LMFDB - Embedded newform 1690.2.d.g.1351.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1690,2,Mod(1351,1690)]

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

lf = mfeigenbasis(mf)

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")

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

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

Self dual:	no
Analytic conductor:	$13.4947179416$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{10})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 25 $ x^4 + 25
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 130)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1351.4
Root			$-1.58114 - 1.58114i$ of defining polynomial
Character	$\chi$	$=$	1690.1351
Dual form			1690.2.d.g.1351.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{2} +3.16228 q^{3} -1.00000 q^{4} -1.00000i q^{5} +3.16228i q^{6} +1.00000i q^{7} -1.00000i q^{8} +7.00000 q^{9} +O(q^{10})$	\(q+1.00000i q^{2} +3.16228 q^{3} -1.00000 q^{4} -1.00000i q^{5} +3.16228i q^{6} +1.00000i q^{7} -1.00000i q^{8} +7.00000 q^{9} +1.00000 q^{10} -2.16228i q^{11} -3.16228 q^{12} -1.00000 q^{14} -3.16228i q^{15} +1.00000 q^{16} +5.16228 q^{17} +7.00000i q^{18} -1.83772i q^{19} +1.00000i q^{20} +3.16228i q^{21} +2.16228 q^{22} -6.00000 q^{23} -3.16228i q^{24} -1.00000 q^{25} +12.6491 q^{27} -1.00000i q^{28} +10.3246 q^{29} +3.16228 q^{30} +7.16228i q^{31} +1.00000i q^{32} -6.83772i q^{33} +5.16228i q^{34} +1.00000 q^{35} -7.00000 q^{36} +0.162278i q^{37} +1.83772 q^{38} -1.00000 q^{40} -10.3246i q^{41} -3.16228 q^{42} -2.00000 q^{43} +2.16228i q^{44} -7.00000i q^{45} -6.00000i q^{46} +3.00000i q^{47} +3.16228 q^{48} +6.00000 q^{49} -1.00000i q^{50} +16.3246 q^{51} -2.16228 q^{53} +12.6491i q^{54} -2.16228 q^{55} +1.00000 q^{56} -5.81139i q^{57} +10.3246i q^{58} +10.3246i q^{59} +3.16228i q^{60} -7.48683 q^{61} -7.16228 q^{62} +7.00000i q^{63} -1.00000 q^{64} +6.83772 q^{66} -2.32456i q^{67} -5.16228 q^{68} -18.9737 q^{69} +1.00000i q^{70} -4.32456i q^{71} -7.00000i q^{72} -2.83772i q^{73} -0.162278 q^{74} -3.16228 q^{75} +1.83772i q^{76} +2.16228 q^{77} -13.4868 q^{79} -1.00000i q^{80} +19.0000 q^{81} +10.3246 q^{82} +9.48683i q^{83} -3.16228i q^{84} -5.16228i q^{85} -2.00000i q^{86} +32.6491 q^{87} -2.16228 q^{88} +7.32456i q^{89} +7.00000 q^{90} +6.00000 q^{92} +22.6491i q^{93} -3.00000 q^{94} -1.83772 q^{95} +3.16228i q^{96} -7.48683i q^{97} +6.00000i q^{98} -15.1359i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{4} + 28 q^{9}+O(q^{10}) $ 4 * q - 4 * q^4 + 28 * q^9	$ 4 q - 4 q^{4} + 28 q^{9} + 4 q^{10} - 4 q^{14} + 4 q^{16} + 8 q^{17} - 4 q^{22} - 24 q^{23} - 4 q^{25} + 16 q^{29} + 4 q^{35} - 28 q^{36} + 20 q^{38} - 4 q^{40} - 8 q^{43} + 24 q^{49} + 40 q^{51} + 4 q^{53} + 4 q^{55} + 4 q^{56} + 8 q^{61} - 16 q^{62} - 4 q^{64} + 40 q^{66} - 8 q^{68} + 12 q^{74} - 4 q^{77} - 16 q^{79} + 76 q^{81} + 16 q^{82} + 80 q^{87} + 4 q^{88} + 28 q^{90} + 24 q^{92} - 12 q^{94} - 20 q^{95}+O(q^{100}) $ 4 * q - 4 * q^4 + 28 * q^9 + 4 * q^10 - 4 * q^14 + 4 * q^16 + 8 * q^17 - 4 * q^22 - 24 * q^23 - 4 * q^25 + 16 * q^29 + 4 * q^35 - 28 * q^36 + 20 * q^38 - 4 * q^40 - 8 * q^43 + 24 * q^49 + 40 * q^51 + 4 * q^53 + 4 * q^55 + 4 * q^56 + 8 * q^61 - 16 * q^62 - 4 * q^64 + 40 * q^66 - 8 * q^68 + 12 * q^74 - 4 * q^77 - 16 * q^79 + 76 * q^81 + 16 * q^82 + 80 * q^87 + 4 * q^88 + 28 * q^90 + 24 * q^92 - 12 * q^94 - 20 * q^95

Display 100 coefficients 10 coefficients

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
$3$	3.16228	1.82574	0.912871	−	0.408248i	$-0.133860\pi$
$3$	3.16228	1.82574	0.912871	+	0.408248i	$0.133860\pi$
$4$	−1.00000	−0.500000
$5$	− 1.00000i	− 0.447214i
$5$	− 1.00000i	− 0.447214i
$6$	3.16228i	1.29099i
$7$	1.00000i	0.377964i	0.981981	+	0.188982i	$0.0605189\pi$
$7$	1.00000i	0.377964i	−0.981981	+	0.188982i	$0.939481\pi$
$8$	− 1.00000i	− 0.353553i
$9$	7.00000	2.33333
$10$	1.00000	0.316228
$11$	− 2.16228i	− 0.651951i	−0.945378	−	0.325976i	$-0.894307\pi$
$11$	− 2.16228i	− 0.651951i	0.945378	−	0.325976i	$-0.105693\pi$
$12$	−3.16228	−0.912871
$13$	0	0
$13$	0	0
$14$	−1.00000	−0.267261
$15$	− 3.16228i	− 0.816497i
$16$	1.00000	0.250000
$17$	5.16228	1.25204	0.626018	−	0.779809i	$-0.284684\pi$
$17$	5.16228	1.25204	0.626018	+	0.779809i	$0.284684\pi$
$18$	7.00000i	1.64992i
$19$	− 1.83772i	− 0.421602i	−0.977529	−	0.210801i	$-0.932393\pi$
$19$	− 1.83772i	− 0.421602i	0.977529	−	0.210801i	$-0.0676073\pi$
$20$	1.00000i	0.223607i
$21$	3.16228i	0.690066i
$22$	2.16228	0.460999
$23$	−6.00000	−1.25109	−0.625543	−	0.780189i	$-0.715123\pi$
$23$	−6.00000	−1.25109	−0.625543	+	0.780189i	$0.715123\pi$
$24$	− 3.16228i	− 0.645497i
$25$	−1.00000	−0.200000
$26$	0	0
$27$	12.6491	2.43432
$28$	− 1.00000i	− 0.188982i
$29$	10.3246	1.91722	0.958611	−	0.284719i	$-0.0919004\pi$
$29$	10.3246	1.91722	0.958611	+	0.284719i	$0.0919004\pi$
$30$	3.16228	0.577350
$31$	7.16228i	1.28638i	0.765705	+	0.643192i	$0.222390\pi$
$31$	7.16228i	1.28638i	−0.765705	+	0.643192i	$0.777610\pi$
$32$	1.00000i	0.176777i
$33$	− 6.83772i	− 1.19029i
$34$	5.16228i	0.885323i
$35$	1.00000	0.169031
$36$	−7.00000	−1.16667
$37$	0.162278i	0.0266783i	0.999911	+	0.0133391i	$0.00424611\pi$
$37$	0.162278i	0.0266783i	−0.999911	+	0.0133391i	$0.995754\pi$
$38$	1.83772	0.298118
$39$	0	0
$40$	−1.00000	−0.158114
$41$	− 10.3246i	− 1.61242i	−0.591626	−	0.806212i	$-0.701514\pi$
$41$	− 10.3246i	− 1.61242i	0.591626	−	0.806212i	$-0.298486\pi$
$42$	−3.16228	−0.487950
$43$	−2.00000	−0.304997	−0.152499	−	0.988304i	$-0.548732\pi$
$43$	−2.00000	−0.304997	−0.152499	+	0.988304i	$0.548732\pi$
$44$	2.16228i	0.325976i
$45$	− 7.00000i	− 1.04350i
$46$	− 6.00000i	− 0.884652i
$47$	3.00000i	0.437595i	0.975770	+	0.218797i	$0.0702134\pi$
$47$	3.00000i	0.437595i	−0.975770	+	0.218797i	$0.929787\pi$
$48$	3.16228	0.456435
$49$	6.00000	0.857143
$50$	− 1.00000i	− 0.141421i
$51$	16.3246	2.28589
$52$	0	0
$53$	−2.16228	−0.297012	−0.148506	−	0.988912i	$-0.547446\pi$
$53$	−2.16228	−0.297012	−0.148506	+	0.988912i	$0.547446\pi$
$54$	12.6491i	1.72133i
$55$	−2.16228	−0.291561
