LMFDB - Embedded newform 2576.2.a.t.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:	$2$
Coefficient field:	$\Q(\zeta_{10})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 1 $ x^2 - x - 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 322)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-0.618034$ 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)

$f(q)$	$=$	$q-1.23607 q^{3} -3.23607 q^{5} +1.00000 q^{7} -1.47214 q^{9} +O(q^{10})$	$q-1.23607 q^{3} -3.23607 q^{5} +1.00000 q^{7} -1.47214 q^{9} -6.47214 q^{13} +4.00000 q^{15} -2.76393 q^{17} -4.47214 q^{19} -1.23607 q^{21} +1.00000 q^{23} +5.47214 q^{25} +5.52786 q^{27} -2.00000 q^{29} -3.23607 q^{31} -3.23607 q^{35} +6.00000 q^{37} +8.00000 q^{39} -10.0000 q^{41} -2.47214 q^{43} +4.76393 q^{45} +11.2361 q^{47} +1.00000 q^{49} +3.41641 q^{51} -6.00000 q^{53} +5.52786 q^{57} -6.76393 q^{59} -1.70820 q^{61} -1.47214 q^{63} +20.9443 q^{65} -4.00000 q^{67} -1.23607 q^{69} -6.47214 q^{71} +13.4164 q^{73} -6.76393 q^{75} +8.94427 q^{79} -2.41641 q^{81} -10.9443 q^{83} +8.94427 q^{85} +2.47214 q^{87} +6.18034 q^{89} -6.47214 q^{91} +4.00000 q^{93} +14.4721 q^{95} -14.1803 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{3} - 2 q^{5} + 2 q^{7} + 6 q^{9}+O(q^{10}) $ 2 * q + 2 * q^3 - 2 * q^5 + 2 * q^7 + 6 * q^9	$ 2 q + 2 q^{3} - 2 q^{5} + 2 q^{7} + 6 q^{9} - 4 q^{13} + 8 q^{15} - 10 q^{17} + 2 q^{21} + 2 q^{23} + 2 q^{25} + 20 q^{27} - 4 q^{29} - 2 q^{31} - 2 q^{35} + 12 q^{37} + 16 q^{39} - 20 q^{41} + 4 q^{43} + 14 q^{45} + 18 q^{47} + 2 q^{49} - 20 q^{51} - 12 q^{53} + 20 q^{57} - 18 q^{59} + 10 q^{61} + 6 q^{63} + 24 q^{65} - 8 q^{67} + 2 q^{69} - 4 q^{71} - 18 q^{75} + 22 q^{81} - 4 q^{83} - 4 q^{87} - 10 q^{89} - 4 q^{91} + 8 q^{93} + 20 q^{95} - 6 q^{97}+O(q^{100}) $ 2 * q + 2 * q^3 - 2 * q^5 + 2 * q^7 + 6 * q^9 - 4 * q^13 + 8 * q^15 - 10 * q^17 + 2 * q^21 + 2 * q^23 + 2 * q^25 + 20 * q^27 - 4 * q^29 - 2 * q^31 - 2 * q^35 + 12 * q^37 + 16 * q^39 - 20 * q^41 + 4 * q^43 + 14 * q^45 + 18 * q^47 + 2 * q^49 - 20 * q^51 - 12 * q^53 + 20 * q^57 - 18 * q^59 + 10 * q^61 + 6 * q^63 + 24 * q^65 - 8 * q^67 + 2 * q^69 - 4 * q^71 - 18 * q^75 + 22 * q^81 - 4 * q^83 - 4 * q^87 - 10 * q^89 - 4 * q^91 + 8 * q^93 + 20 * q^95 - 6 * q^97

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	−1.23607	−0.713644	−0.356822	−	0.934172i	$-0.616140\pi$
$3$	−1.23607	−0.713644	−0.356822	+	0.934172i	$0.616140\pi$
$4$	0	0
$5$	−3.23607	−1.44721	−0.723607	−	0.690212i	$-0.757517\pi$
$5$	−3.23607	−1.44721	−0.723607	+	0.690212i	$0.757517\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	−1.47214	−0.490712
$10$	0	0
$11$	0	0		−	1.00000i	$-0.5\pi$
$11$	0	0			1.00000i	$0.5\pi$
$12$	0	0
$13$	−6.47214	−1.79505	−0.897524	−	0.440966i	$-0.854636\pi$
$13$	−6.47214	−1.79505	−0.897524	+	0.440966i	$0.854636\pi$
$14$	0	0
$15$	4.00000	1.03280
$16$	0	0
$17$	−2.76393	−0.670352	−0.335176	−	0.942156i	$-0.608796\pi$
$17$	−2.76393	−0.670352	−0.335176	+	0.942156i	$0.608796\pi$
$18$	0	0
$19$	−4.47214	−1.02598	−0.512989	−	0.858395i	$-0.671462\pi$
$19$	−4.47214	−1.02598	−0.512989	+	0.858395i	$0.671462\pi$
$20$	0	0
$21$	−1.23607	−0.269732
$22$	0	0
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	0	0
$25$	5.47214	1.09443
$26$	0	0
$27$	5.52786	1.06384
$28$	0	0
$29$	−2.00000	−0.371391	−0.185695	−	0.982607i	$-0.559454\pi$
$29$	−2.00000	−0.371391	−0.185695	+	0.982607i	$0.559454\pi$
$30$	0	0
$31$	−3.23607	−0.581215	−0.290607	−	0.956842i	$-0.593857\pi$
$31$	−3.23607	−0.581215	−0.290607	+	0.956842i	$0.593857\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−3.23607	−0.546995
$36$	0	0
$37$	6.00000	0.986394	0.493197	−	0.869918i	$-0.335828\pi$
$37$	6.00000	0.986394	0.493197	+	0.869918i	$0.335828\pi$
$38$	0	0
$39$	8.00000	1.28103
$40$	0	0
$41$	−10.0000	−1.56174	−0.780869	−	0.624695i	$-0.785223\pi$
$41$	−10.0000	−1.56174	−0.780869	+	0.624695i	$0.785223\pi$
$42$	0	0
$43$	−2.47214	−0.376997	−0.188499	−	0.982073i	$-0.560362\pi$
$43$	−2.47214	−0.376997	−0.188499	+	0.982073i	$0.560362\pi$
$44$	0	0
$45$	4.76393	0.710165
$46$	0	0
$47$	11.2361	1.63895	0.819474	−	0.573116i	$-0.194265\pi$
$47$	11.2361	1.63895	0.819474	+	0.573116i	$0.194265\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	3.41641	0.478393
$52$	0	0
$53$	−6.00000	−0.824163	−0.412082	−	0.911147i	$-0.635198\pi$
$53$	−6.00000	−0.824163	−0.412082	+	0.911147i	$0.635198\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	5.52786	0.732183
$58$	0	0
$59$	−6.76393	−0.880589	−0.440294	−	0.897853i	$-0.645126\pi$
$59$	−6.76393	−0.880589	−0.440294	+	0.897853i	$0.645126\pi$
$60$	0	0
$61$	−1.70820	−0.218713	−0.109357	−	0.994003i	$-0.534879\pi$
$61$	−1.70820	−0.218713	−0.109357	+	0.994003i	$0.534879\pi$
