LMFDB - Embedded newform 32.6.a.d.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [32,6,Mod(1,32)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("32.1"); S:= CuspForms(chi, 6); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(32, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 6, names="a")

Level:	$ N $	$=$	$ 32 = 2^{5} $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	32.a (trivial)

Newform invariants

comment:select newform

sage:traces = [2,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$5.13228223402$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{12})^+$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 3 $ x^2 - 3
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.73205$ of defining polynomial
Character	$\chi$	$=$	32.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+27.7128 q^{3} +46.0000 q^{5} -166.277 q^{7} +525.000 q^{9} +83.1384 q^{11} -42.0000 q^{13} +1274.79 q^{15} +962.000 q^{17} -2078.46 q^{19} -4608.00 q^{21} -3159.26 q^{23} -1009.00 q^{25} +7815.01 q^{27} -2554.00 q^{29} +1995.32 q^{31} +2304.00 q^{33} -7648.74 q^{35} +11950.0 q^{37} -1163.94 q^{39} -5078.00 q^{41} -12553.9 q^{43} +24150.0 q^{45} +12304.5 q^{47} +10841.0 q^{49} +26659.7 q^{51} -19714.0 q^{53} +3824.37 q^{55} -57600.0 q^{57} -8895.81 q^{59} +29318.0 q^{61} -87295.4 q^{63} -1932.00 q^{65} +16877.1 q^{67} -87552.0 q^{69} +80976.8 q^{71} +37914.0 q^{73} -27962.2 q^{75} -13824.0 q^{77} +88791.9 q^{79} +89001.0 q^{81} -39324.5 q^{83} +44252.0 q^{85} -70778.5 q^{87} +13930.0 q^{89} +6983.63 q^{91} +55296.0 q^{93} -95609.2 q^{95} +163602. q^{97} +43647.7 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 92 q^{5} + 1050 q^{9} - 84 q^{13} + 1924 q^{17} - 9216 q^{21} - 2018 q^{25} - 5108 q^{29} + 4608 q^{33} + 23900 q^{37} - 10156 q^{41} + 48300 q^{45} + 21682 q^{49} - 39428 q^{53} - 115200 q^{57} + 58636 q^{61}+ \cdots + 327204 q^{97}+O(q^{100}) $ 2 * q + 92 * q^5 + 1050 * q^9 - 84 * q^13 + 1924 * q^17 - 9216 * q^21 - 2018 * q^25 - 5108 * q^29 + 4608 * q^33 + 23900 * q^37 - 10156 * q^41 + 48300 * q^45 + 21682 * q^49 - 39428 * q^53 - 115200 * q^57 + 58636 * q^61 - 3864 * q^65 - 175104 * q^69 + 75828 * q^73 - 27648 * q^77 + 178002 * q^81 + 88504 * q^85 + 27860 * q^89 + 110592 * q^93 + 327204 * q^97

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
$3$	27.7128	1.77778	0.888889	−	0.458123i	$-0.151478\pi$
$3$	27.7128	1.77778	0.888889	+	0.458123i	$0.151478\pi$
$4$	0	0
$5$	46.0000	0.822873	0.411437	−	0.911438i	$-0.365027\pi$
$5$	46.0000	0.822873	0.411437	+	0.911438i	$0.365027\pi$
$6$	0	0
$7$	−166.277	−1.28259	−0.641293	−	0.767296i	$-0.721602\pi$
$7$	−166.277	−1.28259	−0.641293	+	0.767296i	$0.721602\pi$
$8$	0	0
$9$	525.000	2.16049
$10$	0	0
$11$	83.1384	0.207167	0.103583	−	0.994621i	$-0.466969\pi$
$11$	83.1384	0.207167	0.103583	+	0.994621i	$0.466969\pi$
$12$	0	0
$13$	−42.0000	−0.0689272	−0.0344636	−	0.999406i	$-0.510972\pi$
$13$	−42.0000	−0.0689272	−0.0344636	+	0.999406i	$0.510972\pi$
$14$	0	0
$15$	1274.79	1.46289
$16$	0	0
$17$	962.000	0.807333	0.403667	−	0.914906i	$-0.367736\pi$
$17$	962.000	0.807333	0.403667	+	0.914906i	$0.367736\pi$
$18$	0	0
$19$	−2078.46	−1.32086	−0.660432	−	0.750886i	$-0.729627\pi$
$19$	−2078.46	−1.32086	−0.660432	+	0.750886i	$0.729627\pi$
$20$	0	0
$21$	−4608.00	−2.28015
$22$	0	0
$23$	−3159.26	−1.24528	−0.622638	−	0.782510i	$-0.713939\pi$
$23$	−3159.26	−1.24528	−0.622638	+	0.782510i	$0.713939\pi$
$24$	0	0
$25$	−1009.00	−0.322880
$26$	0	0
$27$	7815.01	2.06310
$28$	0	0
$29$	−2554.00	−0.563931	−0.281965	−	0.959425i	$-0.590986\pi$
$29$	−2554.00	−0.563931	−0.281965	+	0.959425i	$0.590986\pi$
$30$	0	0
$31$	1995.32	0.372914	0.186457	−	0.982463i	$-0.440299\pi$
$31$	1995.32	0.372914	0.186457	+	0.982463i	$0.440299\pi$
$32$	0	0
$33$	2304.00	0.368297
$34$	0	0
$35$	−7648.74	−1.05541
$36$	0	0
$37$	11950.0	1.43504	0.717519	−	0.696539i	$-0.245278\pi$
$37$	11950.0	1.43504	0.717519	+	0.696539i	$0.245278\pi$
$38$	0	0
$39$	−1163.94	−0.122537
$40$	0	0
$41$	−5078.00	−0.471773	−0.235886	−	0.971781i	$-0.575799\pi$
$41$	−5078.00	−0.471773	−0.235886	+	0.971781i	$0.575799\pi$
$42$	0	0
$43$	−12553.9	−1.03540	−0.517699	−	0.855563i	$-0.673211\pi$
$43$	−12553.9	−1.03540	−0.517699	+	0.855563i	$0.673211\pi$
$44$	0	0
$45$	24150.0	1.77781
$46$	0	0
$47$	12304.5	0.812492	0.406246	−	0.913764i	$-0.366838\pi$
$47$	12304.5	0.812492	0.406246	+	0.913764i	$0.366838\pi$
$48$	0	0
$49$	10841.0	0.645029
$50$	0	0
$51$	26659.7	1.43526
$52$	0	0
$53$	−19714.0	−0.964018	−0.482009	−	0.876166i	$-0.660093\pi$
$53$	−19714.0	−0.964018	−0.482009	+	0.876166i	$0.660093\pi$
$54$	0	0
$55$	3824.37	0.170472
$56$	0	0
$57$	−57600.0	−2.34820
$58$	0	0
$59$	−8895.81	−0.332702	−0.166351	−	0.986067i	$-0.553199\pi$
$59$	−8895.81	−0.332702	−0.166351	+	0.986067i	$0.553199\pi$
$60$	0	0
$61$	29318.0	1.00881	0.504405	−	0.863467i	$-0.331712\pi$
$61$	29318.0	1.00881	0.504405	+	0.863467i	$0.331712\pi$
