LMFDB - Embedded newform 525.2.t.d.26.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [525,2,Mod(26,525)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(525, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([3, 0, 5]))

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

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

chi := DirichletCharacter("525.26");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 525 = 3 \cdot 5^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	525.t (of order $6$, degree $2$, 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:	$4.19214610612$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{6})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x + 1 $ x^2 - x + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{6}]$

Embedding invariants

Embedding label			26.1
Root			$0.500000 - 0.866025i$ of defining polynomial
Character	$\chi$	$=$	525.26
Dual form			525.2.t.d.101.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.50000 - 0.866025i) q^{3} +(-1.00000 - 1.73205i) q^{4} +(2.00000 + 1.73205i) q^{7} +(1.50000 - 2.59808i) q^{9} +O(q^{10})$	$q+(1.50000 - 0.866025i) q^{3} +(-1.00000 - 1.73205i) q^{4} +(2.00000 + 1.73205i) q^{7} +(1.50000 - 2.59808i) q^{9} +(-3.00000 - 1.73205i) q^{12} -6.92820i q^{13} +(-2.00000 + 3.46410i) q^{16} +(3.00000 + 1.73205i) q^{19} +(4.50000 + 0.866025i) q^{21} -5.19615i q^{27} +(1.00000 - 5.19615i) q^{28} +(-7.50000 + 4.33013i) q^{31} -6.00000 q^{36} +(5.50000 - 9.52628i) q^{37} +(-6.00000 - 10.3923i) q^{39} -5.00000 q^{43} +6.92820i q^{48} +(1.00000 + 6.92820i) q^{49} +(-12.0000 + 6.92820i) q^{52} +6.00000 q^{57} +(13.5000 + 7.79423i) q^{61} +(7.50000 - 2.59808i) q^{63} +8.00000 q^{64} +(8.00000 + 13.8564i) q^{67} +(-1.50000 + 0.866025i) q^{73} -6.92820i q^{76} +(-8.50000 + 14.7224i) q^{79} +(-4.50000 - 7.79423i) q^{81} +(-3.00000 - 8.66025i) q^{84} +(12.0000 - 13.8564i) q^{91} +(-7.50000 + 12.9904i) q^{93} +5.19615i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 3 q^{3} - 2 q^{4} + 4 q^{7} + 3 q^{9}+O(q^{10}) $ 2 * q + 3 * q^3 - 2 * q^4 + 4 * q^7 + 3 * q^9	$ 2 q + 3 q^{3} - 2 q^{4} + 4 q^{7} + 3 q^{9} - 6 q^{12} - 4 q^{16} + 6 q^{19} + 9 q^{21} + 2 q^{28} - 15 q^{31} - 12 q^{36} + 11 q^{37} - 12 q^{39} - 10 q^{43} + 2 q^{49} - 24 q^{52} + 12 q^{57} + 27 q^{61} + 15 q^{63} + 16 q^{64} + 16 q^{67} - 3 q^{73} - 17 q^{79} - 9 q^{81} - 6 q^{84} + 24 q^{91} - 15 q^{93}+O(q^{100}) $ 2 * q + 3 * q^3 - 2 * q^4 + 4 * q^7 + 3 * q^9 - 6 * q^12 - 4 * q^16 + 6 * q^19 + 9 * q^21 + 2 * q^28 - 15 * q^31 - 12 * q^36 + 11 * q^37 - 12 * q^39 - 10 * q^43 + 2 * q^49 - 24 * q^52 + 12 * q^57 + 27 * q^61 + 15 * q^63 + 16 * q^64 + 16 * q^67 - 3 * q^73 - 17 * q^79 - 9 * q^81 - 6 * q^84 + 24 * q^91 - 15 * q^93

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/525\mathbb{Z}\right)^\times$.

