LMFDB - Embedded newform 5776.2.a.s.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5776,2,Mod(1,5776)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5776, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("5776.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5776 = 2^{4} \cdot 19^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5776.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:	$46.1215922075$
Analytic rank:	$1$
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:	$ 1 $
Twist minimal:	no (minimal twist has level 361)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$1.61803$ of defining polynomial
Character	$\chi$	$=$	5776.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.61803 q^{3} -1.23607 q^{5} -3.00000 q^{7} +3.85410 q^{9} +O(q^{10})$	$q-2.61803 q^{3} -1.23607 q^{5} -3.00000 q^{7} +3.85410 q^{9} -0.618034 q^{11} -1.00000 q^{13} +3.23607 q^{15} +5.23607 q^{17} +7.85410 q^{21} -7.61803 q^{23} -3.47214 q^{25} -2.23607 q^{27} -1.38197 q^{29} +2.14590 q^{31} +1.61803 q^{33} +3.70820 q^{35} +2.14590 q^{37} +2.61803 q^{39} -3.00000 q^{41} +6.85410 q^{43} -4.76393 q^{45} -3.00000 q^{47} +2.00000 q^{49} -13.7082 q^{51} +9.32624 q^{53} +0.763932 q^{55} +15.3262 q^{59} -5.76393 q^{61} -11.5623 q^{63} +1.23607 q^{65} +7.00000 q^{67} +19.9443 q^{69} -1.47214 q^{71} +10.7082 q^{73} +9.09017 q^{75} +1.85410 q^{77} +13.4164 q^{79} -5.70820 q^{81} +0.472136 q^{83} -6.47214 q^{85} +3.61803 q^{87} -12.2361 q^{89} +3.00000 q^{91} -5.61803 q^{93} +7.14590 q^{97} -2.38197 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 3 q^{3} + 2 q^{5} - 6 q^{7} + q^{9}+O(q^{10}) $ 2 * q - 3 * q^3 + 2 * q^5 - 6 * q^7 + q^9	$ 2 q - 3 q^{3} + 2 q^{5} - 6 q^{7} + q^{9} + q^{11} - 2 q^{13} + 2 q^{15} + 6 q^{17} + 9 q^{21} - 13 q^{23} + 2 q^{25} - 5 q^{29} + 11 q^{31} + q^{33} - 6 q^{35} + 11 q^{37} + 3 q^{39} - 6 q^{41} + 7 q^{43} - 14 q^{45} - 6 q^{47} + 4 q^{49} - 14 q^{51} + 3 q^{53} + 6 q^{55} + 15 q^{59} - 16 q^{61} - 3 q^{63} - 2 q^{65} + 14 q^{67} + 22 q^{69} + 6 q^{71} + 8 q^{73} + 7 q^{75} - 3 q^{77} + 2 q^{81} - 8 q^{83} - 4 q^{85} + 5 q^{87} - 20 q^{89} + 6 q^{91} - 9 q^{93} + 21 q^{97} - 7 q^{99}+O(q^{100}) $ 2 * q - 3 * q^3 + 2 * q^5 - 6 * q^7 + q^9 + q^11 - 2 * q^13 + 2 * q^15 + 6 * q^17 + 9 * q^21 - 13 * q^23 + 2 * q^25 - 5 * q^29 + 11 * q^31 + q^33 - 6 * q^35 + 11 * q^37 + 3 * q^39 - 6 * q^41 + 7 * q^43 - 14 * q^45 - 6 * q^47 + 4 * q^49 - 14 * q^51 + 3 * q^53 + 6 * q^55 + 15 * q^59 - 16 * q^61 - 3 * q^63 - 2 * q^65 + 14 * q^67 + 22 * q^69 + 6 * q^71 + 8 * q^73 + 7 * q^75 - 3 * q^77 + 2 * q^81 - 8 * q^83 - 4 * q^85 + 5 * q^87 - 20 * q^89 + 6 * q^91 - 9 * q^93 + 21 * q^97 - 7 * 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
$3$	−2.61803	−1.51152	−0.755761	−	0.654847i	$-0.772733\pi$
$3$	−2.61803	−1.51152	−0.755761	+	0.654847i	$0.772733\pi$
$4$	0	0
$5$	−1.23607	−0.552786	−0.276393	−	0.961045i	$-0.589139\pi$
$5$	−1.23607	−0.552786	−0.276393	+	0.961045i	$0.589139\pi$
$6$	0	0
$7$	−3.00000	−1.13389	−0.566947	−	0.823754i	$-0.691875\pi$
$7$	−3.00000	−1.13389	−0.566947	+	0.823754i	$0.691875\pi$
$8$	0	0
$9$	3.85410	1.28470
$10$	0	0
$11$	−0.618034	−0.186344	−0.0931721	−	0.995650i	$-0.529701\pi$
$11$	−0.618034	−0.186344	−0.0931721	+	0.995650i	$0.529701\pi$
$12$	0	0
$13$	−1.00000	−0.277350	−0.138675	−	0.990338i	$-0.544284\pi$
$13$	−1.00000	−0.277350	−0.138675	+	0.990338i	$0.544284\pi$
$14$	0	0
$15$	3.23607	0.835549
$16$	0	0
$17$	5.23607	1.26993	0.634967	−	0.772540i	$-0.281014\pi$
$17$	5.23607	1.26993	0.634967	+	0.772540i	$0.281014\pi$
$18$	0	0
$19$	0	0
$19$	0	0
$20$	0	0
$21$	7.85410	1.71391
$22$	0	0
$23$	−7.61803	−1.58847	−0.794235	−	0.607611i	$-0.792128\pi$
$23$	−7.61803	−1.58847	−0.794235	+	0.607611i	$0.792128\pi$
$24$	0	0
$25$	−3.47214	−0.694427
$26$	0	0
$27$	−2.23607	−0.430331
$28$	0	0
$29$	−1.38197	−0.256625	−0.128312	−	0.991734i	$-0.540956\pi$
$29$	−1.38197	−0.256625	−0.128312	+	0.991734i	$0.540956\pi$
$30$	0	0
$31$	2.14590	0.385415	0.192707	−	0.981256i	$-0.438273\pi$
$31$	2.14590	0.385415	0.192707	+	0.981256i	$0.438273\pi$
$32$	0	0
$33$	1.61803	0.281664
$34$	0	0
$35$	3.70820	0.626801
$36$	0	0
$37$	2.14590	0.352783	0.176392	−	0.984320i	$-0.443557\pi$
$37$	2.14590	0.352783	0.176392	+	0.984320i	$0.443557\pi$
$38$	0	0
$39$	2.61803	0.419221
$40$	0	0
$41$	−3.00000	−0.468521	−0.234261	−	0.972174i	$-0.575267\pi$
$41$	−3.00000	−0.468521	−0.234261	+	0.972174i	$0.575267\pi$
$42$	0	0
$43$	6.85410	1.04524	0.522620	−	0.852566i	$-0.324955\pi$
$43$	6.85410	1.04524	0.522620	+	0.852566i	$0.324955\pi$
$44$	0	0
$45$	−4.76393	−0.710165
$46$	0	0
$47$	−3.00000	−0.437595	−0.218797	−	0.975770i	$-0.570213\pi$
$47$	−3.00000	−0.437595	−0.218797	+	0.975770i	$0.570213\pi$
$48$	0	0
$49$	2.00000	0.285714
$50$	0	0
$51$	−13.7082	−1.91953
$52$	0	0
$53$	9.32624	1.28106	0.640529	−	0.767934i	$-0.278715\pi$
$53$	9.32624	1.28106	0.640529	+	0.767934i	$0.278715\pi$
$54$	0	0
$55$	0.763932	0.103009
