LMFDB - Embedded newform 2576.2.a.be.1.5

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			1.5
Root			$2.66293$ 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)

$f(q)$	$=$	$q+3.37084 q^{3} +2.66293 q^{5} -1.00000 q^{7} +8.36257 q^{9} +O(q^{10})$	$q+3.37084 q^{3} +2.66293 q^{5} -1.00000 q^{7} +8.36257 q^{9} -3.65046 q^{11} +1.29209 q^{13} +8.97633 q^{15} +4.66293 q^{17} +4.42002 q^{19} -3.37084 q^{21} +1.00000 q^{23} +2.09122 q^{25} +18.0764 q^{27} -5.36257 q^{29} -6.69671 q^{31} -12.3051 q^{33} -2.66293 q^{35} +3.97505 q^{37} +4.35544 q^{39} -10.3389 q^{41} -12.6268 q^{43} +22.2690 q^{45} -3.18840 q^{47} +1.00000 q^{49} +15.7180 q^{51} +6.00000 q^{53} -9.72094 q^{55} +14.8992 q^{57} +12.2014 q^{59} -9.96386 q^{61} -8.36257 q^{63} +3.44076 q^{65} +1.46802 q^{67} +3.37084 q^{69} -13.6872 q^{71} +13.0130 q^{73} +7.04918 q^{75} +3.65046 q^{77} +2.53198 q^{79} +35.8449 q^{81} -4.97926 q^{83} +12.4171 q^{85} -18.0764 q^{87} +2.48049 q^{89} -1.29209 q^{91} -22.5736 q^{93} +11.7702 q^{95} +10.4004 q^{97} -30.5272 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	3.37084	1.94616	0.973078	−	0.230475i	$-0.0740282\pi$
$3$	3.37084	1.94616	0.973078	+	0.230475i	$0.0740282\pi$
$4$	0	0
$5$	2.66293	1.19090	0.595450	−	0.803392i	$-0.296974\pi$
$5$	2.66293	1.19090	0.595450	+	0.803392i	$0.296974\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	8.36257	2.78752
$10$	0	0
$11$	−3.65046	−1.10066	−0.550328	−	0.834949i	$-0.685497\pi$
$11$	−3.65046	−1.10066	−0.550328	+	0.834949i	$0.685497\pi$
$12$	0	0
$13$	1.29209	0.358362	0.179181	−	0.983816i	$-0.442655\pi$
$13$	1.29209	0.358362	0.179181	+	0.983816i	$0.442655\pi$
$14$	0	0
$15$	8.97633	2.31768
$16$	0	0
$17$	4.66293	1.13093	0.565464	−	0.824773i	$-0.308697\pi$
$17$	4.66293	1.13093	0.565464	+	0.824773i	$0.308697\pi$
$18$	0	0
$19$	4.42002	1.01402	0.507011	−	0.861940i	$-0.330750\pi$
$19$	4.42002	1.01402	0.507011	+	0.861940i	$0.330750\pi$
$20$	0	0
$21$	−3.37084	−0.735578
$22$	0	0
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	0	0
$25$	2.09122	0.418244
$26$	0	0
$27$	18.0764	3.47880
$28$	0	0
$29$	−5.36257	−0.995805	−0.497902	−	0.867233i	$-0.665896\pi$
$29$	−5.36257	−0.995805	−0.497902	+	0.867233i	$0.665896\pi$
$30$	0	0
$31$	−6.69671	−1.20276	−0.601382	−	0.798961i	$-0.705383\pi$
$31$	−6.69671	−1.20276	−0.601382	+	0.798961i	$0.705383\pi$
$32$	0	0
$33$	−12.3051	−2.14205
$34$	0	0
$35$	−2.66293	−0.450118
$36$	0	0
$37$	3.97505	0.653495	0.326747	−	0.945112i	$-0.394047\pi$
$37$	3.97505	0.653495	0.326747	+	0.945112i	$0.394047\pi$
$38$	0	0
$39$	4.35544	0.697429
$40$	0	0
$41$	−10.3389	−1.61467	−0.807333	−	0.590096i	$-0.799090\pi$
$41$	−10.3389	−1.61467	−0.807333	+	0.590096i	$0.799090\pi$
$42$	0	0
$43$	−12.6268	−1.92557	−0.962784	−	0.270272i	$-0.912886\pi$
$43$	−12.6268	−1.92557	−0.962784	+	0.270272i	$0.912886\pi$
$44$	0	0
$45$	22.2690	3.31966
$46$	0	0
$47$	−3.18840	−0.465076	−0.232538	−	0.972587i	$-0.574703\pi$
$47$	−3.18840	−0.465076	−0.232538	+	0.972587i	$0.574703\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	15.7180	2.20096
$52$	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
$55$	−9.72094	−1.31077
$56$	0	0
$57$	14.8992	1.97344
$58$	0	0
$59$	12.2014	1.58849	0.794246	−	0.607597i	$-0.207866\pi$
$59$	12.2014	1.58849	0.794246	+	0.607597i	$0.207866\pi$
$60$	0	0
$61$	−9.96386	−1.27574	−0.637871	−	0.770144i	$-0.720185\pi$
$61$	−9.96386	−1.27574	−0.637871	+	0.770144i	$0.720185\pi$
$62$	0	0
$63$	−8.36257	−1.05359
$64$	0	0
$65$	3.44076	0.426774
$66$	0	0
$67$	1.46802	0.179347	0.0896735	−	0.995971i	$-0.471418\pi$
$67$	1.46802	0.179347	0.0896735	+	0.995971i	$0.471418\pi$
$68$	0	0
$69$	3.37084	0.405802
$70$	0	0
$71$	−13.6872	−1.62437	−0.812184	−	0.583402i	$-0.801721\pi$
$71$	−13.6872	−1.62437	−0.812184	+	0.583402i	$0.801721\pi$
$72$	0	0
$73$	13.0130	1.52306	0.761530	−	0.648129i	$-0.224448\pi$
$73$	13.0130	1.52306	0.761530	+	0.648129i	$0.224448\pi$
$74$	0	0
$75$	7.04918	0.813969
$76$	0	0
$77$	3.65046	0.416009
$78$	0	0
$79$	2.53198	0.284870	0.142435	−	0.989804i	$-0.454507\pi$
$79$	2.53198	0.284870	0.142435	+	0.989804i	$0.454507\pi$
$80$	0	0
$81$	35.8449	3.98277
$82$	0	0
$83$	−4.97926	−0.546545	−0.273272	−	0.961937i	$-0.588106\pi$
$83$	−4.97926	−0.546545	−0.273272	+	0.961937i	$0.588106\pi$
$84$	0	0
$85$	12.4171	1.34682
$86$	0	0
$87$	−18.0764	−1.93799
$88$	0	0
$89$	2.48049	0.262932	0.131466	−	0.991321i	$-0.458032\pi$