$62$	0	0
$63$	−87295.4	−2.77102
$64$	0	0
$65$	−1932.00	−0.0567184
$66$	0	0
$67$	16877.1	0.459315	0.229658	−	0.973271i	$-0.426239\pi$
$67$	16877.1	0.459315	0.229658	+	0.973271i	$0.426239\pi$
$68$	0	0
$69$	−87552.0	−2.21383
$70$	0	0
$71$	80976.8	1.90640	0.953202	−	0.302334i	$-0.0977658\pi$
$71$	80976.8	1.90640	0.953202	+	0.302334i	$0.0977658\pi$
$72$	0	0
$73$	37914.0	0.832707	0.416354	−	0.909203i	$-0.363308\pi$
$73$	37914.0	0.832707	0.416354	+	0.909203i	$0.363308\pi$
$74$	0	0
$75$	−27962.2	−0.574009
$76$	0	0
$77$	−13824.0	−0.265709
$78$	0	0
$79$	88791.9	1.60068	0.800342	−	0.599544i	$-0.204651\pi$
$79$	88791.9	1.60068	0.800342	+	0.599544i	$0.204651\pi$
$80$	0	0
$81$	89001.0	1.50724
$82$	0	0
$83$	−39324.5	−0.626567	−0.313284	−	0.949660i	$-0.601429\pi$
$83$	−39324.5	−0.626567	−0.313284	+	0.949660i	$0.601429\pi$
$84$	0	0
$85$	44252.0	0.664333
$86$	0	0
$87$	−70778.5	−1.00254
$88$	0	0
$89$	13930.0	0.186413	0.0932065	−	0.995647i	$-0.470288\pi$
$89$	13930.0	0.186413	0.0932065	+	0.995647i	$0.470288\pi$
$90$	0	0
$91$	6983.63	0.0884052
$92$	0	0
$93$	55296.0	0.662959
$94$	0	0
$95$	−95609.2	−1.08690
$96$	0	0
$97$	163602.	1.76547	0.882733	−	0.469875i	$-0.155701\pi$
$97$	163602.	1.76547	0.882733	+	0.469875i	$0.155701\pi$
$98$	0	0
$99$	43647.7	0.447583
$100$	0	0
$101$	−148562.	−1.44912	−0.724560	−	0.689212i	$-0.757957\pi$
$101$	−148562.	−1.44912	−0.724560	+	0.689212i	$0.757957\pi$
$102$	0	0
$103$	−100598.	−0.934317	−0.467158	−	0.884174i	$-0.654722\pi$
$103$	−100598.	−0.934317	−0.467158	+	0.884174i	$0.654722\pi$
$104$	0	0
$105$	−211968.	−1.87628
$106$	0	0
$107$	194960.	1.64621	0.823105	−	0.567889i	$-0.192240\pi$
$107$	194960.	1.64621	0.823105	+	0.567889i	$0.192240\pi$
$108$	0	0
$109$	123222.	0.993395	0.496698	−	0.867924i	$-0.334546\pi$
$109$	123222.	0.993395	0.496698	+	0.867924i	$0.334546\pi$
$110$	0	0
$111$	331168.	2.55118
$112$	0	0
$113$	−51278.0	−0.377777	−0.188888	−	0.981999i	$-0.560488\pi$
$113$	−51278.0	−0.377777	−0.188888	+	0.981999i	$0.560488\pi$
$114$	0	0
$115$	−145326.	−1.02470
$116$	0	0
$117$	−22050.0	−0.148917
$118$	0	0
$119$	−159958.	−1.03547
$120$	0	0
$121$	−154139.	−0.957082
$122$	0	0
$123$	−140726.	−0.838708
$124$	0	0
$125$	−190164.	−1.08856
$126$	0	0
$127$	−148984.	−0.819654	−0.409827	−	0.912163i	$-0.634411\pi$
$127$	−148984.	−0.819654	−0.409827	+	0.912163i	$0.634411\pi$
$128$	0	0
$129$	−347904.	−1.84071
$130$	0	0
$131$	−221231.	−1.12634	−0.563169	−	0.826342i	$-0.690418\pi$
$131$	−221231.	−1.12634	−0.563169	+	0.826342i	$0.690418\pi$
$132$	0	0
$133$	345600.	1.69412
$134$	0	0
$135$	359491.	1.69767
$136$	0	0
$137$	241034.	1.09718	0.548589	−	0.836092i	$-0.315165\pi$
$137$	241034.	1.09718	0.548589	+	0.836092i	$0.315165\pi$
$138$	0	0
$139$	169020.	0.741997	0.370999	−	0.928633i	$-0.379015\pi$
$139$	169020.	0.741997	0.370999	+	0.928633i	$0.379015\pi$
$140$	0	0
$141$	340992.	1.44443
$142$	0	0
$143$	−3491.81	−0.0142794
$144$	0	0
$145$	−117484.	−0.464044
$146$	0	0
$147$	300435.	1.14672
$148$	0	0
$149$	−103810.	−0.383066	−0.191533	−	0.981486i	$-0.561346\pi$
$149$	−103810.	−0.383066	−0.191533	+	0.981486i	$0.561346\pi$
$150$	0	0
$151$	−101595.	−0.362602	−0.181301	−	0.983428i	$-0.558031\pi$
$151$	−101595.	−0.362602	−0.181301	+	0.983428i	$0.558031\pi$
$152$	0	0
$153$	505050.	1.74424
$154$	0	0
$155$	91784.8	0.306861
$156$	0	0
$157$	−234458.	−0.759130	−0.379565	−	0.925165i	$-0.623926\pi$
$157$	−234458.	−0.759130	−0.379565	+	0.925165i	$0.623926\pi$
$158$	0	0
$159$	−546330.	−1.71381
$160$	0	0
$161$	525312.	1.59718
$162$	0	0
$163$	388672.	1.14581	0.572907	−	0.819620i	$-0.305815\pi$
$163$	388672.	1.14581	0.572907	+	0.819620i	$0.305815\pi$
$164$	0	0
$165$	105984.	0.303061
$166$	0	0
$167$	−67342.1	−0.186851	−0.0934256	−	0.995626i	$-0.529782\pi$
$167$	−67342.1	−0.186851	−0.0934256	+	0.995626i	$0.529782\pi$
$168$	0	0
$169$	−369529.	−0.995249
$170$	0	0
$171$	−1.09119e6	−2.85372
$172$	0	0
$173$	−206282.	−0.524018	−0.262009	−	0.965065i	$-0.584385\pi$
$173$	−206282.	−0.524018	−0.262009	+	0.965065i	$0.584385\pi$
$174$	0	0
$175$	167773.	0.414122
$176$	0	0
$177$	−246528.	−0.591470
$178$	0	0
$179$	198119.	0.462161	0.231081	−	0.972935i	$-0.425774\pi$
$179$	198119.	0.462161	0.231081	+	0.972935i	$0.425774\pi$
$180$	0	0
$181$	−19746.0	−0.0448005	−0.0224002	−	0.999749i	$-0.507131\pi$
$181$	−19746.0	−0.0448005	−0.0224002	+	0.999749i	$0.507131\pi$
$182$	0	0
$183$	812484.	1.79344
$184$	0	0
$185$	549700.	1.18085
$186$	0	0
$187$	79979.2	0.167253
$188$	0	0
$189$	−1.29946e6	−2.64610
$190$	0	0
$191$	701023.	1.39043	0.695215	−	0.718802i	$-0.255309\pi$
$191$	701023.	1.39043	0.695215	+	0.718802i	$0.255309\pi$
$192$	0	0
$193$	−628622.	−1.21478	−0.607388	−	0.794405i	$-0.707783\pi$
$193$	−628622.	−1.21478	−0.607388	+	0.794405i	$0.707783\pi$
$194$	0	0
$195$	−53541.2	−0.100833
$196$	0	0
$197$	−334034.	−0.613232	−0.306616	−	0.951833i	$-0.599197\pi$
$197$	−334034.	−0.613232	−0.306616	+	0.951833i	$0.599197\pi$
