LMFDB - Embedded newform 912.2.d.b.191.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [912,2,Mod(191,912)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(912, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 0, 1, 0]))

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

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

chi := DirichletCharacter("912.191");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 912 = 2^{4} \cdot 3 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	912.d (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:	$7.28235666434$
Analytic rank:	$0$
Dimension:	$24$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			191.2
Character	$\chi$	$=$	912.191
Dual form			912.2.d.b.191.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.72850 + 0.110841i) q^{3} -3.67549i q^{5} -4.87247i q^{7} +(2.97543 - 0.383177i) q^{9} +O(q^{10})$	\(q+(-1.72850 + 0.110841i) q^{3} -3.67549i q^{5} -4.87247i q^{7} +(2.97543 - 0.383177i) q^{9} +0.458180 q^{11} +4.96949 q^{13} +(0.407395 + 6.35309i) q^{15} -3.91518i q^{17} +1.00000i q^{19} +(0.540069 + 8.42206i) q^{21} -2.77039 q^{23} -8.50924 q^{25} +(-5.10056 + 0.992121i) q^{27} +4.21341i q^{29} -3.33770i q^{31} +(-0.791964 + 0.0507851i) q^{33} -17.9087 q^{35} +6.65655 q^{37} +(-8.58977 + 0.550823i) q^{39} -9.60465i q^{41} +6.41172i q^{43} +(-1.40836 - 10.9362i) q^{45} +11.8413 q^{47} -16.7409 q^{49} +(0.433962 + 6.76739i) q^{51} +3.95236i q^{53} -1.68404i q^{55} +(-0.110841 - 1.72850i) q^{57} +2.01813 q^{59} -6.63375 q^{61} +(-1.86702 - 14.4977i) q^{63} -18.2653i q^{65} -1.01112i q^{67} +(4.78862 - 0.307073i) q^{69} -5.71670 q^{71} +9.95226 q^{73} +(14.7082 - 0.943172i) q^{75} -2.23247i q^{77} +6.29894i q^{79} +(8.70635 - 2.28023i) q^{81} -11.6759 q^{83} -14.3902 q^{85} +(-0.467019 - 7.28289i) q^{87} +14.1435i q^{89} -24.2137i q^{91} +(0.369953 + 5.76921i) q^{93} +3.67549 q^{95} -12.6574 q^{97} +(1.36328 - 0.175564i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q + 4 q^{9}+O(q^{10}) $ 24 * q + 4 * q^9	$ 24 q + 4 q^{9} - 12 q^{21} - 64 q^{25} + 12 q^{33} + 64 q^{37} - 16 q^{45} - 8 q^{49} - 24 q^{61} - 8 q^{69} - 16 q^{73} - 4 q^{81} - 8 q^{85} + 32 q^{93} - 8 q^{97}+O(q^{100}) $ 24 * q + 4 * q^9 - 12 * q^21 - 64 * q^25 + 12 * q^33 + 64 * q^37 - 16 * q^45 - 8 * q^49 - 24 * q^61 - 8 * q^69 - 16 * q^73 - 4 * q^81 - 8 * q^85 + 32 * q^93 - 8 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/912\mathbb{Z}\right)^\times$.

$n$	$97$	$229$	$305$	$799$
$\chi(n)$	$1$	$1$	$-1$	$-1$

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
$3$	−1.72850	+	0.110841i	−0.997950	+	0.0639940i
$3$	−1.72850	+	0.110841i	−0.997950	+	0.0639940i
$4$		0			0
$5$		−	3.67549i		−	1.64373i	−0.569682	−	0.821865i	$-0.692934\pi$
$5$		−	3.67549i		−	1.64373i	0.569682	−	0.821865i	$-0.307066\pi$
$6$		0			0
$7$		−	4.87247i		−	1.84162i	−0.390012	−	0.920810i	$-0.627529\pi$
$7$		−	4.87247i		−	1.84162i	0.390012	−	0.920810i	$-0.372471\pi$
$8$		0			0
$9$	2.97543	−	0.383177i	0.991810	−	0.127726i
$10$		0			0
$11$	0.458180			0.138146			0.0690732	−	0.997612i	$-0.477996\pi$
$11$	0.458180			0.138146			0.0690732	+	0.997612i	$0.477996\pi$
$12$		0			0
$13$	4.96949			1.37829			0.689144	−	0.724624i	$-0.257987\pi$
$13$	4.96949			1.37829			0.689144	+	0.724624i	$0.257987\pi$
$14$		0			0
$15$	0.407395	+	6.35309i	0.105189	+	1.64036i
$16$		0			0
$17$		−	3.91518i		−	0.949571i	−0.880102	−	0.474785i	$-0.842526\pi$
$17$		−	3.91518i		−	0.949571i	0.880102	−	0.474785i	$-0.157474\pi$
$18$		0			0
$19$			1.00000i			0.229416i
$19$			1.00000i			0.229416i
$20$		0			0
$21$	0.540069	+	8.42206i	0.117853	+	1.83784i
$22$		0			0
$23$	−2.77039			−0.577666			−0.288833	−	0.957379i	$-0.593267\pi$
$23$	−2.77039			−0.577666			−0.288833	+	0.957379i	$0.593267\pi$
$24$		0			0
$25$	−8.50924			−1.70185
$26$		0			0
$27$	−5.10056	+	0.992121i	−0.981603	+	0.190934i
$28$		0			0
$29$			4.21341i			0.782412i	0.920303	+	0.391206i	$0.127942\pi$
$29$			4.21341i			0.782412i	−0.920303	+	0.391206i	$0.872058\pi$
$30$		0			0
$31$		−	3.33770i		−	0.599468i	−0.954023	−	0.299734i	$-0.903102\pi$
$31$		−	3.33770i		−	0.599468i	0.954023	−	0.299734i	$-0.0968979\pi$
$32$		0			0
$33$	−0.791964	+	0.0507851i	−0.137863	+	0.00884055i
$34$		0			0
$35$	−17.9087			−3.02712
$36$		0			0
$37$	6.65655			1.09433			0.547165	−	0.837025i	$-0.315707\pi$
$37$	6.65655			1.09433			0.547165	+	0.837025i	$0.315707\pi$
$38$		0			0
$39$	−8.58977	+	0.550823i	−1.37546	+	0.0882023i
$40$		0			0
$41$		−	9.60465i		−	1.49999i	−0.661441	−	0.749997i	$-0.730055\pi$
$41$		−	9.60465i		−	1.49999i	0.661441	−	0.749997i	$-0.269945\pi$
$42$		0			0
$43$			6.41172i			0.977779i	0.872346	+	0.488889i	$0.162598\pi$
$43$			6.41172i			0.977779i	−0.872346	+	0.488889i	$0.837402\pi$
$44$		0			0
$45$	−1.40836	−	10.9362i	−0.209947	−	1.63027i
$46$		0			0
$47$	11.8413			1.72723			0.863613	−	0.504155i	$-0.168196\pi$
$47$	11.8413			1.72723			0.863613	+	0.504155i	$0.168196\pi$
$48$		0			0
$49$	−16.7409			−2.39156
$50$		0			0
$51$	0.433962	+	6.76739i	0.0607669	+	0.947625i
$52$		0			0
$53$			3.95236i			0.542898i	0.962453	+	0.271449i	$0.0875029\pi$
$53$			3.95236i			0.542898i	−0.962453	+	0.271449i	$0.912497\pi$
$54$		0			0
$55$		−	1.68404i		−	0.227075i
$56$		0			0
$57$	−0.110841	−	1.72850i	−0.0146812	−	0.228945i
$58$		0			0
$59$	2.01813			0.262738			0.131369	−	0.991334i	$-0.458063\pi$
$59$	2.01813			0.262738			0.131369	+	0.991334i	$0.458063\pi$
$60$		0			0
$61$	−6.63375			−0.849364			−0.424682	−	0.905342i	$-0.639614\pi$
$61$	−6.63375			−0.849364			−0.424682	+	0.905342i	$0.639614\pi$
$62$		0			0
$63$	−1.86702	−	14.4977i	−0.235222	−	1.82654i
$64$		0			0
$65$		−	18.2653i		−	2.26553i
$66$		0			0
$67$		−	1.01112i		−	0.123528i	−0.998091	−	0.0617639i	$-0.980327\pi$
