LMFDB - Embedded newform 7800.2.a.bb.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:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{33}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 8 $ x^2 - x - 8
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1560)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-2.37228$ 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)

$f(q)$	$=$	$q+1.00000 q^{3} +2.37228 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} +2.37228 q^{7} +1.00000 q^{9} +2.37228 q^{11} +1.00000 q^{13} -4.37228 q^{17} +4.74456 q^{19} +2.37228 q^{21} +6.37228 q^{23} +1.00000 q^{27} -2.00000 q^{29} -4.74456 q^{31} +2.37228 q^{33} +0.372281 q^{37} +1.00000 q^{39} +4.37228 q^{41} +4.00000 q^{43} +12.7446 q^{47} -1.37228 q^{49} -4.37228 q^{51} +3.62772 q^{53} +4.74456 q^{57} -8.00000 q^{59} -9.11684 q^{61} +2.37228 q^{63} +4.00000 q^{67} +6.37228 q^{69} -5.62772 q^{71} -7.48913 q^{73} +5.62772 q^{77} +11.8614 q^{79} +1.00000 q^{81} +12.0000 q^{83} -2.00000 q^{87} +7.62772 q^{89} +2.37228 q^{91} -4.74456 q^{93} -8.37228 q^{97} +2.37228 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{3} - q^{7} + 2 q^{9}+O(q^{10}) $ 2 * q + 2 * q^3 - q^7 + 2 * q^9	$ 2 q + 2 q^{3} - q^{7} + 2 q^{9} - q^{11} + 2 q^{13} - 3 q^{17} - 2 q^{19} - q^{21} + 7 q^{23} + 2 q^{27} - 4 q^{29} + 2 q^{31} - q^{33} - 5 q^{37} + 2 q^{39} + 3 q^{41} + 8 q^{43} + 14 q^{47} + 3 q^{49} - 3 q^{51} + 13 q^{53} - 2 q^{57} - 16 q^{59} - q^{61} - q^{63} + 8 q^{67} + 7 q^{69} - 17 q^{71} + 8 q^{73} + 17 q^{77} - 5 q^{79} + 2 q^{81} + 24 q^{83} - 4 q^{87} + 21 q^{89} - q^{91} + 2 q^{93} - 11 q^{97} - q^{99}+O(q^{100}) $ 2 * q + 2 * q^3 - q^7 + 2 * q^9 - q^11 + 2 * q^13 - 3 * q^17 - 2 * q^19 - q^21 + 7 * q^23 + 2 * q^27 - 4 * q^29 + 2 * q^31 - q^33 - 5 * q^37 + 2 * q^39 + 3 * q^41 + 8 * q^43 + 14 * q^47 + 3 * q^49 - 3 * q^51 + 13 * q^53 - 2 * q^57 - 16 * q^59 - q^61 - q^63 + 8 * q^67 + 7 * q^69 - 17 * q^71 + 8 * q^73 + 17 * q^77 - 5 * q^79 + 2 * q^81 + 24 * q^83 - 4 * q^87 + 21 * q^89 - q^91 + 2 * q^93 - 11 * q^97 - 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
$7$	2.37228	0.896638	0.448319	−	0.893874i	$-0.352023\pi$
$7$	2.37228	0.896638	0.448319	+	0.893874i	$0.352023\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	2.37228	0.715270	0.357635	−	0.933862i	$-0.383583\pi$
$11$	2.37228	0.715270	0.357635	+	0.933862i	$0.383583\pi$
$12$	0	0
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−4.37228	−1.06043	−0.530217	−	0.847862i	$-0.677890\pi$
$17$	−4.37228	−1.06043	−0.530217	+	0.847862i	$0.677890\pi$
$18$	0	0
$19$	4.74456	1.08848	0.544239	−	0.838930i	$-0.316819\pi$
$19$	4.74456	1.08848	0.544239	+	0.838930i	$0.316819\pi$
$20$	0	0
$21$	2.37228	0.517674
$22$	0	0
$23$	6.37228	1.32871	0.664356	−	0.747416i	$-0.268706\pi$
$23$	6.37228	1.32871	0.664356	+	0.747416i	$0.268706\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	−2.00000	−0.371391	−0.185695	−	0.982607i	$-0.559454\pi$
$29$	−2.00000	−0.371391	−0.185695	+	0.982607i	$0.559454\pi$
$30$	0	0
$31$	−4.74456	−0.852149	−0.426074	−	0.904688i	$-0.640104\pi$
$31$	−4.74456	−0.852149	−0.426074	+	0.904688i	$0.640104\pi$
$32$	0	0
$33$	2.37228	0.412961
$34$	0	0
$35$	0	0
$36$	0	0
$37$	0.372281	0.0612027	0.0306013	−	0.999532i	$-0.490258\pi$
$37$	0.372281	0.0612027	0.0306013	+	0.999532i	$0.490258\pi$
$38$	0	0
$39$	1.00000	0.160128
$40$	0	0
$41$	4.37228	0.682836	0.341418	−	0.939912i	$-0.389093\pi$
$41$	4.37228	0.682836	0.341418	+	0.939912i	$0.389093\pi$
$42$	0	0
$43$	4.00000	0.609994	0.304997	−	0.952353i	$-0.401344\pi$
$43$	4.00000	0.609994	0.304997	+	0.952353i	$0.401344\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	12.7446	1.85899	0.929493	−	0.368840i	$-0.120245\pi$
$47$	12.7446	1.85899	0.929493	+	0.368840i	$0.120245\pi$
$48$	0	0
$49$	−1.37228	−0.196040
$50$	0	0
$51$	−4.37228	−0.612242
$52$	0	0
$53$	3.62772	0.498305	0.249153	−	0.968464i	$-0.419848\pi$
$53$	3.62772	0.498305	0.249153	+	0.968464i	$0.419848\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	4.74456	0.628433
$58$	0	0
$59$	−8.00000	−1.04151	−0.520756	−	0.853706i	$-0.674350\pi$
$59$	−8.00000	−1.04151	−0.520756	+	0.853706i	$0.674350\pi$
$60$	0	0
$61$	−9.11684	−1.16729	−0.583646	−	0.812008i	$-0.698374\pi$
$61$	−9.11684	−1.16729	−0.583646	+	0.812008i	$0.698374\pi$
$62$	0	0
$63$	2.37228	0.298879
$64$	0	0
$65$	0	0
$66$	0	0
$67$	4.00000	0.488678	0.244339	−	0.969690i	$-0.421429\pi$
$67$	4.00000	0.488678	0.244339	+	0.969690i	$0.421429\pi$
$68$	0	0
$69$	6.37228	0.767133
$70$	0	0
$71$	−5.62772	−0.667887	−0.333944	−	0.942593i	$-0.608380\pi$
