LMFDB - Embedded newform 845.2.c.f.506.2

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:	$6$
Coefficient field:	6.0.153664.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} + 5x^{4} + 6x^{2} + 1 $ x^6 + 5x^4 + 6x^2 + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			506.2
Root			$-1.24698i$ of defining polynomial
Character	$\chi$	$=$	845.506
Dual form			845.2.c.f.506.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.24698i q^{2} -0.198062 q^{3} +0.445042 q^{4} -1.00000i q^{5} +0.246980i q^{6} -0.198062i q^{7} -3.04892i q^{8} -2.96077 q^{9} +O(q^{10})$	\(q-1.24698i q^{2} -0.198062 q^{3} +0.445042 q^{4} -1.00000i q^{5} +0.246980i q^{6} -0.198062i q^{7} -3.04892i q^{8} -2.96077 q^{9} -1.24698 q^{10} -4.04892i q^{11} -0.0881460 q^{12} -0.246980 q^{14} +0.198062i q^{15} -2.91185 q^{16} +5.40581 q^{17} +3.69202i q^{18} +2.18598i q^{19} -0.445042i q^{20} +0.0392287i q^{21} -5.04892 q^{22} -7.23490 q^{23} +0.603875i q^{24} -1.00000 q^{25} +1.18060 q^{27} -0.0881460i q^{28} -7.07606 q^{29} +0.246980 q^{30} +0.0217703i q^{31} -2.46681i q^{32} +0.801938i q^{33} -6.74094i q^{34} -0.198062 q^{35} -1.31767 q^{36} -1.49396i q^{37} +2.72587 q^{38} -3.04892 q^{40} -10.9608i q^{41} +0.0489173 q^{42} +7.55496 q^{43} -1.80194i q^{44} +2.96077i q^{45} +9.02177i q^{46} -1.03923i q^{47} +0.576728 q^{48} +6.96077 q^{49} +1.24698i q^{50} -1.07069 q^{51} -0.554958 q^{53} -1.47219i q^{54} -4.04892 q^{55} -0.603875 q^{56} -0.432960i q^{57} +8.82371i q^{58} -3.89008i q^{59} +0.0881460i q^{60} -5.55496 q^{61} +0.0271471 q^{62} +0.586417i q^{63} -8.89977 q^{64} +1.00000 q^{66} -5.67025i q^{67} +2.40581 q^{68} +1.43296 q^{69} +0.246980i q^{70} -9.52111i q^{71} +9.02715i q^{72} +11.5797i q^{73} -1.86294 q^{74} +0.198062 q^{75} +0.972853i q^{76} -0.801938 q^{77} -15.8291 q^{79} +2.91185i q^{80} +8.64848 q^{81} -13.6679 q^{82} +17.3448i q^{83} +0.0174584i q^{84} -5.40581i q^{85} -9.42088i q^{86} +1.40150 q^{87} -12.3448 q^{88} +3.05429i q^{89} +3.69202 q^{90} -3.21983 q^{92} -0.00431187i q^{93} -1.29590 q^{94} +2.18598 q^{95} +0.488582i q^{96} +5.02177i q^{97} -8.67994i q^{98} +11.9879i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 10 q^{3} + 2 q^{4} + 8 q^{9}+O(q^{10}) $ 6 * q - 10 * q^3 + 2 * q^4 + 8 * q^9	$ 6 q - 10 q^{3} + 2 q^{4} + 8 q^{9} + 2 q^{10} - 8 q^{12} + 8 q^{14} - 10 q^{16} + 6 q^{17} - 12 q^{22} + 4 q^{23} - 6 q^{25} - 16 q^{27} - 12 q^{29} - 8 q^{30} - 10 q^{35} + 26 q^{36} + 38 q^{38} - 18 q^{42} + 46 q^{43} - 2 q^{48} + 16 q^{49} + 18 q^{51} - 4 q^{53} - 6 q^{55} + 14 q^{56} - 34 q^{61} - 12 q^{62} - 8 q^{64} + 6 q^{66} - 12 q^{68} - 30 q^{69} - 22 q^{74} + 10 q^{75} + 4 q^{77} - 74 q^{79} + 54 q^{81} + 4 q^{82} + 20 q^{87} - 28 q^{88} + 12 q^{90} - 22 q^{92} + 20 q^{94} - 16 q^{95}+O(q^{100}) $ 6 * q - 10 * q^3 + 2 * q^4 + 8 * q^9 + 2 * q^10 - 8 * q^12 + 8 * q^14 - 10 * q^16 + 6 * q^17 - 12 * q^22 + 4 * q^23 - 6 * q^25 - 16 * q^27 - 12 * q^29 - 8 * q^30 - 10 * q^35 + 26 * q^36 + 38 * q^38 - 18 * q^42 + 46 * q^43 - 2 * q^48 + 16 * q^49 + 18 * q^51 - 4 * q^53 - 6 * q^55 + 14 * q^56 - 34 * q^61 - 12 * q^62 - 8 * q^64 + 6 * q^66 - 12 * q^68 - 30 * q^69 - 22 * q^74 + 10 * q^75 + 4 * q^77 - 74 * q^79 + 54 * q^81 + 4 * q^82 + 20 * q^87 - 28 * q^88 + 12 * q^90 - 22 * q^92 + 20 * q^94 - 16 * 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 $

$2$	− 1.24698i	− 0.881748i	−0.897569	−	0.440874i	$-0.854669\pi$
$2$	− 1.24698i	− 0.881748i	0.897569	−	0.440874i	$-0.145331\pi$
$3$	−0.198062	−0.114351	−0.0571757	−	0.998364i	$-0.518210\pi$
$3$	−0.198062	−0.114351	−0.0571757	+	0.998364i	$0.518210\pi$
$4$	0.445042	0.222521
$5$	− 1.00000i	− 0.447214i
$5$	− 1.00000i	− 0.447214i
$6$	0.246980i	0.100829i
$7$	− 0.198062i	− 0.0748605i	−0.999299	−	0.0374302i	$-0.988083\pi$
$7$	− 0.198062i	− 0.0748605i	0.999299	−	0.0374302i	$-0.0119172\pi$
$8$	− 3.04892i	− 1.07796i
$9$	−2.96077	−0.986924
$10$	−1.24698	−0.394330
$11$	− 4.04892i	− 1.22079i	−0.792095	−	0.610397i	$-0.791010\pi$
$11$	− 4.04892i	− 1.22079i	0.792095	−	0.610397i	$-0.208990\pi$
$12$	−0.0881460	−0.0254456
$13$	0	0
$13$	0	0
$14$	−0.246980	−0.0660081
$15$	0.198062i	0.0511395i
$16$	−2.91185	−0.727963
$17$	5.40581	1.31110	0.655551	−	0.755151i	$-0.272436\pi$
$17$	5.40581	1.31110	0.655551	+	0.755151i	$0.272436\pi$
$18$	3.69202i	0.870218i
$19$	2.18598i	0.501498i	0.968052	+	0.250749i	$0.0806769\pi$
$19$	2.18598i	0.501498i	−0.968052	+	0.250749i	$0.919323\pi$
$20$	− 0.445042i	− 0.0995144i
$21$	0.0392287i	0.00856040i
$22$	−5.04892	−1.07643
$23$	−7.23490	−1.50858	−0.754290	−	0.656541i	$-0.772019\pi$
$23$	−7.23490	−1.50858	−0.754290	+	0.656541i	$0.772019\pi$
$24$	0.603875i	0.123266i
$25$	−1.00000	−0.200000
$26$	0	0
$27$	1.18060	0.227207
$28$	− 0.0881460i	− 0.0166580i
$29$	−7.07606	−1.31399	−0.656996	−	0.753894i	$-0.728173\pi$
$29$	−7.07606	−1.31399	−0.656996	+	0.753894i	$0.728173\pi$
$30$	0.246980	0.0450921
$31$	0.0217703i	0.00391005i	0.999998	+	0.00195503i	$0.000622305\pi$
$31$	0.0217703i	0.00391005i	−0.999998	+	0.00195503i	$0.999378\pi$
$32$	− 2.46681i	− 0.436075i
$33$	0.801938i	0.139599i
$34$	− 6.74094i	− 1.15606i
$35$	−0.198062	−0.0334786
$36$	−1.31767	−0.219611
$37$	− 1.49396i	− 0.245605i	−0.992431	−	0.122803i	$-0.960812\pi$
$37$	− 1.49396i	− 0.245605i	0.992431	−	0.122803i	$-0.0391882\pi$
$38$	2.72587	0.442195
$39$	0	0
$40$	−3.04892	−0.482076
$41$	− 10.9608i	− 1.71178i	−0.517154	−	0.855892i	$-0.673009\pi$
$41$	− 10.9608i	− 1.71178i	0.517154	−	0.855892i	$-0.326991\pi$
$42$	0.0489173	0.00754811
$43$	7.55496	1.15212	0.576060	−	0.817407i	$-0.304589\pi$
$43$	7.55496	1.15212	0.576060	+	0.817407i	$0.304589\pi$
$44$	− 1.80194i	− 0.271652i
$45$	2.96077i	0.441366i
$46$	9.02177i	1.33019i
$47$	− 1.03923i	− 0.151587i	−0.997124	−	0.0757935i	$-0.975851\pi$
$47$	− 1.03923i	− 0.151587i	0.997124	−	0.0757935i	$-0.0241490\pi$
$48$	0.576728	0.0832436
$49$	6.96077	0.994396
$50$	1.24698i	0.176350i
$51$	−1.07069	−0.149926
$52$	0	0
$53$	−0.554958	−0.0762294	−0.0381147	−	0.999273i	$-0.512135\pi$
$53$	−0.554958	−0.0762294	−0.0381147	+	0.999273i	$0.512135\pi$
$54$	− 1.47219i	− 0.200340i
$55$	−4.04892	−0.545956
