LMFDB - Embedded newform 1725.2.b.s.1174.3

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:	$4$
Coefficient field:	$\Q(\zeta_{8})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 1 $ x^4 + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 345)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1174.3
Root			$-0.707107 - 0.707107i$ of defining polynomial
Character	$\chi$	$=$	1725.1174
Dual form			1725.2.b.s.1174.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.41421i q^{2} -1.00000i q^{3} +1.41421 q^{6} -1.82843i q^{7} +2.82843i q^{8} -1.00000 q^{9} +O(q^{10})$	\(q+1.41421i q^{2} -1.00000i q^{3} +1.41421 q^{6} -1.82843i q^{7} +2.82843i q^{8} -1.00000 q^{9} -2.58579 q^{11} +3.41421i q^{13} +2.58579 q^{14} -4.00000 q^{16} +6.07107i q^{17} -1.41421i q^{18} +6.24264 q^{19} -1.82843 q^{21} -3.65685i q^{22} -1.00000i q^{23} +2.82843 q^{24} -4.82843 q^{26} +1.00000i q^{27} +3.58579 q^{29} -4.17157 q^{31} +2.58579i q^{33} -8.58579 q^{34} -3.00000i q^{37} +8.82843i q^{38} +3.41421 q^{39} -10.4142 q^{41} -2.58579i q^{42} +6.00000i q^{43} +1.41421 q^{46} +8.58579i q^{47} +4.00000i q^{48} +3.65685 q^{49} +6.07107 q^{51} +7.24264i q^{53} -1.41421 q^{54} +5.17157 q^{56} -6.24264i q^{57} +5.07107i q^{58} +6.89949 q^{59} +5.41421 q^{61} -5.89949i q^{62} +1.82843i q^{63} -8.00000 q^{64} -3.65685 q^{66} +11.4853i q^{67} -1.00000 q^{69} +8.89949 q^{71} -2.82843i q^{72} -10.2426i q^{73} +4.24264 q^{74} +4.72792i q^{77} +4.82843i q^{78} +15.3137 q^{79} +1.00000 q^{81} -14.7279i q^{82} +14.0711i q^{83} -8.48528 q^{86} -3.58579i q^{87} -7.31371i q^{88} -10.1421 q^{89} +6.24264 q^{91} +4.17157i q^{93} -12.1421 q^{94} +1.17157i q^{97} +5.17157i q^{98} +2.58579 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{9}+O(q^{10}) $ 4 * q - 4 * q^9	$ 4 q - 4 q^{9} - 16 q^{11} + 16 q^{14} - 16 q^{16} + 8 q^{19} + 4 q^{21} - 8 q^{26} + 20 q^{29} - 28 q^{31} - 40 q^{34} + 8 q^{39} - 36 q^{41} - 8 q^{49} - 4 q^{51} + 32 q^{56} - 12 q^{59} + 16 q^{61} - 32 q^{64} + 8 q^{66} - 4 q^{69} - 4 q^{71} + 16 q^{79} + 4 q^{81} + 16 q^{89} + 8 q^{91} + 8 q^{94} + 16 q^{99}+O(q^{100}) $ 4 * q - 4 * q^9 - 16 * q^11 + 16 * q^14 - 16 * q^16 + 8 * q^19 + 4 * q^21 - 8 * q^26 + 20 * q^29 - 28 * q^31 - 40 * q^34 + 8 * q^39 - 36 * q^41 - 8 * q^49 - 4 * q^51 + 32 * q^56 - 12 * q^59 + 16 * q^61 - 32 * q^64 + 8 * q^66 - 4 * q^69 - 4 * q^71 + 16 * q^79 + 4 * q^81 + 16 * q^89 + 8 * q^91 + 8 * q^94 + 16 * 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 $

$2$	1.41421i	1.00000i	0.866025	+	0.500000i	$0.166667\pi$
$2$	1.41421i	1.00000i	−0.866025	+	0.500000i	$0.833333\pi$
$3$	− 1.00000i	− 0.577350i
$3$	− 1.00000i	− 0.577350i
$4$	0	0
$5$	0	0
$5$	0	0
$6$	1.41421	0.577350
$7$	− 1.82843i	− 0.691080i	−0.938404	−	0.345540i	$-0.887696\pi$
$7$	− 1.82843i	− 0.691080i	0.938404	−	0.345540i	$-0.112304\pi$
$8$	2.82843i	1.00000i
$9$	−1.00000	−0.333333
$10$	0	0
$11$	−2.58579	−0.779644	−0.389822	−	0.920890i	$-0.627463\pi$
$11$	−2.58579	−0.779644	−0.389822	+	0.920890i	$0.627463\pi$
$12$	0	0
$13$	3.41421i	0.946932i	0.880812	+	0.473466i	$0.156997\pi$
$13$	3.41421i	0.946932i	−0.880812	+	0.473466i	$0.843003\pi$
$14$	2.58579	0.691080
$15$	0	0
$16$	−4.00000	−1.00000
$17$	6.07107i	1.47245i	0.676737	+	0.736225i	$0.263394\pi$
$17$	6.07107i	1.47245i	−0.676737	+	0.736225i	$0.736606\pi$
$18$	− 1.41421i	− 0.333333i
$19$	6.24264	1.43216	0.716080	−	0.698018i	$-0.245935\pi$
$19$	6.24264	1.43216	0.716080	+	0.698018i	$0.245935\pi$
$20$	0	0
$21$	−1.82843	−0.398996
$22$	− 3.65685i	− 0.779644i
$23$	− 1.00000i	− 0.208514i
$23$	− 1.00000i	− 0.208514i
$24$	2.82843	0.577350
$25$	0	0
$26$	−4.82843	−0.946932
$27$	1.00000i	0.192450i
$28$	0	0
$29$	3.58579	0.665864	0.332932	−	0.942951i	$-0.391962\pi$
$29$	3.58579	0.665864	0.332932	+	0.942951i	$0.391962\pi$
$30$	0	0
$31$	−4.17157	−0.749237	−0.374618	−	0.927179i	$-0.622226\pi$
$31$	−4.17157	−0.749237	−0.374618	+	0.927179i	$0.622226\pi$
$32$	0	0
$33$	2.58579i	0.450128i
$34$	−8.58579	−1.47245
$35$	0	0
$36$	0	0
$37$	− 3.00000i	− 0.493197i	−0.969118	−	0.246598i	$-0.920687\pi$
$37$	− 3.00000i	− 0.493197i	0.969118	−	0.246598i	$-0.0793129\pi$
$38$	8.82843i	1.43216i
$39$	3.41421	0.546712
$40$	0	0
$41$	−10.4142	−1.62643	−0.813213	−	0.581966i	$-0.802284\pi$
$41$	−10.4142	−1.62643	−0.813213	+	0.581966i	$0.802284\pi$
$42$	− 2.58579i	− 0.398996i
$43$	6.00000i	0.914991i	0.889212	+	0.457496i	$0.151253\pi$
$43$	6.00000i	0.914991i	−0.889212	+	0.457496i	$0.848747\pi$
$44$	0	0
$45$	0	0
$46$	1.41421	0.208514
$47$	8.58579i	1.25237i	0.779676	+	0.626183i	$0.215384\pi$
$47$	8.58579i	1.25237i	−0.779676	+	0.626183i	$0.784616\pi$
$48$	4.00000i	0.577350i
$49$	3.65685	0.522408
$50$	0	0
$51$	6.07107	0.850120
$52$	0	0
$53$	7.24264i	0.994853i	0.867506	+	0.497427i	$0.165722\pi$
$53$	7.24264i	0.994853i	−0.867506	+	0.497427i	$0.834278\pi$
$54$	−1.41421	−0.192450
$55$	0	0
$56$	5.17157	0.691080
$57$	− 6.24264i	− 0.826858i
$58$	5.07107i	0.665864i
$59$	6.89949	0.898238	0.449119	−	0.893472i	$-0.351738\pi$
$59$	6.89949	0.898238	0.449119	+	0.893472i	$0.351738\pi$
$60$	0	0
$61$	5.41421	0.693219	0.346610	−	0.938010i	$-0.387333\pi$
$61$	5.41421	0.693219	0.346610	+	0.938010i	$0.387333\pi$
$62$	− 5.89949i	− 0.749237i
$63$	1.82843i	0.230360i
$64$	−8.00000	−1.00000
$65$	0	0
$66$	−3.65685	−0.450128
