LMFDB - Embedded newform 1352.2.f.a.337.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1352,2,Mod(337,1352)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1352, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1352.337");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1352 = 2^{3} \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1352.f (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$10.7957743533$
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:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 104)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			337.2
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	1352.337
Dual form			1352.2.f.a.337.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} +2.00000i q^{5} +1.00000i q^{7} -2.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} +2.00000i q^{5} +1.00000i q^{7} -2.00000 q^{9} -1.00000i q^{11} +2.00000i q^{15} -3.00000 q^{17} +7.00000i q^{19} +1.00000i q^{21} -1.00000 q^{23} +1.00000 q^{25} -5.00000 q^{27} +3.00000 q^{29} +8.00000i q^{31} -1.00000i q^{33} -2.00000 q^{35} +1.00000i q^{37} +11.0000i q^{41} -11.0000 q^{43} -4.00000i q^{45} -12.0000i q^{47} +6.00000 q^{49} -3.00000 q^{51} -6.00000 q^{53} +2.00000 q^{55} +7.00000i q^{57} +9.00000i q^{59} -9.00000 q^{61} -2.00000i q^{63} -3.00000i q^{67} -1.00000 q^{69} -5.00000i q^{71} +2.00000i q^{73} +1.00000 q^{75} +1.00000 q^{77} -12.0000 q^{79} +1.00000 q^{81} -4.00000i q^{83} -6.00000i q^{85} +3.00000 q^{87} +1.00000i q^{89} +8.00000i q^{93} -14.0000 q^{95} -1.00000i q^{97} +2.00000i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{3} - 4 q^{9}+O(q^{10}) $ 2 * q + 2 * q^3 - 4 * q^9	$ 2 q + 2 q^{3} - 4 q^{9} - 6 q^{17} - 2 q^{23} + 2 q^{25} - 10 q^{27} + 6 q^{29} - 4 q^{35} - 22 q^{43} + 12 q^{49} - 6 q^{51} - 12 q^{53} + 4 q^{55} - 18 q^{61} - 2 q^{69} + 2 q^{75} + 2 q^{77} - 24 q^{79} + 2 q^{81} + 6 q^{87} - 28 q^{95}+O(q^{100}) $ 2 * q + 2 * q^3 - 4 * q^9 - 6 * q^17 - 2 * q^23 + 2 * q^25 - 10 * q^27 + 6 * q^29 - 4 * q^35 - 22 * q^43 + 12 * q^49 - 6 * q^51 - 12 * q^53 + 4 * q^55 - 18 * q^61 - 2 * q^69 + 2 * q^75 + 2 * q^77 - 24 * q^79 + 2 * q^81 + 6 * q^87 - 28 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1352\mathbb{Z}\right)^\times$.

$n$	$677$	$1015$	$1185$
$\chi(n)$	$1$	$1$	$-1$

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
$3$	1.00000	0.577350	0.288675	−	0.957427i	$-0.406785\pi$
$3$	1.00000	0.577350	0.288675	+	0.957427i	$0.406785\pi$
$4$	0	0
$5$	2.00000i	0.894427i	0.894427	+	0.447214i	$0.147584\pi$
$5$	2.00000i	0.894427i	−0.894427	+	0.447214i	$0.852416\pi$
$6$	0	0
$7$	1.00000i	0.377964i	0.981981	+	0.188982i	$0.0605189\pi$
$7$	1.00000i	0.377964i	−0.981981	+	0.188982i	$0.939481\pi$
$8$	0	0
$9$	−2.00000	−0.666667
$10$	0	0
$11$	− 1.00000i	− 0.301511i	−0.988571	−	0.150756i	$-0.951829\pi$
$11$	− 1.00000i	− 0.301511i	0.988571	−	0.150756i	$-0.0481707\pi$
$12$	0	0
$13$	0	0
$13$	0	0
$14$	0	0
$15$	2.00000i	0.516398i
$16$	0	0
$17$	−3.00000	−0.727607	−0.363803	−	0.931476i	$-0.618522\pi$
$17$	−3.00000	−0.727607	−0.363803	+	0.931476i	$0.618522\pi$
$18$	0	0
$19$	7.00000i	1.60591i	0.596040	+	0.802955i	$0.296740\pi$
$19$	7.00000i	1.60591i	−0.596040	+	0.802955i	$0.703260\pi$
$20$	0	0
$21$	1.00000i	0.218218i
$22$	0	0
$23$	−1.00000	−0.208514	−0.104257	−	0.994550i	$-0.533247\pi$
$23$	−1.00000	−0.208514	−0.104257	+	0.994550i	$0.533247\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−5.00000	−0.962250
$28$	0	0
$29$	3.00000	0.557086	0.278543	−	0.960424i	$-0.410149\pi$
$29$	3.00000	0.557086	0.278543	+	0.960424i	$0.410149\pi$
$30$	0	0
$31$	8.00000i	1.43684i	0.695608	+	0.718421i	$0.255135\pi$
$31$	8.00000i	1.43684i	−0.695608	+	0.718421i	$0.744865\pi$
$32$	0	0
$33$	− 1.00000i	− 0.174078i
$34$	0	0
$35$	−2.00000	−0.338062
$36$	0	0
$37$	1.00000i	0.164399i	0.996616	+	0.0821995i	$0.0261945\pi$
$37$	1.00000i	0.164399i	−0.996616	+	0.0821995i	$0.973806\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	11.0000i	1.71791i	0.512050	+	0.858956i	$0.328886\pi$
$41$	11.0000i	1.71791i	−0.512050	+	0.858956i	$0.671114\pi$
$42$	0	0
$43$	−11.0000	−1.67748	−0.838742	−	0.544529i	$-0.816708\pi$
$43$	−11.0000	−1.67748	−0.838742	+	0.544529i	$0.816708\pi$
$44$	0	0
$45$	− 4.00000i	− 0.596285i
$46$	0	0
$47$	− 12.0000i	− 1.75038i	−0.483779	−	0.875190i	$-0.660736\pi$
$47$	− 12.0000i	− 1.75038i	0.483779	−	0.875190i	$-0.339264\pi$
$48$	0	0
$49$	6.00000	0.857143
$50$	0	0
$51$	−3.00000	−0.420084
$52$	0	0
$53$	−6.00000	−0.824163	−0.412082	−	0.911147i	$-0.635198\pi$
$53$	−6.00000	−0.824163	−0.412082	+	0.911147i	$0.635198\pi$
$54$	0	0
$55$	2.00000	0.269680
$56$	0	0
$57$	7.00000i	0.927173i
$58$	0	0
$59$	9.00000i	1.17170i	0.810419	+	0.585850i	$0.199239\pi$
$59$	9.00000i	1.17170i	−0.810419	+	0.585850i	$0.800761\pi$
$60$	0	0
$61$	−9.00000	−1.15233	−0.576166	−	0.817333i	$-0.695452\pi$
$61$	−9.00000	−1.15233	−0.576166	+	0.817333i	$0.695452\pi$
$62$	0	0
