LMFDB - Embedded newform 3344.2.a.t.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3344,2,Mod(1,3344)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3344, 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("3344.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3344 = 2^{4} \cdot 11 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3344.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:	$26.7019744359$
Analytic rank:	$1$
Dimension:	$5$
Coefficient field:	5.5.246832.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 2x^{4} - 5x^{3} + 6x^{2} + 7x - 2 $ x^5 - 2x^4 - 5x^3 + 6x^2 + 7x - 2
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 209)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$0.245526$ of defining polynomial
Character	$\chi$	$=$	3344.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.15766 q^{3} -3.43077 q^{5} -3.93972 q^{7} +1.65548 q^{9} +O(q^{10})$	$q-2.15766 q^{3} -3.43077 q^{5} -3.93972 q^{7} +1.65548 q^{9} -1.00000 q^{11} +3.31182 q^{13} +7.40242 q^{15} +2.80637 q^{17} +1.00000 q^{19} +8.50056 q^{21} -6.88998 q^{23} +6.77018 q^{25} +2.90101 q^{27} +5.67979 q^{29} -2.51864 q^{31} +2.15766 q^{33} +13.5163 q^{35} -6.39893 q^{37} -7.14577 q^{39} +0.560629 q^{41} +9.40080 q^{43} -5.67958 q^{45} +12.1742 q^{47} +8.52137 q^{49} -6.05517 q^{51} +5.68316 q^{53} +3.43077 q^{55} -2.15766 q^{57} -4.35730 q^{59} -3.56412 q^{61} -6.52213 q^{63} -11.3621 q^{65} +9.95563 q^{67} +14.8662 q^{69} +11.4671 q^{71} -8.95834 q^{73} -14.6077 q^{75} +3.93972 q^{77} -8.49105 q^{79} -11.2258 q^{81} +5.21960 q^{83} -9.62799 q^{85} -12.2550 q^{87} -7.28423 q^{89} -13.0476 q^{91} +5.43436 q^{93} -3.43077 q^{95} +10.6574 q^{97} -1.65548 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - q^{3} - 5 q^{5} - 6 q^{7} + 4 q^{9}+O(q^{10}) $ 5 * q - q^3 - 5 * q^5 - 6 * q^7 + 4 * q^9	$ 5 q - q^{3} - 5 q^{5} - 6 q^{7} + 4 q^{9} - 5 q^{11} + 4 q^{13} - 3 q^{15} - 4 q^{17} + 5 q^{19} + 10 q^{21} - 3 q^{23} + 6 q^{25} + 11 q^{27} + 10 q^{29} - 11 q^{31} + q^{33} + 8 q^{35} + q^{37} - 2 q^{39} + 2 q^{41} - 20 q^{43} - 28 q^{45} + 20 q^{47} + 3 q^{49} - 24 q^{51} - 14 q^{53} + 5 q^{55} - q^{57} - 3 q^{59} - 10 q^{61} - 24 q^{63} - 9 q^{67} - 5 q^{69} - 23 q^{71} + 18 q^{75} + 6 q^{77} - 44 q^{79} + q^{81} + 14 q^{83} - 12 q^{85} - 28 q^{87} - 27 q^{89} - 24 q^{91} - 27 q^{93} - 5 q^{95} + 15 q^{97} - 4 q^{99}+O(q^{100}) $ 5 * q - q^3 - 5 * q^5 - 6 * q^7 + 4 * q^9 - 5 * q^11 + 4 * q^13 - 3 * q^15 - 4 * q^17 + 5 * q^19 + 10 * q^21 - 3 * q^23 + 6 * q^25 + 11 * q^27 + 10 * q^29 - 11 * q^31 + q^33 + 8 * q^35 + q^37 - 2 * q^39 + 2 * q^41 - 20 * q^43 - 28 * q^45 + 20 * q^47 + 3 * q^49 - 24 * q^51 - 14 * q^53 + 5 * q^55 - q^57 - 3 * q^59 - 10 * q^61 - 24 * q^63 - 9 * q^67 - 5 * q^69 - 23 * q^71 + 18 * q^75 + 6 * q^77 - 44 * q^79 + q^81 + 14 * q^83 - 12 * q^85 - 28 * q^87 - 27 * q^89 - 24 * q^91 - 27 * q^93 - 5 * q^95 + 15 * q^97 - 4 * q^99

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
$3$	−2.15766	−1.24572	−0.622862	−	0.782332i	$-0.714030\pi$
$3$	−2.15766	−1.24572	−0.622862	+	0.782332i	$0.714030\pi$
$4$	0	0
$5$	−3.43077	−1.53429	−0.767143	−	0.641476i	$-0.778322\pi$
$5$	−3.43077	−1.53429	−0.767143	+	0.641476i	$0.778322\pi$
$6$	0	0
$7$	−3.93972	−1.48907	−0.744537	−	0.667582i	$-0.767329\pi$
$7$	−3.93972	−1.48907	−0.744537	+	0.667582i	$0.767329\pi$
$8$	0	0
$9$	1.65548	0.551827
$10$	0	0
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	3.31182	0.918534	0.459267	−	0.888298i	$-0.348112\pi$
$13$	3.31182	0.918534	0.459267	+	0.888298i	$0.348112\pi$
$14$	0	0
$15$	7.40242	1.91130
$16$	0	0
$17$	2.80637	0.680644	0.340322	−	0.940309i	$-0.389464\pi$
$17$	2.80637	0.680644	0.340322	+	0.940309i	$0.389464\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	8.50056	1.85497
$22$	0	0
$23$	−6.88998	−1.43666	−0.718330	−	0.695702i	$-0.755093\pi$
$23$	−6.88998	−1.43666	−0.718330	+	0.695702i	$0.755093\pi$
$24$	0	0
$25$	6.77018	1.35404
$26$	0	0
$27$	2.90101	0.558299
$28$	0	0
$29$	5.67979	1.05471	0.527355	−	0.849645i	$-0.323184\pi$
$29$	5.67979	1.05471	0.527355	+	0.849645i	$0.323184\pi$
$30$	0	0
$31$	−2.51864	−0.452361	−0.226180	−	0.974085i	$-0.572624\pi$
$31$	−2.51864	−0.452361	−0.226180	+	0.974085i	$0.572624\pi$
$32$	0	0
$33$	2.15766	0.375600
$34$	0	0
$35$	13.5163	2.28466
$36$	0	0
$37$	−6.39893	−1.05198	−0.525989	−	0.850491i	$-0.676305\pi$
$37$	−6.39893	−1.05198	−0.525989	+	0.850491i	$0.676305\pi$
$38$	0	0
$39$	−7.14577	−1.14424
$40$	0	0
$41$	0.560629	0.0875555	0.0437778	−	0.999041i	$-0.486061\pi$
$41$	0.560629	0.0875555	0.0437778	+	0.999041i	$0.486061\pi$
$42$	0	0
$43$	9.40080	1.43361	0.716805	−	0.697274i	$-0.245604\pi$
$43$	9.40080	1.43361	0.716805	+	0.697274i	$0.245604\pi$
$44$	0	0
$45$	−5.67958	−0.846661
$46$	0	0
$47$	12.1742	1.77579	0.887896	−	0.460044i	$-0.152166\pi$
$47$	12.1742	1.77579	0.887896	+	0.460044i	$0.152166\pi$