$n$	$127$	$176$	$451$
$\chi(n)$	$1$	$-1$	$e\left(\frac{5}{6}\right)$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$2$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$3$	1.50000	−	0.866025i	0.866025	−	0.500000i
$3$	1.50000	−	0.866025i	0.866025	−	0.500000i
$4$	−1.00000	−	1.73205i	−0.500000	−	0.866025i
$5$		0			0
$5$		0			0
$6$		0			0
$7$	2.00000	+	1.73205i	0.755929	+	0.654654i
$7$	2.00000	+	1.73205i	0.755929	+	0.654654i
$8$		0			0
$9$	1.50000	−	2.59808i	0.500000	−	0.866025i
$10$		0			0
$11$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$11$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$12$	−3.00000	−	1.73205i	−0.866025	−	0.500000i
$13$		−	6.92820i		−	1.92154i	−0.277350	−	0.960769i	$-0.589456\pi$
$13$		−	6.92820i		−	1.92154i	0.277350	−	0.960769i	$-0.410544\pi$
$14$		0			0
$15$		0			0
$16$	−2.00000	+	3.46410i	−0.500000	+	0.866025i
$17$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$17$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$18$		0			0
$19$	3.00000	+	1.73205i	0.688247	+	0.397360i	0.802955	−	0.596040i	$-0.203260\pi$
$19$	3.00000	+	1.73205i	0.688247	+	0.397360i	−0.114708	+	0.993399i	$0.536593\pi$
$20$		0			0
$21$	4.50000	+	0.866025i	0.981981	+	0.188982i
$22$		0			0
$23$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$23$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$		−	5.19615i		−	1.00000i
$28$	1.00000	−	5.19615i	0.188982	−	0.981981i
$29$		0			0		1.00000			$0$
$29$		0			0		−1.00000			$\pi$
$30$		0			0
$31$	−7.50000	+	4.33013i	−1.34704	+	0.777714i	−0.987829	−	0.155543i	$-0.950287\pi$
$31$	−7.50000	+	4.33013i	−1.34704	+	0.777714i	−0.359211	+	0.933257i	$0.616954\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$	−6.00000			−1.00000
$37$	5.50000	−	9.52628i	0.904194	−	1.56611i	0.0821995	−	0.996616i	$-0.473806\pi$
$37$	5.50000	−	9.52628i	0.904194	−	1.56611i	0.821995	−	0.569495i	$-0.192861\pi$
$38$		0			0
$39$	−6.00000	−	10.3923i	−0.960769	−	1.66410i
$40$		0			0
$41$		0			0			−	1.00000i	$-0.5\pi$
$41$		0			0				1.00000i	$0.5\pi$
$42$		0			0
$43$	−5.00000			−0.762493			−0.381246	−	0.924473i	$-0.624505\pi$
$43$	−5.00000			−0.762493			−0.381246	+	0.924473i	$0.624505\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$47$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$48$			6.92820i			1.00000i
$49$	1.00000	+	6.92820i	0.142857	+	0.989743i
$50$		0			0
$51$		0			0
$52$	−12.0000	+	6.92820i	−1.66410	+	0.960769i
$53$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$53$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	6.00000			0.794719
$58$		0			0
$59$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$59$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$60$		0			0
$61$	13.5000	+	7.79423i	1.72850	+	0.997949i	0.896258	+	0.443533i	$0.146275\pi$
$61$	13.5000	+	7.79423i	1.72850	+	0.997949i	0.832240	+	0.554416i	$0.187058\pi$
$62$		0			0
$63$	7.50000	−	2.59808i	0.944911	−	0.327327i
$64$	8.00000			1.00000
$65$		0			0
$66$		0			0
$67$	8.00000	+	13.8564i	0.977356	+	1.69283i	0.671932	+	0.740613i	$0.265465\pi$
$67$	8.00000	+	13.8564i	0.977356	+	1.69283i	0.305424	+	0.952217i	$0.401202\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		0			0		1.00000			$0$
$71$		0			0		−1.00000			$\pi$
$72$		0			0
$73$	−1.50000	+	0.866025i	−0.175562	+	0.101361i	−0.585206	−	0.810885i	$-0.698986\pi$
$73$	−1.50000	+	0.866025i	−0.175562	+	0.101361i	0.409644	+	0.912245i	$0.365653\pi$
$74$		0			0
$75$		0			0
$76$		−	6.92820i		−	0.794719i
$77$		0			0
$78$		0			0
$79$	−8.50000	+	14.7224i	−0.956325	+	1.65640i	−0.225018	+	0.974355i	$0.572244\pi$
$79$	−8.50000	+	14.7224i	−0.956325	+	1.65640i	−0.731307	+	0.682048i	$0.761089\pi$
$80$		0			0
$81$	−4.50000	−	7.79423i	−0.500000	−	0.866025i
$82$		0			0
$83$		0			0			−	1.00000i	$-0.5\pi$
$83$		0			0				1.00000i	$0.5\pi$
$84$	−3.00000	−	8.66025i	−0.327327	−	0.944911i
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$89$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$90$		0			0
$91$	12.0000	−	13.8564i	1.25794	−	1.45255i
$92$		0			0
$93$	−7.50000	+	12.9904i	−0.777714	+	1.34704i
$94$		0			0
$95$		0			0
$96$		0			0
$97$			5.19615i			0.527589i	0.964579	+	0.263795i	$0.0849741\pi$
$97$			5.19615i			0.527589i	−0.964579	+	0.263795i	$0.915026\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$101$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$102$		0			0
$103$	13.5000	+	7.79423i	1.33019	+	0.767988i	0.985329	−	0.170664i	$-0.0545913\pi$
$103$	13.5000	+	7.79423i	1.33019	+	0.767988i	0.344865	+	0.938652i	$0.387925\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$107$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$108$	−9.00000	+	5.19615i	−0.866025	+	0.500000i
$109$	1.00000	+	1.73205i	0.0957826	+	0.165900i	0.909935	−	0.414751i	$-0.136131\pi$
$109$	1.00000	+	1.73205i	0.0957826	+	0.165900i	−0.814152	+	0.580651i	$0.802798\pi$
$110$		0			0
$111$		−	19.0526i		−	1.80839i
$112$	−10.0000	+	3.46410i	−0.944911	+	0.327327i
$113$		0			0		1.00000			$0$
$113$		0			0		−1.00000			$\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	−18.0000	−	10.3923i	−1.66410	−	0.960769i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−5.50000	+	9.52628i	−0.500000	+	0.866025i
$122$		0			0
$123$		0			0
$124$	15.0000	+	8.66025i	1.34704	+	0.777714i
$125$		0			0
$126$		0			0
$127$	−1.00000			−0.0887357			−0.0443678	−	0.999015i	$-0.514127\pi$
$127$	−1.00000			−0.0887357			−0.0443678	+	0.999015i	$0.514127\pi$
$128$		0			0
$129$	−7.50000	+	4.33013i	−0.660338	+	0.381246i
$130$		0			0
$131$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$131$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$132$		0			0
$133$	3.00000	+	8.66025i	0.260133	+	0.750939i
$134$		0			0
$135$		0			0
$136$		0			0
$137$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$137$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$138$		0			0
$139$			5.19615i			0.440732i	0.975417	+	0.220366i	$0.0707252\pi$
