LMFDB - Embedded newform 8280.2.a.bv.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8280,2,Mod(1,8280)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8280, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("8280.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8280 = 2^{3} \cdot 3^{2} \cdot 5 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8280.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:	$66.1161328736$
Analytic rank:	$0$
Dimension:	$7$
Coefficient field:	$\mathbb{Q}[x]/(x^{7} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{7} - 3x^{6} - 18x^{5} + 46x^{4} + 60x^{3} - 76x^{2} - 51x - 7 $ x^7 - 3x^6 - 18x^5 + 46x^4 + 60x^3 - 76x^2 - 51x - 7
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-0.218266$ of defining polynomial
Character	$\chi$	$=$	8280.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{5} -2.94234 q^{7} +O(q^{10})$	$q-1.00000 q^{5} -2.94234 q^{7} +4.49933 q^{11} -6.14522 q^{13} -7.63197 q^{17} +6.67643 q^{19} -1.00000 q^{23} +1.00000 q^{25} -7.87821 q^{29} -7.74105 q^{31} +2.94234 q^{35} -1.41699 q^{37} -5.34700 q^{41} +4.46746 q^{43} -0.681564 q^{47} +1.65736 q^{49} +6.97447 q^{53} -4.49933 q^{55} +8.08821 q^{59} +13.3518 q^{61} +6.14522 q^{65} +12.6318 q^{67} +2.37925 q^{71} -13.0748 q^{73} -13.2386 q^{77} -4.67532 q^{79} +3.36670 q^{83} +7.63197 q^{85} -13.5564 q^{89} +18.0813 q^{91} -6.67643 q^{95} +6.91181 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q - 7 q^{5} - 6 q^{7}+O(q^{10}) $ 7 * q - 7 * q^5 - 6 * q^7	$ 7 q - 7 q^{5} - 6 q^{7} + 2 q^{11} - 6 q^{13} + 4 q^{17} - 8 q^{19} - 7 q^{23} + 7 q^{25} - 2 q^{29} - 8 q^{31} + 6 q^{35} - 16 q^{37} - 2 q^{41} - 10 q^{43} + 8 q^{47} + 19 q^{49} + 10 q^{53} - 2 q^{55} + 24 q^{59} + 8 q^{61} + 6 q^{65} - 20 q^{67} + 8 q^{71} + 2 q^{73} + 12 q^{77} - 2 q^{79} + 22 q^{83} - 4 q^{85} + 16 q^{89} - 20 q^{91} + 8 q^{95} + 4 q^{97}+O(q^{100}) $ 7 * q - 7 * q^5 - 6 * q^7 + 2 * q^11 - 6 * q^13 + 4 * q^17 - 8 * q^19 - 7 * q^23 + 7 * q^25 - 2 * q^29 - 8 * q^31 + 6 * q^35 - 16 * q^37 - 2 * q^41 - 10 * q^43 + 8 * q^47 + 19 * q^49 + 10 * q^53 - 2 * q^55 + 24 * q^59 + 8 * q^61 + 6 * q^65 - 20 * q^67 + 8 * q^71 + 2 * q^73 + 12 * q^77 - 2 * q^79 + 22 * q^83 - 4 * q^85 + 16 * q^89 - 20 * q^91 + 8 * q^95 + 4 * q^97

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	0
$3$	0	0
$4$	0	0
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	−2.94234	−1.11210	−0.556050	−	0.831149i	$-0.687684\pi$
$7$	−2.94234	−1.11210	−0.556050	+	0.831149i	$0.687684\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	4.49933	1.35660	0.678300	−	0.734785i	$-0.262717\pi$
$11$	4.49933	1.35660	0.678300	+	0.734785i	$0.262717\pi$
$12$	0	0
$13$	−6.14522	−1.70438	−0.852189	−	0.523234i	$-0.824725\pi$
$13$	−6.14522	−1.70438	−0.852189	+	0.523234i	$0.824725\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−7.63197	−1.85103	−0.925513	−	0.378717i	$-0.876365\pi$
$17$	−7.63197	−1.85103	−0.925513	+	0.378717i	$0.876365\pi$
$18$	0	0
$19$	6.67643	1.53168	0.765839	−	0.643032i	$-0.222324\pi$
$19$	6.67643	1.53168	0.765839	+	0.643032i	$0.222324\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−7.87821	−1.46295	−0.731473	−	0.681870i	$-0.761167\pi$
$29$	−7.87821	−1.46295	−0.731473	+	0.681870i	$0.761167\pi$
$30$	0	0
$31$	−7.74105	−1.39033	−0.695167	−	0.718848i	$-0.744670\pi$
$31$	−7.74105	−1.39033	−0.695167	+	0.718848i	$0.744670\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	2.94234	0.497346
$36$	0	0
$37$	−1.41699	−0.232952	−0.116476	−	0.993194i	$-0.537160\pi$
$37$	−1.41699	−0.232952	−0.116476	+	0.993194i	$0.537160\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−5.34700	−0.835060	−0.417530	−	0.908663i	$-0.637104\pi$
$41$	−5.34700	−0.835060	−0.417530	+	0.908663i	$0.637104\pi$
$42$	0	0
$43$	4.46746	0.681281	0.340640	−	0.940194i	$-0.389356\pi$
$43$	4.46746	0.681281	0.340640	+	0.940194i	$0.389356\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−0.681564	−0.0994163	−0.0497082	−	0.998764i	$-0.515829\pi$
$47$	−0.681564	−0.0994163	−0.0497082	+	0.998764i	$0.515829\pi$
$48$	0	0
$49$	1.65736	0.236766
$50$	0	0
$51$	0	0
$52$	0	0
$53$	6.97447	0.958017	0.479008	−	0.877810i	$-0.340996\pi$
$53$	6.97447	0.958017	0.479008	+	0.877810i	$0.340996\pi$
$54$	0	0
$55$	−4.49933	−0.606690
$56$	0	0
$57$	0	0
$58$	0	0
$59$	8.08821	1.05299	0.526497	−	0.850177i	$-0.323505\pi$
$59$	8.08821	1.05299	0.526497	+	0.850177i	$0.323505\pi$
$60$	0	0
$61$	13.3518	1.70952	0.854759	−	0.519026i	$-0.173705\pi$
$61$	13.3518	1.70952	0.854759	+	0.519026i	$0.173705\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	6.14522	0.762221
