LMFDB - Embedded newform 6020.2.a.g.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6020,2,Mod(1,6020)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6020, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("6020.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6020 = 2^{2} \cdot 5 \cdot 7 \cdot 43 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6020.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:	$48.0699420168$
Analytic rank:	$1$
Dimension:	$9$
Coefficient field:	$\mathbb{Q}[x]/(x^{9} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{9} - 16x^{7} + 83x^{5} - 9x^{4} - 160x^{3} + 32x^{2} + 77x - 2 $ x^9 - 16x^7 + 83x^5 - 9x^4 - 160x^3 + 32x^2 + 77x - 2
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$0.0257343$ of defining polynomial
Character	$\chi$	$=$	6020.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.0257343 q^{3} +1.00000 q^{5} -1.00000 q^{7} -2.99934 q^{9} +O(q^{10})$	$q+0.0257343 q^{3} +1.00000 q^{5} -1.00000 q^{7} -2.99934 q^{9} -1.73949 q^{11} +3.34208 q^{13} +0.0257343 q^{15} +5.47819 q^{17} -2.98411 q^{19} -0.0257343 q^{21} +0.452560 q^{23} +1.00000 q^{25} -0.154388 q^{27} -4.73565 q^{29} +0.135699 q^{31} -0.0447646 q^{33} -1.00000 q^{35} -5.71426 q^{37} +0.0860058 q^{39} -5.73877 q^{41} -1.00000 q^{43} -2.99934 q^{45} +6.29425 q^{47} +1.00000 q^{49} +0.140977 q^{51} -2.69526 q^{53} -1.73949 q^{55} -0.0767937 q^{57} -1.27534 q^{59} -11.3439 q^{61} +2.99934 q^{63} +3.34208 q^{65} +11.1441 q^{67} +0.0116463 q^{69} +11.4757 q^{71} -8.79090 q^{73} +0.0257343 q^{75} +1.73949 q^{77} +9.55968 q^{79} +8.99404 q^{81} -6.22335 q^{83} +5.47819 q^{85} -0.121868 q^{87} -9.54554 q^{89} -3.34208 q^{91} +0.00349210 q^{93} -2.98411 q^{95} +8.18810 q^{97} +5.21733 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 9 q + 9 q^{5} - 9 q^{7} + 5 q^{9}+O(q^{10}) $ 9 * q + 9 * q^5 - 9 * q^7 + 5 * q^9	$ 9 q + 9 q^{5} - 9 q^{7} + 5 q^{9} + q^{11} - 14 q^{13} - 11 q^{17} - 2 q^{19} - 6 q^{23} + 9 q^{25} - 6 q^{29} + 6 q^{31} - 14 q^{33} - 9 q^{35} - 20 q^{37} - 6 q^{39} - 6 q^{41} - 9 q^{43} + 5 q^{45} + 9 q^{49} - 8 q^{51} - 31 q^{53} + q^{55} - 16 q^{57} + 2 q^{59} - 13 q^{61} - 5 q^{63} - 14 q^{65} - 10 q^{67} - 18 q^{69} + 12 q^{71} - 32 q^{73} - q^{77} + q^{79} - 27 q^{81} - 10 q^{83} - 11 q^{85} - 5 q^{87} - q^{89} + 14 q^{91} - 49 q^{93} - 2 q^{95} - 28 q^{97} - 14 q^{99}+O(q^{100}) $ 9 * q + 9 * q^5 - 9 * q^7 + 5 * q^9 + q^11 - 14 * q^13 - 11 * q^17 - 2 * q^19 - 6 * q^23 + 9 * q^25 - 6 * q^29 + 6 * q^31 - 14 * q^33 - 9 * q^35 - 20 * q^37 - 6 * q^39 - 6 * q^41 - 9 * q^43 + 5 * q^45 + 9 * q^49 - 8 * q^51 - 31 * q^53 + q^55 - 16 * q^57 + 2 * q^59 - 13 * q^61 - 5 * q^63 - 14 * q^65 - 10 * q^67 - 18 * q^69 + 12 * q^71 - 32 * q^73 - q^77 + q^79 - 27 * q^81 - 10 * q^83 - 11 * q^85 - 5 * q^87 - q^89 + 14 * q^91 - 49 * q^93 - 2 * q^95 - 28 * q^97 - 14 * 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
$3$	0.0257343	0.0148577	0.00742884	−	0.999972i	$-0.497635\pi$
$3$	0.0257343	0.0148577	0.00742884	+	0.999972i	$0.497635\pi$
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	−2.99934	−0.999779
$10$	0	0
$11$	−1.73949	−0.524477	−0.262239	−	0.965003i	$-0.584461\pi$
$11$	−1.73949	−0.524477	−0.262239	+	0.965003i	$0.584461\pi$
$12$	0	0
$13$	3.34208	0.926925	0.463462	−	0.886117i	$-0.346607\pi$
$13$	3.34208	0.926925	0.463462	+	0.886117i	$0.346607\pi$
$14$	0	0
$15$	0.0257343	0.00664456
$16$	0	0
$17$	5.47819	1.32866	0.664328	−	0.747441i	$-0.268718\pi$
$17$	5.47819	1.32866	0.664328	+	0.747441i	$0.268718\pi$
$18$	0	0
$19$	−2.98411	−0.684601	−0.342300	−	0.939591i	$-0.611206\pi$
$19$	−2.98411	−0.684601	−0.342300	+	0.939591i	$0.611206\pi$
$20$	0	0
$21$	−0.0257343	−0.00561568
$22$	0	0
$23$	0.452560	0.0943654	0.0471827	−	0.998886i	$-0.484976\pi$
$23$	0.452560	0.0943654	0.0471827	+	0.998886i	$0.484976\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−0.154388	−0.0297121
$28$	0	0
$29$	−4.73565	−0.879388	−0.439694	−	0.898148i	$-0.644913\pi$
$29$	−4.73565	−0.879388	−0.439694	+	0.898148i	$0.644913\pi$
$30$	0	0
$31$	0.135699	0.0243722	0.0121861	−	0.999926i	$-0.496121\pi$
$31$	0.135699	0.0243722	0.0121861	+	0.999926i	$0.496121\pi$
$32$	0	0
$33$	−0.0447646	−0.00779252
$34$	0	0
$35$	−1.00000	−0.169031
$36$	0	0
$37$	−5.71426	−0.939418	−0.469709	−	0.882821i	$-0.655641\pi$
$37$	−5.71426	−0.939418	−0.469709	+	0.882821i	$0.655641\pi$
$38$	0	0
$39$	0.0860058	0.0137720
$40$	0	0
$41$	−5.73877	−0.896245	−0.448123	−	0.893972i	$-0.647907\pi$
$41$	−5.73877	−0.896245	−0.448123	+	0.893972i	$0.647907\pi$
$42$	0	0
$43$	−1.00000	−0.152499
$43$	−1.00000	−0.152499
$44$	0	0
$45$	−2.99934	−0.447115
$46$	0	0
$47$	6.29425	0.918111	0.459055	−	0.888408i	$-0.348188\pi$
$47$	6.29425	0.918111	0.459055	+	0.888408i	$0.348188\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	0.140977	0.0197408
$52$	0	0
