LMFDB - Embedded newform 2576.2.a.be.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2576,2,Mod(1,2576)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2576, 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("2576.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2576 = 2^{4} \cdot 7 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2576.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:	$20.5694635607$
Analytic rank:	$0$
Dimension:	$5$
Coefficient field:	5.5.3385684.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 2x^{4} - 12x^{3} + 22x^{2} + 20x - 16 $ x^5 - 2x^4 - 12x^3 + 22x^2 + 20x - 16
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 1288)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$0.564619$ of defining polynomial
Character	$\chi$	$=$	2576.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.35133 q^{3} +0.564619 q^{5} -1.00000 q^{7} +8.23144 q^{9} +O(q^{10})$	$q-3.35133 q^{3} +0.564619 q^{5} -1.00000 q^{7} +8.23144 q^{9} +3.02146 q^{11} +5.91595 q^{13} -1.89223 q^{15} +2.56462 q^{17} +6.33695 q^{19} +3.35133 q^{21} +1.00000 q^{23} -4.68121 q^{25} -17.5323 q^{27} -5.23144 q^{29} +4.22210 q^{31} -10.1259 q^{33} -0.564619 q^{35} -5.17216 q^{37} -19.8263 q^{39} +0.660784 q^{41} +4.91369 q^{43} +4.64762 q^{45} -10.0111 q^{47} +1.00000 q^{49} -8.59489 q^{51} +6.00000 q^{53} +1.70597 q^{55} -21.2372 q^{57} +12.2211 q^{59} +5.47831 q^{61} -8.23144 q^{63} +3.34026 q^{65} +8.34095 q^{67} -3.35133 q^{69} -11.0807 q^{71} +6.20998 q^{73} +15.6883 q^{75} -3.02146 q^{77} -4.34095 q^{79} +34.0623 q^{81} -6.99669 q^{83} +1.44803 q^{85} +17.5323 q^{87} +13.9270 q^{89} -5.91595 q^{91} -14.1497 q^{93} +3.57796 q^{95} -16.3069 q^{97} +24.8710 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q + 3 q^{3} + 2 q^{5} - 5 q^{7} + 14 q^{9}+O(q^{10}) $ 5 * q + 3 * q^3 + 2 * q^5 - 5 * q^7 + 14 * q^9	$ 5 q + 3 q^{3} + 2 q^{5} - 5 q^{7} + 14 q^{9} + 2 q^{11} + 9 q^{13} + 2 q^{15} + 12 q^{17} + 12 q^{19} - 3 q^{21} + 5 q^{23} + 3 q^{25} - 3 q^{27} + q^{29} + 3 q^{31} - 16 q^{33} - 2 q^{35} + 2 q^{37} - 21 q^{39} + 19 q^{41} + 14 q^{45} - 17 q^{47} + 5 q^{49} + 8 q^{51} + 30 q^{53} + 2 q^{55} + 14 q^{57} + 14 q^{59} + 2 q^{61} - 14 q^{63} + 30 q^{65} + 2 q^{67} + 3 q^{69} - 43 q^{71} + 17 q^{73} + 39 q^{75} - 2 q^{77} + 18 q^{79} + 73 q^{81} - 2 q^{83} + 32 q^{85} + 3 q^{87} + 16 q^{89} - 9 q^{91} - 27 q^{93} - 12 q^{95} + 18 q^{97} - 4 q^{99}+O(q^{100}) $ 5 * q + 3 * q^3 + 2 * q^5 - 5 * q^7 + 14 * q^9 + 2 * q^11 + 9 * q^13 + 2 * q^15 + 12 * q^17 + 12 * q^19 - 3 * q^21 + 5 * q^23 + 3 * q^25 - 3 * q^27 + q^29 + 3 * q^31 - 16 * q^33 - 2 * q^35 + 2 * q^37 - 21 * q^39 + 19 * q^41 + 14 * q^45 - 17 * q^47 + 5 * q^49 + 8 * q^51 + 30 * q^53 + 2 * q^55 + 14 * q^57 + 14 * q^59 + 2 * q^61 - 14 * q^63 + 30 * q^65 + 2 * q^67 + 3 * q^69 - 43 * q^71 + 17 * q^73 + 39 * q^75 - 2 * q^77 + 18 * q^79 + 73 * q^81 - 2 * q^83 + 32 * q^85 + 3 * q^87 + 16 * q^89 - 9 * q^91 - 27 * q^93 - 12 * q^95 + 18 * 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
$3$	−3.35133	−1.93489	−0.967447	−	0.253074i	$-0.918558\pi$
$3$	−3.35133	−1.93489	−0.967447	+	0.253074i	$0.918558\pi$
$4$	0	0
$5$	0.564619	0.252505	0.126253	−	0.991998i	$-0.459705\pi$
$5$	0.564619	0.252505	0.126253	+	0.991998i	$0.459705\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	8.23144	2.74381
$10$	0	0
$11$	3.02146	0.911005	0.455503	−	0.890234i	$-0.349460\pi$
$11$	3.02146	0.911005	0.455503	+	0.890234i	$0.349460\pi$
$12$	0	0
$13$	5.91595	1.64079	0.820395	−	0.571797i	$-0.193754\pi$
$13$	5.91595	1.64079	0.820395	+	0.571797i	$0.193754\pi$
$14$	0	0
$15$	−1.89223	−0.488571
$16$	0	0
$17$	2.56462	0.622011	0.311006	−	0.950408i	$-0.399334\pi$
$17$	2.56462	0.622011	0.311006	+	0.950408i	$0.399334\pi$
$18$	0	0
$19$	6.33695	1.45380	0.726898	−	0.686745i	$-0.240961\pi$
$19$	6.33695	1.45380	0.726898	+	0.686745i	$0.240961\pi$
$20$	0	0
$21$	3.35133	0.731321
$22$	0	0
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	0	0
$25$	−4.68121	−0.936241
$26$	0	0
$27$	−17.5323	−3.37409
$28$	0	0
$29$	−5.23144	−0.971454	−0.485727	−	0.874110i	$-0.661445\pi$
$29$	−5.23144	−0.971454	−0.485727	+	0.874110i	$0.661445\pi$
$30$	0	0
$31$	4.22210	0.758311	0.379156	−	0.925333i	$-0.376215\pi$
$31$	4.22210	0.758311	0.379156	+	0.925333i	$0.376215\pi$
$32$	0	0
$33$	−10.1259	−1.76270
$34$	0	0
$35$	−0.564619	−0.0954380
$36$	0	0
$37$	−5.17216	−0.850298	−0.425149	−	0.905123i	$-0.639778\pi$
$37$	−5.17216	−0.850298	−0.425149	+	0.905123i	$0.639778\pi$
$38$	0	0
$39$	−19.8263	−3.17475
$40$	0	0
$41$	0.660784	0.103197	0.0515986	−	0.998668i	$-0.483568\pi$
$41$	0.660784	0.103197	0.0515986	+	0.998668i	$0.483568\pi$
$42$	0	0
$43$	4.91369	0.749330	0.374665	−	0.927160i	$-0.377758\pi$
$43$	4.91369	0.749330	0.374665	+	0.927160i	$0.377758\pi$
$44$	0	0
$45$	4.64762	0.692827
$46$	0	0
