LMFDB - Embedded newform 1725.2.b.f.1174.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1725,2,Mod(1174,1725)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1725, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("1725.1174");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1725 = 3 \cdot 5^{2} \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1725.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:	$13.7741943487$
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:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 345)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1174.2
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	1725.1174
Dual form			1725.2.b.f.1174.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{2} -1.00000i q^{3} +1.00000 q^{4} +1.00000 q^{6} +4.00000i q^{7} +3.00000i q^{8} -1.00000 q^{9} +O(q^{10})$	\(q+1.00000i q^{2} -1.00000i q^{3} +1.00000 q^{4} +1.00000 q^{6} +4.00000i q^{7} +3.00000i q^{8} -1.00000 q^{9} +4.00000 q^{11} -1.00000i q^{12} -6.00000i q^{13} -4.00000 q^{14} -1.00000 q^{16} -2.00000i q^{17} -1.00000i q^{18} +4.00000 q^{19} +4.00000 q^{21} +4.00000i q^{22} +1.00000i q^{23} +3.00000 q^{24} +6.00000 q^{26} +1.00000i q^{27} +4.00000i q^{28} +10.0000 q^{29} -8.00000 q^{31} +5.00000i q^{32} -4.00000i q^{33} +2.00000 q^{34} -1.00000 q^{36} +2.00000i q^{37} +4.00000i q^{38} -6.00000 q^{39} +2.00000 q^{41} +4.00000i q^{42} +8.00000i q^{43} +4.00000 q^{44} -1.00000 q^{46} +1.00000i q^{48} -9.00000 q^{49} -2.00000 q^{51} -6.00000i q^{52} +6.00000i q^{53} -1.00000 q^{54} -12.0000 q^{56} -4.00000i q^{57} +10.0000i q^{58} +6.00000 q^{61} -8.00000i q^{62} -4.00000i q^{63} -7.00000 q^{64} +4.00000 q^{66} +8.00000i q^{67} -2.00000i q^{68} +1.00000 q^{69} -4.00000 q^{71} -3.00000i q^{72} -10.0000i q^{73} -2.00000 q^{74} +4.00000 q^{76} +16.0000i q^{77} -6.00000i q^{78} -16.0000 q^{79} +1.00000 q^{81} +2.00000i q^{82} +12.0000i q^{83} +4.00000 q^{84} -8.00000 q^{86} -10.0000i q^{87} +12.0000i q^{88} +10.0000 q^{89} +24.0000 q^{91} +1.00000i q^{92} +8.00000i q^{93} +5.00000 q^{96} -10.0000i q^{97} -9.00000i q^{98} -4.00000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{4} + 2 q^{6} - 2 q^{9}+O(q^{10}) $ 2 * q + 2 * q^4 + 2 * q^6 - 2 * q^9	$ 2 q + 2 q^{4} + 2 q^{6} - 2 q^{9} + 8 q^{11} - 8 q^{14} - 2 q^{16} + 8 q^{19} + 8 q^{21} + 6 q^{24} + 12 q^{26} + 20 q^{29} - 16 q^{31} + 4 q^{34} - 2 q^{36} - 12 q^{39} + 4 q^{41} + 8 q^{44} - 2 q^{46} - 18 q^{49} - 4 q^{51} - 2 q^{54} - 24 q^{56} + 12 q^{61} - 14 q^{64} + 8 q^{66} + 2 q^{69} - 8 q^{71} - 4 q^{74} + 8 q^{76} - 32 q^{79} + 2 q^{81} + 8 q^{84} - 16 q^{86} + 20 q^{89} + 48 q^{91} + 10 q^{96} - 8 q^{99}+O(q^{100}) $ 2 * q + 2 * q^4 + 2 * q^6 - 2 * q^9 + 8 * q^11 - 8 * q^14 - 2 * q^16 + 8 * q^19 + 8 * q^21 + 6 * q^24 + 12 * q^26 + 20 * q^29 - 16 * q^31 + 4 * q^34 - 2 * q^36 - 12 * q^39 + 4 * q^41 + 8 * q^44 - 2 * q^46 - 18 * q^49 - 4 * q^51 - 2 * q^54 - 24 * q^56 + 12 * q^61 - 14 * q^64 + 8 * q^66 + 2 * q^69 - 8 * q^71 - 4 * q^74 + 8 * q^76 - 32 * q^79 + 2 * q^81 + 8 * q^84 - 16 * q^86 + 20 * q^89 + 48 * q^91 + 10 * q^96 - 8 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1725\mathbb{Z}\right)^\times$.

$n$	$277$	$1151$	$1201$
$\chi(n)$	$-1$	$1$	$1$

Coefficient data

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

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

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

$2$	1.00000i	0.707107i	0.935414	+	0.353553i	$0.115027\pi$
$2$	1.00000i	0.707107i	−0.935414	+	0.353553i	$0.884973\pi$
$3$	− 1.00000i	− 0.577350i
$3$	− 1.00000i	− 0.577350i
$4$	1.00000	0.500000
$5$	0	0
$5$	0	0
$6$	1.00000	0.408248
$7$	4.00000i	1.51186i	0.654654	+	0.755929i	$0.272814\pi$
$7$	4.00000i	1.51186i	−0.654654	+	0.755929i	$0.727186\pi$
$8$	3.00000i	1.06066i
$9$	−1.00000	−0.333333
$10$	0	0
$11$	4.00000	1.20605	0.603023	−	0.797724i	$-0.293963\pi$
$11$	4.00000	1.20605	0.603023	+	0.797724i	$0.293963\pi$
$12$	− 1.00000i	− 0.288675i
$13$	− 6.00000i	− 1.66410i	−0.554700	−	0.832050i	$-0.687167\pi$
$13$	− 6.00000i	− 1.66410i	0.554700	−	0.832050i	$-0.312833\pi$
$14$	−4.00000	−1.06904
$15$	0	0
$16$	−1.00000	−0.250000
$17$	− 2.00000i	− 0.485071i	−0.970143	−	0.242536i	$-0.922021\pi$
$17$	− 2.00000i	− 0.485071i	0.970143	−	0.242536i	$-0.0779791\pi$
$18$	− 1.00000i	− 0.235702i
$19$	4.00000	0.917663	0.458831	−	0.888523i	$-0.348268\pi$
$19$	4.00000	0.917663	0.458831	+	0.888523i	$0.348268\pi$
$20$	0	0
$21$	4.00000	0.872872
$22$	4.00000i	0.852803i
$23$	1.00000i	0.208514i
$23$	1.00000i	0.208514i
$24$	3.00000	0.612372
$25$	0	0
$26$	6.00000	1.17670
$27$	1.00000i	0.192450i
$28$	4.00000i	0.755929i
$29$	10.0000	1.85695	0.928477	−	0.371391i	$-0.121119\pi$
$29$	10.0000	1.85695	0.928477	+	0.371391i	$0.121119\pi$
$30$	0	0
$31$	−8.00000	−1.43684	−0.718421	−	0.695608i	$-0.755135\pi$
$31$	−8.00000	−1.43684	−0.718421	+	0.695608i	$0.755135\pi$
$32$	5.00000i	0.883883i
$33$	− 4.00000i	− 0.696311i
$34$	2.00000	0.342997
$35$	0	0
$36$	−1.00000	−0.166667
$37$	2.00000i	0.328798i	0.986394	+	0.164399i	$0.0525685\pi$
$37$	2.00000i	0.328798i	−0.986394	+	0.164399i	$0.947432\pi$
$38$	4.00000i	0.648886i
$39$	−6.00000	−0.960769
$40$	0	0
$41$	2.00000	0.312348	0.156174	−	0.987730i	$-0.450084\pi$
$41$	2.00000	0.312348	0.156174	+	0.987730i	$0.450084\pi$
$42$	4.00000i	0.617213i
$43$	8.00000i	1.21999i	0.792406	+	0.609994i	$0.208828\pi$
$43$	8.00000i	1.21999i	−0.792406	+	0.609994i	$0.791172\pi$
$44$	4.00000	0.603023
$45$	0	0
$46$	−1.00000	−0.147442
$47$	0	0	1.00000			$0$
$47$	0	0	−1.00000			$\pi$
