LMFDB - Embedded newform 1216.2.c.f.609.4

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:	$4$
Coefficient field:	$\Q(i, \sqrt{11})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 5x^{2} + 9 $ x^4 - 5*x^2 + 9
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			609.4
Root			$1.65831 - 0.500000i$ of defining polynomial
Character	$\chi$	$=$	1216.609
Dual form			1216.2.c.f.609.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.00000i q^{3} +3.31662i q^{5} +3.31662 q^{7} -1.00000 q^{9} +O(q^{10})$	$q+2.00000i q^{3} +3.31662i q^{5} +3.31662 q^{7} -1.00000 q^{9} +5.00000i q^{11} -6.63325 q^{15} +5.00000 q^{17} -1.00000i q^{19} +6.63325i q^{21} +6.63325 q^{23} -6.00000 q^{25} +4.00000i q^{27} -6.63325i q^{29} -10.0000 q^{33} +11.0000i q^{35} -6.63325i q^{37} -6.00000 q^{41} -1.00000i q^{43} -3.31662i q^{45} +9.94987 q^{47} +4.00000 q^{49} +10.0000i q^{51} -13.2665i q^{53} -16.5831 q^{55} +2.00000 q^{57} -6.00000i q^{59} -9.94987i q^{61} -3.31662 q^{63} +8.00000i q^{67} +13.2665i q^{69} -6.63325 q^{71} -9.00000 q^{73} -12.0000i q^{75} +16.5831i q^{77} -13.2665 q^{79} -11.0000 q^{81} -4.00000i q^{83} +16.5831i q^{85} +13.2665 q^{87} -4.00000 q^{89} +3.31662 q^{95} -12.0000 q^{97} -5.00000i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{9}+O(q^{10}) $ 4 * q - 4 * q^9	$ 4 q - 4 q^{9} + 20 q^{17} - 24 q^{25} - 40 q^{33} - 24 q^{41} + 16 q^{49} + 8 q^{57} - 36 q^{73} - 44 q^{81} - 16 q^{89} - 48 q^{97}+O(q^{100}) $ 4 * q - 4 * q^9 + 20 * q^17 - 24 * q^25 - 40 * q^33 - 24 * q^41 + 16 * q^49 + 8 * q^57 - 36 * q^73 - 44 * q^81 - 16 * q^89 - 48 * 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
$3$	2.00000i	1.15470i	0.816497	+	0.577350i	$0.195913\pi$
$3$	2.00000i	1.15470i	−0.816497	+	0.577350i	$0.804087\pi$
$4$	0	0
$5$	3.31662i	1.48324i	0.670820	+	0.741620i	$0.265942\pi$
$5$	3.31662i	1.48324i	−0.670820	+	0.741620i	$0.734058\pi$
$6$	0	0
$7$	3.31662	1.25357	0.626783	−	0.779194i	$-0.284371\pi$
$7$	3.31662	1.25357	0.626783	+	0.779194i	$0.284371\pi$
$8$	0	0
$9$	−1.00000	−0.333333
$10$	0	0
$11$	5.00000i	1.50756i	0.657129	+	0.753778i	$0.271771\pi$
$11$	5.00000i	1.50756i	−0.657129	+	0.753778i	$0.728229\pi$
$12$	0	0
$13$	0	0	1.00000			$0$
$13$	0	0	−1.00000			$\pi$
$14$	0	0
$15$	−6.63325	−1.71270
$16$	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
$21$	6.63325i	1.44749i
$22$	0	0
$23$	6.63325	1.38313	0.691564	−	0.722315i	$-0.256922\pi$
$23$	6.63325	1.38313	0.691564	+	0.722315i	$0.256922\pi$
$24$	0	0
$25$	−6.00000	−1.20000
$26$	0	0
$27$	4.00000i	0.769800i
$28$	0	0
$29$	− 6.63325i	− 1.23176i	−0.787839	−	0.615882i	$-0.788800\pi$
$29$	− 6.63325i	− 1.23176i	0.787839	−	0.615882i	$-0.211200\pi$
$30$	0	0
$31$	0	0		−	1.00000i	$-0.5\pi$
$31$	0	0			1.00000i	$0.5\pi$
$32$	0	0
$33$	−10.0000	−1.74078
$34$	0	0
$35$	11.0000i	1.85934i
$36$	0	0
$37$	− 6.63325i	− 1.09050i	−0.838274	−	0.545250i	$-0.816435\pi$
$37$	− 6.63325i	− 1.09050i	0.838274	−	0.545250i	$-0.183565\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−6.00000	−0.937043	−0.468521	−	0.883452i	$-0.655213\pi$
$41$	−6.00000	−0.937043	−0.468521	+	0.883452i	$0.655213\pi$
$42$	0	0
$43$	− 1.00000i	− 0.152499i	−0.997089	−	0.0762493i	$-0.975706\pi$
$43$	− 1.00000i	− 0.152499i	0.997089	−	0.0762493i	$-0.0242945\pi$
$44$	0	0
$45$	− 3.31662i	− 0.494413i
$46$	0	0
$47$	9.94987	1.45134	0.725669	−	0.688044i	$-0.241530\pi$
$47$	9.94987	1.45134	0.725669	+	0.688044i	$0.241530\pi$
$48$	0	0
$49$	4.00000	0.571429
$50$	0	0
$51$	10.0000i	1.40028i
$52$	0	0
$53$	− 13.2665i	− 1.82229i	−0.412082	−	0.911147i	$-0.635198\pi$
$53$	− 13.2665i	− 1.82229i	0.412082	−	0.911147i	$-0.364802\pi$
$54$	0	0
$55$	−16.5831	−2.23607
$56$	0	0
$57$	2.00000	0.264906
$58$	0	0
$59$	− 6.00000i	− 0.781133i	−0.920575	−	0.390567i	$-0.872279\pi$
$59$	− 6.00000i	− 0.781133i	0.920575	−	0.390567i	$-0.127721\pi$
$60$	0	0
$61$	− 9.94987i	− 1.27395i	−0.770884	−	0.636975i	$-0.780185\pi$
$61$	− 9.94987i	− 1.27395i	0.770884	−	0.636975i	$-0.219815\pi$
$62$	0	0
$63$	−3.31662	−0.417855
$64$	0	0
$65$	0	0
$66$	0	0
$67$	8.00000i	0.977356i	0.872464	+	0.488678i	$0.162521\pi$
$67$	8.00000i	0.977356i	−0.872464	+	0.488678i	$0.837479\pi$
$68$	0	0
$69$	13.2665i	1.59710i
$70$	0	0
$71$	−6.63325	−0.787222	−0.393611	−	0.919277i	$-0.628774\pi$
$71$	−6.63325	−0.787222	−0.393611	+	0.919277i	$0.628774\pi$
$72$	0	0
$73$	−9.00000	−1.05337	−0.526685	−	0.850060i	$-0.676565\pi$
$73$	−9.00000	−1.05337	−0.526685	+	0.850060i	$0.676565\pi$
$74$	0	0
$75$	− 12.0000i	− 1.38564i
$76$	0	0
$77$	16.5831i	1.88982i
$78$	0	0
