LMFDB - Embedded newform 8112.2.a.cu.1.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [8112,2,Mod(1,8112)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(8112, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("8112.1"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 8112 = 2^{4} \cdot 3 \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8112.a (trivial)

Newform invariants

comment:select newform

sage:traces = [6,0,-6,0,1,0,5,0,6,0,6,0,0,0,-1,0,9] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(17)] == traces)

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

Self dual:	yes
Analytic conductor:	$64.7746461197$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.6.27700337.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - x^{5} - 19x^{4} + 17x^{3} + 103x^{2} - 71x - 127 $ x^6 - x^5 - 19x^4 + 17x^3 + 103x^2 - 71x - 127
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 4056)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-3.12925$ of defining polynomial
Character	$\chi$	$=$	8112.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} +1.88227 q^{5} -1.29156 q^{7} +1.00000 q^{9} -4.19367 q^{11} -1.88227 q^{15} -1.68421 q^{17} -1.19025 q^{19} +1.29156 q^{21} -3.01499 q^{23} -1.45707 q^{25} -1.00000 q^{27} +3.15000 q^{29} +1.47463 q^{31} +4.19367 q^{33} -2.43106 q^{35} +8.15343 q^{37} -10.0181 q^{41} -10.7587 q^{43} +1.88227 q^{45} +5.31909 q^{47} -5.33187 q^{49} +1.68421 q^{51} +8.29590 q^{53} -7.89361 q^{55} +1.19025 q^{57} +3.03369 q^{59} +7.84487 q^{61} -1.29156 q^{63} -4.50497 q^{67} +3.01499 q^{69} +3.85816 q^{71} +1.76494 q^{73} +1.45707 q^{75} +5.41637 q^{77} +12.6593 q^{79} +1.00000 q^{81} +15.6280 q^{83} -3.17013 q^{85} -3.15000 q^{87} +10.6679 q^{89} -1.47463 q^{93} -2.24036 q^{95} -2.38541 q^{97} -4.19367 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 6 q^{3} + q^{5} + 5 q^{7} + 6 q^{9} + 6 q^{11} - q^{15} + 9 q^{17} - 7 q^{19} - 5 q^{21} - 12 q^{23} + 9 q^{25} - 6 q^{27} + 7 q^{29} + 11 q^{31} - 6 q^{33} - 6 q^{35} + 6 q^{37} + 13 q^{41} - 15 q^{43}+ \cdots + 6 q^{99}+O(q^{100}) $ 6 * q - 6 * q^3 + q^5 + 5 * q^7 + 6 * q^9 + 6 * q^11 - q^15 + 9 * q^17 - 7 * q^19 - 5 * q^21 - 12 * q^23 + 9 * q^25 - 6 * q^27 + 7 * q^29 + 11 * q^31 - 6 * q^33 - 6 * q^35 + 6 * q^37 + 13 * q^41 - 15 * q^43 + q^45 + 9 * q^47 + 13 * q^49 - 9 * q^51 + 22 * q^53 - 3 * q^55 + 7 * q^57 + 7 * q^59 + 25 * q^61 + 5 * q^63 - 5 * q^67 + 12 * q^69 - 8 * q^71 + 15 * q^73 - 9 * q^75 + 45 * q^77 - 14 * q^79 + 6 * q^81 + 13 * q^83 - 35 * q^85 - 7 * q^87 + 33 * q^89 - 11 * q^93 - 47 * q^95 + 50 * q^97 + 6 * q^99

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	0	0
$5$	1.88227	0.841776	0.420888	−	0.907113i	$-0.361719\pi$
$5$	1.88227	0.841776	0.420888	+	0.907113i	$0.361719\pi$
$6$	0	0
$7$	−1.29156	−0.488164	−0.244082	−	0.969755i	$-0.578487\pi$
$7$	−1.29156	−0.488164	−0.244082	+	0.969755i	$0.578487\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−4.19367	−1.26444	−0.632219	−	0.774790i	$-0.717856\pi$
$11$	−4.19367	−1.26444	−0.632219	+	0.774790i	$0.717856\pi$
$12$	0	0
$13$	0	0
$13$	0	0
$14$	0	0
$15$	−1.88227	−0.486000
$16$	0	0
$17$	−1.68421	−0.408480	−0.204240	−	0.978921i	$-0.565472\pi$
$17$	−1.68421	−0.408480	−0.204240	+	0.978921i	$0.565472\pi$
$18$	0	0
$19$	−1.19025	−0.273061	−0.136531	−	0.990636i	$-0.543595\pi$
$19$	−1.19025	−0.273061	−0.136531	+	0.990636i	$0.543595\pi$
$20$	0	0
$21$	1.29156	0.281841
$22$	0	0
$23$	−3.01499	−0.628670	−0.314335	−	0.949312i	$-0.601781\pi$
$23$	−3.01499	−0.628670	−0.314335	+	0.949312i	$0.601781\pi$
$24$	0	0
$25$	−1.45707	−0.291413
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	3.15000	0.584941	0.292471	−	0.956275i	$-0.405523\pi$
$29$	3.15000	0.584941	0.292471	+	0.956275i	$0.405523\pi$
$30$	0	0
$31$	1.47463	0.264851	0.132425	−	0.991193i	$-0.457723\pi$
$31$	1.47463	0.264851	0.132425	+	0.991193i	$0.457723\pi$
$32$	0	0
$33$	4.19367	0.730024
$34$	0	0
$35$	−2.43106	−0.410924
$36$	0	0
$37$	8.15343	1.34041	0.670207	−	0.742174i	$-0.266205\pi$
$37$	8.15343	1.34041	0.670207	+	0.742174i	$0.266205\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−10.0181	−1.56457	−0.782285	−	0.622921i	$-0.785946\pi$
$41$	−10.0181	−1.56457	−0.782285	+	0.622921i	$0.785946\pi$
$42$	0	0
$43$	−10.7587	−1.64069	−0.820345	−	0.571869i	$-0.806218\pi$
$43$	−10.7587	−1.64069	−0.820345	+	0.571869i	$0.806218\pi$
$44$	0	0
$45$	1.88227	0.280592
$46$	0	0
$47$	5.31909	0.775869	0.387934	−	0.921687i	$-0.373189\pi$
$47$	5.31909	0.775869	0.387934	+	0.921687i	$0.373189\pi$
$48$	0	0
$49$	−5.33187	−0.761696
$50$	0	0
$51$	1.68421	0.235836
$52$	0	0
$53$	8.29590	1.13953	0.569765	−	0.821808i	$-0.307034\pi$
$53$	8.29590	1.13953	0.569765	+	0.821808i	$0.307034\pi$