$62$	0	0
$63$	−1.47214	−0.185472
$64$	0	0
$65$	20.9443	2.59782
$66$	0	0
$67$	−4.00000	−0.488678	−0.244339	−	0.969690i	$-0.578571\pi$
$67$	−4.00000	−0.488678	−0.244339	+	0.969690i	$0.578571\pi$
$68$	0	0
$69$	−1.23607	−0.148805
$70$	0	0
$71$	−6.47214	−0.768101	−0.384051	−	0.923312i	$-0.625471\pi$
$71$	−6.47214	−0.768101	−0.384051	+	0.923312i	$0.625471\pi$
$72$	0	0
$73$	13.4164	1.57027	0.785136	−	0.619324i	$-0.212593\pi$
$73$	13.4164	1.57027	0.785136	+	0.619324i	$0.212593\pi$
$74$	0	0
$75$	−6.76393	−0.781032
$76$	0	0
$77$	0	0
$78$	0	0
$79$	8.94427	1.00631	0.503155	−	0.864196i	$-0.332173\pi$
$79$	8.94427	1.00631	0.503155	+	0.864196i	$0.332173\pi$
$80$	0	0
$81$	−2.41641	−0.268490
$82$	0	0
$83$	−10.9443	−1.20129	−0.600645	−	0.799516i	$-0.705089\pi$
$83$	−10.9443	−1.20129	−0.600645	+	0.799516i	$0.705089\pi$
$84$	0	0
$85$	8.94427	0.970143
$86$	0	0
$87$	2.47214	0.265041
$88$	0	0
$89$	6.18034	0.655115	0.327557	−	0.944831i	$-0.393775\pi$
$89$	6.18034	0.655115	0.327557	+	0.944831i	$0.393775\pi$
$90$	0	0
$91$	−6.47214	−0.678464
$92$	0	0
$93$	4.00000	0.414781
$94$	0	0
$95$	14.4721	1.48481
$96$	0	0
$97$	−14.1803	−1.43980	−0.719898	−	0.694080i	$-0.755811\pi$
$97$	−14.1803	−1.43980	−0.719898	+	0.694080i	$0.755811\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−5.52786	−0.550043	−0.275022	−	0.961438i	$-0.588685\pi$
$101$	−5.52786	−0.550043	−0.275022	+	0.961438i	$0.588685\pi$
$102$	0	0
$103$	4.00000	0.394132	0.197066	−	0.980390i	$-0.436859\pi$
$103$	4.00000	0.394132	0.197066	+	0.980390i	$0.436859\pi$
$104$	0	0
$105$	4.00000	0.390360
$106$	0	0
$107$	1.52786	0.147704	0.0738521	−	0.997269i	$-0.476471\pi$
$107$	1.52786	0.147704	0.0738521	+	0.997269i	$0.476471\pi$
$108$	0	0
$109$	12.4721	1.19461	0.597307	−	0.802013i	$-0.296237\pi$
$109$	12.4721	1.19461	0.597307	+	0.802013i	$0.296237\pi$
$110$	0	0
$111$	−7.41641	−0.703934
$112$	0	0
$113$	12.4721	1.17328	0.586640	−	0.809848i	$-0.300450\pi$
$113$	12.4721	1.17328	0.586640	+	0.809848i	$0.300450\pi$
$114$	0	0
$115$	−3.23607	−0.301765
$116$	0	0
$117$	9.52786	0.880851
$118$	0	0
$119$	−2.76393	−0.253369
$120$	0	0
$121$	−11.0000	−1.00000
$122$	0	0
$123$	12.3607	1.11452
$124$	0	0
$125$	−1.52786	−0.136656
$126$	0	0
$127$	19.4164	1.72293	0.861464	−	0.507819i	$-0.169548\pi$
$127$	19.4164	1.72293	0.861464	+	0.507819i	$0.169548\pi$
$128$	0	0
$129$	3.05573	0.269042
$130$	0	0
$131$	17.2361	1.50592	0.752961	−	0.658065i	$-0.228625\pi$
$131$	17.2361	1.50592	0.752961	+	0.658065i	$0.228625\pi$
$132$	0	0
$133$	−4.47214	−0.387783
$134$	0	0
$135$	−17.8885	−1.53960
$136$	0	0
$137$	−9.41641	−0.804498	−0.402249	−	0.915530i	$-0.631771\pi$
$137$	−9.41641	−0.804498	−0.402249	+	0.915530i	$0.631771\pi$
$138$	0	0
$139$	23.1246	1.96140	0.980702	−	0.195509i	$-0.0626358\pi$
$139$	23.1246	1.96140	0.980702	+	0.195509i	$0.0626358\pi$
$140$	0	0
$141$	−13.8885	−1.16963
$142$	0	0
$143$	0	0
$144$	0	0
$145$	6.47214	0.537482
$146$	0	0
$147$	−1.23607	−0.101949
$148$	0	0
$149$	21.4164	1.75450	0.877250	−	0.480033i	$-0.159375\pi$
$149$	21.4164	1.75450	0.877250	+	0.480033i	$0.159375\pi$
$150$	0	0
$151$	9.52786	0.775367	0.387683	−	0.921793i	$-0.373275\pi$
$151$	9.52786	0.775367	0.387683	+	0.921793i	$0.373275\pi$
$152$	0	0
$153$	4.06888	0.328950
$154$	0	0
$155$	10.4721	0.841142
$156$	0	0
$157$	−18.6525	−1.48863	−0.744315	−	0.667829i	$-0.767224\pi$
$157$	−18.6525	−1.48863	−0.744315	+	0.667829i	$0.767224\pi$
$158$	0	0
$159$	7.41641	0.588159
$160$	0	0
$161$	1.00000	0.0788110
$162$	0	0
$163$	6.47214	0.506937	0.253468	−	0.967344i	$-0.418429\pi$
$163$	6.47214	0.506937	0.253468	+	0.967344i	$0.418429\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	6.65248	0.514784	0.257392	−	0.966307i	$-0.417137\pi$
$167$	6.65248	0.514784	0.257392	+	0.966307i	$0.417137\pi$
$168$	0	0
$169$	28.8885	2.22220
$170$	0	0
$171$	6.58359	0.503460
$172$	0	0
$173$	3.05573	0.232323	0.116161	−	0.993230i	$-0.462941\pi$
$173$	3.05573	0.232323	0.116161	+	0.993230i	$0.462941\pi$
$174$	0	0
$175$	5.47214	0.413655
$176$	0	0
$177$	8.36068	0.628427
$178$	0	0
$179$	−16.9443	−1.26647	−0.633237	−	0.773958i	$-0.718274\pi$
$179$	−16.9443	−1.26647	−0.633237	+	0.773958i	$0.718274\pi$
$180$	0	0
$181$	8.76393	0.651418	0.325709	−	0.945470i	$-0.394397\pi$
$181$	8.76393	0.651418	0.325709	+	0.945470i	$0.394397\pi$
$182$	0	0
$183$	2.11146	0.156083
$184$	0	0
$185$	−19.4164	−1.42752
$186$	0	0
$187$	0	0
$188$	0	0
$189$	5.52786	0.402093
$190$	0	0
$191$	−3.05573	−0.221105	−0.110552	−	0.993870i	$-0.535262\pi$
$191$	−3.05573	−0.221105	−0.110552	+	0.993870i	$0.535262\pi$
$192$	0	0
$193$	−5.05573	−0.363919	−0.181960	−	0.983306i	$-0.558244\pi$
$193$	−5.05573	−0.363919	−0.181960	+	0.983306i	$0.558244\pi$
$194$	0	0