$56$	0	0
$57$	0	0
$58$	0	0
$59$	15.3262	1.99531	0.997653	−	0.0684709i	$-0.0218120\pi$
$59$	15.3262	1.99531	0.997653	+	0.0684709i	$0.0218120\pi$
$60$	0	0
$61$	−5.76393	−0.737996	−0.368998	−	0.929430i	$-0.620299\pi$
$61$	−5.76393	−0.737996	−0.368998	+	0.929430i	$0.620299\pi$
$62$	0	0
$63$	−11.5623	−1.45671
$64$	0	0
$65$	1.23607	0.153315
$66$	0	0
$67$	7.00000	0.855186	0.427593	−	0.903971i	$-0.359362\pi$
$67$	7.00000	0.855186	0.427593	+	0.903971i	$0.359362\pi$
$68$	0	0
$69$	19.9443	2.40101
$70$	0	0
$71$	−1.47214	−0.174710	−0.0873552	−	0.996177i	$-0.527842\pi$
$71$	−1.47214	−0.174710	−0.0873552	+	0.996177i	$0.527842\pi$
$72$	0	0
$73$	10.7082	1.25330	0.626650	−	0.779301i	$-0.284425\pi$
$73$	10.7082	1.25330	0.626650	+	0.779301i	$0.284425\pi$
$74$	0	0
$75$	9.09017	1.04964
$76$	0	0
$77$	1.85410	0.211295
$78$	0	0
$79$	13.4164	1.50946	0.754732	−	0.656033i	$-0.227767\pi$
$79$	13.4164	1.50946	0.754732	+	0.656033i	$0.227767\pi$
$80$	0	0
$81$	−5.70820	−0.634245
$82$	0	0
$83$	0.472136	0.0518237	0.0259118	−	0.999664i	$-0.491751\pi$
$83$	0.472136	0.0518237	0.0259118	+	0.999664i	$0.491751\pi$
$84$	0	0
$85$	−6.47214	−0.702002
$86$	0	0
$87$	3.61803	0.387894
$88$	0	0
$89$	−12.2361	−1.29702	−0.648510	−	0.761206i	$-0.724608\pi$
$89$	−12.2361	−1.29702	−0.648510	+	0.761206i	$0.724608\pi$
$90$	0	0
$91$	3.00000	0.314485
$92$	0	0
$93$	−5.61803	−0.582563
$94$	0	0
$95$	0	0
$96$	0	0
$97$	7.14590	0.725556	0.362778	−	0.931876i	$-0.381828\pi$
$97$	7.14590	0.725556	0.362778	+	0.931876i	$0.381828\pi$
$98$	0	0
$99$	−2.38197	−0.239397
$100$	0	0
$101$	13.1803	1.31149	0.655746	−	0.754981i	$-0.272354\pi$
$101$	13.1803	1.31149	0.655746	+	0.754981i	$0.272354\pi$
$102$	0	0
$103$	1.32624	0.130678	0.0653391	−	0.997863i	$-0.479187\pi$
$103$	1.32624	0.130678	0.0653391	+	0.997863i	$0.479187\pi$
$104$	0	0
$105$	−9.70820	−0.947424
$106$	0	0
$107$	10.4164	1.00699	0.503496	−	0.863998i	$-0.332047\pi$
$107$	10.4164	1.00699	0.503496	+	0.863998i	$0.332047\pi$
$108$	0	0
$109$	−16.7082	−1.60036	−0.800178	−	0.599763i	$-0.795262\pi$
$109$	−16.7082	−1.60036	−0.800178	+	0.599763i	$0.795262\pi$
$110$	0	0
$111$	−5.61803	−0.533240
$112$	0	0
$113$	11.2361	1.05700	0.528500	−	0.848933i	$-0.322755\pi$
$113$	11.2361	1.05700	0.528500	+	0.848933i	$0.322755\pi$
$114$	0	0
$115$	9.41641	0.878085
$116$	0	0
$117$	−3.85410	−0.356312
$118$	0	0
$119$	−15.7082	−1.43997
$120$	0	0
$121$	−10.6180	−0.965276
$122$	0	0
$123$	7.85410	0.708181
$124$	0	0
$125$	10.4721	0.936656
$126$	0	0
$127$	−10.2361	−0.908304	−0.454152	−	0.890924i	$-0.650058\pi$
$127$	−10.2361	−0.908304	−0.454152	+	0.890924i	$0.650058\pi$
$128$	0	0
$129$	−17.9443	−1.57991
$130$	0	0
$131$	−3.90983	−0.341603	−0.170802	−	0.985305i	$-0.554636\pi$
$131$	−3.90983	−0.341603	−0.170802	+	0.985305i	$0.554636\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	2.76393	0.237881
$136$	0	0
$137$	−1.47214	−0.125773	−0.0628865	−	0.998021i	$-0.520031\pi$
$137$	−1.47214	−0.125773	−0.0628865	+	0.998021i	$0.520031\pi$
$138$	0	0
$139$	9.79837	0.831087	0.415544	−	0.909573i	$-0.363591\pi$
$139$	9.79837	0.831087	0.415544	+	0.909573i	$0.363591\pi$
$140$	0	0
$141$	7.85410	0.661435
$142$	0	0
$143$	0.618034	0.0516826
$144$	0	0
$145$	1.70820	0.141859
$146$	0	0
$147$	−5.23607	−0.431864
$148$	0	0
$149$	−13.0902	−1.07239	−0.536194	−	0.844095i	$-0.680139\pi$
$149$	−13.0902	−1.07239	−0.536194	+	0.844095i	$0.680139\pi$
$150$	0	0
$151$	9.90983	0.806451	0.403225	−	0.915101i	$-0.367889\pi$
$151$	9.90983	0.806451	0.403225	+	0.915101i	$0.367889\pi$
$152$	0	0
$153$	20.1803	1.63148
$154$	0	0
$155$	−2.65248	−0.213052
$156$	0	0
$157$	−11.1459	−0.889540	−0.444770	−	0.895645i	$-0.646714\pi$
$157$	−11.1459	−0.889540	−0.444770	+	0.895645i	$0.646714\pi$
$158$	0	0
$159$	−24.4164	−1.93635
$160$	0	0
$161$	22.8541	1.80116
$162$	0	0
$163$	−6.23607	−0.488447	−0.244223	−	0.969719i	$-0.578533\pi$
$163$	−6.23607	−0.488447	−0.244223	+	0.969719i	$0.578533\pi$
$164$	0	0
$165$	−2.00000	−0.155700
$166$	0	0
$167$	−15.7639	−1.21985	−0.609925	−	0.792459i	$-0.708800\pi$
$167$	−15.7639	−1.21985	−0.609925	+	0.792459i	$0.708800\pi$
$168$	0	0
$169$	−12.0000	−0.923077
$170$	0	0
$171$	0	0
$172$	0	0
$173$	8.47214	0.644125	0.322062	−	0.946718i	$-0.395624\pi$
$173$	8.47214	0.644125	0.322062	+	0.946718i	$0.395624\pi$
$174$	0	0
$175$	10.4164	0.787406
$176$	0	0
$177$	−40.1246	−3.01595
$178$	0	0
$179$	−7.76393	−0.580304	−0.290152	−	0.956981i	$-0.593706\pi$
$179$	−7.76393	−0.580304	−0.290152	+	0.956981i	$0.593706\pi$
$180$	0	0
$181$	12.0000	0.891953	0.445976	−	0.895045i	$-0.352856\pi$
$181$	12.0000	0.891953	0.445976	+	0.895045i	$0.352856\pi$
$182$	0	0
$183$	15.0902	1.11550
$184$	0	0
$185$	−2.65248	−0.195014
$186$	0	0
$187$	−3.23607	−0.236645
$188$	0	0
$189$	6.70820	0.487950
$190$	0	0
$191$	−9.76393	−0.706493	−0.353247	−	0.935530i	$-0.614922\pi$
$191$	−9.76393	−0.706493	−0.353247	+	0.935530i	$0.614922\pi$
$192$	0	0
$193$	22.9443	1.65156	0.825782	−	0.563989i	$-0.190734\pi$
