LMFDB - Embedded newform 2800.2.a.bn.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2800,2,Mod(1,2800)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2800, 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("2800.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2800 = 2^{4} \cdot 5^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2800.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:	$22.3581125660$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{17}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 4 $ x^2 - x - 4
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 280)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.56155$ of defining polynomial
Character	$\chi$	$=$	2800.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.56155 q^{3} +1.00000 q^{7} -0.561553 q^{9} +O(q^{10})$	$q-1.56155 q^{3} +1.00000 q^{7} -0.561553 q^{9} -1.56155 q^{11} -6.68466 q^{13} -7.56155 q^{17} +7.12311 q^{19} -1.56155 q^{21} +3.12311 q^{23} +5.56155 q^{27} +0.438447 q^{29} -6.24621 q^{31} +2.43845 q^{33} +8.24621 q^{37} +10.4384 q^{39} -1.12311 q^{41} -7.12311 q^{43} +2.43845 q^{47} +1.00000 q^{49} +11.8078 q^{51} +13.1231 q^{53} -11.1231 q^{57} +4.00000 q^{59} -6.87689 q^{61} -0.561553 q^{63} +2.24621 q^{67} -4.87689 q^{69} +4.24621 q^{73} -1.56155 q^{77} -0.684658 q^{79} -7.00000 q^{81} +12.0000 q^{83} -0.684658 q^{87} +5.12311 q^{89} -6.68466 q^{91} +9.75379 q^{93} -1.31534 q^{97} +0.876894 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + q^{3} + 2 q^{7} + 3 q^{9}+O(q^{10}) $ 2 * q + q^3 + 2 * q^7 + 3 * q^9	$ 2 q + q^{3} + 2 q^{7} + 3 q^{9} + q^{11} - q^{13} - 11 q^{17} + 6 q^{19} + q^{21} - 2 q^{23} + 7 q^{27} + 5 q^{29} + 4 q^{31} + 9 q^{33} + 25 q^{39} + 6 q^{41} - 6 q^{43} + 9 q^{47} + 2 q^{49} + 3 q^{51} + 18 q^{53} - 14 q^{57} + 8 q^{59} - 22 q^{61} + 3 q^{63} - 12 q^{67} - 18 q^{69} - 8 q^{73} + q^{77} + 11 q^{79} - 14 q^{81} + 24 q^{83} + 11 q^{87} + 2 q^{89} - q^{91} + 36 q^{93} - 15 q^{97} + 10 q^{99}+O(q^{100}) $ 2 * q + q^3 + 2 * q^7 + 3 * q^9 + q^11 - q^13 - 11 * q^17 + 6 * q^19 + q^21 - 2 * q^23 + 7 * q^27 + 5 * q^29 + 4 * q^31 + 9 * q^33 + 25 * q^39 + 6 * q^41 - 6 * q^43 + 9 * q^47 + 2 * q^49 + 3 * q^51 + 18 * q^53 - 14 * q^57 + 8 * q^59 - 22 * q^61 + 3 * q^63 - 12 * q^67 - 18 * q^69 - 8 * q^73 + q^77 + 11 * q^79 - 14 * q^81 + 24 * q^83 + 11 * q^87 + 2 * q^89 - q^91 + 36 * q^93 - 15 * q^97 + 10 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	−1.56155	−0.901563	−0.450781	−	0.892634i	$-0.648855\pi$
$3$	−1.56155	−0.901563	−0.450781	+	0.892634i	$0.648855\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	−0.561553	−0.187184
$10$	0	0
$11$	−1.56155	−0.470826	−0.235413	−	0.971895i	$-0.575644\pi$
$11$	−1.56155	−0.470826	−0.235413	+	0.971895i	$0.575644\pi$
$12$	0	0
$13$	−6.68466	−1.85399	−0.926995	−	0.375073i	$-0.877618\pi$
$13$	−6.68466	−1.85399	−0.926995	+	0.375073i	$0.877618\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−7.56155	−1.83395	−0.916973	−	0.398949i	$-0.869375\pi$
$17$	−7.56155	−1.83395	−0.916973	+	0.398949i	$0.869375\pi$
$18$	0	0
$19$	7.12311	1.63415	0.817076	−	0.576530i	$-0.195593\pi$
$19$	7.12311	1.63415	0.817076	+	0.576530i	$0.195593\pi$
$20$	0	0
$21$	−1.56155	−0.340759
$22$	0	0
$23$	3.12311	0.651213	0.325606	−	0.945505i	$-0.394432\pi$
$23$	3.12311	0.651213	0.325606	+	0.945505i	$0.394432\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	5.56155	1.07032
$28$	0	0
$29$	0.438447	0.0814176	0.0407088	−	0.999171i	$-0.487038\pi$
$29$	0.438447	0.0814176	0.0407088	+	0.999171i	$0.487038\pi$
$30$	0	0
$31$	−6.24621	−1.12185	−0.560926	−	0.827866i	$-0.689555\pi$
$31$	−6.24621	−1.12185	−0.560926	+	0.827866i	$0.689555\pi$
$32$	0	0
$33$	2.43845	0.424479
$34$	0	0
$35$	0	0
$36$	0	0
$37$	8.24621	1.35567	0.677834	−	0.735215i	$-0.262919\pi$
$37$	8.24621	1.35567	0.677834	+	0.735215i	$0.262919\pi$
$38$	0	0
$39$	10.4384	1.67149
$40$	0	0
$41$	−1.12311	−0.175400	−0.0876998	−	0.996147i	$-0.527952\pi$
$41$	−1.12311	−0.175400	−0.0876998	+	0.996147i	$0.527952\pi$
$42$	0	0
$43$	−7.12311	−1.08626	−0.543132	−	0.839648i	$-0.682762\pi$
$43$	−7.12311	−1.08626	−0.543132	+	0.839648i	$0.682762\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	2.43845	0.355684	0.177842	−	0.984059i	$-0.443088\pi$
$47$	2.43845	0.355684	0.177842	+	0.984059i	$0.443088\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	11.8078	1.65342
$52$	0	0
$53$	13.1231	1.80260	0.901299	−	0.433198i	$-0.142615\pi$
$53$	13.1231	1.80260	0.901299	+	0.433198i	$0.142615\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−11.1231	−1.47329
$58$	0	0
$59$	4.00000	0.520756	0.260378	−	0.965507i	$-0.416153\pi$
$59$	4.00000	0.520756	0.260378	+	0.965507i	$0.416153\pi$
$60$	0	0
$61$	−6.87689	−0.880496	−0.440248	−	0.897876i	$-0.645109\pi$
