LMFDB - Embedded newform 980.4.e.c.589.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [980,4,Mod(589,980)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("980.589"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(980, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 0])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 980 = 2^{2} \cdot 5 \cdot 7^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	980.e (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [2,0,0,0,20,0,0,0,4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(9)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$57.8218718056$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 140)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			589.2
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	980.589
Dual form			980.4.e.c.589.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+5.00000i q^{3} +(10.0000 - 5.00000i) q^{5} +2.00000 q^{9} -65.0000 q^{11} -13.0000i q^{13} +(25.0000 + 50.0000i) q^{15} -113.000i q^{17} +16.0000 q^{19} +186.000i q^{23} +(75.0000 - 100.000i) q^{25} +145.000i q^{27} +57.0000 q^{29} +258.000 q^{31} -325.000i q^{33} +134.000i q^{37} +65.0000 q^{39} +414.000 q^{41} -284.000i q^{43} +(20.0000 - 10.0000i) q^{45} +419.000i q^{47} +565.000 q^{51} -130.000i q^{53} +(-650.000 + 325.000i) q^{55} +80.0000i q^{57} +334.000 q^{59} +56.0000 q^{61} +(-65.0000 - 130.000i) q^{65} -534.000i q^{67} -930.000 q^{69} +952.000 q^{71} -502.000i q^{73} +(500.000 + 375.000i) q^{75} +371.000 q^{79} -671.000 q^{81} -1220.00i q^{83} +(-565.000 - 1130.00i) q^{85} +285.000i q^{87} +880.000 q^{89} +1290.00i q^{93} +(160.000 - 80.0000i) q^{95} -241.000i q^{97} -130.000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 20 q^{5} + 4 q^{9} - 130 q^{11} + 50 q^{15} + 32 q^{19} + 150 q^{25} + 114 q^{29} + 516 q^{31} + 130 q^{39} + 828 q^{41} + 40 q^{45} + 1130 q^{51} - 1300 q^{55} + 668 q^{59} + 112 q^{61} - 130 q^{65}+ \cdots - 260 q^{99}+O(q^{100}) $ 2 * q + 20 * q^5 + 4 * q^9 - 130 * q^11 + 50 * q^15 + 32 * q^19 + 150 * q^25 + 114 * q^29 + 516 * q^31 + 130 * q^39 + 828 * q^41 + 40 * q^45 + 1130 * q^51 - 1300 * q^55 + 668 * q^59 + 112 * q^61 - 130 * q^65 - 1860 * q^69 + 1904 * q^71 + 1000 * q^75 + 742 * q^79 - 1342 * q^81 - 1130 * q^85 + 1760 * q^89 + 320 * q^95 - 260 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/980\mathbb{Z}\right)^\times$.

$n$	$101$	$197$	$491$
$\chi(n)$	$1$	$-1$	$1$

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
$3$			5.00000i			0.962250i	0.876652	+	0.481125i	$0.159772\pi$
$3$			5.00000i			0.962250i	−0.876652	+	0.481125i	$0.840228\pi$
$4$		0			0
$5$	10.0000	−	5.00000i	0.894427	−	0.447214i
$5$	10.0000	−	5.00000i	0.894427	−	0.447214i
$6$		0			0
$7$		0			0
$7$		0			0
$8$		0			0
$9$	2.00000			0.0740741
$10$		0			0
$11$	−65.0000			−1.78166			−0.890829	−	0.454339i	$-0.849876\pi$
$11$	−65.0000			−1.78166			−0.890829	+	0.454339i	$0.849876\pi$
$12$		0			0
$13$		−	13.0000i		−	0.277350i	−0.990338	−	0.138675i	$-0.955716\pi$
$13$		−	13.0000i		−	0.277350i	0.990338	−	0.138675i	$-0.0442844\pi$
$14$		0			0
$15$	25.0000	+	50.0000i	0.430331	+	0.860663i
$16$		0			0
$17$		−	113.000i		−	1.61215i	−0.591814	−	0.806074i	$-0.701588\pi$
$17$		−	113.000i		−	1.61215i	0.591814	−	0.806074i	$-0.298412\pi$
$18$		0			0
$19$	16.0000			0.193192			0.0965961	−	0.995324i	$-0.469204\pi$
$19$	16.0000			0.193192			0.0965961	+	0.995324i	$0.469204\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$			186.000i			1.68625i	0.537720	+	0.843124i	$0.319286\pi$
$23$			186.000i			1.68625i	−0.537720	+	0.843124i	$0.680714\pi$
$24$		0			0
$25$	75.0000	−	100.000i	0.600000	−	0.800000i
$26$		0			0
$27$			145.000i			1.03353i
$28$		0			0
$29$	57.0000			0.364987			0.182494	−	0.983207i	$-0.441583\pi$
$29$	57.0000			0.364987			0.182494	+	0.983207i	$0.441583\pi$
$30$		0			0
$31$	258.000			1.49478			0.747390	−	0.664386i	$-0.231307\pi$
$31$	258.000			1.49478			0.747390	+	0.664386i	$0.231307\pi$
$32$		0			0
$33$		−	325.000i		−	1.71440i
$34$		0			0
$35$		0			0
$36$		0			0
$37$			134.000i			0.595391i	0.954661	+	0.297695i	$0.0962180\pi$
$37$			134.000i			0.595391i	−0.954661	+	0.297695i	$0.903782\pi$
$38$		0			0
$39$	65.0000			0.266880
$40$		0			0
$41$	414.000			1.57697			0.788487	−	0.615051i	$-0.210865\pi$
$41$	414.000			1.57697			0.788487	+	0.615051i	$0.210865\pi$
$42$		0			0
$43$		−	284.000i		−	1.00720i	−0.863937	−	0.503600i	$-0.832009\pi$
$43$		−	284.000i		−	1.00720i	0.863937	−	0.503600i	$-0.167991\pi$
$44$		0			0
$45$	20.0000	−	10.0000i	0.0662539	−	0.0331269i
$46$		0			0
$47$			419.000i			1.30037i	0.759776	+	0.650185i	$0.225309\pi$
$47$			419.000i			1.30037i	−0.759776	+	0.650185i	$0.774691\pi$
$48$		0			0
$49$		0			0
$50$		0			0
$51$	565.000			1.55129
$52$		0			0
$53$		−	130.000i		−	0.336922i	−0.985708	−	0.168461i	$-0.946120\pi$
$53$		−	130.000i		−	0.336922i	0.985708	−	0.168461i	$-0.0538797\pi$
$54$		0			0
$55$	−650.000	+	325.000i	−1.59356	+	0.796782i
$56$		0			0
$57$			80.0000i			0.185899i
$58$		0			0
$59$	334.000			0.737002			0.368501	−	0.929627i	$-0.379871\pi$
$59$	334.000			0.737002			0.368501	+	0.929627i	$0.379871\pi$
$60$		0			0
$61$	56.0000			0.117542			0.0587710	−	0.998271i	$-0.481282\pi$
$61$	56.0000			0.117542			0.0587710	+	0.998271i	$0.481282\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−65.0000	−	130.000i	−0.124035	−	0.248069i
$66$		0			0
$67$		−	534.000i		−	0.973709i	−0.873483	−	0.486854i	$-0.838144\pi$
