LMFDB - Embedded newform 845.2.c.h.506.13

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [845,2,Mod(506,845)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(845, 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("845.506");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 845 = 5 \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	845.c (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$6.74735897080$
Analytic rank:	$0$
Dimension:	$18$
Coefficient field:	$\mathbb{Q}[x]/(x^{18} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{18} + 34x^{16} + 407x^{14} + 2175x^{12} + 5555x^{10} + 6664x^{8} + 3544x^{6} + 681x^{4} + 47x^{2} + 1 $ x^18 + 34x^16 + 407x^14 + 2175x^12 + 5555x^10 + 6664x^8 + 3544x^6 + 681x^4 + 47x^2 + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			506.13
Root			$0.271062i$ of defining polynomial
Character	$\chi$	$=$	845.506
Dual form			845.2.c.h.506.6

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.53088i q^{2} +2.88726 q^{3} -0.343581 q^{4} +1.00000i q^{5} +4.42003i q^{6} -3.86493i q^{7} +2.53577i q^{8} +5.33625 q^{9} +O(q^{10})$	\(q+1.53088i q^{2} +2.88726 q^{3} -0.343581 q^{4} +1.00000i q^{5} +4.42003i q^{6} -3.86493i q^{7} +2.53577i q^{8} +5.33625 q^{9} -1.53088 q^{10} +4.75078i q^{11} -0.992006 q^{12} +5.91673 q^{14} +2.88726i q^{15} -4.56911 q^{16} +0.640094 q^{17} +8.16914i q^{18} -4.70598i q^{19} -0.343581i q^{20} -11.1591i q^{21} -7.27285 q^{22} +0.308077 q^{23} +7.32143i q^{24} -1.00000 q^{25} +6.74536 q^{27} +1.32792i q^{28} +2.48779 q^{29} -4.42003 q^{30} +0.635724i q^{31} -1.92320i q^{32} +13.7167i q^{33} +0.979904i q^{34} +3.86493 q^{35} -1.83343 q^{36} +5.15099i q^{37} +7.20428 q^{38} -2.53577 q^{40} -10.4049i q^{41} +17.0831 q^{42} -8.56187 q^{43} -1.63228i q^{44} +5.33625i q^{45} +0.471628i q^{46} +2.89673i q^{47} -13.1922 q^{48} -7.93772 q^{49} -1.53088i q^{50} +1.84812 q^{51} +1.28060 q^{53} +10.3263i q^{54} -4.75078 q^{55} +9.80059 q^{56} -13.5874i q^{57} +3.80850i q^{58} -4.06969i q^{59} -0.992006i q^{60} +0.335109 q^{61} -0.973215 q^{62} -20.6243i q^{63} -6.19404 q^{64} -20.9986 q^{66} -0.721556i q^{67} -0.219924 q^{68} +0.889497 q^{69} +5.91673i q^{70} +10.3222i q^{71} +13.5315i q^{72} -13.1133i q^{73} -7.88552 q^{74} -2.88726 q^{75} +1.61689i q^{76} +18.3614 q^{77} +10.7026 q^{79} -4.56911i q^{80} +3.46683 q^{81} +15.9287 q^{82} -8.94652i q^{83} +3.83404i q^{84} +0.640094i q^{85} -13.1072i q^{86} +7.18290 q^{87} -12.0469 q^{88} +0.0141672i q^{89} -8.16914 q^{90} -0.105849 q^{92} +1.83550i q^{93} -4.43454 q^{94} +4.70598 q^{95} -5.55278i q^{96} -10.1257i q^{97} -12.1517i q^{98} +25.3514i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 18 q + 14 q^{3} - 34 q^{4} + 32 q^{9}+O(q^{10}) $ 18 * q + 14 * q^3 - 34 * q^4 + 32 * q^9	$ 18 q + 14 q^{3} - 34 q^{4} + 32 q^{9} - 6 q^{10} - 24 q^{12} - 4 q^{14} + 74 q^{16} + 2 q^{17} + 24 q^{22} - 28 q^{23} - 18 q^{25} + 44 q^{27} + 24 q^{29} + 4 q^{30} + 14 q^{35} - 6 q^{36} + 94 q^{38} + 24 q^{40} - 22 q^{42} - 78 q^{43} - 6 q^{48} - 32 q^{49} - 86 q^{51} - 16 q^{53} - 18 q^{55} + 58 q^{56} - 6 q^{61} + 20 q^{62} - 68 q^{64} - 98 q^{66} - 40 q^{68} + 26 q^{69} - 30 q^{74} - 14 q^{75} + 8 q^{77} + 78 q^{79} + 58 q^{81} + 8 q^{82} + 32 q^{87} - 84 q^{88} + 20 q^{90} - 54 q^{92} + 32 q^{94} + 8 q^{95}+O(q^{100}) $ 18 * q + 14 * q^3 - 34 * q^4 + 32 * q^9 - 6 * q^10 - 24 * q^12 - 4 * q^14 + 74 * q^16 + 2 * q^17 + 24 * q^22 - 28 * q^23 - 18 * q^25 + 44 * q^27 + 24 * q^29 + 4 * q^30 + 14 * q^35 - 6 * q^36 + 94 * q^38 + 24 * q^40 - 22 * q^42 - 78 * q^43 - 6 * q^48 - 32 * q^49 - 86 * q^51 - 16 * q^53 - 18 * q^55 + 58 * q^56 - 6 * q^61 + 20 * q^62 - 68 * q^64 - 98 * q^66 - 40 * q^68 + 26 * q^69 - 30 * q^74 - 14 * q^75 + 8 * q^77 + 78 * q^79 + 58 * q^81 + 8 * q^82 + 32 * q^87 - 84 * q^88 + 20 * q^90 - 54 * q^92 + 32 * q^94 + 8 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/845\mathbb{Z}\right)^\times$.

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

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 $

$2$	1.53088i	1.08249i	0.840864	+	0.541246i	$0.182047\pi$
$2$	1.53088i	1.08249i	−0.840864	+	0.541246i	$0.817953\pi$
$3$	2.88726	1.66696	0.833479	−	0.552551i	$-0.186345\pi$
$3$	2.88726	1.66696	0.833479	+	0.552551i	$0.186345\pi$
$4$	−0.343581	−0.171790
$5$	1.00000i	0.447214i
$5$	1.00000i	0.447214i
$6$	4.42003i	1.80447i
$7$	− 3.86493i	− 1.46081i	−0.683015	−	0.730404i	$-0.739332\pi$
$7$	− 3.86493i	− 1.46081i	0.683015	−	0.730404i	$-0.260668\pi$
$8$	2.53577i	0.896531i
$9$	5.33625	1.77875
$10$	−1.53088	−0.484105
$11$	4.75078i	1.43241i	0.697888	+	0.716207i	$0.254123\pi$
$11$	4.75078i	1.43241i	−0.697888	+	0.716207i	$0.745877\pi$
$12$	−0.992006	−0.286368
$13$	0	0
$13$	0	0
$14$	5.91673	1.58131
$15$	2.88726i	0.745487i
$16$	−4.56911	−1.14228
$17$	0.640094	0.155246	0.0776228	−	0.996983i	$-0.475267\pi$
$17$	0.640094	0.155246	0.0776228	+	0.996983i	$0.475267\pi$
$18$	8.16914i	1.92548i
$19$	− 4.70598i	− 1.07963i	−0.841785	−	0.539813i	$-0.818495\pi$
$19$	− 4.70598i	− 1.07963i	0.841785	−	0.539813i	$-0.181505\pi$
$20$	− 0.343581i	− 0.0768270i
$21$	− 11.1591i	− 2.43511i
$22$	−7.27285	−1.55058
$23$	0.308077	0.0642385	0.0321192	−	0.999484i	$-0.489774\pi$
$23$	0.308077	0.0642385	0.0321192	+	0.999484i	$0.489774\pi$
$24$	7.32143i	1.49448i
$25$	−1.00000	−0.200000
$26$	0	0
$27$	6.74536	1.29815
$28$	1.32792i	0.250953i
$29$	2.48779	0.461972	0.230986	−	0.972957i	$-0.425805\pi$
$29$	2.48779	0.461972	0.230986	+	0.972957i	$0.425805\pi$
$30$	−4.42003	−0.806984
$31$	0.635724i	0.114179i	0.998369	+	0.0570897i	$0.0181821\pi$
$31$	0.635724i	0.114179i	−0.998369	+	0.0570897i	$0.981818\pi$
$32$	− 1.92320i	− 0.339977i
$33$	13.7167i	2.38777i
$34$	0.979904i	0.168052i
$35$	3.86493	0.653293
$36$	−1.83343	−0.305572
$37$	5.15099i	0.846817i	0.905939	+	0.423409i	$0.139166\pi$
$37$	5.15099i	0.846817i	−0.905939	+	0.423409i	$0.860834\pi$
$38$	7.20428	1.16869
$39$	0	0
$40$	−2.53577	−0.400941
$41$	− 10.4049i	− 1.62498i	−0.582977	−	0.812488i	$-0.698112\pi$
$41$	− 10.4049i	− 1.62498i	0.582977	−	0.812488i	$-0.301888\pi$
$42$	17.0831	2.63598
$43$	−8.56187	−1.30567	−0.652836	−	0.757499i	$-0.726421\pi$
