LMFDB - Embedded newform 9200.2.a.ct.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [9200,2,Mod(1,9200)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(9200, 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("9200.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 9200 = 2^{4} \cdot 5^{2} \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	9200.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:	$73.4623698596$
Analytic rank:	$0$
Dimension:	$5$
Coefficient field:	5.5.521397.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 9x^{3} - 3x^{2} + 18x + 12 $ x^5 - 9x^3 - 3x^2 + 18*x + 12
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 4600)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-1.36629$ of defining polynomial
Character	$\chi$	$=$	9200.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.36629 q^{3} -3.28093 q^{7} -1.13327 q^{9} -3.49709 q^{11} -3.41420 q^{13} +7.46023 q^{17} -3.49709 q^{19} -4.48269 q^{21} +1.00000 q^{23} -5.64722 q^{27} -3.46268 q^{29} -2.01105 q^{31} -4.77803 q^{33} -0.511497 q^{37} -4.66477 q^{39} -7.07954 q^{41} -2.76452 q^{43} -0.889198 q^{47} +3.76452 q^{49} +10.1928 q^{51} +14.2383 q^{53} -4.77803 q^{57} +4.71325 q^{59} +13.4769 q^{61} +3.71817 q^{63} -2.30025 q^{67} +1.36629 q^{69} +10.6214 q^{71} -3.70765 q^{73} +11.4737 q^{77} +7.97978 q^{79} -4.31592 q^{81} +9.42336 q^{83} -4.73101 q^{87} +0.801390 q^{89} +11.2018 q^{91} -2.74766 q^{93} -11.7722 q^{97} +3.96313 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - 4 q^{7} + 3 q^{9} + 4 q^{13} + 6 q^{17} + 5 q^{23} - 9 q^{27} + 12 q^{29} + 18 q^{31} + 6 q^{33} + 10 q^{37} - 9 q^{39} - 6 q^{41} - 10 q^{43} - 22 q^{47} + 15 q^{49} + 6 q^{51} + 10 q^{53} + 6 q^{57}+ \cdots + 6 q^{99}+O(q^{100}) $ 5 * q - 4 * q^7 + 3 * q^9 + 4 * q^13 + 6 * q^17 + 5 * q^23 - 9 * q^27 + 12 * q^29 + 18 * q^31 + 6 * q^33 + 10 * q^37 - 9 * q^39 - 6 * q^41 - 10 * q^43 - 22 * q^47 + 15 * q^49 + 6 * q^51 + 10 * q^53 + 6 * q^57 + q^59 + 10 * q^61 - 8 * q^67 - 8 * q^71 + 6 * q^73 - 27 * q^81 + 2 * q^83 - 39 * q^87 + 14 * q^89 + 46 * q^91 - 3 * q^93 + 6 * q^97 + 6 * q^99

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
$3$	1.36629	0.788825	0.394413	−	0.918933i	$-0.370948\pi$
$3$	1.36629	0.788825	0.394413	+	0.918933i	$0.370948\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−3.28093	−1.24008	−0.620038	−	0.784572i	$-0.712883\pi$
$7$	−3.28093	−1.24008	−0.620038	+	0.784572i	$0.712883\pi$
$8$	0	0
$9$	−1.13327	−0.377755
$10$	0	0
$11$	−3.49709	−1.05441	−0.527207	−	0.849737i	$-0.676761\pi$
$11$	−3.49709	−1.05441	−0.527207	+	0.849737i	$0.676761\pi$
$12$	0	0
$13$	−3.41420	−0.946928	−0.473464	−	0.880813i	$-0.656997\pi$
$13$	−3.41420	−0.946928	−0.473464	+	0.880813i	$0.656997\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	7.46023	1.80937	0.904685	−	0.426080i	$-0.140106\pi$
$17$	7.46023	1.80937	0.904685	+	0.426080i	$0.140106\pi$
$18$	0	0
$19$	−3.49709	−0.802288	−0.401144	−	0.916015i	$-0.631387\pi$
$19$	−3.49709	−0.802288	−0.401144	+	0.916015i	$0.631387\pi$
$20$	0	0
$21$	−4.48269	−0.978203
$22$	0	0
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	0	0
$25$	0	0
$26$	0	0
$27$	−5.64722	−1.08681
$28$	0	0
$29$	−3.46268	−0.643004	−0.321502	−	0.946909i	$-0.604188\pi$
$29$	−3.46268	−0.643004	−0.321502	+	0.946909i	$0.604188\pi$
$30$	0	0
$31$	−2.01105	−0.361195	−0.180597	−	0.983557i	$-0.557803\pi$
$31$	−2.01105	−0.361195	−0.180597	+	0.983557i	$0.557803\pi$
$32$	0	0
$33$	−4.77803	−0.831748
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−0.511497	−0.0840896	−0.0420448	−	0.999116i	$-0.513387\pi$
$37$	−0.511497	−0.0840896	−0.0420448	+	0.999116i	$0.513387\pi$
$38$	0	0
$39$	−4.66477	−0.746961
$40$	0	0
$41$	−7.07954	−1.10564	−0.552819	−	0.833301i	$-0.686448\pi$
$41$	−7.07954	−1.10564	−0.552819	+	0.833301i	$0.686448\pi$
$42$	0	0
$43$	−2.76452	−0.421586	−0.210793	−	0.977531i	$-0.567605\pi$
$43$	−2.76452	−0.421586	−0.210793	+	0.977531i	$0.567605\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−0.889198	−0.129703	−0.0648514	−	0.997895i	$-0.520657\pi$
$47$	−0.889198	−0.129703	−0.0648514	+	0.997895i	$0.520657\pi$
$48$	0	0
$49$	3.76452	0.537789
$50$	0	0
$51$	10.1928	1.42728
$52$	0	0
$53$	14.2383	1.95577	0.977887	−	0.209132i	$-0.0670640\pi$
$53$	14.2383	1.95577	0.977887	+	0.209132i	$0.0670640\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−4.77803	−0.632865
$58$	0	0
$59$	4.71325	0.613613	0.306807	−	0.951772i	$-0.400739\pi$
$59$	4.71325	0.613613	0.306807	+	0.951772i	$0.400739\pi$
$60$	0	0
$61$	13.4769	1.72554	0.862769	−	0.505599i	$-0.168728\pi$
$61$	13.4769	1.72554	0.862769	+	0.505599i	$0.168728\pi$