$67$		−	534.000i		−	0.973709i	0.873483	−	0.486854i	$-0.161856\pi$
$68$		0			0
$69$	−930.000			−1.62259
$70$		0			0
$71$	952.000			1.59129			0.795645	−	0.605763i	$-0.207132\pi$
$71$	952.000			1.59129			0.795645	+	0.605763i	$0.207132\pi$
$72$		0			0
$73$		−	502.000i		−	0.804858i	−0.915451	−	0.402429i	$-0.868166\pi$
$73$		−	502.000i		−	0.804858i	0.915451	−	0.402429i	$-0.131834\pi$
$74$		0			0
$75$	500.000	+	375.000i	0.769800	+	0.577350i
$76$		0			0
$77$		0			0
$78$		0			0
$79$	371.000			0.528364			0.264182	−	0.964473i	$-0.414898\pi$
$79$	371.000			0.528364			0.264182	+	0.964473i	$0.414898\pi$
$80$		0			0
$81$	−671.000			−0.920439
$82$		0			0
$83$		−	1220.00i		−	1.61340i	−0.590960	−	0.806701i	$-0.701251\pi$
$83$		−	1220.00i		−	1.61340i	0.590960	−	0.806701i	$-0.298749\pi$
$84$		0			0
$85$	−565.000	−	1130.00i	−0.720975	−	1.44195i
$86$		0			0
$87$			285.000i			0.351209i
$88$		0			0
$89$	880.000			1.04809			0.524044	−	0.851691i	$-0.324423\pi$
$89$	880.000			1.04809			0.524044	+	0.851691i	$0.324423\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$			1290.00i			1.43835i
$94$		0			0
$95$	160.000	−	80.0000i	0.172796	−	0.0863982i
$96$		0			0
$97$		−	241.000i		−	0.252266i	−0.992013	−	0.126133i	$-0.959743\pi$
$97$		−	241.000i		−	0.252266i	0.992013	−	0.126133i	$-0.0402567\pi$
$98$		0			0
$99$	−130.000			−0.131975
$100$		0			0
$101$	148.000			0.145807			0.0729037	−	0.997339i	$-0.476773\pi$
$101$	148.000			0.145807			0.0729037	+	0.997339i	$0.476773\pi$
$102$		0			0
$103$			501.000i			0.479272i	0.970863	+	0.239636i	$0.0770281\pi$
$103$			501.000i			0.479272i	−0.970863	+	0.239636i	$0.922972\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		−	744.000i		−	0.672198i	−0.941827	−	0.336099i	$-0.890892\pi$
$107$		−	744.000i		−	0.672198i	0.941827	−	0.336099i	$-0.109108\pi$
$108$		0			0
$109$	−441.000			−0.387524			−0.193762	−	0.981049i	$-0.562069\pi$
$109$	−441.000			−0.387524			−0.193762	+	0.981049i	$0.562069\pi$
$110$		0			0
$111$	−670.000			−0.572915
$112$		0			0
$113$			2118.00i			1.76323i	0.471972	+	0.881614i	$0.343542\pi$
$113$			2118.00i			1.76323i	−0.471972	+	0.881614i	$0.656458\pi$
$114$		0			0
$115$	930.000	+	1860.00i	0.754113	+	1.50823i
$116$		0			0
$117$		−	26.0000i		−	0.0205445i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	2894.00			2.17431
$122$		0			0
$123$			2070.00i			1.51744i
$124$		0			0
$125$	250.000	−	1375.00i	0.178885	−	0.983870i
$126$		0			0
$127$			1794.00i			1.25348i	0.779229	+	0.626739i	$0.215611\pi$
$127$			1794.00i			1.25348i	−0.779229	+	0.626739i	$0.784389\pi$
$128$		0			0
$129$	1420.00			0.969179
$130$		0			0
$131$	494.000			0.329473			0.164737	−	0.986338i	$-0.447323\pi$
$131$	494.000			0.329473			0.164737	+	0.986338i	$0.447323\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$	725.000	+	1450.00i	0.462208	+	0.924416i
$136$		0			0
$137$			964.000i			0.601168i	0.953755	+	0.300584i	$0.0971816\pi$
$137$			964.000i			0.601168i	−0.953755	+	0.300584i	$0.902818\pi$
$138$		0			0
$139$	466.000			0.284357			0.142178	−	0.989841i	$-0.454589\pi$
$139$	466.000			0.284357			0.142178	+	0.989841i	$0.454589\pi$
$140$		0			0
$141$	−2095.00			−1.25128
$142$		0			0
$143$			845.000i			0.494143i
$144$		0			0
$145$	570.000	−	285.000i	0.326455	−	0.163227i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	3246.00			1.78472			0.892358	−	0.451328i	$-0.149050\pi$
$149$	3246.00			1.78472			0.892358	+	0.451328i	$0.149050\pi$
$150$		0			0
$151$	−643.000			−0.346534			−0.173267	−	0.984875i	$-0.555432\pi$
$151$	−643.000			−0.346534			−0.173267	+	0.984875i	$0.555432\pi$
$152$		0			0
$153$		−	226.000i		−	0.119418i
$154$		0			0
$155$	2580.00	−	1290.00i	1.33697	−	0.668486i
$156$		0			0
$157$			1654.00i			0.840787i	0.907342	+	0.420394i	$0.138108\pi$
$157$			1654.00i			0.840787i	−0.907342	+	0.420394i	$0.861892\pi$
$158$		0			0
$159$	650.000			0.324203
$160$		0			0
$161$		0			0
$162$		0			0
$163$			1062.00i			0.510321i	0.966899	+	0.255160i	$0.0821282\pi$
$163$			1062.00i			0.510321i	−0.966899	+	0.255160i	$0.917872\pi$
$164$		0			0
$165$	−1625.00	−	3250.00i	−0.766704	−	1.53341i
$166$		0			0
$167$			2027.00i			0.939245i	0.882867	+	0.469623i	$0.155610\pi$
$167$			2027.00i			0.939245i	−0.882867	+	0.469623i	$0.844390\pi$
$168$		0			0
$169$	2028.00			0.923077
$170$		0			0
$171$	32.0000			0.0143105
$172$		0			0
$173$		−	1579.00i		−	0.693926i	−0.937879	−	0.346963i	$-0.887213\pi$
$173$		−	1579.00i		−	0.693926i	0.937879	−	0.346963i	$-0.112787\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$			1670.00i			0.709180i
$178$		0			0
$179$	−2788.00			−1.16416			−0.582081	−	0.813131i	$-0.697761\pi$
$179$	−2788.00			−1.16416			−0.582081	+	0.813131i	$0.697761\pi$
$180$		0			0
$181$	−2614.00			−1.07346			−0.536732	−	0.843753i	$-0.680341\pi$
$181$	−2614.00			−1.07346			−0.536732	+	0.843753i	$0.680341\pi$
$182$		0			0
$183$			280.000i			0.113105i
$184$		0			0
$185$	670.000	+	1340.00i	0.266267	+	0.532534i
$186$		0			0
$187$			7345.00i			2.87230i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−805.000			−0.304962			−0.152481	−	0.988306i	$-0.548726\pi$
$191$	−805.000			−0.304962			−0.152481	+	0.988306i	$0.548726\pi$
$192$		0			0
$193$		−	3552.00i		−	1.32476i	−0.749168	−	0.662380i	$-0.769547\pi$
$193$		−	3552.00i		−	1.32476i	0.749168	−	0.662380i	$-0.230453\pi$
$194$		0			0
$195$	650.000	−	325.000i	0.238705	−	0.119352i
$196$		0			0
$197$			3462.00i			1.25207i	0.779796	+	0.626034i	$0.215323\pi$
