LMFDB - Embedded newform 3920.2.a.bv.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3920,2,Mod(1,3920)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3920, 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("3920.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3920 = 2^{4} \cdot 5 \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3920.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:	$31.3013575923$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{8})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 2 $ x^2 - 2
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 35)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.41421$ of defining polynomial
Character	$\chi$	$=$	3920.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.41421 q^{3} -1.00000 q^{5} +2.82843 q^{9} +O(q^{10})$	$q+2.41421 q^{3} -1.00000 q^{5} +2.82843 q^{9} -4.82843 q^{11} -0.828427 q^{13} -2.41421 q^{15} +0.828427 q^{17} -2.82843 q^{19} +2.41421 q^{23} +1.00000 q^{25} -0.414214 q^{27} -1.00000 q^{29} -6.00000 q^{31} -11.6569 q^{33} -2.00000 q^{39} +2.17157 q^{41} -6.41421 q^{43} -2.82843 q^{45} +2.00000 q^{47} +2.00000 q^{51} -6.82843 q^{53} +4.82843 q^{55} -6.82843 q^{57} -12.4853 q^{59} +11.4853 q^{61} +0.828427 q^{65} -12.4142 q^{67} +5.82843 q^{69} +12.4853 q^{71} -4.82843 q^{73} +2.41421 q^{75} -9.17157 q^{79} -9.48528 q^{81} -11.7279 q^{83} -0.828427 q^{85} -2.41421 q^{87} -2.65685 q^{89} -14.4853 q^{93} +2.82843 q^{95} -0.343146 q^{97} -13.6569 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{3} - 2 q^{5}+O(q^{10}) $ 2 * q + 2 * q^3 - 2 * q^5	$ 2 q + 2 q^{3} - 2 q^{5} - 4 q^{11} + 4 q^{13} - 2 q^{15} - 4 q^{17} + 2 q^{23} + 2 q^{25} + 2 q^{27} - 2 q^{29} - 12 q^{31} - 12 q^{33} - 4 q^{39} + 10 q^{41} - 10 q^{43} + 4 q^{47} + 4 q^{51} - 8 q^{53} + 4 q^{55} - 8 q^{57} - 8 q^{59} + 6 q^{61} - 4 q^{65} - 22 q^{67} + 6 q^{69} + 8 q^{71} - 4 q^{73} + 2 q^{75} - 24 q^{79} - 2 q^{81} + 2 q^{83} + 4 q^{85} - 2 q^{87} + 6 q^{89} - 12 q^{93} - 12 q^{97} - 16 q^{99}+O(q^{100}) $ 2 * q + 2 * q^3 - 2 * q^5 - 4 * q^11 + 4 * q^13 - 2 * q^15 - 4 * q^17 + 2 * q^23 + 2 * q^25 + 2 * q^27 - 2 * q^29 - 12 * q^31 - 12 * q^33 - 4 * q^39 + 10 * q^41 - 10 * q^43 + 4 * q^47 + 4 * q^51 - 8 * q^53 + 4 * q^55 - 8 * q^57 - 8 * q^59 + 6 * q^61 - 4 * q^65 - 22 * q^67 + 6 * q^69 + 8 * q^71 - 4 * q^73 + 2 * q^75 - 24 * q^79 - 2 * q^81 + 2 * q^83 + 4 * q^85 - 2 * q^87 + 6 * q^89 - 12 * q^93 - 12 * q^97 - 16 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	2.41421	1.39385	0.696923	−	0.717146i	$-0.254552\pi$
$3$	2.41421	1.39385	0.696923	+	0.717146i	$0.254552\pi$
$4$	0	0
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	2.82843	0.942809
$10$	0	0
$11$	−4.82843	−1.45583	−0.727913	−	0.685670i	$-0.759509\pi$
$11$	−4.82843	−1.45583	−0.727913	+	0.685670i	$0.759509\pi$
$12$	0	0
$13$	−0.828427	−0.229764	−0.114882	−	0.993379i	$-0.536649\pi$
$13$	−0.828427	−0.229764	−0.114882	+	0.993379i	$0.536649\pi$
$14$	0	0
$15$	−2.41421	−0.623347
$16$	0	0
$17$	0.828427	0.200923	0.100462	−	0.994941i	$-0.467968\pi$
$17$	0.828427	0.200923	0.100462	+	0.994941i	$0.467968\pi$
$18$	0	0
$19$	−2.82843	−0.648886	−0.324443	−	0.945905i	$-0.605177\pi$
$19$	−2.82843	−0.648886	−0.324443	+	0.945905i	$0.605177\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	2.41421	0.503398	0.251699	−	0.967806i	$-0.419011\pi$
$23$	2.41421	0.503398	0.251699	+	0.967806i	$0.419011\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−0.414214	−0.0797154
$28$	0	0
$29$	−1.00000	−0.185695	−0.0928477	−	0.995680i	$-0.529597\pi$
$29$	−1.00000	−0.185695	−0.0928477	+	0.995680i	$0.529597\pi$
$30$	0	0
$31$	−6.00000	−1.07763	−0.538816	−	0.842424i	$-0.681128\pi$
$31$	−6.00000	−1.07763	−0.538816	+	0.842424i	$0.681128\pi$
$32$	0	0
$33$	−11.6569	−2.02920
$34$	0	0
$35$	0	0
$36$	0	0
$37$	0	0		−	1.00000i	$-0.5\pi$
$37$	0	0			1.00000i	$0.5\pi$
$38$	0	0
$39$	−2.00000	−0.320256
$40$	0	0
$41$	2.17157	0.339143	0.169571	−	0.985518i	$-0.445762\pi$
$41$	2.17157	0.339143	0.169571	+	0.985518i	$0.445762\pi$
$42$	0	0
$43$	−6.41421	−0.978158	−0.489079	−	0.872239i	$-0.662667\pi$
$43$	−6.41421	−0.978158	−0.489079	+	0.872239i	$0.662667\pi$
$44$	0	0
$45$	−2.82843	−0.421637
$46$	0	0
$47$	2.00000	0.291730	0.145865	−	0.989305i	$-0.453403\pi$
$47$	2.00000	0.291730	0.145865	+	0.989305i	$0.453403\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	2.00000	0.280056
$52$	0	0
$53$	−6.82843	−0.937957	−0.468978	−	0.883210i	$-0.655378\pi$
$53$	−6.82843	−0.937957	−0.468978	+	0.883210i	$0.655378\pi$
$54$	0	0
$55$	4.82843	0.651065
$56$	0	0
$57$	−6.82843	−0.904447
$58$	0	0
$59$	−12.4853	−1.62545	−0.812723	−	0.582651i	$-0.802016\pi$
$59$	−12.4853	−1.62545	−0.812723	+	0.582651i	$0.802016\pi$
$60$	0	0
$61$	11.4853	1.47054	0.735270	−	0.677775i	$-0.237055\pi$