$47$	−10.0111	−1.46027	−0.730133	−	0.683305i	$-0.760542\pi$
$47$	−10.0111	−1.46027	−0.730133	+	0.683305i	$0.760542\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−8.59489	−1.20353
$52$	0	0
$53$	6.00000	0.824163	0.412082	−	0.911147i	$-0.364802\pi$
$53$	6.00000	0.824163	0.412082	+	0.911147i	$0.364802\pi$
$54$	0	0
$55$	1.70597	0.230033
$56$	0	0
$57$	−21.2372	−2.81294
$58$	0	0
$59$	12.2211	1.59105	0.795523	−	0.605923i	$-0.207196\pi$
$59$	12.2211	1.59105	0.795523	+	0.605923i	$0.207196\pi$
$60$	0	0
$61$	5.47831	0.701425	0.350713	−	0.936483i	$-0.385939\pi$
$61$	5.47831	0.701425	0.350713	+	0.936483i	$0.385939\pi$
$62$	0	0
$63$	−8.23144	−1.03706
$64$	0	0
$65$	3.34026	0.414308
$66$	0	0
$67$	8.34095	1.01901	0.509505	−	0.860468i	$-0.329829\pi$
$67$	8.34095	1.01901	0.509505	+	0.860468i	$0.329829\pi$
$68$	0	0
$69$	−3.35133	−0.403453
$70$	0	0
$71$	−11.0807	−1.31504	−0.657521	−	0.753436i	$-0.728395\pi$
$71$	−11.0807	−1.31504	−0.657521	+	0.753436i	$0.728395\pi$
$72$	0	0
$73$	6.20998	0.726823	0.363412	−	0.931629i	$-0.381612\pi$
$73$	6.20998	0.726823	0.363412	+	0.931629i	$0.381612\pi$
$74$	0	0
$75$	15.6883	1.81153
$76$	0	0
$77$	−3.02146	−0.344328
$78$	0	0
$79$	−4.34095	−0.488395	−0.244197	−	0.969726i	$-0.578525\pi$
$79$	−4.34095	−0.488395	−0.244197	+	0.969726i	$0.578525\pi$
$80$	0	0
$81$	34.0623	3.78470
$82$	0	0
$83$	−6.99669	−0.767987	−0.383994	−	0.923336i	$-0.625451\pi$
$83$	−6.99669	−0.767987	−0.383994	+	0.923336i	$0.625451\pi$
$84$	0	0
$85$	1.44803	0.157061
$86$	0	0
$87$	17.5323	1.87966
$88$	0	0
$89$	13.9270	1.47626	0.738131	−	0.674657i	$-0.235709\pi$
$89$	13.9270	1.47626	0.738131	+	0.674657i	$0.235709\pi$
$90$	0	0
$91$	−5.91595	−0.620160
$92$	0	0
$93$	−14.1497	−1.46725
$94$	0	0
$95$	3.57796	0.367091
$96$	0	0
$97$	−16.3069	−1.65572	−0.827858	−	0.560938i	$-0.810441\pi$
$97$	−16.3069	−1.65572	−0.827858	+	0.560938i	$0.810441\pi$
$98$	0	0
$99$	24.8710	2.49963
$100$	0	0
$101$	9.72013	0.967189	0.483595	−	0.875292i	$-0.339331\pi$
$101$	9.72013	0.967189	0.483595	+	0.875292i	$0.339331\pi$
$102$	0	0
$103$	5.31949	0.524145	0.262072	−	0.965048i	$-0.415594\pi$
$103$	5.31949	0.524145	0.262072	+	0.965048i	$0.415594\pi$
$104$	0	0
$105$	1.89223	0.184662
$106$	0	0
$107$	−0.659743	−0.0637798	−0.0318899	−	0.999491i	$-0.510153\pi$
$107$	−0.659743	−0.0637798	−0.0318899	+	0.999491i	$0.510153\pi$
$108$	0	0
$109$	−17.3766	−1.66437	−0.832187	−	0.554495i	$-0.812911\pi$
$109$	−17.3766	−1.66437	−0.832187	+	0.554495i	$0.812911\pi$
$110$	0	0
$111$	17.3336	1.64524
$112$	0	0
$113$	−14.5058	−1.36459	−0.682296	−	0.731076i	$-0.739018\pi$
$113$	−14.5058	−1.36459	−0.682296	+	0.731076i	$0.739018\pi$
$114$	0	0
$115$	0.564619	0.0526510
$116$	0	0
$117$	48.6968	4.50202
$118$	0	0
$119$	−2.56462	−0.235098
$120$	0	0
$121$	−1.87076	−0.170069
$122$	0	0
$123$	−2.21451	−0.199675
$124$	0	0
$125$	−5.46619	−0.488911
$126$	0	0
$127$	11.5724	1.02688	0.513442	−	0.858124i	$-0.328370\pi$
$127$	11.5724	1.02688	0.513442	+	0.858124i	$0.328370\pi$
$128$	0	0
$129$	−16.4674	−1.44987
$130$	0	0
$131$	8.58781	0.750321	0.375160	−	0.926960i	$-0.377588\pi$
$131$	8.58781	0.750321	0.375160	+	0.926960i	$0.377588\pi$
$132$	0	0
$133$	−6.33695	−0.549483
$134$	0	0
$135$	−9.89907	−0.851976
$136$	0	0
$137$	−13.8066	−1.17958	−0.589789	−	0.807557i	$-0.700789\pi$
$137$	−13.8066	−1.17958	−0.589789	+	0.807557i	$0.700789\pi$
$138$	0	0
$139$	−14.3554	−1.21761	−0.608805	−	0.793320i	$-0.708351\pi$
$139$	−14.3554	−1.21761	−0.608805	+	0.793320i	$0.708351\pi$
$140$	0	0
$141$	33.5505	2.82546
$142$	0	0
$143$	17.8748	1.49477
$144$	0	0
$145$	−2.95377	−0.245297
$146$	0	0
$147$	−3.35133	−0.276413
$148$	0	0
$149$	15.9934	1.31023	0.655115	−	0.755529i	$-0.272620\pi$
$149$	15.9934	1.31023	0.655115	+	0.755529i	$0.272620\pi$
$150$	0	0
$151$	10.5068	0.855035	0.427518	−	0.904007i	$-0.359388\pi$
$151$	10.5068	0.855035	0.427518	+	0.904007i	$0.359388\pi$
$152$	0	0
$153$	21.1105	1.70668
$154$	0	0
$155$	2.38387	0.191477
$156$	0	0
$157$	5.56131	0.443841	0.221921	−	0.975065i	$-0.428767\pi$
$157$	5.56131	0.443841	0.221921	+	0.975065i	$0.428767\pi$
$158$	0	0
$159$	−20.1080	−1.59467
$160$	0	0
$161$	−1.00000	−0.0788110
$162$	0	0
$163$	10.7082	0.838734	0.419367	−	0.907817i	$-0.362252\pi$
$163$	10.7082	0.838734	0.419367	+	0.907817i	$0.362252\pi$
$164$	0	0
$165$	−5.71729	−0.445090
$166$	0	0
$167$	0.241827	0.0187131	0.00935657	−	0.999956i	$-0.497022\pi$
$167$	0.241827	0.0187131	0.00935657	+	0.999956i	$0.497022\pi$
$168$	0	0
$169$	21.9985	1.69219
$170$	0	0
$171$	52.1622	3.98895
$172$	0	0
$173$	8.16886	0.621067	0.310533	−	0.950562i	$-0.399492\pi$
$173$	8.16886	0.621067	0.310533	+	0.950562i	$0.399492\pi$
$174$	0	0
$175$	4.68121	0.353866
$176$	0	0
$177$	−40.9568	−3.07851
$178$	0	0
$179$	21.9167	1.63813	0.819066	−	0.573699i	$-0.194492\pi$
$179$	21.9167	1.63813	0.819066	+	0.573699i	$0.194492\pi$
$180$	0	0
