LMFDB - Embedded newform 9016.2.a.bd.1.3

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:	$4$
Coefficient field:	4.4.170528.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 9x^{2} + 2 $ x^4 - 9*x^2 + 2
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$0.477491$ 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)

$f(q)$	$=$	$q+0.477491 q^{3} +4.18856 q^{5} -2.77200 q^{9} +O(q^{10})$	$q+0.477491 q^{3} +4.18856 q^{5} -2.77200 q^{9} +2.00000 q^{11} +4.66605 q^{13} +2.00000 q^{15} -3.23357 q^{17} +0.954983 q^{19} -1.00000 q^{23} +12.5440 q^{25} -2.75608 q^{27} -5.77200 q^{29} +7.89962 q^{31} +0.954983 q^{33} -2.00000 q^{37} +2.22800 q^{39} +4.66605 q^{41} -4.00000 q^{43} -11.6107 q^{45} +6.94464 q^{47} -1.54400 q^{51} -6.00000 q^{53} +8.37711 q^{55} +0.455996 q^{57} +6.09852 q^{59} +10.6557 q^{61} +19.5440 q^{65} -6.00000 q^{67} -0.477491 q^{69} +1.77200 q^{71} -3.71106 q^{73} +5.98966 q^{75} +14.0000 q^{79} +7.00000 q^{81} +16.7542 q^{83} -13.5440 q^{85} -2.75608 q^{87} -5.14354 q^{89} +3.77200 q^{93} +4.00000 q^{95} -1.32361 q^{97} -5.54400 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 6 q^{9}+O(q^{10}) $ 4 * q + 6 * q^9	$ 4 q + 6 q^{9} + 8 q^{11} + 8 q^{15} - 4 q^{23} + 16 q^{25} - 6 q^{29} - 8 q^{37} + 26 q^{39} - 16 q^{43} + 28 q^{51} - 24 q^{53} + 36 q^{57} + 44 q^{65} - 24 q^{67} - 10 q^{71} + 56 q^{79} + 28 q^{81} - 20 q^{85} - 2 q^{93} + 16 q^{95} + 12 q^{99}+O(q^{100}) $ 4 * q + 6 * q^9 + 8 * q^11 + 8 * q^15 - 4 * q^23 + 16 * q^25 - 6 * q^29 - 8 * q^37 + 26 * q^39 - 16 * q^43 + 28 * q^51 - 24 * q^53 + 36 * q^57 + 44 * q^65 - 24 * q^67 - 10 * q^71 + 56 * q^79 + 28 * q^81 - 20 * q^85 - 2 * q^93 + 16 * q^95 + 12 * 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
$3$	0.477491	0.275680	0.137840	−	0.990455i	$-0.455984\pi$
$3$	0.477491	0.275680	0.137840	+	0.990455i	$0.455984\pi$
$4$	0	0
$5$	4.18856	1.87318	0.936590	−	0.350428i	$-0.113964\pi$
$5$	4.18856	1.87318	0.936590	+	0.350428i	$0.113964\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	−2.77200	−0.924001
$10$	0	0
$11$	2.00000	0.603023	0.301511	−	0.953463i	$-0.402509\pi$
$11$	2.00000	0.603023	0.301511	+	0.953463i	$0.402509\pi$
$12$	0	0
$13$	4.66605	1.29413	0.647064	−	0.762435i	$-0.275996\pi$
$13$	4.66605	1.29413	0.647064	+	0.762435i	$0.275996\pi$
$14$	0	0
$15$	2.00000	0.516398
$16$	0	0
$17$	−3.23357	−0.784257	−0.392128	−	0.919911i	$-0.628261\pi$
$17$	−3.23357	−0.784257	−0.392128	+	0.919911i	$0.628261\pi$
$18$	0	0
$19$	0.954983	0.219088	0.109544	−	0.993982i	$-0.465061\pi$
$19$	0.954983	0.219088	0.109544	+	0.993982i	$0.465061\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	0	0
$25$	12.5440	2.50880
$26$	0	0
$27$	−2.75608	−0.530408
$28$	0	0
$29$	−5.77200	−1.07183	−0.535917	−	0.844271i	$-0.680034\pi$
$29$	−5.77200	−1.07183	−0.535917	+	0.844271i	$0.680034\pi$
$30$	0	0
$31$	7.89962	1.41881	0.709407	−	0.704799i	$-0.248963\pi$
$31$	7.89962	1.41881	0.709407	+	0.704799i	$0.248963\pi$
$32$	0	0
$33$	0.954983	0.166241
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−2.00000	−0.328798	−0.164399	−	0.986394i	$-0.552568\pi$
$37$	−2.00000	−0.328798	−0.164399	+	0.986394i	$0.552568\pi$
$38$	0	0
$39$	2.22800	0.356765
$40$	0	0
$41$	4.66605	0.728714	0.364357	−	0.931259i	$-0.381289\pi$
$41$	4.66605	0.728714	0.364357	+	0.931259i	$0.381289\pi$
$42$	0	0
$43$	−4.00000	−0.609994	−0.304997	−	0.952353i	$-0.598656\pi$
$43$	−4.00000	−0.609994	−0.304997	+	0.952353i	$0.598656\pi$
$44$	0	0
$45$	−11.6107	−1.73082
$46$	0	0
$47$	6.94464	1.01298	0.506490	−	0.862246i	$-0.330943\pi$
$47$	6.94464	1.01298	0.506490	+	0.862246i	$0.330943\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	−1.54400	−0.216204
$52$	0	0
$53$	−6.00000	−0.824163	−0.412082	−	0.911147i	$-0.635198\pi$
$53$	−6.00000	−0.824163	−0.412082	+	0.911147i	$0.635198\pi$
$54$	0	0
$55$	8.37711	1.12957
$56$	0	0
$57$	0.455996	0.0603982
$58$	0	0
$59$	6.09852	0.793960	0.396980	−	0.917827i	$-0.370058\pi$
$59$	6.09852	0.793960	0.396980	+	0.917827i	$0.370058\pi$
$60$	0	0
$61$	10.6557	1.36432	0.682161	−	0.731202i	$-0.261040\pi$
$61$	10.6557	1.36432	0.682161	+	0.731202i	$0.261040\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	19.5440	2.42414
$66$	0	0
$67$	−6.00000	−0.733017	−0.366508	−	0.930415i	$-0.619447\pi$
$67$	−6.00000	−0.733017	−0.366508	+	0.930415i	$0.619447\pi$
$68$	0	0
$69$	−0.477491	−0.0574832
$70$	0	0
$71$	1.77200	0.210298	0.105149	−	0.994456i	$-0.466468\pi$
$71$	1.77200	0.210298	0.105149	+	0.994456i	$0.466468\pi$
$72$	0	0