$54$	0	0
$55$	−7.89361	−1.06437
$56$	0	0
$57$	1.19025	0.157652
$58$	0	0
$59$	3.03369	0.394953	0.197477	−	0.980308i	$-0.436725\pi$
$59$	3.03369	0.394953	0.197477	+	0.980308i	$0.436725\pi$
$60$	0	0
$61$	7.84487	1.00443	0.502216	−	0.864742i	$-0.332518\pi$
$61$	7.84487	1.00443	0.502216	+	0.864742i	$0.332518\pi$
$62$	0	0
$63$	−1.29156	−0.162721
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−4.50497	−0.550370	−0.275185	−	0.961391i	$-0.588739\pi$
$67$	−4.50497	−0.550370	−0.275185	+	0.961391i	$0.588739\pi$
$68$	0	0
$69$	3.01499	0.362963
$70$	0	0
$71$	3.85816	0.457880	0.228940	−	0.973441i	$-0.426474\pi$
$71$	3.85816	0.457880	0.228940	+	0.973441i	$0.426474\pi$
$72$	0	0
$73$	1.76494	0.206571	0.103285	−	0.994652i	$-0.467064\pi$
$73$	1.76494	0.206571	0.103285	+	0.994652i	$0.467064\pi$
$74$	0	0
$75$	1.45707	0.168248
$76$	0	0
$77$	5.41637	0.617253
$78$	0	0
$79$	12.6593	1.42428	0.712140	−	0.702037i	$-0.247726\pi$
$79$	12.6593	1.42428	0.712140	+	0.702037i	$0.247726\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	15.6280	1.71540	0.857701	−	0.514149i	$-0.171892\pi$
$83$	15.6280	1.71540	0.857701	+	0.514149i	$0.171892\pi$
$84$	0	0
$85$	−3.17013	−0.343849
$86$	0	0
$87$	−3.15000	−0.337716
$88$	0	0
$89$	10.6679	1.13080	0.565398	−	0.824818i	$-0.308722\pi$
$89$	10.6679	1.13080	0.565398	+	0.824818i	$0.308722\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	−1.47463	−0.152912
$94$	0	0
$95$	−2.24036	−0.229856
$96$	0	0
$97$	−2.38541	−0.242201	−0.121101	−	0.992640i	$-0.538642\pi$
$97$	−2.38541	−0.242201	−0.121101	+	0.992640i	$0.538642\pi$
$98$	0	0
$99$	−4.19367	−0.421479
$100$	0	0
$101$	−0.880061	−0.0875693	−0.0437847	−	0.999041i	$-0.513942\pi$
$101$	−0.880061	−0.0875693	−0.0437847	+	0.999041i	$0.513942\pi$
$102$	0	0
$103$	−5.21107	−0.513462	−0.256731	−	0.966483i	$-0.582645\pi$
$103$	−5.21107	−0.513462	−0.256731	+	0.966483i	$0.582645\pi$
$104$	0	0
$105$	2.43106	0.237247
$106$	0	0
$107$	−19.9637	−1.92996	−0.964981	−	0.262319i	$-0.915513\pi$
$107$	−19.9637	−1.92996	−0.964981	+	0.262319i	$0.915513\pi$
$108$	0	0
$109$	−18.5577	−1.77751	−0.888753	−	0.458386i	$-0.848428\pi$
$109$	−18.5577	−1.77751	−0.888753	+	0.458386i	$0.848428\pi$
$110$	0	0
$111$	−8.15343	−0.773889
$112$	0	0
$113$	9.98409	0.939224	0.469612	−	0.882873i	$-0.344394\pi$
$113$	9.98409	0.939224	0.469612	+	0.882873i	$0.344394\pi$
$114$	0	0
$115$	−5.67503	−0.529199
$116$	0	0
$117$	0	0
$118$	0	0
$119$	2.17525	0.199405
$120$	0	0
$121$	6.58685	0.598805
$122$	0	0
$123$	10.0181	0.903305
$124$	0	0
$125$	−12.1539	−1.08708
$126$	0	0
$127$	14.7930	1.31267	0.656335	−	0.754469i	$-0.272106\pi$
$127$	14.7930	1.31267	0.656335	+	0.754469i	$0.272106\pi$
$128$	0	0
$129$	10.7587	0.947252
$130$	0	0
$131$	0.834088	0.0728746	0.0364373	−	0.999336i	$-0.488399\pi$
$131$	0.834088	0.0728746	0.0364373	+	0.999336i	$0.488399\pi$
$132$	0	0
$133$	1.53727	0.133299
$134$	0	0
$135$	−1.88227	−0.162000
$136$	0	0
$137$	6.20377	0.530024	0.265012	−	0.964245i	$-0.414624\pi$
$137$	6.20377	0.530024	0.265012	+	0.964245i	$0.414624\pi$
$138$	0	0
$139$	20.1616	1.71008	0.855042	−	0.518559i	$-0.173531\pi$
$139$	20.1616	1.71008	0.855042	+	0.518559i	$0.173531\pi$
$140$	0	0
$141$	−5.31909	−0.447948
$142$	0	0
$143$	0	0
$144$	0	0
$145$	5.92915	0.492389
$146$	0	0
$147$	5.33187	0.439766
$148$	0	0
$149$	16.2561	1.33175	0.665876	−	0.746063i	$-0.268058\pi$
$149$	16.2561	1.33175	0.665876	+	0.746063i	$0.268058\pi$
$150$	0	0
$151$	18.0930	1.47239	0.736195	−	0.676769i	$-0.236621\pi$
$151$	18.0930	1.47239	0.736195	+	0.676769i	$0.236621\pi$
$152$	0	0
$153$	−1.68421	−0.136160
$154$	0	0
$155$	2.77564	0.222945
$156$	0	0
$157$	15.0559	1.20159	0.600795	−	0.799403i	$-0.294851\pi$
$157$	15.0559	1.20159	0.600795	+	0.799403i	$0.294851\pi$
$158$	0	0
$159$	−8.29590	−0.657907
$160$	0	0
$161$	3.89404	0.306894
$162$	0	0
$163$	−7.31660	−0.573080	−0.286540	−	0.958068i	$-0.592505\pi$
$163$	−7.31660	−0.573080	−0.286540	+	0.958068i	$0.592505\pi$
$164$	0	0
$165$	7.89361	0.614517
$166$	0	0
$167$	11.9586	0.925385	0.462693	−	0.886519i	$-0.346883\pi$
$167$	11.9586	0.925385	0.462693	+	0.886519i	$0.346883\pi$
$168$	0	0
$169$	0	0
$170$	0	0
$171$	−1.19025	−0.0910204
$172$	0	0
$173$	−16.6231	−1.26383	−0.631915	−	0.775038i	$-0.717731\pi$
$173$	−16.6231	−1.26383	−0.631915	+	0.775038i	$0.717731\pi$
$174$	0	0
$175$	1.88189	0.142257
$176$	0	0
$177$	−3.03369	−0.228026
$178$	0	0
$179$	−1.89790	−0.141856	−0.0709278	−	0.997481i	$-0.522596\pi$
$179$	−1.89790	−0.141856	−0.0709278	+	0.997481i	$0.522596\pi$
$180$	0	0
$181$	16.3850	1.21789	0.608944	−	0.793213i	$-0.291594\pi$
$181$	16.3850	1.21789	0.608944	+	0.793213i	$0.291594\pi$
$182$	0	0
$183$	−7.84487	−0.579909
$184$	0	0
$185$	15.3469	1.12833
$186$	0	0