$48$	1.00000i	0.144338i
$49$	−9.00000	−1.28571
$50$	0	0
$51$	−2.00000	−0.280056
$52$	− 6.00000i	− 0.832050i
$53$	6.00000i	0.824163i	0.911147	+	0.412082i	$0.135198\pi$
$53$	6.00000i	0.824163i	−0.911147	+	0.412082i	$0.864802\pi$
$54$	−1.00000	−0.136083
$55$	0	0
$56$	−12.0000	−1.60357
$57$	− 4.00000i	− 0.529813i
$58$	10.0000i	1.31306i
$59$	0	0		−	1.00000i	$-0.5\pi$
$59$	0	0			1.00000i	$0.5\pi$
$60$	0	0
$61$	6.00000	0.768221	0.384111	−	0.923287i	$-0.374508\pi$
$61$	6.00000	0.768221	0.384111	+	0.923287i	$0.374508\pi$
$62$	− 8.00000i	− 1.01600i
$63$	− 4.00000i	− 0.503953i
$64$	−7.00000	−0.875000
$65$	0	0
$66$	4.00000	0.492366
$67$	8.00000i	0.977356i	0.872464	+	0.488678i	$0.162521\pi$
$67$	8.00000i	0.977356i	−0.872464	+	0.488678i	$0.837479\pi$
$68$	− 2.00000i	− 0.242536i
$69$	1.00000	0.120386
$70$	0	0
$71$	−4.00000	−0.474713	−0.237356	−	0.971423i	$-0.576281\pi$
$71$	−4.00000	−0.474713	−0.237356	+	0.971423i	$0.576281\pi$
$72$	− 3.00000i	− 0.353553i
$73$	− 10.0000i	− 1.17041i	−0.810885	−	0.585206i	$-0.801014\pi$
$73$	− 10.0000i	− 1.17041i	0.810885	−	0.585206i	$-0.198986\pi$
$74$	−2.00000	−0.232495
$75$	0	0
$76$	4.00000	0.458831
$77$	16.0000i	1.82337i
$78$	− 6.00000i	− 0.679366i
$79$	−16.0000	−1.80014	−0.900070	−	0.435745i	$-0.856485\pi$
$79$	−16.0000	−1.80014	−0.900070	+	0.435745i	$0.856485\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	2.00000i	0.220863i
$83$	12.0000i	1.31717i	0.752506	+	0.658586i	$0.228845\pi$
$83$	12.0000i	1.31717i	−0.752506	+	0.658586i	$0.771155\pi$
$84$	4.00000	0.436436
$85$	0	0
$86$	−8.00000	−0.862662
$87$	− 10.0000i	− 1.07211i
$88$	12.0000i	1.27920i
$89$	10.0000	1.06000	0.529999	−	0.847998i	$-0.322192\pi$
$89$	10.0000	1.06000	0.529999	+	0.847998i	$0.322192\pi$
$90$	0	0
$91$	24.0000	2.51588
$92$	1.00000i	0.104257i
$93$	8.00000i	0.829561i
$94$	0	0
$95$	0	0
$96$	5.00000	0.510310
$97$	− 10.0000i	− 1.01535i	−0.861550	−	0.507673i	$-0.830506\pi$
$97$	− 10.0000i	− 1.01535i	0.861550	−	0.507673i	$-0.169494\pi$
$98$	− 9.00000i	− 0.909137i
$99$	−4.00000	−0.402015
$100$	0	0
$101$	14.0000	1.39305	0.696526	−	0.717532i	$-0.254728\pi$
$101$	14.0000	1.39305	0.696526	+	0.717532i	$0.254728\pi$
$102$	− 2.00000i	− 0.198030i
$103$	12.0000i	1.18240i	0.806527	+	0.591198i	$0.201345\pi$
$103$	12.0000i	1.18240i	−0.806527	+	0.591198i	$0.798655\pi$
$104$	18.0000	1.76505
$105$	0	0
$106$	−6.00000	−0.582772
$107$	− 12.0000i	− 1.16008i	−0.814587	−	0.580042i	$-0.803036\pi$
$107$	− 12.0000i	− 1.16008i	0.814587	−	0.580042i	$-0.196964\pi$
$108$	1.00000i	0.0962250i
$109$	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
$111$	2.00000	0.189832
$112$	− 4.00000i	− 0.377964i
$113$	2.00000i	0.188144i	0.995565	+	0.0940721i	$0.0299884\pi$
$113$	2.00000i	0.188144i	−0.995565	+	0.0940721i	$0.970012\pi$
$114$	4.00000	0.374634
$115$	0	0
$116$	10.0000	0.928477
$117$	6.00000i	0.554700i
$118$	0	0
$119$	8.00000	0.733359
$120$	0	0
$121$	5.00000	0.454545
$122$	6.00000i	0.543214i
$123$	− 2.00000i	− 0.180334i
$124$	−8.00000	−0.718421
$125$	0	0
$126$	4.00000	0.356348
$127$	0	0	1.00000			$0$
$127$	0	0	−1.00000			$\pi$
$128$	3.00000i	0.265165i
$129$	8.00000	0.704361
$130$	0	0
$131$	8.00000	0.698963	0.349482	−	0.936943i	$-0.386358\pi$
$131$	8.00000	0.698963	0.349482	+	0.936943i	$0.386358\pi$
$132$	− 4.00000i	− 0.348155i
$133$	16.0000i	1.38738i
$134$	−8.00000	−0.691095
$135$	0	0
$136$	6.00000	0.514496
$137$	− 10.0000i	− 0.854358i	−0.904167	−	0.427179i	$-0.859507\pi$
$137$	− 10.0000i	− 0.854358i	0.904167	−	0.427179i	$-0.140493\pi$
$138$	1.00000i	0.0851257i
$139$	−4.00000	−0.339276	−0.169638	−	0.985506i	$-0.554260\pi$
$139$	−4.00000	−0.339276	−0.169638	+	0.985506i	$0.554260\pi$
$140$	0	0
$141$	0	0
$142$	− 4.00000i	− 0.335673i
$143$	− 24.0000i	− 2.00698i
$144$	1.00000	0.0833333
$145$	0	0
$146$	10.0000	0.827606
$147$	9.00000i	0.742307i
$148$	2.00000i	0.164399i
$149$	6.00000	0.491539	0.245770	−	0.969328i	$-0.420959\pi$
$149$	6.00000	0.491539	0.245770	+	0.969328i	$0.420959\pi$
$150$	0	0
$151$	−16.0000	−1.30206	−0.651031	−	0.759051i	$-0.725663\pi$
$151$	−16.0000	−1.30206	−0.651031	+	0.759051i	$0.725663\pi$
$152$	12.0000i	0.973329i
$153$	2.00000i	0.161690i
$154$	−16.0000	−1.28932
$155$	0	0
$156$	−6.00000	−0.480384
$157$	− 14.0000i	− 1.11732i	−0.829396	−	0.558661i	$-0.811315\pi$
$157$	− 14.0000i	− 1.11732i	0.829396	−	0.558661i	$-0.188685\pi$
$158$	− 16.0000i	− 1.27289i
$159$	6.00000	0.475831
$160$	0	0
$161$	−4.00000	−0.315244
$162$	1.00000i	0.0785674i
$163$	− 20.0000i	− 1.56652i	−0.621694	−	0.783260i	$-0.713555\pi$
$163$	− 20.0000i	− 1.56652i	0.621694	−	0.783260i	$-0.286445\pi$
$164$	2.00000	0.156174
$165$	0	0
$166$	−12.0000	−0.931381
$167$	8.00000i	0.619059i	0.950890	+	0.309529i	$0.100171\pi$
$167$	8.00000i	0.619059i	−0.950890	+	0.309529i	$0.899829\pi$
$168$	12.0000i	0.925820i
$169$	−23.0000	−1.76923
$170$	0	0
$171$	−4.00000	−0.305888
$172$	8.00000i	0.609994i
$173$	− 2.00000i	− 0.152057i	−0.997106	−	0.0760286i	$-0.975776\pi$
$173$	− 2.00000i	− 0.152057i	0.997106	−	0.0760286i	$-0.0242240\pi$
$174$	10.0000	0.758098
$175$	0	0
$176$	−4.00000	−0.301511
$177$	0	0
$178$	10.0000i	0.749532i
$179$	−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
$181$	−2.00000	−0.148659	−0.0743294	−	0.997234i	$-0.523682\pi$
$181$	−2.00000	−0.148659	−0.0743294	+	0.997234i	$0.523682\pi$
$182$	24.0000i	1.77900i
$183$	− 6.00000i	− 0.443533i
$184$	−3.00000	−0.221163
$185$	0	0
$186$	−8.00000	−0.586588
$187$	− 8.00000i	− 0.585018i
$188$	0	0
$189$	−4.00000	−0.290957
$190$	0	0
$191$	24.0000	1.73658	0.868290	−	0.496058i	$-0.165220\pi$
