LMFDB - Embedded newform 1675.2.c.a.1274.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1675,2,Mod(1274,1675)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1675, 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("1675.1274");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1675 = 5^{2} \cdot 67 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1675.c (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:	$13.3749423386$
Analytic rank:	$1$
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:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 67)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1274.1
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	1675.1274
Dual form			1675.2.c.a.1274.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.00000i q^{2} -2.00000i q^{3} -2.00000 q^{4} -4.00000 q^{6} +2.00000i q^{7} -1.00000 q^{9} +O(q^{10})$	\(q-2.00000i q^{2} -2.00000i q^{3} -2.00000 q^{4} -4.00000 q^{6} +2.00000i q^{7} -1.00000 q^{9} -4.00000 q^{11} +4.00000i q^{12} +2.00000i q^{13} +4.00000 q^{14} -4.00000 q^{16} -3.00000i q^{17} +2.00000i q^{18} -7.00000 q^{19} +4.00000 q^{21} +8.00000i q^{22} +9.00000i q^{23} +4.00000 q^{26} -4.00000i q^{27} -4.00000i q^{28} +5.00000 q^{29} -10.0000 q^{31} +8.00000i q^{32} +8.00000i q^{33} -6.00000 q^{34} +2.00000 q^{36} +1.00000i q^{37} +14.0000i q^{38} +4.00000 q^{39} -8.00000i q^{42} -2.00000i q^{43} +8.00000 q^{44} +18.0000 q^{46} +1.00000i q^{47} +8.00000i q^{48} +3.00000 q^{49} -6.00000 q^{51} -4.00000i q^{52} +10.0000i q^{53} -8.00000 q^{54} +14.0000i q^{57} -10.0000i q^{58} -9.00000 q^{59} -2.00000 q^{61} +20.0000i q^{62} -2.00000i q^{63} +8.00000 q^{64} +16.0000 q^{66} -1.00000i q^{67} +6.00000i q^{68} +18.0000 q^{69} -7.00000i q^{73} +2.00000 q^{74} +14.0000 q^{76} -8.00000i q^{77} -8.00000i q^{78} +8.00000 q^{79} -11.0000 q^{81} +4.00000i q^{83} -8.00000 q^{84} -4.00000 q^{86} -10.0000i q^{87} -7.00000 q^{89} -4.00000 q^{91} -18.0000i q^{92} +20.0000i q^{93} +2.00000 q^{94} +16.0000 q^{96} -6.00000i q^{98} +4.00000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 4 q^{4} - 8 q^{6} - 2 q^{9}+O(q^{10}) $ 2 * q - 4 * q^4 - 8 * q^6 - 2 * q^9	$ 2 q - 4 q^{4} - 8 q^{6} - 2 q^{9} - 8 q^{11} + 8 q^{14} - 8 q^{16} - 14 q^{19} + 8 q^{21} + 8 q^{26} + 10 q^{29} - 20 q^{31} - 12 q^{34} + 4 q^{36} + 8 q^{39} + 16 q^{44} + 36 q^{46} + 6 q^{49} - 12 q^{51} - 16 q^{54} - 18 q^{59} - 4 q^{61} + 16 q^{64} + 32 q^{66} + 36 q^{69} + 4 q^{74} + 28 q^{76} + 16 q^{79} - 22 q^{81} - 16 q^{84} - 8 q^{86} - 14 q^{89} - 8 q^{91} + 4 q^{94} + 32 q^{96} + 8 q^{99}+O(q^{100}) $ 2 * q - 4 * q^4 - 8 * q^6 - 2 * q^9 - 8 * q^11 + 8 * q^14 - 8 * q^16 - 14 * q^19 + 8 * q^21 + 8 * q^26 + 10 * q^29 - 20 * q^31 - 12 * q^34 + 4 * q^36 + 8 * q^39 + 16 * q^44 + 36 * q^46 + 6 * q^49 - 12 * q^51 - 16 * q^54 - 18 * q^59 - 4 * q^61 + 16 * q^64 + 32 * q^66 + 36 * q^69 + 4 * q^74 + 28 * q^76 + 16 * q^79 - 22 * q^81 - 16 * q^84 - 8 * q^86 - 14 * q^89 - 8 * q^91 + 4 * q^94 + 32 * q^96 + 8 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1675\mathbb{Z}\right)^\times$.

$n$	$202$	$1476$
$\chi(n)$	$-1$	$1$

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	$-0.250000\pi$
$3$	− 2.00000i	− 1.15470i	−0.816497	−	0.577350i	$-0.804087\pi$
$3$	− 2.00000i	− 1.15470i	0.816497	−	0.577350i	$-0.195913\pi$
$4$	−2.00000	−1.00000
$5$	0	0
$5$	0	0
$6$	−4.00000	−1.63299
$7$	2.00000i	0.755929i	0.925820	+	0.377964i	$0.123376\pi$
$7$	2.00000i	0.755929i	−0.925820	+	0.377964i	$0.876624\pi$
$8$	0	0
$9$	−1.00000	−0.333333
$10$	0	0
$11$	−4.00000	−1.20605	−0.603023	−	0.797724i	$-0.706037\pi$
$11$	−4.00000	−1.20605	−0.603023	+	0.797724i	$0.706037\pi$
$12$	4.00000i	1.15470i
$13$	2.00000i	0.554700i	0.960769	+	0.277350i	$0.0894562\pi$
$13$	2.00000i	0.554700i	−0.960769	+	0.277350i	$0.910544\pi$
$14$	4.00000	1.06904
$15$	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	$-0.118522\pi$
$18$	2.00000i	0.471405i
$19$	−7.00000	−1.60591	−0.802955	−	0.596040i	$-0.796740\pi$
$19$	−7.00000	−1.60591	−0.802955	+	0.596040i	$0.796740\pi$
$20$	0	0
$21$	4.00000	0.872872
$22$	8.00000i	1.70561i
$23$	9.00000i	1.87663i	0.345782	+	0.938315i	$0.387614\pi$
$23$	9.00000i	1.87663i	−0.345782	+	0.938315i	$0.612386\pi$
$24$	0	0
$25$	0	0
$26$	4.00000	0.784465
$27$	− 4.00000i	− 0.769800i
$28$	− 4.00000i	− 0.755929i
$29$	5.00000	0.928477	0.464238	−	0.885710i	$-0.346328\pi$
$29$	5.00000	0.928477	0.464238	+	0.885710i	$0.346328\pi$
$30$	0	0
$31$	−10.0000	−1.79605	−0.898027	−	0.439941i	$-0.854999\pi$
$31$	−10.0000	−1.79605	−0.898027	+	0.439941i	$0.854999\pi$
$32$	8.00000i	1.41421i
$33$	8.00000i	1.39262i
$34$	−6.00000	−1.02899
$35$	0	0
$36$	2.00000	0.333333
$37$	1.00000i	0.164399i	0.996616	+	0.0821995i	$0.0261945\pi$
$37$	1.00000i	0.164399i	−0.996616	+	0.0821995i	$0.973806\pi$
$38$	14.0000i	2.27110i
$39$	4.00000	0.640513
$40$	0	0
$41$	0	0		−	1.00000i	$-0.5\pi$
$41$	0	0			1.00000i	$0.5\pi$
$42$	− 8.00000i	− 1.23443i
$43$	− 2.00000i	− 0.304997i	−0.988304	−	0.152499i	$-0.951268\pi$
$43$	− 2.00000i	− 0.304997i	0.988304	−	0.152499i	$-0.0487319\pi$
$44$	8.00000	1.20605
$45$	0	0
$46$	18.0000	2.65396
$47$	1.00000i	0.145865i	0.997337	+	0.0729325i	$0.0232358\pi$
$47$	1.00000i	0.145865i	−0.997337	+	0.0729325i	$0.976764\pi$
$48$	8.00000i	1.15470i
$49$	3.00000	0.428571
$50$	0	0
$51$	−6.00000	−0.840168
$52$	− 4.00000i	− 0.554700i
$53$	10.0000i	1.37361i	0.726844	+	0.686803i	$0.240986\pi$