$63$	− 2.00000i	− 0.251976i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	− 3.00000i	− 0.366508i	−0.983066	−	0.183254i	$-0.941337\pi$
$67$	− 3.00000i	− 0.366508i	0.983066	−	0.183254i	$-0.0586631\pi$
$68$	0	0
$69$	−1.00000	−0.120386
$70$	0	0
$71$	− 5.00000i	− 0.593391i	−0.954972	−	0.296695i	$-0.904115\pi$
$71$	− 5.00000i	− 0.593391i	0.954972	−	0.296695i	$-0.0958846\pi$
$72$	0	0
$73$	2.00000i	0.234082i	0.993127	+	0.117041i	$0.0373409\pi$
$73$	2.00000i	0.234082i	−0.993127	+	0.117041i	$0.962659\pi$
$74$	0	0
$75$	1.00000	0.115470
$76$	0	0
$77$	1.00000	0.113961
$78$	0	0
$79$	−12.0000	−1.35011	−0.675053	−	0.737769i	$-0.735879\pi$
$79$	−12.0000	−1.35011	−0.675053	+	0.737769i	$0.735879\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	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
$85$	− 6.00000i	− 0.650791i
$86$	0	0
$87$	3.00000	0.321634
$88$	0	0
$89$	1.00000i	0.106000i	0.998595	+	0.0529999i	$0.0168783\pi$
$89$	1.00000i	0.106000i	−0.998595	+	0.0529999i	$0.983122\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	8.00000i	0.829561i
$94$	0	0
$95$	−14.0000	−1.43637
$96$	0	0
$97$	− 1.00000i	− 0.101535i	−0.998711	−	0.0507673i	$-0.983833\pi$
$97$	− 1.00000i	− 0.101535i	0.998711	−	0.0507673i	$-0.0161667\pi$
$98$	0	0
$99$	2.00000i	0.201008i
$100$	0	0
$101$	17.0000	1.69156	0.845782	−	0.533529i	$-0.179135\pi$
$101$	17.0000	1.69156	0.845782	+	0.533529i	$0.179135\pi$
$102$	0	0
$103$	8.00000	0.788263	0.394132	−	0.919054i	$-0.371045\pi$
$103$	8.00000	0.788263	0.394132	+	0.919054i	$0.371045\pi$
$104$	0	0
$105$	−2.00000	−0.195180
$106$	0	0
$107$	17.0000	1.64345	0.821726	−	0.569883i	$-0.193011\pi$
$107$	17.0000	1.64345	0.821726	+	0.569883i	$0.193011\pi$
$108$	0	0
$109$	10.0000i	0.957826i	0.877862	+	0.478913i	$0.158969\pi$
$109$	10.0000i	0.957826i	−0.877862	+	0.478913i	$0.841031\pi$
$110$	0	0
$111$	1.00000i	0.0949158i
$112$	0	0
$113$	11.0000	1.03479	0.517396	−	0.855746i	$-0.326901\pi$
$113$	11.0000	1.03479	0.517396	+	0.855746i	$0.326901\pi$
$114$	0	0
$115$	− 2.00000i	− 0.186501i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	− 3.00000i	− 0.275010i
$120$	0	0
$121$	10.0000	0.909091
$122$	0	0
$123$	11.0000i	0.991837i
$124$	0	0
$125$	12.0000i	1.07331i
$126$	0	0
$127$	−13.0000	−1.15356	−0.576782	−	0.816898i	$-0.695692\pi$
$127$	−13.0000	−1.15356	−0.576782	+	0.816898i	$0.695692\pi$
$128$	0	0
$129$	−11.0000	−0.968496
$130$	0	0
$131$	12.0000	1.04844	0.524222	−	0.851581i	$-0.324356\pi$
$131$	12.0000	1.04844	0.524222	+	0.851581i	$0.324356\pi$
$132$	0	0
$133$	−7.00000	−0.606977
$134$	0	0
$135$	− 10.0000i	− 0.860663i
$136$	0	0
$137$	9.00000i	0.768922i	0.923141	+	0.384461i	$0.125613\pi$
$137$	9.00000i	0.768922i	−0.923141	+	0.384461i	$0.874387\pi$
$138$	0	0
$139$	−9.00000	−0.763370	−0.381685	−	0.924292i	$-0.624656\pi$
$139$	−9.00000	−0.763370	−0.381685	+	0.924292i	$0.624656\pi$
$140$	0	0
$141$	− 12.0000i	− 1.01058i
$142$	0	0
$143$	0	0
$144$	0	0
$145$	6.00000i	0.498273i
$146$	0	0
$147$	6.00000	0.494872
$148$	0	0
$149$	− 1.00000i	− 0.0819232i	−0.999161	−	0.0409616i	$-0.986958\pi$
$149$	− 1.00000i	− 0.0819232i	0.999161	−	0.0409616i	$-0.0130421\pi$
$150$	0	0
$151$	− 16.0000i	− 1.30206i	−0.759051	−	0.651031i	$-0.774337\pi$
$151$	− 16.0000i	− 1.30206i	0.759051	−	0.651031i	$-0.225663\pi$
$152$	0	0
$153$	6.00000	0.485071
$154$	0	0
$155$	−16.0000	−1.28515
$156$	0	0
$157$	10.0000	0.798087	0.399043	−	0.916932i	$-0.369342\pi$
$157$	10.0000	0.798087	0.399043	+	0.916932i	$0.369342\pi$
$158$	0	0
$159$	−6.00000	−0.475831
$160$	0	0
$161$	− 1.00000i	− 0.0788110i
$162$	0	0
$163$	5.00000i	0.391630i	0.980641	+	0.195815i	$0.0627352\pi$
$163$	5.00000i	0.391630i	−0.980641	+	0.195815i	$0.937265\pi$
$164$	0	0
$165$	2.00000	0.155700
$166$	0	0
$167$	7.00000i	0.541676i	0.962625	+	0.270838i	$0.0873008\pi$
$167$	7.00000i	0.541676i	−0.962625	+	0.270838i	$0.912699\pi$
$168$	0	0
$169$	0	0
$170$	0	0
$171$	− 14.0000i	− 1.07061i
$172$	0	0
$173$	9.00000	0.684257	0.342129	−	0.939653i	$-0.388852\pi$
$173$	9.00000	0.684257	0.342129	+	0.939653i	$0.388852\pi$
$174$	0	0
$175$	1.00000i	0.0755929i
$176$	0	0
$177$	9.00000i	0.676481i
$178$	0	0
$179$	−9.00000	−0.672692	−0.336346	−	0.941739i	$-0.609191\pi$
$179$	−9.00000	−0.672692	−0.336346	+	0.941739i	$0.609191\pi$
$180$	0	0
$181$	14.0000	1.04061	0.520306	−	0.853980i	$-0.325818\pi$
$181$	14.0000	1.04061	0.520306	+	0.853980i	$0.325818\pi$
$182$	0	0
$183$	−9.00000	−0.665299
$184$	0	0
$185$	−2.00000	−0.147043
$186$	0	0
$187$	3.00000i	0.219382i
$188$	0	0
$189$	− 5.00000i	− 0.363696i
$190$	0	0
$191$	7.00000	0.506502	0.253251	−	0.967401i	$-0.418500\pi$
$191$	7.00000	0.506502	0.253251	+	0.967401i	$0.418500\pi$
$192$	0	0
$193$	5.00000i	0.359908i	0.983675	+	0.179954i	$0.0575949\pi$
$193$	5.00000i	0.359908i	−0.983675	+	0.179954i	$0.942405\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 13.0000i	− 0.926212i	−0.886303	−	0.463106i	$-0.846735\pi$
$197$	− 13.0000i	− 0.926212i	0.886303	−	0.463106i	$-0.153265\pi$
$198$	0	0
$199$	−3.00000	−0.212664	−0.106332	−	0.994331i	$-0.533911\pi$