$198$	0	0
$199$	−318088.	−0.569396	−0.284698	−	0.958617i	$-0.591893\pi$
$199$	−318088.	−0.569396	−0.284698	+	0.958617i	$0.591893\pi$
$200$	0	0
$201$	467712.	0.816561
$202$	0	0
$203$	424671.	0.723290
$204$	0	0
$205$	−233588.	−0.388209
$206$	0	0
$207$	−1.65861e6	−2.69041
$208$	0	0
$209$	−172800.	−0.273639
$210$	0	0
$211$	−736357.	−1.13863	−0.569315	−	0.822120i	$-0.692791\pi$
$211$	−736357.	−1.13863	−0.569315	+	0.822120i	$0.692791\pi$
$212$	0	0
$213$	2.24410e6	3.38916
$214$	0	0
$215$	−577480.	−0.852002
$216$	0	0
$217$	−331776.	−0.478295
$218$	0	0
$219$	1.05070e6	1.48037
$220$	0	0
$221$	−40404.0	−0.0556472
$222$	0	0
$223$	1.31226e6	1.76708	0.883541	−	0.468354i	$-0.155153\pi$
$223$	1.31226e6	1.76708	0.883541	+	0.468354i	$0.155153\pi$
$224$	0	0
$225$	−529725.	−0.697580
$226$	0	0
$227$	−344276.	−0.443448	−0.221724	−	0.975109i	$-0.571168\pi$
$227$	−344276.	−0.443448	−0.221724	+	0.975109i	$0.571168\pi$
$228$	0	0
$229$	−664626.	−0.837507	−0.418754	−	0.908100i	$-0.637533\pi$
$229$	−664626.	−0.837507	−0.418754	+	0.908100i	$0.637533\pi$
$230$	0	0
$231$	−383102.	−0.472372
$232$	0	0
$233$	1.09433e6	1.32056	0.660281	−	0.751019i	$-0.270437\pi$
$233$	1.09433e6	1.32056	0.660281	+	0.751019i	$0.270437\pi$
$234$	0	0
$235$	566006.	0.668577
$236$	0	0
$237$	2.46067e6	2.84566
$238$	0	0
$239$	596269.	0.675223	0.337612	−	0.941285i	$-0.390381\pi$
$239$	596269.	0.675223	0.337612	+	0.941285i	$0.390381\pi$
$240$	0	0
$241$	−64734.0	−0.0717943	−0.0358971	−	0.999355i	$-0.511429\pi$
$241$	−64734.0	−0.0717943	−0.0358971	+	0.999355i	$0.511429\pi$
$242$	0	0
$243$	567420.	0.616437
$244$	0	0
$245$	498686.	0.530777
$246$	0	0
$247$	87295.4	0.0910435
$248$	0	0
$249$	−1.08979e6	−1.11390
$250$	0	0
$251$	−1.29521e6	−1.29765	−0.648824	−	0.760938i	$-0.724739\pi$
$251$	−1.29521e6	−1.29765	−0.648824	+	0.760938i	$0.724739\pi$
$252$	0	0
$253$	−262656.	−0.257980
$254$	0	0
$255$	1.22635e6	1.18104
$256$	0	0
$257$	52226.0	0.0493235	0.0246618	−	0.999696i	$-0.492149\pi$
$257$	52226.0	0.0493235	0.0246618	+	0.999696i	$0.492149\pi$
$258$	0	0
$259$	−1.98701e6	−1.84056
$260$	0	0
$261$	−1.34085e6	−1.21837
$262$	0	0
$263$	−1.17342e6	−1.04607	−0.523037	−	0.852310i	$-0.675201\pi$
$263$	−1.17342e6	−1.04607	−0.523037	+	0.852310i	$0.675201\pi$
$264$	0	0
$265$	−906844.	−0.793264
$266$	0	0
$267$	386039.	0.331401
$268$	0	0
$269$	−359498.	−0.302912	−0.151456	−	0.988464i	$-0.548396\pi$
$269$	−359498.	−0.302912	−0.151456	+	0.988464i	$0.548396\pi$
$270$	0	0
$271$	−1.00331e6	−0.829877	−0.414939	−	0.909849i	$-0.636197\pi$
$271$	−1.00331e6	−0.829877	−0.414939	+	0.909849i	$0.636197\pi$
$272$	0	0
$273$	193536.	0.157165
$274$	0	0
$275$	−83886.7	−0.0668900
$276$	0	0
$277$	−638370.	−0.499888	−0.249944	−	0.968260i	$-0.580412\pi$
$277$	−638370.	−0.499888	−0.249944	+	0.968260i	$0.580412\pi$
$278$	0	0
$279$	1.04754e6	0.805679
$280$	0	0
$281$	−112790.	−0.0852128	−0.0426064	−	0.999092i	$-0.513566\pi$
$281$	−112790.	−0.0852128	−0.0426064	+	0.999092i	$0.513566\pi$
$282$	0	0
$283$	487939.	0.362160	0.181080	−	0.983468i	$-0.442041\pi$
$283$	487939.	0.362160	0.181080	+	0.983468i	$0.442041\pi$
$284$	0	0
$285$	−2.64960e6	−1.93227
$286$	0	0
$287$	844354.	0.605090
$288$	0	0
$289$	−494413.	−0.348213
$290$	0	0
$291$	4.53387e6	3.13861
$292$	0	0
$293$	990350.	0.673938	0.336969	−	0.941516i	$-0.390598\pi$
$293$	990350.	0.673938	0.336969	+	0.941516i	$0.390598\pi$
$294$	0	0
$295$	−409207.	−0.273772
$296$	0	0
$297$	649728.	0.427406
$298$	0	0
$299$	132689.	0.0858335
$300$	0	0
$301$	2.08742e6	1.32799
$302$	0	0
$303$	−4.11707e6	−2.57621
$304$	0	0
$305$	1.34863e6	0.830123
$306$	0	0
$307$	−340618.	−0.206263	−0.103132	−	0.994668i	$-0.532886\pi$
$307$	−340618.	−0.206263	−0.103132	+	0.994668i	$0.532886\pi$
$308$	0	0
$309$	−2.78784e6	−1.66101
$310$	0	0
$311$	1.41418e6	0.829097	0.414548	−	0.910027i	$-0.363940\pi$
$311$	1.41418e6	0.829097	0.414548	+	0.910027i	$0.363940\pi$
$312$	0	0
$313$	940154.	0.542423	0.271212	−	0.962520i	$-0.412576\pi$
$313$	940154.	0.542423	0.271212	+	0.962520i	$0.412576\pi$
$314$	0	0
$315$	−4.01559e6	−2.28020
$316$	0	0
$317$	−2.42825e6	−1.35720	−0.678602	−	0.734506i	$-0.737414\pi$
$317$	−2.42825e6	−1.35720	−0.678602	+	0.734506i	$0.737414\pi$
$318$	0	0
$319$	−212336.	−0.116828
$320$	0	0
$321$	5.40288e6	2.92660
$322$	0	0
$323$	−1.99948e6	−1.06638
$324$	0	0
$325$	42378.0	0.0222552
$326$	0	0
$327$	3.41483e6	1.76604
$328$	0	0
$329$	−2.04595e6	−1.04209
$330$	0	0
$331$	762296.	0.382432	0.191216	−	0.981548i	$-0.438757\pi$
$331$	762296.	0.382432	0.191216	+	0.981548i	$0.438757\pi$
$332$	0	0
$333$	6.27375e6	3.10039
$334$	0	0
$335$	776347.	0.377958
$336$	0	0
$337$	2.97506e6	1.42699	0.713495	−	0.700661i	$-0.247111\pi$
$337$	2.97506e6	1.42699	0.713495	+	0.700661i	$0.247111\pi$
$338$	0	0
$339$	−1.42106e6	−0.671603
$340$	0	0
$341$	165888.	0.0772554
$342$	0	0
$343$	992008.	0.455281
$344$	0	0
$345$	−4.02739e6	−1.82170
$346$	0	0
$347$	2.91159e6	1.29810	0.649048	−	0.760748i	$-0.275167\pi$
