LMFDB - Embedded newform 5776.2.a.bn.1.3

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:	$0$
Dimension:	$3$
Coefficient field:	$\Q(\zeta_{18})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 3x - 1 $ x^3 - 3*x - 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 38)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$1.87939$ 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)

$f(q)$	$=$	$q+1.87939 q^{3} +2.00000 q^{5} -5.06418 q^{7} +0.532089 q^{9} +O(q^{10})$	$q+1.87939 q^{3} +2.00000 q^{5} -5.06418 q^{7} +0.532089 q^{9} +1.41147 q^{11} -1.30541 q^{13} +3.75877 q^{15} +2.38919 q^{17} -9.51754 q^{21} +3.06418 q^{23} -1.00000 q^{25} -4.63816 q^{27} +8.45336 q^{29} -0.369585 q^{31} +2.65270 q^{33} -10.1284 q^{35} +4.82295 q^{37} -2.45336 q^{39} -1.53209 q^{41} +0.758770 q^{43} +1.06418 q^{45} +10.2121 q^{47} +18.6459 q^{49} +4.49020 q^{51} +1.67499 q^{53} +2.82295 q^{55} -0.716881 q^{59} +9.75877 q^{61} -2.69459 q^{63} -2.61081 q^{65} +1.40373 q^{67} +5.75877 q^{69} +6.36959 q^{71} -4.55943 q^{73} -1.87939 q^{75} -7.14796 q^{77} +2.24123 q^{79} -10.3131 q^{81} +3.98545 q^{83} +4.77837 q^{85} +15.8871 q^{87} -10.6459 q^{89} +6.61081 q^{91} -0.694593 q^{93} -1.53209 q^{97} +0.751030 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 6 q^{5} - 6 q^{7} - 3 q^{9}+O(q^{10}) $ 3 * q + 6 * q^5 - 6 * q^7 - 3 * q^9	$ 3 q + 6 q^{5} - 6 q^{7} - 3 q^{9} - 6 q^{11} - 6 q^{13} + 3 q^{17} - 6 q^{21} - 3 q^{25} + 3 q^{27} + 12 q^{29} + 6 q^{31} + 9 q^{33} - 12 q^{35} - 6 q^{37} + 6 q^{39} - 9 q^{43} - 6 q^{45} + 6 q^{47} + 15 q^{49} + 12 q^{51} - 12 q^{55} + 6 q^{59} + 18 q^{61} - 6 q^{63} - 12 q^{65} + 18 q^{67} + 6 q^{69} + 12 q^{71} + 12 q^{73} - 6 q^{77} + 18 q^{79} - 9 q^{81} - 6 q^{83} + 6 q^{85} + 18 q^{87} + 9 q^{89} + 24 q^{91} + 15 q^{99}+O(q^{100}) $ 3 * q + 6 * q^5 - 6 * q^7 - 3 * q^9 - 6 * q^11 - 6 * q^13 + 3 * q^17 - 6 * q^21 - 3 * q^25 + 3 * q^27 + 12 * q^29 + 6 * q^31 + 9 * q^33 - 12 * q^35 - 6 * q^37 + 6 * q^39 - 9 * q^43 - 6 * q^45 + 6 * q^47 + 15 * q^49 + 12 * q^51 - 12 * q^55 + 6 * q^59 + 18 * q^61 - 6 * q^63 - 12 * q^65 + 18 * q^67 + 6 * q^69 + 12 * q^71 + 12 * q^73 - 6 * q^77 + 18 * q^79 - 9 * q^81 - 6 * q^83 + 6 * q^85 + 18 * q^87 + 9 * q^89 + 24 * q^91 + 15 * 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
$3$	1.87939	1.08506	0.542532	−	0.840035i	$-0.317466\pi$
$3$	1.87939	1.08506	0.542532	+	0.840035i	$0.317466\pi$
$4$	0	0
$5$	2.00000	0.894427	0.447214	−	0.894427i	$-0.352416\pi$
$5$	2.00000	0.894427	0.447214	+	0.894427i	$0.352416\pi$
$6$	0	0
$7$	−5.06418	−1.91408	−0.957040	−	0.289957i	$-0.906359\pi$
$7$	−5.06418	−1.91408	−0.957040	+	0.289957i	$0.906359\pi$
$8$	0	0
$9$	0.532089	0.177363
$10$	0	0
$11$	1.41147	0.425575	0.212788	−	0.977098i	$-0.431746\pi$
$11$	1.41147	0.425575	0.212788	+	0.977098i	$0.431746\pi$
$12$	0	0
$13$	−1.30541	−0.362055	−0.181027	−	0.983478i	$-0.557942\pi$
$13$	−1.30541	−0.362055	−0.181027	+	0.983478i	$0.557942\pi$
$14$	0	0
$15$	3.75877	0.970510
$16$	0	0
$17$	2.38919	0.579463	0.289731	−	0.957108i	$-0.406434\pi$
$17$	2.38919	0.579463	0.289731	+	0.957108i	$0.406434\pi$
$18$	0	0
$19$	0	0
$19$	0	0
$20$	0	0
$21$	−9.51754	−2.07690
$22$	0	0
$23$	3.06418	0.638925	0.319463	−	0.947599i	$-0.396498\pi$
$23$	3.06418	0.638925	0.319463	+	0.947599i	$0.396498\pi$
$24$	0	0
$25$	−1.00000	−0.200000
$26$	0	0
$27$	−4.63816	−0.892613
$28$	0	0
$29$	8.45336	1.56975	0.784875	−	0.619654i	$-0.212727\pi$
$29$	8.45336	1.56975	0.784875	+	0.619654i	$0.212727\pi$
$30$	0	0
$31$	−0.369585	−0.0663794	−0.0331897	−	0.999449i	$-0.510567\pi$
$31$	−0.369585	−0.0663794	−0.0331897	+	0.999449i	$0.510567\pi$
$32$	0	0
$33$	2.65270	0.461776
$34$	0	0
$35$	−10.1284	−1.71200
$36$	0	0
$37$	4.82295	0.792888	0.396444	−	0.918059i	$-0.370244\pi$
$37$	4.82295	0.792888	0.396444	+	0.918059i	$0.370244\pi$
$38$	0	0
$39$	−2.45336	−0.392853
$40$	0	0
$41$	−1.53209	−0.239272	−0.119636	−	0.992818i	$-0.538173\pi$
$41$	−1.53209	−0.239272	−0.119636	+	0.992818i	$0.538173\pi$
$42$	0	0
$43$	0.758770	0.115711	0.0578557	−	0.998325i	$-0.481574\pi$
$43$	0.758770	0.115711	0.0578557	+	0.998325i	$0.481574\pi$
$44$	0	0
$45$	1.06418	0.158638
$46$	0	0
$47$	10.2121	1.48959	0.744796	−	0.667292i	$-0.232547\pi$
$47$	10.2121	1.48959	0.744796	+	0.667292i	$0.232547\pi$
$48$	0	0
$49$	18.6459	2.66370
$50$	0	0
$51$	4.49020	0.628754
$52$	0	0
$53$	1.67499	0.230078	0.115039	−	0.993361i	$-0.463301\pi$
$53$	1.67499	0.230078	0.115039	+	0.993361i	$0.463301\pi$
$54$	0	0
$55$	2.82295	0.380646
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−0.716881	−0.0933300	−0.0466650	−	0.998911i	$-0.514859\pi$
$59$	−0.716881	−0.0933300	−0.0466650	+	0.998911i	$0.514859\pi$
$60$	0	0
$61$	9.75877	1.24948	0.624741	−	0.780832i	$-0.285204\pi$
$61$	9.75877	1.24948	0.624741	+	0.780832i	$0.285204\pi$
$62$	0	0
$63$	−2.69459	−0.339487
$64$	0	0
$65$	−2.61081	−0.323832
$66$	0	0
$67$	1.40373	0.171493	0.0857467	−	0.996317i	$-0.472672\pi$