$56$	−0.603875	−0.0806963
$57$	− 0.432960i	− 0.0573470i
$58$	8.82371i	1.15861i
$59$	− 3.89008i	− 0.506446i	−0.967408	−	0.253223i	$-0.918509\pi$
$59$	− 3.89008i	− 0.506446i	0.967408	−	0.253223i	$-0.0814906\pi$
$60$	0.0881460i	0.0113796i
$61$	−5.55496	−0.711240	−0.355620	−	0.934631i	$-0.615730\pi$
$61$	−5.55496	−0.711240	−0.355620	+	0.934631i	$0.615730\pi$
$62$	0.0271471	0.00344768
$63$	0.586417i	0.0738816i
$64$	−8.89977	−1.11247
$65$	0	0
$66$	1.00000	0.123091
$67$	− 5.67025i	− 0.692731i	−0.938100	−	0.346366i	$-0.887416\pi$
$67$	− 5.67025i	− 0.692731i	0.938100	−	0.346366i	$-0.112584\pi$
$68$	2.40581	0.291748
$69$	1.43296	0.172508
$70$	0.246980i	0.0295197i
$71$	− 9.52111i	− 1.12995i	−0.825109	−	0.564974i	$-0.808886\pi$
$71$	− 9.52111i	− 1.12995i	0.825109	−	0.564974i	$-0.191114\pi$
$72$	9.02715i	1.06386i
$73$	11.5797i	1.35530i	0.735383	+	0.677651i	$0.237002\pi$
$73$	11.5797i	1.35530i	−0.735383	+	0.677651i	$0.762998\pi$
$74$	−1.86294	−0.216562
$75$	0.198062	0.0228703
$76$	0.972853i	0.111594i
$77$	−0.801938	−0.0913893
$78$	0	0
$79$	−15.8291	−1.78091	−0.890456	−	0.455070i	$-0.849614\pi$
$79$	−15.8291	−1.78091	−0.890456	+	0.455070i	$0.849614\pi$
$80$	2.91185i	0.325555i
$81$	8.64848	0.960942
$82$	−13.6679	−1.50936
$83$	17.3448i	1.90384i	0.306346	+	0.951920i	$0.400894\pi$
$83$	17.3448i	1.90384i	−0.306346	+	0.951920i	$0.599106\pi$
$84$	0.0174584i	0.00190487i
$85$	− 5.40581i	− 0.586343i
$86$	− 9.42088i	− 1.01588i
$87$	1.40150	0.150257
$88$	−12.3448	−1.31596
$89$	3.05429i	0.323755i	0.986811	+	0.161877i	$0.0517549\pi$
$89$	3.05429i	0.323755i	−0.986811	+	0.161877i	$0.948245\pi$
$90$	3.69202	0.389173
$91$	0	0
$92$	−3.21983	−0.335691
$93$	− 0.00431187i	0 0.000447120i
$94$	−1.29590	−0.133662
$95$	2.18598	0.224277
$96$	0.488582i	0.0498657i
$97$	5.02177i	0.509884i	0.966956	+	0.254942i	$0.0820563\pi$
$97$	5.02177i	0.509884i	−0.966956	+	0.254942i	$0.917944\pi$
$98$	− 8.67994i	− 0.876806i
$99$	11.9879i	1.20483i
$100$	−0.445042	−0.0445042
$101$	12.9487	1.28844	0.644221	−	0.764839i	$-0.277182\pi$
$101$	12.9487	1.28844	0.644221	+	0.764839i	$0.277182\pi$
$102$	1.33513i	0.132197i
$103$	13.4058	1.32091	0.660457	−	0.750864i	$-0.270363\pi$
$103$	13.4058	1.32091	0.660457	+	0.750864i	$0.270363\pi$
$104$	0	0
$105$	0.0392287	0.00382833
$106$	0.692021i	0.0672151i
$107$	0.994623	0.0961539	0.0480769	−	0.998844i	$-0.484691\pi$
$107$	0.994623	0.0961539	0.0480769	+	0.998844i	$0.484691\pi$
$108$	0.525418	0.0505584
$109$	− 19.3056i	− 1.84914i	−0.381013	−	0.924570i	$-0.624425\pi$
$109$	− 19.3056i	− 1.84914i	0.381013	−	0.924570i	$-0.375575\pi$
$110$	5.04892i	0.481395i
$111$	0.295897i	0.0280853i
$112$	0.576728i	0.0544957i
$113$	19.6136	1.84509	0.922544	−	0.385891i	$-0.126106\pi$
$113$	19.6136	1.84509	0.922544	+	0.385891i	$0.126106\pi$
$114$	−0.539893	−0.0505656
$115$	7.23490i	0.674658i
$116$	−3.14914	−0.292391
$117$	0	0
$118$	−4.85086	−0.446557
$119$	− 1.07069i	− 0.0981498i
$120$	0.603875	0.0551260
$121$	−5.39373	−0.490339
$122$	6.92692i	0.627134i
$123$	2.17092i	0.195745i
$124$	0.00968868i	0 0.000870069i
$125$	1.00000i	0.0894427i
$126$	0.731250	0.0651449
$127$	9.76271	0.866300	0.433150	−	0.901322i	$-0.357402\pi$
$127$	9.76271	0.866300	0.433150	+	0.901322i	$0.357402\pi$
$128$	6.16421i	0.544844i
$129$	−1.49635	−0.131746
$130$	0	0
$131$	14.6189	1.27726	0.638631	−	0.769513i	$-0.279501\pi$
$131$	14.6189	1.27726	0.638631	+	0.769513i	$0.279501\pi$
$132$	0.356896i	0.0310638i
$133$	0.432960	0.0375424
$134$	−7.07069	−0.610814
$135$	− 1.18060i	− 0.101610i
$136$	− 16.4819i	− 1.41331i
$137$	5.54288i	0.473560i	0.971563	+	0.236780i	$0.0760920\pi$
$137$	5.54288i	0.473560i	−0.971563	+	0.236780i	$0.923908\pi$
$138$	− 1.78687i	− 0.152109i
$139$	−0.545269	−0.0462492	−0.0231246	−	0.999733i	$-0.507361\pi$
$139$	−0.545269	−0.0462492	−0.0231246	+	0.999733i	$0.507361\pi$
$140$	−0.0881460	−0.00744970
$141$	0.205832i	0.0173342i
$142$	−11.8726	−0.996329
$143$	0	0
$144$	8.62133	0.718444
$145$	7.07606i	0.587635i
$146$	14.4397	1.19504
$147$	−1.37867	−0.113710
$148$	− 0.664874i	− 0.0546523i
$149$	0.768086i	0.0629240i	0.999505	+	0.0314620i	$0.0100163\pi$
$149$	0.768086i	0.0629240i	−0.999505	+	0.0314620i	$0.989984\pi$
$150$	− 0.246980i	− 0.0201658i
$151$	14.0097i	1.14009i	0.821613	+	0.570046i	$0.193075\pi$
$151$	14.0097i	1.14009i	−0.821613	+	0.570046i	$0.806925\pi$
$152$	6.66487	0.540593
$153$	−16.0054	−1.29396
$154$	1.00000i	0.0805823i
$155$	0.0217703	0.00174863
$156$	0	0
$157$	16.2620	1.29785	0.648926	−	0.760851i	$-0.275218\pi$
$157$	16.2620	1.29785	0.648926	+	0.760851i	$0.275218\pi$
$158$	19.7385i	1.57031i
$159$	0.109916	0.00871693
$160$	−2.46681	−0.195019
$161$	1.43296i	0.112933i
$162$	− 10.7845i	− 0.847309i
$163$	− 12.3666i	− 0.968626i	−0.874895	−	0.484313i	$-0.839070\pi$
$163$	− 12.3666i	− 0.968626i	0.874895	−	0.484313i	$-0.160930\pi$
$164$	− 4.87800i	− 0.380908i
$165$	0.801938	0.0624308
$166$	21.6286	1.67871
$167$	− 21.4523i	− 1.66003i	−0.557740	−	0.830016i	$-0.688331\pi$
$167$	− 21.4523i	− 1.66003i	0.557740	−	0.830016i	$-0.311669\pi$
$168$	0.119605	0.00922772
$169$	0	0
$170$	−6.74094	−0.517006
$171$	− 6.47219i	− 0.494941i
$172$	3.36227	0.256371
$173$	2.57002	0.195395	0.0976976	−	0.995216i	$-0.468852\pi$
$173$	2.57002	0.195395	0.0976976	+	0.995216i	$0.468852\pi$
$174$	− 1.74764i	− 0.132489i
$175$	0.198062i	0.0149721i
$176$	11.7899i	0.888694i
$177$	0.770479i	0.0579127i
$178$	3.80864	0.285470
$179$	15.1981	1.13596	0.567978	−	0.823044i	$-0.307726\pi$
$179$	15.1981	1.13596	0.567978	+	0.823044i	$0.307726\pi$
$180$	1.31767i	0.0982131i
$181$	−0.686645	−0.0510379	−0.0255189	−	0.999674i	$-0.508124\pi$
$181$	−0.686645	−0.0510379	−0.0255189	+	0.999674i	$0.508124\pi$
$182$	0	0
$183$	1.10023	0.0813312
$184$	22.0586i	1.62618i
$185$	−1.49396	−0.109838
$186$	−0.00537681	−0.000394247 0
$187$	− 21.8877i	− 1.60059i
$188$	− 0.462500i	− 0.0337313i
$189$	− 0.233833i	− 0.0170089i
$190$	− 2.72587i	− 0.197756i
$191$	3.47219	0.251239	0.125619	−	0.992079i	$-0.459908\pi$
$191$	3.47219	0.251239	0.125619	+	0.992079i	$0.459908\pi$
$192$	1.76271	0.127213
$193$	8.80625i	0.633888i	0.948444	+	0.316944i	$0.102657\pi$
