LMFDB - Embedded newform 575.2.b.a.24.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [575,2,Mod(24,575)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(575, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 0]))

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

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

chi := DirichletCharacter("575.24");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 575 = 5^{2} \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	575.b (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$4.59139811622$
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:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 115)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			24.1
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	575.24
Dual form			575.2.b.a.24.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.00000i q^{2} -2.00000 q^{4} -1.00000i q^{7} +3.00000 q^{9} +O(q^{10})$	$q-2.00000i q^{2} -2.00000 q^{4} -1.00000i q^{7} +3.00000 q^{9} +2.00000 q^{11} -2.00000i q^{13} -2.00000 q^{14} -4.00000 q^{16} -3.00000i q^{17} -6.00000i q^{18} +2.00000 q^{19} -4.00000i q^{22} +1.00000i q^{23} -4.00000 q^{26} +2.00000i q^{28} -7.00000 q^{29} -5.00000 q^{31} +8.00000i q^{32} -6.00000 q^{34} -6.00000 q^{36} -11.0000i q^{37} -4.00000i q^{38} +1.00000 q^{41} -4.00000 q^{44} +2.00000 q^{46} +6.00000 q^{49} +4.00000i q^{52} +11.0000i q^{53} +14.0000i q^{58} +13.0000 q^{59} -8.00000 q^{61} +10.0000i q^{62} -3.00000i q^{63} +8.00000 q^{64} -5.00000i q^{67} +6.00000i q^{68} +5.00000 q^{71} +6.00000i q^{73} -22.0000 q^{74} -4.00000 q^{76} -2.00000i q^{77} +12.0000 q^{79} +9.00000 q^{81} -2.00000i q^{82} +9.00000i q^{83} -4.00000 q^{89} -2.00000 q^{91} -2.00000i q^{92} +14.0000i q^{97} -12.0000i q^{98} +6.00000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 4 q^{4} + 6 q^{9}+O(q^{10}) $ 2 * q - 4 * q^4 + 6 * q^9	$ 2 q - 4 q^{4} + 6 q^{9} + 4 q^{11} - 4 q^{14} - 8 q^{16} + 4 q^{19} - 8 q^{26} - 14 q^{29} - 10 q^{31} - 12 q^{34} - 12 q^{36} + 2 q^{41} - 8 q^{44} + 4 q^{46} + 12 q^{49} + 26 q^{59} - 16 q^{61} + 16 q^{64} + 10 q^{71} - 44 q^{74} - 8 q^{76} + 24 q^{79} + 18 q^{81} - 8 q^{89} - 4 q^{91} + 12 q^{99}+O(q^{100}) $ 2 * q - 4 * q^4 + 6 * q^9 + 4 * q^11 - 4 * q^14 - 8 * q^16 + 4 * q^19 - 8 * q^26 - 14 * q^29 - 10 * q^31 - 12 * q^34 - 12 * q^36 + 2 * q^41 - 8 * q^44 + 4 * q^46 + 12 * q^49 + 26 * q^59 - 16 * q^61 + 16 * q^64 + 10 * q^71 - 44 * q^74 - 8 * q^76 + 24 * q^79 + 18 * q^81 - 8 * q^89 - 4 * q^91 + 12 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/575\mathbb{Z}\right)^\times$.

$n$	$51$	$277$
$\chi(n)$	$1$	$-1$

Coefficient data

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

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

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

$2$	− 2.00000i	− 1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$2$	− 2.00000i	− 1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$3$	0	0	1.00000			$0$
$3$	0	0	−1.00000			$\pi$
$4$	−2.00000	−1.00000
$5$	0	0
$5$	0	0
$6$	0	0
$7$	− 1.00000i	− 0.377964i	−0.981981	−	0.188982i	$-0.939481\pi$
$7$	− 1.00000i	− 0.377964i	0.981981	−	0.188982i	$-0.0605189\pi$
$8$	0	0
$9$	3.00000	1.00000
$10$	0	0
$11$	2.00000	0.603023	0.301511	−	0.953463i	$-0.402509\pi$
$11$	2.00000	0.603023	0.301511	+	0.953463i	$0.402509\pi$
$12$	0	0
$13$	− 2.00000i	− 0.554700i	−0.960769	−	0.277350i	$-0.910544\pi$
$13$	− 2.00000i	− 0.554700i	0.960769	−	0.277350i	$-0.0894562\pi$
$14$	−2.00000	−0.534522
$15$	0	0
$16$	−4.00000	−1.00000
$17$	− 3.00000i	− 0.727607i	−0.931476	−	0.363803i	$-0.881478\pi$
$17$	− 3.00000i	− 0.727607i	0.931476	−	0.363803i	$-0.118522\pi$
$18$	− 6.00000i	− 1.41421i
$19$	2.00000	0.458831	0.229416	−	0.973329i	$-0.426318\pi$
$19$	2.00000	0.458831	0.229416	+	0.973329i	$0.426318\pi$
$20$	0	0
$21$	0	0
$22$	− 4.00000i	− 0.852803i
$23$	1.00000i	0.208514i
$23$	1.00000i	0.208514i
$24$	0	0
$25$	0	0
$26$	−4.00000	−0.784465
$27$	0	0
$28$	2.00000i	0.377964i
$29$	−7.00000	−1.29987	−0.649934	−	0.759991i	$-0.725203\pi$
$29$	−7.00000	−1.29987	−0.649934	+	0.759991i	$0.725203\pi$
$30$	0	0
$31$	−5.00000	−0.898027	−0.449013	−	0.893525i	$-0.648224\pi$
$31$	−5.00000	−0.898027	−0.449013	+	0.893525i	$0.648224\pi$
$32$	8.00000i	1.41421i
$33$	0	0
$34$	−6.00000	−1.02899
$35$	0	0
$36$	−6.00000	−1.00000
$37$	− 11.0000i	− 1.80839i	−0.427121	−	0.904194i	$-0.640472\pi$
$37$	− 11.0000i	− 1.80839i	0.427121	−	0.904194i	$-0.359528\pi$
$38$	− 4.00000i	− 0.648886i
$39$	0	0
$40$	0	0
$41$	1.00000	0.156174	0.0780869	−	0.996947i	$-0.475119\pi$
$41$	1.00000	0.156174	0.0780869	+	0.996947i	$0.475119\pi$
$42$	0	0
$43$	0	0	1.00000			$0$
$43$	0	0	−1.00000			$\pi$
$44$	−4.00000	−0.603023
$45$	0	0
$46$	2.00000	0.294884
$47$	0	0	1.00000			$0$
$47$	0	0	−1.00000			$\pi$
$48$	0	0
$49$	6.00000	0.857143
$50$	0	0
$51$	0	0
$52$	4.00000i	0.554700i
$53$	11.0000i	1.51097i	0.655168	+	0.755483i	$0.272598\pi$
$53$	11.0000i	1.51097i	−0.655168	+	0.755483i	$0.727402\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	0	0
