LMFDB - Embedded newform 2912.2.a.v.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2912,2,Mod(1,2912)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2912, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("2912.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2912 = 2^{5} \cdot 7 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2912.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:	$23.2524370686$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.6.153499364.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 3x^{5} - 7x^{4} + 18x^{3} + 19x^{2} - 25x - 19 $ x^6 - 3x^5 - 7x^4 + 18x^3 + 19x^2 - 25*x - 19
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$3.05415$ of defining polynomial
Character	$\chi$	$=$	2912.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.27366 q^{3} +3.76699 q^{5} +1.00000 q^{7} +2.16955 q^{9} +1.00771 q^{11} -1.00000 q^{13} -8.56486 q^{15} +3.49200 q^{17} -1.76699 q^{19} -2.27366 q^{21} -6.56618 q^{23} +9.19018 q^{25} +1.88817 q^{27} +6.41665 q^{29} +9.86509 q^{31} -2.29119 q^{33} +3.76699 q^{35} +4.41797 q^{37} +2.27366 q^{39} -10.3507 q^{41} +4.31005 q^{43} +8.17265 q^{45} -0.877031 q^{47} +1.00000 q^{49} -7.93964 q^{51} +10.2723 q^{53} +3.79603 q^{55} +4.01753 q^{57} -11.9370 q^{59} -0.644016 q^{61} +2.16955 q^{63} -3.76699 q^{65} -7.22905 q^{67} +14.9293 q^{69} -0.348273 q^{71} -2.38892 q^{73} -20.8954 q^{75} +1.00771 q^{77} -6.05607 q^{79} -10.8017 q^{81} +6.63604 q^{83} +13.1543 q^{85} -14.5893 q^{87} +9.78878 q^{89} -1.00000 q^{91} -22.4299 q^{93} -6.65621 q^{95} -14.2748 q^{97} +2.18627 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 4 q^{3} + 3 q^{5} + 6 q^{7} + 8 q^{9} + 8 q^{11} - 6 q^{13} - 2 q^{15} - 2 q^{17} + 9 q^{19} + 4 q^{21} - 11 q^{23} + 21 q^{25} + 22 q^{27} + 7 q^{29} + q^{31} + 18 q^{33} + 3 q^{35} + 16 q^{37}+ \cdots + 4 q^{99}+O(q^{100}) $ 6 * q + 4 * q^3 + 3 * q^5 + 6 * q^7 + 8 * q^9 + 8 * q^11 - 6 * q^13 - 2 * q^15 - 2 * q^17 + 9 * q^19 + 4 * q^21 - 11 * q^23 + 21 * q^25 + 22 * q^27 + 7 * q^29 + q^31 + 18 * q^33 + 3 * q^35 + 16 * q^37 - 4 * q^39 + q^43 + 29 * q^45 - 11 * q^47 + 6 * q^49 - 8 * q^51 + 23 * q^53 + 18 * q^55 + 10 * q^57 + 10 * q^59 + 10 * q^61 + 8 * q^63 - 3 * q^65 - 8 * q^67 + 6 * q^69 - 14 * q^71 + 11 * q^73 + 14 * q^75 + 8 * q^77 - 3 * q^79 + 30 * q^81 + 17 * q^83 + 6 * q^85 - 36 * q^87 - 5 * q^89 - 6 * q^91 - 16 * q^93 - 45 * q^95 + 27 * q^97 + 4 * 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
$3$	−2.27366	−1.31270	−0.656350	−	0.754456i	$-0.727901\pi$
$3$	−2.27366	−1.31270	−0.656350	+	0.754456i	$0.727901\pi$
$4$	0	0
$5$	3.76699	1.68465	0.842324	−	0.538972i	$-0.181187\pi$
$5$	3.76699	1.68465	0.842324	+	0.538972i	$0.181187\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	2.16955	0.723182
$10$	0	0
$11$	1.00771	0.303836	0.151918	−	0.988393i	$-0.451455\pi$
$11$	1.00771	0.303836	0.151918	+	0.988393i	$0.451455\pi$
$12$	0	0
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	0	0
$15$	−8.56486	−2.21144
$16$	0	0
$17$	3.49200	0.846935	0.423467	−	0.905911i	$-0.360813\pi$
$17$	3.49200	0.846935	0.423467	+	0.905911i	$0.360813\pi$
$18$	0	0
$19$	−1.76699	−0.405374	−0.202687	−	0.979244i	$-0.564967\pi$
$19$	−1.76699	−0.405374	−0.202687	+	0.979244i	$0.564967\pi$
$20$	0	0
$21$	−2.27366	−0.496154
$22$	0	0
$23$	−6.56618	−1.36914	−0.684572	−	0.728946i	$-0.740011\pi$
$23$	−6.56618	−1.36914	−0.684572	+	0.728946i	$0.740011\pi$
$24$	0	0
$25$	9.19018	1.83804
$26$	0	0
$27$	1.88817	0.363379
$28$	0	0
$29$	6.41665	1.19154	0.595771	−	0.803155i	$-0.296847\pi$
$29$	6.41665	1.19154	0.595771	+	0.803155i	$0.296847\pi$
$30$	0	0
$31$	9.86509	1.77182	0.885911	−	0.463856i	$-0.153534\pi$
$31$	9.86509	1.77182	0.885911	+	0.463856i	$0.153534\pi$
$32$	0	0
$33$	−2.29119	−0.398846
$34$	0	0
$35$	3.76699	0.636737
$36$	0	0
$37$	4.41797	0.726309	0.363155	−	0.931729i	$-0.381700\pi$
$37$	4.41797	0.726309	0.363155	+	0.931729i	$0.381700\pi$
$38$	0	0
$39$	2.27366	0.364078
$40$	0	0
$41$	−10.3507	−1.61651	−0.808254	−	0.588835i	$-0.799587\pi$
$41$	−10.3507	−1.61651	−0.808254	+	0.588835i	$0.799587\pi$
$42$	0	0
$43$	4.31005	0.657276	0.328638	−	0.944456i	$-0.393410\pi$
$43$	4.31005	0.657276	0.328638	+	0.944456i	$0.393410\pi$
$44$	0	0
$45$	8.17265	1.21831
$46$	0	0
$47$	−0.877031	−0.127928	−0.0639640	−	0.997952i	$-0.520374\pi$
$47$	−0.877031	−0.127928	−0.0639640	+	0.997952i	$0.520374\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−7.93964	−1.11177
$52$	0	0
$53$	10.2723	1.41101	0.705507	−	0.708703i	$-0.250719\pi$
$53$	10.2723	1.41101	0.705507	+	0.708703i	$0.250719\pi$
$54$	0	0
$55$	3.79603	0.511857
$56$	0	0
$57$	4.01753	0.532135
$58$	0	0
$59$	−11.9370	−1.55406	−0.777032	−	0.629461i	$-0.783276\pi$