$193$	8.80625i	0.633888i	−0.948444	+	0.316944i	$0.897343\pi$
$194$	6.26205	0.449589
$195$	0	0
$196$	3.09783	0.221274
$197$	12.3284i	0.878364i	0.898398	+	0.439182i	$0.144732\pi$
$197$	12.3284i	0.878364i	−0.898398	+	0.439182i	$0.855268\pi$
$198$	14.9487	1.06236
$199$	−6.48188	−0.459488	−0.229744	−	0.973251i	$-0.573789\pi$
$199$	−6.48188	−0.459488	−0.229744	+	0.973251i	$0.573789\pi$
$200$	3.04892i	0.215591i
$201$	1.12306i	0.0792147i
$202$	− 16.1468i	− 1.13608i
$203$	1.40150i	0.0983661i
$204$	−0.476501	−0.0333617
$205$	−10.9608	−0.765533
$206$	− 16.7168i	− 1.16471i
$207$	21.4209	1.48885
$208$	0	0
$209$	8.85086	0.612226
$210$	− 0.0489173i	− 0.00337562i
$211$	−14.6160	−1.00620	−0.503102	−	0.864227i	$-0.667808\pi$
$211$	−14.6160	−1.00620	−0.503102	+	0.864227i	$0.667808\pi$
$212$	−0.246980	−0.0169626
$213$	1.88577i	0.129211i
$214$	− 1.24027i	− 0.0847834i
$215$	− 7.55496i	− 0.515244i
$216$	− 3.59956i	− 0.244919i
$217$	0.00431187	0.000292709 0
$218$	−24.0737	−1.63047
$219$	− 2.29350i	− 0.154981i
$220$	−1.80194	−0.121487
$221$	0	0
$222$	0.368977	0.0247641
$223$	25.4644i	1.70522i	0.522545	+	0.852612i	$0.324983\pi$
$223$	25.4644i	1.70522i	−0.522545	+	0.852612i	$0.675017\pi$
$224$	−0.488582	−0.0326448
$225$	2.96077	0.197385
$226$	− 24.4577i	− 1.62690i
$227$	− 20.7332i	− 1.37611i	−0.725659	−	0.688054i	$-0.758465\pi$
$227$	− 20.7332i	− 1.37611i	0.725659	−	0.688054i	$-0.241535\pi$
$228$	− 0.192685i	− 0.0127609i
$229$	− 7.28621i	− 0.481486i	−0.970589	−	0.240743i	$-0.922609\pi$
$229$	− 7.28621i	− 0.481486i	0.970589	−	0.240743i	$-0.0773911\pi$
$230$	9.02177	0.594878
$231$	0.158834	0.0104505
$232$	21.5743i	1.41642i
$233$	6.06638	0.397421	0.198711	−	0.980058i	$-0.436325\pi$
$233$	6.06638	0.397421	0.198711	+	0.980058i	$0.436325\pi$
$234$	0	0
$235$	−1.03923	−0.0677918
$236$	− 1.73125i	− 0.112695i
$237$	3.13514	0.203650
$238$	−1.33513	−0.0865433
$239$	12.8213i	0.829342i	0.909971	+	0.414671i	$0.136103\pi$
$239$	12.8213i	0.829342i	−0.909971	+	0.414671i	$0.863897\pi$
$240$	− 0.576728i	− 0.0372277i
$241$	− 5.66919i	− 0.365184i	−0.983189	−	0.182592i	$-0.941551\pi$
$241$	− 5.66919i	− 0.365184i	0.983189	−	0.182592i	$-0.0584488\pi$
$242$	6.72587i	0.432356i
$243$	−5.25475	−0.337092
$244$	−2.47219	−0.158266
$245$	− 6.96077i	− 0.444707i
$246$	2.70709	0.172598
$247$	0	0
$248$	0.0663757	0.00421486
$249$	− 3.43535i	− 0.217707i
$250$	1.24698	0.0788659
$251$	−15.3642	−0.969779	−0.484890	−	0.874575i	$-0.661140\pi$
$251$	−15.3642	−0.969779	−0.484890	+	0.874575i	$0.661140\pi$
$252$	0.260980i	0.0164402i
$253$	29.2935i	1.84167i
$254$	− 12.1739i	− 0.763858i
$255$	1.07069i	0.0670491i
$256$	−10.1129	−0.632056
$257$	16.8659	1.05207	0.526034	−	0.850464i	$-0.323679\pi$
$257$	16.8659	1.05207	0.526034	+	0.850464i	$0.323679\pi$
$258$	1.86592i	0.116167i
$259$	−0.295897	−0.0183861
$260$	0	0
$261$	20.9506	1.29681
$262$	− 18.2295i	− 1.12622i
$263$	−6.03923	−0.372395	−0.186197	−	0.982512i	$-0.559616\pi$
$263$	−6.03923	−0.372395	−0.186197	+	0.982512i	$0.559616\pi$
$264$	2.44504	0.150482
$265$	0.554958i	0.0340908i
$266$	− 0.539893i	− 0.0331029i
$267$	− 0.604940i	− 0.0370218i
$268$	− 2.52350i	− 0.154147i
$269$	15.6963	0.957022	0.478511	−	0.878081i	$-0.341177\pi$
$269$	15.6963	0.957022	0.478511	+	0.878081i	$0.341177\pi$
$270$	−1.47219	−0.0895946
$271$	− 15.6310i	− 0.949517i	−0.880116	−	0.474758i	$-0.842535\pi$
$271$	− 15.6310i	− 0.949517i	0.880116	−	0.474758i	$-0.157465\pi$
$272$	−15.7409	−0.954435
$273$	0	0
$274$	6.91185	0.417560
$275$	4.04892i	0.244159i
$276$	0.637727	0.0383867
$277$	17.6189	1.05862	0.529310	−	0.848429i	$-0.322451\pi$
$277$	17.6189	1.05862	0.529310	+	0.848429i	$0.322451\pi$
$278$	0.679940i	0.0407801i
$279$	− 0.0644568i	− 0.00385893i
$280$	0.603875i	0.0360885i
$281$	− 4.04461i	− 0.241281i	−0.992696	−	0.120640i	$-0.961505\pi$
$281$	− 4.04461i	− 0.241281i	0.992696	−	0.120640i	$-0.0384948\pi$
$282$	0.256668	0.0152844
$283$	16.7409	0.995146	0.497573	−	0.867422i	$-0.334225\pi$
$283$	16.7409	0.995146	0.497573	+	0.867422i	$0.334225\pi$
$284$	− 4.23729i	− 0.251437i
$285$	−0.432960	−0.0256464
$286$	0	0
$287$	−2.17092	−0.128145
$288$	7.30367i	0.430373i
$289$	12.2228	0.718989
$290$	8.82371	0.518146
$291$	− 0.994623i	− 0.0583058i
$292$	5.15346i	0.301583i
$293$	− 8.67456i	− 0.506773i	−0.967365	−	0.253387i	$-0.918455\pi$
$293$	− 8.67456i	− 0.506773i	0.967365	−	0.253387i	$-0.0815445\pi$
$294$	1.71917i	0.100264i
$295$	−3.89008	−0.226489
$296$	−4.55496	−0.264752
$297$	− 4.78017i	− 0.277373i
$298$	0.957787	0.0554831
$299$	0	0
$300$	0.0881460	0.00508911
$301$	− 1.49635i	− 0.0862483i
$302$	17.4698	1.00527
$303$	−2.56465	−0.147335
$304$	− 6.36526i	− 0.365073i
$305$	5.55496i	0.318076i
$306$	19.9584i	1.14094i
$307$	− 15.6069i	− 0.890731i	−0.895349	−	0.445365i	$-0.853074\pi$
$307$	− 15.6069i	− 0.890731i	0.895349	−	0.445365i	$-0.146926\pi$
$308$	−0.356896	−0.0203360
$309$	−2.65519	−0.151048
$310$	− 0.0271471i	− 0.00154185i
$311$	−22.0965	−1.25298	−0.626489	−	0.779430i	$-0.715509\pi$
$311$	−22.0965	−1.25298	−0.626489	+	0.779430i	$0.715509\pi$
$312$	0	0
$313$	−19.6015	−1.10794	−0.553971	−	0.832536i	$-0.686888\pi$
$313$	−19.6015	−1.10794	−0.553971	+	0.832536i	$0.686888\pi$
$314$	− 20.2784i	− 1.14438i
$315$	0.586417	0.0330409
$316$	−7.04461	−0.396290
$317$	13.2470i	0.744024i	0.928228	+	0.372012i	$0.121332\pi$
$317$	13.2470i	0.744024i	−0.928228	+	0.372012i	$0.878668\pi$
$318$	− 0.137063i	− 0.00768613i
$319$	28.6504i	1.60411i
$320$	8.89977i	0.497512i
$321$	−0.196997	−0.0109953
$322$	1.78687	0.0995785
$323$	11.8170i	0.657516i
$324$	3.84894	0.213830
$325$	0	0
$326$	−15.4209	−0.854083
$327$	3.82371i	0.211452i
$328$	−33.4185	−1.84523
$329$	−0.205832	−0.0113479
$330$	− 1.00000i	− 0.0550482i
$331$	− 12.2121i	− 0.671236i	−0.941998	−	0.335618i	$-0.891055\pi$
$331$	− 12.2121i	− 0.671236i	0.941998	−	0.335618i	$-0.108945\pi$
$332$	7.71917i	0.423644i
$333$	4.42327i	0.242394i
$334$	−26.7506	−1.46373
$335$	−5.67025	−0.309799
$336$	− 0.114228i	− 0.00623166i
$337$	−5.59611	−0.304839	−0.152420	−	0.988316i	$-0.548707\pi$
$337$	−5.59611	−0.304839	−0.152420	+	0.988316i	$0.548707\pi$
$338$	0	0