$181$	17.7900	1.32232	0.661159	−	0.750246i	$-0.270065\pi$
$181$	17.7900	1.32232	0.661159	+	0.750246i	$0.270065\pi$
$182$	0	0
$183$	−18.3596	−1.35718
$184$	0	0
$185$	−2.92030	−0.214705
$186$	0	0
$187$	7.74890	0.566656
$188$	0	0
$189$	17.5323	1.27529
$190$	0	0
$191$	−19.0041	−1.37509	−0.687543	−	0.726144i	$-0.741311\pi$
$191$	−19.0041	−1.37509	−0.687543	+	0.726144i	$0.741311\pi$
$192$	0	0
$193$	12.6829	0.912938	0.456469	−	0.889739i	$-0.349114\pi$
$193$	12.6829	0.912938	0.456469	+	0.889739i	$0.349114\pi$
$194$	0	0
$195$	−11.1943	−0.801642
$196$	0	0
$197$	−12.3246	−0.878091	−0.439046	−	0.898465i	$-0.644683\pi$
$197$	−12.3246	−0.878091	−0.439046	+	0.898465i	$0.644683\pi$
$198$	0	0
$199$	11.3150	0.802096	0.401048	−	0.916057i	$-0.368646\pi$
$199$	11.3150	0.802096	0.401048	+	0.916057i	$0.368646\pi$
$200$	0	0
$201$	−27.9533	−1.97167
$202$	0	0
$203$	5.23144	0.367175
$204$	0	0
$205$	0.373091	0.0260578
$206$	0	0
$207$	8.23144	0.572125
$208$	0	0
$209$	19.1469	1.32442
$210$	0	0
$211$	0.239785	0.0165075	0.00825375	−	0.999966i	$-0.497373\pi$
$211$	0.239785	0.0165075	0.00825375	+	0.999966i	$0.497373\pi$
$212$	0	0
$213$	37.1353	2.54447
$214$	0	0
$215$	2.77436	0.189210
$216$	0	0
$217$	−4.22210	−0.286615
$218$	0	0
$219$	−20.8117	−1.40633
$220$	0	0
$221$	15.1722	1.02059
$222$	0	0
$223$	−14.8120	−0.991882	−0.495941	−	0.868356i	$-0.665177\pi$
$223$	−14.8120	−0.991882	−0.495941	+	0.868356i	$0.665177\pi$
$224$	0	0
$225$	−38.5331	−2.56887
$226$	0	0
$227$	13.2794	0.881385	0.440693	−	0.897658i	$-0.354733\pi$
$227$	13.2794	0.881385	0.440693	+	0.897658i	$0.354733\pi$
$228$	0	0
$229$	17.8374	1.17873	0.589365	−	0.807867i	$-0.299378\pi$
$229$	17.8374	1.17873	0.589365	+	0.807867i	$0.299378\pi$
$230$	0	0
$231$	10.1259	0.666237
$232$	0	0
$233$	7.27437	0.476560	0.238280	−	0.971197i	$-0.423416\pi$
$233$	7.27437	0.476560	0.238280	+	0.971197i	$0.423416\pi$
$234$	0	0
$235$	−5.65244	−0.368725
$236$	0	0
$237$	14.5480	0.944992
$238$	0	0
$239$	2.93795	0.190040	0.0950200	−	0.995475i	$-0.469708\pi$
$239$	2.93795	0.190040	0.0950200	+	0.995475i	$0.469708\pi$
$240$	0	0
$241$	−12.6792	−0.816741	−0.408371	−	0.912816i	$-0.633903\pi$
$241$	−12.6792	−0.816741	−0.408371	+	0.912816i	$0.633903\pi$
$242$	0	0
$243$	−61.5572	−3.94890
$244$	0	0
$245$	0.564619	0.0360722
$246$	0	0
$247$	37.4891	2.38537
$248$	0	0
$249$	23.4483	1.48597
$250$	0	0
$251$	−2.16886	−0.136897	−0.0684485	−	0.997655i	$-0.521805\pi$
$251$	−2.16886	−0.136897	−0.0684485	+	0.997655i	$0.521805\pi$
$252$	0	0
$253$	3.02146	0.189958
$254$	0	0
$255$	−4.85284	−0.303896
$256$	0	0
$257$	12.3246	0.768787	0.384394	−	0.923169i	$-0.374411\pi$
$257$	12.3246	0.768787	0.384394	+	0.923169i	$0.374411\pi$
$258$	0	0
$259$	5.17216	0.321383
$260$	0	0
$261$	−43.0623	−2.66549
$262$	0	0
$263$	−15.4798	−0.954526	−0.477263	−	0.878760i	$-0.658371\pi$
$263$	−15.4798	−0.954526	−0.477263	+	0.878760i	$0.658371\pi$
$264$	0	0
$265$	3.38771	0.208105
$266$	0	0
$267$	−46.6741	−2.85641
$268$	0	0
$269$	−18.3501	−1.11882	−0.559412	−	0.828890i	$-0.688973\pi$
$269$	−18.3501	−1.11882	−0.559412	+	0.828890i	$0.688973\pi$
$270$	0	0
$271$	−10.5359	−0.640007	−0.320004	−	0.947416i	$-0.603684\pi$
$271$	−10.5359	−0.640007	−0.320004	+	0.947416i	$0.603684\pi$
$272$	0	0
$273$	19.8263	1.19994
$274$	0	0
$275$	−14.1441	−0.852921
$276$	0	0
$277$	−4.12750	−0.247998	−0.123999	−	0.992282i	$-0.539572\pi$
$277$	−4.12750	−0.247998	−0.123999	+	0.992282i	$0.539572\pi$
$278$	0	0
$279$	34.7539	2.08066
$280$	0	0
$281$	−19.1756	−1.14392	−0.571961	−	0.820281i	$-0.693817\pi$
$281$	−19.1756	−1.14392	−0.571961	+	0.820281i	$0.693817\pi$
$282$	0	0
$283$	−20.6317	−1.22643	−0.613215	−	0.789916i	$-0.710124\pi$
$283$	−20.6317	−1.22643	−0.613215	+	0.789916i	$0.710124\pi$
$284$	0	0
$285$	−11.9909	−0.710282
$286$	0	0
$287$	−0.660784	−0.0390048
$288$	0	0
$289$	−10.4227	−0.613102
$290$	0	0
$291$	54.6499	3.20363
$292$	0	0
$293$	5.03411	0.294096	0.147048	−	0.989129i	$-0.453023\pi$
$293$	5.03411	0.294096	0.147048	+	0.989129i	$0.453023\pi$
$294$	0	0
$295$	6.90024	0.401747
$296$	0	0
$297$	−52.9732	−3.07382
$298$	0	0
$299$	5.91595	0.342128
$300$	0	0
$301$	−4.91369	−0.283220
$302$	0	0
$303$	−32.5754	−1.87141
$304$	0	0
$305$	3.09315	0.177113
$306$	0	0
$307$	30.4187	1.73609	0.868043	−	0.496489i	$-0.165378\pi$
$307$	30.4187	1.73609	0.868043	+	0.496489i	$0.165378\pi$
$308$	0	0
$309$	−17.8274	−1.01416
$310$	0	0
$311$	22.6878	1.28651	0.643254	−	0.765653i	$-0.277584\pi$
$311$	22.6878	1.28651	0.643254	+	0.765653i	$0.277584\pi$
$312$	0	0
$313$	−14.4107	−0.814540	−0.407270	−	0.913308i	$-0.633519\pi$
$313$	−14.4107	−0.814540	−0.407270	+	0.913308i	$0.633519\pi$
$314$	0	0
$315$	−4.64762	−0.261864
$316$	0	0
$317$	−4.61358	−0.259125	−0.129562	−	0.991571i	$-0.541357\pi$
$317$	−4.61358	−0.259125	−0.129562	+	0.991571i	$0.541357\pi$
$318$	0	0
$319$	−15.8066	−0.885000
$320$	0	0
