LMFDB - Embedded newform 4840.2.a.ba.1.6

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [4840,2,Mod(1,4840)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("4840.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(4840, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 4840 = 2^{3} \cdot 5 \cdot 11^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4840.a (trivial)

Newform invariants

comment:select newform

sage:traces = [6,0,-3,0,-6,0,-7,0,5,0,0,0,1] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)

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

Self dual:	yes
Analytic conductor:	$38.6475945783$
Analytic rank:	$1$
Dimension:	$6$
Coefficient field:	6.6.25903625.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 3x^{5} - 7x^{4} + 17x^{3} + 16x^{2} - 20x - 5 $ x^6 - 3x^5 - 7x^4 + 17x^3 + 16x^2 - 20*x - 5
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 440)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$-1.71280$ of defining polynomial
Character	$\chi$	$=$	4840.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+1.71280 q^{3} -1.00000 q^{5} -3.70505 q^{7} -0.0663151 q^{9} +1.17750 q^{13} -1.71280 q^{15} +5.40027 q^{17} +3.84752 q^{19} -6.34602 q^{21} -8.68237 q^{23} +1.00000 q^{25} -5.25199 q^{27} +6.87597 q^{29} +8.75541 q^{31} +3.70505 q^{35} +2.25468 q^{37} +2.01682 q^{39} -1.59733 q^{41} -4.11979 q^{43} +0.0663151 q^{45} -12.6800 q^{47} +6.72743 q^{49} +9.24959 q^{51} -12.3795 q^{53} +6.59004 q^{57} -0.393999 q^{59} -1.80085 q^{61} +0.245701 q^{63} -1.17750 q^{65} -14.3809 q^{67} -14.8712 q^{69} -6.85527 q^{71} -10.0551 q^{73} +1.71280 q^{75} +1.11383 q^{79} -8.79666 q^{81} -0.0173212 q^{83} -5.40027 q^{85} +11.7772 q^{87} -8.49434 q^{89} -4.36270 q^{91} +14.9963 q^{93} -3.84752 q^{95} +10.5155 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 3 q^{3} - 6 q^{5} - 7 q^{7} + 5 q^{9} + q^{13} + 3 q^{15} + 6 q^{17} - 7 q^{19} + 14 q^{21} - 9 q^{23} + 6 q^{25} - 21 q^{27} + 10 q^{29} + q^{31} + 7 q^{35} + 3 q^{37} - q^{39} - 6 q^{41} + 18 q^{43}+ \cdots - 33 q^{97}+O(q^{100}) $ 6 * q - 3 * q^3 - 6 * q^5 - 7 * q^7 + 5 * q^9 + q^13 + 3 * q^15 + 6 * q^17 - 7 * q^19 + 14 * q^21 - 9 * q^23 + 6 * q^25 - 21 * q^27 + 10 * q^29 + q^31 + 7 * q^35 + 3 * q^37 - q^39 - 6 * q^41 + 18 * q^43 - 5 * q^45 - 3 * q^47 + 17 * q^49 + 15 * q^51 - 23 * q^53 + 9 * q^57 - 2 * q^59 + 6 * q^61 - 49 * q^63 - q^65 - 22 * q^67 + 2 * q^69 - 13 * q^71 + 10 * q^73 - 3 * q^75 - 22 * q^79 + 10 * q^81 + 10 * q^83 - 6 * q^85 - 3 * q^87 - 25 * q^89 + 12 * q^91 + 19 * q^93 + 7 * q^95 - 33 * q^97

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	1.71280	0.988886	0.494443	−	0.869210i	$-0.335372\pi$
$3$	1.71280	0.988886	0.494443	+	0.869210i	$0.335372\pi$
$4$	0	0
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	−3.70505	−1.40038	−0.700189	−	0.713957i	$-0.746901\pi$
$7$	−3.70505	−1.40038	−0.700189	+	0.713957i	$0.746901\pi$
$8$	0	0
$9$	−0.0663151	−0.0221050
$10$	0	0
$11$	0	0
$11$	0	0
$12$	0	0
$13$	1.17750	0.326580	0.163290	−	0.986578i	$-0.447789\pi$
$13$	1.17750	0.326580	0.163290	+	0.986578i	$0.447789\pi$
$14$	0	0
$15$	−1.71280	−0.442243
$16$	0	0
$17$	5.40027	1.30976	0.654879	−	0.755734i	$-0.272720\pi$
$17$	5.40027	1.30976	0.654879	+	0.755734i	$0.272720\pi$
$18$	0	0
$19$	3.84752	0.882682	0.441341	−	0.897339i	$-0.354503\pi$
$19$	3.84752	0.882682	0.441341	+	0.897339i	$0.354503\pi$
$20$	0	0
$21$	−6.34602	−1.38481
$22$	0	0
$23$	−8.68237	−1.81040	−0.905200	−	0.424986i	$-0.860279\pi$
$23$	−8.68237	−1.81040	−0.905200	+	0.424986i	$0.860279\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−5.25199	−1.01075
$28$	0	0
$29$	6.87597	1.27684	0.638418	−	0.769690i	$-0.279589\pi$
$29$	6.87597	1.27684	0.638418	+	0.769690i	$0.279589\pi$
$30$	0	0
$31$	8.75541	1.57252	0.786259	−	0.617898i	$-0.212015\pi$
$31$	8.75541	1.57252	0.786259	+	0.617898i	$0.212015\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	3.70505	0.626268
$36$	0	0
$37$	2.25468	0.370667	0.185334	−	0.982676i	$-0.440663\pi$
$37$	2.25468	0.370667	0.185334	+	0.982676i	$0.440663\pi$
$38$	0	0
$39$	2.01682	0.322950
$40$	0	0
$41$	−1.59733	−0.249461	−0.124730	−	0.992191i	$-0.539807\pi$
$41$	−1.59733	−0.249461	−0.124730	+	0.992191i	$0.539807\pi$
$42$	0	0
$43$	−4.11979	−0.628262	−0.314131	−	0.949380i	$-0.601713\pi$
$43$	−4.11979	−0.628262	−0.314131	+	0.949380i	$0.601713\pi$
$44$	0	0
$45$	0.0663151	0.00988567
$46$	0	0
$47$	−12.6800	−1.84956	−0.924782	−	0.380497i	$-0.875753\pi$
$47$	−12.6800	−1.84956	−0.924782	+	0.380497i	$0.875753\pi$
$48$	0	0
$49$	6.72743	0.961061
$50$	0	0
$51$	9.24959	1.29520
$52$	0	0
$53$	−12.3795	−1.70046	−0.850229	−	0.526413i	$-0.823536\pi$
$53$	−12.3795	−1.70046	−0.850229	+	0.526413i	$0.823536\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	6.59004	0.872872
