LMFDB - Embedded newform 5780.2.a.q.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5780,2,Mod(1,5780)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5780, 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("5780.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5780 = 2^{2} \cdot 5 \cdot 17^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5780.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:	$46.1535323683$
Analytic rank:	$1$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 24x^{10} + 206x^{8} - 16x^{7} - 776x^{6} + 152x^{5} + 1226x^{4} - 384x^{3} - 588x^{2} + 200x + 34 $ x^12 - 24x^10 + 206x^8 - 16x^7 - 776x^6 + 152x^5 + 1226x^4 - 384x^3 - 588x^2 + 200*x + 34
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 340)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-2.89366$ of defining polynomial
Character	$\chi$	$=$	5780.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.89366 q^{3} -1.00000 q^{5} +3.07222 q^{7} +5.37326 q^{9} +O(q^{10})$	$q-2.89366 q^{3} -1.00000 q^{5} +3.07222 q^{7} +5.37326 q^{9} +1.93238 q^{11} +3.23350 q^{13} +2.89366 q^{15} -4.24104 q^{19} -8.88997 q^{21} -3.08107 q^{23} +1.00000 q^{25} -6.86739 q^{27} -2.19381 q^{29} +0.0478250 q^{31} -5.59165 q^{33} -3.07222 q^{35} -10.2207 q^{37} -9.35664 q^{39} -5.54092 q^{41} +11.4190 q^{43} -5.37326 q^{45} -9.13431 q^{47} +2.43856 q^{49} -0.253720 q^{53} -1.93238 q^{55} +12.2721 q^{57} +13.1530 q^{59} -3.41767 q^{61} +16.5079 q^{63} -3.23350 q^{65} -1.82396 q^{67} +8.91556 q^{69} -8.54892 q^{71} -15.5037 q^{73} -2.89366 q^{75} +5.93670 q^{77} -8.16140 q^{79} +3.75211 q^{81} +15.4122 q^{83} +6.34812 q^{87} +7.13466 q^{89} +9.93403 q^{91} -0.138389 q^{93} +4.24104 q^{95} +8.20149 q^{97} +10.3832 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 12 q^{5} - 8 q^{7} + 12 q^{9}+O(q^{10}) $ 12 * q - 12 * q^5 - 8 * q^7 + 12 * q^9	$ 12 q - 12 q^{5} - 8 q^{7} + 12 q^{9} + 8 q^{13} - 16 q^{21} - 8 q^{23} + 12 q^{25} - 16 q^{29} - 24 q^{31} + 8 q^{35} - 24 q^{37} + 8 q^{39} - 24 q^{41} + 8 q^{43} - 12 q^{45} + 8 q^{47} + 20 q^{49} + 16 q^{53} - 32 q^{57} + 8 q^{59} - 40 q^{61} - 24 q^{63} - 8 q^{65} + 16 q^{67} - 16 q^{69} - 16 q^{71} - 32 q^{73} + 24 q^{77} - 8 q^{79} + 4 q^{81} + 32 q^{83} + 16 q^{87} - 8 q^{89} - 8 q^{91} - 8 q^{93} - 32 q^{97} - 24 q^{99}+O(q^{100}) $ 12 * q - 12 * q^5 - 8 * q^7 + 12 * q^9 + 8 * q^13 - 16 * q^21 - 8 * q^23 + 12 * q^25 - 16 * q^29 - 24 * q^31 + 8 * q^35 - 24 * q^37 + 8 * q^39 - 24 * q^41 + 8 * q^43 - 12 * q^45 + 8 * q^47 + 20 * q^49 + 16 * q^53 - 32 * q^57 + 8 * q^59 - 40 * q^61 - 24 * q^63 - 8 * q^65 + 16 * q^67 - 16 * q^69 - 16 * q^71 - 32 * q^73 + 24 * q^77 - 8 * q^79 + 4 * q^81 + 32 * q^83 + 16 * q^87 - 8 * q^89 - 8 * q^91 - 8 * q^93 - 32 * q^97 - 24 * 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
$3$	−2.89366	−1.67065	−0.835327	−	0.549753i	$-0.814722\pi$
$3$	−2.89366	−1.67065	−0.835327	+	0.549753i	$0.814722\pi$
$4$	0	0
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	3.07222	1.16119	0.580596	−	0.814192i	$-0.302820\pi$
$7$	3.07222	1.16119	0.580596	+	0.814192i	$0.302820\pi$
$8$	0	0
$9$	5.37326	1.79109
$10$	0	0
$11$	1.93238	0.582634	0.291317	−	0.956627i	$-0.405906\pi$
$11$	1.93238	0.582634	0.291317	+	0.956627i	$0.405906\pi$
$12$	0	0
$13$	3.23350	0.896811	0.448406	−	0.893830i	$-0.351992\pi$
$13$	3.23350	0.896811	0.448406	+	0.893830i	$0.351992\pi$
$14$	0	0
$15$	2.89366	0.747139
$16$	0	0
$17$	0	0
$17$	0	0
$18$	0	0
$19$	−4.24104	−0.972960	−0.486480	−	0.873692i	$-0.661719\pi$
$19$	−4.24104	−0.972960	−0.486480	+	0.873692i	$0.661719\pi$
$20$	0	0
$21$	−8.88997	−1.93995
$22$	0	0
$23$	−3.08107	−0.642447	−0.321224	−	0.947003i	$-0.604094\pi$
$23$	−3.08107	−0.642447	−0.321224	+	0.947003i	$0.604094\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−6.86739	−1.32163
$28$	0	0
$29$	−2.19381	−0.407380	−0.203690	−	0.979035i	$-0.565293\pi$
$29$	−2.19381	−0.407380	−0.203690	+	0.979035i	$0.565293\pi$
$30$	0	0
$31$	0.0478250	0.00858963	0.00429481	−	0.999991i	$-0.498633\pi$
$31$	0.0478250	0.00858963	0.00429481	+	0.999991i	$0.498633\pi$
$32$	0	0
$33$	−5.59165	−0.973381
$34$	0	0
$35$	−3.07222	−0.519301
$36$	0	0
$37$	−10.2207	−1.68028	−0.840140	−	0.542370i	$-0.817527\pi$
$37$	−10.2207	−1.68028	−0.840140	+	0.542370i	$0.817527\pi$
$38$	0	0
$39$	−9.35664	−1.49826
$40$	0	0
$41$	−5.54092	−0.865347	−0.432673	−	0.901551i	$-0.642430\pi$
$41$	−5.54092	−0.865347	−0.432673	+	0.901551i	$0.642430\pi$
$42$	0	0
$43$	11.4190	1.74138	0.870692	−	0.491829i	$-0.163672\pi$
$43$	11.4190	1.74138	0.870692	+	0.491829i	$0.163672\pi$
$44$	0	0
$45$	−5.37326	−0.800998
$46$	0	0
$47$	−9.13431	−1.33238	−0.666188	−	0.745784i	$-0.732075\pi$
$47$	−9.13431	−1.33238	−0.666188	+	0.745784i	$0.732075\pi$
$48$	0	0
$49$	2.43856	0.348366
