LMFDB - Embedded newform 8280.2.p.a.1241.14

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8280,2,Mod(1241,8280)]

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, 1, 0, 1]))

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

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

chi := DirichletCharacter("8280.1241");

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.p (of order $2$, degree $1$, minimal)

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:	no
Analytic conductor:	$66.1161328736$
Analytic rank:	$0$
Dimension:	$48$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1241.14
Character	$\chi$	$=$	8280.1241
Dual form			8280.2.p.a.1241.35

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{5} -2.35143i q^{7} +O(q^{10})$	$q-1.00000 q^{5} -2.35143i q^{7} +1.47885 q^{11} +4.67757 q^{13} -5.61193 q^{17} +3.07752i q^{19} +(-4.17933 - 2.35228i) q^{23} +1.00000 q^{25} -2.05045i q^{29} +8.72555 q^{31} +2.35143i q^{35} +3.93677i q^{37} -6.91846i q^{41} -3.78778i q^{43} +10.9770i q^{47} +1.47077 q^{49} +13.7986 q^{53} -1.47885 q^{55} -7.07225i q^{59} +10.2901i q^{61} -4.67757 q^{65} +1.88107i q^{67} -3.91325i q^{71} -6.95228 q^{73} -3.47742i q^{77} -7.99760i q^{79} +0.219311 q^{83} +5.61193 q^{85} -2.65745 q^{89} -10.9990i q^{91} -3.07752i q^{95} +9.88545i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 48 q - 48 q^{5}+O(q^{10}) $ 48 * q - 48 * q^5	$ 48 q - 48 q^{5} - 8 q^{11} + 4 q^{23} + 48 q^{25} + 8 q^{31} - 32 q^{49} + 8 q^{55} - 16 q^{73} - 32 q^{83} - 8 q^{89}+O(q^{100}) $ 48 * q - 48 * q^5 - 8 * q^11 + 4 * q^23 + 48 * q^25 + 8 * q^31 - 32 * q^49 + 8 * q^55 - 16 * q^73 - 32 * q^83 - 8 * q^89

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/8280\mathbb{Z}\right)^\times$.

$n$	$1657$	$2071$	$3961$	$4141$	$4601$
$\chi(n)$	$1$	$1$	$-1$	$1$	$-1$

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
$3$		0			0
$3$		0			0
$4$		0			0
$5$	−1.00000			−0.447214
$5$	−1.00000			−0.447214
$6$		0			0
$7$		−	2.35143i		−	0.888758i	−0.895839	−	0.444379i	$-0.853424\pi$
$7$		−	2.35143i		−	0.888758i	0.895839	−	0.444379i	$-0.146576\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	1.47885			0.445890			0.222945	−	0.974831i	$-0.428433\pi$
$11$	1.47885			0.445890			0.222945	+	0.974831i	$0.428433\pi$
$12$		0			0
$13$	4.67757			1.29732			0.648662	−	0.761077i	$-0.275329\pi$
$13$	4.67757			1.29732			0.648662	+	0.761077i	$0.275329\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−5.61193			−1.36109			−0.680546	−	0.732705i	$-0.738257\pi$
$17$	−5.61193			−1.36109			−0.680546	+	0.732705i	$0.738257\pi$
$18$		0			0
$19$			3.07752i			0.706031i	0.935617	+	0.353016i	$0.114844\pi$
$19$			3.07752i			0.706031i	−0.935617	+	0.353016i	$0.885156\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−4.17933	−	2.35228i	−0.871451	−	0.490483i
$23$	−4.17933	−	2.35228i	−0.871451	−	0.490483i
$24$		0			0
$25$	1.00000			0.200000
$26$		0			0
$27$		0			0
$28$		0			0
$29$		−	2.05045i		−	0.380759i	−0.981711	−	0.190379i	$-0.939028\pi$
$29$		−	2.05045i		−	0.380759i	0.981711	−	0.190379i	$-0.0609718\pi$
$30$		0			0
$31$	8.72555			1.56716			0.783578	−	0.621294i	$-0.213393\pi$
$31$	8.72555			1.56716			0.783578	+	0.621294i	$0.213393\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$			2.35143i			0.397465i
$36$		0			0
$37$			3.93677i			0.647200i	0.946194	+	0.323600i	$0.104893\pi$
$37$			3.93677i			0.647200i	−0.946194	+	0.323600i	$0.895107\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		−	6.91846i		−	1.08048i	−0.841510	−	0.540241i	$-0.818333\pi$
$41$		−	6.91846i		−	1.08048i	0.841510	−	0.540241i	$-0.181667\pi$
$42$		0			0
$43$		−	3.78778i		−	0.577631i	−0.957385	−	0.288816i	$-0.906739\pi$
$43$		−	3.78778i		−	0.577631i	0.957385	−	0.288816i	$-0.0932615\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$			10.9770i			1.60115i	0.599230	+	0.800577i	$0.295474\pi$
$47$			10.9770i			1.60115i	−0.599230	+	0.800577i	$0.704526\pi$
$48$		0			0
$49$	1.47077			0.210110
$50$		0			0
$51$		0			0
$52$		0			0
$53$	13.7986			1.89539			0.947693	−	0.319182i	$-0.103408\pi$
$53$	13.7986			1.89539			0.947693	+	0.319182i	$0.103408\pi$
$54$		0			0
$55$	−1.47885			−0.199408
$56$		0			0
$57$		0			0
$58$		0			0
$59$		−	7.07225i		−	0.920729i	−0.887730	−	0.460365i	$-0.847719\pi$
$59$		−	7.07225i		−	0.920729i	0.887730	−	0.460365i	$-0.152281\pi$
$60$		0			0
$61$			10.2901i			1.31751i	0.752359	+	0.658754i	$0.228916\pi$
$61$			10.2901i			1.31751i	−0.752359	+	0.658754i	$0.771084\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−4.67757			−0.580181
$66$		0			0
$67$			1.88107i			0.229810i	0.993377	+	0.114905i	$0.0366563\pi$
$67$			1.88107i			0.229810i	−0.993377	+	0.114905i	$0.963344\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		−	3.91325i		−	0.464418i	−0.972666	−	0.232209i	$-0.925405\pi$
$71$		−	3.91325i		−	0.464418i	0.972666	−	0.232209i	$-0.0745953\pi$
$72$		0			0
$73$	−6.95228			−0.813703			−0.406852	−	0.913494i	$-0.633373\pi$