$61$	11.4853	1.47054	0.735270	+	0.677775i	$0.237055\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	0.828427	0.102754
$66$	0	0
$67$	−12.4142	−1.51664	−0.758319	−	0.651884i	$-0.773979\pi$
$67$	−12.4142	−1.51664	−0.758319	+	0.651884i	$0.773979\pi$
$68$	0	0
$69$	5.82843	0.701660
$70$	0	0
$71$	12.4853	1.48173	0.740865	−	0.671654i	$-0.234416\pi$
$71$	12.4853	1.48173	0.740865	+	0.671654i	$0.234416\pi$
$72$	0	0
$73$	−4.82843	−0.565125	−0.282562	−	0.959249i	$-0.591184\pi$
$73$	−4.82843	−0.565125	−0.282562	+	0.959249i	$0.591184\pi$
$74$	0	0
$75$	2.41421	0.278769
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−9.17157	−1.03188	−0.515941	−	0.856624i	$-0.672558\pi$
$79$	−9.17157	−1.03188	−0.515941	+	0.856624i	$0.672558\pi$
$80$	0	0
$81$	−9.48528	−1.05392
$82$	0	0
$83$	−11.7279	−1.28731	−0.643653	−	0.765317i	$-0.722582\pi$
$83$	−11.7279	−1.28731	−0.643653	+	0.765317i	$0.722582\pi$
$84$	0	0
$85$	−0.828427	−0.0898555
$86$	0	0
$87$	−2.41421	−0.258831
$88$	0	0
$89$	−2.65685	−0.281626	−0.140813	−	0.990036i	$-0.544972\pi$
$89$	−2.65685	−0.281626	−0.140813	+	0.990036i	$0.544972\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	−14.4853	−1.50205
$94$	0	0
$95$	2.82843	0.290191
$96$	0	0
$97$	−0.343146	−0.0348412	−0.0174206	−	0.999848i	$-0.505545\pi$
$97$	−0.343146	−0.0348412	−0.0174206	+	0.999848i	$0.505545\pi$
$98$	0	0
$99$	−13.6569	−1.37257
$100$	0	0
$101$	−12.3137	−1.22526	−0.612630	−	0.790370i	$-0.709888\pi$
$101$	−12.3137	−1.22526	−0.612630	+	0.790370i	$0.709888\pi$
$102$	0	0
$103$	0.414214	0.0408137	0.0204068	−	0.999792i	$-0.493504\pi$
$103$	0.414214	0.0408137	0.0204068	+	0.999792i	$0.493504\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−2.75736	−0.266564	−0.133282	−	0.991078i	$-0.542552\pi$
$107$	−2.75736	−0.266564	−0.133282	+	0.991078i	$0.542552\pi$
$108$	0	0
$109$	3.48528	0.333829	0.166915	−	0.985971i	$-0.446620\pi$
$109$	3.48528	0.333829	0.166915	+	0.985971i	$0.446620\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	12.4853	1.17452	0.587258	−	0.809400i	$-0.300207\pi$
$113$	12.4853	1.17452	0.587258	+	0.809400i	$0.300207\pi$
$114$	0	0
$115$	−2.41421	−0.225127
$116$	0	0
$117$	−2.34315	−0.216624
$118$	0	0
$119$	0	0
$120$	0	0
$121$	12.3137	1.11943
$122$	0	0
$123$	5.24264	0.472713
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−13.3137	−1.18140	−0.590700	−	0.806891i	$-0.701148\pi$
$127$	−13.3137	−1.18140	−0.590700	+	0.806891i	$0.701148\pi$
$128$	0	0
$129$	−15.4853	−1.36340
$130$	0	0
$131$	3.31371	0.289520	0.144760	−	0.989467i	$-0.453759\pi$
$131$	3.31371	0.289520	0.144760	+	0.989467i	$0.453759\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	0.414214	0.0356498
$136$	0	0
$137$	1.65685	0.141555	0.0707773	−	0.997492i	$-0.477452\pi$
$137$	1.65685	0.141555	0.0707773	+	0.997492i	$0.477452\pi$
$138$	0	0
$139$	12.1421	1.02988	0.514941	−	0.857225i	$-0.327814\pi$
$139$	12.1421	1.02988	0.514941	+	0.857225i	$0.327814\pi$
$140$	0	0
$141$	4.82843	0.406627
$142$	0	0
$143$	4.00000	0.334497
$144$	0	0
$145$	1.00000	0.0830455
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−7.82843	−0.641330	−0.320665	−	0.947193i	$-0.603906\pi$
$149$	−7.82843	−0.641330	−0.320665	+	0.947193i	$0.603906\pi$
$150$	0	0
$151$	−0.343146	−0.0279248	−0.0139624	−	0.999903i	$-0.504445\pi$
$151$	−0.343146	−0.0279248	−0.0139624	+	0.999903i	$0.504445\pi$
$152$	0	0
$153$	2.34315	0.189432
$154$	0	0
$155$	6.00000	0.481932
$156$	0	0
$157$	5.31371	0.424080	0.212040	−	0.977261i	$-0.431989\pi$
$157$	5.31371	0.424080	0.212040	+	0.977261i	$0.431989\pi$
$158$	0	0
$159$	−16.4853	−1.30737
$160$	0	0
$161$	0	0
$162$	0	0
$163$	−23.6569	−1.85295	−0.926474	−	0.376359i	$-0.877176\pi$
$163$	−23.6569	−1.85295	−0.926474	+	0.376359i	$0.877176\pi$
$164$	0	0
$165$	11.6569	0.907485
$166$	0	0
$167$	19.5858	1.51559	0.757797	−	0.652491i	$-0.226276\pi$
$167$	19.5858	1.51559	0.757797	+	0.652491i	$0.226276\pi$
$168$	0	0
$169$	−12.3137	−0.947208
$170$	0	0
$171$	−8.00000	−0.611775
$172$	0	0
$173$	19.3137	1.46839	0.734197	−	0.678936i	$-0.237559\pi$
$173$	19.3137	1.46839	0.734197	+	0.678936i	$0.237559\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−30.1421	−2.26562
$178$	0	0
$179$	10.0000	0.747435	0.373718	−	0.927543i	$-0.378083\pi$
$179$	10.0000	0.747435	0.373718	+	0.927543i	$0.378083\pi$
$180$	0	0
$181$	8.65685	0.643459	0.321729	−	0.946832i	$-0.395736\pi$
$181$	8.65685	0.643459	0.321729	+	0.946832i	$0.395736\pi$
$182$	0	0
$183$	27.7279	2.04971
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−4.00000	−0.292509
$188$	0	0
$189$	0	0
$190$	0	0
$191$	7.17157	0.518917	0.259458	−	0.965754i	$-0.416456\pi$
$191$	7.17157	0.518917	0.259458	+	0.965754i	$0.416456\pi$
$192$	0	0
$193$	−2.00000	−0.143963	−0.0719816	−	0.997406i	$-0.522932\pi$
$193$	−2.00000	−0.143963	−0.0719816	+	0.997406i	$0.522932\pi$
$194$	0	0
$195$	2.00000	0.143223
$196$	0	0
$197$	−23.6569	−1.68548	−0.842741	−	0.538320i	$-0.819059\pi$