$67$	11.4853i	1.40315i	0.712595	+	0.701575i	$0.247520\pi$
$67$	11.4853i	1.40315i	−0.712595	+	0.701575i	$0.752480\pi$
$68$	0	0
$69$	−1.00000	−0.120386
$70$	0	0
$71$	8.89949	1.05618	0.528088	−	0.849190i	$-0.322909\pi$
$71$	8.89949	1.05618	0.528088	+	0.849190i	$0.322909\pi$
$72$	− 2.82843i	− 0.333333i
$73$	− 10.2426i	− 1.19881i	−0.800446	−	0.599405i	$-0.795404\pi$
$73$	− 10.2426i	− 1.19881i	0.800446	−	0.599405i	$-0.204596\pi$
$74$	4.24264	0.493197
$75$	0	0
$76$	0	0
$77$	4.72792i	0.538797i
$78$	4.82843i	0.546712i
$79$	15.3137	1.72293	0.861463	−	0.507820i	$-0.169548\pi$
$79$	15.3137	1.72293	0.861463	+	0.507820i	$0.169548\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	− 14.7279i	− 1.62643i
$83$	14.0711i	1.54450i	0.635319	+	0.772250i	$0.280869\pi$
$83$	14.0711i	1.54450i	−0.635319	+	0.772250i	$0.719131\pi$
$84$	0	0
$85$	0	0
$86$	−8.48528	−0.914991
$87$	− 3.58579i	− 0.384437i
$88$	− 7.31371i	− 0.779644i
$89$	−10.1421	−1.07506	−0.537532	−	0.843243i	$-0.680643\pi$
$89$	−10.1421	−1.07506	−0.537532	+	0.843243i	$0.680643\pi$
$90$	0	0
$91$	6.24264	0.654407
$92$	0	0
$93$	4.17157i	0.432572i
$94$	−12.1421	−1.25237
$95$	0	0
$96$	0	0
$97$	1.17157i	0.118955i	0.998230	+	0.0594776i	$0.0189435\pi$
$97$	1.17157i	0.118955i	−0.998230	+	0.0594776i	$0.981057\pi$
$98$	5.17157i	0.522408i
$99$	2.58579	0.259881
$100$	0	0
$101$	12.5563	1.24940	0.624702	−	0.780863i	$-0.285221\pi$
$101$	12.5563	1.24940	0.624702	+	0.780863i	$0.285221\pi$
$102$	8.58579i	0.850120i
$103$	0.828427i	0.0816274i	0.999167	+	0.0408137i	$0.0129950\pi$
$103$	0.828427i	0.0816274i	−0.999167	+	0.0408137i	$0.987005\pi$
$104$	−9.65685	−0.946932
$105$	0	0
$106$	−10.2426	−0.994853
$107$	4.41421i	0.426738i	0.976972	+	0.213369i	$0.0684437\pi$
$107$	4.41421i	0.426738i	−0.976972	+	0.213369i	$0.931556\pi$
$108$	0	0
$109$	−14.2426	−1.36420	−0.682099	−	0.731260i	$-0.738933\pi$
$109$	−14.2426	−1.36420	−0.682099	+	0.731260i	$0.738933\pi$
$110$	0	0
$111$	−3.00000	−0.284747
$112$	7.31371i	0.691080i
$113$	− 10.4142i	− 0.979687i	−0.871810	−	0.489843i	$-0.837054\pi$
$113$	− 10.4142i	− 0.979687i	0.871810	−	0.489843i	$-0.162946\pi$
$114$	8.82843	0.826858
$115$	0	0
$116$	0	0
$117$	− 3.41421i	− 0.315644i
$118$	9.75736i	0.898238i
$119$	11.1005	1.01758
$120$	0	0
$121$	−4.31371	−0.392155
$122$	7.65685i	0.693219i
$123$	10.4142i	0.939018i
$124$	0	0
$125$	0	0
$126$	−2.58579	−0.230360
$127$	0.242641i	0.0215309i	0.999942	+	0.0107654i	$0.00342681\pi$
$127$	0.242641i	0.0215309i	−0.999942	+	0.0107654i	$0.996573\pi$
$128$	− 11.3137i	− 1.00000i
$129$	6.00000	0.528271
$130$	0	0
$131$	−13.6569	−1.19320	−0.596602	−	0.802537i	$-0.703483\pi$
$131$	−13.6569	−1.19320	−0.596602	+	0.802537i	$0.703483\pi$
$132$	0	0
$133$	− 11.4142i	− 0.989738i
$134$	−16.2426	−1.40315
$135$	0	0
$136$	−17.1716	−1.47245
$137$	− 2.82843i	− 0.241649i	−0.992674	−	0.120824i	$-0.961446\pi$
$137$	− 2.82843i	− 0.241649i	0.992674	−	0.120824i	$-0.0385538\pi$
$138$	− 1.41421i	− 0.120386i
$139$	11.0000	0.933008	0.466504	−	0.884519i	$-0.345513\pi$
$139$	11.0000	0.933008	0.466504	+	0.884519i	$0.345513\pi$
$140$	0	0
$141$	8.58579	0.723054
$142$	12.5858i	1.05618i
$143$	− 8.82843i	− 0.738270i
$144$	4.00000	0.333333
$145$	0	0
$146$	14.4853	1.19881
$147$	− 3.65685i	− 0.301612i
$148$	0	0
$149$	−9.07107	−0.743131	−0.371565	−	0.928407i	$-0.621179\pi$
$149$	−9.07107	−0.743131	−0.371565	+	0.928407i	$0.621179\pi$
$150$	0	0
$151$	−9.31371	−0.757939	−0.378969	−	0.925409i	$-0.623721\pi$
$151$	−9.31371	−0.757939	−0.378969	+	0.925409i	$0.623721\pi$
$152$	17.6569i	1.43216i
$153$	− 6.07107i	− 0.490817i
$154$	−6.68629	−0.538797
$155$	0	0
$156$	0	0
$157$	1.48528i	0.118538i	0.998242	+	0.0592692i	$0.0188770\pi$
$157$	1.48528i	0.118538i	−0.998242	+	0.0592692i	$0.981123\pi$
$158$	21.6569i	1.72293i
$159$	7.24264	0.574379
$160$	0	0
$161$	−1.82843	−0.144100
$162$	1.41421i	0.111111i
$163$	− 2.00000i	− 0.156652i	−0.996928	−	0.0783260i	$-0.975042\pi$
$163$	− 2.00000i	− 0.156652i	0.996928	−	0.0783260i	$-0.0249575\pi$
$164$	0	0
$165$	0	0
$166$	−19.8995	−1.54450
$167$	6.72792i	0.520622i	0.965525	+	0.260311i	$0.0838251\pi$
$167$	6.72792i	0.520622i	−0.965525	+	0.260311i	$0.916175\pi$
$168$	− 5.17157i	− 0.398996i
$169$	1.34315	0.103319
$170$	0	0
$171$	−6.24264	−0.477387
$172$	0	0
$173$	13.6569i	1.03831i	0.854680	+	0.519156i	$0.173754\pi$
$173$	13.6569i	1.03831i	−0.854680	+	0.519156i	$0.826246\pi$
$174$	5.07107	0.384437
$175$	0	0
$176$	10.3431	0.779644
$177$	− 6.89949i	− 0.518598i
$178$	− 14.3431i	− 1.07506i
$179$	7.65685	0.572300	0.286150	−	0.958185i	$-0.407624\pi$
$179$	7.65685	0.572300	0.286150	+	0.958185i	$0.407624\pi$
$180$	0	0
$181$	−6.82843	−0.507553	−0.253776	−	0.967263i	$-0.581673\pi$
$181$	−6.82843	−0.507553	−0.253776	+	0.967263i	$0.581673\pi$
$182$	8.82843i	0.654407i
$183$	− 5.41421i	− 0.400230i
$184$	2.82843	0.208514
$185$	0	0
$186$	−5.89949	−0.432572
$187$	− 15.6985i	− 1.14799i
$188$	0	0
$189$	1.82843	0.132999
$190$	0	0
$191$	10.2426	0.741131	0.370566	−	0.928806i	$-0.379164\pi$
$191$	10.2426	0.741131	0.370566	+	0.928806i	$0.379164\pi$
$192$	8.00000i	0.577350i
$193$	− 8.48528i	− 0.610784i	−0.952227	−	0.305392i	$-0.901213\pi$
$193$	− 8.48528i	− 0.610784i	0.952227	−	0.305392i	$-0.0987875\pi$
$194$	−1.65685	−0.118955
$195$	0	0
$196$	0	0
$197$	− 16.8284i	− 1.19898i	−0.800384	−	0.599488i	$-0.795371\pi$