$58$	0	0
$59$	−0.393999	−0.0512944	−0.0256472	−	0.999671i	$-0.508165\pi$
$59$	−0.393999	−0.0512944	−0.0256472	+	0.999671i	$0.508165\pi$
$60$	0	0
$61$	−1.80085	−0.230575	−0.115288	−	0.993332i	$-0.536779\pi$
$61$	−1.80085	−0.230575	−0.115288	+	0.993332i	$0.536779\pi$
$62$	0	0
$63$	0.245701	0.0309554
$64$	0	0
$65$	−1.17750	−0.146051
$66$	0	0
$67$	−14.3809	−1.75691	−0.878455	−	0.477826i	$-0.841425\pi$
$67$	−14.3809	−1.75691	−0.878455	+	0.477826i	$0.841425\pi$
$68$	0	0
$69$	−14.8712	−1.79028
$70$	0	0
$71$	−6.85527	−0.813571	−0.406785	−	0.913524i	$-0.633350\pi$
$71$	−6.85527	−0.813571	−0.406785	+	0.913524i	$0.633350\pi$
$72$	0	0
$73$	−10.0551	−1.17686	−0.588430	−	0.808548i	$-0.700254\pi$
$73$	−10.0551	−1.17686	−0.588430	+	0.808548i	$0.700254\pi$
$74$	0	0
$75$	1.71280	0.197777
$76$	0	0
$77$	0	0
$78$	0	0
$79$	1.11383	0.125316	0.0626580	−	0.998035i	$-0.480042\pi$
$79$	1.11383	0.125316	0.0626580	+	0.998035i	$0.480042\pi$
$80$	0	0
$81$	−8.79666	−0.977406
$82$	0	0
$83$	−0.0173212	−0.00190125	−0.000950626	−	1.00000i	$-0.500303\pi$
$83$	−0.0173212	−0.00190125	−0.000950626		1.00000i	$0.500303\pi$
$84$	0	0
$85$	−5.40027	−0.585742
$86$	0	0
$87$	11.7772	1.26265
$88$	0	0
$89$	−8.49434	−0.900399	−0.450199	−	0.892928i	$-0.648647\pi$
$89$	−8.49434	−0.900399	−0.450199	+	0.892928i	$0.648647\pi$
$90$	0	0
$91$	−4.36270	−0.457335
$92$	0	0
$93$	14.9963	1.55504
$94$	0	0
$95$	−3.84752	−0.394747
$96$	0	0
$97$	10.5155	1.06768	0.533842	−	0.845584i	$-0.320748\pi$
$97$	10.5155	1.06768	0.533842	+	0.845584i	$0.320748\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−7.17862	−0.714299	−0.357150	−	0.934047i	$-0.616251\pi$
$101$	−7.17862	−0.714299	−0.357150	+	0.934047i	$0.616251\pi$
$102$	0	0
$103$	−3.75016	−0.369515	−0.184757	−	0.982784i	$-0.559150\pi$
$103$	−3.75016	−0.369515	−0.184757	+	0.982784i	$0.559150\pi$
$104$	0	0
$105$	6.34602	0.619308
$106$	0	0
$107$	6.35128	0.614001	0.307001	−	0.951709i	$-0.400675\pi$
$107$	6.35128	0.614001	0.307001	+	0.951709i	$0.400675\pi$
$108$	0	0
$109$	−1.32407	−0.126823	−0.0634115	−	0.997987i	$-0.520198\pi$
$109$	−1.32407	−0.126823	−0.0634115	+	0.997987i	$0.520198\pi$
$110$	0	0
$111$	3.86182	0.366548
$112$	0	0
$113$	5.33169	0.501563	0.250782	−	0.968044i	$-0.419312\pi$
$113$	5.33169	0.501563	0.250782	+	0.968044i	$0.419312\pi$
$114$	0	0
$115$	8.68237	0.809636
$116$	0	0
$117$	−0.0780860	−0.00721905
$118$	0	0
$119$	−20.0083	−1.83416
$120$	0	0
$121$	0	0
$122$	0	0
$123$	−2.73591	−0.246688
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−14.6406	−1.29914	−0.649571	−	0.760301i	$-0.725052\pi$
$127$	−14.6406	−1.29914	−0.649571	+	0.760301i	$0.725052\pi$
$128$	0	0
$129$	−7.05637	−0.621279
$130$	0	0
$131$	−5.93922	−0.518912	−0.259456	−	0.965755i	$-0.583543\pi$
$131$	−5.93922	−0.518912	−0.259456	+	0.965755i	$0.583543\pi$
$132$	0	0
$133$	−14.2553	−1.23609
$134$	0	0
$135$	5.25199	0.452019
$136$	0	0
$137$	14.7698	1.26187	0.630934	−	0.775837i	$-0.282672\pi$
$137$	14.7698	1.26187	0.630934	+	0.775837i	$0.282672\pi$
$138$	0	0
$139$	−1.29252	−0.109630	−0.0548149	−	0.998497i	$-0.517457\pi$
$139$	−1.29252	−0.109630	−0.0548149	+	0.998497i	$0.517457\pi$
$140$	0	0
$141$	−21.7183	−1.82901
$142$	0	0
$143$	0	0
$144$	0	0
$145$	−6.87597	−0.571018
$146$	0	0
$147$	11.5227	0.950379
$148$	0	0
$149$	−23.9060	−1.95846	−0.979229	−	0.202757i	$-0.935010\pi$
$149$	−23.9060	−1.95846	−0.979229	+	0.202757i	$0.935010\pi$
$150$	0	0
$151$	−8.28048	−0.673856	−0.336928	−	0.941530i	$-0.609388\pi$
$151$	−8.28048	−0.673856	−0.336928	+	0.941530i	$0.609388\pi$
$152$	0	0
$153$	−0.358119	−0.0289522
$154$	0	0
$155$	−8.75541	−0.703251
$156$	0	0
$157$	−7.96444	−0.635631	−0.317816	−	0.948153i	$-0.602949\pi$
$157$	−7.96444	−0.635631	−0.317816	+	0.948153i	$0.602949\pi$
$158$	0	0
$159$	−21.2036	−1.68156
$160$	0	0
$161$	32.1687	2.53525
$162$	0	0
$163$	−4.08778	−0.320180	−0.160090	−	0.987102i	$-0.551178\pi$
$163$	−4.08778	−0.320180	−0.160090	+	0.987102i	$0.551178\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	3.98229	0.308159	0.154079	−	0.988058i	$-0.450759\pi$
$167$	3.98229	0.308159	0.154079	+	0.988058i	$0.450759\pi$
$168$	0	0
$169$	−11.6135	−0.893346
$170$	0	0
$171$	−0.255149	−0.0195117
$172$	0	0
$173$	−17.6573	−1.34246	−0.671230	−	0.741250i	$-0.734234\pi$
$173$	−17.6573	−1.34246	−0.671230	+	0.741250i	$0.734234\pi$
$174$	0	0
$175$	−3.70505	−0.280076
$176$	0	0
$177$	−0.674842	−0.0507243
$178$	0	0
$179$	11.9417	0.892562	0.446281	−	0.894893i	$-0.352748\pi$
$179$	11.9417	0.892562	0.446281	+	0.894893i	$0.352748\pi$
$180$	0	0
$181$	−1.03171	−0.0766863	−0.0383431	−	0.999265i	$-0.512208\pi$
$181$	−1.03171	−0.0766863	−0.0383431	+	0.999265i	$0.512208\pi$
$182$	0	0
$183$	−3.08450	−0.228013
$184$	0	0
