LMFDB - Embedded newform 2736.2.a.be.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2736,2,Mod(1,2736)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2736, 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("2736.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2736 = 2^{4} \cdot 3^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2736.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:	$21.8470699930$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.892.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 8x + 10 $ x^3 - x^2 - 8*x + 10
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1368)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.31955$ of defining polynomial
Character	$\chi$	$=$	2736.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.319551 q^{5} +2.61968 q^{7} +O(q^{10})$	$q-0.319551 q^{5} +2.61968 q^{7} +4.31955 q^{11} +6.51757 q^{13} +4.19802 q^{17} +1.00000 q^{19} -0.639102 q^{23} -4.89789 q^{25} -7.87847 q^{29} +6.00000 q^{31} -0.837122 q^{35} +4.00000 q^{37} -12.5176 q^{41} +1.38032 q^{43} +7.55892 q^{47} -0.137255 q^{49} -13.1567 q^{53} -1.38032 q^{55} +13.0351 q^{59} -5.89789 q^{61} -2.08270 q^{65} -11.7569 q^{67} -11.7569 q^{71} +15.1373 q^{73} +11.3159 q^{77} -0.517571 q^{79} +11.8785 q^{83} -1.34148 q^{85} +3.87847 q^{89} +17.0740 q^{91} -0.319551 q^{95} +11.2782 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 2 q^{5} + 2 q^{7}+O(q^{10}) $ 3 * q + 2 * q^5 + 2 * q^7	$ 3 q + 2 q^{5} + 2 q^{7} + 10 q^{11} - 4 q^{13} - 8 q^{17} + 3 q^{19} + 4 q^{23} + 3 q^{25} - 6 q^{29} + 18 q^{31} + 24 q^{35} + 12 q^{37} - 14 q^{41} + 10 q^{43} + 8 q^{47} + 29 q^{49} - 10 q^{53} - 10 q^{55} - 8 q^{59} - 24 q^{65} + 16 q^{73} - 16 q^{77} + 22 q^{79} + 18 q^{83} - 10 q^{85} - 6 q^{89} + 4 q^{91} + 2 q^{95} + 22 q^{97}+O(q^{100}) $ 3 * q + 2 * q^5 + 2 * q^7 + 10 * q^11 - 4 * q^13 - 8 * q^17 + 3 * q^19 + 4 * q^23 + 3 * q^25 - 6 * q^29 + 18 * q^31 + 24 * q^35 + 12 * q^37 - 14 * q^41 + 10 * q^43 + 8 * q^47 + 29 * q^49 - 10 * q^53 - 10 * q^55 - 8 * q^59 - 24 * q^65 + 16 * q^73 - 16 * q^77 + 22 * q^79 + 18 * q^83 - 10 * q^85 - 6 * q^89 + 4 * q^91 + 2 * q^95 + 22 * 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
$5$	−0.319551	−0.142907	−0.0714537	−	0.997444i	$-0.522764\pi$
$5$	−0.319551	−0.142907	−0.0714537	+	0.997444i	$0.522764\pi$
$6$	0	0
$7$	2.61968	0.990148	0.495074	−	0.868851i	$-0.335141\pi$
$7$	2.61968	0.990148	0.495074	+	0.868851i	$0.335141\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	4.31955	1.30239	0.651197	−	0.758909i	$-0.274267\pi$
$11$	4.31955	1.30239	0.651197	+	0.758909i	$0.274267\pi$
$12$	0	0
$13$	6.51757	1.80765	0.903825	−	0.427903i	$-0.140748\pi$
$13$	6.51757	1.80765	0.903825	+	0.427903i	$0.140748\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	4.19802	1.01817	0.509085	−	0.860716i	$-0.329984\pi$
$17$	4.19802	1.01817	0.509085	+	0.860716i	$0.329984\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−0.639102	−0.133262	−0.0666309	−	0.997778i	$-0.521225\pi$
$23$	−0.639102	−0.133262	−0.0666309	+	0.997778i	$0.521225\pi$
$24$	0	0
$25$	−4.89789	−0.979577
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−7.87847	−1.46300	−0.731498	−	0.681844i	$-0.761178\pi$
$29$	−7.87847	−1.46300	−0.731498	+	0.681844i	$0.761178\pi$
$30$	0	0
$31$	6.00000	1.07763	0.538816	−	0.842424i	$-0.318872\pi$
$31$	6.00000	1.07763	0.538816	+	0.842424i	$0.318872\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−0.837122	−0.141499
$36$	0	0
$37$	4.00000	0.657596	0.328798	−	0.944400i	$-0.393356\pi$
$37$	4.00000	0.657596	0.328798	+	0.944400i	$0.393356\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−12.5176	−1.95492	−0.977458	−	0.211129i	$-0.932286\pi$
$41$	−12.5176	−1.95492	−0.977458	+	0.211129i	$0.932286\pi$
$42$	0	0
$43$	1.38032	0.210496	0.105248	−	0.994446i	$-0.466436\pi$
$43$	1.38032	0.210496	0.105248	+	0.994446i	$0.466436\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	7.55892	1.10258	0.551291	−	0.834313i	$-0.314135\pi$
$47$	7.55892	1.10258	0.551291	+	0.834313i	$0.314135\pi$
$48$	0	0
$49$	−0.137255	−0.0196079
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−13.1567	−1.80721	−0.903604	−	0.428369i	$-0.859088\pi$
$53$	−13.1567	−1.80721	−0.903604	+	0.428369i	$0.859088\pi$
$54$	0	0
$55$	−1.38032	−0.186122
$56$	0	0
$57$	0	0
$58$	0	0
$59$	13.0351	1.69703	0.848516	−	0.529171i	$-0.177497\pi$
$59$	13.0351	1.69703	0.848516	+	0.529171i	$0.177497\pi$
$60$	0	0
$61$	−5.89789	−0.755147	−0.377574	−	0.925980i	$-0.623241\pi$
$61$	−5.89789	−0.755147	−0.377574	+	0.925980i	$0.623241\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−2.08270	−0.258327