$197$			3462.00i			1.25207i	−0.779796	+	0.626034i	$0.784677\pi$
$198$		0			0
$199$	−86.0000			−0.0306351			−0.0153175	−	0.999883i	$-0.504876\pi$
$199$	−86.0000			−0.0306351			−0.0153175	+	0.999883i	$0.504876\pi$
$200$		0			0
$201$	2670.00			0.936952
$202$		0			0
$203$		0			0
$204$		0			0
$205$	4140.00	−	2070.00i	1.41049	−	0.705244i
$206$		0			0
$207$			372.000i			0.124907i
$208$		0			0
$209$	−1040.00			−0.344202
$210$		0			0
$211$	−311.000			−0.101470			−0.0507349	−	0.998712i	$-0.516156\pi$
$211$	−311.000			−0.101470			−0.0507349	+	0.998712i	$0.516156\pi$
$212$		0			0
$213$			4760.00i			1.53122i
$214$		0			0
$215$	−1420.00	−	2840.00i	−0.450433	−	0.900867i
$216$		0			0
$217$		0			0
$218$		0			0
$219$	2510.00			0.774475
$220$		0			0
$221$	−1469.00			−0.447130
$222$		0			0
$223$		−	1339.00i		−	0.402090i	−0.979582	−	0.201045i	$-0.935566\pi$
$223$		−	1339.00i		−	0.402090i	0.979582	−	0.201045i	$-0.0644337\pi$
$224$		0			0
$225$	150.000	−	200.000i	0.0444444	−	0.0592593i
$226$		0			0
$227$		−	3685.00i		−	1.07745i	−0.842480	−	0.538727i	$-0.818905\pi$
$227$		−	3685.00i		−	1.07745i	0.842480	−	0.538727i	$-0.181095\pi$
$228$		0			0
$229$	−466.000			−0.134472			−0.0672361	−	0.997737i	$-0.521418\pi$
$229$	−466.000			−0.134472			−0.0672361	+	0.997737i	$0.521418\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$			144.000i			0.0404882i	0.999795	+	0.0202441i	$0.00644434\pi$
$233$			144.000i			0.0404882i	−0.999795	+	0.0202441i	$0.993556\pi$
$234$		0			0
$235$	2095.00	+	4190.00i	0.581544	+	1.16309i
$236$		0			0
$237$			1855.00i			0.508419i
$238$		0			0
$239$	1255.00			0.339662			0.169831	−	0.985473i	$-0.445678\pi$
$239$	1255.00			0.339662			0.169831	+	0.985473i	$0.445678\pi$
$240$		0			0
$241$	−3782.00			−1.01087			−0.505436	−	0.862864i	$-0.668668\pi$
$241$	−3782.00			−1.01087			−0.505436	+	0.862864i	$0.668668\pi$
$242$		0			0
$243$			560.000i			0.147835i
$244$		0			0
$245$		0			0
$246$		0			0
$247$		−	208.000i		−	0.0535819i
$248$		0			0
$249$	6100.00			1.55250
$250$		0			0
$251$	−1146.00			−0.288187			−0.144093	−	0.989564i	$-0.546027\pi$
$251$	−1146.00			−0.288187			−0.144093	+	0.989564i	$0.546027\pi$
$252$		0			0
$253$		−	12090.0i		−	3.00432i
$254$		0			0
$255$	5650.00	−	2825.00i	1.38752	−	0.693758i
$256$		0			0
$257$		−	3946.00i		−	0.957762i	−0.877880	−	0.478881i	$-0.841043\pi$
$257$		−	3946.00i		−	0.957762i	0.877880	−	0.478881i	$-0.158957\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$	114.000			0.0270361
$262$		0			0
$263$		−	2952.00i		−	0.692122i	−0.938212	−	0.346061i	$-0.887519\pi$
$263$		−	2952.00i		−	0.692122i	0.938212	−	0.346061i	$-0.112481\pi$
$264$		0			0
$265$	−650.000	−	1300.00i	−0.150676	−	0.301352i
$266$		0			0
$267$			4400.00i			1.00852i
$268$		0			0
$269$	8210.00			1.86086			0.930432	−	0.366464i	$-0.119432\pi$
$269$	8210.00			1.86086			0.930432	+	0.366464i	$0.119432\pi$
$270$		0			0
$271$	1960.00			0.439341			0.219671	−	0.975574i	$-0.429502\pi$
$271$	1960.00			0.439341			0.219671	+	0.975574i	$0.429502\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−4875.00	+	6500.00i	−1.06899	+	1.42533i
$276$		0			0
$277$			2422.00i			0.525357i	0.964883	+	0.262678i	$0.0846059\pi$
$277$			2422.00i			0.525357i	−0.964883	+	0.262678i	$0.915394\pi$
$278$		0			0
$279$	516.000			0.110724
$280$		0			0
$281$	−5249.00			−1.11434			−0.557169	−	0.830399i	$-0.688113\pi$
$281$	−5249.00			−1.11434			−0.557169	+	0.830399i	$0.688113\pi$
$282$		0			0
$283$		−	6209.00i		−	1.30419i	−0.758136	−	0.652097i	$-0.773890\pi$
$283$		−	6209.00i		−	1.30419i	0.758136	−	0.652097i	$-0.226110\pi$
$284$		0			0
$285$	400.000	+	800.000i	0.0831367	+	0.166273i
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−7856.00			−1.59902
$290$		0			0
$291$	1205.00			0.242743
$292$		0			0
$293$		−	2659.00i		−	0.530172i	−0.964225	−	0.265086i	$-0.914600\pi$
$293$		−	2659.00i		−	0.530172i	0.964225	−	0.265086i	$-0.0854004\pi$
$294$		0			0
$295$	3340.00	−	1670.00i	0.659194	−	0.329597i
$296$		0			0
$297$		−	9425.00i		−	1.84139i
$298$		0			0
$299$	2418.00			0.467681
$300$		0			0
$301$		0			0
$302$		0			0
$303$			740.000i			0.140303i
$304$		0			0
$305$	560.000	−	280.000i	0.105133	−	0.0525664i
$306$		0			0
$307$		−	163.000i		−	0.0303026i	−0.999885	−	0.0151513i	$-0.995177\pi$
$307$		−	163.000i		−	0.0303026i	0.999885	−	0.0151513i	$-0.00482299\pi$
$308$		0			0
$309$	−2505.00			−0.461180
$310$		0			0
$311$	−4516.00			−0.823405			−0.411702	−	0.911318i	$-0.635066\pi$
$311$	−4516.00			−0.823405			−0.411702	+	0.911318i	$0.635066\pi$
$312$		0			0
$313$			763.000i			0.137787i	0.997624	+	0.0688935i	$0.0219469\pi$
$313$			763.000i			0.137787i	−0.997624	+	0.0688935i	$0.978053\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		−	7868.00i		−	1.39404i	−0.717051	−	0.697020i	$-0.754509\pi$
$317$		−	7868.00i		−	1.39404i	0.717051	−	0.697020i	$-0.245491\pi$
$318$		0			0
$319$	−3705.00			−0.650283
$320$		0			0
$321$	3720.00			0.646823
$322$		0			0
$323$		−	1808.00i		−	0.311455i
$324$		0			0
$325$	−1300.00	−	975.000i	−0.221880	−	0.166410i
$326$		0			0
$327$		−	2205.00i		−	0.372895i
$328$		0			0
$329$		0			0
$330$		0			0
$331$	3204.00			0.532048			0.266024	−	0.963966i	$-0.414290\pi$
$331$	3204.00			0.532048			0.266024	+	0.963966i	$0.414290\pi$
$332$		0			0
$333$			268.000i			0.0441030i
$334$		0			0
$335$	−2670.00	−	5340.00i	−0.435456	−	0.870912i
$336$		0			0
$337$			7874.00i			1.27277i	0.771371	+	0.636386i	$0.219571\pi$