$48$	0	0
$49$	8.52137	1.21734
$50$	0	0
$51$	−6.05517	−0.847894
$52$	0	0
$53$	5.68316	0.780643	0.390321	−	0.920679i	$-0.372364\pi$
$53$	5.68316	0.780643	0.390321	+	0.920679i	$0.372364\pi$
$54$	0	0
$55$	3.43077	0.462605
$56$	0	0
$57$	−2.15766	−0.285789
$58$	0	0
$59$	−4.35730	−0.567273	−0.283636	−	0.958932i	$-0.591541\pi$
$59$	−4.35730	−0.567273	−0.283636	+	0.958932i	$0.591541\pi$
$60$	0	0
$61$	−3.56412	−0.456339	−0.228169	−	0.973621i	$-0.573274\pi$
$61$	−3.56412	−0.456339	−0.228169	+	0.973621i	$0.573274\pi$
$62$	0	0
$63$	−6.52213	−0.821711
$64$	0	0
$65$	−11.3621	−1.40929
$66$	0	0
$67$	9.95563	1.21627	0.608137	−	0.793832i	$-0.291917\pi$
$67$	9.95563	1.21627	0.608137	+	0.793832i	$0.291917\pi$
$68$	0	0
$69$	14.8662	1.78968
$70$	0	0
$71$	11.4671	1.36089	0.680447	−	0.732797i	$-0.261786\pi$
$71$	11.4671	1.36089	0.680447	+	0.732797i	$0.261786\pi$
$72$	0	0
$73$	−8.95834	−1.04849	−0.524247	−	0.851566i	$-0.675653\pi$
$73$	−8.95834	−1.04849	−0.524247	+	0.851566i	$0.675653\pi$
$74$	0	0
$75$	−14.6077	−1.68675
$76$	0	0
$77$	3.93972	0.448972
$78$	0	0
$79$	−8.49105	−0.955318	−0.477659	−	0.878545i	$-0.658515\pi$
$79$	−8.49105	−0.955318	−0.477659	+	0.878545i	$0.658515\pi$
$80$	0	0
$81$	−11.2258	−1.24731
$82$	0	0
$83$	5.21960	0.572926	0.286463	−	0.958091i	$-0.407520\pi$
$83$	5.21960	0.572926	0.286463	+	0.958091i	$0.407520\pi$
$84$	0	0
$85$	−9.62799	−1.04430
$86$	0	0
$87$	−12.2550	−1.31388
$88$	0	0
$89$	−7.28423	−0.772127	−0.386064	−	0.922472i	$-0.626166\pi$
$89$	−7.28423	−0.772127	−0.386064	+	0.922472i	$0.626166\pi$
$90$	0	0
$91$	−13.0476	−1.36776
$92$	0	0
$93$	5.43436	0.563517
$94$	0	0
$95$	−3.43077	−0.351989
$96$	0	0
$97$	10.6574	1.08209	0.541045	−	0.840993i	$-0.318029\pi$
$97$	10.6574	1.08209	0.541045	+	0.840993i	$0.318029\pi$
$98$	0	0
$99$	−1.65548	−0.166382
$100$	0	0
$101$	−11.4716	−1.14147	−0.570735	−	0.821134i	$-0.693342\pi$
$101$	−11.4716	−1.14147	−0.570735	+	0.821134i	$0.693342\pi$
$102$	0	0
$103$	−18.3034	−1.80349	−0.901745	−	0.432268i	$-0.857714\pi$
$103$	−18.3034	−1.80349	−0.901745	+	0.432268i	$0.857714\pi$
$104$	0	0
$105$	−29.1634	−2.84606
$106$	0	0
$107$	−1.38838	−0.134220	−0.0671100	−	0.997746i	$-0.521378\pi$
$107$	−1.38838	−0.134220	−0.0671100	+	0.997746i	$0.521378\pi$
$108$	0	0
$109$	−0.412113	−0.0394732	−0.0197366	−	0.999805i	$-0.506283\pi$
$109$	−0.412113	−0.0394732	−0.0197366	+	0.999805i	$0.506283\pi$
$110$	0	0
$111$	13.8067	1.31047
$112$	0	0
$113$	−6.54003	−0.615234	−0.307617	−	0.951510i	$-0.599532\pi$
$113$	−6.54003	−0.615234	−0.307617	+	0.951510i	$0.599532\pi$
$114$	0	0
$115$	23.6379	2.20425
$116$	0	0
$117$	5.48266	0.506872
$118$	0	0
$119$	−11.0563	−1.01353
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	−1.20964	−0.109070
$124$	0	0
$125$	−6.07307	−0.543192
$126$	0	0
$127$	−9.08005	−0.805724	−0.402862	−	0.915261i	$-0.631985\pi$
$127$	−9.08005	−0.805724	−0.402862	+	0.915261i	$0.631985\pi$
$128$	0	0
$129$	−20.2837	−1.78588
$130$	0	0
$131$	10.3876	0.907571	0.453785	−	0.891111i	$-0.350073\pi$
$131$	10.3876	0.907571	0.453785	+	0.891111i	$0.350073\pi$
$132$	0	0
$133$	−3.93972	−0.341617
$134$	0	0
$135$	−9.95269	−0.856591
$136$	0	0
$137$	−0.798293	−0.0682028	−0.0341014	−	0.999418i	$-0.510857\pi$
$137$	−0.798293	−0.0682028	−0.0341014	+	0.999418i	$0.510857\pi$
$138$	0	0
$139$	−5.03184	−0.426795	−0.213398	−	0.976965i	$-0.568453\pi$
$139$	−5.03184	−0.426795	−0.213398	+	0.976965i	$0.568453\pi$
$140$	0	0
$141$	−26.2678	−2.21215
$142$	0	0
$143$	−3.31182	−0.276948
$144$	0	0
$145$	−19.4860	−1.61823
$146$	0	0
$147$	−18.3862	−1.51647
$148$	0	0
$149$	19.8351	1.62496	0.812479	−	0.582991i	$-0.198118\pi$
$149$	19.8351	1.62496	0.812479	+	0.582991i	$0.198118\pi$
$150$	0	0
$151$	−22.5447	−1.83466	−0.917331	−	0.398125i	$-0.869661\pi$
$151$	−22.5447	−1.83466	−0.917331	+	0.398125i	$0.869661\pi$
$152$	0	0
$153$	4.64589	0.375598
$154$	0	0
$155$	8.64087	0.694051
$156$	0	0
$157$	−11.8013	−0.941843	−0.470921	−	0.882175i	$-0.656078\pi$
$157$	−11.8013	−0.941843	−0.470921	+	0.882175i	$0.656078\pi$
$158$	0	0
$159$	−12.2623	−0.972465
$160$	0	0
$161$	27.1446	2.13929
$162$	0	0
$163$	24.8395	1.94558	0.972789	−	0.231691i	$-0.0744259\pi$
$163$	24.8395	1.94558	0.972789	+	0.231691i	$0.0744259\pi$
$164$	0	0
$165$	−7.40242	−0.576278
$166$	0	0
$167$	2.79938	0.216623	0.108311	−	0.994117i	$-0.465456\pi$
$167$	2.79938	0.216623	0.108311	+	0.994117i	$0.465456\pi$
$168$	0	0
$169$	−2.03184	−0.156295
$170$	0	0
$171$	1.65548	0.126598
$172$	0	0
$173$	−6.43926	−0.489568	−0.244784	−	0.969578i	$-0.578717\pi$
$173$	−6.43926	−0.489568	−0.244784	+	0.969578i	$0.578717\pi$
$174$	0	0
$175$	−26.6726	−2.01626
$176$	0	0
$177$	9.40156	0.706665
$178$	0	0
$179$	−12.5241	−0.936095	−0.468048	−	0.883703i	$-0.655042\pi$
$179$	−12.5241	−0.936095	−0.468048	+	0.883703i	$0.655042\pi$
$180$	0	0
$181$	13.7515	1.02214	0.511071	−	0.859538i	$-0.329249\pi$
