LMFDB - Embedded newform 8020.2.a.f.1.10

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8020,2,Mod(1,8020)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8020, 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("8020.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8020 = 2^{2} \cdot 5 \cdot 401 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8020.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:	$64.0400224211$
Analytic rank:	$0$
Dimension:	$37$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.10
Character	$\chi$	$=$	8020.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.70544 q^{3} +1.00000 q^{5} -1.03894 q^{7} -0.0914601 q^{9} +O(q^{10})$	$q-1.70544 q^{3} +1.00000 q^{5} -1.03894 q^{7} -0.0914601 q^{9} +1.24754 q^{11} -4.47261 q^{13} -1.70544 q^{15} +2.95732 q^{17} +0.264785 q^{19} +1.77185 q^{21} -1.16188 q^{23} +1.00000 q^{25} +5.27231 q^{27} -2.37291 q^{29} -3.78592 q^{31} -2.12760 q^{33} -1.03894 q^{35} +10.8644 q^{37} +7.62779 q^{39} -3.50826 q^{41} -5.37679 q^{43} -0.0914601 q^{45} -7.48532 q^{47} -5.92061 q^{49} -5.04354 q^{51} +1.25982 q^{53} +1.24754 q^{55} -0.451576 q^{57} +12.9858 q^{59} +0.329725 q^{61} +0.0950212 q^{63} -4.47261 q^{65} -3.42358 q^{67} +1.98152 q^{69} -13.9478 q^{71} -2.47463 q^{73} -1.70544 q^{75} -1.29611 q^{77} -3.30359 q^{79} -8.71725 q^{81} +9.25472 q^{83} +2.95732 q^{85} +4.04686 q^{87} +3.23642 q^{89} +4.64675 q^{91} +6.45668 q^{93} +0.264785 q^{95} +2.95948 q^{97} -0.114100 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 37 q + 3 q^{3} + 37 q^{5} + 4 q^{7} + 50 q^{9}+O(q^{10}) $ 37 * q + 3 * q^3 + 37 * q^5 + 4 * q^7 + 50 * q^9	$ 37 q + 3 q^{3} + 37 q^{5} + 4 q^{7} + 50 q^{9} + 2 q^{11} + 27 q^{13} + 3 q^{15} + 36 q^{17} - 6 q^{19} + 20 q^{21} + 17 q^{23} + 37 q^{25} + 9 q^{27} + 29 q^{29} + 5 q^{31} + 36 q^{33} + 4 q^{35} + 35 q^{37} + 21 q^{39} + 24 q^{41} + 11 q^{43} + 50 q^{45} + 19 q^{47} + 57 q^{49} + 8 q^{51} + 65 q^{53} + 2 q^{55} + 62 q^{57} - 9 q^{59} + 13 q^{61} + 26 q^{63} + 27 q^{65} + 13 q^{67} + 20 q^{69} + 33 q^{71} + 67 q^{73} + 3 q^{75} + 62 q^{77} + 23 q^{79} + 97 q^{81} + 2 q^{83} + 36 q^{85} + 32 q^{87} + 34 q^{89} + q^{91} + 41 q^{93} - 6 q^{95} + 66 q^{97} - 7 q^{99}+O(q^{100}) $ 37 * q + 3 * q^3 + 37 * q^5 + 4 * q^7 + 50 * q^9 + 2 * q^11 + 27 * q^13 + 3 * q^15 + 36 * q^17 - 6 * q^19 + 20 * q^21 + 17 * q^23 + 37 * q^25 + 9 * q^27 + 29 * q^29 + 5 * q^31 + 36 * q^33 + 4 * q^35 + 35 * q^37 + 21 * q^39 + 24 * q^41 + 11 * q^43 + 50 * q^45 + 19 * q^47 + 57 * q^49 + 8 * q^51 + 65 * q^53 + 2 * q^55 + 62 * q^57 - 9 * q^59 + 13 * q^61 + 26 * q^63 + 27 * q^65 + 13 * q^67 + 20 * q^69 + 33 * q^71 + 67 * q^73 + 3 * q^75 + 62 * q^77 + 23 * q^79 + 97 * q^81 + 2 * q^83 + 36 * q^85 + 32 * q^87 + 34 * q^89 + q^91 + 41 * q^93 - 6 * q^95 + 66 * q^97 - 7 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	−1.70544	−0.984639	−0.492319	−	0.870415i	$-0.663851\pi$
$3$	−1.70544	−0.984639	−0.492319	+	0.870415i	$0.663851\pi$
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	−1.03894	−0.392681	−0.196340	−	0.980536i	$-0.562906\pi$
$7$	−1.03894	−0.392681	−0.196340	+	0.980536i	$0.562906\pi$
$8$	0	0
$9$	−0.0914601	−0.0304867
$10$	0	0
$11$	1.24754	0.376146	0.188073	−	0.982155i	$-0.439776\pi$
$11$	1.24754	0.376146	0.188073	+	0.982155i	$0.439776\pi$
$12$	0	0
$13$	−4.47261	−1.24048	−0.620239	−	0.784413i	$-0.712965\pi$
$13$	−4.47261	−1.24048	−0.620239	+	0.784413i	$0.712965\pi$
$14$	0	0
$15$	−1.70544	−0.440344
$16$	0	0
$17$	2.95732	0.717254	0.358627	−	0.933481i	$-0.383245\pi$
$17$	2.95732	0.717254	0.358627	+	0.933481i	$0.383245\pi$
$18$	0	0
$19$	0.264785	0.0607459	0.0303729	−	0.999539i	$-0.490331\pi$
$19$	0.264785	0.0607459	0.0303729	+	0.999539i	$0.490331\pi$
$20$	0	0
$21$	1.77185	0.386649
$22$	0	0
$23$	−1.16188	−0.242268	−0.121134	−	0.992636i	$-0.538653\pi$
$23$	−1.16188	−0.242268	−0.121134	+	0.992636i	$0.538653\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	5.27231	1.01466
$28$	0	0
$29$	−2.37291	−0.440638	−0.220319	−	0.975428i	$-0.570710\pi$
$29$	−2.37291	−0.440638	−0.220319	+	0.975428i	$0.570710\pi$
$30$	0	0
$31$	−3.78592	−0.679972	−0.339986	−	0.940431i	$-0.610422\pi$
$31$	−3.78592	−0.679972	−0.339986	+	0.940431i	$0.610422\pi$
$32$	0	0
$33$	−2.12760	−0.370368
$34$	0	0
$35$	−1.03894	−0.175612
$36$	0	0
$37$	10.8644	1.78610	0.893048	−	0.449962i	$-0.148562\pi$
$37$	10.8644	1.78610	0.893048	+	0.449962i	$0.148562\pi$
$38$	0	0
$39$	7.62779	1.22142
$40$	0	0
$41$	−3.50826	−0.547897	−0.273949	−	0.961744i	$-0.588330\pi$
$41$	−3.50826	−0.547897	−0.273949	+	0.961744i	$0.588330\pi$
$42$	0	0
$43$	−5.37679	−0.819953	−0.409977	−	0.912096i	$-0.634463\pi$
$43$	−5.37679	−0.819953	−0.409977	+	0.912096i	$0.634463\pi$
$44$	0	0
$45$	−0.0914601	−0.0136341
$46$	0	0
$47$	−7.48532	−1.09185	−0.545923	−	0.837835i	$-0.683821\pi$