$71$	−5.62772	−0.667887	−0.333944	+	0.942593i	$0.608380\pi$
$72$	0	0
$73$	−7.48913	−0.876536	−0.438268	−	0.898844i	$-0.644408\pi$
$73$	−7.48913	−0.876536	−0.438268	+	0.898844i	$0.644408\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	5.62772	0.641338
$78$	0	0
$79$	11.8614	1.33451	0.667256	−	0.744828i	$-0.267469\pi$
$79$	11.8614	1.33451	0.667256	+	0.744828i	$0.267469\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	12.0000	1.31717	0.658586	−	0.752506i	$-0.271155\pi$
$83$	12.0000	1.31717	0.658586	+	0.752506i	$0.271155\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	−2.00000	−0.214423
$88$	0	0
$89$	7.62772	0.808537	0.404268	−	0.914640i	$-0.367526\pi$
$89$	7.62772	0.808537	0.404268	+	0.914640i	$0.367526\pi$
$90$	0	0
$91$	2.37228	0.248683
$92$	0	0
$93$	−4.74456	−0.491988
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−8.37228	−0.850076	−0.425038	−	0.905175i	$-0.639739\pi$
$97$	−8.37228	−0.850076	−0.425038	+	0.905175i	$0.639739\pi$
$98$	0	0
$99$	2.37228	0.238423
$100$	0	0
$101$	−6.74456	−0.671109	−0.335555	−	0.942021i	$-0.608924\pi$
$101$	−6.74456	−0.671109	−0.335555	+	0.942021i	$0.608924\pi$
$102$	0	0
$103$	4.00000	0.394132	0.197066	−	0.980390i	$-0.436859\pi$
$103$	4.00000	0.394132	0.197066	+	0.980390i	$0.436859\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	14.3723	1.38942	0.694710	−	0.719290i	$-0.255533\pi$
$107$	14.3723	1.38942	0.694710	+	0.719290i	$0.255533\pi$
$108$	0	0
$109$	−2.74456	−0.262881	−0.131441	−	0.991324i	$-0.541960\pi$
$109$	−2.74456	−0.262881	−0.131441	+	0.991324i	$0.541960\pi$
$110$	0	0
$111$	0.372281	0.0353354
$112$	0	0
$113$	6.00000	0.564433	0.282216	−	0.959351i	$-0.408930\pi$
$113$	6.00000	0.564433	0.282216	+	0.959351i	$0.408930\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	1.00000	0.0924500
$118$	0	0
$119$	−10.3723	−0.950825
$120$	0	0
$121$	−5.37228	−0.488389
$122$	0	0
$123$	4.37228	0.394235
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−18.2337	−1.61798	−0.808989	−	0.587824i	$-0.799985\pi$
$127$	−18.2337	−1.61798	−0.808989	+	0.587824i	$0.799985\pi$
$128$	0	0
$129$	4.00000	0.352180
$130$	0	0
$131$	−7.25544	−0.633911	−0.316955	−	0.948440i	$-0.602661\pi$
$131$	−7.25544	−0.633911	−0.316955	+	0.948440i	$0.602661\pi$
$132$	0	0
$133$	11.2554	0.975970
$134$	0	0
$135$	0	0
$136$	0	0
$137$	11.4891	0.981582	0.490791	−	0.871277i	$-0.336708\pi$
$137$	11.4891	0.981582	0.490791	+	0.871277i	$0.336708\pi$
$138$	0	0
$139$	1.62772	0.138061	0.0690306	−	0.997615i	$-0.478009\pi$
$139$	1.62772	0.138061	0.0690306	+	0.997615i	$0.478009\pi$
$140$	0	0
$141$	12.7446	1.07329
$142$	0	0
$143$	2.37228	0.198380
$144$	0	0
$145$	0	0
$146$	0	0
$147$	−1.37228	−0.113184
$148$	0	0
$149$	7.62772	0.624887	0.312444	−	0.949936i	$-0.398852\pi$
$149$	7.62772	0.624887	0.312444	+	0.949936i	$0.398852\pi$
$150$	0	0
$151$	0	0		−	1.00000i	$-0.5\pi$
$151$	0	0			1.00000i	$0.5\pi$
$152$	0	0
$153$	−4.37228	−0.353478
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−2.00000	−0.159617	−0.0798087	−	0.996810i	$-0.525431\pi$
$157$	−2.00000	−0.159617	−0.0798087	+	0.996810i	$0.525431\pi$
$158$	0	0
$159$	3.62772	0.287697
$160$	0	0
$161$	15.1168	1.19137
$162$	0	0
$163$	6.37228	0.499116	0.249558	−	0.968360i	$-0.419715\pi$
$163$	6.37228	0.499116	0.249558	+	0.968360i	$0.419715\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	0	0		−	1.00000i	$-0.5\pi$
$167$	0	0			1.00000i	$0.5\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	4.74456	0.362826
$172$	0	0
$173$	15.4891	1.17762	0.588808	−	0.808273i	$-0.299597\pi$
$173$	15.4891	1.17762	0.588808	+	0.808273i	$0.299597\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−8.00000	−0.601317
$178$	0	0
$179$	−10.2337	−0.764902	−0.382451	−	0.923976i	$-0.624920\pi$
$179$	−10.2337	−0.764902	−0.382451	+	0.923976i	$0.624920\pi$
$180$	0	0
$181$	0.372281	0.0276715	0.0138357	−	0.999904i	$-0.495596\pi$
$181$	0.372281	0.0276715	0.0138357	+	0.999904i	$0.495596\pi$
$182$	0	0
$183$	−9.11684	−0.673936
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−10.3723	−0.758496
$188$	0	0
$189$	2.37228	0.172558
$190$	0	0
$191$	−8.00000	−0.578860	−0.289430	−	0.957199i	$-0.593466\pi$
$191$	−8.00000	−0.578860	−0.289430	+	0.957199i	$0.593466\pi$
$192$	0	0
$193$	−3.62772	−0.261129	−0.130564	−	0.991440i	$-0.541679\pi$
$193$	−3.62772	−0.261129	−0.130564	+	0.991440i	$0.541679\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−24.2337	−1.72658	−0.863289	−	0.504709i	$-0.831600\pi$
$197$	−24.2337	−1.72658	−0.863289	+	0.504709i	$0.831600\pi$
$198$	0	0
$199$	−1.48913	−0.105561	−0.0527806	−	0.998606i	$-0.516808\pi$