$79$	−13.2665	−1.49260	−0.746299	−	0.665611i	$-0.768171\pi$
$79$	−13.2665	−1.49260	−0.746299	+	0.665611i	$0.768171\pi$
$80$	0	0
$81$	−11.0000	−1.22222
$82$	0	0
$83$	− 4.00000i	− 0.439057i	−0.975606	−	0.219529i	$-0.929548\pi$
$83$	− 4.00000i	− 0.439057i	0.975606	−	0.219529i	$-0.0704519\pi$
$84$	0	0
$85$	16.5831i	1.79869i
$86$	0	0
$87$	13.2665	1.42232
$88$	0	0
$89$	−4.00000	−0.423999	−0.212000	−	0.977270i	$-0.567998\pi$
$89$	−4.00000	−0.423999	−0.212000	+	0.977270i	$0.567998\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	0	0
$94$	0	0
$95$	3.31662	0.340279
$96$	0	0
$97$	−12.0000	−1.21842	−0.609208	−	0.793011i	$-0.708512\pi$
$97$	−12.0000	−1.21842	−0.609208	+	0.793011i	$0.708512\pi$
$98$	0	0
$99$	− 5.00000i	− 0.502519i
$100$	0	0
$101$	0	0	1.00000			$0$
$101$	0	0	−1.00000			$\pi$
$102$	0	0
$103$	−6.63325	−0.653594	−0.326797	−	0.945095i	$-0.605969\pi$
$103$	−6.63325	−0.653594	−0.326797	+	0.945095i	$0.605969\pi$
$104$	0	0
$105$	−22.0000	−2.14698
$106$	0	0
$107$	− 6.00000i	− 0.580042i	−0.957020	−	0.290021i	$-0.906338\pi$
$107$	− 6.00000i	− 0.580042i	0.957020	−	0.290021i	$-0.0936623\pi$
$108$	0	0
$109$	− 13.2665i	− 1.27070i	−0.772224	−	0.635350i	$-0.780856\pi$
$109$	− 13.2665i	− 1.27070i	0.772224	−	0.635350i	$-0.219144\pi$
$110$	0	0
$111$	13.2665	1.25920
$112$	0	0
$113$	18.0000	1.69330	0.846649	−	0.532152i	$-0.178617\pi$
$113$	18.0000	1.69330	0.846649	+	0.532152i	$0.178617\pi$
$114$	0	0
$115$	22.0000i	2.05151i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	16.5831	1.52017
$120$	0	0
$121$	−14.0000	−1.27273
$122$	0	0
$123$	− 12.0000i	− 1.08200i
$124$	0	0
$125$	− 3.31662i	− 0.296648i
$126$	0	0
$127$	6.63325	0.588606	0.294303	−	0.955712i	$-0.404913\pi$
$127$	6.63325	0.588606	0.294303	+	0.955712i	$0.404913\pi$
$128$	0	0
$129$	2.00000	0.176090
$130$	0	0
$131$	− 15.0000i	− 1.31056i	−0.755388	−	0.655278i	$-0.772551\pi$
$131$	− 15.0000i	− 1.31056i	0.755388	−	0.655278i	$-0.227449\pi$
$132$	0	0
$133$	− 3.31662i	− 0.287588i
$134$	0	0
$135$	−13.2665	−1.14180
$136$	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
$139$	11.0000i	0.933008i	0.884519	+	0.466504i	$0.154487\pi$
$139$	11.0000i	0.933008i	−0.884519	+	0.466504i	$0.845513\pi$
$140$	0	0
$141$	19.8997i	1.67586i
$142$	0	0
$143$	0	0
$144$	0	0
$145$	22.0000	1.82700
$146$	0	0
$147$	8.00000i	0.659829i
$148$	0	0
$149$	3.31662i	0.271708i	0.990729	+	0.135854i	$0.0433779\pi$
$149$	3.31662i	0.271708i	−0.990729	+	0.135854i	$0.956622\pi$
$150$	0	0
$151$	6.63325	0.539806	0.269903	−	0.962887i	$-0.413008\pi$
$151$	6.63325	0.539806	0.269903	+	0.962887i	$0.413008\pi$
$152$	0	0
$153$	−5.00000	−0.404226
$154$	0	0
$155$	0	0
$156$	0	0
$157$	0	0	1.00000			$0$
$157$	0	0	−1.00000			$\pi$
$158$	0	0
$159$	26.5330	2.10420
$160$	0	0
$161$	22.0000	1.73384
$162$	0	0
$163$	− 20.0000i	− 1.56652i	−0.621694	−	0.783260i	$-0.713555\pi$
$163$	− 20.0000i	− 1.56652i	0.621694	−	0.783260i	$-0.286445\pi$
$164$	0	0
$165$	− 33.1662i	− 2.58199i
$166$	0	0
$167$	−6.63325	−0.513296	−0.256648	−	0.966505i	$-0.582618\pi$
$167$	−6.63325	−0.513296	−0.256648	+	0.966505i	$0.582618\pi$
$168$	0	0
$169$	13.0000	1.00000
$170$	0	0
$171$	1.00000i	0.0764719i
$172$	0	0
$173$	19.8997i	1.51295i	0.654023	+	0.756475i	$0.273080\pi$
$173$	19.8997i	1.51295i	−0.654023	+	0.756475i	$0.726920\pi$
$174$	0	0
$175$	−19.8997	−1.50428
$176$	0	0
$177$	12.0000	0.901975
$178$	0	0
$179$	18.0000i	1.34538i	0.739923	+	0.672692i	$0.234862\pi$
$179$	18.0000i	1.34538i	−0.739923	+	0.672692i	$0.765138\pi$
$180$	0	0
$181$	19.8997i	1.47914i	0.673081	+	0.739568i	$0.264970\pi$
$181$	19.8997i	1.47914i	−0.673081	+	0.739568i	$0.735030\pi$
$182$	0	0
$183$	19.8997	1.47103
$184$	0	0
$185$	22.0000	1.61747
$186$	0	0
$187$	25.0000i	1.82818i
$188$	0	0
$189$	13.2665i	0.964996i
$190$	0	0
$191$	−16.5831	−1.19991	−0.599956	−	0.800033i	$-0.704815\pi$
$191$	−16.5831	−1.19991	−0.599956	+	0.800033i	$0.704815\pi$
$192$	0	0
$193$	0	0		−	1.00000i	$-0.5\pi$
$193$	0	0			1.00000i	$0.5\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 26.5330i	− 1.89040i	−0.326495	−	0.945199i	$-0.605868\pi$
$197$	− 26.5330i	− 1.89040i	0.326495	−	0.945199i	$-0.394132\pi$
$198$	0	0
$199$	−16.5831	−1.17555	−0.587773	−	0.809026i	$-0.699995\pi$
$199$	−16.5831	−1.17555	−0.587773	+	0.809026i	$0.699995\pi$
$200$	0	0
$201$	−16.0000	−1.12855
$202$	0	0
$203$	− 22.0000i	− 1.54410i
$204$	0	0
$205$	− 19.8997i	− 1.38986i
$206$	0	0
$207$	−6.63325	−0.461043
$208$	0	0
$209$	5.00000	0.345857
$210$	0	0
$211$	− 14.0000i	− 0.963800i	−0.876226	−	0.481900i	$-0.839947\pi$