$43$	−8.56187	−1.30567	−0.652836	+	0.757499i	$0.726421\pi$
$44$	− 1.63228i	− 0.246075i
$45$	5.33625i	0.795482i
$46$	0.471628i	0.0695377i
$47$	2.89673i	0.422532i	0.977429	+	0.211266i	$0.0677586\pi$
$47$	2.89673i	0.422532i	−0.977429	+	0.211266i	$0.932241\pi$
$48$	−13.1922	−1.90413
$49$	−7.93772	−1.13396
$50$	− 1.53088i	− 0.216499i
$51$	1.84812	0.258788
$52$	0	0
$53$	1.28060	0.175903	0.0879517	−	0.996125i	$-0.471968\pi$
$53$	1.28060	0.175903	0.0879517	+	0.996125i	$0.471968\pi$
$54$	10.3263i	1.40523i
$55$	−4.75078	−0.640595
$56$	9.80059	1.30966
$57$	− 13.5874i	− 1.79969i
$58$	3.80850i	0.500081i
$59$	− 4.06969i	− 0.529828i	−0.964272	−	0.264914i	$-0.914656\pi$
$59$	− 4.06969i	− 0.529828i	0.964272	−	0.264914i	$-0.0853436\pi$
$60$	− 0.992006i	− 0.128067i
$61$	0.335109	0.0429063	0.0214531	−	0.999770i	$-0.493171\pi$
$61$	0.335109	0.0429063	0.0214531	+	0.999770i	$0.493171\pi$
$62$	−0.973215	−0.123598
$63$	− 20.6243i	− 2.59841i
$64$	−6.19404	−0.774255
$65$	0	0
$66$	−20.9986	−2.58475
$67$	− 0.721556i	− 0.0881521i	−0.999028	−	0.0440761i	$-0.985966\pi$
$67$	− 0.721556i	− 0.0881521i	0.999028	−	0.0440761i	$-0.0140344\pi$
$68$	−0.219924	−0.0266697
$69$	0.889497	0.107083
$70$	5.91673i	0.707185i
$71$	10.3222i	1.22502i	0.790463	+	0.612510i	$0.209840\pi$
$71$	10.3222i	1.22502i	−0.790463	+	0.612510i	$0.790160\pi$
$72$	13.5315i	1.59470i
$73$	− 13.1133i	− 1.53480i	−0.641168	−	0.767400i	$-0.721550\pi$
$73$	− 13.1133i	− 1.53480i	0.641168	−	0.767400i	$-0.278450\pi$
$74$	−7.88552	−0.916673
$75$	−2.88726	−0.333392
$76$	1.61689i	0.185470i
$77$	18.3614	2.09248
$78$	0	0
$79$	10.7026	1.20414	0.602069	−	0.798444i	$-0.294343\pi$
$79$	10.7026	1.20414	0.602069	+	0.798444i	$0.294343\pi$
$80$	− 4.56911i	− 0.510842i
$81$	3.46683	0.385204
$82$	15.9287	1.75903
$83$	− 8.94652i	− 0.982008i	−0.871157	−	0.491004i	$-0.836630\pi$
$83$	− 8.94652i	− 0.982008i	0.871157	−	0.491004i	$-0.163370\pi$
$84$	3.83404i	0.418328i
$85$	0.640094i	0.0694279i
$86$	− 13.1072i	− 1.41338i
$87$	7.18290	0.770088
$88$	−12.0469	−1.28420
$89$	0.0141672i	0.00150172i	1.00000		0.000750858i	$0.000239006\pi$
$89$	0.0141672i	0.00150172i	−1.00000		0.000750858i	$0.999761\pi$
$90$	−8.16914	−0.861103
$91$	0	0
$92$	−0.105849	−0.0110356
$93$	1.83550i	0.190332i
$94$	−4.43454	−0.457388
$95$	4.70598	0.482824
$96$	− 5.55278i	− 0.566728i
$97$	− 10.1257i	− 1.02811i	−0.857758	−	0.514054i	$-0.828143\pi$
$97$	− 10.1257i	− 1.02811i	0.857758	−	0.514054i	$-0.171857\pi$
$98$	− 12.1517i	− 1.22750i
$99$	25.3514i	2.54791i
$100$	0.343581	0.0343581
$101$	−12.4363	−1.23746	−0.618730	−	0.785603i	$-0.712353\pi$
$101$	−12.4363	−1.23746	−0.618730	+	0.785603i	$0.712353\pi$
$102$	2.82924i	0.280136i
$103$	−10.2372	−1.00871	−0.504353	−	0.863498i	$-0.668269\pi$
$103$	−10.2372	−1.00871	−0.504353	+	0.863498i	$0.668269\pi$
$104$	0	0
$105$	11.1591	1.08901
$106$	1.96043i	0.190414i
$107$	−15.3401	−1.48298	−0.741490	−	0.670964i	$-0.765881\pi$
$107$	−15.3401	−1.48298	−0.741490	+	0.670964i	$0.765881\pi$
$108$	−2.31758	−0.223009
$109$	0.515675i	0.0493927i	0.999695	+	0.0246963i	$0.00786189\pi$
$109$	0.515675i	0.0493927i	−0.999695	+	0.0246963i	$0.992138\pi$
$110$	− 7.27285i	− 0.693439i
$111$	14.8722i	1.41161i
$112$	17.6593i	1.66865i
$113$	−17.0097	−1.60013	−0.800067	−	0.599911i	$-0.795203\pi$
$113$	−17.0097	−1.60013	−0.800067	+	0.599911i	$0.795203\pi$
$114$	20.8006	1.94815
$115$	0.308077i	0.0287283i
$116$	−0.854759	−0.0793624
$117$	0	0
$118$	6.23019	0.573535
$119$	− 2.47392i	− 0.226784i
$120$	−7.32143	−0.668352
$121$	−11.5699	−1.05181
$122$	0.513010i	0.0464457i
$123$	− 30.0417i	− 2.70877i
$124$	− 0.218423i	− 0.0196149i
$125$	− 1.00000i	− 0.0894427i
$126$	31.5732	2.81276
$127$	21.7877	1.93335	0.966675	−	0.256007i	$-0.0824069\pi$
$127$	21.7877	1.93335	0.966675	+	0.256007i	$0.0824069\pi$
$128$	− 13.3287i	− 1.17810i
$129$	−24.7203	−2.17650
$130$	0	0
$131$	−10.8833	−0.950874	−0.475437	−	0.879750i	$-0.657710\pi$
$131$	−10.8833	−0.950874	−0.475437	+	0.879750i	$0.657710\pi$
$132$	− 4.71280i	− 0.410197i
$133$	−18.1883	−1.57713
$134$	1.10461	0.0954240
$135$	6.74536i	0.580548i
$136$	1.62313i	0.139182i
$137$	− 1.25583i	− 0.107293i	−0.998560	−	0.0536466i	$-0.982916\pi$
$137$	− 1.25583i	− 0.107293i	0.998560	−	0.0536466i	$-0.0170844\pi$
$138$	1.36171i	0.115916i
$139$	18.2380	1.54693	0.773463	−	0.633842i	$-0.218523\pi$
$139$	18.2380	1.54693	0.773463	+	0.633842i	$0.218523\pi$
$140$	−1.32792	−0.112230
$141$	8.36361i	0.704343i
$142$	−15.8020	−1.32608
$143$	0	0
$144$	−24.3819	−2.03183
$145$	2.48779i	0.206600i
$146$	20.0749	1.66141
$147$	−22.9182	−1.89026
$148$	− 1.76978i	− 0.145475i
$149$	21.0429i	1.72390i	0.506991	+	0.861952i	$0.330758\pi$
$149$	21.0429i	1.72390i	−0.506991	+	0.861952i	$0.669242\pi$
$150$	− 4.42003i	− 0.360894i
$151$	16.2381i	1.32144i	0.750632	+	0.660720i	$0.229749\pi$
$151$	16.2381i	1.32144i	−0.750632	+	0.660720i	$0.770251\pi$
$152$	11.9333	0.967918
$153$	3.41570	0.276143
$154$	28.1091i	2.26510i
$155$	−0.635724	−0.0510626
$156$	0	0
$157$	−2.30362	−0.183849	−0.0919246	−	0.995766i	$-0.529302\pi$
$157$	−2.30362	−0.183849	−0.0919246	+	0.995766i	$0.529302\pi$
$158$	16.3844i	1.30347i
$159$	3.69741	0.293224
$160$	1.92320	0.152042
$161$	− 1.19070i	− 0.0938401i
$162$	5.30729i	0.416980i
$163$	− 15.3183i	− 1.19983i	−0.800065	−	0.599913i	$-0.795202\pi$
$163$	− 15.3183i	− 1.19983i	0.800065	−	0.599913i	$-0.204798\pi$
$164$	3.57493i	0.279156i
$165$	−13.7167	−1.06785
$166$	13.6960	1.06302
$167$	− 11.9904i	− 0.927845i	−0.885876	−	0.463922i	$-0.846442\pi$
$167$	− 11.9904i	− 0.927845i	0.885876	−	0.463922i	$-0.153558\pi$
$168$	28.2968	2.18315
$169$	0	0
$170$	−0.979904	−0.0751552
$171$	− 25.1123i	− 1.92039i
$172$	2.94169	0.224302
$173$	9.18608	0.698404	0.349202	−	0.937047i	$-0.386453\pi$
$173$	9.18608	0.698404	0.349202	+	0.937047i	$0.386453\pi$
$174$	10.9961i	0.833615i
$175$	3.86493i	0.292162i
$176$	− 21.7068i	− 1.63622i
$177$	− 11.7502i	− 0.883202i
$178$	−0.0216882	−0.00162560
$179$	2.63815	0.197185	0.0985923	−	0.995128i	$-0.468566\pi$
$179$	2.63815	0.197185	0.0985923	+	0.995128i	$0.468566\pi$
$180$	− 1.83343i	− 0.136656i
$181$	−9.36566	−0.696144	−0.348072	−	0.937468i	$-0.613163\pi$