$73$	−6.95228			−0.813703			−0.406852	+	0.913494i	$0.633373\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$		−	3.47742i		−	0.396288i
$78$		0			0
$79$		−	7.99760i		−	0.899800i	−0.893079	−	0.449900i	$-0.851460\pi$
$79$		−	7.99760i		−	0.899800i	0.893079	−	0.449900i	$-0.148540\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	0.219311			0.0240725			0.0120363	−	0.999928i	$-0.496169\pi$
$83$	0.219311			0.0240725			0.0120363	+	0.999928i	$0.496169\pi$
$84$		0			0
$85$	5.61193			0.608699
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−2.65745			−0.281690			−0.140845	−	0.990032i	$-0.544982\pi$
$89$	−2.65745			−0.281690			−0.140845	+	0.990032i	$0.544982\pi$
$90$		0			0
$91$		−	10.9990i		−	1.15301i
$92$		0			0
$93$		0			0
$94$		0			0
$95$		−	3.07752i		−	0.315747i
$96$		0			0
$97$			9.88545i			1.00372i	0.864950	+	0.501858i	$0.167350\pi$
$97$			9.88545i			1.00372i	−0.864950	+	0.501858i	$0.832650\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$		−	16.7015i		−	1.66186i	−0.556375	−	0.830931i	$-0.687808\pi$
$101$		−	16.7015i		−	1.66186i	0.556375	−	0.830931i	$-0.312192\pi$
$102$		0			0
$103$		−	6.41186i		−	0.631779i	−0.948796	−	0.315890i	$-0.897697\pi$
$103$		−	6.41186i		−	0.631779i	0.948796	−	0.315890i	$-0.102303\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	13.1797			1.27413			0.637065	−	0.770810i	$-0.280148\pi$
$107$	13.1797			1.27413			0.637065	+	0.770810i	$0.280148\pi$
$108$		0			0
$109$		−	0.188623i		−	0.0180668i	−0.999959	−	0.00903339i	$-0.997125\pi$
$109$		−	0.188623i		−	0.0180668i	0.999959	−	0.00903339i	$-0.00287546\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	7.87587			0.740900			0.370450	−	0.928852i	$-0.379204\pi$
$113$	7.87587			0.740900			0.370450	+	0.928852i	$0.379204\pi$
$114$		0			0
$115$	4.17933	+	2.35228i	0.389725	+	0.219351i
$116$		0			0
$117$		0			0
$118$		0			0
$119$			13.1961i			1.20968i
$120$		0			0
$121$	−8.81300			−0.801182
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−1.00000			−0.0894427
$126$		0			0
$127$	3.54279			0.314372			0.157186	−	0.987569i	$-0.449758\pi$
$127$	3.54279			0.314372			0.157186	+	0.987569i	$0.449758\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$		−	12.4354i		−	1.08649i	−0.839576	−	0.543243i	$-0.817196\pi$
$131$		−	12.4354i		−	1.08649i	0.839576	−	0.543243i	$-0.182804\pi$
$132$		0			0
$133$	7.23658			0.627491
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−14.5419			−1.24240			−0.621198	−	0.783653i	$-0.713354\pi$
$137$	−14.5419			−1.24240			−0.621198	+	0.783653i	$0.713354\pi$
$138$		0			0
$139$	20.5864			1.74611			0.873056	−	0.487620i	$-0.162135\pi$
$139$	20.5864			1.74611			0.873056	+	0.487620i	$0.162135\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	6.91742			0.578464
$144$		0			0
$145$			2.05045i			0.170281i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−15.5566			−1.27444			−0.637221	−	0.770681i	$-0.719916\pi$
$149$	−15.5566			−1.27444			−0.637221	+	0.770681i	$0.719916\pi$
$150$		0			0
$151$	5.92860			0.482463			0.241231	−	0.970468i	$-0.422449\pi$
$151$	5.92860			0.482463			0.241231	+	0.970468i	$0.422449\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−8.72555			−0.700853
$156$		0			0
$157$			3.52464i			0.281297i	0.990060	+	0.140649i	$0.0449188\pi$
$157$			3.52464i			0.281297i	−0.990060	+	0.140649i	$0.955081\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−5.53122	+	9.82741i	−0.435921	+	0.774508i
$162$		0			0
$163$	−6.33629			−0.496297			−0.248148	−	0.968722i	$-0.579822\pi$
$163$	−6.33629			−0.496297			−0.248148	+	0.968722i	$0.579822\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$			14.3269i			1.10865i	0.832300	+	0.554325i	$0.187024\pi$
$167$			14.3269i			1.10865i	−0.832300	+	0.554325i	$0.812976\pi$
$168$		0			0
$169$	8.87965			0.683050
$170$		0			0
$171$		0			0
$172$		0			0
$173$			4.53674i			0.344922i	0.985016	+	0.172461i	$0.0551718\pi$
$173$			4.53674i			0.344922i	−0.985016	+	0.172461i	$0.944828\pi$
$174$		0			0
$175$		−	2.35143i		−	0.177752i
$176$		0			0
$177$		0			0
$178$		0			0
$179$		−	2.95952i		−	0.221205i	−0.993865	−	0.110602i	$-0.964722\pi$
$179$		−	2.95952i		−	0.221205i	0.993865	−	0.110602i	$-0.0352780\pi$
$180$		0			0
$181$		−	18.0839i		−	1.34416i	−0.740477	−	0.672081i	$-0.765401\pi$
$181$		−	18.0839i		−	1.34416i	0.740477	−	0.672081i	$-0.234599\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		−	3.93677i		−	0.289437i
$186$		0			0
$187$	−8.29920			−0.606898
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−15.8720			−1.14846			−0.574229	−	0.818695i	$-0.694698\pi$
$191$	−15.8720			−1.14846			−0.574229	+	0.818695i	$0.694698\pi$
$192$		0			0
$193$	20.4693			1.47341			0.736707	−	0.676212i	$-0.236380\pi$
$193$	20.4693			1.47341			0.736707	+	0.676212i	$0.236380\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$			2.02134i			0.144014i	0.997404	+	0.0720071i	$0.0229404\pi$
$197$			2.02134i			0.144014i	−0.997404	+	0.0720071i	$0.977060\pi$
