LMFDB - Embedded newform 2576.2.a.be.1.4

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			1.4
Root			$-3.17666$ 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)

$f(q)$	$=$	$q+2.40396 q^{3} -3.17666 q^{5} -1.00000 q^{7} +2.77901 q^{9} +O(q^{10})$	$q+2.40396 q^{3} -3.17666 q^{5} -1.00000 q^{7} +2.77901 q^{9} +1.28323 q^{11} -3.58061 q^{13} -7.63655 q^{15} -1.17666 q^{17} +8.64286 q^{19} -2.40396 q^{21} +1.00000 q^{23} +5.09115 q^{25} -0.531251 q^{27} +0.220990 q^{29} +5.94936 q^{31} +3.08484 q^{33} +3.17666 q^{35} +5.78685 q^{37} -8.60764 q^{39} +11.8575 q^{41} +8.91978 q^{43} -8.82796 q^{45} +3.77834 q^{47} +1.00000 q^{49} -2.82863 q^{51} +6.00000 q^{53} -4.07639 q^{55} +20.7771 q^{57} -5.28256 q^{59} +5.74312 q^{61} -2.77901 q^{63} +11.3744 q^{65} -9.46553 q^{67} +2.40396 q^{69} -14.8491 q^{71} +2.49578 q^{73} +12.2389 q^{75} -1.28323 q^{77} +13.4655 q^{79} -9.61413 q^{81} -1.26848 q^{83} +3.73783 q^{85} +0.531251 q^{87} -9.35895 q^{89} +3.58061 q^{91} +14.3020 q^{93} -27.4554 q^{95} +8.14963 q^{97} +3.56612 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	2.40396	1.38793	0.693963	−	0.720011i	$-0.255863\pi$
$3$	2.40396	1.38793	0.693963	+	0.720011i	$0.255863\pi$
$4$	0	0
$5$	−3.17666	−1.42064	−0.710322	−	0.703877i	$-0.751451\pi$
$5$	−3.17666	−1.42064	−0.710322	+	0.703877i	$0.751451\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	2.77901	0.926337
$10$	0	0
$11$	1.28323	0.386909	0.193455	−	0.981109i	$-0.438031\pi$
$11$	1.28323	0.386909	0.193455	+	0.981109i	$0.438031\pi$
$12$	0	0
$13$	−3.58061	−0.993084	−0.496542	−	0.868013i	$-0.665397\pi$
$13$	−3.58061	−0.993084	−0.496542	+	0.868013i	$0.665397\pi$
$14$	0	0
$15$	−7.63655	−1.97175
$16$	0	0
$17$	−1.17666	−0.285381	−0.142691	−	0.989767i	$-0.545575\pi$
$17$	−1.17666	−0.285381	−0.142691	+	0.989767i	$0.545575\pi$
$18$	0	0
$19$	8.64286	1.98281	0.991404	−	0.130839i	$-0.0417671\pi$
$19$	8.64286	1.98281	0.991404	+	0.130839i	$0.0417671\pi$
$20$	0	0
$21$	−2.40396	−0.524586
$22$	0	0
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	0	0
$25$	5.09115	1.01823
$26$	0	0
$27$	−0.531251	−0.102239
$28$	0	0
$29$	0.220990	0.0410369	0.0205184	−	0.999789i	$-0.493468\pi$
$29$	0.220990	0.0410369	0.0205184	+	0.999789i	$0.493468\pi$
$30$	0	0
$31$	5.94936	1.06854	0.534268	−	0.845315i	$-0.320587\pi$
$31$	5.94936	1.06854	0.534268	+	0.845315i	$0.320587\pi$
$32$	0	0
$33$	3.08484	0.537001
$34$	0	0
$35$	3.17666	0.536953
$36$	0	0
$37$	5.78685	0.951352	0.475676	−	0.879621i	$-0.342204\pi$
$37$	5.78685	0.951352	0.475676	+	0.879621i	$0.342204\pi$
$38$	0	0
$39$	−8.60764	−1.37833
$40$	0	0
$41$	11.8575	1.85184	0.925918	−	0.377724i	$-0.123293\pi$
$41$	11.8575	1.85184	0.925918	+	0.377724i	$0.123293\pi$
$42$	0	0
$43$	8.91978	1.36025	0.680127	−	0.733095i	$-0.261925\pi$
$43$	8.91978	1.36025	0.680127	+	0.733095i	$0.261925\pi$
$44$	0	0
$45$	−8.82796	−1.31599
$46$	0	0
$47$	3.77834	0.551127	0.275564	−	0.961283i	$-0.411136\pi$
$47$	3.77834	0.551127	0.275564	+	0.961283i	$0.411136\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−2.82863	−0.396088
$52$	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
$55$	−4.07639	−0.549660
$56$	0	0
$57$	20.7771	2.75199
$58$	0	0
$59$	−5.28256	−0.687731	−0.343865	−	0.939019i	$-0.611736\pi$
$59$	−5.28256	−0.687731	−0.343865	+	0.939019i	$0.611736\pi$
$60$	0	0
$61$	5.74312	0.735331	0.367666	−	0.929958i	$-0.380157\pi$
$61$	5.74312	0.735331	0.367666	+	0.929958i	$0.380157\pi$
$62$	0	0
$63$	−2.77901	−0.350122
$64$	0	0
$65$	11.3744	1.41082
$66$	0	0
$67$	−9.46553	−1.15640	−0.578199	−	0.815896i	$-0.696244\pi$
$67$	−9.46553	−1.15640	−0.578199	+	0.815896i	$0.696244\pi$
$68$	0	0
$69$	2.40396	0.289402
$70$	0	0
$71$	−14.8491	−1.76226	−0.881131	−	0.472872i	$-0.843217\pi$
$71$	−14.8491	−1.76226	−0.881131	+	0.472872i	$0.843217\pi$
$72$	0	0
$73$	2.49578	0.292109	0.146054	−	0.989277i	$-0.453343\pi$
$73$	2.49578	0.292109	0.146054	+	0.989277i	$0.453343\pi$
$74$	0	0
$75$	12.2389	1.41323
$76$	0	0
$77$	−1.28323	−0.146238
$78$	0	0
$79$	13.4655	1.51499	0.757495	−	0.652841i	$-0.226423\pi$
$79$	13.4655	1.51499	0.757495	+	0.652841i	$0.226423\pi$
$80$	0	0
$81$	−9.61413	−1.06824
$82$	0	0
$83$	−1.26848	−0.139233	−0.0696167	−	0.997574i	$-0.522178\pi$
$83$	−1.26848	−0.139233	−0.0696167	+	0.997574i	$0.522178\pi$
$84$	0	0
$85$	3.73783	0.405425
$86$	0	0
$87$	0.531251	0.0569561
$88$	0	0
$89$	−9.35895	−0.992047	−0.496023	−	0.868309i	$-0.665207\pi$
$89$	−9.35895	−0.992047	−0.496023	+	0.868309i	$0.665207\pi$