$61$	−6.87689	−0.880496	−0.440248	+	0.897876i	$0.645109\pi$
$62$	0	0
$63$	−0.561553	−0.0707490
$64$	0	0
$65$	0	0
$66$	0	0
$67$	2.24621	0.274418	0.137209	−	0.990542i	$-0.456187\pi$
$67$	2.24621	0.274418	0.137209	+	0.990542i	$0.456187\pi$
$68$	0	0
$69$	−4.87689	−0.587109
$70$	0	0
$71$	0	0		−	1.00000i	$-0.5\pi$
$71$	0	0			1.00000i	$0.5\pi$
$72$	0	0
$73$	4.24621	0.496981	0.248491	−	0.968634i	$-0.420065\pi$
$73$	4.24621	0.496981	0.248491	+	0.968634i	$0.420065\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−1.56155	−0.177955
$78$	0	0
$79$	−0.684658	−0.0770301	−0.0385150	−	0.999258i	$-0.512263\pi$
$79$	−0.684658	−0.0770301	−0.0385150	+	0.999258i	$0.512263\pi$
$80$	0	0
$81$	−7.00000	−0.777778
$82$	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
$87$	−0.684658	−0.0734031
$88$	0	0
$89$	5.12311	0.543048	0.271524	−	0.962432i	$-0.412472\pi$
$89$	5.12311	0.543048	0.271524	+	0.962432i	$0.412472\pi$
$90$	0	0
$91$	−6.68466	−0.700743
$92$	0	0
$93$	9.75379	1.01142
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−1.31534	−0.133553	−0.0667764	−	0.997768i	$-0.521271\pi$
$97$	−1.31534	−0.133553	−0.0667764	+	0.997768i	$0.521271\pi$
$98$	0	0
$99$	0.876894	0.0881312
$100$	0	0
$101$	6.00000	0.597022	0.298511	−	0.954406i	$-0.403510\pi$
$101$	6.00000	0.597022	0.298511	+	0.954406i	$0.403510\pi$
$102$	0	0
$103$	−11.8078	−1.16345	−0.581727	−	0.813384i	$-0.697623\pi$
$103$	−11.8078	−1.16345	−0.581727	+	0.813384i	$0.697623\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	15.1231	1.46201	0.731003	−	0.682374i	$-0.239053\pi$
$107$	15.1231	1.46201	0.731003	+	0.682374i	$0.239053\pi$
$108$	0	0
$109$	−4.43845	−0.425126	−0.212563	−	0.977147i	$-0.568181\pi$
$109$	−4.43845	−0.425126	−0.212563	+	0.977147i	$0.568181\pi$
$110$	0	0
$111$	−12.8769	−1.22222
$112$	0	0
$113$	−8.24621	−0.775738	−0.387869	−	0.921714i	$-0.626789\pi$
$113$	−8.24621	−0.775738	−0.387869	+	0.921714i	$0.626789\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	3.75379	0.347038
$118$	0	0
$119$	−7.56155	−0.693166
$120$	0	0
$121$	−8.56155	−0.778323
$122$	0	0
$123$	1.75379	0.158134
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−6.24621	−0.554262	−0.277131	−	0.960832i	$-0.589384\pi$
$127$	−6.24621	−0.554262	−0.277131	+	0.960832i	$0.589384\pi$
$128$	0	0
$129$	11.1231	0.979335
$130$	0	0
$131$	−15.1231	−1.32131	−0.660656	−	0.750689i	$-0.729722\pi$
$131$	−15.1231	−1.32131	−0.660656	+	0.750689i	$0.729722\pi$
$132$	0	0
$133$	7.12311	0.617652
$134$	0	0
$135$	0	0
$136$	0	0
$137$	7.36932	0.629603	0.314802	−	0.949157i	$-0.398062\pi$
$137$	7.36932	0.629603	0.314802	+	0.949157i	$0.398062\pi$
$138$	0	0
$139$	21.3693	1.81252	0.906261	−	0.422719i	$-0.138924\pi$
$139$	21.3693	1.81252	0.906261	+	0.422719i	$0.138924\pi$
$140$	0	0
$141$	−3.80776	−0.320672
$142$	0	0
$143$	10.4384	0.872907
$144$	0	0
$145$	0	0
$146$	0	0
$147$	−1.56155	−0.128795
$148$	0	0
$149$	−0.246211	−0.0201704	−0.0100852	−	0.999949i	$-0.503210\pi$
$149$	−0.246211	−0.0201704	−0.0100852	+	0.999949i	$0.503210\pi$
$150$	0	0
$151$	19.8078	1.61193	0.805966	−	0.591961i	$-0.201646\pi$
$151$	19.8078	1.61193	0.805966	+	0.591961i	$0.201646\pi$
$152$	0	0
$153$	4.24621	0.343286
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−4.24621	−0.338885	−0.169442	−	0.985540i	$-0.554197\pi$
$157$	−4.24621	−0.338885	−0.169442	+	0.985540i	$0.554197\pi$
$158$	0	0
$159$	−20.4924	−1.62515
$160$	0	0
$161$	3.12311	0.246135
$162$	0	0
$163$	19.6155	1.53641	0.768203	−	0.640206i	$-0.221151\pi$
$163$	19.6155	1.53641	0.768203	+	0.640206i	$0.221151\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	4.19224	0.324405	0.162202	−	0.986757i	$-0.448140\pi$
$167$	4.19224	0.324405	0.162202	+	0.986757i	$0.448140\pi$
$168$	0	0
$169$	31.6847	2.43728
$170$	0	0
$171$	−4.00000	−0.305888
$172$	0	0
$173$	23.1771	1.76212	0.881060	−	0.473004i	$-0.156830\pi$
$173$	23.1771	1.76212	0.881060	+	0.473004i	$0.156830\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−6.24621	−0.469494
$178$	0	0
$179$	4.00000	0.298974	0.149487	−	0.988764i	$-0.452238\pi$
$179$	4.00000	0.298974	0.149487	+	0.988764i	$0.452238\pi$
$180$	0	0
$181$	−5.12311	−0.380797	−0.190399	−	0.981707i	$-0.560978\pi$
$181$	−5.12311	−0.380797	−0.190399	+	0.981707i	$0.560978\pi$
$182$	0	0
$183$	10.7386	0.793823
$184$	0	0
$185$	0	0
$186$	0	0
$187$	11.8078	0.863469
$188$	0	0
$189$	5.56155	0.404543
$190$	0	0
$191$	0.684658	0.0495401	0.0247701	−	0.999693i	$-0.492115\pi$
$191$	0.684658	0.0495401	0.0247701	+	0.999693i	$0.492115\pi$
$192$	0	0
$193$	−13.1231	−0.944622	−0.472311	−	0.881432i	$-0.656580\pi$
$193$	−13.1231	−0.944622	−0.472311	+	0.881432i	$0.656580\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	13.1231	0.934983	0.467491	−	0.883998i	$-0.345158\pi$
