LMFDB - Embedded newform 3920.2.a.ca.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:	$0$
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 490)
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)

$f(q)$	$=$	$q+3.41421 q^{3} -1.00000 q^{5} +8.65685 q^{9} +O(q^{10})$	$q+3.41421 q^{3} -1.00000 q^{5} +8.65685 q^{9} +0.828427 q^{11} +4.82843 q^{13} -3.41421 q^{15} -2.58579 q^{17} +0.585786 q^{19} +1.17157 q^{23} +1.00000 q^{25} +19.3137 q^{27} -4.82843 q^{29} -2.82843 q^{31} +2.82843 q^{33} -7.65685 q^{37} +16.4853 q^{39} +3.07107 q^{41} +8.82843 q^{43} -8.65685 q^{45} +5.17157 q^{47} -8.82843 q^{51} +6.48528 q^{53} -0.828427 q^{55} +2.00000 q^{57} +8.58579 q^{59} -9.31371 q^{61} -4.82843 q^{65} -1.65685 q^{67} +4.00000 q^{69} +4.48528 q^{71} +9.41421 q^{73} +3.41421 q^{75} +6.82843 q^{79} +39.9706 q^{81} -2.24264 q^{83} +2.58579 q^{85} -16.4853 q^{87} -12.7279 q^{89} -9.65685 q^{93} -0.585786 q^{95} +7.75736 q^{97} +7.17157 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 4 q^{3} - 2 q^{5} + 6 q^{9}+O(q^{10}) $ 2 * q + 4 * q^3 - 2 * q^5 + 6 * q^9	$ 2 q + 4 q^{3} - 2 q^{5} + 6 q^{9} - 4 q^{11} + 4 q^{13} - 4 q^{15} - 8 q^{17} + 4 q^{19} + 8 q^{23} + 2 q^{25} + 16 q^{27} - 4 q^{29} - 4 q^{37} + 16 q^{39} - 8 q^{41} + 12 q^{43} - 6 q^{45} + 16 q^{47} - 12 q^{51} - 4 q^{53} + 4 q^{55} + 4 q^{57} + 20 q^{59} + 4 q^{61} - 4 q^{65} + 8 q^{67} + 8 q^{69} - 8 q^{71} + 16 q^{73} + 4 q^{75} + 8 q^{79} + 46 q^{81} + 4 q^{83} + 8 q^{85} - 16 q^{87} - 8 q^{93} - 4 q^{95} + 24 q^{97} + 20 q^{99}+O(q^{100}) $ 2 * q + 4 * q^3 - 2 * q^5 + 6 * q^9 - 4 * q^11 + 4 * q^13 - 4 * q^15 - 8 * q^17 + 4 * q^19 + 8 * q^23 + 2 * q^25 + 16 * q^27 - 4 * q^29 - 4 * q^37 + 16 * q^39 - 8 * q^41 + 12 * q^43 - 6 * q^45 + 16 * q^47 - 12 * q^51 - 4 * q^53 + 4 * q^55 + 4 * q^57 + 20 * q^59 + 4 * q^61 - 4 * q^65 + 8 * q^67 + 8 * q^69 - 8 * q^71 + 16 * q^73 + 4 * q^75 + 8 * q^79 + 46 * q^81 + 4 * q^83 + 8 * q^85 - 16 * q^87 - 8 * q^93 - 4 * q^95 + 24 * q^97 + 20 * 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
$3$	3.41421	1.97120	0.985599	−	0.169102i	$-0.0540867\pi$
$3$	3.41421	1.97120	0.985599	+	0.169102i	$0.0540867\pi$
$4$	0	0
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	8.65685	2.88562
$10$	0	0
$11$	0.828427	0.249780	0.124890	−	0.992171i	$-0.460142\pi$
$11$	0.828427	0.249780	0.124890	+	0.992171i	$0.460142\pi$
$12$	0	0
$13$	4.82843	1.33916	0.669582	−	0.742738i	$-0.266473\pi$
$13$	4.82843	1.33916	0.669582	+	0.742738i	$0.266473\pi$
$14$	0	0
$15$	−3.41421	−0.881546
$16$	0	0
$17$	−2.58579	−0.627145	−0.313573	−	0.949564i	$-0.601526\pi$
$17$	−2.58579	−0.627145	−0.313573	+	0.949564i	$0.601526\pi$
$18$	0	0
$19$	0.585786	0.134389	0.0671943	−	0.997740i	$-0.478595\pi$
$19$	0.585786	0.134389	0.0671943	+	0.997740i	$0.478595\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	1.17157	0.244290	0.122145	−	0.992512i	$-0.461023\pi$
$23$	1.17157	0.244290	0.122145	+	0.992512i	$0.461023\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	19.3137	3.71692
$28$	0	0
$29$	−4.82843	−0.896616	−0.448308	−	0.893879i	$-0.647973\pi$
$29$	−4.82843	−0.896616	−0.448308	+	0.893879i	$0.647973\pi$
$30$	0	0
$31$	−2.82843	−0.508001	−0.254000	−	0.967204i	$-0.581746\pi$
$31$	−2.82843	−0.508001	−0.254000	+	0.967204i	$0.581746\pi$
$32$	0	0
$33$	2.82843	0.492366
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−7.65685	−1.25878	−0.629390	−	0.777090i	$-0.716695\pi$
$37$	−7.65685	−1.25878	−0.629390	+	0.777090i	$0.716695\pi$
$38$	0	0
$39$	16.4853	2.63976
$40$	0	0
$41$	3.07107	0.479620	0.239810	−	0.970820i	$-0.422915\pi$
$41$	3.07107	0.479620	0.239810	+	0.970820i	$0.422915\pi$
$42$	0	0
$43$	8.82843	1.34632	0.673161	−	0.739496i	$-0.264936\pi$
$43$	8.82843	1.34632	0.673161	+	0.739496i	$0.264936\pi$
$44$	0	0
$45$	−8.65685	−1.29049
$46$	0	0
$47$	5.17157	0.754351	0.377176	−	0.926142i	$-0.376895\pi$
$47$	5.17157	0.754351	0.377176	+	0.926142i	$0.376895\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	−8.82843	−1.23623
$52$	0	0
$53$	6.48528	0.890822	0.445411	−	0.895326i	$-0.353058\pi$
$53$	6.48528	0.890822	0.445411	+	0.895326i	$0.353058\pi$
$54$	0	0
$55$	−0.828427	−0.111705
$56$	0	0
$57$	2.00000	0.264906
$58$	0	0
$59$	8.58579	1.11777	0.558887	−	0.829244i	$-0.311229\pi$
$59$	8.58579	1.11777	0.558887	+	0.829244i	$0.311229\pi$
$60$	0	0
$61$	−9.31371	−1.19250	−0.596249	−	0.802799i	$-0.703343\pi$
$61$	−9.31371	−1.19250	−0.596249	+	0.802799i	$0.703343\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−4.82843	−0.598893
$66$	0	0
$67$	−1.65685	−0.202417	−0.101208	−	0.994865i	$-0.532271\pi$
$67$	−1.65685	−0.202417	−0.101208	+	0.994865i	$0.532271\pi$
$68$	0	0
$69$	4.00000	0.481543
$70$	0	0
$71$	4.48528	0.532305	0.266152	−	0.963931i	$-0.414248\pi$