$197$	−23.6569	−1.68548	−0.842741	+	0.538320i	$0.819059\pi$
$198$	0	0
$199$	1.65685	0.117451	0.0587256	−	0.998274i	$-0.481296\pi$
$199$	1.65685	0.117451	0.0587256	+	0.998274i	$0.481296\pi$
$200$	0	0
$201$	−29.9706	−2.11396
$202$	0	0
$203$	0	0
$204$	0	0
$205$	−2.17157	−0.151669
$206$	0	0
$207$	6.82843	0.474608
$208$	0	0
$209$	13.6569	0.944664
$210$	0	0
$211$	−3.51472	−0.241963	−0.120982	−	0.992655i	$-0.538604\pi$
$211$	−3.51472	−0.241963	−0.120982	+	0.992655i	$0.538604\pi$
$212$	0	0
$213$	30.1421	2.06531
$214$	0	0
$215$	6.41421	0.437446
$216$	0	0
$217$	0	0
$218$	0	0
$219$	−11.6569	−0.787697
$220$	0	0
$221$	−0.686292	−0.0461650
$222$	0	0
$223$	11.6569	0.780601	0.390300	−	0.920688i	$-0.372371\pi$
$223$	11.6569	0.780601	0.390300	+	0.920688i	$0.372371\pi$
$224$	0	0
$225$	2.82843	0.188562
$226$	0	0
$227$	26.9706	1.79010	0.895050	−	0.445967i	$-0.147140\pi$
$227$	26.9706	1.79010	0.895050	+	0.445967i	$0.147140\pi$
$228$	0	0
$229$	0.343146	0.0226757	0.0113379	−	0.999936i	$-0.496391\pi$
$229$	0.343146	0.0226757	0.0113379	+	0.999936i	$0.496391\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−11.1716	−0.731874	−0.365937	−	0.930640i	$-0.619251\pi$
$233$	−11.1716	−0.731874	−0.365937	+	0.930640i	$0.619251\pi$
$234$	0	0
$235$	−2.00000	−0.130466
$236$	0	0
$237$	−22.1421	−1.43829
$238$	0	0
$239$	−1.31371	−0.0849767	−0.0424884	−	0.999097i	$-0.513529\pi$
$239$	−1.31371	−0.0849767	−0.0424884	+	0.999097i	$0.513529\pi$
$240$	0	0
$241$	−16.3431	−1.05275	−0.526377	−	0.850251i	$-0.676450\pi$
$241$	−16.3431	−1.05275	−0.526377	+	0.850251i	$0.676450\pi$
$242$	0	0
$243$	−21.6569	−1.38929
$244$	0	0
$245$	0	0
$246$	0	0
$247$	2.34315	0.149091
$248$	0	0
$249$	−28.3137	−1.79431
$250$	0	0
$251$	13.3137	0.840354	0.420177	−	0.907442i	$-0.361968\pi$
$251$	13.3137	0.840354	0.420177	+	0.907442i	$0.361968\pi$
$252$	0	0
$253$	−11.6569	−0.732860
$254$	0	0
$255$	−2.00000	−0.125245
$256$	0	0
$257$	−17.6569	−1.10140	−0.550702	−	0.834702i	$-0.685640\pi$
$257$	−17.6569	−1.10140	−0.550702	+	0.834702i	$0.685640\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	−2.82843	−0.175075
$262$	0	0
$263$	−19.0416	−1.17416	−0.587079	−	0.809530i	$-0.699722\pi$
$263$	−19.0416	−1.17416	−0.587079	+	0.809530i	$0.699722\pi$
$264$	0	0
$265$	6.82843	0.419467
$266$	0	0
$267$	−6.41421	−0.392543
$268$	0	0
$269$	30.4558	1.85693	0.928463	−	0.371425i	$-0.121131\pi$
$269$	30.4558	1.85693	0.928463	+	0.371425i	$0.121131\pi$
$270$	0	0
$271$	−0.485281	−0.0294787	−0.0147394	−	0.999891i	$-0.504692\pi$
$271$	−0.485281	−0.0294787	−0.0147394	+	0.999891i	$0.504692\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−4.82843	−0.291165
$276$	0	0
$277$	−12.1421	−0.729550	−0.364775	−	0.931096i	$-0.618854\pi$
$277$	−12.1421	−0.729550	−0.364775	+	0.931096i	$0.618854\pi$
$278$	0	0
$279$	−16.9706	−1.01600
$280$	0	0
$281$	26.2843	1.56799	0.783994	−	0.620768i	$-0.213179\pi$
$281$	26.2843	1.56799	0.783994	+	0.620768i	$0.213179\pi$
$282$	0	0
$283$	14.0000	0.832214	0.416107	−	0.909316i	$-0.363394\pi$
$283$	14.0000	0.832214	0.416107	+	0.909316i	$0.363394\pi$
$284$	0	0
$285$	6.82843	0.404481
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−16.3137	−0.959630
$290$	0	0
$291$	−0.828427	−0.0485633
$292$	0	0
$293$	16.0000	0.934730	0.467365	−	0.884064i	$-0.345203\pi$
$293$	16.0000	0.934730	0.467365	+	0.884064i	$0.345203\pi$
$294$	0	0
$295$	12.4853	0.726921
$296$	0	0
$297$	2.00000	0.116052
$298$	0	0
$299$	−2.00000	−0.115663
$300$	0	0
$301$	0	0
$302$	0	0
$303$	−29.7279	−1.70782
$304$	0	0
$305$	−11.4853	−0.657645
$306$	0	0
$307$	−13.2426	−0.755797	−0.377899	−	0.925847i	$-0.623353\pi$
$307$	−13.2426	−0.755797	−0.377899	+	0.925847i	$0.623353\pi$
$308$	0	0
$309$	1.00000	0.0568880
$310$	0	0
$311$	−18.8284	−1.06766	−0.533831	−	0.845591i	$-0.679248\pi$
$311$	−18.8284	−1.06766	−0.533831	+	0.845591i	$0.679248\pi$
$312$	0	0
$313$	17.6569	0.998024	0.499012	−	0.866595i	$-0.333696\pi$
$313$	17.6569	0.998024	0.499012	+	0.866595i	$0.333696\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	25.7990	1.44902	0.724508	−	0.689267i	$-0.242067\pi$
$317$	25.7990	1.44902	0.724508	+	0.689267i	$0.242067\pi$
$318$	0	0
$319$	4.82843	0.270340
$320$	0	0
$321$	−6.65685	−0.371549
$322$	0	0
$323$	−2.34315	−0.130376
$324$	0	0
$325$	−0.828427	−0.0459529
$326$	0	0
$327$	8.41421	0.465307
$328$	0	0
$329$	0	0
$330$	0	0
$331$	10.9706	0.602997	0.301498	−	0.953467i	$-0.402513\pi$
$331$	10.9706	0.602997	0.301498	+	0.953467i	$0.402513\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	12.4142	0.678261
$336$	0	0
$337$	14.8284	0.807756	0.403878	−	0.914813i	$-0.367662\pi$
$337$	14.8284	0.807756	0.403878	+	0.914813i	$0.367662\pi$
$338$	0	0
$339$	30.1421	1.63710
$340$	0	0
$341$	28.9706	1.56884
$342$	0	0
$343$	0	0
$344$	0	0
$345$	−5.82843	−0.313792
$346$	0	0
$347$	−22.0711	−1.18484	−0.592418	−	0.805630i	$-0.701827\pi$