$197$	13.1231	0.934983	0.467491	+	0.883998i	$0.345158\pi$
$198$	0	0
$199$	14.2462	1.00989	0.504944	−	0.863152i	$-0.331513\pi$
$199$	14.2462	1.00989	0.504944	+	0.863152i	$0.331513\pi$
$200$	0	0
$201$	−3.50758	−0.247405
$202$	0	0
$203$	0.438447	0.0307730
$204$	0	0
$205$	0	0
$206$	0	0
$207$	−1.75379	−0.121897
$208$	0	0
$209$	−11.1231	−0.769401
$210$	0	0
$211$	17.5616	1.20899	0.604494	−	0.796610i	$-0.293376\pi$
$211$	17.5616	1.20899	0.604494	+	0.796610i	$0.293376\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−6.24621	−0.424020
$218$	0	0
$219$	−6.63068	−0.448060
$220$	0	0
$221$	50.5464	3.40012
$222$	0	0
$223$	24.6847	1.65301	0.826503	−	0.562932i	$-0.190327\pi$
$223$	24.6847	1.65301	0.826503	+	0.562932i	$0.190327\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−11.3153	−0.751026	−0.375513	−	0.926817i	$-0.622533\pi$
$227$	−11.3153	−0.751026	−0.375513	+	0.926817i	$0.622533\pi$
$228$	0	0
$229$	−11.3693	−0.751306	−0.375653	−	0.926760i	$-0.622581\pi$
$229$	−11.3693	−0.751306	−0.375653	+	0.926760i	$0.622581\pi$
$230$	0	0
$231$	2.43845	0.160438
$232$	0	0
$233$	10.8769	0.712569	0.356285	−	0.934378i	$-0.384043\pi$
$233$	10.8769	0.712569	0.356285	+	0.934378i	$0.384043\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	1.06913	0.0694475
$238$	0	0
$239$	18.0540	1.16781	0.583907	−	0.811820i	$-0.301523\pi$
$239$	18.0540	1.16781	0.583907	+	0.811820i	$0.301523\pi$
$240$	0	0
$241$	2.00000	0.128831	0.0644157	−	0.997923i	$-0.479482\pi$
$241$	2.00000	0.128831	0.0644157	+	0.997923i	$0.479482\pi$
$242$	0	0
$243$	−5.75379	−0.369106
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−47.6155	−3.02970
$248$	0	0
$249$	−18.7386	−1.18751
$250$	0	0
$251$	−13.3693	−0.843864	−0.421932	−	0.906628i	$-0.638648\pi$
$251$	−13.3693	−0.843864	−0.421932	+	0.906628i	$0.638648\pi$
$252$	0	0
$253$	−4.87689	−0.306608
$254$	0	0
$255$	0	0
$256$	0	0
$257$	18.4924	1.15353	0.576763	−	0.816912i	$-0.304316\pi$
$257$	18.4924	1.15353	0.576763	+	0.816912i	$0.304316\pi$
$258$	0	0
$259$	8.24621	0.512395
$260$	0	0
$261$	−0.246211	−0.0152401
$262$	0	0
$263$	9.36932	0.577737	0.288868	−	0.957369i	$-0.406721\pi$
$263$	9.36932	0.577737	0.288868	+	0.957369i	$0.406721\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	−8.00000	−0.489592
$268$	0	0
$269$	4.24621	0.258896	0.129448	−	0.991586i	$-0.458679\pi$
$269$	4.24621	0.258896	0.129448	+	0.991586i	$0.458679\pi$
$270$	0	0
$271$	6.24621	0.379430	0.189715	−	0.981839i	$-0.439244\pi$
$271$	6.24621	0.379430	0.189715	+	0.981839i	$0.439244\pi$
$272$	0	0
$273$	10.4384	0.631764
$274$	0	0
$275$	0	0
$276$	0	0
$277$	8.24621	0.495467	0.247733	−	0.968828i	$-0.420314\pi$
$277$	8.24621	0.495467	0.247733	+	0.968828i	$0.420314\pi$
$278$	0	0
$279$	3.50758	0.209993
$280$	0	0
$281$	−19.5616	−1.16694	−0.583472	−	0.812133i	$-0.698306\pi$
$281$	−19.5616	−1.16694	−0.583472	+	0.812133i	$0.698306\pi$
$282$	0	0
$283$	−4.68466	−0.278474	−0.139237	−	0.990259i	$-0.544465\pi$
$283$	−4.68466	−0.278474	−0.139237	+	0.990259i	$0.544465\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−1.12311	−0.0662948
$288$	0	0
$289$	40.1771	2.36336
$290$	0	0
$291$	2.05398	0.120406
$292$	0	0
$293$	−32.0540	−1.87261	−0.936307	−	0.351184i	$-0.885779\pi$
$293$	−32.0540	−1.87261	−0.936307	+	0.351184i	$0.885779\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	−8.68466	−0.503935
$298$	0	0
$299$	−20.8769	−1.20734
$300$	0	0
$301$	−7.12311	−0.410569
$302$	0	0
$303$	−9.36932	−0.538253
$304$	0	0
$305$	0	0
$306$	0	0
$307$	−28.6847	−1.63712	−0.818560	−	0.574421i	$-0.805227\pi$
$307$	−28.6847	−1.63712	−0.818560	+	0.574421i	$0.805227\pi$
$308$	0	0
$309$	18.4384	1.04893
$310$	0	0
$311$	12.8769	0.730182	0.365091	−	0.930972i	$-0.381038\pi$
$311$	12.8769	0.730182	0.365091	+	0.930972i	$0.381038\pi$
$312$	0	0
$313$	−26.6847	−1.50831	−0.754153	−	0.656699i	$-0.771952\pi$
$313$	−26.6847	−1.50831	−0.754153	+	0.656699i	$0.771952\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	10.0000	0.561656	0.280828	−	0.959758i	$-0.409391\pi$
$317$	10.0000	0.561656	0.280828	+	0.959758i	$0.409391\pi$
$318$	0	0
$319$	−0.684658	−0.0383335
$320$	0	0
$321$	−23.6155	−1.31809
$322$	0	0
$323$	−53.8617	−2.99695
$324$	0	0
$325$	0	0
$326$	0	0
$327$	6.93087	0.383278
$328$	0	0
$329$	2.43845	0.134436
$330$	0	0
$331$	12.0000	0.659580	0.329790	−	0.944054i	$-0.393022\pi$
$331$	12.0000	0.659580	0.329790	+	0.944054i	$0.393022\pi$
$332$	0	0
$333$	−4.63068	−0.253760
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−2.00000	−0.108947	−0.0544735	−	0.998515i	$-0.517348\pi$
$337$	−2.00000	−0.108947	−0.0544735	+	0.998515i	$0.517348\pi$
$338$	0	0
$339$	12.8769	0.699377
$340$	0	0
$341$	9.75379	0.528197
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	0	0
$346$	0	0
$347$	15.1231	0.811851	0.405925	−	0.913906i	$-0.366949\pi$