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−0.253720	−0.0348511	−0.0174256	−	0.999848i	$-0.505547\pi$
$53$	−0.253720	−0.0348511	−0.0174256	+	0.999848i	$0.505547\pi$
$54$	0	0
$55$	−1.93238	−0.260562
$56$	0	0
$57$	12.2721	1.62548
$58$	0	0
$59$	13.1530	1.71237	0.856187	−	0.516666i	$-0.172827\pi$
$59$	13.1530	1.71237	0.856187	+	0.516666i	$0.172827\pi$
$60$	0	0
$61$	−3.41767	−0.437588	−0.218794	−	0.975771i	$-0.570212\pi$
$61$	−3.41767	−0.437588	−0.218794	+	0.975771i	$0.570212\pi$
$62$	0	0
$63$	16.5079	2.07979
$64$	0	0
$65$	−3.23350	−0.401066
$66$	0	0
$67$	−1.82396	−0.222832	−0.111416	−	0.993774i	$-0.535539\pi$
$67$	−1.82396	−0.222832	−0.111416	+	0.993774i	$0.535539\pi$
$68$	0	0
$69$	8.91556	1.07331
$70$	0	0
$71$	−8.54892	−1.01457	−0.507285	−	0.861778i	$-0.669351\pi$
$71$	−8.54892	−1.01457	−0.507285	+	0.861778i	$0.669351\pi$
$72$	0	0
$73$	−15.5037	−1.81457	−0.907284	−	0.420519i	$-0.861848\pi$
$73$	−15.5037	−1.81457	−0.907284	+	0.420519i	$0.861848\pi$
$74$	0	0
$75$	−2.89366	−0.334131
$76$	0	0
$77$	5.93670	0.676550
$78$	0	0
$79$	−8.16140	−0.918230	−0.459115	−	0.888377i	$-0.651833\pi$
$79$	−8.16140	−0.918230	−0.459115	+	0.888377i	$0.651833\pi$
$80$	0	0
$81$	3.75211	0.416901
$82$	0	0
$83$	15.4122	1.69171	0.845853	−	0.533416i	$-0.179092\pi$
$83$	15.4122	1.69171	0.845853	+	0.533416i	$0.179092\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	6.34812	0.680590
$88$	0	0
$89$	7.13466	0.756272	0.378136	−	0.925750i	$-0.376565\pi$
$89$	7.13466	0.756272	0.378136	+	0.925750i	$0.376565\pi$
$90$	0	0
$91$	9.93403	1.04137
$92$	0	0
$93$	−0.138389	−0.0143503
$94$	0	0
$95$	4.24104	0.435121
$96$	0	0
$97$	8.20149	0.832735	0.416367	−	0.909196i	$-0.363303\pi$
$97$	8.20149	0.832735	0.416367	+	0.909196i	$0.363303\pi$
$98$	0	0
$99$	10.3832	1.04355
$100$	0	0
$101$	13.8146	1.37460	0.687301	−	0.726373i	$-0.258795\pi$
$101$	13.8146	1.37460	0.687301	+	0.726373i	$0.258795\pi$
$102$	0	0
$103$	−15.7190	−1.54884	−0.774420	−	0.632672i	$-0.781958\pi$
$103$	−15.7190	−1.54884	−0.774420	+	0.632672i	$0.781958\pi$
$104$	0	0
$105$	8.88997	0.867572
$106$	0	0
$107$	11.6782	1.12898	0.564489	−	0.825440i	$-0.309073\pi$
$107$	11.6782	1.12898	0.564489	+	0.825440i	$0.309073\pi$
$108$	0	0
$109$	−15.5905	−1.49330	−0.746650	−	0.665217i	$-0.768339\pi$
$109$	−15.5905	−1.49330	−0.746650	+	0.665217i	$0.768339\pi$
$110$	0	0
$111$	29.5753	2.80717
$112$	0	0
$113$	17.3118	1.62855	0.814277	−	0.580476i	$-0.197134\pi$
$113$	17.3118	1.62855	0.814277	+	0.580476i	$0.197134\pi$
$114$	0	0
$115$	3.08107	0.287311
$116$	0	0
$117$	17.3744	1.60627
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−7.26591	−0.660537
$122$	0	0
$123$	16.0335	1.44569
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−15.3544	−1.36248	−0.681239	−	0.732061i	$-0.738559\pi$
$127$	−15.3544	−1.36248	−0.681239	+	0.732061i	$0.738559\pi$
$128$	0	0
$129$	−33.0427	−2.90925
$130$	0	0
$131$	18.8563	1.64748	0.823739	−	0.566969i	$-0.191884\pi$
$131$	18.8563	1.64748	0.823739	+	0.566969i	$0.191884\pi$
$132$	0	0
$133$	−13.0294	−1.12979
$134$	0	0
$135$	6.86739	0.591051
$136$	0	0
$137$	−1.16327	−0.0993850	−0.0496925	−	0.998765i	$-0.515824\pi$
$137$	−1.16327	−0.0993850	−0.0496925	+	0.998765i	$0.515824\pi$
$138$	0	0
$139$	−8.84611	−0.750317	−0.375159	−	0.926961i	$-0.622412\pi$
$139$	−8.84611	−0.750317	−0.375159	+	0.926961i	$0.622412\pi$
$140$	0	0
$141$	26.4316	2.22594
$142$	0	0
$143$	6.24835	0.522513
$144$	0	0
$145$	2.19381	0.182186
$146$	0	0
$147$	−7.05637	−0.582000
$148$	0	0
$149$	−17.7837	−1.45690	−0.728448	−	0.685101i	$-0.759758\pi$
$149$	−17.7837	−1.45690	−0.728448	+	0.685101i	$0.759758\pi$
$150$	0	0
$151$	−14.9320	−1.21515	−0.607576	−	0.794262i	$-0.707858\pi$
$151$	−14.9320	−1.21515	−0.607576	+	0.794262i	$0.707858\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−0.0478250	−0.00384140
$156$	0	0
$157$	18.4437	1.47197	0.735984	−	0.676999i	$-0.236720\pi$
$157$	18.4437	1.47197	0.735984	+	0.676999i	$0.236720\pi$
$158$	0	0
$159$	0.734179	0.0582242
$160$	0	0
$161$	−9.46573	−0.746004
$162$	0	0
$163$	21.0131	1.64587	0.822936	−	0.568134i	$-0.192334\pi$
$163$	21.0131	1.64587	0.822936	+	0.568134i	$0.192334\pi$
$164$	0	0
$165$	5.59165	0.435309
$166$	0	0
$167$	0.108406	0.00838874	0.00419437	−	0.999991i	$-0.498665\pi$
$167$	0.108406	0.00838874	0.00419437	+	0.999991i	$0.498665\pi$
$168$	0	0
$169$	−2.54449	−0.195730
$170$	0	0
$171$	−22.7882	−1.74266
$172$	0	0
$173$	0.375995	0.0285864	0.0142932	−	0.999898i	$-0.495450\pi$
$173$	0.375995	0.0285864	0.0142932	+	0.999898i	$0.495450\pi$
$174$	0	0
$175$	3.07222	0.232238
$176$	0	0
$177$	−38.0603	−2.86079
$178$	0	0
$179$	−10.4909	−0.784124	−0.392062	−	0.919939i	$-0.628238\pi$
$179$	−10.4909	−0.784124	−0.392062	+	0.919939i	$0.628238\pi$
$180$	0	0
$181$	1.45808	0.108378	0.0541892	−	0.998531i	$-0.482743\pi$
$181$	1.45808	0.108378	0.0541892	+	0.998531i	$0.482743\pi$