$347$	−22.0711	−1.18484	−0.592418	+	0.805630i	$0.701827\pi$
$348$	0	0
$349$	26.6569	1.42691	0.713454	−	0.700702i	$-0.247130\pi$
$349$	26.6569	1.42691	0.713454	+	0.700702i	$0.247130\pi$
$350$	0	0
$351$	0.343146	0.0183158
$352$	0	0
$353$	21.1716	1.12685	0.563425	−	0.826168i	$-0.309484\pi$
$353$	21.1716	1.12685	0.563425	+	0.826168i	$0.309484\pi$
$354$	0	0
$355$	−12.4853	−0.662650
$356$	0	0
$357$	0	0
$358$	0	0
$359$	10.0000	0.527780	0.263890	−	0.964553i	$-0.414994\pi$
$359$	10.0000	0.527780	0.263890	+	0.964553i	$0.414994\pi$
$360$	0	0
$361$	−11.0000	−0.578947
$362$	0	0
$363$	29.7279	1.56031
$364$	0	0
$365$	4.82843	0.252731
$366$	0	0
$367$	−11.2426	−0.586861	−0.293431	−	0.955980i	$-0.594797\pi$
$367$	−11.2426	−0.586861	−0.293431	+	0.955980i	$0.594797\pi$
$368$	0	0
$369$	6.14214	0.319747
$370$	0	0
$371$	0	0
$372$	0	0
$373$	12.9706	0.671590	0.335795	−	0.941935i	$-0.390995\pi$
$373$	12.9706	0.671590	0.335795	+	0.941935i	$0.390995\pi$
$374$	0	0
$375$	−2.41421	−0.124669
$376$	0	0
$377$	0.828427	0.0426662
$378$	0	0
$379$	−21.1716	−1.08751	−0.543755	−	0.839244i	$-0.682998\pi$
$379$	−21.1716	−1.08751	−0.543755	+	0.839244i	$0.682998\pi$
$380$	0	0
$381$	−32.1421	−1.64669
$382$	0	0
$383$	16.8995	0.863524	0.431762	−	0.901988i	$-0.357892\pi$
$383$	16.8995	0.863524	0.431762	+	0.901988i	$0.357892\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−18.1421	−0.922217
$388$	0	0
$389$	12.3431	0.625822	0.312911	−	0.949782i	$-0.398696\pi$
$389$	12.3431	0.625822	0.312911	+	0.949782i	$0.398696\pi$
$390$	0	0
$391$	2.00000	0.101144
$392$	0	0
$393$	8.00000	0.403547
$394$	0	0
$395$	9.17157	0.461472
$396$	0	0
$397$	−28.6274	−1.43677	−0.718384	−	0.695646i	$-0.755118\pi$
$397$	−28.6274	−1.43677	−0.718384	+	0.695646i	$0.755118\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	7.68629	0.383835	0.191918	−	0.981411i	$-0.438529\pi$
$401$	7.68629	0.383835	0.191918	+	0.981411i	$0.438529\pi$
$402$	0	0
$403$	4.97056	0.247601
$404$	0	0
$405$	9.48528	0.471327
$406$	0	0
$407$	0	0
$408$	0	0
$409$	−24.7990	−1.22623	−0.613116	−	0.789993i	$-0.710084\pi$
$409$	−24.7990	−1.22623	−0.613116	+	0.789993i	$0.710084\pi$
$410$	0	0
$411$	4.00000	0.197305
$412$	0	0
$413$	0	0
$414$	0	0
$415$	11.7279	0.575701
$416$	0	0
$417$	29.3137	1.43550
$418$	0	0
$419$	−23.3137	−1.13895	−0.569475	−	0.822009i	$-0.692853\pi$
$419$	−23.3137	−1.13895	−0.569475	+	0.822009i	$0.692853\pi$
$420$	0	0
$421$	−3.48528	−0.169862	−0.0849311	−	0.996387i	$-0.527067\pi$
$421$	−3.48528	−0.169862	−0.0849311	+	0.996387i	$0.527067\pi$
$422$	0	0
$423$	5.65685	0.275046
$424$	0	0
$425$	0.828427	0.0401846
$426$	0	0
$427$	0	0
$428$	0	0
$429$	9.65685	0.466237
$430$	0	0
$431$	21.7990	1.05002	0.525010	−	0.851096i	$-0.324062\pi$
$431$	21.7990	1.05002	0.525010	+	0.851096i	$0.324062\pi$
$432$	0	0
$433$	31.7990	1.52816	0.764081	−	0.645120i	$-0.223193\pi$
$433$	31.7990	1.52816	0.764081	+	0.645120i	$0.223193\pi$
$434$	0	0
$435$	2.41421	0.115753
$436$	0	0
$437$	−6.82843	−0.326648
$438$	0	0
$439$	33.9411	1.61992	0.809961	−	0.586484i	$-0.199488\pi$
$439$	33.9411	1.61992	0.809961	+	0.586484i	$0.199488\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	12.2132	0.580267	0.290133	−	0.956986i	$-0.406300\pi$
$443$	12.2132	0.580267	0.290133	+	0.956986i	$0.406300\pi$
$444$	0	0
$445$	2.65685	0.125947
$446$	0	0
$447$	−18.8995	−0.893915
$448$	0	0
$449$	−1.82843	−0.0862888	−0.0431444	−	0.999069i	$-0.513738\pi$
$449$	−1.82843	−0.0862888	−0.0431444	+	0.999069i	$0.513738\pi$
$450$	0	0
$451$	−10.4853	−0.493733
$452$	0	0
$453$	−0.828427	−0.0389229
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−32.2843	−1.51019	−0.755097	−	0.655613i	$-0.772410\pi$
$457$	−32.2843	−1.51019	−0.755097	+	0.655613i	$0.772410\pi$
$458$	0	0
$459$	−0.343146	−0.0160167
$460$	0	0
$461$	−18.6863	−0.870307	−0.435154	−	0.900356i	$-0.643306\pi$
$461$	−18.6863	−0.870307	−0.435154	+	0.900356i	$0.643306\pi$
$462$	0	0
$463$	−11.0416	−0.513148	−0.256574	−	0.966525i	$-0.582594\pi$
$463$	−11.0416	−0.513148	−0.256574	+	0.966525i	$0.582594\pi$
$464$	0	0
$465$	14.4853	0.671739
$466$	0	0
$467$	−22.8995	−1.05966	−0.529831	−	0.848103i	$-0.677745\pi$
$467$	−22.8995	−1.05966	−0.529831	+	0.848103i	$0.677745\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	12.8284	0.591103
$472$	0	0
$473$	30.9706	1.42403
$474$	0	0
$475$	−2.82843	−0.129777
$476$	0	0
$477$	−19.3137	−0.884314
$478$	0	0
$479$	−24.3431	−1.11227	−0.556133	−	0.831093i	$-0.687716\pi$
$479$	−24.3431	−1.11227	−0.556133	+	0.831093i	$0.687716\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	0	0
$484$	0	0
$485$	0.343146	0.0155814
$486$	0	0
$487$	−15.6569	−0.709480	−0.354740	−	0.934965i	$-0.615431\pi$
$487$	−15.6569	−0.709480	−0.354740	+	0.934965i	$0.615431\pi$
$488$	0	0
$489$	−57.1127	−2.58273
$490$	0	0
$491$	13.3137	0.600839	0.300420	−	0.953807i	$-0.402873\pi$
