LMFDB - Embedded newform 1216.2.c.c.609.1

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

Embedding invariants

Embedding label			609.1
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	1216.609
Dual form			1216.2.c.c.609.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.00000i q^{3} -4.00000i q^{5} +1.00000 q^{7} -6.00000 q^{9} +5.00000i q^{13} -12.0000 q^{15} -5.00000 q^{17} -1.00000i q^{19} -3.00000i q^{21} -3.00000 q^{23} -11.0000 q^{25} +9.00000i q^{27} -7.00000i q^{29} +10.0000 q^{31} -4.00000i q^{35} -2.00000i q^{37} +15.0000 q^{39} -6.00000 q^{41} +4.00000i q^{43} +24.0000i q^{45} +8.00000 q^{47} -6.00000 q^{49} +15.0000i q^{51} -9.00000i q^{53} -3.00000 q^{57} -1.00000i q^{59} +2.00000i q^{61} -6.00000 q^{63} +20.0000 q^{65} -7.00000i q^{67} +9.00000i q^{69} -12.0000 q^{71} +11.0000 q^{73} +33.0000i q^{75} +16.0000 q^{79} +9.00000 q^{81} -14.0000i q^{83} +20.0000i q^{85} -21.0000 q^{87} -4.00000 q^{89} +5.00000i q^{91} -30.0000i q^{93} -4.00000 q^{95} -12.0000 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{7} - 12 q^{9} - 24 q^{15} - 10 q^{17} - 6 q^{23} - 22 q^{25} + 20 q^{31} + 30 q^{39} - 12 q^{41} + 16 q^{47} - 12 q^{49} - 6 q^{57} - 12 q^{63} + 40 q^{65} - 24 q^{71} + 22 q^{73} + 32 q^{79}+ \cdots - 24 q^{97}+O(q^{100}) $ 2 * q + 2 * q^7 - 12 * q^9 - 24 * q^15 - 10 * q^17 - 6 * q^23 - 22 * q^25 + 20 * q^31 + 30 * q^39 - 12 * q^41 + 16 * q^47 - 12 * q^49 - 6 * q^57 - 12 * q^63 + 40 * q^65 - 24 * q^71 + 22 * q^73 + 32 * q^79 + 18 * q^81 - 42 * q^87 - 8 * q^89 - 8 * q^95 - 24 * q^97

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
$3$	− 3.00000i	− 1.73205i	−0.500000	−	0.866025i	$-0.666667\pi$
$3$	− 3.00000i	− 1.73205i	0.500000	−	0.866025i	$-0.333333\pi$
$4$	0	0
$5$	− 4.00000i	− 1.78885i	−0.447214	−	0.894427i	$-0.647584\pi$
$5$	− 4.00000i	− 1.78885i	0.447214	−	0.894427i	$-0.352416\pi$
$6$	0	0
$7$	1.00000	0.377964	0.188982	−	0.981981i	$-0.439481\pi$
$7$	1.00000	0.377964	0.188982	+	0.981981i	$0.439481\pi$
$8$	0	0
$9$	−6.00000	−2.00000
$10$	0	0
$11$	0	0	1.00000			$0$
$11$	0	0	−1.00000			$\pi$
$12$	0	0
$13$	5.00000i	1.38675i	0.720577	+	0.693375i	$0.243877\pi$
$13$	5.00000i	1.38675i	−0.720577	+	0.693375i	$0.756123\pi$
$14$	0	0
$15$	−12.0000	−3.09839
$16$	0	0
$17$	−5.00000	−1.21268	−0.606339	−	0.795206i	$-0.707363\pi$
$17$	−5.00000	−1.21268	−0.606339	+	0.795206i	$0.707363\pi$
$18$	0	0
$19$	− 1.00000i	− 0.229416i
$19$	− 1.00000i	− 0.229416i
$20$	0	0
$21$	− 3.00000i	− 0.654654i
$22$	0	0
$23$	−3.00000	−0.625543	−0.312772	−	0.949828i	$-0.601257\pi$
$23$	−3.00000	−0.625543	−0.312772	+	0.949828i	$0.601257\pi$
$24$	0	0
$25$	−11.0000	−2.20000
$26$	0	0
$27$	9.00000i	1.73205i
$28$	0	0
$29$	− 7.00000i	− 1.29987i	−0.759991	−	0.649934i	$-0.774797\pi$
$29$	− 7.00000i	− 1.29987i	0.759991	−	0.649934i	$-0.225203\pi$
$30$	0	0
$31$	10.0000	1.79605	0.898027	−	0.439941i	$-0.145001\pi$
$31$	10.0000	1.79605	0.898027	+	0.439941i	$0.145001\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	− 4.00000i	− 0.676123i
$36$	0	0
$37$	− 2.00000i	− 0.328798i	−0.986394	−	0.164399i	$-0.947432\pi$
$37$	− 2.00000i	− 0.328798i	0.986394	−	0.164399i	$-0.0525685\pi$
$38$	0	0
$39$	15.0000	2.40192
$40$	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
$43$	4.00000i	0.609994i	0.952353	+	0.304997i	$0.0986555\pi$
$43$	4.00000i	0.609994i	−0.952353	+	0.304997i	$0.901344\pi$
$44$	0	0
$45$	24.0000i	3.57771i
$46$	0	0
$47$	8.00000	1.16692	0.583460	−	0.812142i	$-0.301699\pi$
$47$	8.00000	1.16692	0.583460	+	0.812142i	$0.301699\pi$
$48$	0	0
$49$	−6.00000	−0.857143
$50$	0	0
$51$	15.0000i	2.10042i
$52$	0	0
$53$	− 9.00000i	− 1.23625i	−0.786082	−	0.618123i	$-0.787894\pi$
$53$	− 9.00000i	− 1.23625i	0.786082	−	0.618123i	$-0.212106\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−3.00000	−0.397360
$58$	0	0
$59$	− 1.00000i	− 0.130189i	−0.997879	−	0.0650945i	$-0.979265\pi$
$59$	− 1.00000i	− 0.130189i	0.997879	−	0.0650945i	$-0.0207349\pi$
$60$	0	0
$61$	2.00000i	0.256074i	0.991769	+	0.128037i	$0.0408676\pi$
$61$	2.00000i	0.256074i	−0.991769	+	0.128037i	$0.959132\pi$
$62$	0	0
$63$	−6.00000	−0.755929
$64$	0	0
$65$	20.0000	2.48069
$66$	0	0
$67$	− 7.00000i	− 0.855186i	−0.903971	−	0.427593i	$-0.859362\pi$
$67$	− 7.00000i	− 0.855186i	0.903971	−	0.427593i	$-0.140638\pi$
$68$	0	0
$69$	9.00000i	1.08347i
$70$	0	0
$71$	−12.0000	−1.42414	−0.712069	−	0.702109i	$-0.752242\pi$
$71$	−12.0000	−1.42414	−0.712069	+	0.702109i	$0.752242\pi$
$72$	0	0
$73$	11.0000	1.28745	0.643726	−	0.765256i	$-0.277388\pi$