$181$	13.7515	1.02214	0.511071	+	0.859538i	$0.329249\pi$
$182$	0	0
$183$	7.69015	0.568472
$184$	0	0
$185$	21.9533	1.61404
$186$	0	0
$187$	−2.80637	−0.205222
$188$	0	0
$189$	−11.4292	−0.831348
$190$	0	0
$191$	−3.10678	−0.224799	−0.112399	−	0.993663i	$-0.535854\pi$
$191$	−3.10678	−0.224799	−0.112399	+	0.993663i	$0.535854\pi$
$192$	0	0
$193$	−0.747815	−0.0538289	−0.0269144	−	0.999638i	$-0.508568\pi$
$193$	−0.747815	−0.0538289	−0.0269144	+	0.999638i	$0.508568\pi$
$194$	0	0
$195$	24.5155	1.75559
$196$	0	0
$197$	−3.41798	−0.243521	−0.121761	−	0.992559i	$-0.538854\pi$
$197$	−3.41798	−0.243521	−0.121761	+	0.992559i	$0.538854\pi$
$198$	0	0
$199$	−5.36785	−0.380517	−0.190258	−	0.981734i	$-0.560933\pi$
$199$	−5.36785	−0.380517	−0.190258	+	0.981734i	$0.560933\pi$
$200$	0	0
$201$	−21.4808	−1.51514
$202$	0	0
$203$	−22.3768	−1.57054
$204$	0	0
$205$	−1.92339	−0.134335
$206$	0	0
$207$	−11.4062	−0.792789
$208$	0	0
$209$	−1.00000	−0.0691714
$210$	0	0
$211$	−2.55492	−0.175888	−0.0879441	−	0.996125i	$-0.528030\pi$
$211$	−2.55492	−0.175888	−0.0879441	+	0.996125i	$0.528030\pi$
$212$	0	0
$213$	−24.7421	−1.69530
$214$	0	0
$215$	−32.2520	−2.19957
$216$	0	0
$217$	9.92272	0.673598
$218$	0	0
$219$	19.3290	1.30613
$220$	0	0
$221$	9.29418	0.625194
$222$	0	0
$223$	24.9404	1.67013	0.835066	−	0.550149i	$-0.185429\pi$
$223$	24.9404	1.67013	0.835066	+	0.550149i	$0.185429\pi$
$224$	0	0
$225$	11.2079	0.747194
$226$	0	0
$227$	22.8254	1.51497	0.757487	−	0.652851i	$-0.226427\pi$
$227$	22.8254	1.51497	0.757487	+	0.652851i	$0.226427\pi$
$228$	0	0
$229$	−0.603546	−0.0398834	−0.0199417	−	0.999801i	$-0.506348\pi$
$229$	−0.603546	−0.0398834	−0.0199417	+	0.999801i	$0.506348\pi$
$230$	0	0
$231$	−8.50056	−0.559296
$232$	0	0
$233$	17.3705	1.13798	0.568988	−	0.822346i	$-0.307335\pi$
$233$	17.3705	1.13798	0.568988	+	0.822346i	$0.307335\pi$
$234$	0	0
$235$	−41.7669	−2.72457
$236$	0	0
$237$	18.3208	1.19006
$238$	0	0
$239$	−7.23486	−0.467984	−0.233992	−	0.972238i	$-0.575179\pi$
$239$	−7.23486	−0.467984	−0.233992	+	0.972238i	$0.575179\pi$
$240$	0	0
$241$	−12.2034	−0.786090	−0.393045	−	0.919519i	$-0.628578\pi$
$241$	−12.2034	−0.786090	−0.393045	+	0.919519i	$0.628578\pi$
$242$	0	0
$243$	15.5185	0.995509
$244$	0	0
$245$	−29.2349	−1.86775
$246$	0	0
$247$	3.31182	0.210726
$248$	0	0
$249$	−11.2621	−0.713707
$250$	0	0
$251$	14.0923	0.889499	0.444750	−	0.895655i	$-0.353293\pi$
$251$	14.0923	0.889499	0.444750	+	0.895655i	$0.353293\pi$
$252$	0	0
$253$	6.88998	0.433169
$254$	0	0
$255$	20.7739	1.30091
$256$	0	0
$257$	0.440920	0.0275038	0.0137519	−	0.999905i	$-0.495622\pi$
$257$	0.440920	0.0275038	0.0137519	+	0.999905i	$0.495622\pi$
$258$	0	0
$259$	25.2100	1.56647
$260$	0	0
$261$	9.40279	0.582018
$262$	0	0
$263$	15.0661	0.929016	0.464508	−	0.885569i	$-0.346231\pi$
$263$	15.0661	0.929016	0.464508	+	0.885569i	$0.346231\pi$
$264$	0	0
$265$	−19.4976	−1.19773
$266$	0	0
$267$	15.7169	0.961857
$268$	0	0
$269$	−7.25751	−0.442498	−0.221249	−	0.975217i	$-0.571013\pi$
$269$	−7.25751	−0.442498	−0.221249	+	0.975217i	$0.571013\pi$
$270$	0	0
$271$	16.8878	1.02586	0.512931	−	0.858430i	$-0.328560\pi$
$271$	16.8878	1.02586	0.512931	+	0.858430i	$0.328560\pi$
$272$	0	0
$273$	28.1523	1.70386
$274$	0	0
$275$	−6.77018	−0.408257
$276$	0	0
$277$	−15.3818	−0.924204	−0.462102	−	0.886827i	$-0.652905\pi$
$277$	−15.3818	−0.924204	−0.462102	+	0.886827i	$0.652905\pi$
$278$	0	0
$279$	−4.16956	−0.249625
$280$	0	0
$281$	−25.8974	−1.54491	−0.772456	−	0.635069i	$-0.780972\pi$
$281$	−25.8974	−1.54491	−0.772456	+	0.635069i	$0.780972\pi$
$282$	0	0
$283$	−18.6882	−1.11090	−0.555450	−	0.831550i	$-0.687454\pi$
$283$	−18.6882	−1.11090	−0.555450	+	0.831550i	$0.687454\pi$
$284$	0	0
$285$	7.40242	0.438482
$286$	0	0
$287$	−2.20872	−0.130377
$288$	0	0
$289$	−9.12431	−0.536724
$290$	0	0
$291$	−22.9949	−1.34799
$292$	0	0
$293$	26.7471	1.56258	0.781291	−	0.624167i	$-0.214561\pi$
$293$	26.7471	1.56258	0.781291	+	0.624167i	$0.214561\pi$
$294$	0	0
$295$	14.9489	0.870359
$296$	0	0
$297$	−2.90101	−0.168334
$298$	0	0
$299$	−22.8184	−1.31962
$300$	0	0
$301$	−37.0365	−2.13475
$302$	0	0
$303$	24.7518	1.42196
$304$	0	0
$305$	12.2277	0.700155
$306$	0	0
$307$	−22.6415	−1.29222	−0.646109	−	0.763245i	$-0.723605\pi$
$307$	−22.6415	−1.29222	−0.646109	+	0.763245i	$0.723605\pi$
$308$	0	0
$309$	39.4925	2.24665
$310$	0	0
$311$	1.38723	0.0786628	0.0393314	−	0.999226i	$-0.487477\pi$
$311$	1.38723	0.0786628	0.0393314	+	0.999226i	$0.487477\pi$
$312$	0	0
$313$	12.9018	0.729255	0.364627	−	0.931153i	$-0.381196\pi$
$313$	12.9018	0.729255	0.364627	+	0.931153i	$0.381196\pi$
$314$	0	0
$315$	22.3759	1.26074
$316$	0	0
$317$	−19.0712	−1.07114	−0.535572	−	0.844489i	$-0.679904\pi$
$317$	−19.0712	−1.07114	−0.535572	+	0.844489i	$0.679904\pi$
$318$	0	0
$319$	−5.67979	−0.318007
$320$	0	0
$321$	2.99565	0.167201
$322$	0	0
$323$	2.80637	0.156150
$324$	0	0
$325$	22.4216	1.24373