$90$	0	0
$91$	3.58061	0.375350
$92$	0	0
$93$	14.3020	1.48305
$94$	0	0
$95$	−27.4554	−2.81686
$96$	0	0
$97$	8.14963	0.827469	0.413735	−	0.910397i	$-0.364224\pi$
$97$	8.14963	0.827469	0.413735	+	0.910397i	$0.364224\pi$
$98$	0	0
$99$	3.56612	0.358408
$100$	0	0
$101$	16.5837	1.65014	0.825070	−	0.565030i	$-0.191136\pi$
$101$	16.5837	1.65014	0.825070	+	0.565030i	$0.191136\pi$
$102$	0	0
$103$	−10.7488	−1.05911	−0.529553	−	0.848277i	$-0.677640\pi$
$103$	−10.7488	−1.05911	−0.529553	+	0.848277i	$0.677640\pi$
$104$	0	0
$105$	7.63655	0.745251
$106$	0	0
$107$	7.37438	0.712908	0.356454	−	0.934313i	$-0.383986\pi$
$107$	7.37438	0.712908	0.356454	+	0.934313i	$0.383986\pi$
$108$	0	0
$109$	−10.4778	−1.00359	−0.501796	−	0.864986i	$-0.667327\pi$
$109$	−10.4778	−1.00359	−0.501796	+	0.864986i	$0.667327\pi$
$110$	0	0
$111$	13.9113	1.32041
$112$	0	0
$113$	−0.124485	−0.0117106	−0.00585529	−	0.999983i	$-0.501864\pi$
$113$	−0.124485	−0.0117106	−0.00585529	+	0.999983i	$0.501864\pi$
$114$	0	0
$115$	−3.17666	−0.296225
$116$	0	0
$117$	−9.95056	−0.919930
$118$	0	0
$119$	1.17666	0.107864
$120$	0	0
$121$	−9.35331	−0.850301
$122$	0	0
$123$	28.5050	2.57021
$124$	0	0
$125$	−0.289543	−0.0258975
$126$	0	0
$127$	−11.6865	−1.03701	−0.518505	−	0.855074i	$-0.673511\pi$
$127$	−11.6865	−1.03701	−0.518505	+	0.855074i	$0.673511\pi$
$128$	0	0
$129$	21.4428	1.88793
$130$	0	0
$131$	−3.50141	−0.305920	−0.152960	−	0.988232i	$-0.548881\pi$
$131$	−3.50141	−0.305920	−0.152960	+	0.988232i	$0.548881\pi$
$132$	0	0
$133$	−8.64286	−0.749431
$134$	0	0
$135$	1.68760	0.145246
$136$	0	0
$137$	2.28358	0.195100	0.0975498	−	0.995231i	$-0.468899\pi$
$137$	2.28358	0.195100	0.0975498	+	0.995231i	$0.468899\pi$
$138$	0	0
$139$	21.3520	1.81106	0.905528	−	0.424287i	$-0.139475\pi$
$139$	21.3520	1.81106	0.905528	+	0.424287i	$0.139475\pi$
$140$	0	0
$141$	9.08296	0.764923
$142$	0	0
$143$	−4.59476	−0.384233
$144$	0	0
$145$	−0.702010	−0.0582988
$146$	0	0
$147$	2.40396	0.198275
$148$	0	0
$149$	4.53695	0.371682	0.185841	−	0.982580i	$-0.440499\pi$
$149$	4.53695	0.371682	0.185841	+	0.982580i	$0.440499\pi$
$150$	0	0
$151$	15.3564	1.24969	0.624843	−	0.780750i	$-0.285163\pi$
$151$	15.3564	1.24969	0.624843	+	0.780750i	$0.285163\pi$
$152$	0	0
$153$	−3.26994	−0.264359
$154$	0	0
$155$	−18.8991	−1.51801
$156$	0	0
$157$	−3.90818	−0.311907	−0.155953	−	0.987764i	$-0.549845\pi$
$157$	−3.90818	−0.311907	−0.155953	+	0.987764i	$0.549845\pi$
$158$	0	0
$159$	14.4237	1.14388
$160$	0	0
$161$	−1.00000	−0.0788110
$162$	0	0
$163$	−8.57678	−0.671786	−0.335893	−	0.941900i	$-0.609038\pi$
$163$	−8.57678	−0.671786	−0.335893	+	0.941900i	$0.609038\pi$
$164$	0	0
$165$	−9.79947	−0.762888
$166$	0	0
$167$	6.84058	0.529340	0.264670	−	0.964339i	$-0.414737\pi$
$167$	6.84058	0.529340	0.264670	+	0.964339i	$0.414737\pi$
$168$	0	0
$169$	−0.179206	−0.0137850
$170$	0	0
$171$	24.0186	1.83675
$172$	0	0
$173$	−8.51837	−0.647640	−0.323820	−	0.946119i	$-0.604967\pi$
$173$	−8.51837	−0.647640	−0.323820	+	0.946119i	$0.604967\pi$
$174$	0	0
$175$	−5.09115	−0.384855
$176$	0	0
$177$	−12.6990	−0.954519
$178$	0	0
$179$	−23.2602	−1.73855	−0.869275	−	0.494329i	$-0.835414\pi$
$179$	−23.2602	−1.73855	−0.869275	+	0.494329i	$0.835414\pi$
$180$	0	0
$181$	23.2636	1.72917	0.864586	−	0.502485i	$-0.167581\pi$
$181$	23.2636	1.72917	0.864586	+	0.502485i	$0.167581\pi$
$182$	0	0
$183$	13.8062	1.02059
$184$	0	0
$185$	−18.3828	−1.35153
$186$	0	0
$187$	−1.50992	−0.110417
$188$	0	0
$189$	0.531251	0.0386428
$190$	0	0
$191$	10.9481	0.792175	0.396087	−	0.918213i	$-0.370368\pi$
$191$	10.9481	0.792175	0.396087	+	0.918213i	$0.370368\pi$
$192$	0	0
$193$	−3.69914	−0.266270	−0.133135	−	0.991098i	$-0.542504\pi$
$193$	−3.69914	−0.266270	−0.133135	+	0.991098i	$0.542504\pi$
$194$	0	0
$195$	27.3435	1.95811
$196$	0	0
$197$	14.4649	1.03058	0.515291	−	0.857015i	$-0.327684\pi$
$197$	14.4649	1.03058	0.515291	+	0.857015i	$0.327684\pi$
$198$	0	0
$199$	22.2520	1.57740	0.788702	−	0.614776i	$-0.210753\pi$
$199$	22.2520	1.57740	0.788702	+	0.614776i	$0.210753\pi$
$200$	0	0
$201$	−22.7547	−1.60499
$202$	0	0
$203$	−0.220990	−0.0155105
$204$	0	0
$205$	−37.6673	−2.63080
$206$	0	0
$207$	2.77901	0.193155
$208$	0	0
$209$	11.0908	0.767167
$210$	0	0
$211$	−0.365934	−0.0251919	−0.0125960	−	0.999921i	$-0.504010\pi$
$211$	−0.365934	−0.0251919	−0.0125960	+	0.999921i	$0.504010\pi$
$212$	0	0
$213$	−35.6966	−2.44589
$214$	0	0
$215$	−28.3351	−1.93244
$216$	0	0