$139$			5.19615i			0.440732i	−0.975417	+	0.220366i	$0.929275\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$	6.00000	+	10.3923i	0.500000	+	0.866025i
$145$		0			0
$146$		0			0
$147$	7.50000	+	9.52628i	0.618590	+	0.785714i
$148$	−22.0000			−1.80839
$149$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$149$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$150$		0			0
$151$	−9.50000	−	16.4545i	−0.773099	−	1.33905i	−0.935857	−	0.352381i	$-0.885372\pi$
$151$	−9.50000	−	16.4545i	−0.773099	−	1.33905i	0.162758	−	0.986666i	$-0.447961\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$	−12.0000	+	20.7846i	−0.960769	+	1.66410i
$157$	19.5000	−	11.2583i	1.55627	−	0.898513i	0.558661	−	0.829396i	$-0.311315\pi$
$157$	19.5000	−	11.2583i	1.55627	−	0.898513i	0.997609	−	0.0691164i	$-0.0220180\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	−8.50000	+	14.7224i	−0.665771	+	1.15315i	0.313304	+	0.949653i	$0.398564\pi$
$163$	−8.50000	+	14.7224i	−0.665771	+	1.15315i	−0.979076	+	0.203497i	$0.934769\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0			−	1.00000i	$-0.5\pi$
$167$		0			0				1.00000i	$0.5\pi$
$168$		0			0
$169$	−35.0000			−2.69231
$170$		0			0
$171$	9.00000	−	5.19615i	0.688247	−	0.397360i
$172$	5.00000	+	8.66025i	0.381246	+	0.660338i
$173$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$173$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$		0			0
$179$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$179$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$180$		0			0
$181$			25.9808i			1.93113i	0.260153	+	0.965567i	$0.416227\pi$
$181$			25.9808i			1.93113i	−0.260153	+	0.965567i	$0.583773\pi$
$182$		0			0
$183$	27.0000			1.99590
$184$		0			0
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$	9.00000	−	10.3923i	0.654654	−	0.755929i
$190$		0			0
$191$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$191$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$192$	12.0000	−	6.92820i	0.866025	−	0.500000i
$193$	−12.5000	−	21.6506i	−0.899770	−	1.55845i	−0.827788	−	0.561041i	$-0.810401\pi$
$193$	−12.5000	−	21.6506i	−0.899770	−	1.55845i	−0.0719816	−	0.997406i	$-0.522932\pi$
$194$		0			0
$195$		0			0
$196$	11.0000	−	8.66025i	0.785714	−	0.618590i
$197$		0			0		1.00000			$0$
$197$		0			0		−1.00000			$\pi$
$198$		0			0
$199$	19.5000	−	11.2583i	1.38232	−	0.798082i	0.389885	−	0.920864i	$-0.372515\pi$
$199$	19.5000	−	11.2583i	1.38232	−	0.798082i	0.992434	+	0.122782i	$0.0391815\pi$
$200$		0			0
$201$	24.0000	+	13.8564i	1.69283	+	0.977356i
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$	24.0000	+	13.8564i	1.66410	+	0.960769i
$209$		0			0
$210$		0			0
$211$	−29.0000			−1.99644			−0.998221	−	0.0596196i	$-0.981011\pi$
$211$	−29.0000			−1.99644			−0.998221	+	0.0596196i	$0.981011\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$	−22.5000	−	4.33013i	−1.52740	−	0.293948i
$218$		0			0
$219$	−1.50000	+	2.59808i	−0.101361	+	0.175562i
$220$		0			0
$221$		0			0
$222$		0			0
$223$		−	19.0526i		−	1.27585i	−0.770097	−	0.637927i	$-0.779792\pi$
$223$		−	19.0526i		−	1.27585i	0.770097	−	0.637927i	$-0.220208\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$227$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$228$	−6.00000	−	10.3923i	−0.397360	−	0.688247i
$229$	−18.0000	−	10.3923i	−1.18947	−	0.686743i	−0.231287	−	0.972886i	$-0.574293\pi$
$229$	−18.0000	−	10.3923i	−1.18947	−	0.686743i	−0.958187	+	0.286143i	$0.907627\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$233$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$			29.4449i			1.91265i
$238$		0			0
$239$		0			0		1.00000			$0$
$239$		0			0		−1.00000			$\pi$
$240$		0			0
$241$	−1.50000	+	0.866025i	−0.0966235	+	0.0557856i	−0.547533	−	0.836784i	$-0.684433\pi$
$241$	−1.50000	+	0.866025i	−0.0966235	+	0.0557856i	0.450910	+	0.892570i	$0.351100\pi$
$242$		0			0
$243$	−13.5000	−	7.79423i	−0.866025	−	0.500000i
$244$		−	31.1769i		−	1.99590i
$245$		0			0
$246$		0			0
$247$	12.0000	−	20.7846i	0.763542	−	1.32249i
$248$		0			0
$249$		0			0
$250$		0			0
$251$		0			0			−	1.00000i	$-0.5\pi$
$251$		0			0				1.00000i	$0.5\pi$
$252$	−12.0000	−	10.3923i	−0.755929	−	0.654654i
$253$		0			0
$254$		0			0
$255$		0			0
$256$	−8.00000	−	13.8564i	−0.500000	−	0.866025i
$257$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$257$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$258$		0			0
$259$	27.5000	−	9.52628i	1.70877	−	0.591934i
$260$		0			0
$261$		0			0
$262$		0			0
$263$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$263$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$	16.0000	−	27.7128i	0.977356	−	1.69283i
$269$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$269$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$270$		0			0
$271$	15.0000	+	8.66025i	0.911185	+	0.526073i	0.880812	−	0.473466i	$-0.156997\pi$
$271$	15.0000	+	8.66025i	0.911185	+	0.526073i	0.0303728	+	0.999539i	$0.490331\pi$
$272$		0			0
$273$	6.00000	−	31.1769i	0.363137	−	1.88691i
$274$		0			0
$275$		0			0
$276$		0			0
$277$	−13.0000	−	22.5167i	−0.781094	−	1.35290i	−0.931305	−	0.364241i	$-0.881328\pi$
$277$	−13.0000	−	22.5167i	−0.781094	−	1.35290i	0.150210	−	0.988654i	$-0.452005\pi$
$278$		0			0
$279$			25.9808i			1.55543i
$280$		0			0
$281$		0			0		1.00000			$0$
$281$		0			0		−1.00000			$\pi$
$282$		0			0
$283$	9.00000	−	5.19615i	0.534994	−	0.308879i	−0.208053	−	0.978117i	$-0.566713\pi$
$283$	9.00000	−	5.19615i	0.534994	−	0.308879i	0.743048	+	0.669238i	$0.233379\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	8.50000	−	14.7224i	0.500000	−	0.866025i
$290$		0			0
$291$	4.50000	+	7.79423i	0.263795	+	0.456906i
$292$	3.00000	+	1.73205i	0.175562	+	0.101361i
$293$		0			0			−	1.00000i	$-0.5\pi$
$293$		0			0				1.00000i	$0.5\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$		0			0
$301$	−10.0000	−	8.66025i	−0.576390	−	0.499169i
$302$		0			0
$303$		0			0
$304$	−12.0000	+	6.92820i	−0.688247	+	0.397360i
$305$		0			0
$306$		0			0
$307$			1.73205i			0.0988534i	0.998778	+	0.0494267i	$0.0157394\pi$
