LMFDB - Embedded newform 124.2.d.c.123.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [124,2,Mod(123,124)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(124, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("124.123");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 124 = 2^{2} \cdot 31 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	124.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:	$0.990144985064$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.0.21717639.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - x^{3} + 8 $ x^6 - x^3 + 8
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label			123.1
Root			$-1.18073 + 0.778374i$ of defining polynomial
Character	$\chi$	$=$	124.123
Dual form			124.2.d.c.123.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.18073 - 0.778374i) q^{2} +(0.788267 + 1.83811i) q^{4} -3.93800 q^{5} -5.23296i q^{7} +(0.500000 - 2.78388i) q^{8} -3.00000 q^{9} +O(q^{10})$	\(q+(-1.18073 - 0.778374i) q^{2} +(0.788267 + 1.83811i) q^{4} -3.93800 q^{5} -5.23296i q^{7} +(0.500000 - 2.78388i) q^{8} -3.00000 q^{9} +(4.64974 + 3.06524i) q^{10} +(-4.07320 + 6.17874i) q^{14} +(-2.75727 + 2.89784i) q^{16} +(3.54220 + 2.33512i) q^{18} +0.994032i q^{19} +(-3.10420 - 7.23847i) q^{20} +10.5079 q^{25} +(9.61874 - 4.12497i) q^{28} -5.56776i q^{31} +(5.51120 - 1.27539i) q^{32} +20.6074i q^{35} +(-2.36480 - 5.51432i) q^{36} +(0.773729 - 1.17369i) q^{38} +(-1.96900 + 10.9629i) q^{40} +2.36814 q^{41} +11.8140 q^{45} -11.1355i q^{47} -20.3839 q^{49} +(-12.4070 - 8.17906i) q^{50} +(-14.5679 - 2.61648i) q^{56} -11.4600i q^{59} +(-4.33380 + 6.57405i) q^{62} +15.6989i q^{63} +(-7.50000 - 2.78388i) q^{64} +11.1355i q^{67} +(16.0403 - 24.3319i) q^{70} +3.24490i q^{71} +(-1.50000 + 8.35165i) q^{72} +(-1.82714 + 0.783563i) q^{76} +(10.8581 - 11.4117i) q^{80} +9.00000 q^{81} +(-2.79614 - 1.84330i) q^{82} +(-13.9492 - 9.19572i) q^{90} +(-8.66761 + 13.1481i) q^{94} -3.91450i q^{95} -10.2441 q^{97} +(24.0679 + 15.8663i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 3 q^{8} - 18 q^{9}+O(q^{10}) $ 6 * q + 3 * q^8 - 18 * q^9	$ 6 q + 3 q^{8} - 18 q^{9} + 9 q^{10} - 15 q^{14} - 21 q^{20} + 30 q^{25} + 27 q^{28} + 33 q^{38} - 42 q^{49} - 39 q^{50} - 45 q^{64} - 3 q^{70} - 9 q^{72} + 15 q^{76} + 51 q^{80} + 54 q^{81} + 21 q^{82} - 27 q^{90} + 57 q^{98}+O(q^{100}) $ 6 * q + 3 * q^8 - 18 * q^9 + 9 * q^10 - 15 * q^14 - 21 * q^20 + 30 * q^25 + 27 * q^28 + 33 * q^38 - 42 * q^49 - 39 * q^50 - 45 * q^64 - 3 * q^70 - 9 * q^72 + 15 * q^76 + 51 * q^80 + 54 * q^81 + 21 * q^82 - 27 * q^90 + 57 * q^98

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/124\mathbb{Z}\right)^\times$.

