LMFDB - Embedded newform 4140.2.i.b.1241.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4140,2,Mod(1241,4140)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4140, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("4140.1241");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4140 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4140.i (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:	$33.0580664368$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} + 62x^{14} + 1303x^{12} + 12842x^{10} + 65359x^{8} + 170834x^{6} + 207293x^{4} + 91366x^{2} + 9604 $ x^16 + 62x^14 + 1303x^12 + 12842x^10 + 65359x^8 + 170834x^6 + 207293x^4 + 91366*x^2 + 9604
Coefficient ring:	$\Z[a_1, \ldots, a_{23}]$
Coefficient ring index:	$ 2^{5} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1241.3
Root			$-2.05179i$ of defining polynomial
Character	$\chi$	$=$	4140.1241
Dual form			4140.2.i.b.1241.14

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{5} -4.17184i q^{7} +O(q^{10})$	$q+1.00000 q^{5} -4.17184i q^{7} +4.20061 q^{11} +6.43642 q^{13} +6.40191 q^{17} +3.39011i q^{19} +(4.53475 - 1.56079i) q^{23} +1.00000 q^{25} -4.78406i q^{29} -4.55254 q^{31} -4.17184i q^{35} +0.928309i q^{37} -5.78062i q^{41} +4.60895i q^{43} +9.84878i q^{47} -10.4043 q^{49} -7.85869 q^{53} +4.20061 q^{55} +11.2410i q^{59} +10.7576i q^{61} +6.43642 q^{65} +3.60494i q^{67} +1.57258i q^{71} +13.2296 q^{73} -17.5243i q^{77} +11.4247i q^{79} -12.2920 q^{83} +6.40191 q^{85} -5.66393 q^{89} -26.8517i q^{91} +3.39011i q^{95} -17.8056i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 16 q^{5}+O(q^{10}) $ 16 * q + 16 * q^5	$ 16 q + 16 q^{5} + 8 q^{23} + 16 q^{25} - 8 q^{31} - 40 q^{49} - 4 q^{53} + 8 q^{73} - 20 q^{83} - 32 q^{89}+O(q^{100}) $ 16 * q + 16 * q^5 + 8 * q^23 + 16 * q^25 - 8 * q^31 - 40 * q^49 - 4 * q^53 + 8 * q^73 - 20 * q^83 - 32 * q^89

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/4140\mathbb{Z}\right)^\times$.

$n$	$461$	$1657$	$2071$	$3961$
$\chi(n)$	$-1$	$1$	$1$	$-1$

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
$7$		−	4.17184i		−	1.57681i	−0.615158	−	0.788404i	$-0.710908\pi$
$7$		−	4.17184i		−	1.57681i	0.615158	−	0.788404i	$-0.289092\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	4.20061			1.26653			0.633266	−	0.773934i	$-0.281714\pi$
$11$	4.20061			1.26653			0.633266	+	0.773934i	$0.281714\pi$
$12$		0			0
$13$	6.43642			1.78514			0.892571	−	0.450907i	$-0.148899\pi$
$13$	6.43642			1.78514			0.892571	+	0.450907i	$0.148899\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	6.40191			1.55269			0.776346	−	0.630307i	$-0.217071\pi$
$17$	6.40191			1.55269			0.776346	+	0.630307i	$0.217071\pi$
$18$		0			0
$19$			3.39011i			0.777744i	0.921292	+	0.388872i	$0.127135\pi$
$19$			3.39011i			0.777744i	−0.921292	+	0.388872i	$0.872865\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	4.53475	−	1.56079i	0.945560	−	0.325447i
$23$	4.53475	−	1.56079i	0.945560	−	0.325447i
$24$		0			0
$25$	1.00000			0.200000
$26$		0			0
$27$		0			0
$28$		0			0
$29$		−	4.78406i		−	0.888377i	−0.895933	−	0.444189i	$-0.853492\pi$
$29$		−	4.78406i		−	0.888377i	0.895933	−	0.444189i	$-0.146508\pi$
$30$		0			0
$31$	−4.55254			−0.817660			−0.408830	−	0.912610i	$-0.634063\pi$
$31$	−4.55254			−0.817660			−0.408830	+	0.912610i	$0.634063\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		−	4.17184i		−	0.705170i
$36$		0			0
$37$			0.928309i			0.152613i	0.997084	+	0.0763065i	$0.0243128\pi$
$37$			0.928309i			0.152613i	−0.997084	+	0.0763065i	$0.975687\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		−	5.78062i		−	0.902782i	−0.892326	−	0.451391i	$-0.850928\pi$
$41$		−	5.78062i		−	0.902782i	0.892326	−	0.451391i	$-0.149072\pi$
$42$		0			0
$43$			4.60895i			0.702858i	0.936215	+	0.351429i	$0.114304\pi$
$43$			4.60895i			0.702858i	−0.936215	+	0.351429i	$0.885696\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$			9.84878i			1.43659i	0.695738	+	0.718296i	$0.255078\pi$
$47$			9.84878i			1.43659i	−0.695738	+	0.718296i	$0.744922\pi$
$48$		0			0
$49$	−10.4043			−1.48632
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−7.85869			−1.07947			−0.539737	−	0.841834i	$-0.681476\pi$
$53$	−7.85869			−1.07947			−0.539737	+	0.841834i	$0.681476\pi$
$54$		0			0
$55$	4.20061			0.566411
$56$		0			0
$57$		0			0
$58$		0			0
$59$			11.2410i			1.46345i	0.681599	+	0.731726i	$0.261285\pi$
$59$			11.2410i			1.46345i	−0.681599	+	0.731726i	$0.738715\pi$
$60$		0			0
$61$			10.7576i			1.37737i	0.725059	+	0.688687i	$0.241812\pi$
$61$			10.7576i			1.37737i	−0.725059	+	0.688687i	$0.758188\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	6.43642			0.798340
$66$		0			0
$67$			3.60494i			0.440413i	0.975453	+	0.220207i	$0.0706731\pi$
$67$			3.60494i			0.440413i	−0.975453	+	0.220207i	$0.929327\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$			1.57258i			0.186631i	0.995637	+	0.0933155i	$0.0297465\pi$
$71$			1.57258i			0.186631i	−0.995637	+	0.0933155i	$0.970253\pi$
