LMFDB - Embedded newform 5312.2.a.br.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5312,2,Mod(1,5312)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5312, 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("5312.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5312 = 2^{6} \cdot 83 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5312.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:	$42.4165335537$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 2x^{7} - 18x^{6} + 33x^{5} + 87x^{4} - 127x^{3} - 126x^{2} + 100x - 16 $ x^8 - 2x^7 - 18x^6 + 33x^5 + 87x^4 - 127x^3 - 126x^2 + 100*x - 16
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 664)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$0.297635$ of defining polynomial
Character	$\chi$	$=$	5312.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.297635 q^{3} -3.84665 q^{5} +1.95246 q^{7} -2.91141 q^{9} +O(q^{10})$	$q-0.297635 q^{3} -3.84665 q^{5} +1.95246 q^{7} -2.91141 q^{9} -3.36179 q^{11} -4.30849 q^{13} +1.14490 q^{15} -3.55797 q^{17} +6.21341 q^{19} -0.581121 q^{21} -6.13909 q^{23} +9.79670 q^{25} +1.75944 q^{27} -9.67464 q^{29} -4.49202 q^{31} +1.00058 q^{33} -7.51043 q^{35} -5.91141 q^{37} +1.28236 q^{39} -7.02818 q^{41} +7.98694 q^{43} +11.1992 q^{45} +6.67793 q^{47} -3.18789 q^{49} +1.05898 q^{51} -0.940210 q^{53} +12.9316 q^{55} -1.84933 q^{57} +4.84423 q^{59} -9.67464 q^{61} -5.68442 q^{63} +16.5732 q^{65} -12.8131 q^{67} +1.82721 q^{69} -8.78876 q^{71} -5.27320 q^{73} -2.91584 q^{75} -6.56376 q^{77} +13.0101 q^{79} +8.21057 q^{81} +1.00000 q^{83} +13.6863 q^{85} +2.87951 q^{87} +1.89620 q^{89} -8.41216 q^{91} +1.33698 q^{93} -23.9008 q^{95} +5.58949 q^{97} +9.78755 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 2 q^{3} - 7 q^{5} + q^{7} + 16 q^{9}+O(q^{10}) $ 8 * q - 2 * q^3 - 7 * q^5 + q^7 + 16 * q^9	$ 8 q - 2 q^{3} - 7 q^{5} + q^{7} + 16 q^{9} - 10 q^{11} - 7 q^{13} - 14 q^{15} + 15 q^{17} - 7 q^{19} + 5 q^{21} + 4 q^{23} + 27 q^{25} - 5 q^{27} - 16 q^{29} + 3 q^{31} + 21 q^{33} - 2 q^{35} - 8 q^{37} + 10 q^{39} + 24 q^{41} + q^{43} - 29 q^{45} - 2 q^{47} + 3 q^{49} + 9 q^{51} - 7 q^{53} - 30 q^{55} + 4 q^{57} + 2 q^{59} - 16 q^{61} - 10 q^{63} + 2 q^{65} + 25 q^{67} + 6 q^{69} + 8 q^{71} + 14 q^{73} + 30 q^{75} + 7 q^{77} - 4 q^{79} + 52 q^{81} + 8 q^{83} + 21 q^{85} - 4 q^{87} + 20 q^{89} + 45 q^{91} + 47 q^{93} + 8 q^{95} + 2 q^{97} - 10 q^{99}+O(q^{100}) $ 8 * q - 2 * q^3 - 7 * q^5 + q^7 + 16 * q^9 - 10 * q^11 - 7 * q^13 - 14 * q^15 + 15 * q^17 - 7 * q^19 + 5 * q^21 + 4 * q^23 + 27 * q^25 - 5 * q^27 - 16 * q^29 + 3 * q^31 + 21 * q^33 - 2 * q^35 - 8 * q^37 + 10 * q^39 + 24 * q^41 + q^43 - 29 * q^45 - 2 * q^47 + 3 * q^49 + 9 * q^51 - 7 * q^53 - 30 * q^55 + 4 * q^57 + 2 * q^59 - 16 * q^61 - 10 * q^63 + 2 * q^65 + 25 * q^67 + 6 * q^69 + 8 * q^71 + 14 * q^73 + 30 * q^75 + 7 * q^77 - 4 * q^79 + 52 * q^81 + 8 * q^83 + 21 * q^85 - 4 * q^87 + 20 * q^89 + 45 * q^91 + 47 * q^93 + 8 * q^95 + 2 * q^97 - 10 * 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
$3$	−0.297635	−0.171840	−0.0859198	−	0.996302i	$-0.527383\pi$
$3$	−0.297635	−0.171840	−0.0859198	+	0.996302i	$0.527383\pi$
$4$	0	0
$5$	−3.84665	−1.72027	−0.860136	−	0.510064i	$-0.829622\pi$
$5$	−3.84665	−1.72027	−0.860136	+	0.510064i	$0.829622\pi$
$6$	0	0
$7$	1.95246	0.737961	0.368981	−	0.929437i	$-0.379707\pi$
$7$	1.95246	0.737961	0.368981	+	0.929437i	$0.379707\pi$
$8$	0	0
$9$	−2.91141	−0.970471
$10$	0	0
$11$	−3.36179	−1.01362	−0.506808	−	0.862059i	$-0.669175\pi$
$11$	−3.36179	−1.01362	−0.506808	+	0.862059i	$0.669175\pi$
$12$	0	0
$13$	−4.30849	−1.19496	−0.597480	−	0.801884i	$-0.703831\pi$
$13$	−4.30849	−1.19496	−0.597480	+	0.801884i	$0.703831\pi$
$14$	0	0
$15$	1.14490	0.295611
$16$	0	0
$17$	−3.55797	−0.862934	−0.431467	−	0.902129i	$-0.642004\pi$
$17$	−3.55797	−0.862934	−0.431467	+	0.902129i	$0.642004\pi$
$18$	0	0
$19$	6.21341	1.42545	0.712727	−	0.701441i	$-0.247460\pi$
$19$	6.21341	1.42545	0.712727	+	0.701441i	$0.247460\pi$
$20$	0	0
$21$	−0.581121	−0.126811
$22$	0	0
$23$	−6.13909	−1.28009	−0.640044	−	0.768338i	$-0.721084\pi$
$23$	−6.13909	−1.28009	−0.640044	+	0.768338i	$0.721084\pi$
$24$	0	0
$25$	9.79670	1.95934
$26$	0	0
$27$	1.75944	0.338605
$28$	0	0
$29$	−9.67464	−1.79653	−0.898267	−	0.439449i	$-0.855174\pi$
$29$	−9.67464	−1.79653	−0.898267	+	0.439449i	$0.855174\pi$
$30$	0	0
$31$	−4.49202	−0.806790	−0.403395	−	0.915026i	$-0.632170\pi$
$31$	−4.49202	−0.806790	−0.403395	+	0.915026i	$0.632170\pi$
$32$	0	0
$33$	1.00058	0.174179
$34$	0	0
$35$	−7.51043	−1.26949
$36$	0	0
$37$	−5.91141	−0.971830	−0.485915	−	0.874006i	$-0.661514\pi$
$37$	−5.91141	−0.971830	−0.485915	+	0.874006i	$0.661514\pi$
$38$	0	0
$39$	1.28236	0.205341
$40$	0	0
$41$	−7.02818	−1.09762	−0.548809	−	0.835948i	$-0.684919\pi$
$41$	−7.02818	−1.09762	−0.548809	+	0.835948i	$0.684919\pi$
$42$	0	0