$59$	−11.9370	−1.55406	−0.777032	+	0.629461i	$0.783276\pi$
$60$	0	0
$61$	−0.644016	−0.0824578	−0.0412289	−	0.999150i	$-0.513127\pi$
$61$	−0.644016	−0.0824578	−0.0412289	+	0.999150i	$0.513127\pi$
$62$	0	0
$63$	2.16955	0.273337
$64$	0	0
$65$	−3.76699	−0.467237
$66$	0	0
$67$	−7.22905	−0.883169	−0.441584	−	0.897220i	$-0.645583\pi$
$67$	−7.22905	−0.883169	−0.441584	+	0.897220i	$0.645583\pi$
$68$	0	0
$69$	14.9293	1.79727
$70$	0	0
$71$	−0.348273	−0.0413324	−0.0206662	−	0.999786i	$-0.506579\pi$
$71$	−0.348273	−0.0413324	−0.0206662	+	0.999786i	$0.506579\pi$
$72$	0	0
$73$	−2.38892	−0.279602	−0.139801	−	0.990180i	$-0.544646\pi$
$73$	−2.38892	−0.279602	−0.139801	+	0.990180i	$0.544646\pi$
$74$	0	0
$75$	−20.8954	−2.41279
$76$	0	0
$77$	1.00771	0.114839
$78$	0	0
$79$	−6.05607	−0.681361	−0.340681	−	0.940179i	$-0.610657\pi$
$79$	−6.05607	−0.681361	−0.340681	+	0.940179i	$0.610657\pi$
$80$	0	0
$81$	−10.8017	−1.20019
$82$	0	0
$83$	6.63604	0.728400	0.364200	−	0.931321i	$-0.381343\pi$
$83$	6.63604	0.728400	0.364200	+	0.931321i	$0.381343\pi$
$84$	0	0
$85$	13.1543	1.42679
$86$	0	0
$87$	−14.5893	−1.56414
$88$	0	0
$89$	9.78878	1.03761	0.518804	−	0.854893i	$-0.326377\pi$
$89$	9.78878	1.03761	0.518804	+	0.854893i	$0.326377\pi$
$90$	0	0
$91$	−1.00000	−0.104828
$92$	0	0
$93$	−22.4299	−2.32587
$94$	0	0
$95$	−6.65621	−0.682913
$96$	0	0
$97$	−14.2748	−1.44938	−0.724691	−	0.689074i	$-0.758018\pi$
$97$	−14.2748	−1.44938	−0.724691	+	0.689074i	$0.758018\pi$
$98$	0	0
$99$	2.18627	0.219729
$100$	0	0
$101$	7.41073	0.737396	0.368698	−	0.929549i	$-0.379804\pi$
$101$	7.41073	0.737396	0.368698	+	0.929549i	$0.379804\pi$
$102$	0	0
$103$	15.3406	1.51155	0.755777	−	0.654829i	$-0.227259\pi$
$103$	15.3406	1.51155	0.755777	+	0.654829i	$0.227259\pi$
$104$	0	0
$105$	−8.56486	−0.835845
$106$	0	0
$107$	11.6510	1.12635	0.563174	−	0.826339i	$-0.309580\pi$
$107$	11.6510	1.12635	0.563174	+	0.826339i	$0.309580\pi$
$108$	0	0
$109$	−7.11483	−0.681477	−0.340738	−	0.940158i	$-0.610677\pi$
$109$	−7.11483	−0.681477	−0.340738	+	0.940158i	$0.610677\pi$
$110$	0	0
$111$	−10.0450	−0.953427
$112$	0	0
$113$	16.1712	1.52126	0.760630	−	0.649185i	$-0.224890\pi$
$113$	16.1712	1.52126	0.760630	+	0.649185i	$0.224890\pi$
$114$	0	0
$115$	−24.7347	−2.30652
$116$	0	0
$117$	−2.16955	−0.200575
$118$	0	0
$119$	3.49200	0.320111
$120$	0	0
$121$	−9.98452	−0.907684
$122$	0	0
$123$	23.5340	2.12199
$124$	0	0
$125$	15.7844	1.41180
$126$	0	0
$127$	−2.72216	−0.241553	−0.120776	−	0.992680i	$-0.538538\pi$
$127$	−2.72216	−0.241553	−0.120776	+	0.992680i	$0.538538\pi$
$128$	0	0
$129$	−9.79960	−0.862806
$130$	0	0
$131$	13.9152	1.21578	0.607888	−	0.794023i	$-0.292017\pi$
$131$	13.9152	1.21578	0.607888	+	0.794023i	$0.292017\pi$
$132$	0	0
$133$	−1.76699	−0.153217
$134$	0	0
$135$	7.11272	0.612165
$136$	0	0
$137$	−14.7814	−1.26286	−0.631432	−	0.775431i	$-0.717533\pi$
$137$	−14.7814	−1.26286	−0.631432	+	0.775431i	$0.717533\pi$
$138$	0	0
$139$	20.2431	1.71700	0.858500	−	0.512813i	$-0.171397\pi$
$139$	20.2431	1.71700	0.858500	+	0.512813i	$0.171397\pi$
$140$	0	0
$141$	1.99407	0.167931
$142$	0	0
$143$	−1.00771	−0.0842690
$144$	0	0
$145$	24.1714	2.00733
$146$	0	0
$147$	−2.27366	−0.187529
$148$	0	0
$149$	16.2484	1.33112	0.665559	−	0.746345i	$-0.268193\pi$
$149$	16.2484	1.33112	0.665559	+	0.746345i	$0.268193\pi$
$150$	0	0
$151$	0.268436	0.0218450	0.0109225	−	0.999940i	$-0.496523\pi$
$151$	0.268436	0.0218450	0.0109225	+	0.999940i	$0.496523\pi$
$152$	0	0
$153$	7.57606	0.612488
$154$	0	0
$155$	37.1616	2.98489
$156$	0	0
$157$	−10.4787	−0.836289	−0.418145	−	0.908380i	$-0.637320\pi$
$157$	−10.4787	−0.836289	−0.418145	+	0.908380i	$0.637320\pi$
$158$	0	0
$159$	−23.3559	−1.85224
$160$	0	0
$161$	−6.56618	−0.517487
$162$	0	0
$163$	9.56068	0.748850	0.374425	−	0.927257i	$-0.377840\pi$
$163$	9.56068	0.748850	0.374425	+	0.927257i	$0.377840\pi$
$164$	0	0
$165$	−8.63090	−0.671915
$166$	0	0
$167$	5.51608	0.426847	0.213424	−	0.976960i	$-0.431539\pi$
$167$	5.51608	0.426847	0.213424	+	0.976960i	$0.431539\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	−3.83356	−0.293159
$172$	0	0
$173$	19.8347	1.50800	0.754001	−	0.656874i	$-0.228122\pi$
$173$	19.8347	1.50800	0.754001	+	0.656874i	$0.228122\pi$
$174$	0	0
$175$	9.19018	0.694712
$176$	0	0
$177$	27.1407	2.04002
$178$	0	0
$179$	1.93589	0.144695	0.0723477	−	0.997379i	$-0.476951\pi$
$179$	1.93589	0.144695	0.0723477	+	0.997379i	$0.476951\pi$
$180$	0	0
$181$	7.69934	0.572288	0.286144	−	0.958187i	$-0.407626\pi$
$181$	7.69934	0.572288	0.286144	+	0.958187i	$0.407626\pi$
$182$	0	0
$183$	1.46428	0.108242
$184$	0	0
$185$	16.6424	1.22358
$186$	0	0
$187$	3.51893	0.257329
$188$	0	0
$189$	1.88817	0.137344
$190$	0	0
$191$	−19.3098	−1.39721	−0.698605	−	0.715507i	$-0.746196\pi$