$198$		0			0
$199$		−	4.21021i		−	0.298454i	−0.988803	−	0.149227i	$-0.952321\pi$
$199$		−	4.21021i		−	0.298454i	0.988803	−	0.149227i	$-0.0476785\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−4.82149			−0.338402
$204$		0			0
$205$			6.91846i			0.483206i
$206$		0			0
$207$		0			0
$208$		0			0
$209$			4.55119i			0.314812i
$210$		0			0
$211$	−12.1131			−0.833901			−0.416951	−	0.908929i	$-0.636901\pi$
$211$	−12.1131			−0.833901			−0.416951	+	0.908929i	$0.636901\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$			3.78778i			0.258325i
$216$		0			0
$217$		−	20.5175i		−	1.39282i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−26.2502			−1.76578
$222$		0			0
$223$	−4.13946			−0.277199			−0.138599	−	0.990349i	$-0.544260\pi$
$223$	−4.13946			−0.277199			−0.138599	+	0.990349i	$0.544260\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	14.9769			0.994055			0.497027	−	0.867735i	$-0.334425\pi$
$227$	14.9769			0.994055			0.497027	+	0.867735i	$0.334425\pi$
$228$		0			0
$229$		−	6.14477i		−	0.406058i	−0.979173	−	0.203029i	$-0.934921\pi$
$229$		−	6.14477i		−	0.406058i	0.979173	−	0.203029i	$-0.0650785\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		−	15.5499i		−	1.01871i	−0.860557	−	0.509354i	$-0.829884\pi$
$233$		−	15.5499i		−	1.01871i	0.860557	−	0.509354i	$-0.170116\pi$
$234$		0			0
$235$		−	10.9770i		−	0.716058i
$236$		0			0
$237$		0			0
$238$		0			0
$239$		−	8.52633i		−	0.551522i	−0.961226	−	0.275761i	$-0.911070\pi$
$239$		−	8.52633i		−	0.551522i	0.961226	−	0.275761i	$-0.0889299\pi$
$240$		0			0
$241$		−	5.89346i		−	0.379631i	−0.981820	−	0.189816i	$-0.939211\pi$
$241$		−	5.89346i		−	0.379631i	0.981820	−	0.189816i	$-0.0607890\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−1.47077			−0.0939639
$246$		0			0
$247$			14.3953i			0.915951i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	21.6293			1.36523			0.682616	−	0.730777i	$-0.260842\pi$
$251$	21.6293			1.36523			0.682616	+	0.730777i	$0.260842\pi$
$252$		0			0
$253$	−6.18060	−	3.47866i	−0.388571	−	0.218702i
$254$		0			0
$255$		0			0
$256$		0			0
$257$		−	19.0632i		−	1.18913i	−0.804047	−	0.594566i	$-0.797324\pi$
$257$		−	19.0632i		−	1.18913i	0.804047	−	0.594566i	$-0.202676\pi$
$258$		0			0
$259$	9.25704			0.575204
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−6.89797			−0.425347			−0.212674	−	0.977123i	$-0.568217\pi$
$263$	−6.89797			−0.425347			−0.212674	+	0.977123i	$0.568217\pi$
$264$		0			0
$265$	−13.7986			−0.847643
$266$		0			0
$267$		0			0
$268$		0			0
$269$			10.7798i			0.657256i	0.944459	+	0.328628i	$0.106586\pi$
$269$			10.7798i			0.657256i	−0.944459	+	0.328628i	$0.893414\pi$
$270$		0			0
$271$	7.41048			0.450154			0.225077	−	0.974341i	$-0.427737\pi$
$271$	7.41048			0.450154			0.225077	+	0.974341i	$0.427737\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	1.47885			0.0891780
$276$		0			0
$277$	28.6625			1.72216			0.861080	−	0.508469i	$-0.169788\pi$
$277$	28.6625			1.72216			0.861080	+	0.508469i	$0.169788\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−16.7648			−1.00010			−0.500052	−	0.865996i	$-0.666686\pi$
$281$	−16.7648			−1.00010			−0.500052	+	0.865996i	$0.666686\pi$
$282$		0			0
$283$		−	18.6340i		−	1.10768i	−0.832623	−	0.553840i	$-0.813162\pi$
$283$		−	18.6340i		−	1.10768i	0.832623	−	0.553840i	$-0.186838\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−16.2683			−0.960286
$288$		0			0
$289$	14.4937			0.852573
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−7.09931			−0.414746			−0.207373	−	0.978262i	$-0.566491\pi$
$293$	−7.09931			−0.414746			−0.207373	+	0.978262i	$0.566491\pi$
$294$		0			0
$295$			7.07225i			0.411763i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−19.5491	−	11.0029i	−1.13055	−	0.636316i
$300$		0			0
$301$	−8.90671			−0.513374
$302$		0			0
$303$		0			0
$304$		0			0
$305$		−	10.2901i		−	0.589207i
$306$		0			0
$307$	30.8599			1.76127			0.880635	−	0.473796i	$-0.157116\pi$
$307$	30.8599			1.76127			0.880635	+	0.473796i	$0.157116\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		−	18.7527i		−	1.06337i	−0.846942	−	0.531685i	$-0.821559\pi$
$311$		−	18.7527i		−	1.06337i	0.846942	−	0.531685i	$-0.178441\pi$
$312$		0			0
$313$			10.2600i			0.579929i	0.957037	+	0.289964i	$0.0936434\pi$
$313$			10.2600i			0.579929i	−0.957037	+	0.289964i	$0.906357\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$			2.42607i			0.136262i	0.997676	+	0.0681308i	$0.0217035\pi$
$317$			2.42607i			0.136262i	−0.997676	+	0.0681308i	$0.978296\pi$
$318$		0			0
$319$		−	3.03231i		−	0.169777i
$320$		0			0
$321$		0			0
$322$		0			0
$323$		−	17.2708i		−	0.960974i
$324$		0			0
$325$	4.67757			0.259465
$326$		0			0
$327$		0			0
$328$		0			0
$329$	25.8116			1.42304
$330$		0			0
$331$	4.23343			0.232690			0.116345	−	0.993209i	$-0.462882\pi$
$331$	4.23343			0.232690			0.116345	+	0.993209i	$0.462882\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		−	1.88107i		−	0.102774i
