LMFDB - Embedded newform 2576.2.a.y.1.3

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:	$3$
Coefficient field:	3.3.148.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 3x + 1 $ x^3 - x^2 - 3*x + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1288)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-1.48119$ 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)

$f(q)$	$=$	$q+1.67513 q^{3} +3.28726 q^{5} +1.00000 q^{7} -0.193937 q^{9} +O(q^{10})$	$q+1.67513 q^{3} +3.28726 q^{5} +1.00000 q^{7} -0.193937 q^{9} -2.15633 q^{11} +2.96239 q^{13} +5.50659 q^{15} -0.637519 q^{17} +1.67513 q^{21} -1.00000 q^{23} +5.80606 q^{25} -5.35026 q^{27} +3.19394 q^{29} +5.59991 q^{31} -3.61213 q^{33} +3.28726 q^{35} +10.9624 q^{37} +4.96239 q^{39} +1.22425 q^{41} +5.73813 q^{43} -0.637519 q^{45} +4.63752 q^{47} +1.00000 q^{49} -1.06793 q^{51} -5.22425 q^{53} -7.08840 q^{55} -1.98778 q^{59} +5.80114 q^{61} -0.193937 q^{63} +9.73813 q^{65} -8.08110 q^{67} -1.67513 q^{69} -6.70052 q^{71} +12.0508 q^{73} +9.72592 q^{75} -2.15633 q^{77} -7.76845 q^{79} -8.38058 q^{81} -0.962389 q^{83} -2.09569 q^{85} +5.35026 q^{87} +14.3757 q^{89} +2.96239 q^{91} +9.38058 q^{93} +14.1138 q^{97} +0.418190 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 4 q^{5} + 3 q^{7} - q^{9}+O(q^{10}) $ 3 * q + 4 * q^5 + 3 * q^7 - q^9	$ 3 q + 4 q^{5} + 3 q^{7} - q^{9} + 4 q^{11} - 2 q^{13} - 4 q^{15} + 14 q^{17} - 3 q^{23} + 17 q^{25} - 6 q^{27} + 10 q^{29} - 10 q^{31} - 10 q^{33} + 4 q^{35} + 22 q^{37} + 4 q^{39} + 2 q^{41} + 8 q^{43} + 14 q^{45} - 2 q^{47} + 3 q^{49} - 12 q^{51} - 14 q^{53} - 2 q^{55} + 20 q^{59} + 4 q^{61} - q^{63} + 20 q^{65} + 8 q^{67} + 6 q^{73} - 6 q^{75} + 4 q^{77} - 12 q^{79} - 13 q^{81} + 8 q^{83} + 6 q^{87} + 18 q^{89} - 2 q^{91} + 16 q^{93} + 8 q^{97}+O(q^{100}) $ 3 * q + 4 * q^5 + 3 * q^7 - q^9 + 4 * q^11 - 2 * q^13 - 4 * q^15 + 14 * q^17 - 3 * q^23 + 17 * q^25 - 6 * q^27 + 10 * q^29 - 10 * q^31 - 10 * q^33 + 4 * q^35 + 22 * q^37 + 4 * q^39 + 2 * q^41 + 8 * q^43 + 14 * q^45 - 2 * q^47 + 3 * q^49 - 12 * q^51 - 14 * q^53 - 2 * q^55 + 20 * q^59 + 4 * q^61 - q^63 + 20 * q^65 + 8 * q^67 + 6 * q^73 - 6 * q^75 + 4 * q^77 - 12 * q^79 - 13 * q^81 + 8 * q^83 + 6 * q^87 + 18 * q^89 - 2 * q^91 + 16 * q^93 + 8 * 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
$3$	1.67513	0.967137	0.483569	−	0.875306i	$-0.339340\pi$
$3$	1.67513	0.967137	0.483569	+	0.875306i	$0.339340\pi$
$4$	0	0
$5$	3.28726	1.47011	0.735053	−	0.678009i	$-0.237157\pi$
$5$	3.28726	1.47011	0.735053	+	0.678009i	$0.237157\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	−0.193937	−0.0646455
$10$	0	0
$11$	−2.15633	−0.650157	−0.325078	−	0.945687i	$-0.605391\pi$
$11$	−2.15633	−0.650157	−0.325078	+	0.945687i	$0.605391\pi$
$12$	0	0
$13$	2.96239	0.821619	0.410809	−	0.911721i	$-0.365246\pi$
$13$	2.96239	0.821619	0.410809	+	0.911721i	$0.365246\pi$
$14$	0	0
$15$	5.50659	1.42179
$16$	0	0
$17$	−0.637519	−0.154621	−0.0773106	−	0.997007i	$-0.524633\pi$
$17$	−0.637519	−0.154621	−0.0773106	+	0.997007i	$0.524633\pi$
$18$	0	0
$19$	0	0		−	1.00000i	$-0.5\pi$
$19$	0	0			1.00000i	$0.5\pi$
$20$	0	0
$21$	1.67513	0.365544
$22$	0	0
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	0	0
$25$	5.80606	1.16121
$26$	0	0
$27$	−5.35026	−1.02966
$28$	0	0
$29$	3.19394	0.593099	0.296550	−	0.955017i	$-0.404164\pi$
$29$	3.19394	0.593099	0.296550	+	0.955017i	$0.404164\pi$
$30$	0	0
$31$	5.59991	1.00577	0.502887	−	0.864352i	$-0.332271\pi$
$31$	5.59991	1.00577	0.502887	+	0.864352i	$0.332271\pi$
$32$	0	0
$33$	−3.61213	−0.628791
$34$	0	0
$35$	3.28726	0.555648
$36$	0	0
$37$	10.9624	1.80221	0.901103	−	0.433606i	$-0.142759\pi$
$37$	10.9624	1.80221	0.901103	+	0.433606i	$0.142759\pi$
$38$	0	0
$39$	4.96239	0.794618
$40$	0	0
$41$	1.22425	0.191196	0.0955982	−	0.995420i	$-0.469524\pi$
$41$	1.22425	0.191196	0.0955982	+	0.995420i	$0.469524\pi$
$42$	0	0
$43$	5.73813	0.875057	0.437529	−	0.899204i	$-0.355854\pi$
$43$	5.73813	0.875057	0.437529	+	0.899204i	$0.355854\pi$
$44$	0	0
$45$	−0.637519	−0.0950358
$46$	0	0
$47$	4.63752	0.676452	0.338226	−	0.941065i	$-0.390173\pi$
$47$	4.63752	0.676452	0.338226	+	0.941065i	$0.390173\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−1.06793	−0.149540
$52$	0	0
$53$	−5.22425	−0.717606	−0.358803	−	0.933413i	$-0.616815\pi$
$53$	−5.22425	−0.717606	−0.358803	+	0.933413i	$0.616815\pi$
$54$	0	0
$55$	−7.08840	−0.955799
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−1.98778	−0.258787	−0.129394	−	0.991593i	$-0.541303\pi$
$59$	−1.98778	−0.258787	−0.129394	+	0.991593i	$0.541303\pi$
$60$	0	0
$61$	5.80114	0.742760	0.371380	−	0.928481i	$-0.378885\pi$
$61$	5.80114	0.742760	0.371380	+	0.928481i	$0.378885\pi$
$62$	0	0