$53$	10.0000i	1.37361i	−0.726844	+	0.686803i	$0.759014\pi$
$54$	−8.00000	−1.08866
$55$	0	0
$56$	0	0
$57$	14.0000i	1.85435i
$58$	− 10.0000i	− 1.31306i
$59$	−9.00000	−1.17170	−0.585850	−	0.810419i	$-0.699239\pi$
$59$	−9.00000	−1.17170	−0.585850	+	0.810419i	$0.699239\pi$
$60$	0	0
$61$	−2.00000	−0.256074	−0.128037	−	0.991769i	$-0.540868\pi$
$61$	−2.00000	−0.256074	−0.128037	+	0.991769i	$0.540868\pi$
$62$	20.0000i	2.54000i
$63$	− 2.00000i	− 0.251976i
$64$	8.00000	1.00000
$65$	0	0
$66$	16.0000	1.96946
$67$	− 1.00000i	− 0.122169i
$67$	− 1.00000i	− 0.122169i
$68$	6.00000i	0.727607i
$69$	18.0000	2.16695
$70$	0	0
$71$	0	0		−	1.00000i	$-0.5\pi$
$71$	0	0			1.00000i	$0.5\pi$
$72$	0	0
$73$	− 7.00000i	− 0.819288i	−0.912245	−	0.409644i	$-0.865653\pi$
$73$	− 7.00000i	− 0.819288i	0.912245	−	0.409644i	$-0.134347\pi$
$74$	2.00000	0.232495
$75$	0	0
$76$	14.0000	1.60591
$77$	− 8.00000i	− 0.911685i
$78$	− 8.00000i	− 0.905822i
$79$	8.00000	0.900070	0.450035	−	0.893011i	$-0.351411\pi$
$79$	8.00000	0.900070	0.450035	+	0.893011i	$0.351411\pi$
$80$	0	0
$81$	−11.0000	−1.22222
$82$	0	0
$83$	4.00000i	0.439057i	0.975606	+	0.219529i	$0.0704519\pi$
$83$	4.00000i	0.439057i	−0.975606	+	0.219529i	$0.929548\pi$
$84$	−8.00000	−0.872872
$85$	0	0
$86$	−4.00000	−0.431331
$87$	− 10.0000i	− 1.07211i
$88$	0	0
$89$	−7.00000	−0.741999	−0.370999	−	0.928633i	$-0.620985\pi$
$89$	−7.00000	−0.741999	−0.370999	+	0.928633i	$0.620985\pi$
$90$	0	0
$91$	−4.00000	−0.419314
$92$	− 18.0000i	− 1.87663i
$93$	20.0000i	2.07390i
$94$	2.00000	0.206284
$95$	0	0
$96$	16.0000	1.63299
$97$	0	0	1.00000			$0$
$97$	0	0	−1.00000			$\pi$
$98$	− 6.00000i	− 0.606092i
$99$	4.00000	0.402015
$100$	0	0
$101$	2.00000	0.199007	0.0995037	−	0.995037i	$-0.468274\pi$
$101$	2.00000	0.199007	0.0995037	+	0.995037i	$0.468274\pi$
$102$	12.0000i	1.18818i
$103$	− 16.0000i	− 1.57653i	−0.615338	−	0.788263i	$-0.710980\pi$
$103$	− 16.0000i	− 1.57653i	0.615338	−	0.788263i	$-0.289020\pi$
$104$	0	0
$105$	0	0
$106$	20.0000	1.94257
$107$	7.00000i	0.676716i	0.941018	+	0.338358i	$0.109871\pi$
$107$	7.00000i	0.676716i	−0.941018	+	0.338358i	$0.890129\pi$
$108$	8.00000i	0.769800i
$109$	−2.00000	−0.191565	−0.0957826	−	0.995402i	$-0.530535\pi$
$109$	−2.00000	−0.191565	−0.0957826	+	0.995402i	$0.530535\pi$
$110$	0	0
$111$	2.00000	0.189832
$112$	− 8.00000i	− 0.755929i
$113$	− 12.0000i	− 1.12887i	−0.825479	−	0.564433i	$-0.809095\pi$
$113$	− 12.0000i	− 1.12887i	0.825479	−	0.564433i	$-0.190905\pi$
$114$	28.0000	2.62244
$115$	0	0
$116$	−10.0000	−0.928477
$117$	− 2.00000i	− 0.184900i
$118$	18.0000i	1.65703i
$119$	6.00000	0.550019
$120$	0	0
$121$	5.00000	0.454545
$122$	4.00000i	0.362143i
$123$	0	0
$124$	20.0000	1.79605
$125$	0	0
$126$	−4.00000	−0.356348
$127$	− 7.00000i	− 0.621150i	−0.950549	−	0.310575i	$-0.899478\pi$
$127$	− 7.00000i	− 0.621150i	0.950549	−	0.310575i	$-0.100522\pi$
$128$	0	0
$129$	−4.00000	−0.352180
$130$	0	0
$131$	−12.0000	−1.04844	−0.524222	−	0.851581i	$-0.675644\pi$
$131$	−12.0000	−1.04844	−0.524222	+	0.851581i	$0.675644\pi$
$132$	− 16.0000i	− 1.39262i
$133$	− 14.0000i	− 1.21395i
$134$	−2.00000	−0.172774
$135$	0	0
$136$	0	0
$137$	− 12.0000i	− 1.02523i	−0.858619	−	0.512615i	$-0.828677\pi$
$137$	− 12.0000i	− 1.02523i	0.858619	−	0.512615i	$-0.171323\pi$
$138$	− 36.0000i	− 3.06452i
$139$	−22.0000	−1.86602	−0.933008	−	0.359856i	$-0.882826\pi$
$139$	−22.0000	−1.86602	−0.933008	+	0.359856i	$0.882826\pi$
$140$	0	0
$141$	2.00000	0.168430
$142$	0	0
$143$	− 8.00000i	− 0.668994i
$144$	4.00000	0.333333
$145$	0	0
$146$	−14.0000	−1.15865
$147$	− 6.00000i	− 0.494872i
$148$	− 2.00000i	− 0.164399i
$149$	−21.0000	−1.72039	−0.860194	−	0.509968i	$-0.829657\pi$
$149$	−21.0000	−1.72039	−0.860194	+	0.509968i	$0.829657\pi$
$150$	0	0
$151$	3.00000	0.244137	0.122068	−	0.992522i	$-0.461047\pi$
$151$	3.00000	0.244137	0.122068	+	0.992522i	$0.461047\pi$
$152$	0	0
$153$	3.00000i	0.242536i
$154$	−16.0000	−1.28932
$155$	0	0
$156$	−8.00000	−0.640513
$157$	− 9.00000i	− 0.718278i	−0.933284	−	0.359139i	$-0.883070\pi$
$157$	− 9.00000i	− 0.718278i	0.933284	−	0.359139i	$-0.116930\pi$
$158$	− 16.0000i	− 1.27289i
$159$	20.0000	1.58610
$160$	0	0
$161$	−18.0000	−1.41860
$162$	22.0000i	1.72848i
$163$	19.0000i	1.48819i	0.668071	+	0.744097i	$0.267120\pi$
$163$	19.0000i	1.48819i	−0.668071	+	0.744097i	$0.732880\pi$
$164$	0	0
$165$	0	0
$166$	8.00000	0.620920
$167$	− 24.0000i	− 1.85718i	−0.371113	−	0.928588i	$-0.621024\pi$
$167$	− 24.0000i	− 1.85718i	0.371113	−	0.928588i	$-0.378976\pi$
$168$	0	0
$169$	9.00000	0.692308
$170$	0	0
$171$	7.00000	0.535303
$172$	4.00000i	0.304997i
$173$	11.0000i	0.836315i	0.908375	+	0.418157i	$0.137324\pi$
$173$	11.0000i	0.836315i	−0.908375	+	0.418157i	$0.862676\pi$
$174$	−20.0000	−1.51620
$175$	0	0
$176$	16.0000	1.20605
$177$	18.0000i	1.35296i
$178$	14.0000i	1.04934i
$179$	12.0000	0.896922	0.448461	−	0.893802i	$-0.351972\pi$
$179$	12.0000	0.896922	0.448461	+	0.893802i	$0.351972\pi$
$180$	0	0
$181$	7.00000	0.520306	0.260153	−	0.965567i	$-0.416227\pi$
$181$	7.00000	0.520306	0.260153	+	0.965567i	$0.416227\pi$
$182$	8.00000i	0.592999i
$183$	4.00000i	0.295689i
$184$	0	0
$185$	0	0
$186$	40.0000	2.93294
$187$	12.0000i	0.877527i
$188$	− 2.00000i	− 0.145865i
$189$	8.00000	0.581914
$190$	0	0
$191$	−6.00000	−0.434145	−0.217072	−	0.976156i	$-0.569651\pi$
$191$	−6.00000	−0.434145	−0.217072	+	0.976156i	$0.569651\pi$
$192$	− 16.0000i	− 1.15470i