$66$	0	0
$67$	−11.7569	−1.43634	−0.718169	−	0.695868i	$-0.755020\pi$
$67$	−11.7569	−1.43634	−0.718169	+	0.695868i	$0.755020\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−11.7569	−1.39529	−0.697646	−	0.716443i	$-0.745769\pi$
$71$	−11.7569	−1.39529	−0.697646	+	0.716443i	$0.745769\pi$
$72$	0	0
$73$	15.1373	1.77168	0.885841	−	0.463989i	$-0.153582\pi$
$73$	15.1373	1.77168	0.885841	+	0.463989i	$0.153582\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	11.3159	1.28956
$78$	0	0
$79$	−0.517571	−0.0582313	−0.0291157	−	0.999576i	$-0.509269\pi$
$79$	−0.517571	−0.0582313	−0.0291157	+	0.999576i	$0.509269\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	11.8785	1.30383	0.651916	−	0.758291i	$-0.273966\pi$
$83$	11.8785	1.30383	0.651916	+	0.758291i	$0.273966\pi$
$84$	0	0
$85$	−1.34148	−0.145504
$86$	0	0
$87$	0	0
$88$	0	0
$89$	3.87847	0.411117	0.205558	−	0.978645i	$-0.434099\pi$
$89$	3.87847	0.411117	0.205558	+	0.978645i	$0.434099\pi$
$90$	0	0
$91$	17.0740	1.78984
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−0.319551	−0.0327852
$96$	0	0
$97$	11.2782	1.14513	0.572564	−	0.819860i	$-0.305949\pi$
$97$	11.2782	1.14513	0.572564	+	0.819860i	$0.305949\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	4.51757	0.449515	0.224758	−	0.974415i	$-0.427841\pi$
$101$	4.51757	0.449515	0.224758	+	0.974415i	$0.427841\pi$
$102$	0	0
$103$	19.0351	1.87559	0.937794	−	0.347192i	$-0.112865\pi$
$103$	19.0351	1.87559	0.937794	+	0.347192i	$0.112865\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−7.11784	−0.688107	−0.344054	−	0.938950i	$-0.611800\pi$
$107$	−7.11784	−0.688107	−0.344054	+	0.938950i	$0.611800\pi$
$108$	0	0
$109$	−5.27820	−0.505560	−0.252780	−	0.967524i	$-0.581345\pi$
$109$	−5.27820	−0.505560	−0.252780	+	0.967524i	$0.581345\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	1.39973	0.131676	0.0658379	−	0.997830i	$-0.479028\pi$
$113$	1.39973	0.131676	0.0658379	+	0.997830i	$0.479028\pi$
$114$	0	0
$115$	0.204225	0.0190441
$116$	0	0
$117$	0	0
$118$	0	0
$119$	10.9975	1.00814
$120$	0	0
$121$	7.65852	0.696229
$122$	0	0
$123$	0	0
$124$	0	0
$125$	3.16288	0.282896
$126$	0	0
$127$	9.75694	0.865788	0.432894	−	0.901445i	$-0.357492\pi$
$127$	9.75694	0.865788	0.432894	+	0.901445i	$0.357492\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−19.4374	−1.69825	−0.849126	−	0.528190i	$-0.822871\pi$
$131$	−19.4374	−1.69825	−0.849126	+	0.528190i	$0.822871\pi$
$132$	0	0
$133$	2.61968	0.227155
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−10.9198	−0.932943	−0.466471	−	0.884536i	$-0.654475\pi$
$137$	−10.9198	−0.932943	−0.466471	+	0.884536i	$0.654475\pi$
$138$	0	0
$139$	6.61968	0.561474	0.280737	−	0.959785i	$-0.409421\pi$
$139$	6.61968	0.561474	0.280737	+	0.959785i	$0.409421\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	28.1530	2.35427
$144$	0	0
$145$	2.51757	0.209073
$146$	0	0
$147$	0	0
$148$	0	0
$149$	0.958652	0.0785359	0.0392679	−	0.999229i	$-0.487497\pi$
$149$	0.958652	0.0785359	0.0392679	+	0.999229i	$0.487497\pi$
$150$	0	0
$151$	−0.517571	−0.0421194	−0.0210597	−	0.999778i	$-0.506704\pi$
$151$	−0.517571	−0.0421194	−0.0210597	+	0.999778i	$0.506704\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−1.91730	−0.154002
$156$	0	0
$157$	2.00000	0.159617	0.0798087	−	0.996810i	$-0.474569\pi$
$157$	2.00000	0.159617	0.0798087	+	0.996810i	$0.474569\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−1.67424	−0.131949
$162$	0	0
$163$	−17.0351	−1.33430	−0.667148	−	0.744925i	$-0.732485\pi$
$163$	−17.0351	−1.33430	−0.667148	+	0.744925i	$0.732485\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−12.6391	−0.978043	−0.489022	−	0.872272i	$-0.662646\pi$
$167$	−12.6391	−0.978043	−0.489022	+	0.872272i	$0.662646\pi$
$168$	0	0
$169$	29.4787	2.26760
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−2.60027	−0.197695	−0.0988473	−	0.995103i	$-0.531516\pi$
$173$	−2.60027	−0.197695	−0.0988473	+	0.995103i	$0.531516\pi$
$174$	0	0
$175$	−12.8309	−0.969926
$176$	0	0
$177$	0	0
$178$	0	0
$179$	19.1178	1.42893	0.714467	−	0.699669i	$-0.246669\pi$
$179$	19.1178	1.42893	0.714467	+	0.699669i	$0.246669\pi$
$180$	0	0
$181$	−22.2745	−1.65565	−0.827826	−	0.560985i	$-0.810422\pi$
$181$	−22.2745	−1.65565	−0.827826	+	0.560985i	$0.810422\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−1.27820	−0.0939754
$186$	0	0
$187$	18.1336	1.32606
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−7.55892	−0.546944	−0.273472	−	0.961880i	$-0.588172\pi$
$191$	−7.55892	−0.546944	−0.273472	+	0.961880i	$0.588172\pi$
$192$	0	0
