LMFDB - Embedded newform 1690.2.d.l.1351.12

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1690,2,Mod(1351,1690)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1690, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1690.1351"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 1690 = 2 \cdot 5 \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1690.d (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [12,0,-4,-12,0,0,0,0,32,-12,0,4,0,6] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(14)] == traces)

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

Self dual:	no
Analytic conductor:	$13.4947179416$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} + \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} + 34x^{10} + 453x^{8} + 2990x^{6} + 10094x^{4} + 15876x^{2} + 8281 $ x^12 + 34x^10 + 453x^8 + 2990x^6 + 10094x^4 + 15876*x^2 + 8281
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1351.12
Root			$-2.57254i$ of defining polynomial
Character	$\chi$	$=$	1690.1351
Dual form			1690.2.d.l.1351.6

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+1.00000i q^{2} +2.57254 q^{3} -1.00000 q^{4} +1.00000i q^{5} +2.57254i q^{6} -1.90403i q^{7} -1.00000i q^{8} +3.61798 q^{9} -1.00000 q^{10} -5.17658i q^{11} -2.57254 q^{12} +1.90403 q^{14} +2.57254i q^{15} +1.00000 q^{16} +5.89037 q^{17} +3.61798i q^{18} -8.09045i q^{19} -1.00000i q^{20} -4.89819i q^{21} +5.17658 q^{22} +0.908568 q^{23} -2.57254i q^{24} -1.00000 q^{25} +1.58977 q^{27} +1.90403i q^{28} -5.29954 q^{29} -2.57254 q^{30} -0.0626650i q^{31} +1.00000i q^{32} -13.3170i q^{33} +5.89037i q^{34} +1.90403 q^{35} -3.61798 q^{36} -1.63865i q^{37} +8.09045 q^{38} +1.00000 q^{40} +1.77061i q^{41} +4.89819 q^{42} +8.58977 q^{43} +5.17658i q^{44} +3.61798i q^{45} +0.908568i q^{46} +11.2024i q^{47} +2.57254 q^{48} +3.37468 q^{49} -1.00000i q^{50} +15.1532 q^{51} -8.92525 q^{53} +1.58977i q^{54} +5.17658 q^{55} -1.90403 q^{56} -20.8130i q^{57} -5.29954i q^{58} -10.6767i q^{59} -2.57254i q^{60} -0.681193 q^{61} +0.0626650 q^{62} -6.88873i q^{63} -1.00000 q^{64} +13.3170 q^{66} +14.1492i q^{67} -5.89037 q^{68} +2.33733 q^{69} +1.90403i q^{70} +15.7017i q^{71} -3.61798i q^{72} -0.163089i q^{73} +1.63865 q^{74} -2.57254 q^{75} +8.09045i q^{76} -9.85635 q^{77} +14.7178 q^{79} +1.00000i q^{80} -6.76417 q^{81} -1.77061 q^{82} -1.89246i q^{83} +4.89819i q^{84} +5.89037i q^{85} +8.58977i q^{86} -13.6333 q^{87} -5.17658 q^{88} +0.541298i q^{89} -3.61798 q^{90} -0.908568 q^{92} -0.161208i q^{93} -11.2024 q^{94} +8.09045 q^{95} +2.57254i q^{96} +12.8001i q^{97} +3.37468i q^{98} -18.7287i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 4 q^{3} - 12 q^{4} + 32 q^{9} - 12 q^{10} + 4 q^{12} + 6 q^{14} + 12 q^{16} + 6 q^{17} + 30 q^{22} + 6 q^{23} - 12 q^{25} - 40 q^{27} + 14 q^{29} + 4 q^{30} + 6 q^{35} - 32 q^{36} - 2 q^{38} + 12 q^{40}+ \cdots - 2 q^{95}+O(q^{100}) $ 12 * q - 4 * q^3 - 12 * q^4 + 32 * q^9 - 12 * q^10 + 4 * q^12 + 6 * q^14 + 12 * q^16 + 6 * q^17 + 30 * q^22 + 6 * q^23 - 12 * q^25 - 40 * q^27 + 14 * q^29 + 4 * q^30 + 6 * q^35 - 32 * q^36 - 2 * q^38 + 12 * q^40 - 4 * q^42 + 44 * q^43 - 4 * q^48 - 62 * q^49 + 44 * q^51 - 32 * q^53 + 30 * q^55 - 6 * q^56 + 66 * q^61 - 12 * q^64 + 8 * q^66 - 6 * q^68 + 12 * q^69 + 12 * q^74 + 4 * q^75 + 68 * q^77 + 40 * q^79 - 4 * q^81 - 4 * q^82 - 78 * q^87 - 30 * q^88 - 32 * q^90 - 6 * q^92 + 14 * q^94 - 2 * q^95

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1690\mathbb{Z}\right)^\times$.

$n$	$171$	$677$
$\chi(n)$	$-1$	$1$

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 $

$2$	1.00000i	0.707107i
$2$	1.00000i	0.707107i
$3$	2.57254	1.48526	0.742629	−	0.669703i	$-0.233579\pi$
$3$	2.57254	1.48526	0.742629	+	0.669703i	$0.233579\pi$
$4$	−1.00000	−0.500000
$5$	1.00000i	0.447214i
$5$	1.00000i	0.447214i
$6$	2.57254i	1.05024i
$7$	− 1.90403i	− 0.719655i	−0.933019	−	0.359827i	$-0.882836\pi$
$7$	− 1.90403i	− 0.719655i	0.933019	−	0.359827i	$-0.117164\pi$
$8$	− 1.00000i	− 0.353553i
$9$	3.61798	1.20599
$10$	−1.00000	−0.316228
$11$	− 5.17658i	− 1.56080i	−0.625283	−	0.780398i	$-0.715016\pi$
$11$	− 5.17658i	− 1.56080i	0.625283	−	0.780398i	$-0.284984\pi$
$12$	−2.57254	−0.742629
$13$	0	0
$13$	0	0
$14$	1.90403	0.508873
$15$	2.57254i	0.664228i
$16$	1.00000	0.250000
$17$	5.89037	1.42862	0.714312	−	0.699827i	$-0.246740\pi$
$17$	5.89037	1.42862	0.714312	+	0.699827i	$0.246740\pi$
$18$	3.61798i	0.852765i
$19$	− 8.09045i	− 1.85608i	−0.372485	−	0.928038i	$-0.621494\pi$
$19$	− 8.09045i	− 1.85608i	0.372485	−	0.928038i	$-0.378506\pi$
$20$	− 1.00000i	− 0.223607i
$21$	− 4.89819i	− 1.06887i
$22$	5.17658	1.10365
$23$	0.908568	0.189449	0.0947247	−	0.995504i	$-0.469803\pi$
$23$	0.908568	0.189449	0.0947247	+	0.995504i	$0.469803\pi$
$24$	− 2.57254i	− 0.525118i
$25$	−1.00000	−0.200000
$26$	0	0
$27$	1.58977	0.305952
$28$	1.90403i	0.359827i
$29$	−5.29954	−0.984099	−0.492050	−	0.870567i	$-0.663752\pi$
$29$	−5.29954	−0.984099	−0.492050	+	0.870567i	$0.663752\pi$
$30$	−2.57254	−0.469680
$31$	− 0.0626650i	− 0.0112550i	−0.999984	−	0.00562748i	$-0.998209\pi$
$31$	− 0.0626650i	− 0.0112550i	0.999984	−	0.00562748i	$-0.00179129\pi$
$32$	1.00000i	0.176777i
$33$	− 13.3170i	− 2.31819i
$34$	5.89037i	1.01019i
$35$	1.90403	0.321839
$36$	−3.61798	−0.602996
$37$	− 1.63865i	− 0.269393i	−0.990887	−	0.134696i	$-0.956994\pi$
$37$	− 1.63865i	− 0.269393i	0.990887	−	0.134696i	$-0.0430059\pi$
$38$	8.09045	1.31244
$39$	0	0
$40$	1.00000	0.158114
$41$	1.77061i	0.276522i	0.990396	+	0.138261i	$0.0441513\pi$
$41$	1.77061i	0.276522i	−0.990396	+	0.138261i	$0.955849\pi$
$42$	4.89819	0.755808
$43$	8.58977	1.30993	0.654964	−	0.755660i	$-0.272684\pi$
$43$	8.58977	1.30993	0.654964	+	0.755660i	$0.272684\pi$
$44$	5.17658i	0.780398i
$45$	3.61798i	0.539336i
$46$	0.908568i	0.133961i
$47$	11.2024i	1.63404i	0.576612	+	0.817018i	$0.304374\pi$
$47$	11.2024i	1.63404i	−0.576612	+	0.817018i	$0.695626\pi$
$48$	2.57254	0.371315
$49$	3.37468	0.482097
$50$	− 1.00000i	− 0.141421i
$51$	15.1532	2.12188
