LMFDB - Embedded newform 9680.2.a.da.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [9680,2,Mod(1,9680)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(9680, 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("9680.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 9680 = 2^{4} \cdot 5 \cdot 11^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	9680.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:	$77.2951891566$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.6.22733568.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 8x^{4} - 2x^{3} + 16x^{2} + 8x - 2 $ x^6 - 8x^4 - 2x^3 + 16x^2 + 8x - 2
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 4840)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$-0.821728$ of defining polynomial
Character	$\chi$	$=$	9680.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.32476 q^{3} +1.00000 q^{5} -4.78768 q^{7} +2.40452 q^{9} -0.601547 q^{13} +2.32476 q^{15} -2.64953 q^{17} -2.60155 q^{19} -11.1302 q^{21} -5.17083 q^{23} +1.00000 q^{25} -1.38434 q^{27} +5.44643 q^{29} +10.4577 q^{31} -4.78768 q^{35} +9.39816 q^{37} -1.39845 q^{39} -4.54633 q^{41} +3.48547 q^{43} +2.40452 q^{45} -5.07340 q^{47} +15.9218 q^{49} -6.15952 q^{51} +3.76714 q^{53} -6.04798 q^{57} -11.7599 q^{59} +12.6454 q^{61} -11.5121 q^{63} -0.601547 q^{65} +15.0452 q^{67} -12.0210 q^{69} -0.346489 q^{71} +3.68096 q^{73} +2.32476 q^{75} +8.05322 q^{79} -10.4318 q^{81} +13.0094 q^{83} -2.64953 q^{85} +12.6617 q^{87} +0.462692 q^{89} +2.88001 q^{91} +24.3118 q^{93} -2.60155 q^{95} +7.97011 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 2 q^{3} + 6 q^{5} - 4 q^{7} + 4 q^{9} + 2 q^{15} + 8 q^{17} - 12 q^{19} - 8 q^{21} + 8 q^{23} + 6 q^{25} + 14 q^{27} + 16 q^{29} + 4 q^{31} - 4 q^{35} + 8 q^{37} - 12 q^{39} + 32 q^{41} + 4 q^{43}+ \cdots - 12 q^{95}+O(q^{100}) $ 6 * q + 2 * q^3 + 6 * q^5 - 4 * q^7 + 4 * q^9 + 2 * q^15 + 8 * q^17 - 12 * q^19 - 8 * q^21 + 8 * q^23 + 6 * q^25 + 14 * q^27 + 16 * q^29 + 4 * q^31 - 4 * q^35 + 8 * q^37 - 12 * q^39 + 32 * q^41 + 4 * q^43 + 4 * q^45 + 6 * q^47 + 16 * q^49 - 40 * q^51 + 8 * q^53 - 16 * q^57 - 4 * q^59 + 16 * q^61 - 28 * q^63 + 2 * q^67 + 8 * q^69 + 28 * q^71 + 16 * q^73 + 2 * q^75 - 10 * q^81 - 12 * q^83 + 8 * q^85 + 24 * q^87 + 18 * q^89 + 24 * q^91 + 20 * q^93 - 12 * q^95

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
$3$	2.32476	1.34220	0.671101	−	0.741366i	$-0.265822\pi$
$3$	2.32476	1.34220	0.671101	+	0.741366i	$0.265822\pi$
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	−4.78768	−1.80957	−0.904786	−	0.425867i	$-0.859969\pi$
$7$	−4.78768	−1.80957	−0.904786	+	0.425867i	$0.859969\pi$
$8$	0	0
$9$	2.40452	0.801508
$10$	0	0
$11$	0	0
$11$	0	0
$12$	0	0
$13$	−0.601547	−0.166839	−0.0834195	−	0.996515i	$-0.526584\pi$
$13$	−0.601547	−0.166839	−0.0834195	+	0.996515i	$0.526584\pi$
$14$	0	0
$15$	2.32476	0.600251
$16$	0	0
$17$	−2.64953	−0.642605	−0.321302	−	0.946977i	$-0.604121\pi$
$17$	−2.64953	−0.642605	−0.321302	+	0.946977i	$0.604121\pi$
$18$	0	0
$19$	−2.60155	−0.596836	−0.298418	−	0.954435i	$-0.596459\pi$
$19$	−2.60155	−0.596836	−0.298418	+	0.954435i	$0.596459\pi$
$20$	0	0
$21$	−11.1302	−2.42881
$22$	0	0
$23$	−5.17083	−1.07819	−0.539096	−	0.842244i	$-0.681234\pi$
$23$	−5.17083	−1.07819	−0.539096	+	0.842244i	$0.681234\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−1.38434	−0.266417
$28$	0	0
$29$	5.44643	1.01138	0.505689	−	0.862716i	$-0.331238\pi$
$29$	5.44643	1.01138	0.505689	+	0.862716i	$0.331238\pi$
$30$	0	0
$31$	10.4577	1.87827	0.939133	−	0.343554i	$-0.111631\pi$
$31$	10.4577	1.87827	0.939133	+	0.343554i	$0.111631\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−4.78768	−0.809265
$36$	0	0
$37$	9.39816	1.54505	0.772524	−	0.634985i	$-0.218994\pi$
$37$	9.39816	1.54505	0.772524	+	0.634985i	$0.218994\pi$
$38$	0	0
$39$	−1.39845	−0.223932
$40$	0	0
$41$	−4.54633	−0.710018	−0.355009	−	0.934863i	$-0.615522\pi$
$41$	−4.54633	−0.710018	−0.355009	+	0.934863i	$0.615522\pi$
$42$	0	0
$43$	3.48547	0.531530	0.265765	−	0.964038i	$-0.414376\pi$
$43$	3.48547	0.531530	0.265765	+	0.964038i	$0.414376\pi$
$44$	0	0
$45$	2.40452	0.358445
$46$	0	0
$47$	−5.07340	−0.740031	−0.370016	−	0.929026i	$-0.620648\pi$
$47$	−5.07340	−0.740031	−0.370016	+	0.929026i	$0.620648\pi$
$48$	0	0
$49$	15.9218	2.27455
$50$	0	0
$51$	−6.15952	−0.862506
$52$	0	0
$53$	3.76714	0.517456	0.258728	−	0.965950i	$-0.416697\pi$
$53$	3.76714	0.517456	0.258728	+	0.965950i	$0.416697\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−6.04798	−0.801074
$58$	0	0
$59$	−11.7599	−1.53101	−0.765507	−	0.643428i	$-0.777512\pi$