$53$	−2.69526	−0.370222	−0.185111	−	0.982718i	$-0.559264\pi$
$53$	−2.69526	−0.370222	−0.185111	+	0.982718i	$0.559264\pi$
$54$	0	0
$55$	−1.73949	−0.234553
$56$	0	0
$57$	−0.0767937	−0.0101716
$58$	0	0
$59$	−1.27534	−0.166035	−0.0830173	−	0.996548i	$-0.526456\pi$
$59$	−1.27534	−0.166035	−0.0830173	+	0.996548i	$0.526456\pi$
$60$	0	0
$61$	−11.3439	−1.45243	−0.726216	−	0.687467i	$-0.758723\pi$
$61$	−11.3439	−1.45243	−0.726216	+	0.687467i	$0.758723\pi$
$62$	0	0
$63$	2.99934	0.377881
$64$	0	0
$65$	3.34208	0.414533
$66$	0	0
$67$	11.1441	1.36147	0.680735	−	0.732530i	$-0.261661\pi$
$67$	11.1441	1.36147	0.680735	+	0.732530i	$0.261661\pi$
$68$	0	0
$69$	0.0116463	0.00140205
$70$	0	0
$71$	11.4757	1.36191	0.680955	−	0.732325i	$-0.261565\pi$
$71$	11.4757	1.36191	0.680955	+	0.732325i	$0.261565\pi$
$72$	0	0
$73$	−8.79090	−1.02890	−0.514449	−	0.857521i	$-0.672003\pi$
$73$	−8.79090	−1.02890	−0.514449	+	0.857521i	$0.672003\pi$
$74$	0	0
$75$	0.0257343	0.00297154
$76$	0	0
$77$	1.73949	0.198234
$78$	0	0
$79$	9.55968	1.07555	0.537774	−	0.843089i	$-0.319265\pi$
$79$	9.55968	1.07555	0.537774	+	0.843089i	$0.319265\pi$
$80$	0	0
$81$	8.99404	0.999338
$82$	0	0
$83$	−6.22335	−0.683102	−0.341551	−	0.939863i	$-0.610952\pi$
$83$	−6.22335	−0.683102	−0.341551	+	0.939863i	$0.610952\pi$
$84$	0	0
$85$	5.47819	0.594193
$86$	0	0
$87$	−0.121868	−0.0130657
$88$	0	0
$89$	−9.54554	−1.01183	−0.505913	−	0.862585i	$-0.668844\pi$
$89$	−9.54554	−1.01183	−0.505913	+	0.862585i	$0.668844\pi$
$90$	0	0
$91$	−3.34208	−0.350345
$92$	0	0
$93$	0.00349210	0.000362114 0
$94$	0	0
$95$	−2.98411	−0.306163
$96$	0	0
$97$	8.18810	0.831376	0.415688	−	0.909507i	$-0.363541\pi$
$97$	8.18810	0.831376	0.415688	+	0.909507i	$0.363541\pi$
$98$	0	0
$99$	5.21733	0.524362
$100$	0	0
$101$	−19.4927	−1.93959	−0.969796	−	0.243916i	$-0.921568\pi$
$101$	−19.4927	−1.93959	−0.969796	+	0.243916i	$0.921568\pi$
$102$	0	0
$103$	3.45000	0.339938	0.169969	−	0.985449i	$-0.445633\pi$
$103$	3.45000	0.339938	0.169969	+	0.985449i	$0.445633\pi$
$104$	0	0
$105$	−0.0257343	−0.00251141
$106$	0	0
$107$	−1.18593	−0.114648	−0.0573242	−	0.998356i	$-0.518257\pi$
$107$	−1.18593	−0.114648	−0.0573242	+	0.998356i	$0.518257\pi$
$108$	0	0
$109$	−16.3400	−1.56509	−0.782544	−	0.622595i	$-0.786078\pi$
$109$	−16.3400	−1.56509	−0.782544	+	0.622595i	$0.786078\pi$
$110$	0	0
$111$	−0.147052	−0.0139576
$112$	0	0
$113$	−2.04341	−0.192228	−0.0961138	−	0.995370i	$-0.530641\pi$
$113$	−2.04341	−0.192228	−0.0961138	+	0.995370i	$0.530641\pi$
$114$	0	0
$115$	0.452560	0.0422015
$116$	0	0
$117$	−10.0240	−0.926720
$118$	0	0
$119$	−5.47819	−0.502185
$120$	0	0
$121$	−7.97416	−0.724923
$122$	0	0
$123$	−0.147683	−0.0133161
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	10.0223	0.889337	0.444668	−	0.895695i	$-0.353321\pi$
$127$	10.0223	0.889337	0.444668	+	0.895695i	$0.353321\pi$
$128$	0	0
$129$	−0.0257343	−0.00226577
$130$	0	0
$131$	−17.5186	−1.53060	−0.765301	−	0.643672i	$-0.777410\pi$
$131$	−17.5186	−1.53060	−0.765301	+	0.643672i	$0.777410\pi$
$132$	0	0
$133$	2.98411	0.258755
$134$	0	0
$135$	−0.154388	−0.0132876
$136$	0	0
$137$	5.11529	0.437029	0.218514	−	0.975834i	$-0.429879\pi$
$137$	5.11529	0.437029	0.218514	+	0.975834i	$0.429879\pi$
$138$	0	0
$139$	−8.45881	−0.717467	−0.358733	−	0.933440i	$-0.616791\pi$
$139$	−8.45881	−0.717467	−0.358733	+	0.933440i	$0.616791\pi$
$140$	0	0
$141$	0.161978	0.0136410
$142$	0	0
$143$	−5.81352	−0.486151
$144$	0	0
$145$	−4.73565	−0.393274
$146$	0	0
$147$	0.0257343	0.00212253
$148$	0	0
$149$	−15.5518	−1.27406	−0.637028	−	0.770841i	$-0.719836\pi$
$149$	−15.5518	−1.27406	−0.637028	+	0.770841i	$0.719836\pi$
$150$	0	0
$151$	7.47513	0.608317	0.304159	−	0.952621i	$-0.401625\pi$
$151$	7.47513	0.608317	0.304159	+	0.952621i	$0.401625\pi$
$152$	0	0
$153$	−16.4309	−1.32836
$154$	0	0
$155$	0.135699	0.0108996
$156$	0	0
$157$	9.93035	0.792529	0.396264	−	0.918137i	$-0.370306\pi$
$157$	9.93035	0.792529	0.396264	+	0.918137i	$0.370306\pi$
$158$	0	0
$159$	−0.0693604	−0.00550064
$160$	0	0
$161$	−0.452560	−0.0356668
$162$	0	0
$163$	−14.9709	−1.17261	−0.586307	−	0.810089i	$-0.699419\pi$
$163$	−14.9709	−1.17261	−0.586307	+	0.810089i	$0.699419\pi$
$164$	0	0
$165$	−0.0447646	−0.00348492
$166$	0	0
$167$	−18.7209	−1.44867	−0.724334	−	0.689449i	$-0.757853\pi$
$167$	−18.7209	−1.44867	−0.724334	+	0.689449i	$0.757853\pi$
$168$	0	0
$169$	−1.83053	−0.140810
$170$	0	0
$171$	8.95034	0.684450
$172$	0	0
$173$	6.41087	0.487410	0.243705	−	0.969849i	$-0.421637\pi$
$173$	6.41087	0.487410	0.243705	+	0.969849i	$0.421637\pi$
$174$	0	0
$175$	−1.00000	−0.0755929
$176$	0	0
$177$	−0.0328198	−0.00246689
$178$	0	0
$179$	23.7088	1.77208	0.886039	−	0.463611i	$-0.153447\pi$
$179$	23.7088	1.77208	0.886039	+	0.463611i	$0.153447\pi$
$180$	0	0
$181$	−3.60927	−0.268275	−0.134137	−	0.990963i	$-0.542826\pi$
$181$	−3.60927	−0.268275	−0.134137	+	0.990963i	$0.542826\pi$