$195$	−25.8885	−1.85392
$196$	0	0
$197$	−19.8885	−1.41700	−0.708500	−	0.705711i	$-0.750628\pi$
$197$	−19.8885	−1.41700	−0.708500	+	0.705711i	$0.750628\pi$
$198$	0	0
$199$	12.0000	0.850657	0.425329	−	0.905039i	$-0.360158\pi$
$199$	12.0000	0.850657	0.425329	+	0.905039i	$0.360158\pi$
$200$	0	0
$201$	4.94427	0.348742
$202$	0	0
$203$	−2.00000	−0.140372
$204$	0	0
$205$	32.3607	2.26017
$206$	0	0
$207$	−1.47214	−0.102321
$208$	0	0
$209$	0	0
$210$	0	0
$211$	−16.9443	−1.16649	−0.583246	−	0.812296i	$-0.698218\pi$
$211$	−16.9443	−1.16649	−0.583246	+	0.812296i	$0.698218\pi$
$212$	0	0
$213$	8.00000	0.548151
$214$	0	0
$215$	8.00000	0.545595
$216$	0	0
$217$	−3.23607	−0.219679
$218$	0	0
$219$	−16.5836	−1.12062
$220$	0	0
$221$	17.8885	1.20331
$222$	0	0
$223$	10.6525	0.713343	0.356671	−	0.934230i	$-0.383912\pi$
$223$	10.6525	0.713343	0.356671	+	0.934230i	$0.383912\pi$
$224$	0	0
$225$	−8.05573	−0.537049
$226$	0	0
$227$	21.4164	1.42146	0.710728	−	0.703466i	$-0.248365\pi$
$227$	21.4164	1.42146	0.710728	+	0.703466i	$0.248365\pi$
$228$	0	0
$229$	−15.2361	−1.00683	−0.503414	−	0.864045i	$-0.667923\pi$
$229$	−15.2361	−1.00683	−0.503414	+	0.864045i	$0.667923\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−16.4721	−1.07913	−0.539563	−	0.841945i	$-0.681410\pi$
$233$	−16.4721	−1.07913	−0.539563	+	0.841945i	$0.681410\pi$
$234$	0	0
$235$	−36.3607	−2.37191
$236$	0	0
$237$	−11.0557	−0.718147
$238$	0	0
$239$	−6.47214	−0.418648	−0.209324	−	0.977846i	$-0.567126\pi$
$239$	−6.47214	−0.418648	−0.209324	+	0.977846i	$0.567126\pi$
$240$	0	0
$241$	0.291796	0.0187962	0.00939812	−	0.999956i	$-0.497008\pi$
$241$	0.291796	0.0187962	0.00939812	+	0.999956i	$0.497008\pi$
$242$	0	0
$243$	−13.5967	−0.872232
$244$	0	0
$245$	−3.23607	−0.206745
$246$	0	0
$247$	28.9443	1.84168
$248$	0	0
$249$	13.5279	0.857294
$250$	0	0
$251$	4.47214	0.282279	0.141139	−	0.989990i	$-0.454923\pi$
$251$	4.47214	0.282279	0.141139	+	0.989990i	$0.454923\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	−11.0557	−0.692337
$256$	0	0
$257$	−26.9443	−1.68074	−0.840369	−	0.542015i	$-0.817662\pi$
$257$	−26.9443	−1.68074	−0.840369	+	0.542015i	$0.817662\pi$
$258$	0	0
$259$	6.00000	0.372822
$260$	0	0
$261$	2.94427	0.182246
$262$	0	0
$263$	−12.0000	−0.739952	−0.369976	−	0.929041i	$-0.620634\pi$
$263$	−12.0000	−0.739952	−0.369976	+	0.929041i	$0.620634\pi$
$264$	0	0
$265$	19.4164	1.19274
$266$	0	0
$267$	−7.63932	−0.467519
$268$	0	0
$269$	3.05573	0.186311	0.0931555	−	0.995652i	$-0.470305\pi$
$269$	3.05573	0.186311	0.0931555	+	0.995652i	$0.470305\pi$
$270$	0	0
$271$	15.2361	0.925525	0.462763	−	0.886482i	$-0.346858\pi$
$271$	15.2361	0.925525	0.462763	+	0.886482i	$0.346858\pi$
$272$	0	0
$273$	8.00000	0.484182
$274$	0	0
$275$	0	0
$276$	0	0
$277$	19.8885	1.19499	0.597493	−	0.801874i	$-0.296163\pi$
$277$	19.8885	1.19499	0.597493	+	0.801874i	$0.296163\pi$
$278$	0	0
$279$	4.76393	0.285209
$280$	0	0
$281$	5.41641	0.323116	0.161558	−	0.986863i	$-0.448348\pi$
$281$	5.41641	0.323116	0.161558	+	0.986863i	$0.448348\pi$
$282$	0	0
$283$	−4.47214	−0.265841	−0.132920	−	0.991127i	$-0.542435\pi$
$283$	−4.47214	−0.265841	−0.132920	+	0.991127i	$0.542435\pi$
$284$	0	0
$285$	−17.8885	−1.05963
$286$	0	0
$287$	−10.0000	−0.590281
$288$	0	0
$289$	−9.36068	−0.550628
$290$	0	0
$291$	17.5279	1.02750
$292$	0	0
$293$	−29.1246	−1.70148	−0.850739	−	0.525588i	$-0.823845\pi$
$293$	−29.1246	−1.70148	−0.850739	+	0.525588i	$0.823845\pi$
$294$	0	0
$295$	21.8885	1.27440
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−6.47214	−0.374293
$300$	0	0
$301$	−2.47214	−0.142492
$302$	0	0
$303$	6.83282	0.392535
$304$	0	0
$305$	5.52786	0.316525
$306$	0	0
$307$	−29.2361	−1.66859	−0.834295	−	0.551318i	$-0.814125\pi$
$307$	−29.2361	−1.66859	−0.834295	+	0.551318i	$0.814125\pi$
$308$	0	0
$309$	−4.94427	−0.281270
$310$	0	0
$311$	−25.1246	−1.42469	−0.712343	−	0.701831i	$-0.752366\pi$
$311$	−25.1246	−1.42469	−0.712343	+	0.701831i	$0.752366\pi$
$312$	0	0
$313$	26.1803	1.47980	0.739900	−	0.672717i	$-0.234873\pi$
$313$	26.1803	1.47980	0.739900	+	0.672717i	$0.234873\pi$
$314$	0	0
$315$	4.76393	0.268417
$316$	0	0
$317$	10.0000	0.561656	0.280828	−	0.959758i	$-0.409391\pi$
$317$	10.0000	0.561656	0.280828	+	0.959758i	$0.409391\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	−1.88854	−0.105408
$322$	0	0
$323$	12.3607	0.687767
$324$	0	0
$325$	−35.4164	−1.96455
$326$	0	0
$327$	−15.4164	−0.852529
$328$	0	0
$329$	11.2361	0.619464
$330$	0	0
$331$	30.4721	1.67490	0.837450	−	0.546514i	$-0.184045\pi$
$331$	30.4721	1.67490	0.837450	+	0.546514i	$0.184045\pi$
$332$	0	0
$333$	−8.83282	−0.484035
$334$	0	0
$335$	12.9443	0.707221
$336$	0	0
$337$	−35.8885	−1.95497	−0.977487	−	0.210997i	$-0.932329\pi$