$187$	7.06300	0.516498
$188$	0	0
$189$	1.29156	0.0939471
$190$	0	0
$191$	−13.5859	−0.983042	−0.491521	−	0.870866i	$-0.663559\pi$
$191$	−13.5859	−0.983042	−0.491521	+	0.870866i	$0.663559\pi$
$192$	0	0
$193$	24.8588	1.78938	0.894689	−	0.446689i	$-0.147397\pi$
$193$	24.8588	1.78938	0.894689	+	0.446689i	$0.147397\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	4.47333	0.318711	0.159356	−	0.987221i	$-0.449058\pi$
$197$	4.47333	0.318711	0.159356	+	0.987221i	$0.449058\pi$
$198$	0	0
$199$	4.49522	0.318658	0.159329	−	0.987226i	$-0.449067\pi$
$199$	4.49522	0.318658	0.159329	+	0.987226i	$0.449067\pi$
$200$	0	0
$201$	4.50497	0.317756
$202$	0	0
$203$	−4.06842	−0.285547
$204$	0	0
$205$	−18.8568	−1.31702
$206$	0	0
$207$	−3.01499	−0.209557
$208$	0	0
$209$	4.99150	0.345269
$210$	0	0
$211$	−21.7861	−1.49982	−0.749908	−	0.661542i	$-0.769902\pi$
$211$	−21.7861	−1.49982	−0.749908	+	0.661542i	$0.769902\pi$
$212$	0	0
$213$	−3.85816	−0.264357
$214$	0	0
$215$	−20.2508	−1.38109
$216$	0	0
$217$	−1.90457	−0.129291
$218$	0	0
$219$	−1.76494	−0.119264
$220$	0	0
$221$	0	0
$222$	0	0
$223$	−0.0555395	−0.00371920	−0.00185960	−	0.999998i	$-0.500592\pi$
$223$	−0.0555395	−0.00371920	−0.00185960	+	0.999998i	$0.500592\pi$
$224$	0	0
$225$	−1.45707	−0.0971378
$226$	0	0
$227$	5.24936	0.348412	0.174206	−	0.984709i	$-0.444264\pi$
$227$	5.24936	0.348412	0.174206	+	0.984709i	$0.444264\pi$
$228$	0	0
$229$	−21.6951	−1.43365	−0.716826	−	0.697252i	$-0.754406\pi$
$229$	−21.6951	−1.43365	−0.716826	+	0.697252i	$0.754406\pi$
$230$	0	0
$231$	−5.41637	−0.356371
$232$	0	0
$233$	9.34933	0.612495	0.306248	−	0.951952i	$-0.400926\pi$
$233$	9.34933	0.612495	0.306248	+	0.951952i	$0.400926\pi$
$234$	0	0
$235$	10.0119	0.653107
$236$	0	0
$237$	−12.6593	−0.822309
$238$	0	0
$239$	14.7781	0.955913	0.477957	−	0.878383i	$-0.341378\pi$
$239$	14.7781	0.955913	0.477957	+	0.878383i	$0.341378\pi$
$240$	0	0
$241$	−16.7668	−1.08004	−0.540022	−	0.841651i	$-0.681584\pi$
$241$	−16.7668	−1.08004	−0.540022	+	0.841651i	$0.681584\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	−10.0360	−0.641178
$246$	0	0
$247$	0	0
$248$	0	0
$249$	−15.6280	−0.990387
$250$	0	0
$251$	−1.44048	−0.0909222	−0.0454611	−	0.998966i	$-0.514476\pi$
$251$	−1.44048	−0.0909222	−0.0454611	+	0.998966i	$0.514476\pi$
$252$	0	0
$253$	12.6439	0.794914
$254$	0	0
$255$	3.17013	0.198521
$256$	0	0
$257$	−21.0265	−1.31160	−0.655799	−	0.754935i	$-0.727668\pi$
$257$	−21.0265	−1.31160	−0.655799	+	0.754935i	$0.727668\pi$
$258$	0	0
$259$	−10.5306	−0.654342
$260$	0	0
$261$	3.15000	0.194980
$262$	0	0
$263$	−10.6833	−0.658760	−0.329380	−	0.944197i	$-0.606840\pi$
$263$	−10.6833	−0.658760	−0.329380	+	0.944197i	$0.606840\pi$
$264$	0	0
$265$	15.6151	0.959228
$266$	0	0
$267$	−10.6679	−0.652866
$268$	0	0
$269$	18.3247	1.11728	0.558638	−	0.829412i	$-0.311324\pi$
$269$	18.3247	1.11728	0.558638	+	0.829412i	$0.311324\pi$
$270$	0	0
$271$	3.64076	0.221161	0.110580	−	0.993867i	$-0.464729\pi$
$271$	3.64076	0.221161	0.110580	+	0.993867i	$0.464729\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	6.11045	0.368474
$276$	0	0
$277$	22.3601	1.34349	0.671744	−	0.740783i	$-0.265546\pi$
$277$	22.3601	1.34349	0.671744	+	0.740783i	$0.265546\pi$
$278$	0	0
$279$	1.47463	0.0882836
$280$	0	0
$281$	2.39777	0.143039	0.0715195	−	0.997439i	$-0.477215\pi$
$281$	2.39777	0.143039	0.0715195	+	0.997439i	$0.477215\pi$
$282$	0	0
$283$	1.64168	0.0975879	0.0487939	−	0.998809i	$-0.484462\pi$
$283$	1.64168	0.0975879	0.0487939	+	0.998809i	$0.484462\pi$
$284$	0	0
$285$	2.24036	0.132708
$286$	0	0
$287$	12.9390	0.763766
$288$	0	0
$289$	−14.1635	−0.833144
$290$	0	0
$291$	2.38541	0.139835
$292$	0	0
$293$	1.35348	0.0790709	0.0395355	−	0.999218i	$-0.487412\pi$
$293$	1.35348	0.0790709	0.0395355	+	0.999218i	$0.487412\pi$
$294$	0	0
$295$	5.71022	0.332462
$296$	0	0
$297$	4.19367	0.243341
$298$	0	0
$299$	0	0
$300$	0	0
$301$	13.8955	0.800925
$302$	0	0
$303$	0.880061	0.0505582
$304$	0	0
$305$	14.7661	0.845507
$306$	0	0
$307$	23.9864	1.36898	0.684489	−	0.729023i	$-0.260025\pi$
$307$	23.9864	1.36898	0.684489	+	0.729023i	$0.260025\pi$
$308$	0	0
$309$	5.21107	0.296447
$310$	0	0
$311$	−6.23414	−0.353506	−0.176753	−	0.984255i	$-0.556559\pi$
$311$	−6.23414	−0.353506	−0.176753	+	0.984255i	$0.556559\pi$
$312$	0	0
$313$	−28.7120	−1.62290	−0.811450	−	0.584422i	$-0.801321\pi$
$313$	−28.7120	−1.62290	−0.811450	+	0.584422i	$0.801321\pi$
$314$	0	0
$315$	−2.43106	−0.136975
$316$	0	0
$317$	13.8082	0.775545	0.387772	−	0.921755i	$-0.373245\pi$
$317$	13.8082	0.775545	0.387772	+	0.921755i	$0.373245\pi$
$318$	0	0
$319$	−13.2101	−0.739622
$320$	0	0
$321$	19.9637	1.11426
$322$	0	0
$323$	2.00462	0.111540
$324$	0	0
$325$	0	0
$326$	0	0
$327$	18.5577	1.02624
$328$	0	0