$326$	0	0
$327$	0.889197	0.0491727
$328$	0	0
$329$	−47.9630	−2.64428
$330$	0	0
$331$	12.4616	0.684952	0.342476	−	0.939527i	$-0.388735\pi$
$331$	12.4616	0.684952	0.342476	+	0.939527i	$0.388735\pi$
$332$	0	0
$333$	−10.5933	−0.580510
$334$	0	0
$335$	−34.1555	−1.86611
$336$	0	0
$337$	−0.401035	−0.0218458	−0.0109229	−	0.999940i	$-0.503477\pi$
$337$	−0.401035	−0.0218458	−0.0109229	+	0.999940i	$0.503477\pi$
$338$	0	0
$339$	14.1111	0.766411
$340$	0	0
$341$	2.51864	0.136392
$342$	0	0
$343$	−5.99377	−0.323633
$344$	0	0
$345$	−51.0026	−2.74589
$346$	0	0
$347$	−1.06608	−0.0572304	−0.0286152	−	0.999591i	$-0.509110\pi$
$347$	−1.06608	−0.0572304	−0.0286152	+	0.999591i	$0.509110\pi$
$348$	0	0
$349$	22.2695	1.19206	0.596029	−	0.802963i	$-0.296744\pi$
$349$	22.2695	1.19206	0.596029	+	0.802963i	$0.296744\pi$
$350$	0	0
$351$	9.60762	0.512817
$352$	0	0
$353$	12.3631	0.658023	0.329012	−	0.944326i	$-0.393284\pi$
$353$	12.3631	0.658023	0.329012	+	0.944326i	$0.393284\pi$
$354$	0	0
$355$	−39.3410	−2.08800
$356$	0	0
$357$	23.8557	1.26258
$358$	0	0
$359$	−32.1914	−1.69899	−0.849497	−	0.527593i	$-0.823095\pi$
$359$	−32.1914	−1.69899	−0.849497	+	0.527593i	$0.823095\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	−2.15766	−0.113248
$364$	0	0
$365$	30.7340	1.60869
$366$	0	0
$367$	18.9777	0.990628	0.495314	−	0.868714i	$-0.335053\pi$
$367$	18.9777	0.990628	0.495314	+	0.868714i	$0.335053\pi$
$368$	0	0
$369$	0.928111	0.0483155
$370$	0	0
$371$	−22.3901	−1.16243
$372$	0	0
$373$	1.15238	0.0596678	0.0298339	−	0.999555i	$-0.490502\pi$
$373$	1.15238	0.0596678	0.0298339	+	0.999555i	$0.490502\pi$
$374$	0	0
$375$	13.1036	0.676667
$376$	0	0
$377$	18.8104	0.968787
$378$	0	0
$379$	−13.5365	−0.695325	−0.347663	−	0.937620i	$-0.613025\pi$
$379$	−13.5365	−0.695325	−0.347663	+	0.937620i	$0.613025\pi$
$380$	0	0
$381$	19.5916	1.00371
$382$	0	0
$383$	−13.1671	−0.672809	−0.336404	−	0.941718i	$-0.609211\pi$
$383$	−13.1671	−0.672809	−0.336404	+	0.941718i	$0.609211\pi$
$384$	0	0
$385$	−13.5163	−0.688852
$386$	0	0
$387$	15.5629	0.791105
$388$	0	0
$389$	0.223588	0.0113364	0.00566819	−	0.999984i	$-0.498196\pi$
$389$	0.223588	0.0113364	0.00566819	+	0.999984i	$0.498196\pi$
$390$	0	0
$391$	−19.3358	−0.977854
$392$	0	0
$393$	−22.4129	−1.13058
$394$	0	0
$395$	29.1308	1.46573
$396$	0	0
$397$	−21.6504	−1.08660	−0.543301	−	0.839538i	$-0.682826\pi$
$397$	−21.6504	−1.08660	−0.543301	+	0.839538i	$0.682826\pi$
$398$	0	0
$399$	8.50056	0.425560
$400$	0	0
$401$	1.30180	0.0650089	0.0325045	−	0.999472i	$-0.489652\pi$
$401$	1.30180	0.0650089	0.0325045	+	0.999472i	$0.489652\pi$
$402$	0	0
$403$	−8.34128	−0.415509
$404$	0	0
$405$	38.5132	1.91374
$406$	0	0
$407$	6.39893	0.317183
$408$	0	0
$409$	−2.15631	−0.106622	−0.0533112	−	0.998578i	$-0.516978\pi$
$409$	−2.15631	−0.106622	−0.0533112	+	0.998578i	$0.516978\pi$
$410$	0	0
$411$	1.72244	0.0849618
$412$	0	0
$413$	17.1665	0.844710
$414$	0	0
$415$	−17.9073	−0.879032
$416$	0	0
$417$	10.8570	0.531669
$418$	0	0
$419$	29.6335	1.44769	0.723844	−	0.689963i	$-0.242373\pi$
$419$	29.6335	1.44769	0.723844	+	0.689963i	$0.242373\pi$
$420$	0	0
$421$	−5.25385	−0.256057	−0.128028	−	0.991770i	$-0.540865\pi$
$421$	−5.25385	−0.256057	−0.128028	+	0.991770i	$0.540865\pi$
$422$	0	0
$423$	20.1542	0.979931
$424$	0	0
$425$	18.9996	0.921616
$426$	0	0
$427$	14.0416	0.679522
$428$	0	0
$429$	7.14577	0.345001
$430$	0	0
$431$	−17.8489	−0.859752	−0.429876	−	0.902888i	$-0.641443\pi$
$431$	−17.8489	−0.859752	−0.429876	+	0.902888i	$0.641443\pi$
$432$	0	0
$433$	−0.696383	−0.0334660	−0.0167330	−	0.999860i	$-0.505327\pi$
$433$	−0.696383	−0.0334660	−0.0167330	+	0.999860i	$0.505327\pi$
$434$	0	0
$435$	42.0442	2.01586
$436$	0	0
$437$	−6.88998	−0.329593
$438$	0	0
$439$	24.8210	1.18464	0.592321	−	0.805702i	$-0.298212\pi$
$439$	24.8210	1.18464	0.592321	+	0.805702i	$0.298212\pi$
$440$	0	0
$441$	14.1070	0.671761
$442$	0	0
$443$	3.93424	0.186922	0.0934608	−	0.995623i	$-0.470207\pi$
$443$	3.93424	0.186922	0.0934608	+	0.995623i	$0.470207\pi$
$444$	0	0
$445$	24.9905	1.18466
$446$	0	0
$447$	−42.7974	−2.02425
$448$	0	0
$449$	6.13394	0.289479	0.144739	−	0.989470i	$-0.453766\pi$
$449$	6.13394	0.289479	0.144739	+	0.989470i	$0.453766\pi$
$450$	0	0
$451$	−0.560629	−0.0263990
$452$	0	0
$453$	48.6437	2.28548
$454$	0	0
$455$	44.7634	2.09854
$456$	0	0
$457$	8.40189	0.393024	0.196512	−	0.980501i	$-0.437039\pi$
$457$	8.40189	0.393024	0.196512	+	0.980501i	$0.437039\pi$
$458$	0	0
$459$	8.14129	0.380003
$460$	0	0
$461$	25.9903	1.21049	0.605246	−	0.796039i	$-0.293075\pi$
$461$	25.9903	1.21049	0.605246	+	0.796039i	$0.293075\pi$
$462$	0	0
$463$	−19.3724	−0.900311	−0.450155	−	0.892950i	$-0.648631\pi$
$463$	−19.3724	−0.900311	−0.450155	+	0.892950i	$0.648631\pi$
$464$	0	0
$465$	−18.6440	−0.864596
$466$	0	0
$467$	34.6720	1.60443	0.802214	−	0.597037i	$-0.203655\pi$
$467$	34.6720	1.60443	0.802214	+	0.597037i	$0.203655\pi$