$193$	−0.517571	−0.0372556	−0.0186278	−	0.999826i	$-0.505930\pi$
$193$	−0.517571	−0.0372556	−0.0186278	+	0.999826i	$0.505930\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−11.4824	−0.818089	−0.409045	−	0.912514i	$-0.634138\pi$
$197$	−11.4824	−0.818089	−0.409045	+	0.912514i	$0.634138\pi$
$198$	0	0
$199$	1.38032	0.0978480	0.0489240	−	0.998803i	$-0.484421\pi$
$199$	1.38032	0.0978480	0.0489240	+	0.998803i	$0.484421\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−20.6391	−1.44858
$204$	0	0
$205$	4.00000	0.279372
$206$	0	0
$207$	0	0
$208$	0	0
$209$	4.31955	0.298790
$210$	0	0
$211$	−2.72180	−0.187376	−0.0936881	−	0.995602i	$-0.529866\pi$
$211$	−2.72180	−0.187376	−0.0936881	+	0.995602i	$0.529866\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−0.441081	−0.0300815
$216$	0	0
$217$	15.7181	1.06701
$218$	0	0
$219$	0	0
$220$	0	0
$221$	27.3609	1.84049
$222$	0	0
$223$	9.79577	0.655974	0.327987	−	0.944682i	$-0.393630\pi$
$223$	9.79577	0.655974	0.327987	+	0.944682i	$0.393630\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−12.6391	−0.838887	−0.419443	−	0.907782i	$-0.637775\pi$
$227$	−12.6391	−0.838887	−0.419443	+	0.907782i	$0.637775\pi$
$228$	0	0
$229$	−11.1373	−0.735971	−0.367985	−	0.929832i	$-0.619952\pi$
$229$	−11.1373	−0.735971	−0.367985	+	0.929832i	$0.619952\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	21.8723	1.43290	0.716450	−	0.697639i	$-0.245766\pi$
$233$	21.8723	1.43290	0.716450	+	0.697639i	$0.245766\pi$
$234$	0	0
$235$	−2.41546	−0.157567
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−25.8723	−1.67354	−0.836769	−	0.547556i	$-0.815558\pi$
$239$	−25.8723	−1.67354	−0.836769	+	0.547556i	$0.815558\pi$
$240$	0	0
$241$	9.79577	0.631001	0.315501	−	0.948925i	$-0.397828\pi$
$241$	9.79577	0.631001	0.315501	+	0.948925i	$0.397828\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	0.0438601	0.00280212
$246$	0	0
$247$	6.51757	0.414703
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−3.28441	−0.207310	−0.103655	−	0.994613i	$-0.533054\pi$
$251$	−3.28441	−0.207310	−0.103655	+	0.994613i	$0.533054\pi$
$252$	0	0
$253$	−2.76063	−0.173559
$254$	0	0
$255$	0	0
$256$	0	0
$257$	12.5176	0.780825	0.390412	−	0.920640i	$-0.372332\pi$
$257$	12.5176	0.780825	0.390412	+	0.920640i	$0.372332\pi$
$258$	0	0
$259$	10.4787	0.651117
$260$	0	0
$261$	0	0
$262$	0	0
$263$	24.1980	1.49211	0.746057	−	0.665882i	$-0.231945\pi$
$263$	24.1980	1.49211	0.746057	+	0.665882i	$0.231945\pi$
$264$	0	0
$265$	4.20423	0.258264
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−11.8785	−0.724243	−0.362122	−	0.932131i	$-0.617947\pi$
$269$	−11.8785	−0.724243	−0.362122	+	0.932131i	$0.617947\pi$
$270$	0	0
$271$	6.55641	0.398273	0.199137	−	0.979972i	$-0.436186\pi$
$271$	6.55641	0.398273	0.199137	+	0.979972i	$0.436186\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−21.1567	−1.27580
$276$	0	0
$277$	−16.6585	−1.00091	−0.500457	−	0.865762i	$-0.666835\pi$
$277$	−16.6585	−1.00091	−0.500457	+	0.865762i	$0.666835\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	10.3572	0.617859	0.308930	−	0.951085i	$-0.400029\pi$
$281$	10.3572	0.617859	0.308930	+	0.951085i	$0.400029\pi$
$282$	0	0
$283$	30.4155	1.80801	0.904006	−	0.427520i	$-0.140613\pi$
$283$	30.4155	1.80801	0.904006	+	0.427520i	$0.140613\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−32.7921	−1.93566
$288$	0	0
$289$	0.623376	0.0366692
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−12.2745	−0.717085	−0.358542	−	0.933513i	$-0.616726\pi$
$293$	−12.2745	−0.717085	−0.358542	+	0.933513i	$0.616726\pi$
$294$	0	0
$295$	−4.16539	−0.242518
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−4.16539	−0.240891
$300$	0	0
$301$	3.61599	0.208422
$302$	0	0
$303$	0	0
$304$	0	0
$305$	1.88467	0.107916
$306$	0	0
$307$	−2.72180	−0.155341	−0.0776706	−	0.996979i	$-0.524748\pi$
$307$	−2.72180	−0.155341	−0.0776706	+	0.996979i	$0.524748\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	0.837122	0.0474688	0.0237344	−	0.999718i	$-0.492444\pi$
$311$	0.837122	0.0474688	0.0237344	+	0.999718i	$0.492444\pi$
$312$	0	0
$313$	12.5564	0.709730	0.354865	−	0.934918i	$-0.384527\pi$
$313$	12.5564	0.709730	0.354865	+	0.934918i	$0.384527\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−0.517571	−0.0290697	−0.0145349	−	0.999894i	$-0.504627\pi$
$317$	−0.517571	−0.0290697	−0.0145349	+	0.999894i	$0.504627\pi$
$318$	0	0
$319$	−34.0315	−1.90540
$320$	0	0
$321$	0	0
$322$	0	0
$323$	4.19802	0.233584
$324$	0	0
$325$	−31.9223	−1.77073
$326$	0	0
$327$	0	0
$328$	0	0
$329$	19.8020	1.09172
$330$	0	0
$331$	−18.3133	−1.00659	−0.503296	−	0.864114i	$-0.667880\pi$