$52$	0	0
$53$	−8.92525	−1.22598	−0.612989	−	0.790092i	$-0.710033\pi$
$53$	−8.92525	−1.22598	−0.612989	+	0.790092i	$0.710033\pi$
$54$	1.58977i	0.216341i
$55$	5.17658	0.698009
$56$	−1.90403	−0.254436
$57$	− 20.8130i	− 2.75675i
$58$	− 5.29954i	− 0.695863i
$59$	− 10.6767i	− 1.38999i	−0.719013	−	0.694997i	$-0.755406\pi$
$59$	− 10.6767i	− 1.38999i	0.719013	−	0.694997i	$-0.244594\pi$
$60$	− 2.57254i	− 0.332114i
$61$	−0.681193	−0.0872179	−0.0436089	−	0.999049i	$-0.513886\pi$
$61$	−0.681193	−0.0872179	−0.0436089	+	0.999049i	$0.513886\pi$
$62$	0.0626650	0.00795846
$63$	− 6.88873i	− 0.867898i
$64$	−1.00000	−0.125000
$65$	0	0
$66$	13.3170	1.63921
$67$	14.1492i	1.72860i	0.502975	+	0.864301i	$0.332239\pi$
$67$	14.1492i	1.72860i	−0.502975	+	0.864301i	$0.667761\pi$
$68$	−5.89037	−0.714312
$69$	2.33733	0.281381
$70$	1.90403i	0.227575i
$71$	15.7017i	1.86345i	0.363164	+	0.931725i	$0.381696\pi$
$71$	15.7017i	1.86345i	−0.363164	+	0.931725i	$0.618304\pi$
$72$	− 3.61798i	− 0.426383i
$73$	− 0.163089i	− 0.0190881i	−0.999954	−	0.00954407i	$-0.996962\pi$
$73$	− 0.163089i	− 0.0190881i	0.999954	−	0.00954407i	$-0.00303802\pi$
$74$	1.63865	0.190489
$75$	−2.57254	−0.297052
$76$	8.09045i	0.928038i
$77$	−9.85635	−1.12323
$78$	0	0
$79$	14.7178	1.65589	0.827943	−	0.560813i	$-0.189511\pi$
$79$	14.7178	1.65589	0.827943	+	0.560813i	$0.189511\pi$
$80$	1.00000i	0.111803i
$81$	−6.76417	−0.751575
$82$	−1.77061	−0.195531
$83$	− 1.89246i	− 0.207724i	−0.994592	−	0.103862i	$-0.966880\pi$
$83$	− 1.89246i	− 0.207724i	0.994592	−	0.103862i	$-0.0331201\pi$
$84$	4.89819i	0.534437i
$85$	5.89037i	0.638900i
$86$	8.58977i	0.926259i
$87$	−13.6333	−1.46164
$88$	−5.17658	−0.551825
$89$	0.541298i	0.0573775i	0.999588	+	0.0286887i	$0.00913316\pi$
$89$	0.541298i	0.0573775i	−0.999588	+	0.0286887i	$0.990867\pi$
$90$	−3.61798	−0.381368
$91$	0	0
$92$	−0.908568	−0.0947247
$93$	− 0.161208i	− 0.0167165i
$94$	−11.2024	−1.15544
$95$	8.09045	0.830063
$96$	2.57254i	0.262559i
$97$	12.8001i	1.29966i	0.760081	+	0.649829i	$0.225159\pi$
$97$	12.8001i	1.29966i	−0.760081	+	0.649829i	$0.774841\pi$
$98$	3.37468i	0.340894i
$99$	− 18.7287i	− 1.88231i
$100$	1.00000	0.100000
$101$	−2.59059	−0.257773	−0.128886	−	0.991659i	$-0.541140\pi$
$101$	−2.59059	−0.257773	−0.128886	+	0.991659i	$0.541140\pi$
$102$	15.1532i	1.50039i
$103$	1.77799	0.175191	0.0875953	−	0.996156i	$-0.472082\pi$
$103$	1.77799	0.175191	0.0875953	+	0.996156i	$0.472082\pi$
$104$	0	0
$105$	4.89819	0.478015
$106$	− 8.92525i	− 0.866897i
$107$	3.56829	0.344959	0.172480	−	0.985013i	$-0.444822\pi$
$107$	3.56829	0.344959	0.172480	+	0.985013i	$0.444822\pi$
$108$	−1.58977	−0.152976
$109$	− 11.4449i	− 1.09622i	−0.836405	−	0.548112i	$-0.815347\pi$
$109$	− 11.4449i	− 1.09622i	0.836405	−	0.548112i	$-0.184653\pi$
$110$	5.17658i	0.493567i
$111$	− 4.21550i	− 0.400118i
$112$	− 1.90403i	− 0.179914i
$113$	11.2334	1.05675	0.528373	−	0.849012i	$-0.322802\pi$
$113$	11.2334	1.05675	0.528373	+	0.849012i	$0.322802\pi$
$114$	20.8130	1.94932
$115$	0.908568i	0.0847244i
$116$	5.29954	0.492050
$117$	0	0
$118$	10.6767	0.982874
$119$	− 11.2154i	− 1.02812i
$120$	2.57254	0.234840
$121$	−15.7969	−1.43609
$122$	− 0.681193i	− 0.0616724i
$123$	4.55496i	0.410707i
$124$	0.0626650i	0.00562748i
$125$	− 1.00000i	− 0.0894427i
$126$	6.88873	0.613697
$127$	−10.8735	−0.964870	−0.482435	−	0.875932i	$-0.660248\pi$
$127$	−10.8735	−0.964870	−0.482435	+	0.875932i	$0.660248\pi$
$128$	− 1.00000i	− 0.0883883i
$129$	22.0976	1.94558
$130$	0	0
$131$	1.13042	0.0987653	0.0493827	−	0.998780i	$-0.484275\pi$
$131$	1.13042	0.0987653	0.0493827	+	0.998780i	$0.484275\pi$
$132$	13.3170i	1.15909i
$133$	−15.4044	−1.33573
$134$	−14.1492	−1.22231
$135$	1.58977i	0.136826i
$136$	− 5.89037i	− 0.505095i
$137$	10.9390i	0.934584i	0.884103	+	0.467292i	$0.154770\pi$
$137$	10.9390i	0.934584i	−0.884103	+	0.467292i	$0.845230\pi$
$138$	2.33733i	0.198967i
$139$	−4.30562	−0.365198	−0.182599	−	0.983188i	$-0.558451\pi$
$139$	−4.30562	−0.365198	−0.182599	+	0.983188i	$0.558451\pi$
$140$	−1.90403	−0.160920
$141$	28.8186i	2.42697i
$142$	−15.7017	−1.31766
$143$	0	0
$144$	3.61798	0.301498
$145$	− 5.29954i	− 0.440103i
$146$	0.163089	0.0134974
$147$	8.68150	0.716038
$148$	1.63865i	0.134696i
$149$	− 19.2499i	− 1.57701i	−0.615029	−	0.788505i	$-0.710856\pi$
$149$	− 19.2499i	− 1.57701i	0.615029	−	0.788505i	$-0.289144\pi$
$150$	− 2.57254i	− 0.210047i
$151$	7.91349i	0.643990i	0.946741	+	0.321995i	$0.104354\pi$
$151$	7.91349i	0.643990i	−0.946741	+	0.321995i	$0.895646\pi$
$152$	−8.09045	−0.656222
$153$	21.3112	1.72291
$154$	− 9.85635i	− 0.794247i
$155$	0.0626650	0.00503337
$156$	0	0
$157$	−0.946176	−0.0755131	−0.0377565	−	0.999287i	$-0.512021\pi$
$157$	−0.946176	−0.0755131	−0.0377565	+	0.999287i	$0.512021\pi$
$158$	14.7178i	1.17089i
$159$	−22.9606	−1.82089
$160$	−1.00000	−0.0790569
$161$	− 1.72994i	− 0.136338i
$162$	− 6.76417i	− 0.531444i
$163$	− 4.51269i	− 0.353461i	−0.984259	−	0.176731i	$-0.943448\pi$
$163$	− 4.51269i	− 0.353461i	0.984259	−	0.176731i	$-0.0565521\pi$
$164$	− 1.77061i	− 0.138261i
$165$	13.3170	1.03672
$166$	1.89246	0.146883
$167$	1.96336i	0.151929i	0.997111	+	0.0759647i	$0.0242036\pi$
$167$	1.96336i	0.151929i	−0.997111	+	0.0759647i	$0.975796\pi$
$168$	−4.89819	−0.377904
$169$	0	0
$170$	−5.89037	−0.451771
$171$	− 29.2711i	− 2.23841i
$172$	−8.58977	−0.654964
$173$	−10.2747	−0.781171	−0.390585	−	0.920567i	$-0.627727\pi$
$173$	−10.2747	−0.781171	−0.390585	+	0.920567i	$0.627727\pi$
$174$	− 13.6333i	− 1.03354i
$175$	1.90403i	0.143931i
$176$	− 5.17658i	− 0.390199i
$177$	− 27.4664i	− 2.06450i
$178$	−0.541298	−0.0405720
$179$	−11.3293	−0.846790	−0.423395	−	0.905945i	$-0.639162\pi$
$179$	−11.3293	−0.846790	−0.423395	+	0.905945i	$0.639162\pi$
$180$	− 3.61798i	− 0.269668i
$181$	−16.6110	−1.23468	−0.617342	−	0.786695i	$-0.711791\pi$
$181$	−16.6110	−1.23468	−0.617342	+	0.786695i	$0.711791\pi$
$182$	0	0
$183$	−1.75240	−0.129541
$184$	− 0.908568i	− 0.0669805i
$185$	1.63865	0.120476
$186$	0.161208	0.0118204
$187$	− 30.4919i	− 2.22979i
$188$	− 11.2024i	− 0.817018i
