LMFDB - Embedded newform 9016.2.a.x.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [9016,2,Mod(1,9016)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(9016, 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("9016.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 9016 = 2^{3} \cdot 7^{2} \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	9016.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:	$71.9931224624$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{17}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 4 $ x^2 - x - 4
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1288)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.56155$ of defining polynomial
Character	$\chi$	$=$	9016.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.56155 q^{3} +2.00000 q^{5} -0.561553 q^{9} +O(q^{10})$	$q-1.56155 q^{3} +2.00000 q^{5} -0.561553 q^{9} -3.12311 q^{11} -1.56155 q^{13} -3.12311 q^{15} -6.00000 q^{19} -1.00000 q^{23} -1.00000 q^{25} +5.56155 q^{27} -6.68466 q^{29} +7.56155 q^{31} +4.87689 q^{33} -6.00000 q^{37} +2.43845 q^{39} -6.68466 q^{41} -10.2462 q^{43} -1.12311 q^{45} +11.5616 q^{47} -10.0000 q^{53} -6.24621 q^{55} +9.36932 q^{57} +10.2462 q^{59} -8.24621 q^{61} -3.12311 q^{65} +15.1231 q^{67} +1.56155 q^{69} +1.56155 q^{71} +10.6847 q^{73} +1.56155 q^{75} +17.3693 q^{79} -7.00000 q^{81} -11.3693 q^{83} +10.4384 q^{87} +12.0000 q^{89} -11.8078 q^{93} -12.0000 q^{95} -10.2462 q^{97} +1.75379 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + q^{3} + 4 q^{5} + 3 q^{9}+O(q^{10}) $ 2 * q + q^3 + 4 * q^5 + 3 * q^9	$ 2 q + q^{3} + 4 q^{5} + 3 q^{9} + 2 q^{11} + q^{13} + 2 q^{15} - 12 q^{19} - 2 q^{23} - 2 q^{25} + 7 q^{27} - q^{29} + 11 q^{31} + 18 q^{33} - 12 q^{37} + 9 q^{39} - q^{41} - 4 q^{43} + 6 q^{45} + 19 q^{47} - 20 q^{53} + 4 q^{55} - 6 q^{57} + 4 q^{59} + 2 q^{65} + 22 q^{67} - q^{69} - q^{71} + 9 q^{73} - q^{75} + 10 q^{79} - 14 q^{81} + 2 q^{83} + 25 q^{87} + 24 q^{89} - 3 q^{93} - 24 q^{95} - 4 q^{97} + 20 q^{99}+O(q^{100}) $ 2 * q + q^3 + 4 * q^5 + 3 * q^9 + 2 * q^11 + q^13 + 2 * q^15 - 12 * q^19 - 2 * q^23 - 2 * q^25 + 7 * q^27 - q^29 + 11 * q^31 + 18 * q^33 - 12 * q^37 + 9 * q^39 - q^41 - 4 * q^43 + 6 * q^45 + 19 * q^47 - 20 * q^53 + 4 * q^55 - 6 * q^57 + 4 * q^59 + 2 * q^65 + 22 * q^67 - q^69 - q^71 + 9 * q^73 - q^75 + 10 * q^79 - 14 * q^81 + 2 * q^83 + 25 * q^87 + 24 * q^89 - 3 * q^93 - 24 * q^95 - 4 * 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
$3$	−1.56155	−0.901563	−0.450781	−	0.892634i	$-0.648855\pi$
$3$	−1.56155	−0.901563	−0.450781	+	0.892634i	$0.648855\pi$
$4$	0	0
$5$	2.00000	0.894427	0.447214	−	0.894427i	$-0.352416\pi$
$5$	2.00000	0.894427	0.447214	+	0.894427i	$0.352416\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	−0.561553	−0.187184
$10$	0	0
$11$	−3.12311	−0.941652	−0.470826	−	0.882226i	$-0.656044\pi$
$11$	−3.12311	−0.941652	−0.470826	+	0.882226i	$0.656044\pi$
$12$	0	0
$13$	−1.56155	−0.433097	−0.216548	−	0.976272i	$-0.569480\pi$
$13$	−1.56155	−0.433097	−0.216548	+	0.976272i	$0.569480\pi$
$14$	0	0
$15$	−3.12311	−0.806382
$16$	0	0
$17$	0	0		−	1.00000i	$-0.5\pi$
$17$	0	0			1.00000i	$0.5\pi$
$18$	0	0
$19$	−6.00000	−1.37649	−0.688247	−	0.725476i	$-0.741620\pi$
$19$	−6.00000	−1.37649	−0.688247	+	0.725476i	$0.741620\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	0	0
$25$	−1.00000	−0.200000
$26$	0	0
$27$	5.56155	1.07032
$28$	0	0
$29$	−6.68466	−1.24131	−0.620655	−	0.784084i	$-0.713133\pi$
$29$	−6.68466	−1.24131	−0.620655	+	0.784084i	$0.713133\pi$
$30$	0	0
$31$	7.56155	1.35809	0.679047	−	0.734094i	$-0.262393\pi$
$31$	7.56155	1.35809	0.679047	+	0.734094i	$0.262393\pi$
$32$	0	0
$33$	4.87689	0.848958
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−6.00000	−0.986394	−0.493197	−	0.869918i	$-0.664172\pi$
$37$	−6.00000	−0.986394	−0.493197	+	0.869918i	$0.664172\pi$
$38$	0	0
$39$	2.43845	0.390464
$40$	0	0
$41$	−6.68466	−1.04397	−0.521984	−	0.852955i	$-0.674808\pi$
$41$	−6.68466	−1.04397	−0.521984	+	0.852955i	$0.674808\pi$
$42$	0	0
$43$	−10.2462	−1.56253	−0.781266	−	0.624198i	$-0.785426\pi$
$43$	−10.2462	−1.56253	−0.781266	+	0.624198i	$0.785426\pi$
$44$	0	0
$45$	−1.12311	−0.167423
$46$	0	0
$47$	11.5616	1.68643	0.843213	−	0.537580i	$-0.180661\pi$
$47$	11.5616	1.68643	0.843213	+	0.537580i	$0.180661\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−10.0000	−1.37361	−0.686803	−	0.726844i	$-0.740986\pi$
$53$	−10.0000	−1.37361	−0.686803	+	0.726844i	$0.740986\pi$
$54$	0	0
$55$	−6.24621	−0.842239
$56$	0	0
$57$	9.36932	1.24100
$58$	0	0
$59$	10.2462	1.33394	0.666972	−	0.745083i	$-0.267590\pi$