$191$	−19.3098	−1.39721	−0.698605	+	0.715507i	$0.746196\pi$
$192$	0	0
$193$	−3.42540	−0.246565	−0.123283	−	0.992372i	$-0.539342\pi$
$193$	−3.42540	−0.246565	−0.123283	+	0.992372i	$0.539342\pi$
$194$	0	0
$195$	8.56486	0.613342
$196$	0	0
$197$	−20.6521	−1.47140	−0.735700	−	0.677307i	$-0.763147\pi$
$197$	−20.6521	−1.47140	−0.735700	+	0.677307i	$0.763147\pi$
$198$	0	0
$199$	2.23478	0.158420	0.0792098	−	0.996858i	$-0.474760\pi$
$199$	2.23478	0.158420	0.0792098	+	0.996858i	$0.474760\pi$
$200$	0	0
$201$	16.4364	1.15934
$202$	0	0
$203$	6.41665	0.450360
$204$	0	0
$205$	−38.9909	−2.72324
$206$	0	0
$207$	−14.2456	−0.990140
$208$	0	0
$209$	−1.78061	−0.123167
$210$	0	0
$211$	7.62695	0.525061	0.262530	−	0.964924i	$-0.415443\pi$
$211$	7.62695	0.525061	0.262530	+	0.964924i	$0.415443\pi$
$212$	0	0
$213$	0.791856	0.0542571
$214$	0	0
$215$	16.2359	1.10728
$216$	0	0
$217$	9.86509	0.669686
$218$	0	0
$219$	5.43161	0.367034
$220$	0	0
$221$	−3.49200	−0.234897
$222$	0	0
$223$	23.1439	1.54983	0.774915	−	0.632065i	$-0.217793\pi$
$223$	23.1439	1.54983	0.774915	+	0.632065i	$0.217793\pi$
$224$	0	0
$225$	19.9385	1.32924
$226$	0	0
$227$	−0.775768	−0.0514895	−0.0257448	−	0.999669i	$-0.508196\pi$
$227$	−0.775768	−0.0514895	−0.0257448	+	0.999669i	$0.508196\pi$
$228$	0	0
$229$	18.2298	1.20466	0.602330	−	0.798247i	$-0.294239\pi$
$229$	18.2298	1.20466	0.602330	+	0.798247i	$0.294239\pi$
$230$	0	0
$231$	−2.29119	−0.150750
$232$	0	0
$233$	−11.6407	−0.762610	−0.381305	−	0.924449i	$-0.624525\pi$
$233$	−11.6407	−0.762610	−0.381305	+	0.924449i	$0.624525\pi$
$234$	0	0
$235$	−3.30376	−0.215514
$236$	0	0
$237$	13.7695	0.894423
$238$	0	0
$239$	1.55361	0.100495	0.0502474	−	0.998737i	$-0.483999\pi$
$239$	1.55361	0.100495	0.0502474	+	0.998737i	$0.483999\pi$
$240$	0	0
$241$	−25.7010	−1.65554	−0.827772	−	0.561064i	$-0.810392\pi$
$241$	−25.7010	−1.65554	−0.827772	+	0.561064i	$0.810392\pi$
$242$	0	0
$243$	18.8949	1.21211
$244$	0	0
$245$	3.76699	0.240664
$246$	0	0
$247$	1.76699	0.112431
$248$	0	0
$249$	−15.0881	−0.956171
$250$	0	0
$251$	−2.57009	−0.162222	−0.0811112	−	0.996705i	$-0.525847\pi$
$251$	−2.57009	−0.162222	−0.0811112	+	0.996705i	$0.525847\pi$
$252$	0	0
$253$	−6.61681	−0.415995
$254$	0	0
$255$	−29.9085	−1.87294
$256$	0	0
$257$	−4.21939	−0.263198	−0.131599	−	0.991303i	$-0.542011\pi$
$257$	−4.21939	−0.263198	−0.131599	+	0.991303i	$0.542011\pi$
$258$	0	0
$259$	4.41797	0.274519
$260$	0	0
$261$	13.9212	0.861702
$262$	0	0
$263$	12.3359	0.760664	0.380332	−	0.924850i	$-0.375810\pi$
$263$	12.3359	0.760664	0.380332	+	0.924850i	$0.375810\pi$
$264$	0	0
$265$	38.6958	2.37706
$266$	0	0
$267$	−22.2564	−1.36207
$268$	0	0
$269$	−14.2305	−0.867652	−0.433826	−	0.900997i	$-0.642837\pi$
$269$	−14.2305	−0.867652	−0.433826	+	0.900997i	$0.642837\pi$
$270$	0	0
$271$	−22.8284	−1.38672	−0.693362	−	0.720589i	$-0.743871\pi$
$271$	−22.8284	−1.38672	−0.693362	+	0.720589i	$0.743871\pi$
$272$	0	0
$273$	2.27366	0.137608
$274$	0	0
$275$	9.26104	0.558462
$276$	0	0
$277$	27.7305	1.66617	0.833083	−	0.553148i	$-0.186574\pi$
$277$	27.7305	1.66617	0.833083	+	0.553148i	$0.186574\pi$
$278$	0	0
$279$	21.4028	1.28135
$280$	0	0
$281$	24.0603	1.43531	0.717657	−	0.696396i	$-0.245214\pi$
$281$	24.0603	1.43531	0.717657	+	0.696396i	$0.245214\pi$
$282$	0	0
$283$	17.2902	1.02780	0.513899	−	0.857851i	$-0.328201\pi$
$283$	17.2902	1.02780	0.513899	+	0.857851i	$0.328201\pi$
$284$	0	0
$285$	15.1340	0.896460
$286$	0	0
$287$	−10.3507	−0.610982
$288$	0	0
$289$	−4.80593	−0.282702
$290$	0	0
$291$	32.4560	1.90261
$292$	0	0
$293$	23.6689	1.38275	0.691376	−	0.722495i	$-0.257005\pi$
$293$	23.6689	1.38275	0.691376	+	0.722495i	$0.257005\pi$
$294$	0	0
$295$	−44.9665	−2.61805
$296$	0	0
$297$	1.90273	0.110408
$298$	0	0
$299$	6.56618	0.379732
$300$	0	0
$301$	4.31005	0.248427
$302$	0	0
$303$	−16.8495	−0.967979
$304$	0	0
$305$	−2.42600	−0.138912
$306$	0	0
$307$	28.1383	1.60594	0.802969	−	0.596021i	$-0.203252\pi$
$307$	28.1383	1.60594	0.802969	+	0.596021i	$0.203252\pi$
$308$	0	0
$309$	−34.8793	−1.98422
$310$	0	0
$311$	−6.69873	−0.379850	−0.189925	−	0.981799i	$-0.560825\pi$
$311$	−6.69873	−0.379850	−0.189925	+	0.981799i	$0.560825\pi$
$312$	0	0
$313$	7.00220	0.395788	0.197894	−	0.980223i	$-0.436590\pi$
$313$	7.00220	0.395788	0.197894	+	0.980223i	$0.436590\pi$
$314$	0	0
$315$	8.17265	0.460477
$316$	0	0
$317$	−15.7125	−0.882504	−0.441252	−	0.897383i	$-0.645466\pi$
$317$	−15.7125	−0.882504	−0.441252	+	0.897383i	$0.645466\pi$
$318$	0	0
$319$	6.46612	0.362033
$320$	0	0
$321$	−26.4905	−1.47856
$322$	0	0
$323$	−6.17032	−0.343326
$324$	0	0
$325$	−9.19018	−0.509780
$326$	0	0
$327$	16.1767	0.894575
$328$	0	0
$329$	−0.877031	−0.0483523
$330$	0	0
$331$	−7.19399	−0.395417	−0.197709	−	0.980261i	$-0.563350\pi$