$185$	−2.25468	−0.165767
$186$	0	0
$187$	0	0
$188$	0	0
$189$	19.4589	1.41543
$190$	0	0
$191$	19.0153	1.37590	0.687950	−	0.725758i	$-0.258511\pi$
$191$	19.0153	1.37590	0.687950	+	0.725758i	$0.258511\pi$
$192$	0	0
$193$	−5.40874	−0.389329	−0.194665	−	0.980870i	$-0.562362\pi$
$193$	−5.40874	−0.389329	−0.194665	+	0.980870i	$0.562362\pi$
$194$	0	0
$195$	−2.01682	−0.144428
$196$	0	0
$197$	−4.07618	−0.290416	−0.145208	−	0.989401i	$-0.546385\pi$
$197$	−4.07618	−0.290416	−0.145208	+	0.989401i	$0.546385\pi$
$198$	0	0
$199$	−12.0711	−0.855700	−0.427850	−	0.903850i	$-0.640729\pi$
$199$	−12.0711	−0.855700	−0.427850	+	0.903850i	$0.640729\pi$
$200$	0	0
$201$	−24.6316	−1.73738
$202$	0	0
$203$	−25.4759	−1.78805
$204$	0	0
$205$	1.59733	0.111562
$206$	0	0
$207$	0.575772	0.0400190
$208$	0	0
$209$	0	0
$210$	0	0
$211$	9.31289	0.641126	0.320563	−	0.947227i	$-0.396128\pi$
$211$	9.31289	0.641126	0.320563	+	0.947227i	$0.396128\pi$
$212$	0	0
$213$	−11.7417	−0.804528
$214$	0	0
$215$	4.11979	0.280967
$216$	0	0
$217$	−32.4392	−2.20212
$218$	0	0
$219$	−17.2224	−1.16378
$220$	0	0
$221$	6.35881	0.427740
$222$	0	0
$223$	−4.74364	−0.317658	−0.158829	−	0.987306i	$-0.550772\pi$
$223$	−4.74364	−0.317658	−0.158829	+	0.987306i	$0.550772\pi$
$224$	0	0
$225$	−0.0663151	−0.00442101
$226$	0	0
$227$	−3.51523	−0.233314	−0.116657	−	0.993172i	$-0.537218\pi$
$227$	−3.51523	−0.233314	−0.116657	+	0.993172i	$0.537218\pi$
$228$	0	0
$229$	−9.12613	−0.603072	−0.301536	−	0.953455i	$-0.597499\pi$
$229$	−9.12613	−0.603072	−0.301536	+	0.953455i	$0.597499\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	21.5251	1.41016	0.705079	−	0.709128i	$-0.250911\pi$
$233$	21.5251	1.41016	0.705079	+	0.709128i	$0.250911\pi$
$234$	0	0
$235$	12.6800	0.827150
$236$	0	0
$237$	1.90777	0.123923
$238$	0	0
$239$	−11.3074	−0.731417	−0.365709	−	0.930729i	$-0.619173\pi$
$239$	−11.3074	−0.731417	−0.365709	+	0.930729i	$0.619173\pi$
$240$	0	0
$241$	26.1144	1.68218	0.841089	−	0.540897i	$-0.181915\pi$
$241$	26.1144	1.68218	0.841089	+	0.540897i	$0.181915\pi$
$242$	0	0
$243$	0.689039	0.0442019
$244$	0	0
$245$	−6.72743	−0.429799
$246$	0	0
$247$	4.53045	0.288266
$248$	0	0
$249$	−0.0296678	−0.00188012
$250$	0	0
$251$	−3.71812	−0.234685	−0.117343	−	0.993091i	$-0.537438\pi$
$251$	−3.71812	−0.234685	−0.117343	+	0.993091i	$0.537438\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	−9.24959	−0.579231
$256$	0	0
$257$	−7.09278	−0.442436	−0.221218	−	0.975224i	$-0.571003\pi$
$257$	−7.09278	−0.442436	−0.221218	+	0.975224i	$0.571003\pi$
$258$	0	0
$259$	−8.35372	−0.519075
$260$	0	0
$261$	−0.455981	−0.0282245
$262$	0	0
$263$	−23.3578	−1.44030	−0.720151	−	0.693817i	$-0.755927\pi$
$263$	−23.3578	−1.44030	−0.720151	+	0.693817i	$0.755927\pi$
$264$	0	0
$265$	12.3795	0.760468
$266$	0	0
$267$	−14.5491	−0.890391
$268$	0	0
$269$	−17.0818	−1.04149	−0.520746	−	0.853711i	$-0.674346\pi$
$269$	−17.0818	−1.04149	−0.520746	+	0.853711i	$0.674346\pi$
$270$	0	0
$271$	27.4343	1.66652	0.833258	−	0.552885i	$-0.186473\pi$
$271$	27.4343	1.66652	0.833258	+	0.552885i	$0.186473\pi$
$272$	0	0
$273$	−7.47243	−0.452252
$274$	0	0
$275$	0	0
$276$	0	0
$277$	24.2250	1.45554	0.727771	−	0.685820i	$-0.240556\pi$
$277$	24.2250	1.45554	0.727771	+	0.685820i	$0.240556\pi$
$278$	0	0
$279$	−0.580616	−0.0347605
$280$	0	0
$281$	17.1634	1.02388	0.511941	−	0.859020i	$-0.328927\pi$
$281$	17.1634	1.02388	0.511941	+	0.859020i	$0.328927\pi$
$282$	0	0
$283$	−4.62908	−0.275170	−0.137585	−	0.990490i	$-0.543934\pi$
$283$	−4.62908	−0.275170	−0.137585	+	0.990490i	$0.543934\pi$
$284$	0	0
$285$	−6.59004	−0.390360
$286$	0	0
$287$	5.91819	0.349340
$288$	0	0
$289$	12.1629	0.715466
$290$	0	0
$291$	18.0109	1.05582
$292$	0	0
$293$	18.5006	1.08082	0.540409	−	0.841402i	$-0.318270\pi$
$293$	18.5006	1.08082	0.540409	+	0.841402i	$0.318270\pi$
$294$	0	0
$295$	0.393999	0.0229395
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−10.2235	−0.591240
$300$	0	0
$301$	15.2640	0.879804
$302$	0	0
$303$	−12.2955	−0.706360
$304$	0	0
$305$	1.80085	0.103116
$306$	0	0
$307$	27.2021	1.55250	0.776252	−	0.630422i	$-0.217118\pi$
$307$	27.2021	1.55250	0.776252	+	0.630422i	$0.217118\pi$
$308$	0	0
$309$	−6.42328	−0.365408
$310$	0	0
$311$	32.2598	1.82929	0.914644	−	0.404260i	$-0.132471\pi$
$311$	32.2598	1.82929	0.914644	+	0.404260i	$0.132471\pi$
$312$	0	0
$313$	−0.290748	−0.0164341	−0.00821704	−	0.999966i	$-0.502616\pi$
$313$	−0.290748	−0.0164341	−0.00821704	+	0.999966i	$0.502616\pi$
$314$	0	0
$315$	−0.245701	−0.0138437
$316$	0	0
$317$	22.8346	1.28252	0.641258	−	0.767325i	$-0.278413\pi$
$317$	22.8346	1.28252	0.641258	+	0.767325i	$0.278413\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	10.8785	0.607177
$322$	0	0
$323$	20.7777	1.15610
$324$	0	0
$325$	1.17750	0.0653159
$326$	0	0