$59$	10.2462	1.33394	0.666972	+	0.745083i	$0.267590\pi$
$60$	0	0
$61$	−8.24621	−1.05582	−0.527910	−	0.849301i	$-0.677024\pi$
$61$	−8.24621	−1.05582	−0.527910	+	0.849301i	$0.677024\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−3.12311	−0.387374
$66$	0	0
$67$	15.1231	1.84758	0.923791	−	0.382898i	$-0.125074\pi$
$67$	15.1231	1.84758	0.923791	+	0.382898i	$0.125074\pi$
$68$	0	0
$69$	1.56155	0.187989
$70$	0	0
$71$	1.56155	0.185322	0.0926611	−	0.995698i	$-0.470463\pi$
$71$	1.56155	0.185322	0.0926611	+	0.995698i	$0.470463\pi$
$72$	0	0
$73$	10.6847	1.25054	0.625272	−	0.780407i	$-0.284988\pi$
$73$	10.6847	1.25054	0.625272	+	0.780407i	$0.284988\pi$
$74$	0	0
$75$	1.56155	0.180313
$76$	0	0
$77$	0	0
$78$	0	0
$79$	17.3693	1.95420	0.977100	−	0.212779i	$-0.0682513\pi$
$79$	17.3693	1.95420	0.977100	+	0.212779i	$0.0682513\pi$
$80$	0	0
$81$	−7.00000	−0.777778
$82$	0	0
$83$	−11.3693	−1.24794	−0.623972	−	0.781446i	$-0.714482\pi$
$83$	−11.3693	−1.24794	−0.623972	+	0.781446i	$0.714482\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	10.4384	1.11912
$88$	0	0
$89$	12.0000	1.27200	0.635999	−	0.771690i	$-0.280588\pi$
$89$	12.0000	1.27200	0.635999	+	0.771690i	$0.280588\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	−11.8078	−1.22441
$94$	0	0
$95$	−12.0000	−1.23117
$96$	0	0
$97$	−10.2462	−1.04035	−0.520173	−	0.854061i	$-0.674132\pi$
$97$	−10.2462	−1.04035	−0.520173	+	0.854061i	$0.674132\pi$
$98$	0	0
$99$	1.75379	0.176262
$100$	0	0
$101$	3.12311	0.310761	0.155380	−	0.987855i	$-0.450340\pi$
$101$	3.12311	0.310761	0.155380	+	0.987855i	$0.450340\pi$
$102$	0	0
$103$	−2.24621	−0.221326	−0.110663	−	0.993858i	$-0.535297\pi$
$103$	−2.24621	−0.221326	−0.110663	+	0.993858i	$0.535297\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	17.3693	1.67916	0.839578	−	0.543240i	$-0.182803\pi$
$107$	17.3693	1.67916	0.839578	+	0.543240i	$0.182803\pi$
$108$	0	0
$109$	−5.12311	−0.490705	−0.245352	−	0.969434i	$-0.578904\pi$
$109$	−5.12311	−0.490705	−0.245352	+	0.969434i	$0.578904\pi$
$110$	0	0
$111$	9.36932	0.889296
$112$	0	0
$113$	−7.36932	−0.693247	−0.346624	−	0.938004i	$-0.612672\pi$
$113$	−7.36932	−0.693247	−0.346624	+	0.938004i	$0.612672\pi$
$114$	0	0
$115$	−2.00000	−0.186501
$116$	0	0
$117$	0.876894	0.0810689
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−1.24621	−0.113292
$122$	0	0
$123$	10.4384	0.941203
$124$	0	0
$125$	−12.0000	−1.07331
$126$	0	0
$127$	20.6847	1.83547	0.917733	−	0.397197i	$-0.130017\pi$
$127$	20.6847	1.83547	0.917733	+	0.397197i	$0.130017\pi$
$128$	0	0
$129$	16.0000	1.40872
$130$	0	0
$131$	−15.8078	−1.38113	−0.690565	−	0.723270i	$-0.742638\pi$
$131$	−15.8078	−1.38113	−0.690565	+	0.723270i	$0.742638\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	11.1231	0.957325
$136$	0	0
$137$	7.36932	0.629603	0.314802	−	0.949157i	$-0.398062\pi$
$137$	7.36932	0.629603	0.314802	+	0.949157i	$0.398062\pi$
$138$	0	0
$139$	−10.4384	−0.885378	−0.442689	−	0.896675i	$-0.645975\pi$
$139$	−10.4384	−0.885378	−0.442689	+	0.896675i	$0.645975\pi$
$140$	0	0
$141$	−18.0540	−1.52042
$142$	0	0
$143$	4.87689	0.407826
$144$	0	0
$145$	−13.3693	−1.11026
$146$	0	0
$147$	0	0
$148$	0	0
$149$	14.4924	1.18727	0.593633	−	0.804736i	$-0.297693\pi$
$149$	14.4924	1.18727	0.593633	+	0.804736i	$0.297693\pi$
$150$	0	0
$151$	−10.9309	−0.889542	−0.444771	−	0.895644i	$-0.646715\pi$
$151$	−10.9309	−0.889542	−0.444771	+	0.895644i	$0.646715\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	15.1231	1.21472
$156$	0	0
$157$	13.1231	1.04734	0.523669	−	0.851922i	$-0.324563\pi$
$157$	13.1231	1.04734	0.523669	+	0.851922i	$0.324563\pi$
$158$	0	0
$159$	15.6155	1.23839
$160$	0	0
$161$	0	0
$162$	0	0
$163$	4.68466	0.366931	0.183465	−	0.983026i	$-0.441268\pi$
$163$	4.68466	0.366931	0.183465	+	0.983026i	$0.441268\pi$
$164$	0	0
$165$	9.75379	0.759331
$166$	0	0
$167$	5.12311	0.396438	0.198219	−	0.980158i	$-0.436484\pi$
$167$	5.12311	0.396438	0.198219	+	0.980158i	$0.436484\pi$
$168$	0	0
$169$	−10.5616	−0.812427
$170$	0	0
$171$	3.36932	0.257658
$172$	0	0
$173$	17.3693	1.32056	0.660282	−	0.751017i	$-0.270437\pi$
$173$	17.3693	1.32056	0.660282	+	0.751017i	$0.270437\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−16.0000	−1.20263
$178$	0	0
$179$	2.93087	0.219063	0.109532	−	0.993983i	$-0.465065\pi$
$179$	2.93087	0.219063	0.109532	+	0.993983i	$0.465065\pi$
$180$	0	0
$181$	20.2462	1.50489	0.752445	−	0.658656i	$-0.228875\pi$
$181$	20.2462	1.50489	0.752445	+	0.658656i	$0.228875\pi$
$182$	0	0
$183$	12.8769	0.951887
$184$	0	0
$185$	−12.0000	−0.882258
$186$	0	0
$187$	0	0
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−4.87689	−0.352880	−0.176440	−	0.984311i	$-0.556458\pi$