$47$	−7.48532	−1.09185	−0.545923	+	0.837835i	$0.683821\pi$
$48$	0	0
$49$	−5.92061	−0.845802
$50$	0	0
$51$	−5.04354	−0.706236
$52$	0	0
$53$	1.25982	0.173049	0.0865247	−	0.996250i	$-0.472424\pi$
$53$	1.25982	0.173049	0.0865247	+	0.996250i	$0.472424\pi$
$54$	0	0
$55$	1.24754	0.168218
$56$	0	0
$57$	−0.451576	−0.0598128
$58$	0	0
$59$	12.9858	1.69061	0.845306	−	0.534283i	$-0.179418\pi$
$59$	12.9858	1.69061	0.845306	+	0.534283i	$0.179418\pi$
$60$	0	0
$61$	0.329725	0.0422170	0.0211085	−	0.999777i	$-0.493280\pi$
$61$	0.329725	0.0422170	0.0211085	+	0.999777i	$0.493280\pi$
$62$	0	0
$63$	0.0950212	0.0119715
$64$	0	0
$65$	−4.47261	−0.554759
$66$	0	0
$67$	−3.42358	−0.418257	−0.209129	−	0.977888i	$-0.567063\pi$
$67$	−3.42358	−0.418257	−0.209129	+	0.977888i	$0.567063\pi$
$68$	0	0
$69$	1.98152	0.238546
$70$	0	0
$71$	−13.9478	−1.65530	−0.827652	−	0.561242i	$-0.810324\pi$
$71$	−13.9478	−1.65530	−0.827652	+	0.561242i	$0.810324\pi$
$72$	0	0
$73$	−2.47463	−0.289634	−0.144817	−	0.989458i	$-0.546259\pi$
$73$	−2.47463	−0.289634	−0.144817	+	0.989458i	$0.546259\pi$
$74$	0	0
$75$	−1.70544	−0.196928
$76$	0	0
$77$	−1.29611	−0.147705
$78$	0	0
$79$	−3.30359	−0.371683	−0.185841	−	0.982580i	$-0.559501\pi$
$79$	−3.30359	−0.371683	−0.185841	+	0.982580i	$0.559501\pi$
$80$	0	0
$81$	−8.71725	−0.968584
$82$	0	0
$83$	9.25472	1.01584	0.507919	−	0.861405i	$-0.330415\pi$
$83$	9.25472	1.01584	0.507919	+	0.861405i	$0.330415\pi$
$84$	0	0
$85$	2.95732	0.320766
$86$	0	0
$87$	4.04686	0.433869
$88$	0	0
$89$	3.23642	0.343060	0.171530	−	0.985179i	$-0.445129\pi$
$89$	3.23642	0.343060	0.171530	+	0.985179i	$0.445129\pi$
$90$	0	0
$91$	4.64675	0.487112
$92$	0	0
$93$	6.45668	0.669527
$94$	0	0
$95$	0.264785	0.0271664
$96$	0	0
$97$	2.95948	0.300490	0.150245	−	0.988649i	$-0.451994\pi$
$97$	2.95948	0.300490	0.150245	+	0.988649i	$0.451994\pi$
$98$	0	0
$99$	−0.114100	−0.0114675
$100$	0	0
$101$	−4.92384	−0.489941	−0.244970	−	0.969531i	$-0.578778\pi$
$101$	−4.92384	−0.489941	−0.244970	+	0.969531i	$0.578778\pi$
$102$	0	0
$103$	7.31224	0.720497	0.360248	−	0.932856i	$-0.382692\pi$
$103$	7.31224	0.720497	0.360248	+	0.932856i	$0.382692\pi$
$104$	0	0
$105$	1.77185	0.172915
$106$	0	0
$107$	−5.63925	−0.545167	−0.272583	−	0.962132i	$-0.587878\pi$
$107$	−5.63925	−0.545167	−0.272583	+	0.962132i	$0.587878\pi$
$108$	0	0
$109$	12.5663	1.20363	0.601817	−	0.798634i	$-0.294443\pi$
$109$	12.5663	1.20363	0.601817	+	0.798634i	$0.294443\pi$
$110$	0	0
$111$	−18.5286	−1.75866
$112$	0	0
$113$	11.7634	1.10660	0.553302	−	0.832981i	$-0.313367\pi$
$113$	11.7634	1.10660	0.553302	+	0.832981i	$0.313367\pi$
$114$	0	0
$115$	−1.16188	−0.108346
$116$	0	0
$117$	0.409065	0.0378181
$118$	0	0
$119$	−3.07246	−0.281652
$120$	0	0
$121$	−9.44365	−0.858514
$122$	0	0
$123$	5.98313	0.539481
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	9.74501	0.864730	0.432365	−	0.901699i	$-0.357679\pi$
$127$	9.74501	0.864730	0.432365	+	0.901699i	$0.357679\pi$
$128$	0	0
$129$	9.16982	0.807358
$130$	0	0
$131$	−12.8121	−1.11940	−0.559701	−	0.828695i	$-0.689084\pi$
$131$	−12.8121	−1.11940	−0.559701	+	0.828695i	$0.689084\pi$
$132$	0	0
$133$	−0.275095	−0.0238537
$134$	0	0
$135$	5.27231	0.453768
$136$	0	0
$137$	6.22995	0.532260	0.266130	−	0.963937i	$-0.414255\pi$
$137$	6.22995	0.532260	0.266130	+	0.963937i	$0.414255\pi$
$138$	0	0
$139$	−2.47158	−0.209636	−0.104818	−	0.994491i	$-0.533426\pi$
$139$	−2.47158	−0.209636	−0.104818	+	0.994491i	$0.533426\pi$
$140$	0	0
$141$	12.7658	1.07507
$142$	0	0
$143$	−5.57974	−0.466601
$144$	0	0
$145$	−2.37291	−0.197059
$146$	0	0
$147$	10.0973	0.832809
$148$	0	0
$149$	11.1788	0.915802	0.457901	−	0.889003i	$-0.348601\pi$
$149$	11.1788	0.915802	0.457901	+	0.889003i	$0.348601\pi$
$150$	0	0
$151$	0.434658	0.0353720	0.0176860	−	0.999844i	$-0.494370\pi$
$151$	0.434658	0.0353720	0.0176860	+	0.999844i	$0.494370\pi$
$152$	0	0
$153$	−0.270476	−0.0218667
$154$	0	0
$155$	−3.78592	−0.304093
$156$	0	0
$157$	5.53918	0.442075	0.221037	−	0.975265i	$-0.429056\pi$
$157$	5.53918	0.442075	0.221037	+	0.975265i	$0.429056\pi$
$158$	0	0
$159$	−2.14855	−0.170391
$160$	0	0
$161$	1.20711	0.0951340
$162$	0	0
$163$	12.4778	0.977338	0.488669	−	0.872469i	$-0.337483\pi$
$163$	12.4778	0.977338	0.488669	+	0.872469i	$0.337483\pi$
$164$	0	0
$165$	−2.12760	−0.165634
$166$	0	0
$167$	−2.42245	−0.187455	−0.0937274	−	0.995598i	$-0.529878\pi$
$167$	−2.42245	−0.187455	−0.0937274	+	0.995598i	$0.529878\pi$
$168$	0	0
$169$	7.00423	0.538787
$170$	0	0
$171$	−0.0242173	−0.00185194
$172$	0	0
$173$	−13.8848	−1.05564	−0.527820	−	0.849356i	$-0.676991\pi$
$173$	−13.8848	−1.05564	−0.527820	+	0.849356i	$0.676991\pi$
$174$	0	0
$175$	−1.03894	−0.0785361
$176$	0	0
$177$	−22.1466	−1.66464
$178$	0	0
$179$	−2.17120	−0.162283	−0.0811416	−	0.996703i	$-0.525857\pi$
$179$	−2.17120	−0.162283	−0.0811416	+	0.996703i	$0.525857\pi$
$180$	0	0
$181$	24.8637	1.84810	0.924052	−	0.382266i	$-0.124856\pi$