$211$	− 14.0000i	− 0.963800i	0.876226	−	0.481900i	$-0.160053\pi$
$212$	0	0
$213$	− 13.2665i	− 0.909006i
$214$	0	0
$215$	3.31662	0.226192
$216$	0	0
$217$	0	0
$218$	0	0
$219$	− 18.0000i	− 1.21633i
$220$	0	0
$221$	0	0
$222$	0	0
$223$	19.8997	1.33259	0.666293	−	0.745690i	$-0.267880\pi$
$223$	19.8997	1.33259	0.666293	+	0.745690i	$0.267880\pi$
$224$	0	0
$225$	6.00000	0.400000
$226$	0	0
$227$	− 8.00000i	− 0.530979i	−0.964114	−	0.265489i	$-0.914466\pi$
$227$	− 8.00000i	− 0.530979i	0.964114	−	0.265489i	$-0.0855335\pi$
$228$	0	0
$229$	− 23.2164i	− 1.53418i	−0.641539	−	0.767091i	$-0.721704\pi$
$229$	− 23.2164i	− 1.53418i	0.641539	−	0.767091i	$-0.278296\pi$
$230$	0	0
$231$	−33.1662	−2.18218
$232$	0	0
$233$	3.00000	0.196537	0.0982683	−	0.995160i	$-0.468670\pi$
$233$	3.00000	0.196537	0.0982683	+	0.995160i	$0.468670\pi$
$234$	0	0
$235$	33.0000i	2.15268i
$236$	0	0
$237$	− 26.5330i	− 1.72350i
$238$	0	0
$239$	3.31662	0.214535	0.107267	−	0.994230i	$-0.465790\pi$
$239$	3.31662	0.214535	0.107267	+	0.994230i	$0.465790\pi$
$240$	0	0
$241$	26.0000	1.67481	0.837404	−	0.546585i	$-0.184072\pi$
$241$	26.0000	1.67481	0.837404	+	0.546585i	$0.184072\pi$
$242$	0	0
$243$	− 10.0000i	− 0.641500i
$244$	0	0
$245$	13.2665i	0.847566i
$246$	0	0
$247$	0	0
$248$	0	0
$249$	8.00000	0.506979
$250$	0	0
$251$	5.00000i	0.315597i	0.987471	+	0.157799i	$0.0504397\pi$
$251$	5.00000i	0.315597i	−0.987471	+	0.157799i	$0.949560\pi$
$252$	0	0
$253$	33.1662i	2.08514i
$254$	0	0
$255$	−33.1662	−2.07695
$256$	0	0
$257$	−8.00000	−0.499026	−0.249513	−	0.968371i	$-0.580271\pi$
$257$	−8.00000	−0.499026	−0.249513	+	0.968371i	$0.580271\pi$
$258$	0	0
$259$	− 22.0000i	− 1.36701i
$260$	0	0
$261$	6.63325i	0.410588i
$262$	0	0
$263$	−3.31662	−0.204512	−0.102256	−	0.994758i	$-0.532606\pi$
$263$	−3.31662	−0.204512	−0.102256	+	0.994758i	$0.532606\pi$
$264$	0	0
$265$	44.0000	2.70290
$266$	0	0
$267$	− 8.00000i	− 0.489592i
$268$	0	0
$269$	13.2665i	0.808873i	0.914566	+	0.404436i	$0.132532\pi$
$269$	13.2665i	0.808873i	−0.914566	+	0.404436i	$0.867468\pi$
$270$	0	0
$271$	19.8997	1.20882	0.604412	−	0.796672i	$-0.293408\pi$
$271$	19.8997	1.20882	0.604412	+	0.796672i	$0.293408\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	− 30.0000i	− 1.80907i
$276$	0	0
$277$	− 3.31662i	− 0.199277i	−0.995024	−	0.0996383i	$-0.968231\pi$
$277$	− 3.31662i	− 0.199277i	0.995024	−	0.0996383i	$-0.0317686\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−26.0000	−1.55103	−0.775515	−	0.631329i	$-0.782510\pi$
$281$	−26.0000	−1.55103	−0.775515	+	0.631329i	$0.782510\pi$
$282$	0	0
$283$	11.0000i	0.653882i	0.945045	+	0.326941i	$0.106018\pi$
$283$	11.0000i	0.653882i	−0.945045	+	0.326941i	$0.893982\pi$
$284$	0	0
$285$	6.63325i	0.392920i
$286$	0	0
$287$	−19.8997	−1.17465
$288$	0	0
$289$	8.00000	0.470588
$290$	0	0
$291$	− 24.0000i	− 1.40690i
$292$	0	0
$293$	26.5330i	1.55007i	0.631916	+	0.775037i	$0.282269\pi$
$293$	26.5330i	1.55007i	−0.631916	+	0.775037i	$0.717731\pi$
$294$	0	0
$295$	19.8997	1.15861
$296$	0	0
$297$	−20.0000	−1.16052
$298$	0	0
$299$	0	0
$300$	0	0
$301$	− 3.31662i	− 0.191167i
$302$	0	0
$303$	0	0
$304$	0	0
$305$	33.0000	1.88957
$306$	0	0
$307$	12.0000i	0.684876i	0.939540	+	0.342438i	$0.111253\pi$
$307$	12.0000i	0.684876i	−0.939540	+	0.342438i	$0.888747\pi$
$308$	0	0
$309$	− 13.2665i	− 0.754705i
$310$	0	0
$311$	−16.5831	−0.940343	−0.470171	−	0.882575i	$-0.655808\pi$
$311$	−16.5831	−0.940343	−0.470171	+	0.882575i	$0.655808\pi$
$312$	0	0
$313$	−34.0000	−1.92179	−0.960897	−	0.276907i	$-0.910691\pi$
$313$	−34.0000	−1.92179	−0.960897	+	0.276907i	$0.910691\pi$
$314$	0	0
$315$	− 11.0000i	− 0.619780i
$316$	0	0
$317$	− 6.63325i	− 0.372560i	−0.982497	−	0.186280i	$-0.940357\pi$
$317$	− 6.63325i	− 0.372560i	0.982497	−	0.186280i	$-0.0596432\pi$
$318$	0	0
$319$	33.1662	1.85695
$320$	0	0
$321$	12.0000	0.669775
$322$	0	0
$323$	− 5.00000i	− 0.278207i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	26.5330	1.46728
$328$	0	0
$329$	33.0000	1.81935
$330$	0	0
$331$	4.00000i	0.219860i	0.993939	+	0.109930i	$0.0350627\pi$
$331$	4.00000i	0.219860i	−0.993939	+	0.109930i	$0.964937\pi$
$332$	0	0
$333$	6.63325i	0.363500i
$334$	0	0
$335$	−26.5330	−1.44965
$336$	0	0
$337$	8.00000	0.435788	0.217894	−	0.975972i	$-0.430081\pi$
$337$	8.00000	0.435788	0.217894	+	0.975972i	$0.430081\pi$
$338$	0	0
$339$	36.0000i	1.95525i
$340$	0	0
$341$	0	0
$342$	0	0
$343$	−9.94987	−0.537243
$344$	0	0
$345$	−44.0000	−2.36888
$346$	0	0
$347$	27.0000i	1.44944i	0.689046	+	0.724718i	$0.258030\pi$
$347$	27.0000i	1.44944i	−0.689046	+	0.724718i	$0.741970\pi$