$43$	7.98694	1.21800	0.608998	−	0.793172i	$-0.291572\pi$
$43$	7.98694	1.21800	0.608998	+	0.793172i	$0.291572\pi$
$44$	0	0
$45$	11.1992	1.66948
$46$	0	0
$47$	6.67793	0.974076	0.487038	−	0.873381i	$-0.338077\pi$
$47$	6.67793	0.974076	0.487038	+	0.873381i	$0.338077\pi$
$48$	0	0
$49$	−3.18789	−0.455413
$50$	0	0
$51$	1.05898	0.148286
$52$	0	0
$53$	−0.940210	−0.129148	−0.0645739	−	0.997913i	$-0.520569\pi$
$53$	−0.940210	−0.129148	−0.0645739	+	0.997913i	$0.520569\pi$
$54$	0	0
$55$	12.9316	1.74370
$56$	0	0
$57$	−1.84933	−0.244949
$58$	0	0
$59$	4.84423	0.630666	0.315333	−	0.948981i	$-0.397884\pi$
$59$	4.84423	0.630666	0.315333	+	0.948981i	$0.397884\pi$
$60$	0	0
$61$	−9.67464	−1.23871	−0.619355	−	0.785111i	$-0.712606\pi$
$61$	−9.67464	−1.23871	−0.619355	+	0.785111i	$0.712606\pi$
$62$	0	0
$63$	−5.68442	−0.716170
$64$	0	0
$65$	16.5732	2.05566
$66$	0	0
$67$	−12.8131	−1.56537	−0.782687	−	0.622415i	$-0.786152\pi$
$67$	−12.8131	−1.56537	−0.782687	+	0.622415i	$0.786152\pi$
$68$	0	0
$69$	1.82721	0.219970
$70$	0	0
$71$	−8.78876	−1.04303	−0.521517	−	0.853241i	$-0.674634\pi$
$71$	−8.78876	−1.04303	−0.521517	+	0.853241i	$0.674634\pi$
$72$	0	0
$73$	−5.27320	−0.617181	−0.308591	−	0.951195i	$-0.599857\pi$
$73$	−5.27320	−0.617181	−0.308591	+	0.951195i	$0.599857\pi$
$74$	0	0
$75$	−2.91584	−0.336692
$76$	0	0
$77$	−6.56376	−0.748010
$78$	0	0
$79$	13.0101	1.46375	0.731876	−	0.681438i	$-0.238645\pi$
$79$	13.0101	1.46375	0.731876	+	0.681438i	$0.238645\pi$
$80$	0	0
$81$	8.21057	0.912285
$82$	0	0
$83$	1.00000	0.109764
$83$	1.00000	0.109764
$84$	0	0
$85$	13.6863	1.48448
$86$	0	0
$87$	2.87951	0.308716
$88$	0	0
$89$	1.89620	0.200997	0.100498	−	0.994937i	$-0.467956\pi$
$89$	1.89620	0.200997	0.100498	+	0.994937i	$0.467956\pi$
$90$	0	0
$91$	−8.41216	−0.881834
$92$	0	0
$93$	1.33698	0.138638
$94$	0	0
$95$	−23.9008	−2.45217
$96$	0	0
$97$	5.58949	0.567527	0.283764	−	0.958894i	$-0.408417\pi$
$97$	5.58949	0.567527	0.283764	+	0.958894i	$0.408417\pi$
$98$	0	0
$99$	9.78755	0.983686
$100$	0	0
$101$	9.92663	0.987736	0.493868	−	0.869537i	$-0.335583\pi$
$101$	9.92663	0.987736	0.493868	+	0.869537i	$0.335583\pi$
$102$	0	0
$103$	−17.1172	−1.68660	−0.843302	−	0.537439i	$-0.819392\pi$
$103$	−17.1172	−1.68660	−0.843302	+	0.537439i	$0.819392\pi$
$104$	0	0
$105$	2.23537	0.218149
$106$	0	0
$107$	−4.96471	−0.479957	−0.239978	−	0.970778i	$-0.577140\pi$
$107$	−4.96471	−0.479957	−0.239978	+	0.970778i	$0.577140\pi$
$108$	0	0
$109$	20.2864	1.94309	0.971543	−	0.236862i	$-0.0761190\pi$
$109$	20.2864	1.94309	0.971543	+	0.236862i	$0.0761190\pi$
$110$	0	0
$111$	1.75944	0.166999
$112$	0	0
$113$	−4.69082	−0.441275	−0.220637	−	0.975356i	$-0.570814\pi$
$113$	−4.69082	−0.441275	−0.220637	+	0.975356i	$0.570814\pi$
$114$	0	0
$115$	23.6149	2.20210
$116$	0	0
$117$	12.5438	1.15967
$118$	0	0
$119$	−6.94680	−0.636812
$120$	0	0
$121$	0.301612	0.0274193
$122$	0	0
$123$	2.09183	0.188614
$124$	0	0
$125$	−18.4512	−1.65033
$126$	0	0
$127$	5.30316	0.470579	0.235290	−	0.971925i	$-0.424396\pi$
$127$	5.30316	0.470579	0.235290	+	0.971925i	$0.424396\pi$
$128$	0	0
$129$	−2.37719	−0.209300
$130$	0	0
$131$	−3.57603	−0.312439	−0.156219	−	0.987722i	$-0.549931\pi$
$131$	−3.57603	−0.312439	−0.156219	+	0.987722i	$0.549931\pi$
$132$	0	0
$133$	12.1314	1.05193
$134$	0	0
$135$	−6.76795	−0.582493
$136$	0	0
$137$	3.89497	0.332770	0.166385	−	0.986061i	$-0.446791\pi$
$137$	3.89497	0.332770	0.166385	+	0.986061i	$0.446791\pi$
$138$	0	0
$139$	11.8799	1.00764	0.503819	−	0.863809i	$-0.331928\pi$
$139$	11.8799	1.00764	0.503819	+	0.863809i	$0.331928\pi$
$140$	0	0
$141$	−1.98758	−0.167385
$142$	0	0
$143$	14.4842	1.21123
$144$	0	0
$145$	37.2149	3.09053
$146$	0	0
$147$	0.948828	0.0782580
$148$	0	0
$149$	0.607903	0.0498013	0.0249007	−	0.999690i	$-0.492073\pi$
$149$	0.607903	0.0498013	0.0249007	+	0.999690i	$0.492073\pi$
$150$	0	0
$151$	−18.9340	−1.54083	−0.770415	−	0.637543i	$-0.779951\pi$
$151$	−18.9340	−1.54083	−0.770415	+	0.637543i	$0.779951\pi$
$152$	0	0
$153$	10.3587	0.837453
$154$	0	0
$155$	17.2792	1.38790
$156$	0	0
$157$	17.6682	1.41008	0.705040	−	0.709168i	$-0.250929\pi$
$157$	17.6682	1.41008	0.705040	+	0.709168i	$0.250929\pi$
$158$	0	0
$159$	0.279839	0.0221927
$160$	0	0
$161$	−11.9863	−0.944656
$162$	0	0
$163$	17.8366	1.39707	0.698534	−	0.715577i	$-0.253836\pi$
$163$	17.8366	1.39707	0.698534	+	0.715577i	$0.253836\pi$
$164$	0	0
$165$	−3.84890	−0.299636
$166$	0	0
$167$	1.87561	0.145139	0.0725693	−	0.997363i	$-0.476880\pi$
$167$	1.87561	0.145139	0.0725693	+	0.997363i	$0.476880\pi$
$168$	0	0
$169$	5.56306	0.427928
$170$	0	0
$171$	−18.0898	−1.38336
$172$	0	0
$173$	−2.83264	−0.215361	−0.107681	−	0.994186i	$-0.534342\pi$
$173$	−2.83264	−0.215361	−0.107681	+	0.994186i	$0.534342\pi$
$174$	0	0
$175$	19.1277	1.44592
$176$	0	0
$177$	−1.44181	−0.108373
$178$	0	0
$179$	−9.63502	−0.720155	−0.360078	−	0.932922i	$-0.617250\pi$