$73$	−3.71106	−0.434347	−0.217174	−	0.976133i	$-0.569684\pi$
$73$	−3.71106	−0.434347	−0.217174	+	0.976133i	$0.569684\pi$
$74$	0	0
$75$	5.98966	0.691626
$76$	0	0
$77$	0	0
$78$	0	0
$79$	14.0000	1.57512	0.787562	−	0.616236i	$-0.211343\pi$
$79$	14.0000	1.57512	0.787562	+	0.616236i	$0.211343\pi$
$80$	0	0
$81$	7.00000	0.777778
$82$	0	0
$83$	16.7542	1.83902	0.919508	−	0.393072i	$-0.128588\pi$
$83$	16.7542	1.83902	0.919508	+	0.393072i	$0.128588\pi$
$84$	0	0
$85$	−13.5440	−1.46905
$86$	0	0
$87$	−2.75608	−0.295483
$88$	0	0
$89$	−5.14354	−0.545214	−0.272607	−	0.962125i	$-0.587886\pi$
$89$	−5.14354	−0.545214	−0.272607	+	0.962125i	$0.587886\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	3.77200	0.391138
$94$	0	0
$95$	4.00000	0.410391
$96$	0	0
$97$	−1.32361	−0.134392	−0.0671960	−	0.997740i	$-0.521405\pi$
$97$	−1.32361	−0.134392	−0.0671960	+	0.997740i	$0.521405\pi$
$98$	0	0
$99$	−5.54400	−0.557193
$100$	0	0
$101$	0	0		−	1.00000i	$-0.5\pi$
$101$	0	0			1.00000i	$0.5\pi$
$102$	0	0
$103$	14.8443	1.46265	0.731324	−	0.682030i	$-0.238903\pi$
$103$	14.8443	1.46265	0.731324	+	0.682030i	$0.238903\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−15.5440	−1.50270	−0.751348	−	0.659906i	$-0.770596\pi$
$107$	−15.5440	−1.50270	−0.751348	+	0.659906i	$0.770596\pi$
$108$	0	0
$109$	−13.5440	−1.29728	−0.648640	−	0.761095i	$-0.724662\pi$
$109$	−13.5440	−1.29728	−0.648640	+	0.761095i	$0.724662\pi$
$110$	0	0
$111$	−0.954983	−0.0906430
$112$	0	0
$113$	1.54400	0.145248	0.0726238	−	0.997359i	$-0.476863\pi$
$113$	1.54400	0.145248	0.0726238	+	0.997359i	$0.476863\pi$
$114$	0	0
$115$	−4.18856	−0.390585
$116$	0	0
$117$	−12.9343	−1.19578
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−7.00000	−0.636364
$122$	0	0
$123$	2.22800	0.200892
$124$	0	0
$125$	31.5985	2.82625
$126$	0	0
$127$	−21.3160	−1.89149	−0.945745	−	0.324910i	$-0.894666\pi$
$127$	−21.3160	−1.89149	−0.945745	+	0.324910i	$0.894666\pi$
$128$	0	0
$129$	−1.90997	−0.168163
$130$	0	0
$131$	7.89962	0.690193	0.345097	−	0.938567i	$-0.387846\pi$
$131$	7.89962	0.690193	0.345097	+	0.938567i	$0.387846\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	−11.5440	−0.993550
$136$	0	0
$137$	5.54400	0.473656	0.236828	−	0.971552i	$-0.423892\pi$
$137$	5.54400	0.473656	0.236828	+	0.971552i	$0.423892\pi$
$138$	0	0
$139$	−15.3218	−1.29957	−0.649787	−	0.760116i	$-0.725142\pi$
$139$	−15.3218	−1.29957	−0.649787	+	0.760116i	$0.725142\pi$
$140$	0	0
$141$	3.31601	0.279258
$142$	0	0
$143$	9.33210	0.780389
$144$	0	0
$145$	−24.1764	−2.00774
$146$	0	0
$147$	0	0
$148$	0	0
$149$	21.0880	1.72760	0.863798	−	0.503838i	$-0.168079\pi$
$149$	21.0880	1.72760	0.863798	+	0.503838i	$0.168079\pi$
$150$	0	0
$151$	14.2280	1.15786	0.578929	−	0.815378i	$-0.303471\pi$
$151$	14.2280	1.15786	0.578929	+	0.815378i	$0.303471\pi$
$152$	0	0
$153$	8.96347	0.724654
$154$	0	0
$155$	33.0880	2.65769
$156$	0	0
$157$	−3.23357	−0.258067	−0.129034	−	0.991640i	$-0.541188\pi$
$157$	−3.23357	−0.258067	−0.129034	+	0.991640i	$0.541188\pi$
$158$	0	0
$159$	−2.86495	−0.227205
$160$	0	0
$161$	0	0
$162$	0	0
$163$	13.3160	1.04299	0.521495	−	0.853254i	$-0.325375\pi$
$163$	13.3160	1.04299	0.521495	+	0.853254i	$0.325375\pi$
$164$	0	0
$165$	4.00000	0.311400
$166$	0	0
$167$	8.74574	0.676765	0.338383	−	0.941009i	$-0.390120\pi$
$167$	8.74574	0.676765	0.338383	+	0.941009i	$0.390120\pi$
$168$	0	0
$169$	8.77200	0.674769
$170$	0	0
$171$	−2.64721	−0.202438
$172$	0	0
$173$	−6.46715	−0.491688	−0.245844	−	0.969309i	$-0.579065\pi$
$173$	−6.46715	−0.491688	−0.245844	+	0.969309i	$0.579065\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	2.91199	0.218879
$178$	0	0
$179$	17.7720	1.32834	0.664171	−	0.747581i	$-0.268785\pi$
$179$	17.7720	1.32834	0.664171	+	0.747581i	$0.268785\pi$
$180$	0	0
$181$	−2.27859	−0.169366	−0.0846831	−	0.996408i	$-0.526988\pi$
$181$	−2.27859	−0.169366	−0.0846831	+	0.996408i	$0.526988\pi$
$182$	0	0
$183$	5.08801	0.376116
$184$	0	0
$185$	−8.37711	−0.615898
$186$	0	0
$187$	−6.46715	−0.472925
$188$	0	0
$189$	0	0
$190$	0	0
$191$	3.54400	0.256435	0.128218	−	0.991746i	$-0.459074\pi$
$191$	3.54400	0.256435	0.128218	+	0.991746i	$0.459074\pi$
$192$	0	0
$193$	−12.2280	−0.880191	−0.440095	−	0.897951i	$-0.645055\pi$
$193$	−12.2280	−0.880191	−0.440095	+	0.897951i	$0.645055\pi$
$194$	0	0
$195$	9.33210	0.668285
$196$	0	0
$197$	18.8600	1.34372	0.671860	−	0.740678i	$-0.265496\pi$
$197$	18.8600	1.34372	0.671860	+	0.740678i	$0.265496\pi$
$198$	0	0
$199$	22.2664	1.57842	0.789211	−	0.614122i	$-0.210490\pi$