$348$	0	0
$349$	− 9.94987i	− 0.532605i	−0.963890	−	0.266302i	$-0.914198\pi$
$349$	− 9.94987i	− 0.532605i	0.963890	−	0.266302i	$-0.0858019\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	14.0000	0.745145	0.372572	−	0.928003i	$-0.378476\pi$
$353$	14.0000	0.745145	0.372572	+	0.928003i	$0.378476\pi$
$354$	0	0
$355$	− 22.0000i	− 1.16764i
$356$	0	0
$357$	33.1662i	1.75534i
$358$	0	0
$359$	23.2164	1.22531	0.612657	−	0.790349i	$-0.290101\pi$
$359$	23.2164	1.22531	0.612657	+	0.790349i	$0.290101\pi$
$360$	0	0
$361$	−1.00000	−0.0526316
$362$	0	0
$363$	− 28.0000i	− 1.46962i
$364$	0	0
$365$	− 29.8496i	− 1.56240i
$366$	0	0
$367$	6.63325	0.346253	0.173126	−	0.984900i	$-0.444613\pi$
$367$	6.63325	0.346253	0.173126	+	0.984900i	$0.444613\pi$
$368$	0	0
$369$	6.00000	0.312348
$370$	0	0
$371$	− 44.0000i	− 2.28437i
$372$	0	0
$373$	0	0	1.00000			$0$
$373$	0	0	−1.00000			$\pi$
$374$	0	0
$375$	6.63325	0.342540
$376$	0	0
$377$	0	0
$378$	0	0
$379$	6.00000i	0.308199i	0.988055	+	0.154100i	$0.0492477\pi$
$379$	6.00000i	0.308199i	−0.988055	+	0.154100i	$0.950752\pi$
$380$	0	0
$381$	13.2665i	0.679663i
$382$	0	0
$383$	13.2665	0.677886	0.338943	−	0.940807i	$-0.389931\pi$
$383$	13.2665	0.677886	0.338943	+	0.940807i	$0.389931\pi$
$384$	0	0
$385$	−55.0000	−2.80306
$386$	0	0
$387$	1.00000i	0.0508329i
$388$	0	0
$389$	16.5831i	0.840798i	0.907339	+	0.420399i	$0.138110\pi$
$389$	16.5831i	0.840798i	−0.907339	+	0.420399i	$0.861890\pi$
$390$	0	0
$391$	33.1662	1.67729
$392$	0	0
$393$	30.0000	1.51330
$394$	0	0
$395$	− 44.0000i	− 2.21388i
$396$	0	0
$397$	9.94987i	0.499370i	0.968327	+	0.249685i	$0.0803271\pi$
$397$	9.94987i	0.499370i	−0.968327	+	0.249685i	$0.919673\pi$
$398$	0	0
$399$	6.63325	0.332078
$400$	0	0
$401$	16.0000	0.799002	0.399501	−	0.916733i	$-0.369183\pi$
$401$	16.0000	0.799002	0.399501	+	0.916733i	$0.369183\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	− 36.4829i	− 1.81285i
$406$	0	0
$407$	33.1662	1.64399
$408$	0	0
$409$	0	0		−	1.00000i	$-0.5\pi$
$409$	0	0			1.00000i	$0.5\pi$
$410$	0	0
$411$	10.0000i	0.493264i
$412$	0	0
$413$	− 19.8997i	− 0.979203i
$414$	0	0
$415$	13.2665	0.651227
$416$	0	0
$417$	−22.0000	−1.07734
$418$	0	0
$419$	− 4.00000i	− 0.195413i	−0.995215	−	0.0977064i	$-0.968849\pi$
$419$	− 4.00000i	− 0.195413i	0.995215	−	0.0977064i	$-0.0311506\pi$
$420$	0	0
$421$	13.2665i	0.646570i	0.946302	+	0.323285i	$0.104787\pi$
$421$	13.2665i	0.646570i	−0.946302	+	0.323285i	$0.895213\pi$
$422$	0	0
$423$	−9.94987	−0.483779
$424$	0	0
$425$	−30.0000	−1.45521
$426$	0	0
$427$	− 33.0000i	− 1.59698i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−26.5330	−1.27805	−0.639025	−	0.769186i	$-0.720662\pi$
$431$	−26.5330	−1.27805	−0.639025	+	0.769186i	$0.720662\pi$
$432$	0	0
$433$	−38.0000	−1.82616	−0.913082	−	0.407777i	$-0.866304\pi$
$433$	−38.0000	−1.82616	−0.913082	+	0.407777i	$0.866304\pi$
$434$	0	0
$435$	44.0000i	2.10964i
$436$	0	0
$437$	− 6.63325i	− 0.317311i
$438$	0	0
$439$	−19.8997	−0.949763	−0.474882	−	0.880050i	$-0.657509\pi$
$439$	−19.8997	−0.949763	−0.474882	+	0.880050i	$0.657509\pi$
$440$	0	0
$441$	−4.00000	−0.190476
$442$	0	0
$443$	− 13.0000i	− 0.617649i	−0.951119	−	0.308824i	$-0.900064\pi$
$443$	− 13.0000i	− 0.617649i	0.951119	−	0.308824i	$-0.0999355\pi$
$444$	0	0
$445$	− 13.2665i	− 0.628892i
$446$	0	0
$447$	−6.63325	−0.313742
$448$	0	0
$449$	20.0000	0.943858	0.471929	−	0.881636i	$-0.343558\pi$
$449$	20.0000	0.943858	0.471929	+	0.881636i	$0.343558\pi$
$450$	0	0
$451$	− 30.0000i	− 1.41264i
$452$	0	0
$453$	13.2665i	0.623315i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−29.0000	−1.35656	−0.678281	−	0.734802i	$-0.737275\pi$
$457$	−29.0000	−1.35656	−0.678281	+	0.734802i	$0.737275\pi$
$458$	0	0
$459$	20.0000i	0.933520i
$460$	0	0
$461$	16.5831i	0.772353i	0.922425	+	0.386177i	$0.126204\pi$
$461$	16.5831i	0.772353i	−0.922425	+	0.386177i	$0.873796\pi$
$462$	0	0
$463$	−23.2164	−1.07896	−0.539478	−	0.842000i	$-0.681378\pi$
$463$	−23.2164	−1.07896	−0.539478	+	0.842000i	$0.681378\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	3.00000i	0.138823i	0.997588	+	0.0694117i	$0.0221122\pi$
$467$	3.00000i	0.138823i	−0.997588	+	0.0694117i	$0.977888\pi$
$468$	0	0
$469$	26.5330i	1.22518i
$470$	0	0
$471$	0	0
$472$	0	0
$473$	5.00000	0.229900
$474$	0	0
$475$	6.00000i	0.275299i
$476$	0	0
$477$	13.2665i	0.607431i
$478$	0	0
$479$	6.63325	0.303081	0.151540	−	0.988451i	$-0.451577\pi$
$479$	6.63325	0.303081	0.151540	+	0.988451i	$0.451577\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	44.0000i	2.00207i