$181$	−9.36566	−0.696144	−0.348072	+	0.937468i	$0.613163\pi$
$182$	0	0
$183$	0.967545	0.0715230
$184$	0.781213i	0.0575918i
$185$	−5.15099	−0.378708
$186$	−2.80992	−0.206033
$187$	3.04094i	0.222376i
$188$	− 0.995262i	− 0.0725870i
$189$	− 26.0704i	− 1.89634i
$190$	7.20428i	0.522653i
$191$	−6.30396	−0.456138	−0.228069	−	0.973645i	$-0.573241\pi$
$191$	−6.30396	−0.456138	−0.228069	+	0.973645i	$0.573241\pi$
$192$	−17.8838	−1.29065
$193$	8.22585i	0.592110i	0.955171	+	0.296055i	$0.0956711\pi$
$193$	8.22585i	0.592110i	−0.955171	+	0.296055i	$0.904329\pi$
$194$	15.5012	1.11292
$195$	0	0
$196$	2.72725	0.194803
$197$	0.462516i	0.0329529i	0.999864	+	0.0164764i	$0.00524485\pi$
$197$	0.462516i	0.0329529i	−0.999864	+	0.0164764i	$0.994755\pi$
$198$	−38.8098	−2.75809
$199$	19.3271	1.37007	0.685033	−	0.728512i	$-0.259788\pi$
$199$	19.3271	1.37007	0.685033	+	0.728512i	$0.259788\pi$
$200$	− 2.53577i	− 0.179306i
$201$	− 2.08332i	− 0.146946i
$202$	− 19.0385i	− 1.33954i
$203$	− 9.61516i	− 0.674852i
$204$	−0.634977	−0.0444573
$205$	10.4049	0.726712
$206$	− 15.6719i	− 1.09192i
$207$	1.64398	0.114264
$208$	0	0
$209$	22.3571	1.54647
$210$	17.0831i	1.17885i
$211$	18.8081	1.29480	0.647401	−	0.762149i	$-0.275856\pi$
$211$	18.8081	1.29480	0.647401	+	0.762149i	$0.275856\pi$
$212$	−0.439988	−0.0302185
$213$	29.8029i	2.04206i
$214$	− 23.4837i	− 1.60532i
$215$	− 8.56187i	− 0.583914i
$216$	17.1047i	1.16383i
$217$	2.45703	0.166794
$218$	−0.789434	−0.0534672
$219$	− 37.8616i	− 2.55845i
$220$	1.63228	0.110048
$221$	0	0
$222$	−22.7675	−1.52806
$223$	− 3.50267i	− 0.234556i	−0.993099	−	0.117278i	$-0.962583\pi$
$223$	− 3.50267i	− 0.234556i	0.993099	−	0.117278i	$-0.0374169\pi$
$224$	−7.43305	−0.496642
$225$	−5.33625	−0.355750
$226$	− 26.0397i	− 1.73213i
$227$	24.5730i	1.63097i	0.578781	+	0.815483i	$0.303529\pi$
$227$	24.5730i	1.63097i	−0.578781	+	0.815483i	$0.696471\pi$
$228$	4.66837i	0.309170i
$229$	− 17.8359i	− 1.17863i	−0.807904	−	0.589314i	$-0.799398\pi$
$229$	− 17.8359i	− 1.17863i	0.807904	−	0.589314i	$-0.200602\pi$
$230$	−0.471628	−0.0310982
$231$	53.0142	3.48808
$232$	6.30848i	0.414172i
$233$	−3.03322	−0.198713	−0.0993565	−	0.995052i	$-0.531678\pi$
$233$	−3.03322	−0.198713	−0.0993565	+	0.995052i	$0.531678\pi$
$234$	0	0
$235$	−2.89673	−0.188962
$236$	1.39827i	0.0910194i
$237$	30.9012	2.00725
$238$	3.78727	0.245492
$239$	− 6.93867i	− 0.448825i	−0.974494	−	0.224413i	$-0.927954\pi$
$239$	− 6.93867i	− 0.448825i	0.974494	−	0.224413i	$-0.0720464\pi$
$240$	− 13.1922i	− 0.851553i
$241$	6.59429i	0.424776i	0.977185	+	0.212388i	$0.0681240\pi$
$241$	6.59429i	0.424776i	−0.977185	+	0.212388i	$0.931876\pi$
$242$	− 17.7121i	− 1.13858i
$243$	−10.2264	−0.656027
$244$	−0.115137	−0.00737089
$245$	− 7.93772i	− 0.507122i
$246$	45.9901	2.93222
$247$	0	0
$248$	−1.61205	−0.102365
$249$	− 25.8309i	− 1.63697i
$250$	1.53088	0.0968211
$251$	16.8636	1.06442	0.532211	−	0.846612i	$-0.321361\pi$
$251$	16.8636	1.06442	0.532211	+	0.846612i	$0.321361\pi$
$252$	7.08610i	0.446383i
$253$	1.46361i	0.0920161i
$254$	33.3543i	2.09284i
$255$	1.84812i	0.115733i
$256$	8.01652	0.501033
$257$	10.6593	0.664906	0.332453	−	0.943120i	$-0.392124\pi$
$257$	10.6593	0.664906	0.332453	+	0.943120i	$0.392124\pi$
$258$	− 37.8437i	− 2.35605i
$259$	19.9082	1.23704
$260$	0	0
$261$	13.2755	0.821733
$262$	− 16.6609i	− 1.02931i
$263$	14.2995	0.881745	0.440873	−	0.897570i	$-0.354669\pi$
$263$	14.2995	0.881745	0.440873	+	0.897570i	$0.354669\pi$
$264$	−34.7825	−2.14071
$265$	1.28060i	0.0786664i
$266$	− 27.8441i	− 1.70723i
$267$	0.0409042i	0.00250330i
$268$	0.247913i	0.0151437i
$269$	10.7756	0.657001	0.328500	−	0.944504i	$-0.393457\pi$
$269$	10.7756	0.657001	0.328500	+	0.944504i	$0.393457\pi$
$270$	−10.3263	−0.628439
$271$	19.4273i	1.18012i	0.807358	+	0.590062i	$0.200897\pi$
$271$	19.4273i	1.18012i	−0.807358	+	0.590062i	$0.799103\pi$
$272$	−2.92466	−0.177334
$273$	0	0
$274$	1.92253	0.116144
$275$	− 4.75078i	− 0.286483i
$276$	−0.305614	−0.0183958
$277$	1.04322	0.0626810	0.0313405	−	0.999509i	$-0.490022\pi$
$277$	1.04322	0.0626810	0.0313405	+	0.999509i	$0.490022\pi$
$278$	27.9201i	1.67454i
$279$	3.39239i	0.203097i
$280$	9.80059i	0.585697i
$281$	9.04835i	0.539780i	0.962891	+	0.269890i	$0.0869873\pi$
$281$	9.04835i	0.539780i	−0.962891	+	0.269890i	$0.913013\pi$
$282$	−12.8037	−0.762447
$283$	2.37857	0.141392	0.0706958	−	0.997498i	$-0.477478\pi$
$283$	2.37857	0.141392	0.0706958	+	0.997498i	$0.477478\pi$
$284$	− 3.54651i	− 0.210447i
$285$	13.5874	0.804847
$286$	0	0
$287$	−40.2144	−2.37378
$288$	− 10.2627i	− 0.604735i
$289$	−16.5903	−0.975899
$290$	−3.80850	−0.223643
$291$	− 29.2355i	− 1.71381i
$292$	4.50549i	0.263664i
$293$	14.1290i	0.825423i	0.910862	+	0.412712i	$0.135418\pi$
$293$	14.1290i	0.825423i	−0.910862	+	0.412712i	$0.864582\pi$
$294$	− 35.0850i	− 2.04620i
$295$	4.06969	0.236946
$296$	−13.0617	−0.759198
$297$	32.0457i	1.85948i
$298$	−32.2141	−1.86611
$299$	0	0
$300$	0.992006	0.0572735
$301$	33.0910i	1.90734i
$302$	−24.8586	−1.43045
$303$	−35.9069	−2.06280
$304$	21.5022i	1.23323i
$305$	0.335109i	0.0191883i
$306$	5.22902i	0.298923i
$307$	− 24.3391i	− 1.38910i	−0.719442	−	0.694552i	$-0.755603\pi$
$307$	− 24.3391i	− 1.38910i	0.719442	−	0.694552i	$-0.244397\pi$
$308$	−6.30864	−0.359468
$309$	−29.5576	−1.68147
$310$	− 0.973215i	− 0.0552749i
$311$	−6.14513	−0.348458	−0.174229	−	0.984705i	$-0.555743\pi$
$311$	−6.14513	−0.348458	−0.174229	+	0.984705i	$0.555743\pi$
$312$	0	0
$313$	1.39881	0.0790652	0.0395326	−	0.999218i	$-0.487413\pi$
$313$	1.39881	0.0790652	0.0395326	+	0.999218i	$0.487413\pi$
$314$	− 3.52656i	− 0.199015i
$315$	20.6243	1.16205
$316$	−3.67721	−0.206859
$317$	− 8.92070i	− 0.501036i	−0.968112	−	0.250518i	$-0.919399\pi$
$317$	− 8.92070i	− 0.501036i	0.968112	−	0.250518i	$-0.0806010\pi$
$318$	5.66027i	0.317412i
$319$	11.8190i	0.661735i
$320$	− 6.19404i	− 0.346258i
$321$	−44.2907	−2.47207
$322$	1.82281	0.101581
$323$	− 3.01227i	− 0.167607i
$324$	−1.19114	−0.0661743
$325$	0	0
$326$	23.4505	1.29880
$327$	1.48889i	0.0823356i
$328$	26.3845	1.45684
$329$	11.1957	0.617238
$330$	− 20.9986i	− 1.15593i
$331$	19.7654i	1.08640i	0.839602	+	0.543202i	$0.182788\pi$