$217$	−5.94936	−0.403869
$218$	0	0
$219$	5.99974	0.405425
$220$	0	0
$221$	4.21315	0.283407
$222$	0	0
$223$	−11.6545	−0.780440	−0.390220	−	0.920722i	$-0.627601\pi$
$223$	−11.6545	−0.780440	−0.390220	+	0.920722i	$0.627601\pi$
$224$	0	0
$225$	14.1483	0.943223
$226$	0	0
$227$	3.46901	0.230246	0.115123	−	0.993351i	$-0.463274\pi$
$227$	3.46901	0.230246	0.115123	+	0.993351i	$0.463274\pi$
$228$	0	0
$229$	−7.17070	−0.473853	−0.236926	−	0.971528i	$-0.576140\pi$
$229$	−7.17070	−0.473853	−0.236926	+	0.971528i	$0.576140\pi$
$230$	0	0
$231$	−3.08484	−0.202967
$232$	0	0
$233$	−1.65452	−0.108391	−0.0541957	−	0.998530i	$-0.517259\pi$
$233$	−1.65452	−0.108391	−0.0541957	+	0.998530i	$0.517259\pi$
$234$	0	0
$235$	−12.0025	−0.782955
$236$	0	0
$237$	32.3705	2.10269
$238$	0	0
$239$	−15.1898	−0.982545	−0.491273	−	0.871006i	$-0.663468\pi$
$239$	−15.1898	−0.982545	−0.491273	+	0.871006i	$0.663468\pi$
$240$	0	0
$241$	14.1374	0.910667	0.455333	−	0.890321i	$-0.349520\pi$
$241$	14.1374	0.910667	0.455333	+	0.890321i	$0.349520\pi$
$242$	0	0
$243$	−21.5182	−1.38039
$244$	0	0
$245$	−3.17666	−0.202949
$246$	0	0
$247$	−30.9467	−1.96909
$248$	0	0
$249$	−3.04936	−0.193245
$250$	0	0
$251$	14.5184	0.916391	0.458196	−	0.888851i	$-0.348496\pi$
$251$	14.5184	0.916391	0.458196	+	0.888851i	$0.348496\pi$
$252$	0	0
$253$	1.28323	0.0806762
$254$	0	0
$255$	8.98559	0.562700
$256$	0	0
$257$	−14.4649	−0.902297	−0.451148	−	0.892449i	$-0.648985\pi$
$257$	−14.4649	−0.902297	−0.451148	+	0.892449i	$0.648985\pi$
$258$	0	0
$259$	−5.78685	−0.359577
$260$	0	0
$261$	0.614134	0.0380140
$262$	0	0
$263$	−29.8420	−1.84014	−0.920069	−	0.391755i	$-0.871868\pi$
$263$	−29.8420	−1.84014	−0.920069	+	0.391755i	$0.871868\pi$
$264$	0	0
$265$	−19.0599	−1.17084
$266$	0	0
$267$	−22.4985	−1.37689
$268$	0	0
$269$	−14.0710	−0.857926	−0.428963	−	0.903322i	$-0.641121\pi$
$269$	−14.0710	−0.857926	−0.428963	+	0.903322i	$0.641121\pi$
$270$	0	0
$271$	−22.9170	−1.39211	−0.696053	−	0.717990i	$-0.745062\pi$
$271$	−22.9170	−1.39211	−0.696053	+	0.717990i	$0.745062\pi$
$272$	0	0
$273$	8.60764	0.520958
$274$	0	0
$275$	6.53313	0.393962
$276$	0	0
$277$	−3.25468	−0.195555	−0.0977773	−	0.995208i	$-0.531173\pi$
$277$	−3.25468	−0.195555	−0.0977773	+	0.995208i	$0.531173\pi$
$278$	0	0
$279$	16.5333	0.989824
$280$	0	0
$281$	5.00283	0.298444	0.149222	−	0.988804i	$-0.452323\pi$
$281$	5.00283	0.298444	0.149222	+	0.988804i	$0.452323\pi$
$282$	0	0
$283$	6.96035	0.413750	0.206875	−	0.978367i	$-0.433671\pi$
$283$	6.96035	0.413750	0.206875	+	0.978367i	$0.433671\pi$
$284$	0	0
$285$	−66.0016	−3.90960
$286$	0	0
$287$	−11.8575	−0.699928
$288$	0	0
$289$	−15.6155	−0.918558
$290$	0	0
$291$	19.5914	1.14847
$292$	0	0
$293$	1.84441	0.107752	0.0538758	−	0.998548i	$-0.482842\pi$
$293$	1.84441	0.107752	0.0538758	+	0.998548i	$0.482842\pi$
$294$	0	0
$295$	16.7809	0.977021
$296$	0	0
$297$	−0.681719	−0.0395574
$298$	0	0
$299$	−3.58061	−0.207072
$300$	0	0
$301$	−8.91978	−0.514128
$302$	0	0
$303$	39.8665	2.29027
$304$	0	0
$305$	−18.2439	−1.04464
$306$	0	0
$307$	−19.8946	−1.13544	−0.567721	−	0.823221i	$-0.692175\pi$
$307$	−19.8946	−1.13544	−0.567721	+	0.823221i	$0.692175\pi$
$308$	0	0
$309$	−25.8396	−1.46996
$310$	0	0
$311$	16.2916	0.923812	0.461906	−	0.886929i	$-0.347166\pi$
$311$	16.2916	0.923812	0.461906	+	0.886929i	$0.347166\pi$
$312$	0	0
$313$	−4.32221	−0.244306	−0.122153	−	0.992511i	$-0.538980\pi$
$313$	−4.32221	−0.244306	−0.122153	+	0.992511i	$0.538980\pi$
$314$	0	0
$315$	8.82796	0.497399
$316$	0	0
$317$	15.5121	0.871244	0.435622	−	0.900130i	$-0.356528\pi$
$317$	15.5121	0.871244	0.435622	+	0.900130i	$0.356528\pi$
$318$	0	0
$319$	0.283582	0.0158775
$320$	0	0
$321$	17.7277	0.989463
$322$	0	0
$323$	−10.1697	−0.565856
$324$	0	0
$325$	−18.2294	−1.01119
$326$	0	0
$327$	−25.1882	−1.39291
$328$	0	0
$329$	−3.77834	−0.208306
$330$	0	0
$331$	32.2813	1.77434	0.887170	−	0.461443i	$-0.152668\pi$
$331$	32.2813	1.77434	0.887170	+	0.461443i	$0.152668\pi$
$332$	0	0
$333$	16.0817	0.881272
$334$	0	0
$335$	30.0687	1.64283
$336$	0	0
$337$	18.4778	1.00655	0.503275	−	0.864126i	$-0.332128\pi$
$337$	18.4778	1.00655	0.503275	+	0.864126i	$0.332128\pi$
$338$	0	0
$339$	−0.299257	−0.0162534
$340$	0	0
$341$	7.63441	0.413426
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	−7.63655	−0.411138
$346$	0	0
$347$	−2.75011	−0.147633	−0.0738167	−	0.997272i	$-0.523518\pi$
$347$	−2.75011	−0.147633	−0.0738167	+	0.997272i	$0.523518\pi$
$348$	0	0
