LMFDB - Embedded newform 8020.2.a.c.1.3

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:	$1$
Dimension:	$28$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
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)

$f(q)$	$=$	$q-2.59496 q^{3} -1.00000 q^{5} +1.42420 q^{7} +3.73381 q^{9} +O(q^{10})$	$q-2.59496 q^{3} -1.00000 q^{5} +1.42420 q^{7} +3.73381 q^{9} +3.63795 q^{11} -2.47890 q^{13} +2.59496 q^{15} +4.00900 q^{17} +0.503905 q^{19} -3.69573 q^{21} -8.01771 q^{23} +1.00000 q^{25} -1.90422 q^{27} -1.71938 q^{29} -6.28565 q^{31} -9.44032 q^{33} -1.42420 q^{35} +5.09212 q^{37} +6.43265 q^{39} +1.37887 q^{41} +7.23723 q^{43} -3.73381 q^{45} +7.49079 q^{47} -4.97167 q^{49} -10.4032 q^{51} -7.78413 q^{53} -3.63795 q^{55} -1.30761 q^{57} -5.83373 q^{59} -9.44664 q^{61} +5.31768 q^{63} +2.47890 q^{65} +9.06165 q^{67} +20.8056 q^{69} -9.23923 q^{71} -7.53548 q^{73} -2.59496 q^{75} +5.18115 q^{77} -10.3976 q^{79} -6.26008 q^{81} +6.25543 q^{83} -4.00900 q^{85} +4.46173 q^{87} +2.74123 q^{89} -3.53044 q^{91} +16.3110 q^{93} -0.503905 q^{95} +9.29500 q^{97} +13.5834 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 28 q + 3 q^{3} - 28 q^{5} - 4 q^{7} + 17 q^{9}+O(q^{10}) $ 28 * q + 3 * q^3 - 28 * q^5 - 4 * q^7 + 17 * q^9	$ 28 q + 3 q^{3} - 28 q^{5} - 4 q^{7} + 17 q^{9} + 2 q^{11} + 3 q^{13} - 3 q^{15} - 10 q^{17} - 2 q^{19} - 12 q^{21} - 23 q^{23} + 28 q^{25} + 9 q^{27} - 37 q^{29} - 11 q^{31} + 2 q^{33} + 4 q^{35} - 3 q^{37} - 19 q^{39} - 30 q^{41} + 13 q^{43} - 17 q^{45} - 15 q^{47} + 12 q^{49} - 8 q^{51} - 35 q^{53} - 2 q^{55} - 22 q^{57} - q^{59} - 33 q^{61} - 20 q^{63} - 3 q^{65} + 19 q^{67} - 8 q^{69} - 31 q^{71} + 31 q^{73} + 3 q^{75} - 42 q^{77} - 29 q^{79} - 36 q^{81} + 14 q^{83} + 10 q^{85} - 32 q^{87} - 32 q^{89} - 7 q^{91} - 11 q^{93} + 2 q^{95} + 2 q^{97} - 39 q^{99}+O(q^{100}) $ 28 * q + 3 * q^3 - 28 * q^5 - 4 * q^7 + 17 * q^9 + 2 * q^11 + 3 * q^13 - 3 * q^15 - 10 * q^17 - 2 * q^19 - 12 * q^21 - 23 * q^23 + 28 * q^25 + 9 * q^27 - 37 * q^29 - 11 * q^31 + 2 * q^33 + 4 * q^35 - 3 * q^37 - 19 * q^39 - 30 * q^41 + 13 * q^43 - 17 * q^45 - 15 * q^47 + 12 * q^49 - 8 * q^51 - 35 * q^53 - 2 * q^55 - 22 * q^57 - q^59 - 33 * q^61 - 20 * q^63 - 3 * q^65 + 19 * q^67 - 8 * q^69 - 31 * q^71 + 31 * q^73 + 3 * q^75 - 42 * q^77 - 29 * q^79 - 36 * q^81 + 14 * q^83 + 10 * q^85 - 32 * q^87 - 32 * q^89 - 7 * q^91 - 11 * q^93 + 2 * q^95 + 2 * q^97 - 39 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	−2.59496	−1.49820	−0.749100	−	0.662457i	$-0.769514\pi$
$3$	−2.59496	−1.49820	−0.749100	+	0.662457i	$0.769514\pi$
$4$	0	0
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	1.42420	0.538295	0.269148	−	0.963099i	$-0.413258\pi$
$7$	1.42420	0.538295	0.269148	+	0.963099i	$0.413258\pi$
$8$	0	0
$9$	3.73381	1.24460
$10$	0	0
$11$	3.63795	1.09688	0.548441	−	0.836189i	$-0.315222\pi$
$11$	3.63795	1.09688	0.548441	+	0.836189i	$0.315222\pi$
$12$	0	0
$13$	−2.47890	−0.687523	−0.343762	−	0.939057i	$-0.611701\pi$
$13$	−2.47890	−0.687523	−0.343762	+	0.939057i	$0.611701\pi$
$14$	0	0
$15$	2.59496	0.670016
$16$	0	0
$17$	4.00900	0.972327	0.486163	−	0.873868i	$-0.338396\pi$
$17$	4.00900	0.972327	0.486163	+	0.873868i	$0.338396\pi$
$18$	0	0
$19$	0.503905	0.115604	0.0578018	−	0.998328i	$-0.481591\pi$
$19$	0.503905	0.115604	0.0578018	+	0.998328i	$0.481591\pi$
$20$	0	0
$21$	−3.69573	−0.806474
$22$	0	0
$23$	−8.01771	−1.67181	−0.835904	−	0.548876i	$-0.815056\pi$
$23$	−8.01771	−1.67181	−0.835904	+	0.548876i	$0.815056\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−1.90422	−0.366467
$28$	0	0
$29$	−1.71938	−0.319282	−0.159641	−	0.987175i	$-0.551034\pi$
$29$	−1.71938	−0.319282	−0.159641	+	0.987175i	$0.551034\pi$
$30$	0	0
$31$	−6.28565	−1.12894	−0.564468	−	0.825455i	$-0.690919\pi$
$31$	−6.28565	−1.12894	−0.564468	+	0.825455i	$0.690919\pi$
$32$	0	0
$33$	−9.44032	−1.64335
$34$	0	0
$35$	−1.42420	−0.240733
$36$	0	0
$37$	5.09212	0.837139	0.418569	−	0.908185i	$-0.362532\pi$
$37$	5.09212	0.837139	0.418569	+	0.908185i	$0.362532\pi$
$38$	0	0
$39$	6.43265	1.03005
$40$	0	0
$41$	1.37887	0.215343	0.107672	−	0.994187i	$-0.465661\pi$
$41$	1.37887	0.215343	0.107672	+	0.994187i	$0.465661\pi$
$42$	0	0
$43$	7.23723	1.10367	0.551834	−	0.833954i	$-0.313928\pi$
$43$	7.23723	1.10367	0.551834	+	0.833954i	$0.313928\pi$
$44$	0	0
$45$	−3.73381	−0.556604
$46$	0	0
$47$	7.49079	1.09264	0.546322	−	0.837575i	$-0.316028\pi$
$47$	7.49079	1.09264	0.546322	+	0.837575i	$0.316028\pi$
$48$	0	0
$49$	−4.97167	−0.710238
$50$	0	0
$51$	−10.4032	−1.45674
$52$	0	0
$53$	−7.78413	−1.06923	−0.534617	−	0.845095i	$-0.679544\pi$
$53$	−7.78413	−1.06923	−0.534617	+	0.845095i	$0.679544\pi$
$54$	0	0
$55$	−3.63795	−0.490540
$56$	0	0
$57$	−1.30761	−0.173197
$58$	0	0
$59$	−5.83373	−0.759487	−0.379744	−	0.925092i	$-0.623988\pi$
