LMFDB - Embedded newform 8036.2.a.t.1.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8036,2,Mod(1,8036)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8036, 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("8036.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8036 = 2^{2} \cdot 7^{2} \cdot 41 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8036.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:	$64.1677830643$
Analytic rank:	$0$
Dimension:	$20$
Coefficient field:	$\mathbb{Q}[x]/(x^{20} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} - 4 x^{19} - 30 x^{18} + 128 x^{17} + 348 x^{16} - 1644 x^{15} - 1934 x^{14} + 10948 x^{13} + \cdots + 9 $ x^20 - 4x^19 - 30x^18 + 128x^17 + 348x^16 - 1644x^15 - 1934x^14 + 10948x^13 + 4748x^12 - 40524x^11 - 220x^10 + 82500x^9 - 21992x^8 - 84720x^7 + 37544x^6 + 34656x^5 - 18823x^4 - 2276x^3 + 1130x^2 + 204*x + 9
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$-1.37724$ of defining polynomial
Character	$\chi$	$=$	8036.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.37724 q^{3} +3.83294 q^{5} -1.10321 q^{9} +O(q^{10})$	$q-1.37724 q^{3} +3.83294 q^{5} -1.10321 q^{9} +0.294657 q^{11} +2.83491 q^{13} -5.27887 q^{15} +3.23930 q^{17} -5.79243 q^{19} +5.90009 q^{23} +9.69141 q^{25} +5.65110 q^{27} +3.65123 q^{29} -1.59205 q^{31} -0.405813 q^{33} +5.20818 q^{37} -3.90435 q^{39} -1.00000 q^{41} +5.12051 q^{43} -4.22854 q^{45} -12.1496 q^{47} -4.46130 q^{51} -7.84750 q^{53} +1.12940 q^{55} +7.97757 q^{57} +2.50717 q^{59} +2.24588 q^{61} +10.8660 q^{65} -8.73987 q^{67} -8.12584 q^{69} -1.66646 q^{71} +12.8659 q^{73} -13.3474 q^{75} -0.253450 q^{79} -4.47329 q^{81} +12.7863 q^{83} +12.4160 q^{85} -5.02862 q^{87} -3.82433 q^{89} +2.19263 q^{93} -22.2020 q^{95} +18.4192 q^{97} -0.325068 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q + 4 q^{3} + 8 q^{5} + 16 q^{9}+O(q^{10}) $ 20 * q + 4 * q^3 + 8 * q^5 + 16 * q^9	$ 20 q + 4 q^{3} + 8 q^{5} + 16 q^{9} - 8 q^{11} + 12 q^{13} + 8 q^{15} + 8 q^{17} + 24 q^{19} + 8 q^{23} + 20 q^{25} + 16 q^{27} - 12 q^{29} + 44 q^{33} + 12 q^{37} + 12 q^{39} - 20 q^{41} + 4 q^{43} + 40 q^{45} + 4 q^{47} + 4 q^{51} - 12 q^{53} - 16 q^{55} + 28 q^{57} + 16 q^{59} + 68 q^{61} - 8 q^{65} + 4 q^{67} + 32 q^{69} + 8 q^{71} + 48 q^{73} + 60 q^{75} - 20 q^{79} + 32 q^{81} - 8 q^{83} - 28 q^{85} + 60 q^{89} - 16 q^{93} + 20 q^{95} + 40 q^{97}+O(q^{100}) $ 20 * q + 4 * q^3 + 8 * q^5 + 16 * q^9 - 8 * q^11 + 12 * q^13 + 8 * q^15 + 8 * q^17 + 24 * q^19 + 8 * q^23 + 20 * q^25 + 16 * q^27 - 12 * q^29 + 44 * q^33 + 12 * q^37 + 12 * q^39 - 20 * q^41 + 4 * q^43 + 40 * q^45 + 4 * q^47 + 4 * q^51 - 12 * q^53 - 16 * q^55 + 28 * q^57 + 16 * q^59 + 68 * q^61 - 8 * q^65 + 4 * q^67 + 32 * q^69 + 8 * q^71 + 48 * q^73 + 60 * q^75 - 20 * q^79 + 32 * q^81 - 8 * q^83 - 28 * q^85 + 60 * q^89 - 16 * q^93 + 20 * q^95 + 40 * q^97

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
$3$	−1.37724	−0.795150	−0.397575	−	0.917570i	$-0.630148\pi$
$3$	−1.37724	−0.795150	−0.397575	+	0.917570i	$0.630148\pi$
$4$	0	0
$5$	3.83294	1.71414	0.857071	−	0.515199i	$-0.172282\pi$
$5$	3.83294	1.71414	0.857071	+	0.515199i	$0.172282\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	−1.10321	−0.367737
$10$	0	0
$11$	0.294657	0.0888424	0.0444212	−	0.999013i	$-0.485856\pi$
$11$	0.294657	0.0888424	0.0444212	+	0.999013i	$0.485856\pi$
$12$	0	0
$13$	2.83491	0.786262	0.393131	−	0.919482i	$-0.371392\pi$
$13$	2.83491	0.786262	0.393131	+	0.919482i	$0.371392\pi$
$14$	0	0
$15$	−5.27887	−1.36300
$16$	0	0
$17$	3.23930	0.785646	0.392823	−	0.919614i	$-0.371498\pi$
$17$	3.23930	0.785646	0.392823	+	0.919614i	$0.371498\pi$
$18$	0	0
$19$	−5.79243	−1.32888	−0.664438	−	0.747344i	$-0.731329\pi$
$19$	−5.79243	−1.32888	−0.664438	+	0.747344i	$0.731329\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	5.90009	1.23025	0.615127	−	0.788428i	$-0.289105\pi$
$23$	5.90009	1.23025	0.615127	+	0.788428i	$0.289105\pi$
$24$	0	0
$25$	9.69141	1.93828
$26$	0	0
$27$	5.65110	1.08756
$28$	0	0
$29$	3.65123	0.678016	0.339008	−	0.940783i	$-0.389909\pi$
$29$	3.65123	0.678016	0.339008	+	0.940783i	$0.389909\pi$
$30$	0	0
$31$	−1.59205	−0.285940	−0.142970	−	0.989727i	$-0.545665\pi$
$31$	−1.59205	−0.285940	−0.142970	+	0.989727i	$0.545665\pi$
$32$	0	0
$33$	−0.405813	−0.0706430
$34$	0	0
$35$	0	0
$36$	0	0
$37$	5.20818	0.856219	0.428110	−	0.903727i	$-0.359180\pi$
$37$	5.20818	0.856219	0.428110	+	0.903727i	$0.359180\pi$
$38$	0	0
$39$	−3.90435	−0.625196
$40$	0	0
$41$	−1.00000	−0.156174
$41$	−1.00000	−0.156174
$42$	0	0
$43$	5.12051	0.780871	0.390436	−	0.920630i	$-0.372324\pi$
$43$	5.12051	0.780871	0.390436	+	0.920630i	$0.372324\pi$
$44$	0	0
$45$	−4.22854	−0.630353
$46$	0	0