$327$	−2.26787	−0.125413
$328$	0	0
$329$	46.9800	2.59009
$330$	0	0
$331$	−14.5262	−0.798433	−0.399217	−	0.916857i	$-0.630718\pi$
$331$	−14.5262	−0.798433	−0.399217	+	0.916857i	$0.630718\pi$
$332$	0	0
$333$	−0.149519	−0.00819361
$334$	0	0
$335$	14.3809	0.785714
$336$	0	0
$337$	−14.0343	−0.764498	−0.382249	−	0.924059i	$-0.624850\pi$
$337$	−14.0343	−0.764498	−0.382249	+	0.924059i	$0.624850\pi$
$338$	0	0
$339$	9.13212	0.495989
$340$	0	0
$341$	0	0
$342$	0	0
$343$	1.00990	0.0545297
$344$	0	0
$345$	14.8712	0.800637
$346$	0	0
$347$	27.2726	1.46407	0.732034	−	0.681268i	$-0.238571\pi$
$347$	27.2726	1.46407	0.732034	+	0.681268i	$0.238571\pi$
$348$	0	0
$349$	24.9117	1.33349	0.666747	−	0.745284i	$-0.267686\pi$
$349$	24.9117	1.33349	0.666747	+	0.745284i	$0.267686\pi$
$350$	0	0
$351$	−6.18421	−0.330089
$352$	0	0
$353$	−22.2916	−1.18646	−0.593232	−	0.805031i	$-0.702148\pi$
$353$	−22.2916	−1.18646	−0.593232	+	0.805031i	$0.702148\pi$
$354$	0	0
$355$	6.85527	0.363840
$356$	0	0
$357$	−34.2702	−1.81377
$358$	0	0
$359$	−1.86792	−0.0985850	−0.0492925	−	0.998784i	$-0.515697\pi$
$359$	−1.86792	−0.0985850	−0.0492925	+	0.998784i	$0.515697\pi$
$360$	0	0
$361$	−4.19658	−0.220872
$362$	0	0
$363$	0	0
$364$	0	0
$365$	10.0551	0.526308
$366$	0	0
$367$	2.57693	0.134515	0.0672574	−	0.997736i	$-0.478575\pi$
$367$	2.57693	0.134515	0.0672574	+	0.997736i	$0.478575\pi$
$368$	0	0
$369$	0.105927	0.00551434
$370$	0	0
$371$	45.8668	2.38128
$372$	0	0
$373$	−7.12031	−0.368675	−0.184338	−	0.982863i	$-0.559014\pi$
$373$	−7.12031	−0.368675	−0.184338	+	0.982863i	$0.559014\pi$
$374$	0	0
$375$	−1.71280	−0.0884486
$376$	0	0
$377$	8.09645	0.416989
$378$	0	0
$379$	−28.5424	−1.46612	−0.733061	−	0.680163i	$-0.761909\pi$
$379$	−28.5424	−1.46612	−0.733061	+	0.680163i	$0.761909\pi$
$380$	0	0
$381$	−25.0764	−1.28470
$382$	0	0
$383$	−9.33065	−0.476774	−0.238387	−	0.971170i	$-0.576619\pi$
$383$	−9.33065	−0.476774	−0.238387	+	0.971170i	$0.576619\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0.273204	0.0138877
$388$	0	0
$389$	0.444940	0.0225594	0.0112797	−	0.999936i	$-0.496409\pi$
$389$	0.444940	0.0225594	0.0112797	+	0.999936i	$0.496409\pi$
$390$	0	0
$391$	−46.8872	−2.37119
$392$	0	0
$393$	−10.1727	−0.513145
$394$	0	0
$395$	−1.11383	−0.0560431
$396$	0	0
$397$	1.66941	0.0837850	0.0418925	−	0.999122i	$-0.486661\pi$
$397$	1.66941	0.0837850	0.0418925	+	0.999122i	$0.486661\pi$
$398$	0	0
$399$	−24.4164	−1.22235
$400$	0	0
$401$	33.8566	1.69072	0.845360	−	0.534197i	$-0.179386\pi$
$401$	33.8566	1.69072	0.845360	+	0.534197i	$0.179386\pi$
$402$	0	0
$403$	10.3095	0.513552
$404$	0	0
$405$	8.79666	0.437109
$406$	0	0
$407$	0	0
$408$	0	0
$409$	21.7117	1.07357	0.536787	−	0.843718i	$-0.319638\pi$
$409$	21.7117	1.07357	0.536787	+	0.843718i	$0.319638\pi$
$410$	0	0
$411$	25.2977	1.24784
$412$	0	0
$413$	1.45979	0.0718315
$414$	0	0
$415$	0.0173212	0.000850265 0
$416$	0	0
$417$	−2.21382	−0.108411
$418$	0	0
$419$	−26.5502	−1.29706	−0.648532	−	0.761187i	$-0.724617\pi$
$419$	−26.5502	−1.29706	−0.648532	+	0.761187i	$0.724617\pi$
$420$	0	0
$421$	−10.4845	−0.510984	−0.255492	−	0.966811i	$-0.582237\pi$
$421$	−10.4845	−0.510984	−0.255492	+	0.966811i	$0.582237\pi$
$422$	0	0
$423$	0.840874	0.0408847
$424$	0	0
$425$	5.40027	0.261952
$426$	0	0
$427$	6.67225	0.322893
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−5.41218	−0.260696	−0.130348	−	0.991468i	$-0.541609\pi$
$431$	−5.41218	−0.260696	−0.130348	+	0.991468i	$0.541609\pi$
$432$	0	0
$433$	9.08123	0.436416	0.218208	−	0.975902i	$-0.429979\pi$
$433$	9.08123	0.436416	0.218208	+	0.975902i	$0.429979\pi$
$434$	0	0
$435$	−11.7772	−0.564672
$436$	0	0
$437$	−33.4056	−1.59801
$438$	0	0
$439$	−17.3683	−0.828945	−0.414473	−	0.910062i	$-0.636034\pi$
$439$	−17.3683	−0.828945	−0.414473	+	0.910062i	$0.636034\pi$
$440$	0	0
$441$	−0.446130	−0.0212443
$442$	0	0
$443$	−13.6634	−0.649170	−0.324585	−	0.945857i	$-0.605225\pi$
$443$	−13.6634	−0.649170	−0.324585	+	0.945857i	$0.605225\pi$
$444$	0	0
$445$	8.49434	0.402671
$446$	0	0
$447$	−40.9463	−1.93669
$448$	0	0
$449$	−15.0222	−0.708942	−0.354471	−	0.935067i	$-0.615339\pi$
$449$	−15.0222	−0.708942	−0.354471	+	0.935067i	$0.615339\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	−14.1828	−0.666367
$454$	0	0
$455$	4.36270	0.204526
$456$	0	0
$457$	2.62422	0.122756	0.0613779	−	0.998115i	$-0.480451\pi$
$457$	2.62422	0.122756	0.0613779	+	0.998115i	$0.480451\pi$
$458$	0	0
$459$	−28.3621	−1.32383
$460$	0	0
$461$	−39.2715	−1.82906	−0.914529	−	0.404521i	$-0.867438\pi$
$461$	−39.2715	−1.82906	−0.914529	+	0.404521i	$0.867438\pi$
$462$	0	0
$463$	−3.26421	−0.151701	−0.0758505	−	0.997119i	$-0.524167\pi$
$463$	−3.26421	−0.151701	−0.0758505	+	0.997119i	$0.524167\pi$
$464$	0	0
$465$	−14.9963	−0.695435
$466$	0	0
