LMFDB - Embedded newform 7800.2.a.bp.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7800,2,Mod(1,7800)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7800, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("7800.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7800 = 2^{3} \cdot 3 \cdot 5^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7800.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:	$62.2833135766$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.568.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 6x - 2 $ x^3 - x^2 - 6*x - 2
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 1560)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-1.76156$ of defining polynomial
Character	$\chi$	$=$	7800.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} +1.00000 q^{9} -2.86464 q^{11} +1.00000 q^{13} -5.52311 q^{17} +3.52311 q^{19} -7.52311 q^{23} +1.00000 q^{27} +6.77551 q^{29} +5.72928 q^{31} -2.86464 q^{33} +3.72928 q^{37} +1.00000 q^{39} -10.1170 q^{41} +5.52311 q^{43} -8.65847 q^{47} -7.00000 q^{49} -5.52311 q^{51} +6.77551 q^{53} +3.52311 q^{57} +0.593923 q^{59} -5.25240 q^{61} +10.5048 q^{67} -7.52311 q^{69} -2.38776 q^{71} -5.45856 q^{73} +2.47689 q^{79} +1.00000 q^{81} -8.11704 q^{83} +6.77551 q^{87} -14.1170 q^{89} +5.72928 q^{93} -6.00000 q^{97} -2.86464 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 3 q^{3} + 3 q^{9}+O(q^{10}) $ 3 * q + 3 * q^3 + 3 * q^9	$ 3 q + 3 q^{3} + 3 q^{9} - 6 q^{11} + 3 q^{13} - 4 q^{17} - 2 q^{19} - 10 q^{23} + 3 q^{27} - 10 q^{29} + 12 q^{31} - 6 q^{33} + 6 q^{37} + 3 q^{39} - 10 q^{41} + 4 q^{43} - 16 q^{47} - 21 q^{49} - 4 q^{51} - 10 q^{53} - 2 q^{57} - 6 q^{59} + 2 q^{61} - 4 q^{67} - 10 q^{69} + 8 q^{71} - 6 q^{73} + 20 q^{79} + 3 q^{81} - 4 q^{83} - 10 q^{87} - 22 q^{89} + 12 q^{93} - 18 q^{97} - 6 q^{99}+O(q^{100}) $ 3 * q + 3 * q^3 + 3 * q^9 - 6 * q^11 + 3 * q^13 - 4 * q^17 - 2 * q^19 - 10 * q^23 + 3 * q^27 - 10 * q^29 + 12 * q^31 - 6 * q^33 + 6 * q^37 + 3 * q^39 - 10 * q^41 + 4 * q^43 - 16 * q^47 - 21 * q^49 - 4 * q^51 - 10 * q^53 - 2 * q^57 - 6 * q^59 + 2 * q^61 - 4 * q^67 - 10 * q^69 + 8 * q^71 - 6 * q^73 + 20 * q^79 + 3 * q^81 - 4 * q^83 - 10 * q^87 - 22 * q^89 + 12 * q^93 - 18 * q^97 - 6 * 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
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	0	0		−	1.00000i	$-0.5\pi$
$7$	0	0			1.00000i	$0.5\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−2.86464	−0.863722	−0.431861	−	0.901940i	$-0.642143\pi$
$11$	−2.86464	−0.863722	−0.431861	+	0.901940i	$0.642143\pi$
$12$	0	0
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−5.52311	−1.33955	−0.669776	−	0.742563i	$-0.733610\pi$
$17$	−5.52311	−1.33955	−0.669776	+	0.742563i	$0.733610\pi$
$18$	0	0
$19$	3.52311	0.808258	0.404129	−	0.914702i	$-0.367575\pi$
$19$	3.52311	0.808258	0.404129	+	0.914702i	$0.367575\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−7.52311	−1.56868	−0.784339	−	0.620333i	$-0.786998\pi$
$23$	−7.52311	−1.56868	−0.784339	+	0.620333i	$0.786998\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	6.77551	1.25818	0.629090	−	0.777332i	$-0.283428\pi$
$29$	6.77551	1.25818	0.629090	+	0.777332i	$0.283428\pi$
$30$	0	0
$31$	5.72928	1.02901	0.514505	−	0.857488i	$-0.327976\pi$
$31$	5.72928	1.02901	0.514505	+	0.857488i	$0.327976\pi$
$32$	0	0
$33$	−2.86464	−0.498670
$34$	0	0
$35$	0	0
$36$	0	0
$37$	3.72928	0.613090	0.306545	−	0.951856i	$-0.400827\pi$
$37$	3.72928	0.613090	0.306545	+	0.951856i	$0.400827\pi$
$38$	0	0
$39$	1.00000	0.160128
$40$	0	0
$41$	−10.1170	−1.58002	−0.790008	−	0.613097i	$-0.789924\pi$
$41$	−10.1170	−1.58002	−0.790008	+	0.613097i	$0.789924\pi$
$42$	0	0
$43$	5.52311	0.842267	0.421134	−	0.906999i	$-0.361632\pi$
$43$	5.52311	0.842267	0.421134	+	0.906999i	$0.361632\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−8.65847	−1.26297	−0.631484	−	0.775389i	$-0.717554\pi$
$47$	−8.65847	−1.26297	−0.631484	+	0.775389i	$0.717554\pi$
$48$	0	0
$49$	−7.00000	−1.00000
$50$	0	0
$51$	−5.52311	−0.773391
$52$	0	0
$53$	6.77551	0.930688	0.465344	−	0.885130i	$-0.345931\pi$
$53$	6.77551	0.930688	0.465344	+	0.885130i	$0.345931\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	3.52311	0.466648
$58$	0	0
$59$	0.593923	0.0773221	0.0386611	−	0.999252i	$-0.487691\pi$
$59$	0.593923	0.0773221	0.0386611	+	0.999252i	$0.487691\pi$
$60$	0	0
$61$	−5.25240	−0.672500	−0.336250	−	0.941773i	$-0.609159\pi$
$61$	−5.25240	−0.672500	−0.336250	+	0.941773i	$0.609159\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	0	0
$66$	0	0
