LMFDB - Embedded newform 1216.2.c.h.609.8

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			609.8
Root			$1.26217 + 1.18614i$ of defining polynomial
Character	$\chi$	$=$	1216.609
Dual form			1216.2.c.h.609.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.37228i q^{3} +2.52434i q^{5} -3.31662 q^{7} -8.37228 q^{9} +O(q^{10})$	$q+3.37228i q^{3} +2.52434i q^{5} -3.31662 q^{7} -8.37228 q^{9} -2.37228i q^{11} +5.84096i q^{13} -8.51278 q^{15} +5.00000 q^{17} -1.00000i q^{19} -11.1846i q^{21} -0.792287 q^{23} -1.37228 q^{25} -18.1168i q^{27} -2.67181i q^{29} -3.46410 q^{31} +8.00000 q^{33} -8.37228i q^{35} +10.0974i q^{37} -19.6974 q^{39} +3.62772i q^{43} -21.1345i q^{45} +0.644810 q^{47} +4.00000 q^{49} +16.8614i q^{51} +6.13592i q^{53} +5.98844 q^{55} +3.37228 q^{57} -1.37228i q^{59} -14.5012i q^{61} +27.7677 q^{63} -14.7446 q^{65} -2.62772i q^{67} -2.67181i q^{69} +6.63325 q^{71} +8.48913 q^{73} -4.62772i q^{75} +7.86797i q^{77} +0.294954 q^{79} +35.9783 q^{81} +8.00000i q^{83} +12.6217i q^{85} +9.01011 q^{87} -15.4891 q^{89} -19.3723i q^{91} -11.6819i q^{93} +2.52434 q^{95} -2.74456 q^{97} +19.8614i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	3.37228i	1.94699i	0.228714	+	0.973494i	$0.426548\pi$
$3$	3.37228i	1.94699i	−0.228714	+	0.973494i	$0.573452\pi$
$4$	0	0
$5$	2.52434i	1.12892i	0.825461	+	0.564459i	$0.190915\pi$
$5$	2.52434i	1.12892i	−0.825461	+	0.564459i	$0.809085\pi$
$6$	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
$9$	−8.37228	−2.79076
$10$	0	0
$11$	− 2.37228i	− 0.715270i	−0.933862	−	0.357635i	$-0.883583\pi$
$11$	− 2.37228i	− 0.715270i	0.933862	−	0.357635i	$-0.116417\pi$
$12$	0	0
$13$	5.84096i	1.61999i	0.586436	+	0.809996i	$0.300531\pi$
$13$	5.84096i	1.61999i	−0.586436	+	0.809996i	$0.699469\pi$
$14$	0	0
$15$	−8.51278	−2.19799
$16$	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
$21$	− 11.1846i	− 2.44068i
$22$	0	0
$23$	−0.792287	−0.165203	−0.0826016	−	0.996583i	$-0.526323\pi$
$23$	−0.792287	−0.165203	−0.0826016	+	0.996583i	$0.526323\pi$
$24$	0	0
$25$	−1.37228	−0.274456
$26$	0	0
$27$	− 18.1168i	− 3.48659i
$28$	0	0
$29$	− 2.67181i	− 0.496144i	−0.968742	−	0.248072i	$-0.920203\pi$
$29$	− 2.67181i	− 0.496144i	0.968742	−	0.248072i	$-0.0797969\pi$
$30$	0	0
$31$	−3.46410	−0.622171	−0.311086	−	0.950382i	$-0.600693\pi$
$31$	−3.46410	−0.622171	−0.311086	+	0.950382i	$0.600693\pi$
$32$	0	0
$33$	8.00000	1.39262
$34$	0	0
$35$	− 8.37228i	− 1.41517i
$36$	0	0
$37$	10.0974i	1.65999i	0.557768	+	0.829997i	$0.311658\pi$
$37$	10.0974i	1.65999i	−0.557768	+	0.829997i	$0.688342\pi$
$38$	0	0
$39$	−19.6974	−3.15410
$40$	0	0
$41$	0	0		−	1.00000i	$-0.5\pi$
$41$	0	0			1.00000i	$0.5\pi$
$42$	0	0
$43$	3.62772i	0.553222i	0.960982	+	0.276611i	$0.0892113\pi$
$43$	3.62772i	0.553222i	−0.960982	+	0.276611i	$0.910789\pi$
$44$	0	0
$45$	− 21.1345i	− 3.15054i
$46$	0	0
$47$	0.644810	0.0940552	0.0470276	−	0.998894i	$-0.485025\pi$
$47$	0.644810	0.0940552	0.0470276	+	0.998894i	$0.485025\pi$
$48$	0	0
$49$	4.00000	0.571429
$50$	0	0
$51$	16.8614i	2.36107i
$52$	0	0
$53$	6.13592i	0.842833i	0.906867	+	0.421416i	$0.138467\pi$
$53$	6.13592i	0.842833i	−0.906867	+	0.421416i	$0.861533\pi$
$54$	0	0
$55$	5.98844	0.807481
$56$	0	0
$57$	3.37228	0.446670
$58$	0	0
$59$	− 1.37228i	− 0.178656i	−0.996002	−	0.0893279i	$-0.971528\pi$
$59$	− 1.37228i	− 0.178656i	0.996002	−	0.0893279i	$-0.0284719\pi$
$60$	0	0
$61$	− 14.5012i	− 1.85669i	−0.371719	−	0.928345i	$-0.621232\pi$
$61$	− 14.5012i	− 1.85669i	0.371719	−	0.928345i	$-0.378768\pi$
$62$	0	0
$63$	27.7677	3.49840
$64$	0	0
$65$	−14.7446	−1.82884
$66$	0	0
$67$	− 2.62772i	− 0.321027i	−0.987034	−	0.160513i	$-0.948685\pi$
$67$	− 2.62772i	− 0.321027i	0.987034	−	0.160513i	$-0.0513150\pi$
$68$	0	0
$69$	− 2.67181i	− 0.321649i
$70$	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
$73$	8.48913	0.993577	0.496788	−	0.867872i	$-0.334512\pi$
$73$	8.48913	0.993577	0.496788	+	0.867872i	$0.334512\pi$
$74$	0	0
$75$	− 4.62772i	− 0.534363i
$76$	0	0
$77$	7.86797i	0.896638i
$78$	0	0
$79$	0.294954	0.0331849	0.0165924	−	0.999862i	$-0.494718\pi$
$79$	0.294954	0.0331849	0.0165924	+	0.999862i	$0.494718\pi$
$80$	0	0
$81$	35.9783	3.99758
$82$	0	0
$83$	8.00000i	0.878114i	0.898459	+	0.439057i	$0.144687\pi$
$83$	8.00000i	0.878114i	−0.898459	+	0.439057i	$0.855313\pi$
$84$	0	0
$85$	12.6217i	1.36901i
$86$	0	0
$87$	9.01011	0.965985
$88$	0	0
$89$	−15.4891	−1.64184	−0.820922	−	0.571040i	$-0.806540\pi$