$199$	22.2664	1.57842	0.789211	+	0.614122i	$0.210490\pi$
$200$	0	0
$201$	−2.86495	−0.202078
$202$	0	0
$203$	0	0
$204$	0	0
$205$	19.5440	1.36501
$206$	0	0
$207$	2.77200	0.192667
$208$	0	0
$209$	1.90997	0.132115
$210$	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
$213$	0.846116	0.0579749
$214$	0	0
$215$	−16.7542	−1.14263
$216$	0	0
$217$	0	0
$218$	0	0
$219$	−1.77200	−0.119741
$220$	0	0
$221$	−15.0880	−1.01493
$222$	0	0
$223$	−14.4756	−0.969360	−0.484680	−	0.874691i	$-0.661064\pi$
$223$	−14.4756	−0.969360	−0.484680	+	0.874691i	$0.661064\pi$
$224$	0	0
$225$	−34.7720	−2.31813
$226$	0	0
$227$	−4.55718	−0.302471	−0.151235	−	0.988498i	$-0.548325\pi$
$227$	−4.55718	−0.302471	−0.151235	+	0.988498i	$0.548325\pi$
$228$	0	0
$229$	19.9878	1.32083	0.660415	−	0.750900i	$-0.270380\pi$
$229$	19.9878	1.32083	0.660415	+	0.750900i	$0.270380\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	21.7720	1.42633	0.713166	−	0.700996i	$-0.247261\pi$
$233$	21.7720	1.42633	0.713166	+	0.700996i	$0.247261\pi$
$234$	0	0
$235$	29.0880	1.89749
$236$	0	0
$237$	6.68488	0.434230
$238$	0	0
$239$	−9.77200	−0.632098	−0.316049	−	0.948743i	$-0.602356\pi$
$239$	−9.77200	−0.632098	−0.316049	+	0.948743i	$0.602356\pi$
$240$	0	0
$241$	−10.6557	−0.686394	−0.343197	−	0.939263i	$-0.611510\pi$
$241$	−10.6557	−0.686394	−0.343197	+	0.939263i	$0.611510\pi$
$242$	0	0
$243$	11.6107	0.744826
$244$	0	0
$245$	0	0
$246$	0	0
$247$	4.45600	0.283528
$248$	0	0
$249$	8.00000	0.506979
$250$	0	0
$251$	−6.46715	−0.408203	−0.204101	−	0.978950i	$-0.565427\pi$
$251$	−6.46715	−0.408203	−0.204101	+	0.978950i	$0.565427\pi$
$252$	0	0
$253$	−2.00000	−0.125739
$254$	0	0
$255$	−6.46715	−0.404988
$256$	0	0
$257$	−21.4203	−1.33616	−0.668080	−	0.744089i	$-0.732884\pi$
$257$	−21.4203	−1.33616	−0.668080	+	0.744089i	$0.732884\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	16.0000	0.990375
$262$	0	0
$263$	2.00000	0.123325	0.0616626	−	0.998097i	$-0.480360\pi$
$263$	2.00000	0.123325	0.0616626	+	0.998097i	$0.480360\pi$
$264$	0	0
$265$	−25.1313	−1.54381
$266$	0	0
$267$	−2.45600	−0.150305
$268$	0	0
$269$	−1.80110	−0.109815	−0.0549075	−	0.998491i	$-0.517486\pi$
$269$	−1.80110	−0.109815	−0.0549075	+	0.998491i	$0.517486\pi$
$270$	0	0
$271$	27.4099	1.66503	0.832517	−	0.553999i	$-0.186899\pi$
$271$	27.4099	1.66503	0.832517	+	0.553999i	$0.186899\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	25.0880	1.51286
$276$	0	0
$277$	−9.31601	−0.559745	−0.279872	−	0.960037i	$-0.590292\pi$
$277$	−9.31601	−0.559745	−0.279872	+	0.960037i	$0.590292\pi$
$278$	0	0
$279$	−21.8978	−1.31098
$280$	0	0
$281$	−28.6320	−1.70804	−0.854021	−	0.520238i	$-0.825843\pi$
$281$	−28.6320	−1.70804	−0.854021	+	0.520238i	$0.825843\pi$
$282$	0	0
$283$	0.954983	0.0567678	0.0283839	−	0.999597i	$-0.490964\pi$
$283$	0.954983	0.0567678	0.0283839	+	0.999597i	$0.490964\pi$
$284$	0	0
$285$	1.90997	0.113137
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−6.54400	−0.384941
$290$	0	0
$291$	−0.632011	−0.0370492
$292$	0	0
$293$	−3.23357	−0.188907	−0.0944537	−	0.995529i	$-0.530110\pi$
$293$	−3.23357	−0.188907	−0.0944537	+	0.995529i	$0.530110\pi$
$294$	0	0
$295$	25.5440	1.48723
$296$	0	0
$297$	−5.51216	−0.319848
$298$	0	0
$299$	−4.66605	−0.269845
$300$	0	0
$301$	0	0
$302$	0	0
$303$	0	0
$304$	0	0
$305$	44.6320	2.55562
$306$	0	0
$307$	−33.1398	−1.89139	−0.945695	−	0.325056i	$-0.894617\pi$
$307$	−33.1398	−1.89139	−0.945695	+	0.325056i	$0.894617\pi$
$308$	0	0
$309$	7.08801	0.403223
$310$	0	0
$311$	−11.5018	−0.652208	−0.326104	−	0.945334i	$-0.605736\pi$
$311$	−11.5018	−0.652208	−0.326104	+	0.945334i	$0.605736\pi$
$312$	0	0
$313$	−5.14354	−0.290730	−0.145365	−	0.989378i	$-0.546436\pi$
$313$	−5.14354	−0.290730	−0.145365	+	0.989378i	$0.546436\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	3.54400	0.199051	0.0995255	−	0.995035i	$-0.468268\pi$
$317$	3.54400	0.199051	0.0995255	+	0.995035i	$0.468268\pi$
$318$	0	0
$319$	−11.5440	−0.646340
$320$	0	0
$321$	−7.42213	−0.414263
$322$	0	0
$323$	−3.08801	−0.171821
$324$	0	0
$325$	58.5309	3.24671
$326$	0	0
$327$	−6.46715	−0.357634
$328$	0	0
$329$	0	0
$330$	0	0
$331$	24.8600	1.36643	0.683215	−	0.730217i	$-0.260581\pi$
$331$	24.8600	1.36643	0.683215	+	0.730217i	$0.260581\pi$
$332$	0	0
$333$	5.54400	0.303810
$334$	0	0
$335$	−25.1313	−1.37307
$336$	0	0
$337$	−25.5440	−1.39147	−0.695735	−	0.718298i	$-0.744921\pi$
$337$	−25.5440	−1.39147	−0.695735	+	0.718298i	$0.744921\pi$
$338$	0	0
$339$	0.737249	0.0400419
$340$	0	0
$341$	15.7992	0.855577