$66$	0	0
$67$	12.6318	1.54323	0.771613	−	0.636093i	$-0.219450\pi$
$67$	12.6318	1.54323	0.771613	+	0.636093i	$0.219450\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	2.37925	0.282365	0.141183	−	0.989984i	$-0.454910\pi$
$71$	2.37925	0.282365	0.141183	+	0.989984i	$0.454910\pi$
$72$	0	0
$73$	−13.0748	−1.53030	−0.765148	−	0.643855i	$-0.777334\pi$
$73$	−13.0748	−1.53030	−0.765148	+	0.643855i	$0.777334\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−13.2386	−1.50868
$78$	0	0
$79$	−4.67532	−0.526015	−0.263007	−	0.964794i	$-0.584714\pi$
$79$	−4.67532	−0.526015	−0.263007	+	0.964794i	$0.584714\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	3.36670	0.369543	0.184771	−	0.982782i	$-0.440846\pi$
$83$	3.36670	0.369543	0.184771	+	0.982782i	$0.440846\pi$
$84$	0	0
$85$	7.63197	0.827804
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−13.5564	−1.43697	−0.718486	−	0.695542i	$-0.755164\pi$
$89$	−13.5564	−1.43697	−0.718486	+	0.695542i	$0.755164\pi$
$90$	0	0
$91$	18.0813	1.89544
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−6.67643	−0.684988
$96$	0	0
$97$	6.91181	0.701788	0.350894	−	0.936415i	$-0.385878\pi$
$97$	6.91181	0.701788	0.350894	+	0.936415i	$0.385878\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	9.82227	0.977352	0.488676	−	0.872465i	$-0.337480\pi$
$101$	9.82227	0.977352	0.488676	+	0.872465i	$0.337480\pi$
$102$	0	0
$103$	−10.5674	−1.04123	−0.520616	−	0.853791i	$-0.674298\pi$
$103$	−10.5674	−1.04123	−0.520616	+	0.853791i	$0.674298\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−4.20946	−0.406944	−0.203472	−	0.979081i	$-0.565223\pi$
$107$	−4.20946	−0.406944	−0.203472	+	0.979081i	$0.565223\pi$
$108$	0	0
$109$	−4.43029	−0.424345	−0.212172	−	0.977232i	$-0.568054\pi$
$109$	−4.43029	−0.424345	−0.212172	+	0.977232i	$0.568054\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	14.0685	1.32345	0.661727	−	0.749745i	$-0.269824\pi$
$113$	14.0685	1.32345	0.661727	+	0.749745i	$0.269824\pi$
$114$	0	0
$115$	1.00000	0.0932505
$116$	0	0
$117$	0	0
$118$	0	0
$119$	22.4559	2.05852
$120$	0	0
$121$	9.24401	0.840364
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−18.1071	−1.60675	−0.803373	−	0.595476i	$-0.796963\pi$
$127$	−18.1071	−1.60675	−0.803373	+	0.595476i	$0.796963\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	10.3806	0.906957	0.453478	−	0.891267i	$-0.350183\pi$
$131$	10.3806	0.906957	0.453478	+	0.891267i	$0.350183\pi$
$132$	0	0
$133$	−19.6443	−1.70338
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−0.238791	−0.0204013	−0.0102006	−	0.999948i	$-0.503247\pi$
$137$	−0.238791	−0.0204013	−0.0102006	+	0.999948i	$0.503247\pi$
$138$	0	0
$139$	−14.7185	−1.24840	−0.624201	−	0.781263i	$-0.714576\pi$
$139$	−14.7185	−1.24840	−0.624201	+	0.781263i	$0.714576\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−27.6494	−2.31216
$144$	0	0
$145$	7.87821	0.654249
$146$	0	0
$147$	0	0
$148$	0	0
$149$	21.5341	1.76415	0.882073	−	0.471113i	$-0.156148\pi$
$149$	21.5341	1.76415	0.882073	+	0.471113i	$0.156148\pi$
$150$	0	0
$151$	−5.53707	−0.450600	−0.225300	−	0.974289i	$-0.572336\pi$
$151$	−5.53707	−0.450600	−0.225300	+	0.974289i	$0.572336\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	7.74105	0.621776
$156$	0	0
$157$	−14.8405	−1.18440	−0.592199	−	0.805792i	$-0.701740\pi$
$157$	−14.8405	−1.18440	−0.592199	+	0.805792i	$0.701740\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	2.94234	0.231889
$162$	0	0
$163$	−17.4301	−1.36523	−0.682614	−	0.730779i	$-0.739157\pi$
$163$	−17.4301	−1.36523	−0.682614	+	0.730779i	$0.739157\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	15.0614	1.16549	0.582743	−	0.812657i	$-0.301979\pi$
$167$	15.0614	1.16549	0.582743	+	0.812657i	$0.301979\pi$
$168$	0	0
$169$	24.7638	1.90490
$170$	0	0
$171$	0	0
$172$	0	0
$173$	15.3021	1.16339	0.581697	−	0.813406i	$-0.302389\pi$
$173$	15.3021	1.16339	0.581697	+	0.813406i	$0.302389\pi$
$174$	0	0
$175$	−2.94234	−0.222420
$176$	0	0
$177$	0	0
$178$	0	0
$179$	8.43653	0.630576	0.315288	−	0.948996i	$-0.397899\pi$
$179$	8.43653	0.630576	0.315288	+	0.948996i	$0.397899\pi$
$180$	0	0
$181$	−7.11921	−0.529166	−0.264583	−	0.964363i	$-0.585234\pi$
$181$	−7.11921	−0.529166	−0.264583	+	0.964363i	$0.585234\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	1.41699	0.104179
$186$	0	0
$187$	−34.3388	−2.51110
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−5.72713	−0.414401	−0.207200	−	0.978299i	$-0.566435\pi$
$191$	−5.72713	−0.414401	−0.207200	+	0.978299i	$0.566435\pi$