$307$			1.73205i			0.0988534i	−0.998778	+	0.0494267i	$0.984261\pi$
$308$		0			0
$309$	27.0000			1.53598
$310$		0			0
$311$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$311$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$312$		0			0
$313$	4.50000	+	2.59808i	0.254355	+	0.146852i	0.621757	−	0.783210i	$-0.286419\pi$
$313$	4.50000	+	2.59808i	0.254355	+	0.146852i	−0.367402	+	0.930062i	$0.619753\pi$
$314$		0			0
$315$		0			0
$316$	34.0000			1.91265
$317$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$317$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$		0			0
$322$		0			0
$323$		0			0
$324$	−9.00000	+	15.5885i	−0.500000	+	0.866025i
$325$		0			0
$326$		0			0
$327$	3.00000	+	1.73205i	0.165900	+	0.0957826i
$328$		0			0
$329$		0			0
$330$		0			0
$331$	−15.5000	+	26.8468i	−0.851957	+	1.47563i	0.0274825	+	0.999622i	$0.491251\pi$
$331$	−15.5000	+	26.8468i	−0.851957	+	1.47563i	−0.879440	+	0.476011i	$0.842082\pi$
$332$		0			0
$333$	−16.5000	−	28.5788i	−0.904194	−	1.56611i
$334$		0			0
$335$		0			0
$336$	−12.0000	+	13.8564i	−0.654654	+	0.755929i
$337$	−5.00000			−0.272367			−0.136184	−	0.990684i	$-0.543484\pi$
$337$	−5.00000			−0.272367			−0.136184	+	0.990684i	$0.543484\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$	−10.0000	+	15.5885i	−0.539949	+	0.841698i
$344$		0			0
$345$		0			0
$346$		0			0
$347$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$347$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$348$		0			0
$349$		−	34.6410i		−	1.85429i	−0.374701	−	0.927146i	$-0.622255\pi$
$349$		−	34.6410i		−	1.85429i	0.374701	−	0.927146i	$-0.377745\pi$
$350$		0			0
$351$	−36.0000			−1.92154
$352$		0			0
$353$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$353$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$359$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$360$		0			0
$361$	−3.50000	−	6.06218i	−0.184211	−	0.319062i
$362$		0			0
$363$			19.0526i			1.00000i
$364$	−36.0000	−	6.92820i	−1.88691	−	0.363137i
$365$		0			0
$366$		0			0
$367$	13.5000	−	7.79423i	0.704694	−	0.406855i	−0.104399	−	0.994535i	$-0.533292\pi$
$367$	13.5000	−	7.79423i	0.704694	−	0.406855i	0.809093	+	0.587680i	$0.199959\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$	30.0000			1.55543
$373$	−19.0000	+	32.9090i	−0.983783	+	1.70396i	−0.336557	+	0.941663i	$0.609263\pi$
$373$	−19.0000	+	32.9090i	−0.983783	+	1.70396i	−0.647225	+	0.762299i	$0.724071\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	−8.00000			−0.410932			−0.205466	−	0.978664i	$-0.565871\pi$
$379$	−8.00000			−0.410932			−0.205466	+	0.978664i	$0.565871\pi$
$380$		0			0
$381$	−1.50000	+	0.866025i	−0.0768473	+	0.0443678i
$382$		0			0
$383$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$383$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	−7.50000	+	12.9904i	−0.381246	+	0.660338i
$388$	9.00000	−	5.19615i	0.456906	−	0.263795i
$389$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$389$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$	−16.5000	−	9.52628i	−0.828111	−	0.478110i	0.0250943	−	0.999685i	$-0.492011\pi$
$397$	−16.5000	−	9.52628i	−0.828111	−	0.478110i	−0.853206	+	0.521575i	$0.825345\pi$
$398$		0			0
$399$	12.0000	+	10.3923i	0.600751	+	0.520266i
$400$		0			0
$401$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$401$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$402$		0			0
$403$	30.0000	+	51.9615i	1.49441	+	2.58839i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$	−12.0000	+	6.92820i	−0.593362	+	0.342578i	−0.766426	−	0.642333i	$-0.777967\pi$
$409$	−12.0000	+	6.92820i	−0.593362	+	0.342578i	0.173064	+	0.984911i	$0.444633\pi$
$410$		0			0
$411$		0			0
$412$		−	31.1769i		−	1.53598i
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$	4.50000	+	7.79423i	0.220366	+	0.381685i
$418$		0			0
$419$		0			0			−	1.00000i	$-0.5\pi$
$419$		0			0				1.00000i	$0.5\pi$
$420$		0			0
$421$	41.0000			1.99822			0.999109	−	0.0422075i	$-0.0134391\pi$
$421$	41.0000			1.99822			0.999109	+	0.0422075i	$0.0134391\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$	13.5000	+	38.9711i	0.653311	+	1.88595i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$431$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$432$	18.0000	+	10.3923i	0.866025	+	0.500000i
$433$		−	22.5167i		−	1.08208i	−0.840996	−	0.541041i	$-0.818030\pi$
$433$		−	22.5167i		−	1.08208i	0.840996	−	0.541041i	$-0.181970\pi$
$434$		0			0
$435$		0			0
$436$	2.00000	−	3.46410i	0.0957826	−	0.165900i
$437$		0			0
$438$		0			0
$439$	34.5000	+	19.9186i	1.64660	+	0.950662i	0.978412	+	0.206666i	$0.0662612\pi$
$439$	34.5000	+	19.9186i	1.64660	+	0.950662i	0.668184	+	0.743996i	$0.267072\pi$
$440$		0			0
$441$	19.5000	+	7.79423i	0.928571	+	0.371154i
$442$		0			0
$443$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$443$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$444$	−33.0000	+	19.0526i	−1.56611	+	0.904194i
$445$		0			0
$446$		0			0
$447$		0			0
$448$	16.0000	+	13.8564i	0.755929	+	0.654654i
$449$		0			0		1.00000			$0$
$449$		0			0		−1.00000			$\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$	−28.5000	−	16.4545i	−1.33905	−	0.773099i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−15.5000	+	26.8468i	−0.725059	+	1.25584i	0.233890	+	0.972263i	$0.424854\pi$
$457$	−15.5000	+	26.8468i	−0.725059	+	1.25584i	−0.958950	+	0.283577i	$0.908479\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		0			0			−	1.00000i	$-0.5\pi$
$461$		0			0				1.00000i	$0.5\pi$
$462$		0			0
$463$	−43.0000			−1.99838			−0.999190	−	0.0402476i	$-0.987185\pi$
$463$	−43.0000			−1.99838			−0.999190	+	0.0402476i	$0.987185\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$467$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$468$			41.5692i			1.92154i
$469$	−8.00000	+	41.5692i	−0.369406	+	1.91949i
$470$		0			0
$471$	19.5000	−	33.7750i	0.898513	−	1.55627i
$472$		0			0
$473$		0			0
$474$		0			0
$475$		0			0
$476$		0			0
$477$		0			0
$478$		0			0