$339$	−3.88471	−0.210988
$340$	− 2.40581i	− 0.130474i
$341$	0.0881460	0.00477337
$342$	−8.07069	−0.436413
$343$	− 2.76510i	− 0.149301i
$344$	− 23.0344i	− 1.24193i
$345$	− 1.43296i	− 0.0771480i
$346$	− 3.20477i	− 0.172289i
$347$	−20.8649	−1.12008	−0.560042	−	0.828464i	$-0.689215\pi$
$347$	−20.8649	−1.12008	−0.560042	+	0.828464i	$0.689215\pi$
$348$	0.623727	0.0334353
$349$	17.6082i	0.942545i	0.881988	+	0.471272i	$0.156205\pi$
$349$	17.6082i	0.942545i	−0.881988	+	0.471272i	$0.843795\pi$
$350$	0.246980	0.0132016
$351$	0	0
$352$	−9.98792	−0.532358
$353$	− 26.3043i	− 1.40003i	−0.714126	−	0.700017i	$-0.753176\pi$
$353$	− 26.3043i	− 1.40003i	0.714126	−	0.700017i	$-0.246824\pi$
$354$	0.960771	0.0510644
$355$	−9.52111	−0.505328
$356$	1.35929i	0.0720422i
$357$	0.212063i	0.0112236i
$358$	− 18.9517i	− 1.00163i
$359$	10.9685i	0.578897i	0.957194	+	0.289449i	$0.0934720\pi$
$359$	10.9685i	0.578897i	−0.957194	+	0.289449i	$0.906528\pi$
$360$	9.02715	0.475772
$361$	14.2215	0.748499
$362$	0.856232i	0.0450025i
$363$	1.06829	0.0560709
$364$	0	0
$365$	11.5797	0.606110
$366$	− 1.37196i	− 0.0717136i
$367$	−32.2693	−1.68445	−0.842223	−	0.539129i	$-0.818753\pi$
$367$	−32.2693	−1.68445	−0.842223	+	0.539129i	$0.818753\pi$
$368$	21.0670	1.09819
$369$	32.4523i	1.68940i
$370$	1.86294i	0.0968495i
$371$	0.109916i	0.00570657i
$372$	− 0.00191896i	0 9.94935e-5i
$373$	6.88338	0.356408	0.178204	−	0.983994i	$-0.442971\pi$
$373$	6.88338	0.356408	0.178204	+	0.983994i	$0.442971\pi$
$374$	−27.2935	−1.41131
$375$	− 0.198062i	− 0.0102279i
$376$	−3.16852	−0.163404
$377$	0	0
$378$	−0.291585	−0.0149975
$379$	− 12.2784i	− 0.630701i	−0.948975	−	0.315351i	$-0.897878\pi$
$379$	− 12.2784i	− 0.630701i	0.948975	−	0.315351i	$-0.102122\pi$
$380$	0.972853	0.0499063
$381$	−1.93362	−0.0990626
$382$	− 4.32975i	− 0.221529i
$383$	8.43296i	0.430904i	0.976514	+	0.215452i	$0.0691225\pi$
$383$	8.43296i	0.430904i	−0.976514	+	0.215452i	$0.930877\pi$
$384$	− 1.22090i	− 0.0623037i
$385$	0.801938i	0.0408705i
$386$	10.9812	0.558929
$387$	−22.3685	−1.13705
$388$	2.23490i	0.113460i
$389$	20.4198	1.03533	0.517663	−	0.855585i	$-0.326802\pi$
$389$	20.4198	1.03533	0.517663	+	0.855585i	$0.326802\pi$
$390$	0	0
$391$	−39.1105	−1.97790
$392$	− 21.2228i	− 1.07191i
$393$	−2.89546	−0.146057
$394$	15.3733	0.774495
$395$	15.8291i	0.796448i
$396$	5.33513i	0.268100i
$397$	18.6431i	0.935671i	0.883816	+	0.467835i	$0.154966\pi$
$397$	18.6431i	0.935671i	−0.883816	+	0.467835i	$0.845034\pi$
$398$	8.08277i	0.405153i
$399$	−0.0857531	−0.00429302
$400$	2.91185	0.145593
$401$	30.9831i	1.54722i	0.633660	+	0.773612i	$0.281552\pi$
$401$	30.9831i	1.54722i	−0.633660	+	0.773612i	$0.718448\pi$
$402$	1.40044	0.0698474
$403$	0	0
$404$	5.76271	0.286705
$405$	− 8.64848i	− 0.429746i
$406$	1.74764	0.0867341
$407$	−6.04892	−0.299834
$408$	3.26444i	0.161614i
$409$	23.8073i	1.17720i	0.808426	+	0.588598i	$0.200320\pi$
$409$	23.8073i	1.17720i	−0.808426	+	0.588598i	$0.799680\pi$
$410$	13.6679i	0.675007i
$411$	− 1.09783i	− 0.0541522i
$412$	5.96615	0.293931
$413$	−0.770479	−0.0379128
$414$	− 26.7114i	− 1.31279i
$415$	17.3448	0.851423
$416$	0	0
$417$	0.107997	0.00528865
$418$	− 11.0368i	− 0.539829i
$419$	12.1153	0.591871	0.295935	−	0.955208i	$-0.404369\pi$
$419$	12.1153	0.591871	0.295935	+	0.955208i	$0.404369\pi$
$420$	0.0174584	0.000851883 0
$421$	11.2537i	0.548471i	0.961663	+	0.274236i	$0.0884248\pi$
$421$	11.2537i	0.548471i	−0.961663	+	0.274236i	$0.911575\pi$
$422$	18.2258i	0.887218i
$423$	3.07692i	0.149605i
$424$	1.69202i	0.0821718i
$425$	−5.40581	−0.262220
$426$	2.35152	0.113931
$427$	1.10023i	0.0532437i
$428$	0.442649	0.0213962
$429$	0	0
$430$	−9.42088	−0.454315
$431$	− 17.3575i	− 0.836081i	−0.908428	−	0.418040i	$-0.862717\pi$
$431$	− 17.3575i	− 0.836081i	0.908428	−	0.418040i	$-0.137283\pi$
$432$	−3.43775	−0.165399
$433$	7.17390	0.344756	0.172378	−	0.985031i	$-0.444855\pi$
$433$	7.17390	0.344756	0.172378	+	0.985031i	$0.444855\pi$
$434$	− 0.00537681i	0 0.000258095i
$435$	− 1.40150i	− 0.0671968i
$436$	− 8.59179i	− 0.411472i
$437$	− 15.8153i	− 0.756551i
$438$	−2.85995	−0.136654
$439$	−10.3207	−0.492578	−0.246289	−	0.969196i	$-0.579211\pi$
$439$	−10.3207	−0.492578	−0.246289	+	0.969196i	$0.579211\pi$
$440$	12.3448i	0.588516i
$441$	−20.6093	−0.981393
$442$	0	0
$443$	−30.2271	−1.43613	−0.718067	−	0.695974i	$-0.754973\pi$
$443$	−30.2271	−1.43613	−0.718067	+	0.695974i	$0.754973\pi$
$444$	0.131687i	0.00624957i
$445$	3.05429	0.144787
$446$	31.7536	1.50358
$447$	− 0.152129i	− 0.00719545i
$448$	1.76271i	0.0832802i
$449$	24.7006i	1.16570i	0.812581	+	0.582848i	$0.198062\pi$
$449$	24.7006i	1.16570i	−0.812581	+	0.582848i	$0.801938\pi$
$450$	− 3.69202i	− 0.174044i
$451$	−44.3793	−2.08974
$452$	8.72886	0.410571
$453$	− 2.77479i	− 0.130371i
$454$	−25.8538	−1.21338
$455$	0	0
$456$	−1.32006	−0.0618175
$457$	15.2198i	0.711954i	0.934495	+	0.355977i	$0.115852\pi$
$457$	15.2198i	0.711954i	−0.934495	+	0.355977i	$0.884148\pi$
$458$	−9.08575	−0.424549
$459$	6.38212	0.297892
$460$	3.21983i	0.150125i
$461$	11.7995i	0.549560i	0.961507	+	0.274780i	$0.0886050\pi$
$461$	11.7995i	0.549560i	−0.961507	+	0.274780i	$0.911395\pi$
$462$	− 0.198062i	− 0.00921469i
$463$	− 13.2325i	− 0.614967i	−0.951553	−	0.307483i	$-0.900513\pi$
$463$	− 13.2325i	− 0.614967i	0.951553	−	0.307483i	$-0.0994868\pi$
$464$	20.6045	0.956538
$465$	−0.00431187	−0.000199958 0
$466$	− 7.56465i	− 0.350425i
$467$	8.01507	0.370893	0.185446	−	0.982654i	$-0.440627\pi$
$467$	8.01507	0.370893	0.185446	+	0.982654i	$0.440627\pi$
$468$	0	0
$469$	−1.12306	−0.0518582
$470$	1.29590i	0.0597753i
$471$	−3.22090	−0.148411
$472$	−11.8605	−0.545926
$473$	− 30.5894i	− 1.40650i
$474$	− 3.90946i	− 0.179567i
$475$	− 2.18598i	− 0.100300i
$476$	− 0.476501i	− 0.0218404i
$477$	1.64310	0.0752326
$478$	15.9879	0.731270
$479$	− 6.72886i	− 0.307449i	−0.988114	−	0.153725i	$-0.950873\pi$
$479$	− 6.72886i	− 0.307449i	0.988114	−	0.153725i	$-0.0491269\pi$
$480$	0.488582	0.0223006
$481$	0	0
$482$	−7.06936	−0.322001
$483$	− 0.283815i	− 0.0129140i
$484$	−2.40044	−0.109111
$485$	5.02177	0.228027
$486$	6.55257i	0.297230i