$67$	1.40373	0.171493	0.0857467	+	0.996317i	$0.472672\pi$
$68$	0	0
$69$	5.75877	0.693274
$70$	0	0
$71$	6.36959	0.755931	0.377965	−	0.925820i	$-0.376624\pi$
$71$	6.36959	0.755931	0.377965	+	0.925820i	$0.376624\pi$
$72$	0	0
$73$	−4.55943	−0.533641	−0.266820	−	0.963746i	$-0.585973\pi$
$73$	−4.55943	−0.533641	−0.266820	+	0.963746i	$0.585973\pi$
$74$	0	0
$75$	−1.87939	−0.217013
$76$	0	0
$77$	−7.14796	−0.814585
$78$	0	0
$79$	2.24123	0.252158	0.126079	−	0.992020i	$-0.459761\pi$
$79$	2.24123	0.252158	0.126079	+	0.992020i	$0.459761\pi$
$80$	0	0
$81$	−10.3131	−1.14591
$82$	0	0
$83$	3.98545	0.437460	0.218730	−	0.975785i	$-0.429809\pi$
$83$	3.98545	0.437460	0.218730	+	0.975785i	$0.429809\pi$
$84$	0	0
$85$	4.77837	0.518287
$86$	0	0
$87$	15.8871	1.70328
$88$	0	0
$89$	−10.6459	−1.12846	−0.564231	−	0.825617i	$-0.690827\pi$
$89$	−10.6459	−1.12846	−0.564231	+	0.825617i	$0.690827\pi$
$90$	0	0
$91$	6.61081	0.693002
$92$	0	0
$93$	−0.694593	−0.0720259
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−1.53209	−0.155560	−0.0777800	−	0.996971i	$-0.524783\pi$
$97$	−1.53209	−0.155560	−0.0777800	+	0.996971i	$0.524783\pi$
$98$	0	0
$99$	0.751030	0.0754813
$100$	0	0
$101$	8.82295	0.877916	0.438958	−	0.898508i	$-0.355348\pi$
$101$	8.82295	0.877916	0.438958	+	0.898508i	$0.355348\pi$
$102$	0	0
$103$	7.14796	0.704309	0.352155	−	0.935942i	$-0.385449\pi$
$103$	7.14796	0.704309	0.352155	+	0.935942i	$0.385449\pi$
$104$	0	0
$105$	−19.0351	−1.85763
$106$	0	0
$107$	9.36959	0.905792	0.452896	−	0.891563i	$-0.350391\pi$
$107$	9.36959	0.905792	0.452896	+	0.891563i	$0.350391\pi$
$108$	0	0
$109$	11.0642	1.05976	0.529878	−	0.848074i	$-0.322238\pi$
$109$	11.0642	1.05976	0.529878	+	0.848074i	$0.322238\pi$
$110$	0	0
$111$	9.06418	0.860334
$112$	0	0
$113$	−13.2986	−1.25103	−0.625514	−	0.780213i	$-0.715110\pi$
$113$	−13.2986	−1.25103	−0.625514	+	0.780213i	$0.715110\pi$
$114$	0	0
$115$	6.12836	0.571472
$116$	0	0
$117$	−0.694593	−0.0642151
$118$	0	0
$119$	−12.0993	−1.10914
$120$	0	0
$121$	−9.00774	−0.818886
$122$	0	0
$123$	−2.87939	−0.259625
$124$	0	0
$125$	−12.0000	−1.07331
$126$	0	0
$127$	5.71419	0.507053	0.253526	−	0.967328i	$-0.418410\pi$
$127$	5.71419	0.507053	0.253526	+	0.967328i	$0.418410\pi$
$128$	0	0
$129$	1.42602	0.125554
$130$	0	0
$131$	9.87939	0.863166	0.431583	−	0.902073i	$-0.357955\pi$
$131$	9.87939	0.863166	0.431583	+	0.902073i	$0.357955\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	−9.27631	−0.798378
$136$	0	0
$137$	5.39693	0.461091	0.230545	−	0.973062i	$-0.425949\pi$
$137$	5.39693	0.461091	0.230545	+	0.973062i	$0.425949\pi$
$138$	0	0
$139$	2.33956	0.198439	0.0992193	−	0.995066i	$-0.468365\pi$
$139$	2.33956	0.198439	0.0992193	+	0.995066i	$0.468365\pi$
$140$	0	0
$141$	19.1925	1.61630
$142$	0	0
$143$	−1.84255	−0.154082
$144$	0	0
$145$	16.9067	1.40403
$146$	0	0
$147$	35.0428	2.89028
$148$	0	0
$149$	−14.3696	−1.17720	−0.588601	−	0.808424i	$-0.700321\pi$
$149$	−14.3696	−1.17720	−0.588601	+	0.808424i	$0.700321\pi$
$150$	0	0
$151$	20.8384	1.69581	0.847904	−	0.530150i	$-0.177865\pi$
$151$	20.8384	1.69581	0.847904	+	0.530150i	$0.177865\pi$
$152$	0	0
$153$	1.27126	0.102775
$154$	0	0
$155$	−0.739170	−0.0593716
$156$	0	0
$157$	6.36959	0.508348	0.254174	−	0.967158i	$-0.418196\pi$
$157$	6.36959	0.508348	0.254174	+	0.967158i	$0.418196\pi$
$158$	0	0
$159$	3.14796	0.249649
$160$	0	0
$161$	−15.5175	−1.22295
$162$	0	0
$163$	−4.73143	−0.370594	−0.185297	−	0.982683i	$-0.559325\pi$
$163$	−4.73143	−0.370594	−0.185297	+	0.982683i	$0.559325\pi$
$164$	0	0
$165$	5.30541	0.413025
$166$	0	0
$167$	−17.2763	−1.33688	−0.668441	−	0.743766i	$-0.733038\pi$
$167$	−17.2763	−1.33688	−0.668441	+	0.743766i	$0.733038\pi$
$168$	0	0
$169$	−11.2959	−0.868916
$170$	0	0
$171$	0	0
$172$	0	0
$173$	18.8229	1.43108	0.715541	−	0.698571i	$-0.246180\pi$
$173$	18.8229	1.43108	0.715541	+	0.698571i	$0.246180\pi$
$174$	0	0
$175$	5.06418	0.382816
$176$	0	0
$177$	−1.34730	−0.101269
$178$	0	0
$179$	13.8280	1.03355	0.516777	−	0.856120i	$-0.327132\pi$
$179$	13.8280	1.03355	0.516777	+	0.856120i	$0.327132\pi$
$180$	0	0
$181$	12.2567	0.911034	0.455517	−	0.890227i	$-0.349454\pi$
$181$	12.2567	0.911034	0.455517	+	0.890227i	$0.349454\pi$
$182$	0	0
$183$	18.3405	1.35577
$184$	0	0
$185$	9.64590	0.709180
$186$	0	0
$187$	3.37227	0.246605
$188$	0	0
$189$	23.4884	1.70853
$190$	0	0
$191$	20.0993	1.45433	0.727166	−	0.686462i	$-0.240837\pi$
$191$	20.0993	1.45433	0.727166	+	0.686462i	$0.240837\pi$
$192$	0	0
$193$	16.6851	1.20102	0.600510	−	0.799617i	$-0.294964\pi$
$193$	16.6851	1.20102	0.600510	+	0.799617i	$0.294964\pi$
$194$	0	0
$195$	−4.90673	−0.351378
$196$	0	0
$197$	12.9905	0.925535	0.462768	−	0.886480i	$-0.346856\pi$
$197$	12.9905	0.925535	0.462768	+	0.886480i	$0.346856\pi$
$198$	0	0
$199$	−17.1925	−1.21875	−0.609373	−	0.792884i	$-0.708579\pi$
$199$	−17.1925	−1.21875	−0.609373	+	0.792884i	$0.708579\pi$
$200$	0	0
$201$	2.63816	0.186081
$202$	0	0
$203$	−42.8093	−3.00463
$204$	0	0
$205$	−3.06418	−0.214011