$484$	0	0
$485$	− 39.7995i	− 1.80720i
$486$	0	0
$487$	−33.1662	−1.50291	−0.751453	−	0.659787i	$-0.770647\pi$
$487$	−33.1662	−1.50291	−0.751453	+	0.659787i	$0.770647\pi$
$488$	0	0
$489$	40.0000	1.80886
$490$	0	0
$491$	20.0000i	0.902587i	0.892375	+	0.451294i	$0.149037\pi$
$491$	20.0000i	0.902587i	−0.892375	+	0.451294i	$0.850963\pi$
$492$	0	0
$493$	− 33.1662i	− 1.49373i
$494$	0	0
$495$	16.5831	0.745356
$496$	0	0
$497$	−22.0000	−0.986835
$498$	0	0
$499$	− 3.00000i	− 0.134298i	−0.997743	−	0.0671492i	$-0.978610\pi$
$499$	− 3.00000i	− 0.134298i	0.997743	−	0.0671492i	$-0.0213904\pi$
$500$	0	0
$501$	− 13.2665i	− 0.592703i
$502$	0	0
$503$	−19.8997	−0.887286	−0.443643	−	0.896204i	$-0.646314\pi$
$503$	−19.8997	−0.887286	−0.443643	+	0.896204i	$0.646314\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	26.0000i	1.15470i
$508$	0	0
$509$	26.5330i	1.17605i	0.808841	+	0.588027i	$0.200095\pi$
$509$	26.5330i	1.17605i	−0.808841	+	0.588027i	$0.799905\pi$
$510$	0	0
$511$	−29.8496	−1.32047
$512$	0	0
$513$	4.00000	0.176604
$514$	0	0
$515$	− 22.0000i	− 0.969436i
$516$	0	0
$517$	49.7494i	2.18797i
$518$	0	0
$519$	−39.7995	−1.74700
$520$	0	0
$521$	16.0000	0.700973	0.350486	−	0.936568i	$-0.386016\pi$
$521$	16.0000	0.700973	0.350486	+	0.936568i	$0.386016\pi$
$522$	0	0
$523$	6.00000i	0.262362i	0.991358	+	0.131181i	$0.0418769\pi$
$523$	6.00000i	0.262362i	−0.991358	+	0.131181i	$0.958123\pi$
$524$	0	0
$525$	− 39.7995i	− 1.73699i
$526$	0	0
$527$	0	0
$528$	0	0
$529$	21.0000	0.913043
$530$	0	0
$531$	6.00000i	0.260378i
$532$	0	0
$533$	0	0
$534$	0	0
$535$	19.8997	0.860341
$536$	0	0
$537$	−36.0000	−1.55351
$538$	0	0
$539$	20.0000i	0.861461i
$540$	0	0
$541$	− 9.94987i	− 0.427779i	−0.976858	−	0.213889i	$-0.931387\pi$
$541$	− 9.94987i	− 0.427779i	0.976858	−	0.213889i	$-0.0686132\pi$
$542$	0	0
$543$	−39.7995	−1.70796
$544$	0	0
$545$	44.0000	1.88475
$546$	0	0
$547$	20.0000i	0.855138i	0.903983	+	0.427569i	$0.140630\pi$
$547$	20.0000i	0.855138i	−0.903983	+	0.427569i	$0.859370\pi$
$548$	0	0
$549$	9.94987i	0.424650i
$550$	0	0
$551$	−6.63325	−0.282586
$552$	0	0
$553$	−44.0000	−1.87107
$554$	0	0
$555$	44.0000i	1.86770i
$556$	0	0
$557$	− 3.31662i	− 0.140530i	−0.997528	−	0.0702650i	$-0.977616\pi$
$557$	− 3.31662i	− 0.140530i	0.997528	−	0.0702650i	$-0.0223845\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	−50.0000	−2.11100
$562$	0	0
$563$	14.0000i	0.590030i	0.955493	+	0.295015i	$0.0953246\pi$
$563$	14.0000i	0.590030i	−0.955493	+	0.295015i	$0.904675\pi$
$564$	0	0
$565$	59.6992i	2.51157i
$566$	0	0
$567$	−36.4829	−1.53214
$568$	0	0
$569$	20.0000	0.838444	0.419222	−	0.907884i	$-0.362303\pi$
$569$	20.0000	0.838444	0.419222	+	0.907884i	$0.362303\pi$
$570$	0	0
$571$	44.0000i	1.84134i	0.390339	+	0.920671i	$0.372358\pi$
$571$	44.0000i	1.84134i	−0.390339	+	0.920671i	$0.627642\pi$
$572$	0	0
$573$	− 33.1662i	− 1.38554i
$574$	0	0
$575$	−39.7995	−1.65975
$576$	0	0
$577$	−43.0000	−1.79011	−0.895057	−	0.445952i	$-0.852865\pi$
$577$	−43.0000	−1.79011	−0.895057	+	0.445952i	$0.852865\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	− 13.2665i	− 0.550387i
$582$	0	0
$583$	66.3325	2.74721
$584$	0	0
$585$	0	0
$586$	0	0
$587$	5.00000i	0.206372i	0.994662	+	0.103186i	$0.0329037\pi$
$587$	5.00000i	0.206372i	−0.994662	+	0.103186i	$0.967096\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	53.0660	2.18284
$592$	0	0
$593$	18.0000	0.739171	0.369586	−	0.929197i	$-0.379500\pi$
$593$	18.0000	0.739171	0.369586	+	0.929197i	$0.379500\pi$
$594$	0	0
$595$	55.0000i	2.25478i
$596$	0	0
$597$	− 33.1662i	− 1.35740i
$598$	0	0
$599$	−13.2665	−0.542054	−0.271027	−	0.962572i	$-0.587363\pi$
$599$	−13.2665	−0.542054	−0.271027	+	0.962572i	$0.587363\pi$
$600$	0	0
$601$	42.0000	1.71322	0.856608	−	0.515968i	$-0.172568\pi$
$601$	42.0000	1.71322	0.856608	+	0.515968i	$0.172568\pi$
$602$	0	0
$603$	− 8.00000i	− 0.325785i
$604$	0	0
$605$	− 46.4327i	− 1.88776i
$606$	0	0
$607$	26.5330	1.07694	0.538471	−	0.842644i	$-0.319002\pi$
$607$	26.5330	1.07694	0.538471	+	0.842644i	$0.319002\pi$
$608$	0	0
$609$	44.0000	1.78297
$610$	0	0
$611$	0	0
$612$	0	0
$613$	− 3.31662i	− 0.133957i	−0.997754	−	0.0669786i	$-0.978664\pi$
$613$	− 3.31662i	− 0.133957i	0.997754	−	0.0669786i	$-0.0213359\pi$
$614$	0	0
$615$	39.7995	1.60487
$616$	0	0
$617$	−41.0000	−1.65060	−0.825299	−	0.564696i	$-0.808993\pi$
$617$	−41.0000	−1.65060	−0.825299	+	0.564696i	$0.808993\pi$
$618$	0	0
$619$	4.00000i	0.160774i	0.996764	+	0.0803868i	$0.0256155\pi$
$619$	4.00000i	0.160774i	−0.996764	+	0.0803868i	$0.974384\pi$