$487$	9.27545i	0.420311i	0.977668	+	0.210155i	$0.0673970\pi$
$487$	9.27545i	0.420311i	−0.977668	+	0.210155i	$0.932603\pi$
$488$	16.9366i	0.766684i
$489$	2.44935i	0.110764i
$490$	−8.67994	−0.392120
$491$	−7.94630	−0.358611	−0.179306	−	0.983793i	$-0.557385\pi$
$491$	−7.94630	−0.358611	−0.179306	+	0.983793i	$0.557385\pi$
$492$	0.966148i	0.0435573i
$493$	−38.2519	−1.72278
$494$	0	0
$495$	11.9879	0.538817
$496$	− 0.0633918i	− 0.00284638i
$497$	−1.88577	−0.0845884
$498$	−4.28382	−0.191962
$499$	22.0291i	0.986156i	0.869985	+	0.493078i	$0.164128\pi$
$499$	22.0291i	0.986156i	−0.869985	+	0.493078i	$0.835872\pi$
$500$	0.445042i	0.0199029i
$501$	4.24890i	0.189827i
$502$	19.1588i	0.855101i
$503$	−26.2693	−1.17129	−0.585646	−	0.810567i	$-0.699159\pi$
$503$	−26.2693	−1.17129	−0.585646	+	0.810567i	$0.699159\pi$
$504$	1.78794	0.0796411
$505$	− 12.9487i	− 0.576209i
$506$	36.5284	1.62389
$507$	0	0
$508$	4.34481	0.192770
$509$	− 35.4784i	− 1.57255i	−0.617875	−	0.786277i	$-0.712006\pi$
$509$	− 35.4784i	− 1.57255i	0.617875	−	0.786277i	$-0.287994\pi$
$510$	1.33513	0.0591204
$511$	2.29350	0.101459
$512$	24.9390i	1.10216i
$513$	2.58078i	0.113944i
$514$	− 21.0315i	− 0.927658i
$515$	− 13.4058i	− 0.590731i
$516$	−0.665939	−0.0293163
$517$	−4.20775	−0.185057
$518$	0.368977i	0.0162119i
$519$	−0.509025	−0.0223437
$520$	0	0
$521$	30.2295	1.32438	0.662190	−	0.749336i	$-0.269627\pi$
$521$	30.2295	1.32438	0.662190	+	0.749336i	$0.269627\pi$
$522$	− 26.1250i	− 1.14346i
$523$	2.23191	0.0975948	0.0487974	−	0.998809i	$-0.484461\pi$
$523$	2.23191	0.0975948	0.0487974	+	0.998809i	$0.484461\pi$
$524$	6.50604	0.284218
$525$	− 0.0392287i	− 0.00171208i
$526$	7.53079i	0.328358i
$527$	0.117686i	0.00512648i
$528$	− 2.33513i	− 0.101623i
$529$	29.3437	1.27582
$530$	0.692021	0.0300595
$531$	11.5176i	0.499823i
$532$	0.192685	0.00835397
$533$	0	0
$534$	−0.754348	−0.0326438
$535$	− 0.994623i	− 0.0430013i
$536$	−17.2881	−0.746733
$537$	−3.01016	−0.129898
$538$	− 19.5730i	− 0.843852i
$539$	− 28.1836i	− 1.21395i
$540$	− 0.525418i	− 0.0226104i
$541$	− 17.4306i	− 0.749399i	−0.927146	−	0.374699i	$-0.877746\pi$
$541$	− 17.4306i	− 0.749399i	0.927146	−	0.374699i	$-0.122254\pi$
$542$	−19.4916	−0.837234
$543$	0.135998	0.00583625
$544$	− 13.3351i	− 0.571739i
$545$	−19.3056	−0.826960
$546$	0	0
$547$	2.65578	0.113553	0.0567764	−	0.998387i	$-0.481918\pi$
$547$	2.65578	0.113553	0.0567764	+	0.998387i	$0.481918\pi$
$548$	2.46681i	0.105377i
$549$	16.4470	0.701939
$550$	5.04892	0.215287
$551$	− 15.4681i	− 0.658965i
$552$	− 4.36898i	− 0.185956i
$553$	3.13514i	0.133320i
$554$	− 21.9705i	− 0.933435i
$555$	0.295897	0.0125601
$556$	−0.242668	−0.0102914
$557$	− 30.4547i	− 1.29041i	−0.764010	−	0.645204i	$-0.776772\pi$
$557$	− 30.4547i	− 1.29041i	0.764010	−	0.645204i	$-0.223228\pi$
$558$	−0.0803763	−0.00340260
$559$	0	0
$560$	0.576728	0.0243712
$561$	4.33513i	0.183029i
$562$	−5.04354	−0.212749
$563$	−37.8170	−1.59380	−0.796898	−	0.604113i	$-0.793527\pi$
$563$	−37.8170	−1.59380	−0.796898	+	0.604113i	$0.793527\pi$
$564$	0.0916038i	0.00385722i
$565$	− 19.6136i	− 0.825149i
$566$	− 20.8756i	− 0.877467i
$567$	− 1.71294i	− 0.0719366i
$568$	−29.0291	−1.21803
$569$	−21.3720	−0.895959	−0.447980	−	0.894044i	$-0.647856\pi$
$569$	−21.3720	−0.895959	−0.447980	+	0.894044i	$0.647856\pi$
$570$	0.539893i	0.0226136i
$571$	18.8412	0.788478	0.394239	−	0.919008i	$-0.371008\pi$
$571$	18.8412	0.788478	0.394239	+	0.919008i	$0.371008\pi$
$572$	0	0
$573$	−0.687710	−0.0287295
$574$	2.70709i	0.112992i
$575$	7.23490	0.301716
$576$	26.3502	1.09792
$577$	− 37.6407i	− 1.56700i	−0.621390	−	0.783502i	$-0.713432\pi$
$577$	− 37.6407i	− 1.56700i	0.621390	−	0.783502i	$-0.286568\pi$
$578$	− 15.2416i	− 0.633967i
$579$	− 1.74419i	− 0.0724859i
$580$	3.14914i	0.130761i
$581$	3.43535	0.142522
$582$	−1.24027	−0.0514110
$583$	2.24698i	0.0930604i
$584$	35.3056	1.46096
$585$	0	0
$586$	−10.8170	−0.446846
$587$	39.9627i	1.64944i	0.565544	+	0.824718i	$0.308666\pi$
$587$	39.9627i	1.64944i	−0.565544	+	0.824718i	$0.691334\pi$
$588$	−0.613564	−0.0253030
$589$	−0.0475894	−0.00196089
$590$	4.85086i	0.199707i
$591$	− 2.44179i	− 0.100442i
$592$	4.35019i	0.178792i
$593$	− 28.0116i	− 1.15030i	−0.818048	−	0.575149i	$-0.804944\pi$
$593$	− 28.0116i	− 1.15030i	0.818048	−	0.575149i	$-0.195056\pi$
$594$	−5.96077	−0.244573
$595$	−1.07069	−0.0438939
$596$	0.341830i	0.0140019i
$597$	1.28382	0.0525431
$598$	0	0
$599$	−5.40880	−0.220997	−0.110499	−	0.993876i	$-0.535245\pi$
$599$	−5.40880	−0.220997	−0.110499	+	0.993876i	$0.535245\pi$
$600$	− 0.603875i	− 0.0246531i
$601$	−27.3263	−1.11466	−0.557331	−	0.830290i	$-0.688175\pi$
$601$	−27.3263	−1.11466	−0.557331	+	0.830290i	$0.688175\pi$
$602$	−1.86592	−0.0760492
$603$	16.7883i	0.683673i
$604$	6.23490i	0.253694i
$605$	5.39373i	0.219286i
$606$	3.19806i	0.129912i
$607$	27.8092	1.12874	0.564371	−	0.825521i	$-0.309119\pi$
$607$	27.8092	1.12874	0.564371	+	0.825521i	$0.309119\pi$
$608$	5.39240	0.218691
$609$	− 0.277585i	− 0.0112483i
$610$	6.92692	0.280463
$611$	0	0
$612$	−7.12306	−0.287933
$613$	11.5773i	0.467604i	0.972284	+	0.233802i	$0.0751167\pi$
$613$	11.5773i	0.467604i	−0.972284	+	0.233802i	$0.924883\pi$
$614$	−19.4614	−0.785400
$615$	2.17092	0.0875397
$616$	2.44504i	0.0985135i
$617$	24.7192i	0.995156i	0.867419	+	0.497578i	$0.165777\pi$
$617$	24.7192i	0.995156i	−0.867419	+	0.497578i	$0.834223\pi$
$618$	3.31096i	0.133186i
$619$	10.4983i	0.421961i	0.977490	+	0.210981i	$0.0676657\pi$
$619$	10.4983i	0.421961i	−0.977490	+	0.210981i	$0.932334\pi$
$620$	0.00968868	0.000389107 0
$621$	−8.54155	−0.342761
$622$	27.5539i	1.10481i
$623$	0.604940	0.0242364
$624$	0	0
$625$	1.00000	0.0400000
$626$	24.4426i	0.976925i
$627$	−1.75302	−0.0700089
$628$	7.23729	0.288799
$629$	− 8.07606i	− 0.322014i
$630$	− 0.731250i	− 0.0291337i
$631$	− 33.9124i	− 1.35003i	−0.737803	−	0.675017i	$-0.764136\pi$
$631$	− 33.9124i	− 1.35003i	0.737803	−	0.675017i	$-0.235864\pi$
$632$	48.2616i	1.91974i
$633$	2.89487	0.115061
$634$	16.5187	0.656042
$635$	− 9.76271i	− 0.387421i
$636$	0.0489173	0.00193970
$637$	0	0
$638$	35.7265	1.41442
$639$	28.1898i	1.11517i
$640$	6.16421	0.243662