$468$	0	0
$469$	−39.2224	−1.81112
$470$	0	0
$471$	25.4631	1.17328
$472$	0	0
$473$	−9.40080	−0.432249
$474$	0	0
$475$	6.77018	0.310637
$476$	0	0
$477$	9.40838	0.430780
$478$	0	0
$479$	−23.2352	−1.06164	−0.530821	−	0.847484i	$-0.678117\pi$
$479$	−23.2352	−1.06164	−0.530821	+	0.847484i	$0.678117\pi$
$480$	0	0
$481$	−21.1921	−0.966277
$482$	0	0
$483$	−58.5687	−2.66497
$484$	0	0
$485$	−36.5629	−1.66024
$486$	0	0
$487$	−14.2680	−0.646545	−0.323272	−	0.946306i	$-0.604783\pi$
$487$	−14.2680	−0.646545	−0.323272	+	0.946306i	$0.604783\pi$
$488$	0	0
$489$	−53.5951	−2.42365
$490$	0	0
$491$	−41.6168	−1.87814	−0.939071	−	0.343724i	$-0.888311\pi$
$491$	−41.6168	−1.87814	−0.939071	+	0.343724i	$0.888311\pi$
$492$	0	0
$493$	15.9396	0.717882
$494$	0	0
$495$	5.67958	0.255278
$496$	0	0
$497$	−45.1771	−2.02647
$498$	0	0
$499$	−19.9664	−0.893820	−0.446910	−	0.894579i	$-0.647476\pi$
$499$	−19.9664	−0.893820	−0.446910	+	0.894579i	$0.647476\pi$
$500$	0	0
$501$	−6.04010	−0.269852
$502$	0	0
$503$	−2.31362	−0.103159	−0.0515797	−	0.998669i	$-0.516426\pi$
$503$	−2.31362	−0.103159	−0.0515797	+	0.998669i	$0.516426\pi$
$504$	0	0
$505$	39.3565	1.75134
$506$	0	0
$507$	4.38401	0.194701
$508$	0	0
$509$	−13.2009	−0.585120	−0.292560	−	0.956247i	$-0.594507\pi$
$509$	−13.2009	−0.585120	−0.292560	+	0.956247i	$0.594507\pi$
$510$	0	0
$511$	35.2933	1.56128
$512$	0	0
$513$	2.90101	0.128083
$514$	0	0
$515$	62.7948	2.76707
$516$	0	0
$517$	−12.1742	−0.535421
$518$	0	0
$519$	13.8937	0.609866
$520$	0	0
$521$	−15.6498	−0.685629	−0.342814	−	0.939403i	$-0.611380\pi$
$521$	−15.6498	−0.685629	−0.342814	+	0.939403i	$0.611380\pi$
$522$	0	0
$523$	17.7350	0.775498	0.387749	−	0.921765i	$-0.373253\pi$
$523$	17.7350	0.775498	0.387749	+	0.921765i	$0.373253\pi$
$524$	0	0
$525$	57.5503	2.51170
$526$	0	0
$527$	−7.06822	−0.307897
$528$	0	0
$529$	24.4719	1.06399
$530$	0	0
$531$	−7.21344	−0.313037
$532$	0	0
$533$	1.85670	0.0804227
$534$	0	0
$535$	4.76322	0.205932
$536$	0	0
$537$	27.0227	1.16612
$538$	0	0
$539$	−8.52137	−0.367041
$540$	0	0
$541$	−34.5196	−1.48412	−0.742058	−	0.670336i	$-0.766150\pi$
$541$	−34.5196	−1.48412	−0.742058	+	0.670336i	$0.766150\pi$
$542$	0	0
$543$	−29.6711	−1.27331
$544$	0	0
$545$	1.41386	0.0605632
$546$	0	0
$547$	−1.99561	−0.0853263	−0.0426632	−	0.999090i	$-0.513584\pi$
$547$	−1.99561	−0.0853263	−0.0426632	+	0.999090i	$0.513584\pi$
$548$	0	0
$549$	−5.90034	−0.251820
$550$	0	0
$551$	5.67979	0.241967
$552$	0	0
$553$	33.4523	1.42254
$554$	0	0
$555$	−47.3676	−2.01064
$556$	0	0
$557$	−23.5272	−0.996881	−0.498440	−	0.866924i	$-0.666094\pi$
$557$	−23.5272	−0.996881	−0.498440	+	0.866924i	$0.666094\pi$
$558$	0	0
$559$	31.1338	1.31682
$560$	0	0
$561$	6.05517	0.255650
$562$	0	0
$563$	−14.2122	−0.598972	−0.299486	−	0.954101i	$-0.596815\pi$
$563$	−14.2122	−0.598972	−0.299486	+	0.954101i	$0.596815\pi$
$564$	0	0
$565$	22.4373	0.943945
$566$	0	0
$567$	44.2266	1.85734
$568$	0	0
$569$	22.1906	0.930277	0.465138	−	0.885238i	$-0.346005\pi$
$569$	22.1906	0.930277	0.465138	+	0.885238i	$0.346005\pi$
$570$	0	0
$571$	−8.67954	−0.363227	−0.181614	−	0.983370i	$-0.558132\pi$
$571$	−8.67954	−0.363227	−0.181614	+	0.983370i	$0.558132\pi$
$572$	0	0
$573$	6.70337	0.280037
$574$	0	0
$575$	−46.6464	−1.94529
$576$	0	0
$577$	40.9150	1.70331	0.851657	−	0.524100i	$-0.175598\pi$
$577$	40.9150	1.70331	0.851657	+	0.524100i	$0.175598\pi$
$578$	0	0
$579$	1.61353	0.0670559
$580$	0	0
$581$	−20.5638	−0.853128
$582$	0	0
$583$	−5.68316	−0.235373
$584$	0	0
$585$	−18.8097	−0.777687
$586$	0	0
$587$	21.2684	0.877840	0.438920	−	0.898526i	$-0.355361\pi$
$587$	21.2684	0.877840	0.438920	+	0.898526i	$0.355361\pi$
$588$	0	0
$589$	−2.51864	−0.103779
$590$	0	0
$591$	7.37484	0.303360
$592$	0	0
$593$	−23.0212	−0.945368	−0.472684	−	0.881232i	$-0.656715\pi$
$593$	−23.0212	−0.945368	−0.472684	+	0.881232i	$0.656715\pi$
$594$	0	0
$595$	37.9316	1.55504
$596$	0	0
$597$	11.5820	0.474019
$598$	0	0
$599$	41.6249	1.70075	0.850374	−	0.526179i	$-0.176376\pi$
$599$	41.6249	1.70075	0.850374	+	0.526179i	$0.176376\pi$
$600$	0	0
$601$	−20.8629	−0.851016	−0.425508	−	0.904955i	$-0.639905\pi$
$601$	−20.8629	−0.851016	−0.425508	+	0.904955i	$0.639905\pi$
$602$	0	0
$603$	16.4814	0.671173
$604$	0	0
$605$	−3.43077	−0.139481
$606$	0	0
$607$	0.146703	0.00595450	0.00297725	−	0.999996i	$-0.499052\pi$
$607$	0.146703	0.00595450	0.00297725	+	0.999996i	$0.499052\pi$
$608$	0	0
$609$	48.2813	1.95646
$610$	0	0
$611$	40.3188	1.63113
$612$	0	0
$613$	−18.3393	−0.740716	−0.370358	−	0.928889i	$-0.620765\pi$
$613$	−18.3393	−0.740716	−0.370358	+	0.928889i	$0.620765\pi$
$614$	0	0
$615$	4.15001	0.167345
$616$	0	0
$617$	4.12050	0.165885	0.0829425	−	0.996554i	$-0.473568\pi$
$617$	4.12050	0.165885	0.0829425	+	0.996554i	$0.473568\pi$
$618$	0	0
$619$	−28.5699	−1.14832	−0.574161	−	0.818743i	$-0.694672\pi$
$619$	−28.5699	−1.14832	−0.574161	+	0.818743i	$0.694672\pi$