$59$	−11.7599	−1.53101	−0.765507	+	0.643428i	$0.777512\pi$
$60$	0	0
$61$	12.6454	1.61908	0.809542	−	0.587063i	$-0.199716\pi$
$61$	12.6454	1.61908	0.809542	+	0.587063i	$0.199716\pi$
$62$	0	0
$63$	−11.5121	−1.45039
$64$	0	0
$65$	−0.601547	−0.0746127
$66$	0	0
$67$	15.0452	1.83806	0.919030	−	0.394187i	$-0.128974\pi$
$67$	15.0452	1.83806	0.919030	+	0.394187i	$0.128974\pi$
$68$	0	0
$69$	−12.0210	−1.44715
$70$	0	0
$71$	−0.346489	−0.0411207	−0.0205604	−	0.999789i	$-0.506545\pi$
$71$	−0.346489	−0.0411207	−0.0205604	+	0.999789i	$0.506545\pi$
$72$	0	0
$73$	3.68096	0.430823	0.215412	−	0.976523i	$-0.430891\pi$
$73$	3.68096	0.430823	0.215412	+	0.976523i	$0.430891\pi$
$74$	0	0
$75$	2.32476	0.268441
$76$	0	0
$77$	0	0
$78$	0	0
$79$	8.05322	0.906058	0.453029	−	0.891496i	$-0.350343\pi$
$79$	8.05322	0.906058	0.453029	+	0.891496i	$0.350343\pi$
$80$	0	0
$81$	−10.4318	−1.15909
$82$	0	0
$83$	13.0094	1.42796	0.713981	−	0.700165i	$-0.246890\pi$
$83$	13.0094	1.42796	0.713981	+	0.700165i	$0.246890\pi$
$84$	0	0
$85$	−2.64953	−0.287381
$86$	0	0
$87$	12.6617	1.35747
$88$	0	0
$89$	0.462692	0.0490453	0.0245226	−	0.999699i	$-0.492193\pi$
$89$	0.462692	0.0490453	0.0245226	+	0.999699i	$0.492193\pi$
$90$	0	0
$91$	2.88001	0.301907
$92$	0	0
$93$	24.3118	2.52101
$94$	0	0
$95$	−2.60155	−0.266913
$96$	0	0
$97$	7.97011	0.809242	0.404621	−	0.914484i	$-0.367403\pi$
$97$	7.97011	0.809242	0.404621	+	0.914484i	$0.367403\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−10.9134	−1.08592	−0.542960	−	0.839759i	$-0.682696\pi$
$101$	−10.9134	−1.08592	−0.542960	+	0.839759i	$0.682696\pi$
$102$	0	0
$103$	16.9081	1.66600	0.833001	−	0.553271i	$-0.186621\pi$
$103$	16.9081	1.66600	0.833001	+	0.553271i	$0.186621\pi$
$104$	0	0
$105$	−11.1302	−1.08620
$106$	0	0
$107$	−8.94029	−0.864291	−0.432145	−	0.901804i	$-0.642243\pi$
$107$	−8.94029	−0.864291	−0.432145	+	0.901804i	$0.642243\pi$
$108$	0	0
$109$	8.76627	0.839656	0.419828	−	0.907604i	$-0.362090\pi$
$109$	8.76627	0.839656	0.419828	+	0.907604i	$0.362090\pi$
$110$	0	0
$111$	21.8485	2.07377
$112$	0	0
$113$	−2.49000	−0.234240	−0.117120	−	0.993118i	$-0.537366\pi$
$113$	−2.49000	−0.234240	−0.117120	+	0.993118i	$0.537366\pi$
$114$	0	0
$115$	−5.17083	−0.482182
$116$	0	0
$117$	−1.44643	−0.133723
$118$	0	0
$119$	12.6851	1.16284
$120$	0	0
$121$	0	0
$122$	0	0
$123$	−10.5692	−0.952988
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	9.22655	0.818724	0.409362	−	0.912372i	$-0.365751\pi$
$127$	9.22655	0.818724	0.409362	+	0.912372i	$0.365751\pi$
$128$	0	0
$129$	8.10290	0.713421
$130$	0	0
$131$	−8.09882	−0.707597	−0.353799	−	0.935322i	$-0.615110\pi$
$131$	−8.09882	−0.707597	−0.353799	+	0.935322i	$0.615110\pi$
$132$	0	0
$133$	12.4554	1.08002
$134$	0	0
$135$	−1.38434	−0.119145
$136$	0	0
$137$	6.57195	0.561480	0.280740	−	0.959784i	$-0.409420\pi$
$137$	6.57195	0.561480	0.280740	+	0.959784i	$0.409420\pi$
$138$	0	0
$139$	3.61119	0.306297	0.153149	−	0.988203i	$-0.451059\pi$
$139$	3.61119	0.306297	0.153149	+	0.988203i	$0.451059\pi$
$140$	0	0
$141$	−11.7945	−0.993272
$142$	0	0
$143$	0	0
$144$	0	0
$145$	5.44643	0.452302
$146$	0	0
$147$	37.0145	3.05291
$148$	0	0
$149$	12.1685	0.996886	0.498443	−	0.866922i	$-0.333905\pi$
$149$	12.1685	0.996886	0.498443	+	0.866922i	$0.333905\pi$
$150$	0	0
$151$	−21.5830	−1.75640	−0.878198	−	0.478296i	$-0.841254\pi$
$151$	−21.5830	−1.75640	−0.878198	+	0.478296i	$0.841254\pi$
$152$	0	0
$153$	−6.37085	−0.515053
$154$	0	0
$155$	10.4577	0.839986
$156$	0	0
$157$	7.68096	0.613007	0.306504	−	0.951870i	$-0.400841\pi$
$157$	7.68096	0.613007	0.306504	+	0.951870i	$0.400841\pi$
$158$	0	0
$159$	8.75771	0.694531
$160$	0	0
$161$	24.7562	1.95107
$162$	0	0
$163$	5.21691	0.408620	0.204310	−	0.978906i	$-0.434505\pi$
$163$	5.21691	0.408620	0.204310	+	0.978906i	$0.434505\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	19.2767	1.49168	0.745838	−	0.666127i	$-0.232049\pi$
$167$	19.2767	1.49168	0.745838	+	0.666127i	$0.232049\pi$
$168$	0	0
$169$	−12.6381	−0.972165
$170$	0	0
$171$	−6.25548	−0.478369
$172$	0	0
$173$	19.6210	1.49175	0.745877	−	0.666084i	$-0.232031\pi$
$173$	19.6210	1.49175	0.745877	+	0.666084i	$0.232031\pi$
$174$	0	0
$175$	−4.78768	−0.361914
$176$	0	0
$177$	−27.3391	−2.05493
$178$	0	0
$179$	7.26184	0.542775	0.271388	−	0.962470i	$-0.412517\pi$
$179$	7.26184	0.542775	0.271388	+	0.962470i	$0.412517\pi$
$180$	0	0
$181$	−8.78809	−0.653214	−0.326607	−	0.945160i	$-0.605905\pi$
$181$	−8.78809	−0.653214	−0.326607	+	0.945160i	$0.605905\pi$
$182$	0	0
$183$	29.3977	2.17314
$184$	0	0
$185$	9.39816	0.690967
$186$	0	0
$187$	0	0
$188$	0	0
$189$	6.62778	0.482100
$190$	0	0
$191$	−24.9236	−1.80341	−0.901703	−	0.432355i	$-0.857683\pi$
