LMFDB - Embedded newform 75.2.e.a.32.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [75,2,Mod(32,75)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(75, base_ring=CyclotomicField(4))

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

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

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

chi := DirichletCharacter("75.32");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 75 = 3 \cdot 5^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	75.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:	$0.598878015160$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(i)$
Coefficient field:	$\Q(i, \sqrt{6})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 9 $ x^4 + 9
Coefficient ring:	$\Z[a_1, \ldots, a_{4}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{4}]$

Embedding invariants

Embedding label			32.1
Root			$-1.22474 + 1.22474i$ of defining polynomial
Character	$\chi$	$=$	75.32
Dual form			75.2.e.a.68.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.22474 + 1.22474i) q^{2} +(1.22474 + 1.22474i) q^{3} -1.00000i q^{4} -3.00000 q^{6} +(-1.22474 - 1.22474i) q^{8} +3.00000i q^{9} +O(q^{10})$	$q+(-1.22474 + 1.22474i) q^{2} +(1.22474 + 1.22474i) q^{3} -1.00000i q^{4} -3.00000 q^{6} +(-1.22474 - 1.22474i) q^{8} +3.00000i q^{9} +(1.22474 - 1.22474i) q^{12} +5.00000 q^{16} +(4.89898 - 4.89898i) q^{17} +(-3.67423 - 3.67423i) q^{18} -4.00000i q^{19} +(2.44949 + 2.44949i) q^{23} -3.00000i q^{24} +(-3.67423 + 3.67423i) q^{27} -8.00000 q^{31} +(-3.67423 + 3.67423i) q^{32} +12.0000i q^{34} +3.00000 q^{36} +(4.89898 + 4.89898i) q^{38} -6.00000 q^{46} +(-7.34847 + 7.34847i) q^{47} +(6.12372 + 6.12372i) q^{48} -7.00000i q^{49} +12.0000 q^{51} +(-9.79796 - 9.79796i) q^{53} -9.00000i q^{54} +(4.89898 - 4.89898i) q^{57} +2.00000 q^{61} +(9.79796 - 9.79796i) q^{62} +1.00000i q^{64} +(-4.89898 - 4.89898i) q^{68} +6.00000i q^{69} +(3.67423 - 3.67423i) q^{72} -4.00000 q^{76} +16.0000i q^{79} -9.00000 q^{81} +(2.44949 + 2.44949i) q^{83} +(2.44949 - 2.44949i) q^{92} +(-9.79796 - 9.79796i) q^{93} -18.0000i q^{94} -9.00000 q^{96} +(8.57321 + 8.57321i) q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 12 q^{6}+O(q^{10}) $ 4 * q - 12 * q^6	$ 4 q - 12 q^{6} + 20 q^{16} - 32 q^{31} + 12 q^{36} - 24 q^{46} + 48 q^{51} + 8 q^{61} - 16 q^{76} - 36 q^{81} - 36 q^{96}+O(q^{100}) $ 4 * q - 12 * q^6 + 20 * q^16 - 32 * q^31 + 12 * q^36 - 24 * q^46 + 48 * q^51 + 8 * q^61 - 16 * q^76 - 36 * q^81 - 36 * q^96

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/75\mathbb{Z}\right)^\times$.

$n$	$26$	$52$
$\chi(n)$	$-1$	$e\left(\frac{1}{4}\right)$

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 $

$2$	−1.22474	+	1.22474i	−0.866025	+	0.866025i	−0.992030	−	0.126004i	$-0.959785\pi$
$2$	−1.22474	+	1.22474i	−0.866025	+	0.866025i	0.126004	+	0.992030i	$0.459785\pi$
$3$	1.22474	+	1.22474i	0.707107	+	0.707107i
$3$	1.22474	+	1.22474i	0.707107	+	0.707107i
$4$		−	1.00000i		−	0.500000i
$5$		0			0
$5$		0			0
$6$	−3.00000			−1.22474
$7$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$7$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$8$	−1.22474	−	1.22474i	−0.433013	−	0.433013i
$9$			3.00000i			1.00000i
$10$		0			0
$11$		0			0		1.00000			$0$
$11$		0			0		−1.00000			$\pi$
$12$	1.22474	−	1.22474i	0.353553	−	0.353553i
$13$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$13$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$14$		0			0
$15$		0			0
$16$	5.00000			1.25000
$17$	4.89898	−	4.89898i	1.18818	−	1.18818i	0.210606	−	0.977571i	$-0.432456\pi$
$17$	4.89898	−	4.89898i	1.18818	−	1.18818i	0.977571	−	0.210606i	$-0.0675437\pi$
$18$	−3.67423	−	3.67423i	−0.866025	−	0.866025i
$19$		−	4.00000i		−	0.917663i	−0.888523	−	0.458831i	$-0.848268\pi$
$19$		−	4.00000i		−	0.917663i	0.888523	−	0.458831i	$-0.151732\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	2.44949	+	2.44949i	0.510754	+	0.510754i	0.914757	−	0.404004i	$-0.132382\pi$
$23$	2.44949	+	2.44949i	0.510754	+	0.510754i	−0.404004	+	0.914757i	$0.632382\pi$
$24$		−	3.00000i		−	0.612372i
$25$		0			0
$26$		0			0
$27$	−3.67423	+	3.67423i	−0.707107	+	0.707107i
$28$		0			0
$29$		0			0			−	1.00000i	$-0.5\pi$
$29$		0			0				1.00000i	$0.5\pi$
$30$		0			0
$31$	−8.00000			−1.43684			−0.718421	−	0.695608i	$-0.755135\pi$
$31$	−8.00000			−1.43684			−0.718421	+	0.695608i	$0.755135\pi$
$32$	−3.67423	+	3.67423i	−0.649519	+	0.649519i
$33$		0			0
$34$			12.0000i			2.05798i
$35$		0			0
$36$	3.00000			0.500000
$37$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$37$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$38$	4.89898	+	4.89898i	0.794719	+	0.794719i
$39$		0			0
$40$		0			0
$41$		0			0		1.00000			$0$
$41$		0			0		−1.00000			$\pi$
$42$		0			0
$43$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$43$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$44$		0			0
$45$		0			0
$46$	−6.00000			−0.884652
$47$	−7.34847	+	7.34847i	−1.07188	+	1.07188i	−0.0746766	+	0.997208i	$0.523792\pi$
$47$	−7.34847	+	7.34847i	−1.07188	+	1.07188i	−0.997208	+	0.0746766i	$0.976208\pi$
$48$	6.12372	+	6.12372i	0.883883	+	0.883883i
$49$		−	7.00000i		−	1.00000i
$50$		0			0
$51$	12.0000			1.68034
