LMFDB - Embedded newform 4012.2.a.j.1.10

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4012,2,Mod(1,4012)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4012, 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("4012.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4012 = 2^{2} \cdot 17 \cdot 59 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4012.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:	$32.0359812909$
Analytic rank:	$0$
Dimension:	$21$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.10
Character	$\chi$	$=$	4012.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.210829 q^{3} +0.593138 q^{5} -1.60715 q^{7} -2.95555 q^{9} +O(q^{10})$	$q+0.210829 q^{3} +0.593138 q^{5} -1.60715 q^{7} -2.95555 q^{9} -2.16189 q^{11} +2.74022 q^{13} +0.125051 q^{15} +1.00000 q^{17} +0.223492 q^{19} -0.338835 q^{21} +1.30075 q^{23} -4.64819 q^{25} -1.25560 q^{27} +9.53334 q^{29} +5.20344 q^{31} -0.455789 q^{33} -0.953263 q^{35} -0.872287 q^{37} +0.577718 q^{39} -10.2875 q^{41} +0.790551 q^{43} -1.75305 q^{45} +1.57231 q^{47} -4.41706 q^{49} +0.210829 q^{51} -5.39157 q^{53} -1.28230 q^{55} +0.0471186 q^{57} +1.00000 q^{59} +12.4593 q^{61} +4.75002 q^{63} +1.62533 q^{65} +14.6742 q^{67} +0.274236 q^{69} +13.3754 q^{71} -4.98603 q^{73} -0.979974 q^{75} +3.47448 q^{77} -8.57203 q^{79} +8.60194 q^{81} +10.9011 q^{83} +0.593138 q^{85} +2.00991 q^{87} +6.17014 q^{89} -4.40395 q^{91} +1.09704 q^{93} +0.132561 q^{95} +2.10120 q^{97} +6.38957 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 21 q + 9 q^{3} + q^{5} + 11 q^{7} + 26 q^{9}+O(q^{10}) $ 21 * q + 9 * q^3 + q^5 + 11 * q^7 + 26 * q^9	$ 21 q + 9 q^{3} + q^{5} + 11 q^{7} + 26 q^{9} + 12 q^{11} - 4 q^{13} + 9 q^{15} + 21 q^{17} + 4 q^{19} + 8 q^{21} + 19 q^{23} + 26 q^{25} + 33 q^{27} - q^{29} + 13 q^{31} + 11 q^{33} + 15 q^{35} - 4 q^{37} + 18 q^{39} + 9 q^{41} + 7 q^{43} + 7 q^{45} + 33 q^{47} + 36 q^{49} + 9 q^{51} + 7 q^{53} + 12 q^{55} + 26 q^{57} + 21 q^{59} + 3 q^{61} + 51 q^{63} + q^{65} + 8 q^{69} + 55 q^{71} + 22 q^{73} + 14 q^{75} - 15 q^{77} + 28 q^{79} + 25 q^{81} + 54 q^{83} + q^{85} + 34 q^{87} + 30 q^{89} + 35 q^{91} - 5 q^{93} + 42 q^{95} + 9 q^{97} + 20 q^{99}+O(q^{100}) $ 21 * q + 9 * q^3 + q^5 + 11 * q^7 + 26 * q^9 + 12 * q^11 - 4 * q^13 + 9 * q^15 + 21 * q^17 + 4 * q^19 + 8 * q^21 + 19 * q^23 + 26 * q^25 + 33 * q^27 - q^29 + 13 * q^31 + 11 * q^33 + 15 * q^35 - 4 * q^37 + 18 * q^39 + 9 * q^41 + 7 * q^43 + 7 * q^45 + 33 * q^47 + 36 * q^49 + 9 * q^51 + 7 * q^53 + 12 * q^55 + 26 * q^57 + 21 * q^59 + 3 * q^61 + 51 * q^63 + q^65 + 8 * q^69 + 55 * q^71 + 22 * q^73 + 14 * q^75 - 15 * q^77 + 28 * q^79 + 25 * q^81 + 54 * q^83 + q^85 + 34 * q^87 + 30 * q^89 + 35 * q^91 - 5 * q^93 + 42 * q^95 + 9 * q^97 + 20 * 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
$3$	0.210829	0.121722	0.0608611	−	0.998146i	$-0.480615\pi$
$3$	0.210829	0.121722	0.0608611	+	0.998146i	$0.480615\pi$
$4$	0	0
$5$	0.593138	0.265259	0.132630	−	0.991166i	$-0.457658\pi$
$5$	0.593138	0.265259	0.132630	+	0.991166i	$0.457658\pi$
$6$	0	0
$7$	−1.60715	−0.607447	−0.303723	−	0.952760i	$-0.598230\pi$
$7$	−1.60715	−0.607447	−0.303723	+	0.952760i	$0.598230\pi$
$8$	0	0
$9$	−2.95555	−0.985184
$10$	0	0
$11$	−2.16189	−0.651833	−0.325917	−	0.945399i	$-0.605673\pi$
$11$	−2.16189	−0.651833	−0.325917	+	0.945399i	$0.605673\pi$
$12$	0	0
$13$	2.74022	0.760000	0.380000	−	0.924987i	$-0.375924\pi$
$13$	2.74022	0.760000	0.380000	+	0.924987i	$0.375924\pi$
$14$	0	0
$15$	0.125051	0.0322880
$16$	0	0
$17$	1.00000	0.242536
$17$	1.00000	0.242536
$18$	0	0
$19$	0.223492	0.0512725	0.0256363	−	0.999671i	$-0.491839\pi$
$19$	0.223492	0.0512725	0.0256363	+	0.999671i	$0.491839\pi$
$20$	0	0
$21$	−0.338835	−0.0739398
$22$	0	0
$23$	1.30075	0.271225	0.135613	−	0.990762i	$-0.456700\pi$
$23$	1.30075	0.271225	0.135613	+	0.990762i	$0.456700\pi$
$24$	0	0
$25$	−4.64819	−0.929638
$26$	0	0
$27$	−1.25560	−0.241641
$28$	0	0
$29$	9.53334	1.77030	0.885149	−	0.465308i	$-0.154056\pi$
$29$	9.53334	1.77030	0.885149	+	0.465308i	$0.154056\pi$
$30$	0	0
$31$	5.20344	0.934565	0.467283	−	0.884108i	$-0.345233\pi$
$31$	5.20344	0.934565	0.467283	+	0.884108i	$0.345233\pi$
$32$	0	0
$33$	−0.455789	−0.0793426
$34$	0	0
$35$	−0.953263	−0.161131
$36$	0	0
$37$	−0.872287	−0.143403	−0.0717015	−	0.997426i	$-0.522843\pi$
$37$	−0.872287	−0.143403	−0.0717015	+	0.997426i	$0.522843\pi$
$38$	0	0
$39$	0.577718	0.0925089
$40$	0	0
$41$	−10.2875	−1.60663	−0.803316	−	0.595553i	$-0.796933\pi$
$41$	−10.2875	−1.60663	−0.803316	+	0.595553i	$0.796933\pi$
$42$	0	0
$43$	0.790551	0.120558	0.0602789	−	0.998182i	$-0.480801\pi$
$43$	0.790551	0.120558	0.0602789	+	0.998182i	$0.480801\pi$
$44$	0	0
$45$	−1.75305	−0.261329
$46$	0	0
$47$	1.57231	0.229345	0.114672	−	0.993403i	$-0.463418\pi$
$47$	1.57231	0.229345	0.114672	+	0.993403i	$0.463418\pi$