$62$	0	0
$63$	3.71817	0.468445
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−2.30025	−0.281020	−0.140510	−	0.990079i	$-0.544874\pi$
$67$	−2.30025	−0.281020	−0.140510	+	0.990079i	$0.544874\pi$
$68$	0	0
$69$	1.36629	0.164481
$70$	0	0
$71$	10.6214	1.26053	0.630264	−	0.776381i	$-0.282947\pi$
$71$	10.6214	1.26053	0.630264	+	0.776381i	$0.282947\pi$
$72$	0	0
$73$	−3.70765	−0.433948	−0.216974	−	0.976177i	$-0.569619\pi$
$73$	−3.70765	−0.433948	−0.216974	+	0.976177i	$0.569619\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	11.4737	1.30755
$78$	0	0
$79$	7.97978	0.897796	0.448898	−	0.893583i	$-0.351817\pi$
$79$	7.97978	0.897796	0.448898	+	0.893583i	$0.351817\pi$
$80$	0	0
$81$	−4.31592	−0.479546
$82$	0	0
$83$	9.42336	1.03435	0.517174	−	0.855880i	$-0.326984\pi$
$83$	9.42336	1.03435	0.517174	+	0.855880i	$0.326984\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	−4.73101	−0.507218
$88$	0	0
$89$	0.801390	0.0849471	0.0424736	−	0.999098i	$-0.486476\pi$
$89$	0.801390	0.0849471	0.0424736	+	0.999098i	$0.486476\pi$
$90$	0	0
$91$	11.2018	1.17426
$92$	0	0
$93$	−2.74766	−0.284919
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−11.7722	−1.19529	−0.597644	−	0.801762i	$-0.703896\pi$
$97$	−11.7722	−1.19529	−0.597644	+	0.801762i	$0.703896\pi$
$98$	0	0
$99$	3.96313	0.398310
$100$	0	0
$101$	11.1559	1.11006	0.555028	−	0.831831i	$-0.312707\pi$
$101$	11.1559	1.11006	0.555028	+	0.831831i	$0.312707\pi$
$102$	0	0
$103$	1.28585	0.126698	0.0633491	−	0.997991i	$-0.479822\pi$
$103$	1.28585	0.126698	0.0633491	+	0.997991i	$0.479822\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−13.8975	−1.34352	−0.671759	−	0.740770i	$-0.734461\pi$
$107$	−13.8975	−1.34352	−0.671759	+	0.740770i	$0.734461\pi$
$108$	0	0
$109$	−3.77713	−0.361783	−0.180892	−	0.983503i	$-0.557898\pi$
$109$	−3.77713	−0.361783	−0.180892	+	0.983503i	$0.557898\pi$
$110$	0	0
$111$	−0.698851	−0.0663320
$112$	0	0
$113$	19.3111	1.81663	0.908317	−	0.418282i	$-0.137368\pi$
$113$	19.3111	1.81663	0.908317	+	0.418282i	$0.137368\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	3.86919	0.357707
$118$	0	0
$119$	−24.4765	−2.24376
$120$	0	0
$121$	1.22966	0.111788
$122$	0	0
$123$	−9.67267	−0.872155
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−13.7612	−1.22111	−0.610553	−	0.791975i	$-0.709053\pi$
$127$	−13.7612	−1.22111	−0.610553	+	0.791975i	$0.709053\pi$
$128$	0	0
$129$	−3.77713	−0.332558
$130$	0	0
$131$	7.41665	0.647996	0.323998	−	0.946058i	$-0.394973\pi$
$131$	7.41665	0.647996	0.323998	+	0.946058i	$0.394973\pi$
$132$	0	0
$133$	11.4737	0.994899
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−20.5502	−1.75573	−0.877863	−	0.478912i	$-0.841031\pi$
$137$	−20.5502	−1.75573	−0.877863	+	0.478912i	$0.841031\pi$
$138$	0	0
$139$	5.49652	0.466209	0.233104	−	0.972452i	$-0.425112\pi$
$139$	5.49652	0.466209	0.233104	+	0.972452i	$0.425112\pi$
$140$	0	0
$141$	−1.21490	−0.102313
$142$	0	0
$143$	11.9398	0.998454
$144$	0	0
$145$	0	0
$146$	0	0
$147$	5.14341	0.424221
$148$	0	0
$149$	4.80139	0.393345	0.196673	−	0.980469i	$-0.436986\pi$
$149$	4.80139	0.393345	0.196673	+	0.980469i	$0.436986\pi$
$150$	0	0
$151$	3.26862	0.265997	0.132998	−	0.991116i	$-0.457539\pi$
$151$	3.26862	0.265997	0.132998	+	0.991116i	$0.457539\pi$
$152$	0	0
$153$	−8.45441	−0.683499
$154$	0	0
$155$	0	0
$156$	0	0
$157$	21.8392	1.74296	0.871480	−	0.490430i	$-0.163160\pi$
$157$	21.8392	1.74296	0.871480	+	0.490430i	$0.163160\pi$
$158$	0	0
$159$	19.4535	1.54276
$160$	0	0
$161$	−3.28093	−0.258574
$162$	0	0
$163$	−23.2435	−1.82057	−0.910285	−	0.413982i	$-0.864138\pi$
$163$	−23.2435	−1.82057	−0.910285	+	0.413982i	$0.864138\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	4.13179	0.319728	0.159864	−	0.987139i	$-0.448894\pi$
$167$	4.13179	0.319728	0.159864	+	0.987139i	$0.448894\pi$
$168$	0	0
$169$	−1.34325	−0.103327
$170$	0	0
$171$	3.96313	0.303068
$172$	0	0
$173$	5.88548	0.447465	0.223732	−	0.974651i	$-0.428176\pi$
$173$	5.88548	0.447465	0.223732	+	0.974651i	$0.428176\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	6.43965	0.484034
$178$	0	0
$179$	6.85789	0.512583	0.256292	−	0.966600i	$-0.417499\pi$
$179$	6.85789	0.512583	0.256292	+	0.966600i	$0.417499\pi$
$180$	0	0
$181$	−2.29042	−0.170246	−0.0851229	−	0.996370i	$-0.527128\pi$
$181$	−2.29042	−0.170246	−0.0851229	+	0.996370i	$0.527128\pi$
$182$	0	0
$183$	18.4133	1.36115
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−26.0891	−1.90782
$188$	0	0
$189$	18.5281	1.34772
$190$	0	0
$191$	9.58748	0.693725	0.346863	−	0.937916i	$-0.387247\pi$
$191$	9.58748	0.693725	0.346863	+	0.937916i	$0.387247\pi$
$192$	0	0