$331$	−18.3133	−1.00659	−0.503296	+	0.864114i	$0.667880\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	3.75694	0.205264
$336$	0	0
$337$	−3.27820	−0.178575	−0.0892876	−	0.996006i	$-0.528459\pi$
$337$	−3.27820	−0.178575	−0.0892876	+	0.996006i	$0.528459\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	25.9173	1.40350
$342$	0	0
$343$	−18.6974	−1.00956
$344$	0	0
$345$	0	0
$346$	0	0
$347$	13.5978	0.729966	0.364983	−	0.931014i	$-0.381075\pi$
$347$	13.5978	0.729966	0.364983	+	0.931014i	$0.381075\pi$
$348$	0	0
$349$	16.1724	0.865689	0.432844	−	0.901469i	$-0.357510\pi$
$349$	16.1724	0.865689	0.432844	+	0.901469i	$0.357510\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	6.47874	0.344828	0.172414	−	0.985025i	$-0.444843\pi$
$353$	6.47874	0.344828	0.172414	+	0.985025i	$0.444843\pi$
$354$	0	0
$355$	3.75694	0.199398
$356$	0	0
$357$	0	0
$358$	0	0
$359$	6.67676	0.352386	0.176193	−	0.984356i	$-0.443622\pi$
$359$	6.67676	0.352386	0.176193	+	0.984356i	$0.443622\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−4.83712	−0.253187
$366$	0	0
$367$	−25.0351	−1.30682	−0.653412	−	0.757003i	$-0.726663\pi$
$367$	−25.0351	−1.30682	−0.653412	+	0.757003i	$0.726663\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−34.4663	−1.78940
$372$	0	0
$373$	14.5564	0.753702	0.376851	−	0.926274i	$-0.377007\pi$
$373$	14.5564	0.753702	0.376851	+	0.926274i	$0.377007\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−51.3485	−2.64458
$378$	0	0
$379$	1.23937	0.0636621	0.0318310	−	0.999493i	$-0.489866\pi$
$379$	1.23937	0.0636621	0.0318310	+	0.999493i	$0.489866\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−1.67424	−0.0855499	−0.0427749	−	0.999085i	$-0.513620\pi$
$383$	−1.67424	−0.0855499	−0.0427749	+	0.999085i	$0.513620\pi$
$384$	0	0
$385$	−3.61599	−0.184288
$386$	0	0
$387$	0	0
$388$	0	0
$389$	26.6329	1.35034	0.675171	−	0.737661i	$-0.264070\pi$
$389$	26.6329	1.35034	0.675171	+	0.737661i	$0.264070\pi$
$390$	0	0
$391$	−2.68296	−0.135683
$392$	0	0
$393$	0	0
$394$	0	0
$395$	0.165390	0.00832169
$396$	0	0
$397$	−17.8979	−0.898269	−0.449135	−	0.893464i	$-0.648268\pi$
$397$	−17.8979	−0.898269	−0.449135	+	0.893464i	$0.648268\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	33.7958	1.68768	0.843840	−	0.536595i	$-0.180290\pi$
$401$	33.7958	1.68768	0.843840	+	0.536595i	$0.180290\pi$
$402$	0	0
$403$	39.1054	1.94798
$404$	0	0
$405$	0	0
$406$	0	0
$407$	17.2782	0.856449
$408$	0	0
$409$	−21.5139	−1.06379	−0.531896	−	0.846809i	$-0.678520\pi$
$409$	−21.5139	−1.06379	−0.531896	+	0.846809i	$0.678520\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	34.1480	1.68031
$414$	0	0
$415$	−3.79577	−0.186327
$416$	0	0
$417$	0	0
$418$	0	0
$419$	10.6003	0.517857	0.258928	−	0.965897i	$-0.416631\pi$
$419$	10.6003	0.517857	0.258928	+	0.965897i	$0.416631\pi$
$420$	0	0
$421$	−23.7181	−1.15595	−0.577975	−	0.816055i	$-0.696157\pi$
$421$	−23.7181	−1.15595	−0.577975	+	0.816055i	$0.696157\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−20.5614	−0.997376
$426$	0	0
$427$	−15.4506	−0.747707
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−19.3609	−0.932582	−0.466291	−	0.884631i	$-0.654410\pi$
$431$	−19.3609	−0.932582	−0.466291	+	0.884631i	$0.654410\pi$
$432$	0	0
$433$	−13.7569	−0.661116	−0.330558	−	0.943786i	$-0.607237\pi$
$433$	−13.7569	−0.661116	−0.330558	+	0.943786i	$0.607237\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−0.639102	−0.0305724
$438$	0	0
$439$	−20.2357	−0.965796	−0.482898	−	0.875677i	$-0.660416\pi$
$439$	−20.2357	−0.965796	−0.482898	+	0.875677i	$0.660416\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	6.87596	0.326687	0.163343	−	0.986569i	$-0.447772\pi$
$443$	6.87596	0.326687	0.163343	+	0.986569i	$0.447772\pi$
$444$	0	0
$445$	−1.23937	−0.0587517
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−4.76063	−0.224668	−0.112334	−	0.993670i	$-0.535833\pi$
$449$	−4.76063	−0.224668	−0.112334	+	0.993670i	$0.535833\pi$
$450$	0	0
$451$	−54.0703	−2.54607
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−5.45600	−0.255781
$456$	0	0
$457$	5.82022	0.272258	0.136129	−	0.990691i	$-0.456534\pi$
$457$	5.82022	0.272258	0.136129	+	0.990691i	$0.456534\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−15.0413	−0.700545	−0.350273	−	0.936648i	$-0.613911\pi$
$461$	−15.0413	−0.700545	−0.350273	+	0.936648i	$0.613911\pi$
$462$	0	0
$463$	27.6548	1.28523	0.642614	−	0.766190i	$-0.277850\pi$
$463$	27.6548	1.28523	0.642614	+	0.766190i	$0.277850\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	34.9513	1.61735	0.808676	−	0.588254i	$-0.200185\pi$
$467$	34.9513	1.61735	0.808676	+	0.588254i	$0.200185\pi$