$189$	− 3.02697i	− 0.220180i
$190$	8.09045i	0.586943i
$191$	21.7130	1.57110	0.785550	−	0.618798i	$-0.212380\pi$
$191$	21.7130	1.57110	0.785550	+	0.618798i	$0.212380\pi$
$192$	−2.57254	−0.185657
$193$	− 11.3641i	− 0.818006i	−0.912533	−	0.409003i	$-0.865877\pi$
$193$	− 11.3641i	− 0.818006i	0.912533	−	0.409003i	$-0.134123\pi$
$194$	−12.8001	−0.918997
$195$	0	0
$196$	−3.37468	−0.241048
$197$	17.6398i	1.25678i	0.777898	+	0.628390i	$0.216286\pi$
$197$	17.6398i	1.25678i	−0.777898	+	0.628390i	$0.783714\pi$
$198$	18.7287	1.33099
$199$	23.6305	1.67512	0.837562	−	0.546342i	$-0.183980\pi$
$199$	23.6305	1.67512	0.837562	+	0.546342i	$0.183980\pi$
$200$	1.00000i	0.0707107i
$201$	36.3995i	2.56742i
$202$	− 2.59059i	− 0.182273i
$203$	10.0905i	0.708212i
$204$	−15.1532	−1.06094
$205$	−1.77061	−0.123664
$206$	1.77799i	0.123878i
$207$	3.28718	0.228475
$208$	0	0
$209$	−41.8808	−2.89696
$210$	4.89819i	0.338007i
$211$	−15.2360	−1.04889	−0.524445	−	0.851444i	$-0.675727\pi$
$211$	−15.2360	−1.04889	−0.524445	+	0.851444i	$0.675727\pi$
$212$	8.92525	0.612989
$213$	40.3933i	2.76770i
$214$	3.56829i	0.243923i
$215$	8.58977i	0.585818i
$216$	− 1.58977i	− 0.108170i
$217$	−0.119316	−0.00809969
$218$	11.4449	0.775148
$219$	− 0.419554i	− 0.0283508i
$220$	−5.17658	−0.349005
$221$	0	0
$222$	4.21550	0.282926
$223$	− 2.78265i	− 0.186340i	−0.995650	−	0.0931699i	$-0.970300\pi$
$223$	− 2.78265i	− 0.186340i	0.995650	−	0.0931699i	$-0.0297000\pi$
$224$	1.90403	0.127218
$225$	−3.61798	−0.241198
$226$	11.2334i	0.747232i
$227$	− 3.94347i	− 0.261737i	−0.991400	−	0.130869i	$-0.958223\pi$
$227$	− 3.94347i	− 0.261737i	0.991400	−	0.130869i	$-0.0417766\pi$
$228$	20.8130i	1.37838i
$229$	− 10.3809i	− 0.685986i	−0.939338	−	0.342993i	$-0.888559\pi$
$229$	− 10.3809i	− 0.685986i	0.939338	−	0.342993i	$-0.111441\pi$
$230$	−0.908568	−0.0599092
$231$	−25.3559	−1.66829
$232$	5.29954i	0.347932i
$233$	13.2410	0.867449	0.433725	−	0.901045i	$-0.357199\pi$
$233$	13.2410	0.867449	0.433725	+	0.901045i	$0.357199\pi$
$234$	0	0
$235$	−11.2024	−0.730763
$236$	10.6767i	0.694997i
$237$	37.8623	2.45942
$238$	11.2154	0.726988
$239$	24.3290i	1.57371i	0.617137	+	0.786856i	$0.288292\pi$
$239$	24.3290i	1.57371i	−0.617137	+	0.786856i	$0.711708\pi$
$240$	2.57254i	0.166057i
$241$	11.1983i	0.721349i	0.932692	+	0.360674i	$0.117453\pi$
$241$	11.1983i	0.721349i	−0.932692	+	0.360674i	$0.882547\pi$
$242$	− 15.7969i	− 1.01547i
$243$	−22.1704	−1.42223
$244$	0.681193	0.0436089
$245$	3.37468i	0.215600i
$246$	−4.55496	−0.290414
$247$	0	0
$248$	−0.0626650	−0.00397923
$249$	− 4.86843i	− 0.308524i
$250$	1.00000	0.0632456
$251$	−5.81385	−0.366967	−0.183483	−	0.983023i	$-0.558737\pi$
$251$	−5.81385	−0.366967	−0.183483	+	0.983023i	$0.558737\pi$
$252$	6.88873i	0.433949i
$253$	− 4.70327i	− 0.295692i
$254$	− 10.8735i	− 0.682266i
$255$	15.1532i	0.948932i
$256$	1.00000	0.0625000
$257$	29.1850	1.82051	0.910254	−	0.414050i	$-0.135886\pi$
$257$	29.1850	1.82051	0.910254	+	0.414050i	$0.135886\pi$
$258$	22.0976i	1.37573i
$259$	−3.12004	−0.193870
$260$	0	0
$261$	−19.1736	−1.18682
$262$	1.13042i	0.0698376i
$263$	27.5544	1.69908	0.849538	−	0.527527i	$-0.176881\pi$
$263$	27.5544	1.69908	0.849538	+	0.527527i	$0.176881\pi$
$264$	−13.3170	−0.819603
$265$	− 8.92525i	− 0.548274i
$266$	− 15.4044i	− 0.944507i
$267$	1.39251i	0.0852204i
$268$	− 14.1492i	− 0.864301i
$269$	3.97887	0.242596	0.121298	−	0.992616i	$-0.461294\pi$
$269$	3.97887	0.242596	0.121298	+	0.992616i	$0.461294\pi$
$270$	−1.58977	−0.0967505
$271$	− 13.6399i	− 0.828567i	−0.910148	−	0.414283i	$-0.864032\pi$
$271$	− 13.6399i	− 0.828567i	0.910148	−	0.414283i	$-0.135968\pi$
$272$	5.89037	0.357156
$273$	0	0
$274$	−10.9390	−0.660851
$275$	5.17658i	0.312159i
$276$	−2.33733	−0.140691
$277$	2.30543	0.138520	0.0692600	−	0.997599i	$-0.477936\pi$
$277$	2.30543	0.138520	0.0692600	+	0.997599i	$0.477936\pi$
$278$	− 4.30562i	− 0.258234i
$279$	− 0.226720i	− 0.0135734i
$280$	− 1.90403i	− 0.113787i
$281$	21.0557i	1.25608i	0.778183	+	0.628038i	$0.216142\pi$
$281$	21.0557i	1.25608i	−0.778183	+	0.628038i	$0.783858\pi$
$282$	−28.8186	−1.71612
$283$	2.60164	0.154652	0.0773258	−	0.997006i	$-0.475362\pi$
$283$	2.60164	0.154652	0.0773258	+	0.997006i	$0.475362\pi$
$284$	− 15.7017i	− 0.931725i
$285$	20.8130	1.23286
$286$	0	0
$287$	3.37128	0.199000
$288$	3.61798i	0.213191i
$289$	17.6964	1.04097
$290$	5.29954	0.311200
$291$	32.9289i	1.93033i
$292$	0.163089i	0.00954407i
$293$	− 6.46826i	− 0.377880i	−0.981989	−	0.188940i	$-0.939495\pi$
$293$	− 6.46826i	− 0.377880i	0.981989	−	0.188940i	$-0.0605052\pi$
$294$	8.68150i	0.506316i
$295$	10.6767	0.621624
$296$	−1.63865	−0.0952447
$297$	− 8.22958i	− 0.477529i
$298$	19.2499	1.11511
$299$	0	0
$300$	2.57254	0.148526
$301$	− 16.3552i	− 0.942696i
$302$	−7.91349	−0.455370
$303$	−6.66439	−0.382859
$304$	− 8.09045i	− 0.464019i
$305$	− 0.681193i	− 0.0390050i
$306$	21.3112i	1.21828i
$307$	11.7707i	0.671789i	0.941900	+	0.335894i	$0.109039\pi$
$307$	11.7707i	0.671789i	−0.941900	+	0.335894i	$0.890961\pi$
$308$	9.85635	0.561617
$309$	4.57395	0.260203
$310$	0.0626650i	0.00355913i
$311$	−2.09362	−0.118718	−0.0593591	−	0.998237i	$-0.518906\pi$
$311$	−2.09362	−0.118718	−0.0593591	+	0.998237i	$0.518906\pi$
$312$	0	0
$313$	−24.7645	−1.39977	−0.699885	−	0.714256i	$-0.746765\pi$
$313$	−24.7645	−1.39977	−0.699885	+	0.714256i	$0.746765\pi$
$314$	− 0.946176i	− 0.0533958i
$315$	6.88873	0.388136
$316$	−14.7178	−0.827943
$317$	20.9473i	1.17652i	0.808672	+	0.588259i	$0.200186\pi$
$317$	20.9473i	1.17652i	−0.808672	+	0.588259i	$0.799814\pi$
$318$	− 22.9606i	− 1.28757i
$319$	27.4335i	1.53598i
$320$	− 1.00000i	− 0.0559017i
$321$	9.17957	0.512354
$322$	1.72994	0.0964057
$323$	− 47.6557i	− 2.65164i
$324$	6.76417	0.375787
$325$	0	0
$326$	4.51269	0.249935
$327$	− 29.4426i	− 1.62818i
$328$	1.77061	0.0977653
$329$	21.3297	1.17594
$330$	13.3170i	0.733075i
$331$	− 21.2478i	− 1.16789i	−0.811794	−	0.583944i	$-0.801509\pi$
$331$	− 21.2478i	− 1.16789i	0.811794	−	0.583944i	$-0.198491\pi$
$332$	1.89246i	0.103862i