$331$	−7.19399	−0.395417	−0.197709	+	0.980261i	$0.563350\pi$
$332$	0	0
$333$	9.58499	0.525254
$334$	0	0
$335$	−27.2317	−1.48783
$336$	0	0
$337$	2.72206	0.148280	0.0741399	−	0.997248i	$-0.476379\pi$
$337$	2.72206	0.148280	0.0741399	+	0.997248i	$0.476379\pi$
$338$	0	0
$339$	−36.7679	−1.99696
$340$	0	0
$341$	9.94115	0.538343
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	56.2384	3.02777
$346$	0	0
$347$	−9.85711	−0.529157	−0.264579	−	0.964364i	$-0.585233\pi$
$347$	−9.85711	−0.529157	−0.264579	+	0.964364i	$0.585233\pi$
$348$	0	0
$349$	6.22724	0.333336	0.166668	−	0.986013i	$-0.446699\pi$
$349$	6.22724	0.333336	0.166668	+	0.986013i	$0.446699\pi$
$350$	0	0
$351$	−1.88817	−0.100783
$352$	0	0
$353$	10.9175	0.581080	0.290540	−	0.956863i	$-0.406165\pi$
$353$	10.9175	0.581080	0.290540	+	0.956863i	$0.406165\pi$
$354$	0	0
$355$	−1.31194	−0.0696305
$356$	0	0
$357$	−7.93964	−0.420210
$358$	0	0
$359$	−1.38097	−0.0728850	−0.0364425	−	0.999336i	$-0.511603\pi$
$359$	−1.38097	−0.0728850	−0.0364425	+	0.999336i	$0.511603\pi$
$360$	0	0
$361$	−15.8778	−0.835672
$362$	0	0
$363$	22.7014	1.19152
$364$	0	0
$365$	−8.99904	−0.471031
$366$	0	0
$367$	−0.328800	−0.0171632	−0.00858161	−	0.999963i	$-0.502732\pi$
$367$	−0.328800	−0.0171632	−0.00858161	+	0.999963i	$0.502732\pi$
$368$	0	0
$369$	−22.4563	−1.16903
$370$	0	0
$371$	10.2723	0.533313
$372$	0	0
$373$	−5.18848	−0.268649	−0.134325	−	0.990937i	$-0.542886\pi$
$373$	−5.18848	−0.268649	−0.134325	+	0.990937i	$0.542886\pi$
$374$	0	0
$375$	−35.8883	−1.85326
$376$	0	0
$377$	−6.41665	−0.330474
$378$	0	0
$379$	9.27332	0.476339	0.238169	−	0.971224i	$-0.423453\pi$
$379$	9.27332	0.476339	0.238169	+	0.971224i	$0.423453\pi$
$380$	0	0
$381$	6.18928	0.317086
$382$	0	0
$383$	7.15628	0.365669	0.182834	−	0.983144i	$-0.441473\pi$
$383$	7.15628	0.365669	0.182834	+	0.983144i	$0.441473\pi$
$384$	0	0
$385$	3.79603	0.193464
$386$	0	0
$387$	9.35085	0.475330
$388$	0	0
$389$	−24.6952	−1.25210	−0.626048	−	0.779784i	$-0.715329\pi$
$389$	−24.6952	−1.25210	−0.626048	+	0.779784i	$0.715329\pi$
$390$	0	0
$391$	−22.9291	−1.15957
$392$	0	0
$393$	−31.6385	−1.59595
$394$	0	0
$395$	−22.8131	−1.14785
$396$	0	0
$397$	9.29544	0.466525	0.233262	−	0.972414i	$-0.425060\pi$
$397$	9.29544	0.466525	0.233262	+	0.972414i	$0.425060\pi$
$398$	0	0
$399$	4.01753	0.201128
$400$	0	0
$401$	9.46459	0.472639	0.236320	−	0.971675i	$-0.424059\pi$
$401$	9.46459	0.472639	0.236320	+	0.971675i	$0.424059\pi$
$402$	0	0
$403$	−9.86509	−0.491415
$404$	0	0
$405$	−40.6899	−2.02190
$406$	0	0
$407$	4.45203	0.220679
$408$	0	0
$409$	−4.66688	−0.230762	−0.115381	−	0.993321i	$-0.536809\pi$
$409$	−4.66688	−0.230762	−0.115381	+	0.993321i	$0.536809\pi$
$410$	0	0
$411$	33.6080	1.65776
$412$	0	0
$413$	−11.9370	−0.587381
$414$	0	0
$415$	24.9979	1.22710
$416$	0	0
$417$	−46.0261	−2.25391
$418$	0	0
$419$	12.8304	0.626805	0.313403	−	0.949620i	$-0.398531\pi$
$419$	12.8304	0.626805	0.313403	+	0.949620i	$0.398531\pi$
$420$	0	0
$421$	−19.8843	−0.969104	−0.484552	−	0.874763i	$-0.661017\pi$
$421$	−19.8843	−0.969104	−0.484552	+	0.874763i	$0.661017\pi$
$422$	0	0
$423$	−1.90276	−0.0925153
$424$	0	0
$425$	32.0921	1.55670
$426$	0	0
$427$	−0.644016	−0.0311661
$428$	0	0
$429$	2.29119	0.110620
$430$	0	0
$431$	−21.2340	−1.02281	−0.511404	−	0.859340i	$-0.670874\pi$
$431$	−21.2340	−1.02281	−0.511404	+	0.859340i	$0.670874\pi$
$432$	0	0
$433$	38.9601	1.87230	0.936152	−	0.351596i	$-0.114361\pi$
$433$	38.9601	1.87230	0.936152	+	0.351596i	$0.114361\pi$
$434$	0	0
$435$	−54.9577	−2.63502
$436$	0	0
$437$	11.6023	0.555015
$438$	0	0
$439$	36.3152	1.73323	0.866615	−	0.498977i	$-0.166291\pi$
$439$	36.3152	1.73323	0.866615	+	0.498977i	$0.166291\pi$
$440$	0	0
$441$	2.16955	0.103312
$442$	0	0
$443$	−21.0388	−0.999584	−0.499792	−	0.866145i	$-0.666590\pi$
$443$	−21.0388	−0.999584	−0.499792	+	0.866145i	$0.666590\pi$
$444$	0	0
$445$	36.8742	1.74801
$446$	0	0
$447$	−36.9433	−1.74736
$448$	0	0
$449$	21.4527	1.01242	0.506209	−	0.862411i	$-0.331047\pi$
$449$	21.4527	1.01242	0.506209	+	0.862411i	$0.331047\pi$
$450$	0	0
$451$	−10.4305	−0.491153
$452$	0	0
$453$	−0.610333	−0.0286759
$454$	0	0
$455$	−3.76699	−0.176599
$456$	0	0
$457$	4.51702	0.211297	0.105649	−	0.994404i	$-0.466308\pi$
$457$	4.51702	0.211297	0.105649	+	0.994404i	$0.466308\pi$
$458$	0	0
$459$	6.59350	0.307758
$460$	0	0
$461$	−36.3599	−1.69345	−0.846726	−	0.532029i	$-0.821430\pi$
$461$	−36.3599	−1.69345	−0.846726	+	0.532029i	$0.821430\pi$
$462$	0	0
$463$	26.5410	1.23346	0.616732	−	0.787173i	$-0.288456\pi$
$463$	26.5410	1.23346	0.616732	+	0.787173i	$0.288456\pi$
$464$	0	0
$465$	−84.4931	−3.91827
$466$	0	0
$467$	35.2094	1.62930	0.814648	−	0.579955i	$-0.196930\pi$
$467$	35.2094	1.62930	0.814648	+	0.579955i	$0.196930\pi$
