^37 - 8 * q^38 + 8 * q^41 + 16 * q^43 - 8 * q^46 - 40 * q^47 + 4 * q^49 + 8 * q^58 + 40 * q^62 - 8 * q^64 - 32 * q^67 + 52 * q^77 + 8 * q^79 + 16 * q^83 + 8 * q^89 - 4 * q^91 - 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 $

$2$			1.00000i			0.707107i
$2$			1.00000i			0.707107i
$3$		0			0
$3$		0			0
$4$	−1.00000			−0.500000
$5$		0			0
$5$		0			0
$6$		0			0
$7$	−0.0951965	+	2.64404i	−0.0359809	+	0.999352i
$7$	−0.0951965	+	2.64404i	−0.0359809	+	0.999352i
$8$		−	1.00000i		−	0.353553i
$9$		0			0
$10$		0			0
$11$		−	5.28808i		−	1.59441i	−0.603705	−	0.797207i	$-0.706310\pi$
$11$		−	5.28808i		−	1.59441i	0.603705	−	0.797207i	$-0.293690\pi$
$12$		0			0
$13$		−	2.19039i		−	0.607506i	−0.952751	−	0.303753i	$-0.901760\pi$
$13$		−	2.19039i		−	0.607506i	0.952751	−	0.303753i	$-0.0982397\pi$
$14$	−2.64404	−	0.0951965i	−0.706649	−	0.0254423i
$15$		0			0
$16$	1.00000			0.250000
$17$	1.04544			0.253555			0.126778	−	0.991931i	$-0.459537\pi$
$17$	1.04544			0.253555			0.126778	+	0.991931i	$0.459537\pi$
$18$		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
$22$	5.28808			1.12742
$23$			7.47847i			1.55937i	0.626173	+	0.779684i	$0.284620\pi$
$23$			7.47847i			1.55937i	−0.626173	+	0.779684i	$0.715380\pi$
$24$		0			0
$25$		0			0
$26$	2.19039			0.429571
$27$		0			0
$28$	0.0951965	−	2.64404i	0.0179904	−	0.499676i
$29$		−	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	$-0.695638\pi$
$31$		−	9.09768i		−	1.63399i	0.576643	−	0.816996i	$-0.304362\pi$
$32$			1.00000i			0.176777i
$33$		0			0
$34$			1.04544i			0.179291i
$35$		0			0
$36$		0			0
$37$	0.855043			0.140568			0.0702841	−	0.997527i	$-0.477609\pi$
$37$	0.855043			0.140568			0.0702841	+	0.997527i	$0.477609\pi$
$38$	−6.43303			−1.04358
$39$		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
$43$	0.954564			0.145570			0.0727849	−	0.997348i	$-0.476811\pi$
$43$	0.954564			0.145570			0.0727849	+	0.997348i	$0.476811\pi$
$44$			5.28808i			0.797207i
$45$		0			0
$46$	−7.47847			−1.10264
$47$	11.0092			1.60585			0.802927	−	0.596077i	$-0.203275\pi$
$47$	11.0092			1.60585			0.802927	+	0.596077i	$0.203275\pi$
$48$		0			0
$49$	−6.98188	−	0.503406i	−0.997411	−	0.0719152i
$50$		0			0
$51$		0			0
$52$			2.19039i			0.303753i
$53$		−	3.09768i		−	0.425500i	−0.977107	−	0.212750i	$-0.931758\pi$
$53$		−	3.09768i		−	0.425500i	0.977107	−	0.212750i	$-0.0682419\pi$
$54$		0			0
$55$		0			0
$56$	2.64404	+	0.0951965i	0.353324	+	0.0127212i
$57$		0			0
$58$	7.47847			0.981971
$59$	13.7734			1.79314			0.896569	−	0.442904i	$-0.146052\pi$
$59$	13.7734			1.79314			0.896569	+	0.442904i	$0.146052\pi$
$60$		0			0
$61$			8.05225i			1.03098i	0.856894	+	0.515492i	$0.172391\pi$
$61$			8.05225i			1.03098i	−0.856894	+	0.515492i	$0.827609\pi$
$62$	9.09768			1.15541
$63$		0			0
$64$	−1.00000			−0.125000
$65$		0			0
$66$		0			0
$67$	−5.33535			−0.651817			−0.325908	−	0.945401i	$-0.605670\pi$
$67$	−5.33535			−0.651817			−0.325908	+	0.945401i	$0.605670\pi$
$68$	−1.04544			−0.126778
$69$		0			0
$70$		0			0
$71$			6.43303i			0.763461i	0.924274	+	0.381730i	$0.124672\pi$
$71$			6.43303i			0.763461i	−0.924274	+	0.381730i	$0.875328\pi$
$72$		0			0
$73$		−	4.57615i		−	0.535598i	−0.963475	−	0.267799i	$-0.913704\pi$
$73$		−	4.57615i		−	0.535598i	0.963475	−	0.267799i	$-0.0862963\pi$
$74$			0.855043i			0.0993967i
$75$		0			0
$76$		−	6.43303i		−	0.737920i
$77$	13.9819	+	0.503406i	1.59338	+	0.0573684i
$78$		0			0
$79$	15.6738			1.76344			0.881722	−	0.471769i	$-0.156384\pi$
$79$	15.6738			1.76344			0.881722	+	0.471769i	$0.156384\pi$
$80$		0			0
$81$		0			0
$82$			2.19039i			0.241888i
$83$	4.38079			0.480854			0.240427	−	0.970667i	$-0.422713\pi$
$83$	4.38079			0.480854			0.240427	+	0.970667i	$0.422713\pi$
$84$		0			0
$85$		0			0
$86$			0.954564i			0.102933i
$87$		0			0
$88$	−5.28808			−0.563711
$89$	4.28126			0.453813			0.226907	−	0.973917i	$-0.427139\pi$
$89$	4.28126			0.453813			0.226907	+	0.973917i	$0.427139\pi$
$90$		0			0
$91$	5.79148	+	0.208518i	0.607112	+	0.0218586i
$92$		−	7.47847i		−	0.779684i
$93$		0			0
$94$			11.0092i			1.13551i
$95$		0			0
$96$		0			0
$97$			11.8593i			1.20412i	0.798449	+	0.602062i	$0.205654\pi$
$97$			11.8593i			1.20412i	−0.798449	+	0.602062i	$0.794346\pi$
$98$	0.503406	−	6.98188i	0.0508517	−	0.705276i
$99$		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
$103$		−	16.6757i		−	1.64310i	−0.570134	−	0.821552i	$-0.693109\pi$
$103$		−	16.6757i		−	1.64310i	0.570134	−	0.821552i	$-0.306891\pi$
$104$	−2.19039			−0.214786
$105$		0			0
$106$	3.09768			0.300874
$107$			7.00681i			0.677374i	0.940899	+	0.338687i	$0.109983\pi$
$107$			7.00681i			0.677374i	−0.940899	+	0.338687i	$0.890017\pi$
$108$		0			0
$109$	−4.28991			−0.410899			−0.205450	−	0.978668i	$-0.565866\pi$
$109$	−4.28991			−0.410899			−0.205450	+	0.978668i	$0.565866\pi$