$67$		−	1.01112i		−	0.123528i	0.998091	−	0.0617639i	$-0.0196726\pi$
$68$		0			0
$69$	4.78862	−	0.307073i	0.576482	−	0.0369672i
$70$		0			0
$71$	−5.71670			−0.678448			−0.339224	−	0.940706i	$-0.610164\pi$
$71$	−5.71670			−0.678448			−0.339224	+	0.940706i	$0.610164\pi$
$72$		0			0
$73$	9.95226			1.16482			0.582412	−	0.812894i	$-0.302109\pi$
$73$	9.95226			1.16482			0.582412	+	0.812894i	$0.302109\pi$
$74$		0			0
$75$	14.7082	−	0.943172i	1.69836	−	0.108908i
$76$		0			0
$77$		−	2.23247i		−	0.254413i
$78$		0			0
$79$			6.29894i			0.708687i	0.935115	+	0.354343i	$0.115296\pi$
$79$			6.29894i			0.708687i	−0.935115	+	0.354343i	$0.884704\pi$
$80$		0			0
$81$	8.70635	−	2.28023i	0.967372	−	0.253359i
$82$		0			0
$83$	−11.6759			−1.28160			−0.640799	−	0.767709i	$-0.721397\pi$
$83$	−11.6759			−1.28160			−0.640799	+	0.767709i	$0.721397\pi$
$84$		0			0
$85$	−14.3902			−1.56084
$86$		0			0
$87$	−0.467019	−	7.28289i	−0.0500697	−	0.780808i
$88$		0			0
$89$			14.1435i			1.49921i	0.661885	+	0.749606i	$0.269757\pi$
$89$			14.1435i			1.49921i	−0.661885	+	0.749606i	$0.730243\pi$
$90$		0			0
$91$		−	24.2137i		−	2.53828i
$92$		0			0
$93$	0.369953	+	5.76921i	0.0383624	+	0.598239i
$94$		0			0
$95$	3.67549			0.377098
$96$		0			0
$97$	−12.6574			−1.28516			−0.642581	−	0.766218i	$-0.722136\pi$
$97$	−12.6574			−1.28516			−0.642581	+	0.766218i	$0.722136\pi$
$98$		0			0
$99$	1.36328	−	0.175564i	0.137015	−	0.0176449i
$100$		0			0
$101$			7.76056i			0.772205i	0.922456	+	0.386102i	$0.126179\pi$
$101$			7.76056i			0.772205i	−0.922456	+	0.386102i	$0.873821\pi$
$102$		0			0
$103$		−	8.92512i		−	0.879418i	−0.898140	−	0.439709i	$-0.855082\pi$
$103$		−	8.92512i		−	0.879418i	0.898140	−	0.439709i	$-0.144918\pi$
$104$		0			0
$105$	30.9552	−	1.98502i	3.02092	−	0.193718i
$106$		0			0
$107$	18.6024			1.79836			0.899182	−	0.437574i	$-0.144162\pi$
$107$	18.6024			1.79836			0.899182	+	0.437574i	$0.144162\pi$
$108$		0			0
$109$	−7.73669			−0.741040			−0.370520	−	0.928824i	$-0.620821\pi$
$109$	−7.73669			−0.741040			−0.370520	+	0.928824i	$0.620821\pi$
$110$		0			0
$111$	−11.5059	+	0.737818i	−1.09209	+	0.0700306i
$112$		0			0
$113$			10.0995i			0.950085i	0.879963	+	0.475043i	$0.157567\pi$
$113$			10.0995i			0.950085i	−0.879963	+	0.475043i	$0.842433\pi$
$114$		0			0
$115$			10.1825i			0.949527i
$116$		0			0
$117$	14.7864	−	1.90420i	1.36700	−	0.176043i
$118$		0			0
$119$	−19.0766			−1.74875
$120$		0			0
$121$	−10.7901			−0.980916
$122$		0			0
$123$	1.06459	+	16.6016i	0.0959907	+	1.49692i
$124$		0			0
$125$			12.8982i			1.15365i
$126$		0			0
$127$			4.37994i			0.388657i	0.980936	+	0.194329i	$0.0622528\pi$
$127$			4.37994i			0.388657i	−0.980936	+	0.194329i	$0.937747\pi$
$128$		0			0
$129$	−0.710682	−	11.0827i	−0.0625720	−	0.975775i
$130$		0			0
$131$	5.51958			0.482248			0.241124	−	0.970494i	$-0.422484\pi$
$131$	5.51958			0.482248			0.241124	+	0.970494i	$0.422484\pi$
$132$		0			0
$133$	4.87247			0.422496
$134$		0			0
$135$	3.64653	+	18.7471i	0.313844	+	1.61349i
$136$		0			0
$137$			3.53267i			0.301817i	0.988548	+	0.150908i	$0.0482198\pi$
$137$			3.53267i			0.301817i	−0.988548	+	0.150908i	$0.951780\pi$
$138$		0			0
$139$		−	1.97062i		−	0.167146i	−0.996502	−	0.0835729i	$-0.973367\pi$
$139$		−	1.97062i		−	0.167146i	0.996502	−	0.0835729i	$-0.0266331\pi$
$140$		0			0
$141$	−20.4676	+	1.31250i	−1.72369	+	0.110532i
$142$		0			0
$143$	2.27692			0.190406
$144$		0			0
$145$	15.4864			1.28607
$146$		0			0
$147$	28.9367	−	1.85558i	2.38666	−	0.153046i
$148$		0			0
$149$			9.83262i			0.805520i	0.915306	+	0.402760i	$0.131949\pi$
$149$			9.83262i			0.805520i	−0.915306	+	0.402760i	$0.868051\pi$
$150$		0			0
$151$		−	8.90906i		−	0.725009i	−0.931982	−	0.362505i	$-0.881922\pi$
$151$		−	8.90906i		−	0.725009i	0.931982	−	0.362505i	$-0.118078\pi$
$152$		0			0
$153$	−1.50021	−	11.6493i	−0.121285	−	0.941793i
$154$		0			0
$155$	−12.2677			−0.985364
$156$		0			0
$157$	−0.600145			−0.0478968			−0.0239484	−	0.999713i	$-0.507624\pi$
$157$	−0.600145			−0.0478968			−0.0239484	+	0.999713i	$0.507624\pi$
$158$		0			0
$159$	−0.438083	−	6.83165i	−0.0347422	−	0.541785i
$160$		0			0
$161$			13.4986i			1.06384i
$162$		0			0
$163$		−	1.42612i		−	0.111702i	−0.998439	−	0.0558511i	$-0.982213\pi$
$163$		−	1.42612i		−	0.111702i	0.998439	−	0.0558511i	$-0.0177872\pi$
$164$		0			0
$165$	0.186660	+	2.91086i	0.0145315	+	0.226610i
$166$		0			0
$167$	11.3789			0.880528			0.440264	−	0.897868i	$-0.354885\pi$
$167$	11.3789			0.880528			0.440264	+	0.897868i	$0.354885\pi$
$168$		0			0
$169$	11.6958			0.899679
$170$		0			0
$171$	0.383177	+	2.97543i	0.0293023	+	0.227537i
$172$		0			0
$173$			10.0057i			0.760720i	0.924838	+	0.380360i	$0.124200\pi$
$173$			10.0057i			0.760720i	−0.924838	+	0.380360i	$0.875800\pi$
$174$		0			0
$175$			41.4610i			3.13416i
$176$		0			0
$177$	−3.48833	+	0.223691i	−0.262199	+	0.0168136i
$178$		0			0
$179$	−1.87586			−0.140208			−0.0701041	−	0.997540i	$-0.522333\pi$
$179$	−1.87586			−0.140208			−0.0701041	+	0.997540i	$0.522333\pi$
$180$		0			0
$181$	7.78558			0.578698			0.289349	−	0.957224i	$-0.406561\pi$
$181$	7.78558			0.578698			0.289349	+	0.957224i	$0.406561\pi$
$182$		0			0
$183$	11.4664	−	0.735291i	0.847624	−	0.0543543i
$184$		0			0
$185$		−	24.4661i		−	1.79878i
$186$		0			0
$187$		−	1.79386i		−	0.131180i
$188$		0			0
$189$	4.83408	+	24.8523i	0.351627	+	1.80774i
$190$		0			0
$191$	20.0402			1.45006			0.725030	−	0.688717i	$-0.241826\pi$
$191$	20.0402			1.45006			0.725030	+	0.688717i	$0.241826\pi$
$192$		0			0
$193$	11.3016			0.813505			0.406752	−	0.913538i	$-0.366661\pi$
$193$	11.3016			0.813505			0.406752	+	0.913538i	$0.366661\pi$
$194$		0			0
$195$	2.02455	+	31.5716i	0.144981	+	2.26089i
$196$		0			0