$468$	0	0
$469$	−7.22905	−0.333806
$470$	0	0
$471$	23.8250	1.09780
$472$	0	0
$473$	4.34328	0.199704
$474$	0	0
$475$	−16.2389	−0.745093
$476$	0	0
$477$	22.2863	1.02042
$478$	0	0
$479$	−3.58141	−0.163639	−0.0818194	−	0.996647i	$-0.526073\pi$
$479$	−3.58141	−0.163639	−0.0818194	+	0.996647i	$0.526073\pi$
$480$	0	0
$481$	−4.41797	−0.201442
$482$	0	0
$483$	14.9293	0.679306
$484$	0	0
$485$	−53.7728	−2.44170
$486$	0	0
$487$	2.44618	0.110847	0.0554235	−	0.998463i	$-0.482349\pi$
$487$	2.44618	0.110847	0.0554235	+	0.998463i	$0.482349\pi$
$488$	0	0
$489$	−21.7378	−0.983016
$490$	0	0
$491$	−4.49882	−0.203029	−0.101514	−	0.994834i	$-0.532369\pi$
$491$	−4.49882	−0.203029	−0.101514	+	0.994834i	$0.532369\pi$
$492$	0	0
$493$	22.4069	1.00916
$494$	0	0
$495$	8.23567	0.370166
$496$	0	0
$497$	−0.348273	−0.0156222
$498$	0	0
$499$	−16.8909	−0.756141	−0.378071	−	0.925777i	$-0.623412\pi$
$499$	−16.8909	−0.756141	−0.378071	+	0.925777i	$0.623412\pi$
$500$	0	0
$501$	−12.5417	−0.560322
$502$	0	0
$503$	−28.1487	−1.25509	−0.627543	−	0.778582i	$-0.715940\pi$
$503$	−28.1487	−1.25509	−0.627543	+	0.778582i	$0.715940\pi$
$504$	0	0
$505$	27.9161	1.24225
$506$	0	0
$507$	−2.27366	−0.100977
$508$	0	0
$509$	−26.8484	−1.19003	−0.595017	−	0.803713i	$-0.702855\pi$
$509$	−26.8484	−1.19003	−0.595017	+	0.803713i	$0.702855\pi$
$510$	0	0
$511$	−2.38892	−0.105680
$512$	0	0
$513$	−3.33637	−0.147304
$514$	0	0
$515$	57.7878	2.54643
$516$	0	0
$517$	−0.883793	−0.0388692
$518$	0	0
$519$	−45.0973	−1.97955
$520$	0	0
$521$	9.61803	0.421373	0.210687	−	0.977554i	$-0.432430\pi$
$521$	9.61803	0.421373	0.210687	+	0.977554i	$0.432430\pi$
$522$	0	0
$523$	−32.7593	−1.43246	−0.716231	−	0.697863i	$-0.754134\pi$
$523$	−32.7593	−1.43246	−0.716231	+	0.697863i	$0.754134\pi$
$524$	0	0
$525$	−20.8954	−0.911949
$526$	0	0
$527$	34.4489	1.50062
$528$	0	0
$529$	20.1147	0.874553
$530$	0	0
$531$	−25.8979	−1.12387
$532$	0	0
$533$	10.3507	0.448338
$534$	0	0
$535$	43.8893	1.89750
$536$	0	0
$537$	−4.40157	−0.189942
$538$	0	0
$539$	1.00771	0.0434052
$540$	0	0
$541$	−35.8058	−1.53941	−0.769705	−	0.638400i	$-0.779597\pi$
$541$	−35.8058	−1.53941	−0.769705	+	0.638400i	$0.779597\pi$
$542$	0	0
$543$	−17.5057	−0.751242
$544$	0	0
$545$	−26.8015	−1.14805
$546$	0	0
$547$	−33.5087	−1.43273	−0.716363	−	0.697727i	$-0.754195\pi$
$547$	−33.5087	−1.43273	−0.716363	+	0.697727i	$0.754195\pi$
$548$	0	0
$549$	−1.39722	−0.0596320
$550$	0	0
$551$	−11.3381	−0.483020
$552$	0	0
$553$	−6.05607	−0.257530
$554$	0	0
$555$	−37.8393	−1.60619
$556$	0	0
$557$	−19.5302	−0.827520	−0.413760	−	0.910386i	$-0.635785\pi$
$557$	−19.5302	−0.827520	−0.413760	+	0.910386i	$0.635785\pi$
$558$	0	0
$559$	−4.31005	−0.182296
$560$	0	0
$561$	−8.00086	−0.337796
$562$	0	0
$563$	−33.9566	−1.43110	−0.715549	−	0.698562i	$-0.753824\pi$
$563$	−33.9566	−1.43110	−0.715549	+	0.698562i	$0.753824\pi$
$564$	0	0
$565$	60.9168	2.56279
$566$	0	0
$567$	−10.8017	−0.453629
$568$	0	0
$569$	−25.2262	−1.05754	−0.528769	−	0.848766i	$-0.677346\pi$
$569$	−25.2262	−1.05754	−0.528769	+	0.848766i	$0.677346\pi$
$570$	0	0
$571$	−42.1175	−1.76256	−0.881281	−	0.472594i	$-0.843318\pi$
$571$	−42.1175	−1.76256	−0.881281	+	0.472594i	$0.843318\pi$
$572$	0	0
$573$	43.9041	1.83412
$574$	0	0
$575$	−60.3444	−2.51653
$576$	0	0
$577$	−10.5544	−0.439385	−0.219693	−	0.975569i	$-0.570505\pi$
$577$	−10.5544	−0.439385	−0.219693	+	0.975569i	$0.570505\pi$
$578$	0	0
$579$	7.78820	0.323667
$580$	0	0
$581$	6.63604	0.275309
$582$	0	0
$583$	10.3515	0.428717
$584$	0	0
$585$	−8.17265	−0.337898
$586$	0	0
$587$	−35.1600	−1.45121	−0.725604	−	0.688112i	$-0.758440\pi$
$587$	−35.1600	−1.45121	−0.725604	+	0.688112i	$0.758440\pi$
$588$	0	0
$589$	−17.4315	−0.718251
$590$	0	0
$591$	46.9559	1.93151
$592$	0	0
$593$	−24.7392	−1.01592	−0.507958	−	0.861382i	$-0.669599\pi$
$593$	−24.7392	−1.01592	−0.507958	+	0.861382i	$0.669599\pi$
$594$	0	0
$595$	13.1543	0.539274
$596$	0	0
$597$	−5.08115	−0.207957
$598$	0	0
$599$	−14.2816	−0.583530	−0.291765	−	0.956490i	$-0.594243\pi$
$599$	−14.2816	−0.583530	−0.291765	+	0.956490i	$0.594243\pi$
$600$	0	0
$601$	−10.0470	−0.409825	−0.204912	−	0.978780i	$-0.565691\pi$
$601$	−10.0470	−0.409825	−0.204912	+	0.978780i	$0.565691\pi$
$602$	0	0
$603$	−15.6838	−0.638692
$604$	0	0
$605$	−37.6115	−1.52913
$606$	0	0
$607$	−21.5221	−0.873554	−0.436777	−	0.899570i	$-0.643880\pi$
$607$	−21.5221	−0.873554	−0.436777	+	0.899570i	$0.643880\pi$
$608$	0	0
$609$	−14.5893	−0.591188
$610$	0	0
$611$	0.877031	0.0354809
$612$	0	0
$613$	−40.7770	−1.64697	−0.823483	−	0.567340i	$-0.807972\pi$
$613$	−40.7770	−1.64697	−0.823483	+	0.567340i	$0.807972\pi$
$614$	0	0
$615$	88.6523	3.57480
$616$	0	0
$617$	44.6706	1.79837	0.899186	−	0.437566i	$-0.144159\pi$