$110$		0			0
$111$		0			0
$112$	−0.0951965	+	2.64404i	−0.00899522	+	0.249838i
$113$		−	8.09087i		−	0.761125i	−0.924755	−	0.380563i	$-0.875730\pi$
$113$		−	8.09087i		−	0.761125i	0.924755	−	0.380563i	$-0.124270\pi$
$114$		0			0
$115$		0			0
$116$			7.47847i			0.694358i
$117$		0			0
$118$			13.7734i			1.26794i
$119$	−0.0995218	+	2.76417i	−0.00912314	+	0.253391i
$120$		0			0
$121$	−16.9638			−1.54216
$122$	−8.05225			−0.729016
$123$		0			0
$124$			9.09768i			0.816996i
$125$		0			0
$126$		0			0
$127$	−8.57118			−0.760569			−0.380285	−	0.924870i	$-0.624174\pi$
$127$	−8.57118			−0.760569			−0.380285	+	0.924870i	$0.624174\pi$
$128$		−	1.00000i		−	0.0883883i
$129$		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
$133$	−17.0092	−	0.612402i	−1.47488	−	0.0531020i
$134$		−	5.33535i		−	0.460904i
$135$		0			0
$136$		−	1.04544i		−	0.0896454i
$137$		−	6.86607i		−	0.586608i	−0.956019	−	0.293304i	$-0.905245\pi$
$137$		−	6.86607i		−	0.586608i	0.956019	−	0.293304i	$-0.0947548\pi$
$138$		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
$142$	−6.43303			−0.539848
$143$	−11.5830			−0.968616
$144$		0			0
$145$		0			0
$146$	4.57615			0.378725
$147$		0			0
$148$	−0.855043			−0.0702841
$149$			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	$0.273404\pi$
$152$	6.43303			0.521788
$153$		0			0
$154$	−0.503406	+	13.9819i	−0.0405656	+	1.12669i
$155$		0			0
$156$		0			0
$157$			24.7665i			1.97659i	0.152569	+	0.988293i	$0.451245\pi$
$157$			24.7665i			1.97659i	−0.152569	+	0.988293i	$0.548755\pi$
$158$			15.6738i			1.24694i
$159$		0			0
$160$		0			0
$161$	−19.7734	−	0.711924i	−1.55836	−	0.0561075i
$162$		0			0
$163$	−7.42622			−0.581667			−0.290833	−	0.956774i	$-0.593932\pi$
$163$	−7.42622			−0.581667			−0.290833	+	0.956774i	$0.593932\pi$
$164$	−2.19039			−0.171041
$165$		0			0
$166$			4.38079i			0.340015i
$167$	−0.573779			−0.0444003			−0.0222002	−	0.999754i	$-0.507067\pi$
$167$	−0.573779			−0.0444003			−0.0222002	+	0.999754i	$0.507067\pi$
$168$		0			0
$169$	8.20218			0.630937
$170$		0			0
$171$		0			0
$172$	−0.954564			−0.0727849
$173$	9.96375			0.757530			0.378765	−	0.925493i	$-0.376349\pi$
$173$	9.96375			0.757530			0.378765	+	0.925493i	$0.376349\pi$
$174$		0			0
$175$		0			0
$176$		−	5.28808i		−	0.398604i
$177$		0			0
$178$			4.28126i			0.320894i
$179$		−	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	$0.134176\pi$
$181$			11.0092i			0.818306i	−0.912466	+	0.409153i	$0.865824\pi$
$182$	−0.208518	+	5.79148i	−0.0154564	+	0.429293i
$183$		0			0
$184$	7.47847			0.551320
$185$		0			0
$186$		0			0
$187$		−	5.52834i		−	0.404272i
$188$	−11.0092			−0.802927
$189$		0			0
$190$		0			0
$191$			5.56697i			0.402812i	0.979508	+	0.201406i	$0.0645510\pi$
$191$			5.56697i			0.402812i	−0.979508	+	0.201406i	$0.935449\pi$
$192$		0			0
$193$	8.47166			0.609803			0.304902	−	0.952384i	$-0.401376\pi$
$193$	8.47166			0.609803			0.304902	+	0.952384i	$0.401376\pi$
$194$	−11.8593			−0.851445
$195$		0			0
$196$	6.98188	+	0.503406i	0.498705	+	0.0359576i
$197$		−	12.8661i		−	0.916669i	−0.888780	−	0.458335i	$-0.848446\pi$
$197$		−	12.8661i		−	0.916669i	0.888780	−	0.458335i	$-0.151554\pi$
$198$		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
$202$			11.5830i			0.814975i
$203$	19.7734	+	0.711924i	1.38782	+	0.0499673i
$204$		0			0
$205$		0			0
$206$	16.6757			1.16185
$207$		0			0
$208$		−	2.19039i		−	0.151876i
$209$	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	$0.456034\pi$
$212$			3.09768i			0.212750i
$213$		0			0
$214$	−7.00681			−0.478976
$215$		0			0
$216$		0			0
$217$	24.0546	+	0.866067i	1.63293	+	0.0587925i
$218$		−	4.28991i		−	0.290550i
$219$		0			0
$220$		0			0
$221$		−	2.28991i		−	0.154036i
$222$		0			0
$223$		−	10.8710i		−	0.727979i	−0.931403	−	0.363989i	$-0.881414\pi$
$223$		−	10.8710i		−	0.727979i	0.931403	−	0.363989i	$-0.118586\pi$
$224$	−2.64404	−	0.0951965i	−0.176662	−	0.00636058i
$225$		0			0
$226$	8.09087			0.538197
$227$	14.0909			0.935244			0.467622	−	0.883929i	$-0.345111\pi$
$227$	14.0909			0.935244			0.467622	+	0.883929i	$0.345111\pi$
$228$		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
$232$	−7.47847			−0.490986
$233$		−	20.6806i		−	1.35483i	−0.735599	−	0.677417i	$-0.763099\pi$
$233$		−	20.6806i		−	1.35483i	0.735599	−	0.677417i	$-0.236901\pi$
$234$		0			0
$235$		0			0
$236$	−13.7734			−0.896569
$237$		0			0
$238$	−2.76417	−	0.0995218i	−0.179175	−	0.00645104i
$239$		−	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	$0.119496\pi$
$241$			11.3839i			0.733303i	−0.930358	+	0.366651i	$0.880504\pi$
$242$		−	16.9638i		−	1.09047i
$243$		0			0
$244$		−	8.05225i		−	0.515492i
$245$		0			0
$246$		0			0
$247$	14.0909			0.896581
$248$	−9.09768			−0.577703