$59$	−5.83373	−0.759487	−0.379744	+	0.925092i	$0.623988\pi$
$60$	0	0
$61$	−9.44664	−1.20952	−0.604759	−	0.796408i	$-0.706731\pi$
$61$	−9.44664	−1.20952	−0.604759	+	0.796408i	$0.706731\pi$
$62$	0	0
$63$	5.31768	0.669965
$64$	0	0
$65$	2.47890	0.307470
$66$	0	0
$67$	9.06165	1.10706	0.553528	−	0.832830i	$-0.313281\pi$
$67$	9.06165	1.10706	0.553528	+	0.832830i	$0.313281\pi$
$68$	0	0
$69$	20.8056	2.50470
$70$	0	0
$71$	−9.23923	−1.09649	−0.548247	−	0.836316i	$-0.684705\pi$
$71$	−9.23923	−1.09649	−0.548247	+	0.836316i	$0.684705\pi$
$72$	0	0
$73$	−7.53548	−0.881961	−0.440981	−	0.897517i	$-0.645369\pi$
$73$	−7.53548	−0.881961	−0.440981	+	0.897517i	$0.645369\pi$
$74$	0	0
$75$	−2.59496	−0.299640
$76$	0	0
$77$	5.18115	0.590447
$78$	0	0
$79$	−10.3976	−1.16982	−0.584912	−	0.811096i	$-0.698871\pi$
$79$	−10.3976	−1.16982	−0.584912	+	0.811096i	$0.698871\pi$
$80$	0	0
$81$	−6.26008	−0.695564
$82$	0	0
$83$	6.25543	0.686622	0.343311	−	0.939222i	$-0.388451\pi$
$83$	6.25543	0.686622	0.343311	+	0.939222i	$0.388451\pi$
$84$	0	0
$85$	−4.00900	−0.434838
$86$	0	0
$87$	4.46173	0.478348
$88$	0	0
$89$	2.74123	0.290570	0.145285	−	0.989390i	$-0.453590\pi$
$89$	2.74123	0.290570	0.145285	+	0.989390i	$0.453590\pi$
$90$	0	0
$91$	−3.53044	−0.370091
$92$	0	0
$93$	16.3110	1.69137
$94$	0	0
$95$	−0.503905	−0.0516995
$96$	0	0
$97$	9.29500	0.943764	0.471882	−	0.881662i	$-0.343575\pi$
$97$	9.29500	0.943764	0.471882	+	0.881662i	$0.343575\pi$
$98$	0	0
$99$	13.5834	1.36518
$100$	0	0
$101$	6.15352	0.612298	0.306149	−	0.951984i	$-0.400959\pi$
$101$	6.15352	0.612298	0.306149	+	0.951984i	$0.400959\pi$
$102$	0	0
$103$	−0.805067	−0.0793256	−0.0396628	−	0.999213i	$-0.512628\pi$
$103$	−0.805067	−0.0793256	−0.0396628	+	0.999213i	$0.512628\pi$
$104$	0	0
$105$	3.69573	0.360666
$106$	0	0
$107$	−10.3717	−1.00267	−0.501337	−	0.865252i	$-0.667158\pi$
$107$	−10.3717	−1.00267	−0.501337	+	0.865252i	$0.667158\pi$
$108$	0	0
$109$	4.02337	0.385369	0.192684	−	0.981261i	$-0.438281\pi$
$109$	4.02337	0.385369	0.192684	+	0.981261i	$0.438281\pi$
$110$	0	0
$111$	−13.2138	−1.25420
$112$	0	0
$113$	12.5686	1.18236	0.591178	−	0.806541i	$-0.298663\pi$
$113$	12.5686	1.18236	0.591178	+	0.806541i	$0.298663\pi$
$114$	0	0
$115$	8.01771	0.747655
$116$	0	0
$117$	−9.25575	−0.855695
$118$	0	0
$119$	5.70961	0.523399
$120$	0	0
$121$	2.23465	0.203150
$122$	0	0
$123$	−3.57811	−0.322627
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	1.98446	0.176092	0.0880461	−	0.996116i	$-0.471938\pi$
$127$	1.98446	0.176092	0.0880461	+	0.996116i	$0.471938\pi$
$128$	0	0
$129$	−18.7803	−1.65351
$130$	0	0
$131$	−0.828022	−0.0723446	−0.0361723	−	0.999346i	$-0.511517\pi$
$131$	−0.828022	−0.0723446	−0.0361723	+	0.999346i	$0.511517\pi$
$132$	0	0
$133$	0.717659	0.0622289
$134$	0	0
$135$	1.90422	0.163889
$136$	0	0
$137$	12.2403	1.04576	0.522879	−	0.852407i	$-0.324858\pi$
$137$	12.2403	1.04576	0.522879	+	0.852407i	$0.324858\pi$
$138$	0	0
$139$	19.0082	1.61225	0.806125	−	0.591745i	$-0.201561\pi$
$139$	19.0082	1.61225	0.806125	+	0.591745i	$0.201561\pi$
$140$	0	0
$141$	−19.4383	−1.63700
$142$	0	0
$143$	−9.01811	−0.754132
$144$	0	0
$145$	1.71938	0.142787
$146$	0	0
$147$	12.9013	1.06408
$148$	0	0
$149$	−21.1851	−1.73555	−0.867775	−	0.496958i	$-0.834450\pi$
$149$	−21.1851	−1.73555	−0.867775	+	0.496958i	$0.834450\pi$
$150$	0	0
$151$	16.4754	1.34075	0.670375	−	0.742023i	$-0.266133\pi$
$151$	16.4754	1.34075	0.670375	+	0.742023i	$0.266133\pi$
$152$	0	0
$153$	14.9689	1.21016
$154$	0	0
$155$	6.28565	0.504876
$156$	0	0
$157$	−6.53892	−0.521863	−0.260931	−	0.965357i	$-0.584030\pi$
$157$	−6.53892	−0.521863	−0.260931	+	0.965357i	$0.584030\pi$
$158$	0	0
$159$	20.1995	1.60193
$160$	0	0
$161$	−11.4188	−0.899927
$162$	0	0
$163$	−5.21914	−0.408795	−0.204397	−	0.978888i	$-0.565523\pi$
$163$	−5.21914	−0.408795	−0.204397	+	0.978888i	$0.565523\pi$
$164$	0	0
$165$	9.44032	0.734928
$166$	0	0
$167$	21.5110	1.66457	0.832286	−	0.554346i	$-0.187031\pi$
$167$	21.5110	1.66457	0.832286	+	0.554346i	$0.187031\pi$
$168$	0	0
$169$	−6.85505	−0.527312
$170$	0	0
$171$	1.88149	0.143881
$172$	0	0
$173$	15.3570	1.16757	0.583786	−	0.811907i	$-0.301571\pi$
$173$	15.3570	1.16757	0.583786	+	0.811907i	$0.301571\pi$
$174$	0	0
$175$	1.42420	0.107659
$176$	0	0
$177$	15.1383	1.13786
$178$	0	0
$179$	−0.883490	−0.0660351	−0.0330176	−	0.999455i	$-0.510512\pi$
$179$	−0.883490	−0.0660351	−0.0330176	+	0.999455i	$0.510512\pi$
$180$	0	0
$181$	0.683182	0.0507805	0.0253903	−	0.999678i	$-0.491917\pi$
$181$	0.683182	0.0507805	0.0253903	+	0.999678i	$0.491917\pi$
$182$	0	0
$183$	24.5136	1.81210
$184$	0	0
$185$	−5.09212	−0.374380
$186$	0	0
$187$	14.5845	1.06653
$188$	0	0
$189$	−2.71198	−0.197267
$190$	0	0
$191$	16.2944	1.17902	0.589512	−	0.807760i	$-0.299320\pi$