$47$	−12.1496	−1.77220	−0.886099	−	0.463496i	$-0.846595\pi$
$47$	−12.1496	−1.77220	−0.886099	+	0.463496i	$0.846595\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	−4.46130	−0.624706
$52$	0	0
$53$	−7.84750	−1.07794	−0.538968	−	0.842326i	$-0.681186\pi$
$53$	−7.84750	−1.07794	−0.538968	+	0.842326i	$0.681186\pi$
$54$	0	0
$55$	1.12940	0.152288
$56$	0	0
$57$	7.97757	1.05666
$58$	0	0
$59$	2.50717	0.326406	0.163203	−	0.986593i	$-0.447817\pi$
$59$	2.50717	0.326406	0.163203	+	0.986593i	$0.447817\pi$
$60$	0	0
$61$	2.24588	0.287556	0.143778	−	0.989610i	$-0.454075\pi$
$61$	2.24588	0.287556	0.143778	+	0.989610i	$0.454075\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	10.8660	1.34776
$66$	0	0
$67$	−8.73987	−1.06774	−0.533872	−	0.845565i	$-0.679264\pi$
$67$	−8.73987	−1.06774	−0.533872	+	0.845565i	$0.679264\pi$
$68$	0	0
$69$	−8.12584	−0.978236
$70$	0	0
$71$	−1.66646	−0.197773	−0.0988865	−	0.995099i	$-0.531528\pi$
$71$	−1.66646	−0.197773	−0.0988865	+	0.995099i	$0.531528\pi$
$72$	0	0
$73$	12.8659	1.50584	0.752920	−	0.658112i	$-0.228645\pi$
$73$	12.8659	1.50584	0.752920	+	0.658112i	$0.228645\pi$
$74$	0	0
$75$	−13.3474	−1.54122
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−0.253450	−0.0285154	−0.0142577	−	0.999898i	$-0.504539\pi$
$79$	−0.253450	−0.0285154	−0.0142577	+	0.999898i	$0.504539\pi$
$80$	0	0
$81$	−4.47329	−0.497033
$82$	0	0
$83$	12.7863	1.40348	0.701739	−	0.712435i	$-0.252407\pi$
$83$	12.7863	1.40348	0.701739	+	0.712435i	$0.252407\pi$
$84$	0	0
$85$	12.4160	1.34671
$86$	0	0
$87$	−5.02862	−0.539125
$88$	0	0
$89$	−3.82433	−0.405379	−0.202689	−	0.979243i	$-0.564968\pi$
$89$	−3.82433	−0.405379	−0.202689	+	0.979243i	$0.564968\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	2.19263	0.227365
$94$	0	0
$95$	−22.2020	−2.27788
$96$	0	0
$97$	18.4192	1.87018	0.935091	−	0.354406i	$-0.115317\pi$
$97$	18.4192	1.87018	0.935091	+	0.354406i	$0.115317\pi$
$98$	0	0
$99$	−0.325068	−0.0326706
$100$	0	0
$101$	−6.00514	−0.597533	−0.298767	−	0.954326i	$-0.596575\pi$
$101$	−6.00514	−0.597533	−0.298767	+	0.954326i	$0.596575\pi$
$102$	0	0
$103$	4.01323	0.395435	0.197717	−	0.980259i	$-0.436647\pi$
$103$	4.01323	0.395435	0.197717	+	0.980259i	$0.436647\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	11.8319	1.14384	0.571918	−	0.820311i	$-0.306199\pi$
$107$	11.8319	1.14384	0.571918	+	0.820311i	$0.306199\pi$
$108$	0	0
$109$	−17.7180	−1.69707	−0.848536	−	0.529137i	$-0.822516\pi$
$109$	−17.7180	−1.69707	−0.848536	+	0.529137i	$0.822516\pi$
$110$	0	0
$111$	−7.17291	−0.680823
$112$	0	0
$113$	16.0057	1.50569	0.752845	−	0.658198i	$-0.228681\pi$
$113$	16.0057	1.50569	0.752845	+	0.658198i	$0.228681\pi$
$114$	0	0
$115$	22.6147	2.10883
$116$	0	0
$117$	−3.12750	−0.289137
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−10.9132	−0.992107
$122$	0	0
$123$	1.37724	0.124182
$124$	0	0
$125$	17.9819	1.60835
$126$	0	0
$127$	−19.0524	−1.69062	−0.845312	−	0.534273i	$-0.820585\pi$
$127$	−19.0524	−1.69062	−0.845312	+	0.534273i	$0.820585\pi$
$128$	0	0
$129$	−7.05218	−0.620909
$130$	0	0
$131$	7.45949	0.651739	0.325870	−	0.945415i	$-0.394343\pi$
$131$	7.45949	0.651739	0.325870	+	0.945415i	$0.394343\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	21.6603	1.86422
$136$	0	0
$137$	3.70701	0.316712	0.158356	−	0.987382i	$-0.449381\pi$
$137$	3.70701	0.316712	0.158356	+	0.987382i	$0.449381\pi$
$138$	0	0
$139$	12.7043	1.07756	0.538781	−	0.842446i	$-0.318885\pi$
$139$	12.7043	1.07756	0.538781	+	0.842446i	$0.318885\pi$
$140$	0	0
$141$	16.7329	1.40916
$142$	0	0
$143$	0.835325	0.0698534
$144$	0	0
$145$	13.9949	1.16222
$146$	0	0
$147$	0	0
$148$	0	0
$149$	4.09774	0.335700	0.167850	−	0.985813i	$-0.446318\pi$
$149$	4.09774	0.335700	0.167850	+	0.985813i	$0.446318\pi$
$150$	0	0
$151$	−10.7756	−0.876908	−0.438454	−	0.898754i	$-0.644474\pi$
$151$	−10.7756	−0.876908	−0.438454	+	0.898754i	$0.644474\pi$
$152$	0	0
$153$	−3.57363	−0.288911
$154$	0	0
$155$	−6.10221	−0.490141
$156$	0	0
$157$	8.20658	0.654956	0.327478	−	0.944859i	$-0.393801\pi$
$157$	8.20658	0.654956	0.327478	+	0.944859i	$0.393801\pi$
$158$	0	0
$159$	10.8079	0.857121
$160$	0	0
$161$	0	0
$162$	0	0
$163$	−16.2558	−1.27325	−0.636627	−	0.771172i	$-0.719671\pi$
$163$	−16.2558	−1.27325	−0.636627	+	0.771172i	$0.719671\pi$
$164$	0	0
$165$	−1.55546	−0.121092
$166$	0	0
$167$	20.4805	1.58483	0.792416	−	0.609981i	$-0.208823\pi$
$167$	20.4805	1.58483	0.792416	+	0.609981i	$0.208823\pi$
$168$	0	0
$169$	−4.96330	−0.381792
$170$	0	0
$171$	6.39028	0.488677
$172$	0	0
$173$	−0.934381	−0.0710396	−0.0355198	−	0.999369i	$-0.511309\pi$
$173$	−0.934381	−0.0710396	−0.0355198	+	0.999369i	$0.511309\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−3.45298	−0.259542
$178$	0	0
$179$	−3.62356	−0.270838	−0.135419	−	0.990788i	$-0.543238\pi$
