LMFDB - Embedded newform 3312.2.a.bh.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3312,2,Mod(1,3312)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3312, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("3312.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3312 = 2^{4} \cdot 3^{2} \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3312.a (trivial)

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:	yes
Analytic conductor:	$26.4464531494$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.44688.2
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{3} - 5x^{2} + 6x + 2 $ x^4 - 2x^3 - 5x^2 + 6*x + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 1656)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-1.92812$ of defining polynomial
Character	$\chi$	$=$	3312.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.64575 q^{5} +4.81895 q^{7} +O(q^{10})$	$q+3.64575 q^{5} +4.81895 q^{7} -2.68303 q^{11} -3.85623 q^{13} +0.962718 q^{17} -4.21048 q^{19} +1.00000 q^{23} +8.29150 q^{25} +7.85623 q^{29} -7.85623 q^{31} +17.5687 q^{35} +11.0294 q^{37} +0.564731 q^{41} +5.57655 q^{43} -7.63790 q^{47} +16.2223 q^{49} +9.99215 q^{53} -9.78167 q^{55} +15.0732 q^{59} -0.683033 q^{61} -14.0589 q^{65} +10.7935 q^{67} +3.43527 q^{71} +0.0692033 q^{73} -12.9294 q^{77} -6.89352 q^{79} +1.31697 q^{83} +3.50983 q^{85} +18.0413 q^{89} -18.5830 q^{91} -15.3504 q^{95} -4.34640 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{5} - 2 q^{7}+O(q^{10}) $ 4 * q + 4 * q^5 - 2 * q^7	$ 4 q + 4 q^{5} - 2 q^{7} - 2 q^{11} + 4 q^{13} + 2 q^{17} - 8 q^{19} + 4 q^{23} + 12 q^{25} + 12 q^{29} - 12 q^{31} + 12 q^{35} + 14 q^{37} + 4 q^{41} - 4 q^{43} + 12 q^{47} + 28 q^{49} + 8 q^{53} - 16 q^{55} + 16 q^{59} + 6 q^{61} + 4 q^{65} - 8 q^{67} + 12 q^{71} + 16 q^{73} + 12 q^{77} - 10 q^{79} + 14 q^{83} + 16 q^{85} + 14 q^{89} - 32 q^{91} + 20 q^{95} + 4 q^{97}+O(q^{100}) $ 4 * q + 4 * q^5 - 2 * q^7 - 2 * q^11 + 4 * q^13 + 2 * q^17 - 8 * q^19 + 4 * q^23 + 12 * q^25 + 12 * q^29 - 12 * q^31 + 12 * q^35 + 14 * q^37 + 4 * q^41 - 4 * q^43 + 12 * q^47 + 28 * q^49 + 8 * q^53 - 16 * q^55 + 16 * q^59 + 6 * q^61 + 4 * q^65 - 8 * q^67 + 12 * q^71 + 16 * q^73 + 12 * q^77 - 10 * q^79 + 14 * q^83 + 16 * q^85 + 14 * q^89 - 32 * q^91 + 20 * q^95 + 4 * q^97

Display 100 coefficients 10 coefficients

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
$3$	0	0
$3$	0	0
$4$	0	0
$5$	3.64575	1.63043	0.815215	−	0.579159i	$-0.196619\pi$
$5$	3.64575	1.63043	0.815215	+	0.579159i	$0.196619\pi$
$6$	0	0
$7$	4.81895	1.82139	0.910696	−	0.413076i	$-0.135546\pi$
$7$	4.81895	1.82139	0.910696	+	0.413076i	$0.135546\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−2.68303	−0.808965	−0.404482	−	0.914546i	$-0.632548\pi$
$11$	−2.68303	−0.808965	−0.404482	+	0.914546i	$0.632548\pi$
$12$	0	0
$13$	−3.85623	−1.06953	−0.534763	−	0.845002i	$-0.679599\pi$
$13$	−3.85623	−1.06953	−0.534763	+	0.845002i	$0.679599\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	0.962718	0.233494	0.116747	−	0.993162i	$-0.462753\pi$
$17$	0.962718	0.233494	0.116747	+	0.993162i	$0.462753\pi$
$18$	0	0
$19$	−4.21048	−0.965951	−0.482975	−	0.875634i	$-0.660444\pi$
$19$	−4.21048	−0.965951	−0.482975	+	0.875634i	$0.660444\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	0	0
$25$	8.29150	1.65830
$26$	0	0
$27$	0	0
$28$	0	0
$29$	7.85623	1.45887	0.729433	−	0.684052i	$-0.239784\pi$
$29$	7.85623	1.45887	0.729433	+	0.684052i	$0.239784\pi$
$30$	0	0
$31$	−7.85623	−1.41102	−0.705511	−	0.708699i	$-0.749282\pi$
$31$	−7.85623	−1.41102	−0.705511	+	0.708699i	$0.749282\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	17.5687	2.96965
$36$	0	0
$37$	11.0294	1.81323	0.906614	−	0.421961i	$-0.138658\pi$
$37$	11.0294	1.81323	0.906614	+	0.421961i	$0.138658\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	0.564731	0.0881962	0.0440981	−	0.999027i	$-0.485959\pi$
$41$	0.564731	0.0881962	0.0440981	+	0.999027i	$0.485959\pi$
$42$	0	0
$43$	5.57655	0.850416	0.425208	−	0.905096i	$-0.360201\pi$
$43$	5.57655	0.850416	0.425208	+	0.905096i	$0.360201\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−7.63790	−1.11410	−0.557051	−	0.830478i	$-0.688067\pi$
$47$	−7.63790	−1.11410	−0.557051	+	0.830478i	$0.688067\pi$
$48$	0	0
$49$	16.2223	2.31747
$50$	0	0
$51$	0	0
$52$	0	0
$53$	9.99215	1.37253	0.686264	−	0.727353i	$-0.259250\pi$
$53$	9.99215	1.37253	0.686264	+	0.727353i	$0.259250\pi$
$54$	0	0
$55$	−9.78167	−1.31896
$56$	0	0
$57$	0	0
$58$	0	0
$59$	15.0732	1.96236	0.981180	−	0.193095i	$-0.0618527\pi$
$59$	15.0732	1.96236	0.981180	+	0.193095i	$0.0618527\pi$
$60$	0	0