$193$	− 23.0000i	− 1.65558i	−0.561041	−	0.827788i	$-0.689599\pi$
$193$	− 23.0000i	− 1.65558i	0.561041	−	0.827788i	$-0.310401\pi$
$194$	0	0
$195$	0	0
$196$	−6.00000	−0.428571
$197$	2.00000i	0.142494i	0.997459	+	0.0712470i	$0.0226979\pi$
$197$	2.00000i	0.142494i	−0.997459	+	0.0712470i	$0.977302\pi$
$198$	− 8.00000i	− 0.568535i
$199$	−7.00000	−0.496217	−0.248108	−	0.968732i	$-0.579809\pi$
$199$	−7.00000	−0.496217	−0.248108	+	0.968732i	$0.579809\pi$
$200$	0	0
$201$	−2.00000	−0.141069
$202$	− 4.00000i	− 0.281439i
$203$	10.0000i	0.701862i
$204$	12.0000	0.840168
$205$	0	0
$206$	−32.0000	−2.22955
$207$	− 9.00000i	− 0.625543i
$208$	− 8.00000i	− 0.554700i
$209$	28.0000	1.93680
$210$	0	0
$211$	−12.0000	−0.826114	−0.413057	−	0.910705i	$-0.635539\pi$
$211$	−12.0000	−0.826114	−0.413057	+	0.910705i	$0.635539\pi$
$212$	− 20.0000i	− 1.37361i
$213$	0	0
$214$	14.0000	0.957020
$215$	0	0
$216$	0	0
$217$	− 20.0000i	− 1.35769i
$218$	4.00000i	0.270914i
$219$	−14.0000	−0.946032
$220$	0	0
$221$	6.00000	0.403604
$222$	− 4.00000i	− 0.268462i
$223$	11.0000i	0.736614i	0.929704	+	0.368307i	$0.120063\pi$
$223$	11.0000i	0.736614i	−0.929704	+	0.368307i	$0.879937\pi$
$224$	−16.0000	−1.06904
$225$	0	0
$226$	−24.0000	−1.59646
$227$	− 3.00000i	− 0.199117i	−0.995032	−	0.0995585i	$-0.968257\pi$
$227$	− 3.00000i	− 0.199117i	0.995032	−	0.0995585i	$-0.0317430\pi$
$228$	− 28.0000i	− 1.85435i
$229$	−4.00000	−0.264327	−0.132164	−	0.991228i	$-0.542192\pi$
$229$	−4.00000	−0.264327	−0.132164	+	0.991228i	$0.542192\pi$
$230$	0	0
$231$	−16.0000	−1.05272
$232$	0	0
$233$	10.0000i	0.655122i	0.944830	+	0.327561i	$0.106227\pi$
$233$	10.0000i	0.655122i	−0.944830	+	0.327561i	$0.893773\pi$
$234$	−4.00000	−0.261488
$235$	0	0
$236$	18.0000	1.17170
$237$	− 16.0000i	− 1.03931i
$238$	− 12.0000i	− 0.777844i
$239$	20.0000	1.29369	0.646846	−	0.762620i	$-0.276088\pi$
$239$	20.0000	1.29369	0.646846	+	0.762620i	$0.276088\pi$
$240$	0	0
$241$	−19.0000	−1.22390	−0.611949	−	0.790897i	$-0.709614\pi$
$241$	−19.0000	−1.22390	−0.611949	+	0.790897i	$0.709614\pi$
$242$	− 10.0000i	− 0.642824i
$243$	10.0000i	0.641500i
$244$	4.00000	0.256074
$245$	0	0
$246$	0	0
$247$	− 14.0000i	− 0.890799i
$248$	0	0
$249$	8.00000	0.506979
$250$	0	0
$251$	−2.00000	−0.126239	−0.0631194	−	0.998006i	$-0.520105\pi$
$251$	−2.00000	−0.126239	−0.0631194	+	0.998006i	$0.520105\pi$
$252$	4.00000i	0.251976i
$253$	− 36.0000i	− 2.26330i
$254$	−14.0000	−0.878438
$255$	0	0
$256$	16.0000	1.00000
$257$	1.00000i	0.0623783i	0.999514	+	0.0311891i	$0.00992942\pi$
$257$	1.00000i	0.0623783i	−0.999514	+	0.0311891i	$0.990071\pi$
$258$	8.00000i	0.498058i
$259$	−2.00000	−0.124274
$260$	0	0
$261$	−5.00000	−0.309492
$262$	24.0000i	1.48272i
$263$	16.0000i	0.986602i	0.869859	+	0.493301i	$0.164210\pi$
$263$	16.0000i	0.986602i	−0.869859	+	0.493301i	$0.835790\pi$
$264$	0	0
$265$	0	0
$266$	−28.0000	−1.71679
$267$	14.0000i	0.856786i
$268$	2.00000i	0.122169i
$269$	−2.00000	−0.121942	−0.0609711	−	0.998140i	$-0.519420\pi$
$269$	−2.00000	−0.121942	−0.0609711	+	0.998140i	$0.519420\pi$
$270$	0	0
$271$	8.00000	0.485965	0.242983	−	0.970031i	$-0.421874\pi$
$271$	8.00000	0.485965	0.242983	+	0.970031i	$0.421874\pi$
$272$	12.0000i	0.727607i
$273$	8.00000i	0.484182i
$274$	−24.0000	−1.44989
$275$	0	0
$276$	−36.0000	−2.16695
$277$	2.00000i	0.120168i	0.998193	+	0.0600842i	$0.0191369\pi$
$277$	2.00000i	0.120168i	−0.998193	+	0.0600842i	$0.980863\pi$
$278$	44.0000i	2.63894i
$279$	10.0000	0.598684
$280$	0	0
$281$	−18.0000	−1.07379	−0.536895	−	0.843649i	$-0.680403\pi$
$281$	−18.0000	−1.07379	−0.536895	+	0.843649i	$0.680403\pi$
$282$	− 4.00000i	− 0.238197i
$283$	− 3.00000i	− 0.178331i	−0.996017	−	0.0891657i	$-0.971580\pi$
$283$	− 3.00000i	− 0.178331i	0.996017	−	0.0891657i	$-0.0284201\pi$
$284$	0	0
$285$	0	0
$286$	−16.0000	−0.946100
$287$	0	0
$288$	− 8.00000i	− 0.471405i
$289$	8.00000	0.470588
$290$	0	0
$291$	0	0
$292$	14.0000i	0.819288i
$293$	18.0000i	1.05157i	0.850617	+	0.525786i	$0.176229\pi$
$293$	18.0000i	1.05157i	−0.850617	+	0.525786i	$0.823771\pi$
$294$	−12.0000	−0.699854
$295$	0	0
$296$	0	0
$297$	16.0000i	0.928414i
$298$	42.0000i	2.43299i
$299$	−18.0000	−1.04097
$300$	0	0
$301$	4.00000	0.230556
$302$	− 6.00000i	− 0.345261i
$303$	− 4.00000i	− 0.229794i
$304$	28.0000	1.60591
$305$	0	0
$306$	6.00000	0.342997
$307$	25.0000i	1.42683i	0.700744	+	0.713413i	$0.252851\pi$
$307$	25.0000i	1.42683i	−0.700744	+	0.713413i	$0.747149\pi$
$308$	16.0000i	0.911685i
$309$	−32.0000	−1.82042
$310$	0	0
$311$	18.0000	1.02069	0.510343	−	0.859971i	$-0.329518\pi$
$311$	18.0000	1.02069	0.510343	+	0.859971i	$0.329518\pi$
$312$	0	0
$313$	− 16.0000i	− 0.904373i	−0.891923	−	0.452187i	$-0.850644\pi$
$313$	− 16.0000i	− 0.904373i	0.891923	−	0.452187i	$-0.149356\pi$
$314$	−18.0000	−1.01580
$315$	0	0
$316$	−16.0000	−0.900070
$317$	18.0000i	1.01098i	0.862832	+	0.505490i	$0.168688\pi$
$317$	18.0000i	1.01098i	−0.862832	+	0.505490i	$0.831312\pi$
$318$	− 40.0000i	− 2.24309i
$319$	−20.0000	−1.11979
$320$	0	0
$321$	14.0000	0.781404
$322$	36.0000i	2.00620i
$323$	21.0000i	1.16847i
$324$	22.0000	1.22222
$325$	0	0
$326$	38.0000	2.10463
$327$	4.00000i	0.221201i
$328$	0	0
$329$	−2.00000	−0.110264
$330$	0	0
$331$	−6.00000	−0.329790	−0.164895	−	0.986311i	$-0.552728\pi$
$331$	−6.00000	−0.329790	−0.164895	+	0.986311i	$0.552728\pi$
$332$	− 8.00000i	− 0.439057i
$333$	− 1.00000i	− 0.0547997i
$334$	−48.0000	−2.62644
$335$	0	0
$336$	−16.0000	−0.872872