$192$	0	0
$193$	19.3045	1.38957	0.694783	−	0.719219i	$-0.255500\pi$
$193$	19.3045	1.38957	0.694783	+	0.719219i	$0.255500\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−12.2523	−0.872942	−0.436471	−	0.899718i	$-0.643772\pi$
$197$	−12.2523	−0.872942	−0.436471	+	0.899718i	$0.643772\pi$
$198$	0	0
$199$	25.3334	1.79584	0.897919	−	0.440160i	$-0.145078\pi$
$199$	25.3334	1.79584	0.897919	+	0.440160i	$0.145078\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	23.1804	1.62694
$204$	0	0
$205$	5.34700	0.373450
$206$	0	0
$207$	0	0
$208$	0	0
$209$	30.0395	2.07788
$210$	0	0
$211$	−8.15195	−0.561203	−0.280602	−	0.959824i	$-0.590534\pi$
$211$	−8.15195	−0.561203	−0.280602	+	0.959824i	$0.590534\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−4.46746	−0.304678
$216$	0	0
$217$	22.7768	1.54619
$218$	0	0
$219$	0	0
$220$	0	0
$221$	46.9002	3.15485
$222$	0	0
$223$	−0.208714	−0.0139765	−0.00698825	−	0.999976i	$-0.502224\pi$
$223$	−0.208714	−0.0139765	−0.00698825	+	0.999976i	$0.502224\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	0.562200	0.0373146	0.0186573	−	0.999826i	$-0.494061\pi$
$227$	0.562200	0.0373146	0.0186573	+	0.999826i	$0.494061\pi$
$228$	0	0
$229$	6.46430	0.427173	0.213587	−	0.976924i	$-0.431485\pi$
$229$	6.46430	0.427173	0.213587	+	0.976924i	$0.431485\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	22.5570	1.47776	0.738879	−	0.673838i	$-0.235355\pi$
$233$	22.5570	1.47776	0.738879	+	0.673838i	$0.235355\pi$
$234$	0	0
$235$	0.681564	0.0444603
$236$	0	0
$237$	0	0
$238$	0	0
$239$	1.56994	0.101551	0.0507755	−	0.998710i	$-0.483831\pi$
$239$	1.56994	0.101551	0.0507755	+	0.998710i	$0.483831\pi$
$240$	0	0
$241$	7.94433	0.511739	0.255870	−	0.966711i	$-0.417638\pi$
$241$	7.94433	0.511739	0.255870	+	0.966711i	$0.417638\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−1.65736	−0.105885
$246$	0	0
$247$	−41.0282	−2.61056
$248$	0	0
$249$	0	0
$250$	0	0
$251$	13.2278	0.834931	0.417466	−	0.908693i	$-0.362918\pi$
$251$	13.2278	0.834931	0.417466	+	0.908693i	$0.362918\pi$
$252$	0	0
$253$	−4.49933	−0.282871
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−26.3640	−1.64454	−0.822269	−	0.569098i	$-0.807292\pi$
$257$	−26.3640	−1.64454	−0.822269	+	0.569098i	$0.807292\pi$
$258$	0	0
$259$	4.16926	0.259065
$260$	0	0
$261$	0	0
$262$	0	0
$263$	18.8462	1.16211	0.581053	−	0.813866i	$-0.302641\pi$
$263$	18.8462	1.16211	0.581053	+	0.813866i	$0.302641\pi$
$264$	0	0
$265$	−6.97447	−0.428438
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−10.0443	−0.612414	−0.306207	−	0.951965i	$-0.599060\pi$
$269$	−10.0443	−0.612414	−0.306207	+	0.951965i	$0.599060\pi$
$270$	0	0
$271$	−11.4437	−0.695157	−0.347578	−	0.937651i	$-0.612996\pi$
$271$	−11.4437	−0.695157	−0.347578	+	0.937651i	$0.612996\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	4.49933	0.271320
$276$	0	0
$277$	−15.1177	−0.908334	−0.454167	−	0.890916i	$-0.650063\pi$
$277$	−15.1177	−0.908334	−0.454167	+	0.890916i	$0.650063\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	16.5384	0.986599	0.493300	−	0.869859i	$-0.335791\pi$
$281$	16.5384	0.986599	0.493300	+	0.869859i	$0.335791\pi$
$282$	0	0
$283$	7.14507	0.424731	0.212365	−	0.977190i	$-0.431883\pi$
$283$	7.14507	0.424731	0.212365	+	0.977190i	$0.431883\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	15.7327	0.928671
$288$	0	0
$289$	41.2470	2.42629
$290$	0	0
$291$	0	0
$292$	0	0
$293$	22.6623	1.32394	0.661972	−	0.749529i	$-0.269720\pi$
$293$	22.6623	1.32394	0.661972	+	0.749529i	$0.269720\pi$
$294$	0	0
$295$	−8.08821	−0.470914
$296$	0	0
$297$	0	0
$298$	0	0
$299$	6.14522	0.355387
$300$	0	0
$301$	−13.1448	−0.757652
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−13.3518	−0.764519
$306$	0	0
$307$	4.84641	0.276599	0.138299	−	0.990390i	$-0.455836\pi$
$307$	4.84641	0.276599	0.138299	+	0.990390i	$0.455836\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	12.3309	0.699220	0.349610	−	0.936895i	$-0.386314\pi$
$311$	12.3309	0.699220	0.349610	+	0.936895i	$0.386314\pi$
$312$	0	0
$313$	1.62177	0.0916679	0.0458340	−	0.998949i	$-0.485405\pi$
$313$	1.62177	0.0916679	0.0458340	+	0.998949i	$0.485405\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−3.47742	−0.195312	−0.0976558	−	0.995220i	$-0.531134\pi$
$317$	−3.47742	−0.195312	−0.0976558	+	0.995220i	$0.531134\pi$
$318$	0	0
$319$	−35.4467	−1.98463
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−50.9544	−2.83518
$324$	0	0
$325$	−6.14522	−0.340876
$326$	0	0
$327$	0	0
$328$	0	0
$329$	2.00539	0.110561
$330$	0	0
$331$	24.5216	1.34783	0.673913	−	0.738810i	$-0.264612\pi$