$191$	−4.87689	−0.352880	−0.176440	+	0.984311i	$0.556458\pi$
$192$	0	0
$193$	25.8078	1.85768	0.928842	−	0.370477i	$-0.120806\pi$
$193$	25.8078	1.85768	0.928842	+	0.370477i	$0.120806\pi$
$194$	0	0
$195$	4.87689	0.349242
$196$	0	0
$197$	11.5616	0.823727	0.411863	−	0.911246i	$-0.364878\pi$
$197$	11.5616	0.823727	0.411863	+	0.911246i	$0.364878\pi$
$198$	0	0
$199$	7.12311	0.504944	0.252472	−	0.967604i	$-0.418757\pi$
$199$	7.12311	0.504944	0.252472	+	0.967604i	$0.418757\pi$
$200$	0	0
$201$	−23.6155	−1.66571
$202$	0	0
$203$	0	0
$204$	0	0
$205$	−13.3693	−0.933754
$206$	0	0
$207$	0.561553	0.0390306
$208$	0	0
$209$	18.7386	1.29618
$210$	0	0
$211$	−4.00000	−0.275371	−0.137686	−	0.990476i	$-0.543966\pi$
$211$	−4.00000	−0.275371	−0.137686	+	0.990476i	$0.543966\pi$
$212$	0	0
$213$	−2.43845	−0.167080
$214$	0	0
$215$	−20.4924	−1.39757
$216$	0	0
$217$	0	0
$218$	0	0
$219$	−16.6847	−1.12744
$220$	0	0
$221$	0	0
$222$	0	0
$223$	15.3693	1.02921	0.514603	−	0.857429i	$-0.327939\pi$
$223$	15.3693	1.02921	0.514603	+	0.857429i	$0.327939\pi$
$224$	0	0
$225$	0.561553	0.0374369
$226$	0	0
$227$	27.3693	1.81657	0.908283	−	0.418357i	$-0.137394\pi$
$227$	27.3693	1.81657	0.908283	+	0.418357i	$0.137394\pi$
$228$	0	0
$229$	7.36932	0.486978	0.243489	−	0.969904i	$-0.421708\pi$
$229$	7.36932	0.486978	0.243489	+	0.969904i	$0.421708\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−17.3153	−1.13437	−0.567183	−	0.823592i	$-0.691967\pi$
$233$	−17.3153	−1.13437	−0.567183	+	0.823592i	$0.691967\pi$
$234$	0	0
$235$	23.1231	1.50839
$236$	0	0
$237$	−27.1231	−1.76184
$238$	0	0
$239$	−14.4384	−0.933946	−0.466973	−	0.884272i	$-0.654655\pi$
$239$	−14.4384	−0.933946	−0.466973	+	0.884272i	$0.654655\pi$
$240$	0	0
$241$	11.1231	0.716502	0.358251	−	0.933625i	$-0.383373\pi$
$241$	11.1231	0.716502	0.358251	+	0.933625i	$0.383373\pi$
$242$	0	0
$243$	−5.75379	−0.369106
$244$	0	0
$245$	0	0
$246$	0	0
$247$	9.36932	0.596155
$248$	0	0
$249$	17.7538	1.12510
$250$	0	0
$251$	−17.6155	−1.11188	−0.555941	−	0.831222i	$-0.687642\pi$
$251$	−17.6155	−1.11188	−0.555941	+	0.831222i	$0.687642\pi$
$252$	0	0
$253$	3.12311	0.196348
$254$	0	0
$255$	0	0
$256$	0	0
$257$	3.56155	0.222164	0.111082	−	0.993811i	$-0.464568\pi$
$257$	3.56155	0.222164	0.111082	+	0.993811i	$0.464568\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	3.75379	0.232354
$262$	0	0
$263$	−23.6155	−1.45620	−0.728098	−	0.685473i	$-0.759595\pi$
$263$	−23.6155	−1.45620	−0.728098	+	0.685473i	$0.759595\pi$
$264$	0	0
$265$	−20.0000	−1.22859
$266$	0	0
$267$	−18.7386	−1.14679
$268$	0	0
$269$	7.31534	0.446024	0.223012	−	0.974816i	$-0.428411\pi$
$269$	7.31534	0.446024	0.223012	+	0.974816i	$0.428411\pi$
$270$	0	0
$271$	−2.87689	−0.174759	−0.0873794	−	0.996175i	$-0.527849\pi$
$271$	−2.87689	−0.174759	−0.0873794	+	0.996175i	$0.527849\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	3.12311	0.188330
$276$	0	0
$277$	−18.6847	−1.12265	−0.561326	−	0.827595i	$-0.689709\pi$
$277$	−18.6847	−1.12265	−0.561326	+	0.827595i	$0.689709\pi$
$278$	0	0
$279$	−4.24621	−0.254214
$280$	0	0
$281$	4.63068	0.276243	0.138122	−	0.990415i	$-0.455893\pi$
$281$	4.63068	0.276243	0.138122	+	0.990415i	$0.455893\pi$
$282$	0	0
$283$	−30.0000	−1.78331	−0.891657	−	0.452711i	$-0.850457\pi$
$283$	−30.0000	−1.78331	−0.891657	+	0.452711i	$0.850457\pi$
$284$	0	0
$285$	18.7386	1.10998
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−17.0000	−1.00000
$290$	0	0
$291$	16.0000	0.937937
$292$	0	0
$293$	3.36932	0.196838	0.0984188	−	0.995145i	$-0.468622\pi$
$293$	3.36932	0.196838	0.0984188	+	0.995145i	$0.468622\pi$
$294$	0	0
$295$	20.4924	1.19311
$296$	0	0
$297$	−17.3693	−1.00787
$298$	0	0
$299$	1.56155	0.0903069
$300$	0	0
$301$	0	0
$302$	0	0
$303$	−4.87689	−0.280170
$304$	0	0
$305$	−16.4924	−0.944353
$306$	0	0
$307$	−32.9848	−1.88254	−0.941272	−	0.337649i	$-0.890368\pi$
$307$	−32.9848	−1.88254	−0.941272	+	0.337649i	$0.890368\pi$
$308$	0	0
$309$	3.50758	0.199539
$310$	0	0
$311$	5.31534	0.301405	0.150703	−	0.988579i	$-0.451846\pi$
$311$	5.31534	0.301405	0.150703	+	0.988579i	$0.451846\pi$
$312$	0	0
$313$	−6.24621	−0.353057	−0.176528	−	0.984296i	$-0.556487\pi$
$313$	−6.24621	−0.353057	−0.176528	+	0.984296i	$0.556487\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−3.75379	−0.210834	−0.105417	−	0.994428i	$-0.533618\pi$
$317$	−3.75379	−0.210834	−0.105417	+	0.994428i	$0.533618\pi$
$318$	0	0
$319$	20.8769	1.16888
$320$	0	0
$321$	−27.1231	−1.51386
$322$	0	0
$323$	0	0
$324$	0	0
$325$	1.56155	0.0866194
$326$	0	0
$327$	8.00000	0.442401
$328$	0	0
$329$	0	0
$330$	0	0