$191$	−24.9236	−1.80341	−0.901703	+	0.432355i	$0.857683\pi$
$192$	0	0
$193$	−6.66623	−0.479845	−0.239923	−	0.970792i	$-0.577122\pi$
$193$	−6.66623	−0.479845	−0.239923	+	0.970792i	$0.577122\pi$
$194$	0	0
$195$	−1.39845	−0.100145
$196$	0	0
$197$	−1.66329	−0.118504	−0.0592522	−	0.998243i	$-0.518872\pi$
$197$	−1.66329	−0.118504	−0.0592522	+	0.998243i	$0.518872\pi$
$198$	0	0
$199$	21.0008	1.48870	0.744352	−	0.667787i	$-0.232758\pi$
$199$	21.0008	1.48870	0.744352	+	0.667787i	$0.232758\pi$
$200$	0	0
$201$	34.9765	2.46705
$202$	0	0
$203$	−26.0758	−1.83016
$204$	0	0
$205$	−4.54633	−0.317530
$206$	0	0
$207$	−12.4334	−0.864180
$208$	0	0
$209$	0	0
$210$	0	0
$211$	−11.3170	−0.779095	−0.389548	−	0.921006i	$-0.627369\pi$
$211$	−11.3170	−0.779095	−0.389548	+	0.921006i	$0.627369\pi$
$212$	0	0
$213$	−0.805506	−0.0551924
$214$	0	0
$215$	3.48547	0.237707
$216$	0	0
$217$	−50.0683	−3.39886
$218$	0	0
$219$	8.55735	0.578252
$220$	0	0
$221$	1.59381	0.107212
$222$	0	0
$223$	−0.746956	−0.0500198	−0.0250099	−	0.999687i	$-0.507962\pi$
$223$	−0.746956	−0.0500198	−0.0250099	+	0.999687i	$0.507962\pi$
$224$	0	0
$225$	2.40452	0.160302
$226$	0	0
$227$	13.5057	0.896405	0.448203	−	0.893932i	$-0.352064\pi$
$227$	13.5057	0.896405	0.448203	+	0.893932i	$0.352064\pi$
$228$	0	0
$229$	2.36737	0.156440	0.0782201	−	0.996936i	$-0.475076\pi$
$229$	2.36737	0.156440	0.0782201	+	0.996936i	$0.475076\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−2.15499	−0.141178	−0.0705892	−	0.997505i	$-0.522488\pi$
$233$	−2.15499	−0.141178	−0.0705892	+	0.997505i	$0.522488\pi$
$234$	0	0
$235$	−5.07340	−0.330952
$236$	0	0
$237$	18.7218	1.21611
$238$	0	0
$239$	−28.5539	−1.84700	−0.923500	−	0.383600i	$-0.874684\pi$
$239$	−28.5539	−1.84700	−0.923500	+	0.383600i	$0.874684\pi$
$240$	0	0
$241$	5.62289	0.362202	0.181101	−	0.983465i	$-0.442034\pi$
$241$	5.62289	0.362202	0.181101	+	0.983465i	$0.442034\pi$
$242$	0	0
$243$	−20.0985	−1.28932
$244$	0	0
$245$	15.9218	1.01721
$246$	0	0
$247$	1.56495	0.0995755
$248$	0	0
$249$	30.2437	1.91661
$250$	0	0
$251$	−20.5337	−1.29607	−0.648036	−	0.761609i	$-0.724410\pi$
$251$	−20.5337	−1.29607	−0.648036	+	0.761609i	$0.724410\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	−6.15952	−0.385724
$256$	0	0
$257$	−15.7592	−0.983030	−0.491515	−	0.870869i	$-0.663557\pi$
$257$	−15.7592	−0.983030	−0.491515	+	0.870869i	$0.663557\pi$
$258$	0	0
$259$	−44.9954	−2.79588
$260$	0	0
$261$	13.0961	0.810627
$262$	0	0
$263$	−15.8194	−0.975464	−0.487732	−	0.872993i	$-0.662176\pi$
$263$	−15.8194	−0.975464	−0.487732	+	0.872993i	$0.662176\pi$
$264$	0	0
$265$	3.76714	0.231413
$266$	0	0
$267$	1.07565	0.0658287
$268$	0	0
$269$	18.1701	1.10785	0.553924	−	0.832567i	$-0.313130\pi$
$269$	18.1701	1.10785	0.553924	+	0.832567i	$0.313130\pi$
$270$	0	0
$271$	−24.8599	−1.51013	−0.755065	−	0.655650i	$-0.772395\pi$
$271$	−24.8599	−1.51013	−0.755065	+	0.655650i	$0.772395\pi$
$272$	0	0
$273$	6.69534	0.405221
$274$	0	0
$275$	0	0
$276$	0	0
$277$	23.6352	1.42010	0.710050	−	0.704151i	$-0.248672\pi$
$277$	23.6352	1.42010	0.710050	+	0.704151i	$0.248672\pi$
$278$	0	0
$279$	25.1459	1.50544
$280$	0	0
$281$	−6.70036	−0.399710	−0.199855	−	0.979825i	$-0.564047\pi$
$281$	−6.70036	−0.399710	−0.199855	+	0.979825i	$0.564047\pi$
$282$	0	0
$283$	−7.11292	−0.422819	−0.211410	−	0.977398i	$-0.567805\pi$
$283$	−7.11292	−0.422819	−0.211410	+	0.977398i	$0.567805\pi$
$284$	0	0
$285$	−6.04798	−0.358251
$286$	0	0
$287$	21.7664	1.28483
$288$	0	0
$289$	−9.98001	−0.587059
$290$	0	0
$291$	18.5286	1.08617
$292$	0	0
$293$	27.4950	1.60628	0.803138	−	0.595793i	$-0.203162\pi$
$293$	27.4950	1.60628	0.803138	+	0.595793i	$0.203162\pi$
$294$	0	0
$295$	−11.7599	−0.684690
$296$	0	0
$297$	0	0
$298$	0	0
$299$	3.11049	0.179885
$300$	0	0
$301$	−16.6873	−0.961841
$302$	0	0
$303$	−25.3710	−1.45752
$304$	0	0
$305$	12.6454	0.724076
$306$	0	0
$307$	23.0888	1.31775	0.658875	−	0.752253i	$-0.271033\pi$
$307$	23.0888	1.31775	0.658875	+	0.752253i	$0.271033\pi$
$308$	0	0
$309$	39.3073	2.23611
$310$	0	0
$311$	7.07508	0.401191	0.200595	−	0.979674i	$-0.435712\pi$
$311$	7.07508	0.401191	0.200595	+	0.979674i	$0.435712\pi$
$312$	0	0
$313$	−28.1646	−1.59196	−0.795979	−	0.605325i	$-0.793043\pi$
$313$	−28.1646	−1.59196	−0.795979	+	0.605325i	$0.793043\pi$
$314$	0	0
$315$	−11.5121	−0.648632
$316$	0	0
$317$	23.4408	1.31657	0.658284	−	0.752770i	$-0.271283\pi$
$317$	23.4408	1.31657	0.658284	+	0.752770i	$0.271283\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	−20.7841	−1.16005
$322$	0	0
$323$	6.89287	0.383529
$324$	0	0
$325$	−0.601547	−0.0333678
$326$	0	0
$327$	20.3795	1.12699
$328$	0	0
$329$	24.2898	1.33914
$330$	0	0