$199$	−1.48913	−0.105561	−0.0527806	+	0.998606i	$0.516808\pi$
$200$	0	0
$201$	4.00000	0.282138
$202$	0	0
$203$	−4.74456	−0.333003
$204$	0	0
$205$	0	0
$206$	0	0
$207$	6.37228	0.442904
$208$	0	0
$209$	11.2554	0.778555
$210$	0	0
$211$	−22.9783	−1.58189	−0.790944	−	0.611889i	$-0.790410\pi$
$211$	−22.9783	−1.58189	−0.790944	+	0.611889i	$0.790410\pi$
$212$	0	0
$213$	−5.62772	−0.385605
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−11.2554	−0.764069
$218$	0	0
$219$	−7.48913	−0.506068
$220$	0	0
$221$	−4.37228	−0.294111
$222$	0	0
$223$	26.9783	1.80660	0.903299	−	0.429012i	$-0.141138\pi$
$223$	26.9783	1.80660	0.903299	+	0.429012i	$0.141138\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	7.25544	0.481560	0.240780	−	0.970580i	$-0.422597\pi$
$227$	7.25544	0.481560	0.240780	+	0.970580i	$0.422597\pi$
$228$	0	0
$229$	2.00000	0.132164	0.0660819	−	0.997814i	$-0.478950\pi$
$229$	2.00000	0.132164	0.0660819	+	0.997814i	$0.478950\pi$
$230$	0	0
$231$	5.62772	0.370277
$232$	0	0
$233$	24.3723	1.59668	0.798341	−	0.602206i	$-0.205711\pi$
$233$	24.3723	1.59668	0.798341	+	0.602206i	$0.205711\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	11.8614	0.770481
$238$	0	0
$239$	29.3505	1.89853	0.949264	−	0.314480i	$-0.101830\pi$
$239$	29.3505	1.89853	0.949264	+	0.314480i	$0.101830\pi$
$240$	0	0
$241$	20.9783	1.35133	0.675664	−	0.737210i	$-0.263857\pi$
$241$	20.9783	1.35133	0.675664	+	0.737210i	$0.263857\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	0	0
$246$	0	0
$247$	4.74456	0.301889
$248$	0	0
$249$	12.0000	0.760469
$250$	0	0
$251$	−5.48913	−0.346471	−0.173235	−	0.984880i	$-0.555422\pi$
$251$	−5.48913	−0.346471	−0.173235	+	0.984880i	$0.555422\pi$
$252$	0	0
$253$	15.1168	0.950388
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−10.0000	−0.623783	−0.311891	−	0.950118i	$-0.600963\pi$
$257$	−10.0000	−0.623783	−0.311891	+	0.950118i	$0.600963\pi$
$258$	0	0
$259$	0.883156	0.0548766
$260$	0	0
$261$	−2.00000	−0.123797
$262$	0	0
$263$	−5.48913	−0.338474	−0.169237	−	0.985575i	$-0.554130\pi$
$263$	−5.48913	−0.338474	−0.169237	+	0.985575i	$0.554130\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	7.62772	0.466809
$268$	0	0
$269$	1.25544	0.0765454	0.0382727	−	0.999267i	$-0.487814\pi$
$269$	1.25544	0.0765454	0.0382727	+	0.999267i	$0.487814\pi$
$270$	0	0
$271$	28.7446	1.74611	0.873054	−	0.487624i	$-0.162136\pi$
$271$	28.7446	1.74611	0.873054	+	0.487624i	$0.162136\pi$
$272$	0	0
$273$	2.37228	0.143577
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−20.9783	−1.26046	−0.630230	−	0.776408i	$-0.717040\pi$
$277$	−20.9783	−1.26046	−0.630230	+	0.776408i	$0.717040\pi$
$278$	0	0
$279$	−4.74456	−0.284050
$280$	0	0
$281$	−32.9783	−1.96732	−0.983659	−	0.180043i	$-0.942376\pi$
$281$	−32.9783	−1.96732	−0.983659	+	0.180043i	$0.942376\pi$
$282$	0	0
$283$	−20.0000	−1.18888	−0.594438	−	0.804141i	$-0.702626\pi$
$283$	−20.0000	−1.18888	−0.594438	+	0.804141i	$0.702626\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	10.3723	0.612256
$288$	0	0
$289$	2.11684	0.124520
$290$	0	0
$291$	−8.37228	−0.490792
$292$	0	0
$293$	26.7446	1.56243	0.781217	−	0.624260i	$-0.214599\pi$
$293$	26.7446	1.56243	0.781217	+	0.624260i	$0.214599\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	2.37228	0.137654
$298$	0	0
$299$	6.37228	0.368519
$300$	0	0
$301$	9.48913	0.546944
$302$	0	0
$303$	−6.74456	−0.387465
$304$	0	0
$305$	0	0
$306$	0	0
$307$	−9.62772	−0.549483	−0.274741	−	0.961518i	$-0.588592\pi$
$307$	−9.62772	−0.549483	−0.274741	+	0.961518i	$0.588592\pi$
$308$	0	0
$309$	4.00000	0.227552
$310$	0	0
$311$	0	0		−	1.00000i	$-0.5\pi$
$311$	0	0			1.00000i	$0.5\pi$
$312$	0	0
$313$	6.00000	0.339140	0.169570	−	0.985518i	$-0.445762\pi$
$313$	6.00000	0.339140	0.169570	+	0.985518i	$0.445762\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−16.2337	−0.911775	−0.455887	−	0.890037i	$-0.650678\pi$
$317$	−16.2337	−0.911775	−0.455887	+	0.890037i	$0.650678\pi$
$318$	0	0
$319$	−4.74456	−0.265645
$320$	0	0
$321$	14.3723	0.802183
$322$	0	0
$323$	−20.7446	−1.15426
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−2.74456	−0.151775
$328$	0	0
$329$	30.2337	1.66684
$330$	0	0
$331$	1.48913	0.0818497	0.0409249	−	0.999162i	$-0.486970\pi$
$331$	1.48913	0.0818497	0.0409249	+	0.999162i	$0.486970\pi$
$332$	0	0
$333$	0.372281	0.0204009
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−16.2337	−0.884305	−0.442153	−	0.896940i	$-0.645785\pi$
$337$	−16.2337	−0.884305	−0.442153	+	0.896940i	$0.645785\pi$
$338$	0	0
$339$	6.00000	0.325875
$340$	0	0
$341$	−11.2554	−0.609516
$342$	0	0
$343$	−19.8614	−1.07242
$344$	0	0