$491$	13.3137	0.600839	0.300420	+	0.953807i	$0.402873\pi$
$492$	0	0
$493$	−0.828427	−0.0373105
$494$	0	0
$495$	13.6569	0.613830
$496$	0	0
$497$	0	0
$498$	0	0
$499$	−4.82843	−0.216150	−0.108075	−	0.994143i	$-0.534469\pi$
$499$	−4.82843	−0.216150	−0.108075	+	0.994143i	$0.534469\pi$
$500$	0	0
$501$	47.2843	2.11251
$502$	0	0
$503$	37.8701	1.68854	0.844271	−	0.535916i	$-0.180034\pi$
$503$	37.8701	1.68854	0.844271	+	0.535916i	$0.180034\pi$
$504$	0	0
$505$	12.3137	0.547953
$506$	0	0
$507$	−29.7279	−1.32026
$508$	0	0
$509$	−24.6569	−1.09290	−0.546448	−	0.837493i	$-0.684020\pi$
$509$	−24.6569	−1.09290	−0.546448	+	0.837493i	$0.684020\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	1.17157	0.0517262
$514$	0	0
$515$	−0.414214	−0.0182524
$516$	0	0
$517$	−9.65685	−0.424708
$518$	0	0
$519$	46.6274	2.04672
$520$	0	0
$521$	18.9706	0.831115	0.415558	−	0.909567i	$-0.363586\pi$
$521$	18.9706	0.831115	0.415558	+	0.909567i	$0.363586\pi$
$522$	0	0
$523$	24.3431	1.06445	0.532226	−	0.846602i	$-0.321356\pi$
$523$	24.3431	1.06445	0.532226	+	0.846602i	$0.321356\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−4.97056	−0.216521
$528$	0	0
$529$	−17.1716	−0.746590
$530$	0	0
$531$	−35.3137	−1.53248
$532$	0	0
$533$	−1.79899	−0.0779229
$534$	0	0
$535$	2.75736	0.119211
$536$	0	0
$537$	24.1421	1.04181
$538$	0	0
$539$	0	0
$540$	0	0
$541$	18.6569	0.802121	0.401060	−	0.916052i	$-0.368642\pi$
$541$	18.6569	0.802121	0.401060	+	0.916052i	$0.368642\pi$
$542$	0	0
$543$	20.8995	0.896883
$544$	0	0
$545$	−3.48528	−0.149293
$546$	0	0
$547$	−5.10051	−0.218082	−0.109041	−	0.994037i	$-0.534778\pi$
$547$	−5.10051	−0.218082	−0.109041	+	0.994037i	$0.534778\pi$
$548$	0	0
$549$	32.4853	1.38644
$550$	0	0
$551$	2.82843	0.120495
$552$	0	0
$553$	0	0
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−34.2843	−1.45267	−0.726336	−	0.687340i	$-0.758778\pi$
$557$	−34.2843	−1.45267	−0.726336	+	0.687340i	$0.758778\pi$
$558$	0	0
$559$	5.31371	0.224746
$560$	0	0
$561$	−9.65685	−0.407713
$562$	0	0
$563$	16.2721	0.685786	0.342893	−	0.939374i	$-0.388593\pi$
$563$	16.2721	0.685786	0.342893	+	0.939374i	$0.388593\pi$
$564$	0	0
$565$	−12.4853	−0.525260
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−3.65685	−0.153303	−0.0766517	−	0.997058i	$-0.524423\pi$
$569$	−3.65685	−0.153303	−0.0766517	+	0.997058i	$0.524423\pi$
$570$	0	0
$571$	−14.8284	−0.620550	−0.310275	−	0.950647i	$-0.600421\pi$
$571$	−14.8284	−0.620550	−0.310275	+	0.950647i	$0.600421\pi$
$572$	0	0
$573$	17.3137	0.723291
$574$	0	0
$575$	2.41421	0.100680
$576$	0	0
$577$	−23.9411	−0.996682	−0.498341	−	0.866981i	$-0.666057\pi$
$577$	−23.9411	−0.996682	−0.498341	+	0.866981i	$0.666057\pi$
$578$	0	0
$579$	−4.82843	−0.200663
$580$	0	0
$581$	0	0
$582$	0	0
$583$	32.9706	1.36550
$584$	0	0
$585$	2.34315	0.0968772
$586$	0	0
$587$	22.2843	0.919770	0.459885	−	0.887978i	$-0.347891\pi$
$587$	22.2843	0.919770	0.459885	+	0.887978i	$0.347891\pi$
$588$	0	0
$589$	16.9706	0.699260
$590$	0	0
$591$	−57.1127	−2.34930
$592$	0	0
$593$	−43.7990	−1.79861	−0.899304	−	0.437323i	$-0.855927\pi$
$593$	−43.7990	−1.79861	−0.899304	+	0.437323i	$0.855927\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	4.00000	0.163709
$598$	0	0
$599$	−17.6569	−0.721440	−0.360720	−	0.932674i	$-0.617469\pi$
$599$	−17.6569	−0.721440	−0.360720	+	0.932674i	$0.617469\pi$
$600$	0	0
$601$	−8.34315	−0.340324	−0.170162	−	0.985416i	$-0.554429\pi$
$601$	−8.34315	−0.340324	−0.170162	+	0.985416i	$0.554429\pi$
$602$	0	0
$603$	−35.1127	−1.42990
$604$	0	0
$605$	−12.3137	−0.500623
$606$	0	0
$607$	−4.21320	−0.171009	−0.0855043	−	0.996338i	$-0.527250\pi$
$607$	−4.21320	−0.171009	−0.0855043	+	0.996338i	$0.527250\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−1.65685	−0.0670291
$612$	0	0
$613$	−15.4558	−0.624256	−0.312128	−	0.950040i	$-0.601042\pi$
$613$	−15.4558	−0.624256	−0.312128	+	0.950040i	$0.601042\pi$
$614$	0	0
$615$	−5.24264	−0.211404
$616$	0	0
$617$	−11.3137	−0.455473	−0.227736	−	0.973723i	$-0.573132\pi$
$617$	−11.3137	−0.455473	−0.227736	+	0.973723i	$0.573132\pi$
$618$	0	0
$619$	42.4853	1.70763	0.853814	−	0.520578i	$-0.174284\pi$
$619$	42.4853	1.70763	0.853814	+	0.520578i	$0.174284\pi$
$620$	0	0
$621$	−1.00000	−0.0401286
$622$	0	0
$623$	0	0
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	32.9706	1.31672
$628$	0	0
$629$	0	0
$630$	0	0
$631$	−8.14214	−0.324133	−0.162067	−	0.986780i	$-0.551816\pi$
$631$	−8.14214	−0.324133	−0.162067	+	0.986780i	$0.551816\pi$
$632$	0	0
$633$	−8.48528	−0.337260
$634$	0	0
$635$	13.3137	0.528338
$636$	0	0
$637$	0	0
$638$	0	0
$639$	35.3137	1.39699
$640$	0	0
$641$	14.5147	0.573297	0.286648	−	0.958036i	$-0.407459\pi$
$641$	14.5147	0.573297	0.286648	+	0.958036i	$0.407459\pi$
$642$	0	0
$643$	−30.2843	−1.19430	−0.597148	−	0.802131i	$-0.703699\pi$
$643$	−30.2843	−1.19430	−0.597148	+	0.802131i	$0.703699\pi$
$644$	0	0