$63$	−0.193937	−0.0244337
$64$	0	0
$65$	9.73813	1.20787
$66$	0	0
$67$	−8.08110	−0.987264	−0.493632	−	0.869671i	$-0.664331\pi$
$67$	−8.08110	−0.987264	−0.493632	+	0.869671i	$0.664331\pi$
$68$	0	0
$69$	−1.67513	−0.201662
$70$	0	0
$71$	−6.70052	−0.795206	−0.397603	−	0.917558i	$-0.630158\pi$
$71$	−6.70052	−0.795206	−0.397603	+	0.917558i	$0.630158\pi$
$72$	0	0
$73$	12.0508	1.41044	0.705219	−	0.708990i	$-0.250849\pi$
$73$	12.0508	1.41044	0.705219	+	0.708990i	$0.250849\pi$
$74$	0	0
$75$	9.72592	1.12305
$76$	0	0
$77$	−2.15633	−0.245736
$78$	0	0
$79$	−7.76845	−0.874019	−0.437010	−	0.899457i	$-0.643962\pi$
$79$	−7.76845	−0.874019	−0.437010	+	0.899457i	$0.643962\pi$
$80$	0	0
$81$	−8.38058	−0.931175
$82$	0	0
$83$	−0.962389	−0.105636	−0.0528179	−	0.998604i	$-0.516820\pi$
$83$	−0.962389	−0.105636	−0.0528179	+	0.998604i	$0.516820\pi$
$84$	0	0
$85$	−2.09569	−0.227310
$86$	0	0
$87$	5.35026	0.573608
$88$	0	0
$89$	14.3757	1.52382	0.761908	−	0.647685i	$-0.224263\pi$
$89$	14.3757	1.52382	0.761908	+	0.647685i	$0.224263\pi$
$90$	0	0
$91$	2.96239	0.310543
$92$	0	0
$93$	9.38058	0.972721
$94$	0	0
$95$	0	0
$96$	0	0
$97$	14.1138	1.43304	0.716519	−	0.697567i	$-0.245734\pi$
$97$	14.1138	1.43304	0.716519	+	0.697567i	$0.245734\pi$
$98$	0	0
$99$	0.418190	0.0420297
$100$	0	0
$101$	−0.574515	−0.0571664	−0.0285832	−	0.999591i	$-0.509100\pi$
$101$	−0.574515	−0.0571664	−0.0285832	+	0.999591i	$0.509100\pi$
$102$	0	0
$103$	−5.92478	−0.583786	−0.291893	−	0.956451i	$-0.594285\pi$
$103$	−5.92478	−0.583786	−0.291893	+	0.956451i	$0.594285\pi$
$104$	0	0
$105$	5.50659	0.537388
$106$	0	0
$107$	−12.4387	−1.20249	−0.601245	−	0.799065i	$-0.705329\pi$
$107$	−12.4387	−1.20249	−0.601245	+	0.799065i	$0.705329\pi$
$108$	0	0
$109$	−17.9756	−1.72175	−0.860873	−	0.508819i	$-0.830082\pi$
$109$	−17.9756	−1.72175	−0.860873	+	0.508819i	$0.830082\pi$
$110$	0	0
$111$	18.3634	1.74298
$112$	0	0
$113$	−8.70052	−0.818476	−0.409238	−	0.912428i	$-0.634205\pi$
$113$	−8.70052	−0.818476	−0.409238	+	0.912428i	$0.634205\pi$
$114$	0	0
$115$	−3.28726	−0.306538
$116$	0	0
$117$	−0.574515	−0.0531140
$118$	0	0
$119$	−0.637519	−0.0584413
$120$	0	0
$121$	−6.35026	−0.577297
$122$	0	0
$123$	2.05079	0.184913
$124$	0	0
$125$	2.64974	0.237000
$126$	0	0
$127$	8.96239	0.795283	0.397642	−	0.917541i	$-0.369829\pi$
$127$	8.96239	0.795283	0.397642	+	0.917541i	$0.369829\pi$
$128$	0	0
$129$	9.61213	0.846301
$130$	0	0
$131$	−8.45088	−0.738357	−0.369178	−	0.929359i	$-0.620361\pi$
$131$	−8.45088	−0.738357	−0.369178	+	0.929359i	$0.620361\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	−17.5877	−1.51371
$136$	0	0
$137$	−12.8872	−1.10102	−0.550512	−	0.834827i	$-0.685568\pi$
$137$	−12.8872	−1.10102	−0.550512	+	0.834827i	$0.685568\pi$
$138$	0	0
$139$	−9.98778	−0.847153	−0.423576	−	0.905860i	$-0.639225\pi$
$139$	−9.98778	−0.847153	−0.423576	+	0.905860i	$0.639225\pi$
$140$	0	0
$141$	7.76845	0.654222
$142$	0	0
$143$	−6.38787	−0.534181
$144$	0	0
$145$	10.4993	0.871919
$146$	0	0
$147$	1.67513	0.138162
$148$	0	0
$149$	−1.47627	−0.120941	−0.0604704	−	0.998170i	$-0.519260\pi$
$149$	−1.47627	−0.120941	−0.0604704	+	0.998170i	$0.519260\pi$
$150$	0	0
$151$	−10.1114	−0.822856	−0.411428	−	0.911442i	$-0.634970\pi$
$151$	−10.1114	−0.822856	−0.411428	+	0.911442i	$0.634970\pi$
$152$	0	0
$153$	0.123638	0.00999557
$154$	0	0
$155$	18.4083	1.47859
$156$	0	0
$157$	−3.67513	−0.293307	−0.146654	−	0.989188i	$-0.546850\pi$
$157$	−3.67513	−0.293307	−0.146654	+	0.989188i	$0.546850\pi$
$158$	0	0
$159$	−8.75131	−0.694024
$160$	0	0
$161$	−1.00000	−0.0788110
$162$	0	0
$163$	−20.1622	−1.57923	−0.789613	−	0.613605i	$-0.789719\pi$
$163$	−20.1622	−1.57923	−0.789613	+	0.613605i	$0.789719\pi$
$164$	0	0
$165$	−11.8740	−0.924389
$166$	0	0
$167$	−14.5017	−1.12217	−0.561086	−	0.827757i	$-0.689616\pi$
$167$	−14.5017	−1.12217	−0.561086	+	0.827757i	$0.689616\pi$
$168$	0	0
$169$	−4.22425	−0.324943
$170$	0	0
$171$	0	0
$172$	0	0
$173$	8.07522	0.613948	0.306974	−	0.951718i	$-0.400684\pi$
$173$	8.07522	0.613948	0.306974	+	0.951718i	$0.400684\pi$
$174$	0	0
$175$	5.80606	0.438897
$176$	0	0
$177$	−3.32979	−0.250283
$178$	0	0
$179$	11.6629	0.871727	0.435863	−	0.900013i	$-0.356443\pi$
$179$	11.6629	0.871727	0.435863	+	0.900013i	$0.356443\pi$
$180$	0	0
$181$	−10.1890	−0.757343	−0.378672	−	0.925531i	$-0.623619\pi$
$181$	−10.1890	−0.757343	−0.378672	+	0.925531i	$0.623619\pi$
$182$	0	0
$183$	9.71767	0.718351
$184$	0	0
$185$	36.0362	2.64943
$186$	0	0
$187$	1.37470	0.100528
$188$	0	0
$189$	−5.35026	−0.389174
$190$	0	0
$191$	14.8872	1.07720	0.538599	−	0.842562i	$-0.318954\pi$
$191$	14.8872	1.07720	0.538599	+	0.842562i	$0.318954\pi$
$192$	0	0
$193$	−3.27504	−0.235742	−0.117871	−	0.993029i	$-0.537607\pi$