$337$	−35.8885	−1.95497	−0.977487	+	0.210997i	$0.932329\pi$
$338$	0	0
$339$	−15.4164	−0.837304
$340$	0	0
$341$	0	0
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	4.00000	0.215353
$346$	0	0
$347$	−24.3607	−1.30775	−0.653875	−	0.756603i	$-0.726858\pi$
$347$	−24.3607	−1.30775	−0.653875	+	0.756603i	$0.726858\pi$
$348$	0	0
$349$	−3.05573	−0.163569	−0.0817847	−	0.996650i	$-0.526062\pi$
$349$	−3.05573	−0.163569	−0.0817847	+	0.996650i	$0.526062\pi$
$350$	0	0
$351$	−35.7771	−1.90964
$352$	0	0
$353$	0.472136	0.0251293	0.0125646	−	0.999921i	$-0.496000\pi$
$353$	0.472136	0.0251293	0.0125646	+	0.999921i	$0.496000\pi$
$354$	0	0
$355$	20.9443	1.11161
$356$	0	0
$357$	3.41641	0.180815
$358$	0	0
$359$	−12.0000	−0.633336	−0.316668	−	0.948536i	$-0.602564\pi$
$359$	−12.0000	−0.633336	−0.316668	+	0.948536i	$0.602564\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	13.5967	0.713644
$364$	0	0
$365$	−43.4164	−2.27252
$366$	0	0
$367$	−24.3607	−1.27162	−0.635809	−	0.771847i	$-0.719333\pi$
$367$	−24.3607	−1.27162	−0.635809	+	0.771847i	$0.719333\pi$
$368$	0	0
$369$	14.7214	0.766363
$370$	0	0
$371$	−6.00000	−0.311504
$372$	0	0
$373$	−10.5836	−0.547998	−0.273999	−	0.961730i	$-0.588346\pi$
$373$	−10.5836	−0.547998	−0.273999	+	0.961730i	$0.588346\pi$
$374$	0	0
$375$	1.88854	0.0975240
$376$	0	0
$377$	12.9443	0.666664
$378$	0	0
$379$	17.8885	0.918873	0.459436	−	0.888211i	$-0.348051\pi$
$379$	17.8885	0.918873	0.459436	+	0.888211i	$0.348051\pi$
$380$	0	0
$381$	−24.0000	−1.22956
$382$	0	0
$383$	0.583592	0.0298202	0.0149101	−	0.999889i	$-0.495254\pi$
$383$	0.583592	0.0298202	0.0149101	+	0.999889i	$0.495254\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	3.63932	0.184997
$388$	0	0
$389$	9.41641	0.477431	0.238715	−	0.971090i	$-0.423274\pi$
$389$	9.41641	0.477431	0.238715	+	0.971090i	$0.423274\pi$
$390$	0	0
$391$	−2.76393	−0.139778
$392$	0	0
$393$	−21.3050	−1.07469
$394$	0	0
$395$	−28.9443	−1.45634
$396$	0	0
$397$	14.4721	0.726336	0.363168	−	0.931724i	$-0.381695\pi$
$397$	14.4721	0.726336	0.363168	+	0.931724i	$0.381695\pi$
$398$	0	0
$399$	5.52786	0.276739
$400$	0	0
$401$	−37.7771	−1.88650	−0.943249	−	0.332087i	$-0.892247\pi$
$401$	−37.7771	−1.88650	−0.943249	+	0.332087i	$0.892247\pi$
$402$	0	0
$403$	20.9443	1.04331
$404$	0	0
$405$	7.81966	0.388562
$406$	0	0
$407$	0	0
$408$	0	0
$409$	28.4721	1.40786	0.703928	−	0.710271i	$-0.251428\pi$
$409$	28.4721	1.40786	0.703928	+	0.710271i	$0.251428\pi$
$410$	0	0
$411$	11.6393	0.574125
$412$	0	0
$413$	−6.76393	−0.332831
$414$	0	0
$415$	35.4164	1.73852
$416$	0	0
$417$	−28.5836	−1.39974
$418$	0	0
$419$	−34.3607	−1.67863	−0.839315	−	0.543646i	$-0.817043\pi$
$419$	−34.3607	−1.67863	−0.839315	+	0.543646i	$0.817043\pi$
$420$	0	0
$421$	35.8885	1.74910	0.874550	−	0.484935i	$-0.161157\pi$
$421$	35.8885	1.74910	0.874550	+	0.484935i	$0.161157\pi$
$422$	0	0
$423$	−16.5410	−0.804252
$424$	0	0
$425$	−15.1246	−0.733651
$426$	0	0
$427$	−1.70820	−0.0826658
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−20.9443	−1.00885	−0.504425	−	0.863455i	$-0.668296\pi$
$431$	−20.9443	−1.00885	−0.504425	+	0.863455i	$0.668296\pi$
$432$	0	0
$433$	6.76393	0.325054	0.162527	−	0.986704i	$-0.448036\pi$
$433$	6.76393	0.325054	0.162527	+	0.986704i	$0.448036\pi$
$434$	0	0
$435$	−8.00000	−0.383571
$436$	0	0
$437$	−4.47214	−0.213931
$438$	0	0
$439$	8.76393	0.418280	0.209140	−	0.977886i	$-0.432934\pi$
$439$	8.76393	0.418280	0.209140	+	0.977886i	$0.432934\pi$
$440$	0	0
$441$	−1.47214	−0.0701017
$442$	0	0
$443$	−4.00000	−0.190046	−0.0950229	−	0.995475i	$-0.530292\pi$
$443$	−4.00000	−0.190046	−0.0950229	+	0.995475i	$0.530292\pi$
$444$	0	0
$445$	−20.0000	−0.948091
$446$	0	0
$447$	−26.4721	−1.25209
$448$	0	0
$449$	9.41641	0.444388	0.222194	−	0.975003i	$-0.428678\pi$
$449$	9.41641	0.444388	0.222194	+	0.975003i	$0.428678\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	−11.7771	−0.553336
$454$	0	0
$455$	20.9443	0.981883
$456$	0	0
$457$	22.3607	1.04599	0.522994	−	0.852336i	$-0.324815\pi$
$457$	22.3607	1.04599	0.522994	+	0.852336i	$0.324815\pi$
$458$	0	0
$459$	−15.2786	−0.713146
$460$	0	0
$461$	3.05573	0.142319	0.0711597	−	0.997465i	$-0.477330\pi$
$461$	3.05573	0.142319	0.0711597	+	0.997465i	$0.477330\pi$
$462$	0	0
$463$	11.0557	0.513803	0.256902	−	0.966438i	$-0.417298\pi$
$463$	11.0557	0.513803	0.256902	+	0.966438i	$0.417298\pi$
$464$	0	0
$465$	−12.9443	−0.600276
$466$	0	0
$467$	11.5279	0.533446	0.266723	−	0.963773i	$-0.414059\pi$
$467$	11.5279	0.533446	0.266723	+	0.963773i	$0.414059\pi$
$468$	0	0
$469$	−4.00000	−0.184703
$470$	0	0
$471$	23.0557	1.06235
$472$	0	0
$473$	0	0
$474$	0	0
$475$	−24.4721	−1.12286
$476$	0	0
$477$	8.83282	0.404427
$478$	0	0