$645$	15.4853	0.609732
$646$	0	0
$647$	17.0416	0.669976	0.334988	−	0.942222i	$-0.391268\pi$
$647$	17.0416	0.669976	0.334988	+	0.942222i	$0.391268\pi$
$648$	0	0
$649$	60.2843	2.36636
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−24.8284	−0.971611	−0.485806	−	0.874067i	$-0.661474\pi$
$653$	−24.8284	−0.971611	−0.485806	+	0.874067i	$0.661474\pi$
$654$	0	0
$655$	−3.31371	−0.129477
$656$	0	0
$657$	−13.6569	−0.532805
$658$	0	0
$659$	−26.8284	−1.04509	−0.522544	−	0.852613i	$-0.675017\pi$
$659$	−26.8284	−1.04509	−0.522544	+	0.852613i	$0.675017\pi$
$660$	0	0
$661$	−26.1716	−1.01796	−0.508978	−	0.860779i	$-0.669977\pi$
$661$	−26.1716	−1.01796	−0.508978	+	0.860779i	$0.669977\pi$
$662$	0	0
$663$	−1.65685	−0.0643469
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−2.41421	−0.0934787
$668$	0	0
$669$	28.1421	1.08804
$670$	0	0
$671$	−55.4558	−2.14085
$672$	0	0
$673$	18.3431	0.707076	0.353538	−	0.935420i	$-0.384978\pi$
$673$	18.3431	0.707076	0.353538	+	0.935420i	$0.384978\pi$
$674$	0	0
$675$	−0.414214	−0.0159431
$676$	0	0
$677$	−0.142136	−0.00546272	−0.00273136	−	0.999996i	$-0.500869\pi$
$677$	−0.142136	−0.00546272	−0.00273136	+	0.999996i	$0.500869\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	65.1127	2.49512
$682$	0	0
$683$	43.2426	1.65463	0.827317	−	0.561736i	$-0.189866\pi$
$683$	43.2426	1.65463	0.827317	+	0.561736i	$0.189866\pi$
$684$	0	0
$685$	−1.65685	−0.0633051
$686$	0	0
$687$	0.828427	0.0316065
$688$	0	0
$689$	5.65685	0.215509
$690$	0	0
$691$	−4.82843	−0.183682	−0.0918410	−	0.995774i	$-0.529275\pi$
$691$	−4.82843	−0.183682	−0.0918410	+	0.995774i	$0.529275\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−12.1421	−0.460577
$696$	0	0
$697$	1.79899	0.0681416
$698$	0	0
$699$	−26.9706	−1.02012
$700$	0	0
$701$	−42.7990	−1.61650	−0.808248	−	0.588843i	$-0.799584\pi$
$701$	−42.7990	−1.61650	−0.808248	+	0.588843i	$0.799584\pi$
$702$	0	0
$703$	0	0
$704$	0	0
$705$	−4.82843	−0.181849
$706$	0	0
$707$	0	0
$708$	0	0
$709$	38.3137	1.43890	0.719451	−	0.694543i	$-0.244394\pi$
$709$	38.3137	1.43890	0.719451	+	0.694543i	$0.244394\pi$
$710$	0	0
$711$	−25.9411	−0.972868
$712$	0	0
$713$	−14.4853	−0.542478
$714$	0	0
$715$	−4.00000	−0.149592
$716$	0	0
$717$	−3.17157	−0.118445
$718$	0	0
$719$	−41.1127	−1.53324	−0.766622	−	0.642098i	$-0.778064\pi$
$719$	−41.1127	−1.53324	−0.766622	+	0.642098i	$0.778064\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	−39.4558	−1.46738
$724$	0	0
$725$	−1.00000	−0.0371391
$726$	0	0
$727$	−40.4142	−1.49888	−0.749440	−	0.662072i	$-0.769677\pi$
$727$	−40.4142	−1.49888	−0.749440	+	0.662072i	$0.769677\pi$
$728$	0	0
$729$	−23.8284	−0.882534
$730$	0	0
$731$	−5.31371	−0.196535
$732$	0	0
$733$	22.0000	0.812589	0.406294	−	0.913742i	$-0.366821\pi$
$733$	22.0000	0.812589	0.406294	+	0.913742i	$0.366821\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	59.9411	2.20796
$738$	0	0
$739$	−41.1127	−1.51236	−0.756178	−	0.654367i	$-0.772935\pi$
$739$	−41.1127	−1.51236	−0.756178	+	0.654367i	$0.772935\pi$
$740$	0	0
$741$	5.65685	0.207810
$742$	0	0
$743$	1.92893	0.0707657	0.0353828	−	0.999374i	$-0.488735\pi$
$743$	1.92893	0.0707657	0.0353828	+	0.999374i	$0.488735\pi$
$744$	0	0
$745$	7.82843	0.286811
$746$	0	0
$747$	−33.1716	−1.21368
$748$	0	0
$749$	0	0
$750$	0	0
$751$	41.6569	1.52008	0.760040	−	0.649876i	$-0.225179\pi$
$751$	41.6569	1.52008	0.760040	+	0.649876i	$0.225179\pi$
$752$	0	0
$753$	32.1421	1.17132
$754$	0	0
$755$	0.343146	0.0124884
$756$	0	0
$757$	19.4558	0.707135	0.353567	−	0.935409i	$-0.384969\pi$
$757$	19.4558	0.707135	0.353567	+	0.935409i	$0.384969\pi$
$758$	0	0
$759$	−28.1421	−1.02149
$760$	0	0
$761$	13.3137	0.482622	0.241311	−	0.970448i	$-0.422423\pi$
$761$	13.3137	0.482622	0.241311	+	0.970448i	$0.422423\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	−2.34315	−0.0847166
$766$	0	0
$767$	10.3431	0.373469
$768$	0	0
$769$	−44.6274	−1.60931	−0.804653	−	0.593745i	$-0.797649\pi$
$769$	−44.6274	−1.60931	−0.804653	+	0.593745i	$0.797649\pi$
$770$	0	0
$771$	−42.6274	−1.53519
$772$	0	0
$773$	−25.1127	−0.903241	−0.451620	−	0.892210i	$-0.649154\pi$
$773$	−25.1127	−0.903241	−0.451620	+	0.892210i	$0.649154\pi$
$774$	0	0
$775$	−6.00000	−0.215526
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−6.14214	−0.220065
$780$	0	0
$781$	−60.2843	−2.15714
$782$	0	0
$783$	0.414214	0.0148028
$784$	0	0
$785$	−5.31371	−0.189654
$786$	0	0
$787$	−28.5563	−1.01792	−0.508962	−	0.860789i	$-0.669971\pi$
$787$	−28.5563	−1.01792	−0.508962	+	0.860789i	$0.669971\pi$
$788$	0	0
$789$	−45.9706	−1.63660
$790$	0	0
$791$	0	0
$792$	0	0
$793$	−9.51472	−0.337878
$794$	0	0
$795$	16.4853	0.584673
$796$	0	0
$797$	8.00000	0.283375	0.141687	−	0.989911i	$-0.454747\pi$
$797$	8.00000	0.283375	0.141687	+	0.989911i	$0.454747\pi$
$798$	0	0
$799$	1.65685	0.0586153
$800$	0	0
$801$	−7.51472	−0.265520
$802$	0	0
$803$	23.3137	0.822723
