LMFDB - Embedded newform 100.4.e.c.43.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [100,4,Mod(7,100)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(100, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([2, 1]))

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("100.7");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 100 = 2^{2} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	100.e (of order $4$, 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:	$5.90019100057$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{4}]$

Embedding invariants

Embedding label			43.1
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	100.43
Dual form			100.4.e.c.7.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(2.00000 + 2.00000i) q^{2} +(-7.00000 + 7.00000i) q^{3} +8.00000i q^{4} -28.0000 q^{6} +(9.00000 + 9.00000i) q^{7} +(-16.0000 + 16.0000i) q^{8} -71.0000i q^{9} +O(q^{10})$	\(q+(2.00000 + 2.00000i) q^{2} +(-7.00000 + 7.00000i) q^{3} +8.00000i q^{4} -28.0000 q^{6} +(9.00000 + 9.00000i) q^{7} +(-16.0000 + 16.0000i) q^{8} -71.0000i q^{9} +(-56.0000 - 56.0000i) q^{12} +36.0000i q^{14} -64.0000 q^{16} +(142.000 - 142.000i) q^{18} -126.000 q^{21} +(-67.0000 + 67.0000i) q^{23} -224.000i q^{24} +(308.000 + 308.000i) q^{27} +(-72.0000 + 72.0000i) q^{28} +306.000i q^{29} +(-128.000 - 128.000i) q^{32} +568.000 q^{36} +252.000 q^{41} +(-252.000 - 252.000i) q^{42} +(-297.000 + 297.000i) q^{43} -268.000 q^{46} +(-301.000 - 301.000i) q^{47} +(448.000 - 448.000i) q^{48} -181.000i q^{49} +1232.00i q^{54} -288.000 q^{56} +(-612.000 + 612.000i) q^{58} +952.000 q^{61} +(639.000 - 639.000i) q^{63} -512.000i q^{64} +(549.000 + 549.000i) q^{67} -938.000i q^{69} +(1136.00 + 1136.00i) q^{72} -2395.00 q^{81} +(504.000 + 504.000i) q^{82} +(-77.0000 + 77.0000i) q^{83} -1008.00i q^{84} -1188.00 q^{86} +(-2142.00 - 2142.00i) q^{87} +1386.00i q^{89} +(-536.000 - 536.000i) q^{92} -1204.00i q^{94} +1792.00 q^{96} +(362.000 - 362.000i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 4 q^{2} - 14 q^{3} - 56 q^{6} + 18 q^{7} - 32 q^{8}+O(q^{10}) $ 2 * q + 4 * q^2 - 14 * q^3 - 56 * q^6 + 18 * q^7 - 32 * q^8	$ 2 q + 4 q^{2} - 14 q^{3} - 56 q^{6} + 18 q^{7} - 32 q^{8} - 112 q^{12} - 128 q^{16} + 284 q^{18} - 252 q^{21} - 134 q^{23} + 616 q^{27} - 144 q^{28} - 256 q^{32} + 1136 q^{36} + 504 q^{41} - 504 q^{42} - 594 q^{43} - 536 q^{46} - 602 q^{47} + 896 q^{48} - 576 q^{56} - 1224 q^{58} + 1904 q^{61} + 1278 q^{63} + 1098 q^{67} + 2272 q^{72} - 4790 q^{81} + 1008 q^{82} - 154 q^{83} - 2376 q^{86} - 4284 q^{87} - 1072 q^{92} + 3584 q^{96} + 724 q^{98}+O(q^{100}) $ 2 * q + 4 * q^2 - 14 * q^3 - 56 * q^6 + 18 * q^7 - 32 * q^8 - 112 * q^12 - 128 * q^16 + 284 * q^18 - 252 * q^21 - 134 * q^23 + 616 * q^27 - 144 * q^28 - 256 * q^32 + 1136 * q^36 + 504 * q^41 - 504 * q^42 - 594 * q^43 - 536 * q^46 - 602 * q^47 + 896 * q^48 - 576 * q^56 - 1224 * q^58 + 1904 * q^61 + 1278 * q^63 + 1098 * q^67 + 2272 * q^72 - 4790 * q^81 + 1008 * q^82 - 154 * q^83 - 2376 * q^86 - 4284 * q^87 - 1072 * q^92 + 3584 * q^96 + 724 * q^98

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/100\mathbb{Z}\right)^\times$.

$n$	$51$	$77$
$\chi(n)$	$-1$	$e\left(\frac{3}{4}\right)$

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 $

$2$	2.00000	+	2.00000i	0.707107	+	0.707107i
$2$	2.00000	+	2.00000i	0.707107	+	0.707107i
$3$	−7.00000	+	7.00000i	−1.34715	+	1.34715i	−0.458410	+	0.888741i	$0.651581\pi$
$3$	−7.00000	+	7.00000i	−1.34715	+	1.34715i	−0.888741	+	0.458410i	$0.848419\pi$
$4$			8.00000i			1.00000i
$5$		0			0
$5$		0			0
$6$	−28.0000			−1.90516
$7$	9.00000	+	9.00000i	0.485954	+	0.485954i	0.907027	−	0.421073i	$-0.138346\pi$
$7$	9.00000	+	9.00000i	0.485954	+	0.485954i	−0.421073	+	0.907027i	$0.638346\pi$
$8$	−16.0000	+	16.0000i	−0.707107	+	0.707107i
$9$		−	71.0000i		−	2.62963i
$10$		0			0
$11$		0			0		1.00000			$0$
$11$		0			0		−1.00000			$\pi$
$12$	−56.0000	−	56.0000i	−1.34715	−	1.34715i
$13$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$13$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$14$			36.0000i			0.687243i
$15$		0			0
$16$	−64.0000			−1.00000
$17$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$17$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$18$	142.000	−	142.000i	1.85943	−	1.85943i
$19$		0			0			−	1.00000i	$-0.5\pi$
$19$		0			0				1.00000i	$0.5\pi$
$20$		0			0
$21$	−126.000			−1.30931
$22$		0			0
$23$	−67.0000	+	67.0000i	−0.607412	+	0.607412i	−0.942269	−	0.334857i	$-0.891312\pi$
$23$	−67.0000	+	67.0000i	−0.607412	+	0.607412i	0.334857	+	0.942269i	$0.391312\pi$
$24$		−	224.000i		−	1.90516i
$25$		0			0
$26$		0			0
$27$	308.000	+	308.000i	2.19536	+	2.19536i
$28$	−72.0000	+	72.0000i	−0.485954	+	0.485954i
$29$			306.000i			1.95941i	0.200455	+	0.979703i	$0.435758\pi$
$29$			306.000i			1.95941i	−0.200455	+	0.979703i	$0.564242\pi$
$30$		0			0
$31$		0			0		1.00000			$0$
$31$		0			0		−1.00000			$\pi$
$32$	−128.000	−	128.000i	−0.707107	−	0.707107i
$33$		0			0
$34$		0			0
$35$		0			0
$36$	568.000			2.62963
$37$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$37$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	252.000			0.959897			0.479949	−	0.877297i	$-0.340655\pi$
$41$	252.000			0.959897			0.479949	+	0.877297i	$0.340655\pi$
$42$	−252.000	−	252.000i	−0.925820	−	0.925820i
$43$	−297.000	+	297.000i	−1.05330	+	1.05330i	−0.0548071	+	0.998497i	$0.517454\pi$