$48$	0	0
$49$	−4.41706	−0.631009
$50$	0	0
$51$	0.210829	0.0295220
$52$	0	0
$53$	−5.39157	−0.740590	−0.370295	−	0.928914i	$-0.620743\pi$
$53$	−5.39157	−0.740590	−0.370295	+	0.928914i	$0.620743\pi$
$54$	0	0
$55$	−1.28230	−0.172905
$56$	0	0
$57$	0.0471186	0.00624101
$58$	0	0
$59$	1.00000	0.130189
$59$	1.00000	0.130189
$60$	0	0
$61$	12.4593	1.59525	0.797625	−	0.603154i	$-0.206090\pi$
$61$	12.4593	1.59525	0.797625	+	0.603154i	$0.206090\pi$
$62$	0	0
$63$	4.75002	0.598447
$64$	0	0
$65$	1.62533	0.201597
$66$	0	0
$67$	14.6742	1.79274	0.896368	−	0.443311i	$-0.146196\pi$
$67$	14.6742	1.79274	0.896368	+	0.443311i	$0.146196\pi$
$68$	0	0
$69$	0.274236	0.0330142
$70$	0	0
$71$	13.3754	1.58737	0.793684	−	0.608330i	$-0.208160\pi$
$71$	13.3754	1.58737	0.793684	+	0.608330i	$0.208160\pi$
$72$	0	0
$73$	−4.98603	−0.583571	−0.291786	−	0.956484i	$-0.594249\pi$
$73$	−4.98603	−0.583571	−0.291786	+	0.956484i	$0.594249\pi$
$74$	0	0
$75$	−0.979974	−0.113158
$76$	0	0
$77$	3.47448	0.395954
$78$	0	0
$79$	−8.57203	−0.964429	−0.482214	−	0.876053i	$-0.660167\pi$
$79$	−8.57203	−0.964429	−0.482214	+	0.876053i	$0.660167\pi$
$80$	0	0
$81$	8.60194	0.955771
$82$	0	0
$83$	10.9011	1.19655	0.598277	−	0.801289i	$-0.295852\pi$
$83$	10.9011	1.19655	0.598277	+	0.801289i	$0.295852\pi$
$84$	0	0
$85$	0.593138	0.0643348
$86$	0	0
$87$	2.00991	0.215485
$88$	0	0
$89$	6.17014	0.654033	0.327017	−	0.945019i	$-0.393957\pi$
$89$	6.17014	0.654033	0.327017	+	0.945019i	$0.393957\pi$
$90$	0	0
$91$	−4.40395	−0.461659
$92$	0	0
$93$	1.09704	0.113757
$94$	0	0
$95$	0.132561	0.0136005
$96$	0	0
$97$	2.10120	0.213345	0.106672	−	0.994294i	$-0.465980\pi$
$97$	2.10120	0.213345	0.106672	+	0.994294i	$0.465980\pi$
$98$	0	0
$99$	6.38957	0.642176
$100$	0	0
$101$	6.52283	0.649046	0.324523	−	0.945878i	$-0.394796\pi$
$101$	6.52283	0.649046	0.324523	+	0.945878i	$0.394796\pi$
$102$	0	0
$103$	18.5292	1.82574	0.912870	−	0.408250i	$-0.133861\pi$
$103$	18.5292	1.82574	0.912870	+	0.408250i	$0.133861\pi$
$104$	0	0
$105$	−0.200976	−0.0196132
$106$	0	0
$107$	−3.17480	−0.306919	−0.153460	−	0.988155i	$-0.549042\pi$
$107$	−3.17480	−0.306919	−0.153460	+	0.988155i	$0.549042\pi$
$108$	0	0
$109$	5.60665	0.537020	0.268510	−	0.963277i	$-0.413469\pi$
$109$	5.60665	0.537020	0.268510	+	0.963277i	$0.413469\pi$
$110$	0	0
$111$	−0.183904	−0.0174553
$112$	0	0
$113$	−4.61822	−0.434446	−0.217223	−	0.976122i	$-0.569700\pi$
$113$	−4.61822	−0.434446	−0.217223	+	0.976122i	$0.569700\pi$
$114$	0	0
$115$	0.771524	0.0719450
$116$	0	0
$117$	−8.09885	−0.748739
$118$	0	0
$119$	−1.60715	−0.147327
$120$	0	0
$121$	−6.32625	−0.575113
$122$	0	0
$123$	−2.16890	−0.195563
$124$	0	0
$125$	−5.72270	−0.511854
$126$	0	0
$127$	17.2941	1.53460	0.767301	−	0.641287i	$-0.221599\pi$
$127$	17.2941	1.53460	0.767301	+	0.641287i	$0.221599\pi$
$128$	0	0
$129$	0.166671	0.0146746
$130$	0	0
$131$	3.17040	0.276999	0.138500	−	0.990363i	$-0.455772\pi$
$131$	3.17040	0.276999	0.138500	+	0.990363i	$0.455772\pi$
$132$	0	0
$133$	−0.359185	−0.0311453
$134$	0	0
$135$	−0.744746	−0.0640975
$136$	0	0
$137$	−3.97435	−0.339551	−0.169776	−	0.985483i	$-0.554304\pi$
$137$	−3.97435	−0.339551	−0.169776	+	0.985483i	$0.554304\pi$
$138$	0	0
$139$	−3.51586	−0.298212	−0.149106	−	0.988821i	$-0.547640\pi$
$139$	−3.51586	−0.298212	−0.149106	+	0.988821i	$0.547640\pi$
$140$	0	0
$141$	0.331488	0.0279164
$142$	0	0
$143$	−5.92404	−0.495393
$144$	0	0
$145$	5.65459	0.469588
$146$	0	0
$147$	−0.931245	−0.0768078
$148$	0	0
$149$	16.4781	1.34994	0.674970	−	0.737846i	$-0.264157\pi$
$149$	16.4781	1.34994	0.674970	+	0.737846i	$0.264157\pi$
$150$	0	0
$151$	−11.1808	−0.909879	−0.454939	−	0.890522i	$-0.650339\pi$
$151$	−11.1808	−0.909879	−0.454939	+	0.890522i	$0.650339\pi$
$152$	0	0
$153$	−2.95555	−0.238942
$154$	0	0
$155$	3.08636	0.247902
$156$	0	0
$157$	5.86937	0.468427	0.234213	−	0.972185i	$-0.424749\pi$
$157$	5.86937	0.468427	0.234213	+	0.972185i	$0.424749\pi$
$158$	0	0
$159$	−1.13670	−0.0901463
$160$	0	0
$161$	−2.09051	−0.164755
$162$	0	0
$163$	1.17072	0.0916982	0.0458491	−	0.998948i	$-0.485401\pi$
$163$	1.17072	0.0916982	0.0458491	+	0.998948i	$0.485401\pi$
$164$	0	0
$165$	−0.270346	−0.0210464
$166$	0	0
$167$	1.21000	0.0936326	0.0468163	−	0.998904i	$-0.485092\pi$
$167$	1.21000	0.0936326	0.0468163	+	0.998904i	$0.485092\pi$
$168$	0	0
$169$	−5.49120	−0.422400
$170$	0	0
$171$	−0.660541	−0.0505129
$172$	0	0
$173$	11.9119	0.905644	0.452822	−	0.891601i	$-0.350417\pi$
$173$	11.9119	0.905644	0.452822	+	0.891601i	$0.350417\pi$
$174$	0	0
$175$	7.47035	0.564705
$176$	0	0
$177$	0.210829	0.0158469
$178$	0	0
$179$	17.7725	1.32838	0.664189	−	0.747565i	$-0.268777\pi$
$179$	17.7725	1.32838	0.664189	+	0.747565i	$0.268777\pi$
$180$	0	0
$181$	16.6175	1.23517	0.617586	−	0.786504i	$-0.288111\pi$
$181$	16.6175	1.23517	0.617586	+	0.786504i	$0.288111\pi$
$182$	0	0
$183$	2.62678	0.194177
$184$	0	0