$804$	0	0
$805$	0	0
$806$	0	0
$807$	73.5269	2.58827
$808$	0	0
$809$	−9.62742	−0.338482	−0.169241	−	0.985575i	$-0.554132\pi$
$809$	−9.62742	−0.338482	−0.169241	+	0.985575i	$0.554132\pi$
$810$	0	0
$811$	24.6274	0.864786	0.432393	−	0.901685i	$-0.357669\pi$
$811$	24.6274	0.864786	0.432393	+	0.901685i	$0.357669\pi$
$812$	0	0
$813$	−1.17157	−0.0410889
$814$	0	0
$815$	23.6569	0.828663
$816$	0	0
$817$	18.1421	0.634713
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−19.9411	−0.695950	−0.347975	−	0.937504i	$-0.613131\pi$
$821$	−19.9411	−0.695950	−0.347975	+	0.937504i	$0.613131\pi$
$822$	0	0
$823$	12.0711	0.420771	0.210385	−	0.977619i	$-0.432528\pi$
$823$	12.0711	0.420771	0.210385	+	0.977619i	$0.432528\pi$
$824$	0	0
$825$	−11.6569	−0.405840
$826$	0	0
$827$	−16.2132	−0.563788	−0.281894	−	0.959446i	$-0.590963\pi$
$827$	−16.2132	−0.563788	−0.281894	+	0.959446i	$0.590963\pi$
$828$	0	0
$829$	−6.68629	−0.232225	−0.116112	−	0.993236i	$-0.537043\pi$
$829$	−6.68629	−0.232225	−0.116112	+	0.993236i	$0.537043\pi$
$830$	0	0
$831$	−29.3137	−1.01688
$832$	0	0
$833$	0	0
$834$	0	0
$835$	−19.5858	−0.677794
$836$	0	0
$837$	2.48528	0.0859039
$838$	0	0
$839$	20.8284	0.719077	0.359539	−	0.933130i	$-0.382934\pi$
$839$	20.8284	0.719077	0.359539	+	0.933130i	$0.382934\pi$
$840$	0	0
$841$	−28.0000	−0.965517
$842$	0	0
$843$	63.4558	2.18554
$844$	0	0
$845$	12.3137	0.423604
$846$	0	0
$847$	0	0
$848$	0	0
$849$	33.7990	1.15998
$850$	0	0
$851$	0	0
$852$	0	0
$853$	−53.4558	−1.83029	−0.915147	−	0.403121i	$-0.867925\pi$
$853$	−53.4558	−1.83029	−0.915147	+	0.403121i	$0.867925\pi$
$854$	0	0
$855$	8.00000	0.273594
$856$	0	0
$857$	22.2843	0.761216	0.380608	−	0.924736i	$-0.375715\pi$
$857$	22.2843	0.761216	0.380608	+	0.924736i	$0.375715\pi$
$858$	0	0
$859$	46.6274	1.59091	0.795453	−	0.606015i	$-0.207233\pi$
$859$	46.6274	1.59091	0.795453	+	0.606015i	$0.207233\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	16.5563	0.563585	0.281792	−	0.959475i	$-0.409071\pi$
$863$	16.5563	0.563585	0.281792	+	0.959475i	$0.409071\pi$
$864$	0	0
$865$	−19.3137	−0.656686
$866$	0	0
$867$	−39.3848	−1.33758
$868$	0	0
$869$	44.2843	1.50224
$870$	0	0
$871$	10.2843	0.348469
$872$	0	0
$873$	−0.970563	−0.0328486
$874$	0	0
$875$	0	0
$876$	0	0
$877$	30.8284	1.04100	0.520501	−	0.853861i	$-0.325745\pi$
$877$	30.8284	1.04100	0.520501	+	0.853861i	$0.325745\pi$
$878$	0	0
$879$	38.6274	1.30287
$880$	0	0
$881$	3.82843	0.128983	0.0644915	−	0.997918i	$-0.479457\pi$
$881$	3.82843	0.128983	0.0644915	+	0.997918i	$0.479457\pi$
$882$	0	0
$883$	38.2843	1.28837	0.644184	−	0.764870i	$-0.277197\pi$
$883$	38.2843	1.28837	0.644184	+	0.764870i	$0.277197\pi$
$884$	0	0
$885$	30.1421	1.01322
$886$	0	0
$887$	44.0711	1.47976	0.739881	−	0.672738i	$-0.234882\pi$
$887$	44.0711	1.47976	0.739881	+	0.672738i	$0.234882\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	45.7990	1.53432
$892$	0	0
$893$	−5.65685	−0.189299
$894$	0	0
$895$	−10.0000	−0.334263
$896$	0	0
$897$	−4.82843	−0.161216
$898$	0	0
$899$	6.00000	0.200111
$900$	0	0
$901$	−5.65685	−0.188457
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−8.65685	−0.287764
$906$	0	0
$907$	−28.2132	−0.936804	−0.468402	−	0.883515i	$-0.655170\pi$
$907$	−28.2132	−0.936804	−0.468402	+	0.883515i	$0.655170\pi$
$908$	0	0
$909$	−34.8284	−1.15519
$910$	0	0
$911$	49.7990	1.64991	0.824957	−	0.565195i	$-0.191199\pi$
$911$	49.7990	1.64991	0.824957	+	0.565195i	$0.191199\pi$
$912$	0	0
$913$	56.6274	1.87409
$914$	0	0
$915$	−27.7279	−0.916657
$916$	0	0
$917$	0	0
$918$	0	0
$919$	−19.1127	−0.630470	−0.315235	−	0.949014i	$-0.602083\pi$
$919$	−19.1127	−0.630470	−0.315235	+	0.949014i	$0.602083\pi$
$920$	0	0
$921$	−31.9706	−1.05347
$922$	0	0
$923$	−10.3431	−0.340449
$924$	0	0
$925$	0	0
$926$	0	0
$927$	1.17157	0.0384795
$928$	0	0
$929$	11.4853	0.376820	0.188410	−	0.982090i	$-0.439667\pi$
$929$	11.4853	0.376820	0.188410	+	0.982090i	$0.439667\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	−45.4558	−1.48816
$934$	0	0
$935$	4.00000	0.130814
$936$	0	0
$937$	10.6274	0.347183	0.173591	−	0.984818i	$-0.444463\pi$
$937$	10.6274	0.347183	0.173591	+	0.984818i	$0.444463\pi$
$938$	0	0
$939$	42.6274	1.39109
$940$	0	0
$941$	−10.2843	−0.335258	−0.167629	−	0.985850i	$-0.553611\pi$
$941$	−10.2843	−0.335258	−0.167629	+	0.985850i	$0.553611\pi$
$942$	0	0
$943$	5.24264	0.170724
$944$	0	0
$945$	0	0
$946$	0	0
$947$	43.1838	1.40328	0.701642	−	0.712530i	$-0.252451\pi$
$947$	43.1838	1.40328	0.701642	+	0.712530i	$0.252451\pi$
$948$	0	0
$949$	4.00000	0.129845
$950$	0	0
$951$	62.2843	2.01971
$952$	0	0
$953$	−2.34315	−0.0759019	−0.0379510	−	0.999280i	$-0.512083\pi$
$953$	−2.34315	−0.0759019	−0.0379510	+	0.999280i	$0.512083\pi$
$954$	0	0
$955$	−7.17157	−0.232067
$956$	0	0
$957$	11.6569	0.376813
$958$	0	0
$959$	0	0
$960$	0	0
$961$	5.00000	0.161290
$962$	0	0
$963$	−7.79899	−0.251319