$182$	0	0
$183$	9.88958	0.731059
$184$	0	0
$185$	10.2207	0.751444
$186$	0	0
$187$	0	0
$188$	0	0
$189$	−21.0982	−1.53467
$190$	0	0
$191$	0.656351	0.0474919	0.0237460	−	0.999718i	$-0.492441\pi$
$191$	0.656351	0.0474919	0.0237460	+	0.999718i	$0.492441\pi$
$192$	0	0
$193$	−4.43520	−0.319253	−0.159626	−	0.987177i	$-0.551029\pi$
$193$	−4.43520	−0.319253	−0.159626	+	0.987177i	$0.551029\pi$
$194$	0	0
$195$	9.35664	0.670043
$196$	0	0
$197$	−14.8496	−1.05799	−0.528996	−	0.848624i	$-0.677431\pi$
$197$	−14.8496	−1.05799	−0.528996	+	0.848624i	$0.677431\pi$
$198$	0	0
$199$	−10.1557	−0.719917	−0.359959	−	0.932968i	$-0.617209\pi$
$199$	−10.1557	−0.719917	−0.359959	+	0.932968i	$0.617209\pi$
$200$	0	0
$201$	5.27792	0.372276
$202$	0	0
$203$	−6.73987	−0.473046
$204$	0	0
$205$	5.54092	0.386995
$206$	0	0
$207$	−16.5554	−1.15068
$208$	0	0
$209$	−8.19529	−0.566880
$210$	0	0
$211$	2.27137	0.156367	0.0781837	−	0.996939i	$-0.475088\pi$
$211$	2.27137	0.156367	0.0781837	+	0.996939i	$0.475088\pi$
$212$	0	0
$213$	24.7376	1.69500
$214$	0	0
$215$	−11.4190	−0.778770
$216$	0	0
$217$	0.146929	0.00997421
$218$	0	0
$219$	44.8623	3.03152
$220$	0	0
$221$	0	0
$222$	0	0
$223$	12.7662	0.854888	0.427444	−	0.904042i	$-0.359414\pi$
$223$	12.7662	0.854888	0.427444	+	0.904042i	$0.359414\pi$
$224$	0	0
$225$	5.37326	0.358217
$226$	0	0
$227$	−4.16963	−0.276748	−0.138374	−	0.990380i	$-0.544188\pi$
$227$	−4.16963	−0.276748	−0.138374	+	0.990380i	$0.544188\pi$
$228$	0	0
$229$	−8.48897	−0.560967	−0.280484	−	0.959859i	$-0.590495\pi$
$229$	−8.48897	−0.560967	−0.280484	+	0.959859i	$0.590495\pi$
$230$	0	0
$231$	−17.1788	−1.13028
$232$	0	0
$233$	−3.15334	−0.206582	−0.103291	−	0.994651i	$-0.532937\pi$
$233$	−3.15334	−0.206582	−0.103291	+	0.994651i	$0.532937\pi$
$234$	0	0
$235$	9.13431	0.595857
$236$	0	0
$237$	23.6163	1.53404
$238$	0	0
$239$	9.73351	0.629608	0.314804	−	0.949157i	$-0.398061\pi$
$239$	9.73351	0.629608	0.314804	+	0.949157i	$0.398061\pi$
$240$	0	0
$241$	−7.91940	−0.510134	−0.255067	−	0.966923i	$-0.582097\pi$
$241$	−7.91940	−0.510134	−0.255067	+	0.966923i	$0.582097\pi$
$242$	0	0
$243$	9.74484	0.625132
$244$	0	0
$245$	−2.43856	−0.155794
$246$	0	0
$247$	−13.7134	−0.872562
$248$	0	0
$249$	−44.5976	−2.82626
$250$	0	0
$251$	−1.68179	−0.106154	−0.0530768	−	0.998590i	$-0.516903\pi$
$251$	−1.68179	−0.106154	−0.0530768	+	0.998590i	$0.516903\pi$
$252$	0	0
$253$	−5.95379	−0.374312
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−17.0914	−1.06613	−0.533067	−	0.846073i	$-0.678961\pi$
$257$	−17.0914	−1.06613	−0.533067	+	0.846073i	$0.678961\pi$
$258$	0	0
$259$	−31.4004	−1.95113
$260$	0	0
$261$	−11.7879	−0.729652
$262$	0	0
$263$	−8.36570	−0.515851	−0.257926	−	0.966165i	$-0.583039\pi$
$263$	−8.36570	−0.515851	−0.257926	+	0.966165i	$0.583039\pi$
$264$	0	0
$265$	0.253720	0.0155859
$266$	0	0
$267$	−20.6453	−1.26347
$268$	0	0
$269$	−28.2967	−1.72528	−0.862639	−	0.505819i	$-0.831190\pi$
$269$	−28.2967	−1.72528	−0.862639	+	0.505819i	$0.831190\pi$
$270$	0	0
$271$	−17.4539	−1.06025	−0.530126	−	0.847919i	$-0.677855\pi$
$271$	−17.4539	−1.06025	−0.530126	+	0.847919i	$0.677855\pi$
$272$	0	0
$273$	−28.7457	−1.73977
$274$	0	0
$275$	1.93238	0.116527
$276$	0	0
$277$	−9.49094	−0.570255	−0.285128	−	0.958490i	$-0.592036\pi$
$277$	−9.49094	−0.570255	−0.285128	+	0.958490i	$0.592036\pi$
$278$	0	0
$279$	0.256976	0.0153848
$280$	0	0
$281$	−13.0443	−0.778160	−0.389080	−	0.921204i	$-0.627207\pi$
$281$	−13.0443	−0.778160	−0.389080	+	0.921204i	$0.627207\pi$
$282$	0	0
$283$	3.68418	0.219002	0.109501	−	0.993987i	$-0.465075\pi$
$283$	3.68418	0.219002	0.109501	+	0.993987i	$0.465075\pi$
$284$	0	0
$285$	−12.2721	−0.726937
$286$	0	0
$287$	−17.0230	−1.00483
$288$	0	0
$289$	0	0
$290$	0	0
$291$	−23.7323	−1.39121
$292$	0	0
$293$	−7.28913	−0.425835	−0.212918	−	0.977070i	$-0.568297\pi$
$293$	−7.28913	−0.425835	−0.212918	+	0.977070i	$0.568297\pi$
$294$	0	0
$295$	−13.1530	−0.765797
$296$	0	0
$297$	−13.2704	−0.770027
$298$	0	0
$299$	−9.96263	−0.576154
$300$	0	0
$301$	35.0818	2.02208
$302$	0	0
$303$	−39.9747	−2.29648
$304$	0	0
$305$	3.41767	0.195695
$306$	0	0
$307$	9.86582	0.563072	0.281536	−	0.959551i	$-0.409156\pi$
$307$	9.86582	0.563072	0.281536	+	0.959551i	$0.409156\pi$
$308$	0	0
$309$	45.4854	2.58758
$310$	0	0
$311$	−0.314583	−0.0178384	−0.00891918	−	0.999960i	$-0.502839\pi$
$311$	−0.314583	−0.0178384	−0.00891918	+	0.999960i	$0.502839\pi$
$312$	0	0
$313$	27.7208	1.56687	0.783435	−	0.621474i	$-0.213466\pi$
$313$	27.7208	1.56687	0.783435	+	0.621474i	$0.213466\pi$
$314$	0	0
$315$	−16.5079	−0.930112
$316$	0	0
$317$	−0.787826	−0.0442487	−0.0221244	−	0.999755i	$-0.507043\pi$
$317$	−0.787826	−0.0442487	−0.0221244	+	0.999755i	$0.507043\pi$
$318$	0	0
$319$	−4.23927	−0.237353
$320$	0	0
$321$	−33.7929	−1.88613
$322$	0	0
$323$	0	0
$324$	0	0
$325$	3.23350	0.179362