$181$	24.8637	1.84810	0.924052	+	0.382266i	$0.124856\pi$
$182$	0	0
$183$	−0.562328	−0.0415684
$184$	0	0
$185$	10.8644	0.798766
$186$	0	0
$187$	3.68936	0.269792
$188$	0	0
$189$	−5.47759	−0.398436
$190$	0	0
$191$	−2.42202	−0.175251	−0.0876257	−	0.996153i	$-0.527928\pi$
$191$	−2.42202	−0.175251	−0.0876257	+	0.996153i	$0.527928\pi$
$192$	0	0
$193$	−2.47117	−0.177878	−0.0889392	−	0.996037i	$-0.528348\pi$
$193$	−2.47117	−0.177878	−0.0889392	+	0.996037i	$0.528348\pi$
$194$	0	0
$195$	7.62779	0.546237
$196$	0	0
$197$	−5.77386	−0.411371	−0.205685	−	0.978618i	$-0.565942\pi$
$197$	−5.77386	−0.411371	−0.205685	+	0.978618i	$0.565942\pi$
$198$	0	0
$199$	−4.83937	−0.343054	−0.171527	−	0.985179i	$-0.554870\pi$
$199$	−4.83937	−0.343054	−0.171527	+	0.985179i	$0.554870\pi$
$200$	0	0
$201$	5.83873	0.411832
$202$	0	0
$203$	2.46530	0.173030
$204$	0	0
$205$	−3.50826	−0.245027
$206$	0	0
$207$	0.106265	0.00738596
$208$	0	0
$209$	0.330329	0.0228493
$210$	0	0
$211$	−24.3400	−1.67564	−0.837819	−	0.545949i	$-0.816169\pi$
$211$	−24.3400	−1.67564	−0.837819	+	0.545949i	$0.816169\pi$
$212$	0	0
$213$	23.7873	1.62988
$214$	0	0
$215$	−5.37679	−0.366694
$216$	0	0
$217$	3.93333	0.267012
$218$	0	0
$219$	4.22034	0.285184
$220$	0	0
$221$	−13.2269	−0.889739
$222$	0	0
$223$	21.8137	1.46076	0.730378	−	0.683043i	$-0.239344\pi$
$223$	21.8137	1.46076	0.730378	+	0.683043i	$0.239344\pi$
$224$	0	0
$225$	−0.0914601	−0.00609734
$226$	0	0
$227$	8.91508	0.591715	0.295857	−	0.955232i	$-0.404395\pi$
$227$	8.91508	0.591715	0.295857	+	0.955232i	$0.404395\pi$
$228$	0	0
$229$	−17.2510	−1.13998	−0.569990	−	0.821651i	$-0.693053\pi$
$229$	−17.2510	−1.13998	−0.569990	+	0.821651i	$0.693053\pi$
$230$	0	0
$231$	2.21044	0.145436
$232$	0	0
$233$	2.77959	0.182097	0.0910485	−	0.995846i	$-0.470978\pi$
$233$	2.77959	0.182097	0.0910485	+	0.995846i	$0.470978\pi$
$234$	0	0
$235$	−7.48532	−0.488288
$236$	0	0
$237$	5.63408	0.365973
$238$	0	0
$239$	22.3462	1.44545	0.722726	−	0.691134i	$-0.242889\pi$
$239$	22.3462	1.44545	0.722726	+	0.691134i	$0.242889\pi$
$240$	0	0
$241$	21.3969	1.37829	0.689146	−	0.724622i	$-0.257986\pi$
$241$	21.3969	1.37829	0.689146	+	0.724622i	$0.257986\pi$
$242$	0	0
$243$	−0.950147	−0.0609520
$244$	0	0
$245$	−5.92061	−0.378254
$246$	0	0
$247$	−1.18428	−0.0753540
$248$	0	0
$249$	−15.7834	−1.00023
$250$	0	0
$251$	−15.0744	−0.951485	−0.475743	−	0.879585i	$-0.657821\pi$
$251$	−15.0744	−0.951485	−0.475743	+	0.879585i	$0.657821\pi$
$252$	0	0
$253$	−1.44948	−0.0911282
$254$	0	0
$255$	−5.04354	−0.315838
$256$	0	0
$257$	11.8220	0.737435	0.368717	−	0.929542i	$-0.379797\pi$
$257$	11.8220	0.737435	0.368717	+	0.929542i	$0.379797\pi$
$258$	0	0
$259$	−11.2874	−0.701365
$260$	0	0
$261$	0.217027	0.0134336
$262$	0	0
$263$	12.2808	0.757269	0.378635	−	0.925546i	$-0.376394\pi$
$263$	12.2808	0.757269	0.378635	+	0.925546i	$0.376394\pi$
$264$	0	0
$265$	1.25982	0.0773900
$266$	0	0
$267$	−5.51954	−0.337790
$268$	0	0
$269$	1.37062	0.0835684	0.0417842	−	0.999127i	$-0.486696\pi$
$269$	1.37062	0.0835684	0.0417842	+	0.999127i	$0.486696\pi$
$270$	0	0
$271$	20.8644	1.26742	0.633712	−	0.773569i	$-0.281530\pi$
$271$	20.8644	1.26742	0.633712	+	0.773569i	$0.281530\pi$
$272$	0	0
$273$	−7.92478	−0.479629
$274$	0	0
$275$	1.24754	0.0752292
$276$	0	0
$277$	4.90537	0.294735	0.147368	−	0.989082i	$-0.452920\pi$
$277$	4.90537	0.294735	0.147368	+	0.989082i	$0.452920\pi$
$278$	0	0
$279$	0.346261	0.0207301
$280$	0	0
$281$	18.2408	1.08816	0.544079	−	0.839034i	$-0.316879\pi$
$281$	18.2408	1.08816	0.544079	+	0.839034i	$0.316879\pi$
$282$	0	0
$283$	−4.82060	−0.286555	−0.143278	−	0.989683i	$-0.545764\pi$
$283$	−4.82060	−0.286555	−0.143278	+	0.989683i	$0.545764\pi$
$284$	0	0
$285$	−0.451576	−0.0267491
$286$	0	0
$287$	3.64485	0.215149
$288$	0	0
$289$	−8.25429	−0.485546
$290$	0	0
$291$	−5.04723	−0.295874
$292$	0	0
$293$	16.1770	0.945071	0.472536	−	0.881312i	$-0.343339\pi$
$293$	16.1770	0.945071	0.472536	+	0.881312i	$0.343339\pi$
$294$	0	0
$295$	12.9858	0.756064
$296$	0	0
$297$	6.57740	0.381659
$298$	0	0
$299$	5.19662	0.300528
$300$	0	0
$301$	5.58614	0.321980
$302$	0	0
$303$	8.39734	0.482414
$304$	0	0
$305$	0.329725	0.0188800
$306$	0	0
$307$	−34.3626	−1.96118	−0.980589	−	0.196073i	$-0.937181\pi$
$307$	−34.3626	−1.96118	−0.980589	+	0.196073i	$0.937181\pi$
$308$	0	0
$309$	−12.4706	−0.709429
$310$	0	0
$311$	−32.5692	−1.84683	−0.923416	−	0.383801i	$-0.874615\pi$
$311$	−32.5692	−1.84683	−0.923416	+	0.383801i	$0.874615\pi$
$312$	0	0
$313$	5.45798	0.308503	0.154252	−	0.988032i	$-0.450703\pi$
$313$	5.45798	0.308503	0.154252	+	0.988032i	$0.450703\pi$
$314$	0	0
$315$	0.0950212	0.00535384
$316$	0	0
$317$	4.29720	0.241355	0.120677	−	0.992692i	$-0.461493\pi$
$317$	4.29720	0.241355	0.120677	+	0.992692i	$0.461493\pi$
$318$	0	0
$319$	−2.96029	−0.165744
$320$	0	0
$321$	9.61743	0.536792
$322$	0	0
$323$	0.783053	0.0435703