$620$	0	0
$621$	−19.9879	−0.802087
$622$	0	0
$623$	28.6978	1.14975
$624$	0	0
$625$	−13.0156	−0.520624
$626$	0	0
$627$	2.15766	0.0861685
$628$	0	0
$629$	−17.9577	−0.716022
$630$	0	0
$631$	−33.9323	−1.35082	−0.675412	−	0.737441i	$-0.736034\pi$
$631$	−33.9323	−1.35082	−0.675412	+	0.737441i	$0.736034\pi$
$632$	0	0
$633$	5.51265	0.219108
$634$	0	0
$635$	31.1516	1.23621
$636$	0	0
$637$	28.2213	1.11817
$638$	0	0
$639$	18.9836	0.750979
$640$	0	0
$641$	19.7694	0.780844	0.390422	−	0.920636i	$-0.372329\pi$
$641$	19.7694	0.780844	0.390422	+	0.920636i	$0.372329\pi$
$642$	0	0
$643$	−36.7642	−1.44984	−0.724920	−	0.688833i	$-0.758123\pi$
$643$	−36.7642	−1.44984	−0.724920	+	0.688833i	$0.758123\pi$
$644$	0	0
$645$	69.5887	2.74005
$646$	0	0
$647$	−24.9430	−0.980610	−0.490305	−	0.871551i	$-0.663115\pi$
$647$	−24.9430	−0.980610	−0.490305	+	0.871551i	$0.663115\pi$
$648$	0	0
$649$	4.35730	0.171039
$650$	0	0
$651$	−21.4098	−0.839117
$652$	0	0
$653$	8.60802	0.336858	0.168429	−	0.985714i	$-0.446131\pi$
$653$	8.60802	0.336858	0.168429	+	0.985714i	$0.446131\pi$
$654$	0	0
$655$	−35.6375	−1.39247
$656$	0	0
$657$	−14.8304	−0.578588
$658$	0	0
$659$	−27.7805	−1.08217	−0.541087	−	0.840967i	$-0.681987\pi$
$659$	−27.7805	−1.08217	−0.541087	+	0.840967i	$0.681987\pi$
$660$	0	0
$661$	30.8037	1.19812	0.599062	−	0.800702i	$-0.295540\pi$
$661$	30.8037	1.19812	0.599062	+	0.800702i	$0.295540\pi$
$662$	0	0
$663$	−20.0537	−0.778819
$664$	0	0
$665$	13.5163	0.524138
$666$	0	0
$667$	−39.1336	−1.51526
$668$	0	0
$669$	−53.8128	−2.08052
$670$	0	0
$671$	3.56412	0.137591
$672$	0	0
$673$	15.2015	0.585974	0.292987	−	0.956116i	$-0.405351\pi$
$673$	15.2015	0.585974	0.292987	+	0.956116i	$0.405351\pi$
$674$	0	0
$675$	19.6403	0.755957
$676$	0	0
$677$	−9.49027	−0.364741	−0.182370	−	0.983230i	$-0.558377\pi$
$677$	−9.49027	−0.364741	−0.182370	+	0.983230i	$0.558377\pi$
$678$	0	0
$679$	−41.9870	−1.61131
$680$	0	0
$681$	−49.2493	−1.88724
$682$	0	0
$683$	−15.0960	−0.577631	−0.288816	−	0.957385i	$-0.593261\pi$
$683$	−15.0960	−0.577631	−0.288816	+	0.957385i	$0.593261\pi$
$684$	0	0
$685$	2.73876	0.104643
$686$	0	0
$687$	1.30224	0.0496837
$688$	0	0
$689$	18.8216	0.717047
$690$	0	0
$691$	−3.46673	−0.131881	−0.0659403	−	0.997824i	$-0.521005\pi$
$691$	−3.46673	−0.131881	−0.0659403	+	0.997824i	$0.521005\pi$
$692$	0	0
$693$	6.52213	0.247755
$694$	0	0
$695$	17.2631	0.654826
$696$	0	0
$697$	1.57333	0.0595941
$698$	0	0
$699$	−37.4795	−1.41760
$700$	0	0
$701$	−26.2612	−0.991872	−0.495936	−	0.868359i	$-0.665175\pi$
$701$	−26.2612	−0.991872	−0.495936	+	0.868359i	$0.665175\pi$
$702$	0	0
$703$	−6.39893	−0.241340
$704$	0	0
$705$	90.1187	3.39407
$706$	0	0
$707$	45.1950	1.69973
$708$	0	0
$709$	−20.1488	−0.756704	−0.378352	−	0.925662i	$-0.623509\pi$
$709$	−20.1488	−0.756704	−0.378352	+	0.925662i	$0.623509\pi$
$710$	0	0
$711$	−14.0568	−0.527171
$712$	0	0
$713$	17.3534	0.649889
$714$	0	0
$715$	11.3621	0.424918
$716$	0	0
$717$	15.6104	0.582979
$718$	0	0
$719$	43.4738	1.62130	0.810650	−	0.585532i	$-0.199114\pi$
$719$	43.4738	1.62130	0.810650	+	0.585532i	$0.199114\pi$
$720$	0	0
$721$	72.1103	2.68553
$722$	0	0
$723$	26.3307	0.979250
$724$	0	0
$725$	38.4532	1.42811
$726$	0	0
$727$	−9.55640	−0.354427	−0.177214	−	0.984172i	$-0.556708\pi$
$727$	−9.55640	−0.354427	−0.177214	+	0.984172i	$0.556708\pi$
$728$	0	0
$729$	0.193986	0.00718467
$730$	0	0
$731$	26.3821	0.975777
$732$	0	0
$733$	−20.3072	−0.750065	−0.375032	−	0.927012i	$-0.622368\pi$
$733$	−20.3072	−0.750065	−0.375032	+	0.927012i	$0.622368\pi$
$734$	0	0
$735$	63.0788	2.32670
$736$	0	0
$737$	−9.95563	−0.366720
$738$	0	0
$739$	6.60855	0.243099	0.121550	−	0.992585i	$-0.461214\pi$
$739$	6.60855	0.243099	0.121550	+	0.992585i	$0.461214\pi$
$740$	0	0
$741$	−7.14577	−0.262507
$742$	0	0
$743$	39.1431	1.43602	0.718010	−	0.696033i	$-0.245053\pi$
$743$	39.1431	1.43602	0.718010	+	0.696033i	$0.245053\pi$
$744$	0	0
$745$	−68.0498	−2.49315
$746$	0	0
$747$	8.64096	0.316156
$748$	0	0
$749$	5.46983	0.199863
$750$	0	0
$751$	−43.2173	−1.57702	−0.788511	−	0.615020i	$-0.789148\pi$
$751$	−43.2173	−1.57702	−0.788511	+	0.615020i	$0.789148\pi$
$752$	0	0
$753$	−30.4064	−1.10807
$754$	0	0
$755$	77.3457	2.81490
$756$	0	0
$757$	9.93714	0.361171	0.180586	−	0.983559i	$-0.442201\pi$
$757$	9.93714	0.361171	0.180586	+	0.983559i	$0.442201\pi$
$758$	0	0
$759$	−14.8662	−0.539609
$760$	0	0
$761$	14.5829	0.528630	0.264315	−	0.964436i	$-0.414854\pi$
$761$	14.5829	0.528630	0.264315	+	0.964436i	$0.414854\pi$
$762$	0	0
$763$	1.62361	0.0587785
$764$	0	0
$765$	−15.9390	−0.576275
$766$	0	0
$767$	−14.4306	−0.521059
$768$	0	0
$769$	−11.8924	−0.428853	−0.214426	−	0.976740i	$-0.568788\pi$
$769$	−11.8924	−0.428853	−0.214426	+	0.976740i	$0.568788\pi$
$770$	0	0
$771$	−0.951354	−0.0342622
$772$	0	0
$773$	15.0935	0.542876	0.271438	−	0.962456i	$-0.412501\pi$