$89$	−15.4891	−1.64184	−0.820922	+	0.571040i	$0.806540\pi$
$90$	0	0
$91$	− 19.3723i	− 2.03077i
$92$	0	0
$93$	− 11.6819i	− 1.21136i
$94$	0	0
$95$	2.52434	0.258992
$96$	0	0
$97$	−2.74456	−0.278668	−0.139334	−	0.990245i	$-0.544496\pi$
$97$	−2.74456	−0.278668	−0.139334	+	0.990245i	$0.544496\pi$
$98$	0	0
$99$	19.8614i	1.99615i
$100$	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
$103$	−11.9769	−1.18012	−0.590058	−	0.807361i	$-0.700895\pi$
$103$	−11.9769	−1.18012	−0.590058	+	0.807361i	$0.700895\pi$
$104$	0	0
$105$	28.2337	2.75533
$106$	0	0
$107$	13.3723i	1.29275i	0.763021	+	0.646374i	$0.223715\pi$
$107$	13.3723i	1.29275i	−0.763021	+	0.646374i	$0.776285\pi$
$108$	0	0
$109$	3.96143i	0.379437i	0.981839	+	0.189718i	$0.0607575\pi$
$109$	3.96143i	0.379437i	−0.981839	+	0.189718i	$0.939243\pi$
$110$	0	0
$111$	−34.0511	−3.23199
$112$	0	0
$113$	−14.7446	−1.38705	−0.693526	−	0.720432i	$-0.743944\pi$
$113$	−14.7446	−1.38705	−0.693526	+	0.720432i	$0.743944\pi$
$114$	0	0
$115$	− 2.00000i	− 0.186501i
$116$	0	0
$117$	− 48.9022i	− 4.52101i
$118$	0	0
$119$	−16.5831	−1.52017
$120$	0	0
$121$	5.37228	0.488389
$122$	0	0
$123$	0	0
$124$	0	0
$125$	9.15759i	0.819080i
$126$	0	0
$127$	−12.2718	−1.08895	−0.544475	−	0.838777i	$-0.683271\pi$
$127$	−12.2718	−1.08895	−0.544475	+	0.838777i	$0.683271\pi$
$128$	0	0
$129$	−12.2337	−1.07712
$130$	0	0
$131$	4.37228i	0.382008i	0.981589	+	0.191004i	$0.0611743\pi$
$131$	4.37228i	0.382008i	−0.981589	+	0.191004i	$0.938826\pi$
$132$	0	0
$133$	3.31662i	0.287588i
$134$	0	0
$135$	45.7330	3.93607
$136$	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
$139$	3.62772i	0.307699i	0.988094	+	0.153850i	$0.0491671\pi$
$139$	3.62772i	0.307699i	−0.988094	+	0.153850i	$0.950833\pi$
$140$	0	0
$141$	2.17448i	0.183124i
$142$	0	0
$143$	13.8564	1.15873
$144$	0	0
$145$	6.74456	0.560105
$146$	0	0
$147$	13.4891i	1.11256i
$148$	0	0
$149$	4.69882i	0.384942i	0.981303	+	0.192471i	$0.0616502\pi$
$149$	4.69882i	0.384942i	−0.981303	+	0.192471i	$0.938350\pi$
$150$	0	0
$151$	−15.7359	−1.28057	−0.640286	−	0.768137i	$-0.721184\pi$
$151$	−15.7359	−1.28057	−0.640286	+	0.768137i	$0.721184\pi$
$152$	0	0
$153$	−41.8614	−3.38429
$154$	0	0
$155$	− 8.74456i	− 0.702380i
$156$	0	0
$157$	− 18.6101i	− 1.48525i	−0.669708	−	0.742625i	$-0.733581\pi$
$157$	− 18.6101i	− 1.48525i	0.669708	−	0.742625i	$-0.266419\pi$
$158$	0	0
$159$	−20.6920	−1.64099
$160$	0	0
$161$	2.62772	0.207093
$162$	0	0
$163$	− 10.7446i	− 0.841579i	−0.907158	−	0.420790i	$-0.861753\pi$
$163$	− 10.7446i	− 0.841579i	0.907158	−	0.420790i	$-0.138247\pi$
$164$	0	0
$165$	20.1947i	1.57216i
$166$	0	0
$167$	−6.33830	−0.490472	−0.245236	−	0.969463i	$-0.578865\pi$
$167$	−6.33830	−0.490472	−0.245236	+	0.969463i	$0.578865\pi$
$168$	0	0
$169$	−21.1168	−1.62437
$170$	0	0
$171$	8.37228i	0.640244i
$172$	0	0
$173$	15.1460i	1.15153i	0.817615	+	0.575766i	$0.195296\pi$
$173$	15.1460i	1.15153i	−0.817615	+	0.575766i	$0.804704\pi$
$174$	0	0
$175$	4.55134	0.344049
$176$	0	0
$177$	4.62772	0.347841
$178$	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
$181$	8.21782i	0.610826i	0.952220	+	0.305413i	$0.0987945\pi$
$181$	8.21782i	0.610826i	−0.952220	+	0.305413i	$0.901205\pi$
$182$	0	0
$183$	48.9022	3.61495
$184$	0	0
$185$	−25.4891	−1.87400
$186$	0	0
$187$	− 11.8614i	− 0.867392i
$188$	0	0
$189$	60.0868i	4.37067i
$190$	0	0
$191$	7.07568	0.511978	0.255989	−	0.966680i	$-0.417599\pi$
$191$	7.07568	0.511978	0.255989	+	0.966680i	$0.417599\pi$
$192$	0	0
$193$	3.25544	0.234332	0.117166	−	0.993112i	$-0.462619\pi$
$193$	3.25544	0.234332	0.117166	+	0.993112i	$0.462619\pi$
$194$	0	0
$195$	− 49.7228i	− 3.56072i
$196$	0	0
$197$	− 6.33830i	− 0.451585i	−0.974175	−	0.225792i	$-0.927503\pi$
$197$	− 6.33830i	− 0.451585i	0.974175	−	0.225792i	$-0.0724971\pi$
$198$	0	0
$199$	0.147477	0.0104544	0.00522718	−	0.999986i	$-0.498336\pi$
$199$	0.147477	0.0104544	0.00522718	+	0.999986i	$0.498336\pi$
$200$	0	0
$201$	8.86141	0.625035
$202$	0	0
$203$	8.86141i	0.621949i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	6.63325	0.461043
$208$	0	0
$209$	−2.37228	−0.164094
$210$	0	0
$211$	2.62772i	0.180900i	0.995901	+	0.0904498i	$0.0288305\pi$
$211$	2.62772i	0.180900i	−0.995901	+	0.0904498i	$0.971170\pi$
$212$	0	0
$213$	22.3692i	1.53271i
$214$	0	0
$215$	−9.15759	−0.624542
$216$	0	0
$217$	11.4891	0.779933
$218$	0	0
$219$	28.6277i	1.93448i
$220$	0	0
$221$	29.2048i	1.96453i
$222$	0	0
$223$	6.92820	0.463947	0.231973	−	0.972722i	$-0.425482\pi$