$58$	14.0000i	1.83829i
$59$	13.0000	1.69246	0.846228	−	0.532821i	$-0.178868\pi$
$59$	13.0000	1.69246	0.846228	+	0.532821i	$0.178868\pi$
$60$	0	0
$61$	−8.00000	−1.02430	−0.512148	−	0.858898i	$-0.671150\pi$
$61$	−8.00000	−1.02430	−0.512148	+	0.858898i	$0.671150\pi$
$62$	10.0000i	1.27000i
$63$	− 3.00000i	− 0.377964i
$64$	8.00000	1.00000
$65$	0	0
$66$	0	0
$67$	− 5.00000i	− 0.610847i	−0.952217	−	0.305424i	$-0.901202\pi$
$67$	− 5.00000i	− 0.610847i	0.952217	−	0.305424i	$-0.0987981\pi$
$68$	6.00000i	0.727607i
$69$	0	0
$70$	0	0
$71$	5.00000	0.593391	0.296695	−	0.954972i	$-0.404115\pi$
$71$	5.00000	0.593391	0.296695	+	0.954972i	$0.404115\pi$
$72$	0	0
$73$	6.00000i	0.702247i	0.936329	+	0.351123i	$0.114200\pi$
$73$	6.00000i	0.702247i	−0.936329	+	0.351123i	$0.885800\pi$
$74$	−22.0000	−2.55745
$75$	0	0
$76$	−4.00000	−0.458831
$77$	− 2.00000i	− 0.227921i
$78$	0	0
$79$	12.0000	1.35011	0.675053	−	0.737769i	$-0.264121\pi$
$79$	12.0000	1.35011	0.675053	+	0.737769i	$0.264121\pi$
$80$	0	0
$81$	9.00000	1.00000
$82$	− 2.00000i	− 0.220863i
$83$	9.00000i	0.987878i	0.869496	+	0.493939i	$0.164443\pi$
$83$	9.00000i	0.987878i	−0.869496	+	0.493939i	$0.835557\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−4.00000	−0.423999	−0.212000	−	0.977270i	$-0.567998\pi$
$89$	−4.00000	−0.423999	−0.212000	+	0.977270i	$0.567998\pi$
$90$	0	0
$91$	−2.00000	−0.209657
$92$	− 2.00000i	− 0.208514i
$93$	0	0
$94$	0	0
$95$	0	0
$96$	0	0
$97$	14.0000i	1.42148i	0.703452	+	0.710742i	$0.251641\pi$
$97$	14.0000i	1.42148i	−0.703452	+	0.710742i	$0.748359\pi$
$98$	− 12.0000i	− 1.21218i
$99$	6.00000	0.603023
$100$	0	0
$101$	−5.00000	−0.497519	−0.248759	−	0.968565i	$-0.580023\pi$
$101$	−5.00000	−0.497519	−0.248759	+	0.968565i	$0.580023\pi$
$102$	0	0
$103$	− 8.00000i	− 0.788263i	−0.919054	−	0.394132i	$-0.871045\pi$
$103$	− 8.00000i	− 0.788263i	0.919054	−	0.394132i	$-0.128955\pi$
$104$	0	0
$105$	0	0
$106$	22.0000	2.13683
$107$	15.0000i	1.45010i	0.688694	+	0.725052i	$0.258184\pi$
$107$	15.0000i	1.45010i	−0.688694	+	0.725052i	$0.741816\pi$
$108$	0	0
$109$	10.0000	0.957826	0.478913	−	0.877862i	$-0.341031\pi$
$109$	10.0000	0.957826	0.478913	+	0.877862i	$0.341031\pi$
$110$	0	0
$111$	0	0
$112$	4.00000i	0.377964i
$113$	− 9.00000i	− 0.846649i	−0.905978	−	0.423324i	$-0.860863\pi$
$113$	− 9.00000i	− 0.846649i	0.905978	−	0.423324i	$-0.139137\pi$
$114$	0	0
$115$	0	0
$116$	14.0000	1.29987
$117$	− 6.00000i	− 0.554700i
$118$	− 26.0000i	− 2.39349i
$119$	−3.00000	−0.275010
$120$	0	0
$121$	−7.00000	−0.636364
$122$	16.0000i	1.44857i
$123$	0	0
$124$	10.0000	0.898027
$125$	0	0
$126$	−6.00000	−0.534522
$127$	4.00000i	0.354943i	0.984126	+	0.177471i	$0.0567917\pi$
$127$	4.00000i	0.354943i	−0.984126	+	0.177471i	$0.943208\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	12.0000	1.04844	0.524222	−	0.851581i	$-0.324356\pi$
$131$	12.0000	1.04844	0.524222	+	0.851581i	$0.324356\pi$
$132$	0	0
$133$	− 2.00000i	− 0.173422i
$134$	−10.0000	−0.863868
$135$	0	0
$136$	0	0
$137$	− 6.00000i	− 0.512615i	−0.966595	−	0.256307i	$-0.917494\pi$
$137$	− 6.00000i	− 0.512615i	0.966595	−	0.256307i	$-0.0825059\pi$
$138$	0	0
$139$	19.0000	1.61156	0.805779	−	0.592216i	$-0.201747\pi$
$139$	19.0000	1.61156	0.805779	+	0.592216i	$0.201747\pi$
$140$	0	0
$141$	0	0
$142$	− 10.0000i	− 0.839181i
$143$	− 4.00000i	− 0.334497i
$144$	−12.0000	−1.00000
$145$	0	0
$146$	12.0000	0.993127
$147$	0	0
$148$	22.0000i	1.80839i
$149$	−16.0000	−1.31077	−0.655386	−	0.755295i	$-0.727494\pi$
$149$	−16.0000	−1.31077	−0.655386	+	0.755295i	$0.727494\pi$
$150$	0	0
$151$	12.0000	0.976546	0.488273	−	0.872691i	$-0.337627\pi$
$151$	12.0000	0.976546	0.488273	+	0.872691i	$0.337627\pi$
$152$	0	0
$153$	− 9.00000i	− 0.727607i
$154$	−4.00000	−0.322329
$155$	0	0
$156$	0	0
$157$	− 17.0000i	− 1.35675i	−0.734717	−	0.678374i	$-0.762685\pi$
$157$	− 17.0000i	− 1.35675i	0.734717	−	0.678374i	$-0.237315\pi$
$158$	− 24.0000i	− 1.90934i
$159$	0	0
$160$	0	0
$161$	1.00000	0.0788110
$162$	− 18.0000i	− 1.41421i
$163$	− 18.0000i	− 1.40987i	−0.709273	−	0.704934i	$-0.750976\pi$
$163$	− 18.0000i	− 1.40987i	0.709273	−	0.704934i	$-0.249024\pi$
$164$	−2.00000	−0.156174
$165$	0	0
$166$	18.0000	1.39707
$167$	24.0000i	1.85718i	0.371113	+	0.928588i	$0.378976\pi$
$167$	24.0000i	1.85718i	−0.371113	+	0.928588i	$0.621024\pi$
$168$	0	0
$169$	9.00000	0.692308
$170$	0	0
$171$	6.00000	0.458831
$172$	0	0
$173$	24.0000i	1.82469i	0.409426	+	0.912343i	$0.365729\pi$
$173$	24.0000i	1.82469i	−0.409426	+	0.912343i	$0.634271\pi$
$174$	0	0
$175$	0	0
$176$	−8.00000	−0.603023
$177$	0	0
$178$	8.00000i	0.599625i
$179$	−24.0000	−1.79384	−0.896922	−	0.442189i	$-0.854202\pi$
$179$	−24.0000	−1.79384	−0.896922	+	0.442189i	$0.854202\pi$
$180$	0	0
$181$	16.0000	1.18927	0.594635	−	0.803996i	$-0.297296\pi$
$181$	16.0000	1.18927	0.594635	+	0.803996i	$0.297296\pi$
$182$	4.00000i	0.296500i
$183$	0	0
$184$	0	0
$185$	0	0
$186$	0	0
$187$	− 6.00000i	− 0.438763i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	0	0		−	1.00000i	$-0.5\pi$