$52$		0			0
$53$	−9.79796	−	9.79796i	−1.34585	−	1.34585i	−0.890113	−	0.455740i	$-0.849375\pi$
$53$	−9.79796	−	9.79796i	−1.34585	−	1.34585i	−0.455740	−	0.890113i	$-0.650625\pi$
$54$		−	9.00000i		−	1.22474i
$55$		0			0
$56$		0			0
$57$	4.89898	−	4.89898i	0.648886	−	0.648886i
$58$		0			0
$59$		0			0			−	1.00000i	$-0.5\pi$
$59$		0			0				1.00000i	$0.5\pi$
$60$		0			0
$61$	2.00000			0.256074			0.128037	−	0.991769i	$-0.459132\pi$
$61$	2.00000			0.256074			0.128037	+	0.991769i	$0.459132\pi$
$62$	9.79796	−	9.79796i	1.24434	−	1.24434i
$63$		0			0
$64$			1.00000i			0.125000i
$65$		0			0
$66$		0			0
$67$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$67$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$68$	−4.89898	−	4.89898i	−0.594089	−	0.594089i
$69$			6.00000i			0.722315i
$70$		0			0
$71$		0			0		1.00000			$0$
$71$		0			0		−1.00000			$\pi$
$72$	3.67423	−	3.67423i	0.433013	−	0.433013i
$73$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$73$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$74$		0			0
$75$		0			0
$76$	−4.00000			−0.458831
$77$		0			0
$78$		0			0
$79$			16.0000i			1.80014i	0.435745	+	0.900070i	$0.356485\pi$
$79$			16.0000i			1.80014i	−0.435745	+	0.900070i	$0.643515\pi$
$80$		0			0
$81$	−9.00000			−1.00000
$82$		0			0
$83$	2.44949	+	2.44949i	0.268866	+	0.268866i	0.828643	−	0.559777i	$-0.189113\pi$
$83$	2.44949	+	2.44949i	0.268866	+	0.268866i	−0.559777	+	0.828643i	$0.689113\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$		0			0			−	1.00000i	$-0.5\pi$
$89$		0			0				1.00000i	$0.5\pi$
$90$		0			0
$91$		0			0
$92$	2.44949	−	2.44949i	0.255377	−	0.255377i
$93$	−9.79796	−	9.79796i	−1.01600	−	1.01600i
$94$		−	18.0000i		−	1.85656i
$95$		0			0
$96$	−9.00000			−0.918559
$97$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$97$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$98$	8.57321	+	8.57321i	0.866025	+	0.866025i
$99$		0			0
$100$		0			0
$101$		0			0		1.00000			$0$
$101$		0			0		−1.00000			$\pi$
$102$	−14.6969	+	14.6969i	−1.45521	+	1.45521i
$103$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$103$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$104$		0			0
$105$		0			0
$106$	24.0000			2.33109
$107$	−7.34847	+	7.34847i	−0.710403	+	0.710403i	−0.966620	−	0.256216i	$-0.917524\pi$
$107$	−7.34847	+	7.34847i	−0.710403	+	0.710403i	0.256216	+	0.966620i	$0.417524\pi$
$108$	3.67423	+	3.67423i	0.353553	+	0.353553i
$109$		−	14.0000i		−	1.34096i	−0.741929	−	0.670478i	$-0.766089\pi$
$109$		−	14.0000i		−	1.34096i	0.741929	−	0.670478i	$-0.233911\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	14.6969	+	14.6969i	1.38257	+	1.38257i	0.840027	+	0.542545i	$0.182539\pi$
$113$	14.6969	+	14.6969i	1.38257	+	1.38257i	0.542545	+	0.840027i	$0.317461\pi$
$114$			12.0000i			1.12390i
$115$		0			0
$116$		0			0
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	11.0000			1.00000
$122$	−2.44949	+	2.44949i	−0.221766	+	0.221766i
$123$		0			0
$124$			8.00000i			0.718421i
$125$		0			0
$126$		0			0
$127$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$127$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$128$	−8.57321	−	8.57321i	−0.757772	−	0.757772i
$129$		0			0
$130$		0			0
$131$		0			0		1.00000			$0$
$131$		0			0		−1.00000			$\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$	−12.0000			−1.02899
$137$	4.89898	−	4.89898i	0.418548	−	0.418548i	−0.466155	−	0.884703i	$-0.654361\pi$
$137$	4.89898	−	4.89898i	0.418548	−	0.418548i	0.884703	+	0.466155i	$0.154361\pi$
$138$	−7.34847	−	7.34847i	−0.625543	−	0.625543i
$139$		−	4.00000i		−	0.339276i	−0.985506	−	0.169638i	$-0.945740\pi$
$139$		−	4.00000i		−	0.339276i	0.985506	−	0.169638i	$-0.0542598\pi$
$140$		0			0
$141$	−18.0000			−1.51587
$142$		0			0
$143$		0			0
$144$			15.0000i			1.25000i
$145$		0			0
$146$		0			0
$147$	8.57321	−	8.57321i	0.707107	−	0.707107i
$148$		0			0
$149$		0			0			−	1.00000i	$-0.5\pi$
$149$		0			0				1.00000i	$0.5\pi$
$150$		0			0
$151$	−8.00000			−0.651031			−0.325515	−	0.945537i	$-0.605538\pi$
$151$	−8.00000			−0.651031			−0.325515	+	0.945537i	$0.605538\pi$
$152$	−4.89898	+	4.89898i	−0.397360	+	0.397360i
$153$	14.6969	+	14.6969i	1.18818	+	1.18818i
$154$		0			0
$155$		0			0
$156$		0			0
$157$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$157$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$158$	−19.5959	−	19.5959i	−1.55897	−	1.55897i
$159$		−	24.0000i		−	1.90332i
$160$		0			0
$161$		0			0
$162$	11.0227	−	11.0227i	0.866025	−	0.866025i
$163$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$163$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$164$		0			0
$165$		0			0
$166$	−6.00000			−0.465690
$167$	17.1464	−	17.1464i	1.32683	−	1.32683i	0.418711	−	0.908120i	$-0.362482\pi$
$167$	17.1464	−	17.1464i	1.32683	−	1.32683i	0.908120	−	0.418711i	$-0.137518\pi$
$168$		0			0
$169$			13.0000i			1.00000i
$170$		0			0
$171$	12.0000			0.917663
$172$		0			0
$173$	−9.79796	−	9.79796i	−0.744925	−	0.744925i	0.228596	−	0.973521i	$-0.426586\pi$
$173$	−9.79796	−	9.79796i	−0.744925	−	0.744925i	−0.973521	+	0.228596i	$0.926586\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$		0			0
$179$		0			0			−	1.00000i	$-0.5\pi$
$179$		0			0				1.00000i	$0.5\pi$
$180$		0			0