$67$	10.5048	1.28336	0.641682	−	0.766971i	$-0.278237\pi$
$67$	10.5048	1.28336	0.641682	+	0.766971i	$0.278237\pi$
$68$	0	0
$69$	−7.52311	−0.905677
$70$	0	0
$71$	−2.38776	−0.283374	−0.141687	−	0.989911i	$-0.545253\pi$
$71$	−2.38776	−0.283374	−0.141687	+	0.989911i	$0.545253\pi$
$72$	0	0
$73$	−5.45856	−0.638877	−0.319438	−	0.947607i	$-0.603494\pi$
$73$	−5.45856	−0.638877	−0.319438	+	0.947607i	$0.603494\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	0	0
$78$	0	0
$79$	2.47689	0.278671	0.139336	−	0.990245i	$-0.455503\pi$
$79$	2.47689	0.278671	0.139336	+	0.990245i	$0.455503\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−8.11704	−0.890961	−0.445480	−	0.895292i	$-0.646967\pi$
$83$	−8.11704	−0.890961	−0.445480	+	0.895292i	$0.646967\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	6.77551	0.726411
$88$	0	0
$89$	−14.1170	−1.49640	−0.748201	−	0.663472i	$-0.769082\pi$
$89$	−14.1170	−1.49640	−0.748201	+	0.663472i	$0.769082\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	5.72928	0.594099
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−6.00000	−0.609208	−0.304604	−	0.952479i	$-0.598524\pi$
$97$	−6.00000	−0.609208	−0.304604	+	0.952479i	$0.598524\pi$
$98$	0	0
$99$	−2.86464	−0.287907
$100$	0	0
$101$	−11.7293	−1.16711	−0.583554	−	0.812075i	$-0.698338\pi$
$101$	−11.7293	−1.16711	−0.583554	+	0.812075i	$0.698338\pi$
$102$	0	0
$103$	−14.7755	−1.45587	−0.727937	−	0.685644i	$-0.759521\pi$
$103$	−14.7755	−1.45587	−0.727937	+	0.685644i	$0.759521\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−1.72928	−0.167176	−0.0835880	−	0.996500i	$-0.526638\pi$
$107$	−1.72928	−0.167176	−0.0835880	+	0.996500i	$0.526638\pi$
$108$	0	0
$109$	12.0279	1.15206	0.576032	−	0.817427i	$-0.304600\pi$
$109$	12.0279	1.15206	0.576032	+	0.817427i	$0.304600\pi$
$110$	0	0
$111$	3.72928	0.353968
$112$	0	0
$113$	7.25240	0.682248	0.341124	−	0.940018i	$-0.389192\pi$
$113$	7.25240	0.682248	0.341124	+	0.940018i	$0.389192\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	1.00000	0.0924500
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−2.79383	−0.253985
$122$	0	0
$123$	−10.1170	−0.912223
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−17.2524	−1.53090	−0.765451	−	0.643494i	$-0.777484\pi$
$127$	−17.2524	−1.53090	−0.765451	+	0.643494i	$0.777484\pi$
$128$	0	0
$129$	5.52311	0.486283
$130$	0	0
$131$	−8.77551	−0.766720	−0.383360	−	0.923599i	$-0.625233\pi$
$131$	−8.77551	−0.766720	−0.383360	+	0.923599i	$0.625233\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−19.4340	−1.66036	−0.830179	−	0.557497i	$-0.811762\pi$
$137$	−19.4340	−1.66036	−0.830179	+	0.557497i	$0.811762\pi$
$138$	0	0
$139$	7.58767	0.643577	0.321789	−	0.946812i	$-0.395716\pi$
$139$	7.58767	0.643577	0.321789	+	0.946812i	$0.395716\pi$
$140$	0	0
$141$	−8.65847	−0.729175
$142$	0	0
$143$	−2.86464	−0.239553
$144$	0	0
$145$	0	0
$146$	0	0
$147$	−7.00000	−0.577350
$148$	0	0
$149$	−1.40608	−0.115190	−0.0575952	−	0.998340i	$-0.518343\pi$
$149$	−1.40608	−0.115190	−0.0575952	+	0.998340i	$0.518343\pi$
$150$	0	0
$151$	−1.31695	−0.107172	−0.0535858	−	0.998563i	$-0.517065\pi$
$151$	−1.31695	−0.107172	−0.0535858	+	0.998563i	$0.517065\pi$
$152$	0	0
$153$	−5.52311	−0.446517
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−20.7110	−1.65291	−0.826457	−	0.562999i	$-0.809647\pi$
$157$	−20.7110	−1.65291	−0.826457	+	0.562999i	$0.809647\pi$
$158$	0	0
$159$	6.77551	0.537333
$160$	0	0
$161$	0	0
$162$	0	0
$163$	−2.27072	−0.177856	−0.0889282	−	0.996038i	$-0.528344\pi$
$163$	−2.27072	−0.177856	−0.0889282	+	0.996038i	$0.528344\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−7.34153	−0.568104	−0.284052	−	0.958809i	$-0.591679\pi$
$167$	−7.34153	−0.568104	−0.284052	+	0.958809i	$0.591679\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	3.52311	0.269419
$172$	0	0
$173$	16.0925	1.22349	0.611743	−	0.791056i	$-0.290468\pi$
$173$	16.0925	1.22349	0.611743	+	0.791056i	$0.290468\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	0.593923	0.0446420
$178$	0	0
$179$	15.0462	1.12461	0.562304	−	0.826931i	$-0.309915\pi$
$179$	15.0462	1.12461	0.562304	+	0.826931i	$0.309915\pi$
$180$	0	0
$181$	5.58767	0.415328	0.207664	−	0.978200i	$-0.433414\pi$
$181$	5.58767	0.415328	0.207664	+	0.978200i	$0.433414\pi$
$182$	0	0
$183$	−5.25240	−0.388268
$184$	0	0
$185$	0	0
$186$	0	0
$187$	15.8217	1.15700
$188$	0	0
$189$	0	0
$190$	0	0
$191$	17.5510	1.26995	0.634974	−	0.772534i	$-0.281011\pi$
$191$	17.5510	1.26995	0.634974	+	0.772534i	$0.281011\pi$
$192$	0	0
$193$	−13.8217	−0.994911	−0.497455	−	0.867490i	$-0.665732\pi$