$89$	2.48049	0.262932	0.131466	+	0.991321i	$0.458032\pi$
$90$	0	0
$91$	−1.29209	−0.135448
$92$	0	0
$93$	−22.5736	−2.34077
$94$	0	0
$95$	11.7702	1.20760
$96$	0	0
$97$	10.4004	1.05600	0.528001	−	0.849244i	$-0.322942\pi$
$97$	10.4004	1.05600	0.528001	+	0.849244i	$0.322942\pi$
$98$	0	0
$99$	−30.5272	−3.06810
$100$	0	0
$101$	−5.44015	−0.541315	−0.270657	−	0.962676i	$-0.587241\pi$
$101$	−5.44015	−0.541315	−0.270657	+	0.962676i	$0.587241\pi$
$102$	0	0
$103$	5.11848	0.504339	0.252169	−	0.967683i	$-0.418856\pi$
$103$	5.11848	0.504339	0.252169	+	0.967683i	$0.418856\pi$
$104$	0	0
$105$	−8.97633	−0.876000
$106$	0	0
$107$	−0.559240	−0.0540638	−0.0270319	−	0.999635i	$-0.508606\pi$
$107$	−0.559240	−0.0540638	−0.0270319	+	0.999635i	$0.508606\pi$
$108$	0	0
$109$	−0.0983526	−0.00942047	−0.00471024	−	0.999989i	$-0.501499\pi$
$109$	−0.0983526	−0.00942047	−0.00471024	+	0.999989i	$0.501499\pi$
$110$	0	0
$111$	13.3993	1.27180
$112$	0	0
$113$	−1.42422	−0.133980	−0.0669898	−	0.997754i	$-0.521339\pi$
$113$	−1.42422	−0.133980	−0.0669898	+	0.997754i	$0.521339\pi$
$114$	0	0
$115$	2.66293	0.248320
$116$	0	0
$117$	10.8052	0.998943
$118$	0	0
$119$	−4.66293	−0.427451
$120$	0	0
$121$	2.32587	0.211443
$122$	0	0
$123$	−34.8508	−3.14239
$124$	0	0
$125$	−7.74589	−0.692813
$126$	0	0
$127$	4.83059	0.428646	0.214323	−	0.976763i	$-0.431246\pi$
$127$	4.83059	0.428646	0.214323	+	0.976763i	$0.431246\pi$
$128$	0	0
$129$	−42.5629	−3.74746
$130$	0	0
$131$	−13.8584	−1.21082	−0.605408	−	0.795916i	$-0.706990\pi$
$131$	−13.8584	−1.21082	−0.605408	+	0.795916i	$0.706990\pi$
$132$	0	0
$133$	−4.42002	−0.383264
$134$	0	0
$135$	48.1362	4.14291
$136$	0	0
$137$	21.5759	1.84335	0.921675	−	0.387962i	$-0.126821\pi$
$137$	21.5759	1.84335	0.921675	+	0.387962i	$0.126821\pi$
$138$	0	0
$139$	10.7617	0.912796	0.456398	−	0.889776i	$-0.349139\pi$
$139$	10.7617	0.912796	0.456398	+	0.889776i	$0.349139\pi$
$140$	0	0
$141$	−10.7476	−0.905110
$142$	0	0
$143$	−4.71674	−0.394433
$144$	0	0
$145$	−14.2802	−1.18590
$146$	0	0
$147$	3.37084	0.278022
$148$	0	0
$149$	11.9585	0.979680	0.489840	−	0.871812i	$-0.337055\pi$
$149$	11.9585	0.979680	0.489840	+	0.871812i	$0.337055\pi$
$150$	0	0
$151$	−13.4739	−1.09649	−0.548246	−	0.836317i	$-0.684704\pi$
$151$	−13.4739	−1.09649	−0.548246	+	0.836317i	$0.684704\pi$
$152$	0	0
$153$	38.9941	3.15249
$154$	0	0
$155$	−17.8329	−1.43237
$156$	0	0
$157$	5.64219	0.450296	0.225148	−	0.974325i	$-0.427713\pi$
$157$	5.64219	0.450296	0.225148	+	0.974325i	$0.427713\pi$
$158$	0	0
$159$	20.2250	1.60395
$160$	0	0
$161$	−1.00000	−0.0788110
$162$	0	0
$163$	12.1979	0.955417	0.477708	−	0.878518i	$-0.341468\pi$
$163$	12.1979	0.955417	0.477708	+	0.878518i	$0.341468\pi$
$164$	0	0
$165$	−32.7678	−2.55097
$166$	0	0
$167$	0.523713	0.0405261	0.0202630	−	0.999795i	$-0.493550\pi$
$167$	0.523713	0.0405261	0.0202630	+	0.999795i	$0.493550\pi$
$168$	0	0
$169$	−11.3305	−0.871577
$170$	0	0
$171$	36.9627	2.82661
$172$	0	0
$173$	−2.99580	−0.227766	−0.113883	−	0.993494i	$-0.536329\pi$
$173$	−2.99580	−0.227766	−0.113883	+	0.993494i	$0.536329\pi$
$174$	0	0
$175$	−2.09122	−0.158081
$176$	0	0
$177$	41.1291	3.09145
$178$	0	0
$179$	−3.11952	−0.233164	−0.116582	−	0.993181i	$-0.537194\pi$
$179$	−3.11952	−0.233164	−0.116582	+	0.993181i	$0.537194\pi$
$180$	0	0
$181$	−25.7039	−1.91056	−0.955278	−	0.295710i	$-0.904444\pi$
$181$	−25.7039	−1.91056	−0.955278	+	0.295710i	$0.904444\pi$
$182$	0	0
$183$	−33.5866	−2.48279
$184$	0	0
$185$	10.5853	0.778247
$186$	0	0
$187$	−17.0219	−1.24476
$188$	0	0
$189$	−18.0764	−1.31486
$190$	0	0
$191$	−0.609133	−0.0440753	−0.0220377	−	0.999757i	$-0.507015\pi$
$191$	−0.609133	−0.0440753	−0.0220377	+	0.999757i	$0.507015\pi$
$192$	0	0
$193$	−11.9621	−0.861051	−0.430526	−	0.902578i	$-0.641672\pi$
$193$	−11.9621	−0.861051	−0.430526	+	0.902578i	$0.641672\pi$
$194$	0	0
$195$	11.5983	0.830568
$196$	0	0
$197$	17.1705	1.22335	0.611675	−	0.791109i	$-0.290496\pi$
$197$	17.1705	1.22335	0.611675	+	0.791109i	$0.290496\pi$
$198$	0	0
$199$	−14.7193	−1.04342	−0.521711	−	0.853122i	$-0.674706\pi$
$199$	−14.7193	−1.04342	−0.521711	+	0.853122i	$0.674706\pi$
$200$	0	0
$201$	4.94846	0.349037
$202$	0	0
$203$	5.36257	0.376379
$204$	0	0
$205$	−27.5318	−1.92291
$206$	0	0
$207$	8.36257	0.581239
$208$	0	0
$209$	−16.1351	−1.11609
$210$	0	0
$211$	−13.4668	−0.927095	−0.463547	−	0.886072i	$-0.653424\pi$
$211$	−13.4668	−0.927095	−0.463547	+	0.886072i	$0.653424\pi$
$212$	0	0