$181$	22.0000			1.63525			0.817624	−	0.575753i	$-0.195291\pi$
$181$	22.0000			1.63525			0.817624	+	0.575753i	$0.195291\pi$
$182$		0			0
$183$	2.44949	+	2.44949i	0.181071	+	0.181071i
$184$		−	6.00000i		−	0.442326i
$185$		0			0
$186$	24.0000			1.75977
$187$		0			0
$188$	7.34847	+	7.34847i	0.535942	+	0.535942i
$189$		0			0
$190$		0			0
$191$		0			0		1.00000			$0$
$191$		0			0		−1.00000			$\pi$
$192$	−1.22474	+	1.22474i	−0.0883883	+	0.0883883i
$193$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$193$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$194$		0			0
$195$		0			0
$196$	−7.00000			−0.500000
$197$	−19.5959	+	19.5959i	−1.39615	+	1.39615i	−0.585424	+	0.810727i	$0.699072\pi$
$197$	−19.5959	+	19.5959i	−1.39615	+	1.39615i	−0.810727	+	0.585424i	$0.800928\pi$
$198$		0			0
$199$			16.0000i			1.13421i	0.823646	+	0.567105i	$0.191937\pi$
$199$			16.0000i			1.13421i	−0.823646	+	0.567105i	$0.808063\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		0			0
$204$		−	12.0000i		−	0.840168i
$205$		0			0
$206$		0			0
$207$	−7.34847	+	7.34847i	−0.510754	+	0.510754i
$208$		0			0
$209$		0			0
$210$		0			0
$211$	−28.0000			−1.92760			−0.963800	−	0.266627i	$-0.914091\pi$
$211$	−28.0000			−1.92760			−0.963800	+	0.266627i	$0.914091\pi$
$212$	−9.79796	+	9.79796i	−0.672927	+	0.672927i
$213$		0			0
$214$		−	18.0000i		−	1.23045i
$215$		0			0
$216$	9.00000			0.612372
$217$		0			0
$218$	17.1464	+	17.1464i	1.16130	+	1.16130i
$219$		0			0
$220$		0			0
$221$		0			0
$222$		0			0
$223$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$223$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$224$		0			0
$225$		0			0
$226$	−36.0000			−2.39468
$227$	17.1464	−	17.1464i	1.13805	−	1.13805i	0.149249	−	0.988800i	$-0.452314\pi$
$227$	17.1464	−	17.1464i	1.13805	−	1.13805i	0.988800	−	0.149249i	$-0.0476855\pi$
$228$	−4.89898	−	4.89898i	−0.324443	−	0.324443i
$229$			26.0000i			1.71813i	0.511868	+	0.859064i	$0.328954\pi$
$229$			26.0000i			1.71813i	−0.511868	+	0.859064i	$0.671046\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	14.6969	+	14.6969i	0.962828	+	0.962828i	0.999333	−	0.0365050i	$-0.0116225\pi$
$233$	14.6969	+	14.6969i	0.962828	+	0.962828i	−0.0365050	+	0.999333i	$0.511622\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	−19.5959	+	19.5959i	−1.27289	+	1.27289i
$238$		0			0
$239$		0			0			−	1.00000i	$-0.5\pi$
$239$		0			0				1.00000i	$0.5\pi$
$240$		0			0
$241$	2.00000			0.128831			0.0644157	−	0.997923i	$-0.479482\pi$
$241$	2.00000			0.128831			0.0644157	+	0.997923i	$0.479482\pi$
$242$	−13.4722	+	13.4722i	−0.866025	+	0.866025i
$243$	−11.0227	−	11.0227i	−0.707107	−	0.707107i
$244$		−	2.00000i		−	0.128037i
$245$		0			0
$246$		0			0
$247$		0			0
$248$	9.79796	+	9.79796i	0.622171	+	0.622171i
$249$			6.00000i			0.380235i
$250$		0			0
$251$		0			0		1.00000			$0$
$251$		0			0		−1.00000			$\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$	19.0000			1.18750
$257$	4.89898	−	4.89898i	0.305590	−	0.305590i	−0.537606	−	0.843196i	$-0.680671\pi$
$257$	4.89898	−	4.89898i	0.305590	−	0.305590i	0.843196	+	0.537606i	$0.180671\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−22.0454	−	22.0454i	−1.35938	−	1.35938i	−0.874683	−	0.484695i	$-0.838931\pi$
$263$	−22.0454	−	22.0454i	−1.35938	−	1.35938i	−0.484695	−	0.874683i	$-0.661069\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$		0			0
$269$		0			0			−	1.00000i	$-0.5\pi$
$269$		0			0				1.00000i	$0.5\pi$
$270$		0			0
$271$	32.0000			1.94386			0.971931	−	0.235267i	$-0.0755965\pi$
$271$	32.0000			1.94386			0.971931	+	0.235267i	$0.0755965\pi$
$272$	24.4949	−	24.4949i	1.48522	−	1.48522i
$273$		0			0
$274$			12.0000i			0.724947i
$275$		0			0
$276$	6.00000			0.361158
$277$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$277$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$278$	4.89898	+	4.89898i	0.293821	+	0.293821i
$279$		−	24.0000i		−	1.43684i
$280$		0			0
$281$		0			0		1.00000			$0$
$281$		0			0		−1.00000			$\pi$
$282$	22.0454	−	22.0454i	1.31278	−	1.31278i
$283$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$283$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$	−11.0227	−	11.0227i	−0.649519	−	0.649519i
$289$		−	31.0000i		−	1.82353i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−9.79796	−	9.79796i	−0.572403	−	0.572403i	0.360396	−	0.932799i	$-0.382641\pi$
$293$	−9.79796	−	9.79796i	−0.572403	−	0.572403i	−0.932799	+	0.360396i	$0.882641\pi$
$294$			21.0000i			1.22474i
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$		0			0
$301$		0			0
$302$	9.79796	−	9.79796i	0.563809	−	0.563809i
$303$		0			0
$304$		−	20.0000i		−	1.14708i
$305$		0			0
$306$	−36.0000			−2.05798
$307$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$307$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		0			0		1.00000			$0$
$311$		0			0		−1.00000			$\pi$
$312$		0			0
$313$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$313$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$314$		0			0
$315$		0			0
$316$	16.0000			0.900070
$317$	−19.5959	+	19.5959i	−1.10062	+	1.10062i	−0.106280	+	0.994336i	$0.533894\pi$
$317$	−19.5959	+	19.5959i	−1.10062	+	1.10062i	−0.994336	+	0.106280i	$0.966106\pi$
$318$	29.3939	+	29.3939i	1.64833	+	1.64833i
$319$		0			0