$324$	0	0
$325$	−4.47261	−0.248096
$326$	0	0
$327$	−21.4312	−1.18515
$328$	0	0
$329$	7.77677	0.428747
$330$	0	0
$331$	12.7563	0.701149	0.350575	−	0.936535i	$-0.385986\pi$
$331$	12.7563	0.701149	0.350575	+	0.936535i	$0.385986\pi$
$332$	0	0
$333$	−0.993659	−0.0544522
$334$	0	0
$335$	−3.42358	−0.187050
$336$	0	0
$337$	10.9831	0.598286	0.299143	−	0.954208i	$-0.403299\pi$
$337$	10.9831	0.598286	0.299143	+	0.954208i	$0.403299\pi$
$338$	0	0
$339$	−20.0618	−1.08961
$340$	0	0
$341$	−4.72307	−0.255769
$342$	0	0
$343$	13.4237	0.724811
$344$	0	0
$345$	1.98152	0.106681
$346$	0	0
$347$	29.2286	1.56907	0.784537	−	0.620082i	$-0.212901\pi$
$347$	29.2286	1.56907	0.784537	+	0.620082i	$0.212901\pi$
$348$	0	0
$349$	14.9388	0.799656	0.399828	−	0.916590i	$-0.369070\pi$
$349$	14.9388	0.799656	0.399828	+	0.916590i	$0.369070\pi$
$350$	0	0
$351$	−23.5810	−1.25866
$352$	0	0
$353$	−3.70996	−0.197461	−0.0987307	−	0.995114i	$-0.531478\pi$
$353$	−3.70996	−0.197461	−0.0987307	+	0.995114i	$0.531478\pi$
$354$	0	0
$355$	−13.9478	−0.740274
$356$	0	0
$357$	5.23991	0.277325
$358$	0	0
$359$	−31.5876	−1.66713	−0.833564	−	0.552423i	$-0.813703\pi$
$359$	−31.5876	−1.66713	−0.833564	+	0.552423i	$0.813703\pi$
$360$	0	0
$361$	−18.9299	−0.996310
$362$	0	0
$363$	16.1056	0.845326
$364$	0	0
$365$	−2.47463	−0.129528
$366$	0	0
$367$	29.7342	1.55211	0.776056	−	0.630664i	$-0.217217\pi$
$367$	29.7342	1.55211	0.776056	+	0.630664i	$0.217217\pi$
$368$	0	0
$369$	0.320866	0.0167036
$370$	0	0
$371$	−1.30887	−0.0679531
$372$	0	0
$373$	26.4303	1.36851	0.684255	−	0.729243i	$-0.260128\pi$
$373$	26.4303	1.36851	0.684255	+	0.729243i	$0.260128\pi$
$374$	0	0
$375$	−1.70544	−0.0880688
$376$	0	0
$377$	10.6131	0.546602
$378$	0	0
$379$	14.7764	0.759011	0.379505	−	0.925190i	$-0.376094\pi$
$379$	14.7764	0.759011	0.379505	+	0.925190i	$0.376094\pi$
$380$	0	0
$381$	−16.6196	−0.851447
$382$	0	0
$383$	13.5329	0.691498	0.345749	−	0.938327i	$-0.387625\pi$
$383$	13.5329	0.691498	0.345749	+	0.938327i	$0.387625\pi$
$384$	0	0
$385$	−1.29611	−0.0660558
$386$	0	0
$387$	0.491762	0.0249977
$388$	0	0
$389$	8.65637	0.438895	0.219448	−	0.975624i	$-0.429575\pi$
$389$	8.65637	0.438895	0.219448	+	0.975624i	$0.429575\pi$
$390$	0	0
$391$	−3.43604	−0.173768
$392$	0	0
$393$	21.8504	1.10221
$394$	0	0
$395$	−3.30359	−0.166222
$396$	0	0
$397$	19.6287	0.985139	0.492569	−	0.870273i	$-0.336058\pi$
$397$	19.6287	0.985139	0.492569	+	0.870273i	$0.336058\pi$
$398$	0	0
$399$	0.469159	0.0234873
$400$	0	0
$401$	1.00000	0.0499376
$401$	1.00000	0.0499376
$402$	0	0
$403$	16.9330	0.843491
$404$	0	0
$405$	−8.71725	−0.433164
$406$	0	0
$407$	13.5537	0.671833
$408$	0	0
$409$	22.5314	1.11411	0.557054	−	0.830476i	$-0.311932\pi$
$409$	22.5314	1.11411	0.557054	+	0.830476i	$0.311932\pi$
$410$	0	0
$411$	−10.6248	−0.524084
$412$	0	0
$413$	−13.4914	−0.663870
$414$	0	0
$415$	9.25472	0.454296
$416$	0	0
$417$	4.21513	0.206416
$418$	0	0
$419$	−36.9477	−1.80501	−0.902507	−	0.430675i	$-0.858276\pi$
$419$	−36.9477	−1.80501	−0.902507	+	0.430675i	$0.858276\pi$
$420$	0	0
$421$	36.4356	1.77576	0.887881	−	0.460073i	$-0.152177\pi$
$421$	36.4356	1.77576	0.887881	+	0.460073i	$0.152177\pi$
$422$	0	0
$423$	0.684608	0.0332868
$424$	0	0
$425$	2.95732	0.143451
$426$	0	0
$427$	−0.342563	−0.0165778
$428$	0	0
$429$	9.51594	0.459434
$430$	0	0
$431$	−5.85860	−0.282199	−0.141100	−	0.989995i	$-0.545064\pi$
$431$	−5.85860	−0.282199	−0.141100	+	0.989995i	$0.545064\pi$
$432$	0	0
$433$	18.0506	0.867455	0.433728	−	0.901044i	$-0.357198\pi$
$433$	18.0506	0.867455	0.433728	+	0.901044i	$0.357198\pi$
$434$	0	0
$435$	4.04686	0.194032
$436$	0	0
$437$	−0.307648	−0.0147168
$438$	0	0
$439$	4.26849	0.203724	0.101862	−	0.994799i	$-0.467520\pi$
$439$	4.26849	0.203724	0.101862	+	0.994799i	$0.467520\pi$
$440$	0	0
$441$	0.541500	0.0257857
$442$	0	0
$443$	−28.0788	−1.33406	−0.667031	−	0.745030i	$-0.732435\pi$
$443$	−28.0788	−1.33406	−0.667031	+	0.745030i	$0.732435\pi$
$444$	0	0
$445$	3.23642	0.153421
$446$	0	0
$447$	−19.0648	−0.901734
$448$	0	0
$449$	4.09066	0.193050	0.0965250	−	0.995331i	$-0.469227\pi$
$449$	4.09066	0.193050	0.0965250	+	0.995331i	$0.469227\pi$
$450$	0	0
$451$	−4.37667	−0.206090
$452$	0	0
$453$	−0.741285	−0.0348286
$454$	0	0
$455$	4.64675	0.217843
$456$	0	0
$457$	29.2733	1.36935	0.684674	−	0.728849i	$-0.259945\pi$
$457$	29.2733	1.36935	0.684674	+	0.728849i	$0.259945\pi$
$458$	0	0
$459$	15.5919	0.727767
$460$	0	0
$461$	20.5121	0.955342	0.477671	−	0.878539i	$-0.341481\pi$
$461$	20.5121	0.955342	0.477671	+	0.878539i	$0.341481\pi$
$462$	0	0
$463$	7.94546	0.369257	0.184628	−	0.982808i	$-0.440892\pi$
$463$	7.94546	0.369257	0.184628	+	0.982808i	$0.440892\pi$
$464$	0	0
$465$	6.45668	0.299421
$466$	0	0
$467$	−24.1027	−1.11534	−0.557671	−	0.830062i	$-0.688305\pi$
$467$	−24.1027	−1.11534	−0.557671	+	0.830062i	$0.688305\pi$
$468$	0	0
$469$	3.55688	0.164242