$349$	−12.5413	−0.671321	−0.335660	−	0.941983i	$-0.608959\pi$
$349$	−12.5413	−0.671321	−0.335660	+	0.941983i	$0.608959\pi$
$350$	0	0
$351$	1.90221	0.101532
$352$	0	0
$353$	−10.9378	−0.582158	−0.291079	−	0.956699i	$-0.594014\pi$
$353$	−10.9378	−0.582158	−0.291079	+	0.956699i	$0.594014\pi$
$354$	0	0
$355$	47.1705	2.50355
$356$	0	0
$357$	2.82863	0.149707
$358$	0	0
$359$	9.65974	0.509822	0.254911	−	0.966965i	$-0.417954\pi$
$359$	9.65974	0.509822	0.254911	+	0.966965i	$0.417954\pi$
$360$	0	0
$361$	55.6990	2.93152
$362$	0	0
$363$	−22.4850	−1.18015
$364$	0	0
$365$	−7.92823	−0.414982
$366$	0	0
$367$	−2.96191	−0.154611	−0.0773053	−	0.997007i	$-0.524632\pi$
$367$	−2.96191	−0.154611	−0.0773053	+	0.997007i	$0.524632\pi$
$368$	0	0
$369$	32.9522	1.71542
$370$	0	0
$371$	−6.00000	−0.311504
$372$	0	0
$373$	7.76861	0.402243	0.201122	−	0.979566i	$-0.435541\pi$
$373$	7.76861	0.402243	0.201122	+	0.979566i	$0.435541\pi$
$374$	0	0
$375$	−0.696049	−0.0359438
$376$	0	0
$377$	−0.791281	−0.0407530
$378$	0	0
$379$	20.6934	1.06295	0.531475	−	0.847074i	$-0.321638\pi$
$379$	20.6934	1.06295	0.531475	+	0.847074i	$0.321638\pi$
$380$	0	0
$381$	−28.0939	−1.43929
$382$	0	0
$383$	11.4567	0.585412	0.292706	−	0.956203i	$-0.405444\pi$
$383$	11.4567	0.585412	0.292706	+	0.956203i	$0.405444\pi$
$384$	0	0
$385$	4.07639	0.207752
$386$	0	0
$387$	24.7882	1.26005
$388$	0	0
$389$	−26.1105	−1.32386	−0.661928	−	0.749567i	$-0.730262\pi$
$389$	−26.1105	−1.32386	−0.661928	+	0.749567i	$0.730262\pi$
$390$	0	0
$391$	−1.17666	−0.0595061
$392$	0	0
$393$	−8.41725	−0.424594
$394$	0	0
$395$	−42.7754	−2.15226
$396$	0	0
$397$	38.7763	1.94613	0.973064	−	0.230536i	$-0.0740481\pi$
$397$	38.7763	1.94613	0.973064	+	0.230536i	$0.0740481\pi$
$398$	0	0
$399$	−20.7771	−1.04015
$400$	0	0
$401$	−30.0967	−1.50296	−0.751478	−	0.659758i	$-0.770659\pi$
$401$	−30.0967	−1.50296	−0.751478	+	0.659758i	$0.770659\pi$
$402$	0	0
$403$	−21.3023	−1.06115
$404$	0	0
$405$	30.5408	1.51758
$406$	0	0
$407$	7.42587	0.368087
$408$	0	0
$409$	22.9270	1.13367	0.566835	−	0.823832i	$-0.308168\pi$
$409$	22.9270	1.13367	0.566835	+	0.823832i	$0.308168\pi$
$410$	0	0
$411$	5.48963	0.270784
$412$	0	0
$413$	5.28256	0.259938
$414$	0	0
$415$	4.02951	0.197801
$416$	0	0
$417$	51.3294	2.51361
$418$	0	0
$419$	31.9316	1.55996	0.779981	−	0.625803i	$-0.215229\pi$
$419$	31.9316	1.55996	0.779981	+	0.625803i	$0.215229\pi$
$420$	0	0
$421$	16.1811	0.788617	0.394309	−	0.918978i	$-0.370984\pi$
$421$	16.1811	0.788617	0.394309	+	0.918978i	$0.370984\pi$
$422$	0	0
$423$	10.5000	0.510529
$424$	0	0
$425$	−5.99053	−0.290583
$426$	0	0
$427$	−5.74312	−0.277929
$428$	0	0
$429$	−11.0456	−0.533287
$430$	0	0
$431$	−17.6158	−0.848525	−0.424262	−	0.905539i	$-0.639467\pi$
$431$	−17.6158	−0.848525	−0.424262	+	0.905539i	$0.639467\pi$
$432$	0	0
$433$	25.8773	1.24358	0.621792	−	0.783182i	$-0.286405\pi$
$433$	25.8773	1.24358	0.621792	+	0.783182i	$0.286405\pi$
$434$	0	0
$435$	−1.68760	−0.0809144
$436$	0	0
$437$	8.64286	0.413444
$438$	0	0
$439$	−5.31778	−0.253804	−0.126902	−	0.991915i	$-0.540503\pi$
$439$	−5.31778	−0.253804	−0.126902	+	0.991915i	$0.540503\pi$
$440$	0	0
$441$	2.77901	0.132334
$442$	0	0
$443$	−16.3536	−0.776982	−0.388491	−	0.921453i	$-0.627003\pi$
$443$	−16.3536	−0.776982	−0.388491	+	0.921453i	$0.627003\pi$
$444$	0	0
$445$	29.7302	1.40935
$446$	0	0
$447$	10.9066	0.515866
$448$	0	0
$449$	−7.71960	−0.364310	−0.182155	−	0.983270i	$-0.558307\pi$
$449$	−7.71960	−0.364310	−0.182155	+	0.983270i	$0.558307\pi$
$450$	0	0
$451$	15.2160	0.716493
$452$	0	0
$453$	36.9161	1.73447
$454$	0	0
$455$	−11.3744	−0.533239
$456$	0	0
$457$	4.64882	0.217463	0.108731	−	0.994071i	$-0.465321\pi$
$457$	4.64882	0.217463	0.108731	+	0.994071i	$0.465321\pi$
$458$	0	0
$459$	0.625100	0.0291772
$460$	0	0
$461$	−8.99713	−0.419038	−0.209519	−	0.977805i	$-0.567190\pi$
$461$	−8.99713	−0.419038	−0.209519	+	0.977805i	$0.567190\pi$
$462$	0	0
$463$	29.8339	1.38650	0.693250	−	0.720697i	$-0.256178\pi$
$463$	29.8339	1.38650	0.693250	+	0.720697i	$0.256178\pi$
$464$	0	0
$465$	−45.4325	−2.10688
$466$	0	0
$467$	−16.9730	−0.785415	−0.392708	−	0.919663i	$-0.628462\pi$
$467$	−16.9730	−0.785415	−0.392708	+	0.919663i	$0.628462\pi$
$468$	0	0
$469$	9.46553	0.437077
$470$	0	0
$471$	−9.39510	−0.432903
$472$	0	0
$473$	11.4462	0.526295
$474$	0	0
$475$	44.0021	2.01895
$476$	0	0
$477$	16.6741	0.763453
$478$	0	0
$479$	−21.4799	−0.981443	−0.490722	−	0.871316i	$-0.663267\pi$
$479$	−21.4799	−0.981443	−0.490722	+	0.871316i	$0.663267\pi$
