LMFDB - Embedded newform 1216.2.c.h.609.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1216,2,Mod(609,1216)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1216, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1216.609");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1216 = 2^{6} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1216.c (of order $2$, degree $1$, 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:	$9.70980888579$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.303595776.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 5x^{6} + 16x^{4} + 45x^{2} + 81 $ x^8 + 5x^6 + 16x^4 + 45*x^2 + 81
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			609.5
Root			$0.396143 + 1.68614i$ of defining polynomial
Character	$\chi$	$=$	1216.609
Dual form			1216.2.c.h.609.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.37228i q^{3} -0.792287i q^{5} -3.31662 q^{7} -2.62772 q^{9} +O(q^{10})$	$q+2.37228i q^{3} -0.792287i q^{5} -3.31662 q^{7} -2.62772 q^{9} -3.37228i q^{11} -4.10891i q^{13} +1.87953 q^{15} +5.00000 q^{17} +1.00000i q^{19} -7.86797i q^{21} -2.52434 q^{23} +4.37228 q^{25} +0.883156i q^{27} -5.98844i q^{29} +3.46410 q^{31} +8.00000 q^{33} +2.62772i q^{35} -3.16915i q^{37} +9.74749 q^{39} -9.37228i q^{43} +2.08191i q^{45} +9.30506 q^{47} +4.00000 q^{49} +11.8614i q^{51} +9.45254i q^{53} -2.67181 q^{55} -2.37228 q^{57} -4.37228i q^{59} -4.55134i q^{61} +8.71516 q^{63} -3.25544 q^{65} +8.37228i q^{67} -5.98844i q^{69} +6.63325 q^{71} -14.4891 q^{73} +10.3723i q^{75} +11.1846i q^{77} -13.5615 q^{79} -9.97825 q^{81} -8.00000i q^{83} -3.96143i q^{85} +14.2063 q^{87} +7.48913 q^{89} +13.6277i q^{91} +8.21782i q^{93} +0.792287 q^{95} +8.74456 q^{97} +8.86141i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 44 q^{9}+O(q^{10}) $ 8 * q - 44 * q^9	$ 8 q - 44 q^{9} + 40 q^{17} + 12 q^{25} + 64 q^{33} + 32 q^{49} + 4 q^{57} - 72 q^{65} - 24 q^{73} + 104 q^{81} - 32 q^{89} + 24 q^{97}+O(q^{100}) $ 8 * q - 44 * q^9 + 40 * q^17 + 12 * q^25 + 64 * q^33 + 32 * q^49 + 4 * q^57 - 72 * q^65 - 24 * q^73 + 104 * q^81 - 32 * q^89 + 24 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1216\mathbb{Z}\right)^\times$.

$n$	$191$	$705$	$837$
$\chi(n)$	$1$	$1$	$-1$

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 $

$2$	0	0
$2$	0	0
$3$	2.37228i	1.36964i	0.728714	+	0.684819i	$0.240119\pi$
$3$	2.37228i	1.36964i	−0.728714	+	0.684819i	$0.759881\pi$
$4$	0	0
$5$	− 0.792287i	− 0.354322i	−0.984182	−	0.177161i	$-0.943309\pi$
$5$	− 0.792287i	− 0.354322i	0.984182	−	0.177161i	$-0.0566913\pi$
$6$	0	0
$7$	−3.31662	−1.25357	−0.626783	−	0.779194i	$-0.715629\pi$
$7$	−3.31662	−1.25357	−0.626783	+	0.779194i	$0.715629\pi$
$8$	0	0
$9$	−2.62772	−0.875906
$10$	0	0
$11$	− 3.37228i	− 1.01678i	−0.861127	−	0.508391i	$-0.830241\pi$
$11$	− 3.37228i	− 1.01678i	0.861127	−	0.508391i	$-0.169759\pi$
$12$	0	0
$13$	− 4.10891i	− 1.13961i	−0.821781	−	0.569804i	$-0.807019\pi$
$13$	− 4.10891i	− 1.13961i	0.821781	−	0.569804i	$-0.192981\pi$
$14$	0	0
$15$	1.87953	0.485292
$16$	0	0
$17$	5.00000	1.21268	0.606339	−	0.795206i	$-0.292637\pi$
$17$	5.00000	1.21268	0.606339	+	0.795206i	$0.292637\pi$
$18$	0	0
$19$	1.00000i	0.229416i
$19$	1.00000i	0.229416i
$20$	0	0
$21$	− 7.86797i	− 1.71693i
$22$	0	0
$23$	−2.52434	−0.526361	−0.263180	−	0.964747i	$-0.584771\pi$
$23$	−2.52434	−0.526361	−0.263180	+	0.964747i	$0.584771\pi$
$24$	0	0
$25$	4.37228	0.874456
$26$	0	0
$27$	0.883156i	0.169963i
$28$	0	0
$29$	− 5.98844i	− 1.11203i	−0.831174	−	0.556013i	$-0.812331\pi$
$29$	− 5.98844i	− 1.11203i	0.831174	−	0.556013i	$-0.187669\pi$
$30$	0	0
$31$	3.46410	0.622171	0.311086	−	0.950382i	$-0.399307\pi$
$31$	3.46410	0.622171	0.311086	+	0.950382i	$0.399307\pi$
$32$	0	0
$33$	8.00000	1.39262
$34$	0	0
$35$	2.62772i	0.444166i
$36$	0	0
$37$	− 3.16915i	− 0.521005i	−0.965473	−	0.260502i	$-0.916112\pi$
$37$	− 3.16915i	− 0.521005i	0.965473	−	0.260502i	$-0.0838882\pi$
$38$	0	0
$39$	9.74749	1.56085
$40$	0	0
$41$	0	0		−	1.00000i	$-0.5\pi$
$41$	0	0			1.00000i	$0.5\pi$
$42$	0	0
$43$	− 9.37228i	− 1.42926i	−0.699503	−	0.714630i	$-0.746595\pi$
$43$	− 9.37228i	− 1.42926i	0.699503	−	0.714630i	$-0.253405\pi$
$44$	0	0
$45$	2.08191i	0.310352i
$46$	0	0
$47$	9.30506	1.35728	0.678642	−	0.734470i	$-0.262569\pi$
$47$	9.30506	1.35728	0.678642	+	0.734470i	$0.262569\pi$
$48$	0	0
$49$	4.00000	0.571429
$50$	0	0
$51$	11.8614i	1.66093i
$52$	0	0
$53$	9.45254i	1.29841i	0.760615	+	0.649203i	$0.224898\pi$
$53$	9.45254i	1.29841i	−0.760615	+	0.649203i	$0.775102\pi$
$54$	0	0
$55$	−2.67181	−0.360267
$56$	0	0
$57$	−2.37228	−0.314216
$58$	0	0
$59$	− 4.37228i	− 0.569223i	−0.958643	−	0.284611i	$-0.908135\pi$
$59$	− 4.37228i	− 0.569223i	0.958643	−	0.284611i	$-0.0918645\pi$
$60$	0	0
$61$	− 4.55134i	− 0.582740i	−0.956610	−	0.291370i	$-0.905889\pi$