$191$	0	0			1.00000i	$0.5\pi$
$192$	0	0
$193$	16.0000i	1.15171i	0.817554	+	0.575853i	$0.195330\pi$
$193$	16.0000i	1.15171i	−0.817554	+	0.575853i	$0.804670\pi$
$194$	28.0000	2.01028
$195$	0	0
$196$	−12.0000	−0.857143
$197$	2.00000i	0.142494i	0.997459	+	0.0712470i	$0.0226979\pi$
$197$	2.00000i	0.142494i	−0.997459	+	0.0712470i	$0.977302\pi$
$198$	− 12.0000i	− 0.852803i
$199$	−26.0000	−1.84309	−0.921546	−	0.388270i	$-0.873073\pi$
$199$	−26.0000	−1.84309	−0.921546	+	0.388270i	$0.873073\pi$
$200$	0	0
$201$	0	0
$202$	10.0000i	0.703598i
$203$	7.00000i	0.491304i
$204$	0	0
$205$	0	0
$206$	−16.0000	−1.11477
$207$	3.00000i	0.208514i
$208$	8.00000i	0.554700i
$209$	4.00000	0.276686
$210$	0	0
$211$	15.0000	1.03264	0.516321	−	0.856395i	$-0.327301\pi$
$211$	15.0000	1.03264	0.516321	+	0.856395i	$0.327301\pi$
$212$	− 22.0000i	− 1.51097i
$213$	0	0
$214$	30.0000	2.05076
$215$	0	0
$216$	0	0
$217$	5.00000i	0.339422i
$218$	− 20.0000i	− 1.35457i
$219$	0	0
$220$	0	0
$221$	−6.00000	−0.403604
$222$	0	0
$223$	14.0000i	0.937509i	0.883328	+	0.468755i	$0.155297\pi$
$223$	14.0000i	0.937509i	−0.883328	+	0.468755i	$0.844703\pi$
$224$	8.00000	0.534522
$225$	0	0
$226$	−18.0000	−1.19734
$227$	− 4.00000i	− 0.265489i	−0.991150	−	0.132745i	$-0.957621\pi$
$227$	− 4.00000i	− 0.265489i	0.991150	−	0.132745i	$-0.0423790\pi$
$228$	0	0
$229$	−8.00000	−0.528655	−0.264327	−	0.964433i	$-0.585150\pi$
$229$	−8.00000	−0.528655	−0.264327	+	0.964433i	$0.585150\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	4.00000i	0.262049i	0.991379	+	0.131024i	$0.0418266\pi$
$233$	4.00000i	0.262049i	−0.991379	+	0.131024i	$0.958173\pi$
$234$	−12.0000	−0.784465
$235$	0	0
$236$	−26.0000	−1.69246
$237$	0	0
$238$	6.00000i	0.388922i
$239$	29.0000	1.87585	0.937927	−	0.346833i	$-0.112743\pi$
$239$	29.0000	1.87585	0.937927	+	0.346833i	$0.112743\pi$
$240$	0	0
$241$	−18.0000	−1.15948	−0.579741	−	0.814801i	$-0.696846\pi$
$241$	−18.0000	−1.15948	−0.579741	+	0.814801i	$0.696846\pi$
$242$	14.0000i	0.899954i
$243$	0	0
$244$	16.0000	1.02430
$245$	0	0
$246$	0	0
$247$	− 4.00000i	− 0.254514i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−26.0000	−1.64111	−0.820553	−	0.571571i	$-0.806334\pi$
$251$	−26.0000	−1.64111	−0.820553	+	0.571571i	$0.806334\pi$
$252$	6.00000i	0.377964i
$253$	2.00000i	0.125739i
$254$	8.00000	0.501965
$255$	0	0
$256$	16.0000	1.00000
$257$	8.00000i	0.499026i	0.968371	+	0.249513i	$0.0802706\pi$
$257$	8.00000i	0.499026i	−0.968371	+	0.249513i	$0.919729\pi$
$258$	0	0
$259$	−11.0000	−0.683507
$260$	0	0
$261$	−21.0000	−1.29987
$262$	− 24.0000i	− 1.48272i
$263$	19.0000i	1.17159i	0.810459	+	0.585795i	$0.199218\pi$
$263$	19.0000i	1.17159i	−0.810459	+	0.585795i	$0.800782\pi$
$264$	0	0
$265$	0	0
$266$	−4.00000	−0.245256
$267$	0	0
$268$	10.0000i	0.610847i
$269$	−21.0000	−1.28039	−0.640196	−	0.768211i	$-0.721147\pi$
$269$	−21.0000	−1.28039	−0.640196	+	0.768211i	$0.721147\pi$
$270$	0	0
$271$	13.0000	0.789694	0.394847	−	0.918747i	$-0.370798\pi$
$271$	13.0000	0.789694	0.394847	+	0.918747i	$0.370798\pi$
$272$	12.0000i	0.727607i
$273$	0	0
$274$	−12.0000	−0.724947
$275$	0	0
$276$	0	0
$277$	− 6.00000i	− 0.360505i	−0.983620	−	0.180253i	$-0.942309\pi$
$277$	− 6.00000i	− 0.360505i	0.983620	−	0.180253i	$-0.0576915\pi$
$278$	− 38.0000i	− 2.27909i
$279$	−15.0000	−0.898027
$280$	0	0
$281$	−6.00000	−0.357930	−0.178965	−	0.983855i	$-0.557275\pi$
$281$	−6.00000	−0.357930	−0.178965	+	0.983855i	$0.557275\pi$
$282$	0	0
$283$	11.0000i	0.653882i	0.945045	+	0.326941i	$0.106018\pi$
$283$	11.0000i	0.653882i	−0.945045	+	0.326941i	$0.893982\pi$
$284$	−10.0000	−0.593391
$285$	0	0
$286$	−8.00000	−0.473050
$287$	− 1.00000i	− 0.0590281i
$288$	24.0000i	1.41421i
$289$	8.00000	0.470588
$290$	0	0
$291$	0	0
$292$	− 12.0000i	− 0.702247i
$293$	9.00000i	0.525786i	0.964825	+	0.262893i	$0.0846766\pi$
$293$	9.00000i	0.525786i	−0.964825	+	0.262893i	$0.915323\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	0	0
$298$	32.0000i	1.85371i
$299$	2.00000	0.115663
$300$	0	0
$301$	0	0
$302$	− 24.0000i	− 1.38104i
$303$	0	0
$304$	−8.00000	−0.458831
$305$	0	0
$306$	−18.0000	−1.02899
$307$	26.0000i	1.48390i	0.670456	+	0.741949i	$0.266098\pi$
$307$	26.0000i	1.48390i	−0.670456	+	0.741949i	$0.733902\pi$
$308$	4.00000i	0.227921i
$309$	0	0
$310$	0	0
$311$	−8.00000	−0.453638	−0.226819	−	0.973937i	$-0.572833\pi$
$311$	−8.00000	−0.453638	−0.226819	+	0.973937i	$0.572833\pi$
$312$	0	0
$313$	− 3.00000i	− 0.169570i	−0.996399	−	0.0847850i	$-0.972980\pi$
$313$	− 3.00000i	− 0.169570i	0.996399	−	0.0847850i	$-0.0270203\pi$
$314$	−34.0000	−1.91873
$315$	0	0
$316$	−24.0000	−1.35011
$317$	− 12.0000i	− 0.673987i	−0.941507	−	0.336994i	$-0.890590\pi$
$317$	− 12.0000i	− 0.673987i	0.941507	−	0.336994i	$-0.109410\pi$
$318$	0	0
$319$	−14.0000	−0.783850
$320$	0	0
$321$	0	0
$322$	− 2.00000i	− 0.111456i
$323$	− 6.00000i	− 0.333849i
$324$	−18.0000	−1.00000
$325$	0	0
$326$	−36.0000	−1.99386
$327$	0	0
$328$	0	0
$329$	0	0
$330$	0	0
$331$	11.0000	0.604615	0.302307	−	0.953211i	$-0.402243\pi$
$331$	11.0000	0.604615	0.302307	+	0.953211i	$0.402243\pi$