$331$	0.192236	0.0105662	0.00528312	−	0.999986i	$-0.498318\pi$
$331$	0.192236	0.0105662	0.00528312	+	0.999986i	$0.498318\pi$
$332$	0	0
$333$	3.36932	0.184637
$334$	0	0
$335$	30.2462	1.65253
$336$	0	0
$337$	2.87689	0.156714	0.0783572	−	0.996925i	$-0.475033\pi$
$337$	2.87689	0.156714	0.0783572	+	0.996925i	$0.475033\pi$
$338$	0	0
$339$	11.5076	0.625006
$340$	0	0
$341$	−23.6155	−1.27885
$342$	0	0
$343$	0	0
$344$	0	0
$345$	3.12311	0.168142
$346$	0	0
$347$	−32.4924	−1.74428	−0.872142	−	0.489252i	$-0.837270\pi$
$347$	−32.4924	−1.74428	−0.872142	+	0.489252i	$0.837270\pi$
$348$	0	0
$349$	3.80776	0.203825	0.101912	−	0.994793i	$-0.467504\pi$
$349$	3.80776	0.203825	0.101912	+	0.994793i	$0.467504\pi$
$350$	0	0
$351$	−8.68466	−0.463553
$352$	0	0
$353$	−16.0540	−0.854467	−0.427233	−	0.904141i	$-0.640512\pi$
$353$	−16.0540	−0.854467	−0.427233	+	0.904141i	$0.640512\pi$
$354$	0	0
$355$	3.12311	0.165757
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−4.87689	−0.257393	−0.128696	−	0.991684i	$-0.541079\pi$
$359$	−4.87689	−0.257393	−0.128696	+	0.991684i	$0.541079\pi$
$360$	0	0
$361$	17.0000	0.894737
$362$	0	0
$363$	1.94602	0.102140
$364$	0	0
$365$	21.3693	1.11852
$366$	0	0
$367$	4.00000	0.208798	0.104399	−	0.994535i	$-0.466708\pi$
$367$	4.00000	0.208798	0.104399	+	0.994535i	$0.466708\pi$
$368$	0	0
$369$	3.75379	0.195414
$370$	0	0
$371$	0	0
$372$	0	0
$373$	18.0000	0.932005	0.466002	−	0.884783i	$-0.345694\pi$
$373$	18.0000	0.932005	0.466002	+	0.884783i	$0.345694\pi$
$374$	0	0
$375$	18.7386	0.967659
$376$	0	0
$377$	10.4384	0.537607
$378$	0	0
$379$	30.2462	1.55364	0.776822	−	0.629721i	$-0.216831\pi$
$379$	30.2462	1.55364	0.776822	+	0.629721i	$0.216831\pi$
$380$	0	0
$381$	−32.3002	−1.65479
$382$	0	0
$383$	−3.50758	−0.179229	−0.0896144	−	0.995977i	$-0.528563\pi$
$383$	−3.50758	−0.179229	−0.0896144	+	0.995977i	$0.528563\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	5.75379	0.292482
$388$	0	0
$389$	15.3693	0.779255	0.389628	−	0.920972i	$-0.372604\pi$
$389$	15.3693	0.779255	0.389628	+	0.920972i	$0.372604\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	24.6847	1.24518
$394$	0	0
$395$	34.7386	1.74789
$396$	0	0
$397$	−1.56155	−0.0783721	−0.0391860	−	0.999232i	$-0.512476\pi$
$397$	−1.56155	−0.0783721	−0.0391860	+	0.999232i	$0.512476\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	10.4924	0.523967	0.261983	−	0.965072i	$-0.415623\pi$
$401$	10.4924	0.523967	0.261983	+	0.965072i	$0.415623\pi$
$402$	0	0
$403$	−11.8078	−0.588187
$404$	0	0
$405$	−14.0000	−0.695666
$406$	0	0
$407$	18.7386	0.928840
$408$	0	0
$409$	8.43845	0.417254	0.208627	−	0.977995i	$-0.433100\pi$
$409$	8.43845	0.417254	0.208627	+	0.977995i	$0.433100\pi$
$410$	0	0
$411$	−11.5076	−0.567627
$412$	0	0
$413$	0	0
$414$	0	0
$415$	−22.7386	−1.11620
$416$	0	0
$417$	16.3002	0.798224
$418$	0	0
$419$	10.8769	0.531371	0.265686	−	0.964060i	$-0.414402\pi$
$419$	10.8769	0.531371	0.265686	+	0.964060i	$0.414402\pi$
$420$	0	0
$421$	−8.63068	−0.420634	−0.210317	−	0.977633i	$-0.567450\pi$
$421$	−8.63068	−0.420634	−0.210317	+	0.977633i	$0.567450\pi$
$422$	0	0
$423$	−6.49242	−0.315672
$424$	0	0
$425$	0	0
$426$	0	0
$427$	0	0
$428$	0	0
$429$	−7.61553	−0.367681
$430$	0	0
$431$	26.7386	1.28795	0.643977	−	0.765045i	$-0.277283\pi$
$431$	26.7386	1.28795	0.643977	+	0.765045i	$0.277283\pi$
$432$	0	0
$433$	15.1231	0.726770	0.363385	−	0.931639i	$-0.381621\pi$
$433$	15.1231	0.726770	0.363385	+	0.931639i	$0.381621\pi$
$434$	0	0
$435$	20.8769	1.00097
$436$	0	0
$437$	6.00000	0.287019
$438$	0	0
$439$	−4.05398	−0.193486	−0.0967428	−	0.995309i	$-0.530842\pi$
$439$	−4.05398	−0.193486	−0.0967428	+	0.995309i	$0.530842\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−12.6847	−0.602666	−0.301333	−	0.953519i	$-0.597432\pi$
$443$	−12.6847	−0.602666	−0.301333	+	0.953519i	$0.597432\pi$
$444$	0	0
$445$	24.0000	1.13771
$446$	0	0
$447$	−22.6307	−1.07039
$448$	0	0
$449$	−30.4924	−1.43903	−0.719513	−	0.694479i	$-0.755635\pi$
$449$	−30.4924	−1.43903	−0.719513	+	0.694479i	$0.755635\pi$
$450$	0	0
$451$	20.8769	0.983055
$452$	0	0
$453$	17.0691	0.801978
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−33.1231	−1.54943	−0.774717	−	0.632308i	$-0.782108\pi$
$457$	−33.1231	−1.54943	−0.774717	+	0.632308i	$0.782108\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−29.1771	−1.35891	−0.679456	−	0.733716i	$-0.737784\pi$
$461$	−29.1771	−1.35891	−0.679456	+	0.733716i	$0.737784\pi$
$462$	0	0
$463$	4.49242	0.208781	0.104390	−	0.994536i	$-0.466711\pi$
$463$	4.49242	0.208781	0.104390	+	0.994536i	$0.466711\pi$
$464$	0	0
$465$	−23.6155	−1.09514
$466$	0	0