$329$	−6.86992	−0.378751
$330$	0	0
$331$	−7.06203	−0.388165	−0.194082	−	0.980985i	$-0.562173\pi$
$331$	−7.06203	−0.388165	−0.194082	+	0.980985i	$0.562173\pi$
$332$	0	0
$333$	8.15343	0.446805
$334$	0	0
$335$	−8.47956	−0.463288
$336$	0	0
$337$	−15.9635	−0.869585	−0.434792	−	0.900531i	$-0.643178\pi$
$337$	−15.9635	−0.869585	−0.434792	+	0.900531i	$0.643178\pi$
$338$	0	0
$339$	−9.98409	−0.542261
$340$	0	0
$341$	−6.18410	−0.334888
$342$	0	0
$343$	15.9274	0.859996
$344$	0	0
$345$	5.67503	0.305533
$346$	0	0
$347$	−17.1822	−0.922388	−0.461194	−	0.887299i	$-0.652579\pi$
$347$	−17.1822	−0.922388	−0.461194	+	0.887299i	$0.652579\pi$
$348$	0	0
$349$	28.2999	1.51486	0.757429	−	0.652918i	$-0.226455\pi$
$349$	28.2999	1.51486	0.757429	+	0.652918i	$0.226455\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	0.313161	0.0166679	0.00833393	−	0.999965i	$-0.497347\pi$
$353$	0.313161	0.0166679	0.00833393	+	0.999965i	$0.497347\pi$
$354$	0	0
$355$	7.26210	0.385432
$356$	0	0
$357$	−2.17525	−0.115127
$358$	0	0
$359$	−12.3342	−0.650977	−0.325488	−	0.945546i	$-0.605529\pi$
$359$	−12.3342	−0.650977	−0.325488	+	0.945546i	$0.605529\pi$
$360$	0	0
$361$	−17.5833	−0.925438
$362$	0	0
$363$	−6.58685	−0.345720
$364$	0	0
$365$	3.32209	0.173886
$366$	0	0
$367$	9.08368	0.474164	0.237082	−	0.971490i	$-0.423809\pi$
$367$	9.08368	0.474164	0.237082	+	0.971490i	$0.423809\pi$
$368$	0	0
$369$	−10.0181	−0.521523
$370$	0	0
$371$	−10.7146	−0.556277
$372$	0	0
$373$	−16.5431	−0.856569	−0.428285	−	0.903644i	$-0.640882\pi$
$373$	−16.5431	−0.856569	−0.428285	+	0.903644i	$0.640882\pi$
$374$	0	0
$375$	12.1539	0.627626
$376$	0	0
$377$	0	0
$378$	0	0
$379$	−10.4212	−0.535299	−0.267650	−	0.963516i	$-0.586247\pi$
$379$	−10.4212	−0.535299	−0.267650	+	0.963516i	$0.586247\pi$
$380$	0	0
$381$	−14.7930	−0.757871
$382$	0	0
$383$	15.5767	0.795931	0.397965	−	0.917400i	$-0.369716\pi$
$383$	15.5767	0.795931	0.397965	+	0.917400i	$0.369716\pi$
$384$	0	0
$385$	10.1951	0.519589
$386$	0	0
$387$	−10.7587	−0.546896
$388$	0	0
$389$	13.0121	0.659742	0.329871	−	0.944026i	$-0.392995\pi$
$389$	13.0121	0.659742	0.329871	+	0.944026i	$0.392995\pi$
$390$	0	0
$391$	5.07787	0.256799
$392$	0	0
$393$	−0.834088	−0.0420742
$394$	0	0
$395$	23.8282	1.19892
$396$	0	0
$397$	−33.9449	−1.70364	−0.851822	−	0.523831i	$-0.824502\pi$
$397$	−33.9449	−1.70364	−0.851822	+	0.523831i	$0.824502\pi$
$398$	0	0
$399$	−1.53727	−0.0769600
$400$	0	0
$401$	37.0787	1.85162	0.925811	−	0.377988i	$-0.123384\pi$
$401$	37.0787	1.85162	0.925811	+	0.377988i	$0.123384\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	1.88227	0.0935307
$406$	0	0
$407$	−34.1928	−1.69487
$408$	0	0
$409$	31.0280	1.53424	0.767118	−	0.641505i	$-0.221690\pi$
$409$	31.0280	1.53424	0.767118	+	0.641505i	$0.221690\pi$
$410$	0	0
$411$	−6.20377	−0.306009
$412$	0	0
$413$	−3.91819	−0.192802
$414$	0	0
$415$	29.4162	1.44398
$416$	0	0
$417$	−20.1616	−0.987317
$418$	0	0
$419$	27.5226	1.34457	0.672284	−	0.740293i	$-0.265313\pi$
$419$	27.5226	1.34457	0.672284	+	0.740293i	$0.265313\pi$
$420$	0	0
$421$	−10.2919	−0.501599	−0.250799	−	0.968039i	$-0.580693\pi$
$421$	−10.2919	−0.501599	−0.250799	+	0.968039i	$0.580693\pi$
$422$	0	0
$423$	5.31909	0.258623
$424$	0	0
$425$	2.45400	0.119036
$426$	0	0
$427$	−10.1321	−0.490327
$428$	0	0
$429$	0	0
$430$	0	0
$431$	30.9340	1.49004	0.745020	−	0.667043i	$-0.232440\pi$
$431$	30.9340	1.49004	0.745020	+	0.667043i	$0.232440\pi$
$432$	0	0
$433$	9.59342	0.461030	0.230515	−	0.973069i	$-0.425959\pi$
$433$	9.59342	0.461030	0.230515	+	0.973069i	$0.425959\pi$
$434$	0	0
$435$	−5.92915	−0.284281
$436$	0	0
$437$	3.58859	0.171665
$438$	0	0
$439$	−18.3735	−0.876918	−0.438459	−	0.898751i	$-0.644476\pi$
$439$	−18.3735	−0.876918	−0.438459	+	0.898751i	$0.644476\pi$
$440$	0	0
$441$	−5.33187	−0.253899
$442$	0	0
$443$	16.7144	0.794125	0.397062	−	0.917792i	$-0.370030\pi$
$443$	16.7144	0.794125	0.397062	+	0.917792i	$0.370030\pi$
$444$	0	0
$445$	20.0799	0.951878
$446$	0	0
$447$	−16.2561	−0.768887
$448$	0	0
$449$	32.4482	1.53132	0.765662	−	0.643243i	$-0.222412\pi$
$449$	32.4482	1.53132	0.765662	+	0.643243i	$0.222412\pi$
$450$	0	0
$451$	42.0127	1.97830
$452$	0	0
$453$	−18.0930	−0.850085
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−16.5767	−0.775427	−0.387713	−	0.921780i	$-0.626735\pi$
$457$	−16.5767	−0.775427	−0.387713	+	0.921780i	$0.626735\pi$
$458$	0	0
$459$	1.68421	0.0786120
$460$	0	0
$461$	−2.43380	−0.113353	−0.0566767	−	0.998393i	$-0.518050\pi$
$461$	−2.43380	−0.113353	−0.0566767	+	0.998393i	$0.518050\pi$
$462$	0	0
$463$	30.9325	1.43755	0.718777	−	0.695240i	$-0.244702\pi$
$463$	30.9325	1.43755	0.718777	+	0.695240i	$0.244702\pi$
$464$	0	0
$465$	−2.77564	−0.128717
$466$	0	0