$337$			7874.00i			1.27277i	−0.771371	+	0.636386i	$0.780429\pi$
$338$		0			0
$339$	−10590.0			−1.69667
$340$		0			0
$341$	−16770.0			−2.66319
$342$		0			0
$343$		0			0
$344$		0			0
$345$	−9300.00	+	4650.00i	−1.45129	+	0.725645i
$346$		0			0
$347$		−	2522.00i		−	0.390167i	−0.980787	−	0.195084i	$-0.937502\pi$
$347$		−	2522.00i		−	0.390167i	0.980787	−	0.195084i	$-0.0624978\pi$
$348$		0			0
$349$	−4404.00			−0.675475			−0.337737	−	0.941240i	$-0.609662\pi$
$349$	−4404.00			−0.675475			−0.337737	+	0.941240i	$0.609662\pi$
$350$		0			0
$351$	1885.00			0.286649
$352$		0			0
$353$			7629.00i			1.15029i	0.818053	+	0.575143i	$0.195053\pi$
$353$			7629.00i			1.15029i	−0.818053	+	0.575143i	$0.804947\pi$
$354$		0			0
$355$	9520.00	−	4760.00i	1.42329	−	0.711647i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−6020.00			−0.885024			−0.442512	−	0.896763i	$-0.645913\pi$
$359$	−6020.00			−0.885024			−0.442512	+	0.896763i	$0.645913\pi$
$360$		0			0
$361$	−6603.00			−0.962677
$362$		0			0
$363$			14470.0i			2.09223i
$364$		0			0
$365$	−2510.00	−	5020.00i	−0.359944	−	0.719887i
$366$		0			0
$367$			1331.00i			0.189312i	0.995510	+	0.0946562i	$0.0301752\pi$
$367$			1331.00i			0.189312i	−0.995510	+	0.0946562i	$0.969825\pi$
$368$		0			0
$369$	828.000			0.116813
$370$		0			0
$371$		0			0
$372$		0			0
$373$		−	8616.00i		−	1.19603i	−0.801485	−	0.598016i	$-0.795956\pi$
$373$		−	8616.00i		−	1.19603i	0.801485	−	0.598016i	$-0.204044\pi$
$374$		0			0
$375$	6875.00	+	1250.00i	0.946729	+	0.172133i
$376$		0			0
$377$		−	741.000i		−	0.101229i
$378$		0			0
$379$	5572.00			0.755183			0.377592	−	0.925972i	$-0.376752\pi$
$379$	5572.00			0.755183			0.377592	+	0.925972i	$0.376752\pi$
$380$		0			0
$381$	−8970.00			−1.20616
$382$		0			0
$383$			2352.00i			0.313790i	0.987615	+	0.156895i	$0.0501484\pi$
$383$			2352.00i			0.313790i	−0.987615	+	0.156895i	$0.949852\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		−	568.000i		−	0.0746074i
$388$		0			0
$389$	6831.00			0.890348			0.445174	−	0.895444i	$-0.353142\pi$
$389$	6831.00			0.890348			0.445174	+	0.895444i	$0.353142\pi$
$390$		0			0
$391$	21018.0			2.71848
$392$		0			0
$393$			2470.00i			0.317036i
$394$		0			0
$395$	3710.00	−	1855.00i	0.472583	−	0.236292i
$396$		0			0
$397$			3499.00i			0.442342i	0.975235	+	0.221171i	$0.0709879\pi$
$397$			3499.00i			0.442342i	−0.975235	+	0.221171i	$0.929012\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	6813.00			0.848441			0.424221	−	0.905559i	$-0.360548\pi$
$401$	6813.00			0.848441			0.424221	+	0.905559i	$0.360548\pi$
$402$		0			0
$403$		−	3354.00i		−	0.414577i
$404$		0			0
$405$	−6710.00	+	3355.00i	−0.823266	+	0.411633i
$406$		0			0
$407$		−	8710.00i		−	1.06078i
$408$		0			0
$409$	8020.00			0.969593			0.484796	−	0.874627i	$-0.338894\pi$
$409$	8020.00			0.969593			0.484796	+	0.874627i	$0.338894\pi$
$410$		0			0
$411$	−4820.00			−0.578475
$412$		0			0
$413$		0			0
$414$		0			0
$415$	−6100.00	−	12200.0i	−0.721535	−	1.44307i
$416$		0			0
$417$			2330.00i			0.273623i
$418$		0			0
$419$	8538.00			0.995486			0.497743	−	0.867325i	$-0.334162\pi$
$419$	8538.00			0.995486			0.497743	+	0.867325i	$0.334162\pi$
$420$		0			0
$421$	−5735.00			−0.663912			−0.331956	−	0.943295i	$-0.607708\pi$
$421$	−5735.00			−0.663912			−0.331956	+	0.943295i	$0.607708\pi$
$422$		0			0
$423$			838.000i			0.0963238i
$424$		0			0
$425$	−11300.0	−	8475.00i	−1.28972	−	0.967289i
$426$		0			0
$427$		0			0
$428$		0			0
$429$	−4225.00			−0.475489
$430$		0			0
$431$	−6945.00			−0.776169			−0.388085	−	0.921624i	$-0.626863\pi$
$431$	−6945.00			−0.776169			−0.388085	+	0.921624i	$0.626863\pi$
$432$		0			0
$433$			6374.00i			0.707425i	0.935354	+	0.353712i	$0.115081\pi$
$433$			6374.00i			0.707425i	−0.935354	+	0.353712i	$0.884919\pi$
$434$		0			0
$435$	1425.00	+	2850.00i	0.157066	+	0.314131i
$436$		0			0
$437$			2976.00i			0.325770i
$438$		0			0
$439$	−13758.0			−1.49575			−0.747874	−	0.663841i	$-0.768925\pi$
$439$	−13758.0			−1.49575			−0.747874	+	0.663841i	$0.768925\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$		−	10812.0i		−	1.15958i	−0.814766	−	0.579790i	$-0.803135\pi$
$443$		−	10812.0i		−	1.15958i	0.814766	−	0.579790i	$-0.196865\pi$
$444$		0			0
$445$	8800.00	−	4400.00i	0.937438	−	0.468719i
$446$		0			0
$447$			16230.0i			1.71734i
$448$		0			0
$449$	−10203.0			−1.07240			−0.536202	−	0.844090i	$-0.680141\pi$
$449$	−10203.0			−1.07240			−0.536202	+	0.844090i	$0.680141\pi$
$450$		0			0
$451$	−26910.0			−2.80963
$452$		0			0
$453$		−	3215.00i		−	0.333452i
$454$		0			0
$455$		0			0
$456$		0			0
$457$			14198.0i			1.45329i	0.687012	+	0.726646i	$0.258922\pi$
$457$			14198.0i			1.45329i	−0.687012	+	0.726646i	$0.741078\pi$
$458$		0			0
$459$	16385.0			1.66620
$460$		0			0
$461$	15876.0			1.60395			0.801973	−	0.597360i	$-0.203784\pi$
$461$	15876.0			1.60395			0.801973	+	0.597360i	$0.203784\pi$
$462$		0			0
$463$			16524.0i			1.65861i	0.558798	+	0.829304i	$0.311263\pi$
$463$			16524.0i			1.65861i	−0.558798	+	0.829304i	$0.688737\pi$
$464$		0			0
$465$	6450.00	+	12900.0i	0.643251	+	1.28650i
$466$		0			0
$467$			7193.00i			0.712746i	0.934344	+	0.356373i	$0.115987\pi$
$467$			7193.00i			0.712746i	−0.934344	+	0.356373i	$0.884013\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$	−8270.00			−0.809048
$472$		0			0
$473$			18460.0i			1.79449i
$474$		0			0
$475$	1200.00	−	1600.00i	0.115915	−	0.154554i
$476$		0			0
$477$		−	260.000i		−	0.0249572i