$191$	16.2944	1.17902	0.589512	+	0.807760i	$0.299320\pi$
$192$	0	0
$193$	17.1368	1.23353	0.616767	−	0.787146i	$-0.288442\pi$
$193$	17.1368	1.23353	0.616767	+	0.787146i	$0.288442\pi$
$194$	0	0
$195$	−6.43265	−0.460651
$196$	0	0
$197$	0.178606	0.0127251	0.00636256	−	0.999980i	$-0.497975\pi$
$197$	0.178606	0.0127251	0.00636256	+	0.999980i	$0.497975\pi$
$198$	0	0
$199$	7.07367	0.501439	0.250720	−	0.968060i	$-0.419333\pi$
$199$	7.07367	0.501439	0.250720	+	0.968060i	$0.419333\pi$
$200$	0	0
$201$	−23.5146	−1.65859
$202$	0	0
$203$	−2.44874	−0.171868
$204$	0	0
$205$	−1.37887	−0.0963044
$206$	0	0
$207$	−29.9366	−2.08074
$208$	0	0
$209$	1.83318	0.126804
$210$	0	0
$211$	−7.17620	−0.494030	−0.247015	−	0.969012i	$-0.579450\pi$
$211$	−7.17620	−0.494030	−0.247015	+	0.969012i	$0.579450\pi$
$212$	0	0
$213$	23.9754	1.64277
$214$	0	0
$215$	−7.23723	−0.493575
$216$	0	0
$217$	−8.95200	−0.607701
$218$	0	0
$219$	19.5543	1.32135
$220$	0	0
$221$	−9.93792	−0.668497
$222$	0	0
$223$	−8.11856	−0.543659	−0.271829	−	0.962345i	$-0.587629\pi$
$223$	−8.11856	−0.543659	−0.271829	+	0.962345i	$0.587629\pi$
$224$	0	0
$225$	3.73381	0.248921
$226$	0	0
$227$	8.82846	0.585966	0.292983	−	0.956118i	$-0.405352\pi$
$227$	8.82846	0.585966	0.292983	+	0.956118i	$0.405352\pi$
$228$	0	0
$229$	−14.3450	−0.947946	−0.473973	−	0.880539i	$-0.657181\pi$
$229$	−14.3450	−0.947946	−0.473973	+	0.880539i	$0.657181\pi$
$230$	0	0
$231$	−13.4449	−0.884607
$232$	0	0
$233$	−18.5730	−1.21676	−0.608378	−	0.793648i	$-0.708179\pi$
$233$	−18.5730	−1.21676	−0.608378	+	0.793648i	$0.708179\pi$
$234$	0	0
$235$	−7.49079	−0.488645
$236$	0	0
$237$	26.9814	1.75263
$238$	0	0
$239$	−28.3897	−1.83638	−0.918188	−	0.396145i	$-0.870348\pi$
$239$	−28.3897	−1.83638	−0.918188	+	0.396145i	$0.870348\pi$
$240$	0	0
$241$	−7.16168	−0.461324	−0.230662	−	0.973034i	$-0.574089\pi$
$241$	−7.16168	−0.461324	−0.230662	+	0.973034i	$0.574089\pi$
$242$	0	0
$243$	21.9573	1.40856
$244$	0	0
$245$	4.97167	0.317628
$246$	0	0
$247$	−1.24913	−0.0794802
$248$	0	0
$249$	−16.2326	−1.02870
$250$	0	0
$251$	4.47845	0.282677	0.141339	−	0.989961i	$-0.454859\pi$
$251$	4.47845	0.282677	0.141339	+	0.989961i	$0.454859\pi$
$252$	0	0
$253$	−29.1680	−1.83378
$254$	0	0
$255$	10.4032	0.651474
$256$	0	0
$257$	−8.09408	−0.504895	−0.252447	−	0.967611i	$-0.581235\pi$
$257$	−8.09408	−0.504895	−0.252447	+	0.967611i	$0.581235\pi$
$258$	0	0
$259$	7.25217	0.450628
$260$	0	0
$261$	−6.41986	−0.397380
$262$	0	0
$263$	−18.5791	−1.14564	−0.572820	−	0.819682i	$-0.694150\pi$
$263$	−18.5791	−1.14564	−0.572820	+	0.819682i	$0.694150\pi$
$264$	0	0
$265$	7.78413	0.478176
$266$	0	0
$267$	−7.11339	−0.435332
$268$	0	0
$269$	−5.47249	−0.333664	−0.166832	−	0.985985i	$-0.553354\pi$
$269$	−5.47249	−0.333664	−0.166832	+	0.985985i	$0.553354\pi$
$270$	0	0
$271$	11.1710	0.678589	0.339295	−	0.940680i	$-0.389812\pi$
$271$	11.1710	0.678589	0.339295	+	0.940680i	$0.389812\pi$
$272$	0	0
$273$	9.16135	0.554470
$274$	0	0
$275$	3.63795	0.219376
$276$	0	0
$277$	−23.5462	−1.41475	−0.707377	−	0.706837i	$-0.750122\pi$
$277$	−23.5462	−1.41475	−0.707377	+	0.706837i	$0.750122\pi$
$278$	0	0
$279$	−23.4695	−1.40508
$280$	0	0
$281$	−5.88015	−0.350780	−0.175390	−	0.984499i	$-0.556119\pi$
$281$	−5.88015	−0.350780	−0.175390	+	0.984499i	$0.556119\pi$
$282$	0	0
$283$	−17.4270	−1.03593	−0.517964	−	0.855403i	$-0.673310\pi$
$283$	−17.4270	−1.03593	−0.517964	+	0.855403i	$0.673310\pi$
$284$	0	0
$285$	1.30761	0.0774562
$286$	0	0
$287$	1.96378	0.115918
$288$	0	0
$289$	−0.927879	−0.0545811
$290$	0	0
$291$	−24.1201	−1.41395
$292$	0	0
$293$	−25.5468	−1.49246	−0.746229	−	0.665689i	$-0.768138\pi$
$293$	−25.5468	−1.49246	−0.746229	+	0.665689i	$0.768138\pi$
$294$	0	0
$295$	5.83373	0.339653
$296$	0	0
$297$	−6.92743	−0.401971
$298$	0	0
$299$	19.8751	1.14941
$300$	0	0
$301$	10.3072	0.594099
$302$	0	0
$303$	−15.9681	−0.917345
$304$	0	0
$305$	9.44664	0.540913
$306$	0	0
$307$	0.0681924	0.00389194	0.00194597	−	0.999998i	$-0.499381\pi$
$307$	0.0681924	0.00389194	0.00194597	+	0.999998i	$0.499381\pi$
$308$	0	0
$309$	2.08912	0.118846
$310$	0	0
$311$	−33.9157	−1.92318	−0.961592	−	0.274483i	$-0.911493\pi$
$311$	−33.9157	−1.92318	−0.961592	+	0.274483i	$0.911493\pi$
$312$	0	0
$313$	3.72134	0.210343	0.105171	−	0.994454i	$-0.466461\pi$
$313$	3.72134	0.210343	0.105171	+	0.994454i	$0.466461\pi$
$314$	0	0
$315$	−5.31768	−0.299617
$316$	0	0
$317$	1.50585	0.0845769	0.0422885	−	0.999105i	$-0.486535\pi$
$317$	1.50585	0.0845769	0.0422885	+	0.999105i	$0.486535\pi$
$318$	0	0
$319$	−6.25503	−0.350214
$320$	0	0
$321$	26.9142	1.50221
$322$	0	0
$323$	2.02016	0.112404
$324$	0	0
$325$	−2.47890	−0.137505
$326$	0	0
$327$	−10.4405	−0.577360
$328$	0	0
$329$	10.6684	0.588165
$330$	0	0
$331$	2.61045	0.143483	0.0717417	−	0.997423i	$-0.477144\pi$