$n$	$63$	$65$
$\chi(n)$	$-1$	$-1$

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 $

$2$	−1.18073	−	0.778374i	−0.834905	−	0.550394i
$2$	−1.18073	−	0.778374i	−0.834905	−	0.550394i
$3$		0			0			−	1.00000i	$-0.5\pi$
$3$		0			0				1.00000i	$0.5\pi$
$4$	0.788267	+	1.83811i	0.394134	+	0.919053i
$5$	−3.93800			−1.76113			−0.880564	−	0.473927i	$-0.842836\pi$
$5$	−3.93800			−1.76113			−0.880564	+	0.473927i	$0.842836\pi$
$6$		0			0
$7$		−	5.23296i		−	1.97787i	−0.148339	−	0.988937i	$-0.547393\pi$
$7$		−	5.23296i		−	1.97787i	0.148339	−	0.988937i	$-0.452607\pi$
$8$	0.500000	−	2.78388i	0.176777	−	0.984251i
$9$	−3.00000			−1.00000
$10$	4.64974	+	3.06524i	1.47038	+	0.969314i
$11$		0			0			−	1.00000i	$-0.5\pi$
$11$		0			0				1.00000i	$0.5\pi$
$12$		0			0
$13$		0			0		1.00000			$0$
$13$		0			0		−1.00000			$\pi$
$14$	−4.07320	+	6.17874i	−1.08861	+	1.65134i
$15$		0			0
$16$	−2.75727	+	2.89784i	−0.689317	+	0.724460i
$17$		0			0		1.00000			$0$
$17$		0			0		−1.00000			$\pi$
$18$	3.54220	+	2.33512i	0.834905	+	0.550394i
$19$			0.994032i			0.228047i	0.993478	+	0.114023i	$0.0363739\pi$
$19$			0.994032i			0.228047i	−0.993478	+	0.114023i	$0.963626\pi$
$20$	−3.10420	−	7.23847i	−0.694120	−	1.61857i
$21$		0			0
$22$		0			0
$23$		0			0			−	1.00000i	$-0.5\pi$
$23$		0			0				1.00000i	$0.5\pi$
$24$		0			0
$25$	10.5079			2.10157
$26$		0			0
$27$		0			0
$28$	9.61874	−	4.12497i	1.81777	−	0.779546i
$29$		0			0		1.00000			$0$
$29$		0			0		−1.00000			$\pi$
$30$		0			0
$31$		−	5.56776i		−	1.00000i
$31$		−	5.56776i		−	1.00000i
$32$	5.51120	−	1.27539i	0.974253	−	0.225459i
$33$		0			0
$34$		0			0
$35$			20.6074i			3.48329i
$36$	−2.36480	−	5.51432i	−0.394134	−	0.919053i
$37$		0			0		1.00000			$0$
$37$		0			0		−1.00000			$\pi$
$38$	0.773729	−	1.17369i	0.125515	−	0.190397i
$39$		0			0
$40$	−1.96900	+	10.9629i	−0.311327	+	1.73339i
$41$	2.36814			0.369841			0.184920	−	0.982754i	$-0.440797\pi$
$41$	2.36814			0.369841			0.184920	+	0.982754i	$0.440797\pi$
$42$		0			0
$43$		0			0			−	1.00000i	$-0.5\pi$
$43$		0			0				1.00000i	$0.5\pi$
$44$		0			0
$45$	11.8140			1.76113
$46$		0			0
$47$		−	11.1355i		−	1.62428i	−0.583460	−	0.812142i	$-0.698301\pi$
$47$		−	11.1355i		−	1.62428i	0.583460	−	0.812142i	$-0.301699\pi$
$48$		0			0
$49$	−20.3839			−2.91198
$50$	−12.4070	−	8.17906i	−1.75462	−	1.15669i
$51$		0			0
$52$		0			0
$53$		0			0		1.00000			$0$
$53$		0			0		−1.00000			$\pi$
$54$		0			0
$55$		0			0
$56$	−14.5679	−	2.61648i	−1.94672	−	0.349642i
$57$		0			0
$58$		0			0
$59$		−	11.4600i		−	1.49196i	−0.665969	−	0.745979i	$-0.731982\pi$
$59$		−	11.4600i		−	1.49196i	0.665969	−	0.745979i	$-0.268018\pi$
$60$		0			0
$61$		0			0		1.00000			$0$
$61$		0			0		−1.00000			$\pi$
$62$	−4.33380	+	6.57405i	−0.550394	+	0.834905i