$206$	0	0
$207$	1.63041	0.113322
$208$	0	0
$209$	0	0
$210$	0	0
$211$	16.1088	1.10897	0.554486	−	0.832193i	$-0.312915\pi$
$211$	16.1088	1.10897	0.554486	+	0.832193i	$0.312915\pi$
$212$	0	0
$213$	11.9709	0.820233
$214$	0	0
$215$	1.51754	0.103495
$216$	0	0
$217$	1.87164	0.127056
$218$	0	0
$219$	−8.56893	−0.579034
$220$	0	0
$221$	−3.11886	−0.209797
$222$	0	0
$223$	−4.08378	−0.273470	−0.136735	−	0.990608i	$-0.543661\pi$
$223$	−4.08378	−0.273470	−0.136735	+	0.990608i	$0.543661\pi$
$224$	0	0
$225$	−0.532089	−0.0354726
$226$	0	0
$227$	−13.6604	−0.906676	−0.453338	−	0.891339i	$-0.649767\pi$
$227$	−13.6604	−0.906676	−0.453338	+	0.891339i	$0.649767\pi$
$228$	0	0
$229$	−5.22163	−0.345055	−0.172527	−	0.985005i	$-0.555193\pi$
$229$	−5.22163	−0.345055	−0.172527	+	0.985005i	$0.555193\pi$
$230$	0	0
$231$	−13.4338	−0.883877
$232$	0	0
$233$	−27.0428	−1.77163	−0.885817	−	0.464035i	$-0.846401\pi$
$233$	−27.0428	−1.77163	−0.885817	+	0.464035i	$0.846401\pi$
$234$	0	0
$235$	20.4243	1.33233
$236$	0	0
$237$	4.21213	0.273607
$238$	0	0
$239$	0.285807	0.0184873	0.00924366	−	0.999957i	$-0.497058\pi$
$239$	0.285807	0.0184873	0.00924366	+	0.999957i	$0.497058\pi$
$240$	0	0
$241$	3.10101	0.199754	0.0998769	−	0.995000i	$-0.468155\pi$
$241$	3.10101	0.199754	0.0998769	+	0.995000i	$0.468155\pi$
$242$	0	0
$243$	−5.46791	−0.350767
$244$	0	0
$245$	37.2918	2.38249
$246$	0	0
$247$	0	0
$248$	0	0
$249$	7.49020	0.474672
$250$	0	0
$251$	12.6578	0.798950	0.399475	−	0.916744i	$-0.369192\pi$
$251$	12.6578	0.798950	0.399475	+	0.916744i	$0.369192\pi$
$252$	0	0
$253$	4.32501	0.271911
$254$	0	0
$255$	8.98040	0.562374
$256$	0	0
$257$	−30.3928	−1.89585	−0.947926	−	0.318492i	$-0.896824\pi$
$257$	−30.3928	−1.89585	−0.947926	+	0.318492i	$0.896824\pi$
$258$	0	0
$259$	−24.4243	−1.51765
$260$	0	0
$261$	4.49794	0.278416
$262$	0	0
$263$	−21.9026	−1.35057	−0.675286	−	0.737556i	$-0.735980\pi$
$263$	−21.9026	−1.35057	−0.675286	+	0.737556i	$0.735980\pi$
$264$	0	0
$265$	3.34998	0.205788
$266$	0	0
$267$	−20.0077	−1.22445
$268$	0	0
$269$	−1.43376	−0.0874181	−0.0437090	−	0.999044i	$-0.513917\pi$
$269$	−1.43376	−0.0874181	−0.0437090	+	0.999044i	$0.513917\pi$
$270$	0	0
$271$	−16.1729	−0.982436	−0.491218	−	0.871037i	$-0.663448\pi$
$271$	−16.1729	−0.982436	−0.491218	+	0.871037i	$0.663448\pi$
$272$	0	0
$273$	12.4243	0.751951
$274$	0	0
$275$	−1.41147	−0.0851151
$276$	0	0
$277$	11.1088	0.667460	0.333730	−	0.942669i	$-0.391693\pi$
$277$	11.1088	0.667460	0.333730	+	0.942669i	$0.391693\pi$
$278$	0	0
$279$	−0.196652	−0.0117733
$280$	0	0
$281$	4.24897	0.253472	0.126736	−	0.991936i	$-0.459550\pi$
$281$	4.24897	0.253472	0.126736	+	0.991936i	$0.459550\pi$
$282$	0	0
$283$	5.45605	0.324329	0.162164	−	0.986764i	$-0.448153\pi$
$283$	5.45605	0.324329	0.162164	+	0.986764i	$0.448153\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	7.75877	0.457986
$288$	0	0
$289$	−11.2918	−0.664223
$290$	0	0
$291$	−2.87939	−0.168793
$292$	0	0
$293$	17.8135	1.04067	0.520337	−	0.853961i	$-0.325807\pi$
$293$	17.8135	1.04067	0.520337	+	0.853961i	$0.325807\pi$
$294$	0	0
$295$	−1.43376	−0.0834769
$296$	0	0
$297$	−6.54664	−0.379874
$298$	0	0
$299$	−4.00000	−0.231326
$300$	0	0
$301$	−3.84255	−0.221481
$302$	0	0
$303$	16.5817	0.952595
$304$	0	0
$305$	19.5175	1.11757
$306$	0	0
$307$	28.6587	1.63564	0.817819	−	0.575476i	$-0.195183\pi$
$307$	28.6587	1.63564	0.817819	+	0.575476i	$0.195183\pi$
$308$	0	0
$309$	13.4338	0.764220
$310$	0	0
$311$	−15.8135	−0.896699	−0.448349	−	0.893858i	$-0.647988\pi$
$311$	−15.8135	−0.896699	−0.448349	+	0.893858i	$0.647988\pi$
$312$	0	0
$313$	13.1402	0.742729	0.371364	−	0.928487i	$-0.378890\pi$
$313$	13.1402	0.742729	0.371364	+	0.928487i	$0.378890\pi$
$314$	0	0
$315$	−5.38919	−0.303646
$316$	0	0
$317$	−21.4047	−1.20221	−0.601103	−	0.799172i	$-0.705272\pi$
$317$	−21.4047	−1.20221	−0.601103	+	0.799172i	$0.705272\pi$
$318$	0	0
$319$	11.9317	0.668047
$320$	0	0
$321$	17.6091	0.982842
$322$	0	0
$323$	0	0
$324$	0	0
$325$	1.30541	0.0724110
$326$	0	0
$327$	20.7939	1.14990
$328$	0	0
$329$	−51.7161	−2.85120
$330$	0	0
$331$	25.3979	1.39599	0.697996	−	0.716101i	$-0.254075\pi$
$331$	25.3979	1.39599	0.697996	+	0.716101i	$0.254075\pi$
$332$	0	0
$333$	2.56624	0.140629
$334$	0	0
$335$	2.80747	0.153388
$336$	0	0
$337$	26.3773	1.43686	0.718432	−	0.695597i	$-0.244860\pi$
$337$	26.3773	1.43686	0.718432	+	0.695597i	$0.244860\pi$
$338$	0	0
$339$	−24.9932	−1.35744
$340$	0	0
$341$	−0.521660	−0.0282495
$342$	0	0
$343$	−58.9769	−3.18445
$344$	0	0
$345$	11.5175	0.620084
$346$	0	0
$347$	25.6313	1.37596	0.687981	−	0.725728i	$-0.258497\pi$
$347$	25.6313	1.37596	0.687981	+	0.725728i	$0.258497\pi$
$348$	0	0
$349$	−5.84255	−0.312744	−0.156372	−	0.987698i	$-0.549980\pi$
$349$	−5.84255	−0.312744	−0.156372	+	0.987698i	$0.549980\pi$
$350$	0	0
$351$	6.05468	0.323175
$352$	0	0
$353$	−22.4097	−1.19275	−0.596375	−	0.802706i	$-0.703393\pi$
$353$	−22.4097	−1.19275	−0.596375	+	0.802706i	$0.703393\pi$
$354$	0	0