$43$	−297.000	+	297.000i	−1.05330	+	1.05330i	−0.998497	+	0.0548071i	$0.982546\pi$
$44$		0			0
$45$		0			0
$46$	−268.000			−0.859010
$47$	−301.000	−	301.000i	−0.934157	−	0.934157i	0.0638057	−	0.997962i	$-0.479676\pi$
$47$	−301.000	−	301.000i	−0.934157	−	0.934157i	−0.997962	+	0.0638057i	$0.979676\pi$
$48$	448.000	−	448.000i	1.34715	−	1.34715i
$49$		−	181.000i		−	0.527697i
$50$		0			0
$51$		0			0
$52$		0			0
$53$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$53$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$54$			1232.00i			3.10470i
$55$		0			0
$56$	−288.000			−0.687243
$57$		0			0
$58$	−612.000	+	612.000i	−1.38551	+	1.38551i
$59$		0			0			−	1.00000i	$-0.5\pi$
$59$		0			0				1.00000i	$0.5\pi$
$60$		0			0
$61$	952.000			1.99821			0.999107	−	0.0422409i	$-0.0134497\pi$
$61$	952.000			1.99821			0.999107	+	0.0422409i	$0.0134497\pi$
$62$		0			0
$63$	639.000	−	639.000i	1.27788	−	1.27788i
$64$		−	512.000i		−	1.00000i
$65$		0			0
$66$		0			0
$67$	549.000	+	549.000i	1.00106	+	1.00106i	0.999999	+	0.00106064i	$0.000337611\pi$
$67$	549.000	+	549.000i	1.00106	+	1.00106i	0.00106064	+	0.999999i	$0.499662\pi$
$68$		0			0
$69$		−	938.000i		−	1.63655i
$70$		0			0
$71$		0			0		1.00000			$0$
$71$		0			0		−1.00000			$\pi$
$72$	1136.00	+	1136.00i	1.85943	+	1.85943i
$73$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$73$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$		0			0
$78$		0			0
$79$		0			0			−	1.00000i	$-0.5\pi$
$79$		0			0				1.00000i	$0.5\pi$
$80$		0			0
$81$	−2395.00			−3.28532
$82$	504.000	+	504.000i	0.678750	+	0.678750i
$83$	−77.0000	+	77.0000i	−0.101829	+	0.101829i	−0.756186	−	0.654357i	$-0.772940\pi$
$83$	−77.0000	+	77.0000i	−0.101829	+	0.101829i	0.654357	+	0.756186i	$0.272940\pi$
$84$		−	1008.00i		−	1.30931i
$85$		0			0
$86$	−1188.00			−1.48960
$87$	−2142.00	−	2142.00i	−2.63961	−	2.63961i
$88$		0			0
$89$			1386.00i			1.65074i	0.564593	+	0.825369i	$0.309033\pi$
$89$			1386.00i			1.65074i	−0.564593	+	0.825369i	$0.690967\pi$
$90$		0			0
$91$		0			0
$92$	−536.000	−	536.000i	−0.607412	−	0.607412i
$93$		0			0
$94$		−	1204.00i		−	1.32110i
$95$		0			0
$96$	1792.00			1.90516
$97$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$97$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$98$	362.000	−	362.000i	0.373138	−	0.373138i
$99$		0			0
$100$		0			0
$101$	−378.000			−0.372400			−0.186200	−	0.982512i	$-0.559617\pi$
$101$	−378.000			−0.372400			−0.186200	+	0.982512i	$0.559617\pi$
$102$		0			0
$103$	1323.00	−	1323.00i	1.26562	−	1.26562i	0.317295	−	0.948327i	$-0.397225\pi$
$103$	1323.00	−	1323.00i	1.26562	−	1.26562i	0.948327	−	0.317295i	$-0.102775\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−221.000	−	221.000i	−0.199672	−	0.199672i	0.600188	−	0.799859i	$-0.295093\pi$
$107$	−221.000	−	221.000i	−0.199672	−	0.199672i	−0.799859	+	0.600188i	$0.795093\pi$
$108$	−2464.00	+	2464.00i	−2.19536	+	2.19536i
$109$			1136.00i			0.998248i	0.866530	+	0.499124i	$0.166345\pi$
$109$			1136.00i			0.998248i	−0.866530	+	0.499124i	$0.833655\pi$
$110$		0			0
$111$		0			0
$112$	−576.000	−	576.000i	−0.485954	−	0.485954i
$113$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$113$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$114$		0			0
$115$		0			0
$116$	−2448.00			−1.95941
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	1331.00			1.00000
$122$	1904.00	+	1904.00i	1.41295	+	1.41295i
$123$	−1764.00	+	1764.00i	−1.29313	+	1.29313i
$124$		0			0
$125$		0			0
$126$	2556.00			1.80720
$127$	1089.00	+	1089.00i	0.760891	+	0.760891i	0.976483	−	0.215593i	$-0.0691683\pi$
$127$	1089.00	+	1089.00i	0.760891	+	0.760891i	−0.215593	+	0.976483i	$0.569168\pi$
$128$	1024.00	−	1024.00i	0.707107	−	0.707107i
$129$		−	4158.00i		−	2.83792i
$130$		0			0
$131$		0			0		1.00000			$0$
$131$		0			0		−1.00000			$\pi$
$132$		0			0
$133$		0			0
$134$			2196.00i			1.41571i
$135$		0			0
$136$		0			0
$137$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$137$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$138$	1876.00	−	1876.00i	1.15722	−	1.15722i
$139$		0			0			−	1.00000i	$-0.5\pi$
$139$		0			0				1.00000i	$0.5\pi$
$140$		0			0
$141$	4214.00			2.51690
$142$		0			0
$143$		0			0
$144$			4544.00i			2.62963i
$145$		0			0
$146$		0			0
$147$	1267.00	+	1267.00i	0.710887	+	0.710887i
$148$		0			0
$149$			3096.00i			1.70224i	0.524969	+	0.851121i	$0.324077\pi$
$149$			3096.00i			1.70224i	−0.524969	+	0.851121i	$0.675923\pi$
$150$		0			0
$151$		0			0		1.00000			$0$
$151$		0			0		−1.00000			$\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$		0			0
$157$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$157$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−1206.00			−0.590349
$162$	−4790.00	−	4790.00i	−2.32307	−	2.32307i
$163$	2943.00	−	2943.00i	1.41419	−	1.41419i	0.703336	−	0.710858i	$-0.251693\pi$
$163$	2943.00	−	2943.00i	1.41419	−	1.41419i	0.710858	−	0.703336i	$-0.248307\pi$
$164$			2016.00i			0.959897i
$165$		0			0
$166$	−308.000			−0.144009
$167$	1309.00	+	1309.00i	0.606548	+	0.606548i	0.942042	−	0.335494i	$-0.108903\pi$
$167$	1309.00	+	1309.00i	0.606548	+	0.606548i	−0.335494	+	0.942042i	$0.608903\pi$
$168$	2016.00	−	2016.00i	0.925820	−	0.925820i
$169$		−	2197.00i		−	1.00000i
$170$		0			0
$171$		0			0
$172$	−2376.00	−	2376.00i	−1.05330	−	1.05330i