$63$			15.6989i			1.97787i
$64$	−7.50000	−	2.78388i	−0.937500	−	0.347985i
$65$		0			0
$66$		0			0
$67$			11.1355i			1.36042i	0.733017	+	0.680211i	$0.238112\pi$
$67$			11.1355i			1.36042i	−0.733017	+	0.680211i	$0.761888\pi$
$68$		0			0
$69$		0			0
$70$	16.0403	−	24.3319i	1.91718	−	2.90822i
$71$			3.24490i			0.385098i	0.981287	+	0.192549i	$0.0616755\pi$
$71$			3.24490i			0.385098i	−0.981287	+	0.192549i	$0.938325\pi$
$72$	−1.50000	+	8.35165i	−0.176777	+	0.984251i
$73$		0			0		1.00000			$0$
$73$		0			0		−1.00000			$\pi$
$74$		0			0
$75$		0			0
$76$	−1.82714	+	0.783563i	−0.209587	+	0.0898809i
$77$		0			0
$78$		0			0
$79$		0			0			−	1.00000i	$-0.5\pi$
$79$		0			0				1.00000i	$0.5\pi$
$80$	10.8581	−	11.4117i	1.21398	−	1.27587i
$81$	9.00000			1.00000
$82$	−2.79614	−	1.84330i	−0.308782	−	0.203558i
$83$		0			0			−	1.00000i	$-0.5\pi$
$83$		0			0				1.00000i	$0.5\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$		0			0		1.00000			$0$
$89$		0			0		−1.00000			$\pi$
$90$	−13.9492	−	9.19572i	−1.47038	−	0.969314i
$91$		0			0
$92$		0			0
$93$		0			0
$94$	−8.66761	+	13.1481i	−0.893995	+	1.35612i
$95$		−	3.91450i		−	0.401620i
$96$		0			0
$97$	−10.2441			−1.04014			−0.520068	−	0.854125i	$-0.674093\pi$
$97$	−10.2441			−1.04014			−0.520068	+	0.854125i	$0.674093\pi$
$98$	24.0679	+	15.8663i	2.43123	+	1.60274i
$99$		0			0
$100$	8.28301	+	19.3146i	0.828301	+	1.93146i
$101$	−7.07774			−0.704261			−0.352131	−	0.935951i	$-0.614543\pi$
$101$	−7.07774			−0.704261			−0.352131	+	0.935951i	$0.614543\pi$
$102$		0			0
$103$		−	13.7108i		−	1.35097i	−0.737375	−	0.675483i	$-0.763935\pi$
$103$		−	13.7108i		−	1.35097i	0.737375	−	0.675483i	$-0.236065\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		−	19.9378i		−	1.92746i	−0.266878	−	0.963730i	$-0.585992\pi$
$107$		−	19.9378i		−	1.92746i	0.266878	−	0.963730i	$-0.414008\pi$
$108$		0			0
$109$	14.9537			1.43231			0.716155	−	0.697942i	$-0.245901\pi$
$109$	14.9537			1.43231			0.716155	+	0.697942i	$0.245901\pi$
$110$		0			0
$111$		0			0
$112$	15.1643	+	14.4287i	1.43289	+	1.36338i
$113$	21.2599			1.99996			0.999981	−	0.00618051i	$-0.00196733\pi$
$113$	21.2599			1.99996			0.999981	+	0.00618051i	$0.00196733\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$		0			0
$118$	−8.92013	+	13.5312i	−0.821165	+	1.24564i
$119$		0			0
$120$		0			0
$121$	−11.0000			−1.00000
$122$		0			0
$123$		0			0
$124$	10.2341	−	4.38889i	0.919053	−	0.394134i
$125$	−21.6900			−1.94001
$126$	12.2196	−	18.5362i	1.08861	−	1.65134i
$127$		0			0			−	1.00000i	$-0.5\pi$
$127$		0			0				1.00000i	$0.5\pi$
$128$	6.68861	+	9.12483i	0.591195	+	0.806529i
$129$		0			0
$130$		0			0
$131$			11.1355i			0.972916i	0.873704	+	0.486458i	$0.161711\pi$
$131$			11.1355i			0.972916i	−0.873704	+	0.486458i	$0.838289\pi$