$342$	0	0
$343$	0	0
$344$	0	0
$345$	−2.00000	−0.107676
$346$	0	0
$347$	−15.0880	−0.809967	−0.404983	−	0.914324i	$-0.632723\pi$
$347$	−15.0880	−0.809967	−0.404983	+	0.914324i	$0.632723\pi$
$348$	0	0
$349$	−11.1332	−0.595946	−0.297973	−	0.954574i	$-0.596311\pi$
$349$	−11.1332	−0.595946	−0.297973	+	0.954574i	$0.596311\pi$
$350$	0	0
$351$	−12.8600	−0.686417
$352$	0	0
$353$	−14.9531	−0.795875	−0.397937	−	0.917413i	$-0.630274\pi$
$353$	−14.9531	−0.795875	−0.397937	+	0.917413i	$0.630274\pi$
$354$	0	0
$355$	7.42213	0.393926
$356$	0	0
$357$	0	0
$358$	0	0
$359$	18.0000	0.950004	0.475002	−	0.879985i	$-0.342447\pi$
$359$	18.0000	0.950004	0.475002	+	0.879985i	$0.342447\pi$
$360$	0	0
$361$	−18.0880	−0.952000
$362$	0	0
$363$	−3.34244	−0.175433
$364$	0	0
$365$	−15.5440	−0.813610
$366$	0	0
$367$	4.55718	0.237883	0.118941	−	0.992901i	$-0.462050\pi$
$367$	4.55718	0.237883	0.118941	+	0.992901i	$0.462050\pi$
$368$	0	0
$369$	−12.9343	−0.673332
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−17.0880	−0.884783	−0.442392	−	0.896822i	$-0.645870\pi$
$373$	−17.0880	−0.884783	−0.442392	+	0.896822i	$0.645870\pi$
$374$	0	0
$375$	15.0880	0.779141
$376$	0	0
$377$	−26.9324	−1.38709
$378$	0	0
$379$	29.5440	1.51757	0.758787	−	0.651339i	$-0.225792\pi$
$379$	29.5440	1.51757	0.758787	+	0.651339i	$0.225792\pi$
$380$	0	0
$381$	−10.1782	−0.521446
$382$	0	0
$383$	−8.37711	−0.428050	−0.214025	−	0.976828i	$-0.568657\pi$
$383$	−8.37711	−0.428050	−0.214025	+	0.976828i	$0.568657\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	11.0880	0.563635
$388$	0	0
$389$	10.4560	0.530140	0.265070	−	0.964229i	$-0.414605\pi$
$389$	10.4560	0.530140	0.265070	+	0.964229i	$0.414605\pi$
$390$	0	0
$391$	3.23357	0.163529
$392$	0	0
$393$	3.77200	0.190272
$394$	0	0
$395$	58.6398	2.95049
$396$	0	0
$397$	−0.846116	−0.0424653	−0.0212327	−	0.999775i	$-0.506759\pi$
$397$	−0.846116	−0.0424653	−0.0212327	+	0.999775i	$0.506759\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−13.0880	−0.653584	−0.326792	−	0.945096i	$-0.605968\pi$
$401$	−13.0880	−0.653584	−0.326792	+	0.945096i	$0.605968\pi$
$402$	0	0
$403$	36.8600	1.83613
$404$	0	0
$405$	29.3199	1.45692
$406$	0	0
$407$	−4.00000	−0.198273
$408$	0	0
$409$	16.8631	0.833826	0.416913	−	0.908946i	$-0.363112\pi$
$409$	16.8631	0.833826	0.416913	+	0.908946i	$0.363112\pi$
$410$	0	0
$411$	2.64721	0.130577
$412$	0	0
$413$	0	0
$414$	0	0
$415$	70.1760	3.44481
$416$	0	0
$417$	−7.31601	−0.358266
$418$	0	0
$419$	6.46715	0.315941	0.157970	−	0.987444i	$-0.449505\pi$
$419$	6.46715	0.315941	0.157970	+	0.987444i	$0.449505\pi$
$420$	0	0
$421$	−33.5440	−1.63483	−0.817417	−	0.576046i	$-0.804595\pi$
$421$	−33.5440	−1.63483	−0.817417	+	0.576046i	$0.804595\pi$
$422$	0	0
$423$	−19.2505	−0.935994
$424$	0	0
$425$	−40.5620	−1.96754
$426$	0	0
$427$	0	0
$428$	0	0
$429$	4.45600	0.215138
$430$	0	0
$431$	24.0000	1.15604	0.578020	−	0.816023i	$-0.303826\pi$
$431$	24.0000	1.15604	0.578020	+	0.816023i	$0.303826\pi$
$432$	0	0
$433$	−14.4756	−0.695655	−0.347827	−	0.937559i	$-0.613080\pi$
$433$	−14.4756	−0.695655	−0.347827	+	0.937559i	$0.613080\pi$
$434$	0	0
$435$	−11.5440	−0.553493
$436$	0	0
$437$	−0.954983	−0.0456830
$438$	0	0
$439$	−28.2560	−1.34859	−0.674294	−	0.738463i	$-0.735552\pi$
$439$	−28.2560	−1.34859	−0.674294	+	0.738463i	$0.735552\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−32.4040	−1.53956	−0.769781	−	0.638309i	$-0.779634\pi$
$443$	−32.4040	−1.53956	−0.769781	+	0.638309i	$0.779634\pi$
$444$	0	0
$445$	−21.5440	−1.02128
$446$	0	0
$447$	10.0693	0.476264
$448$	0	0
$449$	−19.5440	−0.922339	−0.461169	−	0.887312i	$-0.652570\pi$
$449$	−19.5440	−0.922339	−0.461169	+	0.887312i	$0.652570\pi$
$450$	0	0
$451$	9.33210	0.439431
$452$	0	0
$453$	6.79375	0.319198
$454$	0	0
$455$	0	0
$456$	0	0
$457$	14.4560	0.676223	0.338111	−	0.941106i	$-0.390212\pi$
$457$	14.4560	0.676223	0.338111	+	0.941106i	$0.390212\pi$
$458$	0	0
$459$	8.91199	0.415976
$460$	0	0
$461$	7.31326	0.340613	0.170306	−	0.985391i	$-0.445524\pi$
$461$	7.31326	0.340613	0.170306	+	0.985391i	$0.445524\pi$
$462$	0	0
$463$	35.0880	1.63068	0.815339	−	0.578984i	$-0.196551\pi$
$463$	35.0880	1.63068	0.815339	+	0.578984i	$0.196551\pi$
$464$	0	0
$465$	15.7992	0.732672
$466$	0	0
$467$	−32.5535	−1.50639	−0.753197	−	0.657795i	$-0.771490\pi$
$467$	−32.5535	−1.50639	−0.753197	+	0.657795i	$0.771490\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	−1.54400	−0.0711439
$472$	0	0
$473$	−8.00000	−0.367840
$474$	0	0
$475$	11.9793	0.549648
$476$	0	0
$477$	16.6320	0.761527