$355$	12.7392	0.676125
$356$	0	0
$357$	−22.7392	−1.20348
$358$	0	0
$359$	−9.61680	−0.507555	−0.253778	−	0.967263i	$-0.581673\pi$
$359$	−9.61680	−0.507555	−0.253778	+	0.967263i	$0.581673\pi$
$360$	0	0
$361$	0	0
$362$	0	0
$363$	−16.9290	−0.888543
$364$	0	0
$365$	−9.11886	−0.477303
$366$	0	0
$367$	−23.3601	−1.21939	−0.609693	−	0.792637i	$-0.708707\pi$
$367$	−23.3601	−1.21939	−0.609693	+	0.792637i	$0.708707\pi$
$368$	0	0
$369$	−0.815207	−0.0424380
$370$	0	0
$371$	−8.48246	−0.440387
$372$	0	0
$373$	−25.9418	−1.34322	−0.671608	−	0.740907i	$-0.734396\pi$
$373$	−25.9418	−1.34322	−0.671608	+	0.740907i	$0.734396\pi$
$374$	0	0
$375$	−22.5526	−1.16461
$376$	0	0
$377$	−11.0351	−0.568336
$378$	0	0
$379$	−9.47834	−0.486870	−0.243435	−	0.969917i	$-0.578274\pi$
$379$	−9.47834	−0.486870	−0.243435	+	0.969917i	$0.578274\pi$
$380$	0	0
$381$	10.7392	0.550184
$382$	0	0
$383$	13.4884	0.689227	0.344614	−	0.938745i	$-0.388010\pi$
$383$	13.4884	0.689227	0.344614	+	0.938745i	$0.388010\pi$
$384$	0	0
$385$	−14.2959	−0.728587
$386$	0	0
$387$	0.403733	0.0205229
$388$	0	0
$389$	−25.1480	−1.27505	−0.637526	−	0.770429i	$-0.720042\pi$
$389$	−25.1480	−1.27505	−0.637526	+	0.770429i	$0.720042\pi$
$390$	0	0
$391$	7.32089	0.370233
$392$	0	0
$393$	18.5672	0.936590
$394$	0	0
$395$	4.48246	0.225537
$396$	0	0
$397$	6.56624	0.329550	0.164775	−	0.986331i	$-0.447310\pi$
$397$	6.56624	0.329550	0.164775	+	0.986331i	$0.447310\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−29.5107	−1.47370	−0.736848	−	0.676059i	$-0.763687\pi$
$401$	−29.5107	−1.47370	−0.736848	+	0.676059i	$0.763687\pi$
$402$	0	0
$403$	0.482459	0.0240330
$404$	0	0
$405$	−20.6263	−1.02493
$406$	0	0
$407$	6.80747	0.337434
$408$	0	0
$409$	21.1310	1.04486	0.522431	−	0.852681i	$-0.325025\pi$
$409$	21.1310	1.04486	0.522431	+	0.852681i	$0.325025\pi$
$410$	0	0
$411$	10.1429	0.500313
$412$	0	0
$413$	3.63041	0.178641
$414$	0	0
$415$	7.97090	0.391276
$416$	0	0
$417$	4.39693	0.215318
$418$	0	0
$419$	27.8830	1.36217	0.681087	−	0.732202i	$-0.261508\pi$
$419$	27.8830	1.36217	0.681087	+	0.732202i	$0.261508\pi$
$420$	0	0
$421$	−0.0445774	−0.00217257	−0.00108629	−	0.999999i	$-0.500346\pi$
$421$	−0.0445774	−0.00217257	−0.00108629	+	0.999999i	$0.500346\pi$
$422$	0	0
$423$	5.43376	0.264199
$424$	0	0
$425$	−2.38919	−0.115893
$426$	0	0
$427$	−49.4201	−2.39161
$428$	0	0
$429$	−3.46286	−0.167188
$430$	0	0
$431$	11.7879	0.567802	0.283901	−	0.958854i	$-0.408371\pi$
$431$	11.7879	0.567802	0.283901	+	0.958854i	$0.408371\pi$
$432$	0	0
$433$	27.6459	1.32858	0.664289	−	0.747476i	$-0.268735\pi$
$433$	27.6459	1.32858	0.664289	+	0.747476i	$0.268735\pi$
$434$	0	0
$435$	31.7743	1.52346
$436$	0	0
$437$	0	0
$438$	0	0
$439$	−37.4492	−1.78735	−0.893677	−	0.448710i	$-0.851884\pi$
$439$	−37.4492	−1.78735	−0.893677	+	0.448710i	$0.851884\pi$
$440$	0	0
$441$	9.92127	0.472442
$442$	0	0
$443$	−20.9881	−0.997177	−0.498588	−	0.866839i	$-0.666148\pi$
$443$	−20.9881	−0.997177	−0.498588	+	0.866839i	$0.666148\pi$
$444$	0	0
$445$	−21.2918	−1.00933
$446$	0	0
$447$	−27.0060	−1.27734
$448$	0	0
$449$	−21.8949	−1.03328	−0.516641	−	0.856202i	$-0.672818\pi$
$449$	−21.8949	−1.03328	−0.516641	+	0.856202i	$0.672818\pi$
$450$	0	0
$451$	−2.16250	−0.101828
$452$	0	0
$453$	39.1634	1.84006
$454$	0	0
$455$	13.2216	0.619840
$456$	0	0
$457$	−13.0496	−0.610436	−0.305218	−	0.952283i	$-0.598729\pi$
$457$	−13.0496	−0.610436	−0.305218	+	0.952283i	$0.598729\pi$
$458$	0	0
$459$	−11.0814	−0.517236
$460$	0	0
$461$	4.40879	0.205338	0.102669	−	0.994716i	$-0.467262\pi$
$461$	4.40879	0.205338	0.102669	+	0.994716i	$0.467262\pi$
$462$	0	0
$463$	−26.6655	−1.23925	−0.619625	−	0.784898i	$-0.712715\pi$
$463$	−26.6655	−1.23925	−0.619625	+	0.784898i	$0.712715\pi$
$464$	0	0
$465$	−1.38919	−0.0644219
$466$	0	0
$467$	−30.1138	−1.39350	−0.696750	−	0.717314i	$-0.745371\pi$
$467$	−30.1138	−1.39350	−0.696750	+	0.717314i	$0.745371\pi$
$468$	0	0
$469$	−7.10876	−0.328252
$470$	0	0
$471$	11.9709	0.551590
$472$	0	0
$473$	1.07098	0.0492439
$474$	0	0
$475$	0	0
$476$	0	0
$477$	0.891245	0.0408073
$478$	0	0
$479$	−8.16756	−0.373185	−0.186593	−	0.982437i	$-0.559744\pi$
$479$	−8.16756	−0.373185	−0.186593	+	0.982437i	$0.559744\pi$
$480$	0	0
$481$	−6.29591	−0.287069
$482$	0	0
$483$	−29.1634	−1.32698
$484$	0	0
$485$	−3.06418	−0.139137
$486$	0	0
$487$	−26.5871	−1.20478	−0.602388	−	0.798203i	$-0.705784\pi$
$487$	−26.5871	−1.20478	−0.602388	+	0.798203i	$0.705784\pi$
$488$	0	0
$489$	−8.89218	−0.402118
$490$	0	0
$491$	−38.8289	−1.75233	−0.876163	−	0.482016i	$-0.839905\pi$
$491$	−38.8289	−1.75233	−0.876163	+	0.482016i	$0.839905\pi$
$492$	0	0
$493$	20.1967	0.909611
$494$	0	0
$495$	1.50206	0.0675125
$496$	0	0
$497$	−32.2567	−1.44691
$498$	0	0
$499$	−20.6587	−0.924810	−0.462405	−	0.886669i	$-0.653013\pi$
$499$	−20.6587	−0.924810	−0.462405	+	0.886669i	$0.653013\pi$
$500$	0	0
$501$	−32.4688	−1.45060
$502$	0	0
$503$	−0.0736733	−0.00328493	−0.00164246	−	0.999999i	$-0.500523\pi$