$173$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$173$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$174$		−	8568.00i		−	3.73298i
$175$		0			0
$176$		0			0
$177$		0			0
$178$	−2772.00	+	2772.00i	−1.16725	+	1.16725i
$179$		0			0			−	1.00000i	$-0.5\pi$
$179$		0			0				1.00000i	$0.5\pi$
$180$		0			0
$181$	−1078.00			−0.442691			−0.221346	−	0.975195i	$-0.571045\pi$
$181$	−1078.00			−0.442691			−0.221346	+	0.975195i	$0.571045\pi$
$182$		0			0
$183$	−6664.00	+	6664.00i	−2.69190	+	2.69190i
$184$		−	2144.00i		−	0.859010i
$185$		0			0
$186$		0			0
$187$		0			0
$188$	2408.00	−	2408.00i	0.934157	−	0.934157i
$189$			5544.00i			2.13369i
$190$		0			0
$191$		0			0		1.00000			$0$
$191$		0			0		−1.00000			$\pi$
$192$	3584.00	+	3584.00i	1.34715	+	1.34715i
$193$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$193$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$194$		0			0
$195$		0			0
$196$	1448.00			0.527697
$197$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$197$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$198$		0			0
$199$		0			0			−	1.00000i	$-0.5\pi$
$199$		0			0				1.00000i	$0.5\pi$
$200$		0			0
$201$	−7686.00			−2.69716
$202$	−756.000	−	756.000i	−0.263327	−	0.263327i
$203$	−2754.00	+	2754.00i	−0.952182	+	0.952182i
$204$		0			0
$205$		0			0
$206$	5292.00			1.78986
$207$	4757.00	+	4757.00i	1.59727	+	1.59727i
$208$		0			0
$209$		0			0
$210$		0			0
$211$		0			0		1.00000			$0$
$211$		0			0		−1.00000			$\pi$
$212$		0			0
$213$		0			0
$214$		−	884.000i		−	0.282378i
$215$		0			0
$216$	−9856.00			−3.10470
$217$		0			0
$218$	−2272.00	+	2272.00i	−0.705868	+	0.705868i
$219$		0			0
$220$		0			0
$221$		0			0
$222$		0			0
$223$	4473.00	−	4473.00i	1.34320	−	1.34320i	0.450352	−	0.892851i	$-0.351299\pi$
$223$	4473.00	−	4473.00i	1.34320	−	1.34320i	0.892851	−	0.450352i	$-0.148701\pi$
$224$		−	2304.00i		−	0.687243i
$225$		0			0
$226$		0			0
$227$	−4081.00	−	4081.00i	−1.19324	−	1.19324i	−0.976152	−	0.217088i	$-0.930344\pi$
$227$	−4081.00	−	4081.00i	−1.19324	−	1.19324i	−0.217088	−	0.976152i	$-0.569656\pi$
$228$		0			0
$229$		−	6874.00i		−	1.98361i	−0.127761	−	0.991805i	$-0.540779\pi$
$229$		−	6874.00i		−	1.98361i	0.127761	−	0.991805i	$-0.459221\pi$
$230$		0			0
$231$		0			0
$232$	−4896.00	−	4896.00i	−1.38551	−	1.38551i
$233$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$233$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$		0			0			−	1.00000i	$-0.5\pi$
$239$		0			0				1.00000i	$0.5\pi$
$240$		0			0
$241$	−1708.00			−0.456523			−0.228261	−	0.973600i	$-0.573304\pi$
$241$	−1708.00			−0.456523			−0.228261	+	0.973600i	$0.573304\pi$
$242$	2662.00	+	2662.00i	0.707107	+	0.707107i
$243$	8449.00	−	8449.00i	2.23047	−	2.23047i
$244$			7616.00i			1.99821i
$245$		0			0
$246$	−7056.00			−1.82876
$247$		0			0
$248$		0			0
$249$		−	1078.00i		−	0.274359i
$250$		0			0
$251$		0			0		1.00000			$0$
$251$		0			0		−1.00000			$\pi$
$252$	5112.00	+	5112.00i	1.27788	+	1.27788i
$253$		0			0
$254$			4356.00i			1.07606i
$255$		0			0
$256$	4096.00			1.00000
$257$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$257$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$258$	8316.00	−	8316.00i	2.00671	−	2.00671i
$259$		0			0
$260$		0			0
$261$	21726.0			5.15251
$262$		0			0
$263$	−6017.00	+	6017.00i	−1.41074	+	1.41074i	−0.655826	+	0.754912i	$0.727679\pi$
$263$	−6017.00	+	6017.00i	−1.41074	+	1.41074i	−0.754912	+	0.655826i	$0.772321\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$	−9702.00	−	9702.00i	−2.22379	−	2.22379i
$268$	−4392.00	+	4392.00i	−1.00106	+	1.00106i
$269$		−	5544.00i		−	1.25659i	−0.777974	−	0.628297i	$-0.783752\pi$
$269$		−	5544.00i		−	1.25659i	0.777974	−	0.628297i	$-0.216248\pi$
$270$		0			0
$271$		0			0		1.00000			$0$
$271$		0			0		−1.00000			$\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		0			0
$276$	7504.00			1.63655
$277$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$277$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−8388.00			−1.78073			−0.890367	−	0.455244i	$-0.849552\pi$
$281$	−8388.00			−1.78073			−0.890367	+	0.455244i	$0.849552\pi$
$282$	8428.00	+	8428.00i	1.77972	+	1.77972i
$283$	693.000	−	693.000i	0.145564	−	0.145564i	−0.630569	−	0.776133i	$-0.717178\pi$
$283$	693.000	−	693.000i	0.145564	−	0.145564i	0.776133	+	0.630569i	$0.217178\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	2268.00	+	2268.00i	0.466466	+	0.466466i
$288$	−9088.00	+	9088.00i	−1.85943	+	1.85943i
$289$			4913.00i			1.00000i
$290$		0			0
$291$		0			0
$292$		0			0
$293$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$293$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$294$			5068.00i			1.00535i
$295$		0			0
$296$		0			0
$297$		0			0
$298$	−6192.00	+	6192.00i	−1.20367	+	1.20367i
$299$		0			0
$300$		0			0
$301$	−5346.00			−1.02372
$302$		0			0
$303$	2646.00	−	2646.00i	0.501679	−	0.501679i
$304$		0			0
$305$		0			0
$306$		0			0
$307$	2709.00	+	2709.00i	0.503618	+	0.503618i	0.912560	−	0.408942i	$-0.134102\pi$
$307$	2709.00	+	2709.00i	0.503618	+	0.503618i	−0.408942	+	0.912560i	$0.634102\pi$
$308$		0			0
$309$			18522.0i			3.40997i
$310$		0			0
$311$		0			0		1.00000			$0$
$311$		0			0		−1.00000			$\pi$
$312$		0			0
$313$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$313$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$317$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$	3094.00			0.537976
$322$	−2412.00	−	2412.00i	−0.417439	−	0.417439i