$321$	2.21102	0.123407
$322$	0	0
$323$	16.2519	0.904278
$324$	0	0
$325$	−27.6938	−1.53618
$326$	0	0
$327$	58.2347	3.22039
$328$	0	0
$329$	10.0111	0.551929
$330$	0	0
$331$	−13.4472	−0.739126	−0.369563	−	0.929206i	$-0.620493\pi$
$331$	−13.4472	−0.739126	−0.369563	+	0.929206i	$0.620493\pi$
$332$	0	0
$333$	−42.5744	−2.33306
$334$	0	0
$335$	4.70945	0.257305
$336$	0	0
$337$	25.3766	1.38235	0.691175	−	0.722687i	$-0.257093\pi$
$337$	25.3766	1.38235	0.691175	+	0.722687i	$0.257093\pi$
$338$	0	0
$339$	48.6138	2.64034
$340$	0	0
$341$	12.7569	0.690825
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	−1.89223	−0.101874
$346$	0	0
$347$	−25.1656	−1.35096	−0.675479	−	0.737379i	$-0.736063\pi$
$347$	−25.1656	−1.35096	−0.675479	+	0.737379i	$0.736063\pi$
$348$	0	0
$349$	20.0306	1.07221	0.536106	−	0.844151i	$-0.319895\pi$
$349$	20.0306	1.07221	0.536106	+	0.844151i	$0.319895\pi$
$350$	0	0
$351$	−103.720	−5.53618
$352$	0	0
$353$	−3.74710	−0.199438	−0.0997189	−	0.995016i	$-0.531794\pi$
$353$	−3.74710	−0.199438	−0.0997189	+	0.995016i	$0.531794\pi$
$354$	0	0
$355$	−6.25639	−0.332055
$356$	0	0
$357$	8.59489	0.454890
$358$	0	0
$359$	14.8422	0.783343	0.391671	−	0.920105i	$-0.371897\pi$
$359$	14.8422	0.783343	0.391671	+	0.920105i	$0.371897\pi$
$360$	0	0
$361$	21.1570	1.11352
$362$	0	0
$363$	6.26955	0.329066
$364$	0	0
$365$	3.50627	0.183527
$366$	0	0
$367$	2.14732	0.112089	0.0560447	−	0.998428i	$-0.482151\pi$
$367$	2.14732	0.112089	0.0560447	+	0.998428i	$0.482151\pi$
$368$	0	0
$369$	5.43920	0.283154
$370$	0	0
$371$	−6.00000	−0.311504
$372$	0	0
$373$	−26.8173	−1.38855	−0.694273	−	0.719712i	$-0.744274\pi$
$373$	−26.8173	−1.38855	−0.694273	+	0.719712i	$0.744274\pi$
$374$	0	0
$375$	18.3190	0.945990
$376$	0	0
$377$	−30.9490	−1.59395
$378$	0	0
$379$	32.2012	1.65406	0.827032	−	0.562155i	$-0.190028\pi$
$379$	32.2012	1.65406	0.827032	+	0.562155i	$0.190028\pi$
$380$	0	0
$381$	−38.7829	−1.98691
$382$	0	0
$383$	18.9071	0.966107	0.483053	−	0.875591i	$-0.339528\pi$
$383$	18.9071	0.966107	0.483053	+	0.875591i	$0.339528\pi$
$384$	0	0
$385$	−1.70597	−0.0869445
$386$	0	0
$387$	40.4467	2.05602
$388$	0	0
$389$	4.86868	0.246852	0.123426	−	0.992354i	$-0.460612\pi$
$389$	4.86868	0.246852	0.123426	+	0.992354i	$0.460612\pi$
$390$	0	0
$391$	2.56462	0.129698
$392$	0	0
$393$	−28.7806	−1.45179
$394$	0	0
$395$	−2.45098	−0.123322
$396$	0	0
$397$	−10.3934	−0.521632	−0.260816	−	0.965389i	$-0.583992\pi$
$397$	−10.3934	−0.521632	−0.260816	+	0.965389i	$0.583992\pi$
$398$	0	0
$399$	21.2372	1.06319
$400$	0	0
$401$	25.7254	1.28467	0.642333	−	0.766425i	$-0.277967\pi$
$401$	25.7254	1.28467	0.642333	+	0.766425i	$0.277967\pi$
$402$	0	0
$403$	24.9777	1.24423
$404$	0	0
$405$	19.2322	0.955656
$406$	0	0
$407$	−15.6275	−0.774626
$408$	0	0
$409$	35.8592	1.77312	0.886561	−	0.462611i	$-0.153087\pi$
$409$	35.8592	1.77312	0.886561	+	0.462611i	$0.153087\pi$
$410$	0	0
$411$	46.2706	2.28236
$412$	0	0
$413$	−12.2211	−0.601359
$414$	0	0
$415$	−3.95046	−0.193921
$416$	0	0
$417$	48.1098	2.35594
$418$	0	0
$419$	−14.6858	−0.717449	−0.358724	−	0.933444i	$-0.616788\pi$
$419$	−14.6858	−0.717449	−0.358724	+	0.933444i	$0.616788\pi$
$420$	0	0
$421$	−21.3297	−1.03955	−0.519773	−	0.854304i	$-0.673984\pi$
$421$	−21.3297	−1.03955	−0.519773	+	0.854304i	$0.673984\pi$
$422$	0	0
$423$	−82.4056	−4.00670
$424$	0	0
$425$	−12.0055	−0.582353
$426$	0	0
$427$	−5.47831	−0.265114
$428$	0	0
$429$	−59.9045	−2.89222
$430$	0	0
$431$	5.40534	0.260366	0.130183	−	0.991490i	$-0.458444\pi$
$431$	5.40534	0.260366	0.130183	+	0.991490i	$0.458444\pi$
$432$	0	0
$433$	−14.0959	−0.677405	−0.338703	−	0.940893i	$-0.609988\pi$
$433$	−14.0959	−0.677405	−0.338703	+	0.940893i	$0.609988\pi$
$434$	0	0
$435$	9.89907	0.474624
$436$	0	0
$437$	6.33695	0.303138
$438$	0	0
$439$	25.7104	1.22709	0.613546	−	0.789659i	$-0.289742\pi$
$439$	25.7104	1.22709	0.613546	+	0.789659i	$0.289742\pi$
$440$	0	0
$441$	8.23144	0.391973
$442$	0	0
$443$	−35.6825	−1.69533	−0.847663	−	0.530535i	$-0.821991\pi$
$443$	−35.6825	−1.69533	−0.847663	+	0.530535i	$0.821991\pi$
$444$	0	0
$445$	7.86346	0.372764
$446$	0	0
$447$	−53.5992	−2.53515
$448$	0	0
$449$	18.1971	0.858774	0.429387	−	0.903121i	$-0.358730\pi$
$449$	18.1971	0.858774	0.429387	+	0.903121i	$0.358730\pi$
$450$	0	0
$451$	1.99653	0.0940131
$452$	0	0
$453$	−35.2120	−1.65440
$454$	0	0
$455$	−3.34026	−0.156594
$456$	0	0
$457$	23.6097	1.10442	0.552209	−	0.833706i	$-0.313785\pi$
$457$	23.6097	1.10442	0.552209	+	0.833706i	$0.313785\pi$
$458$	0	0
$459$	−44.9637	−2.09873
$460$	0	0
$461$	9.63671	0.448826	0.224413	−	0.974494i	$-0.427954\pi$
$461$	9.63671	0.448826	0.224413	+	0.974494i	$0.427954\pi$
$462$	0	0
$463$	5.07171	0.235702	0.117851	−	0.993031i	$-0.462399\pi$
$463$	5.07171	0.235702	0.117851	+	0.993031i	$0.462399\pi$
$464$	0	0
$465$	−7.98916	−0.370488
$466$	0	0
$467$	3.74229	0.173172	0.0865862	−	0.996244i	$-0.472404\pi$