$56$	1.00000	0.133631
$57$	− 5.81139i	− 0.769737i
$58$	10.3246i	1.35568i
$59$	10.3246i	1.34414i	0.740486	+	0.672071i	$0.234595\pi$
$59$	10.3246i	1.34414i	−0.740486	+	0.672071i	$0.765405\pi$
$60$	3.16228i	0.408248i
$61$	−7.48683	−0.958591	−0.479295	−	0.877654i	$-0.659108\pi$
$61$	−7.48683	−0.958591	−0.479295	+	0.877654i	$0.659108\pi$
$62$	−7.16228	−0.909610
$63$	7.00000i	0.881917i
$64$	−1.00000	−0.125000
$65$	0	0
$66$	6.83772	0.841665
$67$	− 2.32456i	− 0.283990i	−0.989867	−	0.141995i	$-0.954648\pi$
$67$	− 2.32456i	− 0.283990i	0.989867	−	0.141995i	$-0.0453516\pi$
$68$	−5.16228	−0.626018
$69$	−18.9737	−2.28416
$70$	1.00000i	0.119523i
$71$	− 4.32456i	− 0.513230i	−0.966514	−	0.256615i	$-0.917393\pi$
$71$	− 4.32456i	− 0.513230i	0.966514	−	0.256615i	$-0.0826073\pi$
$72$	− 7.00000i	− 0.824958i
$73$	− 2.83772i	− 0.332130i	−0.986115	−	0.166065i	$-0.946894\pi$
$73$	− 2.83772i	− 0.332130i	0.986115	−	0.166065i	$-0.0531062\pi$
$74$	−0.162278	−0.0188644
$75$	−3.16228	−0.365148
$76$	1.83772i	0.210801i
$77$	2.16228	0.246414
$78$	0	0
$79$	−13.4868	−1.51739	−0.758694	−	0.651448i	$-0.774162\pi$
$79$	−13.4868	−1.51739	−0.758694	+	0.651448i	$0.774162\pi$
$80$	− 1.00000i	− 0.111803i
$81$	19.0000	2.11111
$82$	10.3246	1.14016
$83$	9.48683i	1.04132i	0.853766	+	0.520658i	$0.174313\pi$
$83$	9.48683i	1.04132i	−0.853766	+	0.520658i	$0.825687\pi$
$84$	− 3.16228i	− 0.345033i
$85$	− 5.16228i	− 0.559928i
$86$	− 2.00000i	− 0.215666i
$87$	32.6491	3.50035
$88$	−2.16228	−0.230500
$89$	7.32456i	0.776401i	0.921575	+	0.388201i	$0.126903\pi$
$89$	7.32456i	0.776401i	−0.921575	+	0.388201i	$0.873097\pi$
$90$	7.00000	0.737865
$91$	0	0
$92$	6.00000	0.625543
$93$	22.6491i	2.34860i
$94$	−3.00000	−0.309426
$95$	−1.83772	−0.188546
$96$	3.16228i	0.322749i
$97$	− 7.48683i	− 0.760173i	−0.924951	−	0.380086i	$-0.875894\pi$
$97$	− 7.48683i	− 0.760173i	0.924951	−	0.380086i	$-0.124106\pi$
$98$	6.00000i	0.606092i
$99$	− 15.1359i	− 1.52122i
$100$	1.00000	0.100000
$101$	11.1623	1.11069	0.555344	−	0.831621i	$-0.312587\pi$
$101$	11.1623	1.11069	0.555344	+	0.831621i	$0.312587\pi$
$102$	16.3246i	1.61637i
$103$	−0.675445	−0.0665535	−0.0332768	−	0.999446i	$-0.510594\pi$
$103$	−0.675445	−0.0665535	−0.0332768	+	0.999446i	$0.510594\pi$
$104$	0	0
$105$	3.16228	0.308607
$106$	− 2.16228i	− 0.210019i
$107$	−3.48683	−0.337085	−0.168542	−	0.985694i	$-0.553906\pi$
$107$	−3.48683	−0.337085	−0.168542	+	0.985694i	$0.553906\pi$
$108$	−12.6491	−1.21716
$109$	10.6491i	1.02000i	0.860174	+	0.510000i	$0.170355\pi$
$109$	10.6491i	1.02000i	−0.860174	+	0.510000i	$0.829645\pi$
$110$	− 2.16228i	− 0.206165i
$111$	0.513167i	0.0487077i
$112$	1.00000i	0.0944911i
$113$	−8.64911	−0.813640	−0.406820	−	0.913508i	$-0.633362\pi$
$113$	−8.64911	−0.813640	−0.406820	+	0.913508i	$0.633362\pi$
$114$	5.81139	0.544286
$115$	6.00000i	0.559503i
$116$	−10.3246	−0.958611
$117$	0	0
$118$	−10.3246	−0.950452
$119$	5.16228i	0.473225i
$120$	−3.16228	−0.288675
$121$	6.32456	0.574960
$122$	− 7.48683i	− 0.677826i
$123$	− 32.6491i	− 2.94387i
$124$	− 7.16228i	− 0.643192i
$125$	1.00000i	0.0894427i
$126$	−7.00000	−0.623610
$127$	−3.32456	−0.295007	−0.147503	−	0.989062i	$-0.547124\pi$
$127$	−3.32456	−0.295007	−0.147503	+	0.989062i	$0.547124\pi$
$128$	− 1.00000i	− 0.0883883i
$129$	−6.32456	−0.556846
$130$	0	0
$131$	−10.8114	−0.944595	−0.472298	−	0.881439i	$-0.656575\pi$
$131$	−10.8114	−0.944595	−0.472298	+	0.881439i	$0.656575\pi$
$132$	6.83772i	0.595147i
$133$	1.83772	0.159351
$134$	2.32456	0.200811
$135$	− 12.6491i	− 1.08866i
$136$	− 5.16228i	− 0.442662i
$137$	− 15.4868i	− 1.32313i	−0.749888	−	0.661565i	$-0.769893\pi$
$137$	− 15.4868i	− 1.32313i	0.749888	−	0.661565i	$-0.230107\pi$
$138$	− 18.9737i	− 1.61515i
$139$	−12.1623	−1.03159	−0.515795	−	0.856712i	$-0.672504\pi$
$139$	−12.1623	−1.03159	−0.515795	+	0.856712i	$0.672504\pi$
$140$	−1.00000	−0.0845154
$141$	9.48683i	0.798935i
$142$	4.32456	0.362909
$143$	0	0
$144$	7.00000	0.583333
$145$	− 10.3246i	− 0.857408i
$146$	2.83772	0.234852
$147$	18.9737	1.56492
$148$	− 0.162278i	− 0.0133391i
$149$	16.3246i	1.33736i	0.743550	+	0.668680i	$0.233140\pi$
$149$	16.3246i	1.33736i	−0.743550	+	0.668680i	$0.766860\pi$
$150$	− 3.16228i	− 0.258199i
$151$	4.83772i	0.393688i	0.980435	+	0.196844i	$0.0630693\pi$
$151$	4.83772i	0.393688i	−0.980435	+	0.196844i	$0.936931\pi$
$152$	−1.83772	−0.149059
$153$	36.1359	2.92142
$154$	2.16228i	0.174241i
$155$	7.16228	0.575288
$156$	0	0
$157$	−12.1623	−0.970655	−0.485328	−	0.874332i	$-0.661300\pi$
$157$	−12.1623	−0.970655	−0.485328	+	0.874332i	$0.661300\pi$
$158$	− 13.4868i	− 1.07295i
$159$	−6.83772	−0.542267
$160$	1.00000	0.0790569
$161$	− 6.00000i	− 0.472866i
$162$	19.0000i	1.49278i
$163$	− 17.4868i	− 1.36967i	−0.728696	−	0.684837i	$-0.759873\pi$
$163$	− 17.4868i	− 1.36967i	0.728696	−	0.684837i	$-0.240127\pi$
$164$	10.3246i	0.806212i
$165$	−6.83772	−0.532316
$166$	−9.48683	−0.736321
$167$	− 13.3246i	− 1.03109i	−0.856864	−	0.515543i	$-0.827590\pi$
$167$	− 13.3246i	− 1.03109i	0.856864	−	0.515543i	$-0.172410\pi$
$168$	3.16228	0.243975
$169$	0	0
$170$	5.16228	0.395929
$171$	− 12.8641i	− 0.983739i
$172$	2.00000	0.152499
$173$	−2.16228	−0.164395	−0.0821975	−	0.996616i	$-0.526194\pi$
$173$	−2.16228	−0.164395	−0.0821975	+	0.996616i	$0.526194\pi$
$174$	32.6491i	2.47512i
$175$	− 1.00000i	− 0.0755929i
$176$	− 2.16228i	− 0.162988i
$177$	32.6491i	2.45406i
$178$	−7.32456	−0.548999
$179$	−22.3246	−1.66862	−0.834308	−	0.551299i	$-0.814132\pi$
$179$	−22.3246	−1.66862	−0.834308	+	0.551299i	$0.814132\pi$
$180$	7.00000i	0.521749i
$181$	−9.81139	−0.729275	−0.364637	−	0.931150i	$-0.618807\pi$
$181$	−9.81139	−0.729275	−0.364637	+	0.931150i	$0.618807\pi$
$182$	0	0
$183$	−23.6754	−1.75014
$184$	6.00000i	0.442326i
$185$	0.162278	0.0119309
$186$	−22.6491	−1.66071
$187$	− 11.1623i	− 0.816267i
$188$	− 3.00000i	− 0.218797i
$189$	12.6491i	0.920087i
$190$	− 1.83772i	− 0.133322i
$191$	9.48683	0.686443	0.343222	−	0.939254i	$-0.388482\pi$
$191$	9.48683	0.686443	0.343222	+	0.939254i	$0.388482\pi$
$192$	−3.16228	−0.228218
$193$	4.00000i	0.287926i	0.989583	+	0.143963i	$0.0459847\pi$