$193$	13.9410	1.00350	0.501749	−	0.865013i	$-0.332690\pi$
$193$	13.9410	1.00350	0.501749	+	0.865013i	$0.332690\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	15.1011	1.07591	0.537954	−	0.842974i	$-0.319197\pi$
$197$	15.1011	1.07591	0.537954	+	0.842974i	$0.319197\pi$
$198$	0	0
$199$	−14.5128	−1.02879	−0.514394	−	0.857554i	$-0.671983\pi$
$199$	−14.5128	−1.02879	−0.514394	+	0.857554i	$0.671983\pi$
$200$	0	0
$201$	−3.14280	−0.221676
$202$	0	0
$203$	11.3608	0.797374
$204$	0	0
$205$	0	0
$206$	0	0
$207$	−1.13327	−0.0787674
$208$	0	0
$209$	12.2297	0.845944
$210$	0	0
$211$	23.4846	1.61674	0.808372	−	0.588672i	$-0.200349\pi$
$211$	23.4846	1.61674	0.808372	+	0.588672i	$0.200349\pi$
$212$	0	0
$213$	14.5119	0.994336
$214$	0	0
$215$	0	0
$216$	0	0
$217$	6.59811	0.447909
$218$	0	0
$219$	−5.06571	−0.342309
$220$	0	0
$221$	−25.4707	−1.71334
$222$	0	0
$223$	8.20720	0.549595	0.274797	−	0.961502i	$-0.411389\pi$
$223$	8.20720	0.549595	0.274797	+	0.961502i	$0.411389\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−16.9942	−1.12794	−0.563972	−	0.825794i	$-0.690727\pi$
$227$	−16.9942	−1.12794	−0.563972	+	0.825794i	$0.690727\pi$
$228$	0	0
$229$	19.6818	1.30061	0.650306	−	0.759672i	$-0.274641\pi$
$229$	19.6818	1.30061	0.650306	+	0.759672i	$0.274641\pi$
$230$	0	0
$231$	15.6764	1.03143
$232$	0	0
$233$	24.8799	1.62993	0.814967	−	0.579507i	$-0.196755\pi$
$233$	24.8799	1.62993	0.814967	+	0.579507i	$0.196755\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	10.9027	0.708204
$238$	0	0
$239$	0.0718483	0.00464748	0.00232374	−	0.999997i	$-0.499260\pi$
$239$	0.0718483	0.00464748	0.00232374	+	0.999997i	$0.499260\pi$
$240$	0	0
$241$	−8.50658	−0.547957	−0.273979	−	0.961736i	$-0.588340\pi$
$241$	−8.50658	−0.547957	−0.273979	+	0.961736i	$0.588340\pi$
$242$	0	0
$243$	11.0449	0.708530
$244$	0	0
$245$	0	0
$246$	0	0
$247$	11.9398	0.759709
$248$	0	0
$249$	12.8750	0.815920
$250$	0	0
$251$	19.8858	1.25518	0.627591	−	0.778543i	$-0.284041\pi$
$251$	19.8858	1.25518	0.627591	+	0.778543i	$0.284041\pi$
$252$	0	0
$253$	−3.49709	−0.219860
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−11.4062	−0.711498	−0.355749	−	0.934582i	$-0.615774\pi$
$257$	−11.4062	−0.711498	−0.355749	+	0.934582i	$0.615774\pi$
$258$	0	0
$259$	1.67819	0.104277
$260$	0	0
$261$	3.92414	0.242898
$262$	0	0
$263$	13.3903	0.825679	0.412840	−	0.910804i	$-0.364537\pi$
$263$	13.3903	0.825679	0.412840	+	0.910804i	$0.364537\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	1.09493	0.0670084
$268$	0	0
$269$	13.8103	0.842027	0.421014	−	0.907054i	$-0.361674\pi$
$269$	13.8103	0.842027	0.421014	+	0.907054i	$0.361674\pi$
$270$	0	0
$271$	−26.7574	−1.62540	−0.812699	−	0.582683i	$-0.802003\pi$
$271$	−26.7574	−1.62540	−0.812699	+	0.582683i	$0.802003\pi$
$272$	0	0
$273$	15.3048	0.926288
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−1.80667	−0.108552	−0.0542760	−	0.998526i	$-0.517285\pi$
$277$	−1.80667	−0.108552	−0.0542760	+	0.998526i	$0.517285\pi$
$278$	0	0
$279$	2.27905	0.136443
$280$	0	0
$281$	27.1348	1.61873	0.809364	−	0.587308i	$-0.199812\pi$
$281$	27.1348	1.61873	0.809364	+	0.587308i	$0.199812\pi$
$282$	0	0
$283$	−20.9875	−1.24758	−0.623788	−	0.781594i	$-0.714407\pi$
$283$	−20.9875	−1.24758	−0.623788	+	0.781594i	$0.714407\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	23.2275	1.37108
$288$	0	0
$289$	38.6550	2.27382
$290$	0	0
$291$	−16.0842	−0.942873
$292$	0	0
$293$	0.00671252	0.000392149 0	0.000196075	−	1.00000i	$-0.499938\pi$
$293$	0.00671252	0.000392149 0	0.000196075		1.00000i	$0.499938\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	19.7489	1.14594
$298$	0	0
$299$	−3.41420	−0.197448
$300$	0	0
$301$	9.07022	0.522799
$302$	0	0
$303$	15.2422	0.875641
$304$	0	0
$305$	0	0
$306$	0	0
$307$	19.9767	1.14013	0.570067	−	0.821598i	$-0.306917\pi$
$307$	19.9767	1.14013	0.570067	+	0.821598i	$0.306917\pi$
$308$	0	0
$309$	1.75683	0.0999427
$310$	0	0
$311$	−23.8601	−1.35298	−0.676491	−	0.736451i	$-0.736500\pi$
$311$	−23.8601	−1.35298	−0.676491	+	0.736451i	$0.736500\pi$
$312$	0	0
$313$	−17.4406	−0.985803	−0.492901	−	0.870085i	$-0.664064\pi$
$313$	−17.4406	−0.985803	−0.492901	+	0.870085i	$0.664064\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	1.35581	0.0761499	0.0380750	−	0.999275i	$-0.487877\pi$
$317$	1.35581	0.0761499	0.0380750	+	0.999275i	$0.487877\pi$
$318$	0	0
$319$	12.1093	0.677992
$320$	0	0
$321$	−18.9879	−1.05980
$322$	0	0
$323$	−26.0891	−1.45164
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−5.16063	−0.285384
$328$	0	0
$329$	2.91740	0.160841
$330$	0	0
$331$	−17.4050	−0.956667	−0.478333	−	0.878178i	$-0.658759\pi$
$331$	−17.4050	−0.956667	−0.478333	+	0.878178i	$0.658759\pi$