$470$	0	0
$471$	−9.44677	−0.435284
$472$	0	0
$473$	−6.70774	−0.308422
$474$	0	0
$475$	0.264785	0.0121492
$476$	0	0
$477$	−0.115223	−0.00527571
$478$	0	0
$479$	40.0509	1.82997	0.914986	−	0.403485i	$-0.132201\pi$
$479$	40.0509	1.82997	0.914986	+	0.403485i	$0.132201\pi$
$480$	0	0
$481$	−48.5922	−2.21561
$482$	0	0
$483$	−2.05867	−0.0936726
$484$	0	0
$485$	2.95948	0.134383
$486$	0	0
$487$	18.5151	0.838998	0.419499	−	0.907756i	$-0.362206\pi$
$487$	18.5151	0.838998	0.419499	+	0.907756i	$0.362206\pi$
$488$	0	0
$489$	−21.2802	−0.962325
$490$	0	0
$491$	15.4837	0.698771	0.349385	−	0.936979i	$-0.386390\pi$
$491$	15.4837	0.698771	0.349385	+	0.936979i	$0.386390\pi$
$492$	0	0
$493$	−7.01744	−0.316050
$494$	0	0
$495$	−0.114100	−0.00512840
$496$	0	0
$497$	14.4909	0.650006
$498$	0	0
$499$	−28.3315	−1.26829	−0.634146	−	0.773213i	$-0.718648\pi$
$499$	−28.3315	−1.26829	−0.634146	+	0.773213i	$0.718648\pi$
$500$	0	0
$501$	4.13135	0.184575
$502$	0	0
$503$	4.97020	0.221610	0.110805	−	0.993842i	$-0.464657\pi$
$503$	4.97020	0.221610	0.110805	+	0.993842i	$0.464657\pi$
$504$	0	0
$505$	−4.92384	−0.219108
$506$	0	0
$507$	−11.9453	−0.530511
$508$	0	0
$509$	4.98686	0.221039	0.110519	−	0.993874i	$-0.464749\pi$
$509$	4.98686	0.221039	0.110519	+	0.993874i	$0.464749\pi$
$510$	0	0
$511$	2.57098	0.113734
$512$	0	0
$513$	1.39603	0.0616363
$514$	0	0
$515$	7.31224	0.322216
$516$	0	0
$517$	−9.33821	−0.410694
$518$	0	0
$519$	23.6797	1.03942
$520$	0	0
$521$	17.2374	0.755184	0.377592	−	0.925972i	$-0.376752\pi$
$521$	17.2374	0.755184	0.377592	+	0.925972i	$0.376752\pi$
$522$	0	0
$523$	−18.1672	−0.794398	−0.397199	−	0.917733i	$-0.630018\pi$
$523$	−18.1672	−0.794398	−0.397199	+	0.917733i	$0.630018\pi$
$524$	0	0
$525$	1.77185	0.0773297
$526$	0	0
$527$	−11.1962	−0.487713
$528$	0	0
$529$	−21.6500	−0.941306
$530$	0	0
$531$	−1.18769	−0.0515412
$532$	0	0
$533$	15.6911	0.679655
$534$	0	0
$535$	−5.63925	−0.243806
$536$	0	0
$537$	3.70286	0.159790
$538$	0	0
$539$	−7.38618	−0.318145
$540$	0	0
$541$	12.6109	0.542183	0.271092	−	0.962554i	$-0.412615\pi$
$541$	12.6109	0.542183	0.271092	+	0.962554i	$0.412615\pi$
$542$	0	0
$543$	−42.4037	−1.81972
$544$	0	0
$545$	12.5663	0.538282
$546$	0	0
$547$	−7.86168	−0.336141	−0.168071	−	0.985775i	$-0.553754\pi$
$547$	−7.86168	−0.336141	−0.168071	+	0.985775i	$0.553754\pi$
$548$	0	0
$549$	−0.0301567	−0.00128706
$550$	0	0
$551$	−0.628311	−0.0267670
$552$	0	0
$553$	3.43221	0.145953
$554$	0	0
$555$	−18.5286	−0.786496
$556$	0	0
$557$	27.1398	1.14995	0.574974	−	0.818172i	$-0.305012\pi$
$557$	27.1398	1.14995	0.574974	+	0.818172i	$0.305012\pi$
$558$	0	0
$559$	24.0483	1.01713
$560$	0	0
$561$	−6.29199	−0.265648
$562$	0	0
$563$	−31.8505	−1.34234	−0.671168	−	0.741305i	$-0.734207\pi$
$563$	−31.8505	−1.34234	−0.671168	+	0.741305i	$0.734207\pi$
$564$	0	0
$565$	11.7634	0.494889
$566$	0	0
$567$	9.05667	0.380344
$568$	0	0
$569$	40.4465	1.69560	0.847802	−	0.530313i	$-0.177926\pi$
$569$	40.4465	1.69560	0.847802	+	0.530313i	$0.177926\pi$
$570$	0	0
$571$	47.0510	1.96902	0.984511	−	0.175324i	$-0.0560974\pi$
$571$	47.0510	1.96902	0.984511	+	0.175324i	$0.0560974\pi$
$572$	0	0
$573$	4.13062	0.172559
$574$	0	0
$575$	−1.16188	−0.0484536
$576$	0	0
$577$	−11.7884	−0.490758	−0.245379	−	0.969427i	$-0.578912\pi$
$577$	−11.7884	−0.490758	−0.245379	+	0.969427i	$0.578912\pi$
$578$	0	0
$579$	4.21444	0.175146
$580$	0	0
$581$	−9.61506	−0.398900
$582$	0	0
$583$	1.57167	0.0650919
$584$	0	0
$585$	0.409065	0.0169128
$586$	0	0
$587$	0.0190257	0.000785275 0	0.000392637	−	1.00000i	$-0.499875\pi$
$587$	0.0190257	0.000785275 0	0.000392637		1.00000i	$0.499875\pi$
$588$	0	0
$589$	−1.00246	−0.0413055
$590$	0	0
$591$	9.84700	0.405052
$592$	0	0
$593$	37.6448	1.54589	0.772944	−	0.634475i	$-0.218784\pi$
$593$	37.6448	1.54589	0.772944	+	0.634475i	$0.218784\pi$
$594$	0	0
$595$	−3.07246	−0.125959
$596$	0	0
$597$	8.25327	0.337784
$598$	0	0
$599$	−16.9824	−0.693881	−0.346940	−	0.937887i	$-0.612779\pi$
$599$	−16.9824	−0.693881	−0.346940	+	0.937887i	$0.612779\pi$
$600$	0	0
$601$	−2.46937	−0.100728	−0.0503638	−	0.998731i	$-0.516038\pi$
$601$	−2.46937	−0.100728	−0.0503638	+	0.998731i	$0.516038\pi$
$602$	0	0
$603$	0.313121	0.0127513
$604$	0	0
$605$	−9.44365	−0.383939
$606$	0	0
$607$	−8.34548	−0.338733	−0.169366	−	0.985553i	$-0.554172\pi$
$607$	−8.34548	−0.338733	−0.169366	+	0.985553i	$0.554172\pi$
$608$	0	0
$609$	−4.20443	−0.170372
$610$	0	0
$611$	33.4789	1.35441
$612$	0	0
$613$	13.7942	0.557143	0.278571	−	0.960416i	$-0.410139\pi$
$613$	13.7942	0.557143	0.278571	+	0.960416i	$0.410139\pi$
$614$	0	0
$615$	5.98313	0.241263
$616$	0	0
$617$	−34.8627	−1.40352	−0.701760	−	0.712414i	$-0.747602\pi$
$617$	−34.8627	−1.40352	−0.701760	+	0.712414i	$0.747602\pi$
$618$	0	0
$619$	35.5689	1.42964	0.714818	−	0.699311i	$-0.246510\pi$
$619$	35.5689	1.42964	0.714818	+	0.699311i	$0.246510\pi$
$620$	0	0