$199$	−3.00000	−0.212664	−0.106332	+	0.994331i	$0.533911\pi$
$200$	0	0
$201$	− 3.00000i	− 0.211604i
$202$	0	0
$203$	3.00000i	0.210559i
$204$	0	0
$205$	−22.0000	−1.53655
$206$	0	0
$207$	2.00000	0.139010
$208$	0	0
$209$	7.00000	0.484200
$210$	0	0
$211$	1.00000	0.0688428	0.0344214	−	0.999407i	$-0.489041\pi$
$211$	1.00000	0.0688428	0.0344214	+	0.999407i	$0.489041\pi$
$212$	0	0
$213$	− 5.00000i	− 0.342594i
$214$	0	0
$215$	− 22.0000i	− 1.50039i
$216$	0	0
$217$	−8.00000	−0.543075
$218$	0	0
$219$	2.00000i	0.135147i
$220$	0	0
$221$	0	0
$222$	0	0
$223$	1.00000i	0.0669650i	0.999439	+	0.0334825i	$0.0106598\pi$
$223$	1.00000i	0.0669650i	−0.999439	+	0.0334825i	$0.989340\pi$
$224$	0	0
$225$	−2.00000	−0.133333
$226$	0	0
$227$	− 1.00000i	− 0.0663723i	−0.999449	−	0.0331862i	$-0.989435\pi$
$227$	− 1.00000i	− 0.0663723i	0.999449	−	0.0331862i	$-0.0105654\pi$
$228$	0	0
$229$	6.00000i	0.396491i	0.980152	+	0.198246i	$0.0635244\pi$
$229$	6.00000i	0.396491i	−0.980152	+	0.198246i	$0.936476\pi$
$230$	0	0
$231$	1.00000	0.0657952
$232$	0	0
$233$	26.0000	1.70332	0.851658	−	0.524097i	$-0.175597\pi$
$233$	26.0000	1.70332	0.851658	+	0.524097i	$0.175597\pi$
$234$	0	0
$235$	24.0000	1.56559
$236$	0	0
$237$	−12.0000	−0.779484
$238$	0	0
$239$	− 24.0000i	− 1.55243i	−0.630468	−	0.776215i	$-0.717137\pi$
$239$	− 24.0000i	− 1.55243i	0.630468	−	0.776215i	$-0.282863\pi$
$240$	0	0
$241$	− 11.0000i	− 0.708572i	−0.935137	−	0.354286i	$-0.884724\pi$
$241$	− 11.0000i	− 0.708572i	0.935137	−	0.354286i	$-0.115276\pi$
$242$	0	0
$243$	16.0000	1.02640
$244$	0	0
$245$	12.0000i	0.766652i
$246$	0	0
$247$	0	0
$248$	0	0
$249$	− 4.00000i	− 0.253490i
$250$	0	0
$251$	13.0000	0.820553	0.410276	−	0.911961i	$-0.365432\pi$
$251$	13.0000	0.820553	0.410276	+	0.911961i	$0.365432\pi$
$252$	0	0
$253$	1.00000i	0.0628695i
$254$	0	0
$255$	− 6.00000i	− 0.375735i
$256$	0	0
$257$	13.0000	0.810918	0.405459	−	0.914113i	$-0.367112\pi$
$257$	13.0000	0.810918	0.405459	+	0.914113i	$0.367112\pi$
$258$	0	0
$259$	−1.00000	−0.0621370
$260$	0	0
$261$	−6.00000	−0.371391
$262$	0	0
$263$	−31.0000	−1.91154	−0.955771	−	0.294112i	$-0.904976\pi$
$263$	−31.0000	−1.91154	−0.955771	+	0.294112i	$0.904976\pi$
$264$	0	0
$265$	− 12.0000i	− 0.737154i
$266$	0	0
$267$	1.00000i	0.0611990i
$268$	0	0
$269$	−5.00000	−0.304855	−0.152428	−	0.988315i	$-0.548709\pi$
$269$	−5.00000	−0.304855	−0.152428	+	0.988315i	$0.548709\pi$
$270$	0	0
$271$	7.00000i	0.425220i	0.977137	+	0.212610i	$0.0681963\pi$
$271$	7.00000i	0.425220i	−0.977137	+	0.212610i	$0.931804\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	− 1.00000i	− 0.0603023i
$276$	0	0
$277$	21.0000	1.26177	0.630884	−	0.775877i	$-0.282692\pi$
$277$	21.0000	1.26177	0.630884	+	0.775877i	$0.282692\pi$
$278$	0	0
$279$	− 16.0000i	− 0.957895i
$280$	0	0
$281$	− 14.0000i	− 0.835170i	−0.908638	−	0.417585i	$-0.862877\pi$
$281$	− 14.0000i	− 0.835170i	0.908638	−	0.417585i	$-0.137123\pi$
$282$	0	0
$283$	−25.0000	−1.48610	−0.743048	−	0.669238i	$-0.766621\pi$
$283$	−25.0000	−1.48610	−0.743048	+	0.669238i	$0.766621\pi$
$284$	0	0
$285$	−14.0000	−0.829288
$286$	0	0
$287$	−11.0000	−0.649309
$288$	0	0
$289$	−8.00000	−0.470588
$290$	0	0
$291$	− 1.00000i	− 0.0586210i
$292$	0	0
$293$	9.00000i	0.525786i	0.964825	+	0.262893i	$0.0846766\pi$
$293$	9.00000i	0.525786i	−0.964825	+	0.262893i	$0.915323\pi$
$294$	0	0
$295$	−18.0000	−1.04800
$296$	0	0
$297$	5.00000i	0.290129i
$298$	0	0
$299$	0	0
$300$	0	0
$301$	− 11.0000i	− 0.634029i
$302$	0	0
$303$	17.0000	0.976624
$304$	0	0
$305$	− 18.0000i	− 1.03068i
$306$	0	0
$307$	20.0000i	1.14146i	0.821138	+	0.570730i	$0.193340\pi$
$307$	20.0000i	1.14146i	−0.821138	+	0.570730i	$0.806660\pi$
$308$	0	0
$309$	8.00000	0.455104
$310$	0	0
$311$	−8.00000	−0.453638	−0.226819	−	0.973937i	$-0.572833\pi$
$311$	−8.00000	−0.453638	−0.226819	+	0.973937i	$0.572833\pi$
$312$	0	0
$313$	30.0000	1.69570	0.847850	−	0.530236i	$-0.177897\pi$
$313$	30.0000	1.69570	0.847850	+	0.530236i	$0.177897\pi$
$314$	0	0
$315$	4.00000	0.225374
$316$	0	0
$317$	− 2.00000i	− 0.112331i	−0.998421	−	0.0561656i	$-0.982113\pi$
$317$	− 2.00000i	− 0.112331i	0.998421	−	0.0561656i	$-0.0178875\pi$
$318$	0	0
$319$	− 3.00000i	− 0.167968i
$320$	0	0
$321$	17.0000	0.948847
$322$	0	0
$323$	− 21.0000i	− 1.16847i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	10.0000i	0.553001i
$328$	0	0
$329$	12.0000	0.661581
$330$	0	0
$331$	27.0000i	1.48405i	0.670370	+	0.742027i	$0.266135\pi$
$331$	27.0000i	1.48405i	−0.670370	+	0.742027i	$0.733865\pi$
$332$	0	0
$333$	− 2.00000i	− 0.109599i
$334$	0	0
$335$	6.00000	0.327815
$336$	0	0
$337$	−6.00000	−0.326841	−0.163420	−	0.986557i	$-0.552253\pi$
$337$	−6.00000	−0.326841	−0.163420	+	0.986557i	$0.552253\pi$
$338$	0	0
$339$	11.0000	0.597438
$340$	0	0
$341$	8.00000	0.433224
$342$	0	0
$343$	13.0000i	0.701934i
$344$	0	0
$345$	− 2.00000i	− 0.107676i
$346$	0	0
$347$	27.0000	1.44944	0.724718	−	0.689046i	$-0.241970\pi$
$347$	27.0000	1.44944	0.724718	+	0.689046i	$0.241970\pi$
$348$	0	0