$347$	2.91159e6	1.29810	0.649048	+	0.760748i	$0.275167\pi$
$348$	0	0
$349$	1.60641e6	0.705979	0.352989	−	0.935627i	$-0.385165\pi$
$349$	1.60641e6	0.705979	0.352989	+	0.935627i	$0.385165\pi$
$350$	0	0
$351$	−328231.	−0.142204
$352$	0	0
$353$	−4.22819e6	−1.80600	−0.903000	−	0.429641i	$-0.858640\pi$
$353$	−4.22819e6	−1.80600	−0.903000	+	0.429641i	$0.858640\pi$
$354$	0	0
$355$	3.72493e6	1.56873
$356$	0	0
$357$	−4.43290e6	−1.84084
$358$	0	0
$359$	−2.07464e6	−0.849583	−0.424792	−	0.905291i	$-0.639653\pi$
$359$	−2.07464e6	−0.849583	−0.424792	+	0.905291i	$0.639653\pi$
$360$	0	0
$361$	1.84390e6	0.744680
$362$	0	0
$363$	−4.27163e6	−1.70148
$364$	0	0
$365$	1.74404e6	0.685212
$366$	0	0
$367$	2.33087e6	0.903343	0.451672	−	0.892184i	$-0.350828\pi$
$367$	2.33087e6	0.903343	0.451672	+	0.892184i	$0.350828\pi$
$368$	0	0
$369$	−2.66595e6	−1.01926
$370$	0	0
$371$	3.27798e6	1.23644
$372$	0	0
$373$	1.50630e6	0.560583	0.280292	−	0.959915i	$-0.409569\pi$
$373$	1.50630e6	0.560583	0.280292	+	0.959915i	$0.409569\pi$
$374$	0	0
$375$	−5.26998e6	−1.93522
$376$	0	0
$377$	107268.	0.0388702
$378$	0	0
$379$	843772.	0.301736	0.150868	−	0.988554i	$-0.451793\pi$
$379$	843772.	0.301736	0.150868	+	0.988554i	$0.451793\pi$
$380$	0	0
$381$	−4.12877e6	−1.45716
$382$	0	0
$383$	−3.58360e6	−1.24831	−0.624155	−	0.781300i	$-0.714557\pi$
$383$	−3.58360e6	−1.24831	−0.624155	+	0.781300i	$0.714557\pi$
$384$	0	0
$385$	−635904.	−0.218645
$386$	0	0
$387$	−6.59080e6	−2.23697
$388$	0	0
$389$	786446.	0.263509	0.131754	−	0.991282i	$-0.457939\pi$
$389$	786446.	0.263509	0.131754	+	0.991282i	$0.457939\pi$
$390$	0	0
$391$	−3.03921e6	−1.00535
$392$	0	0
$393$	−6.13094e6	−2.00238
$394$	0	0
$395$	4.08443e6	1.31716
$396$	0	0
$397$	1.16889e6	0.372217	0.186108	−	0.982529i	$-0.440412\pi$
$397$	1.16889e6	0.372217	0.186108	+	0.982529i	$0.440412\pi$
$398$	0	0
$399$	9.57755e6	3.01177
$400$	0	0
$401$	−4.58400e6	−1.42359	−0.711793	−	0.702390i	$-0.752116\pi$
$401$	−4.58400e6	−1.42359	−0.711793	+	0.702390i	$0.752116\pi$
$402$	0	0
$403$	−83803.5	−0.0257039
$404$	0	0
$405$	4.09405e6	1.24027
$406$	0	0
$407$	993504.	0.297292
$408$	0	0
$409$	4.31356e6	1.27505	0.637526	−	0.770429i	$-0.279958\pi$
$409$	4.31356e6	1.27505	0.637526	+	0.770429i	$0.279958\pi$
$410$	0	0
$411$	6.67973e6	1.95054
$412$	0	0
$413$	1.47917e6	0.426719
$414$	0	0
$415$	−1.80893e6	−0.515585
$416$	0	0
$417$	4.68403e6	1.31911
$418$	0	0
$419$	−3.63257e6	−1.01083	−0.505416	−	0.862876i	$-0.668661\pi$
$419$	−3.63257e6	−1.01083	−0.505416	+	0.862876i	$0.668661\pi$
$420$	0	0
$421$	−5.01688e6	−1.37952	−0.689761	−	0.724037i	$-0.742284\pi$
$421$	−5.01688e6	−1.37952	−0.689761	+	0.724037i	$0.742284\pi$
$422$	0	0
$423$	6.45986e6	1.75538
$424$	0	0
$425$	−970658.	−0.260672
$426$	0	0
$427$	−4.87491e6	−1.29389
$428$	0	0
$429$	−96768.0	−0.0253857
$430$	0	0
$431$	−189888.	−0.0492385	−0.0246192	−	0.999697i	$-0.507837\pi$
$431$	−189888.	−0.0492385	−0.0246192	+	0.999697i	$0.507837\pi$
$432$	0	0
$433$	5.21687e6	1.33718	0.668590	−	0.743631i	$-0.266898\pi$
$433$	5.21687e6	1.33718	0.668590	+	0.743631i	$0.266898\pi$
$434$	0	0
$435$	−3.25581e6	−0.824966
$436$	0	0
$437$	6.56640e6	1.64484
$438$	0	0
$439$	−3.36661e6	−0.833741	−0.416871	−	0.908966i	$-0.636873\pi$
$439$	−3.36661e6	−0.833741	−0.416871	+	0.908966i	$0.636873\pi$
$440$	0	0
$441$	5.69152e6	1.39358
$442$	0	0
$443$	−4.42654e6	−1.07166	−0.535828	−	0.844327i	$-0.680000\pi$
$443$	−4.42654e6	−1.07166	−0.535828	+	0.844327i	$0.680000\pi$
$444$	0	0
$445$	640780.	0.153394
$446$	0	0
$447$	−2.87687e6	−0.681006
$448$	0	0
$449$	443506.	0.103821	0.0519103	−	0.998652i	$-0.483469\pi$
$449$	443506.	0.103821	0.0519103	+	0.998652i	$0.483469\pi$
$450$	0	0
$451$	−422177.	−0.0977357
$452$	0	0
$453$	−2.81549e6	−0.644627
$454$	0	0
$455$	321247.	0.0727462
$456$	0	0
$457$	590538.	0.132269	0.0661344	−	0.997811i	$-0.478933\pi$
$457$	590538.	0.132269	0.0661344	+	0.997811i	$0.478933\pi$
$458$	0	0
$459$	7.51804e6	1.66561
$460$	0	0
$461$	−4.79463e6	−1.05076	−0.525380	−	0.850868i	$-0.676077\pi$
$461$	−4.79463e6	−1.05076	−0.525380	+	0.850868i	$0.676077\pi$
$462$	0	0
$463$	1.40038e6	0.303595	0.151798	−	0.988412i	$-0.451494\pi$
$463$	1.40038e6	0.303595	0.151798	+	0.988412i	$0.451494\pi$
$464$	0	0
$465$	2.54362e6	0.545531
$466$	0	0
$467$	3.31681e6	0.703766	0.351883	−	0.936044i	$-0.385542\pi$
$467$	3.31681e6	0.703766	0.351883	+	0.936044i	$0.385542\pi$
$468$	0	0
$469$	−2.80627e6	−0.589112
$470$	0	0
$471$	−6.49749e6	−1.34956
$472$	0	0
$473$	−1.04371e6	−0.214500
$474$	0	0
$475$	2.09717e6	0.426480
$476$	0	0
$477$	−1.03498e7	−2.08275
$478$	0	0
$479$	−5.00360e6	−0.996424	−0.498212	−	0.867055i	$-0.666010\pi$
$479$	−5.00360e6	−0.996424	−0.498212	+	0.867055i	$0.666010\pi$
$480$	0	0
$481$	−501900.	−0.0989133
$482$	0	0
$483$	1.45579e7	2.83942
$484$	0	0
$485$	7.52569e6	1.45275
$486$	0	0
$487$	5.90133e6	1.12753	0.563764	−	0.825936i	$-0.309353\pi$
$487$	5.90133e6	1.12753	0.563764	+	0.825936i	$0.309353\pi$