$333$	− 5.92861i	− 0.324886i
$334$	−1.96336	−0.107430
$335$	−14.1492	−0.773054
$336$	− 4.89819i	− 0.267218i
$337$	−11.8333	−0.644603	−0.322301	−	0.946637i	$-0.604456\pi$
$337$	−11.8333	−0.644603	−0.322301	+	0.946637i	$0.604456\pi$
$338$	0	0
$339$	28.8983	1.56954
$340$	− 5.89037i	− 0.319450i
$341$	−0.324390	−0.0175667
$342$	29.2711	1.58280
$343$	− 19.7537i	− 1.06660i
$344$	− 8.58977i	− 0.463130i
$345$	2.33733i	0.125838i
$346$	− 10.2747i	− 0.552371i
$347$	−21.3904	−1.14830	−0.574149	−	0.818751i	$-0.694667\pi$
$347$	−21.3904	−1.14830	−0.574149	+	0.818751i	$0.694667\pi$
$348$	13.6333	0.730821
$349$	18.5446i	0.992670i	0.868131	+	0.496335i	$0.165321\pi$
$349$	18.5446i	0.992670i	−0.868131	+	0.496335i	$0.834679\pi$
$350$	−1.90403	−0.101775
$351$	0	0
$352$	5.17658	0.275912
$353$	11.2265i	0.597527i	0.954327	+	0.298764i	$0.0965742\pi$
$353$	11.2265i	0.597527i	−0.954327	+	0.298764i	$0.903426\pi$
$354$	27.4664	1.45982
$355$	−15.7017	−0.833360
$356$	− 0.541298i	− 0.0286887i
$357$	− 28.8522i	− 1.52702i
$358$	− 11.3293i	− 0.598771i
$359$	7.97148i	0.420719i	0.977624	+	0.210359i	$0.0674634\pi$
$359$	7.97148i	0.420719i	−0.977624	+	0.210359i	$0.932537\pi$
$360$	3.61798	0.190684
$361$	−46.4554	−2.44502
$362$	− 16.6110i	− 0.873054i
$363$	−40.6383	−2.13296
$364$	0	0
$365$	0.163089	0.00853648
$366$	− 1.75240i	− 0.0915994i
$367$	−20.4864	−1.06938	−0.534690	−	0.845048i	$-0.679572\pi$
$367$	−20.4864	−1.06938	−0.534690	+	0.845048i	$0.679572\pi$
$368$	0.908568	0.0473624
$369$	6.40601i	0.333484i
$370$	1.63865i	0.0851895i
$371$	16.9939i	0.882281i
$372$	0.161208i	0.00835826i
$373$	14.4319	0.747254	0.373627	−	0.927579i	$-0.378114\pi$
$373$	14.4319	0.747254	0.373627	+	0.927579i	$0.378114\pi$
$374$	30.4919	1.57670
$375$	− 2.57254i	− 0.132846i
$376$	11.2024	0.577719
$377$	0	0
$378$	3.02697	0.155691
$379$	0.607528i	0.0312066i	0.999878	+	0.0156033i	$0.00496689\pi$
$379$	0.607528i	0.0312066i	−0.999878	+	0.0156033i	$0.995033\pi$
$380$	−8.09045	−0.415031
$381$	−27.9726	−1.43308
$382$	21.7130i	1.11094i
$383$	14.6778i	0.749998i	0.927025	+	0.374999i	$0.122357\pi$
$383$	14.6778i	0.749998i	−0.927025	+	0.374999i	$0.877643\pi$
$384$	− 2.57254i	− 0.131280i
$385$	− 9.85635i	− 0.502326i
$386$	11.3641	0.578418
$387$	31.0776	1.57976
$388$	− 12.8001i	− 0.649829i
$389$	−18.6299	−0.944572	−0.472286	−	0.881445i	$-0.656571\pi$
$389$	−18.6299	−0.944572	−0.472286	+	0.881445i	$0.656571\pi$
$390$	0	0
$391$	5.35180	0.270652
$392$	− 3.37468i	− 0.170447i
$393$	2.90806	0.146692
$394$	−17.6398	−0.888678
$395$	14.7178i	0.740535i
$396$	18.7287i	0.941154i
$397$	− 7.96619i	− 0.399811i	−0.979815	−	0.199906i	$-0.935936\pi$
$397$	− 7.96619i	− 0.399811i	0.979815	−	0.199906i	$-0.0640636\pi$
$398$	23.6305i	1.18449i
$399$	−39.6286	−1.98391
$400$	−1.00000	−0.0500000
$401$	4.76351i	0.237878i	0.992902	+	0.118939i	$0.0379493\pi$
$401$	4.76351i	0.237878i	−0.992902	+	0.118939i	$0.962051\pi$
$402$	−36.3995	−1.81544
$403$	0	0
$404$	2.59059	0.128886
$405$	− 6.76417i	− 0.336114i
$406$	−10.0905	−0.500782
$407$	−8.48261	−0.420467
$408$	− 15.1532i	− 0.750196i
$409$	− 9.59521i	− 0.474452i	−0.971454	−	0.237226i	$-0.923762\pi$
$409$	− 9.59521i	− 0.474452i	0.971454	−	0.237226i	$-0.0762383\pi$
$410$	− 1.77061i	− 0.0874440i
$411$	28.1411i	1.38810i
$412$	−1.77799	−0.0875953
$413$	−20.3288	−1.00032
$414$	3.28718i	0.161556i
$415$	1.89246	0.0928971
$416$	0	0
$417$	−11.0764	−0.542413
$418$	− 41.8808i	− 2.04846i
$419$	17.0041	0.830707	0.415353	−	0.909660i	$-0.363658\pi$
$419$	17.0041	0.830707	0.415353	+	0.909660i	$0.363658\pi$
$420$	−4.89819	−0.239007
$421$	− 11.3647i	− 0.553882i	−0.960887	−	0.276941i	$-0.910679\pi$
$421$	− 11.3647i	− 0.553882i	0.960887	−	0.276941i	$-0.0893207\pi$
$422$	− 15.2360i	− 0.741677i
$423$	40.5300i	1.97064i
$424$	8.92525i	0.433449i
$425$	−5.89037	−0.285725
$426$	−40.3933	−1.95706
$427$	1.29701i	0.0627668i
$428$	−3.56829	−0.172480
$429$	0	0
$430$	−8.58977	−0.414236
$431$	22.6058i	1.08888i	0.838799	+	0.544441i	$0.183258\pi$
$431$	22.6058i	1.08888i	−0.838799	+	0.544441i	$0.816742\pi$
$432$	1.58977	0.0764880
$433$	−2.99497	−0.143929	−0.0719645	−	0.997407i	$-0.522927\pi$
$433$	−2.99497	−0.143929	−0.0719645	+	0.997407i	$0.522927\pi$
$434$	− 0.119316i	− 0.00572734i
$435$	− 13.6333i	− 0.653666i
$436$	11.4449i	0.548112i
$437$	− 7.35072i	− 0.351633i
$438$	0.419554	0.0200471
$439$	−13.4489	−0.641881	−0.320940	−	0.947099i	$-0.603999\pi$
$439$	−13.4489	−0.641881	−0.320940	+	0.947099i	$0.603999\pi$
$440$	− 5.17658i	− 0.246784i
$441$	12.2095	0.581405
$442$	0	0
$443$	−16.6206	−0.789671	−0.394835	−	0.918752i	$-0.629198\pi$
$443$	−16.6206	−0.789671	−0.394835	+	0.918752i	$0.629198\pi$
$444$	4.21550i	0.200059i
$445$	−0.541298	−0.0256600
$446$	2.78265	0.131762
$447$	− 49.5211i	− 2.34227i
$448$	1.90403i	0.0899569i
$449$	− 14.0974i	− 0.665298i	−0.943051	−	0.332649i	$-0.892058\pi$
$449$	− 14.0974i	− 0.665298i	0.943051	−	0.332649i	$-0.107942\pi$
$450$	− 3.61798i	− 0.170553i
$451$	9.16567	0.431595
$452$	−11.2334	−0.528373
$453$	20.3578i	0.956492i
$454$	3.94347	0.185076
$455$	0	0
$456$	−20.8130	−0.974660
$457$	− 26.2785i	− 1.22926i	−0.788817	−	0.614628i	$-0.789306\pi$
$457$	− 26.2785i	− 1.22926i	0.788817	−	0.614628i	$-0.210694\pi$
$458$	10.3809	0.485066
$459$	9.36435	0.437090
$460$	− 0.908568i	− 0.0423622i
$461$	− 2.52721i	− 0.117704i	−0.998267	−	0.0588520i	$-0.981256\pi$
$461$	− 2.52721i	− 0.117704i	0.998267	−	0.0588520i	$-0.0187440\pi$
$462$	− 25.3559i	− 1.17966i
$463$	− 20.9477i	− 0.973522i	−0.873535	−	0.486761i	$-0.838178\pi$
$463$	− 20.9477i	− 0.973522i	0.873535	−	0.486761i	$-0.161822\pi$
$464$	−5.29954	−0.246025
$465$	0.161208	0.00747586
$466$	13.2410i	0.613379i
$467$	19.2552	0.891023	0.445511	−	0.895276i	$-0.353022\pi$
$467$	19.2552	0.891023	0.445511	+	0.895276i	$0.353022\pi$
$468$	0	0
$469$	26.9405	1.24400
$470$	− 11.2024i	− 0.516728i
$471$	−2.43408	−0.112156
$472$	−10.6767	−0.491437
$473$	− 44.4656i	− 2.04453i
$474$	37.8623i	1.73907i
$475$	8.09045i	0.371215i
$476$	11.2154i	0.514058i
$477$	−32.2914	−1.47852
$478$	−24.3290	−1.11278
