LMFDB - Embedded newform 9800.2.a.cn.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [9800,2,Mod(1,9800)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(9800, 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("9800.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 9800 = 2^{3} \cdot 5^{2} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	9800.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:	$78.2533939809$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{2}, \sqrt{5})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 6x^{2} + 4 $ x^4 - 6*x^2 + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$0.874032$ of defining polynomial
Character	$\chi$	$=$	9800.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.874032 q^{3} -2.23607 q^{9} +O(q^{10})$	$q+0.874032 q^{3} -2.23607 q^{9} -1.00000 q^{11} +2.28825 q^{13} -1.74806 q^{17} -3.16228 q^{19} +1.00000 q^{23} -4.57649 q^{27} +4.70820 q^{29} +6.86474 q^{31} -0.874032 q^{33} -4.23607 q^{37} +2.00000 q^{39} -0.206331 q^{41} +2.23607 q^{43} -1.41421 q^{47} -1.52786 q^{51} +3.23607 q^{53} -2.76393 q^{57} +1.62054 q^{59} -1.95440 q^{61} -1.47214 q^{67} +0.874032 q^{69} -6.70820 q^{71} -12.8554 q^{73} +5.94427 q^{79} +2.70820 q^{81} -1.20788 q^{83} +4.11512 q^{87} -5.11667 q^{89} +6.00000 q^{93} -2.95595 q^{97} +2.23607 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q+O(q^{10}) $ 4 * q	$ 4 q - 4 q^{11} + 4 q^{23} - 8 q^{29} - 8 q^{37} + 8 q^{39} - 24 q^{51} + 4 q^{53} - 20 q^{57} + 12 q^{67} - 12 q^{79} - 16 q^{81} + 24 q^{93}+O(q^{100}) $ 4 * q - 4 * q^11 + 4 * q^23 - 8 * q^29 - 8 * q^37 + 8 * q^39 - 24 * q^51 + 4 * q^53 - 20 * q^57 + 12 * q^67 - 12 * q^79 - 16 * q^81 + 24 * q^93

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	0.874032	0.504623	0.252311	−	0.967646i	$-0.418809\pi$
$3$	0.874032	0.504623	0.252311	+	0.967646i	$0.418809\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	−2.23607	−0.745356
$10$	0	0
$11$	−1.00000	−0.301511	−0.150756	−	0.988571i	$-0.548171\pi$
$11$	−1.00000	−0.301511	−0.150756	+	0.988571i	$0.548171\pi$
$12$	0	0
$13$	2.28825	0.634645	0.317323	−	0.948318i	$-0.397216\pi$
$13$	2.28825	0.634645	0.317323	+	0.948318i	$0.397216\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−1.74806	−0.423968	−0.211984	−	0.977273i	$-0.567992\pi$
$17$	−1.74806	−0.423968	−0.211984	+	0.977273i	$0.567992\pi$
$18$	0	0
$19$	−3.16228	−0.725476	−0.362738	−	0.931891i	$-0.618158\pi$
$19$	−3.16228	−0.725476	−0.362738	+	0.931891i	$0.618158\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	1.00000	0.208514	0.104257	−	0.994550i	$-0.466753\pi$
$23$	1.00000	0.208514	0.104257	+	0.994550i	$0.466753\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	−4.57649	−0.880746
$28$	0	0
$29$	4.70820	0.874292	0.437146	−	0.899391i	$-0.355989\pi$
$29$	4.70820	0.874292	0.437146	+	0.899391i	$0.355989\pi$
$30$	0	0
$31$	6.86474	1.23294	0.616472	−	0.787377i	$-0.288562\pi$
$31$	6.86474	1.23294	0.616472	+	0.787377i	$0.288562\pi$
$32$	0	0
$33$	−0.874032	−0.152149
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−4.23607	−0.696405	−0.348203	−	0.937419i	$-0.613208\pi$
$37$	−4.23607	−0.696405	−0.348203	+	0.937419i	$0.613208\pi$
$38$	0	0
$39$	2.00000	0.320256
$40$	0	0
$41$	−0.206331	−0.0322235	−0.0161117	−	0.999870i	$-0.505129\pi$
$41$	−0.206331	−0.0322235	−0.0161117	+	0.999870i	$0.505129\pi$
$42$	0	0
$43$	2.23607	0.340997	0.170499	−	0.985358i	$-0.445462\pi$
$43$	2.23607	0.340997	0.170499	+	0.985358i	$0.445462\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−1.41421	−0.206284	−0.103142	−	0.994667i	$-0.532890\pi$
$47$	−1.41421	−0.206284	−0.103142	+	0.994667i	$0.532890\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	−1.52786	−0.213944
$52$	0	0
$53$	3.23607	0.444508	0.222254	−	0.974989i	$-0.428659\pi$
$53$	3.23607	0.444508	0.222254	+	0.974989i	$0.428659\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−2.76393	−0.366092
$58$	0	0
$59$	1.62054	0.210977	0.105488	−	0.994421i	$-0.466359\pi$
$59$	1.62054	0.210977	0.105488	+	0.994421i	$0.466359\pi$
$60$	0	0
$61$	−1.95440	−0.250235	−0.125117	−	0.992142i	$-0.539931\pi$
$61$	−1.95440	−0.250235	−0.125117	+	0.992142i	$0.539931\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−1.47214	−0.179850	−0.0899250	−	0.995949i	$-0.528663\pi$
$67$	−1.47214	−0.179850	−0.0899250	+	0.995949i	$0.528663\pi$
$68$	0	0
$69$	0.874032	0.105221
$70$	0	0
$71$	−6.70820	−0.796117	−0.398059	−	0.917360i	$-0.630316\pi$
