LMFDB - Embedded newform 4020.2.a.f.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4020,2,Mod(1,4020)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4020, 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("4020.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4020 = 2^{2} \cdot 3 \cdot 5 \cdot 67 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4020.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:	$32.0998616126$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	$\mathbb{Q}[x]/(x^{6} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - x^{5} - 28x^{4} - 12x^{3} + 209x^{2} + 360x + 144 $ x^6 - x^5 - 28x^4 - 12x^3 + 209x^2 + 360x + 144
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-3.20623$ of defining polynomial
Character	$\chi$	$=$	4020.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} +1.00000 q^{5} -3.20623 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} +1.00000 q^{5} -3.20623 q^{7} +1.00000 q^{9} -4.50584 q^{11} +4.52637 q^{13} -1.00000 q^{15} -5.03221 q^{17} -0.977396 q^{19} +3.20623 q^{21} +0.658064 q^{23} +1.00000 q^{25} -1.00000 q^{27} -8.92117 q^{29} +3.82598 q^{31} +4.50584 q^{33} -3.20623 q^{35} +1.34194 q^{37} -4.52637 q^{39} +8.69234 q^{41} +6.73467 q^{43} +1.00000 q^{45} -7.39273 q^{47} +3.27989 q^{49} +5.03221 q^{51} +7.43506 q^{53} -4.50584 q^{55} +0.977396 q^{57} -2.52637 q^{59} +2.97740 q^{61} -3.20623 q^{63} +4.52637 q^{65} +1.00000 q^{67} -0.658064 q^{69} +2.16596 q^{71} -7.94377 q^{73} -1.00000 q^{75} +14.4467 q^{77} -6.32597 q^{79} +1.00000 q^{81} +3.79672 q^{83} -5.03221 q^{85} +8.92117 q^{87} +2.65518 q^{89} -14.5126 q^{91} -3.82598 q^{93} -0.977396 q^{95} +14.0755 q^{97} -4.50584 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 6 q^{3} + 6 q^{5} + q^{7} + 6 q^{9}+O(q^{10}) $ 6 * q - 6 * q^3 + 6 * q^5 + q^7 + 6 * q^9	$ 6 q - 6 q^{3} + 6 q^{5} + q^{7} + 6 q^{9} - 7 q^{11} + 7 q^{13} - 6 q^{15} + 10 q^{17} - 3 q^{19} - q^{21} - 4 q^{23} + 6 q^{25} - 6 q^{27} + 9 q^{29} + 3 q^{31} + 7 q^{33} + q^{35} + 16 q^{37} - 7 q^{39} + 7 q^{41} + 3 q^{43} + 6 q^{45} + q^{47} + 15 q^{49} - 10 q^{51} + 7 q^{53} - 7 q^{55} + 3 q^{57} + 5 q^{59} + 15 q^{61} + q^{63} + 7 q^{65} + 6 q^{67} + 4 q^{69} - 12 q^{71} + 12 q^{73} - 6 q^{75} + 9 q^{77} + 9 q^{79} + 6 q^{81} - 2 q^{83} + 10 q^{85} - 9 q^{87} + 10 q^{89} - 5 q^{91} - 3 q^{93} - 3 q^{95} + 37 q^{97} - 7 q^{99}+O(q^{100}) $ 6 * q - 6 * q^3 + 6 * q^5 + q^7 + 6 * q^9 - 7 * q^11 + 7 * q^13 - 6 * q^15 + 10 * q^17 - 3 * q^19 - q^21 - 4 * q^23 + 6 * q^25 - 6 * q^27 + 9 * q^29 + 3 * q^31 + 7 * q^33 + q^35 + 16 * q^37 - 7 * q^39 + 7 * q^41 + 3 * q^43 + 6 * q^45 + q^47 + 15 * q^49 - 10 * q^51 + 7 * q^53 - 7 * q^55 + 3 * q^57 + 5 * q^59 + 15 * q^61 + q^63 + 7 * q^65 + 6 * q^67 + 4 * q^69 - 12 * q^71 + 12 * q^73 - 6 * q^75 + 9 * q^77 + 9 * q^79 + 6 * q^81 - 2 * q^83 + 10 * q^85 - 9 * q^87 + 10 * q^89 - 5 * q^91 - 3 * q^93 - 3 * q^95 + 37 * 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
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	−3.20623	−1.21184	−0.605920	−	0.795526i	$-0.707195\pi$
$7$	−3.20623	−1.21184	−0.605920	+	0.795526i	$0.707195\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−4.50584	−1.35856	−0.679280	−	0.733879i	$-0.737708\pi$
$11$	−4.50584	−1.35856	−0.679280	+	0.733879i	$0.737708\pi$
$12$	0	0
$13$	4.52637	1.25539	0.627695	−	0.778459i	$-0.283998\pi$
$13$	4.52637	1.25539	0.627695	+	0.778459i	$0.283998\pi$
$14$	0	0
$15$	−1.00000	−0.258199
$16$	0	0
$17$	−5.03221	−1.22049	−0.610245	−	0.792213i	$-0.708929\pi$
$17$	−5.03221	−1.22049	−0.610245	+	0.792213i	$0.708929\pi$
$18$	0	0
$19$	−0.977396	−0.224230	−0.112115	−	0.993695i	$-0.535763\pi$
$19$	−0.977396	−0.224230	−0.112115	+	0.993695i	$0.535763\pi$
$20$	0	0
$21$	3.20623	0.699656
$22$	0	0
$23$	0.658064	0.137216	0.0686079	−	0.997644i	$-0.478144\pi$
$23$	0.658064	0.137216	0.0686079	+	0.997644i	$0.478144\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	−8.92117	−1.65662	−0.828310	−	0.560271i	$-0.810697\pi$
$29$	−8.92117	−1.65662	−0.828310	+	0.560271i	$0.810697\pi$
$30$	0	0
$31$	3.82598	0.687167	0.343583	−	0.939122i	$-0.388359\pi$
$31$	3.82598	0.687167	0.343583	+	0.939122i	$0.388359\pi$
$32$	0	0
$33$	4.50584	0.784365
$34$	0	0
$35$	−3.20623	−0.541951
$36$	0	0
$37$	1.34194	0.220613	0.110306	−	0.993898i	$-0.464817\pi$
$37$	1.34194	0.220613	0.110306	+	0.993898i	$0.464817\pi$
$38$	0	0
$39$	−4.52637	−0.724800
$40$	0	0
$41$	8.69234	1.35752	0.678758	−	0.734362i	$-0.262519\pi$
$41$	8.69234	1.35752	0.678758	+	0.734362i	$0.262519\pi$
$42$	0	0
$43$	6.73467	1.02703	0.513513	−	0.858082i	$-0.328344\pi$
$43$	6.73467	1.02703	0.513513	+	0.858082i	$0.328344\pi$
$44$	0	0
$45$	1.00000	0.149071
$46$	0	0
$47$	−7.39273	−1.07834	−0.539170	−	0.842197i	$-0.681262\pi$
$47$	−7.39273	−1.07834	−0.539170	+	0.842197i	$0.681262\pi$