$467$	3.74229	0.173172	0.0865862	+	0.996244i	$0.472404\pi$
$468$	0	0
$469$	−8.34095	−0.385149
$470$	0	0
$471$	−18.6378	−0.858785
$472$	0	0
$473$	14.8465	0.682644
$474$	0	0
$475$	−29.6646	−1.36110
$476$	0	0
$477$	49.3886	2.26135
$478$	0	0
$479$	−39.8571	−1.82112	−0.910558	−	0.413381i	$-0.864348\pi$
$479$	−39.8571	−1.82112	−0.910558	+	0.413381i	$0.864348\pi$
$480$	0	0
$481$	−30.5983	−1.39516
$482$	0	0
$483$	3.35133	0.152491
$484$	0	0
$485$	−9.20718	−0.418077
$486$	0	0
$487$	31.5043	1.42760	0.713798	−	0.700351i	$-0.246973\pi$
$487$	31.5043	1.42760	0.713798	+	0.700351i	$0.246973\pi$
$488$	0	0
$489$	−35.8869	−1.62286
$490$	0	0
$491$	−30.9985	−1.39894	−0.699471	−	0.714661i	$-0.746581\pi$
$491$	−30.9985	−1.39894	−0.699471	+	0.714661i	$0.746581\pi$
$492$	0	0
$493$	−13.4167	−0.604256
$494$	0	0
$495$	14.0426	0.631169
$496$	0	0
$497$	11.0807	0.497039
$498$	0	0
$499$	−14.7688	−0.661142	−0.330571	−	0.943781i	$-0.607241\pi$
$499$	−14.7688	−0.661142	−0.330571	+	0.943781i	$0.607241\pi$
$500$	0	0
$501$	−0.810443	−0.0362079
$502$	0	0
$503$	−34.1176	−1.52123	−0.760615	−	0.649203i	$-0.775103\pi$
$503$	−34.1176	−1.52123	−0.760615	+	0.649203i	$0.775103\pi$
$504$	0	0
$505$	5.48817	0.244220
$506$	0	0
$507$	−73.7243	−3.27421
$508$	0	0
$509$	−37.7272	−1.67223	−0.836115	−	0.548554i	$-0.815178\pi$
$509$	−37.7272	−1.67223	−0.836115	+	0.548554i	$0.815178\pi$
$510$	0	0
$511$	−6.20998	−0.274713
$512$	0	0
$513$	−111.101	−4.90525
$514$	0	0
$515$	3.00348	0.132349
$516$	0	0
$517$	−30.2481	−1.33031
$518$	0	0
$519$	−27.3766	−1.20170
$520$	0	0
$521$	21.1711	0.927521	0.463760	−	0.885961i	$-0.346500\pi$
$521$	21.1711	0.927521	0.463760	+	0.885961i	$0.346500\pi$
$522$	0	0
$523$	−7.02093	−0.307004	−0.153502	−	0.988148i	$-0.549055\pi$
$523$	−7.02093	−0.307004	−0.153502	+	0.988148i	$0.549055\pi$
$524$	0	0
$525$	−15.6883	−0.684693
$526$	0	0
$527$	10.8281	0.471678
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	100.597	4.36553
$532$	0	0
$533$	3.90917	0.169325
$534$	0	0
$535$	−0.372503	−0.0161047
$536$	0	0
$537$	−73.4502	−3.16961
$538$	0	0
$539$	3.02146	0.130144
$540$	0	0
$541$	24.7983	1.06616	0.533080	−	0.846065i	$-0.321034\pi$
$541$	24.7983	1.06616	0.533080	+	0.846065i	$0.321034\pi$
$542$	0	0
$543$	−59.6201	−2.55854
$544$	0	0
$545$	−9.81114	−0.420263
$546$	0	0
$547$	4.24629	0.181558	0.0907792	−	0.995871i	$-0.471064\pi$
$547$	4.24629	0.181558	0.0907792	+	0.995871i	$0.471064\pi$
$548$	0	0
$549$	45.0944	1.92458
$550$	0	0
$551$	−33.1514	−1.41230
$552$	0	0
$553$	4.34095	0.184596
$554$	0	0
$555$	9.78690	0.415431
$556$	0	0
$557$	25.4957	1.08029	0.540144	−	0.841573i	$-0.318370\pi$
$557$	25.4957	1.08029	0.540144	+	0.841573i	$0.318370\pi$
$558$	0	0
$559$	29.0691	1.22949
$560$	0	0
$561$	−25.9692	−1.09642
$562$	0	0
$563$	24.0042	1.01166	0.505829	−	0.862634i	$-0.331187\pi$
$563$	24.0042	1.01166	0.505829	+	0.862634i	$0.331187\pi$
$564$	0	0
$565$	−8.19025	−0.344566
$566$	0	0
$567$	−34.0623	−1.43048
$568$	0	0
$569$	22.6634	0.950097	0.475049	−	0.879960i	$-0.342430\pi$
$569$	22.6634	0.950097	0.475049	+	0.879960i	$0.342430\pi$
$570$	0	0
$571$	26.7501	1.11946	0.559729	−	0.828676i	$-0.310905\pi$
$571$	26.7501	1.11946	0.559729	+	0.828676i	$0.310905\pi$
$572$	0	0
$573$	63.6890	2.66065
$574$	0	0
$575$	−4.68121	−0.195220
$576$	0	0
$577$	−35.4392	−1.47535	−0.737677	−	0.675154i	$-0.764077\pi$
$577$	−35.4392	−1.47535	−0.737677	+	0.675154i	$0.764077\pi$
$578$	0	0
$579$	−42.5048	−1.76644
$580$	0	0
$581$	6.99669	0.290272
$582$	0	0
$583$	18.1288	0.750817
$584$	0	0
$585$	27.4951	1.13678
$586$	0	0
$587$	−5.76374	−0.237895	−0.118948	−	0.992901i	$-0.537952\pi$
$587$	−5.76374	−0.237895	−0.118948	+	0.992901i	$0.537952\pi$
$588$	0	0
$589$	26.7552	1.10243
$590$	0	0
$591$	41.3038	1.69901
$592$	0	0
$593$	−0.559874	−0.0229913	−0.0114956	−	0.999934i	$-0.503659\pi$
$593$	−0.559874	−0.0229913	−0.0114956	+	0.999934i	$0.503659\pi$
$594$	0	0
$595$	−1.44803	−0.0593635
$596$	0	0
$597$	−37.9202	−1.55197
$598$	0	0
$599$	−4.91369	−0.200768	−0.100384	−	0.994949i	$-0.532007\pi$
$599$	−4.91369	−0.200768	−0.100384	+	0.994949i	$0.532007\pi$
$600$	0	0
$601$	22.6548	0.924107	0.462053	−	0.886852i	$-0.347113\pi$
$601$	22.6548	0.924107	0.462053	+	0.886852i	$0.347113\pi$
$602$	0	0
$603$	68.6580	2.79597
$604$	0	0
$605$	−1.05627	−0.0429434
$606$	0	0
$607$	27.2438	1.10579	0.552896	−	0.833250i	$-0.313523\pi$
$607$	27.2438	1.10579	0.552896	+	0.833250i	$0.313523\pi$
$608$	0	0
$609$	−17.5323	−0.710445
$610$	0	0
$611$	−59.2251	−2.39599
$612$	0	0
$613$	22.9105	0.925348	0.462674	−	0.886528i	$-0.346890\pi$
$613$	22.9105	0.925348	0.462674	+	0.886528i	$0.346890\pi$
$614$	0	0
$615$	−1.25035	−0.0504191
$616$	0	0
$617$	38.1555	1.53608	0.768040	−	0.640401i	$-0.221232\pi$
$617$	38.1555	1.53608	0.768040	+	0.640401i	$0.221232\pi$
$618$	0	0
$619$	10.1568	0.408235	0.204118	−	0.978946i	$-0.434567\pi$