$132$		0			0
$133$	5.20173			0.451047
$134$	8.66761	−	13.1481i	0.748767	−	1.13582i
$135$		0			0
$136$		0			0
$137$		0			0		1.00000			$0$
$137$		0			0		−1.00000			$\pi$
$138$		0			0
$139$		0			0			−	1.00000i	$-0.5\pi$
$139$		0			0				1.00000i	$0.5\pi$
$140$	−37.8786	+	16.2442i	−3.20133	+	1.37288i
$141$		0			0
$142$	2.52574	−	3.83136i	0.211956	−	0.321521i
$143$		0			0
$144$	8.27181	−	8.69352i	0.689317	−	0.724460i
$145$		0			0
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−10.0000			−0.819232			−0.409616	−	0.912258i	$-0.634337\pi$
$149$	−10.0000			−0.819232			−0.409616	+	0.912258i	$0.634337\pi$
$150$		0			0
$151$		0			0			−	1.00000i	$-0.5\pi$
$151$		0			0				1.00000i	$0.5\pi$
$152$	2.76727	+	0.497016i	0.224455	+	0.0403133i
$153$		0			0
$154$		0			0
$155$			21.9259i			1.76113i
$156$		0			0
$157$	−22.8298			−1.82201			−0.911006	−	0.412392i	$-0.864693\pi$
$157$	−22.8298			−1.82201			−0.911006	+	0.412392i	$0.864693\pi$
$158$		0			0
$159$		0			0
$160$	−21.7031	+	5.02249i	−1.71578	+	0.397063i
$161$		0			0
$162$	−10.6266	−	7.00537i	−0.834905	−	0.550394i
$163$		−	7.48382i		−	0.586178i	−0.956085	−	0.293089i	$-0.905317\pi$
$163$		−	7.48382i		−	0.586178i	0.956085	−	0.293089i	$-0.0946833\pi$
$164$	1.86672	+	4.35288i	0.145767	+	0.339903i
$165$		0			0
$166$		0			0
$167$		0			0			−	1.00000i	$-0.5\pi$
$167$		0			0				1.00000i	$0.5\pi$
$168$		0			0
$169$	13.0000			1.00000
$170$		0			0
$171$		−	2.98210i		−	0.228047i
$172$		0			0
$173$	14.0000			1.06440			0.532200	−	0.846619i	$-0.321365\pi$
$173$	14.0000			1.06440			0.532200	+	0.846619i	$0.321365\pi$
$174$		0			0
$175$		−	54.9873i		−	4.15665i
$176$		0			0
$177$		0			0
$178$		0			0
$179$		0			0			−	1.00000i	$-0.5\pi$
$179$		0			0				1.00000i	$0.5\pi$
$180$	9.31260	+	21.7154i	0.694120	+	1.61857i
$181$		0			0		1.00000			$0$
$181$		0			0		−1.00000			$\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		0			0
$186$		0			0
$187$		0			0
$188$	20.4683	−	8.77777i	1.49280	−	0.640185i
$189$		0			0
$190$	−3.04695	+	4.62199i	−0.221049	+	0.335314i
$191$			24.1767i			1.74937i	0.484694	+	0.874684i	$0.338931\pi$
$191$			24.1767i			1.74937i	−0.484694	+	0.874684i	$0.661069\pi$
$192$		0			0
$193$	18.1201			1.30432			0.652158	−	0.758083i	$-0.273864\pi$
$193$	18.1201			1.30432			0.652158	+	0.758083i	$0.273864\pi$
$194$	12.0956	+	7.97378i	0.868414	+	0.572484i
$195$		0			0
$196$	−16.0679	−	37.4677i	−1.14771	−	2.67627i
$197$		0			0		1.00000			$0$
$197$		0			0		−1.00000			$\pi$
$198$		0			0
$199$		0			0			−	1.00000i	$-0.5\pi$
$199$		0			0				1.00000i	$0.5\pi$
$200$	5.25394	−	29.2527i	0.371509	−	2.06848i
$201$		0			0
$202$	8.35693	+	5.50913i	0.587992	+	0.387621i
$203$		0			0
$204$		0			0
$205$	−9.32573			−0.651337