$479$	22.9230i	1.04738i	0.851910	+	0.523689i	$0.175445\pi$
$479$	22.9230i	1.04738i	−0.851910	+	0.523689i	$0.824555\pi$
$480$	−2.57254	−0.117420
$481$	0	0
$482$	−11.1983	−0.510071
$483$	− 4.45034i	− 0.202498i
$484$	15.7969	0.718043
$485$	−12.8001	−0.581225
$486$	− 22.1704i	− 1.00567i
$487$	− 27.0722i	− 1.22676i	−0.789789	−	0.613379i	$-0.789810\pi$
$487$	− 27.0722i	− 1.22676i	0.789789	−	0.613379i	$-0.210190\pi$
$488$	0.681193i	0.0308362i
$489$	− 11.6091i	− 0.524981i
$490$	−3.37468	−0.152452
$491$	14.7213	0.664363	0.332182	−	0.943215i	$-0.392215\pi$
$491$	14.7213	0.664363	0.332182	+	0.943215i	$0.392215\pi$
$492$	− 4.55496i	− 0.205353i
$493$	−31.2162	−1.40591
$494$	0	0
$495$	18.7287	0.841794
$496$	− 0.0626650i	− 0.00281374i
$497$	29.8965	1.34104
$498$	4.86843	0.218160
$499$	− 4.24391i	− 0.189984i	−0.995478	−	0.0949918i	$-0.969718\pi$
$499$	− 4.24391i	− 0.189984i	0.995478	−	0.0949918i	$-0.0302825\pi$
$500$	1.00000i	0.0447214i
$501$	5.05083i	0.225654i
$502$	− 5.81385i	− 0.259485i
$503$	−9.57492	−0.426925	−0.213462	−	0.976951i	$-0.568474\pi$
$503$	−9.57492	−0.426925	−0.213462	+	0.976951i	$0.568474\pi$
$504$	−6.88873	−0.306848
$505$	− 2.59059i	− 0.115280i
$506$	4.70327	0.209086
$507$	0	0
$508$	10.8735	0.482435
$509$	33.5783i	1.48833i	0.667995	+	0.744165i	$0.267153\pi$
$509$	33.5783i	1.48833i	−0.667995	+	0.744165i	$0.732847\pi$
$510$	−15.1532	−0.670996
$511$	−0.310526	−0.0137369
$512$	1.00000i	0.0441942i
$513$	− 12.8620i	− 0.567870i
$514$	29.1850i	1.28729i
$515$	1.77799i	0.0783476i
$516$	−22.0976	−0.972791
$517$	57.9900	2.55040
$518$	− 3.12004i	− 0.137087i
$519$	−26.4321	−1.16024
$520$	0	0
$521$	2.45328	0.107480	0.0537400	−	0.998555i	$-0.482886\pi$
$521$	2.45328	0.107480	0.0537400	+	0.998555i	$0.482886\pi$
$522$	− 19.1736i	− 0.839206i
$523$	−28.1579	−1.23126	−0.615630	−	0.788035i	$-0.711098\pi$
$523$	−28.1579	−1.23126	−0.615630	+	0.788035i	$0.711098\pi$
$524$	−1.13042	−0.0493827
$525$	4.89819i	0.213775i
$526$	27.5544i	1.20143i
$527$	− 0.369120i	− 0.0160791i
$528$	− 13.3170i	− 0.579546i
$529$	−22.1745	−0.964109
$530$	8.92525	0.387688
$531$	− 38.6282i	− 1.67632i
$532$	15.4044	0.667867
$533$	0	0
$534$	−1.39251	−0.0602599
$535$	3.56829i	0.154271i
$536$	14.1492	0.611153
$537$	−29.1450	−1.25770
$538$	3.97887i	0.171541i
$539$	− 17.4693i	− 0.752455i
$540$	− 1.58977i	− 0.0684130i
$541$	− 21.4643i	− 0.922820i	−0.887187	−	0.461410i	$-0.847344\pi$
$541$	− 21.4643i	− 0.922820i	0.887187	−	0.461410i	$-0.152656\pi$
$542$	13.6399	0.585885
$543$	−42.7325	−1.83383
$544$	5.89037i	0.252547i
$545$	11.4449	0.490247
$546$	0	0
$547$	17.0076	0.727192	0.363596	−	0.931557i	$-0.381549\pi$
$547$	17.0076	0.727192	0.363596	+	0.931557i	$0.381549\pi$
$548$	− 10.9390i	− 0.467292i
$549$	−2.46454	−0.105184
$550$	−5.17658	−0.220730
$551$	42.8756i	1.82656i
$552$	− 2.33733i	− 0.0994834i
$553$	− 28.0232i	− 1.19167i
$554$	2.30543i	0.0979484i
$555$	4.21550	0.178938
$556$	4.30562	0.182599
$557$	− 3.62480i	− 0.153588i	−0.997047	−	0.0767938i	$-0.975532\pi$
$557$	− 3.62480i	− 0.153588i	0.997047	−	0.0767938i	$-0.0244683\pi$
$558$	0.226720	0.00959784
$559$	0	0
$560$	1.90403	0.0804599
$561$	− 78.4418i	− 3.31182i
$562$	−21.0557	−0.888180
$563$	40.7538	1.71757	0.858783	−	0.512339i	$-0.171221\pi$
$563$	40.7538	1.71757	0.858783	+	0.512339i	$0.171221\pi$
$564$	− 28.8186i	− 1.21348i
$565$	11.2334i	0.472591i
$566$	2.60164i	0.109355i
$567$	12.8792i	0.540874i
$568$	15.7017	0.658829
$569$	−31.7297	−1.33018	−0.665089	−	0.746764i	$-0.731606\pi$
$569$	−31.7297	−1.33018	−0.665089	+	0.746764i	$0.731606\pi$
$570$	20.8130i	0.871762i
$571$	14.5697	0.609722	0.304861	−	0.952397i	$-0.401390\pi$
$571$	14.5697	0.609722	0.304861	+	0.952397i	$0.401390\pi$
$572$	0	0
$573$	55.8577	2.33349
$574$	3.37128i	0.140715i
$575$	−0.908568	−0.0378899
$576$	−3.61798	−0.150749
$577$	− 5.78013i	− 0.240630i	−0.992736	−	0.120315i	$-0.961610\pi$
$577$	− 5.78013i	− 0.240630i	0.992736	−	0.120315i	$-0.0383905\pi$
$578$	17.6964i	0.736075i
$579$	− 29.2347i	− 1.21495i
$580$	5.29954i	0.220051i
$581$	−3.60329	−0.149490
$582$	−32.9289	−1.36495
$583$	46.2023i	1.91350i
$584$	−0.163089	−0.00674868
$585$	0	0
$586$	6.46826	0.267201
$587$	− 1.46614i	− 0.0605139i	−0.999542	−	0.0302570i	$-0.990367\pi$
$587$	− 1.46614i	− 0.0605139i	0.999542	−	0.0302570i	$-0.00963256\pi$
$588$	−8.68150	−0.358019
$589$	−0.506988	−0.0208901
$590$	10.6767i	0.439554i
$591$	45.3790i	1.86664i
$592$	− 1.63865i	− 0.0673482i
$593$	− 25.7264i	− 1.05646i	−0.849103	−	0.528228i	$-0.822857\pi$
$593$	− 25.7264i	− 1.05646i	0.849103	−	0.528228i	$-0.177143\pi$
$594$	8.22958	0.337664
$595$	11.2154	0.459788
$596$	19.2499i	0.788505i
$597$	60.7906	2.48799
$598$	0	0
$599$	−26.7207	−1.09178	−0.545890	−	0.837857i	$-0.683808\pi$
$599$	−26.7207	−1.09178	−0.545890	+	0.837857i	$0.683808\pi$
$600$	2.57254i	0.105024i
$601$	26.9562	1.09956	0.549782	−	0.835308i	$-0.314711\pi$
$601$	26.9562	1.09956	0.549782	+	0.835308i	$0.314711\pi$
$602$	16.3552	0.666587
$603$	51.1915i	2.08468i
$604$	− 7.91349i	− 0.321995i
$605$	− 15.7969i	− 0.642237i
$606$	− 6.66439i	− 0.270722i
$607$	−32.3286	−1.31218	−0.656088	−	0.754684i	$-0.727790\pi$
$607$	−32.3286	−1.31218	−0.656088	+	0.754684i	$0.727790\pi$
$608$	8.09045	0.328111
$609$	25.9582i	1.05188i
$610$	0.681193	0.0275807
$611$	0	0
$612$	−21.3112	−0.861455
$613$	10.2117i	0.412447i	0.978505	+	0.206224i	$0.0661174\pi$
$613$	10.2117i	0.412447i	−0.978505	+	0.206224i	$0.933883\pi$
$614$	−11.7707	−0.475027
$615$	−4.55496	−0.183674
$616$	9.85635i	0.397123i
$617$	− 32.6071i	− 1.31271i	−0.754452	−	0.656356i	$-0.772097\pi$
$617$	− 32.6071i	− 1.31271i	0.754452	−	0.656356i	$-0.227903\pi$
$618$	4.57395i	0.183991i
$619$	8.49492i	0.341440i	0.985320	+	0.170720i	$0.0546093\pi$
$619$	8.49492i	0.341440i	−0.985320	+	0.170720i	$0.945391\pi$
$620$	−0.0626650	−0.00251669
$621$	1.44442	0.0579625
$622$	− 2.09362i	− 0.0839465i
$623$	1.03065	0.0412920
$624$	0	0
$625$	1.00000	0.0400000
$626$	− 24.7645i	− 0.989787i
$627$	−107.740	−4.30273
$628$	0.946176	0.0377565
$629$	− 9.65226i	− 0.384861i