$71$	−6.70820	−0.796117	−0.398059	+	0.917360i	$0.630316\pi$
$72$	0	0
$73$	−12.8554	−1.50462	−0.752308	−	0.658812i	$-0.771059\pi$
$73$	−12.8554	−1.50462	−0.752308	+	0.658812i	$0.771059\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	0	0
$78$	0	0
$79$	5.94427	0.668783	0.334391	−	0.942434i	$-0.391469\pi$
$79$	5.94427	0.668783	0.334391	+	0.942434i	$0.391469\pi$
$80$	0	0
$81$	2.70820	0.300912
$82$	0	0
$83$	−1.20788	−0.132582	−0.0662912	−	0.997800i	$-0.521117\pi$
$83$	−1.20788	−0.132582	−0.0662912	+	0.997800i	$0.521117\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	4.11512	0.441187
$88$	0	0
$89$	−5.11667	−0.542366	−0.271183	−	0.962528i	$-0.587415\pi$
$89$	−5.11667	−0.542366	−0.271183	+	0.962528i	$0.587415\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	6.00000	0.622171
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−2.95595	−0.300131	−0.150065	−	0.988676i	$-0.547948\pi$
$97$	−2.95595	−0.300131	−0.150065	+	0.988676i	$0.547948\pi$
$98$	0	0
$99$	2.23607	0.224733
$100$	0	0
$101$	12.1877	1.21273	0.606363	−	0.795188i	$-0.292628\pi$
$101$	12.1877	1.21273	0.606363	+	0.795188i	$0.292628\pi$
$102$	0	0
$103$	−2.08191	−0.205137	−0.102569	−	0.994726i	$-0.532706\pi$
$103$	−2.08191	−0.205137	−0.102569	+	0.994726i	$0.532706\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−13.4164	−1.29701	−0.648507	−	0.761209i	$-0.724606\pi$
$107$	−13.4164	−1.29701	−0.648507	+	0.761209i	$0.724606\pi$
$108$	0	0
$109$	−11.4721	−1.09883	−0.549416	−	0.835549i	$-0.685150\pi$
$109$	−11.4721	−1.09883	−0.549416	+	0.835549i	$0.685150\pi$
$110$	0	0
$111$	−3.70246	−0.351422
$112$	0	0
$113$	7.18034	0.675470	0.337735	−	0.941241i	$-0.390339\pi$
$113$	7.18034	0.675470	0.337735	+	0.941241i	$0.390339\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	−5.11667	−0.473037
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−10.0000	−0.909091
$122$	0	0
$123$	−0.180340	−0.0162607
$124$	0	0
$125$	0	0
$126$	0	0
$127$	18.1246	1.60830	0.804150	−	0.594427i	$-0.202621\pi$
$127$	18.1246	1.60830	0.804150	+	0.594427i	$0.202621\pi$
$128$	0	0
$129$	1.95440	0.172075
$130$	0	0
$131$	−14.8098	−1.29394	−0.646971	−	0.762515i	$-0.723964\pi$
$131$	−14.8098	−1.29394	−0.646971	+	0.762515i	$0.723964\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−18.9443	−1.61852	−0.809259	−	0.587452i	$-0.800131\pi$
$137$	−18.9443	−1.61852	−0.809259	+	0.587452i	$0.800131\pi$
$138$	0	0
$139$	−13.5231	−1.14702	−0.573509	−	0.819199i	$-0.694418\pi$
$139$	−13.5231	−1.14702	−0.573509	+	0.819199i	$0.694418\pi$
$140$	0	0
$141$	−1.23607	−0.104096
$142$	0	0
$143$	−2.28825	−0.191353
$144$	0	0
$145$	0	0
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−15.6525	−1.28230	−0.641150	−	0.767415i	$-0.721543\pi$
$149$	−15.6525	−1.28230	−0.641150	+	0.767415i	$0.721543\pi$
$150$	0	0
$151$	−18.2361	−1.48403	−0.742015	−	0.670383i	$-0.766130\pi$
$151$	−18.2361	−1.48403	−0.742015	+	0.670383i	$0.766130\pi$
$152$	0	0
$153$	3.90879	0.316007
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−0.461370	−0.0368213	−0.0184107	−	0.999831i	$-0.505861\pi$
$157$	−0.461370	−0.0368213	−0.0184107	+	0.999831i	$0.505861\pi$
$158$	0	0
$159$	2.82843	0.224309
$160$	0	0
$161$	0	0
$162$	0	0
$163$	8.76393	0.686444	0.343222	−	0.939254i	$-0.388482\pi$
$163$	8.76393	0.686444	0.343222	+	0.939254i	$0.388482\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−7.27740	−0.563142	−0.281571	−	0.959540i	$-0.590856\pi$
$167$	−7.27740	−0.563142	−0.281571	+	0.959540i	$0.590856\pi$
$168$	0	0
$169$	−7.76393	−0.597226
$170$	0	0
$171$	7.07107	0.540738
$172$	0	0
$173$	16.8918	1.28426	0.642128	−	0.766597i	$-0.278052\pi$
$173$	16.8918	1.28426	0.642128	+	0.766597i	$0.278052\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	1.41641	0.106464
$178$	0	0
$179$	−6.18034	−0.461940	−0.230970	−	0.972961i	$-0.574190\pi$
$179$	−6.18034	−0.461940	−0.230970	+	0.972961i	$0.574190\pi$
$180$	0	0
$181$	−10.4397	−0.775975	−0.387988	−	0.921665i	$-0.626830\pi$
$181$	−10.4397	−0.775975	−0.387988	+	0.921665i	$0.626830\pi$
$182$	0	0
$183$	−1.70820	−0.126274
$184$	0	0
$185$	0	0
$186$	0	0
$187$	1.74806	0.127831
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−21.2361	−1.53659	−0.768294	−	0.640097i	$-0.778894\pi$
$191$	−21.2361	−1.53659	−0.768294	+	0.640097i	$0.778894\pi$
$192$	0	0
$193$	−2.23607	−0.160956	−0.0804778	−	0.996756i	$-0.525645\pi$
$193$	−2.23607	−0.160956	−0.0804778	+	0.996756i	$0.525645\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	0.416408	0.0296678	0.0148339	−	0.999890i	$-0.495278\pi$