$193$	−13.8217	−0.994911	−0.497455	+	0.867490i	$0.665732\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−26.3511	−1.87744	−0.938719	−	0.344682i	$-0.887987\pi$
$197$	−26.3511	−1.87744	−0.938719	+	0.344682i	$0.887987\pi$
$198$	0	0
$199$	−23.0741	−1.63568	−0.817841	−	0.575444i	$-0.804829\pi$
$199$	−23.0741	−1.63568	−0.817841	+	0.575444i	$0.804829\pi$
$200$	0	0
$201$	10.5048	0.740951
$202$	0	0
$203$	0	0
$204$	0	0
$205$	0	0
$206$	0	0
$207$	−7.52311	−0.522893
$208$	0	0
$209$	−10.0925	−0.698110
$210$	0	0
$211$	1.52311	0.104856	0.0524278	−	0.998625i	$-0.483304\pi$
$211$	1.52311	0.104856	0.0524278	+	0.998625i	$0.483304\pi$
$212$	0	0
$213$	−2.38776	−0.163606
$214$	0	0
$215$	0	0
$216$	0	0
$217$	0	0
$218$	0	0
$219$	−5.45856	−0.368856
$220$	0	0
$221$	−5.52311	−0.371525
$222$	0	0
$223$	−13.3169	−0.891769	−0.445884	−	0.895091i	$-0.647111\pi$
$223$	−13.3169	−0.891769	−0.445884	+	0.895091i	$0.647111\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	6.80009	0.451338	0.225669	−	0.974204i	$-0.427543\pi$
$227$	6.80009	0.451338	0.225669	+	0.974204i	$0.427543\pi$
$228$	0	0
$229$	1.52311	0.100650	0.0503251	−	0.998733i	$-0.483974\pi$
$229$	1.52311	0.100650	0.0503251	+	0.998733i	$0.483974\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	10.7110	0.701698	0.350849	−	0.936432i	$-0.385893\pi$
$233$	10.7110	0.701698	0.350849	+	0.936432i	$0.385893\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	2.47689	0.160891
$238$	0	0
$239$	4.65847	0.301332	0.150666	−	0.988585i	$-0.451858\pi$
$239$	4.65847	0.301332	0.150666	+	0.988585i	$0.451858\pi$
$240$	0	0
$241$	5.22449	0.336539	0.168269	−	0.985741i	$-0.446182\pi$
$241$	5.22449	0.336539	0.168269	+	0.985741i	$0.446182\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	0	0
$246$	0	0
$247$	3.52311	0.224170
$248$	0	0
$249$	−8.11704	−0.514396
$250$	0	0
$251$	5.85838	0.369778	0.184889	−	0.982759i	$-0.440807\pi$
$251$	5.85838	0.369778	0.184889	+	0.982759i	$0.440807\pi$
$252$	0	0
$253$	21.5510	1.35490
$254$	0	0
$255$	0	0
$256$	0	0
$257$	2.29862	0.143384	0.0716921	−	0.997427i	$-0.477160\pi$
$257$	2.29862	0.143384	0.0716921	+	0.997427i	$0.477160\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	6.77551	0.419394
$262$	0	0
$263$	31.5789	1.94724	0.973620	−	0.228175i	$-0.0732760\pi$
$263$	31.5789	1.94724	0.973620	+	0.228175i	$0.0732760\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	−14.1170	−0.863949
$268$	0	0
$269$	−30.3632	−1.85128	−0.925638	−	0.378411i	$-0.876471\pi$
$269$	−30.3632	−1.85128	−0.925638	+	0.378411i	$0.876471\pi$
$270$	0	0
$271$	28.5972	1.73716	0.868580	−	0.495550i	$-0.165033\pi$
$271$	28.5972	1.73716	0.868580	+	0.495550i	$0.165033\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−1.79383	−0.107781	−0.0538905	−	0.998547i	$-0.517162\pi$
$277$	−1.79383	−0.107781	−0.0538905	+	0.998547i	$0.517162\pi$
$278$	0	0
$279$	5.72928	0.343003
$280$	0	0
$281$	32.2095	1.92146	0.960729	−	0.277489i	$-0.0895024\pi$
$281$	32.2095	1.92146	0.960729	+	0.277489i	$0.0895024\pi$
$282$	0	0
$283$	−18.0925	−1.07548	−0.537742	−	0.843109i	$-0.680723\pi$
$283$	−18.0925	−1.07548	−0.537742	+	0.843109i	$0.680723\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	0	0
$288$	0	0
$289$	13.5048	0.794400
$290$	0	0
$291$	−6.00000	−0.351726
$292$	0	0
$293$	12.6218	0.737375	0.368688	−	0.929553i	$-0.379807\pi$
$293$	12.6218	0.737375	0.368688	+	0.929553i	$0.379807\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	−2.86464	−0.166223
$298$	0	0
$299$	−7.52311	−0.435073
$300$	0	0
$301$	0	0
$302$	0	0
$303$	−11.7293	−0.673830
$304$	0	0
$305$	0	0
$306$	0	0
$307$	−20.0000	−1.14146	−0.570730	−	0.821138i	$-0.693340\pi$
$307$	−20.0000	−1.14146	−0.570730	+	0.821138i	$0.693340\pi$
$308$	0	0
$309$	−14.7755	−0.840549
$310$	0	0
$311$	−31.8217	−1.80445	−0.902223	−	0.431271i	$-0.858065\pi$
$311$	−31.8217	−1.80445	−0.902223	+	0.431271i	$0.858065\pi$
$312$	0	0
$313$	−11.7938	−0.666627	−0.333313	−	0.942816i	$-0.608167\pi$
$313$	−11.7938	−0.666627	−0.333313	+	0.942816i	$0.608167\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−11.4340	−0.642197	−0.321098	−	0.947046i	$-0.604052\pi$
$317$	−11.4340	−0.642197	−0.321098	+	0.947046i	$0.604052\pi$
$318$	0	0
$319$	−19.4094	−1.08672
$320$	0	0
$321$	−1.72928	−0.0965191
$322$	0	0
$323$	−19.4586	−1.08270
$324$	0	0
$325$	0	0
$326$	0	0
$327$	12.0279	0.665145
$328$	0	0
$329$	0	0
$330$	0	0
$331$	2.20617	0.121262	0.0606310	−	0.998160i	$-0.480689\pi$
$331$	2.20617	0.121262	0.0606310	+	0.998160i	$0.480689\pi$
$332$	0	0
$333$	3.72928	0.204363
$334$	0	0
$335$	0	0
$336$	0	0
$337$	16.8034	0.915340	0.457670	−	0.889122i	$-0.348684\pi$