$182$	0	0
$183$	−0.291926	−0.0215798
$184$	0	0
$185$	−5.71426	−0.420121
$186$	0	0
$187$	−9.52929	−0.696851
$188$	0	0
$189$	0.154388	0.0112301
$190$	0	0
$191$	4.35247	0.314934	0.157467	−	0.987524i	$-0.449667\pi$
$191$	4.35247	0.314934	0.157467	+	0.987524i	$0.449667\pi$
$192$	0	0
$193$	−3.16192	−0.227600	−0.113800	−	0.993504i	$-0.536302\pi$
$193$	−3.16192	−0.227600	−0.113800	+	0.993504i	$0.536302\pi$
$194$	0	0
$195$	0.0860058	0.00615900
$196$	0	0
$197$	−17.5811	−1.25260	−0.626302	−	0.779581i	$-0.715432\pi$
$197$	−17.5811	−1.25260	−0.626302	+	0.779581i	$0.715432\pi$
$198$	0	0
$199$	−24.5233	−1.73841	−0.869204	−	0.494453i	$-0.835368\pi$
$199$	−24.5233	−1.73841	−0.869204	+	0.494453i	$0.835368\pi$
$200$	0	0
$201$	0.286785	0.0202283
$202$	0	0
$203$	4.73565	0.332377
$204$	0	0
$205$	−5.73877	−0.400813
$206$	0	0
$207$	−1.35738	−0.0943445
$208$	0	0
$209$	5.19084	0.359058
$210$	0	0
$211$	−20.4326	−1.40664	−0.703320	−	0.710874i	$-0.748300\pi$
$211$	−20.4326	−1.40664	−0.703320	+	0.710874i	$0.748300\pi$
$212$	0	0
$213$	0.295318	0.0202348
$214$	0	0
$215$	−1.00000	−0.0681994
$216$	0	0
$217$	−0.135699	−0.00921183
$218$	0	0
$219$	−0.226227	−0.0152870
$220$	0	0
$221$	18.3085	1.23156
$222$	0	0
$223$	−13.2175	−0.885109	−0.442555	−	0.896742i	$-0.645928\pi$
$223$	−13.2175	−0.885109	−0.442555	+	0.896742i	$0.645928\pi$
$224$	0	0
$225$	−2.99934	−0.199956
$226$	0	0
$227$	26.5963	1.76526	0.882628	−	0.470073i	$-0.155772\pi$
$227$	26.5963	1.76526	0.882628	+	0.470073i	$0.155772\pi$
$228$	0	0
$229$	8.52867	0.563590	0.281795	−	0.959475i	$-0.409070\pi$
$229$	8.52867	0.563590	0.281795	+	0.959475i	$0.409070\pi$
$230$	0	0
$231$	0.0447646	0.00294530
$232$	0	0
$233$	−16.3482	−1.07100	−0.535501	−	0.844534i	$-0.679877\pi$
$233$	−16.3482	−1.07100	−0.535501	+	0.844534i	$0.679877\pi$
$234$	0	0
$235$	6.29425	0.410592
$236$	0	0
$237$	0.246011	0.0159801
$238$	0	0
$239$	−26.1847	−1.69375	−0.846873	−	0.531796i	$-0.821517\pi$
$239$	−26.1847	−1.69375	−0.846873	+	0.531796i	$0.821517\pi$
$240$	0	0
$241$	−15.3980	−0.991872	−0.495936	−	0.868359i	$-0.665175\pi$
$241$	−15.3980	−0.991872	−0.495936	+	0.868359i	$0.665175\pi$
$242$	0	0
$243$	0.694620	0.0445599
$244$	0	0
$245$	1.00000	0.0638877
$246$	0	0
$247$	−9.97310	−0.634573
$248$	0	0
$249$	−0.160153	−0.0101493
$250$	0	0
$251$	0.551500	0.0348104	0.0174052	−	0.999849i	$-0.494459\pi$
$251$	0.551500	0.0348104	0.0174052	+	0.999849i	$0.494459\pi$
$252$	0	0
$253$	−0.787226	−0.0494925
$254$	0	0
$255$	0.140977	0.00882833
$256$	0	0
$257$	−15.6231	−0.974540	−0.487270	−	0.873251i	$-0.662007\pi$
$257$	−15.6231	−0.974540	−0.487270	+	0.873251i	$0.662007\pi$
$258$	0	0
$259$	5.71426	0.355067
$260$	0	0
$261$	14.2038	0.879194
$262$	0	0
$263$	−6.23189	−0.384275	−0.192138	−	0.981368i	$-0.561542\pi$
$263$	−6.23189	−0.384275	−0.192138	+	0.981368i	$0.561542\pi$
$264$	0	0
$265$	−2.69526	−0.165568
$266$	0	0
$267$	−0.245647	−0.0150334
$268$	0	0
$269$	30.7581	1.87535	0.937676	−	0.347510i	$-0.112973\pi$
$269$	30.7581	1.87535	0.937676	+	0.347510i	$0.112973\pi$
$270$	0	0
$271$	7.71069	0.468391	0.234195	−	0.972190i	$-0.424754\pi$
$271$	7.71069	0.468391	0.234195	+	0.972190i	$0.424754\pi$
$272$	0	0
$273$	−0.0860058	−0.00520531
$274$	0	0
$275$	−1.73949	−0.104895
$276$	0	0
$277$	−24.1355	−1.45016	−0.725080	−	0.688665i	$-0.758197\pi$
$277$	−24.1355	−1.45016	−0.725080	+	0.688665i	$0.758197\pi$
$278$	0	0
$279$	−0.407006	−0.0243668
$280$	0	0
$281$	24.1945	1.44333	0.721663	−	0.692245i	$-0.243378\pi$
$281$	24.1945	1.44333	0.721663	+	0.692245i	$0.243378\pi$
$282$	0	0
$283$	3.86792	0.229924	0.114962	−	0.993370i	$-0.463325\pi$
$283$	3.86792	0.229924	0.114962	+	0.993370i	$0.463325\pi$
$284$	0	0
$285$	−0.0767937	−0.00454887
$286$	0	0
$287$	5.73877	0.338749
$288$	0	0
$289$	13.0106	0.765329
$290$	0	0
$291$	0.210715	0.0123523
$292$	0	0
$293$	−11.1325	−0.650366	−0.325183	−	0.945651i	$-0.605426\pi$
$293$	−11.1325	−0.650366	−0.325183	+	0.945651i	$0.605426\pi$
$294$	0	0
$295$	−1.27534	−0.0742529
$296$	0	0
$297$	0.268558	0.0155833
$298$	0	0
$299$	1.51249	0.0874696
$300$	0	0
$301$	1.00000	0.0576390
$302$	0	0
$303$	−0.501629	−0.0288178
$304$	0	0
$305$	−11.3439	−0.649547
$306$	0	0
$307$	4.41944	0.252231	0.126115	−	0.992016i	$-0.459749\pi$
$307$	4.41944	0.252231	0.126115	+	0.992016i	$0.459749\pi$
$308$	0	0
$309$	0.0887831	0.00505069
$310$	0	0
$311$	13.3613	0.757652	0.378826	−	0.925468i	$-0.376328\pi$
$311$	13.3613	0.757652	0.378826	+	0.925468i	$0.376328\pi$
$312$	0	0
$313$	5.74354	0.324644	0.162322	−	0.986738i	$-0.448102\pi$
$313$	5.74354	0.324644	0.162322	+	0.986738i	$0.448102\pi$
$314$	0	0
$315$	2.99934	0.168994
$316$	0	0
$317$	18.8387	1.05809	0.529045	−	0.848594i	$-0.322550\pi$
$317$	18.8387	1.05809	0.529045	+	0.848594i	$0.322550\pi$
$318$	0	0
$319$	8.23764	0.461219
$320$	0	0
$321$	−0.0305191	−0.00170341
$322$	0	0
$323$	−16.3475	−0.909599
$324$	0	0