$206$	−10.6721	+	16.1888i	−0.743564	+	1.12793i
$207$		0			0
$208$		0			0
$209$		0			0
$210$		0			0
$211$			17.9497i			1.23571i	0.786291	+	0.617856i	$0.211998\pi$
$211$			17.9497i			1.23571i	−0.786291	+	0.617856i	$0.788002\pi$
$212$		0			0
$213$		0			0
$214$	−15.5191	+	23.5413i	−1.06086	+	1.60925i
$215$		0			0
$216$		0			0
$217$	−29.1359			−1.97787
$218$	−17.6564	−	11.6396i	−1.19584	−	0.788334i
$219$		0			0
$220$		0			0
$221$		0			0
$222$		0			0
$223$		0			0			−	1.00000i	$-0.5\pi$
$223$		0			0				1.00000i	$0.5\pi$
$224$	−6.67407	−	28.8399i	−0.445930	−	1.92695i
$225$	−31.5236			−2.10157
$226$	−25.1023	−	16.5481i	−1.66978	−	1.10077i
$227$			11.1355i			0.739091i	0.929213	+	0.369546i	$0.120487\pi$
$227$			11.1355i			0.739091i	−0.929213	+	0.369546i	$0.879513\pi$
$228$		0			0
$229$		0			0		1.00000			$0$
$229$		0			0		−1.00000			$\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	8.64761			0.566524			0.283262	−	0.959043i	$-0.408583\pi$
$233$	8.64761			0.566524			0.283262	+	0.959043i	$0.408583\pi$
$234$		0			0
$235$			43.8518i			2.86057i
$236$	21.0646	−	9.03351i	1.37119	−	0.588031i
$237$		0			0
$238$		0			0
$239$		0			0			−	1.00000i	$-0.5\pi$
$239$		0			0				1.00000i	$0.5\pi$
$240$		0			0
$241$		0			0		1.00000			$0$
$241$		0			0		−1.00000			$\pi$
$242$	12.9881	+	8.56212i	0.834905	+	0.550394i
$243$		0			0
$244$		0			0
$245$	80.2718			5.12838
$246$		0			0
$247$		0			0
$248$	−15.5000	−	2.78388i	−0.984251	−	0.176777i
$249$		0			0
$250$	25.6101	+	16.8829i	1.61973	+	1.06777i
$251$		0			0			−	1.00000i	$-0.5\pi$
$251$		0			0				1.00000i	$0.5\pi$
$252$	−28.8562	+	12.3749i	−1.81777	+	0.779546i
$253$		0			0
$254$		0			0
$255$		0			0
$256$	−0.794934	−	15.9802i	−0.0496834	−	0.998765i
$257$	24.3996			1.52201			0.761003	−	0.648748i	$-0.224707\pi$
$257$	24.3996			1.52201			0.761003	+	0.648748i	$0.224707\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$		0			0
$262$	8.66761	−	13.1481i	0.535487	−	0.812292i
$263$		0			0			−	1.00000i	$-0.5\pi$
$263$		0			0				1.00000i	$0.5\pi$
$264$		0			0
$265$		0			0
$266$	−6.14186	−	4.04889i	−0.376582	−	0.248254i
$267$		0			0
$268$	−20.4683	+	8.77777i	−1.25030	+	0.536188i
$269$		0			0		1.00000			$0$
$269$		0			0		−1.00000			$\pi$
$270$		0			0
$271$		0			0			−	1.00000i	$-0.5\pi$
$271$		0			0				1.00000i	$0.5\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		0			0
$276$		0			0
$277$		0			0		1.00000			$0$
$277$		0			0		−1.00000			$\pi$
$278$		0			0
$279$			16.7033i			1.00000i
$280$	57.3686	+	10.3037i	3.42843	+	0.615764i
$281$	−0.771601			−0.0460298			−0.0230149	−	0.999735i	$-0.507327\pi$
$281$	−0.771601			−0.0460298			−0.0230149	+	0.999735i	$0.507327\pi$
$282$		0			0
$283$		−	33.4066i		−	1.98582i	−0.118888	−	0.992908i	$-0.537933\pi$