$326$	0	0
$327$	45.1136	2.49479
$328$	0	0
$329$	−28.0627	−1.54714
$330$	0	0
$331$	1.95619	0.107522	0.0537609	−	0.998554i	$-0.482879\pi$
$331$	1.95619	0.107522	0.0537609	+	0.998554i	$0.482879\pi$
$332$	0	0
$333$	−54.9187	−3.00952
$334$	0	0
$335$	1.82396	0.0996537
$336$	0	0
$337$	−12.7942	−0.696947	−0.348473	−	0.937319i	$-0.613300\pi$
$337$	−12.7942	−0.696947	−0.348473	+	0.937319i	$0.613300\pi$
$338$	0	0
$339$	−50.0944	−2.72075
$340$	0	0
$341$	0.0924161	0.00500461
$342$	0	0
$343$	−14.0138	−0.756672
$344$	0	0
$345$	−8.91556	−0.479997
$346$	0	0
$347$	10.2807	0.551898	0.275949	−	0.961172i	$-0.411008\pi$
$347$	10.2807	0.551898	0.275949	+	0.961172i	$0.411008\pi$
$348$	0	0
$349$	−31.3578	−1.67854	−0.839272	−	0.543712i	$-0.817018\pi$
$349$	−31.3578	−1.67854	−0.839272	+	0.543712i	$0.817018\pi$
$350$	0	0
$351$	−22.2057	−1.18525
$352$	0	0
$353$	25.2045	1.34150	0.670750	−	0.741684i	$-0.265972\pi$
$353$	25.2045	1.34150	0.670750	+	0.741684i	$0.265972\pi$
$354$	0	0
$355$	8.54892	0.453729
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−12.1830	−0.642994	−0.321497	−	0.946911i	$-0.604186\pi$
$359$	−12.1830	−0.642994	−0.321497	+	0.946911i	$0.604186\pi$
$360$	0	0
$361$	−1.01362	−0.0533482
$362$	0	0
$363$	21.0251	1.10353
$364$	0	0
$365$	15.5037	0.811499
$366$	0	0
$367$	27.7911	1.45068	0.725342	−	0.688388i	$-0.241681\pi$
$367$	27.7911	1.45068	0.725342	+	0.688388i	$0.241681\pi$
$368$	0	0
$369$	−29.7728	−1.54991
$370$	0	0
$371$	−0.779485	−0.0404689
$372$	0	0
$373$	−13.0506	−0.675734	−0.337867	−	0.941194i	$-0.609705\pi$
$373$	−13.0506	−0.675734	−0.337867	+	0.941194i	$0.609705\pi$
$374$	0	0
$375$	2.89366	0.149428
$376$	0	0
$377$	−7.09367	−0.365343
$378$	0	0
$379$	−6.49556	−0.333655	−0.166827	−	0.985986i	$-0.553352\pi$
$379$	−6.49556	−0.333655	−0.166827	+	0.985986i	$0.553352\pi$
$380$	0	0
$381$	44.4303	2.27623
$382$	0	0
$383$	31.8439	1.62715	0.813573	−	0.581463i	$-0.197519\pi$
$383$	31.8439	1.62715	0.813573	+	0.581463i	$0.197519\pi$
$384$	0	0
$385$	−5.93670	−0.302562
$386$	0	0
$387$	61.3573	3.11897
$388$	0	0
$389$	−10.5081	−0.532782	−0.266391	−	0.963865i	$-0.585831\pi$
$389$	−10.5081	−0.532782	−0.266391	+	0.963865i	$0.585831\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	−54.5635	−2.75237
$394$	0	0
$395$	8.16140	0.410645
$396$	0	0
$397$	−20.0776	−1.00766	−0.503832	−	0.863802i	$-0.668077\pi$
$397$	−20.0776	−1.00766	−0.503832	+	0.863802i	$0.668077\pi$
$398$	0	0
$399$	37.7027	1.88749
$400$	0	0
$401$	5.67231	0.283261	0.141631	−	0.989920i	$-0.454765\pi$
$401$	5.67231	0.283261	0.141631	+	0.989920i	$0.454765\pi$
$402$	0	0
$403$	0.154642	0.00770327
$404$	0	0
$405$	−3.75211	−0.186444
$406$	0	0
$407$	−19.7503	−0.978988
$408$	0	0
$409$	−35.8010	−1.77024	−0.885122	−	0.465359i	$-0.845925\pi$
$409$	−35.8010	−1.77024	−0.885122	+	0.465359i	$0.845925\pi$
$410$	0	0
$411$	3.36611	0.166038
$412$	0	0
$413$	40.4090	1.98840
$414$	0	0
$415$	−15.4122	−0.756554
$416$	0	0
$417$	25.5976	1.25352
$418$	0	0
$419$	1.18084	0.0576880	0.0288440	−	0.999584i	$-0.490817\pi$
$419$	1.18084	0.0576880	0.0288440	+	0.999584i	$0.490817\pi$
$420$	0	0
$421$	2.75111	0.134081	0.0670405	−	0.997750i	$-0.478644\pi$
$421$	2.75111	0.134081	0.0670405	+	0.997750i	$0.478644\pi$
$422$	0	0
$423$	−49.0810	−2.38640
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−10.4999	−0.508124
$428$	0	0
$429$	−18.0806	−0.872938
$430$	0	0
$431$	−5.88188	−0.283320	−0.141660	−	0.989915i	$-0.545244\pi$
$431$	−5.88188	−0.283320	−0.141660	+	0.989915i	$0.545244\pi$
$432$	0	0
$433$	−31.0185	−1.49066	−0.745328	−	0.666698i	$-0.767707\pi$
$433$	−31.0185	−1.49066	−0.745328	+	0.666698i	$0.767707\pi$
$434$	0	0
$435$	−6.34812	−0.304369
$436$	0	0
$437$	13.0669	0.625075
$438$	0	0
$439$	20.6842	0.987202	0.493601	−	0.869688i	$-0.335680\pi$
$439$	20.6842	0.987202	0.493601	+	0.869688i	$0.335680\pi$
$440$	0	0
$441$	13.1030	0.623954
$442$	0	0
$443$	−9.67739	−0.459787	−0.229893	−	0.973216i	$-0.573838\pi$
$443$	−9.67739	−0.459787	−0.229893	+	0.973216i	$0.573838\pi$
$444$	0	0
$445$	−7.13466	−0.338215
$446$	0	0
$447$	51.4599	2.43397
$448$	0	0
$449$	−6.44566	−0.304189	−0.152095	−	0.988366i	$-0.548602\pi$
$449$	−6.44566	−0.304189	−0.152095	+	0.988366i	$0.548602\pi$
$450$	0	0
$451$	−10.7072	−0.504181
$452$	0	0
$453$	43.2082	2.03010
$454$	0	0
$455$	−9.93403	−0.465715
$456$	0	0
$457$	17.1381	0.801687	0.400843	−	0.916147i	$-0.368717\pi$
$457$	17.1381	0.801687	0.400843	+	0.916147i	$0.368717\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	23.5436	1.09653	0.548267	−	0.836303i	$-0.315288\pi$
$461$	23.5436	1.09653	0.548267	+	0.836303i	$0.315288\pi$
$462$	0	0
$463$	22.5574	1.04833	0.524167	−	0.851616i	$-0.324377\pi$
$463$	22.5574	1.04833	0.524167	+	0.851616i	$0.324377\pi$
$464$	0	0
$465$	0.138389	0.00641765
$466$	0	0
$467$	−19.0935	−0.883540	−0.441770	−	0.897128i	$-0.645649\pi$