$345$	0	0
$346$	0	0
$347$	7.86141	0.422022	0.211011	−	0.977484i	$-0.432324\pi$
$347$	7.86141	0.422022	0.211011	+	0.977484i	$0.432324\pi$
$348$	0	0
$349$	32.2337	1.72543	0.862715	−	0.505691i	$-0.168762\pi$
$349$	32.2337	1.72543	0.862715	+	0.505691i	$0.168762\pi$
$350$	0	0
$351$	1.00000	0.0533761
$352$	0	0
$353$	−14.0000	−0.745145	−0.372572	−	0.928003i	$-0.621524\pi$
$353$	−14.0000	−0.745145	−0.372572	+	0.928003i	$0.621524\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	−10.3723	−0.548959
$358$	0	0
$359$	8.00000	0.422224	0.211112	−	0.977462i	$-0.432292\pi$
$359$	8.00000	0.422224	0.211112	+	0.977462i	$0.432292\pi$
$360$	0	0
$361$	3.51087	0.184783
$362$	0	0
$363$	−5.37228	−0.281972
$364$	0	0
$365$	0	0
$366$	0	0
$367$	−13.4891	−0.704127	−0.352063	−	0.935976i	$-0.614520\pi$
$367$	−13.4891	−0.704127	−0.352063	+	0.935976i	$0.614520\pi$
$368$	0	0
$369$	4.37228	0.227612
$370$	0	0
$371$	8.60597	0.446800
$372$	0	0
$373$	26.7446	1.38478	0.692390	−	0.721523i	$-0.256558\pi$
$373$	26.7446	1.38478	0.692390	+	0.721523i	$0.256558\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−2.00000	−0.103005
$378$	0	0
$379$	3.25544	0.167221	0.0836103	−	0.996499i	$-0.473355\pi$
$379$	3.25544	0.167221	0.0836103	+	0.996499i	$0.473355\pi$
$380$	0	0
$381$	−18.2337	−0.934140
$382$	0	0
$383$	−20.7446	−1.06000	−0.529999	−	0.847998i	$-0.677808\pi$
$383$	−20.7446	−1.06000	−0.529999	+	0.847998i	$0.677808\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	4.00000	0.203331
$388$	0	0
$389$	−8.23369	−0.417465	−0.208732	−	0.977973i	$-0.566934\pi$
$389$	−8.23369	−0.417465	−0.208732	+	0.977973i	$0.566934\pi$
$390$	0	0
$391$	−27.8614	−1.40901
$392$	0	0
$393$	−7.25544	−0.365988
$394$	0	0
$395$	0	0
$396$	0	0
$397$	27.3505	1.37268	0.686342	−	0.727279i	$-0.259215\pi$
$397$	27.3505	1.37268	0.686342	+	0.727279i	$0.259215\pi$
$398$	0	0
$399$	11.2554	0.563477
$400$	0	0
$401$	10.0000	0.499376	0.249688	−	0.968326i	$-0.419672\pi$
$401$	10.0000	0.499376	0.249688	+	0.968326i	$0.419672\pi$
$402$	0	0
$403$	−4.74456	−0.236343
$404$	0	0
$405$	0	0
$406$	0	0
$407$	0.883156	0.0437764
$408$	0	0
$409$	−32.9783	−1.63067	−0.815335	−	0.578990i	$-0.803447\pi$
$409$	−32.9783	−1.63067	−0.815335	+	0.578990i	$0.803447\pi$
$410$	0	0
$411$	11.4891	0.566717
$412$	0	0
$413$	−18.9783	−0.933859
$414$	0	0
$415$	0	0
$416$	0	0
$417$	1.62772	0.0797097
$418$	0	0
$419$	2.23369	0.109123	0.0545614	−	0.998510i	$-0.482624\pi$
$419$	2.23369	0.109123	0.0545614	+	0.998510i	$0.482624\pi$
$420$	0	0
$421$	−29.7228	−1.44860	−0.724301	−	0.689484i	$-0.757837\pi$
$421$	−29.7228	−1.44860	−0.724301	+	0.689484i	$0.757837\pi$
$422$	0	0
$423$	12.7446	0.619662
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−21.6277	−1.04664
$428$	0	0
$429$	2.37228	0.114535
$430$	0	0
$431$	−26.9783	−1.29950	−0.649748	−	0.760149i	$-0.725126\pi$
$431$	−26.9783	−1.29950	−0.649748	+	0.760149i	$0.725126\pi$
$432$	0	0
$433$	28.2337	1.35682	0.678412	−	0.734681i	$-0.262668\pi$
$433$	28.2337	1.35682	0.678412	+	0.734681i	$0.262668\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	30.2337	1.44627
$438$	0	0
$439$	40.6060	1.93802	0.969009	−	0.247027i	$-0.0794537\pi$
$439$	40.6060	1.93802	0.969009	+	0.247027i	$0.0794537\pi$
$440$	0	0
$441$	−1.37228	−0.0653467
$442$	0	0
$443$	33.3505	1.58453	0.792266	−	0.610176i	$-0.208901\pi$
$443$	33.3505	1.58453	0.792266	+	0.610176i	$0.208901\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	7.62772	0.360779
$448$	0	0
$449$	4.37228	0.206341	0.103170	−	0.994664i	$-0.467101\pi$
$449$	4.37228	0.206341	0.103170	+	0.994664i	$0.467101\pi$
$450$	0	0
$451$	10.3723	0.488412
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−22.8832	−1.07043	−0.535214	−	0.844716i	$-0.679769\pi$
$457$	−22.8832	−1.07043	−0.535214	+	0.844716i	$0.679769\pi$
$458$	0	0
$459$	−4.37228	−0.204081
$460$	0	0
$461$	31.3505	1.46014	0.730070	−	0.683372i	$-0.239487\pi$
$461$	31.3505	1.46014	0.730070	+	0.683372i	$0.239487\pi$
$462$	0	0
$463$	−8.60597	−0.399953	−0.199977	−	0.979801i	$-0.564087\pi$
$463$	−8.60597	−0.399953	−0.199977	+	0.979801i	$0.564087\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	41.3505	1.91347	0.956737	−	0.290953i	$-0.0939725\pi$
$467$	41.3505	1.91347	0.956737	+	0.290953i	$0.0939725\pi$
$468$	0	0
$469$	9.48913	0.438167
$470$	0	0
$471$	−2.00000	−0.0921551
$472$	0	0
$473$	9.48913	0.436310
$474$	0	0
$475$	0	0
$476$	0	0
$477$	3.62772	0.166102
$478$	0	0
$479$	31.1168	1.42176	0.710882	−	0.703311i	$-0.248296\pi$
$479$	31.1168	1.42176	0.710882	+	0.703311i	$0.248296\pi$
$480$	0	0
$481$	0.372281	0.0169746
$482$	0	0
$483$	15.1168	0.687840