$61$	− 4.55134i	− 0.582740i	0.956610	−	0.291370i	$-0.0941110\pi$
$62$	0	0
$63$	8.71516	1.09801
$64$	0	0
$65$	−3.25544	−0.403787
$66$	0	0
$67$	8.37228i	1.02284i	0.859332	+	0.511418i	$0.170880\pi$
$67$	8.37228i	1.02284i	−0.859332	+	0.511418i	$0.829120\pi$
$68$	0	0
$69$	− 5.98844i	− 0.720923i
$70$	0	0
$71$	6.63325	0.787222	0.393611	−	0.919277i	$-0.371226\pi$
$71$	6.63325	0.787222	0.393611	+	0.919277i	$0.371226\pi$
$72$	0	0
$73$	−14.4891	−1.69582	−0.847912	−	0.530137i	$-0.822140\pi$
$73$	−14.4891	−1.69582	−0.847912	+	0.530137i	$0.822140\pi$
$74$	0	0
$75$	10.3723i	1.19769i
$76$	0	0
$77$	11.1846i	1.27460i
$78$	0	0
$79$	−13.5615	−1.52578	−0.762891	−	0.646527i	$-0.776221\pi$
$79$	−13.5615	−1.52578	−0.762891	+	0.646527i	$0.776221\pi$
$80$	0	0
$81$	−9.97825	−1.10869
$82$	0	0
$83$	− 8.00000i	− 0.878114i	−0.898459	−	0.439057i	$-0.855313\pi$
$83$	− 8.00000i	− 0.878114i	0.898459	−	0.439057i	$-0.144687\pi$
$84$	0	0
$85$	− 3.96143i	− 0.429678i
$86$	0	0
$87$	14.2063	1.52307
$88$	0	0
$89$	7.48913	0.793846	0.396923	−	0.917852i	$-0.370078\pi$
$89$	7.48913	0.793846	0.396923	+	0.917852i	$0.370078\pi$
$90$	0	0
$91$	13.6277i	1.42857i
$92$	0	0
$93$	8.21782i	0.852149i
$94$	0	0
$95$	0.792287	0.0812869
$96$	0	0
$97$	8.74456	0.887876	0.443938	−	0.896058i	$-0.353581\pi$
$97$	8.74456	0.887876	0.443938	+	0.896058i	$0.353581\pi$
$98$	0	0
$99$	8.86141i	0.890605i
$100$	0	0
$101$	− 13.8564i	− 1.37876i	−0.724398	−	0.689382i	$-0.757882\pi$
$101$	− 13.8564i	− 1.37876i	0.724398	−	0.689382i	$-0.242118\pi$
$102$	0	0
$103$	5.34363	0.526523	0.263262	−	0.964724i	$-0.415202\pi$
$103$	5.34363	0.526523	0.263262	+	0.964724i	$0.415202\pi$
$104$	0	0
$105$	−6.23369	−0.608346
$106$	0	0
$107$	− 7.62772i	− 0.737399i	−0.929549	−	0.368700i	$-0.879803\pi$
$107$	− 7.62772i	− 0.737399i	0.929549	−	0.368700i	$-0.120197\pi$
$108$	0	0
$109$	− 12.6217i	− 1.20894i	−0.796628	−	0.604469i	$-0.793385\pi$
$109$	− 12.6217i	− 1.20894i	0.796628	−	0.604469i	$-0.206615\pi$
$110$	0	0
$111$	7.51811	0.713587
$112$	0	0
$113$	−3.25544	−0.306246	−0.153123	−	0.988207i	$-0.548933\pi$
$113$	−3.25544	−0.306246	−0.153123	+	0.988207i	$0.548933\pi$
$114$	0	0
$115$	2.00000i	0.186501i
$116$	0	0
$117$	10.7971i	0.998189i
$118$	0	0
$119$	−16.5831	−1.52017
$120$	0	0
$121$	−0.372281	−0.0338438
$122$	0	0
$123$	0	0
$124$	0	0
$125$	− 7.42554i	− 0.664160i
$126$	0	0
$127$	18.9051	1.67755	0.838777	−	0.544475i	$-0.183271\pi$
$127$	18.9051	1.67755	0.838777	+	0.544475i	$0.183271\pi$
$128$	0	0
$129$	22.2337	1.95757
$130$	0	0
$131$	1.37228i	0.119897i	0.998201	+	0.0599484i	$0.0190936\pi$
$131$	1.37228i	0.119897i	−0.998201	+	0.0599484i	$0.980906\pi$
$132$	0	0
$133$	− 3.31662i	− 0.287588i
$134$	0	0
$135$	0.699713	0.0602217
$136$	0	0
$137$	5.00000	0.427179	0.213589	−	0.976924i	$-0.431485\pi$
$137$	5.00000	0.427179	0.213589	+	0.976924i	$0.431485\pi$
$138$	0	0
$139$	− 9.37228i	− 0.794947i	−0.917614	−	0.397473i	$-0.869887\pi$
$139$	− 9.37228i	− 0.794947i	0.917614	−	0.397473i	$-0.130113\pi$
$140$	0	0
$141$	22.0742i	1.85899i
$142$	0	0
$143$	−13.8564	−1.15873
$144$	0	0
$145$	−4.74456	−0.394014
$146$	0	0
$147$	9.48913i	0.782650i
$148$	0	0
$149$	21.2819i	1.74348i	0.489965	+	0.871742i	$0.337010\pi$
$149$	21.2819i	1.74348i	−0.489965	+	0.871742i	$0.662990\pi$
$150$	0	0
$151$	22.3692	1.82038	0.910189	−	0.414193i	$-0.135936\pi$
$151$	22.3692	1.82038	0.910189	+	0.414193i	$0.135936\pi$
$152$	0	0
$153$	−13.1386	−1.06219
$154$	0	0
$155$	− 2.74456i	− 0.220449i
$156$	0	0
$157$	1.28962i	0.102923i	0.998675	+	0.0514615i	$0.0163879\pi$
$157$	1.28962i	0.102923i	−0.998675	+	0.0514615i	$0.983612\pi$
$158$	0	0
$159$	−22.4241	−1.77835
$160$	0	0
$161$	8.37228	0.659828
$162$	0	0
$163$	− 0.744563i	− 0.0583186i	−0.999575	−	0.0291593i	$-0.990717\pi$
$163$	− 0.744563i	− 0.0583186i	0.999575	−	0.0291593i	$-0.00928302\pi$
$164$	0	0
$165$	− 6.33830i	− 0.493436i
$166$	0	0
$167$	−20.1947	−1.56271	−0.781356	−	0.624085i	$-0.785472\pi$
$167$	−20.1947	−1.56271	−0.781356	+	0.624085i	$0.785472\pi$
$168$	0	0
$169$	−3.88316	−0.298704
$170$	0	0
$171$	− 2.62772i	− 0.200947i
$172$	0	0
$173$	− 4.75372i	− 0.361419i	−0.983537	−	0.180709i	$-0.942161\pi$
$173$	− 4.75372i	− 0.361419i	0.983537	−	0.180709i	$-0.0578393\pi$
$174$	0	0
$175$	−14.5012	−1.09619
$176$	0	0
$177$	10.3723	0.779628
$178$	0	0
$179$	− 22.9783i	− 1.71748i	−0.512416	−	0.858738i	$-0.671249\pi$
$179$	− 22.9783i	− 1.71748i	0.512416	−	0.858738i	$-0.328751\pi$
$180$	0	0
$181$	− 11.6819i	− 0.868311i	−0.900838	−	0.434155i	$-0.857047\pi$
$181$	− 11.6819i	− 0.868311i	0.900838	−	0.434155i	$-0.142953\pi$
$182$	0	0
$183$	10.7971	0.798142
$184$	0	0
$185$	−2.51087	−0.184603
$186$	0	0
$187$	− 16.8614i	− 1.23303i
$188$	0	0
$189$	− 2.92910i	− 0.213060i
$190$	0	0
$191$	−13.7089	−0.991943	−0.495972	−	0.868339i	$-0.665188\pi$
$191$	−13.7089	−0.991943	−0.495972	+	0.868339i	$0.665188\pi$