$197$		−	13.9523i		−	0.994059i	−0.867734	−	0.497029i	$-0.834424\pi$
$197$		−	13.9523i		−	0.994059i	0.867734	−	0.497029i	$-0.165576\pi$
$198$		0			0
$199$		−	16.3458i		−	1.15873i	−0.815070	−	0.579363i	$-0.803301\pi$
$199$		−	16.3458i		−	1.15873i	0.815070	−	0.579363i	$-0.196699\pi$
$200$		0			0
$201$	0.112073	+	1.74772i	0.00790504	+	0.123275i
$202$		0			0
$203$	20.5297			1.44090
$204$		0			0
$205$	−35.3018			−2.46559
$206$		0			0
$207$	−8.24310	+	1.06155i	−0.572935	+	0.0737828i
$208$		0			0
$209$			0.458180i			0.0316930i
$210$		0			0
$211$		−	11.2045i		−	0.771348i	−0.922635	−	0.385674i	$-0.873969\pi$
$211$		−	11.2045i		−	0.771348i	0.922635	−	0.385674i	$-0.126031\pi$
$212$		0			0
$213$	9.88133	−	0.633645i	0.677057	−	0.0434166i
$214$		0			0
$215$	23.5662			1.60720
$216$		0			0
$217$	−16.2628			−1.10399
$218$		0			0
$219$	−17.2025	+	1.10312i	−1.16244	+	0.0745418i
$220$		0			0
$221$		−	19.4565i		−	1.30878i
$222$		0			0
$223$			20.4542i			1.36971i	0.728677	+	0.684857i	$0.240135\pi$
$223$			20.4542i			1.36971i	−0.728677	+	0.684857i	$0.759865\pi$
$224$		0			0
$225$	−25.3186	+	3.26055i	−1.68791	+	0.217370i
$226$		0			0
$227$	1.32690			0.0880698			0.0440349	−	0.999030i	$-0.485979\pi$
$227$	1.32690			0.0880698			0.0440349	+	0.999030i	$0.485979\pi$
$228$		0			0
$229$	12.7298			0.841209			0.420605	−	0.907244i	$-0.361818\pi$
$229$	12.7298			0.841209			0.420605	+	0.907244i	$0.361818\pi$
$230$		0			0
$231$	0.247449	+	3.85882i	0.0162809	+	0.253892i
$232$		0			0
$233$		−	17.1683i		−	1.12473i	−0.826889	−	0.562366i	$-0.809891\pi$
$233$		−	17.1683i		−	1.12473i	0.826889	−	0.562366i	$-0.190109\pi$
$234$		0			0
$235$		−	43.5225i		−	2.83909i
$236$		0			0
$237$	−0.698181	−	10.8877i	−0.0453517	−	0.707234i
$238$		0			0
$239$	14.3625			0.929032			0.464516	−	0.885565i	$-0.346228\pi$
$239$	14.3625			0.929032			0.464516	+	0.885565i	$0.346228\pi$
$240$		0			0
$241$	8.11757			0.522899			0.261449	−	0.965217i	$-0.415800\pi$
$241$	8.11757			0.522899			0.261449	+	0.965217i	$0.415800\pi$
$242$		0			0
$243$	−14.7962	+	4.90640i	−0.949176	+	0.314746i
$244$		0			0
$245$			61.5311i			3.93108i
$246$		0			0
$247$			4.96949i			0.316201i
$248$		0			0
$249$	20.1818	−	1.29417i	1.27897	−	0.0820146i
$250$		0			0
$251$	3.96058			0.249990			0.124995	−	0.992157i	$-0.460109\pi$
$251$	3.96058			0.249990			0.124995	+	0.992157i	$0.460109\pi$
$252$		0			0
$253$	−1.26934			−0.0798026
$254$		0			0
$255$	24.8735	−	1.59502i	1.55764	−	0.0998843i
$256$		0			0
$257$		−	12.1159i		−	0.755771i	−0.925852	−	0.377885i	$-0.876651\pi$
$257$		−	12.1159i		−	0.755771i	0.925852	−	0.377885i	$-0.123349\pi$
$258$		0			0
$259$		−	32.4338i		−	2.01534i
$260$		0			0
$261$	1.61448	+	12.5367i	0.0999341	+	0.776003i
$262$		0			0
$263$	23.7968			1.46737			0.733686	−	0.679488i	$-0.237798\pi$
$263$	23.7968			1.46737			0.733686	+	0.679488i	$0.237798\pi$
$264$		0			0
$265$	14.5269			0.892378
$266$		0			0
$267$	−1.56768	−	24.4471i	−0.0959406	−	1.49614i
$268$		0			0
$269$			9.75296i			0.594648i	0.954777	+	0.297324i	$0.0960942\pi$
$269$			9.75296i			0.594648i	−0.954777	+	0.297324i	$0.903906\pi$
$270$		0			0
$271$		−	12.0706i		−	0.733239i	−0.930371	−	0.366620i	$-0.880515\pi$
$271$		−	12.0706i		−	0.733239i	0.930371	−	0.366620i	$-0.119485\pi$
$272$		0			0
$273$	2.68387	+	41.8533i	0.162435	+	2.53308i
$274$		0			0
$275$	−3.89876			−0.235104
$276$		0			0
$277$	−29.0483			−1.74534			−0.872672	−	0.488306i	$-0.837615\pi$
$277$	−29.0483			−1.74534			−0.872672	+	0.488306i	$0.837615\pi$
$278$		0			0
$279$	−1.27893	−	9.93108i	−0.0765675	−	0.594558i
$280$		0			0
$281$		−	25.0496i		−	1.49433i	−0.664637	−	0.747166i	$-0.731414\pi$
$281$		−	25.0496i		−	1.49433i	0.664637	−	0.747166i	$-0.268586\pi$
$282$		0			0
$283$		−	4.04411i		−	0.240397i	−0.992750	−	0.120199i	$-0.961647\pi$
$283$		−	4.04411i		−	0.240397i	0.992750	−	0.120199i	$-0.0383532\pi$
$284$		0			0
$285$	−6.35309	+	0.407395i	−0.376325	+	0.0241320i
$286$		0			0
$287$	−46.7983			−2.76242
$288$		0			0
$289$	1.67136			0.0983151
$290$		0			0
$291$	21.8783	−	1.40296i	1.28253	−	0.0822427i
$292$		0			0
$293$			12.1605i			0.710426i	0.934785	+	0.355213i	$0.115592\pi$
$293$			12.1605i			0.710426i	−0.934785	+	0.355213i	$0.884408\pi$
$294$		0			0
$295$		−	7.41760i		−	0.431870i
$296$		0			0
$297$	−2.33697	+	0.454570i	−0.135605	+	0.0263768i
$298$		0			0
$299$	−13.7674			−0.796191
$300$		0			0
$301$	31.2409			1.80070
$302$		0			0
$303$	−0.860188	−	13.4141i	−0.0494165	−	0.770622i
$304$		0			0
$305$			24.3823i			1.39613i
$306$		0			0
$307$			10.2313i			0.583934i	0.956428	+	0.291967i	$0.0943097\pi$
$307$			10.2313i			0.583934i	−0.956428	+	0.291967i	$0.905690\pi$
$308$		0			0
$309$	0.989268	+	15.4271i	0.0562775	+	0.877616i
$310$		0			0
$311$	−1.47072			−0.0833970			−0.0416985	−	0.999130i	$-0.513277\pi$
$311$	−1.47072			−0.0833970			−0.0416985	+	0.999130i	$0.513277\pi$
$312$		0			0
$313$	−16.1089			−0.910527			−0.455264	−	0.890357i	$-0.650455\pi$
$313$	−16.1089			−0.910527			−0.455264	+	0.890357i	$0.650455\pi$
$314$		0			0
$315$	−53.2861	+	6.86221i	−3.00233	+	0.386642i
$316$		0			0
$317$		−	24.4406i		−	1.37272i	−0.727260	−	0.686362i	$-0.759207\pi$
$317$		−	24.4406i		−	1.37272i	0.727260	−	0.686362i	$-0.240793\pi$
$318$		0			0
$319$			1.93050i			0.108087i
$320$		0			0
$321$	−32.1543	+	2.06191i	−1.79468	+	0.115085i
$322$		0			0
$323$	3.91518			0.217847
$324$		0			0
$325$	−42.2866			−2.34564
$326$		0			0
$327$	13.3729	−	0.857542i	0.739522	−	0.0474222i
$328$		0			0
$329$		−	57.6962i		−	3.18089i
$330$		0			0
$331$			22.4550i			1.23424i	0.786870	+	0.617119i	$0.211700\pi$
$331$			22.4550i			1.23424i	−0.786870	+	0.617119i	$0.788300\pi$
$332$		0			0
$333$	19.8061	−	2.55064i	1.08537	−	0.139774i
$334$		0			0
$335$	−3.71636			−0.203046
$336$		0			0