$468$	0	0
$469$	−30.7995	−1.42219
$470$	0	0
$471$	0	0
$472$	0	0
$473$	5.96234	0.274149
$474$	0	0
$475$	−4.89789	−0.224730
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−9.67424	−0.442028	−0.221014	−	0.975271i	$-0.570937\pi$
$479$	−9.67424	−0.442028	−0.221014	+	0.975271i	$0.570937\pi$
$480$	0	0
$481$	26.0703	1.18870
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−3.60396	−0.163647
$486$	0	0
$487$	−20.5176	−0.929740	−0.464870	−	0.885379i	$-0.653899\pi$
$487$	−20.5176	−0.929740	−0.464870	+	0.885379i	$0.653899\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	36.9136	1.66589	0.832944	−	0.553357i	$-0.186654\pi$
$491$	36.9136	1.66589	0.832944	+	0.553357i	$0.186654\pi$
$492$	0	0
$493$	−33.0740	−1.48958
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−30.7995	−1.38154
$498$	0	0
$499$	−9.17609	−0.410778	−0.205389	−	0.978680i	$-0.565846\pi$
$499$	−9.17609	−0.410778	−0.205389	+	0.978680i	$0.565846\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−32.1530	−1.43363	−0.716815	−	0.697263i	$-0.754401\pi$
$503$	−32.1530	−1.43363	−0.716815	+	0.697263i	$0.754401\pi$
$504$	0	0
$505$	−1.44359	−0.0642391
$506$	0	0
$507$	0	0
$508$	0	0
$509$	16.9136	0.749683	0.374841	−	0.927089i	$-0.377697\pi$
$509$	16.9136	0.749683	0.374841	+	0.927089i	$0.377697\pi$
$510$	0	0
$511$	39.6548	1.75423
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−6.08270	−0.268036
$516$	0	0
$517$	32.6511	1.43600
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−10.0388	−0.439809	−0.219905	−	0.975521i	$-0.570575\pi$
$521$	−10.0388	−0.439809	−0.219905	+	0.975521i	$0.570575\pi$
$522$	0	0
$523$	−34.2745	−1.49872	−0.749360	−	0.662163i	$-0.769639\pi$
$523$	−34.2745	−1.49872	−0.749360	+	0.662163i	$0.769639\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	25.1881	1.09721
$528$	0	0
$529$	−22.5915	−0.982241
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−81.5842	−3.53380
$534$	0	0
$535$	2.27451	0.0983357
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−0.592882	−0.0255372
$540$	0	0
$541$	−28.1724	−1.21123	−0.605613	−	0.795759i	$-0.707072\pi$
$541$	−28.1724	−1.21123	−0.605613	+	0.795759i	$0.707072\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	1.68665	0.0722483
$546$	0	0
$547$	7.55271	0.322931	0.161465	−	0.986878i	$-0.448378\pi$
$547$	7.55271	0.322931	0.161465	+	0.986878i	$0.448378\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−7.87847	−0.335634
$552$	0	0
$553$	−1.35587	−0.0576576
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−23.4374	−0.993074	−0.496537	−	0.868016i	$-0.665395\pi$
$557$	−23.4374	−0.993074	−0.496537	+	0.868016i	$0.665395\pi$
$558$	0	0
$559$	8.99631	0.380503
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−34.3133	−1.44614	−0.723068	−	0.690777i	$-0.757268\pi$
$563$	−34.3133	−1.44614	−0.723068	+	0.690777i	$0.757268\pi$
$564$	0	0
$565$	−0.447286	−0.0188175
$566$	0	0
$567$	0	0
$568$	0	0
$569$	23.8785	1.00104	0.500519	−	0.865726i	$-0.333143\pi$
$569$	23.8785	1.00104	0.500519	+	0.865726i	$0.333143\pi$
$570$	0	0
$571$	9.03514	0.378109	0.189054	−	0.981967i	$-0.439458\pi$
$571$	9.03514	0.378109	0.189054	+	0.981967i	$0.439458\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	3.13025	0.130540
$576$	0	0
$577$	−18.8554	−0.784959	−0.392479	−	0.919761i	$-0.628383\pi$
$577$	−18.8554	−0.784959	−0.392479	+	0.919761i	$0.628383\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	31.1178	1.29099
$582$	0	0
$583$	−56.8309	−2.35370
$584$	0	0
$585$	0	0
$586$	0	0
$587$	3.68045	0.151908	0.0759542	−	0.997111i	$-0.475800\pi$
$587$	3.68045	0.151908	0.0759542	+	0.997111i	$0.475800\pi$
$588$	0	0
$589$	6.00000	0.247226
$590$	0	0
$591$	0	0
$592$	0	0
$593$	23.5139	0.965599	0.482800	−	0.875731i	$-0.339620\pi$
$593$	23.5139	0.965599	0.482800	+	0.875731i	$0.339620\pi$
$594$	0	0
$595$	−3.51426	−0.144070
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−6.31335	−0.257956	−0.128978	−	0.991647i	$-0.541170\pi$
$599$	−6.31335	−0.257956	−0.128978	+	0.991647i	$0.541170\pi$
$600$	0	0
$601$	19.4824	0.794705	0.397352	−	0.917666i	$-0.369929\pi$
$601$	19.4824	0.794705	0.397352	+	0.917666i	$0.369929\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−2.44729	−0.0994963
$606$	0	0
$607$	37.7569	1.53251	0.766253	−	0.642538i	$-0.222119\pi$
$607$	37.7569	1.53251	0.766253	+	0.642538i	$0.222119\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	49.2658	1.99308
$612$	0	0
$613$	−4.65852	−0.188156	−0.0940779	−	0.995565i	$-0.529990\pi$
$613$	−4.65852	−0.188156	−0.0940779	+	0.995565i	$0.529990\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	36.3510	1.46344	0.731718	−	0.681607i	$-0.238719\pi$