$347$	15.1231	0.811851	0.405925	+	0.913906i	$0.366949\pi$
$348$	0	0
$349$	−11.7538	−0.629166	−0.314583	−	0.949230i	$-0.601865\pi$
$349$	−11.7538	−0.629166	−0.314583	+	0.949230i	$0.601865\pi$
$350$	0	0
$351$	−37.1771	−1.98437
$352$	0	0
$353$	2.19224	0.116681	0.0583405	−	0.998297i	$-0.481419\pi$
$353$	2.19224	0.116681	0.0583405	+	0.998297i	$0.481419\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	11.8078	0.624933
$358$	0	0
$359$	−32.0000	−1.68890	−0.844448	−	0.535638i	$-0.820071\pi$
$359$	−32.0000	−1.68890	−0.844448	+	0.535638i	$0.820071\pi$
$360$	0	0
$361$	31.7386	1.67045
$362$	0	0
$363$	13.3693	0.701707
$364$	0	0
$365$	0	0
$366$	0	0
$367$	14.9309	0.779385	0.389693	−	0.920945i	$-0.372581\pi$
$367$	14.9309	0.779385	0.389693	+	0.920945i	$0.372581\pi$
$368$	0	0
$369$	0.630683	0.0328321
$370$	0	0
$371$	13.1231	0.681318
$372$	0	0
$373$	−15.3693	−0.795793	−0.397897	−	0.917430i	$-0.630260\pi$
$373$	−15.3693	−0.795793	−0.397897	+	0.917430i	$0.630260\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−2.93087	−0.150947
$378$	0	0
$379$	−32.4924	−1.66902	−0.834512	−	0.550990i	$-0.814250\pi$
$379$	−32.4924	−1.66902	−0.834512	+	0.550990i	$0.814250\pi$
$380$	0	0
$381$	9.75379	0.499702
$382$	0	0
$383$	−9.75379	−0.498395	−0.249198	−	0.968453i	$-0.580167\pi$
$383$	−9.75379	−0.498395	−0.249198	+	0.968453i	$0.580167\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	4.00000	0.203331
$388$	0	0
$389$	22.3002	1.13066	0.565332	−	0.824863i	$-0.308748\pi$
$389$	22.3002	1.13066	0.565332	+	0.824863i	$0.308748\pi$
$390$	0	0
$391$	−23.6155	−1.19429
$392$	0	0
$393$	23.6155	1.19125
$394$	0	0
$395$	0	0
$396$	0	0
$397$	23.1771	1.16322	0.581612	−	0.813466i	$-0.302422\pi$
$397$	23.1771	1.16322	0.581612	+	0.813466i	$0.302422\pi$
$398$	0	0
$399$	−11.1231	−0.556852
$400$	0	0
$401$	−12.9309	−0.645737	−0.322868	−	0.946444i	$-0.604647\pi$
$401$	−12.9309	−0.645737	−0.322868	+	0.946444i	$0.604647\pi$
$402$	0	0
$403$	41.7538	2.07990
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−12.8769	−0.638284
$408$	0	0
$409$	−18.4924	−0.914391	−0.457196	−	0.889366i	$-0.651146\pi$
$409$	−18.4924	−0.914391	−0.457196	+	0.889366i	$0.651146\pi$
$410$	0	0
$411$	−11.5076	−0.567627
$412$	0	0
$413$	4.00000	0.196827
$414$	0	0
$415$	0	0
$416$	0	0
$417$	−33.3693	−1.63410
$418$	0	0
$419$	18.2462	0.891386	0.445693	−	0.895186i	$-0.352957\pi$
$419$	18.2462	0.891386	0.445693	+	0.895186i	$0.352957\pi$
$420$	0	0
$421$	3.56155	0.173579	0.0867897	−	0.996227i	$-0.472339\pi$
$421$	3.56155	0.173579	0.0867897	+	0.996227i	$0.472339\pi$
$422$	0	0
$423$	−1.36932	−0.0665785
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−6.87689	−0.332796
$428$	0	0
$429$	−16.3002	−0.786980
$430$	0	0
$431$	22.9309	1.10454	0.552271	−	0.833665i	$-0.313762\pi$
$431$	22.9309	1.10454	0.552271	+	0.833665i	$0.313762\pi$
$432$	0	0
$433$	−19.7538	−0.949307	−0.474653	−	0.880173i	$-0.657427\pi$
$433$	−19.7538	−0.949307	−0.474653	+	0.880173i	$0.657427\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	22.2462	1.06418
$438$	0	0
$439$	−19.1231	−0.912696	−0.456348	−	0.889801i	$-0.650843\pi$
$439$	−19.1231	−0.912696	−0.456348	+	0.889801i	$0.650843\pi$
$440$	0	0
$441$	−0.561553	−0.0267406
$442$	0	0
$443$	−19.6155	−0.931962	−0.465981	−	0.884795i	$-0.654298\pi$
$443$	−19.6155	−0.931962	−0.465981	+	0.884795i	$0.654298\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	0.384472	0.0181849
$448$	0	0
$449$	−21.3153	−1.00593	−0.502967	−	0.864306i	$-0.667758\pi$
$449$	−21.3153	−1.00593	−0.502967	+	0.864306i	$0.667758\pi$
$450$	0	0
$451$	1.75379	0.0825827
$452$	0	0
$453$	−30.9309	−1.45326
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−8.63068	−0.403726	−0.201863	−	0.979414i	$-0.564700\pi$
$457$	−8.63068	−0.403726	−0.201863	+	0.979414i	$0.564700\pi$
$458$	0	0
$459$	−42.0540	−1.96291
$460$	0	0
$461$	18.8769	0.879185	0.439592	−	0.898197i	$-0.355123\pi$
$461$	18.8769	0.879185	0.439592	+	0.898197i	$0.355123\pi$
$462$	0	0
$463$	−6.24621	−0.290286	−0.145143	−	0.989411i	$-0.546364\pi$
$463$	−6.24621	−0.290286	−0.145143	+	0.989411i	$0.546364\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	25.5616	1.18285	0.591424	−	0.806361i	$-0.298566\pi$
$467$	25.5616	1.18285	0.591424	+	0.806361i	$0.298566\pi$
$468$	0	0
$469$	2.24621	0.103720
$470$	0	0
$471$	6.63068	0.305526
$472$	0	0
$473$	11.1231	0.511441
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−7.36932	−0.337418
$478$	0	0
$479$	−17.3693	−0.793624	−0.396812	−	0.917900i	$-0.629884\pi$
$479$	−17.3693	−0.793624	−0.396812	+	0.917900i	$0.629884\pi$
$480$	0	0
$481$	−55.1231	−2.51340
$482$	0	0
$483$	−4.87689	−0.221906
$484$	0	0
$485$	0	0
$486$	0	0
$487$	3.12311	0.141521	0.0707607	−	0.997493i	$-0.477457\pi$