$193$	−3.27504	−0.235742	−0.117871	+	0.993029i	$0.537607\pi$
$194$	0	0
$195$	16.3127	1.16817
$196$	0	0
$197$	−18.6253	−1.32700	−0.663499	−	0.748177i	$-0.730929\pi$
$197$	−18.6253	−1.32700	−0.663499	+	0.748177i	$0.730929\pi$
$198$	0	0
$199$	1.92478	0.136444	0.0682219	−	0.997670i	$-0.478267\pi$
$199$	1.92478	0.136444	0.0682219	+	0.997670i	$0.478267\pi$
$200$	0	0
$201$	−13.5369	−0.954820
$202$	0	0
$203$	3.19394	0.224170
$204$	0	0
$205$	4.02444	0.281079
$206$	0	0
$207$	0.193937	0.0134795
$208$	0	0
$209$	0	0
$210$	0	0
$211$	26.8872	1.85099	0.925494	−	0.378761i	$-0.123650\pi$
$211$	26.8872	1.85099	0.925494	+	0.378761i	$0.123650\pi$
$212$	0	0
$213$	−11.2243	−0.769073
$214$	0	0
$215$	18.8627	1.28643
$216$	0	0
$217$	5.59991	0.380147
$218$	0	0
$219$	20.1866	1.36409
$220$	0	0
$221$	−1.88858	−0.127040
$222$	0	0
$223$	1.98778	0.133112	0.0665558	−	0.997783i	$-0.478799\pi$
$223$	1.98778	0.133112	0.0665558	+	0.997783i	$0.478799\pi$
$224$	0	0
$225$	−1.12601	−0.0750672
$226$	0	0
$227$	−3.68735	−0.244738	−0.122369	−	0.992485i	$-0.539049\pi$
$227$	−3.68735	−0.244738	−0.122369	+	0.992485i	$0.539049\pi$
$228$	0	0
$229$	−15.3987	−1.01757	−0.508787	−	0.860893i	$-0.669906\pi$
$229$	−15.3987	−1.01757	−0.508787	+	0.860893i	$0.669906\pi$
$230$	0	0
$231$	−3.61213	−0.237660
$232$	0	0
$233$	−6.73084	−0.440952	−0.220476	−	0.975392i	$-0.570761\pi$
$233$	−6.73084	−0.440952	−0.220476	+	0.975392i	$0.570761\pi$
$234$	0	0
$235$	15.2447	0.994456
$236$	0	0
$237$	−13.0132	−0.845296
$238$	0	0
$239$	−14.8872	−0.962971	−0.481485	−	0.876454i	$-0.659903\pi$
$239$	−14.8872	−0.962971	−0.481485	+	0.876454i	$0.659903\pi$
$240$	0	0
$241$	−15.3380	−0.988010	−0.494005	−	0.869459i	$-0.664468\pi$
$241$	−15.3380	−0.988010	−0.494005	+	0.869459i	$0.664468\pi$
$242$	0	0
$243$	2.01222	0.129084
$244$	0	0
$245$	3.28726	0.210015
$246$	0	0
$247$	0	0
$248$	0	0
$249$	−1.61213	−0.102164
$250$	0	0
$251$	12.3733	0.780995	0.390497	−	0.920604i	$-0.372303\pi$
$251$	12.3733	0.780995	0.390497	+	0.920604i	$0.372303\pi$
$252$	0	0
$253$	2.15633	0.135567
$254$	0	0
$255$	−3.51056	−0.219840
$256$	0	0
$257$	10.6253	0.662788	0.331394	−	0.943492i	$-0.392481\pi$
$257$	10.6253	0.662788	0.331394	+	0.943492i	$0.392481\pi$
$258$	0	0
$259$	10.9624	0.681170
$260$	0	0
$261$	−0.619421	−0.0383412
$262$	0	0
$263$	7.61801	0.469746	0.234873	−	0.972026i	$-0.424533\pi$
$263$	7.61801	0.469746	0.234873	+	0.972026i	$0.424533\pi$
$264$	0	0
$265$	−17.1735	−1.05496
$266$	0	0
$267$	24.0811	1.47374
$268$	0	0
$269$	1.22425	0.0746441	0.0373220	−	0.999303i	$-0.488117\pi$
$269$	1.22425	0.0746441	0.0373220	+	0.999303i	$0.488117\pi$
$270$	0	0
$271$	4.96476	0.301588	0.150794	−	0.988565i	$-0.451817\pi$
$271$	4.96476	0.301588	0.150794	+	0.988565i	$0.451817\pi$
$272$	0	0
$273$	4.96239	0.300337
$274$	0	0
$275$	−12.5198	−0.754970
$276$	0	0
$277$	12.1319	0.728934	0.364467	−	0.931216i	$-0.381251\pi$
$277$	12.1319	0.728934	0.364467	+	0.931216i	$0.381251\pi$
$278$	0	0
$279$	−1.08603	−0.0650187
$280$	0	0
$281$	−14.7513	−0.879989	−0.439995	−	0.898000i	$-0.645020\pi$
$281$	−14.7513	−0.879989	−0.439995	+	0.898000i	$0.645020\pi$
$282$	0	0
$283$	28.9380	1.72018	0.860091	−	0.510140i	$-0.170406\pi$
$283$	28.9380	1.72018	0.860091	+	0.510140i	$0.170406\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	1.22425	0.0722654
$288$	0	0
$289$	−16.5936	−0.976092
$290$	0	0
$291$	23.6424	1.38594
$292$	0	0
$293$	5.15140	0.300948	0.150474	−	0.988614i	$-0.451920\pi$
$293$	5.15140	0.300948	0.150474	+	0.988614i	$0.451920\pi$
$294$	0	0
$295$	−6.53435	−0.380445
$296$	0	0
$297$	11.5369	0.669439
$298$	0	0
$299$	−2.96239	−0.171319
$300$	0	0
$301$	5.73813	0.330741
$302$	0	0
$303$	−0.962389	−0.0552878
$304$	0	0
$305$	19.0698	1.09194
$306$	0	0
$307$	14.5867	0.832509	0.416254	−	0.909248i	$-0.363343\pi$
$307$	14.5867	0.832509	0.416254	+	0.909248i	$0.363343\pi$
$308$	0	0
$309$	−9.92478	−0.564601
$310$	0	0
$311$	15.1368	0.858330	0.429165	−	0.903226i	$-0.358808\pi$
$311$	15.1368	0.858330	0.429165	+	0.903226i	$0.358808\pi$
$312$	0	0
$313$	11.2727	0.637169	0.318584	−	0.947894i	$-0.396793\pi$
$313$	11.2727	0.637169	0.318584	+	0.947894i	$0.396793\pi$
$314$	0	0
$315$	−0.637519	−0.0359202
$316$	0	0
$317$	−11.4460	−0.642869	−0.321434	−	0.946932i	$-0.604165\pi$
$317$	−11.4460	−0.642869	−0.321434	+	0.946932i	$0.604165\pi$
$318$	0	0
$319$	−6.88717	−0.385607
$320$	0	0
$321$	−20.8364	−1.16297
$322$	0	0
$323$	0	0
$324$	0	0
$325$	17.1998	0.954074
$326$	0	0
$327$	−30.1114	−1.66517
$328$	0	0
$329$	4.63752	0.255675
$330$	0	0
$331$	11.8740	0.652654	0.326327	−	0.945257i	$-0.394189\pi$
$331$	11.8740	0.652654	0.326327	+	0.945257i	$0.394189\pi$
$332$	0	0
$333$	−2.12601	−0.116505
$334$	0	0
$335$	−26.5647	−1.45138
$336$	0	0
$337$	22.1260	1.20528	0.602640	−	0.798013i	$-0.294115\pi$