$336$		0			0
$337$		−	21.5688i		−	1.17493i	−0.809251	−	0.587463i	$-0.800127\pi$
$337$		−	21.5688i		−	1.17493i	0.809251	−	0.587463i	$-0.199873\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	12.9038			0.698779
$342$		0			0
$343$		−	19.9184i		−	1.07549i
$344$		0			0
$345$		0			0
$346$		0			0
$347$			26.3767i			1.41598i	0.706224	+	0.707988i	$0.250397\pi$
$347$			26.3767i			1.41598i	−0.706224	+	0.707988i	$0.749603\pi$
$348$		0			0
$349$	−21.3172			−1.14108			−0.570542	−	0.821268i	$-0.693267\pi$
$349$	−21.3172			−1.14108			−0.570542	+	0.821268i	$0.693267\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$		−	16.8537i		−	0.897032i	−0.893775	−	0.448516i	$-0.851953\pi$
$353$		−	16.8537i		−	0.897032i	0.893775	−	0.448516i	$-0.148047\pi$
$354$		0			0
$355$			3.91325i			0.207694i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−7.81434			−0.412425			−0.206213	−	0.978507i	$-0.566114\pi$
$359$	−7.81434			−0.412425			−0.206213	+	0.978507i	$0.566114\pi$
$360$		0			0
$361$	9.52888			0.501520
$362$		0			0
$363$		0			0
$364$		0			0
$365$	6.95228			0.363899
$366$		0			0
$367$		−	13.7083i		−	0.715568i	−0.933804	−	0.357784i	$-0.883532\pi$
$367$		−	13.7083i		−	0.715568i	0.933804	−	0.357784i	$-0.116468\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		−	32.4465i		−	1.68454i
$372$		0			0
$373$		−	23.7328i		−	1.22884i	−0.788980	−	0.614419i	$-0.789391\pi$
$373$		−	23.7328i		−	1.22884i	0.788980	−	0.614419i	$-0.210609\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		−	9.59112i		−	0.493968i
$378$		0			0
$379$		−	26.6280i		−	1.36779i	−0.729582	−	0.683894i	$-0.760285\pi$
$379$		−	26.6280i		−	1.36779i	0.729582	−	0.683894i	$-0.239715\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−21.7247			−1.11008			−0.555040	−	0.831823i	$-0.687297\pi$
$383$	−21.7247			−1.11008			−0.555040	+	0.831823i	$0.687297\pi$
$384$		0			0
$385$			3.47742i			0.177225i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	11.8114			0.598861			0.299430	−	0.954118i	$-0.403203\pi$
$389$	11.8114			0.598861			0.299430	+	0.954118i	$0.403203\pi$
$390$		0			0
$391$	23.4541	+	13.2008i	1.18612	+	0.667593i
$392$		0			0
$393$		0			0
$394$		0			0
$395$			7.99760i			0.402403i
$396$		0			0
$397$	−6.82731			−0.342653			−0.171327	−	0.985214i	$-0.554805\pi$
$397$	−6.82731			−0.342653			−0.171327	+	0.985214i	$0.554805\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	6.11871			0.305554			0.152777	−	0.988261i	$-0.451178\pi$
$401$	6.11871			0.305554			0.152777	+	0.988261i	$0.451178\pi$
$402$		0			0
$403$	40.8144			2.03311
$404$		0			0
$405$		0			0
$406$		0			0
$407$			5.82189i			0.288580i
$408$		0			0
$409$	−18.1145			−0.895706			−0.447853	−	0.894107i	$-0.647811\pi$
$409$	−18.1145			−0.895706			−0.447853	+	0.894107i	$0.647811\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−16.6299			−0.818305
$414$		0			0
$415$	−0.219311			−0.0107656
$416$		0			0
$417$		0			0
$418$		0			0
$419$	22.8561			1.11659			0.558296	−	0.829642i	$-0.311455\pi$
$419$	22.8561			1.11659			0.558296	+	0.829642i	$0.311455\pi$
$420$		0			0
$421$		−	22.6095i		−	1.10192i	−0.834532	−	0.550959i	$-0.814262\pi$
$421$		−	22.6095i		−	1.10192i	0.834532	−	0.550959i	$-0.185738\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−5.61193			−0.272219
$426$		0			0
$427$	24.1964			1.17094
$428$		0			0
$429$		0			0
$430$		0			0
$431$	5.34963			0.257682			0.128841	−	0.991665i	$-0.458874\pi$
$431$	5.34963			0.257682			0.128841	+	0.991665i	$0.458874\pi$
$432$		0			0
$433$		−	12.2176i		−	0.587138i	−0.955938	−	0.293569i	$-0.905157\pi$
$433$		−	12.2176i		−	0.587138i	0.955938	−	0.293569i	$-0.0948431\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	7.23917	−	12.8620i	0.346297	−	0.615271i
$438$		0			0
$439$	3.39359			0.161967			0.0809837	−	0.996715i	$-0.474194\pi$
$439$	3.39359			0.161967			0.0809837	+	0.996715i	$0.474194\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$		−	11.5646i		−	0.549450i	−0.961523	−	0.274725i	$-0.911413\pi$
$443$		−	11.5646i		−	0.549450i	0.961523	−	0.274725i	$-0.0885869\pi$
$444$		0			0
$445$	2.65745			0.125975
$446$		0			0
$447$		0			0
$448$		0			0
$449$		−	36.3617i		−	1.71602i	−0.513636	−	0.858008i	$-0.671702\pi$
$449$		−	36.3617i		−	1.71602i	0.513636	−	0.858008i	$-0.328298\pi$
$450$		0			0
$451$		−	10.2314i		−	0.481776i
$452$		0			0
$453$		0			0
$454$		0			0
$455$			10.9990i			0.515640i
$456$		0			0
$457$		−	25.3964i		−	1.18799i	−0.804467	−	0.593997i	$-0.797549\pi$
$457$		−	25.3964i		−	1.18799i	0.804467	−	0.593997i	$-0.202451\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		−	23.5239i		−	1.09562i	−0.836604	−	0.547808i	$-0.815463\pi$
$461$		−	23.5239i		−	1.09562i	0.836604	−	0.547808i	$-0.184537\pi$
$462$		0			0
$463$	14.5555			0.676450			0.338225	−	0.941065i	$-0.390173\pi$
$463$	14.5555			0.676450			0.338225	+	0.941065i	$0.390173\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	11.5490			0.534424			0.267212	−	0.963638i	$-0.413898\pi$