$331$	−5.08798	−0.279661	−0.139830	−	0.990175i	$-0.544656\pi$
$331$	−5.08798	−0.279661	−0.139830	+	0.990175i	$0.544656\pi$
$332$	0	0
$333$	22.5981	1.23837
$334$	0	0
$335$	15.0452	0.822006
$336$	0	0
$337$	25.4172	1.38456	0.692280	−	0.721629i	$-0.256606\pi$
$337$	25.4172	1.38456	0.692280	+	0.721629i	$0.256606\pi$
$338$	0	0
$339$	−5.78867	−0.314398
$340$	0	0
$341$	0	0
$342$	0	0
$343$	−42.7149	−2.30639
$344$	0	0
$345$	−12.0210	−0.647186
$346$	0	0
$347$	17.8827	0.959995	0.479997	−	0.877270i	$-0.340638\pi$
$347$	17.8827	0.959995	0.479997	+	0.877270i	$0.340638\pi$
$348$	0	0
$349$	6.29515	0.336971	0.168486	−	0.985704i	$-0.446112\pi$
$349$	6.29515	0.336971	0.168486	+	0.985704i	$0.446112\pi$
$350$	0	0
$351$	0.832746	0.0444487
$352$	0	0
$353$	3.17636	0.169060	0.0845302	−	0.996421i	$-0.473061\pi$
$353$	3.17636	0.169060	0.0845302	+	0.996421i	$0.473061\pi$
$354$	0	0
$355$	−0.346489	−0.0183898
$356$	0	0
$357$	29.4898	1.56077
$358$	0	0
$359$	5.12020	0.270234	0.135117	−	0.990830i	$-0.456859\pi$
$359$	5.12020	0.270234	0.135117	+	0.990830i	$0.456859\pi$
$360$	0	0
$361$	−12.2320	−0.643787
$362$	0	0
$363$	0	0
$364$	0	0
$365$	3.68096	0.192670
$366$	0	0
$367$	10.8346	0.565564	0.282782	−	0.959184i	$-0.408743\pi$
$367$	10.8346	0.565564	0.282782	+	0.959184i	$0.408743\pi$
$368$	0	0
$369$	−10.9318	−0.569085
$370$	0	0
$371$	−18.0358	−0.936374
$372$	0	0
$373$	36.5605	1.89303	0.946516	−	0.322656i	$-0.104576\pi$
$373$	36.5605	1.89303	0.946516	+	0.322656i	$0.104576\pi$
$374$	0	0
$375$	2.32476	0.120050
$376$	0	0
$377$	−3.27628	−0.168737
$378$	0	0
$379$	−0.681538	−0.0350083	−0.0175041	−	0.999847i	$-0.505572\pi$
$379$	−0.681538	−0.0350083	−0.0175041	+	0.999847i	$0.505572\pi$
$380$	0	0
$381$	21.4495	1.09889
$382$	0	0
$383$	−36.6876	−1.87465	−0.937323	−	0.348461i	$-0.886705\pi$
$383$	−36.6876	−1.87465	−0.937323	+	0.348461i	$0.886705\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	8.38090	0.426025
$388$	0	0
$389$	36.9171	1.87177	0.935887	−	0.352301i	$-0.114601\pi$
$389$	36.9171	1.87177	0.935887	+	0.352301i	$0.114601\pi$
$390$	0	0
$391$	13.7002	0.692851
$392$	0	0
$393$	−18.8278	−0.949739
$394$	0	0
$395$	8.05322	0.405201
$396$	0	0
$397$	−29.6085	−1.48601	−0.743003	−	0.669288i	$-0.766599\pi$
$397$	−29.6085	−1.48601	−0.743003	+	0.669288i	$0.766599\pi$
$398$	0	0
$399$	28.9558	1.44960
$400$	0	0
$401$	−17.1919	−0.858523	−0.429262	−	0.903180i	$-0.641226\pi$
$401$	−17.1919	−0.858523	−0.429262	+	0.903180i	$0.641226\pi$
$402$	0	0
$403$	−6.29082	−0.313368
$404$	0	0
$405$	−10.4318	−0.518362
$406$	0	0
$407$	0	0
$408$	0	0
$409$	7.36760	0.364304	0.182152	−	0.983270i	$-0.441694\pi$
$409$	7.36760	0.364304	0.182152	+	0.983270i	$0.441694\pi$
$410$	0	0
$411$	15.2782	0.753619
$412$	0	0
$413$	56.3028	2.77048
$414$	0	0
$415$	13.0094	0.638604
$416$	0	0
$417$	8.39516	0.411113
$418$	0	0
$419$	−19.3728	−0.946425	−0.473212	−	0.880948i	$-0.656906\pi$
$419$	−19.3728	−0.946425	−0.473212	+	0.880948i	$0.656906\pi$
$420$	0	0
$421$	17.2101	0.838769	0.419384	−	0.907809i	$-0.362246\pi$
$421$	17.2101	0.838769	0.419384	+	0.907809i	$0.362246\pi$
$422$	0	0
$423$	−12.1991	−0.593141
$424$	0	0
$425$	−2.64953	−0.128521
$426$	0	0
$427$	−60.5423	−2.92985
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−30.8744	−1.48717	−0.743584	−	0.668643i	$-0.766876\pi$
$431$	−30.8744	−1.48717	−0.743584	+	0.668643i	$0.766876\pi$
$432$	0	0
$433$	2.48737	0.119536	0.0597678	−	0.998212i	$-0.480964\pi$
$433$	2.48737	0.119536	0.0597678	+	0.998212i	$0.480964\pi$
$434$	0	0
$435$	12.6617	0.607080
$436$	0	0
$437$	13.4522	0.643504
$438$	0	0
$439$	14.6203	0.697790	0.348895	−	0.937162i	$-0.386557\pi$
$439$	14.6203	0.697790	0.348895	+	0.937162i	$0.386557\pi$
$440$	0	0
$441$	38.2844	1.82307
$442$	0	0
$443$	32.0885	1.52457	0.762285	−	0.647241i	$-0.224077\pi$
$443$	32.0885	1.52457	0.762285	+	0.647241i	$0.224077\pi$
$444$	0	0
$445$	0.462692	0.0219337
$446$	0	0
$447$	28.2890	1.33802
$448$	0	0
$449$	11.2208	0.529544	0.264772	−	0.964311i	$-0.414703\pi$
$449$	11.2208	0.529544	0.264772	+	0.964311i	$0.414703\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	−50.1753	−2.35744
$454$	0	0
$455$	2.88001	0.135017
$456$	0	0
$457$	7.28861	0.340947	0.170473	−	0.985362i	$-0.445470\pi$
$457$	7.28861	0.340947	0.170473	+	0.985362i	$0.445470\pi$
$458$	0	0
$459$	3.66785	0.171200
$460$	0	0
$461$	−8.56686	−0.398999	−0.199499	−	0.979898i	$-0.563932\pi$
$461$	−8.56686	−0.398999	−0.199499	+	0.979898i	$0.563932\pi$
$462$	0	0
$463$	22.6373	1.05204	0.526022	−	0.850471i	$-0.323683\pi$
$463$	22.6373	1.05204	0.526022	+	0.850471i	$0.323683\pi$
$464$	0	0
$465$	24.3118	1.12743
$466$	0	0
$467$	31.1970	1.44363	0.721813	−	0.692089i	$-0.243309\pi$