$73$	11.0000	1.28745	0.643726	+	0.765256i	$0.277388\pi$
$74$	0	0
$75$	33.0000i	3.81051i
$76$	0	0
$77$	0	0
$78$	0	0
$79$	16.0000	1.80014	0.900070	−	0.435745i	$-0.143515\pi$
$79$	16.0000	1.80014	0.900070	+	0.435745i	$0.143515\pi$
$80$	0	0
$81$	9.00000	1.00000
$82$	0	0
$83$	− 14.0000i	− 1.53670i	−0.640030	−	0.768350i	$-0.721078\pi$
$83$	− 14.0000i	− 1.53670i	0.640030	−	0.768350i	$-0.278922\pi$
$84$	0	0
$85$	20.0000i	2.16930i
$86$	0	0
$87$	−21.0000	−2.25144
$88$	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
$91$	5.00000i	0.524142i
$92$	0	0
$93$	− 30.0000i	− 3.11086i
$94$	0	0
$95$	−4.00000	−0.410391
$96$	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
$99$	0	0
$100$	0	0
$101$	0	0	1.00000			$0$
$101$	0	0	−1.00000			$\pi$
$102$	0	0
$103$	−2.00000	−0.197066	−0.0985329	−	0.995134i	$-0.531415\pi$
$103$	−2.00000	−0.197066	−0.0985329	+	0.995134i	$0.531415\pi$
$104$	0	0
$105$	−12.0000	−1.17108
$106$	0	0
$107$	9.00000i	0.870063i	0.900415	+	0.435031i	$0.143263\pi$
$107$	9.00000i	0.870063i	−0.900415	+	0.435031i	$0.856737\pi$
$108$	0	0
$109$	− 9.00000i	− 0.862044i	−0.902342	−	0.431022i	$-0.858153\pi$
$109$	− 9.00000i	− 0.862044i	0.902342	−	0.431022i	$-0.141847\pi$
$110$	0	0
$111$	−6.00000	−0.569495
$112$	0	0
$113$	8.00000	0.752577	0.376288	−	0.926503i	$-0.377200\pi$
$113$	8.00000	0.752577	0.376288	+	0.926503i	$0.377200\pi$
$114$	0	0
$115$	12.0000i	1.11901i
$116$	0	0
$117$	− 30.0000i	− 2.77350i
$118$	0	0
$119$	−5.00000	−0.458349
$120$	0	0
$121$	11.0000	1.00000
$122$	0	0
$123$	18.0000i	1.62301i
$124$	0	0
$125$	24.0000i	2.14663i
$126$	0	0
$127$	2.00000	0.177471	0.0887357	−	0.996055i	$-0.471717\pi$
$127$	2.00000	0.177471	0.0887357	+	0.996055i	$0.471717\pi$
$128$	0	0
$129$	12.0000	1.05654
$130$	0	0
$131$	0	0	1.00000			$0$
$131$	0	0	−1.00000			$\pi$
$132$	0	0
$133$	− 1.00000i	− 0.0867110i
$134$	0	0
$135$	36.0000	3.09839
$136$	0	0
$137$	−15.0000	−1.28154	−0.640768	−	0.767734i	$-0.721384\pi$
$137$	−15.0000	−1.28154	−0.640768	+	0.767734i	$0.721384\pi$
$138$	0	0
$139$	− 14.0000i	− 1.18746i	−0.804663	−	0.593732i	$-0.797654\pi$
$139$	− 14.0000i	− 1.18746i	0.804663	−	0.593732i	$-0.202346\pi$
$140$	0	0
$141$	− 24.0000i	− 2.02116i
$142$	0	0
$143$	0	0
$144$	0	0
$145$	−28.0000	−2.32527
$146$	0	0
$147$	18.0000i	1.48461i
$148$	0	0
$149$	6.00000i	0.491539i	0.969328	+	0.245770i	$0.0790407\pi$
$149$	6.00000i	0.491539i	−0.969328	+	0.245770i	$0.920959\pi$
$150$	0	0
$151$	−8.00000	−0.651031	−0.325515	−	0.945537i	$-0.605538\pi$
$151$	−8.00000	−0.651031	−0.325515	+	0.945537i	$0.605538\pi$
$152$	0	0
$153$	30.0000	2.42536
$154$	0	0
$155$	− 40.0000i	− 3.21288i
$156$	0	0
$157$	20.0000i	1.59617i	0.602542	+	0.798087i	$0.294154\pi$
$157$	20.0000i	1.59617i	−0.602542	+	0.798087i	$0.705846\pi$
$158$	0	0
$159$	−27.0000	−2.14124
$160$	0	0
$161$	−3.00000	−0.236433
$162$	0	0
$163$	− 10.0000i	− 0.783260i	−0.920123	−	0.391630i	$-0.871911\pi$
$163$	− 10.0000i	− 0.783260i	0.920123	−	0.391630i	$-0.128089\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−2.00000	−0.154765	−0.0773823	−	0.997001i	$-0.524656\pi$
$167$	−2.00000	−0.154765	−0.0773823	+	0.997001i	$0.524656\pi$
$168$	0	0
$169$	−12.0000	−0.923077
$170$	0	0
$171$	6.00000i	0.458831i
$172$	0	0
$173$	− 14.0000i	− 1.06440i	−0.846619	−	0.532200i	$-0.821365\pi$
$173$	− 14.0000i	− 1.06440i	0.846619	−	0.532200i	$-0.178635\pi$
$174$	0	0
$175$	−11.0000	−0.831522
$176$	0	0
$177$	−3.00000	−0.225494
$178$	0	0
$179$	− 12.0000i	− 0.896922i	−0.893802	−	0.448461i	$-0.851972\pi$
$179$	− 12.0000i	− 0.896922i	0.893802	−	0.448461i	$-0.148028\pi$
$180$	0	0
$181$	6.00000i	0.445976i	0.974821	+	0.222988i	$0.0715812\pi$
$181$	6.00000i	0.445976i	−0.974821	+	0.222988i	$0.928419\pi$
$182$	0	0
$183$	6.00000	0.443533
$184$	0	0
$185$	−8.00000	−0.588172
$186$	0	0
$187$	0	0
$188$	0	0
$189$	9.00000i	0.654654i
$190$	0	0
$191$	−5.00000	−0.361787	−0.180894	−	0.983503i	$-0.557899\pi$
$191$	−5.00000	−0.361787	−0.180894	+	0.983503i	$0.557899\pi$
$192$	0	0
$193$	20.0000	1.43963	0.719816	−	0.694165i	$-0.244226\pi$
$193$	20.0000	1.43963	0.719816	+	0.694165i	$0.244226\pi$
$194$	0	0
$195$	− 60.0000i	− 4.29669i
$196$	0	0
$197$	2.00000i	0.142494i	0.997459	+	0.0712470i	$0.0226979\pi$
$197$	2.00000i	0.142494i	−0.997459	+	0.0712470i	$0.977302\pi$
$198$	0	0
$199$	5.00000	0.354441	0.177220	−	0.984171i	$-0.443289\pi$
$199$	5.00000	0.354441	0.177220	+	0.984171i	$0.443289\pi$
$200$	0	0
$201$	−21.0000	−1.48123
$202$	0	0
$203$	− 7.00000i	− 0.491304i
$204$	0	0
$205$	24.0000i	1.67623i
$206$	0	0
$207$	18.0000	1.25109
$208$	0	0