$61$	−0.683033	−0.0874534	−0.0437267	−	0.999044i	$-0.513923\pi$
$61$	−0.683033	−0.0874534	−0.0437267	+	0.999044i	$0.513923\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−14.0589	−1.74379
$66$	0	0
$67$	10.7935	1.31863	0.659317	−	0.751865i	$-0.270845\pi$
$67$	10.7935	1.31863	0.659317	+	0.751865i	$0.270845\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	3.43527	0.407691	0.203846	−	0.979003i	$-0.434656\pi$
$71$	3.43527	0.407691	0.203846	+	0.979003i	$0.434656\pi$
$72$	0	0
$73$	0.0692033	0.00809963	0.00404981	−	0.999992i	$-0.498711\pi$
$73$	0.0692033	0.00809963	0.00404981	+	0.999992i	$0.498711\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−12.9294	−1.47344
$78$	0	0
$79$	−6.89352	−0.775581	−0.387791	−	0.921748i	$-0.626762\pi$
$79$	−6.89352	−0.775581	−0.387791	+	0.921748i	$0.626762\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	1.31697	0.144556	0.0722780	−	0.997385i	$-0.476973\pi$
$83$	1.31697	0.144556	0.0722780	+	0.997385i	$0.476973\pi$
$84$	0	0
$85$	3.50983	0.380695
$86$	0	0
$87$	0	0
$88$	0	0
$89$	18.0413	1.91237	0.956184	−	0.292765i	$-0.0945754\pi$
$89$	18.0413	1.91237	0.956184	+	0.292765i	$0.0945754\pi$
$90$	0	0
$91$	−18.5830	−1.94803
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−15.3504	−1.57491
$96$	0	0
$97$	−4.34640	−0.441310	−0.220655	−	0.975352i	$-0.570820\pi$
$97$	−4.34640	−0.441310	−0.220655	+	0.975352i	$0.570820\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	2.21833	0.220732	0.110366	−	0.993891i	$-0.464798\pi$
$101$	2.21833	0.220732	0.110366	+	0.993891i	$0.464798\pi$
$102$	0	0
$103$	−4.54711	−0.448040	−0.224020	−	0.974584i	$-0.571918\pi$
$103$	−4.54711	−0.448040	−0.224020	+	0.974584i	$0.571918\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	0.336631	0.0325434	0.0162717	−	0.999868i	$-0.494820\pi$
$107$	0.336631	0.0325434	0.0162717	+	0.999868i	$0.494820\pi$
$108$	0	0
$109$	−11.2660	−1.07909	−0.539545	−	0.841957i	$-0.681404\pi$
$109$	−11.2660	−1.07909	−0.539545	+	0.841957i	$0.681404\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−19.9667	−1.87831	−0.939154	−	0.343496i	$-0.888389\pi$
$113$	−19.9667	−1.87831	−0.939154	+	0.343496i	$0.888389\pi$
$114$	0	0
$115$	3.64575	0.339968
$116$	0	0
$117$	0	0
$118$	0	0
$119$	4.63929	0.425283
$120$	0	0
$121$	−3.80133	−0.345576
$122$	0	0
$123$	0	0
$124$	0	0
$125$	12.0000	1.07331
$126$	0	0
$127$	−4.94510	−0.438807	−0.219403	−	0.975634i	$-0.570411\pi$
$127$	−4.94510	−0.438807	−0.219403	+	0.975634i	$0.570411\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	10.3464	0.903969	0.451985	−	0.892026i	$-0.350716\pi$
$131$	10.3464	0.903969	0.451985	+	0.892026i	$0.350716\pi$
$132$	0	0
$133$	−20.2901	−1.75938
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−4.82431	−0.412169	−0.206084	−	0.978534i	$-0.566072\pi$
$137$	−4.82431	−0.412169	−0.206084	+	0.978534i	$0.566072\pi$
$138$	0	0
$139$	−4.00000	−0.339276	−0.169638	−	0.985506i	$-0.554260\pi$
$139$	−4.00000	−0.339276	−0.169638	+	0.985506i	$0.554260\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	10.3464	0.865210
$144$	0	0
$145$	28.6419	2.37858
$146$	0	0
$147$	0	0
$148$	0	0
$149$	4.20512	0.344497	0.172249	−	0.985054i	$-0.444897\pi$
$149$	4.20512	0.344497	0.172249	+	0.985054i	$0.444897\pi$
$150$	0	0
$151$	−2.91113	−0.236905	−0.118452	−	0.992960i	$-0.537793\pi$
$151$	−2.91113	−0.236905	−0.118452	+	0.992960i	$0.537793\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−28.6419	−2.30057
$156$	0	0
$157$	−21.6124	−1.72486	−0.862430	−	0.506176i	$-0.831059\pi$
$157$	−21.6124	−1.72486	−0.862430	+	0.506176i	$0.831059\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	4.81895	0.379787
$162$	0	0
$163$	−22.0236	−1.72502	−0.862512	−	0.506036i	$-0.831110\pi$
$163$	−22.0236	−1.72502	−0.862512	+	0.506036i	$0.831110\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−0.415605	−0.0321605	−0.0160802	−	0.999871i	$-0.505119\pi$
$167$	−0.415605	−0.0321605	−0.0160802	+	0.999871i	$0.505119\pi$
$168$	0	0
$169$	1.87054	0.143888
$170$	0	0
$171$	0	0
$172$	0	0
$173$	18.8759	1.43511	0.717554	−	0.696503i	$-0.245262\pi$
$173$	18.8759	1.43511	0.717554	+	0.696503i	$0.245262\pi$
$174$	0	0
$175$	39.9564	3.02042
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−8.87590	−0.663416	−0.331708	−	0.943382i	$-0.607625\pi$
$179$	−8.87590	−0.663416	−0.331708	+	0.943382i	$0.607625\pi$
$180$	0	0
$181$	1.66337	0.123637	0.0618186	−	0.998087i	$-0.480310\pi$
$181$	1.66337	0.123637	0.0618186	+	0.998087i	$0.480310\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	40.2106	2.95634
$186$	0	0
$187$	−2.58301	−0.188888