$480$	0	0
$481$	−20.7205	−0.944772
$482$	0	0
$483$	−2.40396	−0.109384
$484$	0	0
$485$	−25.8886	−1.17554
$486$	0	0
$487$	−5.05472	−0.229051	−0.114526	−	0.993420i	$-0.536535\pi$
$487$	−5.05472	−0.229051	−0.114526	+	0.993420i	$0.536535\pi$
$488$	0	0
$489$	−20.6182	−0.932388
$490$	0	0
$491$	−8.82079	−0.398077	−0.199038	−	0.979992i	$-0.563782\pi$
$491$	−8.82079	−0.398077	−0.199038	+	0.979992i	$0.563782\pi$
$492$	0	0
$493$	−0.260030	−0.0117111
$494$	0	0
$495$	−11.3283	−0.509170
$496$	0	0
$497$	14.8491	0.666073
$498$	0	0
$499$	8.56621	0.383476	0.191738	−	0.981446i	$-0.438588\pi$
$499$	8.56621	0.383476	0.191738	+	0.981446i	$0.438588\pi$
$500$	0	0
$501$	16.4445	0.734684
$502$	0	0
$503$	−39.2371	−1.74950	−0.874749	−	0.484577i	$-0.838974\pi$
$503$	−39.2371	−1.74950	−0.874749	+	0.484577i	$0.838974\pi$
$504$	0	0
$505$	−52.6807	−2.34426
$506$	0	0
$507$	−0.430803	−0.0191326
$508$	0	0
$509$	34.5444	1.53115	0.765576	−	0.643345i	$-0.222454\pi$
$509$	34.5444	1.53115	0.765576	+	0.643345i	$0.222454\pi$
$510$	0	0
$511$	−2.49578	−0.110407
$512$	0	0
$513$	−4.59153	−0.202721
$514$	0	0
$515$	34.1451	1.50461
$516$	0	0
$517$	4.84849	0.213236
$518$	0	0
$519$	−20.4778	−0.898876
$520$	0	0
$521$	−7.25936	−0.318038	−0.159019	−	0.987275i	$-0.550833\pi$
$521$	−7.25936	−0.318038	−0.159019	+	0.987275i	$0.550833\pi$
$522$	0	0
$523$	−12.1756	−0.532403	−0.266202	−	0.963917i	$-0.585769\pi$
$523$	−12.1756	−0.532403	−0.266202	+	0.963917i	$0.585769\pi$
$524$	0	0
$525$	−12.2389	−0.534149
$526$	0	0
$527$	−7.00035	−0.304940
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	−14.6803	−0.637070
$532$	0	0
$533$	−42.4573	−1.83903
$534$	0	0
$535$	−23.4259	−1.01279
$536$	0	0
$537$	−55.9166	−2.41298
$538$	0	0
$539$	1.28323	0.0552728
$540$	0	0
$541$	3.86136	0.166013	0.0830064	−	0.996549i	$-0.473548\pi$
$541$	3.86136	0.166013	0.0830064	+	0.996549i	$0.473548\pi$
$542$	0	0
$543$	55.9248	2.39996
$544$	0	0
$545$	33.2844	1.42575
$546$	0	0
$547$	−14.4008	−0.615734	−0.307867	−	0.951429i	$-0.599615\pi$
$547$	−14.4008	−0.615734	−0.307867	+	0.951429i	$0.599615\pi$
$548$	0	0
$549$	15.9602	0.681164
$550$	0	0
$551$	1.90999	0.0813682
$552$	0	0
$553$	−13.4655	−0.572612
$554$	0	0
$555$	−44.1915	−1.87583
$556$	0	0
$557$	−31.4837	−1.33401	−0.667003	−	0.745055i	$-0.732423\pi$
$557$	−31.4837	−1.33401	−0.667003	+	0.745055i	$0.732423\pi$
$558$	0	0
$559$	−31.9383	−1.35085
$560$	0	0
$561$	−3.62979	−0.153250
$562$	0	0
$563$	−24.8956	−1.04922	−0.524612	−	0.851342i	$-0.675790\pi$
$563$	−24.8956	−1.04922	−0.524612	+	0.851342i	$0.675790\pi$
$564$	0	0
$565$	0.395447	0.0166366
$566$	0	0
$567$	9.61413	0.403756
$568$	0	0
$569$	−18.2697	−0.765908	−0.382954	−	0.923768i	$-0.625093\pi$
$569$	−18.2697	−0.765908	−0.382954	+	0.923768i	$0.625093\pi$
$570$	0	0
$571$	−15.2422	−0.637868	−0.318934	−	0.947777i	$-0.603325\pi$
$571$	−15.2422	−0.637868	−0.318934	+	0.947777i	$0.603325\pi$
$572$	0	0
$573$	26.3187	1.09948
$574$	0	0
$575$	5.09115	0.212316
$576$	0	0
$577$	−29.9355	−1.24623	−0.623115	−	0.782130i	$-0.714133\pi$
$577$	−29.9355	−1.24623	−0.623115	+	0.782130i	$0.714133\pi$
$578$	0	0
$579$	−8.89257	−0.369563
$580$	0	0
$581$	1.26848	0.0526253
$582$	0	0
$583$	7.69940	0.318876
$584$	0	0
$585$	31.6095	1.30689
$586$	0	0
$587$	8.60945	0.355350	0.177675	−	0.984089i	$-0.443142\pi$
$587$	8.60945	0.355350	0.177675	+	0.984089i	$0.443142\pi$
$588$	0	0
$589$	51.4194	2.11870
$590$	0	0
$591$	34.7730	1.43037
$592$	0	0
$593$	32.8468	1.34886	0.674428	−	0.738341i	$-0.264390\pi$
$593$	32.8468	1.34886	0.674428	+	0.738341i	$0.264390\pi$
$594$	0	0
$595$	−3.73783	−0.153236
$596$	0	0
$597$	53.4929	2.18932
$598$	0	0
$599$	−8.91978	−0.364452	−0.182226	−	0.983257i	$-0.558330\pi$
$599$	−8.91978	−0.364452	−0.182226	+	0.983257i	$0.558330\pi$
$600$	0	0
$601$	−38.6987	−1.57855	−0.789277	−	0.614038i	$-0.789544\pi$
$601$	−38.6987	−1.57855	−0.789277	+	0.614038i	$0.789544\pi$
$602$	0	0
$603$	−26.3048	−1.07121
$604$	0	0
$605$	29.7123	1.20798
$606$	0	0
$607$	−34.5713	−1.40321	−0.701603	−	0.712568i	$-0.747532\pi$
$607$	−34.5713	−1.40321	−0.701603	+	0.712568i	$0.747532\pi$
$608$	0	0
$609$	−0.531251	−0.0215274
$610$	0	0
$611$	−13.5288	−0.547315
$612$	0	0
$613$	2.24075	0.0905030	0.0452515	−	0.998976i	$-0.485591\pi$
$613$	2.24075	0.0905030	0.0452515	+	0.998976i	$0.485591\pi$
$614$	0	0
$615$	−90.5506	−3.65135
$616$	0	0
$617$	−26.8581	−1.08126	−0.540632	−	0.841259i	$-0.681815\pi$
$617$	−26.8581	−1.08126	−0.540632	+	0.841259i	$0.681815\pi$