$337$	−11.4268			−0.622460			−0.311230	−	0.950335i	$-0.600741\pi$
$337$	−11.4268			−0.622460			−0.311230	+	0.950335i	$0.600741\pi$
$338$		0			0
$339$	−1.11944	−	17.4571i	−0.0607998	−	0.948138i
$340$		0			0
$341$		−	1.52927i		−	0.0828144i
$342$		0			0
$343$			47.4623i			2.56273i
$344$		0			0
$345$	−1.12864	−	17.6005i	−0.0607641	−	0.947581i
$346$		0			0
$347$	3.80109			0.204053			0.102027	−	0.994782i	$-0.467467\pi$
$347$	3.80109			0.204053			0.102027	+	0.994782i	$0.467467\pi$
$348$		0			0
$349$	9.12591			0.488499			0.244249	−	0.969712i	$-0.421458\pi$
$349$	9.12591			0.488499			0.244249	+	0.969712i	$0.421458\pi$
$350$		0			0
$351$	−25.3472	+	4.93034i	−1.35293	+	0.263162i
$352$		0			0
$353$			32.0681i			1.70682i	0.521244	+	0.853408i	$0.325468\pi$
$353$			32.0681i			1.70682i	−0.521244	+	0.853408i	$0.674532\pi$
$354$		0			0
$355$			21.0117i			1.11519i
$356$		0			0
$357$	32.9739	−	2.11447i	1.74516	−	0.111909i
$358$		0			0
$359$	−21.5410			−1.13689			−0.568445	−	0.822721i	$-0.692455\pi$
$359$	−21.5410			−1.13689			−0.568445	+	0.822721i	$0.692455\pi$
$360$		0			0
$361$	−1.00000			−0.0526316
$362$		0			0
$363$	18.6506	−	1.19598i	0.978905	−	0.0627727i
$364$		0			0
$365$		−	36.5795i		−	1.91466i
$366$		0			0
$367$		−	24.3688i		−	1.27204i	−0.771671	−	0.636022i	$-0.780579\pi$
$367$		−	24.3688i		−	1.27204i	0.771671	−	0.636022i	$-0.219421\pi$
$368$		0			0
$369$	−3.68028	−	28.5779i	−0.191588	−	1.48771i
$370$		0			0
$371$	19.2577			0.999812
$372$		0			0
$373$	18.3460			0.949920			0.474960	−	0.880007i	$-0.342463\pi$
$373$	18.3460			0.949920			0.474960	+	0.880007i	$0.342463\pi$
$374$		0			0
$375$	−1.42965	−	22.2945i	−0.0738266	−	1.15128i
$376$		0			0
$377$			20.9385i			1.07839i
$378$		0			0
$379$			1.29049i			0.0662881i	0.999451	+	0.0331441i	$0.0105520\pi$
$379$			1.29049i			0.0662881i	−0.999451	+	0.0331441i	$0.989448\pi$
$380$		0			0
$381$	−0.485477	−	7.57073i	−0.0248717	−	0.387860i
$382$		0			0
$383$	21.9739			1.12281			0.561407	−	0.827540i	$-0.310260\pi$
$383$	21.9739			1.12281			0.561407	+	0.827540i	$0.310260\pi$
$384$		0			0
$385$	−8.20541			−0.418187
$386$		0			0
$387$	2.45683	+	19.0776i	0.124888	+	0.969770i
$388$		0			0
$389$		−	26.1642i		−	1.32658i	−0.748364	−	0.663288i	$-0.769160\pi$
$389$		−	26.1642i		−	1.32658i	0.748364	−	0.663288i	$-0.230840\pi$
$390$		0			0
$391$			10.8466i			0.548535i
$392$		0			0
$393$	−9.54060	+	0.611795i	−0.481260	+	0.0308610i
$394$		0			0
$395$	23.1517			1.16489
$396$		0			0
$397$	31.3935			1.57559			0.787797	−	0.615935i	$-0.211222\pi$
$397$	31.3935			1.57559			0.787797	+	0.615935i	$0.211222\pi$
$398$		0			0
$399$	−8.42206	+	0.540069i	−0.421630	+	0.0270373i
$400$		0			0
$401$			13.5738i			0.677841i	0.940815	+	0.338921i	$0.110062\pi$
$401$			13.5738i			0.677841i	−0.940815	+	0.338921i	$0.889938\pi$
$402$		0			0
$403$		−	16.5867i		−	0.826240i
$404$		0			0
$405$	−8.38098	−	32.0001i	−0.416454	−	1.59010i
$406$		0			0
$407$	3.04990			0.151178
$408$		0			0
$409$	−19.8998			−0.983980			−0.491990	−	0.870601i	$-0.663730\pi$
$409$	−19.8998			−0.983980			−0.491990	+	0.870601i	$0.663730\pi$
$410$		0			0
$411$	−0.391565	−	6.10623i	−0.0193145	−	0.301198i
$412$		0			0
$413$		−	9.83325i		−	0.483863i
$414$		0			0
$415$			42.9147i			2.10660i
$416$		0			0
$417$	0.218425	+	3.40622i	0.0106963	+	0.166803i
$418$		0			0
$419$	−1.94929			−0.0952291			−0.0476146	−	0.998866i	$-0.515162\pi$
$419$	−1.94929			−0.0952291			−0.0476146	+	0.998866i	$0.515162\pi$
$420$		0			0
$421$	−26.5881			−1.29583			−0.647913	−	0.761715i	$-0.724358\pi$
$421$	−26.5881			−1.29583			−0.647913	+	0.761715i	$0.724358\pi$
$422$		0			0
$423$	35.2328	−	4.53730i	1.71308	−	0.220611i
$424$		0			0
$425$			33.3152i			1.61603i
$426$		0			0
$427$			32.3227i			1.56421i
$428$		0			0
$429$	−3.93566	+	0.252376i	−0.190015	+	0.0121848i
$430$		0			0
$431$	−40.7664			−1.96365			−0.981825	−	0.189789i	$-0.939220\pi$
$431$	−40.7664			−1.96365			−0.981825	+	0.189789i	$0.939220\pi$
$432$		0			0
$433$	41.5783			1.99813			0.999063	−	0.0432833i	$-0.0137818\pi$
$433$	41.5783			1.99813			0.999063	+	0.0432833i	$0.0137818\pi$
$434$		0			0
$435$	−26.7682	+	1.71652i	−1.28344	+	0.0823010i
$436$		0			0
$437$		−	2.77039i		−	0.132526i
$438$		0			0
$439$			21.5230i			1.02724i	0.858019	+	0.513619i	$0.171695\pi$
$439$			21.5230i			1.02724i	−0.858019	+	0.513619i	$0.828305\pi$
$440$		0			0
$441$	−49.8114	+	6.41474i	−2.37197	+	0.305464i
$442$		0			0
$443$	−16.5866			−0.788053			−0.394026	−	0.919099i	$-0.628918\pi$
$443$	−16.5866			−0.788053			−0.394026	+	0.919099i	$0.628918\pi$
$444$		0			0
$445$	51.9844			2.46430
$446$		0			0
$447$	−1.08986	−	16.9957i	−0.0515485	−	0.803869i
$448$		0			0
$449$		−	29.4916i		−	1.39179i	−0.718142	−	0.695897i	$-0.755007\pi$
$449$		−	29.4916i		−	1.39179i	0.718142	−	0.695897i	$-0.244993\pi$
$450$		0			0
$451$		−	4.40066i		−	0.207219i
$452$		0			0
$453$	0.987489	+	15.3993i	0.0463963	+	0.723523i
$454$		0			0
$455$	−88.9972			−4.17225
$456$		0			0
$457$	−10.1563			−0.475090			−0.237545	−	0.971377i	$-0.576343\pi$
$457$	−10.1563			−0.475090			−0.237545	+	0.971377i	$0.576343\pi$
$458$		0			0
$459$	3.88433	+	19.9696i	0.181305	+	0.932102i
$460$		0			0
$461$		−	19.6327i		−	0.914385i	−0.889368	−	0.457192i	$-0.848855\pi$
$461$		−	19.6327i		−	0.914385i	0.889368	−	0.457192i	$-0.151145\pi$
$462$		0			0
$463$			20.8892i			0.970802i	0.874292	+	0.485401i	$0.161326\pi$
$463$			20.8892i			0.970802i	−0.874292	+	0.485401i	$0.838674\pi$
$464$		0			0
$465$	21.2047	−	1.35976i	0.983344	−	0.0630574i
$466$		0			0
$467$	−7.59395			−0.351406			−0.175703	−	0.984443i	$-0.556220\pi$
$467$	−7.59395			−0.351406			−0.175703	+	0.984443i	$0.556220\pi$
$468$		0			0
$469$	−4.92664			−0.227491
$470$		0			0
$471$	1.03735	−	0.0665207i	0.0477986	−	0.00306511i
$472$		0			0