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−4.92941	−0.356679	−0.178340	−	0.983969i	$-0.557073\pi$
$191$	−4.92941	−0.356679	−0.178340	+	0.983969i	$0.557073\pi$
$192$	0	0
$193$	−6.35176	−0.457210	−0.228605	−	0.973519i	$-0.573416\pi$
$193$	−6.35176	−0.457210	−0.228605	+	0.973519i	$0.573416\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−6.20264	−0.441919	−0.220960	−	0.975283i	$-0.570919\pi$
$197$	−6.20264	−0.441919	−0.220960	+	0.975283i	$0.570919\pi$
$198$	0	0
$199$	−8.66983	−0.614588	−0.307294	−	0.951615i	$-0.599423\pi$
$199$	−8.66983	−0.614588	−0.307294	+	0.951615i	$0.599423\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	37.8588	2.65717
$204$	0	0
$205$	2.05887	0.143798
$206$	0	0
$207$	0	0
$208$	0	0
$209$	11.2969	0.781420
$210$	0	0
$211$	−11.9843	−0.825034	−0.412517	−	0.910950i	$-0.635350\pi$
$211$	−11.9843	−0.825034	−0.412517	+	0.910950i	$0.635350\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	20.3307	1.38654
$216$	0	0
$217$	−37.8588	−2.57002
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−3.71247	−0.249728
$222$	0	0
$223$	−24.0771	−1.61232	−0.806162	−	0.591694i	$-0.798459\pi$
$223$	−24.0771	−1.61232	−0.806162	+	0.591694i	$0.798459\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	14.9196	0.990251	0.495126	−	0.868821i	$-0.335122\pi$
$227$	14.9196	0.990251	0.495126	+	0.868821i	$0.335122\pi$
$228$	0	0
$229$	15.2660	1.00881	0.504404	−	0.863468i	$-0.331712\pi$
$229$	15.2660	1.00881	0.504404	+	0.863468i	$0.331712\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	11.2758	0.738703	0.369351	−	0.929290i	$-0.379580\pi$
$233$	11.2758	0.738703	0.369351	+	0.929290i	$0.379580\pi$
$234$	0	0
$235$	−27.8459	−1.81647
$236$	0	0
$237$	0	0
$238$	0	0
$239$	19.2758	1.24685	0.623424	−	0.781884i	$-0.285741\pi$
$239$	19.2758	1.24685	0.623424	+	0.781884i	$0.285741\pi$
$240$	0	0
$241$	18.6419	1.20083	0.600414	−	0.799689i	$-0.295002\pi$
$241$	18.6419	1.20083	0.600414	+	0.799689i	$0.295002\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	59.1425	3.77847
$246$	0	0
$247$	16.2366	1.03311
$248$	0	0
$249$	0	0
$250$	0	0
$251$	2.44643	0.154417	0.0772086	−	0.997015i	$-0.475399\pi$
$251$	2.44643	0.154417	0.0772086	+	0.997015i	$0.475399\pi$
$252$	0	0
$253$	−2.68303	−0.168681
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−26.9986	−1.68413	−0.842064	−	0.539378i	$-0.818659\pi$
$257$	−26.9986	−1.68413	−0.842064	+	0.539378i	$0.818659\pi$
$258$	0	0
$259$	53.1503	3.30260
$260$	0	0
$261$	0	0
$262$	0	0
$263$	3.71247	0.228921	0.114460	−	0.993428i	$-0.463486\pi$
$263$	3.71247	0.228921	0.114460	+	0.993428i	$0.463486\pi$
$264$	0	0
$265$	36.4289	2.23781
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−23.8053	−1.45144	−0.725718	−	0.687993i	$-0.758492\pi$
$269$	−23.8053	−1.45144	−0.725718	+	0.687993i	$0.758492\pi$
$270$	0	0
$271$	−9.63790	−0.585461	−0.292730	−	0.956195i	$-0.594564\pi$
$271$	−9.63790	−0.585461	−0.292730	+	0.956195i	$0.594564\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−22.2464	−1.34151
$276$	0	0
$277$	−8.63393	−0.518763	−0.259381	−	0.965775i	$-0.583519\pi$
$277$	−8.63393	−0.518763	−0.259381	+	0.965775i	$0.583519\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−20.9118	−1.24749	−0.623746	−	0.781627i	$-0.714390\pi$
$281$	−20.9118	−1.24749	−0.623746	+	0.781627i	$0.714390\pi$
$282$	0	0
$283$	23.4863	1.39612	0.698058	−	0.716042i	$-0.254048\pi$
$283$	23.4863	1.39612	0.698058	+	0.716042i	$0.254048\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	2.72141	0.160640
$288$	0	0
$289$	−16.0732	−0.945481
$290$	0	0
$291$	0	0
$292$	0	0
$293$	17.0471	0.995899	0.497950	−	0.867206i	$-0.334086\pi$
$293$	17.0471	0.995899	0.497950	+	0.867206i	$0.334086\pi$
$294$	0	0
$295$	54.9530	3.19949
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−3.85623	−0.223012
$300$	0	0
$301$	26.8731	1.54894
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−2.49017	−0.142587
$306$	0	0
$307$	20.1334	1.14908	0.574538	−	0.818478i	$-0.305182\pi$
$307$	20.1334	1.14908	0.574538	+	0.818478i	$0.305182\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	8.96338	0.508266	0.254133	−	0.967169i	$-0.418210\pi$
$311$	8.96338	0.508266	0.254133	+	0.967169i	$0.418210\pi$
$312$	0	0
$313$	19.6222	1.10911	0.554556	−	0.832146i	$-0.312888\pi$
$313$	19.6222	1.10911	0.554556	+	0.832146i	$0.312888\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−21.8642	−1.22801	−0.614007	−	0.789300i	$-0.710443\pi$
$317$	−21.8642	−1.22801	−0.614007	+	0.789300i	$0.710443\pi$
$318$	0	0
$319$	−21.0785	−1.18017
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−4.05351	−0.225543
$324$	0	0
$325$	−31.9740	−1.77360
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−36.8067	−2.02922