$193$	22.9443	1.65156	0.825782	+	0.563989i	$0.190734\pi$
$194$	0	0
$195$	−3.23607	−0.231740
$196$	0	0
$197$	3.00000	0.213741	0.106871	−	0.994273i	$-0.465917\pi$
$197$	3.00000	0.213741	0.106871	+	0.994273i	$0.465917\pi$
$198$	0	0
$199$	−13.4164	−0.951064	−0.475532	−	0.879698i	$-0.657744\pi$
$199$	−13.4164	−0.951064	−0.475532	+	0.879698i	$0.657744\pi$
$200$	0	0
$201$	−18.3262	−1.29263
$202$	0	0
$203$	4.14590	0.290985
$204$	0	0
$205$	3.70820	0.258992
$206$	0	0
$207$	−29.3607	−2.04071
$208$	0	0
$209$	0	0
$210$	0	0
$211$	−2.85410	−0.196484	−0.0982422	−	0.995163i	$-0.531322\pi$
$211$	−2.85410	−0.196484	−0.0982422	+	0.995163i	$0.531322\pi$
$212$	0	0
$213$	3.85410	0.264079
$214$	0	0
$215$	−8.47214	−0.577795
$216$	0	0
$217$	−6.43769	−0.437019
$218$	0	0
$219$	−28.0344	−1.89439
$220$	0	0
$221$	−5.23607	−0.352216
$222$	0	0
$223$	−19.6525	−1.31603	−0.658014	−	0.753006i	$-0.728603\pi$
$223$	−19.6525	−1.31603	−0.658014	+	0.753006i	$0.728603\pi$
$224$	0	0
$225$	−13.3820	−0.892131
$226$	0	0
$227$	10.4164	0.691361	0.345681	−	0.938352i	$-0.387648\pi$
$227$	10.4164	0.691361	0.345681	+	0.938352i	$0.387648\pi$
$228$	0	0
$229$	−11.3820	−0.752141	−0.376071	−	0.926591i	$-0.622725\pi$
$229$	−11.3820	−0.752141	−0.376071	+	0.926591i	$0.622725\pi$
$230$	0	0
$231$	−4.85410	−0.319376
$232$	0	0
$233$	13.4721	0.882589	0.441294	−	0.897362i	$-0.354519\pi$
$233$	13.4721	0.882589	0.441294	+	0.897362i	$0.354519\pi$
$234$	0	0
$235$	3.70820	0.241897
$236$	0	0
$237$	−35.1246	−2.28159
$238$	0	0
$239$	−15.3262	−0.991372	−0.495686	−	0.868502i	$-0.665083\pi$
$239$	−15.3262	−0.991372	−0.495686	+	0.868502i	$0.665083\pi$
$240$	0	0
$241$	−19.1803	−1.23551	−0.617757	−	0.786369i	$-0.711959\pi$
$241$	−19.1803	−1.23551	−0.617757	+	0.786369i	$0.711959\pi$
$242$	0	0
$243$	21.6525	1.38901
$244$	0	0
$245$	−2.47214	−0.157939
$246$	0	0
$247$	0	0
$248$	0	0
$249$	−1.23607	−0.0783326
$250$	0	0
$251$	−19.3607	−1.22204	−0.611018	−	0.791617i	$-0.709240\pi$
$251$	−19.3607	−1.22204	−0.611018	+	0.791617i	$0.709240\pi$
$252$	0	0
$253$	4.70820	0.296002
$254$	0	0
$255$	16.9443	1.06109
$256$	0	0
$257$	−24.3607	−1.51958	−0.759789	−	0.650170i	$-0.774698\pi$
$257$	−24.3607	−1.51958	−0.759789	+	0.650170i	$0.774698\pi$
$258$	0	0
$259$	−6.43769	−0.400019
$260$	0	0
$261$	−5.32624	−0.329686
$262$	0	0
$263$	−2.94427	−0.181552	−0.0907758	−	0.995871i	$-0.528935\pi$
$263$	−2.94427	−0.181552	−0.0907758	+	0.995871i	$0.528935\pi$
$264$	0	0
$265$	−11.5279	−0.708151
$266$	0	0
$267$	32.0344	1.96048
$268$	0	0
$269$	14.6738	0.894675	0.447338	−	0.894365i	$-0.352372\pi$
$269$	14.6738	0.894675	0.447338	+	0.894365i	$0.352372\pi$
$270$	0	0
$271$	−7.85410	−0.477103	−0.238551	−	0.971130i	$-0.576673\pi$
$271$	−7.85410	−0.477103	−0.238551	+	0.971130i	$0.576673\pi$
$272$	0	0
$273$	−7.85410	−0.475352
$274$	0	0
$275$	2.14590	0.129403
$276$	0	0
$277$	−15.4164	−0.926282	−0.463141	−	0.886285i	$-0.653278\pi$
$277$	−15.4164	−0.926282	−0.463141	+	0.886285i	$0.653278\pi$
$278$	0	0
$279$	8.27051	0.495142
$280$	0	0
$281$	8.50658	0.507460	0.253730	−	0.967275i	$-0.418343\pi$
$281$	8.50658	0.507460	0.253730	+	0.967275i	$0.418343\pi$
$282$	0	0
$283$	3.03444	0.180379	0.0901894	−	0.995925i	$-0.471253\pi$
$283$	3.03444	0.180379	0.0901894	+	0.995925i	$0.471253\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	9.00000	0.531253
$288$	0	0
$289$	10.4164	0.612730
$290$	0	0
$291$	−18.7082	−1.09669
$292$	0	0
$293$	−16.8541	−0.984627	−0.492314	−	0.870418i	$-0.663849\pi$
$293$	−16.8541	−0.984627	−0.492314	+	0.870418i	$0.663849\pi$
$294$	0	0
$295$	−18.9443	−1.10298
$296$	0	0
$297$	1.38197	0.0801898
$298$	0	0
$299$	7.61803	0.440562
$300$	0	0
$301$	−20.5623	−1.18519
$302$	0	0
$303$	−34.5066	−1.98235
$304$	0	0
$305$	7.12461	0.407954
$306$	0	0
$307$	1.67376	0.0955266	0.0477633	−	0.998859i	$-0.484791\pi$
$307$	1.67376	0.0955266	0.0477633	+	0.998859i	$0.484791\pi$
$308$	0	0
$309$	−3.47214	−0.197523
$310$	0	0
$311$	18.6525	1.05768	0.528842	−	0.848720i	$-0.322626\pi$
$311$	18.6525	1.05768	0.528842	+	0.848720i	$0.322626\pi$
$312$	0	0
$313$	4.32624	0.244533	0.122267	−	0.992497i	$-0.460984\pi$
$313$	4.32624	0.244533	0.122267	+	0.992497i	$0.460984\pi$
$314$	0	0
$315$	14.2918	0.805251
$316$	0	0
$317$	18.0000	1.01098	0.505490	−	0.862832i	$-0.331312\pi$
$317$	18.0000	1.01098	0.505490	+	0.862832i	$0.331312\pi$
$318$	0	0
$319$	0.854102	0.0478205
$320$	0	0
$321$	−27.2705	−1.52209
$322$	0	0
$323$	0	0
$324$	0	0
$325$	3.47214	0.192599
$326$	0	0
$327$	43.7426	2.41897
$328$	0	0
$329$	9.00000	0.496186
$330$	0	0
$331$	6.94427	0.381692	0.190846	−	0.981620i	$-0.438877\pi$
$331$	6.94427	0.381692	0.190846	+	0.981620i	$0.438877\pi$
$332$	0	0
$333$	8.27051	0.453221
$334$	0	0
$335$	−8.65248	−0.472735
$336$	0	0
$337$	23.1246	1.25968	0.629839	−	0.776726i	$-0.283121\pi$
$337$	23.1246	1.25968	0.629839	+	0.776726i	$0.283121\pi$
$338$	0	0
$339$	−29.4164	−1.59768
$340$	0	0
$341$	−1.32624	−0.0718198
$342$	0	0
$343$	15.0000	0.809924