$197$	− 16.8284i	− 1.19898i	0.800384	−	0.599488i	$-0.204629\pi$
$198$	3.65685i	0.259881i
$199$	−10.4853	−0.743282	−0.371641	−	0.928377i	$-0.621205\pi$
$199$	−10.4853	−0.743282	−0.371641	+	0.928377i	$0.621205\pi$
$200$	0	0
$201$	11.4853	0.810109
$202$	17.7574i	1.24940i
$203$	− 6.55635i	− 0.460166i
$204$	0	0
$205$	0	0
$206$	−1.17157	−0.0816274
$207$	1.00000i	0.0695048i
$208$	− 13.6569i	− 0.946932i
$209$	−16.1421	−1.11657
$210$	0	0
$211$	0.656854	0.0452197	0.0226099	−	0.999744i	$-0.492802\pi$
$211$	0.656854	0.0452197	0.0226099	+	0.999744i	$0.492802\pi$
$212$	0	0
$213$	− 8.89949i	− 0.609783i
$214$	−6.24264	−0.426738
$215$	0	0
$216$	−2.82843	−0.192450
$217$	7.62742i	0.517783i
$218$	− 20.1421i	− 1.36420i
$219$	−10.2426	−0.692134
$220$	0	0
$221$	−20.7279	−1.39431
$222$	− 4.24264i	− 0.284747i
$223$	− 14.0000i	− 0.937509i	−0.883328	−	0.468755i	$-0.844703\pi$
$223$	− 14.0000i	− 0.937509i	0.883328	−	0.468755i	$-0.155297\pi$
$224$	0	0
$225$	0	0
$226$	14.7279	0.979687
$227$	− 17.6569i	− 1.17193i	−0.810338	−	0.585963i	$-0.800716\pi$
$227$	− 17.6569i	− 1.17193i	0.810338	−	0.585963i	$-0.199284\pi$
$228$	0	0
$229$	−23.7990	−1.57268	−0.786341	−	0.617793i	$-0.788027\pi$
$229$	−23.7990	−1.57268	−0.786341	+	0.617793i	$0.788027\pi$
$230$	0	0
$231$	4.72792	0.311074
$232$	10.1421i	0.665864i
$233$	9.65685i	0.632642i	0.948652	+	0.316321i	$0.102448\pi$
$233$	9.65685i	0.632642i	−0.948652	+	0.316321i	$0.897552\pi$
$234$	4.82843	0.315644
$235$	0	0
$236$	0	0
$237$	− 15.3137i	− 0.994732i
$238$	15.6985i	1.01758i
$239$	1.58579	0.102576	0.0512880	−	0.998684i	$-0.483667\pi$
$239$	1.58579	0.102576	0.0512880	+	0.998684i	$0.483667\pi$
$240$	0	0
$241$	23.4142	1.50824	0.754121	−	0.656735i	$-0.228063\pi$
$241$	23.4142	1.50824	0.754121	+	0.656735i	$0.228063\pi$
$242$	− 6.10051i	− 0.392155i
$243$	− 1.00000i	− 0.0641500i
$244$	0	0
$245$	0	0
$246$	−14.7279	−0.939018
$247$	21.3137i	1.35616i
$248$	− 11.7990i	− 0.749237i
$249$	14.0711	0.891718
$250$	0	0
$251$	−20.6274	−1.30199	−0.650996	−	0.759082i	$-0.725648\pi$
$251$	−20.6274	−1.30199	−0.650996	+	0.759082i	$0.725648\pi$
$252$	0	0
$253$	2.58579i	0.162567i
$254$	−0.343146	−0.0215309
$255$	0	0
$256$	0	0
$257$	− 30.0416i	− 1.87395i	−0.349403	−	0.936973i	$-0.613615\pi$
$257$	− 30.0416i	− 1.87395i	0.349403	−	0.936973i	$-0.386385\pi$
$258$	8.48528i	0.528271i
$259$	−5.48528	−0.340839
$260$	0	0
$261$	−3.58579	−0.221955
$262$	− 19.3137i	− 1.19320i
$263$	3.72792i	0.229874i	0.993373	+	0.114937i	$0.0366665\pi$
$263$	3.72792i	0.229874i	−0.993373	+	0.114937i	$0.963333\pi$
$264$	−7.31371	−0.450128
$265$	0	0
$266$	16.1421	0.989738
$267$	10.1421i	0.620689i
$268$	0	0
$269$	26.5563	1.61917	0.809585	−	0.587003i	$-0.199692\pi$
$269$	26.5563	1.61917	0.809585	+	0.587003i	$0.199692\pi$
$270$	0	0
$271$	−6.51472	−0.395741	−0.197870	−	0.980228i	$-0.563403\pi$
$271$	−6.51472	−0.395741	−0.197870	+	0.980228i	$0.563403\pi$
$272$	− 24.2843i	− 1.47245i
$273$	− 6.24264i	− 0.377822i
$274$	4.00000	0.241649
$275$	0	0
$276$	0	0
$277$	− 25.3137i	− 1.52095i	−0.649365	−	0.760477i	$-0.724965\pi$
$277$	− 25.3137i	− 1.52095i	0.649365	−	0.760477i	$-0.275035\pi$
$278$	15.5563i	0.933008i
$279$	4.17157	0.249746
$280$	0	0
$281$	17.4142	1.03884	0.519422	−	0.854518i	$-0.326147\pi$
$281$	17.4142	1.03884	0.519422	+	0.854518i	$0.326147\pi$
$282$	12.1421i	0.723054i
$283$	− 28.6569i	− 1.70347i	−0.523970	−	0.851737i	$-0.675550\pi$
$283$	− 28.6569i	− 1.70347i	0.523970	−	0.851737i	$-0.324450\pi$
$284$	0	0
$285$	0	0
$286$	12.4853	0.738270
$287$	19.0416i	1.12399i
$288$	0	0
$289$	−19.8579	−1.16811
$290$	0	0
$291$	1.17157	0.0686788
$292$	0	0
$293$	− 3.10051i	− 0.181133i	−0.995890	−	0.0905667i	$-0.971132\pi$
$293$	− 3.10051i	− 0.181133i	0.995890	−	0.0905667i	$-0.0288678\pi$
$294$	5.17157	0.301612
$295$	0	0
$296$	8.48528	0.493197
$297$	− 2.58579i	− 0.150043i
$298$	− 12.8284i	− 0.743131i
$299$	3.41421	0.197449
$300$	0	0
$301$	10.9706	0.632333
$302$	− 13.1716i	− 0.757939i
$303$	− 12.5563i	− 0.721343i
$304$	−24.9706	−1.43216
$305$	0	0
$306$	8.58579	0.490817
$307$	− 27.2132i	− 1.55314i	−0.630031	−	0.776570i	$-0.716958\pi$
$307$	− 27.2132i	− 1.55314i	0.630031	−	0.776570i	$-0.283042\pi$
$308$	0	0
$309$	0.828427	0.0471276
$310$	0	0
$311$	17.6569	1.00123	0.500614	−	0.865671i	$-0.333108\pi$
$311$	17.6569	1.00123	0.500614	+	0.865671i	$0.333108\pi$
$312$	9.65685i	0.546712i
$313$	25.9706i	1.46794i	0.679180	+	0.733971i	$0.262335\pi$
$313$	25.9706i	1.46794i	−0.679180	+	0.733971i	$0.737665\pi$
$314$	−2.10051	−0.118538
$315$	0	0
$316$	0	0
$317$	29.5563i	1.66005i	0.557726	+	0.830025i	$0.311674\pi$
$317$	29.5563i	1.66005i	−0.557726	+	0.830025i	$0.688326\pi$
$318$	10.2426i	0.574379i
$319$	−9.27208	−0.519137
$320$	0	0
$321$	4.41421	0.246377
$322$	− 2.58579i	− 0.144100i
$323$	37.8995i	2.10878i
$324$	0	0
$325$	0	0
$326$	2.82843	0.156652
$327$	14.2426i	0.787620i
$328$	− 29.4558i	− 1.62643i
$329$	15.6985	0.865485
$330$	0	0
$331$	1.00000	0.0549650	0.0274825	−	0.999622i	$-0.491251\pi$
$331$	1.00000	0.0549650	0.0274825	+	0.999622i	$0.491251\pi$
$332$	0	0
$333$	3.00000i	0.164399i
$334$	−9.51472	−0.520622
$335$	0	0
$336$	7.31371	0.398996
$337$	− 30.9706i	− 1.68707i	−0.537071	−	0.843537i	$-0.680469\pi$
$337$	− 30.9706i	− 1.68707i	0.537071	−	0.843537i	$-0.319531\pi$
$338$	1.89949i	0.103319i
$339$	−10.4142	−0.565622