$478$		0			0
$479$	−7810.00			−0.744985			−0.372493	−	0.928035i	$-0.621497\pi$
$479$	−7810.00			−0.744985			−0.372493	+	0.928035i	$0.621497\pi$
$480$		0			0
$481$	1742.00			0.165132
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−1205.00	−	2410.00i	−0.112817	−	0.225634i
$486$		0			0
$487$			11514.0i			1.07135i	0.844423	+	0.535677i	$0.179943\pi$
$487$			11514.0i			1.07135i	−0.844423	+	0.535677i	$0.820057\pi$
$488$		0			0
$489$	−5310.00			−0.491056
$490$		0			0
$491$	−6435.00			−0.591461			−0.295731	−	0.955271i	$-0.595563\pi$
$491$	−6435.00			−0.591461			−0.295731	+	0.955271i	$0.595563\pi$
$492$		0			0
$493$		−	6441.00i		−	0.588414i
$494$		0			0
$495$	−1300.00	+	650.000i	−0.118042	+	0.0590209i
$496$		0			0
$497$		0			0
$498$		0			0
$499$	−18023.0			−1.61687			−0.808437	−	0.588582i	$-0.799686\pi$
$499$	−18023.0			−1.61687			−0.808437	+	0.588582i	$0.799686\pi$
$500$		0			0
$501$	−10135.0			−0.903789
$502$		0			0
$503$		−	19809.0i		−	1.75594i	−0.478712	−	0.877972i	$-0.658896\pi$
$503$		−	19809.0i		−	1.75594i	0.478712	−	0.877972i	$-0.341104\pi$
$504$		0			0
$505$	1480.00	−	740.000i	0.130414	−	0.0652071i
$506$		0			0
$507$			10140.0i			0.888231i
$508$		0			0
$509$	−598.000			−0.0520744			−0.0260372	−	0.999661i	$-0.508289\pi$
$509$	−598.000			−0.0520744			−0.0260372	+	0.999661i	$0.508289\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$			2320.00i			0.199670i
$514$		0			0
$515$	2505.00	+	5010.00i	0.214337	+	0.428674i
$516$		0			0
$517$		−	27235.0i		−	2.31682i
$518$		0			0
$519$	7895.00			0.667730
$520$		0			0
$521$	−11124.0			−0.935415			−0.467708	−	0.883883i	$-0.654920\pi$
$521$	−11124.0			−0.935415			−0.467708	+	0.883883i	$0.654920\pi$
$522$		0			0
$523$		−	3876.00i		−	0.324064i	−0.986785	−	0.162032i	$-0.948195\pi$
$523$		−	3876.00i		−	0.324064i	0.986785	−	0.162032i	$-0.0518048\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		−	29154.0i		−	2.40981i
$528$		0			0
$529$	−22429.0			−1.84343
$530$		0			0
$531$	668.000			0.0545927
$532$		0			0
$533$		−	5382.00i		−	0.437374i
$534$		0			0
$535$	−3720.00	−	7440.00i	−0.300616	−	0.601232i
$536$		0			0
$537$		−	13940.0i		−	1.12021i
$538$		0			0
$539$		0			0
$540$		0			0
$541$	−5321.00			−0.422861			−0.211430	−	0.977393i	$-0.567812\pi$
$541$	−5321.00			−0.422861			−0.211430	+	0.977393i	$0.567812\pi$
$542$		0			0
$543$		−	13070.0i		−	1.03294i
$544$		0			0
$545$	−4410.00	+	2205.00i	−0.346612	+	0.173306i
$546$		0			0
$547$		−	496.000i		−	0.0387704i	−0.999812	−	0.0193852i	$-0.993829\pi$
$547$		−	496.000i		−	0.0387704i	0.999812	−	0.0193852i	$-0.00617089\pi$
$548$		0			0
$549$	112.000			0.00870682
$550$		0			0
$551$	912.000			0.0705127
$552$		0			0
$553$		0			0
$554$		0			0
$555$	−6700.00	+	3350.00i	−0.512431	+	0.256215i
$556$		0			0
$557$		−	2292.00i		−	0.174354i	−0.996193	−	0.0871770i	$-0.972215\pi$
$557$		−	2292.00i		−	0.174354i	0.996193	−	0.0871770i	$-0.0277846\pi$
$558$		0			0
$559$	−3692.00			−0.279347
$560$		0			0
$561$	−36725.0			−2.76387
$562$		0			0
$563$		−	14568.0i		−	1.09053i	−0.838264	−	0.545265i	$-0.816429\pi$
$563$		−	14568.0i		−	1.09053i	0.838264	−	0.545265i	$-0.183571\pi$
$564$		0			0
$565$	10590.0	+	21180.0i	0.788539	+	1.57708i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−498.000			−0.0366911			−0.0183456	−	0.999832i	$-0.505840\pi$
$569$	−498.000			−0.0366911			−0.0183456	+	0.999832i	$0.505840\pi$
$570$		0			0
$571$	−2300.00			−0.168567			−0.0842837	−	0.996442i	$-0.526860\pi$
$571$	−2300.00			−0.168567			−0.0842837	+	0.996442i	$0.526860\pi$
$572$		0			0
$573$		−	4025.00i		−	0.293450i
$574$		0			0
$575$	18600.0	+	13950.0i	1.34900	+	1.01175i
$576$		0			0
$577$		−	13347.0i		−	0.962986i	−0.876450	−	0.481493i	$-0.840095\pi$
$577$		−	13347.0i		−	0.962986i	0.876450	−	0.481493i	$-0.159905\pi$
$578$		0			0
$579$	17760.0			1.27475
$580$		0			0
$581$		0			0
$582$		0			0
$583$			8450.00i			0.600280i
$584$		0			0
$585$	−130.000	−	260.000i	−0.00918776	−	0.0183755i
$586$		0			0
$587$			15764.0i			1.10843i	0.832373	+	0.554216i	$0.186982\pi$
$587$			15764.0i			1.10843i	−0.832373	+	0.554216i	$0.813018\pi$
$588$		0			0
$589$	4128.00			0.288780
$590$		0			0
$591$	−17310.0			−1.20480
$592$		0			0
$593$		−	4913.00i		−	0.340224i	−0.985425	−	0.170112i	$-0.945587\pi$
$593$		−	4913.00i		−	0.340224i	0.985425	−	0.170112i	$-0.0544129\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		−	430.000i		−	0.0294786i
$598$		0			0
$599$	26089.0			1.77958			0.889789	−	0.456371i	$-0.150851\pi$
$599$	26089.0			1.77958			0.889789	+	0.456371i	$0.150851\pi$
$600$		0			0
$601$	−28440.0			−1.93027			−0.965135	−	0.261754i	$-0.915699\pi$
$601$	−28440.0			−1.93027			−0.965135	+	0.261754i	$0.915699\pi$
$602$		0			0
$603$		−	1068.00i		−	0.0721266i
$604$		0			0
$605$	28940.0	−	14470.0i	1.94476	−	0.972379i
$606$		0			0
$607$		−	22235.0i		−	1.48681i	−0.668844	−	0.743403i	$-0.733211\pi$
$607$		−	22235.0i		−	1.48681i	0.668844	−	0.743403i	$-0.266789\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	5447.00			0.360658
$612$		0			0
$613$			16612.0i			1.09454i	0.836956	+	0.547269i	$0.184333\pi$
$613$			16612.0i			1.09454i	−0.836956	+	0.547269i	$0.815667\pi$
$614$		0			0
$615$	10350.0	+	20700.0i	0.678622	+	1.35724i
$616$		0			0
$617$		−	29814.0i		−	1.94533i	−0.232220	−	0.972663i	$-0.574599\pi$
$617$		−	29814.0i		−	1.94533i	0.232220	−	0.972663i	$-0.425401\pi$
$618$		0			0
$619$	24606.0			1.59774			0.798868	−	0.601506i	$-0.205433\pi$