$320$		0			0
$321$	−18.0000			−1.00466
$322$		0			0
$323$	−19.5959	−	19.5959i	−1.09035	−	1.09035i
$324$			9.00000i			0.500000i
$325$		0			0
$326$		0			0
$327$	17.1464	−	17.1464i	0.948200	−	0.948200i
$328$		0			0
$329$		0			0
$330$		0			0
$331$	−28.0000			−1.53902			−0.769510	−	0.638635i	$-0.779499\pi$
$331$	−28.0000			−1.53902			−0.769510	+	0.638635i	$0.779499\pi$
$332$	2.44949	−	2.44949i	0.134433	−	0.134433i
$333$		0			0
$334$			42.0000i			2.29814i
$335$		0			0
$336$		0			0
$337$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$337$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$338$	−15.9217	−	15.9217i	−0.866025	−	0.866025i
$339$			36.0000i			1.95525i
$340$		0			0
$341$		0			0
$342$	−14.6969	+	14.6969i	−0.794719	+	0.794719i
$343$		0			0
$344$		0			0
$345$		0			0
$346$	24.0000			1.29025
$347$	−7.34847	+	7.34847i	−0.394486	+	0.394486i	−0.876283	−	0.481797i	$-0.839984\pi$
$347$	−7.34847	+	7.34847i	−0.394486	+	0.394486i	0.481797	+	0.876283i	$0.339984\pi$
$348$		0			0
$349$		−	34.0000i		−	1.81998i	−0.414632	−	0.909989i	$-0.636090\pi$
$349$		−	34.0000i		−	1.81998i	0.414632	−	0.909989i	$-0.363910\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	14.6969	+	14.6969i	0.782239	+	0.782239i	0.980208	−	0.197969i	$-0.0634346\pi$
$353$	14.6969	+	14.6969i	0.782239	+	0.782239i	−0.197969	+	0.980208i	$0.563435\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$		0			0			−	1.00000i	$-0.5\pi$
$359$		0			0				1.00000i	$0.5\pi$
$360$		0			0
$361$	3.00000			0.157895
$362$	−26.9444	+	26.9444i	−1.41617	+	1.41617i
$363$	13.4722	+	13.4722i	0.707107	+	0.707107i
$364$		0			0
$365$		0			0
$366$	−6.00000			−0.313625
$367$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$367$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$368$	12.2474	+	12.2474i	0.638442	+	0.638442i
$369$		0			0
$370$		0			0
$371$		0			0
$372$	−9.79796	+	9.79796i	−0.508001	+	0.508001i
$373$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$373$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$374$		0			0
$375$		0			0
$376$	18.0000			0.928279
$377$		0			0
$378$		0			0
$379$		−	4.00000i		−	0.205466i	−0.994709	−	0.102733i	$-0.967241\pi$
$379$		−	4.00000i		−	0.205466i	0.994709	−	0.102733i	$-0.0327588\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	26.9444	+	26.9444i	1.37679	+	1.37679i	0.849982	+	0.526812i	$0.176613\pi$
$383$	26.9444	+	26.9444i	1.37679	+	1.37679i	0.526812	+	0.849982i	$0.323387\pi$
$384$		−	21.0000i		−	1.07165i
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$		0			0			−	1.00000i	$-0.5\pi$
$389$		0			0				1.00000i	$0.5\pi$
$390$		0			0
$391$	24.0000			1.21373
$392$	−8.57321	+	8.57321i	−0.433013	+	0.433013i
$393$		0			0
$394$		−	48.0000i		−	2.41821i
$395$		0			0
$396$		0			0
$397$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$397$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$398$	−19.5959	−	19.5959i	−0.982255	−	0.982255i
$399$		0			0
$400$		0			0
$401$		0			0		1.00000			$0$
$401$		0			0		−1.00000			$\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$	−14.6969	−	14.6969i	−0.727607	−	0.727607i
$409$			26.0000i			1.28562i	0.766027	+	0.642809i	$0.222231\pi$
$409$			26.0000i			1.28562i	−0.766027	+	0.642809i	$0.777769\pi$
$410$		0			0
$411$	12.0000			0.591916
$412$		0			0
$413$		0			0
$414$		−	18.0000i		−	0.884652i
$415$		0			0
$416$		0			0
$417$	4.89898	−	4.89898i	0.239904	−	0.239904i
$418$		0			0
$419$		0			0			−	1.00000i	$-0.5\pi$
$419$		0			0				1.00000i	$0.5\pi$
$420$		0			0
$421$	−38.0000			−1.85201			−0.926003	−	0.377515i	$-0.876779\pi$
$421$	−38.0000			−1.85201			−0.926003	+	0.377515i	$0.876779\pi$
$422$	34.2929	−	34.2929i	1.66935	−	1.66935i
$423$	−22.0454	−	22.0454i	−1.07188	−	1.07188i
$424$			24.0000i			1.16554i
$425$		0			0
$426$		0			0
$427$		0			0
$428$	7.34847	+	7.34847i	0.355202	+	0.355202i
$429$		0			0
$430$		0			0
$431$		0			0		1.00000			$0$
$431$		0			0		−1.00000			$\pi$
$432$	−18.3712	+	18.3712i	−0.883883	+	0.883883i
$433$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$433$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$434$		0			0
$435$		0			0
$436$	−14.0000			−0.670478
$437$	9.79796	−	9.79796i	0.468700	−	0.468700i
$438$		0			0
$439$			16.0000i			0.763638i	0.924237	+	0.381819i	$0.124702\pi$
$439$			16.0000i			0.763638i	−0.924237	+	0.381819i	$0.875298\pi$
$440$		0			0
$441$	21.0000			1.00000
$442$		0			0
$443$	26.9444	+	26.9444i	1.28017	+	1.28017i	0.940572	+	0.339595i	$0.110290\pi$
$443$	26.9444	+	26.9444i	1.28017	+	1.28017i	0.339595	+	0.940572i	$0.389710\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$		0			0
$449$		0			0			−	1.00000i	$-0.5\pi$
$449$		0			0				1.00000i	$0.5\pi$
$450$		0			0
$451$		0			0
$452$	14.6969	−	14.6969i	0.691286	−	0.691286i
$453$	−9.79796	−	9.79796i	−0.460348	−	0.460348i
$454$			42.0000i			1.97116i
$455$		0			0
$456$	−12.0000			−0.561951
$457$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$457$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$458$	−31.8434	−	31.8434i	−1.48794	−	1.48794i
$459$			36.0000i			1.68034i
$460$		0			0
$461$		0			0		1.00000			$0$
$461$		0			0		−1.00000			$\pi$
$462$		0			0
$463$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$463$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$464$		0			0