$249$		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
$253$	39.5467			2.48628
$254$		−	8.57118i		−	0.537804i
$255$		0			0
$256$	1.00000			0.0625000
$257$	−2.18779			−0.136471			−0.0682354	−	0.997669i	$-0.521737\pi$
$257$	−2.18779			−0.136471			−0.0682354	+	0.997669i	$0.521737\pi$
$258$		0			0
$259$	−0.0813970	+	2.26077i	−0.00505777	+	0.140477i
$260$		0			0
$261$		0			0
$262$			5.90048i			0.364533i
$263$			17.3876i			1.07217i	0.844166	+	0.536083i	$0.180096\pi$
$263$			17.3876i			1.07217i	−0.844166	+	0.536083i	$0.819904\pi$
$264$		0			0
$265$		0			0
$266$	0.612402	−	17.0092i	0.0375488	−	1.04290i
$267$		0			0
$268$	5.33535			0.325908
$269$	20.3445			1.24043			0.620214	−	0.784433i	$-0.287046\pi$
$269$	20.3445			1.24043			0.620214	+	0.784433i	$0.287046\pi$
$270$		0			0
$271$			1.47847i			0.0898106i	0.998991	+	0.0449053i	$0.0142986\pi$
$271$			1.47847i			0.0898106i	−0.998991	+	0.0449053i	$0.985701\pi$
$272$	1.04544			0.0633888
$273$		0			0
$274$	6.86607			0.414794
$275$		0			0
$276$		0			0
$277$	−10.7642			−0.646756			−0.323378	−	0.946270i	$-0.604819\pi$
$277$	−10.7642			−0.646756			−0.323378	+	0.946270i	$0.604819\pi$
$278$	−13.9115			−0.834356
$279$		0			0
$280$		0			0
$281$		−	2.42806i		−	0.144846i	−0.997374	−	0.0724230i	$-0.976927\pi$
$281$		−	2.42806i		−	0.144846i	0.997374	−	0.0724230i	$-0.0230731\pi$
$282$		0			0
$283$			24.5898i			1.46171i	0.682532	+	0.730855i	$0.260879\pi$
$283$			24.5898i			1.46171i	−0.682532	+	0.730855i	$0.739121\pi$
$284$		−	6.43303i		−	0.381730i
$285$		0			0
$286$		−	11.5830i		−	0.684915i
$287$	−0.208518	+	5.79148i	−0.0123084	+	0.341860i
$288$		0			0
$289$	−15.9071			−0.935710
$290$		0			0
$291$		0			0
$292$			4.57615i			0.267799i
$293$	−12.6707			−0.740230			−0.370115	−	0.928986i	$-0.620682\pi$
$293$	−12.6707			−0.740230			−0.370115	+	0.928986i	$0.620682\pi$
$294$		0			0
$295$		0			0
$296$		−	0.855043i		−	0.0496983i
$297$		0			0
$298$	−15.9638			−0.924755
$299$	16.3808			0.947325
$300$		0			0
$301$	−0.0908711	+	2.52390i	−0.00523773	+	0.145475i
$302$			16.0546i			0.923840i
$303$		0			0
$304$			6.43303i			0.368960i
$305$		0			0
$306$		0			0
$307$		−	0.866067i		−	0.0494291i	−0.999695	−	0.0247145i	$-0.992132\pi$
$307$		−	0.866067i		−	0.0494291i	0.999695	−	0.0247145i	$-0.00786768\pi$
$308$	−13.9819	−	0.503406i	−0.796691	−	0.0286842i
$309$		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
$313$		−	5.80096i		−	0.327889i	−0.986470	−	0.163945i	$-0.947578\pi$
$313$		−	5.80096i		−	0.327889i	0.986470	−	0.163945i	$-0.0524219\pi$
$314$	−24.7665			−1.39766
$315$		0			0
$316$	−15.6738			−0.881722
$317$			28.1591i			1.58157i	0.612092	+	0.790787i	$0.290328\pi$
$317$			28.1591i			1.58157i	−0.612092	+	0.790787i	$0.709672\pi$
$318$		0			0
$319$	−39.5467			−2.21419
$320$		0			0
$321$		0			0
$322$	0.711924	−	19.7734i	0.0396740	−	1.10193i
$323$			6.72532i			0.374207i
$324$		0			0
$325$		0			0
$326$		−	7.42622i		−	0.411300i
$327$		0			0
$328$		−	2.19039i		−	0.120944i
$329$	−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	$0.680891\pi$
$332$	−4.38079			−0.240427
$333$		0			0
$334$		−	0.573779i		−	0.0313958i
$335$		0			0
$336$		0			0
$337$	28.3992			1.54700			0.773500	−	0.633796i	$-0.218504\pi$
$337$	28.3992			1.54700			0.773500	+	0.633796i	$0.218504\pi$
$338$			8.20218i			0.446140i
$339$		0			0
$340$		0			0
$341$	−48.1092			−2.60526
$342$		0			0
$343$	1.99567	−	18.4124i	0.107756	−	0.994177i
$344$		−	0.954564i		−	0.0514667i
$345$		0			0
$346$			9.96375i			0.535655i
$347$		−	12.5898i		−	0.675855i	−0.941172	−	0.337927i	$-0.890274\pi$
$347$		−	12.5898i		−	0.675855i	0.941172	−	0.337927i	$-0.109726\pi$
$348$		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
$352$	5.28808			0.281855
$353$	15.1363			0.805624			0.402812	−	0.915283i	$-0.368033\pi$
$353$	15.1363			0.805624			0.402812	+	0.915283i	$0.368033\pi$
$354$		0			0
$355$		0			0
$356$	−4.28126			−0.226907
$357$		0			0
$358$	1.77336			0.0937249
$359$		−	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
$362$	−11.0092			−0.578630
$363$		0			0
$364$	−5.79148	−	0.208518i	−0.303556	−	0.0109293i
$365$		0			0
$366$		0			0
$367$			11.6825i			0.609821i	0.952381	+	0.304910i	$0.0986265\pi$
$367$			11.6825i			0.609821i	−0.952381	+	0.304910i	$0.901373\pi$
$368$			7.47847i			0.389842i
$369$		0			0
$370$		0			0
$371$	8.19039	+	0.294888i	0.425224	+	0.0153098i
$372$		0			0
$373$	−12.8550			−0.665609			−0.332804	−	0.942996i	$-0.607995\pi$
$373$	−12.8550			−0.665609			−0.332804	+	0.942996i	$0.607995\pi$
$374$	5.52834			0.285864
$375$		0			0
$376$		−	11.0092i		−	0.567755i
$377$	−16.3808			−0.843653
$378$		0			0
$379$	25.0751			1.28802			0.644010	−	0.765017i	$-0.277270\pi$
$379$	25.0751			1.28802			0.644010	+	0.765017i	$0.277270\pi$
$380$		0			0
$381$		0			0
$382$	−5.56697			−0.284831