$193$	4.00000i	0.287926i	−0.989583	+	0.143963i	$0.954015\pi$
$194$	7.48683	0.537523
$195$	0	0
$196$	−6.00000	−0.428571
$197$	− 18.4868i	− 1.31713i	−0.752523	−	0.658566i	$-0.771163\pi$
$197$	− 18.4868i	− 1.31713i	0.752523	−	0.658566i	$-0.228837\pi$
$198$	15.1359	1.07566
$199$	1.35089	0.0957620	0.0478810	−	0.998853i	$-0.484753\pi$
$199$	1.35089	0.0957620	0.0478810	+	0.998853i	$0.484753\pi$
$200$	1.00000i	0.0707107i
$201$	− 7.35089i	− 0.518492i
$202$	11.1623i	0.785375i
$203$	10.3246i	0.724642i
$204$	−16.3246	−1.14295
$205$	−10.3246	−0.721098
$206$	− 0.675445i	− 0.0470605i
$207$	−42.0000	−2.91920
$208$	0	0
$209$	−3.97367	−0.274864
$210$	3.16228i	0.218218i
$211$	11.8377	0.814942	0.407471	−	0.913218i	$-0.366411\pi$
$211$	11.8377	0.814942	0.407471	+	0.913218i	$0.366411\pi$
$212$	2.16228	0.148506
$213$	− 13.6754i	− 0.937026i
$214$	− 3.48683i	− 0.238355i
$215$	2.00000i	0.136399i
$216$	− 12.6491i	− 0.860663i
$217$	−7.16228	−0.486207
$218$	−10.6491	−0.721249
$219$	− 8.97367i	− 0.606384i
$220$	2.16228	0.145781
$221$	0	0
$222$	−0.513167	−0.0344415
$223$	3.32456i	0.222629i	0.993785	+	0.111314i	$0.0355060\pi$
$223$	3.32456i	0.222629i	−0.993785	+	0.111314i	$0.964494\pi$
$224$	−1.00000	−0.0668153
$225$	−7.00000	−0.466667
$226$	− 8.64911i	− 0.575330i
$227$	10.3246i	0.685265i	0.939470	+	0.342632i	$0.111319\pi$
$227$	10.3246i	0.685265i	−0.939470	+	0.342632i	$0.888681\pi$
$228$	5.81139i	0.384869i
$229$	− 2.83772i	− 0.187522i	−0.995595	−	0.0937610i	$-0.970111\pi$
$229$	− 2.83772i	− 0.187522i	0.995595	−	0.0937610i	$-0.0298890\pi$
$230$	−6.00000	−0.395628
$231$	6.83772	0.449889
$232$	− 10.3246i	− 0.677840i
$233$	−9.48683	−0.621503	−0.310752	−	0.950491i	$-0.600581\pi$
$233$	−9.48683	−0.621503	−0.310752	+	0.950491i	$0.600581\pi$
$234$	0	0
$235$	3.00000	0.195698
$236$	− 10.3246i	− 0.672071i
$237$	−42.6491	−2.77036
$238$	−5.16228	−0.334621
$239$	11.1623i	0.722028i	0.932560	+	0.361014i	$0.117569\pi$
$239$	11.1623i	0.722028i	−0.932560	+	0.361014i	$0.882431\pi$
$240$	− 3.16228i	− 0.204124i
$241$	25.9737i	1.67311i	0.547882	+	0.836555i	$0.315434\pi$
$241$	25.9737i	1.67311i	−0.547882	+	0.836555i	$0.684566\pi$
$242$	6.32456i	0.406558i
$243$	22.1359	1.42002
$244$	7.48683	0.479295
$245$	− 6.00000i	− 0.383326i
$246$	32.6491	2.08163
$247$	0	0
$248$	7.16228	0.454805
$249$	30.0000i	1.90117i
$250$	−1.00000	−0.0632456
$251$	−12.4868	−0.788162	−0.394081	−	0.919076i	$-0.628937\pi$
$251$	−12.4868	−0.788162	−0.394081	+	0.919076i	$0.628937\pi$
$252$	− 7.00000i	− 0.440959i
$253$	12.9737i	0.815647i
$254$	− 3.32456i	− 0.208601i
$255$	− 16.3246i	− 1.02228i
$256$	1.00000	0.0625000
$257$	0	0		−	1.00000i	$-0.5\pi$
$257$	0	0			1.00000i	$0.5\pi$
$258$	− 6.32456i	− 0.393750i
$259$	−0.162278	−0.0100834
$260$	0	0
$261$	72.2719	4.47352
$262$	− 10.8114i	− 0.667930i
$263$	−11.6491	−0.718315	−0.359157	−	0.933277i	$-0.616936\pi$
$263$	−11.6491	−0.718315	−0.359157	+	0.933277i	$0.616936\pi$
$264$	−6.83772	−0.420833
$265$	2.16228i	0.132828i
$266$	1.83772i	0.112678i
$267$	23.1623i	1.41751i
$268$	2.32456i	0.141995i
$269$	2.51317	0.153230	0.0766152	−	0.997061i	$-0.475589\pi$
$269$	2.51317	0.153230	0.0766152	+	0.997061i	$0.475589\pi$
$270$	12.6491	0.769800
$271$	− 6.32456i	− 0.384189i	−0.981376	−	0.192095i	$-0.938472\pi$
$271$	− 6.32456i	− 0.384189i	0.981376	−	0.192095i	$-0.0615281\pi$
$272$	5.16228	0.313009
$273$	0	0
$274$	15.4868	0.935594
$275$	2.16228i	0.130390i
$276$	18.9737	1.14208
$277$	−1.51317	−0.0909174	−0.0454587	−	0.998966i	$-0.514475\pi$
$277$	−1.51317	−0.0909174	−0.0454587	+	0.998966i	$0.514475\pi$
$278$	− 12.1623i	− 0.729445i
$279$	50.1359i	3.00156i
$280$	− 1.00000i	− 0.0597614i
$281$	18.9737i	1.13187i	0.824448	+	0.565937i	$0.191485\pi$
$281$	18.9737i	1.13187i	−0.824448	+	0.565937i	$0.808515\pi$
$282$	−9.48683	−0.564933
$283$	22.0000	1.30776	0.653882	−	0.756596i	$-0.273139\pi$
$283$	22.0000	1.30776	0.653882	+	0.756596i	$0.273139\pi$
$284$	4.32456i	0.256615i
$285$	−5.81139	−0.344237
$286$	0	0
$287$	10.3246	0.609439
$288$	7.00000i	0.412479i
$289$	9.64911	0.567595
$290$	10.3246	0.606279
$291$	− 23.6754i	− 1.38788i
$292$	2.83772i	0.166065i
$293$	21.8377i	1.27577i	0.770130	+	0.637887i	$0.220191\pi$
$293$	21.8377i	1.27577i	−0.770130	+	0.637887i	$0.779809\pi$
$294$	18.9737i	1.10657i
$295$	10.3246	0.601119
$296$	0.162278	0.00943220
$297$	− 27.3509i	− 1.58706i
$298$	−16.3246	−0.945656
$299$	0	0
$300$	3.16228	0.182574
$301$	− 2.00000i	− 0.115278i
$302$	−4.83772	−0.278380
$303$	35.2982	2.02783
$304$	− 1.83772i	− 0.105401i
$305$	7.48683i	0.428695i
$306$	36.1359i	2.06575i
$307$	− 15.6754i	− 0.894645i	−0.894373	−	0.447322i	$-0.852378\pi$
$307$	− 15.6754i	− 0.894645i	0.894373	−	0.447322i	$-0.147622\pi$
$308$	−2.16228	−0.123207
$309$	−2.13594	−0.121510
$310$	7.16228i	0.406790i
$311$	3.48683	0.197720	0.0988601	−	0.995101i	$-0.468480\pi$
$311$	3.48683	0.197720	0.0988601	+	0.995101i	$0.468480\pi$
$312$	0	0
$313$	−8.32456	−0.470532	−0.235266	−	0.971931i	$-0.575596\pi$
$313$	−8.32456	−0.470532	−0.235266	+	0.971931i	$0.575596\pi$
$314$	− 12.1623i	− 0.686357i
$315$	7.00000	0.394405
$316$	13.4868	0.758694
$317$	12.4868i	0.701330i	0.936501	+	0.350665i	$0.114044\pi$
$317$	12.4868i	0.701330i	−0.936501	+	0.350665i	$0.885956\pi$
$318$	− 6.83772i	− 0.383440i
$319$	− 22.3246i	− 1.24994i
$320$	1.00000i	0.0559017i
$321$	−11.0263	−0.615430
$322$	6.00000	0.334367
$323$	− 9.48683i	− 0.527862i
$324$	−19.0000	−1.05556
$325$	0	0
$326$	17.4868	0.968506
$327$	33.6754i	1.86226i
$328$	−10.3246	−0.570078
$329$	−3.00000	−0.165395
$330$	− 6.83772i	− 0.376404i
$331$	− 24.6491i	− 1.35484i	−0.735598	−	0.677419i	$-0.763099\pi$
$331$	− 24.6491i	− 1.35484i	0.735598	−	0.677419i	$-0.236901\pi$
$332$	− 9.48683i	− 0.520658i
$333$	1.13594i	0.0622493i
$334$	13.3246	0.729087
$335$	−2.32456	−0.127004
$336$	3.16228i	0.172516i
$337$	−17.4868	−0.952568	−0.476284	−	0.879291i	$-0.658017\pi$
$337$	−17.4868	−0.952568	−0.476284	+	0.879291i	$0.658017\pi$
$338$	0	0
$339$	−27.3509	−1.48550
$340$	5.16228i	0.279964i
$341$	15.4868	0.838659
$342$	12.8641	0.695609
$343$	13.0000i	0.701934i
$344$	2.00000i	0.107833i
$345$	18.9737i	1.02151i