$323$		0			0
$324$		−	19160.0i		−	3.28532i
$325$		0			0
$326$	11772.0			1.99997
$327$	−7952.00	−	7952.00i	−1.34479	−	1.34479i
$328$	−4032.00	+	4032.00i	−0.678750	+	0.678750i
$329$		−	5418.00i		−	0.907915i
$330$		0			0
$331$		0			0		1.00000			$0$
$331$		0			0		−1.00000			$\pi$
$332$	−616.000	−	616.000i	−0.101829	−	0.101829i
$333$		0			0
$334$			5236.00i			0.857788i
$335$		0			0
$336$	8064.00			1.30931
$337$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$337$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$338$	4394.00	−	4394.00i	0.707107	−	0.707107i
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$	4716.00	−	4716.00i	0.742391	−	0.742391i
$344$		−	9504.00i		−	1.48960i
$345$		0			0
$346$		0			0
$347$	−7871.00	−	7871.00i	−1.21769	−	1.21769i	−0.968440	−	0.249247i	$-0.919817\pi$
$347$	−7871.00	−	7871.00i	−1.21769	−	1.21769i	−0.249247	−	0.968440i	$-0.580183\pi$
$348$	17136.0	−	17136.0i	2.63961	−	2.63961i
$349$			9646.00i			1.47948i	0.672893	+	0.739740i	$0.265052\pi$
$349$			9646.00i			1.47948i	−0.672893	+	0.739740i	$0.734948\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$353$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$354$		0			0
$355$		0			0
$356$	−11088.0			−1.65074
$357$		0			0
$358$		0			0
$359$		0			0			−	1.00000i	$-0.5\pi$
$359$		0			0				1.00000i	$0.5\pi$
$360$		0			0
$361$	−6859.00			−1.00000
$362$	−2156.00	−	2156.00i	−0.313030	−	0.313030i
$363$	−9317.00	+	9317.00i	−1.34715	+	1.34715i
$364$		0			0
$365$		0			0
$366$	−26656.0			−3.80692
$367$	9639.00	+	9639.00i	1.37099	+	1.37099i	0.858985	+	0.512000i	$0.171095\pi$
$367$	9639.00	+	9639.00i	1.37099	+	1.37099i	0.512000	+	0.858985i	$0.328905\pi$
$368$	4288.00	−	4288.00i	0.607412	−	0.607412i
$369$		−	17892.0i		−	2.52417i
$370$		0			0
$371$		0			0
$372$		0			0
$373$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$373$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$374$		0			0
$375$		0			0
$376$	9632.00			1.32110
$377$		0			0
$378$	−11088.0	+	11088.0i	−1.50874	+	1.50874i
$379$		0			0			−	1.00000i	$-0.5\pi$
$379$		0			0				1.00000i	$0.5\pi$
$380$		0			0
$381$	−15246.0			−2.05007
$382$		0			0
$383$	8113.00	−	8113.00i	1.08239	−	1.08239i	0.0861026	−	0.996286i	$-0.472559\pi$
$383$	8113.00	−	8113.00i	1.08239	−	1.08239i	0.996286	−	0.0861026i	$-0.0274413\pi$
$384$			14336.0i			1.90516i
$385$		0			0
$386$		0			0
$387$	21087.0	+	21087.0i	2.76980	+	2.76980i
$388$		0			0
$389$		−	14184.0i		−	1.84873i	−0.381505	−	0.924367i	$-0.624594\pi$
$389$		−	14184.0i		−	1.84873i	0.381505	−	0.924367i	$-0.375406\pi$
$390$		0			0
$391$		0			0
$392$	2896.00	+	2896.00i	0.373138	+	0.373138i
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$397$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	15822.0			1.97036			0.985178	−	0.171534i	$-0.0548722\pi$
$401$	15822.0			1.97036			0.985178	+	0.171534i	$0.0548722\pi$
$402$	−15372.0	−	15372.0i	−1.90718	−	1.90718i
$403$		0			0
$404$		−	3024.00i		−	0.372400i
$405$		0			0
$406$	−11016.0			−1.34659
$407$		0			0
$408$		0			0
$409$		−	4844.00i		−	0.585624i	−0.956170	−	0.292812i	$-0.905409\pi$
$409$		−	4844.00i		−	0.585624i	0.956170	−	0.292812i	$-0.0945911\pi$
$410$		0			0
$411$		0			0
$412$	10584.0	+	10584.0i	1.26562	+	1.26562i
$413$		0			0
$414$			19028.0i			2.25888i
$415$		0			0
$416$		0			0
$417$		0			0
$418$		0			0
$419$		0			0			−	1.00000i	$-0.5\pi$
$419$		0			0				1.00000i	$0.5\pi$
$420$		0			0
$421$	9592.00			1.11042			0.555208	−	0.831711i	$-0.312638\pi$
$421$	9592.00			1.11042			0.555208	+	0.831711i	$0.312638\pi$
$422$		0			0
$423$	−21371.0	+	21371.0i	−2.45649	+	2.45649i
$424$		0			0
$425$		0			0
$426$		0			0
$427$	8568.00	+	8568.00i	0.971041	+	0.971041i
$428$	1768.00	−	1768.00i	0.199672	−	0.199672i
$429$		0			0
$430$		0			0
$431$		0			0		1.00000			$0$
$431$		0			0		−1.00000			$\pi$
$432$	−19712.0	−	19712.0i	−2.19536	−	2.19536i
$433$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$433$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$434$		0			0
$435$		0			0
$436$	−9088.00			−0.998248
$437$		0			0
$438$		0			0
$439$		0			0			−	1.00000i	$-0.5\pi$
$439$		0			0				1.00000i	$0.5\pi$
$440$		0			0
$441$	−12851.0			−1.38765
$442$		0			0
$443$	−10237.0	+	10237.0i	−1.09791	+	1.09791i	−0.103256	+	0.994655i	$0.532926\pi$
$443$	−10237.0	+	10237.0i	−1.09791	+	1.09791i	−0.994655	+	0.103256i	$0.967074\pi$
$444$		0			0
$445$		0			0
$446$	17892.0			1.89958
$447$	−21672.0	−	21672.0i	−2.29318	−	2.29318i
$448$	4608.00	−	4608.00i	0.485954	−	0.485954i
$449$			1836.00i			0.192976i	0.995334	+	0.0964880i	$0.0307609\pi$
$449$			1836.00i			0.192976i	−0.995334	+	0.0964880i	$0.969239\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$		0			0
$454$		−	16324.0i		−	1.68750i
$455$		0			0
$456$		0			0
$457$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$457$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$458$	13748.0	−	13748.0i	1.40262	−	1.40262i
$459$		0			0
$460$		0			0
$461$	16002.0			1.61668			0.808338	−	0.588719i	$-0.200368\pi$
$461$	16002.0			1.61668			0.808338	+	0.588719i	$0.200368\pi$
$462$		0			0
$463$	11043.0	−	11043.0i	1.10845	−	1.10845i	0.115094	−	0.993355i	$-0.463283\pi$
$463$	11043.0	−	11043.0i	1.10845	−	1.10845i	0.993355	−	0.115094i	$-0.0367169\pi$
$464$		−	19584.0i		−	1.95941i
$465$		0			0
$466$		0			0
$467$	7889.00	+	7889.00i	0.781712	+	0.781712i	0.980120	−	0.198408i	$-0.0635770\pi$