$467$	31.1970	1.44363	0.721813	+	0.692089i	$0.243309\pi$
$468$	0	0
$469$	−72.0314	−3.32610
$470$	0	0
$471$	17.8564	0.822780
$472$	0	0
$473$	0	0
$474$	0	0
$475$	−2.60155	−0.119367
$476$	0	0
$477$	9.05817	0.414745
$478$	0	0
$479$	−33.1292	−1.51371	−0.756856	−	0.653582i	$-0.773265\pi$
$479$	−33.1292	−1.51371	−0.756856	+	0.653582i	$0.773265\pi$
$480$	0	0
$481$	−5.65343	−0.257774
$482$	0	0
$483$	57.5524	2.61873
$484$	0	0
$485$	7.97011	0.361904
$486$	0	0
$487$	6.89985	0.312662	0.156331	−	0.987705i	$-0.450033\pi$
$487$	6.89985	0.312662	0.156331	+	0.987705i	$0.450033\pi$
$488$	0	0
$489$	12.1281	0.548451
$490$	0	0
$491$	−3.53178	−0.159387	−0.0796935	−	0.996819i	$-0.525394\pi$
$491$	−3.53178	−0.159387	−0.0796935	+	0.996819i	$0.525394\pi$
$492$	0	0
$493$	−14.4305	−0.649916
$494$	0	0
$495$	0	0
$496$	0	0
$497$	1.65888	0.0744109
$498$	0	0
$499$	4.67978	0.209496	0.104748	−	0.994499i	$-0.466596\pi$
$499$	4.67978	0.209496	0.104748	+	0.994499i	$0.466596\pi$
$500$	0	0
$501$	44.8138	2.00213
$502$	0	0
$503$	−35.5622	−1.58564	−0.792821	−	0.609455i	$-0.791388\pi$
$503$	−35.5622	−1.58564	−0.792821	+	0.609455i	$0.791388\pi$
$504$	0	0
$505$	−10.9134	−0.485638
$506$	0	0
$507$	−29.3807	−1.30484
$508$	0	0
$509$	−10.7168	−0.475012	−0.237506	−	0.971386i	$-0.576330\pi$
$509$	−10.7168	−0.475012	−0.237506	+	0.971386i	$0.576330\pi$
$510$	0	0
$511$	−17.6232	−0.779606
$512$	0	0
$513$	3.60143	0.159007
$514$	0	0
$515$	16.9081	0.745059
$516$	0	0
$517$	0	0
$518$	0	0
$519$	45.6141	2.00224
$520$	0	0
$521$	−2.19519	−0.0961729	−0.0480865	−	0.998843i	$-0.515312\pi$
$521$	−2.19519	−0.0961729	−0.0480865	+	0.998843i	$0.515312\pi$
$522$	0	0
$523$	−2.21848	−0.0970075	−0.0485038	−	0.998823i	$-0.515445\pi$
$523$	−2.21848	−0.0970075	−0.0485038	+	0.998823i	$0.515445\pi$
$524$	0	0
$525$	−11.1302	−0.485762
$526$	0	0
$527$	−27.7081	−1.20698
$528$	0	0
$529$	3.73746	0.162498
$530$	0	0
$531$	−28.2771	−1.22712
$532$	0	0
$533$	2.73483	0.118459
$534$	0	0
$535$	−8.94029	−0.386523
$536$	0	0
$537$	16.8821	0.728514
$538$	0	0
$539$	0	0
$540$	0	0
$541$	38.8481	1.67021	0.835106	−	0.550089i	$-0.185406\pi$
$541$	38.8481	1.67021	0.835106	+	0.550089i	$0.185406\pi$
$542$	0	0
$543$	−20.4302	−0.876745
$544$	0	0
$545$	8.76627	0.375506
$546$	0	0
$547$	12.4573	0.532637	0.266319	−	0.963885i	$-0.414193\pi$
$547$	12.4573	0.532637	0.266319	+	0.963885i	$0.414193\pi$
$548$	0	0
$549$	30.4063	1.29771
$550$	0	0
$551$	−14.1691	−0.603626
$552$	0	0
$553$	−38.5562	−1.63958
$554$	0	0
$555$	21.8485	0.927417
$556$	0	0
$557$	−33.9789	−1.43973	−0.719867	−	0.694112i	$-0.755797\pi$
$557$	−33.9789	−1.43973	−0.719867	+	0.694112i	$0.755797\pi$
$558$	0	0
$559$	−2.09667	−0.0886799
$560$	0	0
$561$	0	0
$562$	0	0
$563$	23.9594	1.00977	0.504884	−	0.863187i	$-0.331535\pi$
$563$	23.9594	1.00977	0.504884	+	0.863187i	$0.331535\pi$
$564$	0	0
$565$	−2.49000	−0.104755
$566$	0	0
$567$	49.9443	2.09746
$568$	0	0
$569$	7.52879	0.315623	0.157812	−	0.987469i	$-0.449556\pi$
$569$	7.52879	0.315623	0.157812	+	0.987469i	$0.449556\pi$
$570$	0	0
$571$	−13.5353	−0.566433	−0.283216	−	0.959056i	$-0.591401\pi$
$571$	−13.5353	−0.566433	−0.283216	+	0.959056i	$0.591401\pi$
$572$	0	0
$573$	−57.9414	−2.42054
$574$	0	0
$575$	−5.17083	−0.215638
$576$	0	0
$577$	27.3683	1.13936	0.569678	−	0.821868i	$-0.307068\pi$
$577$	27.3683	1.13936	0.569678	+	0.821868i	$0.307068\pi$
$578$	0	0
$579$	−15.4974	−0.644050
$580$	0	0
$581$	−62.2846	−2.58400
$582$	0	0
$583$	0	0
$584$	0	0
$585$	−1.44643	−0.0598027
$586$	0	0
$587$	−17.1927	−0.709618	−0.354809	−	0.934939i	$-0.615454\pi$
$587$	−17.1927	−0.709618	−0.354809	+	0.934939i	$0.615454\pi$
$588$	0	0
$589$	−27.2063	−1.12102
$590$	0	0
$591$	−3.86675	−0.159057
$592$	0	0
$593$	38.4574	1.57925	0.789627	−	0.613587i	$-0.210274\pi$
$593$	38.4574	1.57925	0.789627	+	0.613587i	$0.210274\pi$
$594$	0	0
$595$	12.6851	0.520037
$596$	0	0
$597$	48.8218	1.99814
$598$	0	0
$599$	9.63410	0.393639	0.196819	−	0.980440i	$-0.436939\pi$
$599$	9.63410	0.393639	0.196819	+	0.980440i	$0.436939\pi$
$600$	0	0
$601$	−20.0426	−0.817555	−0.408777	−	0.912634i	$-0.634045\pi$
$601$	−20.0426	−0.817555	−0.408777	+	0.912634i	$0.634045\pi$
$602$	0	0
$603$	36.1765	1.47322
$604$	0	0
$605$	0	0
$606$	0	0
$607$	17.4596	0.708662	0.354331	−	0.935120i	$-0.384709\pi$
$607$	17.4596	0.708662	0.354331	+	0.935120i	$0.384709\pi$
$608$	0	0
$609$	−60.6200	−2.45644
$610$	0	0
$611$	3.05189	0.123466
$612$	0	0
$613$	−20.9105	−0.844569	−0.422284	−	0.906463i	$-0.638772\pi$
$613$	−20.9105	−0.844569	−0.422284	+	0.906463i	$0.638772\pi$
$614$	0	0
$615$	−10.5692	−0.426189
$616$	0	0
$617$	−41.2590	−1.66103	−0.830513	−	0.557000i	$-0.811952\pi$