$209$	0	0
$210$	0	0
$211$	1.00000i	0.0688428i	0.999407	+	0.0344214i	$0.0109588\pi$
$211$	1.00000i	0.0688428i	−0.999407	+	0.0344214i	$0.989041\pi$
$212$	0	0
$213$	36.0000i	2.46668i
$214$	0	0
$215$	16.0000	1.09119
$216$	0	0
$217$	10.0000	0.678844
$218$	0	0
$219$	− 33.0000i	− 2.22993i
$220$	0	0
$221$	− 25.0000i	− 1.68168i
$222$	0	0
$223$	−14.0000	−0.937509	−0.468755	−	0.883328i	$-0.655297\pi$
$223$	−14.0000	−0.937509	−0.468755	+	0.883328i	$0.655297\pi$
$224$	0	0
$225$	66.0000	4.40000
$226$	0	0
$227$	17.0000i	1.12833i	0.825662	+	0.564165i	$0.190802\pi$
$227$	17.0000i	1.12833i	−0.825662	+	0.564165i	$0.809198\pi$
$228$	0	0
$229$	− 22.0000i	− 1.45380i	−0.686743	−	0.726900i	$-0.740960\pi$
$229$	− 22.0000i	− 1.45380i	0.686743	−	0.726900i	$-0.259040\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	18.0000	1.17922	0.589610	−	0.807688i	$-0.299282\pi$
$233$	18.0000	1.17922	0.589610	+	0.807688i	$0.299282\pi$
$234$	0	0
$235$	− 32.0000i	− 2.08745i
$236$	0	0
$237$	− 48.0000i	− 3.11794i
$238$	0	0
$239$	11.0000	0.711531	0.355765	−	0.934575i	$-0.384220\pi$
$239$	11.0000	0.711531	0.355765	+	0.934575i	$0.384220\pi$
$240$	0	0
$241$	−14.0000	−0.901819	−0.450910	−	0.892570i	$-0.648900\pi$
$241$	−14.0000	−0.901819	−0.450910	+	0.892570i	$0.648900\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	24.0000i	1.53330i
$246$	0	0
$247$	5.00000	0.318142
$248$	0	0
$249$	−42.0000	−2.66164
$250$	0	0
$251$	− 20.0000i	− 1.26239i	−0.775625	−	0.631194i	$-0.782565\pi$
$251$	− 20.0000i	− 1.26239i	0.775625	−	0.631194i	$-0.217435\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	60.0000	3.75735
$256$	0	0
$257$	12.0000	0.748539	0.374270	−	0.927320i	$-0.377893\pi$
$257$	12.0000	0.748539	0.374270	+	0.927320i	$0.377893\pi$
$258$	0	0
$259$	− 2.00000i	− 0.124274i
$260$	0	0
$261$	42.0000i	2.59973i
$262$	0	0
$263$	24.0000	1.47990	0.739952	−	0.672660i	$-0.234848\pi$
$263$	24.0000	1.47990	0.739952	+	0.672660i	$0.234848\pi$
$264$	0	0
$265$	−36.0000	−2.21146
$266$	0	0
$267$	12.0000i	0.734388i
$268$	0	0
$269$	14.0000i	0.853595i	0.904347	+	0.426798i	$0.140358\pi$
$269$	14.0000i	0.853595i	−0.904347	+	0.426798i	$0.859642\pi$
$270$	0	0
$271$	11.0000	0.668202	0.334101	−	0.942537i	$-0.391567\pi$
$271$	11.0000	0.668202	0.334101	+	0.942537i	$0.391567\pi$
$272$	0	0
$273$	15.0000	0.907841
$274$	0	0
$275$	0	0
$276$	0	0
$277$	− 6.00000i	− 0.360505i	−0.983620	−	0.180253i	$-0.942309\pi$
$277$	− 6.00000i	− 0.360505i	0.983620	−	0.180253i	$-0.0576915\pi$
$278$	0	0
$279$	−60.0000	−3.59211
$280$	0	0
$281$	−16.0000	−0.954480	−0.477240	−	0.878773i	$-0.658363\pi$
$281$	−16.0000	−0.954480	−0.477240	+	0.878773i	$0.658363\pi$
$282$	0	0
$283$	6.00000i	0.356663i	0.983970	+	0.178331i	$0.0570699\pi$
$283$	6.00000i	0.356663i	−0.983970	+	0.178331i	$0.942930\pi$
$284$	0	0
$285$	12.0000i	0.710819i
$286$	0	0
$287$	−6.00000	−0.354169
$288$	0	0
$289$	8.00000	0.470588
$290$	0	0
$291$	36.0000i	2.11036i
$292$	0	0
$293$	− 7.00000i	− 0.408944i	−0.978872	−	0.204472i	$-0.934452\pi$
$293$	− 7.00000i	− 0.408944i	0.978872	−	0.204472i	$-0.0655478\pi$
$294$	0	0
$295$	−4.00000	−0.232889
$296$	0	0
$297$	0	0
$298$	0	0
$299$	− 15.0000i	− 0.867472i
$300$	0	0
$301$	4.00000i	0.230556i
$302$	0	0
$303$	0	0
$304$	0	0
$305$	8.00000	0.458079
$306$	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
$309$	6.00000i	0.341328i
$310$	0	0
$311$	−35.0000	−1.98467	−0.992334	−	0.123585i	$-0.960561\pi$
$311$	−35.0000	−1.98467	−0.992334	+	0.123585i	$0.960561\pi$
$312$	0	0
$313$	−9.00000	−0.508710	−0.254355	−	0.967111i	$-0.581863\pi$
$313$	−9.00000	−0.508710	−0.254355	+	0.967111i	$0.581863\pi$
$314$	0	0
$315$	24.0000i	1.35225i
$316$	0	0
$317$	23.0000i	1.29181i	0.763418	+	0.645904i	$0.223520\pi$
$317$	23.0000i	1.29181i	−0.763418	+	0.645904i	$0.776480\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	27.0000	1.50699
$322$	0	0
$323$	5.00000i	0.278207i
$324$	0	0
$325$	− 55.0000i	− 3.05085i
$326$	0	0
$327$	−27.0000	−1.49310
$328$	0	0
$329$	8.00000	0.441054
$330$	0	0
$331$	− 11.0000i	− 0.604615i	−0.953211	−	0.302307i	$-0.902243\pi$
$331$	− 11.0000i	− 0.604615i	0.953211	−	0.302307i	$-0.0977569\pi$
$332$	0	0
$333$	12.0000i	0.657596i
$334$	0	0
$335$	−28.0000	−1.52980
$336$	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
$339$	− 24.0000i	− 1.30350i
$340$	0	0
$341$	0	0
$342$	0	0
$343$	−13.0000	−0.701934
$344$	0	0
$345$	36.0000	1.93817
$346$	0	0
$347$	2.00000i	0.107366i	0.998558	+	0.0536828i	$0.0170960\pi$
$347$	2.00000i	0.107366i	−0.998558	+	0.0536828i	$0.982904\pi$
$348$	0	0