$467$	−25.1521	−1.16390	−0.581949	−	0.813225i	$-0.697710\pi$
$467$	−25.1521	−1.16390	−0.581949	+	0.813225i	$0.697710\pi$
$468$	0	0
$469$	53.2821	2.46034
$470$	0	0
$471$	−13.6415	−0.628567
$472$	0	0
$473$	0	0
$474$	0	0
$475$	3.84752	0.176536
$476$	0	0
$477$	0.820949	0.0375887
$478$	0	0
$479$	−29.8987	−1.36611	−0.683053	−	0.730369i	$-0.739348\pi$
$479$	−29.8987	−1.36611	−0.683053	+	0.730369i	$0.739348\pi$
$480$	0	0
$481$	2.65489	0.121052
$482$	0	0
$483$	55.0985	2.50707
$484$	0	0
$485$	−10.5155	−0.477483
$486$	0	0
$487$	31.2658	1.41679	0.708393	−	0.705818i	$-0.249420\pi$
$487$	31.2658	1.41679	0.708393	+	0.705818i	$0.249420\pi$
$488$	0	0
$489$	−7.00156	−0.316621
$490$	0	0
$491$	−8.15752	−0.368144	−0.184072	−	0.982913i	$-0.558928\pi$
$491$	−8.15752	−0.368144	−0.184072	+	0.982913i	$0.558928\pi$
$492$	0	0
$493$	37.1321	1.67235
$494$	0	0
$495$	0	0
$496$	0	0
$497$	25.3991	1.13931
$498$	0	0
$499$	8.52746	0.381742	0.190871	−	0.981615i	$-0.438869\pi$
$499$	8.52746	0.381742	0.190871	+	0.981615i	$0.438869\pi$
$500$	0	0
$501$	6.82087	0.304734
$502$	0	0
$503$	−9.83853	−0.438678	−0.219339	−	0.975649i	$-0.570390\pi$
$503$	−9.83853	−0.438678	−0.219339	+	0.975649i	$0.570390\pi$
$504$	0	0
$505$	7.17862	0.319444
$506$	0	0
$507$	−19.8916	−0.883417
$508$	0	0
$509$	−0.488846	−0.0216677	−0.0108339	−	0.999941i	$-0.503449\pi$
$509$	−0.488846	−0.0216677	−0.0108339	+	0.999941i	$0.503449\pi$
$510$	0	0
$511$	37.2547	1.64805
$512$	0	0
$513$	−20.2071	−0.892167
$514$	0	0
$515$	3.75016	0.165252
$516$	0	0
$517$	0	0
$518$	0	0
$519$	−30.2434	−1.32754
$520$	0	0
$521$	11.1536	0.488650	0.244325	−	0.969693i	$-0.421434\pi$
$521$	11.1536	0.488650	0.244325	+	0.969693i	$0.421434\pi$
$522$	0	0
$523$	−3.36279	−0.147045	−0.0735223	−	0.997294i	$-0.523424\pi$
$523$	−3.36279	−0.147045	−0.0735223	+	0.997294i	$0.523424\pi$
$524$	0	0
$525$	−6.34602	−0.276963
$526$	0	0
$527$	47.2816	2.05962
$528$	0	0
$529$	52.3836	2.27755
$530$	0	0
$531$	0.0261281	0.00113386
$532$	0	0
$533$	−1.88085	−0.0814688
$534$	0	0
$535$	−6.35128	−0.274590
$536$	0	0
$537$	20.4537	0.882642
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−2.63681	−0.113365	−0.0566827	−	0.998392i	$-0.518052\pi$
$541$	−2.63681	−0.113365	−0.0566827	+	0.998392i	$0.518052\pi$
$542$	0	0
$543$	−1.76711	−0.0758340
$544$	0	0
$545$	1.32407	0.0567170
$546$	0	0
$547$	−38.2751	−1.63652	−0.818262	−	0.574845i	$-0.805062\pi$
$547$	−38.2751	−1.63652	−0.818262	+	0.574845i	$0.805062\pi$
$548$	0	0
$549$	0.119424	0.00509687
$550$	0	0
$551$	26.4555	1.12704
$552$	0	0
$553$	−4.12681	−0.175490
$554$	0	0
$555$	−3.86182	−0.163925
$556$	0	0
$557$	18.1087	0.767289	0.383645	−	0.923481i	$-0.374669\pi$
$557$	18.1087	0.767289	0.383645	+	0.923481i	$0.374669\pi$
$558$	0	0
$559$	−4.85105	−0.205177
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−3.56855	−0.150396	−0.0751982	−	0.997169i	$-0.523959\pi$
$563$	−3.56855	−0.150396	−0.0751982	+	0.997169i	$0.523959\pi$
$564$	0	0
$565$	−5.33169	−0.224306
$566$	0	0
$567$	32.5921	1.36874
$568$	0	0
$569$	40.3751	1.69261	0.846307	−	0.532696i	$-0.178821\pi$
$569$	40.3751	1.69261	0.846307	+	0.532696i	$0.178821\pi$
$570$	0	0
$571$	−36.3873	−1.52276	−0.761381	−	0.648305i	$-0.775478\pi$
$571$	−36.3873	−1.52276	−0.761381	+	0.648305i	$0.775478\pi$
$572$	0	0
$573$	32.5694	1.36061
$574$	0	0
$575$	−8.68237	−0.362080
$576$	0	0
$577$	−1.48422	−0.0617887	−0.0308944	−	0.999523i	$-0.509836\pi$
$577$	−1.48422	−0.0617887	−0.0308944	+	0.999523i	$0.509836\pi$
$578$	0	0
$579$	−9.26409	−0.385002
$580$	0	0
$581$	0.0641761	0.00266247
$582$	0	0
$583$	0	0
$584$	0	0
$585$	0.0780860	0.00322846
$586$	0	0
$587$	−39.3563	−1.62441	−0.812204	−	0.583374i	$-0.801732\pi$
$587$	−39.3563	−1.62441	−0.812204	+	0.583374i	$0.801732\pi$
$588$	0	0
$589$	33.6866	1.38803
$590$	0	0
$591$	−6.98168	−0.287188
$592$	0	0
$593$	−27.2990	−1.12104	−0.560518	−	0.828142i	$-0.689398\pi$
$593$	−27.2990	−1.12104	−0.560518	+	0.828142i	$0.689398\pi$
$594$	0	0
$595$	20.0083	0.820260
$596$	0	0
$597$	−20.6755	−0.846190
$598$	0	0
$599$	21.1311	0.863393	0.431696	−	0.902019i	$-0.357915\pi$
$599$	21.1311	0.863393	0.431696	+	0.902019i	$0.357915\pi$
$600$	0	0
$601$	2.86260	0.116768	0.0583840	−	0.998294i	$-0.481405\pi$
$601$	2.86260	0.116768	0.0583840	+	0.998294i	$0.481405\pi$
$602$	0	0
$603$	0.953672	0.0388365
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−31.9944	−1.29861	−0.649306	−	0.760527i	$-0.724941\pi$
$607$	−31.9944	−1.29861	−0.649306	+	0.760527i	$0.724941\pi$
$608$	0	0
$609$	−43.6350	−1.76818
$610$	0	0
$611$	−14.9307	−0.604030
$612$	0	0
$613$	35.6008	1.43790	0.718952	−	0.695060i	$-0.244622\pi$
$613$	35.6008	1.43790	0.718952	+	0.695060i	$0.244622\pi$
$614$	0	0
$615$	2.73591	0.110322
$616$	0	0
$617$	−36.9894	−1.48914	−0.744568	−	0.667546i	$-0.767345\pi$