$617$	36.3510	1.46344	0.731718	+	0.681607i	$0.238719\pi$
$618$	0	0
$619$	28.0000	1.12542	0.562708	−	0.826656i	$-0.309760\pi$
$619$	28.0000	1.12542	0.562708	+	0.826656i	$0.309760\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	10.1604	0.407066
$624$	0	0
$625$	23.4787	0.939149
$626$	0	0
$627$	0	0
$628$	0	0
$629$	16.7921	0.669544
$630$	0	0
$631$	−35.4506	−1.41127	−0.705633	−	0.708577i	$-0.749337\pi$
$631$	−35.4506	−1.41127	−0.705633	+	0.708577i	$0.749337\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−3.11784	−0.123728
$636$	0	0
$637$	−0.894572	−0.0354442
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−23.8785	−0.943143	−0.471571	−	0.881828i	$-0.656313\pi$
$641$	−23.8785	−0.943143	−0.471571	+	0.881828i	$0.656313\pi$
$642$	0	0
$643$	3.93672	0.155249	0.0776246	−	0.996983i	$-0.475266\pi$
$643$	3.93672	0.155249	0.0776246	+	0.996983i	$0.475266\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	15.8020	0.621240	0.310620	−	0.950534i	$-0.399463\pi$
$647$	15.8020	0.621240	0.310620	+	0.950534i	$0.399463\pi$
$648$	0	0
$649$	56.3060	2.21020
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−43.2720	−1.69336	−0.846682	−	0.532099i	$-0.821403\pi$
$653$	−43.2720	−1.69336	−0.846682	+	0.532099i	$0.821403\pi$
$654$	0	0
$655$	6.21123	0.242693
$656$	0	0
$657$	0	0
$658$	0	0
$659$	21.8396	0.850751	0.425376	−	0.905017i	$-0.360142\pi$
$659$	21.8396	0.850751	0.425376	+	0.905017i	$0.360142\pi$
$660$	0	0
$661$	−9.03514	−0.351426	−0.175713	−	0.984441i	$-0.556223\pi$
$661$	−9.03514	−0.351426	−0.175713	+	0.984441i	$0.556223\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−0.837122	−0.0324622
$666$	0	0
$667$	5.03514	0.194962
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−25.4762	−0.983499
$672$	0	0
$673$	9.96116	0.383975	0.191987	−	0.981397i	$-0.438507\pi$
$673$	9.96116	0.383975	0.191987	+	0.981397i	$0.438507\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−6.35721	−0.244327	−0.122164	−	0.992510i	$-0.538983\pi$
$677$	−6.35721	−0.244327	−0.122164	+	0.992510i	$0.538983\pi$
$678$	0	0
$679$	29.5453	1.13385
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−37.1881	−1.42296	−0.711482	−	0.702704i	$-0.751976\pi$
$683$	−37.1881	−1.42296	−0.711482	+	0.702704i	$0.751976\pi$
$684$	0	0
$685$	3.48944	0.133325
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−85.7496	−3.26680
$690$	0	0
$691$	−32.8942	−1.25135	−0.625677	−	0.780082i	$-0.715177\pi$
$691$	−32.8942	−1.25135	−0.625677	+	0.780082i	$0.715177\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−2.11533	−0.0802389
$696$	0	0
$697$	−52.5490	−1.99044
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−16.4399	−0.620926	−0.310463	−	0.950585i	$-0.600484\pi$
$701$	−16.4399	−0.620926	−0.310463	+	0.950585i	$0.600484\pi$
$702$	0	0
$703$	4.00000	0.150863
$704$	0	0
$705$	0	0
$706$	0	0
$707$	11.8346	0.445086
$708$	0	0
$709$	−41.5139	−1.55909	−0.779543	−	0.626348i	$-0.784549\pi$
$709$	−41.5139	−1.55909	−0.779543	+	0.626348i	$0.784549\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−3.83461	−0.143607
$714$	0	0
$715$	−8.99631	−0.336443
$716$	0	0
$717$	0	0
$718$	0	0
$719$	32.8371	1.22462	0.612309	−	0.790619i	$-0.290241\pi$
$719$	32.8371	1.22462	0.612309	+	0.790619i	$0.290241\pi$
$720$	0	0
$721$	49.8661	1.85711
$722$	0	0
$723$	0	0
$724$	0	0
$725$	38.5879	1.43312
$726$	0	0
$727$	−4.89419	−0.181516	−0.0907578	−	0.995873i	$-0.528929\pi$
$727$	−4.89419	−0.181516	−0.0907578	+	0.995873i	$0.528929\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	5.79459	0.214321
$732$	0	0
$733$	−6.00000	−0.221615	−0.110808	−	0.993842i	$-0.535344\pi$
$733$	−6.00000	−0.221615	−0.110808	+	0.993842i	$0.535344\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−50.7847	−1.87068
$738$	0	0
$739$	−19.4506	−0.715502	−0.357751	−	0.933817i	$-0.616456\pi$
$739$	−19.4506	−0.715502	−0.357751	+	0.933817i	$0.616456\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	23.5915	0.865490	0.432745	−	0.901516i	$-0.357545\pi$
$743$	23.5915	0.865490	0.432745	+	0.901516i	$0.357545\pi$
$744$	0	0
$745$	−0.306338	−0.0112234
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−18.6465	−0.681328
$750$	0	0
$751$	−4.76063	−0.173718	−0.0868590	−	0.996221i	$-0.527683\pi$
$751$	−4.76063	−0.173718	−0.0868590	+	0.996221i	$0.527683\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	0.165390	0.00601917
$756$	0	0
$757$	−46.9330	−1.70581	−0.852905	−	0.522066i	$-0.825161\pi$
$757$	−46.9330	−1.70581	−0.852905	+	0.522066i	$0.825161\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	33.3108	1.20752	0.603758	−	0.797167i	$-0.293669\pi$