$479$	34.8328	1.59155	0.795776	−	0.605591i	$-0.207063\pi$
$479$	34.8328	1.59155	0.795776	+	0.605591i	$0.207063\pi$
$480$	0	0
$481$	−38.8328	−1.77062
$482$	0	0
$483$	−1.23607	−0.0562430
$484$	0	0
$485$	45.8885	2.08369
$486$	0	0
$487$	0	0		−	1.00000i	$-0.5\pi$
$487$	0	0			1.00000i	$0.5\pi$
$488$	0	0
$489$	−8.00000	−0.361773
$490$	0	0
$491$	7.05573	0.318421	0.159210	−	0.987245i	$-0.449105\pi$
$491$	7.05573	0.318421	0.159210	+	0.987245i	$0.449105\pi$
$492$	0	0
$493$	5.52786	0.248962
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−6.47214	−0.290315
$498$	0	0
$499$	8.94427	0.400401	0.200200	−	0.979755i	$-0.435841\pi$
$499$	8.94427	0.400401	0.200200	+	0.979755i	$0.435841\pi$
$500$	0	0
$501$	−8.22291	−0.367373
$502$	0	0
$503$	4.94427	0.220454	0.110227	−	0.993906i	$-0.464842\pi$
$503$	4.94427	0.220454	0.110227	+	0.993906i	$0.464842\pi$
$504$	0	0
$505$	17.8885	0.796030
$506$	0	0
$507$	−35.7082	−1.58586
$508$	0	0
$509$	33.8885	1.50208	0.751042	−	0.660255i	$-0.229552\pi$
$509$	33.8885	1.50208	0.751042	+	0.660255i	$0.229552\pi$
$510$	0	0
$511$	13.4164	0.593507
$512$	0	0
$513$	−24.7214	−1.09147
$514$	0	0
$515$	−12.9443	−0.570393
$516$	0	0
$517$	0	0
$518$	0	0
$519$	−3.77709	−0.165796
$520$	0	0
$521$	11.7082	0.512946	0.256473	−	0.966551i	$-0.417440\pi$
$521$	11.7082	0.512946	0.256473	+	0.966551i	$0.417440\pi$
$522$	0	0
$523$	−10.5836	−0.462788	−0.231394	−	0.972860i	$-0.574329\pi$
$523$	−10.5836	−0.462788	−0.231394	+	0.972860i	$0.574329\pi$
$524$	0	0
$525$	−6.76393	−0.295202
$526$	0	0
$527$	8.94427	0.389619
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	9.95743	0.432116
$532$	0	0
$533$	64.7214	2.80339
$534$	0	0
$535$	−4.94427	−0.213760
$536$	0	0
$537$	20.9443	0.903812
$538$	0	0
$539$	0	0
$540$	0	0
$541$	3.88854	0.167182	0.0835908	−	0.996500i	$-0.473361\pi$
$541$	3.88854	0.167182	0.0835908	+	0.996500i	$0.473361\pi$
$542$	0	0
$543$	−10.8328	−0.464881
$544$	0	0
$545$	−40.3607	−1.72886
$546$	0	0
$547$	22.4721	0.960839	0.480420	−	0.877039i	$-0.340484\pi$
$547$	22.4721	0.960839	0.480420	+	0.877039i	$0.340484\pi$
$548$	0	0
$549$	2.51471	0.107325
$550$	0	0
$551$	8.94427	0.381039
$552$	0	0
$553$	8.94427	0.380349
$554$	0	0
$555$	24.0000	1.01874
$556$	0	0
$557$	−17.4164	−0.737957	−0.368978	−	0.929438i	$-0.620292\pi$
$557$	−17.4164	−0.737957	−0.368978	+	0.929438i	$0.620292\pi$
$558$	0	0
$559$	16.0000	0.676728
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−6.58359	−0.277465	−0.138733	−	0.990330i	$-0.544303\pi$
$563$	−6.58359	−0.277465	−0.138733	+	0.990330i	$0.544303\pi$
$564$	0	0
$565$	−40.3607	−1.69799
$566$	0	0
$567$	−2.41641	−0.101480
$568$	0	0
$569$	−2.00000	−0.0838444	−0.0419222	−	0.999121i	$-0.513348\pi$
$569$	−2.00000	−0.0838444	−0.0419222	+	0.999121i	$0.513348\pi$
$570$	0	0
$571$	33.3050	1.39377	0.696884	−	0.717183i	$-0.254569\pi$
$571$	33.3050	1.39377	0.696884	+	0.717183i	$0.254569\pi$
$572$	0	0
$573$	3.77709	0.157790
$574$	0	0
$575$	5.47214	0.228204
$576$	0	0
$577$	−25.4164	−1.05810	−0.529049	−	0.848591i	$-0.677451\pi$
$577$	−25.4164	−1.05810	−0.529049	+	0.848591i	$0.677451\pi$
$578$	0	0
$579$	6.24922	0.259709
$580$	0	0
$581$	−10.9443	−0.454045
$582$	0	0
$583$	0	0
$584$	0	0
$585$	−30.8328	−1.27478
$586$	0	0
$587$	−39.7082	−1.63893	−0.819466	−	0.573127i	$-0.805730\pi$
$587$	−39.7082	−1.63893	−0.819466	+	0.573127i	$0.805730\pi$
$588$	0	0
$589$	14.4721	0.596314
$590$	0	0
$591$	24.5836	1.01123
$592$	0	0
$593$	−38.0000	−1.56047	−0.780236	−	0.625485i	$-0.784901\pi$
$593$	−38.0000	−1.56047	−0.780236	+	0.625485i	$0.784901\pi$
$594$	0	0
$595$	8.94427	0.366679
$596$	0	0
$597$	−14.8328	−0.607067
$598$	0	0
$599$	−38.8328	−1.58667	−0.793333	−	0.608788i	$-0.791656\pi$
$599$	−38.8328	−1.58667	−0.793333	+	0.608788i	$0.791656\pi$
$600$	0	0
$601$	−4.47214	−0.182422	−0.0912111	−	0.995832i	$-0.529074\pi$
$601$	−4.47214	−0.182422	−0.0912111	+	0.995832i	$0.529074\pi$
$602$	0	0
$603$	5.88854	0.239800
$604$	0	0
$605$	35.5967	1.44721
$606$	0	0
$607$	−0.763932	−0.0310070	−0.0155035	−	0.999880i	$-0.504935\pi$
$607$	−0.763932	−0.0310070	−0.0155035	+	0.999880i	$0.504935\pi$
$608$	0	0
$609$	2.47214	0.100176
$610$	0	0
$611$	−72.7214	−2.94199
$612$	0	0
$613$	−36.2492	−1.46409	−0.732046	−	0.681255i	$-0.761434\pi$
$613$	−36.2492	−1.46409	−0.732046	+	0.681255i	$0.761434\pi$
$614$	0	0
$615$	−40.0000	−1.61296
$616$	0	0
$617$	9.05573	0.364570	0.182285	−	0.983246i	$-0.441651\pi$
$617$	9.05573	0.364570	0.182285	+	0.983246i	$0.441651\pi$
$618$	0	0
$619$	−1.05573	−0.0424333	−0.0212166	−	0.999775i	$-0.506754\pi$
$619$	−1.05573	−0.0424333	−0.0212166	+	0.999775i	$0.506754\pi$
$620$	0	0
$621$	5.52786	0.221826
$622$	0	0
$623$	6.18034	0.247610
$624$	0	0
$625$	−22.4164	−0.896656
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−16.5836	−0.661231