$465$		0			0
$466$	−36.0000			−1.66767
$467$	17.1464	−	17.1464i	0.793442	−	0.793442i	−0.188610	−	0.982052i	$-0.560398\pi$
$467$	17.1464	−	17.1464i	0.793442	−	0.793442i	0.982052	+	0.188610i	$0.0603982\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		0			0
$472$		0			0
$473$		0			0
$474$		−	48.0000i		−	2.20471i
$475$		0			0
$476$		0			0
$477$	29.3939	−	29.3939i	1.34585	−	1.34585i
$478$		0			0
$479$		0			0			−	1.00000i	$-0.5\pi$
$479$		0			0				1.00000i	$0.5\pi$
$480$		0			0
$481$		0			0
$482$	−2.44949	+	2.44949i	−0.111571	+	0.111571i
$483$		0			0
$484$		−	11.0000i		−	0.500000i
$485$		0			0
$486$	27.0000			1.22474
$487$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$487$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$488$	−2.44949	−	2.44949i	−0.110883	−	0.110883i
$489$		0			0
$490$		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
$496$	−40.0000			−1.79605
$497$		0			0
$498$	−7.34847	−	7.34847i	−0.329293	−	0.329293i
$499$		−	44.0000i		−	1.96971i	−0.173379	−	0.984855i	$-0.555468\pi$
$499$		−	44.0000i		−	1.96971i	0.173379	−	0.984855i	$-0.444532\pi$
$500$		0			0
$501$	42.0000			1.87642
$502$		0			0
$503$	2.44949	+	2.44949i	0.109217	+	0.109217i	0.759604	−	0.650386i	$-0.225393\pi$
$503$	2.44949	+	2.44949i	0.109217	+	0.109217i	−0.650386	+	0.759604i	$0.725393\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	−15.9217	+	15.9217i	−0.707107	+	0.707107i
$508$		0			0
$509$		0			0			−	1.00000i	$-0.5\pi$
$509$		0			0				1.00000i	$0.5\pi$
$510$		0			0
$511$		0			0
$512$	−6.12372	+	6.12372i	−0.270633	+	0.270633i
$513$	14.6969	+	14.6969i	0.648886	+	0.648886i
$514$			12.0000i			0.529297i
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$		−	24.0000i		−	1.05348i
$520$		0			0
$521$		0			0		1.00000			$0$
$521$		0			0		−1.00000			$\pi$
$522$		0			0
$523$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$523$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$524$		0			0
$525$		0			0
$526$	54.0000			2.35451
$527$	−39.1918	+	39.1918i	−1.70722	+	1.70722i
$528$		0			0
$529$		−	11.0000i		−	0.478261i
$530$		0			0
$531$		0			0
$532$		0			0
$533$		0			0
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	2.00000			0.0859867			0.0429934	−	0.999075i	$-0.486311\pi$
$541$	2.00000			0.0859867			0.0429934	+	0.999075i	$0.486311\pi$
$542$	−39.1918	+	39.1918i	−1.68343	+	1.68343i
$543$	26.9444	+	26.9444i	1.15629	+	1.15629i
$544$			36.0000i			1.54349i
$545$		0			0
$546$		0			0
$547$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$547$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$548$	−4.89898	−	4.89898i	−0.209274	−	0.209274i
$549$			6.00000i			0.256074i
$550$		0			0
$551$		0			0
$552$	7.34847	−	7.34847i	0.312772	−	0.312772i
$553$		0			0
$554$		0			0
$555$		0			0
$556$	−4.00000			−0.169638
$557$	29.3939	−	29.3939i	1.24546	−	1.24546i	0.287754	−	0.957704i	$-0.407091\pi$
$557$	29.3939	−	29.3939i	1.24546	−	1.24546i	0.957704	−	0.287754i	$-0.0929086\pi$
$558$	29.3939	+	29.3939i	1.24434	+	1.24434i
$559$		0			0
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−22.0454	−	22.0454i	−0.929103	−	0.929103i	0.0685449	−	0.997648i	$-0.478164\pi$
$563$	−22.0454	−	22.0454i	−0.929103	−	0.929103i	−0.997648	+	0.0685449i	$0.978164\pi$
$564$			18.0000i			0.757937i
$565$		0			0
$566$		0			0
$567$		0			0
$568$		0			0
$569$		0			0			−	1.00000i	$-0.5\pi$
$569$		0			0				1.00000i	$0.5\pi$
$570$		0			0
$571$	−28.0000			−1.17176			−0.585882	−	0.810397i	$-0.699252\pi$
$571$	−28.0000			−1.17176			−0.585882	+	0.810397i	$0.699252\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$	−3.00000			−0.125000
$577$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$577$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$578$	37.9671	+	37.9671i	1.57922	+	1.57922i
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$	24.0000			0.991431
$587$	−31.8434	+	31.8434i	−1.31432	+	1.31432i	−0.396116	+	0.918201i	$0.629642\pi$
$587$	−31.8434	+	31.8434i	−1.31432	+	1.31432i	−0.918201	+	0.396116i	$0.870358\pi$
$588$	−8.57321	−	8.57321i	−0.353553	−	0.353553i
$589$			32.0000i			1.31854i
$590$		0			0
$591$	−48.0000			−1.97446
$592$		0			0
$593$	−34.2929	−	34.2929i	−1.40824	−	1.40824i	−0.769049	−	0.639190i	$-0.779270\pi$
$593$	−34.2929	−	34.2929i	−1.40824	−	1.40824i	−0.639190	−	0.769049i	$-0.720730\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	−19.5959	+	19.5959i	−0.802008	+	0.802008i
$598$		0			0
$599$		0			0			−	1.00000i	$-0.5\pi$
$599$		0			0				1.00000i	$0.5\pi$
$600$		0			0
$601$	−38.0000			−1.55005			−0.775026	−	0.631929i	$-0.782263\pi$
$601$	−38.0000			−1.55005			−0.775026	+	0.631929i	$0.782263\pi$
$602$		0			0
$603$		0			0
$604$			8.00000i			0.325515i
$605$		0			0
$606$		0			0
$607$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$607$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$608$	14.6969	+	14.6969i	0.596040	+	0.596040i
$609$		0			0
$610$		0			0
$611$		0			0
$612$	14.6969	−	14.6969i	0.594089	−	0.594089i
$613$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$613$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	4.89898	−	4.89898i	0.197225	−	0.197225i	−0.601584	−	0.798810i	$-0.705464\pi$
$617$	4.89898	−	4.89898i	0.197225	−	0.197225i	0.798810	+	0.601584i	$0.205464\pi$