$349$	− 28.0000i	− 1.49881i	−0.662114	−	0.749403i	$-0.730341\pi$
$349$	− 28.0000i	− 1.49881i	0.662114	−	0.749403i	$-0.269659\pi$
$350$	0	0
$351$	−45.0000	−2.40192
$352$	0	0
$353$	29.0000	1.54351	0.771757	−	0.635917i	$-0.219378\pi$
$353$	29.0000	1.54351	0.771757	+	0.635917i	$0.219378\pi$
$354$	0	0
$355$	48.0000i	2.54758i
$356$	0	0
$357$	15.0000i	0.793884i
$358$	0	0
$359$	27.0000	1.42501	0.712503	−	0.701669i	$-0.247562\pi$
$359$	27.0000	1.42501	0.712503	+	0.701669i	$0.247562\pi$
$360$	0	0
$361$	−1.00000	−0.0526316
$362$	0	0
$363$	− 33.0000i	− 1.73205i
$364$	0	0
$365$	− 44.0000i	− 2.30307i
$366$	0	0
$367$	32.0000	1.67039	0.835193	−	0.549957i	$-0.185356\pi$
$367$	32.0000	1.67039	0.835193	+	0.549957i	$0.185356\pi$
$368$	0	0
$369$	36.0000	1.87409
$370$	0	0
$371$	− 9.00000i	− 0.467257i
$372$	0	0
$373$	− 15.0000i	− 0.776671i	−0.921518	−	0.388335i	$-0.873050\pi$
$373$	− 15.0000i	− 0.776671i	0.921518	−	0.388335i	$-0.126950\pi$
$374$	0	0
$375$	72.0000	3.71806
$376$	0	0
$377$	35.0000	1.80259
$378$	0	0
$379$	11.0000i	0.565032i	0.959263	+	0.282516i	$0.0911690\pi$
$379$	11.0000i	0.565032i	−0.959263	+	0.282516i	$0.908831\pi$
$380$	0	0
$381$	− 6.00000i	− 0.307389i
$382$	0	0
$383$	−6.00000	−0.306586	−0.153293	−	0.988181i	$-0.548988\pi$
$383$	−6.00000	−0.306586	−0.153293	+	0.988181i	$0.548988\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	− 24.0000i	− 1.21999i
$388$	0	0
$389$	0	0	1.00000			$0$
$389$	0	0	−1.00000			$\pi$
$390$	0	0
$391$	15.0000	0.758583
$392$	0	0
$393$	0	0
$394$	0	0
$395$	− 64.0000i	− 3.22019i
$396$	0	0
$397$	− 2.00000i	− 0.100377i	−0.998740	−	0.0501886i	$-0.984018\pi$
$397$	− 2.00000i	− 0.100377i	0.998740	−	0.0501886i	$-0.0159822\pi$
$398$	0	0
$399$	−3.00000	−0.150188
$400$	0	0
$401$	−4.00000	−0.199750	−0.0998752	−	0.995000i	$-0.531844\pi$
$401$	−4.00000	−0.199750	−0.0998752	+	0.995000i	$0.531844\pi$
$402$	0	0
$403$	50.0000i	2.49068i
$404$	0	0
$405$	− 36.0000i	− 1.78885i
$406$	0	0
$407$	0	0
$408$	0	0
$409$	−30.0000	−1.48340	−0.741702	−	0.670729i	$-0.765981\pi$
$409$	−30.0000	−1.48340	−0.741702	+	0.670729i	$0.765981\pi$
$410$	0	0
$411$	45.0000i	2.21969i
$412$	0	0
$413$	− 1.00000i	− 0.0492068i
$414$	0	0
$415$	−56.0000	−2.74893
$416$	0	0
$417$	−42.0000	−2.05675
$418$	0	0
$419$	− 24.0000i	− 1.17248i	−0.810139	−	0.586238i	$-0.800608\pi$
$419$	− 24.0000i	− 1.17248i	0.810139	−	0.586238i	$-0.199392\pi$
$420$	0	0
$421$	− 21.0000i	− 1.02348i	−0.859141	−	0.511739i	$-0.829002\pi$
$421$	− 21.0000i	− 1.02348i	0.859141	−	0.511739i	$-0.170998\pi$
$422$	0	0
$423$	−48.0000	−2.33384
$424$	0	0
$425$	55.0000	2.66789
$426$	0	0
$427$	2.00000i	0.0967868i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	22.0000	1.05970	0.529851	−	0.848091i	$-0.322248\pi$
$431$	22.0000	1.05970	0.529851	+	0.848091i	$0.322248\pi$
$432$	0	0
$433$	2.00000	0.0961139	0.0480569	−	0.998845i	$-0.484697\pi$
$433$	2.00000	0.0961139	0.0480569	+	0.998845i	$0.484697\pi$
$434$	0	0
$435$	84.0000i	4.02749i
$436$	0	0
$437$	3.00000i	0.143509i
$438$	0	0
$439$	−16.0000	−0.763638	−0.381819	−	0.924237i	$-0.624702\pi$
$439$	−16.0000	−0.763638	−0.381819	+	0.924237i	$0.624702\pi$
$440$	0	0
$441$	36.0000	1.71429
$442$	0	0
$443$	32.0000i	1.52037i	0.649709	+	0.760183i	$0.274891\pi$
$443$	32.0000i	1.52037i	−0.649709	+	0.760183i	$0.725109\pi$
$444$	0	0
$445$	16.0000i	0.758473i
$446$	0	0
$447$	18.0000	0.851371
$448$	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
$451$	0	0
$452$	0	0
$453$	24.0000i	1.12762i
$454$	0	0
$455$	20.0000	0.937614
$456$	0	0
$457$	11.0000	0.514558	0.257279	−	0.966337i	$-0.417174\pi$
$457$	11.0000	0.514558	0.257279	+	0.966337i	$0.417174\pi$
$458$	0	0
$459$	− 45.0000i	− 2.10042i
$460$	0	0
$461$	20.0000i	0.931493i	0.884918	+	0.465746i	$0.154214\pi$
$461$	20.0000i	0.931493i	−0.884918	+	0.465746i	$0.845786\pi$
$462$	0	0
$463$	8.00000	0.371792	0.185896	−	0.982569i	$-0.440481\pi$
$463$	8.00000	0.371792	0.185896	+	0.982569i	$0.440481\pi$
$464$	0	0
$465$	−120.000	−5.56487
$466$	0	0
$467$	− 12.0000i	− 0.555294i	−0.960683	−	0.277647i	$-0.910445\pi$
$467$	− 12.0000i	− 0.555294i	0.960683	−	0.277647i	$-0.0895545\pi$
$468$	0	0
$469$	− 7.00000i	− 0.323230i
$470$	0	0
$471$	60.0000	2.76465
$472$	0	0
$473$	0	0
$474$	0	0
$475$	11.0000i	0.504715i
$476$	0	0
$477$	54.0000i	2.47249i
$478$	0	0
$479$	12.0000	0.548294	0.274147	−	0.961688i	$-0.411605\pi$
$479$	12.0000	0.548294	0.274147	+	0.961688i	$0.411605\pi$
$480$	0	0
$481$	10.0000	0.455961
$482$	0	0
$483$	9.00000i	0.409514i
$484$	0	0
$485$	48.0000i	2.17957i
$486$	0	0