$213$	−46.1373	−3.16127
$214$	0	0
$215$	−33.6243	−2.29316
$216$	0	0
$217$	6.69671	0.454602
$218$	0	0
$219$	43.8649	2.96411
$220$	0	0
$221$	6.02495	0.405282
$222$	0	0
$223$	4.56458	0.305667	0.152834	−	0.988252i	$-0.451160\pi$
$223$	4.56458	0.305667	0.152834	+	0.988252i	$0.451160\pi$
$224$	0	0
$225$	17.4880	1.16587
$226$	0	0
$227$	−15.7885	−1.04792	−0.523960	−	0.851743i	$-0.675546\pi$
$227$	−15.7885	−1.04792	−0.523960	+	0.851743i	$0.675546\pi$
$228$	0	0
$229$	−13.1670	−0.870103	−0.435051	−	0.900406i	$-0.643270\pi$
$229$	−13.1670	−0.870103	−0.435051	+	0.900406i	$0.643270\pi$
$230$	0	0
$231$	12.3051	0.809618
$232$	0	0
$233$	−5.93835	−0.389034	−0.194517	−	0.980899i	$-0.562314\pi$
$233$	−5.93835	−0.389034	−0.194517	+	0.980899i	$0.562314\pi$
$234$	0	0
$235$	−8.49050	−0.553859
$236$	0	0
$237$	8.53491	0.554402
$238$	0	0
$239$	22.4314	1.45097	0.725483	−	0.688240i	$-0.241616\pi$
$239$	22.4314	1.45097	0.725483	+	0.688240i	$0.241616\pi$
$240$	0	0
$241$	21.5206	1.38627	0.693133	−	0.720810i	$-0.256230\pi$
$241$	21.5206	1.38627	0.693133	+	0.720810i	$0.256230\pi$
$242$	0	0
$243$	66.5983	4.27229
$244$	0	0
$245$	2.66293	0.170129
$246$	0	0
$247$	5.71108	0.363387
$248$	0	0
$249$	−16.7843	−1.06366
$250$	0	0
$251$	8.99580	0.567810	0.283905	−	0.958852i	$-0.408370\pi$
$251$	8.99580	0.567810	0.283905	+	0.958852i	$0.408370\pi$
$252$	0	0
$253$	−3.65046	−0.229503
$254$	0	0
$255$	41.8560	2.62113
$256$	0	0
$257$	−17.1705	−1.07107	−0.535534	−	0.844514i	$-0.679890\pi$
$257$	−17.1705	−1.07107	−0.535534	+	0.844514i	$0.679890\pi$
$258$	0	0
$259$	−3.97505	−0.246998
$260$	0	0
$261$	−44.8449	−2.77583
$262$	0	0
$263$	16.7631	1.03366	0.516828	−	0.856089i	$-0.327113\pi$
$263$	16.7631	1.03366	0.516828	+	0.856089i	$0.327113\pi$
$264$	0	0
$265$	15.9776	0.981497
$266$	0	0
$267$	8.36134	0.511706
$268$	0	0
$269$	−23.5990	−1.43885	−0.719427	−	0.694568i	$-0.755595\pi$
$269$	−23.5990	−1.43885	−0.719427	+	0.694568i	$0.755595\pi$
$270$	0	0
$271$	−22.2447	−1.35127	−0.675633	−	0.737238i	$-0.736130\pi$
$271$	−22.2447	−1.35127	−0.675633	+	0.737238i	$0.736130\pi$
$272$	0	0
$273$	−4.35544	−0.263603
$274$	0	0
$275$	−7.63392	−0.460343
$276$	0	0
$277$	−26.1968	−1.57401	−0.787005	−	0.616946i	$-0.788370\pi$
$277$	−26.1968	−1.57401	−0.787005	+	0.616946i	$0.788370\pi$
$278$	0	0
$279$	−56.0017	−3.35274
$280$	0	0
$281$	25.7168	1.53414	0.767069	−	0.641565i	$-0.221715\pi$
$281$	25.7168	1.53414	0.767069	+	0.641565i	$0.221715\pi$
$282$	0	0
$283$	−9.72935	−0.578350	−0.289175	−	0.957276i	$-0.593381\pi$
$283$	−9.72935	−0.578350	−0.289175	+	0.957276i	$0.593381\pi$
$284$	0	0
$285$	39.6755	2.35018
$286$	0	0
$287$	10.3389	0.610286
$288$	0	0
$289$	4.74296	0.278998
$290$	0	0
$291$	35.0581	2.05514
$292$	0	0
$293$	11.4296	0.667722	0.333861	−	0.942622i	$-0.391648\pi$
$293$	11.4296	0.667722	0.333861	+	0.942622i	$0.391648\pi$
$294$	0	0
$295$	32.4916	1.89174
$296$	0	0
$297$	−65.9871	−3.82896
$298$	0	0
$299$	1.29209	0.0747237
$300$	0	0
$301$	12.6268	0.727796
$302$	0	0
$303$	−18.3379	−1.05348
$304$	0	0
$305$	−26.5331	−1.51928
$306$	0	0
$307$	9.62392	0.549266	0.274633	−	0.961549i	$-0.411444\pi$
$307$	9.62392	0.549266	0.274633	+	0.961549i	$0.411444\pi$
$308$	0	0
$309$	17.2536	0.981522
$310$	0	0
$311$	−30.1795	−1.71132	−0.855660	−	0.517538i	$-0.826849\pi$
$311$	−30.1795	−1.71132	−0.855660	+	0.517538i	$0.826849\pi$
$312$	0	0
$313$	−3.52792	−0.199410	−0.0997048	−	0.995017i	$-0.531790\pi$
$313$	−3.52792	−0.199410	−0.0997048	+	0.995017i	$0.531790\pi$
$314$	0	0
$315$	−22.2690	−1.25472
$316$	0	0
$317$	−2.40055	−0.134828	−0.0674142	−	0.997725i	$-0.521475\pi$
$317$	−2.40055	−0.134828	−0.0674142	+	0.997725i	$0.521475\pi$
$318$	0	0
$319$	19.5759	1.09604
$320$	0	0
$321$	−1.88511	−0.105217
$322$	0	0
$323$	20.6103	1.14679
$324$	0	0
$325$	2.70205	0.149883
$326$	0	0
$327$	−0.331531	−0.0183337
$328$	0	0
$329$	3.18840	0.175782
$330$	0	0
$331$	15.8861	0.873182	0.436591	−	0.899660i	$-0.356186\pi$
$331$	15.8861	0.873182	0.436591	+	0.899660i	$0.356186\pi$
$332$	0	0
$333$	33.2417	1.82163
$334$	0	0
$335$	3.90924	0.213584
$336$	0	0
$337$	8.09835	0.441145	0.220573	−	0.975371i	$-0.429207\pi$
$337$	8.09835	0.441145	0.220573	+	0.975371i	$0.429207\pi$
$338$	0	0
$339$	−4.80083	−0.260745
$340$	0	0
$341$	24.4461	1.32383
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	8.97633	0.483269
$346$	0	0
$347$	−11.9835	−0.643306	−0.321653	−	0.946858i	$-0.604238\pi$