$179$	−3.62356	−0.270838	−0.135419	+	0.990788i	$0.543238\pi$
$180$	0	0
$181$	3.44901	0.256363	0.128181	−	0.991751i	$-0.459086\pi$
$181$	3.44901	0.256363	0.128181	+	0.991751i	$0.459086\pi$
$182$	0	0
$183$	−3.09312	−0.228650
$184$	0	0
$185$	19.9626	1.46768
$186$	0	0
$187$	0.954482	0.0697987
$188$	0	0
$189$	0	0
$190$	0	0
$191$	5.69472	0.412056	0.206028	−	0.978546i	$-0.433946\pi$
$191$	5.69472	0.412056	0.206028	+	0.978546i	$0.433946\pi$
$192$	0	0
$193$	23.3923	1.68382	0.841909	−	0.539620i	$-0.181432\pi$
$193$	23.3923	1.68382	0.841909	+	0.539620i	$0.181432\pi$
$194$	0	0
$195$	−14.9651	−1.07167
$196$	0	0
$197$	3.84256	0.273771	0.136885	−	0.990587i	$-0.456291\pi$
$197$	3.84256	0.273771	0.136885	+	0.990587i	$0.456291\pi$
$198$	0	0
$199$	4.82239	0.341850	0.170925	−	0.985284i	$-0.445324\pi$
$199$	4.82239	0.341850	0.170925	+	0.985284i	$0.445324\pi$
$200$	0	0
$201$	12.0369	0.849017
$202$	0	0
$203$	0	0
$204$	0	0
$205$	−3.83294	−0.267704
$206$	0	0
$207$	−6.50904	−0.452410
$208$	0	0
$209$	−1.70678	−0.118060
$210$	0	0
$211$	−7.14339	−0.491771	−0.245886	−	0.969299i	$-0.579079\pi$
$211$	−7.14339	−0.491771	−0.245886	+	0.969299i	$0.579079\pi$
$212$	0	0
$213$	2.29512	0.157259
$214$	0	0
$215$	19.6266	1.33852
$216$	0	0
$217$	0	0
$218$	0	0
$219$	−17.7194	−1.19737
$220$	0	0
$221$	9.18312	0.617724
$222$	0	0
$223$	−22.2017	−1.48674	−0.743368	−	0.668882i	$-0.766773\pi$
$223$	−22.2017	−1.48674	−0.743368	+	0.668882i	$0.766773\pi$
$224$	0	0
$225$	−10.6917	−0.712777
$226$	0	0
$227$	−16.6344	−1.10407	−0.552033	−	0.833822i	$-0.686148\pi$
$227$	−16.6344	−1.10407	−0.552033	+	0.833822i	$0.686148\pi$
$228$	0	0
$229$	20.2931	1.34100	0.670502	−	0.741908i	$-0.266079\pi$
$229$	20.2931	1.34100	0.670502	+	0.741908i	$0.266079\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	23.0911	1.51275	0.756374	−	0.654140i	$-0.226969\pi$
$233$	23.0911	1.51275	0.756374	+	0.654140i	$0.226969\pi$
$234$	0	0
$235$	−46.5686	−3.03780
$236$	0	0
$237$	0.349062	0.0226740
$238$	0	0
$239$	3.75798	0.243083	0.121542	−	0.992586i	$-0.461216\pi$
$239$	3.75798	0.243083	0.121542	+	0.992586i	$0.461216\pi$
$240$	0	0
$241$	−15.6661	−1.00914	−0.504572	−	0.863370i	$-0.668350\pi$
$241$	−15.6661	−1.00914	−0.504572	+	0.863370i	$0.668350\pi$
$242$	0	0
$243$	−10.7925	−0.692340
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−16.4210	−1.04484
$248$	0	0
$249$	−17.6098	−1.11597
$250$	0	0
$251$	9.71267	0.613058	0.306529	−	0.951861i	$-0.400832\pi$
$251$	9.71267	0.613058	0.306529	+	0.951861i	$0.400832\pi$
$252$	0	0
$253$	1.73850	0.109299
$254$	0	0
$255$	−17.0999	−1.07084
$256$	0	0
$257$	9.93296	0.619601	0.309800	−	0.950802i	$-0.399738\pi$
$257$	9.93296	0.619601	0.309800	+	0.950802i	$0.399738\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	−4.02808	−0.249332
$262$	0	0
$263$	4.98671	0.307493	0.153747	−	0.988110i	$-0.450866\pi$
$263$	4.98671	0.307493	0.153747	+	0.988110i	$0.450866\pi$
$264$	0	0
$265$	−30.0790	−1.84774
$266$	0	0
$267$	5.26703	0.322337
$268$	0	0
$269$	−28.7677	−1.75400	−0.876999	−	0.480492i	$-0.840458\pi$
$269$	−28.7677	−1.75400	−0.876999	+	0.480492i	$0.840458\pi$
$270$	0	0
$271$	−6.85014	−0.416116	−0.208058	−	0.978116i	$-0.566714\pi$
$271$	−6.85014	−0.416116	−0.208058	+	0.978116i	$0.566714\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	2.85564	0.172201
$276$	0	0
$277$	20.6436	1.24035	0.620176	−	0.784463i	$-0.287061\pi$
$277$	20.6436	1.24035	0.620176	+	0.784463i	$0.287061\pi$
$278$	0	0
$279$	1.75636	0.105151
$280$	0	0
$281$	−25.0914	−1.49683	−0.748414	−	0.663232i	$-0.769184\pi$
$281$	−25.0914	−1.49683	−0.748414	+	0.663232i	$0.769184\pi$
$282$	0	0
$283$	20.9648	1.24623	0.623115	−	0.782130i	$-0.285867\pi$
$283$	20.9648	1.24623	0.623115	+	0.782130i	$0.285867\pi$
$284$	0	0
$285$	30.5775	1.81126
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−6.50692	−0.382760
$290$	0	0
$291$	−25.3676	−1.48708
$292$	0	0
$293$	7.12226	0.416087	0.208043	−	0.978120i	$-0.433290\pi$
$293$	7.12226	0.416087	0.208043	+	0.978120i	$0.433290\pi$
$294$	0	0
$295$	9.60983	0.559506
$296$	0	0
$297$	1.66514	0.0966210
$298$	0	0
$299$	16.7262	0.967302
$300$	0	0
$301$	0	0
$302$	0	0
$303$	8.27051	0.475128
$304$	0	0
$305$	8.60832	0.492911
$306$	0	0
$307$	2.75757	0.157383	0.0786915	−	0.996899i	$-0.474926\pi$
$307$	2.75757	0.157383	0.0786915	+	0.996899i	$0.474926\pi$
$308$	0	0
$309$	−5.52717	−0.314430
$310$	0	0
$311$	−2.57795	−0.146182	−0.0730911	−	0.997325i	$-0.523286\pi$
$311$	−2.57795	−0.146182	−0.0730911	+	0.997325i	$0.523286\pi$
$312$	0	0
$313$	−9.32344	−0.526992	−0.263496	−	0.964660i	$-0.584876\pi$
$313$	−9.32344	−0.526992	−0.263496	+	0.964660i	$0.584876\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	1.32284	0.0742984	0.0371492	−	0.999310i	$-0.488172\pi$