$641$	−11.8170	−0.466744	−0.233372	−	0.972388i	$-0.574976\pi$
$641$	−11.8170	−0.466744	−0.233372	+	0.972388i	$0.574976\pi$
$642$	0.245652i	0.00969510i
$643$	− 26.6746i	− 1.05194i	−0.850502	−	0.525971i	$-0.823702\pi$
$643$	− 26.6746i	− 1.05194i	0.850502	−	0.525971i	$-0.176298\pi$
$644$	0.637727i	0.0251300i
$645$	1.49635i	0.0589188i
$646$	14.7356	0.579763
$647$	−32.6485	−1.28354	−0.641772	−	0.766895i	$-0.721800\pi$
$647$	−32.6485	−1.28354	−0.641772	+	0.766895i	$0.721800\pi$
$648$	− 26.3685i	− 1.03585i
$649$	−15.7506	−0.618266
$650$	0	0
$651$	−0.000854018 0	−3.34716e−5 0
$652$	− 5.50365i	− 0.215539i
$653$	4.06936	0.159246	0.0796232	−	0.996825i	$-0.474628\pi$
$653$	4.06936	0.159246	0.0796232	+	0.996825i	$0.474628\pi$
$654$	4.76809	0.186447
$655$	− 14.6189i	− 0.571209i
$656$	31.9162i	1.24612i
$657$	− 34.2849i	− 1.33758i
$658$	0.256668i	0.0100060i
$659$	−21.9909	−0.856644	−0.428322	−	0.903626i	$-0.640895\pi$
$659$	−21.9909	−0.856644	−0.428322	+	0.903626i	$0.640895\pi$
$660$	0.356896	0.0138922
$661$	11.8562i	0.461154i	0.973054	+	0.230577i	$0.0740614\pi$
$661$	11.8562i	0.461154i	−0.973054	+	0.230577i	$0.925939\pi$
$662$	−15.2282	−0.591861
$663$	0	0
$664$	52.8829	2.05225
$665$	− 0.432960i	− 0.0167895i
$666$	5.51573	0.213730
$667$	51.1946	1.98226
$668$	− 9.54719i	− 0.369392i
$669$	− 5.04354i	− 0.194995i
$670$	7.07069i	0.273164i
$671$	22.4916i	0.868277i
$672$	0.0967697	0.00373297
$673$	24.1943	0.932623	0.466312	−	0.884621i	$-0.345583\pi$
$673$	24.1943	0.932623	0.466312	+	0.884621i	$0.345583\pi$
$674$	6.97823i	0.268791i
$675$	−1.18060	−0.0454415
$676$	0	0
$677$	25.3806	0.975455	0.487728	−	0.872996i	$-0.337826\pi$
$677$	25.3806	0.975455	0.487728	+	0.872996i	$0.337826\pi$
$678$	4.84415i	0.186038i
$679$	0.994623	0.0381701
$680$	−16.4819	−0.632051
$681$	4.10646i	0.157360i
$682$	− 0.109916i	− 0.00420891i
$683$	19.2741i	0.737504i	0.929528	+	0.368752i	$0.120215\pi$
$683$	19.2741i	0.737504i	−0.929528	+	0.368752i	$0.879785\pi$
$684$	− 2.88040i	− 0.110135i
$685$	5.54288	0.211782
$686$	−3.44803	−0.131646
$687$	1.44312i	0.0550586i
$688$	−21.9989	−0.838702
$689$	0	0
$690$	−1.78687	−0.0680251
$691$	33.8049i	1.28600i	0.765866	+	0.643000i	$0.222310\pi$
$691$	33.8049i	1.28600i	−0.765866	+	0.643000i	$0.777690\pi$
$692$	1.14377	0.0434795
$693$	2.37435	0.0901943
$694$	26.0180i	0.987632i
$695$	0.545269i	0.0206832i
$696$	− 4.27306i	− 0.161970i
$697$	− 59.2519i	− 2.24433i
$698$	21.9571	0.831087
$699$	−1.20152	−0.0454457
$700$	0.0881460i	0.00333161i
$701$	−15.5907	−0.588854	−0.294427	−	0.955674i	$-0.595129\pi$
$701$	−15.5907	−0.588854	−0.294427	+	0.955674i	$0.595129\pi$
$702$	0	0
$703$	3.26577	0.123171
$704$	36.0344i	1.35810i
$705$	0.205832	0.00775208
$706$	−32.8009	−1.23448
$707$	− 2.56465i	− 0.0964535i
$708$	0.342895i	0.0128868i
$709$	− 6.09485i	− 0.228897i	−0.993429	−	0.114448i	$-0.963490\pi$
$709$	− 6.09485i	− 0.228897i	0.993429	−	0.114448i	$-0.0365101\pi$
$710$	11.8726i	0.445572i
$711$	46.8663	1.75762
$712$	9.31229	0.348993
$713$	− 0.157506i	− 0.00589863i
$714$	0.264438	0.00989634
$715$	0	0
$716$	6.76377	0.252774
$717$	− 2.53942i	− 0.0948363i
$718$	13.6775	0.510442
$719$	−26.5961	−0.991867	−0.495934	−	0.868360i	$-0.665174\pi$
$719$	−26.5961	−0.991867	−0.495934	+	0.868360i	$0.665174\pi$
$720$	− 8.62133i	− 0.321298i
$721$	− 2.65519i	− 0.0988843i
$722$	− 17.7339i	− 0.659988i
$723$	1.12285i	0.0417593i
$724$	−0.305586	−0.0113570
$725$	7.07606	0.262798
$726$	− 1.33214i	− 0.0494404i
$727$	9.91915	0.367881	0.183940	−	0.982937i	$-0.441115\pi$
$727$	9.91915	0.367881	0.183940	+	0.982937i	$0.441115\pi$
$728$	0	0
$729$	−24.9047	−0.922395
$730$	− 14.4397i	− 0.534436i
$731$	40.8407	1.51055
$732$	0.489647	0.0180979
$733$	− 26.2194i	− 0.968434i	−0.874948	−	0.484217i	$-0.839105\pi$
$733$	− 26.2194i	− 0.968434i	0.874948	−	0.484217i	$-0.160895\pi$
$734$	40.2392i	1.48526i
$735$	1.37867i	0.0508529i
$736$	17.8471i	0.657854i
$737$	−22.9584	−0.845683
$738$	40.4674	1.48963
$739$	3.50471i	0.128923i	0.997920	+	0.0644615i	$0.0205330\pi$
$739$	3.50471i	0.128923i	−0.997920	+	0.0644615i	$0.979467\pi$
$740$	−0.664874	−0.0244413
$741$	0	0
$742$	0.137063	0.00503175
$743$	− 17.7138i	− 0.649856i	−0.945739	−	0.324928i	$-0.894660\pi$
$743$	− 17.7138i	− 0.649856i	0.945739	−	0.324928i	$-0.105340\pi$
$744$	−0.0131465	−0.000481975 0
$745$	0.768086	0.0281405
$746$	− 8.58343i	− 0.314262i
$747$	− 51.3540i	− 1.87895i
$748$	− 9.74094i	− 0.356164i
$749$	− 0.196997i	− 0.00719813i
$750$	−0.246980	−0.00901842
$751$	−50.1879	−1.83138	−0.915691	−	0.401883i	$-0.868356\pi$
$751$	−50.1879	−1.83138	−0.915691	+	0.401883i	$0.868356\pi$
$752$	3.02608i	0.110350i
$753$	3.04307	0.110896
$754$	0	0
$755$	14.0097	0.509865
$756$	− 0.104066i	− 0.00378483i
$757$	37.8256	1.37480	0.687398	−	0.726281i	$-0.258753\pi$
$757$	37.8256	1.37480	0.687398	+	0.726281i	$0.258753\pi$
$758$	−15.3110	−0.556119
$759$	− 5.80194i	− 0.210597i
$760$	− 6.66487i	− 0.241760i
$761$	13.3080i	0.482414i	0.970474	+	0.241207i	$0.0775432\pi$
$761$	13.3080i	0.482414i	−0.970474	+	0.241207i	$0.922457\pi$
$762$	2.41119i	0.0873482i
$763$	−3.82371	−0.138428
$764$	1.54527	0.0559059
$765$	16.0054i	0.578676i
$766$	10.5157	0.379949
$767$	0	0
$768$	2.00298	0.0722765
$769$	9.45904i	0.341102i	0.985349	+	0.170551i	$0.0545547\pi$
$769$	9.45904i	0.341102i	−0.985349	+	0.170551i	$0.945445\pi$
$770$	1.00000	0.0360375
$771$	−3.34050	−0.120305
$772$	3.91915i	0.141053i
$773$	− 50.2137i	− 1.80606i	−0.429575	−	0.903031i	$-0.641337\pi$
$773$	− 50.2137i	− 1.80606i	0.429575	−	0.903031i	$-0.358663\pi$
$774$	27.8931i	1.00260i
$775$	− 0.0217703i	0 0.000782011i
$776$	15.3110	0.549632
$777$	0.0586060	0.00210248
$778$	− 25.4631i	− 0.912896i
$779$	23.9600	0.858457
$780$	0	0
$781$	−38.5502	−1.37943
$782$	48.7700i	1.74401i
$783$	−8.35403	−0.298549
$784$	−20.2687	−0.723884
$785$	− 16.2620i	− 0.580417i
$786$	3.61058i	0.128785i
$787$	− 5.07175i	− 0.180788i	−0.995906	−	0.0903942i	$-0.971187\pi$
$787$	− 5.07175i	− 0.180788i	0.995906	−	0.0903942i	$-0.0288127\pi$
$788$	5.48666i	0.195454i
$789$	1.19614	0.0425838
$790$	19.7385	0.702266
$791$	− 3.88471i	− 0.138124i
$792$	36.5502	1.29875
$793$	0	0