$488$	0	0
$489$	1.07712e7	2.03700
$490$	0	0
$491$	−8.15480e6	−1.52654	−0.763272	−	0.646077i	$-0.776409\pi$
$491$	−8.15480e6	−1.52654	−0.763272	+	0.646077i	$0.776409\pi$
$492$	0	0
$493$	−2.45695e6	−0.455280
$494$	0	0
$495$	2.00779e6	0.368304
$496$	0	0
$497$	−1.34646e7	−2.44513
$498$	0	0
$499$	−470979.	−0.0846741	−0.0423370	−	0.999103i	$-0.513480\pi$
$499$	−470979.	−0.0846741	−0.0423370	+	0.999103i	$0.513480\pi$
$500$	0	0
$501$	−1.86624e6	−0.332180
$502$	0	0
$503$	9.23452e6	1.62740	0.813700	−	0.581285i	$-0.197450\pi$
$503$	9.23452e6	1.62740	0.813700	+	0.581285i	$0.197450\pi$
$504$	0	0
$505$	−6.83385e6	−1.19244
$506$	0	0
$507$	−1.02407e7	−1.76933
$508$	0	0
$509$	−2.95283e6	−0.505177	−0.252588	−	0.967574i	$-0.581282\pi$
$509$	−2.95283e6	−0.505177	−0.252588	+	0.967574i	$0.581282\pi$
$510$	0	0
$511$	−6.30422e6	−1.06802
$512$	0	0
$513$	−1.62432e7	−2.72507
$514$	0	0
$515$	−4.62749e6	−0.768824
$516$	0	0
$517$	1.02298e6	0.168321
$518$	0	0
$519$	−5.71665e6	−0.931587
$520$	0	0
$521$	1.04570e7	1.68776	0.843882	−	0.536529i	$-0.180265\pi$
$521$	1.04570e7	1.68776	0.843882	+	0.536529i	$0.180265\pi$
$522$	0	0
$523$	−5.77571e6	−0.923318	−0.461659	−	0.887058i	$-0.652746\pi$
$523$	−5.77571e6	−0.923318	−0.461659	+	0.887058i	$0.652746\pi$
$524$	0	0
$525$	4.64947e6	0.736216
$526$	0	0
$527$	1.91950e6	0.301066
$528$	0	0
$529$	3.54459e6	0.550714
$530$	0	0
$531$	−4.67030e6	−0.718801
$532$	0	0
$533$	213276.	0.0325180
$534$	0	0
$535$	8.96814e6	1.35462
$536$	0	0
$537$	5.49043e6	0.821620
$538$	0	0
$539$	901304.	0.133629
$540$	0	0
$541$	−3.86409e6	−0.567615	−0.283808	−	0.958881i	$-0.591598\pi$
$541$	−3.86409e6	−0.567615	−0.283808	+	0.958881i	$0.591598\pi$
$542$	0	0
$543$	−547217.	−0.0796453
$544$	0	0
$545$	5.66821e6	0.817438
$546$	0	0
$547$	8.49633e6	1.21412	0.607062	−	0.794654i	$-0.292348\pi$
$547$	8.49633e6	1.21412	0.607062	+	0.794654i	$0.292348\pi$
$548$	0	0
$549$	1.53920e7	2.17953
$550$	0	0
$551$	5.30839e6	0.744876
$552$	0	0
$553$	−1.47640e7	−2.05302
$554$	0	0
$555$	1.52337e7	2.09930
$556$	0	0
$557$	6.33253e6	0.864848	0.432424	−	0.901671i	$-0.357658\pi$
$557$	6.33253e6	0.864848	0.432424	+	0.901671i	$0.357658\pi$
$558$	0	0
$559$	527264.	0.0713672
$560$	0	0
$561$	2.21645e6	0.297338
$562$	0	0
$563$	8.47605e6	1.12700	0.563498	−	0.826117i	$-0.309455\pi$
$563$	8.47605e6	1.12700	0.563498	+	0.826117i	$0.309455\pi$
$564$	0	0
$565$	−2.35879e6	−0.310862
$566$	0	0
$567$	−1.47988e7	−1.93317
$568$	0	0
$569$	1.69279e6	0.219191	0.109596	−	0.993976i	$-0.465044\pi$
$569$	1.69279e6	0.219191	0.109596	+	0.993976i	$0.465044\pi$
$570$	0	0
$571$	−4.50901e6	−0.578750	−0.289375	−	0.957216i	$-0.593448\pi$
$571$	−4.50901e6	−0.578750	−0.289375	+	0.957216i	$0.593448\pi$
$572$	0	0
$573$	1.94273e7	2.47188
$574$	0	0
$575$	3.18769e6	0.402075
$576$	0	0
$577$	−4.07577e6	−0.509648	−0.254824	−	0.966987i	$-0.582018\pi$
$577$	−4.07577e6	−0.509648	−0.254824	+	0.966987i	$0.582018\pi$
$578$	0	0
$579$	−1.74209e7	−2.15960
$580$	0	0
$581$	6.53875e6	0.803627
$582$	0	0
$583$	−1.63899e6	−0.199712
$584$	0	0
$585$	−1.01430e6	−0.122540
$586$	0	0
$587$	8.15563e6	0.976928	0.488464	−	0.872584i	$-0.337557\pi$
$587$	8.15563e6	0.976928	0.488464	+	0.872584i	$0.337557\pi$
$588$	0	0
$589$	−4.14720e6	−0.492569
$590$	0	0
$591$	−9.25702e6	−1.09019
$592$	0	0
$593$	−528622.	−0.0617317	−0.0308659	−	0.999524i	$-0.509826\pi$
$593$	−528622.	−0.0617317	−0.0308659	+	0.999524i	$0.509826\pi$
$594$	0	0
$595$	−7.35808e6	−0.852064
$596$	0	0
$597$	−8.81510e6	−1.01226
$598$	0	0
$599$	−4.84381e6	−0.551595	−0.275797	−	0.961216i	$-0.588942\pi$
$599$	−4.84381e6	−0.551595	−0.275797	+	0.961216i	$0.588942\pi$
$600$	0	0
$601$	−2.11804e6	−0.239193	−0.119596	−	0.992823i	$-0.538160\pi$
$601$	−2.11804e6	−0.239193	−0.119596	+	0.992823i	$0.538160\pi$
$602$	0	0
$603$	8.86048e6	0.992348
$604$	0	0
$605$	−7.09039e6	−0.787557
$606$	0	0
$607$	6.93109e6	0.763536	0.381768	−	0.924258i	$-0.375315\pi$
$607$	6.93109e6	0.763536	0.381768	+	0.924258i	$0.375315\pi$
$608$	0	0
$609$	1.17688e7	1.28585
$610$	0	0
$611$	−516789.	−0.0560028
$612$	0	0
$613$	−3.21599e6	−0.345671	−0.172836	−	0.984951i	$-0.555293\pi$
$613$	−3.21599e6	−0.345671	−0.172836	+	0.984951i	$0.555293\pi$
$614$	0	0
$615$	−6.47338e6	−0.690150
$616$	0	0
$617$	1.46326e7	1.54742	0.773709	−	0.633541i	$-0.218399\pi$
$617$	1.46326e7	1.54742	0.773709	+	0.633541i	$0.218399\pi$
$618$	0	0
$619$	2.42706e6	0.254597	0.127299	−	0.991864i	$-0.459369\pi$
$619$	2.42706e6	0.254597	0.127299	+	0.991864i	$0.459369\pi$
$620$	0	0
$621$	−2.46897e7	−2.56913
$622$	0	0
$623$	−2.31624e6	−0.239091
$624$	0	0
$625$	−5.59442e6	−0.572869
$626$	0	0
$627$	−4.78877e6	−0.486469
$628$	0	0
$629$	1.14959e7	1.15855
$630$	0	0
$631$	1.62235e7	1.62207	0.811036	−	0.584996i	$-0.198904\pi$
$631$	1.62235e7	1.62207	0.811036	+	0.584996i	$0.198904\pi$
$632$	0	0
$633$	−2.04065e7	−2.02423
$634$	0	0
$635$	−6.85327e6	−0.674471
$636$	0	0
$637$	−455322.	−0.0444601
$638$	0	0