$467$	−19.0935	−0.883540	−0.441770	+	0.897128i	$0.645649\pi$
$468$	0	0
$469$	−5.60362	−0.258751
$470$	0	0
$471$	−53.3698	−2.45915
$472$	0	0
$473$	22.0659	1.01459
$474$	0	0
$475$	−4.24104	−0.194592
$476$	0	0
$477$	−1.36330	−0.0624214
$478$	0	0
$479$	7.85013	0.358681	0.179341	−	0.983787i	$-0.442604\pi$
$479$	7.85013	0.358681	0.179341	+	0.983787i	$0.442604\pi$
$480$	0	0
$481$	−33.0487	−1.50689
$482$	0	0
$483$	27.3906	1.24632
$484$	0	0
$485$	−8.20149	−0.372410
$486$	0	0
$487$	−7.89845	−0.357913	−0.178956	−	0.983857i	$-0.557272\pi$
$487$	−7.89845	−0.357913	−0.178956	+	0.983857i	$0.557272\pi$
$488$	0	0
$489$	−60.8047	−2.74968
$490$	0	0
$491$	31.4268	1.41827	0.709136	−	0.705072i	$-0.249085\pi$
$491$	31.4268	1.41827	0.709136	+	0.705072i	$0.249085\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	−10.3832	−0.466689
$496$	0	0
$497$	−26.2642	−1.17811
$498$	0	0
$499$	−26.0000	−1.16392	−0.581959	−	0.813218i	$-0.697714\pi$
$499$	−26.0000	−1.16392	−0.581959	+	0.813218i	$0.697714\pi$
$500$	0	0
$501$	−0.313691	−0.0140147
$502$	0	0
$503$	−11.9666	−0.533563	−0.266781	−	0.963757i	$-0.585960\pi$
$503$	−11.9666	−0.533563	−0.266781	+	0.963757i	$0.585960\pi$
$504$	0	0
$505$	−13.8146	−0.614740
$506$	0	0
$507$	7.36288	0.326997
$508$	0	0
$509$	−28.2076	−1.25028	−0.625141	−	0.780512i	$-0.714959\pi$
$509$	−28.2076	−1.25028	−0.625141	+	0.780512i	$0.714959\pi$
$510$	0	0
$511$	−47.6308	−2.10706
$512$	0	0
$513$	29.1249	1.28589
$514$	0	0
$515$	15.7190	0.692662
$516$	0	0
$517$	−17.6510	−0.776288
$518$	0	0
$519$	−1.08800	−0.0477580
$520$	0	0
$521$	33.0179	1.44654	0.723270	−	0.690565i	$-0.242638\pi$
$521$	33.0179	1.44654	0.723270	+	0.690565i	$0.242638\pi$
$522$	0	0
$523$	5.81329	0.254198	0.127099	−	0.991890i	$-0.459433\pi$
$523$	5.81329	0.254198	0.127099	+	0.991890i	$0.459433\pi$
$524$	0	0
$525$	−8.88997	−0.387990
$526$	0	0
$527$	0	0
$528$	0	0
$529$	−13.5070	−0.587262
$530$	0	0
$531$	70.6744	3.06701
$532$	0	0
$533$	−17.9166	−0.776052
$534$	0	0
$535$	−11.6782	−0.504895
$536$	0	0
$537$	30.3570	1.31000
$538$	0	0
$539$	4.71223	0.202970
$540$	0	0
$541$	−17.3728	−0.746913	−0.373456	−	0.927648i	$-0.621827\pi$
$541$	−17.3728	−0.746913	−0.373456	+	0.927648i	$0.621827\pi$
$542$	0	0
$543$	−4.21919	−0.181063
$544$	0	0
$545$	15.5905	0.667824
$546$	0	0
$547$	19.4774	0.832791	0.416396	−	0.909183i	$-0.363293\pi$
$547$	19.4774	0.832791	0.416396	+	0.909183i	$0.363293\pi$
$548$	0	0
$549$	−18.3640	−0.783758
$550$	0	0
$551$	9.30401	0.396364
$552$	0	0
$553$	−25.0737	−1.06624
$554$	0	0
$555$	−29.5753	−1.25540
$556$	0	0
$557$	−25.5903	−1.08430	−0.542148	−	0.840283i	$-0.682389\pi$
$557$	−25.5903	−1.08430	−0.542148	+	0.840283i	$0.682389\pi$
$558$	0	0
$559$	36.9234	1.56169
$560$	0	0
$561$	0	0
$562$	0	0
$563$	30.8871	1.30174	0.650868	−	0.759191i	$-0.274405\pi$
$563$	30.8871	1.30174	0.650868	+	0.759191i	$0.274405\pi$
$564$	0	0
$565$	−17.3118	−0.728312
$566$	0	0
$567$	11.5273	0.484103
$568$	0	0
$569$	−0.478098	−0.0200429	−0.0100214	−	0.999950i	$-0.503190\pi$
$569$	−0.478098	−0.0200429	−0.0100214	+	0.999950i	$0.503190\pi$
$570$	0	0
$571$	−27.9457	−1.16949	−0.584746	−	0.811216i	$-0.698806\pi$
$571$	−27.9457	−1.16949	−0.584746	+	0.811216i	$0.698806\pi$
$572$	0	0
$573$	−1.89926	−0.0793425
$574$	0	0
$575$	−3.08107	−0.128489
$576$	0	0
$577$	22.6033	0.940989	0.470495	−	0.882403i	$-0.344075\pi$
$577$	22.6033	0.940989	0.470495	+	0.882403i	$0.344075\pi$
$578$	0	0
$579$	12.8340	0.533361
$580$	0	0
$581$	47.3497	1.96439
$582$	0	0
$583$	−0.490284	−0.0203055
$584$	0	0
$585$	−17.3744	−0.718344
$586$	0	0
$587$	−38.6428	−1.59496	−0.797480	−	0.603346i	$-0.793834\pi$
$587$	−38.6428	−1.59496	−0.797480	+	0.603346i	$0.793834\pi$
$588$	0	0
$589$	−0.202828	−0.00835737
$590$	0	0
$591$	42.9697	1.76754
$592$	0	0
$593$	−33.1530	−1.36143	−0.680715	−	0.732548i	$-0.738331\pi$
$593$	−33.1530	−1.36143	−0.680715	+	0.732548i	$0.738331\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	29.3871	1.20273
$598$	0	0
$599$	34.3322	1.40278	0.701388	−	0.712780i	$-0.252564\pi$
$599$	34.3322	1.40278	0.701388	+	0.712780i	$0.252564\pi$
$600$	0	0
$601$	−6.25684	−0.255222	−0.127611	−	0.991824i	$-0.540731\pi$
$601$	−6.25684	−0.255222	−0.127611	+	0.991824i	$0.540731\pi$
$602$	0	0
$603$	−9.80062	−0.399112
$604$	0	0
$605$	7.26591	0.295401
$606$	0	0
$607$	−6.26712	−0.254374	−0.127187	−	0.991879i	$-0.540595\pi$
$607$	−6.26712	−0.254374	−0.127187	+	0.991879i	$0.540595\pi$
$608$	0	0
$609$	19.5029	0.790296
$610$	0	0
$611$	−29.5358	−1.19489
$612$	0	0
$613$	−5.74422	−0.232007	−0.116003	−	0.993249i	$-0.537008\pi$
$613$	−5.74422	−0.232007	−0.116003	+	0.993249i	$0.537008\pi$
$614$	0	0
$615$	−16.0335	−0.646534
$616$	0	0
$617$	28.4752	1.14637	0.573184	−	0.819427i	$-0.305708\pi$
$617$	28.4752	1.14637	0.573184	+	0.819427i	$0.305708\pi$
$618$	0	0