$337$	16.8034	0.915340	0.457670	+	0.889122i	$0.348684\pi$
$338$	0	0
$339$	7.25240	0.393896
$340$	0	0
$341$	−16.4123	−0.888778
$342$	0	0
$343$	0	0
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−9.96336	−0.534861	−0.267430	−	0.963577i	$-0.586175\pi$
$347$	−9.96336	−0.534861	−0.267430	+	0.963577i	$0.586175\pi$
$348$	0	0
$349$	−21.3449	−1.14256	−0.571282	−	0.820754i	$-0.693554\pi$
$349$	−21.3449	−1.14256	−0.571282	+	0.820754i	$0.693554\pi$
$350$	0	0
$351$	1.00000	0.0533761
$352$	0	0
$353$	−33.3973	−1.77756	−0.888781	−	0.458333i	$-0.848447\pi$
$353$	−33.3973	−1.77756	−0.888781	+	0.458333i	$0.848447\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	0	0
$358$	0	0
$359$	21.9754	1.15982	0.579909	−	0.814681i	$-0.303088\pi$
$359$	21.9754	1.15982	0.579909	+	0.814681i	$0.303088\pi$
$360$	0	0
$361$	−6.58767	−0.346719
$362$	0	0
$363$	−2.79383	−0.146638
$364$	0	0
$365$	0	0
$366$	0	0
$367$	−29.2803	−1.52842	−0.764210	−	0.644968i	$-0.776871\pi$
$367$	−29.2803	−1.52842	−0.764210	+	0.644968i	$0.776871\pi$
$368$	0	0
$369$	−10.1170	−0.526672
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−2.41233	−0.124906	−0.0624530	−	0.998048i	$-0.519892\pi$
$373$	−2.41233	−0.124906	−0.0624530	+	0.998048i	$0.519892\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	6.77551	0.348957
$378$	0	0
$379$	19.5231	1.00284	0.501418	−	0.865205i	$-0.332812\pi$
$379$	19.5231	1.00284	0.501418	+	0.865205i	$0.332812\pi$
$380$	0	0
$381$	−17.2524	−0.883867
$382$	0	0
$383$	23.7047	1.21125	0.605627	−	0.795749i	$-0.292922\pi$
$383$	23.7047	1.21125	0.605627	+	0.795749i	$0.292922\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	5.52311	0.280756
$388$	0	0
$389$	−14.0000	−0.709828	−0.354914	−	0.934899i	$-0.615490\pi$
$389$	−14.0000	−0.709828	−0.354914	+	0.934899i	$0.615490\pi$
$390$	0	0
$391$	41.5510	2.10133
$392$	0	0
$393$	−8.77551	−0.442666
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−28.8680	−1.44884	−0.724421	−	0.689358i	$-0.757893\pi$
$397$	−28.8680	−1.44884	−0.724421	+	0.689358i	$0.757893\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	20.0804	1.00277	0.501383	−	0.865225i	$-0.332825\pi$
$401$	20.0804	1.00277	0.501383	+	0.865225i	$0.332825\pi$
$402$	0	0
$403$	5.72928	0.285396
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−10.6831	−0.529539
$408$	0	0
$409$	−13.6926	−0.677057	−0.338529	−	0.940956i	$-0.609929\pi$
$409$	−13.6926	−0.677057	−0.338529	+	0.940956i	$0.609929\pi$
$410$	0	0
$411$	−19.4340	−0.958608
$412$	0	0
$413$	0	0
$414$	0	0
$415$	0	0
$416$	0	0
$417$	7.58767	0.371570
$418$	0	0
$419$	−16.3632	−0.799393	−0.399697	−	0.916647i	$-0.630885\pi$
$419$	−16.3632	−0.799393	−0.399697	+	0.916647i	$0.630885\pi$
$420$	0	0
$421$	24.9817	1.21753	0.608766	−	0.793350i	$-0.291665\pi$
$421$	24.9817	1.21753	0.608766	+	0.793350i	$0.291665\pi$
$422$	0	0
$423$	−8.65847	−0.420989
$424$	0	0
$425$	0	0
$426$	0	0
$427$	0	0
$428$	0	0
$429$	−2.86464	−0.138306
$430$	0	0
$431$	−27.9388	−1.34576	−0.672882	−	0.739750i	$-0.734944\pi$
$431$	−27.9388	−1.34576	−0.672882	+	0.739750i	$0.734944\pi$
$432$	0	0
$433$	26.7110	1.28365	0.641823	−	0.766852i	$-0.278178\pi$
$433$	26.7110	1.28365	0.641823	+	0.766852i	$0.278178\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−26.5048	−1.26790
$438$	0	0
$439$	−5.00958	−0.239094	−0.119547	−	0.992829i	$-0.538144\pi$
$439$	−5.00958	−0.239094	−0.119547	+	0.992829i	$0.538144\pi$
$440$	0	0
$441$	−7.00000	−0.333333
$442$	0	0
$443$	19.6926	0.935625	0.467813	−	0.883828i	$-0.345042\pi$
$443$	19.6926	0.935625	0.467813	+	0.883828i	$0.345042\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	−1.40608	−0.0665052
$448$	0	0
$449$	38.7143	1.82704	0.913520	−	0.406794i	$-0.133353\pi$
$449$	38.7143	1.82704	0.913520	+	0.406794i	$0.133353\pi$
$450$	0	0
$451$	28.9817	1.36469
$452$	0	0
$453$	−1.31695	−0.0618756
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−10.9538	−0.512396	−0.256198	−	0.966624i	$-0.582470\pi$
$457$	−10.9538	−0.512396	−0.256198	+	0.966624i	$0.582470\pi$
$458$	0	0
$459$	−5.52311	−0.257797
$460$	0	0
$461$	−0.864641	−0.0402703	−0.0201352	−	0.999797i	$-0.506410\pi$
$461$	−0.864641	−0.0402703	−0.0201352	+	0.999797i	$0.506410\pi$
$462$	0	0
$463$	−16.0000	−0.743583	−0.371792	−	0.928316i	$-0.621256\pi$
$463$	−16.0000	−0.743583	−0.371792	+	0.928316i	$0.621256\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−20.7755	−0.961376	−0.480688	−	0.876892i	$-0.659613\pi$
$467$	−20.7755	−0.961376	−0.480688	+	0.876892i	$0.659613\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	−20.7110	−0.954311
$472$	0	0
$473$	−15.8217	−0.727484