$467$	7889.00	+	7889.00i	0.781712	+	0.781712i	−0.198408	+	0.980120i	$0.563577\pi$
$468$		0			0
$469$			9882.00i			0.972939i
$470$		0			0
$471$		0			0
$472$		0			0
$473$		0			0
$474$		0			0
$475$		0			0
$476$		0			0
$477$		0			0
$478$		0			0
$479$		0			0			−	1.00000i	$-0.5\pi$
$479$		0			0				1.00000i	$0.5\pi$
$480$		0			0
$481$		0			0
$482$	−3416.00	−	3416.00i	−0.322810	−	0.322810i
$483$	8442.00	−	8442.00i	0.795288	−	0.795288i
$484$			10648.0i			1.00000i
$485$		0			0
$486$	33796.0			3.15436
$487$	−81.0000	−	81.0000i	−0.00753688	−	0.00753688i	0.703328	−	0.710865i	$-0.251696\pi$
$487$	−81.0000	−	81.0000i	−0.00753688	−	0.00753688i	−0.710865	+	0.703328i	$0.751696\pi$
$488$	−15232.0	+	15232.0i	−1.41295	+	1.41295i
$489$			41202.0i			3.81026i
$490$		0			0
$491$		0			0		1.00000			$0$
$491$		0			0		−1.00000			$\pi$
$492$	−14112.0	−	14112.0i	−1.29313	−	1.29313i
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$	2156.00	−	2156.00i	0.194001	−	0.194001i
$499$		0			0			−	1.00000i	$-0.5\pi$
$499$		0			0				1.00000i	$0.5\pi$
$500$		0			0
$501$	−18326.0			−1.63422
$502$		0			0
$503$	12803.0	−	12803.0i	1.13491	−	1.13491i	0.145556	−	0.989350i	$-0.453503\pi$
$503$	12803.0	−	12803.0i	1.13491	−	1.13491i	0.989350	−	0.145556i	$-0.0464970\pi$
$504$			20448.0i			1.80720i
$505$		0			0
$506$		0			0
$507$	15379.0	+	15379.0i	1.34715	+	1.34715i
$508$	−8712.00	+	8712.00i	−0.760891	+	0.760891i
$509$			8946.00i			0.779026i	0.921021	+	0.389513i	$0.127357\pi$
$509$			8946.00i			0.779026i	−0.921021	+	0.389513i	$0.872643\pi$
$510$		0			0
$511$		0			0
$512$	8192.00	+	8192.00i	0.707107	+	0.707107i
$513$		0			0
$514$		0			0
$515$		0			0
$516$	33264.0			2.83792
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$	8442.00			0.709886			0.354943	−	0.934888i	$-0.384500\pi$
$521$	8442.00			0.709886			0.354943	+	0.934888i	$0.384500\pi$
$522$	43452.0	+	43452.0i	3.64338	+	3.64338i
$523$	−14427.0	+	14427.0i	−1.20621	+	1.20621i	−0.233967	+	0.972245i	$0.575171\pi$
$523$	−14427.0	+	14427.0i	−1.20621	+	1.20621i	−0.972245	+	0.233967i	$0.924829\pi$
$524$		0			0
$525$		0			0
$526$	−24068.0			−1.99508
$527$		0			0
$528$		0			0
$529$			3189.00i			0.262102i
$530$		0			0
$531$		0			0
$532$		0			0
$533$		0			0
$534$		−	38808.0i		−	3.14492i
$535$		0			0
$536$	−17568.0			−1.41571
$537$		0			0
$538$	11088.0	−	11088.0i	0.888546	−	0.888546i
$539$		0			0
$540$		0			0
$541$	6802.00			0.540556			0.270278	−	0.962782i	$-0.412884\pi$
$541$	6802.00			0.540556			0.270278	+	0.962782i	$0.412884\pi$
$542$		0			0
$543$	7546.00	−	7546.00i	0.596372	−	0.596372i
$544$		0			0
$545$		0			0
$546$		0			0
$547$	−17721.0	−	17721.0i	−1.38518	−	1.38518i	−0.835129	−	0.550055i	$-0.814607\pi$
$547$	−17721.0	−	17721.0i	−1.38518	−	1.38518i	−0.550055	−	0.835129i	$-0.685393\pi$
$548$		0			0
$549$		−	67592.0i		−	5.25457i
$550$		0			0
$551$		0			0
$552$	15008.0	+	15008.0i	1.15722	+	1.15722i
$553$		0			0
$554$		0			0
$555$		0			0
$556$		0			0
$557$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$557$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$	−16776.0	−	16776.0i	−1.25917	−	1.25917i
$563$	−1687.00	+	1687.00i	−0.126285	+	0.126285i	−0.767425	−	0.641139i	$-0.778462\pi$
$563$	−1687.00	+	1687.00i	−0.126285	+	0.126285i	0.641139	+	0.767425i	$0.278462\pi$
$564$			33712.0i			2.51690i
$565$		0			0
$566$	2772.00			0.205858
$567$	−21555.0	−	21555.0i	−1.59652	−	1.59652i
$568$		0			0
$569$			14796.0i			1.09012i	0.838396	+	0.545062i	$0.183494\pi$
$569$			14796.0i			1.09012i	−0.838396	+	0.545062i	$0.816506\pi$
$570$		0			0
$571$		0			0		1.00000			$0$
$571$		0			0		−1.00000			$\pi$
$572$		0			0
$573$		0			0
$574$			9072.00i			0.659683i
$575$		0			0
$576$	−36352.0			−2.62963
$577$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$577$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$578$	−9826.00	+	9826.00i	−0.707107	+	0.707107i
$579$		0			0
$580$		0			0
$581$	−1386.00			−0.0989690
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−11641.0	−	11641.0i	−0.818527	−	0.818527i	0.167367	−	0.985895i	$-0.446473\pi$
$587$	−11641.0	−	11641.0i	−0.818527	−	0.818527i	−0.985895	+	0.167367i	$0.946473\pi$
$588$	−10136.0	+	10136.0i	−0.710887	+	0.710887i
$589$		0			0
$590$		0			0
$591$		0			0
$592$		0			0
$593$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$593$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$594$		0			0
$595$		0			0
$596$	−24768.0			−1.70224
$597$		0			0
$598$		0			0
$599$		0			0			−	1.00000i	$-0.5\pi$
$599$		0			0				1.00000i	$0.5\pi$
$600$		0			0
$601$	28532.0			1.93651			0.968257	−	0.249958i	$-0.0804168\pi$
$601$	28532.0			1.93651			0.968257	+	0.249958i	$0.0804168\pi$
$602$	−10692.0	−	10692.0i	−0.723876	−	0.723876i
$603$	38979.0	−	38979.0i	2.63242	−	2.63242i
$604$		0			0
$605$		0			0
$606$	10584.0			0.709481
$607$	−11781.0	−	11781.0i	−0.787769	−	0.787769i	0.193359	−	0.981128i	$-0.438062\pi$
$607$	−11781.0	−	11781.0i	−0.787769	−	0.787769i	−0.981128	+	0.193359i	$0.938062\pi$
$608$		0			0
$609$		−	38556.0i		−	2.56546i
$610$		0			0
$611$		0			0
$612$		0			0
$613$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$613$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$614$			10836.0i			0.712224i
$615$		0			0
$616$		0			0
$617$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$617$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$618$	−37044.0	+	37044.0i	−2.41121	+	2.41121i