$619$	24606.0			1.59774			0.798868	+	0.601506i	$0.205433\pi$
$620$		0			0
$621$	−26970.0			−1.74278
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−4375.00	−	15000.0i	−0.280000	−	0.960000i
$626$		0			0
$627$		−	5200.00i		−	0.331209i
$628$		0			0
$629$	15142.0			0.959859
$630$		0			0
$631$	−12501.0			−0.788680			−0.394340	−	0.918965i	$-0.629027\pi$
$631$	−12501.0			−0.788680			−0.394340	+	0.918965i	$0.629027\pi$
$632$		0			0
$633$		−	1555.00i		−	0.0976393i
$634$		0			0
$635$	8970.00	+	17940.0i	0.560573	+	1.12115i
$636$		0			0
$637$		0			0
$638$		0			0
$639$	1904.00			0.117873
$640$		0			0
$641$	10998.0			0.677683			0.338842	−	0.940843i	$-0.389965\pi$
$641$	10998.0			0.677683			0.338842	+	0.940843i	$0.389965\pi$
$642$		0			0
$643$		−	23147.0i		−	1.41964i	−0.704383	−	0.709820i	$-0.748776\pi$
$643$		−	23147.0i		−	1.41964i	0.704383	−	0.709820i	$-0.251224\pi$
$644$		0			0
$645$	14200.0	−	7100.00i	0.866860	−	0.433430i
$646$		0			0
$647$			18632.0i			1.13215i	0.824355	+	0.566074i	$0.191538\pi$
$647$			18632.0i			1.13215i	−0.824355	+	0.566074i	$0.808462\pi$
$648$		0			0
$649$	−21710.0			−1.31308
$650$		0			0
$651$		0			0
$652$		0			0
$653$			31292.0i			1.87527i	0.347623	+	0.937634i	$0.386989\pi$
$653$			31292.0i			1.87527i	−0.347623	+	0.937634i	$0.613011\pi$
$654$		0			0
$655$	4940.00	−	2470.00i	0.294690	−	0.147345i
$656$		0			0
$657$		−	1004.00i		−	0.0596191i
$658$		0			0
$659$	24909.0			1.47241			0.736204	−	0.676760i	$-0.236616\pi$
$659$	24909.0			1.47241			0.736204	+	0.676760i	$0.236616\pi$
$660$		0			0
$661$	−18844.0			−1.10885			−0.554423	−	0.832235i	$-0.687061\pi$
$661$	−18844.0			−1.10885			−0.554423	+	0.832235i	$0.687061\pi$
$662$		0			0
$663$		−	7345.00i		−	0.430251i
$664$		0			0
$665$		0			0
$666$		0			0
$667$			10602.0i			0.615459i
$668$		0			0
$669$	6695.00			0.386911
$670$		0			0
$671$	−3640.00			−0.209420
$672$		0			0
$673$		−	14184.0i		−	0.812412i	−0.913782	−	0.406206i	$-0.866852\pi$
$673$		−	14184.0i		−	0.812412i	0.913782	−	0.406206i	$-0.133148\pi$
$674$		0			0
$675$	14500.0	+	10875.0i	0.826823	+	0.620117i
$676$		0			0
$677$			15177.0i			0.861595i	0.902449	+	0.430797i	$0.141768\pi$
$677$			15177.0i			0.861595i	−0.902449	+	0.430797i	$0.858232\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$	18425.0			1.03678
$682$		0			0
$683$			2624.00i			0.147005i	0.997295	+	0.0735026i	$0.0234177\pi$
$683$			2624.00i			0.147005i	−0.997295	+	0.0735026i	$0.976582\pi$
$684$		0			0
$685$	4820.00	+	9640.00i	0.268851	+	0.537701i
$686$		0			0
$687$		−	2330.00i		−	0.129396i
$688$		0			0
$689$	−1690.00			−0.0934454
$690$		0			0
$691$	25112.0			1.38250			0.691249	−	0.722617i	$-0.257061\pi$
$691$	25112.0			1.38250			0.691249	+	0.722617i	$0.257061\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	4660.00	−	2330.00i	0.254337	−	0.127168i
$696$		0			0
$697$		−	46782.0i		−	2.54232i
$698$		0			0
$699$	−720.000			−0.0389598
$700$		0			0
$701$	927.000			0.0499462			0.0249731	−	0.999688i	$-0.492050\pi$
$701$	927.000			0.0499462			0.0249731	+	0.999688i	$0.492050\pi$
$702$		0			0
$703$			2144.00i			0.115025i
$704$		0			0
$705$	−20950.0	+	10475.0i	−1.11918	+	0.559591i
$706$		0			0
$707$		0			0
$708$		0			0
$709$	15209.0			0.805622			0.402811	−	0.915283i	$-0.368033\pi$
$709$	15209.0			0.805622			0.402811	+	0.915283i	$0.368033\pi$
$710$		0			0
$711$	742.000			0.0391381
$712$		0			0
$713$			47988.0i			2.52057i
$714$		0			0
$715$	4225.00	+	8450.00i	0.220987	+	0.441975i
$716$		0			0
$717$			6275.00i			0.326840i
$718$		0			0
$719$	22018.0			1.14205			0.571024	−	0.820933i	$-0.306546\pi$
$719$	22018.0			1.14205			0.571024	+	0.820933i	$0.306546\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$		−	18910.0i		−	0.972712i
$724$		0			0
$725$	4275.00	−	5700.00i	0.218992	−	0.291990i
$726$		0			0
$727$		−	16324.0i		−	0.832770i	−0.909188	−	0.416385i	$-0.863297\pi$
$727$		−	16324.0i		−	0.832770i	0.909188	−	0.416385i	$-0.136703\pi$
$728$		0			0
$729$	−20917.0			−1.06269
$730$		0			0
$731$	−32092.0			−1.62376
$732$		0			0
$733$		−	5805.00i		−	0.292514i	−0.989247	−	0.146257i	$-0.953277\pi$
$733$		−	5805.00i		−	0.292514i	0.989247	−	0.146257i	$-0.0467226\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$			34710.0i			1.73482i
$738$		0			0
$739$	−37703.0			−1.87676			−0.938381	−	0.345602i	$-0.887675\pi$
$739$	−37703.0			−1.87676			−0.938381	+	0.345602i	$0.887675\pi$
$740$		0			0
$741$	1040.00			0.0515592
$742$		0			0
$743$			36542.0i			1.80430i	0.431421	+	0.902151i	$0.358012\pi$
$743$			36542.0i			1.80430i	−0.431421	+	0.902151i	$0.641988\pi$
$744$		0			0
$745$	32460.0	−	16230.0i	1.59630	−	0.798149i
$746$		0			0
$747$		−	2440.00i		−	0.119511i
$748$		0			0
$749$		0			0
$750$		0			0
$751$	24505.0			1.19068			0.595340	−	0.803474i	$-0.297018\pi$
$751$	24505.0			1.19068			0.595340	+	0.803474i	$0.297018\pi$
$752$		0			0
$753$		−	5730.00i		−	0.277308i
$754$		0			0
$755$	−6430.00	+	3215.00i	−0.309949	+	0.154975i
$756$		0			0
$757$		−	21568.0i		−	1.03554i	−0.855520	−	0.517769i	$-0.826763\pi$
$757$		−	21568.0i		−	1.03554i	0.855520	−	0.517769i	$-0.173237\pi$
$758$		0			0
$759$	60450.0			2.89090
$760$		0			0
$761$	9150.00			0.435857			0.217929	−	0.975965i	$-0.430070\pi$
$761$	9150.00			0.435857			0.217929	+	0.975965i	$0.430070\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$	−1130.00	−	2260.00i	−0.0534055	−	0.106811i
$766$		0			0
$767$		−	4342.00i		−	0.204407i
$768$		0			0
$769$	−34848.0			−1.63414			−0.817068	−	0.576541i	$-0.804402\pi$