$347$	−11.9835	−0.643306	−0.321653	+	0.946858i	$0.604238\pi$
$348$	0	0
$349$	−20.8915	−1.11829	−0.559147	−	0.829068i	$-0.688871\pi$
$349$	−20.8915	−1.11829	−0.559147	+	0.829068i	$0.688871\pi$
$350$	0	0
$351$	23.3564	1.24667
$352$	0	0
$353$	−10.2879	−0.547569	−0.273785	−	0.961791i	$-0.588276\pi$
$353$	−10.2879	−0.547569	−0.273785	+	0.961791i	$0.588276\pi$
$354$	0	0
$355$	−36.4480	−1.93446
$356$	0	0
$357$	−15.7180	−0.831886
$358$	0	0
$359$	−30.9455	−1.63324	−0.816621	−	0.577174i	$-0.804156\pi$
$359$	−30.9455	−1.63324	−0.816621	+	0.577174i	$0.804156\pi$
$360$	0	0
$361$	0.536558	0.0282399
$362$	0	0
$363$	7.84014	0.411501
$364$	0	0
$365$	34.6529	1.81381
$366$	0	0
$367$	11.0935	0.579078	0.289539	−	0.957166i	$-0.406498\pi$
$367$	11.0935	0.579078	0.289539	+	0.957166i	$0.406498\pi$
$368$	0	0
$369$	−86.4598	−4.50092
$370$	0	0
$371$	−6.00000	−0.311504
$372$	0	0
$373$	22.9252	1.18702	0.593512	−	0.804825i	$-0.297741\pi$
$373$	22.9252	1.18702	0.593512	+	0.804825i	$0.297741\pi$
$374$	0	0
$375$	−26.1102	−1.34832
$376$	0	0
$377$	−6.92894	−0.356859
$378$	0	0
$379$	8.61380	0.442461	0.221230	−	0.975222i	$-0.428993\pi$
$379$	8.61380	0.442461	0.221230	+	0.975222i	$0.428993\pi$
$380$	0	0
$381$	16.2832	0.834211
$382$	0	0
$383$	−2.66828	−0.136343	−0.0681713	−	0.997674i	$-0.521716\pi$
$383$	−2.66828	−0.136343	−0.0681713	+	0.997674i	$0.521716\pi$
$384$	0	0
$385$	9.72094	0.495425
$386$	0	0
$387$	−105.592	−5.36757
$388$	0	0
$389$	22.4704	1.13930	0.569648	−	0.821889i	$-0.307080\pi$
$389$	22.4704	1.13930	0.569648	+	0.821889i	$0.307080\pi$
$390$	0	0
$391$	4.66293	0.235815
$392$	0	0
$393$	−46.7145	−2.35644
$394$	0	0
$395$	6.74250	0.339252
$396$	0	0
$397$	−0.154724	−0.00776538	−0.00388269	−	0.999992i	$-0.501236\pi$
$397$	−0.154724	−0.00776538	−0.00388269	+	0.999992i	$0.501236\pi$
$398$	0	0
$399$	−14.8992	−0.745892
$400$	0	0
$401$	15.9860	0.798304	0.399152	−	0.916885i	$-0.369305\pi$
$401$	15.9860	0.798304	0.399152	+	0.916885i	$0.369305\pi$
$402$	0	0
$403$	−8.65277	−0.431025
$404$	0	0
$405$	95.4526	4.74308
$406$	0	0
$407$	−14.5108	−0.719273
$408$	0	0
$409$	30.0439	1.48558	0.742788	−	0.669527i	$-0.233503\pi$
$409$	30.0439	1.48558	0.742788	+	0.669527i	$0.233503\pi$
$410$	0	0
$411$	72.7288	3.58745
$412$	0	0
$413$	−12.2014	−0.600393
$414$	0	0
$415$	−13.2594	−0.650880
$416$	0	0
$417$	36.2760	1.77644
$418$	0	0
$419$	−20.3077	−0.992095	−0.496048	−	0.868295i	$-0.665216\pi$
$419$	−20.3077	−0.992095	−0.496048	+	0.868295i	$0.665216\pi$
$420$	0	0
$421$	37.4595	1.82567	0.912833	−	0.408333i	$-0.133890\pi$
$421$	37.4595	1.82567	0.912833	+	0.408333i	$0.133890\pi$
$422$	0	0
$423$	−26.6632	−1.29641
$424$	0	0
$425$	9.75123	0.473004
$426$	0	0
$427$	9.96386	0.482185
$428$	0	0
$429$	−15.8994	−0.767629
$430$	0	0
$431$	−21.4834	−1.03482	−0.517409	−	0.855738i	$-0.673103\pi$
$431$	−21.4834	−1.03482	−0.517409	+	0.855738i	$0.673103\pi$
$432$	0	0
$433$	8.51530	0.409219	0.204610	−	0.978844i	$-0.434407\pi$
$433$	8.51530	0.409219	0.204610	+	0.978844i	$0.434407\pi$
$434$	0	0
$435$	−48.1362	−2.30796
$436$	0	0
$437$	4.42002	0.211438
$438$	0	0
$439$	3.42597	0.163513	0.0817564	−	0.996652i	$-0.473947\pi$
$439$	3.42597	0.163513	0.0817564	+	0.996652i	$0.473947\pi$
$440$	0	0
$441$	8.36257	0.398218
$442$	0	0
$443$	33.1907	1.57694	0.788470	−	0.615073i	$-0.210874\pi$
$443$	33.1907	1.57694	0.788470	+	0.615073i	$0.210874\pi$
$444$	0	0
$445$	6.60539	0.313125
$446$	0	0
$447$	40.3103	1.90661
$448$	0	0
$449$	−33.3673	−1.57470	−0.787350	−	0.616507i	$-0.788547\pi$
$449$	−33.3673	−1.57470	−0.787350	+	0.616507i	$0.788547\pi$
$450$	0	0
$451$	37.7418	1.77719
$452$	0	0
$453$	−45.4185	−2.13395
$454$	0	0
$455$	−3.44076	−0.161305
$456$	0	0
$457$	−11.4100	−0.533735	−0.266868	−	0.963733i	$-0.585989\pi$
$457$	−11.4100	−0.533735	−0.266868	+	0.963733i	$0.585989\pi$
$458$	0	0
$459$	84.2890	3.93427
$460$	0	0
$461$	−3.68193	−0.171484	−0.0857422	−	0.996317i	$-0.527326\pi$
$461$	−3.68193	−0.171484	−0.0857422	+	0.996317i	$0.527326\pi$
$462$	0	0
$463$	28.4893	1.32401	0.662005	−	0.749499i	$-0.269706\pi$
$463$	28.4893	1.32401	0.662005	+	0.749499i	$0.269706\pi$
$464$	0	0
$465$	−60.1119	−2.78762
$466$	0	0
$467$	−25.0633	−1.15979	−0.579897	−	0.814690i	$-0.696907\pi$
$467$	−25.0633	−1.15979	−0.579897	+	0.814690i	$0.696907\pi$
$468$	0	0
$469$	−1.46802	−0.0677868
$470$	0	0
$471$	19.0189	0.876346
$472$	0	0
$473$	46.0936	2.11939
$474$	0	0
$475$	9.24324	0.424109
$476$	0	0
$477$	50.1754	2.29738