$346$	− 2.16228i	− 0.116245i
$347$	2.51317	0.134914	0.0674569	−	0.997722i	$-0.478511\pi$
$347$	2.51317	0.134914	0.0674569	+	0.997722i	$0.478511\pi$
$348$	−32.6491	−1.75018
$349$	− 17.4868i	− 0.936049i	−0.883716	−	0.468024i	$-0.844966\pi$
$349$	− 17.4868i	− 0.936049i	0.883716	−	0.468024i	$-0.155034\pi$
$350$	1.00000	0.0534522
$351$	0	0
$352$	2.16228	0.115250
$353$	5.16228i	0.274760i	0.990518	+	0.137380i	$0.0438682\pi$
$353$	5.16228i	0.274760i	−0.990518	+	0.137380i	$0.956132\pi$
$354$	−32.6491	−1.73528
$355$	−4.32456	−0.229524
$356$	− 7.32456i	− 0.388201i
$357$	16.3246i	0.863987i
$358$	− 22.3246i	− 1.17989i
$359$	6.00000i	0.316668i	0.987386	+	0.158334i	$0.0506123\pi$
$359$	6.00000i	0.316668i	−0.987386	+	0.158334i	$0.949388\pi$
$360$	−7.00000	−0.368932
$361$	15.6228	0.822251
$362$	− 9.81139i	− 0.515675i
$363$	20.0000	1.04973
$364$	0	0
$365$	−2.83772	−0.148533
$366$	− 23.6754i	− 1.23754i
$367$	4.64911	0.242682	0.121341	−	0.992611i	$-0.461281\pi$
$367$	4.64911	0.242682	0.121341	+	0.992611i	$0.461281\pi$
$368$	−6.00000	−0.312772
$369$	− 72.2719i	− 3.76232i
$370$	0.162278i	0.00843641i
$371$	− 2.16228i	− 0.112260i
$372$	− 22.6491i	− 1.17430i
$373$	−30.6491	−1.58695	−0.793475	−	0.608602i	$-0.791730\pi$
$373$	−30.6491	−1.58695	−0.793475	+	0.608602i	$0.791730\pi$
$374$	11.1623	0.577188
$375$	3.16228i	0.163299i
$376$	3.00000	0.154713
$377$	0	0
$378$	−12.6491	−0.650600
$379$	− 38.8114i	− 1.99361i	−0.0798917	−	0.996804i	$-0.525457\pi$
$379$	− 38.8114i	− 1.99361i	0.0798917	−	0.996804i	$-0.474543\pi$
$380$	1.83772	0.0942732
$381$	−10.5132	−0.538606
$382$	9.48683i	0.485389i
$383$	30.0000i	1.53293i	0.642287	+	0.766464i	$0.277986\pi$
$383$	30.0000i	1.53293i	−0.642287	+	0.766464i	$0.722014\pi$
$384$	− 3.16228i	− 0.161374i
$385$	− 2.16228i	− 0.110200i
$386$	−4.00000	−0.203595
$387$	−14.0000	−0.711660
$388$	7.48683i	0.380086i
$389$	−12.8377	−0.650898	−0.325449	−	0.945560i	$-0.605516\pi$
$389$	−12.8377	−0.650898	−0.325449	+	0.945560i	$0.605516\pi$
$390$	0	0
$391$	−30.9737	−1.56641
$392$	− 6.00000i	− 0.303046i
$393$	−34.1886	−1.72459
$394$	18.4868	0.931353
$395$	13.4868i	0.678596i
$396$	15.1359i	0.760610i
$397$	− 38.4868i	− 1.93160i	−0.259290	−	0.965799i	$-0.583489\pi$
$397$	− 38.4868i	− 1.93160i	0.259290	−	0.965799i	$-0.416511\pi$
$398$	1.35089i	0.0677140i
$399$	5.81139	0.290933
$400$	−1.00000	−0.0500000
$401$	− 21.9737i	− 1.09731i	−0.836048	−	0.548656i	$-0.815140\pi$
$401$	− 21.9737i	− 1.09731i	0.836048	−	0.548656i	$-0.184860\pi$
$402$	7.35089	0.366629
$403$	0	0
$404$	−11.1623	−0.555344
$405$	− 19.0000i	− 0.944118i
$406$	−10.3246	−0.512399
$407$	0.350889	0.0173929
$408$	− 16.3246i	− 0.808186i
$409$	− 21.6491i	− 1.07048i	−0.844700	−	0.535240i	$-0.820221\pi$
$409$	− 21.6491i	− 1.07048i	0.844700	−	0.535240i	$-0.179779\pi$
$410$	− 10.3246i	− 0.509893i
$411$	− 48.9737i	− 2.41569i
$412$	0.675445	0.0332768
$413$	−10.3246	−0.508038
$414$	− 42.0000i	− 2.06419i
$415$	9.48683	0.465690
$416$	0	0
$417$	−38.4605	−1.88342
$418$	− 3.97367i	− 0.194358i
$419$	14.6491	0.715656	0.357828	−	0.933788i	$-0.383517\pi$
$419$	14.6491	0.715656	0.357828	+	0.933788i	$0.383517\pi$
$420$	−3.16228	−0.154303
$421$	− 3.16228i	− 0.154120i	−0.997026	−	0.0770600i	$-0.975447\pi$
$421$	− 3.16228i	− 0.154120i	0.997026	−	0.0770600i	$-0.0245533\pi$
$422$	11.8377i	0.576251i
$423$	21.0000i	1.02105i
$424$	2.16228i	0.105009i
$425$	−5.16228	−0.250407
$426$	13.6754	0.662577
$427$	− 7.48683i	− 0.362313i
$428$	3.48683	0.168542
$429$	0	0
$430$	−2.00000	−0.0964486
$431$	15.4868i	0.745974i	0.927836	+	0.372987i	$0.121666\pi$
$431$	15.4868i	0.745974i	−0.927836	+	0.372987i	$0.878334\pi$
$432$	12.6491	0.608581
$433$	−6.32456	−0.303939	−0.151969	−	0.988385i	$-0.548562\pi$
$433$	−6.32456	−0.303939	−0.151969	+	0.988385i	$0.548562\pi$
$434$	− 7.16228i	− 0.343800i
$435$	− 32.6491i	− 1.56541i
$436$	− 10.6491i	− 0.510000i
$437$	11.0263i	0.527461i
$438$	8.97367	0.428778
$439$	−3.81139	−0.181908	−0.0909538	−	0.995855i	$-0.528992\pi$
$439$	−3.81139	−0.181908	−0.0909538	+	0.995855i	$0.528992\pi$
$440$	2.16228i	0.103083i
$441$	42.0000	2.00000
$442$	0	0
$443$	26.6491	1.26614	0.633069	−	0.774096i	$-0.281795\pi$
$443$	26.6491	1.26614	0.633069	+	0.774096i	$0.281795\pi$
$444$	− 0.513167i	− 0.0243538i
$445$	7.32456	0.347217
$446$	−3.32456	−0.157422
$447$	51.6228i	2.44167i
$448$	− 1.00000i	− 0.0472456i
$449$	2.02633i	0.0956286i	0.998856	+	0.0478143i	$0.0152256\pi$
$449$	2.02633i	0.0956286i	−0.998856	+	0.0478143i	$0.984774\pi$
$450$	− 7.00000i	− 0.329983i
$451$	−22.3246	−1.05122
$452$	8.64911	0.406820
$453$	15.2982i	0.718773i
$454$	−10.3246	−0.484555
$455$	0	0
$456$	−5.81139	−0.272143
$457$	24.4605i	1.14421i	0.820179	+	0.572107i	$0.193874\pi$
$457$	24.4605i	1.14421i	−0.820179	+	0.572107i	$0.806126\pi$
$458$	2.83772	0.132598
$459$	65.2982	3.04786
$460$	− 6.00000i	− 0.279751i
$461$	21.4868i	1.00074i	0.865811	+	0.500371i	$0.166803\pi$
$461$	21.4868i	1.00074i	−0.865811	+	0.500371i	$0.833197\pi$
$462$	6.83772i	0.318120i
$463$	− 12.3246i	− 0.572771i	−0.958115	−	0.286385i	$-0.907546\pi$
$463$	− 12.3246i	− 0.572771i	0.958115	−	0.286385i	$-0.0924537\pi$
$464$	10.3246	0.479305
$465$	22.6491	1.05033
$466$	− 9.48683i	− 0.439469i
$467$	−32.6491	−1.51082	−0.755410	−	0.655252i	$-0.772562\pi$
$467$	−32.6491	−1.51082	−0.755410	+	0.655252i	$0.772562\pi$
$468$	0	0
$469$	2.32456	0.107338
$470$	3.00000i	0.138380i
$471$	−38.4605	−1.77217
$472$	10.3246	0.475226
$473$	4.32456i	0.198843i
$474$	− 42.6491i	− 1.95894i
$475$	1.83772i	0.0843205i
$476$	− 5.16228i	− 0.236613i
$477$	−15.1359	−0.693027
$478$	−11.1623	−0.510551
$479$	12.1359i	0.554505i	0.960797	+	0.277253i	$0.0894239\pi$
$479$	12.1359i	0.554505i	−0.960797	+	0.277253i	$0.910576\pi$
$480$	3.16228	0.144338
$481$	0	0
$482$	−25.9737	−1.18307
$483$	− 18.9737i	− 0.863332i
$484$	−6.32456	−0.287480
$485$	−7.48683	−0.339960
$486$	22.1359i	1.00411i
$487$	− 1.00000i	− 0.0453143i	−0.999743	−	0.0226572i	$-0.992787\pi$
$487$	− 1.00000i	− 0.0453143i	0.999743	−	0.0226572i	$-0.00721262\pi$
$488$	7.48683i	0.338913i
$489$	− 55.2982i	− 2.50067i
$490$	6.00000	0.271052