$474$	0	0
$475$	0	0
$476$	0	0
$477$	6.77551	0.310229
$478$	0	0
$479$	20.6585	0.943910	0.471955	−	0.881623i	$-0.343549\pi$
$479$	20.6585	0.943910	0.471955	+	0.881623i	$0.343549\pi$
$480$	0	0
$481$	3.72928	0.170041
$482$	0	0
$483$	0	0
$484$	0	0
$485$	0	0
$486$	0	0
$487$	16.3632	0.741486	0.370743	−	0.928735i	$-0.379103\pi$
$487$	16.3632	0.741486	0.370743	+	0.928735i	$0.379103\pi$
$488$	0	0
$489$	−2.27072	−0.102685
$490$	0	0
$491$	26.7389	1.20671	0.603354	−	0.797473i	$-0.293831\pi$
$491$	26.7389	1.20671	0.603354	+	0.797473i	$0.293831\pi$
$492$	0	0
$493$	−37.4219	−1.68540
$494$	0	0
$495$	0	0
$496$	0	0
$497$	0	0
$498$	0	0
$499$	4.24281	0.189934	0.0949672	−	0.995480i	$-0.469725\pi$
$499$	4.24281	0.189934	0.0949672	+	0.995480i	$0.469725\pi$
$500$	0	0
$501$	−7.34153	−0.327995
$502$	0	0
$503$	0.889220	0.0396484	0.0198242	−	0.999803i	$-0.493689\pi$
$503$	0.889220	0.0396484	0.0198242	+	0.999803i	$0.493689\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	1.00000	0.0444116
$508$	0	0
$509$	−11.3694	−0.503941	−0.251971	−	0.967735i	$-0.581079\pi$
$509$	−11.3694	−0.503941	−0.251971	+	0.967735i	$0.581079\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	3.52311	0.155549
$514$	0	0
$515$	0	0
$516$	0	0
$517$	24.8034	1.09085
$518$	0	0
$519$	16.0925	0.706380
$520$	0	0
$521$	3.18785	0.139662	0.0698310	−	0.997559i	$-0.477754\pi$
$521$	3.18785	0.139662	0.0698310	+	0.997559i	$0.477754\pi$
$522$	0	0
$523$	37.1666	1.62518	0.812591	−	0.582835i	$-0.198056\pi$
$523$	37.1666	1.62518	0.812591	+	0.582835i	$0.198056\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−31.6435	−1.37841
$528$	0	0
$529$	33.5972	1.46075
$530$	0	0
$531$	0.593923	0.0257740
$532$	0	0
$533$	−10.1170	−0.438218
$534$	0	0
$535$	0	0
$536$	0	0
$537$	15.0462	0.649293
$538$	0	0
$539$	20.0525	0.863722
$540$	0	0
$541$	−8.02791	−0.345147	−0.172573	−	0.984997i	$-0.555208\pi$
$541$	−8.02791	−0.345147	−0.172573	+	0.984997i	$0.555208\pi$
$542$	0	0
$543$	5.58767	0.239790
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−31.2032	−1.33415	−0.667077	−	0.744989i	$-0.732455\pi$
$547$	−31.2032	−1.33415	−0.667077	+	0.744989i	$0.732455\pi$
$548$	0	0
$549$	−5.25240	−0.224167
$550$	0	0
$551$	23.8709	1.01693
$552$	0	0
$553$	0	0
$554$	0	0
$555$	0	0
$556$	0	0
$557$	29.2191	1.23805	0.619026	−	0.785370i	$-0.287528\pi$
$557$	29.2191	1.23805	0.619026	+	0.785370i	$0.287528\pi$
$558$	0	0
$559$	5.52311	0.233603
$560$	0	0
$561$	15.8217	0.667994
$562$	0	0
$563$	−30.5048	−1.28562	−0.642812	−	0.766024i	$-0.722232\pi$
$563$	−30.5048	−1.28562	−0.642812	+	0.766024i	$0.722232\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−12.1416	−0.509003	−0.254502	−	0.967072i	$-0.581911\pi$
$569$	−12.1416	−0.509003	−0.254502	+	0.967072i	$0.581911\pi$
$570$	0	0
$571$	−8.85258	−0.370469	−0.185234	−	0.982694i	$-0.559304\pi$
$571$	−8.85258	−0.370469	−0.185234	+	0.982694i	$0.559304\pi$
$572$	0	0
$573$	17.5510	0.733204
$574$	0	0
$575$	0	0
$576$	0	0
$577$	30.2341	1.25866	0.629330	−	0.777138i	$-0.283329\pi$
$577$	30.2341	1.25866	0.629330	+	0.777138i	$0.283329\pi$
$578$	0	0
$579$	−13.8217	−0.574412
$580$	0	0
$581$	0	0
$582$	0	0
$583$	−19.4094	−0.803855
$584$	0	0
$585$	0	0
$586$	0	0
$587$	23.0342	0.950722	0.475361	−	0.879791i	$-0.342318\pi$
$587$	23.0342	0.950722	0.475361	+	0.879791i	$0.342318\pi$
$588$	0	0
$589$	20.1849	0.831705
$590$	0	0
$591$	−26.3511	−1.08394
$592$	0	0
$593$	−27.9754	−1.14881	−0.574406	−	0.818570i	$-0.694767\pi$
$593$	−27.9754	−1.14881	−0.574406	+	0.818570i	$0.694767\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−23.0741	−0.944361
$598$	0	0
$599$	22.7389	0.929085	0.464542	−	0.885551i	$-0.346219\pi$
$599$	22.7389	0.929085	0.464542	+	0.885551i	$0.346219\pi$
$600$	0	0
$601$	−11.2158	−0.457500	−0.228750	−	0.973485i	$-0.573464\pi$
$601$	−11.2158	−0.457500	−0.228750	+	0.973485i	$0.573464\pi$
$602$	0	0
$603$	10.5048	0.427788
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−23.7293	−0.963142	−0.481571	−	0.876407i	$-0.659934\pi$
$607$	−23.7293	−0.963142	−0.481571	+	0.876407i	$0.659934\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−8.65847	−0.350284
$612$	0	0
$613$	18.2341	0.736467	0.368234	−	0.929733i	$-0.379963\pi$
$613$	18.2341	0.736467	0.368234	+	0.929733i	$0.379963\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−0.0245796	−0.000989536 0	−0.000494768	−	1.00000i	$-0.500157\pi$
$617$	−0.0245796	−0.000989536 0	−0.000494768		1.00000i	$0.500157\pi$
$618$	0	0
$619$	31.2158	1.25467	0.627334	−	0.778751i	$-0.284146\pi$
$619$	31.2158	1.25467	0.627334	+	0.778751i	$0.284146\pi$
$620$	0	0
$621$	−7.52311	−0.301892