$478$	0	0
$479$	16.6375	0.760186	0.380093	−	0.924948i	$-0.375892\pi$
$479$	16.6375	0.760186	0.380093	+	0.924948i	$0.375892\pi$
$480$	0	0
$481$	5.13614	0.234188
$482$	0	0
$483$	−3.37084	−0.153379
$484$	0	0
$485$	27.6956	1.25759
$486$	0	0
$487$	−14.9063	−0.675468	−0.337734	−	0.941242i	$-0.609660\pi$
$487$	−14.9063	−0.675468	−0.337734	+	0.941242i	$0.609660\pi$
$488$	0	0
$489$	41.1173	1.85939
$490$	0	0
$491$	2.33050	0.105174	0.0525869	−	0.998616i	$-0.483253\pi$
$491$	2.33050	0.105174	0.0525869	+	0.998616i	$0.483253\pi$
$492$	0	0
$493$	−25.0053	−1.12618
$494$	0	0
$495$	−81.2921	−3.65381
$496$	0	0
$497$	13.6872	0.613953
$498$	0	0
$499$	36.5640	1.63683	0.818414	−	0.574630i	$-0.194854\pi$
$499$	36.5640	1.63683	0.818414	+	0.574630i	$0.194854\pi$
$500$	0	0
$501$	1.76535	0.0788701
$502$	0	0
$503$	35.7820	1.59544	0.797721	−	0.603027i	$-0.206039\pi$
$503$	35.7820	1.59544	0.797721	+	0.603027i	$0.206039\pi$
$504$	0	0
$505$	−14.4868	−0.644652
$506$	0	0
$507$	−38.1933	−1.69622
$508$	0	0
$509$	3.32374	0.147322	0.0736610	−	0.997283i	$-0.476532\pi$
$509$	3.32374	0.147322	0.0736610	+	0.997283i	$0.476532\pi$
$510$	0	0
$511$	−13.0130	−0.575663
$512$	0	0
$513$	79.8979	3.52758
$514$	0	0
$515$	13.6302	0.600617
$516$	0	0
$517$	11.6391	0.511888
$518$	0	0
$519$	−10.0984	−0.443268
$520$	0	0
$521$	−13.7678	−0.603177	−0.301588	−	0.953438i	$-0.597517\pi$
$521$	−13.7678	−0.603177	−0.301588	+	0.953438i	$0.597517\pi$
$522$	0	0
$523$	30.4402	1.33106	0.665529	−	0.746372i	$-0.268206\pi$
$523$	30.4402	1.33106	0.665529	+	0.746372i	$0.268206\pi$
$524$	0	0
$525$	−7.04918	−0.307651
$526$	0	0
$527$	−31.2263	−1.36024
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	102.035	4.42796
$532$	0	0
$533$	−13.3588	−0.578635
$534$	0	0
$535$	−1.48922	−0.0643846
$536$	0	0
$537$	−10.5154	−0.453773
$538$	0	0
$539$	−3.65046	−0.157237
$540$	0	0
$541$	3.25354	0.139880	0.0699402	−	0.997551i	$-0.477719\pi$
$541$	3.25354	0.139880	0.0699402	+	0.997551i	$0.477719\pi$
$542$	0	0
$543$	−86.6437	−3.71824
$544$	0	0
$545$	−0.261907	−0.0112188
$546$	0	0
$547$	−6.32937	−0.270624	−0.135312	−	0.990803i	$-0.543204\pi$
$547$	−6.32937	−0.270624	−0.135312	+	0.990803i	$0.543204\pi$
$548$	0	0
$549$	−83.3235	−3.55616
$550$	0	0
$551$	−23.7027	−1.00977
$552$	0	0
$553$	−2.53198	−0.107671
$554$	0	0
$555$	35.6814	1.51459
$556$	0	0
$557$	−2.24744	−0.0952271	−0.0476136	−	0.998866i	$-0.515162\pi$
$557$	−2.24744	−0.0952271	−0.0476136	+	0.998866i	$0.515162\pi$
$558$	0	0
$559$	−16.3150	−0.690051
$560$	0	0
$561$	−57.3780	−2.42250
$562$	0	0
$563$	−32.1534	−1.35510	−0.677552	−	0.735475i	$-0.736959\pi$
$563$	−32.1534	−1.35510	−0.677552	+	0.735475i	$0.736959\pi$
$564$	0	0
$565$	−3.79261	−0.159556
$566$	0	0
$567$	−35.8449	−1.50534
$568$	0	0
$569$	−40.0603	−1.67941	−0.839707	−	0.543040i	$-0.817273\pi$
$569$	−40.0603	−1.67941	−0.839707	+	0.543040i	$0.817273\pi$
$570$	0	0
$571$	25.7952	1.07949	0.539747	−	0.841827i	$-0.318520\pi$
$571$	25.7952	1.07949	0.539747	+	0.841827i	$0.318520\pi$
$572$	0	0
$573$	−2.05329	−0.0857774
$574$	0	0
$575$	2.09122	0.0872100
$576$	0	0
$577$	−16.0179	−0.666832	−0.333416	−	0.942780i	$-0.608201\pi$
$577$	−16.0179	−0.666832	−0.333416	+	0.942780i	$0.608201\pi$
$578$	0	0
$579$	−40.3224	−1.67574
$580$	0	0
$581$	4.97926	0.206574
$582$	0	0
$583$	−21.9028	−0.907120
$584$	0	0
$585$	28.7736	1.18964
$586$	0	0
$587$	−20.4159	−0.842655	−0.421328	−	0.906909i	$-0.638436\pi$
$587$	−20.4159	−0.842655	−0.421328	+	0.906909i	$0.638436\pi$
$588$	0	0
$589$	−29.5996	−1.21963
$590$	0	0
$591$	57.8791	2.38083
$592$	0	0
$593$	−4.00255	−0.164365	−0.0821826	−	0.996617i	$-0.526189\pi$
$593$	−4.00255	−0.164365	−0.0821826	+	0.996617i	$0.526189\pi$
$594$	0	0
$595$	−12.4171	−0.509051
$596$	0	0
$597$	−49.6164	−2.03066
$598$	0	0
$599$	12.6268	0.515917	0.257958	−	0.966156i	$-0.416950\pi$
$599$	12.6268	0.515917	0.257958	+	0.966156i	$0.416950\pi$
$600$	0	0
$601$	−21.4014	−0.872983	−0.436491	−	0.899708i	$-0.643779\pi$
$601$	−21.4014	−0.872983	−0.436491	+	0.899708i	$0.643779\pi$
$602$	0	0
$603$	12.2764	0.499934
$604$	0	0
$605$	6.19364	0.251807
$606$	0	0
$607$	30.9291	1.25537	0.627687	−	0.778466i	$-0.284002\pi$
$607$	30.9291	1.25537	0.627687	+	0.778466i	$0.284002\pi$
$608$	0	0
$609$	18.0764	0.732492
$610$	0	0
$611$	−4.11971	−0.166666
$612$	0	0
$613$	−34.4100	−1.38981	−0.694904	−	0.719102i	$-0.744553\pi$
$613$	−34.4100	−1.38981	−0.694904	+	0.719102i	$0.744553\pi$