$618$	0	0
$619$	19.2854	0.775144	0.387572	−	0.921839i	$-0.373314\pi$
$619$	19.2854	0.775144	0.387572	+	0.921839i	$0.373314\pi$
$620$	0	0
$621$	−0.531251	−0.0213184
$622$	0	0
$623$	9.35895	0.374958
$624$	0	0
$625$	−24.5360	−0.981438
$626$	0	0
$627$	26.6618	1.06477
$628$	0	0
$629$	−6.80913	−0.271498
$630$	0	0
$631$	32.3778	1.28894	0.644470	−	0.764630i	$-0.277078\pi$
$631$	32.3778	1.28894	0.644470	+	0.764630i	$0.277078\pi$
$632$	0	0
$633$	−0.879689	−0.0349645
$634$	0	0
$635$	37.1241	1.47322
$636$	0	0
$637$	−3.58061	−0.141869
$638$	0	0
$639$	−41.2658	−1.63245
$640$	0	0
$641$	−18.2338	−0.720191	−0.360096	−	0.932915i	$-0.617256\pi$
$641$	−18.2338	−0.720191	−0.360096	+	0.932915i	$0.617256\pi$
$642$	0	0
$643$	48.1530	1.89897	0.949484	−	0.313814i	$-0.101607\pi$
$643$	48.1530	1.89897	0.949484	+	0.313814i	$0.101607\pi$
$644$	0	0
$645$	−68.1163	−2.68208
$646$	0	0
$647$	−43.0105	−1.69092	−0.845459	−	0.534041i	$-0.820673\pi$
$647$	−43.0105	−1.69092	−0.845459	+	0.534041i	$0.820673\pi$
$648$	0	0
$649$	−6.77876	−0.266089
$650$	0	0
$651$	−14.3020	−0.560539
$652$	0	0
$653$	33.8499	1.32465	0.662324	−	0.749218i	$-0.269570\pi$
$653$	33.8499	1.32465	0.662324	+	0.749218i	$0.269570\pi$
$654$	0	0
$655$	11.1228	0.434603
$656$	0	0
$657$	6.93579	0.270591
$658$	0	0
$659$	−21.2857	−0.829174	−0.414587	−	0.910010i	$-0.636074\pi$
$659$	−21.2857	−0.829174	−0.414587	+	0.910010i	$0.636074\pi$
$660$	0	0
$661$	16.0951	0.626026	0.313013	−	0.949749i	$-0.398662\pi$
$661$	16.0951	0.626026	0.313013	+	0.949749i	$0.398662\pi$
$662$	0	0
$663$	10.1282	0.393348
$664$	0	0
$665$	27.4554	1.06467
$666$	0	0
$667$	0.220990	0.00855678
$668$	0	0
$669$	−28.0168	−1.08319
$670$	0	0
$671$	7.36976	0.284507
$672$	0	0
$673$	−5.46527	−0.210671	−0.105335	−	0.994437i	$-0.533592\pi$
$673$	−5.46527	−0.210671	−0.105335	+	0.994437i	$0.533592\pi$
$674$	0	0
$675$	−2.70468	−0.104103
$676$	0	0
$677$	17.0710	0.656091	0.328045	−	0.944662i	$-0.393610\pi$
$677$	17.0710	0.656091	0.328045	+	0.944662i	$0.393610\pi$
$678$	0	0
$679$	−8.14963	−0.312754
$680$	0	0
$681$	8.33935	0.319565
$682$	0	0
$683$	26.1812	1.00180	0.500898	−	0.865506i	$-0.333003\pi$
$683$	26.1812	1.00180	0.500898	+	0.865506i	$0.333003\pi$
$684$	0	0
$685$	−7.25416	−0.277167
$686$	0	0
$687$	−17.2380	−0.657672
$688$	0	0
$689$	−21.4837	−0.818463
$690$	0	0
$691$	5.29496	0.201430	0.100715	−	0.994915i	$-0.467887\pi$
$691$	5.29496	0.201430	0.100715	+	0.994915i	$0.467887\pi$
$692$	0	0
$693$	−3.56612	−0.135466
$694$	0	0
$695$	−67.8281	−2.57287
$696$	0	0
$697$	−13.9522	−0.528479
$698$	0	0
$699$	−3.97741	−0.150439
$700$	0	0
$701$	35.9206	1.35670	0.678351	−	0.734738i	$-0.262695\pi$
$701$	35.9206	1.35670	0.678351	+	0.734738i	$0.262695\pi$
$702$	0	0
$703$	50.0149	1.88635
$704$	0	0
$705$	−28.8534	−1.08668
$706$	0	0
$707$	−16.5837	−0.623694
$708$	0	0
$709$	30.4778	1.14462	0.572309	−	0.820038i	$-0.306048\pi$
$709$	30.4778	1.14462	0.572309	+	0.820038i	$0.306048\pi$
$710$	0	0
$711$	37.4208	1.40339
$712$	0	0
$713$	5.94936	0.222805
$714$	0	0
$715$	14.5960	0.545859
$716$	0	0
$717$	−36.5156	−1.36370
$718$	0	0
$719$	−21.9208	−0.817508	−0.408754	−	0.912645i	$-0.634037\pi$
$719$	−21.9208	−0.817508	−0.408754	+	0.912645i	$0.634037\pi$
$720$	0	0
$721$	10.7488	0.400305
$722$	0	0
$723$	33.9856	1.26394
$724$	0	0
$725$	1.12509	0.0417849
$726$	0	0
$727$	−44.4217	−1.64751	−0.823755	−	0.566946i	$-0.808125\pi$
$727$	−44.4217	−1.64751	−0.823755	+	0.566946i	$0.808125\pi$
$728$	0	0
$729$	−22.8865	−0.847647
$730$	0	0
$731$	−10.4955	−0.388191
$732$	0	0
$733$	−7.48274	−0.276381	−0.138191	−	0.990406i	$-0.544129\pi$
$733$	−7.48274	−0.276381	−0.138191	+	0.990406i	$0.544129\pi$
$734$	0	0
$735$	−7.63655	−0.281678
$736$	0	0
$737$	−12.1465	−0.447421
$738$	0	0
$739$	−39.8499	−1.46590	−0.732951	−	0.680282i	$-0.761857\pi$
$739$	−39.8499	−1.46590	−0.732951	+	0.680282i	$0.761857\pi$
$740$	0	0
$741$	−74.3946	−2.73295
$742$	0	0
$743$	32.4494	1.19045	0.595227	−	0.803558i	$-0.297062\pi$
$743$	32.4494	1.19045	0.595227	+	0.803558i	$0.297062\pi$
$744$	0	0
$745$	−14.4123	−0.528027
$746$	0	0
$747$	−3.52511	−0.128977
$748$	0	0
$749$	−7.37438	−0.269454
$750$	0	0
$751$	−14.5280	−0.530135	−0.265068	−	0.964230i	$-0.585394\pi$
$751$	−14.5280	−0.530135	−0.265068	+	0.964230i	$0.585394\pi$
$752$	0	0
$753$	34.9015	1.27188
$754$	0	0
$755$	−48.7820	−1.77536
$756$	0	0
$757$	16.3583	0.594552	0.297276	−	0.954792i	$-0.403922\pi$
$757$	16.3583	0.594552	0.297276	+	0.954792i	$0.403922\pi$