$344$	0	0
$345$	−24.6525	−1.32724
$346$	0	0
$347$	−1.41641	−0.0760368	−0.0380184	−	0.999277i	$-0.512105\pi$
$347$	−1.41641	−0.0760368	−0.0380184	+	0.999277i	$0.512105\pi$
$348$	0	0
$349$	25.9787	1.39061	0.695304	−	0.718715i	$-0.255270\pi$
$349$	25.9787	1.39061	0.695304	+	0.718715i	$0.255270\pi$
$350$	0	0
$351$	2.23607	0.119352
$352$	0	0
$353$	−31.4508	−1.67396	−0.836980	−	0.547234i	$-0.815681\pi$
$353$	−31.4508	−1.67396	−0.836980	+	0.547234i	$0.815681\pi$
$354$	0	0
$355$	1.81966	0.0965775
$356$	0	0
$357$	41.1246	2.17655
$358$	0	0
$359$	22.0344	1.16293	0.581467	−	0.813570i	$-0.302479\pi$
$359$	22.0344	1.16293	0.581467	+	0.813570i	$0.302479\pi$
$360$	0	0
$361$	0	0
$362$	0	0
$363$	27.7984	1.45904
$364$	0	0
$365$	−13.2361	−0.692807
$366$	0	0
$367$	−1.94427	−0.101490	−0.0507451	−	0.998712i	$-0.516160\pi$
$367$	−1.94427	−0.101490	−0.0507451	+	0.998712i	$0.516160\pi$
$368$	0	0
$369$	−11.5623	−0.601910
$370$	0	0
$371$	−27.9787	−1.45258
$372$	0	0
$373$	3.47214	0.179780	0.0898902	−	0.995952i	$-0.471348\pi$
$373$	3.47214	0.179780	0.0898902	+	0.995952i	$0.471348\pi$
$374$	0	0
$375$	−27.4164	−1.41578
$376$	0	0
$377$	1.38197	0.0711749
$378$	0	0
$379$	−25.1246	−1.29056	−0.645282	−	0.763944i	$-0.723260\pi$
$379$	−25.1246	−1.29056	−0.645282	+	0.763944i	$0.723260\pi$
$380$	0	0
$381$	26.7984	1.37292
$382$	0	0
$383$	−2.61803	−0.133775	−0.0668876	−	0.997761i	$-0.521307\pi$
$383$	−2.61803	−0.133775	−0.0668876	+	0.997761i	$0.521307\pi$
$384$	0	0
$385$	−2.29180	−0.116801
$386$	0	0
$387$	26.4164	1.34282
$388$	0	0
$389$	−24.2705	−1.23056	−0.615282	−	0.788307i	$-0.710958\pi$
$389$	−24.2705	−1.23056	−0.615282	+	0.788307i	$0.710958\pi$
$390$	0	0
$391$	−39.8885	−2.01725
$392$	0	0
$393$	10.2361	0.516341
$394$	0	0
$395$	−16.5836	−0.834411
$396$	0	0
$397$	−2.52786	−0.126870	−0.0634349	−	0.997986i	$-0.520206\pi$
$397$	−2.52786	−0.126870	−0.0634349	+	0.997986i	$0.520206\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−0.111456	−0.00556586	−0.00278293	−	0.999996i	$-0.500886\pi$
$401$	−0.111456	−0.00556586	−0.00278293	+	0.999996i	$0.500886\pi$
$402$	0	0
$403$	−2.14590	−0.106895
$404$	0	0
$405$	7.05573	0.350602
$406$	0	0
$407$	−1.32624	−0.0657392
$408$	0	0
$409$	−21.7082	−1.07340	−0.536701	−	0.843773i	$-0.680330\pi$
$409$	−21.7082	−1.07340	−0.536701	+	0.843773i	$0.680330\pi$
$410$	0	0
$411$	3.85410	0.190109
$412$	0	0
$413$	−45.9787	−2.26246
$414$	0	0
$415$	−0.583592	−0.0286474
$416$	0	0
$417$	−25.6525	−1.25621
$418$	0	0
$419$	−8.94427	−0.436956	−0.218478	−	0.975842i	$-0.570109\pi$
$419$	−8.94427	−0.436956	−0.218478	+	0.975842i	$0.570109\pi$
$420$	0	0
$421$	−18.5279	−0.902993	−0.451496	−	0.892273i	$-0.649110\pi$
$421$	−18.5279	−0.902993	−0.451496	+	0.892273i	$0.649110\pi$
$422$	0	0
$423$	−11.5623	−0.562179
$424$	0	0
$425$	−18.1803	−0.881876
$426$	0	0
$427$	17.2918	0.836809
$428$	0	0
$429$	−1.61803	−0.0781194
$430$	0	0
$431$	3.65248	0.175934	0.0879668	−	0.996123i	$-0.471963\pi$
$431$	3.65248	0.175934	0.0879668	+	0.996123i	$0.471963\pi$
$432$	0	0
$433$	−23.5623	−1.13233	−0.566166	−	0.824291i	$-0.691574\pi$
$433$	−23.5623	−1.13233	−0.566166	+	0.824291i	$0.691574\pi$
$434$	0	0
$435$	−4.47214	−0.214423
$436$	0	0
$437$	0	0
$438$	0	0
$439$	−14.5967	−0.696665	−0.348332	−	0.937371i	$-0.613252\pi$
$439$	−14.5967	−0.696665	−0.348332	+	0.937371i	$0.613252\pi$
$440$	0	0
$441$	7.70820	0.367057
$442$	0	0
$443$	34.4164	1.63517	0.817586	−	0.575806i	$-0.195312\pi$
$443$	34.4164	1.63517	0.817586	+	0.575806i	$0.195312\pi$
$444$	0	0
$445$	15.1246	0.716975
$446$	0	0
$447$	34.2705	1.62094
$448$	0	0
$449$	−32.8885	−1.55211	−0.776053	−	0.630667i	$-0.782781\pi$
$449$	−32.8885	−1.55211	−0.776053	+	0.630667i	$0.782781\pi$
$450$	0	0
$451$	1.85410	0.0873063
$452$	0	0
$453$	−25.9443	−1.21897
$454$	0	0
$455$	−3.70820	−0.173843
$456$	0	0
$457$	6.29180	0.294318	0.147159	−	0.989113i	$-0.452987\pi$
$457$	6.29180	0.294318	0.147159	+	0.989113i	$0.452987\pi$
$458$	0	0
$459$	−11.7082	−0.546492
$460$	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
$463$	5.27051	0.244941	0.122471	−	0.992472i	$-0.460918\pi$
$463$	5.27051	0.244941	0.122471	+	0.992472i	$0.460918\pi$
$464$	0	0
$465$	6.94427	0.322033
$466$	0	0
$467$	−1.94427	−0.0899702	−0.0449851	−	0.998988i	$-0.514324\pi$
$467$	−1.94427	−0.0899702	−0.0449851	+	0.998988i	$0.514324\pi$
$468$	0	0
$469$	−21.0000	−0.969690
$470$	0	0
$471$	29.1803	1.34456
$472$	0	0
$473$	−4.23607	−0.194775
$474$	0	0
$475$	0	0
$476$	0	0
$477$	35.9443	1.64578
$478$	0	0
$479$	−11.9098	−0.544174	−0.272087	−	0.962273i	$-0.587714\pi$
$479$	−11.9098	−0.544174	−0.272087	+	0.962273i	$0.587714\pi$
$480$	0	0
$481$	−2.14590	−0.0978445
$482$	0	0
$483$	−59.8328	−2.72249
$484$	0	0
$485$	−8.83282	−0.401078
$486$	0	0
$487$	18.1803	0.823830	0.411915	−	0.911222i	$-0.364860\pi$
$487$	18.1803	0.823830	0.411915	+	0.911222i	$0.364860\pi$
$488$	0	0
$489$	16.3262	0.738298
$490$	0	0
$491$	−30.2148	−1.36357	−0.681787	−	0.731551i	$-0.738797\pi$