$487$	3.12311	0.141521	0.0707607	+	0.997493i	$0.477457\pi$
$488$	0	0
$489$	−30.6307	−1.38517
$490$	0	0
$491$	3.31534	0.149619	0.0748096	−	0.997198i	$-0.476165\pi$
$491$	3.31534	0.149619	0.0748096	+	0.997198i	$0.476165\pi$
$492$	0	0
$493$	−3.31534	−0.149315
$494$	0	0
$495$	0	0
$496$	0	0
$497$	0	0
$498$	0	0
$499$	0.192236	0.00860566	0.00430283	−	0.999991i	$-0.498630\pi$
$499$	0.192236	0.00860566	0.00430283	+	0.999991i	$0.498630\pi$
$500$	0	0
$501$	−6.54640	−0.292471
$502$	0	0
$503$	−29.1771	−1.30094	−0.650471	−	0.759531i	$-0.725428\pi$
$503$	−29.1771	−1.30094	−0.650471	+	0.759531i	$0.725428\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	−49.4773	−2.19736
$508$	0	0
$509$	32.7386	1.45111	0.725557	−	0.688162i	$-0.241582\pi$
$509$	32.7386	1.45111	0.725557	+	0.688162i	$0.241582\pi$
$510$	0	0
$511$	4.24621	0.187841
$512$	0	0
$513$	39.6155	1.74907
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−3.80776	−0.167465
$518$	0	0
$519$	−36.1922	−1.58866
$520$	0	0
$521$	−12.2462	−0.536516	−0.268258	−	0.963347i	$-0.586448\pi$
$521$	−12.2462	−0.536516	−0.268258	+	0.963347i	$0.586448\pi$
$522$	0	0
$523$	−12.0000	−0.524723	−0.262362	−	0.964970i	$-0.584501\pi$
$523$	−12.0000	−0.524723	−0.262362	+	0.964970i	$0.584501\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	47.2311	2.05742
$528$	0	0
$529$	−13.2462	−0.575922
$530$	0	0
$531$	−2.24621	−0.0974773
$532$	0	0
$533$	7.50758	0.325189
$534$	0	0
$535$	0	0
$536$	0	0
$537$	−6.24621	−0.269544
$538$	0	0
$539$	−1.56155	−0.0672608
$540$	0	0
$541$	41.4233	1.78093	0.890463	−	0.455055i	$-0.150380\pi$
$541$	41.4233	1.78093	0.890463	+	0.455055i	$0.150380\pi$
$542$	0	0
$543$	8.00000	0.343313
$544$	0	0
$545$	0	0
$546$	0	0
$547$	12.0000	0.513083	0.256541	−	0.966533i	$-0.417417\pi$
$547$	12.0000	0.513083	0.256541	+	0.966533i	$0.417417\pi$
$548$	0	0
$549$	3.86174	0.164815
$550$	0	0
$551$	3.12311	0.133049
$552$	0	0
$553$	−0.684658	−0.0291146
$554$	0	0
$555$	0	0
$556$	0	0
$557$	17.6155	0.746394	0.373197	−	0.927752i	$-0.378262\pi$
$557$	17.6155	0.746394	0.373197	+	0.927752i	$0.378262\pi$
$558$	0	0
$559$	47.6155	2.01392
$560$	0	0
$561$	−18.4384	−0.778472
$562$	0	0
$563$	−7.50758	−0.316407	−0.158203	−	0.987407i	$-0.550570\pi$
$563$	−7.50758	−0.316407	−0.158203	+	0.987407i	$0.550570\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	−7.00000	−0.293972
$568$	0	0
$569$	18.9848	0.795886	0.397943	−	0.917410i	$-0.369724\pi$
$569$	18.9848	0.795886	0.397943	+	0.917410i	$0.369724\pi$
$570$	0	0
$571$	−20.0000	−0.836974	−0.418487	−	0.908223i	$-0.637439\pi$
$571$	−20.0000	−0.836974	−0.418487	+	0.908223i	$0.637439\pi$
$572$	0	0
$573$	−1.06913	−0.0446636
$574$	0	0
$575$	0	0
$576$	0	0
$577$	3.56155	0.148269	0.0741347	−	0.997248i	$-0.476381\pi$
$577$	3.56155	0.148269	0.0741347	+	0.997248i	$0.476381\pi$
$578$	0	0
$579$	20.4924	0.851636
$580$	0	0
$581$	12.0000	0.497844
$582$	0	0
$583$	−20.4924	−0.848709
$584$	0	0
$585$	0	0
$586$	0	0
$587$	10.2462	0.422906	0.211453	−	0.977388i	$-0.432180\pi$
$587$	10.2462	0.422906	0.211453	+	0.977388i	$0.432180\pi$
$588$	0	0
$589$	−44.4924	−1.83328
$590$	0	0
$591$	−20.4924	−0.842946
$592$	0	0
$593$	−37.4233	−1.53679	−0.768395	−	0.639976i	$-0.778944\pi$
$593$	−37.4233	−1.53679	−0.768395	+	0.639976i	$0.778944\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−22.2462	−0.910477
$598$	0	0
$599$	46.9309	1.91754	0.958772	−	0.284178i	$-0.0917205\pi$
$599$	46.9309	1.91754	0.958772	+	0.284178i	$0.0917205\pi$
$600$	0	0
$601$	−22.0000	−0.897399	−0.448699	−	0.893683i	$-0.648113\pi$
$601$	−22.0000	−0.897399	−0.448699	+	0.893683i	$0.648113\pi$
$602$	0	0
$603$	−1.26137	−0.0513668
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−8.68466	−0.352499	−0.176250	−	0.984345i	$-0.556397\pi$
$607$	−8.68466	−0.352499	−0.176250	+	0.984345i	$0.556397\pi$
$608$	0	0
$609$	−0.684658	−0.0277438
$610$	0	0
$611$	−16.3002	−0.659435
$612$	0	0
$613$	−16.7386	−0.676067	−0.338034	−	0.941134i	$-0.609762\pi$
$613$	−16.7386	−0.676067	−0.338034	+	0.941134i	$0.609762\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	15.7538	0.634224	0.317112	−	0.948388i	$-0.397287\pi$
$617$	15.7538	0.634224	0.317112	+	0.948388i	$0.397287\pi$
$618$	0	0
$619$	−10.6307	−0.427283	−0.213642	−	0.976912i	$-0.568532\pi$
$619$	−10.6307	−0.427283	−0.213642	+	0.976912i	$0.568532\pi$
$620$	0	0
$621$	17.3693	0.697007
$622$	0	0
$623$	5.12311	0.205253
$624$	0	0
$625$	0	0
$626$	0	0
$627$	17.3693	0.693664
$628$	0	0
$629$	−62.3542	−2.48622
$630$	0	0
$631$	27.4233	1.09170	0.545852	−	0.837882i	$-0.316206\pi$
$631$	27.4233	1.09170	0.545852	+	0.837882i	$0.316206\pi$
$632$	0	0
$633$	−27.4233	−1.08998
$634$	0	0
$635$	0	0
$636$	0	0
$637$	−6.68466	−0.264856
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−22.9848	−0.907847	−0.453923	−	0.891041i	$-0.649976\pi$