$331$	19.7654i	1.08640i	−0.839602	+	0.543202i	$0.817212\pi$
$332$	3.07385i	0.168700i
$333$	27.4870i	1.50628i
$334$	18.3558	1.00439
$335$	0.721556	0.0394228
$336$	50.9870i	2.78157i
$337$	−20.2361	−1.10233	−0.551166	−	0.834396i	$-0.685817\pi$
$337$	−20.2361	−1.10233	−0.551166	+	0.834396i	$0.685817\pi$
$338$	0	0
$339$	−49.1112	−2.66736
$340$	− 0.219924i	− 0.0119271i
$341$	−3.02019	−0.163552
$342$	38.4438	2.07880
$343$	3.62422i	0.195689i
$344$	− 21.7109i	− 1.17058i
$345$	0.889497i	0.0478889i
$346$	14.0627i	0.756018i
$347$	29.2678	1.57118	0.785589	−	0.618749i	$-0.212360\pi$
$347$	29.2678	1.57118	0.785589	+	0.618749i	$0.212360\pi$
$348$	−2.46791	−0.132294
$349$	0.195008i	0.0104386i	0.999986	+	0.00521928i	$0.00166136\pi$
$349$	0.195008i	0.0104386i	−0.999986	+	0.00521928i	$0.998339\pi$
$350$	−5.91673	−0.316263
$351$	0	0
$352$	9.13671	0.486988
$353$	12.2379i	0.651358i	0.945480	+	0.325679i	$0.105593\pi$
$353$	12.2379i	0.651358i	−0.945480	+	0.325679i	$0.894407\pi$
$354$	17.9881	0.956059
$355$	−10.3222	−0.547846
$356$	− 0.00486757i	0 0.000257981i
$357$	− 7.14285i	− 0.378040i
$358$	4.03868i	0.213451i
$359$	2.43702i	0.128621i	0.997930	+	0.0643104i	$0.0204848\pi$
$359$	2.43702i	0.128621i	−0.997930	+	0.0643104i	$0.979515\pi$
$360$	−13.5315	−0.713174
$361$	−3.14628	−0.165593
$362$	− 14.3377i	− 0.753571i
$363$	−33.4053	−1.75332
$364$	0	0
$365$	13.1133	0.686384
$366$	1.48119i	0.0774231i
$367$	−28.8213	−1.50446	−0.752229	−	0.658901i	$-0.771022\pi$
$367$	−28.8213	−1.50446	−0.752229	+	0.658901i	$0.771022\pi$
$368$	−1.40764	−0.0733783
$369$	− 55.5233i	− 2.89043i
$370$	− 7.88552i	− 0.409949i
$371$	− 4.94942i	− 0.256961i
$372$	− 0.630643i	− 0.0326973i
$373$	3.72479	0.192862	0.0964311	−	0.995340i	$-0.469257\pi$
$373$	3.72479	0.192862	0.0964311	+	0.995340i	$0.469257\pi$
$374$	−4.65531	−0.240720
$375$	− 2.88726i	− 0.149097i
$376$	−7.34546	−0.378813
$377$	0	0
$378$	39.9105	2.05278
$379$	27.1916i	1.39674i	0.715737	+	0.698370i	$0.246091\pi$
$379$	27.1916i	1.39674i	−0.715737	+	0.698370i	$0.753909\pi$
$380$	−1.61689	−0.0829445
$381$	62.9068	3.22281
$382$	− 9.65057i	− 0.493766i
$383$	33.2900i	1.70104i	0.525945	+	0.850519i	$0.323712\pi$
$383$	33.2900i	1.70104i	−0.525945	+	0.850519i	$0.676288\pi$
$384$	− 38.4834i	− 1.96385i
$385$	18.3614i	0.935786i
$386$	−12.5928	−0.640955
$387$	−45.6883	−2.32247
$388$	3.47899i	0.176619i
$389$	23.5358	1.19331	0.596655	−	0.802498i	$-0.296496\pi$
$389$	23.5358	1.19331	0.596655	+	0.802498i	$0.296496\pi$
$390$	0	0
$391$	0.197198	0.00997274
$392$	− 20.1282i	− 1.01663i
$393$	−31.4227	−1.58507
$394$	−0.708054	−0.0356713
$395$	10.7026i	0.538507i
$396$	− 8.71024i	− 0.437706i
$397$	− 22.4582i	− 1.12714i	−0.826067	−	0.563572i	$-0.809427\pi$
$397$	− 22.4582i	− 1.12714i	0.826067	−	0.563572i	$-0.190573\pi$
$398$	29.5875i	1.48309i
$399$	−52.5143	−2.62901
$400$	4.56911	0.228456
$401$	− 0.0227333i	− 0.00113525i	−1.00000		0.000567624i	$-0.999819\pi$
$401$	− 0.0227333i	− 0.00113525i	1.00000		0.000567624i	$-0.000180680\pi$
$402$	3.18930	0.159068
$403$	0	0
$404$	4.27288	0.212584
$405$	3.46683i	0.172268i
$406$	14.7196	0.730523
$407$	−24.4712	−1.21299
$408$	4.68640i	0.232011i
$409$	28.7890i	1.42352i	0.702421	+	0.711761i	$0.252102\pi$
$409$	28.7890i	1.42352i	−0.702421	+	0.711761i	$0.747898\pi$
$410$	15.9287i	0.786660i
$411$	− 3.62592i	− 0.178853i
$412$	3.51732	0.173286
$413$	−15.7291	−0.773977
$414$	2.51672i	0.123690i
$415$	8.94652	0.439167
$416$	0	0
$417$	52.6578	2.57866
$418$	34.2259i	1.67404i
$419$	17.4064	0.850358	0.425179	−	0.905109i	$-0.360211\pi$
$419$	17.4064	0.850358	0.425179	+	0.905109i	$0.360211\pi$
$420$	−3.83404	−0.187082
$421$	30.1642i	1.47011i	0.678007	+	0.735056i	$0.262844\pi$
$421$	30.1642i	1.47011i	−0.678007	+	0.735056i	$0.737156\pi$
$422$	28.7929i	1.40161i
$423$	15.4577i	0.751579i
$424$	3.24730i	0.157703i
$425$	−0.640094	−0.0310491
$426$	−45.6245	−2.21051
$427$	− 1.29517i	− 0.0626778i
$428$	5.27055	0.254762
$429$	0	0
$430$	13.1072	0.632083
$431$	− 10.1531i	− 0.489056i	−0.969642	−	0.244528i	$-0.921367\pi$
$431$	− 10.1531i	− 0.489056i	0.969642	−	0.244528i	$-0.0786331\pi$
$432$	−30.8203	−1.48284
$433$	−22.2687	−1.07017	−0.535083	−	0.844799i	$-0.679720\pi$
$433$	−22.2687	−1.07017	−0.535083	+	0.844799i	$0.679720\pi$
$434$	3.76141i	0.180554i
$435$	7.18290i	0.344394i
$436$	− 0.177176i	− 0.00848519i
$437$	− 1.44981i	− 0.0693536i
$438$	57.9614	2.76950
$439$	−37.8809	−1.80796	−0.903980	−	0.427576i	$-0.859368\pi$
$439$	−37.8809	−1.80796	−0.903980	+	0.427576i	$0.859368\pi$
$440$	− 12.0469i	− 0.574313i
$441$	−42.3577	−2.01703
$442$	0	0
$443$	−0.918139	−0.0436221	−0.0218110	−	0.999762i	$-0.506943\pi$
$443$	−0.918139	−0.0436221	−0.0218110	+	0.999762i	$0.506943\pi$
$444$	− 5.10981i	− 0.242501i
$445$	−0.0141672	−0.000671588 0
$446$	5.36216	0.253906
$447$	60.7563i	2.87368i
$448$	23.9396i	1.13104i
$449$	34.7819i	1.64146i	0.571317	+	0.820730i	$0.306433\pi$
$449$	34.7819i	1.64146i	−0.571317	+	0.820730i	$0.693567\pi$
$450$	− 8.16914i	− 0.385097i
$451$	49.4315	2.32764
$452$	5.84419	0.274888
$453$	46.8837i	2.20279i
$454$	−37.6182	−1.76551
$455$	0	0
$456$	34.4545	1.61348
$457$	13.7197i	0.641781i	0.947116	+	0.320890i	$0.103982\pi$
$457$	13.7197i	0.641781i	−0.947116	+	0.320890i	$0.896018\pi$
$458$	27.3045	1.27586
$459$	4.31767	0.201531
$460$	− 0.105849i	− 0.00493525i
$461$	− 32.5057i	− 1.51394i	−0.653449	−	0.756971i	$-0.726679\pi$
$461$	− 32.5057i	− 1.51394i	0.653449	−	0.756971i	$-0.273321\pi$
$462$	81.1582i	3.77582i
$463$	34.0054i	1.58037i	0.612871	+	0.790183i	$0.290015\pi$
$463$	34.0054i	1.58037i	−0.612871	+	0.790183i	$0.709985\pi$
$464$	−11.3670	−0.527701
$465$	−1.83550	−0.0851192
$466$	− 4.64349i	− 0.215105i
$467$	0.0693333	0.00320836	0.00160418	−	0.999999i	$-0.499489\pi$
$467$	0.0693333	0.00320836	0.00160418	+	0.999999i	$0.499489\pi$
$468$	0	0
$469$	−2.78877	−0.128773
$470$	− 4.43454i	− 0.204550i
$471$	−6.65115	−0.306469
$472$	10.3198	0.475007
$473$	− 40.6755i	− 1.87026i
$474$	47.3059i	2.17283i
$475$	4.70598i	0.215925i
$476$	0.849992i	0.0389593i
$477$	6.83358	0.312888
$478$	10.6222	0.485850
$479$	9.16068i	0.418562i	0.977856	+	0.209281i	$0.0671124\pi$
$479$	9.16068i	0.418562i	−0.977856	+	0.209281i	$0.932888\pi$