$773$	15.0935	0.542876	0.271438	+	0.962456i	$0.412501\pi$
$774$	0	0
$775$	−17.0516	−0.612513
$776$	0	0
$777$	−54.3945	−1.95139
$778$	0	0
$779$	0.560629	0.0200866
$780$	0	0
$781$	−11.4671	−0.410325
$782$	0	0
$783$	16.4771	0.588844
$784$	0	0
$785$	40.4874	1.44506
$786$	0	0
$787$	5.98214	0.213240	0.106620	−	0.994300i	$-0.465997\pi$
$787$	5.98214	0.213240	0.106620	+	0.994300i	$0.465997\pi$
$788$	0	0
$789$	−32.5075	−1.15730
$790$	0	0
$791$	25.7659	0.916128
$792$	0	0
$793$	−11.8037	−0.419163
$794$	0	0
$795$	42.0692	1.49204
$796$	0	0
$797$	24.2718	0.859751	0.429876	−	0.902888i	$-0.358557\pi$
$797$	24.2718	0.859751	0.429876	+	0.902888i	$0.358557\pi$
$798$	0	0
$799$	34.1653	1.20868
$800$	0	0
$801$	−12.0589	−0.426081
$802$	0	0
$803$	8.95834	0.316133
$804$	0	0
$805$	−93.1268	−3.28229
$806$	0	0
$807$	15.6592	0.551230
$808$	0	0
$809$	−17.9557	−0.631289	−0.315644	−	0.948878i	$-0.602221\pi$
$809$	−17.9557	−0.631289	−0.315644	+	0.948878i	$0.602221\pi$
$810$	0	0
$811$	−5.48300	−0.192534	−0.0962671	−	0.995356i	$-0.530690\pi$
$811$	−5.48300	−0.192534	−0.0962671	+	0.995356i	$0.530690\pi$
$812$	0	0
$813$	−36.4381	−1.27794
$814$	0	0
$815$	−85.2185	−2.98508
$816$	0	0
$817$	9.40080	0.328892
$818$	0	0
$819$	−21.6001	−0.754770
$820$	0	0
$821$	52.6324	1.83688	0.918441	−	0.395558i	$-0.129449\pi$
$821$	52.6324	1.83688	0.918441	+	0.395558i	$0.129449\pi$
$822$	0	0
$823$	20.4784	0.713831	0.356916	−	0.934137i	$-0.383828\pi$
$823$	20.4784	0.713831	0.356916	+	0.934137i	$0.383828\pi$
$824$	0	0
$825$	14.6077	0.508575
$826$	0	0
$827$	0.359755	0.0125099	0.00625496	−	0.999980i	$-0.498009\pi$
$827$	0.359755	0.0125099	0.00625496	+	0.999980i	$0.498009\pi$
$828$	0	0
$829$	22.7422	0.789870	0.394935	−	0.918709i	$-0.370767\pi$
$829$	22.7422	0.789870	0.394935	+	0.918709i	$0.370767\pi$
$830$	0	0
$831$	33.1887	1.15130
$832$	0	0
$833$	23.9141	0.828574
$834$	0	0
$835$	−9.60403	−0.332361
$836$	0	0
$837$	−7.30659	−0.252553
$838$	0	0
$839$	−13.1616	−0.454388	−0.227194	−	0.973849i	$-0.572955\pi$
$839$	−13.1616	−0.454388	−0.227194	+	0.973849i	$0.572955\pi$
$840$	0	0
$841$	3.25998	0.112413
$842$	0	0
$843$	55.8778	1.92453
$844$	0	0
$845$	6.97077	0.239802
$846$	0	0
$847$	−3.93972	−0.135370
$848$	0	0
$849$	40.3228	1.38387
$850$	0	0
$851$	44.0885	1.51133
$852$	0	0
$853$	−36.9102	−1.26378	−0.631891	−	0.775057i	$-0.717721\pi$
$853$	−36.9102	−1.26378	−0.631891	+	0.775057i	$0.717721\pi$
$854$	0	0
$855$	−5.67958	−0.194237
$856$	0	0
$857$	24.3615	0.832174	0.416087	−	0.909325i	$-0.363401\pi$
$857$	24.3615	0.832174	0.416087	+	0.909325i	$0.363401\pi$
$858$	0	0
$859$	7.69750	0.262635	0.131318	−	0.991340i	$-0.458079\pi$
$859$	7.69750	0.262635	0.131318	+	0.991340i	$0.458079\pi$
$860$	0	0
$861$	4.76566	0.162413
$862$	0	0
$863$	29.0225	0.987937	0.493969	−	0.869480i	$-0.335546\pi$
$863$	29.0225	0.987937	0.493969	+	0.869480i	$0.335546\pi$
$864$	0	0
$865$	22.0916	0.751137
$866$	0	0
$867$	19.6871	0.668610
$868$	0	0
$869$	8.49105	0.288039
$870$	0	0
$871$	32.9713	1.11719
$872$	0	0
$873$	17.6431	0.597127
$874$	0	0
$875$	23.9262	0.808852
$876$	0	0
$877$	−36.0010	−1.21567	−0.607834	−	0.794064i	$-0.707961\pi$
$877$	−36.0010	−1.21567	−0.607834	+	0.794064i	$0.707961\pi$
$878$	0	0
$879$	−57.7111	−1.94655
$880$	0	0
$881$	−15.3587	−0.517448	−0.258724	−	0.965951i	$-0.583302\pi$
$881$	−15.3587	−0.517448	−0.258724	+	0.965951i	$0.583302\pi$
$882$	0	0
$883$	−40.2870	−1.35577	−0.677883	−	0.735170i	$-0.737102\pi$
$883$	−40.2870	−1.35577	−0.677883	+	0.735170i	$0.737102\pi$
$884$	0	0
$885$	−32.2546	−1.08423
$886$	0	0
$887$	7.09855	0.238346	0.119173	−	0.992874i	$-0.461976\pi$
$887$	7.09855	0.238346	0.119173	+	0.992874i	$0.461976\pi$
$888$	0	0
$889$	35.7728	1.19978
$890$	0	0
$891$	11.2258	0.376079
$892$	0	0
$893$	12.1742	0.407395
$894$	0	0
$895$	42.9673	1.43624
$896$	0	0
$897$	49.2342	1.64388
$898$	0	0
$899$	−14.3053	−0.477110
$900$	0	0
$901$	15.9490	0.531339
$902$	0	0
$903$	79.9121	2.65931
$904$	0	0
$905$	−47.1783	−1.56826
$906$	0	0
$907$	−28.4435	−0.944452	−0.472226	−	0.881478i	$-0.656549\pi$
$907$	−28.4435	−0.944452	−0.472226	+	0.881478i	$0.656549\pi$
$908$	0	0
$909$	−18.9911	−0.629895
$910$	0	0
$911$	−49.5740	−1.64246	−0.821230	−	0.570598i	$-0.806712\pi$
$911$	−49.5740	−1.64246	−0.821230	+	0.570598i	$0.806712\pi$
$912$	0	0
$913$	−5.21960	−0.172744
$914$	0	0
$915$	−26.3831	−0.872199
$916$	0	0
$917$	−40.9243	−1.35144
$918$	0	0
$919$	−17.8421	−0.588556	−0.294278	−	0.955720i	$-0.595079\pi$
$919$	−17.8421	−0.588556	−0.294278	+	0.955720i	$0.595079\pi$
$920$	0	0
$921$	48.8526	1.60975
$922$	0	0
$923$	37.9770	1.25003
$924$	0	0
$925$	−43.3219	−1.42441
$926$	0	0
$927$	−30.3010	−0.995215
$928$	0	0
$929$	36.0244	1.18192	0.590961	−	0.806700i	$-0.298749\pi$
$929$	36.0244	1.18192	0.590961	+	0.806700i	$0.298749\pi$
$930$	0	0
$931$	8.52137	0.279277
$932$	0	0
$933$	−2.99317	−0.0979921
$934$	0	0
$935$	9.62799	0.314869
$936$	0	0