$337$	22.1260	1.20528	0.602640	+	0.798013i	$0.294115\pi$
$338$	0	0
$339$	−14.5745	−0.791579
$340$	0	0
$341$	−12.0752	−0.653910
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	−5.50659	−0.296465
$346$	0	0
$347$	−29.3357	−1.57482	−0.787411	−	0.616429i	$-0.788579\pi$
$347$	−29.3357	−1.57482	−0.787411	+	0.616429i	$0.788579\pi$
$348$	0	0
$349$	−30.1622	−1.61455	−0.807273	−	0.590178i	$-0.799057\pi$
$349$	−30.1622	−1.61455	−0.807273	+	0.590178i	$0.799057\pi$
$350$	0	0
$351$	−15.8496	−0.845987
$352$	0	0
$353$	32.3634	1.72253	0.861266	−	0.508154i	$-0.169672\pi$
$353$	32.3634	1.72253	0.861266	+	0.508154i	$0.169672\pi$
$354$	0	0
$355$	−22.0263	−1.16904
$356$	0	0
$357$	−1.06793	−0.0565208
$358$	0	0
$359$	1.84367	0.0973054	0.0486527	−	0.998816i	$-0.484507\pi$
$359$	1.84367	0.0973054	0.0486527	+	0.998816i	$0.484507\pi$
$360$	0	0
$361$	−19.0000	−1.00000
$362$	0	0
$363$	−10.6375	−0.558325
$364$	0	0
$365$	39.6140	2.07349
$366$	0	0
$367$	−23.0376	−1.20255	−0.601277	−	0.799041i	$-0.705341\pi$
$367$	−23.0376	−1.20255	−0.601277	+	0.799041i	$0.705341\pi$
$368$	0	0
$369$	−0.237428	−0.0123600
$370$	0	0
$371$	−5.22425	−0.271230
$372$	0	0
$373$	−11.7743	−0.609652	−0.304826	−	0.952408i	$-0.598598\pi$
$373$	−11.7743	−0.609652	−0.304826	+	0.952408i	$0.598598\pi$
$374$	0	0
$375$	4.43866	0.229211
$376$	0	0
$377$	9.46168	0.487301
$378$	0	0
$379$	37.0191	1.90154	0.950771	−	0.309896i	$-0.100294\pi$
$379$	37.0191	1.90154	0.950771	+	0.309896i	$0.100294\pi$
$380$	0	0
$381$	15.0132	0.769148
$382$	0	0
$383$	22.9887	1.17467	0.587335	−	0.809344i	$-0.300177\pi$
$383$	22.9887	1.17467	0.587335	+	0.809344i	$0.300177\pi$
$384$	0	0
$385$	−7.08840	−0.361258
$386$	0	0
$387$	−1.11283	−0.0565685
$388$	0	0
$389$	3.17347	0.160901	0.0804506	−	0.996759i	$-0.474364\pi$
$389$	3.17347	0.160901	0.0804506	+	0.996759i	$0.474364\pi$
$390$	0	0
$391$	0.637519	0.0322407
$392$	0	0
$393$	−14.1563	−0.714092
$394$	0	0
$395$	−25.5369	−1.28490
$396$	0	0
$397$	30.2882	1.52012	0.760061	−	0.649852i	$-0.225169\pi$
$397$	30.2882	1.52012	0.760061	+	0.649852i	$0.225169\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	0.201231	0.0100490	0.00502449	−	0.999987i	$-0.498401\pi$
$401$	0.201231	0.0100490	0.00502449	+	0.999987i	$0.498401\pi$
$402$	0	0
$403$	16.5891	0.826362
$404$	0	0
$405$	−27.5491	−1.36893
$406$	0	0
$407$	−23.6385	−1.17172
$408$	0	0
$409$	−7.73813	−0.382626	−0.191313	−	0.981529i	$-0.561275\pi$
$409$	−7.73813	−0.382626	−0.191313	+	0.981529i	$0.561275\pi$
$410$	0	0
$411$	−21.5877	−1.06484
$412$	0	0
$413$	−1.98778	−0.0978123
$414$	0	0
$415$	−3.16362	−0.155296
$416$	0	0
$417$	−16.7308	−0.819313
$418$	0	0
$419$	−24.2228	−1.18336	−0.591682	−	0.806172i	$-0.701536\pi$
$419$	−24.2228	−1.18336	−0.591682	+	0.806172i	$0.701536\pi$
$420$	0	0
$421$	−34.9986	−1.70573	−0.852863	−	0.522134i	$-0.825136\pi$
$421$	−34.9986	−1.70573	−0.852863	+	0.522134i	$0.825136\pi$
$422$	0	0
$423$	−0.899385	−0.0437296
$424$	0	0
$425$	−3.70148	−0.179548
$426$	0	0
$427$	5.80114	0.280737
$428$	0	0
$429$	−10.7005	−0.516626
$430$	0	0
$431$	−18.5501	−0.893526	−0.446763	−	0.894652i	$-0.647423\pi$
$431$	−18.5501	−0.893526	−0.446763	+	0.894652i	$0.647423\pi$
$432$	0	0
$433$	−33.5247	−1.61109	−0.805547	−	0.592532i	$-0.798128\pi$
$433$	−33.5247	−1.61109	−0.805547	+	0.592532i	$0.798128\pi$
$434$	0	0
$435$	17.5877	0.843265
$436$	0	0
$437$	0	0
$438$	0	0
$439$	−22.6982	−1.08332	−0.541662	−	0.840596i	$-0.682205\pi$
$439$	−22.6982	−1.08332	−0.541662	+	0.840596i	$0.682205\pi$
$440$	0	0
$441$	−0.193937	−0.00923507
$442$	0	0
$443$	9.46168	0.449538	0.224769	−	0.974412i	$-0.427837\pi$
$443$	9.46168	0.449538	0.224769	+	0.974412i	$0.427837\pi$
$444$	0	0
$445$	47.2565	2.24017
$446$	0	0
$447$	−2.47295	−0.116966
$448$	0	0
$449$	−21.8945	−1.03326	−0.516632	−	0.856208i	$-0.672814\pi$
$449$	−21.8945	−1.03326	−0.516632	+	0.856208i	$0.672814\pi$
$450$	0	0
$451$	−2.63989	−0.124308
$452$	0	0
$453$	−16.9380	−0.795814
$454$	0	0
$455$	9.73813	0.456531
$456$	0	0
$457$	1.93937	0.0907197	0.0453598	−	0.998971i	$-0.485557\pi$
$457$	1.93937	0.0907197	0.0453598	+	0.998971i	$0.485557\pi$
$458$	0	0
$459$	3.41090	0.159207
$460$	0	0
$461$	42.4749	1.97825	0.989126	−	0.147073i	$-0.0469853\pi$
$461$	42.4749	1.97825	0.989126	+	0.147073i	$0.0469853\pi$
$462$	0	0
$463$	7.50071	0.348587	0.174294	−	0.984694i	$-0.444236\pi$
$463$	7.50071	0.348587	0.174294	+	0.984694i	$0.444236\pi$
$464$	0	0
$465$	30.8364	1.43000
$466$	0	0
$467$	18.5501	0.858395	0.429198	−	0.903211i	$-0.358796\pi$
$467$	18.5501	0.858395	0.429198	+	0.903211i	$0.358796\pi$
$468$	0	0
$469$	−8.08110	−0.373151
$470$	0	0
$471$	−6.15633	−0.283668
$472$	0	0
$473$	−12.3733	−0.568924
$474$	0	0
$475$	0	0
$476$	0	0
$477$	1.01317	0.0463900