$480$	5.55278	0.253449
$481$	0	0
$482$	−10.0950	−0.459816
$483$	− 3.43785i	− 0.156428i
$484$	3.97520	0.180691
$485$	10.1257	0.459784
$486$	− 15.6554i	− 0.710144i
$487$	− 27.2691i	− 1.23568i	−0.786303	−	0.617841i	$-0.788008\pi$
$487$	− 27.2691i	− 1.23568i	0.786303	−	0.617841i	$-0.211992\pi$
$488$	0.849759i	0.0384668i
$489$	− 44.2280i	− 2.00006i
$490$	12.1517	0.548956
$491$	17.9855	0.811676	0.405838	−	0.913945i	$-0.366980\pi$
$491$	17.9855	0.811676	0.405838	+	0.913945i	$0.366980\pi$
$492$	10.3218i	0.465341i
$493$	1.59242	0.0717191
$494$	0	0
$495$	−25.3514	−1.13946
$496$	− 2.90470i	− 0.130425i
$497$	39.8946	1.78952
$498$	39.5439	1.77200
$499$	12.9963i	0.581792i	0.956755	+	0.290896i	$0.0939534\pi$
$499$	12.9963i	0.581792i	−0.956755	+	0.290896i	$0.906047\pi$
$500$	0.343581i	0.0153654i
$501$	− 34.6194i	− 1.54668i
$502$	25.8161i	1.15223i
$503$	−6.92255	−0.308661	−0.154331	−	0.988019i	$-0.549322\pi$
$503$	−6.92255	−0.308661	−0.154331	+	0.988019i	$0.549322\pi$
$504$	52.2984	2.32956
$505$	− 12.4363i	− 0.553409i
$506$	−2.24060	−0.0996068
$507$	0	0
$508$	−7.48585	−0.332131
$509$	1.39866i	0.0619945i	0.999519	+	0.0309972i	$0.00986831\pi$
$509$	1.39866i	0.0619945i	−0.999519	+	0.0309972i	$0.990132\pi$
$510$	−2.82924	−0.125281
$511$	−50.6822	−2.24205
$512$	− 14.3851i	− 0.635739i
$513$	− 31.7436i	− 1.40151i
$514$	16.3180i	0.719756i
$515$	− 10.2372i	− 0.451107i
$516$	8.49343	0.373902
$517$	−13.7617	−0.605241
$518$	30.4770i	1.33908i
$519$	26.5226	1.16421
$520$	0	0
$521$	−19.4736	−0.853154	−0.426577	−	0.904451i	$-0.640281\pi$
$521$	−19.4736	−0.853154	−0.426577	+	0.904451i	$0.640281\pi$
$522$	20.3231i	0.889520i
$523$	−8.05342	−0.352152	−0.176076	−	0.984377i	$-0.556340\pi$
$523$	−8.05342	−0.352152	−0.176076	+	0.984377i	$0.556340\pi$
$524$	3.73928	0.163351
$525$	11.1591i	0.487021i
$526$	21.8908i	0.954483i
$527$	0.406923i	0.0177259i
$528$	− 62.6732i	− 2.72750i
$529$	−22.9051	−0.995873
$530$	−1.96043	−0.0851558
$531$	− 21.7169i	− 0.942432i
$532$	6.24916	0.270935
$533$	0	0
$534$	−0.0626193	−0.00270980
$535$	− 15.3401i	− 0.663209i
$536$	1.82970	0.0790311
$537$	7.61702	0.328699
$538$	16.4961i	0.711199i
$539$	− 37.7103i	− 1.62430i
$540$	− 2.31758i	− 0.0997327i
$541$	7.04652i	0.302954i	0.988461	+	0.151477i	$0.0484029\pi$
$541$	7.04652i	0.302954i	−0.988461	+	0.151477i	$0.951597\pi$
$542$	−29.7408	−1.27748
$543$	−27.0411	−1.16044
$544$	− 1.23103i	− 0.0527800i
$545$	−0.515675	−0.0220891
$546$	0	0
$547$	39.5338	1.69034	0.845172	−	0.534494i	$-0.179498\pi$
$547$	39.5338	1.69034	0.845172	+	0.534494i	$0.179498\pi$
$548$	0.431481i	0.0184319i
$549$	1.78822	0.0763196
$550$	7.27285	0.310115
$551$	− 11.7075i	− 0.498757i
$552$	2.25556i	0.0960031i
$553$	− 41.3649i	− 1.75901i
$554$	1.59704i	0.0678518i
$555$	−14.8722	−0.631291
$556$	−6.26622	−0.265747
$557$	8.81508i	0.373507i	0.982407	+	0.186753i	$0.0597965\pi$
$557$	8.81508i	0.373507i	−0.982407	+	0.186753i	$0.940203\pi$
$558$	−5.19332	−0.219851
$559$	0	0
$560$	−17.6593	−0.746243
$561$	8.77999i	0.370691i
$562$	−13.8519	−0.584307
$563$	−10.7679	−0.453813	−0.226907	−	0.973917i	$-0.572861\pi$
$563$	−10.7679	−0.453813	−0.226907	+	0.973917i	$0.572861\pi$
$564$	− 2.87358i	− 0.120999i
$565$	− 17.0097i	− 0.715602i
$566$	3.64130i	0.153055i
$567$	− 13.3991i	− 0.562709i
$568$	−26.1748	−1.09827
$569$	22.5894	0.946995	0.473498	−	0.880795i	$-0.342991\pi$
$569$	22.5894	0.946995	0.473498	+	0.880795i	$0.342991\pi$
$570$	20.8006i	0.871241i
$571$	5.95207	0.249087	0.124543	−	0.992214i	$-0.460253\pi$
$571$	5.95207	0.249087	0.124543	+	0.992214i	$0.460253\pi$
$572$	0	0
$573$	−18.2011	−0.760364
$574$	− 61.5632i	− 2.56960i
$575$	−0.308077	−0.0128477
$576$	−33.0530	−1.37721
$577$	− 21.0425i	− 0.876010i	−0.898972	−	0.438005i	$-0.855685\pi$
$577$	− 21.0425i	− 0.876010i	0.898972	−	0.438005i	$-0.144315\pi$
$578$	− 25.3977i	− 1.05640i
$579$	23.7501i	0.987022i
$580$	− 0.854759i	− 0.0354919i
$581$	−34.5777	−1.43452
$582$	44.7559	1.85519
$583$	6.08383i	0.251966i
$584$	33.2524	1.37600
$585$	0	0
$586$	−21.6297	−0.893515
$587$	19.8969i	0.821232i	0.911808	+	0.410616i	$0.134686\pi$
$587$	19.8969i	0.821232i	−0.911808	+	0.410616i	$0.865314\pi$
$588$	7.87427	0.324729
$589$	2.99171	0.123271
$590$	6.23019i	0.256493i
$591$	1.33540i	0.0549311i
$592$	− 23.5354i	− 0.967301i
$593$	− 35.4858i	− 1.45723i	−0.684925	−	0.728614i	$-0.740165\pi$
$593$	− 35.4858i	− 1.45723i	0.684925	−	0.728614i	$-0.259835\pi$
$594$	−49.0580	−2.01287
$595$	2.47392	0.101421
$596$	− 7.22995i	− 0.296150i
$597$	55.8024	2.28384
$598$	0	0
$599$	12.5055	0.510962	0.255481	−	0.966814i	$-0.417766\pi$
$599$	12.5055	0.510962	0.255481	+	0.966814i	$0.417766\pi$
$600$	− 7.32143i	− 0.298896i
$601$	−10.9255	−0.445661	−0.222831	−	0.974857i	$-0.571530\pi$
$601$	−10.9255	−0.445661	−0.222831	+	0.974857i	$0.571530\pi$
$602$	−50.6583	−2.06468
$603$	− 3.85041i	− 0.156801i
$604$	− 5.57911i	− 0.227011i
$605$	− 11.5699i	− 0.470383i
$606$	− 54.9690i	− 2.23296i
$607$	−2.90532	−0.117923	−0.0589616	−	0.998260i	$-0.518779\pi$
$607$	−2.90532	−0.117923	−0.0589616	+	0.998260i	$0.518779\pi$
$608$	−9.05056	−0.367049
$609$	− 27.7614i	− 1.12495i
$610$	−0.513010	−0.0207712
$611$	0	0
$612$	−1.17357	−0.0474388
$613$	19.2621i	0.777990i	0.921240	+	0.388995i	$0.127178\pi$
$613$	19.2621i	0.777990i	−0.921240	+	0.388995i	$0.872822\pi$
$614$	37.2601	1.50370
$615$	30.0417	1.21140
$616$	46.5604i	1.87597i
$617$	0.902301i	0.0363253i	0.999835	+	0.0181626i	$0.00578166\pi$
$617$	0.902301i	0.0363253i	−0.999835	+	0.0181626i	$0.994218\pi$
$618$	− 45.2489i	− 1.82018i
$619$	23.8784i	0.959752i	0.877336	+	0.479876i	$0.159318\pi$
$619$	23.8784i	0.959752i	−0.877336	+	0.479876i	$0.840682\pi$
$620$	0.218423	0.00877207
$621$	2.07809	0.0833909
$622$	− 9.40743i	− 0.377204i
$623$	0.0547552	0.00219372
$624$	0	0
$625$	1.00000	0.0400000
$626$	2.14140i	0.0855875i
$627$	64.5506	2.57790
$628$	0.791481	0.0315835
$629$	3.29712i	0.131465i
$630$	31.5732i	1.25791i
$631$	45.4774i	1.81043i	0.424958	+	0.905213i	$0.360289\pi$
$631$	45.4774i	1.81043i	−0.424958	+	0.905213i	$0.639711\pi$
$632$	27.1394i	1.07955i
$633$	54.3038	2.15838
$634$	13.6565	0.542368
$635$	21.7877i	0.864620i
$636$	−1.27036	−0.0503730
$637$	0	0