$491$	−30.2148	−1.36357	−0.681787	+	0.731551i	$0.738797\pi$
$492$	0	0
$493$	−7.23607	−0.325896
$494$	0	0
$495$	2.94427	0.132335
$496$	0	0
$497$	4.41641	0.198103
$498$	0	0
$499$	15.1246	0.677071	0.338535	−	0.940954i	$-0.390069\pi$
$499$	15.1246	0.677071	0.338535	+	0.940954i	$0.390069\pi$
$500$	0	0
$501$	41.2705	1.84383
$502$	0	0
$503$	17.8328	0.795126	0.397563	−	0.917575i	$-0.369856\pi$
$503$	17.8328	0.795126	0.397563	+	0.917575i	$0.369856\pi$
$504$	0	0
$505$	−16.2918	−0.724975
$506$	0	0
$507$	31.4164	1.39525
$508$	0	0
$509$	−27.0344	−1.19828	−0.599140	−	0.800644i	$-0.704491\pi$
$509$	−27.0344	−1.19828	−0.599140	+	0.800644i	$0.704491\pi$
$510$	0	0
$511$	−32.1246	−1.42111
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−1.63932	−0.0722371
$516$	0	0
$517$	1.85410	0.0815433
$518$	0	0
$519$	−22.1803	−0.973609
$520$	0	0
$521$	−27.2705	−1.19474	−0.597371	−	0.801965i	$-0.703788\pi$
$521$	−27.2705	−1.19474	−0.597371	+	0.801965i	$0.703788\pi$
$522$	0	0
$523$	−22.4164	−0.980201	−0.490101	−	0.871666i	$-0.663040\pi$
$523$	−22.4164	−0.980201	−0.490101	+	0.871666i	$0.663040\pi$
$524$	0	0
$525$	−27.2705	−1.19018
$526$	0	0
$527$	11.2361	0.489451
$528$	0	0
$529$	35.0344	1.52324
$530$	0	0
$531$	59.0689	2.56337
$532$	0	0
$533$	3.00000	0.129944
$534$	0	0
$535$	−12.8754	−0.556652
$536$	0	0
$537$	20.3262	0.877142
$538$	0	0
$539$	−1.23607	−0.0532412
$540$	0	0
$541$	23.8328	1.02465	0.512326	−	0.858791i	$-0.328784\pi$
$541$	23.8328	1.02465	0.512326	+	0.858791i	$0.328784\pi$
$542$	0	0
$543$	−31.4164	−1.34821
$544$	0	0
$545$	20.6525	0.884655
$546$	0	0
$547$	−37.9230	−1.62147	−0.810735	−	0.585413i	$-0.800932\pi$
$547$	−37.9230	−1.62147	−0.810735	+	0.585413i	$0.800932\pi$
$548$	0	0
$549$	−22.2148	−0.948104
$550$	0	0
$551$	0	0
$552$	0	0
$553$	−40.2492	−1.71157
$554$	0	0
$555$	6.94427	0.294768
$556$	0	0
$557$	−23.1803	−0.982183	−0.491091	−	0.871108i	$-0.663402\pi$
$557$	−23.1803	−0.982183	−0.491091	+	0.871108i	$0.663402\pi$
$558$	0	0
$559$	−6.85410	−0.289898
$560$	0	0
$561$	8.47214	0.357694
$562$	0	0
$563$	−20.8328	−0.877999	−0.438999	−	0.898487i	$-0.644667\pi$
$563$	−20.8328	−0.877999	−0.438999	+	0.898487i	$0.644667\pi$
$564$	0	0
$565$	−13.8885	−0.584295
$566$	0	0
$567$	17.1246	0.719166
$568$	0	0
$569$	28.0902	1.17760	0.588801	−	0.808278i	$-0.299600\pi$
$569$	28.0902	1.17760	0.588801	+	0.808278i	$0.299600\pi$
$570$	0	0
$571$	−22.3262	−0.934324	−0.467162	−	0.884172i	$-0.654724\pi$
$571$	−22.3262	−0.934324	−0.467162	+	0.884172i	$0.654724\pi$
$572$	0	0
$573$	25.5623	1.06788
$574$	0	0
$575$	26.4508	1.10308
$576$	0	0
$577$	28.1246	1.17084	0.585421	−	0.810729i	$-0.300929\pi$
$577$	28.1246	1.17084	0.585421	+	0.810729i	$0.300929\pi$
$578$	0	0
$579$	−60.0689	−2.49638
$580$	0	0
$581$	−1.41641	−0.0587625
$582$	0	0
$583$	−5.76393	−0.238718
$584$	0	0
$585$	4.76393	0.196964
$586$	0	0
$587$	−38.1246	−1.57357	−0.786786	−	0.617226i	$-0.788256\pi$
$587$	−38.1246	−1.57357	−0.786786	+	0.617226i	$0.788256\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	−7.85410	−0.323075
$592$	0	0
$593$	−12.7082	−0.521863	−0.260932	−	0.965357i	$-0.584030\pi$
$593$	−12.7082	−0.521863	−0.260932	+	0.965357i	$0.584030\pi$
$594$	0	0
$595$	19.4164	0.795995
$596$	0	0
$597$	35.1246	1.43755
$598$	0	0
$599$	−1.58359	−0.0647038	−0.0323519	−	0.999477i	$-0.510300\pi$
$599$	−1.58359	−0.0647038	−0.0323519	+	0.999477i	$0.510300\pi$
$600$	0	0
$601$	33.7082	1.37499	0.687493	−	0.726191i	$-0.258711\pi$
$601$	33.7082	1.37499	0.687493	+	0.726191i	$0.258711\pi$
$602$	0	0
$603$	26.9787	1.09866
$604$	0	0
$605$	13.1246	0.533591
$606$	0	0
$607$	−27.2705	−1.10688	−0.553438	−	0.832890i	$-0.686684\pi$
$607$	−27.2705	−1.10688	−0.553438	+	0.832890i	$0.686684\pi$
$608$	0	0
$609$	−10.8541	−0.439830
$610$	0	0
$611$	3.00000	0.121367
$612$	0	0
$613$	−2.05573	−0.0830301	−0.0415150	−	0.999138i	$-0.513218\pi$
$613$	−2.05573	−0.0830301	−0.0415150	+	0.999138i	$0.513218\pi$
$614$	0	0
$615$	−9.70820	−0.391473
$616$	0	0
$617$	23.3262	0.939079	0.469539	−	0.882911i	$-0.344420\pi$
$617$	23.3262	0.939079	0.469539	+	0.882911i	$0.344420\pi$
$618$	0	0
$619$	30.1246	1.21081	0.605405	−	0.795917i	$-0.293011\pi$
$619$	30.1246	1.21081	0.605405	+	0.795917i	$0.293011\pi$
$620$	0	0
$621$	17.0344	0.683569
$622$	0	0
$623$	36.7082	1.47068
$624$	0	0
$625$	4.41641	0.176656
$626$	0	0
$627$	0	0
$628$	0	0
$629$	11.2361	0.448011
$630$	0	0
$631$	15.3607	0.611499	0.305750	−	0.952112i	$-0.401093\pi$
$631$	15.3607	0.611499	0.305750	+	0.952112i	$0.401093\pi$
$632$	0	0
$633$	7.47214	0.296991
$634$	0	0
$635$	12.6525	0.502098
$636$	0	0
$637$	−2.00000	−0.0792429
$638$	0	0
$639$	−5.67376	−0.224451
$640$	0	0
$641$	−1.49342	−0.0589866	−0.0294933	−	0.999565i	$-0.509389\pi$
$641$	−1.49342	−0.0589866	−0.0294933	+	0.999565i	$0.509389\pi$
$642$	0	0
$643$	37.7082	1.48707	0.743533	−	0.668699i	$-0.233149\pi$
$643$	37.7082	1.48707	0.743533	+	0.668699i	$0.233149\pi$
$644$	0	0
$645$	22.1803	0.873350
$646$	0	0