$72$		0			0
$73$	13.2296			1.54841			0.774205	−	0.632935i	$-0.218150\pi$
$73$	13.2296			1.54841			0.774205	+	0.632935i	$0.218150\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$		−	17.5243i		−	1.99708i
$78$		0			0
$79$			11.4247i			1.28538i	0.766125	+	0.642692i	$0.222182\pi$
$79$			11.4247i			1.28538i	−0.766125	+	0.642692i	$0.777818\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−12.2920			−1.34923			−0.674613	−	0.738171i	$-0.735690\pi$
$83$	−12.2920			−1.34923			−0.674613	+	0.738171i	$0.735690\pi$
$84$		0			0
$85$	6.40191			0.694385
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−5.66393			−0.600376			−0.300188	−	0.953880i	$-0.597049\pi$
$89$	−5.66393			−0.600376			−0.300188	+	0.953880i	$0.597049\pi$
$90$		0			0
$91$		−	26.8517i		−	2.81483i
$92$		0			0
$93$		0			0
$94$		0			0
$95$			3.39011i			0.347818i
$96$		0			0
$97$		−	17.8056i		−	1.80788i	−0.427658	−	0.903941i	$-0.640661\pi$
$97$		−	17.8056i		−	1.80788i	0.427658	−	0.903941i	$-0.359339\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$		−	11.9342i		−	1.18750i	−0.804650	−	0.593749i	$-0.797647\pi$
$101$		−	11.9342i		−	1.18750i	0.804650	−	0.593749i	$-0.202353\pi$
$102$		0			0
$103$			19.6520i			1.93637i	0.250240	+	0.968184i	$0.419490\pi$
$103$			19.6520i			1.93637i	−0.250240	+	0.968184i	$0.580510\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	1.89095			0.182805			0.0914023	−	0.995814i	$-0.470865\pi$
$107$	1.89095			0.182805			0.0914023	+	0.995814i	$0.470865\pi$
$108$		0			0
$109$		−	8.86484i		−	0.849097i	−0.905405	−	0.424549i	$-0.860433\pi$
$109$		−	8.86484i		−	0.849097i	0.905405	−	0.424549i	$-0.139567\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	0.685740			0.0645090			0.0322545	−	0.999480i	$-0.489731\pi$
$113$	0.685740			0.0645090			0.0322545	+	0.999480i	$0.489731\pi$
$114$		0			0
$115$	4.53475	−	1.56079i	0.422867	−	0.145544i
$116$		0			0
$117$		0			0
$118$		0			0
$119$		−	26.7078i		−	2.44830i
$120$		0			0
$121$	6.64516			0.604106
$122$		0			0
$123$		0			0
$124$		0			0
$125$	1.00000			0.0894427
$126$		0			0
$127$	−17.7752			−1.57729			−0.788646	−	0.614847i	$-0.789218\pi$
$127$	−17.7752			−1.57729			−0.788646	+	0.614847i	$0.789218\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$		−	15.4583i		−	1.35060i	−0.737544	−	0.675299i	$-0.764014\pi$
$131$		−	15.4583i		−	1.35060i	0.737544	−	0.675299i	$-0.235986\pi$
$132$		0			0
$133$	14.1430			1.22635
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−12.3010			−1.05094			−0.525471	−	0.850811i	$-0.676111\pi$
$137$	−12.3010			−1.05094			−0.525471	+	0.850811i	$0.676111\pi$
$138$		0			0
$139$	−4.73131			−0.401304			−0.200652	−	0.979663i	$-0.564306\pi$
$139$	−4.73131			−0.401304			−0.200652	+	0.979663i	$0.564306\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	27.0369			2.26094
$144$		0			0
$145$		−	4.78406i		−	0.397294i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−11.3747			−0.931852			−0.465926	−	0.884824i	$-0.654279\pi$
$149$	−11.3747			−0.931852			−0.465926	+	0.884824i	$0.654279\pi$
$150$		0			0
$151$	10.1643			0.827162			0.413581	−	0.910467i	$-0.364278\pi$
$151$	10.1643			0.827162			0.413581	+	0.910467i	$0.364278\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−4.55254			−0.365669
$156$		0			0
$157$			13.0914i			1.04481i	0.852698	+	0.522405i	$0.174965\pi$
$157$			13.0914i			1.04481i	−0.852698	+	0.522405i	$0.825035\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−6.51136	−	18.9182i	−0.513167	−	1.49097i
$162$		0			0
$163$	23.7790			1.86252			0.931258	−	0.364361i	$-0.118712\pi$
$163$	23.7790			1.86252			0.931258	+	0.364361i	$0.118712\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$			14.5647i			1.12705i	0.826098	+	0.563526i	$0.190555\pi$
$167$			14.5647i			1.12705i	−0.826098	+	0.563526i	$0.809445\pi$
$168$		0			0
$169$	28.4275			2.18673
$170$		0			0
$171$		0			0
$172$		0			0
$173$		−	3.01471i		−	0.229204i	−0.993411	−	0.114602i	$-0.963441\pi$
$173$		−	3.01471i		−	0.229204i	0.993411	−	0.114602i	$-0.0365593\pi$
$174$		0			0
$175$		−	4.17184i		−	0.315361i
$176$		0			0
$177$		0			0
$178$		0			0
$179$		−	25.1269i		−	1.87807i	−0.343817	−	0.939037i	$-0.611720\pi$
$179$		−	25.1269i		−	1.87807i	0.343817	−	0.939037i	$-0.388280\pi$
$180$		0			0
$181$		−	6.87620i		−	0.511104i	−0.966795	−	0.255552i	$-0.917743\pi$
$181$		−	6.87620i		−	0.511104i	0.966795	−	0.255552i	$-0.0822572\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$			0.928309i			0.0682506i
$186$		0			0
$187$	26.8920			1.96654
$188$		0			0
$189$		0			0
$190$		0			0
$191$	26.4573			1.91438			0.957191	−	0.289456i	$-0.0934743\pi$
$191$	26.4573			1.91438			0.957191	+	0.289456i	$0.0934743\pi$
$192$		0			0
$193$	−24.8464			−1.78849			−0.894243	−	0.447581i	$-0.852285\pi$
$193$	−24.8464			−1.78849			−0.894243	+	0.447581i	$0.852285\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		−	23.7237i		−	1.69024i	−0.534575	−	0.845121i	$-0.679528\pi$