$337$	− 22.0000i	− 1.19842i	−0.800593	−	0.599208i	$-0.795482\pi$
$337$	− 22.0000i	− 1.19842i	0.800593	−	0.599208i	$-0.204518\pi$
$338$	− 18.0000i	− 0.979071i
$339$	−24.0000	−1.30350
$340$	0	0
$341$	40.0000	2.16612
$342$	− 14.0000i	− 0.757033i
$343$	20.0000i	1.07990i
$344$	0	0
$345$	0	0
$346$	22.0000	1.18273
$347$	30.0000i	1.61048i	0.592946	+	0.805242i	$0.297965\pi$
$347$	30.0000i	1.61048i	−0.592946	+	0.805242i	$0.702035\pi$
$348$	20.0000i	1.07211i
$349$	−6.00000	−0.321173	−0.160586	−	0.987022i	$-0.551338\pi$
$349$	−6.00000	−0.321173	−0.160586	+	0.987022i	$0.551338\pi$
$350$	0	0
$351$	8.00000	0.427008
$352$	− 32.0000i	− 1.70561i
$353$	0	0	1.00000			$0$
$353$	0	0	−1.00000			$\pi$
$354$	36.0000	1.91338
$355$	0	0
$356$	14.0000	0.741999
$357$	− 12.0000i	− 0.635107i
$358$	− 24.0000i	− 1.26844i
$359$	−19.0000	−1.00278	−0.501391	−	0.865221i	$-0.667178\pi$
$359$	−19.0000	−1.00278	−0.501391	+	0.865221i	$0.667178\pi$
$360$	0	0
$361$	30.0000	1.57895
$362$	− 14.0000i	− 0.735824i
$363$	− 10.0000i	− 0.524864i
$364$	8.00000	0.419314
$365$	0	0
$366$	8.00000	0.418167
$367$	8.00000i	0.417597i	0.977959	+	0.208798i	$0.0669552\pi$
$367$	8.00000i	0.417597i	−0.977959	+	0.208798i	$0.933045\pi$
$368$	− 36.0000i	− 1.87663i
$369$	0	0
$370$	0	0
$371$	−20.0000	−1.03835
$372$	− 40.0000i	− 2.07390i
$373$	0	0	1.00000			$0$
$373$	0	0	−1.00000			$\pi$
$374$	24.0000	1.24101
$375$	0	0
$376$	0	0
$377$	10.0000i	0.515026i
$378$	− 16.0000i	− 0.822951i
$379$	18.0000	0.924598	0.462299	−	0.886724i	$-0.347025\pi$
$379$	18.0000	0.924598	0.462299	+	0.886724i	$0.347025\pi$
$380$	0	0
$381$	−14.0000	−0.717242
$382$	12.0000i	0.613973i
$383$	− 16.0000i	− 0.817562i	−0.912633	−	0.408781i	$-0.865954\pi$
$383$	− 16.0000i	− 0.817562i	0.912633	−	0.408781i	$-0.134046\pi$
$384$	0	0
$385$	0	0
$386$	−46.0000	−2.34134
$387$	2.00000i	0.101666i
$388$	0	0
$389$	10.0000	0.507020	0.253510	−	0.967333i	$-0.418415\pi$
$389$	10.0000	0.507020	0.253510	+	0.967333i	$0.418415\pi$
$390$	0	0
$391$	27.0000	1.36545
$392$	0	0
$393$	24.0000i	1.21064i
$394$	4.00000	0.201517
$395$	0	0
$396$	−8.00000	−0.402015
$397$	31.0000i	1.55585i	0.628360	+	0.777923i	$0.283727\pi$
$397$	31.0000i	1.55585i	−0.628360	+	0.777923i	$0.716273\pi$
$398$	14.0000i	0.701757i
$399$	−28.0000	−1.40175
$400$	0	0
$401$	−20.0000	−0.998752	−0.499376	−	0.866385i	$-0.666437\pi$
$401$	−20.0000	−0.998752	−0.499376	+	0.866385i	$0.666437\pi$
$402$	4.00000i	0.199502i
$403$	− 20.0000i	− 0.996271i
$404$	−4.00000	−0.199007
$405$	0	0
$406$	20.0000	0.992583
$407$	− 4.00000i	− 0.198273i
$408$	0	0
$409$	−8.00000	−0.395575	−0.197787	−	0.980245i	$-0.563376\pi$
$409$	−8.00000	−0.395575	−0.197787	+	0.980245i	$0.563376\pi$
$410$	0	0
$411$	−24.0000	−1.18383
$412$	32.0000i	1.57653i
$413$	− 18.0000i	− 0.885722i
$414$	−18.0000	−0.884652
$415$	0	0
$416$	−16.0000	−0.784465
$417$	44.0000i	2.15469i
$418$	− 56.0000i	− 2.73905i
$419$	9.00000	0.439679	0.219839	−	0.975536i	$-0.429447\pi$
$419$	9.00000	0.439679	0.219839	+	0.975536i	$0.429447\pi$
$420$	0	0
$421$	−21.0000	−1.02348	−0.511739	−	0.859141i	$-0.670998\pi$
$421$	−21.0000	−1.02348	−0.511739	+	0.859141i	$0.670998\pi$
$422$	24.0000i	1.16830i
$423$	− 1.00000i	− 0.0486217i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	− 4.00000i	− 0.193574i
$428$	− 14.0000i	− 0.676716i
$429$	−16.0000	−0.772487
$430$	0	0
$431$	−9.00000	−0.433515	−0.216757	−	0.976226i	$-0.569548\pi$
$431$	−9.00000	−0.433515	−0.216757	+	0.976226i	$0.569548\pi$
$432$	16.0000i	0.769800i
$433$	− 18.0000i	− 0.865025i	−0.901628	−	0.432512i	$-0.857627\pi$
$433$	− 18.0000i	− 0.865025i	0.901628	−	0.432512i	$-0.142373\pi$
$434$	−40.0000	−1.92006
$435$	0	0
$436$	4.00000	0.191565
$437$	− 63.0000i	− 3.01370i
$438$	28.0000i	1.33789i
$439$	−23.0000	−1.09773	−0.548865	−	0.835911i	$-0.684940\pi$
$439$	−23.0000	−1.09773	−0.548865	+	0.835911i	$0.684940\pi$
$440$	0	0
$441$	−3.00000	−0.142857
$442$	− 12.0000i	− 0.570782i
$443$	2.00000i	0.0950229i	0.998871	+	0.0475114i	$0.0151291\pi$
$443$	2.00000i	0.0950229i	−0.998871	+	0.0475114i	$0.984871\pi$
$444$	−4.00000	−0.189832
$445$	0	0
$446$	22.0000	1.04173
$447$	42.0000i	1.98653i
$448$	16.0000i	0.755929i
$449$	−39.0000	−1.84052	−0.920262	−	0.391303i	$-0.872024\pi$
$449$	−39.0000	−1.84052	−0.920262	+	0.391303i	$0.872024\pi$
$450$	0	0
$451$	0	0
$452$	24.0000i	1.12887i
$453$	− 6.00000i	− 0.281905i
$454$	−6.00000	−0.281594
$455$	0	0
$456$	0	0
$457$	− 29.0000i	− 1.35656i	−0.734802	−	0.678281i	$-0.762725\pi$
$457$	− 29.0000i	− 1.35656i	0.734802	−	0.678281i	$-0.237275\pi$
$458$	8.00000i	0.373815i
$459$	−12.0000	−0.560112
$460$	0	0
$461$	37.0000	1.72326	0.861631	−	0.507535i	$-0.169443\pi$
$461$	37.0000	1.72326	0.861631	+	0.507535i	$0.169443\pi$
$462$	32.0000i	1.48877i
$463$	− 8.00000i	− 0.371792i	−0.982569	−	0.185896i	$-0.940481\pi$
$463$	− 8.00000i	− 0.371792i	0.982569	−	0.185896i	$-0.0595187\pi$
$464$	−20.0000	−0.928477
$465$	0	0
$466$	20.0000	0.926482
$467$	12.0000i	0.555294i	0.960683	+	0.277647i	$0.0895545\pi$
$467$	12.0000i	0.555294i	−0.960683	+	0.277647i	$0.910445\pi$
$468$	4.00000i	0.184900i
$469$	2.00000	0.0923514
$470$	0	0
$471$	−18.0000	−0.829396
$472$	0	0
$473$	8.00000i	0.367840i
$474$	−32.0000	−1.46981
$475$	0	0
$476$	−12.0000	−0.550019
$477$	− 10.0000i	− 0.457869i
$478$	− 40.0000i	− 1.82956i
$479$	29.0000	1.32504	0.662522	−	0.749043i	$-0.269486\pi$
$479$	29.0000	1.32504	0.662522	+	0.749043i	$0.269486\pi$
$480$	0	0
$481$	−2.00000	−0.0911922