$71$	4.48528	0.532305	0.266152	+	0.963931i	$0.414248\pi$
$72$	0	0
$73$	9.41421	1.10185	0.550925	−	0.834555i	$-0.314275\pi$
$73$	9.41421	1.10185	0.550925	+	0.834555i	$0.314275\pi$
$74$	0	0
$75$	3.41421	0.394239
$76$	0	0
$77$	0	0
$78$	0	0
$79$	6.82843	0.768258	0.384129	−	0.923279i	$-0.374502\pi$
$79$	6.82843	0.768258	0.384129	+	0.923279i	$0.374502\pi$
$80$	0	0
$81$	39.9706	4.44117
$82$	0	0
$83$	−2.24264	−0.246162	−0.123081	−	0.992397i	$-0.539277\pi$
$83$	−2.24264	−0.246162	−0.123081	+	0.992397i	$0.539277\pi$
$84$	0	0
$85$	2.58579	0.280468
$86$	0	0
$87$	−16.4853	−1.76741
$88$	0	0
$89$	−12.7279	−1.34916	−0.674579	−	0.738203i	$-0.735675\pi$
$89$	−12.7279	−1.34916	−0.674579	+	0.738203i	$0.735675\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	−9.65685	−1.00137
$94$	0	0
$95$	−0.585786	−0.0601004
$96$	0	0
$97$	7.75736	0.787641	0.393820	−	0.919187i	$-0.371153\pi$
$97$	7.75736	0.787641	0.393820	+	0.919187i	$0.371153\pi$
$98$	0	0
$99$	7.17157	0.720770
$100$	0	0
$101$	−13.3137	−1.32476	−0.662382	−	0.749166i	$-0.730454\pi$
$101$	−13.3137	−1.32476	−0.662382	+	0.749166i	$0.730454\pi$
$102$	0	0
$103$	−14.8284	−1.46109	−0.730544	−	0.682865i	$-0.760734\pi$
$103$	−14.8284	−1.46109	−0.730544	+	0.682865i	$0.760734\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−9.65685	−0.933563	−0.466782	−	0.884373i	$-0.654587\pi$
$107$	−9.65685	−0.933563	−0.466782	+	0.884373i	$0.654587\pi$
$108$	0	0
$109$	2.48528	0.238047	0.119023	−	0.992891i	$-0.462024\pi$
$109$	2.48528	0.238047	0.119023	+	0.992891i	$0.462024\pi$
$110$	0	0
$111$	−26.1421	−2.48130
$112$	0	0
$113$	−15.3137	−1.44059	−0.720296	−	0.693667i	$-0.755994\pi$
$113$	−15.3137	−1.44059	−0.720296	+	0.693667i	$0.755994\pi$
$114$	0	0
$115$	−1.17157	−0.109250
$116$	0	0
$117$	41.7990	3.86432
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−10.3137	−0.937610
$122$	0	0
$123$	10.4853	0.945426
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−2.82843	−0.250982	−0.125491	−	0.992095i	$-0.540051\pi$
$127$	−2.82843	−0.250982	−0.125491	+	0.992095i	$0.540051\pi$
$128$	0	0
$129$	30.1421	2.65387
$130$	0	0
$131$	−6.24264	−0.545422	−0.272711	−	0.962096i	$-0.587920\pi$
$131$	−6.24264	−0.545422	−0.272711	+	0.962096i	$0.587920\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	−19.3137	−1.66226
$136$	0	0
$137$	−16.0000	−1.36697	−0.683486	−	0.729964i	$-0.739537\pi$
$137$	−16.0000	−1.36697	−0.683486	+	0.729964i	$0.739537\pi$
$138$	0	0
$139$	19.8995	1.68785	0.843927	−	0.536459i	$-0.180238\pi$
$139$	19.8995	1.68785	0.843927	+	0.536459i	$0.180238\pi$
$140$	0	0
$141$	17.6569	1.48698
$142$	0	0
$143$	4.00000	0.334497
$144$	0	0
$145$	4.82843	0.400979
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−6.00000	−0.491539	−0.245770	−	0.969328i	$-0.579041\pi$
$149$	−6.00000	−0.491539	−0.245770	+	0.969328i	$0.579041\pi$
$150$	0	0
$151$	11.3137	0.920697	0.460348	−	0.887738i	$-0.347725\pi$
$151$	11.3137	0.920697	0.460348	+	0.887738i	$0.347725\pi$
$152$	0	0
$153$	−22.3848	−1.80970
$154$	0	0
$155$	2.82843	0.227185
$156$	0	0
$157$	−6.48528	−0.517582	−0.258791	−	0.965933i	$-0.583324\pi$
$157$	−6.48528	−0.517582	−0.258791	+	0.965933i	$0.583324\pi$
$158$	0	0
$159$	22.1421	1.75599
$160$	0	0
$161$	0	0
$162$	0	0
$163$	20.1421	1.57765	0.788827	−	0.614615i	$-0.210689\pi$
$163$	20.1421	1.57765	0.788827	+	0.614615i	$0.210689\pi$
$164$	0	0
$165$	−2.82843	−0.220193
$166$	0	0
$167$	15.7990	1.22256	0.611281	−	0.791413i	$-0.290654\pi$
$167$	15.7990	1.22256	0.611281	+	0.791413i	$0.290654\pi$
$168$	0	0
$169$	10.3137	0.793362
$170$	0	0
$171$	5.07107	0.387794
$172$	0	0
$173$	8.82843	0.671213	0.335606	−	0.942002i	$-0.391059\pi$
$173$	8.82843	0.671213	0.335606	+	0.942002i	$0.391059\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	29.3137	2.20335
$178$	0	0
$179$	−4.00000	−0.298974	−0.149487	−	0.988764i	$-0.547762\pi$
$179$	−4.00000	−0.298974	−0.149487	+	0.988764i	$0.547762\pi$
$180$	0	0
$181$	2.48528	0.184730	0.0923648	−	0.995725i	$-0.470557\pi$
$181$	2.48528	0.184730	0.0923648	+	0.995725i	$0.470557\pi$
$182$	0	0
$183$	−31.7990	−2.35065
$184$	0	0
$185$	7.65685	0.562943
$186$	0	0
$187$	−2.14214	−0.156648
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−10.1421	−0.733859	−0.366930	−	0.930249i	$-0.619591\pi$
$191$	−10.1421	−0.733859	−0.366930	+	0.930249i	$0.619591\pi$
$192$	0	0
$193$	5.65685	0.407189	0.203595	−	0.979055i	$-0.434738\pi$
$193$	5.65685	0.407189	0.203595	+	0.979055i	$0.434738\pi$
$194$	0	0
$195$	−16.4853	−1.18054
$196$	0	0
$197$	−25.7990	−1.83810	−0.919051	−	0.394139i	$-0.871043\pi$
$197$	−25.7990	−1.83810	−0.919051	+	0.394139i	$0.871043\pi$
$198$	0	0
$199$	16.4853	1.16861	0.584305	−	0.811534i	$-0.301367\pi$
$199$	16.4853	1.16861	0.584305	+	0.811534i	$0.301367\pi$
$200$	0	0