$197$		−	23.7237i		−	1.69024i	0.534575	−	0.845121i	$-0.320472\pi$
$198$		0			0
$199$			10.0072i			0.709393i	0.934982	+	0.354696i	$0.115416\pi$
$199$			10.0072i			0.709393i	−0.934982	+	0.354696i	$0.884584\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−19.9583			−1.40080
$204$		0			0
$205$		−	5.78062i		−	0.403736i
$206$		0			0
$207$		0			0
$208$		0			0
$209$			14.2405i			0.985038i
$210$		0			0
$211$	−18.6560			−1.28433			−0.642165	−	0.766567i	$-0.721964\pi$
$211$	−18.6560			−1.28433			−0.642165	+	0.766567i	$0.721964\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$			4.60895i			0.314327i
$216$		0			0
$217$			18.9925i			1.28929i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	41.2054			2.77178
$222$		0			0
$223$	−13.5257			−0.905746			−0.452873	−	0.891575i	$-0.649601\pi$
$223$	−13.5257			−0.905746			−0.452873	+	0.891575i	$0.649601\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	0.569486			0.0377981			0.0188991	−	0.999821i	$-0.493984\pi$
$227$	0.569486			0.0377981			0.0188991	+	0.999821i	$0.493984\pi$
$228$		0			0
$229$			18.3911i			1.21532i	0.794199	+	0.607658i	$0.207891\pi$
$229$			18.3911i			1.21532i	−0.794199	+	0.607658i	$0.792109\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$			5.49416i			0.359934i	0.983673	+	0.179967i	$0.0575992\pi$
$233$			5.49416i			0.359934i	−0.983673	+	0.179967i	$0.942401\pi$
$234$		0			0
$235$			9.84878i			0.642463i
$236$		0			0
$237$		0			0
$238$		0			0
$239$		−	3.24212i		−	0.209715i	−0.994487	−	0.104858i	$-0.966561\pi$
$239$		−	3.24212i		−	0.209715i	0.994487	−	0.104858i	$-0.0334387\pi$
$240$		0			0
$241$		−	5.36945i		−	0.345877i	−0.984933	−	0.172938i	$-0.944674\pi$
$241$		−	5.36945i		−	0.345877i	0.984933	−	0.172938i	$-0.0553261\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−10.4043			−0.664703
$246$		0			0
$247$			21.8202i			1.38838i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	0.517147			0.0326420			0.0163210	−	0.999867i	$-0.494805\pi$
$251$	0.517147			0.0326420			0.0163210	+	0.999867i	$0.494805\pi$
$252$		0			0
$253$	19.0487	−	6.55627i	1.19758	−	0.412189i
$254$		0			0
$255$		0			0
$256$		0			0
$257$		−	20.6191i		−	1.28619i	−0.765788	−	0.643093i	$-0.777651\pi$
$257$		−	20.6191i		−	1.28619i	0.765788	−	0.643093i	$-0.222349\pi$
$258$		0			0
$259$	3.87276			0.240641
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−2.74278			−0.169127			−0.0845634	−	0.996418i	$-0.526950\pi$
$263$	−2.74278			−0.169127			−0.0845634	+	0.996418i	$0.526950\pi$
$264$		0			0
$265$	−7.85869			−0.482756
$266$		0			0
$267$		0			0
$268$		0			0
$269$			21.3685i			1.30286i	0.758708	+	0.651430i	$0.225831\pi$
$269$			21.3685i			1.30286i	−0.758708	+	0.651430i	$0.774169\pi$
$270$		0			0
$271$	9.39239			0.570547			0.285274	−	0.958446i	$-0.407916\pi$
$271$	9.39239			0.570547			0.285274	+	0.958446i	$0.407916\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	4.20061			0.253307
$276$		0			0
$277$	−29.8742			−1.79497			−0.897484	−	0.441048i	$-0.854607\pi$
$277$	−29.8742			−1.79497			−0.897484	+	0.441048i	$0.854607\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	4.27445			0.254992			0.127496	−	0.991839i	$-0.459306\pi$
$281$	4.27445			0.254992			0.127496	+	0.991839i	$0.459306\pi$
$282$		0			0
$283$		−	24.3976i		−	1.45029i	−0.688597	−	0.725144i	$-0.741773\pi$
$283$		−	24.3976i		−	1.45029i	0.688597	−	0.725144i	$-0.258227\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−24.1158			−1.42351
$288$		0			0
$289$	23.9845			1.41085
$290$		0			0
$291$		0			0
$292$		0			0
$293$	8.80428			0.514352			0.257176	−	0.966365i	$-0.417208\pi$
$293$	8.80428			0.514352			0.257176	+	0.966365i	$0.417208\pi$
$294$		0			0
$295$			11.2410i			0.654476i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	29.1875	−	10.0459i	1.68796	−	0.580969i
$300$		0			0
$301$	19.2278			1.10827
$302$		0			0
$303$		0			0
$304$		0			0
$305$			10.7576i			0.615980i
$306$		0			0
$307$	−20.0431			−1.14392			−0.571960	−	0.820281i	$-0.693817\pi$
$307$	−20.0431			−1.14392			−0.571960	+	0.820281i	$0.693817\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$			9.09416i			0.515683i	0.966187	+	0.257841i	$0.0830112\pi$
$311$			9.09416i			0.515683i	−0.966187	+	0.257841i	$0.916989\pi$
$312$		0			0
$313$		−	6.90583i		−	0.390341i	−0.980769	−	0.195170i	$-0.937474\pi$
$313$		−	6.90583i		−	0.390341i	0.980769	−	0.195170i	$-0.0625259\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		−	15.3902i		−	0.864401i	−0.901778	−	0.432200i	$-0.857737\pi$
$317$		−	15.3902i		−	0.864401i	0.901778	−	0.432200i	$-0.142263\pi$
$318$		0			0
$319$		−	20.0960i		−	1.12516i
$320$		0			0
$321$		0			0
$322$		0			0
$323$			21.7032i			1.20760i
$324$		0			0
$325$	6.43642			0.357028
$326$		0			0
$327$		0			0
$328$		0			0
$329$	41.0875			2.26523
$330$		0			0
$331$	−21.2122			−1.16593			−0.582965	−	0.812497i	$-0.698108\pi$
$331$	−21.2122			−1.16593			−0.582965	+	0.812497i	$0.698108\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$			3.60494i			0.196959i