$340$	0	0
$341$	10.7868	0.584138
$342$	− 8.82843i	− 0.477387i
$343$	− 19.4853i	− 1.05211i
$344$	−16.9706	−0.914991
$345$	0	0
$346$	−19.3137	−1.03831
$347$	− 13.1716i	− 0.707087i	−0.935418	−	0.353544i	$-0.884977\pi$
$347$	− 13.1716i	− 0.707087i	0.935418	−	0.353544i	$-0.115023\pi$
$348$	0	0
$349$	32.1127	1.71895	0.859477	−	0.511175i	$-0.170790\pi$
$349$	32.1127	1.71895	0.859477	+	0.511175i	$0.170790\pi$
$350$	0	0
$351$	−3.41421	−0.182237
$352$	0	0
$353$	− 34.5269i	− 1.83768i	−0.394628	−	0.918841i	$-0.629126\pi$
$353$	− 34.5269i	− 1.83768i	0.394628	−	0.918841i	$-0.370874\pi$
$354$	9.75736	0.518598
$355$	0	0
$356$	0	0
$357$	− 11.1005i	− 0.587501i
$358$	10.8284i	0.572300i
$359$	10.9289	0.576807	0.288403	−	0.957509i	$-0.406876\pi$
$359$	10.9289	0.576807	0.288403	+	0.957509i	$0.406876\pi$
$360$	0	0
$361$	19.9706	1.05108
$362$	− 9.65685i	− 0.507553i
$363$	4.31371i	0.226411i
$364$	0	0
$365$	0	0
$366$	7.65685	0.400230
$367$	13.9706i	0.729257i	0.931153	+	0.364629i	$0.118804\pi$
$367$	13.9706i	0.729257i	−0.931153	+	0.364629i	$0.881196\pi$
$368$	4.00000i	0.208514i
$369$	10.4142	0.542142
$370$	0	0
$371$	13.2426	0.687524
$372$	0	0
$373$	− 12.9706i	− 0.671590i	−0.941935	−	0.335795i	$-0.890995\pi$
$373$	− 12.9706i	− 0.671590i	0.941935	−	0.335795i	$-0.109005\pi$
$374$	22.2010	1.14799
$375$	0	0
$376$	−24.2843	−1.25237
$377$	12.2426i	0.630528i
$378$	2.58579i	0.132999i
$379$	−28.9706	−1.48812	−0.744059	−	0.668114i	$-0.767102\pi$
$379$	−28.9706	−1.48812	−0.744059	+	0.668114i	$0.767102\pi$
$380$	0	0
$381$	0.242641	0.0124309
$382$	14.4853i	0.741131i
$383$	14.4142i	0.736532i	0.929720	+	0.368266i	$0.120048\pi$
$383$	14.4142i	0.736532i	−0.929720	+	0.368266i	$0.879952\pi$
$384$	−11.3137	−0.577350
$385$	0	0
$386$	12.0000	0.610784
$387$	− 6.00000i	− 0.304997i
$388$	0	0
$389$	−14.4853	−0.734433	−0.367216	−	0.930136i	$-0.619689\pi$
$389$	−14.4853	−0.734433	−0.367216	+	0.930136i	$0.619689\pi$
$390$	0	0
$391$	6.07107	0.307027
$392$	10.3431i	0.522408i
$393$	13.6569i	0.688897i
$394$	23.7990	1.19898
$395$	0	0
$396$	0	0
$397$	15.3137i	0.768573i	0.923214	+	0.384286i	$0.125553\pi$
$397$	15.3137i	0.768573i	−0.923214	+	0.384286i	$0.874447\pi$
$398$	− 14.8284i	− 0.743282i
$399$	−11.4142	−0.571425
$400$	0	0
$401$	−32.1421	−1.60510	−0.802551	−	0.596584i	$-0.796524\pi$
$401$	−32.1421	−1.60510	−0.802551	+	0.596584i	$0.796524\pi$
$402$	16.2426i	0.810109i
$403$	− 14.2426i	− 0.709476i
$404$	0	0
$405$	0	0
$406$	9.27208	0.460166
$407$	7.75736i	0.384518i
$408$	17.1716i	0.850120i
$409$	−1.48528	−0.0734424	−0.0367212	−	0.999326i	$-0.511691\pi$
$409$	−1.48528	−0.0734424	−0.0367212	+	0.999326i	$0.511691\pi$
$410$	0	0
$411$	−2.82843	−0.139516
$412$	0	0
$413$	− 12.6152i	− 0.620755i
$414$	−1.41421	−0.0695048
$415$	0	0
$416$	0	0
$417$	− 11.0000i	− 0.538672i
$418$	− 22.8284i	− 1.11657i
$419$	24.0416	1.17451	0.587255	−	0.809402i	$-0.300208\pi$
$419$	24.0416	1.17451	0.587255	+	0.809402i	$0.300208\pi$
$420$	0	0
$421$	−1.75736	−0.0856485	−0.0428242	−	0.999083i	$-0.513636\pi$
$421$	−1.75736	−0.0856485	−0.0428242	+	0.999083i	$0.513636\pi$
$422$	0.928932i	0.0452197i
$423$	− 8.58579i	− 0.417455i
$424$	−20.4853	−0.994853
$425$	0	0
$426$	12.5858	0.609783
$427$	− 9.89949i	− 0.479070i
$428$	0	0
$429$	−8.82843	−0.426240
$430$	0	0
$431$	−27.7990	−1.33903	−0.669515	−	0.742798i	$-0.733498\pi$
$431$	−27.7990	−1.33903	−0.669515	+	0.742798i	$0.733498\pi$
$432$	− 4.00000i	− 0.192450i
$433$	− 1.82843i	− 0.0878686i	−0.999034	−	0.0439343i	$-0.986011\pi$
$433$	− 1.82843i	− 0.0878686i	0.999034	−	0.0439343i	$-0.0139892\pi$
$434$	−10.7868	−0.517783
$435$	0	0
$436$	0	0
$437$	− 6.24264i	− 0.298626i
$438$	− 14.4853i	− 0.692134i
$439$	18.4853	0.882254	0.441127	−	0.897445i	$-0.354579\pi$
$439$	18.4853	0.882254	0.441127	+	0.897445i	$0.354579\pi$
$440$	0	0
$441$	−3.65685	−0.174136
$442$	− 29.3137i	− 1.39431i
$443$	30.5269i	1.45038i	0.688550	+	0.725189i	$0.258247\pi$
$443$	30.5269i	1.45038i	−0.688550	+	0.725189i	$0.741753\pi$
$444$	0	0
$445$	0	0
$446$	19.7990	0.937509
$447$	9.07107i	0.429047i
$448$	14.6274i	0.691080i
$449$	13.2426	0.624959	0.312479	−	0.949925i	$-0.398840\pi$
$449$	13.2426	0.624959	0.312479	+	0.949925i	$0.398840\pi$
$450$	0	0
$451$	26.9289	1.26803
$452$	0	0
$453$	9.31371i	0.437596i
$454$	24.9706	1.17193
$455$	0	0
$456$	17.6569	0.826858
$457$	31.0000i	1.45012i	0.688686	+	0.725059i	$0.258188\pi$
$457$	31.0000i	1.45012i	−0.688686	+	0.725059i	$0.741812\pi$
$458$	− 33.6569i	− 1.57268i
$459$	−6.07107	−0.283373
$460$	0	0
$461$	−9.51472	−0.443145	−0.221572	−	0.975144i	$-0.571119\pi$
$461$	−9.51472	−0.443145	−0.221572	+	0.975144i	$0.571119\pi$
$462$	6.68629i	0.311074i
$463$	17.0711i	0.793360i	0.917957	+	0.396680i	$0.129838\pi$
$463$	17.0711i	0.793360i	−0.917957	+	0.396680i	$0.870162\pi$
$464$	−14.3431	−0.665864
$465$	0	0
$466$	−13.6569	−0.632642
$467$	− 3.58579i	− 0.165930i	−0.996552	−	0.0829652i	$-0.973561\pi$
$467$	− 3.58579i	− 0.165930i	0.996552	−	0.0829652i	$-0.0264390\pi$
$468$	0	0
$469$	21.0000	0.969690
$470$	0	0
$471$	1.48528	0.0684382
$472$	19.5147i	0.898238i
$473$	− 15.5147i	− 0.713368i
$474$	21.6569	0.994732
$475$	0	0
$476$	0	0
$477$	− 7.24264i	− 0.331618i
$478$	2.24264i	0.102576i
$479$	18.7279	0.855701	0.427850	−	0.903850i	$-0.359271\pi$
$479$	18.7279	0.855701	0.427850	+	0.903850i	$0.359271\pi$