$617$	−41.2590	−1.66103	−0.830513	+	0.557000i	$0.811952\pi$
$618$	0	0
$619$	−0.151406	−0.00608551	−0.00304276	−	0.999995i	$-0.500969\pi$
$619$	−0.151406	−0.00608551	−0.00304276	+	0.999995i	$0.500969\pi$
$620$	0	0
$621$	7.15819	0.287248
$622$	0	0
$623$	−2.21522	−0.0887510
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−24.9007	−0.992855
$630$	0	0
$631$	−9.21832	−0.366976	−0.183488	−	0.983022i	$-0.558739\pi$
$631$	−9.21832	−0.366976	−0.183488	+	0.983022i	$0.558739\pi$
$632$	0	0
$633$	−26.3094	−1.04570
$634$	0	0
$635$	9.22655	0.366145
$636$	0	0
$637$	−9.57773	−0.379483
$638$	0	0
$639$	−0.833142	−0.0329586
$640$	0	0
$641$	−19.8665	−0.784678	−0.392339	−	0.919821i	$-0.628334\pi$
$641$	−19.8665	−0.784678	−0.392339	+	0.919821i	$0.628334\pi$
$642$	0	0
$643$	17.8982	0.705838	0.352919	−	0.935654i	$-0.385189\pi$
$643$	17.8982	0.705838	0.352919	+	0.935654i	$0.385189\pi$
$644$	0	0
$645$	8.10290	0.319051
$646$	0	0
$647$	22.6373	0.889964	0.444982	−	0.895539i	$-0.353210\pi$
$647$	22.6373	0.889964	0.444982	+	0.895539i	$0.353210\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	−116.397	−4.56195
$652$	0	0
$653$	11.1498	0.436325	0.218163	−	0.975912i	$-0.429994\pi$
$653$	11.1498	0.436325	0.218163	+	0.975912i	$0.429994\pi$
$654$	0	0
$655$	−8.09882	−0.316447
$656$	0	0
$657$	8.85095	0.345308
$658$	0	0
$659$	18.0304	0.702366	0.351183	−	0.936307i	$-0.385779\pi$
$659$	18.0304	0.702366	0.351183	+	0.936307i	$0.385779\pi$
$660$	0	0
$661$	−16.1373	−0.627668	−0.313834	−	0.949478i	$-0.601613\pi$
$661$	−16.1373	−0.627668	−0.313834	+	0.949478i	$0.601613\pi$
$662$	0	0
$663$	3.70524	0.143900
$664$	0	0
$665$	12.4554	0.482998
$666$	0	0
$667$	−28.1626	−1.09046
$668$	0	0
$669$	−1.73649	−0.0671368
$670$	0	0
$671$	0	0
$672$	0	0
$673$	−35.5210	−1.36923	−0.684617	−	0.728903i	$-0.740031\pi$
$673$	−35.5210	−1.36923	−0.684617	+	0.728903i	$0.740031\pi$
$674$	0	0
$675$	−1.38434	−0.0532833
$676$	0	0
$677$	−32.4138	−1.24576	−0.622882	−	0.782315i	$-0.714039\pi$
$677$	−32.4138	−1.24576	−0.622882	+	0.782315i	$0.714039\pi$
$678$	0	0
$679$	−38.1583	−1.46438
$680$	0	0
$681$	31.3976	1.20316
$682$	0	0
$683$	−7.36687	−0.281885	−0.140943	−	0.990018i	$-0.545013\pi$
$683$	−7.36687	−0.281885	−0.140943	+	0.990018i	$0.545013\pi$
$684$	0	0
$685$	6.57195	0.251101
$686$	0	0
$687$	5.50357	0.209974
$688$	0	0
$689$	−2.26611	−0.0863319
$690$	0	0
$691$	0.768429	0.0292324	0.0146162	−	0.999893i	$-0.495347\pi$
$691$	0.768429	0.0292324	0.0146162	+	0.999893i	$0.495347\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	3.61119	0.136980
$696$	0	0
$697$	12.0456	0.456261
$698$	0	0
$699$	−5.00985	−0.189490
$700$	0	0
$701$	−7.95905	−0.300609	−0.150305	−	0.988640i	$-0.548025\pi$
$701$	−7.95905	−0.300609	−0.150305	+	0.988640i	$0.548025\pi$
$702$	0	0
$703$	−24.4498	−0.922140
$704$	0	0
$705$	−11.7945	−0.444205
$706$	0	0
$707$	52.2496	1.96505
$708$	0	0
$709$	15.3307	0.575755	0.287877	−	0.957667i	$-0.407050\pi$
$709$	15.3307	0.575755	0.287877	+	0.957667i	$0.407050\pi$
$710$	0	0
$711$	19.3642	0.726212
$712$	0	0
$713$	−54.0752	−2.02513
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−66.3811	−2.47905
$718$	0	0
$719$	−46.2627	−1.72531	−0.862654	−	0.505794i	$-0.831200\pi$
$719$	−46.2627	−1.72531	−0.862654	+	0.505794i	$0.831200\pi$
$720$	0	0
$721$	−80.9504	−3.01475
$722$	0	0
$723$	13.0719	0.486148
$724$	0	0
$725$	5.44643	0.202275
$726$	0	0
$727$	−24.0817	−0.893140	−0.446570	−	0.894749i	$-0.647355\pi$
$727$	−24.0817	−0.893140	−0.446570	+	0.894749i	$0.647355\pi$
$728$	0	0
$729$	−15.4288	−0.571437
$730$	0	0
$731$	−9.23485	−0.341563
$732$	0	0
$733$	−13.0668	−0.482632	−0.241316	−	0.970447i	$-0.577579\pi$
$733$	−13.0668	−0.482632	−0.241316	+	0.970447i	$0.577579\pi$
$734$	0	0
$735$	37.0145	1.36530
$736$	0	0
$737$	0	0
$738$	0	0
$739$	33.5963	1.23586	0.617930	−	0.786233i	$-0.287972\pi$
$739$	33.5963	1.23586	0.617930	+	0.786233i	$0.287972\pi$
$740$	0	0
$741$	3.63814	0.133650
$742$	0	0
$743$	27.1661	0.996626	0.498313	−	0.866997i	$-0.333953\pi$
$743$	27.1661	0.996626	0.498313	+	0.866997i	$0.333953\pi$
$744$	0	0
$745$	12.1685	0.445821
$746$	0	0
$747$	31.2813	1.14452
$748$	0	0
$749$	42.8032	1.56400
$750$	0	0
$751$	−13.5581	−0.494742	−0.247371	−	0.968921i	$-0.579567\pi$
$751$	−13.5581	−0.494742	−0.247371	+	0.968921i	$0.579567\pi$
$752$	0	0
$753$	−47.7359	−1.73959
$754$	0	0
$755$	−21.5830	−0.785485
$756$	0	0
$757$	32.1498	1.16850	0.584252	−	0.811572i	$-0.301388\pi$
$757$	32.1498	1.16850	0.584252	+	0.811572i	$0.301388\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	9.70988	0.351983	0.175991	−	0.984392i	$-0.443687\pi$
$761$	9.70988	0.351983	0.175991	+	0.984392i	$0.443687\pi$
$762$	0	0