$491$	−17.5132	−0.790358	−0.395179	−	0.918604i	$-0.629317\pi$
$491$	−17.5132	−0.790358	−0.395179	+	0.918604i	$0.629317\pi$
$492$	32.6491i	1.47194i
$493$	53.2982	2.40043
$494$	0	0
$495$	−15.1359	−0.680310
$496$	7.16228i	0.321596i
$497$	4.32456	0.193983
$498$	−30.0000	−1.34433
$499$	− 22.0000i	− 0.984855i	−0.870353	−	0.492428i	$-0.836110\pi$
$499$	− 22.0000i	− 0.984855i	0.870353	−	0.492428i	$-0.163890\pi$
$500$	− 1.00000i	− 0.0447214i
$501$	− 42.1359i	− 1.88250i
$502$	− 12.4868i	− 0.557315i
$503$	−14.2982	−0.637526	−0.318763	−	0.947834i	$-0.603267\pi$
$503$	−14.2982	−0.637526	−0.318763	+	0.947834i	$0.603267\pi$
$504$	7.00000	0.311805
$505$	− 11.1623i	− 0.496715i
$506$	−12.9737	−0.576750
$507$	0	0
$508$	3.32456	0.147503
$509$	− 0.973666i	− 0.0431570i	−0.999767	−	0.0215785i	$-0.993131\pi$
$509$	− 0.973666i	− 0.0431570i	0.999767	−	0.0215785i	$-0.00686918\pi$
$510$	16.3246	0.722863
$511$	2.83772	0.125533
$512$	1.00000i	0.0441942i
$513$	− 23.2456i	− 1.02632i
$514$	0	0
$515$	0.675445i	0.0297636i
$516$	6.32456	0.278423
$517$	6.48683	0.285291
$518$	− 0.162278i	− 0.00713007i
$519$	−6.83772	−0.300143
$520$	0	0
$521$	22.6754	0.993429	0.496715	−	0.867914i	$-0.334540\pi$
$521$	22.6754	0.993429	0.496715	+	0.867914i	$0.334540\pi$
$522$	72.2719i	3.16325i
$523$	−39.2982	−1.71839	−0.859196	−	0.511647i	$-0.829035\pi$
$523$	−39.2982	−1.71839	−0.859196	+	0.511647i	$0.829035\pi$
$524$	10.8114	0.472298
$525$	− 3.16228i	− 0.138013i
$526$	− 11.6491i	− 0.507925i
$527$	36.9737i	1.61060i
$528$	− 6.83772i	− 0.297574i
$529$	13.0000	0.565217
$530$	−2.16228	−0.0939233
$531$	72.2719i	3.13633i
$532$	−1.83772	−0.0796754
$533$	0	0
$534$	−23.1623	−1.00233
$535$	3.48683i	0.150749i
$536$	−2.32456	−0.100405
$537$	−70.5964	−3.04646
$538$	2.51317i	0.108350i
$539$	− 12.9737i	− 0.558815i
$540$	12.6491i	0.544331i
$541$	− 26.8377i	− 1.15384i	−0.816799	−	0.576922i	$-0.804254\pi$
$541$	− 26.8377i	− 1.15384i	0.816799	−	0.576922i	$-0.195746\pi$
$542$	6.32456	0.271663
$543$	−31.0263	−1.33147
$544$	5.16228i	0.221331i
$545$	10.6491	0.456158
$546$	0	0
$547$	−6.64911	−0.284295	−0.142148	−	0.989845i	$-0.545401\pi$
$547$	−6.64911	−0.284295	−0.142148	+	0.989845i	$0.545401\pi$
$548$	15.4868i	0.661565i
$549$	−52.4078	−2.23671
$550$	−2.16228	−0.0921998
$551$	− 18.9737i	− 0.808305i
$552$	18.9737i	0.807573i
$553$	− 13.4868i	− 0.573518i
$554$	− 1.51317i	− 0.0642883i
$555$	0.513167	0.0217827
$556$	12.1623	0.515795
$557$	− 15.8377i	− 0.671066i	−0.942028	−	0.335533i	$-0.891084\pi$
$557$	− 15.8377i	− 0.671066i	0.942028	−	0.335533i	$-0.108916\pi$
$558$	−50.1359	−2.12242
$559$	0	0
$560$	1.00000	0.0422577
$561$	− 35.2982i	− 1.49029i
$562$	−18.9737	−0.800356
$563$	24.0000	1.01148	0.505740	−	0.862686i	$-0.331220\pi$
$563$	24.0000	1.01148	0.505740	+	0.862686i	$0.331220\pi$
$564$	− 9.48683i	− 0.399468i
$565$	8.64911i	0.363871i
$566$	22.0000i	0.924729i
$567$	19.0000i	0.797925i
$568$	−4.32456	−0.181454
$569$	33.9737	1.42425	0.712125	−	0.702053i	$-0.247733\pi$
$569$	33.9737	1.42425	0.712125	+	0.702053i	$0.247733\pi$
$570$	− 5.81139i	− 0.243412i
$571$	25.1359	1.05191	0.525953	−	0.850513i	$-0.323709\pi$
$571$	25.1359	1.05191	0.525953	+	0.850513i	$0.323709\pi$
$572$	0	0
$573$	30.0000	1.25327
$574$	10.3246i	0.430939i
$575$	6.00000	0.250217
$576$	−7.00000	−0.291667
$577$	7.16228i	0.298170i	0.988824	+	0.149085i	$0.0476327\pi$
$577$	7.16228i	0.298170i	−0.988824	+	0.149085i	$0.952367\pi$
$578$	9.64911i	0.401350i
$579$	12.6491i	0.525679i
$580$	10.3246i	0.428704i
$581$	−9.48683	−0.393580
$582$	23.6754	0.981379
$583$	4.67544i	0.193637i
$584$	−2.83772	−0.117426
$585$	0	0
$586$	−21.8377	−0.902108
$587$	12.9737i	0.535481i	0.963491	+	0.267740i	$0.0862769\pi$
$587$	12.9737i	0.535481i	−0.963491	+	0.267740i	$0.913723\pi$
$588$	−18.9737	−0.782461
$589$	13.1623	0.542342
$590$	10.3246i	0.425055i
$591$	− 58.4605i	− 2.40474i
$592$	0.162278i	0.00666957i
$593$	3.35089i	0.137605i	0.997630	+	0.0688023i	$0.0219178\pi$
$593$	3.35089i	0.137605i	−0.997630	+	0.0688023i	$0.978082\pi$
$594$	27.3509	1.12222
$595$	5.16228	0.211633
$596$	− 16.3246i	− 0.668680i
$597$	4.27189	0.174837
$598$	0	0
$599$	43.9473	1.79564	0.897820	−	0.440363i	$-0.145150\pi$
$599$	43.9473	1.79564	0.897820	+	0.440363i	$0.145150\pi$
$600$	3.16228i	0.129099i
$601$	34.2982	1.39905	0.699527	−	0.714606i	$-0.253394\pi$
$601$	34.2982	1.39905	0.699527	+	0.714606i	$0.253394\pi$
$602$	2.00000	0.0815139
$603$	− 16.2719i	− 0.662642i
$604$	− 4.83772i	− 0.196844i
$605$	− 6.32456i	− 0.257130i
$606$	35.2982i	1.43389i
$607$	5.00000	0.202944	0.101472	−	0.994838i	$-0.467645\pi$
$607$	5.00000	0.202944	0.101472	+	0.994838i	$0.467645\pi$
$608$	1.83772	0.0745295
$609$	32.6491i	1.32301i
$610$	−7.48683	−0.303133
$611$	0	0
$612$	−36.1359	−1.46071
$613$	20.4868i	0.827455i	0.910401	+	0.413728i	$0.135773\pi$
$613$	20.4868i	0.827455i	−0.910401	+	0.413728i	$0.864227\pi$
$614$	15.6754	0.632609
$615$	−32.6491	−1.31654
$616$	− 2.16228i	− 0.0871206i
$617$	− 31.9473i	− 1.28615i	−0.765803	−	0.643076i	$-0.777658\pi$
$617$	− 31.9473i	− 1.28615i	0.765803	−	0.643076i	$-0.222342\pi$
$618$	− 2.13594i	− 0.0859203i
$619$	29.4605i	1.18412i	0.805895	+	0.592059i	$0.201685\pi$
$619$	29.4605i	1.18412i	−0.805895	+	0.592059i	$0.798315\pi$
$620$	−7.16228	−0.287644
$621$	−75.8947	−3.04555
$622$	3.48683i	0.139809i
$623$	−7.32456	−0.293452
$624$	0	0
$625$	1.00000	0.0400000
$626$	− 8.32456i	− 0.332716i
$627$	−12.5658	−0.501831
$628$	12.1623	0.485328
$629$	0.837722i	0.0334022i
$630$	7.00000i	0.278887i
$631$	14.3246i	0.570252i	0.958490	+	0.285126i	$0.0920354\pi$
$631$	14.3246i	0.570252i	−0.958490	+	0.285126i	$0.907965\pi$
$632$	13.4868i	0.536477i
$633$	37.4342	1.48787
$634$	−12.4868	−0.495915
$635$	3.32456i	0.131931i
$636$	6.83772	0.271133
$637$	0	0
$638$	22.3246	0.883838
$639$	− 30.2719i	− 1.19754i
$640$	−1.00000	−0.0395285
$641$	9.97367	0.393936	0.196968	−	0.980410i	$-0.436890\pi$
$641$	9.97367	0.393936	0.196968	+	0.980410i	$0.436890\pi$
$642$	− 11.0263i	− 0.435175i
$643$	12.4605i	0.491394i	0.969347	+	0.245697i	$0.0790168\pi$
$643$	12.4605i	0.491394i	−0.969347	+	0.245697i	$0.920983\pi$