$480$	0	0
$481$	10.2426	0.467024
$482$	33.1127i	1.50824i
$483$	1.82843i	0.0831963i
$484$	0	0
$485$	0	0
$486$	1.41421	0.0641500
$487$	− 10.3848i	− 0.470579i	−0.971925	−	0.235290i	$-0.924396\pi$
$487$	− 10.3848i	− 0.470579i	0.971925	−	0.235290i	$-0.0756038\pi$
$488$	15.3137i	0.693219i
$489$	−2.00000	−0.0904431
$490$	0	0
$491$	2.07107	0.0934660	0.0467330	−	0.998907i	$-0.485119\pi$
$491$	2.07107	0.0934660	0.0467330	+	0.998907i	$0.485119\pi$
$492$	0	0
$493$	21.7696i	0.980451i
$494$	−30.1421	−1.35616
$495$	0	0
$496$	16.6863	0.749237
$497$	− 16.2721i	− 0.729902i
$498$	19.8995i	0.891718i
$499$	13.0000	0.581960	0.290980	−	0.956729i	$-0.406019\pi$
$499$	13.0000	0.581960	0.290980	+	0.956729i	$0.406019\pi$
$500$	0	0
$501$	6.72792	0.300581
$502$	− 29.1716i	− 1.30199i
$503$	− 14.2721i	− 0.636361i	−0.948030	−	0.318180i	$-0.896928\pi$
$503$	− 14.2721i	− 0.636361i	0.948030	−	0.318180i	$-0.103072\pi$
$504$	−5.17157	−0.230360
$505$	0	0
$506$	−3.65685	−0.162567
$507$	− 1.34315i	− 0.0596512i
$508$	0	0
$509$	31.7990	1.40947	0.704733	−	0.709473i	$-0.251067\pi$
$509$	31.7990	1.40947	0.704733	+	0.709473i	$0.251067\pi$
$510$	0	0
$511$	−18.7279	−0.828474
$512$	− 22.6274i	− 1.00000i
$513$	6.24264i	0.275619i
$514$	42.4853	1.87395
$515$	0	0
$516$	0	0
$517$	− 22.2010i	− 0.976399i
$518$	− 7.75736i	− 0.340839i
$519$	13.6569	0.599469
$520$	0	0
$521$	−0.443651	−0.0194367	−0.00971835	−	0.999953i	$-0.503093\pi$
$521$	−0.443651	−0.0194367	−0.00971835	+	0.999953i	$0.503093\pi$
$522$	− 5.07107i	− 0.221955i
$523$	12.0000i	0.524723i	0.964970	+	0.262362i	$0.0845013\pi$
$523$	12.0000i	0.524723i	−0.964970	+	0.262362i	$0.915499\pi$
$524$	0	0
$525$	0	0
$526$	−5.27208	−0.229874
$527$	− 25.3259i	− 1.10321i
$528$	− 10.3431i	− 0.450128i
$529$	−1.00000	−0.0434783
$530$	0	0
$531$	−6.89949	−0.299413
$532$	0	0
$533$	− 35.5563i	− 1.54012i
$534$	−14.3431	−0.620689
$535$	0	0
$536$	−32.4853	−1.40315
$537$	− 7.65685i	− 0.330418i
$538$	37.5563i	1.61917i
$539$	−9.45584	−0.407292
$540$	0	0
$541$	34.1421	1.46789	0.733943	−	0.679212i	$-0.237678\pi$
$541$	34.1421	1.46789	0.733943	+	0.679212i	$0.237678\pi$
$542$	− 9.21320i	− 0.395741i
$543$	6.82843i	0.293036i
$544$	0	0
$545$	0	0
$546$	8.82843	0.377822
$547$	20.9706i	0.896637i	0.893874	+	0.448318i	$0.147977\pi$
$547$	20.9706i	0.896637i	−0.893874	+	0.448318i	$0.852023\pi$
$548$	0	0
$549$	−5.41421	−0.231073
$550$	0	0
$551$	22.3848	0.953624
$552$	− 2.82843i	− 0.120386i
$553$	− 28.0000i	− 1.19068i
$554$	35.7990	1.52095
$555$	0	0
$556$	0	0
$557$	− 12.8995i	− 0.546569i	−0.961933	−	0.273285i	$-0.911890\pi$
$557$	− 12.8995i	− 0.546569i	0.961933	−	0.273285i	$-0.0881101\pi$
$558$	5.89949i	0.249746i
$559$	−20.4853	−0.866435
$560$	0	0
$561$	−15.6985	−0.662791
$562$	24.6274i	1.03884i
$563$	1.10051i	0.0463808i	0.999731	+	0.0231904i	$0.00738239\pi$
$563$	1.10051i	0.0463808i	−0.999731	+	0.0231904i	$0.992618\pi$
$564$	0	0
$565$	0	0
$566$	40.5269	1.70347
$567$	− 1.82843i	− 0.0767867i
$568$	25.1716i	1.05618i
$569$	16.1421	0.676714	0.338357	−	0.941018i	$-0.390129\pi$
$569$	16.1421	0.676714	0.338357	+	0.941018i	$0.390129\pi$
$570$	0	0
$571$	3.27208	0.136932	0.0684661	−	0.997653i	$-0.478190\pi$
$571$	3.27208	0.136932	0.0684661	+	0.997653i	$0.478190\pi$
$572$	0	0
$573$	− 10.2426i	− 0.427892i
$574$	−26.9289	−1.12399
$575$	0	0
$576$	8.00000	0.333333
$577$	36.9706i	1.53910i	0.638584	+	0.769552i	$0.279521\pi$
$577$	36.9706i	1.53910i	−0.638584	+	0.769552i	$0.720479\pi$
$578$	− 28.0833i	− 1.16811i
$579$	−8.48528	−0.352636
$580$	0	0
$581$	25.7279	1.06737
$582$	1.65685i	0.0686788i
$583$	− 18.7279i	− 0.775631i
$584$	28.9706	1.19881
$585$	0	0
$586$	4.38478	0.181133
$587$	− 40.6274i	− 1.67687i	−0.544999	−	0.838436i	$-0.683470\pi$
$587$	− 40.6274i	− 1.67687i	0.544999	−	0.838436i	$-0.316530\pi$
$588$	0	0
$589$	−26.0416	−1.07303
$590$	0	0
$591$	−16.8284	−0.692229
$592$	12.0000i	0.493197i
$593$	− 2.44365i	− 0.100349i	−0.998740	−	0.0501744i	$-0.984022\pi$
$593$	− 2.44365i	− 0.100349i	0.998740	−	0.0501744i	$-0.0159777\pi$
$594$	3.65685	0.150043
$595$	0	0
$596$	0	0
$597$	10.4853i	0.429134i
$598$	4.82843i	0.197449i
$599$	−33.1716	−1.35535	−0.677677	−	0.735360i	$-0.737013\pi$
$599$	−33.1716	−1.35535	−0.677677	+	0.735360i	$0.737013\pi$
$600$	0	0
$601$	−18.3137	−0.747032	−0.373516	−	0.927624i	$-0.621848\pi$
$601$	−18.3137	−0.747032	−0.373516	+	0.927624i	$0.621848\pi$
$602$	15.5147i	0.632333i
$603$	− 11.4853i	− 0.467717i
$604$	0	0
$605$	0	0
$606$	17.7574	0.721343
$607$	38.3848i	1.55799i	0.627030	+	0.778995i	$0.284270\pi$
$607$	38.3848i	1.55799i	−0.627030	+	0.778995i	$0.715730\pi$
$608$	0	0
$609$	−6.55635	−0.265677
$610$	0	0
$611$	−29.3137	−1.18591
$612$	0	0
$613$	− 32.6274i	− 1.31781i	−0.752227	−	0.658904i	$-0.771020\pi$
$613$	− 32.6274i	− 1.31781i	0.752227	−	0.658904i	$-0.228980\pi$
$614$	38.4853	1.55314
$615$	0	0
$616$	−13.3726	−0.538797
$617$	− 4.89949i	− 0.197246i	−0.995125	−	0.0986231i	$-0.968556\pi$
$617$	− 4.89949i	− 0.197246i	0.995125	−	0.0986231i	$-0.0314438\pi$
$618$	1.17157i	0.0471276i
$619$	48.2843	1.94071	0.970354	−	0.241687i	$-0.0777006\pi$
$619$	48.2843	1.94071	0.970354	+	0.241687i	$0.0777006\pi$
$620$	0	0
$621$	1.00000	0.0401286
$622$	24.9706i	1.00123i
$623$	18.5442i	0.742956i
$624$	−13.6569	−0.546712
$625$	0	0
$626$	−36.7279	−1.46794
$627$	16.1421i	0.644655i