$192$	0	0
$193$	14.7446	1.06134	0.530668	−	0.847580i	$-0.321941\pi$
$193$	14.7446	1.06134	0.530668	+	0.847580i	$0.321941\pi$
$194$	0	0
$195$	− 7.72281i	− 0.553042i
$196$	0	0
$197$	20.1947i	1.43881i	0.694589	+	0.719406i	$0.255586\pi$
$197$	20.1947i	1.43881i	−0.694589	+	0.719406i	$0.744414\pi$
$198$	0	0
$199$	−6.78073	−0.480673	−0.240336	−	0.970690i	$-0.577258\pi$
$199$	−6.78073	−0.480673	−0.240336	+	0.970690i	$0.577258\pi$
$200$	0	0
$201$	−19.8614	−1.40092
$202$	0	0
$203$	19.8614i	1.39400i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	6.63325	0.461043
$208$	0	0
$209$	3.37228	0.233266
$210$	0	0
$211$	− 8.37228i	− 0.576372i	−0.957575	−	0.288186i	$-0.906948\pi$
$211$	− 8.37228i	− 0.576372i	0.957575	−	0.288186i	$-0.0930521\pi$
$212$	0	0
$213$	15.7359i	1.07821i
$214$	0	0
$215$	−7.42554	−0.506417
$216$	0	0
$217$	−11.4891	−0.779933
$218$	0	0
$219$	− 34.3723i	− 2.32266i
$220$	0	0
$221$	− 20.5446i	− 1.38198i
$222$	0	0
$223$	−6.92820	−0.463947	−0.231973	−	0.972722i	$-0.574518\pi$
$223$	−6.92820	−0.463947	−0.231973	+	0.972722i	$0.574518\pi$
$224$	0	0
$225$	−11.4891	−0.765942
$226$	0	0
$227$	3.11684i	0.206872i	0.994636	+	0.103436i	$0.0329837\pi$
$227$	3.11684i	0.206872i	−0.994636	+	0.103436i	$0.967016\pi$
$228$	0	0
$229$	− 3.96143i	− 0.261779i	−0.991397	−	0.130889i	$-0.958217\pi$
$229$	− 3.96143i	− 0.261779i	0.991397	−	0.130889i	$-0.0417833\pi$
$230$	0	0
$231$	−26.5330	−1.74574
$232$	0	0
$233$	−28.1168	−1.84200	−0.920998	−	0.389568i	$-0.872624\pi$
$233$	−28.1168	−1.84200	−0.920998	+	0.389568i	$0.872624\pi$
$234$	0	0
$235$	− 7.37228i	− 0.480915i
$236$	0	0
$237$	− 32.1716i	− 2.08977i
$238$	0	0
$239$	−24.1012	−1.55898	−0.779490	−	0.626415i	$-0.784522\pi$
$239$	−24.1012	−1.55898	−0.779490	+	0.626415i	$0.784522\pi$
$240$	0	0
$241$	8.00000	0.515325	0.257663	−	0.966235i	$-0.417048\pi$
$241$	8.00000	0.515325	0.257663	+	0.966235i	$0.417048\pi$
$242$	0	0
$243$	− 21.0217i	− 1.34855i
$244$	0	0
$245$	− 3.16915i	− 0.202469i
$246$	0	0
$247$	4.10891	0.261444
$248$	0	0
$249$	18.9783	1.20270
$250$	0	0
$251$	10.8614i	0.685566i	0.939415	+	0.342783i	$0.111370\pi$
$251$	10.8614i	0.685566i	−0.939415	+	0.342783i	$0.888630\pi$
$252$	0	0
$253$	8.51278i	0.535194i
$254$	0	0
$255$	9.39764	0.588503
$256$	0	0
$257$	18.2337	1.13739	0.568693	−	0.822550i	$-0.307449\pi$
$257$	18.2337	1.13739	0.568693	+	0.822550i	$0.307449\pi$
$258$	0	0
$259$	10.5109i	0.653114i
$260$	0	0
$261$	15.7359i	0.974030i
$262$	0	0
$263$	−24.1561	−1.48953	−0.744766	−	0.667326i	$-0.767439\pi$
$263$	−24.1561	−1.48953	−0.744766	+	0.667326i	$0.767439\pi$
$264$	0	0
$265$	7.48913	0.460053
$266$	0	0
$267$	17.7663i	1.08728i
$268$	0	0
$269$	8.51278i	0.519033i	0.965739	+	0.259517i	$0.0835632\pi$
$269$	8.51278i	0.519033i	−0.965739	+	0.259517i	$0.916437\pi$
$270$	0	0
$271$	−7.57301	−0.460028	−0.230014	−	0.973187i	$-0.573877\pi$
$271$	−7.57301	−0.460028	−0.230014	+	0.973187i	$0.573877\pi$
$272$	0	0
$273$	−32.3288	−1.95663
$274$	0	0
$275$	− 14.7446i	− 0.889131i
$276$	0	0
$277$	− 13.0641i	− 0.784947i	−0.919763	−	0.392473i	$-0.871619\pi$
$277$	− 13.0641i	− 0.784947i	0.919763	−	0.392473i	$-0.128381\pi$
$278$	0	0
$279$	−9.10268	−0.544963
$280$	0	0
$281$	−28.2337	−1.68428	−0.842140	−	0.539258i	$-0.818705\pi$
$281$	−28.2337	−1.68428	−0.842140	+	0.539258i	$0.818705\pi$
$282$	0	0
$283$	23.3723i	1.38934i	0.719330	+	0.694669i	$0.244449\pi$
$283$	23.3723i	1.38934i	−0.719330	+	0.694669i	$0.755551\pi$
$284$	0	0
$285$	1.87953i	0.111334i
$286$	0	0
$287$	0	0
$288$	0	0
$289$	8.00000	0.470588
$290$	0	0
$291$	20.7446i	1.21607i
$292$	0	0
$293$	28.0627i	1.63944i	0.572765	+	0.819719i	$0.305871\pi$
$293$	28.0627i	1.63944i	−0.572765	+	0.819719i	$0.694129\pi$
$294$	0	0
$295$	−3.46410	−0.201688
$296$	0	0
$297$	2.97825	0.172816
$298$	0	0
$299$	10.3723i	0.599845i
$300$	0	0
$301$	31.0843i	1.79167i
$302$	0	0
$303$	32.8713	1.88841
$304$	0	0
$305$	−3.60597	−0.206477
$306$	0	0
$307$	− 26.7446i	− 1.52639i	−0.646167	−	0.763196i	$-0.723629\pi$
$307$	− 26.7446i	− 1.52639i	0.646167	−	0.763196i	$-0.276371\pi$
$308$	0	0
$309$	12.6766i	0.721146i
$310$	0	0
$311$	−23.2164	−1.31648	−0.658240	−	0.752808i	$-0.728699\pi$
$311$	−23.2164	−1.31648	−0.658240	+	0.752808i	$0.728699\pi$
$312$	0	0
$313$	21.1168	1.19359	0.596797	−	0.802392i	$-0.296440\pi$
$313$	21.1168	1.19359	0.596797	+	0.802392i	$0.296440\pi$
$314$	0	0
$315$	− 6.90491i	− 0.389047i
$316$	0	0
$317$	18.2603i	1.02560i	0.858508	+	0.512800i	$0.171392\pi$
$317$	18.2603i	1.02560i	−0.858508	+	0.512800i	$0.828608\pi$
$318$	0	0
$319$	−20.1947	−1.13069
$320$	0	0
$321$	18.0951	1.00997
$322$	0	0
$323$	5.00000i	0.278207i
$324$	0	0
$325$	− 17.9653i	− 0.996537i
$326$	0	0
$327$	29.9422	1.65581
$328$	0	0
$329$	−30.8614	−1.70144
$330$	0	0
$331$	17.8614i	0.981752i	0.871230	+	0.490876i	$0.163323\pi$