$482$	38.0000i	1.73085i
$483$	36.0000i	1.63806i
$484$	−10.0000	−0.454545
$485$	0	0
$486$	20.0000	0.907218
$487$	− 2.00000i	− 0.0906287i	−0.998973	−	0.0453143i	$-0.985571\pi$
$487$	− 2.00000i	− 0.0906287i	0.998973	−	0.0453143i	$-0.0144289\pi$
$488$	0	0
$489$	38.0000	1.71842
$490$	0	0
$491$	−9.00000	−0.406164	−0.203082	−	0.979162i	$-0.565096\pi$
$491$	−9.00000	−0.406164	−0.203082	+	0.979162i	$0.565096\pi$
$492$	0	0
$493$	− 15.0000i	− 0.675566i
$494$	−28.0000	−1.25978
$495$	0	0
$496$	40.0000	1.79605
$497$	0	0
$498$	− 16.0000i	− 0.716977i
$499$	−24.0000	−1.07439	−0.537194	−	0.843459i	$-0.680516\pi$
$499$	−24.0000	−1.07439	−0.537194	+	0.843459i	$0.680516\pi$
$500$	0	0
$501$	−48.0000	−2.14448
$502$	4.00000i	0.178529i
$503$	− 2.00000i	− 0.0891756i	−0.999005	−	0.0445878i	$-0.985803\pi$
$503$	− 2.00000i	− 0.0891756i	0.999005	−	0.0445878i	$-0.0141974\pi$
$504$	0	0
$505$	0	0
$506$	−72.0000	−3.20079
$507$	− 18.0000i	− 0.799408i
$508$	14.0000i	0.621150i
$509$	−9.00000	−0.398918	−0.199459	−	0.979906i	$-0.563918\pi$
$509$	−9.00000	−0.398918	−0.199459	+	0.979906i	$0.563918\pi$
$510$	0	0
$511$	14.0000	0.619324
$512$	− 32.0000i	− 1.41421i
$513$	28.0000i	1.23623i
$514$	2.00000	0.0882162
$515$	0	0
$516$	8.00000	0.352180
$517$	− 4.00000i	− 0.175920i
$518$	4.00000i	0.175750i
$519$	22.0000	0.965693
$520$	0	0
$521$	4.00000	0.175243	0.0876216	−	0.996154i	$-0.472073\pi$
$521$	4.00000	0.175243	0.0876216	+	0.996154i	$0.472073\pi$
$522$	10.0000i	0.437688i
$523$	− 19.0000i	− 0.830812i	−0.909636	−	0.415406i	$-0.863640\pi$
$523$	− 19.0000i	− 0.830812i	0.909636	−	0.415406i	$-0.136360\pi$
$524$	24.0000	1.04844
$525$	0	0
$526$	32.0000	1.39527
$527$	30.0000i	1.30682i
$528$	− 32.0000i	− 1.39262i
$529$	−58.0000	−2.52174
$530$	0	0
$531$	9.00000	0.390567
$532$	28.0000i	1.21395i
$533$	0	0
$534$	28.0000	1.21168
$535$	0	0
$536$	0	0
$537$	− 24.0000i	− 1.03568i
$538$	4.00000i	0.172452i
$539$	−12.0000	−0.516877
$540$	0	0
$541$	14.0000	0.601907	0.300954	−	0.953639i	$-0.402695\pi$
$541$	14.0000	0.601907	0.300954	+	0.953639i	$0.402695\pi$
$542$	− 16.0000i	− 0.687259i
$543$	− 14.0000i	− 0.600798i
$544$	24.0000	1.02899
$545$	0	0
$546$	16.0000	0.684737
$547$	− 38.0000i	− 1.62476i	−0.583127	−	0.812381i	$-0.698171\pi$
$547$	− 38.0000i	− 1.62476i	0.583127	−	0.812381i	$-0.301829\pi$
$548$	24.0000i	1.02523i
$549$	2.00000	0.0853579
$550$	0	0
$551$	−35.0000	−1.49105
$552$	0	0
$553$	16.0000i	0.680389i
$554$	4.00000	0.169944
$555$	0	0
$556$	44.0000	1.86602
$557$	2.00000i	0.0847427i	0.999102	+	0.0423714i	$0.0134913\pi$
$557$	2.00000i	0.0847427i	−0.999102	+	0.0423714i	$0.986509\pi$
$558$	− 20.0000i	− 0.846668i
$559$	4.00000	0.169182
$560$	0	0
$561$	24.0000	1.01328
$562$	36.0000i	1.51857i
$563$	− 24.0000i	− 1.01148i	−0.862686	−	0.505740i	$-0.831220\pi$
$563$	− 24.0000i	− 1.01148i	0.862686	−	0.505740i	$-0.168780\pi$
$564$	−4.00000	−0.168430
$565$	0	0
$566$	−6.00000	−0.252199
$567$	− 22.0000i	− 0.923913i
$568$	0	0
$569$	−9.00000	−0.377300	−0.188650	−	0.982044i	$-0.560411\pi$
$569$	−9.00000	−0.377300	−0.188650	+	0.982044i	$0.560411\pi$
$570$	0	0
$571$	−23.0000	−0.962520	−0.481260	−	0.876578i	$-0.659821\pi$
$571$	−23.0000	−0.962520	−0.481260	+	0.876578i	$0.659821\pi$
$572$	16.0000i	0.668994i
$573$	12.0000i	0.501307i
$574$	0	0
$575$	0	0
$576$	−8.00000	−0.333333
$577$	24.0000i	0.999133i	0.866276	+	0.499567i	$0.166507\pi$
$577$	24.0000i	0.999133i	−0.866276	+	0.499567i	$0.833493\pi$
$578$	− 16.0000i	− 0.665512i
$579$	−46.0000	−1.91169
$580$	0	0
$581$	−8.00000	−0.331896
$582$	0	0
$583$	− 40.0000i	− 1.65663i
$584$	0	0
$585$	0	0
$586$	36.0000	1.48715
$587$	− 12.0000i	− 0.495293i	−0.968850	−	0.247647i	$-0.920343\pi$
$587$	− 12.0000i	− 0.495293i	0.968850	−	0.247647i	$-0.0796572\pi$
$588$	12.0000i	0.494872i
$589$	70.0000	2.88430
$590$	0	0
$591$	4.00000	0.164538
$592$	− 4.00000i	− 0.164399i
$593$	18.0000i	0.739171i	0.929197	+	0.369586i	$0.120500\pi$
$593$	18.0000i	0.739171i	−0.929197	+	0.369586i	$0.879500\pi$
$594$	32.0000	1.31298
$595$	0	0
$596$	42.0000	1.72039
$597$	14.0000i	0.572982i
$598$	36.0000i	1.47215i
$599$	32.0000	1.30748	0.653742	−	0.756717i	$-0.273198\pi$
$599$	32.0000	1.30748	0.653742	+	0.756717i	$0.273198\pi$
$600$	0	0
$601$	−5.00000	−0.203954	−0.101977	−	0.994787i	$-0.532517\pi$
$601$	−5.00000	−0.203954	−0.101977	+	0.994787i	$0.532517\pi$
$602$	− 8.00000i	− 0.326056i
$603$	1.00000i	0.0407231i
$604$	−6.00000	−0.244137
$605$	0	0
$606$	−8.00000	−0.324978
$607$	− 8.00000i	− 0.324710i	−0.986732	−	0.162355i	$-0.948091\pi$
$607$	− 8.00000i	− 0.324710i	0.986732	−	0.162355i	$-0.0519090\pi$
$608$	− 56.0000i	− 2.27110i
$609$	20.0000	0.810441
$610$	0	0
$611$	−2.00000	−0.0809113
$612$	− 6.00000i	− 0.242536i
$613$	39.0000i	1.57520i	0.616190	+	0.787598i	$0.288675\pi$
$613$	39.0000i	1.57520i	−0.616190	+	0.787598i	$0.711325\pi$
$614$	50.0000	2.01784
$615$	0	0
$616$	0	0
$617$	1.00000i	0.0402585i	0.999797	+	0.0201292i	$0.00640777\pi$
$617$	1.00000i	0.0402585i	−0.999797	+	0.0201292i	$0.993592\pi$
$618$	64.0000i	2.57446i
$619$	−20.0000	−0.803868	−0.401934	−	0.915669i	$-0.631662\pi$
$619$	−20.0000	−0.803868	−0.401934	+	0.915669i	$0.631662\pi$
$620$	0	0
$621$	36.0000	1.44463
$622$	− 36.0000i	− 1.44347i
$623$	− 14.0000i	− 0.560898i
$624$	−16.0000	−0.640513
$625$	0	0
$626$	−32.0000	−1.27898
$627$	− 56.0000i	− 2.23642i
$628$	18.0000i	0.718278i
$629$	3.00000	0.119618
$630$	0	0
$631$	−50.0000	−1.99047	−0.995234	−	0.0975126i	$-0.968911\pi$