$317$	1.32284	0.0742984	0.0371492	+	0.999310i	$0.488172\pi$
$318$	0	0
$319$	1.07586	0.0602366
$320$	0	0
$321$	−16.2954	−0.909522
$322$	0	0
$323$	−18.7634	−1.04403
$324$	0	0
$325$	27.4742	1.52400
$326$	0	0
$327$	24.4019	1.34943
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−3.42585	−0.188302	−0.0941508	−	0.995558i	$-0.530014\pi$
$331$	−3.42585	−0.188302	−0.0941508	+	0.995558i	$0.530014\pi$
$332$	0	0
$333$	−5.74572	−0.314863
$334$	0	0
$335$	−33.4994	−1.83027
$336$	0	0
$337$	−11.2175	−0.611054	−0.305527	−	0.952183i	$-0.598833\pi$
$337$	−11.2175	−0.611054	−0.305527	+	0.952183i	$0.598833\pi$
$338$	0	0
$339$	−22.0437	−1.19725
$340$	0	0
$341$	−0.469107	−0.0254036
$342$	0	0
$343$	0	0
$344$	0	0
$345$	−31.1458	−1.67684
$346$	0	0
$347$	0.348754	0.0187221	0.00936106	−	0.999956i	$-0.497020\pi$
$347$	0.348754	0.0187221	0.00936106	+	0.999956i	$0.497020\pi$
$348$	0	0
$349$	−22.5913	−1.20929	−0.604643	−	0.796496i	$-0.706684\pi$
$349$	−22.5913	−1.20929	−0.604643	+	0.796496i	$0.706684\pi$
$350$	0	0
$351$	16.0204	0.855104
$352$	0	0
$353$	−5.99615	−0.319143	−0.159571	−	0.987186i	$-0.551011\pi$
$353$	−5.99615	−0.319143	−0.159571	+	0.987186i	$0.551011\pi$
$354$	0	0
$355$	−6.38745	−0.339011
$356$	0	0
$357$	0	0
$358$	0	0
$359$	20.0664	1.05907	0.529533	−	0.848289i	$-0.322367\pi$
$359$	20.0664	1.05907	0.529533	+	0.848289i	$0.322367\pi$
$360$	0	0
$361$	14.5523	0.765911
$362$	0	0
$363$	15.0301	0.788874
$364$	0	0
$365$	49.3142	2.58122
$366$	0	0
$367$	−0.869278	−0.0453759	−0.0226880	−	0.999743i	$-0.507222\pi$
$367$	−0.869278	−0.0453759	−0.0226880	+	0.999743i	$0.507222\pi$
$368$	0	0
$369$	1.10321	0.0574309
$370$	0	0
$371$	0	0
$372$	0	0
$373$	20.9446	1.08447	0.542235	−	0.840227i	$-0.317578\pi$
$373$	20.9446	1.08447	0.542235	+	0.840227i	$0.317578\pi$
$374$	0	0
$375$	−24.7653	−1.27888
$376$	0	0
$377$	10.3509	0.533098
$378$	0	0
$379$	22.3371	1.14738	0.573689	−	0.819073i	$-0.305512\pi$
$379$	22.3371	1.14738	0.573689	+	0.819073i	$0.305512\pi$
$380$	0	0
$381$	26.2397	1.34430
$382$	0	0
$383$	23.5563	1.20367	0.601835	−	0.798621i	$-0.294437\pi$
$383$	23.5563	1.20367	0.601835	+	0.798621i	$0.294437\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−5.64901	−0.287155
$388$	0	0
$389$	−10.8318	−0.549195	−0.274597	−	0.961559i	$-0.588545\pi$
$389$	−10.8318	−0.549195	−0.274597	+	0.961559i	$0.588545\pi$
$390$	0	0
$391$	19.1122	0.966544
$392$	0	0
$393$	−10.2735	−0.518230
$394$	0	0
$395$	−0.971458	−0.0488794
$396$	0	0
$397$	−18.1223	−0.909533	−0.454766	−	0.890611i	$-0.650277\pi$
$397$	−18.1223	−0.909533	−0.454766	+	0.890611i	$0.650277\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	6.39251	0.319227	0.159613	−	0.987180i	$-0.448975\pi$
$401$	6.39251	0.319227	0.159613	+	0.987180i	$0.448975\pi$
$402$	0	0
$403$	−4.51330	−0.224824
$404$	0	0
$405$	−17.1459	−0.851984
$406$	0	0
$407$	1.53463	0.0760686
$408$	0	0
$409$	20.3931	1.00837	0.504187	−	0.863595i	$-0.331792\pi$
$409$	20.3931	1.00837	0.504187	+	0.863595i	$0.331792\pi$
$410$	0	0
$411$	−5.10545	−0.251833
$412$	0	0
$413$	0	0
$414$	0	0
$415$	49.0090	2.40576
$416$	0	0
$417$	−17.4968	−0.856823
$418$	0	0
$419$	30.9267	1.51087	0.755435	−	0.655224i	$-0.227426\pi$
$419$	30.9267	1.51087	0.755435	+	0.655224i	$0.227426\pi$
$420$	0	0
$421$	−17.5442	−0.855050	−0.427525	−	0.904004i	$-0.640614\pi$
$421$	−17.5442	−0.855050	−0.427525	+	0.904004i	$0.640614\pi$
$422$	0	0
$423$	13.4035	0.651703
$424$	0	0
$425$	31.3934	1.52280
$426$	0	0
$427$	0	0
$428$	0	0
$429$	−1.15044	−0.0555439
$430$	0	0
$431$	−4.18368	−0.201521	−0.100760	−	0.994911i	$-0.532128\pi$
$431$	−4.18368	−0.201521	−0.100760	+	0.994911i	$0.532128\pi$
$432$	0	0
$433$	34.6744	1.66635	0.833173	−	0.553013i	$-0.186522\pi$
$433$	34.6744	1.66635	0.833173	+	0.553013i	$0.186522\pi$
$434$	0	0
$435$	−19.2744	−0.924136
$436$	0	0
$437$	−34.1759	−1.63485
$438$	0	0
$439$	15.0297	0.717327	0.358663	−	0.933467i	$-0.383233\pi$
$439$	15.0297	0.717327	0.358663	+	0.933467i	$0.383233\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−3.16103	−0.150185	−0.0750925	−	0.997177i	$-0.523925\pi$
$443$	−3.16103	−0.150185	−0.0750925	+	0.997177i	$0.523925\pi$
$444$	0	0
$445$	−14.6584	−0.694876
$446$	0	0
$447$	−5.64357	−0.266932
$448$	0	0
$449$	19.8314	0.935901	0.467951	−	0.883755i	$-0.344992\pi$
$449$	19.8314	0.935901	0.467951	+	0.883755i	$0.344992\pi$
$450$	0	0
$451$	−0.294657	−0.0138748
$452$	0	0
$453$	14.8406	0.697273
$454$	0	0
$455$	0	0
$456$	0	0
$457$	21.4151	1.00176	0.500878	−	0.865518i	$-0.333011\pi$
$457$	21.4151	1.00176	0.500878	+	0.865518i	$0.333011\pi$
$458$	0	0
$459$	18.3056	0.854434
$460$	0	0
$461$	11.5514	0.538000	0.269000	−	0.963140i	$-0.413307\pi$
$461$	11.5514	0.538000	0.269000	+	0.963140i	$0.413307\pi$
$462$	0	0
$463$	−23.9717	−1.11406	−0.557029	−	0.830493i	$-0.688059\pi$