$331$	17.8614i	0.981752i	−0.871230	+	0.490876i	$0.836677\pi$
$332$	0	0
$333$	8.32763i	0.456351i
$334$	0	0
$335$	6.63325	0.362413
$336$	0	0
$337$	−24.7446	−1.34792	−0.673961	−	0.738767i	$-0.735408\pi$
$337$	−24.7446	−1.34792	−0.673961	+	0.738767i	$0.735408\pi$
$338$	0	0
$339$	− 7.72281i	− 0.419446i
$340$	0	0
$341$	− 11.6819i	− 0.632612i
$342$	0	0
$343$	9.94987	0.537243
$344$	0	0
$345$	−4.74456	−0.255439
$346$	0	0
$347$	− 19.8832i	− 1.06738i	−0.845679	−	0.533692i	$-0.820804\pi$
$347$	− 19.8832i	− 1.06738i	0.845679	−	0.533692i	$-0.179196\pi$
$348$	0	0
$349$	23.1615i	1.23981i	0.784679	+	0.619903i	$0.212828\pi$
$349$	23.1615i	1.23981i	−0.784679	+	0.619903i	$0.787172\pi$
$350$	0	0
$351$	3.62881	0.193692
$352$	0	0
$353$	21.1168	1.12394	0.561968	−	0.827159i	$-0.310044\pi$
$353$	21.1168	1.12394	0.561968	+	0.827159i	$0.310044\pi$
$354$	0	0
$355$	− 5.25544i	− 0.278930i
$356$	0	0
$357$	− 39.3398i	− 2.08208i
$358$	0	0
$359$	16.5831	0.875224	0.437612	−	0.899164i	$-0.355824\pi$
$359$	16.5831	0.875224	0.437612	+	0.899164i	$0.355824\pi$
$360$	0	0
$361$	−1.00000	−0.0526316
$362$	0	0
$363$	− 0.883156i	− 0.0463537i
$364$	0	0
$365$	11.4795i	0.600867i
$366$	0	0
$367$	30.5870	1.59663	0.798314	−	0.602241i	$-0.205725\pi$
$367$	30.5870	1.59663	0.798314	+	0.602241i	$0.205725\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	− 31.3505i	− 1.62764i
$372$	0	0
$373$	24.8935i	1.28894i	0.764631	+	0.644469i	$0.222921\pi$
$373$	24.8935i	1.28894i	−0.764631	+	0.644469i	$0.777079\pi$
$374$	0	0
$375$	17.6155	0.909659
$376$	0	0
$377$	−24.6060	−1.26727
$378$	0	0
$379$	15.3505i	0.788504i	0.919002	+	0.394252i	$0.128996\pi$
$379$	15.3505i	0.788504i	−0.919002	+	0.394252i	$0.871004\pi$
$380$	0	0
$381$	44.8482i	2.29764i
$382$	0	0
$383$	7.92287	0.404840	0.202420	−	0.979299i	$-0.435119\pi$
$383$	7.92287	0.404840	0.202420	+	0.979299i	$0.435119\pi$
$384$	0	0
$385$	8.86141	0.451619
$386$	0	0
$387$	24.6277i	1.25190i
$388$	0	0
$389$	− 12.1793i	− 0.617513i	−0.951141	−	0.308756i	$-0.900087\pi$
$389$	− 12.1793i	− 0.617513i	0.951141	−	0.308756i	$-0.0999128\pi$
$390$	0	0
$391$	−12.6217	−0.638306
$392$	0	0
$393$	−3.25544	−0.164215
$394$	0	0
$395$	10.7446i	0.540618i
$396$	0	0
$397$	− 9.30506i	− 0.467008i	−0.972356	−	0.233504i	$-0.924981\pi$
$397$	− 9.30506i	− 0.467008i	0.972356	−	0.233504i	$-0.0750192\pi$
$398$	0	0
$399$	7.86797	0.393891
$400$	0	0
$401$	−21.7228	−1.08479	−0.542393	−	0.840125i	$-0.682482\pi$
$401$	−21.7228	−1.08479	−0.542393	+	0.840125i	$0.682482\pi$
$402$	0	0
$403$	− 14.2337i	− 0.709030i
$404$	0	0
$405$	7.90564i	0.392834i
$406$	0	0
$407$	−10.6873	−0.529748
$408$	0	0
$409$	−6.00000	−0.296681	−0.148340	−	0.988936i	$-0.547393\pi$
$409$	−6.00000	−0.296681	−0.148340	+	0.988936i	$0.547393\pi$
$410$	0	0
$411$	11.8614i	0.585080i
$412$	0	0
$413$	14.5012i	0.713558i
$414$	0	0
$415$	−6.33830	−0.311135
$416$	0	0
$417$	22.2337	1.08879
$418$	0	0
$419$	− 34.2337i	− 1.67243i	−0.548405	−	0.836213i	$-0.684765\pi$
$419$	− 34.2337i	− 1.67243i	0.548405	−	0.836213i	$-0.315235\pi$
$420$	0	0
$421$	35.9855i	1.75383i	0.480647	+	0.876914i	$0.340402\pi$
$421$	35.9855i	1.75383i	−0.480647	+	0.876914i	$0.659598\pi$
$422$	0	0
$423$	−24.4511	−1.18885
$424$	0	0
$425$	21.8614	1.06043
$426$	0	0
$427$	15.0951i	0.730503i
$428$	0	0
$429$	− 32.8713i	− 1.58704i
$430$	0	0
$431$	11.3870	0.548491	0.274246	−	0.961660i	$-0.411572\pi$
$431$	11.3870	0.548491	0.274246	+	0.961660i	$0.411572\pi$
$432$	0	0
$433$	21.4891	1.03270	0.516351	−	0.856377i	$-0.327290\pi$
$433$	21.4891	1.03270	0.516351	+	0.856377i	$0.327290\pi$
$434$	0	0
$435$	− 11.2554i	− 0.539657i
$436$	0	0
$437$	− 2.52434i	− 0.120755i
$438$	0	0
$439$	−6.92820	−0.330665	−0.165333	−	0.986238i	$-0.552870\pi$
$439$	−6.92820	−0.330665	−0.165333	+	0.986238i	$0.552870\pi$
$440$	0	0
$441$	−10.5109	−0.500518
$442$	0	0
$443$	10.3505i	0.491769i	0.969299	+	0.245884i	$0.0790783\pi$
$443$	10.3505i	0.491769i	−0.969299	+	0.245884i	$0.920922\pi$
$444$	0	0
$445$	− 5.93354i	− 0.281277i
$446$	0	0
$447$	−50.4868	−2.38794
$448$	0	0
$449$	14.5109	0.684811	0.342405	−	0.939552i	$-0.388758\pi$
$449$	14.5109	0.684811	0.342405	+	0.939552i	$0.388758\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	53.0660i	2.49326i
$454$	0	0
$455$	10.7971	0.506174
$456$	0	0
$457$	31.0000	1.45012	0.725059	−	0.688686i	$-0.241812\pi$
$457$	31.0000	1.45012	0.725059	+	0.688686i	$0.241812\pi$
$458$	0	0
$459$	4.41578i	0.206111i
$460$	0	0
$461$	11.1846i	0.520918i	0.965485	+	0.260459i	$0.0838739\pi$
$461$	11.1846i	0.520918i	−0.965485	+	0.260459i	$0.916126\pi$
$462$	0	0
$463$	8.31040	0.386217	0.193108	−	0.981177i	$-0.438143\pi$
$463$	8.31040	0.386217	0.193108	+	0.981177i	$0.438143\pi$
$464$	0	0
$465$	6.51087	0.301935
$466$	0	0
$467$	28.1168i	1.30109i	0.759467	+	0.650546i	$0.225460\pi$
$467$	28.1168i	1.30109i	−0.759467	+	0.650546i	$0.774540\pi$