$619$	10.1568	0.408235	0.204118	+	0.978946i	$0.434567\pi$
$620$	0	0
$621$	−17.5323	−0.703547
$622$	0	0
$623$	−13.9270	−0.557975
$624$	0	0
$625$	20.3197	0.812789
$626$	0	0
$627$	−64.1675	−2.56260
$628$	0	0
$629$	−13.2646	−0.528895
$630$	0	0
$631$	−33.1845	−1.32105	−0.660526	−	0.750803i	$-0.729667\pi$
$631$	−33.1845	−1.32105	−0.660526	+	0.750803i	$0.729667\pi$
$632$	0	0
$633$	−0.803601	−0.0319403
$634$	0	0
$635$	6.53399	0.259293
$636$	0	0
$637$	5.91595	0.234399
$638$	0	0
$639$	−91.2105	−3.60823
$640$	0	0
$641$	16.3302	0.645003	0.322501	−	0.946569i	$-0.395476\pi$
$641$	16.3302	0.645003	0.322501	+	0.946569i	$0.395476\pi$
$642$	0	0
$643$	20.5874	0.811889	0.405945	−	0.913898i	$-0.366943\pi$
$643$	20.5874	0.811889	0.405945	+	0.913898i	$0.366943\pi$
$644$	0	0
$645$	−9.29781	−0.366101
$646$	0	0
$647$	−36.8017	−1.44682	−0.723412	−	0.690417i	$-0.757427\pi$
$647$	−36.8017	−1.44682	−0.723412	+	0.690417i	$0.757427\pi$
$648$	0	0
$649$	36.9255	1.44945
$650$	0	0
$651$	14.1497	0.554569
$652$	0	0
$653$	3.07621	0.120382	0.0601908	−	0.998187i	$-0.480829\pi$
$653$	3.07621	0.120382	0.0601908	+	0.998187i	$0.480829\pi$
$654$	0	0
$655$	4.84884	0.189460
$656$	0	0
$657$	51.1171	1.99427
$658$	0	0
$659$	−16.6739	−0.649523	−0.324762	−	0.945796i	$-0.605284\pi$
$659$	−16.6739	−0.649523	−0.324762	+	0.945796i	$0.605284\pi$
$660$	0	0
$661$	−30.1360	−1.17215	−0.586077	−	0.810256i	$-0.699328\pi$
$661$	−30.1360	−1.17215	−0.586077	+	0.810256i	$0.699328\pi$
$662$	0	0
$663$	−50.8470	−1.97473
$664$	0	0
$665$	−3.57796	−0.138747
$666$	0	0
$667$	−5.23144	−0.202562
$668$	0	0
$669$	49.6398	1.91919
$670$	0	0
$671$	16.5525	0.639002
$672$	0	0
$673$	39.1527	1.50922	0.754612	−	0.656171i	$-0.227825\pi$
$673$	39.1527	1.50922	0.754612	+	0.656171i	$0.227825\pi$
$674$	0	0
$675$	82.0724	3.15897
$676$	0	0
$677$	11.0912	0.426269	0.213135	−	0.977023i	$-0.431633\pi$
$677$	11.0912	0.426269	0.213135	+	0.977023i	$0.431633\pi$
$678$	0	0
$679$	16.3069	0.625802
$680$	0	0
$681$	−44.5037	−1.70539
$682$	0	0
$683$	−5.03573	−0.192687	−0.0963435	−	0.995348i	$-0.530715\pi$
$683$	−5.03573	−0.192687	−0.0963435	+	0.995348i	$0.530715\pi$
$684$	0	0
$685$	−7.79547	−0.297849
$686$	0	0
$687$	−59.7791	−2.28072
$688$	0	0
$689$	35.4957	1.35228
$690$	0	0
$691$	10.6690	0.405869	0.202934	−	0.979192i	$-0.434952\pi$
$691$	10.6690	0.405869	0.202934	+	0.979192i	$0.434952\pi$
$692$	0	0
$693$	−24.8710	−0.944771
$694$	0	0
$695$	−8.10533	−0.307453
$696$	0	0
$697$	1.69466	0.0641898
$698$	0	0
$699$	−24.3788	−0.922092
$700$	0	0
$701$	4.90916	0.185416	0.0927082	−	0.995693i	$-0.470448\pi$
$701$	4.90916	0.185416	0.0927082	+	0.995693i	$0.470448\pi$
$702$	0	0
$703$	−32.7757	−1.23616
$704$	0	0
$705$	18.9432	0.713443
$706$	0	0
$707$	−9.72013	−0.365563
$708$	0	0
$709$	37.3766	1.40371	0.701853	−	0.712322i	$-0.252356\pi$
$709$	37.3766	1.40371	0.701853	+	0.712322i	$0.252356\pi$
$710$	0	0
$711$	−35.7323	−1.34006
$712$	0	0
$713$	4.22210	0.158119
$714$	0	0
$715$	10.0925	0.377437
$716$	0	0
$717$	−9.84604	−0.367707
$718$	0	0
$719$	−19.3281	−0.720818	−0.360409	−	0.932794i	$-0.617363\pi$
$719$	−19.3281	−0.720818	−0.360409	+	0.932794i	$0.617363\pi$
$720$	0	0
$721$	−5.31949	−0.198108
$722$	0	0
$723$	42.4924	1.58031
$724$	0	0
$725$	24.4895	0.909516
$726$	0	0
$727$	−7.06309	−0.261956	−0.130978	−	0.991385i	$-0.541812\pi$
$727$	−7.06309	−0.261956	−0.130978	+	0.991385i	$0.541812\pi$
$728$	0	0
$729$	104.112	3.85600
$730$	0	0
$731$	12.6017	0.466092
$732$	0	0
$733$	−34.2349	−1.26450	−0.632248	−	0.774766i	$-0.717868\pi$
$733$	−34.2349	−1.26450	−0.632248	+	0.774766i	$0.717868\pi$
$734$	0	0
$735$	−1.89223	−0.0697958
$736$	0	0
$737$	25.2019	0.928323
$738$	0	0
$739$	−9.07621	−0.333874	−0.166937	−	0.985968i	$-0.553388\pi$
$739$	−9.07621	−0.333874	−0.166937	+	0.985968i	$0.553388\pi$
$740$	0	0
$741$	−125.639	−4.61545
$742$	0	0
$743$	2.49444	0.0915120	0.0457560	−	0.998953i	$-0.485430\pi$
$743$	2.49444	0.0915120	0.0457560	+	0.998953i	$0.485430\pi$
$744$	0	0
$745$	9.03016	0.330840
$746$	0	0
$747$	−57.5929	−2.10721
$748$	0	0
$749$	0.659743	0.0241065
$750$	0	0
$751$	−30.7237	−1.12112	−0.560561	−	0.828113i	$-0.689415\pi$
$751$	−30.7237	−1.12112	−0.560561	+	0.828113i	$0.689415\pi$
$752$	0	0
$753$	7.26857	0.264881
$754$	0	0
$755$	5.93236	0.215901
$756$	0	0
$757$	−3.82436	−0.138999	−0.0694993	−	0.997582i	$-0.522140\pi$
$757$	−3.82436	−0.138999	−0.0694993	+	0.997582i	$0.522140\pi$
$758$	0	0
$759$	−10.1259	−0.367548
$760$	0	0
$761$	−8.28561	−0.300353	−0.150177	−	0.988659i	$-0.547984\pi$
$761$	−8.28561	−0.300353	−0.150177	+	0.988659i	$0.547984\pi$
$762$	0	0
$763$	17.3766	0.629074
$764$	0	0
$765$	11.9194	0.430946
$766$	0	0
$767$	72.2992	2.61057
$768$	0	0
$769$	18.5172	0.667746	0.333873	−	0.942618i	$-0.391644\pi$
$769$	18.5172	0.667746	0.333873	+	0.942618i	$0.391644\pi$
$770$	0	0
$771$	−41.3038	−1.48752