$794$	23.2476	0.825025
$795$	− 0.109916i	− 0.00389833i
$796$	−2.88471	−0.102246
$797$	9.07798	0.321559	0.160779	−	0.986990i	$-0.448599\pi$
$797$	9.07798	0.321559	0.160779	+	0.986990i	$0.448599\pi$
$798$	0.106932i	0.00378536i
$799$	− 5.61788i	− 0.198746i
$800$	2.46681i	0.0872150i
$801$	− 9.04307i	− 0.319521i
$802$	38.6353	1.36426
$803$	46.8853	1.65455
$804$	0.499810i	0.0176269i
$805$	1.43296	0.0505052
$806$	0	0
$807$	−3.10885	−0.109437
$808$	− 39.4795i	− 1.38888i
$809$	33.2553	1.16920	0.584598	−	0.811323i	$-0.301252\pi$
$809$	33.2553	1.16920	0.584598	+	0.811323i	$0.301252\pi$
$810$	−10.7845	−0.378928
$811$	21.0258i	0.738316i	0.929367	+	0.369158i	$0.120354\pi$
$811$	21.0258i	0.738316i	−0.929367	+	0.369158i	$0.879646\pi$
$812$	0.623727i	0.0218885i
$813$	3.09592i	0.108579i
$814$	7.54288i	0.264378i
$815$	−12.3666	−0.433183
$816$	3.11769	0.109141
$817$	16.5150i	0.577786i
$818$	29.6872	1.03799
$819$	0	0
$820$	−4.87800	−0.170347
$821$	13.7463i	0.479750i	0.970804	+	0.239875i	$0.0771064\pi$
$821$	13.7463i	0.479750i	−0.970804	+	0.239875i	$0.922894\pi$
$822$	−1.36898	−0.0477486
$823$	−37.3159	−1.30075	−0.650375	−	0.759614i	$-0.725388\pi$
$823$	−37.3159	−1.30075	−0.650375	+	0.759614i	$0.725388\pi$
$824$	− 40.8732i	− 1.42389i
$825$	− 0.801938i	− 0.0279199i
$826$	0.960771i	0.0334295i
$827$	36.5478i	1.27089i	0.772146	+	0.635445i	$0.219183\pi$
$827$	36.5478i	1.27089i	−0.772146	+	0.635445i	$0.780817\pi$
$828$	9.53319	0.331301
$829$	32.4566	1.12727	0.563633	−	0.826025i	$-0.309403\pi$
$829$	32.4566	1.12727	0.563633	+	0.826025i	$0.309403\pi$
$830$	− 21.6286i	− 0.750741i
$831$	−3.48965	−0.121055
$832$	0	0
$833$	37.6286	1.30375
$834$	− 0.134670i	− 0.00466326i
$835$	−21.4523	−0.742389
$836$	3.93900	0.136233
$837$	0.0257021i	0 0.000888393i
$838$	− 15.1075i	− 0.521881i
$839$	− 12.5670i	− 0.433862i	−0.976187	−	0.216931i	$-0.930395\pi$
$839$	− 12.5670i	− 0.433862i	0.976187	−	0.216931i	$-0.0696047\pi$
$840$	− 0.119605i	− 0.00412676i
$841$	21.0707	0.726575
$842$	14.0331	0.483613
$843$	0.801084i	0.0275908i
$844$	−6.50471	−0.223901
$845$	0	0
$846$	3.83685	0.131914
$847$	1.06829i	0.0367070i
$848$	1.61596	0.0554922
$849$	−3.31575	−0.113796
$850$	6.74094i	0.231212i
$851$	10.8086i	0.370515i
$852$	0.839247i	0.0287521i
$853$	38.3220i	1.31212i	0.754709	+	0.656060i	$0.227778\pi$
$853$	38.3220i	1.31212i	−0.754709	+	0.656060i	$0.772222\pi$
$854$	1.37196	0.0469476
$855$	−6.47219	−0.221344
$856$	− 3.03252i	− 0.103650i
$857$	−35.9081	−1.22660	−0.613299	−	0.789851i	$-0.710158\pi$
$857$	−35.9081	−1.22660	−0.613299	+	0.789851i	$0.710158\pi$
$858$	0	0
$859$	−46.2881	−1.57933	−0.789665	−	0.613538i	$-0.789746\pi$
$859$	−46.2881	−1.57933	−0.789665	+	0.613538i	$0.789746\pi$
$860$	− 3.36227i	− 0.114653i
$861$	0.429976	0.0146536
$862$	−21.6444	−0.737212
$863$	7.86725i	0.267804i	0.990995	+	0.133902i	$0.0427508\pi$
$863$	7.86725i	0.267804i	−0.990995	+	0.133902i	$0.957249\pi$
$864$	− 2.91233i	− 0.0990794i
$865$	− 2.57002i	− 0.0873834i
$866$	− 8.94571i	− 0.303987i
$867$	−2.42088	−0.0822174
$868$	0.00191896	6.51338e−5 0
$869$	64.0907i	2.17413i
$870$	−1.74764	−0.0592507
$871$	0	0
$872$	−58.8611	−1.99329
$873$	− 14.8683i	− 0.503216i
$874$	−19.7214	−0.667087
$875$	0.198062	0.00669573
$876$	− 1.02071i	− 0.0344864i
$877$	41.3913i	1.39769i	0.715275	+	0.698843i	$0.246301\pi$
$877$	41.3913i	1.39769i	−0.715275	+	0.698843i	$0.753699\pi$
$878$	12.8696i	0.434329i
$879$	1.71810i	0.0579502i
$880$	11.7899	0.397436
$881$	−38.8558	−1.30908	−0.654542	−	0.756026i	$-0.727138\pi$
$881$	−38.8558	−1.30908	−0.654542	+	0.756026i	$0.727138\pi$
$882$	25.6993i	0.865341i
$883$	6.22713	0.209560	0.104780	−	0.994495i	$-0.466586\pi$
$883$	6.22713	0.209560	0.104780	+	0.994495i	$0.466586\pi$
$884$	0	0
$885$	0.770479	0.0258994
$886$	37.6926i	1.26631i
$887$	36.3274	1.21975	0.609877	−	0.792496i	$-0.291219\pi$
$887$	36.3274	1.21975	0.609877	+	0.792496i	$0.291219\pi$
$888$	0.902165	0.0302747
$889$	− 1.93362i	− 0.0648517i
$890$	− 3.80864i	− 0.127666i
$891$	− 35.0170i	− 1.17311i
$892$	11.3327i	0.379448i
$893$	2.27173	0.0760207
$894$	−0.189702	−0.00634457
$895$	− 15.1981i	− 0.508015i
$896$	1.22090	0.0407873
$897$	0	0
$898$	30.8012	1.02785
$899$	− 0.154048i	− 0.00513778i
$900$	1.31767	0.0439222
$901$	−3.00000	−0.0999445
$902$	55.3400i	1.84262i
$903$	0.296371i	0.00986261i
$904$	− 59.8001i	− 1.98892i
$905$	0.686645i	0.0228248i
$906$	−3.46011	−0.114954
$907$	4.34721	0.144347	0.0721733	−	0.997392i	$-0.477007\pi$
$907$	4.34721	0.144347	0.0721733	+	0.997392i	$0.477007\pi$
$908$	− 9.22713i	− 0.306213i
$909$	−38.3381	−1.27159
$910$	0	0
$911$	−21.8866	−0.725136	−0.362568	−	0.931957i	$-0.618100\pi$
$911$	−21.8866	−0.725136	−0.362568	+	0.931957i	$0.618100\pi$
$912$	1.26072i	0.0417465i
$913$	70.2277	2.32420
$914$	18.9788	0.627764
$915$	− 1.10023i	− 0.0363724i
$916$	− 3.24267i	− 0.107141i
$917$	− 2.89546i	− 0.0956165i
$918$	− 7.95838i	− 0.262666i
$919$	25.6292	0.845430	0.422715	−	0.906263i	$-0.361077\pi$
$919$	25.6292	0.845430	0.422715	+	0.906263i	$0.361077\pi$
$920$	22.0586	0.727251
$921$	3.09113i	0.101856i
$922$	14.7138	0.484573
$923$	0	0
$924$	0.0706876	0.00232545
$925$	1.49396i	0.0491211i
$926$	−16.5007	−0.542245
$927$	−39.6915	−1.30364
$928$	17.4553i	0.572999i
$929$	− 13.6635i	− 0.448286i	−0.974556	−	0.224143i	$-0.928042\pi$
$929$	− 13.6635i	− 0.448286i	0.974556	−	0.224143i	$-0.0719583\pi$
$930$	0.00537681i	0 0.000176313i
$931$	15.2161i	0.498688i
$932$	2.69979	0.0884346
$933$	4.37648	0.143280
$934$	− 9.99462i	− 0.327034i
$935$	−21.8877	−0.715804
$936$	0	0
$937$	17.1381	0.559878	0.279939	−	0.960018i	$-0.409686\pi$
$937$	17.1381	0.559878	0.279939	+	0.960018i	$0.409686\pi$
$938$	1.40044i	0.0457259i
$939$	3.88231	0.126695
$940$	−0.462500	−0.0150851
$941$	2.58775i	0.0843581i	0.999110	+	0.0421790i	$0.0134300\pi$
$941$	2.58775i	0.0843581i	−0.999110	+	0.0421790i	$0.986570\pi$
$942$	4.01639i	0.130861i
$943$	79.3001i	2.58237i
$944$	11.3274i	0.368674i
$945$	−0.233833	−0.00760659
$946$	−38.1444	−1.24018
$947$	− 11.2078i	− 0.364203i	−0.983280	−	0.182101i	$-0.941710\pi$
$947$	− 11.2078i	− 0.364203i	0.983280	−	0.182101i	$-0.0582899\pi$