$628$	0	0
$629$	18.2132	0.726208
$630$	0	0
$631$	33.2132	1.32220	0.661098	−	0.750299i	$-0.270091\pi$
$631$	33.2132	1.32220	0.661098	+	0.750299i	$0.270091\pi$
$632$	43.3137i	1.72293i
$633$	− 0.656854i	− 0.0261076i
$634$	−41.7990	−1.66005
$635$	0	0
$636$	0	0
$637$	12.4853i	0.494685i
$638$	− 13.1127i	− 0.519137i
$639$	−8.89949	−0.352059
$640$	0	0
$641$	4.92893	0.194681	0.0973406	−	0.995251i	$-0.468966\pi$
$641$	4.92893	0.194681	0.0973406	+	0.995251i	$0.468966\pi$
$642$	6.24264i	0.246377i
$643$	33.0000i	1.30139i	0.759338	+	0.650696i	$0.225523\pi$
$643$	33.0000i	1.30139i	−0.759338	+	0.650696i	$0.774477\pi$
$644$	0	0
$645$	0	0
$646$	−53.5980	−2.10878
$647$	− 21.2132i	− 0.833977i	−0.908912	−	0.416989i	$-0.863086\pi$
$647$	− 21.2132i	− 0.833977i	0.908912	−	0.416989i	$-0.136914\pi$
$648$	2.82843i	0.111111i
$649$	−17.8406	−0.700306
$650$	0	0
$651$	7.62742	0.298942
$652$	0	0
$653$	− 2.58579i	− 0.101190i	−0.998719	−	0.0505948i	$-0.983888\pi$
$653$	− 2.58579i	− 0.101190i	0.998719	−	0.0505948i	$-0.0161117\pi$
$654$	−20.1421	−0.787620
$655$	0	0
$656$	41.6569	1.62643
$657$	10.2426i	0.399603i
$658$	22.2010i	0.865485i
$659$	−2.38478	−0.0928977	−0.0464488	−	0.998921i	$-0.514790\pi$
$659$	−2.38478	−0.0928977	−0.0464488	+	0.998921i	$0.514790\pi$
$660$	0	0
$661$	11.4558	0.445581	0.222790	−	0.974866i	$-0.428483\pi$
$661$	11.4558	0.445581	0.222790	+	0.974866i	$0.428483\pi$
$662$	1.41421i	0.0549650i
$663$	20.7279i	0.805006i
$664$	−39.7990	−1.54450
$665$	0	0
$666$	−4.24264	−0.164399
$667$	− 3.58579i	− 0.138842i
$668$	0	0
$669$	−14.0000	−0.541271
$670$	0	0
$671$	−14.0000	−0.540464
$672$	0	0
$673$	41.6985i	1.60736i	0.595063	+	0.803679i	$0.297127\pi$
$673$	41.6985i	1.60736i	−0.595063	+	0.803679i	$0.702873\pi$
$674$	43.7990	1.68707
$675$	0	0
$676$	0	0
$677$	19.2426i	0.739555i	0.929120	+	0.369777i	$0.120566\pi$
$677$	19.2426i	0.739555i	−0.929120	+	0.369777i	$0.879434\pi$
$678$	− 14.7279i	− 0.565622i
$679$	2.14214	0.0822076
$680$	0	0
$681$	−17.6569	−0.676612
$682$	15.2548i	0.584138i
$683$	− 32.1838i	− 1.23148i	−0.787950	−	0.615739i	$-0.788858\pi$
$683$	− 32.1838i	− 1.23148i	0.787950	−	0.615739i	$-0.211142\pi$
$684$	0	0
$685$	0	0
$686$	27.5563	1.05211
$687$	23.7990i	0.907988i
$688$	− 24.0000i	− 0.914991i
$689$	−24.7279	−0.942059
$690$	0	0
$691$	−15.6569	−0.595615	−0.297807	−	0.954626i	$-0.596255\pi$
$691$	−15.6569	−0.595615	−0.297807	+	0.954626i	$0.596255\pi$
$692$	0	0
$693$	− 4.72792i	− 0.179599i
$694$	18.6274	0.707087
$695$	0	0
$696$	10.1421	0.384437
$697$	− 63.2254i	− 2.39483i
$698$	45.4142i	1.71895i
$699$	9.65685	0.365256
$700$	0	0
$701$	−36.3848	−1.37423	−0.687117	−	0.726547i	$-0.741124\pi$
$701$	−36.3848	−1.37423	−0.687117	+	0.726547i	$0.741124\pi$
$702$	− 4.82843i	− 0.182237i
$703$	− 18.7279i	− 0.706337i
$704$	20.6863	0.779644
$705$	0	0
$706$	48.8284	1.83768
$707$	− 22.9584i	− 0.863438i
$708$	0	0
$709$	−16.7279	−0.628230	−0.314115	−	0.949385i	$-0.601708\pi$
$709$	−16.7279	−0.628230	−0.314115	+	0.949385i	$0.601708\pi$
$710$	0	0
$711$	−15.3137	−0.574309
$712$	− 28.6863i	− 1.07506i
$713$	4.17157i	0.156227i
$714$	15.6985	0.587501
$715$	0	0
$716$	0	0
$717$	− 1.58579i	− 0.0592223i
$718$	15.4558i	0.576807i
$719$	−36.2132	−1.35052	−0.675262	−	0.737578i	$-0.735970\pi$
$719$	−36.2132	−1.35052	−0.675262	+	0.737578i	$0.735970\pi$
$720$	0	0
$721$	1.51472	0.0564111
$722$	28.2426i	1.05108i
$723$	− 23.4142i	− 0.870784i
$724$	0	0
$725$	0	0
$726$	−6.10051	−0.226411
$727$	23.0000i	0.853023i	0.904482	+	0.426511i	$0.140258\pi$
$727$	23.0000i	0.853023i	−0.904482	+	0.426511i	$0.859742\pi$
$728$	17.6569i	0.654407i
$729$	−1.00000	−0.0370370
$730$	0	0
$731$	−36.4264	−1.34728
$732$	0	0
$733$	− 17.4853i	− 0.645834i	−0.946427	−	0.322917i	$-0.895337\pi$
$733$	− 17.4853i	− 0.645834i	0.946427	−	0.322917i	$-0.104663\pi$
$734$	−19.7574	−0.729257
$735$	0	0
$736$	0	0
$737$	− 29.6985i	− 1.09396i
$738$	14.7279i	0.542142i
$739$	53.6274	1.97272	0.986358	−	0.164613i	$-0.0526376\pi$
$739$	53.6274	1.97272	0.986358	+	0.164613i	$0.0526376\pi$
$740$	0	0
$741$	21.3137	0.782979
$742$	18.7279i	0.687524i
$743$	− 25.6569i	− 0.941259i	−0.882331	−	0.470629i	$-0.844027\pi$
$743$	− 25.6569i	− 0.941259i	0.882331	−	0.470629i	$-0.155973\pi$
$744$	−11.7990	−0.432572
$745$	0	0
$746$	18.3431	0.671590
$747$	− 14.0711i	− 0.514833i
$748$	0	0
$749$	8.07107	0.294910
$750$	0	0
$751$	−9.27208	−0.338343	−0.169171	−	0.985587i	$-0.554109\pi$
$751$	−9.27208	−0.338343	−0.169171	+	0.985587i	$0.554109\pi$
$752$	− 34.3431i	− 1.25237i
$753$	20.6274i	0.751705i
$754$	−17.3137	−0.630528
$755$	0	0
$756$	0	0
$757$	− 27.0000i	− 0.981332i	−0.871348	−	0.490666i	$-0.836754\pi$
$757$	− 27.0000i	− 0.981332i	0.871348	−	0.490666i	$-0.163246\pi$
$758$	− 40.9706i	− 1.48812i
$759$	2.58579	0.0938581
$760$	0	0
$761$	−19.9289	−0.722423	−0.361212	−	0.932484i	$-0.617637\pi$
$761$	−19.9289	−0.722423	−0.361212	+	0.932484i	$0.617637\pi$
$762$	0.343146i	0.0124309i
$763$	26.0416i	0.942770i
$764$	0	0
$765$	0	0
$766$	−20.3848	−0.736532
$767$	23.5563i	0.850570i
$768$	0	0
$769$	−20.0416	−0.722720	−0.361360	−	0.932426i	$-0.617687\pi$
$769$	−20.0416	−0.722720	−0.361360	+	0.932426i	$0.617687\pi$
$770$	0	0
$771$	−30.0416	−1.08192
$772$	0	0
$773$	− 3.17157i	− 0.114074i	−0.998372	−	0.0570368i	$-0.981835\pi$
$773$	− 3.17157i	− 0.114074i	0.998372	−	0.0570368i	$-0.0181652\pi$