$473$			2.93772i			0.135077i
$474$		0			0
$475$		−	8.50924i		−	0.390431i
$476$		0			0
$477$	1.51445	+	11.7600i	0.0693421	+	0.538452i
$478$		0			0
$479$	−17.9802			−0.821538			−0.410769	−	0.911739i	$-0.634740\pi$
$479$	−17.9802			−0.821538			−0.410769	+	0.911739i	$0.634740\pi$
$480$		0			0
$481$	33.0797			1.50830
$482$		0			0
$483$	−1.49620	−	23.3324i	−0.0680795	−	1.06166i
$484$		0			0
$485$			46.5221i			2.11246i
$486$		0			0
$487$			13.4070i			0.607530i	0.952747	+	0.303765i	$0.0982437\pi$
$487$			13.4070i			0.607530i	−0.952747	+	0.303765i	$0.901756\pi$
$488$		0			0
$489$	0.158072	+	2.46505i	0.00714828	+	0.111473i
$490$		0			0
$491$	10.3770			0.468309			0.234155	−	0.972199i	$-0.424768\pi$
$491$	10.3770			0.468309			0.234155	+	0.972199i	$0.424768\pi$
$492$		0			0
$493$	16.4963			0.742955
$494$		0			0
$495$	−0.645285	−	5.01073i	−0.0290034	−	0.225216i
$496$		0			0
$497$			27.8545i			1.24944i
$498$		0			0
$499$			10.4778i			0.469051i	0.972110	+	0.234526i	$0.0753537\pi$
$499$			10.4778i			0.469051i	−0.972110	+	0.234526i	$0.924646\pi$
$500$		0			0
$501$	−19.6685	+	1.26125i	−0.878724	+	0.0563486i
$502$		0			0
$503$	4.24048			0.189074			0.0945369	−	0.995521i	$-0.469863\pi$
$503$	4.24048			0.189074			0.0945369	+	0.995521i	$0.469863\pi$
$504$		0			0
$505$	28.5239			1.26930
$506$		0			0
$507$	−20.2163	+	1.29638i	−0.897835	+	0.0575741i
$508$		0			0
$509$		−	0.417908i		−	0.0185234i	−0.999957	−	0.00926172i	$-0.997052\pi$
$509$		−	0.417908i		−	0.0185234i	0.999957	−	0.00926172i	$-0.00294814\pi$
$510$		0			0
$511$		−	48.4921i		−	2.14516i
$512$		0			0
$513$	−0.992121	−	5.10056i	−0.0438032	−	0.225195i
$514$		0			0
$515$	−32.8042			−1.44553
$516$		0			0
$517$	5.42543			0.238610
$518$		0			0
$519$	−1.10904	−	17.2949i	−0.0486816	−	0.759161i
$520$		0			0
$521$		−	1.40766i		−	0.0616706i	−0.999524	−	0.0308353i	$-0.990183\pi$
$521$		−	1.40766i		−	0.0616706i	0.999524	−	0.0308353i	$-0.00981674\pi$
$522$		0			0
$523$		−	36.8671i		−	1.61209i	−0.591856	−	0.806044i	$-0.701605\pi$
$523$		−	36.8671i		−	1.61209i	0.591856	−	0.806044i	$-0.298395\pi$
$524$		0			0
$525$	−4.59557	−	71.6653i	−0.200567	−	3.12773i
$526$		0			0
$527$	−13.0677			−0.569237
$528$		0			0
$529$	−15.3249			−0.666302
$530$		0			0
$531$	6.00479	−	0.773300i	0.260586	−	0.0335583i
$532$		0			0
$533$		−	47.7302i		−	2.06742i
$534$		0			0
$535$		−	68.3731i		−	2.95603i
$536$		0			0
$537$	3.24242	−	0.207922i	0.139921	−	0.00897248i
$538$		0			0
$539$	−7.67036			−0.330386
$540$		0			0
$541$	8.35415			0.359173			0.179587	−	0.983742i	$-0.442524\pi$
$541$	8.35415			0.359173			0.179587	+	0.983742i	$0.442524\pi$
$542$		0			0
$543$	−13.4574	+	0.862961i	−0.577512	+	0.0370332i
$544$		0			0
$545$			28.4361i			1.21807i
$546$		0			0
$547$		−	3.21551i		−	0.137485i	−0.997634	−	0.0687427i	$-0.978101\pi$
$547$		−	3.21551i		−	0.137485i	0.997634	−	0.0687427i	$-0.0218988\pi$
$548$		0			0
$549$	−19.7382	+	2.54190i	−0.842408	+	0.108486i
$550$		0			0
$551$	−4.21341			−0.179498
$552$		0			0
$553$	30.6914			1.30513
$554$		0			0
$555$	2.71185	+	42.2897i	0.115111	+	1.79510i
$556$		0			0
$557$		−	3.46405i		−	0.146777i	−0.997303	−	0.0733883i	$-0.976619\pi$
$557$		−	3.46405i		−	0.146777i	0.997303	−	0.0733883i	$-0.0233812\pi$
$558$		0			0
$559$			31.8630i			1.34766i
$560$		0			0
$561$	0.198833	+	3.10068i	0.00839473	+	0.130911i
$562$		0			0
$563$	20.5544			0.866264			0.433132	−	0.901330i	$-0.357408\pi$
$563$	20.5544			0.866264			0.433132	+	0.901330i	$0.357408\pi$
$564$		0			0
$565$	37.1208			1.56168
$566$		0			0
$567$	−11.1104	−	42.4214i	−0.466591	−	1.78153i
$568$		0			0
$569$		−	24.5397i		−	1.02876i	−0.857563	−	0.514379i	$-0.828022\pi$
$569$		−	24.5397i		−	1.02876i	0.857563	−	0.514379i	$-0.171978\pi$
$570$		0			0
$571$		−	33.1631i		−	1.38783i	−0.720056	−	0.693916i	$-0.755884\pi$
$571$		−	33.1631i		−	1.38783i	0.720056	−	0.693916i	$-0.244116\pi$
$572$		0			0
$573$	−34.6396	+	2.22128i	−1.44709	+	0.0927952i
$574$		0			0
$575$	23.5739			0.983100
$576$		0			0
$577$	33.9733			1.41433			0.707163	−	0.707051i	$-0.249975\pi$
$577$	33.9733			1.41433			0.707163	+	0.707051i	$0.249975\pi$
$578$		0			0
$579$	−19.5348	+	1.25268i	−0.811837	+	0.0520595i
$580$		0			0
$581$			56.8905i			2.36021i
$582$		0			0
$583$			1.81089i			0.0749995i
$584$		0			0
$585$	−6.99885	−	54.3472i	−0.289367	−	2.24698i
$586$		0			0
$587$	−42.7301			−1.76366			−0.881830	−	0.471567i	$-0.843688\pi$
$587$	−42.7301			−1.76366			−0.881830	+	0.471567i	$0.843688\pi$
$588$		0			0
$589$	3.33770			0.137527
$590$		0			0
$591$	1.54648	+	24.1165i	0.0636138	+	0.992021i
$592$		0			0
$593$		−	34.6793i		−	1.42411i	−0.702125	−	0.712054i	$-0.747765\pi$
$593$		−	34.6793i		−	1.42411i	0.702125	−	0.712054i	$-0.252235\pi$
$594$		0			0
$595$			70.1158i			2.87447i
$596$		0			0
$597$	1.81179	+	28.2538i	0.0741516	+	1.15635i
$598$		0			0
$599$	−17.5797			−0.718288			−0.359144	−	0.933282i	$-0.616931\pi$
$599$	−17.5797			−0.718288			−0.359144	+	0.933282i	$0.616931\pi$
$600$		0			0
$601$	−9.17032			−0.374065			−0.187033	−	0.982354i	$-0.559887\pi$
$601$	−9.17032			−0.374065			−0.187033	+	0.982354i	$0.559887\pi$
$602$		0			0
$603$	−0.387438	−	3.00851i	−0.0157777	−	0.122516i
$604$		0			0
$605$			39.6588i			1.61236i
$606$		0			0
$607$		−	13.7473i		−	0.557987i	−0.960293	−	0.278993i	$-0.909999\pi$
$607$		−	13.7473i		−	0.557987i	0.960293	−	0.278993i	$-0.0900007\pi$
$608$		0			0
$609$	−35.4856	+	2.27553i	−1.43795	+	0.0922093i
$610$		0			0
$611$	58.8450			2.38062
$612$		0			0
$613$	−27.1115			−1.09502			−0.547511	−	0.836799i	$-0.684425\pi$
$613$	−27.1115			−1.09502			−0.547511	+	0.836799i	$0.684425\pi$
$614$		0			0
$615$	61.0192	−	3.91289i	2.46053	−	0.157783i
$616$		0			0
$617$			34.7987i			1.40094i	0.713681	+	0.700471i	$0.247027\pi$