$763$	−41.9701	−1.51942
$764$	0	0
$765$	−6.37085	−0.230339
$766$	0	0
$767$	7.07415	0.255433
$768$	0	0
$769$	15.6081	0.562841	0.281421	−	0.959584i	$-0.409194\pi$
$769$	15.6081	0.562841	0.281421	+	0.959584i	$0.409194\pi$
$770$	0	0
$771$	−36.6363	−1.31943
$772$	0	0
$773$	−21.4874	−0.772849	−0.386425	−	0.922321i	$-0.626290\pi$
$773$	−21.4874	−0.772849	−0.386425	+	0.922321i	$0.626290\pi$
$774$	0	0
$775$	10.4577	0.375653
$776$	0	0
$777$	−104.604	−3.75263
$778$	0	0
$779$	11.8275	0.423764
$780$	0	0
$781$	0	0
$782$	0	0
$783$	−7.53972	−0.269448
$784$	0	0
$785$	7.68096	0.274145
$786$	0	0
$787$	−6.04684	−0.215547	−0.107773	−	0.994175i	$-0.534372\pi$
$787$	−6.04684	−0.215547	−0.107773	+	0.994175i	$0.534372\pi$
$788$	0	0
$789$	−36.7763	−1.30927
$790$	0	0
$791$	11.9213	0.423874
$792$	0	0
$793$	−7.60682	−0.270126
$794$	0	0
$795$	8.75771	0.310604
$796$	0	0
$797$	32.0918	1.13675	0.568376	−	0.822769i	$-0.307572\pi$
$797$	32.0918	1.13675	0.568376	+	0.822769i	$0.307572\pi$
$798$	0	0
$799$	13.4421	0.475547
$800$	0	0
$801$	1.11255	0.0393102
$802$	0	0
$803$	0	0
$804$	0	0
$805$	24.7562	0.872543
$806$	0	0
$807$	42.2411	1.48696
$808$	0	0
$809$	−2.20394	−0.0774865	−0.0387432	−	0.999249i	$-0.512335\pi$
$809$	−2.20394	−0.0774865	−0.0387432	+	0.999249i	$0.512335\pi$
$810$	0	0
$811$	9.59190	0.336817	0.168409	−	0.985717i	$-0.446137\pi$
$811$	9.59190	0.336817	0.168409	+	0.985717i	$0.446137\pi$
$812$	0	0
$813$	−57.7933	−2.02690
$814$	0	0
$815$	5.21691	0.182741
$816$	0	0
$817$	−9.06762	−0.317236
$818$	0	0
$819$	6.92505	0.241981
$820$	0	0
$821$	14.2376	0.496894	0.248447	−	0.968645i	$-0.420080\pi$
$821$	14.2376	0.496894	0.248447	+	0.968645i	$0.420080\pi$
$822$	0	0
$823$	0.430172	0.0149949	0.00749743	−	0.999972i	$-0.497613\pi$
$823$	0.430172	0.0149949	0.00749743	+	0.999972i	$0.497613\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−35.6560	−1.23988	−0.619940	−	0.784649i	$-0.712843\pi$
$827$	−35.6560	−1.23988	−0.619940	+	0.784649i	$0.712843\pi$
$828$	0	0
$829$	10.9221	0.379339	0.189669	−	0.981848i	$-0.439258\pi$
$829$	10.9221	0.379339	0.189669	+	0.981848i	$0.439258\pi$
$830$	0	0
$831$	54.9462	1.90606
$832$	0	0
$833$	−42.1853	−1.46164
$834$	0	0
$835$	19.2767	0.667098
$836$	0	0
$837$	−14.4771	−0.500401
$838$	0	0
$839$	−30.0181	−1.03634	−0.518169	−	0.855278i	$-0.673386\pi$
$839$	−30.0181	−1.03634	−0.518169	+	0.855278i	$0.673386\pi$
$840$	0	0
$841$	0.663634	0.0228839
$842$	0	0
$843$	−15.5768	−0.536492
$844$	0	0
$845$	−12.6381	−0.434765
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−16.5359	−0.567509
$850$	0	0
$851$	−48.5963	−1.66586
$852$	0	0
$853$	20.5878	0.704912	0.352456	−	0.935828i	$-0.385347\pi$
$853$	20.5878	0.704912	0.352456	+	0.935828i	$0.385347\pi$
$854$	0	0
$855$	−6.25548	−0.213933
$856$	0	0
$857$	−7.44802	−0.254419	−0.127210	−	0.991876i	$-0.540602\pi$
$857$	−7.44802	−0.254419	−0.127210	+	0.991876i	$0.540602\pi$
$858$	0	0
$859$	53.0056	1.80853	0.904263	−	0.426975i	$-0.140421\pi$
$859$	53.0056	1.80853	0.904263	+	0.426975i	$0.140421\pi$
$860$	0	0
$861$	50.6017	1.72450
$862$	0	0
$863$	29.1040	0.990713	0.495356	−	0.868690i	$-0.335037\pi$
$863$	29.1040	0.990713	0.495356	+	0.868690i	$0.335037\pi$
$864$	0	0
$865$	19.6210	0.667132
$866$	0	0
$867$	−23.2012	−0.787953
$868$	0	0
$869$	0	0
$870$	0	0
$871$	−9.05037	−0.306660
$872$	0	0
$873$	19.1643	0.648614
$874$	0	0
$875$	−4.78768	−0.161853
$876$	0	0
$877$	−28.1013	−0.948912	−0.474456	−	0.880279i	$-0.657355\pi$
$877$	−28.1013	−0.948912	−0.474456	+	0.880279i	$0.657355\pi$
$878$	0	0
$879$	63.9194	2.15595
$880$	0	0
$881$	29.5508	0.995591	0.497796	−	0.867294i	$-0.334143\pi$
$881$	29.5508	0.995591	0.497796	+	0.867294i	$0.334143\pi$
$882$	0	0
$883$	−38.9274	−1.31001	−0.655005	−	0.755624i	$-0.727334\pi$
$883$	−38.9274	−1.31001	−0.655005	+	0.755624i	$0.727334\pi$
$884$	0	0
$885$	−27.3391	−0.918993
$886$	0	0
$887$	26.2960	0.882933	0.441467	−	0.897278i	$-0.354458\pi$
$887$	26.2960	0.882933	0.441467	+	0.897278i	$0.354458\pi$
$888$	0	0
$889$	−44.1737	−1.48154
$890$	0	0
$891$	0	0
$892$	0	0
$893$	13.1987	0.441677
$894$	0	0
$895$	7.26184	0.242736
$896$	0	0
$897$	7.23116	0.241441
$898$	0	0
$899$	56.9574	1.89964
$900$	0	0
$901$	−9.98113	−0.332520
$902$	0	0
$903$	−38.7941	−1.29099
$904$	0	0
$905$	−8.78809	−0.292126
$906$	0	0
$907$	−25.5540	−0.848505	−0.424253	−	0.905544i	$-0.639463\pi$
$907$	−25.5540	−0.848505	−0.424253	+	0.905544i	$0.639463\pi$
$908$	0	0
$909$	−26.2414	−0.870373
$910$	0	0
$911$	48.0753	1.59281	0.796403	−	0.604766i	$-0.206733\pi$
$911$	48.0753	1.59281	0.796403	+	0.604766i	$0.206733\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	29.3977	0.971857
$916$	0	0
$917$	38.7745	1.28045