$197$	0.416408	0.0296678	0.0148339	+	0.999890i	$0.495278\pi$
$198$	0	0
$199$	3.03476	0.215128	0.107564	−	0.994198i	$-0.465695\pi$
$199$	3.03476	0.215128	0.107564	+	0.994198i	$0.465695\pi$
$200$	0	0
$201$	−1.28669	−0.0907564
$202$	0	0
$203$	0	0
$204$	0	0
$205$	0	0
$206$	0	0
$207$	−2.23607	−0.155417
$208$	0	0
$209$	3.16228	0.218739
$210$	0	0
$211$	−10.4721	−0.720932	−0.360466	−	0.932772i	$-0.617382\pi$
$211$	−10.4721	−0.720932	−0.360466	+	0.932772i	$0.617382\pi$
$212$	0	0
$213$	−5.86319	−0.401739
$214$	0	0
$215$	0	0
$216$	0	0
$217$	0	0
$218$	0	0
$219$	−11.2361	−0.759263
$220$	0	0
$221$	−4.00000	−0.269069
$222$	0	0
$223$	−7.19859	−0.482053	−0.241027	−	0.970519i	$-0.577484\pi$
$223$	−7.19859	−0.482053	−0.241027	+	0.970519i	$0.577484\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	15.5563	1.03251	0.516256	−	0.856435i	$-0.327325\pi$
$227$	15.5563	1.03251	0.516256	+	0.856435i	$0.327325\pi$
$228$	0	0
$229$	2.74962	0.181700	0.0908499	−	0.995865i	$-0.471042\pi$
$229$	2.74962	0.181700	0.0908499	+	0.995865i	$0.471042\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	1.58359	0.103745	0.0518723	−	0.998654i	$-0.483481\pi$
$233$	1.58359	0.103745	0.0518723	+	0.998654i	$0.483481\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	5.19548	0.337483
$238$	0	0
$239$	4.76393	0.308153	0.154077	−	0.988059i	$-0.450760\pi$
$239$	4.76393	0.308153	0.154077	+	0.988059i	$0.450760\pi$
$240$	0	0
$241$	28.2055	1.81687	0.908437	−	0.418022i	$-0.137277\pi$
$241$	28.2055	1.81687	0.908437	+	0.418022i	$0.137277\pi$
$242$	0	0
$243$	16.0965	1.03259
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−7.23607	−0.460420
$248$	0	0
$249$	−1.05573	−0.0669040
$250$	0	0
$251$	4.57649	0.288866	0.144433	−	0.989515i	$-0.453864\pi$
$251$	4.57649	0.288866	0.144433	+	0.989515i	$0.453864\pi$
$252$	0	0
$253$	−1.00000	−0.0628695
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−21.6746	−1.35202	−0.676011	−	0.736891i	$-0.736293\pi$
$257$	−21.6746	−1.35202	−0.676011	+	0.736891i	$0.736293\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	−10.5279	−0.651658
$262$	0	0
$263$	16.7082	1.03027	0.515136	−	0.857108i	$-0.327741\pi$
$263$	16.7082	1.03027	0.515136	+	0.857108i	$0.327741\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	−4.47214	−0.273690
$268$	0	0
$269$	−0.127520	−0.00777500	−0.00388750	−	0.999992i	$-0.501237\pi$
$269$	−0.127520	−0.00777500	−0.00388750	+	0.999992i	$0.501237\pi$
$270$	0	0
$271$	−6.60970	−0.401511	−0.200755	−	0.979641i	$-0.564340\pi$
$271$	−6.60970	−0.401511	−0.200755	+	0.979641i	$0.564340\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−20.9443	−1.25842	−0.629210	−	0.777236i	$-0.716621\pi$
$277$	−20.9443	−1.25842	−0.629210	+	0.777236i	$0.716621\pi$
$278$	0	0
$279$	−15.3500	−0.918982
$280$	0	0
$281$	−11.6525	−0.695128	−0.347564	−	0.937656i	$-0.612991\pi$
$281$	−11.6525	−0.695128	−0.347564	+	0.937656i	$0.612991\pi$
$282$	0	0
$283$	−31.0339	−1.84477	−0.922387	−	0.386268i	$-0.873764\pi$
$283$	−31.0339	−1.84477	−0.922387	+	0.386268i	$0.873764\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−13.9443	−0.820251
$290$	0	0
$291$	−2.58359	−0.151453
$292$	0	0
$293$	21.2619	1.24213	0.621067	−	0.783757i	$-0.286699\pi$
$293$	21.2619	1.24213	0.621067	+	0.783757i	$0.286699\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	4.57649	0.265555
$298$	0	0
$299$	2.28825	0.132333
$300$	0	0
$301$	0	0
$302$	0	0
$303$	10.6525	0.611969
$304$	0	0
$305$	0	0
$306$	0	0
$307$	−23.7078	−1.35307	−0.676537	−	0.736408i	$-0.736520\pi$
$307$	−23.7078	−1.35307	−0.676537	+	0.736408i	$0.736520\pi$
$308$	0	0
$309$	−1.81966	−0.103517
$310$	0	0
$311$	24.3755	1.38221	0.691103	−	0.722756i	$-0.257125\pi$
$311$	24.3755	1.38221	0.691103	+	0.722756i	$0.257125\pi$
$312$	0	0
$313$	2.08191	0.117677	0.0588384	−	0.998268i	$-0.481260\pi$
$313$	2.08191	0.117677	0.0588384	+	0.998268i	$0.481260\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−19.4721	−1.09366	−0.546832	−	0.837242i	$-0.684166\pi$
$317$	−19.4721	−1.09366	−0.546832	+	0.837242i	$0.684166\pi$
$318$	0	0
$319$	−4.70820	−0.263609
$320$	0	0
$321$	−11.7264	−0.654502
$322$	0	0
$323$	5.52786	0.307579
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−10.0270	−0.554495
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−15.0000	−0.824475	−0.412237	−	0.911077i	$-0.635253\pi$
$331$	−15.0000	−0.824475	−0.412237	+	0.911077i	$0.635253\pi$