$617$	−36.9894	−1.48914	−0.744568	+	0.667546i	$0.767345\pi$
$618$	0	0
$619$	33.9911	1.36622	0.683109	−	0.730317i	$-0.260628\pi$
$619$	33.9911	1.36622	0.683109	+	0.730317i	$0.260628\pi$
$620$	0	0
$621$	45.5997	1.82985
$622$	0	0
$623$	31.4720	1.26090
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	12.1759	0.485485
$630$	0	0
$631$	2.88653	0.114911	0.0574555	−	0.998348i	$-0.481701\pi$
$631$	2.88653	0.114911	0.0574555	+	0.998348i	$0.481701\pi$
$632$	0	0
$633$	15.9511	0.634000
$634$	0	0
$635$	14.6406	0.580994
$636$	0	0
$637$	7.92154	0.313863
$638$	0	0
$639$	0.454608	0.0179840
$640$	0	0
$641$	11.6233	0.459092	0.229546	−	0.973298i	$-0.426276\pi$
$641$	11.6233	0.459092	0.229546	+	0.973298i	$0.426276\pi$
$642$	0	0
$643$	−27.0864	−1.06818	−0.534091	−	0.845427i	$-0.679346\pi$
$643$	−27.0864	−1.06818	−0.534091	+	0.845427i	$0.679346\pi$
$644$	0	0
$645$	7.05637	0.277844
$646$	0	0
$647$	1.66070	0.0652888	0.0326444	−	0.999467i	$-0.489607\pi$
$647$	1.66070	0.0652888	0.0326444	+	0.999467i	$0.489607\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	−55.5620	−2.17764
$652$	0	0
$653$	−40.2905	−1.57669	−0.788345	−	0.615234i	$-0.789062\pi$
$653$	−40.2905	−1.57669	−0.788345	+	0.615234i	$0.789062\pi$
$654$	0	0
$655$	5.93922	0.232065
$656$	0	0
$657$	0.666804	0.0260145
$658$	0	0
$659$	−33.7115	−1.31321	−0.656607	−	0.754233i	$-0.728009\pi$
$659$	−33.7115	−1.31321	−0.656607	+	0.754233i	$0.728009\pi$
$660$	0	0
$661$	42.4279	1.65026	0.825128	−	0.564946i	$-0.191103\pi$
$661$	42.4279	1.65026	0.825128	+	0.564946i	$0.191103\pi$
$662$	0	0
$663$	10.8914	0.422986
$664$	0	0
$665$	14.2553	0.552796
$666$	0	0
$667$	−59.6998	−2.31158
$668$	0	0
$669$	−8.12491	−0.314127
$670$	0	0
$671$	0	0
$672$	0	0
$673$	38.1273	1.46970	0.734850	−	0.678230i	$-0.237253\pi$
$673$	38.1273	1.46970	0.734850	+	0.678230i	$0.237253\pi$
$674$	0	0
$675$	−5.25199	−0.202149
$676$	0	0
$677$	12.0095	0.461561	0.230781	−	0.973006i	$-0.425872\pi$
$677$	12.0095	0.461561	0.230781	+	0.973006i	$0.425872\pi$
$678$	0	0
$679$	−38.9604	−1.49516
$680$	0	0
$681$	−6.02089	−0.230721
$682$	0	0
$683$	37.6564	1.44088	0.720441	−	0.693516i	$-0.243939\pi$
$683$	37.6564	1.44088	0.720441	+	0.693516i	$0.243939\pi$
$684$	0	0
$685$	−14.7698	−0.564324
$686$	0	0
$687$	−15.6312	−0.596369
$688$	0	0
$689$	−14.5769	−0.555335
$690$	0	0
$691$	−9.93572	−0.377973	−0.188986	−	0.981980i	$-0.560520\pi$
$691$	−9.93572	−0.377973	−0.188986	+	0.981980i	$0.560520\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	1.29252	0.0490280
$696$	0	0
$697$	−8.62601	−0.326733
$698$	0	0
$699$	36.8683	1.39449
$700$	0	0
$701$	−10.0853	−0.380915	−0.190458	−	0.981695i	$-0.560997\pi$
$701$	−10.0853	−0.380915	−0.190458	+	0.981695i	$0.560997\pi$
$702$	0	0
$703$	8.67494	0.327181
$704$	0	0
$705$	21.7183	0.817957
$706$	0	0
$707$	26.5972	1.00029
$708$	0	0
$709$	5.21637	0.195905	0.0979524	−	0.995191i	$-0.468771\pi$
$709$	5.21637	0.195905	0.0979524	+	0.995191i	$0.468771\pi$
$710$	0	0
$711$	−0.0738640	−0.00277012
$712$	0	0
$713$	−76.0177	−2.84689
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−19.3674	−0.723288
$718$	0	0
$719$	−7.90947	−0.294973	−0.147487	−	0.989064i	$-0.547118\pi$
$719$	−7.90947	−0.294973	−0.147487	+	0.989064i	$0.547118\pi$
$720$	0	0
$721$	13.8946	0.517460
$722$	0	0
$723$	44.7288	1.66348
$724$	0	0
$725$	6.87597	0.255367
$726$	0	0
$727$	10.2281	0.379338	0.189669	−	0.981848i	$-0.439258\pi$
$727$	10.2281	0.379338	0.189669	+	0.981848i	$0.439258\pi$
$728$	0	0
$729$	27.5702	1.02112
$730$	0	0
$731$	−22.2480	−0.822871
$732$	0	0
$733$	15.3672	0.567601	0.283801	−	0.958883i	$-0.408405\pi$
$733$	15.3672	0.567601	0.283801	+	0.958883i	$0.408405\pi$
$734$	0	0
$735$	−11.5227	−0.425023
$736$	0	0
$737$	0	0
$738$	0	0
$739$	28.2071	1.03762	0.518808	−	0.854891i	$-0.326376\pi$
$739$	28.2071	1.03762	0.518808	+	0.854891i	$0.326376\pi$
$740$	0	0
$741$	7.75976	0.285062
$742$	0	0
$743$	−44.1059	−1.61809	−0.809044	−	0.587748i	$-0.800015\pi$
$743$	−44.1059	−1.61809	−0.809044	+	0.587748i	$0.800015\pi$
$744$	0	0
$745$	23.9060	0.875849
$746$	0	0
$747$	0.00114866	4.20272e−5 0
$748$	0	0
$749$	−23.5318	−0.859834
$750$	0	0
$751$	−43.1816	−1.57572	−0.787860	−	0.615854i	$-0.788811\pi$
$751$	−43.1816	−1.57572	−0.787860	+	0.615854i	$0.788811\pi$
$752$	0	0
$753$	−6.36839	−0.232077
$754$	0	0
$755$	8.28048	0.301358
$756$	0	0
$757$	19.2800	0.700743	0.350371	−	0.936611i	$-0.386055\pi$
$757$	19.2800	0.700743	0.350371	+	0.936611i	$0.386055\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−34.9095	−1.26547	−0.632733	−	0.774370i	$-0.718067\pi$
$761$	−34.9095	−1.26547	−0.632733	+	0.774370i	$0.718067\pi$
$762$	0	0
$763$	4.90576	0.177600
$764$	0	0
$765$	0.358119	0.0129478
$766$	0	0
$767$	−0.463934	−0.0167517
$768$	0	0