$48$	0	0
$49$	3.27989	0.468555
$50$	0	0
$51$	5.03221	0.704650
$52$	0	0
$53$	7.43506	1.02128	0.510642	−	0.859794i	$-0.329408\pi$
$53$	7.43506	1.02128	0.510642	+	0.859794i	$0.329408\pi$
$54$	0	0
$55$	−4.50584	−0.607567
$56$	0	0
$57$	0.977396	0.129459
$58$	0	0
$59$	−2.52637	−0.328906	−0.164453	−	0.986385i	$-0.552586\pi$
$59$	−2.52637	−0.328906	−0.164453	+	0.986385i	$0.552586\pi$
$60$	0	0
$61$	2.97740	0.381217	0.190608	−	0.981666i	$-0.438954\pi$
$61$	2.97740	0.381217	0.190608	+	0.981666i	$0.438954\pi$
$62$	0	0
$63$	−3.20623	−0.403947
$64$	0	0
$65$	4.52637	0.561428
$66$	0	0
$67$	1.00000	0.122169
$67$	1.00000	0.122169
$68$	0	0
$69$	−0.658064	−0.0792216
$70$	0	0
$71$	2.16596	0.257053	0.128526	−	0.991706i	$-0.458975\pi$
$71$	2.16596	0.257053	0.128526	+	0.991706i	$0.458975\pi$
$72$	0	0
$73$	−7.94377	−0.929748	−0.464874	−	0.885377i	$-0.653900\pi$
$73$	−7.94377	−0.929748	−0.464874	+	0.885377i	$0.653900\pi$
$74$	0	0
$75$	−1.00000	−0.115470
$76$	0	0
$77$	14.4467	1.64636
$78$	0	0
$79$	−6.32597	−0.711727	−0.355864	−	0.934538i	$-0.615813\pi$
$79$	−6.32597	−0.711727	−0.355864	+	0.934538i	$0.615813\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	3.79672	0.416744	0.208372	−	0.978050i	$-0.433184\pi$
$83$	3.79672	0.416744	0.208372	+	0.978050i	$0.433184\pi$
$84$	0	0
$85$	−5.03221	−0.545820
$86$	0	0
$87$	8.92117	0.956450
$88$	0	0
$89$	2.65518	0.281449	0.140724	−	0.990049i	$-0.455057\pi$
$89$	2.65518	0.281449	0.140724	+	0.990049i	$0.455057\pi$
$90$	0	0
$91$	−14.5126	−1.52133
$92$	0	0
$93$	−3.82598	−0.396736
$94$	0	0
$95$	−0.977396	−0.100279
$96$	0	0
$97$	14.0755	1.42915	0.714573	−	0.699561i	$-0.246621\pi$
$97$	14.0755	1.42915	0.714573	+	0.699561i	$0.246621\pi$
$98$	0	0
$99$	−4.50584	−0.452853
$100$	0	0
$101$	5.11572	0.509034	0.254517	−	0.967068i	$-0.418084\pi$
$101$	5.11572	0.509034	0.254517	+	0.967068i	$0.418084\pi$
$102$	0	0
$103$	10.0508	0.990334	0.495167	−	0.868798i	$-0.335107\pi$
$103$	10.0508	0.990334	0.495167	+	0.868798i	$0.335107\pi$
$104$	0	0
$105$	3.20623	0.312896
$106$	0	0
$107$	−4.18650	−0.404725	−0.202362	−	0.979311i	$-0.564862\pi$
$107$	−4.18650	−0.404725	−0.202362	+	0.979311i	$0.564862\pi$
$108$	0	0
$109$	5.80248	0.555776	0.277888	−	0.960613i	$-0.410366\pi$
$109$	5.80248	0.555776	0.277888	+	0.960613i	$0.410366\pi$
$110$	0	0
$111$	−1.34194	−0.127371
$112$	0	0
$113$	0.789646	0.0742836	0.0371418	−	0.999310i	$-0.488175\pi$
$113$	0.789646	0.0742836	0.0371418	+	0.999310i	$0.488175\pi$
$114$	0	0
$115$	0.658064	0.0613648
$116$	0	0
$117$	4.52637	0.418463
$118$	0	0
$119$	16.1344	1.47904
$120$	0	0
$121$	9.30255	0.845686
$122$	0	0
$123$	−8.69234	−0.783762
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	−0.123529	−0.0109614	−0.00548070	−	0.999985i	$-0.501745\pi$
$127$	−0.123529	−0.0109614	−0.00548070	+	0.999985i	$0.501745\pi$
$128$	0	0
$129$	−6.73467	−0.592954
$130$	0	0
$131$	3.03221	0.264925	0.132463	−	0.991188i	$-0.457712\pi$
$131$	3.03221	0.264925	0.132463	+	0.991188i	$0.457712\pi$
$132$	0	0
$133$	3.13375	0.271731
$134$	0	0
$135$	−1.00000	−0.0860663
$136$	0	0
$137$	21.2283	1.81366	0.906829	−	0.421499i	$-0.138496\pi$
$137$	21.2283	1.81366	0.906829	+	0.421499i	$0.138496\pi$
$138$	0	0
$139$	−8.71701	−0.739367	−0.369683	−	0.929158i	$-0.620534\pi$
$139$	−8.71701	−0.739367	−0.369683	+	0.929158i	$0.620534\pi$
$140$	0	0
$141$	7.39273	0.622580
$142$	0	0
$143$	−20.3951	−1.70552
$144$	0	0
$145$	−8.92117	−0.740863
$146$	0	0
$147$	−3.27989	−0.270520
$148$	0	0
$149$	13.9907	1.14616	0.573081	−	0.819498i	$-0.305748\pi$
$149$	13.9907	1.14616	0.573081	+	0.819498i	$0.305748\pi$
$150$	0	0
$151$	18.5685	1.51109	0.755543	−	0.655099i	$-0.227373\pi$
$151$	18.5685	1.51109	0.755543	+	0.655099i	$0.227373\pi$
$152$	0	0
$153$	−5.03221	−0.406830
$154$	0	0
$155$	3.82598	0.307310
$156$	0	0
$157$	9.82321	0.783978	0.391989	−	0.919970i	$-0.371787\pi$
$157$	9.82321	0.783978	0.391989	+	0.919970i	$0.371787\pi$
$158$	0	0
$159$	−7.43506	−0.589638
$160$	0	0
$161$	−2.10990	−0.166284
$162$	0	0
$163$	9.52844	0.746325	0.373162	−	0.927766i	$-0.378273\pi$
$163$	9.52844	0.746325	0.373162	+	0.927766i	$0.378273\pi$
$164$	0	0
$165$	4.50584	0.350779
$166$	0	0
$167$	−15.0566	−1.16512	−0.582558	−	0.812789i	$-0.697948\pi$
$167$	−15.0566	−1.16512	−0.582558	+	0.812789i	$0.697948\pi$
$168$	0	0
$169$	7.48807	0.576005
$170$	0	0
$171$	−0.977396	−0.0747434
$172$	0	0
$173$	21.3542	1.62353	0.811763	−	0.583987i	$-0.198508\pi$
$173$	21.3542	1.62353	0.811763	+	0.583987i	$0.198508\pi$
$174$	0	0
$175$	−3.20623	−0.242368
$176$	0	0
$177$	2.52637	0.189894
$178$	0	0
$179$	−5.09920	−0.381132	−0.190566	−	0.981674i	$-0.561032\pi$
$179$	−5.09920	−0.381132	−0.190566	+	0.981674i	$0.561032\pi$
$180$	0	0
$181$	−13.9723	−1.03855	−0.519276	−	0.854607i	$-0.673798\pi$
$181$	−13.9723	−1.03855	−0.519276	+	0.854607i	$0.673798\pi$
$182$	0	0