$336$		0			0
$337$			20.7369i			1.12961i	0.825224	+	0.564806i	$0.191049\pi$
$337$			20.7369i			1.12961i	−0.825224	+	0.564806i	$0.808951\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−19.1235			−1.03559
$342$		0			0
$343$			14.2020i			0.766836i
$344$		0			0
$345$		0			0
$346$		0			0
$347$		−	9.30653i		−	0.499601i	−0.968297	−	0.249800i	$-0.919635\pi$
$347$		−	9.30653i		−	0.499601i	0.968297	−	0.249800i	$-0.0803650\pi$
$348$		0			0
$349$	−13.0800			−0.700154			−0.350077	−	0.936721i	$-0.613845\pi$
$349$	−13.0800			−0.700154			−0.350077	+	0.936721i	$0.613845\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$		−	13.1943i		−	0.702262i	−0.936326	−	0.351131i	$-0.885797\pi$
$353$		−	13.1943i		−	0.702262i	0.936326	−	0.351131i	$-0.114203\pi$
$354$		0			0
$355$			1.57258i			0.0834640i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−16.2853			−0.859504			−0.429752	−	0.902947i	$-0.641399\pi$
$359$	−16.2853			−0.859504			−0.429752	+	0.902947i	$0.641399\pi$
$360$		0			0
$361$	7.50718			0.395115
$362$		0			0
$363$		0			0
$364$		0			0
$365$	13.2296			0.692470
$366$		0			0
$367$			27.0571i			1.41237i	0.708027	+	0.706185i	$0.249585\pi$
$367$			27.0571i			1.41237i	−0.708027	+	0.706185i	$0.750415\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$			32.7852i			1.70212i
$372$		0			0
$373$			13.5776i			0.703020i	0.936184	+	0.351510i	$0.114332\pi$
$373$			13.5776i			0.703020i	−0.936184	+	0.351510i	$0.885668\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		−	30.7922i		−	1.58588i
$378$		0			0
$379$		−	4.68440i		−	0.240621i	−0.992736	−	0.120311i	$-0.961611\pi$
$379$		−	4.68440i		−	0.240621i	0.992736	−	0.120311i	$-0.0383891\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−23.3913			−1.19524			−0.597621	−	0.801779i	$-0.703887\pi$
$383$	−23.3913			−1.19524			−0.597621	+	0.801779i	$0.703887\pi$
$384$		0			0
$385$		−	17.5243i		−	0.893121i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−8.72665			−0.442459			−0.221229	−	0.975222i	$-0.571007\pi$
$389$	−8.72665			−0.442459			−0.221229	+	0.975222i	$0.571007\pi$
$390$		0			0
$391$	29.0311	−	9.99203i	1.46816	−	0.505319i
$392$		0			0
$393$		0			0
$394$		0			0
$395$			11.4247i			0.574841i
$396$		0			0
$397$	28.5307			1.43192			0.715958	−	0.698144i	$-0.245990\pi$
$397$	28.5307			1.43192			0.715958	+	0.698144i	$0.245990\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−9.47565			−0.473192			−0.236596	−	0.971608i	$-0.576032\pi$
$401$	−9.47565			−0.473192			−0.236596	+	0.971608i	$0.576032\pi$
$402$		0			0
$403$	−29.3021			−1.45964
$404$		0			0
$405$		0			0
$406$		0			0
$407$			3.89947i			0.193289i
$408$		0			0
$409$	−39.9828			−1.97702			−0.988511	−	0.151152i	$-0.951702\pi$
$409$	−39.9828			−1.97702			−0.988511	+	0.151152i	$0.951702\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	46.8956			2.30758
$414$		0			0
$415$	−12.2920			−0.603393
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−11.3468			−0.554326			−0.277163	−	0.960823i	$-0.589394\pi$
$419$	−11.3468			−0.554326			−0.277163	+	0.960823i	$0.589394\pi$
$420$		0			0
$421$		−	22.5165i		−	1.09739i	−0.836023	−	0.548695i	$-0.815125\pi$
$421$		−	22.5165i		−	1.09739i	0.836023	−	0.548695i	$-0.184875\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	6.40191			0.310538
$426$		0			0
$427$	44.8791			2.17185
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−2.19322			−0.105644			−0.0528219	−	0.998604i	$-0.516822\pi$
$431$	−2.19322			−0.105644			−0.0528219	+	0.998604i	$0.516822\pi$
$432$		0			0
$433$			21.4123i			1.02901i	0.857487	+	0.514505i	$0.172024\pi$
$433$			21.4123i			1.02901i	−0.857487	+	0.514505i	$0.827976\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	5.29124	+	15.3733i	0.253114	+	0.735404i
$438$		0			0
$439$	5.89159			0.281190			0.140595	−	0.990067i	$-0.455098\pi$
$439$	5.89159			0.281190			0.140595	+	0.990067i	$0.455098\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$		−	33.6439i		−	1.59847i	−0.601020	−	0.799234i	$-0.705239\pi$
$443$		−	33.6439i		−	1.59847i	0.601020	−	0.799234i	$-0.294761\pi$
$444$		0			0
$445$	−5.66393			−0.268496
$446$		0			0
$447$		0			0
$448$		0			0
$449$		−	9.68309i		−	0.456973i	−0.973547	−	0.228487i	$-0.926622\pi$
$449$		−	9.68309i		−	0.456973i	0.973547	−	0.228487i	$-0.0733777\pi$
$450$		0			0
$451$		−	24.2822i		−	1.14340i
$452$		0			0
$453$		0			0
$454$		0			0
$455$		−	26.8517i		−	1.25883i
$456$		0			0
$457$			22.0173i			1.02993i	0.857212	+	0.514964i	$0.172195\pi$
$457$			22.0173i			1.02993i	−0.857212	+	0.514964i	$0.827805\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$			33.5085i			1.56065i	0.625376	+	0.780323i	$0.284945\pi$
$461$			33.5085i			1.56065i	−0.625376	+	0.780323i	$0.715055\pi$
$462$		0			0
$463$	−20.3584			−0.946134			−0.473067	−	0.881026i	$-0.656853\pi$
$463$	−20.3584			−0.946134			−0.473067	+	0.881026i	$0.656853\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−11.3175			−0.523712			−0.261856	−	0.965107i	$-0.584334\pi$