$332$	0	0
$333$	9.47214	0.519070
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−12.6525	−0.689224	−0.344612	−	0.938745i	$-0.611990\pi$
$337$	−12.6525	−0.689224	−0.344612	+	0.938745i	$0.611990\pi$
$338$	0	0
$339$	6.27585	0.340857
$340$	0	0
$341$	−6.86474	−0.371746
$342$	0	0
$343$	0	0
$344$	0	0
$345$	0	0
$346$	0	0
$347$	24.8885	1.33609	0.668044	−	0.744122i	$-0.267132\pi$
$347$	24.8885	1.33609	0.668044	+	0.744122i	$0.267132\pi$
$348$	0	0
$349$	23.6290	1.26483	0.632415	−	0.774630i	$-0.282064\pi$
$349$	23.6290	1.26483	0.632415	+	0.774630i	$0.282064\pi$
$350$	0	0
$351$	−10.4721	−0.558961
$352$	0	0
$353$	−10.1058	−0.537879	−0.268939	−	0.963157i	$-0.586673\pi$
$353$	−10.1058	−0.537879	−0.268939	+	0.963157i	$0.586673\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	0	0
$358$	0	0
$359$	5.18034	0.273408	0.136704	−	0.990612i	$-0.456349\pi$
$359$	5.18034	0.273408	0.136704	+	0.990612i	$0.456349\pi$
$360$	0	0
$361$	−9.00000	−0.473684
$362$	0	0
$363$	−8.74032	−0.458748
$364$	0	0
$365$	0	0
$366$	0	0
$367$	34.4025	1.79580	0.897898	−	0.440204i	$-0.145094\pi$
$367$	34.4025	1.79580	0.897898	+	0.440204i	$0.145094\pi$
$368$	0	0
$369$	0.461370	0.0240180
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−1.18034	−0.0611157	−0.0305578	−	0.999533i	$-0.509728\pi$
$373$	−1.18034	−0.0611157	−0.0305578	+	0.999533i	$0.509728\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	10.7735	0.554865
$378$	0	0
$379$	−11.1803	−0.574295	−0.287148	−	0.957886i	$-0.592707\pi$
$379$	−11.1803	−0.574295	−0.287148	+	0.957886i	$0.592707\pi$
$380$	0	0
$381$	15.8415	0.811584
$382$	0	0
$383$	−5.11667	−0.261450	−0.130725	−	0.991419i	$-0.541730\pi$
$383$	−5.11667	−0.261450	−0.130725	+	0.991419i	$0.541730\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−5.00000	−0.254164
$388$	0	0
$389$	−0.819660	−0.0415584	−0.0207792	−	0.999784i	$-0.506615\pi$
$389$	−0.819660	−0.0415584	−0.0207792	+	0.999784i	$0.506615\pi$
$390$	0	0
$391$	−1.74806	−0.0884034
$392$	0	0
$393$	−12.9443	−0.652952
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−21.8809	−1.09817	−0.549086	−	0.835766i	$-0.685024\pi$
$397$	−21.8809	−1.09817	−0.549086	+	0.835766i	$0.685024\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−20.7082	−1.03412	−0.517059	−	0.855950i	$-0.672973\pi$
$401$	−20.7082	−1.03412	−0.517059	+	0.855950i	$0.672973\pi$
$402$	0	0
$403$	15.7082	0.782481
$404$	0	0
$405$	0	0
$406$	0	0
$407$	4.23607	0.209974
$408$	0	0
$409$	5.57804	0.275816	0.137908	−	0.990445i	$-0.455962\pi$
$409$	5.57804	0.275816	0.137908	+	0.990445i	$0.455962\pi$
$410$	0	0
$411$	−16.5579	−0.816741
$412$	0	0
$413$	0	0
$414$	0	0
$415$	0	0
$416$	0	0
$417$	−11.8197	−0.578811
$418$	0	0
$419$	−34.7363	−1.69698	−0.848491	−	0.529210i	$-0.822488\pi$
$419$	−34.7363	−1.69698	−0.848491	+	0.529210i	$0.822488\pi$
$420$	0	0
$421$	23.0689	1.12431	0.562154	−	0.827032i	$-0.309973\pi$
$421$	23.0689	1.12431	0.562154	+	0.827032i	$0.309973\pi$
$422$	0	0
$423$	3.16228	0.153755
$424$	0	0
$425$	0	0
$426$	0	0
$427$	0	0
$428$	0	0
$429$	−2.00000	−0.0965609
$430$	0	0
$431$	−30.1803	−1.45374	−0.726868	−	0.686777i	$-0.759025\pi$
$431$	−30.1803	−1.45374	−0.726868	+	0.686777i	$0.759025\pi$
$432$	0	0
$433$	24.3755	1.17141	0.585705	−	0.810524i	$-0.300818\pi$
$433$	24.3755	1.17141	0.585705	+	0.810524i	$0.300818\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−3.16228	−0.151272
$438$	0	0
$439$	−23.6777	−1.13007	−0.565037	−	0.825066i	$-0.691138\pi$
$439$	−23.6777	−1.13007	−0.565037	+	0.825066i	$0.691138\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	36.1803	1.71898	0.859490	−	0.511153i	$-0.170781\pi$
$443$	36.1803	1.71898	0.859490	+	0.511153i	$0.170781\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	−13.6808	−0.647078
$448$	0	0
$449$	1.29180	0.0609636	0.0304818	−	0.999535i	$-0.490296\pi$
$449$	1.29180	0.0609636	0.0304818	+	0.999535i	$0.490296\pi$
$450$	0	0
$451$	0.206331	0.00971575
$452$	0	0
$453$	−15.9389	−0.748875
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−20.2361	−0.946603	−0.473302	−	0.880900i	$-0.656938\pi$
$457$	−20.2361	−0.946603	−0.473302	+	0.880900i	$0.656938\pi$
$458$	0	0
$459$	8.00000	0.373408
$460$	0	0
$461$	−42.1413	−1.96271	−0.981357	−	0.192193i	$-0.938440\pi$
$461$	−42.1413	−1.96271	−0.981357	+	0.192193i	$0.938440\pi$
$462$	0	0
$463$	13.4164	0.623513	0.311757	−	0.950162i	$-0.399083\pi$
$463$	13.4164	0.623513	0.311757	+	0.950162i	$0.399083\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	26.7124	1.23610	0.618052	−	0.786137i	$-0.287922\pi$