$647$	1.47214	0.0578756	0.0289378	−	0.999581i	$-0.490788\pi$
$647$	1.47214	0.0578756	0.0289378	+	0.999581i	$0.490788\pi$
$648$	0	0
$649$	−9.47214	−0.371814
$650$	0	0
$651$	16.8541	0.660564
$652$	0	0
$653$	−3.43769	−0.134527	−0.0672637	−	0.997735i	$-0.521427\pi$
$653$	−3.43769	−0.134527	−0.0672637	+	0.997735i	$0.521427\pi$
$654$	0	0
$655$	4.83282	0.188834
$656$	0	0
$657$	41.2705	1.61012
$658$	0	0
$659$	−45.7771	−1.78322	−0.891611	−	0.452802i	$-0.850424\pi$
$659$	−45.7771	−1.78322	−0.891611	+	0.452802i	$0.850424\pi$
$660$	0	0
$661$	−21.4164	−0.833002	−0.416501	−	0.909135i	$-0.636744\pi$
$661$	−21.4164	−0.833002	−0.416501	+	0.909135i	$0.636744\pi$
$662$	0	0
$663$	13.7082	0.532383
$664$	0	0
$665$	0	0
$666$	0	0
$667$	10.5279	0.407641
$668$	0	0
$669$	51.4508	1.98920
$670$	0	0
$671$	3.56231	0.137521
$672$	0	0
$673$	−6.12461	−0.236086	−0.118043	−	0.993008i	$-0.537662\pi$
$673$	−6.12461	−0.236086	−0.118043	+	0.993008i	$0.537662\pi$
$674$	0	0
$675$	7.76393	0.298834
$676$	0	0
$677$	11.7426	0.451307	0.225653	−	0.974208i	$-0.427548\pi$
$677$	11.7426	0.451307	0.225653	+	0.974208i	$0.427548\pi$
$678$	0	0
$679$	−21.4377	−0.822703
$680$	0	0
$681$	−27.2705	−1.04501
$682$	0	0
$683$	21.6525	0.828509	0.414254	−	0.910161i	$-0.364042\pi$
$683$	21.6525	0.828509	0.414254	+	0.910161i	$0.364042\pi$
$684$	0	0
$685$	1.81966	0.0695256
$686$	0	0
$687$	29.7984	1.13688
$688$	0	0
$689$	−9.32624	−0.355301
$690$	0	0
$691$	16.8197	0.639850	0.319925	−	0.947443i	$-0.396342\pi$
$691$	16.8197	0.639850	0.319925	+	0.947443i	$0.396342\pi$
$692$	0	0
$693$	7.14590	0.271450
$694$	0	0
$695$	−12.1115	−0.459414
$696$	0	0
$697$	−15.7082	−0.594991
$698$	0	0
$699$	−35.2705	−1.33405
$700$	0	0
$701$	38.6312	1.45908	0.729540	−	0.683938i	$-0.239734\pi$
$701$	38.6312	1.45908	0.729540	+	0.683938i	$0.239734\pi$
$702$	0	0
$703$	0	0
$704$	0	0
$705$	−9.70820	−0.365632
$706$	0	0
$707$	−39.5410	−1.48709
$708$	0	0
$709$	43.4164	1.63054	0.815269	−	0.579083i	$-0.196589\pi$
$709$	43.4164	1.63054	0.815269	+	0.579083i	$0.196589\pi$
$710$	0	0
$711$	51.7082	1.93921
$712$	0	0
$713$	−16.3475	−0.612220
$714$	0	0
$715$	−0.763932	−0.0285694
$716$	0	0
$717$	40.1246	1.49848
$718$	0	0
$719$	−17.9656	−0.670002	−0.335001	−	0.942218i	$-0.608737\pi$
$719$	−17.9656	−0.670002	−0.335001	+	0.942218i	$0.608737\pi$
$720$	0	0
$721$	−3.97871	−0.148175
$722$	0	0
$723$	50.2148	1.86751
$724$	0	0
$725$	4.79837	0.178207
$726$	0	0
$727$	−42.0689	−1.56025	−0.780124	−	0.625625i	$-0.784844\pi$
$727$	−42.0689	−1.56025	−0.780124	+	0.625625i	$0.784844\pi$
$728$	0	0
$729$	−39.5623	−1.46527
$730$	0	0
$731$	35.8885	1.32739
$732$	0	0
$733$	40.9574	1.51280	0.756399	−	0.654111i	$-0.226957\pi$
$733$	40.9574	1.51280	0.756399	+	0.654111i	$0.226957\pi$
$734$	0	0
$735$	6.47214	0.238728
$736$	0	0
$737$	−4.32624	−0.159359
$738$	0	0
$739$	25.0000	0.919640	0.459820	−	0.888012i	$-0.347914\pi$
$739$	25.0000	0.919640	0.459820	+	0.888012i	$0.347914\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−41.3607	−1.51738	−0.758688	−	0.651454i	$-0.774159\pi$
$743$	−41.3607	−1.51738	−0.758688	+	0.651454i	$0.774159\pi$
$744$	0	0
$745$	16.1803	0.592802
$746$	0	0
$747$	1.81966	0.0665779
$748$	0	0
$749$	−31.2492	−1.14182
$750$	0	0
$751$	23.8541	0.870449	0.435224	−	0.900322i	$-0.356669\pi$
$751$	23.8541	0.870449	0.435224	+	0.900322i	$0.356669\pi$
$752$	0	0
$753$	50.6869	1.84713
$754$	0	0
$755$	−12.2492	−0.445795
$756$	0	0
$757$	−15.7426	−0.572176	−0.286088	−	0.958203i	$-0.592355\pi$
$757$	−15.7426	−0.572176	−0.286088	+	0.958203i	$0.592355\pi$
$758$	0	0
$759$	−12.3262	−0.447414
$760$	0	0
$761$	−30.8885	−1.11971	−0.559854	−	0.828591i	$-0.689143\pi$
$761$	−30.8885	−1.11971	−0.559854	+	0.828591i	$0.689143\pi$
$762$	0	0
$763$	50.1246	1.81463
$764$	0	0
$765$	−24.9443	−0.901862
$766$	0	0
$767$	−15.3262	−0.553398
$768$	0	0
$769$	−41.6312	−1.50126	−0.750630	−	0.660723i	$-0.770250\pi$
$769$	−41.6312	−1.50126	−0.750630	+	0.660723i	$0.770250\pi$
$770$	0	0
$771$	63.7771	2.29688
$772$	0	0
$773$	28.9230	1.04029	0.520144	−	0.854079i	$-0.325878\pi$
$773$	28.9230	1.04029	0.520144	+	0.854079i	$0.325878\pi$
$774$	0	0
$775$	−7.45085	−0.267642
$776$	0	0
$777$	16.8541	0.604638
$778$	0	0
$779$	0	0
$780$	0	0
$781$	0.909830	0.0325563
$782$	0	0
$783$	3.09017	0.110434
$784$	0	0
$785$	13.7771	0.491725
$786$	0	0
$787$	−38.0000	−1.35455	−0.677277	−	0.735728i	$-0.736840\pi$
$787$	−38.0000	−1.35455	−0.677277	+	0.735728i	$0.736840\pi$
$788$	0	0
$789$	7.70820	0.274419
$790$	0	0
$791$	−33.7082	−1.19853
$792$	0	0
$793$	5.76393	0.204683
$794$	0	0
$795$	30.1803	1.07039
$796$	0	0
$797$	−33.7082	−1.19401	−0.597003	−	0.802239i	$-0.703642\pi$
$797$	−33.7082	−1.19401	−0.597003	+	0.802239i	$0.703642\pi$
$798$	0	0
$799$	−15.7082	−0.555716
$800$	0	0
$801$	−47.1591	−1.66628
$802$	0	0
$803$	−6.61803	−0.233545
$804$	0	0
$805$	−28.2492	−0.995654
$806$	0	0
$807$	−38.4164	−1.35232
$808$	0	0
$809$	−0.201626	−0.00708880	−0.00354440	−	0.999994i	$-0.501128\pi$