$620$	0	0
$621$	26.5330i	1.06473i
$622$	0	0
$623$	−13.2665	−0.531511
$624$	0	0
$625$	−19.0000	−0.760000
$626$	0	0
$627$	10.0000i	0.399362i
$628$	0	0
$629$	− 33.1662i	− 1.32242i
$630$	0	0
$631$	9.94987	0.396098	0.198049	−	0.980192i	$-0.436539\pi$
$631$	9.94987	0.396098	0.198049	+	0.980192i	$0.436539\pi$
$632$	0	0
$633$	28.0000	1.11290
$634$	0	0
$635$	22.0000i	0.873043i
$636$	0	0
$637$	0	0
$638$	0	0
$639$	6.63325	0.262407
$640$	0	0
$641$	−24.0000	−0.947943	−0.473972	−	0.880540i	$-0.657180\pi$
$641$	−24.0000	−0.947943	−0.473972	+	0.880540i	$0.657180\pi$
$642$	0	0
$643$	− 5.00000i	− 0.197181i	−0.995128	−	0.0985904i	$-0.968567\pi$
$643$	− 5.00000i	− 0.197181i	0.995128	−	0.0985904i	$-0.0314334\pi$
$644$	0	0
$645$	6.63325i	0.261184i
$646$	0	0
$647$	−16.5831	−0.651950	−0.325975	−	0.945378i	$-0.605693\pi$
$647$	−16.5831	−0.651950	−0.325975	+	0.945378i	$0.605693\pi$
$648$	0	0
$649$	30.0000	1.17760
$650$	0	0
$651$	0	0
$652$	0	0
$653$	36.4829i	1.42769i	0.700306	+	0.713843i	$0.253047\pi$
$653$	36.4829i	1.42769i	−0.700306	+	0.713843i	$0.746953\pi$
$654$	0	0
$655$	49.7494	1.94387
$656$	0	0
$657$	9.00000	0.351123
$658$	0	0
$659$	− 26.0000i	− 1.01282i	−0.862294	−	0.506408i	$-0.830973\pi$
$659$	− 26.0000i	− 1.01282i	0.862294	−	0.506408i	$-0.169027\pi$
$660$	0	0
$661$	− 39.7995i	− 1.54802i	−0.633173	−	0.774011i	$-0.718248\pi$
$661$	− 39.7995i	− 1.54802i	0.633173	−	0.774011i	$-0.281752\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	11.0000	0.426562
$666$	0	0
$667$	− 44.0000i	− 1.70369i
$668$	0	0
$669$	39.7995i	1.53874i
$670$	0	0
$671$	49.7494	1.92055
$672$	0	0
$673$	−34.0000	−1.31060	−0.655302	−	0.755367i	$-0.727459\pi$
$673$	−34.0000	−1.31060	−0.655302	+	0.755367i	$0.727459\pi$
$674$	0	0
$675$	− 24.0000i	− 0.923760i
$676$	0	0
$677$	19.8997i	0.764809i	0.923995	+	0.382405i	$0.124904\pi$
$677$	19.8997i	0.764809i	−0.923995	+	0.382405i	$0.875096\pi$
$678$	0	0
$679$	−39.7995	−1.52736
$680$	0	0
$681$	16.0000	0.613121
$682$	0	0
$683$	− 24.0000i	− 0.918334i	−0.888350	−	0.459167i	$-0.848148\pi$
$683$	− 24.0000i	− 0.918334i	0.888350	−	0.459167i	$-0.151852\pi$
$684$	0	0
$685$	16.5831i	0.633609i
$686$	0	0
$687$	46.4327	1.77152
$688$	0	0
$689$	0	0
$690$	0	0
$691$	31.0000i	1.17930i	0.807661	+	0.589648i	$0.200733\pi$
$691$	31.0000i	1.17930i	−0.807661	+	0.589648i	$0.799267\pi$
$692$	0	0
$693$	− 16.5831i	− 0.629941i
$694$	0	0
$695$	−36.4829	−1.38387
$696$	0	0
$697$	−30.0000	−1.13633
$698$	0	0
$699$	6.00000i	0.226941i
$700$	0	0
$701$	0	0	1.00000			$0$
$701$	0	0	−1.00000			$\pi$
$702$	0	0
$703$	−6.63325	−0.250178
$704$	0	0
$705$	−66.0000	−2.48570
$706$	0	0
$707$	0	0
$708$	0	0
$709$	13.2665i	0.498234i	0.968473	+	0.249117i	$0.0801403\pi$
$709$	13.2665i	0.498234i	−0.968473	+	0.249117i	$0.919860\pi$
$710$	0	0
$711$	13.2665	0.497533
$712$	0	0
$713$	0	0
$714$	0	0
$715$	0	0
$716$	0	0
$717$	6.63325i	0.247723i
$718$	0	0
$719$	−3.31662	−0.123689	−0.0618446	−	0.998086i	$-0.519698\pi$
$719$	−3.31662	−0.123689	−0.0618446	+	0.998086i	$0.519698\pi$
$720$	0	0
$721$	−22.0000	−0.819323
$722$	0	0
$723$	52.0000i	1.93390i
$724$	0	0
$725$	39.7995i	1.47812i
$726$	0	0
$727$	−16.5831	−0.615034	−0.307517	−	0.951543i	$-0.599498\pi$
$727$	−16.5831	−0.615034	−0.307517	+	0.951543i	$0.599498\pi$
$728$	0	0
$729$	−13.0000	−0.481481
$730$	0	0
$731$	− 5.00000i	− 0.184932i
$732$	0	0
$733$	0	0	1.00000			$0$
$733$	0	0	−1.00000			$\pi$
$734$	0	0
$735$	−26.5330	−0.978684
$736$	0	0
$737$	−40.0000	−1.47342
$738$	0	0
$739$	45.0000i	1.65535i	0.561206	+	0.827676i	$0.310337\pi$
$739$	45.0000i	1.65535i	−0.561206	+	0.827676i	$0.689663\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	53.0660	1.94680	0.973401	−	0.229107i	$-0.0735805\pi$
$743$	53.0660	1.94680	0.973401	+	0.229107i	$0.0735805\pi$
$744$	0	0
$745$	−11.0000	−0.403009
$746$	0	0
$747$	4.00000i	0.146352i
$748$	0	0
$749$	− 19.8997i	− 0.727121i
$750$	0	0
$751$	−39.7995	−1.45230	−0.726152	−	0.687534i	$-0.758693\pi$
$751$	−39.7995	−1.45230	−0.726152	+	0.687534i	$0.758693\pi$
$752$	0	0
$753$	−10.0000	−0.364420
$754$	0	0
$755$	22.0000i	0.800662i
$756$	0	0
$757$	9.94987i	0.361634i	0.983517	+	0.180817i	$0.0578742\pi$
$757$	9.94987i	0.361634i	−0.983517	+	0.180817i	$0.942126\pi$
$758$	0	0
$759$	−66.3325	−2.40772
$760$	0	0
$761$	1.00000	0.0362500	0.0181250	−	0.999836i	$-0.494230\pi$
$761$	1.00000	0.0362500	0.0181250	+	0.999836i	$0.494230\pi$
$762$	0	0
$763$	− 44.0000i	− 1.59291i
$764$	0	0
$765$	− 16.5831i	− 0.599564i
$766$	0	0
$767$	0	0
$768$	0	0
$769$	33.0000	1.19001	0.595005	−	0.803722i	$-0.297150\pi$