$467$	26.7124	1.23610	0.618052	+	0.786137i	$0.287922\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	−0.403252	−0.0185809
$472$	0	0
$473$	−2.23607	−0.102815
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−7.23607	−0.331317
$478$	0	0
$479$	27.5865	1.26046	0.630229	−	0.776409i	$-0.282961\pi$
$479$	27.5865	1.26046	0.630229	+	0.776409i	$0.282961\pi$
$480$	0	0
$481$	−9.69316	−0.441970
$482$	0	0
$483$	0	0
$484$	0	0
$485$	0	0
$486$	0	0
$487$	13.0000	0.589086	0.294543	−	0.955638i	$-0.404833\pi$
$487$	13.0000	0.589086	0.294543	+	0.955638i	$0.404833\pi$
$488$	0	0
$489$	7.65996	0.346395
$490$	0	0
$491$	14.1246	0.637435	0.318717	−	0.947850i	$-0.396748\pi$
$491$	14.1246	0.637435	0.318717	+	0.947850i	$0.396748\pi$
$492$	0	0
$493$	−8.23024	−0.370671
$494$	0	0
$495$	0	0
$496$	0	0
$497$	0	0
$498$	0	0
$499$	−13.0557	−0.584455	−0.292227	−	0.956349i	$-0.594396\pi$
$499$	−13.0557	−0.584455	−0.292227	+	0.956349i	$0.594396\pi$
$500$	0	0
$501$	−6.36068	−0.284174
$502$	0	0
$503$	4.49768	0.200542	0.100271	−	0.994960i	$-0.468029\pi$
$503$	4.49768	0.200542	0.100271	+	0.994960i	$0.468029\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	−6.78593	−0.301374
$508$	0	0
$509$	−29.0795	−1.28893	−0.644463	−	0.764636i	$-0.722919\pi$
$509$	−29.0795	−1.28893	−0.644463	+	0.764636i	$0.722919\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	14.4721	0.638960
$514$	0	0
$515$	0	0
$516$	0	0
$517$	1.41421	0.0621970
$518$	0	0
$519$	14.7639	0.648065
$520$	0	0
$521$	40.1869	1.76062	0.880309	−	0.474400i	$-0.157335\pi$
$521$	40.1869	1.76062	0.880309	+	0.474400i	$0.157335\pi$
$522$	0	0
$523$	30.7000	1.34242	0.671209	−	0.741268i	$-0.265775\pi$
$523$	30.7000	1.34242	0.671209	+	0.741268i	$0.265775\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−12.0000	−0.522728
$528$	0	0
$529$	−22.0000	−0.956522
$530$	0	0
$531$	−3.62365	−0.157253
$532$	0	0
$533$	−0.472136	−0.0204505
$534$	0	0
$535$	0	0
$536$	0	0
$537$	−5.40182	−0.233106
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−42.0132	−1.80629	−0.903143	−	0.429339i	$-0.858746\pi$
$541$	−42.0132	−1.80629	−0.903143	+	0.429339i	$0.858746\pi$
$542$	0	0
$543$	−9.12461	−0.391575
$544$	0	0
$545$	0	0
$546$	0	0
$547$	4.34752	0.185887	0.0929434	−	0.995671i	$-0.470372\pi$
$547$	4.34752	0.185887	0.0929434	+	0.995671i	$0.470372\pi$
$548$	0	0
$549$	4.37016	0.186514
$550$	0	0
$551$	−14.8886	−0.634278
$552$	0	0
$553$	0	0
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−32.2361	−1.36589	−0.682943	−	0.730472i	$-0.739300\pi$
$557$	−32.2361	−1.36589	−0.682943	+	0.730472i	$0.739300\pi$
$558$	0	0
$559$	5.11667	0.216412
$560$	0	0
$561$	1.52786	0.0645065
$562$	0	0
$563$	−12.7279	−0.536418	−0.268209	−	0.963361i	$-0.586432\pi$
$563$	−12.7279	−0.536418	−0.268209	+	0.963361i	$0.586432\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	0	0
$568$	0	0
$569$	12.8885	0.540316	0.270158	−	0.962816i	$-0.412924\pi$
$569$	12.8885	0.540316	0.270158	+	0.962816i	$0.412924\pi$
$570$	0	0
$571$	−30.1246	−1.26068	−0.630338	−	0.776321i	$-0.717084\pi$
$571$	−30.1246	−1.26068	−0.630338	+	0.776321i	$0.717084\pi$
$572$	0	0
$573$	−18.5610	−0.775397
$574$	0	0
$575$	0	0
$576$	0	0
$577$	23.8840	0.994304	0.497152	−	0.867663i	$-0.334379\pi$
$577$	23.8840	0.994304	0.497152	+	0.867663i	$0.334379\pi$
$578$	0	0
$579$	−1.95440	−0.0812219
$580$	0	0
$581$	0	0
$582$	0	0
$583$	−3.23607	−0.134024
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−9.77198	−0.403333	−0.201666	−	0.979454i	$-0.564636\pi$
$587$	−9.77198	−0.403333	−0.201666	+	0.979454i	$0.564636\pi$
$588$	0	0
$589$	−21.7082	−0.894471
$590$	0	0
$591$	0.363954	0.0149711
$592$	0	0
$593$	6.73722	0.276664	0.138332	−	0.990386i	$-0.455826\pi$
$593$	6.73722	0.276664	0.138332	+	0.990386i	$0.455826\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	2.65248	0.108559
$598$	0	0
$599$	25.9443	1.06005	0.530027	−	0.847981i	$-0.322182\pi$
$599$	25.9443	1.06005	0.530027	+	0.847981i	$0.322182\pi$
$600$	0	0
$601$	−1.49302	−0.0609018	−0.0304509	−	0.999536i	$-0.509694\pi$
$601$	−1.49302	−0.0609018	−0.0304509	+	0.999536i	$0.509694\pi$
$602$	0	0
$603$	3.29180	0.134052
$604$	0	0
$605$	0	0
$606$	0	0
$607$	10.7735	0.437284	0.218642	−	0.975805i	$-0.429837\pi$
$607$	10.7735	0.437284	0.218642	+	0.975805i	$0.429837\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−3.23607	−0.130917
$612$	0	0
$613$	−6.05573	−0.244589	−0.122294	−	0.992494i	$-0.539025\pi$