$630$	0	0
$631$	2.11146	0.0840557	0.0420279	−	0.999116i	$-0.486618\pi$
$631$	2.11146	0.0840557	0.0420279	+	0.999116i	$0.486618\pi$
$632$	0	0
$633$	20.9443	0.832460
$634$	0	0
$635$	−62.8328	−2.49344
$636$	0	0
$637$	−6.47214	−0.256435
$638$	0	0
$639$	9.52786	0.376916
$640$	0	0
$641$	34.9443	1.38022	0.690108	−	0.723707i	$-0.257563\pi$
$641$	34.9443	1.38022	0.690108	+	0.723707i	$0.257563\pi$
$642$	0	0
$643$	−44.2492	−1.74502	−0.872510	−	0.488597i	$-0.837509\pi$
$643$	−44.2492	−1.74502	−0.872510	+	0.488597i	$0.837509\pi$
$644$	0	0
$645$	−9.88854	−0.389361
$646$	0	0
$647$	−1.34752	−0.0529766	−0.0264883	−	0.999649i	$-0.508432\pi$
$647$	−1.34752	−0.0529766	−0.0264883	+	0.999649i	$0.508432\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	4.00000	0.156772
$652$	0	0
$653$	29.7771	1.16527	0.582634	−	0.812735i	$-0.302022\pi$
$653$	29.7771	1.16527	0.582634	+	0.812735i	$0.302022\pi$
$654$	0	0
$655$	−55.7771	−2.17939
$656$	0	0
$657$	−19.7508	−0.770551
$658$	0	0
$659$	46.2492	1.80161	0.900807	−	0.434220i	$-0.142976\pi$
$659$	46.2492	1.80161	0.900807	+	0.434220i	$0.142976\pi$
$660$	0	0
$661$	5.12461	0.199324	0.0996621	−	0.995021i	$-0.468224\pi$
$661$	5.12461	0.199324	0.0996621	+	0.995021i	$0.468224\pi$
$662$	0	0
$663$	−22.1115	−0.858738
$664$	0	0
$665$	14.4721	0.561205
$666$	0	0
$667$	−2.00000	−0.0774403
$668$	0	0
$669$	−13.1672	−0.509073
$670$	0	0
$671$	0	0
$672$	0	0
$673$	−29.4164	−1.13392	−0.566960	−	0.823746i	$-0.691880\pi$
$673$	−29.4164	−1.13392	−0.566960	+	0.823746i	$0.691880\pi$
$674$	0	0
$675$	30.2492	1.16429
$676$	0	0
$677$	23.2361	0.893035	0.446517	−	0.894775i	$-0.352664\pi$
$677$	23.2361	0.893035	0.446517	+	0.894775i	$0.352664\pi$
$678$	0	0
$679$	−14.1803	−0.544191
$680$	0	0
$681$	−26.4721	−1.01441
$682$	0	0
$683$	4.00000	0.153056	0.0765279	−	0.997067i	$-0.475617\pi$
$683$	4.00000	0.153056	0.0765279	+	0.997067i	$0.475617\pi$
$684$	0	0
$685$	30.4721	1.16428
$686$	0	0
$687$	18.8328	0.718517
$688$	0	0
$689$	38.8328	1.47941
$690$	0	0
$691$	−49.0132	−1.86455	−0.932274	−	0.361753i	$-0.882179\pi$
$691$	−49.0132	−1.86455	−0.932274	+	0.361753i	$0.882179\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−74.8328	−2.83857
$696$	0	0
$697$	27.6393	1.04691
$698$	0	0
$699$	20.3607	0.770112
$700$	0	0
$701$	−21.0557	−0.795264	−0.397632	−	0.917545i	$-0.630168\pi$
$701$	−21.0557	−0.795264	−0.397632	+	0.917545i	$0.630168\pi$
$702$	0	0
$703$	−26.8328	−1.01202
$704$	0	0
$705$	44.9443	1.69270
$706$	0	0
$707$	−5.52786	−0.207897
$708$	0	0
$709$	14.0000	0.525781	0.262891	−	0.964826i	$-0.415324\pi$
$709$	14.0000	0.525781	0.262891	+	0.964826i	$0.415324\pi$
$710$	0	0
$711$	−13.1672	−0.493808
$712$	0	0
$713$	−3.23607	−0.121192
$714$	0	0
$715$	0	0
$716$	0	0
$717$	8.00000	0.298765
$718$	0	0
$719$	32.5410	1.21358	0.606788	−	0.794864i	$-0.292458\pi$
$719$	32.5410	1.21358	0.606788	+	0.794864i	$0.292458\pi$
$720$	0	0
$721$	4.00000	0.148968
$722$	0	0
$723$	−0.360680	−0.0134138
$724$	0	0
$725$	−10.9443	−0.406460
$726$	0	0
$727$	−7.05573	−0.261682	−0.130841	−	0.991403i	$-0.541768\pi$
$727$	−7.05573	−0.261682	−0.130841	+	0.991403i	$0.541768\pi$
$728$	0	0
$729$	24.0557	0.890953
$730$	0	0
$731$	6.83282	0.252721
$732$	0	0
$733$	3.59675	0.132849	0.0664245	−	0.997791i	$-0.478841\pi$
$733$	3.59675	0.132849	0.0664245	+	0.997791i	$0.478841\pi$
$734$	0	0
$735$	4.00000	0.147542
$736$	0	0
$737$	0	0
$738$	0	0
$739$	−24.3607	−0.896122	−0.448061	−	0.894003i	$-0.647885\pi$
$739$	−24.3607	−0.896122	−0.448061	+	0.894003i	$0.647885\pi$
$740$	0	0
$741$	−35.7771	−1.31430
$742$	0	0
$743$	48.9443	1.79559	0.897796	−	0.440412i	$-0.145168\pi$
$743$	48.9443	1.79559	0.897796	+	0.440412i	$0.145168\pi$
$744$	0	0
$745$	−69.3050	−2.53914
$746$	0	0
$747$	16.1115	0.589487
$748$	0	0
$749$	1.52786	0.0558269
$750$	0	0
$751$	40.9443	1.49408	0.747039	−	0.664780i	$-0.231475\pi$
$751$	40.9443	1.49408	0.747039	+	0.664780i	$0.231475\pi$
$752$	0	0
$753$	−5.52786	−0.201447
$754$	0	0
$755$	−30.8328	−1.12212
$756$	0	0
$757$	23.5279	0.855135	0.427567	−	0.903983i	$-0.359371\pi$
$757$	23.5279	0.855135	0.427567	+	0.903983i	$0.359371\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−0.472136	−0.0171149	−0.00855746	−	0.999963i	$-0.502724\pi$
$761$	−0.472136	−0.0171149	−0.00855746	+	0.999963i	$0.502724\pi$
$762$	0	0
$763$	12.4721	0.451522
$764$	0	0
$765$	−13.1672	−0.476061
$766$	0	0
$767$	43.7771	1.58070
$768$	0	0
$769$	21.2361	0.765792	0.382896	−	0.923791i	$-0.374927\pi$
$769$	21.2361	0.765792	0.382896	+	0.923791i	$0.374927\pi$
$770$	0	0
$771$	33.3050	1.19945
$772$	0	0
$773$	16.1803	0.581966	0.290983	−	0.956728i	$-0.406018\pi$
$773$	16.1803	0.581966	0.290983	+	0.956728i	$0.406018\pi$
$774$	0	0