$617$			34.7987i			1.40094i	−0.713681	+	0.700471i	$0.752973\pi$
$618$		0			0
$619$		−	13.8015i		−	0.554728i	−0.960765	−	0.277364i	$-0.910539\pi$
$619$		−	13.8015i		−	0.554728i	0.960765	−	0.277364i	$-0.0894608\pi$
$620$		0			0
$621$	14.1305	−	2.74856i	0.567039	−	0.110296i
$622$		0			0
$623$	68.9139			2.76098
$624$		0			0
$625$	4.86097			0.194439
$626$		0			0
$627$	−0.0507851	−	0.791964i	−0.00202816	−	0.0316280i
$628$		0			0
$629$		−	26.0616i		−	1.03914i
$630$		0			0
$631$		−	30.5490i		−	1.21614i	−0.793885	−	0.608068i	$-0.791945\pi$
$631$		−	30.5490i		−	1.21614i	0.793885	−	0.608068i	$-0.208055\pi$
$632$		0			0
$633$	1.24191	+	19.3669i	0.0493616	+	0.769766i
$634$		0			0
$635$	16.0984			0.638847
$636$		0			0
$637$	−83.1939			−3.29626
$638$		0			0
$639$	−17.0096	+	2.19051i	−0.672891	+	0.0866553i
$640$		0			0
$641$		−	23.6474i		−	0.934014i	−0.884254	−	0.467007i	$-0.845332\pi$
$641$		−	23.6474i		−	0.934014i	0.884254	−	0.467007i	$-0.154668\pi$
$642$		0			0
$643$			0.248940i			0.00981723i	0.999988	+	0.00490862i	$0.00156247\pi$
$643$			0.248940i			0.00981723i	−0.999988	+	0.00490862i	$0.998438\pi$
$644$		0			0
$645$	−40.7343	+	2.61210i	−1.60391	+	0.102851i
$646$		0			0
$647$	2.50401			0.0984429			0.0492215	−	0.998788i	$-0.484326\pi$
$647$	2.50401			0.0984429			0.0492215	+	0.998788i	$0.484326\pi$
$648$		0			0
$649$	0.924665			0.0362963
$650$		0			0
$651$	28.1103	−	1.80259i	1.10173	−	0.0706489i
$652$		0			0
$653$			2.07925i			0.0813672i	0.999172	+	0.0406836i	$0.0129536\pi$
$653$			2.07925i			0.0813672i	−0.999172	+	0.0406836i	$0.987046\pi$
$654$		0			0
$655$		−	20.2872i		−	0.792686i
$656$		0			0
$657$	29.6122	−	3.81348i	1.15528	−	0.148778i
$658$		0			0
$659$	−44.2887			−1.72524			−0.862622	−	0.505849i	$-0.831179\pi$
$659$	−44.2887			−1.72524			−0.862622	+	0.505849i	$0.831179\pi$
$660$		0			0
$661$	−21.5567			−0.838458			−0.419229	−	0.907881i	$-0.637700\pi$
$661$	−21.5567			−0.838458			−0.419229	+	0.907881i	$0.637700\pi$
$662$		0			0
$663$	2.15657	+	33.6305i	0.0837543	+	1.30610i
$664$		0			0
$665$		−	17.9087i		−	0.694470i
$666$		0			0
$667$		−	11.6728i		−	0.451973i
$668$		0			0
$669$	−2.26716	−	35.3551i	−0.0876536	−	1.36691i
$670$		0			0
$671$	−3.03945			−0.117337
$672$		0			0
$673$	2.09131			0.0806139			0.0403069	−	0.999187i	$-0.487166\pi$
$673$	2.09131			0.0806139			0.0403069	+	0.999187i	$0.487166\pi$
$674$		0			0
$675$	43.4019	−	8.44220i	1.67054	−	0.324940i
$676$		0			0
$677$		−	31.8723i		−	1.22495i	−0.790489	−	0.612476i	$-0.790174\pi$
$677$		−	31.8723i		−	1.22495i	0.790489	−	0.612476i	$-0.209826\pi$
$678$		0			0
$679$			61.6727i			2.36678i
$680$		0			0
$681$	−2.29356	+	0.147075i	−0.0878893	+	0.00563594i
$682$		0			0
$683$	−24.9455			−0.954512			−0.477256	−	0.878764i	$-0.658369\pi$
$683$	−24.9455			−0.954512			−0.477256	+	0.878764i	$0.658369\pi$
$684$		0			0
$685$	12.9843			0.496105
$686$		0			0
$687$	−22.0035	+	1.41098i	−0.839485	+	0.0538324i
$688$		0			0
$689$			19.6412i			0.748270i
$690$		0			0
$691$		−	9.21814i		−	0.350674i	−0.984508	−	0.175337i	$-0.943898\pi$
$691$		−	9.21814i		−	0.350674i	0.984508	−	0.175337i	$-0.0561016\pi$
$692$		0			0
$693$	−0.855430	−	6.64255i	−0.0324951	−	0.252329i
$694$		0			0
$695$	−7.24299			−0.274742
$696$		0			0
$697$	−37.6039			−1.42435
$698$		0			0
$699$	1.90295	+	29.6754i	0.0719761	+	1.12243i
$700$		0			0
$701$			4.80868i			0.181621i	0.995868	+	0.0908106i	$0.0289458\pi$
$701$			4.80868i			0.181621i	−0.995868	+	0.0908106i	$0.971054\pi$
$702$		0			0
$703$			6.65655i			0.251057i
$704$		0			0
$705$	4.82407	+	75.2286i	0.181685	+	2.83327i
$706$		0			0
$707$	37.8131			1.42211
$708$		0			0
$709$	34.2498			1.28628			0.643139	−	0.765750i	$-0.277632\pi$
$709$	34.2498			1.28628			0.643139	+	0.765750i	$0.277632\pi$
$710$		0			0
$711$	2.41361	+	18.7421i	0.0905175	+	0.702882i
$712$		0			0
$713$			9.24672i			0.346292i
$714$		0			0
$715$		−	8.36880i		−	0.312976i
$716$		0			0
$717$	−24.8256	+	1.59195i	−0.927128	+	0.0594525i
$718$		0			0
$719$	−14.7212			−0.549009			−0.274504	−	0.961586i	$-0.588514\pi$
$719$	−14.7212			−0.549009			−0.274504	+	0.961586i	$0.588514\pi$
$720$		0			0
$721$	−43.4873			−1.61955
$722$		0			0
$723$	−14.0312	+	0.899759i	−0.521827	+	0.0334624i
$724$		0			0
$725$		−	35.8530i		−	1.33155i
$726$		0			0
$727$		−	9.15451i		−	0.339522i	−0.985485	−	0.169761i	$-0.945700\pi$
$727$		−	9.15451i		−	0.339522i	0.985485	−	0.169761i	$-0.0542996\pi$
$728$		0			0
$729$	25.0314	−	10.1207i	0.927089	−	0.374842i
$730$		0			0
$731$	25.1031			0.928470
$732$		0			0
$733$	47.7844			1.76496			0.882479	−	0.470353i	$-0.155873\pi$
$733$	47.7844			1.76496			0.882479	+	0.470353i	$0.155873\pi$
$734$		0			0
$735$	−6.82017	−	106.357i	−0.251566	−	3.92302i
$736$		0			0
$737$		−	0.463274i		−	0.0170649i
$738$		0			0
$739$			33.8896i			1.24665i	0.781963	+	0.623325i	$0.214219\pi$
$739$			33.8896i			1.24665i	−0.781963	+	0.623325i	$0.785781\pi$
$740$		0			0
$741$	−0.550823	−	8.58977i	−0.0202350	−	0.315553i
$742$		0			0
$743$	16.8773			0.619169			0.309584	−	0.950872i	$-0.399810\pi$
$743$	16.8773			0.619169			0.309584	+	0.950872i	$0.399810\pi$
$744$		0			0
$745$	36.1397			1.32406
$746$		0			0
$747$	−34.7408	+	4.47394i	−1.27110	+	0.163693i
$748$		0			0
$749$		−	90.6397i		−	3.31190i
$750$		0			0
$751$			46.3239i			1.69038i	0.534465	+	0.845191i	$0.320513\pi$
$751$			46.3239i			1.69038i	−0.534465	+	0.845191i	$0.679487\pi$
$752$		0			0
$753$	−6.84586	+	0.438994i	−0.249477	+	0.0159978i
$754$		0			0
$755$	−32.7452			−1.19172
$756$		0			0
$757$	−20.1404			−0.732014			−0.366007	−	0.930612i	$-0.619275\pi$
$757$	−20.1404			−0.732014			−0.366007	+	0.930612i	$0.619275\pi$
$758$		0			0
$759$	2.19405	−	0.140695i	0.0796390	−	0.00510689i
$760$		0			0
$761$			23.5191i			0.852565i	0.904590	+	0.426283i	$0.140177\pi$