$349$	9.00000i	0.481759i	0.970555	+	0.240879i	$0.0774359\pi$
$349$	9.00000i	0.481759i	−0.970555	+	0.240879i	$0.922564\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	− 25.0000i	− 1.33062i	−0.746569	−	0.665308i	$-0.768300\pi$
$353$	− 25.0000i	− 1.33062i	0.746569	−	0.665308i	$-0.231700\pi$
$354$	0	0
$355$	10.0000	0.530745
$356$	0	0
$357$	− 3.00000i	− 0.158777i
$358$	0	0
$359$	24.0000i	1.26667i	0.773877	+	0.633336i	$0.218315\pi$
$359$	24.0000i	1.26667i	−0.773877	+	0.633336i	$0.781685\pi$
$360$	0	0
$361$	−30.0000	−1.57895
$362$	0	0
$363$	10.0000	0.524864
$364$	0	0
$365$	−4.00000	−0.209370
$366$	0	0
$367$	−15.0000	−0.782994	−0.391497	−	0.920179i	$-0.628043\pi$
$367$	−15.0000	−0.782994	−0.391497	+	0.920179i	$0.628043\pi$
$368$	0	0
$369$	− 22.0000i	− 1.14527i
$370$	0	0
$371$	− 6.00000i	− 0.311504i
$372$	0	0
$373$	11.0000	0.569558	0.284779	−	0.958593i	$-0.408080\pi$
$373$	11.0000	0.569558	0.284779	+	0.958593i	$0.408080\pi$
$374$	0	0
$375$	12.0000i	0.619677i
$376$	0	0
$377$	0	0
$378$	0	0
$379$	− 15.0000i	− 0.770498i	−0.922813	−	0.385249i	$-0.874116\pi$
$379$	− 15.0000i	− 0.770498i	0.922813	−	0.385249i	$-0.125884\pi$
$380$	0	0
$381$	−13.0000	−0.666010
$382$	0	0
$383$	− 33.0000i	− 1.68622i	−0.537740	−	0.843111i	$-0.680722\pi$
$383$	− 33.0000i	− 1.68622i	0.537740	−	0.843111i	$-0.319278\pi$
$384$	0	0
$385$	2.00000i	0.101929i
$386$	0	0
$387$	22.0000	1.11832
$388$	0	0
$389$	−6.00000	−0.304212	−0.152106	−	0.988364i	$-0.548606\pi$
$389$	−6.00000	−0.304212	−0.152106	+	0.988364i	$0.548606\pi$
$390$	0	0
$391$	3.00000	0.151717
$392$	0	0
$393$	12.0000	0.605320
$394$	0	0
$395$	− 24.0000i	− 1.20757i
$396$	0	0
$397$	− 19.0000i	− 0.953583i	−0.879017	−	0.476791i	$-0.841800\pi$
$397$	− 19.0000i	− 0.953583i	0.879017	−	0.476791i	$-0.158200\pi$
$398$	0	0
$399$	−7.00000	−0.350438
$400$	0	0
$401$	− 27.0000i	− 1.34832i	−0.738587	−	0.674158i	$-0.764507\pi$
$401$	− 27.0000i	− 1.34832i	0.738587	−	0.674158i	$-0.235493\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	2.00000i	0.0993808i
$406$	0	0
$407$	1.00000	0.0495682
$408$	0	0
$409$	35.0000i	1.73064i	0.501221	+	0.865319i	$0.332884\pi$
$409$	35.0000i	1.73064i	−0.501221	+	0.865319i	$0.667116\pi$
$410$	0	0
$411$	9.00000i	0.443937i
$412$	0	0
$413$	−9.00000	−0.442861
$414$	0	0
$415$	8.00000	0.392705
$416$	0	0
$417$	−9.00000	−0.440732
$418$	0	0
$419$	21.0000	1.02592	0.512959	−	0.858413i	$-0.328549\pi$
$419$	21.0000	1.02592	0.512959	+	0.858413i	$0.328549\pi$
$420$	0	0
$421$	− 22.0000i	− 1.07221i	−0.844150	−	0.536107i	$-0.819894\pi$
$421$	− 22.0000i	− 1.07221i	0.844150	−	0.536107i	$-0.180106\pi$
$422$	0	0
$423$	24.0000i	1.16692i
$424$	0	0
$425$	−3.00000	−0.145521
$426$	0	0
$427$	− 9.00000i	− 0.435541i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	− 11.0000i	− 0.529851i	−0.964269	−	0.264926i	$-0.914653\pi$
$431$	− 11.0000i	− 0.529851i	0.964269	−	0.264926i	$-0.0853474\pi$
$432$	0	0
$433$	−19.0000	−0.913082	−0.456541	−	0.889702i	$-0.650912\pi$
$433$	−19.0000	−0.913082	−0.456541	+	0.889702i	$0.650912\pi$
$434$	0	0
$435$	6.00000i	0.287678i
$436$	0	0
$437$	− 7.00000i	− 0.334855i
$438$	0	0
$439$	35.0000	1.67046	0.835229	−	0.549902i	$-0.185335\pi$
$439$	35.0000	1.67046	0.835229	+	0.549902i	$0.185335\pi$
$440$	0	0
$441$	−12.0000	−0.571429
$442$	0	0
$443$	12.0000	0.570137	0.285069	−	0.958507i	$-0.407984\pi$
$443$	12.0000	0.570137	0.285069	+	0.958507i	$0.407984\pi$
$444$	0	0
$445$	−2.00000	−0.0948091
$446$	0	0
$447$	− 1.00000i	− 0.0472984i
$448$	0	0
$449$	21.0000i	0.991051i	0.868593	+	0.495526i	$0.165025\pi$
$449$	21.0000i	0.991051i	−0.868593	+	0.495526i	$0.834975\pi$
$450$	0	0
$451$	11.0000	0.517970
$452$	0	0
$453$	− 16.0000i	− 0.751746i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	7.00000i	0.327446i	0.986506	+	0.163723i	$0.0523504\pi$
$457$	7.00000i	0.327446i	−0.986506	+	0.163723i	$0.947650\pi$
$458$	0	0
$459$	15.0000	0.700140
$460$	0	0
$461$	3.00000i	0.139724i	0.997557	+	0.0698620i	$0.0222559\pi$
$461$	3.00000i	0.139724i	−0.997557	+	0.0698620i	$0.977744\pi$
$462$	0	0
$463$	− 24.0000i	− 1.11537i	−0.830051	−	0.557687i	$-0.811689\pi$
$463$	− 24.0000i	− 1.11537i	0.830051	−	0.557687i	$-0.188311\pi$
$464$	0	0
$465$	−16.0000	−0.741982
$466$	0	0
$467$	8.00000	0.370196	0.185098	−	0.982720i	$-0.440740\pi$
$467$	8.00000	0.370196	0.185098	+	0.982720i	$0.440740\pi$
$468$	0	0
$469$	3.00000	0.138527
$470$	0	0
$471$	10.0000	0.460776
$472$	0	0
$473$	11.0000i	0.505781i
$474$	0	0
$475$	7.00000i	0.321182i
$476$	0	0
$477$	12.0000	0.549442
$478$	0	0
$479$	27.0000i	1.23366i	0.787096	+	0.616831i	$0.211584\pi$
$479$	27.0000i	1.23366i	−0.787096	+	0.616831i	$0.788416\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	− 1.00000i	− 0.0455016i
$484$	0	0
$485$	2.00000	0.0908153
$486$	0	0
$487$	− 13.0000i	− 0.589086i	−0.955638	−	0.294543i	$-0.904833\pi$
$487$	− 13.0000i	− 0.589086i	0.955638	−	0.294543i	$-0.0951675\pi$
$488$	0	0
$489$	5.00000i	0.226108i
$490$	0	0
$491$	3.00000	0.135388	0.0676941	−	0.997706i	$-0.478436\pi$