$644$	6.00000i	0.236433i
$645$	6.32456i	0.249029i
$646$	9.48683	0.373254
$647$	15.9737	0.627990	0.313995	−	0.949425i	$-0.398333\pi$
$647$	15.9737	0.627990	0.313995	+	0.949425i	$0.398333\pi$
$648$	− 19.0000i	− 0.746390i
$649$	22.3246	0.876315
$650$	0	0
$651$	−22.6491	−0.887689
$652$	17.4868i	0.684837i
$653$	17.5132	0.685343	0.342672	−	0.939455i	$-0.388668\pi$
$653$	17.5132	0.685343	0.342672	+	0.939455i	$0.388668\pi$
$654$	−33.6754	−1.31681
$655$	10.8114i	0.422436i
$656$	− 10.3246i	− 0.403106i
$657$	− 19.8641i	− 0.774971i
$658$	− 3.00000i	− 0.116952i
$659$	23.2982	0.907570	0.453785	−	0.891111i	$-0.350073\pi$
$659$	23.2982	0.907570	0.453785	+	0.891111i	$0.350073\pi$
$660$	6.83772	0.266158
$661$	10.8377i	0.421539i	0.977536	+	0.210769i	$0.0675969\pi$
$661$	10.8377i	0.421539i	−0.977536	+	0.210769i	$0.932403\pi$
$662$	24.6491	0.958015
$663$	0	0
$664$	9.48683	0.368161
$665$	− 1.83772i	− 0.0712638i
$666$	−1.13594	−0.0440169
$667$	−61.9473	−2.39861
$668$	13.3246i	0.515543i
$669$	10.5132i	0.406463i
$670$	− 2.32456i	− 0.0898054i
$671$	16.1886i	0.624954i
$672$	−3.16228	−0.121988
$673$	−14.9737	−0.577192	−0.288596	−	0.957451i	$-0.593189\pi$
$673$	−14.9737	−0.577192	−0.288596	+	0.957451i	$0.593189\pi$
$674$	− 17.4868i	− 0.673568i
$675$	−12.6491	−0.486864
$676$	0	0
$677$	−12.9737	−0.498618	−0.249309	−	0.968424i	$-0.580204\pi$
$677$	−12.9737	−0.498618	−0.249309	+	0.968424i	$0.580204\pi$
$678$	− 27.3509i	− 1.05040i
$679$	7.48683	0.287318
$680$	−5.16228	−0.197964
$681$	32.6491i	1.25112i
$682$	15.4868i	0.593021i
$683$	− 19.6754i	− 0.752860i	−0.926445	−	0.376430i	$-0.877152\pi$
$683$	− 19.6754i	− 0.752860i	0.926445	−	0.376430i	$-0.122848\pi$
$684$	12.8641i	0.491869i
$685$	−15.4868	−0.591721
$686$	−13.0000	−0.496342
$687$	− 8.97367i	− 0.342367i
$688$	−2.00000	−0.0762493
$689$	0	0
$690$	−18.9737	−0.722315
$691$	− 37.1359i	− 1.41272i	−0.707854	−	0.706359i	$-0.750336\pi$
$691$	− 37.1359i	− 1.41272i	0.707854	−	0.706359i	$-0.249664\pi$
$692$	2.16228	0.0821975
$693$	15.1359	0.574967
$694$	2.51317i	0.0953985i
$695$	12.1623i	0.461341i
$696$	− 32.6491i	− 1.23756i
$697$	− 53.2982i	− 2.01881i
$698$	17.4868	0.661886
$699$	−30.0000	−1.13470
$700$	1.00000i	0.0377964i
$701$	23.1623	0.874827	0.437414	−	0.899260i	$-0.355895\pi$
$701$	23.1623	0.874827	0.437414	+	0.899260i	$0.355895\pi$
$702$	0	0
$703$	0.298221	0.0112476
$704$	2.16228i	0.0814939i
$705$	9.48683	0.357295
$706$	−5.16228	−0.194285
$707$	11.1623i	0.419801i
$708$	− 32.6491i	− 1.22703i
$709$	− 41.4868i	− 1.55807i	−0.626980	−	0.779035i	$-0.715709\pi$
$709$	− 41.4868i	− 1.55807i	0.626980	−	0.779035i	$-0.284291\pi$
$710$	− 4.32456i	− 0.162298i
$711$	−94.4078	−3.54057
$712$	7.32456	0.274499
$713$	− 42.9737i	− 1.60938i
$714$	−16.3246	−0.610931
$715$	0	0
$716$	22.3246	0.834308
$717$	35.2982i	1.31824i
$718$	−6.00000	−0.223918
$719$	−12.0000	−0.447524	−0.223762	−	0.974644i	$-0.571834\pi$
$719$	−12.0000	−0.447524	−0.223762	+	0.974644i	$0.571834\pi$
$720$	− 7.00000i	− 0.260875i
$721$	− 0.675445i	− 0.0251549i
$722$	15.6228i	0.581420i
$723$	82.1359i	3.05467i
$724$	9.81139	0.364637
$725$	−10.3246	−0.383444
$726$	20.0000i	0.742270i
$727$	20.6754	0.766810	0.383405	−	0.923580i	$-0.374751\pi$
$727$	20.6754	0.766810	0.383405	+	0.923580i	$0.374751\pi$
$728$	0	0
$729$	13.0000	0.481481
$730$	− 2.83772i	− 0.105029i
$731$	−10.3246	−0.381867
$732$	23.6754	0.875070
$733$	8.48683i	0.313468i	0.987641	+	0.156734i	$0.0500966\pi$
$733$	8.48683i	0.313468i	−0.987641	+	0.156734i	$0.949903\pi$
$734$	4.64911i	0.171602i
$735$	− 18.9737i	− 0.699854i
$736$	− 6.00000i	− 0.221163i
$737$	−5.02633	−0.185147
$738$	72.2719	2.66036
$739$	44.8114i	1.64841i	0.566289	+	0.824207i	$0.308379\pi$
$739$	44.8114i	1.64841i	−0.566289	+	0.824207i	$0.691621\pi$
$740$	−0.162278	−0.00596545
$741$	0	0
$742$	2.16228	0.0793797
$743$	− 18.0000i	− 0.660356i	−0.943919	−	0.330178i	$-0.892891\pi$
$743$	− 18.0000i	− 0.660356i	0.943919	−	0.330178i	$-0.107109\pi$
$744$	22.6491	0.830357
$745$	16.3246	0.598085
$746$	− 30.6491i	− 1.12214i
$747$	66.4078i	2.42974i
$748$	11.1623i	0.408133i
$749$	− 3.48683i	− 0.127406i
$750$	−3.16228	−0.115470
$751$	34.9737	1.27621	0.638104	−	0.769951i	$-0.279719\pi$
$751$	34.9737	1.27621	0.638104	+	0.769951i	$0.279719\pi$
$752$	3.00000i	0.109399i
$753$	−39.4868	−1.43898
$754$	0	0
$755$	4.83772	0.176063
$756$	− 12.6491i	− 0.460044i
$757$	34.1623	1.24165	0.620825	−	0.783950i	$-0.286798\pi$
$757$	34.1623	1.24165	0.620825	+	0.783950i	$0.286798\pi$
$758$	38.8114	1.40969
$759$	41.0263i	1.48916i
$760$	1.83772i	0.0666612i
$761$	8.29822i	0.300810i	0.988624	+	0.150405i	$0.0480578\pi$
$761$	8.29822i	0.300810i	−0.988624	+	0.150405i	$0.951942\pi$
$762$	− 10.5132i	− 0.380852i
$763$	−10.6491	−0.385524
$764$	−9.48683	−0.343222
$765$	− 36.1359i	− 1.30650i
$766$	−30.0000	−1.08394
$767$	0	0
$768$	3.16228	0.114109
$769$	6.32456i	0.228069i	0.993477	+	0.114035i	$0.0363775\pi$
$769$	6.32456i	0.228069i	−0.993477	+	0.114035i	$0.963623\pi$
$770$	2.16228	0.0779231
$771$	0	0
$772$	− 4.00000i	− 0.143963i
$773$	47.7851i	1.71871i	0.511380	+	0.859354i	$0.329134\pi$
$773$	47.7851i	1.71871i	−0.511380	+	0.859354i	$0.670866\pi$
$774$	− 14.0000i	− 0.503220i
$775$	− 7.16228i	− 0.257277i
$776$	−7.48683	−0.268762
$777$	−0.513167	−0.0184098
$778$	− 12.8377i	− 0.460255i
$779$	−18.9737	−0.679802
$780$	0	0
$781$	−9.35089	−0.334601
$782$	− 30.9737i	− 1.10762i
$783$	130.596	4.66714
$784$	6.00000	0.214286
$785$	12.1623i	0.434090i
$786$	− 34.1886i	− 1.21947i
$787$	46.1359i	1.64457i	0.569077	+	0.822284i	$0.307301\pi$
$787$	46.1359i	1.64457i	−0.569077	+	0.822284i	$0.692699\pi$
$788$	18.4868i	0.658566i
$789$	−36.8377	−1.31146
$790$	−13.4868	−0.479840
$791$	− 8.64911i	− 0.307527i
$792$	−15.1359	−0.537832
$793$	0	0
$794$	38.4868	1.36585
$795$	6.83772i	0.242509i
$796$	−1.35089	−0.0478810
$797$	13.6754	0.484409	0.242205	−	0.970225i	$-0.422129\pi$
$797$	13.6754	0.484409	0.242205	+	0.970225i	$0.422129\pi$
$798$	5.81139i	0.205721i
$799$	15.4868i	0.547885i
$800$	− 1.00000i	− 0.0353553i
$801$	51.2719i	1.81160i
$802$	21.9737	0.775917
$803$	−6.13594	−0.216533
$804$	7.35089i	0.259246i