$331$	2.61045	0.143483	0.0717417	+	0.997423i	$0.477144\pi$
$332$	0	0
$333$	19.0130	1.04191
$334$	0	0
$335$	−9.06165	−0.495091
$336$	0	0
$337$	−27.0266	−1.47223	−0.736117	−	0.676855i	$-0.763342\pi$
$337$	−27.0266	−1.47223	−0.736117	+	0.676855i	$0.763342\pi$
$338$	0	0
$339$	−32.6151	−1.77141
$340$	0	0
$341$	−22.8669	−1.23831
$342$	0	0
$343$	−17.0500	−0.920613
$344$	0	0
$345$	−20.8056	−1.12014
$346$	0	0
$347$	−7.73203	−0.415077	−0.207539	−	0.978227i	$-0.566545\pi$
$347$	−7.73203	−0.415077	−0.207539	+	0.978227i	$0.566545\pi$
$348$	0	0
$349$	4.20935	0.225322	0.112661	−	0.993634i	$-0.464063\pi$
$349$	4.20935	0.225322	0.112661	+	0.993634i	$0.464063\pi$
$350$	0	0
$351$	4.72036	0.251954
$352$	0	0
$353$	−16.7101	−0.889391	−0.444696	−	0.895682i	$-0.646688\pi$
$353$	−16.7101	−0.889391	−0.444696	+	0.895682i	$0.646688\pi$
$354$	0	0
$355$	9.23923	0.490367
$356$	0	0
$357$	−14.8162	−0.784157
$358$	0	0
$359$	−35.9142	−1.89548	−0.947740	−	0.319042i	$-0.896639\pi$
$359$	−35.9142	−1.89548	−0.947740	+	0.319042i	$0.896639\pi$
$360$	0	0
$361$	−18.7461	−0.986636
$362$	0	0
$363$	−5.79882	−0.304359
$364$	0	0
$365$	7.53548	0.394425
$366$	0	0
$367$	−18.6666	−0.974388	−0.487194	−	0.873294i	$-0.661980\pi$
$367$	−18.6666	−0.974388	−0.487194	+	0.873294i	$0.661980\pi$
$368$	0	0
$369$	5.14844	0.268017
$370$	0	0
$371$	−11.0861	−0.575563
$372$	0	0
$373$	4.92470	0.254991	0.127496	−	0.991839i	$-0.459306\pi$
$373$	4.92470	0.254991	0.127496	+	0.991839i	$0.459306\pi$
$374$	0	0
$375$	2.59496	0.134003
$376$	0	0
$377$	4.26218	0.219514
$378$	0	0
$379$	25.0669	1.28760	0.643801	−	0.765193i	$-0.277356\pi$
$379$	25.0669	1.28760	0.643801	+	0.765193i	$0.277356\pi$
$380$	0	0
$381$	−5.14959	−0.263822
$382$	0	0
$383$	3.05637	0.156173	0.0780867	−	0.996947i	$-0.475119\pi$
$383$	3.05637	0.156173	0.0780867	+	0.996947i	$0.475119\pi$
$384$	0	0
$385$	−5.18115	−0.264056
$386$	0	0
$387$	27.0225	1.37363
$388$	0	0
$389$	−2.17911	−0.110485	−0.0552426	−	0.998473i	$-0.517593\pi$
$389$	−2.17911	−0.110485	−0.0552426	+	0.998473i	$0.517593\pi$
$390$	0	0
$391$	−32.1430	−1.62554
$392$	0	0
$393$	2.14868	0.108387
$394$	0	0
$395$	10.3976	0.523162
$396$	0	0
$397$	30.3900	1.52523	0.762616	−	0.646852i	$-0.223915\pi$
$397$	30.3900	1.52523	0.762616	+	0.646852i	$0.223915\pi$
$398$	0	0
$399$	−1.86230	−0.0932314
$400$	0	0
$401$	1.00000	0.0499376
$401$	1.00000	0.0499376
$402$	0	0
$403$	15.5815	0.776170
$404$	0	0
$405$	6.26008	0.311066
$406$	0	0
$407$	18.5248	0.918243
$408$	0	0
$409$	6.84265	0.338347	0.169174	−	0.985586i	$-0.445890\pi$
$409$	6.84265	0.338347	0.169174	+	0.985586i	$0.445890\pi$
$410$	0	0
$411$	−31.7630	−1.56676
$412$	0	0
$413$	−8.30838	−0.408828
$414$	0	0
$415$	−6.25543	−0.307067
$416$	0	0
$417$	−49.3254	−2.41547
$418$	0	0
$419$	27.9516	1.36553	0.682764	−	0.730639i	$-0.260778\pi$
$419$	27.9516	1.36553	0.682764	+	0.730639i	$0.260778\pi$
$420$	0	0
$421$	2.33475	0.113789	0.0568944	−	0.998380i	$-0.481880\pi$
$421$	2.33475	0.113789	0.0568944	+	0.998380i	$0.481880\pi$
$422$	0	0
$423$	27.9692	1.35991
$424$	0	0
$425$	4.00900	0.194465
$426$	0	0
$427$	−13.4539	−0.651078
$428$	0	0
$429$	23.4016	1.12984
$430$	0	0
$431$	−12.6286	−0.608298	−0.304149	−	0.952624i	$-0.598372\pi$
$431$	−12.6286	−0.608298	−0.304149	+	0.952624i	$0.598372\pi$
$432$	0	0
$433$	−16.4901	−0.792465	−0.396233	−	0.918150i	$-0.629683\pi$
$433$	−16.4901	−0.792465	−0.396233	+	0.918150i	$0.629683\pi$
$434$	0	0
$435$	−4.46173	−0.213924
$436$	0	0
$437$	−4.04016	−0.193267
$438$	0	0
$439$	−17.2439	−0.823006	−0.411503	−	0.911408i	$-0.634996\pi$
$439$	−17.2439	−0.823006	−0.411503	+	0.911408i	$0.634996\pi$
$440$	0	0
$441$	−18.5633	−0.883965
$442$	0	0
$443$	17.1801	0.816252	0.408126	−	0.912926i	$-0.366182\pi$
$443$	17.1801	0.816252	0.408126	+	0.912926i	$0.366182\pi$
$444$	0	0
$445$	−2.74123	−0.129947
$446$	0	0
$447$	54.9744	2.60020
$448$	0	0
$449$	−37.3687	−1.76354	−0.881769	−	0.471682i	$-0.843647\pi$
$449$	−37.3687	−1.76354	−0.881769	+	0.471682i	$0.843647\pi$
$450$	0	0
$451$	5.01625	0.236206
$452$	0	0
$453$	−42.7530	−2.00871
$454$	0	0
$455$	3.53044	0.165510
$456$	0	0
$457$	−24.3362	−1.13840	−0.569200	−	0.822199i	$-0.692747\pi$
$457$	−24.3362	−1.13840	−0.569200	+	0.822199i	$0.692747\pi$
$458$	0	0
$459$	−7.63401	−0.356325
$460$	0	0
$461$	17.0938	0.796137	0.398069	−	0.917356i	$-0.369681\pi$
$461$	17.0938	0.796137	0.398069	+	0.917356i	$0.369681\pi$
$462$	0	0
$463$	−12.5515	−0.583316	−0.291658	−	0.956523i	$-0.594207\pi$
$463$	−12.5515	−0.583316	−0.291658	+	0.956523i	$0.594207\pi$
$464$	0	0
$465$	−16.3110	−0.756405
$466$	0	0
$467$	32.5500	1.50624	0.753118	−	0.657886i	$-0.228549\pi$
$467$	32.5500	1.50624	0.753118	+	0.657886i	$0.228549\pi$
$468$	0	0
$469$	12.9056	0.595923
$470$	0	0
$471$	16.9682	0.781855
$472$	0	0
$473$	26.3286	1.21059
$474$	0	0