$775$	−17.7082	−0.636097
$776$	0	0
$777$	−7.41641	−0.266062
$778$	0	0
$779$	44.7214	1.60231
$780$	0	0
$781$	0	0
$782$	0	0
$783$	−11.0557	−0.395099
$784$	0	0
$785$	60.3607	2.15437
$786$	0	0
$787$	35.8885	1.27929	0.639644	−	0.768671i	$-0.279082\pi$
$787$	35.8885	1.27929	0.639644	+	0.768671i	$0.279082\pi$
$788$	0	0
$789$	14.8328	0.528062
$790$	0	0
$791$	12.4721	0.443458
$792$	0	0
$793$	11.0557	0.392600
$794$	0	0
$795$	−24.0000	−0.851192
$796$	0	0
$797$	−13.3475	−0.472794	−0.236397	−	0.971657i	$-0.575967\pi$
$797$	−13.3475	−0.472794	−0.236397	+	0.971657i	$0.575967\pi$
$798$	0	0
$799$	−31.0557	−1.09867
$800$	0	0
$801$	−9.09830	−0.321473
$802$	0	0
$803$	0	0
$804$	0	0
$805$	−3.23607	−0.114056
$806$	0	0
$807$	−3.77709	−0.132960
$808$	0	0
$809$	−32.8328	−1.15434	−0.577170	−	0.816624i	$-0.695843\pi$
$809$	−32.8328	−1.15434	−0.577170	+	0.816624i	$0.695843\pi$
$810$	0	0
$811$	−21.5967	−0.758364	−0.379182	−	0.925322i	$-0.623795\pi$
$811$	−21.5967	−0.758364	−0.379182	+	0.925322i	$0.623795\pi$
$812$	0	0
$813$	−18.8328	−0.660496
$814$	0	0
$815$	−20.9443	−0.733646
$816$	0	0
$817$	11.0557	0.386791
$818$	0	0
$819$	9.52786	0.332931
$820$	0	0
$821$	−2.94427	−0.102756	−0.0513779	−	0.998679i	$-0.516361\pi$
$821$	−2.94427	−0.102756	−0.0513779	+	0.998679i	$0.516361\pi$
$822$	0	0
$823$	11.4164	0.397951	0.198975	−	0.980004i	$-0.436239\pi$
$823$	11.4164	0.397951	0.198975	+	0.980004i	$0.436239\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	12.5836	0.437574	0.218787	−	0.975773i	$-0.429790\pi$
$827$	12.5836	0.437574	0.218787	+	0.975773i	$0.429790\pi$
$828$	0	0
$829$	13.8885	0.482369	0.241185	−	0.970479i	$-0.422464\pi$
$829$	13.8885	0.482369	0.241185	+	0.970479i	$0.422464\pi$
$830$	0	0
$831$	−24.5836	−0.852795
$832$	0	0
$833$	−2.76393	−0.0957646
$834$	0	0
$835$	−21.5279	−0.745002
$836$	0	0
$837$	−17.8885	−0.618319
$838$	0	0
$839$	−28.3607	−0.979119	−0.489560	−	0.871970i	$-0.662842\pi$
$839$	−28.3607	−0.979119	−0.489560	+	0.871970i	$0.662842\pi$
$840$	0	0
$841$	−25.0000	−0.862069
$842$	0	0
$843$	−6.69505	−0.230590
$844$	0	0
$845$	−93.4853	−3.21599
$846$	0	0
$847$	−11.0000	−0.377964
$848$	0	0
$849$	5.52786	0.189716
$850$	0	0
$851$	6.00000	0.205677
$852$	0	0
$853$	11.4164	0.390890	0.195445	−	0.980715i	$-0.437385\pi$
$853$	11.4164	0.390890	0.195445	+	0.980715i	$0.437385\pi$
$854$	0	0
$855$	−21.3050	−0.728614
$856$	0	0
$857$	−11.8885	−0.406105	−0.203052	−	0.979168i	$-0.565086\pi$
$857$	−11.8885	−0.406105	−0.203052	+	0.979168i	$0.565086\pi$
$858$	0	0
$859$	−2.76393	−0.0943041	−0.0471521	−	0.998888i	$-0.515015\pi$
$859$	−2.76393	−0.0943041	−0.0471521	+	0.998888i	$0.515015\pi$
$860$	0	0
$861$	12.3607	0.421251
$862$	0	0
$863$	3.41641	0.116296	0.0581479	−	0.998308i	$-0.481480\pi$
$863$	3.41641	0.116296	0.0581479	+	0.998308i	$0.481480\pi$
$864$	0	0
$865$	−9.88854	−0.336221
$866$	0	0
$867$	11.5704	0.392953
$868$	0	0
$869$	0	0
$870$	0	0
$871$	25.8885	0.877200
$872$	0	0
$873$	20.8754	0.706525
$874$	0	0
$875$	−1.52786	−0.0516512
$876$	0	0
$877$	26.9443	0.909843	0.454922	−	0.890531i	$-0.349667\pi$
$877$	26.9443	0.909843	0.454922	+	0.890531i	$0.349667\pi$
$878$	0	0
$879$	36.0000	1.21425
$880$	0	0
$881$	−21.2361	−0.715461	−0.357731	−	0.933825i	$-0.616449\pi$
$881$	−21.2361	−0.715461	−0.357731	+	0.933825i	$0.616449\pi$
$882$	0	0
$883$	−12.0000	−0.403832	−0.201916	−	0.979403i	$-0.564717\pi$
$883$	−12.0000	−0.403832	−0.201916	+	0.979403i	$0.564717\pi$
$884$	0	0
$885$	−27.0557	−0.909468
$886$	0	0
$887$	35.5967	1.19522	0.597611	−	0.801786i	$-0.296117\pi$
$887$	35.5967	1.19522	0.597611	+	0.801786i	$0.296117\pi$
$888$	0	0
$889$	19.4164	0.651205
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−50.2492	−1.68153
$894$	0	0
$895$	54.8328	1.83286
$896$	0	0
$897$	8.00000	0.267112
$898$	0	0
$899$	6.47214	0.215858
$900$	0	0
$901$	16.5836	0.552480
$902$	0	0
$903$	3.05573	0.101688
$904$	0	0
$905$	−28.3607	−0.942741
$906$	0	0
$907$	1.52786	0.0507319	0.0253659	−	0.999678i	$-0.491925\pi$
$907$	1.52786	0.0507319	0.0253659	+	0.999678i	$0.491925\pi$
$908$	0	0
$909$	8.13777	0.269913
$910$	0	0
$911$	−21.8885	−0.725200	−0.362600	−	0.931945i	$-0.618111\pi$
$911$	−21.8885	−0.725200	−0.362600	+	0.931945i	$0.618111\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	−6.83282	−0.225886
$916$	0	0
$917$	17.2361	0.569185
$918$	0	0
$919$	10.1115	0.333546	0.166773	−	0.985995i	$-0.446665\pi$
$919$	10.1115	0.333546	0.166773	+	0.985995i	$0.446665\pi$
$920$	0	0
$921$	36.1378	1.19078
$922$	0	0
$923$	41.8885	1.37878
$924$	0	0
$925$	32.8328	1.07954
$926$	0	0
$927$	−5.88854	−0.193405
$928$	0	0
$929$	−34.0000	−1.11550	−0.557752	−	0.830008i	$-0.688336\pi$
$929$	−34.0000	−1.11550	−0.557752	+	0.830008i	$0.688336\pi$
$930$	0	0