$769$	−34848.0			−1.63414			−0.817068	+	0.576541i	$0.804402\pi$
$770$		0			0
$771$	19730.0			0.921606
$772$		0			0
$773$		−	12617.0i		−	0.587066i	−0.955949	−	0.293533i	$-0.905169\pi$
$773$		−	12617.0i		−	0.587066i	0.955949	−	0.293533i	$-0.0948310\pi$
$774$		0			0
$775$	19350.0	−	25800.0i	0.896868	−	1.19582i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	6624.00			0.304659
$780$		0			0
$781$	−61880.0			−2.83514
$782$		0			0
$783$			8265.00i			0.377225i
$784$		0			0
$785$	8270.00	+	16540.0i	0.376011	+	0.752023i
$786$		0			0
$787$		−	6949.00i		−	0.314746i	−0.987539	−	0.157373i	$-0.949698\pi$
$787$		−	6949.00i		−	0.314746i	0.987539	−	0.157373i	$-0.0503025\pi$
$788$		0			0
$789$	14760.0			0.665995
$790$		0			0
$791$		0			0
$792$		0			0
$793$		−	728.000i		−	0.0326003i
$794$		0			0
$795$	6500.00	−	3250.00i	0.289976	−	0.144988i
$796$		0			0
$797$		−	3081.00i		−	0.136932i	−0.997653	−	0.0684659i	$-0.978190\pi$
$797$		−	3081.00i		−	0.136932i	0.997653	−	0.0684659i	$-0.0218104\pi$
$798$		0			0
$799$	47347.0			2.09639
$800$		0			0
$801$	1760.00			0.0776361
$802$		0			0
$803$			32630.0i			1.43398i
$804$		0			0
$805$		0			0
$806$		0			0
$807$			41050.0i			1.79062i
$808$		0			0
$809$	23739.0			1.03167			0.515834	−	0.856689i	$-0.327482\pi$
$809$	23739.0			1.03167			0.515834	+	0.856689i	$0.327482\pi$
$810$		0			0
$811$	24226.0			1.04894			0.524470	−	0.851429i	$-0.324264\pi$
$811$	24226.0			1.04894			0.524470	+	0.851429i	$0.324264\pi$
$812$		0			0
$813$			9800.00i			0.422756i
$814$		0			0
$815$	5310.00	+	10620.0i	0.228222	+	0.456445i
$816$		0			0
$817$		−	4544.00i		−	0.194583i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	25269.0			1.07417			0.537085	−	0.843528i	$-0.319525\pi$
$821$	25269.0			1.07417			0.537085	+	0.843528i	$0.319525\pi$
$822$		0			0
$823$		−	12602.0i		−	0.533752i	−0.963731	−	0.266876i	$-0.914009\pi$
$823$		−	12602.0i		−	0.533752i	0.963731	−	0.266876i	$-0.0859915\pi$
$824$		0			0
$825$	−32500.0	−	24375.0i	−1.37152	−	1.02864i
$826$		0			0
$827$		−	23274.0i		−	0.978617i	−0.872111	−	0.489309i	$-0.837249\pi$
$827$		−	23274.0i		−	0.978617i	0.872111	−	0.489309i	$-0.162751\pi$
$828$		0			0
$829$	−17062.0			−0.714822			−0.357411	−	0.933947i	$-0.616341\pi$
$829$	−17062.0			−0.714822			−0.357411	+	0.933947i	$0.616341\pi$
$830$		0			0
$831$	−12110.0			−0.505525
$832$		0			0
$833$		0			0
$834$		0			0
$835$	10135.0	+	20270.0i	0.420043	+	0.840087i
$836$		0			0
$837$			37410.0i			1.54490i
$838$		0			0
$839$	−5134.00			−0.211258			−0.105629	−	0.994406i	$-0.533686\pi$
$839$	−5134.00			−0.211258			−0.105629	+	0.994406i	$0.533686\pi$
$840$		0			0
$841$	−21140.0			−0.866784
$842$		0			0
$843$		−	26245.0i		−	1.07227i
$844$		0			0
$845$	20280.0	−	10140.0i	0.825625	−	0.412813i
$846$		0			0
$847$		0			0
$848$		0			0
$849$	31045.0			1.25496
$850$		0			0
$851$	−24924.0			−1.00398
$852$		0			0
$853$		−	25202.0i		−	1.01161i	−0.862649	−	0.505803i	$-0.831196\pi$
$853$		−	25202.0i		−	1.01161i	0.862649	−	0.505803i	$-0.168804\pi$
$854$		0			0
$855$	320.000	−	160.000i	0.0127997	−	0.00639986i
$856$		0			0
$857$			34234.0i			1.36454i	0.731100	+	0.682270i	$0.239007\pi$
$857$			34234.0i			1.36454i	−0.731100	+	0.682270i	$0.760993\pi$
$858$		0			0
$859$	14904.0			0.591988			0.295994	−	0.955190i	$-0.404349\pi$
$859$	14904.0			0.591988			0.295994	+	0.955190i	$0.404349\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$			18054.0i			0.712127i	0.934462	+	0.356063i	$0.115881\pi$
$863$			18054.0i			0.712127i	−0.934462	+	0.356063i	$0.884119\pi$
$864$		0			0
$865$	−7895.00	−	15790.0i	−0.310333	−	0.620666i
$866$		0			0
$867$		−	39280.0i		−	1.53866i
$868$		0			0
$869$	−24115.0			−0.941364
$870$		0			0
$871$	−6942.00			−0.270058
$872$		0			0
$873$		−	482.000i		−	0.0186864i
$874$		0			0
$875$		0			0
$876$		0			0
$877$		−	31440.0i		−	1.21055i	−0.796016	−	0.605276i	$-0.793063\pi$
$877$		−	31440.0i		−	1.21055i	0.796016	−	0.605276i	$-0.206937\pi$
$878$		0			0
$879$	13295.0			0.510158
$880$		0			0
$881$	37248.0			1.42442			0.712212	−	0.701965i	$-0.247694\pi$
$881$	37248.0			1.42442			0.712212	+	0.701965i	$0.247694\pi$
$882$		0			0
$883$			47324.0i			1.80360i	0.432153	+	0.901800i	$0.357754\pi$
$883$			47324.0i			1.80360i	−0.432153	+	0.901800i	$0.642246\pi$
$884$		0			0
$885$	8350.00	+	16700.0i	0.317155	+	0.634310i
$886$		0			0
$887$		−	612.000i		−	0.0231668i	−0.999933	−	0.0115834i	$-0.996313\pi$
$887$		−	612.000i		−	0.0231668i	0.999933	−	0.0115834i	$-0.00368719\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$	43615.0			1.63991
$892$		0			0
$893$			6704.00i			0.251222i
$894$		0			0
$895$	−27880.0	+	13940.0i	−1.04126	+	0.520629i
$896$		0			0
$897$			12090.0i			0.450026i
$898$		0			0
$899$	14706.0			0.545576
$900$		0			0
$901$	−14690.0			−0.543169
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−26140.0	+	13070.0i	−0.960136	+	0.480068i
$906$		0			0
$907$			13666.0i			0.500300i	0.968207	+	0.250150i	$0.0804799\pi$
$907$			13666.0i			0.500300i	−0.968207	+	0.250150i	$0.919520\pi$
$908$		0			0
$909$	296.000			0.0108006
$910$		0			0
$911$	−45468.0			−1.65359			−0.826796	−	0.562502i	$-0.809839\pi$
$911$	−45468.0			−1.65359			−0.826796	+	0.562502i	$0.809839\pi$
$912$		0			0
$913$			79300.0i			2.87453i
$914$		0			0
$915$	1400.00	+	2800.00i	0.0505820	+	0.101164i
$916$		0			0
$917$		0			0
$918$		0			0
$919$	−38789.0			−1.39231			−0.696154	−	0.717892i	$-0.745107\pi$