$774$	8.48528	0.304997
$775$	0	0
$776$	−3.31371	−0.118955
$777$	5.48528i	0.196783i
$778$	− 20.4853i	− 0.734433i
$779$	−65.0122	−2.32930
$780$	0	0
$781$	−23.0122	−0.823441
$782$	8.58579i	0.307027i
$783$	3.58579i	0.128146i
$784$	−14.6274	−0.522408
$785$	0	0
$786$	−19.3137	−0.688897
$787$	− 3.62742i	− 0.129303i	−0.997908	−	0.0646517i	$-0.979406\pi$
$787$	− 3.62742i	− 0.129303i	0.997908	−	0.0646517i	$-0.0205936\pi$
$788$	0	0
$789$	3.72792	0.132718
$790$	0	0
$791$	−19.0416	−0.677042
$792$	7.31371i	0.259881i
$793$	18.4853i	0.656432i
$794$	−21.6569	−0.768573
$795$	0	0
$796$	0	0
$797$	− 16.7574i	− 0.593576i	−0.954943	−	0.296788i	$-0.904085\pi$
$797$	− 16.7574i	− 0.593576i	0.954943	−	0.296788i	$-0.0959155\pi$
$798$	− 16.1421i	− 0.571425i
$799$	−52.1249	−1.84405
$800$	0	0
$801$	10.1421	0.358355
$802$	− 45.4558i	− 1.60510i
$803$	26.4853i	0.934645i
$804$	0	0
$805$	0	0
$806$	20.1421	0.709476
$807$	− 26.5563i	− 0.934828i
$808$	35.5147i	1.24940i
$809$	5.87006	0.206380	0.103190	−	0.994662i	$-0.467095\pi$
$809$	5.87006	0.206380	0.103190	+	0.994662i	$0.467095\pi$
$810$	0	0
$811$	39.2843	1.37946	0.689729	−	0.724068i	$-0.257730\pi$
$811$	39.2843	1.37946	0.689729	+	0.724068i	$0.257730\pi$
$812$	0	0
$813$	6.51472i	0.228481i
$814$	−10.9706	−0.384518
$815$	0	0
$816$	−24.2843	−0.850120
$817$	37.4558i	1.31041i
$818$	− 2.10051i	− 0.0734424i
$819$	−6.24264	−0.218136
$820$	0	0
$821$	−43.7990	−1.52860	−0.764298	−	0.644864i	$-0.776914\pi$
$821$	−43.7990	−1.52860	−0.764298	+	0.644864i	$0.776914\pi$
$822$	− 4.00000i	− 0.139516i
$823$	45.6569i	1.59150i	0.605627	+	0.795749i	$0.292923\pi$
$823$	45.6569i	1.59150i	−0.605627	+	0.795749i	$0.707077\pi$
$824$	−2.34315	−0.0816274
$825$	0	0
$826$	17.8406	0.620755
$827$	50.0122i	1.73909i	0.493850	+	0.869547i	$0.335589\pi$
$827$	50.0122i	1.73909i	−0.493850	+	0.869547i	$0.664411\pi$
$828$	0	0
$829$	−23.4853	−0.815678	−0.407839	−	0.913054i	$-0.633717\pi$
$829$	−23.4853	−0.815678	−0.407839	+	0.913054i	$0.633717\pi$
$830$	0	0
$831$	−25.3137	−0.878123
$832$	− 27.3137i	− 0.946932i
$833$	22.2010i	0.769219i
$834$	15.5563	0.538672
$835$	0	0
$836$	0	0
$837$	− 4.17157i	− 0.144191i
$838$	34.0000i	1.17451i
$839$	4.62742	0.159756	0.0798781	−	0.996805i	$-0.474547\pi$
$839$	4.62742	0.159756	0.0798781	+	0.996805i	$0.474547\pi$
$840$	0	0
$841$	−16.1421	−0.556625
$842$	− 2.48528i	− 0.0856485i
$843$	− 17.4142i	− 0.599777i
$844$	0	0
$845$	0	0
$846$	12.1421	0.417455
$847$	7.88730i	0.271011i
$848$	− 28.9706i	− 0.994853i
$849$	−28.6569	−0.983501
$850$	0	0
$851$	−3.00000	−0.102839
$852$	0	0
$853$	− 31.4558i	− 1.07703i	−0.842617	−	0.538514i	$-0.818986\pi$
$853$	− 31.4558i	− 1.07703i	0.842617	−	0.538514i	$-0.181014\pi$
$854$	14.0000	0.479070
$855$	0	0
$856$	−12.4853	−0.426738
$857$	− 21.5980i	− 0.737773i	−0.929474	−	0.368886i	$-0.879739\pi$
$857$	− 21.5980i	− 0.737773i	0.929474	−	0.368886i	$-0.120261\pi$
$858$	− 12.4853i	− 0.426240i
$859$	57.0000	1.94481	0.972407	−	0.233289i	$-0.0749488\pi$
$859$	57.0000	1.94481	0.972407	+	0.233289i	$0.0749488\pi$
$860$	0	0
$861$	19.0416	0.648937
$862$	− 39.3137i	− 1.33903i
$863$	11.1716i	0.380285i	0.981757	+	0.190142i	$0.0608950\pi$
$863$	11.1716i	0.380285i	−0.981757	+	0.190142i	$0.939105\pi$
$864$	0	0
$865$	0	0
$866$	2.58579	0.0878686
$867$	19.8579i	0.674408i
$868$	0	0
$869$	−39.5980	−1.34327
$870$	0	0
$871$	−39.2132	−1.32869
$872$	− 40.2843i	− 1.36420i
$873$	− 1.17157i	− 0.0396517i
$874$	8.82843	0.298626
$875$	0	0
$876$	0	0
$877$	1.51472i	0.0511484i	0.999673	+	0.0255742i	$0.00814141\pi$
$877$	1.51472i	0.0511484i	−0.999673	+	0.0255742i	$0.991859\pi$
$878$	26.1421i	0.882254i
$879$	−3.10051	−0.104577
$880$	0	0
$881$	19.4142	0.654081	0.327041	−	0.945010i	$-0.393949\pi$
$881$	19.4142	0.654081	0.327041	+	0.945010i	$0.393949\pi$
$882$	− 5.17157i	− 0.174136i
$883$	− 1.89949i	− 0.0639231i	−0.999489	−	0.0319615i	$-0.989825\pi$
$883$	− 1.89949i	− 0.0639231i	0.999489	−	0.0319615i	$-0.0101754\pi$
$884$	0	0
$885$	0	0
$886$	−43.1716	−1.45038
$887$	− 21.7990i	− 0.731938i	−0.930627	−	0.365969i	$-0.880738\pi$
$887$	− 21.7990i	− 0.731938i	0.930627	−	0.365969i	$-0.119262\pi$
$888$	− 8.48528i	− 0.284747i
$889$	0.443651	0.0148796
$890$	0	0
$891$	−2.58579	−0.0866271
$892$	0	0
$893$	53.5980i	1.79359i
$894$	−12.8284	−0.429047
$895$	0	0
$896$	−20.6863	−0.691080
$897$	− 3.41421i	− 0.113997i
$898$	18.7279i	0.624959i
$899$	−14.9584	−0.498890
$900$	0	0
$901$	−43.9706	−1.46487
$902$	38.0833i	1.26803i
$903$	− 10.9706i	− 0.365077i
$904$	29.4558	0.979687
$905$	0	0
$906$	−13.1716	−0.437596
$907$	− 31.1421i	− 1.03406i	−0.855968	−	0.517029i	$-0.827038\pi$
$907$	− 31.1421i	− 1.03406i	0.855968	−	0.517029i	$-0.172962\pi$
$908$	0	0
$909$	−12.5563	−0.416468
$910$	0	0
$911$	−5.65685	−0.187420	−0.0937100	−	0.995600i	$-0.529873\pi$
$911$	−5.65685	−0.187420	−0.0937100	+	0.995600i	$0.529873\pi$
$912$	24.9706i	0.826858i
$913$	− 36.3848i	− 1.20416i
$914$	−43.8406	−1.45012
$915$	0	0
$916$	0	0
$917$	24.9706i	0.824601i
$918$	− 8.58579i	− 0.283373i
$919$	16.0000	0.527791	0.263896	−	0.964551i	$-0.414993\pi$
$919$	16.0000	0.527791	0.263896	+	0.964551i	$0.414993\pi$
$920$	0	0
$921$	−27.2132	−0.896706
$922$	− 13.4558i	− 0.443145i
$923$	30.3848i	1.00013i
$924$	0	0
$925$	0	0
$926$	−24.1421	−0.793360
$927$	− 0.828427i	− 0.0272091i
$928$	0	0