$183$	−2.97740	−0.220095
$184$	0	0
$185$	1.34194	0.0986611
$186$	0	0
$187$	22.6743	1.65811
$188$	0	0
$189$	3.20623	0.233219
$190$	0	0
$191$	−27.3284	−1.97742	−0.988709	−	0.149851i	$-0.952120\pi$
$191$	−27.3284	−1.97742	−0.988709	+	0.149851i	$0.952120\pi$
$192$	0	0
$193$	8.75145	0.629943	0.314971	−	0.949101i	$-0.398005\pi$
$193$	8.75145	0.629943	0.314971	+	0.949101i	$0.398005\pi$
$194$	0	0
$195$	−4.52637	−0.324140
$196$	0	0
$197$	1.40951	0.100423	0.0502117	−	0.998739i	$-0.484010\pi$
$197$	1.40951	0.100423	0.0502117	+	0.998739i	$0.484010\pi$
$198$	0	0
$199$	−22.5515	−1.59863	−0.799315	−	0.600912i	$-0.794804\pi$
$199$	−22.5515	−1.59863	−0.799315	+	0.600912i	$0.794804\pi$
$200$	0	0
$201$	−1.00000	−0.0705346
$202$	0	0
$203$	28.6033	2.00756
$204$	0	0
$205$	8.69234	0.607099
$206$	0	0
$207$	0.658064	0.0457386
$208$	0	0
$209$	4.40399	0.304630
$210$	0	0
$211$	4.57053	0.314648	0.157324	−	0.987547i	$-0.449713\pi$
$211$	4.57053	0.314648	0.157324	+	0.987547i	$0.449713\pi$
$212$	0	0
$213$	−2.16596	−0.148409
$214$	0	0
$215$	6.73467	0.459300
$216$	0	0
$217$	−12.2670	−0.832736
$218$	0	0
$219$	7.94377	0.536790
$220$	0	0
$221$	−22.7777	−1.53219
$222$	0	0
$223$	20.5097	1.37343	0.686716	−	0.726926i	$-0.259052\pi$
$223$	20.5097	1.37343	0.686716	+	0.726926i	$0.259052\pi$
$224$	0	0
$225$	1.00000	0.0666667
$226$	0	0
$227$	2.02525	0.134420	0.0672102	−	0.997739i	$-0.478590\pi$
$227$	2.02525	0.134420	0.0672102	+	0.997739i	$0.478590\pi$
$228$	0	0
$229$	−7.58048	−0.500932	−0.250466	−	0.968125i	$-0.580584\pi$
$229$	−7.58048	−0.500932	−0.250466	+	0.968125i	$0.580584\pi$
$230$	0	0
$231$	−14.4467	−0.950525
$232$	0	0
$233$	2.58559	0.169388	0.0846939	−	0.996407i	$-0.473009\pi$
$233$	2.58559	0.169388	0.0846939	+	0.996407i	$0.473009\pi$
$234$	0	0
$235$	−7.39273	−0.482248
$236$	0	0
$237$	6.32597	0.410916
$238$	0	0
$239$	2.21115	0.143027	0.0715137	−	0.997440i	$-0.477217\pi$
$239$	2.21115	0.143027	0.0715137	+	0.997440i	$0.477217\pi$
$240$	0	0
$241$	3.04450	0.196113	0.0980567	−	0.995181i	$-0.468737\pi$
$241$	3.04450	0.196113	0.0980567	+	0.995181i	$0.468737\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	3.27989	0.209544
$246$	0	0
$247$	−4.42406	−0.281496
$248$	0	0
$249$	−3.79672	−0.240607
$250$	0	0
$251$	3.20230	0.202127	0.101064	−	0.994880i	$-0.467775\pi$
$251$	3.20230	0.202127	0.101064	+	0.994880i	$0.467775\pi$
$252$	0	0
$253$	−2.96513	−0.186416
$254$	0	0
$255$	5.03221	0.315129
$256$	0	0
$257$	13.2652	0.827458	0.413729	−	0.910400i	$-0.364226\pi$
$257$	13.2652	0.827458	0.413729	+	0.910400i	$0.364226\pi$
$258$	0	0
$259$	−4.30255	−0.267347
$260$	0	0
$261$	−8.92117	−0.552206
$262$	0	0
$263$	23.0612	1.42202	0.711008	−	0.703184i	$-0.248239\pi$
$263$	23.0612	1.42202	0.711008	+	0.703184i	$0.248239\pi$
$264$	0	0
$265$	7.43506	0.456732
$266$	0	0
$267$	−2.65518	−0.162495
$268$	0	0
$269$	13.8749	0.845971	0.422985	−	0.906137i	$-0.360982\pi$
$269$	13.8749	0.845971	0.422985	+	0.906137i	$0.360982\pi$
$270$	0	0
$271$	31.1034	1.88940	0.944698	−	0.327942i	$-0.106355\pi$
$271$	31.1034	1.88940	0.944698	+	0.327942i	$0.106355\pi$
$272$	0	0
$273$	14.5126	0.878341
$274$	0	0
$275$	−4.50584	−0.271712
$276$	0	0
$277$	−5.83847	−0.350800	−0.175400	−	0.984497i	$-0.556122\pi$
$277$	−5.83847	−0.350800	−0.175400	+	0.984497i	$0.556122\pi$
$278$	0	0
$279$	3.82598	0.229056
$280$	0	0
$281$	16.7473	0.999058	0.499529	−	0.866297i	$-0.333506\pi$
$281$	16.7473	0.999058	0.499529	+	0.866297i	$0.333506\pi$
$282$	0	0
$283$	11.4655	0.681551	0.340776	−	0.940145i	$-0.389310\pi$
$283$	11.4655	0.681551	0.340776	+	0.940145i	$0.389310\pi$
$284$	0	0
$285$	0.977396	0.0578960
$286$	0	0
$287$	−27.8696	−1.64509
$288$	0	0
$289$	8.32314	0.489596
$290$	0	0
$291$	−14.0755	−0.825118
$292$	0	0
$293$	−19.7530	−1.15398	−0.576991	−	0.816751i	$-0.695773\pi$
$293$	−19.7530	−1.15398	−0.576991	+	0.816751i	$0.695773\pi$
$294$	0	0
$295$	−2.52637	−0.147091
$296$	0	0
$297$	4.50584	0.261455
$298$	0	0
$299$	2.97864	0.172259
$300$	0	0
$301$	−21.5929	−1.24459
$302$	0	0
$303$	−5.11572	−0.293891
$304$	0	0
$305$	2.97740	0.170485
$306$	0	0
$307$	8.85648	0.505466	0.252733	−	0.967536i	$-0.418671\pi$
$307$	8.85648	0.505466	0.252733	+	0.967536i	$0.418671\pi$
$308$	0	0
$309$	−10.0508	−0.571770
$310$	0	0
$311$	−30.4905	−1.72896	−0.864479	−	0.502669i	$-0.832351\pi$
$311$	−30.4905	−1.72896	−0.864479	+	0.502669i	$0.832351\pi$
$312$	0	0
$313$	−5.31308	−0.300313	−0.150157	−	0.988662i	$-0.547978\pi$
$313$	−5.31308	−0.300313	−0.150157	+	0.988662i	$0.547978\pi$
$314$	0	0
$315$	−3.20623	−0.180650
$316$	0	0
$317$	−23.6219	−1.32674	−0.663368	−	0.748293i	$-0.730874\pi$
$317$	−23.6219	−1.32674	−0.663368	+	0.748293i	$0.730874\pi$
$318$	0	0
$319$	40.1973	2.25062
$320$	0	0
$321$	4.18650	0.233668
$322$	0	0
$323$	4.91846	0.273671
$324$	0	0
$325$	4.52637	0.251078
$326$	0	0
$327$	−5.80248	−0.320878
$328$	0	0