$478$	0	0
$479$	12.9343	0.590983	0.295491	−	0.955345i	$-0.404517\pi$
$479$	12.9343	0.590983	0.295491	+	0.955345i	$0.404517\pi$
$480$	0	0
$481$	−9.33210	−0.425507
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−5.54400	−0.251740
$486$	0	0
$487$	24.4040	1.10585	0.552926	−	0.833231i	$-0.313511\pi$
$487$	24.4040	1.10585	0.552926	+	0.833231i	$0.313511\pi$
$488$	0	0
$489$	6.35828	0.287531
$490$	0	0
$491$	−9.77200	−0.441004	−0.220502	−	0.975386i	$-0.570770\pi$
$491$	−9.77200	−0.441004	−0.220502	+	0.975386i	$0.570770\pi$
$492$	0	0
$493$	18.6642	0.840593
$494$	0	0
$495$	−23.2214	−1.04372
$496$	0	0
$497$	0	0
$498$	0	0
$499$	−35.9480	−1.60925	−0.804627	−	0.593780i	$-0.797635\pi$
$499$	−35.9480	−1.60925	−0.804627	+	0.593780i	$0.797635\pi$
$500$	0	0
$501$	4.17601	0.186571
$502$	0	0
$503$	−26.0863	−1.16313	−0.581566	−	0.813499i	$-0.697560\pi$
$503$	−26.0863	−1.16313	−0.581566	+	0.813499i	$0.697560\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	4.18856	0.186020
$508$	0	0
$509$	27.8874	1.23609	0.618044	−	0.786143i	$-0.287925\pi$
$509$	27.8874	1.23609	0.618044	+	0.786143i	$0.287925\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	−2.63201	−0.116206
$514$	0	0
$515$	62.1760	2.73980
$516$	0	0
$517$	13.8893	0.610850
$518$	0	0
$519$	−3.08801	−0.135548
$520$	0	0
$521$	41.2992	1.80935	0.904675	−	0.426101i	$-0.140113\pi$
$521$	41.2992	1.80935	0.904675	+	0.426101i	$0.140113\pi$
$522$	0	0
$523$	21.3114	0.931883	0.465941	−	0.884816i	$-0.345716\pi$
$523$	21.3114	0.931883	0.465941	+	0.884816i	$0.345716\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−25.5440	−1.11271
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	−16.9051	−0.733620
$532$	0	0
$533$	21.7720	0.943050
$534$	0	0
$535$	−65.1069	−2.81482
$536$	0	0
$537$	8.48598	0.366197
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−40.4040	−1.73710	−0.868552	−	0.495598i	$-0.834949\pi$
$541$	−40.4040	−1.73710	−0.868552	+	0.495598i	$0.834949\pi$
$542$	0	0
$543$	−1.08801	−0.0466909
$544$	0	0
$545$	−56.7298	−2.43004
$546$	0	0
$547$	−16.4040	−0.701385	−0.350692	−	0.936491i	$-0.614054\pi$
$547$	−16.4040	−0.701385	−0.350692	+	0.936491i	$0.614054\pi$
$548$	0	0
$549$	−29.5376	−1.26064
$550$	0	0
$551$	−5.51216	−0.234826
$552$	0	0
$553$	0	0
$554$	0	0
$555$	−4.00000	−0.169791
$556$	0	0
$557$	26.4560	1.12098	0.560488	−	0.828162i	$-0.310613\pi$
$557$	26.4560	1.12098	0.560488	+	0.828162i	$0.310613\pi$
$558$	0	0
$559$	−18.6642	−0.789411
$560$	0	0
$561$	−3.08801	−0.130376
$562$	0	0
$563$	−19.4014	−0.817673	−0.408837	−	0.912608i	$-0.634065\pi$
$563$	−19.4014	−0.817673	−0.408837	+	0.912608i	$0.634065\pi$
$564$	0	0
$565$	6.46715	0.272075
$566$	0	0
$567$	0	0
$568$	0	0
$569$	30.0000	1.25767	0.628833	−	0.777541i	$-0.283533\pi$
$569$	30.0000	1.25767	0.628833	+	0.777541i	$0.283533\pi$
$570$	0	0
$571$	−12.9120	−0.540350	−0.270175	−	0.962811i	$-0.587082\pi$
$571$	−12.9120	−0.540350	−0.270175	+	0.962811i	$0.587082\pi$
$572$	0	0
$573$	1.69223	0.0706940
$574$	0	0
$575$	−12.5440	−0.523121
$576$	0	0
$577$	−2.01883	−0.0840451	−0.0420226	−	0.999117i	$-0.513380\pi$
$577$	−2.01883	−0.0840451	−0.0420226	+	0.999117i	$0.513380\pi$
$578$	0	0
$579$	−5.83877	−0.242651
$580$	0	0
$581$	0	0
$582$	0	0
$583$	−12.0000	−0.496989
$584$	0	0
$585$	−54.1760	−2.23990
$586$	0	0
$587$	45.9653	1.89719	0.948594	−	0.316494i	$-0.102506\pi$
$587$	45.9653	1.89719	0.948594	+	0.316494i	$0.102506\pi$
$588$	0	0
$589$	7.54400	0.310845
$590$	0	0
$591$	9.00549	0.370437
$592$	0	0
$593$	18.6642	0.766446	0.383223	−	0.923656i	$-0.374814\pi$
$593$	18.6642	0.766446	0.383223	+	0.923656i	$0.374814\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	10.6320	0.435139
$598$	0	0
$599$	22.1760	0.906087	0.453044	−	0.891488i	$-0.350338\pi$
$599$	22.1760	0.906087	0.453044	+	0.891488i	$0.350338\pi$
$600$	0	0
$601$	−28.1052	−1.14643	−0.573217	−	0.819404i	$-0.694305\pi$
$601$	−28.1052	−1.14643	−0.573217	+	0.819404i	$0.694305\pi$
$602$	0	0
$603$	16.6320	0.677308
$604$	0	0
$605$	−29.3199	−1.19202
$606$	0	0
$607$	6.09852	0.247531	0.123766	−	0.992311i	$-0.460503\pi$
$607$	6.09852	0.247531	0.123766	+	0.992311i	$0.460503\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	32.4040	1.31093
$612$	0	0
$613$	47.7200	1.92739	0.963697	−	0.266999i	$-0.0860322\pi$
$613$	47.7200	1.92739	0.963697	+	0.266999i	$0.0860322\pi$
$614$	0	0
$615$	9.33210	0.376306
$616$	0	0
$617$	−25.0880	−1.01001	−0.505003	−	0.863118i	$-0.668508\pi$
$617$	−25.0880	−1.01001	−0.505003	+	0.863118i	$0.668508\pi$
$618$	0	0