$201$	−5.65685	−0.399004
$202$	0	0
$203$	0	0
$204$	0	0
$205$	−3.07107	−0.214493
$206$	0	0
$207$	10.1421	0.704927
$208$	0	0
$209$	0.485281	0.0335676
$210$	0	0
$211$	−18.6274	−1.28236	−0.641182	−	0.767389i	$-0.721556\pi$
$211$	−18.6274	−1.28236	−0.641182	+	0.767389i	$0.721556\pi$
$212$	0	0
$213$	15.3137	1.04928
$214$	0	0
$215$	−8.82843	−0.602094
$216$	0	0
$217$	0	0
$218$	0	0
$219$	32.1421	2.17196
$220$	0	0
$221$	−12.4853	−0.839851
$222$	0	0
$223$	−7.31371	−0.489762	−0.244881	−	0.969553i	$-0.578749\pi$
$223$	−7.31371	−0.489762	−0.244881	+	0.969553i	$0.578749\pi$
$224$	0	0
$225$	8.65685	0.577124
$226$	0	0
$227$	18.2426	1.21081	0.605403	−	0.795919i	$-0.293012\pi$
$227$	18.2426	1.21081	0.605403	+	0.795919i	$0.293012\pi$
$228$	0	0
$229$	16.1421	1.06670	0.533351	−	0.845894i	$-0.320932\pi$
$229$	16.1421	1.06670	0.533351	+	0.845894i	$0.320932\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	23.3137	1.52733	0.763666	−	0.645612i	$-0.223397\pi$
$233$	23.3137	1.52733	0.763666	+	0.645612i	$0.223397\pi$
$234$	0	0
$235$	−5.17157	−0.337356
$236$	0	0
$237$	23.3137	1.51439
$238$	0	0
$239$	−1.65685	−0.107173	−0.0535865	−	0.998563i	$-0.517065\pi$
$239$	−1.65685	−0.107173	−0.0535865	+	0.998563i	$0.517065\pi$
$240$	0	0
$241$	13.4142	0.864085	0.432043	−	0.901853i	$-0.357793\pi$
$241$	13.4142	0.864085	0.432043	+	0.901853i	$0.357793\pi$
$242$	0	0
$243$	78.5269	5.03750
$244$	0	0
$245$	0	0
$246$	0	0
$247$	2.82843	0.179969
$248$	0	0
$249$	−7.65685	−0.485233
$250$	0	0
$251$	0.585786	0.0369745	0.0184873	−	0.999829i	$-0.494115\pi$
$251$	0.585786	0.0369745	0.0184873	+	0.999829i	$0.494115\pi$
$252$	0	0
$253$	0.970563	0.0610188
$254$	0	0
$255$	8.82843	0.552858
$256$	0	0
$257$	9.89949	0.617514	0.308757	−	0.951141i	$-0.400087\pi$
$257$	9.89949	0.617514	0.308757	+	0.951141i	$0.400087\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	−41.7990	−2.58729
$262$	0	0
$263$	−28.0000	−1.72655	−0.863277	−	0.504730i	$-0.831592\pi$
$263$	−28.0000	−1.72655	−0.863277	+	0.504730i	$0.831592\pi$
$264$	0	0
$265$	−6.48528	−0.398388
$266$	0	0
$267$	−43.4558	−2.65945
$268$	0	0
$269$	−18.4853	−1.12707	−0.563534	−	0.826093i	$-0.690559\pi$
$269$	−18.4853	−1.12707	−0.563534	+	0.826093i	$0.690559\pi$
$270$	0	0
$271$	12.0000	0.728948	0.364474	−	0.931214i	$-0.381249\pi$
$271$	12.0000	0.728948	0.364474	+	0.931214i	$0.381249\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0.828427	0.0499560
$276$	0	0
$277$	−8.14214	−0.489214	−0.244607	−	0.969622i	$-0.578659\pi$
$277$	−8.14214	−0.489214	−0.244607	+	0.969622i	$0.578659\pi$
$278$	0	0
$279$	−24.4853	−1.46590
$280$	0	0
$281$	8.00000	0.477240	0.238620	−	0.971113i	$-0.423305\pi$
$281$	8.00000	0.477240	0.238620	+	0.971113i	$0.423305\pi$
$282$	0	0
$283$	−2.24264	−0.133311	−0.0666556	−	0.997776i	$-0.521233\pi$
$283$	−2.24264	−0.133311	−0.0666556	+	0.997776i	$0.521233\pi$
$284$	0	0
$285$	−2.00000	−0.118470
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−10.3137	−0.606689
$290$	0	0
$291$	26.4853	1.55259
$292$	0	0
$293$	8.34315	0.487412	0.243706	−	0.969849i	$-0.421637\pi$
$293$	8.34315	0.487412	0.243706	+	0.969849i	$0.421637\pi$
$294$	0	0
$295$	−8.58579	−0.499884
$296$	0	0
$297$	16.0000	0.928414
$298$	0	0
$299$	5.65685	0.327144
$300$	0	0
$301$	0	0
$302$	0	0
$303$	−45.4558	−2.61137
$304$	0	0
$305$	9.31371	0.533301
$306$	0	0
$307$	14.9289	0.852039	0.426020	−	0.904714i	$-0.359915\pi$
$307$	14.9289	0.852039	0.426020	+	0.904714i	$0.359915\pi$
$308$	0	0
$309$	−50.6274	−2.88009
$310$	0	0
$311$	4.00000	0.226819	0.113410	−	0.993548i	$-0.463823\pi$
$311$	4.00000	0.226819	0.113410	+	0.993548i	$0.463823\pi$
$312$	0	0
$313$	−14.3848	−0.813076	−0.406538	−	0.913634i	$-0.633264\pi$
$313$	−14.3848	−0.813076	−0.406538	+	0.913634i	$0.633264\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	10.4853	0.588912	0.294456	−	0.955665i	$-0.404862\pi$
$317$	10.4853	0.588912	0.294456	+	0.955665i	$0.404862\pi$
$318$	0	0
$319$	−4.00000	−0.223957
$320$	0	0
$321$	−32.9706	−1.84024
$322$	0	0
$323$	−1.51472	−0.0842812
$324$	0	0
$325$	4.82843	0.267833
$326$	0	0
$327$	8.48528	0.469237
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−33.7990	−1.85776	−0.928880	−	0.370380i	$-0.879227\pi$
$331$	−33.7990	−1.85776	−0.928880	+	0.370380i	$0.879227\pi$
$332$	0	0
$333$	−66.2843	−3.63236
$334$	0	0
$335$	1.65685	0.0905236
$336$	0	0
$337$	6.00000	0.326841	0.163420	−	0.986557i	$-0.447747\pi$
$337$	6.00000	0.326841	0.163420	+	0.986557i	$0.447747\pi$
$338$	0	0
$339$	−52.2843	−2.83969
$340$	0	0
$341$	−2.34315	−0.126888
$342$	0	0
$343$	0	0
$344$	0	0
$345$	−4.00000	−0.215353
$346$	0	0
$347$	3.17157	0.170259	0.0851295	−	0.996370i	$-0.472870\pi$
$347$	3.17157	0.170259	0.0851295	+	0.996370i	$0.472870\pi$
$348$	0	0
$349$	−2.48528	−0.133034	−0.0665170	−	0.997785i	$-0.521189\pi$