$630$	6.88873i	0.274454i
$631$	− 41.4992i	− 1.65206i	−0.563628	−	0.826029i	$-0.690595\pi$
$631$	− 41.4992i	− 1.65206i	0.563628	−	0.826029i	$-0.309405\pi$
$632$	− 14.7178i	− 0.585444i
$633$	−39.1953	−1.55787
$634$	−20.9473	−0.831924
$635$	− 10.8735i	− 0.431503i
$636$	22.9606	0.910447
$637$	0	0
$638$	−27.4335	−1.08610
$639$	56.8084i	2.24731i
$640$	1.00000	0.0395285
$641$	−17.9976	−0.710862	−0.355431	−	0.934703i	$-0.615666\pi$
$641$	−17.9976	−0.710862	−0.355431	+	0.934703i	$0.615666\pi$
$642$	9.17957i	0.362289i
$643$	24.9350i	0.983342i	0.870781	+	0.491671i	$0.163614\pi$
$643$	24.9350i	0.983342i	−0.870781	+	0.491671i	$0.836386\pi$
$644$	1.72994i	0.0681691i
$645$	22.0976i	0.870091i
$646$	47.6557	1.87499
$647$	31.5515	1.24042	0.620209	−	0.784437i	$-0.287048\pi$
$647$	31.5515	1.24042	0.620209	+	0.784437i	$0.287048\pi$
$648$	6.76417i	0.265722i
$649$	−55.2690	−2.16950
$650$	0	0
$651$	−0.306945	−0.0120301
$652$	4.51269i	0.176731i
$653$	8.31180	0.325266	0.162633	−	0.986687i	$-0.448001\pi$
$653$	8.31180	0.325266	0.162633	+	0.986687i	$0.448001\pi$
$654$	29.4426	1.15130
$655$	1.13042i	0.0441692i
$656$	1.77061i	0.0691305i
$657$	− 0.590053i	− 0.0230202i
$658$	21.3297i	0.831517i
$659$	−4.11647	−0.160355	−0.0801774	−	0.996781i	$-0.525549\pi$
$659$	−4.11647	−0.160355	−0.0801774	+	0.996781i	$0.525549\pi$
$660$	−13.3170	−0.518362
$661$	39.4880i	1.53591i	0.640506	+	0.767953i	$0.278724\pi$
$661$	39.4880i	1.53591i	−0.640506	+	0.767953i	$0.721276\pi$
$662$	21.2478	0.825821
$663$	0	0
$664$	−1.89246	−0.0734416
$665$	− 15.4044i	− 0.597359i
$666$	5.92861	0.229729
$667$	−4.81499	−0.186437
$668$	− 1.96336i	− 0.0759647i
$669$	− 7.15847i	− 0.276763i
$670$	− 14.1492i	− 0.546632i
$671$	3.52625i	0.136129i
$672$	4.89819	0.188952
$673$	30.1004	1.16028	0.580142	−	0.814515i	$-0.302997\pi$
$673$	30.1004	1.16028	0.580142	+	0.814515i	$0.302997\pi$
$674$	− 11.8333i	− 0.455803i
$675$	−1.58977	−0.0611904
$676$	0	0
$677$	15.8446	0.608959	0.304479	−	0.952519i	$-0.401518\pi$
$677$	15.8446	0.608959	0.304479	+	0.952519i	$0.401518\pi$
$678$	28.8983i	1.10983i
$679$	24.3718	0.935305
$680$	5.89037	0.225885
$681$	− 10.1447i	− 0.388747i
$682$	− 0.324390i	− 0.0124215i
$683$	− 3.74741i	− 0.143391i	−0.997427	−	0.0716954i	$-0.977159\pi$
$683$	− 3.74741i	− 0.143391i	0.997427	−	0.0716954i	$-0.0228409\pi$
$684$	29.2711i	1.11921i
$685$	−10.9390	−0.417959
$686$	19.7537	0.754199
$687$	− 26.7052i	− 1.01887i
$688$	8.58977	0.327482
$689$	0	0
$690$	−2.33733	−0.0889806
$691$	49.7932i	1.89422i	0.320905	+	0.947111i	$0.396013\pi$
$691$	49.7932i	1.89422i	−0.320905	+	0.947111i	$0.603987\pi$
$692$	10.2747	0.390585
$693$	−35.6600	−1.35461
$694$	− 21.3904i	− 0.811969i
$695$	− 4.30562i	− 0.163321i
$696$	13.6333i	0.516768i
$697$	10.4295i	0.395046i
$698$	−18.5446	−0.701924
$699$	34.0631	1.28839
$700$	− 1.90403i	− 0.0719655i
$701$	−32.9089	−1.24295	−0.621476	−	0.783434i	$-0.713466\pi$
$701$	−32.9089	−1.24295	−0.621476	+	0.783434i	$0.713466\pi$
$702$	0	0
$703$	−13.2574	−0.500014
$704$	5.17658i	0.195100i
$705$	−28.8186	−1.08537
$706$	−11.2265	−0.422515
$707$	4.93255i	0.185508i
$708$	27.4664i	1.03225i
$709$	23.1678i	0.870083i	0.900410	+	0.435042i	$0.143266\pi$
$709$	23.1678i	0.870083i	−0.900410	+	0.435042i	$0.856734\pi$
$710$	− 15.7017i	− 0.589275i
$711$	53.2488	1.99699
$712$	0.541298	0.0202860
$713$	− 0.0569354i	− 0.00213225i
$714$	28.8522	1.07977
$715$	0	0
$716$	11.3293	0.423395
$717$	62.5874i	2.33737i
$718$	−7.97148	−0.297493
$719$	−4.76483	−0.177698	−0.0888491	−	0.996045i	$-0.528319\pi$
$719$	−4.76483	−0.177698	−0.0888491	+	0.996045i	$0.528319\pi$
$720$	3.61798i	0.134834i
$721$	− 3.38534i	− 0.126077i
$722$	− 46.4554i	− 1.72889i
$723$	28.8082i	1.07139i
$724$	16.6110	0.617342
$725$	5.29954	0.196820
$726$	− 40.6383i	− 1.50823i
$727$	−42.5004	−1.57625	−0.788127	−	0.615513i	$-0.788949\pi$
$727$	−42.5004	−1.57625	−0.788127	+	0.615513i	$0.788949\pi$
$728$	0	0
$729$	−36.7419	−1.36081
$730$	0.163089i	0.00603620i
$731$	50.5969	1.87139
$732$	1.75240	0.0647705
$733$	− 10.3609i	− 0.382689i	−0.981523	−	0.191345i	$-0.938715\pi$
$733$	− 10.3609i	− 0.382689i	0.981523	−	0.191345i	$-0.0612848\pi$
$734$	− 20.4864i	− 0.756166i
$735$	8.68150i	0.320222i
$736$	0.908568i	0.0334903i
$737$	73.2445	2.69800
$738$	−6.40601	−0.235808
$739$	25.9157i	0.953324i	0.879087	+	0.476662i	$0.158154\pi$
$739$	25.9157i	0.953324i	−0.879087	+	0.476662i	$0.841846\pi$
$740$	−1.63865	−0.0602380
$741$	0	0
$742$	−16.9939	−0.623867
$743$	9.84397i	0.361140i	0.983562	+	0.180570i	$0.0577943\pi$
$743$	9.84397i	0.361140i	−0.983562	+	0.180570i	$0.942206\pi$
$744$	−0.161208	−0.00591018
$745$	19.2499	0.705260
$746$	14.4319i	0.528388i
$747$	− 6.84687i	− 0.250514i
$748$	30.4919i	1.11490i
$749$	− 6.79412i	− 0.248252i
$750$	2.57254	0.0939360
$751$	48.3807	1.76544	0.882718	−	0.469903i	$-0.155711\pi$
$751$	48.3807	1.76544	0.882718	+	0.469903i	$0.155711\pi$
$752$	11.2024i	0.408509i
$753$	−14.9564	−0.545041
$754$	0	0
$755$	−7.91349	−0.288001
$756$	3.02697i	0.110090i
$757$	−10.3630	−0.376649	−0.188324	−	0.982107i	$-0.560306\pi$
$757$	−10.3630	−0.376649	−0.188324	+	0.982107i	$0.560306\pi$
$758$	−0.607528	−0.0220664
$759$	− 12.0994i	− 0.439179i
$760$	− 8.09045i	− 0.293471i
$761$	24.0420i	0.871521i	0.900063	+	0.435760i	$0.143520\pi$
$761$	24.0420i	0.871521i	−0.900063	+	0.435760i	$0.856480\pi$
$762$	− 27.9726i	− 1.01334i
$763$	−21.7915	−0.788904
$764$	−21.7130	−0.785550
$765$	21.3112i	0.770509i
$766$	−14.6778	−0.530329
$767$	0	0
$768$	2.57254	0.0928286
$769$	11.4605i	0.413277i	0.978417	+	0.206639i	$0.0662524\pi$
$769$	11.4605i	0.413277i	−0.978417	+	0.206639i	$0.933748\pi$
$770$	9.85635	0.355198
$771$	75.0796	2.70393
$772$	11.3641i	0.409003i
$773$	− 0.507988i	− 0.0182711i	−0.999958	−	0.00913553i	$-0.997092\pi$
$773$	− 0.507988i	− 0.0182711i	0.999958	−	0.00913553i	$-0.00290797\pi$
$774$	31.0776i	1.11706i
$775$	0.0626650i	0.00225099i
$776$	12.8001	0.459498
$777$	−8.02643	−0.287947
$778$	− 18.6299i	− 0.667914i
$779$	14.3250	0.513246
$780$	0	0
$781$	81.2811	2.90847
$782$	5.35180i	0.191380i