$948$	1.39527	0.0453163
$949$	0	0
$950$	−2.72587	−0.0884390
$951$	− 2.62373i	− 0.0850802i
$952$	−3.26444	−0.105801
$953$	38.2591	1.23933	0.619666	−	0.784865i	$-0.287268\pi$
$953$	38.2591	1.23933	0.619666	+	0.784865i	$0.287268\pi$
$954$	− 2.04892i	− 0.0663361i
$955$	− 3.47219i	− 0.112357i
$956$	5.70602i	0.184546i
$957$	− 5.67456i	− 0.183433i
$958$	−8.39075	−0.271093
$959$	1.09783	0.0354509
$960$	− 1.76271i	− 0.0568912i
$961$	30.9995	0.999985
$962$	0	0
$963$	−2.94485	−0.0948965
$964$	− 2.52303i	− 0.0812612i
$965$	8.80625	0.283483
$966$	−0.353912	−0.0113869
$967$	− 5.26875i	− 0.169432i	−0.996405	−	0.0847158i	$-0.973002\pi$
$967$	− 5.26875i	− 0.169432i	0.996405	−	0.0847158i	$-0.0269982\pi$
$968$	16.4450i	0.528564i
$969$	− 2.34050i	− 0.0751878i
$970$	− 6.26205i	− 0.201062i
$971$	−7.31203	−0.234654	−0.117327	−	0.993093i	$-0.537433\pi$
$971$	−7.31203	−0.234654	−0.117327	+	0.993093i	$0.537433\pi$
$972$	−2.33858	−0.0750101
$973$	0.107997i	0.00346223i
$974$	11.5663	0.370608
$975$	0	0
$976$	16.1752	0.517756
$977$	31.3661i	1.00349i	0.865015	+	0.501745i	$0.167309\pi$
$977$	31.3661i	1.00349i	−0.865015	+	0.501745i	$0.832691\pi$
$978$	3.05429	0.0976656
$979$	12.3666	0.395238
$980$	− 3.09783i	− 0.0989567i
$981$	57.1594i	1.82496i
$982$	9.90887i	0.316205i
$983$	35.9724i	1.14734i	0.819086	+	0.573670i	$0.194481\pi$
$983$	35.9724i	1.14734i	−0.819086	+	0.573670i	$0.805519\pi$
$984$	6.61894	0.211004
$985$	12.3284	0.392816
$986$	47.6993i	1.51906i
$987$	0.0407675	0.00129765
$988$	0	0
$989$	−54.6594	−1.73807
$990$	− 14.9487i	− 0.475101i
$991$	21.2476	0.674951	0.337476	−	0.941334i	$-0.390427\pi$
$991$	21.2476	0.674951	0.337476	+	0.941334i	$0.390427\pi$
$992$	0.0537032	0.00170508
$993$	2.41875i	0.0767567i
$994$	2.35152i	0.0745857i
$995$	6.48188i	0.205489i
$996$	− 1.52888i	− 0.0484443i
$997$	38.4040	1.21627	0.608134	−	0.793835i	$-0.291918\pi$
$997$	38.4040	1.21627	0.608134	+	0.793835i	$0.291918\pi$
$998$	27.4698	0.869541
$999$	− 1.76377i	− 0.0558033i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	845.2.c.f.506.2		6
13.2	odd	12		845.2.e.j.191.3		6
13.3	even	3		845.2.m.i.316.5		12
13.4	even	6		845.2.m.i.361.5		12
13.5	odd	4		845.2.a.j.1.1	yes	3
13.6	odd	12		845.2.e.j.146.3		6
13.7	odd	12		845.2.e.l.146.1		6
13.8	odd	4		845.2.a.h.1.3	✓	3
13.9	even	3		845.2.m.i.361.2		12
13.10	even	6		845.2.m.i.316.2		12
13.11	odd	12		845.2.e.l.191.1		6
13.12	even	2	inner	845.2.c.f.506.5		6
39.5	even	4		7605.2.a.br.1.3		3
39.8	even	4		7605.2.a.by.1.1		3
65.34	odd	4		4225.2.a.bf.1.1		3
65.44	odd	4		4225.2.a.bd.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
845.2.a.h.1.3	✓	3	13.8	odd	4
845.2.a.j.1.1	yes	3	13.5	odd	4
845.2.c.f.506.2		6	1.1	even	1	trivial
845.2.c.f.506.5		6	13.12	even	2	inner
845.2.e.j.146.3		6	13.6	odd	12
845.2.e.j.191.3		6	13.2	odd	12
845.2.e.l.146.1		6	13.7	odd	12
845.2.e.l.191.1		6	13.11	odd	12
845.2.m.i.316.2		12	13.10	even	6
845.2.m.i.316.5		12	13.3	even	3
845.2.m.i.361.2		12	13.9	even	3
845.2.m.i.361.5		12	13.4	even	6
4225.2.a.bd.1.3		3	65.44	odd	4
4225.2.a.bf.1.1		3	65.34	odd	4
7605.2.a.br.1.3		3	39.5	even	4
7605.2.a.by.1.1		3	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)

\(f(q)\)	\(=\)	\(q-1.24698i q^{2} -0.198062 q^{3} +0.445042 q^{4} -1.00000i q^{5} +0.246980i q^{6} -0.198062i q^{7} -3.04892i q^{8} -2.96077 q^{9} +O(q^{10})\)	\(q-1.24698i q^{2} -0.198062 q^{3} +0.445042 q^{4} -1.00000i q^{5} +0.246980i q^{6} -0.198062i q^{7} -3.04892i q^{8} -2.96077 q^{9} -1.24698 q^{10} -4.04892i q^{11} -0.0881460 q^{12} -0.246980 q^{14} +0.198062i q^{15} -2.91185 q^{16} +5.40581 q^{17} +3.69202i q^{18} +2.18598i q^{19} -0.445042i q^{20} +0.0392287i q^{21} -5.04892 q^{22} -7.23490 q^{23} +0.603875i q^{24} -1.00000 q^{25} +1.18060 q^{27} -0.0881460i q^{28} -7.07606 q^{29} +0.246980 q^{30} +0.0217703i q^{31} -2.46681i q^{32} +0.801938i q^{33} -6.74094i q^{34} -0.198062 q^{35} -1.31767 q^{36} -1.49396i q^{37} +2.72587 q^{38} -3.04892 q^{40} -10.9608i q^{41} +0.0489173 q^{42} +7.55496 q^{43} -1.80194i q^{44} +2.96077i q^{45} +9.02177i q^{46} -1.03923i q^{47} +0.576728 q^{48} +6.96077 q^{49} +1.24698i q^{50} -1.07069 q^{51} -0.554958 q^{53} -1.47219i q^{54} -4.04892 q^{55} -0.603875 q^{56} -0.432960i q^{57} +8.82371i q^{58} -3.89008i q^{59} +0.0881460i q^{60} -5.55496 q^{61} +0.0271471 q^{62} +0.586417i q^{63} -8.89977 q^{64} +1.00000 q^{66} -5.67025i q^{67} +2.40581 q^{68} +1.43296 q^{69} +0.246980i q^{70} -9.52111i q^{71} +9.02715i q^{72} +11.5797i q^{73} -1.86294 q^{74} +0.198062 q^{75} +0.972853i q^{76} -0.801938 q^{77} -15.8291 q^{79} +2.91185i q^{80} +8.64848 q^{81} -13.6679 q^{82} +17.3448i q^{83} +0.0174584i q^{84} -5.40581i q^{85} -9.42088i q^{86} +1.40150 q^{87} -12.3448 q^{88} +3.05429i q^{89} +3.69202 q^{90} -3.21983 q^{92} -0.00431187i q^{93} -1.29590 q^{94} +2.18598 q^{95} +0.488582i q^{96} +5.02177i q^{97} -8.67994i q^{98} +11.9879i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 10 q^{3} + 2 q^{4} + 8 q^{9}+O(q^{10}) \) 6 * q - 10 * q^3 + 2 * q^4 + 8 * q^9	\( 6 q - 10 q^{3} + 2 q^{4} + 8 q^{9} + 2 q^{10} - 8 q^{12} + 8 q^{14} - 10 q^{16} + 6 q^{17} - 12 q^{22} + 4 q^{23} - 6 q^{25} - 16 q^{27} - 12 q^{29} - 8 q^{30} - 10 q^{35} + 26 q^{36} + 38 q^{38} - 18 q^{42} + 46 q^{43} - 2 q^{48} + 16 q^{49} + 18 q^{51} - 4 q^{53} - 6 q^{55} + 14 q^{56} - 34 q^{61} - 12 q^{62} - 8 q^{64} + 6 q^{66} - 12 q^{68} - 30 q^{69} - 22 q^{74} + 10 q^{75} + 4 q^{77} - 74 q^{79} + 54 q^{81} + 4 q^{82} + 20 q^{87} - 28 q^{88} + 12 q^{90} - 22 q^{92} + 20 q^{94} - 16 q^{95}+O(q^{100}) \) 6 * q - 10 * q^3 + 2 * q^4 + 8 * q^9 + 2 * q^10 - 8 * q^12 + 8 * q^14 - 10 * q^16 + 6 * q^17 - 12 * q^22 + 4 * q^23 - 6 * q^25 - 16 * q^27 - 12 * q^29 - 8 * q^30 - 10 * q^35 + 26 * q^36 + 38 * q^38 - 18 * q^42 + 46 * q^43 - 2 * q^48 + 16 * q^49 + 18 * q^51 - 4 * q^53 - 6 * q^55 + 14 * q^56 - 34 * q^61 - 12 * q^62 - 8 * q^64 + 6 * q^66 - 12 * q^68 - 30 * q^69 - 22 * q^74 + 10 * q^75 + 4 * q^77 - 74 * q^79 + 54 * q^81 + 4 * q^82 + 20 * q^87 - 28 * q^88 + 12 * q^90 - 22 * q^92 + 20 * q^94 - 16 * q^95

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