$467$	11.5490			0.534424			0.267212	+	0.963638i	$0.413898\pi$
$468$		0			0
$469$	4.42321			0.204245
$470$		0			0
$471$		0			0
$472$		0			0
$473$		−	5.60156i		−	0.257560i
$474$		0			0
$475$			3.07752i			0.141206i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	30.6337			1.39969			0.699844	−	0.714295i	$-0.253253\pi$
$479$	30.6337			1.39969			0.699844	+	0.714295i	$0.253253\pi$
$480$		0			0
$481$			18.4145i			0.839629i
$482$		0			0
$483$		0			0
$484$		0			0
$485$		−	9.88545i		−	0.448875i
$486$		0			0
$487$	16.1826			0.733306			0.366653	−	0.930358i	$-0.380504\pi$
$487$	16.1826			0.733306			0.366653	+	0.930358i	$0.380504\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$			32.6496i			1.47346i	0.676189	+	0.736729i	$0.263630\pi$
$491$			32.6496i			1.47346i	−0.676189	+	0.736729i	$0.736370\pi$
$492$		0			0
$493$			11.5070i			0.518248i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−9.20175			−0.412755
$498$		0			0
$499$	−15.9644			−0.714663			−0.357332	−	0.933978i	$-0.616313\pi$
$499$	−15.9644			−0.714663			−0.357332	+	0.933978i	$0.616313\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	34.4127			1.53439			0.767193	−	0.641416i	$-0.221653\pi$
$503$	34.4127			1.53439			0.767193	+	0.641416i	$0.221653\pi$
$504$		0			0
$505$			16.7015i			0.743207i
$506$		0			0
$507$		0			0
$508$		0			0
$509$		−	33.0486i		−	1.46485i	−0.680846	−	0.732427i	$-0.738388\pi$
$509$		−	33.0486i		−	1.46485i	0.680846	−	0.732427i	$-0.261612\pi$
$510$		0			0
$511$			16.3478i			0.723185i
$512$		0			0
$513$		0			0
$514$		0			0
$515$			6.41186i			0.282540i
$516$		0			0
$517$			16.2333i			0.713939i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	20.5894			0.902037			0.451019	−	0.892515i	$-0.351061\pi$
$521$	20.5894			0.902037			0.451019	+	0.892515i	$0.351061\pi$
$522$		0			0
$523$			23.3782i			1.02226i	0.859504	+	0.511129i	$0.170772\pi$
$523$			23.3782i			1.02226i	−0.859504	+	0.511129i	$0.829228\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−48.9672			−2.13304
$528$		0			0
$529$	11.9336	+	19.6619i	0.518852	+	0.854864i
$530$		0			0
$531$		0			0
$532$		0			0
$533$		−	32.3616i		−	1.40173i
$534$		0			0
$535$	−13.1797			−0.569808
$536$		0			0
$537$		0			0
$538$		0			0
$539$	2.17504			0.0936858
$540$		0			0
$541$	−34.3436			−1.47655			−0.738274	−	0.674501i	$-0.764359\pi$
$541$	−34.3436			−1.47655			−0.738274	+	0.674501i	$0.764359\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$			0.188623i			0.00807971i
$546$		0			0
$547$	−21.7771			−0.931124			−0.465562	−	0.885015i	$-0.654148\pi$
$547$	−21.7771			−0.931124			−0.465562	+	0.885015i	$0.654148\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	6.31030			0.268828
$552$		0			0
$553$	−18.8058			−0.799705
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−28.1226			−1.19159			−0.595797	−	0.803135i	$-0.703164\pi$
$557$	−28.1226			−1.19159			−0.595797	+	0.803135i	$0.703164\pi$
$558$		0			0
$559$		−	17.7176i		−	0.749375i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	18.0761			0.761818			0.380909	−	0.924613i	$-0.375611\pi$
$563$	18.0761			0.761818			0.380909	+	0.924613i	$0.375611\pi$
$564$		0			0
$565$	−7.87587			−0.331340
$566$		0			0
$567$		0			0
$568$		0			0
$569$	17.6190			0.738627			0.369314	−	0.929305i	$-0.379593\pi$
$569$	17.6190			0.738627			0.369314	+	0.929305i	$0.379593\pi$
$570$		0			0
$571$		−	2.78326i		−	0.116476i	−0.998303	−	0.0582379i	$-0.981452\pi$
$571$		−	2.78326i		−	0.116476i	0.998303	−	0.0582379i	$-0.0185482\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−4.17933	−	2.35228i	−0.174290	−	0.0980967i
$576$		0			0
$577$	23.5675			0.981126			0.490563	−	0.871406i	$-0.336791\pi$
$577$	23.5675			0.981126			0.490563	+	0.871406i	$0.336791\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$		−	0.515695i		−	0.0213946i
$582$		0			0
$583$	20.4061			0.845134
$584$		0			0
$585$		0			0
$586$		0			0
$587$			25.7309i			1.06203i	0.847364	+	0.531013i	$0.178188\pi$
$587$			25.7309i			1.06203i	−0.847364	+	0.531013i	$0.821812\pi$
$588$		0			0
$589$			26.8531i			1.10646i
$590$		0			0
$591$		0			0
$592$		0			0
$593$			45.7373i			1.87821i	0.343637	+	0.939103i	$0.388341\pi$
$593$			45.7373i			1.87821i	−0.343637	+	0.939103i	$0.611659\pi$
$594$		0			0
$595$		−	13.1961i		−	0.540986i
$596$		0			0
$597$		0			0
$598$		0			0
$599$			43.9583i			1.79609i	0.439906	+	0.898044i	$0.355012\pi$
$599$			43.9583i			1.79609i	−0.439906	+	0.898044i	$0.644988\pi$
$600$		0			0
$601$	24.7342			1.00893			0.504465	−	0.863432i	$-0.331690\pi$
$601$	24.7342			1.00893			0.504465	+	0.863432i	$0.331690\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	8.81300			0.358300
$606$		0			0
$607$	31.0124			1.25875			0.629376	−	0.777101i	$-0.283310\pi$
$607$	31.0124			1.25875			0.629376	+	0.777101i	$0.283310\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$			51.3455i			2.07722i
$612$		0			0
$613$			10.0359i			0.405347i	0.979246	+	0.202674i	$0.0649630\pi$