$619$		0			0			−	1.00000i	$-0.5\pi$
$619$		0			0				1.00000i	$0.5\pi$
$620$		0			0
$621$	−41272.0			−2.66697
$622$		0			0
$623$	−12474.0	+	12474.0i	−0.802183	+	0.802183i
$624$		0			0
$625$		0			0
$626$		0			0
$627$		0			0
$628$		0			0
$629$		0			0
$630$		0			0
$631$		0			0		1.00000			$0$
$631$		0			0		−1.00000			$\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		0			0
$636$		0			0
$637$		0			0
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−21348.0			−1.31544			−0.657719	−	0.753264i	$-0.728478\pi$
$641$	−21348.0			−1.31544			−0.657719	+	0.753264i	$0.728478\pi$
$642$	6188.00	+	6188.00i	0.380406	+	0.380406i
$643$	−21987.0	+	21987.0i	−1.34850	+	1.34850i	−0.461199	+	0.887297i	$0.652580\pi$
$643$	−21987.0	+	21987.0i	−1.34850	+	1.34850i	−0.887297	+	0.461199i	$0.847420\pi$
$644$		−	9648.00i		−	0.590349i
$645$		0			0
$646$		0			0
$647$	20839.0	+	20839.0i	1.26625	+	1.26625i	0.948009	+	0.318244i	$0.103093\pi$
$647$	20839.0	+	20839.0i	1.26625	+	1.26625i	0.318244	+	0.948009i	$0.396907\pi$
$648$	38320.0	−	38320.0i	2.32307	−	2.32307i
$649$		0			0
$650$		0			0
$651$		0			0
$652$	23544.0	+	23544.0i	1.41419	+	1.41419i
$653$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$653$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$654$		−	31808.0i		−	1.90182i
$655$		0			0
$656$	−16128.0			−0.959897
$657$		0			0
$658$	10836.0	−	10836.0i	0.641993	−	0.641993i
$659$		0			0			−	1.00000i	$-0.5\pi$
$659$		0			0				1.00000i	$0.5\pi$
$660$		0			0
$661$	−30688.0			−1.80579			−0.902893	−	0.429865i	$-0.858561\pi$
$661$	−30688.0			−1.80579			−0.902893	+	0.429865i	$0.858561\pi$
$662$		0			0
$663$		0			0
$664$		−	2464.00i		−	0.144009i
$665$		0			0
$666$		0			0
$667$	−20502.0	−	20502.0i	−1.19017	−	1.19017i
$668$	−10472.0	+	10472.0i	−0.606548	+	0.606548i
$669$			62622.0i			3.61899i
$670$		0			0
$671$		0			0
$672$	16128.0	+	16128.0i	0.925820	+	0.925820i
$673$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$673$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$674$		0			0
$675$		0			0
$676$	17576.0			1.00000
$677$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$677$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$	57134.0			3.21495
$682$		0			0
$683$	−3937.00	+	3937.00i	−0.220564	+	0.220564i	−0.808736	−	0.588172i	$-0.799848\pi$
$683$	−3937.00	+	3937.00i	−0.220564	+	0.220564i	0.588172	+	0.808736i	$0.299848\pi$
$684$		0			0
$685$		0			0
$686$	18864.0			1.04990
$687$	48118.0	+	48118.0i	2.67222	+	2.67222i
$688$	19008.0	−	19008.0i	1.05330	−	1.05330i
$689$		0			0
$690$		0			0
$691$		0			0		1.00000			$0$
$691$		0			0		−1.00000			$\pi$
$692$		0			0
$693$		0			0
$694$		−	31484.0i		−	1.72207i
$695$		0			0
$696$	68544.0			3.73298
$697$		0			0
$698$	−19292.0	+	19292.0i	−1.04615	+	1.04615i
$699$		0			0
$700$		0			0
$701$	−9648.00			−0.519829			−0.259914	−	0.965632i	$-0.583694\pi$
$701$	−9648.00			−0.519829			−0.259914	+	0.965632i	$0.583694\pi$
$702$		0			0
$703$		0			0
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−3402.00	−	3402.00i	−0.180969	−	0.180969i
$708$		0			0
$709$			506.000i			0.0268029i	0.999910	+	0.0134014i	$0.00426594\pi$
$709$			506.000i			0.0268029i	−0.999910	+	0.0134014i	$0.995734\pi$
$710$		0			0
$711$		0			0
$712$	−22176.0	−	22176.0i	−1.16725	−	1.16725i
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$		0			0			−	1.00000i	$-0.5\pi$
$719$		0			0				1.00000i	$0.5\pi$
$720$		0			0
$721$	23814.0			1.23007
$722$	−13718.0	−	13718.0i	−0.707107	−	0.707107i
$723$	11956.0	−	11956.0i	0.615005	−	0.615005i
$724$		−	8624.00i		−	0.442691i
$725$		0			0
$726$	−37268.0			−1.90516
$727$	6489.00	+	6489.00i	0.331037	+	0.331037i	0.852980	−	0.521943i	$-0.174793\pi$
$727$	6489.00	+	6489.00i	0.331037	+	0.331037i	−0.521943	+	0.852980i	$0.674793\pi$
$728$		0			0
$729$			53621.0i			2.72423i
$730$		0			0
$731$		0			0
$732$	−53312.0	−	53312.0i	−2.69190	−	2.69190i
$733$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$733$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$734$			38556.0i			1.93887i
$735$		0			0
$736$	17152.0			0.859010
$737$		0			0
$738$	35784.0	−	35784.0i	1.78486	−	1.78486i
$739$		0			0			−	1.00000i	$-0.5\pi$
$739$		0			0				1.00000i	$0.5\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	28633.0	−	28633.0i	1.41379	−	1.41379i	0.689504	−	0.724282i	$-0.257829\pi$
$743$	28633.0	−	28633.0i	1.41379	−	1.41379i	0.724282	−	0.689504i	$-0.242171\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$	5467.00	+	5467.00i	0.267774	+	0.267774i
$748$		0			0
$749$		−	3978.00i		−	0.194063i
$750$		0			0
$751$		0			0		1.00000			$0$
$751$		0			0		−1.00000			$\pi$
$752$	19264.0	+	19264.0i	0.934157	+	0.934157i
$753$		0			0
$754$		0			0
$755$		0			0
$756$	−44352.0			−2.13369
$757$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$757$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−21798.0			−1.03834			−0.519170	−	0.854671i	$-0.673759\pi$
$761$	−21798.0			−1.03834			−0.519170	+	0.854671i	$0.673759\pi$
$762$	−30492.0	−	30492.0i	−1.44962	−	1.44962i
$763$	−10224.0	+	10224.0i	−0.485103	+	0.485103i
$764$		0			0
$765$		0			0
$766$	32452.0			1.53073
$767$		0			0
$768$	−28672.0	+	28672.0i	−1.34715	+	1.34715i
$769$		−	29554.0i		−	1.38588i	−0.720994	−	0.692942i	$-0.756314\pi$
$769$		−	29554.0i		−	1.38588i	0.720994	−	0.692942i	$-0.243686\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$773$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$774$			84348.0i			3.91709i