$484$	0	0
$485$	0	0
$486$	0	0
$487$	15.1168	0.685010	0.342505	−	0.939516i	$-0.388725\pi$
$487$	15.1168	0.685010	0.342505	+	0.939516i	$0.388725\pi$
$488$	0	0
$489$	6.37228	0.288165
$490$	0	0
$491$	12.0000	0.541552	0.270776	−	0.962642i	$-0.412720\pi$
$491$	12.0000	0.541552	0.270776	+	0.962642i	$0.412720\pi$
$492$	0	0
$493$	8.74456	0.393835
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−13.3505	−0.598853
$498$	0	0
$499$	−3.25544	−0.145733	−0.0728667	−	0.997342i	$-0.523215\pi$
$499$	−3.25544	−0.145733	−0.0728667	+	0.997342i	$0.523215\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−22.9783	−1.02455	−0.512275	−	0.858822i	$-0.671197\pi$
$503$	−22.9783	−1.02455	−0.512275	+	0.858822i	$0.671197\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	1.00000	0.0444116
$508$	0	0
$509$	6.13859	0.272088	0.136044	−	0.990703i	$-0.456561\pi$
$509$	6.13859	0.272088	0.136044	+	0.990703i	$0.456561\pi$
$510$	0	0
$511$	−17.7663	−0.785935
$512$	0	0
$513$	4.74456	0.209478
$514$	0	0
$515$	0	0
$516$	0	0
$517$	30.2337	1.32968
$518$	0	0
$519$	15.4891	0.679897
$520$	0	0
$521$	0.510875	0.0223818	0.0111909	−	0.999937i	$-0.496438\pi$
$521$	0.510875	0.0223818	0.0111909	+	0.999937i	$0.496438\pi$
$522$	0	0
$523$	−28.0000	−1.22435	−0.612177	−	0.790721i	$-0.709706\pi$
$523$	−28.0000	−1.22435	−0.612177	+	0.790721i	$0.709706\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	20.7446	0.903647
$528$	0	0
$529$	17.6060	0.765477
$530$	0	0
$531$	−8.00000	−0.347170
$532$	0	0
$533$	4.37228	0.189385
$534$	0	0
$535$	0	0
$536$	0	0
$537$	−10.2337	−0.441616
$538$	0	0
$539$	−3.25544	−0.140222
$540$	0	0
$541$	0.510875	0.0219642	0.0109821	−	0.999940i	$-0.496504\pi$
$541$	0.510875	0.0219642	0.0109821	+	0.999940i	$0.496504\pi$
$542$	0	0
$543$	0.372281	0.0159761
$544$	0	0
$545$	0	0
$546$	0	0
$547$	38.9783	1.66659	0.833295	−	0.552829i	$-0.186452\pi$
$547$	38.9783	1.66659	0.833295	+	0.552829i	$0.186452\pi$
$548$	0	0
$549$	−9.11684	−0.389097
$550$	0	0
$551$	−9.48913	−0.404250
$552$	0	0
$553$	28.1386	1.19657
$554$	0	0
$555$	0	0
$556$	0	0
$557$	32.9783	1.39733	0.698667	−	0.715447i	$-0.253777\pi$
$557$	32.9783	1.39733	0.698667	+	0.715447i	$0.253777\pi$
$558$	0	0
$559$	4.00000	0.169182
$560$	0	0
$561$	−10.3723	−0.437918
$562$	0	0
$563$	−19.1168	−0.805679	−0.402839	−	0.915271i	$-0.631977\pi$
$563$	−19.1168	−0.805679	−0.402839	+	0.915271i	$0.631977\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	2.37228	0.0996265
$568$	0	0
$569$	33.7228	1.41373	0.706867	−	0.707347i	$-0.250108\pi$
$569$	33.7228	1.41373	0.706867	+	0.707347i	$0.250108\pi$
$570$	0	0
$571$	17.3505	0.726097	0.363049	−	0.931770i	$-0.381736\pi$
$571$	17.3505	0.726097	0.363049	+	0.931770i	$0.381736\pi$
$572$	0	0
$573$	−8.00000	−0.334205
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−22.8832	−0.952638	−0.476319	−	0.879272i	$-0.658029\pi$
$577$	−22.8832	−0.952638	−0.476319	+	0.879272i	$0.658029\pi$
$578$	0	0
$579$	−3.62772	−0.150763
$580$	0	0
$581$	28.4674	1.18103
$582$	0	0
$583$	8.60597	0.356423
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−8.74456	−0.360927	−0.180463	−	0.983582i	$-0.557760\pi$
$587$	−8.74456	−0.360927	−0.180463	+	0.983582i	$0.557760\pi$
$588$	0	0
$589$	−22.5109	−0.927544
$590$	0	0
$591$	−24.2337	−0.996841
$592$	0	0
$593$	22.7446	0.934007	0.467004	−	0.884255i	$-0.345333\pi$
$593$	22.7446	0.934007	0.467004	+	0.884255i	$0.345333\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−1.48913	−0.0609458
$598$	0	0
$599$	−17.4891	−0.714586	−0.357293	−	0.933992i	$-0.616300\pi$
$599$	−17.4891	−0.714586	−0.357293	+	0.933992i	$0.616300\pi$
$600$	0	0
$601$	2.60597	0.106300	0.0531499	−	0.998587i	$-0.483074\pi$
$601$	2.60597	0.106300	0.0531499	+	0.998587i	$0.483074\pi$
$602$	0	0
$603$	4.00000	0.162893
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−28.0000	−1.13648	−0.568242	−	0.822861i	$-0.692376\pi$
$607$	−28.0000	−1.13648	−0.568242	+	0.822861i	$0.692376\pi$
$608$	0	0
$609$	−4.74456	−0.192259
$610$	0	0
$611$	12.7446	0.515590
$612$	0	0
$613$	−13.8614	−0.559857	−0.279928	−	0.960021i	$-0.590311\pi$
$613$	−13.8614	−0.559857	−0.279928	+	0.960021i	$0.590311\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	22.4674	0.904502	0.452251	−	0.891891i	$-0.350621\pi$
$617$	22.4674	0.904502	0.452251	+	0.891891i	$0.350621\pi$
$618$	0	0
$619$	−15.7228	−0.631953	−0.315977	−	0.948767i	$-0.602332\pi$
$619$	−15.7228	−0.631953	−0.315977	+	0.948767i	$0.602332\pi$
$620$	0	0
$621$	6.37228	0.255711
$622$	0	0
$623$	18.0951	0.724965
$624$	0	0
$625$	0	0
$626$	0	0
$627$	11.2554	0.449499
$628$	0	0
$629$	−1.62772	−0.0649014
$630$	0	0