$929$	−41.8701	−1.37371	−0.686856	−	0.726794i	$-0.741010\pi$
$929$	−41.8701	−1.37371	−0.686856	+	0.726794i	$0.741010\pi$
$930$	0	0
$931$	22.8284	0.748171
$932$	0	0
$933$	− 17.6569i	− 0.578059i
$934$	5.07107	0.165930
$935$	0	0
$936$	9.65685	0.315644
$937$	59.7990i	1.95355i	0.214272	+	0.976774i	$0.431262\pi$
$937$	59.7990i	1.95355i	−0.214272	+	0.976774i	$0.568738\pi$
$938$	29.6985i	0.969690i
$939$	25.9706	0.847517
$940$	0	0
$941$	−7.41421	−0.241696	−0.120848	−	0.992671i	$-0.538561\pi$
$941$	−7.41421	−0.241696	−0.120848	+	0.992671i	$0.538561\pi$
$942$	2.10051i	0.0684382i
$943$	10.4142i	0.339133i
$944$	−27.5980	−0.898238
$945$	0	0
$946$	21.9411	0.713368
$947$	3.79899i	0.123451i	0.998093	+	0.0617253i	$0.0196603\pi$
$947$	3.79899i	0.123451i	−0.998093	+	0.0617253i	$0.980340\pi$
$948$	0	0
$949$	34.9706	1.13519
$950$	0	0
$951$	29.5563	0.958430
$952$	31.3970i	1.01758i
$953$	7.37258i	0.238821i	0.992845	+	0.119411i	$0.0381005\pi$
$953$	7.37258i	0.238821i	−0.992845	+	0.119411i	$0.961899\pi$
$954$	10.2426	0.331618
$955$	0	0
$956$	0	0
$957$	9.27208i	0.299724i
$958$	26.4853i	0.855701i
$959$	−5.17157	−0.166999
$960$	0	0
$961$	−13.5980	−0.438645
$962$	14.4853i	0.467024i
$963$	− 4.41421i	− 0.142246i
$964$	0	0
$965$	0	0
$966$	−2.58579	−0.0831963
$967$	− 0.443651i	− 0.0142668i	−0.999975	−	0.00713342i	$-0.997729\pi$
$967$	− 0.443651i	− 0.0142668i	0.999975	−	0.00713342i	$-0.00227066\pi$
$968$	− 12.2010i	− 0.392155i
$969$	37.8995	1.21751
$970$	0	0
$971$	30.4264	0.976430	0.488215	−	0.872723i	$-0.337648\pi$
$971$	30.4264	0.976430	0.488215	+	0.872723i	$0.337648\pi$
$972$	0	0
$973$	− 20.1127i	− 0.644784i
$974$	14.6863	0.470579
$975$	0	0
$976$	−21.6569	−0.693219
$977$	15.5858i	0.498633i	0.968422	+	0.249317i	$0.0802060\pi$
$977$	15.5858i	0.498633i	−0.968422	+	0.249317i	$0.919794\pi$
$978$	− 2.82843i	− 0.0904431i
$979$	26.2254	0.838167
$980$	0	0
$981$	14.2426	0.454733
$982$	2.92893i	0.0934660i
$983$	24.0122i	0.765870i	0.923775	+	0.382935i	$0.125087\pi$
$983$	24.0122i	0.765870i	−0.923775	+	0.382935i	$0.874913\pi$
$984$	−29.4558	−0.939018
$985$	0	0
$986$	−30.7868	−0.980451
$987$	− 15.6985i	− 0.499688i
$988$	0	0
$989$	6.00000	0.190789
$990$	0	0
$991$	1.34315	0.0426664	0.0213332	−	0.999772i	$-0.493209\pi$
$991$	1.34315	0.0426664	0.0213332	+	0.999772i	$0.493209\pi$
$992$	0	0
$993$	− 1.00000i	− 0.0317340i
$994$	23.0122	0.729902
$995$	0	0
$996$	0	0
$997$	34.1421i	1.08129i	0.841250	+	0.540646i	$0.181820\pi$
$997$	34.1421i	1.08129i	−0.841250	+	0.540646i	$0.818180\pi$
$998$	18.3848i	0.581960i
$999$	3.00000	0.0949158

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1725.2.b.s.1174.3		4
5.2	odd	4		345.2.a.h.1.1	✓	2
5.3	odd	4		1725.2.a.z.1.2		2
5.4	even	2	inner	1725.2.b.s.1174.2		4
15.2	even	4		1035.2.a.j.1.2		2
15.8	even	4		5175.2.a.bj.1.1		2
20.7	even	4		5520.2.a.bm.1.1		2
115.22	even	4		7935.2.a.q.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
345.2.a.h.1.1	✓	2	5.2	odd	4
1035.2.a.j.1.2		2	15.2	even	4
1725.2.a.z.1.2		2	5.3	odd	4
1725.2.b.s.1174.2		4	5.4	even	2	inner
1725.2.b.s.1174.3		4	1.1	even	1	trivial
5175.2.a.bj.1.1		2	15.8	even	4
5520.2.a.bm.1.1		2	20.7	even	4
7935.2.a.q.1.1		2	115.22	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)

\(f(q)\)	\(=\)	\(q+1.41421i q^{2} -1.00000i q^{3} +1.41421 q^{6} -1.82843i q^{7} +2.82843i q^{8} -1.00000 q^{9} +O(q^{10})\)	\(q+1.41421i q^{2} -1.00000i q^{3} +1.41421 q^{6} -1.82843i q^{7} +2.82843i q^{8} -1.00000 q^{9} -2.58579 q^{11} +3.41421i q^{13} +2.58579 q^{14} -4.00000 q^{16} +6.07107i q^{17} -1.41421i q^{18} +6.24264 q^{19} -1.82843 q^{21} -3.65685i q^{22} -1.00000i q^{23} +2.82843 q^{24} -4.82843 q^{26} +1.00000i q^{27} +3.58579 q^{29} -4.17157 q^{31} +2.58579i q^{33} -8.58579 q^{34} -3.00000i q^{37} +8.82843i q^{38} +3.41421 q^{39} -10.4142 q^{41} -2.58579i q^{42} +6.00000i q^{43} +1.41421 q^{46} +8.58579i q^{47} +4.00000i q^{48} +3.65685 q^{49} +6.07107 q^{51} +7.24264i q^{53} -1.41421 q^{54} +5.17157 q^{56} -6.24264i q^{57} +5.07107i q^{58} +6.89949 q^{59} +5.41421 q^{61} -5.89949i q^{62} +1.82843i q^{63} -8.00000 q^{64} -3.65685 q^{66} +11.4853i q^{67} -1.00000 q^{69} +8.89949 q^{71} -2.82843i q^{72} -10.2426i q^{73} +4.24264 q^{74} +4.72792i q^{77} +4.82843i q^{78} +15.3137 q^{79} +1.00000 q^{81} -14.7279i q^{82} +14.0711i q^{83} -8.48528 q^{86} -3.58579i q^{87} -7.31371i q^{88} -10.1421 q^{89} +6.24264 q^{91} +4.17157i q^{93} -12.1421 q^{94} +1.17157i q^{97} +5.17157i q^{98} +2.58579 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{9}+O(q^{10}) \) 4 * q - 4 * q^9	\( 4 q - 4 q^{9} - 16 q^{11} + 16 q^{14} - 16 q^{16} + 8 q^{19} + 4 q^{21} - 8 q^{26} + 20 q^{29} - 28 q^{31} - 40 q^{34} + 8 q^{39} - 36 q^{41} - 8 q^{49} - 4 q^{51} + 32 q^{56} - 12 q^{59} + 16 q^{61} - 32 q^{64} + 8 q^{66} - 4 q^{69} - 4 q^{71} + 16 q^{79} + 4 q^{81} + 16 q^{89} + 8 q^{91} + 8 q^{94} + 16 q^{99}+O(q^{100}) \) 4 * q - 4 * q^9 - 16 * q^11 + 16 * q^14 - 16 * q^16 + 8 * q^19 + 4 * q^21 - 8 * q^26 + 20 * q^29 - 28 * q^31 - 40 * q^34 + 8 * q^39 - 36 * q^41 - 8 * q^49 - 4 * q^51 + 32 * q^56 - 12 * q^59 + 16 * q^61 - 32 * q^64 + 8 * q^66 - 4 * q^69 - 4 * q^71 + 16 * q^79 + 4 * q^81 + 16 * q^89 + 8 * q^91 + 8 * q^94 + 16 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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