$479$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$479$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$480$		0			0
$481$	−66.0000	−	38.1051i	−3.00934	−	1.73744i
$482$		0			0
$483$		0			0
$484$	22.0000			1.00000
$485$		0			0
$486$		0			0
$487$	−12.5000	−	21.6506i	−0.566429	−	0.981084i	−0.996915	−	0.0784867i	$-0.974991\pi$
$487$	−12.5000	−	21.6506i	−0.566429	−	0.981084i	0.430486	−	0.902597i	$-0.358342\pi$
$488$		0			0
$489$			29.4449i			1.33154i
$490$		0			0
$491$		0			0		1.00000			$0$
$491$		0			0		−1.00000			$\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$		−	34.6410i		−	1.55543i
$497$		0			0
$498$		0			0
$499$	16.0000	−	27.7128i	0.716258	−	1.24060i	−0.246214	−	0.969216i	$-0.579187\pi$
$499$	16.0000	−	27.7128i	0.716258	−	1.24060i	0.962472	−	0.271380i	$-0.0874801\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		0			0			−	1.00000i	$-0.5\pi$
$503$		0			0				1.00000i	$0.5\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	−52.5000	+	30.3109i	−2.33161	+	1.34615i
$508$	1.00000	+	1.73205i	0.0443678	+	0.0768473i
$509$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$509$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$510$		0			0
$511$	−4.50000	−	0.866025i	−0.199068	−	0.0383107i
$512$		0			0
$513$	9.00000	−	15.5885i	0.397360	−	0.688247i
$514$		0			0
$515$		0			0
$516$	15.0000	+	8.66025i	0.660338	+	0.381246i
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$521$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$522$		0			0
$523$	25.5000	+	14.7224i	1.11504	+	0.643767i	0.940129	−	0.340818i	$-0.110704\pi$
$523$	25.5000	+	14.7224i	1.11504	+	0.643767i	0.174908	+	0.984585i	$0.444037\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		0			0
$528$		0			0
$529$	−11.5000	−	19.9186i	−0.500000	−	0.866025i
$530$		0			0
$531$		0			0
$532$	12.0000	−	13.8564i	0.520266	−	0.600751i
$533$		0			0
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	23.0000	−	39.8372i	0.988847	−	1.71273i	0.365444	−	0.930834i	$-0.380917\pi$
$541$	23.0000	−	39.8372i	0.988847	−	1.71273i	0.623404	−	0.781900i	$-0.285749\pi$
$542$		0			0
$543$	22.5000	+	38.9711i	0.965567	+	1.67241i
$544$		0			0
$545$		0			0
$546$		0			0
$547$	−40.0000			−1.71028			−0.855138	−	0.518400i	$-0.826528\pi$
$547$	−40.0000			−1.71028			−0.855138	+	0.518400i	$0.826528\pi$
$548$		0			0
$549$	40.5000	−	23.3827i	1.72850	−	0.997949i
$550$		0			0
$551$		0			0
$552$		0			0
$553$	−42.5000	+	14.7224i	−1.80728	+	0.626061i
$554$		0			0
$555$		0			0
$556$	9.00000	−	5.19615i	0.381685	−	0.220366i
$557$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$557$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$558$		0			0
$559$			34.6410i			1.46516i
$560$		0			0
$561$		0			0
$562$		0			0
$563$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$563$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	4.50000	−	23.3827i	0.188982	−	0.981981i
$568$		0			0
$569$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$569$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$570$		0			0
$571$	8.00000	+	13.8564i	0.334790	+	0.579873i	0.983444	−	0.181210i	$-0.0580014\pi$
$571$	8.00000	+	13.8564i	0.334790	+	0.579873i	−0.648655	+	0.761083i	$0.724668\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$	12.0000	−	20.7846i	0.500000	−	0.866025i
$577$	−28.5000	+	16.4545i	−1.18647	+	0.685009i	−0.957503	−	0.288425i	$-0.906868\pi$
$577$	−28.5000	+	16.4545i	−1.18647	+	0.685009i	−0.228968	+	0.973434i	$0.573535\pi$
$578$		0			0
$579$	−37.5000	−	21.6506i	−1.55845	−	0.899770i
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$		0			0
$587$		0			0			−	1.00000i	$-0.5\pi$
$587$		0			0				1.00000i	$0.5\pi$
$588$	9.00000	−	22.5167i	0.371154	−	0.928571i
$589$	−30.0000			−1.23613
$590$		0			0
$591$		0			0
$592$	22.0000	+	38.1051i	0.904194	+	1.56611i
$593$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$593$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	19.5000	−	33.7750i	0.798082	−	1.38232i
$598$		0			0
$599$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$599$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$600$		0			0
$601$			41.5692i			1.69564i	0.530281	+	0.847822i	$0.322086\pi$
$601$			41.5692i			1.69564i	−0.530281	+	0.847822i	$0.677914\pi$
$602$		0			0
$603$	48.0000			1.95471
$604$	−19.0000	+	32.9090i	−0.773099	+	1.33905i
$605$		0			0
$606$		0			0
$607$	34.5000	+	19.9186i	1.40031	+	0.808470i	0.994424	−	0.105453i	$-0.0336291\pi$
$607$	34.5000	+	19.9186i	1.40031	+	0.808470i	0.405887	+	0.913923i	$0.366962\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	5.00000	+	8.66025i	0.201948	+	0.349784i	0.949156	−	0.314806i	$-0.101939\pi$
$613$	5.00000	+	8.66025i	0.201948	+	0.349784i	−0.747208	+	0.664590i	$0.768606\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		0			0		1.00000			$0$
$617$		0			0		−1.00000			$\pi$
$618$		0			0
$619$	−7.50000	+	4.33013i	−0.301450	+	0.174042i	−0.643094	−	0.765787i	$-0.722350\pi$
$619$	−7.50000	+	4.33013i	−0.301450	+	0.174042i	0.341644	+	0.939829i	$0.389016\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$	48.0000			1.92154
$625$		0			0
$626$		0			0
$627$		0			0
$628$	−39.0000	−	22.5167i	−1.55627	−	0.898513i
$629$		0			0
$630$		0			0
$631$	−1.00000			−0.0398094			−0.0199047	−	0.999802i	$-0.506336\pi$
$631$	−1.00000			−0.0398094			−0.0199047	+	0.999802i	$0.506336\pi$
$632$		0			0
$633$	−43.5000	+	25.1147i	−1.72897	+	0.998221i
$634$		0			0
$635$		0			0
$636$		0			0
$637$	48.0000	−	6.92820i	1.90183	−	0.274505i
$638$		0			0
$639$		0			0
$640$		0			0
$641$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$641$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$642$		0			0
$643$		−	19.0526i		−	0.751360i	−0.926750	−	0.375680i	$-0.877409\pi$
$643$		−	19.0526i		−	0.751360i	0.926750	−	0.375680i	$-0.122591\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$647$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$	−37.5000	+	12.9904i	−1.46974	+	0.509133i
$652$	34.0000			1.33154