$638$	−18.0934	−0.716323
$639$	55.0819i	2.17901i
$640$	13.3287	0.526864
$641$	−38.7591	−1.53089	−0.765447	−	0.643499i	$-0.777482\pi$
$641$	−38.7591	−1.53089	−0.765447	+	0.643499i	$0.777482\pi$
$642$	− 67.8036i	− 2.67599i
$643$	− 14.5186i	− 0.572559i	−0.958146	−	0.286279i	$-0.907581\pi$
$643$	− 14.5186i	− 0.572559i	0.958146	−	0.286279i	$-0.0924185\pi$
$644$	0.409101i	0.0161208i
$645$	− 24.7203i	− 0.973361i
$646$	4.61141	0.181434
$647$	5.86833	0.230708	0.115354	−	0.993324i	$-0.463200\pi$
$647$	5.86833	0.230708	0.115354	+	0.993324i	$0.463200\pi$
$648$	8.79110i	0.345347i
$649$	19.3342	0.758933
$650$	0	0
$651$	7.09408	0.278039
$652$	5.26309i	0.206119i
$653$	−13.9308	−0.545154	−0.272577	−	0.962134i	$-0.587876\pi$
$653$	−13.9308	−0.545154	−0.272577	+	0.962134i	$0.587876\pi$
$654$	−2.27930	−0.0891276
$655$	− 10.8833i	− 0.425244i
$656$	47.5413i	1.85618i
$657$	− 69.9761i	− 2.73003i
$658$	17.1392i	0.668156i
$659$	32.4177	1.26281	0.631406	−	0.775452i	$-0.282478\pi$
$659$	32.4177	1.26281	0.631406	+	0.775452i	$0.282478\pi$
$660$	4.71280	0.183446
$661$	− 42.0999i	− 1.63750i	−0.574154	−	0.818748i	$-0.694669\pi$
$661$	− 42.0999i	− 1.63750i	0.574154	−	0.818748i	$-0.305331\pi$
$662$	−30.2584	−1.17603
$663$	0	0
$664$	22.6863	0.880400
$665$	− 18.1883i	− 0.705313i
$666$	−42.0791	−1.63053
$667$	0.766432	0.0296764
$668$	4.11967i	0.159395i
$669$	− 10.1131i	− 0.390996i
$670$	1.10461i	0.0426749i
$671$	1.59203i	0.0614595i
$672$	−21.4611	−0.827881
$673$	−9.29450	−0.358277	−0.179138	−	0.983824i	$-0.557331\pi$
$673$	−9.29450	−0.358277	−0.179138	+	0.983824i	$0.557331\pi$
$674$	− 30.9790i	− 1.19327i
$675$	−6.74536	−0.259629
$676$	0	0
$677$	−45.6554	−1.75468	−0.877340	−	0.479870i	$-0.840684\pi$
$677$	−45.6554	−1.75468	−0.877340	+	0.479870i	$0.840684\pi$
$678$	− 75.1832i	− 2.88739i
$679$	−39.1351	−1.50187
$680$	−1.62313	−0.0622443
$681$	70.9485i	2.71875i
$682$	− 4.62353i	− 0.177044i
$683$	12.5289i	0.479403i	0.970847	+	0.239702i	$0.0770496\pi$
$683$	12.5289i	0.479403i	−0.970847	+	0.239702i	$0.922950\pi$
$684$	8.62811i	0.329904i
$685$	1.25583	0.0479830
$686$	−5.54822	−0.211832
$687$	− 51.4968i	− 1.96472i
$688$	39.1201	1.49144
$689$	0	0
$690$	−1.36171	−0.0518394
$691$	− 5.09243i	− 0.193725i	−0.995298	−	0.0968625i	$-0.969119\pi$
$691$	− 5.09243i	− 0.193725i	0.995298	−	0.0968625i	$-0.0308807\pi$
$692$	−3.15616	−0.119979
$693$	97.9813	3.72200
$694$	44.8054i	1.70079i
$695$	18.2380i	0.691806i
$696$	18.2142i	0.690408i
$697$	− 6.66013i	− 0.252270i
$698$	−0.298534	−0.0112997
$699$	−8.75769	−0.331246
$700$	− 1.32792i	− 0.0501906i
$701$	33.0018	1.24646	0.623230	−	0.782039i	$-0.285820\pi$
$701$	33.0018	1.24646	0.623230	+	0.782039i	$0.285820\pi$
$702$	0	0
$703$	24.2405	0.914246
$704$	− 29.4265i	− 1.10905i
$705$	−8.36361	−0.314992
$706$	−18.7347	−0.705090
$707$	48.0656i	1.80769i
$708$	4.03716i	0.151726i
$709$	9.50378i	0.356922i	0.983947	+	0.178461i	$0.0571118\pi$
$709$	9.50378i	0.356922i	−0.983947	+	0.178461i	$0.942888\pi$
$710$	− 15.8020i	− 0.593039i
$711$	57.1119	2.14186
$712$	−0.0359247	−0.00134633
$713$	0.195852i	0.00733472i
$714$	10.9348	0.409225
$715$	0	0
$716$	−0.906418	−0.0338744
$717$	− 20.0337i	− 0.748174i
$718$	−3.73077	−0.139231
$719$	−32.8065	−1.22348	−0.611739	−	0.791060i	$-0.709530\pi$
$719$	−32.8065	−1.22348	−0.611739	+	0.791060i	$0.709530\pi$
$720$	− 24.3819i	− 0.908662i
$721$	39.5663i	1.47353i
$722$	− 4.81656i	− 0.179254i
$723$	19.0394i	0.708083i
$724$	3.21786	0.119591
$725$	−2.48779	−0.0923944
$726$	− 51.1393i	− 1.89796i
$727$	23.3248	0.865070	0.432535	−	0.901617i	$-0.357619\pi$
$727$	23.3248	0.865070	0.432535	+	0.901617i	$0.357619\pi$
$728$	0	0
$729$	−39.9269	−1.47877
$730$	20.0749i	0.743005i
$731$	−5.48040	−0.202700
$732$	−0.332430	−0.0122870
$733$	− 41.5378i	− 1.53423i	−0.641507	−	0.767117i	$-0.721690\pi$
$733$	− 41.5378i	− 1.53423i	0.641507	−	0.767117i	$-0.278310\pi$
$734$	− 44.1218i	− 1.62857i
$735$	− 22.9182i	− 0.845352i
$736$	− 0.592494i	− 0.0218396i
$737$	3.42795	0.126270
$738$	84.9993	3.12887
$739$	− 1.05857i	− 0.0389401i	−0.999810	−	0.0194701i	$-0.993802\pi$
$739$	− 1.05857i	− 0.0389401i	0.999810	−	0.0194701i	$-0.00619790\pi$
$740$	1.76978	0.0650584
$741$	0	0
$742$	7.57695	0.278158
$743$	19.0752i	0.699800i	0.936787	+	0.349900i	$0.113784\pi$
$743$	19.0752i	0.699800i	−0.936787	+	0.349900i	$0.886216\pi$
$744$	−4.65441	−0.170639
$745$	−21.0429	−0.770953
$746$	5.70219i	0.208772i
$747$	− 47.7409i	− 1.74675i
$748$	− 1.04481i	− 0.0382021i
$749$	59.2884i	2.16635i
$750$	4.42003	0.161397
$751$	−46.7588	−1.70625	−0.853127	−	0.521703i	$-0.825297\pi$
$751$	−46.7588	−1.70625	−0.853127	+	0.521703i	$0.825297\pi$
$752$	− 13.2355i	− 0.482649i
$753$	48.6896	1.77435
$754$	0	0
$755$	−16.2381	−0.590966
$756$	8.95729i	0.325773i
$757$	15.0982	0.548752	0.274376	−	0.961623i	$-0.411529\pi$
$757$	15.0982	0.548752	0.274376	+	0.961623i	$0.411529\pi$
$758$	−41.6270	−1.51196
$759$	4.22581i	0.153387i
$760$	11.9333i	0.432866i
$761$	− 20.5311i	− 0.744253i	−0.928182	−	0.372127i	$-0.878629\pi$
$761$	− 20.5311i	− 0.744253i	0.928182	−	0.372127i	$-0.121371\pi$
$762$	96.3025i	3.48867i
$763$	1.99305	0.0721532
$764$	2.16592	0.0783602
$765$	3.41570i	0.123495i
$766$	−50.9628	−1.84136
$767$	0	0
$768$	23.1458	0.835201
$769$	− 33.9725i	− 1.22508i	−0.790440	−	0.612540i	$-0.790148\pi$
$769$	− 33.9725i	− 1.22508i	0.790440	−	0.612540i	$-0.209852\pi$
$770$	−28.1091	−1.01298
$771$	30.7760	1.10837
$772$	− 2.82625i	− 0.101719i
$773$	− 10.8481i	− 0.390177i	−0.980786	−	0.195089i	$-0.937501\pi$
$773$	− 10.8481i	− 0.390177i	0.980786	−	0.195089i	$-0.0624994\pi$
$774$	− 69.9431i	− 2.51405i
$775$	− 0.635724i	− 0.0228359i
$776$	25.6764	0.921731
$777$	57.4802	2.06209
$778$	36.0303i	1.29175i
$779$	−48.9654	−1.75437
$780$	0	0
$781$	−49.0385	−1.75474
$782$	0.301886i	0.0107954i
$783$	16.7811	0.599707
$784$	36.2683	1.29530
$785$	− 2.30362i	− 0.0822198i
$786$	− 48.1043i	− 1.71582i
$787$	9.91801i	0.353539i	0.984252	+	0.176769i	$0.0565647\pi$
$787$	9.91801i	0.353539i	−0.984252	+	0.176769i	$0.943435\pi$
$788$	− 0.158912i	− 0.00566099i
$789$	41.2863	1.46983
$790$	−16.3844	−0.582930
$791$	65.7412i	2.33749i
$792$	−64.2853	−2.28428
$793$	0	0