$330$	0	0
$331$	−5.87451	−0.322892	−0.161446	−	0.986882i	$-0.551616\pi$
$331$	−5.87451	−0.322892	−0.161446	+	0.986882i	$0.551616\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	39.3504	2.14994
$336$	0	0
$337$	−26.2561	−1.43026	−0.715132	−	0.698990i	$-0.753633\pi$
$337$	−26.2561	−1.43026	−0.715132	+	0.698990i	$0.753633\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	21.0785	1.14147
$342$	0	0
$343$	44.4418	2.39963
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−16.5776	−0.889935	−0.444967	−	0.895547i	$-0.646785\pi$
$347$	−16.5776	−0.889935	−0.444967	+	0.895547i	$0.646785\pi$
$348$	0	0
$349$	−14.0535	−0.752267	−0.376134	−	0.926565i	$-0.622747\pi$
$349$	−14.0535	−0.752267	−0.376134	+	0.926565i	$0.622747\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−25.3504	−1.34926	−0.674632	−	0.738154i	$-0.735698\pi$
$353$	−25.3504	−1.34926	−0.674632	+	0.738154i	$0.735698\pi$
$354$	0	0
$355$	12.5241	0.664712
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−2.78306	−0.146884	−0.0734422	−	0.997299i	$-0.523398\pi$
$359$	−2.78306	−0.146884	−0.0734422	+	0.997299i	$0.523398\pi$
$360$	0	0
$361$	−1.27184	−0.0669389
$362$	0	0
$363$	0	0
$364$	0	0
$365$	0.252298	0.0132059
$366$	0	0
$367$	−5.47652	−0.285872	−0.142936	−	0.989732i	$-0.545654\pi$
$367$	−5.47652	−0.285872	−0.142936	+	0.989732i	$0.545654\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	48.1517	2.49991
$372$	0	0
$373$	−11.1786	−0.578804	−0.289402	−	0.957208i	$-0.593456\pi$
$373$	−11.1786	−0.578804	−0.289402	+	0.957208i	$0.593456\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−30.2955	−1.56030
$378$	0	0
$379$	15.6354	0.803137	0.401569	−	0.915829i	$-0.368465\pi$
$379$	15.6354	0.803137	0.401569	+	0.915829i	$0.368465\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−15.7125	−0.802870	−0.401435	−	0.915888i	$-0.631488\pi$
$383$	−15.7125	−0.802870	−0.401435	+	0.915888i	$0.631488\pi$
$384$	0	0
$385$	−47.1374	−2.40234
$386$	0	0
$387$	0	0
$388$	0	0
$389$	0.215842	0.0109436	0.00547182	−	0.999985i	$-0.498258\pi$
$389$	0.215842	0.0109436	0.00547182	+	0.999985i	$0.498258\pi$
$390$	0	0
$391$	0.962718	0.0486868
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−25.1320	−1.26453
$396$	0	0
$397$	−0.870538	−0.0436910	−0.0218455	−	0.999761i	$-0.506954\pi$
$397$	−0.870538	−0.0436910	−0.0218455	+	0.999761i	$0.506954\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	0.962718	0.0480759	0.0240379	−	0.999711i	$-0.492348\pi$
$401$	0.962718	0.0480759	0.0240379	+	0.999711i	$0.492348\pi$
$402$	0	0
$403$	30.2955	1.50912
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−29.5923	−1.46684
$408$	0	0
$409$	−16.2420	−0.803113	−0.401557	−	0.915834i	$-0.631531\pi$
$409$	−16.2420	−0.803113	−0.401557	+	0.915834i	$0.631531\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	72.6369	3.57423
$414$	0	0
$415$	4.80133	0.235688
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−15.5250	−0.758444	−0.379222	−	0.925306i	$-0.623808\pi$
$419$	−15.5250	−0.758444	−0.379222	+	0.925306i	$0.623808\pi$
$420$	0	0
$421$	0.246374	0.0120075	0.00600377	−	0.999982i	$-0.498089\pi$
$421$	0.246374	0.0120075	0.00600377	+	0.999982i	$0.498089\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	7.98238	0.387202
$426$	0	0
$427$	−3.29150	−0.159287
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−5.50447	−0.265141	−0.132571	−	0.991174i	$-0.542323\pi$
$431$	−5.50447	−0.265141	−0.132571	+	0.991174i	$0.542323\pi$
$432$	0	0
$433$	9.62499	0.462547	0.231274	−	0.972889i	$-0.425711\pi$
$433$	9.62499	0.462547	0.231274	+	0.972889i	$0.425711\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−4.21048	−0.201415
$438$	0	0
$439$	12.6928	0.605794	0.302897	−	0.953023i	$-0.402046\pi$
$439$	12.6928	0.605794	0.302897	+	0.953023i	$0.402046\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−30.2955	−1.43938	−0.719691	−	0.694295i	$-0.755716\pi$
$443$	−30.2955	−1.43938	−0.719691	+	0.694295i	$0.755716\pi$
$444$	0	0
$445$	65.7739	3.11798
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−17.4067	−0.821471	−0.410736	−	0.911755i	$-0.634728\pi$
$449$	−17.4067	−0.821471	−0.410736	+	0.911755i	$0.634728\pi$
$450$	0	0
$451$	−1.51519	−0.0713476
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−67.7490	−3.17612
$456$	0	0
$457$	−32.3464	−1.51310	−0.756550	−	0.653935i	$-0.773117\pi$
$457$	−32.3464	−1.51310	−0.756550	+	0.653935i	$0.773117\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−39.2302	−1.82713	−0.913567	−	0.406688i	$-0.866684\pi$
$461$	−39.2302	−1.82713	−0.913567	+	0.406688i	$0.866684\pi$
$462$	0	0