$639$	4.25128e7	4.11877
$640$	0	0
$641$	−1.22399e7	−1.17662	−0.588308	−	0.808637i	$-0.700206\pi$
$641$	−1.22399e7	−1.17662	−0.588308	+	0.808637i	$0.700206\pi$
$642$	0	0
$643$	2.40461e6	0.229360	0.114680	−	0.993402i	$-0.463416\pi$
$643$	2.40461e6	0.229360	0.114680	+	0.993402i	$0.463416\pi$
$644$	0	0
$645$	−1.60036e7	−1.51467
$646$	0	0
$647$	−1.39515e7	−1.31026	−0.655132	−	0.755514i	$-0.727387\pi$
$647$	−1.39515e7	−1.31026	−0.655132	+	0.755514i	$0.727387\pi$
$648$	0	0
$649$	−739584.	−0.0689248
$650$	0	0
$651$	−9.19445e6	−0.850302
$652$	0	0
$653$	−1.59391e7	−1.46279	−0.731395	−	0.681954i	$-0.761130\pi$
$653$	−1.59391e7	−1.46279	−0.731395	+	0.681954i	$0.761130\pi$
$654$	0	0
$655$	−1.01766e7	−0.926833
$656$	0	0
$657$	1.99048e7	1.79906
$658$	0	0
$659$	3.36470e6	0.301809	0.150905	−	0.988548i	$-0.451781\pi$
$659$	3.36470e6	0.301809	0.150905	+	0.988548i	$0.451781\pi$
$660$	0	0
$661$	−8.51194e6	−0.757748	−0.378874	−	0.925448i	$-0.623689\pi$
$661$	−8.51194e6	−0.757748	−0.378874	+	0.925448i	$0.623689\pi$
$662$	0	0
$663$	−1.11971e6	−0.0989284
$664$	0	0
$665$	1.58976e7	1.39405
$666$	0	0
$667$	8.06875e6	0.702250
$668$	0	0
$669$	3.63663e7	3.14148
$670$	0	0
$671$	2.43745e6	0.208992
$672$	0	0
$673$	1.15169e7	0.980161	0.490080	−	0.871677i	$-0.336967\pi$
$673$	1.15169e7	0.980161	0.490080	+	0.871677i	$0.336967\pi$
$674$	0	0
$675$	−7.88535e6	−0.666134
$676$	0	0
$677$	−2.32692e7	−1.95124	−0.975618	−	0.219475i	$-0.929565\pi$
$677$	−2.32692e7	−1.95124	−0.975618	+	0.219475i	$0.929565\pi$
$678$	0	0
$679$	−2.72032e7	−2.26436
$680$	0	0
$681$	−9.54086e6	−0.788352
$682$	0	0
$683$	−1.90399e7	−1.56176	−0.780880	−	0.624682i	$-0.785229\pi$
$683$	−1.90399e7	−1.56176	−0.780880	+	0.624682i	$0.785229\pi$
$684$	0	0
$685$	1.10876e7	0.902838
$686$	0	0
$687$	−1.84187e7	−1.48890
$688$	0	0
$689$	827988.	0.0664471
$690$	0	0
$691$	7.15780e6	0.570275	0.285138	−	0.958487i	$-0.407961\pi$
$691$	7.15780e6	0.570275	0.285138	+	0.958487i	$0.407961\pi$
$692$	0	0
$693$	−7.25760e6	−0.574064
$694$	0	0
$695$	7.77494e6	0.610569
$696$	0	0
$697$	−4.88504e6	−0.380878
$698$	0	0
$699$	3.03270e7	2.34766
$700$	0	0
$701$	8.77684e6	0.674595	0.337297	−	0.941398i	$-0.390487\pi$
$701$	8.77684e6	0.674595	0.337297	+	0.941398i	$0.390487\pi$
$702$	0	0
$703$	−2.48376e7	−1.89549
$704$	0	0
$705$	1.56856e7	1.18858
$706$	0	0
$707$	2.47024e7	1.85862
$708$	0	0
$709$	8.08481e6	0.604024	0.302012	−	0.953304i	$-0.402342\pi$
$709$	8.08481e6	0.604024	0.302012	+	0.953304i	$0.402342\pi$
$710$	0	0
$711$	4.66157e7	3.45827
$712$	0	0
$713$	−6.30374e6	−0.464381
$714$	0	0
$715$	−160623.	−0.0117502
$716$	0	0
$717$	1.65243e7	1.20040
$718$	0	0
$719$	−1.42855e7	−1.03056	−0.515280	−	0.857022i	$-0.672312\pi$
$719$	−1.42855e7	−1.03056	−0.515280	+	0.857022i	$0.672312\pi$
$720$	0	0
$721$	1.67270e7	1.19834
$722$	0	0
$723$	−1.79396e6	−0.127634
$724$	0	0
$725$	2.57699e6	0.182082
$726$	0	0
$727$	1.17227e7	0.822605	0.411302	−	0.911499i	$-0.365074\pi$
$727$	1.17227e7	0.822605	0.411302	+	0.911499i	$0.365074\pi$
$728$	0	0
$729$	−5.90244e6	−0.411351
$730$	0	0
$731$	−1.20769e7	−0.835912
$732$	0	0
$733$	1.81661e7	1.24882	0.624412	−	0.781095i	$-0.285339\pi$
$733$	1.81661e7	1.24882	0.624412	+	0.781095i	$0.285339\pi$
$734$	0	0
$735$	1.38200e7	0.943603
$736$	0	0
$737$	1.40314e6	0.0951549
$738$	0	0
$739$	−1.27312e7	−0.857550	−0.428775	−	0.903411i	$-0.641055\pi$
$739$	−1.27312e7	−0.857550	−0.428775	+	0.903411i	$0.641055\pi$
$740$	0	0
$741$	2.41920e6	0.161855
$742$	0	0
$743$	−1.82988e6	−0.121605	−0.0608023	−	0.998150i	$-0.519366\pi$
$743$	−1.82988e6	−0.121605	−0.0608023	+	0.998150i	$0.519366\pi$
$744$	0	0
$745$	−4.77526e6	−0.315215
$746$	0	0
$747$	−2.06454e7	−1.35369
$748$	0	0
$749$	−3.24173e7	−2.11141
$750$	0	0
$751$	5.64577e6	0.365278	0.182639	−	0.983180i	$-0.441536\pi$
$751$	5.64577e6	0.365278	0.182639	+	0.983180i	$0.441536\pi$
$752$	0	0
$753$	−3.58940e7	−2.30693
$754$	0	0
$755$	−4.67338e6	−0.298376
$756$	0	0
$757$	139230.	0.00883066	0.00441533	−	0.999990i	$-0.498595\pi$
$757$	139230.	0.00883066	0.00441533	+	0.999990i	$0.498595\pi$
$758$	0	0
$759$	−7.27894e6	−0.458631
$760$	0	0
$761$	1.36251e7	0.852863	0.426432	−	0.904520i	$-0.359770\pi$
$761$	1.36251e7	0.852863	0.426432	+	0.904520i	$0.359770\pi$
$762$	0	0
$763$	−2.04890e7	−1.27412
$764$	0	0
$765$	2.32323e7	1.43529
$766$	0	0
$767$	373624.	0.0229322
$768$	0	0
$769$	−1.06840e7	−0.651504	−0.325752	−	0.945455i	$-0.605618\pi$
$769$	−1.06840e7	−0.651504	−0.325752	+	0.945455i	$0.605618\pi$
$770$	0	0
$771$	1.44733e6	0.0876862
$772$	0	0
$773$	2.22367e7	1.33851	0.669255	−	0.743033i	$-0.266614\pi$
$773$	2.22367e7	1.33851	0.669255	+	0.743033i	$0.266614\pi$
$774$	0	0
$775$	−2.01328e6	−0.120407
$776$	0	0
$777$	−5.50656e7	−3.27211
$778$	0	0
$779$	1.05544e7	0.623148
$780$	0	0
$781$	6.73229e6	0.394944
$782$	0	0
$783$	−1.99595e7	−1.16345
$784$	0	0
$785$	−1.07851e7	−0.624667
$786$	0	0
$787$	2.81333e7	1.61914	0.809569	−	0.587025i	$-0.199701\pi$