$467$	−11.3175			−0.523712			−0.261856	+	0.965107i	$0.584334\pi$
$468$		0			0
$469$	15.0392			0.694447
$470$		0			0
$471$		0			0
$472$		0			0
$473$			19.3604i			0.890192i
$474$		0			0
$475$			3.39011i			0.155549i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	18.3747			0.839560			0.419780	−	0.907626i	$-0.362107\pi$
$479$	18.3747			0.839560			0.419780	+	0.907626i	$0.362107\pi$
$480$		0			0
$481$			5.97499i			0.272436i
$482$		0			0
$483$		0			0
$484$		0			0
$485$		−	17.8056i		−	0.808509i
$486$		0			0
$487$	19.0275			0.862220			0.431110	−	0.902299i	$-0.358122\pi$
$487$	19.0275			0.862220			0.431110	+	0.902299i	$0.358122\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$		−	29.4261i		−	1.32798i	−0.747741	−	0.663990i	$-0.768862\pi$
$491$		−	29.4261i		−	1.32798i	0.747741	−	0.663990i	$-0.231138\pi$
$492$		0			0
$493$		−	30.6271i		−	1.37938i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	6.56056			0.294281
$498$		0			0
$499$	2.09849			0.0939414			0.0469707	−	0.998896i	$-0.485043\pi$
$499$	2.09849			0.0939414			0.0469707	+	0.998896i	$0.485043\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	0.868705			0.0387337			0.0193668	−	0.999812i	$-0.493835\pi$
$503$	0.868705			0.0387337			0.0193668	+	0.999812i	$0.493835\pi$
$504$		0			0
$505$		−	11.9342i		−	0.531065i
$506$		0			0
$507$		0			0
$508$		0			0
$509$		−	7.02421i		−	0.311343i	−0.987809	−	0.155671i	$-0.950246\pi$
$509$		−	7.02421i		−	0.311343i	0.987809	−	0.155671i	$-0.0497541\pi$
$510$		0			0
$511$		−	55.1918i		−	2.44154i
$512$		0			0
$513$		0			0
$514$		0			0
$515$			19.6520i			0.865970i
$516$		0			0
$517$			41.3709i			1.81949i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	5.25255			0.230118			0.115059	−	0.993359i	$-0.463294\pi$
$521$	5.25255			0.230118			0.115059	+	0.993359i	$0.463294\pi$
$522$		0			0
$523$		−	12.1218i		−	0.530048i	−0.964242	−	0.265024i	$-0.914620\pi$
$523$		−	12.1218i		−	0.530048i	0.964242	−	0.265024i	$-0.0853799\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−29.1450			−1.26957
$528$		0			0
$529$	18.1279	−	14.1556i	0.788169	−	0.615459i
$530$		0			0
$531$		0			0
$532$		0			0
$533$		−	37.2065i		−	1.61159i
$534$		0			0
$535$	1.89095			0.0817527
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−43.7043			−1.88248
$540$		0			0
$541$	7.80470			0.335550			0.167775	−	0.985825i	$-0.446342\pi$
$541$	7.80470			0.335550			0.167775	+	0.985825i	$0.446342\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		−	8.86484i		−	0.379728i
$546$		0			0
$547$	−20.5213			−0.877426			−0.438713	−	0.898627i	$-0.644565\pi$
$547$	−20.5213			−0.877426			−0.438713	+	0.898627i	$0.644565\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	16.2185			0.690930
$552$		0			0
$553$	47.6622			2.02680
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−31.8044			−1.34760			−0.673798	−	0.738915i	$-0.735338\pi$
$557$	−31.8044			−1.34760			−0.673798	+	0.738915i	$0.735338\pi$
$558$		0			0
$559$			29.6651i			1.25470i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−24.4734			−1.03143			−0.515715	−	0.856760i	$-0.672474\pi$
$563$	−24.4734			−1.03143			−0.515715	+	0.856760i	$0.672474\pi$
$564$		0			0
$565$	0.685740			0.0288493
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−4.88157			−0.204646			−0.102323	−	0.994751i	$-0.532628\pi$
$569$	−4.88157			−0.204646			−0.102323	+	0.994751i	$0.532628\pi$
$570$		0			0
$571$		−	15.7980i		−	0.661125i	−0.943784	−	0.330563i	$-0.892762\pi$
$571$		−	15.7980i		−	0.661125i	0.943784	−	0.330563i	$-0.107238\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	4.53475	−	1.56079i	0.189112	−	0.0650894i
$576$		0			0
$577$	8.85625			0.368691			0.184345	−	0.982862i	$-0.440984\pi$
$577$	8.85625			0.368691			0.184345	+	0.982862i	$0.440984\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$			51.2804i			2.12747i
$582$		0			0
$583$	−33.0113			−1.36719
$584$		0			0
$585$		0			0
$586$		0			0
$587$			20.0979i			0.829529i	0.909929	+	0.414764i	$0.136136\pi$
$587$			20.0979i			0.829529i	−0.909929	+	0.414764i	$0.863864\pi$
$588$		0			0
$589$		−	15.4336i		−	0.635930i
$590$		0			0
$591$		0			0
$592$		0			0
$593$		−	31.4585i		−	1.29184i	−0.763403	−	0.645922i	$-0.776473\pi$
$593$		−	31.4585i		−	1.29184i	0.763403	−	0.645922i	$-0.223527\pi$
$594$		0			0
$595$		−	26.7078i		−	1.09491i
$596$		0			0
$597$		0			0
$598$		0			0
$599$			17.0146i			0.695200i	0.937643	+	0.347600i	$0.113003\pi$
$599$			17.0146i			0.695200i	−0.937643	+	0.347600i	$0.886997\pi$
$600$		0			0
$601$	−14.0735			−0.574071			−0.287036	−	0.957920i	$-0.592670\pi$
$601$	−14.0735			−0.574071			−0.287036	+	0.957920i	$0.592670\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	6.64516			0.270164
$606$		0			0
$607$	−16.1413			−0.655156			−0.327578	−	0.944824i	$-0.606232\pi$
$607$	−16.1413			−0.655156			−0.327578	+	0.944824i	$0.606232\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$			63.3909i			2.56452i
$612$		0			0
$613$		−	4.98085i		−	0.201175i	−0.994928	−	0.100587i	$-0.967928\pi$