$467$	−19.4582	−0.900417	−0.450208	−	0.892924i	$-0.648650\pi$
$467$	−19.4582	−0.900417	−0.450208	+	0.892924i	$0.648650\pi$
$468$	0	0
$469$	5.81844	0.268670
$470$	0	0
$471$	−15.0559	−0.693738
$472$	0	0
$473$	45.1185	2.07455
$474$	0	0
$475$	1.73427	0.0795737
$476$	0	0
$477$	8.29590	0.379843
$478$	0	0
$479$	21.0622	0.962356	0.481178	−	0.876623i	$-0.340209\pi$
$479$	21.0622	0.962356	0.481178	+	0.876623i	$0.340209\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	−3.89404	−0.177185
$484$	0	0
$485$	−4.48997	−0.203879
$486$	0	0
$487$	37.6092	1.70424	0.852118	−	0.523349i	$-0.175318\pi$
$487$	37.6092	1.70424	0.852118	+	0.523349i	$0.175318\pi$
$488$	0	0
$489$	7.31660	0.330868
$490$	0	0
$491$	−36.5335	−1.64873	−0.824367	−	0.566056i	$-0.808469\pi$
$491$	−36.5335	−1.64873	−0.824367	+	0.566056i	$0.808469\pi$
$492$	0	0
$493$	−5.30526	−0.238937
$494$	0	0
$495$	−7.89361	−0.354791
$496$	0	0
$497$	−4.98305	−0.223520
$498$	0	0
$499$	−22.4840	−1.00652	−0.503262	−	0.864134i	$-0.667867\pi$
$499$	−22.4840	−1.00652	−0.503262	+	0.864134i	$0.667867\pi$
$500$	0	0
$501$	−11.9586	−0.534271
$502$	0	0
$503$	2.21863	0.0989237	0.0494619	−	0.998776i	$-0.484249\pi$
$503$	2.21863	0.0989237	0.0494619	+	0.998776i	$0.484249\pi$
$504$	0	0
$505$	−1.65651	−0.0737137
$506$	0	0
$507$	0	0
$508$	0	0
$509$	28.2456	1.25196	0.625982	−	0.779837i	$-0.284698\pi$
$509$	28.2456	1.25196	0.625982	+	0.779837i	$0.284698\pi$
$510$	0	0
$511$	−2.27953	−0.100840
$512$	0	0
$513$	1.19025	0.0525507
$514$	0	0
$515$	−9.80863	−0.432220
$516$	0	0
$517$	−22.3065	−0.981038
$518$	0	0
$519$	16.6231	0.729673
$520$	0	0
$521$	28.1688	1.23410	0.617050	−	0.786924i	$-0.288328\pi$
$521$	28.1688	1.23410	0.617050	+	0.786924i	$0.288328\pi$
$522$	0	0
$523$	−4.70961	−0.205937	−0.102968	−	0.994685i	$-0.532834\pi$
$523$	−4.70961	−0.205937	−0.102968	+	0.994685i	$0.532834\pi$
$524$	0	0
$525$	−1.88189	−0.0821323
$526$	0	0
$527$	−2.48358	−0.108186
$528$	0	0
$529$	−13.9098	−0.604774
$530$	0	0
$531$	3.03369	0.131651
$532$	0	0
$533$	0	0
$534$	0	0
$535$	−37.5770	−1.62460
$536$	0	0
$537$	1.89790	0.0819004
$538$	0	0
$539$	22.3601	0.963118
$540$	0	0
$541$	−3.17651	−0.136569	−0.0682844	−	0.997666i	$-0.521753\pi$
$541$	−3.17651	−0.136569	−0.0682844	+	0.997666i	$0.521753\pi$
$542$	0	0
$543$	−16.3850	−0.703148
$544$	0	0
$545$	−34.9306	−1.49626
$546$	0	0
$547$	25.4331	1.08744	0.543721	−	0.839266i	$-0.317015\pi$
$547$	25.4331	1.08744	0.543721	+	0.839266i	$0.317015\pi$
$548$	0	0
$549$	7.84487	0.334811
$550$	0	0
$551$	−3.74928	−0.159725
$552$	0	0
$553$	−16.3502	−0.695282
$554$	0	0
$555$	−15.3469	−0.651441
$556$	0	0
$557$	−20.1670	−0.854505	−0.427252	−	0.904132i	$-0.640518\pi$
$557$	−20.1670	−0.854505	−0.427252	+	0.904132i	$0.640518\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	−7.06300	−0.298200
$562$	0	0
$563$	18.2498	0.769136	0.384568	−	0.923097i	$-0.374350\pi$
$563$	18.2498	0.769136	0.384568	+	0.923097i	$0.374350\pi$
$564$	0	0
$565$	18.7927	0.790616
$566$	0	0
$567$	−1.29156	−0.0542404
$568$	0	0
$569$	27.4857	1.15226	0.576129	−	0.817359i	$-0.304562\pi$
$569$	27.4857	1.15226	0.576129	+	0.817359i	$0.304562\pi$
$570$	0	0
$571$	0.0250476	0.00104821	0.000524106	−	1.00000i	$-0.499833\pi$
$571$	0.0250476	0.00104821	0.000524106		1.00000i	$0.499833\pi$
$572$	0	0
$573$	13.5859	0.567559
$574$	0	0
$575$	4.39305	0.183203
$576$	0	0
$577$	21.0269	0.875362	0.437681	−	0.899130i	$-0.355800\pi$
$577$	21.0269	0.875362	0.437681	+	0.899130i	$0.355800\pi$
$578$	0	0
$579$	−24.8588	−1.03310
$580$	0	0
$581$	−20.1846	−0.837397
$582$	0	0
$583$	−34.7902	−1.44086
$584$	0	0
$585$	0	0
$586$	0	0
$587$	29.4551	1.21574	0.607872	−	0.794035i	$-0.292024\pi$
$587$	29.4551	1.21574	0.607872	+	0.794035i	$0.292024\pi$
$588$	0	0
$589$	−1.75517	−0.0723206
$590$	0	0
$591$	−4.47333	−0.184008
$592$	0	0
$593$	−3.93477	−0.161582	−0.0807909	−	0.996731i	$-0.525745\pi$
$593$	−3.93477	−0.161582	−0.0807909	+	0.996731i	$0.525745\pi$
$594$	0	0
$595$	4.09441	0.167854
$596$	0	0
$597$	−4.49522	−0.183977
$598$	0	0
$599$	−35.1931	−1.43795	−0.718975	−	0.695036i	$-0.755388\pi$
$599$	−35.1931	−1.43795	−0.718975	+	0.695036i	$0.755388\pi$
$600$	0	0
$601$	−14.9780	−0.610967	−0.305484	−	0.952197i	$-0.598818\pi$
$601$	−14.9780	−0.610967	−0.305484	+	0.952197i	$0.598818\pi$
$602$	0	0
$603$	−4.50497	−0.183457
$604$	0	0
$605$	12.3982	0.504059
$606$	0	0
$607$	24.6169	0.999170	0.499585	−	0.866265i	$-0.333486\pi$
$607$	24.6169	0.999170	0.499585	+	0.866265i	$0.333486\pi$
$608$	0	0
$609$	4.06842	0.164861
$610$	0	0
$611$	0	0
$612$	0	0
$613$	36.1429	1.45980	0.729900	−	0.683554i	$-0.239567\pi$
$613$	36.1429	1.45980	0.729900	+	0.683554i	$0.239567\pi$
$614$	0	0
$615$	18.8568	0.760380
$616$	0	0
$617$	−12.0808	−0.486356	−0.243178	−	0.969982i	$-0.578190\pi$