$331$	24.5216	1.34783	0.673913	+	0.738810i	$0.264612\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−12.6318	−0.690151
$336$	0	0
$337$	−11.0840	−0.603786	−0.301893	−	0.953342i	$-0.597618\pi$
$337$	−11.0840	−0.603786	−0.301893	+	0.953342i	$0.597618\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−34.8296	−1.88613
$342$	0	0
$343$	15.7199	0.848792
$344$	0	0
$345$	0	0
$346$	0	0
$347$	33.2478	1.78484	0.892419	−	0.451209i	$-0.149007\pi$
$347$	33.2478	1.78484	0.892419	+	0.451209i	$0.149007\pi$
$348$	0	0
$349$	19.2781	1.03193	0.515967	−	0.856608i	$-0.327433\pi$
$349$	19.2781	1.03193	0.515967	+	0.856608i	$0.327433\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−21.5048	−1.14459	−0.572293	−	0.820049i	$-0.693946\pi$
$353$	−21.5048	−1.14459	−0.572293	+	0.820049i	$0.693946\pi$
$354$	0	0
$355$	−2.37925	−0.126278
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−29.4723	−1.55549	−0.777745	−	0.628580i	$-0.783636\pi$
$359$	−29.4723	−1.55549	−0.777745	+	0.628580i	$0.783636\pi$
$360$	0	0
$361$	25.5748	1.34604
$362$	0	0
$363$	0	0
$364$	0	0
$365$	13.0748	0.684369
$366$	0	0
$367$	4.12840	0.215501	0.107750	−	0.994178i	$-0.465635\pi$
$367$	4.12840	0.215501	0.107750	+	0.994178i	$0.465635\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−20.5213	−1.06541
$372$	0	0
$373$	24.8842	1.28845	0.644227	−	0.764834i	$-0.277179\pi$
$373$	24.8842	1.28845	0.644227	+	0.764834i	$0.277179\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	48.4133	2.49341
$378$	0	0
$379$	31.4010	1.61296	0.806479	−	0.591262i	$-0.201370\pi$
$379$	31.4010	1.61296	0.806479	+	0.591262i	$0.201370\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	9.51143	0.486012	0.243006	−	0.970025i	$-0.421867\pi$
$383$	9.51143	0.486012	0.243006	+	0.970025i	$0.421867\pi$
$384$	0	0
$385$	13.2386	0.674700
$386$	0	0
$387$	0	0
$388$	0	0
$389$	30.7578	1.55948	0.779740	−	0.626103i	$-0.215351\pi$
$389$	30.7578	1.55948	0.779740	+	0.626103i	$0.215351\pi$
$390$	0	0
$391$	7.63197	0.385965
$392$	0	0
$393$	0	0
$394$	0	0
$395$	4.67532	0.235241
$396$	0	0
$397$	13.4067	0.672865	0.336433	−	0.941708i	$-0.390780\pi$
$397$	13.4067	0.672865	0.336433	+	0.941708i	$0.390780\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−15.0502	−0.751570	−0.375785	−	0.926707i	$-0.622627\pi$
$401$	−15.0502	−0.751570	−0.375785	+	0.926707i	$0.622627\pi$
$402$	0	0
$403$	47.5705	2.36965
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−6.37551	−0.316022
$408$	0	0
$409$	−9.83384	−0.486252	−0.243126	−	0.969995i	$-0.578173\pi$
$409$	−9.83384	−0.486252	−0.243126	+	0.969995i	$0.578173\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−23.7982	−1.17104
$414$	0	0
$415$	−3.36670	−0.165265
$416$	0	0
$417$	0	0
$418$	0	0
$419$	5.51239	0.269298	0.134649	−	0.990893i	$-0.457009\pi$
$419$	5.51239	0.269298	0.134649	+	0.990893i	$0.457009\pi$
$420$	0	0
$421$	20.7665	1.01209	0.506047	−	0.862506i	$-0.331106\pi$
$421$	20.7665	1.01209	0.506047	+	0.862506i	$0.331106\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−7.63197	−0.370205
$426$	0	0
$427$	−39.2854	−1.90115
$428$	0	0
$429$	0	0
$430$	0	0
$431$	25.1384	1.21087	0.605436	−	0.795894i	$-0.292999\pi$
$431$	25.1384	1.21087	0.605436	+	0.795894i	$0.292999\pi$
$432$	0	0
$433$	−17.4179	−0.837053	−0.418527	−	0.908205i	$-0.637453\pi$
$433$	−17.4179	−0.837053	−0.418527	+	0.908205i	$0.637453\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−6.67643	−0.319377
$438$	0	0
$439$	0.830895	0.0396564	0.0198282	−	0.999803i	$-0.493688\pi$
$439$	0.830895	0.0396564	0.0198282	+	0.999803i	$0.493688\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−6.73582	−0.320028	−0.160014	−	0.987115i	$-0.551154\pi$
$443$	−6.73582	−0.320028	−0.160014	+	0.987115i	$0.551154\pi$
$444$	0	0
$445$	13.5564	0.642633
$446$	0	0
$447$	0	0
$448$	0	0
$449$	15.4042	0.726968	0.363484	−	0.931600i	$-0.381587\pi$
$449$	15.4042	0.726968	0.363484	+	0.931600i	$0.381587\pi$
$450$	0	0
$451$	−24.0579	−1.13284
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−18.0813	−0.847666
$456$	0	0
$457$	−22.9088	−1.07163	−0.535815	−	0.844335i	$-0.679996\pi$
$457$	−22.9088	−1.07163	−0.535815	+	0.844335i	$0.679996\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−3.68038	−0.171412	−0.0857062	−	0.996320i	$-0.527315\pi$
$461$	−3.68038	−0.171412	−0.0857062	+	0.996320i	$0.527315\pi$
$462$	0	0
$463$	−6.01377	−0.279483	−0.139742	−	0.990188i	$-0.544627\pi$
$463$	−6.01377	−0.279483	−0.139742	+	0.990188i	$0.544627\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	14.6763	0.679138	0.339569	−	0.940581i	$-0.389719\pi$