$964$	0	0
$965$	2.00000	0.0643823
$966$	0	0
$967$	27.5269	0.885206	0.442603	−	0.896718i	$-0.354055\pi$
$967$	27.5269	0.885206	0.442603	+	0.896718i	$0.354055\pi$
$968$	0	0
$969$	−5.65685	−0.181724
$970$	0	0
$971$	−24.0000	−0.770197	−0.385098	−	0.922876i	$-0.625832\pi$
$971$	−24.0000	−0.770197	−0.385098	+	0.922876i	$0.625832\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	−2.00000	−0.0640513
$976$	0	0
$977$	21.3137	0.681886	0.340943	−	0.940084i	$-0.389254\pi$
$977$	21.3137	0.681886	0.340943	+	0.940084i	$0.389254\pi$
$978$	0	0
$979$	12.8284	0.409998
$980$	0	0
$981$	9.85786	0.314737
$982$	0	0
$983$	−14.2132	−0.453331	−0.226665	−	0.973973i	$-0.572782\pi$
$983$	−14.2132	−0.453331	−0.226665	+	0.973973i	$0.572782\pi$
$984$	0	0
$985$	23.6569	0.753770
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−15.4853	−0.492403
$990$	0	0
$991$	15.6569	0.497356	0.248678	−	0.968586i	$-0.420004\pi$
$991$	15.6569	0.497356	0.248678	+	0.968586i	$0.420004\pi$
$992$	0	0
$993$	26.4853	0.840485
$994$	0	0
$995$	−1.65685	−0.0525258
$996$	0	0
$997$	17.4558	0.552832	0.276416	−	0.961038i	$-0.410853\pi$
$997$	17.4558	0.552832	0.276416	+	0.961038i	$0.410853\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3920.2.a.bv.1.2		2
4.3	odd	2		245.2.a.g.1.1		2
7.3	odd	6		560.2.q.k.401.2		4
7.5	odd	6		560.2.q.k.81.2		4
7.6	odd	2		3920.2.a.bq.1.1		2
12.11	even	2		2205.2.a.q.1.2		2
20.3	even	4		1225.2.b.h.99.3		4
20.7	even	4		1225.2.b.h.99.2		4
20.19	odd	2		1225.2.a.m.1.2		2
28.3	even	6		35.2.e.a.16.2	yes	4
28.11	odd	6		245.2.e.e.226.2		4
28.19	even	6		35.2.e.a.11.2	✓	4
28.23	odd	6		245.2.e.e.116.2		4
28.27	even	2		245.2.a.h.1.1		2
84.47	odd	6		315.2.j.e.46.1		4
84.59	odd	6		315.2.j.e.226.1		4
84.83	odd	2		2205.2.a.n.1.2		2
140.3	odd	12		175.2.k.a.149.2		8
140.19	even	6		175.2.e.c.151.1		4
140.27	odd	4		1225.2.b.g.99.2		4
140.47	odd	12		175.2.k.a.74.2		8
140.59	even	6		175.2.e.c.51.1		4
140.83	odd	4		1225.2.b.g.99.3		4
140.87	odd	12		175.2.k.a.149.3		8
140.103	odd	12		175.2.k.a.74.3		8
140.139	even	2		1225.2.a.k.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
35.2.e.a.11.2	✓	4	28.19	even	6
35.2.e.a.16.2	yes	4	28.3	even	6
175.2.e.c.51.1		4	140.59	even	6
175.2.e.c.151.1		4	140.19	even	6
175.2.k.a.74.2		8	140.47	odd	12
175.2.k.a.74.3		8	140.103	odd	12
175.2.k.a.149.2		8	140.3	odd	12
175.2.k.a.149.3		8	140.87	odd	12
245.2.a.g.1.1		2	4.3	odd	2
245.2.a.h.1.1		2	28.27	even	2
245.2.e.e.116.2		4	28.23	odd	6
245.2.e.e.226.2		4	28.11	odd	6
315.2.j.e.46.1		4	84.47	odd	6
315.2.j.e.226.1		4	84.59	odd	6
560.2.q.k.81.2		4	7.5	odd	6
560.2.q.k.401.2		4	7.3	odd	6
1225.2.a.k.1.2		2	140.139	even	2
1225.2.a.m.1.2		2	20.19	odd	2
1225.2.b.g.99.2		4	140.27	odd	4
1225.2.b.g.99.3		4	140.83	odd	4
1225.2.b.h.99.2		4	20.7	even	4
1225.2.b.h.99.3		4	20.3	even	4
2205.2.a.n.1.2		2	84.83	odd	2
2205.2.a.q.1.2		2	12.11	even	2
3920.2.a.bq.1.1		2	7.6	odd	2
3920.2.a.bv.1.2		2	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 3920 = 2^{4} \cdot 5 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3920.a (trivial)

\(f(q)\)	\(=\)	\(q+2.41421 q^{3} -1.00000 q^{5} +2.82843 q^{9} +O(q^{10})\)	\(q+2.41421 q^{3} -1.00000 q^{5} +2.82843 q^{9} -4.82843 q^{11} -0.828427 q^{13} -2.41421 q^{15} +0.828427 q^{17} -2.82843 q^{19} +2.41421 q^{23} +1.00000 q^{25} -0.414214 q^{27} -1.00000 q^{29} -6.00000 q^{31} -11.6569 q^{33} -2.00000 q^{39} +2.17157 q^{41} -6.41421 q^{43} -2.82843 q^{45} +2.00000 q^{47} +2.00000 q^{51} -6.82843 q^{53} +4.82843 q^{55} -6.82843 q^{57} -12.4853 q^{59} +11.4853 q^{61} +0.828427 q^{65} -12.4142 q^{67} +5.82843 q^{69} +12.4853 q^{71} -4.82843 q^{73} +2.41421 q^{75} -9.17157 q^{79} -9.48528 q^{81} -11.7279 q^{83} -0.828427 q^{85} -2.41421 q^{87} -2.65685 q^{89} -14.4853 q^{93} +2.82843 q^{95} -0.343146 q^{97} -13.6569 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{3} - 2 q^{5}+O(q^{10}) \) 2 * q + 2 * q^3 - 2 * q^5	\( 2 q + 2 q^{3} - 2 q^{5} - 4 q^{11} + 4 q^{13} - 2 q^{15} - 4 q^{17} + 2 q^{23} + 2 q^{25} + 2 q^{27} - 2 q^{29} - 12 q^{31} - 12 q^{33} - 4 q^{39} + 10 q^{41} - 10 q^{43} + 4 q^{47} + 4 q^{51} - 8 q^{53} + 4 q^{55} - 8 q^{57} - 8 q^{59} + 6 q^{61} - 4 q^{65} - 22 q^{67} + 6 q^{69} + 8 q^{71} - 4 q^{73} + 2 q^{75} - 24 q^{79} - 2 q^{81} + 2 q^{83} + 4 q^{85} - 2 q^{87} + 6 q^{89} - 12 q^{93} - 12 q^{97} - 16 q^{99}+O(q^{100}) \) 2 * q + 2 * q^3 - 2 * q^5 - 4 * q^11 + 4 * q^13 - 2 * q^15 - 4 * q^17 + 2 * q^23 + 2 * q^25 + 2 * q^27 - 2 * q^29 - 12 * q^31 - 12 * q^33 - 4 * q^39 + 10 * q^41 - 10 * q^43 + 4 * q^47 + 4 * q^51 - 8 * q^53 + 4 * q^55 - 8 * q^57 - 8 * q^59 + 6 * q^61 - 4 * q^65 - 22 * q^67 + 6 * q^69 + 8 * q^71 - 4 * q^73 + 2 * q^75 - 24 * q^79 - 2 * q^81 + 2 * q^83 + 4 * q^85 - 2 * q^87 + 6 * q^89 - 12 * q^93 - 12 * q^97 - 16 * q^99

Introduction

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