$478$	0	0
$479$	−27.9102	−1.27525	−0.637625	−	0.770347i	$-0.720083\pi$
$479$	−27.9102	−1.27525	−0.637625	+	0.770347i	$0.720083\pi$
$480$	0	0
$481$	32.4749	1.48073
$482$	0	0
$483$	−1.67513	−0.0762211
$484$	0	0
$485$	46.3957	2.10672
$486$	0	0
$487$	−20.2520	−0.917706	−0.458853	−	0.888512i	$-0.651740\pi$
$487$	−20.2520	−0.917706	−0.458853	+	0.888512i	$0.651740\pi$
$488$	0	0
$489$	−33.7743	−1.52733
$490$	0	0
$491$	−15.0738	−0.680271	−0.340136	−	0.940376i	$-0.610473\pi$
$491$	−15.0738	−0.680271	−0.340136	+	0.940376i	$0.610473\pi$
$492$	0	0
$493$	−2.03620	−0.0917057
$494$	0	0
$495$	1.37470	0.0617881
$496$	0	0
$497$	−6.70052	−0.300560
$498$	0	0
$499$	−4.90175	−0.219433	−0.109716	−	0.993963i	$-0.534994\pi$
$499$	−4.90175	−0.219433	−0.109716	+	0.993963i	$0.534994\pi$
$500$	0	0
$501$	−24.2922	−1.08529
$502$	0	0
$503$	−4.98683	−0.222352	−0.111176	−	0.993801i	$-0.535462\pi$
$503$	−4.98683	−0.222352	−0.111176	+	0.993801i	$0.535462\pi$
$504$	0	0
$505$	−1.88858	−0.0840407
$506$	0	0
$507$	−7.07618	−0.314264
$508$	0	0
$509$	−11.6121	−0.514698	−0.257349	−	0.966318i	$-0.582849\pi$
$509$	−11.6121	−0.514698	−0.257349	+	0.966318i	$0.582849\pi$
$510$	0	0
$511$	12.0508	0.533095
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−19.4763	−0.858227
$516$	0	0
$517$	−10.0000	−0.439799
$518$	0	0
$519$	13.5271	0.593772
$520$	0	0
$521$	18.6375	0.816525	0.408262	−	0.912865i	$-0.366135\pi$
$521$	18.6375	0.816525	0.408262	+	0.912865i	$0.366135\pi$
$522$	0	0
$523$	39.6385	1.73327	0.866635	−	0.498943i	$-0.166278\pi$
$523$	39.6385	1.73327	0.866635	+	0.498943i	$0.166278\pi$
$524$	0	0
$525$	9.72592	0.424474
$526$	0	0
$527$	−3.57005	−0.155514
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	0.385503	0.0167294
$532$	0	0
$533$	3.62672	0.157090
$534$	0	0
$535$	−40.8891	−1.76779
$536$	0	0
$537$	19.5369	0.843080
$538$	0	0
$539$	−2.15633	−0.0928795
$540$	0	0
$541$	23.3561	1.00416	0.502079	−	0.864821i	$-0.332568\pi$
$541$	23.3561	1.00416	0.502079	+	0.864821i	$0.332568\pi$
$542$	0	0
$543$	−17.0679	−0.732455
$544$	0	0
$545$	−59.0903	−2.53115
$546$	0	0
$547$	20.9624	0.896287	0.448144	−	0.893962i	$-0.352085\pi$
$547$	20.9624	0.896287	0.448144	+	0.893962i	$0.352085\pi$
$548$	0	0
$549$	−1.12505	−0.0480161
$550$	0	0
$551$	0	0
$552$	0	0
$553$	−7.76845	−0.330348
$554$	0	0
$555$	60.3653	2.56237
$556$	0	0
$557$	1.62672	0.0689261	0.0344631	−	0.999406i	$-0.489028\pi$
$557$	1.62672	0.0689261	0.0344631	+	0.999406i	$0.489028\pi$
$558$	0	0
$559$	16.9986	0.718964
$560$	0	0
$561$	2.30280	0.0972243
$562$	0	0
$563$	−18.9116	−0.797029	−0.398515	−	0.917162i	$-0.630474\pi$
$563$	−18.9116	−0.797029	−0.398515	+	0.917162i	$0.630474\pi$
$564$	0	0
$565$	−28.6009	−1.20325
$566$	0	0
$567$	−8.38058	−0.351951
$568$	0	0
$569$	−0.111420	−0.00467095	−0.00233548	−	0.999997i	$-0.500743\pi$
$569$	−0.111420	−0.00467095	−0.00233548	+	0.999997i	$0.500743\pi$
$570$	0	0
$571$	37.0640	1.55108	0.775539	−	0.631299i	$-0.217478\pi$
$571$	37.0640	1.55108	0.775539	+	0.631299i	$0.217478\pi$
$572$	0	0
$573$	24.9380	1.04180
$574$	0	0
$575$	−5.80606	−0.242130
$576$	0	0
$577$	−11.8594	−0.493713	−0.246857	−	0.969052i	$-0.579398\pi$
$577$	−11.8594	−0.493713	−0.246857	+	0.969052i	$0.579398\pi$
$578$	0	0
$579$	−5.48612	−0.227995
$580$	0	0
$581$	−0.962389	−0.0399266
$582$	0	0
$583$	11.2652	0.466556
$584$	0	0
$585$	−1.88858	−0.0780832
$586$	0	0
$587$	−27.2873	−1.12627	−0.563133	−	0.826366i	$-0.690404\pi$
$587$	−27.2873	−1.12627	−0.563133	+	0.826366i	$0.690404\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	−31.1998	−1.28339
$592$	0	0
$593$	3.08840	0.126825	0.0634126	−	0.997987i	$-0.479802\pi$
$593$	3.08840	0.126825	0.0634126	+	0.997987i	$0.479802\pi$
$594$	0	0
$595$	−2.09569	−0.0859149
$596$	0	0
$597$	3.22425	0.131960
$598$	0	0
$599$	−8.58910	−0.350941	−0.175471	−	0.984485i	$-0.556145\pi$
$599$	−8.58910	−0.350941	−0.175471	+	0.984485i	$0.556145\pi$
$600$	0	0
$601$	−17.8134	−0.726622	−0.363311	−	0.931668i	$-0.618354\pi$
$601$	−17.8134	−0.726622	−0.363311	+	0.931668i	$0.618354\pi$
$602$	0	0
$603$	1.56722	0.0638222
$604$	0	0
$605$	−20.8749	−0.848687
$606$	0	0
$607$	16.4871	0.669190	0.334595	−	0.942362i	$-0.391401\pi$
$607$	16.4871	0.669190	0.334595	+	0.942362i	$0.391401\pi$
$608$	0	0
$609$	5.35026	0.216804
$610$	0	0
$611$	13.7381	0.555785
$612$	0	0
$613$	−27.7499	−1.12081	−0.560404	−	0.828220i	$-0.689354\pi$
$613$	−27.7499	−1.12081	−0.560404	+	0.828220i	$0.689354\pi$
$614$	0	0
$615$	6.74146	0.271842
$616$	0	0
$617$	15.5877	0.627537	0.313768	−	0.949500i	$-0.398408\pi$
$617$	15.5877	0.627537	0.313768	+	0.949500i	$0.398408\pi$
$618$	0	0
$619$	3.41090	0.137095	0.0685477	−	0.997648i	$-0.478163\pi$
$619$	3.41090	0.137095	0.0685477	+	0.997648i	$0.478163\pi$