$621$	−6.12578	−0.245819
$622$	0	0
$623$	−3.36244	−0.134713
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	−0.563358	−0.0224983
$628$	0	0
$629$	32.1294	1.28108
$630$	0	0
$631$	34.6299	1.37860	0.689298	−	0.724478i	$-0.257919\pi$
$631$	34.6299	1.37860	0.689298	+	0.724478i	$0.257919\pi$
$632$	0	0
$633$	41.5106	1.64990
$634$	0	0
$635$	9.74501	0.386719
$636$	0	0
$637$	26.4806	1.04920
$638$	0	0
$639$	1.27567	0.0504648
$640$	0	0
$641$	−16.4660	−0.650367	−0.325183	−	0.945651i	$-0.605426\pi$
$641$	−16.4660	−0.650367	−0.325183	+	0.945651i	$0.605426\pi$
$642$	0	0
$643$	12.6618	0.499332	0.249666	−	0.968332i	$-0.419679\pi$
$643$	12.6618	0.499332	0.249666	+	0.968332i	$0.419679\pi$
$644$	0	0
$645$	9.16982	0.361061
$646$	0	0
$647$	34.9919	1.37567	0.687837	−	0.725865i	$-0.258560\pi$
$647$	34.9919	1.37567	0.687837	+	0.725865i	$0.258560\pi$
$648$	0	0
$649$	16.2003	0.635917
$650$	0	0
$651$	−6.70807	−0.262910
$652$	0	0
$653$	36.0998	1.41269	0.706347	−	0.707865i	$-0.250342\pi$
$653$	36.0998	1.41269	0.706347	+	0.707865i	$0.250342\pi$
$654$	0	0
$655$	−12.8121	−0.500611
$656$	0	0
$657$	0.226330	0.00882997
$658$	0	0
$659$	0.541141	0.0210798	0.0105399	−	0.999944i	$-0.496645\pi$
$659$	0.541141	0.0210798	0.0105399	+	0.999944i	$0.496645\pi$
$660$	0	0
$661$	−7.65857	−0.297884	−0.148942	−	0.988846i	$-0.547587\pi$
$661$	−7.65857	−0.297884	−0.148942	+	0.988846i	$0.547587\pi$
$662$	0	0
$663$	22.5578	0.876071
$664$	0	0
$665$	−0.275095	−0.0106677
$666$	0	0
$667$	2.75703	0.106753
$668$	0	0
$669$	−37.2021	−1.43832
$670$	0	0
$671$	0.411344	0.0158797
$672$	0	0
$673$	35.9836	1.38707	0.693533	−	0.720425i	$-0.256053\pi$
$673$	35.9836	1.38707	0.693533	+	0.720425i	$0.256053\pi$
$674$	0	0
$675$	5.27231	0.202931
$676$	0	0
$677$	5.18804	0.199393	0.0996963	−	0.995018i	$-0.468213\pi$
$677$	5.18804	0.199393	0.0996963	+	0.995018i	$0.468213\pi$
$678$	0	0
$679$	−3.07471	−0.117997
$680$	0	0
$681$	−15.2042	−0.582625
$682$	0	0
$683$	3.89639	0.149091	0.0745456	−	0.997218i	$-0.476249\pi$
$683$	3.89639	0.149091	0.0745456	+	0.997218i	$0.476249\pi$
$684$	0	0
$685$	6.22995	0.238034
$686$	0	0
$687$	29.4207	1.12247
$688$	0	0
$689$	−5.63467	−0.214664
$690$	0	0
$691$	−25.7879	−0.981017	−0.490509	−	0.871436i	$-0.663189\pi$
$691$	−25.7879	−0.981017	−0.490509	+	0.871436i	$0.663189\pi$
$692$	0	0
$693$	0.118542	0.00450305
$694$	0	0
$695$	−2.47158	−0.0937522
$696$	0	0
$697$	−10.3750	−0.392982
$698$	0	0
$699$	−4.74044	−0.179300
$700$	0	0
$701$	−15.5620	−0.587770	−0.293885	−	0.955841i	$-0.594948\pi$
$701$	−15.5620	−0.587770	−0.293885	+	0.955841i	$0.594948\pi$
$702$	0	0
$703$	2.87673	0.108498
$704$	0	0
$705$	12.7658	0.480788
$706$	0	0
$707$	5.11555	0.192390
$708$	0	0
$709$	22.2090	0.834075	0.417038	−	0.908889i	$-0.363068\pi$
$709$	22.2090	0.834075	0.417038	+	0.908889i	$0.363068\pi$
$710$	0	0
$711$	0.302147	0.0113314
$712$	0	0
$713$	4.39878	0.164735
$714$	0	0
$715$	−5.57974	−0.208670
$716$	0	0
$717$	−38.1101	−1.42325
$718$	0	0
$719$	11.3486	0.423230	0.211615	−	0.977353i	$-0.432128\pi$
$719$	11.3486	0.423230	0.211615	+	0.977353i	$0.432128\pi$
$720$	0	0
$721$	−7.59695	−0.282925
$722$	0	0
$723$	−36.4911	−1.35712
$724$	0	0
$725$	−2.37291	−0.0881276
$726$	0	0
$727$	10.9526	0.406211	0.203105	−	0.979157i	$-0.434897\pi$
$727$	10.9526	0.406211	0.203105	+	0.979157i	$0.434897\pi$
$728$	0	0
$729$	27.7722	1.02860
$730$	0	0
$731$	−15.9009	−0.588115
$732$	0	0
$733$	7.49978	0.277011	0.138505	−	0.990362i	$-0.455770\pi$
$733$	7.49978	0.277011	0.138505	+	0.990362i	$0.455770\pi$
$734$	0	0
$735$	10.0973	0.372444
$736$	0	0
$737$	−4.27104	−0.157326
$738$	0	0
$739$	−16.1186	−0.592934	−0.296467	−	0.955043i	$-0.595808\pi$
$739$	−16.1186	−0.592934	−0.296467	+	0.955043i	$0.595808\pi$
$740$	0	0
$741$	2.01972	0.0741964
$742$	0	0
$743$	22.5604	0.827659	0.413830	−	0.910354i	$-0.364191\pi$
$743$	22.5604	0.827659	0.413830	+	0.910354i	$0.364191\pi$
$744$	0	0
$745$	11.1788	0.409559
$746$	0	0
$747$	−0.846438	−0.0309695
$748$	0	0
$749$	5.85882	0.214077
$750$	0	0
$751$	−28.9217	−1.05537	−0.527684	−	0.849441i	$-0.676940\pi$
$751$	−28.9217	−1.05537	−0.527684	+	0.849441i	$0.676940\pi$
$752$	0	0
$753$	25.7085	0.936869
$754$	0	0
$755$	0.434658	0.0158188
$756$	0	0
$757$	35.0020	1.27217	0.636085	−	0.771619i	$-0.280553\pi$
$757$	35.0020	1.27217	0.636085	+	0.771619i	$0.280553\pi$
$758$	0	0
$759$	2.47201	0.0897284
$760$	0	0
$761$	−7.27185	−0.263604	−0.131802	−	0.991276i	$-0.542076\pi$
$761$	−7.27185	−0.263604	−0.131802	+	0.991276i	$0.542076\pi$
$762$	0	0
$763$	−13.0556	−0.472644
$764$	0	0
$765$	−0.270476	−0.00977910
$766$	0	0
$767$	−58.0806	−2.09717
$768$	0	0
$769$	28.1027	1.01341	0.506704	−	0.862120i	$-0.330864\pi$
$769$	28.1027	1.01341	0.506704	+	0.862120i	$0.330864\pi$
$770$	0	0
$771$	−20.1617	−0.726107
$772$	0	0
$773$	−4.65665	−0.167488	−0.0837440	−	0.996487i	$-0.526688\pi$
$773$	−4.65665	−0.167488	−0.0837440	+	0.996487i	$0.526688\pi$