$467$	14.6763	0.679138	0.339569	+	0.940581i	$0.389719\pi$
$468$	0	0
$469$	−37.1672	−1.71622
$470$	0	0
$471$	0	0
$472$	0	0
$473$	20.1006	0.924226
$474$	0	0
$475$	6.67643	0.306336
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−15.9369	−0.728177	−0.364088	−	0.931364i	$-0.618619\pi$
$479$	−15.9369	−0.728177	−0.364088	+	0.931364i	$0.618619\pi$
$480$	0	0
$481$	8.70771	0.397038
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−6.91181	−0.313849
$486$	0	0
$487$	30.4205	1.37849	0.689243	−	0.724530i	$-0.257943\pi$
$487$	30.4205	1.37849	0.689243	+	0.724530i	$0.257943\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−2.22937	−0.100610	−0.0503051	−	0.998734i	$-0.516019\pi$
$491$	−2.22937	−0.100610	−0.0503051	+	0.998734i	$0.516019\pi$
$492$	0	0
$493$	60.1262	2.70795
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−7.00057	−0.314018
$498$	0	0
$499$	13.2451	0.592934	0.296467	−	0.955043i	$-0.404192\pi$
$499$	13.2451	0.592934	0.296467	+	0.955043i	$0.404192\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−23.7476	−1.05885	−0.529427	−	0.848355i	$-0.677593\pi$
$503$	−23.7476	−1.05885	−0.529427	+	0.848355i	$0.677593\pi$
$504$	0	0
$505$	−9.82227	−0.437085
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−3.46613	−0.153633	−0.0768167	−	0.997045i	$-0.524476\pi$
$509$	−3.46613	−0.153633	−0.0768167	+	0.997045i	$0.524476\pi$
$510$	0	0
$511$	38.4706	1.70184
$512$	0	0
$513$	0	0
$514$	0	0
$515$	10.5674	0.465653
$516$	0	0
$517$	−3.06658	−0.134868
$518$	0	0
$519$	0	0
$520$	0	0
$521$	29.1587	1.27747	0.638733	−	0.769428i	$-0.279459\pi$
$521$	29.1587	1.27747	0.638733	+	0.769428i	$0.279459\pi$
$522$	0	0
$523$	6.32137	0.276414	0.138207	−	0.990403i	$-0.455866\pi$
$523$	6.32137	0.276414	0.138207	+	0.990403i	$0.455866\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	59.0795	2.57354
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	0	0
$532$	0	0
$533$	32.8585	1.42326
$534$	0	0
$535$	4.20946	0.181991
$536$	0	0
$537$	0	0
$538$	0	0
$539$	7.45703	0.321197
$540$	0	0
$541$	−21.6883	−0.932452	−0.466226	−	0.884666i	$-0.654387\pi$
$541$	−21.6883	−0.932452	−0.466226	+	0.884666i	$0.654387\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	4.43029	0.189773
$546$	0	0
$547$	−36.4275	−1.55753	−0.778764	−	0.627317i	$-0.784153\pi$
$547$	−36.4275	−1.55753	−0.778764	+	0.627317i	$0.784153\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−52.5983	−2.24076
$552$	0	0
$553$	13.7564	0.584981
$554$	0	0
$555$	0	0
$556$	0	0
$557$	13.2796	0.562677	0.281338	−	0.959609i	$-0.409222\pi$
$557$	13.2796	0.562677	0.281338	+	0.959609i	$0.409222\pi$
$558$	0	0
$559$	−27.4535	−1.16116
$560$	0	0
$561$	0	0
$562$	0	0
$563$	14.6900	0.619111	0.309555	−	0.950881i	$-0.399820\pi$
$563$	14.6900	0.619111	0.309555	+	0.950881i	$0.399820\pi$
$564$	0	0
$565$	−14.0685	−0.591866
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−12.8599	−0.539115	−0.269557	−	0.962984i	$-0.586877\pi$
$569$	−12.8599	−0.539115	−0.269557	+	0.962984i	$0.586877\pi$
$570$	0	0
$571$	25.3157	1.05943	0.529715	−	0.848176i	$-0.322299\pi$
$571$	25.3157	1.05943	0.529715	+	0.848176i	$0.322299\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−1.00000	−0.0417029
$576$	0	0
$577$	−23.1352	−0.963130	−0.481565	−	0.876410i	$-0.659931\pi$
$577$	−23.1352	−0.963130	−0.481565	+	0.876410i	$0.659931\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−9.90597	−0.410969
$582$	0	0
$583$	31.3805	1.29965
$584$	0	0
$585$	0	0
$586$	0	0
$587$	22.8357	0.942531	0.471266	−	0.881991i	$-0.343797\pi$
$587$	22.8357	0.942531	0.471266	+	0.881991i	$0.343797\pi$
$588$	0	0
$589$	−51.6826	−2.12955
$590$	0	0
$591$	0	0
$592$	0	0
$593$	18.9815	0.779478	0.389739	−	0.920925i	$-0.372565\pi$
$593$	18.9815	0.779478	0.389739	+	0.920925i	$0.372565\pi$
$594$	0	0
$595$	−22.4559	−0.920600
$596$	0	0
$597$	0	0
$598$	0	0
$599$	3.64101	0.148768	0.0743839	−	0.997230i	$-0.476301\pi$
$599$	3.64101	0.148768	0.0743839	+	0.997230i	$0.476301\pi$
$600$	0	0
$601$	−44.1677	−1.80164	−0.900820	−	0.434193i	$-0.857034\pi$
$601$	−44.1677	−1.80164	−0.900820	+	0.434193i	$0.857034\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−9.24401	−0.375822
$606$	0	0
$607$	21.3293	0.865730	0.432865	−	0.901459i	$-0.357503\pi$
$607$	21.3293	0.865730	0.432865	+	0.901459i	$0.357503\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	4.18836	0.169443
$612$	0	0
$613$	19.0200	0.768211	0.384106	−	0.923289i	$-0.374510\pi$
$613$	19.0200	0.768211	0.384106	+	0.923289i	$0.374510\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	10.3460	0.416516	0.208258	−	0.978074i	$-0.433221\pi$