$775$		0			0
$776$		0			0
$777$		0			0
$778$	28368.0	−	28368.0i	1.30725	−	1.30725i
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$	−94248.0	+	94248.0i	−4.30159	+	4.30159i
$784$			11584.0i			0.527697i
$785$		0			0
$786$		0			0
$787$	−24381.0	−	24381.0i	−1.10431	−	1.10431i	−0.993885	−	0.110421i	$-0.964780\pi$
$787$	−24381.0	−	24381.0i	−1.10431	−	1.10431i	−0.110421	−	0.993885i	$-0.535220\pi$
$788$		0			0
$789$		−	84238.0i		−	3.80095i
$790$		0			0
$791$		0			0
$792$		0			0
$793$		0			0
$794$		0			0
$795$		0			0
$796$		0			0
$797$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$797$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$	98406.0			4.34083
$802$	31644.0	+	31644.0i	1.39325	+	1.39325i
$803$		0			0
$804$		−	61488.0i		−	2.69716i
$805$		0			0
$806$		0			0
$807$	38808.0	+	38808.0i	1.69282	+	1.69282i
$808$	6048.00	−	6048.00i	0.263327	−	0.263327i
$809$			26406.0i			1.14757i	0.819005	+	0.573786i	$0.194526\pi$
$809$			26406.0i			1.14757i	−0.819005	+	0.573786i	$0.805474\pi$
$810$		0			0
$811$		0			0		1.00000			$0$
$811$		0			0		−1.00000			$\pi$
$812$	−22032.0	−	22032.0i	−0.952182	−	0.952182i
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$		0			0
$818$	9688.00	−	9688.00i	0.414099	−	0.414099i
$819$		0			0
$820$		0			0
$821$	7632.00			0.324432			0.162216	−	0.986755i	$-0.448136\pi$
$821$	7632.00			0.324432			0.162216	+	0.986755i	$0.448136\pi$
$822$		0			0
$823$	−22347.0	+	22347.0i	−0.946498	+	0.946498i	−0.998640	−	0.0521422i	$-0.983395\pi$
$823$	−22347.0	+	22347.0i	−0.946498	+	0.946498i	0.0521422	+	0.998640i	$0.483395\pi$
$824$			42336.0i			1.78986i
$825$		0			0
$826$		0			0
$827$	32729.0	+	32729.0i	1.37618	+	1.37618i	0.850981	+	0.525197i	$0.176008\pi$
$827$	32729.0	+	32729.0i	1.37618	+	1.37618i	0.525197	+	0.850981i	$0.323992\pi$
$828$	−38056.0	+	38056.0i	−1.59727	+	1.59727i
$829$		−	36344.0i		−	1.52265i	−0.648369	−	0.761326i	$-0.724548\pi$
$829$		−	36344.0i		−	1.52265i	0.648369	−	0.761326i	$-0.275452\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		0			0
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$		0			0
$839$		0			0			−	1.00000i	$-0.5\pi$
$839$		0			0				1.00000i	$0.5\pi$
$840$		0			0
$841$	−69247.0			−2.83927
$842$	19184.0	+	19184.0i	0.785183	+	0.785183i
$843$	58716.0	−	58716.0i	2.39892	−	2.39892i
$844$		0			0
$845$		0			0
$846$	−85484.0			−3.47400
$847$	11979.0	+	11979.0i	0.485954	+	0.485954i
$848$		0			0
$849$			9702.00i			0.392193i
$850$		0			0
$851$		0			0
$852$		0			0
$853$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$853$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$854$			34272.0i			1.37326i
$855$		0			0
$856$	7072.00			0.282378
$857$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$857$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$858$		0			0
$859$		0			0			−	1.00000i	$-0.5\pi$
$859$		0			0				1.00000i	$0.5\pi$
$860$		0			0
$861$	−31752.0			−1.25680
$862$		0			0
$863$	31693.0	−	31693.0i	1.25011	−	1.25011i	0.294436	−	0.955671i	$-0.404868\pi$
$863$	31693.0	−	31693.0i	1.25011	−	1.25011i	0.955671	−	0.294436i	$-0.0951316\pi$
$864$		−	78848.0i		−	3.10470i
$865$		0			0
$866$		0			0
$867$	−34391.0	−	34391.0i	−1.34715	−	1.34715i
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$	−18176.0	−	18176.0i	−0.705868	−	0.705868i
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$877$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−29988.0			−1.14679			−0.573395	−	0.819279i	$-0.694374\pi$
$881$	−29988.0			−1.14679			−0.573395	+	0.819279i	$0.694374\pi$
$882$	−25702.0	−	25702.0i	−0.981215	−	0.981215i
$883$	−15327.0	+	15327.0i	−0.584139	+	0.584139i	−0.936038	−	0.351899i	$-0.885536\pi$
$883$	−15327.0	+	15327.0i	−0.584139	+	0.584139i	0.351899	+	0.936038i	$0.385536\pi$
$884$		0			0
$885$		0			0
$886$	−40948.0			−1.55268
$887$	2849.00	+	2849.00i	0.107847	+	0.107847i	0.758971	−	0.651124i	$-0.225702\pi$
$887$	2849.00	+	2849.00i	0.107847	+	0.107847i	−0.651124	+	0.758971i	$0.725702\pi$
$888$		0			0
$889$			19602.0i			0.739516i
$890$		0			0
$891$		0			0
$892$	35784.0	+	35784.0i	1.34320	+	1.34320i
$893$		0			0
$894$		−	86688.0i		−	3.24304i
$895$		0			0
$896$	18432.0			0.687243
$897$		0			0
$898$	−3672.00	+	3672.00i	−0.136455	+	0.136455i
$899$		0			0
$900$		0			0
$901$		0			0
$902$		0			0
$903$	37422.0	−	37422.0i	1.37910	−	1.37910i
$904$		0			0
$905$		0			0
$906$		0			0
$907$	8109.00	+	8109.00i	0.296863	+	0.296863i	0.839784	−	0.542921i	$-0.182682\pi$
$907$	8109.00	+	8109.00i	0.296863	+	0.296863i	−0.542921	+	0.839784i	$0.682682\pi$
$908$	32648.0	−	32648.0i	1.19324	−	1.19324i
$909$			26838.0i			0.979274i
$910$		0			0
$911$		0			0		1.00000			$0$
$911$		0			0		−1.00000			$\pi$
$912$		0			0
$913$		0			0
$914$		0			0
$915$		0			0
$916$	54992.0			1.98361
$917$		0			0
$918$		0			0
$919$		0			0			−	1.00000i	$-0.5\pi$
$919$		0			0				1.00000i	$0.5\pi$
$920$		0			0
$921$	−37926.0			−1.35690
$922$	32004.0	+	32004.0i	1.14316	+	1.14316i
$923$		0			0
$924$		0			0
$925$		0			0
$926$	44172.0			1.56758
$927$	−93933.0	−	93933.0i	−3.32812	−	3.32812i
$928$	39168.0	−	39168.0i	1.38551	−	1.38551i
$929$			53676.0i			1.89564i	0.318801	+	0.947822i	$0.396720\pi$
$929$			53676.0i			1.89564i	−0.318801	+	0.947822i	$0.603280\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$		0			0
$934$			31556.0i			1.10551i
$935$		0			0
$936$		0			0