$774$	0	0
$775$	−3.78592	−0.135994
$776$	0	0
$777$	19.2500	0.690591
$778$	0	0
$779$	−0.928934	−0.0332825
$780$	0	0
$781$	−17.4004	−0.622636
$782$	0	0
$783$	−12.5107	−0.447097
$784$	0	0
$785$	5.53918	0.197702
$786$	0	0
$787$	−34.8416	−1.24197	−0.620985	−	0.783823i	$-0.713267\pi$
$787$	−34.8416	−1.24197	−0.620985	+	0.783823i	$0.713267\pi$
$788$	0	0
$789$	−20.9443	−0.745636
$790$	0	0
$791$	−12.2214	−0.434542
$792$	0	0
$793$	−1.47473	−0.0523692
$794$	0	0
$795$	−2.14855	−0.0762012
$796$	0	0
$797$	−20.9674	−0.742704	−0.371352	−	0.928492i	$-0.621106\pi$
$797$	−20.9674	−0.742704	−0.371352	+	0.928492i	$0.621106\pi$
$798$	0	0
$799$	−22.1365	−0.783131
$800$	0	0
$801$	−0.296004	−0.0104588
$802$	0	0
$803$	−3.08719	−0.108945
$804$	0	0
$805$	1.20711	0.0425452
$806$	0	0
$807$	−2.33752	−0.0822847
$808$	0	0
$809$	48.6343	1.70989	0.854946	−	0.518717i	$-0.173590\pi$
$809$	48.6343	1.70989	0.854946	+	0.518717i	$0.173590\pi$
$810$	0	0
$811$	−16.7943	−0.589727	−0.294864	−	0.955539i	$-0.595274\pi$
$811$	−16.7943	−0.589727	−0.294864	+	0.955539i	$0.595274\pi$
$812$	0	0
$813$	−35.5831	−1.24796
$814$	0	0
$815$	12.4778	0.437079
$816$	0	0
$817$	−1.42370	−0.0498088
$818$	0	0
$819$	−0.424993	−0.0148504
$820$	0	0
$821$	24.0295	0.838636	0.419318	−	0.907839i	$-0.362269\pi$
$821$	24.0295	0.838636	0.419318	+	0.907839i	$0.362269\pi$
$822$	0	0
$823$	−20.2743	−0.706717	−0.353359	−	0.935488i	$-0.614960\pi$
$823$	−20.2743	−0.706717	−0.353359	+	0.935488i	$0.614960\pi$
$824$	0	0
$825$	−2.12760	−0.0740736
$826$	0	0
$827$	−31.3412	−1.08984	−0.544920	−	0.838488i	$-0.683440\pi$
$827$	−31.3412	−1.08984	−0.544920	+	0.838488i	$0.683440\pi$
$828$	0	0
$829$	−8.81916	−0.306302	−0.153151	−	0.988203i	$-0.548942\pi$
$829$	−8.81916	−0.306302	−0.153151	+	0.988203i	$0.548942\pi$
$830$	0	0
$831$	−8.36584	−0.290208
$832$	0	0
$833$	−17.5091	−0.606655
$834$	0	0
$835$	−2.42245	−0.0838323
$836$	0	0
$837$	−19.9606	−0.689938
$838$	0	0
$839$	20.3130	0.701284	0.350642	−	0.936510i	$-0.385963\pi$
$839$	20.3130	0.701284	0.350642	+	0.936510i	$0.385963\pi$
$840$	0	0
$841$	−23.3693	−0.805838
$842$	0	0
$843$	−31.1087	−1.07144
$844$	0	0
$845$	7.00423	0.240953
$846$	0	0
$847$	9.81135	0.337122
$848$	0	0
$849$	8.22127	0.282153
$850$	0	0
$851$	−12.6231	−0.432714
$852$	0	0
$853$	−42.3454	−1.44988	−0.724940	−	0.688812i	$-0.758133\pi$
$853$	−42.3454	−1.44988	−0.724940	+	0.688812i	$0.758133\pi$
$854$	0	0
$855$	−0.0242173	−0.000828214 0
$856$	0	0
$857$	−15.2312	−0.520287	−0.260143	−	0.965570i	$-0.583770\pi$
$857$	−15.2312	−0.520287	−0.260143	+	0.965570i	$0.583770\pi$
$858$	0	0
$859$	14.3949	0.491147	0.245573	−	0.969378i	$-0.421024\pi$
$859$	14.3949	0.491147	0.245573	+	0.969378i	$0.421024\pi$
$860$	0	0
$861$	−6.21609	−0.211844
$862$	0	0
$863$	−19.7061	−0.670804	−0.335402	−	0.942075i	$-0.608872\pi$
$863$	−19.7061	−0.670804	−0.335402	+	0.942075i	$0.608872\pi$
$864$	0	0
$865$	−13.8848	−0.472097
$866$	0	0
$867$	14.0772	0.478088
$868$	0	0
$869$	−4.12134	−0.139807
$870$	0	0
$871$	15.3124	0.518839
$872$	0	0
$873$	−0.270675	−0.00916095
$874$	0	0
$875$	−1.03894	−0.0351224
$876$	0	0
$877$	−22.8570	−0.771826	−0.385913	−	0.922535i	$-0.626114\pi$
$877$	−22.8570	−0.771826	−0.385913	+	0.922535i	$0.626114\pi$
$878$	0	0
$879$	−27.5890	−0.930554
$880$	0	0
$881$	−44.6516	−1.50435	−0.752175	−	0.658964i	$-0.770995\pi$
$881$	−44.6516	−1.50435	−0.752175	+	0.658964i	$0.770995\pi$
$882$	0	0
$883$	−8.86586	−0.298360	−0.149180	−	0.988810i	$-0.547663\pi$
$883$	−8.86586	−0.298360	−0.149180	+	0.988810i	$0.547663\pi$
$884$	0	0
$885$	−22.1466	−0.744450
$886$	0	0
$887$	44.1170	1.48130	0.740652	−	0.671889i	$-0.234517\pi$
$887$	44.1170	1.48130	0.740652	+	0.671889i	$0.234517\pi$
$888$	0	0
$889$	−10.1244	−0.339563
$890$	0	0
$891$	−10.8751	−0.364329
$892$	0	0
$893$	−1.98200	−0.0663252
$894$	0	0
$895$	−2.17120	−0.0725753
$896$	0	0
$897$	−8.86255	−0.295912
$898$	0	0
$899$	8.98365	0.299622
$900$	0	0
$901$	3.72568	0.124120
$902$	0	0
$903$	−9.52685	−0.317034
$904$	0	0
$905$	24.8637	0.826498
$906$	0	0
$907$	−1.91674	−0.0636442	−0.0318221	−	0.999494i	$-0.510131\pi$
$907$	−1.91674	−0.0636442	−0.0318221	+	0.999494i	$0.510131\pi$
$908$	0	0
$909$	0.450335	0.0149367
$910$	0	0
$911$	33.0610	1.09536	0.547680	−	0.836688i	$-0.315511\pi$
$911$	33.0610	1.09536	0.547680	+	0.836688i	$0.315511\pi$
$912$	0	0
$913$	11.5456	0.382103
$914$	0	0
$915$	−0.562328	−0.0185900
$916$	0	0
$917$	13.3110	0.439567
$918$	0	0
$919$	−46.7397	−1.54180	−0.770900	−	0.636956i	$-0.780193\pi$
$919$	−46.7397	−1.54180	−0.770900	+	0.636956i	$0.780193\pi$
$920$	0	0
$921$	58.6035	1.93105
$922$	0	0
$923$	62.3832	2.05337
$924$	0	0
$925$	10.8644	0.357219
$926$	0	0
$927$	−0.668779	−0.0219656
$928$	0	0
$929$	−29.2846	−0.960797	−0.480398	−	0.877050i	$-0.659508\pi$
$929$	−29.2846	−0.960797	−0.480398	+	0.877050i	$0.659508\pi$
$930$	0	0
$931$	−1.56769	−0.0513790