$772$	0	0
$773$	−38.3816	−1.38049	−0.690245	−	0.723575i	$-0.742497\pi$
$773$	−38.3816	−1.38049	−0.690245	+	0.723575i	$0.742497\pi$
$774$	0	0
$775$	−19.7645	−0.709962
$776$	0	0
$777$	−17.3336	−0.621841
$778$	0	0
$779$	4.18736	0.150028
$780$	0	0
$781$	−33.4800	−1.19801
$782$	0	0
$783$	91.7193	3.27778
$784$	0	0
$785$	3.14002	0.112072
$786$	0	0
$787$	30.7811	1.09723	0.548615	−	0.836075i	$-0.315155\pi$
$787$	30.7811	1.09723	0.548615	+	0.836075i	$0.315155\pi$
$788$	0	0
$789$	51.8780	1.84691
$790$	0	0
$791$	14.5058	0.515767
$792$	0	0
$793$	32.4094	1.15089
$794$	0	0
$795$	−11.3534	−0.402662
$796$	0	0
$797$	−44.5481	−1.57797	−0.788987	−	0.614410i	$-0.789394\pi$
$797$	−44.5481	−1.57797	−0.788987	+	0.614410i	$0.789394\pi$
$798$	0	0
$799$	−25.6746	−0.908302
$800$	0	0
$801$	114.640	4.05059
$802$	0	0
$803$	18.7632	0.662140
$804$	0	0
$805$	−0.564619	−0.0199002
$806$	0	0
$807$	61.4972	2.16480
$808$	0	0
$809$	14.4047	0.506444	0.253222	−	0.967408i	$-0.418510\pi$
$809$	14.4047	0.506444	0.253222	+	0.967408i	$0.418510\pi$
$810$	0	0
$811$	5.44095	0.191058	0.0955288	−	0.995427i	$-0.469546\pi$
$811$	5.44095	0.191058	0.0955288	+	0.995427i	$0.469546\pi$
$812$	0	0
$813$	35.3092	1.23835
$814$	0	0
$815$	6.04607	0.211785
$816$	0	0
$817$	31.1378	1.08937
$818$	0	0
$819$	−48.6968	−1.70160
$820$	0	0
$821$	−42.9942	−1.50051	−0.750254	−	0.661149i	$-0.770069\pi$
$821$	−42.9942	−1.50051	−0.750254	+	0.661149i	$0.770069\pi$
$822$	0	0
$823$	−8.28375	−0.288753	−0.144377	−	0.989523i	$-0.546118\pi$
$823$	−8.28375	−0.288753	−0.144377	+	0.989523i	$0.546118\pi$
$824$	0	0
$825$	47.4016	1.65031
$826$	0	0
$827$	13.4427	0.467449	0.233725	−	0.972303i	$-0.424909\pi$
$827$	13.4427	0.467449	0.233725	+	0.972303i	$0.424909\pi$
$828$	0	0
$829$	11.5819	0.402255	0.201127	−	0.979565i	$-0.435539\pi$
$829$	11.5819	0.402255	0.201127	+	0.979565i	$0.435539\pi$
$830$	0	0
$831$	13.8326	0.479849
$832$	0	0
$833$	2.56462	0.0888588
$834$	0	0
$835$	0.136540	0.00472516
$836$	0	0
$837$	−74.0231	−2.55861
$838$	0	0
$839$	−6.68353	−0.230741	−0.115371	−	0.993323i	$-0.536806\pi$
$839$	−6.68353	−0.230741	−0.115371	+	0.993323i	$0.536806\pi$
$840$	0	0
$841$	−1.63202	−0.0562765
$842$	0	0
$843$	64.2639	2.21337
$844$	0	0
$845$	12.4208	0.427287
$846$	0	0
$847$	1.87076	0.0642802
$848$	0	0
$849$	69.1439	2.37301
$850$	0	0
$851$	−5.17216	−0.177299
$852$	0	0
$853$	46.4304	1.58974	0.794872	−	0.606777i	$-0.207538\pi$
$853$	46.4304	1.58974	0.794872	+	0.606777i	$0.207538\pi$
$854$	0	0
$855$	29.4518	1.00723
$856$	0	0
$857$	0.324595	0.0110880	0.00554399	−	0.999985i	$-0.498235\pi$
$857$	0.324595	0.0110880	0.00554399	+	0.999985i	$0.498235\pi$
$858$	0	0
$859$	50.3392	1.71755	0.858775	−	0.512353i	$-0.171226\pi$
$859$	50.3392	1.71755	0.858775	+	0.512353i	$0.171226\pi$
$860$	0	0
$861$	2.21451	0.0754702
$862$	0	0
$863$	9.59108	0.326484	0.163242	−	0.986586i	$-0.447805\pi$
$863$	9.59108	0.326484	0.163242	+	0.986586i	$0.447805\pi$
$864$	0	0
$865$	4.61229	0.156823
$866$	0	0
$867$	34.9301	1.18629
$868$	0	0
$869$	−13.1160	−0.444930
$870$	0	0
$871$	49.3447	1.67198
$872$	0	0
$873$	−134.229	−4.54298
$874$	0	0
$875$	5.46619	0.184791
$876$	0	0
$877$	19.8180	0.669206	0.334603	−	0.942359i	$-0.391398\pi$
$877$	19.8180	0.669206	0.334603	+	0.942359i	$0.391398\pi$
$878$	0	0
$879$	−16.8710	−0.569045
$880$	0	0
$881$	−22.4723	−0.757111	−0.378555	−	0.925579i	$-0.623579\pi$
$881$	−22.4723	−0.757111	−0.378555	+	0.925579i	$0.623579\pi$
$882$	0	0
$883$	32.1407	1.08162	0.540811	−	0.841144i	$-0.318118\pi$
$883$	32.1407	1.08162	0.540811	+	0.841144i	$0.318118\pi$
$884$	0	0
$885$	−23.1250	−0.777338
$886$	0	0
$887$	26.0502	0.874681	0.437341	−	0.899296i	$-0.355920\pi$
$887$	26.0502	0.874681	0.437341	+	0.899296i	$0.355920\pi$
$888$	0	0
$889$	−11.5724	−0.388126
$890$	0	0
$891$	102.918	3.44788
$892$	0	0
$893$	−63.4397	−2.12293
$894$	0	0
$895$	12.3746	0.413637
$896$	0	0
$897$	−19.8263	−0.661982
$898$	0	0
$899$	−22.0877	−0.736665
$900$	0	0
$901$	15.3877	0.512639
$902$	0	0
$903$	16.4674	0.548001
$904$	0	0
$905$	10.0445	0.333892
$906$	0	0
$907$	3.21553	0.106770	0.0533850	−	0.998574i	$-0.482999\pi$
$907$	3.21553	0.106770	0.0533850	+	0.998574i	$0.482999\pi$
$908$	0	0
$909$	80.0107	2.65379
$910$	0	0
$911$	−42.4913	−1.40780	−0.703900	−	0.710299i	$-0.748560\pi$
$911$	−42.4913	−1.40780	−0.703900	+	0.710299i	$0.748560\pi$
$912$	0	0
$913$	−21.1403	−0.699640
$914$	0	0
$915$	−10.3662	−0.342696
$916$	0	0
$917$	−8.58781	−0.283595
$918$	0	0
$919$	4.08700	0.134818	0.0674089	−	0.997725i	$-0.478527\pi$
$919$	4.08700	0.134818	0.0674089	+	0.997725i	$0.478527\pi$
$920$	0	0
$921$	−101.943	−3.35914
$922$	0	0
$923$	−65.5531	−2.15771
$924$	0	0
$925$	24.2120	0.796084
$926$	0	0
$927$	43.7870	1.43816
$928$	0	0
$929$	11.1847	0.366958	0.183479	−	0.983024i	$-0.441264\pi$
$929$	11.1847	0.366958	0.183479	+	0.983024i	$0.441264\pi$
$930$	0	0