$631$	−41.4891	−1.65166	−0.825828	−	0.563922i	$-0.809292\pi$
$631$	−41.4891	−1.65166	−0.825828	+	0.563922i	$0.809292\pi$
$632$	0	0
$633$	−22.9783	−0.913303
$634$	0	0
$635$	0	0
$636$	0	0
$637$	−1.37228	−0.0543718
$638$	0	0
$639$	−5.62772	−0.222629
$640$	0	0
$641$	−12.2337	−0.483202	−0.241601	−	0.970376i	$-0.577672\pi$
$641$	−12.2337	−0.483202	−0.241601	+	0.970376i	$0.577672\pi$
$642$	0	0
$643$	−11.1168	−0.438406	−0.219203	−	0.975679i	$-0.570346\pi$
$643$	−11.1168	−0.438406	−0.219203	+	0.975679i	$0.570346\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−23.8614	−0.938089	−0.469044	−	0.883175i	$-0.655402\pi$
$647$	−23.8614	−0.938089	−0.469044	+	0.883175i	$0.655402\pi$
$648$	0	0
$649$	−18.9783	−0.744961
$650$	0	0
$651$	−11.2554	−0.441135
$652$	0	0
$653$	30.0000	1.17399	0.586995	−	0.809590i	$-0.300311\pi$
$653$	30.0000	1.17399	0.586995	+	0.809590i	$0.300311\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	−7.48913	−0.292179
$658$	0	0
$659$	−37.4891	−1.46037	−0.730184	−	0.683250i	$-0.760566\pi$
$659$	−37.4891	−1.46037	−0.730184	+	0.683250i	$0.760566\pi$
$660$	0	0
$661$	0.233688	0.00908941	0.00454470	−	0.999990i	$-0.498553\pi$
$661$	0.233688	0.00908941	0.00454470	+	0.999990i	$0.498553\pi$
$662$	0	0
$663$	−4.37228	−0.169805
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−12.7446	−0.493471
$668$	0	0
$669$	26.9783	1.04304
$670$	0	0
$671$	−21.6277	−0.834929
$672$	0	0
$673$	31.4891	1.21382	0.606908	−	0.794772i	$-0.292410\pi$
$673$	31.4891	1.21382	0.606908	+	0.794772i	$0.292410\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	11.6277	0.446890	0.223445	−	0.974717i	$-0.428270\pi$
$677$	11.6277	0.446890	0.223445	+	0.974717i	$0.428270\pi$
$678$	0	0
$679$	−19.8614	−0.762211
$680$	0	0
$681$	7.25544	0.278029
$682$	0	0
$683$	−46.9783	−1.79757	−0.898786	−	0.438387i	$-0.855550\pi$
$683$	−46.9783	−1.79757	−0.898786	+	0.438387i	$0.855550\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	2.00000	0.0763048
$688$	0	0
$689$	3.62772	0.138205
$690$	0	0
$691$	23.7228	0.902458	0.451229	−	0.892408i	$-0.350986\pi$
$691$	23.7228	0.902458	0.451229	+	0.892408i	$0.350986\pi$
$692$	0	0
$693$	5.62772	0.213779
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−19.1168	−0.724102
$698$	0	0
$699$	24.3723	0.921844
$700$	0	0
$701$	−5.25544	−0.198495	−0.0992476	−	0.995063i	$-0.531644\pi$
$701$	−5.25544	−0.198495	−0.0992476	+	0.995063i	$0.531644\pi$
$702$	0	0
$703$	1.76631	0.0666177
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−16.0000	−0.601742
$708$	0	0
$709$	−50.7446	−1.90575	−0.952876	−	0.303360i	$-0.901892\pi$
$709$	−50.7446	−1.90575	−0.952876	+	0.303360i	$0.901892\pi$
$710$	0	0
$711$	11.8614	0.444838
$712$	0	0
$713$	−30.2337	−1.13226
$714$	0	0
$715$	0	0
$716$	0	0
$717$	29.3505	1.09612
$718$	0	0
$719$	5.02175	0.187280	0.0936398	−	0.995606i	$-0.470150\pi$
$719$	5.02175	0.187280	0.0936398	+	0.995606i	$0.470150\pi$
$720$	0	0
$721$	9.48913	0.353393
$722$	0	0
$723$	20.9783	0.780190
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−0.744563	−0.0276143	−0.0138072	−	0.999905i	$-0.504395\pi$
$727$	−0.744563	−0.0276143	−0.0138072	+	0.999905i	$0.504395\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−17.4891	−0.646859
$732$	0	0
$733$	−20.0951	−0.742229	−0.371115	−	0.928587i	$-0.621024\pi$
$733$	−20.0951	−0.742229	−0.371115	+	0.928587i	$0.621024\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	9.48913	0.349536
$738$	0	0
$739$	−6.23369	−0.229310	−0.114655	−	0.993405i	$-0.536576\pi$
$739$	−6.23369	−0.229310	−0.114655	+	0.993405i	$0.536576\pi$
$740$	0	0
$741$	4.74456	0.174296
$742$	0	0
$743$	−22.2337	−0.815675	−0.407837	−	0.913055i	$-0.633717\pi$
$743$	−22.2337	−0.815675	−0.407837	+	0.913055i	$0.633717\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	12.0000	0.439057
$748$	0	0
$749$	34.0951	1.24581
$750$	0	0
$751$	−19.8614	−0.724753	−0.362377	−	0.932032i	$-0.618035\pi$
$751$	−19.8614	−0.724753	−0.362377	+	0.932032i	$0.618035\pi$
$752$	0	0
$753$	−5.48913	−0.200035
$754$	0	0
$755$	0	0
$756$	0	0
$757$	50.7446	1.84434	0.922171	−	0.386782i	$-0.126413\pi$
$757$	50.7446	1.84434	0.922171	+	0.386782i	$0.126413\pi$
$758$	0	0
$759$	15.1168	0.548707
$760$	0	0
$761$	−23.4891	−0.851480	−0.425740	−	0.904846i	$-0.639986\pi$
$761$	−23.4891	−0.851480	−0.425740	+	0.904846i	$0.639986\pi$
$762$	0	0
$763$	−6.51087	−0.235709
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−8.00000	−0.288863
$768$	0	0
$769$	−28.2337	−1.01813	−0.509066	−	0.860727i	$-0.670009\pi$
$769$	−28.2337	−1.01813	−0.509066	+	0.860727i	$0.670009\pi$
$770$	0	0
$771$	−10.0000	−0.360141