$325$	3.34208	0.185385
$326$	0	0
$327$	−0.420498	−0.0232536
$328$	0	0
$329$	−6.29425	−0.347013
$330$	0	0
$331$	−19.6920	−1.08237	−0.541186	−	0.840903i	$-0.682025\pi$
$331$	−19.6920	−1.08237	−0.541186	+	0.840903i	$0.682025\pi$
$332$	0	0
$333$	17.1390	0.939211
$334$	0	0
$335$	11.1441	0.608868
$336$	0	0
$337$	20.0821	1.09394	0.546972	−	0.837151i	$-0.315781\pi$
$337$	20.0821	1.09394	0.546972	+	0.837151i	$0.315781\pi$
$338$	0	0
$339$	−0.0525856	−0.00285606
$340$	0	0
$341$	−0.236047	−0.0127827
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	0.0116463	0.000627016 0
$346$	0	0
$347$	−25.5581	−1.37203	−0.686015	−	0.727587i	$-0.740642\pi$
$347$	−25.5581	−1.37203	−0.686015	+	0.727587i	$0.740642\pi$
$348$	0	0
$349$	−19.4472	−1.04098	−0.520492	−	0.853867i	$-0.674251\pi$
$349$	−19.4472	−1.04098	−0.520492	+	0.853867i	$0.674251\pi$
$350$	0	0
$351$	−0.515978	−0.0275409
$352$	0	0
$353$	13.1168	0.698139	0.349070	−	0.937097i	$-0.386498\pi$
$353$	13.1168	0.698139	0.349070	+	0.937097i	$0.386498\pi$
$354$	0	0
$355$	11.4757	0.609065
$356$	0	0
$357$	−0.140977	−0.00746130
$358$	0	0
$359$	0.984995	0.0519861	0.0259930	−	0.999662i	$-0.491725\pi$
$359$	0.984995	0.0519861	0.0259930	+	0.999662i	$0.491725\pi$
$360$	0	0
$361$	−10.0951	−0.531322
$362$	0	0
$363$	−0.205209	−0.0107707
$364$	0	0
$365$	−8.79090	−0.460137
$366$	0	0
$367$	−12.9276	−0.674817	−0.337408	−	0.941358i	$-0.609550\pi$
$367$	−12.9276	−0.674817	−0.337408	+	0.941358i	$0.609550\pi$
$368$	0	0
$369$	17.2125	0.896047
$370$	0	0
$371$	2.69526	0.139931
$372$	0	0
$373$	−25.5533	−1.32310	−0.661551	−	0.749900i	$-0.730101\pi$
$373$	−25.5533	−1.32310	−0.661551	+	0.749900i	$0.730101\pi$
$374$	0	0
$375$	0.0257343	0.00132891
$376$	0	0
$377$	−15.8269	−0.815127
$378$	0	0
$379$	30.4134	1.56223	0.781115	−	0.624387i	$-0.214651\pi$
$379$	30.4134	1.56223	0.781115	+	0.624387i	$0.214651\pi$
$380$	0	0
$381$	0.257917	0.0132135
$382$	0	0
$383$	−5.50718	−0.281404	−0.140702	−	0.990052i	$-0.544936\pi$
$383$	−5.50718	−0.281404	−0.140702	+	0.990052i	$0.544936\pi$
$384$	0	0
$385$	1.73949	0.0886529
$386$	0	0
$387$	2.99934	0.152465
$388$	0	0
$389$	−26.5669	−1.34700	−0.673498	−	0.739189i	$-0.735209\pi$
$389$	−26.5669	−1.34700	−0.673498	+	0.739189i	$0.735209\pi$
$390$	0	0
$391$	2.47921	0.125379
$392$	0	0
$393$	−0.450827	−0.0227412
$394$	0	0
$395$	9.55968	0.481000
$396$	0	0
$397$	27.2566	1.36797	0.683985	−	0.729496i	$-0.260245\pi$
$397$	27.2566	1.36797	0.683985	+	0.729496i	$0.260245\pi$
$398$	0	0
$399$	0.0767937	0.00384449
$400$	0	0
$401$	−22.2340	−1.11031	−0.555157	−	0.831745i	$-0.687342\pi$
$401$	−22.2340	−1.11031	−0.555157	+	0.831745i	$0.687342\pi$
$402$	0	0
$403$	0.453515	0.0225912
$404$	0	0
$405$	8.99404	0.446917
$406$	0	0
$407$	9.93992	0.492704
$408$	0	0
$409$	13.9041	0.687512	0.343756	−	0.939059i	$-0.388301\pi$
$409$	13.9041	0.687512	0.343756	+	0.939059i	$0.388301\pi$
$410$	0	0
$411$	0.131638	0.00649323
$412$	0	0
$413$	1.27534	0.0627551
$414$	0	0
$415$	−6.22335	−0.305492
$416$	0	0
$417$	−0.217681	−0.0106599
$418$	0	0
$419$	−31.4316	−1.53554	−0.767768	−	0.640729i	$-0.778632\pi$
$419$	−31.4316	−1.53554	−0.767768	+	0.640729i	$0.778632\pi$
$420$	0	0
$421$	−20.2755	−0.988167	−0.494084	−	0.869414i	$-0.664496\pi$
$421$	−20.2755	−0.988167	−0.494084	+	0.869414i	$0.664496\pi$
$422$	0	0
$423$	−18.8786	−0.917908
$424$	0	0
$425$	5.47819	0.265731
$426$	0	0
$427$	11.3439	0.548968
$428$	0	0
$429$	−0.149607	−0.00722308
$430$	0	0
$431$	37.4653	1.80464	0.902320	−	0.431067i	$-0.141863\pi$
$431$	37.4653	1.80464	0.902320	+	0.431067i	$0.141863\pi$
$432$	0	0
$433$	35.0148	1.68270	0.841351	−	0.540489i	$-0.181761\pi$
$433$	35.0148	1.68270	0.841351	+	0.540489i	$0.181761\pi$
$434$	0	0
$435$	−0.121868	−0.00584314
$436$	0	0
$437$	−1.35049	−0.0646026
$438$	0	0
$439$	25.1251	1.19916	0.599578	−	0.800317i	$-0.295335\pi$
$439$	25.1251	1.19916	0.599578	+	0.800317i	$0.295335\pi$
$440$	0	0
$441$	−2.99934	−0.142826
$442$	0	0
$443$	−9.20288	−0.437242	−0.218621	−	0.975810i	$-0.570156\pi$
$443$	−9.20288	−0.437242	−0.218621	+	0.975810i	$0.570156\pi$
$444$	0	0
$445$	−9.54554	−0.452502
$446$	0	0
$447$	−0.400215	−0.0189295
$448$	0	0
$449$	35.1030	1.65661	0.828306	−	0.560276i	$-0.189305\pi$
$449$	35.1030	1.65661	0.828306	+	0.560276i	$0.189305\pi$
$450$	0	0
$451$	9.98256	0.470060
$452$	0	0
$453$	0.192367	0.00903818
$454$	0	0
$455$	−3.34208	−0.156679
$456$	0	0
$457$	−23.1024	−1.08069	−0.540343	−	0.841445i	$-0.681706\pi$
$457$	−23.1024	−1.08069	−0.540343	+	0.841445i	$0.681706\pi$
$458$	0	0
$459$	−0.845770	−0.0394772
$460$	0	0
$461$	−25.8902	−1.20583	−0.602914	−	0.797806i	$-0.705994\pi$
$461$	−25.8902	−1.20583	−0.602914	+	0.797806i	$0.705994\pi$
$462$	0	0
$463$	−37.1088	−1.72459	−0.862296	−	0.506404i	$-0.830974\pi$
$463$	−37.1088	−1.72459	−0.862296	+	0.506404i	$0.830974\pi$
$464$	0	0
$465$	0.00349210	0.000161942 0
$466$	0	0