$761$	33.3108	1.20752	0.603758	+	0.797167i	$0.293669\pi$
$762$	0	0
$763$	−13.8272	−0.500579
$764$	0	0
$765$	0	0
$766$	0	0
$767$	84.9575	3.06764
$768$	0	0
$769$	−14.9330	−0.538499	−0.269249	−	0.963070i	$-0.586776\pi$
$769$	−14.9330	−0.538499	−0.269249	+	0.963070i	$0.586776\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	32.5176	1.16958	0.584788	−	0.811186i	$-0.301178\pi$
$773$	32.5176	1.16958	0.584788	+	0.811186i	$0.301178\pi$
$774$	0	0
$775$	−29.3873	−1.05562
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−12.5176	−0.448489
$780$	0	0
$781$	−50.7847	−1.81722
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−0.639102	−0.0228105
$786$	0	0
$787$	−14.5564	−0.518880	−0.259440	−	0.965759i	$-0.583538\pi$
$787$	−14.5564	−0.518880	−0.259440	+	0.965759i	$0.583538\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	3.66686	0.130379
$792$	0	0
$793$	−38.4399	−1.36504
$794$	0	0
$795$	0	0
$796$	0	0
$797$	23.2394	0.823181	0.411590	−	0.911369i	$-0.364973\pi$
$797$	23.2394	0.823181	0.411590	+	0.911369i	$0.364973\pi$
$798$	0	0
$799$	31.7325	1.12262
$800$	0	0
$801$	0	0
$802$	0	0
$803$	65.3861	2.30743
$804$	0	0
$805$	0.535006	0.0188565
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−39.7896	−1.39893	−0.699463	−	0.714668i	$-0.746578\pi$
$809$	−39.7896	−1.39893	−0.699463	+	0.714668i	$0.746578\pi$
$810$	0	0
$811$	−39.4750	−1.38616	−0.693078	−	0.720862i	$-0.743746\pi$
$811$	−39.4750	−1.38616	−0.693078	+	0.720862i	$0.743746\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	5.44359	0.190681
$816$	0	0
$817$	1.38032	0.0482911
$818$	0	0
$819$	0	0
$820$	0	0
$821$	16.3196	0.569556	0.284778	−	0.958593i	$-0.408080\pi$
$821$	16.3196	0.569556	0.284778	+	0.958593i	$0.408080\pi$
$822$	0	0
$823$	−16.9719	−0.591602	−0.295801	−	0.955250i	$-0.595587\pi$
$823$	−16.9719	−0.591602	−0.295801	+	0.955250i	$0.595587\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	25.1128	0.873258	0.436629	−	0.899642i	$-0.356172\pi$
$827$	25.1128	0.873258	0.436629	+	0.899642i	$0.356172\pi$
$828$	0	0
$829$	−35.5915	−1.23615	−0.618073	−	0.786121i	$-0.712086\pi$
$829$	−35.5915	−1.23615	−0.618073	+	0.786121i	$0.712086\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−0.576201	−0.0199642
$834$	0	0
$835$	4.03884	0.139770
$836$	0	0
$837$	0	0
$838$	0	0
$839$	29.8396	1.03018	0.515089	−	0.857137i	$-0.327759\pi$
$839$	29.8396	1.03018	0.515089	+	0.857137i	$0.327759\pi$
$840$	0	0
$841$	33.0703	1.14035
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−9.41995	−0.324056
$846$	0	0
$847$	20.0629	0.689369
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−2.55641	−0.0876325
$852$	0	0
$853$	14.0777	0.482010	0.241005	−	0.970524i	$-0.422523\pi$
$853$	14.0777	0.482010	0.241005	+	0.970524i	$0.422523\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−30.9963	−1.05881	−0.529407	−	0.848368i	$-0.677585\pi$
$857$	−30.9963	−1.05881	−0.529407	+	0.848368i	$0.677585\pi$
$858$	0	0
$859$	43.2464	1.47555	0.737774	−	0.675048i	$-0.235877\pi$
$859$	43.2464	1.47555	0.737774	+	0.675048i	$0.235877\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−12.9575	−0.441077	−0.220539	−	0.975378i	$-0.570782\pi$
$863$	−12.9575	−0.441077	−0.220539	+	0.975378i	$0.570782\pi$
$864$	0	0
$865$	0.830917	0.0282520
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−2.23568	−0.0758401
$870$	0	0
$871$	−76.6267	−2.59640
$872$	0	0
$873$	0	0
$874$	0	0
$875$	8.28574	0.280109
$876$	0	0
$877$	−11.7181	−0.395692	−0.197846	−	0.980233i	$-0.563395\pi$
$877$	−11.7181	−0.395692	−0.197846	+	0.980233i	$0.563395\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	28.3510	0.955169	0.477585	−	0.878586i	$-0.341512\pi$
$881$	28.3510	0.955169	0.477585	+	0.878586i	$0.341512\pi$
$882$	0	0
$883$	−22.3378	−0.751726	−0.375863	−	0.926675i	$-0.622654\pi$
$883$	−22.3378	−0.751726	−0.375863	+	0.926675i	$0.622654\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−36.5490	−1.22720	−0.613598	−	0.789619i	$-0.710278\pi$
$887$	−36.5490	−1.22720	−0.613598	+	0.789619i	$0.710278\pi$
$888$	0	0
$889$	25.5601	0.857258
$890$	0	0
$891$	0	0
$892$	0	0
$893$	7.55892	0.252950
$894$	0	0
$895$	−6.10912	−0.204205
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−47.2708	−1.57657
$900$	0	0
$901$	−55.2320	−1.84004
$902$	0	0
$903$	0	0
$904$	0	0
$905$	7.11784	0.236605
$906$	0	0
$907$	−47.3097	−1.57089	−0.785446	−	0.618931i	$-0.787566\pi$
$907$	−47.3097	−1.57089	−0.785446	+	0.618931i	$0.787566\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−37.8396	−1.25368	−0.626842	−	0.779147i	$-0.715653\pi$
$911$	−37.8396	−1.25368	−0.626842	+	0.779147i	$0.715653\pi$