$619$	−14.6429	−0.588548	−0.294274	−	0.955721i	$-0.595078\pi$
$619$	−14.6429	−0.588548	−0.294274	+	0.955721i	$0.595078\pi$
$620$	0	0
$621$	21.1589	0.849077
$622$	0	0
$623$	21.9193	0.878177
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	23.7144	0.947061
$628$	0	0
$629$	0	0
$630$	0	0
$631$	6.01702	0.239534	0.119767	−	0.992802i	$-0.461785\pi$
$631$	6.01702	0.239534	0.119767	+	0.992802i	$0.461785\pi$
$632$	0	0
$633$	−6.57256	−0.261236
$634$	0	0
$635$	15.3544	0.609319
$636$	0	0
$637$	7.88509	0.312419
$638$	0	0
$639$	−45.9355	−1.81718
$640$	0	0
$641$	12.3688	0.488537	0.244268	−	0.969708i	$-0.421452\pi$
$641$	12.3688	0.488537	0.244268	+	0.969708i	$0.421452\pi$
$642$	0	0
$643$	36.3388	1.43306	0.716530	−	0.697556i	$-0.245729\pi$
$643$	36.3388	1.43306	0.716530	+	0.697556i	$0.245729\pi$
$644$	0	0
$645$	33.0427	1.30106
$646$	0	0
$647$	4.44024	0.174564	0.0872820	−	0.996184i	$-0.472182\pi$
$647$	4.44024	0.174564	0.0872820	+	0.996184i	$0.472182\pi$
$648$	0	0
$649$	25.4166	0.997688
$650$	0	0
$651$	−0.425163	−0.0166634
$652$	0	0
$653$	−16.1724	−0.632874	−0.316437	−	0.948613i	$-0.602487\pi$
$653$	−16.1724	−0.632874	−0.316437	+	0.948613i	$0.602487\pi$
$654$	0	0
$655$	−18.8563	−0.736775
$656$	0	0
$657$	−83.3052	−3.25005
$658$	0	0
$659$	−33.7527	−1.31482	−0.657410	−	0.753533i	$-0.728348\pi$
$659$	−33.7527	−1.31482	−0.657410	+	0.753533i	$0.728348\pi$
$660$	0	0
$661$	23.0915	0.898156	0.449078	−	0.893492i	$-0.351752\pi$
$661$	23.0915	0.898156	0.449078	+	0.893492i	$0.351752\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	13.0294	0.505259
$666$	0	0
$667$	6.75927	0.261720
$668$	0	0
$669$	−36.9410	−1.42822
$670$	0	0
$671$	−6.60424	−0.254954
$672$	0	0
$673$	−4.65627	−0.179486	−0.0897431	−	0.995965i	$-0.528605\pi$
$673$	−4.65627	−0.179486	−0.0897431	+	0.995965i	$0.528605\pi$
$674$	0	0
$675$	−6.86739	−0.264326
$676$	0	0
$677$	−29.6815	−1.14075	−0.570376	−	0.821384i	$-0.693202\pi$
$677$	−29.6815	−1.14075	−0.570376	+	0.821384i	$0.693202\pi$
$678$	0	0
$679$	25.1968	0.966965
$680$	0	0
$681$	12.0655	0.462350
$682$	0	0
$683$	−40.9178	−1.56568	−0.782838	−	0.622225i	$-0.786229\pi$
$683$	−40.9178	−1.56568	−0.782838	+	0.622225i	$0.786229\pi$
$684$	0	0
$685$	1.16327	0.0444463
$686$	0	0
$687$	24.5642	0.937182
$688$	0	0
$689$	−0.820404	−0.0312549
$690$	0	0
$691$	25.4257	0.967238	0.483619	−	0.875279i	$-0.339322\pi$
$691$	25.4257	0.967238	0.483619	+	0.875279i	$0.339322\pi$
$692$	0	0
$693$	31.8994	1.21176
$694$	0	0
$695$	8.84611	0.335552
$696$	0	0
$697$	0	0
$698$	0	0
$699$	9.12468	0.345127
$700$	0	0
$701$	−30.0302	−1.13422	−0.567112	−	0.823641i	$-0.691939\pi$
$701$	−30.0302	−1.13422	−0.567112	+	0.823641i	$0.691939\pi$
$702$	0	0
$703$	43.3465	1.63485
$704$	0	0
$705$	−26.4316	−0.995471
$706$	0	0
$707$	42.4415	1.59618
$708$	0	0
$709$	−27.8627	−1.04640	−0.523202	−	0.852209i	$-0.675263\pi$
$709$	−27.8627	−1.04640	−0.523202	+	0.852209i	$0.675263\pi$
$710$	0	0
$711$	−43.8533	−1.64463
$712$	0	0
$713$	−0.147352	−0.00551838
$714$	0	0
$715$	−6.24835	−0.233675
$716$	0	0
$717$	−28.1654	−1.05186
$718$	0	0
$719$	−18.3201	−0.683224	−0.341612	−	0.939841i	$-0.610973\pi$
$719$	−18.3201	−0.683224	−0.341612	+	0.939841i	$0.610973\pi$
$720$	0	0
$721$	−48.2923	−1.79850
$722$	0	0
$723$	22.9160	0.852257
$724$	0	0
$725$	−2.19381	−0.0814759
$726$	0	0
$727$	−22.1041	−0.819795	−0.409898	−	0.912132i	$-0.634436\pi$
$727$	−22.1041	−0.819795	−0.409898	+	0.912132i	$0.634436\pi$
$728$	0	0
$729$	−39.4546	−1.46128
$730$	0	0
$731$	0	0
$732$	0	0
$733$	43.9027	1.62158	0.810791	−	0.585335i	$-0.199037\pi$
$733$	43.9027	1.62158	0.810791	+	0.585335i	$0.199037\pi$
$734$	0	0
$735$	7.05637	0.260278
$736$	0	0
$737$	−3.52459	−0.129830
$738$	0	0
$739$	6.48640	0.238606	0.119303	−	0.992858i	$-0.461934\pi$
$739$	6.48640	0.238606	0.119303	+	0.992858i	$0.461934\pi$
$740$	0	0
$741$	39.6818	1.45775
$742$	0	0
$743$	−2.08346	−0.0764347	−0.0382173	−	0.999269i	$-0.512168\pi$
$743$	−2.08346	−0.0764347	−0.0382173	+	0.999269i	$0.512168\pi$
$744$	0	0
$745$	17.7837	0.651544
$746$	0	0
$747$	82.8135	3.02999
$748$	0	0
$749$	35.8782	1.31096
$750$	0	0
$751$	−27.0726	−0.987893	−0.493946	−	0.869492i	$-0.664446\pi$
$751$	−27.0726	−0.987893	−0.493946	+	0.869492i	$0.664446\pi$
$752$	0	0
$753$	4.86653	0.177346
$754$	0	0
$755$	14.9320	0.543432
$756$	0	0
$757$	13.5772	0.493471	0.246736	−	0.969083i	$-0.420642\pi$
$757$	13.5772	0.493471	0.246736	+	0.969083i	$0.420642\pi$
$758$	0	0
$759$	17.2282	0.625345
$760$	0	0
$761$	13.2665	0.480909	0.240454	−	0.970660i	$-0.422704\pi$
$761$	13.2665	0.480909	0.240454	+	0.970660i	$0.422704\pi$
$762$	0	0
$763$	−47.8975	−1.73401
$764$	0	0
$765$	0	0
$766$	0	0
$767$	42.5302	1.53568
$768$	0	0
$769$	−16.8371	−0.607162	−0.303581	−	0.952806i	$-0.598182\pi$
$769$	−16.8371	−0.607162	−0.303581	+	0.952806i	$0.598182\pi$
$770$	0	0
$771$	49.4568	1.78114