$467$	−22.2204	−1.02824	−0.514118	−	0.857719i	$-0.671881\pi$
$467$	−22.2204	−1.02824	−0.514118	+	0.857719i	$0.671881\pi$
$468$	0	0
$469$	−11.1441	−0.514587
$470$	0	0
$471$	0.255550	0.0117751
$472$	0	0
$473$	1.73949	0.0799821
$474$	0	0
$475$	−2.98411	−0.136920
$476$	0	0
$477$	8.08399	0.370140
$478$	0	0
$479$	−5.20350	−0.237754	−0.118877	−	0.992909i	$-0.537929\pi$
$479$	−5.20350	−0.237754	−0.118877	+	0.992909i	$0.537929\pi$
$480$	0	0
$481$	−19.0975	−0.870770
$482$	0	0
$483$	−0.0116463	−0.000529925 0
$484$	0	0
$485$	8.18810	0.371803
$486$	0	0
$487$	13.7428	0.622745	0.311373	−	0.950288i	$-0.399211\pi$
$487$	13.7428	0.622745	0.311373	+	0.950288i	$0.399211\pi$
$488$	0	0
$489$	−0.385266	−0.0174223
$490$	0	0
$491$	41.7224	1.88291	0.941454	−	0.337142i	$-0.109460\pi$
$491$	41.7224	1.88291	0.941454	+	0.337142i	$0.109460\pi$
$492$	0	0
$493$	−25.9428	−1.16840
$494$	0	0
$495$	5.21733	0.234502
$496$	0	0
$497$	−11.4757	−0.514754
$498$	0	0
$499$	32.8169	1.46909	0.734544	−	0.678562i	$-0.237396\pi$
$499$	32.8169	1.46909	0.734544	+	0.678562i	$0.237396\pi$
$500$	0	0
$501$	−0.481769	−0.0215239
$502$	0	0
$503$	−42.2547	−1.88405	−0.942023	−	0.335550i	$-0.891078\pi$
$503$	−42.2547	−1.88405	−0.942023	+	0.335550i	$0.891078\pi$
$504$	0	0
$505$	−19.4927	−0.867412
$506$	0	0
$507$	−0.0471074	−0.00209211
$508$	0	0
$509$	23.6248	1.04715	0.523576	−	0.851979i	$-0.324598\pi$
$509$	23.6248	1.04715	0.523576	+	0.851979i	$0.324598\pi$
$510$	0	0
$511$	8.79090	0.388887
$512$	0	0
$513$	0.460711	0.0203409
$514$	0	0
$515$	3.45000	0.152025
$516$	0	0
$517$	−10.9488	−0.481528
$518$	0	0
$519$	0.164979	0.00724178
$520$	0	0
$521$	37.7042	1.65185	0.825925	−	0.563780i	$-0.190654\pi$
$521$	37.7042	1.65185	0.825925	+	0.563780i	$0.190654\pi$
$522$	0	0
$523$	−4.04231	−0.176758	−0.0883789	−	0.996087i	$-0.528169\pi$
$523$	−4.04231	−0.176758	−0.0883789	+	0.996087i	$0.528169\pi$
$524$	0	0
$525$	−0.0257343	−0.00112314
$526$	0	0
$527$	0.743383	0.0323823
$528$	0	0
$529$	−22.7952	−0.991095
$530$	0	0
$531$	3.82516	0.165998
$532$	0	0
$533$	−19.1794	−0.830752
$534$	0	0
$535$	−1.18593	−0.0512723
$536$	0	0
$537$	0.610128	0.0263290
$538$	0	0
$539$	−1.73949	−0.0749254
$540$	0	0
$541$	5.30920	0.228260	0.114130	−	0.993466i	$-0.463592\pi$
$541$	5.30920	0.228260	0.114130	+	0.993466i	$0.463592\pi$
$542$	0	0
$543$	−0.0928818	−0.00398594
$544$	0	0
$545$	−16.3400	−0.699929
$546$	0	0
$547$	32.9007	1.40673	0.703366	−	0.710828i	$-0.251679\pi$
$547$	32.9007	1.40673	0.703366	+	0.710828i	$0.251679\pi$
$548$	0	0
$549$	34.0240	1.45211
$550$	0	0
$551$	14.1317	0.602030
$552$	0	0
$553$	−9.55968	−0.406519
$554$	0	0
$555$	−0.147052	−0.00624202
$556$	0	0
$557$	−18.3145	−0.776010	−0.388005	−	0.921657i	$-0.626836\pi$
$557$	−18.3145	−0.776010	−0.388005	+	0.921657i	$0.626836\pi$
$558$	0	0
$559$	−3.34208	−0.141355
$560$	0	0
$561$	−0.245229	−0.0103536
$562$	0	0
$563$	−22.9753	−0.968294	−0.484147	−	0.874987i	$-0.660870\pi$
$563$	−22.9753	−0.968294	−0.484147	+	0.874987i	$0.660870\pi$
$564$	0	0
$565$	−2.04341	−0.0859668
$566$	0	0
$567$	−8.99404	−0.377714
$568$	0	0
$569$	−20.6203	−0.864448	−0.432224	−	0.901766i	$-0.642271\pi$
$569$	−20.6203	−0.864448	−0.432224	+	0.901766i	$0.642271\pi$
$570$	0	0
$571$	40.2601	1.68483	0.842417	−	0.538827i	$-0.181132\pi$
$571$	40.2601	1.68483	0.842417	+	0.538827i	$0.181132\pi$
$572$	0	0
$573$	0.112008	0.00467919
$574$	0	0
$575$	0.452560	0.0188731
$576$	0	0
$577$	−15.9970	−0.665962	−0.332981	−	0.942933i	$-0.608055\pi$
$577$	−15.9970	−0.665962	−0.332981	+	0.942933i	$0.608055\pi$
$578$	0	0
$579$	−0.0813696	−0.00338161
$580$	0	0
$581$	6.22335	0.258188
$582$	0	0
$583$	4.68839	0.194173
$584$	0	0
$585$	−10.0240	−0.414442
$586$	0	0
$587$	−30.6731	−1.26602	−0.633008	−	0.774145i	$-0.718180\pi$
$587$	−30.6731	−1.26602	−0.633008	+	0.774145i	$0.718180\pi$
$588$	0	0
$589$	−0.404939	−0.0166852
$590$	0	0
$591$	−0.452437	−0.0186108
$592$	0	0
$593$	−16.0582	−0.659433	−0.329716	−	0.944080i	$-0.606953\pi$
$593$	−16.0582	−0.659433	−0.329716	+	0.944080i	$0.606953\pi$
$594$	0	0
$595$	−5.47819	−0.224584
$596$	0	0
$597$	−0.631088	−0.0258287
$598$	0	0
$599$	−19.2177	−0.785214	−0.392607	−	0.919706i	$-0.628427\pi$
$599$	−19.2177	−0.785214	−0.392607	+	0.919706i	$0.628427\pi$
$600$	0	0
$601$	−8.88973	−0.362620	−0.181310	−	0.983426i	$-0.558034\pi$
$601$	−8.88973	−0.362620	−0.181310	+	0.983426i	$0.558034\pi$
$602$	0	0
$603$	−33.4250	−1.36117
$604$	0	0
$605$	−7.97416	−0.324196
$606$	0	0
$607$	−8.54919	−0.347001	−0.173500	−	0.984834i	$-0.555508\pi$
$607$	−8.54919	−0.347001	−0.173500	+	0.984834i	$0.555508\pi$
$608$	0	0
$609$	0.121868	0.00493836
$610$	0	0
$611$	21.0359	0.851020
$612$	0	0
$613$	−35.4241	−1.43077	−0.715383	−	0.698732i	$-0.753748\pi$
$613$	−35.4241	−1.43077	−0.715383	+	0.698732i	$0.753748\pi$
$614$	0	0
$615$	−0.147683	−0.00595515
$616$	0	0
$617$	−26.3012	−1.05885	−0.529423	−	0.848358i	$-0.677591\pi$