$912$	0	0
$913$	51.3097	1.69810
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−50.9198	−1.68152
$918$	0	0
$919$	43.5915	1.43795	0.718976	−	0.695035i	$-0.244611\pi$
$919$	43.5915	1.43795	0.718976	+	0.695035i	$0.244611\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−76.6267	−2.52220
$924$	0	0
$925$	−19.5915	−0.644166
$926$	0	0
$927$	0	0
$928$	0	0
$929$	23.7569	0.779440	0.389720	−	0.920933i	$-0.372572\pi$
$929$	23.7569	0.779440	0.389720	+	0.920933i	$0.372572\pi$
$930$	0	0
$931$	−0.137255	−0.00449836
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−5.79459	−0.189504
$936$	0	0
$937$	−45.4894	−1.48608	−0.743038	−	0.669250i	$-0.766616\pi$
$937$	−45.4894	−1.48608	−0.743038	+	0.669250i	$0.766616\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−48.7482	−1.58915	−0.794573	−	0.607168i	$-0.792305\pi$
$941$	−48.7482	−1.58915	−0.794573	+	0.607168i	$0.792305\pi$
$942$	0	0
$943$	8.00000	0.260516
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−5.94876	−0.193309	−0.0966543	−	0.995318i	$-0.530814\pi$
$947$	−5.94876	−0.193309	−0.0966543	+	0.995318i	$0.530814\pi$
$948$	0	0
$949$	98.6581	3.20258
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−14.4473	−0.467993	−0.233997	−	0.972237i	$-0.575181\pi$
$953$	−14.4473	−0.467993	−0.233997	+	0.972237i	$0.575181\pi$
$954$	0	0
$955$	2.41546	0.0781624
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−28.6065	−0.923751
$960$	0	0
$961$	5.00000	0.161290
$962$	0	0
$963$	0	0
$964$	0	0
$965$	0.165390	0.00532410
$966$	0	0
$967$	−18.0703	−0.581101	−0.290551	−	0.956860i	$-0.593838\pi$
$967$	−18.0703	−0.581101	−0.290551	+	0.956860i	$0.593838\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−40.0000	−1.28366	−0.641831	−	0.766846i	$-0.721825\pi$
$971$	−40.0000	−1.28366	−0.641831	+	0.766846i	$0.721825\pi$
$972$	0	0
$973$	17.3415	0.555942
$974$	0	0
$975$	0	0
$976$	0	0
$977$	13.1567	0.420919	0.210460	−	0.977603i	$-0.432504\pi$
$977$	13.1567	0.420919	0.210460	+	0.977603i	$0.432504\pi$
$978$	0	0
$979$	16.7532	0.535436
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−6.23568	−0.198887	−0.0994436	−	0.995043i	$-0.531706\pi$
$983$	−6.23568	−0.198887	−0.0994436	+	0.995043i	$0.531706\pi$
$984$	0	0
$985$	3.66922	0.116911
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−0.882162	−0.0280511
$990$	0	0
$991$	47.8661	1.52052	0.760258	−	0.649622i	$-0.225073\pi$
$991$	47.8661	1.52052	0.760258	+	0.649622i	$0.225073\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−0.441081	−0.0139832
$996$	0	0
$997$	44.9256	1.42281	0.711405	−	0.702783i	$-0.248059\pi$
$997$	44.9256	1.42281	0.711405	+	0.702783i	$0.248059\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2736.2.a.be.1.2		3
3.2	odd	2		2736.2.a.bc.1.2		3
4.3	odd	2		1368.2.a.o.1.2	yes	3
12.11	even	2		1368.2.a.m.1.2	✓	3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1368.2.a.m.1.2	✓	3	12.11	even	2
1368.2.a.o.1.2	yes	3	4.3	odd	2
2736.2.a.bc.1.2		3	3.2	odd	2
2736.2.a.be.1.2		3	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:	\( N \)	\(=\)	\( 2736 = 2^{4} \cdot 3^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2736.a (trivial)

\(f(q)\)	\(=\)	\(q-0.319551 q^{5} +2.61968 q^{7} +O(q^{10})\)	\(q-0.319551 q^{5} +2.61968 q^{7} +4.31955 q^{11} +6.51757 q^{13} +4.19802 q^{17} +1.00000 q^{19} -0.639102 q^{23} -4.89789 q^{25} -7.87847 q^{29} +6.00000 q^{31} -0.837122 q^{35} +4.00000 q^{37} -12.5176 q^{41} +1.38032 q^{43} +7.55892 q^{47} -0.137255 q^{49} -13.1567 q^{53} -1.38032 q^{55} +13.0351 q^{59} -5.89789 q^{61} -2.08270 q^{65} -11.7569 q^{67} -11.7569 q^{71} +15.1373 q^{73} +11.3159 q^{77} -0.517571 q^{79} +11.8785 q^{83} -1.34148 q^{85} +3.87847 q^{89} +17.0740 q^{91} -0.319551 q^{95} +11.2782 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 2 q^{5} + 2 q^{7}+O(q^{10}) \) 3 * q + 2 * q^5 + 2 * q^7	\( 3 q + 2 q^{5} + 2 q^{7} + 10 q^{11} - 4 q^{13} - 8 q^{17} + 3 q^{19} + 4 q^{23} + 3 q^{25} - 6 q^{29} + 18 q^{31} + 24 q^{35} + 12 q^{37} - 14 q^{41} + 10 q^{43} + 8 q^{47} + 29 q^{49} - 10 q^{53} - 10 q^{55} - 8 q^{59} - 24 q^{65} + 16 q^{73} - 16 q^{77} + 22 q^{79} + 18 q^{83} - 10 q^{85} - 6 q^{89} + 4 q^{91} + 2 q^{95} + 22 q^{97}+O(q^{100}) \) 3 * q + 2 * q^5 + 2 * q^7 + 10 * q^11 - 4 * q^13 - 8 * q^17 + 3 * q^19 + 4 * q^23 + 3 * q^25 - 6 * q^29 + 18 * q^31 + 24 * q^35 + 12 * q^37 - 14 * q^41 + 10 * q^43 + 8 * q^47 + 29 * q^49 - 10 * q^53 - 10 * q^55 - 8 * q^59 - 24 * q^65 + 16 * q^73 - 16 * q^77 + 22 * q^79 + 18 * q^83 - 10 * q^85 - 6 * q^89 + 4 * q^91 + 2 * q^95 + 22 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more