$349$	−2.48528	−0.133034	−0.0665170	+	0.997785i	$0.521189\pi$
$350$	0	0
$351$	93.2548	4.97757
$352$	0	0
$353$	2.38478	0.126929	0.0634644	−	0.997984i	$-0.479785\pi$
$353$	2.38478	0.126929	0.0634644	+	0.997984i	$0.479785\pi$
$354$	0	0
$355$	−4.48528	−0.238054
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−28.2843	−1.49279	−0.746393	−	0.665505i	$-0.768216\pi$
$359$	−28.2843	−1.49279	−0.746393	+	0.665505i	$0.768216\pi$
$360$	0	0
$361$	−18.6569	−0.981940
$362$	0	0
$363$	−35.2132	−1.84821
$364$	0	0
$365$	−9.41421	−0.492762
$366$	0	0
$367$	24.9706	1.30345	0.651726	−	0.758454i	$-0.274045\pi$
$367$	24.9706	1.30345	0.651726	+	0.758454i	$0.274045\pi$
$368$	0	0
$369$	26.5858	1.38400
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−30.4853	−1.57847	−0.789234	−	0.614093i	$-0.789522\pi$
$373$	−30.4853	−1.57847	−0.789234	+	0.614093i	$0.789522\pi$
$374$	0	0
$375$	−3.41421	−0.176309
$376$	0	0
$377$	−23.3137	−1.20072
$378$	0	0
$379$	−34.4853	−1.77139	−0.885695	−	0.464268i	$-0.846318\pi$
$379$	−34.4853	−1.77139	−0.885695	+	0.464268i	$0.846318\pi$
$380$	0	0
$381$	−9.65685	−0.494736
$382$	0	0
$383$	−32.4853	−1.65992	−0.829960	−	0.557823i	$-0.811637\pi$
$383$	−32.4853	−1.65992	−0.829960	+	0.557823i	$0.811637\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	76.4264	3.88497
$388$	0	0
$389$	28.1421	1.42686	0.713431	−	0.700725i	$-0.247140\pi$
$389$	28.1421	1.42686	0.713431	+	0.700725i	$0.247140\pi$
$390$	0	0
$391$	−3.02944	−0.153205
$392$	0	0
$393$	−21.3137	−1.07513
$394$	0	0
$395$	−6.82843	−0.343575
$396$	0	0
$397$	−33.7990	−1.69632	−0.848161	−	0.529738i	$-0.822290\pi$
$397$	−33.7990	−1.69632	−0.848161	+	0.529738i	$0.822290\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−6.00000	−0.299626	−0.149813	−	0.988714i	$-0.547867\pi$
$401$	−6.00000	−0.299626	−0.149813	+	0.988714i	$0.547867\pi$
$402$	0	0
$403$	−13.6569	−0.680296
$404$	0	0
$405$	−39.9706	−1.98615
$406$	0	0
$407$	−6.34315	−0.314418
$408$	0	0
$409$	10.5858	0.523433	0.261717	−	0.965145i	$-0.415711\pi$
$409$	10.5858	0.523433	0.261717	+	0.965145i	$0.415711\pi$
$410$	0	0
$411$	−54.6274	−2.69457
$412$	0	0
$413$	0	0
$414$	0	0
$415$	2.24264	0.110087
$416$	0	0
$417$	67.9411	3.32709
$418$	0	0
$419$	−20.8701	−1.01957	−0.509785	−	0.860302i	$-0.670275\pi$
$419$	−20.8701	−1.01957	−0.509785	+	0.860302i	$0.670275\pi$
$420$	0	0
$421$	17.3137	0.843819	0.421909	−	0.906638i	$-0.361360\pi$
$421$	17.3137	0.843819	0.421909	+	0.906638i	$0.361360\pi$
$422$	0	0
$423$	44.7696	2.17677
$424$	0	0
$425$	−2.58579	−0.125429
$426$	0	0
$427$	0	0
$428$	0	0
$429$	13.6569	0.659359
$430$	0	0
$431$	22.3431	1.07623	0.538116	−	0.842871i	$-0.319136\pi$
$431$	22.3431	1.07623	0.538116	+	0.842871i	$0.319136\pi$
$432$	0	0
$433$	−10.5858	−0.508720	−0.254360	−	0.967110i	$-0.581865\pi$
$433$	−10.5858	−0.508720	−0.254360	+	0.967110i	$0.581865\pi$
$434$	0	0
$435$	16.4853	0.790409
$436$	0	0
$437$	0.686292	0.0328298
$438$	0	0
$439$	−24.9706	−1.19178	−0.595890	−	0.803066i	$-0.703201\pi$
$439$	−24.9706	−1.19178	−0.595890	+	0.803066i	$0.703201\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−3.02944	−0.143933	−0.0719665	−	0.997407i	$-0.522927\pi$
$443$	−3.02944	−0.143933	−0.0719665	+	0.997407i	$0.522927\pi$
$444$	0	0
$445$	12.7279	0.603361
$446$	0	0
$447$	−20.4853	−0.968921
$448$	0	0
$449$	−16.6274	−0.784696	−0.392348	−	0.919817i	$-0.628337\pi$
$449$	−16.6274	−0.784696	−0.392348	+	0.919817i	$0.628337\pi$
$450$	0	0
$451$	2.54416	0.119800
$452$	0	0
$453$	38.6274	1.81487
$454$	0	0
$455$	0	0
$456$	0	0
$457$	21.6569	1.01306	0.506532	−	0.862221i	$-0.330927\pi$
$457$	21.6569	1.01306	0.506532	+	0.862221i	$0.330927\pi$
$458$	0	0
$459$	−49.9411	−2.33105
$460$	0	0
$461$	12.8284	0.597479	0.298740	−	0.954335i	$-0.403434\pi$
$461$	12.8284	0.597479	0.298740	+	0.954335i	$0.403434\pi$
$462$	0	0
$463$	16.9706	0.788689	0.394344	−	0.918963i	$-0.370972\pi$
$463$	16.9706	0.788689	0.394344	+	0.918963i	$0.370972\pi$
$464$	0	0
$465$	9.65685	0.447826
$466$	0	0
$467$	−15.8995	−0.735741	−0.367870	−	0.929877i	$-0.619913\pi$
$467$	−15.8995	−0.735741	−0.367870	+	0.929877i	$0.619913\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	−22.1421	−1.02026
$472$	0	0
$473$	7.31371	0.336285
$474$	0	0
$475$	0.585786	0.0268777
$476$	0	0
$477$	56.1421	2.57057
$478$	0	0
$479$	−17.1716	−0.784589	−0.392295	−	0.919840i	$-0.628319\pi$
$479$	−17.1716	−0.784589	−0.392295	+	0.919840i	$0.628319\pi$
$480$	0	0
$481$	−36.9706	−1.68571
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−7.75736	−0.352244
$486$	0	0
$487$	−31.7990	−1.44095	−0.720475	−	0.693481i	$-0.756076\pi$
$487$	−31.7990	−1.44095	−0.720475	+	0.693481i	$0.756076\pi$
$488$	0	0
$489$	68.7696	3.10987
$490$	0	0
$491$	−32.2843	−1.45697	−0.728484	−	0.685062i	$-0.759775\pi$