$931$	6.33695	0.207685
$932$	0	0
$933$	−76.0345	−2.48926
$934$	0	0
$935$	4.37517	0.143083
$936$	0	0
$937$	53.8285	1.75850	0.879251	−	0.476359i	$-0.158044\pi$
$937$	53.8285	1.75850	0.879251	+	0.476359i	$0.158044\pi$
$938$	0	0
$939$	48.2950	1.57605
$940$	0	0
$941$	25.1060	0.818432	0.409216	−	0.912438i	$-0.365802\pi$
$941$	25.1060	0.818432	0.409216	+	0.912438i	$0.365802\pi$
$942$	0	0
$943$	0.660784	0.0215181
$944$	0	0
$945$	9.89907	0.322017
$946$	0	0
$947$	−16.7754	−0.545127	−0.272564	−	0.962138i	$-0.587872\pi$
$947$	−16.7754	−0.545127	−0.272564	+	0.962138i	$0.587872\pi$
$948$	0	0
$949$	36.7379	1.19256
$950$	0	0
$951$	15.4617	0.501379
$952$	0	0
$953$	−5.62750	−0.182293	−0.0911463	−	0.995838i	$-0.529053\pi$
$953$	−5.62750	−0.182293	−0.0911463	+	0.995838i	$0.529053\pi$
$954$	0	0
$955$	−10.7301	−0.347216
$956$	0	0
$957$	52.9732	1.71238
$958$	0	0
$959$	13.8066	0.445839
$960$	0	0
$961$	−13.1739	−0.424964
$962$	0	0
$963$	−5.43064	−0.175000
$964$	0	0
$965$	7.16102	0.230521
$966$	0	0
$967$	−52.2832	−1.68131	−0.840657	−	0.541568i	$-0.817831\pi$
$967$	−52.2832	−1.68131	−0.840657	+	0.541568i	$0.817831\pi$
$968$	0	0
$969$	−54.4654	−1.74968
$970$	0	0
$971$	−50.0208	−1.60525	−0.802623	−	0.596487i	$-0.796563\pi$
$971$	−50.0208	−1.60525	−0.802623	+	0.596487i	$0.796563\pi$
$972$	0	0
$973$	14.3554	0.460213
$974$	0	0
$975$	92.8112	2.97234
$976$	0	0
$977$	−56.3837	−1.80387	−0.901937	−	0.431869i	$-0.857854\pi$
$977$	−56.3837	−1.80387	−0.901937	+	0.431869i	$0.857854\pi$
$978$	0	0
$979$	42.0800	1.34488
$980$	0	0
$981$	−143.034	−4.56673
$982$	0	0
$983$	−3.06160	−0.0976499	−0.0488249	−	0.998807i	$-0.515548\pi$
$983$	−3.06160	−0.0976499	−0.0488249	+	0.998807i	$0.515548\pi$
$984$	0	0
$985$	−6.95870	−0.221722
$986$	0	0
$987$	−33.5505	−1.06792
$988$	0	0
$989$	4.91369	0.156246
$990$	0	0
$991$	43.3129	1.37588	0.687939	−	0.725768i	$-0.258515\pi$
$991$	43.3129	1.37588	0.687939	+	0.725768i	$0.258515\pi$
$992$	0	0
$993$	45.0661	1.43013
$994$	0	0
$995$	6.38863	0.202533
$996$	0	0
$997$	−5.99623	−0.189903	−0.0949513	−	0.995482i	$-0.530270\pi$
$997$	−5.99623	−0.189903	−0.0949513	+	0.995482i	$0.530270\pi$
$998$	0	0
$999$	90.6800	2.86899

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2576.2.a.be.1.1		5
4.3	odd	2		1288.2.a.p.1.5	✓	5
28.27	even	2		9016.2.a.bg.1.1		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1288.2.a.p.1.5	✓	5	4.3	odd	2
2576.2.a.be.1.1		5	1.1	even	1	trivial
9016.2.a.bg.1.1		5	28.27	even	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:	\( N \)	\(=\)	\( 2576 = 2^{4} \cdot 7 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2576.a (trivial)

\(f(q)\)	\(=\)	\(q-3.35133 q^{3} +0.564619 q^{5} -1.00000 q^{7} +8.23144 q^{9} +O(q^{10})\)	\(q-3.35133 q^{3} +0.564619 q^{5} -1.00000 q^{7} +8.23144 q^{9} +3.02146 q^{11} +5.91595 q^{13} -1.89223 q^{15} +2.56462 q^{17} +6.33695 q^{19} +3.35133 q^{21} +1.00000 q^{23} -4.68121 q^{25} -17.5323 q^{27} -5.23144 q^{29} +4.22210 q^{31} -10.1259 q^{33} -0.564619 q^{35} -5.17216 q^{37} -19.8263 q^{39} +0.660784 q^{41} +4.91369 q^{43} +4.64762 q^{45} -10.0111 q^{47} +1.00000 q^{49} -8.59489 q^{51} +6.00000 q^{53} +1.70597 q^{55} -21.2372 q^{57} +12.2211 q^{59} +5.47831 q^{61} -8.23144 q^{63} +3.34026 q^{65} +8.34095 q^{67} -3.35133 q^{69} -11.0807 q^{71} +6.20998 q^{73} +15.6883 q^{75} -3.02146 q^{77} -4.34095 q^{79} +34.0623 q^{81} -6.99669 q^{83} +1.44803 q^{85} +17.5323 q^{87} +13.9270 q^{89} -5.91595 q^{91} -14.1497 q^{93} +3.57796 q^{95} -16.3069 q^{97} +24.8710 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + 3 q^{3} + 2 q^{5} - 5 q^{7} + 14 q^{9}+O(q^{10}) \) 5 * q + 3 * q^3 + 2 * q^5 - 5 * q^7 + 14 * q^9	\( 5 q + 3 q^{3} + 2 q^{5} - 5 q^{7} + 14 q^{9} + 2 q^{11} + 9 q^{13} + 2 q^{15} + 12 q^{17} + 12 q^{19} - 3 q^{21} + 5 q^{23} + 3 q^{25} - 3 q^{27} + q^{29} + 3 q^{31} - 16 q^{33} - 2 q^{35} + 2 q^{37} - 21 q^{39} + 19 q^{41} + 14 q^{45} - 17 q^{47} + 5 q^{49} + 8 q^{51} + 30 q^{53} + 2 q^{55} + 14 q^{57} + 14 q^{59} + 2 q^{61} - 14 q^{63} + 30 q^{65} + 2 q^{67} + 3 q^{69} - 43 q^{71} + 17 q^{73} + 39 q^{75} - 2 q^{77} + 18 q^{79} + 73 q^{81} - 2 q^{83} + 32 q^{85} + 3 q^{87} + 16 q^{89} - 9 q^{91} - 27 q^{93} - 12 q^{95} + 18 q^{97} - 4 q^{99}+O(q^{100}) \) 5 * q + 3 * q^3 + 2 * q^5 - 5 * q^7 + 14 * q^9 + 2 * q^11 + 9 * q^13 + 2 * q^15 + 12 * q^17 + 12 * q^19 - 3 * q^21 + 5 * q^23 + 3 * q^25 - 3 * q^27 + q^29 + 3 * q^31 - 16 * q^33 - 2 * q^35 + 2 * q^37 - 21 * q^39 + 19 * q^41 + 14 * q^45 - 17 * q^47 + 5 * q^49 + 8 * q^51 + 30 * q^53 + 2 * q^55 + 14 * q^57 + 14 * q^59 + 2 * q^61 - 14 * q^63 + 30 * q^65 + 2 * q^67 + 3 * q^69 - 43 * q^71 + 17 * q^73 + 39 * q^75 - 2 * q^77 + 18 * q^79 + 73 * q^81 - 2 * q^83 + 32 * q^85 + 3 * q^87 + 16 * q^89 - 9 * q^91 - 27 * q^93 - 12 * q^95 + 18 * q^97 - 4 * q^99

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