$185$	−0.517386	−0.0380390
$186$	0	0
$187$	−2.16189	−0.158093
$188$	0	0
$189$	2.01795	0.146784
$190$	0	0
$191$	15.5972	1.12857	0.564285	−	0.825580i	$-0.309152\pi$
$191$	15.5972	1.12857	0.564285	+	0.825580i	$0.309152\pi$
$192$	0	0
$193$	−5.87660	−0.423007	−0.211503	−	0.977377i	$-0.567836\pi$
$193$	−5.87660	−0.423007	−0.211503	+	0.977377i	$0.567836\pi$
$194$	0	0
$195$	0.342666	0.0245388
$196$	0	0
$197$	9.74123	0.694034	0.347017	−	0.937859i	$-0.387195\pi$
$197$	9.74123	0.694034	0.347017	+	0.937859i	$0.387195\pi$
$198$	0	0
$199$	2.11856	0.150181	0.0750904	−	0.997177i	$-0.476075\pi$
$199$	2.11856	0.150181	0.0750904	+	0.997177i	$0.476075\pi$
$200$	0	0
$201$	3.09375	0.218216
$202$	0	0
$203$	−15.3215	−1.07536
$204$	0	0
$205$	−6.10188	−0.426174
$206$	0	0
$207$	−3.84443	−0.267207
$208$	0	0
$209$	−0.483164	−0.0334211
$210$	0	0
$211$	−1.90833	−0.131375	−0.0656873	−	0.997840i	$-0.520924\pi$
$211$	−1.90833	−0.131375	−0.0656873	+	0.997840i	$0.520924\pi$
$212$	0	0
$213$	2.81993	0.193218
$214$	0	0
$215$	0.468906	0.0319791
$216$	0	0
$217$	−8.36272	−0.567699
$218$	0	0
$219$	−1.05120	−0.0710336
$220$	0	0
$221$	2.74022	0.184327
$222$	0	0
$223$	−8.06665	−0.540183	−0.270092	−	0.962835i	$-0.587054\pi$
$223$	−8.06665	−0.540183	−0.270092	+	0.962835i	$0.587054\pi$
$224$	0	0
$225$	13.7380	0.915864
$226$	0	0
$227$	3.86232	0.256351	0.128176	−	0.991752i	$-0.459088\pi$
$227$	3.86232	0.256351	0.128176	+	0.991752i	$0.459088\pi$
$228$	0	0
$229$	−3.91566	−0.258754	−0.129377	−	0.991595i	$-0.541298\pi$
$229$	−3.91566	−0.258754	−0.129377	+	0.991595i	$0.541298\pi$
$230$	0	0
$231$	0.732522	0.0481964
$232$	0	0
$233$	−6.11670	−0.400719	−0.200359	−	0.979722i	$-0.564211\pi$
$233$	−6.11670	−0.400719	−0.200359	+	0.979722i	$0.564211\pi$
$234$	0	0
$235$	0.932595	0.0608358
$236$	0	0
$237$	−1.80723	−0.117392
$238$	0	0
$239$	−12.1743	−0.787488	−0.393744	−	0.919220i	$-0.628820\pi$
$239$	−12.1743	−0.787488	−0.393744	+	0.919220i	$0.628820\pi$
$240$	0	0
$241$	−6.42874	−0.414111	−0.207056	−	0.978329i	$-0.566388\pi$
$241$	−6.42874	−0.414111	−0.207056	+	0.978329i	$0.566388\pi$
$242$	0	0
$243$	5.58035	0.357980
$244$	0	0
$245$	−2.61992	−0.167381
$246$	0	0
$247$	0.612416	0.0389671
$248$	0	0
$249$	2.29828	0.145647
$250$	0	0
$251$	−18.6413	−1.17663	−0.588316	−	0.808631i	$-0.700209\pi$
$251$	−18.6413	−1.17663	−0.588316	+	0.808631i	$0.700209\pi$
$252$	0	0
$253$	−2.81208	−0.176794
$254$	0	0
$255$	0.125051	0.00783098
$256$	0	0
$257$	−21.4043	−1.33516	−0.667580	−	0.744538i	$-0.732670\pi$
$257$	−21.4043	−1.33516	−0.667580	+	0.744538i	$0.732670\pi$
$258$	0	0
$259$	1.40190	0.0871097
$260$	0	0
$261$	−28.1763	−1.74407
$262$	0	0
$263$	−10.9973	−0.678123	−0.339061	−	0.940764i	$-0.610109\pi$
$263$	−10.9973	−0.678123	−0.339061	+	0.940764i	$0.610109\pi$
$264$	0	0
$265$	−3.19795	−0.196448
$266$	0	0
$267$	1.30085	0.0796104
$268$	0	0
$269$	−12.5296	−0.763945	−0.381973	−	0.924174i	$-0.624755\pi$
$269$	−12.5296	−0.763945	−0.381973	+	0.924174i	$0.624755\pi$
$270$	0	0
$271$	23.7858	1.44489	0.722444	−	0.691430i	$-0.243019\pi$
$271$	23.7858	1.44489	0.722444	+	0.691430i	$0.243019\pi$
$272$	0	0
$273$	−0.928481	−0.0561942
$274$	0	0
$275$	10.0489	0.605969
$276$	0	0
$277$	−2.09464	−0.125855	−0.0629273	−	0.998018i	$-0.520044\pi$
$277$	−2.09464	−0.125855	−0.0629273	+	0.998018i	$0.520044\pi$
$278$	0	0
$279$	−15.3790	−0.920718
$280$	0	0
$281$	9.34392	0.557412	0.278706	−	0.960377i	$-0.410095\pi$
$281$	9.34392	0.557412	0.278706	+	0.960377i	$0.410095\pi$
$282$	0	0
$283$	−13.6470	−0.811228	−0.405614	−	0.914044i	$-0.632942\pi$
$283$	−13.6470	−0.811228	−0.405614	+	0.914044i	$0.632942\pi$
$284$	0	0
$285$	0.0279478	0.00165549
$286$	0	0
$287$	16.5335	0.975943
$288$	0	0
$289$	1.00000	0.0588235
$290$	0	0
$291$	0.442994	0.0259688
$292$	0	0
$293$	−4.65952	−0.272212	−0.136106	−	0.990694i	$-0.543459\pi$
$293$	−4.65952	−0.272212	−0.136106	+	0.990694i	$0.543459\pi$
$294$	0	0
$295$	0.593138	0.0345338
$296$	0	0
$297$	2.71447	0.157510
$298$	0	0
$299$	3.56434	0.206131
$300$	0	0
$301$	−1.27054	−0.0732325
$302$	0	0
$303$	1.37520	0.0790033
$304$	0	0
$305$	7.39008	0.423155
$306$	0	0
$307$	32.7598	1.86970	0.934851	−	0.355041i	$-0.115533\pi$
$307$	32.7598	1.86970	0.934851	+	0.355041i	$0.115533\pi$
$308$	0	0
$309$	3.90651	0.222233
$310$	0	0
$311$	9.28888	0.526724	0.263362	−	0.964697i	$-0.415169\pi$
$311$	9.28888	0.526724	0.263362	+	0.964697i	$0.415169\pi$
$312$	0	0
$313$	20.1699	1.14007	0.570034	−	0.821621i	$-0.306930\pi$
$313$	20.1699	1.14007	0.570034	+	0.821621i	$0.306930\pi$
$314$	0	0
$315$	2.81742	0.158743
$316$	0	0
$317$	6.01915	0.338069	0.169034	−	0.985610i	$-0.445935\pi$
$317$	6.01915	0.338069	0.169034	+	0.985610i	$0.445935\pi$
$318$	0	0
$319$	−20.6100	−1.15394
$320$	0	0
$321$	−0.669340	−0.0373589
$322$	0	0
$323$	0.223492	0.0124354
$324$	0	0
$325$	−12.7370	−0.706524
$326$	0	0
$327$	1.18205	0.0653673
$328$	0	0