$613$	−6.05573	−0.244589	−0.122294	+	0.992494i	$0.539025\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−48.3050	−1.94468	−0.972342	−	0.233561i	$-0.924962\pi$
$617$	−48.3050	−1.94468	−0.972342	+	0.233561i	$0.924962\pi$
$618$	0	0
$619$	30.8276	1.23906	0.619532	−	0.784971i	$-0.287322\pi$
$619$	30.8276	1.23906	0.619532	+	0.784971i	$0.287322\pi$
$620$	0	0
$621$	−4.57649	−0.183648
$622$	0	0
$623$	0	0
$624$	0	0
$625$	0	0
$626$	0	0
$627$	2.76393	0.110381
$628$	0	0
$629$	7.40492	0.295253
$630$	0	0
$631$	−16.5279	−0.657964	−0.328982	−	0.944336i	$-0.606706\pi$
$631$	−16.5279	−0.657964	−0.328982	+	0.944336i	$0.606706\pi$
$632$	0	0
$633$	−9.15298	−0.363798
$634$	0	0
$635$	0	0
$636$	0	0
$637$	0	0
$638$	0	0
$639$	15.0000	0.593391
$640$	0	0
$641$	26.3050	1.03898	0.519492	−	0.854476i	$-0.326121\pi$
$641$	26.3050	1.03898	0.519492	+	0.854476i	$0.326121\pi$
$642$	0	0
$643$	32.7031	1.28969	0.644843	−	0.764315i	$-0.276923\pi$
$643$	32.7031	1.28969	0.644843	+	0.764315i	$0.276923\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	48.9272	1.92353	0.961763	−	0.273883i	$-0.0883079\pi$
$647$	48.9272	1.92353	0.961763	+	0.273883i	$0.0883079\pi$
$648$	0	0
$649$	−1.62054	−0.0636119
$650$	0	0
$651$	0	0
$652$	0	0
$653$	4.47214	0.175008	0.0875041	−	0.996164i	$-0.472111\pi$
$653$	4.47214	0.175008	0.0875041	+	0.996164i	$0.472111\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	28.7456	1.12147
$658$	0	0
$659$	−1.41641	−0.0551754	−0.0275877	−	0.999619i	$-0.508783\pi$
$659$	−1.41641	−0.0551754	−0.0275877	+	0.999619i	$0.508783\pi$
$660$	0	0
$661$	6.14833	0.239142	0.119571	−	0.992826i	$-0.461848\pi$
$661$	6.14833	0.239142	0.119571	+	0.992826i	$0.461848\pi$
$662$	0	0
$663$	−3.49613	−0.135778
$664$	0	0
$665$	0	0
$666$	0	0
$667$	4.70820	0.182302
$668$	0	0
$669$	−6.29180	−0.243255
$670$	0	0
$671$	1.95440	0.0754486
$672$	0	0
$673$	14.0689	0.542316	0.271158	−	0.962535i	$-0.412593\pi$
$673$	14.0689	0.542316	0.271158	+	0.962535i	$0.412593\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−2.49458	−0.0958744	−0.0479372	−	0.998850i	$-0.515265\pi$
$677$	−2.49458	−0.0958744	−0.0479372	+	0.998850i	$0.515265\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	13.5967	0.521029
$682$	0	0
$683$	33.9443	1.29884	0.649421	−	0.760429i	$-0.275011\pi$
$683$	33.9443	1.29884	0.649421	+	0.760429i	$0.275011\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	2.40325	0.0916898
$688$	0	0
$689$	7.40492	0.282105
$690$	0	0
$691$	−20.9281	−0.796141	−0.398070	−	0.917355i	$-0.630320\pi$
$691$	−20.9281	−0.796141	−0.398070	+	0.917355i	$0.630320\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	0	0
$696$	0	0
$697$	0.360680	0.0136617
$698$	0	0
$699$	1.38411	0.0523519
$700$	0	0
$701$	17.3050	0.653599	0.326800	−	0.945094i	$-0.394030\pi$
$701$	17.3050	0.653599	0.326800	+	0.945094i	$0.394030\pi$
$702$	0	0
$703$	13.3956	0.505225
$704$	0	0
$705$	0	0
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−9.05573	−0.340095	−0.170048	−	0.985436i	$-0.554392\pi$
$709$	−9.05573	−0.340095	−0.170048	+	0.985436i	$0.554392\pi$
$710$	0	0
$711$	−13.2918	−0.498481
$712$	0	0
$713$	6.86474	0.257086
$714$	0	0
$715$	0	0
$716$	0	0
$717$	4.16383	0.155501
$718$	0	0
$719$	25.9172	0.966549	0.483274	−	0.875469i	$-0.339447\pi$
$719$	25.9172	0.966549	0.483274	+	0.875469i	$0.339447\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	24.6525	0.916835
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−39.5980	−1.46861	−0.734304	−	0.678821i	$-0.762491\pi$
$727$	−39.5980	−1.46861	−0.734304	+	0.678821i	$0.762491\pi$
$728$	0	0
$729$	5.94427	0.220158
$730$	0	0
$731$	−3.90879	−0.144572
$732$	0	0
$733$	−6.48218	−0.239425	−0.119712	−	0.992809i	$-0.538197\pi$
$733$	−6.48218	−0.239425	−0.119712	+	0.992809i	$0.538197\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	1.47214	0.0542268
$738$	0	0
$739$	−4.63932	−0.170660	−0.0853301	−	0.996353i	$-0.527194\pi$
$739$	−4.63932	−0.170660	−0.0853301	+	0.996353i	$0.527194\pi$
$740$	0	0
$741$	−6.32456	−0.232338
$742$	0	0
$743$	−0.180340	−0.00661603	−0.00330801	−	0.999995i	$-0.501053\pi$
$743$	−0.180340	−0.00661603	−0.00330801	+	0.999995i	$0.501053\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	2.70091	0.0988210
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−41.8885	−1.52853	−0.764267	−	0.644900i	$-0.776899\pi$
$751$	−41.8885	−1.52853	−0.764267	+	0.644900i	$0.776899\pi$
$752$	0	0
$753$	4.00000	0.145768