$641$	−22.9848	−0.907847	−0.453923	+	0.891041i	$0.649976\pi$
$642$	0	0
$643$	−30.0540	−1.18521	−0.592607	−	0.805492i	$-0.701901\pi$
$643$	−30.0540	−1.18521	−0.592607	+	0.805492i	$0.701901\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	8.00000	0.314512	0.157256	−	0.987558i	$-0.449735\pi$
$647$	8.00000	0.314512	0.157256	+	0.987558i	$0.449735\pi$
$648$	0	0
$649$	−6.24621	−0.245185
$650$	0	0
$651$	9.75379	0.382281
$652$	0	0
$653$	38.4924	1.50632	0.753162	−	0.657835i	$-0.228527\pi$
$653$	38.4924	1.50632	0.753162	+	0.657835i	$0.228527\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	−2.38447	−0.0930271
$658$	0	0
$659$	0.192236	0.00748845	0.00374422	−	0.999993i	$-0.498808\pi$
$659$	0.192236	0.00748845	0.00374422	+	0.999993i	$0.498808\pi$
$660$	0	0
$661$	−17.6155	−0.685165	−0.342582	−	0.939488i	$-0.611302\pi$
$661$	−17.6155	−0.685165	−0.342582	+	0.939488i	$0.611302\pi$
$662$	0	0
$663$	−78.9309	−3.06542
$664$	0	0
$665$	0	0
$666$	0	0
$667$	1.36932	0.0530202
$668$	0	0
$669$	−38.5464	−1.49029
$670$	0	0
$671$	10.7386	0.414560
$672$	0	0
$673$	−41.6155	−1.60416	−0.802080	−	0.597216i	$-0.796273\pi$
$673$	−41.6155	−1.60416	−0.802080	+	0.597216i	$0.796273\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	1.31534	0.0505527	0.0252763	−	0.999681i	$-0.491953\pi$
$677$	1.31534	0.0505527	0.0252763	+	0.999681i	$0.491953\pi$
$678$	0	0
$679$	−1.31534	−0.0504782
$680$	0	0
$681$	17.6695	0.677097
$682$	0	0
$683$	44.9848	1.72130	0.860649	−	0.509199i	$-0.170058\pi$
$683$	44.9848	1.72130	0.860649	+	0.509199i	$0.170058\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	17.7538	0.677349
$688$	0	0
$689$	−87.7235	−3.34200
$690$	0	0
$691$	−16.4924	−0.627401	−0.313701	−	0.949522i	$-0.601569\pi$
$691$	−16.4924	−0.627401	−0.313701	+	0.949522i	$0.601569\pi$
$692$	0	0
$693$	0.876894	0.0333105
$694$	0	0
$695$	0	0
$696$	0	0
$697$	8.49242	0.321673
$698$	0	0
$699$	−16.9848	−0.642426
$700$	0	0
$701$	−9.31534	−0.351836	−0.175918	−	0.984405i	$-0.556289\pi$
$701$	−9.31534	−0.351836	−0.175918	+	0.984405i	$0.556289\pi$
$702$	0	0
$703$	58.7386	2.21537
$704$	0	0
$705$	0	0
$706$	0	0
$707$	6.00000	0.225653
$708$	0	0
$709$	−4.05398	−0.152250	−0.0761251	−	0.997098i	$-0.524255\pi$
$709$	−4.05398	−0.152250	−0.0761251	+	0.997098i	$0.524255\pi$
$710$	0	0
$711$	0.384472	0.0144188
$712$	0	0
$713$	−19.5076	−0.730565
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−28.1922	−1.05286
$718$	0	0
$719$	23.6155	0.880711	0.440355	−	0.897824i	$-0.354852\pi$
$719$	23.6155	0.880711	0.440355	+	0.897824i	$0.354852\pi$
$720$	0	0
$721$	−11.8078	−0.439744
$722$	0	0
$723$	−3.12311	−0.116150
$724$	0	0
$725$	0	0
$726$	0	0
$727$	48.9848	1.81675	0.908374	−	0.418159i	$-0.137325\pi$
$727$	48.9848	1.81675	0.908374	+	0.418159i	$0.137325\pi$
$728$	0	0
$729$	29.9848	1.11055
$730$	0	0
$731$	53.8617	1.99215
$732$	0	0
$733$	−16.4384	−0.607168	−0.303584	−	0.952805i	$-0.598183\pi$
$733$	−16.4384	−0.607168	−0.303584	+	0.952805i	$0.598183\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−3.50758	−0.129203
$738$	0	0
$739$	6.43845	0.236842	0.118421	−	0.992963i	$-0.462217\pi$
$739$	6.43845	0.236842	0.118421	+	0.992963i	$0.462217\pi$
$740$	0	0
$741$	74.3542	2.73147
$742$	0	0
$743$	−24.0000	−0.880475	−0.440237	−	0.897881i	$-0.645106\pi$
$743$	−24.0000	−0.880475	−0.440237	+	0.897881i	$0.645106\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	−6.73863	−0.246554
$748$	0	0
$749$	15.1231	0.552586
$750$	0	0
$751$	31.3153	1.14271	0.571357	−	0.820702i	$-0.306417\pi$
$751$	31.3153	1.14271	0.571357	+	0.820702i	$0.306417\pi$
$752$	0	0
$753$	20.8769	0.760796
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−15.3693	−0.558607	−0.279304	−	0.960203i	$-0.590104\pi$
$757$	−15.3693	−0.558607	−0.279304	+	0.960203i	$0.590104\pi$
$758$	0	0
$759$	7.61553	0.276426
$760$	0	0
$761$	26.0000	0.942499	0.471250	−	0.882000i	$-0.343803\pi$
$761$	26.0000	0.942499	0.471250	+	0.882000i	$0.343803\pi$
$762$	0	0
$763$	−4.43845	−0.160683
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−26.7386	−0.965476
$768$	0	0
$769$	42.9848	1.55007	0.775037	−	0.631916i	$-0.217731\pi$
$769$	42.9848	1.55007	0.775037	+	0.631916i	$0.217731\pi$
$770$	0	0
$771$	−28.8769	−1.03998
$772$	0	0
$773$	29.8078	1.07211	0.536055	−	0.844183i	$-0.319914\pi$
$773$	29.8078	1.07211	0.536055	+	0.844183i	$0.319914\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	−12.8769	−0.461956
$778$	0	0
$779$	−8.00000	−0.286630
$780$	0	0
$781$	0	0
$782$	0	0
$783$	2.43845	0.0871430
$784$	0	0
$785$	0	0
$786$	0	0
$787$	33.1771	1.18264	0.591318	−	0.806439i	$-0.298608\pi$
$787$	33.1771	1.18264	0.591318	+	0.806439i	$0.298608\pi$
$788$	0	0
$789$	−14.6307	−0.520866
$790$	0	0
$791$	−8.24621	−0.293202
$792$	0	0
$793$	45.9697	1.63243