$179$	−9.63502	−0.720155	−0.360078	+	0.932922i	$0.617250\pi$
$180$	0	0
$181$	14.0440	1.04388	0.521941	−	0.852982i	$-0.325208\pi$
$181$	14.0440	1.04388	0.521941	+	0.852982i	$0.325208\pi$
$182$	0	0
$183$	2.87951	0.212859
$184$	0	0
$185$	22.7391	1.67181
$186$	0	0
$187$	11.9611	0.874685
$188$	0	0
$189$	3.43524	0.249877
$190$	0	0
$191$	11.7727	0.851846	0.425923	−	0.904759i	$-0.359950\pi$
$191$	11.7727	0.851846	0.425923	+	0.904759i	$0.359950\pi$
$192$	0	0
$193$	−21.4497	−1.54399	−0.771993	−	0.635631i	$-0.780740\pi$
$193$	−21.4497	−1.54399	−0.771993	+	0.635631i	$0.780740\pi$
$194$	0	0
$195$	−4.93277	−0.353243
$196$	0	0
$197$	−9.87874	−0.703831	−0.351916	−	0.936032i	$-0.614470\pi$
$197$	−9.87874	−0.703831	−0.351916	+	0.936032i	$0.614470\pi$
$198$	0	0
$199$	−15.5924	−1.10531	−0.552656	−	0.833409i	$-0.686386\pi$
$199$	−15.5924	−1.10531	−0.552656	+	0.833409i	$0.686386\pi$
$200$	0	0
$201$	3.81364	0.268993
$202$	0	0
$203$	−18.8894	−1.32577
$204$	0	0
$205$	27.0349	1.88820
$206$	0	0
$207$	17.8734	1.24229
$208$	0	0
$209$	−20.8882	−1.44486
$210$	0	0
$211$	−10.5994	−0.729695	−0.364848	−	0.931067i	$-0.618879\pi$
$211$	−10.5994	−0.729695	−0.364848	+	0.931067i	$0.618879\pi$
$212$	0	0
$213$	2.61584	0.179234
$214$	0	0
$215$	−30.7229	−2.09529
$216$	0	0
$217$	−8.77049	−0.595380
$218$	0	0
$219$	1.56949	0.106056
$220$	0	0
$221$	15.3295	1.03117
$222$	0	0
$223$	21.2459	1.42273	0.711365	−	0.702822i	$-0.248077\pi$
$223$	21.2459	1.42273	0.711365	+	0.702822i	$0.248077\pi$
$224$	0	0
$225$	−28.5222	−1.90148
$226$	0	0
$227$	−1.67282	−0.111029	−0.0555144	−	0.998458i	$-0.517680\pi$
$227$	−1.67282	−0.111029	−0.0555144	+	0.998458i	$0.517680\pi$
$228$	0	0
$229$	−13.6746	−0.903641	−0.451821	−	0.892109i	$-0.649225\pi$
$229$	−13.6746	−0.903641	−0.451821	+	0.892109i	$0.649225\pi$
$230$	0	0
$231$	1.95360	0.128538
$232$	0	0
$233$	15.0677	0.987120	0.493560	−	0.869712i	$-0.335695\pi$
$233$	15.0677	0.987120	0.493560	+	0.869712i	$0.335695\pi$
$234$	0	0
$235$	−25.6876	−1.67568
$236$	0	0
$237$	−3.87226	−0.251530
$238$	0	0
$239$	3.13774	0.202963	0.101482	−	0.994837i	$-0.467642\pi$
$239$	3.13774	0.202963	0.101482	+	0.994837i	$0.467642\pi$
$240$	0	0
$241$	21.0347	1.35497	0.677483	−	0.735538i	$-0.263071\pi$
$241$	21.0347	1.35497	0.677483	+	0.735538i	$0.263071\pi$
$242$	0	0
$243$	−7.72208	−0.495372
$244$	0	0
$245$	12.2627	0.783435
$246$	0	0
$247$	−26.7704	−1.70336
$248$	0	0
$249$	−0.297635	−0.0188618
$250$	0	0
$251$	−23.7003	−1.49595	−0.747976	−	0.663726i	$-0.768974\pi$
$251$	−23.7003	−1.49595	−0.747976	+	0.663726i	$0.768974\pi$
$252$	0	0
$253$	20.6383	1.29752
$254$	0	0
$255$	−4.07351	−0.255093
$256$	0	0
$257$	−8.78673	−0.548101	−0.274050	−	0.961715i	$-0.588364\pi$
$257$	−8.78673	−0.548101	−0.274050	+	0.961715i	$0.588364\pi$
$258$	0	0
$259$	−11.5418	−0.717173
$260$	0	0
$261$	28.1669	1.74349
$262$	0	0
$263$	9.33838	0.575829	0.287915	−	0.957656i	$-0.407038\pi$
$263$	9.33838	0.575829	0.287915	+	0.957656i	$0.407038\pi$
$264$	0	0
$265$	3.61666	0.222169
$266$	0	0
$267$	−0.564375	−0.0345392
$268$	0	0
$269$	−26.1829	−1.59640	−0.798198	−	0.602395i	$-0.794213\pi$
$269$	−26.1829	−1.59640	−0.798198	+	0.602395i	$0.794213\pi$
$270$	0	0
$271$	17.8483	1.08421	0.542103	−	0.840312i	$-0.317628\pi$
$271$	17.8483	1.08421	0.542103	+	0.840312i	$0.317628\pi$
$272$	0	0
$273$	2.50375	0.151534
$274$	0	0
$275$	−32.9344	−1.98602
$276$	0	0
$277$	14.1906	0.852631	0.426315	−	0.904575i	$-0.359811\pi$
$277$	14.1906	0.852631	0.426315	+	0.904575i	$0.359811\pi$
$278$	0	0
$279$	13.0781	0.782967
$280$	0	0
$281$	10.1483	0.605397	0.302699	−	0.953086i	$-0.402112\pi$
$281$	10.1483	0.605397	0.302699	+	0.953086i	$0.402112\pi$
$282$	0	0
$283$	10.5364	0.626327	0.313163	−	0.949699i	$-0.398611\pi$
$283$	10.5364	0.626327	0.313163	+	0.949699i	$0.398611\pi$
$284$	0	0
$285$	7.11371	0.421380
$286$	0	0
$287$	−13.7223	−0.809999
$288$	0	0
$289$	−4.34085	−0.255344
$290$	0	0
$291$	−1.66363	−0.0975236
$292$	0	0
$293$	−3.49114	−0.203954	−0.101977	−	0.994787i	$-0.532517\pi$
$293$	−3.49114	−0.203954	−0.101977	+	0.994787i	$0.532517\pi$
$294$	0	0
$295$	−18.6341	−1.08492
$296$	0	0
$297$	−5.91487	−0.343216
$298$	0	0
$299$	26.4502	1.52965
$300$	0	0
$301$	15.5942	0.898834
$302$	0	0
$303$	−2.95451	−0.169732
$304$	0	0
$305$	37.2149	2.13092
$306$	0	0
$307$	4.48031	0.255705	0.127852	−	0.991793i	$-0.459192\pi$
$307$	4.48031	0.255705	0.127852	+	0.991793i	$0.459192\pi$
$308$	0	0
$309$	5.09466	0.289825
$310$	0	0
$311$	26.9209	1.52654	0.763271	−	0.646079i	$-0.223592\pi$
$311$	26.9209	1.52654	0.763271	+	0.646079i	$0.223592\pi$
$312$	0	0
$313$	15.7811	0.892000	0.446000	−	0.895033i	$-0.352848\pi$
$313$	15.7811	0.892000	0.446000	+	0.895033i	$0.352848\pi$
$314$	0	0
$315$	21.8660	1.23201
$316$	0	0
$317$	13.6425	0.766242	0.383121	−	0.923698i	$-0.374849\pi$
$317$	13.6425	0.766242	0.383121	+	0.923698i	$0.374849\pi$
$318$	0	0
$319$	32.5241	1.82100
$320$	0	0
$321$	1.47767	0.0824756