$613$			10.0359i			0.405347i	−0.979246	+	0.202674i	$0.935037\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−41.1182			−1.65536			−0.827679	−	0.561202i	$-0.810339\pi$
$617$	−41.1182			−1.65536			−0.827679	+	0.561202i	$0.810339\pi$
$618$		0			0
$619$		−	41.8690i		−	1.68286i	−0.540369	−	0.841428i	$-0.681715\pi$
$619$		−	41.8690i		−	1.68286i	0.540369	−	0.841428i	$-0.318285\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$			6.24882i			0.250354i
$624$		0			0
$625$	1.00000			0.0400000
$626$		0			0
$627$		0			0
$628$		0			0
$629$		−	22.0929i		−	0.880900i
$630$		0			0
$631$			44.1153i			1.75620i	0.478476	+	0.878101i	$0.341189\pi$
$631$			44.1153i			1.75620i	−0.478476	+	0.878101i	$0.658811\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−3.54279			−0.140591
$636$		0			0
$637$	6.87961			0.272580
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−11.6145			−0.458746			−0.229373	−	0.973339i	$-0.573668\pi$
$641$	−11.6145			−0.458746			−0.229373	+	0.973339i	$0.573668\pi$
$642$		0			0
$643$			6.64529i			0.262065i	0.991378	+	0.131032i	$0.0418292\pi$
$643$			6.64529i			0.262065i	−0.991378	+	0.131032i	$0.958171\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		−	24.5141i		−	0.963750i	−0.876240	−	0.481875i	$-0.839956\pi$
$647$		−	24.5141i		−	0.963750i	0.876240	−	0.481875i	$-0.160044\pi$
$648$		0			0
$649$		−	10.4588i		−	0.410544i
$650$		0			0
$651$		0			0
$652$		0			0
$653$			1.23591i			0.0483650i	0.999708	+	0.0241825i	$0.00769828\pi$
$653$			1.23591i			0.0483650i	−0.999708	+	0.0241825i	$0.992302\pi$
$654$		0			0
$655$			12.4354i			0.485891i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−10.7206			−0.417617			−0.208808	−	0.977957i	$-0.566959\pi$
$659$	−10.7206			−0.417617			−0.208808	+	0.977957i	$0.566959\pi$
$660$		0			0
$661$		−	2.62911i		−	0.102260i	−0.998692	−	0.0511302i	$-0.983718\pi$
$661$		−	2.62911i		−	0.102260i	0.998692	−	0.0511302i	$-0.0162823\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−7.23658			−0.280622
$666$		0			0
$667$	−4.82322	+	8.56951i	−0.186756	+	0.331813i
$668$		0			0
$669$		0			0
$670$		0			0
$671$			15.2175i			0.587463i
$672$		0			0
$673$	−29.5342			−1.13846			−0.569229	−	0.822179i	$-0.692758\pi$
$673$	−29.5342			−1.13846			−0.569229	+	0.822179i	$0.692758\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−28.6805			−1.10228			−0.551140	−	0.834413i	$-0.685807\pi$
$677$	−28.6805			−1.10228			−0.551140	+	0.834413i	$0.685807\pi$
$678$		0			0
$679$	23.2450			0.892060
$680$		0			0
$681$		0			0
$682$		0			0
$683$			23.7660i			0.909380i	0.890650	+	0.454690i	$0.150250\pi$
$683$			23.7660i			0.909380i	−0.890650	+	0.454690i	$0.849750\pi$
$684$		0			0
$685$	14.5419			0.555617
$686$		0			0
$687$		0			0
$688$		0			0
$689$	64.5440			2.45893
$690$		0			0
$691$	−14.5310			−0.552787			−0.276393	−	0.961045i	$-0.589139\pi$
$691$	−14.5310			−0.552787			−0.276393	+	0.961045i	$0.589139\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−20.5864			−0.780885
$696$		0			0
$697$			38.8259i			1.47064i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−5.63840			−0.212960			−0.106480	−	0.994315i	$-0.533958\pi$
$701$	−5.63840			−0.212960			−0.106480	+	0.994315i	$0.533958\pi$
$702$		0			0
$703$	−12.1155			−0.456944
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−39.2725			−1.47699
$708$		0			0
$709$		−	29.2398i		−	1.09812i	−0.835782	−	0.549062i	$-0.814985\pi$
$709$		−	29.2398i		−	1.09812i	0.835782	−	0.549062i	$-0.185015\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−36.4670	−	20.5249i	−1.36570	−	0.768664i
$714$		0			0
$715$	−6.91742			−0.258697
$716$		0			0
$717$		0			0
$718$		0			0
$719$		−	36.1032i		−	1.34642i	−0.739451	−	0.673210i	$-0.764915\pi$
$719$		−	36.1032i		−	1.34642i	0.739451	−	0.673210i	$-0.235085\pi$
$720$		0			0
$721$	−15.0771			−0.561499
$722$		0			0
$723$		0			0
$724$		0			0
$725$		−	2.05045i		−	0.0761518i
$726$		0			0
$727$		−	6.45374i		−	0.239356i	−0.992813	−	0.119678i	$-0.961814\pi$
$727$		−	6.45374i		−	0.239356i	0.992813	−	0.119678i	$-0.0381862\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$			21.2568i			0.786210i
$732$		0			0
$733$		−	28.3379i		−	1.04669i	−0.852122	−	0.523343i	$-0.824685\pi$
$733$		−	28.3379i		−	1.04669i	0.852122	−	0.523343i	$-0.175315\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$			2.78182i			0.102470i
$738$		0			0
$739$	0.291688			0.0107299			0.00536496	−	0.999986i	$-0.498292\pi$
$739$	0.291688			0.0107299			0.00536496	+	0.999986i	$0.498292\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	27.5249			1.00979			0.504895	−	0.863181i	$-0.331531\pi$
$743$	27.5249			1.00979			0.504895	+	0.863181i	$0.331531\pi$
$744$		0			0
$745$	15.5566			0.569948
$746$		0			0
$747$		0			0
$748$		0			0
$749$		−	30.9912i		−	1.13239i
$750$		0			0
$751$		−	38.7724i		−	1.41483i	−0.706800	−	0.707413i	$-0.749862\pi$
$751$		−	38.7724i		−	1.41483i	0.706800	−	0.707413i	$-0.250138\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−5.92860			−0.215764