$463$	−23.9717	−1.11406	−0.557029	+	0.830493i	$0.688059\pi$
$464$	0	0
$465$	8.40421	0.389736
$466$	0	0
$467$	41.1512	1.90425	0.952126	−	0.305706i	$-0.0988925\pi$
$467$	41.1512	1.90425	0.952126	+	0.305706i	$0.0988925\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	−11.3024	−0.520788
$472$	0	0
$473$	1.50879	0.0693744
$474$	0	0
$475$	−56.1368	−2.57573
$476$	0	0
$477$	8.65744	0.396397
$478$	0	0
$479$	−2.86717	−0.131004	−0.0655021	−	0.997852i	$-0.520865\pi$
$479$	−2.86717	−0.131004	−0.0655021	+	0.997852i	$0.520865\pi$
$480$	0	0
$481$	14.7647	0.673213
$482$	0	0
$483$	0	0
$484$	0	0
$485$	70.5995	3.20576
$486$	0	0
$487$	−14.7224	−0.667134	−0.333567	−	0.942726i	$-0.608252\pi$
$487$	−14.7224	−0.667134	−0.333567	+	0.942726i	$0.608252\pi$
$488$	0	0
$489$	22.3881	1.01243
$490$	0	0
$491$	−22.4449	−1.01293	−0.506463	−	0.862262i	$-0.669047\pi$
$491$	−22.4449	−1.01293	−0.506463	+	0.862262i	$0.669047\pi$
$492$	0	0
$493$	11.8274	0.532681
$494$	0	0
$495$	−1.24597	−0.0560021
$496$	0	0
$497$	0	0
$498$	0	0
$499$	11.1361	0.498520	0.249260	−	0.968437i	$-0.419813\pi$
$499$	11.1361	0.498520	0.249260	+	0.968437i	$0.419813\pi$
$500$	0	0
$501$	−28.2066	−1.26018
$502$	0	0
$503$	−19.3862	−0.864389	−0.432194	−	0.901781i	$-0.642261\pi$
$503$	−19.3862	−0.864389	−0.432194	+	0.901781i	$0.642261\pi$
$504$	0	0
$505$	−23.0173	−1.02426
$506$	0	0
$507$	6.83565	0.303582
$508$	0	0
$509$	9.36124	0.414930	0.207465	−	0.978242i	$-0.433479\pi$
$509$	9.36124	0.414930	0.207465	+	0.978242i	$0.433479\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	−32.7337	−1.44523
$514$	0	0
$515$	15.3824	0.677831
$516$	0	0
$517$	−3.57996	−0.157446
$518$	0	0
$519$	1.28687	0.0564871
$520$	0	0
$521$	21.1882	0.928272	0.464136	−	0.885764i	$-0.346365\pi$
$521$	21.1882	0.928272	0.464136	+	0.885764i	$0.346365\pi$
$522$	0	0
$523$	2.91654	0.127531	0.0637657	−	0.997965i	$-0.479689\pi$
$523$	2.91654	0.127531	0.0637657	+	0.997965i	$0.479689\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−5.15712	−0.224648
$528$	0	0
$529$	11.8111	0.513525
$530$	0	0
$531$	−2.76594	−0.120032
$532$	0	0
$533$	−2.83491	−0.122793
$534$	0	0
$535$	45.3511	1.96070
$536$	0	0
$537$	4.99051	0.215357
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−19.1177	−0.821935	−0.410968	−	0.911650i	$-0.634809\pi$
$541$	−19.1177	−0.821935	−0.410968	+	0.911650i	$0.634809\pi$
$542$	0	0
$543$	−4.75011	−0.203847
$544$	0	0
$545$	−67.9118	−2.90902
$546$	0	0
$547$	1.62679	0.0695566	0.0347783	−	0.999395i	$-0.488927\pi$
$547$	1.62679	0.0695566	0.0347783	+	0.999395i	$0.488927\pi$
$548$	0	0
$549$	−2.47768	−0.105745
$550$	0	0
$551$	−21.1495	−0.900999
$552$	0	0
$553$	0	0
$554$	0	0
$555$	−27.4933	−1.16703
$556$	0	0
$557$	−9.02361	−0.382342	−0.191171	−	0.981557i	$-0.561229\pi$
$557$	−9.02361	−0.382342	−0.191171	+	0.981557i	$0.561229\pi$
$558$	0	0
$559$	14.5162	0.613969
$560$	0	0
$561$	−1.31455	−0.0555004
$562$	0	0
$563$	39.7373	1.67473	0.837363	−	0.546647i	$-0.184096\pi$
$563$	39.7373	1.67473	0.837363	+	0.546647i	$0.184096\pi$
$564$	0	0
$565$	61.3488	2.58096
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−22.4800	−0.942410	−0.471205	−	0.882024i	$-0.656181\pi$
$569$	−22.4800	−0.942410	−0.471205	+	0.882024i	$0.656181\pi$
$570$	0	0
$571$	−20.3494	−0.851594	−0.425797	−	0.904819i	$-0.640006\pi$
$571$	−20.3494	−0.851594	−0.425797	+	0.904819i	$0.640006\pi$
$572$	0	0
$573$	−7.84300	−0.327646
$574$	0	0
$575$	57.1802	2.38458
$576$	0	0
$577$	27.0963	1.12803	0.564017	−	0.825763i	$-0.309255\pi$
$577$	27.0963	1.12803	0.564017	+	0.825763i	$0.309255\pi$
$578$	0	0
$579$	−32.2169	−1.33889
$580$	0	0
$581$	0	0
$582$	0	0
$583$	−2.31232	−0.0957664
$584$	0	0
$585$	−11.9875	−0.495623
$586$	0	0
$587$	32.7604	1.35217	0.676083	−	0.736825i	$-0.263676\pi$
$587$	32.7604	1.35217	0.676083	+	0.736825i	$0.263676\pi$
$588$	0	0
$589$	9.22182	0.379979
$590$	0	0
$591$	−5.29212	−0.217689
$592$	0	0
$593$	35.3446	1.45143	0.725714	−	0.687996i	$-0.241509\pi$
$593$	35.3446	1.45143	0.725714	+	0.687996i	$0.241509\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−6.64159	−0.271822
$598$	0	0
$599$	20.9366	0.855446	0.427723	−	0.903910i	$-0.359316\pi$
$599$	20.9366	0.855446	0.427723	+	0.903910i	$0.359316\pi$
$600$	0	0
$601$	25.8264	1.05348	0.526741	−	0.850026i	$-0.323414\pi$
$601$	25.8264	1.05348	0.526741	+	0.850026i	$0.323414\pi$
$602$	0	0
$603$	9.64191	0.392649
$604$	0	0
$605$	−41.8295	−1.70061
$606$	0	0
$607$	−17.9433	−0.728297	−0.364149	−	0.931341i	$-0.618640\pi$
$607$	−17.9433	−0.728297	−0.364149	+	0.931341i	$0.618640\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−34.4429	−1.39341
$612$	0	0
$613$	14.1662	0.572166	0.286083	−	0.958205i	$-0.407647\pi$
$613$	14.1662	0.572166	0.286083	+	0.958205i	$0.407647\pi$