$622$	0	0
$623$	0	0
$624$	0	0
$625$	0	0
$626$	0	0
$627$	−10.0925	−0.403054
$628$	0	0
$629$	−20.5972	−0.821266
$630$	0	0
$631$	13.4460	0.535279	0.267639	−	0.963519i	$-0.413756\pi$
$631$	13.4460	0.535279	0.267639	+	0.963519i	$0.413756\pi$
$632$	0	0
$633$	1.52311	0.0605384
$634$	0	0
$635$	0	0
$636$	0	0
$637$	−7.00000	−0.277350
$638$	0	0
$639$	−2.38776	−0.0944581
$640$	0	0
$641$	2.54144	0.100381	0.0501904	−	0.998740i	$-0.484017\pi$
$641$	2.54144	0.100381	0.0501904	+	0.998740i	$0.484017\pi$
$642$	0	0
$643$	−13.9634	−0.550661	−0.275330	−	0.961350i	$-0.588787\pi$
$643$	−13.9634	−0.550661	−0.275330	+	0.961350i	$0.588787\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	30.8034	1.21101	0.605504	−	0.795843i	$-0.292972\pi$
$647$	30.8034	1.21101	0.605504	+	0.795843i	$0.292972\pi$
$648$	0	0
$649$	−1.70138	−0.0667848
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−2.00000	−0.0782660	−0.0391330	−	0.999234i	$-0.512460\pi$
$653$	−2.00000	−0.0782660	−0.0391330	+	0.999234i	$0.512460\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	−5.45856	−0.212959
$658$	0	0
$659$	−4.05581	−0.157992	−0.0789960	−	0.996875i	$-0.525171\pi$
$659$	−4.05581	−0.157992	−0.0789960	+	0.996875i	$0.525171\pi$
$660$	0	0
$661$	−32.9325	−1.28093	−0.640463	−	0.767989i	$-0.721258\pi$
$661$	−32.9325	−1.28093	−0.640463	+	0.767989i	$0.721258\pi$
$662$	0	0
$663$	−5.52311	−0.214500
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−50.9729	−1.97368
$668$	0	0
$669$	−13.3169	−0.514863
$670$	0	0
$671$	15.0462	0.580853
$672$	0	0
$673$	−9.42192	−0.363188	−0.181594	−	0.983374i	$-0.558126\pi$
$673$	−9.42192	−0.363188	−0.181594	+	0.983374i	$0.558126\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−42.2341	−1.62319	−0.811594	−	0.584222i	$-0.801400\pi$
$677$	−42.2341	−1.62319	−0.811594	+	0.584222i	$0.801400\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	6.80009	0.260580
$682$	0	0
$683$	18.7509	0.717484	0.358742	−	0.933437i	$-0.383206\pi$
$683$	18.7509	0.717484	0.358742	+	0.933437i	$0.383206\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	1.52311	0.0581104
$688$	0	0
$689$	6.77551	0.258126
$690$	0	0
$691$	45.2524	1.72148	0.860741	−	0.509043i	$-0.170001\pi$
$691$	45.2524	1.72148	0.860741	+	0.509043i	$0.170001\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	0	0
$696$	0	0
$697$	55.8776	2.11651
$698$	0	0
$699$	10.7110	0.405126
$700$	0	0
$701$	−37.4586	−1.41479	−0.707395	−	0.706818i	$-0.750130\pi$
$701$	−37.4586	−1.41479	−0.707395	+	0.706818i	$0.750130\pi$
$702$	0	0
$703$	13.1387	0.495535
$704$	0	0
$705$	0	0
$706$	0	0
$707$	0	0
$708$	0	0
$709$	11.8988	0.446869	0.223435	−	0.974719i	$-0.428273\pi$
$709$	11.8988	0.446869	0.223435	+	0.974719i	$0.428273\pi$
$710$	0	0
$711$	2.47689	0.0928905
$712$	0	0
$713$	−43.1020	−1.61418
$714$	0	0
$715$	0	0
$716$	0	0
$717$	4.65847	0.173974
$718$	0	0
$719$	25.7851	0.961622	0.480811	−	0.876824i	$-0.340342\pi$
$719$	25.7851	0.961622	0.480811	+	0.876824i	$0.340342\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	5.22449	0.194301
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−34.8034	−1.29079	−0.645394	−	0.763850i	$-0.723307\pi$
$727$	−34.8034	−1.29079	−0.645394	+	0.763850i	$0.723307\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−30.5048	−1.12826
$732$	0	0
$733$	−19.5510	−0.722133	−0.361067	−	0.932540i	$-0.617587\pi$
$733$	−19.5510	−0.722133	−0.361067	+	0.932540i	$0.617587\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−30.0925	−1.10847
$738$	0	0
$739$	12.2986	0.452412	0.226206	−	0.974079i	$-0.427368\pi$
$739$	12.2986	0.452412	0.226206	+	0.974079i	$0.427368\pi$
$740$	0	0
$741$	3.52311	0.129425
$742$	0	0
$743$	−40.9975	−1.50405	−0.752027	−	0.659133i	$-0.770924\pi$
$743$	−40.9975	−1.50405	−0.752027	+	0.659133i	$0.770924\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	−8.11704	−0.296987
$748$	0	0
$749$	0	0
$750$	0	0
$751$	33.5510	1.22429	0.612147	−	0.790744i	$-0.290306\pi$
$751$	33.5510	1.22429	0.612147	+	0.790744i	$0.290306\pi$
$752$	0	0
$753$	5.85838	0.213491
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−25.1020	−0.912349	−0.456175	−	0.889890i	$-0.650781\pi$
$757$	−25.1020	−0.912349	−0.456175	+	0.889890i	$0.650781\pi$
$758$	0	0
$759$	21.5510	0.782253
$760$	0	0
$761$	−39.9021	−1.44645	−0.723226	−	0.690612i	$-0.757341\pi$
$761$	−39.9021	−1.44645	−0.723226	+	0.690612i	$0.757341\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	0	0
$766$	0	0
$767$	0.593923	0.0214453
$768$	0	0
$769$	−15.5510	−0.560784	−0.280392	−	0.959886i	$-0.590464\pi$
$769$	−15.5510	−0.560784	−0.280392	+	0.959886i	$0.590464\pi$