$383$	23.0092			1.17571			0.587857	−	0.808965i	$-0.299972\pi$
$383$	23.0092			1.17571			0.587857	+	0.808965i	$0.299972\pi$
$384$		0			0
$385$		0			0
$386$			8.47166i			0.431196i
$387$		0			0
$388$		−	11.8593i		−	0.602062i
$389$		−	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
$392$	−0.503406	+	6.98188i	−0.0254258	+	0.352638i
$393$		0			0
$394$	12.8661			0.648183
$395$		0			0
$396$		0			0
$397$			13.9141i			0.698329i	0.937061	+	0.349164i	$0.113535\pi$
$397$			13.9141i			0.698329i	−0.937061	+	0.349164i	$0.886465\pi$
$398$	17.5830			0.881354
$399$		0			0
$400$		0			0
$401$			14.6197i			0.730075i	0.930993	+	0.365038i	$0.118944\pi$
$401$			14.6197i			0.730075i	−0.930993	+	0.365038i	$0.881056\pi$
$402$		0			0
$403$	−19.9275			−0.992660
$404$	−11.5830			−0.576274
$405$		0			0
$406$	−0.711924	+	19.7734i	−0.0353322	+	0.981335i
$407$		−	4.52153i		−	0.224124i
$408$		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
$412$			16.6757i			0.821552i
$413$	−1.31117	+	36.4173i	−0.0645187	+	1.79198i
$414$		0			0
$415$		0			0
$416$	2.19039			0.107393
$417$		0			0
$418$			34.0184i			1.66389i
$419$	2.60743			0.127381			0.0636906	−	0.997970i	$-0.479713\pi$
$419$	2.60743			0.127381			0.0636906	+	0.997970i	$0.479713\pi$
$420$		0			0
$421$	0.852443			0.0415455			0.0207728	−	0.999784i	$-0.493387\pi$
$421$	0.852443			0.0415455			0.0207728	+	0.999784i	$0.493387\pi$
$422$			4.00000i			0.194717i
$423$		0			0
$424$	−3.09768			−0.150437
$425$		0			0
$426$		0			0
$427$	−21.2905	−	0.766545i	−1.03032	−	0.0370957i
$428$		−	7.00681i		−	0.338687i
$429$		0			0
$430$		0			0
$431$		−	20.2476i		−	0.975293i	−0.873041	−	0.487647i	$-0.837855\pi$
$431$		−	20.2476i		−	0.975293i	0.873041	−	0.487647i	$-0.162145\pi$
$432$		0			0
$433$		−	30.6444i		−	1.47268i	−0.676614	−	0.736338i	$-0.736553\pi$
$433$		−	30.6444i		−	1.47268i	0.676614	−	0.736338i	$-0.263447\pi$
$434$	−0.866067	+	24.0546i	−0.0415726	+	1.15466i
$435$		0			0
$436$	4.28991			0.205450
$437$	−48.1092			−2.30138
$438$		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
$442$	2.28991			0.108920
$443$		−	36.3083i		−	1.72506i	−0.506007	−	0.862529i	$-0.668879\pi$
$443$		−	36.3083i		−	1.72506i	0.506007	−	0.862529i	$-0.331121\pi$
$444$		0			0
$445$		0			0
$446$	10.8710			0.514759
$447$		0			0
$448$	0.0951965	−	2.64404i	0.00449761	−	0.124919i
$449$		−	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
$452$			8.09087i			0.380563i
$453$		0			0
$454$			14.0909i			0.661317i
$455$		0			0
$456$		0			0
$457$	−25.3477			−1.18571			−0.592857	−	0.805308i	$-0.702000\pi$
$457$	−25.3477			−1.18571			−0.592857	+	0.805308i	$0.702000\pi$
$458$	−14.5239			−0.678657
$459$		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
$463$	21.2655			0.988289			0.494145	−	0.869380i	$-0.335481\pi$
$463$	21.2655			0.988289			0.494145	+	0.869380i	$0.335481\pi$
$464$		−	7.47847i		−	0.347179i
$465$		0			0
$466$	20.6806			0.958013
$467$	13.1424			0.608156			0.304078	−	0.952647i	$-0.401652\pi$
$467$	13.1424			0.608156			0.304078	+	0.952647i	$0.401652\pi$
$468$		0			0
$469$	0.507906	−	14.1069i	0.0234529	−	0.651395i
$470$		0			0
$471$		0			0
$472$		−	13.7734i		−	0.633970i
$473$		−	5.04781i		−	0.232099i
$474$		0			0
$475$		0			0
$476$	0.0995218	−	2.76417i	0.00456157	−	0.126696i
$477$		0			0
$478$	5.56697			0.254627
$479$	−26.2899			−1.20122			−0.600608	−	0.799543i	$-0.705075\pi$
$479$	−26.2899			−1.20122			−0.600608	+	0.799543i	$0.705075\pi$
$480$		0			0
$481$		−	1.87288i		−	0.0853959i
$482$	−11.3839			−0.518523
$483$		0			0
$484$	16.9638			0.771080
$485$		0			0
$486$		0			0
$487$	−39.5381			−1.79164			−0.895820	−	0.444416i	$-0.853411\pi$
$487$	−39.5381			−1.79164			−0.895820	+	0.444416i	$0.853411\pi$
$488$	8.05225			0.364508
$489$		0			0
$490$		0			0
$491$			4.62105i			0.208545i	0.994549	+	0.104273i	$0.0332514\pi$
$491$			4.62105i			0.208545i	−0.994549	+	0.104273i	$0.966749\pi$
$492$		0			0
$493$		−	7.81826i		−	0.352117i
$494$			14.0909i			0.633978i
$495$		0			0
$496$		−	9.09768i		−	0.408498i
$497$	−17.0092	−	0.612402i	−0.762966	−	0.0274700i
$498$		0			0
$499$	31.0152			1.38843			0.694216	−	0.719766i	$-0.255751\pi$
$499$	31.0152			1.38843			0.694216	+	0.719766i	$0.255751\pi$
$500$		0			0
$501$		0			0
$502$			1.71874i			0.0767109i
$503$	−28.3385			−1.26355			−0.631775	−	0.775152i	$-0.717673\pi$
$503$	−28.3385			−1.26355			−0.631775	+	0.775152i	$0.717673\pi$
$504$		0			0
$505$		0			0
$506$			39.5467i			1.75807i
$507$		0			0
$508$	8.57118			0.380285
$509$	−22.3808			−0.992011			−0.496005	−	0.868319i	$-0.665200\pi$
$509$	−22.3808			−0.992011			−0.496005	+	0.868319i	$0.665200\pi$
$510$		0			0
$511$	12.0995	+	0.435633i	0.535251	+	0.0192713i
$512$			1.00000i			0.0441942i
$513$		0			0
$514$		−	2.18779i		−	0.0964994i
$515$		0			0
$516$		0			0