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−23.9443	−0.870269	−0.435135	−	0.900365i	$-0.643299\pi$
$757$	−23.9443	−0.870269	−0.435135	+	0.900365i	$0.643299\pi$
$758$	0	0
$759$	−0.874032	−0.0317254
$760$	0	0
$761$	0.667701	0.0242041	0.0121021	−	0.999927i	$-0.496148\pi$
$761$	0.667701	0.0242041	0.0121021	+	0.999927i	$0.496148\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	0	0
$766$	0	0
$767$	3.70820	0.133895
$768$	0	0
$769$	42.8878	1.54657	0.773286	−	0.634057i	$-0.218612\pi$
$769$	42.8878	1.54657	0.773286	+	0.634057i	$0.218612\pi$
$770$	0	0
$771$	−18.9443	−0.682261
$772$	0	0
$773$	−15.7627	−0.566944	−0.283472	−	0.958980i	$-0.591486\pi$
$773$	−15.7627	−0.566944	−0.283472	+	0.958980i	$0.591486\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	0	0
$778$	0	0
$779$	0.652476	0.0233774
$780$	0	0
$781$	6.70820	0.240038
$782$	0	0
$783$	−21.5471	−0.770029
$784$	0	0
$785$	0	0
$786$	0	0
$787$	20.2604	0.722204	0.361102	−	0.932526i	$-0.382401\pi$
$787$	20.2604	0.722204	0.361102	+	0.932526i	$0.382401\pi$
$788$	0	0
$789$	14.6035	0.519899
$790$	0	0
$791$	0	0
$792$	0	0
$793$	−4.47214	−0.158810
$794$	0	0
$795$	0	0
$796$	0	0
$797$	23.7078	0.839773	0.419886	−	0.907577i	$-0.362070\pi$
$797$	23.7078	0.839773	0.419886	+	0.907577i	$0.362070\pi$
$798$	0	0
$799$	2.47214	0.0874579
$800$	0	0
$801$	11.4412	0.404256
$802$	0	0
$803$	12.8554	0.453659
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−0.111456	−0.00392344
$808$	0	0
$809$	−18.7082	−0.657745	−0.328873	−	0.944374i	$-0.606669\pi$
$809$	−18.7082	−0.657745	−0.328873	+	0.944374i	$0.606669\pi$
$810$	0	0
$811$	29.2858	1.02836	0.514182	−	0.857681i	$-0.328096\pi$
$811$	29.2858	1.02836	0.514182	+	0.857681i	$0.328096\pi$
$812$	0	0
$813$	−5.77709	−0.202611
$814$	0	0
$815$	0	0
$816$	0	0
$817$	−7.07107	−0.247385
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−9.41641	−0.328635	−0.164317	−	0.986408i	$-0.552542\pi$
$821$	−9.41641	−0.328635	−0.164317	+	0.986408i	$0.552542\pi$
$822$	0	0
$823$	10.3050	0.359208	0.179604	−	0.983739i	$-0.442518\pi$
$823$	10.3050	0.359208	0.179604	+	0.983739i	$0.442518\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	15.7639	0.548166	0.274083	−	0.961706i	$-0.411626\pi$
$827$	15.7639	0.548166	0.274083	+	0.961706i	$0.411626\pi$
$828$	0	0
$829$	−27.8716	−0.968021	−0.484011	−	0.875062i	$-0.660820\pi$
$829$	−27.8716	−0.968021	−0.484011	+	0.875062i	$0.660820\pi$
$830$	0	0
$831$	−18.3060	−0.635027
$832$	0	0
$833$	0	0
$834$	0	0
$835$	0	0
$836$	0	0
$837$	−31.4164	−1.08591
$838$	0	0
$839$	−19.5927	−0.676414	−0.338207	−	0.941072i	$-0.609820\pi$
$839$	−19.5927	−0.676414	−0.338207	+	0.941072i	$0.609820\pi$
$840$	0	0
$841$	−6.83282	−0.235614
$842$	0	0
$843$	−10.1846	−0.350778
$844$	0	0
$845$	0	0
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−27.1246	−0.930914
$850$	0	0
$851$	−4.23607	−0.145211
$852$	0	0
$853$	41.8561	1.43313	0.716563	−	0.697522i	$-0.245714\pi$
$853$	41.8561	1.43313	0.716563	+	0.697522i	$0.245714\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−11.2650	−0.384805	−0.192403	−	0.981316i	$-0.561628\pi$
$857$	−11.2650	−0.384805	−0.192403	+	0.981316i	$0.561628\pi$
$858$	0	0
$859$	4.08502	0.139379	0.0696895	−	0.997569i	$-0.477799\pi$
$859$	4.08502	0.139379	0.0696895	+	0.997569i	$0.477799\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−16.1246	−0.548888	−0.274444	−	0.961603i	$-0.588494\pi$
$863$	−16.1246	−0.548888	−0.274444	+	0.961603i	$0.588494\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	−12.1877	−0.413917
$868$	0	0
$869$	−5.94427	−0.201646
$870$	0	0
$871$	−3.36861	−0.114141
$872$	0	0
$873$	6.60970	0.223704
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−1.70820	−0.0576819	−0.0288410	−	0.999584i	$-0.509182\pi$
$877$	−1.70820	−0.0576819	−0.0288410	+	0.999584i	$0.509182\pi$
$878$	0	0
$879$	18.5836	0.626809
$880$	0	0
$881$	−29.4922	−0.993616	−0.496808	−	0.867861i	$-0.665495\pi$
$881$	−29.4922	−0.993616	−0.496808	+	0.867861i	$0.665495\pi$
$882$	0	0
$883$	−17.3607	−0.584233	−0.292117	−	0.956383i	$-0.594360\pi$
$883$	−17.3607	−0.584233	−0.292117	+	0.956383i	$0.594360\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−29.8260	−1.00146	−0.500730	−	0.865604i	$-0.666935\pi$
$887$	−29.8260	−1.00146	−0.500730	+	0.865604i	$0.666935\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	−2.70820	−0.0907282
$892$	0	0
$893$	4.47214	0.149654
$894$	0	0
$895$	0	0
$896$	0	0
$897$	2.00000	0.0667781
$898$	0	0
$899$	32.3206	1.07795
$900$	0	0