$787$	2.81333e7	1.61914	0.809569	+	0.587025i	$0.199701\pi$
$788$	0	0
$789$	−3.25187e7	−1.85969
$790$	0	0
$791$	8.52635e6	0.484531
$792$	0	0
$793$	−1.23136e6	−0.0695346
$794$	0	0
$795$	−2.51312e7	−1.41025
$796$	0	0
$797$	−2.43450e7	−1.35758	−0.678789	−	0.734333i	$-0.737495\pi$
$797$	−2.43450e7	−1.35758	−0.678789	+	0.734333i	$0.737495\pi$
$798$	0	0
$799$	1.18369e7	0.655951
$800$	0	0
$801$	7.31325e6	0.402744
$802$	0	0
$803$	3.15211e6	0.172509
$804$	0	0
$805$	2.41644e7	1.31427
$806$	0	0
$807$	−9.96270e6	−0.538509
$808$	0	0
$809$	2.30810e7	1.23989	0.619945	−	0.784645i	$-0.287155\pi$
$809$	2.30810e7	1.23989	0.619945	+	0.784645i	$0.287155\pi$
$810$	0	0
$811$	3.33586e7	1.78096	0.890482	−	0.455019i	$-0.150368\pi$
$811$	3.33586e7	1.78096	0.890482	+	0.455019i	$0.150368\pi$
$812$	0	0
$813$	−2.78047e7	−1.47534
$814$	0	0
$815$	1.78789e7	0.942860
$816$	0	0
$817$	2.60928e7	1.36762
$818$	0	0
$819$	3.66641e6	0.190999
$820$	0	0
$821$	−2.03134e7	−1.05178	−0.525890	−	0.850553i	$-0.676268\pi$
$821$	−2.03134e7	−1.05178	−0.525890	+	0.850553i	$0.676268\pi$
$822$	0	0
$823$	6.90531e6	0.355372	0.177686	−	0.984087i	$-0.443139\pi$
$823$	6.90531e6	0.355372	0.177686	+	0.984087i	$0.443139\pi$
$824$	0	0
$825$	−2.32474e6	−0.118916
$826$	0	0
$827$	3.12498e7	1.58885	0.794427	−	0.607360i	$-0.207771\pi$
$827$	3.12498e7	1.58885	0.794427	+	0.607360i	$0.207771\pi$
$828$	0	0
$829$	2.32015e7	1.17254	0.586272	−	0.810114i	$-0.300595\pi$
$829$	2.32015e7	1.17254	0.586272	+	0.810114i	$0.300595\pi$
$830$	0	0
$831$	−1.76910e7	−0.888690
$832$	0	0
$833$	1.04290e7	0.520753
$834$	0	0
$835$	−3.09774e6	−0.153755
$836$	0	0
$837$	1.55935e7	0.769359
$838$	0	0
$839$	−2.91651e7	−1.43041	−0.715203	−	0.698917i	$-0.753666\pi$
$839$	−2.91651e7	−1.43041	−0.715203	+	0.698917i	$0.753666\pi$
$840$	0	0
$841$	−1.39882e7	−0.681982
$842$	0	0
$843$	−3.12573e6	−0.151489
$844$	0	0
$845$	−1.69983e7	−0.818964
$846$	0	0
$847$	2.56298e7	1.22754
$848$	0	0
$849$	1.35222e7	0.643839
$850$	0	0
$851$	−3.77532e7	−1.78702
$852$	0	0
$853$	−9.23146e6	−0.434408	−0.217204	−	0.976126i	$-0.569694\pi$
$853$	−9.23146e6	−0.434408	−0.217204	+	0.976126i	$0.569694\pi$
$854$	0	0
$855$	−5.01948e7	−2.34825
$856$	0	0
$857$	−7.34077e6	−0.341421	−0.170710	−	0.985321i	$-0.554606\pi$
$857$	−7.34077e6	−0.341421	−0.170710	+	0.985321i	$0.554606\pi$
$858$	0	0
$859$	−1.57274e7	−0.727233	−0.363616	−	0.931549i	$-0.618458\pi$
$859$	−1.57274e7	−0.727233	−0.363616	+	0.931549i	$0.618458\pi$
$860$	0	0
$861$	2.33994e7	1.07572
$862$	0	0
$863$	−5.72990e6	−0.261891	−0.130945	−	0.991390i	$-0.541801\pi$
$863$	−5.72990e6	−0.261891	−0.130945	+	0.991390i	$0.541801\pi$
$864$	0	0
$865$	−9.48897e6	−0.431200
$866$	0	0
$867$	−1.37016e7	−0.619046
$868$	0	0
$869$	7.38202e6	0.331608
$870$	0	0
$871$	−708838.	−0.0316593
$872$	0	0
$873$	8.58910e7	3.81428
$874$	0	0
$875$	3.16199e7	1.39618
$876$	0	0
$877$	3.57123e7	1.56790	0.783950	−	0.620823i	$-0.213202\pi$
$877$	3.57123e7	1.56790	0.783950	+	0.620823i	$0.213202\pi$
$878$	0	0
$879$	2.74454e7	1.19811
$880$	0	0
$881$	−1.77000e6	−0.0768304	−0.0384152	−	0.999262i	$-0.512231\pi$
$881$	−1.77000e6	−0.0768304	−0.0384152	+	0.999262i	$0.512231\pi$
$882$	0	0
$883$	−2.70394e7	−1.16706	−0.583532	−	0.812090i	$-0.698330\pi$
$883$	−2.70394e7	−1.16706	−0.583532	+	0.812090i	$0.698330\pi$
$884$	0	0
$885$	−1.13403e7	−0.486705
$886$	0	0
$887$	3.07893e7	1.31399	0.656994	−	0.753896i	$-0.271828\pi$
$887$	3.07893e7	1.31399	0.656994	+	0.753896i	$0.271828\pi$
$888$	0	0
$889$	2.47726e7	1.05128
$890$	0	0
$891$	7.39940e6	0.312250
$892$	0	0
$893$	−2.55744e7	−1.07319
$894$	0	0
$895$	9.11347e6	0.380300
$896$	0	0
$897$	3.67718e6	0.152593
$898$	0	0
$899$	−5.09605e6	−0.210298
$900$	0	0
$901$	−1.89649e7	−0.778284
$902$	0	0
$903$	5.78484e7	2.36087
$904$	0	0
$905$	−908316.	−0.0368651
$906$	0	0
$907$	−1.94729e7	−0.785983	−0.392992	−	0.919542i	$-0.628560\pi$
$907$	−1.94729e7	−0.785983	−0.392992	+	0.919542i	$0.628560\pi$
$908$	0	0
$909$	−7.79950e7	−3.13081
$910$	0	0
$911$	2.44863e7	0.977522	0.488761	−	0.872418i	$-0.337449\pi$
$911$	2.44863e7	0.977522	0.488761	+	0.872418i	$0.337449\pi$
$912$	0	0
$913$	−3.26938e6	−0.129804
$914$	0	0
$915$	3.73743e7	1.47577
$916$	0	0
$917$	3.67857e7	1.44463
$918$	0	0
$919$	−2.05656e7	−0.803254	−0.401627	−	0.915803i	$-0.631555\pi$
$919$	−2.05656e7	−0.803254	−0.401627	+	0.915803i	$0.631555\pi$
$920$	0	0
$921$	−9.43949e6	−0.366690
$922$	0	0
$923$	−3.40103e6	−0.131403
$924$	0	0
$925$	−1.20576e7	−0.463345
$926$	0	0
$927$	−5.28137e7	−2.01859
$928$	0	0
$929$	−4.50171e7	−1.71135	−0.855674	−	0.517514i	$-0.826857\pi$
$929$	−4.50171e7	−1.71135	−0.855674	+	0.517514i	$0.826857\pi$
$930$	0	0
$931$	−2.25326e7	−0.851995
$932$	0	0
$933$	3.91910e7	1.47395
$934$	0	0
$935$	3.67904e6	0.137628
$936$	0	0
$937$	2.31151e7	0.860097	0.430048	−	0.902806i	$-0.358496\pi$
$937$	2.31151e7	0.860097	0.430048	+	0.902806i	$0.358496\pi$
$938$	0	0
$939$	2.60543e7	0.964308
$940$	0	0