$756$		0			0
$757$			48.5522i			1.76466i	0.470633	+	0.882329i	$0.344026\pi$
$757$			48.5522i			1.76466i	−0.470633	+	0.882329i	$0.655974\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$			29.1443i			1.05648i	0.849095	+	0.528241i	$0.177148\pi$
$761$			29.1443i			1.05648i	−0.849095	+	0.528241i	$0.822852\pi$
$762$		0			0
$763$	−0.443534			−0.0160570
$764$		0			0
$765$		0			0
$766$		0			0
$767$		−	33.0810i		−	1.19448i
$768$		0			0
$769$		−	11.7605i		−	0.424093i	−0.977260	−	0.212046i	$-0.931987\pi$
$769$		−	11.7605i		−	0.424093i	0.977260	−	0.212046i	$-0.0680128\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−1.51910			−0.0546381			−0.0273191	−	0.999627i	$-0.508697\pi$
$773$	−1.51910			−0.0546381			−0.0273191	+	0.999627i	$0.508697\pi$
$774$		0			0
$775$	8.72555			0.313431
$776$		0			0
$777$		0			0
$778$		0			0
$779$	21.2917			0.762854
$780$		0			0
$781$		−	5.78711i		−	0.207079i
$782$		0			0
$783$		0			0
$784$		0			0
$785$		−	3.52464i		−	0.125800i
$786$		0			0
$787$			54.3428i			1.93711i	0.248799	+	0.968555i	$0.419964\pi$
$787$			54.3428i			1.93711i	−0.248799	+	0.968555i	$0.580036\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		−	18.5196i		−	0.658480i
$792$		0			0
$793$			48.1325i			1.70923i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−16.7650			−0.593848			−0.296924	−	0.954901i	$-0.595961\pi$
$797$	−16.7650			−0.593848			−0.296924	+	0.954901i	$0.595961\pi$
$798$		0			0
$799$		−	61.6019i		−	2.17932i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−10.2814			−0.362822
$804$		0			0
$805$	5.53122	−	9.82741i	0.194950	−	0.346371i
$806$		0			0
$807$		0			0
$808$		0			0
$809$		−	36.7731i		−	1.29287i	−0.762967	−	0.646437i	$-0.776258\pi$
$809$		−	36.7731i		−	1.29287i	0.762967	−	0.646437i	$-0.223742\pi$
$810$		0			0
$811$	41.3004			1.45025			0.725127	−	0.688615i	$-0.241781\pi$
$811$	41.3004			1.45025			0.725127	+	0.688615i	$0.241781\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	6.33629			0.221951
$816$		0			0
$817$	11.6570			0.407826
$818$		0			0
$819$		0			0
$820$		0			0
$821$		−	27.5461i		−	0.961366i	−0.876894	−	0.480683i	$-0.840389\pi$
$821$		−	27.5461i		−	0.961366i	0.876894	−	0.480683i	$-0.159611\pi$
$822$		0			0
$823$	26.5082			0.924019			0.462010	−	0.886875i	$-0.347129\pi$
$823$	26.5082			0.924019			0.462010	+	0.886875i	$0.347129\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	27.5092			0.956589			0.478294	−	0.878200i	$-0.341255\pi$
$827$	27.5092			0.956589			0.478294	+	0.878200i	$0.341255\pi$
$828$		0			0
$829$	−27.6937			−0.961840			−0.480920	−	0.876764i	$-0.659697\pi$
$829$	−27.6937			−0.961840			−0.480920	+	0.876764i	$0.659697\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−8.25384			−0.285979
$834$		0			0
$835$		−	14.3269i		−	0.495803i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	18.8369			0.650324			0.325162	−	0.945658i	$-0.394581\pi$
$839$	18.8369			0.650324			0.325162	+	0.945658i	$0.394581\pi$
$840$		0			0
$841$	24.7957			0.855023
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−8.87965			−0.305469
$846$		0			0
$847$			20.7232i			0.712057i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	9.26036	−	16.4530i	0.317441	−	0.564003i
$852$		0			0
$853$	−31.9474			−1.09386			−0.546929	−	0.837179i	$-0.684203\pi$
$853$	−31.9474			−1.09386			−0.546929	+	0.837179i	$0.684203\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$			51.2928i			1.75213i	0.482193	+	0.876065i	$0.339840\pi$
$857$			51.2928i			1.75213i	−0.482193	+	0.876065i	$0.660160\pi$
$858$		0			0
$859$	10.6747			0.364216			0.182108	−	0.983279i	$-0.441708\pi$
$859$	10.6747			0.364216			0.182108	+	0.983279i	$0.441708\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		−	7.87601i		−	0.268103i	−0.990974	−	0.134051i	$-0.957201\pi$
$863$		−	7.87601i		−	0.268103i	0.990974	−	0.134051i	$-0.0427987\pi$
$864$		0			0
$865$		−	4.53674i		−	0.154254i
$866$		0			0
$867$		0			0
$868$		0			0
$869$		−	11.8273i		−	0.401212i
$870$		0			0
$871$			8.79885i			0.298138i
$872$		0			0
$873$		0			0
$874$		0			0
$875$			2.35143i			0.0794929i
$876$		0			0
$877$	51.5619			1.74112			0.870561	−	0.492060i	$-0.163756\pi$
$877$	51.5619			1.74112			0.870561	+	0.492060i	$0.163756\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	13.4425			0.452888			0.226444	−	0.974024i	$-0.427290\pi$
$881$	13.4425			0.452888			0.226444	+	0.974024i	$0.427290\pi$
$882$		0			0
$883$	7.39070			0.248717			0.124358	−	0.992237i	$-0.460313\pi$
$883$	7.39070			0.248717			0.124358	+	0.992237i	$0.460313\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$			40.6676i			1.36549i	0.730659	+	0.682743i	$0.239213\pi$
$887$			40.6676i			1.36549i	−0.730659	+	0.682743i	$0.760787\pi$
$888$		0			0
$889$		−	8.33062i		−	0.279400i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−33.7818			−1.13046
$894$		0			0
$895$			2.95952i			0.0989257i
$896$		0			0
$897$		0			0
$898$		0			0