$517$		−	58.2174i		−	2.56040i
$518$	−2.26077	−	0.0813970i	−0.0993323	−	0.00357638i
$519$		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
$523$			3.51472i			0.153688i	0.997043	+	0.0768440i	$0.0244843\pi$
$523$			3.51472i			0.153688i	−0.997043	+	0.0768440i	$0.975516\pi$
$524$	−5.90048			−0.257764
$525$		0			0
$526$	−17.3876			−0.758135
$527$		−	9.51104i		−	0.414307i
$528$		0			0
$529$	−32.9275			−1.43163
$530$		0			0
$531$		0			0
$532$	17.0092	+	0.612402i	0.737442	+	0.0265510i
$533$		−	4.79782i		−	0.207817i
$534$		0			0
$535$		0			0
$536$			5.33535i			0.230452i
$537$		0			0
$538$			20.3445i			0.877115i
$539$	−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	$0.272486\pi$
$542$	−1.47847			−0.0635057
$543$		0			0
$544$			1.04544i			0.0448227i
$545$		0			0
$546$		0			0
$547$	6.86369			0.293470			0.146735	−	0.989176i	$-0.453123\pi$
$547$	6.86369			0.293470			0.146735	+	0.989176i	$0.453123\pi$
$548$			6.86607i			0.293304i
$549$		0			0
$550$		0			0
$551$	48.1092			2.04952
$552$		0			0
$553$	−1.49209	+	41.4422i	−0.0634503	+	1.76230i
$554$		−	10.7642i		−	0.457326i
$555$		0			0
$556$		−	13.9115i		−	0.589979i
$557$		−	1.81826i		−	0.0770421i	−0.999258	−	0.0385210i	$-0.987735\pi$
$557$		−	1.81826i		−	0.0770421i	0.999258	−	0.0385210i	$-0.0122647\pi$
$558$		0			0
$559$		−	2.09087i		−	0.0884344i
$560$		0			0
$561$		0			0
$562$	2.42806			0.102422
$563$	34.0184			1.43370			0.716852	−	0.697226i	$-0.245582\pi$
$563$	34.0184			1.43370			0.716852	+	0.697226i	$0.245582\pi$
$564$		0			0
$565$		0			0
$566$	−24.5898			−1.03359
$567$		0			0
$568$	6.43303			0.269924
$569$			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	$0.427294\pi$
$572$	11.5830			0.484308
$573$		0			0
$574$	−5.79148	−	0.208518i	−0.241732	−	0.00870336i
$575$		0			0
$576$		0			0
$577$		−	39.1523i		−	1.62993i	−0.579509	−	0.814966i	$-0.696756\pi$
$577$		−	39.1523i		−	1.62993i	0.579509	−	0.814966i	$-0.303244\pi$
$578$		−	15.9071i		−	0.661647i
$579$		0			0
$580$		0			0
$581$	−0.417035	+	11.5830i	−0.0173015	+	0.480542i
$582$		0			0
$583$	−16.3808			−0.678423
$584$	−4.57615			−0.189363
$585$		0			0
$586$		−	12.6707i		−	0.523422i
$587$	5.52834			0.228179			0.114090	−	0.993470i	$-0.463605\pi$
$587$	5.52834			0.228179			0.114090	+	0.993470i	$0.463605\pi$
$588$		0			0
$589$	58.5257			2.41151
$590$		0			0
$591$		0			0
$592$	0.855043			0.0351420
$593$	−18.6830			−0.767220			−0.383610	−	0.923495i	$-0.625319\pi$
$593$	−18.6830			−0.767220			−0.383610	+	0.923495i	$0.625319\pi$
$594$		0			0
$595$		0			0
$596$		−	15.9638i		−	0.653901i
$597$		0			0
$598$			16.3808i			0.669860i
$599$		−	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	$0.171189\pi$
$601$			25.1161i			1.02451i	−0.858835	+	0.512253i	$0.828811\pi$
$602$	−2.52390	−	0.0908711i	−0.102867	−	0.00370363i
$603$		0			0
$604$	−16.0546			−0.653253
$605$		0			0
$606$		0			0
$607$		−	31.4336i		−	1.27585i	−0.770099	−	0.637925i	$-0.779793\pi$
$607$		−	31.4336i		−	1.27585i	0.770099	−	0.637925i	$-0.220207\pi$
$608$	−6.43303			−0.260894
$609$		0			0
$610$		0			0
$611$		−	24.1144i		−	0.975566i
$612$		0			0
$613$	−39.4532			−1.59350			−0.796751	−	0.604308i	$-0.793450\pi$
$613$	−39.4532			−1.59350			−0.796751	+	0.604308i	$0.793450\pi$
$614$	0.866067			0.0349516
$615$		0			0
$616$	0.503406	−	13.9819i	0.0202828	−	0.563346i
$617$		−	39.7421i		−	1.59996i	−0.600029	−	0.799978i	$-0.704844\pi$
$617$		−	39.7421i		−	1.59996i	0.600029	−	0.799978i	$-0.295156\pi$
$618$		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
$622$		−	9.71009i		−	0.389339i
$623$	−0.407561	+	11.3198i	−0.0163286	+	0.453519i
$624$		0			0
$625$		0			0
$626$	5.80096			0.231853
$627$		0			0
$628$		−	24.7665i		−	0.988293i
$629$	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	$0.605233\pi$
$632$		−	15.6738i		−	0.623472i
$633$		0			0
$634$	−28.1591			−1.11834
$635$		0			0
$636$		0			0
$637$	−1.10266	+	15.2931i	−0.0436889	+	0.605933i
$638$		−	39.5467i		−	1.56567i
$639$		0			0
$640$		0			0
$641$			12.5289i			0.494861i	0.968906	+	0.247430i	$0.0795862\pi$
$641$			12.5289i			0.494861i	−0.968906	+	0.247430i	$0.920414\pi$
$642$		0			0
$643$			14.3992i			0.567847i	0.958847	+	0.283924i	$0.0916362\pi$
$643$			14.3992i			0.567847i	−0.958847	+	0.283924i	$0.908364\pi$
$644$	19.7734	+	0.711924i	0.779179	+	0.0280537i
$645$		0			0
$646$	−6.72532			−0.264604
$647$	31.7708			1.24904			0.624519	−	0.781010i	$-0.285295\pi$
$647$	31.7708			1.24904			0.624519	+	0.781010i	$0.285295\pi$
$648$		0			0
$649$		−	72.8346i		−	2.85901i
$650$		0			0
$651$		0			0
$652$	7.42622			0.290833
$653$		−	20.7389i		−	0.811578i	−0.913967	−	0.405789i	$-0.866997\pi$
$653$		−	20.7389i		−	0.811578i	0.913967	−	0.405789i	$-0.133003\pi$
$654$		0			0
$655$		0			0
$656$	2.19039			0.0855205
$657$		0			0