$223$	6.92820	0.463947	0.231973	+	0.972722i	$0.425482\pi$
$224$	0	0
$225$	11.4891	0.765942
$226$	0	0
$227$	14.1168i	0.936968i	0.883472	+	0.468484i	$0.155200\pi$
$227$	14.1168i	0.936968i	−0.883472	+	0.468484i	$0.844800\pi$
$228$	0	0
$229$	12.6217i	0.834065i	0.908892	+	0.417032i	$0.136930\pi$
$229$	12.6217i	0.834065i	−0.908892	+	0.417032i	$0.863070\pi$
$230$	0	0
$231$	−26.5330	−1.74574
$232$	0	0
$233$	−10.8832	−0.712979	−0.356490	−	0.934299i	$-0.616026\pi$
$233$	−10.8832	−0.712979	−0.356490	+	0.934299i	$0.616026\pi$
$234$	0	0
$235$	1.62772i	0.106181i
$236$	0	0
$237$	0.994667i	0.0646105i
$238$	0	0
$239$	17.4680	1.12991	0.564955	−	0.825122i	$-0.308894\pi$
$239$	17.4680	1.12991	0.564955	+	0.825122i	$0.308894\pi$
$240$	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
$243$	66.9783i	4.29666i
$244$	0	0
$245$	10.0974i	0.645096i
$246$	0	0
$247$	5.84096	0.371652
$248$	0	0
$249$	−26.9783	−1.70968
$250$	0	0
$251$	17.8614i	1.12740i	0.825979	+	0.563701i	$0.190623\pi$
$251$	17.8614i	1.12740i	−0.825979	+	0.563701i	$0.809377\pi$
$252$	0	0
$253$	1.87953i	0.118165i
$254$	0	0
$255$	−42.5639	−2.66545
$256$	0	0
$257$	−16.2337	−1.01263	−0.506315	−	0.862349i	$-0.668993\pi$
$257$	−16.2337	−1.01263	−0.506315	+	0.862349i	$0.668993\pi$
$258$	0	0
$259$	− 33.4891i	− 2.08091i
$260$	0	0
$261$	22.3692i	1.38462i
$262$	0	0
$263$	−18.9600	−1.16912	−0.584561	−	0.811349i	$-0.698733\pi$
$263$	−18.9600	−1.16912	−0.584561	+	0.811349i	$0.698733\pi$
$264$	0	0
$265$	−15.4891	−0.951489
$266$	0	0
$267$	− 52.2337i	− 3.19665i
$268$	0	0
$269$	1.87953i	0.114597i	0.998357	+	0.0572984i	$0.0182486\pi$
$269$	1.87953i	0.114597i	−0.998357	+	0.0572984i	$0.981751\pi$
$270$	0	0
$271$	−2.37686	−0.144384	−0.0721920	−	0.997391i	$-0.522999\pi$
$271$	−2.37686	−0.144384	−0.0721920	+	0.997391i	$0.522999\pi$
$272$	0	0
$273$	65.3288	3.95388
$274$	0	0
$275$	3.25544i	0.196310i
$276$	0	0
$277$	− 16.3807i	− 0.984224i	−0.870532	−	0.492112i	$-0.836225\pi$
$277$	− 16.3807i	− 0.984224i	0.870532	−	0.492112i	$-0.163775\pi$
$278$	0	0
$279$	29.0024	1.73633
$280$	0	0
$281$	6.23369	0.371871	0.185935	−	0.982562i	$-0.440469\pi$
$281$	6.23369	0.371871	0.185935	+	0.982562i	$0.440469\pi$
$282$	0	0
$283$	− 17.6277i	− 1.04786i	−0.851762	−	0.523930i	$-0.824466\pi$
$283$	− 17.6277i	− 1.04786i	0.851762	−	0.523930i	$-0.175534\pi$
$284$	0	0
$285$	8.51278i	0.504253i
$286$	0	0
$287$	0	0
$288$	0	0
$289$	8.00000	0.470588
$290$	0	0
$291$	− 9.25544i	− 0.542563i
$292$	0	0
$293$	4.84630i	0.283124i	0.989929	+	0.141562i	$0.0452124\pi$
$293$	4.84630i	0.283124i	−0.989929	+	0.141562i	$0.954788\pi$
$294$	0	0
$295$	3.46410	0.201688
$296$	0	0
$297$	−42.9783	−2.49385
$298$	0	0
$299$	− 4.62772i	− 0.267628i
$300$	0	0
$301$	− 12.0318i	− 0.693500i
$302$	0	0
$303$	46.7277	2.68444
$304$	0	0
$305$	36.6060	2.09605
$306$	0	0
$307$	15.2554i	0.870674i	0.900268	+	0.435337i	$0.143371\pi$
$307$	15.2554i	0.870674i	−0.900268	+	0.435337i	$0.856629\pi$
$308$	0	0
$309$	− 40.3894i	− 2.29767i
$310$	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
$313$	3.88316	0.219489	0.109744	−	0.993960i	$-0.464997\pi$
$313$	3.88316	0.219489	0.109744	+	0.993960i	$0.464997\pi$
$314$	0	0
$315$	70.0951i	3.94941i
$316$	0	0
$317$	21.5769i	1.21188i	0.795511	+	0.605940i	$0.207203\pi$
$317$	21.5769i	1.21188i	−0.795511	+	0.605940i	$0.792797\pi$
$318$	0	0
$319$	−6.33830	−0.354876
$320$	0	0
$321$	−45.0951	−2.51696
$322$	0	0
$323$	− 5.00000i	− 0.278207i
$324$	0	0
$325$	− 8.01544i	− 0.444617i
$326$	0	0
$327$	−13.3591	−0.738758
$328$	0	0
$329$	−2.13859	−0.117904
$330$	0	0
$331$	10.8614i	0.596997i	0.954410	+	0.298498i	$0.0964858\pi$
$331$	10.8614i	0.596997i	−0.954410	+	0.298498i	$0.903514\pi$
$332$	0	0
$333$	− 84.5379i	− 4.63265i
$334$	0	0
$335$	6.63325	0.362413
$336$	0	0
$337$	−13.2554	−0.722070	−0.361035	−	0.932552i	$-0.617576\pi$
$337$	−13.2554	−0.722070	−0.361035	+	0.932552i	$0.617576\pi$
$338$	0	0
$339$	− 49.7228i	− 2.70057i
$340$	0	0
$341$	8.21782i	0.445020i
$342$	0	0
$343$	9.94987	0.537243
$344$	0	0
$345$	6.74456	0.363115
$346$	0	0
$347$	37.1168i	1.99254i	0.0863104	+	0.996268i	$0.472492\pi$
$347$	37.1168i	1.99254i	−0.0863104	+	0.996268i	$0.527508\pi$
$348$	0	0
$349$	13.2116i	0.707201i	0.935397	+	0.353600i	$0.115043\pi$
$349$	13.2116i	0.707201i	−0.935397	+	0.353600i	$0.884957\pi$
$350$	0	0
$351$	105.820	5.64824
$352$	0	0
$353$	3.88316	0.206680	0.103340	−	0.994646i	$-0.467047\pi$
$353$	3.88316	0.206680	0.103340	+	0.994646i	$0.467047\pi$
$354$	0	0