$329$	−2.52694	−0.139315
$330$	0	0
$331$	−21.2561	−1.16834	−0.584171	−	0.811630i	$-0.698580\pi$
$331$	−21.2561	−1.16834	−0.584171	+	0.811630i	$0.698580\pi$
$332$	0	0
$333$	2.57809	0.141278
$334$	0	0
$335$	8.70381	0.475540
$336$	0	0
$337$	15.0769	0.821293	0.410646	−	0.911795i	$-0.365303\pi$
$337$	15.0769	0.821293	0.410646	+	0.911795i	$0.365303\pi$
$338$	0	0
$339$	−0.973657	−0.0528818
$340$	0	0
$341$	−11.2492	−0.609181
$342$	0	0
$343$	18.3490	0.990751
$344$	0	0
$345$	0.162660	0.00875731
$346$	0	0
$347$	22.9744	1.23333	0.616667	−	0.787224i	$-0.288483\pi$
$347$	22.9744	1.23333	0.616667	+	0.787224i	$0.288483\pi$
$348$	0	0
$349$	−18.0195	−0.964561	−0.482281	−	0.876017i	$-0.660191\pi$
$349$	−18.0195	−0.964561	−0.482281	+	0.876017i	$0.660191\pi$
$350$	0	0
$351$	−3.44063	−0.183647
$352$	0	0
$353$	−19.6715	−1.04701	−0.523504	−	0.852023i	$-0.675375\pi$
$353$	−19.6715	−1.04701	−0.523504	+	0.852023i	$0.675375\pi$
$354$	0	0
$355$	7.93346	0.421064
$356$	0	0
$357$	−0.338835	−0.0179330
$358$	0	0
$359$	10.8560	0.572957	0.286479	−	0.958087i	$-0.407515\pi$
$359$	10.8560	0.572957	0.286479	+	0.958087i	$0.407515\pi$
$360$	0	0
$361$	−18.9501	−0.997371
$362$	0	0
$363$	−1.33376	−0.0700041
$364$	0	0
$365$	−2.95740	−0.154798
$366$	0	0
$367$	−15.8608	−0.827927	−0.413964	−	0.910293i	$-0.635856\pi$
$367$	−15.8608	−0.827927	−0.413964	+	0.910293i	$0.635856\pi$
$368$	0	0
$369$	30.4051	1.58283
$370$	0	0
$371$	8.66508	0.449869
$372$	0	0
$373$	−20.2611	−1.04908	−0.524541	−	0.851385i	$-0.675763\pi$
$373$	−20.2611	−1.04908	−0.524541	+	0.851385i	$0.675763\pi$
$374$	0	0
$375$	−1.20651	−0.0623041
$376$	0	0
$377$	26.1234	1.34543
$378$	0	0
$379$	11.4987	0.590650	0.295325	−	0.955397i	$-0.404572\pi$
$379$	11.4987	0.590650	0.295325	+	0.955397i	$0.404572\pi$
$380$	0	0
$381$	3.64610	0.186795
$382$	0	0
$383$	−2.83703	−0.144966	−0.0724828	−	0.997370i	$-0.523092\pi$
$383$	−2.83703	−0.144966	−0.0724828	+	0.997370i	$0.523092\pi$
$384$	0	0
$385$	2.06085	0.105030
$386$	0	0
$387$	−2.33651	−0.118772
$388$	0	0
$389$	36.3209	1.84154	0.920772	−	0.390101i	$-0.127560\pi$
$389$	36.3209	1.84154	0.920772	+	0.390101i	$0.127560\pi$
$390$	0	0
$391$	1.30075	0.0657818
$392$	0	0
$393$	0.668413	0.0337170
$394$	0	0
$395$	−5.08439	−0.255824
$396$	0	0
$397$	−4.45751	−0.223716	−0.111858	−	0.993724i	$-0.535680\pi$
$397$	−4.45751	−0.223716	−0.111858	+	0.993724i	$0.535680\pi$
$398$	0	0
$399$	−0.0757268	−0.00379108
$400$	0	0
$401$	24.1653	1.20676	0.603378	−	0.797456i	$-0.293821\pi$
$401$	24.1653	1.20676	0.603378	+	0.797456i	$0.293821\pi$
$402$	0	0
$403$	14.2586	0.710269
$404$	0	0
$405$	5.10213	0.253527
$406$	0	0
$407$	1.88578	0.0934749
$408$	0	0
$409$	11.5263	0.569941	0.284970	−	0.958536i	$-0.408016\pi$
$409$	11.5263	0.569941	0.284970	+	0.958536i	$0.408016\pi$
$410$	0	0
$411$	−0.837908	−0.0413310
$412$	0	0
$413$	−1.60715	−0.0790828
$414$	0	0
$415$	6.46587	0.317397
$416$	0	0
$417$	−0.741247	−0.0362990
$418$	0	0
$419$	13.3882	0.654055	0.327027	−	0.945015i	$-0.393953\pi$
$419$	13.3882	0.654055	0.327027	+	0.945015i	$0.393953\pi$
$420$	0	0
$421$	1.22942	0.0599182	0.0299591	−	0.999551i	$-0.490462\pi$
$421$	1.22942	0.0599182	0.0299591	+	0.999551i	$0.490462\pi$
$422$	0	0
$423$	−4.64704	−0.225947
$424$	0	0
$425$	−4.64819	−0.225470
$426$	0	0
$427$	−20.0240	−0.969029
$428$	0	0
$429$	−1.24896	−0.0603004
$430$	0	0
$431$	16.9328	0.815626	0.407813	−	0.913065i	$-0.366292\pi$
$431$	16.9328	0.815626	0.407813	+	0.913065i	$0.366292\pi$
$432$	0	0
$433$	−22.4558	−1.07916	−0.539579	−	0.841935i	$-0.681417\pi$
$433$	−22.4558	−1.07916	−0.539579	+	0.841935i	$0.681417\pi$
$434$	0	0
$435$	1.19215	0.0571593
$436$	0	0
$437$	0.290707	0.0139064
$438$	0	0
$439$	−1.22131	−0.0582902	−0.0291451	−	0.999575i	$-0.509278\pi$
$439$	−1.22131	−0.0582902	−0.0291451	+	0.999575i	$0.509278\pi$
$440$	0	0
$441$	13.0548	0.621659
$442$	0	0
$443$	2.62732	0.124828	0.0624138	−	0.998050i	$-0.480120\pi$
$443$	2.62732	0.124828	0.0624138	+	0.998050i	$0.480120\pi$
$444$	0	0
$445$	3.65974	0.173488
$446$	0	0
$447$	3.47407	0.164318
$448$	0	0
$449$	7.30513	0.344750	0.172375	−	0.985031i	$-0.444856\pi$
$449$	7.30513	0.344750	0.172375	+	0.985031i	$0.444856\pi$
$450$	0	0
$451$	22.2403	1.04726
$452$	0	0
$453$	−2.35723	−0.110753
$454$	0	0
$455$	−2.61215	−0.122459
$456$	0	0
$457$	−24.1907	−1.13159	−0.565797	−	0.824545i	$-0.691431\pi$
$457$	−24.1907	−1.13159	−0.565797	+	0.824545i	$0.691431\pi$
$458$	0	0
$459$	−1.25560	−0.0586066
$460$	0	0
$461$	23.0754	1.07473	0.537365	−	0.843350i	$-0.319420\pi$
$461$	23.0754	1.07473	0.537365	+	0.843350i	$0.319420\pi$
$462$	0	0
$463$	−8.04086	−0.373690	−0.186845	−	0.982389i	$-0.559826\pi$
$463$	−8.04086	−0.373690	−0.186845	+	0.982389i	$0.559826\pi$
$464$	0	0
$465$	0.650694	0.0301752
$466$	0	0
$467$	8.93397	0.413415	0.206707	−	0.978403i	$-0.433725\pi$
$467$	8.93397	0.413415	0.206707	+	0.978403i	$0.433725\pi$
$468$	0	0