$491$	3.00000	0.135388	0.0676941	+	0.997706i	$0.478436\pi$
$492$	0	0
$493$	−9.00000	−0.405340
$494$	0	0
$495$	−4.00000	−0.179787
$496$	0	0
$497$	5.00000	0.224281
$498$	0	0
$499$	36.0000i	1.61158i	0.592200	+	0.805791i	$0.298259\pi$
$499$	36.0000i	1.61158i	−0.592200	+	0.805791i	$0.701741\pi$
$500$	0	0
$501$	7.00000i	0.312737i
$502$	0	0
$503$	−25.0000	−1.11469	−0.557347	−	0.830279i	$-0.688181\pi$
$503$	−25.0000	−1.11469	−0.557347	+	0.830279i	$0.688181\pi$
$504$	0	0
$505$	34.0000i	1.51298i
$506$	0	0
$507$	0	0
$508$	0	0
$509$	− 1.00000i	− 0.0443242i	−0.999754	−	0.0221621i	$-0.992945\pi$
$509$	− 1.00000i	− 0.0443242i	0.999754	−	0.0221621i	$-0.00705500\pi$
$510$	0	0
$511$	−2.00000	−0.0884748
$512$	0	0
$513$	− 35.0000i	− 1.54529i
$514$	0	0
$515$	16.0000i	0.705044i
$516$	0	0
$517$	−12.0000	−0.527759
$518$	0	0
$519$	9.00000	0.395056
$520$	0	0
$521$	−42.0000	−1.84005	−0.920027	−	0.391856i	$-0.871833\pi$
$521$	−42.0000	−1.84005	−0.920027	+	0.391856i	$0.871833\pi$
$522$	0	0
$523$	−19.0000	−0.830812	−0.415406	−	0.909636i	$-0.636360\pi$
$523$	−19.0000	−0.830812	−0.415406	+	0.909636i	$0.636360\pi$
$524$	0	0
$525$	1.00000i	0.0436436i
$526$	0	0
$527$	− 24.0000i	− 1.04546i
$528$	0	0
$529$	−22.0000	−0.956522
$530$	0	0
$531$	− 18.0000i	− 0.781133i
$532$	0	0
$533$	0	0
$534$	0	0
$535$	34.0000i	1.46995i
$536$	0	0
$537$	−9.00000	−0.388379
$538$	0	0
$539$	− 6.00000i	− 0.258438i
$540$	0	0
$541$	− 30.0000i	− 1.28980i	−0.764267	−	0.644900i	$-0.776899\pi$
$541$	− 30.0000i	− 1.28980i	0.764267	−	0.644900i	$-0.223101\pi$
$542$	0	0
$543$	14.0000	0.600798
$544$	0	0
$545$	−20.0000	−0.856706
$546$	0	0
$547$	12.0000	0.513083	0.256541	−	0.966533i	$-0.417417\pi$
$547$	12.0000	0.513083	0.256541	+	0.966533i	$0.417417\pi$
$548$	0	0
$549$	18.0000	0.768221
$550$	0	0
$551$	21.0000i	0.894630i
$552$	0	0
$553$	− 12.0000i	− 0.510292i
$554$	0	0
$555$	−2.00000	−0.0848953
$556$	0	0
$557$	− 19.0000i	− 0.805056i	−0.915408	−	0.402528i	$-0.868132\pi$
$557$	− 19.0000i	− 0.805056i	0.915408	−	0.402528i	$-0.131868\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	3.00000i	0.126660i
$562$	0	0
$563$	9.00000	0.379305	0.189652	−	0.981851i	$-0.439264\pi$
$563$	9.00000	0.379305	0.189652	+	0.981851i	$0.439264\pi$
$564$	0	0
$565$	22.0000i	0.925547i
$566$	0	0
$567$	1.00000i	0.0419961i
$568$	0	0
$569$	17.0000	0.712677	0.356339	−	0.934357i	$-0.384025\pi$
$569$	17.0000	0.712677	0.356339	+	0.934357i	$0.384025\pi$
$570$	0	0
$571$	20.0000	0.836974	0.418487	−	0.908223i	$-0.362561\pi$
$571$	20.0000	0.836974	0.418487	+	0.908223i	$0.362561\pi$
$572$	0	0
$573$	7.00000	0.292429
$574$	0	0
$575$	−1.00000	−0.0417029
$576$	0	0
$577$	22.0000i	0.915872i	0.888985	+	0.457936i	$0.151411\pi$
$577$	22.0000i	0.915872i	−0.888985	+	0.457936i	$0.848589\pi$
$578$	0	0
$579$	5.00000i	0.207793i
$580$	0	0
$581$	4.00000	0.165948
$582$	0	0
$583$	6.00000i	0.248495i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	21.0000i	0.866763i	0.901211	+	0.433381i	$0.142680\pi$
$587$	21.0000i	0.866763i	−0.901211	+	0.433381i	$0.857320\pi$
$588$	0	0
$589$	−56.0000	−2.30744
$590$	0	0
$591$	− 13.0000i	− 0.534749i
$592$	0	0
$593$	− 14.0000i	− 0.574911i	−0.957794	−	0.287456i	$-0.907191\pi$
$593$	− 14.0000i	− 0.574911i	0.957794	−	0.287456i	$-0.0928094\pi$
$594$	0	0
$595$	6.00000	0.245976
$596$	0	0
$597$	−3.00000	−0.122782
$598$	0	0
$599$	−44.0000	−1.79779	−0.898896	−	0.438163i	$-0.855629\pi$
$599$	−44.0000	−1.79779	−0.898896	+	0.438163i	$0.855629\pi$
$600$	0	0
$601$	−21.0000	−0.856608	−0.428304	−	0.903635i	$-0.640889\pi$
$601$	−21.0000	−0.856608	−0.428304	+	0.903635i	$0.640889\pi$
$602$	0	0
$603$	6.00000i	0.244339i
$604$	0	0
$605$	20.0000i	0.813116i
$606$	0	0
$607$	−13.0000	−0.527654	−0.263827	−	0.964570i	$-0.584985\pi$
$607$	−13.0000	−0.527654	−0.263827	+	0.964570i	$0.584985\pi$
$608$	0	0
$609$	3.00000i	0.121566i
$610$	0	0
$611$	0	0
$612$	0	0
$613$	39.0000i	1.57520i	0.616190	+	0.787598i	$0.288675\pi$
$613$	39.0000i	1.57520i	−0.616190	+	0.787598i	$0.711325\pi$
$614$	0	0
$615$	−22.0000	−0.887126
$616$	0	0
$617$	47.0000i	1.89215i	0.323949	+	0.946074i	$0.394989\pi$
$617$	47.0000i	1.89215i	−0.323949	+	0.946074i	$0.605011\pi$
$618$	0	0
$619$	20.0000i	0.803868i	0.915669	+	0.401934i	$0.131662\pi$
$619$	20.0000i	0.803868i	−0.915669	+	0.401934i	$0.868338\pi$
$620$	0	0
$621$	5.00000	0.200643
$622$	0	0
$623$	−1.00000	−0.0400642
$624$	0	0
$625$	−19.0000	−0.760000
$626$	0	0
$627$	7.00000	0.279553
$628$	0	0
$629$	− 3.00000i	− 0.119618i
$630$	0	0
$631$	− 27.0000i	− 1.07485i	−0.843311	−	0.537427i	$-0.819397\pi$
$631$	− 27.0000i	− 1.07485i	0.843311	−	0.537427i	$-0.180603\pi$
$632$	0	0
$633$	1.00000	0.0397464
$634$	0	0
$635$	− 26.0000i	− 1.03178i
$636$	0	0
$637$	0	0
$638$	0	0
$639$	10.0000i	0.395594i
$640$	0	0
$641$	−11.0000	−0.434474	−0.217237	−	0.976119i	$-0.569704\pi$
$641$	−11.0000	−0.434474	−0.217237	+	0.976119i	$0.569704\pi$
$642$	0	0
$643$	19.0000i	0.749287i	0.927169	+	0.374643i	$0.122235\pi$
$643$	19.0000i	0.749287i	−0.927169	+	0.374643i	$0.877765\pi$