$355$	16.7446i	0.888709i
$356$	0	0
$357$	− 55.9230i	− 2.95976i
$358$	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
$363$	18.1168i	0.950888i
$364$	0	0
$365$	21.4294i	1.12167i
$366$	0	0
$367$	−4.05401	−0.211618	−0.105809	−	0.994386i	$-0.533743\pi$
$367$	−4.05401	−0.211618	−0.105809	+	0.994386i	$0.533743\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	− 20.3505i	− 1.05655i
$372$	0	0
$373$	14.9436i	0.773753i	0.922132	+	0.386876i	$0.126446\pi$
$373$	14.9436i	0.773753i	−0.922132	+	0.386876i	$0.873554\pi$
$374$	0	0
$375$	−30.8820	−1.59474
$376$	0	0
$377$	15.6060	0.803748
$378$	0	0
$379$	36.3505i	1.86720i	0.358315	+	0.933601i	$0.383351\pi$
$379$	36.3505i	1.86720i	−0.358315	+	0.933601i	$0.616649\pi$
$380$	0	0
$381$	− 41.3841i	− 2.12017i
$382$	0	0
$383$	25.2434	1.28988	0.644938	−	0.764235i	$-0.276883\pi$
$383$	25.2434	1.28988	0.644938	+	0.764235i	$0.276883\pi$
$384$	0	0
$385$	−19.8614	−1.01223
$386$	0	0
$387$	− 30.3723i	− 1.54391i
$388$	0	0
$389$	24.3036i	1.23224i	0.787651	+	0.616121i	$0.211297\pi$
$389$	24.3036i	1.23224i	−0.787651	+	0.616121i	$0.788703\pi$
$390$	0	0
$391$	−3.96143	−0.200338
$392$	0	0
$393$	−14.7446	−0.743765
$394$	0	0
$395$	0.744563i	0.0374630i
$396$	0	0
$397$	0.644810i	0.0323621i	0.999869	+	0.0161810i	$0.00515081\pi$
$397$	0.644810i	0.0323621i	−0.999869	+	0.0161810i	$0.994849\pi$
$398$	0	0
$399$	−11.1846	−0.559930
$400$	0	0
$401$	35.7228	1.78391	0.891956	−	0.452122i	$-0.149333\pi$
$401$	35.7228	1.78391	0.891956	+	0.452122i	$0.149333\pi$
$402$	0	0
$403$	− 20.2337i	− 1.00791i
$404$	0	0
$405$	90.8213i	4.51294i
$406$	0	0
$407$	23.9538	1.18734
$408$	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
$411$	16.8614i	0.831712i
$412$	0	0
$413$	4.55134i	0.223957i
$414$	0	0
$415$	−20.1947	−0.991319
$416$	0	0
$417$	−12.2337	−0.599086
$418$	0	0
$419$	− 0.233688i	− 0.0114164i	−0.999984	−	0.00570820i	$-0.998183\pi$
$419$	− 0.233688i	− 0.0114164i	0.999984	−	0.00570820i	$-0.00181699\pi$
$420$	0	0
$421$	− 20.3971i	− 0.994093i	−0.867724	−	0.497046i	$-0.834418\pi$
$421$	− 20.3971i	− 0.994093i	0.867724	−	0.497046i	$-0.165582\pi$
$422$	0	0
$423$	−5.39853	−0.262486
$424$	0	0
$425$	−6.86141	−0.332827
$426$	0	0
$427$	48.0951i	2.32748i
$428$	0	0
$429$	46.7277i	2.25603i
$430$	0	0
$431$	21.7793	1.04907	0.524535	−	0.851389i	$-0.324239\pi$
$431$	21.7793	1.04907	0.524535	+	0.851389i	$0.324239\pi$
$432$	0	0
$433$	−1.48913	−0.0715628	−0.0357814	−	0.999360i	$-0.511392\pi$
$433$	−1.48913	−0.0715628	−0.0357814	+	0.999360i	$0.511392\pi$
$434$	0	0
$435$	22.7446i	1.09052i
$436$	0	0
$437$	0.792287i	0.0379002i
$438$	0	0
$439$	6.92820	0.330665	0.165333	−	0.986238i	$-0.447130\pi$
$439$	6.92820	0.330665	0.165333	+	0.986238i	$0.447130\pi$
$440$	0	0
$441$	−33.4891	−1.59472
$442$	0	0
$443$	41.3505i	1.96462i	0.187254	+	0.982312i	$0.440041\pi$
$443$	41.3505i	1.96462i	−0.187254	+	0.982312i	$0.559959\pi$
$444$	0	0
$445$	− 39.0998i	− 1.85351i
$446$	0	0
$447$	−15.8457	−0.749478
$448$	0	0
$449$	37.4891	1.76922	0.884611	−	0.466330i	$-0.154424\pi$
$449$	37.4891	1.76922	0.884611	+	0.466330i	$0.154424\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	− 53.0660i	− 2.49326i
$454$	0	0
$455$	48.9022	2.29257
$456$	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
$459$	− 90.5842i	− 4.22811i
$460$	0	0
$461$	7.86797i	0.366448i	0.983071	+	0.183224i	$0.0586533\pi$
$461$	7.86797i	0.366448i	−0.983071	+	0.183224i	$0.941347\pi$
$462$	0	0
$463$	−31.5268	−1.46517	−0.732587	−	0.680674i	$-0.761687\pi$
$463$	−31.5268	−1.46517	−0.732587	+	0.680674i	$0.761687\pi$
$464$	0	0
$465$	29.4891	1.36753
$466$	0	0
$467$	− 10.8832i	− 0.503612i	−0.967778	−	0.251806i	$-0.918975\pi$
$467$	− 10.8832i	− 0.503612i	0.967778	−	0.251806i	$-0.0810246\pi$
$468$	0	0
$469$	8.71516i	0.402429i
$470$	0	0
$471$	62.7586	2.89176
$472$	0	0
$473$	8.60597	0.395703
$474$	0	0
$475$	1.37228i	0.0629646i
$476$	0	0
$477$	− 51.3716i	− 2.35214i
$478$	0	0
$479$	−27.4179	−1.25275	−0.626377	−	0.779520i	$-0.715463\pi$
$479$	−27.4179	−1.25275	−0.626377	+	0.779520i	$0.715463\pi$
$480$	0	0
$481$	−58.9783	−2.68918
$482$	0	0
$483$	8.86141i	0.403208i
$484$	0	0
$485$	− 6.92820i	− 0.314594i
$486$	0	0
$487$	−22.6641	−1.02701	−0.513505	−	0.858087i	$-0.671653\pi$
$487$	−22.6641	−1.02701	−0.513505	+	0.858087i	$0.671653\pi$
$488$	0	0
$489$	36.2337	1.63854
$490$	0	0
$491$	30.9783i	1.39803i	0.715108	+	0.699014i	$0.246378\pi$
$491$	30.9783i	1.39803i	−0.715108	+	0.699014i	$0.753622\pi$
$492$	0	0