$491$	−32.2843	−1.45697	−0.728484	+	0.685062i	$0.759775\pi$
$492$	0	0
$493$	12.4853	0.562309
$494$	0	0
$495$	−7.17157	−0.322338
$496$	0	0
$497$	0	0
$498$	0	0
$499$	−30.3431	−1.35835	−0.679173	−	0.733978i	$-0.737661\pi$
$499$	−30.3431	−1.35835	−0.679173	+	0.733978i	$0.737661\pi$
$500$	0	0
$501$	53.9411	2.40991
$502$	0	0
$503$	17.6569	0.787280	0.393640	−	0.919265i	$-0.371216\pi$
$503$	17.6569	0.787280	0.393640	+	0.919265i	$0.371216\pi$
$504$	0	0
$505$	13.3137	0.592452
$506$	0	0
$507$	35.2132	1.56387
$508$	0	0
$509$	−5.79899	−0.257036	−0.128518	−	0.991707i	$-0.541022\pi$
$509$	−5.79899	−0.257036	−0.128518	+	0.991707i	$0.541022\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	11.3137	0.499512
$514$	0	0
$515$	14.8284	0.653419
$516$	0	0
$517$	4.28427	0.188422
$518$	0	0
$519$	30.1421	1.32309
$520$	0	0
$521$	−19.0711	−0.835519	−0.417759	−	0.908558i	$-0.637184\pi$
$521$	−19.0711	−0.835519	−0.417759	+	0.908558i	$0.637184\pi$
$522$	0	0
$523$	23.8995	1.04505	0.522526	−	0.852623i	$-0.324990\pi$
$523$	23.8995	1.04505	0.522526	+	0.852623i	$0.324990\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	7.31371	0.318590
$528$	0	0
$529$	−21.6274	−0.940322
$530$	0	0
$531$	74.3259	3.22547
$532$	0	0
$533$	14.8284	0.642290
$534$	0	0
$535$	9.65685	0.417502
$536$	0	0
$537$	−13.6569	−0.589337
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−14.9706	−0.643635	−0.321817	−	0.946802i	$-0.604294\pi$
$541$	−14.9706	−0.643635	−0.321817	+	0.946802i	$0.604294\pi$
$542$	0	0
$543$	8.48528	0.364138
$544$	0	0
$545$	−2.48528	−0.106458
$546$	0	0
$547$	10.4853	0.448318	0.224159	−	0.974553i	$-0.428036\pi$
$547$	10.4853	0.448318	0.224159	+	0.974553i	$0.428036\pi$
$548$	0	0
$549$	−80.6274	−3.44109
$550$	0	0
$551$	−2.82843	−0.120495
$552$	0	0
$553$	0	0
$554$	0	0
$555$	26.1421	1.10967
$556$	0	0
$557$	15.1716	0.642840	0.321420	−	0.946937i	$-0.395840\pi$
$557$	15.1716	0.642840	0.321420	+	0.946937i	$0.395840\pi$
$558$	0	0
$559$	42.6274	1.80295
$560$	0	0
$561$	−7.31371	−0.308785
$562$	0	0
$563$	−36.5858	−1.54191	−0.770954	−	0.636891i	$-0.780220\pi$
$563$	−36.5858	−1.54191	−0.770954	+	0.636891i	$0.780220\pi$
$564$	0	0
$565$	15.3137	0.644253
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−29.3137	−1.22889	−0.614447	−	0.788958i	$-0.710621\pi$
$569$	−29.3137	−1.22889	−0.614447	+	0.788958i	$0.710621\pi$
$570$	0	0
$571$	2.20101	0.0921094	0.0460547	−	0.998939i	$-0.485335\pi$
$571$	2.20101	0.0921094	0.0460547	+	0.998939i	$0.485335\pi$
$572$	0	0
$573$	−34.6274	−1.44658
$574$	0	0
$575$	1.17157	0.0488580
$576$	0	0
$577$	−6.10051	−0.253967	−0.126984	−	0.991905i	$-0.540530\pi$
$577$	−6.10051	−0.253967	−0.126984	+	0.991905i	$0.540530\pi$
$578$	0	0
$579$	19.3137	0.802650
$580$	0	0
$581$	0	0
$582$	0	0
$583$	5.37258	0.222510
$584$	0	0
$585$	−41.7990	−1.72818
$586$	0	0
$587$	17.0711	0.704598	0.352299	−	0.935887i	$-0.385400\pi$
$587$	17.0711	0.704598	0.352299	+	0.935887i	$0.385400\pi$
$588$	0	0
$589$	−1.65685	−0.0682695
$590$	0	0
$591$	−88.0833	−3.62326
$592$	0	0
$593$	−3.27208	−0.134368	−0.0671841	−	0.997741i	$-0.521401\pi$
$593$	−3.27208	−0.134368	−0.0671841	+	0.997741i	$0.521401\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	56.2843	2.30356
$598$	0	0
$599$	10.8284	0.442438	0.221219	−	0.975224i	$-0.428997\pi$
$599$	10.8284	0.442438	0.221219	+	0.975224i	$0.428997\pi$
$600$	0	0
$601$	−6.58579	−0.268640	−0.134320	−	0.990938i	$-0.542885\pi$
$601$	−6.58579	−0.268640	−0.134320	+	0.990938i	$0.542885\pi$
$602$	0	0
$603$	−14.3431	−0.584098
$604$	0	0
$605$	10.3137	0.419312
$606$	0	0
$607$	16.2843	0.660958	0.330479	−	0.943813i	$-0.392790\pi$
$607$	16.2843	0.660958	0.330479	+	0.943813i	$0.392790\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	24.9706	1.01020
$612$	0	0
$613$	−12.3431	−0.498535	−0.249267	−	0.968435i	$-0.580190\pi$
$613$	−12.3431	−0.498535	−0.249267	+	0.968435i	$0.580190\pi$
$614$	0	0
$615$	−10.4853	−0.422807
$616$	0	0
$617$	−33.3137	−1.34116	−0.670580	−	0.741837i	$-0.733955\pi$
$617$	−33.3137	−1.34116	−0.670580	+	0.741837i	$0.733955\pi$
$618$	0	0
$619$	29.0711	1.16846	0.584232	−	0.811586i	$-0.301396\pi$
$619$	29.0711	1.16846	0.584232	+	0.811586i	$0.301396\pi$
$620$	0	0
$621$	22.6274	0.908007
$622$	0	0
$623$	0	0
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	1.65685	0.0661684
$628$	0	0
$629$	19.7990	0.789437
$630$	0	0
$631$	12.4853	0.497031	0.248516	−	0.968628i	$-0.420057\pi$
$631$	12.4853	0.497031	0.248516	+	0.968628i	$0.420057\pi$
$632$	0	0
$633$	−63.5980	−2.52779
$634$	0	0
$635$	2.82843	0.112243
$636$	0	0
$637$	0	0
$638$	0	0
$639$	38.8284	1.53603
$640$	0	0
$641$	−24.6274	−0.972724	−0.486362	−	0.873757i	$-0.661676\pi$
$641$	−24.6274	−0.972724	−0.486362	+	0.873757i	$0.661676\pi$
$642$	0	0