$901$	−5.65685	−0.188457
$902$	0	0
$903$	0	0
$904$	0	0
$905$	0	0
$906$	0	0
$907$	32.0000	1.06254	0.531271	−	0.847202i	$-0.321714\pi$
$907$	32.0000	1.06254	0.531271	+	0.847202i	$0.321714\pi$
$908$	0	0
$909$	−27.2526	−0.903912
$910$	0	0
$911$	9.36068	0.310133	0.155067	−	0.987904i	$-0.450441\pi$
$911$	9.36068	0.310133	0.155067	+	0.987904i	$0.450441\pi$
$912$	0	0
$913$	1.20788	0.0399751
$914$	0	0
$915$	0	0
$916$	0	0
$917$	0	0
$918$	0	0
$919$	−2.59675	−0.0856588	−0.0428294	−	0.999082i	$-0.513637\pi$
$919$	−2.59675	−0.0856588	−0.0428294	+	0.999082i	$0.513637\pi$
$920$	0	0
$921$	−20.7214	−0.682792
$922$	0	0
$923$	−15.3500	−0.505252
$924$	0	0
$925$	0	0
$926$	0	0
$927$	4.65530	0.152900
$928$	0	0
$929$	−9.35931	−0.307069	−0.153535	−	0.988143i	$-0.549066\pi$
$929$	−9.35931	−0.307069	−0.153535	+	0.988143i	$0.549066\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	21.3050	0.697493
$934$	0	0
$935$	0	0
$936$	0	0
$937$	53.2185	1.73857	0.869287	−	0.494308i	$-0.164578\pi$
$937$	53.2185	1.73857	0.869287	+	0.494308i	$0.164578\pi$
$938$	0	0
$939$	1.81966	0.0593824
$940$	0	0
$941$	−22.5973	−0.736651	−0.368326	−	0.929697i	$-0.620069\pi$
$941$	−22.5973	−0.736651	−0.368326	+	0.929697i	$0.620069\pi$
$942$	0	0
$943$	−0.206331	−0.00671906
$944$	0	0
$945$	0	0
$946$	0	0
$947$	29.8885	0.971247	0.485624	−	0.874168i	$-0.338593\pi$
$947$	29.8885	0.971247	0.485624	+	0.874168i	$0.338593\pi$
$948$	0	0
$949$	−29.4164	−0.954897
$950$	0	0
$951$	−17.0193	−0.551888
$952$	0	0
$953$	−18.2361	−0.590724	−0.295362	−	0.955385i	$-0.595440\pi$
$953$	−18.2361	−0.590724	−0.295362	+	0.955385i	$0.595440\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	−4.11512	−0.133023
$958$	0	0
$959$	0	0
$960$	0	0
$961$	16.1246	0.520149
$962$	0	0
$963$	30.0000	0.966736
$964$	0	0
$965$	0	0
$966$	0	0
$967$	5.52786	0.177764	0.0888821	−	0.996042i	$-0.471671\pi$
$967$	5.52786	0.177764	0.0888821	+	0.996042i	$0.471671\pi$
$968$	0	0
$969$	4.83153	0.155211
$970$	0	0
$971$	32.6243	1.04696	0.523482	−	0.852037i	$-0.324633\pi$
$971$	32.6243	1.04696	0.523482	+	0.852037i	$0.324633\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−5.40325	−0.172865	−0.0864327	−	0.996258i	$-0.527547\pi$
$977$	−5.40325	−0.172865	−0.0864327	+	0.996258i	$0.527547\pi$
$978$	0	0
$979$	5.11667	0.163530
$980$	0	0
$981$	25.6525	0.819021
$982$	0	0
$983$	54.9480	1.75257	0.876284	−	0.481794i	$-0.160015\pi$
$983$	54.9480	1.75257	0.876284	+	0.481794i	$0.160015\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	0	0
$988$	0	0
$989$	2.23607	0.0711028
$990$	0	0
$991$	−14.3475	−0.455764	−0.227882	−	0.973689i	$-0.573180\pi$
$991$	−14.3475	−0.455764	−0.227882	+	0.973689i	$0.573180\pi$
$992$	0	0
$993$	−13.1105	−0.416049
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−20.3879	−0.645691	−0.322845	−	0.946452i	$-0.604639\pi$
$997$	−20.3879	−0.645691	−0.322845	+	0.946452i	$0.604639\pi$
$998$	0	0
$999$	19.3863	0.613356

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9800.2.a.cn.1.3	yes	4
5.4	even	2		9800.2.a.cm.1.2	✓	4
7.6	odd	2	inner	9800.2.a.cn.1.2	yes	4
35.34	odd	2		9800.2.a.cm.1.3	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
9800.2.a.cm.1.2	✓	4	5.4	even	2
9800.2.a.cm.1.3	yes	4	35.34	odd	2
9800.2.a.cn.1.2	yes	4	7.6	odd	2	inner
9800.2.a.cn.1.3	yes	4	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 9800 = 2^{3} \cdot 5^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9800.a (trivial)

\(f(q)\)	\(=\)	\(q+0.874032 q^{3} -2.23607 q^{9} +O(q^{10})\)	\(q+0.874032 q^{3} -2.23607 q^{9} -1.00000 q^{11} +2.28825 q^{13} -1.74806 q^{17} -3.16228 q^{19} +1.00000 q^{23} -4.57649 q^{27} +4.70820 q^{29} +6.86474 q^{31} -0.874032 q^{33} -4.23607 q^{37} +2.00000 q^{39} -0.206331 q^{41} +2.23607 q^{43} -1.41421 q^{47} -1.52786 q^{51} +3.23607 q^{53} -2.76393 q^{57} +1.62054 q^{59} -1.95440 q^{61} -1.47214 q^{67} +0.874032 q^{69} -6.70820 q^{71} -12.8554 q^{73} +5.94427 q^{79} +2.70820 q^{81} -1.20788 q^{83} +4.11512 q^{87} -5.11667 q^{89} +6.00000 q^{93} -2.95595 q^{97} +2.23607 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q+O(q^{10}) \) 4 * q	\( 4 q - 4 q^{11} + 4 q^{23} - 8 q^{29} - 8 q^{37} + 8 q^{39} - 24 q^{51} + 4 q^{53} - 20 q^{57} + 12 q^{67} - 12 q^{79} - 16 q^{81} + 24 q^{93}+O(q^{100}) \) 4 * q - 4 * q^11 + 4 * q^23 - 8 * q^29 - 8 * q^37 + 8 * q^39 - 24 * q^51 + 4 * q^53 - 20 * q^57 + 12 * q^67 - 12 * q^79 - 16 * q^81 + 24 * q^93

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