$493$	− 13.3591i	− 0.601662i
$494$	0	0
$495$	−50.1369	−2.25349
$496$	0	0
$497$	−22.0000	−0.986835
$498$	0	0
$499$	4.37228i	0.195730i	0.995200	+	0.0978651i	$0.0312014\pi$
$499$	4.37228i	0.195730i	−0.995200	+	0.0978651i	$0.968799\pi$
$500$	0	0
$501$	− 21.3745i	− 0.954943i
$502$	0	0
$503$	11.4795	0.511848	0.255924	−	0.966697i	$-0.417620\pi$
$503$	11.4795	0.511848	0.255924	+	0.966697i	$0.417620\pi$
$504$	0	0
$505$	34.9783	1.55651
$506$	0	0
$507$	− 71.2119i	− 3.16263i
$508$	0	0
$509$	− 37.8102i	− 1.67591i	−0.545742	−	0.837953i	$-0.683752\pi$
$509$	− 37.8102i	− 1.67591i	0.545742	−	0.837953i	$-0.316248\pi$
$510$	0	0
$511$	−28.1552	−1.24551
$512$	0	0
$513$	−18.1168	−0.799878
$514$	0	0
$515$	− 30.2337i	− 1.33226i
$516$	0	0
$517$	− 1.52967i	− 0.0672749i
$518$	0	0
$519$	−51.0767	−2.24202
$520$	0	0
$521$	−5.76631	−0.252627	−0.126313	−	0.991990i	$-0.540314\pi$
$521$	−5.76631	−0.252627	−0.126313	+	0.991990i	$0.540314\pi$
$522$	0	0
$523$	− 25.3723i	− 1.10945i	−0.832033	−	0.554726i	$-0.812823\pi$
$523$	− 25.3723i	− 1.10945i	0.832033	−	0.554726i	$-0.187177\pi$
$524$	0	0
$525$	15.3484i	0.669859i
$526$	0	0
$527$	−17.3205	−0.754493
$528$	0	0
$529$	−22.3723	−0.972708
$530$	0	0
$531$	11.4891i	0.498586i
$532$	0	0
$533$	0	0
$534$	0	0
$535$	−33.7562	−1.45941
$536$	0	0
$537$	77.4891	3.34390
$538$	0	0
$539$	− 9.48913i	− 0.408726i
$540$	0	0
$541$	− 23.6039i	− 1.01481i	−0.861707	−	0.507405i	$-0.830605\pi$
$541$	− 23.6039i	− 1.01481i	0.861707	−	0.507405i	$-0.169395\pi$
$542$	0	0
$543$	−27.7128	−1.18927
$544$	0	0
$545$	−10.0000	−0.428353
$546$	0	0
$547$	2.51087i	0.107357i	0.998558	+	0.0536786i	$0.0170947\pi$
$547$	2.51087i	0.107357i	−0.998558	+	0.0536786i	$0.982905\pi$
$548$	0	0
$549$	121.408i	5.18158i
$550$	0	0
$551$	−2.67181	−0.113823
$552$	0	0
$553$	−0.978251	−0.0415994
$554$	0	0
$555$	− 85.9565i	− 3.64865i
$556$	0	0
$557$	25.1885i	1.06727i	0.845715	+	0.533635i	$0.179174\pi$
$557$	25.1885i	1.06727i	−0.845715	+	0.533635i	$0.820826\pi$
$558$	0	0
$559$	−21.1894	−0.896215
$560$	0	0
$561$	40.0000	1.68880
$562$	0	0
$563$	13.4891i	0.568499i	0.958750	+	0.284249i	$0.0917444\pi$
$563$	13.4891i	0.568499i	−0.958750	+	0.284249i	$0.908256\pi$
$564$	0	0
$565$	− 37.2203i	− 1.56587i
$566$	0	0
$567$	−119.326	−5.01124
$568$	0	0
$569$	−26.9783	−1.13099	−0.565494	−	0.824753i	$-0.691314\pi$
$569$	−26.9783	−1.13099	−0.565494	+	0.824753i	$0.691314\pi$
$570$	0	0
$571$	42.9783i	1.79858i	0.437349	+	0.899292i	$0.355917\pi$
$571$	42.9783i	1.79858i	−0.437349	+	0.899292i	$0.644083\pi$
$572$	0	0
$573$	23.8612i	0.996815i
$574$	0	0
$575$	1.08724	0.0453411
$576$	0	0
$577$	33.4674	1.39327	0.696633	−	0.717428i	$-0.254681\pi$
$577$	33.4674	1.39327	0.696633	+	0.717428i	$0.254681\pi$
$578$	0	0
$579$	10.9783i	0.456241i
$580$	0	0
$581$	− 26.5330i	− 1.10077i
$582$	0	0
$583$	14.5561	0.602853
$584$	0	0
$585$	123.446	5.10385
$586$	0	0
$587$	− 31.8614i	− 1.31506i	−0.753428	−	0.657530i	$-0.771601\pi$
$587$	− 31.8614i	− 1.31506i	0.753428	−	0.657530i	$-0.228399\pi$
$588$	0	0
$589$	3.46410i	0.142736i
$590$	0	0
$591$	21.3745	0.879230
$592$	0	0
$593$	4.97825	0.204432	0.102216	−	0.994762i	$-0.467407\pi$
$593$	4.97825	0.204432	0.102216	+	0.994762i	$0.467407\pi$
$594$	0	0
$595$	− 41.8614i	− 1.71615i
$596$	0	0
$597$	0.497333i	0.0203545i
$598$	0	0
$599$	39.6897	1.62168	0.810838	−	0.585270i	$-0.199012\pi$
$599$	39.6897	1.62168	0.810838	+	0.585270i	$0.199012\pi$
$600$	0	0
$601$	−22.4674	−0.916463	−0.458232	−	0.888833i	$-0.651517\pi$
$601$	−22.4674	−0.916463	−0.458232	+	0.888833i	$0.651517\pi$
$602$	0	0
$603$	22.0000i	0.895909i
$604$	0	0
$605$	13.5615i	0.551351i
$606$	0	0
$607$	38.8048	1.57504	0.787520	−	0.616289i	$-0.211365\pi$
$607$	38.8048	1.57504	0.787520	+	0.616289i	$0.211365\pi$
$608$	0	0
$609$	−29.8832	−1.21093
$610$	0	0
$611$	3.76631i	0.152369i
$612$	0	0
$613$	− 18.5552i	− 0.749439i	−0.927138	−	0.374719i	$-0.877739\pi$
$613$	− 18.5552i	− 0.749439i	0.927138	−	0.374719i	$-0.122261\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−40.8397	−1.64414	−0.822071	−	0.569384i	$-0.807182\pi$
$617$	−40.8397	−1.64414	−0.822071	+	0.569384i	$0.807182\pi$
$618$	0	0
$619$	− 16.2337i	− 0.652487i	−0.945286	−	0.326244i	$-0.894217\pi$
$619$	− 16.2337i	− 0.652487i	0.945286	−	0.326244i	$-0.105783\pi$
$620$	0	0
$621$	14.3537i	0.575996i
$622$	0	0
$623$	51.3716	2.05816
$624$	0	0
$625$	−29.9783	−1.19913
$626$	0	0
$627$	− 8.00000i	− 0.319489i
$628$	0	0
$629$	50.4868i	2.01304i
$630$	0	0