$769$	33.0000	1.19001	0.595005	+	0.803722i	$0.297150\pi$
$770$	0	0
$771$	− 16.0000i	− 0.576226i
$772$	0	0
$773$	6.63325i	0.238581i	0.992859	+	0.119291i	$0.0380620\pi$
$773$	6.63325i	0.238581i	−0.992859	+	0.119291i	$0.961938\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	44.0000	1.57849
$778$	0	0
$779$	6.00000i	0.214972i
$780$	0	0
$781$	− 33.1662i	− 1.18678i
$782$	0	0
$783$	26.5330	0.948212
$784$	0	0
$785$	0	0
$786$	0	0
$787$	28.0000i	0.998092i	0.866575	+	0.499046i	$0.166316\pi$
$787$	28.0000i	0.998092i	−0.866575	+	0.499046i	$0.833684\pi$
$788$	0	0
$789$	− 6.63325i	− 0.236150i
$790$	0	0
$791$	59.6992	2.12266
$792$	0	0
$793$	0	0
$794$	0	0
$795$	88.0000i	3.12104i
$796$	0	0
$797$	26.5330i	0.939847i	0.882707	+	0.469924i	$0.155719\pi$
$797$	26.5330i	0.939847i	−0.882707	+	0.469924i	$0.844281\pi$
$798$	0	0
$799$	49.7494	1.76001
$800$	0	0
$801$	4.00000	0.141333
$802$	0	0
$803$	− 45.0000i	− 1.58802i
$804$	0	0
$805$	72.9657i	2.57170i
$806$	0	0
$807$	−26.5330	−0.934006
$808$	0	0
$809$	−39.0000	−1.37117	−0.685583	−	0.727994i	$-0.740453\pi$
$809$	−39.0000	−1.37117	−0.685583	+	0.727994i	$0.740453\pi$
$810$	0	0
$811$	28.0000i	0.983213i	0.870817	+	0.491606i	$0.163590\pi$
$811$	28.0000i	0.983213i	−0.870817	+	0.491606i	$0.836410\pi$
$812$	0	0
$813$	39.7995i	1.39583i
$814$	0	0
$815$	66.3325	2.32353
$816$	0	0
$817$	−1.00000	−0.0349856
$818$	0	0
$819$	0	0
$820$	0	0
$821$	− 9.94987i	− 0.347253i	−0.984812	−	0.173627i	$-0.944451\pi$
$821$	− 9.94987i	− 0.347253i	0.984812	−	0.173627i	$-0.0555486\pi$
$822$	0	0
$823$	−29.8496	−1.04049	−0.520246	−	0.854016i	$-0.674160\pi$
$823$	−29.8496	−1.04049	−0.520246	+	0.854016i	$0.674160\pi$
$824$	0	0
$825$	60.0000	2.08893
$826$	0	0
$827$	− 32.0000i	− 1.11275i	−0.830932	−	0.556375i	$-0.812192\pi$
$827$	− 32.0000i	− 1.11275i	0.830932	−	0.556375i	$-0.187808\pi$
$828$	0	0
$829$	0	0	1.00000			$0$
$829$	0	0	−1.00000			$\pi$
$830$	0	0
$831$	6.63325	0.230105
$832$	0	0
$833$	20.0000	0.692959
$834$	0	0
$835$	− 22.0000i	− 0.761341i
$836$	0	0
$837$	0	0
$838$	0	0
$839$	46.4327	1.60304	0.801518	−	0.597970i	$-0.204026\pi$
$839$	46.4327	1.60304	0.801518	+	0.597970i	$0.204026\pi$
$840$	0	0
$841$	−15.0000	−0.517241
$842$	0	0
$843$	− 52.0000i	− 1.79098i
$844$	0	0
$845$	43.1161i	1.48324i
$846$	0	0
$847$	−46.4327	−1.59545
$848$	0	0
$849$	−22.0000	−0.755038
$850$	0	0
$851$	− 44.0000i	− 1.50830i
$852$	0	0
$853$	− 53.0660i	− 1.81695i	−0.417945	−	0.908473i	$-0.637249\pi$
$853$	− 53.0660i	− 1.81695i	0.417945	−	0.908473i	$-0.362751\pi$
$854$	0	0
$855$	−3.31662	−0.113426
$856$	0	0
$857$	−10.0000	−0.341593	−0.170797	−	0.985306i	$-0.554634\pi$
$857$	−10.0000	−0.341593	−0.170797	+	0.985306i	$0.554634\pi$
$858$	0	0
$859$	− 41.0000i	− 1.39890i	−0.714681	−	0.699451i	$-0.753428\pi$
$859$	− 41.0000i	− 1.39890i	0.714681	−	0.699451i	$-0.246572\pi$
$860$	0	0
$861$	− 39.7995i	− 1.35636i
$862$	0	0
$863$	−33.1662	−1.12899	−0.564496	−	0.825436i	$-0.690929\pi$
$863$	−33.1662	−1.12899	−0.564496	+	0.825436i	$0.690929\pi$
$864$	0	0
$865$	−66.0000	−2.24407
$866$	0	0
$867$	16.0000i	0.543388i
$868$	0	0
$869$	− 66.3325i	− 2.25018i
$870$	0	0
$871$	0	0
$872$	0	0
$873$	12.0000	0.406138
$874$	0	0
$875$	− 11.0000i	− 0.371868i
$876$	0	0
$877$	− 6.63325i	− 0.223989i	−0.993709	−	0.111994i	$-0.964276\pi$
$877$	− 6.63325i	− 0.223989i	0.993709	−	0.111994i	$-0.0357239\pi$
$878$	0	0
$879$	−53.0660	−1.78987
$880$	0	0
$881$	51.0000	1.71823	0.859117	−	0.511780i	$-0.171014\pi$
$881$	51.0000	1.71823	0.859117	+	0.511780i	$0.171014\pi$
$882$	0	0
$883$	− 31.0000i	− 1.04323i	−0.853180	−	0.521617i	$-0.825329\pi$
$883$	− 31.0000i	− 1.04323i	0.853180	−	0.521617i	$-0.174671\pi$
$884$	0	0
$885$	39.7995i	1.33785i
$886$	0	0
$887$	19.8997	0.668168	0.334084	−	0.942543i	$-0.391573\pi$
$887$	19.8997	0.668168	0.334084	+	0.942543i	$0.391573\pi$
$888$	0	0
$889$	22.0000	0.737856
$890$	0	0
$891$	− 55.0000i	− 1.84257i
$892$	0	0
$893$	− 9.94987i	− 0.332960i
$894$	0	0
$895$	−59.6992	−1.99553
$896$	0	0
$897$	0	0
$898$	0	0
$899$	0	0
$900$	0	0
$901$	− 66.3325i	− 2.20986i
$902$	0	0
$903$	6.63325	0.220741
$904$	0	0
$905$	−66.0000	−2.19391
$906$	0	0
$907$	32.0000i	1.06254i	0.847202	+	0.531271i	$0.178286\pi$
$907$	32.0000i	1.06254i	−0.847202	+	0.531271i	$0.821714\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−19.8997	−0.659308	−0.329654	−	0.944102i	$-0.606932\pi$
$911$	−19.8997	−0.659308	−0.329654	+	0.944102i	$0.606932\pi$
$912$	0	0
$913$	20.0000	0.661903
$914$	0	0
$915$	66.0000i	2.18189i
$916$	0	0
$917$	− 49.7494i	− 1.64287i
$918$	0	0