$614$	0	0
$615$	5.27887	0.212865
$616$	0	0
$617$	39.7328	1.59958	0.799791	−	0.600279i	$-0.204944\pi$
$617$	39.7328	1.59958	0.799791	+	0.600279i	$0.204944\pi$
$618$	0	0
$619$	−6.92778	−0.278451	−0.139226	−	0.990261i	$-0.544461\pi$
$619$	−6.92778	−0.278451	−0.139226	+	0.990261i	$0.544461\pi$
$620$	0	0
$621$	33.3420	1.33797
$622$	0	0
$623$	0	0
$624$	0	0
$625$	20.4663	0.818653
$626$	0	0
$627$	2.35065	0.0938757
$628$	0	0
$629$	16.8709	0.672686
$630$	0	0
$631$	39.4682	1.57121	0.785603	−	0.618731i	$-0.212353\pi$
$631$	39.4682	1.57121	0.785603	+	0.618731i	$0.212353\pi$
$632$	0	0
$633$	9.83816	0.391032
$634$	0	0
$635$	−73.0265	−2.89797
$636$	0	0
$637$	0	0
$638$	0	0
$639$	1.83846	0.0727284
$640$	0	0
$641$	−9.71499	−0.383719	−0.191860	−	0.981422i	$-0.561452\pi$
$641$	−9.71499	−0.383719	−0.191860	+	0.981422i	$0.561452\pi$
$642$	0	0
$643$	−39.6059	−1.56191	−0.780953	−	0.624590i	$-0.785266\pi$
$643$	−39.6059	−1.56191	−0.780953	+	0.624590i	$0.785266\pi$
$644$	0	0
$645$	−27.0305	−1.06433
$646$	0	0
$647$	−14.3703	−0.564955	−0.282478	−	0.959274i	$-0.591156\pi$
$647$	−14.3703	−0.564955	−0.282478	+	0.959274i	$0.591156\pi$
$648$	0	0
$649$	0.738755	0.0289987
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−35.7027	−1.39716	−0.698578	−	0.715534i	$-0.746184\pi$
$653$	−35.7027	−1.39716	−0.698578	+	0.715534i	$0.746184\pi$
$654$	0	0
$655$	28.5918	1.11717
$656$	0	0
$657$	−14.1938	−0.553753
$658$	0	0
$659$	−7.16822	−0.279234	−0.139617	−	0.990206i	$-0.544587\pi$
$659$	−7.16822	−0.279234	−0.139617	+	0.990206i	$0.544587\pi$
$660$	0	0
$661$	4.78101	0.185960	0.0929798	−	0.995668i	$-0.470361\pi$
$661$	4.78101	0.185960	0.0929798	+	0.995668i	$0.470361\pi$
$662$	0	0
$663$	−12.6474	−0.491183
$664$	0	0
$665$	0	0
$666$	0	0
$667$	21.5426	0.834132
$668$	0	0
$669$	30.5771	1.18218
$670$	0	0
$671$	0.661764	0.0255471
$672$	0	0
$673$	16.1792	0.623664	0.311832	−	0.950137i	$-0.399057\pi$
$673$	16.1792	0.623664	0.311832	+	0.950137i	$0.399057\pi$
$674$	0	0
$675$	54.7672	2.10799
$676$	0	0
$677$	−46.0847	−1.77118	−0.885590	−	0.464468i	$-0.846246\pi$
$677$	−46.0847	−1.77118	−0.885590	+	0.464468i	$0.846246\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	22.9096	0.877897
$682$	0	0
$683$	−45.3568	−1.73553	−0.867765	−	0.496975i	$-0.834444\pi$
$683$	−45.3568	−1.73553	−0.867765	+	0.496975i	$0.834444\pi$
$684$	0	0
$685$	14.2087	0.542888
$686$	0	0
$687$	−27.9484	−1.06630
$688$	0	0
$689$	−22.2469	−0.847541
$690$	0	0
$691$	−46.4521	−1.76712	−0.883561	−	0.468315i	$-0.844861\pi$
$691$	−46.4521	−1.76712	−0.883561	+	0.468315i	$0.844861\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	48.6946	1.84709
$696$	0	0
$697$	−3.23930	−0.122697
$698$	0	0
$699$	−31.8020	−1.20286
$700$	0	0
$701$	47.7384	1.80305	0.901527	−	0.432723i	$-0.142447\pi$
$701$	47.7384	1.80305	0.901527	+	0.432723i	$0.142447\pi$
$702$	0	0
$703$	−30.1680	−1.13781
$704$	0	0
$705$	64.1361	2.41550
$706$	0	0
$707$	0	0
$708$	0	0
$709$	20.3344	0.763674	0.381837	−	0.924230i	$-0.375292\pi$
$709$	20.3344	0.763674	0.381837	+	0.924230i	$0.375292\pi$
$710$	0	0
$711$	0.279609	0.0104862
$712$	0	0
$713$	−9.39321	−0.351779
$714$	0	0
$715$	3.20175	0.119739
$716$	0	0
$717$	−5.17564	−0.193288
$718$	0	0
$719$	46.1440	1.72088	0.860441	−	0.509550i	$-0.170188\pi$
$719$	46.1440	1.72088	0.860441	+	0.509550i	$0.170188\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	21.5760	0.802421
$724$	0	0
$725$	35.3855	1.31419
$726$	0	0
$727$	−43.4046	−1.60979	−0.804894	−	0.593418i	$-0.797778\pi$
$727$	−43.4046	−1.60979	−0.804894	+	0.593418i	$0.797778\pi$
$728$	0	0
$729$	28.2838	1.04755
$730$	0	0
$731$	16.5869	0.613488
$732$	0	0
$733$	−32.0809	−1.18493	−0.592467	−	0.805594i	$-0.701846\pi$
$733$	−32.0809	−1.18493	−0.592467	+	0.805594i	$0.701846\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−2.57526	−0.0948610
$738$	0	0
$739$	−27.2107	−1.00096	−0.500481	−	0.865748i	$-0.666843\pi$
$739$	−27.2107	−1.00096	−0.500481	+	0.865748i	$0.666843\pi$
$740$	0	0
$741$	22.6157	0.830808
$742$	0	0
$743$	21.4440	0.786704	0.393352	−	0.919388i	$-0.371315\pi$
$743$	21.4440	0.786704	0.393352	+	0.919388i	$0.371315\pi$
$744$	0	0
$745$	15.7064	0.575437
$746$	0	0
$747$	−14.1060	−0.516110
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−31.8901	−1.16369	−0.581843	−	0.813301i	$-0.697668\pi$
$751$	−31.8901	−1.16369	−0.581843	+	0.813301i	$0.697668\pi$
$752$	0	0
$753$	−13.3767	−0.487473
$754$	0	0
$755$	−41.3023	−1.50314
$756$	0	0
$757$	13.2405	0.481234	0.240617	−	0.970620i	$-0.422650\pi$
$757$	13.2405	0.481234	0.240617	+	0.970620i	$0.422650\pi$
$758$	0	0
$759$	−2.39433	−0.0869088
$760$	0	0
$761$	−10.8471	−0.393208	−0.196604	−	0.980483i	$-0.562991\pi$
$761$	−10.8471	−0.393208	−0.196604	+	0.980483i	$0.562991\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	−13.6975	−0.495234