$191$	24.0000	1.73658	0.868290	+	0.496058i	$0.165220\pi$
$192$	7.00000i	0.505181i
$193$	6.00000i	0.431889i	0.976406	+	0.215945i	$0.0692831\pi$
$193$	6.00000i	0.431889i	−0.976406	+	0.215945i	$0.930717\pi$
$194$	10.0000	0.717958
$195$	0	0
$196$	−9.00000	−0.642857
$197$	− 6.00000i	− 0.427482i	−0.976890	−	0.213741i	$-0.931435\pi$
$197$	− 6.00000i	− 0.427482i	0.976890	−	0.213741i	$-0.0685649\pi$
$198$	− 4.00000i	− 0.284268i
$199$	−24.0000	−1.70131	−0.850657	−	0.525720i	$-0.823796\pi$
$199$	−24.0000	−1.70131	−0.850657	+	0.525720i	$0.823796\pi$
$200$	0	0
$201$	8.00000	0.564276
$202$	14.0000i	0.985037i
$203$	40.0000i	2.80745i
$204$	−2.00000	−0.140028
$205$	0	0
$206$	−12.0000	−0.836080
$207$	− 1.00000i	− 0.0695048i
$208$	6.00000i	0.416025i
$209$	16.0000	1.10674
$210$	0	0
$211$	−4.00000	−0.275371	−0.137686	−	0.990476i	$-0.543966\pi$
$211$	−4.00000	−0.275371	−0.137686	+	0.990476i	$0.543966\pi$
$212$	6.00000i	0.412082i
$213$	4.00000i	0.274075i
$214$	12.0000	0.820303
$215$	0	0
$216$	−3.00000	−0.204124
$217$	− 32.0000i	− 2.17230i
$218$	10.0000i	0.677285i
$219$	−10.0000	−0.675737
$220$	0	0
$221$	−12.0000	−0.807207
$222$	2.00000i	0.134231i
$223$	− 24.0000i	− 1.60716i	−0.595198	−	0.803579i	$-0.702926\pi$
$223$	− 24.0000i	− 1.60716i	0.595198	−	0.803579i	$-0.297074\pi$
$224$	−20.0000	−1.33631
$225$	0	0
$226$	−2.00000	−0.133038
$227$	− 12.0000i	− 0.796468i	−0.917284	−	0.398234i	$-0.869623\pi$
$227$	− 12.0000i	− 0.796468i	0.917284	−	0.398234i	$-0.130377\pi$
$228$	− 4.00000i	− 0.264906i
$229$	−6.00000	−0.396491	−0.198246	−	0.980152i	$-0.563524\pi$
$229$	−6.00000	−0.396491	−0.198246	+	0.980152i	$0.563524\pi$
$230$	0	0
$231$	16.0000	1.05272
$232$	30.0000i	1.96960i
$233$	2.00000i	0.131024i	0.997852	+	0.0655122i	$0.0208681\pi$
$233$	2.00000i	0.131024i	−0.997852	+	0.0655122i	$0.979132\pi$
$234$	−6.00000	−0.392232
$235$	0	0
$236$	0	0
$237$	16.0000i	1.03931i
$238$	8.00000i	0.518563i
$239$	−20.0000	−1.29369	−0.646846	−	0.762620i	$-0.723912\pi$
$239$	−20.0000	−1.29369	−0.646846	+	0.762620i	$0.723912\pi$
$240$	0	0
$241$	−14.0000	−0.901819	−0.450910	−	0.892570i	$-0.648900\pi$
$241$	−14.0000	−0.901819	−0.450910	+	0.892570i	$0.648900\pi$
$242$	5.00000i	0.321412i
$243$	− 1.00000i	− 0.0641500i
$244$	6.00000	0.384111
$245$	0	0
$246$	2.00000	0.127515
$247$	− 24.0000i	− 1.52708i
$248$	− 24.0000i	− 1.52400i
$249$	12.0000	0.760469
$250$	0	0
$251$	−12.0000	−0.757433	−0.378717	−	0.925513i	$-0.623635\pi$
$251$	−12.0000	−0.757433	−0.378717	+	0.925513i	$0.623635\pi$
$252$	− 4.00000i	− 0.251976i
$253$	4.00000i	0.251478i
$254$	0	0
$255$	0	0
$256$	−17.0000	−1.06250
$257$	30.0000i	1.87135i	0.352865	+	0.935674i	$0.385208\pi$
$257$	30.0000i	1.87135i	−0.352865	+	0.935674i	$0.614792\pi$
$258$	8.00000i	0.498058i
$259$	−8.00000	−0.497096
$260$	0	0
$261$	−10.0000	−0.618984
$262$	8.00000i	0.494242i
$263$	0	0	1.00000			$0$
$263$	0	0	−1.00000			$\pi$
$264$	12.0000	0.738549
$265$	0	0
$266$	−16.0000	−0.981023
$267$	− 10.0000i	− 0.611990i
$268$	8.00000i	0.488678i
$269$	2.00000	0.121942	0.0609711	−	0.998140i	$-0.480580\pi$
$269$	2.00000	0.121942	0.0609711	+	0.998140i	$0.480580\pi$
$270$	0	0
$271$	16.0000	0.971931	0.485965	−	0.873978i	$-0.338468\pi$
$271$	16.0000	0.971931	0.485965	+	0.873978i	$0.338468\pi$
$272$	2.00000i	0.121268i
$273$	− 24.0000i	− 1.45255i
$274$	10.0000	0.604122
$275$	0	0
$276$	1.00000	0.0601929
$277$	− 10.0000i	− 0.600842i	−0.953807	−	0.300421i	$-0.902873\pi$
$277$	− 10.0000i	− 0.600842i	0.953807	−	0.300421i	$-0.0971271\pi$
$278$	− 4.00000i	− 0.239904i
$279$	8.00000	0.478947
$280$	0	0
$281$	6.00000	0.357930	0.178965	−	0.983855i	$-0.442725\pi$
$281$	6.00000	0.357930	0.178965	+	0.983855i	$0.442725\pi$
$282$	0	0
$283$	24.0000i	1.42665i	0.700832	+	0.713326i	$0.252812\pi$
$283$	24.0000i	1.42665i	−0.700832	+	0.713326i	$0.747188\pi$
$284$	−4.00000	−0.237356
$285$	0	0
$286$	24.0000	1.41915
$287$	8.00000i	0.472225i
$288$	− 5.00000i	− 0.294628i
$289$	13.0000	0.764706
$290$	0	0
$291$	−10.0000	−0.586210
$292$	− 10.0000i	− 0.585206i
$293$	− 26.0000i	− 1.51894i	−0.650545	−	0.759468i	$-0.725459\pi$
$293$	− 26.0000i	− 1.51894i	0.650545	−	0.759468i	$-0.274541\pi$
$294$	−9.00000	−0.524891
$295$	0	0
$296$	−6.00000	−0.348743
$297$	4.00000i	0.232104i
$298$	6.00000i	0.347571i
$299$	6.00000	0.346989
$300$	0	0
$301$	−32.0000	−1.84445
$302$	− 16.0000i	− 0.920697i
$303$	− 14.0000i	− 0.804279i
$304$	−4.00000	−0.229416
$305$	0	0
$306$	−2.00000	−0.114332
$307$	− 28.0000i	− 1.59804i	−0.601302	−	0.799022i	$-0.705351\pi$
$307$	− 28.0000i	− 1.59804i	0.601302	−	0.799022i	$-0.294649\pi$
$308$	16.0000i	0.911685i
$309$	12.0000	0.682656
$310$	0	0
$311$	12.0000	0.680458	0.340229	−	0.940343i	$-0.389495\pi$
$311$	12.0000	0.680458	0.340229	+	0.940343i	$0.389495\pi$
$312$	− 18.0000i	− 1.01905i
$313$	− 14.0000i	− 0.791327i	−0.918396	−	0.395663i	$-0.870515\pi$
$313$	− 14.0000i	− 0.791327i	0.918396	−	0.395663i	$-0.129485\pi$
$314$	14.0000	0.790066
$315$	0	0
$316$	−16.0000	−0.900070
$317$	− 6.00000i	− 0.336994i	−0.985702	−	0.168497i	$-0.946109\pi$
$317$	− 6.00000i	− 0.336994i	0.985702	−	0.168497i	$-0.0538913\pi$
$318$	6.00000i	0.336463i
$319$	40.0000	2.23957
$320$	0	0
$321$	−12.0000	−0.669775
$322$	− 4.00000i	− 0.222911i
$323$	− 8.00000i	− 0.445132i
$324$	1.00000	0.0555556
$325$	0	0
$326$	20.0000	1.10770
$327$	− 10.0000i	− 0.553001i
$328$	6.00000i	0.331295i
$329$	0	0
$330$	0	0
$331$	12.0000	0.659580	0.329790	−	0.944054i	$-0.393022\pi$
$331$	12.0000	0.659580	0.329790	+	0.944054i	$0.393022\pi$
$332$	12.0000i	0.658586i
$333$	− 2.00000i	− 0.109599i
$334$	−8.00000	−0.437741