$332$	0	0
$333$	0.579662	0.0317653
$334$	0	0
$335$	0	0
$336$	0	0
$337$	22.4648	1.22373	0.611867	−	0.790961i	$-0.290419\pi$
$337$	22.4648	1.22373	0.611867	+	0.790961i	$0.290419\pi$
$338$	0	0
$339$	26.3845	1.43301
$340$	0	0
$341$	7.03282	0.380849
$342$	0	0
$343$	10.6154	0.573177
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−2.97296	−0.159597	−0.0797984	−	0.996811i	$-0.525428\pi$
$347$	−2.97296	−0.159597	−0.0797984	+	0.996811i	$0.525428\pi$
$348$	0	0
$349$	−4.36975	−0.233908	−0.116954	−	0.993137i	$-0.537313\pi$
$349$	−4.36975	−0.233908	−0.116954	+	0.993137i	$0.537313\pi$
$350$	0	0
$351$	19.2807	1.02913
$352$	0	0
$353$	31.4419	1.67348	0.836742	−	0.547598i	$-0.184458\pi$
$353$	31.4419	1.67348	0.836742	+	0.547598i	$0.184458\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	−33.4419	−1.76993
$358$	0	0
$359$	−27.7912	−1.46676	−0.733381	−	0.679818i	$-0.762059\pi$
$359$	−27.7912	−1.46676	−0.733381	+	0.679818i	$0.762059\pi$
$360$	0	0
$361$	−6.77034	−0.356333
$362$	0	0
$363$	1.68007	0.0881809
$364$	0	0
$365$	0	0
$366$	0	0
$367$	−13.9061	−0.725890	−0.362945	−	0.931811i	$-0.618229\pi$
$367$	−13.9061	−0.725890	−0.362945	+	0.931811i	$0.618229\pi$
$368$	0	0
$369$	8.02299	0.417660
$370$	0	0
$371$	−46.7148	−2.42531
$372$	0	0
$373$	−26.2799	−1.36072	−0.680362	−	0.732876i	$-0.738177\pi$
$373$	−26.2799	−1.36072	−0.680362	+	0.732876i	$0.738177\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	11.8223	0.608879
$378$	0	0
$379$	3.06792	0.157588	0.0787942	−	0.996891i	$-0.474893\pi$
$379$	3.06792	0.157588	0.0787942	+	0.996891i	$0.474893\pi$
$380$	0	0
$381$	−18.8017	−0.963239
$382$	0	0
$383$	−28.8872	−1.47607	−0.738034	−	0.674763i	$-0.764246\pi$
$383$	−28.8872	−1.47607	−0.738034	+	0.674763i	$0.764246\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	3.13294	0.159256
$388$	0	0
$389$	24.4697	1.24066	0.620331	−	0.784340i	$-0.286998\pi$
$389$	24.4697	1.24066	0.620331	+	0.784340i	$0.286998\pi$
$390$	0	0
$391$	7.46023	0.377280
$392$	0	0
$393$	10.1333	0.511156
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−8.44828	−0.424007	−0.212004	−	0.977269i	$-0.567999\pi$
$397$	−8.44828	−0.424007	−0.212004	+	0.977269i	$0.567999\pi$
$398$	0	0
$399$	15.6764	0.784801
$400$	0	0
$401$	20.3352	1.01549	0.507746	−	0.861507i	$-0.330479\pi$
$401$	20.3352	1.01549	0.507746	+	0.861507i	$0.330479\pi$
$402$	0	0
$403$	6.86611	0.342025
$404$	0	0
$405$	0	0
$406$	0	0
$407$	1.78875	0.0886652
$408$	0	0
$409$	6.05655	0.299477	0.149738	−	0.988726i	$-0.452157\pi$
$409$	6.05655	0.299477	0.149738	+	0.988726i	$0.452157\pi$
$410$	0	0
$411$	−28.0775	−1.38496
$412$	0	0
$413$	−15.4639	−0.760927
$414$	0	0
$415$	0	0
$416$	0	0
$417$	7.50982	0.367757
$418$	0	0
$419$	−7.02434	−0.343162	−0.171581	−	0.985170i	$-0.554887\pi$
$419$	−7.02434	−0.343162	−0.171581	+	0.985170i	$0.554887\pi$
$420$	0	0
$421$	7.17652	0.349762	0.174881	−	0.984590i	$-0.444046\pi$
$421$	7.17652	0.349762	0.174881	+	0.984590i	$0.444046\pi$
$422$	0	0
$423$	1.00770	0.0489959
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−44.2167	−2.13980
$428$	0	0
$429$	16.3131	0.787605
$430$	0	0
$431$	−11.2944	−0.544034	−0.272017	−	0.962292i	$-0.587691\pi$
$431$	−11.2944	−0.544034	−0.272017	+	0.962292i	$0.587691\pi$
$432$	0	0
$433$	−22.2266	−1.06814	−0.534072	−	0.845439i	$-0.679339\pi$
$433$	−22.2266	−1.06814	−0.534072	+	0.845439i	$0.679339\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−3.49709	−0.167289
$438$	0	0
$439$	30.2573	1.44410	0.722052	−	0.691839i	$-0.243199\pi$
$439$	30.2573	1.44410	0.722052	+	0.691839i	$0.243199\pi$
$440$	0	0
$441$	−4.26620	−0.203153
$442$	0	0
$443$	−14.0307	−0.666620	−0.333310	−	0.942817i	$-0.608166\pi$
$443$	−14.0307	−0.666620	−0.333310	+	0.942817i	$0.608166\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	6.56007	0.310281
$448$	0	0
$449$	8.70110	0.410630	0.205315	−	0.978696i	$-0.434178\pi$
$449$	8.70110	0.410630	0.205315	+	0.978696i	$0.434178\pi$
$450$	0	0
$451$	24.7578	1.16580
$452$	0	0
$453$	4.46587	0.209825
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−33.8379	−1.58287	−0.791434	−	0.611255i	$-0.790665\pi$
$457$	−33.8379	−1.58287	−0.791434	+	0.611255i	$0.790665\pi$
$458$	0	0
$459$	−42.1295	−1.96644
$460$	0	0
$461$	39.6512	1.84674	0.923370	−	0.383912i	$-0.125423\pi$
$461$	39.6512	1.84674	0.923370	+	0.383912i	$0.125423\pi$
$462$	0	0
$463$	34.8490	1.61957	0.809784	−	0.586727i	$-0.199584\pi$
$463$	34.8490	1.61957	0.809784	+	0.586727i	$0.199584\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−13.9442	−0.645258	−0.322629	−	0.946525i	$-0.604567\pi$
$467$	−13.9442	−0.645258	−0.322629	+	0.946525i	$0.604567\pi$