$467$	11.7538	0.543900	0.271950	−	0.962311i	$-0.412331\pi$
$467$	11.7538	0.543900	0.271950	+	0.962311i	$0.412331\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	−20.4924	−0.944241
$472$	0	0
$473$	32.0000	1.47136
$474$	0	0
$475$	6.00000	0.275299
$476$	0	0
$477$	5.61553	0.257117
$478$	0	0
$479$	2.24621	0.102632	0.0513160	−	0.998682i	$-0.483658\pi$
$479$	2.24621	0.102632	0.0513160	+	0.998682i	$0.483658\pi$
$480$	0	0
$481$	9.36932	0.427204
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−20.4924	−0.930513
$486$	0	0
$487$	−18.0540	−0.818104	−0.409052	−	0.912511i	$-0.634140\pi$
$487$	−18.0540	−0.818104	−0.409052	+	0.912511i	$0.634140\pi$
$488$	0	0
$489$	−7.31534	−0.330811
$490$	0	0
$491$	34.9309	1.57641	0.788204	−	0.615414i	$-0.211011\pi$
$491$	34.9309	1.57641	0.788204	+	0.615414i	$0.211011\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	3.50758	0.157654
$496$	0	0
$497$	0	0
$498$	0	0
$499$	3.31534	0.148415	0.0742075	−	0.997243i	$-0.476357\pi$
$499$	3.31534	0.148415	0.0742075	+	0.997243i	$0.476357\pi$
$500$	0	0
$501$	−8.00000	−0.357414
$502$	0	0
$503$	44.1080	1.96668	0.983338	−	0.181786i	$-0.0581877\pi$
$503$	44.1080	1.96668	0.983338	+	0.181786i	$0.0581877\pi$
$504$	0	0
$505$	6.24621	0.277953
$506$	0	0
$507$	16.4924	0.732454
$508$	0	0
$509$	24.3002	1.07709	0.538543	−	0.842598i	$-0.318975\pi$
$509$	24.3002	1.07709	0.538543	+	0.842598i	$0.318975\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	−33.3693	−1.47329
$514$	0	0
$515$	−4.49242	−0.197960
$516$	0	0
$517$	−36.1080	−1.58803
$518$	0	0
$519$	−27.1231	−1.19057
$520$	0	0
$521$	32.4924	1.42352	0.711759	−	0.702423i	$-0.247899\pi$
$521$	32.4924	1.42352	0.711759	+	0.702423i	$0.247899\pi$
$522$	0	0
$523$	37.1231	1.62328	0.811640	−	0.584158i	$-0.198575\pi$
$523$	37.1231	1.62328	0.811640	+	0.584158i	$0.198575\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	0	0
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	−5.75379	−0.249693
$532$	0	0
$533$	10.4384	0.452139
$534$	0	0
$535$	34.7386	1.50188
$536$	0	0
$537$	−4.57671	−0.197500
$538$	0	0
$539$	0	0
$540$	0	0
$541$	16.0540	0.690214	0.345107	−	0.938563i	$-0.387843\pi$
$541$	16.0540	0.690214	0.345107	+	0.938563i	$0.387843\pi$
$542$	0	0
$543$	−31.6155	−1.35675
$544$	0	0
$545$	−10.2462	−0.438899
$546$	0	0
$547$	9.17708	0.392384	0.196192	−	0.980566i	$-0.437142\pi$
$547$	9.17708	0.392384	0.196192	+	0.980566i	$0.437142\pi$
$548$	0	0
$549$	4.63068	0.197633
$550$	0	0
$551$	40.1080	1.70866
$552$	0	0
$553$	0	0
$554$	0	0
$555$	18.7386	0.795411
$556$	0	0
$557$	25.6155	1.08536	0.542682	−	0.839938i	$-0.317409\pi$
$557$	25.6155	1.08536	0.542682	+	0.839938i	$0.317409\pi$
$558$	0	0
$559$	16.0000	0.676728
$560$	0	0
$561$	0	0
$562$	0	0
$563$	11.8617	0.499913	0.249956	−	0.968257i	$-0.419584\pi$
$563$	11.8617	0.499913	0.249956	+	0.968257i	$0.419584\pi$
$564$	0	0
$565$	−14.7386	−0.620059
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−12.7386	−0.534031	−0.267016	−	0.963692i	$-0.586038\pi$
$569$	−12.7386	−0.534031	−0.267016	+	0.963692i	$0.586038\pi$
$570$	0	0
$571$	16.4924	0.690186	0.345093	−	0.938568i	$-0.387847\pi$
$571$	16.4924	0.690186	0.345093	+	0.938568i	$0.387847\pi$
$572$	0	0
$573$	7.61553	0.318143
$574$	0	0
$575$	1.00000	0.0417029
$576$	0	0
$577$	8.93087	0.371797	0.185898	−	0.982569i	$-0.440480\pi$
$577$	8.93087	0.371797	0.185898	+	0.982569i	$0.440480\pi$
$578$	0	0
$579$	−40.3002	−1.67482
$580$	0	0
$581$	0	0
$582$	0	0
$583$	31.2311	1.29346
$584$	0	0
$585$	1.75379	0.0725102
$586$	0	0
$587$	4.19224	0.173032	0.0865160	−	0.996250i	$-0.472427\pi$
$587$	4.19224	0.173032	0.0865160	+	0.996250i	$0.472427\pi$
$588$	0	0
$589$	−45.3693	−1.86941
$590$	0	0
$591$	−18.0540	−0.742641
$592$	0	0
$593$	44.2462	1.81697	0.908487	−	0.417913i	$-0.137238\pi$
$593$	44.2462	1.81697	0.908487	+	0.417913i	$0.137238\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−11.1231	−0.455238
$598$	0	0
$599$	24.9848	1.02085	0.510427	−	0.859921i	$-0.329488\pi$
$599$	24.9848	1.02085	0.510427	+	0.859921i	$0.329488\pi$
$600$	0	0
$601$	2.68466	0.109510	0.0547548	−	0.998500i	$-0.482562\pi$
$601$	2.68466	0.109510	0.0547548	+	0.998500i	$0.482562\pi$
$602$	0	0
$603$	−8.49242	−0.345838
$604$	0	0
$605$	−2.49242	−0.101331
$606$	0	0
$607$	−12.6307	−0.512664	−0.256332	−	0.966589i	$-0.582514\pi$
$607$	−12.6307	−0.512664	−0.256332	+	0.966589i	$0.582514\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−18.0540	−0.730386
$612$	0	0
$613$	41.1231	1.66095	0.830473	−	0.557058i	$-0.188070\pi$
$613$	41.1231	1.66095	0.830473	+	0.557058i	$0.188070\pi$
$614$	0	0
$615$	20.8769	0.841838
$616$	0	0
$617$	−34.0000	−1.36879	−0.684394	−	0.729112i	$-0.739933\pi$