$761$			23.5191i			0.852565i	−0.904590	+	0.426283i	$0.859823\pi$
$762$		0			0
$763$			37.6968i			1.36471i
$764$		0			0
$765$	−42.8171	+	5.51400i	−1.54805	+	0.199359i
$766$		0			0
$767$	10.0291			0.362128
$768$		0			0
$769$	9.43505			0.340237			0.170118	−	0.985424i	$-0.445585\pi$
$769$	9.43505			0.340237			0.170118	+	0.985424i	$0.445585\pi$
$770$		0			0
$771$	1.34294	+	20.9424i	0.0483648	+	0.754222i
$772$		0			0
$773$			30.5433i			1.09857i	0.835636	+	0.549283i	$0.185099\pi$
$773$			30.5433i			1.09857i	−0.835636	+	0.549283i	$0.814901\pi$
$774$		0			0
$775$			28.4013i			1.02020i
$776$		0			0
$777$	3.59500	+	56.0619i	0.128970	+	2.01121i
$778$		0			0
$779$	9.60465			0.344122
$780$		0			0
$781$	−2.61928			−0.0937252
$782$		0			0
$783$	−4.18022	−	21.4908i	−0.149389	−	0.768017i
$784$		0			0
$785$			2.20583i			0.0787294i
$786$		0			0
$787$			8.32720i			0.296833i	0.988925	+	0.148416i	$0.0474175\pi$
$787$			8.32720i			0.296833i	−0.988925	+	0.148416i	$0.952582\pi$
$788$		0			0
$789$	−41.1328	+	2.63766i	−1.46437	+	0.0939031i
$790$		0			0
$791$	49.2097			1.74970
$792$		0			0
$793$	−32.9663			−1.17067
$794$		0			0
$795$	−25.1097	+	1.61017i	−0.890549	+	0.0571069i
$796$		0			0
$797$			4.69531i			0.166317i	0.996536	+	0.0831583i	$0.0265007\pi$
$797$			4.69531i			0.166317i	−0.996536	+	0.0831583i	$0.973499\pi$
$798$		0			0
$799$		−	46.3607i		−	1.64012i
$800$		0			0
$801$	5.41948	+	42.0831i	0.191488	+	1.48693i
$802$		0			0
$803$	4.55993			0.160916
$804$		0			0
$805$	49.6141			1.74867
$806$		0			0
$807$	−1.08103	−	16.8580i	−0.0380539	−	0.593429i
$808$		0			0
$809$			29.1013i			1.02315i	0.859239	+	0.511574i	$0.170937\pi$
$809$			29.1013i			1.02315i	−0.859239	+	0.511574i	$0.829063\pi$
$810$		0			0
$811$			46.6250i			1.63722i	0.574347	+	0.818612i	$0.305256\pi$
$811$			46.6250i			1.63722i	−0.574347	+	0.818612i	$0.694744\pi$
$812$		0			0
$813$	1.33792	+	20.8641i	0.0469229	+	0.731736i
$814$		0			0
$815$	−5.24169			−0.183608
$816$		0			0
$817$	−6.41172			−0.224318
$818$		0			0
$819$	−9.27813	−	72.0461i	−0.324204	−	2.51749i
$820$		0			0
$821$			47.8969i			1.67161i	0.549024	+	0.835806i	$0.314999\pi$
$821$			47.8969i			1.67161i	−0.549024	+	0.835806i	$0.685001\pi$
$822$		0			0
$823$			20.6418i			0.719528i	0.933043	+	0.359764i	$0.117143\pi$
$823$			20.6418i			0.719528i	−0.933043	+	0.359764i	$0.882857\pi$
$824$		0			0
$825$	6.73902	−	0.432143i	0.234622	−	0.0150453i
$826$		0			0
$827$	27.9435			0.971691			0.485845	−	0.874045i	$-0.338512\pi$
$827$	27.9435			0.971691			0.485845	+	0.874045i	$0.338512\pi$
$828$		0			0
$829$	4.10149			0.142451			0.0712254	−	0.997460i	$-0.477309\pi$
$829$	4.10149			0.142451			0.0712254	+	0.997460i	$0.477309\pi$
$830$		0			0
$831$	50.2100	−	3.21974i	1.74177	−	0.111692i
$832$		0			0
$833$			65.5438i			2.27096i
$834$		0			0
$835$		−	41.8232i		−	1.44735i
$836$		0			0
$837$	3.31140	+	17.0241i	0.114459	+	0.588440i
$838$		0			0
$839$	11.4642			0.395789			0.197894	−	0.980223i	$-0.436590\pi$
$839$	11.4642			0.395789			0.197894	+	0.980223i	$0.436590\pi$
$840$		0			0
$841$	11.2471			0.387832
$842$		0			0
$843$	2.77652	+	43.2982i	0.0956284	+	1.49127i
$844$		0			0
$845$		−	42.9879i		−	1.47883i
$846$		0			0
$847$			52.5743i			1.80647i
$848$		0			0
$849$	0.448253	+	6.99025i	0.0153840	+	0.239905i
$850$		0			0
$851$	−18.4412			−0.632158
$852$		0			0
$853$	−7.34278			−0.251412			−0.125706	−	0.992068i	$-0.540120\pi$
$853$	−7.34278			−0.251412			−0.125706	+	0.992068i	$0.540120\pi$
$854$		0			0
$855$	10.9362	−	1.40836i	0.374009	−	0.0481651i
$856$		0			0
$857$			36.6827i			1.25306i	0.779398	+	0.626529i	$0.215525\pi$
$857$			36.6827i			1.25306i	−0.779398	+	0.626529i	$0.784475\pi$
$858$		0			0
$859$		−	21.4036i		−	0.730283i	−0.930952	−	0.365141i	$-0.881021\pi$
$859$		−	21.4036i		−	0.730283i	0.930952	−	0.365141i	$-0.118979\pi$
$860$		0			0
$861$	80.8909	−	5.18717i	2.75676	−	0.176778i
$862$		0			0
$863$	43.1955			1.47039			0.735196	−	0.677855i	$-0.237090\pi$
$863$	43.1955			1.47039			0.735196	+	0.677855i	$0.237090\pi$
$864$		0			0
$865$	36.7759			1.25042
$866$		0			0
$867$	−2.88894	+	0.185255i	−0.0981136	+	0.00629158i
$868$		0			0
$869$			2.88605i			0.0979025i
$870$		0			0
$871$		−	5.02474i		−	0.170257i
$872$		0			0
$873$	−37.6611	+	4.85002i	−1.27464	+	0.164148i
$874$		0			0
$875$	62.8460			2.12458
$876$		0			0
$877$	35.5936			1.20191			0.600955	−	0.799283i	$-0.294787\pi$
$877$	35.5936			1.20191			0.600955	+	0.799283i	$0.294787\pi$
$878$		0			0
$879$	−1.34788	−	21.0195i	−0.0454630	−	0.708970i
$880$		0			0
$881$		−	1.10230i		−	0.0371376i	−0.999828	−	0.0185688i	$-0.994089\pi$
$881$		−	1.10230i		−	0.0371376i	0.999828	−	0.0185688i	$-0.00591097\pi$
$882$		0			0
$883$		−	18.7841i		−	0.632134i	−0.948737	−	0.316067i	$-0.897638\pi$
$883$		−	18.7841i		−	0.632134i	0.948737	−	0.316067i	$-0.102362\pi$
$884$		0			0
$885$	0.822174	+	12.8213i	0.0276371	+	0.430984i
$886$		0			0
$887$	24.5227			0.823391			0.411696	−	0.911321i	$-0.364937\pi$
$887$	24.5227			0.823391			0.411696	+	0.911321i	$0.364937\pi$
$888$		0			0
$889$	21.3411			0.715758
$890$		0			0
$891$	3.98908	−	1.04476i	0.133639	−	0.0350007i
$892$		0			0
$893$			11.8413i			0.396253i
$894$		0			0
$895$			6.89470i			0.230464i
$896$		0			0
$897$	23.7970	−	1.52599i	0.794559	−	0.0509515i
$898$		0			0
$899$	14.0631			0.469031
$900$		0			0
$901$	15.4742			0.515520
$902$		0			0
$903$	−53.9999	+	3.46277i	−1.79701	+	0.115234i
$904$		0			0
$905$		−	28.6158i		−	0.951223i
$906$		0			0
$907$		−	25.1303i		−	0.834437i	−0.908806	−	0.417218i	$-0.863005\pi$
$907$		−	25.1303i		−	0.834437i	0.908806	−	0.417218i	$-0.136995\pi$
$908$		0			0
$909$	2.97367	+	23.0910i	0.0986304	+	0.765880i
$910$		0			0
$911$	40.0888			1.32820			0.664100	−	0.747644i	$-0.268815\pi$