$322$	0	0
$323$	−22.1071	−1.23007
$324$	0	0
$325$	−42.2089	−2.34133
$326$	0	0
$327$	−6.03794	−0.333899
$328$	0	0
$329$	13.0384	0.718831
$330$	0	0
$331$	4.95696	0.272459	0.136229	−	0.990677i	$-0.456502\pi$
$331$	4.95696	0.272459	0.136229	+	0.990677i	$0.456502\pi$
$332$	0	0
$333$	17.2106	0.943133
$334$	0	0
$335$	49.2876	2.69287
$336$	0	0
$337$	−7.29444	−0.397354	−0.198677	−	0.980065i	$-0.563664\pi$
$337$	−7.29444	−0.397354	−0.198677	+	0.980065i	$0.563664\pi$
$338$	0	0
$339$	1.39615	0.0758285
$340$	0	0
$341$	15.1012	0.817776
$342$	0	0
$343$	−19.8915	−1.07404
$344$	0	0
$345$	−7.02862	−0.378408
$346$	0	0
$347$	−1.57397	−0.0844949	−0.0422474	−	0.999107i	$-0.513452\pi$
$347$	−1.57397	−0.0844949	−0.0422474	+	0.999107i	$0.513452\pi$
$348$	0	0
$349$	−26.8257	−1.43595	−0.717973	−	0.696071i	$-0.754930\pi$
$349$	−26.8257	−1.43595	−0.717973	+	0.696071i	$0.754930\pi$
$350$	0	0
$351$	−7.58054	−0.404619
$352$	0	0
$353$	−13.4802	−0.717480	−0.358740	−	0.933438i	$-0.616794\pi$
$353$	−13.4802	−0.717480	−0.358740	+	0.933438i	$0.616794\pi$
$354$	0	0
$355$	33.8072	1.79430
$356$	0	0
$357$	2.06761	0.109429
$358$	0	0
$359$	17.5146	0.924387	0.462194	−	0.886779i	$-0.347062\pi$
$359$	17.5146	0.924387	0.462194	+	0.886779i	$0.347062\pi$
$360$	0	0
$361$	19.6065	1.03192
$362$	0	0
$363$	−0.0897702	−0.00471171
$364$	0	0
$365$	20.2841	1.06172
$366$	0	0
$367$	−35.0719	−1.83074	−0.915368	−	0.402618i	$-0.868100\pi$
$367$	−35.0719	−1.83074	−0.915368	+	0.402618i	$0.868100\pi$
$368$	0	0
$369$	20.4619	1.06521
$370$	0	0
$371$	−1.83572	−0.0953061
$372$	0	0
$373$	10.1613	0.526134	0.263067	−	0.964778i	$-0.415266\pi$
$373$	10.1613	0.526134	0.263067	+	0.964778i	$0.415266\pi$
$374$	0	0
$375$	5.49172	0.283591
$376$	0	0
$377$	41.6831	2.14679
$378$	0	0
$379$	3.67281	0.188659	0.0943297	−	0.995541i	$-0.469929\pi$
$379$	3.67281	0.188659	0.0943297	+	0.995541i	$0.469929\pi$
$380$	0	0
$381$	−1.57841	−0.0808642
$382$	0	0
$383$	−2.03677	−0.104074	−0.0520369	−	0.998645i	$-0.516571\pi$
$383$	−2.03677	−0.104074	−0.0520369	+	0.998645i	$0.516571\pi$
$384$	0	0
$385$	25.2485	1.28678
$386$	0	0
$387$	−23.2533	−1.18203
$388$	0	0
$389$	36.2283	1.83685	0.918424	−	0.395598i	$-0.129463\pi$
$389$	36.2283	1.83685	0.918424	+	0.395598i	$0.129463\pi$
$390$	0	0
$391$	21.8427	1.10463
$392$	0	0
$393$	1.06435	0.0536894
$394$	0	0
$395$	−50.0453	−2.51805
$396$	0	0
$397$	−9.63505	−0.483569	−0.241785	−	0.970330i	$-0.577733\pi$
$397$	−9.63505	−0.483569	−0.241785	+	0.970330i	$0.577733\pi$
$398$	0	0
$399$	−3.61074	−0.180763
$400$	0	0
$401$	−20.2376	−1.01062	−0.505310	−	0.862938i	$-0.668622\pi$
$401$	−20.2376	−1.01062	−0.505310	+	0.862938i	$0.668622\pi$
$402$	0	0
$403$	19.3538	0.964082
$404$	0	0
$405$	−31.5832	−1.56938
$406$	0	0
$407$	19.8729	0.985064
$408$	0	0
$409$	−17.7170	−0.876049	−0.438025	−	0.898963i	$-0.644322\pi$
$409$	−17.7170	−0.876049	−0.438025	+	0.898963i	$0.644322\pi$
$410$	0	0
$411$	−1.15928	−0.0571830
$412$	0	0
$413$	9.45818	0.465407
$414$	0	0
$415$	−3.84665	−0.188824
$416$	0	0
$417$	−3.53586	−0.173152
$418$	0	0
$419$	25.9831	1.26936	0.634679	−	0.772776i	$-0.281132\pi$
$419$	25.9831	1.26936	0.634679	+	0.772776i	$0.281132\pi$
$420$	0	0
$421$	−32.7959	−1.59837	−0.799186	−	0.601084i	$-0.794736\pi$
$421$	−32.7959	−1.59837	−0.799186	+	0.601084i	$0.794736\pi$
$422$	0	0
$423$	−19.4422	−0.945313
$424$	0	0
$425$	−34.8563	−1.69078
$426$	0	0
$427$	−18.8894	−0.914120
$428$	0	0
$429$	−4.31101	−0.208137
$430$	0	0
$431$	22.6017	1.08869	0.544343	−	0.838862i	$-0.316779\pi$
$431$	22.6017	1.08869	0.544343	+	0.838862i	$0.316779\pi$
$432$	0	0
$433$	−13.9384	−0.669837	−0.334919	−	0.942247i	$-0.608709\pi$
$433$	−13.9384	−0.669837	−0.334919	+	0.942247i	$0.608709\pi$
$434$	0	0
$435$	−11.0765	−0.531075
$436$	0	0
$437$	−38.1447	−1.82471
$438$	0	0
$439$	−0.493599	−0.0235582	−0.0117791	−	0.999931i	$-0.503749\pi$
$439$	−0.493599	−0.0235582	−0.0117791	+	0.999931i	$0.503749\pi$
$440$	0	0
$441$	9.28128	0.441966
$442$	0	0
$443$	2.85429	0.135611	0.0678056	−	0.997699i	$-0.478400\pi$
$443$	2.85429	0.135611	0.0678056	+	0.997699i	$0.478400\pi$
$444$	0	0
$445$	−7.29401	−0.345769
$446$	0	0
$447$	−0.180933	−0.00855784
$448$	0	0
$449$	16.1490	0.762118	0.381059	−	0.924551i	$-0.375560\pi$
$449$	16.1490	0.762118	0.381059	+	0.924551i	$0.375560\pi$
$450$	0	0
$451$	23.6273	1.11256
$452$	0	0
$453$	5.63543	0.264775
$454$	0	0
$455$	32.3586	1.51699
$456$	0	0
$457$	16.1732	0.756552	0.378276	−	0.925693i	$-0.376517\pi$
$457$	16.1732	0.756552	0.378276	+	0.925693i	$0.376517\pi$
$458$	0	0
$459$	−6.26004	−0.292194
$460$	0	0
$461$	−16.0789	−0.748868	−0.374434	−	0.927253i	$-0.622163\pi$
$461$	−16.0789	−0.748868	−0.374434	+	0.927253i	$0.622163\pi$
$462$	0	0
$463$	−6.53936	−0.303910	−0.151955	−	0.988387i	$-0.548557\pi$
$463$	−6.53936	−0.303910	−0.151955	+	0.988387i	$0.548557\pi$
$464$	0	0
$465$	−5.14289	−0.238496
$466$	0	0