$468$	0	0
$469$	7.54697	0.348486
$470$	0	0
$471$	29.8386	1.37489
$472$	0	0
$473$	9.66780	0.444526
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−16.1357	−0.738804
$478$	0	0
$479$	32.4103	1.48086	0.740432	−	0.672131i	$-0.234621\pi$
$479$	32.4103	1.48086	0.740432	+	0.672131i	$0.234621\pi$
$480$	0	0
$481$	1.74635	0.0796268
$482$	0	0
$483$	−4.48269	−0.203969
$484$	0	0
$485$	0	0
$486$	0	0
$487$	−25.1999	−1.14192	−0.570959	−	0.820979i	$-0.693428\pi$
$487$	−25.1999	−1.14192	−0.570959	+	0.820979i	$0.693428\pi$
$488$	0	0
$489$	−31.7572	−1.43611
$490$	0	0
$491$	32.9247	1.48587	0.742936	−	0.669363i	$-0.233433\pi$
$491$	32.9247	1.48587	0.742936	+	0.669363i	$0.233433\pi$
$492$	0	0
$493$	−25.8324	−1.16343
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−34.8481	−1.56315
$498$	0	0
$499$	−12.4047	−0.555309	−0.277654	−	0.960681i	$-0.589557\pi$
$499$	−12.4047	−0.555309	−0.277654	+	0.960681i	$0.589557\pi$
$500$	0	0
$501$	5.64521	0.252209
$502$	0	0
$503$	3.74517	0.166989	0.0834945	−	0.996508i	$-0.473392\pi$
$503$	3.74517	0.166989	0.0834945	+	0.996508i	$0.473392\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	−1.83526	−0.0815069
$508$	0	0
$509$	−5.84783	−0.259201	−0.129600	−	0.991566i	$-0.541369\pi$
$509$	−5.84783	−0.259201	−0.129600	+	0.991566i	$0.541369\pi$
$510$	0	0
$511$	12.1646	0.538128
$512$	0	0
$513$	19.7489	0.871933
$514$	0	0
$515$	0	0
$516$	0	0
$517$	3.10961	0.136760
$518$	0	0
$519$	8.04124	0.352971
$520$	0	0
$521$	18.3808	0.805278	0.402639	−	0.915359i	$-0.368093\pi$
$521$	18.3808	0.805278	0.402639	+	0.915359i	$0.368093\pi$
$522$	0	0
$523$	−23.4541	−1.02558	−0.512789	−	0.858515i	$-0.671388\pi$
$523$	−23.4541	−1.02558	−0.512789	+	0.858515i	$0.671388\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−15.0029	−0.653535
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	−5.34137	−0.231796
$532$	0	0
$533$	24.1710	1.04696
$534$	0	0
$535$	0	0
$536$	0	0
$537$	9.36984	0.404338
$538$	0	0
$539$	−13.1649	−0.567052
$540$	0	0
$541$	3.03474	0.130474	0.0652368	−	0.997870i	$-0.479220\pi$
$541$	3.03474	0.130474	0.0652368	+	0.997870i	$0.479220\pi$
$542$	0	0
$543$	−3.12937	−0.134294
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−16.4383	−0.702851	−0.351426	−	0.936216i	$-0.614303\pi$
$547$	−16.4383	−0.702851	−0.351426	+	0.936216i	$0.614303\pi$
$548$	0	0
$549$	−15.2729	−0.651830
$550$	0	0
$551$	12.1093	0.515875
$552$	0	0
$553$	−26.1811	−1.11334
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−6.72395	−0.284903	−0.142451	−	0.989802i	$-0.545498\pi$
$557$	−6.72395	−0.284903	−0.142451	+	0.989802i	$0.545498\pi$
$558$	0	0
$559$	9.43863	0.399212
$560$	0	0
$561$	−35.6452	−1.50494
$562$	0	0
$563$	42.0044	1.77027	0.885137	−	0.465331i	$-0.154065\pi$
$563$	42.0044	1.77027	0.885137	+	0.465331i	$0.154065\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	14.1602	0.594674
$568$	0	0
$569$	−39.1152	−1.63980	−0.819898	−	0.572510i	$-0.805970\pi$
$569$	−39.1152	−1.63980	−0.819898	+	0.572510i	$0.805970\pi$
$570$	0	0
$571$	13.2607	0.554944	0.277472	−	0.960734i	$-0.410503\pi$
$571$	13.2607	0.554944	0.277472	+	0.960734i	$0.410503\pi$
$572$	0	0
$573$	13.0992	0.547228
$574$	0	0
$575$	0	0
$576$	0	0
$577$	9.31588	0.387825	0.193913	−	0.981019i	$-0.437882\pi$
$577$	9.31588	0.387825	0.193913	+	0.981019i	$0.437882\pi$
$578$	0	0
$579$	19.0474	0.791584
$580$	0	0
$581$	−30.9174	−1.28267
$582$	0	0
$583$	−49.7925	−2.06220
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−7.80336	−0.322079	−0.161040	−	0.986948i	$-0.551485\pi$
$587$	−7.80336	−0.322079	−0.161040	+	0.986948i	$0.551485\pi$
$588$	0	0
$589$	7.03282	0.289782
$590$	0	0
$591$	20.6324	0.848704
$592$	0	0
$593$	12.8383	0.527204	0.263602	−	0.964632i	$-0.415089\pi$
$593$	12.8383	0.527204	0.263602	+	0.964632i	$0.415089\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−19.8287	−0.811534
$598$	0	0
$599$	10.4957	0.428843	0.214422	−	0.976741i	$-0.431213\pi$
$599$	10.4957	0.428843	0.214422	+	0.976741i	$0.431213\pi$
$600$	0	0
$601$	−28.1978	−1.15021	−0.575106	−	0.818079i	$-0.695039\pi$
$601$	−28.1978	−1.15021	−0.575106	+	0.818079i	$0.695039\pi$
$602$	0	0
$603$	2.60679	0.106157
$604$	0	0
$605$	0	0
$606$	0	0
$607$	46.5662	1.89007	0.945033	−	0.326976i	$-0.106030\pi$
$607$	46.5662	1.89007	0.945033	+	0.326976i	$0.106030\pi$
$608$	0	0
$609$	15.5221	0.628989
$610$	0	0
$611$	3.03590	0.122819
$612$	0	0
$613$	−30.3120	−1.22429	−0.612145	−	0.790746i	$-0.709693\pi$
$613$	−30.3120	−1.22429	−0.612145	+	0.790746i	$0.709693\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−24.1879	−0.973768	−0.486884	−	0.873467i	$-0.661866\pi$
$617$	−24.1879	−0.973768	−0.486884	+	0.873467i	$0.661866\pi$