$329$	23.7028	1.30678
$330$	0	0
$331$	27.0210	1.48521	0.742604	−	0.669730i	$-0.233590\pi$
$331$	27.0210	1.48521	0.742604	+	0.669730i	$0.233590\pi$
$332$	0	0
$333$	1.34194	0.0735376
$334$	0	0
$335$	1.00000	0.0546358
$336$	0	0
$337$	32.1234	1.74987	0.874936	−	0.484239i	$-0.160904\pi$
$337$	32.1234	1.74987	0.874936	+	0.484239i	$0.160904\pi$
$338$	0	0
$339$	−0.789646	−0.0428877
$340$	0	0
$341$	−17.2393	−0.933558
$342$	0	0
$343$	11.9275	0.644026
$344$	0	0
$345$	−0.658064	−0.0354290
$346$	0	0
$347$	15.7752	0.846859	0.423429	−	0.905929i	$-0.360826\pi$
$347$	15.7752	0.846859	0.423429	+	0.905929i	$0.360826\pi$
$348$	0	0
$349$	−27.4656	−1.47020	−0.735100	−	0.677958i	$-0.762865\pi$
$349$	−27.4656	−1.47020	−0.735100	+	0.677958i	$0.762865\pi$
$350$	0	0
$351$	−4.52637	−0.241600
$352$	0	0
$353$	−18.0740	−0.961984	−0.480992	−	0.876725i	$-0.659723\pi$
$353$	−18.0740	−0.961984	−0.480992	+	0.876725i	$0.659723\pi$
$354$	0	0
$355$	2.16596	0.114957
$356$	0	0
$357$	−16.1344	−0.853923
$358$	0	0
$359$	16.9814	0.896242	0.448121	−	0.893973i	$-0.352093\pi$
$359$	16.9814	0.896242	0.448121	+	0.893973i	$0.352093\pi$
$360$	0	0
$361$	−18.0447	−0.949721
$362$	0	0
$363$	−9.30255	−0.488257
$364$	0	0
$365$	−7.94377	−0.415796
$366$	0	0
$367$	−13.1872	−0.688364	−0.344182	−	0.938903i	$-0.611844\pi$
$367$	−13.1872	−0.688364	−0.344182	+	0.938903i	$0.611844\pi$
$368$	0	0
$369$	8.69234	0.452505
$370$	0	0
$371$	−23.8385	−1.23763
$372$	0	0
$373$	32.4456	1.67997	0.839984	−	0.542611i	$-0.182564\pi$
$373$	32.4456	1.67997	0.839984	+	0.542611i	$0.182564\pi$
$374$	0	0
$375$	−1.00000	−0.0516398
$376$	0	0
$377$	−40.3805	−2.07970
$378$	0	0
$379$	7.58028	0.389373	0.194687	−	0.980866i	$-0.437631\pi$
$379$	7.58028	0.389373	0.194687	+	0.980866i	$0.437631\pi$
$380$	0	0
$381$	0.123529	0.00632857
$382$	0	0
$383$	27.2580	1.39282	0.696409	−	0.717645i	$-0.254780\pi$
$383$	27.2580	1.39282	0.696409	+	0.717645i	$0.254780\pi$
$384$	0	0
$385$	14.4467	0.736273
$386$	0	0
$387$	6.73467	0.342342
$388$	0	0
$389$	12.3643	0.626894	0.313447	−	0.949606i	$-0.398516\pi$
$389$	12.3643	0.626894	0.313447	+	0.949606i	$0.398516\pi$
$390$	0	0
$391$	−3.31152	−0.167471
$392$	0	0
$393$	−3.03221	−0.152955
$394$	0	0
$395$	−6.32597	−0.318294
$396$	0	0
$397$	19.6986	0.988646	0.494323	−	0.869278i	$-0.335416\pi$
$397$	19.6986	0.988646	0.494323	+	0.869278i	$0.335416\pi$
$398$	0	0
$399$	−3.13375	−0.156884
$400$	0	0
$401$	−21.8682	−1.09204	−0.546022	−	0.837771i	$-0.683858\pi$
$401$	−21.8682	−1.09204	−0.546022	+	0.837771i	$0.683858\pi$
$402$	0	0
$403$	17.3178	0.862663
$404$	0	0
$405$	1.00000	0.0496904
$406$	0	0
$407$	−6.04654	−0.299716
$408$	0	0
$409$	23.7300	1.17337	0.586687	−	0.809814i	$-0.300432\pi$
$409$	23.7300	1.17337	0.586687	+	0.809814i	$0.300432\pi$
$410$	0	0
$411$	−21.2283	−1.04712
$412$	0	0
$413$	8.10013	0.398581
$414$	0	0
$415$	3.79672	0.186373
$416$	0	0
$417$	8.71701	0.426874
$418$	0	0
$419$	11.2896	0.551531	0.275766	−	0.961225i	$-0.411069\pi$
$419$	11.2896	0.551531	0.275766	+	0.961225i	$0.411069\pi$
$420$	0	0
$421$	−27.6162	−1.34593	−0.672966	−	0.739674i	$-0.734980\pi$
$421$	−27.6162	−1.34593	−0.672966	+	0.739674i	$0.734980\pi$
$422$	0	0
$423$	−7.39273	−0.359447
$424$	0	0
$425$	−5.03221	−0.244098
$426$	0	0
$427$	−9.54621	−0.461973
$428$	0	0
$429$	20.3951	0.984685
$430$	0	0
$431$	−33.3523	−1.60652	−0.803261	−	0.595627i	$-0.796904\pi$
$431$	−33.3523	−1.60652	−0.803261	+	0.595627i	$0.796904\pi$
$432$	0	0
$433$	23.4743	1.12810	0.564052	−	0.825739i	$-0.309242\pi$
$433$	23.4743	1.12810	0.564052	+	0.825739i	$0.309242\pi$
$434$	0	0
$435$	8.92117	0.427737
$436$	0	0
$437$	−0.643189	−0.0307679
$438$	0	0
$439$	6.65815	0.317776	0.158888	−	0.987297i	$-0.449209\pi$
$439$	6.65815	0.317776	0.158888	+	0.987297i	$0.449209\pi$
$440$	0	0
$441$	3.27989	0.156185
$442$	0	0
$443$	30.5877	1.45327	0.726634	−	0.687025i	$-0.241084\pi$
$443$	30.5877	1.45327	0.726634	+	0.687025i	$0.241084\pi$
$444$	0	0
$445$	2.65518	0.125868
$446$	0	0
$447$	−13.9907	−0.661737
$448$	0	0
$449$	−15.5929	−0.735873	−0.367936	−	0.929851i	$-0.619936\pi$
$449$	−15.5929	−0.735873	−0.367936	+	0.929851i	$0.619936\pi$
$450$	0	0
$451$	−39.1662	−1.84427
$452$	0	0
$453$	−18.5685	−0.872426
$454$	0	0
$455$	−14.5126	−0.680360
$456$	0	0
$457$	22.5812	1.05630	0.528152	−	0.849150i	$-0.322885\pi$
$457$	22.5812	1.05630	0.528152	+	0.849150i	$0.322885\pi$
$458$	0	0
$459$	5.03221	0.234883
$460$	0	0
$461$	19.7037	0.917695	0.458847	−	0.888515i	$-0.348262\pi$
$461$	19.7037	0.917695	0.458847	+	0.888515i	$0.348262\pi$
$462$	0	0
$463$	13.3254	0.619285	0.309643	−	0.950853i	$-0.399791\pi$
$463$	13.3254	0.619285	0.309643	+	0.950853i	$0.399791\pi$
$464$	0	0
$465$	−3.82598	−0.177426
$466$	0	0
$467$	34.2109	1.58309	0.791547	−	0.611109i	$-0.209276\pi$
$467$	34.2109	1.58309	0.791547	+	0.611109i	$0.209276\pi$
$468$	0	0
$469$	−3.20623	−0.148050