$503$	−0.0736733	−0.00328493	−0.00164246	+	0.999999i	$0.500523\pi$
$504$	0	0
$505$	17.6459	0.785232
$506$	0	0
$507$	−21.2294	−0.942829
$508$	0	0
$509$	15.0196	0.665732	0.332866	−	0.942974i	$-0.391984\pi$
$509$	15.0196	0.665732	0.332866	+	0.942974i	$0.391984\pi$
$510$	0	0
$511$	23.0898	1.02143
$512$	0	0
$513$	0	0
$514$	0	0
$515$	14.2959	0.629953
$516$	0	0
$517$	14.4142	0.633934
$518$	0	0
$519$	35.3756	1.55282
$520$	0	0
$521$	45.1712	1.97899	0.989493	−	0.144583i	$-0.0461842\pi$
$521$	45.1712	1.97899	0.989493	+	0.144583i	$0.0461842\pi$
$522$	0	0
$523$	−0.167556	−0.00732672	−0.00366336	−	0.999993i	$-0.501166\pi$
$523$	−0.167556	−0.00732672	−0.00366336	+	0.999993i	$0.501166\pi$
$524$	0	0
$525$	9.51754	0.415380
$526$	0	0
$527$	−0.883007	−0.0384644
$528$	0	0
$529$	−13.6108	−0.591775
$530$	0	0
$531$	−0.381445	−0.0165533
$532$	0	0
$533$	2.00000	0.0866296
$534$	0	0
$535$	18.7392	0.810165
$536$	0	0
$537$	25.9881	1.12147
$538$	0	0
$539$	26.3182	1.13361
$540$	0	0
$541$	−0.980400	−0.0421507	−0.0210753	−	0.999778i	$-0.506709\pi$
$541$	−0.980400	−0.0421507	−0.0210753	+	0.999778i	$0.506709\pi$
$542$	0	0
$543$	23.0351	0.988530
$544$	0	0
$545$	22.1284	0.947875
$546$	0	0
$547$	−28.4047	−1.21450	−0.607248	−	0.794512i	$-0.707727\pi$
$547$	−28.4047	−1.21450	−0.607248	+	0.794512i	$0.707727\pi$
$548$	0	0
$549$	5.19253	0.221612
$550$	0	0
$551$	0	0
$552$	0	0
$553$	−11.3500	−0.482650
$554$	0	0
$555$	18.1284	0.769506
$556$	0	0
$557$	35.6323	1.50979	0.754894	−	0.655847i	$-0.227688\pi$
$557$	35.6323	1.50979	0.754894	+	0.655847i	$0.227688\pi$
$558$	0	0
$559$	−0.990505	−0.0418939
$560$	0	0
$561$	6.33780	0.267582
$562$	0	0
$563$	−8.62773	−0.363615	−0.181808	−	0.983334i	$-0.558195\pi$
$563$	−8.62773	−0.363615	−0.181808	+	0.983334i	$0.558195\pi$
$564$	0	0
$565$	−26.5972	−1.11895
$566$	0	0
$567$	52.2276	2.19335
$568$	0	0
$569$	22.3310	0.936164	0.468082	−	0.883685i	$-0.344945\pi$
$569$	22.3310	0.936164	0.468082	+	0.883685i	$0.344945\pi$
$570$	0	0
$571$	9.56448	0.400261	0.200131	−	0.979769i	$-0.435863\pi$
$571$	9.56448	0.400261	0.200131	+	0.979769i	$0.435863\pi$
$572$	0	0
$573$	37.7743	1.57804
$574$	0	0
$575$	−3.06418	−0.127785
$576$	0	0
$577$	22.4757	0.935674	0.467837	−	0.883815i	$-0.345033\pi$
$577$	22.4757	0.935674	0.467837	+	0.883815i	$0.345033\pi$
$578$	0	0
$579$	31.3577	1.30318
$580$	0	0
$581$	−20.1830	−0.837334
$582$	0	0
$583$	2.36421	0.0979155
$584$	0	0
$585$	−1.38919	−0.0574357
$586$	0	0
$587$	−4.14796	−0.171204	−0.0856022	−	0.996329i	$-0.527281\pi$
$587$	−4.14796	−0.171204	−0.0856022	+	0.996329i	$0.527281\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	24.4142	1.00426
$592$	0	0
$593$	10.4175	0.427794	0.213897	−	0.976856i	$-0.431384\pi$
$593$	10.4175	0.427794	0.213897	+	0.976856i	$0.431384\pi$
$594$	0	0
$595$	−24.1985	−0.992043
$596$	0	0
$597$	−32.3114	−1.32242
$598$	0	0
$599$	39.6560	1.62030	0.810150	−	0.586222i	$-0.199386\pi$
$599$	39.6560	1.62030	0.810150	+	0.586222i	$0.199386\pi$
$600$	0	0
$601$	23.8648	0.973467	0.486734	−	0.873551i	$-0.338188\pi$
$601$	23.8648	0.973467	0.486734	+	0.873551i	$0.338188\pi$
$602$	0	0
$603$	0.746911	0.0304166
$604$	0	0
$605$	−18.0155	−0.732433
$606$	0	0
$607$	29.9317	1.21489	0.607445	−	0.794362i	$-0.292194\pi$
$607$	29.9317	1.21489	0.607445	+	0.794362i	$0.292194\pi$
$608$	0	0
$609$	−80.4552	−3.26021
$610$	0	0
$611$	−13.3310	−0.539314
$612$	0	0
$613$	−35.5776	−1.43697	−0.718483	−	0.695545i	$-0.755163\pi$
$613$	−35.5776	−1.43697	−0.718483	+	0.695545i	$0.755163\pi$
$614$	0	0
$615$	−5.75877	−0.232216
$616$	0	0
$617$	12.0324	0.484406	0.242203	−	0.970226i	$-0.422130\pi$
$617$	12.0324	0.484406	0.242203	+	0.970226i	$0.422130\pi$
$618$	0	0
$619$	17.1129	0.687824	0.343912	−	0.939002i	$-0.388248\pi$
$619$	17.1129	0.687824	0.343912	+	0.939002i	$0.388248\pi$
$620$	0	0
$621$	−14.2121	−0.570313
$622$	0	0
$623$	53.9127	2.15997
$624$	0	0
$625$	−19.0000	−0.760000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	11.5229	0.459449
$630$	0	0
$631$	−18.3405	−0.730123	−0.365062	−	0.930983i	$-0.618952\pi$
$631$	−18.3405	−0.730123	−0.365062	+	0.930983i	$0.618952\pi$
$632$	0	0
$633$	30.2746	1.20331
$634$	0	0
$635$	11.4284	0.453522
$636$	0	0
$637$	−24.3405	−0.964405
$638$	0	0
$639$	3.38919	0.134074
$640$	0	0
$641$	−31.1780	−1.23146	−0.615728	−	0.787959i	$-0.711138\pi$
$641$	−31.1780	−1.23146	−0.615728	+	0.787959i	$0.711138\pi$
$642$	0	0
$643$	31.9495	1.25997	0.629984	−	0.776608i	$-0.283062\pi$
$643$	31.9495	1.25997	0.629984	+	0.776608i	$0.283062\pi$
$644$	0	0
$645$	2.85204	0.112299
$646$	0	0
$647$	−2.99588	−0.117780	−0.0588901	−	0.998264i	$-0.518756\pi$
$647$	−2.99588	−0.117780	−0.0588901	+	0.998264i	$0.518756\pi$
$648$	0	0
$649$	−1.01186	−0.0397190
$650$	0	0
$651$	3.51754	0.137863
$652$	0	0
$653$	0.935822	0.0366216	0.0183108	−	0.999832i	$-0.494171\pi$
$653$	0.935822	0.0366216	0.0183108	+	0.999832i	$0.494171\pi$
$654$	0	0
$655$	19.7588	0.772039
$656$	0	0
$657$	−2.42602	−0.0946481
$658$	0	0
$659$	12.2371	0.476690	0.238345	−	0.971181i	$-0.423395\pi$