$475$	0.503905	0.0231207
$476$	0	0
$477$	−29.0645	−1.33077
$478$	0	0
$479$	10.7692	0.492059	0.246030	−	0.969262i	$-0.420874\pi$
$479$	10.7692	0.492059	0.246030	+	0.969262i	$0.420874\pi$
$480$	0	0
$481$	−12.6229	−0.575553
$482$	0	0
$483$	29.6313	1.34827
$484$	0	0
$485$	−9.29500	−0.422064
$486$	0	0
$487$	−19.3407	−0.876411	−0.438206	−	0.898875i	$-0.644386\pi$
$487$	−19.3407	−0.876411	−0.438206	+	0.898875i	$0.644386\pi$
$488$	0	0
$489$	13.5435	0.612456
$490$	0	0
$491$	26.9486	1.21617	0.608086	−	0.793871i	$-0.291938\pi$
$491$	26.9486	1.21617	0.608086	+	0.793871i	$0.291938\pi$
$492$	0	0
$493$	−6.89302	−0.310446
$494$	0	0
$495$	−13.5834	−0.610529
$496$	0	0
$497$	−13.1585	−0.590238
$498$	0	0
$499$	6.53278	0.292447	0.146224	−	0.989252i	$-0.453288\pi$
$499$	6.53278	0.292447	0.146224	+	0.989252i	$0.453288\pi$
$500$	0	0
$501$	−55.8202	−2.49386
$502$	0	0
$503$	13.6923	0.610508	0.305254	−	0.952271i	$-0.401259\pi$
$503$	13.6923	0.610508	0.305254	+	0.952271i	$0.401259\pi$
$504$	0	0
$505$	−6.15352	−0.273828
$506$	0	0
$507$	17.7886	0.790019
$508$	0	0
$509$	6.57046	0.291230	0.145615	−	0.989341i	$-0.453484\pi$
$509$	6.57046	0.291230	0.145615	+	0.989341i	$0.453484\pi$
$510$	0	0
$511$	−10.7320	−0.474756
$512$	0	0
$513$	−0.959543	−0.0423649
$514$	0	0
$515$	0.805067	0.0354755
$516$	0	0
$517$	27.2511	1.19850
$518$	0	0
$519$	−39.8508	−1.74926
$520$	0	0
$521$	12.0777	0.529135	0.264568	−	0.964367i	$-0.414771\pi$
$521$	12.0777	0.529135	0.264568	+	0.964367i	$0.414771\pi$
$522$	0	0
$523$	20.9073	0.914212	0.457106	−	0.889412i	$-0.348886\pi$
$523$	20.9073	0.914212	0.457106	+	0.889412i	$0.348886\pi$
$524$	0	0
$525$	−3.69573	−0.161295
$526$	0	0
$527$	−25.1992	−1.09769
$528$	0	0
$529$	41.2836	1.79494
$530$	0	0
$531$	−21.7821	−0.945261
$532$	0	0
$533$	−3.41808	−0.148053
$534$	0	0
$535$	10.3717	0.448409
$536$	0	0
$537$	2.29262	0.0989338
$538$	0	0
$539$	−18.0866	−0.779047
$540$	0	0
$541$	−20.2377	−0.870088	−0.435044	−	0.900409i	$-0.643267\pi$
$541$	−20.2377	−0.870088	−0.435044	+	0.900409i	$0.643267\pi$
$542$	0	0
$543$	−1.77283	−0.0760794
$544$	0	0
$545$	−4.02337	−0.172342
$546$	0	0
$547$	5.53845	0.236807	0.118404	−	0.992966i	$-0.462222\pi$
$547$	5.53845	0.236807	0.118404	+	0.992966i	$0.462222\pi$
$548$	0	0
$549$	−35.2720	−1.50537
$550$	0	0
$551$	−0.866406	−0.0369101
$552$	0	0
$553$	−14.8083	−0.629711
$554$	0	0
$555$	13.2138	0.560896
$556$	0	0
$557$	−11.7355	−0.497247	−0.248624	−	0.968600i	$-0.579978\pi$
$557$	−11.7355	−0.497247	−0.248624	+	0.968600i	$0.579978\pi$
$558$	0	0
$559$	−17.9404	−0.758797
$560$	0	0
$561$	−37.8463	−1.59787
$562$	0	0
$563$	24.3519	1.02631	0.513155	−	0.858296i	$-0.328477\pi$
$563$	24.3519	1.02631	0.513155	+	0.858296i	$0.328477\pi$
$564$	0	0
$565$	−12.5686	−0.528766
$566$	0	0
$567$	−8.91558	−0.374419
$568$	0	0
$569$	−37.7052	−1.58069	−0.790343	−	0.612665i	$-0.790098\pi$
$569$	−37.7052	−1.58069	−0.790343	+	0.612665i	$0.790098\pi$
$570$	0	0
$571$	20.7929	0.870157	0.435078	−	0.900393i	$-0.356721\pi$
$571$	20.7929	0.870157	0.435078	+	0.900393i	$0.356721\pi$
$572$	0	0
$573$	−42.2834	−1.76641
$574$	0	0
$575$	−8.01771	−0.334362
$576$	0	0
$577$	−35.9303	−1.49580	−0.747898	−	0.663813i	$-0.768937\pi$
$577$	−35.9303	−1.49580	−0.747898	+	0.663813i	$0.768937\pi$
$578$	0	0
$579$	−44.4693	−1.84808
$580$	0	0
$581$	8.90895	0.369606
$582$	0	0
$583$	−28.3183	−1.17282
$584$	0	0
$585$	9.25575	0.382678
$586$	0	0
$587$	−45.9010	−1.89454	−0.947268	−	0.320443i	$-0.896168\pi$
$587$	−45.9010	−1.89454	−0.947268	+	0.320443i	$0.896168\pi$
$588$	0	0
$589$	−3.16737	−0.130509
$590$	0	0
$591$	−0.463474	−0.0190648
$592$	0	0
$593$	−26.0219	−1.06859	−0.534295	−	0.845298i	$-0.679423\pi$
$593$	−26.0219	−1.06859	−0.534295	+	0.845298i	$0.679423\pi$
$594$	0	0
$595$	−5.70961	−0.234071
$596$	0	0
$597$	−18.3559	−0.751256
$598$	0	0
$599$	−29.5387	−1.20692	−0.603460	−	0.797393i	$-0.706212\pi$
$599$	−29.5387	−1.20692	−0.603460	+	0.797393i	$0.706212\pi$
$600$	0	0
$601$	3.37231	0.137559	0.0687797	−	0.997632i	$-0.478089\pi$
$601$	3.37231	0.137559	0.0687797	+	0.997632i	$0.478089\pi$
$602$	0	0
$603$	33.8345	1.37785
$604$	0	0
$605$	−2.23465	−0.0908513
$606$	0	0
$607$	13.9276	0.565302	0.282651	−	0.959223i	$-0.408786\pi$
$607$	13.9276	0.565302	0.282651	+	0.959223i	$0.408786\pi$
$608$	0	0
$609$	6.35438	0.257493
$610$	0	0
$611$	−18.5689	−0.751218
$612$	0	0
$613$	−28.3379	−1.14456	−0.572279	−	0.820059i	$-0.693941\pi$
$613$	−28.3379	−1.14456	−0.572279	+	0.820059i	$0.693941\pi$
$614$	0	0
$615$	3.57811	0.144283
$616$	0	0
$617$	8.43683	0.339654	0.169827	−	0.985474i	$-0.445679\pi$
$617$	8.43683	0.339654	0.169827	+	0.985474i	$0.445679\pi$
$618$	0	0
$619$	0.781251	0.0314011	0.0157006	−	0.999877i	$-0.495002\pi$
$619$	0.781251	0.0314011	0.0157006	+	0.999877i	$0.495002\pi$
$620$	0	0
$621$	15.2674	0.612662