$620$	0	0
$621$	5.35026	0.214699
$622$	0	0
$623$	14.3757	0.575948
$624$	0	0
$625$	−20.3199	−0.812798
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−6.98874	−0.278659
$630$	0	0
$631$	−36.6458	−1.45885	−0.729423	−	0.684063i	$-0.760211\pi$
$631$	−36.6458	−1.45885	−0.729423	+	0.684063i	$0.760211\pi$
$632$	0	0
$633$	45.0395	1.79016
$634$	0	0
$635$	29.4617	1.16915
$636$	0	0
$637$	2.96239	0.117374
$638$	0	0
$639$	1.29948	0.0514065
$640$	0	0
$641$	−48.3146	−1.90831	−0.954155	−	0.299312i	$-0.903243\pi$
$641$	−48.3146	−1.90831	−0.954155	+	0.299312i	$0.903243\pi$
$642$	0	0
$643$	−29.1900	−1.15114	−0.575570	−	0.817753i	$-0.695220\pi$
$643$	−29.1900	−1.15114	−0.575570	+	0.817753i	$0.695220\pi$
$644$	0	0
$645$	31.5975	1.24415
$646$	0	0
$647$	−34.7127	−1.36470	−0.682349	−	0.731026i	$-0.739042\pi$
$647$	−34.7127	−1.36470	−0.682349	+	0.731026i	$0.739042\pi$
$648$	0	0
$649$	4.28630	0.168252
$650$	0	0
$651$	9.38058	0.367654
$652$	0	0
$653$	18.2228	0.713115	0.356557	−	0.934273i	$-0.383950\pi$
$653$	18.2228	0.713115	0.356557	+	0.934273i	$0.383950\pi$
$654$	0	0
$655$	−27.7802	−1.08546
$656$	0	0
$657$	−2.33709	−0.0911785
$658$	0	0
$659$	−4.00000	−0.155818	−0.0779089	−	0.996960i	$-0.524824\pi$
$659$	−4.00000	−0.155818	−0.0779089	+	0.996960i	$0.524824\pi$
$660$	0	0
$661$	−21.9878	−0.855226	−0.427613	−	0.903962i	$-0.640645\pi$
$661$	−21.9878	−0.855226	−0.427613	+	0.903962i	$0.640645\pi$
$662$	0	0
$663$	−3.16362	−0.122865
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−3.19394	−0.123670
$668$	0	0
$669$	3.32979	0.128737
$670$	0	0
$671$	−12.5091	−0.482910
$672$	0	0
$673$	−2.47882	−0.0955517	−0.0477758	−	0.998858i	$-0.515213\pi$
$673$	−2.47882	−0.0955517	−0.0477758	+	0.998858i	$0.515213\pi$
$674$	0	0
$675$	−31.0640	−1.19565
$676$	0	0
$677$	−4.37565	−0.168170	−0.0840850	−	0.996459i	$-0.526797\pi$
$677$	−4.37565	−0.168170	−0.0840850	+	0.996459i	$0.526797\pi$
$678$	0	0
$679$	14.1138	0.541638
$680$	0	0
$681$	−6.17679	−0.236695
$682$	0	0
$683$	44.7875	1.71375	0.856873	−	0.515527i	$-0.172404\pi$
$683$	44.7875	1.71375	0.856873	+	0.515527i	$0.172404\pi$
$684$	0	0
$685$	−42.3634	−1.61862
$686$	0	0
$687$	−25.7948	−0.984133
$688$	0	0
$689$	−15.4763	−0.589599
$690$	0	0
$691$	8.76353	0.333380	0.166690	−	0.986009i	$-0.446692\pi$
$691$	8.76353	0.333380	0.166690	+	0.986009i	$0.446692\pi$
$692$	0	0
$693$	0.418190	0.0158857
$694$	0	0
$695$	−32.8324	−1.24540
$696$	0	0
$697$	−0.780486	−0.0295630
$698$	0	0
$699$	−11.2750	−0.426461
$700$	0	0
$701$	−26.9135	−1.01651	−0.508255	−	0.861207i	$-0.669709\pi$
$701$	−26.9135	−1.01651	−0.508255	+	0.861207i	$0.669709\pi$
$702$	0	0
$703$	0	0
$704$	0	0
$705$	25.5369	0.961775
$706$	0	0
$707$	−0.574515	−0.0216069
$708$	0	0
$709$	29.3503	1.10227	0.551136	−	0.834415i	$-0.314195\pi$
$709$	29.3503	1.10227	0.551136	+	0.834415i	$0.314195\pi$
$710$	0	0
$711$	1.50659	0.0565014
$712$	0	0
$713$	−5.59991	−0.209718
$714$	0	0
$715$	−20.9986	−0.785303
$716$	0	0
$717$	−24.9380	−0.931325
$718$	0	0
$719$	−6.81431	−0.254131	−0.127065	−	0.991894i	$-0.540556\pi$
$719$	−6.81431	−0.254131	−0.127065	+	0.991894i	$0.540556\pi$
$720$	0	0
$721$	−5.92478	−0.220650
$722$	0	0
$723$	−25.6932	−0.955541
$724$	0	0
$725$	18.5442	0.688714
$726$	0	0
$727$	−14.5501	−0.539633	−0.269816	−	0.962912i	$-0.586963\pi$
$727$	−14.5501	−0.539633	−0.269816	+	0.962912i	$0.586963\pi$
$728$	0	0
$729$	28.5125	1.05602
$730$	0	0
$731$	−3.65817	−0.135302
$732$	0	0
$733$	30.4626	1.12516	0.562582	−	0.826742i	$-0.309808\pi$
$733$	30.4626	1.12516	0.562582	+	0.826742i	$0.309808\pi$
$734$	0	0
$735$	5.50659	0.203114
$736$	0	0
$737$	17.4255	0.641876
$738$	0	0
$739$	27.4471	1.00966	0.504829	−	0.863219i	$-0.331556\pi$
$739$	27.4471	1.00966	0.504829	+	0.863219i	$0.331556\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−22.0665	−0.809542	−0.404771	−	0.914418i	$-0.632649\pi$
$743$	−22.0665	−0.809542	−0.404771	+	0.914418i	$0.632649\pi$
$744$	0	0
$745$	−4.85288	−0.177796
$746$	0	0
$747$	0.186642	0.00682889
$748$	0	0
$749$	−12.4387	−0.454499
$750$	0	0
$751$	−18.6312	−0.679861	−0.339931	−	0.940450i	$-0.610404\pi$
$751$	−18.6312	−0.679861	−0.339931	+	0.940450i	$0.610404\pi$
$752$	0	0
$753$	20.7269	0.755329
$754$	0	0
$755$	−33.2388	−1.20969
$756$	0	0
$757$	50.5402	1.83692	0.918458	−	0.395519i	$-0.129435\pi$
$757$	50.5402	1.83692	0.918458	+	0.395519i	$0.129435\pi$
$758$	0	0
$759$	3.61213	0.131112
$760$	0	0
$761$	5.62198	0.203796	0.101898	−	0.994795i	$-0.467508\pi$
$761$	5.62198	0.203796	0.101898	+	0.994795i	$0.467508\pi$
$762$	0	0
$763$	−17.9756	−0.650759
$764$	0	0
$765$	0.406431	0.0146945
$766$	0	0
$767$	−5.88858	−0.212624
$768$	0	0
$769$	−3.12696	−0.112761	−0.0563806	−	0.998409i	$-0.517956\pi$