$487$	−30.0000	−1.35943	−0.679715	−	0.733476i	$-0.737896\pi$
$487$	−30.0000	−1.35943	−0.679715	+	0.733476i	$0.737896\pi$
$488$	0	0
$489$	−30.0000	−1.35665
$490$	0	0
$491$	0	0	1.00000			$0$
$491$	0	0	−1.00000			$\pi$
$492$	0	0
$493$	35.0000i	1.57632i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−12.0000	−0.538274
$498$	0	0
$499$	− 28.0000i	− 1.25345i	−0.779240	−	0.626726i	$-0.784395\pi$
$499$	− 28.0000i	− 1.25345i	0.779240	−	0.626726i	$-0.215605\pi$
$500$	0	0
$501$	6.00000i	0.268060i
$502$	0	0
$503$	29.0000	1.29305	0.646523	−	0.762894i	$-0.276222\pi$
$503$	29.0000	1.29305	0.646523	+	0.762894i	$0.276222\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	36.0000i	1.59882i
$508$	0	0
$509$	− 22.0000i	− 0.975133i	−0.873086	−	0.487566i	$-0.837885\pi$
$509$	− 22.0000i	− 0.975133i	0.873086	−	0.487566i	$-0.162115\pi$
$510$	0	0
$511$	11.0000	0.486611
$512$	0	0
$513$	9.00000	0.397360
$514$	0	0
$515$	8.00000i	0.352522i
$516$	0	0
$517$	0	0
$518$	0	0
$519$	−42.0000	−1.84360
$520$	0	0
$521$	6.00000	0.262865	0.131432	−	0.991325i	$-0.458042\pi$
$521$	6.00000	0.262865	0.131432	+	0.991325i	$0.458042\pi$
$522$	0	0
$523$	1.00000i	0.0437269i	0.999761	+	0.0218635i	$0.00695991\pi$
$523$	1.00000i	0.0437269i	−0.999761	+	0.0218635i	$0.993040\pi$
$524$	0	0
$525$	33.0000i	1.44024i
$526$	0	0
$527$	−50.0000	−2.17803
$528$	0	0
$529$	−14.0000	−0.608696
$530$	0	0
$531$	6.00000i	0.260378i
$532$	0	0
$533$	− 30.0000i	− 1.29944i
$534$	0	0
$535$	36.0000	1.55642
$536$	0	0
$537$	−36.0000	−1.55351
$538$	0	0
$539$	0	0
$540$	0	0
$541$	22.0000i	0.945854i	0.881102	+	0.472927i	$0.156803\pi$
$541$	22.0000i	0.945854i	−0.881102	+	0.472927i	$0.843197\pi$
$542$	0	0
$543$	18.0000	0.772454
$544$	0	0
$545$	−36.0000	−1.54207
$546$	0	0
$547$	− 20.0000i	− 0.855138i	−0.903983	−	0.427569i	$-0.859370\pi$
$547$	− 20.0000i	− 0.855138i	0.903983	−	0.427569i	$-0.140630\pi$
$548$	0	0
$549$	− 12.0000i	− 0.512148i
$550$	0	0
$551$	−7.00000	−0.298210
$552$	0	0
$553$	16.0000	0.680389
$554$	0	0
$555$	24.0000i	1.01874i
$556$	0	0
$557$	24.0000i	1.01691i	0.861088	+	0.508456i	$0.169784\pi$
$557$	24.0000i	1.01691i	−0.861088	+	0.508456i	$0.830216\pi$
$558$	0	0
$559$	−20.0000	−0.845910
$560$	0	0
$561$	0	0
$562$	0	0
$563$	24.0000i	1.01148i	0.862686	+	0.505740i	$0.168780\pi$
$563$	24.0000i	1.01148i	−0.862686	+	0.505740i	$0.831220\pi$
$564$	0	0
$565$	− 32.0000i	− 1.34625i
$566$	0	0
$567$	9.00000	0.377964
$568$	0	0
$569$	−30.0000	−1.25767	−0.628833	−	0.777541i	$-0.716467\pi$
$569$	−30.0000	−1.25767	−0.628833	+	0.777541i	$0.716467\pi$
$570$	0	0
$571$	− 26.0000i	− 1.08807i	−0.839064	−	0.544033i	$-0.816897\pi$
$571$	− 26.0000i	− 1.08807i	0.839064	−	0.544033i	$-0.183103\pi$
$572$	0	0
$573$	15.0000i	0.626634i
$574$	0	0
$575$	33.0000	1.37620
$576$	0	0
$577$	−33.0000	−1.37381	−0.686904	−	0.726748i	$-0.741031\pi$
$577$	−33.0000	−1.37381	−0.686904	+	0.726748i	$0.741031\pi$
$578$	0	0
$579$	− 60.0000i	− 2.49351i
$580$	0	0
$581$	− 14.0000i	− 0.580818i
$582$	0	0
$583$	0	0
$584$	0	0
$585$	−120.000	−4.96139
$586$	0	0
$587$	− 20.0000i	− 0.825488i	−0.910847	−	0.412744i	$-0.864570\pi$
$587$	− 20.0000i	− 0.825488i	0.910847	−	0.412744i	$-0.135430\pi$
$588$	0	0
$589$	− 10.0000i	− 0.412043i
$590$	0	0
$591$	6.00000	0.246807
$592$	0	0
$593$	−2.00000	−0.0821302	−0.0410651	−	0.999156i	$-0.513075\pi$
$593$	−2.00000	−0.0821302	−0.0410651	+	0.999156i	$0.513075\pi$
$594$	0	0
$595$	20.0000i	0.819920i
$596$	0	0
$597$	− 15.0000i	− 0.613909i
$598$	0	0
$599$	−24.0000	−0.980613	−0.490307	−	0.871550i	$-0.663115\pi$
$599$	−24.0000	−0.980613	−0.490307	+	0.871550i	$0.663115\pi$
$600$	0	0
$601$	2.00000	0.0815817	0.0407909	−	0.999168i	$-0.487012\pi$
$601$	2.00000	0.0815817	0.0407909	+	0.999168i	$0.487012\pi$
$602$	0	0
$603$	42.0000i	1.71037i
$604$	0	0
$605$	− 44.0000i	− 1.78885i
$606$	0	0
$607$	−32.0000	−1.29884	−0.649420	−	0.760430i	$-0.724988\pi$
$607$	−32.0000	−1.29884	−0.649420	+	0.760430i	$0.724988\pi$
$608$	0	0
$609$	−21.0000	−0.850963
$610$	0	0
$611$	40.0000i	1.61823i
$612$	0	0
$613$	− 6.00000i	− 0.242338i	−0.992632	−	0.121169i	$-0.961336\pi$
$613$	− 6.00000i	− 0.242338i	0.992632	−	0.121169i	$-0.0386643\pi$
$614$	0	0
$615$	72.0000	2.90332
$616$	0	0
$617$	−6.00000	−0.241551	−0.120775	−	0.992680i	$-0.538538\pi$
$617$	−6.00000	−0.241551	−0.120775	+	0.992680i	$0.538538\pi$
$618$	0	0
$619$	34.0000i	1.36658i	0.730149	+	0.683288i	$0.239451\pi$
$619$	34.0000i	1.36658i	−0.730149	+	0.683288i	$0.760549\pi$
$620$	0	0
$621$	− 27.0000i	− 1.08347i
$622$	0	0
$623$	−4.00000	−0.160257
$624$	0	0