$622$	0	0
$623$	3.90405	0.156413
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	−4.75702	−0.189977
$628$	0	0
$629$	20.4143	0.813972
$630$	0	0
$631$	−3.47116	−0.138185	−0.0690923	−	0.997610i	$-0.522010\pi$
$631$	−3.47116	−0.138185	−0.0690923	+	0.997610i	$0.522010\pi$
$632$	0	0
$633$	18.6220	0.740156
$634$	0	0
$635$	−1.98446	−0.0787509
$636$	0	0
$637$	12.3243	0.488305
$638$	0	0
$639$	−34.4976	−1.36470
$640$	0	0
$641$	−0.174923	−0.00690906	−0.00345453	−	0.999994i	$-0.501100\pi$
$641$	−0.174923	−0.00690906	−0.00345453	+	0.999994i	$0.501100\pi$
$642$	0	0
$643$	23.1130	0.911489	0.455745	−	0.890111i	$-0.349373\pi$
$643$	23.1130	0.911489	0.455745	+	0.890111i	$0.349373\pi$
$644$	0	0
$645$	18.7803	0.739474
$646$	0	0
$647$	4.79588	0.188545	0.0942727	−	0.995546i	$-0.469947\pi$
$647$	4.79588	0.188545	0.0942727	+	0.995546i	$0.469947\pi$
$648$	0	0
$649$	−21.2228	−0.833068
$650$	0	0
$651$	23.2301	0.910458
$652$	0	0
$653$	−49.6716	−1.94380	−0.971900	−	0.235393i	$-0.924362\pi$
$653$	−49.6716	−1.94380	−0.971900	+	0.235393i	$0.924362\pi$
$654$	0	0
$655$	0.828022	0.0323535
$656$	0	0
$657$	−28.1361	−1.09769
$658$	0	0
$659$	−27.5862	−1.07461	−0.537303	−	0.843389i	$-0.680557\pi$
$659$	−27.5862	−1.07461	−0.537303	+	0.843389i	$0.680557\pi$
$660$	0	0
$661$	−47.2481	−1.83774	−0.918869	−	0.394562i	$-0.870896\pi$
$661$	−47.2481	−1.83774	−0.918869	+	0.394562i	$0.870896\pi$
$662$	0	0
$663$	25.7885	1.00154
$664$	0	0
$665$	−0.717659	−0.0278296
$666$	0	0
$667$	13.7855	0.533778
$668$	0	0
$669$	21.0673	0.814510
$670$	0	0
$671$	−34.3664	−1.32670
$672$	0	0
$673$	−16.9272	−0.652497	−0.326249	−	0.945284i	$-0.605785\pi$
$673$	−16.9272	−0.652497	−0.326249	+	0.945284i	$0.605785\pi$
$674$	0	0
$675$	−1.90422	−0.0732933
$676$	0	0
$677$	−33.8684	−1.30167	−0.650835	−	0.759219i	$-0.725581\pi$
$677$	−33.8684	−1.30167	−0.650835	+	0.759219i	$0.725581\pi$
$678$	0	0
$679$	13.2379	0.508024
$680$	0	0
$681$	−22.9095	−0.877894
$682$	0	0
$683$	28.7464	1.09995	0.549975	−	0.835181i	$-0.314637\pi$
$683$	28.7464	1.09995	0.549975	+	0.835181i	$0.314637\pi$
$684$	0	0
$685$	−12.2403	−0.467677
$686$	0	0
$687$	37.2248	1.42021
$688$	0	0
$689$	19.2961	0.735123
$690$	0	0
$691$	9.21658	0.350615	0.175308	−	0.984514i	$-0.443908\pi$
$691$	9.21658	0.350615	0.175308	+	0.984514i	$0.443908\pi$
$692$	0	0
$693$	19.3454	0.734872
$694$	0	0
$695$	−19.0082	−0.721020
$696$	0	0
$697$	5.52789	0.209384
$698$	0	0
$699$	48.1961	1.82294
$700$	0	0
$701$	−31.6359	−1.19487	−0.597435	−	0.801917i	$-0.703813\pi$
$701$	−31.6359	−1.19487	−0.597435	+	0.801917i	$0.703813\pi$
$702$	0	0
$703$	2.56594	0.0967763
$704$	0	0
$705$	19.4383	0.732088
$706$	0	0
$707$	8.76382	0.329597
$708$	0	0
$709$	35.0993	1.31818	0.659091	−	0.752063i	$-0.270941\pi$
$709$	35.0993	1.31818	0.659091	+	0.752063i	$0.270941\pi$
$710$	0	0
$711$	−38.8228	−1.45597
$712$	0	0
$713$	50.3965	1.88736
$714$	0	0
$715$	9.01811	0.337258
$716$	0	0
$717$	73.6701	2.75126
$718$	0	0
$719$	16.6574	0.621215	0.310608	−	0.950538i	$-0.399467\pi$
$719$	16.6574	0.621215	0.310608	+	0.950538i	$0.399467\pi$
$720$	0	0
$721$	−1.14657	−0.0427006
$722$	0	0
$723$	18.5843	0.691156
$724$	0	0
$725$	−1.71938	−0.0638564
$726$	0	0
$727$	−23.6433	−0.876881	−0.438441	−	0.898760i	$-0.644469\pi$
$727$	−23.6433	−0.876881	−0.438441	+	0.898760i	$0.644469\pi$
$728$	0	0
$729$	−38.1981	−1.41474
$730$	0	0
$731$	29.0141	1.07312
$732$	0	0
$733$	21.0606	0.777890	0.388945	−	0.921261i	$-0.372840\pi$
$733$	21.0606	0.777890	0.388945	+	0.921261i	$0.372840\pi$
$734$	0	0
$735$	−12.9013	−0.475871
$736$	0	0
$737$	32.9658	1.21431
$738$	0	0
$739$	34.1042	1.25454	0.627271	−	0.778801i	$-0.284172\pi$
$739$	34.1042	1.25454	0.627271	+	0.778801i	$0.284172\pi$
$740$	0	0
$741$	3.24144	0.119077
$742$	0	0
$743$	−11.3882	−0.417794	−0.208897	−	0.977938i	$-0.566987\pi$
$743$	−11.3882	−0.417794	−0.208897	+	0.977938i	$0.566987\pi$
$744$	0	0
$745$	21.1851	0.776161
$746$	0	0
$747$	23.3566	0.854573
$748$	0	0
$749$	−14.7714	−0.539735
$750$	0	0
$751$	38.6841	1.41160	0.705802	−	0.708409i	$-0.250587\pi$
$751$	38.6841	1.41160	0.705802	+	0.708409i	$0.250587\pi$
$752$	0	0
$753$	−11.6214	−0.423507
$754$	0	0
$755$	−16.4754	−0.599601
$756$	0	0
$757$	−25.4459	−0.924848	−0.462424	−	0.886659i	$-0.653020\pi$
$757$	−25.4459	−0.924848	−0.462424	+	0.886659i	$0.653020\pi$
$758$	0	0
$759$	75.6897	2.74736
$760$	0	0
$761$	−18.8715	−0.684093	−0.342046	−	0.939683i	$-0.611120\pi$
$761$	−18.8715	−0.684093	−0.342046	+	0.939683i	$0.611120\pi$
$762$	0	0
$763$	5.73007	0.207442
$764$	0	0
$765$	−14.9689	−0.541201
$766$	0	0
$767$	14.4612	0.522165
$768$	0	0
$769$	3.80388	0.137171	0.0685857	−	0.997645i	$-0.478151\pi$
$769$	3.80388	0.137171	0.0685857	+	0.997645i	$0.478151\pi$
$770$	0	0
$771$	21.0038	0.756434
$772$	0	0
$773$	−39.6538	−1.42625	−0.713123	−	0.701039i	$-0.752720\pi$