$617$	−34.0000	−1.36879	−0.684394	+	0.729112i	$0.739933\pi$
$618$	0	0
$619$	−11.3693	−0.456971	−0.228486	−	0.973547i	$-0.573377\pi$
$619$	−11.3693	−0.456971	−0.228486	+	0.973547i	$0.573377\pi$
$620$	0	0
$621$	−5.56155	−0.223177
$622$	0	0
$623$	0	0
$624$	0	0
$625$	−19.0000	−0.760000
$626$	0	0
$627$	−29.2614	−1.16859
$628$	0	0
$629$	0	0
$630$	0	0
$631$	34.7386	1.38292	0.691462	−	0.722413i	$-0.256967\pi$
$631$	34.7386	1.38292	0.691462	+	0.722413i	$0.256967\pi$
$632$	0	0
$633$	6.24621	0.248265
$634$	0	0
$635$	41.3693	1.64169
$636$	0	0
$637$	0	0
$638$	0	0
$639$	−0.876894	−0.0346894
$640$	0	0
$641$	−23.8617	−0.942482	−0.471241	−	0.882004i	$-0.656194\pi$
$641$	−23.8617	−0.942482	−0.471241	+	0.882004i	$0.656194\pi$
$642$	0	0
$643$	45.2311	1.78374	0.891869	−	0.452293i	$-0.149394\pi$
$643$	45.2311	1.78374	0.891869	+	0.452293i	$0.149394\pi$
$644$	0	0
$645$	32.0000	1.26000
$646$	0	0
$647$	25.3153	0.995249	0.497624	−	0.867393i	$-0.334206\pi$
$647$	25.3153	0.995249	0.497624	+	0.867393i	$0.334206\pi$
$648$	0	0
$649$	−32.0000	−1.25611
$650$	0	0
$651$	0	0
$652$	0	0
$653$	18.6847	0.731187	0.365594	−	0.930775i	$-0.380866\pi$
$653$	18.6847	0.731187	0.365594	+	0.930775i	$0.380866\pi$
$654$	0	0
$655$	−31.6155	−1.23532
$656$	0	0
$657$	−6.00000	−0.234082
$658$	0	0
$659$	−2.24621	−0.0875000	−0.0437500	−	0.999043i	$-0.513930\pi$
$659$	−2.24621	−0.0875000	−0.0437500	+	0.999043i	$0.513930\pi$
$660$	0	0
$661$	−6.49242	−0.252526	−0.126263	−	0.991997i	$-0.540298\pi$
$661$	−6.49242	−0.252526	−0.126263	+	0.991997i	$0.540298\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	6.68466	0.258831
$668$	0	0
$669$	−24.0000	−0.927894
$670$	0	0
$671$	25.7538	0.994214
$672$	0	0
$673$	35.1771	1.35598	0.677988	−	0.735073i	$-0.262852\pi$
$673$	35.1771	1.35598	0.677988	+	0.735073i	$0.262852\pi$
$674$	0	0
$675$	−5.56155	−0.214064
$676$	0	0
$677$	−14.0000	−0.538064	−0.269032	−	0.963131i	$-0.586704\pi$
$677$	−14.0000	−0.538064	−0.269032	+	0.963131i	$0.586704\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	−42.7386	−1.63775
$682$	0	0
$683$	−14.0540	−0.537760	−0.268880	−	0.963174i	$-0.586654\pi$
$683$	−14.0540	−0.537760	−0.268880	+	0.963174i	$0.586654\pi$
$684$	0	0
$685$	14.7386	0.563134
$686$	0	0
$687$	−11.5076	−0.439041
$688$	0	0
$689$	15.6155	0.594904
$690$	0	0
$691$	−38.2462	−1.45495	−0.727477	−	0.686132i	$-0.759307\pi$
$691$	−38.2462	−1.45495	−0.727477	+	0.686132i	$0.759307\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−20.8769	−0.791906
$696$	0	0
$697$	0	0
$698$	0	0
$699$	27.0388	1.02270
$700$	0	0
$701$	−12.6307	−0.477054	−0.238527	−	0.971136i	$-0.576665\pi$
$701$	−12.6307	−0.477054	−0.238527	+	0.971136i	$0.576665\pi$
$702$	0	0
$703$	36.0000	1.35777
$704$	0	0
$705$	−36.1080	−1.35990
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−17.1231	−0.643072	−0.321536	−	0.946897i	$-0.604199\pi$
$709$	−17.1231	−0.643072	−0.321536	+	0.946897i	$0.604199\pi$
$710$	0	0
$711$	−9.75379	−0.365796
$712$	0	0
$713$	−7.56155	−0.283182
$714$	0	0
$715$	9.75379	0.364771
$716$	0	0
$717$	22.5464	0.842011
$718$	0	0
$719$	33.6155	1.25365	0.626824	−	0.779161i	$-0.284355\pi$
$719$	33.6155	1.25365	0.626824	+	0.779161i	$0.284355\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	−17.3693	−0.645972
$724$	0	0
$725$	6.68466	0.248262
$726$	0	0
$727$	23.1231	0.857589	0.428794	−	0.903402i	$-0.358939\pi$
$727$	23.1231	0.857589	0.428794	+	0.903402i	$0.358939\pi$
$728$	0	0
$729$	29.9848	1.11055
$730$	0	0
$731$	0	0
$732$	0	0
$733$	46.1080	1.70304	0.851518	−	0.524325i	$-0.175682\pi$
$733$	46.1080	1.70304	0.851518	+	0.524325i	$0.175682\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−47.2311	−1.73978
$738$	0	0
$739$	−38.0540	−1.39984	−0.699919	−	0.714222i	$-0.746781\pi$
$739$	−38.0540	−1.39984	−0.699919	+	0.714222i	$0.746781\pi$
$740$	0	0
$741$	−14.6307	−0.537472
$742$	0	0
$743$	42.7386	1.56793	0.783964	−	0.620806i	$-0.213195\pi$
$743$	42.7386	1.56793	0.783964	+	0.620806i	$0.213195\pi$
$744$	0	0
$745$	28.9848	1.06192
$746$	0	0
$747$	6.38447	0.233596
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−16.9848	−0.619786	−0.309893	−	0.950771i	$-0.600293\pi$
$751$	−16.9848	−0.619786	−0.309893	+	0.950771i	$0.600293\pi$
$752$	0	0
$753$	27.5076	1.00243
$754$	0	0
$755$	−21.8617	−0.795630
$756$	0	0
$757$	20.2462	0.735861	0.367931	−	0.929853i	$-0.380066\pi$
$757$	20.2462	0.735861	0.367931	+	0.929853i	$0.380066\pi$
$758$	0	0
$759$	−4.87689	−0.177020
$760$	0	0
$761$	−3.06913	−0.111256	−0.0556279	−	0.998452i	$-0.517716\pi$
$761$	−3.06913	−0.111256	−0.0556279	+	0.998452i	$0.517716\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−16.0000	−0.577727