$631$	−50.0000	−1.99047	−0.995234	+	0.0975126i	$0.968911\pi$
$632$	0	0
$633$	24.0000i	0.953914i
$634$	36.0000	1.42974
$635$	0	0
$636$	−40.0000	−1.58610
$637$	6.00000i	0.237729i
$638$	40.0000i	1.58362i
$639$	0	0
$640$	0	0
$641$	20.0000	0.789953	0.394976	−	0.918691i	$-0.370753\pi$
$641$	20.0000	0.789953	0.394976	+	0.918691i	$0.370753\pi$
$642$	− 28.0000i	− 1.10507i
$643$	12.0000i	0.473234i	0.971603	+	0.236617i	$0.0760386\pi$
$643$	12.0000i	0.473234i	−0.971603	+	0.236617i	$0.923961\pi$
$644$	36.0000	1.41860
$645$	0	0
$646$	42.0000	1.65247
$647$	− 6.00000i	− 0.235884i	−0.993020	−	0.117942i	$-0.962370\pi$
$647$	− 6.00000i	− 0.235884i	0.993020	−	0.117942i	$-0.0376297\pi$
$648$	0	0
$649$	36.0000	1.41312
$650$	0	0
$651$	−40.0000	−1.56772
$652$	− 38.0000i	− 1.48819i
$653$	− 18.0000i	− 0.704394i	−0.935926	−	0.352197i	$-0.885435\pi$
$653$	− 18.0000i	− 0.704394i	0.935926	−	0.352197i	$-0.114565\pi$
$654$	8.00000	0.312825
$655$	0	0
$656$	0	0
$657$	7.00000i	0.273096i
$658$	4.00000i	0.155936i
$659$	9.00000	0.350590	0.175295	−	0.984516i	$-0.443912\pi$
$659$	9.00000	0.350590	0.175295	+	0.984516i	$0.443912\pi$
$660$	0	0
$661$	−18.0000	−0.700119	−0.350059	−	0.936727i	$-0.613839\pi$
$661$	−18.0000	−0.700119	−0.350059	+	0.936727i	$0.613839\pi$
$662$	12.0000i	0.466393i
$663$	− 12.0000i	− 0.466041i
$664$	0	0
$665$	0	0
$666$	−2.00000	−0.0774984
$667$	45.0000i	1.74241i
$668$	48.0000i	1.85718i
$669$	22.0000	0.850569
$670$	0	0
$671$	8.00000	0.308837
$672$	32.0000i	1.23443i
$673$	− 14.0000i	− 0.539660i	−0.962908	−	0.269830i	$-0.913032\pi$
$673$	− 14.0000i	− 0.539660i	0.962908	−	0.269830i	$-0.0869676\pi$
$674$	−44.0000	−1.69482
$675$	0	0
$676$	−18.0000	−0.692308
$677$	− 12.0000i	− 0.461197i	−0.973049	−	0.230599i	$-0.925932\pi$
$677$	− 12.0000i	− 0.461197i	0.973049	−	0.230599i	$-0.0740685\pi$
$678$	48.0000i	1.84343i
$679$	0	0
$680$	0	0
$681$	−6.00000	−0.229920
$682$	− 80.0000i	− 3.06336i
$683$	− 30.0000i	− 1.14792i	−0.818884	−	0.573959i	$-0.805407\pi$
$683$	− 30.0000i	− 1.14792i	0.818884	−	0.573959i	$-0.194593\pi$
$684$	−14.0000	−0.535303
$685$	0	0
$686$	40.0000	1.52721
$687$	8.00000i	0.305219i
$688$	8.00000i	0.304997i
$689$	−20.0000	−0.761939
$690$	0	0
$691$	15.0000	0.570627	0.285313	−	0.958434i	$-0.407902\pi$
$691$	15.0000	0.570627	0.285313	+	0.958434i	$0.407902\pi$
$692$	− 22.0000i	− 0.836315i
$693$	8.00000i	0.303895i
$694$	60.0000	2.27757
$695$	0	0
$696$	0	0
$697$	0	0
$698$	12.0000i	0.454207i
$699$	20.0000	0.756469
$700$	0	0
$701$	20.0000	0.755390	0.377695	−	0.925930i	$-0.376717\pi$
$701$	20.0000	0.755390	0.377695	+	0.925930i	$0.376717\pi$
$702$	− 16.0000i	− 0.603881i
$703$	− 7.00000i	− 0.264010i
$704$	−32.0000	−1.20605
$705$	0	0
$706$	0	0
$707$	4.00000i	0.150435i
$708$	− 36.0000i	− 1.35296i
$709$	10.0000	0.375558	0.187779	−	0.982211i	$-0.439871\pi$
$709$	10.0000	0.375558	0.187779	+	0.982211i	$0.439871\pi$
$710$	0	0
$711$	−8.00000	−0.300023
$712$	0	0
$713$	− 90.0000i	− 3.37053i
$714$	−24.0000	−0.898177
$715$	0	0
$716$	−24.0000	−0.896922
$717$	− 40.0000i	− 1.49383i
$718$	38.0000i	1.41815i
$719$	−25.0000	−0.932343	−0.466171	−	0.884694i	$-0.654367\pi$
$719$	−25.0000	−0.932343	−0.466171	+	0.884694i	$0.654367\pi$
$720$	0	0
$721$	32.0000	1.19174
$722$	− 60.0000i	− 2.23297i
$723$	38.0000i	1.41324i
$724$	−14.0000	−0.520306
$725$	0	0
$726$	−20.0000	−0.742270
$727$	2.00000i	0.0741759i	0.999312	+	0.0370879i	$0.0118082\pi$
$727$	2.00000i	0.0741759i	−0.999312	+	0.0370879i	$0.988192\pi$
$728$	0	0
$729$	−13.0000	−0.481481
$730$	0	0
$731$	−6.00000	−0.221918
$732$	− 8.00000i	− 0.295689i
$733$	0	0	1.00000			$0$
$733$	0	0	−1.00000			$\pi$
$734$	16.0000	0.590571
$735$	0	0
$736$	−72.0000	−2.65396
$737$	4.00000i	0.147342i
$738$	0	0
$739$	20.0000	0.735712	0.367856	−	0.929883i	$-0.380092\pi$
$739$	20.0000	0.735712	0.367856	+	0.929883i	$0.380092\pi$
$740$	0	0
$741$	−28.0000	−1.02861
$742$	40.0000i	1.46845i
$743$	− 24.0000i	− 0.880475i	−0.897881	−	0.440237i	$-0.854894\pi$
$743$	− 24.0000i	− 0.880475i	0.897881	−	0.440237i	$-0.145106\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	− 4.00000i	− 0.146352i
$748$	− 24.0000i	− 0.877527i
$749$	−14.0000	−0.511549
$750$	0	0
$751$	35.0000	1.27717	0.638584	−	0.769552i	$-0.279520\pi$
$751$	35.0000	1.27717	0.638584	+	0.769552i	$0.279520\pi$
$752$	− 4.00000i	− 0.145865i
$753$	4.00000i	0.145768i
$754$	20.0000	0.728357
$755$	0	0
$756$	−16.0000	−0.581914
$757$	− 40.0000i	− 1.45382i	−0.686730	−	0.726912i	$-0.740955\pi$
$757$	− 40.0000i	− 1.45382i	0.686730	−	0.726912i	$-0.259045\pi$
$758$	− 36.0000i	− 1.30758i
$759$	−72.0000	−2.61343
$760$	0	0
$761$	47.0000	1.70375	0.851874	−	0.523746i	$-0.175466\pi$
$761$	47.0000	1.70375	0.851874	+	0.523746i	$0.175466\pi$
$762$	28.0000i	1.01433i
$763$	− 4.00000i	− 0.144810i
$764$	12.0000	0.434145
$765$	0	0
$766$	−32.0000	−1.15621
$767$	− 18.0000i	− 0.649942i
$768$	− 32.0000i	− 1.15470i
$769$	18.0000	0.649097	0.324548	−	0.945869i	$-0.394788\pi$
$769$	18.0000	0.649097	0.324548	+	0.945869i	$0.394788\pi$
$770$	0	0
$771$	2.00000	0.0720282
$772$	46.0000i	1.65558i
$773$	9.00000i	0.323708i	0.986815	+	0.161854i	$0.0517473\pi$
$773$	9.00000i	0.323708i	−0.986815	+	0.161854i	$0.948253\pi$
$774$	4.00000	0.143777
$775$	0	0
$776$	0	0
$777$	4.00000i	0.143499i
$778$	− 20.0000i	− 0.717035i
$779$	0	0
$780$	0	0
$781$	0	0
$782$	− 54.0000i	− 1.93104i
$783$	− 20.0000i	− 0.714742i
$784$	−12.0000	−0.428571
$785$	0	0
$786$	48.0000	1.71210