$468$	0	0
$469$	− 27.7677i	− 1.28219i
$470$	0	0
$471$	−3.05934	−0.140967
$472$	0	0
$473$	−31.6060	−1.45324
$474$	0	0
$475$	4.37228i	0.200614i
$476$	0	0
$477$	− 24.8386i	− 1.13728i
$478$	0	0
$479$	14.1514	0.646592	0.323296	−	0.946298i	$-0.395209\pi$
$479$	14.1514	0.646592	0.323296	+	0.946298i	$0.395209\pi$
$480$	0	0
$481$	−13.0217	−0.593741
$482$	0	0
$483$	19.8614i	0.903725i
$484$	0	0
$485$	− 6.92820i	− 0.314594i
$486$	0	0
$487$	29.2974	1.32759	0.663796	−	0.747914i	$-0.268944\pi$
$487$	29.2974	1.32759	0.663796	+	0.747914i	$0.268944\pi$
$488$	0	0
$489$	1.76631	0.0798754
$490$	0	0
$491$	14.9783i	0.675959i	0.941153	+	0.337979i	$0.109743\pi$
$491$	14.9783i	0.675959i	−0.941153	+	0.337979i	$0.890257\pi$
$492$	0	0
$493$	− 29.9422i	− 1.34853i
$494$	0	0
$495$	7.02078	0.315560
$496$	0	0
$497$	−22.0000	−0.986835
$498$	0	0
$499$	1.37228i	0.0614317i	0.999528	+	0.0307159i	$0.00977870\pi$
$499$	1.37228i	0.0614317i	−0.999528	+	0.0307159i	$0.990221\pi$
$500$	0	0
$501$	− 47.9075i	− 2.14035i
$502$	0	0
$503$	−21.4294	−0.955491	−0.477745	−	0.878498i	$-0.658546\pi$
$503$	−21.4294	−0.955491	−0.477745	+	0.878498i	$0.658546\pi$
$504$	0	0
$505$	−10.9783	−0.488526
$506$	0	0
$507$	− 9.21194i	− 0.409117i
$508$	0	0
$509$	− 24.5437i	− 1.08788i	−0.839124	−	0.543939i	$-0.816932\pi$
$509$	− 24.5437i	− 1.08788i	0.839124	−	0.543939i	$-0.183068\pi$
$510$	0	0
$511$	48.0550	2.12583
$512$	0	0
$513$	−0.883156	−0.0389923
$514$	0	0
$515$	− 4.23369i	− 0.186559i
$516$	0	0
$517$	− 31.3793i	− 1.38006i
$518$	0	0
$519$	11.2772	0.495013
$520$	0	0
$521$	−40.2337	−1.76267	−0.881335	−	0.472492i	$-0.843355\pi$
$521$	−40.2337	−1.76267	−0.881335	+	0.472492i	$0.843355\pi$
$522$	0	0
$523$	19.6277i	0.858260i	0.903243	+	0.429130i	$0.141180\pi$
$523$	19.6277i	0.858260i	−0.903243	+	0.429130i	$0.858820\pi$
$524$	0	0
$525$	− 34.4010i	− 1.50138i
$526$	0	0
$527$	17.3205	0.754493
$528$	0	0
$529$	−16.6277	−0.722944
$530$	0	0
$531$	11.4891i	0.498586i
$532$	0	0
$533$	0	0
$534$	0	0
$535$	−6.04334	−0.261276
$536$	0	0
$537$	54.5109	2.35232
$538$	0	0
$539$	− 13.4891i	− 0.581018i
$540$	0	0
$541$	− 33.5538i	− 1.44259i	−0.692628	−	0.721295i	$-0.743547\pi$
$541$	− 33.5538i	− 1.44259i	0.692628	−	0.721295i	$-0.256453\pi$
$542$	0	0
$543$	27.7128	1.18927
$544$	0	0
$545$	−10.0000	−0.428353
$546$	0	0
$547$	− 25.4891i	− 1.08984i	−0.838489	−	0.544918i	$-0.816561\pi$
$547$	− 25.4891i	− 1.08984i	0.838489	−	0.544918i	$-0.183439\pi$
$548$	0	0
$549$	11.9596i	0.510425i
$550$	0	0
$551$	5.98844	0.255116
$552$	0	0
$553$	44.9783	1.91267
$554$	0	0
$555$	− 5.95650i	− 0.252839i
$556$	0	0
$557$	28.5051i	1.20780i	0.797060	+	0.603900i	$0.206387\pi$
$557$	28.5051i	1.20780i	−0.797060	+	0.603900i	$0.793613\pi$
$558$	0	0
$559$	−38.5099	−1.62879
$560$	0	0
$561$	40.0000	1.68880
$562$	0	0
$563$	9.48913i	0.399919i	0.979804	+	0.199959i	$0.0640811\pi$
$563$	9.48913i	0.399919i	−0.979804	+	0.199959i	$0.935919\pi$
$564$	0	0
$565$	2.57924i	0.108509i
$566$	0	0
$567$	33.0941	1.38982
$568$	0	0
$569$	18.9783	0.795610	0.397805	−	0.917470i	$-0.369772\pi$
$569$	18.9783	0.795610	0.397805	+	0.917470i	$0.369772\pi$
$570$	0	0
$571$	2.97825i	0.124636i	0.998056	+	0.0623180i	$0.0198493\pi$
$571$	2.97825i	0.124636i	−0.998056	+	0.0623180i	$0.980151\pi$
$572$	0	0
$573$	− 32.5214i	− 1.35860i
$574$	0	0
$575$	−11.0371	−0.460280
$576$	0	0
$577$	−35.4674	−1.47653	−0.738263	−	0.674513i	$-0.764354\pi$
$577$	−35.4674	−1.47653	−0.738263	+	0.674513i	$0.764354\pi$
$578$	0	0
$579$	34.9783i	1.45365i
$580$	0	0
$581$	26.5330i	1.10077i
$582$	0	0
$583$	31.8766	1.32020
$584$	0	0
$585$	8.55437	0.353680
$586$	0	0
$587$	3.13859i	0.129544i	0.997900	+	0.0647718i	$0.0206319\pi$
$587$	3.13859i	0.129544i	−0.997900	+	0.0647718i	$0.979368\pi$
$588$	0	0
$589$	3.46410i	0.142736i
$590$	0	0
$591$	−47.9075	−1.97065
$592$	0	0
$593$	−40.9783	−1.68278	−0.841388	−	0.540432i	$-0.818261\pi$
$593$	−40.9783	−1.68278	−0.841388	+	0.540432i	$0.818261\pi$
$594$	0	0
$595$	13.1386i	0.538630i
$596$	0	0
$597$	− 16.0858i	− 0.658348i
$598$	0	0
$599$	−33.0564	−1.35065	−0.675325	−	0.737520i	$-0.735997\pi$
$599$	−33.0564	−1.35065	−0.675325	+	0.737520i	$0.735997\pi$
$600$	0	0
$601$	46.4674	1.89544	0.947722	−	0.319097i	$-0.103380\pi$
$601$	46.4674	1.89544	0.947722	+	0.319097i	$0.103380\pi$
$602$	0	0
$603$	− 22.0000i	− 0.895909i
$604$	0	0
$605$	0.294954i	0.0119916i
$606$	0	0
$607$	7.62792	0.309608	0.154804	−	0.987945i	$-0.450525\pi$
$607$	7.62792	0.309608	0.154804	+	0.987945i	$0.450525\pi$
$608$	0	0
$609$	−47.1168	−1.90927
$610$	0	0
$611$	− 38.2337i	− 1.54677i
$612$	0	0
$613$	− 35.1383i	− 1.41922i	−0.704592	−	0.709612i	$-0.748870\pi$
$613$	− 35.1383i	− 1.41922i	0.704592	−	0.709612i	$-0.251130\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	33.8397	1.36233	0.681167	−	0.732128i	$-0.261473\pi$
$617$	33.8397	1.36233	0.681167	+	0.732128i	$0.261473\pi$