$772$	0	0
$773$	−31.5020	−1.13305	−0.566524	−	0.824045i	$-0.691712\pi$
$773$	−31.5020	−1.13305	−0.566524	+	0.824045i	$0.691712\pi$
$774$	0	0
$775$	0.0478250	0.00171793
$776$	0	0
$777$	90.8620	3.25966
$778$	0	0
$779$	23.4992	0.841948
$780$	0	0
$781$	−16.5198	−0.591123
$782$	0	0
$783$	15.0657	0.538405
$784$	0	0
$785$	−18.4437	−0.658284
$786$	0	0
$787$	−18.6313	−0.664134	−0.332067	−	0.943256i	$-0.607746\pi$
$787$	−18.6313	−0.664134	−0.332067	+	0.943256i	$0.607746\pi$
$788$	0	0
$789$	24.2075	0.861809
$790$	0	0
$791$	53.1857	1.89106
$792$	0	0
$793$	−11.0510	−0.392434
$794$	0	0
$795$	−0.734179	−0.0260387
$796$	0	0
$797$	14.5989	0.517118	0.258559	−	0.965995i	$-0.416752\pi$
$797$	14.5989	0.517118	0.258559	+	0.965995i	$0.416752\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	38.3363	1.35455
$802$	0	0
$803$	−29.9590	−1.05723
$804$	0	0
$805$	9.46573	0.333623
$806$	0	0
$807$	81.8809	2.88234
$808$	0	0
$809$	32.0433	1.12658	0.563292	−	0.826258i	$-0.309535\pi$
$809$	32.0433	1.12658	0.563292	+	0.826258i	$0.309535\pi$
$810$	0	0
$811$	10.7673	0.378091	0.189045	−	0.981968i	$-0.439461\pi$
$811$	10.7673	0.378091	0.189045	+	0.981968i	$0.439461\pi$
$812$	0	0
$813$	50.5057	1.77131
$814$	0	0
$815$	−21.0131	−0.736056
$816$	0	0
$817$	−48.4284	−1.69430
$818$	0	0
$819$	53.3781	1.86518
$820$	0	0
$821$	−9.75346	−0.340398	−0.170199	−	0.985410i	$-0.554441\pi$
$821$	−9.75346	−0.340398	−0.170199	+	0.985410i	$0.554441\pi$
$822$	0	0
$823$	2.57587	0.0897891	0.0448946	−	0.998992i	$-0.485705\pi$
$823$	2.57587	0.0897891	0.0448946	+	0.998992i	$0.485705\pi$
$824$	0	0
$825$	−5.59165	−0.194676
$826$	0	0
$827$	−27.5075	−0.956528	−0.478264	−	0.878216i	$-0.658734\pi$
$827$	−27.5075	−0.956528	−0.478264	+	0.878216i	$0.658734\pi$
$828$	0	0
$829$	9.61699	0.334012	0.167006	−	0.985956i	$-0.446590\pi$
$829$	9.61699	0.334012	0.167006	+	0.985956i	$0.446590\pi$
$830$	0	0
$831$	27.4635	0.952699
$832$	0	0
$833$	0	0
$834$	0	0
$835$	−0.108406	−0.00375156
$836$	0	0
$837$	−0.328433	−0.0113523
$838$	0	0
$839$	52.9554	1.82822	0.914111	−	0.405463i	$-0.132890\pi$
$839$	52.9554	1.82822	0.914111	+	0.405463i	$0.132890\pi$
$840$	0	0
$841$	−24.1872	−0.834042
$842$	0	0
$843$	37.7458	1.30004
$844$	0	0
$845$	2.54449	0.0875331
$846$	0	0
$847$	−22.3225	−0.767010
$848$	0	0
$849$	−10.6608	−0.365876
$850$	0	0
$851$	31.4908	1.07949
$852$	0	0
$853$	−28.2185	−0.966183	−0.483092	−	0.875570i	$-0.660486\pi$
$853$	−28.2185	−0.966183	−0.483092	+	0.875570i	$0.660486\pi$
$854$	0	0
$855$	22.7882	0.779339
$856$	0	0
$857$	1.95611	0.0668194	0.0334097	−	0.999442i	$-0.489363\pi$
$857$	1.95611	0.0668194	0.0334097	+	0.999442i	$0.489363\pi$
$858$	0	0
$859$	14.8662	0.507228	0.253614	−	0.967305i	$-0.418381\pi$
$859$	14.8662	0.507228	0.253614	+	0.967305i	$0.418381\pi$
$860$	0	0
$861$	49.2586	1.67873
$862$	0	0
$863$	−48.4548	−1.64942	−0.824711	−	0.565555i	$-0.808662\pi$
$863$	−48.4548	−1.64942	−0.824711	+	0.565555i	$0.808662\pi$
$864$	0	0
$865$	−0.375995	−0.0127842
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−15.7709	−0.534992
$870$	0	0
$871$	−5.89778	−0.199839
$872$	0	0
$873$	44.0687	1.49150
$874$	0	0
$875$	−3.07222	−0.103860
$876$	0	0
$877$	−21.1605	−0.714541	−0.357270	−	0.934001i	$-0.616293\pi$
$877$	−21.1605	−0.714541	−0.357270	+	0.934001i	$0.616293\pi$
$878$	0	0
$879$	21.0922	0.711423
$880$	0	0
$881$	19.2849	0.649724	0.324862	−	0.945761i	$-0.394682\pi$
$881$	19.2849	0.649724	0.324862	+	0.945761i	$0.394682\pi$
$882$	0	0
$883$	−0.604599	−0.0203464	−0.0101732	−	0.999948i	$-0.503238\pi$
$883$	−0.604599	−0.0203464	−0.0101732	+	0.999948i	$0.503238\pi$
$884$	0	0
$885$	38.0603	1.27938
$886$	0	0
$887$	−39.7673	−1.33525	−0.667627	−	0.744496i	$-0.732690\pi$
$887$	−39.7673	−1.33525	−0.667627	+	0.744496i	$0.732690\pi$
$888$	0	0
$889$	−47.1720	−1.58210
$890$	0	0
$891$	7.25051	0.242901
$892$	0	0
$893$	38.7389	1.29635
$894$	0	0
$895$	10.4909	0.350671
$896$	0	0
$897$	28.8284	0.962553
$898$	0	0
$899$	−0.104919	−0.00349924
$900$	0	0
$901$	0	0
$902$	0	0
$903$	−101.515	−3.37820
$904$	0	0
$905$	−1.45808	−0.0484683
$906$	0	0
$907$	43.7695	1.45334	0.726671	−	0.686986i	$-0.241067\pi$
$907$	43.7695	1.45334	0.726671	+	0.686986i	$0.241067\pi$
$908$	0	0
$909$	74.2292	2.46203
$910$	0	0
$911$	45.6778	1.51337	0.756687	−	0.653777i	$-0.226817\pi$
$911$	45.6778	1.51337	0.756687	+	0.653777i	$0.226817\pi$
$912$	0	0
$913$	29.7822	0.985646
$914$	0	0
$915$	−9.88958	−0.326939
$916$	0	0
$917$	57.9306	1.91304
$918$	0	0
$919$	13.1429	0.433544	0.216772	−	0.976222i	$-0.430447\pi$
$919$	13.1429	0.433544	0.216772	+	0.976222i	$0.430447\pi$
$920$	0	0
$921$	−28.5483	−0.940699
$922$	0	0
$923$	−27.6429	−0.909877
$924$	0	0
$925$	−10.2207	−0.336056
$926$	0	0
$927$	−84.4622	−2.77410
$928$	0	0
$929$	−4.33338	−0.142174	−0.0710868	−	0.997470i	$-0.522647\pi$
$929$	−4.33338	−0.142174	−0.0710868	+	0.997470i	$0.522647\pi$