$467$	32.9651	1.52544	0.762720	−	0.646728i	$-0.223863\pi$
$467$	32.9651	1.52544	0.762720	+	0.646728i	$0.223863\pi$
$468$	0	0
$469$	−25.0172	−1.15519
$470$	0	0
$471$	−5.25869	−0.242307
$472$	0	0
$473$	−26.8504	−1.23458
$474$	0	0
$475$	60.8709	2.79295
$476$	0	0
$477$	2.73734	0.125334
$478$	0	0
$479$	26.1201	1.19346	0.596729	−	0.802443i	$-0.296467\pi$
$479$	26.1201	1.19346	0.596729	+	0.802443i	$0.296467\pi$
$480$	0	0
$481$	25.4693	1.16130
$482$	0	0
$483$	3.56755	0.162329
$484$	0	0
$485$	−21.5008	−0.976301
$486$	0	0
$487$	−19.4977	−0.883524	−0.441762	−	0.897132i	$-0.645646\pi$
$487$	−19.4977	−0.883524	−0.441762	+	0.897132i	$0.645646\pi$
$488$	0	0
$489$	−5.30878	−0.240071
$490$	0	0
$491$	−10.3626	−0.467658	−0.233829	−	0.972278i	$-0.575126\pi$
$491$	−10.3626	−0.467658	−0.233829	+	0.972278i	$0.575126\pi$
$492$	0	0
$493$	34.4221	1.55029
$494$	0	0
$495$	−37.6493	−1.69221
$496$	0	0
$497$	−17.1597	−0.769718
$498$	0	0
$499$	−28.4833	−1.27509	−0.637543	−	0.770415i	$-0.720049\pi$
$499$	−28.4833	−1.27509	−0.637543	+	0.770415i	$0.720049\pi$
$500$	0	0
$501$	−0.558245	−0.0249406
$502$	0	0
$503$	27.6186	1.23145	0.615726	−	0.787960i	$-0.288863\pi$
$503$	27.6186	1.23145	0.615726	+	0.787960i	$0.288863\pi$
$504$	0	0
$505$	−38.1842	−1.69918
$506$	0	0
$507$	−1.65576	−0.0735349
$508$	0	0
$509$	14.4135	0.638868	0.319434	−	0.947608i	$-0.396507\pi$
$509$	14.4135	0.638868	0.319434	+	0.947608i	$0.396507\pi$
$510$	0	0
$511$	−10.2957	−0.455456
$512$	0	0
$513$	10.9321	0.482666
$514$	0	0
$515$	65.8437	2.90142
$516$	0	0
$517$	−22.4498	−0.987340
$518$	0	0
$519$	0.843092	0.0370076
$520$	0	0
$521$	−34.2567	−1.50081	−0.750407	−	0.660976i	$-0.770142\pi$
$521$	−34.2567	−1.50081	−0.750407	+	0.660976i	$0.770142\pi$
$522$	0	0
$523$	38.1449	1.66796	0.833979	−	0.551796i	$-0.186057\pi$
$523$	38.1449	1.66796	0.833979	+	0.551796i	$0.186057\pi$
$524$	0	0
$525$	−5.69306	−0.248466
$526$	0	0
$527$	15.9825	0.696207
$528$	0	0
$529$	14.6884	0.638627
$530$	0	0
$531$	−14.1036	−0.612043
$532$	0	0
$533$	30.2808	1.31161
$534$	0	0
$535$	19.0975	0.825657
$536$	0	0
$537$	2.86772	0.123751
$538$	0	0
$539$	10.7170	0.461615
$540$	0	0
$541$	10.0253	0.431020	0.215510	−	0.976502i	$-0.430859\pi$
$541$	10.0253	0.431020	0.215510	+	0.976502i	$0.430859\pi$
$542$	0	0
$543$	−4.17998	−0.179380
$544$	0	0
$545$	−78.0347	−3.34264
$546$	0	0
$547$	−16.9306	−0.723902	−0.361951	−	0.932197i	$-0.617889\pi$
$547$	−16.9306	−0.723902	−0.361951	+	0.932197i	$0.617889\pi$
$548$	0	0
$549$	28.1669	1.20213
$550$	0	0
$551$	−60.1125	−2.56088
$552$	0	0
$553$	25.4017	1.08019
$554$	0	0
$555$	−6.76795	−0.287284
$556$	0	0
$557$	−29.9200	−1.26775	−0.633875	−	0.773435i	$-0.718537\pi$
$557$	−29.9200	−1.26775	−0.633875	+	0.773435i	$0.718537\pi$
$558$	0	0
$559$	−34.4116	−1.45546
$560$	0	0
$561$	−3.56005	−0.150305
$562$	0	0
$563$	−28.9551	−1.22031	−0.610156	−	0.792281i	$-0.708893\pi$
$563$	−28.9551	−1.22031	−0.610156	+	0.792281i	$0.708893\pi$
$564$	0	0
$565$	18.0439	0.759113
$566$	0	0
$567$	16.0308	0.673231
$568$	0	0
$569$	1.68034	0.0704436	0.0352218	−	0.999380i	$-0.488786\pi$
$569$	1.68034	0.0704436	0.0352218	+	0.999380i	$0.488786\pi$
$570$	0	0
$571$	−34.5579	−1.44620	−0.723101	−	0.690742i	$-0.757284\pi$
$571$	−34.5579	−1.44620	−0.723101	+	0.690742i	$0.757284\pi$
$572$	0	0
$573$	−3.50398	−0.146381
$574$	0	0
$575$	−60.1428	−2.50813
$576$	0	0
$577$	21.8924	0.911391	0.455696	−	0.890136i	$-0.349391\pi$
$577$	21.8924	0.911391	0.455696	+	0.890136i	$0.349391\pi$
$578$	0	0
$579$	6.38419	0.265318
$580$	0	0
$581$	1.95246	0.0810018
$582$	0	0
$583$	3.16079	0.130906
$584$	0	0
$585$	−48.2515	−1.99496
$586$	0	0
$587$	−20.6431	−0.852033	−0.426017	−	0.904715i	$-0.640083\pi$
$587$	−20.6431	−0.852033	−0.426017	+	0.904715i	$0.640083\pi$
$588$	0	0
$589$	−27.9108	−1.15004
$590$	0	0
$591$	2.94026	0.120946
$592$	0	0
$593$	−12.3731	−0.508101	−0.254051	−	0.967191i	$-0.581763\pi$
$593$	−12.3731	−0.508101	−0.254051	+	0.967191i	$0.581763\pi$
$594$	0	0
$595$	26.7219	1.09549
$596$	0	0
$597$	4.64083	0.189936
$598$	0	0
$599$	23.1498	0.945874	0.472937	−	0.881096i	$-0.343194\pi$
$599$	23.1498	0.945874	0.472937	+	0.881096i	$0.343194\pi$
$600$	0	0
$601$	3.86867	0.157806	0.0789031	−	0.996882i	$-0.474858\pi$
$601$	3.86867	0.157806	0.0789031	+	0.996882i	$0.474858\pi$
$602$	0	0
$603$	37.3044	1.51915
$604$	0	0
$605$	−1.16019	−0.0471686
$606$	0	0
$607$	0.269972	0.0109578	0.00547892	−	0.999985i	$-0.498256\pi$
$607$	0.269972	0.0109578	0.00547892	+	0.999985i	$0.498256\pi$
$608$	0	0
$609$	5.62213	0.227820
$610$	0	0
$611$	−28.7718	−1.16398
$612$	0	0
$613$	5.98773	0.241842	0.120921	−	0.992662i	$-0.461415\pi$
$613$	5.98773	0.241842	0.120921	+	0.992662i	$0.461415\pi$
$614$	0	0
$615$	−8.04654	−0.324468
$616$	0	0
$617$	6.35394	0.255800	0.127900	−	0.991787i	$-0.459176\pi$
$617$	6.35394	0.255800	0.127900	+	0.991787i	$0.459176\pi$
$618$	0	0
$619$	−40.5269	−1.62891	−0.814457	−	0.580224i	$-0.802965\pi$