$617$	−26.3012	−1.05885	−0.529423	+	0.848358i	$0.677591\pi$
$618$	0	0
$619$	−36.5460	−1.46891	−0.734453	−	0.678659i	$-0.762561\pi$
$619$	−36.5460	−1.46891	−0.734453	+	0.678659i	$0.762561\pi$
$620$	0	0
$621$	−0.0698701	−0.00280379
$622$	0	0
$623$	9.54554	0.382434
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	0.133582	0.00533476
$628$	0	0
$629$	−31.3038	−1.24816
$630$	0	0
$631$	−12.9343	−0.514907	−0.257453	−	0.966291i	$-0.582883\pi$
$631$	−12.9343	−0.514907	−0.257453	+	0.966291i	$0.582883\pi$
$632$	0	0
$633$	−0.525818	−0.0208994
$634$	0	0
$635$	10.0223	0.397724
$636$	0	0
$637$	3.34208	0.132418
$638$	0	0
$639$	−34.4194	−1.36161
$640$	0	0
$641$	−12.9861	−0.512922	−0.256461	−	0.966555i	$-0.582557\pi$
$641$	−12.9861	−0.512922	−0.256461	+	0.966555i	$0.582557\pi$
$642$	0	0
$643$	−27.3135	−1.07714	−0.538569	−	0.842581i	$-0.681035\pi$
$643$	−27.3135	−1.07714	−0.538569	+	0.842581i	$0.681035\pi$
$644$	0	0
$645$	−0.0257343	−0.00101329
$646$	0	0
$647$	31.5593	1.24072	0.620362	−	0.784315i	$-0.286986\pi$
$647$	31.5593	1.24072	0.620362	+	0.784315i	$0.286986\pi$
$648$	0	0
$649$	2.21844	0.0870814
$650$	0	0
$651$	−0.00349210	−0.000136866 0
$652$	0	0
$653$	44.7558	1.75143	0.875715	−	0.482829i	$-0.160391\pi$
$653$	44.7558	1.75143	0.875715	+	0.482829i	$0.160391\pi$
$654$	0	0
$655$	−17.5186	−0.684506
$656$	0	0
$657$	26.3669	1.02867
$658$	0	0
$659$	11.6949	0.455567	0.227784	−	0.973712i	$-0.426852\pi$
$659$	11.6949	0.455567	0.227784	+	0.973712i	$0.426852\pi$
$660$	0	0
$661$	25.5253	0.992821	0.496410	−	0.868088i	$-0.334651\pi$
$661$	25.5253	0.992821	0.496410	+	0.868088i	$0.334651\pi$
$662$	0	0
$663$	0.471156	0.0182982
$664$	0	0
$665$	2.98411	0.115719
$666$	0	0
$667$	−2.14317	−0.0829838
$668$	0	0
$669$	−0.340143	−0.0131507
$670$	0	0
$671$	19.7326	0.761768
$672$	0	0
$673$	−24.6593	−0.950545	−0.475273	−	0.879839i	$-0.657651\pi$
$673$	−24.6593	−0.950545	−0.475273	+	0.879839i	$0.657651\pi$
$674$	0	0
$675$	−0.154388	−0.00594242
$676$	0	0
$677$	4.91464	0.188885	0.0944426	−	0.995530i	$-0.469893\pi$
$677$	4.91464	0.188885	0.0944426	+	0.995530i	$0.469893\pi$
$678$	0	0
$679$	−8.18810	−0.314231
$680$	0	0
$681$	0.684435	0.0262276
$682$	0	0
$683$	38.9191	1.48920	0.744599	−	0.667512i	$-0.232641\pi$
$683$	38.9191	1.48920	0.744599	+	0.667512i	$0.232641\pi$
$684$	0	0
$685$	5.11529	0.195445
$686$	0	0
$687$	0.219479	0.00837364
$688$	0	0
$689$	−9.00775	−0.343168
$690$	0	0
$691$	14.3279	0.545058	0.272529	−	0.962148i	$-0.412140\pi$
$691$	14.3279	0.545058	0.272529	+	0.962148i	$0.412140\pi$
$692$	0	0
$693$	−5.21733	−0.198190
$694$	0	0
$695$	−8.45881	−0.320861
$696$	0	0
$697$	−31.4381	−1.19080
$698$	0	0
$699$	−0.420708	−0.0159126
$700$	0	0
$701$	0.983020	0.0371282	0.0185641	−	0.999828i	$-0.494091\pi$
$701$	0.983020	0.0371282	0.0185641	+	0.999828i	$0.494091\pi$
$702$	0	0
$703$	17.0519	0.643126
$704$	0	0
$705$	0.161978	0.00610044
$706$	0	0
$707$	19.4927	0.733097
$708$	0	0
$709$	−14.9364	−0.560949	−0.280475	−	0.959861i	$-0.590492\pi$
$709$	−14.9364	−0.560949	−0.280475	+	0.959861i	$0.590492\pi$
$710$	0	0
$711$	−28.6727	−1.07531
$712$	0	0
$713$	0.0614118	0.00229989
$714$	0	0
$715$	−5.81352	−0.217413
$716$	0	0
$717$	−0.673843	−0.0251651
$718$	0	0
$719$	18.0025	0.671378	0.335689	−	0.941973i	$-0.391031\pi$
$719$	18.0025	0.671378	0.335689	+	0.941973i	$0.391031\pi$
$720$	0	0
$721$	−3.45000	−0.128485
$722$	0	0
$723$	−0.396256	−0.0147369
$724$	0	0
$725$	−4.73565	−0.175878
$726$	0	0
$727$	30.0504	1.11451	0.557254	−	0.830342i	$-0.311855\pi$
$727$	30.0504	1.11451	0.557254	+	0.830342i	$0.311855\pi$
$728$	0	0
$729$	−26.9642	−0.998676
$730$	0	0
$731$	−5.47819	−0.202618
$732$	0	0
$733$	0.120442	0.00444861	0.00222431	−	0.999998i	$-0.499292\pi$
$733$	0.120442	0.00444861	0.00222431	+	0.999998i	$0.499292\pi$
$734$	0	0
$735$	0.0257343	0.000949222 0
$736$	0	0
$737$	−19.3851	−0.714060
$738$	0	0
$739$	3.50333	0.128872	0.0644361	−	0.997922i	$-0.479475\pi$
$739$	3.50333	0.128872	0.0644361	+	0.997922i	$0.479475\pi$
$740$	0	0
$741$	−0.256650	−0.00942829
$742$	0	0
$743$	14.5135	0.532449	0.266224	−	0.963911i	$-0.414224\pi$
$743$	14.5135	0.532449	0.266224	+	0.963911i	$0.414224\pi$
$744$	0	0
$745$	−15.5518	−0.569775
$746$	0	0
$747$	18.6659	0.682951
$748$	0	0
$749$	1.18593	0.0433330
$750$	0	0
$751$	−13.0580	−0.476493	−0.238247	−	0.971205i	$-0.576573\pi$
$751$	−13.0580	−0.476493	−0.238247	+	0.971205i	$0.576573\pi$
$752$	0	0
$753$	0.0141924	0.000517201 0
$754$	0	0
$755$	7.47513	0.272048
$756$	0	0
$757$	36.6173	1.33088	0.665439	−	0.746452i	$-0.268244\pi$
$757$	36.6173	1.33088	0.665439	+	0.746452i	$0.268244\pi$
$758$	0	0
$759$	−0.0202587	−0.000735344 0
$760$	0	0
$761$	35.2665	1.27841	0.639206	−	0.769036i	$-0.279263\pi$
$761$	35.2665	1.27841	0.639206	+	0.769036i	$0.279263\pi$
$762$	0	0
$763$	16.3400	0.591548
$764$	0	0
$765$	−16.4309	−0.594062
$766$	0	0
$767$	−4.26227	−0.153902
$768$	0	0