$618$		0			0
$619$		−	44.0000i		−	1.76851i	−0.467005	−	0.884255i	$-0.654667\pi$
$619$		−	44.0000i		−	1.76851i	0.467005	−	0.884255i	$-0.345333\pi$
$620$		0			0
$621$	−18.0000			−0.722315
$622$		0			0
$623$		0			0
$624$		0			0
$625$		0			0
$626$		0			0
$627$		0			0
$628$		0			0
$629$		0			0
$630$		0			0
$631$	32.0000			1.27390			0.636950	−	0.770905i	$-0.280196\pi$
$631$	32.0000			1.27390			0.636950	+	0.770905i	$0.280196\pi$
$632$	19.5959	−	19.5959i	0.779484	−	0.779484i
$633$	−34.2929	−	34.2929i	−1.36302	−	1.36302i
$634$		−	48.0000i		−	1.90632i
$635$		0			0
$636$	−24.0000			−0.951662
$637$		0			0
$638$		0			0
$639$		0			0
$640$		0			0
$641$		0			0		1.00000			$0$
$641$		0			0		−1.00000			$\pi$
$642$	22.0454	−	22.0454i	0.870063	−	0.870063i
$643$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$643$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$644$		0			0
$645$		0			0
$646$	48.0000			1.88853
$647$	17.1464	−	17.1464i	0.674096	−	0.674096i	−0.284562	−	0.958658i	$-0.591848\pi$
$647$	17.1464	−	17.1464i	0.674096	−	0.674096i	0.958658	+	0.284562i	$0.0918482\pi$
$648$	11.0227	+	11.0227i	0.433013	+	0.433013i
$649$		0			0
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−9.79796	−	9.79796i	−0.383424	−	0.383424i	0.488910	−	0.872334i	$-0.337395\pi$
$653$	−9.79796	−	9.79796i	−0.383424	−	0.383424i	−0.872334	+	0.488910i	$0.837395\pi$
$654$			42.0000i			1.64233i
$655$		0			0
$656$		0			0
$657$		0			0
$658$		0			0
$659$		0			0			−	1.00000i	$-0.5\pi$
$659$		0			0				1.00000i	$0.5\pi$
$660$		0			0
$661$	22.0000			0.855701			0.427850	−	0.903850i	$-0.359271\pi$
$661$	22.0000			0.855701			0.427850	+	0.903850i	$0.359271\pi$
$662$	34.2929	−	34.2929i	1.33283	−	1.33283i
$663$		0			0
$664$		−	6.00000i		−	0.232845i
$665$		0			0
$666$		0			0
$667$		0			0
$668$	−17.1464	−	17.1464i	−0.663415	−	0.663415i
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$673$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$674$		0			0
$675$		0			0
$676$	13.0000			0.500000
$677$	29.3939	−	29.3939i	1.12970	−	1.12970i	0.139473	−	0.990226i	$-0.455459\pi$
$677$	29.3939	−	29.3939i	1.12970	−	1.12970i	0.990226	−	0.139473i	$-0.0445407\pi$
$678$	−44.0908	−	44.0908i	−1.69330	−	1.69330i
$679$		0			0
$680$		0			0
$681$	42.0000			1.60944
$682$		0			0
$683$	26.9444	+	26.9444i	1.03100	+	1.03100i	0.999504	+	0.0314944i	$0.0100266\pi$
$683$	26.9444	+	26.9444i	1.03100	+	1.03100i	0.0314944	+	0.999504i	$0.489973\pi$
$684$		−	12.0000i		−	0.458831i
$685$		0			0
$686$		0			0
$687$	−31.8434	+	31.8434i	−1.21490	+	1.21490i
$688$		0			0
$689$		0			0
$690$		0			0
$691$	52.0000			1.97817			0.989087	−	0.147335i	$-0.0470696\pi$
$691$	52.0000			1.97817			0.989087	+	0.147335i	$0.0470696\pi$
$692$	−9.79796	+	9.79796i	−0.372463	+	0.372463i
$693$		0			0
$694$		−	18.0000i		−	0.683271i
$695$		0			0
$696$		0			0
$697$		0			0
$698$	41.6413	+	41.6413i	1.57615	+	1.57615i
$699$			36.0000i			1.36165i
$700$		0			0
$701$		0			0		1.00000			$0$
$701$		0			0		−1.00000			$\pi$
$702$		0			0
$703$		0			0
$704$		0			0
$705$		0			0
$706$	−36.0000			−1.35488
$707$		0			0
$708$		0			0
$709$			26.0000i			0.976450i	0.872718	+	0.488225i	$0.162356\pi$
$709$			26.0000i			0.976450i	−0.872718	+	0.488225i	$0.837644\pi$
$710$		0			0
$711$	−48.0000			−1.80014
$712$		0			0
$713$	−19.5959	−	19.5959i	−0.733873	−	0.733873i
$714$		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
$721$		0			0
$722$	−3.67423	+	3.67423i	−0.136741	+	0.136741i
$723$	2.44949	+	2.44949i	0.0910975	+	0.0910975i
$724$		−	22.0000i		−	0.817624i
$725$		0			0
$726$	−33.0000			−1.22474
$727$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$727$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$728$		0			0
$729$		−	27.0000i		−	1.00000i
$730$		0			0
$731$		0			0
$732$	2.44949	−	2.44949i	0.0905357	−	0.0905357i
$733$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$733$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$734$		0			0
$735$		0			0
$736$	−18.0000			−0.663489
$737$		0			0
$738$		0			0
$739$		−	4.00000i		−	0.147142i	−0.997290	−	0.0735712i	$-0.976560\pi$
$739$		−	4.00000i		−	0.147142i	0.997290	−	0.0735712i	$-0.0234396\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−22.0454	−	22.0454i	−0.808768	−	0.808768i	0.175680	−	0.984447i	$-0.443788\pi$
$743$	−22.0454	−	22.0454i	−0.808768	−	0.808768i	−0.984447	+	0.175680i	$0.943788\pi$
$744$			24.0000i			0.879883i
$745$		0			0
$746$		0			0
$747$	−7.34847	+	7.34847i	−0.268866	+	0.268866i
$748$		0			0
$749$		0			0
$750$		0			0
$751$	−8.00000			−0.291924			−0.145962	−	0.989290i	$-0.546628\pi$
$751$	−8.00000			−0.291924			−0.145962	+	0.989290i	$0.546628\pi$
$752$	−36.7423	+	36.7423i	−1.33986	+	1.33986i
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$757$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$758$	4.89898	+	4.89898i	0.177939	+	0.177939i
$759$		0			0
$760$		0			0
$761$		0			0		1.00000			$0$
$761$		0			0		−1.00000			$\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$		0			0
$766$	−66.0000			−2.38468
$767$		0			0
$768$	23.2702	+	23.2702i	0.839689	+	0.839689i
$769$			46.0000i			1.65880i	0.558653	+	0.829401i	$0.311318\pi$
$769$			46.0000i			1.65880i	−0.558653	+	0.829401i	$0.688682\pi$