$643$	4.78680	0.188773	0.0943864	−	0.995536i	$-0.469911\pi$
$643$	4.78680	0.188773	0.0943864	+	0.995536i	$0.469911\pi$
$644$	0	0
$645$	−30.1421	−1.18685
$646$	0	0
$647$	−23.1127	−0.908654	−0.454327	−	0.890835i	$-0.650120\pi$
$647$	−23.1127	−0.908654	−0.454327	+	0.890835i	$0.650120\pi$
$648$	0	0
$649$	7.11270	0.279198
$650$	0	0
$651$	0	0
$652$	0	0
$653$	4.34315	0.169960	0.0849802	−	0.996383i	$-0.472917\pi$
$653$	4.34315	0.169960	0.0849802	+	0.996383i	$0.472917\pi$
$654$	0	0
$655$	6.24264	0.243920
$656$	0	0
$657$	81.4975	3.17952
$658$	0	0
$659$	27.1716	1.05845	0.529227	−	0.848480i	$-0.322482\pi$
$659$	27.1716	1.05845	0.529227	+	0.848480i	$0.322482\pi$
$660$	0	0
$661$	38.2843	1.48909	0.744543	−	0.667575i	$-0.232668\pi$
$661$	38.2843	1.48909	0.744543	+	0.667575i	$0.232668\pi$
$662$	0	0
$663$	−42.6274	−1.65551
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−5.65685	−0.219034
$668$	0	0
$669$	−24.9706	−0.965418
$670$	0	0
$671$	−7.71573	−0.297862
$672$	0	0
$673$	−48.0000	−1.85026	−0.925132	−	0.379646i	$-0.876046\pi$
$673$	−48.0000	−1.85026	−0.925132	+	0.379646i	$0.876046\pi$
$674$	0	0
$675$	19.3137	0.743385
$676$	0	0
$677$	−39.4558	−1.51641	−0.758206	−	0.652015i	$-0.773924\pi$
$677$	−39.4558	−1.51641	−0.758206	+	0.652015i	$0.773924\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	62.2843	2.38674
$682$	0	0
$683$	−33.6569	−1.28784	−0.643922	−	0.765091i	$-0.722694\pi$
$683$	−33.6569	−1.28784	−0.643922	+	0.765091i	$0.722694\pi$
$684$	0	0
$685$	16.0000	0.611329
$686$	0	0
$687$	55.1127	2.10268
$688$	0	0
$689$	31.3137	1.19296
$690$	0	0
$691$	1.75736	0.0668531	0.0334265	−	0.999441i	$-0.489358\pi$
$691$	1.75736	0.0668531	0.0334265	+	0.999441i	$0.489358\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−19.8995	−0.754831
$696$	0	0
$697$	−7.94113	−0.300792
$698$	0	0
$699$	79.5980	3.01067
$700$	0	0
$701$	2.48528	0.0938678	0.0469339	−	0.998898i	$-0.485055\pi$
$701$	2.48528	0.0938678	0.0469339	+	0.998898i	$0.485055\pi$
$702$	0	0
$703$	−4.48528	−0.169166
$704$	0	0
$705$	−17.6569	−0.664996
$706$	0	0
$707$	0	0
$708$	0	0
$709$	45.1127	1.69424	0.847121	−	0.531399i	$-0.178334\pi$
$709$	45.1127	1.69424	0.847121	+	0.531399i	$0.178334\pi$
$710$	0	0
$711$	59.1127	2.21690
$712$	0	0
$713$	−3.31371	−0.124099
$714$	0	0
$715$	−4.00000	−0.149592
$716$	0	0
$717$	−5.65685	−0.211259
$718$	0	0
$719$	41.4558	1.54604	0.773021	−	0.634380i	$-0.218745\pi$
$719$	41.4558	1.54604	0.773021	+	0.634380i	$0.218745\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	45.7990	1.70328
$724$	0	0
$725$	−4.82843	−0.179323
$726$	0	0
$727$	−3.51472	−0.130354	−0.0651768	−	0.997874i	$-0.520761\pi$
$727$	−3.51472	−0.130354	−0.0651768	+	0.997874i	$0.520761\pi$
$728$	0	0
$729$	148.196	5.48874
$730$	0	0
$731$	−22.8284	−0.844340
$732$	0	0
$733$	−34.0000	−1.25582	−0.627909	−	0.778287i	$-0.716089\pi$
$733$	−34.0000	−1.25582	−0.627909	+	0.778287i	$0.716089\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−1.37258	−0.0505597
$738$	0	0
$739$	−3.17157	−0.116668	−0.0583341	−	0.998297i	$-0.518579\pi$
$739$	−3.17157	−0.116668	−0.0583341	+	0.998297i	$0.518579\pi$
$740$	0	0
$741$	9.65685	0.354753
$742$	0	0
$743$	−51.7990	−1.90032	−0.950160	−	0.311762i	$-0.899081\pi$
$743$	−51.7990	−1.90032	−0.950160	+	0.311762i	$0.899081\pi$
$744$	0	0
$745$	6.00000	0.219823
$746$	0	0
$747$	−19.4142	−0.710329
$748$	0	0
$749$	0	0
$750$	0	0
$751$	39.3137	1.43458	0.717289	−	0.696776i	$-0.245383\pi$
$751$	39.3137	1.43458	0.717289	+	0.696776i	$0.245383\pi$
$752$	0	0
$753$	2.00000	0.0728841
$754$	0	0
$755$	−11.3137	−0.411748
$756$	0	0
$757$	3.65685	0.132911	0.0664553	−	0.997789i	$-0.478831\pi$
$757$	3.65685	0.132911	0.0664553	+	0.997789i	$0.478831\pi$
$758$	0	0
$759$	3.31371	0.120280
$760$	0	0
$761$	−22.3848	−0.811448	−0.405724	−	0.913996i	$-0.632980\pi$
$761$	−22.3848	−0.811448	−0.405724	+	0.913996i	$0.632980\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	22.3848	0.809323
$766$	0	0
$767$	41.4558	1.49688
$768$	0	0
$769$	−19.5563	−0.705220	−0.352610	−	0.935770i	$-0.614706\pi$
$769$	−19.5563	−0.705220	−0.352610	+	0.935770i	$0.614706\pi$
$770$	0	0
$771$	33.7990	1.21724
$772$	0	0
$773$	2.00000	0.0719350	0.0359675	−	0.999353i	$-0.488549\pi$
$773$	2.00000	0.0719350	0.0359675	+	0.999353i	$0.488549\pi$
$774$	0	0
$775$	−2.82843	−0.101600
$776$	0	0
$777$	0	0
$778$	0	0
$779$	1.79899	0.0644555
$780$	0	0
$781$	3.71573	0.132959
$782$	0	0
$783$	−93.2548	−3.33266
$784$	0	0
$785$	6.48528	0.231470
$786$	0	0
$787$	−1.27208	−0.0453447	−0.0226723	−	0.999743i	$-0.507217\pi$
$787$	−1.27208	−0.0453447	−0.0226723	+	0.999743i	$0.507217\pi$
$788$	0	0
$789$	−95.5980	−3.40338
$790$	0	0
$791$	0	0
$792$	0	0
$793$	−44.9706	−1.59695
$794$	0	0
$795$	−22.1421	−0.785301
$796$	0	0