$930$	0	0
$931$	−10.3420	−0.338947
$932$	0	0
$933$	0.910295	0.0298017
$934$	0	0
$935$	0	0
$936$	0	0
$937$	17.9407	0.586098	0.293049	−	0.956097i	$-0.405330\pi$
$937$	17.9407	0.586098	0.293049	+	0.956097i	$0.405330\pi$
$938$	0	0
$939$	−80.2144	−2.61770
$940$	0	0
$941$	−58.0350	−1.89189	−0.945943	−	0.324332i	$-0.894861\pi$
$941$	−58.0350	−1.89189	−0.945943	+	0.324332i	$0.894861\pi$
$942$	0	0
$943$	17.0720	0.555939
$944$	0	0
$945$	21.0982	0.686324
$946$	0	0
$947$	1.61431	0.0524579	0.0262289	−	0.999656i	$-0.491650\pi$
$947$	1.61431	0.0524579	0.0262289	+	0.999656i	$0.491650\pi$
$948$	0	0
$949$	−50.1311	−1.62732
$950$	0	0
$951$	2.27970	0.0739243
$952$	0	0
$953$	20.5222	0.664781	0.332390	−	0.943142i	$-0.392145\pi$
$953$	20.5222	0.664781	0.332390	+	0.943142i	$0.392145\pi$
$954$	0	0
$955$	−0.656351	−0.0212390
$956$	0	0
$957$	12.2670	0.396535
$958$	0	0
$959$	−3.57383	−0.115405
$960$	0	0
$961$	−30.9977	−0.999926
$962$	0	0
$963$	62.7502	2.02210
$964$	0	0
$965$	4.43520	0.142774
$966$	0	0
$967$	−28.1972	−0.906760	−0.453380	−	0.891317i	$-0.649782\pi$
$967$	−28.1972	−0.906760	−0.453380	+	0.891317i	$0.649782\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−55.8946	−1.79374	−0.896872	−	0.442290i	$-0.854166\pi$
$971$	−55.8946	−1.79374	−0.896872	+	0.442290i	$0.854166\pi$
$972$	0	0
$973$	−27.1772	−0.871262
$974$	0	0
$975$	−9.35664	−0.299652
$976$	0	0
$977$	−22.8402	−0.730722	−0.365361	−	0.930866i	$-0.619054\pi$
$977$	−22.8402	−0.730722	−0.365361	+	0.930866i	$0.619054\pi$
$978$	0	0
$979$	13.7869	0.440630
$980$	0	0
$981$	−83.7718	−2.67463
$982$	0	0
$983$	−20.5614	−0.655808	−0.327904	−	0.944711i	$-0.606342\pi$
$983$	−20.5614	−0.655808	−0.327904	+	0.944711i	$0.606342\pi$
$984$	0	0
$985$	14.8496	0.473148
$986$	0	0
$987$	81.2037	2.58474
$988$	0	0
$989$	−35.1828	−1.11875
$990$	0	0
$991$	32.5805	1.03495	0.517477	−	0.855697i	$-0.326871\pi$
$991$	32.5805	1.03495	0.517477	+	0.855697i	$0.326871\pi$
$992$	0	0
$993$	−5.66053	−0.179632
$994$	0	0
$995$	10.1557	0.321957
$996$	0	0
$997$	30.0291	0.951031	0.475516	−	0.879707i	$-0.342262\pi$
$997$	30.0291	0.951031	0.475516	+	0.879707i	$0.342262\pi$
$998$	0	0
$999$	70.1898	2.22071

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5780.2.a.q.1.1		12
17.3	odd	16		340.2.u.a.281.1	yes	24
17.4	even	4		5780.2.c.j.5201.23		24
17.6	odd	16		340.2.u.a.121.1	✓	24
17.13	even	4		5780.2.c.j.5201.2		24
17.16	even	2		5780.2.a.r.1.12		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
340.2.u.a.121.1	✓	24	17.6	odd	16
340.2.u.a.281.1	yes	24	17.3	odd	16
5780.2.a.q.1.1		12	1.1	even	1	trivial
5780.2.a.r.1.12		12	17.16	even	2
5780.2.c.j.5201.2		24	17.13	even	4
5780.2.c.j.5201.23		24	17.4	even	4

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:	\( N \)	\(=\)	\( 5780 = 2^{2} \cdot 5 \cdot 17^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5780.a (trivial)

\(f(q)\)	\(=\)	\(q-2.89366 q^{3} -1.00000 q^{5} +3.07222 q^{7} +5.37326 q^{9} +O(q^{10})\)	\(q-2.89366 q^{3} -1.00000 q^{5} +3.07222 q^{7} +5.37326 q^{9} +1.93238 q^{11} +3.23350 q^{13} +2.89366 q^{15} -4.24104 q^{19} -8.88997 q^{21} -3.08107 q^{23} +1.00000 q^{25} -6.86739 q^{27} -2.19381 q^{29} +0.0478250 q^{31} -5.59165 q^{33} -3.07222 q^{35} -10.2207 q^{37} -9.35664 q^{39} -5.54092 q^{41} +11.4190 q^{43} -5.37326 q^{45} -9.13431 q^{47} +2.43856 q^{49} -0.253720 q^{53} -1.93238 q^{55} +12.2721 q^{57} +13.1530 q^{59} -3.41767 q^{61} +16.5079 q^{63} -3.23350 q^{65} -1.82396 q^{67} +8.91556 q^{69} -8.54892 q^{71} -15.5037 q^{73} -2.89366 q^{75} +5.93670 q^{77} -8.16140 q^{79} +3.75211 q^{81} +15.4122 q^{83} +6.34812 q^{87} +7.13466 q^{89} +9.93403 q^{91} -0.138389 q^{93} +4.24104 q^{95} +8.20149 q^{97} +10.3832 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 12 q^{5} - 8 q^{7} + 12 q^{9}+O(q^{10}) \) 12 * q - 12 * q^5 - 8 * q^7 + 12 * q^9	\( 12 q - 12 q^{5} - 8 q^{7} + 12 q^{9} + 8 q^{13} - 16 q^{21} - 8 q^{23} + 12 q^{25} - 16 q^{29} - 24 q^{31} + 8 q^{35} - 24 q^{37} + 8 q^{39} - 24 q^{41} + 8 q^{43} - 12 q^{45} + 8 q^{47} + 20 q^{49} + 16 q^{53} - 32 q^{57} + 8 q^{59} - 40 q^{61} - 24 q^{63} - 8 q^{65} + 16 q^{67} - 16 q^{69} - 16 q^{71} - 32 q^{73} + 24 q^{77} - 8 q^{79} + 4 q^{81} + 32 q^{83} + 16 q^{87} - 8 q^{89} - 8 q^{91} - 8 q^{93} - 32 q^{97} - 24 q^{99}+O(q^{100}) \) 12 * q - 12 * q^5 - 8 * q^7 + 12 * q^9 + 8 * q^13 - 16 * q^21 - 8 * q^23 + 12 * q^25 - 16 * q^29 - 24 * q^31 + 8 * q^35 - 24 * q^37 + 8 * q^39 - 24 * q^41 + 8 * q^43 - 12 * q^45 + 8 * q^47 + 20 * q^49 + 16 * q^53 - 32 * q^57 + 8 * q^59 - 40 * q^61 - 24 * q^63 - 8 * q^65 + 16 * q^67 - 16 * q^69 - 16 * q^71 - 32 * q^73 + 24 * q^77 - 8 * q^79 + 4 * q^81 + 32 * q^83 + 16 * q^87 - 8 * q^89 - 8 * q^91 - 8 * q^93 - 32 * q^97 - 24 * q^99

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