$335$	0	0
$336$	−4.00000	−0.218218
$337$	− 26.0000i	− 1.41631i	−0.706057	−	0.708155i	$-0.749528\pi$
$337$	− 26.0000i	− 1.41631i	0.706057	−	0.708155i	$-0.250472\pi$
$338$	− 23.0000i	− 1.25104i
$339$	2.00000	0.108625
$340$	0	0
$341$	−32.0000	−1.73290
$342$	− 4.00000i	− 0.216295i
$343$	− 8.00000i	− 0.431959i
$344$	−24.0000	−1.29399
$345$	0	0
$346$	2.00000	0.107521
$347$	− 12.0000i	− 0.644194i	−0.946707	−	0.322097i	$-0.895612\pi$
$347$	− 12.0000i	− 0.644194i	0.946707	−	0.322097i	$-0.104388\pi$
$348$	− 10.0000i	− 0.536056i
$349$	−14.0000	−0.749403	−0.374701	−	0.927146i	$-0.622255\pi$
$349$	−14.0000	−0.749403	−0.374701	+	0.927146i	$0.622255\pi$
$350$	0	0
$351$	6.00000	0.320256
$352$	20.0000i	1.06600i
$353$	− 6.00000i	− 0.319348i	−0.987170	−	0.159674i	$-0.948956\pi$
$353$	− 6.00000i	− 0.319348i	0.987170	−	0.159674i	$-0.0510443\pi$
$354$	0	0
$355$	0	0
$356$	10.0000	0.529999
$357$	− 8.00000i	− 0.423405i
$358$	− 24.0000i	− 1.26844i
$359$	−8.00000	−0.422224	−0.211112	−	0.977462i	$-0.567708\pi$
$359$	−8.00000	−0.422224	−0.211112	+	0.977462i	$0.567708\pi$
$360$	0	0
$361$	−3.00000	−0.157895
$362$	− 2.00000i	− 0.105118i
$363$	− 5.00000i	− 0.262432i
$364$	24.0000	1.25794
$365$	0	0
$366$	6.00000	0.313625
$367$	28.0000i	1.46159i	0.682598	+	0.730794i	$0.260850\pi$
$367$	28.0000i	1.46159i	−0.682598	+	0.730794i	$0.739150\pi$
$368$	− 1.00000i	− 0.0521286i
$369$	−2.00000	−0.104116
$370$	0	0
$371$	−24.0000	−1.24602
$372$	8.00000i	0.414781i
$373$	− 10.0000i	− 0.517780i	−0.965907	−	0.258890i	$-0.916643\pi$
$373$	− 10.0000i	− 0.517780i	0.965907	−	0.258890i	$-0.0833568\pi$
$374$	8.00000	0.413670
$375$	0	0
$376$	0	0
$377$	− 60.0000i	− 3.09016i
$378$	− 4.00000i	− 0.205738i
$379$	−4.00000	−0.205466	−0.102733	−	0.994709i	$-0.532759\pi$
$379$	−4.00000	−0.205466	−0.102733	+	0.994709i	$0.532759\pi$
$380$	0	0
$381$	0	0
$382$	24.0000i	1.22795i
$383$	− 32.0000i	− 1.63512i	−0.575841	−	0.817562i	$-0.695325\pi$
$383$	− 32.0000i	− 1.63512i	0.575841	−	0.817562i	$-0.304675\pi$
$384$	3.00000	0.153093
$385$	0	0
$386$	−6.00000	−0.305392
$387$	− 8.00000i	− 0.406663i
$388$	− 10.0000i	− 0.507673i
$389$	14.0000	0.709828	0.354914	−	0.934899i	$-0.384510\pi$
$389$	14.0000	0.709828	0.354914	+	0.934899i	$0.384510\pi$
$390$	0	0
$391$	2.00000	0.101144
$392$	− 27.0000i	− 1.36371i
$393$	− 8.00000i	− 0.403547i
$394$	6.00000	0.302276
$395$	0	0
$396$	−4.00000	−0.201008
$397$	30.0000i	1.50566i	0.658217	+	0.752828i	$0.271311\pi$
$397$	30.0000i	1.50566i	−0.658217	+	0.752828i	$0.728689\pi$
$398$	− 24.0000i	− 1.20301i
$399$	16.0000	0.801002
$400$	0	0
$401$	−10.0000	−0.499376	−0.249688	−	0.968326i	$-0.580328\pi$
$401$	−10.0000	−0.499376	−0.249688	+	0.968326i	$0.580328\pi$
$402$	8.00000i	0.399004i
$403$	48.0000i	2.39105i
$404$	14.0000	0.696526
$405$	0	0
$406$	−40.0000	−1.98517
$407$	8.00000i	0.396545i
$408$	− 6.00000i	− 0.297044i
$409$	−10.0000	−0.494468	−0.247234	−	0.968956i	$-0.579522\pi$
$409$	−10.0000	−0.494468	−0.247234	+	0.968956i	$0.579522\pi$
$410$	0	0
$411$	−10.0000	−0.493264
$412$	12.0000i	0.591198i
$413$	0	0
$414$	1.00000	0.0491473
$415$	0	0
$416$	30.0000	1.47087
$417$	4.00000i	0.195881i
$418$	16.0000i	0.782586i
$419$	28.0000	1.36789	0.683945	−	0.729534i	$-0.260263\pi$
$419$	28.0000	1.36789	0.683945	+	0.729534i	$0.260263\pi$
$420$	0	0
$421$	−10.0000	−0.487370	−0.243685	−	0.969854i	$-0.578356\pi$
$421$	−10.0000	−0.487370	−0.243685	+	0.969854i	$0.578356\pi$
$422$	− 4.00000i	− 0.194717i
$423$	0	0
$424$	−18.0000	−0.874157
$425$	0	0
$426$	−4.00000	−0.193801
$427$	24.0000i	1.16144i
$428$	− 12.0000i	− 0.580042i
$429$	−24.0000	−1.15873
$430$	0	0
$431$	0	0		−	1.00000i	$-0.5\pi$
$431$	0	0			1.00000i	$0.5\pi$
$432$	− 1.00000i	− 0.0481125i
$433$	− 14.0000i	− 0.672797i	−0.941720	−	0.336399i	$-0.890791\pi$
$433$	− 14.0000i	− 0.672797i	0.941720	−	0.336399i	$-0.109209\pi$
$434$	32.0000	1.53605
$435$	0	0
$436$	10.0000	0.478913
$437$	4.00000i	0.191346i
$438$	− 10.0000i	− 0.477818i
$439$	16.0000	0.763638	0.381819	−	0.924237i	$-0.375298\pi$
$439$	16.0000	0.763638	0.381819	+	0.924237i	$0.375298\pi$
$440$	0	0
$441$	9.00000	0.428571
$442$	− 12.0000i	− 0.570782i
$443$	12.0000i	0.570137i	0.958507	+	0.285069i	$0.0920164\pi$
$443$	12.0000i	0.570137i	−0.958507	+	0.285069i	$0.907984\pi$
$444$	2.00000	0.0949158
$445$	0	0
$446$	24.0000	1.13643
$447$	− 6.00000i	− 0.283790i
$448$	− 28.0000i	− 1.32288i
$449$	14.0000	0.660701	0.330350	−	0.943858i	$-0.392833\pi$
$449$	14.0000	0.660701	0.330350	+	0.943858i	$0.392833\pi$
$450$	0	0
$451$	8.00000	0.376705
$452$	2.00000i	0.0940721i
$453$	16.0000i	0.751746i
$454$	12.0000	0.563188
$455$	0	0
$456$	12.0000	0.561951
$457$	− 18.0000i	− 0.842004i	−0.907060	−	0.421002i	$-0.861678\pi$
$457$	− 18.0000i	− 0.842004i	0.907060	−	0.421002i	$-0.138322\pi$
$458$	− 6.00000i	− 0.280362i
$459$	2.00000	0.0933520
$460$	0	0
$461$	−34.0000	−1.58354	−0.791769	−	0.610821i	$-0.790840\pi$
$461$	−34.0000	−1.58354	−0.791769	+	0.610821i	$0.790840\pi$
$462$	16.0000i	0.744387i
$463$	32.0000i	1.48717i	0.668644	+	0.743583i	$0.266875\pi$
$463$	32.0000i	1.48717i	−0.668644	+	0.743583i	$0.733125\pi$
$464$	−10.0000	−0.464238
$465$	0	0
$466$	−2.00000	−0.0926482
$467$	20.0000i	0.925490i	0.886492	+	0.462745i	$0.153135\pi$
$467$	20.0000i	0.925490i	−0.886492	+	0.462745i	$0.846865\pi$
$468$	6.00000i	0.277350i
$469$	−32.0000	−1.47762
$470$	0	0
$471$	−14.0000	−0.645086
$472$	0	0
$473$	32.0000i	1.47136i
$474$	−16.0000	−0.734904
$475$	0	0
$476$	8.00000	0.366679
$477$	− 6.00000i	− 0.274721i
$478$	− 20.0000i	− 0.914779i
$479$	−40.0000	−1.82765	−0.913823	−	0.406112i	$-0.866884\pi$