$644$	0	0
$645$	− 22.0000i	− 0.866249i
$646$	0	0
$647$	3.00000	0.117942	0.0589711	−	0.998260i	$-0.481218\pi$
$647$	3.00000	0.117942	0.0589711	+	0.998260i	$0.481218\pi$
$648$	0	0
$649$	9.00000	0.353281
$650$	0	0
$651$	−8.00000	−0.313545
$652$	0	0
$653$	−9.00000	−0.352197	−0.176099	−	0.984373i	$-0.556348\pi$
$653$	−9.00000	−0.352197	−0.176099	+	0.984373i	$0.556348\pi$
$654$	0	0
$655$	24.0000i	0.937758i
$656$	0	0
$657$	− 4.00000i	− 0.156055i
$658$	0	0
$659$	−29.0000	−1.12968	−0.564840	−	0.825201i	$-0.691062\pi$
$659$	−29.0000	−1.12968	−0.564840	+	0.825201i	$0.691062\pi$
$660$	0	0
$661$	25.0000i	0.972387i	0.873851	+	0.486194i	$0.161615\pi$
$661$	25.0000i	0.972387i	−0.873851	+	0.486194i	$0.838385\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	− 14.0000i	− 0.542897i
$666$	0	0
$667$	−3.00000	−0.116160
$668$	0	0
$669$	1.00000i	0.0386622i
$670$	0	0
$671$	9.00000i	0.347441i
$672$	0	0
$673$	−39.0000	−1.50334	−0.751670	−	0.659540i	$-0.770751\pi$
$673$	−39.0000	−1.50334	−0.751670	+	0.659540i	$0.770751\pi$
$674$	0	0
$675$	−5.00000	−0.192450
$676$	0	0
$677$	−6.00000	−0.230599	−0.115299	−	0.993331i	$-0.536783\pi$
$677$	−6.00000	−0.230599	−0.115299	+	0.993331i	$0.536783\pi$
$678$	0	0
$679$	1.00000	0.0383765
$680$	0	0
$681$	− 1.00000i	− 0.0383201i
$682$	0	0
$683$	9.00000i	0.344375i	0.985064	+	0.172188i	$0.0550836\pi$
$683$	9.00000i	0.344375i	−0.985064	+	0.172188i	$0.944916\pi$
$684$	0	0
$685$	−18.0000	−0.687745
$686$	0	0
$687$	6.00000i	0.228914i
$688$	0	0
$689$	0	0
$690$	0	0
$691$	25.0000i	0.951045i	0.879704	+	0.475522i	$0.157741\pi$
$691$	25.0000i	0.951045i	−0.879704	+	0.475522i	$0.842259\pi$
$692$	0	0
$693$	−2.00000	−0.0759737
$694$	0	0
$695$	− 18.0000i	− 0.682779i
$696$	0	0
$697$	− 33.0000i	− 1.24996i
$698$	0	0
$699$	26.0000	0.983410
$700$	0	0
$701$	22.0000	0.830929	0.415464	−	0.909610i	$-0.363619\pi$
$701$	22.0000	0.830929	0.415464	+	0.909610i	$0.363619\pi$
$702$	0	0
$703$	−7.00000	−0.264010
$704$	0	0
$705$	24.0000	0.903892
$706$	0	0
$707$	17.0000i	0.639351i
$708$	0	0
$709$	29.0000i	1.08912i	0.838723	+	0.544559i	$0.183303\pi$
$709$	29.0000i	1.08912i	−0.838723	+	0.544559i	$0.816697\pi$
$710$	0	0
$711$	24.0000	0.900070
$712$	0	0
$713$	− 8.00000i	− 0.299602i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	− 24.0000i	− 0.896296i
$718$	0	0
$719$	−39.0000	−1.45445	−0.727227	−	0.686397i	$-0.759191\pi$
$719$	−39.0000	−1.45445	−0.727227	+	0.686397i	$0.759191\pi$
$720$	0	0
$721$	8.00000i	0.297936i
$722$	0	0
$723$	− 11.0000i	− 0.409094i
$724$	0	0
$725$	3.00000	0.111417
$726$	0	0
$727$	8.00000	0.296704	0.148352	−	0.988935i	$-0.452603\pi$
$727$	8.00000	0.296704	0.148352	+	0.988935i	$0.452603\pi$
$728$	0	0
$729$	13.0000	0.481481
$730$	0	0
$731$	33.0000	1.22055
$732$	0	0
$733$	46.0000i	1.69905i	0.527549	+	0.849524i	$0.323111\pi$
$733$	46.0000i	1.69905i	−0.527549	+	0.849524i	$0.676889\pi$
$734$	0	0
$735$	12.0000i	0.442627i
$736$	0	0
$737$	−3.00000	−0.110506
$738$	0	0
$739$	11.0000i	0.404642i	0.979319	+	0.202321i	$0.0648484\pi$
$739$	11.0000i	0.404642i	−0.979319	+	0.202321i	$0.935152\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	29.0000i	1.06391i	0.846774	+	0.531953i	$0.178542\pi$
$743$	29.0000i	1.06391i	−0.846774	+	0.531953i	$0.821458\pi$
$744$	0	0
$745$	2.00000	0.0732743
$746$	0	0
$747$	8.00000i	0.292705i
$748$	0	0
$749$	17.0000i	0.621166i
$750$	0	0
$751$	−17.0000	−0.620339	−0.310169	−	0.950681i	$-0.600386\pi$
$751$	−17.0000	−0.620339	−0.310169	+	0.950681i	$0.600386\pi$
$752$	0	0
$753$	13.0000	0.473746
$754$	0	0
$755$	32.0000	1.16460
$756$	0	0
$757$	47.0000	1.70824	0.854122	−	0.520073i	$-0.174095\pi$
$757$	47.0000	1.70824	0.854122	+	0.520073i	$0.174095\pi$
$758$	0	0
$759$	1.00000i	0.0362977i
$760$	0	0
$761$	− 3.00000i	− 0.108750i	−0.998521	−	0.0543750i	$-0.982683\pi$
$761$	− 3.00000i	− 0.108750i	0.998521	−	0.0543750i	$-0.0173166\pi$
$762$	0	0
$763$	−10.0000	−0.362024
$764$	0	0
$765$	12.0000i	0.433861i
$766$	0	0
$767$	0	0
$768$	0	0
$769$	− 5.00000i	− 0.180305i	−0.995928	−	0.0901523i	$-0.971265\pi$
$769$	− 5.00000i	− 0.180305i	0.995928	−	0.0901523i	$-0.0287354\pi$
$770$	0	0
$771$	13.0000	0.468184
$772$	0	0
$773$	3.00000i	0.107903i	0.998544	+	0.0539513i	$0.0171816\pi$
$773$	3.00000i	0.107903i	−0.998544	+	0.0539513i	$0.982818\pi$
$774$	0	0
$775$	8.00000i	0.287368i
$776$	0	0
$777$	−1.00000	−0.0358748
$778$	0	0
$779$	−77.0000	−2.75881
$780$	0	0
$781$	−5.00000	−0.178914
$782$	0	0
$783$	−15.0000	−0.536056
$784$	0	0
$785$	20.0000i	0.713831i
$786$	0	0
$787$	21.0000i	0.748569i	0.927314	+	0.374285i	$0.122112\pi$
$787$	21.0000i	0.748569i	−0.927314	+	0.374285i	$0.877888\pi$
$788$	0	0
$789$	−31.0000	−1.10363
$790$	0	0
$791$	11.0000i	0.391115i
$792$	0	0
$793$	0	0
$794$	0	0
$795$	− 12.0000i	− 0.425596i
$796$	0	0
$797$	−23.0000	−0.814702	−0.407351	−	0.913272i	$-0.633547\pi$
$797$	−23.0000	−0.814702	−0.407351	+	0.913272i	$0.633547\pi$
$798$	0	0
$799$	36.0000i	1.27359i
$800$	0	0
$801$	− 2.00000i	− 0.0706665i
$802$	0	0
$803$	2.00000	0.0705785