$658$	−29.1087	−	1.04804i	−1.13478	−	0.0408567i
$659$			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	$-0.775067\pi$
$661$		−	33.3863i		−	1.29858i	0.760542	−	0.649288i	$-0.224933\pi$
$662$		−	19.5830i		−	0.761114i
$663$		0			0
$664$		−	4.38079i		−	0.170007i
$665$		0			0
$666$		0			0
$667$	55.9275			2.16552
$668$	0.573779			0.0222002
$669$		0			0
$670$		0			0
$671$	42.5809			1.64382
$672$		0			0
$673$	18.6943			0.720611			0.360306	−	0.932834i	$-0.382672\pi$
$673$	18.6943			0.720611			0.360306	+	0.932834i	$0.382672\pi$
$674$			28.3992i			1.09389i
$675$		0			0
$676$	−8.20218			−0.315468
$677$	−36.5861			−1.40612			−0.703059	−	0.711131i	$-0.748183\pi$
$677$	−36.5861			−1.40612			−0.703059	+	0.711131i	$0.748183\pi$
$678$		0			0
$679$	−31.3563	−	1.12896i	−1.20335	−	0.0433255i
$680$		0			0
$681$		0			0
$682$		−	48.1092i		−	1.84220i
$683$			36.9207i			1.41273i	0.707847	+	0.706365i	$0.249666\pi$
$683$			36.9207i			1.41273i	−0.707847	+	0.706365i	$0.750334\pi$
$684$		0			0
$685$		0			0
$686$	18.4124	+	1.99567i	0.702990	+	0.0761952i
$687$		0			0
$688$	0.954564			0.0363924
$689$	−6.78514			−0.258493
$690$		0			0
$691$			5.20823i			0.198130i	0.995081	+	0.0990652i	$0.0315852\pi$
$691$			5.20823i			0.198130i	−0.995081	+	0.0990652i	$0.968415\pi$
$692$	−9.96375			−0.378765
$693$		0			0
$694$	12.5898			0.477901
$695$		0			0
$696$		0			0
$697$	2.28991			0.0867367
$698$	20.9956			0.794694
$699$		0			0
$700$		0			0
$701$			13.2831i			0.501696i	0.968027	+	0.250848i	$0.0807094\pi$
$701$			13.2831i			0.501696i	−0.968027	+	0.250848i	$0.919291\pi$
$702$		0			0
$703$			5.50052i			0.207456i
$704$			5.28808i			0.199302i
$705$		0			0
$706$			15.1363i			0.569662i
$707$	−1.10266	+	30.6258i	−0.0414697	+	1.15180i
$708$		0			0
$709$	−4.09087			−0.153636			−0.0768179	−	0.997045i	$-0.524476\pi$
$709$	−4.09087			−0.153636			−0.0768179	+	0.997045i	$0.524476\pi$
$710$		0			0
$711$		0			0
$712$		−	4.28126i		−	0.160447i
$713$	68.0367			2.54800
$714$		0			0
$715$		0			0
$716$			1.77336i			0.0662735i
$717$		0			0
$718$	33.1909			1.23867
$719$	−16.1817			−0.603477			−0.301739	−	0.953391i	$-0.597567\pi$
$719$	−16.1817			−0.603477			−0.301739	+	0.953391i	$0.597567\pi$
$720$		0			0
$721$	44.0911	+	1.58747i	1.64204	+	0.0591203i
$722$		−	22.3839i		−	0.833043i
$723$		0			0
$724$		−	11.0092i		−	0.409153i
$725$		0			0
$726$		0			0
$727$		−	40.7849i		−	1.51263i	−0.654208	−	0.756314i	$-0.726998\pi$
$727$		−	40.7849i		−	1.51263i	0.654208	−	0.756314i	$-0.273002\pi$
$728$	0.208518	−	5.79148i	0.00772818	−	0.214647i
$729$		0			0
$730$		0			0
$731$	0.997936			0.0369100
$732$		0			0
$733$			39.2481i			1.44966i	0.688926	+	0.724832i	$0.258083\pi$
$733$			39.2481i			1.44966i	−0.688926	+	0.724832i	$0.741917\pi$
$734$	−11.6825			−0.431208
$735$		0			0
$736$	−7.47847			−0.275660
$737$			28.2137i			1.03927i
$738$		0			0
$739$	−16.7305			−0.615442			−0.307721	−	0.951477i	$-0.599566\pi$
$739$	−16.7305			−0.615442			−0.307721	+	0.951477i	$0.599566\pi$
$740$		0			0
$741$		0			0
$742$	−0.294888	+	8.19039i	−0.0108257	+	0.300679i
$743$		−	35.7185i		−	1.31039i	−0.755462	−	0.655193i	$-0.772588\pi$
$743$		−	35.7185i		−	1.31039i	0.755462	−	0.655193i	$-0.227412\pi$
$744$		0			0
$745$		0			0
$746$		−	12.8550i		−	0.470657i
$747$		0			0
$748$			5.52834i			0.202136i
$749$	−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	$0.553345\pi$
$752$	11.0092			0.401464
$753$		0			0
$754$		−	16.3808i		−	0.596553i
$755$		0			0
$756$		0			0
$757$	−48.2027			−1.75196			−0.875979	−	0.482350i	$-0.839783\pi$
$757$	−48.2027			−1.75196			−0.875979	+	0.482350i	$0.839783\pi$
$758$			25.0751i			0.910767i
$759$		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
$763$	0.408385	−	11.3427i	0.0147845	−	0.410633i
$764$		−	5.56697i		−	0.201406i
$765$		0			0
$766$			23.0092i			0.831356i
$767$		−	30.1691i		−	1.08934i
$768$		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
$772$	−8.47166			−0.304902
$773$	−12.2726			−0.441415			−0.220708	−	0.975340i	$-0.570837\pi$
$773$	−12.2726			−0.441415			−0.220708	+	0.975340i	$0.570837\pi$
$774$		0			0
$775$		0			0
$776$	11.8593			0.425722
$777$		0			0
$778$	10.1591			0.364222
$779$			14.0909i			0.504858i
$780$		0			0
$781$	34.0184			1.21727
$782$	−7.81826			−0.279580
$783$		0			0
$784$	−6.98188	−	0.503406i	−0.249353	−	0.0179788i
$785$		0			0
$786$		0			0
$787$		−	21.4286i		−	0.763847i	−0.924194	−	0.381923i	$-0.875262\pi$
$787$		−	21.4286i		−	0.763847i	0.924194	−	0.381923i	$-0.124738\pi$
$788$			12.8661i			0.458335i
$789$		0			0
$790$		0			0
$791$	21.3926	+	0.770222i	0.760632	+	0.0273859i
$792$		0			0
$793$	17.6376			0.626329
$794$	−13.9141			−0.493793
$795$		0			0
$796$			17.5830i			0.623212i
$797$	−9.96375			−0.352934			−0.176467	−	0.984307i	$-0.556467\pi$