$470$	0	0
$471$	−9.82321	−0.452630
$472$	0	0
$473$	−30.3453	−1.39528
$474$	0	0
$475$	−0.977396	−0.0448460
$476$	0	0
$477$	7.43506	0.340428
$478$	0	0
$479$	40.0805	1.83133	0.915663	−	0.401948i	$-0.131667\pi$
$479$	40.0805	1.83133	0.915663	+	0.401948i	$0.131667\pi$
$480$	0	0
$481$	6.07411	0.276955
$482$	0	0
$483$	2.10990	0.0960039
$484$	0	0
$485$	14.0755	0.639134
$486$	0	0
$487$	−10.5585	−0.478452	−0.239226	−	0.970964i	$-0.576894\pi$
$487$	−10.5585	−0.478452	−0.239226	+	0.970964i	$0.576894\pi$
$488$	0	0
$489$	−9.52844	−0.430891
$490$	0	0
$491$	3.65959	0.165155	0.0825775	−	0.996585i	$-0.473685\pi$
$491$	3.65959	0.165155	0.0825775	+	0.996585i	$0.473685\pi$
$492$	0	0
$493$	44.8932	2.02189
$494$	0	0
$495$	−4.50584	−0.202522
$496$	0	0
$497$	−6.94457	−0.311506
$498$	0	0
$499$	−14.9465	−0.669098	−0.334549	−	0.942378i	$-0.608584\pi$
$499$	−14.9465	−0.669098	−0.334549	+	0.942378i	$0.608584\pi$
$500$	0	0
$501$	15.0566	0.672680
$502$	0	0
$503$	−42.9454	−1.91484	−0.957421	−	0.288697i	$-0.906778\pi$
$503$	−42.9454	−1.91484	−0.957421	+	0.288697i	$0.906778\pi$
$504$	0	0
$505$	5.11572	0.227647
$506$	0	0
$507$	−7.48807	−0.332557
$508$	0	0
$509$	−31.2716	−1.38609	−0.693045	−	0.720894i	$-0.743731\pi$
$509$	−31.2716	−1.38609	−0.693045	+	0.720894i	$0.743731\pi$
$510$	0	0
$511$	25.4695	1.12671
$512$	0	0
$513$	0.977396	0.0431531
$514$	0	0
$515$	10.0508	0.442891
$516$	0	0
$517$	33.3104	1.46499
$518$	0	0
$519$	−21.3542	−0.937344
$520$	0	0
$521$	−38.6135	−1.69169	−0.845844	−	0.533431i	$-0.820902\pi$
$521$	−38.6135	−1.69169	−0.845844	+	0.533431i	$0.820902\pi$
$522$	0	0
$523$	34.0161	1.48742	0.743709	−	0.668503i	$-0.233065\pi$
$523$	34.0161	1.48742	0.743709	+	0.668503i	$0.233065\pi$
$524$	0	0
$525$	3.20623	0.139931
$526$	0	0
$527$	−19.2532	−0.838681
$528$	0	0
$529$	−22.5670	−0.981172
$530$	0	0
$531$	−2.52637	−0.109635
$532$	0	0
$533$	39.3448	1.70421
$534$	0	0
$535$	−4.18650	−0.180998
$536$	0	0
$537$	5.09920	0.220047
$538$	0	0
$539$	−14.7786	−0.636560
$540$	0	0
$541$	40.6243	1.74658	0.873288	−	0.487204i	$-0.161983\pi$
$541$	40.6243	1.74658	0.873288	+	0.487204i	$0.161983\pi$
$542$	0	0
$543$	13.9723	0.599608
$544$	0	0
$545$	5.80248	0.248551
$546$	0	0
$547$	−0.309058	−0.0132144	−0.00660719	−	0.999978i	$-0.502103\pi$
$547$	−0.309058	−0.0132144	−0.00660719	+	0.999978i	$0.502103\pi$
$548$	0	0
$549$	2.97740	0.127072
$550$	0	0
$551$	8.71952	0.371464
$552$	0	0
$553$	20.2825	0.862499
$554$	0	0
$555$	−1.34194	−0.0569620
$556$	0	0
$557$	−36.5906	−1.55039	−0.775197	−	0.631719i	$-0.782349\pi$
$557$	−36.5906	−1.55039	−0.775197	+	0.631719i	$0.782349\pi$
$558$	0	0
$559$	30.4836	1.28932
$560$	0	0
$561$	−22.6743	−0.957310
$562$	0	0
$563$	−34.6857	−1.46183	−0.730915	−	0.682469i	$-0.760906\pi$
$563$	−34.6857	−1.46183	−0.730915	+	0.682469i	$0.760906\pi$
$564$	0	0
$565$	0.789646	0.0332206
$566$	0	0
$567$	−3.20623	−0.134649
$568$	0	0
$569$	−0.105977	−0.00444279	−0.00222140	−	0.999998i	$-0.500707\pi$
$569$	−0.105977	−0.00444279	−0.00222140	+	0.999998i	$0.500707\pi$
$570$	0	0
$571$	25.8039	1.07986	0.539929	−	0.841710i	$-0.318451\pi$
$571$	25.8039	1.07986	0.539929	+	0.841710i	$0.318451\pi$
$572$	0	0
$573$	27.3284	1.14166
$574$	0	0
$575$	0.658064	0.0274432
$576$	0	0
$577$	−13.6825	−0.569611	−0.284806	−	0.958585i	$-0.591929\pi$
$577$	−13.6825	−0.569611	−0.284806	+	0.958585i	$0.591929\pi$
$578$	0	0
$579$	−8.75145	−0.363698
$580$	0	0
$581$	−12.1731	−0.505026
$582$	0	0
$583$	−33.5011	−1.38748
$584$	0	0
$585$	4.52637	0.187143
$586$	0	0
$587$	−31.9752	−1.31976	−0.659878	−	0.751373i	$-0.729392\pi$
$587$	−31.9752	−1.31976	−0.659878	+	0.751373i	$0.729392\pi$
$588$	0	0
$589$	−3.73950	−0.154084
$590$	0	0
$591$	−1.40951	−0.0579795
$592$	0	0
$593$	−1.47738	−0.0606688	−0.0303344	−	0.999540i	$-0.509657\pi$
$593$	−1.47738	−0.0606688	−0.0303344	+	0.999540i	$0.509657\pi$
$594$	0	0
$595$	16.1344	0.661446
$596$	0	0
$597$	22.5515	0.922970
$598$	0	0
$599$	−13.5756	−0.554683	−0.277342	−	0.960771i	$-0.589453\pi$
$599$	−13.5756	−0.554683	−0.277342	+	0.960771i	$0.589453\pi$
$600$	0	0
$601$	−26.2070	−1.06900	−0.534502	−	0.845167i	$-0.679501\pi$
$601$	−26.2070	−1.06900	−0.534502	+	0.845167i	$0.679501\pi$
$602$	0	0
$603$	1.00000	0.0407231
$604$	0	0
$605$	9.30255	0.378202
$606$	0	0
$607$	7.03894	0.285702	0.142851	−	0.989744i	$-0.454373\pi$
$607$	7.03894	0.285702	0.142851	+	0.989744i	$0.454373\pi$
$608$	0	0
$609$	−28.6033	−1.15906
$610$	0	0
$611$	−33.4623	−1.35374
$612$	0	0
$613$	−2.18938	−0.0884284	−0.0442142	−	0.999022i	$-0.514078\pi$
$613$	−2.18938	−0.0884284	−0.0442142	+	0.999022i	$0.514078\pi$
$614$	0	0
$615$	−8.69234	−0.350509
$616$	0	0
$617$	5.29721	0.213258	0.106629	−	0.994299i	$-0.465994\pi$
$617$	5.29721	0.213258	0.106629	+	0.994299i	$0.465994\pi$
$618$	0	0
$619$	−18.9138	−0.760210	−0.380105	−	0.924943i	$-0.624112\pi$