$283$		−	33.4066i		−	1.98582i	0.118888	−	0.992908i	$-0.462067\pi$
$284$	−5.96446	+	2.55785i	−0.353926	+	0.151780i
$285$		0			0
$286$		0			0
$287$		−	12.3924i		−	0.731498i
$288$	−16.5336	+	3.82617i	−0.974253	+	0.225459i
$289$	17.0000			1.00000
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−26.0000			−1.51894			−0.759468	−	0.650545i	$-0.774541\pi$
$293$	−26.0000			−1.51894			−0.759468	+	0.650545i	$0.774541\pi$
$294$		0			0
$295$			45.1293i			2.62753i
$296$		0			0
$297$		0			0
$298$	11.8073	+	7.78374i	0.683981	+	0.450900i
$299$		0			0
$300$		0			0
$301$		0			0
$302$		0			0
$303$		0			0
$304$	−2.88055	−	2.74081i	−0.165211	−	0.157197i
$305$		0			0
$306$		0			0
$307$		−	32.3918i		−	1.84870i	−0.381549	−	0.924349i	$-0.624609\pi$
$307$		−	32.3918i		−	1.84870i	0.381549	−	0.924349i	$-0.375391\pi$
$308$		0			0
$309$		0			0
$310$	17.0665	−	25.8886i	0.969314	−	1.47038i
$311$			28.1529i			1.59640i	0.602391	+	0.798201i	$0.294215\pi$
$311$			28.1529i			1.59640i	−0.602391	+	0.798201i	$0.705785\pi$
$312$		0			0
$313$		0			0		1.00000			$0$
$313$		0			0		−1.00000			$\pi$
$314$	26.9559	+	17.7701i	1.52121	+	1.00282i
$315$		−	61.8223i		−	3.48329i
$316$		0			0
$317$	−16.5503			−0.929556			−0.464778	−	0.885427i	$-0.653866\pi$
$317$	−16.5503			−0.929556			−0.464778	+	0.885427i	$0.653866\pi$
$318$		0			0
$319$		0			0
$320$	29.5350	+	10.9629i	1.65106	+	0.612847i
$321$		0			0
$322$		0			0
$323$		0			0
$324$	7.09441	+	16.5430i	0.394134	+	0.919053i
$325$		0			0
$326$	−5.82522	+	8.83641i	−0.322629	+	0.489403i
$327$		0			0
$328$	1.18407	−	6.59261i	0.0653792	−	0.364016i
$329$	−58.2718			−3.21263
$330$		0			0
$331$		0			0			−	1.00000i	$-0.5\pi$
$331$		0			0				1.00000i	$0.5\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		−	43.8518i		−	2.39588i
$336$		0			0
$337$		0			0		1.00000			$0$
$337$		0			0		−1.00000			$\pi$
$338$	−15.3495	−	10.1189i	−0.834905	−	0.550394i
$339$		0			0
$340$		0			0
$341$		0			0
$342$	−2.32119	+	3.52106i	−0.125515	+	0.190397i
$343$			70.0373i			3.78166i
$344$		0			0
$345$		0			0
$346$	−16.5303	−	10.8972i	−0.888673	−	0.585839i
$347$		0			0			−	1.00000i	$-0.5\pi$
$347$		0			0				1.00000i	$0.5\pi$
$348$		0			0
$349$	30.0000			1.60586			0.802932	−	0.596071i	$-0.203272\pi$
$349$	30.0000			1.60586			0.802932	+	0.596071i	$0.203272\pi$
$350$	−42.8007	+	64.9254i	−2.28779	+	3.47041i
$351$		0			0
$352$		0			0
$353$		0			0		1.00000			$0$
$353$		0			0		−1.00000			$\pi$
$354$		0			0
$355$		−	12.7784i		−	0.678208i
$356$		0			0
$357$		0			0
$358$		0			0
$359$		−	22.1887i		−	1.17107i	−0.810646	−	0.585537i	$-0.800884\pi$
$359$		−	22.1887i		−	1.17107i	0.810646	−	0.585537i	$-0.199116\pi$
$360$	5.90701	−	32.8888i	0.311327	−	1.73339i
$361$	18.0119			0.947995
$362$		0			0
$363$		0			0