$332$	− 18.0000i	− 0.987878i
$333$	− 33.0000i	− 1.80839i
$334$	48.0000	2.62644
$335$	0	0
$336$	0	0
$337$	− 10.0000i	− 0.544735i	−0.962193	−	0.272367i	$-0.912193\pi$
$337$	− 10.0000i	− 0.544735i	0.962193	−	0.272367i	$-0.0878066\pi$
$338$	− 18.0000i	− 0.979071i
$339$	0	0
$340$	0	0
$341$	−10.0000	−0.541530
$342$	− 12.0000i	− 0.648886i
$343$	− 13.0000i	− 0.701934i
$344$	0	0
$345$	0	0
$346$	48.0000	2.58050
$347$	12.0000i	0.644194i	0.946707	+	0.322097i	$0.104388\pi$
$347$	12.0000i	0.644194i	−0.946707	+	0.322097i	$0.895612\pi$
$348$	0	0
$349$	−11.0000	−0.588817	−0.294408	−	0.955680i	$-0.595123\pi$
$349$	−11.0000	−0.588817	−0.294408	+	0.955680i	$0.595123\pi$
$350$	0	0
$351$	0	0
$352$	16.0000i	0.852803i
$353$	− 8.00000i	− 0.425797i	−0.977074	−	0.212899i	$-0.931710\pi$
$353$	− 8.00000i	− 0.425797i	0.977074	−	0.212899i	$-0.0682904\pi$
$354$	0	0
$355$	0	0
$356$	8.00000	0.423999
$357$	0	0
$358$	48.0000i	2.53688i
$359$	12.0000	0.633336	0.316668	−	0.948536i	$-0.397436\pi$
$359$	12.0000	0.633336	0.316668	+	0.948536i	$0.397436\pi$
$360$	0	0
$361$	−15.0000	−0.789474
$362$	− 32.0000i	− 1.68188i
$363$	0	0
$364$	4.00000	0.209657
$365$	0	0
$366$	0	0
$367$	13.0000i	0.678594i	0.940679	+	0.339297i	$0.110189\pi$
$367$	13.0000i	0.678594i	−0.940679	+	0.339297i	$0.889811\pi$
$368$	− 4.00000i	− 0.208514i
$369$	3.00000	0.156174
$370$	0	0
$371$	11.0000	0.571092
$372$	0	0
$373$	− 34.0000i	− 1.76045i	−0.474554	−	0.880227i	$-0.657390\pi$
$373$	− 34.0000i	− 1.76045i	0.474554	−	0.880227i	$-0.342610\pi$
$374$	−12.0000	−0.620505
$375$	0	0
$376$	0	0
$377$	14.0000i	0.721037i
$378$	0	0
$379$	−12.0000	−0.616399	−0.308199	−	0.951322i	$-0.599726\pi$
$379$	−12.0000	−0.616399	−0.308199	+	0.951322i	$0.599726\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	11.0000i	0.562074i	0.959697	+	0.281037i	$0.0906783\pi$
$383$	11.0000i	0.562074i	−0.959697	+	0.281037i	$0.909322\pi$
$384$	0	0
$385$	0	0
$386$	32.0000	1.62876
$387$	0	0
$388$	− 28.0000i	− 1.42148i
$389$	−10.0000	−0.507020	−0.253510	−	0.967333i	$-0.581585\pi$
$389$	−10.0000	−0.507020	−0.253510	+	0.967333i	$0.581585\pi$
$390$	0	0
$391$	3.00000	0.151717
$392$	0	0
$393$	0	0
$394$	4.00000	0.201517
$395$	0	0
$396$	−12.0000	−0.603023
$397$	− 4.00000i	− 0.200754i	−0.994949	−	0.100377i	$-0.967995\pi$
$397$	− 4.00000i	− 0.200754i	0.994949	−	0.100377i	$-0.0320049\pi$
$398$	52.0000i	2.60652i
$399$	0	0
$400$	0	0
$401$	2.00000	0.0998752	0.0499376	−	0.998752i	$-0.484098\pi$
$401$	2.00000	0.0998752	0.0499376	+	0.998752i	$0.484098\pi$
$402$	0	0
$403$	10.0000i	0.498135i
$404$	10.0000	0.497519
$405$	0	0
$406$	14.0000	0.694808
$407$	− 22.0000i	− 1.09050i
$408$	0	0
$409$	11.0000	0.543915	0.271957	−	0.962309i	$-0.412329\pi$
$409$	11.0000	0.543915	0.271957	+	0.962309i	$0.412329\pi$
$410$	0	0
$411$	0	0
$412$	16.0000i	0.788263i
$413$	− 13.0000i	− 0.639688i
$414$	6.00000	0.294884
$415$	0	0
$416$	16.0000	0.784465
$417$	0	0
$418$	− 8.00000i	− 0.391293i
$419$	18.0000	0.879358	0.439679	−	0.898155i	$-0.355092\pi$
$419$	18.0000	0.879358	0.439679	+	0.898155i	$0.355092\pi$
$420$	0	0
$421$	−2.00000	−0.0974740	−0.0487370	−	0.998812i	$-0.515520\pi$
$421$	−2.00000	−0.0974740	−0.0487370	+	0.998812i	$0.515520\pi$
$422$	− 30.0000i	− 1.46038i
$423$	0	0
$424$	0	0
$425$	0	0
$426$	0	0
$427$	8.00000i	0.387147i
$428$	− 30.0000i	− 1.45010i
$429$	0	0
$430$	0	0
$431$	28.0000	1.34871	0.674356	−	0.738406i	$-0.264421\pi$
$431$	28.0000	1.34871	0.674356	+	0.738406i	$0.264421\pi$
$432$	0	0
$433$	− 1.00000i	− 0.0480569i	−0.999711	−	0.0240285i	$-0.992351\pi$
$433$	− 1.00000i	− 0.0480569i	0.999711	−	0.0240285i	$-0.00764923\pi$
$434$	10.0000	0.480015
$435$	0	0
$436$	−20.0000	−0.957826
$437$	2.00000i	0.0956730i
$438$	0	0
$439$	−8.00000	−0.381819	−0.190910	−	0.981608i	$-0.561144\pi$
$439$	−8.00000	−0.381819	−0.190910	+	0.981608i	$0.561144\pi$
$440$	0	0
$441$	18.0000	0.857143
$442$	12.0000i	0.570782i
$443$	− 32.0000i	− 1.52037i	−0.649709	−	0.760183i	$-0.725109\pi$
$443$	− 32.0000i	− 1.52037i	0.649709	−	0.760183i	$-0.274891\pi$
$444$	0	0
$445$	0	0
$446$	28.0000	1.32584
$447$	0	0
$448$	− 8.00000i	− 0.377964i
$449$	21.0000	0.991051	0.495526	−	0.868593i	$-0.334975\pi$
$449$	21.0000	0.991051	0.495526	+	0.868593i	$0.334975\pi$
$450$	0	0
$451$	2.00000	0.0941763
$452$	18.0000i	0.846649i
$453$	0	0
$454$	−8.00000	−0.375459
$455$	0	0
$456$	0	0
$457$	− 25.0000i	− 1.16945i	−0.811231	−	0.584725i	$-0.801202\pi$
$457$	− 25.0000i	− 1.16945i	0.811231	−	0.584725i	$-0.198798\pi$
$458$	16.0000i	0.747631i
$459$	0	0
$460$	0	0
$461$	−2.00000	−0.0931493	−0.0465746	−	0.998915i	$-0.514831\pi$
$461$	−2.00000	−0.0931493	−0.0465746	+	0.998915i	$0.514831\pi$
$462$	0	0
$463$	− 10.0000i	− 0.464739i	−0.972628	−	0.232370i	$-0.925352\pi$
$463$	− 10.0000i	− 0.464739i	0.972628	−	0.232370i	$-0.0746479\pi$
$464$	28.0000	1.29987
$465$	0	0
$466$	8.00000	0.370593
$467$	− 33.0000i	− 1.52706i	−0.645774	−	0.763529i	$-0.723465\pi$
$467$	− 33.0000i	− 1.52706i	0.645774	−	0.763529i	$-0.276535\pi$
$468$	12.0000i	0.554700i
$469$	−5.00000	−0.230879