$619$	−40.5269	−1.62891	−0.814457	+	0.580224i	$0.802965\pi$
$620$	0	0
$621$	−10.8014	−0.433444
$622$	0	0
$623$	3.70225	0.148328
$624$	0	0
$625$	21.9918	0.879671
$626$	0	0
$627$	6.21704	0.248285
$628$	0	0
$629$	21.0326	0.838626
$630$	0	0
$631$	5.95010	0.236870	0.118435	−	0.992962i	$-0.462212\pi$
$631$	5.95010	0.236870	0.118435	+	0.992962i	$0.462212\pi$
$632$	0	0
$633$	3.15476	0.125390
$634$	0	0
$635$	−20.3994	−0.809525
$636$	0	0
$637$	13.7350	0.544200
$638$	0	0
$639$	25.5877	1.01223
$640$	0	0
$641$	20.4627	0.808229	0.404114	−	0.914708i	$-0.367580\pi$
$641$	20.4627	0.808229	0.404114	+	0.914708i	$0.367580\pi$
$642$	0	0
$643$	−30.7374	−1.21217	−0.606083	−	0.795401i	$-0.707260\pi$
$643$	−30.7374	−1.21217	−0.606083	+	0.795401i	$0.707260\pi$
$644$	0	0
$645$	9.14421	0.360053
$646$	0	0
$647$	−21.2440	−0.835188	−0.417594	−	0.908634i	$-0.637127\pi$
$647$	−21.2440	−0.835188	−0.417594	+	0.908634i	$0.637127\pi$
$648$	0	0
$649$	−16.2853	−0.639253
$650$	0	0
$651$	2.61040	0.102310
$652$	0	0
$653$	−6.26029	−0.244984	−0.122492	−	0.992470i	$-0.539089\pi$
$653$	−6.26029	−0.244984	−0.122492	+	0.992470i	$0.539089\pi$
$654$	0	0
$655$	13.7557	0.537480
$656$	0	0
$657$	15.3525	0.598957
$658$	0	0
$659$	11.2457	0.438071	0.219035	−	0.975717i	$-0.429709\pi$
$659$	11.2457	0.438071	0.219035	+	0.975717i	$0.429709\pi$
$660$	0	0
$661$	37.9620	1.47655	0.738276	−	0.674499i	$-0.235640\pi$
$661$	37.9620	1.47655	0.738276	+	0.674499i	$0.235640\pi$
$662$	0	0
$663$	−4.56258	−0.177196
$664$	0	0
$665$	−46.6654	−1.80961
$666$	0	0
$667$	59.3935	2.29972
$668$	0	0
$669$	−6.32352	−0.244481
$670$	0	0
$671$	32.5241	1.25558
$672$	0	0
$673$	19.4720	0.750590	0.375295	−	0.926905i	$-0.377541\pi$
$673$	19.4720	0.750590	0.375295	+	0.926905i	$0.377541\pi$
$674$	0	0
$675$	17.2367	0.663442
$676$	0	0
$677$	−45.4654	−1.74738	−0.873689	−	0.486485i	$-0.838279\pi$
$677$	−45.4654	−1.74738	−0.873689	+	0.486485i	$0.838279\pi$
$678$	0	0
$679$	10.9133	0.418813
$680$	0	0
$681$	0.497889	0.0190791
$682$	0	0
$683$	13.8810	0.531143	0.265572	−	0.964091i	$-0.414439\pi$
$683$	13.8810	0.531143	0.265572	+	0.964091i	$0.414439\pi$
$684$	0	0
$685$	−14.9826	−0.572455
$686$	0	0
$687$	4.07003	0.155281
$688$	0	0
$689$	4.05088	0.154326
$690$	0	0
$691$	−36.3175	−1.38158	−0.690791	−	0.723055i	$-0.742737\pi$
$691$	−36.3175	−1.38158	−0.690791	+	0.723055i	$0.742737\pi$
$692$	0	0
$693$	19.1098	0.725922
$694$	0	0
$695$	−45.6977	−1.73341
$696$	0	0
$697$	25.0061	0.947172
$698$	0	0
$699$	−4.48468	−0.169626
$700$	0	0
$701$	−13.8289	−0.522310	−0.261155	−	0.965297i	$-0.584103\pi$
$701$	−13.8289	−0.522310	−0.261155	+	0.965297i	$0.584103\pi$
$702$	0	0
$703$	−36.7300	−1.38530
$704$	0	0
$705$	7.64554	0.287948
$706$	0	0
$707$	19.3814	0.728911
$708$	0	0
$709$	−47.6187	−1.78836	−0.894179	−	0.447709i	$-0.852240\pi$
$709$	−47.6187	−1.78836	−0.894179	+	0.447709i	$0.852240\pi$
$710$	0	0
$711$	−37.8778	−1.42053
$712$	0	0
$713$	27.5769	1.03276
$714$	0	0
$715$	−55.7157	−2.08365
$716$	0	0
$717$	−0.933900	−0.0348771
$718$	0	0
$719$	−11.8666	−0.442551	−0.221276	−	0.975211i	$-0.571022\pi$
$719$	−11.8666	−0.442551	−0.221276	+	0.975211i	$0.571022\pi$
$720$	0	0
$721$	−33.4206	−1.24465
$722$	0	0
$723$	−6.26067	−0.232837
$724$	0	0
$725$	−94.7795	−3.52002
$726$	0	0
$727$	−40.3613	−1.49692	−0.748459	−	0.663181i	$-0.769206\pi$
$727$	−40.3613	−1.49692	−0.748459	+	0.663181i	$0.769206\pi$
$728$	0	0
$729$	−22.3333	−0.827161
$730$	0	0
$731$	−28.4173	−1.05105
$732$	0	0
$733$	20.1559	0.744474	0.372237	−	0.928138i	$-0.378591\pi$
$733$	20.1559	0.744474	0.372237	+	0.928138i	$0.378591\pi$
$734$	0	0
$735$	−3.64981	−0.134625
$736$	0	0
$737$	43.0751	1.58669
$738$	0	0
$739$	14.2713	0.524979	0.262490	−	0.964935i	$-0.415456\pi$
$739$	14.2713	0.524979	0.262490	+	0.964935i	$0.415456\pi$
$740$	0	0
$741$	7.96780	0.292705
$742$	0	0
$743$	−9.60118	−0.352233	−0.176117	−	0.984369i	$-0.556354\pi$
$743$	−9.60118	−0.352233	−0.176117	+	0.984369i	$0.556354\pi$
$744$	0	0
$745$	−2.33839	−0.0856719
$746$	0	0
$747$	−2.91141	−0.106523
$748$	0	0
$749$	−9.69341	−0.354190
$750$	0	0
$751$	23.5839	0.860588	0.430294	−	0.902689i	$-0.358410\pi$
$751$	23.5839	0.860588	0.430294	+	0.902689i	$0.358410\pi$
$752$	0	0
$753$	7.05404	0.257064
$754$	0	0
$755$	72.8326	2.65065
$756$	0	0
$757$	−34.7513	−1.26306	−0.631529	−	0.775352i	$-0.717572\pi$
$757$	−34.7513	−1.26306	−0.631529	+	0.775352i	$0.717572\pi$
$758$	0	0
$759$	−6.14268	−0.222965
$760$	0	0
$761$	−29.3425	−1.06366	−0.531832	−	0.846850i	$-0.678496\pi$
$761$	−29.3425	−1.06366	−0.531832	+	0.846850i	$0.678496\pi$
$762$	0	0
$763$	39.6085	1.43392
$764$	0	0
$765$	−39.8463	−1.44065
$766$	0	0
$767$	−20.8713	−0.753620
$768$	0	0
$769$	−23.0896	−0.832634	−0.416317	−	0.909220i	$-0.636679\pi$
$769$	−23.0896	−0.832634	−0.416317	+	0.909220i	$0.636679\pi$
$770$	0	0
$771$	2.61524	0.0941854
$772$	0	0
$773$	14.8924	0.535642	0.267821	−	0.963469i	$-0.413696\pi$