$772$	0	0
$773$	−5.25544	−0.189025	−0.0945125	−	0.995524i	$-0.530129\pi$
$773$	−5.25544	−0.189025	−0.0945125	+	0.995524i	$0.530129\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	0.883156	0.0316830
$778$	0	0
$779$	20.7446	0.743251
$780$	0	0
$781$	−13.3505	−0.477720
$782$	0	0
$783$	−2.00000	−0.0714742
$784$	0	0
$785$	0	0
$786$	0	0
$787$	−38.9783	−1.38942	−0.694712	−	0.719288i	$-0.744468\pi$
$787$	−38.9783	−1.38942	−0.694712	+	0.719288i	$0.744468\pi$
$788$	0	0
$789$	−5.48913	−0.195418
$790$	0	0
$791$	14.2337	0.506092
$792$	0	0
$793$	−9.11684	−0.323749
$794$	0	0
$795$	0	0
$796$	0	0
$797$	16.3723	0.579936	0.289968	−	0.957036i	$-0.406355\pi$
$797$	16.3723	0.579936	0.289968	+	0.957036i	$0.406355\pi$
$798$	0	0
$799$	−55.7228	−1.97133
$800$	0	0
$801$	7.62772	0.269512
$802$	0	0
$803$	−17.7663	−0.626960
$804$	0	0
$805$	0	0
$806$	0	0
$807$	1.25544	0.0441935
$808$	0	0
$809$	−34.4674	−1.21181	−0.605904	−	0.795538i	$-0.707189\pi$
$809$	−34.4674	−1.21181	−0.605904	+	0.795538i	$0.707189\pi$
$810$	0	0
$811$	6.51087	0.228628	0.114314	−	0.993445i	$-0.463533\pi$
$811$	6.51087	0.228628	0.114314	+	0.993445i	$0.463533\pi$
$812$	0	0
$813$	28.7446	1.00812
$814$	0	0
$815$	0	0
$816$	0	0
$817$	18.9783	0.663965
$818$	0	0
$819$	2.37228	0.0828942
$820$	0	0
$821$	−24.0951	−0.840925	−0.420462	−	0.907310i	$-0.638132\pi$
$821$	−24.0951	−0.840925	−0.420462	+	0.907310i	$0.638132\pi$
$822$	0	0
$823$	44.0000	1.53374	0.766872	−	0.641800i	$-0.221812\pi$
$823$	44.0000	1.53374	0.766872	+	0.641800i	$0.221812\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−2.23369	−0.0776729	−0.0388365	−	0.999246i	$-0.512365\pi$
$827$	−2.23369	−0.0776729	−0.0388365	+	0.999246i	$0.512365\pi$
$828$	0	0
$829$	24.9783	0.867531	0.433765	−	0.901026i	$-0.357185\pi$
$829$	24.9783	0.867531	0.433765	+	0.901026i	$0.357185\pi$
$830$	0	0
$831$	−20.9783	−0.727727
$832$	0	0
$833$	6.00000	0.207888
$834$	0	0
$835$	0	0
$836$	0	0
$837$	−4.74456	−0.163996
$838$	0	0
$839$	−5.62772	−0.194290	−0.0971452	−	0.995270i	$-0.530971\pi$
$839$	−5.62772	−0.194290	−0.0971452	+	0.995270i	$0.530971\pi$
$840$	0	0
$841$	−25.0000	−0.862069
$842$	0	0
$843$	−32.9783	−1.13583
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−12.7446	−0.437908
$848$	0	0
$849$	−20.0000	−0.686398
$850$	0	0
$851$	2.37228	0.0813208
$852$	0	0
$853$	21.1168	0.723027	0.361513	−	0.932367i	$-0.382260\pi$
$853$	21.1168	0.723027	0.361513	+	0.932367i	$0.382260\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−18.6060	−0.635568	−0.317784	−	0.948163i	$-0.602939\pi$
$857$	−18.6060	−0.635568	−0.317784	+	0.948163i	$0.602939\pi$
$858$	0	0
$859$	−12.8832	−0.439568	−0.219784	−	0.975549i	$-0.570535\pi$
$859$	−12.8832	−0.439568	−0.219784	+	0.975549i	$0.570535\pi$
$860$	0	0
$861$	10.3723	0.353486
$862$	0	0
$863$	−35.2554	−1.20011	−0.600055	−	0.799959i	$-0.704854\pi$
$863$	−35.2554	−1.20011	−0.600055	+	0.799959i	$0.704854\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	2.11684	0.0718918
$868$	0	0
$869$	28.1386	0.954536
$870$	0	0
$871$	4.00000	0.135535
$872$	0	0
$873$	−8.37228	−0.283359
$874$	0	0
$875$	0	0
$876$	0	0
$877$	32.9783	1.11360	0.556798	−	0.830648i	$-0.312030\pi$
$877$	32.9783	1.11360	0.556798	+	0.830648i	$0.312030\pi$
$878$	0	0
$879$	26.7446	0.902072
$880$	0	0
$881$	18.0000	0.606435	0.303218	−	0.952921i	$-0.401939\pi$
$881$	18.0000	0.606435	0.303218	+	0.952921i	$0.401939\pi$
$882$	0	0
$883$	−38.9783	−1.31172	−0.655861	−	0.754881i	$-0.727694\pi$
$883$	−38.9783	−1.31172	−0.655861	+	0.754881i	$0.727694\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−1.35053	−0.0453464	−0.0226732	−	0.999743i	$-0.507218\pi$
$887$	−1.35053	−0.0453464	−0.0226732	+	0.999743i	$0.507218\pi$
$888$	0	0
$889$	−43.2554	−1.45074
$890$	0	0
$891$	2.37228	0.0794744
$892$	0	0
$893$	60.4674	2.02346
$894$	0	0
$895$	0	0
$896$	0	0
$897$	6.37228	0.212764
$898$	0	0
$899$	9.48913	0.316480
$900$	0	0
$901$	−15.8614	−0.528420
$902$	0	0
$903$	9.48913	0.315778
$904$	0	0
$905$	0	0
$906$	0	0
$907$	−48.7446	−1.61854	−0.809268	−	0.587439i	$-0.800136\pi$
$907$	−48.7446	−1.61854	−0.809268	+	0.587439i	$0.800136\pi$
$908$	0	0
$909$	−6.74456	−0.223703
$910$	0	0
$911$	−24.0000	−0.795155	−0.397578	−	0.917568i	$-0.630149\pi$
$911$	−24.0000	−0.795155	−0.397578	+	0.917568i	$0.630149\pi$
$912$	0	0
$913$	28.4674	0.942133
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−17.2119	−0.568388
$918$	0	0
$919$	−20.1386	−0.664311	−0.332155	−	0.943225i	$-0.607776\pi$
$919$	−20.1386	−0.664311	−0.332155	+	0.943225i	$0.607776\pi$
$920$	0	0
$921$	−9.62772	−0.317244
$922$	0	0
$923$	−5.62772	−0.185239
$924$	0	0
$925$	0	0