$617$	−12.0808	−0.486356	−0.243178	+	0.969982i	$0.578190\pi$
$618$	0	0
$619$	−23.1257	−0.929500	−0.464750	−	0.885442i	$-0.653856\pi$
$619$	−23.1257	−0.929500	−0.464750	+	0.885442i	$0.653856\pi$
$620$	0	0
$621$	3.01499	0.120988
$622$	0	0
$623$	−13.7782	−0.552014
$624$	0	0
$625$	−15.5916	−0.623665
$626$	0	0
$627$	−4.99150	−0.199341
$628$	0	0
$629$	−13.7320	−0.547533
$630$	0	0
$631$	43.7565	1.74192	0.870960	−	0.491354i	$-0.163498\pi$
$631$	43.7565	1.74192	0.870960	+	0.491354i	$0.163498\pi$
$632$	0	0
$633$	21.7861	0.865920
$634$	0	0
$635$	27.8445	1.10497
$636$	0	0
$637$	0	0
$638$	0	0
$639$	3.85816	0.152627
$640$	0	0
$641$	−7.76562	−0.306724	−0.153362	−	0.988170i	$-0.549010\pi$
$641$	−7.76562	−0.306724	−0.153362	+	0.988170i	$0.549010\pi$
$642$	0	0
$643$	9.44419	0.372442	0.186221	−	0.982508i	$-0.440376\pi$
$643$	9.44419	0.372442	0.186221	+	0.982508i	$0.440376\pi$
$644$	0	0
$645$	20.2508	0.797374
$646$	0	0
$647$	41.8606	1.64571	0.822856	−	0.568250i	$-0.192379\pi$
$647$	41.8606	1.64571	0.822856	+	0.568250i	$0.192379\pi$
$648$	0	0
$649$	−12.7223	−0.499394
$650$	0	0
$651$	1.90457	0.0746460
$652$	0	0
$653$	−43.8823	−1.71725	−0.858624	−	0.512605i	$-0.828680\pi$
$653$	−43.8823	−1.71725	−0.858624	+	0.512605i	$0.828680\pi$
$654$	0	0
$655$	1.56998	0.0613441
$656$	0	0
$657$	1.76494	0.0688569
$658$	0	0
$659$	27.3533	1.06553	0.532767	−	0.846262i	$-0.321152\pi$
$659$	27.3533	1.06553	0.532767	+	0.846262i	$0.321152\pi$
$660$	0	0
$661$	−46.8238	−1.82123	−0.910617	−	0.413251i	$-0.864393\pi$
$661$	−46.8238	−1.82123	−0.910617	+	0.413251i	$0.864393\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	2.89356	0.112208
$666$	0	0
$667$	−9.49724	−0.367735
$668$	0	0
$669$	0.0555395	0.00214728
$670$	0	0
$671$	−32.8988	−1.27004
$672$	0	0
$673$	36.2574	1.39762	0.698811	−	0.715307i	$-0.253713\pi$
$673$	36.2574	1.39762	0.698811	+	0.715307i	$0.253713\pi$
$674$	0	0
$675$	1.45707	0.0560825
$676$	0	0
$677$	51.1630	1.96636	0.983178	−	0.182652i	$-0.0584682\pi$
$677$	51.1630	1.96636	0.983178	+	0.182652i	$0.0584682\pi$
$678$	0	0
$679$	3.08089	0.118234
$680$	0	0
$681$	−5.24936	−0.201156
$682$	0	0
$683$	30.8240	1.17945	0.589724	−	0.807605i	$-0.299237\pi$
$683$	30.8240	1.17945	0.589724	+	0.807605i	$0.299237\pi$
$684$	0	0
$685$	11.6772	0.446161
$686$	0	0
$687$	21.6951	0.827720
$688$	0	0
$689$	0	0
$690$	0	0
$691$	−4.12305	−0.156848	−0.0784241	−	0.996920i	$-0.524989\pi$
$691$	−4.12305	−0.156848	−0.0784241	+	0.996920i	$0.524989\pi$
$692$	0	0
$693$	5.41637	0.205751
$694$	0	0
$695$	37.9495	1.43951
$696$	0	0
$697$	16.8726	0.639095
$698$	0	0
$699$	−9.34933	−0.353624
$700$	0	0
$701$	6.97922	0.263602	0.131801	−	0.991276i	$-0.457924\pi$
$701$	6.97922	0.263602	0.131801	+	0.991276i	$0.457924\pi$
$702$	0	0
$703$	−9.70459	−0.366015
$704$	0	0
$705$	−10.0119	−0.377072
$706$	0	0
$707$	1.13665	0.0427482
$708$	0	0
$709$	−23.1455	−0.869247	−0.434623	−	0.900612i	$-0.643118\pi$
$709$	−23.1455	−0.869247	−0.434623	+	0.900612i	$0.643118\pi$
$710$	0	0
$711$	12.6593	0.474760
$712$	0	0
$713$	−4.44599	−0.166504
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−14.7781	−0.551897
$718$	0	0
$719$	−21.8249	−0.813930	−0.406965	−	0.913444i	$-0.633413\pi$
$719$	−21.8249	−0.813930	−0.406965	+	0.913444i	$0.633413\pi$
$720$	0	0
$721$	6.73041	0.250653
$722$	0	0
$723$	16.7668	0.623563
$724$	0	0
$725$	−4.58976	−0.170460
$726$	0	0
$727$	43.0916	1.59818	0.799090	−	0.601211i	$-0.205315\pi$
$727$	43.0916	1.59818	0.799090	+	0.601211i	$0.205315\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	18.1199	0.670189
$732$	0	0
$733$	−23.5077	−0.868277	−0.434138	−	0.900846i	$-0.642947\pi$
$733$	−23.5077	−0.868277	−0.434138	+	0.900846i	$0.642947\pi$
$734$	0	0
$735$	10.0360	0.370184
$736$	0	0
$737$	18.8923	0.695909
$738$	0	0
$739$	23.9187	0.879862	0.439931	−	0.898032i	$-0.355003\pi$
$739$	23.9187	0.879862	0.439931	+	0.898032i	$0.355003\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	0.613067	0.0224912	0.0112456	−	0.999937i	$-0.496420\pi$
$743$	0.613067	0.0224912	0.0112456	+	0.999937i	$0.496420\pi$
$744$	0	0
$745$	30.5983	1.12104
$746$	0	0
$747$	15.6280	0.571800
$748$	0	0
$749$	25.7843	0.942137
$750$	0	0
$751$	−0.945022	−0.0344843	−0.0172422	−	0.999851i	$-0.505489\pi$
$751$	−0.945022	−0.0344843	−0.0172422	+	0.999851i	$0.505489\pi$
$752$	0	0
$753$	1.44048	0.0524939
$754$	0	0
$755$	34.0559	1.23942
$756$	0	0
$757$	−14.8014	−0.537966	−0.268983	−	0.963145i	$-0.586688\pi$
$757$	−14.8014	−0.537966	−0.268983	+	0.963145i	$0.586688\pi$
$758$	0	0
$759$	−12.6439	−0.458944
$760$	0	0
$761$	−11.6688	−0.422994	−0.211497	−	0.977379i	$-0.567834\pi$
$761$	−11.6688	−0.422994	−0.211497	+	0.977379i	$0.567834\pi$
$762$	0	0
$763$	23.9684	0.867714
$764$	0	0
$765$	−3.17013	−0.114616
$766$	0	0