$613$		−	4.98085i		−	0.201175i	0.994928	−	0.100587i	$-0.0320722\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	23.4544			0.944240			0.472120	−	0.881534i	$-0.343489\pi$
$617$	23.4544			0.944240			0.472120	+	0.881534i	$0.343489\pi$
$618$		0			0
$619$		−	12.4098i		−	0.498793i	−0.968401	−	0.249397i	$-0.919768\pi$
$619$		−	12.4098i		−	0.498793i	0.968401	−	0.249397i	$-0.0802322\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$			23.6290i			0.946677i
$624$		0			0
$625$	1.00000			0.0400000
$626$		0			0
$627$		0			0
$628$		0			0
$629$			5.94295i			0.236961i
$630$		0			0
$631$			21.8805i			0.871050i	0.900177	+	0.435525i	$0.143437\pi$
$631$			21.8805i			0.871050i	−0.900177	+	0.435525i	$0.856563\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−17.7752			−0.705387
$636$		0			0
$637$	−66.9662			−2.65330
$638$		0			0
$639$		0			0
$640$		0			0
$641$	35.1839			1.38968			0.694841	−	0.719163i	$-0.255475\pi$
$641$	35.1839			1.38968			0.694841	+	0.719163i	$0.255475\pi$
$642$		0			0
$643$		−	33.6148i		−	1.32564i	−0.748779	−	0.662819i	$-0.769360\pi$
$643$		−	33.6148i		−	1.32564i	0.748779	−	0.662819i	$-0.230640\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$			5.38280i			0.211620i	0.994386	+	0.105810i	$0.0337435\pi$
$647$			5.38280i			0.211620i	−0.994386	+	0.105810i	$0.966256\pi$
$648$		0			0
$649$			47.2191i			1.85351i
$650$		0			0
$651$		0			0
$652$		0			0
$653$		−	33.7884i		−	1.32224i	−0.750279	−	0.661122i	$-0.770081\pi$
$653$		−	33.7884i		−	1.32224i	0.750279	−	0.661122i	$-0.229919\pi$
$654$		0			0
$655$		−	15.4583i		−	0.604006i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−46.7635			−1.82165			−0.910823	−	0.412797i	$-0.864552\pi$
$659$	−46.7635			−1.82165			−0.910823	+	0.412797i	$0.864552\pi$
$660$		0			0
$661$		−	19.7099i		−	0.766628i	−0.923618	−	0.383314i	$-0.874783\pi$
$661$		−	19.7099i		−	0.766628i	0.923618	−	0.383314i	$-0.125217\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	14.1430			0.548441
$666$		0			0
$667$	−7.46690	−	21.6945i	−0.289120	−	0.840014i
$668$		0			0
$669$		0			0
$670$		0			0
$671$			45.1887i			1.74449i
$672$		0			0
$673$	−16.3424			−0.629954			−0.314977	−	0.949099i	$-0.601997\pi$
$673$	−16.3424			−0.629954			−0.314977	+	0.949099i	$0.601997\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−12.5403			−0.481961			−0.240981	−	0.970530i	$-0.577469\pi$
$677$	−12.5403			−0.481961			−0.240981	+	0.970530i	$0.577469\pi$
$678$		0			0
$679$	−74.2820			−2.85068
$680$		0			0
$681$		0			0
$682$		0			0
$683$			27.8122i			1.06421i	0.846680	+	0.532103i	$0.178598\pi$
$683$			27.8122i			1.06421i	−0.846680	+	0.532103i	$0.821402\pi$
$684$		0			0
$685$	−12.3010			−0.469996
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−50.5819			−1.92702
$690$		0			0
$691$	−0.800445			−0.0304504			−0.0152252	−	0.999884i	$-0.504847\pi$
$691$	−0.800445			−0.0304504			−0.0152252	+	0.999884i	$0.504847\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−4.73131			−0.179469
$696$		0			0
$697$		−	37.0071i		−	1.40174i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	8.31676			0.314120			0.157060	−	0.987589i	$-0.449798\pi$
$701$	8.31676			0.314120			0.157060	+	0.987589i	$0.449798\pi$
$702$		0			0
$703$	−3.14707			−0.118694
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−49.7876			−1.87246
$708$		0			0
$709$		−	35.0975i		−	1.31812i	−0.752092	−	0.659058i	$-0.770955\pi$
$709$		−	35.0975i		−	1.31812i	0.752092	−	0.659058i	$-0.229045\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−20.6446	+	7.10555i	−0.773147	+	0.266105i
$714$		0			0
$715$	27.0369			1.01112
$716$		0			0
$717$		0			0
$718$		0			0
$719$			1.30524i			0.0486774i	0.999704	+	0.0243387i	$0.00774801\pi$
$719$			1.30524i			0.0486774i	−0.999704	+	0.0243387i	$0.992252\pi$
$720$		0			0
$721$	81.9849			3.05328
$722$		0			0
$723$		0			0
$724$		0			0
$725$		−	4.78406i		−	0.177675i
$726$		0			0
$727$			12.7651i			0.473431i	0.971579	+	0.236715i	$0.0760709\pi$
$727$			12.7651i			0.473431i	−0.971579	+	0.236715i	$0.923929\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$			29.5061i			1.09132i
$732$		0			0
$733$			27.8670i			1.02929i	0.857403	+	0.514646i	$0.172077\pi$
$733$			27.8670i			1.02929i	−0.857403	+	0.514646i	$0.827923\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$			15.1429i			0.557798i
$738$		0			0
$739$	−21.3622			−0.785821			−0.392911	−	0.919577i	$-0.628532\pi$
$739$	−21.3622			−0.785821			−0.392911	+	0.919577i	$0.628532\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−12.3219			−0.452048			−0.226024	−	0.974122i	$-0.572573\pi$
$743$	−12.3219			−0.452048			−0.226024	+	0.974122i	$0.572573\pi$
$744$		0			0
$745$	−11.3747			−0.416737
$746$		0			0
$747$		0			0
$748$		0			0
$749$		−	7.88873i		−	0.288248i
$750$		0			0
$751$		−	32.0721i		−	1.17033i	−0.810915	−	0.585164i	$-0.801030\pi$
$751$		−	32.0721i		−	1.17033i	0.810915	−	0.585164i	$-0.198970\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	10.1643			0.369918