$758$	0	0
$759$	3.08484	0.111972
$760$	0	0
$761$	−19.0610	−0.690962	−0.345481	−	0.938426i	$-0.612284\pi$
$761$	−19.0610	−0.690962	−0.345481	+	0.938426i	$0.612284\pi$
$762$	0	0
$763$	10.4778	0.379322
$764$	0	0
$765$	10.3875	0.375560
$766$	0	0
$767$	18.9148	0.682974
$768$	0	0
$769$	45.2577	1.63203	0.816017	−	0.578029i	$-0.196178\pi$
$769$	45.2577	1.63203	0.816017	+	0.578029i	$0.196178\pi$
$770$	0	0
$771$	−34.7730	−1.25232
$772$	0	0
$773$	−22.5210	−0.810026	−0.405013	−	0.914311i	$-0.632733\pi$
$773$	−22.5210	−0.810026	−0.405013	+	0.914311i	$0.632733\pi$
$774$	0	0
$775$	30.2890	1.08801
$776$	0	0
$777$	−13.9113	−0.499066
$778$	0	0
$779$	102.483	3.67183
$780$	0	0
$781$	−19.0548	−0.681836
$782$	0	0
$783$	−0.117401	−0.00419558
$784$	0	0
$785$	12.4149	0.443108
$786$	0	0
$787$	36.5416	1.30257	0.651283	−	0.758835i	$-0.274231\pi$
$787$	36.5416	1.30257	0.651283	+	0.758835i	$0.274231\pi$
$788$	0	0
$789$	−71.7390	−2.55398
$790$	0	0
$791$	0.124485	0.00442619
$792$	0	0
$793$	−20.5639	−0.730246
$794$	0	0
$795$	−45.8193	−1.62504
$796$	0	0
$797$	32.1954	1.14042	0.570209	−	0.821499i	$-0.306862\pi$
$797$	32.1954	1.14042	0.570209	+	0.821499i	$0.306862\pi$
$798$	0	0
$799$	−4.44581	−0.157281
$800$	0	0
$801$	−26.0086	−0.918969
$802$	0	0
$803$	3.20266	0.113020
$804$	0	0
$805$	3.17666	0.111962
$806$	0	0
$807$	−33.8262	−1.19074
$808$	0	0
$809$	8.11626	0.285353	0.142676	−	0.989769i	$-0.454429\pi$
$809$	8.11626	0.285353	0.142676	+	0.989769i	$0.454429\pi$
$810$	0	0
$811$	1.40779	0.0494341	0.0247171	−	0.999694i	$-0.492132\pi$
$811$	1.40779	0.0494341	0.0247171	+	0.999694i	$0.492132\pi$
$812$	0	0
$813$	−55.0914	−1.93214
$814$	0	0
$815$	27.2455	0.954368
$816$	0	0
$817$	77.0924	2.69712
$818$	0	0
$819$	9.95056	0.347701
$820$	0	0
$821$	13.9823	0.487987	0.243993	−	0.969777i	$-0.421542\pi$
$821$	13.9823	0.487987	0.243993	+	0.969777i	$0.421542\pi$
$822$	0	0
$823$	−23.4324	−0.816804	−0.408402	−	0.912802i	$-0.633914\pi$
$823$	−23.4324	−0.816804	−0.408402	+	0.912802i	$0.633914\pi$
$824$	0	0
$825$	15.7054	0.546790
$826$	0	0
$827$	−38.2972	−1.33172	−0.665862	−	0.746075i	$-0.731936\pi$
$827$	−38.2972	−1.33172	−0.665862	+	0.746075i	$0.731936\pi$
$828$	0	0
$829$	−30.4560	−1.05778	−0.528890	−	0.848691i	$-0.677391\pi$
$829$	−30.4560	−1.05778	−0.528890	+	0.848691i	$0.677391\pi$
$830$	0	0
$831$	−7.82411	−0.271415
$832$	0	0
$833$	−1.17666	−0.0407687
$834$	0	0
$835$	−21.7302	−0.752004
$836$	0	0
$837$	−3.16060	−0.109246
$838$	0	0
$839$	−50.9466	−1.75887	−0.879436	−	0.476017i	$-0.842080\pi$
$839$	−50.9466	−1.75887	−0.879436	+	0.476017i	$0.842080\pi$
$840$	0	0
$841$	−28.9512	−0.998316
$842$	0	0
$843$	12.0266	0.414218
$844$	0	0
$845$	0.569275	0.0195836
$846$	0	0
$847$	9.35331	0.321384
$848$	0	0
$849$	16.7324	0.574254
$850$	0	0
$851$	5.78685	0.198371
$852$	0	0
$853$	42.9728	1.47136	0.735681	−	0.677328i	$-0.236862\pi$
$853$	42.9728	1.47136	0.735681	+	0.677328i	$0.236862\pi$
$854$	0	0
$855$	−76.2988	−2.60936
$856$	0	0
$857$	−26.4649	−0.904024	−0.452012	−	0.892012i	$-0.649294\pi$
$857$	−26.4649	−0.904024	−0.452012	+	0.892012i	$0.649294\pi$
$858$	0	0
$859$	−36.4760	−1.24454	−0.622272	−	0.782801i	$-0.713790\pi$
$859$	−36.4760	−1.24454	−0.622272	+	0.782801i	$0.713790\pi$
$860$	0	0
$861$	−28.5050	−0.971448
$862$	0	0
$863$	−28.0272	−0.954057	−0.477029	−	0.878888i	$-0.658286\pi$
$863$	−28.0272	−0.954057	−0.477029	+	0.878888i	$0.658286\pi$
$864$	0	0
$865$	27.0599	0.920066
$866$	0	0
$867$	−37.5389	−1.27489
$868$	0	0
$869$	17.2794	0.586164
$870$	0	0
$871$	33.8924	1.14840
$872$	0	0
$873$	22.6479	0.766515
$874$	0	0
$875$	0.289543	0.00978834
$876$	0	0
$877$	−40.6085	−1.37125	−0.685626	−	0.727954i	$-0.740471\pi$
$877$	−40.6085	−1.37125	−0.685626	+	0.727954i	$0.740471\pi$
$878$	0	0
$879$	4.43388	0.149551
$880$	0	0
$881$	5.45202	0.183683	0.0918416	−	0.995774i	$-0.470725\pi$
$881$	5.45202	0.183683	0.0918416	+	0.995774i	$0.470725\pi$
$882$	0	0
$883$	15.5750	0.524142	0.262071	−	0.965049i	$-0.415595\pi$
$883$	15.5750	0.524142	0.262071	+	0.965049i	$0.415595\pi$
$884$	0	0
$885$	40.3405	1.35603
$886$	0	0
$887$	0.109316	0.00367047	0.00183523	−	0.999998i	$-0.499416\pi$
$887$	0.109316	0.00367047	0.00183523	+	0.999998i	$0.499416\pi$
$888$	0	0
$889$	11.6865	0.391953
$890$	0	0
$891$	−12.3372	−0.413311
$892$	0	0
$893$	32.6556	1.09278
$894$	0	0
$895$	73.8897	2.46986
$896$	0	0
$897$	−8.60764	−0.287401
$898$	0	0
$899$	1.31475	0.0438494
$900$	0	0
$901$	−7.05994	−0.235201
$902$	0	0