$773$	−39.6538	−1.42625	−0.713123	+	0.701039i	$0.752720\pi$
$774$	0	0
$775$	−6.28565	−0.225787
$776$	0	0
$777$	−18.8191	−0.675131
$778$	0	0
$779$	0.694818	0.0248944
$780$	0	0
$781$	−33.6118	−1.20273
$782$	0	0
$783$	3.27408	0.117006
$784$	0	0
$785$	6.53892	0.233384
$786$	0	0
$787$	20.5552	0.732714	0.366357	−	0.930474i	$-0.380605\pi$
$787$	20.5552	0.732714	0.366357	+	0.930474i	$0.380605\pi$
$788$	0	0
$789$	48.2121	1.71640
$790$	0	0
$791$	17.9002	0.636457
$792$	0	0
$793$	23.4173	0.831572
$794$	0	0
$795$	−20.1995	−0.716403
$796$	0	0
$797$	−44.3768	−1.57191	−0.785953	−	0.618286i	$-0.787827\pi$
$797$	−44.3768	−1.57191	−0.785953	+	0.618286i	$0.787827\pi$
$798$	0	0
$799$	30.0306	1.06241
$800$	0	0
$801$	10.2353	0.361645
$802$	0	0
$803$	−27.4137	−0.967407
$804$	0	0
$805$	11.4188	0.402459
$806$	0	0
$807$	14.2009	0.499895
$808$	0	0
$809$	−15.9158	−0.559570	−0.279785	−	0.960063i	$-0.590263\pi$
$809$	−15.9158	−0.559570	−0.279785	+	0.960063i	$0.590263\pi$
$810$	0	0
$811$	−16.4178	−0.576507	−0.288254	−	0.957554i	$-0.593075\pi$
$811$	−16.4178	−0.576507	−0.288254	+	0.957554i	$0.593075\pi$
$812$	0	0
$813$	−28.9883	−1.01666
$814$	0	0
$815$	5.21914	0.182818
$816$	0	0
$817$	3.64687	0.127588
$818$	0	0
$819$	−13.1820	−0.460617
$820$	0	0
$821$	5.85873	0.204471	0.102236	−	0.994760i	$-0.467400\pi$
$821$	5.85873	0.204471	0.102236	+	0.994760i	$0.467400\pi$
$822$	0	0
$823$	−23.3572	−0.814182	−0.407091	−	0.913388i	$-0.633457\pi$
$823$	−23.3572	−0.814182	−0.407091	+	0.913388i	$0.633457\pi$
$824$	0	0
$825$	−9.44032	−0.328670
$826$	0	0
$827$	45.9011	1.59614	0.798069	−	0.602566i	$-0.205855\pi$
$827$	45.9011	1.59614	0.798069	+	0.602566i	$0.205855\pi$
$828$	0	0
$829$	10.4465	0.362822	0.181411	−	0.983407i	$-0.441934\pi$
$829$	10.4465	0.362822	0.181411	+	0.983407i	$0.441934\pi$
$830$	0	0
$831$	61.1014	2.11958
$832$	0	0
$833$	−19.9314	−0.690583
$834$	0	0
$835$	−21.5110	−0.744420
$836$	0	0
$837$	11.9692	0.413717
$838$	0	0
$839$	−21.0564	−0.726947	−0.363473	−	0.931605i	$-0.618409\pi$
$839$	−21.0564	−0.726947	−0.363473	+	0.931605i	$0.618409\pi$
$840$	0	0
$841$	−26.0437	−0.898059
$842$	0	0
$843$	15.2588	0.525539
$844$	0	0
$845$	6.85505	0.235821
$846$	0	0
$847$	3.18258	0.109355
$848$	0	0
$849$	45.2224	1.55203
$850$	0	0
$851$	−40.8271	−1.39954
$852$	0	0
$853$	−2.24512	−0.0768713	−0.0384356	−	0.999261i	$-0.512237\pi$
$853$	−2.24512	−0.0768713	−0.0384356	+	0.999261i	$0.512237\pi$
$854$	0	0
$855$	−1.88149	−0.0643454
$856$	0	0
$857$	−25.8802	−0.884052	−0.442026	−	0.897002i	$-0.645740\pi$
$857$	−25.8802	−0.884052	−0.442026	+	0.897002i	$0.645740\pi$
$858$	0	0
$859$	−46.6926	−1.59313	−0.796565	−	0.604553i	$-0.793352\pi$
$859$	−46.6926	−1.59313	−0.796565	+	0.604553i	$0.793352\pi$
$860$	0	0
$861$	−5.09593	−0.173669
$862$	0	0
$863$	23.2973	0.793050	0.396525	−	0.918024i	$-0.370216\pi$
$863$	23.2973	0.793050	0.396525	+	0.918024i	$0.370216\pi$
$864$	0	0
$865$	−15.3570	−0.522154
$866$	0	0
$867$	2.40781	0.0817735
$868$	0	0
$869$	−37.8260	−1.28316
$870$	0	0
$871$	−22.4629	−0.761127
$872$	0	0
$873$	34.7058	1.17461
$874$	0	0
$875$	−1.42420	−0.0481466
$876$	0	0
$877$	43.6745	1.47478	0.737391	−	0.675466i	$-0.236058\pi$
$877$	43.6745	1.47478	0.737391	+	0.675466i	$0.236058\pi$
$878$	0	0
$879$	66.2929	2.23600
$880$	0	0
$881$	−54.8569	−1.84818	−0.924088	−	0.382179i	$-0.875174\pi$
$881$	−54.8569	−1.84818	−0.924088	+	0.382179i	$0.875174\pi$
$882$	0	0
$883$	33.9440	1.14231	0.571154	−	0.820843i	$-0.306496\pi$
$883$	33.9440	1.14231	0.571154	+	0.820843i	$0.306496\pi$
$884$	0	0
$885$	−15.1383	−0.508868
$886$	0	0
$887$	−16.9930	−0.570568	−0.285284	−	0.958443i	$-0.592088\pi$
$887$	−16.9930	−0.570568	−0.285284	+	0.958443i	$0.592088\pi$
$888$	0	0
$889$	2.82626	0.0947897
$890$	0	0
$891$	−22.7738	−0.762952
$892$	0	0
$893$	3.77464	0.126314
$894$	0	0
$895$	0.883490	0.0295318
$896$	0	0
$897$	−51.5751	−1.72204
$898$	0	0
$899$	10.8075	0.360449
$900$	0	0
$901$	−31.2066	−1.03964
$902$	0	0
$903$	−26.7469	−0.890080
$904$	0	0
$905$	−0.683182	−0.0227097
$906$	0	0
$907$	18.8580	0.626169	0.313085	−	0.949725i	$-0.398638\pi$
$907$	18.8580	0.626169	0.313085	+	0.949725i	$0.398638\pi$
$908$	0	0
$909$	22.9761	0.762069
$910$	0	0
$911$	39.7194	1.31596	0.657982	−	0.753034i	$-0.271410\pi$
$911$	39.7194	1.31596	0.657982	+	0.753034i	$0.271410\pi$
$912$	0	0
$913$	22.7569	0.753143
$914$	0	0
$915$	−24.5136	−0.810396
$916$	0	0
$917$	−1.17927	−0.0389428
$918$	0	0
$919$	−40.4926	−1.33573	−0.667864	−	0.744283i	$-0.732791\pi$
$919$	−40.4926	−1.33573	−0.667864	+	0.744283i	$0.732791\pi$
$920$	0	0
$921$	−0.176956	−0.00583091
$922$	0	0
$923$	22.9031	0.753866
$924$	0	0
$925$	5.09212	0.167428
$926$	0	0
$927$	−3.00597	−0.0987290
$928$	0	0
$929$	−30.5170	−1.00123	−0.500616	−	0.865670i	$-0.666893\pi$