$659$	12.2371	0.476690	0.238345	+	0.971181i	$0.423395\pi$
$660$	0	0
$661$	−11.9554	−0.465012	−0.232506	−	0.972595i	$-0.574693\pi$
$661$	−11.9554	−0.465012	−0.232506	+	0.972595i	$0.574693\pi$
$662$	0	0
$663$	−5.86154	−0.227643
$664$	0	0
$665$	0	0
$666$	0	0
$667$	25.9026	1.00295
$668$	0	0
$669$	−7.67499	−0.296732
$670$	0	0
$671$	13.7743	0.531749
$672$	0	0
$673$	44.8634	1.72936	0.864679	−	0.502325i	$-0.167522\pi$
$673$	44.8634	1.72936	0.864679	+	0.502325i	$0.167522\pi$
$674$	0	0
$675$	4.63816	0.178523
$676$	0	0
$677$	0.945927	0.0363549	0.0181775	−	0.999835i	$-0.494214\pi$
$677$	0.945927	0.0363549	0.0181775	+	0.999835i	$0.494214\pi$
$678$	0	0
$679$	7.75877	0.297754
$680$	0	0
$681$	−25.6732	−0.983801
$682$	0	0
$683$	5.92221	0.226607	0.113303	−	0.993560i	$-0.463857\pi$
$683$	5.92221	0.226607	0.113303	+	0.993560i	$0.463857\pi$
$684$	0	0
$685$	10.7939	0.412412
$686$	0	0
$687$	−9.81345	−0.374407
$688$	0	0
$689$	−2.18655	−0.0833008
$690$	0	0
$691$	−0.206148	−0.00784222	−0.00392111	−	0.999992i	$-0.501248\pi$
$691$	−0.206148	−0.00784222	−0.00392111	+	0.999992i	$0.501248\pi$
$692$	0	0
$693$	−3.80335	−0.144477
$694$	0	0
$695$	4.67911	0.177489
$696$	0	0
$697$	−3.66044	−0.138649
$698$	0	0
$699$	−50.8239	−1.92234
$700$	0	0
$701$	4.36959	0.165037	0.0825185	−	0.996590i	$-0.473704\pi$
$701$	4.36959	0.165037	0.0825185	+	0.996590i	$0.473704\pi$
$702$	0	0
$703$	0	0
$704$	0	0
$705$	38.3851	1.44567
$706$	0	0
$707$	−44.6810	−1.68040
$708$	0	0
$709$	8.75465	0.328788	0.164394	−	0.986395i	$-0.447433\pi$
$709$	8.75465	0.328788	0.164394	+	0.986395i	$0.447433\pi$
$710$	0	0
$711$	1.19253	0.0447235
$712$	0	0
$713$	−1.13247	−0.0424115
$714$	0	0
$715$	−3.68510	−0.137815
$716$	0	0
$717$	0.537141	0.0200599
$718$	0	0
$719$	27.6067	1.02956	0.514778	−	0.857324i	$-0.327874\pi$
$719$	27.6067	1.02956	0.514778	+	0.857324i	$0.327874\pi$
$720$	0	0
$721$	−36.1985	−1.34810
$722$	0	0
$723$	5.82800	0.216746
$724$	0	0
$725$	−8.45336	−0.313950
$726$	0	0
$727$	−28.6364	−1.06207	−0.531033	−	0.847351i	$-0.678196\pi$
$727$	−28.6364	−1.06207	−0.531033	+	0.847351i	$0.678196\pi$
$728$	0	0
$729$	20.6631	0.765301
$730$	0	0
$731$	1.81284	0.0670504
$732$	0	0
$733$	36.9614	1.36520	0.682600	−	0.730792i	$-0.260849\pi$
$733$	36.9614	1.36520	0.682600	+	0.730792i	$0.260849\pi$
$734$	0	0
$735$	70.0856	2.58515
$736$	0	0
$737$	1.98133	0.0729833
$738$	0	0
$739$	−46.0265	−1.69311	−0.846556	−	0.532299i	$-0.821328\pi$
$739$	−46.0265	−1.69311	−0.846556	+	0.532299i	$0.821328\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−26.4107	−0.968913	−0.484456	−	0.874815i	$-0.660983\pi$
$743$	−26.4107	−0.968913	−0.484456	+	0.874815i	$0.660983\pi$
$744$	0	0
$745$	−28.7392	−1.05292
$746$	0	0
$747$	2.12061	0.0775892
$748$	0	0
$749$	−47.4492	−1.73376
$750$	0	0
$751$	27.1771	0.991705	0.495852	−	0.868407i	$-0.334856\pi$
$751$	27.1771	0.991705	0.495852	+	0.868407i	$0.334856\pi$
$752$	0	0
$753$	23.7888	0.866912
$754$	0	0
$755$	41.6769	1.51678
$756$	0	0
$757$	19.4047	0.705275	0.352637	−	0.935760i	$-0.385285\pi$
$757$	19.4047	0.705275	0.352637	+	0.935760i	$0.385285\pi$
$758$	0	0
$759$	8.12836	0.295041
$760$	0	0
$761$	40.2645	1.45959	0.729793	−	0.683669i	$-0.239617\pi$
$761$	40.2645	1.45959	0.729793	+	0.683669i	$0.239617\pi$
$762$	0	0
$763$	−56.0310	−2.02846
$764$	0	0
$765$	2.54252	0.0919249
$766$	0	0
$767$	0.935822	0.0337906
$768$	0	0
$769$	−38.9418	−1.40428	−0.702139	−	0.712040i	$-0.747771\pi$
$769$	−38.9418	−1.40428	−0.702139	+	0.712040i	$0.747771\pi$
$770$	0	0
$771$	−57.1198	−2.05712
$772$	0	0
$773$	−6.76289	−0.243244	−0.121622	−	0.992576i	$-0.538810\pi$
$773$	−6.76289	−0.243244	−0.121622	+	0.992576i	$0.538810\pi$
$774$	0	0
$775$	0.369585	0.0132759
$776$	0	0
$777$	−45.9026	−1.64675
$778$	0	0
$779$	0	0
$780$	0	0
$781$	8.99050	0.321706
$782$	0	0
$783$	−39.2080	−1.40118
$784$	0	0
$785$	12.7392	0.454680
$786$	0	0
$787$	−5.80478	−0.206918	−0.103459	−	0.994634i	$-0.532991\pi$
$787$	−5.80478	−0.206918	−0.103459	+	0.994634i	$0.532991\pi$
$788$	0	0
$789$	−41.1634	−1.46546
$790$	0	0
$791$	67.3465	2.39456
$792$	0	0
$793$	−12.7392	−0.452381
$794$	0	0
$795$	6.29591	0.223293
$796$	0	0
$797$	−39.2181	−1.38918	−0.694589	−	0.719407i	$-0.744414\pi$
$797$	−39.2181	−1.38918	−0.694589	+	0.719407i	$0.744414\pi$
$798$	0	0
$799$	24.3987	0.863163
$800$	0	0
$801$	−5.66456	−0.200148
$802$	0	0
$803$	−6.43552	−0.227104
$804$	0	0
$805$	−31.0351	−1.09384
$806$	0	0
$807$	−2.69459	−0.0948542
$808$	0	0
$809$	7.00269	0.246201	0.123101	−	0.992394i	$-0.460716\pi$
$809$	7.00269	0.246201	0.123101	+	0.992394i	$0.460716\pi$
$810$	0	0
$811$	−43.6851	−1.53399	−0.766996	−	0.641652i	$-0.778249\pi$
$811$	−43.6851	−1.53399	−0.766996	+	0.641652i	$0.778249\pi$
$812$	0	0
$813$	−30.3952	−1.06601
$814$	0	0
$815$	−9.46286	−0.331469
$816$	0	0
$817$	0	0
$818$	0	0
$819$	3.51754	0.122913
$820$	0	0
$821$	12.3851	0.432242	0.216121	−	0.976367i	$-0.430659\pi$
$821$	12.3851	0.432242	0.216121	+	0.976367i	$0.430659\pi$