$653$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$653$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$			5.19615i			0.202721i
$658$		0			0
$659$		0			0		1.00000			$0$
$659$		0			0		−1.00000			$\pi$
$660$		0			0
$661$	−43.5000	+	25.1147i	−1.69195	+	0.976850i	−0.739014	+	0.673690i	$0.764708\pi$
$661$	−43.5000	+	25.1147i	−1.69195	+	0.976850i	−0.952940	+	0.303160i	$0.901958\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$	−16.5000	−	28.5788i	−0.637927	−	1.10492i
$670$		0			0
$671$		0			0
$672$		0			0
$673$	13.0000			0.501113			0.250557	−	0.968102i	$-0.419386\pi$
$673$	13.0000			0.501113			0.250557	+	0.968102i	$0.419386\pi$
$674$		0			0
$675$		0			0
$676$	35.0000	+	60.6218i	1.34615	+	2.33161i
$677$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$677$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$678$		0			0
$679$	−9.00000	+	10.3923i	−0.345388	+	0.398820i
$680$		0			0
$681$		0			0
$682$		0			0
$683$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$683$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$684$	−18.0000	−	10.3923i	−0.688247	−	0.397360i
$685$		0			0
$686$		0			0
$687$	−36.0000			−1.37349
$688$	10.0000	−	17.3205i	0.381246	−	0.660338i
$689$		0			0
$690$		0			0
$691$	−28.5000	−	16.4545i	−1.08419	−	0.625958i	−0.152167	−	0.988355i	$-0.548625\pi$
$691$	−28.5000	−	16.4545i	−1.08419	−	0.625958i	−0.932024	+	0.362397i	$0.881959\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$		0			0
$701$		0			0		1.00000			$0$
$701$		0			0		−1.00000			$\pi$
$702$		0			0
$703$	33.0000	−	19.0526i	1.24462	−	0.718581i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	26.5000	−	45.8993i	0.995228	−	1.72379i	0.413114	−	0.910679i	$-0.364441\pi$
$709$	26.5000	−	45.8993i	0.995228	−	1.72379i	0.582115	−	0.813107i	$-0.302225\pi$
$710$		0			0
$711$	25.5000	+	44.1673i	0.956325	+	1.65640i
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$719$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$720$		0			0
$721$	13.5000	+	38.9711i	0.502766	+	1.45136i
$722$		0			0
$723$	−1.50000	+	2.59808i	−0.0557856	+	0.0966235i
$724$	45.0000	−	25.9808i	1.67241	−	0.965567i
$725$		0			0
$726$		0			0
$727$		−	31.1769i		−	1.15629i	−0.815935	−	0.578144i	$-0.803777\pi$
$727$		−	31.1769i		−	1.15629i	0.815935	−	0.578144i	$-0.196223\pi$
$728$		0			0
$729$	−27.0000			−1.00000
$730$		0			0
$731$		0			0
$732$	−27.0000	−	46.7654i	−0.997949	−	1.72850i
$733$	−28.5000	−	16.4545i	−1.05267	−	0.607760i	−0.129275	−	0.991609i	$-0.541265\pi$
$733$	−28.5000	−	16.4545i	−1.05267	−	0.607760i	−0.923396	+	0.383849i	$0.874598\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$	18.5000	+	32.0429i	0.680534	+	1.17872i	0.974818	+	0.223001i	$0.0715853\pi$
$739$	18.5000	+	32.0429i	0.680534	+	1.17872i	−0.294285	+	0.955718i	$0.595081\pi$
$740$		0			0
$741$		−	41.5692i		−	1.52708i
$742$		0			0
$743$		0			0		1.00000			$0$
$743$		0			0		−1.00000			$\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	5.50000	−	9.52628i	0.200698	−	0.347619i	−0.748056	−	0.663636i	$-0.769012\pi$
$751$	5.50000	−	9.52628i	0.200698	−	0.347619i	0.948753	+	0.316017i	$0.102346\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$	−27.0000	−	5.19615i	−0.981981	−	0.188982i
$757$	−29.0000			−1.05402			−0.527011	−	0.849858i	$-0.676688\pi$
$757$	−29.0000			−1.05402			−0.527011	+	0.849858i	$0.676688\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$761$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$762$		0			0
$763$	−1.00000	+	5.19615i	−0.0362024	+	0.188113i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$	−24.0000	−	13.8564i	−0.866025	−	0.500000i
$769$			25.9808i			0.936890i	0.883493	+	0.468445i	$0.155186\pi$
$769$			25.9808i			0.936890i	−0.883493	+	0.468445i	$0.844814\pi$
$770$		0			0
$771$		0			0
$772$	−25.0000	+	43.3013i	−0.899770	+	1.55845i
$773$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$773$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$	33.0000	−	38.1051i	1.18387	−	1.36701i
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$	−26.0000	−	10.3923i	−0.928571	−	0.371154i
$785$		0			0
$786$		0			0
$787$	40.5000	−	23.3827i	1.44367	−	0.833503i	0.445577	−	0.895244i	$-0.352999\pi$
$787$	40.5000	−	23.3827i	1.44367	−	0.833503i	0.998092	+	0.0617409i	$0.0196653\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	54.0000	−	93.5307i	1.91760	−	3.32137i
$794$		0			0
$795$		0			0
$796$	−39.0000	−	22.5167i	−1.38232	−	0.798082i
$797$		0			0			−	1.00000i	$-0.5\pi$
$797$		0			0				1.00000i	$0.5\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$		0			0
$802$		0			0
$803$		0			0
$804$		−	55.4256i		−	1.95471i
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$809$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$810$		0			0
$811$			53.6936i			1.88544i	0.333590	+	0.942718i	$0.391740\pi$
$811$			53.6936i			1.88544i	−0.333590	+	0.942718i	$0.608260\pi$
$812$		0			0
$813$	30.0000			1.05215
$814$		0			0
$815$		0			0
$816$		0			0
$817$	−15.0000	−	8.66025i	−0.524784	−	0.302984i
$818$		0			0
$819$	−18.0000	−	51.9615i	−0.628971	−	1.81568i
$820$		0			0
$821$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$821$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$822$		0			0
$823$	−23.5000	−	40.7032i	−0.819159	−	1.41882i	−0.906303	−	0.422628i	$-0.861108\pi$
$823$	−23.5000	−	40.7032i	−0.819159	−	1.41882i	0.0871445	−	0.996196i	$-0.472226\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0		1.00000			$0$
$827$		0			0		−1.00000			$\pi$
$828$		0			0
$829$	19.5000	−	11.2583i	0.677263	−	0.391018i	−0.121560	−	0.992584i	$-0.538790\pi$
$829$	19.5000	−	11.2583i	0.677263	−	0.391018i	0.798823	+	0.601566i	$0.205456\pi$
$830$		0			0
$831$	−39.0000	−	22.5167i	−1.35290	−	0.781094i
$832$		−	55.4256i		−	1.92154i
$833$		0			0
$834$		0			0
$835$		0			0
$836$		0			0
$837$	22.5000	+	38.9711i	0.777714	+	1.34704i
$838$		0			0
$839$		0			0			−	1.00000i	$-0.5\pi$
$839$		0			0				1.00000i	$0.5\pi$
$840$		0			0
$841$	29.0000			1.00000
$842$		0			0
$843$		0			0