$797$	−9.96375			−0.352934			−0.176467	+	0.984307i	$0.556467\pi$
$798$		0			0
$799$	11.5094			0.407173
$800$		0			0
$801$		0			0
$802$	−14.6197			−0.516241
$803$	−24.1990			−0.853966
$804$		0			0
$805$		0			0
$806$		−	19.9275i		−	0.701916i
$807$		0			0
$808$		−	11.5830i		−	0.407487i
$809$			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	$-0.530699\pi$
$811$		−	56.6914i		−	1.99071i	0.0962936	−	0.995353i	$-0.469301\pi$
$812$	−19.7734	−	0.711924i	−0.693909	−	0.0249836i
$813$		0			0
$814$	4.52153			0.158480
$815$		0			0
$816$		0			0
$817$			6.14075i			0.214837i
$818$	12.0000			0.419570
$819$		0			0
$820$		0			0
$821$		−	45.3198i		−	1.58167i	−0.612027	−	0.790837i	$-0.709646\pi$
$821$		−	45.3198i		−	1.58167i	0.612027	−	0.790837i	$-0.290354\pi$
$822$		0			0
$823$	−29.6462			−1.03340			−0.516701	−	0.856166i	$-0.672840\pi$
$823$	−29.6462			−1.03340			−0.516701	+	0.856166i	$0.672840\pi$
$824$	−16.6757			−0.580925
$825$		0			0
$826$	−36.4173	−	1.31117i	−1.26712	−	0.0456216i
$827$			42.1129i			1.46441i	0.681084	+	0.732205i	$0.261509\pi$
$827$			42.1129i			1.46441i	−0.681084	+	0.732205i	$0.738491\pi$
$828$		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
$832$			2.19039i			0.0759382i
$833$	−7.29910	−	0.526279i	−0.252899	−	0.0182345i
$834$		0			0
$835$		0			0
$836$	−34.0184			−1.17655
$837$		0			0
$838$			2.60743i			0.0900721i
$839$	−41.4385			−1.43062			−0.715309	−	0.698809i	$-0.753714\pi$
$839$	−41.4385			−1.43062			−0.715309	+	0.698809i	$0.753714\pi$
$840$		0			0
$841$	−26.9275			−0.928535
$842$			0.852443i			0.0293771i
$843$		0			0
$844$	−4.00000			−0.137686
$845$		0			0
$846$		0			0
$847$	1.61489	−	44.8528i	0.0554882	−	1.54116i
$848$		−	3.09768i		−	0.106375i
$849$		0			0
$850$		0			0
$851$			6.39441i			0.219198i
$852$		0			0
$853$			30.4866i			1.04384i	0.852994	+	0.521920i	$0.174784\pi$
$853$			30.4866i			1.04384i	−0.852994	+	0.521920i	$0.825216\pi$
$854$	0.766545	−	21.2905i	0.0262306	−	0.728544i
$855$		0			0
$856$	7.00681			0.239488
$857$	−5.22718			−0.178557			−0.0892785	−	0.996007i	$-0.528456\pi$
$857$	−5.22718			−0.178557			−0.0892785	+	0.996007i	$0.528456\pi$
$858$		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
$862$	20.2476			0.689636
$863$		−	30.9259i		−	1.05273i	−0.850259	−	0.526365i	$-0.823555\pi$
$863$		−	30.9259i		−	1.05273i	0.850259	−	0.526365i	$-0.176445\pi$
$864$		0			0
$865$		0			0
$866$	30.6444			1.04134
$867$		0			0
$868$	−24.0546	−	0.866067i	−0.816467	−	0.0293962i
$869$		−	82.8844i		−	2.81166i
$870$		0			0
$871$			11.6865i			0.395982i
$872$			4.28991i			0.145275i
$873$		0			0
$874$		−	48.1092i		−	1.62732i
$875$		0			0
$876$		0			0
$877$	−1.79836			−0.0607262			−0.0303631	−	0.999539i	$-0.509666\pi$
$877$	−1.79836			−0.0607262			−0.0303631	+	0.999539i	$0.509666\pi$
$878$	22.6308			0.763752
$879$		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
$883$	52.5013			1.76681			0.883404	−	0.468611i	$-0.155246\pi$
$883$	52.5013			1.76681			0.883404	+	0.468611i	$0.155246\pi$
$884$			2.28991i			0.0770182i
$885$		0			0
$886$	36.3083			1.21980
$887$	−39.1720			−1.31527			−0.657633	−	0.753338i	$-0.728442\pi$
$887$	−39.1720			−1.31527			−0.657633	+	0.753338i	$0.728442\pi$
$888$		0			0
$889$	0.815946	−	22.6625i	0.0273659	−	0.760077i
$890$		0			0
$891$		0			0
$892$			10.8710i			0.363989i
$893$			70.8225i			2.36998i
$894$		0			0
$895$		0			0
$896$	2.64404	+	0.0951965i	0.0883311	+	0.00318029i
$897$		0			0
$898$	25.6764			0.856834
$899$	−68.0367			−2.26915
$900$		0			0
$901$		−	3.23843i		−	0.107888i
$902$	11.5830			0.385671
$903$		0			0
$904$	−8.09087			−0.269098
$905$		0			0
$906$		0			0
$907$	47.0822			1.56334			0.781669	−	0.623693i	$-0.214369\pi$
$907$	47.0822			1.56334			0.781669	+	0.623693i	$0.214369\pi$
$908$	−14.0909			−0.467622
$909$		0			0
$910$		0			0
$911$			40.6504i			1.34681i	0.739274	+	0.673405i	$0.235169\pi$
$911$			40.6504i			1.34681i	−0.739274	+	0.673405i	$0.764831\pi$
$912$		0			0
$913$		−	23.1659i		−	0.766680i
$914$		−	25.3477i		−	0.838426i
$915$		0			0
$916$		−	14.5239i		−	0.479883i
$917$	−0.561705	+	15.6011i	−0.0185491	+	0.515193i
$918$		0			0
$919$	44.4900			1.46759			0.733795	−	0.679371i	$-0.237747\pi$
$919$	44.4900			1.46759			0.733795	+	0.679371i	$0.237747\pi$
$920$		0			0
$921$		0			0
$922$		−	12.7253i		−	0.419086i
$923$	14.0909			0.463807
$924$		0			0
$925$		0			0
$926$			21.2655i			0.698826i
$927$		0			0
$928$	7.47847			0.245493
$929$	53.7371			1.76306			0.881529	−	0.472131i	$-0.156515\pi$
$929$	53.7371			1.76306			0.881529	+	0.472131i	$0.156515\pi$
$930$		0			0
$931$	3.23843	−	44.9146i	0.106135	−	1.47202i
$932$			20.6806i			0.677417i
$933$		0			0
$934$			13.1424i			0.430031i
$935$		0			0
$936$		0			0
$937$			44.0682i			1.43965i	0.694157	+	0.719823i	$0.255777\pi$