$787$	− 52.0000i	− 1.85360i	−0.375555	−	0.926800i	$-0.622548\pi$
$787$	− 52.0000i	− 1.85360i	0.375555	−	0.926800i	$-0.377452\pi$
$788$	− 4.00000i	− 0.142494i
$789$	32.0000	1.13923
$790$	0	0
$791$	24.0000	0.853342
$792$	0	0
$793$	− 4.00000i	− 0.142044i
$794$	62.0000	2.20030
$795$	0	0
$796$	14.0000	0.496217
$797$	6.00000i	0.212531i	0.994338	+	0.106265i	$0.0338893\pi$
$797$	6.00000i	0.212531i	−0.994338	+	0.106265i	$0.966111\pi$
$798$	56.0000i	1.98238i
$799$	3.00000	0.106132
$800$	0	0
$801$	7.00000	0.247333
$802$	40.0000i	1.41245i
$803$	28.0000i	0.988099i
$804$	4.00000	0.141069
$805$	0	0
$806$	−40.0000	−1.40894
$807$	4.00000i	0.140807i
$808$	0	0
$809$	6.00000	0.210949	0.105474	−	0.994422i	$-0.466364\pi$
$809$	6.00000	0.210949	0.105474	+	0.994422i	$0.466364\pi$
$810$	0	0
$811$	38.0000	1.33436	0.667180	−	0.744896i	$-0.267501\pi$
$811$	38.0000	1.33436	0.667180	+	0.744896i	$0.267501\pi$
$812$	− 20.0000i	− 0.701862i
$813$	− 16.0000i	− 0.561144i
$814$	−8.00000	−0.280400
$815$	0	0
$816$	24.0000	0.840168
$817$	14.0000i	0.489798i
$818$	16.0000i	0.559427i
$819$	4.00000	0.139771
$820$	0	0
$821$	−25.0000	−0.872506	−0.436253	−	0.899824i	$-0.643695\pi$
$821$	−25.0000	−0.872506	−0.436253	+	0.899824i	$0.643695\pi$
$822$	48.0000i	1.67419i
$823$	51.0000i	1.77775i	0.458151	+	0.888874i	$0.348512\pi$
$823$	51.0000i	1.77775i	−0.458151	+	0.888874i	$0.651488\pi$
$824$	0	0
$825$	0	0
$826$	−36.0000	−1.25260
$827$	15.0000i	0.521601i	0.965393	+	0.260801i	$0.0839865\pi$
$827$	15.0000i	0.521601i	−0.965393	+	0.260801i	$0.916014\pi$
$828$	18.0000i	0.625543i
$829$	55.0000	1.91023	0.955114	−	0.296237i	$-0.0957318\pi$
$829$	55.0000	1.91023	0.955114	+	0.296237i	$0.0957318\pi$
$830$	0	0
$831$	4.00000	0.138758
$832$	16.0000i	0.554700i
$833$	− 9.00000i	− 0.311832i
$834$	88.0000	3.04719
$835$	0	0
$836$	−56.0000	−1.93680
$837$	40.0000i	1.38260i
$838$	− 18.0000i	− 0.621800i
$839$	23.0000	0.794048	0.397024	−	0.917808i	$-0.370043\pi$
$839$	23.0000	0.794048	0.397024	+	0.917808i	$0.370043\pi$
$840$	0	0
$841$	−4.00000	−0.137931
$842$	42.0000i	1.44742i
$843$	36.0000i	1.23991i
$844$	24.0000	0.826114
$845$	0	0
$846$	−2.00000	−0.0687614
$847$	10.0000i	0.343604i
$848$	− 40.0000i	− 1.37361i
$849$	−6.00000	−0.205919
$850$	0	0
$851$	−9.00000	−0.308516
$852$	0	0
$853$	29.0000i	0.992941i	0.868054	+	0.496471i	$0.165371\pi$
$853$	29.0000i	0.992941i	−0.868054	+	0.496471i	$0.834629\pi$
$854$	−8.00000	−0.273754
$855$	0	0
$856$	0	0
$857$	6.00000i	0.204956i	0.994735	+	0.102478i	$0.0326771\pi$
$857$	6.00000i	0.204956i	−0.994735	+	0.102478i	$0.967323\pi$
$858$	32.0000i	1.09246i
$859$	8.00000	0.272956	0.136478	−	0.990643i	$-0.456422\pi$
$859$	8.00000	0.272956	0.136478	+	0.990643i	$0.456422\pi$
$860$	0	0
$861$	0	0
$862$	18.0000i	0.613082i
$863$	1.00000i	0.0340404i	0.999855	+	0.0170202i	$0.00541796\pi$
$863$	1.00000i	0.0340404i	−0.999855	+	0.0170202i	$0.994582\pi$
$864$	32.0000	1.08866
$865$	0	0
$866$	−36.0000	−1.22333
$867$	− 16.0000i	− 0.543388i
$868$	40.0000i	1.35769i
$869$	−32.0000	−1.08553
$870$	0	0
$871$	2.00000	0.0677674
$872$	0	0
$873$	0	0
$874$	−126.000	−4.26201
$875$	0	0
$876$	28.0000	0.946032
$877$	− 41.0000i	− 1.38447i	−0.721671	−	0.692236i	$-0.756626\pi$
$877$	− 41.0000i	− 1.38447i	0.721671	−	0.692236i	$-0.243374\pi$
$878$	46.0000i	1.55242i
$879$	36.0000	1.21425
$880$	0	0
$881$	−57.0000	−1.92038	−0.960189	−	0.279350i	$-0.909881\pi$
$881$	−57.0000	−1.92038	−0.960189	+	0.279350i	$0.909881\pi$
$882$	6.00000i	0.202031i
$883$	− 44.0000i	− 1.48072i	−0.672212	−	0.740359i	$-0.734656\pi$
$883$	− 44.0000i	− 1.48072i	0.672212	−	0.740359i	$-0.265344\pi$
$884$	−12.0000	−0.403604
$885$	0	0
$886$	4.00000	0.134383
$887$	31.0000i	1.04088i	0.853899	+	0.520439i	$0.174232\pi$
$887$	31.0000i	1.04088i	−0.853899	+	0.520439i	$0.825768\pi$
$888$	0	0
$889$	14.0000	0.469545
$890$	0	0
$891$	44.0000	1.47406
$892$	− 22.0000i	− 0.736614i
$893$	− 7.00000i	− 0.234246i
$894$	84.0000	2.80938
$895$	0	0
$896$	0	0
$897$	36.0000i	1.20201i
$898$	78.0000i	2.60289i
$899$	−50.0000	−1.66759
$900$	0	0
$901$	30.0000	0.999445
$902$	0	0
$903$	− 8.00000i	− 0.266223i
$904$	0	0
$905$	0	0
$906$	−12.0000	−0.398673
$907$	43.0000i	1.42779i	0.700252	+	0.713896i	$0.253071\pi$
$907$	43.0000i	1.42779i	−0.700252	+	0.713896i	$0.746929\pi$
$908$	6.00000i	0.199117i
$909$	−2.00000	−0.0663358
$910$	0	0
$911$	−15.0000	−0.496972	−0.248486	−	0.968635i	$-0.579933\pi$
$911$	−15.0000	−0.496972	−0.248486	+	0.968635i	$0.579933\pi$
$912$	− 56.0000i	− 1.85435i
$913$	− 16.0000i	− 0.529523i
$914$	−58.0000	−1.91847
$915$	0	0
$916$	8.00000	0.264327
$917$	− 24.0000i	− 0.792550i
$918$	24.0000i	0.792118i
$919$	−12.0000	−0.395843	−0.197922	−	0.980218i	$-0.563419\pi$
$919$	−12.0000	−0.395843	−0.197922	+	0.980218i	$0.563419\pi$
$920$	0	0
$921$	50.0000	1.64756
$922$	− 74.0000i	− 2.43706i
$923$	0	0
$924$	32.0000	1.05272
$925$	0	0
$926$	−16.0000	−0.525793
$927$	16.0000i	0.525509i
$928$	40.0000i	1.31306i
$929$	−50.0000	−1.64045	−0.820223	−	0.572043i	$-0.806151\pi$
$929$	−50.0000	−1.64045	−0.820223	+	0.572043i	$0.806151\pi$
$930$	0	0
$931$	−21.0000	−0.688247
$932$	− 20.0000i	− 0.655122i
$933$	− 36.0000i	− 1.17859i
$934$	24.0000	0.785304
$935$	0	0
$936$	0	0
$937$	− 30.0000i	− 0.980057i	−0.871706	−	0.490029i	$-0.836986\pi$
$937$	− 30.0000i	− 0.980057i	0.871706	−	0.490029i	$-0.163014\pi$
$938$	− 4.00000i	− 0.130605i
$939$	−32.0000	−1.04428
$940$	0	0
$941$	8.00000	0.260793	0.130396	−	0.991462i	$-0.458375\pi$