$617$	44.6706	1.79837	0.899186	+	0.437566i	$0.144159\pi$
$618$	0	0
$619$	0.537823	0.0216169	0.0108085	−	0.999942i	$-0.496559\pi$
$619$	0.537823	0.0216169	0.0108085	+	0.999942i	$0.496559\pi$
$620$	0	0
$621$	−12.3981	−0.497518
$622$	0	0
$623$	9.78878	0.392179
$624$	0	0
$625$	13.5085	0.540341
$626$	0	0
$627$	4.04851	0.161682
$628$	0	0
$629$	15.4276	0.615137
$630$	0	0
$631$	−10.4583	−0.416339	−0.208169	−	0.978093i	$-0.566750\pi$
$631$	−10.4583	−0.416339	−0.208169	+	0.978093i	$0.566750\pi$
$632$	0	0
$633$	−17.3411	−0.689247
$634$	0	0
$635$	−10.2543	−0.406931
$636$	0	0
$637$	−1.00000	−0.0396214
$638$	0	0
$639$	−0.755595	−0.0298909
$640$	0	0
$641$	31.0787	1.22753	0.613767	−	0.789487i	$-0.289653\pi$
$641$	31.0787	1.22753	0.613767	+	0.789487i	$0.289653\pi$
$642$	0	0
$643$	20.7212	0.817165	0.408582	−	0.912721i	$-0.366023\pi$
$643$	20.7212	0.817165	0.408582	+	0.912721i	$0.366023\pi$
$644$	0	0
$645$	−36.9149	−1.45352
$646$	0	0
$647$	−39.2883	−1.54458	−0.772290	−	0.635270i	$-0.780889\pi$
$647$	−39.2883	−1.54458	−0.772290	+	0.635270i	$0.780889\pi$
$648$	0	0
$649$	−12.0290	−0.472181
$650$	0	0
$651$	−22.4299	−0.879097
$652$	0	0
$653$	−48.3671	−1.89275	−0.946376	−	0.323066i	$-0.895286\pi$
$653$	−48.3671	−1.89275	−0.946376	+	0.323066i	$0.895286\pi$
$654$	0	0
$655$	52.4183	2.04815
$656$	0	0
$657$	−5.18288	−0.202203
$658$	0	0
$659$	3.93287	0.153203	0.0766015	−	0.997062i	$-0.475593\pi$
$659$	3.93287	0.153203	0.0766015	+	0.997062i	$0.475593\pi$
$660$	0	0
$661$	1.22198	0.0475296	0.0237648	−	0.999718i	$-0.492435\pi$
$661$	1.22198	0.0475296	0.0237648	+	0.999718i	$0.492435\pi$
$662$	0	0
$663$	7.93964	0.308350
$664$	0	0
$665$	−6.65621	−0.258117
$666$	0	0
$667$	−42.1329	−1.63139
$668$	0	0
$669$	−52.6215	−2.03446
$670$	0	0
$671$	−0.648982	−0.0250537
$672$	0	0
$673$	−7.41799	−0.285942	−0.142971	−	0.989727i	$-0.545666\pi$
$673$	−7.41799	−0.285942	−0.142971	+	0.989727i	$0.545666\pi$
$674$	0	0
$675$	17.3526	0.667904
$676$	0	0
$677$	−34.2053	−1.31461	−0.657307	−	0.753623i	$-0.728305\pi$
$677$	−34.2053	−1.31461	−0.657307	+	0.753623i	$0.728305\pi$
$678$	0	0
$679$	−14.2748	−0.547815
$680$	0	0
$681$	1.76384	0.0675903
$682$	0	0
$683$	26.2853	1.00578	0.502889	−	0.864351i	$-0.332271\pi$
$683$	26.2853	1.00578	0.502889	+	0.864351i	$0.332271\pi$
$684$	0	0
$685$	−55.6815	−2.12748
$686$	0	0
$687$	−41.4485	−1.58136
$688$	0	0
$689$	−10.2723	−0.391345
$690$	0	0
$691$	−33.3052	−1.26699	−0.633496	−	0.773746i	$-0.718380\pi$
$691$	−33.3052	−1.26699	−0.633496	+	0.773746i	$0.718380\pi$
$692$	0	0
$693$	2.18627	0.0830497
$694$	0	0
$695$	76.2556	2.89254
$696$	0	0
$697$	−36.1446	−1.36908
$698$	0	0
$699$	26.4671	1.00108
$700$	0	0
$701$	10.7139	0.404659	0.202329	−	0.979318i	$-0.435149\pi$
$701$	10.7139	0.404659	0.202329	+	0.979318i	$0.435149\pi$
$702$	0	0
$703$	−7.80649	−0.294427
$704$	0	0
$705$	7.51164	0.282905
$706$	0	0
$707$	7.41073	0.278709
$708$	0	0
$709$	11.0894	0.416470	0.208235	−	0.978079i	$-0.433228\pi$
$709$	11.0894	0.416470	0.208235	+	0.978079i	$0.433228\pi$
$710$	0	0
$711$	−13.1389	−0.492748
$712$	0	0
$713$	−64.7759	−2.42588
$714$	0	0
$715$	−3.79603	−0.141964
$716$	0	0
$717$	−3.53239	−0.131920
$718$	0	0
$719$	−14.7869	−0.551458	−0.275729	−	0.961235i	$-0.588919\pi$
$719$	−14.7869	−0.551458	−0.275729	+	0.961235i	$0.588919\pi$
$720$	0	0
$721$	15.3406	0.571314
$722$	0	0
$723$	58.4353	2.17323
$724$	0	0
$725$	58.9702	2.19010
$726$	0	0
$727$	26.2320	0.972891	0.486446	−	0.873711i	$-0.338293\pi$
$727$	26.2320	0.972891	0.486446	+	0.873711i	$0.338293\pi$
$728$	0	0
$729$	−10.5556	−0.390948
$730$	0	0
$731$	15.0507	0.556670
$732$	0	0
$733$	−49.7996	−1.83939	−0.919695	−	0.392634i	$-0.871564\pi$
$733$	−49.7996	−1.83939	−0.919695	+	0.392634i	$0.871564\pi$
$734$	0	0
$735$	−8.56486	−0.315920
$736$	0	0
$737$	−7.28479	−0.268339
$738$	0	0
$739$	19.2289	0.707345	0.353673	−	0.935369i	$-0.384933\pi$
$739$	19.2289	0.707345	0.353673	+	0.935369i	$0.384933\pi$
$740$	0	0
$741$	−4.01753	−0.147588
$742$	0	0
$743$	1.43628	0.0526920	0.0263460	−	0.999653i	$-0.491613\pi$
$743$	1.43628	0.0526920	0.0263460	+	0.999653i	$0.491613\pi$
$744$	0	0
$745$	61.2073	2.24246
$746$	0	0
$747$	14.3972	0.526766
$748$	0	0
$749$	11.6510	0.425719
$750$	0	0
$751$	22.7155	0.828902	0.414451	−	0.910072i	$-0.363974\pi$
$751$	22.7155	0.828902	0.414451	+	0.910072i	$0.363974\pi$
$752$	0	0
$753$	5.84351	0.212949
$754$	0	0
$755$	1.01119	0.0368011
$756$	0	0
$757$	44.2660	1.60888	0.804438	−	0.594037i	$-0.202467\pi$
$757$	44.2660	1.60888	0.804438	+	0.594037i	$0.202467\pi$
$758$	0	0
$759$	15.0444	0.546077
$760$	0	0
$761$	30.8439	1.11809	0.559046	−	0.829137i	$-0.311168\pi$
$761$	30.8439	1.11809	0.559046	+	0.829137i	$0.311168\pi$
$762$	0	0
$763$	−7.11483	−0.257574
$764$	0	0
$765$	28.5389	1.03183
$766$	0	0