$619$	−18.9138	−0.760210	−0.380105	+	0.924943i	$0.624112\pi$
$620$	0	0
$621$	−0.658064	−0.0264072
$622$	0	0
$623$	−8.51312	−0.341071
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	−4.40399	−0.175878
$628$	0	0
$629$	−6.75290	−0.269256
$630$	0	0
$631$	2.44000	0.0971349	0.0485674	−	0.998820i	$-0.484534\pi$
$631$	2.44000	0.0971349	0.0485674	+	0.998820i	$0.484534\pi$
$632$	0	0
$633$	−4.57053	−0.181662
$634$	0	0
$635$	−0.123529	−0.00490209
$636$	0	0
$637$	14.8460	0.588220
$638$	0	0
$639$	2.16596	0.0856842
$640$	0	0
$641$	14.8101	0.584964	0.292482	−	0.956271i	$-0.405519\pi$
$641$	14.8101	0.584964	0.292482	+	0.956271i	$0.405519\pi$
$642$	0	0
$643$	−16.1889	−0.638429	−0.319215	−	0.947682i	$-0.603419\pi$
$643$	−16.1889	−0.638429	−0.319215	+	0.947682i	$0.603419\pi$
$644$	0	0
$645$	−6.73467	−0.265177
$646$	0	0
$647$	−6.95116	−0.273278	−0.136639	−	0.990621i	$-0.543630\pi$
$647$	−6.95116	−0.273278	−0.136639	+	0.990621i	$0.543630\pi$
$648$	0	0
$649$	11.3834	0.446839
$650$	0	0
$651$	12.2670	0.480780
$652$	0	0
$653$	−2.57695	−0.100844	−0.0504219	−	0.998728i	$-0.516057\pi$
$653$	−2.57695	−0.100844	−0.0504219	+	0.998728i	$0.516057\pi$
$654$	0	0
$655$	3.03221	0.118478
$656$	0	0
$657$	−7.94377	−0.309916
$658$	0	0
$659$	−29.0012	−1.12973	−0.564864	−	0.825184i	$-0.691071\pi$
$659$	−29.0012	−1.12973	−0.564864	+	0.825184i	$0.691071\pi$
$660$	0	0
$661$	−28.4668	−1.10723	−0.553615	−	0.832773i	$-0.686752\pi$
$661$	−28.4668	−1.10723	−0.553615	+	0.832773i	$0.686752\pi$
$662$	0	0
$663$	22.7777	0.884611
$664$	0	0
$665$	3.13375	0.121522
$666$	0	0
$667$	−5.87070	−0.227314
$668$	0	0
$669$	−20.5097	−0.792951
$670$	0	0
$671$	−13.4157	−0.517906
$672$	0	0
$673$	−43.7337	−1.68581	−0.842906	−	0.538061i	$-0.819157\pi$
$673$	−43.7337	−1.68581	−0.842906	+	0.538061i	$0.819157\pi$
$674$	0	0
$675$	−1.00000	−0.0384900
$676$	0	0
$677$	−50.1963	−1.92920	−0.964600	−	0.263719i	$-0.915051\pi$
$677$	−50.1963	−1.92920	−0.964600	+	0.263719i	$0.915051\pi$
$678$	0	0
$679$	−45.1291	−1.73190
$680$	0	0
$681$	−2.02525	−0.0776077
$682$	0	0
$683$	−14.1490	−0.541396	−0.270698	−	0.962664i	$-0.587254\pi$
$683$	−14.1490	−0.541396	−0.270698	+	0.962664i	$0.587254\pi$
$684$	0	0
$685$	21.2283	0.811092
$686$	0	0
$687$	7.58048	0.289213
$688$	0	0
$689$	33.6538	1.28211
$690$	0	0
$691$	−4.60163	−0.175054	−0.0875272	−	0.996162i	$-0.527896\pi$
$691$	−4.60163	−0.175054	−0.0875272	+	0.996162i	$0.527896\pi$
$692$	0	0
$693$	14.4467	0.548786
$694$	0	0
$695$	−8.71701	−0.330655
$696$	0	0
$697$	−43.7417	−1.65683
$698$	0	0
$699$	−2.58559	−0.0977960
$700$	0	0
$701$	23.4844	0.886992	0.443496	−	0.896276i	$-0.353738\pi$
$701$	23.4844	0.886992	0.443496	+	0.896276i	$0.353738\pi$
$702$	0	0
$703$	−1.31160	−0.0494681
$704$	0	0
$705$	7.39273	0.278426
$706$	0	0
$707$	−16.4022	−0.616867
$708$	0	0
$709$	32.4138	1.21733	0.608664	−	0.793428i	$-0.291706\pi$
$709$	32.4138	1.21733	0.608664	+	0.793428i	$0.291706\pi$
$710$	0	0
$711$	−6.32597	−0.237242
$712$	0	0
$713$	2.51774	0.0942902
$714$	0	0
$715$	−20.3951	−0.762733
$716$	0	0
$717$	−2.21115	−0.0825769
$718$	0	0
$719$	18.2775	0.681637	0.340818	−	0.940129i	$-0.389296\pi$
$719$	18.2775	0.681637	0.340818	+	0.940129i	$0.389296\pi$
$720$	0	0
$721$	−32.2251	−1.20013
$722$	0	0
$723$	−3.04450	−0.113226
$724$	0	0
$725$	−8.92117	−0.331324
$726$	0	0
$727$	26.0004	0.964300	0.482150	−	0.876089i	$-0.339856\pi$
$727$	26.0004	0.964300	0.482150	+	0.876089i	$0.339856\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−33.8902	−1.25348
$732$	0	0
$733$	−18.0630	−0.667173	−0.333586	−	0.942720i	$-0.608259\pi$
$733$	−18.0630	−0.667173	−0.333586	+	0.942720i	$0.608259\pi$
$734$	0	0
$735$	−3.27989	−0.120980
$736$	0	0
$737$	−4.50584	−0.165975
$738$	0	0
$739$	−41.4072	−1.52319	−0.761594	−	0.648055i	$-0.775583\pi$
$739$	−41.4072	−1.52319	−0.761594	+	0.648055i	$0.775583\pi$
$740$	0	0
$741$	4.42406	0.162522
$742$	0	0
$743$	−8.85941	−0.325020	−0.162510	−	0.986707i	$-0.551959\pi$
$743$	−8.85941	−0.325020	−0.162510	+	0.986707i	$0.551959\pi$
$744$	0	0
$745$	13.9907	0.512580
$746$	0	0
$747$	3.79672	0.138915
$748$	0	0
$749$	13.4229	0.490461
$750$	0	0
$751$	1.04841	0.0382571	0.0191285	−	0.999817i	$-0.493911\pi$
$751$	1.04841	0.0382571	0.0191285	+	0.999817i	$0.493911\pi$
$752$	0	0
$753$	−3.20230	−0.116698
$754$	0	0
$755$	18.5685	0.675778
$756$	0	0
$757$	−42.6015	−1.54838	−0.774188	−	0.632956i	$-0.781842\pi$
$757$	−42.6015	−1.54838	−0.774188	+	0.632956i	$0.781842\pi$
$758$	0	0
$759$	2.96513	0.107627
$760$	0	0
$761$	2.88766	0.104678	0.0523389	−	0.998629i	$-0.483332\pi$
$761$	2.88766	0.104678	0.0523389	+	0.998629i	$0.483332\pi$
$762$	0	0
$763$	−18.6041	−0.673512
$764$	0	0
$765$	−5.03221	−0.181940
$766$	0	0
$767$	−11.4353	−0.412905
$768$	0	0
$769$	−6.04878	−0.218124	−0.109062	−	0.994035i	$-0.534785\pi$
$769$	−6.04878	−0.218124	−0.109062	+	0.994035i	$0.534785\pi$
$770$	0	0