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−17.8078	−0.630783	−0.315392	−	0.948962i	$-0.602136\pi$
$797$	−17.8078	−0.630783	−0.315392	+	0.948962i	$0.602136\pi$
$798$	0	0
$799$	−18.4384	−0.652305
$800$	0	0
$801$	−2.87689	−0.101650
$802$	0	0
$803$	−6.63068	−0.233992
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−6.63068	−0.233411
$808$	0	0
$809$	4.05398	0.142530	0.0712651	−	0.997457i	$-0.477296\pi$
$809$	4.05398	0.142530	0.0712651	+	0.997457i	$0.477296\pi$
$810$	0	0
$811$	27.6155	0.969712	0.484856	−	0.874594i	$-0.338872\pi$
$811$	27.6155	0.969712	0.484856	+	0.874594i	$0.338872\pi$
$812$	0	0
$813$	−9.75379	−0.342080
$814$	0	0
$815$	0	0
$816$	0	0
$817$	−50.7386	−1.77512
$818$	0	0
$819$	3.75379	0.131168
$820$	0	0
$821$	54.3002	1.89509	0.947545	−	0.319623i	$-0.103556\pi$
$821$	54.3002	1.89509	0.947545	+	0.319623i	$0.103556\pi$
$822$	0	0
$823$	−17.7538	−0.618858	−0.309429	−	0.950923i	$-0.600138\pi$
$823$	−17.7538	−0.618858	−0.309429	+	0.950923i	$0.600138\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	24.8769	0.865054	0.432527	−	0.901621i	$-0.357622\pi$
$827$	24.8769	0.865054	0.432527	+	0.901621i	$0.357622\pi$
$828$	0	0
$829$	5.61553	0.195035	0.0975177	−	0.995234i	$-0.468910\pi$
$829$	5.61553	0.195035	0.0975177	+	0.995234i	$0.468910\pi$
$830$	0	0
$831$	−12.8769	−0.446695
$832$	0	0
$833$	−7.56155	−0.261992
$834$	0	0
$835$	0	0
$836$	0	0
$837$	−34.7386	−1.20074
$838$	0	0
$839$	−19.1231	−0.660203	−0.330101	−	0.943945i	$-0.607083\pi$
$839$	−19.1231	−0.660203	−0.330101	+	0.943945i	$0.607083\pi$
$840$	0	0
$841$	−28.8078	−0.993371
$842$	0	0
$843$	30.5464	1.05207
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−8.56155	−0.294178
$848$	0	0
$849$	7.31534	0.251062
$850$	0	0
$851$	25.7538	0.882829
$852$	0	0
$853$	−15.7538	−0.539399	−0.269700	−	0.962944i	$-0.586924\pi$
$853$	−15.7538	−0.539399	−0.269700	+	0.962944i	$0.586924\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−52.7386	−1.80152	−0.900759	−	0.434320i	$-0.856989\pi$
$857$	−52.7386	−1.80152	−0.900759	+	0.434320i	$0.856989\pi$
$858$	0	0
$859$	−36.9848	−1.26191	−0.630953	−	0.775821i	$-0.717336\pi$
$859$	−36.9848	−1.26191	−0.630953	+	0.775821i	$0.717336\pi$
$860$	0	0
$861$	1.75379	0.0597690
$862$	0	0
$863$	28.4924	0.969893	0.484947	−	0.874544i	$-0.338839\pi$
$863$	28.4924	0.969893	0.484947	+	0.874544i	$0.338839\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	−62.7386	−2.13072
$868$	0	0
$869$	1.06913	0.0362678
$870$	0	0
$871$	−15.0152	−0.508769
$872$	0	0
$873$	0.738634	0.0249990
$874$	0	0
$875$	0	0
$876$	0	0
$877$	13.5076	0.456118	0.228059	−	0.973647i	$-0.426762\pi$
$877$	13.5076	0.456118	0.228059	+	0.973647i	$0.426762\pi$
$878$	0	0
$879$	50.0540	1.68828
$880$	0	0
$881$	54.1080	1.82294	0.911472	−	0.411363i	$-0.134947\pi$
$881$	54.1080	1.82294	0.911472	+	0.411363i	$0.134947\pi$
$882$	0	0
$883$	21.7538	0.732073	0.366037	−	0.930600i	$-0.380714\pi$
$883$	21.7538	0.732073	0.366037	+	0.930600i	$0.380714\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−36.4924	−1.22530	−0.612648	−	0.790356i	$-0.709896\pi$
$887$	−36.4924	−1.22530	−0.612648	+	0.790356i	$0.709896\pi$
$888$	0	0
$889$	−6.24621	−0.209491
$890$	0	0
$891$	10.9309	0.366198
$892$	0	0
$893$	17.3693	0.581242
$894$	0	0
$895$	0	0
$896$	0	0
$897$	32.6004	1.08849
$898$	0	0
$899$	−2.73863	−0.0913385
$900$	0	0
$901$	−99.2311	−3.30587
$902$	0	0
$903$	11.1231	0.370154
$904$	0	0
$905$	0	0
$906$	0	0
$907$	−48.1080	−1.59740	−0.798699	−	0.601731i	$-0.794478\pi$
$907$	−48.1080	−1.59740	−0.798699	+	0.601731i	$0.794478\pi$
$908$	0	0
$909$	−3.36932	−0.111753
$910$	0	0
$911$	−8.00000	−0.265052	−0.132526	−	0.991180i	$-0.542309\pi$
$911$	−8.00000	−0.265052	−0.132526	+	0.991180i	$0.542309\pi$
$912$	0	0
$913$	−18.7386	−0.620158
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−15.1231	−0.499409
$918$	0	0
$919$	−2.43845	−0.0804370	−0.0402185	−	0.999191i	$-0.512805\pi$
$919$	−2.43845	−0.0804370	−0.0402185	+	0.999191i	$0.512805\pi$
$920$	0	0
$921$	44.7926	1.47597
$922$	0	0
$923$	0	0
$924$	0	0
$925$	0	0
$926$	0	0
$927$	6.63068	0.217780
$928$	0	0
$929$	−2.87689	−0.0943878	−0.0471939	−	0.998886i	$-0.515028\pi$
$929$	−2.87689	−0.0943878	−0.0471939	+	0.998886i	$0.515028\pi$
$930$	0	0
$931$	7.12311	0.233450
$932$	0	0
$933$	−20.1080	−0.658305
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−37.8078	−1.23513	−0.617563	−	0.786521i	$-0.711880\pi$
$937$	−37.8078	−1.23513	−0.617563	+	0.786521i	$0.711880\pi$
$938$	0	0
$939$	41.6695	1.35983
$940$	0	0
$941$	28.6307	0.933334	0.466667	−	0.884433i	$-0.345455\pi$
$941$	28.6307	0.933334	0.466667	+	0.884433i	$0.345455\pi$
$942$	0	0
$943$	−3.50758	−0.114222
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−35.2311	−1.14486	−0.572428	−	0.819955i	$-0.693998\pi$