$805$	−6.00000	−0.211472
$806$	0	0
$807$	7.94733	0.279759
$808$	− 11.1623i	− 0.392688i
$809$	10.3246	0.362992	0.181496	−	0.983392i	$-0.441906\pi$
$809$	10.3246	0.362992	0.181496	+	0.983392i	$0.441906\pi$
$810$	19.0000	0.667592
$811$	− 6.16228i	− 0.216387i	−0.994130	−	0.108193i	$-0.965493\pi$
$811$	− 6.16228i	− 0.216387i	0.994130	−	0.108193i	$-0.0345066\pi$
$812$	− 10.3246i	− 0.362321i
$813$	− 20.0000i	− 0.701431i
$814$	0.350889i	0.0122987i
$815$	−17.4868	−0.612537
$816$	16.3246	0.571474
$817$	3.67544i	0.128588i
$818$	21.6491	0.756943
$819$	0	0
$820$	10.3246	0.360549
$821$	− 5.16228i	− 0.180165i	−0.995934	−	0.0900824i	$-0.971287\pi$
$821$	− 5.16228i	− 0.180165i	0.995934	−	0.0900824i	$-0.0287130\pi$
$822$	48.9737	1.70815
$823$	−47.0000	−1.63832	−0.819159	−	0.573567i	$-0.805559\pi$
$823$	−47.0000	−1.63832	−0.819159	+	0.573567i	$0.805559\pi$
$824$	0.675445i	0.0235302i
$825$	6.83772i	0.238059i
$826$	− 10.3246i	− 0.359237i
$827$	51.4868i	1.79037i	0.445692	+	0.895186i	$0.352958\pi$
$827$	51.4868i	1.79037i	−0.445692	+	0.895186i	$0.647042\pi$
$828$	42.0000	1.45960
$829$	−32.0000	−1.11141	−0.555703	−	0.831381i	$-0.687551\pi$
$829$	−32.0000	−1.11141	−0.555703	+	0.831381i	$0.687551\pi$
$830$	9.48683i	0.329293i
$831$	−4.78505	−0.165992
$832$	0	0
$833$	30.9737	1.07317
$834$	− 38.4605i	− 1.33178i
$835$	−13.3246	−0.461115
$836$	3.97367	0.137432
$837$	90.5964i	3.13147i
$838$	14.6491i	0.506045i
$839$	23.1623i	0.799651i	0.916591	+	0.399825i	$0.130929\pi$
$839$	23.1623i	0.799651i	−0.916591	+	0.399825i	$0.869071\pi$
$840$	− 3.16228i	− 0.109109i
$841$	77.5964	2.67574
$842$	3.16228	0.108979
$843$	60.0000i	2.06651i
$844$	−11.8377	−0.407471
$845$	0	0
$846$	−21.0000	−0.721995
$847$	6.32456i	0.217314i
$848$	−2.16228	−0.0742529
$849$	69.5701	2.38764
$850$	− 5.16228i	− 0.177065i
$851$	− 0.973666i	− 0.0333768i
$852$	13.6754i	0.468513i
$853$	− 56.2719i	− 1.92671i	−0.268225	−	0.963356i	$-0.586437\pi$
$853$	− 56.2719i	− 1.92671i	0.268225	−	0.963356i	$-0.413563\pi$
$854$	7.48683	0.256194
$855$	−12.8641	−0.439941
$856$	3.48683i	0.119177i
$857$	−30.1359	−1.02942	−0.514712	−	0.857363i	$-0.672101\pi$
$857$	−30.1359	−1.02942	−0.514712	+	0.857363i	$0.672101\pi$
$858$	0	0
$859$	47.1359	1.60826	0.804129	−	0.594455i	$-0.202632\pi$
$859$	47.1359	1.60826	0.804129	+	0.594455i	$0.202632\pi$
$860$	− 2.00000i	− 0.0681994i
$861$	32.6491	1.11268
$862$	−15.4868	−0.527484
$863$	− 18.0000i	− 0.612727i	−0.951915	−	0.306364i	$-0.900888\pi$
$863$	− 18.0000i	− 0.612727i	0.951915	−	0.306364i	$-0.0991123\pi$
$864$	12.6491i	0.430331i
$865$	2.16228i	0.0735196i
$866$	− 6.32456i	− 0.214917i
$867$	30.5132	1.03628
$868$	7.16228	0.243104
$869$	29.1623i	0.989263i
$870$	32.6491	1.10691
$871$	0	0
$872$	10.6491	0.360624
$873$	− 52.4078i	− 1.77374i
$874$	−11.0263	−0.372971
$875$	−1.00000	−0.0338062
$876$	8.97367i	0.303192i
$877$	− 23.6754i	− 0.799463i	−0.916632	−	0.399731i	$-0.869103\pi$
$877$	− 23.6754i	− 0.799463i	0.916632	−	0.399731i	$-0.130897\pi$
$878$	− 3.81139i	− 0.128628i
$879$	69.0569i	2.32923i
$880$	−2.16228	−0.0728904
$881$	57.9737	1.95318	0.976591	−	0.215104i	$-0.0690089\pi$
$881$	57.9737	1.95318	0.976591	+	0.215104i	$0.0690089\pi$
$882$	42.0000i	1.41421i
$883$	−35.4868	−1.19423	−0.597114	−	0.802157i	$-0.703686\pi$
$883$	−35.4868	−1.19423	−0.597114	+	0.802157i	$0.703686\pi$
$884$	0	0
$885$	32.6491	1.09749
$886$	26.6491i	0.895294i
$887$	36.6228	1.22967	0.614836	−	0.788655i	$-0.289222\pi$
$887$	36.6228	1.22967	0.614836	+	0.788655i	$0.289222\pi$
$888$	0.513167	0.0172208
$889$	− 3.32456i	− 0.111502i
$890$	7.32456i	0.245520i
$891$	− 41.0833i	− 1.37634i
$892$	− 3.32456i	− 0.111314i
$893$	5.51317	0.184491
$894$	−51.6228	−1.72652
$895$	22.3246i	0.746228i
$896$	1.00000	0.0334077
$897$	0	0
$898$	−2.02633	−0.0676196
$899$	73.9473i	2.46628i
$900$	7.00000	0.233333
$901$	−11.1623	−0.371869
$902$	− 22.3246i	− 0.743326i
$903$	− 6.32456i	− 0.210468i
$904$	8.64911i	0.287665i
$905$	9.81139i	0.326142i
$906$	−15.2982	−0.508249
$907$	16.1359	0.535785	0.267893	−	0.963449i	$-0.413673\pi$
$907$	16.1359	0.535785	0.267893	+	0.963449i	$0.413673\pi$
$908$	− 10.3246i	− 0.342632i
$909$	78.1359	2.59161
$910$	0	0
$911$	−17.2982	−0.573116	−0.286558	−	0.958063i	$-0.592511\pi$
$911$	−17.2982	−0.573116	−0.286558	+	0.958063i	$0.592511\pi$
$912$	− 5.81139i	− 0.192434i
$913$	20.5132	0.678887
$914$	−24.4605	−0.809081
$915$	23.6754i	0.782686i
$916$	2.83772i	0.0937610i
$917$	− 10.8114i	− 0.357023i
$918$	65.2982i	2.15516i
$919$	−11.6754	−0.385137	−0.192569	−	0.981283i	$-0.561682\pi$
$919$	−11.6754	−0.385137	−0.192569	+	0.981283i	$0.561682\pi$
$920$	6.00000	0.197814
$921$	− 49.5701i	− 1.63339i
$922$	−21.4868	−0.707631
$923$	0	0
$924$	−6.83772	−0.224945
$925$	− 0.162278i	− 0.00533566i
$926$	12.3246	0.405010
$927$	−4.72811	−0.155292
$928$	10.3246i	0.338920i
$929$	54.0000i	1.77168i	0.463988	+	0.885841i	$0.346418\pi$
$929$	54.0000i	1.77168i	−0.463988	+	0.885841i	$0.653582\pi$
$930$	22.6491i	0.742694i
$931$	− 11.0263i	− 0.361374i
$932$	9.48683	0.310752
$933$	11.0263	0.360986
$934$	− 32.6491i	− 1.06831i
$935$	−11.1623	−0.365046
$936$	0	0
$937$	27.6754	0.904117	0.452059	−	0.891988i	$-0.350690\pi$
$937$	27.6754	0.904117	0.452059	+	0.891988i	$0.350690\pi$
$938$	2.32456i	0.0758994i
$939$	−26.3246	−0.859069
$940$	−3.00000	−0.0978492
$941$	22.3246i	0.727760i	0.931446	+	0.363880i	$0.118548\pi$
$941$	22.3246i	0.727760i	−0.931446	+	0.363880i	$0.881452\pi$
$942$	− 38.4605i	− 1.25311i
$943$	61.9473i	2.01728i
$944$	10.3246i	0.336036i
$945$	12.6491	0.411476
$946$	−4.32456	−0.140603
$947$	1.81139i	0.0588622i	0.999567	+	0.0294311i	$0.00936956\pi$
$947$	1.81139i	0.0588622i	−0.999567	+	0.0294311i	$0.990630\pi$
$948$	42.6491	1.38518
$949$	0	0
$950$	−1.83772	−0.0596236
$951$	39.4868i	1.28045i
$952$	5.16228	0.167310
$953$	2.51317	0.0814095	0.0407047	−	0.999171i	$-0.487040\pi$
$953$	2.51317	0.0814095	0.0407047	+	0.999171i	$0.487040\pi$
$954$	− 15.1359i	− 0.490044i
$955$	− 9.48683i	− 0.306987i
$956$	− 11.1623i	− 0.361014i
$957$	− 70.5964i	− 2.28206i
$958$	−12.1359	−0.392095
$959$	15.4868	0.500096
$960$	3.16228i	0.102062i