$631$	37.8651	1.50738	0.753692	−	0.657227i	$-0.228271\pi$
$631$	37.8651	1.50738	0.753692	+	0.657227i	$0.228271\pi$
$632$	0	0
$633$	−8.86141	−0.352209
$634$	0	0
$635$	− 30.9783i	− 1.22933i
$636$	0	0
$637$	23.3639i	0.925709i
$638$	0	0
$639$	−55.5354	−2.19695
$640$	0	0
$641$	−10.9783	−0.433615	−0.216807	−	0.976214i	$-0.569564\pi$
$641$	−10.9783	−0.433615	−0.216807	+	0.976214i	$0.569564\pi$
$642$	0	0
$643$	− 36.3723i	− 1.43438i	−0.696876	−	0.717191i	$-0.745427\pi$
$643$	− 36.3723i	− 1.43438i	0.696876	−	0.717191i	$-0.254573\pi$
$644$	0	0
$645$	− 30.8820i	− 1.21598i
$646$	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
$649$	−3.25544	−0.127787
$650$	0	0
$651$	38.7446i	1.51852i
$652$	0	0
$653$	− 35.3956i	− 1.38514i	−0.721352	−	0.692569i	$-0.756479\pi$
$653$	− 35.3956i	− 1.38514i	0.721352	−	0.692569i	$-0.243521\pi$
$654$	0	0
$655$	−11.0371	−0.431256
$656$	0	0
$657$	−71.0733	−2.77284
$658$	0	0
$659$	46.8614i	1.82546i	0.408562	+	0.912731i	$0.366030\pi$
$659$	46.8614i	1.82546i	−0.408562	+	0.912731i	$0.633970\pi$
$660$	0	0
$661$	10.5947i	0.412085i	0.978543	+	0.206043i	$0.0660586\pi$
$661$	10.5947i	0.412085i	−0.978543	+	0.206043i	$0.933941\pi$
$662$	0	0
$663$	−98.4868	−3.82491
$664$	0	0
$665$	−8.37228	−0.324663
$666$	0	0
$667$	2.11684i	0.0819645i
$668$	0	0
$669$	23.3639i	0.903299i
$670$	0	0
$671$	−34.4010	−1.32803
$672$	0	0
$673$	16.7446	0.645455	0.322728	−	0.946492i	$-0.395400\pi$
$673$	16.7446	0.645455	0.322728	+	0.946492i	$0.395400\pi$
$674$	0	0
$675$	24.8614i	0.956916i
$676$	0	0
$677$	− 5.84096i	− 0.224486i	−0.993681	−	0.112243i	$-0.964196\pi$
$677$	− 5.84096i	− 0.224486i	0.993681	−	0.112243i	$-0.0358036\pi$
$678$	0	0
$679$	9.10268	0.349329
$680$	0	0
$681$	−47.6060	−1.82426
$682$	0	0
$683$	− 12.0000i	− 0.459167i	−0.973289	−	0.229584i	$-0.926264\pi$
$683$	− 12.0000i	− 0.459167i	0.973289	−	0.229584i	$-0.0737364\pi$
$684$	0	0
$685$	12.6217i	0.482250i
$686$	0	0
$687$	−42.5639	−1.62391
$688$	0	0
$689$	−35.8397	−1.36538
$690$	0	0
$691$	− 17.8614i	− 0.679480i	−0.940519	−	0.339740i	$-0.889661\pi$
$691$	− 17.8614i	− 0.679480i	0.940519	−	0.339740i	$-0.110339\pi$
$692$	0	0
$693$	− 65.8728i	− 2.50230i
$694$	0	0
$695$	−9.15759	−0.347367
$696$	0	0
$697$	0	0
$698$	0	0
$699$	− 36.7011i	− 1.38816i
$700$	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
$703$	10.0974	0.380829
$704$	0	0
$705$	−5.48913	−0.206732
$706$	0	0
$707$	45.9565i	1.72837i
$708$	0	0
$709$	− 45.7330i	− 1.71754i	−0.512361	−	0.858770i	$-0.671229\pi$
$709$	− 45.7330i	− 1.71754i	0.512361	−	0.858770i	$-0.328771\pi$
$710$	0	0
$711$	−2.46943	−0.0926110
$712$	0	0
$713$	2.74456	0.102785
$714$	0	0
$715$	34.9783i	1.30811i
$716$	0	0
$717$	58.9070i	2.19992i
$718$	0	0
$719$	40.5369	1.51177	0.755885	−	0.654704i	$-0.227207\pi$
$719$	40.5369	1.51177	0.755885	+	0.654704i	$0.227207\pi$
$720$	0	0
$721$	39.7228	1.47935
$722$	0	0
$723$	26.9783i	1.00333i
$724$	0	0
$725$	3.66648i	0.136170i
$726$	0	0
$727$	14.0039	0.519375	0.259688	−	0.965693i	$-0.416380\pi$
$727$	14.0039	0.519375	0.259688	+	0.965693i	$0.416380\pi$
$728$	0	0
$729$	−117.935	−4.36795
$730$	0	0
$731$	18.1386i	0.670880i
$732$	0	0
$733$	− 21.1894i	− 0.782647i	−0.920253	−	0.391324i	$-0.872017\pi$
$733$	− 21.1894i	− 0.782647i	0.920253	−	0.391324i	$-0.127983\pi$
$734$	0	0
$735$	−34.0511	−1.25599
$736$	0	0
$737$	−6.23369	−0.229621
$738$	0	0
$739$	12.6060i	0.463718i	0.972749	+	0.231859i	$0.0744808\pi$
$739$	12.6060i	0.463718i	−0.972749	+	0.231859i	$0.925519\pi$
$740$	0	0
$741$	19.6974i	0.723601i
$742$	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
$745$	−11.8614	−0.434568
$746$	0	0
$747$	− 66.9783i	− 2.45061i
$748$	0	0
$749$	− 44.3508i	− 1.62054i
$750$	0	0
$751$	9.50744	0.346932	0.173466	−	0.984840i	$-0.444503\pi$
$751$	9.50744	0.346932	0.173466	+	0.984840i	$0.444503\pi$
$752$	0	0
$753$	−60.2337	−2.19504
$754$	0	0
$755$	− 39.7228i	− 1.44566i
$756$	0	0
$757$	18.8502i	0.685121i	0.939496	+	0.342561i	$0.111294\pi$
$757$	18.8502i	0.685121i	−0.939496	+	0.342561i	$0.888706\pi$
$758$	0	0
$759$	−6.33830	−0.230066
$760$	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
$763$	− 13.1386i	− 0.475649i
$764$	0	0
$765$	− 105.672i	− 3.82059i
$766$	0	0
$767$	8.01544	0.289421
$768$	0	0
$769$	37.4674	1.35111	0.675554	−	0.737310i	$-0.263904\pi$
$769$	37.4674	1.35111	0.675554	+	0.737310i	$0.263904\pi$
$770$	0	0
$771$	− 54.7446i	− 1.97158i
$772$	0	0
$773$	− 24.1561i	− 0.868836i	−0.900711	−	0.434418i	$-0.856954\pi$