$470$	0	0
$471$	0	0
$472$	0	0
$473$	0	0
$474$	0	0
$475$	0	0
$476$	6.00000	0.275010
$477$	33.0000i	1.51097i
$478$	− 58.0000i	− 2.65286i
$479$	−12.0000	−0.548294	−0.274147	−	0.961688i	$-0.588395\pi$
$479$	−12.0000	−0.548294	−0.274147	+	0.961688i	$0.588395\pi$
$480$	0	0
$481$	−22.0000	−1.00311
$482$	36.0000i	1.63976i
$483$	0	0
$484$	14.0000	0.636364
$485$	0	0
$486$	0	0
$487$	28.0000i	1.26880i	0.773004	+	0.634401i	$0.218753\pi$
$487$	28.0000i	1.26880i	−0.773004	+	0.634401i	$0.781247\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	9.00000	0.406164	0.203082	−	0.979162i	$-0.434904\pi$
$491$	9.00000	0.406164	0.203082	+	0.979162i	$0.434904\pi$
$492$	0	0
$493$	21.0000i	0.945792i
$494$	−8.00000	−0.359937
$495$	0	0
$496$	20.0000	0.898027
$497$	− 5.00000i	− 0.224281i
$498$	0	0
$499$	−1.00000	−0.0447661	−0.0223831	−	0.999749i	$-0.507125\pi$
$499$	−1.00000	−0.0447661	−0.0223831	+	0.999749i	$0.507125\pi$
$500$	0	0
$501$	0	0
$502$	52.0000i	2.32087i
$503$	− 33.0000i	− 1.47140i	−0.677309	−	0.735699i	$-0.736854\pi$
$503$	− 33.0000i	− 1.47140i	0.677309	−	0.735699i	$-0.263146\pi$
$504$	0	0
$505$	0	0
$506$	4.00000	0.177822
$507$	0	0
$508$	− 8.00000i	− 0.354943i
$509$	−18.0000	−0.797836	−0.398918	−	0.916987i	$-0.630614\pi$
$509$	−18.0000	−0.797836	−0.398918	+	0.916987i	$0.630614\pi$
$510$	0	0
$511$	6.00000	0.265424
$512$	− 32.0000i	− 1.41421i
$513$	0	0
$514$	16.0000	0.705730
$515$	0	0
$516$	0	0
$517$	0	0
$518$	22.0000i	0.966625i
$519$	0	0
$520$	0	0
$521$	−42.0000	−1.84005	−0.920027	−	0.391856i	$-0.871833\pi$
$521$	−42.0000	−1.84005	−0.920027	+	0.391856i	$0.871833\pi$
$522$	42.0000i	1.83829i
$523$	− 4.00000i	− 0.174908i	−0.996169	−	0.0874539i	$-0.972127\pi$
$523$	− 4.00000i	− 0.174908i	0.996169	−	0.0874539i	$-0.0278730\pi$
$524$	−24.0000	−1.04844
$525$	0	0
$526$	38.0000	1.65688
$527$	15.0000i	0.653410i
$528$	0	0
$529$	−1.00000	−0.0434783
$530$	0	0
$531$	39.0000	1.69246
$532$	4.00000i	0.173422i
$533$	− 2.00000i	− 0.0866296i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	0	0
$538$	42.0000i	1.81075i
$539$	12.0000	0.516877
$540$	0	0
$541$	−26.0000	−1.11783	−0.558914	−	0.829226i	$-0.688782\pi$
$541$	−26.0000	−1.11783	−0.558914	+	0.829226i	$0.688782\pi$
$542$	− 26.0000i	− 1.11680i
$543$	0	0
$544$	24.0000	1.02899
$545$	0	0
$546$	0	0
$547$	− 24.0000i	− 1.02617i	−0.858339	−	0.513083i	$-0.828503\pi$
$547$	− 24.0000i	− 1.02617i	0.858339	−	0.513083i	$-0.171497\pi$
$548$	12.0000i	0.512615i
$549$	−24.0000	−1.02430
$550$	0	0
$551$	−14.0000	−0.596420
$552$	0	0
$553$	− 12.0000i	− 0.510292i
$554$	−12.0000	−0.509831
$555$	0	0
$556$	−38.0000	−1.61156
$557$	− 11.0000i	− 0.466085i	−0.972467	−	0.233042i	$-0.925132\pi$
$557$	− 11.0000i	− 0.466085i	0.972467	−	0.233042i	$-0.0748681\pi$
$558$	30.0000i	1.27000i
$559$	0	0
$560$	0	0
$561$	0	0
$562$	12.0000i	0.506189i
$563$	31.0000i	1.30649i	0.757145	+	0.653247i	$0.226594\pi$
$563$	31.0000i	1.30649i	−0.757145	+	0.653247i	$0.773406\pi$
$564$	0	0
$565$	0	0
$566$	22.0000	0.924729
$567$	− 9.00000i	− 0.377964i
$568$	0	0
$569$	6.00000	0.251533	0.125767	−	0.992060i	$-0.459861\pi$
$569$	6.00000	0.251533	0.125767	+	0.992060i	$0.459861\pi$
$570$	0	0
$571$	−16.0000	−0.669579	−0.334790	−	0.942293i	$-0.608665\pi$
$571$	−16.0000	−0.669579	−0.334790	+	0.942293i	$0.608665\pi$
$572$	8.00000i	0.334497i
$573$	0	0
$574$	−2.00000	−0.0834784
$575$	0	0
$576$	24.0000	1.00000
$577$	− 20.0000i	− 0.832611i	−0.909225	−	0.416305i	$-0.863325\pi$
$577$	− 20.0000i	− 0.832611i	0.909225	−	0.416305i	$-0.136675\pi$
$578$	− 16.0000i	− 0.665512i
$579$	0	0
$580$	0	0
$581$	9.00000	0.373383
$582$	0	0
$583$	22.0000i	0.911147i
$584$	0	0
$585$	0	0
$586$	18.0000	0.743573
$587$	− 2.00000i	− 0.0825488i	−0.999148	−	0.0412744i	$-0.986858\pi$
$587$	− 2.00000i	− 0.0825488i	0.999148	−	0.0412744i	$-0.0131418\pi$
$588$	0	0
$589$	−10.0000	−0.412043
$590$	0	0
$591$	0	0
$592$	44.0000i	1.80839i
$593$	− 4.00000i	− 0.164260i	−0.996622	−	0.0821302i	$-0.973828\pi$
$593$	− 4.00000i	− 0.164260i	0.996622	−	0.0821302i	$-0.0261723\pi$
$594$	0	0
$595$	0	0
$596$	32.0000	1.31077
$597$	0	0
$598$	− 4.00000i	− 0.163572i
$599$	24.0000	0.980613	0.490307	−	0.871550i	$-0.336885\pi$
$599$	24.0000	0.980613	0.490307	+	0.871550i	$0.336885\pi$
$600$	0	0
$601$	−21.0000	−0.856608	−0.428304	−	0.903635i	$-0.640889\pi$
$601$	−21.0000	−0.856608	−0.428304	+	0.903635i	$0.640889\pi$
$602$	0	0
$603$	− 15.0000i	− 0.610847i
$604$	−24.0000	−0.976546
$605$	0	0
$606$	0	0
$607$	− 32.0000i	− 1.29884i	−0.760430	−	0.649420i	$-0.775012\pi$
$607$	− 32.0000i	− 1.29884i	0.760430	−	0.649420i	$-0.224988\pi$
$608$	16.0000i	0.648886i
$609$	0	0
$610$	0	0
$611$	0	0
$612$	18.0000i	0.727607i
$613$	− 26.0000i	− 1.05013i	−0.851062	−	0.525065i	$-0.824041\pi$
$613$	− 26.0000i	− 1.05013i	0.851062	−	0.525065i	$-0.175959\pi$
$614$	52.0000	2.09855
$615$	0	0
$616$	0	0
$617$	− 13.0000i	− 0.523360i	−0.965155	−	0.261680i	$-0.915723\pi$
$617$	− 13.0000i	− 0.523360i	0.965155	−	0.261680i	$-0.0842766\pi$
$618$	0	0
$619$	−36.0000	−1.44696	−0.723481	−	0.690344i	$-0.757459\pi$