$769$	−3.12696	−0.112761	−0.0563806	+	0.998409i	$0.517956\pi$
$770$	0	0
$771$	17.7988	0.641007
$772$	0	0
$773$	−40.2106	−1.44628	−0.723138	−	0.690704i	$-0.757301\pi$
$773$	−40.2106	−1.44628	−0.723138	+	0.690704i	$0.757301\pi$
$774$	0	0
$775$	32.5134	1.16792
$776$	0	0
$777$	18.3634	0.658785
$778$	0	0
$779$	0	0
$780$	0	0
$781$	14.4485	0.517008
$782$	0	0
$783$	−17.0884	−0.610689
$784$	0	0
$785$	−12.0811	−0.431193
$786$	0	0
$787$	46.0019	1.63979	0.819896	−	0.572513i	$-0.194031\pi$
$787$	46.0019	1.63979	0.819896	+	0.572513i	$0.194031\pi$
$788$	0	0
$789$	12.7612	0.454309
$790$	0	0
$791$	−8.70052	−0.309355
$792$	0	0
$793$	17.1852	0.610265
$794$	0	0
$795$	−28.7678	−1.02029
$796$	0	0
$797$	55.0127	1.94865	0.974325	−	0.225145i	$-0.0722857\pi$
$797$	55.0127	1.94865	0.974325	+	0.225145i	$0.0722857\pi$
$798$	0	0
$799$	−2.95651	−0.104594
$800$	0	0
$801$	−2.78797	−0.0985079
$802$	0	0
$803$	−25.9854	−0.917005
$804$	0	0
$805$	−3.28726	−0.115861
$806$	0	0
$807$	2.05079	0.0721911
$808$	0	0
$809$	−15.4255	−0.542331	−0.271166	−	0.962533i	$-0.587409\pi$
$809$	−15.4255	−0.542331	−0.271166	+	0.962533i	$0.587409\pi$
$810$	0	0
$811$	25.1006	0.881402	0.440701	−	0.897654i	$-0.354730\pi$
$811$	25.1006	0.881402	0.440701	+	0.897654i	$0.354730\pi$
$812$	0	0
$813$	8.31662	0.291677
$814$	0	0
$815$	−66.2784	−2.32163
$816$	0	0
$817$	0	0
$818$	0	0
$819$	−0.574515	−0.0200752
$820$	0	0
$821$	10.8930	0.380170	0.190085	−	0.981768i	$-0.439124\pi$
$821$	10.8930	0.380170	0.190085	+	0.981768i	$0.439124\pi$
$822$	0	0
$823$	49.1608	1.71364	0.856819	−	0.515618i	$-0.172438\pi$
$823$	49.1608	1.71364	0.856819	+	0.515618i	$0.172438\pi$
$824$	0	0
$825$	−20.9722	−0.730160
$826$	0	0
$827$	−49.7255	−1.72912	−0.864562	−	0.502527i	$-0.832404\pi$
$827$	−49.7255	−1.72912	−0.864562	+	0.502527i	$0.832404\pi$
$828$	0	0
$829$	14.8364	0.515289	0.257644	−	0.966240i	$-0.417054\pi$
$829$	14.8364	0.515289	0.257644	+	0.966240i	$0.417054\pi$
$830$	0	0
$831$	20.3225	0.704980
$832$	0	0
$833$	−0.637519	−0.0220887
$834$	0	0
$835$	−47.6707	−1.64971
$836$	0	0
$837$	−29.9610	−1.03560
$838$	0	0
$839$	19.4109	0.670139	0.335069	−	0.942193i	$-0.391240\pi$
$839$	19.4109	0.670139	0.335069	+	0.942193i	$0.391240\pi$
$840$	0	0
$841$	−18.7988	−0.648233
$842$	0	0
$843$	−24.7104	−0.851070
$844$	0	0
$845$	−13.8862	−0.477700
$846$	0	0
$847$	−6.35026	−0.218198
$848$	0	0
$849$	48.4749	1.66365
$850$	0	0
$851$	−10.9624	−0.375786
$852$	0	0
$853$	−39.5877	−1.35546	−0.677728	−	0.735312i	$-0.737035\pi$
$853$	−39.5877	−1.35546	−0.677728	+	0.735312i	$0.737035\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−36.9234	−1.26128	−0.630639	−	0.776076i	$-0.717207\pi$
$857$	−36.9234	−1.26128	−0.630639	+	0.776076i	$0.717207\pi$
$858$	0	0
$859$	1.50025	0.0511878	0.0255939	−	0.999672i	$-0.491852\pi$
$859$	1.50025	0.0511878	0.0255939	+	0.999672i	$0.491852\pi$
$860$	0	0
$861$	2.05079	0.0698906
$862$	0	0
$863$	56.9789	1.93958	0.969792	−	0.243934i	$-0.0784379\pi$
$863$	56.9789	1.93958	0.969792	+	0.243934i	$0.0784379\pi$
$864$	0	0
$865$	26.5453	0.902569
$866$	0	0
$867$	−27.7964	−0.944015
$868$	0	0
$869$	16.7513	0.568249
$870$	0	0
$871$	−23.9394	−0.811154
$872$	0	0
$873$	−2.73718	−0.0926395
$874$	0	0
$875$	2.64974	0.0895775
$876$	0	0
$877$	21.4920	0.725733	0.362867	−	0.931841i	$-0.381798\pi$
$877$	21.4920	0.725733	0.362867	+	0.931841i	$0.381798\pi$
$878$	0	0
$879$	8.62927	0.291058
$880$	0	0
$881$	−28.3004	−0.953466	−0.476733	−	0.879048i	$-0.658179\pi$
$881$	−28.3004	−0.953466	−0.476733	+	0.879048i	$0.658179\pi$
$882$	0	0
$883$	15.6629	0.527099	0.263549	−	0.964646i	$-0.415107\pi$
$883$	15.6629	0.527099	0.263549	+	0.964646i	$0.415107\pi$
$884$	0	0
$885$	−10.9459	−0.367942
$886$	0	0
$887$	21.5247	0.722728	0.361364	−	0.932425i	$-0.382311\pi$
$887$	21.5247	0.722728	0.361364	+	0.932425i	$0.382311\pi$
$888$	0	0
$889$	8.96239	0.300589
$890$	0	0
$891$	18.0713	0.605410
$892$	0	0
$893$	0	0
$894$	0	0
$895$	38.3390	1.28153
$896$	0	0
$897$	−4.96239	−0.165689
$898$	0	0
$899$	17.8858	0.596523
$900$	0	0
$901$	3.33056	0.110957
$902$	0	0
$903$	9.61213	0.319872
$904$	0	0
$905$	−33.4939	−1.11338
$906$	0	0
$907$	−39.2243	−1.30242	−0.651210	−	0.758898i	$-0.725738\pi$
$907$	−39.2243	−1.30242	−0.651210	+	0.758898i	$0.725738\pi$
$908$	0	0
$909$	0.111420	0.00369555
$910$	0	0
$911$	42.9029	1.42144	0.710718	−	0.703477i	$-0.248370\pi$
$911$	42.9029	1.42144	0.710718	+	0.703477i	$0.248370\pi$
$912$	0	0
$913$	2.07522	0.0686799
$914$	0	0
$915$	31.9445	1.05605
$916$	0	0
$917$	−8.45088	−0.279073
$918$	0	0
$919$	17.4215	0.574683	0.287341	−	0.957828i	$-0.407229\pi$
$919$	17.4215	0.574683	0.287341	+	0.957828i	$0.407229\pi$
$920$	0	0
$921$	24.4347	0.805150
$922$	0	0
$923$	−19.8496	−0.653356
$924$	0	0
$925$	63.6483	2.09274
$926$	0	0