$773$	− 24.1561i	− 0.868836i	0.900711	−	0.434418i	$-0.143046\pi$
$774$	0	0
$775$	4.75372	0.170759
$776$	0	0
$777$	112.935	4.05151
$778$	0	0
$779$	0	0
$780$	0	0
$781$	− 15.7359i	− 0.563076i
$782$	0	0
$783$	−48.4048	−1.72985
$784$	0	0
$785$	46.9783	1.67673
$786$	0	0
$787$	19.6060i	0.698877i	0.936959	+	0.349439i	$0.113628\pi$
$787$	19.6060i	0.698877i	−0.936959	+	0.349439i	$0.886372\pi$
$788$	0	0
$789$	− 63.9384i	− 2.27627i
$790$	0	0
$791$	48.9022	1.73876
$792$	0	0
$793$	84.7011	3.00782
$794$	0	0
$795$	− 52.2337i	− 1.85254i
$796$	0	0
$797$	− 14.6487i	− 0.518883i	−0.965759	−	0.259442i	$-0.916461\pi$
$797$	− 14.6487i	− 0.518883i	0.965759	−	0.259442i	$-0.0835385\pi$
$798$	0	0
$799$	3.22405	0.114059
$800$	0	0
$801$	129.679	4.58199
$802$	0	0
$803$	− 20.1386i	− 0.710676i
$804$	0	0
$805$	6.63325i	0.233791i
$806$	0	0
$807$	−6.33830	−0.223119
$808$	0	0
$809$	−37.9783	−1.33524	−0.667622	−	0.744500i	$-0.732688\pi$
$809$	−37.9783	−1.33524	−0.667622	+	0.744500i	$0.732688\pi$
$810$	0	0
$811$	28.8614i	1.01346i	0.862105	+	0.506731i	$0.169146\pi$
$811$	28.8614i	1.01346i	−0.862105	+	0.506731i	$0.830854\pi$
$812$	0	0
$813$	− 8.01544i	− 0.281114i
$814$	0	0
$815$	27.1229	0.950074
$816$	0	0
$817$	3.62772	0.126918
$818$	0	0
$819$	162.190i	5.66738i
$820$	0	0
$821$	27.0680i	0.944680i	0.881416	+	0.472340i	$0.156591\pi$
$821$	27.0680i	0.944680i	−0.881416	+	0.472340i	$0.843409\pi$
$822$	0	0
$823$	−7.37063	−0.256924	−0.128462	−	0.991714i	$-0.541004\pi$
$823$	−7.37063	−0.256924	−0.128462	+	0.991714i	$0.541004\pi$
$824$	0	0
$825$	−10.9783	−0.382214
$826$	0	0
$827$	40.3505i	1.40313i	0.712608	+	0.701563i	$0.247514\pi$
$827$	40.3505i	1.40313i	−0.712608	+	0.701563i	$0.752486\pi$
$828$	0	0
$829$	3.26172i	0.113284i	0.998395	+	0.0566421i	$0.0180394\pi$
$829$	3.26172i	0.113284i	−0.998395	+	0.0566421i	$0.981961\pi$
$830$	0	0
$831$	55.2405	1.91627
$832$	0	0
$833$	20.0000	0.692959
$834$	0	0
$835$	− 16.0000i	− 0.553703i
$836$	0	0
$837$	62.7586i	2.16925i
$838$	0	0
$839$	9.80240	0.338416	0.169208	−	0.985580i	$-0.445879\pi$
$839$	9.80240	0.338416	0.169208	+	0.985580i	$0.445879\pi$
$840$	0	0
$841$	21.8614	0.753842
$842$	0	0
$843$	21.0217i	0.724028i
$844$	0	0
$845$	− 53.3060i	− 1.83378i
$846$	0	0
$847$	−17.8178	−0.612228
$848$	0	0
$849$	59.4456	2.04017
$850$	0	0
$851$	− 8.00000i	− 0.274236i
$852$	0	0
$853$	− 40.3894i	− 1.38291i	−0.722421	−	0.691453i	$-0.756971\pi$
$853$	− 40.3894i	− 1.38291i	0.722421	−	0.691453i	$-0.243029\pi$
$854$	0	0
$855$	−21.1345	−0.722783
$856$	0	0
$857$	22.2337	0.759488	0.379744	−	0.925092i	$-0.376012\pi$
$857$	22.2337	0.759488	0.379744	+	0.925092i	$0.376012\pi$
$858$	0	0
$859$	− 8.60597i	− 0.293632i	−0.989164	−	0.146816i	$-0.953097\pi$
$859$	− 8.60597i	− 0.293632i	0.989164	−	0.146816i	$-0.0469025\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−2.76439	−0.0941009	−0.0470504	−	0.998893i	$-0.514982\pi$
$863$	−2.76439	−0.0941009	−0.0470504	+	0.998893i	$0.514982\pi$
$864$	0	0
$865$	−38.2337	−1.29998
$866$	0	0
$867$	26.9783i	0.916229i
$868$	0	0
$869$	− 0.699713i	− 0.0237361i
$870$	0	0
$871$	15.3484	0.520061
$872$	0	0
$873$	22.9783	0.777696
$874$	0	0
$875$	− 30.3723i	− 1.02677i
$876$	0	0
$877$	− 1.38219i	− 0.0466734i	−0.999728	−	0.0233367i	$-0.992571\pi$
$877$	− 1.38219i	− 0.0466734i	0.999728	−	0.0233367i	$-0.00742897\pi$
$878$	0	0
$879$	−16.3431	−0.551238
$880$	0	0
$881$	−4.37228	−0.147306	−0.0736530	−	0.997284i	$-0.523466\pi$
$881$	−4.37228	−0.147306	−0.0736530	+	0.997284i	$0.523466\pi$
$882$	0	0
$883$	18.8832i	0.635469i	0.948180	+	0.317734i	$0.102922\pi$
$883$	18.8832i	0.635469i	−0.948180	+	0.317734i	$0.897078\pi$
$884$	0	0
$885$	11.6819i	0.392684i
$886$	0	0
$887$	−45.0333	−1.51207	−0.756035	−	0.654531i	$-0.772866\pi$
$887$	−45.0333	−1.51207	−0.756035	+	0.654531i	$0.772866\pi$
$888$	0	0
$889$	40.7011	1.36507
$890$	0	0
$891$	− 85.3505i	− 2.85935i
$892$	0	0
$893$	− 0.644810i	− 0.0215777i
$894$	0	0
$895$	58.0049	1.93889
$896$	0	0
$897$	15.6060	0.521068
$898$	0	0
$899$	9.25544i	0.308686i
$900$	0	0
$901$	30.6796i	1.02209i
$902$	0	0
$903$	40.5746	1.35024
$904$	0	0
$905$	−20.7446	−0.689573
$906$	0	0
$907$	− 46.8614i	− 1.55601i	−0.628260	−	0.778004i	$-0.716232\pi$
$907$	− 46.8614i	− 1.55601i	0.628260	−	0.778004i	$-0.283768\pi$
$908$	0	0
$909$	116.010i	3.84780i
$910$	0	0
$911$	−47.6126	−1.57747	−0.788737	−	0.614730i	$-0.789265\pi$
$911$	−47.6126	−1.57747	−0.788737	+	0.614730i	$0.789265\pi$
$912$	0	0
$913$	18.9783	0.628088
$914$	0	0
$915$	123.446i	4.08099i