$479$	−40.0000	−1.82765	−0.913823	+	0.406112i	$0.866884\pi$
$480$	0	0
$481$	12.0000	0.547153
$482$	− 14.0000i	− 0.637683i
$483$	4.00000i	0.182006i
$484$	5.00000	0.227273
$485$	0	0
$486$	1.00000	0.0453609
$487$	24.0000i	1.08754i	0.839233	+	0.543772i	$0.183004\pi$
$487$	24.0000i	1.08754i	−0.839233	+	0.543772i	$0.816996\pi$
$488$	18.0000i	0.814822i
$489$	−20.0000	−0.904431
$490$	0	0
$491$	16.0000	0.722070	0.361035	−	0.932552i	$-0.382424\pi$
$491$	16.0000	0.722070	0.361035	+	0.932552i	$0.382424\pi$
$492$	− 2.00000i	− 0.0901670i
$493$	− 20.0000i	− 0.900755i
$494$	24.0000	1.07981
$495$	0	0
$496$	8.00000	0.359211
$497$	− 16.0000i	− 0.717698i
$498$	12.0000i	0.537733i
$499$	−4.00000	−0.179065	−0.0895323	−	0.995984i	$-0.528537\pi$
$499$	−4.00000	−0.179065	−0.0895323	+	0.995984i	$0.528537\pi$
$500$	0	0
$501$	8.00000	0.357414
$502$	− 12.0000i	− 0.535586i
$503$	− 16.0000i	− 0.713405i	−0.934218	−	0.356702i	$-0.883901\pi$
$503$	− 16.0000i	− 0.713405i	0.934218	−	0.356702i	$-0.116099\pi$
$504$	12.0000	0.534522
$505$	0	0
$506$	−4.00000	−0.177822
$507$	23.0000i	1.02147i
$508$	0	0
$509$	−22.0000	−0.975133	−0.487566	−	0.873086i	$-0.662115\pi$
$509$	−22.0000	−0.975133	−0.487566	+	0.873086i	$0.662115\pi$
$510$	0	0
$511$	40.0000	1.76950
$512$	− 11.0000i	− 0.486136i
$513$	4.00000i	0.176604i
$514$	−30.0000	−1.32324
$515$	0	0
$516$	8.00000	0.352180
$517$	0	0
$518$	− 8.00000i	− 0.351500i
$519$	−2.00000	−0.0877903
$520$	0	0
$521$	6.00000	0.262865	0.131432	−	0.991325i	$-0.458042\pi$
$521$	6.00000	0.262865	0.131432	+	0.991325i	$0.458042\pi$
$522$	− 10.0000i	− 0.437688i
$523$	− 8.00000i	− 0.349816i	−0.984585	−	0.174908i	$-0.944037\pi$
$523$	− 8.00000i	− 0.349816i	0.984585	−	0.174908i	$-0.0559627\pi$
$524$	8.00000	0.349482
$525$	0	0
$526$	0	0
$527$	16.0000i	0.696971i
$528$	4.00000i	0.174078i
$529$	−1.00000	−0.0434783
$530$	0	0
$531$	0	0
$532$	16.0000i	0.693688i
$533$	− 12.0000i	− 0.519778i
$534$	10.0000	0.432742
$535$	0	0
$536$	−24.0000	−1.03664
$537$	24.0000i	1.03568i
$538$	2.00000i	0.0862261i
$539$	−36.0000	−1.55063
$540$	0	0
$541$	−34.0000	−1.46177	−0.730887	−	0.682498i	$-0.760893\pi$
$541$	−34.0000	−1.46177	−0.730887	+	0.682498i	$0.760893\pi$
$542$	16.0000i	0.687259i
$543$	2.00000i	0.0858282i
$544$	10.0000	0.428746
$545$	0	0
$546$	24.0000	1.02711
$547$	− 28.0000i	− 1.19719i	−0.801050	−	0.598597i	$-0.795725\pi$
$547$	− 28.0000i	− 1.19719i	0.801050	−	0.598597i	$-0.204275\pi$
$548$	− 10.0000i	− 0.427179i
$549$	−6.00000	−0.256074
$550$	0	0
$551$	40.0000	1.70406
$552$	3.00000i	0.127688i
$553$	− 64.0000i	− 2.72156i
$554$	10.0000	0.424859
$555$	0	0
$556$	−4.00000	−0.169638
$557$	2.00000i	0.0847427i	0.999102	+	0.0423714i	$0.0134913\pi$
$557$	2.00000i	0.0847427i	−0.999102	+	0.0423714i	$0.986509\pi$
$558$	8.00000i	0.338667i
$559$	48.0000	2.03018
$560$	0	0
$561$	−8.00000	−0.337760
$562$	6.00000i	0.253095i
$563$	− 12.0000i	− 0.505740i	−0.967500	−	0.252870i	$-0.918626\pi$
$563$	− 12.0000i	− 0.505740i	0.967500	−	0.252870i	$-0.0813744\pi$
$564$	0	0
$565$	0	0
$566$	−24.0000	−1.00880
$567$	4.00000i	0.167984i
$568$	− 12.0000i	− 0.503509i
$569$	−30.0000	−1.25767	−0.628833	−	0.777541i	$-0.716467\pi$
$569$	−30.0000	−1.25767	−0.628833	+	0.777541i	$0.716467\pi$
$570$	0	0
$571$	44.0000	1.84134	0.920671	−	0.390339i	$-0.127642\pi$
$571$	44.0000	1.84134	0.920671	+	0.390339i	$0.127642\pi$
$572$	− 24.0000i	− 1.00349i
$573$	− 24.0000i	− 1.00261i
$574$	−8.00000	−0.333914
$575$	0	0
$576$	7.00000	0.291667
$577$	− 38.0000i	− 1.58196i	−0.611842	−	0.790980i	$-0.709571\pi$
$577$	− 38.0000i	− 1.58196i	0.611842	−	0.790980i	$-0.290429\pi$
$578$	13.0000i	0.540729i
$579$	6.00000	0.249351
$580$	0	0
$581$	−48.0000	−1.99138
$582$	− 10.0000i	− 0.414513i
$583$	24.0000i	0.993978i
$584$	30.0000	1.24141
$585$	0	0
$586$	26.0000	1.07405
$587$	− 36.0000i	− 1.48588i	−0.669359	−	0.742940i	$-0.733431\pi$
$587$	− 36.0000i	− 1.48588i	0.669359	−	0.742940i	$-0.266569\pi$
$588$	9.00000i	0.371154i
$589$	−32.0000	−1.31854
$590$	0	0
$591$	−6.00000	−0.246807
$592$	− 2.00000i	− 0.0821995i
$593$	− 46.0000i	− 1.88899i	−0.328521	−	0.944497i	$-0.606550\pi$
$593$	− 46.0000i	− 1.88899i	0.328521	−	0.944497i	$-0.393450\pi$
$594$	−4.00000	−0.164122
$595$	0	0
$596$	6.00000	0.245770
$597$	24.0000i	0.982255i
$598$	6.00000i	0.245358i
$599$	−28.0000	−1.14405	−0.572024	−	0.820237i	$-0.693842\pi$
$599$	−28.0000	−1.14405	−0.572024	+	0.820237i	$0.693842\pi$
$600$	0	0
$601$	10.0000	0.407909	0.203954	−	0.978980i	$-0.434621\pi$
$601$	10.0000	0.407909	0.203954	+	0.978980i	$0.434621\pi$
$602$	− 32.0000i	− 1.30422i
$603$	− 8.00000i	− 0.325785i
$604$	−16.0000	−0.651031
$605$	0	0
$606$	14.0000	0.568711
$607$	16.0000i	0.649420i	0.945814	+	0.324710i	$0.105267\pi$
$607$	16.0000i	0.649420i	−0.945814	+	0.324710i	$0.894733\pi$
$608$	20.0000i	0.811107i
$609$	40.0000	1.62088
$610$	0	0
$611$	0	0
$612$	2.00000i	0.0808452i
$613$	6.00000i	0.242338i	0.992632	+	0.121169i	$0.0386643\pi$
$613$	6.00000i	0.242338i	−0.992632	+	0.121169i	$0.961336\pi$
$614$	28.0000	1.12999
$615$	0	0
$616$	−48.0000	−1.93398
$617$	6.00000i	0.241551i	0.992680	+	0.120775i	$0.0385381\pi$
$617$	6.00000i	0.241551i	−0.992680	+	0.120775i	$0.961462\pi$
$618$	12.0000i	0.482711i
$619$	4.00000	0.160774	0.0803868	−	0.996764i	$-0.474384\pi$
$619$	4.00000	0.160774	0.0803868	+	0.996764i	$0.474384\pi$
$620$	0	0
$621$	−1.00000	−0.0401286
$622$	12.0000i	0.481156i
$623$	40.0000i	1.60257i
$624$	6.00000	0.240192
$625$	0	0
$626$	14.0000	0.559553
$627$	− 16.0000i	− 0.638978i
$628$	− 14.0000i	− 0.558661i
$629$	4.00000	0.159490
$630$	0	0