$926$	0	0
$927$	4.00000	0.131377
$928$	0	0
$929$	50.6060	1.66033	0.830164	−	0.557519i	$-0.188247\pi$
$929$	50.6060	1.66033	0.830164	+	0.557519i	$0.188247\pi$
$930$	0	0
$931$	−6.51087	−0.213385
$932$	0	0
$933$	0	0
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−19.4891	−0.636682	−0.318341	−	0.947976i	$-0.603126\pi$
$937$	−19.4891	−0.636682	−0.318341	+	0.947976i	$0.603126\pi$
$938$	0	0
$939$	6.00000	0.195803
$940$	0	0
$941$	31.6277	1.03103	0.515517	−	0.856879i	$-0.327600\pi$
$941$	31.6277	1.03103	0.515517	+	0.856879i	$0.327600\pi$
$942$	0	0
$943$	27.8614	0.907292
$944$	0	0
$945$	0	0
$946$	0	0
$947$	1.02175	0.0332024	0.0166012	−	0.999862i	$-0.494715\pi$
$947$	1.02175	0.0332024	0.0166012	+	0.999862i	$0.494715\pi$
$948$	0	0
$949$	−7.48913	−0.243107
$950$	0	0
$951$	−16.2337	−0.526413
$952$	0	0
$953$	8.64947	0.280184	0.140092	−	0.990139i	$-0.455260\pi$
$953$	8.64947	0.280184	0.140092	+	0.990139i	$0.455260\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	−4.74456	−0.153370
$958$	0	0
$959$	27.2554	0.880124
$960$	0	0
$961$	−8.48913	−0.273843
$962$	0	0
$963$	14.3723	0.463140
$964$	0	0
$965$	0	0
$966$	0	0
$967$	10.9783	0.353037	0.176518	−	0.984297i	$-0.443517\pi$
$967$	10.9783	0.353037	0.176518	+	0.984297i	$0.443517\pi$
$968$	0	0
$969$	−20.7446	−0.666411
$970$	0	0
$971$	29.4891	0.946351	0.473176	−	0.880968i	$-0.343108\pi$
$971$	29.4891	0.946351	0.473176	+	0.880968i	$0.343108\pi$
$972$	0	0
$973$	3.86141	0.123791
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−49.2554	−1.57582	−0.787911	−	0.615790i	$-0.788837\pi$
$977$	−49.2554	−1.57582	−0.787911	+	0.615790i	$0.788837\pi$
$978$	0	0
$979$	18.0951	0.578322
$980$	0	0
$981$	−2.74456	−0.0876271
$982$	0	0
$983$	17.4891	0.557816	0.278908	−	0.960318i	$-0.410027\pi$
$983$	17.4891	0.557816	0.278908	+	0.960318i	$0.410027\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	30.2337	0.962349
$988$	0	0
$989$	25.4891	0.810507
$990$	0	0
$991$	24.6060	0.781634	0.390817	−	0.920468i	$-0.372192\pi$
$991$	24.6060	0.781634	0.390817	+	0.920468i	$0.372192\pi$
$992$	0	0
$993$	1.48913	0.0472560
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−33.7228	−1.06801	−0.534006	−	0.845481i	$-0.679314\pi$
$997$	−33.7228	−1.06801	−0.534006	+	0.845481i	$0.679314\pi$
$998$	0	0
$999$	0.372281	0.0117785

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7800.2.a.bb.1.2		2
5.4	even	2		1560.2.a.n.1.1	✓	2
15.14	odd	2		4680.2.a.bf.1.1		2
20.19	odd	2		3120.2.a.bd.1.2		2
60.59	even	2		9360.2.a.co.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1560.2.a.n.1.1	✓	2	5.4	even	2
3120.2.a.bd.1.2		2	20.19	odd	2
4680.2.a.bf.1.1		2	15.14	odd	2
7800.2.a.bb.1.2		2	1.1	even	1	trivial
9360.2.a.co.1.2		2	60.59	even	2

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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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} +2.37228 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} +2.37228 q^{7} +1.00000 q^{9} +2.37228 q^{11} +1.00000 q^{13} -4.37228 q^{17} +4.74456 q^{19} +2.37228 q^{21} +6.37228 q^{23} +1.00000 q^{27} -2.00000 q^{29} -4.74456 q^{31} +2.37228 q^{33} +0.372281 q^{37} +1.00000 q^{39} +4.37228 q^{41} +4.00000 q^{43} +12.7446 q^{47} -1.37228 q^{49} -4.37228 q^{51} +3.62772 q^{53} +4.74456 q^{57} -8.00000 q^{59} -9.11684 q^{61} +2.37228 q^{63} +4.00000 q^{67} +6.37228 q^{69} -5.62772 q^{71} -7.48913 q^{73} +5.62772 q^{77} +11.8614 q^{79} +1.00000 q^{81} +12.0000 q^{83} -2.00000 q^{87} +7.62772 q^{89} +2.37228 q^{91} -4.74456 q^{93} -8.37228 q^{97} +2.37228 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{3} - q^{7} + 2 q^{9}+O(q^{10}) \) 2 * q + 2 * q^3 - q^7 + 2 * q^9	\( 2 q + 2 q^{3} - q^{7} + 2 q^{9} - q^{11} + 2 q^{13} - 3 q^{17} - 2 q^{19} - q^{21} + 7 q^{23} + 2 q^{27} - 4 q^{29} + 2 q^{31} - q^{33} - 5 q^{37} + 2 q^{39} + 3 q^{41} + 8 q^{43} + 14 q^{47} + 3 q^{49} - 3 q^{51} + 13 q^{53} - 2 q^{57} - 16 q^{59} - q^{61} - q^{63} + 8 q^{67} + 7 q^{69} - 17 q^{71} + 8 q^{73} + 17 q^{77} - 5 q^{79} + 2 q^{81} + 24 q^{83} - 4 q^{87} + 21 q^{89} - q^{91} + 2 q^{93} - 11 q^{97} - q^{99}+O(q^{100}) \) 2 * q + 2 * q^3 - q^7 + 2 * q^9 - q^11 + 2 * q^13 - 3 * q^17 - 2 * q^19 - q^21 + 7 * q^23 + 2 * q^27 - 4 * q^29 + 2 * q^31 - q^33 - 5 * q^37 + 2 * q^39 + 3 * q^41 + 8 * q^43 + 14 * q^47 + 3 * q^49 - 3 * q^51 + 13 * q^53 - 2 * q^57 - 16 * q^59 - q^61 - q^63 + 8 * q^67 + 7 * q^69 - 17 * q^71 + 8 * q^73 + 17 * q^77 - 5 * q^79 + 2 * q^81 + 24 * q^83 - 4 * q^87 + 21 * q^89 - q^91 + 2 * q^93 - 11 * q^97 - q^99

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