$767$	0	0
$768$	0	0
$769$	2.64412	0.0953494	0.0476747	−	0.998863i	$-0.484819\pi$
$769$	2.64412	0.0953494	0.0476747	+	0.998863i	$0.484819\pi$
$770$	0	0
$771$	21.0265	0.757252
$772$	0	0
$773$	−0.0776248	−0.00279197	−0.00139599	−	0.999999i	$-0.500444\pi$
$773$	−0.0776248	−0.00279197	−0.00139599	+	0.999999i	$0.500444\pi$
$774$	0	0
$775$	−2.14863	−0.0771811
$776$	0	0
$777$	10.5306	0.377784
$778$	0	0
$779$	11.9240	0.427223
$780$	0	0
$781$	−16.1799	−0.578961
$782$	0	0
$783$	−3.15000	−0.112572
$784$	0	0
$785$	28.3392	1.01147
$786$	0	0
$787$	24.3434	0.867749	0.433874	−	0.900973i	$-0.357146\pi$
$787$	24.3434	0.867749	0.433874	+	0.900973i	$0.357146\pi$
$788$	0	0
$789$	10.6833	0.380335
$790$	0	0
$791$	−12.8950	−0.458495
$792$	0	0
$793$	0	0
$794$	0	0
$795$	−15.6151	−0.553811
$796$	0	0
$797$	−46.4833	−1.64652	−0.823261	−	0.567663i	$-0.807848\pi$
$797$	−46.4833	−1.64652	−0.823261	+	0.567663i	$0.807848\pi$
$798$	0	0
$799$	−8.95844	−0.316927
$800$	0	0
$801$	10.6679	0.376932
$802$	0	0
$803$	−7.40158	−0.261196
$804$	0	0
$805$	7.32964	0.258336
$806$	0	0
$807$	−18.3247	−0.645060
$808$	0	0
$809$	17.4581	0.613796	0.306898	−	0.951742i	$-0.400709\pi$
$809$	17.4581	0.613796	0.306898	+	0.951742i	$0.400709\pi$
$810$	0	0
$811$	13.3281	0.468012	0.234006	−	0.972235i	$-0.424817\pi$
$811$	13.3281	0.468012	0.234006	+	0.972235i	$0.424817\pi$
$812$	0	0
$813$	−3.64076	−0.127687
$814$	0	0
$815$	−13.7718	−0.482405
$816$	0	0
$817$	12.8055	0.448009
$818$	0	0
$819$	0	0
$820$	0	0
$821$	11.0294	0.384928	0.192464	−	0.981304i	$-0.438352\pi$
$821$	11.0294	0.384928	0.192464	+	0.981304i	$0.438352\pi$
$822$	0	0
$823$	12.1161	0.422340	0.211170	−	0.977449i	$-0.432273\pi$
$823$	12.1161	0.422340	0.211170	+	0.977449i	$0.432273\pi$
$824$	0	0
$825$	−6.11045	−0.212739
$826$	0	0
$827$	−23.6413	−0.822087	−0.411044	−	0.911616i	$-0.634836\pi$
$827$	−23.6413	−0.822087	−0.411044	+	0.911616i	$0.634836\pi$
$828$	0	0
$829$	−43.4519	−1.50915	−0.754574	−	0.656215i	$-0.772157\pi$
$829$	−43.4519	−1.50915	−0.754574	+	0.656215i	$0.772157\pi$
$830$	0	0
$831$	−22.3601	−0.775663
$832$	0	0
$833$	8.97997	0.311138
$834$	0	0
$835$	22.5093	0.778967
$836$	0	0
$837$	−1.47463	−0.0509706
$838$	0	0
$839$	17.2176	0.594416	0.297208	−	0.954813i	$-0.403945\pi$
$839$	17.2176	0.594416	0.297208	+	0.954813i	$0.403945\pi$
$840$	0	0
$841$	−19.0775	−0.657844
$842$	0	0
$843$	−2.39777	−0.0825836
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−8.50731	−0.292315
$848$	0	0
$849$	−1.64168	−0.0563424
$850$	0	0
$851$	−24.5825	−0.842678
$852$	0	0
$853$	19.5329	0.668793	0.334397	−	0.942432i	$-0.391467\pi$
$853$	19.5329	0.668793	0.334397	+	0.942432i	$0.391467\pi$
$854$	0	0
$855$	−2.24036	−0.0766188
$856$	0	0
$857$	−19.9607	−0.681843	−0.340922	−	0.940092i	$-0.610739\pi$
$857$	−19.9607	−0.681843	−0.340922	+	0.940092i	$0.610739\pi$
$858$	0	0
$859$	−36.0681	−1.23063	−0.615315	−	0.788282i	$-0.710971\pi$
$859$	−36.0681	−1.23063	−0.615315	+	0.788282i	$0.710971\pi$
$860$	0	0
$861$	−12.9390	−0.440960
$862$	0	0
$863$	3.06724	0.104410	0.0522051	−	0.998636i	$-0.483375\pi$
$863$	3.06724	0.104410	0.0522051	+	0.998636i	$0.483375\pi$
$864$	0	0
$865$	−31.2891	−1.06386
$866$	0	0
$867$	14.1635	0.481016
$868$	0	0
$869$	−53.0888	−1.80091
$870$	0	0
$871$	0	0
$872$	0	0
$873$	−2.38541	−0.0807337
$874$	0	0
$875$	15.6975	0.530673
$876$	0	0
$877$	−26.5958	−0.898075	−0.449038	−	0.893513i	$-0.648233\pi$
$877$	−26.5958	−0.898075	−0.449038	+	0.893513i	$0.648233\pi$
$878$	0	0
$879$	−1.35348	−0.0456516
$880$	0	0
$881$	56.4634	1.90230	0.951151	−	0.308727i	$-0.0999030\pi$
$881$	56.4634	1.90230	0.951151	+	0.308727i	$0.0999030\pi$
$882$	0	0
$883$	−29.6728	−0.998568	−0.499284	−	0.866438i	$-0.666404\pi$
$883$	−29.6728	−0.998568	−0.499284	+	0.866438i	$0.666404\pi$
$884$	0	0
$885$	−5.71022	−0.191947
$886$	0	0
$887$	42.4301	1.42466	0.712332	−	0.701843i	$-0.247639\pi$
$887$	42.4301	1.42466	0.712332	+	0.701843i	$0.247639\pi$
$888$	0	0
$889$	−19.1061	−0.640798
$890$	0	0
$891$	−4.19367	−0.140493
$892$	0	0
$893$	−6.33103	−0.211860
$894$	0	0
$895$	−3.57236	−0.119411
$896$	0	0
$897$	0	0
$898$	0	0
$899$	4.64508	0.154922
$900$	0	0
$901$	−13.9720	−0.465475
$902$	0	0
$903$	−13.8955	−0.462414
$904$	0	0
$905$	30.8410	1.02519
$906$	0	0
$907$	11.1465	0.370114	0.185057	−	0.982728i	$-0.440753\pi$
$907$	11.1465	0.370114	0.185057	+	0.982728i	$0.440753\pi$
$908$	0	0
$909$	−0.880061	−0.0291898
$910$	0	0
$911$	44.4798	1.47368	0.736841	−	0.676066i	$-0.236317\pi$
$911$	44.4798	1.47368	0.736841	+	0.676066i	$0.236317\pi$
$912$	0	0
$913$	−65.5389	−2.16902
$914$	0	0
$915$	−14.7661	−0.488154
$916$	0	0
$917$	−1.07727	−0.0355747
$918$	0	0
$919$	31.2712	1.03154	0.515771	−	0.856726i	$-0.327505\pi$
$919$	31.2712	1.03154	0.515771	+	0.856726i	$0.327505\pi$