$773$	14.8924	0.535642	0.267821	+	0.963469i	$0.413696\pi$
$774$	0	0
$775$	−44.0069	−1.58078
$776$	0	0
$777$	3.43524	0.123239
$778$	0	0
$779$	−43.6690	−1.56460
$780$	0	0
$781$	29.5459	1.05724
$782$	0	0
$783$	−17.0220	−0.608315
$784$	0	0
$785$	−67.9635	−2.42572
$786$	0	0
$787$	27.6419	0.985329	0.492664	−	0.870219i	$-0.336023\pi$
$787$	27.6419	0.985329	0.492664	+	0.870219i	$0.336023\pi$
$788$	0	0
$789$	−2.77943	−0.0989503
$790$	0	0
$791$	−9.15864	−0.325644
$792$	0	0
$793$	41.6831	1.48021
$794$	0	0
$795$	−1.07644	−0.0381775
$796$	0	0
$797$	21.0693	0.746313	0.373156	−	0.927768i	$-0.378275\pi$
$797$	21.0693	0.746313	0.373156	+	0.927768i	$0.378275\pi$
$798$	0	0
$799$	−23.7599	−0.840564
$800$	0	0
$801$	−5.52062	−0.195061
$802$	0	0
$803$	17.7274	0.625586
$804$	0	0
$805$	46.1072	1.62507
$806$	0	0
$807$	7.79293	0.274324
$808$	0	0
$809$	−20.7319	−0.728894	−0.364447	−	0.931224i	$-0.618742\pi$
$809$	−20.7319	−0.728894	−0.364447	+	0.931224i	$0.618742\pi$
$810$	0	0
$811$	−6.37186	−0.223746	−0.111873	−	0.993723i	$-0.535685\pi$
$811$	−6.37186	−0.223746	−0.111873	+	0.993723i	$0.535685\pi$
$812$	0	0
$813$	−5.31227	−0.186310
$814$	0	0
$815$	−68.6110	−2.40334
$816$	0	0
$817$	49.6261	1.73620
$818$	0	0
$819$	24.4913	0.855794
$820$	0	0
$821$	9.34818	0.326254	0.163127	−	0.986605i	$-0.447842\pi$
$821$	9.34818	0.326254	0.163127	+	0.986605i	$0.447842\pi$
$822$	0	0
$823$	−41.2177	−1.43676	−0.718380	−	0.695651i	$-0.755116\pi$
$823$	−41.2177	−1.43676	−0.718380	+	0.695651i	$0.755116\pi$
$824$	0	0
$825$	9.80242	0.341277
$826$	0	0
$827$	−51.7970	−1.80116	−0.900579	−	0.434692i	$-0.856857\pi$
$827$	−51.7970	−1.80116	−0.900579	+	0.434692i	$0.856857\pi$
$828$	0	0
$829$	25.3785	0.881432	0.440716	−	0.897647i	$-0.354725\pi$
$829$	25.3785	0.881432	0.440716	+	0.897647i	$0.354725\pi$
$830$	0	0
$831$	−4.22362	−0.146516
$832$	0	0
$833$	11.3424	0.392992
$834$	0	0
$835$	−7.21479	−0.249678
$836$	0	0
$837$	−7.90345	−0.273183
$838$	0	0
$839$	27.7612	0.958423	0.479211	−	0.877699i	$-0.340923\pi$
$839$	27.7612	0.958423	0.479211	+	0.877699i	$0.340923\pi$
$840$	0	0
$841$	64.5986	2.22754
$842$	0	0
$843$	−3.02049	−0.104031
$844$	0	0
$845$	−21.3991	−0.736153
$846$	0	0
$847$	0.588886	0.0202343
$848$	0	0
$849$	−3.13601	−0.107628
$850$	0	0
$851$	36.2907	1.24403
$852$	0	0
$853$	−30.2122	−1.03445	−0.517223	−	0.855851i	$-0.673034\pi$
$853$	−30.2122	−1.03445	−0.517223	+	0.855851i	$0.673034\pi$
$854$	0	0
$855$	69.5851	2.37976
$856$	0	0
$857$	−48.9691	−1.67275	−0.836376	−	0.548156i	$-0.815330\pi$
$857$	−48.9691	−1.67275	−0.836376	+	0.548156i	$0.815330\pi$
$858$	0	0
$859$	−21.9356	−0.748434	−0.374217	−	0.927341i	$-0.622089\pi$
$859$	−21.9356	−0.748434	−0.374217	+	0.927341i	$0.622089\pi$
$860$	0	0
$861$	4.08422	0.139190
$862$	0	0
$863$	26.2136	0.892322	0.446161	−	0.894953i	$-0.352791\pi$
$863$	26.2136	0.892322	0.446161	+	0.894953i	$0.352791\pi$
$864$	0	0
$865$	10.8962	0.370480
$866$	0	0
$867$	1.29199	0.0438782
$868$	0	0
$869$	−43.7372	−1.48368
$870$	0	0
$871$	55.2053	1.87056
$872$	0	0
$873$	−16.2733	−0.550769
$874$	0	0
$875$	−36.0253	−1.21788
$876$	0	0
$877$	31.1584	1.05214	0.526072	−	0.850440i	$-0.323664\pi$
$877$	31.1584	1.05214	0.526072	+	0.850440i	$0.323664\pi$
$878$	0	0
$879$	1.03908	0.0350474
$880$	0	0
$881$	16.0918	0.542148	0.271074	−	0.962559i	$-0.412621\pi$
$881$	16.0918	0.542148	0.271074	+	0.962559i	$0.412621\pi$
$882$	0	0
$883$	27.5401	0.926800	0.463400	−	0.886149i	$-0.346629\pi$
$883$	27.5401	0.926800	0.463400	+	0.886149i	$0.346629\pi$
$884$	0	0
$885$	5.54614	0.186432
$886$	0	0
$887$	23.8811	0.801848	0.400924	−	0.916111i	$-0.368689\pi$
$887$	23.8811	0.801848	0.400924	+	0.916111i	$0.368689\pi$
$888$	0	0
$889$	10.3542	0.347269
$890$	0	0
$891$	−27.6022	−0.924708
$892$	0	0
$893$	41.4927	1.38850
$894$	0	0
$895$	37.0625	1.23886
$896$	0	0
$897$	−7.87250	−0.262855
$898$	0	0
$899$	43.4586	1.44943
$900$	0	0
$901$	3.34524	0.111446
$902$	0	0
$903$	−4.64137	−0.154455
$904$	0	0
$905$	−54.0223	−1.79576
$906$	0	0
$907$	−18.5303	−0.615289	−0.307645	−	0.951501i	$-0.599541\pi$
$907$	−18.5303	−0.615289	−0.307645	+	0.951501i	$0.599541\pi$
$908$	0	0
$909$	−28.9005	−0.958570
$910$	0	0
$911$	44.0016	1.45784	0.728920	−	0.684599i	$-0.240023\pi$
$911$	44.0016	1.45784	0.728920	+	0.684599i	$0.240023\pi$
$912$	0	0
$913$	−3.36179	−0.111259
$914$	0	0
$915$	−11.0765	−0.366176
$916$	0	0
$917$	−6.98206	−0.230568
$918$	0	0
$919$	21.0668	0.694931	0.347465	−	0.937693i	$-0.387042\pi$
$919$	21.0668	0.694931	0.347465	+	0.937693i	$0.387042\pi$
$920$	0	0
$921$	−1.33349	−0.0439402
$922$	0	0
$923$	37.8662	1.24638
$924$	0	0
$925$	−57.9123	−1.90415
$926$	0	0
$927$	49.8352	1.63680
$928$	0	0
$929$	3.43686	0.112760	0.0563798	−	0.998409i	$-0.482044\pi$
$929$	3.43686	0.112760	0.0563798	+	0.998409i	$0.482044\pi$
$930$	0	0
$931$	−19.8077	−0.649171
$932$	0	0
$933$	−8.01258	−0.262320
$934$	0	0
$935$	−46.0103	−1.50470
$936$	0	0