$625$	41.0000	1.64000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	10.0000i	0.398726i
$630$	0	0
$631$	8.00000	0.318475	0.159237	−	0.987240i	$-0.449096\pi$
$631$	8.00000	0.318475	0.159237	+	0.987240i	$0.449096\pi$
$632$	0	0
$633$	3.00000	0.119239
$634$	0	0
$635$	− 8.00000i	− 0.317470i
$636$	0	0
$637$	− 30.0000i	− 1.18864i
$638$	0	0
$639$	72.0000	2.84828
$640$	0	0
$641$	46.0000	1.81689	0.908445	−	0.418004i	$-0.137270\pi$
$641$	46.0000	1.81689	0.908445	+	0.418004i	$0.137270\pi$
$642$	0	0
$643$	− 20.0000i	− 0.788723i	−0.918955	−	0.394362i	$-0.870966\pi$
$643$	− 20.0000i	− 0.788723i	0.918955	−	0.394362i	$-0.129034\pi$
$644$	0	0
$645$	− 48.0000i	− 1.89000i
$646$	0	0
$647$	25.0000	0.982851	0.491426	−	0.870919i	$-0.336476\pi$
$647$	25.0000	0.982851	0.491426	+	0.870919i	$0.336476\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	− 30.0000i	− 1.17579i
$652$	0	0
$653$	6.00000i	0.234798i	0.993085	+	0.117399i	$0.0374557\pi$
$653$	6.00000i	0.234798i	−0.993085	+	0.117399i	$0.962544\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	−66.0000	−2.57491
$658$	0	0
$659$	− 21.0000i	− 0.818044i	−0.912525	−	0.409022i	$-0.865870\pi$
$659$	− 21.0000i	− 0.818044i	0.912525	−	0.409022i	$-0.134130\pi$
$660$	0	0
$661$	13.0000i	0.505641i	0.967513	+	0.252821i	$0.0813583\pi$
$661$	13.0000i	0.505641i	−0.967513	+	0.252821i	$0.918642\pi$
$662$	0	0
$663$	−75.0000	−2.91276
$664$	0	0
$665$	−4.00000	−0.155113
$666$	0	0
$667$	21.0000i	0.813123i
$668$	0	0
$669$	42.0000i	1.62381i
$670$	0	0
$671$	0	0
$672$	0	0
$673$	−14.0000	−0.539660	−0.269830	−	0.962908i	$-0.586968\pi$
$673$	−14.0000	−0.539660	−0.269830	+	0.962908i	$0.586968\pi$
$674$	0	0
$675$	− 99.0000i	− 3.81051i
$676$	0	0
$677$	1.00000i	0.0384331i	0.999815	+	0.0192166i	$0.00611720\pi$
$677$	1.00000i	0.0384331i	−0.999815	+	0.0192166i	$0.993883\pi$
$678$	0	0
$679$	−12.0000	−0.460518
$680$	0	0
$681$	51.0000	1.95432
$682$	0	0
$683$	16.0000i	0.612223i	0.951996	+	0.306111i	$0.0990280\pi$
$683$	16.0000i	0.612223i	−0.951996	+	0.306111i	$0.900972\pi$
$684$	0	0
$685$	60.0000i	2.29248i
$686$	0	0
$687$	−66.0000	−2.51806
$688$	0	0
$689$	45.0000	1.71436
$690$	0	0
$691$	6.00000i	0.228251i	0.993466	+	0.114125i	$0.0364066\pi$
$691$	6.00000i	0.228251i	−0.993466	+	0.114125i	$0.963593\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−56.0000	−2.12420
$696$	0	0
$697$	30.0000	1.13633
$698$	0	0
$699$	− 54.0000i	− 2.04247i
$700$	0	0
$701$	20.0000i	0.755390i	0.925930	+	0.377695i	$0.123283\pi$
$701$	20.0000i	0.755390i	−0.925930	+	0.377695i	$0.876717\pi$
$702$	0	0
$703$	−2.00000	−0.0754314
$704$	0	0
$705$	−96.0000	−3.61557
$706$	0	0
$707$	0	0
$708$	0	0
$709$	14.0000i	0.525781i	0.964826	+	0.262891i	$0.0846758\pi$
$709$	14.0000i	0.525781i	−0.964826	+	0.262891i	$0.915324\pi$
$710$	0	0
$711$	−96.0000	−3.60028
$712$	0	0
$713$	−30.0000	−1.12351
$714$	0	0
$715$	0	0
$716$	0	0
$717$	− 33.0000i	− 1.23241i
$718$	0	0
$719$	39.0000	1.45445	0.727227	−	0.686397i	$-0.240809\pi$
$719$	39.0000	1.45445	0.727227	+	0.686397i	$0.240809\pi$
$720$	0	0
$721$	−2.00000	−0.0744839
$722$	0	0
$723$	42.0000i	1.56200i
$724$	0	0
$725$	77.0000i	2.85971i
$726$	0	0
$727$	25.0000	0.927199	0.463599	−	0.886045i	$-0.346558\pi$
$727$	25.0000	0.927199	0.463599	+	0.886045i	$0.346558\pi$
$728$	0	0
$729$	27.0000	1.00000
$730$	0	0
$731$	− 20.0000i	− 0.739727i
$732$	0	0
$733$	− 40.0000i	− 1.47743i	−0.674016	−	0.738717i	$-0.735432\pi$
$733$	− 40.0000i	− 1.47743i	0.674016	−	0.738717i	$-0.264568\pi$
$734$	0	0
$735$	72.0000	2.65576
$736$	0	0
$737$	0	0
$738$	0	0
$739$	40.0000i	1.47142i	0.677295	+	0.735712i	$0.263152\pi$
$739$	40.0000i	1.47142i	−0.677295	+	0.735712i	$0.736848\pi$
$740$	0	0
$741$	− 15.0000i	− 0.551039i
$742$	0	0
$743$	6.00000	0.220119	0.110059	−	0.993925i	$-0.464896\pi$
$743$	6.00000	0.220119	0.110059	+	0.993925i	$0.464896\pi$
$744$	0	0
$745$	24.0000	0.879292
$746$	0	0
$747$	84.0000i	3.07340i
$748$	0	0
$749$	9.00000i	0.328853i
$750$	0	0
$751$	−22.0000	−0.802791	−0.401396	−	0.915905i	$-0.631475\pi$
$751$	−22.0000	−0.802791	−0.401396	+	0.915905i	$0.631475\pi$
$752$	0	0
$753$	−60.0000	−2.18652
$754$	0	0
$755$	32.0000i	1.16460i
$756$	0	0
$757$	− 2.00000i	− 0.0726912i	−0.999339	−	0.0363456i	$-0.988428\pi$
$757$	− 2.00000i	− 0.0726912i	0.999339	−	0.0363456i	$-0.0115717\pi$
$758$	0	0
$759$	0	0
$760$	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
$763$	− 9.00000i	− 0.325822i
$764$	0	0
$765$	− 120.000i	− 4.33861i
$766$	0	0
$767$	5.00000	0.180540
$768$	0	0
$769$	−7.00000	−0.252426	−0.126213	−	0.992003i	$-0.540282\pi$