$619$	−36.0000	−1.44696	−0.723481	+	0.690344i	$0.757459\pi$
$620$	0	0
$621$	0	0
$622$	16.0000i	0.641542i
$623$	4.00000i	0.160257i
$624$	0	0
$625$	0	0
$626$	−6.00000	−0.239808
$627$	0	0
$628$	34.0000i	1.35675i
$629$	−33.0000	−1.31580
$630$	0	0
$631$	40.0000	1.59237	0.796187	−	0.605050i	$-0.206847\pi$
$631$	40.0000	1.59237	0.796187	+	0.605050i	$0.206847\pi$
$632$	0	0
$633$	0	0
$634$	−24.0000	−0.953162
$635$	0	0
$636$	0	0
$637$	− 12.0000i	− 0.475457i
$638$	28.0000i	1.10853i
$639$	15.0000	0.593391
$640$	0	0
$641$	14.0000	0.552967	0.276483	−	0.961019i	$-0.410831\pi$
$641$	14.0000	0.552967	0.276483	+	0.961019i	$0.410831\pi$
$642$	0	0
$643$	25.0000i	0.985904i	0.870057	+	0.492952i	$0.164082\pi$
$643$	25.0000i	0.985904i	−0.870057	+	0.492952i	$0.835918\pi$
$644$	−2.00000	−0.0788110
$645$	0	0
$646$	−12.0000	−0.472134
$647$	50.0000i	1.96570i	0.184399	+	0.982851i	$0.440966\pi$
$647$	50.0000i	1.96570i	−0.184399	+	0.982851i	$0.559034\pi$
$648$	0	0
$649$	26.0000	1.02059
$650$	0	0
$651$	0	0
$652$	36.0000i	1.40987i
$653$	18.0000i	0.704394i	0.935926	+	0.352197i	$0.114565\pi$
$653$	18.0000i	0.704394i	−0.935926	+	0.352197i	$0.885435\pi$
$654$	0	0
$655$	0	0
$656$	−4.00000	−0.156174
$657$	18.0000i	0.702247i
$658$	0	0
$659$	30.0000	1.16863	0.584317	−	0.811525i	$-0.301362\pi$
$659$	30.0000	1.16863	0.584317	+	0.811525i	$0.301362\pi$
$660$	0	0
$661$	−14.0000	−0.544537	−0.272268	−	0.962221i	$-0.587774\pi$
$661$	−14.0000	−0.544537	−0.272268	+	0.962221i	$0.587774\pi$
$662$	− 22.0000i	− 0.855054i
$663$	0	0
$664$	0	0
$665$	0	0
$666$	−66.0000	−2.55745
$667$	− 7.00000i	− 0.271041i
$668$	− 48.0000i	− 1.85718i
$669$	0	0
$670$	0	0
$671$	−16.0000	−0.617673
$672$	0	0
$673$	28.0000i	1.07932i	0.841883	+	0.539660i	$0.181447\pi$
$673$	28.0000i	1.07932i	−0.841883	+	0.539660i	$0.818553\pi$
$674$	−20.0000	−0.770371
$675$	0	0
$676$	−18.0000	−0.692308
$677$	27.0000i	1.03769i	0.854867	+	0.518847i	$0.173639\pi$
$677$	27.0000i	1.03769i	−0.854867	+	0.518847i	$0.826361\pi$
$678$	0	0
$679$	14.0000	0.537271
$680$	0	0
$681$	0	0
$682$	20.0000i	0.765840i
$683$	− 28.0000i	− 1.07139i	−0.844411	−	0.535695i	$-0.820050\pi$
$683$	− 28.0000i	− 1.07139i	0.844411	−	0.535695i	$-0.179950\pi$
$684$	−12.0000	−0.458831
$685$	0	0
$686$	−26.0000	−0.992685
$687$	0	0
$688$	0	0
$689$	22.0000	0.838133
$690$	0	0
$691$	−44.0000	−1.67384	−0.836919	−	0.547326i	$-0.815646\pi$
$691$	−44.0000	−1.67384	−0.836919	+	0.547326i	$0.815646\pi$
$692$	− 48.0000i	− 1.82469i
$693$	− 6.00000i	− 0.227921i
$694$	24.0000	0.911028
$695$	0	0
$696$	0	0
$697$	− 3.00000i	− 0.113633i
$698$	22.0000i	0.832712i
$699$	0	0
$700$	0	0
$701$	0	0		−	1.00000i	$-0.5\pi$
$701$	0	0			1.00000i	$0.5\pi$
$702$	0	0
$703$	− 22.0000i	− 0.829746i
$704$	16.0000	0.603023
$705$	0	0
$706$	−16.0000	−0.602168
$707$	5.00000i	0.188044i
$708$	0	0
$709$	−12.0000	−0.450669	−0.225335	−	0.974281i	$-0.572348\pi$
$709$	−12.0000	−0.450669	−0.225335	+	0.974281i	$0.572348\pi$
$710$	0	0
$711$	36.0000	1.35011
$712$	0	0
$713$	− 5.00000i	− 0.187251i
$714$	0	0
$715$	0	0
$716$	48.0000	1.79384
$717$	0	0
$718$	− 24.0000i	− 0.895672i
$719$	37.0000	1.37987	0.689934	−	0.723873i	$-0.257640\pi$
$719$	37.0000	1.37987	0.689934	+	0.723873i	$0.257640\pi$
$720$	0	0
$721$	−8.00000	−0.297936
$722$	30.0000i	1.11648i
$723$	0	0
$724$	−32.0000	−1.18927
$725$	0	0
$726$	0	0
$727$	− 19.0000i	− 0.704671i	−0.935874	−	0.352335i	$-0.885388\pi$
$727$	− 19.0000i	− 0.704671i	0.935874	−	0.352335i	$-0.114612\pi$
$728$	0	0
$729$	27.0000	1.00000
$730$	0	0
$731$	0	0
$732$	0	0
$733$	11.0000i	0.406294i	0.979148	+	0.203147i	$0.0651170\pi$
$733$	11.0000i	0.406294i	−0.979148	+	0.203147i	$0.934883\pi$
$734$	26.0000	0.959678
$735$	0	0
$736$	−8.00000	−0.294884
$737$	− 10.0000i	− 0.368355i
$738$	− 6.00000i	− 0.220863i
$739$	−23.0000	−0.846069	−0.423034	−	0.906114i	$-0.639035\pi$
$739$	−23.0000	−0.846069	−0.423034	+	0.906114i	$0.639035\pi$
$740$	0	0
$741$	0	0
$742$	− 22.0000i	− 0.807645i
$743$	16.0000i	0.586983i	0.955962	+	0.293492i	$0.0948173\pi$
$743$	16.0000i	0.586983i	−0.955962	+	0.293492i	$0.905183\pi$
$744$	0	0
$745$	0	0
$746$	−68.0000	−2.48966
$747$	27.0000i	0.987878i
$748$	12.0000i	0.438763i
$749$	15.0000	0.548088
$750$	0	0
$751$	−22.0000	−0.802791	−0.401396	−	0.915905i	$-0.631475\pi$
$751$	−22.0000	−0.802791	−0.401396	+	0.915905i	$0.631475\pi$
$752$	0	0
$753$	0	0
$754$	28.0000	1.01970
$755$	0	0
$756$	0	0
$757$	− 37.0000i	− 1.34479i	−0.740193	−	0.672394i	$-0.765266\pi$
$757$	− 37.0000i	− 1.34479i	0.740193	−	0.672394i	$-0.234734\pi$
$758$	24.0000i	0.871719i
$759$	0	0
$760$	0	0
$761$	45.0000	1.63125	0.815624	−	0.578582i	$-0.196394\pi$
$761$	45.0000	1.63125	0.815624	+	0.578582i	$0.196394\pi$
$762$	0	0
$763$	− 10.0000i	− 0.362024i
$764$	0	0
$765$	0	0
$766$	22.0000	0.794892
$767$	− 26.0000i	− 0.938806i
$768$	0	0
$769$	−32.0000	−1.15395	−0.576975	−	0.816762i	$-0.695767\pi$
$769$	−32.0000	−1.15395	−0.576975	+	0.816762i	$0.695767\pi$
$770$	0	0
$771$	0	0
$772$	− 32.0000i	− 1.15171i
$773$	− 10.0000i	− 0.359675i	−0.983696	−	0.179838i	$-0.942443\pi$