$617$	10.3460	0.416516	0.208258	+	0.978074i	$0.433221\pi$
$618$	0	0
$619$	−13.4924	−0.542305	−0.271152	−	0.962536i	$-0.587405\pi$
$619$	−13.4924	−0.542305	−0.271152	+	0.962536i	$0.587405\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	39.8874	1.59806
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	10.8144	0.431199
$630$	0	0
$631$	10.4061	0.414261	0.207130	−	0.978313i	$-0.433588\pi$
$631$	10.4061	0.414261	0.207130	+	0.978313i	$0.433588\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	18.1071	0.718559
$636$	0	0
$637$	−10.1849	−0.403539
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−35.4650	−1.40078	−0.700391	−	0.713759i	$-0.746991\pi$
$641$	−35.4650	−1.40078	−0.700391	+	0.713759i	$0.746991\pi$
$642$	0	0
$643$	−31.4519	−1.24034	−0.620171	−	0.784467i	$-0.712937\pi$
$643$	−31.4519	−1.24034	−0.620171	+	0.784467i	$0.712937\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	30.8797	1.21400	0.607002	−	0.794700i	$-0.292372\pi$
$647$	30.8797	1.21400	0.607002	+	0.794700i	$0.292372\pi$
$648$	0	0
$649$	36.3915	1.42849
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−2.38136	−0.0931899	−0.0465949	−	0.998914i	$-0.514837\pi$
$653$	−2.38136	−0.0931899	−0.0465949	+	0.998914i	$0.514837\pi$
$654$	0	0
$655$	−10.3806	−0.405603
$656$	0	0
$657$	0	0
$658$	0	0
$659$	29.5210	1.14997	0.574987	−	0.818163i	$-0.305007\pi$
$659$	29.5210	1.14997	0.574987	+	0.818163i	$0.305007\pi$
$660$	0	0
$661$	−28.4064	−1.10488	−0.552440	−	0.833553i	$-0.686303\pi$
$661$	−28.4064	−1.10488	−0.552440	+	0.833553i	$0.686303\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	19.6443	0.761775
$666$	0	0
$667$	7.87821	0.305045
$668$	0	0
$669$	0	0
$670$	0	0
$671$	60.0740	2.31913
$672$	0	0
$673$	−19.4362	−0.749210	−0.374605	−	0.927185i	$-0.622222\pi$
$673$	−19.4362	−0.749210	−0.374605	+	0.927185i	$0.622222\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−7.62254	−0.292958	−0.146479	−	0.989214i	$-0.546794\pi$
$677$	−7.62254	−0.292958	−0.146479	+	0.989214i	$0.546794\pi$
$678$	0	0
$679$	−20.3369	−0.780458
$680$	0	0
$681$	0	0
$682$	0	0
$683$	45.3098	1.73373	0.866865	−	0.498543i	$-0.166131\pi$
$683$	45.3098	1.73373	0.866865	+	0.498543i	$0.166131\pi$
$684$	0	0
$685$	0.238791	0.00912373
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−42.8597	−1.63282
$690$	0	0
$691$	28.6251	1.08895	0.544475	−	0.838777i	$-0.316729\pi$
$691$	28.6251	1.08895	0.544475	+	0.838777i	$0.316729\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	14.7185	0.558303
$696$	0	0
$697$	40.8081	1.54572
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−29.3111	−1.10707	−0.553533	−	0.832827i	$-0.686721\pi$
$701$	−29.3111	−1.10707	−0.553533	+	0.832827i	$0.686721\pi$
$702$	0	0
$703$	−9.46043	−0.356807
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−28.9005	−1.08691
$708$	0	0
$709$	−26.9218	−1.01107	−0.505534	−	0.862807i	$-0.668704\pi$
$709$	−26.9218	−1.01107	−0.505534	+	0.862807i	$0.668704\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	7.74105	0.289905
$714$	0	0
$715$	27.6494	1.03403
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−8.80096	−0.328220	−0.164110	−	0.986442i	$-0.552475\pi$
$719$	−8.80096	−0.328220	−0.164110	+	0.986442i	$0.552475\pi$
$720$	0	0
$721$	31.0927	1.15795
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−7.87821	−0.292589
$726$	0	0
$727$	−30.8395	−1.14377	−0.571887	−	0.820332i	$-0.693789\pi$
$727$	−30.8395	−1.14377	−0.571887	+	0.820332i	$0.693789\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−34.0955	−1.26107
$732$	0	0
$733$	26.3106	0.971805	0.485902	−	0.874013i	$-0.338491\pi$
$733$	26.3106	0.971805	0.485902	+	0.874013i	$0.338491\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	56.8349	2.09354
$738$	0	0
$739$	−48.2273	−1.77407	−0.887036	−	0.461701i	$-0.847239\pi$
$739$	−48.2273	−1.77407	−0.887036	+	0.461701i	$0.847239\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−20.3761	−0.747528	−0.373764	−	0.927524i	$-0.621933\pi$
$743$	−20.3761	−0.747528	−0.373764	+	0.927524i	$0.621933\pi$
$744$	0	0
$745$	−21.5341	−0.788950
$746$	0	0
$747$	0	0
$748$	0	0
$749$	12.3857	0.452562
$750$	0	0
$751$	−48.9084	−1.78469	−0.892346	−	0.451351i	$-0.850942\pi$
$751$	−48.9084	−1.78469	−0.892346	+	0.451351i	$0.850942\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	5.53707	0.201515
$756$	0	0
$757$	−31.1382	−1.13174	−0.565869	−	0.824495i	$-0.691459\pi$
$757$	−31.1382	−1.13174	−0.565869	+	0.824495i	$0.691459\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−24.8444	−0.900609	−0.450305	−	0.892875i	$-0.648685\pi$