$920$	0	0
$921$	−23.9864	−0.790380
$922$	0	0
$923$	0	0
$924$	0	0
$925$	−11.8801	−0.390615
$926$	0	0
$927$	−5.21107	−0.171154
$928$	0	0
$929$	19.2303	0.630927	0.315464	−	0.948938i	$-0.397840\pi$
$929$	19.2303	0.630927	0.315464	+	0.948938i	$0.397840\pi$
$930$	0	0
$931$	6.34625	0.207990
$932$	0	0
$933$	6.23414	0.204097
$934$	0	0
$935$	13.2945	0.434775
$936$	0	0
$937$	−40.1991	−1.31325	−0.656624	−	0.754218i	$-0.728016\pi$
$937$	−40.1991	−1.31325	−0.656624	+	0.754218i	$0.728016\pi$
$938$	0	0
$939$	28.7120	0.936982
$940$	0	0
$941$	57.3886	1.87082	0.935408	−	0.353570i	$-0.115032\pi$
$941$	57.3886	1.87082	0.935408	+	0.353570i	$0.115032\pi$
$942$	0	0
$943$	30.2046	0.983597
$944$	0	0
$945$	2.43106	0.0790824
$946$	0	0
$947$	12.1040	0.393328	0.196664	−	0.980471i	$-0.436989\pi$
$947$	12.1040	0.393328	0.196664	+	0.980471i	$0.436989\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	−13.8082	−0.447761
$952$	0	0
$953$	23.5095	0.761547	0.380773	−	0.924668i	$-0.375658\pi$
$953$	23.5095	0.761547	0.380773	+	0.924668i	$0.375658\pi$
$954$	0	0
$955$	−25.5723	−0.827501
$956$	0	0
$957$	13.2101	0.427021
$958$	0	0
$959$	−8.01254	−0.258738
$960$	0	0
$961$	−28.8255	−0.929854
$962$	0	0
$963$	−19.9637	−0.643321
$964$	0	0
$965$	46.7910	1.50626
$966$	0	0
$967$	−45.1235	−1.45107	−0.725537	−	0.688183i	$-0.758409\pi$
$967$	−45.1235	−1.45107	−0.725537	+	0.688183i	$0.758409\pi$
$968$	0	0
$969$	−2.00462	−0.0643977
$970$	0	0
$971$	−39.7833	−1.27671	−0.638353	−	0.769744i	$-0.720384\pi$
$971$	−39.7833	−1.27671	−0.638353	+	0.769744i	$0.720384\pi$
$972$	0	0
$973$	−26.0399	−0.834801
$974$	0	0
$975$	0	0
$976$	0	0
$977$	10.4379	0.333937	0.166968	−	0.985962i	$-0.446602\pi$
$977$	10.4379	0.333937	0.166968	+	0.985962i	$0.446602\pi$
$978$	0	0
$979$	−44.7377	−1.42982
$980$	0	0
$981$	−18.5577	−0.592502
$982$	0	0
$983$	1.56154	0.0498052	0.0249026	−	0.999690i	$-0.492072\pi$
$983$	1.56154	0.0498052	0.0249026	+	0.999690i	$0.492072\pi$
$984$	0	0
$985$	8.42000	0.268283
$986$	0	0
$987$	6.86992	0.218672
$988$	0	0
$989$	32.4375	1.03145
$990$	0	0
$991$	3.79782	0.120642	0.0603209	−	0.998179i	$-0.480788\pi$
$991$	3.79782	0.120642	0.0603209	+	0.998179i	$0.480788\pi$
$992$	0	0
$993$	7.06203	0.224107
$994$	0	0
$995$	8.46121	0.268238
$996$	0	0
$997$	−19.2540	−0.609780	−0.304890	−	0.952388i	$-0.598620\pi$
$997$	−19.2540	−0.609780	−0.304890	+	0.952388i	$0.598620\pi$
$998$	0	0
$999$	−8.15343	−0.257963

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8112.2.a.cu.1.4		6
4.3	odd	2		4056.2.a.bi.1.4	yes	6
13.12	even	2		8112.2.a.ct.1.3		6
52.31	even	4		4056.2.c.r.337.5		12
52.47	even	4		4056.2.c.r.337.8		12
52.51	odd	2		4056.2.a.bh.1.3	✓	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4056.2.a.bh.1.3	✓	6	52.51	odd	2
4056.2.a.bi.1.4	yes	6	4.3	odd	2
4056.2.c.r.337.5		12	52.31	even	4
4056.2.c.r.337.8		12	52.47	even	4
8112.2.a.ct.1.3		6	13.12	even	2
8112.2.a.cu.1.4		6	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 8112 = 2^{4} \cdot 3 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8112.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} +1.88227 q^{5} -1.29156 q^{7} +1.00000 q^{9} -4.19367 q^{11} -1.88227 q^{15} -1.68421 q^{17} -1.19025 q^{19} +1.29156 q^{21} -3.01499 q^{23} -1.45707 q^{25} -1.00000 q^{27} +3.15000 q^{29} +1.47463 q^{31} +4.19367 q^{33} -2.43106 q^{35} +8.15343 q^{37} -10.0181 q^{41} -10.7587 q^{43} +1.88227 q^{45} +5.31909 q^{47} -5.33187 q^{49} +1.68421 q^{51} +8.29590 q^{53} -7.89361 q^{55} +1.19025 q^{57} +3.03369 q^{59} +7.84487 q^{61} -1.29156 q^{63} -4.50497 q^{67} +3.01499 q^{69} +3.85816 q^{71} +1.76494 q^{73} +1.45707 q^{75} +5.41637 q^{77} +12.6593 q^{79} +1.00000 q^{81} +15.6280 q^{83} -3.17013 q^{85} -3.15000 q^{87} +10.6679 q^{89} -1.47463 q^{93} -2.24036 q^{95} -2.38541 q^{97} -4.19367 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 6 q^{3} + q^{5} + 5 q^{7} + 6 q^{9} + 6 q^{11} - q^{15} + 9 q^{17} - 7 q^{19} - 5 q^{21} - 12 q^{23} + 9 q^{25} - 6 q^{27} + 7 q^{29} + 11 q^{31} - 6 q^{33} - 6 q^{35} + 6 q^{37} + 13 q^{41} - 15 q^{43}+ \cdots + 6 q^{99}+O(q^{100}) \) 6 * q - 6 * q^3 + q^5 + 5 * q^7 + 6 * q^9 + 6 * q^11 - q^15 + 9 * q^17 - 7 * q^19 - 5 * q^21 - 12 * q^23 + 9 * q^25 - 6 * q^27 + 7 * q^29 + 11 * q^31 - 6 * q^33 - 6 * q^35 + 6 * q^37 + 13 * q^41 - 15 * q^43 + q^45 + 9 * q^47 + 13 * q^49 - 9 * q^51 + 22 * q^53 - 3 * q^55 + 7 * q^57 + 7 * q^59 + 25 * q^61 + 5 * q^63 - 5 * q^67 + 12 * q^69 - 8 * q^71 + 15 * q^73 - 9 * q^75 + 45 * q^77 - 14 * q^79 + 6 * q^81 + 13 * q^83 - 35 * q^85 - 7 * q^87 + 33 * q^89 - 11 * q^93 - 47 * q^95 + 50 * q^97 + 6 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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