$469$	−23.5836	−1.08899
$470$	0	0
$471$	1.23744	0.0570180
$472$	0	0
$473$	−1.70908	−0.0785836
$474$	0	0
$475$	−1.03883	−0.0476649
$476$	0	0
$477$	15.9351	0.729617
$478$	0	0
$479$	−18.1804	−0.830685	−0.415342	−	0.909665i	$-0.636338\pi$
$479$	−18.1804	−0.830685	−0.415342	+	0.909665i	$0.636338\pi$
$480$	0	0
$481$	−2.39026	−0.108986
$482$	0	0
$483$	−0.440740	−0.0200543
$484$	0	0
$485$	1.24630	0.0565916
$486$	0	0
$487$	−12.2143	−0.553485	−0.276742	−	0.960944i	$-0.589255\pi$
$487$	−12.2143	−0.553485	−0.276742	+	0.960944i	$0.589255\pi$
$488$	0	0
$489$	0.246823	0.0111617
$490$	0	0
$491$	8.04038	0.362857	0.181429	−	0.983404i	$-0.441928\pi$
$491$	8.04038	0.362857	0.181429	+	0.983404i	$0.441928\pi$
$492$	0	0
$493$	9.53334	0.429360
$494$	0	0
$495$	3.78989	0.170343
$496$	0	0
$497$	−21.4963	−0.964242
$498$	0	0
$499$	6.22892	0.278845	0.139422	−	0.990233i	$-0.455475\pi$
$499$	6.22892	0.278845	0.139422	+	0.990233i	$0.455475\pi$
$500$	0	0
$501$	0.255103	0.0113972
$502$	0	0
$503$	−29.6866	−1.32366	−0.661831	−	0.749653i	$-0.730220\pi$
$503$	−29.6866	−1.32366	−0.661831	+	0.749653i	$0.730220\pi$
$504$	0	0
$505$	3.86894	0.172165
$506$	0	0
$507$	−1.15771	−0.0514155
$508$	0	0
$509$	−22.6036	−1.00189	−0.500944	−	0.865479i	$-0.667014\pi$
$509$	−22.6036	−1.00189	−0.500944	+	0.865479i	$0.667014\pi$
$510$	0	0
$511$	8.01332	0.354488
$512$	0	0
$513$	−0.280617	−0.0123895
$514$	0	0
$515$	10.9904	0.484295
$516$	0	0
$517$	−3.39915	−0.149495
$518$	0	0
$519$	2.51137	0.110237
$520$	0	0
$521$	−16.4142	−0.719117	−0.359559	−	0.933122i	$-0.617073\pi$
$521$	−16.4142	−0.719117	−0.359559	+	0.933122i	$0.617073\pi$
$522$	0	0
$523$	−0.586971	−0.0256664	−0.0128332	−	0.999918i	$-0.504085\pi$
$523$	−0.586971	−0.0256664	−0.0128332	+	0.999918i	$0.504085\pi$
$524$	0	0
$525$	1.57497	0.0687372
$526$	0	0
$527$	5.20344	0.226665
$528$	0	0
$529$	−21.3080	−0.926437
$530$	0	0
$531$	−2.95555	−0.128260
$532$	0	0
$533$	−28.1899	−1.22104
$534$	0	0
$535$	−1.88309	−0.0814132
$536$	0	0
$537$	3.74696	0.161693
$538$	0	0
$539$	9.54918	0.411312
$540$	0	0
$541$	−15.3768	−0.661100	−0.330550	−	0.943788i	$-0.607234\pi$
$541$	−15.3768	−0.661100	−0.330550	+	0.943788i	$0.607234\pi$
$542$	0	0
$543$	3.50346	0.150348
$544$	0	0
$545$	3.32552	0.142449
$546$	0	0
$547$	−1.42778	−0.0610476	−0.0305238	−	0.999534i	$-0.509718\pi$
$547$	−1.42778	−0.0610476	−0.0305238	+	0.999534i	$0.509718\pi$
$548$	0	0
$549$	−36.8241	−1.57161
$550$	0	0
$551$	2.13062	0.0907676
$552$	0	0
$553$	13.7766	0.585839
$554$	0	0
$555$	−0.109080	−0.00463019
$556$	0	0
$557$	−16.2277	−0.687588	−0.343794	−	0.939045i	$-0.611712\pi$
$557$	−16.2277	−0.687588	−0.343794	+	0.939045i	$0.611712\pi$
$558$	0	0
$559$	2.16628	0.0916240
$560$	0	0
$561$	−0.455789	−0.0192434
$562$	0	0
$563$	−9.26313	−0.390394	−0.195197	−	0.980764i	$-0.562535\pi$
$563$	−9.26313	−0.390394	−0.195197	+	0.980764i	$0.562535\pi$
$564$	0	0
$565$	−2.73924	−0.115241
$566$	0	0
$567$	−13.8246	−0.580580
$568$	0	0
$569$	6.09581	0.255549	0.127775	−	0.991803i	$-0.459217\pi$
$569$	6.09581	0.255549	0.127775	+	0.991803i	$0.459217\pi$
$570$	0	0
$571$	32.2341	1.34896	0.674478	−	0.738295i	$-0.264369\pi$
$571$	32.2341	1.34896	0.674478	+	0.738295i	$0.264369\pi$
$572$	0	0
$573$	3.28834	0.137372
$574$	0	0
$575$	−6.04613	−0.252141
$576$	0	0
$577$	−35.9991	−1.49866	−0.749331	−	0.662195i	$-0.769625\pi$
$577$	−35.9991	−1.49866	−0.749331	+	0.662195i	$0.769625\pi$
$578$	0	0
$579$	−1.23896	−0.0514893
$580$	0	0
$581$	−17.5198	−0.726843
$582$	0	0
$583$	11.6560	0.482741
$584$	0	0
$585$	−4.80374	−0.198610
$586$	0	0
$587$	14.8205	0.611706	0.305853	−	0.952079i	$-0.401058\pi$
$587$	14.8205	0.611706	0.305853	+	0.952079i	$0.401058\pi$
$588$	0	0
$589$	1.16293	0.0479175
$590$	0	0
$591$	2.05373	0.0844794
$592$	0	0
$593$	−8.50425	−0.349228	−0.174614	−	0.984637i	$-0.555868\pi$
$593$	−8.50425	−0.349228	−0.174614	+	0.984637i	$0.555868\pi$
$594$	0	0
$595$	−0.953263	−0.0390800
$596$	0	0
$597$	0.446654	0.0182803
$598$	0	0
$599$	−0.238737	−0.00975453	−0.00487726	−	0.999988i	$-0.501552\pi$
$599$	−0.238737	−0.00975453	−0.00487726	+	0.999988i	$0.501552\pi$
$600$	0	0
$601$	39.6483	1.61729	0.808645	−	0.588298i	$-0.200202\pi$
$601$	39.6483	1.61729	0.808645	+	0.588298i	$0.200202\pi$
$602$	0	0
$603$	−43.3703	−1.76617
$604$	0	0
$605$	−3.75234	−0.152554
$606$	0	0
$607$	−2.87459	−0.116676	−0.0583380	−	0.998297i	$-0.518580\pi$
$607$	−2.87459	−0.116676	−0.0583380	+	0.998297i	$0.518580\pi$
$608$	0	0
$609$	−3.23023	−0.130895
$610$	0	0
$611$	4.30847	0.174302
$612$	0	0
$613$	32.8915	1.32848	0.664238	−	0.747521i	$-0.268756\pi$
$613$	32.8915	1.32848	0.664238	+	0.747521i	$0.268756\pi$
$614$	0	0
$615$	−1.28646	−0.0518749
$616$	0	0
$617$	−6.11432	−0.246153	−0.123077	−	0.992397i	$-0.539276\pi$
$617$	−6.11432	−0.246153	−0.123077	+	0.992397i	$0.539276\pi$
$618$	0	0
$619$	−27.4603	−1.10372	−0.551861	−	0.833936i	$-0.686082\pi$