$761$	−24.8444	−0.900609	−0.450305	+	0.892875i	$0.648685\pi$
$762$	0	0
$763$	13.0354	0.471914
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−49.7038	−1.79470
$768$	0	0
$769$	37.9919	1.37002	0.685012	−	0.728532i	$-0.259797\pi$
$769$	37.9919	1.37002	0.685012	+	0.728532i	$0.259797\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	15.7071	0.564944	0.282472	−	0.959276i	$-0.408846\pi$
$773$	15.7071	0.564944	0.282472	+	0.959276i	$0.408846\pi$
$774$	0	0
$775$	−7.74105	−0.278067
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−35.6989	−1.27904
$780$	0	0
$781$	10.7051	0.383057
$782$	0	0
$783$	0	0
$784$	0	0
$785$	14.8405	0.529679
$786$	0	0
$787$	−41.9105	−1.49395	−0.746974	−	0.664853i	$-0.768494\pi$
$787$	−41.9105	−1.49395	−0.746974	+	0.664853i	$0.768494\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−41.3943	−1.47181
$792$	0	0
$793$	−82.0495	−2.91366
$794$	0	0
$795$	0	0
$796$	0	0
$797$	8.99867	0.318749	0.159375	−	0.987218i	$-0.449052\pi$
$797$	8.99867	0.318749	0.159375	+	0.987218i	$0.449052\pi$
$798$	0	0
$799$	5.20168	0.184022
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−58.8281	−2.07600
$804$	0	0
$805$	−2.94234	−0.103704
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−27.3360	−0.961082	−0.480541	−	0.876972i	$-0.659560\pi$
$809$	−27.3360	−0.961082	−0.480541	+	0.876972i	$0.659560\pi$
$810$	0	0
$811$	−34.0481	−1.19559	−0.597795	−	0.801649i	$-0.703956\pi$
$811$	−34.0481	−1.19559	−0.597795	+	0.801649i	$0.703956\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	17.4301	0.610549
$816$	0	0
$817$	29.8267	1.04350
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−34.7114	−1.21144	−0.605718	−	0.795679i	$-0.707114\pi$
$821$	−34.7114	−1.21144	−0.605718	+	0.795679i	$0.707114\pi$
$822$	0	0
$823$	9.86480	0.343865	0.171933	−	0.985109i	$-0.444999\pi$
$823$	9.86480	0.343865	0.171933	+	0.985109i	$0.444999\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−52.3071	−1.81890	−0.909448	−	0.415817i	$-0.863496\pi$
$827$	−52.3071	−1.81890	−0.909448	+	0.415817i	$0.863496\pi$
$828$	0	0
$829$	28.4502	0.988117	0.494058	−	0.869429i	$-0.335513\pi$
$829$	28.4502	0.988117	0.494058	+	0.869429i	$0.335513\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−12.6489	−0.438260
$834$	0	0
$835$	−15.0614	−0.521221
$836$	0	0
$837$	0	0
$838$	0	0
$839$	7.68172	0.265202	0.132601	−	0.991169i	$-0.457667\pi$
$839$	7.68172	0.265202	0.132601	+	0.991169i	$0.457667\pi$
$840$	0	0
$841$	33.0661	1.14021
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−24.7638	−0.851899
$846$	0	0
$847$	−27.1990	−0.934569
$848$	0	0
$849$	0	0
$850$	0	0
$851$	1.41699	0.0485738
$852$	0	0
$853$	26.4875	0.906913	0.453457	−	0.891278i	$-0.350191\pi$
$853$	26.4875	0.906913	0.453457	+	0.891278i	$0.350191\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	13.9037	0.474941	0.237470	−	0.971395i	$-0.423682\pi$
$857$	13.9037	0.474941	0.237470	+	0.971395i	$0.423682\pi$
$858$	0	0
$859$	−23.1983	−0.791514	−0.395757	−	0.918355i	$-0.629518\pi$
$859$	−23.1983	−0.791514	−0.395757	+	0.918355i	$0.629518\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−25.0610	−0.853086	−0.426543	−	0.904467i	$-0.640269\pi$
$863$	−25.0610	−0.853086	−0.426543	+	0.904467i	$0.640269\pi$
$864$	0	0
$865$	−15.3021	−0.520286
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−21.0358	−0.713592
$870$	0	0
$871$	−77.6255	−2.63024
$872$	0	0
$873$	0	0
$874$	0	0
$875$	2.94234	0.0994692
$876$	0	0
$877$	−6.07171	−0.205027	−0.102513	−	0.994732i	$-0.532688\pi$
$877$	−6.07171	−0.205027	−0.102513	+	0.994732i	$0.532688\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−11.5784	−0.390086	−0.195043	−	0.980795i	$-0.562485\pi$
$881$	−11.5784	−0.390086	−0.195043	+	0.980795i	$0.562485\pi$
$882$	0	0
$883$	−4.51421	−0.151915	−0.0759576	−	0.997111i	$-0.524201\pi$
$883$	−4.51421	−0.151915	−0.0759576	+	0.997111i	$0.524201\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−22.8462	−0.767099	−0.383550	−	0.923520i	$-0.625298\pi$
$887$	−22.8462	−0.767099	−0.383550	+	0.923520i	$0.625298\pi$
$888$	0	0
$889$	53.2773	1.78686
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−4.55042	−0.152274
$894$	0	0
$895$	−8.43653	−0.282002
$896$	0	0
$897$	0	0
$898$	0	0
$899$	60.9856	2.03398
$900$	0	0
$901$	−53.2289	−1.77331
$902$	0	0
$903$	0	0
$904$	0	0
$905$	7.11921	0.236650
$906$	0	0
$907$	8.71708	0.289446	0.144723	−	0.989472i	$-0.453771\pi$
$907$	8.71708	0.289446	0.144723	+	0.989472i	$0.453771\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	40.4405	1.33985	0.669926	−	0.742428i	$-0.266326\pi$