$961$	−20.2982	−0.654781
$962$	0	0
$963$	−24.4078	−0.786531
$964$	− 25.9737i	− 0.836555i
$965$	4.00000	0.128765
$966$	18.9737	0.610468
$967$	− 16.6228i	− 0.534552i	−0.963620	−	0.267276i	$-0.913876\pi$
$967$	− 16.6228i	− 0.534552i	0.963620	−	0.267276i	$-0.0861236\pi$
$968$	− 6.32456i	− 0.203279i
$969$	− 30.0000i	− 0.963739i
$970$	− 7.48683i	− 0.240388i
$971$	59.7851	1.91859	0.959297	−	0.282400i	$-0.0911304\pi$
$971$	59.7851	1.91859	0.959297	+	0.282400i	$0.0911304\pi$
$972$	−22.1359	−0.710011
$973$	− 12.1623i	− 0.389905i
$974$	1.00000	0.0320421
$975$	0	0
$976$	−7.48683	−0.239648
$977$	18.9737i	0.607021i	0.952828	+	0.303511i	$0.0981588\pi$
$977$	18.9737i	0.607021i	−0.952828	+	0.303511i	$0.901841\pi$
$978$	55.2982	1.76824
$979$	15.8377	0.506176
$980$	6.00000i	0.191663i
$981$	74.5438i	2.38000i
$982$	− 17.5132i	− 0.558868i
$983$	− 30.3509i	− 0.968043i	−0.875056	−	0.484022i	$-0.839176\pi$
$983$	− 30.3509i	− 0.968043i	0.875056	−	0.484022i	$-0.160824\pi$
$984$	−32.6491	−1.04082
$985$	−18.4868	−0.589039
$986$	53.2982i	1.69736i
$987$	−9.48683	−0.301969
$988$	0	0
$989$	12.0000	0.381578
$990$	− 15.1359i	− 0.481052i
$991$	−39.2982	−1.24835	−0.624175	−	0.781285i	$-0.714565\pi$
$991$	−39.2982	−1.24835	−0.624175	+	0.781285i	$0.714565\pi$
$992$	−7.16228	−0.227403
$993$	− 77.9473i	− 2.47358i
$994$	4.32456i	0.137167i
$995$	− 1.35089i	− 0.0428261i
$996$	− 30.0000i	− 0.950586i
$997$	11.8377	0.374904	0.187452	−	0.982274i	$-0.439977\pi$
$997$	11.8377	0.374904	0.187452	+	0.982274i	$0.439977\pi$
$998$	22.0000	0.696398
$999$	2.05267i	0.0649435i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1690.2.d.g.1351.4		4
13.2	odd	12		130.2.e.c.61.1	✓	4
13.3	even	3		1690.2.l.k.1161.1		8
13.4	even	6		1690.2.l.k.361.1		8
13.5	odd	4		1690.2.a.n.1.2		2
13.6	odd	12		130.2.e.c.81.1	yes	4
13.7	odd	12		1690.2.e.m.991.1		4
13.8	odd	4		1690.2.a.k.1.2		2
13.9	even	3		1690.2.l.k.361.3		8
13.10	even	6		1690.2.l.k.1161.3		8
13.11	odd	12		1690.2.e.m.191.1		4
13.12	even	2	inner	1690.2.d.g.1351.2		4
39.2	even	12		1170.2.i.q.451.2		4
39.32	even	12		1170.2.i.q.991.2		4
52.15	even	12		1040.2.q.m.321.2		4
52.19	even	12		1040.2.q.m.81.2		4
65.2	even	12		650.2.o.g.399.2		8
65.19	odd	12		650.2.e.h.601.2		4
65.28	even	12		650.2.o.g.399.3		8
65.32	even	12		650.2.o.g.549.3		8
65.34	odd	4		8450.2.a.bj.1.1		2
65.44	odd	4		8450.2.a.bc.1.1		2
65.54	odd	12		650.2.e.h.451.2		4
65.58	even	12		650.2.o.g.549.2		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
130.2.e.c.61.1	✓	4	13.2	odd	12
130.2.e.c.81.1	yes	4	13.6	odd	12
650.2.e.h.451.2		4	65.54	odd	12
650.2.e.h.601.2		4	65.19	odd	12
650.2.o.g.399.2		8	65.2	even	12
650.2.o.g.399.3		8	65.28	even	12
650.2.o.g.549.2		8	65.58	even	12
650.2.o.g.549.3		8	65.32	even	12
1040.2.q.m.81.2		4	52.19	even	12
1040.2.q.m.321.2		4	52.15	even	12
1170.2.i.q.451.2		4	39.2	even	12
1170.2.i.q.991.2		4	39.32	even	12
1690.2.a.k.1.2		2	13.8	odd	4
1690.2.a.n.1.2		2	13.5	odd	4
1690.2.d.g.1351.2		4	13.12	even	2	inner
1690.2.d.g.1351.4		4	1.1	even	1	trivial
1690.2.e.m.191.1		4	13.11	odd	12
1690.2.e.m.991.1		4	13.7	odd	12
1690.2.l.k.361.1		8	13.4	even	6
1690.2.l.k.361.3		8	13.9	even	3
1690.2.l.k.1161.1		8	13.3	even	3
1690.2.l.k.1161.3		8	13.10	even	6
8450.2.a.bc.1.1		2	65.44	odd	4
8450.2.a.bj.1.1		2	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 $\F_{q}$

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

\(f(q)\)	\(=\)	\(q+1.00000i q^{2} +3.16228 q^{3} -1.00000 q^{4} -1.00000i q^{5} +3.16228i q^{6} +1.00000i q^{7} -1.00000i q^{8} +7.00000 q^{9} +O(q^{10})\)	\(q+1.00000i q^{2} +3.16228 q^{3} -1.00000 q^{4} -1.00000i q^{5} +3.16228i q^{6} +1.00000i q^{7} -1.00000i q^{8} +7.00000 q^{9} +1.00000 q^{10} -2.16228i q^{11} -3.16228 q^{12} -1.00000 q^{14} -3.16228i q^{15} +1.00000 q^{16} +5.16228 q^{17} +7.00000i q^{18} -1.83772i q^{19} +1.00000i q^{20} +3.16228i q^{21} +2.16228 q^{22} -6.00000 q^{23} -3.16228i q^{24} -1.00000 q^{25} +12.6491 q^{27} -1.00000i q^{28} +10.3246 q^{29} +3.16228 q^{30} +7.16228i q^{31} +1.00000i q^{32} -6.83772i q^{33} +5.16228i q^{34} +1.00000 q^{35} -7.00000 q^{36} +0.162278i q^{37} +1.83772 q^{38} -1.00000 q^{40} -10.3246i q^{41} -3.16228 q^{42} -2.00000 q^{43} +2.16228i q^{44} -7.00000i q^{45} -6.00000i q^{46} +3.00000i q^{47} +3.16228 q^{48} +6.00000 q^{49} -1.00000i q^{50} +16.3246 q^{51} -2.16228 q^{53} +12.6491i q^{54} -2.16228 q^{55} +1.00000 q^{56} -5.81139i q^{57} +10.3246i q^{58} +10.3246i q^{59} +3.16228i q^{60} -7.48683 q^{61} -7.16228 q^{62} +7.00000i q^{63} -1.00000 q^{64} +6.83772 q^{66} -2.32456i q^{67} -5.16228 q^{68} -18.9737 q^{69} +1.00000i q^{70} -4.32456i q^{71} -7.00000i q^{72} -2.83772i q^{73} -0.162278 q^{74} -3.16228 q^{75} +1.83772i q^{76} +2.16228 q^{77} -13.4868 q^{79} -1.00000i q^{80} +19.0000 q^{81} +10.3246 q^{82} +9.48683i q^{83} -3.16228i q^{84} -5.16228i q^{85} -2.00000i q^{86} +32.6491 q^{87} -2.16228 q^{88} +7.32456i q^{89} +7.00000 q^{90} +6.00000 q^{92} +22.6491i q^{93} -3.00000 q^{94} -1.83772 q^{95} +3.16228i q^{96} -7.48683i q^{97} +6.00000i q^{98} -15.1359i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{4} + 28 q^{9}+O(q^{10}) \) 4 * q - 4 * q^4 + 28 * q^9	\( 4 q - 4 q^{4} + 28 q^{9} + 4 q^{10} - 4 q^{14} + 4 q^{16} + 8 q^{17} - 4 q^{22} - 24 q^{23} - 4 q^{25} + 16 q^{29} + 4 q^{35} - 28 q^{36} + 20 q^{38} - 4 q^{40} - 8 q^{43} + 24 q^{49} + 40 q^{51} + 4 q^{53} + 4 q^{55} + 4 q^{56} + 8 q^{61} - 16 q^{62} - 4 q^{64} + 40 q^{66} - 8 q^{68} + 12 q^{74} - 4 q^{77} - 16 q^{79} + 76 q^{81} + 16 q^{82} + 80 q^{87} + 4 q^{88} + 28 q^{90} + 24 q^{92} - 12 q^{94} - 20 q^{95}+O(q^{100}) \) 4 * q - 4 * q^4 + 28 * q^9 + 4 * q^10 - 4 * q^14 + 4 * q^16 + 8 * q^17 - 4 * q^22 - 24 * q^23 - 4 * q^25 + 16 * q^29 + 4 * q^35 - 28 * q^36 + 20 * q^38 - 4 * q^40 - 8 * q^43 + 24 * q^49 + 40 * q^51 + 4 * q^53 + 4 * q^55 + 4 * q^56 + 8 * q^61 - 16 * q^62 - 4 * q^64 + 40 * q^66 - 8 * q^68 + 12 * q^74 - 4 * q^77 - 16 * q^79 + 76 * q^81 + 16 * q^82 + 80 * q^87 + 4 * q^88 + 28 * q^90 + 24 * q^92 - 12 * q^94 - 20 * q^95

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