$809$	−0.201626	−0.00708880	−0.00354440	+	0.999994i	$0.501128\pi$
$810$	0	0
$811$	23.9787	0.842007	0.421003	−	0.907059i	$-0.361678\pi$
$811$	23.9787	0.842007	0.421003	+	0.907059i	$0.361678\pi$
$812$	0	0
$813$	20.5623	0.721152
$814$	0	0
$815$	7.70820	0.270007
$816$	0	0
$817$	0	0
$818$	0	0
$819$	11.5623	0.404020
$820$	0	0
$821$	−53.0476	−1.85137	−0.925687	−	0.378290i	$-0.876512\pi$
$821$	−53.0476	−1.85137	−0.925687	+	0.378290i	$0.876512\pi$
$822$	0	0
$823$	−23.5967	−0.822531	−0.411265	−	0.911516i	$-0.634913\pi$
$823$	−23.5967	−0.822531	−0.411265	+	0.911516i	$0.634913\pi$
$824$	0	0
$825$	−5.61803	−0.195595
$826$	0	0
$827$	16.5967	0.577125	0.288563	−	0.957461i	$-0.406823\pi$
$827$	16.5967	0.577125	0.288563	+	0.957461i	$0.406823\pi$
$828$	0	0
$829$	20.3262	0.705959	0.352980	−	0.935631i	$-0.385168\pi$
$829$	20.3262	0.705959	0.352980	+	0.935631i	$0.385168\pi$
$830$	0	0
$831$	40.3607	1.40010
$832$	0	0
$833$	10.4721	0.362838
$834$	0	0
$835$	19.4853	0.674316
$836$	0	0
$837$	−4.79837	−0.165856
$838$	0	0
$839$	−39.7984	−1.37399	−0.686996	−	0.726661i	$-0.741071\pi$
$839$	−39.7984	−1.37399	−0.686996	+	0.726661i	$0.741071\pi$
$840$	0	0
$841$	−27.0902	−0.934144
$842$	0	0
$843$	−22.2705	−0.767037
$844$	0	0
$845$	14.8328	0.510264
$846$	0	0
$847$	31.8541	1.09452
$848$	0	0
$849$	−7.94427	−0.272647
$850$	0	0
$851$	−16.3475	−0.560386
$852$	0	0
$853$	17.2918	0.592060	0.296030	−	0.955179i	$-0.404337\pi$
$853$	17.2918	0.592060	0.296030	+	0.955179i	$0.404337\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−8.38197	−0.286323	−0.143161	−	0.989699i	$-0.545727\pi$
$857$	−8.38197	−0.286323	−0.143161	+	0.989699i	$0.545727\pi$
$858$	0	0
$859$	−18.5410	−0.632611	−0.316306	−	0.948657i	$-0.602443\pi$
$859$	−18.5410	−0.632611	−0.316306	+	0.948657i	$0.602443\pi$
$860$	0	0
$861$	−23.5623	−0.803001
$862$	0	0
$863$	44.9443	1.52992	0.764960	−	0.644077i	$-0.222759\pi$
$863$	44.9443	1.52992	0.764960	+	0.644077i	$0.222759\pi$
$864$	0	0
$865$	−10.4721	−0.356063
$866$	0	0
$867$	−27.2705	−0.926155
$868$	0	0
$869$	−8.29180	−0.281280
$870$	0	0
$871$	−7.00000	−0.237186
$872$	0	0
$873$	27.5410	0.932122
$874$	0	0
$875$	−31.4164	−1.06207
$876$	0	0
$877$	34.1803	1.15419	0.577094	−	0.816678i	$-0.304187\pi$
$877$	34.1803	1.15419	0.577094	+	0.816678i	$0.304187\pi$
$878$	0	0
$879$	44.1246	1.48829
$880$	0	0
$881$	−23.4508	−0.790079	−0.395040	−	0.918664i	$-0.629269\pi$
$881$	−23.4508	−0.790079	−0.395040	+	0.918664i	$0.629269\pi$
$882$	0	0
$883$	−13.9230	−0.468546	−0.234273	−	0.972171i	$-0.575271\pi$
$883$	−13.9230	−0.468546	−0.234273	+	0.972171i	$0.575271\pi$
$884$	0	0
$885$	49.5967	1.66718
$886$	0	0
$887$	−58.6525	−1.96936	−0.984679	−	0.174378i	$-0.944208\pi$
$887$	−58.6525	−1.96936	−0.984679	+	0.174378i	$0.944208\pi$
$888$	0	0
$889$	30.7082	1.02992
$890$	0	0
$891$	3.52786	0.118188
$892$	0	0
$893$	0	0
$894$	0	0
$895$	9.59675	0.320784
$896$	0	0
$897$	−19.9443	−0.665920
$898$	0	0
$899$	−2.96556	−0.0989069
$900$	0	0
$901$	48.8328	1.62686
$902$	0	0
$903$	53.8328	1.79144
$904$	0	0
$905$	−14.8328	−0.493059
$906$	0	0
$907$	17.5279	0.582003	0.291002	−	0.956723i	$-0.406012\pi$
$907$	17.5279	0.582003	0.291002	+	0.956723i	$0.406012\pi$
$908$	0	0
$909$	50.7984	1.68488
$910$	0	0
$911$	−5.61803	−0.186134	−0.0930669	−	0.995660i	$-0.529667\pi$
$911$	−5.61803	−0.186134	−0.0930669	+	0.995660i	$0.529667\pi$
$912$	0	0
$913$	−0.291796	−0.00965704
$914$	0	0
$915$	−18.6525	−0.616632
$916$	0	0
$917$	11.7295	0.387342
$918$	0	0
$919$	−36.7082	−1.21089	−0.605446	−	0.795886i	$-0.707005\pi$
$919$	−36.7082	−1.21089	−0.605446	+	0.795886i	$0.707005\pi$
$920$	0	0
$921$	−4.38197	−0.144391
$922$	0	0
$923$	1.47214	0.0484559
$924$	0	0
$925$	−7.45085	−0.244982
$926$	0	0
$927$	5.11146	0.167882
$928$	0	0
$929$	18.6180	0.610838	0.305419	−	0.952218i	$-0.401203\pi$
$929$	18.6180	0.610838	0.305419	+	0.952218i	$0.401203\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	−48.8328	−1.59871
$934$	0	0
$935$	4.00000	0.130814
$936$	0	0
$937$	20.4377	0.667670	0.333835	−	0.942631i	$-0.391657\pi$
$937$	20.4377	0.667670	0.333835	+	0.942631i	$0.391657\pi$
$938$	0	0
$939$	−11.3262	−0.369618
$940$	0	0
$941$	49.6869	1.61975	0.809874	−	0.586604i	$-0.199536\pi$
$941$	49.6869	1.61975	0.809874	+	0.586604i	$0.199536\pi$
$942$	0	0
$943$	22.8541	0.744232
$944$	0	0
$945$	−8.29180	−0.269732
$946$	0	0
$947$	1.34752	0.0437887	0.0218943	−	0.999760i	$-0.493030\pi$
$947$	1.34752	0.0437887	0.0218943	+	0.999760i	$0.493030\pi$
$948$	0	0
$949$	−10.7082	−0.347603
$950$	0	0
$951$	−47.1246	−1.52812
$952$	0	0
$953$	30.7082	0.994736	0.497368	−	0.867540i	$-0.334300\pi$
$953$	30.7082	0.994736	0.497368	+	0.867540i	$0.334300\pi$
$954$	0	0
$955$	12.0689	0.390540
$956$	0	0
$957$	−2.23607	−0.0722818
$958$	0	0
$959$	4.41641	0.142613
$960$	0	0
$961$	−26.3951	−0.851456
$962$	0	0
$963$	40.1459	1.29368
$964$	0	0
$965$	−28.3607	−0.912963
$966$	0	0
$967$	60.5410	1.94687	0.973434	−	0.228968i	$-0.0735351\pi$