$614$	0	0
$615$	−92.8054	−3.74228
$616$	0	0
$617$	10.3118	0.415137	0.207569	−	0.978220i	$-0.433445\pi$
$617$	10.3118	0.415137	0.207569	+	0.978220i	$0.433445\pi$
$618$	0	0
$619$	−19.0729	−0.766604	−0.383302	−	0.923623i	$-0.625213\pi$
$619$	−19.0729	−0.766604	−0.383302	+	0.923623i	$0.625213\pi$
$620$	0	0
$621$	18.0764	0.725380
$622$	0	0
$623$	−2.48049	−0.0993788
$624$	0	0
$625$	−31.0829	−1.24332
$626$	0	0
$627$	−54.3889	−2.17208
$628$	0	0
$629$	18.5354	0.739055
$630$	0	0
$631$	9.09935	0.362239	0.181120	−	0.983461i	$-0.442028\pi$
$631$	9.09935	0.362239	0.181120	+	0.983461i	$0.442028\pi$
$632$	0	0
$633$	−45.3945	−1.80427
$634$	0	0
$635$	12.8635	0.510474
$636$	0	0
$637$	1.29209	0.0511946
$638$	0	0
$639$	−114.460	−4.52796
$640$	0	0
$641$	1.76910	0.0698751	0.0349376	−	0.999389i	$-0.488877\pi$
$641$	1.76910	0.0698751	0.0349376	+	0.999389i	$0.488877\pi$
$642$	0	0
$643$	36.9758	1.45818	0.729091	−	0.684417i	$-0.239943\pi$
$643$	36.9758	1.45818	0.729091	+	0.684417i	$0.239943\pi$
$644$	0	0
$645$	−113.342	−4.46285
$646$	0	0
$647$	31.9514	1.25614	0.628070	−	0.778157i	$-0.283845\pi$
$647$	31.9514	1.25614	0.628070	+	0.778157i	$0.283845\pi$
$648$	0	0
$649$	−44.5409	−1.74838
$650$	0	0
$651$	22.5736	0.884727
$652$	0	0
$653$	−20.1506	−0.788554	−0.394277	−	0.918992i	$-0.629005\pi$
$653$	−20.1506	−0.788554	−0.394277	+	0.918992i	$0.629005\pi$
$654$	0	0
$655$	−36.9040	−1.44196
$656$	0	0
$657$	108.822	4.24557
$658$	0	0
$659$	−12.8400	−0.500177	−0.250088	−	0.968223i	$-0.580460\pi$
$659$	−12.8400	−0.500177	−0.250088	+	0.968223i	$0.580460\pi$
$660$	0	0
$661$	−36.3917	−1.41547	−0.707736	−	0.706477i	$-0.750283\pi$
$661$	−36.3917	−1.41547	−0.707736	+	0.706477i	$0.750283\pi$
$662$	0	0
$663$	20.3091	0.788742
$664$	0	0
$665$	−11.7702	−0.456430
$666$	0	0
$667$	−5.36257	−0.207640
$668$	0	0
$669$	15.3865	0.594876
$670$	0	0
$671$	36.3727	1.40415
$672$	0	0
$673$	−32.3969	−1.24881	−0.624404	−	0.781102i	$-0.714658\pi$
$673$	−32.3969	−1.24881	−0.624404	+	0.781102i	$0.714658\pi$
$674$	0	0
$675$	37.8017	1.45499
$676$	0	0
$677$	0.409437	0.0157359	0.00786797	−	0.999969i	$-0.497496\pi$
$677$	0.409437	0.0157359	0.00786797	+	0.999969i	$0.497496\pi$
$678$	0	0
$679$	−10.4004	−0.399131
$680$	0	0
$681$	−53.2205	−2.03941
$682$	0	0
$683$	−42.7844	−1.63710	−0.818549	−	0.574436i	$-0.805221\pi$
$683$	−42.7844	−1.63710	−0.818549	+	0.574436i	$0.805221\pi$
$684$	0	0
$685$	57.4551	2.19525
$686$	0	0
$687$	−44.3840	−1.69336
$688$	0	0
$689$	7.75256	0.295349
$690$	0	0
$691$	35.0574	1.33365	0.666823	−	0.745216i	$-0.267654\pi$
$691$	35.0574	1.33365	0.666823	+	0.745216i	$0.267654\pi$
$692$	0	0
$693$	30.5272	1.15963
$694$	0	0
$695$	28.6577	1.08705
$696$	0	0
$697$	−48.2096	−1.82607
$698$	0	0
$699$	−20.0172	−0.757121
$700$	0	0
$701$	−38.4646	−1.45279	−0.726393	−	0.687279i	$-0.758805\pi$
$701$	−38.4646	−1.45279	−0.726393	+	0.687279i	$0.758805\pi$
$702$	0	0
$703$	17.5698	0.662658
$704$	0	0
$705$	−28.6201	−1.07790
$706$	0	0
$707$	5.44015	0.204598
$708$	0	0
$709$	20.0984	0.754809	0.377405	−	0.926048i	$-0.376817\pi$
$709$	20.0984	0.754809	0.377405	+	0.926048i	$0.376817\pi$
$710$	0	0
$711$	21.1739	0.794082
$712$	0	0
$713$	−6.69671	−0.250794
$714$	0	0
$715$	−12.5604	−0.469731
$716$	0	0
$717$	75.6127	2.82381
$718$	0	0
$719$	−37.1505	−1.38548	−0.692740	−	0.721187i	$-0.743597\pi$
$719$	−37.1505	−1.38548	−0.692740	+	0.721187i	$0.743597\pi$
$720$	0	0
$721$	−5.11848	−0.190622
$722$	0	0
$723$	72.5426	2.69789
$724$	0	0
$725$	−11.2143	−0.416490
$726$	0	0
$727$	23.3295	0.865245	0.432622	−	0.901575i	$-0.357588\pi$
$727$	23.3295	0.865245	0.432622	+	0.901575i	$0.357588\pi$
$728$	0	0
$729$	116.958	4.33177
$730$	0	0
$731$	−58.8779	−2.17768
$732$	0	0
$733$	51.5089	1.90253	0.951263	−	0.308382i	$-0.0997874\pi$
$733$	51.5089	1.90253	0.951263	+	0.308382i	$0.0997874\pi$
$734$	0	0
$735$	8.97633	0.331097
$736$	0	0
$737$	−5.35895	−0.197399
$738$	0	0
$739$	14.1506	0.520538	0.260269	−	0.965536i	$-0.416189\pi$
$739$	14.1506	0.520538	0.260269	+	0.965536i	$0.416189\pi$
$740$	0	0
$741$	19.2511	0.707208
$742$	0	0
$743$	10.7463	0.394245	0.197123	−	0.980379i	$-0.436840\pi$
$743$	10.7463	0.394245	0.197123	+	0.980379i	$0.436840\pi$
$744$	0	0
$745$	31.8447	1.16670
$746$	0	0
$747$	−41.6394	−1.52351
$748$	0	0
$749$	0.559240	0.0204342
$750$	0	0
$751$	33.6208	1.22684	0.613420	−	0.789757i	$-0.289793\pi$
$751$	33.6208	1.22684	0.613420	+	0.789757i	$0.289793\pi$
$752$	0	0
$753$	30.3234	1.10505