$773$	− 10.0000i	− 0.359675i	0.983696	−	0.179838i	$-0.0575572\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	0	0
$778$	20.0000i	0.717035i
$779$	2.00000	0.0716574
$780$	0	0
$781$	10.0000	0.357828
$782$	− 6.00000i	− 0.214560i
$783$	0	0
$784$	−24.0000	−0.857143
$785$	0	0
$786$	0	0
$787$	7.00000i	0.249523i	0.992187	+	0.124762i	$0.0398166\pi$
$787$	7.00000i	0.249523i	−0.992187	+	0.124762i	$0.960183\pi$
$788$	− 4.00000i	− 0.142494i
$789$	0	0
$790$	0	0
$791$	−9.00000	−0.320003
$792$	0	0
$793$	16.0000i	0.568177i
$794$	−8.00000	−0.283909
$795$	0	0
$796$	52.0000	1.84309
$797$	− 53.0000i	− 1.87736i	−0.344795	−	0.938678i	$-0.612051\pi$
$797$	− 53.0000i	− 1.87736i	0.344795	−	0.938678i	$-0.387949\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	−12.0000	−0.423999
$802$	− 4.00000i	− 0.141245i
$803$	12.0000i	0.423471i
$804$	0	0
$805$	0	0
$806$	20.0000	0.704470
$807$	0	0
$808$	0	0
$809$	−15.0000	−0.527372	−0.263686	−	0.964609i	$-0.584938\pi$
$809$	−15.0000	−0.527372	−0.263686	+	0.964609i	$0.584938\pi$
$810$	0	0
$811$	15.0000	0.526721	0.263361	−	0.964697i	$-0.415169\pi$
$811$	15.0000	0.526721	0.263361	+	0.964697i	$0.415169\pi$
$812$	− 14.0000i	− 0.491304i
$813$	0	0
$814$	−44.0000	−1.54220
$815$	0	0
$816$	0	0
$817$	0	0
$818$	− 22.0000i	− 0.769212i
$819$	−6.00000	−0.209657
$820$	0	0
$821$	10.0000	0.349002	0.174501	−	0.984657i	$-0.444169\pi$
$821$	10.0000	0.349002	0.174501	+	0.984657i	$0.444169\pi$
$822$	0	0
$823$	− 16.0000i	− 0.557725i	−0.960331	−	0.278862i	$-0.910043\pi$
$823$	− 16.0000i	− 0.557725i	0.960331	−	0.278862i	$-0.0899574\pi$
$824$	0	0
$825$	0	0
$826$	−26.0000	−0.904656
$827$	29.0000i	1.00843i	0.863579	+	0.504214i	$0.168218\pi$
$827$	29.0000i	1.00843i	−0.863579	+	0.504214i	$0.831782\pi$
$828$	− 6.00000i	− 0.208514i
$829$	−45.0000	−1.56291	−0.781457	−	0.623959i	$-0.785523\pi$
$829$	−45.0000	−1.56291	−0.781457	+	0.623959i	$0.785523\pi$
$830$	0	0
$831$	0	0
$832$	− 16.0000i	− 0.554700i
$833$	− 18.0000i	− 0.623663i
$834$	0	0
$835$	0	0
$836$	−8.00000	−0.276686
$837$	0	0
$838$	− 36.0000i	− 1.24360i
$839$	−54.0000	−1.86429	−0.932144	−	0.362089i	$-0.882064\pi$
$839$	−54.0000	−1.86429	−0.932144	+	0.362089i	$0.882064\pi$
$840$	0	0
$841$	20.0000	0.689655
$842$	4.00000i	0.137849i
$843$	0	0
$844$	−30.0000	−1.03264
$845$	0	0
$846$	0	0
$847$	7.00000i	0.240523i
$848$	− 44.0000i	− 1.51097i
$849$	0	0
$850$	0	0
$851$	11.0000	0.377075
$852$	0	0
$853$	− 14.0000i	− 0.479351i	−0.970853	−	0.239675i	$-0.922959\pi$
$853$	− 14.0000i	− 0.479351i	0.970853	−	0.239675i	$-0.0770410\pi$
$854$	16.0000	0.547509
$855$	0	0
$856$	0	0
$857$	− 22.0000i	− 0.751506i	−0.926720	−	0.375753i	$-0.877384\pi$
$857$	− 22.0000i	− 0.751506i	0.926720	−	0.375753i	$-0.122616\pi$
$858$	0	0
$859$	5.00000	0.170598	0.0852989	−	0.996355i	$-0.472815\pi$
$859$	5.00000	0.170598	0.0852989	+	0.996355i	$0.472815\pi$
$860$	0	0
$861$	0	0
$862$	− 56.0000i	− 1.90737i
$863$	− 18.0000i	− 0.612727i	−0.951915	−	0.306364i	$-0.900888\pi$
$863$	− 18.0000i	− 0.612727i	0.951915	−	0.306364i	$-0.0991123\pi$
$864$	0	0
$865$	0	0
$866$	−2.00000	−0.0679628
$867$	0	0
$868$	− 10.0000i	− 0.339422i
$869$	24.0000	0.814144
$870$	0	0
$871$	−10.0000	−0.338837
$872$	0	0
$873$	42.0000i	1.42148i
$874$	4.00000	0.135302
$875$	0	0
$876$	0	0
$877$	2.00000i	0.0675352i	0.999430	+	0.0337676i	$0.0107506\pi$
$877$	2.00000i	0.0675352i	−0.999430	+	0.0337676i	$0.989249\pi$
$878$	16.0000i	0.539974i
$879$	0	0
$880$	0	0
$881$	−32.0000	−1.07811	−0.539054	−	0.842271i	$-0.681218\pi$
$881$	−32.0000	−1.07811	−0.539054	+	0.842271i	$0.681218\pi$
$882$	− 36.0000i	− 1.21218i
$883$	− 56.0000i	− 1.88455i	−0.334840	−	0.942275i	$-0.608682\pi$
$883$	− 56.0000i	− 1.88455i	0.334840	−	0.942275i	$-0.391318\pi$
$884$	12.0000	0.403604
$885$	0	0
$886$	−64.0000	−2.15012
$887$	26.0000i	0.872995i	0.899706	+	0.436497i	$0.143781\pi$
$887$	26.0000i	0.872995i	−0.899706	+	0.436497i	$0.856219\pi$
$888$	0	0
$889$	4.00000	0.134156
$890$	0	0
$891$	18.0000	0.603023
$892$	− 28.0000i	− 0.937509i
$893$	0	0
$894$	0	0
$895$	0	0
$896$	0	0
$897$	0	0
$898$	− 42.0000i	− 1.40156i
$899$	35.0000	1.16732
$900$	0	0
$901$	33.0000	1.09939
$902$	− 4.00000i	− 0.133185i
$903$	0	0
$904$	0	0
$905$	0	0
$906$	0	0
$907$	− 5.00000i	− 0.166022i	−0.996549	−	0.0830111i	$-0.973546\pi$
$907$	− 5.00000i	− 0.166022i	0.996549	−	0.0830111i	$-0.0264537\pi$
$908$	8.00000i	0.265489i
$909$	−15.0000	−0.497519
$910$	0	0
$911$	−40.0000	−1.32526	−0.662630	−	0.748947i	$-0.730560\pi$
$911$	−40.0000	−1.32526	−0.662630	+	0.748947i	$0.730560\pi$
$912$	0	0
$913$	18.0000i	0.595713i
$914$	−50.0000	−1.65385
$915$	0	0
$916$	16.0000	0.528655
$917$	− 12.0000i	− 0.396275i
$918$	0	0
$919$	4.00000	0.131948	0.0659739	−	0.997821i	$-0.478985\pi$
$919$	4.00000	0.131948	0.0659739	+	0.997821i	$0.478985\pi$
$920$	0	0
$921$	0	0
$922$	4.00000i	0.131733i
$923$	− 10.0000i	− 0.329154i
$924$	0	0
$925$	0	0
$926$	−20.0000	−0.657241
$927$	− 24.0000i	− 0.788263i
$928$	− 56.0000i	− 1.83829i
$929$	−27.0000	−0.885841	−0.442921	−	0.896561i	$-0.646058\pi$
$929$	−27.0000	−0.885841	−0.442921	+	0.896561i	$0.646058\pi$