$618$	0	0
$619$	− 18.2337i	− 0.732874i	−0.930443	−	0.366437i	$-0.880578\pi$
$619$	− 18.2337i	− 0.732874i	0.930443	−	0.366437i	$-0.119422\pi$
$620$	0	0
$621$	− 2.22938i	− 0.0894621i
$622$	0	0
$623$	−24.8386	−0.995138
$624$	0	0
$625$	15.9783	0.639130
$626$	0	0
$627$	8.00000i	0.319489i
$628$	0	0
$629$	− 15.8457i	− 0.631811i
$630$	0	0
$631$	11.8843	0.473107	0.236553	−	0.971619i	$-0.423982\pi$
$631$	11.8843	0.473107	0.236553	+	0.971619i	$0.423982\pi$
$632$	0	0
$633$	19.8614	0.789420
$634$	0	0
$635$	− 14.9783i	− 0.594394i
$636$	0	0
$637$	− 16.4356i	− 0.651204i
$638$	0	0
$639$	−17.4303	−0.689533
$640$	0	0
$641$	34.9783	1.38156	0.690779	−	0.723066i	$-0.257268\pi$
$641$	34.9783	1.38156	0.690779	+	0.723066i	$0.257268\pi$
$642$	0	0
$643$	30.6277i	1.20784i	0.797045	+	0.603920i	$0.206395\pi$
$643$	30.6277i	1.20784i	−0.797045	+	0.603920i	$0.793605\pi$
$644$	0	0
$645$	− 17.6155i	− 0.693608i
$646$	0	0
$647$	16.5831	0.651950	0.325975	−	0.945378i	$-0.394307\pi$
$647$	16.5831	0.651950	0.325975	+	0.945378i	$0.394307\pi$
$648$	0	0
$649$	−14.7446	−0.578775
$650$	0	0
$651$	− 27.2554i	− 1.06822i
$652$	0	0
$653$	47.5200i	1.85960i	0.368064	+	0.929800i	$0.380021\pi$
$653$	47.5200i	1.85960i	−0.368064	+	0.929800i	$0.619979\pi$
$654$	0	0
$655$	1.08724	0.0424820
$656$	0	0
$657$	38.0733	1.48538
$658$	0	0
$659$	− 18.1386i	− 0.706579i	−0.935514	−	0.353290i	$-0.885063\pi$
$659$	− 18.1386i	− 0.706579i	0.935514	−	0.353290i	$-0.114937\pi$
$660$	0	0
$661$	− 19.2549i	− 0.748930i	−0.927241	−	0.374465i	$-0.877826\pi$
$661$	− 19.2549i	− 0.748930i	0.927241	−	0.374465i	$-0.122174\pi$
$662$	0	0
$663$	48.7375	1.89281
$664$	0	0
$665$	−2.62772	−0.101899
$666$	0	0
$667$	15.1168i	0.585327i
$668$	0	0
$669$	− 16.4356i	− 0.635439i
$670$	0	0
$671$	−15.3484	−0.592519
$672$	0	0
$673$	5.25544	0.202582	0.101291	−	0.994857i	$-0.467703\pi$
$673$	5.25544	0.202582	0.101291	+	0.994857i	$0.467703\pi$
$674$	0	0
$675$	3.86141i	0.148626i
$676$	0	0
$677$	4.10891i	0.157918i	0.996878	+	0.0789592i	$0.0251597\pi$
$677$	4.10891i	0.157918i	−0.996878	+	0.0789592i	$0.974840\pi$
$678$	0	0
$679$	−29.0024	−1.11301
$680$	0	0
$681$	−7.39403	−0.283340
$682$	0	0
$683$	12.0000i	0.459167i	0.973289	+	0.229584i	$0.0737364\pi$
$683$	12.0000i	0.459167i	−0.973289	+	0.229584i	$0.926264\pi$
$684$	0	0
$685$	− 3.96143i	− 0.151359i
$686$	0	0
$687$	9.39764	0.358542
$688$	0	0
$689$	38.8397	1.47967
$690$	0	0
$691$	− 10.8614i	− 0.413187i	−0.978427	−	0.206594i	$-0.933762\pi$
$691$	− 10.8614i	− 0.413187i	0.978427	−	0.206594i	$-0.0662378\pi$
$692$	0	0
$693$	− 29.3900i	− 1.11643i
$694$	0	0
$695$	−7.42554	−0.281667
$696$	0	0
$697$	0	0
$698$	0	0
$699$	− 66.7011i	− 2.52287i
$700$	0	0
$701$	27.7128i	1.04670i	0.852118	+	0.523349i	$0.175318\pi$
$701$	27.7128i	1.04670i	−0.852118	+	0.523349i	$0.824682\pi$
$702$	0	0
$703$	3.16915	0.119527
$704$	0	0
$705$	17.4891	0.658679
$706$	0	0
$707$	45.9565i	1.72837i
$708$	0	0
$709$	0.699713i	0.0262783i	0.999914	+	0.0131391i	$0.00418244\pi$
$709$	0.699713i	0.0262783i	−0.999914	+	0.0131391i	$0.995818\pi$
$710$	0	0
$711$	35.6357	1.33644
$712$	0	0
$713$	−8.74456	−0.327486
$714$	0	0
$715$	10.9783i	0.410563i
$716$	0	0
$717$	− 57.1749i	− 2.13524i
$718$	0	0
$719$	5.89587	0.219879	0.109939	−	0.993938i	$-0.464934\pi$
$719$	5.89587	0.219879	0.109939	+	0.993938i	$0.464934\pi$
$720$	0	0
$721$	−17.7228	−0.660032
$722$	0	0
$723$	18.9783i	0.705809i
$724$	0	0
$725$	− 26.1831i	− 0.972417i
$726$	0	0
$727$	−20.6371	−0.765389	−0.382694	−	0.923875i	$-0.625004\pi$
$727$	−20.6371	−0.765389	−0.382694	+	0.923875i	$0.625004\pi$
$728$	0	0
$729$	19.9348	0.738324
$730$	0	0
$731$	− 46.8614i	− 1.73323i
$732$	0	0
$733$	38.5099i	1.42239i	0.702992	+	0.711197i	$0.251847\pi$
$733$	38.5099i	1.42239i	−0.702992	+	0.711197i	$0.748153\pi$
$734$	0	0
$735$	7.51811	0.277310
$736$	0	0
$737$	28.2337	1.04000
$738$	0	0
$739$	27.6060i	1.01550i	0.861504	+	0.507751i	$0.169523\pi$
$739$	27.6060i	1.01550i	−0.861504	+	0.507751i	$0.830477\pi$
$740$	0	0
$741$	9.74749i	0.358083i
$742$	0	0
$743$	6.63325	0.243350	0.121675	−	0.992570i	$-0.461173\pi$
$743$	6.63325	0.243350	0.121675	+	0.992570i	$0.461173\pi$
$744$	0	0
$745$	16.8614	0.617754
$746$	0	0
$747$	21.0217i	0.769146i
$748$	0	0
$749$	25.2983i	0.924379i
$750$	0	0
$751$	30.2921	1.10537	0.552686	−	0.833389i	$-0.313603\pi$
$751$	30.2921	1.10537	0.552686	+	0.833389i	$0.313603\pi$
$752$	0	0
$753$	−25.7663	−0.938977
$754$	0	0
$755$	− 17.7228i	− 0.644999i
$756$	0	0
$757$	48.6998i	1.77002i	0.465568	+	0.885012i	$0.345850\pi$
$757$	48.6998i	1.77002i	−0.465568	+	0.885012i	$0.654150\pi$
$758$	0	0
$759$	−20.1947	−0.733021
$760$	0	0
$761$	−11.0000	−0.398750	−0.199375	−	0.979923i	$-0.563891\pi$
$761$	−11.0000	−0.398750	−0.199375	+	0.979923i	$0.563891\pi$
$762$	0	0
$763$	41.8614i	1.51548i
$764$	0	0
$765$	10.4095i	0.376358i