$916$	0	0
$917$	− 14.5012i	− 0.478872i
$918$	0	0
$919$	8.71516	0.287486	0.143743	−	0.989615i	$-0.454086\pi$
$919$	8.71516	0.287486	0.143743	+	0.989615i	$0.454086\pi$
$920$	0	0
$921$	−51.4456	−1.69519
$922$	0	0
$923$	38.7446i	1.27529i
$924$	0	0
$925$	− 13.8564i	− 0.455596i
$926$	0	0
$927$	100.274	3.29342
$928$	0	0
$929$	−27.8832	−0.914817	−0.457408	−	0.889257i	$-0.651222\pi$
$929$	−27.8832	−0.914817	−0.457408	+	0.889257i	$0.651222\pi$
$930$	0	0
$931$	− 4.00000i	− 0.131095i
$932$	0	0
$933$	− 78.2921i	− 2.56317i
$934$	0	0
$935$	29.9422	0.979215
$936$	0	0
$937$	−21.9783	−0.717998	−0.358999	−	0.933338i	$-0.616882\pi$
$937$	−21.9783	−0.717998	−0.358999	+	0.933338i	$0.616882\pi$
$938$	0	0
$939$	13.0951i	0.427342i
$940$	0	0
$941$	10.8896i	0.354992i	0.984122	+	0.177496i	$0.0567997\pi$
$941$	10.8896i	0.354992i	−0.984122	+	0.177496i	$0.943200\pi$
$942$	0	0
$943$	0	0
$944$	0	0
$945$	−151.679	−4.93413
$946$	0	0
$947$	43.2119i	1.40420i	0.712079	+	0.702100i	$0.247754\pi$
$947$	43.2119i	1.40420i	−0.712079	+	0.702100i	$0.752246\pi$
$948$	0	0
$949$	49.5847i	1.60959i
$950$	0	0
$951$	−72.7634	−2.35951
$952$	0	0
$953$	26.2337	0.849793	0.424896	−	0.905242i	$-0.360311\pi$
$953$	26.2337	0.849793	0.424896	+	0.905242i	$0.360311\pi$
$954$	0	0
$955$	17.8614i	0.577982i
$956$	0	0
$957$	− 21.3745i	− 0.690940i
$958$	0	0
$959$	−16.5831	−0.535497
$960$	0	0
$961$	−19.0000	−0.612903
$962$	0	0
$963$	− 111.957i	− 3.60775i
$964$	0	0
$965$	8.21782i	0.264541i
$966$	0	0
$967$	55.1307	1.77288	0.886441	−	0.462841i	$-0.153170\pi$
$967$	55.1307	1.77288	0.886441	+	0.462841i	$0.153170\pi$
$968$	0	0
$969$	16.8614	0.541666
$970$	0	0
$971$	4.00000i	0.128366i	0.997938	+	0.0641831i	$0.0204442\pi$
$971$	4.00000i	0.128366i	−0.997938	+	0.0641831i	$0.979556\pi$
$972$	0	0
$973$	− 12.0318i	− 0.385721i
$974$	0	0
$975$	27.0303	0.865663
$976$	0	0
$977$	27.4891	0.879455	0.439728	−	0.898131i	$-0.355075\pi$
$977$	27.4891	0.879455	0.439728	+	0.898131i	$0.355075\pi$
$978$	0	0
$979$	36.7446i	1.17436i
$980$	0	0
$981$	− 33.1662i	− 1.05892i
$982$	0	0
$983$	52.6612	1.67963	0.839816	−	0.542871i	$-0.182663\pi$
$983$	52.6612	1.67963	0.839816	+	0.542871i	$0.182663\pi$
$984$	0	0
$985$	16.0000	0.509802
$986$	0	0
$987$	− 7.21194i	− 0.229559i
$988$	0	0
$989$	− 2.87419i	− 0.0913941i
$990$	0	0
$991$	−7.92287	−0.251678	−0.125839	−	0.992051i	$-0.540162\pi$
$991$	−7.92287	−0.251678	−0.125839	+	0.992051i	$0.540162\pi$
$992$	0	0
$993$	−36.6277	−1.16235
$994$	0	0
$995$	0.372281i	0.0118021i
$996$	0	0
$997$	− 32.8164i	− 1.03931i	−0.854378	−	0.519653i	$-0.826061\pi$
$997$	− 32.8164i	− 1.03931i	0.854378	−	0.519653i	$-0.173939\pi$
$998$	0	0
$999$	182.932	5.78772

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1216.2.c.h.609.8	yes	8
4.3	odd	2	inner	1216.2.c.h.609.2	yes	8
8.3	odd	2	inner	1216.2.c.h.609.7	yes	8
8.5	even	2	inner	1216.2.c.h.609.1	✓	8
16.3	odd	4		4864.2.a.bl.1.4		4
16.5	even	4		4864.2.a.bl.1.3		4
16.11	odd	4		4864.2.a.bi.1.1		4
16.13	even	4		4864.2.a.bi.1.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1216.2.c.h.609.1	✓	8	8.5	even	2	inner
1216.2.c.h.609.2	yes	8	4.3	odd	2	inner
1216.2.c.h.609.7	yes	8	8.3	odd	2	inner
1216.2.c.h.609.8	yes	8	1.1	even	1	trivial
4864.2.a.bi.1.1		4	16.11	odd	4
4864.2.a.bi.1.2		4	16.13	even	4
4864.2.a.bl.1.3		4	16.5	even	4
4864.2.a.bl.1.4		4	16.3	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)

\(f(q)\)	\(=\)	\(q+3.37228i q^{3} +2.52434i q^{5} -3.31662 q^{7} -8.37228 q^{9} +O(q^{10})\)	\(q+3.37228i q^{3} +2.52434i q^{5} -3.31662 q^{7} -8.37228 q^{9} -2.37228i q^{11} +5.84096i q^{13} -8.51278 q^{15} +5.00000 q^{17} -1.00000i q^{19} -11.1846i q^{21} -0.792287 q^{23} -1.37228 q^{25} -18.1168i q^{27} -2.67181i q^{29} -3.46410 q^{31} +8.00000 q^{33} -8.37228i q^{35} +10.0974i q^{37} -19.6974 q^{39} +3.62772i q^{43} -21.1345i q^{45} +0.644810 q^{47} +4.00000 q^{49} +16.8614i q^{51} +6.13592i q^{53} +5.98844 q^{55} +3.37228 q^{57} -1.37228i q^{59} -14.5012i q^{61} +27.7677 q^{63} -14.7446 q^{65} -2.62772i q^{67} -2.67181i q^{69} +6.63325 q^{71} +8.48913 q^{73} -4.62772i q^{75} +7.86797i q^{77} +0.294954 q^{79} +35.9783 q^{81} +8.00000i q^{83} +12.6217i q^{85} +9.01011 q^{87} -15.4891 q^{89} -19.3723i q^{91} -11.6819i q^{93} +2.52434 q^{95} -2.74456 q^{97} +19.8614i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 44 q^{9}+O(q^{10}) \) 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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