$756$		0			0
$757$			21.4898i			0.781060i	0.920590	+	0.390530i	$0.127708\pi$
$757$			21.4898i			0.781060i	−0.920590	+	0.390530i	$0.872292\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		−	21.5693i		−	0.781885i	−0.920415	−	0.390943i	$-0.872149\pi$
$761$		−	21.5693i		−	0.781885i	0.920415	−	0.390943i	$-0.127851\pi$
$762$		0			0
$763$	−36.9827			−1.33886
$764$		0			0
$765$		0			0
$766$		0			0
$767$			72.3517i			2.61247i
$768$		0			0
$769$			48.4197i			1.74606i	0.487668	+	0.873029i	$0.337848\pi$
$769$			48.4197i			1.74606i	−0.487668	+	0.873029i	$0.662152\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	45.1642			1.62444			0.812222	−	0.583349i	$-0.198258\pi$
$773$	45.1642			1.62444			0.812222	+	0.583349i	$0.198258\pi$
$774$		0			0
$775$	−4.55254			−0.163532
$776$		0			0
$777$		0			0
$778$		0			0
$779$	19.5969			0.702133
$780$		0			0
$781$			6.60581i			0.236374i
$782$		0			0
$783$		0			0
$784$		0			0
$785$			13.0914i			0.467253i
$786$		0			0
$787$			15.1787i			0.541061i	0.962711	+	0.270530i	$0.0871991\pi$
$787$			15.1787i			0.541061i	−0.962711	+	0.270530i	$0.912801\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		−	2.86080i		−	0.101718i
$792$		0			0
$793$			69.2406i			2.45881i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−8.26669			−0.292821			−0.146411	−	0.989224i	$-0.546772\pi$
$797$	−8.26669			−0.292821			−0.146411	+	0.989224i	$0.546772\pi$
$798$		0			0
$799$			63.0510i			2.23059i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	55.5725			1.96111
$804$		0			0
$805$	−6.51136	−	18.9182i	−0.229495	−	0.666781i
$806$		0			0
$807$		0			0
$808$		0			0
$809$		−	8.85482i		−	0.311319i	−0.987811	−	0.155659i	$-0.950250\pi$
$809$		−	8.85482i		−	0.311319i	0.987811	−	0.155659i	$-0.0497502\pi$
$810$		0			0
$811$	7.93004			0.278462			0.139231	−	0.990260i	$-0.455537\pi$
$811$	7.93004			0.278462			0.139231	+	0.990260i	$0.455537\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	23.7790			0.832942
$816$		0			0
$817$	−15.6248			−0.546643
$818$		0			0
$819$		0			0
$820$		0			0
$821$		−	12.9054i		−	0.450401i	−0.974312	−	0.225201i	$-0.927696\pi$
$821$		−	12.9054i		−	0.450401i	0.974312	−	0.225201i	$-0.0723038\pi$
$822$		0			0
$823$	−2.41441			−0.0841609			−0.0420805	−	0.999114i	$-0.513399\pi$
$823$	−2.41441			−0.0841609			−0.0420805	+	0.999114i	$0.513399\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	38.5702			1.34122			0.670609	−	0.741811i	$-0.266033\pi$
$827$	38.5702			1.34122			0.670609	+	0.741811i	$0.266033\pi$
$828$		0			0
$829$	8.89651			0.308989			0.154494	−	0.987994i	$-0.450625\pi$
$829$	8.89651			0.308989			0.154494	+	0.987994i	$0.450625\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−66.6071			−2.30780
$834$		0			0
$835$			14.5647i			0.504033i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	19.5300			0.674250			0.337125	−	0.941460i	$-0.390546\pi$
$839$	19.5300			0.674250			0.337125	+	0.941460i	$0.390546\pi$
$840$		0			0
$841$	6.11278			0.210786
$842$		0			0
$843$		0			0
$844$		0			0
$845$	28.4275			0.977936
$846$		0			0
$847$		−	27.7226i		−	0.952558i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	1.44889	+	4.20965i	0.0496674	+	0.144305i
$852$		0			0
$853$	−48.1455			−1.64847			−0.824236	−	0.566246i	$-0.808395\pi$
$853$	−48.1455			−1.64847			−0.824236	+	0.566246i	$0.808395\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$			8.94639i			0.305603i	0.988257	+	0.152801i	$0.0488295\pi$
$857$			8.94639i			0.305603i	−0.988257	+	0.152801i	$0.951171\pi$
$858$		0			0
$859$	−40.1902			−1.37127			−0.685635	−	0.727945i	$-0.740476\pi$
$859$	−40.1902			−1.37127			−0.685635	+	0.727945i	$0.740476\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		−	26.7374i		−	0.910150i	−0.890453	−	0.455075i	$-0.849612\pi$
$863$		−	26.7374i		−	0.910150i	0.890453	−	0.455075i	$-0.150388\pi$
$864$		0			0
$865$		−	3.01471i		−	0.102503i
$866$		0			0
$867$		0			0
$868$		0			0
$869$			47.9909i			1.62798i
$870$		0			0
$871$			23.2029i			0.786200i
$872$		0			0
$873$		0			0
$874$		0			0
$875$		−	4.17184i		−	0.141034i
$876$		0			0
$877$	20.8008			0.702395			0.351197	−	0.936301i	$-0.385775\pi$
$877$	20.8008			0.702395			0.351197	+	0.936301i	$0.385775\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−30.5858			−1.03046			−0.515231	−	0.857051i	$-0.672294\pi$
$881$	−30.5858			−1.03046			−0.515231	+	0.857051i	$0.672294\pi$
$882$		0			0
$883$	2.94171			0.0989965			0.0494983	−	0.998774i	$-0.484238\pi$
$883$	2.94171			0.0989965			0.0494983	+	0.998774i	$0.484238\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$			22.1245i			0.742868i	0.928459	+	0.371434i	$0.121134\pi$
$887$			22.1245i			0.742868i	−0.928459	+	0.371434i	$0.878866\pi$
$888$		0			0
$889$			74.1553i			2.48709i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−33.3884			−1.11730
$894$		0			0
$895$		−	25.1269i		−	0.839900i
$896$		0			0
$897$		0			0
$898$		0			0
$899$			21.7796i			0.726391i
$900$		0			0