$930$	0	0
$931$	12.0000	0.393284
$932$	− 8.00000i	− 0.262049i
$933$	0	0
$934$	−66.0000	−2.15959
$935$	0	0
$936$	0	0
$937$	22.0000i	0.718709i	0.933201	+	0.359354i	$0.117003\pi$
$937$	22.0000i	0.718709i	−0.933201	+	0.359354i	$0.882997\pi$
$938$	10.0000i	0.326512i
$939$	0	0
$940$	0	0
$941$	−24.0000	−0.782378	−0.391189	−	0.920310i	$-0.627936\pi$
$941$	−24.0000	−0.782378	−0.391189	+	0.920310i	$0.627936\pi$
$942$	0	0
$943$	1.00000i	0.0325645i
$944$	−52.0000	−1.69246
$945$	0	0
$946$	0	0
$947$	52.0000i	1.68977i	0.534946	+	0.844886i	$0.320332\pi$
$947$	52.0000i	1.68977i	−0.534946	+	0.844886i	$0.679668\pi$
$948$	0	0
$949$	12.0000	0.389536
$950$	0	0
$951$	0	0
$952$	0	0
$953$	26.0000i	0.842223i	0.907009	+	0.421111i	$0.138360\pi$
$953$	26.0000i	0.842223i	−0.907009	+	0.421111i	$0.861640\pi$
$954$	66.0000	2.13683
$955$	0	0
$956$	−58.0000	−1.87585
$957$	0	0
$958$	24.0000i	0.775405i
$959$	−6.00000	−0.193750
$960$	0	0
$961$	−6.00000	−0.193548
$962$	44.0000i	1.41862i
$963$	45.0000i	1.45010i
$964$	36.0000	1.15948
$965$	0	0
$966$	0	0
$967$	− 44.0000i	− 1.41494i	−0.706741	−	0.707472i	$-0.749835\pi$
$967$	− 44.0000i	− 1.41494i	0.706741	−	0.707472i	$-0.250165\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	12.0000	0.385098	0.192549	−	0.981287i	$-0.438325\pi$
$971$	12.0000	0.385098	0.192549	+	0.981287i	$0.438325\pi$
$972$	0	0
$973$	− 19.0000i	− 0.609112i
$974$	56.0000	1.79436
$975$	0	0
$976$	32.0000	1.02430
$977$	39.0000i	1.24772i	0.781536	+	0.623860i	$0.214437\pi$
$977$	39.0000i	1.24772i	−0.781536	+	0.623860i	$0.785563\pi$
$978$	0	0
$979$	−8.00000	−0.255681
$980$	0	0
$981$	30.0000	0.957826
$982$	− 18.0000i	− 0.574403i
$983$	9.00000i	0.287055i	0.989646	+	0.143528i	$0.0458446\pi$
$983$	9.00000i	0.287055i	−0.989646	+	0.143528i	$0.954155\pi$
$984$	0	0
$985$	0	0
$986$	42.0000	1.33755
$987$	0	0
$988$	8.00000i	0.254514i
$989$	0	0
$990$	0	0
$991$	−1.00000	−0.0317660	−0.0158830	−	0.999874i	$-0.505056\pi$
$991$	−1.00000	−0.0317660	−0.0158830	+	0.999874i	$0.505056\pi$
$992$	− 40.0000i	− 1.27000i
$993$	0	0
$994$	−10.0000	−0.317181
$995$	0	0
$996$	0	0
$997$	40.0000i	1.26681i	0.773819	+	0.633406i	$0.218344\pi$
$997$	40.0000i	1.26681i	−0.773819	+	0.633406i	$0.781656\pi$
$998$	2.00000i	0.0633089i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	575.2.b.a.24.1		2
5.2	odd	4		115.2.a.a.1.1	✓	1
5.3	odd	4		575.2.a.b.1.1		1
5.4	even	2	inner	575.2.b.a.24.2		2
15.2	even	4		1035.2.a.b.1.1		1
15.8	even	4		5175.2.a.y.1.1		1
20.3	even	4		9200.2.a.t.1.1		1
20.7	even	4		1840.2.a.d.1.1		1
35.27	even	4		5635.2.a.j.1.1		1
40.27	even	4		7360.2.a.n.1.1		1
40.37	odd	4		7360.2.a.q.1.1		1
115.22	even	4		2645.2.a.c.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
115.2.a.a.1.1	✓	1	5.2	odd	4
575.2.a.b.1.1		1	5.3	odd	4
575.2.b.a.24.1		2	1.1	even	1	trivial
575.2.b.a.24.2		2	5.4	even	2	inner
1035.2.a.b.1.1		1	15.2	even	4
1840.2.a.d.1.1		1	20.7	even	4
2645.2.a.c.1.1		1	115.22	even	4
5175.2.a.y.1.1		1	15.8	even	4
5635.2.a.j.1.1		1	35.27	even	4
7360.2.a.n.1.1		1	40.27	even	4
7360.2.a.q.1.1		1	40.37	odd	4
9200.2.a.t.1.1		1	20.3	even	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 575 = 5^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	575.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-2.00000i q^{2} -2.00000 q^{4} -1.00000i q^{7} +3.00000 q^{9} +O(q^{10})\)	\(q-2.00000i q^{2} -2.00000 q^{4} -1.00000i q^{7} +3.00000 q^{9} +2.00000 q^{11} -2.00000i q^{13} -2.00000 q^{14} -4.00000 q^{16} -3.00000i q^{17} -6.00000i q^{18} +2.00000 q^{19} -4.00000i q^{22} +1.00000i q^{23} -4.00000 q^{26} +2.00000i q^{28} -7.00000 q^{29} -5.00000 q^{31} +8.00000i q^{32} -6.00000 q^{34} -6.00000 q^{36} -11.0000i q^{37} -4.00000i q^{38} +1.00000 q^{41} -4.00000 q^{44} +2.00000 q^{46} +6.00000 q^{49} +4.00000i q^{52} +11.0000i q^{53} +14.0000i q^{58} +13.0000 q^{59} -8.00000 q^{61} +10.0000i q^{62} -3.00000i q^{63} +8.00000 q^{64} -5.00000i q^{67} +6.00000i q^{68} +5.00000 q^{71} +6.00000i q^{73} -22.0000 q^{74} -4.00000 q^{76} -2.00000i q^{77} +12.0000 q^{79} +9.00000 q^{81} -2.00000i q^{82} +9.00000i q^{83} -4.00000 q^{89} -2.00000 q^{91} -2.00000i q^{92} +14.0000i q^{97} -12.0000i q^{98} +6.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 4 q^{4} + 6 q^{9}+O(q^{10}) \) 2 * q - 4 * q^4 + 6 * q^9	\( 2 q - 4 q^{4} + 6 q^{9} + 4 q^{11} - 4 q^{14} - 8 q^{16} + 4 q^{19} - 8 q^{26} - 14 q^{29} - 10 q^{31} - 12 q^{34} - 12 q^{36} + 2 q^{41} - 8 q^{44} + 4 q^{46} + 12 q^{49} + 26 q^{59} - 16 q^{61} + 16 q^{64} + 10 q^{71} - 44 q^{74} - 8 q^{76} + 24 q^{79} + 18 q^{81} - 8 q^{89} - 4 q^{91} + 12 q^{99}+O(q^{100}) \) 2 * q - 4 * q^4 + 6 * q^9 + 4 * q^11 - 4 * q^14 - 8 * q^16 + 4 * q^19 - 8 * q^26 - 14 * q^29 - 10 * q^31 - 12 * q^34 - 12 * q^36 + 2 * q^41 - 8 * q^44 + 4 * q^46 + 12 * q^49 + 26 * q^59 - 16 * q^61 + 16 * q^64 + 10 * q^71 - 44 * q^74 - 8 * q^76 + 24 * q^79 + 18 * q^81 - 8 * q^89 - 4 * q^91 + 12 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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