$927$	1.14903	0.0377391
$928$	0	0
$929$	50.1740	1.64615	0.823077	−	0.567930i	$-0.192255\pi$
$929$	50.1740	1.64615	0.823077	+	0.567930i	$0.192255\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	25.3561	0.830123
$934$	0	0
$935$	4.51899	0.147787
$936$	0	0
$937$	−1.18569	−0.0387347	−0.0193674	−	0.999812i	$-0.506165\pi$
$937$	−1.18569	−0.0387347	−0.0193674	+	0.999812i	$0.506165\pi$
$938$	0	0
$939$	18.8832	0.616230
$940$	0	0
$941$	−26.9962	−0.880051	−0.440026	−	0.897985i	$-0.645031\pi$
$941$	−26.9962	−0.880051	−0.440026	+	0.897985i	$0.645031\pi$
$942$	0	0
$943$	−1.22425	−0.0398672
$944$	0	0
$945$	−17.5877	−0.572128
$946$	0	0
$947$	−41.7499	−1.35669	−0.678345	−	0.734744i	$-0.737302\pi$
$947$	−41.7499	−1.35669	−0.678345	+	0.734744i	$0.737302\pi$
$948$	0	0
$949$	35.6991	1.15884
$950$	0	0
$951$	−19.1735	−0.621742
$952$	0	0
$953$	−15.6121	−0.505726	−0.252863	−	0.967502i	$-0.581372\pi$
$953$	−15.6121	−0.505726	−0.252863	+	0.967502i	$0.581372\pi$
$954$	0	0
$955$	48.9380	1.58359
$956$	0	0
$957$	−11.5369	−0.372935
$958$	0	0
$959$	−12.8872	−0.416148
$960$	0	0
$961$	0.358971	0.0115797
$962$	0	0
$963$	2.41231	0.0777356
$964$	0	0
$965$	−10.7659	−0.346566
$966$	0	0
$967$	34.6107	1.11301	0.556503	−	0.830846i	$-0.312143\pi$
$967$	34.6107	1.11301	0.556503	+	0.830846i	$0.312143\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	25.7988	0.827922	0.413961	−	0.910295i	$-0.364145\pi$
$971$	25.7988	0.827922	0.413961	+	0.910295i	$0.364145\pi$
$972$	0	0
$973$	−9.98778	−0.320194
$974$	0	0
$975$	28.8119	0.922721
$976$	0	0
$977$	13.2487	0.423863	0.211932	−	0.977285i	$-0.432025\pi$
$977$	13.2487	0.423863	0.211932	+	0.977285i	$0.432025\pi$
$978$	0	0
$979$	−30.9986	−0.990719
$980$	0	0
$981$	3.48612	0.111303
$982$	0	0
$983$	1.01126	0.0322543	0.0161272	−	0.999870i	$-0.494866\pi$
$983$	1.01126	0.0322543	0.0161272	+	0.999870i	$0.494866\pi$
$984$	0	0
$985$	−61.2262	−1.95083
$986$	0	0
$987$	7.76845	0.247273
$988$	0	0
$989$	−5.73813	−0.182462
$990$	0	0
$991$	27.5778	0.876039	0.438019	−	0.898965i	$-0.355680\pi$
$991$	27.5778	0.876039	0.438019	+	0.898965i	$0.355680\pi$
$992$	0	0
$993$	19.8905	0.631206
$994$	0	0
$995$	6.32724	0.200587
$996$	0	0
$997$	−3.40105	−0.107712	−0.0538561	−	0.998549i	$-0.517151\pi$
$997$	−3.40105	−0.107712	−0.0538561	+	0.998549i	$0.517151\pi$
$998$	0	0
$999$	−58.6516	−1.85566

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2576.2.a.y.1.3		3
4.3	odd	2		1288.2.a.l.1.1	✓	3
28.27	even	2		9016.2.a.z.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1288.2.a.l.1.1	✓	3	4.3	odd	2
2576.2.a.y.1.3		3	1.1	even	1	trivial
9016.2.a.z.1.3		3	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)

\(f(q)\)	\(=\)	\(q+1.67513 q^{3} +3.28726 q^{5} +1.00000 q^{7} -0.193937 q^{9} +O(q^{10})\)	\(q+1.67513 q^{3} +3.28726 q^{5} +1.00000 q^{7} -0.193937 q^{9} -2.15633 q^{11} +2.96239 q^{13} +5.50659 q^{15} -0.637519 q^{17} +1.67513 q^{21} -1.00000 q^{23} +5.80606 q^{25} -5.35026 q^{27} +3.19394 q^{29} +5.59991 q^{31} -3.61213 q^{33} +3.28726 q^{35} +10.9624 q^{37} +4.96239 q^{39} +1.22425 q^{41} +5.73813 q^{43} -0.637519 q^{45} +4.63752 q^{47} +1.00000 q^{49} -1.06793 q^{51} -5.22425 q^{53} -7.08840 q^{55} -1.98778 q^{59} +5.80114 q^{61} -0.193937 q^{63} +9.73813 q^{65} -8.08110 q^{67} -1.67513 q^{69} -6.70052 q^{71} +12.0508 q^{73} +9.72592 q^{75} -2.15633 q^{77} -7.76845 q^{79} -8.38058 q^{81} -0.962389 q^{83} -2.09569 q^{85} +5.35026 q^{87} +14.3757 q^{89} +2.96239 q^{91} +9.38058 q^{93} +14.1138 q^{97} +0.418190 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 4 q^{5} + 3 q^{7} - q^{9}+O(q^{10}) \) 3 * q + 4 * q^5 + 3 * q^7 - q^9	\( 3 q + 4 q^{5} + 3 q^{7} - q^{9} + 4 q^{11} - 2 q^{13} - 4 q^{15} + 14 q^{17} - 3 q^{23} + 17 q^{25} - 6 q^{27} + 10 q^{29} - 10 q^{31} - 10 q^{33} + 4 q^{35} + 22 q^{37} + 4 q^{39} + 2 q^{41} + 8 q^{43} + 14 q^{45} - 2 q^{47} + 3 q^{49} - 12 q^{51} - 14 q^{53} - 2 q^{55} + 20 q^{59} + 4 q^{61} - q^{63} + 20 q^{65} + 8 q^{67} + 6 q^{73} - 6 q^{75} + 4 q^{77} - 12 q^{79} - 13 q^{81} + 8 q^{83} + 6 q^{87} + 18 q^{89} - 2 q^{91} + 16 q^{93} + 8 q^{97}+O(q^{100}) \) 3 * q + 4 * q^5 + 3 * q^7 - q^9 + 4 * q^11 - 2 * q^13 - 4 * q^15 + 14 * q^17 - 3 * q^23 + 17 * q^25 - 6 * q^27 + 10 * q^29 - 10 * q^31 - 10 * q^33 + 4 * q^35 + 22 * q^37 + 4 * q^39 + 2 * q^41 + 8 * q^43 + 14 * q^45 - 2 * q^47 + 3 * q^49 - 12 * q^51 - 14 * q^53 - 2 * q^55 + 20 * q^59 + 4 * q^61 - q^63 + 20 * q^65 + 8 * q^67 + 6 * q^73 - 6 * q^75 + 4 * q^77 - 12 * q^79 - 13 * q^81 + 8 * q^83 + 6 * q^87 + 18 * q^89 - 2 * q^91 + 16 * q^93 + 8 * q^97

Introduction

L-functions

Modular forms

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Database

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