$767$	11.9370	0.431020
$768$	0	0
$769$	21.0441	0.758871	0.379435	−	0.925218i	$-0.376118\pi$
$769$	21.0441	0.758871	0.379435	+	0.925218i	$0.376118\pi$
$770$	0	0
$771$	9.59347	0.345500
$772$	0	0
$773$	38.2194	1.37466	0.687328	−	0.726347i	$-0.258783\pi$
$773$	38.2194	1.37466	0.687328	+	0.726347i	$0.258783\pi$
$774$	0	0
$775$	90.6619	3.25667
$776$	0	0
$777$	−10.0450	−0.360361
$778$	0	0
$779$	18.2895	0.655291
$780$	0	0
$781$	−0.350959	−0.0125583
$782$	0	0
$783$	12.1157	0.432981
$784$	0	0
$785$	−39.4730	−1.40885
$786$	0	0
$787$	45.2105	1.61158	0.805790	−	0.592201i	$-0.201741\pi$
$787$	45.2105	1.61158	0.805790	+	0.592201i	$0.201741\pi$
$788$	0	0
$789$	−28.0477	−0.998524
$790$	0	0
$791$	16.1712	0.574983
$792$	0	0
$793$	0.644016	0.0228697
$794$	0	0
$795$	−87.9812	−3.12037
$796$	0	0
$797$	14.0273	0.496872	0.248436	−	0.968648i	$-0.420083\pi$
$797$	14.0273	0.496872	0.248436	+	0.968648i	$0.420083\pi$
$798$	0	0
$799$	−3.06259	−0.108347
$800$	0	0
$801$	21.2372	0.750380
$802$	0	0
$803$	−2.40734	−0.0849533
$804$	0	0
$805$	−24.7347	−0.871784
$806$	0	0
$807$	32.3555	1.13897
$808$	0	0
$809$	−17.5529	−0.617128	−0.308564	−	0.951204i	$-0.599848\pi$
$809$	−17.5529	−0.617128	−0.308564	+	0.951204i	$0.599848\pi$
$810$	0	0
$811$	13.4770	0.473243	0.236621	−	0.971602i	$-0.423960\pi$
$811$	13.4770	0.473243	0.236621	+	0.971602i	$0.423960\pi$
$812$	0	0
$813$	51.9040	1.82035
$814$	0	0
$815$	36.0150	1.26155
$816$	0	0
$817$	−7.61579	−0.266443
$818$	0	0
$819$	−2.16955	−0.0758101
$820$	0	0
$821$	−34.3848	−1.20004	−0.600019	−	0.799986i	$-0.704840\pi$
$821$	−34.3848	−1.20004	−0.600019	+	0.799986i	$0.704840\pi$
$822$	0	0
$823$	−52.2121	−1.82000	−0.910000	−	0.414608i	$-0.863919\pi$
$823$	−52.2121	−1.82000	−0.910000	+	0.414608i	$0.863919\pi$
$824$	0	0
$825$	−21.0565	−0.733093
$826$	0	0
$827$	0.628626	0.0218595	0.0109297	−	0.999940i	$-0.496521\pi$
$827$	0.628626	0.0218595	0.0109297	+	0.999940i	$0.496521\pi$
$828$	0	0
$829$	43.4741	1.50992	0.754959	−	0.655772i	$-0.227657\pi$
$829$	43.4741	1.50992	0.754959	+	0.655772i	$0.227657\pi$
$830$	0	0
$831$	−63.0499	−2.18718
$832$	0	0
$833$	3.49200	0.120991
$834$	0	0
$835$	20.7790	0.719087
$836$	0	0
$837$	18.6270	0.643843
$838$	0	0
$839$	−12.2311	−0.422265	−0.211132	−	0.977457i	$-0.567715\pi$
$839$	−12.2311	−0.422265	−0.211132	+	0.977457i	$0.567715\pi$
$840$	0	0
$841$	12.1734	0.419771
$842$	0	0
$843$	−54.7049	−1.88414
$844$	0	0
$845$	3.76699	0.129588
$846$	0	0
$847$	−9.98452	−0.343072
$848$	0	0
$849$	−39.3122	−1.34919
$850$	0	0
$851$	−29.0092	−0.994422
$852$	0	0
$853$	23.7523	0.813263	0.406631	−	0.913592i	$-0.366703\pi$
$853$	23.7523	0.813263	0.406631	+	0.913592i	$0.366703\pi$
$854$	0	0
$855$	−14.4410	−0.493870
$856$	0	0
$857$	−35.3466	−1.20742	−0.603708	−	0.797205i	$-0.706311\pi$
$857$	−35.3466	−1.20742	−0.603708	+	0.797205i	$0.706311\pi$
$858$	0	0
$859$	23.0835	0.787598	0.393799	−	0.919197i	$-0.371161\pi$
$859$	23.0835	0.787598	0.393799	+	0.919197i	$0.371161\pi$
$860$	0	0
$861$	23.5340	0.802037
$862$	0	0
$863$	50.1962	1.70870	0.854350	−	0.519698i	$-0.173956\pi$
$863$	50.1962	1.70870	0.854350	+	0.519698i	$0.173956\pi$
$864$	0	0
$865$	74.7169	2.54045
$866$	0	0
$867$	10.9271	0.371103
$868$	0	0
$869$	−6.10277	−0.207022
$870$	0	0
$871$	7.22905	0.244947
$872$	0	0
$873$	−30.9698	−1.04817
$874$	0	0
$875$	15.7844	0.533609
$876$	0	0
$877$	9.66080	0.326222	0.163111	−	0.986608i	$-0.447847\pi$
$877$	9.66080	0.326222	0.163111	+	0.986608i	$0.447847\pi$
$878$	0	0
$879$	−53.8152	−1.81514
$880$	0	0
$881$	53.5130	1.80290	0.901450	−	0.432883i	$-0.142504\pi$
$881$	53.5130	1.80290	0.901450	+	0.432883i	$0.142504\pi$
$882$	0	0
$883$	1.94035	0.0652981	0.0326490	−	0.999467i	$-0.489606\pi$
$883$	1.94035	0.0652981	0.0326490	+	0.999467i	$0.489606\pi$
$884$	0	0
$885$	102.239	3.43672
$886$	0	0
$887$	20.4933	0.688099	0.344049	−	0.938952i	$-0.388201\pi$
$887$	20.4933	0.688099	0.344049	+	0.938952i	$0.388201\pi$
$888$	0	0
$889$	−2.72216	−0.0912983
$890$	0	0
$891$	−10.8850	−0.364661
$892$	0	0
$893$	1.54970	0.0518587
$894$	0	0
$895$	7.29248	0.243761
$896$	0	0
$897$	−14.9293	−0.498474
$898$	0	0
$899$	63.3008	2.11120
$900$	0	0
$901$	35.8710	1.19504
$902$	0	0
$903$	−9.79960	−0.326110
$904$	0	0
$905$	29.0033	0.964103
$906$	0	0
$907$	−24.5229	−0.814269	−0.407135	−	0.913368i	$-0.633472\pi$
$907$	−24.5229	−0.814269	−0.407135	+	0.913368i	$0.633472\pi$
$908$	0	0
$909$	16.0779	0.533271
$910$	0	0
$911$	−52.8441	−1.75080	−0.875402	−	0.483395i	$-0.839403\pi$
$911$	−52.8441	−1.75080	−0.875402	+	0.483395i	$0.839403\pi$
$912$	0	0
$913$	6.68720	0.221314
$914$	0	0
$915$	5.51591	0.182350
$916$	0	0
$917$	13.9152	0.459520
$918$	0	0
$919$	−10.1225	−0.333911	−0.166955	−	0.985964i	$-0.553394\pi$
$919$	−10.1225	−0.333911	−0.166955	+	0.985964i	$0.553394\pi$