$901$	−50.3107			−1.67609
$902$		0			0
$903$		0			0
$904$		0			0
$905$		−	6.87620i		−	0.228573i
$906$		0			0
$907$			2.82874i			0.0939268i	0.998897	+	0.0469634i	$0.0149544\pi$
$907$			2.82874i			0.0939268i	−0.998897	+	0.0469634i	$0.985046\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	8.59649			0.284815			0.142407	−	0.989808i	$-0.454516\pi$
$911$	8.59649			0.284815			0.142407	+	0.989808i	$0.454516\pi$
$912$		0			0
$913$	−51.6341			−1.70884
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−64.4896			−2.12963
$918$		0			0
$919$		−	22.7687i		−	0.751069i	−0.926808	−	0.375535i	$-0.877459\pi$
$919$		−	22.7687i		−	0.751069i	0.926808	−	0.375535i	$-0.122541\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$			10.1218i			0.333163i
$924$		0			0
$925$			0.928309i			0.0305226i
$926$		0			0
$927$		0			0
$928$		0			0
$929$		−	47.5113i		−	1.55880i	−0.626530	−	0.779398i	$-0.715525\pi$
$929$		−	47.5113i		−	1.55880i	0.626530	−	0.779398i	$-0.284475\pi$
$930$		0			0
$931$		−	35.2715i		−	1.15598i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	26.8920			0.879462
$936$		0			0
$937$			9.02484i			0.294829i	0.989075	+	0.147414i	$0.0470951\pi$
$937$			9.02484i			0.294829i	−0.989075	+	0.147414i	$0.952905\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	30.7944			1.00387			0.501935	−	0.864905i	$-0.332622\pi$
$941$	30.7944			1.00387			0.501935	+	0.864905i	$0.332622\pi$
$942$		0			0
$943$	−9.02233	−	26.2137i	−0.293807	−	0.853634i
$944$		0			0
$945$		0			0
$946$		0			0
$947$			14.5788i			0.473747i	0.971541	+	0.236873i	$0.0761227\pi$
$947$			14.5788i			0.473747i	−0.971541	+	0.236873i	$0.923877\pi$
$948$		0			0
$949$	85.1514			2.76413
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−0.378923			−0.0122745			−0.00613726	−	0.999981i	$-0.501954\pi$
$953$	−0.378923			−0.0122745			−0.00613726	+	0.999981i	$0.501954\pi$
$954$		0			0
$955$	26.4573			0.856138
$956$		0			0
$957$		0			0
$958$		0			0
$959$			51.3177i			1.65713i
$960$		0			0
$961$	−10.2744			−0.331432
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−24.8464			−0.799835
$966$		0			0
$967$	−14.4117			−0.463448			−0.231724	−	0.972782i	$-0.574437\pi$
$967$	−14.4117			−0.463448			−0.231724	+	0.972782i	$0.574437\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	33.4666			1.07399			0.536997	−	0.843584i	$-0.319558\pi$
$971$	33.4666			1.07399			0.536997	+	0.843584i	$0.319558\pi$
$972$		0			0
$973$			19.7383i			0.632780i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	45.1191			1.44349			0.721744	−	0.692160i	$-0.243341\pi$
$977$	45.1191			1.44349			0.721744	+	0.692160i	$0.243341\pi$
$978$		0			0
$979$	−23.7920			−0.760395
$980$		0			0
$981$		0			0
$982$		0			0
$983$	10.4895			0.334564			0.167282	−	0.985909i	$-0.446501\pi$
$983$	10.4895			0.334564			0.167282	+	0.985909i	$0.446501\pi$
$984$		0			0
$985$		−	23.7237i		−	0.755899i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	7.19359	+	20.9004i	0.228743	+	0.664594i
$990$		0			0
$991$	38.9245			1.23648			0.618238	−	0.785991i	$-0.287847\pi$
$991$	38.9245			1.23648			0.618238	+	0.785991i	$0.287847\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$			10.0072i			0.317250i
$996$		0			0
$997$	−8.23825			−0.260908			−0.130454	−	0.991454i	$-0.541643\pi$
$997$	−8.23825			−0.260908			−0.130454	+	0.991454i	$0.541643\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4140.2.i.b.1241.3	yes	16
3.2	odd	2		4140.2.i.a.1241.3	✓	16
23.22	odd	2		4140.2.i.a.1241.14	yes	16
69.68	even	2	inner	4140.2.i.b.1241.14	yes	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4140.2.i.a.1241.3	✓	16	3.2	odd	2
4140.2.i.a.1241.14	yes	16	23.22	odd	2
4140.2.i.b.1241.3	yes	16	1.1	even	1	trivial
4140.2.i.b.1241.14	yes	16	69.68	even	2	inner

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:	\( N \)	\(=\)	\( 4140 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4140.i (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.00000 q^{5} -4.17184i q^{7} +O(q^{10})\)	\(q+1.00000 q^{5} -4.17184i q^{7} +4.20061 q^{11} +6.43642 q^{13} +6.40191 q^{17} +3.39011i q^{19} +(4.53475 - 1.56079i) q^{23} +1.00000 q^{25} -4.78406i q^{29} -4.55254 q^{31} -4.17184i q^{35} +0.928309i q^{37} -5.78062i q^{41} +4.60895i q^{43} +9.84878i q^{47} -10.4043 q^{49} -7.85869 q^{53} +4.20061 q^{55} +11.2410i q^{59} +10.7576i q^{61} +6.43642 q^{65} +3.60494i q^{67} +1.57258i q^{71} +13.2296 q^{73} -17.5243i q^{77} +11.4247i q^{79} -12.2920 q^{83} +6.40191 q^{85} -5.66393 q^{89} -26.8517i q^{91} +3.39011i q^{95} -17.8056i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 16 q^{5}+O(q^{10}) \) 16 * q + 16 * q^5	\( 16 q + 16 q^{5} + 8 q^{23} + 16 q^{25} - 8 q^{31} - 40 q^{49} - 4 q^{53} + 8 q^{73} - 20 q^{83} - 32 q^{89}+O(q^{100}) \) 16 * q + 16 * q^5 + 8 * q^23 + 16 * q^25 - 8 * q^31 - 40 * q^49 - 4 * q^53 + 8 * q^73 - 20 * q^83 - 32 * q^89

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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