$631$	8.00000	0.318475	0.159237	−	0.987240i	$-0.449096\pi$
$631$	8.00000	0.318475	0.159237	+	0.987240i	$0.449096\pi$
$632$	− 48.0000i	− 1.90934i
$633$	4.00000i	0.158986i
$634$	6.00000	0.238290
$635$	0	0
$636$	6.00000	0.237915
$637$	54.0000i	2.13956i
$638$	40.0000i	1.58362i
$639$	4.00000	0.158238
$640$	0	0
$641$	−26.0000	−1.02694	−0.513469	−	0.858108i	$-0.671640\pi$
$641$	−26.0000	−1.02694	−0.513469	+	0.858108i	$0.671640\pi$
$642$	− 12.0000i	− 0.473602i
$643$	− 16.0000i	− 0.630978i	−0.948929	−	0.315489i	$-0.897831\pi$
$643$	− 16.0000i	− 0.630978i	0.948929	−	0.315489i	$-0.102169\pi$
$644$	−4.00000	−0.157622
$645$	0	0
$646$	8.00000	0.314756
$647$	8.00000i	0.314512i	0.987558	+	0.157256i	$0.0502649\pi$
$647$	8.00000i	0.314512i	−0.987558	+	0.157256i	$0.949735\pi$
$648$	3.00000i	0.117851i
$649$	0	0
$650$	0	0
$651$	−32.0000	−1.25418
$652$	− 20.0000i	− 0.783260i
$653$	6.00000i	0.234798i	0.993085	+	0.117399i	$0.0374557\pi$
$653$	6.00000i	0.234798i	−0.993085	+	0.117399i	$0.962544\pi$
$654$	10.0000	0.391031
$655$	0	0
$656$	−2.00000	−0.0780869
$657$	10.0000i	0.390137i
$658$	0	0
$659$	−4.00000	−0.155818	−0.0779089	−	0.996960i	$-0.524824\pi$
$659$	−4.00000	−0.155818	−0.0779089	+	0.996960i	$0.524824\pi$
$660$	0	0
$661$	14.0000	0.544537	0.272268	−	0.962221i	$-0.412226\pi$
$661$	14.0000	0.544537	0.272268	+	0.962221i	$0.412226\pi$
$662$	12.0000i	0.466393i
$663$	12.0000i	0.466041i
$664$	−36.0000	−1.39707
$665$	0	0
$666$	2.00000	0.0774984
$667$	10.0000i	0.387202i
$668$	8.00000i	0.309529i
$669$	−24.0000	−0.927894
$670$	0	0
$671$	24.0000	0.926510
$672$	20.0000i	0.771517i
$673$	46.0000i	1.77317i	0.462566	+	0.886585i	$0.346929\pi$
$673$	46.0000i	1.77317i	−0.462566	+	0.886585i	$0.653071\pi$
$674$	26.0000	1.00148
$675$	0	0
$676$	−23.0000	−0.884615
$677$	− 6.00000i	− 0.230599i	−0.993331	−	0.115299i	$-0.963217\pi$
$677$	− 6.00000i	− 0.230599i	0.993331	−	0.115299i	$-0.0367827\pi$
$678$	2.00000i	0.0768095i
$679$	40.0000	1.53506
$680$	0	0
$681$	−12.0000	−0.459841
$682$	− 32.0000i	− 1.22534i
$683$	4.00000i	0.153056i	0.997067	+	0.0765279i	$0.0243834\pi$
$683$	4.00000i	0.153056i	−0.997067	+	0.0765279i	$0.975617\pi$
$684$	−4.00000	−0.152944
$685$	0	0
$686$	8.00000	0.305441
$687$	6.00000i	0.228914i
$688$	− 8.00000i	− 0.304997i
$689$	36.0000	1.37149
$690$	0	0
$691$	−20.0000	−0.760836	−0.380418	−	0.924815i	$-0.624220\pi$
$691$	−20.0000	−0.760836	−0.380418	+	0.924815i	$0.624220\pi$
$692$	− 2.00000i	− 0.0760286i
$693$	− 16.0000i	− 0.607790i
$694$	12.0000	0.455514
$695$	0	0
$696$	30.0000	1.13715
$697$	− 4.00000i	− 0.151511i
$698$	− 14.0000i	− 0.529908i
$699$	2.00000	0.0756469
$700$	0	0
$701$	34.0000	1.28416	0.642081	−	0.766637i	$-0.278071\pi$
$701$	34.0000	1.28416	0.642081	+	0.766637i	$0.278071\pi$
$702$	6.00000i	0.226455i
$703$	8.00000i	0.301726i
$704$	−28.0000	−1.05529
$705$	0	0
$706$	6.00000	0.225813
$707$	56.0000i	2.10610i
$708$	0	0
$709$	42.0000	1.57734	0.788672	−	0.614815i	$-0.210769\pi$
$709$	42.0000	1.57734	0.788672	+	0.614815i	$0.210769\pi$
$710$	0	0
$711$	16.0000	0.600047
$712$	30.0000i	1.12430i
$713$	− 8.00000i	− 0.299602i
$714$	8.00000	0.299392
$715$	0	0
$716$	−24.0000	−0.896922
$717$	20.0000i	0.746914i
$718$	− 8.00000i	− 0.298557i
$719$	−36.0000	−1.34257	−0.671287	−	0.741198i	$-0.734258\pi$
$719$	−36.0000	−1.34257	−0.671287	+	0.741198i	$0.734258\pi$
$720$	0	0
$721$	−48.0000	−1.78761
$722$	− 3.00000i	− 0.111648i
$723$	14.0000i	0.520666i
$724$	−2.00000	−0.0743294
$725$	0	0
$726$	5.00000	0.185567
$727$	− 12.0000i	− 0.445055i	−0.974926	−	0.222528i	$-0.928569\pi$
$727$	− 12.0000i	− 0.445055i	0.974926	−	0.222528i	$-0.0714308\pi$
$728$	72.0000i	2.66850i
$729$	−1.00000	−0.0370370
$730$	0	0
$731$	16.0000	0.591781
$732$	− 6.00000i	− 0.221766i
$733$	6.00000i	0.221615i	0.993842	+	0.110808i	$0.0353437\pi$
$733$	6.00000i	0.221615i	−0.993842	+	0.110808i	$0.964656\pi$
$734$	−28.0000	−1.03350
$735$	0	0
$736$	−5.00000	−0.184302
$737$	32.0000i	1.17874i
$738$	− 2.00000i	− 0.0736210i
$739$	−12.0000	−0.441427	−0.220714	−	0.975339i	$-0.570839\pi$
$739$	−12.0000	−0.441427	−0.220714	+	0.975339i	$0.570839\pi$
$740$	0	0
$741$	−24.0000	−0.881662
$742$	− 24.0000i	− 0.881068i
$743$	8.00000i	0.293492i	0.989174	+	0.146746i	$0.0468799\pi$
$743$	8.00000i	0.293492i	−0.989174	+	0.146746i	$0.953120\pi$
$744$	−24.0000	−0.879883
$745$	0	0
$746$	10.0000	0.366126
$747$	− 12.0000i	− 0.439057i
$748$	− 8.00000i	− 0.292509i
$749$	48.0000	1.75388
$750$	0	0
$751$	−24.0000	−0.875772	−0.437886	−	0.899030i	$-0.644273\pi$
$751$	−24.0000	−0.875772	−0.437886	+	0.899030i	$0.644273\pi$
$752$	0	0
$753$	12.0000i	0.437304i
$754$	60.0000	2.18507
$755$	0	0
$756$	−4.00000	−0.145479
$757$	− 38.0000i	− 1.38113i	−0.723269	−	0.690567i	$-0.757361\pi$
$757$	− 38.0000i	− 1.38113i	0.723269	−	0.690567i	$-0.242639\pi$
$758$	− 4.00000i	− 0.145287i
$759$	4.00000	0.145191
$760$	0	0
$761$	−30.0000	−1.08750	−0.543750	−	0.839248i	$-0.682996\pi$
$761$	−30.0000	−1.08750	−0.543750	+	0.839248i	$0.682996\pi$
$762$	0	0
$763$	40.0000i	1.44810i
$764$	24.0000	0.868290
$765$	0	0
$766$	32.0000	1.15621
$767$	0	0
$768$	17.0000i	0.613435i
$769$	−42.0000	−1.51456	−0.757279	−	0.653091i	$-0.773472\pi$
$769$	−42.0000	−1.51456	−0.757279	+	0.653091i	$0.773472\pi$
$770$	0	0
$771$	30.0000	1.08042
$772$	6.00000i	0.215945i
$773$	38.0000i	1.36677i	0.730061	+	0.683383i	$0.239492\pi$
$773$	38.0000i	1.36677i	−0.730061	+	0.683383i	$0.760508\pi$
$774$	8.00000	0.287554
$775$	0	0
$776$	30.0000	1.07694
$777$	8.00000i	0.286998i
$778$	14.0000i	0.501924i
$779$	8.00000	0.286630
$780$	0	0
$781$	−16.0000	−0.572525
$782$	2.00000i	0.0715199i