$783$	−8.42506	−0.301087
$784$	3.37468	0.120524
$785$	− 0.946176i	− 0.0337705i
$786$	2.90806i	0.103727i
$787$	27.4018i	0.976769i	0.872629	+	0.488384i	$0.162414\pi$
$787$	27.4018i	0.976769i	−0.872629	+	0.488384i	$0.837586\pi$
$788$	− 17.6398i	− 0.628390i
$789$	70.8849	2.52357
$790$	−14.7178	−0.523637
$791$	− 21.3886i	− 0.760493i
$792$	−18.7287	−0.665497
$793$	0	0
$794$	7.96619	0.282709
$795$	− 22.9606i	− 0.814328i
$796$	−23.6305	−0.837562
$797$	2.46568	0.0873388	0.0436694	−	0.999046i	$-0.486095\pi$
$797$	2.46568	0.0873388	0.0436694	+	0.999046i	$0.486095\pi$
$798$	− 39.6286i	− 1.40284i
$799$	65.9862i	2.33442i
$800$	− 1.00000i	− 0.0353553i
$801$	1.95840i	0.0691968i
$802$	−4.76351	−0.168205
$803$	−0.844244	−0.0297927
$804$	− 36.3995i	− 1.28371i
$805$	1.72994	0.0609723
$806$	0	0
$807$	10.2358	0.360318
$808$	2.59059i	0.0911365i
$809$	1.54477	0.0543112	0.0271556	−	0.999631i	$-0.491355\pi$
$809$	1.54477	0.0543112	0.0271556	+	0.999631i	$0.491355\pi$
$810$	6.76417	0.237669
$811$	26.1714i	0.919003i	0.888177	+	0.459502i	$0.151972\pi$
$811$	26.1714i	0.919003i	−0.888177	+	0.459502i	$0.848028\pi$
$812$	− 10.0905i	− 0.354106i
$813$	− 35.0893i	− 1.23064i
$814$	− 8.48261i	− 0.297315i
$815$	4.51269	0.158073
$816$	15.1532	0.530469
$817$	− 69.4951i	− 2.43133i
$818$	9.59521	0.335489
$819$	0	0
$820$	1.77061	0.0618322
$821$	41.5128i	1.44881i	0.689377	+	0.724403i	$0.257884\pi$
$821$	41.5128i	1.44881i	−0.689377	+	0.724403i	$0.742116\pi$
$822$	−28.1411	−0.981534
$823$	17.6447	0.615056	0.307528	−	0.951539i	$-0.400498\pi$
$823$	17.6447	0.615056	0.307528	+	0.951539i	$0.400498\pi$
$824$	− 1.77799i	− 0.0619392i
$825$	13.3170i	0.463637i
$826$	− 20.3288i	− 0.707330i
$827$	− 30.1723i	− 1.04919i	−0.851351	−	0.524596i	$-0.824216\pi$
$827$	− 30.1723i	− 1.04919i	0.851351	−	0.524596i	$-0.175784\pi$
$828$	−3.28718	−0.114237
$829$	−19.3250	−0.671185	−0.335593	−	0.942007i	$-0.608937\pi$
$829$	−19.3250	−0.671185	−0.335593	+	0.942007i	$0.608937\pi$
$830$	1.89246i	0.0656882i
$831$	5.93082	0.205738
$832$	0	0
$833$	19.8781	0.688735
$834$	− 11.0764i	− 0.383544i
$835$	−1.96336	−0.0679449
$836$	41.8808	1.44848
$837$	− 0.0996231i	− 0.00344348i
$838$	17.0041i	0.587398i
$839$	3.98927i	0.137725i	0.997626	+	0.0688625i	$0.0219370\pi$
$839$	3.98927i	0.137725i	−0.997626	+	0.0688625i	$0.978063\pi$
$840$	− 4.89819i	− 0.169004i
$841$	−0.914902	−0.0315483
$842$	11.3647	0.391654
$843$	54.1666i	1.86560i
$844$	15.2360	0.524445
$845$	0	0
$846$	−40.5300	−1.39345
$847$	30.0778i	1.03349i
$848$	−8.92525	−0.306494
$849$	6.69283	0.229697
$850$	− 5.89037i	− 0.202038i
$851$	− 1.48883i	− 0.0510363i
$852$	− 40.3933i	− 1.38385i
$853$	− 0.0825760i	− 0.00282735i	−0.999999	−	0.00141367i	$-0.999550\pi$
$853$	− 0.0825760i	− 0.00282735i	0.999999	−	0.00141367i	$-0.000449986\pi$
$854$	−1.29701	−0.0443828
$855$	29.2711	1.00105
$856$	− 3.56829i	− 0.121962i
$857$	33.2315	1.13517	0.567584	−	0.823316i	$-0.307878\pi$
$857$	33.2315	1.13517	0.567584	+	0.823316i	$0.307878\pi$
$858$	0	0
$859$	−11.0394	−0.376658	−0.188329	−	0.982106i	$-0.560307\pi$
$859$	−11.0394	−0.376658	−0.188329	+	0.982106i	$0.560307\pi$
$860$	− 8.58977i	− 0.292909i
$861$	8.67277	0.295567
$862$	−22.6058	−0.769957
$863$	22.7798i	0.775432i	0.921779	+	0.387716i	$0.126736\pi$
$863$	22.7798i	0.775432i	−0.921779	+	0.387716i	$0.873264\pi$
$864$	1.58977i	0.0540852i
$865$	− 10.2747i	− 0.349350i
$866$	− 2.99497i	− 0.101773i
$867$	45.5248	1.54610
$868$	0.119316	0.00404984
$869$	− 76.1880i	− 2.58450i
$870$	13.6333	0.462212
$871$	0	0
$872$	−11.4449	−0.387574
$873$	46.3106i	1.56738i
$874$	7.35072	0.248642
$875$	−1.90403	−0.0643679
$876$	0.419554i	0.0141754i
$877$	− 33.3282i	− 1.12541i	−0.826657	−	0.562706i	$-0.809760\pi$
$877$	− 33.3282i	− 1.12541i	0.826657	−	0.562706i	$-0.190240\pi$
$878$	− 13.4489i	− 0.453878i
$879$	− 16.6399i	− 0.561249i
$880$	5.17658	0.174502
$881$	−20.8588	−0.702750	−0.351375	−	0.936235i	$-0.614286\pi$
$881$	−20.8588	−0.702750	−0.351375	+	0.936235i	$0.614286\pi$
$882$	12.2095i	0.411115i
$883$	38.2565	1.28743	0.643716	−	0.765264i	$-0.277392\pi$
$883$	38.2565	1.28743	0.643716	+	0.765264i	$0.277392\pi$
$884$	0	0
$885$	27.4664	0.923272
$886$	− 16.6206i	− 0.558382i
$887$	42.9534	1.44223	0.721117	−	0.692814i	$-0.243629\pi$
$887$	42.9534	1.44223	0.721117	+	0.692814i	$0.243629\pi$
$888$	−4.21550	−0.141463
$889$	20.7035i	0.694373i
$890$	− 0.541298i	− 0.0181444i
$891$	35.0152i	1.17305i
$892$	2.78265i	0.0931699i
$893$	90.6324	3.03290
$894$	49.5211	1.65623
$895$	− 11.3293i	− 0.378696i
$896$	−1.90403	−0.0636091
$897$	0	0
$898$	14.0974	0.470437
$899$	0.332095i	0.0110760i
$900$	3.61798	0.120599
$901$	−52.5730	−1.75146
$902$	9.16567i	0.305184i
$903$	− 42.0744i	− 1.40015i
$904$	− 11.2334i	− 0.373616i
$905$	− 16.6110i	− 0.552168i
$906$	−20.3578	−0.676342
$907$	15.3490	0.509656	0.254828	−	0.966986i	$-0.417981\pi$
$907$	15.3490	0.509656	0.254828	+	0.966986i	$0.417981\pi$
$908$	3.94347i	0.130869i
$909$	−9.37268	−0.310872
$910$	0	0
$911$	−7.48652	−0.248040	−0.124020	−	0.992280i	$-0.539579\pi$
$911$	−7.48652	−0.248040	−0.124020	+	0.992280i	$0.539579\pi$
$912$	− 20.8130i	− 0.689188i
$913$	−9.79646	−0.324215
$914$	26.2785	0.869216
$915$	− 1.75240i	− 0.0579325i
$916$	10.3809i	0.342993i
$917$	− 2.15235i	− 0.0710769i
$918$	9.36435i	0.309070i
$919$	−12.4412	−0.410396	−0.205198	−	0.978720i	$-0.565784\pi$
$919$	−12.4412	−0.410396	−0.205198	+	0.978720i	$0.565784\pi$
$920$	0.908568	0.0299546
$921$	30.2806i	0.997780i
$922$	2.52721	0.0832293
$923$	0	0
$924$	25.3559	0.834147
$925$	1.63865i	0.0538785i
$926$	20.9477	0.688384
$927$	6.43273	0.211278
$928$	− 5.29954i	− 0.173966i
$929$	22.5855i	0.741007i	0.928831	+	0.370504i	$0.120815\pi$
$929$	22.5855i	0.741007i	−0.928831	+	0.370504i	$0.879185\pi$
$930$	0.161208i	0.00528623i
$931$	− 27.3027i	− 0.894809i
$932$	−13.2410	−0.433725
$933$	−5.38592	−0.176327
$934$	19.2552i	0.630048i
$935$	30.4919	0.997193
$936$	0	0
$937$	−20.5615	−0.671714	−0.335857	−	0.941913i	$-0.609026\pi$
$937$	−20.5615	−0.671714	−0.335857	+	0.941913i	$0.609026\pi$