$619$	−27.4603	−1.10372	−0.551861	+	0.833936i	$0.686082\pi$
$620$	0	0
$621$	−1.63323	−0.0655392
$622$	0	0
$623$	−9.91635	−0.397290
$624$	0	0
$625$	19.8466	0.793863
$626$	0	0
$627$	−0.101865	−0.00406810
$628$	0	0
$629$	−0.872287	−0.0347803
$630$	0	0
$631$	27.3530	1.08890	0.544452	−	0.838792i	$-0.316738\pi$
$631$	27.3530	1.08890	0.544452	+	0.838792i	$0.316738\pi$
$632$	0	0
$633$	−0.402331	−0.0159912
$634$	0	0
$635$	10.2578	0.407067
$636$	0	0
$637$	−12.1037	−0.479566
$638$	0	0
$639$	−39.5317	−1.56385
$640$	0	0
$641$	−12.7937	−0.505320	−0.252660	−	0.967555i	$-0.581305\pi$
$641$	−12.7937	−0.505320	−0.252660	+	0.967555i	$0.581305\pi$
$642$	0	0
$643$	5.09751	0.201026	0.100513	−	0.994936i	$-0.467952\pi$
$643$	5.09751	0.201026	0.100513	+	0.994936i	$0.467952\pi$
$644$	0	0
$645$	0.0988590	0.00389257
$646$	0	0
$647$	−21.8955	−0.860801	−0.430401	−	0.902638i	$-0.641628\pi$
$647$	−21.8955	−0.860801	−0.430401	+	0.902638i	$0.641628\pi$
$648$	0	0
$649$	−2.16189	−0.0848615
$650$	0	0
$651$	−1.76311	−0.0691016
$652$	0	0
$653$	9.79473	0.383297	0.191649	−	0.981464i	$-0.438617\pi$
$653$	9.79473	0.383297	0.191649	+	0.981464i	$0.438617\pi$
$654$	0	0
$655$	1.88048	0.0734766
$656$	0	0
$657$	14.7365	0.574925
$658$	0	0
$659$	−13.2834	−0.517448	−0.258724	−	0.965951i	$-0.583302\pi$
$659$	−13.2834	−0.517448	−0.258724	+	0.965951i	$0.583302\pi$
$660$	0	0
$661$	24.1604	0.939730	0.469865	−	0.882738i	$-0.344303\pi$
$661$	24.1604	0.939730	0.469865	+	0.882738i	$0.344303\pi$
$662$	0	0
$663$	0.577718	0.0224367
$664$	0	0
$665$	−0.213046	−0.00826159
$666$	0	0
$667$	12.4005	0.480149
$668$	0	0
$669$	−1.70069	−0.0657523
$670$	0	0
$671$	−26.9356	−1.03984
$672$	0	0
$673$	6.92266	0.266849	0.133424	−	0.991059i	$-0.457403\pi$
$673$	6.92266	0.266849	0.133424	+	0.991059i	$0.457403\pi$
$674$	0	0
$675$	5.83628	0.224639
$676$	0	0
$677$	35.1813	1.35213	0.676064	−	0.736843i	$-0.263684\pi$
$677$	35.1813	1.35213	0.676064	+	0.736843i	$0.263684\pi$
$678$	0	0
$679$	−3.37695	−0.129595
$680$	0	0
$681$	0.814289	0.0312036
$682$	0	0
$683$	31.8420	1.21840	0.609201	−	0.793016i	$-0.291490\pi$
$683$	31.8420	1.21840	0.609201	+	0.793016i	$0.291490\pi$
$684$	0	0
$685$	−2.35733	−0.0900691
$686$	0	0
$687$	−0.825536	−0.0314962
$688$	0	0
$689$	−14.7741	−0.562848
$690$	0	0
$691$	30.6805	1.16714	0.583570	−	0.812063i	$-0.301655\pi$
$691$	30.6805	1.16714	0.583570	+	0.812063i	$0.301655\pi$
$692$	0	0
$693$	−10.2690	−0.390087
$694$	0	0
$695$	−2.08539	−0.0791034
$696$	0	0
$697$	−10.2875	−0.389665
$698$	0	0
$699$	−1.28958	−0.0487764
$700$	0	0
$701$	0.915987	0.0345964	0.0172982	−	0.999850i	$-0.494494\pi$
$701$	0.915987	0.0345964	0.0172982	+	0.999850i	$0.494494\pi$
$702$	0	0
$703$	−0.194949	−0.00735264
$704$	0	0
$705$	0.196618	0.00740507
$706$	0	0
$707$	−10.4832	−0.394261
$708$	0	0
$709$	−33.6536	−1.26389	−0.631943	−	0.775015i	$-0.717742\pi$
$709$	−33.6536	−1.26389	−0.631943	+	0.775015i	$0.717742\pi$
$710$	0	0
$711$	25.3351	0.950139
$712$	0	0
$713$	6.76838	0.253478
$714$	0	0
$715$	−3.51377	−0.131408
$716$	0	0
$717$	−2.56669	−0.0958548
$718$	0	0
$719$	−26.4066	−0.984800	−0.492400	−	0.870369i	$-0.663880\pi$
$719$	−26.4066	−0.984800	−0.492400	+	0.870369i	$0.663880\pi$
$720$	0	0
$721$	−29.7793	−1.10904
$722$	0	0
$723$	−1.35537	−0.0504066
$724$	0	0
$725$	−44.3128	−1.64574
$726$	0	0
$727$	7.84585	0.290987	0.145493	−	0.989359i	$-0.453523\pi$
$727$	7.84585	0.290987	0.145493	+	0.989359i	$0.453523\pi$
$728$	0	0
$729$	−24.6293	−0.912196
$730$	0	0
$731$	0.790551	0.0292396
$732$	0	0
$733$	2.84239	0.104986	0.0524930	−	0.998621i	$-0.483283\pi$
$733$	2.84239	0.104986	0.0524930	+	0.998621i	$0.483283\pi$
$734$	0	0
$735$	−0.552357	−0.0203740
$736$	0	0
$737$	−31.7239	−1.16857
$738$	0	0
$739$	−15.2148	−0.559686	−0.279843	−	0.960046i	$-0.590282\pi$
$739$	−15.2148	−0.559686	−0.279843	+	0.960046i	$0.590282\pi$
$740$	0	0
$741$	0.129115	0.00474317
$742$	0	0
$743$	37.9123	1.39087	0.695433	−	0.718591i	$-0.255213\pi$
$743$	37.9123	1.39087	0.695433	+	0.718591i	$0.255213\pi$
$744$	0	0
$745$	9.77379	0.358084
$746$	0	0
$747$	−32.2188	−1.17883
$748$	0	0
$749$	5.10239	0.186437
$750$	0	0
$751$	13.2100	0.482040	0.241020	−	0.970520i	$-0.422518\pi$
$751$	13.2100	0.482040	0.241020	+	0.970520i	$0.422518\pi$
$752$	0	0
$753$	−3.93014	−0.143222
$754$	0	0
$755$	−6.63174	−0.241354
$756$	0	0
$757$	−17.8034	−0.647077	−0.323538	−	0.946215i	$-0.604872\pi$
$757$	−17.8034	−0.647077	−0.323538	+	0.946215i	$0.604872\pi$
$758$	0	0
$759$	−0.592868	−0.0215197
$760$	0	0
$761$	−22.8616	−0.828732	−0.414366	−	0.910110i	$-0.635997\pi$
$761$	−22.8616	−0.828732	−0.414366	+	0.910110i	$0.635997\pi$
$762$	0	0
$763$	−9.01074	−0.326211
$764$	0	0
$765$	−1.75305	−0.0633816
$766$	0	0
$767$	2.74022	0.0989435
$768$	0	0
$769$	−27.3365	−0.985778	−0.492889	−	0.870092i	$-0.664059\pi$
$769$	−27.3365	−0.985778	−0.492889	+	0.870092i	$0.664059\pi$
$770$	0	0