$364$		0			0
$365$		0			0
$366$		0			0
$367$		0			0			−	1.00000i	$-0.5\pi$
$367$		0			0				1.00000i	$0.5\pi$
$368$		0			0
$369$	−7.10441			−0.369841
$370$		0			0
$371$		0			0
$372$		0			0
$373$	−38.5818			−1.99769			−0.998844	−	0.0480672i	$-0.984694\pi$
$373$	−38.5818			−1.99769			−0.998844	+	0.0480672i	$0.984694\pi$
$374$		0			0
$375$		0			0
$376$	−31.0000	−	5.56776i	−1.59870	−	0.287136i
$377$		0			0
$378$		0			0
$379$		−	33.4066i		−	1.71598i	−0.513665	−	0.857991i	$-0.671713\pi$
$379$		−	33.4066i		−	1.71598i	0.513665	−	0.857991i	$-0.328287\pi$
$380$	7.19527	−	3.08568i	0.369110	−	0.158292i
$381$		0			0
$382$	18.8185	−	28.5463i	0.962841	−	1.46056i
$383$		0			0			−	1.00000i	$-0.5\pi$
$383$		0			0				1.00000i	$0.5\pi$
$384$		0			0
$385$		0			0
$386$	−21.3951	−	14.1043i	−1.08898	−	0.717888i
$387$		0			0
$388$	−8.07512	−	18.8298i	−0.409952	−	0.955939i
$389$		0			0		1.00000			$0$
$389$		0			0		−1.00000			$\pi$
$390$		0			0
$391$		0			0
$392$	−10.1919	+	56.7463i	−0.514771	+	2.86612i
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$	18.0935			0.908086			0.454043	−	0.890980i	$-0.349981\pi$
$397$	18.0935			0.908086			0.454043	+	0.890980i	$0.349981\pi$
$398$		0			0
$399$		0			0
$400$	−28.9730	+	30.4501i	−1.44865	+	1.52251i
$401$		0			0		1.00000			$0$
$401$		0			0		−1.00000			$\pi$
$402$		0			0
$403$		0			0
$404$	−5.57915	−	13.0096i	−0.277573	−	0.647254i
$405$	−35.4420			−1.76113
$406$		0			0
$407$		0			0
$408$		0			0
$409$		0			0		1.00000			$0$
$409$		0			0		−1.00000			$\pi$
$410$	11.0112	+	7.25890i	0.543805	+	0.358492i
$411$		0			0
$412$	25.2019	−	10.8078i	1.24161	−	0.532462i
$413$	−59.9695			−2.95091
$414$		0			0
$415$		0			0
$416$		0			0
$417$		0			0
$418$		0			0
$419$		−	40.8697i		−	1.99661i	−0.0581712	−	0.998307i	$-0.518527\pi$
$419$		−	40.8697i		−	1.99661i	0.0581712	−	0.998307i	$-0.481473\pi$
$420$		0			0
$421$	−10.2175			−0.497969			−0.248985	−	0.968507i	$-0.580097\pi$
$421$	−10.2175			−0.497969			−0.248985	+	0.968507i	$0.580097\pi$
$422$	13.9716	−	21.1939i	0.680128	−	1.03170i
$423$			33.4066i			1.62428i
$424$		0			0
$425$		0			0
$426$		0			0
$427$		0			0
$428$	36.6478	−	15.7163i	1.77144	−	0.759677i
$429$		0			0
$430$		0			0
$431$		−	11.1355i		−	0.536380i	−0.963366	−	0.268190i	$-0.913575\pi$
$431$		−	11.1355i		−	0.536380i	0.963366	−	0.268190i	$-0.0864254\pi$
$432$		0			0
$433$		0			0		1.00000			$0$
$433$		0			0		−1.00000			$\pi$
$434$	34.4018	+	22.6786i	1.65134	+	1.08861i
$435$		0			0
$436$	11.7876	+	27.4866i	0.564521	+	1.31637i
$437$		0			0
$438$		0			0
$439$			19.6750i			0.939037i	0.882923	+	0.469519i	$0.155572\pi$
$439$			19.6750i			0.939037i	−0.882923	+	0.469519i	$0.844428\pi$
$440$		0			0
$441$	61.1516			2.91198
$442$		0			0