$769$	−7.00000	−0.252426	−0.126213	+	0.992003i	$0.540282\pi$
$770$	0	0
$771$	− 36.0000i	− 1.29651i
$772$	0	0
$773$	17.0000i	0.611448i	0.952120	+	0.305724i	$0.0988984\pi$
$773$	17.0000i	0.611448i	−0.952120	+	0.305724i	$0.901102\pi$
$774$	0	0
$775$	−110.000	−3.95132
$776$	0	0
$777$	−6.00000	−0.215249
$778$	0	0
$779$	6.00000i	0.214972i
$780$	0	0
$781$	0	0
$782$	0	0
$783$	63.0000	2.25144
$784$	0	0
$785$	80.0000	2.85532
$786$	0	0
$787$	13.0000i	0.463400i	0.972787	+	0.231700i	$0.0744288\pi$
$787$	13.0000i	0.463400i	−0.972787	+	0.231700i	$0.925571\pi$
$788$	0	0
$789$	− 72.0000i	− 2.56327i
$790$	0	0
$791$	8.00000	0.284447
$792$	0	0
$793$	−10.0000	−0.355110
$794$	0	0
$795$	108.000i	3.83037i
$796$	0	0
$797$	− 27.0000i	− 0.956389i	−0.878254	−	0.478195i	$-0.841291\pi$
$797$	− 27.0000i	− 0.956389i	0.878254	−	0.478195i	$-0.158709\pi$
$798$	0	0
$799$	−40.0000	−1.41510
$800$	0	0
$801$	24.0000	0.847998
$802$	0	0
$803$	0	0
$804$	0	0
$805$	12.0000i	0.422944i
$806$	0	0
$807$	42.0000	1.47847
$808$	0	0
$809$	−29.0000	−1.01959	−0.509793	−	0.860297i	$-0.670278\pi$
$809$	−29.0000	−1.01959	−0.509793	+	0.860297i	$0.670278\pi$
$810$	0	0
$811$	− 7.00000i	− 0.245803i	−0.992419	−	0.122902i	$-0.960780\pi$
$811$	− 7.00000i	− 0.245803i	0.992419	−	0.122902i	$-0.0392200\pi$
$812$	0	0
$813$	− 33.0000i	− 1.15736i
$814$	0	0
$815$	−40.0000	−1.40114
$816$	0	0
$817$	4.00000	0.139942
$818$	0	0
$819$	− 30.0000i	− 1.04828i
$820$	0	0
$821$	2.00000i	0.0698005i	0.999391	+	0.0349002i	$0.0111113\pi$
$821$	2.00000i	0.0698005i	−0.999391	+	0.0349002i	$0.988889\pi$
$822$	0	0
$823$	51.0000	1.77775	0.888874	−	0.458151i	$-0.151488\pi$
$823$	51.0000	1.77775	0.888874	+	0.458151i	$0.151488\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	43.0000i	1.49526i	0.664117	+	0.747628i	$0.268807\pi$
$827$	43.0000i	1.49526i	−0.664117	+	0.747628i	$0.731193\pi$
$828$	0	0
$829$	− 5.00000i	− 0.173657i	−0.996223	−	0.0868286i	$-0.972327\pi$
$829$	− 5.00000i	− 0.173657i	0.996223	−	0.0868286i	$-0.0276732\pi$
$830$	0	0
$831$	−18.0000	−0.624413
$832$	0	0
$833$	30.0000	1.03944
$834$	0	0
$835$	8.00000i	0.276851i
$836$	0	0
$837$	90.0000i	3.11086i
$838$	0	0
$839$	−16.0000	−0.552381	−0.276191	−	0.961103i	$-0.589072\pi$
$839$	−16.0000	−0.552381	−0.276191	+	0.961103i	$0.589072\pi$
$840$	0	0
$841$	−20.0000	−0.689655
$842$	0	0
$843$	48.0000i	1.65321i
$844$	0	0
$845$	48.0000i	1.65125i
$846$	0	0
$847$	11.0000	0.377964
$848$	0	0
$849$	18.0000	0.617758
$850$	0	0
$851$	6.00000i	0.205677i
$852$	0	0
$853$	− 26.0000i	− 0.890223i	−0.895475	−	0.445112i	$-0.853164\pi$
$853$	− 26.0000i	− 0.890223i	0.895475	−	0.445112i	$-0.146836\pi$
$854$	0	0
$855$	24.0000	0.820783
$856$	0	0
$857$	20.0000	0.683187	0.341593	−	0.939848i	$-0.389033\pi$
$857$	20.0000	0.683187	0.341593	+	0.939848i	$0.389033\pi$
$858$	0	0
$859$	14.0000i	0.477674i	0.971060	+	0.238837i	$0.0767661\pi$
$859$	14.0000i	0.477674i	−0.971060	+	0.238837i	$0.923234\pi$
$860$	0	0
$861$	18.0000i	0.613438i
$862$	0	0
$863$	10.0000	0.340404	0.170202	−	0.985409i	$-0.445558\pi$
$863$	10.0000	0.340404	0.170202	+	0.985409i	$0.445558\pi$
$864$	0	0
$865$	−56.0000	−1.90406
$866$	0	0
$867$	− 24.0000i	− 0.815083i
$868$	0	0
$869$	0	0
$870$	0	0
$871$	35.0000	1.18593
$872$	0	0
$873$	72.0000	2.43683
$874$	0	0
$875$	24.0000i	0.811348i
$876$	0	0
$877$	− 37.0000i	− 1.24940i	−0.780864	−	0.624701i	$-0.785221\pi$
$877$	− 37.0000i	− 1.24940i	0.780864	−	0.624701i	$-0.214779\pi$
$878$	0	0
$879$	−21.0000	−0.708312
$880$	0	0
$881$	−14.0000	−0.471672	−0.235836	−	0.971793i	$-0.575783\pi$
$881$	−14.0000	−0.471672	−0.235836	+	0.971793i	$0.575783\pi$
$882$	0	0
$883$	34.0000i	1.14419i	0.820187	+	0.572096i	$0.193869\pi$
$883$	34.0000i	1.14419i	−0.820187	+	0.572096i	$0.806131\pi$
$884$	0	0
$885$	12.0000i	0.403376i
$886$	0	0
$887$	26.0000	0.872995	0.436497	−	0.899706i	$-0.356219\pi$
$887$	26.0000	0.872995	0.436497	+	0.899706i	$0.356219\pi$
$888$	0	0
$889$	2.00000	0.0670778
$890$	0	0
$891$	0	0
$892$	0	0
$893$	− 8.00000i	− 0.267710i
$894$	0	0
$895$	−48.0000	−1.60446
$896$	0	0
$897$	−45.0000	−1.50251
$898$	0	0
$899$	− 70.0000i	− 2.33463i
$900$	0	0
$901$	45.0000i	1.49917i
$902$	0	0
$903$	12.0000	0.399335
$904$	0	0
$905$	24.0000	0.797787
$906$	0	0
$907$	37.0000i	1.22856i	0.789086	+	0.614282i	$0.210554\pi$
$907$	37.0000i	1.22856i	−0.789086	+	0.614282i	$0.789446\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	24.0000	0.795155	0.397578	−	0.917568i	$-0.369851\pi$
$911$	24.0000	0.795155	0.397578	+	0.917568i	$0.369851\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	− 24.0000i	− 0.793416i