$770$	0	0
$771$	2.29862	0.0827830
$772$	0	0
$773$	−27.7972	−0.999794	−0.499897	−	0.866085i	$-0.666629\pi$
$773$	−27.7972	−0.999794	−0.499897	+	0.866085i	$0.666629\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−35.6435	−1.27706
$780$	0	0
$781$	6.84006	0.244757
$782$	0	0
$783$	6.77551	0.242137
$784$	0	0
$785$	0	0
$786$	0	0
$787$	30.6339	1.09198	0.545990	−	0.837792i	$-0.316154\pi$
$787$	30.6339	1.09198	0.545990	+	0.837792i	$0.316154\pi$
$788$	0	0
$789$	31.5789	1.12424
$790$	0	0
$791$	0	0
$792$	0	0
$793$	−5.25240	−0.186518
$794$	0	0
$795$	0	0
$796$	0	0
$797$	18.6464	0.660490	0.330245	−	0.943895i	$-0.392869\pi$
$797$	18.6464	0.660490	0.330245	+	0.943895i	$0.392869\pi$
$798$	0	0
$799$	47.8217	1.69181
$800$	0	0
$801$	−14.1170	−0.498801
$802$	0	0
$803$	15.6368	0.551812
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−30.3632	−1.06883
$808$	0	0
$809$	−54.8313	−1.92777	−0.963883	−	0.266325i	$-0.914191\pi$
$809$	−54.8313	−1.92777	−0.963883	+	0.266325i	$0.914191\pi$
$810$	0	0
$811$	5.97209	0.209709	0.104854	−	0.994488i	$-0.466562\pi$
$811$	5.97209	0.209709	0.104854	+	0.994488i	$0.466562\pi$
$812$	0	0
$813$	28.5972	1.00295
$814$	0	0
$815$	0	0
$816$	0	0
$817$	19.4586	0.680769
$818$	0	0
$819$	0	0
$820$	0	0
$821$	30.2866	1.05701	0.528504	−	0.848931i	$-0.322753\pi$
$821$	30.2866	1.05701	0.528504	+	0.848931i	$0.322753\pi$
$822$	0	0
$823$	31.2437	1.08909	0.544543	−	0.838733i	$-0.316703\pi$
$823$	31.2437	1.08909	0.544543	+	0.838733i	$0.316703\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	21.9754	0.764160	0.382080	−	0.924129i	$-0.375208\pi$
$827$	21.9754	0.764160	0.382080	+	0.924129i	$0.375208\pi$
$828$	0	0
$829$	−11.7014	−0.406406	−0.203203	−	0.979137i	$-0.565135\pi$
$829$	−11.7014	−0.406406	−0.203203	+	0.979137i	$0.565135\pi$
$830$	0	0
$831$	−1.79383	−0.0622274
$832$	0	0
$833$	38.6618	1.33955
$834$	0	0
$835$	0	0
$836$	0	0
$837$	5.72928	0.198033
$838$	0	0
$839$	11.7047	0.404091	0.202046	−	0.979376i	$-0.435241\pi$
$839$	11.7047	0.404091	0.202046	+	0.979376i	$0.435241\pi$
$840$	0	0
$841$	16.9075	0.583019
$842$	0	0
$843$	32.2095	1.10935
$844$	0	0
$845$	0	0
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−18.0925	−0.620932
$850$	0	0
$851$	−28.0558	−0.961741
$852$	0	0
$853$	15.3169	0.524442	0.262221	−	0.965008i	$-0.415545\pi$
$853$	15.3169	0.524442	0.262221	+	0.965008i	$0.415545\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−20.7476	−0.708725	−0.354362	−	0.935108i	$-0.615302\pi$
$857$	−20.7476	−0.708725	−0.354362	+	0.935108i	$0.615302\pi$
$858$	0	0
$859$	53.0375	1.80962	0.904808	−	0.425820i	$-0.140014\pi$
$859$	53.0375	1.80962	0.904808	+	0.425820i	$0.140014\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−10.2095	−0.347535	−0.173768	−	0.984787i	$-0.555594\pi$
$863$	−10.2095	−0.347535	−0.173768	+	0.984787i	$0.555594\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	13.5048	0.458647
$868$	0	0
$869$	−7.09539	−0.240695
$870$	0	0
$871$	10.5048	0.355941
$872$	0	0
$873$	−6.00000	−0.203069
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−9.28030	−0.313374	−0.156687	−	0.987648i	$-0.550081\pi$
$877$	−9.28030	−0.313374	−0.156687	+	0.987648i	$0.550081\pi$
$878$	0	0
$879$	12.6218	0.425724
$880$	0	0
$881$	−22.4923	−0.757784	−0.378892	−	0.925441i	$-0.623695\pi$
$881$	−22.4923	−0.757784	−0.378892	+	0.925441i	$0.623695\pi$
$882$	0	0
$883$	28.5972	0.962374	0.481187	−	0.876618i	$-0.340206\pi$
$883$	28.5972	0.962374	0.481187	+	0.876618i	$0.340206\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	34.2062	1.14853	0.574265	−	0.818669i	$-0.305288\pi$
$887$	34.2062	1.14853	0.574265	+	0.818669i	$0.305288\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	−2.86464	−0.0959691
$892$	0	0
$893$	−30.5048	−1.02080
$894$	0	0
$895$	0	0
$896$	0	0
$897$	−7.52311	−0.251189
$898$	0	0
$899$	38.8188	1.29468
$900$	0	0
$901$	−37.4219	−1.24670
$902$	0	0
$903$	0	0
$904$	0	0
$905$	0	0
$906$	0	0
$907$	25.9634	0.862099	0.431050	−	0.902328i	$-0.358143\pi$
$907$	25.9634	0.862099	0.431050	+	0.902328i	$0.358143\pi$
$908$	0	0
$909$	−11.7293	−0.389036
$910$	0	0
$911$	−51.9267	−1.72041	−0.860204	−	0.509949i	$-0.829664\pi$
$911$	−51.9267	−1.72041	−0.860204	+	0.509949i	$0.829664\pi$
$912$	0	0
$913$	23.2524	0.769542
$914$	0	0
$915$	0	0
$916$	0	0
$917$	0	0
$918$	0	0
$919$	−11.6156	−0.383162	−0.191581	−	0.981477i	$-0.561362\pi$
$919$	−11.6156	−0.383162	−0.191581	+	0.981477i	$0.561362\pi$
$920$	0	0
$921$	−20.0000	−0.659022
$922$	0	0
$923$	−2.38776	−0.0785939
$924$	0	0
$925$	0	0
$926$	0	0
$927$	−14.7755	−0.485291
$928$	0	0