$797$	41.7990	1.48060	0.740298	−	0.672279i	$-0.234684\pi$
$797$	41.7990	1.48060	0.740298	+	0.672279i	$0.234684\pi$
$798$	0	0
$799$	−13.3726	−0.473088
$800$	0	0
$801$	−110.184	−3.89315
$802$	0	0
$803$	7.79899	0.275220
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−63.1127	−2.22167
$808$	0	0
$809$	3.02944	0.106509	0.0532547	−	0.998581i	$-0.483040\pi$
$809$	3.02944	0.106509	0.0532547	+	0.998581i	$0.483040\pi$
$810$	0	0
$811$	32.5858	1.14424	0.572121	−	0.820169i	$-0.306121\pi$
$811$	32.5858	1.14424	0.572121	+	0.820169i	$0.306121\pi$
$812$	0	0
$813$	40.9706	1.43690
$814$	0	0
$815$	−20.1421	−0.705548
$816$	0	0
$817$	5.17157	0.180930
$818$	0	0
$819$	0	0
$820$	0	0
$821$	17.3137	0.604253	0.302126	−	0.953268i	$-0.402304\pi$
$821$	17.3137	0.604253	0.302126	+	0.953268i	$0.402304\pi$
$822$	0	0
$823$	−20.2843	−0.707065	−0.353533	−	0.935422i	$-0.615020\pi$
$823$	−20.2843	−0.707065	−0.353533	+	0.935422i	$0.615020\pi$
$824$	0	0
$825$	2.82843	0.0984732
$826$	0	0
$827$	5.37258	0.186823	0.0934115	−	0.995628i	$-0.470223\pi$
$827$	5.37258	0.186823	0.0934115	+	0.995628i	$0.470223\pi$
$828$	0	0
$829$	5.02944	0.174680	0.0873398	−	0.996179i	$-0.472163\pi$
$829$	5.02944	0.174680	0.0873398	+	0.996179i	$0.472163\pi$
$830$	0	0
$831$	−27.7990	−0.964336
$832$	0	0
$833$	0	0
$834$	0	0
$835$	−15.7990	−0.546747
$836$	0	0
$837$	−54.6274	−1.88820
$838$	0	0
$839$	−42.1421	−1.45491	−0.727454	−	0.686156i	$-0.759297\pi$
$839$	−42.1421	−1.45491	−0.727454	+	0.686156i	$0.759297\pi$
$840$	0	0
$841$	−5.68629	−0.196079
$842$	0	0
$843$	27.3137	0.940734
$844$	0	0
$845$	−10.3137	−0.354802
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−7.65685	−0.262783
$850$	0	0
$851$	−8.97056	−0.307507
$852$	0	0
$853$	43.1716	1.47817	0.739083	−	0.673614i	$-0.235259\pi$
$853$	43.1716	1.47817	0.739083	+	0.673614i	$0.235259\pi$
$854$	0	0
$855$	−5.07107	−0.173427
$856$	0	0
$857$	4.92893	0.168369	0.0841846	−	0.996450i	$-0.473171\pi$
$857$	4.92893	0.168369	0.0841846	+	0.996450i	$0.473171\pi$
$858$	0	0
$859$	−7.21320	−0.246111	−0.123056	−	0.992400i	$-0.539269\pi$
$859$	−7.21320	−0.246111	−0.123056	+	0.992400i	$0.539269\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	4.97056	0.169200	0.0846000	−	0.996415i	$-0.473039\pi$
$863$	4.97056	0.169200	0.0846000	+	0.996415i	$0.473039\pi$
$864$	0	0
$865$	−8.82843	−0.300176
$866$	0	0
$867$	−35.2132	−1.19590
$868$	0	0
$869$	5.65685	0.191896
$870$	0	0
$871$	−8.00000	−0.271070
$872$	0	0
$873$	67.1543	2.27283
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−30.2843	−1.02263	−0.511314	−	0.859394i	$-0.670841\pi$
$877$	−30.2843	−1.02263	−0.511314	+	0.859394i	$0.670841\pi$
$878$	0	0
$879$	28.4853	0.960785
$880$	0	0
$881$	2.38478	0.0803452	0.0401726	−	0.999193i	$-0.487209\pi$
$881$	2.38478	0.0803452	0.0401726	+	0.999193i	$0.487209\pi$
$882$	0	0
$883$	41.6569	1.40186	0.700932	−	0.713228i	$-0.252767\pi$
$883$	41.6569	1.40186	0.700932	+	0.713228i	$0.252767\pi$
$884$	0	0
$885$	−29.3137	−0.985370
$886$	0	0
$887$	−55.1127	−1.85050	−0.925252	−	0.379354i	$-0.876146\pi$
$887$	−55.1127	−1.85050	−0.925252	+	0.379354i	$0.876146\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	33.1127	1.10932
$892$	0	0
$893$	3.02944	0.101376
$894$	0	0
$895$	4.00000	0.133705
$896$	0	0
$897$	19.3137	0.644866
$898$	0	0
$899$	13.6569	0.455482
$900$	0	0
$901$	−16.7696	−0.558675
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−2.48528	−0.0826135
$906$	0	0
$907$	−0.284271	−0.00943907	−0.00471954	−	0.999989i	$-0.501502\pi$
$907$	−0.284271	−0.00943907	−0.00471954	+	0.999989i	$0.501502\pi$
$908$	0	0
$909$	−115.255	−3.82276
$910$	0	0
$911$	−36.2843	−1.20215	−0.601076	−	0.799192i	$-0.705261\pi$
$911$	−36.2843	−1.20215	−0.601076	+	0.799192i	$0.705261\pi$
$912$	0	0
$913$	−1.85786	−0.0614863
$914$	0	0
$915$	31.7990	1.05124
$916$	0	0
$917$	0	0
$918$	0	0
$919$	−15.5147	−0.511783	−0.255892	−	0.966705i	$-0.582369\pi$
$919$	−15.5147	−0.511783	−0.255892	+	0.966705i	$0.582369\pi$
$920$	0	0
$921$	50.9706	1.67954
$922$	0	0
$923$	21.6569	0.712844
$924$	0	0
$925$	−7.65685	−0.251756
$926$	0	0
$927$	−128.368	−4.21614
$928$	0	0
$929$	17.2132	0.564747	0.282373	−	0.959305i	$-0.408878\pi$
$929$	17.2132	0.564747	0.282373	+	0.959305i	$0.408878\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	13.6569	0.447105
$934$	0	0
$935$	2.14214	0.0700553
$936$	0	0
$937$	−20.2426	−0.661298	−0.330649	−	0.943754i	$-0.607268\pi$
$937$	−20.2426	−0.661298	−0.330649	+	0.943754i	$0.607268\pi$
$938$	0	0
$939$	−49.1127	−1.60273
$940$	0	0
$941$	50.0000	1.62995	0.814977	−	0.579494i	$-0.196750\pi$
$941$	50.0000	1.62995	0.814977	+	0.579494i	$0.196750\pi$
$942$	0	0
$943$	3.59798	0.117166
$944$	0	0
$945$	0	0
$946$	0	0
$947$	4.82843	0.156903	0.0784514	−	0.996918i	$-0.475002\pi$
$947$	4.82843	0.156903	0.0784514	+	0.996918i	$0.475002\pi$