$618$	0	0
$619$	4.29534	0.172644	0.0863221	−	0.996267i	$-0.472489\pi$
$619$	4.29534	0.172644	0.0863221	+	0.996267i	$0.472489\pi$
$620$	0	0
$621$	−5.64722	−0.226615
$622$	0	0
$623$	−2.62931	−0.105341
$624$	0	0
$625$	0	0
$626$	0	0
$627$	16.7092	0.667301
$628$	0	0
$629$	−3.81588	−0.152149
$630$	0	0
$631$	−12.3857	−0.493066	−0.246533	−	0.969134i	$-0.579291\pi$
$631$	−12.3857	−0.493066	−0.246533	+	0.969134i	$0.579291\pi$
$632$	0	0
$633$	32.0866	1.27533
$634$	0	0
$635$	0	0
$636$	0	0
$637$	−12.8528	−0.509248
$638$	0	0
$639$	−12.0369	−0.476171
$640$	0	0
$641$	−5.21653	−0.206041	−0.103020	−	0.994679i	$-0.532851\pi$
$641$	−5.21653	−0.206041	−0.103020	+	0.994679i	$0.532851\pi$
$642$	0	0
$643$	12.5639	0.495471	0.247735	−	0.968828i	$-0.420314\pi$
$643$	12.5639	0.495471	0.247735	+	0.968828i	$0.420314\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−22.7423	−0.894094	−0.447047	−	0.894510i	$-0.647524\pi$
$647$	−22.7423	−0.894094	−0.447047	+	0.894510i	$0.647524\pi$
$648$	0	0
$649$	−16.4827	−0.647002
$650$	0	0
$651$	9.01490	0.353322
$652$	0	0
$653$	−9.36997	−0.366675	−0.183338	−	0.983050i	$-0.558690\pi$
$653$	−9.36997	−0.366675	−0.183338	+	0.983050i	$0.558690\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	4.20175	0.163926
$658$	0	0
$659$	15.4086	0.600235	0.300117	−	0.953902i	$-0.402974\pi$
$659$	15.4086	0.600235	0.300117	+	0.953902i	$0.402974\pi$
$660$	0	0
$661$	43.1386	1.67790	0.838948	−	0.544211i	$-0.183171\pi$
$661$	43.1386	1.67790	0.838948	+	0.544211i	$0.183171\pi$
$662$	0	0
$663$	−34.8002	−1.35153
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−3.46268	−0.134076
$668$	0	0
$669$	11.2134	0.433534
$670$	0	0
$671$	−47.1299	−1.81943
$672$	0	0
$673$	12.6356	0.487066	0.243533	−	0.969893i	$-0.421694\pi$
$673$	12.6356	0.487066	0.243533	+	0.969893i	$0.421694\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−10.9139	−0.419454	−0.209727	−	0.977760i	$-0.567258\pi$
$677$	−10.9139	−0.419454	−0.209727	+	0.977760i	$0.567258\pi$
$678$	0	0
$679$	38.6238	1.48225
$680$	0	0
$681$	−23.2189	−0.889750
$682$	0	0
$683$	17.5334	0.670896	0.335448	−	0.942059i	$-0.391112\pi$
$683$	17.5334	0.670896	0.335448	+	0.942059i	$0.391112\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	26.8910	1.02596
$688$	0	0
$689$	−48.6122	−1.85198
$690$	0	0
$691$	52.2318	1.98699	0.993496	−	0.113871i	$-0.0363250\pi$
$691$	52.2318	1.98699	0.993496	+	0.113871i	$0.0363250\pi$
$692$	0	0
$693$	−13.0028	−0.493935
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−52.8150	−2.00051
$698$	0	0
$699$	33.9930	1.28573
$700$	0	0
$701$	26.0270	0.983026	0.491513	−	0.870870i	$-0.336444\pi$
$701$	26.0270	0.983026	0.491513	+	0.870870i	$0.336444\pi$
$702$	0	0
$703$	1.78875	0.0674641
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−36.6019	−1.37655
$708$	0	0
$709$	−34.7971	−1.30683	−0.653415	−	0.757000i	$-0.726664\pi$
$709$	−34.7971	−1.30683	−0.653415	+	0.757000i	$0.726664\pi$
$710$	0	0
$711$	−9.04321	−0.339147
$712$	0	0
$713$	−2.01105	−0.0753143
$714$	0	0
$715$	0	0
$716$	0	0
$717$	0.0981652	0.00366605
$718$	0	0
$719$	−4.40110	−0.164133	−0.0820667	−	0.996627i	$-0.526152\pi$
$719$	−4.40110	−0.164133	−0.0820667	+	0.996627i	$0.526152\pi$
$720$	0	0
$721$	−4.21878	−0.157115
$722$	0	0
$723$	−11.6224	−0.432242
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−0.744921	−0.0276276	−0.0138138	−	0.999905i	$-0.504397\pi$
$727$	−0.744921	−0.0276276	−0.0138138	+	0.999905i	$0.504397\pi$
$728$	0	0
$729$	28.0382	1.03845
$730$	0	0
$731$	−20.6240	−0.762805
$732$	0	0
$733$	−7.35631	−0.271711	−0.135856	−	0.990729i	$-0.543378\pi$
$733$	−7.35631	−0.271711	−0.135856	+	0.990729i	$0.543378\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	8.04419	0.296311
$738$	0	0
$739$	−13.2496	−0.487394	−0.243697	−	0.969851i	$-0.578360\pi$
$739$	−13.2496	−0.487394	−0.243697	+	0.969851i	$0.578360\pi$
$740$	0	0
$741$	16.3131	0.599278
$742$	0	0
$743$	28.8314	1.05772	0.528861	−	0.848708i	$-0.322619\pi$
$743$	28.8314	1.05772	0.528861	+	0.848708i	$0.322619\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	−10.6792	−0.390730
$748$	0	0
$749$	45.5966	1.66607
$750$	0	0
$751$	24.3962	0.890228	0.445114	−	0.895474i	$-0.353163\pi$
$751$	24.3962	0.890228	0.445114	+	0.895474i	$0.353163\pi$
$752$	0	0
$753$	27.1697	0.990120
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−9.49825	−0.345220	−0.172610	−	0.984990i	$-0.555220\pi$
$757$	−9.49825	−0.345220	−0.172610	+	0.984990i	$0.555220\pi$
$758$	0	0
$759$	−4.77803	−0.173431
$760$	0	0
$761$	−8.43957	−0.305934	−0.152967	−	0.988231i	$-0.548883\pi$
$761$	−8.43957	−0.305934	−0.152967	+	0.988231i	$0.548883\pi$