$794$	34.3807	1.22013
$795$	3.69741i	0.131134i
$796$	−6.64044	−0.235364
$797$	33.0159	1.16948	0.584742	−	0.811219i	$-0.301196\pi$
$797$	33.0159	1.16948	0.584742	+	0.811219i	$0.301196\pi$
$798$	− 80.3929i	− 2.84588i
$799$	1.85418i	0.0655962i
$800$	1.92320i	0.0679955i
$801$	0.0755996i	0.00267118i
$802$	0.0348019	0.00122890
$803$	62.2986	2.19847
$804$	0.715788i	0.0252439i
$805$	1.19070	0.0419666
$806$	0	0
$807$	31.1120	1.09519
$808$	− 31.5357i	− 1.10942i
$809$	−11.0248	−0.387610	−0.193805	−	0.981040i	$-0.562083\pi$
$809$	−11.0248	−0.387610	−0.193805	+	0.981040i	$0.562083\pi$
$810$	−5.30729	−0.186479
$811$	31.2682i	1.09797i	0.835831	+	0.548987i	$0.184986\pi$
$811$	31.2682i	1.09797i	−0.835831	+	0.548987i	$0.815014\pi$
$812$	3.30359i	0.115933i
$813$	56.0916i	1.96722i
$814$	− 37.4624i	− 1.31306i
$815$	15.3183	0.536578
$816$	−8.44425	−0.295608
$817$	40.2920i	1.40964i
$818$	−44.0723	−1.54095
$819$	0	0
$820$	−3.57493	−0.124842
$821$	12.3719i	0.431782i	0.976417	+	0.215891i	$0.0692656\pi$
$821$	12.3719i	0.431782i	−0.976417	+	0.215891i	$0.930734\pi$
$822$	5.55083	0.193607
$823$	47.6643	1.66147	0.830737	−	0.556665i	$-0.187919\pi$
$823$	47.6643	1.66147	0.830737	+	0.556665i	$0.187919\pi$
$824$	− 25.9593i	− 0.904336i
$825$	− 13.7167i	− 0.477555i
$826$	− 24.0793i	− 0.837825i
$827$	− 18.9966i	− 0.660575i	−0.943880	−	0.330287i	$-0.892854\pi$
$827$	− 18.9966i	− 0.660575i	0.943880	−	0.330287i	$-0.107146\pi$
$828$	−0.564839	−0.0196295
$829$	44.5434	1.54706	0.773528	−	0.633762i	$-0.218490\pi$
$829$	44.5434	1.54706	0.773528	+	0.633762i	$0.218490\pi$
$830$	13.6960i	0.475395i
$831$	3.01205	0.104487
$832$	0	0
$833$	−5.08088	−0.176042
$834$	80.6125i	2.79138i
$835$	11.9904	0.414945
$836$	−7.68147	−0.265669
$837$	4.28819i	0.148222i
$838$	26.6470i	0.920506i
$839$	23.6568i	0.816724i	0.912820	+	0.408362i	$0.133900\pi$
$839$	23.6568i	0.816724i	−0.912820	+	0.408362i	$0.866100\pi$
$840$	28.2968i	0.976333i
$841$	−22.8109	−0.786582
$842$	−46.1776	−1.59138
$843$	26.1249i	0.899790i
$844$	−6.46210	−0.222435
$845$	0	0
$846$	−23.6638	−0.813579
$847$	44.7169i	1.53649i
$848$	−5.85119	−0.200931
$849$	6.86756	0.235694
$850$	− 0.979904i	− 0.0336104i
$851$	1.58690i	0.0543983i
$852$	− 10.2397i	− 0.350806i
$853$	1.85306i	0.0634474i	0.999497	+	0.0317237i	$0.0100997\pi$
$853$	1.85306i	0.0634474i	−0.999497	+	0.0317237i	$0.989900\pi$
$854$	1.98275	0.0678483
$855$	25.1123	0.858823
$856$	− 38.8989i	− 1.32954i
$857$	−32.9142	−1.12433	−0.562163	−	0.827026i	$-0.690031\pi$
$857$	−32.9142	−1.12433	−0.562163	+	0.827026i	$0.690031\pi$
$858$	0	0
$859$	31.9147	1.08892	0.544458	−	0.838788i	$-0.316735\pi$
$859$	31.9147	1.08892	0.544458	+	0.838788i	$0.316735\pi$
$860$	2.94169i	0.100311i
$861$	−116.109	−3.95699
$862$	15.5431	0.529400
$863$	− 11.5883i	− 0.394469i	−0.980356	−	0.197235i	$-0.936804\pi$
$863$	− 11.5883i	− 0.394469i	0.980356	−	0.197235i	$-0.0631961\pi$
$864$	− 12.9727i	− 0.441340i
$865$	9.18608i	0.312336i
$866$	− 34.0906i	− 1.15845i
$867$	−47.9004	−1.62678
$868$	−0.844189	−0.0286537
$869$	50.8457i	1.72482i
$870$	−10.9961	−0.372804
$871$	0	0
$872$	−1.30763	−0.0442821
$873$	− 54.0333i	− 1.82875i
$874$	2.21947	0.0750748
$875$	−3.86493	−0.130659
$876$	13.0085i	0.439517i
$877$	− 48.0221i	− 1.62159i	−0.585328	−	0.810796i	$-0.699034\pi$
$877$	− 48.0221i	− 1.62159i	0.585328	−	0.810796i	$-0.300966\pi$
$878$	− 57.9910i	− 1.95710i
$879$	40.7940i	1.37595i
$880$	21.7068	0.731738
$881$	32.4131	1.09202	0.546012	−	0.837777i	$-0.316145\pi$
$881$	32.4131	1.09202	0.546012	+	0.837777i	$0.316145\pi$
$882$	− 64.8443i	− 2.18342i
$883$	−53.6030	−1.80389	−0.901943	−	0.431856i	$-0.857859\pi$
$883$	−53.6030	−1.80389	−0.901943	+	0.431856i	$0.857859\pi$
$884$	0	0
$885$	11.7502	0.394980
$886$	− 1.40556i	− 0.0472206i
$887$	−39.8878	−1.33930	−0.669650	−	0.742677i	$-0.733556\pi$
$887$	−39.8878	−1.33930	−0.669650	+	0.742677i	$0.733556\pi$
$888$	−37.7126	−1.26555
$889$	− 84.2082i	− 2.82425i
$890$	− 0.0216882i	0 0.000726989i
$891$	16.4702i	0.551771i
$892$	1.20345i	0.0402946i
$893$	13.6320	0.456177
$894$	−93.0104	−3.11073
$895$	2.63815i	0.0881837i
$896$	−51.5146	−1.72098
$897$	0	0
$898$	−53.2468	−1.77687
$899$	1.58155i	0.0527477i
$900$	1.83343	0.0611145
$901$	0.819702	0.0273082
$902$	75.6735i	2.51965i
$903$	95.5424i	3.17945i
$904$	− 43.1326i	− 1.43457i
$905$	− 9.36566i	− 0.311325i
$906$	−71.7731	−2.38450
$907$	−33.5672	−1.11458	−0.557290	−	0.830318i	$-0.688159\pi$
$907$	−33.5672	−1.11458	−0.557290	+	0.830318i	$0.688159\pi$
$908$	− 8.44281i	− 0.280185i
$909$	−66.3634	−2.20113
$910$	0	0
$911$	−25.2448	−0.836399	−0.418199	−	0.908355i	$-0.637339\pi$
$911$	−25.2448	−0.836399	−0.418199	+	0.908355i	$0.637339\pi$
$912$	62.0823i	2.05575i
$913$	42.5029	1.40664
$914$	−21.0032	−0.694723
$915$	0.967545i	0.0319861i
$916$	6.12807i	0.202477i
$917$	42.0631i	1.38904i
$918$	6.60981i	0.218156i
$919$	30.1551	0.994726	0.497363	−	0.867543i	$-0.334302\pi$
$919$	30.1551	0.994726	0.497363	+	0.867543i	$0.334302\pi$
$920$	−0.781213	−0.0257558
$921$	− 70.2732i	− 2.31558i
$922$	49.7622	1.63883
$923$	0	0
$924$	−18.2147	−0.599219
$925$	− 5.15099i	− 0.169363i
$926$	−52.0581	−1.71074
$927$	−54.6285	−1.79424
$928$	− 4.78453i	− 0.157060i
$929$	− 34.6023i	− 1.13526i	−0.823283	−	0.567632i	$-0.807860\pi$
$929$	− 34.6023i	− 1.13526i	0.823283	−	0.567632i	$-0.192140\pi$
$930$	− 2.80992i	− 0.0921410i
$931$	37.3548i	1.22425i
$932$	1.04216	0.0341370
$933$	−17.7426	−0.580866
$934$	0.106141i	0.00347303i
$935$	−3.04094	−0.0994495
$936$	0	0
$937$	−15.7809	−0.515541	−0.257771	−	0.966206i	$-0.582988\pi$
$937$	−15.7809	−0.515541	−0.257771	+	0.966206i	$0.582988\pi$
$938$	− 4.26926i	− 0.139396i
$939$	4.03871	0.131798
$940$	0.995262	0.0324619
$941$	− 22.5246i	− 0.734280i	−0.930166	−	0.367140i	$-0.880337\pi$
$941$	− 22.5246i	− 0.734280i	0.930166	−	0.367140i	$-0.119663\pi$
$942$	− 10.1821i	− 0.331750i
$943$	− 3.20552i	− 0.104386i
$944$	18.5949i	0.605211i
$945$	26.0704	0.848070
$946$	62.2692	2.02455
$947$	5.97656i	0.194212i	0.995274	+	0.0971060i	$0.0309586\pi$
$947$	5.97656i	0.194212i	−0.995274	+	0.0971060i	$0.969041\pi$
$948$	−10.6171	−0.344826
$949$	0	0
$950$	−7.20428	−0.233738
$951$	− 25.7563i	− 0.835207i