$769$	−10.7367	−0.387177	−0.193588	−	0.981083i	$-0.562013\pi$
$769$	−10.7367	−0.387177	−0.193588	+	0.981083i	$0.562013\pi$
$770$	0	0
$771$	−12.1485	−0.437518
$772$	0	0
$773$	31.6402	1.13802	0.569010	−	0.822331i	$-0.307327\pi$
$773$	31.6402	1.13802	0.569010	+	0.822331i	$0.307327\pi$
$774$	0	0
$775$	8.75541	0.314503
$776$	0	0
$777$	−14.3082	−0.513306
$778$	0	0
$779$	−6.14576	−0.220195
$780$	0	0
$781$	0	0
$782$	0	0
$783$	−36.1125	−1.29056
$784$	0	0
$785$	7.96444	0.284263
$786$	0	0
$787$	−15.6444	−0.557662	−0.278831	−	0.960340i	$-0.589947\pi$
$787$	−15.6444	−0.557662	−0.278831	+	0.960340i	$0.589947\pi$
$788$	0	0
$789$	−40.0072	−1.42429
$790$	0	0
$791$	−19.7542	−0.702379
$792$	0	0
$793$	−2.12050	−0.0753011
$794$	0	0
$795$	21.2036	0.752016
$796$	0	0
$797$	−4.97978	−0.176393	−0.0881964	−	0.996103i	$-0.528110\pi$
$797$	−4.97978	−0.176393	−0.0881964	+	0.996103i	$0.528110\pi$
$798$	0	0
$799$	−68.4753	−2.42248
$800$	0	0
$801$	0.563303	0.0199033
$802$	0	0
$803$	0	0
$804$	0	0
$805$	−32.1687	−1.13380
$806$	0	0
$807$	−29.2576	−1.02992
$808$	0	0
$809$	−30.8902	−1.08604	−0.543021	−	0.839719i	$-0.682720\pi$
$809$	−30.8902	−1.08604	−0.543021	+	0.839719i	$0.682720\pi$
$810$	0	0
$811$	−4.47943	−0.157294	−0.0786471	−	0.996903i	$-0.525060\pi$
$811$	−4.47943	−0.157294	−0.0786471	+	0.996903i	$0.525060\pi$
$812$	0	0
$813$	46.9895	1.64799
$814$	0	0
$815$	4.08778	0.143189
$816$	0	0
$817$	−15.8510	−0.554555
$818$	0	0
$819$	0.289313	0.0101094
$820$	0	0
$821$	−12.4808	−0.435583	−0.217791	−	0.975995i	$-0.569885\pi$
$821$	−12.4808	−0.435583	−0.217791	+	0.975995i	$0.569885\pi$
$822$	0	0
$823$	16.5038	0.575288	0.287644	−	0.957737i	$-0.407128\pi$
$823$	16.5038	0.575288	0.287644	+	0.957737i	$0.407128\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	45.1967	1.57164	0.785822	−	0.618453i	$-0.212240\pi$
$827$	45.1967	1.57164	0.785822	+	0.618453i	$0.212240\pi$
$828$	0	0
$829$	−11.3874	−0.395502	−0.197751	−	0.980252i	$-0.563364\pi$
$829$	−11.3874	−0.395502	−0.197751	+	0.980252i	$0.563364\pi$
$830$	0	0
$831$	41.4927	1.43936
$832$	0	0
$833$	36.3299	1.25876
$834$	0	0
$835$	−3.98229	−0.137813
$836$	0	0
$837$	−45.9833	−1.58941
$838$	0	0
$839$	−14.1100	−0.487131	−0.243565	−	0.969884i	$-0.578317\pi$
$839$	−14.1100	−0.487131	−0.243565	+	0.969884i	$0.578317\pi$
$840$	0	0
$841$	18.2790	0.630310
$842$	0	0
$843$	29.3975	1.01250
$844$	0	0
$845$	11.6135	0.399516
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−7.92870	−0.272112
$850$	0	0
$851$	−19.5760	−0.671056
$852$	0	0
$853$	20.9805	0.718360	0.359180	−	0.933268i	$-0.383056\pi$
$853$	20.9805	0.718360	0.359180	+	0.933268i	$0.383056\pi$
$854$	0	0
$855$	0.255149	0.00872590
$856$	0	0
$857$	−25.0649	−0.856201	−0.428101	−	0.903731i	$-0.640817\pi$
$857$	−25.0649	−0.856201	−0.428101	+	0.903731i	$0.640817\pi$
$858$	0	0
$859$	−38.0496	−1.29823	−0.649117	−	0.760689i	$-0.724861\pi$
$859$	−38.0496	−1.29823	−0.649117	+	0.760689i	$0.724861\pi$
$860$	0	0
$861$	10.1367	0.345457
$862$	0	0
$863$	−8.58888	−0.292369	−0.146184	−	0.989257i	$-0.546699\pi$
$863$	−8.58888	−0.292369	−0.146184	+	0.989257i	$0.546699\pi$
$864$	0	0
$865$	17.6573	0.600366
$866$	0	0
$867$	20.8327	0.707514
$868$	0	0
$869$	0	0
$870$	0	0
$871$	−16.9335	−0.573771
$872$	0	0
$873$	−0.697334	−0.0236012
$874$	0	0
$875$	3.70505	0.125254
$876$	0	0
$877$	10.5663	0.356798	0.178399	−	0.983958i	$-0.442908\pi$
$877$	10.5663	0.356798	0.178399	+	0.983958i	$0.442908\pi$
$878$	0	0
$879$	31.6879	1.06881
$880$	0	0
$881$	−35.5069	−1.19626	−0.598129	−	0.801400i	$-0.704089\pi$
$881$	−35.5069	−1.19626	−0.598129	+	0.801400i	$0.704089\pi$
$882$	0	0
$883$	−34.0160	−1.14473	−0.572364	−	0.820000i	$-0.693974\pi$
$883$	−34.0160	−1.14473	−0.572364	+	0.820000i	$0.693974\pi$
$884$	0	0
$885$	0.674842	0.0226846
$886$	0	0
$887$	9.99766	0.335689	0.167844	−	0.985814i	$-0.446319\pi$
$887$	9.99766	0.335689	0.167844	+	0.985814i	$0.446319\pi$
$888$	0	0
$889$	54.2442	1.81929
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−48.7865	−1.63258
$894$	0	0
$895$	−11.9417	−0.399166
$896$	0	0
$897$	−17.5108	−0.584668
$898$	0	0
$899$	60.2019	2.00785
$900$	0	0
$901$	−66.8528	−2.22719
$902$	0	0
$903$	26.1442	0.870026
$904$	0	0
$905$	1.03171	0.0342952
$906$	0	0
$907$	14.3972	0.478051	0.239025	−	0.971013i	$-0.423172\pi$
$907$	14.3972	0.478051	0.239025	+	0.971013i	$0.423172\pi$
$908$	0	0
$909$	0.476051	0.0157896
$910$	0	0
$911$	−16.1765	−0.535951	−0.267975	−	0.963426i	$-0.586355\pi$
$911$	−16.1765	−0.535951	−0.267975	+	0.963426i	$0.586355\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	3.08450	0.101970
$916$	0	0
$917$	22.0051	0.726674
$918$	0	0
$919$	18.1790	0.599671	0.299836	−	0.953991i	$-0.403068\pi$
$919$	18.1790	0.599671	0.299836	+	0.953991i	$0.403068\pi$
$920$	0	0
$921$	46.5917	1.53525
$922$	0	0