$762$	0	0
$763$	12.3925	0.448639
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−16.0920	−0.581048
$768$	0	0
$769$	8.85911	0.319468	0.159734	−	0.987160i	$-0.448936\pi$
$769$	8.85911	0.319468	0.159734	+	0.987160i	$0.448936\pi$
$770$	0	0
$771$	−15.5841	−0.561247
$772$	0	0
$773$	−32.7351	−1.17740	−0.588699	−	0.808352i	$-0.700360\pi$
$773$	−32.7351	−1.17740	−0.588699	+	0.808352i	$0.700360\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	2.29288	0.0822567
$778$	0	0
$779$	24.7578	0.887041
$780$	0	0
$781$	−37.1440	−1.32912
$782$	0	0
$783$	19.5545	0.698822
$784$	0	0
$785$	0	0
$786$	0	0
$787$	0.736862	0.0262663	0.0131331	−	0.999914i	$-0.495819\pi$
$787$	0.736862	0.0262663	0.0131331	+	0.999914i	$0.495819\pi$
$788$	0	0
$789$	18.2949	0.651316
$790$	0	0
$791$	−63.3584	−2.25276
$792$	0	0
$793$	−46.0127	−1.63396
$794$	0	0
$795$	0	0
$796$	0	0
$797$	41.7353	1.47834	0.739170	−	0.673519i	$-0.235218\pi$
$797$	41.7353	1.47834	0.739170	+	0.673519i	$0.235218\pi$
$798$	0	0
$799$	−6.63362	−0.234681
$800$	0	0
$801$	−0.908187	−0.0320892
$802$	0	0
$803$	12.9660	0.457560
$804$	0	0
$805$	0	0
$806$	0	0
$807$	18.8688	0.664212
$808$	0	0
$809$	21.3182	0.749506	0.374753	−	0.927125i	$-0.377727\pi$
$809$	21.3182	0.749506	0.374753	+	0.927125i	$0.377727\pi$
$810$	0	0
$811$	−24.3126	−0.853732	−0.426866	−	0.904315i	$-0.640382\pi$
$811$	−24.3126	−0.853732	−0.426866	+	0.904315i	$0.640382\pi$
$812$	0	0
$813$	−36.5583	−1.28216
$814$	0	0
$815$	0	0
$816$	0	0
$817$	9.66780	0.338233
$818$	0	0
$819$	−12.6946	−0.443584
$820$	0	0
$821$	49.3157	1.72113	0.860566	−	0.509340i	$-0.170110\pi$
$821$	49.3157	1.72113	0.860566	+	0.509340i	$0.170110\pi$
$822$	0	0
$823$	41.5427	1.44809	0.724043	−	0.689755i	$-0.242282\pi$
$823$	41.5427	1.44809	0.724043	+	0.689755i	$0.242282\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−0.306510	−0.0106584	−0.00532919	−	0.999986i	$-0.501696\pi$
$827$	−0.306510	−0.0106584	−0.00532919	+	0.999986i	$0.501696\pi$
$828$	0	0
$829$	17.5423	0.609268	0.304634	−	0.952469i	$-0.401466\pi$
$829$	17.5423	0.609268	0.304634	+	0.952469i	$0.401466\pi$
$830$	0	0
$831$	−2.46842	−0.0856285
$832$	0	0
$833$	28.0842	0.973060
$834$	0	0
$835$	0	0
$836$	0	0
$837$	11.3568	0.392549
$838$	0	0
$839$	44.1480	1.52416	0.762078	−	0.647485i	$-0.224179\pi$
$839$	44.1480	1.52416	0.762078	+	0.647485i	$0.224179\pi$
$840$	0	0
$841$	−17.0098	−0.586546
$842$	0	0
$843$	37.0739	1.27689
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−4.03445	−0.138625
$848$	0	0
$849$	−28.6749	−0.984119
$850$	0	0
$851$	−0.511497	−0.0175339
$852$	0	0
$853$	41.2098	1.41100	0.705499	−	0.708711i	$-0.250723\pi$
$853$	41.2098	1.41100	0.705499	+	0.708711i	$0.250723\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−19.3245	−0.660114	−0.330057	−	0.943961i	$-0.607068\pi$
$857$	−19.3245	−0.660114	−0.330057	+	0.943961i	$0.607068\pi$
$858$	0	0
$859$	16.2174	0.553330	0.276665	−	0.960966i	$-0.410771\pi$
$859$	16.2174	0.553330	0.276665	+	0.960966i	$0.410771\pi$
$860$	0	0
$861$	31.7354	1.08154
$862$	0	0
$863$	−56.9036	−1.93702	−0.968510	−	0.248974i	$-0.919907\pi$
$863$	−56.9036	−1.93702	−0.968510	+	0.248974i	$0.919907\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	52.8137	1.79365
$868$	0	0
$869$	−27.9061	−0.946648
$870$	0	0
$871$	7.85351	0.266106
$872$	0	0
$873$	13.3410	0.451526
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−33.4569	−1.12976	−0.564880	−	0.825173i	$-0.691077\pi$
$877$	−33.4569	−1.12976	−0.564880	+	0.825173i	$0.691077\pi$
$878$	0	0
$879$	0.00917121	0.000309337 0
$880$	0	0
$881$	−5.99700	−0.202044	−0.101022	−	0.994884i	$-0.532211\pi$
$881$	−5.99700	−0.202044	−0.101022	+	0.994884i	$0.532211\pi$
$882$	0	0
$883$	51.3210	1.72709	0.863544	−	0.504273i	$-0.168239\pi$
$883$	51.3210	1.72709	0.863544	+	0.504273i	$0.168239\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	27.1121	0.910336	0.455168	−	0.890405i	$-0.349579\pi$
$887$	27.1121	0.910336	0.455168	+	0.890405i	$0.349579\pi$
$888$	0	0
$889$	45.1495	1.51426
$890$	0	0
$891$	15.0932	0.505640
$892$	0	0
$893$	3.10961	0.104059
$894$	0	0
$895$	0	0
$896$	0	0
$897$	−4.66477	−0.155752
$898$	0	0
$899$	6.96362	0.232250
$900$	0	0
$901$	106.221	3.53872
$902$	0	0
$903$	12.3925	0.412397
$904$	0	0
$905$	0	0
$906$	0	0
$907$	40.6194	1.34875	0.674373	−	0.738391i	$-0.264414\pi$
$907$	40.6194	1.34875	0.674373	+	0.738391i	$0.264414\pi$
$908$	0	0
$909$	−12.6426	−0.419329
$910$	0	0
$911$	−32.2475	−1.06841	−0.534203	−	0.845356i	$-0.679388\pi$
$911$	−32.2475	−1.06841	−0.534203	+	0.845356i	$0.679388\pi$
$912$	0	0
$913$	−32.9544	−1.09063
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−24.3335	−0.803565