$766$	0	0
$767$	7.10760	0.256641
$768$	0	0
$769$	19.1030	0.688870	0.344435	−	0.938810i	$-0.388070\pi$
$769$	19.1030	0.688870	0.344435	+	0.938810i	$0.388070\pi$
$770$	0	0
$771$	−13.6801	−0.492675
$772$	0	0
$773$	21.8868	0.787214	0.393607	−	0.919279i	$-0.371227\pi$
$773$	21.8868	0.787214	0.393607	+	0.919279i	$0.371227\pi$
$774$	0	0
$775$	−15.4292	−0.554232
$776$	0	0
$777$	0	0
$778$	0	0
$779$	5.79243	0.207536
$780$	0	0
$781$	−0.491035	−0.0175706
$782$	0	0
$783$	20.6335	0.737380
$784$	0	0
$785$	31.4553	1.12269
$786$	0	0
$787$	28.7321	1.02419	0.512094	−	0.858930i	$-0.328870\pi$
$787$	28.7321	1.02419	0.512094	+	0.858930i	$0.328870\pi$
$788$	0	0
$789$	−6.86789	−0.244503
$790$	0	0
$791$	0	0
$792$	0	0
$793$	6.36686	0.226094
$794$	0	0
$795$	41.4259	1.46923
$796$	0	0
$797$	−39.5018	−1.39922	−0.699612	−	0.714523i	$-0.746644\pi$
$797$	−39.5018	−1.39922	−0.699612	+	0.714523i	$0.746644\pi$
$798$	0	0
$799$	−39.3562	−1.39232
$800$	0	0
$801$	4.21905	0.149073
$802$	0	0
$803$	3.79103	0.133782
$804$	0	0
$805$	0	0
$806$	0	0
$807$	39.6200	1.39469
$808$	0	0
$809$	15.9020	0.559086	0.279543	−	0.960133i	$-0.409817\pi$
$809$	15.9020	0.559086	0.279543	+	0.960133i	$0.409817\pi$
$810$	0	0
$811$	18.2762	0.641763	0.320882	−	0.947119i	$-0.396021\pi$
$811$	18.2762	0.641763	0.320882	+	0.947119i	$0.396021\pi$
$812$	0	0
$813$	9.43428	0.330875
$814$	0	0
$815$	−62.3075	−2.18254
$816$	0	0
$817$	−29.6602	−1.03768
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−47.6782	−1.66398	−0.831990	−	0.554790i	$-0.812799\pi$
$821$	−47.6782	−1.66398	−0.831990	+	0.554790i	$0.812799\pi$
$822$	0	0
$823$	21.5891	0.752549	0.376275	−	0.926508i	$-0.377205\pi$
$823$	21.5891	0.752549	0.376275	+	0.926508i	$0.377205\pi$
$824$	0	0
$825$	−3.93290	−0.136926
$826$	0	0
$827$	−27.9750	−0.972786	−0.486393	−	0.873740i	$-0.661688\pi$
$827$	−27.9750	−0.972786	−0.486393	+	0.873740i	$0.661688\pi$
$828$	0	0
$829$	−42.6324	−1.48068	−0.740342	−	0.672231i	$-0.765336\pi$
$829$	−42.6324	−1.48068	−0.740342	+	0.672231i	$0.765336\pi$
$830$	0	0
$831$	−28.4311	−0.986265
$832$	0	0
$833$	0	0
$834$	0	0
$835$	78.5006	2.71663
$836$	0	0
$837$	−8.99682	−0.310975
$838$	0	0
$839$	44.3683	1.53176	0.765881	−	0.642982i	$-0.222303\pi$
$839$	44.3683	1.53176	0.765881	+	0.642982i	$0.222303\pi$
$840$	0	0
$841$	−15.6685	−0.540294
$842$	0	0
$843$	34.5569	1.19020
$844$	0	0
$845$	−19.0240	−0.654446
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−28.8736	−0.990940
$850$	0	0
$851$	30.7287	1.05337
$852$	0	0
$853$	41.9873	1.43762	0.718809	−	0.695207i	$-0.244687\pi$
$853$	41.9873	1.43762	0.718809	+	0.695207i	$0.244687\pi$
$854$	0	0
$855$	24.4935	0.837661
$856$	0	0
$857$	−14.8736	−0.508073	−0.254037	−	0.967195i	$-0.581758\pi$
$857$	−14.8736	−0.508073	−0.254037	+	0.967195i	$0.581758\pi$
$858$	0	0
$859$	27.4283	0.935841	0.467920	−	0.883771i	$-0.345003\pi$
$859$	27.4283	0.935841	0.467920	+	0.883771i	$0.345003\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−18.1036	−0.616253	−0.308126	−	0.951345i	$-0.599702\pi$
$863$	−18.1036	−0.616253	−0.308126	+	0.951345i	$0.599702\pi$
$864$	0	0
$865$	−3.58142	−0.121772
$866$	0	0
$867$	8.96159	0.304352
$868$	0	0
$869$	−0.0746808	−0.00253337
$870$	0	0
$871$	−24.7767	−0.839527
$872$	0	0
$873$	−20.3202	−0.687735
$874$	0	0
$875$	0	0
$876$	0	0
$877$	48.2964	1.63085	0.815427	−	0.578860i	$-0.196502\pi$
$877$	48.2964	1.63085	0.815427	+	0.578860i	$0.196502\pi$
$878$	0	0
$879$	−9.80906	−0.330851
$880$	0	0
$881$	24.5426	0.826861	0.413431	−	0.910536i	$-0.364330\pi$
$881$	24.5426	0.826861	0.413431	+	0.910536i	$0.364330\pi$
$882$	0	0
$883$	15.2037	0.511646	0.255823	−	0.966724i	$-0.417654\pi$
$883$	15.2037	0.511646	0.255823	+	0.966724i	$0.417654\pi$
$884$	0	0
$885$	−13.2350	−0.444891
$886$	0	0
$887$	−9.71937	−0.326344	−0.163172	−	0.986598i	$-0.552173\pi$
$887$	−9.71937	−0.326344	−0.163172	+	0.986598i	$0.552173\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	−1.31809	−0.0441576
$892$	0	0
$893$	70.3756	2.35503
$894$	0	0
$895$	−13.8889	−0.464254
$896$	0	0
$897$	−23.0360	−0.769150
$898$	0	0
$899$	−5.81292	−0.193872
$900$	0	0
$901$	−25.4204	−0.846877
$902$	0	0
$903$	0	0
$904$	0	0
$905$	13.2198	0.439442
$906$	0	0
$907$	−26.8243	−0.890688	−0.445344	−	0.895360i	$-0.646919\pi$
$907$	−26.8243	−0.890688	−0.445344	+	0.895360i	$0.646919\pi$
$908$	0	0
$909$	6.62493	0.219735
$910$	0	0
$911$	−44.8553	−1.48612	−0.743060	−	0.669224i	$-0.766627\pi$
$911$	−44.8553	−1.48612	−0.743060	+	0.669224i	$0.766627\pi$
$912$	0	0
$913$	3.76757	0.124688
$914$	0	0
$915$	−11.8557	−0.391938
$916$	0	0
$917$	0	0
$918$	0	0
$919$	−5.26072	−0.173535	−0.0867675	−	0.996229i	$-0.527654\pi$
$919$	−5.26072	−0.173535	−0.0867675	+	0.996229i	$0.527654\pi$
$920$	0	0