$771$	−13.2652	−0.477733
$772$	0	0
$773$	−20.1585	−0.725050	−0.362525	−	0.931974i	$-0.618085\pi$
$773$	−20.1585	−0.725050	−0.362525	+	0.931974i	$0.618085\pi$
$774$	0	0
$775$	3.82598	0.137433
$776$	0	0
$777$	4.30255	0.154353
$778$	0	0
$779$	−8.49586	−0.304396
$780$	0	0
$781$	−9.75947	−0.349221
$782$	0	0
$783$	8.92117	0.318817
$784$	0	0
$785$	9.82321	0.350605
$786$	0	0
$787$	−12.1758	−0.434022	−0.217011	−	0.976169i	$-0.569631\pi$
$787$	−12.1758	−0.434022	−0.217011	+	0.976169i	$0.569631\pi$
$788$	0	0
$789$	−23.0612	−0.821001
$790$	0	0
$791$	−2.53178	−0.0900198
$792$	0	0
$793$	13.4768	0.478576
$794$	0	0
$795$	−7.43506	−0.263694
$796$	0	0
$797$	−10.0293	−0.355256	−0.177628	−	0.984098i	$-0.556842\pi$
$797$	−10.0293	−0.355256	−0.177628	+	0.984098i	$0.556842\pi$
$798$	0	0
$799$	37.2018	1.31610
$800$	0	0
$801$	2.65518	0.0938163
$802$	0	0
$803$	35.7933	1.26312
$804$	0	0
$805$	−2.10990	−0.0743643
$806$	0	0
$807$	−13.8749	−0.488421
$808$	0	0
$809$	17.0590	0.599763	0.299882	−	0.953976i	$-0.403053\pi$
$809$	17.0590	0.599763	0.299882	+	0.953976i	$0.403053\pi$
$810$	0	0
$811$	−43.0346	−1.51115	−0.755575	−	0.655062i	$-0.772643\pi$
$811$	−43.0346	−1.51115	−0.755575	+	0.655062i	$0.772643\pi$
$812$	0	0
$813$	−31.1034	−1.09084
$814$	0	0
$815$	9.52844	0.333767
$816$	0	0
$817$	−6.58244	−0.230290
$818$	0	0
$819$	−14.5126	−0.507111
$820$	0	0
$821$	−5.32830	−0.185959	−0.0929794	−	0.995668i	$-0.529639\pi$
$821$	−5.32830	−0.185959	−0.0929794	+	0.995668i	$0.529639\pi$
$822$	0	0
$823$	−9.89946	−0.345074	−0.172537	−	0.985003i	$-0.555196\pi$
$823$	−9.89946	−0.345074	−0.172537	+	0.985003i	$0.555196\pi$
$824$	0	0
$825$	4.50584	0.156873
$826$	0	0
$827$	−10.9295	−0.380056	−0.190028	−	0.981779i	$-0.560858\pi$
$827$	−10.9295	−0.380056	−0.190028	+	0.981779i	$0.560858\pi$
$828$	0	0
$829$	−17.8573	−0.620209	−0.310105	−	0.950702i	$-0.600364\pi$
$829$	−17.8573	−0.620209	−0.310105	+	0.950702i	$0.600364\pi$
$830$	0	0
$831$	5.83847	0.202534
$832$	0	0
$833$	−16.5051	−0.571867
$834$	0	0
$835$	−15.0566	−0.521056
$836$	0	0
$837$	−3.82598	−0.132245
$838$	0	0
$839$	19.0130	0.656402	0.328201	−	0.944608i	$-0.393558\pi$
$839$	19.0130	0.656402	0.328201	+	0.944608i	$0.393558\pi$
$840$	0	0
$841$	50.5872	1.74439
$842$	0	0
$843$	−16.7473	−0.576806
$844$	0	0
$845$	7.48807	0.257597
$846$	0	0
$847$	−29.8261	−1.02484
$848$	0	0
$849$	−11.4655	−0.393494
$850$	0	0
$851$	0.883080	0.0302716
$852$	0	0
$853$	21.6416	0.740995	0.370498	−	0.928833i	$-0.379187\pi$
$853$	21.6416	0.740995	0.370498	+	0.928833i	$0.379187\pi$
$854$	0	0
$855$	−0.977396	−0.0334262
$856$	0	0
$857$	15.6163	0.533442	0.266721	−	0.963774i	$-0.414060\pi$
$857$	15.6163	0.533442	0.266721	+	0.963774i	$0.414060\pi$
$858$	0	0
$859$	−7.71300	−0.263164	−0.131582	−	0.991305i	$-0.542006\pi$
$859$	−7.71300	−0.263164	−0.131582	+	0.991305i	$0.542006\pi$
$860$	0	0
$861$	27.8696	0.949793
$862$	0	0
$863$	−18.1023	−0.616208	−0.308104	−	0.951353i	$-0.599694\pi$
$863$	−18.1023	−0.616208	−0.308104	+	0.951353i	$0.599694\pi$
$864$	0	0
$865$	21.3542	0.726063
$866$	0	0
$867$	−8.32314	−0.282669
$868$	0	0
$869$	28.5038	0.966925
$870$	0	0
$871$	4.52637	0.153370
$872$	0	0
$873$	14.0755	0.476382
$874$	0	0
$875$	−3.20623	−0.108390
$876$	0	0
$877$	17.2995	0.584164	0.292082	−	0.956393i	$-0.405652\pi$
$877$	17.2995	0.584164	0.292082	+	0.956393i	$0.405652\pi$
$878$	0	0
$879$	19.7530	0.666251
$880$	0	0
$881$	26.3412	0.887459	0.443729	−	0.896161i	$-0.353655\pi$
$881$	26.3412	0.887459	0.443729	+	0.896161i	$0.353655\pi$
$882$	0	0
$883$	−39.1845	−1.31866	−0.659332	−	0.751852i	$-0.729161\pi$
$883$	−39.1845	−1.31866	−0.659332	+	0.751852i	$0.729161\pi$
$884$	0	0
$885$	2.52637	0.0849232
$886$	0	0
$887$	45.3412	1.52241	0.761205	−	0.648512i	$-0.224608\pi$
$887$	45.3412	1.52241	0.761205	+	0.648512i	$0.224608\pi$
$888$	0	0
$889$	0.396061	0.0132835
$890$	0	0
$891$	−4.50584	−0.150951
$892$	0	0
$893$	7.22563	0.241796
$894$	0	0
$895$	−5.09920	−0.170448
$896$	0	0
$897$	−2.97864	−0.0994540
$898$	0	0
$899$	−34.1322	−1.13837
$900$	0	0
$901$	−37.4148	−1.24647
$902$	0	0
$903$	21.5929	0.718565
$904$	0	0
$905$	−13.9723	−0.464454
$906$	0	0
$907$	9.06439	0.300978	0.150489	−	0.988612i	$-0.451915\pi$
$907$	9.06439	0.300978	0.150489	+	0.988612i	$0.451915\pi$
$908$	0	0
$909$	5.11572	0.169678
$910$	0	0
$911$	−18.4259	−0.610478	−0.305239	−	0.952276i	$-0.598736\pi$
$911$	−18.4259	−0.610478	−0.305239	+	0.952276i	$0.598736\pi$
$912$	0	0
$913$	−17.1074	−0.566171
$914$	0	0
$915$	−2.97740	−0.0984297
$916$	0	0
$917$	−9.72195	−0.321047
$918$	0	0
$919$	29.1006	0.959940	0.479970	−	0.877285i	$-0.340648\pi$
$919$	29.1006	0.959940	0.479970	+	0.877285i	$0.340648\pi$
$920$	0	0
$921$	−8.85648	−0.291831
$922$	0	0
$923$	9.80396	0.322701
$924$	0	0
$925$	1.34194	0.0441226
$926$	0	0
$927$	10.0508	0.330111
$928$	0	0
$929$	−12.1144	−0.397462	−0.198731	−	0.980054i	$-0.563682\pi$