$937$			44.0682i			1.43965i	−0.694157	+	0.719823i	$0.744223\pi$
$938$	14.1069	+	0.507906i	0.460606	+	0.0165837i
$939$		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
$943$			16.3808i			0.533432i
$944$	13.7734			0.448285
$945$		0			0
$946$	5.04781			0.164118
$947$			30.9244i			1.00491i	0.864604	+	0.502453i	$0.167569\pi$
$947$			30.9244i			1.00491i	−0.864604	+	0.502453i	$0.832431\pi$
$948$		0			0
$949$	−10.0236			−0.325379
$950$		0			0
$951$		0			0
$952$	2.76417	+	0.0995218i	0.0895873	+	0.00322552i
$953$			31.8414i			1.03144i	0.856756	+	0.515722i	$0.172476\pi$
$953$			31.8414i			1.03144i	−0.856756	+	0.515722i	$0.827524\pi$
$954$		0			0
$955$		0			0
$956$			5.56697i			0.180049i
$957$		0			0
$958$		−	26.2899i		−	0.849389i
$959$	18.1541	+	0.653625i	0.586228	+	0.0211067i
$960$		0			0
$961$	−51.7678			−1.66993
$962$	1.87288			0.0603841
$963$		0			0
$964$		−	11.3839i		−	0.366651i
$965$		0			0
$966$		0			0
$967$	−32.6804			−1.05093			−0.525466	−	0.850815i	$-0.676109\pi$
$967$	−32.6804			−1.05093			−0.525466	+	0.850815i	$0.676109\pi$
$968$			16.9638i			0.545236i
$969$		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
$973$	−36.7825	−	1.32433i	−1.17919	−	0.0424559i
$974$		−	39.5381i		−	1.26688i
$975$		0			0
$976$			8.05225i			0.257746i
$977$			2.48528i			0.0795112i	0.999209	+	0.0397556i	$0.0126579\pi$
$977$			2.48528i			0.0795112i	−0.999209	+	0.0397556i	$0.987342\pi$
$978$		0			0
$979$		−	22.6397i		−	0.723566i
$980$		0			0
$981$		0			0
$982$	−4.62105			−0.147464
$983$	49.4135			1.57605			0.788024	−	0.615645i	$-0.211104\pi$
$983$	49.4135			1.57605			0.788024	+	0.615645i	$0.211104\pi$
$984$		0			0
$985$		0			0
$986$	7.81826			0.248984
$987$		0			0
$988$	−14.0909			−0.448290
$989$			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	$0.861005\pi$
$992$	9.09768			0.288852
$993$		0			0
$994$	0.612402	−	17.0092i	0.0194242	−	0.539499i
$995$		0			0
$996$		0			0
$997$		−	16.6794i		−	0.528240i	−0.964490	−	0.264120i	$-0.914918\pi$
$997$		−	16.6794i		−	0.528240i	0.964490	−	0.264120i	$-0.0850816\pi$
$998$			31.0152i			0.981770i
$999$		0			0

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
630.2.b.a.251.3	✓	8	35.34	odd	2
630.2.b.a.251.7	yes	8	15.14	odd	2
630.2.b.b.251.3	yes	8	5.4	even	2
630.2.b.b.251.7	yes	8	105.104	even	2
3150.2.b.e.251.2		8	3.2	odd	2
3150.2.b.e.251.6		8	7.6	odd	2
3150.2.b.f.251.2		8	21.20	even	2	inner
3150.2.b.f.251.6		8	1.1	even	1	trivial
3150.2.d.a.3149.1		8	5.2	odd	4
3150.2.d.a.3149.2		8	105.83	odd	4
3150.2.d.c.3149.7		8	35.27	even	4
3150.2.d.c.3149.8		8	15.8	even	4
3150.2.d.d.3149.1		8	15.2	even	4
3150.2.d.d.3149.2		8	35.13	even	4
3150.2.d.f.3149.7		8	105.62	odd	4
3150.2.d.f.3149.8		8	5.3	odd	4
5040.2.f.f.881.3		8	140.139	even	2
5040.2.f.f.881.4		8	60.59	even	2
5040.2.f.i.881.3		8	420.419	odd	2
5040.2.f.i.881.4		8	20.19	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 3150 = 2 \cdot 3^{2} \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3150.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{2} -1.00000 q^{4} +(-0.0951965 + 2.64404i) q^{7} -1.00000i q^{8} +O(q^{10})\)	\(q+1.00000i q^{2} -1.00000 q^{4} +(-0.0951965 + 2.64404i) q^{7} -1.00000i q^{8} -5.28808i q^{11} -2.19039i q^{13} +(-2.64404 - 0.0951965i) q^{14} +1.00000 q^{16} +1.04544 q^{17} +6.43303i q^{19} +5.28808 q^{22} +7.47847i q^{23} +2.19039 q^{26} +(0.0951965 - 2.64404i) q^{28} -7.47847i q^{29} -9.09768i q^{31} +1.00000i q^{32} +1.04544i q^{34} +0.855043 q^{37} -6.43303 q^{38} +2.19039 q^{41} +0.954564 q^{43} +5.28808i q^{44} -7.47847 q^{46} +11.0092 q^{47} +(-6.98188 - 0.503406i) q^{49} +2.19039i q^{52} -3.09768i q^{53} +(2.64404 + 0.0951965i) q^{56} +7.47847 q^{58} +13.7734 q^{59} +8.05225i q^{61} +9.09768 q^{62} -1.00000 q^{64} -5.33535 q^{67} -1.04544 q^{68} +6.43303i q^{71} -4.57615i q^{73} +0.855043i q^{74} -6.43303i q^{76} +(13.9819 + 0.503406i) q^{77} +15.6738 q^{79} +2.19039i q^{82} +4.38079 q^{83} +0.954564i q^{86} -5.28808 q^{88} +4.28126 q^{89} +(5.79148 + 0.208518i) q^{91} -7.47847i q^{92} +11.0092i q^{94} +11.8593i q^{97} +(0.503406 - 6.98188i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 8 q^{4} + 4 q^{7}+O(q^{10}) \) 8 * q - 8 * q^4 + 4 * q^7	\( 8 q - 8 q^{4} + 4 q^{7} + 8 q^{16} + 8 q^{26} - 4 q^{28} + 8 q^{37} - 8 q^{38} + 8 q^{41} + 16 q^{43} - 8 q^{46} - 40 q^{47} + 4 q^{49} + 8 q^{58} + 40 q^{62} - 8 q^{64} - 32 q^{67} + 52 q^{77} + 8 q^{79} + 16 q^{83} + 8 q^{89} - 4 q^{91} - 4 q^{98}+O(q^{100}) \) 8 * q - 8 * q^4 + 4 * q^7 + 8 * q^16 + 8 * q^26 - 4 * q^28 + 8 * q^37 - 8 * q^38 + 8 * q^41 + 16 * q^43 - 8 * q^46 - 40 * q^47 + 4 * q^49 + 8 * q^58 + 40 * q^62 - 8 * q^64 - 32 * q^67 + 52 * q^77 + 8 * q^79 + 16 * q^83 + 8 * q^89 - 4 * q^91 - 4 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more