$921$	−3.79784	−0.125143
$922$	0	0
$923$	−4.72427	−0.155501
$924$	0	0
$925$	50.4746	1.65959
$926$	0	0
$927$	−4.42743	−0.145416
$928$	0	0
$929$	−28.4683	−0.934015	−0.467007	−	0.884253i	$-0.654668\pi$
$929$	−28.4683	−0.934015	−0.467007	+	0.884253i	$0.654668\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	3.55046	0.116237
$934$	0	0
$935$	3.65847	0.119645
$936$	0	0
$937$	25.1023	0.820058	0.410029	−	0.912072i	$-0.365519\pi$
$937$	25.1023	0.820058	0.410029	+	0.912072i	$0.365519\pi$
$938$	0	0
$939$	12.8406	0.419038
$940$	0	0
$941$	54.2707	1.76917	0.884586	−	0.466376i	$-0.154441\pi$
$941$	54.2707	1.76917	0.884586	+	0.466376i	$0.154441\pi$
$942$	0	0
$943$	−5.90009	−0.192133
$944$	0	0
$945$	0	0
$946$	0	0
$947$	40.9659	1.33121	0.665607	−	0.746302i	$-0.268173\pi$
$947$	40.9659	1.33121	0.665607	+	0.746302i	$0.268173\pi$
$948$	0	0
$949$	36.4737	1.18398
$950$	0	0
$951$	−1.82187	−0.0590783
$952$	0	0
$953$	−5.07815	−0.164497	−0.0822487	−	0.996612i	$-0.526210\pi$
$953$	−5.07815	−0.164497	−0.0822487	+	0.996612i	$0.526210\pi$
$954$	0	0
$955$	21.8275	0.706322
$956$	0	0
$957$	−1.48172	−0.0478971
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−28.4654	−0.918238
$962$	0	0
$963$	−13.0531	−0.420631
$964$	0	0
$965$	89.6614	2.88630
$966$	0	0
$967$	47.0412	1.51274	0.756372	−	0.654142i	$-0.226970\pi$
$967$	47.0412	1.51274	0.756372	+	0.654142i	$0.226970\pi$
$968$	0	0
$969$	25.8418	0.830157
$970$	0	0
$971$	1.37496	0.0441246	0.0220623	−	0.999757i	$-0.492977\pi$
$971$	1.37496	0.0441246	0.0220623	+	0.999757i	$0.492977\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	−37.8386	−1.21181
$976$	0	0
$977$	−17.6707	−0.565334	−0.282667	−	0.959218i	$-0.591219\pi$
$977$	−17.6707	−0.565334	−0.282667	+	0.959218i	$0.591219\pi$
$978$	0	0
$979$	−1.12687	−0.0360148
$980$	0	0
$981$	19.5466	0.624076
$982$	0	0
$983$	−5.49225	−0.175175	−0.0875877	−	0.996157i	$-0.527916\pi$
$983$	−5.49225	−0.175175	−0.0875877	+	0.996157i	$0.527916\pi$
$984$	0	0
$985$	14.7283	0.469282
$986$	0	0
$987$	0	0
$988$	0	0
$989$	30.2115	0.960670
$990$	0	0
$991$	−13.1735	−0.418470	−0.209235	−	0.977865i	$-0.567097\pi$
$991$	−13.1735	−0.418470	−0.209235	+	0.977865i	$0.567097\pi$
$992$	0	0
$993$	4.71821	0.149728
$994$	0	0
$995$	18.4839	0.585980
$996$	0	0
$997$	41.1141	1.30210	0.651049	−	0.759036i	$-0.274329\pi$
$997$	41.1141	1.30210	0.651049	+	0.759036i	$0.274329\pi$
$998$	0	0
$999$	29.4320	0.931186

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8036.2.a.t.1.6	yes	20
7.6	odd	2		8036.2.a.s.1.15	✓	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8036.2.a.s.1.15	✓	20	7.6	odd	2
8036.2.a.t.1.6	yes	20	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:	\( N \)	\(=\)	\( 8036 = 2^{2} \cdot 7^{2} \cdot 41 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8036.a (trivial)

\(f(q)\)	\(=\)	\(q-1.37724 q^{3} +3.83294 q^{5} -1.10321 q^{9} +O(q^{10})\)	\(q-1.37724 q^{3} +3.83294 q^{5} -1.10321 q^{9} +0.294657 q^{11} +2.83491 q^{13} -5.27887 q^{15} +3.23930 q^{17} -5.79243 q^{19} +5.90009 q^{23} +9.69141 q^{25} +5.65110 q^{27} +3.65123 q^{29} -1.59205 q^{31} -0.405813 q^{33} +5.20818 q^{37} -3.90435 q^{39} -1.00000 q^{41} +5.12051 q^{43} -4.22854 q^{45} -12.1496 q^{47} -4.46130 q^{51} -7.84750 q^{53} +1.12940 q^{55} +7.97757 q^{57} +2.50717 q^{59} +2.24588 q^{61} +10.8660 q^{65} -8.73987 q^{67} -8.12584 q^{69} -1.66646 q^{71} +12.8659 q^{73} -13.3474 q^{75} -0.253450 q^{79} -4.47329 q^{81} +12.7863 q^{83} +12.4160 q^{85} -5.02862 q^{87} -3.82433 q^{89} +2.19263 q^{93} -22.2020 q^{95} +18.4192 q^{97} -0.325068 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q + 4 q^{3} + 8 q^{5} + 16 q^{9}+O(q^{10}) \) 20 * q + 4 * q^3 + 8 * q^5 + 16 * q^9	\( 20 q + 4 q^{3} + 8 q^{5} + 16 q^{9} - 8 q^{11} + 12 q^{13} + 8 q^{15} + 8 q^{17} + 24 q^{19} + 8 q^{23} + 20 q^{25} + 16 q^{27} - 12 q^{29} + 44 q^{33} + 12 q^{37} + 12 q^{39} - 20 q^{41} + 4 q^{43} + 40 q^{45} + 4 q^{47} + 4 q^{51} - 12 q^{53} - 16 q^{55} + 28 q^{57} + 16 q^{59} + 68 q^{61} - 8 q^{65} + 4 q^{67} + 32 q^{69} + 8 q^{71} + 48 q^{73} + 60 q^{75} - 20 q^{79} + 32 q^{81} - 8 q^{83} - 28 q^{85} + 60 q^{89} - 16 q^{93} + 20 q^{95} + 40 q^{97}+O(q^{100}) \) 20 * q + 4 * q^3 + 8 * q^5 + 16 * q^9 - 8 * q^11 + 12 * q^13 + 8 * q^15 + 8 * q^17 + 24 * q^19 + 8 * q^23 + 20 * q^25 + 16 * q^27 - 12 * q^29 + 44 * q^33 + 12 * q^37 + 12 * q^39 - 20 * q^41 + 4 * q^43 + 40 * q^45 + 4 * q^47 + 4 * q^51 - 12 * q^53 - 16 * q^55 + 28 * q^57 + 16 * q^59 + 68 * q^61 - 8 * q^65 + 4 * q^67 + 32 * q^69 + 8 * q^71 + 48 * q^73 + 60 * q^75 - 20 * q^79 + 32 * q^81 - 8 * q^83 - 28 * q^85 + 60 * q^89 - 16 * q^93 + 20 * q^95 + 40 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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