$929$	−30.5170	−1.00123	−0.500616	+	0.865670i	$0.666893\pi$
$930$	0	0
$931$	−2.50524	−0.0821061
$932$	0	0
$933$	88.0099	2.88131
$934$	0	0
$935$	−14.5845	−0.476966
$936$	0	0
$937$	0.327421	0.0106964	0.00534819	−	0.999986i	$-0.498298\pi$
$937$	0.327421	0.0106964	0.00534819	+	0.999986i	$0.498298\pi$
$938$	0	0
$939$	−9.65672	−0.315135
$940$	0	0
$941$	−14.4811	−0.472069	−0.236035	−	0.971745i	$-0.575848\pi$
$941$	−14.4811	−0.472069	−0.236035	+	0.971745i	$0.575848\pi$
$942$	0	0
$943$	−11.0554	−0.360012
$944$	0	0
$945$	2.71198	0.0882206
$946$	0	0
$947$	4.50133	0.146273	0.0731367	−	0.997322i	$-0.476699\pi$
$947$	4.50133	0.146273	0.0731367	+	0.997322i	$0.476699\pi$
$948$	0	0
$949$	18.6797	0.606369
$950$	0	0
$951$	−3.90762	−0.126713
$952$	0	0
$953$	−23.7032	−0.767822	−0.383911	−	0.923370i	$-0.625423\pi$
$953$	−23.7032	−0.767822	−0.383911	+	0.923370i	$0.625423\pi$
$954$	0	0
$955$	−16.2944	−0.527276
$956$	0	0
$957$	16.2315	0.524691
$958$	0	0
$959$	17.4326	0.562927
$960$	0	0
$961$	8.50942	0.274497
$962$	0	0
$963$	−38.7261	−1.24793
$964$	0	0
$965$	−17.1368	−0.551653
$966$	0	0
$967$	36.7675	1.18236	0.591181	−	0.806539i	$-0.298662\pi$
$967$	36.7675	1.18236	0.591181	+	0.806539i	$0.298662\pi$
$968$	0	0
$969$	−5.24222	−0.168404
$970$	0	0
$971$	−28.8370	−0.925422	−0.462711	−	0.886509i	$-0.653123\pi$
$971$	−28.8370	−0.925422	−0.462711	+	0.886509i	$0.653123\pi$
$972$	0	0
$973$	27.0713	0.867867
$974$	0	0
$975$	6.43265	0.206010
$976$	0	0
$977$	26.2326	0.839256	0.419628	−	0.907696i	$-0.362160\pi$
$977$	26.2326	0.839256	0.419628	+	0.907696i	$0.362160\pi$
$978$	0	0
$979$	9.97245	0.318721
$980$	0	0
$981$	15.0225	0.479632
$982$	0	0
$983$	−47.6140	−1.51865	−0.759325	−	0.650711i	$-0.774471\pi$
$983$	−47.6140	−1.51865	−0.759325	+	0.650711i	$0.774471\pi$
$984$	0	0
$985$	−0.178606	−0.00569085
$986$	0	0
$987$	−27.6839	−0.881189
$988$	0	0
$989$	−58.0260	−1.84512
$990$	0	0
$991$	25.5023	0.810108	0.405054	−	0.914293i	$-0.367253\pi$
$991$	25.5023	0.810108	0.405054	+	0.914293i	$0.367253\pi$
$992$	0	0
$993$	−6.77401	−0.214967
$994$	0	0
$995$	−7.07367	−0.224250
$996$	0	0
$997$	−20.3487	−0.644449	−0.322225	−	0.946663i	$-0.604431\pi$
$997$	−20.3487	−0.644449	−0.322225	+	0.946663i	$0.604431\pi$
$998$	0	0
$999$	−9.69649	−0.306783

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8020.2.a.c.1.3	✓	28

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8020.2.a.c.1.3	✓	28	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)

\(f(q)\)	\(=\)	\(q-2.59496 q^{3} -1.00000 q^{5} +1.42420 q^{7} +3.73381 q^{9} +O(q^{10})\)	\(q-2.59496 q^{3} -1.00000 q^{5} +1.42420 q^{7} +3.73381 q^{9} +3.63795 q^{11} -2.47890 q^{13} +2.59496 q^{15} +4.00900 q^{17} +0.503905 q^{19} -3.69573 q^{21} -8.01771 q^{23} +1.00000 q^{25} -1.90422 q^{27} -1.71938 q^{29} -6.28565 q^{31} -9.44032 q^{33} -1.42420 q^{35} +5.09212 q^{37} +6.43265 q^{39} +1.37887 q^{41} +7.23723 q^{43} -3.73381 q^{45} +7.49079 q^{47} -4.97167 q^{49} -10.4032 q^{51} -7.78413 q^{53} -3.63795 q^{55} -1.30761 q^{57} -5.83373 q^{59} -9.44664 q^{61} +5.31768 q^{63} +2.47890 q^{65} +9.06165 q^{67} +20.8056 q^{69} -9.23923 q^{71} -7.53548 q^{73} -2.59496 q^{75} +5.18115 q^{77} -10.3976 q^{79} -6.26008 q^{81} +6.25543 q^{83} -4.00900 q^{85} +4.46173 q^{87} +2.74123 q^{89} -3.53044 q^{91} +16.3110 q^{93} -0.503905 q^{95} +9.29500 q^{97} +13.5834 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 28 q + 3 q^{3} - 28 q^{5} - 4 q^{7} + 17 q^{9}+O(q^{10}) \) 28 * q + 3 * q^3 - 28 * q^5 - 4 * q^7 + 17 * q^9	\( 28 q + 3 q^{3} - 28 q^{5} - 4 q^{7} + 17 q^{9} + 2 q^{11} + 3 q^{13} - 3 q^{15} - 10 q^{17} - 2 q^{19} - 12 q^{21} - 23 q^{23} + 28 q^{25} + 9 q^{27} - 37 q^{29} - 11 q^{31} + 2 q^{33} + 4 q^{35} - 3 q^{37} - 19 q^{39} - 30 q^{41} + 13 q^{43} - 17 q^{45} - 15 q^{47} + 12 q^{49} - 8 q^{51} - 35 q^{53} - 2 q^{55} - 22 q^{57} - q^{59} - 33 q^{61} - 20 q^{63} - 3 q^{65} + 19 q^{67} - 8 q^{69} - 31 q^{71} + 31 q^{73} + 3 q^{75} - 42 q^{77} - 29 q^{79} - 36 q^{81} + 14 q^{83} + 10 q^{85} - 32 q^{87} - 32 q^{89} - 7 q^{91} - 11 q^{93} + 2 q^{95} + 2 q^{97} - 39 q^{99}+O(q^{100}) \) 28 * q + 3 * q^3 - 28 * q^5 - 4 * q^7 + 17 * q^9 + 2 * q^11 + 3 * q^13 - 3 * q^15 - 10 * q^17 - 2 * q^19 - 12 * q^21 - 23 * q^23 + 28 * q^25 + 9 * q^27 - 37 * q^29 - 11 * q^31 + 2 * q^33 + 4 * q^35 - 3 * q^37 - 19 * q^39 - 30 * q^41 + 13 * q^43 - 17 * q^45 - 15 * q^47 + 12 * q^49 - 8 * q^51 - 35 * q^53 - 2 * q^55 - 22 * q^57 - q^59 - 33 * q^61 - 20 * q^63 - 3 * q^65 + 19 * q^67 - 8 * q^69 - 31 * q^71 + 31 * q^73 + 3 * q^75 - 42 * q^77 - 29 * q^79 - 36 * q^81 + 14 * q^83 + 10 * q^85 - 32 * q^87 - 32 * q^89 - 7 * q^91 - 11 * q^93 + 2 * q^95 + 2 * q^97 - 39 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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