$804$	0	0
$805$	2.00000	0.0704907
$806$	0	0
$807$	−5.00000	−0.176008
$808$	0	0
$809$	27.0000	0.949269	0.474635	−	0.880183i	$-0.342580\pi$
$809$	27.0000	0.949269	0.474635	+	0.880183i	$0.342580\pi$
$810$	0	0
$811$	32.0000i	1.12367i	0.827249	+	0.561836i	$0.189905\pi$
$811$	32.0000i	1.12367i	−0.827249	+	0.561836i	$0.810095\pi$
$812$	0	0
$813$	7.00000i	0.245501i
$814$	0	0
$815$	−10.0000	−0.350285
$816$	0	0
$817$	− 77.0000i	− 2.69389i
$818$	0	0
$819$	0	0
$820$	0	0
$821$	− 5.00000i	− 0.174501i	−0.996186	−	0.0872506i	$-0.972192\pi$
$821$	− 5.00000i	− 0.174501i	0.996186	−	0.0872506i	$-0.0278081\pi$
$822$	0	0
$823$	29.0000	1.01088	0.505438	−	0.862863i	$-0.331331\pi$
$823$	29.0000	1.01088	0.505438	+	0.862863i	$0.331331\pi$
$824$	0	0
$825$	− 1.00000i	− 0.0348155i
$826$	0	0
$827$	− 20.0000i	− 0.695468i	−0.937593	−	0.347734i	$-0.886951\pi$
$827$	− 20.0000i	− 0.695468i	0.937593	−	0.347734i	$-0.113049\pi$
$828$	0	0
$829$	−3.00000	−0.104194	−0.0520972	−	0.998642i	$-0.516591\pi$
$829$	−3.00000	−0.104194	−0.0520972	+	0.998642i	$0.516591\pi$
$830$	0	0
$831$	21.0000	0.728482
$832$	0	0
$833$	−18.0000	−0.623663
$834$	0	0
$835$	−14.0000	−0.484490
$836$	0	0
$837$	− 40.0000i	− 1.38260i
$838$	0	0
$839$	− 27.0000i	− 0.932144i	−0.884747	−	0.466072i	$-0.845669\pi$
$839$	− 27.0000i	− 0.932144i	0.884747	−	0.466072i	$-0.154331\pi$
$840$	0	0
$841$	−20.0000	−0.689655
$842$	0	0
$843$	− 14.0000i	− 0.482186i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	10.0000i	0.343604i
$848$	0	0
$849$	−25.0000	−0.857998
$850$	0	0
$851$	− 1.00000i	− 0.0342796i
$852$	0	0
$853$	− 10.0000i	− 0.342393i	−0.985237	−	0.171197i	$-0.945237\pi$
$853$	− 10.0000i	− 0.342393i	0.985237	−	0.171197i	$-0.0547634\pi$
$854$	0	0
$855$	28.0000	0.957580
$856$	0	0
$857$	18.0000	0.614868	0.307434	−	0.951569i	$-0.400530\pi$
$857$	18.0000	0.614868	0.307434	+	0.951569i	$0.400530\pi$
$858$	0	0
$859$	20.0000	0.682391	0.341196	−	0.939992i	$-0.389168\pi$
$859$	20.0000	0.682391	0.341196	+	0.939992i	$0.389168\pi$
$860$	0	0
$861$	−11.0000	−0.374879
$862$	0	0
$863$	− 16.0000i	− 0.544646i	−0.962206	−	0.272323i	$-0.912208\pi$
$863$	− 16.0000i	− 0.544646i	0.962206	−	0.272323i	$-0.0877920\pi$
$864$	0	0
$865$	18.0000i	0.612018i
$866$	0	0
$867$	−8.00000	−0.271694
$868$	0	0
$869$	12.0000i	0.407072i
$870$	0	0
$871$	0	0
$872$	0	0
$873$	2.00000i	0.0676897i
$874$	0	0
$875$	−12.0000	−0.405674
$876$	0	0
$877$	− 1.00000i	− 0.0337676i	−0.999857	−	0.0168838i	$-0.994625\pi$
$877$	− 1.00000i	− 0.0337676i	0.999857	−	0.0168838i	$-0.00537454\pi$
$878$	0	0
$879$	9.00000i	0.303562i
$880$	0	0
$881$	45.0000	1.51609	0.758044	−	0.652203i	$-0.226155\pi$
$881$	45.0000	1.51609	0.758044	+	0.652203i	$0.226155\pi$
$882$	0	0
$883$	12.0000	0.403832	0.201916	−	0.979403i	$-0.435283\pi$
$883$	12.0000	0.403832	0.201916	+	0.979403i	$0.435283\pi$
$884$	0	0
$885$	−18.0000	−0.605063
$886$	0	0
$887$	−27.0000	−0.906571	−0.453286	−	0.891365i	$-0.649748\pi$
$887$	−27.0000	−0.906571	−0.453286	+	0.891365i	$0.649748\pi$
$888$	0	0
$889$	− 13.0000i	− 0.436006i
$890$	0	0
$891$	− 1.00000i	− 0.0335013i
$892$	0	0
$893$	84.0000	2.81095
$894$	0	0
$895$	− 18.0000i	− 0.601674i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	24.0000i	0.800445i
$900$	0	0
$901$	18.0000	0.599667
$902$	0	0
$903$	− 11.0000i	− 0.366057i
$904$	0	0
$905$	28.0000i	0.930751i
$906$	0	0
$907$	−25.0000	−0.830111	−0.415056	−	0.909796i	$-0.636238\pi$
$907$	−25.0000	−0.830111	−0.415056	+	0.909796i	$0.636238\pi$
$908$	0	0
$909$	−34.0000	−1.12771
$910$	0	0
$911$	−16.0000	−0.530104	−0.265052	−	0.964234i	$-0.585389\pi$
$911$	−16.0000	−0.530104	−0.265052	+	0.964234i	$0.585389\pi$
$912$	0	0
$913$	−4.00000	−0.132381
$914$	0	0
$915$	− 18.0000i	− 0.595062i
$916$	0	0
$917$	12.0000i	0.396275i
$918$	0	0
$919$	15.0000	0.494804	0.247402	−	0.968913i	$-0.420423\pi$
$919$	15.0000	0.494804	0.247402	+	0.968913i	$0.420423\pi$
$920$	0	0
$921$	20.0000i	0.659022i
$922$	0	0
$923$	0	0
$924$	0	0
$925$	1.00000i	0.0328798i
$926$	0	0
$927$	−16.0000	−0.525509
$928$	0	0
$929$	− 5.00000i	− 0.164045i	−0.996630	−	0.0820223i	$-0.973862\pi$
$929$	− 5.00000i	− 0.164045i	0.996630	−	0.0820223i	$-0.0261379\pi$
$930$	0	0
$931$	42.0000i	1.37649i
$932$	0	0
$933$	−8.00000	−0.261908
$934$	0	0
$935$	−6.00000	−0.196221
$936$	0	0
$937$	−2.00000	−0.0653372	−0.0326686	−	0.999466i	$-0.510401\pi$
$937$	−2.00000	−0.0653372	−0.0326686	+	0.999466i	$0.510401\pi$
$938$	0	0
$939$	30.0000	0.979013
$940$	0	0
$941$	14.0000i	0.456387i	0.973616	+	0.228193i	$0.0732819\pi$
$941$	14.0000i	0.456387i	−0.973616	+	0.228193i	$0.926718\pi$
$942$	0	0
$943$	− 11.0000i	− 0.358209i
$944$	0	0
$945$	10.0000	0.325300
$946$	0	0
$947$	39.0000i	1.26733i	0.773608	+	0.633665i	$0.218450\pi$
$947$	39.0000i	1.26733i	−0.773608	+	0.633665i	$0.781550\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	− 2.00000i	− 0.0648544i
$952$	0	0
$953$	9.00000	0.291539	0.145769	−	0.989319i	$-0.453434\pi$
$953$	9.00000	0.291539	0.145769	+	0.989319i	$0.453434\pi$
$954$	0	0