$844$	29.0000	+	50.2295i	0.998221	+	1.72897i
$845$		0			0
$846$		0			0
$847$	−27.5000	+	9.52628i	−0.944911	+	0.327327i
$848$		0			0
$849$	9.00000	−	15.5885i	0.308879	−	0.534994i
$850$		0			0
$851$		0			0
$852$		0			0
$853$		−	46.7654i		−	1.60122i	−0.599189	−	0.800608i	$-0.704510\pi$
$853$		−	46.7654i		−	1.60122i	0.599189	−	0.800608i	$-0.295490\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$857$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$858$		0			0
$859$	15.0000	+	8.66025i	0.511793	+	0.295484i	0.733571	−	0.679613i	$-0.237852\pi$
$859$	15.0000	+	8.66025i	0.511793	+	0.295484i	−0.221777	+	0.975097i	$0.571186\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$863$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$		−	29.4449i		−	1.00000i
$868$	15.0000	+	43.3013i	0.509133	+	1.46974i
$869$		0			0
$870$		0			0
$871$	96.0000	−	55.4256i	3.25284	−	1.87803i
$872$		0			0
$873$	13.5000	+	7.79423i	0.456906	+	0.263795i
$874$		0			0
$875$		0			0
$876$	6.00000			0.202721
$877$	−29.5000	+	51.0955i	−0.996144	+	1.72537i	−0.422095	+	0.906552i	$0.638705\pi$
$877$	−29.5000	+	51.0955i	−0.996144	+	1.72537i	−0.574049	+	0.818821i	$0.694628\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		0			0			−	1.00000i	$-0.5\pi$
$881$		0			0				1.00000i	$0.5\pi$
$882$		0			0
$883$	−8.00000			−0.269221			−0.134611	−	0.990899i	$-0.542978\pi$
$883$	−8.00000			−0.269221			−0.134611	+	0.990899i	$0.542978\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$887$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$888$		0			0
$889$	−2.00000	−	1.73205i	−0.0670778	−	0.0580911i
$890$		0			0
$891$		0			0
$892$	−33.0000	+	19.0526i	−1.10492	+	0.637927i
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$		0			0
$900$		0			0
$901$		0			0
$902$		0			0
$903$	−22.5000	−	4.33013i	−0.748753	−	0.144098i
$904$		0			0
$905$		0			0
$906$		0			0
$907$	−9.50000	−	16.4545i	−0.315442	−	0.546362i	0.664089	−	0.747653i	$-0.268820\pi$
$907$	−9.50000	−	16.4545i	−0.315442	−	0.546362i	−0.979531	+	0.201291i	$0.935486\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		0			0		1.00000			$0$
$911$		0			0		−1.00000			$\pi$
$912$	−12.0000	+	20.7846i	−0.397360	+	0.688247i
$913$		0			0
$914$		0			0
$915$		0			0
$916$			41.5692i			1.37349i
$917$		0			0
$918$		0			0
$919$	−0.500000	+	0.866025i	−0.0164935	+	0.0285675i	−0.874154	−	0.485648i	$-0.838584\pi$
$919$	−0.500000	+	0.866025i	−0.0164935	+	0.0285675i	0.857661	+	0.514216i	$0.171917\pi$
$920$		0			0
$921$	1.50000	+	2.59808i	0.0494267	+	0.0856095i
$922$		0			0
$923$		0			0
$924$		0			0
$925$		0			0
$926$		0			0
$927$	40.5000	−	23.3827i	1.33019	−	0.767988i
$928$		0			0
$929$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$929$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$930$		0			0
$931$	−9.00000	+	22.5167i	−0.294963	+	0.737954i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$			50.2295i			1.64093i	0.571700	+	0.820463i	$0.306284\pi$
$937$			50.2295i			1.64093i	−0.571700	+	0.820463i	$0.693716\pi$
$938$		0			0
$939$	9.00000			0.293704
$940$		0			0
$941$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$941$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$947$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$948$	51.0000	−	29.4449i	1.65640	−	0.956325i
$949$	6.00000	+	10.3923i	0.194768	+	0.337348i
$950$		0			0
$951$		0			0
$952$		0			0
$953$		0			0		1.00000			$0$
$953$		0			0		−1.00000			$\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	22.0000	−	38.1051i	0.709677	−	1.22920i
$962$		0			0
$963$		0			0
$964$	3.00000	+	1.73205i	0.0966235	+	0.0557856i
$965$		0			0
$966$		0			0
$967$	41.0000			1.31847			0.659236	−	0.751936i	$-0.270880\pi$
$967$	41.0000			1.31847			0.659236	+	0.751936i	$0.270880\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$971$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$972$			31.1769i			1.00000i
$973$	−9.00000	+	10.3923i	−0.288527	+	0.333162i
$974$		0			0
$975$		0			0
$976$	−54.0000	+	31.1769i	−1.72850	+	0.997949i
$977$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$977$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$	6.00000			0.191565
$982$		0			0
$983$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$983$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$	−48.0000			−1.52708
$989$		0			0
$990$		0			0
$991$	8.50000	+	14.7224i	0.270011	+	0.467673i	0.968864	−	0.247592i	$-0.0796392\pi$
$991$	8.50000	+	14.7224i	0.270011	+	0.467673i	−0.698853	+	0.715265i	$0.746306\pi$
$992$		0			0
$993$			53.6936i			1.70391i
$994$		0			0
$995$		0			0
$996$		0			0
$997$	19.5000	−	11.2583i	0.617571	−	0.356555i	−0.158352	−	0.987383i	$-0.550618\pi$
$997$	19.5000	−	11.2583i	0.617571	−	0.356555i	0.775923	+	0.630828i	$0.217285\pi$
$998$		0			0
$999$	−49.5000	−	28.5788i	−1.56611	−	0.904194i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	525.2.t.d.26.1	yes	2
3.2	odd	2	CM	525.2.t.d.26.1	yes	2
5.2	odd	4		525.2.q.c.299.1		4
5.3	odd	4		525.2.q.c.299.2		4
5.4	even	2		525.2.t.b.26.1	✓	2
7.3	odd	6	inner	525.2.t.d.101.1	yes	2
15.2	even	4		525.2.q.c.299.1		4
15.8	even	4		525.2.q.c.299.2		4
15.14	odd	2		525.2.t.b.26.1	✓	2
21.17	even	6	inner	525.2.t.d.101.1	yes	2
35.3	even	12		525.2.q.c.374.1		4
35.17	even	12		525.2.q.c.374.2		4
35.24	odd	6		525.2.t.b.101.1	yes	2
105.17	odd	12		525.2.q.c.374.2		4
105.38	odd	12		525.2.q.c.374.1		4
105.59	even	6		525.2.t.b.101.1	yes	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
525.2.q.c.299.1		4	5.2	odd	4
525.2.q.c.299.1		4	15.2	even	4
525.2.q.c.299.2		4	5.3	odd	4
525.2.q.c.299.2		4	15.8	even	4
525.2.q.c.374.1		4	35.3	even	12
525.2.q.c.374.1		4	105.38	odd	12
525.2.q.c.374.2		4	35.17	even	12
525.2.q.c.374.2		4	105.17	odd	12
525.2.t.b.26.1	✓	2	5.4	even	2
525.2.t.b.26.1	✓	2	15.14	odd	2
525.2.t.b.101.1	yes	2	35.24	odd	6
525.2.t.b.101.1	yes	2	105.59	even	6
525.2.t.d.26.1	yes	2	1.1	even	1	trivial
525.2.t.d.26.1	yes	2	3.2	odd	2	CM
525.2.t.d.101.1	yes	2	7.3	odd	6	inner
525.2.t.d.101.1	yes	2	21.17	even	6	inner