$916$	0	0
$917$	0	0
$918$	0	0
$919$	55.0000	1.81428	0.907141	−	0.420826i	$-0.138260\pi$
$919$	55.0000	1.81428	0.907141	+	0.420826i	$0.138260\pi$
$920$	0	0
$921$	36.0000	1.18624
$922$	0	0
$923$	− 60.0000i	− 1.97492i
$924$	0	0
$925$	22.0000i	0.723356i
$926$	0	0
$927$	12.0000	0.394132
$928$	0	0
$929$	29.0000	0.951459	0.475730	−	0.879592i	$-0.342184\pi$
$929$	29.0000	0.951459	0.475730	+	0.879592i	$0.342184\pi$
$930$	0	0
$931$	6.00000i	0.196642i
$932$	0	0
$933$	105.000i	3.43755i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	31.0000	1.01273	0.506363	−	0.862320i	$-0.330990\pi$
$937$	31.0000	1.01273	0.506363	+	0.862320i	$0.330990\pi$
$938$	0	0
$939$	27.0000i	0.881112i
$940$	0	0
$941$	− 51.0000i	− 1.66255i	−0.555860	−	0.831276i	$-0.687611\pi$
$941$	− 51.0000i	− 1.66255i	0.555860	−	0.831276i	$-0.312389\pi$
$942$	0	0
$943$	18.0000	0.586161
$944$	0	0
$945$	36.0000	1.17108
$946$	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
$949$	55.0000i	1.78538i
$950$	0	0
$951$	69.0000	2.23748
$952$	0	0
$953$	6.00000	0.194359	0.0971795	−	0.995267i	$-0.469018\pi$
$953$	6.00000	0.194359	0.0971795	+	0.995267i	$0.469018\pi$
$954$	0	0
$955$	20.0000i	0.647185i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−15.0000	−0.484375
$960$	0	0
$961$	69.0000	2.22581
$962$	0	0
$963$	− 54.0000i	− 1.74013i
$964$	0	0
$965$	− 80.0000i	− 2.57529i
$966$	0	0
$967$	40.0000	1.28631	0.643157	−	0.765735i	$-0.277624\pi$
$967$	40.0000	1.28631	0.643157	+	0.765735i	$0.277624\pi$
$968$	0	0
$969$	15.0000	0.481869
$970$	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
$973$	− 14.0000i	− 0.448819i
$974$	0	0
$975$	−165.000	−5.28423
$976$	0	0
$977$	26.0000	0.831814	0.415907	−	0.909407i	$-0.363464\pi$
$977$	26.0000	0.831814	0.415907	+	0.909407i	$0.363464\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	54.0000i	1.72409i
$982$	0	0
$983$	6.00000	0.191370	0.0956851	−	0.995412i	$-0.469496\pi$
$983$	6.00000	0.191370	0.0956851	+	0.995412i	$0.469496\pi$
$984$	0	0
$985$	8.00000	0.254901
$986$	0	0
$987$	− 24.0000i	− 0.763928i
$988$	0	0
$989$	− 12.0000i	− 0.381578i
$990$	0	0
$991$	40.0000	1.27064	0.635321	−	0.772248i	$-0.280868\pi$
$991$	40.0000	1.27064	0.635321	+	0.772248i	$0.280868\pi$
$992$	0	0
$993$	−33.0000	−1.04722
$994$	0	0
$995$	− 20.0000i	− 0.634043i
$996$	0	0
$997$	2.00000i	0.0633406i	0.999498	+	0.0316703i	$0.0100827\pi$
$997$	2.00000i	0.0633406i	−0.999498	+	0.0316703i	$0.989917\pi$
$998$	0	0
$999$	18.0000	0.569495

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1216.2.c.c.609.1	yes	2
4.3	odd	2		1216.2.c.b.609.2	yes	2
8.3	odd	2		1216.2.c.b.609.1	✓	2
8.5	even	2	inner	1216.2.c.c.609.2	yes	2
16.3	odd	4		4864.2.a.a.1.1		1
16.5	even	4		4864.2.a.b.1.1		1
16.11	odd	4		4864.2.a.p.1.1		1
16.13	even	4		4864.2.a.o.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1216.2.c.b.609.1	✓	2	8.3	odd	2
1216.2.c.b.609.2	yes	2	4.3	odd	2
1216.2.c.c.609.1	yes	2	1.1	even	1	trivial
1216.2.c.c.609.2	yes	2	8.5	even	2	inner
4864.2.a.a.1.1		1	16.3	odd	4
4864.2.a.b.1.1		1	16.5	even	4
4864.2.a.o.1.1		1	16.13	even	4
4864.2.a.p.1.1		1	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)

\(f(q)\)	\(=\)	\(q-3.00000i q^{3} -4.00000i q^{5} +1.00000 q^{7} -6.00000 q^{9} +5.00000i q^{13} -12.0000 q^{15} -5.00000 q^{17} -1.00000i q^{19} -3.00000i q^{21} -3.00000 q^{23} -11.0000 q^{25} +9.00000i q^{27} -7.00000i q^{29} +10.0000 q^{31} -4.00000i q^{35} -2.00000i q^{37} +15.0000 q^{39} -6.00000 q^{41} +4.00000i q^{43} +24.0000i q^{45} +8.00000 q^{47} -6.00000 q^{49} +15.0000i q^{51} -9.00000i q^{53} -3.00000 q^{57} -1.00000i q^{59} +2.00000i q^{61} -6.00000 q^{63} +20.0000 q^{65} -7.00000i q^{67} +9.00000i q^{69} -12.0000 q^{71} +11.0000 q^{73} +33.0000i q^{75} +16.0000 q^{79} +9.00000 q^{81} -14.0000i q^{83} +20.0000i q^{85} -21.0000 q^{87} -4.00000 q^{89} +5.00000i q^{91} -30.0000i q^{93} -4.00000 q^{95} -12.0000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{7} - 12 q^{9} - 24 q^{15} - 10 q^{17} - 6 q^{23} - 22 q^{25} + 20 q^{31} + 30 q^{39} - 12 q^{41} + 16 q^{47} - 12 q^{49} - 6 q^{57} - 12 q^{63} + 40 q^{65} - 24 q^{71} + 22 q^{73} + 32 q^{79}+ \cdots - 24 q^{97}+O(q^{100}) \) 2 * q + 2 * q^7 - 12 * q^9 - 24 * q^15 - 10 * q^17 - 6 * q^23 - 22 * q^25 + 20 * q^31 + 30 * q^39 - 12 * q^41 + 16 * q^47 - 12 * q^49 - 6 * q^57 - 12 * q^63 + 40 * q^65 - 24 * q^71 + 22 * q^73 + 32 * q^79 + 18 * q^81 - 42 * q^87 - 8 * q^89 - 8 * q^95 - 24 * q^97

Introduction

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