$770$		0			0
$771$	12.0000			0.432169
$772$		0			0
$773$	39.1918	+	39.1918i	1.40963	+	1.40963i	0.761686	+	0.647947i	$0.224372\pi$
$773$	39.1918	+	39.1918i	1.40963	+	1.40963i	0.647947	+	0.761686i	$0.275628\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$		0			0
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$	−29.3939	+	29.3939i	−1.05112	+	1.05112i
$783$		0			0
$784$		−	35.0000i		−	1.25000i
$785$		0			0
$786$		0			0
$787$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$787$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$788$	19.5959	+	19.5959i	0.698076	+	0.698076i
$789$		−	54.0000i		−	1.92245i
$790$		0			0
$791$		0			0
$792$		0			0
$793$		0			0
$794$		0			0
$795$		0			0
$796$	16.0000			0.567105
$797$	−19.5959	+	19.5959i	−0.694123	+	0.694123i	−0.963136	−	0.269013i	$-0.913302\pi$
$797$	−19.5959	+	19.5959i	−0.694123	+	0.694123i	0.269013	+	0.963136i	$0.413302\pi$
$798$		0			0
$799$			72.0000i			2.54718i
$800$		0			0
$801$		0			0
$802$		0			0
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$		0			0			−	1.00000i	$-0.5\pi$
$809$		0			0				1.00000i	$0.5\pi$
$810$		0			0
$811$	52.0000			1.82597			0.912983	−	0.407997i	$-0.133772\pi$
$811$	52.0000			1.82597			0.912983	+	0.407997i	$0.133772\pi$
$812$		0			0
$813$	39.1918	+	39.1918i	1.37452	+	1.37452i
$814$		0			0
$815$		0			0
$816$	60.0000			2.10042
$817$		0			0
$818$	−31.8434	−	31.8434i	−1.11338	−	1.11338i
$819$		0			0
$820$		0			0
$821$		0			0		1.00000			$0$
$821$		0			0		−1.00000			$\pi$
$822$	−14.6969	+	14.6969i	−0.512615	+	0.512615i
$823$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$823$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−31.8434	+	31.8434i	−1.10730	+	1.10730i	−0.113799	+	0.993504i	$0.536302\pi$
$827$	−31.8434	+	31.8434i	−1.10730	+	1.10730i	−0.993504	+	0.113799i	$0.963698\pi$
$828$	7.34847	+	7.34847i	0.255377	+	0.255377i
$829$		−	34.0000i		−	1.18087i	−0.807086	−	0.590434i	$-0.798956\pi$
$829$		−	34.0000i		−	1.18087i	0.807086	−	0.590434i	$-0.201044\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−34.2929	−	34.2929i	−1.18818	−	1.18818i
$834$			12.0000i			0.415526i
$835$		0			0
$836$		0			0
$837$	29.3939	−	29.3939i	1.01600	−	1.01600i
$838$		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
$842$	46.5403	−	46.5403i	1.60388	−	1.60388i
$843$		0			0
$844$			28.0000i			0.963800i
$845$		0			0
$846$	54.0000			1.85656
$847$		0			0
$848$	−48.9898	−	48.9898i	−1.68232	−	1.68232i
$849$		0			0
$850$		0			0
$851$		0			0
$852$		0			0
$853$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$853$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$854$		0			0
$855$		0			0
$856$	18.0000			0.615227
$857$	4.89898	−	4.89898i	0.167346	−	0.167346i	−0.618466	−	0.785812i	$-0.712245\pi$
$857$	4.89898	−	4.89898i	0.167346	−	0.167346i	0.785812	+	0.618466i	$0.212245\pi$
$858$		0			0
$859$		−	44.0000i		−	1.50126i	−0.660722	−	0.750630i	$-0.729750\pi$
$859$		−	44.0000i		−	1.50126i	0.660722	−	0.750630i	$-0.270250\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	26.9444	+	26.9444i	0.917198	+	0.917198i	0.996825	−	0.0796271i	$-0.0253730\pi$
$863$	26.9444	+	26.9444i	0.917198	+	0.917198i	−0.0796271	+	0.996825i	$0.525373\pi$
$864$		−	27.0000i		−	0.918559i
$865$		0			0
$866$		0			0
$867$	37.9671	−	37.9671i	1.28943	−	1.28943i
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$	−17.1464	+	17.1464i	−0.580651	+	0.580651i
$873$		0			0
$874$			24.0000i			0.811812i
$875$		0			0
$876$		0			0
$877$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$877$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$878$	−19.5959	−	19.5959i	−0.661330	−	0.661330i
$879$		−	24.0000i		−	0.809500i
$880$		0			0
$881$		0			0		1.00000			$0$
$881$		0			0		−1.00000			$\pi$
$882$	−25.7196	+	25.7196i	−0.866025	+	0.866025i
$883$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$883$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$884$		0			0
$885$		0			0
$886$	−66.0000			−2.21731
$887$	41.6413	−	41.6413i	1.39818	−	1.39818i	0.592911	−	0.805268i	$-0.297979\pi$
$887$	41.6413	−	41.6413i	1.39818	−	1.39818i	0.805268	−	0.592911i	$-0.202021\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$		0			0
$892$		0			0
$893$	29.3939	+	29.3939i	0.983629	+	0.983629i
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$		0			0
$900$		0			0
$901$	−96.0000			−3.19822
$902$		0			0
$903$		0			0
$904$		−	36.0000i		−	1.19734i
$905$		0			0
$906$	24.0000			0.797347
$907$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$907$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$908$	−17.1464	−	17.1464i	−0.569024	−	0.569024i
$909$		0			0
$910$		0			0
$911$		0			0		1.00000			$0$
$911$		0			0		−1.00000			$\pi$
$912$	24.4949	−	24.4949i	0.811107	−	0.811107i
$913$		0			0
$914$		0			0
$915$		0			0
$916$	26.0000			0.859064
$917$		0			0
$918$	−44.0908	−	44.0908i	−1.45521	−	1.45521i
$919$			56.0000i			1.84727i	0.383274	+	0.923635i	$0.374797\pi$
$919$			56.0000i			1.84727i	−0.383274	+	0.923635i	$0.625203\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		0			0
$924$		0			0
$925$		0			0
$926$		0			0
$927$		0			0
$928$		0			0
$929$		0			0			−	1.00000i	$-0.5\pi$
$929$		0			0				1.00000i	$0.5\pi$
$930$		0			0
$931$	−28.0000			−0.917663