$937$	−3.50371	−0.114461	−0.0572307	−	0.998361i	$-0.518227\pi$
$937$	−3.50371	−0.114461	−0.0572307	+	0.998361i	$0.518227\pi$
$938$	0	0
$939$	−27.8377	−0.908450
$940$	0	0
$941$	44.2074	1.44112	0.720560	−	0.693392i	$-0.243885\pi$
$941$	44.2074	1.44112	0.720560	+	0.693392i	$0.243885\pi$
$942$	0	0
$943$	−3.86272	−0.125788
$944$	0	0
$945$	39.2108	1.27553
$946$	0	0
$947$	−37.9515	−1.23326	−0.616629	−	0.787254i	$-0.711502\pi$
$947$	−37.9515	−1.23326	−0.616629	+	0.787254i	$0.711502\pi$
$948$	0	0
$949$	−29.6684	−0.963077
$950$	0	0
$951$	41.1491	1.33435
$952$	0	0
$953$	13.0303	0.422094	0.211047	−	0.977476i	$-0.432313\pi$
$953$	13.0303	0.422094	0.211047	+	0.977476i	$0.432313\pi$
$954$	0	0
$955$	10.6586	0.344906
$956$	0	0
$957$	12.2550	0.396149
$958$	0	0
$959$	3.14505	0.101559
$960$	0	0
$961$	−24.6565	−0.795370
$962$	0	0
$963$	−2.29844	−0.0740662
$964$	0	0
$965$	2.56558	0.0825889
$966$	0	0
$967$	−24.0504	−0.773409	−0.386705	−	0.922204i	$-0.626387\pi$
$967$	−24.0504	−0.773409	−0.386705	+	0.922204i	$0.626387\pi$
$968$	0	0
$969$	−6.05517	−0.194520
$970$	0	0
$971$	−54.1335	−1.73723	−0.868614	−	0.495490i	$-0.834988\pi$
$971$	−54.1335	−1.73723	−0.868614	+	0.495490i	$0.834988\pi$
$972$	0	0
$973$	19.8240	0.635529
$974$	0	0
$975$	−48.3781	−1.54934
$976$	0	0
$977$	−50.2507	−1.60766	−0.803832	−	0.594857i	$-0.797209\pi$
$977$	−50.2507	−1.60766	−0.803832	+	0.594857i	$0.797209\pi$
$978$	0	0
$979$	7.28423	0.232805
$980$	0	0
$981$	−0.682245	−0.0217824
$982$	0	0
$983$	−6.64633	−0.211985	−0.105993	−	0.994367i	$-0.533802\pi$
$983$	−6.64633	−0.211985	−0.105993	+	0.994367i	$0.533802\pi$
$984$	0	0
$985$	11.7263	0.373631
$986$	0	0
$987$	103.488	3.29405
$988$	0	0
$989$	−64.7714	−2.05961
$990$	0	0
$991$	−51.7993	−1.64546	−0.822729	−	0.568434i	$-0.807549\pi$
$991$	−51.7993	−1.64546	−0.822729	+	0.568434i	$0.807549\pi$
$992$	0	0
$993$	−26.8879	−0.853260
$994$	0	0
$995$	18.4159	0.583822
$996$	0	0
$997$	−14.2042	−0.449851	−0.224925	−	0.974376i	$-0.572214\pi$
$997$	−14.2042	−0.449851	−0.224925	+	0.974376i	$0.572214\pi$
$998$	0	0
$999$	−18.5633	−0.587318

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3344.2.a.t.1.2		5
4.3	odd	2		209.2.a.c.1.1	✓	5
12.11	even	2		1881.2.a.k.1.5		5
20.19	odd	2		5225.2.a.h.1.5		5
44.43	even	2		2299.2.a.n.1.5		5
76.75	even	2		3971.2.a.h.1.5		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
209.2.a.c.1.1	✓	5	4.3	odd	2
1881.2.a.k.1.5		5	12.11	even	2
2299.2.a.n.1.5		5	44.43	even	2
3344.2.a.t.1.2		5	1.1	even	1	trivial
3971.2.a.h.1.5		5	76.75	even	2
5225.2.a.h.1.5		5	20.19	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:	\( N \)	\(=\)	\( 3344 = 2^{4} \cdot 11 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3344.a (trivial)

\(f(q)\)	\(=\)	\(q-2.15766 q^{3} -3.43077 q^{5} -3.93972 q^{7} +1.65548 q^{9} +O(q^{10})\)	\(q-2.15766 q^{3} -3.43077 q^{5} -3.93972 q^{7} +1.65548 q^{9} -1.00000 q^{11} +3.31182 q^{13} +7.40242 q^{15} +2.80637 q^{17} +1.00000 q^{19} +8.50056 q^{21} -6.88998 q^{23} +6.77018 q^{25} +2.90101 q^{27} +5.67979 q^{29} -2.51864 q^{31} +2.15766 q^{33} +13.5163 q^{35} -6.39893 q^{37} -7.14577 q^{39} +0.560629 q^{41} +9.40080 q^{43} -5.67958 q^{45} +12.1742 q^{47} +8.52137 q^{49} -6.05517 q^{51} +5.68316 q^{53} +3.43077 q^{55} -2.15766 q^{57} -4.35730 q^{59} -3.56412 q^{61} -6.52213 q^{63} -11.3621 q^{65} +9.95563 q^{67} +14.8662 q^{69} +11.4671 q^{71} -8.95834 q^{73} -14.6077 q^{75} +3.93972 q^{77} -8.49105 q^{79} -11.2258 q^{81} +5.21960 q^{83} -9.62799 q^{85} -12.2550 q^{87} -7.28423 q^{89} -13.0476 q^{91} +5.43436 q^{93} -3.43077 q^{95} +10.6574 q^{97} -1.65548 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - q^{3} - 5 q^{5} - 6 q^{7} + 4 q^{9}+O(q^{10}) \) 5 * q - q^3 - 5 * q^5 - 6 * q^7 + 4 * q^9	\( 5 q - q^{3} - 5 q^{5} - 6 q^{7} + 4 q^{9} - 5 q^{11} + 4 q^{13} - 3 q^{15} - 4 q^{17} + 5 q^{19} + 10 q^{21} - 3 q^{23} + 6 q^{25} + 11 q^{27} + 10 q^{29} - 11 q^{31} + q^{33} + 8 q^{35} + q^{37} - 2 q^{39} + 2 q^{41} - 20 q^{43} - 28 q^{45} + 20 q^{47} + 3 q^{49} - 24 q^{51} - 14 q^{53} + 5 q^{55} - q^{57} - 3 q^{59} - 10 q^{61} - 24 q^{63} - 9 q^{67} - 5 q^{69} - 23 q^{71} + 18 q^{75} + 6 q^{77} - 44 q^{79} + q^{81} + 14 q^{83} - 12 q^{85} - 28 q^{87} - 27 q^{89} - 24 q^{91} - 27 q^{93} - 5 q^{95} + 15 q^{97} - 4 q^{99}+O(q^{100}) \) 5 * q - q^3 - 5 * q^5 - 6 * q^7 + 4 * q^9 - 5 * q^11 + 4 * q^13 - 3 * q^15 - 4 * q^17 + 5 * q^19 + 10 * q^21 - 3 * q^23 + 6 * q^25 + 11 * q^27 + 10 * q^29 - 11 * q^31 + q^33 + 8 * q^35 + q^37 - 2 * q^39 + 2 * q^41 - 20 * q^43 - 28 * q^45 + 20 * q^47 + 3 * q^49 - 24 * q^51 - 14 * q^53 + 5 * q^55 - q^57 - 3 * q^59 - 10 * q^61 - 24 * q^63 - 9 * q^67 - 5 * q^69 - 23 * q^71 + 18 * q^75 + 6 * q^77 - 44 * q^79 + q^81 + 14 * q^83 - 12 * q^85 - 28 * q^87 - 27 * q^89 - 24 * q^91 - 27 * q^93 - 5 * q^95 + 15 * q^97 - 4 * q^99

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