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:	\( N \)	\(=\)	\( 525 = 3 \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	525.t (of order \(6\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(1.50000 - 0.866025i) q^{3} +(-1.00000 - 1.73205i) q^{4} +(2.00000 + 1.73205i) q^{7} +(1.50000 - 2.59808i) q^{9} +O(q^{10})\)	\(q+(1.50000 - 0.866025i) q^{3} +(-1.00000 - 1.73205i) q^{4} +(2.00000 + 1.73205i) q^{7} +(1.50000 - 2.59808i) q^{9} +(-3.00000 - 1.73205i) q^{12} -6.92820i q^{13} +(-2.00000 + 3.46410i) q^{16} +(3.00000 + 1.73205i) q^{19} +(4.50000 + 0.866025i) q^{21} -5.19615i q^{27} +(1.00000 - 5.19615i) q^{28} +(-7.50000 + 4.33013i) q^{31} -6.00000 q^{36} +(5.50000 - 9.52628i) q^{37} +(-6.00000 - 10.3923i) q^{39} -5.00000 q^{43} +6.92820i q^{48} +(1.00000 + 6.92820i) q^{49} +(-12.0000 + 6.92820i) q^{52} +6.00000 q^{57} +(13.5000 + 7.79423i) q^{61} +(7.50000 - 2.59808i) q^{63} +8.00000 q^{64} +(8.00000 + 13.8564i) q^{67} +(-1.50000 + 0.866025i) q^{73} -6.92820i q^{76} +(-8.50000 + 14.7224i) q^{79} +(-4.50000 - 7.79423i) q^{81} +(-3.00000 - 8.66025i) q^{84} +(12.0000 - 13.8564i) q^{91} +(-7.50000 + 12.9904i) q^{93} +5.19615i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 3 q^{3} - 2 q^{4} + 4 q^{7} + 3 q^{9}+O(q^{10}) \) 2 * q + 3 * q^3 - 2 * q^4 + 4 * q^7 + 3 * q^9	\( 2 q + 3 q^{3} - 2 q^{4} + 4 q^{7} + 3 q^{9} - 6 q^{12} - 4 q^{16} + 6 q^{19} + 9 q^{21} + 2 q^{28} - 15 q^{31} - 12 q^{36} + 11 q^{37} - 12 q^{39} - 10 q^{43} + 2 q^{49} - 24 q^{52} + 12 q^{57} + 27 q^{61} + 15 q^{63} + 16 q^{64} + 16 q^{67} - 3 q^{73} - 17 q^{79} - 9 q^{81} - 6 q^{84} + 24 q^{91} - 15 q^{93}+O(q^{100}) \) 2 * q + 3 * q^3 - 2 * q^4 + 4 * q^7 + 3 * q^9 - 6 * q^12 - 4 * q^16 + 6 * q^19 + 9 * q^21 + 2 * q^28 - 15 * q^31 - 12 * q^36 + 11 * q^37 - 12 * q^39 - 10 * q^43 + 2 * q^49 - 24 * q^52 + 12 * q^57 + 27 * q^61 + 15 * q^63 + 16 * q^64 + 16 * q^67 - 3 * q^73 - 17 * q^79 - 9 * q^81 - 6 * q^84 + 24 * q^91 - 15 * q^93

Introduction

L-functions

Modular forms

Varieties

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Representations

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Database

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