$463$	−34.7007	−1.61268	−0.806340	−	0.591452i	$-0.798555\pi$
$463$	−34.7007	−1.61268	−0.806340	+	0.591452i	$0.798555\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−21.9079	−1.01378	−0.506889	−	0.862012i	$-0.669204\pi$
$467$	−21.9079	−1.01378	−0.506889	+	0.862012i	$0.669204\pi$
$468$	0	0
$469$	52.0133	2.40175
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−14.9621	−0.687956
$474$	0	0
$475$	−34.9112	−1.60184
$476$	0	0
$477$	0	0
$478$	0	0
$479$	42.0079	1.91939	0.959696	−	0.281041i	$-0.0906796\pi$
$479$	42.0079	1.91939	0.959696	+	0.281041i	$0.0906796\pi$
$480$	0	0
$481$	−42.5321	−1.93930
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−15.8459	−0.719525
$486$	0	0
$487$	−1.89020	−0.0856532	−0.0428266	−	0.999083i	$-0.513636\pi$
$487$	−1.89020	−0.0856532	−0.0428266	+	0.999083i	$0.513636\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−17.8562	−0.805841	−0.402920	−	0.915235i	$-0.632005\pi$
$491$	−17.8562	−0.805841	−0.402920	+	0.915235i	$0.632005\pi$
$492$	0	0
$493$	7.56334	0.340636
$494$	0	0
$495$	0	0
$496$	0	0
$497$	16.5544	0.742566
$498$	0	0
$499$	−12.1227	−0.542687	−0.271344	−	0.962483i	$-0.587468\pi$
$499$	−12.1227	−0.542687	−0.271344	+	0.962483i	$0.587468\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	31.3740	1.39890	0.699449	−	0.714683i	$-0.253429\pi$
$503$	31.3740	1.39890	0.699449	+	0.714683i	$0.253429\pi$
$504$	0	0
$505$	8.08748	0.359888
$506$	0	0
$507$	0	0
$508$	0	0
$509$	13.0222	0.577201	0.288600	−	0.957450i	$-0.406810\pi$
$509$	13.0222	0.577201	0.288600	+	0.957450i	$0.406810\pi$
$510$	0	0
$511$	0.333487	0.0147526
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−16.5776	−0.730498
$516$	0	0
$517$	20.4927	0.901270
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−1.14708	−0.0502545	−0.0251272	−	0.999684i	$-0.507999\pi$
$521$	−1.14708	−0.0502545	−0.0251272	+	0.999684i	$0.507999\pi$
$522$	0	0
$523$	−36.4314	−1.59303	−0.796517	−	0.604616i	$-0.793326\pi$
$523$	−36.4314	−1.59303	−0.796517	+	0.604616i	$0.793326\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−7.56334	−0.329464
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−2.17773	−0.0943282
$534$	0	0
$535$	1.22727	0.0530597
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−43.5250	−1.87475
$540$	0	0
$541$	−22.0642	−0.948615	−0.474308	−	0.880359i	$-0.657302\pi$
$541$	−22.0642	−0.948615	−0.474308	+	0.880359i	$0.657302\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−41.0732	−1.75938
$546$	0	0
$547$	32.8942	1.40645	0.703227	−	0.710966i	$-0.251742\pi$
$547$	32.8942	1.40645	0.703227	+	0.710966i	$0.251742\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−33.0785	−1.40919
$552$	0	0
$553$	−33.2195	−1.41264
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−13.7046	−0.580683	−0.290342	−	0.956923i	$-0.593769\pi$
$557$	−13.7046	−0.580683	−0.290342	+	0.956923i	$0.593769\pi$
$558$	0	0
$559$	−21.5045	−0.909542
$560$	0	0
$561$	0	0
$562$	0	0
$563$	3.52496	0.148559	0.0742797	−	0.997237i	$-0.476334\pi$
$563$	3.52496	0.148559	0.0742797	+	0.997237i	$0.476334\pi$
$564$	0	0
$565$	−72.7936	−3.06245
$566$	0	0
$567$	0	0
$568$	0	0
$569$	22.3131	0.935413	0.467707	−	0.883884i	$-0.345080\pi$
$569$	22.3131	0.935413	0.467707	+	0.883884i	$0.345080\pi$
$570$	0	0
$571$	−14.7935	−0.619088	−0.309544	−	0.950885i	$-0.600176\pi$
$571$	−14.7935	−0.619088	−0.309544	+	0.950885i	$0.600176\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	8.29150	0.345780
$576$	0	0
$577$	31.6968	1.31955	0.659777	−	0.751461i	$-0.270651\pi$
$577$	31.6968	1.31955	0.659777	+	0.751461i	$0.270651\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	6.34640	0.263293
$582$	0	0
$583$	−26.8093	−1.11033
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−22.2080	−0.916622	−0.458311	−	0.888792i	$-0.651545\pi$
$587$	−22.2080	−0.916622	−0.458311	+	0.888792i	$0.651545\pi$
$588$	0	0
$589$	33.0785	1.36298
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−16.7504	−0.687857	−0.343928	−	0.938996i	$-0.611758\pi$
$593$	−16.7504	−0.687857	−0.343928	+	0.938996i	$0.611758\pi$
$594$	0	0
$595$	16.9137	0.693395
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−38.5830	−1.57646	−0.788229	−	0.615381i	$-0.789002\pi$
$599$	−38.5830	−1.57646	−0.788229	+	0.615381i	$0.789002\pi$
$600$	0	0
$601$	18.5308	0.755886	0.377943	−	0.925829i	$-0.376632\pi$
$601$	18.5308	0.755886	0.377943	+	0.925829i	$0.376632\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−13.8587	−0.563437
$606$	0	0
$607$	−12.5138	−0.507920	−0.253960	−	0.967215i	$-0.581733\pi$
$607$	−12.5138	−0.507920	−0.253960	+	0.967215i	$0.581733\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	29.4535	1.19156