$919$	−33.1662	−1.09405	−0.547027	−	0.837115i	$-0.684240\pi$
$919$	−33.1662	−1.09405	−0.547027	+	0.837115i	$0.684240\pi$
$920$	0	0
$921$	−24.0000	−0.790827
$922$	0	0
$923$	0	0
$924$	0	0
$925$	39.7995i	1.30860i
$926$	0	0
$927$	6.63325	0.217865
$928$	0	0
$929$	34.0000	1.11550	0.557752	−	0.830008i	$-0.311664\pi$
$929$	34.0000	1.11550	0.557752	+	0.830008i	$0.311664\pi$
$930$	0	0
$931$	− 4.00000i	− 0.131095i
$932$	0	0
$933$	− 33.1662i	− 1.08581i
$934$	0	0
$935$	−82.9156	−2.71163
$936$	0	0
$937$	1.00000	0.0326686	0.0163343	−	0.999867i	$-0.494800\pi$
$937$	1.00000	0.0326686	0.0163343	+	0.999867i	$0.494800\pi$
$938$	0	0
$939$	− 68.0000i	− 2.21910i
$940$	0	0
$941$	46.4327i	1.51366i	0.653609	+	0.756832i	$0.273254\pi$
$941$	46.4327i	1.51366i	−0.653609	+	0.756832i	$0.726746\pi$
$942$	0	0
$943$	−39.7995	−1.29605
$944$	0	0
$945$	−44.0000	−1.43132
$946$	0	0
$947$	36.0000i	1.16984i	0.811090	+	0.584921i	$0.198875\pi$
$947$	36.0000i	1.16984i	−0.811090	+	0.584921i	$0.801125\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	13.2665	0.430196
$952$	0	0
$953$	−24.0000	−0.777436	−0.388718	−	0.921357i	$-0.627082\pi$
$953$	−24.0000	−0.777436	−0.388718	+	0.921357i	$0.627082\pi$
$954$	0	0
$955$	− 55.0000i	− 1.77976i
$956$	0	0
$957$	66.3325i	2.14423i
$958$	0	0
$959$	16.5831	0.535497
$960$	0	0
$961$	−31.0000	−1.00000
$962$	0	0
$963$	6.00000i	0.193347i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	33.1662	1.06655	0.533277	−	0.845940i	$-0.320960\pi$
$967$	33.1662	1.06655	0.533277	+	0.845940i	$0.320960\pi$
$968$	0	0
$969$	10.0000	0.321246
$970$	0	0
$971$	− 32.0000i	− 1.02693i	−0.858111	−	0.513464i	$-0.828362\pi$
$971$	− 32.0000i	− 1.02693i	0.858111	−	0.513464i	$-0.171638\pi$
$972$	0	0
$973$	36.4829i	1.16959i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	16.0000	0.511885	0.255943	−	0.966692i	$-0.417614\pi$
$977$	16.0000	0.511885	0.255943	+	0.966692i	$0.417614\pi$
$978$	0	0
$979$	− 20.0000i	− 0.639203i
$980$	0	0
$981$	13.2665i	0.423567i
$982$	0	0
$983$	−13.2665	−0.423136	−0.211568	−	0.977363i	$-0.567857\pi$
$983$	−13.2665	−0.423136	−0.211568	+	0.977363i	$0.567857\pi$
$984$	0	0
$985$	88.0000	2.80391
$986$	0	0
$987$	66.0000i	2.10080i
$988$	0	0
$989$	− 6.63325i	− 0.210925i
$990$	0	0
$991$	−33.1662	−1.05356	−0.526780	−	0.850001i	$-0.676601\pi$
$991$	−33.1662	−1.05356	−0.526780	+	0.850001i	$0.676601\pi$
$992$	0	0
$993$	−8.00000	−0.253872
$994$	0	0
$995$	− 55.0000i	− 1.74362i
$996$	0	0
$997$	− 43.1161i	− 1.36550i	−0.730651	−	0.682751i	$-0.760784\pi$
$997$	− 43.1161i	− 1.36550i	0.730651	−	0.682751i	$-0.239216\pi$
$998$	0	0
$999$	26.5330	0.839467

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1216.2.c.f.609.4	yes	4
4.3	odd	2	inner	1216.2.c.f.609.2	yes	4
8.3	odd	2	inner	1216.2.c.f.609.3	yes	4
8.5	even	2	inner	1216.2.c.f.609.1	✓	4
16.3	odd	4		4864.2.a.y.1.2		2
16.5	even	4		4864.2.a.y.1.1		2
16.11	odd	4		4864.2.a.r.1.1		2
16.13	even	4		4864.2.a.r.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1216.2.c.f.609.1	✓	4	8.5	even	2	inner
1216.2.c.f.609.2	yes	4	4.3	odd	2	inner
1216.2.c.f.609.3	yes	4	8.3	odd	2	inner
1216.2.c.f.609.4	yes	4	1.1	even	1	trivial
4864.2.a.r.1.1		2	16.11	odd	4
4864.2.a.r.1.2		2	16.13	even	4
4864.2.a.y.1.1		2	16.5	even	4
4864.2.a.y.1.2		2	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)

\(f(q)\)	\(=\)	\(q+2.00000i q^{3} +3.31662i q^{5} +3.31662 q^{7} -1.00000 q^{9} +O(q^{10})\)	\(q+2.00000i q^{3} +3.31662i q^{5} +3.31662 q^{7} -1.00000 q^{9} +5.00000i q^{11} -6.63325 q^{15} +5.00000 q^{17} -1.00000i q^{19} +6.63325i q^{21} +6.63325 q^{23} -6.00000 q^{25} +4.00000i q^{27} -6.63325i q^{29} -10.0000 q^{33} +11.0000i q^{35} -6.63325i q^{37} -6.00000 q^{41} -1.00000i q^{43} -3.31662i q^{45} +9.94987 q^{47} +4.00000 q^{49} +10.0000i q^{51} -13.2665i q^{53} -16.5831 q^{55} +2.00000 q^{57} -6.00000i q^{59} -9.94987i q^{61} -3.31662 q^{63} +8.00000i q^{67} +13.2665i q^{69} -6.63325 q^{71} -9.00000 q^{73} -12.0000i q^{75} +16.5831i q^{77} -13.2665 q^{79} -11.0000 q^{81} -4.00000i q^{83} +16.5831i q^{85} +13.2665 q^{87} -4.00000 q^{89} +3.31662 q^{95} -12.0000 q^{97} -5.00000i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{9}+O(q^{10}) \) 4 * q - 4 * q^9	\( 4 q - 4 q^{9} + 20 q^{17} - 24 q^{25} - 40 q^{33} - 24 q^{41} + 16 q^{49} + 8 q^{57} - 36 q^{73} - 44 q^{81} - 16 q^{89} - 48 q^{97}+O(q^{100}) \) 4 * q - 4 * q^9 + 20 * q^17 - 24 * q^25 - 40 * q^33 - 24 * q^41 + 16 * q^49 + 8 * q^57 - 36 * q^73 - 44 * q^81 - 16 * q^89 - 48 * q^97

Introduction

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