$619$	−12.9343	−0.519873	−0.259937	−	0.965626i	$-0.583702\pi$
$619$	−12.9343	−0.519873	−0.259937	+	0.965626i	$0.583702\pi$
$620$	0	0
$621$	2.75608	0.110598
$622$	0	0
$623$	0	0
$624$	0	0
$625$	69.6320	2.78528
$626$	0	0
$627$	0.911993	0.0364215
$628$	0	0
$629$	6.46715	0.257862
$630$	0	0
$631$	−13.5440	−0.539178	−0.269589	−	0.962975i	$-0.586888\pi$
$631$	−13.5440	−0.539178	−0.269589	+	0.962975i	$0.586888\pi$
$632$	0	0
$633$	−1.90997	−0.0759143
$634$	0	0
$635$	−89.2833	−3.54310
$636$	0	0
$637$	0	0
$638$	0	0
$639$	−4.91199	−0.194315
$640$	0	0
$641$	10.4560	0.412987	0.206494	−	0.978448i	$-0.433795\pi$
$641$	10.4560	0.412987	0.206494	+	0.978448i	$0.433795\pi$
$642$	0	0
$643$	−11.9793	−0.472418	−0.236209	−	0.971702i	$-0.575905\pi$
$643$	−11.9793	−0.472418	−0.236209	+	0.971702i	$0.575905\pi$
$644$	0	0
$645$	−8.00000	−0.315000
$646$	0	0
$647$	−23.6989	−0.931698	−0.465849	−	0.884864i	$-0.654251\pi$
$647$	−23.6989	−0.931698	−0.465849	+	0.884864i	$0.654251\pi$
$648$	0	0
$649$	12.1970	0.478776
$650$	0	0
$651$	0	0
$652$	0	0
$653$	6.86001	0.268453	0.134226	−	0.990951i	$-0.457145\pi$
$653$	6.86001	0.268453	0.134226	+	0.990951i	$0.457145\pi$
$654$	0	0
$655$	33.0880	1.29286
$656$	0	0
$657$	10.2871	0.401337
$658$	0	0
$659$	−19.0880	−0.743563	−0.371782	−	0.928320i	$-0.621253\pi$
$659$	−19.0880	−0.743563	−0.371782	+	0.928320i	$0.621253\pi$
$660$	0	0
$661$	−33.8771	−1.31767	−0.658833	−	0.752289i	$-0.728950\pi$
$661$	−33.8771	−1.31767	−0.658833	+	0.752289i	$0.728950\pi$
$662$	0	0
$663$	−7.20440	−0.279796
$664$	0	0
$665$	0	0
$666$	0	0
$667$	5.77200	0.223493
$668$	0	0
$669$	−6.91199	−0.267233
$670$	0	0
$671$	21.3114	0.822718
$672$	0	0
$673$	−10.2280	−0.394260	−0.197130	−	0.980377i	$-0.563162\pi$
$673$	−10.2280	−0.394260	−0.197130	+	0.980377i	$0.563162\pi$
$674$	0	0
$675$	−34.5723	−1.33069
$676$	0	0
$677$	36.5243	1.40374	0.701871	−	0.712304i	$-0.252348\pi$
$677$	36.5243	1.40374	0.701871	+	0.712304i	$0.252348\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	−2.17601	−0.0833851
$682$	0	0
$683$	1.31601	0.0503556	0.0251778	−	0.999683i	$-0.491985\pi$
$683$	1.31601	0.0503556	0.0251778	+	0.999683i	$0.491985\pi$
$684$	0	0
$685$	23.2214	0.887243
$686$	0	0
$687$	9.54400	0.364126
$688$	0	0
$689$	−27.9963	−1.06657
$690$	0	0
$691$	39.6070	1.50672	0.753360	−	0.657608i	$-0.228432\pi$
$691$	39.6070	1.50672	0.753360	+	0.657608i	$0.228432\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−64.1760	−2.43434
$696$	0	0
$697$	−15.0880	−0.571499
$698$	0	0
$699$	10.3959	0.393211
$700$	0	0
$701$	−5.54400	−0.209394	−0.104697	−	0.994504i	$-0.533387\pi$
$701$	−5.54400	−0.209394	−0.104697	+	0.994504i	$0.533387\pi$
$702$	0	0
$703$	−1.90997	−0.0720357
$704$	0	0
$705$	13.8893	0.523100
$706$	0	0
$707$	0	0
$708$	0	0
$709$	43.7200	1.64194	0.820970	−	0.570971i	$-0.193433\pi$
$709$	43.7200	1.64194	0.820970	+	0.570971i	$0.193433\pi$
$710$	0	0
$711$	−38.8080	−1.45541
$712$	0	0
$713$	−7.89962	−0.295843
$714$	0	0
$715$	39.0880	1.46181
$716$	0	0
$717$	−4.66605	−0.174257
$718$	0	0
$719$	−24.7627	−0.923493	−0.461747	−	0.887012i	$-0.652777\pi$
$719$	−24.7627	−0.923493	−0.461747	+	0.887012i	$0.652777\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	−5.08801	−0.189225
$724$	0	0
$725$	−72.4040	−2.68902
$726$	0	0
$727$	9.33210	0.346108	0.173054	−	0.984912i	$-0.444636\pi$
$727$	9.33210	0.346108	0.173054	+	0.984912i	$0.444636\pi$
$728$	0	0
$729$	−15.4560	−0.572444
$730$	0	0
$731$	12.9343	0.478392
$732$	0	0
$733$	45.1191	1.66651	0.833257	−	0.552886i	$-0.186474\pi$
$733$	45.1191	1.66651	0.833257	+	0.552886i	$0.186474\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−12.0000	−0.442026
$738$	0	0
$739$	3.13999	0.115506	0.0577532	−	0.998331i	$-0.481606\pi$
$739$	3.13999	0.115506	0.0577532	+	0.998331i	$0.481606\pi$
$740$	0	0
$741$	2.12770	0.0781630
$742$	0	0
$743$	−6.45600	−0.236848	−0.118424	−	0.992963i	$-0.537784\pi$
$743$	−6.45600	−0.236848	−0.118424	+	0.992963i	$0.537784\pi$
$744$	0	0
$745$	88.3283	3.23610
$746$	0	0
$747$	−46.4427	−1.69925
$748$	0	0
$749$	0	0
$750$	0	0
$751$	21.5440	0.786152	0.393076	−	0.919506i	$-0.371411\pi$
$751$	21.5440	0.786152	0.393076	+	0.919506i	$0.371411\pi$
$752$	0	0
$753$	−3.08801	−0.112533
$754$	0	0
$755$	59.5948	2.16888
$756$	0	0
$757$	−14.0000	−0.508839	−0.254419	−	0.967094i	$-0.581884\pi$
$757$	−14.0000	−0.508839	−0.254419	+	0.967094i	$0.581884\pi$
$758$	0	0
$759$	−0.954983	−0.0346637
$760$	0	0
$761$	−20.4653	−0.741866	−0.370933	−	0.928660i	$-0.620962\pi$