$903$	−21.4428	−0.713571
$904$	0	0
$905$	−73.9005	−2.45654
$906$	0	0
$907$	−52.6342	−1.74769	−0.873845	−	0.486205i	$-0.838381\pi$
$907$	−52.6342	−1.74769	−0.873845	+	0.486205i	$0.838381\pi$
$908$	0	0
$909$	46.0863	1.52859
$910$	0	0
$911$	−47.7704	−1.58270	−0.791352	−	0.611361i	$-0.790622\pi$
$911$	−47.7704	−1.58270	−0.791352	+	0.611361i	$0.790622\pi$
$912$	0	0
$913$	−1.62775	−0.0538707
$914$	0	0
$915$	−43.8576	−1.44989
$916$	0	0
$917$	3.50141	0.115627
$918$	0	0
$919$	−25.7597	−0.849734	−0.424867	−	0.905256i	$-0.639679\pi$
$919$	−25.7597	−0.849734	−0.424867	+	0.905256i	$0.639679\pi$
$920$	0	0
$921$	−47.8257	−1.57591
$922$	0	0
$923$	53.1689	1.75007
$924$	0	0
$925$	29.4617	0.968694
$926$	0	0
$927$	−29.8709	−0.981089
$928$	0	0
$929$	54.7345	1.79578	0.897891	−	0.440218i	$-0.145099\pi$
$929$	54.7345	1.79578	0.897891	+	0.440218i	$0.145099\pi$
$930$	0	0
$931$	8.64286	0.283258
$932$	0	0
$933$	39.1643	1.28218
$934$	0	0
$935$	4.79651	0.156863
$936$	0	0
$937$	−46.5112	−1.51945	−0.759727	−	0.650242i	$-0.774667\pi$
$937$	−46.5112	−1.51945	−0.759727	+	0.650242i	$0.774667\pi$
$938$	0	0
$939$	−10.3904	−0.339078
$940$	0	0
$941$	27.7308	0.904000	0.452000	−	0.892018i	$-0.350711\pi$
$941$	27.7308	0.904000	0.452000	+	0.892018i	$0.350711\pi$
$942$	0	0
$943$	11.8575	0.386135
$944$	0	0
$945$	−1.68760	−0.0548977
$946$	0	0
$947$	−4.89684	−0.159126	−0.0795630	−	0.996830i	$-0.525352\pi$
$947$	−4.89684	−0.159126	−0.0795630	+	0.996830i	$0.525352\pi$
$948$	0	0
$949$	−8.93641	−0.290088
$950$	0	0
$951$	37.2903	1.20922
$952$	0	0
$953$	17.4259	0.564479	0.282240	−	0.959344i	$-0.408923\pi$
$953$	17.4259	0.564479	0.282240	+	0.959344i	$0.408923\pi$
$954$	0	0
$955$	−34.7783	−1.12540
$956$	0	0
$957$	0.681719	0.0220368
$958$	0	0
$959$	−2.28358	−0.0737407
$960$	0	0
$961$	4.39484	0.141769
$962$	0	0
$963$	20.4935	0.660393
$964$	0	0
$965$	11.7509	0.378275
$966$	0	0
$967$	42.3906	1.36319	0.681595	−	0.731730i	$-0.261287\pi$
$967$	42.3906	1.36319	0.681595	+	0.731730i	$0.261287\pi$
$968$	0	0
$969$	−24.4475	−0.785366
$970$	0	0
$971$	51.7001	1.65914	0.829568	−	0.558406i	$-0.188587\pi$
$971$	51.7001	1.65914	0.829568	+	0.558406i	$0.188587\pi$
$972$	0	0
$973$	−21.3520	−0.684515
$974$	0	0
$975$	−43.8228	−1.40345
$976$	0	0
$977$	−47.7276	−1.52694	−0.763470	−	0.645843i	$-0.776506\pi$
$977$	−47.7276	−1.52694	−0.763470	+	0.645843i	$0.776506\pi$
$978$	0	0
$979$	−12.0097	−0.383832
$980$	0	0
$981$	−29.1179	−0.929663
$982$	0	0
$983$	59.1353	1.88612	0.943062	−	0.332617i	$-0.107932\pi$
$983$	59.1353	1.88612	0.943062	+	0.332617i	$0.107932\pi$
$984$	0	0
$985$	−45.9501	−1.46409
$986$	0	0
$987$	−9.08296	−0.289114
$988$	0	0
$989$	8.91978	0.283632
$990$	0	0
$991$	15.7882	0.501528	0.250764	−	0.968048i	$-0.419318\pi$
$991$	15.7882	0.501528	0.250764	+	0.968048i	$0.419318\pi$
$992$	0	0
$993$	77.6028	2.46265
$994$	0	0
$995$	−70.6870	−2.24093
$996$	0	0
$997$	2.67881	0.0848389	0.0424194	−	0.999100i	$-0.486493\pi$
$997$	2.67881	0.0848389	0.0424194	+	0.999100i	$0.486493\pi$
$998$	0	0
$999$	−3.07427	−0.0972656

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2576.2.a.be.1.4		5
4.3	odd	2		1288.2.a.p.1.2	✓	5
28.27	even	2		9016.2.a.bg.1.4		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1288.2.a.p.1.2	✓	5	4.3	odd	2
2576.2.a.be.1.4		5	1.1	even	1	trivial
9016.2.a.bg.1.4		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)

\(f(q)\)	\(=\)	\(q+2.40396 q^{3} -3.17666 q^{5} -1.00000 q^{7} +2.77901 q^{9} +O(q^{10})\)	\(q+2.40396 q^{3} -3.17666 q^{5} -1.00000 q^{7} +2.77901 q^{9} +1.28323 q^{11} -3.58061 q^{13} -7.63655 q^{15} -1.17666 q^{17} +8.64286 q^{19} -2.40396 q^{21} +1.00000 q^{23} +5.09115 q^{25} -0.531251 q^{27} +0.220990 q^{29} +5.94936 q^{31} +3.08484 q^{33} +3.17666 q^{35} +5.78685 q^{37} -8.60764 q^{39} +11.8575 q^{41} +8.91978 q^{43} -8.82796 q^{45} +3.77834 q^{47} +1.00000 q^{49} -2.82863 q^{51} +6.00000 q^{53} -4.07639 q^{55} +20.7771 q^{57} -5.28256 q^{59} +5.74312 q^{61} -2.77901 q^{63} +11.3744 q^{65} -9.46553 q^{67} +2.40396 q^{69} -14.8491 q^{71} +2.49578 q^{73} +12.2389 q^{75} -1.28323 q^{77} +13.4655 q^{79} -9.61413 q^{81} -1.26848 q^{83} +3.73783 q^{85} +0.531251 q^{87} -9.35895 q^{89} +3.58061 q^{91} +14.3020 q^{93} -27.4554 q^{95} +8.14963 q^{97} +3.56612 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + 3 q^{3} + 2 q^{5} - 5 q^{7} + 14 q^{9}+O(q^{10}) \) 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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