$911$	40.4405	1.33985	0.669926	+	0.742428i	$0.266326\pi$
$912$	0	0
$913$	15.1479	0.501322
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−30.5432	−1.00863
$918$	0	0
$919$	17.7589	0.585813	0.292907	−	0.956141i	$-0.405377\pi$
$919$	17.7589	0.585813	0.292907	+	0.956141i	$0.405377\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−14.6210	−0.481257
$924$	0	0
$925$	−1.41699	−0.0465903
$926$	0	0
$927$	0	0
$928$	0	0
$929$	25.0961	0.823376	0.411688	−	0.911325i	$-0.364939\pi$
$929$	25.0961	0.823376	0.411688	+	0.911325i	$0.364939\pi$
$930$	0	0
$931$	11.0653	0.362650
$932$	0	0
$933$	0	0
$934$	0	0
$935$	34.3388	1.12300
$936$	0	0
$937$	−16.3849	−0.535272	−0.267636	−	0.963520i	$-0.586242\pi$
$937$	−16.3849	−0.535272	−0.267636	+	0.963520i	$0.586242\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	38.6967	1.26148	0.630738	−	0.775996i	$-0.282752\pi$
$941$	38.6967	1.26148	0.630738	+	0.775996i	$0.282752\pi$
$942$	0	0
$943$	5.34700	0.174122
$944$	0	0
$945$	0	0
$946$	0	0
$947$	12.8362	0.417121	0.208561	−	0.978009i	$-0.433122\pi$
$947$	12.8362	0.417121	0.208561	+	0.978009i	$0.433122\pi$
$948$	0	0
$949$	80.3479	2.60820
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−18.4904	−0.598963	−0.299482	−	0.954102i	$-0.596814\pi$
$953$	−18.4904	−0.598963	−0.299482	+	0.954102i	$0.596814\pi$
$954$	0	0
$955$	5.72713	0.185326
$956$	0	0
$957$	0	0
$958$	0	0
$959$	0.702604	0.0226883
$960$	0	0
$961$	28.9239	0.933028
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−19.3045	−0.621433
$966$	0	0
$967$	22.0111	0.707830	0.353915	−	0.935278i	$-0.384850\pi$
$967$	22.0111	0.707830	0.353915	+	0.935278i	$0.384850\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	17.1953	0.551823	0.275912	−	0.961183i	$-0.411020\pi$
$971$	17.1953	0.551823	0.275912	+	0.961183i	$0.411020\pi$
$972$	0	0
$973$	43.3067	1.38835
$974$	0	0
$975$	0	0
$976$	0	0
$977$	53.6088	1.71510	0.857549	−	0.514403i	$-0.171986\pi$
$977$	53.6088	1.71510	0.857549	+	0.514403i	$0.171986\pi$
$978$	0	0
$979$	−60.9946	−1.94940
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−41.4581	−1.32231	−0.661154	−	0.750250i	$-0.729933\pi$
$983$	−41.4581	−1.32231	−0.661154	+	0.750250i	$0.729933\pi$
$984$	0	0
$985$	12.2523	0.390392
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−4.46746	−0.142057
$990$	0	0
$991$	−39.7669	−1.26324	−0.631619	−	0.775279i	$-0.717609\pi$
$991$	−39.7669	−1.26324	−0.631619	+	0.775279i	$0.717609\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−25.3334	−0.803123
$996$	0	0
$997$	−36.0661	−1.14222	−0.571112	−	0.820872i	$-0.693488\pi$
$997$	−36.0661	−1.14222	−0.571112	+	0.820872i	$0.693488\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8280.2.a.bv.1.2	✓	7
3.2	odd	2		8280.2.a.bw.1.2	yes	7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8280.2.a.bv.1.2	✓	7	1.1	even	1	trivial
8280.2.a.bw.1.2	yes	7	3.2	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 8280 = 2^{3} \cdot 3^{2} \cdot 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8280.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{5} -2.94234 q^{7} +O(q^{10})\)	\(q-1.00000 q^{5} -2.94234 q^{7} +4.49933 q^{11} -6.14522 q^{13} -7.63197 q^{17} +6.67643 q^{19} -1.00000 q^{23} +1.00000 q^{25} -7.87821 q^{29} -7.74105 q^{31} +2.94234 q^{35} -1.41699 q^{37} -5.34700 q^{41} +4.46746 q^{43} -0.681564 q^{47} +1.65736 q^{49} +6.97447 q^{53} -4.49933 q^{55} +8.08821 q^{59} +13.3518 q^{61} +6.14522 q^{65} +12.6318 q^{67} +2.37925 q^{71} -13.0748 q^{73} -13.2386 q^{77} -4.67532 q^{79} +3.36670 q^{83} +7.63197 q^{85} -13.5564 q^{89} +18.0813 q^{91} -6.67643 q^{95} +6.91181 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q - 7 q^{5} - 6 q^{7}+O(q^{10}) \) 7 * q - 7 * q^5 - 6 * q^7	\( 7 q - 7 q^{5} - 6 q^{7} + 2 q^{11} - 6 q^{13} + 4 q^{17} - 8 q^{19} - 7 q^{23} + 7 q^{25} - 2 q^{29} - 8 q^{31} + 6 q^{35} - 16 q^{37} - 2 q^{41} - 10 q^{43} + 8 q^{47} + 19 q^{49} + 10 q^{53} - 2 q^{55} + 24 q^{59} + 8 q^{61} + 6 q^{65} - 20 q^{67} + 8 q^{71} + 2 q^{73} + 12 q^{77} - 2 q^{79} + 22 q^{83} - 4 q^{85} + 16 q^{89} - 20 q^{91} + 8 q^{95} + 4 q^{97}+O(q^{100}) \) 7 * q - 7 * q^5 - 6 * q^7 + 2 * q^11 - 6 * q^13 + 4 * q^17 - 8 * q^19 - 7 * q^23 + 7 * q^25 - 2 * q^29 - 8 * q^31 + 6 * q^35 - 16 * q^37 - 2 * q^41 - 10 * q^43 + 8 * q^47 + 19 * q^49 + 10 * q^53 - 2 * q^55 + 24 * q^59 + 8 * q^61 + 6 * q^65 - 20 * q^67 + 8 * q^71 + 2 * q^73 + 12 * q^77 - 2 * q^79 + 22 * q^83 - 4 * q^85 + 16 * q^89 - 20 * q^91 + 8 * q^95 + 4 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more