$932$	0	0
$933$	55.5450	1.81846
$934$	0	0
$935$	3.68936	0.120655
$936$	0	0
$937$	9.84570	0.321645	0.160822	−	0.986983i	$-0.448585\pi$
$937$	9.84570	0.321645	0.160822	+	0.986983i	$0.448585\pi$
$938$	0	0
$939$	−9.30828	−0.303764
$940$	0	0
$941$	40.1728	1.30959	0.654797	−	0.755804i	$-0.272754\pi$
$941$	40.1728	1.30959	0.654797	+	0.755804i	$0.272754\pi$
$942$	0	0
$943$	4.07616	0.132738
$944$	0	0
$945$	−5.47759	−0.178186
$946$	0	0
$947$	38.9735	1.26647	0.633235	−	0.773960i	$-0.281727\pi$
$947$	38.9735	1.26647	0.633235	+	0.773960i	$0.281727\pi$
$948$	0	0
$949$	11.0681	0.359284
$950$	0	0
$951$	−7.32864	−0.237647
$952$	0	0
$953$	−12.0204	−0.389380	−0.194690	−	0.980865i	$-0.562370\pi$
$953$	−12.0204	−0.389380	−0.194690	+	0.980865i	$0.562370\pi$
$954$	0	0
$955$	−2.42202	−0.0783748
$956$	0	0
$957$	5.04861	0.163198
$958$	0	0
$959$	−6.47252	−0.209008
$960$	0	0
$961$	−16.6668	−0.537638
$962$	0	0
$963$	0.515767	0.0166203
$964$	0	0
$965$	−2.47117	−0.0795497
$966$	0	0
$967$	−31.3940	−1.00956	−0.504781	−	0.863248i	$-0.668427\pi$
$967$	−31.3940	−1.00956	−0.504781	+	0.863248i	$0.668427\pi$
$968$	0	0
$969$	−1.33545	−0.0429010
$970$	0	0
$971$	−15.3196	−0.491629	−0.245815	−	0.969317i	$-0.579055\pi$
$971$	−15.3196	−0.491629	−0.245815	+	0.969317i	$0.579055\pi$
$972$	0	0
$973$	2.56781	0.0823201
$974$	0	0
$975$	7.62779	0.244285
$976$	0	0
$977$	32.8928	1.05233	0.526167	−	0.850381i	$-0.323629\pi$
$977$	32.8928	1.05233	0.526167	+	0.850381i	$0.323629\pi$
$978$	0	0
$979$	4.03756	0.129041
$980$	0	0
$981$	−1.14932	−0.0366949
$982$	0	0
$983$	36.6876	1.17015	0.585077	−	0.810978i	$-0.301064\pi$
$983$	36.6876	1.17015	0.585077	+	0.810978i	$0.301064\pi$
$984$	0	0
$985$	−5.77386	−0.183971
$986$	0	0
$987$	−13.2628	−0.422161
$988$	0	0
$989$	6.24717	0.198649
$990$	0	0
$991$	−34.0692	−1.08224	−0.541121	−	0.840944i	$-0.682000\pi$
$991$	−34.0692	−1.08224	−0.541121	+	0.840944i	$0.682000\pi$
$992$	0	0
$993$	−21.7551	−0.690378
$994$	0	0
$995$	−4.83937	−0.153418
$996$	0	0
$997$	−22.9353	−0.726369	−0.363184	−	0.931717i	$-0.618310\pi$
$997$	−22.9353	−0.726369	−0.363184	+	0.931717i	$0.618310\pi$
$998$	0	0
$999$	57.2805	1.81227

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8020.2.a.f.1.10	✓	37

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8020.2.a.f.1.10	✓	37	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 8020 = 2^{2} \cdot 5 \cdot 401 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8020.a (trivial)

\(f(q)\)	\(=\)	\(q-1.70544 q^{3} +1.00000 q^{5} -1.03894 q^{7} -0.0914601 q^{9} +O(q^{10})\)	\(q-1.70544 q^{3} +1.00000 q^{5} -1.03894 q^{7} -0.0914601 q^{9} +1.24754 q^{11} -4.47261 q^{13} -1.70544 q^{15} +2.95732 q^{17} +0.264785 q^{19} +1.77185 q^{21} -1.16188 q^{23} +1.00000 q^{25} +5.27231 q^{27} -2.37291 q^{29} -3.78592 q^{31} -2.12760 q^{33} -1.03894 q^{35} +10.8644 q^{37} +7.62779 q^{39} -3.50826 q^{41} -5.37679 q^{43} -0.0914601 q^{45} -7.48532 q^{47} -5.92061 q^{49} -5.04354 q^{51} +1.25982 q^{53} +1.24754 q^{55} -0.451576 q^{57} +12.9858 q^{59} +0.329725 q^{61} +0.0950212 q^{63} -4.47261 q^{65} -3.42358 q^{67} +1.98152 q^{69} -13.9478 q^{71} -2.47463 q^{73} -1.70544 q^{75} -1.29611 q^{77} -3.30359 q^{79} -8.71725 q^{81} +9.25472 q^{83} +2.95732 q^{85} +4.04686 q^{87} +3.23642 q^{89} +4.64675 q^{91} +6.45668 q^{93} +0.264785 q^{95} +2.95948 q^{97} -0.114100 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 37 q + 3 q^{3} + 37 q^{5} + 4 q^{7} + 50 q^{9}+O(q^{10}) \) 37 * q + 3 * q^3 + 37 * q^5 + 4 * q^7 + 50 * q^9	\( 37 q + 3 q^{3} + 37 q^{5} + 4 q^{7} + 50 q^{9} + 2 q^{11} + 27 q^{13} + 3 q^{15} + 36 q^{17} - 6 q^{19} + 20 q^{21} + 17 q^{23} + 37 q^{25} + 9 q^{27} + 29 q^{29} + 5 q^{31} + 36 q^{33} + 4 q^{35} + 35 q^{37} + 21 q^{39} + 24 q^{41} + 11 q^{43} + 50 q^{45} + 19 q^{47} + 57 q^{49} + 8 q^{51} + 65 q^{53} + 2 q^{55} + 62 q^{57} - 9 q^{59} + 13 q^{61} + 26 q^{63} + 27 q^{65} + 13 q^{67} + 20 q^{69} + 33 q^{71} + 67 q^{73} + 3 q^{75} + 62 q^{77} + 23 q^{79} + 97 q^{81} + 2 q^{83} + 36 q^{85} + 32 q^{87} + 34 q^{89} + q^{91} + 41 q^{93} - 6 q^{95} + 66 q^{97} - 7 q^{99}+O(q^{100}) \) 37 * q + 3 * q^3 + 37 * q^5 + 4 * q^7 + 50 * q^9 + 2 * q^11 + 27 * q^13 + 3 * q^15 + 36 * q^17 - 6 * q^19 + 20 * q^21 + 17 * q^23 + 37 * q^25 + 9 * q^27 + 29 * q^29 + 5 * q^31 + 36 * q^33 + 4 * q^35 + 35 * q^37 + 21 * q^39 + 24 * q^41 + 11 * q^43 + 50 * q^45 + 19 * q^47 + 57 * q^49 + 8 * q^51 + 65 * q^53 + 2 * q^55 + 62 * q^57 - 9 * q^59 + 13 * q^61 + 26 * q^63 + 27 * q^65 + 13 * q^67 + 20 * q^69 + 33 * q^71 + 67 * q^73 + 3 * q^75 + 62 * q^77 + 23 * q^79 + 97 * q^81 + 2 * q^83 + 36 * q^85 + 32 * q^87 + 34 * q^89 + q^91 + 41 * q^93 - 6 * q^95 + 66 * q^97 - 7 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

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Groups

Database

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