$612$	0	0
$613$	32.3955	1.30844	0.654221	−	0.756303i	$-0.272997\pi$
$613$	32.3955	1.30844	0.654221	+	0.756303i	$0.272997\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	16.3367	0.657692	0.328846	−	0.944384i	$-0.393340\pi$
$617$	16.3367	0.657692	0.328846	+	0.944384i	$0.393340\pi$
$618$	0	0
$619$	−18.1007	−0.727528	−0.363764	−	0.931491i	$-0.618509\pi$
$619$	−18.1007	−0.727528	−0.363764	+	0.931491i	$0.618509\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	86.9399	3.48317
$624$	0	0
$625$	2.29150	0.0916601
$626$	0	0
$627$	0	0
$628$	0	0
$629$	10.6182	0.423377
$630$	0	0
$631$	37.1381	1.47844	0.739221	−	0.673463i	$-0.235194\pi$
$631$	37.1381	1.47844	0.739221	+	0.673463i	$0.235194\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−18.0286	−0.715444
$636$	0	0
$637$	−62.5570	−2.47860
$638$	0	0
$639$	0	0
$640$	0	0
$641$	27.1091	1.07074	0.535372	−	0.844616i	$-0.320171\pi$
$641$	27.1091	1.07074	0.535372	+	0.844616i	$0.320171\pi$
$642$	0	0
$643$	−18.4235	−0.726550	−0.363275	−	0.931682i	$-0.618341\pi$
$643$	−18.4235	−0.726550	−0.363275	+	0.931682i	$0.618341\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	43.6276	1.71518	0.857588	−	0.514337i	$-0.171962\pi$
$647$	43.6276	1.71518	0.857588	+	0.514337i	$0.171962\pi$
$648$	0	0
$649$	−40.4418	−1.58748
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−2.76201	−0.108086	−0.0540428	−	0.998539i	$-0.517211\pi$
$653$	−2.76201	−0.108086	−0.0540428	+	0.998539i	$0.517211\pi$
$654$	0	0
$655$	37.7204	1.47386
$656$	0	0
$657$	0	0
$658$	0	0
$659$	42.4320	1.65292	0.826459	−	0.562997i	$-0.190352\pi$
$659$	42.4320	1.65292	0.826459	+	0.562997i	$0.190352\pi$
$660$	0	0
$661$	35.1276	1.36631	0.683153	−	0.730275i	$-0.260608\pi$
$661$	35.1276	1.36631	0.683153	+	0.730275i	$0.260608\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−73.9727	−2.86854
$666$	0	0
$667$	7.85623	0.304195
$668$	0	0
$669$	0	0
$670$	0	0
$671$	1.83260	0.0707467
$672$	0	0
$673$	19.9843	0.770338	0.385169	−	0.922846i	$-0.374143\pi$
$673$	19.9843	0.770338	0.385169	+	0.922846i	$0.374143\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	35.0864	1.34848	0.674240	−	0.738513i	$-0.264472\pi$
$677$	35.0864	1.34848	0.674240	+	0.738513i	$0.264472\pi$
$678$	0	0
$679$	−20.9451	−0.803799
$680$	0	0
$681$	0	0
$682$	0	0
$683$	9.51519	0.364089	0.182044	−	0.983290i	$-0.441729\pi$
$683$	9.51519	0.364089	0.182044	+	0.983290i	$0.441729\pi$
$684$	0	0
$685$	−17.5882	−0.672012
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−38.5321	−1.46796
$690$	0	0
$691$	−3.14238	−0.119542	−0.0597708	−	0.998212i	$-0.519037\pi$
$691$	−3.14238	−0.119542	−0.0597708	+	0.998212i	$0.519037\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−14.5830	−0.553165
$696$	0	0
$697$	0.543677	0.0205932
$698$	0	0
$699$	0	0
$700$	0	0
$701$	33.5555	1.26737	0.633687	−	0.773590i	$-0.281541\pi$
$701$	33.5555	1.26737	0.633687	+	0.773590i	$0.281541\pi$
$702$	0	0
$703$	−46.4392	−1.75149
$704$	0	0
$705$	0	0
$706$	0	0
$707$	10.6900	0.402040
$708$	0	0
$709$	5.95090	0.223491	0.111745	−	0.993737i	$-0.464356\pi$
$709$	5.95090	0.223491	0.111745	+	0.993737i	$0.464356\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−7.85623	−0.294218
$714$	0	0
$715$	37.7204	1.41066
$716$	0	0
$717$	0	0
$718$	0	0
$719$	14.9348	0.556973	0.278486	−	0.960440i	$-0.410167\pi$
$719$	14.9348	0.556973	0.278486	+	0.960440i	$0.410167\pi$
$720$	0	0
$721$	−21.9123	−0.816058
$722$	0	0
$723$	0	0
$724$	0	0
$725$	65.1400	2.41924
$726$	0	0
$727$	29.1783	1.08216	0.541081	−	0.840970i	$-0.318015\pi$
$727$	29.1783	1.08216	0.541081	+	0.840970i	$0.318015\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	5.36865	0.198567
$732$	0	0
$733$	9.21789	0.340470	0.170235	−	0.985403i	$-0.445547\pi$
$733$	9.21789	0.340470	0.170235	+	0.985403i	$0.445547\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−28.9593	−1.06673
$738$	0	0
$739$	−42.6900	−1.57038	−0.785189	−	0.619256i	$-0.787434\pi$
$739$	−42.6900	−1.57038	−0.785189	+	0.619256i	$0.787434\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	22.2848	0.817548	0.408774	−	0.912636i	$-0.365956\pi$
$743$	22.2848	0.817548	0.408774	+	0.912636i	$0.365956\pi$
$744$	0	0
$745$	15.3308	0.561678
$746$	0	0
$747$	0	0
$748$	0	0
$749$	1.62221	0.0592743
$750$	0	0
$751$	5.56268	0.202985	0.101493	−	0.994836i	$-0.467638\pi$
$751$	5.56268	0.202985	0.101493	+	0.994836i	$0.467638\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−10.6133	−0.386256
$756$	0	0
$757$	18.3052	0.665315	0.332658	−	0.943048i	$-0.392055\pi$
$757$	18.3052	0.665315	0.332658	+	0.943048i	$0.392055\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	20.9137	0.758121	0.379061	−	0.925372i	$-0.376247\pi$