$948$	0	0
$949$	45.4558	1.47556
$950$	0	0
$951$	35.7990	1.16086
$952$	0	0
$953$	0.343146	0.0111156	0.00555779	−	0.999985i	$-0.498231\pi$
$953$	0.343146	0.0111156	0.00555779	+	0.999985i	$0.498231\pi$
$954$	0	0
$955$	10.1421	0.328192
$956$	0	0
$957$	−13.6569	−0.441463
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−23.0000	−0.741935
$962$	0	0
$963$	−83.5980	−2.69391
$964$	0	0
$965$	−5.65685	−0.182101
$966$	0	0
$967$	37.4558	1.20450	0.602249	−	0.798308i	$-0.294271\pi$
$967$	37.4558	1.20450	0.602249	+	0.798308i	$0.294271\pi$
$968$	0	0
$969$	−5.17157	−0.166135
$970$	0	0
$971$	−33.3553	−1.07042	−0.535212	−	0.844718i	$-0.679768\pi$
$971$	−33.3553	−1.07042	−0.535212	+	0.844718i	$0.679768\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	16.4853	0.527952
$976$	0	0
$977$	−12.6863	−0.405870	−0.202935	−	0.979192i	$-0.565048\pi$
$977$	−12.6863	−0.405870	−0.202935	+	0.979192i	$0.565048\pi$
$978$	0	0
$979$	−10.5442	−0.336993
$980$	0	0
$981$	21.5147	0.686912
$982$	0	0
$983$	−12.2010	−0.389152	−0.194576	−	0.980887i	$-0.562333\pi$
$983$	−12.2010	−0.389152	−0.194576	+	0.980887i	$0.562333\pi$
$984$	0	0
$985$	25.7990	0.822024
$986$	0	0
$987$	0	0
$988$	0	0
$989$	10.3431	0.328893
$990$	0	0
$991$	−44.7696	−1.42215	−0.711076	−	0.703115i	$-0.751792\pi$
$991$	−44.7696	−1.42215	−0.711076	+	0.703115i	$0.751792\pi$
$992$	0	0
$993$	−115.397	−3.66201
$994$	0	0
$995$	−16.4853	−0.522619
$996$	0	0
$997$	18.2843	0.579069	0.289534	−	0.957168i	$-0.406500\pi$
$997$	18.2843	0.579069	0.289534	+	0.957168i	$0.406500\pi$
$998$	0	0
$999$	−147.882	−4.67879

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3920.2.a.ca.1.2		2
4.3	odd	2		490.2.a.l.1.1	✓	2
7.6	odd	2		3920.2.a.bm.1.1		2
12.11	even	2		4410.2.a.by.1.2		2
20.3	even	4		2450.2.c.w.99.3		4
20.7	even	4		2450.2.c.w.99.2		4
20.19	odd	2		2450.2.a.bs.1.2		2
28.3	even	6		490.2.e.i.471.1		4
28.11	odd	6		490.2.e.j.471.2		4
28.19	even	6		490.2.e.i.361.1		4
28.23	odd	6		490.2.e.j.361.2		4
28.27	even	2		490.2.a.m.1.2	yes	2
84.83	odd	2		4410.2.a.bt.1.2		2
140.27	odd	4		2450.2.c.t.99.1		4
140.83	odd	4		2450.2.c.t.99.4		4
140.139	even	2		2450.2.a.bn.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
490.2.a.l.1.1	✓	2	4.3	odd	2
490.2.a.m.1.2	yes	2	28.27	even	2
490.2.e.i.361.1		4	28.19	even	6
490.2.e.i.471.1		4	28.3	even	6
490.2.e.j.361.2		4	28.23	odd	6
490.2.e.j.471.2		4	28.11	odd	6
2450.2.a.bn.1.1		2	140.139	even	2
2450.2.a.bs.1.2		2	20.19	odd	2
2450.2.c.t.99.1		4	140.27	odd	4
2450.2.c.t.99.4		4	140.83	odd	4
2450.2.c.w.99.2		4	20.7	even	4
2450.2.c.w.99.3		4	20.3	even	4
3920.2.a.bm.1.1		2	7.6	odd	2
3920.2.a.ca.1.2		2	1.1	even	1	trivial
4410.2.a.bt.1.2		2	84.83	odd	2
4410.2.a.by.1.2		2	12.11	even	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:	\( N \)	\(=\)	\( 3920 = 2^{4} \cdot 5 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3920.a (trivial)

\(f(q)\)	\(=\)	\(q+3.41421 q^{3} -1.00000 q^{5} +8.65685 q^{9} +O(q^{10})\)	\(q+3.41421 q^{3} -1.00000 q^{5} +8.65685 q^{9} +0.828427 q^{11} +4.82843 q^{13} -3.41421 q^{15} -2.58579 q^{17} +0.585786 q^{19} +1.17157 q^{23} +1.00000 q^{25} +19.3137 q^{27} -4.82843 q^{29} -2.82843 q^{31} +2.82843 q^{33} -7.65685 q^{37} +16.4853 q^{39} +3.07107 q^{41} +8.82843 q^{43} -8.65685 q^{45} +5.17157 q^{47} -8.82843 q^{51} +6.48528 q^{53} -0.828427 q^{55} +2.00000 q^{57} +8.58579 q^{59} -9.31371 q^{61} -4.82843 q^{65} -1.65685 q^{67} +4.00000 q^{69} +4.48528 q^{71} +9.41421 q^{73} +3.41421 q^{75} +6.82843 q^{79} +39.9706 q^{81} -2.24264 q^{83} +2.58579 q^{85} -16.4853 q^{87} -12.7279 q^{89} -9.65685 q^{93} -0.585786 q^{95} +7.75736 q^{97} +7.17157 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 4 q^{3} - 2 q^{5} + 6 q^{9}+O(q^{10}) \) 2 * q + 4 * q^3 - 2 * q^5 + 6 * q^9	\( 2 q + 4 q^{3} - 2 q^{5} + 6 q^{9} - 4 q^{11} + 4 q^{13} - 4 q^{15} - 8 q^{17} + 4 q^{19} + 8 q^{23} + 2 q^{25} + 16 q^{27} - 4 q^{29} - 4 q^{37} + 16 q^{39} - 8 q^{41} + 12 q^{43} - 6 q^{45} + 16 q^{47} - 12 q^{51} - 4 q^{53} + 4 q^{55} + 4 q^{57} + 20 q^{59} + 4 q^{61} - 4 q^{65} + 8 q^{67} + 8 q^{69} - 8 q^{71} + 16 q^{73} + 4 q^{75} + 8 q^{79} + 46 q^{81} + 4 q^{83} + 8 q^{85} - 16 q^{87} - 8 q^{93} - 4 q^{95} + 24 q^{97} + 20 q^{99}+O(q^{100}) \) 2 * q + 4 * q^3 - 2 * q^5 + 6 * q^9 - 4 * q^11 + 4 * q^13 - 4 * q^15 - 8 * q^17 + 4 * q^19 + 8 * q^23 + 2 * q^25 + 16 * q^27 - 4 * q^29 - 4 * q^37 + 16 * q^39 - 8 * q^41 + 12 * q^43 - 6 * q^45 + 16 * q^47 - 12 * q^51 - 4 * q^53 + 4 * q^55 + 4 * q^57 + 20 * q^59 + 4 * q^61 - 4 * q^65 + 8 * q^67 + 8 * q^69 - 8 * q^71 + 16 * q^73 + 4 * q^75 + 8 * q^79 + 46 * q^81 + 4 * q^83 + 8 * q^85 - 16 * q^87 - 8 * q^93 - 4 * q^95 + 24 * q^97 + 20 * q^99

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