$920$	0	0
$921$	−63.9770	−2.10812
$922$	0	0
$923$	0.348273	0.0114635
$924$	0	0
$925$	40.6019	1.33498
$926$	0	0
$927$	33.2821	1.09313
$928$	0	0
$929$	−33.3120	−1.09293	−0.546466	−	0.837482i	$-0.684027\pi$
$929$	−33.3120	−1.09293	−0.546466	+	0.837482i	$0.684027\pi$
$930$	0	0
$931$	−1.76699	−0.0579106
$932$	0	0
$933$	15.2307	0.498630
$934$	0	0
$935$	13.2557	0.433509
$936$	0	0
$937$	35.2254	1.15076	0.575381	−	0.817885i	$-0.304854\pi$
$937$	35.2254	1.15076	0.575381	+	0.817885i	$0.304854\pi$
$938$	0	0
$939$	−15.9206	−0.519551
$940$	0	0
$941$	−39.1903	−1.27757	−0.638784	−	0.769386i	$-0.720562\pi$
$941$	−39.1903	−1.27757	−0.638784	+	0.769386i	$0.720562\pi$
$942$	0	0
$943$	67.9645	2.21323
$944$	0	0
$945$	7.11272	0.231377
$946$	0	0
$947$	17.6716	0.574249	0.287125	−	0.957893i	$-0.407301\pi$
$947$	17.6716	0.574249	0.287125	+	0.957893i	$0.407301\pi$
$948$	0	0
$949$	2.38892	0.0775477
$950$	0	0
$951$	35.7250	1.15846
$952$	0	0
$953$	−31.3877	−1.01675	−0.508374	−	0.861136i	$-0.669753\pi$
$953$	−31.3877	−1.01675	−0.508374	+	0.861136i	$0.669753\pi$
$954$	0	0
$955$	−72.7399	−2.35381
$956$	0	0
$957$	−14.7018	−0.475241
$958$	0	0
$959$	−14.7814	−0.477318
$960$	0	0
$961$	66.3199	2.13935
$962$	0	0
$963$	25.2774	0.814554
$964$	0	0
$965$	−12.9034	−0.415376
$966$	0	0
$967$	−53.5803	−1.72303	−0.861513	−	0.507735i	$-0.830483\pi$
$967$	−53.5803	−1.72303	−0.861513	+	0.507735i	$0.830483\pi$
$968$	0	0
$969$	14.0292	0.450684
$970$	0	0
$971$	40.7993	1.30931	0.654655	−	0.755928i	$-0.272814\pi$
$971$	40.7993	1.30931	0.654655	+	0.755928i	$0.272814\pi$
$972$	0	0
$973$	20.2431	0.648965
$974$	0	0
$975$	20.8954	0.669188
$976$	0	0
$977$	−26.3071	−0.841638	−0.420819	−	0.907145i	$-0.638257\pi$
$977$	−26.3071	−0.841638	−0.420819	+	0.907145i	$0.638257\pi$
$978$	0	0
$979$	9.86426	0.315263
$980$	0	0
$981$	−15.4359	−0.492832
$982$	0	0
$983$	3.60971	0.115132	0.0575659	−	0.998342i	$-0.481666\pi$
$983$	3.60971	0.115132	0.0575659	+	0.998342i	$0.481666\pi$
$984$	0	0
$985$	−77.7961	−2.47879
$986$	0	0
$987$	1.99407	0.0634720
$988$	0	0
$989$	−28.3005	−0.899905
$990$	0	0
$991$	−37.2680	−1.18386	−0.591929	−	0.805990i	$-0.701633\pi$
$991$	−37.2680	−1.18386	−0.591929	+	0.805990i	$0.701633\pi$
$992$	0	0
$993$	16.3567	0.519064
$994$	0	0
$995$	8.41840	0.266881
$996$	0	0
$997$	−4.41273	−0.139752	−0.0698762	−	0.997556i	$-0.522260\pi$
$997$	−4.41273	−0.139752	−0.0698762	+	0.997556i	$0.522260\pi$
$998$	0	0
$999$	8.34188	0.263926

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2912.2.a.v.1.1	yes	6
4.3	odd	2		2912.2.a.u.1.6	✓	6
8.3	odd	2		5824.2.a.cn.1.1		6
8.5	even	2		5824.2.a.cm.1.6		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2912.2.a.u.1.6	✓	6	4.3	odd	2
2912.2.a.v.1.1	yes	6	1.1	even	1	trivial
5824.2.a.cm.1.6		6	8.5	even	2
5824.2.a.cn.1.1		6	8.3	odd	2

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:	\( N \)	\(=\)	\( 2912 = 2^{5} \cdot 7 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2912.a (trivial)

\(f(q)\)	\(=\)	\(q-2.27366 q^{3} +3.76699 q^{5} +1.00000 q^{7} +2.16955 q^{9} +1.00771 q^{11} -1.00000 q^{13} -8.56486 q^{15} +3.49200 q^{17} -1.76699 q^{19} -2.27366 q^{21} -6.56618 q^{23} +9.19018 q^{25} +1.88817 q^{27} +6.41665 q^{29} +9.86509 q^{31} -2.29119 q^{33} +3.76699 q^{35} +4.41797 q^{37} +2.27366 q^{39} -10.3507 q^{41} +4.31005 q^{43} +8.17265 q^{45} -0.877031 q^{47} +1.00000 q^{49} -7.93964 q^{51} +10.2723 q^{53} +3.79603 q^{55} +4.01753 q^{57} -11.9370 q^{59} -0.644016 q^{61} +2.16955 q^{63} -3.76699 q^{65} -7.22905 q^{67} +14.9293 q^{69} -0.348273 q^{71} -2.38892 q^{73} -20.8954 q^{75} +1.00771 q^{77} -6.05607 q^{79} -10.8017 q^{81} +6.63604 q^{83} +13.1543 q^{85} -14.5893 q^{87} +9.78878 q^{89} -1.00000 q^{91} -22.4299 q^{93} -6.65621 q^{95} -14.2748 q^{97} +2.18627 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 4 q^{3} + 3 q^{5} + 6 q^{7} + 8 q^{9} + 8 q^{11} - 6 q^{13} - 2 q^{15} - 2 q^{17} + 9 q^{19} + 4 q^{21} - 11 q^{23} + 21 q^{25} + 22 q^{27} + 7 q^{29} + q^{31} + 18 q^{33} + 3 q^{35} + 16 q^{37}+ \cdots + 4 q^{99}+O(q^{100}) \) 6 * q + 4 * q^3 + 3 * q^5 + 6 * q^7 + 8 * q^9 + 8 * q^11 - 6 * q^13 - 2 * q^15 - 2 * q^17 + 9 * q^19 + 4 * q^21 - 11 * q^23 + 21 * q^25 + 22 * q^27 + 7 * q^29 + q^31 + 18 * q^33 + 3 * q^35 + 16 * q^37 - 4 * q^39 + q^43 + 29 * q^45 - 11 * q^47 + 6 * q^49 - 8 * q^51 + 23 * q^53 + 18 * q^55 + 10 * q^57 + 10 * q^59 + 10 * q^61 + 8 * q^63 - 3 * q^65 - 8 * q^67 + 6 * q^69 - 14 * q^71 + 11 * q^73 + 14 * q^75 + 8 * q^77 - 3 * q^79 + 30 * q^81 + 17 * q^83 + 6 * q^85 - 36 * q^87 - 5 * q^89 - 6 * q^91 - 16 * q^93 - 45 * q^95 + 27 * q^97 + 4 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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