$937$	21.9722	0.717800	0.358900	−	0.933376i	$-0.383152\pi$
$937$	21.9722	0.717800	0.358900	+	0.933376i	$0.383152\pi$
$938$	0	0
$939$	−4.69700	−0.153281
$940$	0	0
$941$	−43.3096	−1.41185	−0.705926	−	0.708286i	$-0.749469\pi$
$941$	−43.3096	−1.41185	−0.705926	+	0.708286i	$0.749469\pi$
$942$	0	0
$943$	43.1466	1.40505
$944$	0	0
$945$	−13.2142	−0.429857
$946$	0	0
$947$	38.9590	1.26600	0.632999	−	0.774153i	$-0.281824\pi$
$947$	38.9590	1.26600	0.632999	+	0.774153i	$0.281824\pi$
$948$	0	0
$949$	22.7195	0.737507
$950$	0	0
$951$	−4.06050	−0.131671
$952$	0	0
$953$	−16.1725	−0.523877	−0.261939	−	0.965085i	$-0.584362\pi$
$953$	−16.1725	−0.523877	−0.261939	+	0.965085i	$0.584362\pi$
$954$	0	0
$955$	−45.2856	−1.46541
$956$	0	0
$957$	−9.68029	−0.312919
$958$	0	0
$959$	7.60478	0.245571
$960$	0	0
$961$	−10.8218	−0.349089
$962$	0	0
$963$	14.4543	0.465784
$964$	0	0
$965$	82.5096	2.65608
$966$	0	0
$967$	−7.34049	−0.236054	−0.118027	−	0.993010i	$-0.537657\pi$
$967$	−7.34049	−0.236054	−0.118027	+	0.993010i	$0.537657\pi$
$968$	0	0
$969$	6.57985	0.211375
$970$	0	0
$971$	29.6682	0.952097	0.476049	−	0.879419i	$-0.342069\pi$
$971$	29.6682	0.952097	0.476049	+	0.879419i	$0.342069\pi$
$972$	0	0
$973$	23.1950	0.743598
$974$	0	0
$975$	12.5628	0.402333
$976$	0	0
$977$	−15.1712	−0.485369	−0.242684	−	0.970105i	$-0.578028\pi$
$977$	−15.1712	−0.485369	−0.242684	+	0.970105i	$0.578028\pi$
$978$	0	0
$979$	−6.37462	−0.203734
$980$	0	0
$981$	−59.0622	−1.88571
$982$	0	0
$983$	−23.3016	−0.743206	−0.371603	−	0.928392i	$-0.621192\pi$
$983$	−23.3016	−0.743206	−0.371603	+	0.928392i	$0.621192\pi$
$984$	0	0
$985$	38.0000	1.21078
$986$	0	0
$987$	−3.88068	−0.123524
$988$	0	0
$989$	−49.0325	−1.55914
$990$	0	0
$991$	−17.2039	−0.546501	−0.273251	−	0.961943i	$-0.588099\pi$
$991$	−17.2039	−0.546501	−0.273251	+	0.961943i	$0.588099\pi$
$992$	0	0
$993$	−1.47536	−0.0468192
$994$	0	0
$995$	59.9783	1.90144
$996$	0	0
$997$	−22.5624	−0.714559	−0.357279	−	0.933998i	$-0.616296\pi$
$997$	−22.5624	−0.714559	−0.357279	+	0.933998i	$0.616296\pi$
$998$	0	0
$999$	−10.4008	−0.329066

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5312.2.a.br.1.5		8
4.3	odd	2		5312.2.a.bs.1.4		8
8.3	odd	2		1328.2.a.n.1.5		8
8.5	even	2		664.2.a.f.1.4	✓	8
24.5	odd	2		5976.2.a.s.1.2		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
664.2.a.f.1.4	✓	8	8.5	even	2
1328.2.a.n.1.5		8	8.3	odd	2
5312.2.a.br.1.5		8	1.1	even	1	trivial
5312.2.a.bs.1.4		8	4.3	odd	2
5976.2.a.s.1.2		8	24.5	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:	\( N \)	\(=\)	\( 5312 = 2^{6} \cdot 83 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5312.a (trivial)

\(f(q)\)	\(=\)	\(q-0.297635 q^{3} -3.84665 q^{5} +1.95246 q^{7} -2.91141 q^{9} +O(q^{10})\)	\(q-0.297635 q^{3} -3.84665 q^{5} +1.95246 q^{7} -2.91141 q^{9} -3.36179 q^{11} -4.30849 q^{13} +1.14490 q^{15} -3.55797 q^{17} +6.21341 q^{19} -0.581121 q^{21} -6.13909 q^{23} +9.79670 q^{25} +1.75944 q^{27} -9.67464 q^{29} -4.49202 q^{31} +1.00058 q^{33} -7.51043 q^{35} -5.91141 q^{37} +1.28236 q^{39} -7.02818 q^{41} +7.98694 q^{43} +11.1992 q^{45} +6.67793 q^{47} -3.18789 q^{49} +1.05898 q^{51} -0.940210 q^{53} +12.9316 q^{55} -1.84933 q^{57} +4.84423 q^{59} -9.67464 q^{61} -5.68442 q^{63} +16.5732 q^{65} -12.8131 q^{67} +1.82721 q^{69} -8.78876 q^{71} -5.27320 q^{73} -2.91584 q^{75} -6.56376 q^{77} +13.0101 q^{79} +8.21057 q^{81} +1.00000 q^{83} +13.6863 q^{85} +2.87951 q^{87} +1.89620 q^{89} -8.41216 q^{91} +1.33698 q^{93} -23.9008 q^{95} +5.58949 q^{97} +9.78755 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 2 q^{3} - 7 q^{5} + q^{7} + 16 q^{9}+O(q^{10}) \) 8 * q - 2 * q^3 - 7 * q^5 + q^7 + 16 * q^9	\( 8 q - 2 q^{3} - 7 q^{5} + q^{7} + 16 q^{9} - 10 q^{11} - 7 q^{13} - 14 q^{15} + 15 q^{17} - 7 q^{19} + 5 q^{21} + 4 q^{23} + 27 q^{25} - 5 q^{27} - 16 q^{29} + 3 q^{31} + 21 q^{33} - 2 q^{35} - 8 q^{37} + 10 q^{39} + 24 q^{41} + q^{43} - 29 q^{45} - 2 q^{47} + 3 q^{49} + 9 q^{51} - 7 q^{53} - 30 q^{55} + 4 q^{57} + 2 q^{59} - 16 q^{61} - 10 q^{63} + 2 q^{65} + 25 q^{67} + 6 q^{69} + 8 q^{71} + 14 q^{73} + 30 q^{75} + 7 q^{77} - 4 q^{79} + 52 q^{81} + 8 q^{83} + 21 q^{85} - 4 q^{87} + 20 q^{89} + 45 q^{91} + 47 q^{93} + 8 q^{95} + 2 q^{97} - 10 q^{99}+O(q^{100}) \) 8 * q - 2 * q^3 - 7 * q^5 + q^7 + 16 * q^9 - 10 * q^11 - 7 * q^13 - 14 * q^15 + 15 * q^17 - 7 * q^19 + 5 * q^21 + 4 * q^23 + 27 * q^25 - 5 * q^27 - 16 * q^29 + 3 * q^31 + 21 * q^33 - 2 * q^35 - 8 * q^37 + 10 * q^39 + 24 * q^41 + q^43 - 29 * q^45 - 2 * q^47 + 3 * q^49 + 9 * q^51 - 7 * q^53 - 30 * q^55 + 4 * q^57 + 2 * q^59 - 16 * q^61 - 10 * q^63 + 2 * q^65 + 25 * q^67 + 6 * q^69 + 8 * q^71 + 14 * q^73 + 30 * q^75 + 7 * q^77 - 4 * q^79 + 52 * q^81 + 8 * q^83 + 21 * q^85 - 4 * q^87 + 20 * q^89 + 45 * q^91 + 47 * q^93 + 8 * q^95 + 2 * q^97 - 10 * q^99

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