$899$		−	17.8913i		−	0.596708i
$900$		0			0
$901$	−77.4369			−2.57980
$902$		0			0
$903$		0			0
$904$		0			0
$905$			18.0839i			0.601128i
$906$		0			0
$907$			5.18375i			0.172124i	0.996290	+	0.0860619i	$0.0274283\pi$
$907$			5.18375i			0.172124i	−0.996290	+	0.0860619i	$0.972572\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−24.9465			−0.826514			−0.413257	−	0.910614i	$-0.635609\pi$
$911$	−24.9465			−0.826514			−0.413257	+	0.910614i	$0.635609\pi$
$912$		0			0
$913$	0.324328			0.0107337
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−29.2410			−0.965623
$918$		0			0
$919$		−	7.47864i		−	0.246697i	−0.992363	−	0.123349i	$-0.960637\pi$
$919$		−	7.47864i		−	0.246697i	0.992363	−	0.123349i	$-0.0393634\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		−	18.3045i		−	0.602500i
$924$		0			0
$925$			3.93677i			0.129440i
$926$		0			0
$927$		0			0
$928$		0			0
$929$			14.5726i			0.478112i	0.971006	+	0.239056i	$0.0768380\pi$
$929$			14.5726i			0.478112i	−0.971006	+	0.239056i	$0.923162\pi$
$930$		0			0
$931$			4.52631i			0.148344i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	8.29920			0.271413
$936$		0			0
$937$			4.61659i			0.150817i	0.997153	+	0.0754087i	$0.0240262\pi$
$937$			4.61659i			0.150817i	−0.997153	+	0.0754087i	$0.975974\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−58.4105			−1.90413			−0.952064	−	0.305900i	$-0.901043\pi$
$941$	−58.4105			−1.90413			−0.952064	+	0.305900i	$0.901043\pi$
$942$		0			0
$943$	−16.2741	+	28.9145i	−0.529958	+	0.941586i
$944$		0			0
$945$		0			0
$946$		0			0
$947$		−	36.9387i		−	1.20035i	−0.799870	−	0.600173i	$-0.795098\pi$
$947$		−	36.9387i		−	1.20035i	0.799870	−	0.600173i	$-0.204902\pi$
$948$		0			0
$949$	−32.5198			−1.05564
$950$		0			0
$951$		0			0
$952$		0			0
$953$	26.1374			0.846673			0.423337	−	0.905972i	$-0.360859\pi$
$953$	26.1374			0.846673			0.423337	+	0.905972i	$0.360859\pi$
$954$		0			0
$955$	15.8720			0.513606
$956$		0			0
$957$		0			0
$958$		0			0
$959$			34.1942i			1.10419i
$960$		0			0
$961$	45.1353			1.45598
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−20.4693			−0.658931
$966$		0			0
$967$	47.5947			1.53054			0.765272	−	0.643707i	$-0.222604\pi$
$967$	47.5947			1.53054			0.765272	+	0.643707i	$0.222604\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−11.2077			−0.359673			−0.179837	−	0.983696i	$-0.557557\pi$
$971$	−11.2077			−0.359673			−0.179837	+	0.983696i	$0.557557\pi$
$972$		0			0
$973$		−	48.4074i		−	1.55187i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	36.3290			1.16227			0.581134	−	0.813808i	$-0.302609\pi$
$977$	36.3290			1.16227			0.581134	+	0.813808i	$0.302609\pi$
$978$		0			0
$979$	−3.92998			−0.125603
$980$		0			0
$981$		0			0
$982$		0			0
$983$	21.0966			0.672877			0.336438	−	0.941706i	$-0.390778\pi$
$983$	21.0966			0.672877			0.336438	+	0.941706i	$0.390778\pi$
$984$		0			0
$985$		−	2.02134i		−	0.0644051i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−8.90991	+	15.8304i	−0.283319	+	0.503377i
$990$		0			0
$991$	24.3276			0.772793			0.386396	−	0.922333i	$-0.373720\pi$
$991$	24.3276			0.772793			0.386396	+	0.922333i	$0.373720\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$			4.21021i			0.133473i
$996$		0			0
$997$	−34.0113			−1.07715			−0.538575	−	0.842578i	$-0.681037\pi$
$997$	−34.0113			−1.07715			−0.538575	+	0.842578i	$0.681037\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8280.2.p.a.1241.14	✓	48
3.2	odd	2		8280.2.p.b.1241.14	yes	48
23.22	odd	2		8280.2.p.b.1241.35	yes	48
69.68	even	2	inner	8280.2.p.a.1241.35	yes	48

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8280.2.p.a.1241.14	✓	48	1.1	even	1	trivial
8280.2.p.a.1241.35	yes	48	69.68	even	2	inner
8280.2.p.b.1241.14	yes	48	3.2	odd	2
8280.2.p.b.1241.35	yes	48	23.22	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.p (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.00000 q^{5} -2.35143i q^{7} +O(q^{10})\)	\(q-1.00000 q^{5} -2.35143i q^{7} +1.47885 q^{11} +4.67757 q^{13} -5.61193 q^{17} +3.07752i q^{19} +(-4.17933 - 2.35228i) q^{23} +1.00000 q^{25} -2.05045i q^{29} +8.72555 q^{31} +2.35143i q^{35} +3.93677i q^{37} -6.91846i q^{41} -3.78778i q^{43} +10.9770i q^{47} +1.47077 q^{49} +13.7986 q^{53} -1.47885 q^{55} -7.07225i q^{59} +10.2901i q^{61} -4.67757 q^{65} +1.88107i q^{67} -3.91325i q^{71} -6.95228 q^{73} -3.47742i q^{77} -7.99760i q^{79} +0.219311 q^{83} +5.61193 q^{85} -2.65745 q^{89} -10.9990i q^{91} -3.07752i q^{95} +9.88545i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q - 48 q^{5}+O(q^{10}) \) 48 * q - 48 * q^5	\( 48 q - 48 q^{5} - 8 q^{11} + 4 q^{23} + 48 q^{25} + 8 q^{31} - 32 q^{49} + 8 q^{55} - 16 q^{73} - 32 q^{83} - 8 q^{89}+O(q^{100}) \) 48 * q - 48 * q^5 - 8 * q^11 + 4 * q^23 + 48 * q^25 + 8 * q^31 - 32 * q^49 + 8 * q^55 - 16 * q^73 - 32 * q^83 - 8 * q^89

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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