$771$	−4.51264	−0.162519
$772$	0	0
$773$	−50.1402	−1.80342	−0.901708	−	0.432345i	$-0.857686\pi$
$773$	−50.1402	−1.80342	−0.901708	+	0.432345i	$0.857686\pi$
$774$	0	0
$775$	−24.1866	−0.868807
$776$	0	0
$777$	0.295561	0.0106032
$778$	0	0
$779$	−2.29916	−0.0823761
$780$	0	0
$781$	−28.9161	−1.03470
$782$	0	0
$783$	−11.9701	−0.427777
$784$	0	0
$785$	3.48135	0.124255
$786$	0	0
$787$	−13.1614	−0.469153	−0.234576	−	0.972098i	$-0.575370\pi$
$787$	−13.1614	−0.469153	−0.234576	+	0.972098i	$0.575370\pi$
$788$	0	0
$789$	−2.31855	−0.0825427
$790$	0	0
$791$	7.42219	0.263903
$792$	0	0
$793$	34.1412	1.21239
$794$	0	0
$795$	−0.674220	−0.0239121
$796$	0	0
$797$	−10.9258	−0.387012	−0.193506	−	0.981099i	$-0.561986\pi$
$797$	−10.9258	−0.387012	−0.193506	+	0.981099i	$0.561986\pi$
$798$	0	0
$799$	1.57231	0.0556243
$800$	0	0
$801$	−18.2362	−0.644343
$802$	0	0
$803$	10.7792	0.380391
$804$	0	0
$805$	−1.23996	−0.0437028
$806$	0	0
$807$	−2.64161	−0.0929891
$808$	0	0
$809$	−2.83160	−0.0995537	−0.0497769	−	0.998760i	$-0.515851\pi$
$809$	−2.83160	−0.0995537	−0.0497769	+	0.998760i	$0.515851\pi$
$810$	0	0
$811$	−26.5715	−0.933052	−0.466526	−	0.884507i	$-0.654495\pi$
$811$	−26.5715	−0.933052	−0.466526	+	0.884507i	$0.654495\pi$
$812$	0	0
$813$	5.01475	0.175875
$814$	0	0
$815$	0.694401	0.0243238
$816$	0	0
$817$	0.176682	0.00618131
$818$	0	0
$819$	13.0161	0.454819
$820$	0	0
$821$	−28.1265	−0.981620	−0.490810	−	0.871267i	$-0.663299\pi$
$821$	−28.1265	−0.981620	−0.490810	+	0.871267i	$0.663299\pi$
$822$	0	0
$823$	−14.9977	−0.522787	−0.261393	−	0.965232i	$-0.584182\pi$
$823$	−14.9977	−0.522787	−0.261393	+	0.965232i	$0.584182\pi$
$824$	0	0
$825$	2.11859	0.0737599
$826$	0	0
$827$	17.2541	0.599982	0.299991	−	0.953942i	$-0.403016\pi$
$827$	17.2541	0.599982	0.299991	+	0.953942i	$0.403016\pi$
$828$	0	0
$829$	−30.3405	−1.05377	−0.526885	−	0.849937i	$-0.676640\pi$
$829$	−30.3405	−1.05377	−0.526885	+	0.849937i	$0.676640\pi$
$830$	0	0
$831$	−0.441611	−0.0153193
$832$	0	0
$833$	−4.41706	−0.153042
$834$	0	0
$835$	0.717697	0.0248369
$836$	0	0
$837$	−6.53346	−0.225829
$838$	0	0
$839$	10.9363	0.377562	0.188781	−	0.982019i	$-0.439546\pi$
$839$	10.9363	0.377562	0.188781	+	0.982019i	$0.439546\pi$
$840$	0	0
$841$	61.8847	2.13395
$842$	0	0
$843$	1.96997	0.0678494
$844$	0	0
$845$	−3.25704	−0.112046
$846$	0	0
$847$	10.1672	0.349351
$848$	0	0
$849$	−2.87718	−0.0987445
$850$	0	0
$851$	−1.13463	−0.0388945
$852$	0	0
$853$	−29.6155	−1.01402	−0.507008	−	0.861941i	$-0.669249\pi$
$853$	−29.6155	−1.01402	−0.507008	+	0.861941i	$0.669249\pi$
$854$	0	0
$855$	−0.391792	−0.0133990
$856$	0	0
$857$	−9.49896	−0.324478	−0.162239	−	0.986751i	$-0.551872\pi$
$857$	−9.49896	−0.324478	−0.162239	+	0.986751i	$0.551872\pi$
$858$	0	0
$859$	−1.47166	−0.0502122	−0.0251061	−	0.999685i	$-0.507992\pi$
$859$	−1.47166	−0.0502122	−0.0251061	+	0.999685i	$0.507992\pi$
$860$	0	0
$861$	3.48575	0.118794
$862$	0	0
$863$	55.5601	1.89129	0.945644	−	0.325203i	$-0.105432\pi$
$863$	55.5601	1.89129	0.945644	+	0.325203i	$0.105432\pi$
$864$	0	0
$865$	7.06539	0.240230
$866$	0	0
$867$	0.210829	0.00716014
$868$	0	0
$869$	18.5318	0.628647
$870$	0	0
$871$	40.2104	1.36248
$872$	0	0
$873$	−6.21021	−0.210184
$874$	0	0
$875$	9.19726	0.310924
$876$	0	0
$877$	43.8841	1.48186	0.740930	−	0.671582i	$-0.234385\pi$
$877$	43.8841	1.48186	0.740930	+	0.671582i	$0.234385\pi$
$878$	0	0
$879$	−0.982364	−0.0331343
$880$	0	0
$881$	23.2458	0.783171	0.391585	−	0.920142i	$-0.371927\pi$
$881$	23.2458	0.783171	0.391585	+	0.920142i	$0.371927\pi$
$882$	0	0
$883$	−14.9691	−0.503749	−0.251874	−	0.967760i	$-0.581047\pi$
$883$	−14.9691	−0.503749	−0.251874	+	0.967760i	$0.581047\pi$
$884$	0	0
$885$	0.125051	0.00420354
$886$	0	0
$887$	−9.06482	−0.304367	−0.152183	−	0.988352i	$-0.548630\pi$
$887$	−9.06482	−0.304367	−0.152183	+	0.988352i	$0.548630\pi$
$888$	0	0
$889$	−27.7942	−0.932189
$890$	0	0
$891$	−18.5964	−0.623003
$892$	0	0
$893$	0.351398	0.0117591
$894$	0	0
$895$	10.5415	0.352365
$896$	0	0
$897$	0.751467	0.0250908
$898$	0	0
$899$	49.6062	1.65446
$900$	0	0
$901$	−5.39157	−0.179619
$902$	0	0
$903$	−0.267866	−0.00891403
$904$	0	0
$905$	9.85649	0.327641
$906$	0	0
$907$	−50.9640	−1.69223	−0.846116	−	0.532999i	$-0.821065\pi$
$907$	−50.9640	−1.69223	−0.846116	+	0.532999i	$0.821065\pi$
$908$	0	0
$909$	−19.2785	−0.639429
$910$	0	0
$911$	−10.8780	−0.360404	−0.180202	−	0.983630i	$-0.557675\pi$
$911$	−10.8780	−0.360404	−0.180202	+	0.983630i	$0.557675\pi$
$912$	0	0
$913$	−23.5670	−0.779954
$914$	0	0
$915$	1.55804	0.0515074
$916$	0	0
$917$	−5.09532	−0.168262
$918$	0	0
$919$	49.9281	1.64698	0.823489	−	0.567333i	$-0.192025\pi$
$919$	49.9281	1.64698	0.823489	+	0.567333i	$0.192025\pi$
$920$	0	0
$921$	6.90673	0.227584
$922$	0	0
$923$	36.6515	1.20640
$924$	0	0
$925$	4.05455	0.133313
$926$	0	0
$927$	−54.7641	−1.79869
$928$	0	0
$929$	−14.2299	−0.466867	−0.233433	−	0.972373i	$-0.574996\pi$