$766$	0	0
$767$	−17.9653	−0.648690
$768$	0	0
$769$	−31.4674	−1.13474	−0.567371	−	0.823462i	$-0.692040\pi$
$769$	−31.4674	−1.13474	−0.567371	+	0.823462i	$0.692040\pi$
$770$	0	0
$771$	43.2554i	1.55781i
$772$	0	0
$773$	18.9600i	0.681943i	0.940074	+	0.340972i	$0.110756\pi$
$773$	18.9600i	0.681943i	−0.940074	+	0.340972i	$0.889244\pi$
$774$	0	0
$775$	15.1460	0.544061
$776$	0	0
$777$	−24.9348	−0.894529
$778$	0	0
$779$	0	0
$780$	0	0
$781$	− 22.3692i	− 0.800432i
$782$	0	0
$783$	5.28873	0.189004
$784$	0	0
$785$	1.02175	0.0364678
$786$	0	0
$787$	20.6060i	0.734523i	0.930118	+	0.367262i	$0.119705\pi$
$787$	20.6060i	0.734523i	−0.930118	+	0.367262i	$0.880295\pi$
$788$	0	0
$789$	− 57.3052i	− 2.04012i
$790$	0	0
$791$	10.7971	0.383899
$792$	0	0
$793$	−18.7011	−0.664094
$794$	0	0
$795$	17.7663i	0.630106i
$796$	0	0
$797$	− 11.3321i	− 0.401402i	−0.979652	−	0.200701i	$-0.935678\pi$
$797$	− 11.3321i	− 0.401402i	0.979652	−	0.200701i	$-0.0643220\pi$
$798$	0	0
$799$	46.5253	1.64595
$800$	0	0
$801$	−19.6793	−0.695334
$802$	0	0
$803$	48.8614i	1.72428i
$804$	0	0
$805$	− 6.63325i	− 0.233791i
$806$	0	0
$807$	−20.1947	−0.710887
$808$	0	0
$809$	7.97825	0.280500	0.140250	−	0.990116i	$-0.455209\pi$
$809$	7.97825	0.280500	0.140250	+	0.990116i	$0.455209\pi$
$810$	0	0
$811$	− 0.138593i	− 0.00486667i	−0.999997	−	0.00243334i	$-0.999225\pi$
$811$	− 0.138593i	− 0.00486667i	0.999997	−	0.00243334i	$-0.000774556\pi$
$812$	0	0
$813$	− 17.9653i	− 0.630071i
$814$	0	0
$815$	−0.589907	−0.0206636
$816$	0	0
$817$	9.37228	0.327895
$818$	0	0
$819$	− 35.8098i	− 1.25130i
$820$	0	0
$821$	37.0179i	1.29193i	0.763366	+	0.645966i	$0.223545\pi$
$821$	37.0179i	1.29193i	−0.763366	+	0.645966i	$0.776455\pi$
$822$	0	0
$823$	27.2704	0.950586	0.475293	−	0.879828i	$-0.342342\pi$
$823$	27.2704	0.950586	0.475293	+	0.879828i	$0.342342\pi$
$824$	0	0
$825$	34.9783	1.21779
$826$	0	0
$827$	11.3505i	0.394697i	0.980333	+	0.197348i	$0.0632330\pi$
$827$	11.3505i	0.394697i	−0.980333	+	0.197348i	$0.936767\pi$
$828$	0	0
$829$	33.1113i	1.15000i	0.818152	+	0.575002i	$0.194999\pi$
$829$	33.1113i	1.15000i	−0.818152	+	0.575002i	$0.805001\pi$
$830$	0	0
$831$	30.9918	1.07509
$832$	0	0
$833$	20.0000	0.692959
$834$	0	0
$835$	16.0000i	0.553703i
$836$	0	0
$837$	3.05934i	0.105746i
$838$	0	0
$839$	16.7306	0.577604	0.288802	−	0.957389i	$-0.406743\pi$
$839$	16.7306	0.577604	0.288802	+	0.957389i	$0.406743\pi$
$840$	0	0
$841$	−6.86141	−0.236600
$842$	0	0
$843$	− 66.9783i	− 2.30685i
$844$	0	0
$845$	3.07657i	0.105837i
$846$	0	0
$847$	1.23472	0.0424254
$848$	0	0
$849$	−55.4456	−1.90289
$850$	0	0
$851$	8.00000i	0.274236i
$852$	0	0
$853$	12.6766i	0.434038i	0.976167	+	0.217019i	$0.0696334\pi$
$853$	12.6766i	0.434038i	−0.976167	+	0.217019i	$0.930367\pi$
$854$	0	0
$855$	−2.08191	−0.0711997
$856$	0	0
$857$	−12.2337	−0.417895	−0.208947	−	0.977927i	$-0.567004\pi$
$857$	−12.2337	−0.417895	−0.208947	+	0.977927i	$0.567004\pi$
$858$	0	0
$859$	− 31.6060i	− 1.07838i	−0.842184	−	0.539191i	$-0.818730\pi$
$859$	− 31.6060i	− 1.07838i	0.842184	−	0.539191i	$-0.181270\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	49.1971	1.67469	0.837345	−	0.546675i	$-0.184107\pi$
$863$	49.1971	1.67469	0.837345	+	0.546675i	$0.184107\pi$
$864$	0	0
$865$	−3.76631	−0.128058
$866$	0	0
$867$	18.9783i	0.644535i
$868$	0	0
$869$	45.7330i	1.55139i
$870$	0	0
$871$	34.4010	1.16563
$872$	0	0
$873$	−22.9783	−0.777696
$874$	0	0
$875$	24.6277i	0.832569i
$876$	0	0
$877$	− 24.5986i	− 0.830635i	−0.909677	−	0.415317i	$-0.863671\pi$
$877$	− 24.5986i	− 0.830635i	0.909677	−	0.415317i	$-0.136329\pi$
$878$	0	0
$879$	−66.5725	−2.24544
$880$	0	0
$881$	1.37228	0.0462333	0.0231167	−	0.999733i	$-0.492641\pi$
$881$	1.37228	0.0462333	0.0231167	+	0.999733i	$0.492641\pi$
$882$	0	0
$883$	− 36.1168i	− 1.21543i	−0.794156	−	0.607714i	$-0.792087\pi$
$883$	− 36.1168i	− 1.21543i	0.794156	−	0.607714i	$-0.207913\pi$
$884$	0	0
$885$	− 8.21782i	− 0.276239i
$886$	0	0
$887$	45.0333	1.51207	0.756035	−	0.654531i	$-0.227134\pi$
$887$	45.0333	1.51207	0.756035	+	0.654531i	$0.227134\pi$
$888$	0	0
$889$	−62.7011	−2.10293
$890$	0	0
$891$	33.6495i	1.12730i
$892$	0	0
$893$	9.30506i	0.311382i
$894$	0	0
$895$	−18.2054	−0.608538
$896$	0	0
$897$	−24.6060	−0.821569
$898$	0	0
$899$	− 20.7446i	− 0.691870i
$900$	0	0
$901$	47.2627i	1.57455i
$902$	0	0
$903$	−73.7408	−2.45394
$904$	0	0
$905$	−9.25544	−0.307661
$906$	0	0
$907$	18.1386i	0.602282i	0.953580	+	0.301141i	$0.0973675\pi$
$907$	18.1386i	0.602282i	−0.953580	+	0.301141i	$0.902632\pi$
$908$	0	0
$909$	36.4107i	1.20767i
$910$	0	0
$911$	7.81306	0.258858	0.129429	−	0.991589i	$-0.458686\pi$
$911$	7.81306	0.258858	0.129429	+	0.991589i	$0.458686\pi$
$912$	0	0
$913$	−26.9783	−0.892850
$914$	0	0
$915$	− 8.55437i	− 0.282799i
$916$	0	0
$917$	− 4.55134i	− 0.150299i
$918$	0	0