$947$	−35.2311	−1.14486	−0.572428	+	0.819955i	$0.693998\pi$
$948$	0	0
$949$	−28.3845	−0.921399
$950$	0	0
$951$	−15.6155	−0.506368
$952$	0	0
$953$	51.8617	1.67997	0.839983	−	0.542612i	$-0.182565\pi$
$953$	51.8617	1.67997	0.839983	+	0.542612i	$0.182565\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	1.06913	0.0345601
$958$	0	0
$959$	7.36932	0.237968
$960$	0	0
$961$	8.01515	0.258553
$962$	0	0
$963$	−8.49242	−0.273664
$964$	0	0
$965$	0	0
$966$	0	0
$967$	44.1080	1.41842	0.709208	−	0.704999i	$-0.249053\pi$
$967$	44.1080	1.41842	0.709208	+	0.704999i	$0.249053\pi$
$968$	0	0
$969$	84.1080	2.70194
$970$	0	0
$971$	22.7386	0.729717	0.364859	−	0.931063i	$-0.381117\pi$
$971$	22.7386	0.729717	0.364859	+	0.931063i	$0.381117\pi$
$972$	0	0
$973$	21.3693	0.685069
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−10.9848	−0.351436	−0.175718	−	0.984441i	$-0.556225\pi$
$977$	−10.9848	−0.351436	−0.175718	+	0.984441i	$0.556225\pi$
$978$	0	0
$979$	−8.00000	−0.255681
$980$	0	0
$981$	2.49242	0.0795769
$982$	0	0
$983$	−29.1771	−0.930604	−0.465302	−	0.885152i	$-0.654054\pi$
$983$	−29.1771	−0.930604	−0.465302	+	0.885152i	$0.654054\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	−3.80776	−0.121202
$988$	0	0
$989$	−22.2462	−0.707388
$990$	0	0
$991$	20.4924	0.650963	0.325482	−	0.945548i	$-0.394474\pi$
$991$	20.4924	0.650963	0.325482	+	0.945548i	$0.394474\pi$
$992$	0	0
$993$	−18.7386	−0.594653
$994$	0	0
$995$	0	0
$996$	0	0
$997$	56.9309	1.80302	0.901509	−	0.432760i	$-0.142460\pi$
$997$	56.9309	1.80302	0.901509	+	0.432760i	$0.142460\pi$
$998$	0	0
$999$	45.8617	1.45100

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2800.2.a.bn.1.1		2
4.3	odd	2		1400.2.a.p.1.2		2
5.2	odd	4		2800.2.g.u.449.3		4
5.3	odd	4		2800.2.g.u.449.2		4
5.4	even	2		560.2.a.g.1.2		2
15.14	odd	2		5040.2.a.bq.1.2		2
20.3	even	4		1400.2.g.k.449.3		4
20.7	even	4		1400.2.g.k.449.2		4
20.19	odd	2		280.2.a.d.1.1	✓	2
28.27	even	2		9800.2.a.by.1.1		2
35.34	odd	2		3920.2.a.bu.1.1		2
40.19	odd	2		2240.2.a.be.1.2		2
40.29	even	2		2240.2.a.bi.1.1		2
60.59	even	2		2520.2.a.w.1.1		2
140.19	even	6		1960.2.q.u.361.1		4
140.39	odd	6		1960.2.q.s.961.2		4
140.59	even	6		1960.2.q.u.961.1		4
140.79	odd	6		1960.2.q.s.361.2		4
140.139	even	2		1960.2.a.r.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
280.2.a.d.1.1	✓	2	20.19	odd	2
560.2.a.g.1.2		2	5.4	even	2
1400.2.a.p.1.2		2	4.3	odd	2
1400.2.g.k.449.2		4	20.7	even	4
1400.2.g.k.449.3		4	20.3	even	4
1960.2.a.r.1.2		2	140.139	even	2
1960.2.q.s.361.2		4	140.79	odd	6
1960.2.q.s.961.2		4	140.39	odd	6
1960.2.q.u.361.1		4	140.19	even	6
1960.2.q.u.961.1		4	140.59	even	6
2240.2.a.be.1.2		2	40.19	odd	2
2240.2.a.bi.1.1		2	40.29	even	2
2520.2.a.w.1.1		2	60.59	even	2
2800.2.a.bn.1.1		2	1.1	even	1	trivial
2800.2.g.u.449.2		4	5.3	odd	4
2800.2.g.u.449.3		4	5.2	odd	4
3920.2.a.bu.1.1		2	35.34	odd	2
5040.2.a.bq.1.2		2	15.14	odd	2
9800.2.a.by.1.1		2	28.27	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 2800 = 2^{4} \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2800.a (trivial)

\(f(q)\)	\(=\)	\(q-1.56155 q^{3} +1.00000 q^{7} -0.561553 q^{9} +O(q^{10})\)	\(q-1.56155 q^{3} +1.00000 q^{7} -0.561553 q^{9} -1.56155 q^{11} -6.68466 q^{13} -7.56155 q^{17} +7.12311 q^{19} -1.56155 q^{21} +3.12311 q^{23} +5.56155 q^{27} +0.438447 q^{29} -6.24621 q^{31} +2.43845 q^{33} +8.24621 q^{37} +10.4384 q^{39} -1.12311 q^{41} -7.12311 q^{43} +2.43845 q^{47} +1.00000 q^{49} +11.8078 q^{51} +13.1231 q^{53} -11.1231 q^{57} +4.00000 q^{59} -6.87689 q^{61} -0.561553 q^{63} +2.24621 q^{67} -4.87689 q^{69} +4.24621 q^{73} -1.56155 q^{77} -0.684658 q^{79} -7.00000 q^{81} +12.0000 q^{83} -0.684658 q^{87} +5.12311 q^{89} -6.68466 q^{91} +9.75379 q^{93} -1.31534 q^{97} +0.876894 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + q^{3} + 2 q^{7} + 3 q^{9}+O(q^{10}) \) 2 * q + q^3 + 2 * q^7 + 3 * q^9	\( 2 q + q^{3} + 2 q^{7} + 3 q^{9} + q^{11} - q^{13} - 11 q^{17} + 6 q^{19} + q^{21} - 2 q^{23} + 7 q^{27} + 5 q^{29} + 4 q^{31} + 9 q^{33} + 25 q^{39} + 6 q^{41} - 6 q^{43} + 9 q^{47} + 2 q^{49} + 3 q^{51} + 18 q^{53} - 14 q^{57} + 8 q^{59} - 22 q^{61} + 3 q^{63} - 12 q^{67} - 18 q^{69} - 8 q^{73} + q^{77} + 11 q^{79} - 14 q^{81} + 24 q^{83} + 11 q^{87} + 2 q^{89} - q^{91} + 36 q^{93} - 15 q^{97} + 10 q^{99}+O(q^{100}) \) 2 * q + q^3 + 2 * q^7 + 3 * q^9 + q^11 - q^13 - 11 * q^17 + 6 * q^19 + q^21 - 2 * q^23 + 7 * q^27 + 5 * q^29 + 4 * q^31 + 9 * q^33 + 25 * q^39 + 6 * q^41 - 6 * q^43 + 9 * q^47 + 2 * q^49 + 3 * q^51 + 18 * q^53 - 14 * q^57 + 8 * q^59 - 22 * q^61 + 3 * q^63 - 12 * q^67 - 18 * q^69 - 8 * q^73 + q^77 + 11 * q^79 - 14 * q^81 + 24 * q^83 + 11 * q^87 + 2 * q^89 - q^91 + 36 * q^93 - 15 * q^97 + 10 * q^99

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