$761$	−20.4653	−0.741866	−0.370933	+	0.928660i	$0.620962\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	37.5440	1.35741
$766$	0	0
$767$	28.4560	1.02749
$768$	0	0
$769$	−33.1398	−1.19505	−0.597526	−	0.801849i	$-0.703850\pi$
$769$	−33.1398	−1.19505	−0.597526	+	0.801849i	$0.703850\pi$
$770$	0	0
$771$	−10.2280	−0.368352
$772$	0	0
$773$	−42.4719	−1.52761	−0.763804	−	0.645448i	$-0.776671\pi$
$773$	−42.4719	−1.52761	−0.763804	+	0.645448i	$0.776671\pi$
$774$	0	0
$775$	99.0929	3.55952
$776$	0	0
$777$	0	0
$778$	0	0
$779$	4.45600	0.159653
$780$	0	0
$781$	3.54400	0.126814
$782$	0	0
$783$	15.9081	0.568509
$784$	0	0
$785$	−13.5440	−0.483406
$786$	0	0
$787$	20.5742	0.733389	0.366695	−	0.930341i	$-0.380489\pi$
$787$	20.5742	0.733389	0.366695	+	0.930341i	$0.380489\pi$
$788$	0	0
$789$	0.954983	0.0339983
$790$	0	0
$791$	0	0
$792$	0	0
$793$	49.7200	1.76561
$794$	0	0
$795$	−12.0000	−0.425596
$796$	0	0
$797$	−31.9671	−1.13233	−0.566167	−	0.824291i	$-0.691574\pi$
$797$	−31.9671	−1.13233	−0.566167	+	0.824291i	$0.691574\pi$
$798$	0	0
$799$	−22.4560	−0.794436
$800$	0	0
$801$	14.2579	0.503778
$802$	0	0
$803$	−7.42213	−0.261921
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−0.860009	−0.0302738
$808$	0	0
$809$	13.0880	0.460150	0.230075	−	0.973173i	$-0.426103\pi$
$809$	13.0880	0.460150	0.230075	+	0.973173i	$0.426103\pi$
$810$	0	0
$811$	−6.94464	−0.243859	−0.121930	−	0.992539i	$-0.538908\pi$
$811$	−6.94464	−0.243859	−0.121930	+	0.992539i	$0.538908\pi$
$812$	0	0
$813$	13.0880	0.459016
$814$	0	0
$815$	55.7748	1.95371
$816$	0	0
$817$	−3.81993	−0.133643
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−15.5440	−0.542489	−0.271245	−	0.962510i	$-0.587435\pi$
$821$	−15.5440	−0.542489	−0.271245	+	0.962510i	$0.587435\pi$
$822$	0	0
$823$	−9.31601	−0.324736	−0.162368	−	0.986730i	$-0.551913\pi$
$823$	−9.31601	−0.324736	−0.162368	+	0.986730i	$0.551913\pi$
$824$	0	0
$825$	11.9793	0.417066
$826$	0	0
$827$	−18.1760	−0.632042	−0.316021	−	0.948752i	$-0.602347\pi$
$827$	−18.1760	−0.632042	−0.316021	+	0.948752i	$0.602347\pi$
$828$	0	0
$829$	−8.37711	−0.290949	−0.145475	−	0.989362i	$-0.546471\pi$
$829$	−8.37711	−0.290949	−0.145475	+	0.989362i	$0.546471\pi$
$830$	0	0
$831$	−4.44831	−0.154310
$832$	0	0
$833$	0	0
$834$	0	0
$835$	36.6320	1.26770
$836$	0	0
$837$	−21.7720	−0.752550
$838$	0	0
$839$	−14.1070	−0.487028	−0.243514	−	0.969897i	$-0.578300\pi$
$839$	−14.1070	−0.487028	−0.243514	+	0.969897i	$0.578300\pi$
$840$	0	0
$841$	4.31601	0.148828
$842$	0	0
$843$	−13.6715	−0.470873
$844$	0	0
$845$	36.7420	1.26396
$846$	0	0
$847$	0	0
$848$	0	0
$849$	0.455996	0.0156498
$850$	0	0
$851$	2.00000	0.0685591
$852$	0	0
$853$	44.5328	1.52477	0.762387	−	0.647122i	$-0.224027\pi$
$853$	44.5328	1.52477	0.762387	+	0.647122i	$0.224027\pi$
$854$	0	0
$855$	−11.0880	−0.379202
$856$	0	0
$857$	−40.0845	−1.36926	−0.684630	−	0.728891i	$-0.740036\pi$
$857$	−40.0845	−1.36926	−0.684630	+	0.728891i	$0.740036\pi$
$858$	0	0
$859$	−4.29742	−0.146626	−0.0733131	−	0.997309i	$-0.523357\pi$
$859$	−4.29742	−0.146626	−0.0733131	+	0.997309i	$0.523357\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	29.3160	0.997928	0.498964	−	0.866623i	$-0.333714\pi$
$863$	29.3160	0.997928	0.498964	+	0.866623i	$0.333714\pi$
$864$	0	0
$865$	−27.0880	−0.921020
$866$	0	0
$867$	−3.12471	−0.106121
$868$	0	0
$869$	28.0000	0.949835
$870$	0	0
$871$	−27.9963	−0.948618
$872$	0	0
$873$	3.66904	0.124178
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−36.4560	−1.23103	−0.615516	−	0.788125i	$-0.711052\pi$
$877$	−36.4560	−1.23103	−0.615516	+	0.788125i	$0.711052\pi$
$878$	0	0
$879$	−1.54400	−0.0520780
$880$	0	0
$881$	55.4062	1.86668	0.933341	−	0.358990i	$-0.116879\pi$
$881$	55.4062	1.86668	0.933341	+	0.358990i	$0.116879\pi$
$882$	0	0
$883$	−44.0000	−1.48072	−0.740359	−	0.672212i	$-0.765344\pi$
$883$	−44.0000	−1.48072	−0.740359	+	0.672212i	$0.765344\pi$
$884$	0	0
$885$	12.1970	0.409999
$886$	0	0
$887$	3.34244	0.112228	0.0561141	−	0.998424i	$-0.482129\pi$
$887$	3.34244	0.112228	0.0561141	+	0.998424i	$0.482129\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	14.0000	0.469018
$892$	0	0
$893$	6.63201	0.221932
$894$	0	0
$895$	74.4390	2.48822
$896$	0	0
$897$	−2.22800	−0.0743907
$898$	0	0
$899$	−45.5966	−1.52073
$900$	0	0
$901$	19.4014	0.646356
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−9.54400	−0.317253
$906$	0	0
$907$	−0.455996	−0.0151411	−0.00757055	−	0.999971i	$-0.502410\pi$
$907$	−0.455996	−0.0151411	−0.00757055	+	0.999971i	$0.502410\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	45.0880	1.49383	0.746916	−	0.664918i	$-0.231534\pi$