$941$	−664042.	−0.0244468	−0.0122234	−	0.999925i	$-0.503891\pi$
$941$	−664042.	−0.0244468	−0.0122234	+	0.999925i	$0.503891\pi$
$942$	0	0
$943$	1.60427e7	0.587488
$944$	0	0
$945$	−5.97750e7	−2.17741
$946$	0	0
$947$	−5.17375e7	−1.87469	−0.937347	−	0.348398i	$-0.886726\pi$
$947$	−5.17375e7	−1.87469	−0.937347	+	0.348398i	$0.886726\pi$
$948$	0	0
$949$	−1.59239e6	−0.0573962
$950$	0	0
$951$	−6.72936e7	−2.41281
$952$	0	0
$953$	2.89030e7	1.03089	0.515444	−	0.856923i	$-0.327627\pi$
$953$	2.89030e7	1.03089	0.515444	+	0.856923i	$0.327627\pi$
$954$	0	0
$955$	3.22471e7	1.14415
$956$	0	0
$957$	−5.88442e6	−0.207694
$958$	0	0
$959$	−4.00784e7	−1.40723
$960$	0	0
$961$	−2.46478e7	−0.860935
$962$	0	0
$963$	1.02354e8	3.55663
$964$	0	0
$965$	−2.89166e7	−0.999607
$966$	0	0
$967$	−2.36554e7	−0.813512	−0.406756	−	0.913537i	$-0.633340\pi$
$967$	−2.36554e7	−0.813512	−0.406756	+	0.913537i	$0.633340\pi$
$968$	0	0
$969$	−5.54112e7	−1.89578
$970$	0	0
$971$	−3.15207e7	−1.07287	−0.536436	−	0.843941i	$-0.680230\pi$
$971$	−3.15207e7	−1.07287	−0.536436	+	0.843941i	$0.680230\pi$
$972$	0	0
$973$	−2.81042e7	−0.951676
$974$	0	0
$975$	1.17441e6	0.0395649
$976$	0	0
$977$	4.82778e6	0.161812	0.0809060	−	0.996722i	$-0.474219\pi$
$977$	4.82778e6	0.161812	0.0809060	+	0.996722i	$0.474219\pi$
$978$	0	0
$979$	1.15812e6	0.0386186
$980$	0	0
$981$	6.46916e7	2.14622
$982$	0	0
$983$	2.81641e7	0.929636	0.464818	−	0.885406i	$-0.346120\pi$
$983$	2.81641e7	0.929636	0.464818	+	0.885406i	$0.346120\pi$
$984$	0	0
$985$	−1.53656e7	−0.504612
$986$	0	0
$987$	−5.66991e7	−1.85261
$988$	0	0
$989$	3.96611e7	1.28936
$990$	0	0
$991$	−2.95301e7	−0.955171	−0.477585	−	0.878585i	$-0.658488\pi$
$991$	−2.95301e7	−0.955171	−0.477585	+	0.878585i	$0.658488\pi$
$992$	0	0
$993$	2.11254e7	0.679879
$994$	0	0
$995$	−1.46320e7	−0.468540
$996$	0	0
$997$	−5.16306e7	−1.64501	−0.822507	−	0.568756i	$-0.807425\pi$
$997$	−5.16306e7	−1.64501	−0.822507	+	0.568756i	$0.807425\pi$
$998$	0	0
$999$	9.33894e7	2.96063

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	32.6.a.d.1.2	yes	2
3.2	odd	2		288.6.a.l.1.1		2
4.3	odd	2	inner	32.6.a.d.1.1	✓	2
5.2	odd	4		800.6.c.d.449.2		4
5.3	odd	4		800.6.c.d.449.4		4
5.4	even	2		800.6.a.k.1.1		2
8.3	odd	2		64.6.a.h.1.2		2
8.5	even	2		64.6.a.h.1.1		2
12.11	even	2		288.6.a.l.1.2		2
16.3	odd	4		256.6.b.l.129.1		4
16.5	even	4		256.6.b.l.129.2		4
16.11	odd	4		256.6.b.l.129.4		4
16.13	even	4		256.6.b.l.129.3		4
20.3	even	4		800.6.c.d.449.1		4
20.7	even	4		800.6.c.d.449.3		4
20.19	odd	2		800.6.a.k.1.2		2
24.5	odd	2		576.6.a.bp.1.1		2
24.11	even	2		576.6.a.bp.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
32.6.a.d.1.1	✓	2	4.3	odd	2	inner
32.6.a.d.1.2	yes	2	1.1	even	1	trivial
64.6.a.h.1.1		2	8.5	even	2
64.6.a.h.1.2		2	8.3	odd	2
256.6.b.l.129.1		4	16.3	odd	4
256.6.b.l.129.2		4	16.5	even	4
256.6.b.l.129.3		4	16.13	even	4
256.6.b.l.129.4		4	16.11	odd	4
288.6.a.l.1.1		2	3.2	odd	2
288.6.a.l.1.2		2	12.11	even	2
576.6.a.bp.1.1		2	24.5	odd	2
576.6.a.bp.1.2		2	24.11	even	2
800.6.a.k.1.1		2	5.4	even	2
800.6.a.k.1.2		2	20.19	odd	2
800.6.c.d.449.1		4	20.3	even	4
800.6.c.d.449.2		4	5.2	odd	4
800.6.c.d.449.3		4	20.7	even	4
800.6.c.d.449.4		4	5.3	odd	4

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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 32 = 2^{5} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	32.a (trivial)

\(f(q)\)	\(=\)	\(q+27.7128 q^{3} +46.0000 q^{5} -166.277 q^{7} +525.000 q^{9} +83.1384 q^{11} -42.0000 q^{13} +1274.79 q^{15} +962.000 q^{17} -2078.46 q^{19} -4608.00 q^{21} -3159.26 q^{23} -1009.00 q^{25} +7815.01 q^{27} -2554.00 q^{29} +1995.32 q^{31} +2304.00 q^{33} -7648.74 q^{35} +11950.0 q^{37} -1163.94 q^{39} -5078.00 q^{41} -12553.9 q^{43} +24150.0 q^{45} +12304.5 q^{47} +10841.0 q^{49} +26659.7 q^{51} -19714.0 q^{53} +3824.37 q^{55} -57600.0 q^{57} -8895.81 q^{59} +29318.0 q^{61} -87295.4 q^{63} -1932.00 q^{65} +16877.1 q^{67} -87552.0 q^{69} +80976.8 q^{71} +37914.0 q^{73} -27962.2 q^{75} -13824.0 q^{77} +88791.9 q^{79} +89001.0 q^{81} -39324.5 q^{83} +44252.0 q^{85} -70778.5 q^{87} +13930.0 q^{89} +6983.63 q^{91} +55296.0 q^{93} -95609.2 q^{95} +163602. q^{97} +43647.7 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 92 q^{5} + 1050 q^{9} - 84 q^{13} + 1924 q^{17} - 9216 q^{21} - 2018 q^{25} - 5108 q^{29} + 4608 q^{33} + 23900 q^{37} - 10156 q^{41} + 48300 q^{45} + 21682 q^{49} - 39428 q^{53} - 115200 q^{57} + 58636 q^{61}+ \cdots + 327204 q^{97}+O(q^{100}) \) 2 * q + 92 * q^5 + 1050 * q^9 - 84 * q^13 + 1924 * q^17 - 9216 * q^21 - 2018 * q^25 - 5108 * q^29 + 4608 * q^33 + 23900 * q^37 - 10156 * q^41 + 48300 * q^45 + 21682 * q^49 - 39428 * q^53 - 115200 * q^57 + 58636 * q^61 - 3864 * q^65 - 175104 * q^69 + 75828 * q^73 - 27648 * q^77 + 178002 * q^81 + 88504 * q^85 + 27860 * q^89 + 110592 * q^93 + 327204 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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