$938$	26.9405i	0.879639i
$939$	−63.7076	−2.07902
$940$	11.2024	0.365382
$941$	35.6763i	1.16301i	0.813542	+	0.581506i	$0.197536\pi$
$941$	35.6763i	1.16301i	−0.813542	+	0.581506i	$0.802464\pi$
$942$	− 2.43408i	− 0.0793066i
$943$	1.60871i	0.0523870i
$944$	− 10.6767i	− 0.347498i
$945$	3.02697	0.0984674
$946$	44.4656	1.44570
$947$	16.0381i	0.521167i	0.965451	+	0.260584i	$0.0839150\pi$
$947$	16.0381i	0.521167i	−0.965451	+	0.260584i	$0.916085\pi$
$948$	−37.8623	−1.22971
$949$	0	0
$950$	−8.09045	−0.262489
$951$	53.8879i	1.74743i
$952$	−11.2154	−0.363494
$953$	35.6022	1.15327	0.576634	−	0.817002i	$-0.304366\pi$
$953$	35.6022	1.15327	0.576634	+	0.817002i	$0.304366\pi$
$954$	− 32.2914i	− 1.04547i
$955$	21.7130i	0.702617i
$956$	− 24.3290i	− 0.786856i
$957$	70.5738i	2.28133i
$958$	−22.9230	−0.740608
$959$	20.8282	0.672578
$960$	− 2.57254i	− 0.0830285i
$961$	30.9961	0.999873
$962$	0	0
$963$	12.9100	0.416018
$964$	− 11.1983i	− 0.360674i
$965$	11.3641	0.365824
$966$	4.45034	0.143187
$967$	8.36719i	0.269071i	0.990909	+	0.134535i	$0.0429542\pi$
$967$	8.36719i	0.269071i	−0.990909	+	0.134535i	$0.957046\pi$
$968$	15.7969i	0.507733i
$969$	− 122.596i	− 3.93836i
$970$	− 12.8001i	− 0.410988i
$971$	29.0504	0.932270	0.466135	−	0.884714i	$-0.345646\pi$
$971$	29.0504	0.932270	0.466135	+	0.884714i	$0.345646\pi$
$972$	22.1704	0.711117
$973$	8.19802i	0.262816i
$974$	27.0722	0.867449
$975$	0	0
$976$	−0.681193	−0.0218045
$977$	− 14.1557i	− 0.452880i	−0.974025	−	0.226440i	$-0.927291\pi$
$977$	− 14.1557i	− 0.452880i	0.974025	−	0.226440i	$-0.0727087\pi$
$978$	11.6091	0.371218
$979$	2.80207	0.0895546
$980$	− 3.37468i	− 0.107800i
$981$	− 41.4075i	− 1.32204i
$982$	14.7213i	0.469776i
$983$	7.74156i	0.246917i	0.992350	+	0.123459i	$0.0393986\pi$
$983$	7.74156i	0.246917i	−0.992350	+	0.123459i	$0.960601\pi$
$984$	4.55496	0.145207
$985$	−17.6398	−0.562049
$986$	− 31.2162i	− 0.994127i
$987$	54.8715	1.74658
$988$	0	0
$989$	7.80439	0.248165
$990$	18.7287i	0.595238i
$991$	−20.8876	−0.663517	−0.331759	−	0.943364i	$-0.607642\pi$
$991$	−20.8876	−0.663517	−0.331759	+	0.943364i	$0.607642\pi$
$992$	0.0626650	0.00198961
$993$	− 54.6610i	− 1.73461i
$994$	29.8965i	0.948259i
$995$	23.6305i	0.749138i
$996$	4.86843i	0.154262i
$997$	−33.3691	−1.05681	−0.528406	−	0.848992i	$-0.677210\pi$
$997$	−33.3691	−1.05681	−0.528406	+	0.848992i	$0.677210\pi$
$998$	4.24391	0.134339
$999$	− 2.60509i	− 0.0824213i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1690.2.d.l.1351.12		12
13.2	odd	12		1690.2.e.u.191.1		12
13.3	even	3		1690.2.l.n.1161.6		24
13.4	even	6		1690.2.l.n.361.6		24
13.5	odd	4		1690.2.a.w.1.6	yes	6
13.6	odd	12		1690.2.e.u.991.1		12
13.7	odd	12		1690.2.e.v.991.1		12
13.8	odd	4		1690.2.a.v.1.6	✓	6
13.9	even	3		1690.2.l.n.361.10		24
13.10	even	6		1690.2.l.n.1161.10		24
13.11	odd	12		1690.2.e.v.191.1		12
13.12	even	2	inner	1690.2.d.l.1351.6		12
65.34	odd	4		8450.2.a.cq.1.1		6
65.44	odd	4		8450.2.a.cp.1.1		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1690.2.a.v.1.6	✓	6	13.8	odd	4
1690.2.a.w.1.6	yes	6	13.5	odd	4
1690.2.d.l.1351.6		12	13.12	even	2	inner
1690.2.d.l.1351.12		12	1.1	even	1	trivial
1690.2.e.u.191.1		12	13.2	odd	12
1690.2.e.u.991.1		12	13.6	odd	12
1690.2.e.v.191.1		12	13.11	odd	12
1690.2.e.v.991.1		12	13.7	odd	12
1690.2.l.n.361.6		24	13.4	even	6
1690.2.l.n.361.10		24	13.9	even	3
1690.2.l.n.1161.6		24	13.3	even	3
1690.2.l.n.1161.10		24	13.10	even	6
8450.2.a.cp.1.1		6	65.44	odd	4
8450.2.a.cq.1.1		6	65.34	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 1690 = 2 \cdot 5 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1690.d (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{2} +2.57254 q^{3} -1.00000 q^{4} +1.00000i q^{5} +2.57254i q^{6} -1.90403i q^{7} -1.00000i q^{8} +3.61798 q^{9} -1.00000 q^{10} -5.17658i q^{11} -2.57254 q^{12} +1.90403 q^{14} +2.57254i q^{15} +1.00000 q^{16} +5.89037 q^{17} +3.61798i q^{18} -8.09045i q^{19} -1.00000i q^{20} -4.89819i q^{21} +5.17658 q^{22} +0.908568 q^{23} -2.57254i q^{24} -1.00000 q^{25} +1.58977 q^{27} +1.90403i q^{28} -5.29954 q^{29} -2.57254 q^{30} -0.0626650i q^{31} +1.00000i q^{32} -13.3170i q^{33} +5.89037i q^{34} +1.90403 q^{35} -3.61798 q^{36} -1.63865i q^{37} +8.09045 q^{38} +1.00000 q^{40} +1.77061i q^{41} +4.89819 q^{42} +8.58977 q^{43} +5.17658i q^{44} +3.61798i q^{45} +0.908568i q^{46} +11.2024i q^{47} +2.57254 q^{48} +3.37468 q^{49} -1.00000i q^{50} +15.1532 q^{51} -8.92525 q^{53} +1.58977i q^{54} +5.17658 q^{55} -1.90403 q^{56} -20.8130i q^{57} -5.29954i q^{58} -10.6767i q^{59} -2.57254i q^{60} -0.681193 q^{61} +0.0626650 q^{62} -6.88873i q^{63} -1.00000 q^{64} +13.3170 q^{66} +14.1492i q^{67} -5.89037 q^{68} +2.33733 q^{69} +1.90403i q^{70} +15.7017i q^{71} -3.61798i q^{72} -0.163089i q^{73} +1.63865 q^{74} -2.57254 q^{75} +8.09045i q^{76} -9.85635 q^{77} +14.7178 q^{79} +1.00000i q^{80} -6.76417 q^{81} -1.77061 q^{82} -1.89246i q^{83} +4.89819i q^{84} +5.89037i q^{85} +8.58977i q^{86} -13.6333 q^{87} -5.17658 q^{88} +0.541298i q^{89} -3.61798 q^{90} -0.908568 q^{92} -0.161208i q^{93} -11.2024 q^{94} +8.09045 q^{95} +2.57254i q^{96} +12.8001i q^{97} +3.37468i q^{98} -18.7287i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 4 q^{3} - 12 q^{4} + 32 q^{9} - 12 q^{10} + 4 q^{12} + 6 q^{14} + 12 q^{16} + 6 q^{17} + 30 q^{22} + 6 q^{23} - 12 q^{25} - 40 q^{27} + 14 q^{29} + 4 q^{30} + 6 q^{35} - 32 q^{36} - 2 q^{38} + 12 q^{40}+ \cdots - 2 q^{95}+O(q^{100}) \) 12 * q - 4 * q^3 - 12 * q^4 + 32 * q^9 - 12 * q^10 + 4 * q^12 + 6 * q^14 + 12 * q^16 + 6 * q^17 + 30 * q^22 + 6 * q^23 - 12 * q^25 - 40 * q^27 + 14 * q^29 + 4 * q^30 + 6 * q^35 - 32 * q^36 - 2 * q^38 + 12 * q^40 - 4 * q^42 + 44 * q^43 - 4 * q^48 - 62 * q^49 + 44 * q^51 - 32 * q^53 + 30 * q^55 - 6 * q^56 + 66 * q^61 - 12 * q^64 + 8 * q^66 - 6 * q^68 + 12 * q^69 + 12 * q^74 + 4 * q^75 + 68 * q^77 + 40 * q^79 - 4 * q^81 - 4 * q^82 - 78 * q^87 - 30 * q^88 - 32 * q^90 - 6 * q^92 + 14 * q^94 - 2 * q^95

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