$754$	0	0
$755$	−35.8802	−1.30581
$756$	0	0
$757$	−2.34488	−0.0852259	−0.0426130	−	0.999092i	$-0.513568\pi$
$757$	−2.34488	−0.0852259	−0.0426130	+	0.999092i	$0.513568\pi$
$758$	0	0
$759$	−12.3051	−0.446648
$760$	0	0
$761$	−46.9892	−1.70336	−0.851679	−	0.524064i	$-0.824415\pi$
$761$	−46.9892	−1.70336	−0.851679	+	0.524064i	$0.824415\pi$
$762$	0	0
$763$	0.0983526	0.00356060
$764$	0	0
$765$	103.839	3.75430
$766$	0	0
$767$	15.7654	0.569255
$768$	0	0
$769$	8.12609	0.293034	0.146517	−	0.989208i	$-0.453194\pi$
$769$	8.12609	0.293034	0.146517	+	0.989208i	$0.453194\pi$
$770$	0	0
$771$	−57.8791	−2.08447
$772$	0	0
$773$	44.8815	1.61428	0.807138	−	0.590363i	$-0.201015\pi$
$773$	44.8815	1.61428	0.807138	+	0.590363i	$0.201015\pi$
$774$	0	0
$775$	−14.0043	−0.503050
$776$	0	0
$777$	−13.3993	−0.480696
$778$	0	0
$779$	−45.6981	−1.63731
$780$	0	0
$781$	49.9645	1.78787
$782$	0	0
$783$	−96.9359	−3.46421
$784$	0	0
$785$	15.0248	0.536258
$786$	0	0
$787$	7.02660	0.250471	0.125236	−	0.992127i	$-0.460031\pi$
$787$	7.02660	0.250471	0.125236	+	0.992127i	$0.460031\pi$
$788$	0	0
$789$	56.5057	2.01166
$790$	0	0
$791$	1.42422	0.0506395
$792$	0	0
$793$	−12.8742	−0.457177
$794$	0	0
$795$	53.8580	1.91015
$796$	0	0
$797$	9.81272	0.347585	0.173792	−	0.984782i	$-0.444398\pi$
$797$	9.81272	0.347585	0.173792	+	0.984782i	$0.444398\pi$
$798$	0	0
$799$	−14.8673	−0.525967
$800$	0	0
$801$	20.7433	0.732928
$802$	0	0
$803$	−47.5036	−1.67636
$804$	0	0
$805$	−2.66293	−0.0938561
$806$	0	0
$807$	−79.5483	−2.80023
$808$	0	0
$809$	−29.8343	−1.04892	−0.524459	−	0.851436i	$-0.675732\pi$
$809$	−29.8343	−1.04892	−0.524459	+	0.851436i	$0.675732\pi$
$810$	0	0
$811$	18.2767	0.641782	0.320891	−	0.947116i	$-0.396018\pi$
$811$	18.2767	0.641782	0.320891	+	0.947116i	$0.396018\pi$
$812$	0	0
$813$	−74.9832	−2.62978
$814$	0	0
$815$	32.4823	1.13781
$816$	0	0
$817$	−55.8106	−1.95257
$818$	0	0
$819$	−10.8052	−0.377565
$820$	0	0
$821$	−5.10804	−0.178272	−0.0891359	−	0.996019i	$-0.528411\pi$
$821$	−5.10804	−0.178272	−0.0891359	+	0.996019i	$0.528411\pi$
$822$	0	0
$823$	29.6659	1.03409	0.517044	−	0.855959i	$-0.327032\pi$
$823$	29.6659	1.03409	0.517044	+	0.855959i	$0.327032\pi$
$824$	0	0
$825$	−25.7327	−0.895899
$826$	0	0
$827$	30.7538	1.06941	0.534707	−	0.845038i	$-0.320422\pi$
$827$	30.7538	1.06941	0.534707	+	0.845038i	$0.320422\pi$
$828$	0	0
$829$	39.1419	1.35945	0.679727	−	0.733466i	$-0.262098\pi$
$829$	39.1419	1.35945	0.679727	+	0.733466i	$0.262098\pi$
$830$	0	0
$831$	−88.3051	−3.06327
$832$	0	0
$833$	4.66293	0.161561
$834$	0	0
$835$	1.39461	0.0482626
$836$	0	0
$837$	−121.052	−4.18418
$838$	0	0
$839$	26.7169	0.922371	0.461186	−	0.887304i	$-0.347424\pi$
$839$	26.7169	0.922371	0.461186	+	0.887304i	$0.347424\pi$
$840$	0	0
$841$	−0.242815	−0.00837293
$842$	0	0
$843$	86.6873	2.98567
$844$	0	0
$845$	−30.1724	−1.03796
$846$	0	0
$847$	−2.32587	−0.0799178
$848$	0	0
$849$	−32.7961	−1.12556
$850$	0	0
$851$	3.97505	0.136263
$852$	0	0
$853$	10.0575	0.344362	0.172181	−	0.985065i	$-0.444919\pi$
$853$	10.0575	0.344362	0.172181	+	0.985065i	$0.444919\pi$
$854$	0	0
$855$	98.4293	3.36621
$856$	0	0
$857$	−29.1705	−0.996446	−0.498223	−	0.867049i	$-0.666014\pi$
$857$	−29.1705	−0.996446	−0.498223	+	0.867049i	$0.666014\pi$
$858$	0	0
$859$	50.7538	1.73170	0.865848	−	0.500307i	$-0.166779\pi$
$859$	50.7538	1.73170	0.865848	+	0.500307i	$0.166779\pi$
$860$	0	0
$861$	34.8508	1.18771
$862$	0	0
$863$	24.9492	0.849279	0.424640	−	0.905362i	$-0.360401\pi$
$863$	24.9492	0.849279	0.424640	+	0.905362i	$0.360401\pi$
$864$	0	0
$865$	−7.97761	−0.271247
$866$	0	0
$867$	15.9878	0.542973
$868$	0	0
$869$	−9.24290	−0.313544
$870$	0	0
$871$	1.89682	0.0642712
$872$	0	0
$873$	86.9742	2.94363
$874$	0	0
$875$	7.74589	0.261859
$876$	0	0
$877$	−36.8980	−1.24596	−0.622978	−	0.782239i	$-0.714078\pi$
$877$	−36.8980	−1.24596	−0.622978	+	0.782239i	$0.714078\pi$
$878$	0	0
$879$	38.5272	1.29949
$880$	0	0
$881$	−20.3456	−0.685459	−0.342730	−	0.939434i	$-0.611351\pi$
$881$	−20.3456	−0.685459	−0.342730	+	0.939434i	$0.611351\pi$
$882$	0	0
$883$	37.0520	1.24690	0.623450	−	0.781863i	$-0.285730\pi$
$883$	37.0520	1.24690	0.623450	+	0.781863i	$0.285730\pi$
$884$	0	0
$885$	109.524	3.68161
$886$	0	0
$887$	−40.3619	−1.35522	−0.677610	−	0.735421i	$-0.736984\pi$
$887$	−40.3619	−1.35522	−0.677610	+	0.735421i	$0.736984\pi$
$888$	0	0
$889$	−4.83059	−0.162013
$890$	0	0
$891$	−130.850	−4.38365