$783$	10.0000i	0.357371i
$784$	9.00000	0.321429
$785$	0	0
$786$	8.00000	0.285351
$787$	− 16.0000i	− 0.570338i	−0.958477	−	0.285169i	$-0.907950\pi$
$787$	− 16.0000i	− 0.570338i	0.958477	−	0.285169i	$-0.0920498\pi$
$788$	− 6.00000i	− 0.213741i
$789$	0	0
$790$	0	0
$791$	−8.00000	−0.284447
$792$	− 12.0000i	− 0.426401i
$793$	− 36.0000i	− 1.27840i
$794$	−30.0000	−1.06466
$795$	0	0
$796$	−24.0000	−0.850657
$797$	2.00000i	0.0708436i	0.999372	+	0.0354218i	$0.0112775\pi$
$797$	2.00000i	0.0708436i	−0.999372	+	0.0354218i	$0.988723\pi$
$798$	16.0000i	0.566394i
$799$	0	0
$800$	0	0
$801$	−10.0000	−0.353333
$802$	− 10.0000i	− 0.353112i
$803$	− 40.0000i	− 1.41157i
$804$	8.00000	0.282138
$805$	0	0
$806$	−48.0000	−1.69073
$807$	− 2.00000i	− 0.0704033i
$808$	42.0000i	1.47755i
$809$	−10.0000	−0.351581	−0.175791	−	0.984428i	$-0.556248\pi$
$809$	−10.0000	−0.351581	−0.175791	+	0.984428i	$0.556248\pi$
$810$	0	0
$811$	44.0000	1.54505	0.772524	−	0.634985i	$-0.218994\pi$
$811$	44.0000	1.54505	0.772524	+	0.634985i	$0.218994\pi$
$812$	40.0000i	1.40372i
$813$	− 16.0000i	− 0.561144i
$814$	−8.00000	−0.280400
$815$	0	0
$816$	2.00000	0.0700140
$817$	32.0000i	1.11954i
$818$	− 10.0000i	− 0.349642i
$819$	−24.0000	−0.838628
$820$	0	0
$821$	22.0000	0.767805	0.383903	−	0.923374i	$-0.374580\pi$
$821$	22.0000	0.767805	0.383903	+	0.923374i	$0.374580\pi$
$822$	− 10.0000i	− 0.348790i
$823$	− 16.0000i	− 0.557725i	−0.960331	−	0.278862i	$-0.910043\pi$
$823$	− 16.0000i	− 0.557725i	0.960331	−	0.278862i	$-0.0899574\pi$
$824$	−36.0000	−1.25412
$825$	0	0
$826$	0	0
$827$	− 52.0000i	− 1.80822i	−0.427303	−	0.904109i	$-0.640536\pi$
$827$	− 52.0000i	− 1.80822i	0.427303	−	0.904109i	$-0.359464\pi$
$828$	− 1.00000i	− 0.0347524i
$829$	2.00000	0.0694629	0.0347314	−	0.999397i	$-0.488942\pi$
$829$	2.00000	0.0694629	0.0347314	+	0.999397i	$0.488942\pi$
$830$	0	0
$831$	−10.0000	−0.346896
$832$	42.0000i	1.45609i
$833$	18.0000i	0.623663i
$834$	−4.00000	−0.138509
$835$	0	0
$836$	16.0000	0.553372
$837$	− 8.00000i	− 0.276520i
$838$	28.0000i	0.967244i
$839$	0	0		−	1.00000i	$-0.5\pi$
$839$	0	0			1.00000i	$0.5\pi$
$840$	0	0
$841$	71.0000	2.44828
$842$	− 10.0000i	− 0.344623i
$843$	− 6.00000i	− 0.206651i
$844$	−4.00000	−0.137686
$845$	0	0
$846$	0	0
$847$	20.0000i	0.687208i
$848$	− 6.00000i	− 0.206041i
$849$	24.0000	0.823678
$850$	0	0
$851$	−2.00000	−0.0685591
$852$	4.00000i	0.137038i
$853$	26.0000i	0.890223i	0.895475	+	0.445112i	$0.146836\pi$
$853$	26.0000i	0.890223i	−0.895475	+	0.445112i	$0.853164\pi$
$854$	−24.0000	−0.821263
$855$	0	0
$856$	36.0000	1.23045
$857$	14.0000i	0.478231i	0.970991	+	0.239115i	$0.0768574\pi$
$857$	14.0000i	0.478231i	−0.970991	+	0.239115i	$0.923143\pi$
$858$	− 24.0000i	− 0.819346i
$859$	52.0000	1.77422	0.887109	−	0.461561i	$-0.152710\pi$
$859$	52.0000	1.77422	0.887109	+	0.461561i	$0.152710\pi$
$860$	0	0
$861$	8.00000	0.272639
$862$	0	0
$863$	8.00000i	0.272323i	0.990687	+	0.136162i	$0.0434766\pi$
$863$	8.00000i	0.272323i	−0.990687	+	0.136162i	$0.956523\pi$
$864$	−5.00000	−0.170103
$865$	0	0
$866$	14.0000	0.475739
$867$	− 13.0000i	− 0.441503i
$868$	− 32.0000i	− 1.08615i
$869$	−64.0000	−2.17105
$870$	0	0
$871$	48.0000	1.62642
$872$	30.0000i	1.01593i
$873$	10.0000i	0.338449i
$874$	−4.00000	−0.135302
$875$	0	0
$876$	−10.0000	−0.337869
$877$	54.0000i	1.82345i	0.410801	+	0.911725i	$0.365249\pi$
$877$	54.0000i	1.82345i	−0.410801	+	0.911725i	$0.634751\pi$
$878$	16.0000i	0.539974i
$879$	−26.0000	−0.876958
$880$	0	0
$881$	30.0000	1.01073	0.505363	−	0.862907i	$-0.331359\pi$
$881$	30.0000	1.01073	0.505363	+	0.862907i	$0.331359\pi$
$882$	9.00000i	0.303046i
$883$	− 4.00000i	− 0.134611i	−0.997732	−	0.0673054i	$-0.978560\pi$
$883$	− 4.00000i	− 0.134611i	0.997732	−	0.0673054i	$-0.0214402\pi$
$884$	−12.0000	−0.403604
$885$	0	0
$886$	−12.0000	−0.403148
$887$	48.0000i	1.61168i	0.592132	+	0.805841i	$0.298286\pi$
$887$	48.0000i	1.61168i	−0.592132	+	0.805841i	$0.701714\pi$
$888$	6.00000i	0.201347i
$889$	0	0
$890$	0	0
$891$	4.00000	0.134005
$892$	− 24.0000i	− 0.803579i
$893$	0	0
$894$	6.00000	0.200670
$895$	0	0
$896$	−12.0000	−0.400892
$897$	− 6.00000i	− 0.200334i
$898$	14.0000i	0.467186i
$899$	−80.0000	−2.66815
$900$	0	0
$901$	12.0000	0.399778
$902$	8.00000i	0.266371i
$903$	32.0000i	1.06489i
$904$	−6.00000	−0.199557
$905$	0	0
$906$	−16.0000	−0.531564
$907$	− 40.0000i	− 1.32818i	−0.747653	−	0.664089i	$-0.768820\pi$
$907$	− 40.0000i	− 1.32818i	0.747653	−	0.664089i	$-0.231180\pi$
$908$	− 12.0000i	− 0.398234i
$909$	−14.0000	−0.464351
$910$	0	0
$911$	−8.00000	−0.265052	−0.132526	−	0.991180i	$-0.542309\pi$
$911$	−8.00000	−0.265052	−0.132526	+	0.991180i	$0.542309\pi$
$912$	4.00000i	0.132453i
$913$	48.0000i	1.58857i
$914$	18.0000	0.595387
$915$	0	0
$916$	−6.00000	−0.198246
$917$	32.0000i	1.05673i
$918$	2.00000i	0.0660098i
$919$	−16.0000	−0.527791	−0.263896	−	0.964551i	$-0.585007\pi$
$919$	−16.0000	−0.527791	−0.263896	+	0.964551i	$0.585007\pi$
$920$	0	0
$921$	−28.0000	−0.922631
$922$	− 34.0000i	− 1.11973i
$923$	24.0000i	0.789970i
$924$	16.0000	0.526361
$925$	0	0
$926$	−32.0000	−1.05159
$927$	− 12.0000i	− 0.394132i
$928$	50.0000i	1.64133i
$929$	−18.0000	−0.590561	−0.295280	−	0.955411i	$-0.595413\pi$
$929$	−18.0000	−0.590561	−0.295280	+	0.955411i	$0.595413\pi$
$930$	0	0
$931$	−36.0000	−1.17985
$932$	2.00000i	0.0655122i
$933$	− 12.0000i	− 0.392862i
$934$	−20.0000	−0.654420
$935$	0	0
$936$	−18.0000	−0.588348
$937$	− 2.00000i	− 0.0653372i	−0.999466	−	0.0326686i	$-0.989599\pi$
$937$	− 2.00000i	− 0.0653372i	0.999466	−	0.0326686i	$-0.0104006\pi$
$938$	− 32.0000i	− 1.04484i