$937$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$937$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$938$	−19764.0	+	19764.0i	−0.687972	+	0.687972i
$939$		0			0
$940$		0			0
$941$	−44478.0			−1.54085			−0.770426	−	0.637530i	$-0.779956\pi$
$941$	−44478.0			−1.54085			−0.770426	+	0.637530i	$0.779956\pi$
$942$		0			0
$943$	−16884.0	+	16884.0i	−0.583053	+	0.583053i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−36851.0	−	36851.0i	−1.26452	−	1.26452i	−0.948881	−	0.315635i	$-0.897782\pi$
$947$	−36851.0	−	36851.0i	−1.26452	−	1.26452i	−0.315635	−	0.948881i	$-0.602218\pi$
$948$		0			0
$949$		0			0
$950$		0			0
$951$		0			0
$952$		0			0
$953$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$953$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	29791.0			1.00000
$962$		0			0
$963$	−15691.0	+	15691.0i	−0.525063	+	0.525063i
$964$		−	13664.0i		−	0.456523i
$965$		0			0
$966$	33768.0			1.12471
$967$	23859.0	+	23859.0i	0.793437	+	0.793437i	0.982051	−	0.188614i	$-0.0603995\pi$
$967$	23859.0	+	23859.0i	0.793437	+	0.793437i	−0.188614	+	0.982051i	$0.560400\pi$
$968$	−21296.0	+	21296.0i	−0.707107	+	0.707107i
$969$		0			0
$970$		0			0
$971$		0			0		1.00000			$0$
$971$		0			0		−1.00000			$\pi$
$972$	67592.0	+	67592.0i	2.23047	+	2.23047i
$973$		0			0
$974$		−	324.000i		−	0.0106588i
$975$		0			0
$976$	−60928.0			−1.99821
$977$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$977$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$978$	−82404.0	+	82404.0i	−2.69426	+	2.69426i
$979$		0			0
$980$		0			0
$981$	80656.0			2.62502
$982$		0			0
$983$	36743.0	−	36743.0i	1.19219	−	1.19219i	0.215735	−	0.976452i	$-0.430785\pi$
$983$	36743.0	−	36743.0i	1.19219	−	1.19219i	0.976452	−	0.215735i	$-0.0692147\pi$
$984$		−	56448.0i		−	1.82876i
$985$		0			0
$986$		0			0
$987$	37926.0	+	37926.0i	1.22310	+	1.22310i
$988$		0			0
$989$		−	39798.0i		−	1.27958i
$990$		0			0
$991$		0			0		1.00000			$0$
$991$		0			0		−1.00000			$\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		0			0
$996$	8624.00			0.274359
$997$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$997$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	100.4.e.c.43.1	yes	2
4.3	odd	2		100.4.e.b.43.1	yes	2
5.2	odd	4		100.4.e.b.7.1	✓	2
5.3	odd	4	inner	100.4.e.c.7.1	yes	2
5.4	even	2		100.4.e.b.43.1	yes	2
20.3	even	4		100.4.e.b.7.1	✓	2
20.7	even	4	inner	100.4.e.c.7.1	yes	2
20.19	odd	2	CM	100.4.e.c.43.1	yes	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
100.4.e.b.7.1	✓	2	5.2	odd	4
100.4.e.b.7.1	✓	2	20.3	even	4
100.4.e.b.43.1	yes	2	4.3	odd	2
100.4.e.b.43.1	yes	2	5.4	even	2
100.4.e.c.7.1	yes	2	5.3	odd	4	inner
100.4.e.c.7.1	yes	2	20.7	even	4	inner
100.4.e.c.43.1	yes	2	1.1	even	1	trivial
100.4.e.c.43.1	yes	2	20.19	odd	2	CM

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:	\( N \)	\(=\)	\( 100 = 2^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	100.e (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(2.00000 + 2.00000i) q^{2} +(-7.00000 + 7.00000i) q^{3} +8.00000i q^{4} -28.0000 q^{6} +(9.00000 + 9.00000i) q^{7} +(-16.0000 + 16.0000i) q^{8} -71.0000i q^{9} +O(q^{10})\)	\(q+(2.00000 + 2.00000i) q^{2} +(-7.00000 + 7.00000i) q^{3} +8.00000i q^{4} -28.0000 q^{6} +(9.00000 + 9.00000i) q^{7} +(-16.0000 + 16.0000i) q^{8} -71.0000i q^{9} +(-56.0000 - 56.0000i) q^{12} +36.0000i q^{14} -64.0000 q^{16} +(142.000 - 142.000i) q^{18} -126.000 q^{21} +(-67.0000 + 67.0000i) q^{23} -224.000i q^{24} +(308.000 + 308.000i) q^{27} +(-72.0000 + 72.0000i) q^{28} +306.000i q^{29} +(-128.000 - 128.000i) q^{32} +568.000 q^{36} +252.000 q^{41} +(-252.000 - 252.000i) q^{42} +(-297.000 + 297.000i) q^{43} -268.000 q^{46} +(-301.000 - 301.000i) q^{47} +(448.000 - 448.000i) q^{48} -181.000i q^{49} +1232.00i q^{54} -288.000 q^{56} +(-612.000 + 612.000i) q^{58} +952.000 q^{61} +(639.000 - 639.000i) q^{63} -512.000i q^{64} +(549.000 + 549.000i) q^{67} -938.000i q^{69} +(1136.00 + 1136.00i) q^{72} -2395.00 q^{81} +(504.000 + 504.000i) q^{82} +(-77.0000 + 77.0000i) q^{83} -1008.00i q^{84} -1188.00 q^{86} +(-2142.00 - 2142.00i) q^{87} +1386.00i q^{89} +(-536.000 - 536.000i) q^{92} -1204.00i q^{94} +1792.00 q^{96} +(362.000 - 362.000i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 4 q^{2} - 14 q^{3} - 56 q^{6} + 18 q^{7} - 32 q^{8}+O(q^{10}) \) 2 * q + 4 * q^2 - 14 * q^3 - 56 * q^6 + 18 * q^7 - 32 * q^8	\( 2 q + 4 q^{2} - 14 q^{3} - 56 q^{6} + 18 q^{7} - 32 q^{8} - 112 q^{12} - 128 q^{16} + 284 q^{18} - 252 q^{21} - 134 q^{23} + 616 q^{27} - 144 q^{28} - 256 q^{32} + 1136 q^{36} + 504 q^{41} - 504 q^{42} - 594 q^{43} - 536 q^{46} - 602 q^{47} + 896 q^{48} - 576 q^{56} - 1224 q^{58} + 1904 q^{61} + 1278 q^{63} + 1098 q^{67} + 2272 q^{72} - 4790 q^{81} + 1008 q^{82} - 154 q^{83} - 2376 q^{86} - 4284 q^{87} - 1072 q^{92} + 3584 q^{96} + 724 q^{98}+O(q^{100}) \) 2 * q + 4 * q^2 - 14 * q^3 - 56 * q^6 + 18 * q^7 - 32 * q^8 - 112 * q^12 - 128 * q^16 + 284 * q^18 - 252 * q^21 - 134 * q^23 + 616 * q^27 - 144 * q^28 - 256 * q^32 + 1136 * q^36 + 504 * q^41 - 504 * q^42 - 594 * q^43 - 536 * q^46 - 602 * q^47 + 896 * q^48 - 576 * q^56 - 1224 * q^58 + 1904 * q^61 + 1278 * q^63 + 1098 * q^67 + 2272 * q^72 - 4790 * q^81 + 1008 * q^82 - 154 * q^83 - 2376 * q^86 - 4284 * q^87 - 1072 * q^92 + 3584 * q^96 + 724 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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