$941$	8.00000	0.260793	0.130396	+	0.991462i	$0.458375\pi$
$942$	36.0000i	1.17294i
$943$	0	0
$944$	36.0000	1.17170
$945$	0	0
$946$	16.0000	0.520205
$947$	− 9.00000i	− 0.292461i	−0.989251	−	0.146230i	$-0.953286\pi$
$947$	− 9.00000i	− 0.292461i	0.989251	−	0.146230i	$-0.0467141\pi$
$948$	32.0000i	1.03931i
$949$	14.0000	0.454459
$950$	0	0
$951$	36.0000	1.16738
$952$	0	0
$953$	− 7.00000i	− 0.226752i	−0.993552	−	0.113376i	$-0.963833\pi$
$953$	− 7.00000i	− 0.226752i	0.993552	−	0.113376i	$-0.0361665\pi$
$954$	−20.0000	−0.647524
$955$	0	0
$956$	−40.0000	−1.29369
$957$	40.0000i	1.29302i
$958$	− 58.0000i	− 1.87389i
$959$	24.0000	0.775000
$960$	0	0
$961$	69.0000	2.22581
$962$	4.00000i	0.128965i
$963$	− 7.00000i	− 0.225572i
$964$	38.0000	1.22390
$965$	0	0
$966$	72.0000	2.31656
$967$	0	0	1.00000			$0$
$967$	0	0	−1.00000			$\pi$
$968$	0	0
$969$	42.0000	1.34923
$970$	0	0
$971$	−15.0000	−0.481373	−0.240686	−	0.970603i	$-0.577373\pi$
$971$	−15.0000	−0.481373	−0.240686	+	0.970603i	$0.577373\pi$
$972$	− 20.0000i	− 0.641500i
$973$	− 44.0000i	− 1.41058i
$974$	−4.00000	−0.128168
$975$	0	0
$976$	8.00000	0.256074
$977$	13.0000i	0.415907i	0.978139	+	0.207953i	$0.0666802\pi$
$977$	13.0000i	0.415907i	−0.978139	+	0.207953i	$0.933320\pi$
$978$	− 76.0000i	− 2.43021i
$979$	28.0000	0.894884
$980$	0	0
$981$	2.00000	0.0638551
$982$	18.0000i	0.574403i
$983$	16.0000i	0.510321i	0.966899	+	0.255160i	$0.0821283\pi$
$983$	16.0000i	0.510321i	−0.966899	+	0.255160i	$0.917872\pi$
$984$	0	0
$985$	0	0
$986$	−30.0000	−0.955395
$987$	4.00000i	0.127321i
$988$	28.0000i	0.890799i
$989$	18.0000	0.572367
$990$	0	0
$991$	44.0000	1.39771	0.698853	−	0.715265i	$-0.253694\pi$
$991$	44.0000	1.39771	0.698853	+	0.715265i	$0.253694\pi$
$992$	− 80.0000i	− 2.54000i
$993$	12.0000i	0.380808i
$994$	0	0
$995$	0	0
$996$	−16.0000	−0.506979
$997$	− 6.00000i	− 0.190022i	−0.995476	−	0.0950110i	$-0.969711\pi$
$997$	− 6.00000i	− 0.190022i	0.995476	−	0.0950110i	$-0.0302886\pi$
$998$	48.0000i	1.51941i
$999$	4.00000	0.126554

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1675.2.c.a.1274.1		2
5.2	odd	4		67.2.a.a.1.1	✓	1
5.3	odd	4		1675.2.a.a.1.1		1
5.4	even	2	inner	1675.2.c.a.1274.2		2
15.2	even	4		603.2.a.a.1.1		1
20.7	even	4		1072.2.a.b.1.1		1
35.27	even	4		3283.2.a.e.1.1		1
40.27	even	4		4288.2.a.a.1.1		1
40.37	odd	4		4288.2.a.e.1.1		1
55.32	even	4		8107.2.a.a.1.1		1
60.47	odd	4		9648.2.a.g.1.1		1
335.267	even	4		4489.2.a.a.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
67.2.a.a.1.1	✓	1	5.2	odd	4
603.2.a.a.1.1		1	15.2	even	4
1072.2.a.b.1.1		1	20.7	even	4
1675.2.a.a.1.1		1	5.3	odd	4
1675.2.c.a.1274.1		2	1.1	even	1	trivial
1675.2.c.a.1274.2		2	5.4	even	2	inner
3283.2.a.e.1.1		1	35.27	even	4
4288.2.a.a.1.1		1	40.27	even	4
4288.2.a.e.1.1		1	40.37	odd	4
4489.2.a.a.1.1		1	335.267	even	4
8107.2.a.a.1.1		1	55.32	even	4
9648.2.a.g.1.1		1	60.47	odd	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1675 = 5^{2} \cdot 67 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1675.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-2.00000i q^{2} -2.00000i q^{3} -2.00000 q^{4} -4.00000 q^{6} +2.00000i q^{7} -1.00000 q^{9} +O(q^{10})\)	\(q-2.00000i q^{2} -2.00000i q^{3} -2.00000 q^{4} -4.00000 q^{6} +2.00000i q^{7} -1.00000 q^{9} -4.00000 q^{11} +4.00000i q^{12} +2.00000i q^{13} +4.00000 q^{14} -4.00000 q^{16} -3.00000i q^{17} +2.00000i q^{18} -7.00000 q^{19} +4.00000 q^{21} +8.00000i q^{22} +9.00000i q^{23} +4.00000 q^{26} -4.00000i q^{27} -4.00000i q^{28} +5.00000 q^{29} -10.0000 q^{31} +8.00000i q^{32} +8.00000i q^{33} -6.00000 q^{34} +2.00000 q^{36} +1.00000i q^{37} +14.0000i q^{38} +4.00000 q^{39} -8.00000i q^{42} -2.00000i q^{43} +8.00000 q^{44} +18.0000 q^{46} +1.00000i q^{47} +8.00000i q^{48} +3.00000 q^{49} -6.00000 q^{51} -4.00000i q^{52} +10.0000i q^{53} -8.00000 q^{54} +14.0000i q^{57} -10.0000i q^{58} -9.00000 q^{59} -2.00000 q^{61} +20.0000i q^{62} -2.00000i q^{63} +8.00000 q^{64} +16.0000 q^{66} -1.00000i q^{67} +6.00000i q^{68} +18.0000 q^{69} -7.00000i q^{73} +2.00000 q^{74} +14.0000 q^{76} -8.00000i q^{77} -8.00000i q^{78} +8.00000 q^{79} -11.0000 q^{81} +4.00000i q^{83} -8.00000 q^{84} -4.00000 q^{86} -10.0000i q^{87} -7.00000 q^{89} -4.00000 q^{91} -18.0000i q^{92} +20.0000i q^{93} +2.00000 q^{94} +16.0000 q^{96} -6.00000i q^{98} +4.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 4 q^{4} - 8 q^{6} - 2 q^{9}+O(q^{10}) \) 2 * q - 4 * q^4 - 8 * q^6 - 2 * q^9	\( 2 q - 4 q^{4} - 8 q^{6} - 2 q^{9} - 8 q^{11} + 8 q^{14} - 8 q^{16} - 14 q^{19} + 8 q^{21} + 8 q^{26} + 10 q^{29} - 20 q^{31} - 12 q^{34} + 4 q^{36} + 8 q^{39} + 16 q^{44} + 36 q^{46} + 6 q^{49} - 12 q^{51} - 16 q^{54} - 18 q^{59} - 4 q^{61} + 16 q^{64} + 32 q^{66} + 36 q^{69} + 4 q^{74} + 28 q^{76} + 16 q^{79} - 22 q^{81} - 16 q^{84} - 8 q^{86} - 14 q^{89} - 8 q^{91} + 4 q^{94} + 32 q^{96} + 8 q^{99}+O(q^{100}) \) 2 * q - 4 * q^4 - 8 * q^6 - 2 * q^9 - 8 * q^11 + 8 * q^14 - 8 * q^16 - 14 * q^19 + 8 * q^21 + 8 * q^26 + 10 * q^29 - 20 * q^31 - 12 * q^34 + 4 * q^36 + 8 * q^39 + 16 * q^44 + 36 * q^46 + 6 * q^49 - 12 * q^51 - 16 * q^54 - 18 * q^59 - 4 * q^61 + 16 * q^64 + 32 * q^66 + 36 * q^69 + 4 * q^74 + 28 * q^76 + 16 * q^79 - 22 * q^81 - 16 * q^84 - 8 * q^86 - 14 * q^89 - 8 * q^91 + 4 * q^94 + 32 * q^96 + 8 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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