$761$	20.9137	0.758121	0.379061	+	0.925372i	$0.376247\pi$
$762$	0	0
$763$	−54.2905	−1.96545
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−58.1257	−2.09880
$768$	0	0
$769$	−1.51241	−0.0545390	−0.0272695	−	0.999628i	$-0.508681\pi$
$769$	−1.51241	−0.0545390	−0.0272695	+	0.999628i	$0.508681\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−37.3662	−1.34397	−0.671984	−	0.740566i	$-0.734557\pi$
$773$	−37.3662	−1.34397	−0.671984	+	0.740566i	$0.734557\pi$
$774$	0	0
$775$	−65.1400	−2.33990
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−2.37779	−0.0851932
$780$	0	0
$781$	−9.21694	−0.329808
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−78.7936	−2.81226
$786$	0	0
$787$	25.5256	0.909890	0.454945	−	0.890520i	$-0.349659\pi$
$787$	25.5256	0.909890	0.454945	+	0.890520i	$0.349659\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−96.2185	−3.42114
$792$	0	0
$793$	2.63393	0.0935337
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−29.7764	−1.05473	−0.527367	−	0.849637i	$-0.676821\pi$
$797$	−29.7764	−1.05473	−0.527367	+	0.849637i	$0.676821\pi$
$798$	0	0
$799$	−7.35315	−0.260136
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−0.185675	−0.00655232
$804$	0	0
$805$	17.5687	0.619215
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−49.6379	−1.74518	−0.872588	−	0.488457i	$-0.837560\pi$
$809$	−49.6379	−1.74518	−0.872588	+	0.488457i	$0.837560\pi$
$810$	0	0
$811$	45.8538	1.61015	0.805073	−	0.593176i	$-0.202126\pi$
$811$	45.8538	1.61015	0.805073	+	0.593176i	$0.202126\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−80.2927	−2.81253
$816$	0	0
$817$	−23.4800	−0.821460
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−32.5138	−1.13474	−0.567370	−	0.823463i	$-0.692039\pi$
$821$	−32.5138	−1.13474	−0.567370	+	0.823463i	$0.692039\pi$
$822$	0	0
$823$	−5.41163	−0.188638	−0.0943189	−	0.995542i	$-0.530067\pi$
$823$	−5.41163	−0.188638	−0.0943189	+	0.995542i	$0.530067\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−12.2384	−0.425572	−0.212786	−	0.977099i	$-0.568254\pi$
$827$	−12.2384	−0.425572	−0.212786	+	0.977099i	$0.568254\pi$
$828$	0	0
$829$	41.6169	1.44541	0.722706	−	0.691155i	$-0.242898\pi$
$829$	41.6169	1.44541	0.722706	+	0.691155i	$0.242898\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	15.6175	0.541115
$834$	0	0
$835$	−1.51519	−0.0524354
$836$	0	0
$837$	0	0
$838$	0	0
$839$	36.4525	1.25848	0.629241	−	0.777210i	$-0.283366\pi$
$839$	36.4525	1.25848	0.629241	+	0.777210i	$0.283366\pi$
$840$	0	0
$841$	32.7204	1.12829
$842$	0	0
$843$	0	0
$844$	0	0
$845$	6.81952	0.234598
$846$	0	0
$847$	−18.3184	−0.629429
$848$	0	0
$849$	0	0
$850$	0	0
$851$	11.0294	0.378084
$852$	0	0
$853$	26.2561	0.898993	0.449497	−	0.893282i	$-0.351603\pi$
$853$	26.2561	0.898993	0.449497	+	0.893282i	$0.351603\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−18.1463	−0.619867	−0.309934	−	0.950758i	$-0.600307\pi$
$857$	−18.1463	−0.619867	−0.309934	+	0.950758i	$0.600307\pi$
$858$	0	0
$859$	31.1374	1.06239	0.531197	−	0.847248i	$-0.321742\pi$
$859$	31.1374	1.06239	0.531197	+	0.847248i	$0.321742\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	18.9240	0.644182	0.322091	−	0.946709i	$-0.395614\pi$
$863$	18.9240	0.644182	0.322091	+	0.946709i	$0.395614\pi$
$864$	0	0
$865$	68.8168	2.33984
$866$	0	0
$867$	0	0
$868$	0	0
$869$	18.4955	0.627418
$870$	0	0
$871$	−41.6222	−1.41031
$872$	0	0
$873$	0	0
$874$	0	0
$875$	57.8274	1.95492
$876$	0	0
$877$	−35.0471	−1.18346	−0.591729	−	0.806137i	$-0.701554\pi$
$877$	−35.0471	−1.18346	−0.591729	+	0.806137i	$0.701554\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−39.1600	−1.31933	−0.659667	−	0.751558i	$-0.729303\pi$
$881$	−39.1600	−1.31933	−0.659667	+	0.751558i	$0.729303\pi$
$882$	0	0
$883$	26.8548	0.903737	0.451869	−	0.892084i	$-0.350758\pi$
$883$	26.8548	0.903737	0.451869	+	0.892084i	$0.350758\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	18.1139	0.608205	0.304102	−	0.952639i	$-0.401643\pi$
$887$	18.1139	0.608205	0.304102	+	0.952639i	$0.401643\pi$
$888$	0	0
$889$	−23.8302	−0.799239
$890$	0	0
$891$	0	0
$892$	0	0
$893$	32.1593	1.07617
$894$	0	0
$895$	−32.3593	−1.08165
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−61.7204	−2.05849
$900$	0	0
$901$	9.61963	0.320476
$902$	0	0
$903$	0	0
$904$	0	0
$905$	6.06423	0.201582
$906$	0	0
$907$	−8.95553	−0.297363	−0.148682	−	0.988885i	$-0.547503\pi$
$907$	−8.95553	−0.297363	−0.148682	+	0.988885i	$0.547503\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−6.89008	−0.228278	−0.114139	−	0.993465i	$-0.536411\pi$