$911$	45.0880	1.49383	0.746916	+	0.664918i	$0.231534\pi$
$912$	0	0
$913$	33.5085	1.10897
$914$	0	0
$915$	21.3114	0.704533
$916$	0	0
$917$	0	0
$918$	0	0
$919$	−48.6320	−1.60422	−0.802111	−	0.597175i	$-0.796290\pi$
$919$	−48.6320	−1.60422	−0.802111	+	0.597175i	$0.796290\pi$
$920$	0	0
$921$	−15.8240	−0.521418
$922$	0	0
$923$	8.26825	0.272153
$924$	0	0
$925$	−25.0880	−0.824889
$926$	0	0
$927$	−41.1483	−1.35149
$928$	0	0
$929$	−36.4823	−1.19694	−0.598472	−	0.801144i	$-0.704225\pi$
$929$	−36.4823	−1.19694	−0.598472	+	0.801144i	$0.704225\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	−5.49202	−0.179801
$934$	0	0
$935$	−27.0880	−0.885873
$936$	0	0
$937$	8.74574	0.285711	0.142855	−	0.989744i	$-0.454372\pi$
$937$	8.74574	0.285711	0.142855	+	0.989744i	$0.454372\pi$
$938$	0	0
$939$	−2.45600	−0.0801484
$940$	0	0
$941$	−41.2992	−1.34632	−0.673158	−	0.739499i	$-0.735063\pi$
$941$	−41.2992	−1.34632	−0.673158	+	0.739499i	$0.735063\pi$
$942$	0	0
$943$	−4.66605	−0.151947
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−18.6840	−0.607148	−0.303574	−	0.952808i	$-0.598180\pi$
$947$	−18.6840	−0.607148	−0.303574	+	0.952808i	$0.598180\pi$
$948$	0	0
$949$	−17.3160	−0.562101
$950$	0	0
$951$	1.69223	0.0548744
$952$	0	0
$953$	−10.9120	−0.353474	−0.176737	−	0.984258i	$-0.556554\pi$
$953$	−10.9120	−0.353474	−0.176737	+	0.984258i	$0.556554\pi$
$954$	0	0
$955$	14.8443	0.480349
$956$	0	0
$957$	−5.51216	−0.178183
$958$	0	0
$959$	0	0
$960$	0	0
$961$	31.4040	1.01303
$962$	0	0
$963$	43.0880	1.38849
$964$	0	0
$965$	−51.2177	−1.64875
$966$	0	0
$967$	36.4040	1.17067	0.585337	−	0.810790i	$-0.300962\pi$
$967$	36.4040	1.17067	0.585337	+	0.810790i	$0.300962\pi$
$968$	0	0
$969$	−1.47450	−0.0473677
$970$	0	0
$971$	9.33210	0.299481	0.149741	−	0.988725i	$-0.452156\pi$
$971$	9.33210	0.299481	0.149741	+	0.988725i	$0.452156\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	27.9480	0.895053
$976$	0	0
$977$	6.00000	0.191957	0.0959785	−	0.995383i	$-0.469402\pi$
$977$	6.00000	0.191957	0.0959785	+	0.995383i	$0.469402\pi$
$978$	0	0
$979$	−10.2871	−0.328776
$980$	0	0
$981$	37.5440	1.19869
$982$	0	0
$983$	−36.3734	−1.16013	−0.580066	−	0.814570i	$-0.696973\pi$
$983$	−36.3734	−1.16013	−0.580066	+	0.814570i	$0.696973\pi$
$984$	0	0
$985$	78.9962	2.51703
$986$	0	0
$987$	0	0
$988$	0	0
$989$	4.00000	0.127193
$990$	0	0
$991$	12.9120	0.410163	0.205081	−	0.978745i	$-0.434254\pi$
$991$	12.9120	0.410163	0.205081	+	0.978745i	$0.434254\pi$
$992$	0	0
$993$	11.8704	0.376697
$994$	0	0
$995$	93.2640	2.95667
$996$	0	0
$997$	−2.64721	−0.0838381	−0.0419191	−	0.999121i	$-0.513347\pi$
$997$	−2.64721	−0.0838381	−0.0419191	+	0.999121i	$0.513347\pi$
$998$	0	0
$999$	5.51216	0.174397

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9016.2.a.bd.1.3	yes	4
7.6	odd	2	inner	9016.2.a.bd.1.2	✓	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
9016.2.a.bd.1.2	✓	4	7.6	odd	2	inner
9016.2.a.bd.1.3	yes	4	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)

\(f(q)\)	\(=\)	\(q+0.477491 q^{3} +4.18856 q^{5} -2.77200 q^{9} +O(q^{10})\)	\(q+0.477491 q^{3} +4.18856 q^{5} -2.77200 q^{9} +2.00000 q^{11} +4.66605 q^{13} +2.00000 q^{15} -3.23357 q^{17} +0.954983 q^{19} -1.00000 q^{23} +12.5440 q^{25} -2.75608 q^{27} -5.77200 q^{29} +7.89962 q^{31} +0.954983 q^{33} -2.00000 q^{37} +2.22800 q^{39} +4.66605 q^{41} -4.00000 q^{43} -11.6107 q^{45} +6.94464 q^{47} -1.54400 q^{51} -6.00000 q^{53} +8.37711 q^{55} +0.455996 q^{57} +6.09852 q^{59} +10.6557 q^{61} +19.5440 q^{65} -6.00000 q^{67} -0.477491 q^{69} +1.77200 q^{71} -3.71106 q^{73} +5.98966 q^{75} +14.0000 q^{79} +7.00000 q^{81} +16.7542 q^{83} -13.5440 q^{85} -2.75608 q^{87} -5.14354 q^{89} +3.77200 q^{93} +4.00000 q^{95} -1.32361 q^{97} -5.54400 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 6 q^{9}+O(q^{10}) \) 4 * q + 6 * q^9	\( 4 q + 6 q^{9} + 8 q^{11} + 8 q^{15} - 4 q^{23} + 16 q^{25} - 6 q^{29} - 8 q^{37} + 26 q^{39} - 16 q^{43} + 28 q^{51} - 24 q^{53} + 36 q^{57} + 44 q^{65} - 24 q^{67} - 10 q^{71} + 56 q^{79} + 28 q^{81} - 20 q^{85} - 2 q^{93} + 16 q^{95} + 12 q^{99}+O(q^{100}) \) 4 * q + 6 * q^9 + 8 * q^11 + 8 * q^15 - 4 * q^23 + 16 * q^25 - 6 * q^29 - 8 * q^37 + 26 * q^39 - 16 * q^43 + 28 * q^51 - 24 * q^53 + 36 * q^57 + 44 * q^65 - 24 * q^67 - 10 * q^71 + 56 * q^79 + 28 * q^81 - 20 * q^85 - 2 * q^93 + 16 * q^95 + 12 * q^99

Introduction

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