$967$	60.5410	1.94687	0.973434	+	0.228968i	$0.0735351\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	14.5066	0.465538	0.232769	−	0.972532i	$-0.425221\pi$
$971$	14.5066	0.465538	0.232769	+	0.972532i	$0.425221\pi$
$972$	0	0
$973$	−29.3951	−0.942364
$974$	0	0
$975$	−9.09017	−0.291118
$976$	0	0
$977$	55.3607	1.77115	0.885573	−	0.464501i	$-0.153766\pi$
$977$	55.3607	1.77115	0.885573	+	0.464501i	$0.153766\pi$
$978$	0	0
$979$	7.56231	0.241692
$980$	0	0
$981$	−64.3951	−2.05598
$982$	0	0
$983$	−30.3820	−0.969034	−0.484517	−	0.874782i	$-0.661005\pi$
$983$	−30.3820	−0.969034	−0.484517	+	0.874782i	$0.661005\pi$
$984$	0	0
$985$	−3.70820	−0.118153
$986$	0	0
$987$	−23.5623	−0.749996
$988$	0	0
$989$	−52.2148	−1.66033
$990$	0	0
$991$	48.4508	1.53909	0.769546	−	0.638591i	$-0.220483\pi$
$991$	48.4508	1.53909	0.769546	+	0.638591i	$0.220483\pi$
$992$	0	0
$993$	−18.1803	−0.576936
$994$	0	0
$995$	16.5836	0.525735
$996$	0	0
$997$	26.2918	0.832670	0.416335	−	0.909211i	$-0.363314\pi$
$997$	26.2918	0.832670	0.416335	+	0.909211i	$0.363314\pi$
$998$	0	0
$999$	−4.79837	−0.151814

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5776.2.a.s.1.1		2
4.3	odd	2		361.2.a.f.1.1	yes	2
12.11	even	2		3249.2.a.i.1.2		2
19.18	odd	2		5776.2.a.bg.1.2		2
20.19	odd	2		9025.2.a.n.1.2		2
76.3	even	18		361.2.e.i.28.2		12
76.7	odd	6		361.2.c.d.68.2		4
76.11	odd	6		361.2.c.d.292.2		4
76.15	even	18		361.2.e.i.54.2		12
76.23	odd	18		361.2.e.j.54.1		12
76.27	even	6		361.2.c.g.292.1		4
76.31	even	6		361.2.c.g.68.1		4
76.35	odd	18		361.2.e.j.28.1		12
76.43	odd	18		361.2.e.j.234.1		12
76.47	odd	18		361.2.e.j.62.2		12
76.51	even	18		361.2.e.i.245.2		12
76.55	odd	18		361.2.e.j.99.2		12
76.59	even	18		361.2.e.i.99.1		12
76.63	odd	18		361.2.e.j.245.1		12
76.67	even	18		361.2.e.i.62.1		12
76.71	even	18		361.2.e.i.234.2		12
76.75	even	2		361.2.a.c.1.2	✓	2
228.227	odd	2		3249.2.a.o.1.1		2
380.379	even	2		9025.2.a.s.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
361.2.a.c.1.2	✓	2	76.75	even	2
361.2.a.f.1.1	yes	2	4.3	odd	2
361.2.c.d.68.2		4	76.7	odd	6
361.2.c.d.292.2		4	76.11	odd	6
361.2.c.g.68.1		4	76.31	even	6
361.2.c.g.292.1		4	76.27	even	6
361.2.e.i.28.2		12	76.3	even	18
361.2.e.i.54.2		12	76.15	even	18
361.2.e.i.62.1		12	76.67	even	18
361.2.e.i.99.1		12	76.59	even	18
361.2.e.i.234.2		12	76.71	even	18
361.2.e.i.245.2		12	76.51	even	18
361.2.e.j.28.1		12	76.35	odd	18
361.2.e.j.54.1		12	76.23	odd	18
361.2.e.j.62.2		12	76.47	odd	18
361.2.e.j.99.2		12	76.55	odd	18
361.2.e.j.234.1		12	76.43	odd	18
361.2.e.j.245.1		12	76.63	odd	18
3249.2.a.i.1.2		2	12.11	even	2
3249.2.a.o.1.1		2	228.227	odd	2
5776.2.a.s.1.1		2	1.1	even	1	trivial
5776.2.a.bg.1.2		2	19.18	odd	2
9025.2.a.n.1.2		2	20.19	odd	2
9025.2.a.s.1.1		2	380.379	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:	\( N \)	\(=\)	\( 5776 = 2^{4} \cdot 19^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5776.a (trivial)

\(f(q)\)	\(=\)	\(q-2.61803 q^{3} -1.23607 q^{5} -3.00000 q^{7} +3.85410 q^{9} +O(q^{10})\)	\(q-2.61803 q^{3} -1.23607 q^{5} -3.00000 q^{7} +3.85410 q^{9} -0.618034 q^{11} -1.00000 q^{13} +3.23607 q^{15} +5.23607 q^{17} +7.85410 q^{21} -7.61803 q^{23} -3.47214 q^{25} -2.23607 q^{27} -1.38197 q^{29} +2.14590 q^{31} +1.61803 q^{33} +3.70820 q^{35} +2.14590 q^{37} +2.61803 q^{39} -3.00000 q^{41} +6.85410 q^{43} -4.76393 q^{45} -3.00000 q^{47} +2.00000 q^{49} -13.7082 q^{51} +9.32624 q^{53} +0.763932 q^{55} +15.3262 q^{59} -5.76393 q^{61} -11.5623 q^{63} +1.23607 q^{65} +7.00000 q^{67} +19.9443 q^{69} -1.47214 q^{71} +10.7082 q^{73} +9.09017 q^{75} +1.85410 q^{77} +13.4164 q^{79} -5.70820 q^{81} +0.472136 q^{83} -6.47214 q^{85} +3.61803 q^{87} -12.2361 q^{89} +3.00000 q^{91} -5.61803 q^{93} +7.14590 q^{97} -2.38197 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 3 q^{3} + 2 q^{5} - 6 q^{7} + q^{9}+O(q^{10}) \) 2 * q - 3 * q^3 + 2 * q^5 - 6 * q^7 + q^9	\( 2 q - 3 q^{3} + 2 q^{5} - 6 q^{7} + q^{9} + q^{11} - 2 q^{13} + 2 q^{15} + 6 q^{17} + 9 q^{21} - 13 q^{23} + 2 q^{25} - 5 q^{29} + 11 q^{31} + q^{33} - 6 q^{35} + 11 q^{37} + 3 q^{39} - 6 q^{41} + 7 q^{43} - 14 q^{45} - 6 q^{47} + 4 q^{49} - 14 q^{51} + 3 q^{53} + 6 q^{55} + 15 q^{59} - 16 q^{61} - 3 q^{63} - 2 q^{65} + 14 q^{67} + 22 q^{69} + 6 q^{71} + 8 q^{73} + 7 q^{75} - 3 q^{77} + 2 q^{81} - 8 q^{83} - 4 q^{85} + 5 q^{87} - 20 q^{89} + 6 q^{91} - 9 q^{93} + 21 q^{97} - 7 q^{99}+O(q^{100}) \) 2 * q - 3 * q^3 + 2 * q^5 - 6 * q^7 + q^9 + q^11 - 2 * q^13 + 2 * q^15 + 6 * q^17 + 9 * q^21 - 13 * q^23 + 2 * q^25 - 5 * q^29 + 11 * q^31 + q^33 - 6 * q^35 + 11 * q^37 + 3 * q^39 - 6 * q^41 + 7 * q^43 - 14 * q^45 - 6 * q^47 + 4 * q^49 - 14 * q^51 + 3 * q^53 + 6 * q^55 + 15 * q^59 - 16 * q^61 - 3 * q^63 - 2 * q^65 + 14 * q^67 + 22 * q^69 + 6 * q^71 + 8 * q^73 + 7 * q^75 - 3 * q^77 + 2 * q^81 - 8 * q^83 - 4 * q^85 + 5 * q^87 - 20 * q^89 + 6 * q^91 - 9 * q^93 + 21 * q^97 - 7 * q^99

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