$911$	−6.89008	−0.228278	−0.114139	+	0.993465i	$0.536411\pi$
$912$	0	0
$913$	−3.53347	−0.116941
$914$	0	0
$915$	0	0
$916$	0	0
$917$	49.8588	1.64648
$918$	0	0
$919$	−15.0779	−0.497373	−0.248687	−	0.968584i	$-0.579999\pi$
$919$	−15.0779	−0.497373	−0.248687	+	0.968584i	$0.579999\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−13.2472	−0.436037
$924$	0	0
$925$	91.4506	3.00688
$926$	0	0
$927$	0	0
$928$	0	0
$929$	28.6717	0.940690	0.470345	−	0.882483i	$-0.344130\pi$
$929$	28.6717	0.940690	0.470345	+	0.882483i	$0.344130\pi$
$930$	0	0
$931$	−68.3037	−2.23856
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−9.41699	−0.307969
$936$	0	0
$937$	32.0079	1.04565	0.522827	−	0.852439i	$-0.324877\pi$
$937$	32.0079	1.04565	0.522827	+	0.852439i	$0.324877\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−13.8587	−0.451781	−0.225891	−	0.974153i	$-0.572529\pi$
$941$	−13.8587	−0.451781	−0.225891	+	0.974153i	$0.572529\pi$
$942$	0	0
$943$	0.564731	0.0183902
$944$	0	0
$945$	0	0
$946$	0	0
$947$	40.8080	1.32608	0.663040	−	0.748584i	$-0.269266\pi$
$947$	40.8080	1.32608	0.663040	+	0.748584i	$0.269266\pi$
$948$	0	0
$949$	−0.266864	−0.00866277
$950$	0	0
$951$	0	0
$952$	0	0
$953$	4.74181	0.153602	0.0768011	−	0.997046i	$-0.475529\pi$
$953$	4.74181	0.153602	0.0768011	+	0.997046i	$0.475529\pi$
$954$	0	0
$955$	−17.9714	−0.581541
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−23.2481	−0.750721
$960$	0	0
$961$	30.7204	0.990981
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−23.1569	−0.745448
$966$	0	0
$967$	−30.1517	−0.969614	−0.484807	−	0.874621i	$-0.661110\pi$
$967$	−30.1517	−0.969614	−0.484807	+	0.874621i	$0.661110\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	32.6035	1.04630	0.523148	−	0.852242i	$-0.324758\pi$
$971$	32.6035	1.04630	0.523148	+	0.852242i	$0.324758\pi$
$972$	0	0
$973$	−19.2758	−0.617954
$974$	0	0
$975$	0	0
$976$	0	0
$977$	56.4349	1.80551	0.902757	−	0.430152i	$-0.141540\pi$
$977$	56.4349	1.80551	0.902757	+	0.430152i	$0.141540\pi$
$978$	0	0
$979$	−48.4053	−1.54704
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−53.4838	−1.70587	−0.852934	−	0.522018i	$-0.825179\pi$
$983$	−53.4838	−1.70587	−0.852934	+	0.522018i	$0.825179\pi$
$984$	0	0
$985$	−22.6133	−0.720519
$986$	0	0
$987$	0	0
$988$	0	0
$989$	5.57655	0.177324
$990$	0	0
$991$	24.8838	0.790461	0.395231	−	0.918582i	$-0.370665\pi$
$991$	24.8838	0.790461	0.395231	+	0.918582i	$0.370665\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−31.6080	−1.00204
$996$	0	0
$997$	28.6365	0.906928	0.453464	−	0.891275i	$-0.350188\pi$
$997$	28.6365	0.906928	0.453464	+	0.891275i	$0.350188\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3312.2.a.bh.1.4		4
3.2	odd	2		3312.2.a.bg.1.2		4
4.3	odd	2		1656.2.a.p.1.3	yes	4
12.11	even	2		1656.2.a.o.1.1	✓	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1656.2.a.o.1.1	✓	4	12.11	even	2
1656.2.a.p.1.3	yes	4	4.3	odd	2
3312.2.a.bg.1.2		4	3.2	odd	2
3312.2.a.bh.1.4		4	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 3312 = 2^{4} \cdot 3^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3312.a (trivial)

\(f(q)\)	\(=\)	\(q+3.64575 q^{5} +4.81895 q^{7} +O(q^{10})\)	\(q+3.64575 q^{5} +4.81895 q^{7} -2.68303 q^{11} -3.85623 q^{13} +0.962718 q^{17} -4.21048 q^{19} +1.00000 q^{23} +8.29150 q^{25} +7.85623 q^{29} -7.85623 q^{31} +17.5687 q^{35} +11.0294 q^{37} +0.564731 q^{41} +5.57655 q^{43} -7.63790 q^{47} +16.2223 q^{49} +9.99215 q^{53} -9.78167 q^{55} +15.0732 q^{59} -0.683033 q^{61} -14.0589 q^{65} +10.7935 q^{67} +3.43527 q^{71} +0.0692033 q^{73} -12.9294 q^{77} -6.89352 q^{79} +1.31697 q^{83} +3.50983 q^{85} +18.0413 q^{89} -18.5830 q^{91} -15.3504 q^{95} -4.34640 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{5} - 2 q^{7}+O(q^{10}) \) 4 * q + 4 * q^5 - 2 * q^7	\( 4 q + 4 q^{5} - 2 q^{7} - 2 q^{11} + 4 q^{13} + 2 q^{17} - 8 q^{19} + 4 q^{23} + 12 q^{25} + 12 q^{29} - 12 q^{31} + 12 q^{35} + 14 q^{37} + 4 q^{41} - 4 q^{43} + 12 q^{47} + 28 q^{49} + 8 q^{53} - 16 q^{55} + 16 q^{59} + 6 q^{61} + 4 q^{65} - 8 q^{67} + 12 q^{71} + 16 q^{73} + 12 q^{77} - 10 q^{79} + 14 q^{83} + 16 q^{85} + 14 q^{89} - 32 q^{91} + 20 q^{95} + 4 q^{97}+O(q^{100}) \) 4 * q + 4 * q^5 - 2 * q^7 - 2 * q^11 + 4 * q^13 + 2 * q^17 - 8 * q^19 + 4 * q^23 + 12 * q^25 + 12 * q^29 - 12 * q^31 + 12 * q^35 + 14 * q^37 + 4 * q^41 - 4 * q^43 + 12 * q^47 + 28 * q^49 + 8 * q^53 - 16 * q^55 + 16 * q^59 + 6 * q^61 + 4 * q^65 - 8 * q^67 + 12 * q^71 + 16 * q^73 + 12 * q^77 - 10 * q^79 + 14 * q^83 + 16 * q^85 + 14 * q^89 - 32 * q^91 + 20 * q^95 + 4 * q^97

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