$768$	0	0
$769$	21.8617	0.788354	0.394177	−	0.919034i	$-0.371030\pi$
$769$	21.8617	0.788354	0.394177	+	0.919034i	$0.371030\pi$
$770$	0	0
$771$	−5.56155	−0.200294
$772$	0	0
$773$	−30.1080	−1.08291	−0.541454	−	0.840730i	$-0.682126\pi$
$773$	−30.1080	−1.08291	−0.541454	+	0.840730i	$0.682126\pi$
$774$	0	0
$775$	−7.56155	−0.271619
$776$	0	0
$777$	0	0
$778$	0	0
$779$	40.1080	1.43702
$780$	0	0
$781$	−4.87689	−0.174509
$782$	0	0
$783$	−37.1771	−1.32860
$784$	0	0
$785$	26.2462	0.936767
$786$	0	0
$787$	−31.8617	−1.13575	−0.567874	−	0.823115i	$-0.692234\pi$
$787$	−31.8617	−1.13575	−0.567874	+	0.823115i	$0.692234\pi$
$788$	0	0
$789$	36.8769	1.31285
$790$	0	0
$791$	0	0
$792$	0	0
$793$	12.8769	0.457272
$794$	0	0
$795$	31.2311	1.10765
$796$	0	0
$797$	14.0000	0.495905	0.247953	−	0.968772i	$-0.420242\pi$
$797$	14.0000	0.495905	0.247953	+	0.968772i	$0.420242\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	−6.73863	−0.238098
$802$	0	0
$803$	−33.3693	−1.17758
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−11.4233	−0.402119
$808$	0	0
$809$	−8.24621	−0.289921	−0.144961	−	0.989437i	$-0.546306\pi$
$809$	−8.24621	−0.289921	−0.144961	+	0.989437i	$0.546306\pi$
$810$	0	0
$811$	−1.56155	−0.0548335	−0.0274168	−	0.999624i	$-0.508728\pi$
$811$	−1.56155	−0.0548335	−0.0274168	+	0.999624i	$0.508728\pi$
$812$	0	0
$813$	4.49242	0.157556
$814$	0	0
$815$	9.36932	0.328193
$816$	0	0
$817$	61.4773	2.15082
$818$	0	0
$819$	0	0
$820$	0	0
$821$	20.2462	0.706598	0.353299	−	0.935511i	$-0.385060\pi$
$821$	20.2462	0.706598	0.353299	+	0.935511i	$0.385060\pi$
$822$	0	0
$823$	25.5616	0.891020	0.445510	−	0.895277i	$-0.353022\pi$
$823$	25.5616	0.891020	0.445510	+	0.895277i	$0.353022\pi$
$824$	0	0
$825$	−4.87689	−0.169792
$826$	0	0
$827$	−4.49242	−0.156217	−0.0781084	−	0.996945i	$-0.524888\pi$
$827$	−4.49242	−0.156217	−0.0781084	+	0.996945i	$0.524888\pi$
$828$	0	0
$829$	25.8617	0.898215	0.449108	−	0.893478i	$-0.351742\pi$
$829$	25.8617	0.898215	0.449108	+	0.893478i	$0.351742\pi$
$830$	0	0
$831$	29.1771	1.01214
$832$	0	0
$833$	0	0
$834$	0	0
$835$	10.2462	0.354585
$836$	0	0
$837$	42.0540	1.45360
$838$	0	0
$839$	−12.4924	−0.431286	−0.215643	−	0.976472i	$-0.569185\pi$
$839$	−12.4924	−0.431286	−0.215643	+	0.976472i	$0.569185\pi$
$840$	0	0
$841$	15.6847	0.540850
$842$	0	0
$843$	−7.23106	−0.249051
$844$	0	0
$845$	−21.1231	−0.726657
$846$	0	0
$847$	0	0
$848$	0	0
$849$	46.8466	1.60777
$850$	0	0
$851$	6.00000	0.205677
$852$	0	0
$853$	−40.8769	−1.39960	−0.699799	−	0.714340i	$-0.746727\pi$
$853$	−40.8769	−1.39960	−0.699799	+	0.714340i	$0.746727\pi$
$854$	0	0
$855$	6.73863	0.230456
$856$	0	0
$857$	−4.93087	−0.168435	−0.0842177	−	0.996447i	$-0.526839\pi$
$857$	−4.93087	−0.168435	−0.0842177	+	0.996447i	$0.526839\pi$
$858$	0	0
$859$	8.68466	0.296317	0.148158	−	0.988964i	$-0.452665\pi$
$859$	8.68466	0.296317	0.148158	+	0.988964i	$0.452665\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−11.3153	−0.385179	−0.192589	−	0.981279i	$-0.561689\pi$
$863$	−11.3153	−0.385179	−0.192589	+	0.981279i	$0.561689\pi$
$864$	0	0
$865$	34.7386	1.18115
$866$	0	0
$867$	26.5464	0.901563
$868$	0	0
$869$	−54.2462	−1.84018
$870$	0	0
$871$	−23.6155	−0.800182
$872$	0	0
$873$	5.75379	0.194736
$874$	0	0
$875$	0	0
$876$	0	0
$877$	41.2311	1.39227	0.696137	−	0.717909i	$-0.254901\pi$
$877$	41.2311	1.39227	0.696137	+	0.717909i	$0.254901\pi$
$878$	0	0
$879$	−5.26137	−0.177461
$880$	0	0
$881$	18.7386	0.631321	0.315660	−	0.948872i	$-0.397774\pi$
$881$	18.7386	0.631321	0.315660	+	0.948872i	$0.397774\pi$
$882$	0	0
$883$	−41.4773	−1.39582	−0.697911	−	0.716185i	$-0.745887\pi$
$883$	−41.4773	−1.39582	−0.697911	+	0.716185i	$0.745887\pi$
$884$	0	0
$885$	−32.0000	−1.07567
$886$	0	0
$887$	21.8078	0.732233	0.366117	−	0.930569i	$-0.380687\pi$
$887$	21.8078	0.732233	0.366117	+	0.930569i	$0.380687\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	21.8617	0.732396
$892$	0	0
$893$	−69.3693	−2.32136
$894$	0	0
$895$	5.86174	0.195936
$896$	0	0
$897$	−2.43845	−0.0814174
$898$	0	0
$899$	−50.5464	−1.68582
$900$	0	0
$901$	0	0
$902$	0	0
$903$	0	0
$904$	0	0
$905$	40.4924	1.34601
$906$	0	0
$907$	8.38447	0.278402	0.139201	−	0.990264i	$-0.455547\pi$
$907$	8.38447	0.278402	0.139201	+	0.990264i	$0.455547\pi$
$908$	0	0
$909$	−1.75379	−0.0581695
$910$	0	0
$911$	35.1231	1.16368	0.581840	−	0.813303i	$-0.302333\pi$
$911$	35.1231	1.16368	0.581840	+	0.813303i	$0.302333\pi$
$912$	0	0
$913$	35.5076	1.17513
$914$	0	0
$915$	25.7538	0.851394
$916$	0	0
$917$	0	0
$918$	0	0
$919$	24.0000	0.791687	0.395843	−	0.918318i	$-0.370452\pi$