$952$	6.27330	0.203319
$953$	0.933220	0.0302300	0.0151150	−	0.999886i	$-0.495189\pi$
$953$	0.933220	0.0302300	0.0151150	+	0.999886i	$0.495189\pi$
$954$	10.4614i	0.338699i
$955$	− 6.30396i	− 0.203991i
$956$	2.38400i	0.0771039i
$957$	34.1244i	1.10308i
$958$	−14.0239	−0.453091
$959$	−4.85372	−0.156735
$960$	− 17.8838i	− 0.577197i
$961$	30.5959	0.986963
$962$	0	0
$963$	−81.8585	−2.63785
$964$	− 2.26567i	− 0.0729724i
$965$	−8.22585	−0.264800
$966$	5.26292	0.169332
$967$	− 39.4196i	− 1.26765i	−0.773477	−	0.633825i	$-0.781484\pi$
$967$	− 39.4196i	− 1.26765i	0.773477	−	0.633825i	$-0.218516\pi$
$968$	− 29.3386i	− 0.942979i
$969$	− 8.69720i	− 0.279394i
$970$	15.5012i	0.497713i
$971$	−37.6269	−1.20750	−0.603752	−	0.797172i	$-0.706328\pi$
$971$	−37.6269	−1.20750	−0.603752	+	0.797172i	$0.706328\pi$
$972$	3.51361	0.112699
$973$	− 70.4886i	− 2.25976i
$974$	41.7456	1.33762
$975$	0	0
$976$	−1.53115	−0.0490109
$977$	− 62.0293i	− 1.98449i	−0.124282	−	0.992247i	$-0.539663\pi$
$977$	− 62.0293i	− 1.98449i	0.124282	−	0.992247i	$-0.460337\pi$
$978$	67.7076	2.16505
$979$	−0.0673051	−0.00215108
$980$	2.72725i	0.0871187i
$981$	2.75177i	0.0878573i
$982$	27.5336i	0.878633i
$983$	27.9591i	0.891757i	0.895093	+	0.445879i	$0.147109\pi$
$983$	27.9591i	0.891757i	−0.895093	+	0.445879i	$0.852891\pi$
$984$	76.1789	2.42849
$985$	−0.462516	−0.0147370
$986$	2.43780i	0.0776354i
$987$	32.3248	1.02891
$988$	0	0
$989$	−2.63771	−0.0838744
$990$	− 38.8098i	− 1.23346i
$991$	56.5077	1.79503	0.897513	−	0.440988i	$-0.145372\pi$
$991$	56.5077	1.79503	0.897513	+	0.440988i	$0.145372\pi$
$992$	1.22263	0.0388184
$993$	57.0678i	1.81099i
$994$	61.0738i	1.93714i
$995$	19.3271i	0.612712i
$996$	8.87500i	0.281215i
$997$	−6.01115	−0.190375	−0.0951875	−	0.995459i	$-0.530345\pi$
$997$	−6.01115	−0.190375	−0.0951875	+	0.995459i	$0.530345\pi$
$998$	−19.8957	−0.629786
$999$	34.7453i	1.09929i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	845.2.c.h.506.13		18
13.2	odd	12		845.2.e.o.191.4		18
13.3	even	3		845.2.m.j.316.6		36
13.4	even	6		845.2.m.j.361.6		36
13.5	odd	4		845.2.a.o.1.6	yes	9
13.6	odd	12		845.2.e.o.146.4		18
13.7	odd	12		845.2.e.p.146.6		18
13.8	odd	4		845.2.a.n.1.4	✓	9
13.9	even	3		845.2.m.j.361.13		36
13.10	even	6		845.2.m.j.316.13		36
13.11	odd	12		845.2.e.p.191.6		18
13.12	even	2	inner	845.2.c.h.506.6		18
39.5	even	4		7605.2.a.cp.1.4		9
39.8	even	4		7605.2.a.cs.1.6		9
65.34	odd	4		4225.2.a.bt.1.6		9
65.44	odd	4		4225.2.a.bs.1.4		9

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
845.2.a.n.1.4	✓	9	13.8	odd	4
845.2.a.o.1.6	yes	9	13.5	odd	4
845.2.c.h.506.6		18	13.12	even	2	inner
845.2.c.h.506.13		18	1.1	even	1	trivial
845.2.e.o.146.4		18	13.6	odd	12
845.2.e.o.191.4		18	13.2	odd	12
845.2.e.p.146.6		18	13.7	odd	12
845.2.e.p.191.6		18	13.11	odd	12
845.2.m.j.316.6		36	13.3	even	3
845.2.m.j.316.13		36	13.10	even	6
845.2.m.j.361.6		36	13.4	even	6
845.2.m.j.361.13		36	13.9	even	3
4225.2.a.bs.1.4		9	65.44	odd	4
4225.2.a.bt.1.6		9	65.34	odd	4
7605.2.a.cp.1.4		9	39.5	even	4
7605.2.a.cs.1.6		9	39.8	even	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 845 = 5 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	845.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.53088i q^{2} +2.88726 q^{3} -0.343581 q^{4} +1.00000i q^{5} +4.42003i q^{6} -3.86493i q^{7} +2.53577i q^{8} +5.33625 q^{9} +O(q^{10})\)	\(q+1.53088i q^{2} +2.88726 q^{3} -0.343581 q^{4} +1.00000i q^{5} +4.42003i q^{6} -3.86493i q^{7} +2.53577i q^{8} +5.33625 q^{9} -1.53088 q^{10} +4.75078i q^{11} -0.992006 q^{12} +5.91673 q^{14} +2.88726i q^{15} -4.56911 q^{16} +0.640094 q^{17} +8.16914i q^{18} -4.70598i q^{19} -0.343581i q^{20} -11.1591i q^{21} -7.27285 q^{22} +0.308077 q^{23} +7.32143i q^{24} -1.00000 q^{25} +6.74536 q^{27} +1.32792i q^{28} +2.48779 q^{29} -4.42003 q^{30} +0.635724i q^{31} -1.92320i q^{32} +13.7167i q^{33} +0.979904i q^{34} +3.86493 q^{35} -1.83343 q^{36} +5.15099i q^{37} +7.20428 q^{38} -2.53577 q^{40} -10.4049i q^{41} +17.0831 q^{42} -8.56187 q^{43} -1.63228i q^{44} +5.33625i q^{45} +0.471628i q^{46} +2.89673i q^{47} -13.1922 q^{48} -7.93772 q^{49} -1.53088i q^{50} +1.84812 q^{51} +1.28060 q^{53} +10.3263i q^{54} -4.75078 q^{55} +9.80059 q^{56} -13.5874i q^{57} +3.80850i q^{58} -4.06969i q^{59} -0.992006i q^{60} +0.335109 q^{61} -0.973215 q^{62} -20.6243i q^{63} -6.19404 q^{64} -20.9986 q^{66} -0.721556i q^{67} -0.219924 q^{68} +0.889497 q^{69} +5.91673i q^{70} +10.3222i q^{71} +13.5315i q^{72} -13.1133i q^{73} -7.88552 q^{74} -2.88726 q^{75} +1.61689i q^{76} +18.3614 q^{77} +10.7026 q^{79} -4.56911i q^{80} +3.46683 q^{81} +15.9287 q^{82} -8.94652i q^{83} +3.83404i q^{84} +0.640094i q^{85} -13.1072i q^{86} +7.18290 q^{87} -12.0469 q^{88} +0.0141672i q^{89} -8.16914 q^{90} -0.105849 q^{92} +1.83550i q^{93} -4.43454 q^{94} +4.70598 q^{95} -5.55278i q^{96} -10.1257i q^{97} -12.1517i q^{98} +25.3514i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 18 q + 14 q^{3} - 34 q^{4} + 32 q^{9}+O(q^{10}) \) 18 * q + 14 * q^3 - 34 * q^4 + 32 * q^9	\( 18 q + 14 q^{3} - 34 q^{4} + 32 q^{9} - 6 q^{10} - 24 q^{12} - 4 q^{14} + 74 q^{16} + 2 q^{17} + 24 q^{22} - 28 q^{23} - 18 q^{25} + 44 q^{27} + 24 q^{29} + 4 q^{30} + 14 q^{35} - 6 q^{36} + 94 q^{38} + 24 q^{40} - 22 q^{42} - 78 q^{43} - 6 q^{48} - 32 q^{49} - 86 q^{51} - 16 q^{53} - 18 q^{55} + 58 q^{56} - 6 q^{61} + 20 q^{62} - 68 q^{64} - 98 q^{66} - 40 q^{68} + 26 q^{69} - 30 q^{74} - 14 q^{75} + 8 q^{77} + 78 q^{79} + 58 q^{81} + 8 q^{82} + 32 q^{87} - 84 q^{88} + 20 q^{90} - 54 q^{92} + 32 q^{94} + 8 q^{95}+O(q^{100}) \) 18 * q + 14 * q^3 - 34 * q^4 + 32 * q^9 - 6 * q^10 - 24 * q^12 - 4 * q^14 + 74 * q^16 + 2 * q^17 + 24 * q^22 - 28 * q^23 - 18 * q^25 + 44 * q^27 + 24 * q^29 + 4 * q^30 + 14 * q^35 - 6 * q^36 + 94 * q^38 + 24 * q^40 - 22 * q^42 - 78 * q^43 - 6 * q^48 - 32 * q^49 - 86 * q^51 - 16 * q^53 - 18 * q^55 + 58 * q^56 - 6 * q^61 + 20 * q^62 - 68 * q^64 - 98 * q^66 - 40 * q^68 + 26 * q^69 - 30 * q^74 - 14 * q^75 + 8 * q^77 + 78 * q^79 + 58 * q^81 + 8 * q^82 + 32 * q^87 - 84 * q^88 + 20 * q^90 - 54 * q^92 + 32 * q^94 + 8 * q^95

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