$923$	−8.07207	−0.265696
$924$	0	0
$925$	2.25468	0.0741335
$926$	0	0
$927$	0.248692	0.00816813
$928$	0	0
$929$	−24.3272	−0.798150	−0.399075	−	0.916918i	$-0.630669\pi$
$929$	−24.3272	−0.798150	−0.399075	+	0.916918i	$0.630669\pi$
$930$	0	0
$931$	25.8839	0.848311
$932$	0	0
$933$	55.2547	1.80896
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−24.1898	−0.790246	−0.395123	−	0.918628i	$-0.629298\pi$
$937$	−24.1898	−0.790246	−0.395123	+	0.918628i	$0.629298\pi$
$938$	0	0
$939$	−0.497994	−0.0162514
$940$	0	0
$941$	6.12261	0.199591	0.0997956	−	0.995008i	$-0.468181\pi$
$941$	6.12261	0.199591	0.0997956	+	0.995008i	$0.468181\pi$
$942$	0	0
$943$	13.8686	0.451624
$944$	0	0
$945$	−19.4589	−0.632998
$946$	0	0
$947$	30.4758	0.990330	0.495165	−	0.868799i	$-0.335108\pi$
$947$	30.4758	0.990330	0.495165	+	0.868799i	$0.335108\pi$
$948$	0	0
$949$	−11.8399	−0.384338
$950$	0	0
$951$	39.1110	1.26826
$952$	0	0
$953$	−12.3055	−0.398614	−0.199307	−	0.979937i	$-0.563869\pi$
$953$	−12.3055	−0.398614	−0.199307	+	0.979937i	$0.563869\pi$
$954$	0	0
$955$	−19.0153	−0.615321
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−54.7228	−1.76709
$960$	0	0
$961$	45.6571	1.47281
$962$	0	0
$963$	−0.421186	−0.0135725
$964$	0	0
$965$	5.40874	0.174113
$966$	0	0
$967$	−22.5927	−0.726532	−0.363266	−	0.931685i	$-0.618338\pi$
$967$	−22.5927	−0.726532	−0.363266	+	0.931685i	$0.618338\pi$
$968$	0	0
$969$	35.5880	1.14325
$970$	0	0
$971$	43.2544	1.38810	0.694050	−	0.719927i	$-0.255825\pi$
$971$	43.2544	1.38810	0.694050	+	0.719927i	$0.255825\pi$
$972$	0	0
$973$	4.78884	0.153523
$974$	0	0
$975$	2.01682	0.0645900
$976$	0	0
$977$	15.0741	0.482263	0.241131	−	0.970492i	$-0.422482\pi$
$977$	15.0741	0.482263	0.241131	+	0.970492i	$0.422482\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	0.0878059	0.00280343
$982$	0	0
$983$	−41.9188	−1.33700	−0.668502	−	0.743711i	$-0.733064\pi$
$983$	−41.9188	−1.33700	−0.668502	+	0.743711i	$0.733064\pi$
$984$	0	0
$985$	4.07618	0.129878
$986$	0	0
$987$	80.4673	2.56130
$988$	0	0
$989$	35.7695	1.13740
$990$	0	0
$991$	29.5068	0.937314	0.468657	−	0.883380i	$-0.344738\pi$
$991$	29.5068	0.937314	0.468657	+	0.883380i	$0.344738\pi$
$992$	0	0
$993$	−24.8805	−0.789559
$994$	0	0
$995$	12.0711	0.382681
$996$	0	0
$997$	−48.6155	−1.53967	−0.769835	−	0.638243i	$-0.779661\pi$
$997$	−48.6155	−1.53967	−0.769835	+	0.638243i	$0.779661\pi$
$998$	0	0
$999$	−11.8416	−0.374650

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4840.2.a.ba.1.6		6
4.3	odd	2		9680.2.a.dd.1.1		6
11.2	odd	10		440.2.y.c.81.3	✓	12
11.6	odd	10		440.2.y.c.201.3	yes	12
11.10	odd	2		4840.2.a.bb.1.6		6
44.35	even	10		880.2.bo.i.81.1		12
44.39	even	10		880.2.bo.i.641.1		12
44.43	even	2		9680.2.a.dc.1.1		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
440.2.y.c.81.3	✓	12	11.2	odd	10
440.2.y.c.201.3	yes	12	11.6	odd	10
880.2.bo.i.81.1		12	44.35	even	10
880.2.bo.i.641.1		12	44.39	even	10
4840.2.a.ba.1.6		6	1.1	even	1	trivial
4840.2.a.bb.1.6		6	11.10	odd	2
9680.2.a.dc.1.1		6	44.43	even	2
9680.2.a.dd.1.1		6	4.3	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 4840 = 2^{3} \cdot 5 \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4840.a (trivial)

\(f(q)\)	\(=\)	\(q+1.71280 q^{3} -1.00000 q^{5} -3.70505 q^{7} -0.0663151 q^{9} +1.17750 q^{13} -1.71280 q^{15} +5.40027 q^{17} +3.84752 q^{19} -6.34602 q^{21} -8.68237 q^{23} +1.00000 q^{25} -5.25199 q^{27} +6.87597 q^{29} +8.75541 q^{31} +3.70505 q^{35} +2.25468 q^{37} +2.01682 q^{39} -1.59733 q^{41} -4.11979 q^{43} +0.0663151 q^{45} -12.6800 q^{47} +6.72743 q^{49} +9.24959 q^{51} -12.3795 q^{53} +6.59004 q^{57} -0.393999 q^{59} -1.80085 q^{61} +0.245701 q^{63} -1.17750 q^{65} -14.3809 q^{67} -14.8712 q^{69} -6.85527 q^{71} -10.0551 q^{73} +1.71280 q^{75} +1.11383 q^{79} -8.79666 q^{81} -0.0173212 q^{83} -5.40027 q^{85} +11.7772 q^{87} -8.49434 q^{89} -4.36270 q^{91} +14.9963 q^{93} -3.84752 q^{95} +10.5155 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 3 q^{3} - 6 q^{5} - 7 q^{7} + 5 q^{9} + q^{13} + 3 q^{15} + 6 q^{17} - 7 q^{19} + 14 q^{21} - 9 q^{23} + 6 q^{25} - 21 q^{27} + 10 q^{29} + q^{31} + 7 q^{35} + 3 q^{37} - q^{39} - 6 q^{41} + 18 q^{43}+ \cdots - 33 q^{97}+O(q^{100}) \) 6 * q - 3 * q^3 - 6 * q^5 - 7 * q^7 + 5 * q^9 + q^13 + 3 * q^15 + 6 * q^17 - 7 * q^19 + 14 * q^21 - 9 * q^23 + 6 * q^25 - 21 * q^27 + 10 * q^29 + q^31 + 7 * q^35 + 3 * q^37 - q^39 - 6 * q^41 + 18 * q^43 - 5 * q^45 - 3 * q^47 + 17 * q^49 + 15 * q^51 - 23 * q^53 + 9 * q^57 - 2 * q^59 + 6 * q^61 - 49 * q^63 - q^65 - 22 * q^67 + 2 * q^69 - 13 * q^71 + 10 * q^73 - 3 * q^75 - 22 * q^79 + 10 * q^81 + 10 * q^83 - 6 * q^85 - 3 * q^87 - 25 * q^89 + 12 * q^91 + 19 * q^93 + 7 * q^95 - 33 * q^97

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