$939$	−14.0000	−0.456873
$940$	0	0
$941$	−14.0000	−0.456387	−0.228193	−	0.973616i	$-0.573282\pi$
$941$	−14.0000	−0.456387	−0.228193	+	0.973616i	$0.573282\pi$
$942$	− 14.0000i	− 0.456145i
$943$	2.00000i	0.0651290i
$944$	0	0
$945$	0	0
$946$	−32.0000	−1.04041
$947$	4.00000i	0.129983i	0.997886	+	0.0649913i	$0.0207020\pi$
$947$	4.00000i	0.129983i	−0.997886	+	0.0649913i	$0.979298\pi$
$948$	16.0000i	0.519656i
$949$	−60.0000	−1.94768
$950$	0	0
$951$	−6.00000	−0.194563
$952$	24.0000i	0.777844i
$953$	42.0000i	1.36051i	0.732974	+	0.680257i	$0.238132\pi$
$953$	42.0000i	1.36051i	−0.732974	+	0.680257i	$0.761868\pi$
$954$	6.00000	0.194257
$955$	0	0
$956$	−20.0000	−0.646846
$957$	− 40.0000i	− 1.29302i
$958$	− 40.0000i	− 1.29234i
$959$	40.0000	1.29167
$960$	0	0
$961$	33.0000	1.06452
$962$	12.0000i	0.386896i
$963$	12.0000i	0.386695i
$964$	−14.0000	−0.450910
$965$	0	0
$966$	−4.00000	−0.128698
$967$	− 32.0000i	− 1.02905i	−0.857475	−	0.514525i	$-0.827968\pi$
$967$	− 32.0000i	− 1.02905i	0.857475	−	0.514525i	$-0.172032\pi$
$968$	15.0000i	0.482118i
$969$	−8.00000	−0.256997
$970$	0	0
$971$	4.00000	0.128366	0.0641831	−	0.997938i	$-0.479556\pi$
$971$	4.00000	0.128366	0.0641831	+	0.997938i	$0.479556\pi$
$972$	− 1.00000i	− 0.0320750i
$973$	− 16.0000i	− 0.512936i
$974$	−24.0000	−0.769010
$975$	0	0
$976$	−6.00000	−0.192055
$977$	− 50.0000i	− 1.59964i	−0.600239	−	0.799821i	$-0.704928\pi$
$977$	− 50.0000i	− 1.59964i	0.600239	−	0.799821i	$-0.295072\pi$
$978$	− 20.0000i	− 0.639529i
$979$	40.0000	1.27841
$980$	0	0
$981$	−10.0000	−0.319275
$982$	16.0000i	0.510581i
$983$	− 8.00000i	− 0.255160i	−0.991828	−	0.127580i	$-0.959279\pi$
$983$	− 8.00000i	− 0.255160i	0.991828	−	0.127580i	$-0.0407210\pi$
$984$	6.00000	0.191273
$985$	0	0
$986$	20.0000	0.636930
$987$	0	0
$988$	− 24.0000i	− 0.763542i
$989$	−8.00000	−0.254385
$990$	0	0
$991$	−40.0000	−1.27064	−0.635321	−	0.772248i	$-0.719132\pi$
$991$	−40.0000	−1.27064	−0.635321	+	0.772248i	$0.719132\pi$
$992$	− 40.0000i	− 1.27000i
$993$	− 12.0000i	− 0.380808i
$994$	16.0000	0.507489
$995$	0	0
$996$	12.0000	0.380235
$997$	− 2.00000i	− 0.0633406i	−0.999498	−	0.0316703i	$-0.989917\pi$
$997$	− 2.00000i	− 0.0633406i	0.999498	−	0.0316703i	$-0.0100827\pi$
$998$	− 4.00000i	− 0.126618i
$999$	−2.00000	−0.0632772

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1725.2.b.f.1174.2		2
5.2	odd	4		1725.2.a.b.1.1		1
5.3	odd	4		345.2.a.e.1.1	✓	1
5.4	even	2	inner	1725.2.b.f.1174.1		2
15.2	even	4		5175.2.a.q.1.1		1
15.8	even	4		1035.2.a.c.1.1		1
20.3	even	4		5520.2.a.a.1.1		1
115.68	even	4		7935.2.a.j.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
345.2.a.e.1.1	✓	1	5.3	odd	4
1035.2.a.c.1.1		1	15.8	even	4
1725.2.a.b.1.1		1	5.2	odd	4
1725.2.b.f.1174.1		2	5.4	even	2	inner
1725.2.b.f.1174.2		2	1.1	even	1	trivial
5175.2.a.q.1.1		1	15.2	even	4
5520.2.a.a.1.1		1	20.3	even	4
7935.2.a.j.1.1		1	115.68	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:	\( N \)	\(=\)	\( 1725 = 3 \cdot 5^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1725.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{2} -1.00000i q^{3} +1.00000 q^{4} +1.00000 q^{6} +4.00000i q^{7} +3.00000i q^{8} -1.00000 q^{9} +O(q^{10})\)	\(q+1.00000i q^{2} -1.00000i q^{3} +1.00000 q^{4} +1.00000 q^{6} +4.00000i q^{7} +3.00000i q^{8} -1.00000 q^{9} +4.00000 q^{11} -1.00000i q^{12} -6.00000i q^{13} -4.00000 q^{14} -1.00000 q^{16} -2.00000i q^{17} -1.00000i q^{18} +4.00000 q^{19} +4.00000 q^{21} +4.00000i q^{22} +1.00000i q^{23} +3.00000 q^{24} +6.00000 q^{26} +1.00000i q^{27} +4.00000i q^{28} +10.0000 q^{29} -8.00000 q^{31} +5.00000i q^{32} -4.00000i q^{33} +2.00000 q^{34} -1.00000 q^{36} +2.00000i q^{37} +4.00000i q^{38} -6.00000 q^{39} +2.00000 q^{41} +4.00000i q^{42} +8.00000i q^{43} +4.00000 q^{44} -1.00000 q^{46} +1.00000i q^{48} -9.00000 q^{49} -2.00000 q^{51} -6.00000i q^{52} +6.00000i q^{53} -1.00000 q^{54} -12.0000 q^{56} -4.00000i q^{57} +10.0000i q^{58} +6.00000 q^{61} -8.00000i q^{62} -4.00000i q^{63} -7.00000 q^{64} +4.00000 q^{66} +8.00000i q^{67} -2.00000i q^{68} +1.00000 q^{69} -4.00000 q^{71} -3.00000i q^{72} -10.0000i q^{73} -2.00000 q^{74} +4.00000 q^{76} +16.0000i q^{77} -6.00000i q^{78} -16.0000 q^{79} +1.00000 q^{81} +2.00000i q^{82} +12.0000i q^{83} +4.00000 q^{84} -8.00000 q^{86} -10.0000i q^{87} +12.0000i q^{88} +10.0000 q^{89} +24.0000 q^{91} +1.00000i q^{92} +8.00000i q^{93} +5.00000 q^{96} -10.0000i q^{97} -9.00000i q^{98} -4.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{4} + 2 q^{6} - 2 q^{9}+O(q^{10}) \) 2 * q + 2 * q^4 + 2 * q^6 - 2 * q^9	\( 2 q + 2 q^{4} + 2 q^{6} - 2 q^{9} + 8 q^{11} - 8 q^{14} - 2 q^{16} + 8 q^{19} + 8 q^{21} + 6 q^{24} + 12 q^{26} + 20 q^{29} - 16 q^{31} + 4 q^{34} - 2 q^{36} - 12 q^{39} + 4 q^{41} + 8 q^{44} - 2 q^{46} - 18 q^{49} - 4 q^{51} - 2 q^{54} - 24 q^{56} + 12 q^{61} - 14 q^{64} + 8 q^{66} + 2 q^{69} - 8 q^{71} - 4 q^{74} + 8 q^{76} - 32 q^{79} + 2 q^{81} + 8 q^{84} - 16 q^{86} + 20 q^{89} + 48 q^{91} + 10 q^{96} - 8 q^{99}+O(q^{100}) \) 2 * q + 2 * q^4 + 2 * q^6 - 2 * q^9 + 8 * q^11 - 8 * q^14 - 2 * q^16 + 8 * q^19 + 8 * q^21 + 6 * q^24 + 12 * q^26 + 20 * q^29 - 16 * q^31 + 4 * q^34 - 2 * q^36 - 12 * q^39 + 4 * q^41 + 8 * q^44 - 2 * q^46 - 18 * q^49 - 4 * q^51 - 2 * q^54 - 24 * q^56 + 12 * q^61 - 14 * q^64 + 8 * q^66 + 2 * q^69 - 8 * q^71 - 4 * q^74 + 8 * q^76 - 32 * q^79 + 2 * q^81 + 8 * q^84 - 16 * q^86 + 20 * q^89 + 48 * q^91 + 10 * q^96 - 8 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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