$929$	−12.1144	−0.397462	−0.198731	+	0.980054i	$0.563682\pi$
$930$	0	0
$931$	−3.20575	−0.105064
$932$	0	0
$933$	30.4905	0.998214
$934$	0	0
$935$	22.6743	0.741529
$936$	0	0
$937$	−18.1693	−0.593564	−0.296782	−	0.954945i	$-0.595913\pi$
$937$	−18.1693	−0.593564	−0.296782	+	0.954945i	$0.595913\pi$
$938$	0	0
$939$	5.31308	0.173386
$940$	0	0
$941$	38.8154	1.26534	0.632672	−	0.774420i	$-0.281958\pi$
$941$	38.8154	1.26534	0.632672	+	0.774420i	$0.281958\pi$
$942$	0	0
$943$	5.72011	0.186273
$944$	0	0
$945$	3.20623	0.104299
$946$	0	0
$947$	18.1439	0.589598	0.294799	−	0.955559i	$-0.404747\pi$
$947$	18.1439	0.589598	0.294799	+	0.955559i	$0.404747\pi$
$948$	0	0
$949$	−35.9565	−1.16720
$950$	0	0
$951$	23.6219	0.765991
$952$	0	0
$953$	−16.4330	−0.532316	−0.266158	−	0.963929i	$-0.585754\pi$
$953$	−16.4330	−0.532316	−0.266158	+	0.963929i	$0.585754\pi$
$954$	0	0
$955$	−27.3284	−0.884328
$956$	0	0
$957$	−40.1973	−1.29939
$958$	0	0
$959$	−68.0628	−2.19786
$960$	0	0
$961$	−16.3618	−0.527802
$962$	0	0
$963$	−4.18650	−0.134908
$964$	0	0
$965$	8.75145	0.281719
$966$	0	0
$967$	−13.5182	−0.434716	−0.217358	−	0.976092i	$-0.569744\pi$
$967$	−13.5182	−0.434716	−0.217358	+	0.976092i	$0.569744\pi$
$968$	0	0
$969$	−4.91846	−0.158004
$970$	0	0
$971$	−10.9829	−0.352457	−0.176229	−	0.984349i	$-0.556390\pi$
$971$	−10.9829	−0.352457	−0.176229	+	0.984349i	$0.556390\pi$
$972$	0	0
$973$	27.9487	0.895994
$974$	0	0
$975$	−4.52637	−0.144960
$976$	0	0
$977$	10.3170	0.330069	0.165035	−	0.986288i	$-0.447226\pi$
$977$	10.3170	0.330069	0.165035	+	0.986288i	$0.447226\pi$
$978$	0	0
$979$	−11.9638	−0.382365
$980$	0	0
$981$	5.80248	0.185259
$982$	0	0
$983$	32.0054	1.02081	0.510407	−	0.859933i	$-0.329495\pi$
$983$	32.0054	1.02081	0.510407	+	0.859933i	$0.329495\pi$
$984$	0	0
$985$	1.40951	0.0449107
$986$	0	0
$987$	−23.7028	−0.754467
$988$	0	0
$989$	4.43184	0.140924
$990$	0	0
$991$	4.66987	0.148343	0.0741717	−	0.997245i	$-0.476369\pi$
$991$	4.66987	0.148343	0.0741717	+	0.997245i	$0.476369\pi$
$992$	0	0
$993$	−27.0210	−0.857486
$994$	0	0
$995$	−22.5515	−0.714929
$996$	0	0
$997$	13.6260	0.431539	0.215770	−	0.976444i	$-0.430774\pi$
$997$	13.6260	0.431539	0.215770	+	0.976444i	$0.430774\pi$
$998$	0	0
$999$	−1.34194	−0.0424570

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4020.2.a.f.1.1	✓	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4020.2.a.f.1.1	✓	6	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:	\( N \)	\(=\)	\( 4020 = 2^{2} \cdot 3 \cdot 5 \cdot 67 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4020.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} +1.00000 q^{5} -3.20623 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} +1.00000 q^{5} -3.20623 q^{7} +1.00000 q^{9} -4.50584 q^{11} +4.52637 q^{13} -1.00000 q^{15} -5.03221 q^{17} -0.977396 q^{19} +3.20623 q^{21} +0.658064 q^{23} +1.00000 q^{25} -1.00000 q^{27} -8.92117 q^{29} +3.82598 q^{31} +4.50584 q^{33} -3.20623 q^{35} +1.34194 q^{37} -4.52637 q^{39} +8.69234 q^{41} +6.73467 q^{43} +1.00000 q^{45} -7.39273 q^{47} +3.27989 q^{49} +5.03221 q^{51} +7.43506 q^{53} -4.50584 q^{55} +0.977396 q^{57} -2.52637 q^{59} +2.97740 q^{61} -3.20623 q^{63} +4.52637 q^{65} +1.00000 q^{67} -0.658064 q^{69} +2.16596 q^{71} -7.94377 q^{73} -1.00000 q^{75} +14.4467 q^{77} -6.32597 q^{79} +1.00000 q^{81} +3.79672 q^{83} -5.03221 q^{85} +8.92117 q^{87} +2.65518 q^{89} -14.5126 q^{91} -3.82598 q^{93} -0.977396 q^{95} +14.0755 q^{97} -4.50584 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 6 q^{3} + 6 q^{5} + q^{7} + 6 q^{9}+O(q^{10}) \) 6 * q - 6 * q^3 + 6 * q^5 + q^7 + 6 * q^9	\( 6 q - 6 q^{3} + 6 q^{5} + q^{7} + 6 q^{9} - 7 q^{11} + 7 q^{13} - 6 q^{15} + 10 q^{17} - 3 q^{19} - q^{21} - 4 q^{23} + 6 q^{25} - 6 q^{27} + 9 q^{29} + 3 q^{31} + 7 q^{33} + q^{35} + 16 q^{37} - 7 q^{39} + 7 q^{41} + 3 q^{43} + 6 q^{45} + q^{47} + 15 q^{49} - 10 q^{51} + 7 q^{53} - 7 q^{55} + 3 q^{57} + 5 q^{59} + 15 q^{61} + q^{63} + 7 q^{65} + 6 q^{67} + 4 q^{69} - 12 q^{71} + 12 q^{73} - 6 q^{75} + 9 q^{77} + 9 q^{79} + 6 q^{81} - 2 q^{83} + 10 q^{85} - 9 q^{87} + 10 q^{89} - 5 q^{91} - 3 q^{93} - 3 q^{95} + 37 q^{97} - 7 q^{99}+O(q^{100}) \) 6 * q - 6 * q^3 + 6 * q^5 + q^7 + 6 * q^9 - 7 * q^11 + 7 * q^13 - 6 * q^15 + 10 * q^17 - 3 * q^19 - q^21 - 4 * q^23 + 6 * q^25 - 6 * q^27 + 9 * q^29 + 3 * q^31 + 7 * q^33 + q^35 + 16 * q^37 - 7 * q^39 + 7 * q^41 + 3 * q^43 + 6 * q^45 + q^47 + 15 * q^49 - 10 * q^51 + 7 * q^53 - 7 * q^55 + 3 * q^57 + 5 * q^59 + 15 * q^61 + q^63 + 7 * q^65 + 6 * q^67 + 4 * q^69 - 12 * q^71 + 12 * q^73 - 6 * q^75 + 9 * q^77 + 9 * q^79 + 6 * q^81 - 2 * q^83 + 10 * q^85 - 9 * q^87 + 10 * q^89 - 5 * q^91 - 3 * q^93 - 3 * q^95 + 37 * q^97 - 7 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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