$919$	24.0000	0.791687	0.395843	+	0.918318i	$0.370452\pi$
$920$	0	0
$921$	51.5076	1.69723
$922$	0	0
$923$	−2.43845	−0.0802625
$924$	0	0
$925$	6.00000	0.197279
$926$	0	0
$927$	1.26137	0.0414287
$928$	0	0
$929$	−31.6695	−1.03904	−0.519521	−	0.854457i	$-0.673890\pi$
$929$	−31.6695	−1.03904	−0.519521	+	0.854457i	$0.673890\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	−8.30019	−0.271736
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−29.8617	−0.975541	−0.487770	−	0.872972i	$-0.662190\pi$
$937$	−29.8617	−0.975541	−0.487770	+	0.872972i	$0.662190\pi$
$938$	0	0
$939$	9.75379	0.318303
$940$	0	0
$941$	−7.36932	−0.240233	−0.120116	−	0.992760i	$-0.538327\pi$
$941$	−7.36932	−0.240233	−0.120116	+	0.992760i	$0.538327\pi$
$942$	0	0
$943$	6.68466	0.217682
$944$	0	0
$945$	0	0
$946$	0	0
$947$	31.8078	1.03361	0.516807	−	0.856102i	$-0.327121\pi$
$947$	31.8078	1.03361	0.516807	+	0.856102i	$0.327121\pi$
$948$	0	0
$949$	−16.6847	−0.541607
$950$	0	0
$951$	5.86174	0.190080
$952$	0	0
$953$	0.246211	0.00797556	0.00398778	−	0.999992i	$-0.498731\pi$
$953$	0.246211	0.00797556	0.00398778	+	0.999992i	$0.498731\pi$
$954$	0	0
$955$	−9.75379	−0.315625
$956$	0	0
$957$	−32.6004	−1.05382
$958$	0	0
$959$	0	0
$960$	0	0
$961$	26.1771	0.844422
$962$	0	0
$963$	−9.75379	−0.314311
$964$	0	0
$965$	51.6155	1.66156
$966$	0	0
$967$	−31.3153	−1.00703	−0.503517	−	0.863985i	$-0.667961\pi$
$967$	−31.3153	−1.00703	−0.503517	+	0.863985i	$0.667961\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	10.4924	0.336718	0.168359	−	0.985726i	$-0.446153\pi$
$971$	10.4924	0.336718	0.168359	+	0.985726i	$0.446153\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	−2.43845	−0.0780928
$976$	0	0
$977$	−35.7538	−1.14387	−0.571933	−	0.820301i	$-0.693806\pi$
$977$	−35.7538	−1.14387	−0.571933	+	0.820301i	$0.693806\pi$
$978$	0	0
$979$	−37.4773	−1.19778
$980$	0	0
$981$	2.87689	0.0918522
$982$	0	0
$983$	4.87689	0.155549	0.0777744	−	0.996971i	$-0.475219\pi$
$983$	4.87689	0.155549	0.0777744	+	0.996971i	$0.475219\pi$
$984$	0	0
$985$	23.1231	0.736763
$986$	0	0
$987$	0	0
$988$	0	0
$989$	10.2462	0.325811
$990$	0	0
$991$	2.24621	0.0713533	0.0356766	−	0.999363i	$-0.488641\pi$
$991$	2.24621	0.0713533	0.0356766	+	0.999363i	$0.488641\pi$
$992$	0	0
$993$	−0.300187	−0.00952613
$994$	0	0
$995$	14.2462	0.451635
$996$	0	0
$997$	52.6004	1.66587	0.832935	−	0.553370i	$-0.186659\pi$
$997$	52.6004	1.66587	0.832935	+	0.553370i	$0.186659\pi$
$998$	0	0
$999$	−33.3693	−1.05576

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9016.2.a.x.1.1		2
7.6	odd	2		1288.2.a.j.1.2	✓	2
28.27	even	2		2576.2.a.r.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1288.2.a.j.1.2	✓	2	7.6	odd	2
2576.2.a.r.1.1		2	28.27	even	2
9016.2.a.x.1.1		2	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 9016 = 2^{3} \cdot 7^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9016.a (trivial)

\(f(q)\)	\(=\)	\(q-1.56155 q^{3} +2.00000 q^{5} -0.561553 q^{9} +O(q^{10})\)	\(q-1.56155 q^{3} +2.00000 q^{5} -0.561553 q^{9} -3.12311 q^{11} -1.56155 q^{13} -3.12311 q^{15} -6.00000 q^{19} -1.00000 q^{23} -1.00000 q^{25} +5.56155 q^{27} -6.68466 q^{29} +7.56155 q^{31} +4.87689 q^{33} -6.00000 q^{37} +2.43845 q^{39} -6.68466 q^{41} -10.2462 q^{43} -1.12311 q^{45} +11.5616 q^{47} -10.0000 q^{53} -6.24621 q^{55} +9.36932 q^{57} +10.2462 q^{59} -8.24621 q^{61} -3.12311 q^{65} +15.1231 q^{67} +1.56155 q^{69} +1.56155 q^{71} +10.6847 q^{73} +1.56155 q^{75} +17.3693 q^{79} -7.00000 q^{81} -11.3693 q^{83} +10.4384 q^{87} +12.0000 q^{89} -11.8078 q^{93} -12.0000 q^{95} -10.2462 q^{97} +1.75379 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + q^{3} + 4 q^{5} + 3 q^{9}+O(q^{10}) \) 2 * q + q^3 + 4 * q^5 + 3 * q^9	\( 2 q + q^{3} + 4 q^{5} + 3 q^{9} + 2 q^{11} + q^{13} + 2 q^{15} - 12 q^{19} - 2 q^{23} - 2 q^{25} + 7 q^{27} - q^{29} + 11 q^{31} + 18 q^{33} - 12 q^{37} + 9 q^{39} - q^{41} - 4 q^{43} + 6 q^{45} + 19 q^{47} - 20 q^{53} + 4 q^{55} - 6 q^{57} + 4 q^{59} + 2 q^{65} + 22 q^{67} - q^{69} - q^{71} + 9 q^{73} - q^{75} + 10 q^{79} - 14 q^{81} + 2 q^{83} + 25 q^{87} + 24 q^{89} - 3 q^{93} - 24 q^{95} - 4 q^{97} + 20 q^{99}+O(q^{100}) \) 2 * q + q^3 + 4 * q^5 + 3 * q^9 + 2 * q^11 + q^13 + 2 * q^15 - 12 * q^19 - 2 * q^23 - 2 * q^25 + 7 * q^27 - q^29 + 11 * q^31 + 18 * q^33 - 12 * q^37 + 9 * q^39 - q^41 - 4 * q^43 + 6 * q^45 + 19 * q^47 - 20 * q^53 + 4 * q^55 - 6 * q^57 + 4 * q^59 + 2 * q^65 + 22 * q^67 - q^69 - q^71 + 9 * q^73 - q^75 + 10 * q^79 - 14 * q^81 + 2 * q^83 + 25 * q^87 + 24 * q^89 - 3 * q^93 - 24 * q^95 - 4 * q^97 + 20 * q^99

Introduction

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