$822$	0	0
$823$	12.8283	0.447167	0.223584	−	0.974685i	$-0.428224\pi$
$823$	12.8283	0.447167	0.223584	+	0.974685i	$0.428224\pi$
$824$	0	0
$825$	−2.65270	−0.0923553
$826$	0	0
$827$	−25.0966	−0.872693	−0.436347	−	0.899779i	$-0.643728\pi$
$827$	−25.0966	−0.872693	−0.436347	+	0.899779i	$0.643728\pi$
$828$	0	0
$829$	28.3269	0.983833	0.491917	−	0.870642i	$-0.336297\pi$
$829$	28.3269	0.983833	0.491917	+	0.870642i	$0.336297\pi$
$830$	0	0
$831$	20.8776	0.724237
$832$	0	0
$833$	44.5485	1.54351
$834$	0	0
$835$	−34.5526	−1.19574
$836$	0	0
$837$	1.71419	0.0592512
$838$	0	0
$839$	30.3304	1.04712	0.523561	−	0.851988i	$-0.324603\pi$
$839$	30.3304	1.04712	0.523561	+	0.851988i	$0.324603\pi$
$840$	0	0
$841$	42.4593	1.46412
$842$	0	0
$843$	7.98545	0.275034
$844$	0	0
$845$	−22.5918	−0.777182
$846$	0	0
$847$	45.6168	1.56741
$848$	0	0
$849$	10.2540	0.351917
$850$	0	0
$851$	14.7784	0.506596
$852$	0	0
$853$	−43.6323	−1.49394	−0.746970	−	0.664857i	$-0.768492\pi$
$853$	−43.6323	−1.49394	−0.746970	+	0.664857i	$0.768492\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−31.5098	−1.07635	−0.538177	−	0.842832i	$-0.680887\pi$
$857$	−31.5098	−1.07635	−0.538177	+	0.842832i	$0.680887\pi$
$858$	0	0
$859$	18.6979	0.637964	0.318982	−	0.947761i	$-0.396659\pi$
$859$	18.6979	0.637964	0.318982	+	0.947761i	$0.396659\pi$
$860$	0	0
$861$	14.5817	0.496944
$862$	0	0
$863$	−10.6263	−0.361723	−0.180862	−	0.983509i	$-0.557889\pi$
$863$	−10.6263	−0.361723	−0.180862	+	0.983509i	$0.557889\pi$
$864$	0	0
$865$	37.6459	1.28000
$866$	0	0
$867$	−21.2216	−0.720724
$868$	0	0
$869$	3.16344	0.107312
$870$	0	0
$871$	−1.83244	−0.0620900
$872$	0	0
$873$	−0.815207	−0.0275906
$874$	0	0
$875$	60.7701	2.05441
$876$	0	0
$877$	44.8985	1.51611	0.758057	−	0.652188i	$-0.226149\pi$
$877$	44.8985	1.51611	0.758057	+	0.652188i	$0.226149\pi$
$878$	0	0
$879$	33.4783	1.12920
$880$	0	0
$881$	−18.0164	−0.606988	−0.303494	−	0.952833i	$-0.598153\pi$
$881$	−18.0164	−0.606988	−0.303494	+	0.952833i	$0.598153\pi$
$882$	0	0
$883$	46.5981	1.56815	0.784076	−	0.620665i	$-0.213137\pi$
$883$	46.5981	1.56815	0.784076	+	0.620665i	$0.213137\pi$
$884$	0	0
$885$	−2.69459	−0.0905777
$886$	0	0
$887$	−7.07966	−0.237712	−0.118856	−	0.992912i	$-0.537923\pi$
$887$	−7.07966	−0.237712	−0.118856	+	0.992912i	$0.537923\pi$
$888$	0	0
$889$	−28.9377	−0.970539
$890$	0	0
$891$	−14.5567	−0.487669
$892$	0	0
$893$	0	0
$894$	0	0
$895$	27.6560	0.924438
$896$	0	0
$897$	−7.51754	−0.251003
$898$	0	0
$899$	−3.12424	−0.104199
$900$	0	0
$901$	4.00187	0.133322
$902$	0	0
$903$	−7.22163	−0.240321
$904$	0	0
$905$	24.5134	0.814854
$906$	0	0
$907$	−15.1156	−0.501904	−0.250952	−	0.968000i	$-0.580744\pi$
$907$	−15.1156	−0.501904	−0.250952	+	0.968000i	$0.580744\pi$
$908$	0	0
$909$	4.69459	0.155710
$910$	0	0
$911$	12.8366	0.425294	0.212647	−	0.977129i	$-0.431792\pi$
$911$	12.8366	0.425294	0.212647	+	0.977129i	$0.431792\pi$
$912$	0	0
$913$	5.62536	0.186172
$914$	0	0
$915$	36.6810	1.21264
$916$	0	0
$917$	−50.0310	−1.65217
$918$	0	0
$919$	20.6791	0.682141	0.341070	−	0.940038i	$-0.389211\pi$
$919$	20.6791	0.682141	0.341070	+	0.940038i	$0.389211\pi$
$920$	0	0
$921$	53.8607	1.77477
$922$	0	0
$923$	−8.31490	−0.273688
$924$	0	0
$925$	−4.82295	−0.158578
$926$	0	0
$927$	3.80335	0.124918
$928$	0	0
$929$	−18.2499	−0.598760	−0.299380	−	0.954134i	$-0.596780\pi$
$929$	−18.2499	−0.598760	−0.299380	+	0.954134i	$0.596780\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	−29.7196	−0.972975
$934$	0	0
$935$	6.74455	0.220570
$936$	0	0
$937$	−4.74691	−0.155075	−0.0775374	−	0.996989i	$-0.524706\pi$
$937$	−4.74691	−0.155075	−0.0775374	+	0.996989i	$0.524706\pi$
$938$	0	0
$939$	24.6955	0.805908
$940$	0	0
$941$	47.0215	1.53286	0.766428	−	0.642330i	$-0.222032\pi$
$941$	47.0215	1.53286	0.766428	+	0.642330i	$0.222032\pi$
$942$	0	0
$943$	−4.69459	−0.152877
$944$	0	0
$945$	46.9769	1.52816
$946$	0	0
$947$	−36.9959	−1.20220	−0.601102	−	0.799172i	$-0.705272\pi$
$947$	−36.9959	−1.20220	−0.601102	+	0.799172i	$0.705272\pi$
$948$	0	0
$949$	5.95191	0.193207
$950$	0	0
$951$	−40.2276	−1.30447
$952$	0	0
$953$	−37.7093	−1.22152	−0.610761	−	0.791815i	$-0.709136\pi$
$953$	−37.7093	−1.22152	−0.610761	+	0.791815i	$0.709136\pi$
$954$	0	0
$955$	40.1985	1.30079
$956$	0	0
$957$	22.4243	0.724874
$958$	0	0
$959$	−27.3310	−0.882564
$960$	0	0
$961$	−30.8634	−0.995594
$962$	0	0
$963$	4.98545	0.160654
$964$	0	0
$965$	33.3702	1.07422
$966$	0	0
$967$	−40.5134	−1.30282	−0.651412	−	0.758724i	$-0.725823\pi$
$967$	−40.5134	−1.30282	−0.651412	+	0.758724i	$0.725823\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	14.0779	0.451781	0.225891	−	0.974153i	$-0.427471\pi$
$971$	14.0779	0.451781	0.225891	+	0.974153i	$0.427471\pi$
$972$	0	0
$973$	−11.8479	−0.379827
$974$	0	0
$975$	2.45336	0.0785705
$976$	0	0
$977$	−35.2057	−1.12633	−0.563164	−	0.826345i	$-0.690416\pi$
$977$	−35.2057	−1.12633	−0.563164	+	0.826345i	$0.690416\pi$
$978$	0	0
$979$	−15.0264	−0.480246
$980$	0	0
$981$	5.88713	0.187961