$919$	27.7677	0.915972	0.457986	−	0.888959i	$-0.348571\pi$
$919$	27.7677	0.915972	0.457986	+	0.888959i	$0.348571\pi$
$920$	0	0
$921$	63.4456	2.09060
$922$	0	0
$923$	− 27.2554i	− 0.897124i
$924$	0	0
$925$	− 13.8564i	− 0.455596i
$926$	0	0
$927$	−14.0416	−0.461185
$928$	0	0
$929$	−45.1168	−1.48024	−0.740118	−	0.672477i	$-0.765230\pi$
$929$	−45.1168	−1.48024	−0.740118	+	0.672477i	$0.765230\pi$
$930$	0	0
$931$	4.00000i	0.131095i
$932$	0	0
$933$	− 55.0758i	− 1.80310i
$934$	0	0
$935$	−13.3591	−0.436888
$936$	0	0
$937$	23.9783	0.783335	0.391668	−	0.920107i	$-0.371898\pi$
$937$	23.9783	0.783335	0.391668	+	0.920107i	$0.371898\pi$
$938$	0	0
$939$	50.0951i	1.63479i
$940$	0	0
$941$	− 5.69349i	− 0.185602i	−0.995685	−	0.0928012i	$-0.970418\pi$
$941$	− 5.69349i	− 0.185602i	0.995685	−	0.0928012i	$-0.0295821\pi$
$942$	0	0
$943$	0	0
$944$	0	0
$945$	−2.32069	−0.0754919
$946$	0	0
$947$	37.2119i	1.20923i	0.796520	+	0.604613i	$0.206672\pi$
$947$	37.2119i	1.20923i	−0.796520	+	0.604613i	$0.793328\pi$
$948$	0	0
$949$	59.5345i	1.93257i
$950$	0	0
$951$	−43.3185	−1.40470
$952$	0	0
$953$	−8.23369	−0.266715	−0.133358	−	0.991068i	$-0.542576\pi$
$953$	−8.23369	−0.266715	−0.133358	+	0.991068i	$0.542576\pi$
$954$	0	0
$955$	10.8614i	0.351467i
$956$	0	0
$957$	− 47.9075i	− 1.54863i
$958$	0	0
$959$	−16.5831	−0.535497
$960$	0	0
$961$	−19.0000	−0.612903
$962$	0	0
$963$	20.0435i	0.645893i
$964$	0	0
$965$	− 11.6819i	− 0.376054i
$966$	0	0
$967$	−41.8642	−1.34626	−0.673131	−	0.739524i	$-0.735051\pi$
$967$	−41.8642	−1.34626	−0.673131	+	0.739524i	$0.735051\pi$
$968$	0	0
$969$	−11.8614	−0.381043
$970$	0	0
$971$	− 4.00000i	− 0.128366i	−0.997938	−	0.0641831i	$-0.979556\pi$
$971$	− 4.00000i	− 0.128366i	0.997938	−	0.0641831i	$-0.0204442\pi$
$972$	0	0
$973$	31.0843i	0.996518i
$974$	0	0
$975$	42.6188	1.36489
$976$	0	0
$977$	4.51087	0.144316	0.0721578	−	0.997393i	$-0.477011\pi$
$977$	4.51087	0.144316	0.0721578	+	0.997393i	$0.477011\pi$
$978$	0	0
$979$	− 25.2554i	− 0.807167i
$980$	0	0
$981$	33.1662i	1.05892i
$982$	0	0
$983$	−6.22849	−0.198658	−0.0993290	−	0.995055i	$-0.531670\pi$
$983$	−6.22849	−0.198658	−0.0993290	+	0.995055i	$0.531670\pi$
$984$	0	0
$985$	16.0000	0.509802
$986$	0	0
$987$	− 73.2119i	− 2.33036i
$988$	0	0
$989$	23.6588i	0.752306i
$990$	0	0
$991$	−25.2434	−0.801882	−0.400941	−	0.916104i	$-0.631317\pi$
$991$	−25.2434	−0.801882	−0.400941	+	0.916104i	$0.631317\pi$
$992$	0	0
$993$	−42.3723	−1.34464
$994$	0	0
$995$	5.37228i	0.170313i
$996$	0	0
$997$	10.2997i	0.326196i	0.986610	+	0.163098i	$0.0521486\pi$
$997$	10.2997i	0.326196i	−0.986610	+	0.163098i	$0.947851\pi$
$998$	0	0
$999$	2.79885	0.0885518

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1216.2.c.h.609.5	yes	8
4.3	odd	2	inner	1216.2.c.h.609.3	✓	8
8.3	odd	2	inner	1216.2.c.h.609.6	yes	8
8.5	even	2	inner	1216.2.c.h.609.4	yes	8
16.3	odd	4		4864.2.a.bi.1.3		4
16.5	even	4		4864.2.a.bi.1.4		4
16.11	odd	4		4864.2.a.bl.1.2		4
16.13	even	4		4864.2.a.bl.1.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1216.2.c.h.609.3	✓	8	4.3	odd	2	inner
1216.2.c.h.609.4	yes	8	8.5	even	2	inner
1216.2.c.h.609.5	yes	8	1.1	even	1	trivial
1216.2.c.h.609.6	yes	8	8.3	odd	2	inner
4864.2.a.bi.1.3		4	16.3	odd	4
4864.2.a.bi.1.4		4	16.5	even	4
4864.2.a.bl.1.1		4	16.13	even	4
4864.2.a.bl.1.2		4	16.11	odd	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1216 = 2^{6} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1216.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+2.37228i q^{3} -0.792287i q^{5} -3.31662 q^{7} -2.62772 q^{9} +O(q^{10})\)	\(q+2.37228i q^{3} -0.792287i q^{5} -3.31662 q^{7} -2.62772 q^{9} -3.37228i q^{11} -4.10891i q^{13} +1.87953 q^{15} +5.00000 q^{17} +1.00000i q^{19} -7.86797i q^{21} -2.52434 q^{23} +4.37228 q^{25} +0.883156i q^{27} -5.98844i q^{29} +3.46410 q^{31} +8.00000 q^{33} +2.62772i q^{35} -3.16915i q^{37} +9.74749 q^{39} -9.37228i q^{43} +2.08191i q^{45} +9.30506 q^{47} +4.00000 q^{49} +11.8614i q^{51} +9.45254i q^{53} -2.67181 q^{55} -2.37228 q^{57} -4.37228i q^{59} -4.55134i q^{61} +8.71516 q^{63} -3.25544 q^{65} +8.37228i q^{67} -5.98844i q^{69} +6.63325 q^{71} -14.4891 q^{73} +10.3723i q^{75} +11.1846i q^{77} -13.5615 q^{79} -9.97825 q^{81} -8.00000i q^{83} -3.96143i q^{85} +14.2063 q^{87} +7.48913 q^{89} +13.6277i q^{91} +8.21782i q^{93} +0.792287 q^{95} +8.74456 q^{97} +8.86141i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 44 q^{9}+O(q^{10}) \) 8 * q - 44 * q^9	\( 8 q - 44 q^{9} + 40 q^{17} + 12 q^{25} + 64 q^{33} + 32 q^{49} + 4 q^{57} - 72 q^{65} - 24 q^{73} + 104 q^{81} - 32 q^{89} + 24 q^{97}+O(q^{100}) \) 8 * q - 44 * q^9 + 40 * q^17 + 12 * q^25 + 64 * q^33 + 32 * q^49 + 4 * q^57 - 72 * q^65 - 24 * q^73 + 104 * q^81 - 32 * q^89 + 24 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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