$892$	0	0
$893$	−14.0928	−0.471597
$894$	0	0
$895$	−8.30707	−0.277675
$896$	0	0
$897$	4.35544	0.145424
$898$	0	0
$899$	35.9116	1.19772
$900$	0	0
$901$	27.9776	0.932069
$902$	0	0
$903$	42.5629	1.41641
$904$	0	0
$905$	−68.4478	−2.27528
$906$	0	0
$907$	−21.4193	−0.711216	−0.355608	−	0.934635i	$-0.615726\pi$
$907$	−21.4193	−0.711216	−0.355608	+	0.934635i	$0.615726\pi$
$908$	0	0
$909$	−45.4936	−1.50893
$910$	0	0
$911$	−10.9631	−0.363223	−0.181612	−	0.983370i	$-0.558131\pi$
$911$	−10.9631	−0.363223	−0.181612	+	0.983370i	$0.558131\pi$
$912$	0	0
$913$	18.1766	0.601557
$914$	0	0
$915$	−89.4389	−2.95676
$916$	0	0
$917$	13.8584	0.457645
$918$	0	0
$919$	14.6541	0.483393	0.241696	−	0.970352i	$-0.422296\pi$
$919$	14.6541	0.483393	0.241696	+	0.970352i	$0.422296\pi$
$920$	0	0
$921$	32.4407	1.06896
$922$	0	0
$923$	−17.6851	−0.582112
$924$	0	0
$925$	8.31272	0.273320
$926$	0	0
$927$	42.8037	1.40586
$928$	0	0
$929$	38.2249	1.25412	0.627060	−	0.778971i	$-0.284258\pi$
$929$	38.2249	1.25412	0.627060	+	0.778971i	$0.284258\pi$
$930$	0	0
$931$	4.42002	0.144860
$932$	0	0
$933$	−101.730	−3.33050
$934$	0	0
$935$	−45.3281	−1.48239
$936$	0	0
$937$	31.9783	1.04469	0.522343	−	0.852735i	$-0.325058\pi$
$937$	31.9783	1.04469	0.522343	+	0.852735i	$0.325058\pi$
$938$	0	0
$939$	−11.8920	−0.388082
$940$	0	0
$941$	17.1564	0.559281	0.279641	−	0.960105i	$-0.409785\pi$
$941$	17.1564	0.559281	0.279641	+	0.960105i	$0.409785\pi$
$942$	0	0
$943$	−10.3389	−0.336681
$944$	0	0
$945$	−48.1362	−1.56587
$946$	0	0
$947$	30.5225	0.991847	0.495923	−	0.868366i	$-0.334830\pi$
$947$	30.5225	0.991847	0.495923	+	0.868366i	$0.334830\pi$
$948$	0	0
$949$	16.8141	0.545807
$950$	0	0
$951$	−8.09188	−0.262397
$952$	0	0
$953$	−4.51078	−0.146119	−0.0730593	−	0.997328i	$-0.523276\pi$
$953$	−4.51078	−0.146119	−0.0730593	+	0.997328i	$0.523276\pi$
$954$	0	0
$955$	−1.62208	−0.0524893
$956$	0	0
$957$	65.9871	2.13306
$958$	0	0
$959$	−21.5759	−0.696721
$960$	0	0
$961$	13.8459	0.446643
$962$	0	0
$963$	−4.67669	−0.150704
$964$	0	0
$965$	−31.8543	−1.02543
$966$	0	0
$967$	4.69283	0.150911	0.0754556	−	0.997149i	$-0.475959\pi$
$967$	4.69283	0.150911	0.0754556	+	0.997149i	$0.475959\pi$
$968$	0	0
$969$	69.4739	2.23182
$970$	0	0
$971$	−37.7615	−1.21182	−0.605912	−	0.795532i	$-0.707192\pi$
$971$	−37.7615	−1.21182	−0.605912	+	0.795532i	$0.707192\pi$
$972$	0	0
$973$	−10.7617	−0.345005
$974$	0	0
$975$	9.10819	0.291696
$976$	0	0
$977$	12.8910	0.412419	0.206209	−	0.978508i	$-0.433887\pi$
$977$	12.8910	0.412419	0.206209	+	0.978508i	$0.433887\pi$
$978$	0	0
$979$	−9.05494	−0.289397
$980$	0	0
$981$	−0.822481	−0.0262598
$982$	0	0
$983$	34.5543	1.10211	0.551056	−	0.834469i	$-0.314225\pi$
$983$	34.5543	1.10211	0.551056	+	0.834469i	$0.314225\pi$
$984$	0	0
$985$	45.7240	1.45689
$986$	0	0
$987$	10.7476	0.342099
$988$	0	0
$989$	−12.6268	−0.401509
$990$	0	0
$991$	39.0770	1.24132	0.620661	−	0.784079i	$-0.286864\pi$
$991$	39.0770	1.24132	0.620661	+	0.784079i	$0.286864\pi$
$992$	0	0
$993$	53.5497	1.69935
$994$	0	0
$995$	−39.1965	−1.24261
$996$	0	0
$997$	40.2494	1.27471	0.637355	−	0.770570i	$-0.280028\pi$
$997$	40.2494	1.27471	0.637355	+	0.770570i	$0.280028\pi$
$998$	0	0
$999$	71.8546	2.27338

Twists

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

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

\(f(q)\)	\(=\)	\(q+3.37084 q^{3} +2.66293 q^{5} -1.00000 q^{7} +8.36257 q^{9} +O(q^{10})\)	\(q+3.37084 q^{3} +2.66293 q^{5} -1.00000 q^{7} +8.36257 q^{9} -3.65046 q^{11} +1.29209 q^{13} +8.97633 q^{15} +4.66293 q^{17} +4.42002 q^{19} -3.37084 q^{21} +1.00000 q^{23} +2.09122 q^{25} +18.0764 q^{27} -5.36257 q^{29} -6.69671 q^{31} -12.3051 q^{33} -2.66293 q^{35} +3.97505 q^{37} +4.35544 q^{39} -10.3389 q^{41} -12.6268 q^{43} +22.2690 q^{45} -3.18840 q^{47} +1.00000 q^{49} +15.7180 q^{51} +6.00000 q^{53} -9.72094 q^{55} +14.8992 q^{57} +12.2014 q^{59} -9.96386 q^{61} -8.36257 q^{63} +3.44076 q^{65} +1.46802 q^{67} +3.37084 q^{69} -13.6872 q^{71} +13.0130 q^{73} +7.04918 q^{75} +3.65046 q^{77} +2.53198 q^{79} +35.8449 q^{81} -4.97926 q^{83} +12.4171 q^{85} -18.0764 q^{87} +2.48049 q^{89} -1.29209 q^{91} -22.5736 q^{93} +11.7702 q^{95} +10.4004 q^{97} -30.5272 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + 3 q^{3} + 2 q^{5} - 5 q^{7} + 14 q^{9}+O(q^{10}) \) 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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