$982$	0	0
$983$	−22.4397	−0.715717	−0.357858	−	0.933776i	$-0.616493\pi$
$983$	−22.4397	−0.715717	−0.357858	+	0.933776i	$0.616493\pi$
$984$	0	0
$985$	25.9810	0.827824
$986$	0	0
$987$	−97.1944	−3.09373
$988$	0	0
$989$	2.32501	0.0739309
$990$	0	0
$991$	−27.1034	−0.860967	−0.430484	−	0.902598i	$-0.641657\pi$
$991$	−27.1034	−0.860967	−0.430484	+	0.902598i	$0.641657\pi$
$992$	0	0
$993$	47.7324	1.51474
$994$	0	0
$995$	−34.3851	−1.09008
$996$	0	0
$997$	27.9472	0.885096	0.442548	−	0.896745i	$-0.354075\pi$
$997$	27.9472	0.885096	0.442548	+	0.896745i	$0.354075\pi$
$998$	0	0
$999$	−22.3696	−0.707742

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5776.2.a.bn.1.3		3
4.3	odd	2		722.2.a.l.1.1		3
12.11	even	2		6498.2.a.bl.1.3		3
19.9	even	9		304.2.u.c.81.1		6
19.17	even	9		304.2.u.c.289.1		6
19.18	odd	2		5776.2.a.bo.1.1		3
76.3	even	18		722.2.e.l.389.1		6
76.7	odd	6		722.2.c.k.429.3		6
76.11	odd	6		722.2.c.k.653.3		6
76.15	even	18		722.2.e.a.415.1		6
76.23	odd	18		722.2.e.m.415.1		6
76.27	even	6		722.2.c.l.653.1		6
76.31	even	6		722.2.c.l.429.1		6
76.35	odd	18		722.2.e.b.389.1		6
76.43	odd	18		722.2.e.m.595.1		6
76.47	odd	18		38.2.e.a.5.1	✓	6
76.51	even	18		722.2.e.l.245.1		6
76.55	odd	18		38.2.e.a.23.1	yes	6
76.59	even	18		722.2.e.k.99.1		6
76.63	odd	18		722.2.e.b.245.1		6
76.67	even	18		722.2.e.k.423.1		6
76.71	even	18		722.2.e.a.595.1		6
76.75	even	2		722.2.a.k.1.3		3
228.47	even	18		342.2.u.c.271.1		6
228.131	even	18		342.2.u.c.289.1		6
228.227	odd	2		6498.2.a.bq.1.3		3
380.47	even	36		950.2.u.b.499.1		12
380.123	even	36		950.2.u.b.499.2		12
380.199	odd	18		950.2.l.d.651.1		6
380.207	even	36		950.2.u.b.99.2		12
380.283	even	36		950.2.u.b.99.1		12
380.359	odd	18		950.2.l.d.251.1		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
38.2.e.a.5.1	✓	6	76.47	odd	18
38.2.e.a.23.1	yes	6	76.55	odd	18
304.2.u.c.81.1		6	19.9	even	9
304.2.u.c.289.1		6	19.17	even	9
342.2.u.c.271.1		6	228.47	even	18
342.2.u.c.289.1		6	228.131	even	18
722.2.a.k.1.3		3	76.75	even	2
722.2.a.l.1.1		3	4.3	odd	2
722.2.c.k.429.3		6	76.7	odd	6
722.2.c.k.653.3		6	76.11	odd	6
722.2.c.l.429.1		6	76.31	even	6
722.2.c.l.653.1		6	76.27	even	6
722.2.e.a.415.1		6	76.15	even	18
722.2.e.a.595.1		6	76.71	even	18
722.2.e.b.245.1		6	76.63	odd	18
722.2.e.b.389.1		6	76.35	odd	18
722.2.e.k.99.1		6	76.59	even	18
722.2.e.k.423.1		6	76.67	even	18
722.2.e.l.245.1		6	76.51	even	18
722.2.e.l.389.1		6	76.3	even	18
722.2.e.m.415.1		6	76.23	odd	18
722.2.e.m.595.1		6	76.43	odd	18
950.2.l.d.251.1		6	380.359	odd	18
950.2.l.d.651.1		6	380.199	odd	18
950.2.u.b.99.1		12	380.283	even	36
950.2.u.b.99.2		12	380.207	even	36
950.2.u.b.499.1		12	380.47	even	36
950.2.u.b.499.2		12	380.123	even	36
5776.2.a.bn.1.3		3	1.1	even	1	trivial
5776.2.a.bo.1.1		3	19.18	odd	2
6498.2.a.bl.1.3		3	12.11	even	2
6498.2.a.bq.1.3		3	228.227	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:	\( N \)	\(=\)	\( 5776 = 2^{4} \cdot 19^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5776.a (trivial)

\(f(q)\)	\(=\)	\(q+1.87939 q^{3} +2.00000 q^{5} -5.06418 q^{7} +0.532089 q^{9} +O(q^{10})\)	\(q+1.87939 q^{3} +2.00000 q^{5} -5.06418 q^{7} +0.532089 q^{9} +1.41147 q^{11} -1.30541 q^{13} +3.75877 q^{15} +2.38919 q^{17} -9.51754 q^{21} +3.06418 q^{23} -1.00000 q^{25} -4.63816 q^{27} +8.45336 q^{29} -0.369585 q^{31} +2.65270 q^{33} -10.1284 q^{35} +4.82295 q^{37} -2.45336 q^{39} -1.53209 q^{41} +0.758770 q^{43} +1.06418 q^{45} +10.2121 q^{47} +18.6459 q^{49} +4.49020 q^{51} +1.67499 q^{53} +2.82295 q^{55} -0.716881 q^{59} +9.75877 q^{61} -2.69459 q^{63} -2.61081 q^{65} +1.40373 q^{67} +5.75877 q^{69} +6.36959 q^{71} -4.55943 q^{73} -1.87939 q^{75} -7.14796 q^{77} +2.24123 q^{79} -10.3131 q^{81} +3.98545 q^{83} +4.77837 q^{85} +15.8871 q^{87} -10.6459 q^{89} +6.61081 q^{91} -0.694593 q^{93} -1.53209 q^{97} +0.751030 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 6 q^{5} - 6 q^{7} - 3 q^{9}+O(q^{10}) \) 3 * q + 6 * q^5 - 6 * q^7 - 3 * q^9	\( 3 q + 6 q^{5} - 6 q^{7} - 3 q^{9} - 6 q^{11} - 6 q^{13} + 3 q^{17} - 6 q^{21} - 3 q^{25} + 3 q^{27} + 12 q^{29} + 6 q^{31} + 9 q^{33} - 12 q^{35} - 6 q^{37} + 6 q^{39} - 9 q^{43} - 6 q^{45} + 6 q^{47} + 15 q^{49} + 12 q^{51} - 12 q^{55} + 6 q^{59} + 18 q^{61} - 6 q^{63} - 12 q^{65} + 18 q^{67} + 6 q^{69} + 12 q^{71} + 12 q^{73} - 6 q^{77} + 18 q^{79} - 9 q^{81} - 6 q^{83} + 6 q^{85} + 18 q^{87} + 9 q^{89} + 24 q^{91} + 15 q^{99}+O(q^{100}) \) 3 * q + 6 * q^5 - 6 * q^7 - 3 * q^9 - 6 * q^11 - 6 * q^13 + 3 * q^17 - 6 * q^21 - 3 * q^25 + 3 * q^27 + 12 * q^29 + 6 * q^31 + 9 * q^33 - 12 * q^35 - 6 * q^37 + 6 * q^39 - 9 * q^43 - 6 * q^45 + 6 * q^47 + 15 * q^49 + 12 * q^51 - 12 * q^55 + 6 * q^59 + 18 * q^61 - 6 * q^63 - 12 * q^65 + 18 * q^67 + 6 * q^69 + 12 * q^71 + 12 * q^73 - 6 * q^77 + 18 * q^79 - 9 * q^81 - 6 * q^83 + 6 * q^85 + 18 * q^87 + 9 * q^89 + 24 * q^91 + 15 * q^99

Introduction

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