$769$	4.94774	0.178420	0.0892100	−	0.996013i	$-0.471566\pi$
$769$	4.94774	0.178420	0.0892100	+	0.996013i	$0.471566\pi$
$770$	0	0
$771$	−0.402048	−0.0144794
$772$	0	0
$773$	8.43010	0.303210	0.151605	−	0.988441i	$-0.451556\pi$
$773$	8.43010	0.303210	0.151605	+	0.988441i	$0.451556\pi$
$774$	0	0
$775$	0.135699	0.00487444
$776$	0	0
$777$	0.147052	0.00527547
$778$	0	0
$779$	17.1251	0.613570
$780$	0	0
$781$	−19.9619	−0.714291
$782$	0	0
$783$	0.731130	0.0261284
$784$	0	0
$785$	9.93035	0.354430
$786$	0	0
$787$	45.2313	1.61232	0.806161	−	0.591696i	$-0.201542\pi$
$787$	45.2313	1.61232	0.806161	+	0.591696i	$0.201542\pi$
$788$	0	0
$789$	−0.160373	−0.00570944
$790$	0	0
$791$	2.04341	0.0726552
$792$	0	0
$793$	−37.9120	−1.34629
$794$	0	0
$795$	−0.0693604	−0.00245996
$796$	0	0
$797$	−43.5599	−1.54297	−0.771486	−	0.636246i	$-0.780486\pi$
$797$	−43.5599	−1.54297	−0.771486	+	0.636246i	$0.780486\pi$
$798$	0	0
$799$	34.4811	1.21985
$800$	0	0
$801$	28.6303	1.01160
$802$	0	0
$803$	15.2917	0.539633
$804$	0	0
$805$	−0.452560	−0.0159507
$806$	0	0
$807$	0.791536	0.0278634
$808$	0	0
$809$	52.8484	1.85805	0.929026	−	0.370015i	$-0.120648\pi$
$809$	52.8484	1.85805	0.929026	+	0.370015i	$0.120648\pi$
$810$	0	0
$811$	40.1294	1.40913	0.704566	−	0.709638i	$-0.251142\pi$
$811$	40.1294	1.40913	0.704566	+	0.709638i	$0.251142\pi$
$812$	0	0
$813$	0.198429	0.00695920
$814$	0	0
$815$	−14.9709	−0.524409
$816$	0	0
$817$	2.98411	0.104401
$818$	0	0
$819$	10.0240	0.350267
$820$	0	0
$821$	−31.4956	−1.09921	−0.549603	−	0.835426i	$-0.685221\pi$
$821$	−31.4956	−1.09921	−0.549603	+	0.835426i	$0.685221\pi$
$822$	0	0
$823$	9.53195	0.332263	0.166131	−	0.986104i	$-0.446872\pi$
$823$	9.53195	0.332263	0.166131	+	0.986104i	$0.446872\pi$
$824$	0	0
$825$	−0.0447646	−0.00155850
$826$	0	0
$827$	−4.69984	−0.163429	−0.0817146	−	0.996656i	$-0.526040\pi$
$827$	−4.69984	−0.163429	−0.0817146	+	0.996656i	$0.526040\pi$
$828$	0	0
$829$	−30.4362	−1.05709	−0.528546	−	0.848905i	$-0.677262\pi$
$829$	−30.4362	−1.05709	−0.528546	+	0.848905i	$0.677262\pi$
$830$	0	0
$831$	−0.621109	−0.0215460
$832$	0	0
$833$	5.47819	0.189808
$834$	0	0
$835$	−18.7209	−0.647864
$836$	0	0
$837$	−0.0209503	−0.000724149 0
$838$	0	0
$839$	−1.38493	−0.0478130	−0.0239065	−	0.999714i	$-0.507610\pi$
$839$	−1.38493	−0.0478130	−0.0239065	+	0.999714i	$0.507610\pi$
$840$	0	0
$841$	−6.57363	−0.226677
$842$	0	0
$843$	0.622629	0.0214445
$844$	0	0
$845$	−1.83053	−0.0629723
$846$	0	0
$847$	7.97416	0.273995
$848$	0	0
$849$	0.0995380	0.00341613
$850$	0	0
$851$	−2.58605	−0.0886485
$852$	0	0
$853$	45.1642	1.54639	0.773196	−	0.634167i	$-0.218657\pi$
$853$	45.1642	1.54639	0.773196	+	0.634167i	$0.218657\pi$
$854$	0	0
$855$	8.95034	0.306095
$856$	0	0
$857$	−18.3286	−0.626092	−0.313046	−	0.949738i	$-0.601349\pi$
$857$	−18.3286	−0.626092	−0.313046	+	0.949738i	$0.601349\pi$
$858$	0	0
$859$	33.2628	1.13491	0.567457	−	0.823403i	$-0.307927\pi$
$859$	33.2628	1.13491	0.567457	+	0.823403i	$0.307927\pi$
$860$	0	0
$861$	0.147683	0.00503302
$862$	0	0
$863$	13.8038	0.469887	0.234944	−	0.972009i	$-0.424509\pi$
$863$	13.8038	0.469887	0.234944	+	0.972009i	$0.424509\pi$
$864$	0	0
$865$	6.41087	0.217976
$866$	0	0
$867$	0.334818	0.0113710
$868$	0	0
$869$	−16.6290	−0.564101
$870$	0	0
$871$	37.2445	1.26198
$872$	0	0
$873$	−24.5589	−0.831193
$874$	0	0
$875$	−1.00000	−0.0338062
$876$	0	0
$877$	4.69493	0.158537	0.0792683	−	0.996853i	$-0.474742\pi$
$877$	4.69493	0.158537	0.0792683	+	0.996853i	$0.474742\pi$
$878$	0	0
$879$	−0.286486	−0.00966294
$880$	0	0
$881$	1.84626	0.0622019	0.0311010	−	0.999516i	$-0.490099\pi$
$881$	1.84626	0.0622019	0.0311010	+	0.999516i	$0.490099\pi$
$882$	0	0
$883$	−43.1432	−1.45189	−0.725943	−	0.687755i	$-0.758596\pi$
$883$	−43.1432	−1.45189	−0.725943	+	0.687755i	$0.758596\pi$
$884$	0	0
$885$	−0.0328198	−0.00110323
$886$	0	0
$887$	21.3188	0.715814	0.357907	−	0.933757i	$-0.383490\pi$
$887$	21.3188	0.715814	0.357907	+	0.933757i	$0.383490\pi$
$888$	0	0
$889$	−10.0223	−0.336138
$890$	0	0
$891$	−15.6451	−0.524130
$892$	0	0
$893$	−18.7827	−0.628539
$894$	0	0
$895$	23.7088	0.792497
$896$	0	0
$897$	0.0389228	0.00129960
$898$	0	0
$899$	−0.642621	−0.0214326
$900$	0	0
$901$	−14.7651	−0.491898
$902$	0	0
$903$	0.0257343	0.000856382 0
$904$	0	0
$905$	−3.60927	−0.119976
$906$	0	0
$907$	−34.0390	−1.13025	−0.565123	−	0.825006i	$-0.691171\pi$
$907$	−34.0390	−1.13025	−0.565123	+	0.825006i	$0.691171\pi$
$908$	0	0
$909$	58.4651	1.93916
$910$	0	0
$911$	20.4153	0.676390	0.338195	−	0.941076i	$-0.390184\pi$
$911$	20.4153	0.676390	0.338195	+	0.941076i	$0.390184\pi$
$912$	0	0
$913$	10.8255	0.358271
$914$	0	0
$915$	−0.291926	−0.00965076
$916$	0	0
$917$	17.5186	0.578514
$918$	0	0
$919$	−48.5428	−1.60128	−0.800640	−	0.599146i	$-0.795507\pi$
$919$	−48.5428	−1.60128	−0.800640	+	0.599146i	$0.795507\pi$
$920$	0	0
$921$	0.113731	0.00374757
$922$	0	0
$923$	38.3525	1.26239