$918$	0	0
$919$	−8.02354	−0.264672	−0.132336	−	0.991205i	$-0.542248\pi$
$919$	−8.02354	−0.264672	−0.132336	+	0.991205i	$0.542248\pi$
$920$	0	0
$921$	53.6761	1.76869
$922$	0	0
$923$	0.208430	0.00686054
$924$	0	0
$925$	9.39816	0.309010
$926$	0	0
$927$	40.6559	1.33531
$928$	0	0
$929$	47.7554	1.56680	0.783402	−	0.621516i	$-0.213483\pi$
$929$	47.7554	1.56680	0.783402	+	0.621516i	$0.213483\pi$
$930$	0	0
$931$	−41.4214	−1.35753
$932$	0	0
$933$	16.4479	0.538480
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−13.5515	−0.442709	−0.221354	−	0.975193i	$-0.571048\pi$
$937$	−13.5515	−0.442709	−0.221354	+	0.975193i	$0.571048\pi$
$938$	0	0
$939$	−65.4760	−2.13673
$940$	0	0
$941$	−11.7661	−0.383564	−0.191782	−	0.981438i	$-0.561427\pi$
$941$	−11.7661	−0.383564	−0.191782	+	0.981438i	$0.561427\pi$
$942$	0	0
$943$	23.5083	0.765536
$944$	0	0
$945$	6.62778	0.215602
$946$	0	0
$947$	−7.85540	−0.255266	−0.127633	−	0.991821i	$-0.540738\pi$
$947$	−7.85540	−0.255266	−0.127633	+	0.991821i	$0.540738\pi$
$948$	0	0
$949$	−2.21427	−0.0718782
$950$	0	0
$951$	54.4944	1.76710
$952$	0	0
$953$	35.7034	1.15655	0.578274	−	0.815843i	$-0.303726\pi$
$953$	35.7034	1.15655	0.578274	+	0.815843i	$0.303726\pi$
$954$	0	0
$955$	−24.9236	−0.806508
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−31.4644	−1.01604
$960$	0	0
$961$	78.3643	2.52788
$962$	0	0
$963$	−21.4971	−0.692736
$964$	0	0
$965$	−6.66623	−0.214593
$966$	0	0
$967$	12.5681	0.404162	0.202081	−	0.979369i	$-0.435230\pi$
$967$	12.5681	0.404162	0.202081	+	0.979369i	$0.435230\pi$
$968$	0	0
$969$	16.0243	0.514774
$970$	0	0
$971$	−1.33283	−0.0427724	−0.0213862	−	0.999771i	$-0.506808\pi$
$971$	−1.33283	−0.0427724	−0.0213862	+	0.999771i	$0.506808\pi$
$972$	0	0
$973$	−17.2892	−0.554267
$974$	0	0
$975$	−1.39845	−0.0447864
$976$	0	0
$977$	−44.0415	−1.40901	−0.704506	−	0.709698i	$-0.748831\pi$
$977$	−44.0415	−1.40901	−0.704506	+	0.709698i	$0.748831\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	21.0787	0.672991
$982$	0	0
$983$	13.3425	0.425560	0.212780	−	0.977100i	$-0.431748\pi$
$983$	13.3425	0.425560	0.212780	+	0.977100i	$0.431748\pi$
$984$	0	0
$985$	−1.66329	−0.0529968
$986$	0	0
$987$	56.4680	1.79740
$988$	0	0
$989$	−18.0228	−0.573091
$990$	0	0
$991$	−8.69302	−0.276143	−0.138071	−	0.990422i	$-0.544090\pi$
$991$	−8.69302	−0.276143	−0.138071	+	0.990422i	$0.544090\pi$
$992$	0	0
$993$	−11.8283	−0.375361
$994$	0	0
$995$	21.0008	0.665769
$996$	0	0
$997$	−19.1952	−0.607918	−0.303959	−	0.952685i	$-0.598309\pi$
$997$	−19.1952	−0.607918	−0.303959	+	0.952685i	$0.598309\pi$
$998$	0	0
$999$	−13.0103	−0.411626

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9680.2.a.da.1.5		6
4.3	odd	2		4840.2.a.bd.1.2	yes	6
11.10	odd	2		9680.2.a.db.1.5		6
44.43	even	2		4840.2.a.bc.1.2	✓	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4840.2.a.bc.1.2	✓	6	44.43	even	2
4840.2.a.bd.1.2	yes	6	4.3	odd	2
9680.2.a.da.1.5		6	1.1	even	1	trivial
9680.2.a.db.1.5		6	11.10	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 9680 = 2^{4} \cdot 5 \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9680.a (trivial)

\(f(q)\)	\(=\)	\(q+2.32476 q^{3} +1.00000 q^{5} -4.78768 q^{7} +2.40452 q^{9} -0.601547 q^{13} +2.32476 q^{15} -2.64953 q^{17} -2.60155 q^{19} -11.1302 q^{21} -5.17083 q^{23} +1.00000 q^{25} -1.38434 q^{27} +5.44643 q^{29} +10.4577 q^{31} -4.78768 q^{35} +9.39816 q^{37} -1.39845 q^{39} -4.54633 q^{41} +3.48547 q^{43} +2.40452 q^{45} -5.07340 q^{47} +15.9218 q^{49} -6.15952 q^{51} +3.76714 q^{53} -6.04798 q^{57} -11.7599 q^{59} +12.6454 q^{61} -11.5121 q^{63} -0.601547 q^{65} +15.0452 q^{67} -12.0210 q^{69} -0.346489 q^{71} +3.68096 q^{73} +2.32476 q^{75} +8.05322 q^{79} -10.4318 q^{81} +13.0094 q^{83} -2.64953 q^{85} +12.6617 q^{87} +0.462692 q^{89} +2.88001 q^{91} +24.3118 q^{93} -2.60155 q^{95} +7.97011 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 2 q^{3} + 6 q^{5} - 4 q^{7} + 4 q^{9} + 2 q^{15} + 8 q^{17} - 12 q^{19} - 8 q^{21} + 8 q^{23} + 6 q^{25} + 14 q^{27} + 16 q^{29} + 4 q^{31} - 4 q^{35} + 8 q^{37} - 12 q^{39} + 32 q^{41} + 4 q^{43}+ \cdots - 12 q^{95}+O(q^{100}) \) 6 * q + 2 * q^3 + 6 * q^5 - 4 * q^7 + 4 * q^9 + 2 * q^15 + 8 * q^17 - 12 * q^19 - 8 * q^21 + 8 * q^23 + 6 * q^25 + 14 * q^27 + 16 * q^29 + 4 * q^31 - 4 * q^35 + 8 * q^37 - 12 * q^39 + 32 * q^41 + 4 * q^43 + 4 * q^45 + 6 * q^47 + 16 * q^49 - 40 * q^51 + 8 * q^53 - 16 * q^57 - 4 * q^59 + 16 * q^61 - 28 * q^63 + 2 * q^67 + 8 * q^69 + 28 * q^71 + 16 * q^73 + 2 * q^75 - 10 * q^81 - 12 * q^83 + 8 * q^85 + 24 * q^87 + 18 * q^89 + 24 * q^91 + 20 * q^93 - 12 * q^95

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