$929$	5.83380	0.191401	0.0957004	−	0.995410i	$-0.469491\pi$
$929$	5.83380	0.191401	0.0957004	+	0.995410i	$0.469491\pi$
$930$	0	0
$931$	−24.6618	−0.808258
$932$	0	0
$933$	−31.8217	−1.04180
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−27.4373	−0.896338	−0.448169	−	0.893949i	$-0.647924\pi$
$937$	−27.4373	−0.896338	−0.448169	+	0.893949i	$0.647924\pi$
$938$	0	0
$939$	−11.7938	−0.384877
$940$	0	0
$941$	−34.9571	−1.13957	−0.569784	−	0.821794i	$-0.692973\pi$
$941$	−34.9571	−1.13957	−0.569784	+	0.821794i	$0.692973\pi$
$942$	0	0
$943$	76.1116	2.47854
$944$	0	0
$945$	0	0
$946$	0	0
$947$	53.3848	1.73477	0.867387	−	0.497634i	$-0.165798\pi$
$947$	53.3848	1.73477	0.867387	+	0.497634i	$0.165798\pi$
$948$	0	0
$949$	−5.45856	−0.177192
$950$	0	0
$951$	−11.4340	−0.370772
$952$	0	0
$953$	−22.7110	−0.735680	−0.367840	−	0.929889i	$-0.619903\pi$
$953$	−22.7110	−0.735680	−0.367840	+	0.929889i	$0.619903\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	−19.4094	−0.627417
$958$	0	0
$959$	0	0
$960$	0	0
$961$	1.82467	0.0588603
$962$	0	0
$963$	−1.72928	−0.0557253
$964$	0	0
$965$	0	0
$966$	0	0
$967$	28.8313	0.927153	0.463576	−	0.886057i	$-0.346566\pi$
$967$	28.8313	0.927153	0.463576	+	0.886057i	$0.346566\pi$
$968$	0	0
$969$	−19.4586	−0.625099
$970$	0	0
$971$	19.0462	0.611223	0.305611	−	0.952156i	$-0.401139\pi$
$971$	19.0462	0.611223	0.305611	+	0.952156i	$0.401139\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	0	0
$976$	0	0
$977$	30.1729	0.965315	0.482658	−	0.875809i	$-0.339672\pi$
$977$	30.1729	0.965315	0.482658	+	0.875809i	$0.339672\pi$
$978$	0	0
$979$	40.4402	1.29248
$980$	0	0
$981$	12.0279	0.384022
$982$	0	0
$983$	18.9292	0.603747	0.301874	−	0.953348i	$-0.402388\pi$
$983$	18.9292	0.603747	0.301874	+	0.953348i	$0.402388\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−41.5510	−1.32125
$990$	0	0
$991$	−25.5510	−0.811655	−0.405827	−	0.913950i	$-0.633017\pi$
$991$	−25.5510	−0.811655	−0.405827	+	0.913950i	$0.633017\pi$
$992$	0	0
$993$	2.20617	0.0700106
$994$	0	0
$995$	0	0
$996$	0	0
$997$	4.16952	0.132050	0.0660251	−	0.997818i	$-0.478968\pi$
$997$	4.16952	0.132050	0.0660251	+	0.997818i	$0.478968\pi$
$998$	0	0
$999$	3.72928	0.117989

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7800.2.a.bp.1.2		3
5.2	odd	4		1560.2.l.c.1249.2	✓	6
5.3	odd	4		1560.2.l.c.1249.5	yes	6
5.4	even	2		7800.2.a.bj.1.2		3
15.2	even	4		4680.2.l.e.2809.4		6
15.8	even	4		4680.2.l.e.2809.3		6
20.3	even	4		3120.2.l.m.1249.2		6
20.7	even	4		3120.2.l.m.1249.5		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1560.2.l.c.1249.2	✓	6	5.2	odd	4
1560.2.l.c.1249.5	yes	6	5.3	odd	4
3120.2.l.m.1249.2		6	20.3	even	4
3120.2.l.m.1249.5		6	20.7	even	4
4680.2.l.e.2809.3		6	15.8	even	4
4680.2.l.e.2809.4		6	15.2	even	4
7800.2.a.bj.1.2		3	5.4	even	2
7800.2.a.bp.1.2		3	1.1	even	1	trivial

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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:	\( N \)	\(=\)	\( 7800 = 2^{3} \cdot 3 \cdot 5^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7800.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} +1.00000 q^{9} -2.86464 q^{11} +1.00000 q^{13} -5.52311 q^{17} +3.52311 q^{19} -7.52311 q^{23} +1.00000 q^{27} +6.77551 q^{29} +5.72928 q^{31} -2.86464 q^{33} +3.72928 q^{37} +1.00000 q^{39} -10.1170 q^{41} +5.52311 q^{43} -8.65847 q^{47} -7.00000 q^{49} -5.52311 q^{51} +6.77551 q^{53} +3.52311 q^{57} +0.593923 q^{59} -5.25240 q^{61} +10.5048 q^{67} -7.52311 q^{69} -2.38776 q^{71} -5.45856 q^{73} +2.47689 q^{79} +1.00000 q^{81} -8.11704 q^{83} +6.77551 q^{87} -14.1170 q^{89} +5.72928 q^{93} -6.00000 q^{97} -2.86464 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 3 q^{3} + 3 q^{9}+O(q^{10}) \) 3 * q + 3 * q^3 + 3 * q^9	\( 3 q + 3 q^{3} + 3 q^{9} - 6 q^{11} + 3 q^{13} - 4 q^{17} - 2 q^{19} - 10 q^{23} + 3 q^{27} - 10 q^{29} + 12 q^{31} - 6 q^{33} + 6 q^{37} + 3 q^{39} - 10 q^{41} + 4 q^{43} - 16 q^{47} - 21 q^{49} - 4 q^{51} - 10 q^{53} - 2 q^{57} - 6 q^{59} + 2 q^{61} - 4 q^{67} - 10 q^{69} + 8 q^{71} - 6 q^{73} + 20 q^{79} + 3 q^{81} - 4 q^{83} - 10 q^{87} - 22 q^{89} + 12 q^{93} - 18 q^{97} - 6 q^{99}+O(q^{100}) \) 3 * q + 3 * q^3 + 3 * q^9 - 6 * q^11 + 3 * q^13 - 4 * q^17 - 2 * q^19 - 10 * q^23 + 3 * q^27 - 10 * q^29 + 12 * q^31 - 6 * q^33 + 6 * q^37 + 3 * q^39 - 10 * q^41 + 4 * q^43 - 16 * q^47 - 21 * q^49 - 4 * q^51 - 10 * q^53 - 2 * q^57 - 6 * q^59 + 2 * q^61 - 4 * q^67 - 10 * q^69 + 8 * q^71 - 6 * q^73 + 20 * q^79 + 3 * q^81 - 4 * q^83 - 10 * q^87 - 22 * q^89 + 12 * q^93 - 18 * q^97 - 6 * q^99

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