$918$	0	0
$919$	30.4042	1.00294	0.501470	−	0.865175i	$-0.332793\pi$
$919$	30.4042	1.00294	0.501470	+	0.865175i	$0.332793\pi$
$920$	0	0
$921$	27.2939	0.899366
$922$	0	0
$923$	−36.2636	−1.19363
$924$	0	0
$925$	0	0
$926$	0	0
$927$	−1.45720	−0.0478609
$928$	0	0
$929$	−25.2834	−0.829520	−0.414760	−	0.909931i	$-0.636134\pi$
$929$	−25.2834	−0.829520	−0.414760	+	0.909931i	$0.636134\pi$
$930$	0	0
$931$	−13.1649	−0.431462
$932$	0	0
$933$	−32.5997	−1.06727
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−22.1769	−0.724487	−0.362243	−	0.932084i	$-0.617989\pi$
$937$	−22.1769	−0.724487	−0.362243	+	0.932084i	$0.617989\pi$
$938$	0	0
$939$	−23.8289	−0.777626
$940$	0	0
$941$	4.23228	0.137968	0.0689842	−	0.997618i	$-0.478024\pi$
$941$	4.23228	0.137968	0.0689842	+	0.997618i	$0.478024\pi$
$942$	0	0
$943$	−7.07954	−0.230542
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−45.2374	−1.47002	−0.735009	−	0.678058i	$-0.762822\pi$
$947$	−45.2374	−1.47002	−0.735009	+	0.678058i	$0.762822\pi$
$948$	0	0
$949$	12.6587	0.410917
$950$	0	0
$951$	1.85242	0.0600690
$952$	0	0
$953$	−43.7170	−1.41613	−0.708066	−	0.706146i	$-0.750432\pi$
$953$	−43.7170	−1.41613	−0.708066	+	0.706146i	$0.750432\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	16.5448	0.534817
$958$	0	0
$959$	67.4240	2.17723
$960$	0	0
$961$	−26.9557	−0.869538
$962$	0	0
$963$	15.7495	0.507521
$964$	0	0
$965$	0	0
$966$	0	0
$967$	5.58482	0.179596	0.0897979	−	0.995960i	$-0.471378\pi$
$967$	5.58482	0.179596	0.0897979	+	0.995960i	$0.471378\pi$
$968$	0	0
$969$	−35.6452	−1.14509
$970$	0	0
$971$	−39.1206	−1.25544	−0.627720	−	0.778439i	$-0.716012\pi$
$971$	−39.1206	−1.25544	−0.627720	+	0.778439i	$0.716012\pi$
$972$	0	0
$973$	−18.0337	−0.578135
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−52.3050	−1.67339	−0.836693	−	0.547672i	$-0.815514\pi$
$977$	−52.3050	−1.67339	−0.836693	+	0.547672i	$0.815514\pi$
$978$	0	0
$979$	−2.80253	−0.0895694
$980$	0	0
$981$	4.28049	0.136665
$982$	0	0
$983$	−22.4395	−0.715710	−0.357855	−	0.933777i	$-0.616492\pi$
$983$	−22.4395	−0.715710	−0.357855	+	0.933777i	$0.616492\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	3.98600	0.126876
$988$	0	0
$989$	−2.76452	−0.0879067
$990$	0	0
$991$	0.851383	0.0270451	0.0135225	−	0.999909i	$-0.495696\pi$
$991$	0.851383	0.0270451	0.0135225	+	0.999909i	$0.495696\pi$
$992$	0	0
$993$	−23.7802	−0.754643
$994$	0	0
$995$	0	0
$996$	0	0
$997$	12.1784	0.385693	0.192846	−	0.981229i	$-0.438228\pi$
$997$	12.1784	0.385693	0.192846	+	0.981229i	$0.438228\pi$
$998$	0	0
$999$	2.88853	0.0913892

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9200.2.a.ct.1.4		5
4.3	odd	2		4600.2.a.bf.1.2	yes	5
5.4	even	2		9200.2.a.cv.1.2		5
20.3	even	4		4600.2.e.w.4049.4		10
20.7	even	4		4600.2.e.w.4049.7		10
20.19	odd	2		4600.2.a.bd.1.4	✓	5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4600.2.a.bd.1.4	✓	5	20.19	odd	2
4600.2.a.bf.1.2	yes	5	4.3	odd	2
4600.2.e.w.4049.4		10	20.3	even	4
4600.2.e.w.4049.7		10	20.7	even	4
9200.2.a.ct.1.4		5	1.1	even	1	trivial
9200.2.a.cv.1.2		5	5.4	even	2

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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:	\( N \)	\(=\)	\( 9200 = 2^{4} \cdot 5^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9200.a (trivial)

\(f(q)\)	\(=\)	\(q+1.36629 q^{3} -3.28093 q^{7} -1.13327 q^{9} -3.49709 q^{11} -3.41420 q^{13} +7.46023 q^{17} -3.49709 q^{19} -4.48269 q^{21} +1.00000 q^{23} -5.64722 q^{27} -3.46268 q^{29} -2.01105 q^{31} -4.77803 q^{33} -0.511497 q^{37} -4.66477 q^{39} -7.07954 q^{41} -2.76452 q^{43} -0.889198 q^{47} +3.76452 q^{49} +10.1928 q^{51} +14.2383 q^{53} -4.77803 q^{57} +4.71325 q^{59} +13.4769 q^{61} +3.71817 q^{63} -2.30025 q^{67} +1.36629 q^{69} +10.6214 q^{71} -3.70765 q^{73} +11.4737 q^{77} +7.97978 q^{79} -4.31592 q^{81} +9.42336 q^{83} -4.73101 q^{87} +0.801390 q^{89} +11.2018 q^{91} -2.74766 q^{93} -11.7722 q^{97} +3.96313 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - 4 q^{7} + 3 q^{9} + 4 q^{13} + 6 q^{17} + 5 q^{23} - 9 q^{27} + 12 q^{29} + 18 q^{31} + 6 q^{33} + 10 q^{37} - 9 q^{39} - 6 q^{41} - 10 q^{43} - 22 q^{47} + 15 q^{49} + 6 q^{51} + 10 q^{53} + 6 q^{57}+ \cdots + 6 q^{99}+O(q^{100}) \) 5 * q - 4 * q^7 + 3 * q^9 + 4 * q^13 + 6 * q^17 + 5 * q^23 - 9 * q^27 + 12 * q^29 + 18 * q^31 + 6 * q^33 + 10 * q^37 - 9 * q^39 - 6 * q^41 - 10 * q^43 - 22 * q^47 + 15 * q^49 + 6 * q^51 + 10 * q^53 + 6 * q^57 + q^59 + 10 * q^61 - 8 * q^67 - 8 * q^71 + 6 * q^73 - 27 * q^81 + 2 * q^83 - 39 * q^87 + 14 * q^89 + 46 * q^91 - 3 * q^93 + 6 * q^97 + 6 * q^99

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