$911$	40.0888			1.32820			0.664100	+	0.747644i	$0.268815\pi$
$912$		0			0
$913$	−5.34967			−0.177048
$914$		0			0
$915$	−2.70256	−	42.1448i	−0.0893437	−	1.39326i
$916$		0			0
$917$		−	26.8940i		−	0.888117i
$918$		0			0
$919$			53.3270i			1.75909i	0.475811	+	0.879547i	$0.342154\pi$
$919$			53.3270i			1.75909i	−0.475811	+	0.879547i	$0.657846\pi$
$920$		0			0
$921$	−1.13405	−	17.6849i	−0.0373683	−	0.582737i
$922$		0			0
$923$	−28.4091			−0.935097
$924$		0			0
$925$	−56.6422			−1.86238
$926$		0			0
$927$	−3.41990	−	26.5561i	−0.112324	−	0.872215i
$928$		0			0
$929$			9.39055i			0.308094i	0.988064	+	0.154047i	$0.0492307\pi$
$929$			9.39055i			0.308094i	−0.988064	+	0.154047i	$0.950769\pi$
$930$		0			0
$931$		−	16.7409i		−	0.548662i
$932$		0			0
$933$	2.54214	−	0.163016i	0.0832260	−	0.00533691i
$934$		0			0
$935$	−6.59331			−0.215624
$936$		0			0
$937$	−14.7801			−0.482846			−0.241423	−	0.970420i	$-0.577614\pi$
$937$	−14.7801			−0.482846			−0.241423	+	0.970420i	$0.577614\pi$
$938$		0			0
$939$	27.8442	−	1.78552i	0.908661	−	0.0582683i
$940$		0			0
$941$		−	30.3640i		−	0.989837i	−0.868939	−	0.494919i	$-0.835198\pi$
$941$		−	30.3640i		−	0.989837i	0.868939	−	0.494919i	$-0.164802\pi$
$942$		0			0
$943$			26.6086i			0.866496i
$944$		0			0
$945$	91.3444	−	17.7676i	2.97143	−	0.577980i
$946$		0			0
$947$	3.93072			0.127731			0.0638656	−	0.997959i	$-0.479657\pi$
$947$	3.93072			0.127731			0.0638656	+	0.997959i	$0.479657\pi$
$948$		0			0
$949$	49.4577			1.60546
$950$		0			0
$951$	2.70902	+	42.2457i	0.0878461	+	1.36991i
$952$		0			0
$953$		−	46.9232i		−	1.51999i	−0.649929	−	0.759995i	$-0.725201\pi$
$953$		−	46.9232i		−	1.51999i	0.649929	−	0.759995i	$-0.274799\pi$
$954$		0			0
$955$		−	73.6577i		−	2.38351i
$956$		0			0
$957$	−0.213979	−	3.33687i	−0.00691695	−	0.107866i
$958$		0			0
$959$	17.2128			0.555831
$960$		0			0
$961$	19.8598			0.640638
$962$		0			0
$963$	55.3502	−	7.12803i	1.78364	−	0.229697i
$964$		0			0
$965$		−	41.5388i		−	1.33718i
$966$		0			0
$967$		−	6.63554i		−	0.213385i	−0.994292	−	0.106692i	$-0.965974\pi$
$967$		−	6.63554i		−	0.213385i	0.994292	−	0.106692i	$-0.0340260\pi$
$968$		0			0
$969$	−6.76739	+	0.433962i	−0.217400	+	0.0139409i
$970$		0			0
$971$	−50.1573			−1.60962			−0.804812	−	0.593530i	$-0.797734\pi$
$971$	−50.1573			−1.60962			−0.804812	+	0.593530i	$0.797734\pi$
$972$		0			0
$973$	−9.60177			−0.307819
$974$		0			0
$975$	73.0924	−	4.68708i	2.34083	−	0.150107i
$976$		0			0
$977$			20.3448i			0.650889i	0.945561	+	0.325444i	$0.105514\pi$
$977$			20.3448i			0.650889i	−0.945561	+	0.325444i	$0.894486\pi$
$978$		0			0
$979$			6.48028i			0.207111i
$980$		0			0
$981$	−23.0200	+	2.96452i	−0.734971	+	0.0946499i
$982$		0			0
$983$	52.2578			1.66676			0.833382	−	0.552698i	$-0.186402\pi$
$983$	52.2578			1.66676			0.833382	+	0.552698i	$0.186402\pi$
$984$		0			0
$985$	−51.2815			−1.63396
$986$		0			0
$987$	6.39510	+	99.7279i	0.203558	+	3.17437i
$988$		0			0
$989$		−	17.7630i		−	0.564830i
$990$		0			0
$991$			6.47055i			0.205544i	0.994705	+	0.102772i	$0.0327712\pi$
$991$			6.47055i			0.205544i	−0.994705	+	0.102772i	$0.967229\pi$
$992$		0			0
$993$	−2.48893	−	38.8135i	−0.0789839	−	1.23171i
$994$		0			0
$995$	−60.0790			−1.90463
$996$		0			0
$997$	−13.1233			−0.415619			−0.207810	−	0.978169i	$-0.566633\pi$
$997$	−13.1233			−0.415619			−0.207810	+	0.978169i	$0.566633\pi$
$998$		0			0
$999$	−33.9521	+	6.60411i	−1.07420	+	0.208945i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	912.2.d.b.191.2	yes	24
3.2	odd	2	inner	912.2.d.b.191.24	yes	24
4.3	odd	2	inner	912.2.d.b.191.23	yes	24
12.11	even	2	inner	912.2.d.b.191.1	✓	24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
912.2.d.b.191.1	✓	24	12.11	even	2	inner
912.2.d.b.191.2	yes	24	1.1	even	1	trivial
912.2.d.b.191.23	yes	24	4.3	odd	2	inner
912.2.d.b.191.24	yes	24	3.2	odd	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:	\( N \)	\(=\)	\( 912 = 2^{4} \cdot 3 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	912.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(-1.72850 + 0.110841i) q^{3} -3.67549i q^{5} -4.87247i q^{7} +(2.97543 - 0.383177i) q^{9} +O(q^{10})\)	\(q+(-1.72850 + 0.110841i) q^{3} -3.67549i q^{5} -4.87247i q^{7} +(2.97543 - 0.383177i) q^{9} +0.458180 q^{11} +4.96949 q^{13} +(0.407395 + 6.35309i) q^{15} -3.91518i q^{17} +1.00000i q^{19} +(0.540069 + 8.42206i) q^{21} -2.77039 q^{23} -8.50924 q^{25} +(-5.10056 + 0.992121i) q^{27} +4.21341i q^{29} -3.33770i q^{31} +(-0.791964 + 0.0507851i) q^{33} -17.9087 q^{35} +6.65655 q^{37} +(-8.58977 + 0.550823i) q^{39} -9.60465i q^{41} +6.41172i q^{43} +(-1.40836 - 10.9362i) q^{45} +11.8413 q^{47} -16.7409 q^{49} +(0.433962 + 6.76739i) q^{51} +3.95236i q^{53} -1.68404i q^{55} +(-0.110841 - 1.72850i) q^{57} +2.01813 q^{59} -6.63375 q^{61} +(-1.86702 - 14.4977i) q^{63} -18.2653i q^{65} -1.01112i q^{67} +(4.78862 - 0.307073i) q^{69} -5.71670 q^{71} +9.95226 q^{73} +(14.7082 - 0.943172i) q^{75} -2.23247i q^{77} +6.29894i q^{79} +(8.70635 - 2.28023i) q^{81} -11.6759 q^{83} -14.3902 q^{85} +(-0.467019 - 7.28289i) q^{87} +14.1435i q^{89} -24.2137i q^{91} +(0.369953 + 5.76921i) q^{93} +3.67549 q^{95} -12.6574 q^{97} +(1.36328 - 0.175564i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q + 4 q^{9}+O(q^{10}) \) 24 * q + 4 * q^9	\( 24 q + 4 q^{9} - 12 q^{21} - 64 q^{25} + 12 q^{33} + 64 q^{37} - 16 q^{45} - 8 q^{49} - 24 q^{61} - 8 q^{69} - 16 q^{73} - 4 q^{81} - 8 q^{85} + 32 q^{93} - 8 q^{97}+O(q^{100}) \) 24 * q + 4 * q^9 - 12 * q^21 - 64 * q^25 + 12 * q^33 + 64 * q^37 - 16 * q^45 - 8 * q^49 - 24 * q^61 - 8 * q^69 - 16 * q^73 - 4 * q^81 - 8 * q^85 + 32 * q^93 - 8 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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