$919$	−38789.0			−1.39231			−0.696154	+	0.717892i	$0.745107\pi$
$920$		0			0
$921$	815.000			0.0291587
$922$		0			0
$923$		−	12376.0i		−	0.441345i
$924$		0			0
$925$	13400.0	+	10050.0i	0.476313	+	0.357235i
$926$		0			0
$927$			1002.00i			0.0355016i
$928$		0			0
$929$	10830.0			0.382477			0.191238	−	0.981544i	$-0.438750\pi$
$929$	10830.0			0.382477			0.191238	+	0.981544i	$0.438750\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$		−	22580.0i		−	0.792322i
$934$		0			0
$935$	36725.0	+	73450.0i	1.28453	+	2.56906i
$936$		0			0
$937$			707.000i			0.0246496i	0.999924	+	0.0123248i	$0.00392321\pi$
$937$			707.000i			0.0246496i	−0.999924	+	0.0123248i	$0.996077\pi$
$938$		0			0
$939$	−3815.00			−0.132586
$940$		0			0
$941$	25152.0			0.871341			0.435670	−	0.900106i	$-0.356511\pi$
$941$	25152.0			0.871341			0.435670	+	0.900106i	$0.356511\pi$
$942$		0			0
$943$			77004.0i			2.65917i
$944$		0			0
$945$		0			0
$946$		0			0
$947$		−	3582.00i		−	0.122914i	−0.998110	−	0.0614569i	$-0.980425\pi$
$947$		−	3582.00i		−	0.122914i	0.998110	−	0.0614569i	$-0.0195747\pi$
$948$		0			0
$949$	−6526.00			−0.223228
$950$		0			0
$951$	39340.0			1.34142
$952$		0			0
$953$		−	19530.0i		−	0.663839i	−0.943308	−	0.331920i	$-0.892304\pi$
$953$		−	19530.0i		−	0.663839i	0.943308	−	0.331920i	$-0.107696\pi$
$954$		0			0
$955$	−8050.00	+	4025.00i	−0.272766	+	0.136383i
$956$		0			0
$957$		−	18525.0i		−	0.625735i
$958$		0			0
$959$		0			0
$960$		0			0
$961$	36773.0			1.23437
$962$		0			0
$963$		−	1488.00i		−	0.0497925i
$964$		0			0
$965$	−17760.0	−	35520.0i	−0.592450	−	1.18490i
$966$		0			0
$967$			8952.00i			0.297701i	0.988860	+	0.148851i	$0.0475573\pi$
$967$			8952.00i			0.297701i	−0.988860	+	0.148851i	$0.952443\pi$
$968$		0			0
$969$	9040.00			0.299697
$970$		0			0
$971$	−930.000			−0.0307365			−0.0153682	−	0.999882i	$-0.504892\pi$
$971$	−930.000			−0.0307365			−0.0153682	+	0.999882i	$0.504892\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$	4875.00	−	6500.00i	0.160128	−	0.213504i
$976$		0			0
$977$		−	7068.00i		−	0.231449i	−0.993281	−	0.115724i	$-0.963081\pi$
$977$		−	7068.00i		−	0.231449i	0.993281	−	0.115724i	$-0.0369189\pi$
$978$		0			0
$979$	−57200.0			−1.86733
$980$		0			0
$981$	−882.000			−0.0287055
$982$		0			0
$983$			18315.0i			0.594260i	0.954837	+	0.297130i	$0.0960295\pi$
$983$			18315.0i			0.594260i	−0.954837	+	0.297130i	$0.903970\pi$
$984$		0			0
$985$	17310.0	+	34620.0i	0.559942	+	1.11988i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	52824.0			1.69839
$990$		0			0
$991$	−41736.0			−1.33783			−0.668914	−	0.743340i	$-0.733241\pi$
$991$	−41736.0			−1.33783			−0.668914	+	0.743340i	$0.733241\pi$
$992$		0			0
$993$			16020.0i			0.511963i
$994$		0			0
$995$	−860.000	+	430.000i	−0.0274008	+	0.0137004i
$996$		0			0
$997$		−	2161.00i		−	0.0686455i	−0.999411	−	0.0343227i	$-0.989073\pi$
$997$		−	2161.00i		−	0.0686455i	0.999411	−	0.0343227i	$-0.0109274\pi$
$998$		0			0
$999$	−19430.0			−0.615353

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	980.4.e.c.589.2		2
5.4	even	2	inner	980.4.e.c.589.1		2
7.6	odd	2		140.4.e.a.29.1	✓	2
21.20	even	2		1260.4.k.c.1009.1		2
28.27	even	2		560.4.g.a.449.2		2
35.13	even	4		700.4.a.l.1.1		1
35.27	even	4		700.4.a.d.1.1		1
35.34	odd	2		140.4.e.a.29.2	yes	2
105.104	even	2		1260.4.k.c.1009.2		2
140.139	even	2		560.4.g.a.449.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
140.4.e.a.29.1	✓	2	7.6	odd	2
140.4.e.a.29.2	yes	2	35.34	odd	2
560.4.g.a.449.1		2	140.139	even	2
560.4.g.a.449.2		2	28.27	even	2
700.4.a.d.1.1		1	35.27	even	4
700.4.a.l.1.1		1	35.13	even	4
980.4.e.c.589.1		2	5.4	even	2	inner
980.4.e.c.589.2		2	1.1	even	1	trivial
1260.4.k.c.1009.1		2	21.20	even	2
1260.4.k.c.1009.2		2	105.104	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 980 = 2^{2} \cdot 5 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	980.e (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+5.00000i q^{3} +(10.0000 - 5.00000i) q^{5} +2.00000 q^{9} -65.0000 q^{11} -13.0000i q^{13} +(25.0000 + 50.0000i) q^{15} -113.000i q^{17} +16.0000 q^{19} +186.000i q^{23} +(75.0000 - 100.000i) q^{25} +145.000i q^{27} +57.0000 q^{29} +258.000 q^{31} -325.000i q^{33} +134.000i q^{37} +65.0000 q^{39} +414.000 q^{41} -284.000i q^{43} +(20.0000 - 10.0000i) q^{45} +419.000i q^{47} +565.000 q^{51} -130.000i q^{53} +(-650.000 + 325.000i) q^{55} +80.0000i q^{57} +334.000 q^{59} +56.0000 q^{61} +(-65.0000 - 130.000i) q^{65} -534.000i q^{67} -930.000 q^{69} +952.000 q^{71} -502.000i q^{73} +(500.000 + 375.000i) q^{75} +371.000 q^{79} -671.000 q^{81} -1220.00i q^{83} +(-565.000 - 1130.00i) q^{85} +285.000i q^{87} +880.000 q^{89} +1290.00i q^{93} +(160.000 - 80.0000i) q^{95} -241.000i q^{97} -130.000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 20 q^{5} + 4 q^{9} - 130 q^{11} + 50 q^{15} + 32 q^{19} + 150 q^{25} + 114 q^{29} + 516 q^{31} + 130 q^{39} + 828 q^{41} + 40 q^{45} + 1130 q^{51} - 1300 q^{55} + 668 q^{59} + 112 q^{61} - 130 q^{65}+ \cdots - 260 q^{99}+O(q^{100}) \) 2 * q + 20 * q^5 + 4 * q^9 - 130 * q^11 + 50 * q^15 + 32 * q^19 + 150 * q^25 + 114 * q^29 + 516 * q^31 + 130 * q^39 + 828 * q^41 + 40 * q^45 + 1130 * q^51 - 1300 * q^55 + 668 * q^59 + 112 * q^61 - 130 * q^65 - 1860 * q^69 + 1904 * q^71 + 1000 * q^75 + 742 * q^79 - 1342 * q^81 - 1130 * q^85 + 1760 * q^89 + 320 * q^95 - 260 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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