$955$	14.0000i	0.453029i
$956$	0	0
$957$	− 3.00000i	− 0.0969762i
$958$	0	0
$959$	−9.00000	−0.290625
$960$	0	0
$961$	−33.0000	−1.06452
$962$	0	0
$963$	−34.0000	−1.09563
$964$	0	0
$965$	−10.0000	−0.321911
$966$	0	0
$967$	− 16.0000i	− 0.514525i	−0.966342	−	0.257263i	$-0.917179\pi$
$967$	− 16.0000i	− 0.514525i	0.966342	−	0.257263i	$-0.0828206\pi$
$968$	0	0
$969$	− 21.0000i	− 0.674617i
$970$	0	0
$971$	27.0000	0.866471	0.433236	−	0.901281i	$-0.357372\pi$
$971$	27.0000	0.866471	0.433236	+	0.901281i	$0.357372\pi$
$972$	0	0
$973$	− 9.00000i	− 0.288527i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 9.00000i	− 0.287936i	−0.989582	−	0.143968i	$-0.954014\pi$
$977$	− 9.00000i	− 0.287936i	0.989582	−	0.143968i	$-0.0459862\pi$
$978$	0	0
$979$	1.00000	0.0319601
$980$	0	0
$981$	− 20.0000i	− 0.638551i
$982$	0	0
$983$	− 32.0000i	− 1.02064i	−0.859984	−	0.510321i	$-0.829527\pi$
$983$	− 32.0000i	− 1.02064i	0.859984	−	0.510321i	$-0.170473\pi$
$984$	0	0
$985$	26.0000	0.828429
$986$	0	0
$987$	12.0000	0.381964
$988$	0	0
$989$	11.0000	0.349780
$990$	0	0
$991$	13.0000	0.412959	0.206479	−	0.978451i	$-0.433799\pi$
$991$	13.0000	0.412959	0.206479	+	0.978451i	$0.433799\pi$
$992$	0	0
$993$	27.0000i	0.856819i
$994$	0	0
$995$	− 6.00000i	− 0.190213i
$996$	0	0
$997$	−25.0000	−0.791758	−0.395879	−	0.918303i	$-0.629560\pi$
$997$	−25.0000	−0.791758	−0.395879	+	0.918303i	$0.629560\pi$
$998$	0	0
$999$	− 5.00000i	− 0.158193i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1352.2.f.a.337.2		2
4.3	odd	2		2704.2.f.c.337.2		2
13.2	odd	12		104.2.i.a.9.1	✓	2
13.3	even	3		1352.2.o.b.1161.2		4
13.4	even	6		1352.2.o.b.361.1		4
13.5	odd	4		1352.2.a.c.1.1		1
13.6	odd	12		104.2.i.a.81.1	yes	2
13.7	odd	12		1352.2.i.a.1329.1		2
13.8	odd	4		1352.2.a.a.1.1		1
13.9	even	3		1352.2.o.b.361.2		4
13.10	even	6		1352.2.o.b.1161.1		4
13.11	odd	12		1352.2.i.a.529.1		2
13.12	even	2	inner	1352.2.f.a.337.1		2
39.2	even	12		936.2.t.c.217.1		2
39.32	even	12		936.2.t.c.289.1		2
52.15	even	12		208.2.i.c.113.1		2
52.19	even	12		208.2.i.c.81.1		2
52.31	even	4		2704.2.a.e.1.1		1
52.47	even	4		2704.2.a.c.1.1		1
52.51	odd	2		2704.2.f.c.337.1		2
104.19	even	12		832.2.i.d.705.1		2
104.45	odd	12		832.2.i.g.705.1		2
104.67	even	12		832.2.i.d.321.1		2
104.93	odd	12		832.2.i.g.321.1		2
156.71	odd	12		1872.2.t.d.289.1		2
156.119	odd	12		1872.2.t.d.1153.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
104.2.i.a.9.1	✓	2	13.2	odd	12
104.2.i.a.81.1	yes	2	13.6	odd	12
208.2.i.c.81.1		2	52.19	even	12
208.2.i.c.113.1		2	52.15	even	12
832.2.i.d.321.1		2	104.67	even	12
832.2.i.d.705.1		2	104.19	even	12
832.2.i.g.321.1		2	104.93	odd	12
832.2.i.g.705.1		2	104.45	odd	12
936.2.t.c.217.1		2	39.2	even	12
936.2.t.c.289.1		2	39.32	even	12
1352.2.a.a.1.1		1	13.8	odd	4
1352.2.a.c.1.1		1	13.5	odd	4
1352.2.f.a.337.1		2	13.12	even	2	inner
1352.2.f.a.337.2		2	1.1	even	1	trivial
1352.2.i.a.529.1		2	13.11	odd	12
1352.2.i.a.1329.1		2	13.7	odd	12
1352.2.o.b.361.1		4	13.4	even	6
1352.2.o.b.361.2		4	13.9	even	3
1352.2.o.b.1161.1		4	13.10	even	6
1352.2.o.b.1161.2		4	13.3	even	3
1872.2.t.d.289.1		2	156.71	odd	12
1872.2.t.d.1153.1		2	156.119	odd	12
2704.2.a.c.1.1		1	52.47	even	4
2704.2.a.e.1.1		1	52.31	even	4
2704.2.f.c.337.1		2	52.51	odd	2
2704.2.f.c.337.2		2	4.3	odd	2

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:	\( N \)	\(=\)	\( 1352 = 2^{3} \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1352.f (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} +2.00000i q^{5} +1.00000i q^{7} -2.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} +2.00000i q^{5} +1.00000i q^{7} -2.00000 q^{9} -1.00000i q^{11} +2.00000i q^{15} -3.00000 q^{17} +7.00000i q^{19} +1.00000i q^{21} -1.00000 q^{23} +1.00000 q^{25} -5.00000 q^{27} +3.00000 q^{29} +8.00000i q^{31} -1.00000i q^{33} -2.00000 q^{35} +1.00000i q^{37} +11.0000i q^{41} -11.0000 q^{43} -4.00000i q^{45} -12.0000i q^{47} +6.00000 q^{49} -3.00000 q^{51} -6.00000 q^{53} +2.00000 q^{55} +7.00000i q^{57} +9.00000i q^{59} -9.00000 q^{61} -2.00000i q^{63} -3.00000i q^{67} -1.00000 q^{69} -5.00000i q^{71} +2.00000i q^{73} +1.00000 q^{75} +1.00000 q^{77} -12.0000 q^{79} +1.00000 q^{81} -4.00000i q^{83} -6.00000i q^{85} +3.00000 q^{87} +1.00000i q^{89} +8.00000i q^{93} -14.0000 q^{95} -1.00000i q^{97} +2.00000i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{3} - 4 q^{9}+O(q^{10}) \) 2 * q + 2 * q^3 - 4 * q^9	\( 2 q + 2 q^{3} - 4 q^{9} - 6 q^{17} - 2 q^{23} + 2 q^{25} - 10 q^{27} + 6 q^{29} - 4 q^{35} - 22 q^{43} + 12 q^{49} - 6 q^{51} - 12 q^{53} + 4 q^{55} - 18 q^{61} - 2 q^{69} + 2 q^{75} + 2 q^{77} - 24 q^{79} + 2 q^{81} + 6 q^{87} - 28 q^{95}+O(q^{100}) \) 2 * q + 2 * q^3 - 4 * q^9 - 6 * q^17 - 2 * q^23 + 2 * q^25 - 10 * q^27 + 6 * q^29 - 4 * q^35 - 22 * q^43 + 12 * q^49 - 6 * q^51 - 12 * q^53 + 4 * q^55 - 18 * q^61 - 2 * q^69 + 2 * q^75 + 2 * q^77 - 24 * q^79 + 2 * q^81 + 6 * q^87 - 28 * q^95

Introduction

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