$932$	14.6969	−	14.6969i	0.481414	−	0.481414i
$933$		0			0
$934$			42.0000i			1.37428i
$935$		0			0
$936$		0			0
$937$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$937$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$		0			0		1.00000			$0$
$941$		0			0		−1.00000			$\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$	41.6413	−	41.6413i	1.35316	−	1.35316i	0.471060	−	0.882101i	$-0.343871\pi$
$947$	41.6413	−	41.6413i	1.35316	−	1.35316i	0.882101	−	0.471060i	$-0.156129\pi$
$948$	19.5959	+	19.5959i	0.636446	+	0.636446i
$949$		0			0
$950$		0			0
$951$	−48.0000			−1.55651
$952$		0			0
$953$	14.6969	+	14.6969i	0.476081	+	0.476081i	0.903876	−	0.427795i	$-0.140710\pi$
$953$	14.6969	+	14.6969i	0.476081	+	0.476081i	−0.427795	+	0.903876i	$0.640710\pi$
$954$			72.0000i			2.33109i
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	33.0000			1.06452
$962$		0			0
$963$	−22.0454	−	22.0454i	−0.710403	−	0.710403i
$964$		−	2.00000i		−	0.0644157i
$965$		0			0
$966$		0			0
$967$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$967$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$968$	−13.4722	−	13.4722i	−0.433013	−	0.433013i
$969$		−	48.0000i		−	1.54198i
$970$		0			0
$971$		0			0		1.00000			$0$
$971$		0			0		−1.00000			$\pi$
$972$	−11.0227	+	11.0227i	−0.353553	+	0.353553i
$973$		0			0
$974$		0			0
$975$		0			0
$976$	10.0000			0.320092
$977$	−44.0908	+	44.0908i	−1.41059	+	1.41059i	−0.654710	+	0.755880i	$0.727209\pi$
$977$	−44.0908	+	44.0908i	−1.41059	+	1.41059i	−0.755880	+	0.654710i	$0.772791\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$	42.0000			1.34096
$982$		0			0
$983$	2.44949	+	2.44949i	0.0781266	+	0.0781266i	0.745090	−	0.666964i	$-0.232406\pi$
$983$	2.44949	+	2.44949i	0.0781266	+	0.0781266i	−0.666964	+	0.745090i	$0.732406\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$	32.0000			1.01651			0.508257	−	0.861206i	$-0.330290\pi$
$991$	32.0000			1.01651			0.508257	+	0.861206i	$0.330290\pi$
$992$	29.3939	−	29.3939i	0.933257	−	0.933257i
$993$	−34.2929	−	34.2929i	−1.08825	−	1.08825i
$994$		0			0
$995$		0			0
$996$	6.00000			0.190117
$997$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$997$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$998$	53.8888	+	53.8888i	1.70582	+	1.70582i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	75.2.e.a.32.1	✓	4
3.2	odd	2	inner	75.2.e.a.32.2	yes	4
4.3	odd	2		1200.2.v.g.257.1		4
5.2	odd	4	inner	75.2.e.a.68.1	yes	4
5.3	odd	4	inner	75.2.e.a.68.2	yes	4
5.4	even	2	inner	75.2.e.a.32.2	yes	4
12.11	even	2		1200.2.v.g.257.2		4
15.2	even	4	inner	75.2.e.a.68.2	yes	4
15.8	even	4	inner	75.2.e.a.68.1	yes	4
15.14	odd	2	CM	75.2.e.a.32.1	✓	4
20.3	even	4		1200.2.v.g.593.2		4
20.7	even	4		1200.2.v.g.593.1		4
20.19	odd	2		1200.2.v.g.257.2		4
60.23	odd	4		1200.2.v.g.593.1		4
60.47	odd	4		1200.2.v.g.593.2		4
60.59	even	2		1200.2.v.g.257.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
75.2.e.a.32.1	✓	4	1.1	even	1	trivial
75.2.e.a.32.1	✓	4	15.14	odd	2	CM
75.2.e.a.32.2	yes	4	3.2	odd	2	inner
75.2.e.a.32.2	yes	4	5.4	even	2	inner
75.2.e.a.68.1	yes	4	5.2	odd	4	inner
75.2.e.a.68.1	yes	4	15.8	even	4	inner
75.2.e.a.68.2	yes	4	5.3	odd	4	inner
75.2.e.a.68.2	yes	4	15.2	even	4	inner
1200.2.v.g.257.1		4	4.3	odd	2
1200.2.v.g.257.1		4	60.59	even	2
1200.2.v.g.257.2		4	12.11	even	2
1200.2.v.g.257.2		4	20.19	odd	2
1200.2.v.g.593.1		4	20.7	even	4
1200.2.v.g.593.1		4	60.23	odd	4
1200.2.v.g.593.2		4	20.3	even	4
1200.2.v.g.593.2		4	60.47	odd	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 75 = 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	75.e (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-1.22474 + 1.22474i) q^{2} +(1.22474 + 1.22474i) q^{3} -1.00000i q^{4} -3.00000 q^{6} +(-1.22474 - 1.22474i) q^{8} +3.00000i q^{9} +O(q^{10})\)	\(q+(-1.22474 + 1.22474i) q^{2} +(1.22474 + 1.22474i) q^{3} -1.00000i q^{4} -3.00000 q^{6} +(-1.22474 - 1.22474i) q^{8} +3.00000i q^{9} +(1.22474 - 1.22474i) q^{12} +5.00000 q^{16} +(4.89898 - 4.89898i) q^{17} +(-3.67423 - 3.67423i) q^{18} -4.00000i q^{19} +(2.44949 + 2.44949i) q^{23} -3.00000i q^{24} +(-3.67423 + 3.67423i) q^{27} -8.00000 q^{31} +(-3.67423 + 3.67423i) q^{32} +12.0000i q^{34} +3.00000 q^{36} +(4.89898 + 4.89898i) q^{38} -6.00000 q^{46} +(-7.34847 + 7.34847i) q^{47} +(6.12372 + 6.12372i) q^{48} -7.00000i q^{49} +12.0000 q^{51} +(-9.79796 - 9.79796i) q^{53} -9.00000i q^{54} +(4.89898 - 4.89898i) q^{57} +2.00000 q^{61} +(9.79796 - 9.79796i) q^{62} +1.00000i q^{64} +(-4.89898 - 4.89898i) q^{68} +6.00000i q^{69} +(3.67423 - 3.67423i) q^{72} -4.00000 q^{76} +16.0000i q^{79} -9.00000 q^{81} +(2.44949 + 2.44949i) q^{83} +(2.44949 - 2.44949i) q^{92} +(-9.79796 - 9.79796i) q^{93} -18.0000i q^{94} -9.00000 q^{96} +(8.57321 + 8.57321i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 12 q^{6}+O(q^{10}) \) 4 * q - 12 * q^6	\( 4 q - 12 q^{6} + 20 q^{16} - 32 q^{31} + 12 q^{36} - 24 q^{46} + 48 q^{51} + 8 q^{61} - 16 q^{76} - 36 q^{81} - 36 q^{96}+O(q^{100}) \) 4 * q - 12 * q^6 + 20 * q^16 - 32 * q^31 + 12 * q^36 - 24 * q^46 + 48 * q^51 + 8 * q^61 - 16 * q^76 - 36 * q^81 - 36 * q^96

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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