$931$	−4.47214	−0.146568
$932$	0	0
$933$	31.0557	1.01672
$934$	0	0
$935$	0	0
$936$	0	0
$937$	50.7639	1.65839	0.829193	−	0.558963i	$-0.188801\pi$
$937$	50.7639	1.65839	0.829193	+	0.558963i	$0.188801\pi$
$938$	0	0
$939$	−32.3607	−1.05605
$940$	0	0
$941$	19.5967	0.638836	0.319418	−	0.947614i	$-0.396513\pi$
$941$	19.5967	0.638836	0.319418	+	0.947614i	$0.396513\pi$
$942$	0	0
$943$	−10.0000	−0.325645
$944$	0	0
$945$	−17.8885	−0.581914
$946$	0	0
$947$	−42.2492	−1.37292	−0.686458	−	0.727170i	$-0.740835\pi$
$947$	−42.2492	−1.37292	−0.686458	+	0.727170i	$0.740835\pi$
$948$	0	0
$949$	−86.8328	−2.81871
$950$	0	0
$951$	−12.3607	−0.400823
$952$	0	0
$953$	45.7771	1.48287	0.741433	−	0.671027i	$-0.234147\pi$
$953$	45.7771	1.48287	0.741433	+	0.671027i	$0.234147\pi$
$954$	0	0
$955$	9.88854	0.319986
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−9.41641	−0.304072
$960$	0	0
$961$	−20.5279	−0.662189
$962$	0	0
$963$	−2.24922	−0.0724802
$964$	0	0
$965$	16.3607	0.526669
$966$	0	0
$967$	9.88854	0.317994	0.158997	−	0.987279i	$-0.449174\pi$
$967$	9.88854	0.317994	0.158997	+	0.987279i	$0.449174\pi$
$968$	0	0
$969$	−15.2786	−0.490821
$970$	0	0
$971$	35.8885	1.15172	0.575859	−	0.817549i	$-0.304668\pi$
$971$	35.8885	1.15172	0.575859	+	0.817549i	$0.304668\pi$
$972$	0	0
$973$	23.1246	0.741341
$974$	0	0
$975$	43.7771	1.40199
$976$	0	0
$977$	−12.4721	−0.399019	−0.199509	−	0.979896i	$-0.563935\pi$
$977$	−12.4721	−0.399019	−0.199509	+	0.979896i	$0.563935\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	−18.3607	−0.586211
$982$	0	0
$983$	13.5279	0.431472	0.215736	−	0.976452i	$-0.430785\pi$
$983$	13.5279	0.431472	0.215736	+	0.976452i	$0.430785\pi$
$984$	0	0
$985$	64.3607	2.05070
$986$	0	0
$987$	−13.8885	−0.442077
$988$	0	0
$989$	−2.47214	−0.0786094
$990$	0	0
$991$	−21.3050	−0.676774	−0.338387	−	0.941007i	$-0.609881\pi$
$991$	−21.3050	−0.676774	−0.338387	+	0.941007i	$0.609881\pi$
$992$	0	0
$993$	−37.6656	−1.19528
$994$	0	0
$995$	−38.8328	−1.23108
$996$	0	0
$997$	10.8328	0.343079	0.171539	−	0.985177i	$-0.445126\pi$
$997$	10.8328	0.343079	0.171539	+	0.985177i	$0.445126\pi$
$998$	0	0
$999$	33.1672	1.04936

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2576.2.a.t.1.1		2
4.3	odd	2		322.2.a.e.1.2	✓	2
12.11	even	2		2898.2.a.bd.1.2		2
20.19	odd	2		8050.2.a.bf.1.1		2
28.27	even	2		2254.2.a.k.1.1		2
92.91	even	2		7406.2.a.j.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
322.2.a.e.1.2	✓	2	4.3	odd	2
2254.2.a.k.1.1		2	28.27	even	2
2576.2.a.t.1.1		2	1.1	even	1	trivial
2898.2.a.bd.1.2		2	12.11	even	2
7406.2.a.j.1.2		2	92.91	even	2
8050.2.a.bf.1.1		2	20.19	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 2576 = 2^{4} \cdot 7 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2576.a (trivial)

\(f(q)\)	\(=\)	\(q-1.23607 q^{3} -3.23607 q^{5} +1.00000 q^{7} -1.47214 q^{9} +O(q^{10})\)	\(q-1.23607 q^{3} -3.23607 q^{5} +1.00000 q^{7} -1.47214 q^{9} -6.47214 q^{13} +4.00000 q^{15} -2.76393 q^{17} -4.47214 q^{19} -1.23607 q^{21} +1.00000 q^{23} +5.47214 q^{25} +5.52786 q^{27} -2.00000 q^{29} -3.23607 q^{31} -3.23607 q^{35} +6.00000 q^{37} +8.00000 q^{39} -10.0000 q^{41} -2.47214 q^{43} +4.76393 q^{45} +11.2361 q^{47} +1.00000 q^{49} +3.41641 q^{51} -6.00000 q^{53} +5.52786 q^{57} -6.76393 q^{59} -1.70820 q^{61} -1.47214 q^{63} +20.9443 q^{65} -4.00000 q^{67} -1.23607 q^{69} -6.47214 q^{71} +13.4164 q^{73} -6.76393 q^{75} +8.94427 q^{79} -2.41641 q^{81} -10.9443 q^{83} +8.94427 q^{85} +2.47214 q^{87} +6.18034 q^{89} -6.47214 q^{91} +4.00000 q^{93} +14.4721 q^{95} -14.1803 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{3} - 2 q^{5} + 2 q^{7} + 6 q^{9}+O(q^{10}) \) 2 * q + 2 * q^3 - 2 * q^5 + 2 * q^7 + 6 * q^9	\( 2 q + 2 q^{3} - 2 q^{5} + 2 q^{7} + 6 q^{9} - 4 q^{13} + 8 q^{15} - 10 q^{17} + 2 q^{21} + 2 q^{23} + 2 q^{25} + 20 q^{27} - 4 q^{29} - 2 q^{31} - 2 q^{35} + 12 q^{37} + 16 q^{39} - 20 q^{41} + 4 q^{43} + 14 q^{45} + 18 q^{47} + 2 q^{49} - 20 q^{51} - 12 q^{53} + 20 q^{57} - 18 q^{59} + 10 q^{61} + 6 q^{63} + 24 q^{65} - 8 q^{67} + 2 q^{69} - 4 q^{71} - 18 q^{75} + 22 q^{81} - 4 q^{83} - 4 q^{87} - 10 q^{89} - 4 q^{91} + 8 q^{93} + 20 q^{95} - 6 q^{97}+O(q^{100}) \) 2 * q + 2 * q^3 - 2 * q^5 + 2 * q^7 + 6 * q^9 - 4 * q^13 + 8 * q^15 - 10 * q^17 + 2 * q^21 + 2 * q^23 + 2 * q^25 + 20 * q^27 - 4 * q^29 - 2 * q^31 - 2 * q^35 + 12 * q^37 + 16 * q^39 - 20 * q^41 + 4 * q^43 + 14 * q^45 + 18 * q^47 + 2 * q^49 - 20 * q^51 - 12 * q^53 + 20 * q^57 - 18 * q^59 + 10 * q^61 + 6 * q^63 + 24 * q^65 - 8 * q^67 + 2 * q^69 - 4 * q^71 - 18 * q^75 + 22 * q^81 - 4 * q^83 - 4 * q^87 - 10 * q^89 - 4 * q^91 + 8 * q^93 + 20 * q^95 - 6 * q^97

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