$924$	0	0
$925$	−5.71426	−0.187884
$926$	0	0
$927$	−10.3477	−0.339863
$928$	0	0
$929$	−2.27015	−0.0744812	−0.0372406	−	0.999306i	$-0.511857\pi$
$929$	−2.27015	−0.0744812	−0.0372406	+	0.999306i	$0.511857\pi$
$930$	0	0
$931$	−2.98411	−0.0978001
$932$	0	0
$933$	0.343844	0.0112570
$934$	0	0
$935$	−9.52929	−0.311641
$936$	0	0
$937$	12.6810	0.414271	0.207136	−	0.978312i	$-0.433586\pi$
$937$	12.6810	0.414271	0.207136	+	0.978312i	$0.433586\pi$
$938$	0	0
$939$	0.147806	0.00482346
$940$	0	0
$941$	15.1521	0.493943	0.246972	−	0.969023i	$-0.420565\pi$
$941$	15.1521	0.493943	0.246972	+	0.969023i	$0.420565\pi$
$942$	0	0
$943$	−2.59714	−0.0845745
$944$	0	0
$945$	0.154388	0.00502226
$946$	0	0
$947$	43.5881	1.41642	0.708211	−	0.706001i	$-0.249503\pi$
$947$	43.5881	1.41642	0.708211	+	0.706001i	$0.249503\pi$
$948$	0	0
$949$	−29.3799	−0.953711
$950$	0	0
$951$	0.484801	0.0157208
$952$	0	0
$953$	38.8864	1.25965	0.629827	−	0.776735i	$-0.283126\pi$
$953$	38.8864	1.25965	0.629827	+	0.776735i	$0.283126\pi$
$954$	0	0
$955$	4.35247	0.140843
$956$	0	0
$957$	0.211989	0.00685265
$958$	0	0
$959$	−5.11529	−0.165181
$960$	0	0
$961$	−30.9816	−0.999406
$962$	0	0
$963$	3.55701	0.114623
$964$	0	0
$965$	−3.16192	−0.101786
$966$	0	0
$967$	−34.6570	−1.11450	−0.557248	−	0.830346i	$-0.688143\pi$
$967$	−34.6570	−1.11450	−0.557248	+	0.830346i	$0.688143\pi$
$968$	0	0
$969$	−0.420691	−0.0135145
$970$	0	0
$971$	36.6259	1.17538	0.587691	−	0.809086i	$-0.300037\pi$
$971$	36.6259	1.17538	0.587691	+	0.809086i	$0.300037\pi$
$972$	0	0
$973$	8.45881	0.271177
$974$	0	0
$975$	0.0860058	0.00275439
$976$	0	0
$977$	56.9934	1.82338	0.911690	−	0.410878i	$-0.134778\pi$
$977$	56.9934	1.82338	0.911690	+	0.410878i	$0.134778\pi$
$978$	0	0
$979$	16.6044	0.530680
$980$	0	0
$981$	49.0092	1.56474
$982$	0	0
$983$	52.7904	1.68375	0.841876	−	0.539672i	$-0.181452\pi$
$983$	52.7904	1.68375	0.841876	+	0.539672i	$0.181452\pi$
$984$	0	0
$985$	−17.5811	−0.560181
$986$	0	0
$987$	−0.161978	−0.00515581
$988$	0	0
$989$	−0.452560	−0.0143906
$990$	0	0
$991$	51.3821	1.63221	0.816103	−	0.577906i	$-0.196130\pi$
$991$	51.3821	1.63221	0.816103	+	0.577906i	$0.196130\pi$
$992$	0	0
$993$	−0.506760	−0.0160816
$994$	0	0
$995$	−24.5233	−0.777440
$996$	0	0
$997$	42.6362	1.35030	0.675150	−	0.737680i	$-0.264079\pi$
$997$	42.6362	1.35030	0.675150	+	0.737680i	$0.264079\pi$
$998$	0	0
$999$	0.882216	0.0279121

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6020.2.a.g.1.5	✓	9

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6020.2.a.g.1.5	✓	9	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:	\( N \)	\(=\)	\( 6020 = 2^{2} \cdot 5 \cdot 7 \cdot 43 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6020.a (trivial)

\(f(q)\)	\(=\)	\(q+0.0257343 q^{3} +1.00000 q^{5} -1.00000 q^{7} -2.99934 q^{9} +O(q^{10})\)	\(q+0.0257343 q^{3} +1.00000 q^{5} -1.00000 q^{7} -2.99934 q^{9} -1.73949 q^{11} +3.34208 q^{13} +0.0257343 q^{15} +5.47819 q^{17} -2.98411 q^{19} -0.0257343 q^{21} +0.452560 q^{23} +1.00000 q^{25} -0.154388 q^{27} -4.73565 q^{29} +0.135699 q^{31} -0.0447646 q^{33} -1.00000 q^{35} -5.71426 q^{37} +0.0860058 q^{39} -5.73877 q^{41} -1.00000 q^{43} -2.99934 q^{45} +6.29425 q^{47} +1.00000 q^{49} +0.140977 q^{51} -2.69526 q^{53} -1.73949 q^{55} -0.0767937 q^{57} -1.27534 q^{59} -11.3439 q^{61} +2.99934 q^{63} +3.34208 q^{65} +11.1441 q^{67} +0.0116463 q^{69} +11.4757 q^{71} -8.79090 q^{73} +0.0257343 q^{75} +1.73949 q^{77} +9.55968 q^{79} +8.99404 q^{81} -6.22335 q^{83} +5.47819 q^{85} -0.121868 q^{87} -9.54554 q^{89} -3.34208 q^{91} +0.00349210 q^{93} -2.98411 q^{95} +8.18810 q^{97} +5.21733 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 9 q + 9 q^{5} - 9 q^{7} + 5 q^{9}+O(q^{10}) \) 9 * q + 9 * q^5 - 9 * q^7 + 5 * q^9	\( 9 q + 9 q^{5} - 9 q^{7} + 5 q^{9} + q^{11} - 14 q^{13} - 11 q^{17} - 2 q^{19} - 6 q^{23} + 9 q^{25} - 6 q^{29} + 6 q^{31} - 14 q^{33} - 9 q^{35} - 20 q^{37} - 6 q^{39} - 6 q^{41} - 9 q^{43} + 5 q^{45} + 9 q^{49} - 8 q^{51} - 31 q^{53} + q^{55} - 16 q^{57} + 2 q^{59} - 13 q^{61} - 5 q^{63} - 14 q^{65} - 10 q^{67} - 18 q^{69} + 12 q^{71} - 32 q^{73} - q^{77} + q^{79} - 27 q^{81} - 10 q^{83} - 11 q^{85} - 5 q^{87} - q^{89} + 14 q^{91} - 49 q^{93} - 2 q^{95} - 28 q^{97} - 14 q^{99}+O(q^{100}) \) 9 * q + 9 * q^5 - 9 * q^7 + 5 * q^9 + q^11 - 14 * q^13 - 11 * q^17 - 2 * q^19 - 6 * q^23 + 9 * q^25 - 6 * q^29 + 6 * q^31 - 14 * q^33 - 9 * q^35 - 20 * q^37 - 6 * q^39 - 6 * q^41 - 9 * q^43 + 5 * q^45 + 9 * q^49 - 8 * q^51 - 31 * q^53 + q^55 - 16 * q^57 + 2 * q^59 - 13 * q^61 - 5 * q^63 - 14 * q^65 - 10 * q^67 - 18 * q^69 + 12 * q^71 - 32 * q^73 - q^77 + q^79 - 27 * q^81 - 10 * q^83 - 11 * q^85 - 5 * q^87 - q^89 + 14 * q^91 - 49 * q^93 - 2 * q^95 - 28 * q^97 - 14 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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