$929$	−14.2299	−0.466867	−0.233433	+	0.972373i	$0.574996\pi$
$930$	0	0
$931$	−0.987176	−0.0323534
$932$	0	0
$933$	1.95837	0.0641140
$934$	0	0
$935$	−1.28230	−0.0419356
$936$	0	0
$937$	−49.3361	−1.61174	−0.805870	−	0.592093i	$-0.798302\pi$
$937$	−49.3361	−1.61174	−0.805870	+	0.592093i	$0.798302\pi$
$938$	0	0
$939$	4.25239	0.138772
$940$	0	0
$941$	−17.6031	−0.573846	−0.286923	−	0.957954i	$-0.592632\pi$
$941$	−17.6031	−0.573846	−0.286923	+	0.957954i	$0.592632\pi$
$942$	0	0
$943$	−13.3814	−0.435759
$944$	0	0
$945$	1.19692	0.0389358
$946$	0	0
$947$	−32.1310	−1.04412	−0.522058	−	0.852910i	$-0.674836\pi$
$947$	−32.1310	−1.04412	−0.522058	+	0.852910i	$0.674836\pi$
$948$	0	0
$949$	−13.6628	−0.443514
$950$	0	0
$951$	1.26901	0.0411505
$952$	0	0
$953$	35.6229	1.15394	0.576969	−	0.816766i	$-0.304235\pi$
$953$	35.6229	1.15394	0.576969	+	0.816766i	$0.304235\pi$
$954$	0	0
$955$	9.25126	0.299364
$956$	0	0
$957$	−4.34519	−0.140460
$958$	0	0
$959$	6.38738	0.206259
$960$	0	0
$961$	−3.92422	−0.126588
$962$	0	0
$963$	9.38328	0.302372
$964$	0	0
$965$	−3.48563	−0.112206
$966$	0	0
$967$	−22.6763	−0.729219	−0.364610	−	0.931160i	$-0.618798\pi$
$967$	−22.6763	−0.729219	−0.364610	+	0.931160i	$0.618798\pi$
$968$	0	0
$969$	0.0471186	0.00151367
$970$	0	0
$971$	5.92233	0.190057	0.0950284	−	0.995475i	$-0.469706\pi$
$971$	5.92233	0.190057	0.0950284	+	0.995475i	$0.469706\pi$
$972$	0	0
$973$	5.65053	0.181148
$974$	0	0
$975$	−2.68534	−0.0859998
$976$	0	0
$977$	32.2394	1.03143	0.515714	−	0.856761i	$-0.327527\pi$
$977$	32.2394	1.03143	0.515714	+	0.856761i	$0.327527\pi$
$978$	0	0
$979$	−13.3391	−0.426321
$980$	0	0
$981$	−16.5707	−0.529063
$982$	0	0
$983$	−4.75385	−0.151624	−0.0758121	−	0.997122i	$-0.524155\pi$
$983$	−4.75385	−0.151624	−0.0758121	+	0.997122i	$0.524155\pi$
$984$	0	0
$985$	5.77789	0.184099
$986$	0	0
$987$	−0.532753	−0.0169577
$988$	0	0
$989$	1.02831	0.0326983
$990$	0	0
$991$	−16.4437	−0.522351	−0.261175	−	0.965291i	$-0.584110\pi$
$991$	−16.4437	−0.522351	−0.261175	+	0.965291i	$0.584110\pi$
$992$	0	0
$993$	−4.48141	−0.142213
$994$	0	0
$995$	1.25660	0.0398368
$996$	0	0
$997$	27.1390	0.859499	0.429750	−	0.902948i	$-0.358602\pi$
$997$	27.1390	0.859499	0.429750	+	0.902948i	$0.358602\pi$
$998$	0	0
$999$	1.09525	0.0346521

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4012.2.a.j.1.10	✓	21

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4012.2.a.j.1.10	✓	21	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:	\( N \)	\(=\)	\( 4012 = 2^{2} \cdot 17 \cdot 59 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4012.a (trivial)

\(f(q)\)	\(=\)	\(q+0.210829 q^{3} +0.593138 q^{5} -1.60715 q^{7} -2.95555 q^{9} +O(q^{10})\)	\(q+0.210829 q^{3} +0.593138 q^{5} -1.60715 q^{7} -2.95555 q^{9} -2.16189 q^{11} +2.74022 q^{13} +0.125051 q^{15} +1.00000 q^{17} +0.223492 q^{19} -0.338835 q^{21} +1.30075 q^{23} -4.64819 q^{25} -1.25560 q^{27} +9.53334 q^{29} +5.20344 q^{31} -0.455789 q^{33} -0.953263 q^{35} -0.872287 q^{37} +0.577718 q^{39} -10.2875 q^{41} +0.790551 q^{43} -1.75305 q^{45} +1.57231 q^{47} -4.41706 q^{49} +0.210829 q^{51} -5.39157 q^{53} -1.28230 q^{55} +0.0471186 q^{57} +1.00000 q^{59} +12.4593 q^{61} +4.75002 q^{63} +1.62533 q^{65} +14.6742 q^{67} +0.274236 q^{69} +13.3754 q^{71} -4.98603 q^{73} -0.979974 q^{75} +3.47448 q^{77} -8.57203 q^{79} +8.60194 q^{81} +10.9011 q^{83} +0.593138 q^{85} +2.00991 q^{87} +6.17014 q^{89} -4.40395 q^{91} +1.09704 q^{93} +0.132561 q^{95} +2.10120 q^{97} +6.38957 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 21 q + 9 q^{3} + q^{5} + 11 q^{7} + 26 q^{9}+O(q^{10}) \) 21 * q + 9 * q^3 + q^5 + 11 * q^7 + 26 * q^9	\( 21 q + 9 q^{3} + q^{5} + 11 q^{7} + 26 q^{9} + 12 q^{11} - 4 q^{13} + 9 q^{15} + 21 q^{17} + 4 q^{19} + 8 q^{21} + 19 q^{23} + 26 q^{25} + 33 q^{27} - q^{29} + 13 q^{31} + 11 q^{33} + 15 q^{35} - 4 q^{37} + 18 q^{39} + 9 q^{41} + 7 q^{43} + 7 q^{45} + 33 q^{47} + 36 q^{49} + 9 q^{51} + 7 q^{53} + 12 q^{55} + 26 q^{57} + 21 q^{59} + 3 q^{61} + 51 q^{63} + q^{65} + 8 q^{69} + 55 q^{71} + 22 q^{73} + 14 q^{75} - 15 q^{77} + 28 q^{79} + 25 q^{81} + 54 q^{83} + q^{85} + 34 q^{87} + 30 q^{89} + 35 q^{91} - 5 q^{93} + 42 q^{95} + 9 q^{97} + 20 q^{99}+O(q^{100}) \) 21 * q + 9 * q^3 + q^5 + 11 * q^7 + 26 * q^9 + 12 * q^11 - 4 * q^13 + 9 * q^15 + 21 * q^17 + 4 * q^19 + 8 * q^21 + 19 * q^23 + 26 * q^25 + 33 * q^27 - q^29 + 13 * q^31 + 11 * q^33 + 15 * q^35 - 4 * q^37 + 18 * q^39 + 9 * q^41 + 7 * q^43 + 7 * q^45 + 33 * q^47 + 36 * q^49 + 9 * q^51 + 7 * q^53 + 12 * q^55 + 26 * q^57 + 21 * q^59 + 3 * q^61 + 51 * q^63 + q^65 + 8 * q^69 + 55 * q^71 + 22 * q^73 + 14 * q^75 - 15 * q^77 + 28 * q^79 + 25 * q^81 + 54 * q^83 + q^85 + 34 * q^87 + 30 * q^89 + 35 * q^91 - 5 * q^93 + 42 * q^95 + 9 * q^97 + 20 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more