$443$			5.49576i			0.261111i	0.991441	+	0.130556i	$0.0416761\pi$
$443$			5.49576i			0.261111i	−0.991441	+	0.130556i	$0.958324\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$	−14.5679	+	39.2472i	−0.688271	+	1.85426i
$449$		0			0		1.00000			$0$
$449$		0			0		−1.00000			$\pi$
$450$	37.2210	+	24.5372i	1.75462	+	1.15669i
$451$		0			0
$452$	16.7585	+	39.0779i	0.788252	+	1.83807i
$453$		0			0
$454$	8.66761	−	13.1481i	0.406791	−	0.617071i
$455$		0			0
$456$		0			0
$457$		0			0		1.00000			$0$
$457$		0			0		−1.00000			$\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		0			0		1.00000			$0$
$461$		0			0		−1.00000			$\pi$
$462$		0			0
$463$		0			0			−	1.00000i	$-0.5\pi$
$463$		0			0				1.00000i	$0.5\pi$
$464$		0			0
$465$		0			0
$466$	−10.2105	−	6.73107i	−0.472994	−	0.311811i
$467$		−	15.9617i		−	0.738619i	−0.929307	−	0.369309i	$-0.879594\pi$
$467$		−	15.9617i		−	0.738619i	0.929307	−	0.369309i	$-0.120406\pi$
$468$		0			0
$469$	58.2718			2.69074
$470$	34.1331	−	51.7773i	1.57444	−	2.38831i
$471$		0			0
$472$	−31.9032	−	5.72998i	−1.46846	−	0.263744i
$473$		0			0
$474$		0			0
$475$			10.4452i			0.479257i
$476$		0			0
$477$		0			0
$478$		0			0
$479$			32.6546i			1.49203i	0.665931	+	0.746013i	$0.268034\pi$
$479$			32.6546i			1.49203i	−0.665931	+	0.746013i	$0.731966\pi$
$480$		0			0
$481$		0			0
$482$		0			0
$483$		0			0
$484$	−8.67094	−	20.2192i	−0.394134	−	0.919053i
$485$	40.3415			1.83181
$486$		0			0
$487$		0			0			−	1.00000i	$-0.5\pi$
$487$		0			0				1.00000i	$0.5\pi$
$488$		0			0
$489$		0			0
$490$	−94.7797	−	62.4815i	−4.28171	−	2.82263i
$491$		0			0			−	1.00000i	$-0.5\pi$
$491$		0			0				1.00000i	$0.5\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$	16.1345	+	15.3518i	0.724460	+	0.689317i
$497$	16.9804			0.761676
$498$		0			0
$499$		0			0			−	1.00000i	$-0.5\pi$
$499$		0			0				1.00000i	$0.5\pi$
$500$	−17.0975	−	39.8686i	−0.764625	−	1.78298i
$501$		0			0
$502$		0			0
$503$		−	38.6188i		−	1.72193i	−0.508667	−	0.860963i	$-0.669862\pi$
$503$		−	38.6188i		−	1.72193i	0.508667	−	0.860963i	$-0.330138\pi$
$504$	43.7038	+	7.84944i	1.94672	+	0.349642i
$505$	27.8722			1.24029
$506$		0			0
$507$		0			0
$508$		0			0
$509$		0			0		1.00000			$0$
$509$		0			0		−1.00000			$\pi$
$510$		0			0
$511$		0			0
$512$	−11.5000	+	19.4872i	−0.508233	+	0.861220i
$513$		0			0
$514$	−28.8095	−	18.9920i	−1.27073	−	0.837703i
$515$			53.9932i			2.37923i
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$	10.0000			0.438108			0.219054	−	0.975713i	$-0.429703\pi$
$521$	10.0000			0.438108			0.219054	+	0.975713i	$0.429703\pi$
$522$		0			0
$523$		0			0			−	1.00000i	$-0.5\pi$
$523$		0			0				1.00000i	$0.5\pi$
$524$	−20.4683	+	8.77777i	−0.894161	+	0.383459i
$525$		0			0
$526$		0			0
$527$		0			0
$528$		0			0
$529$	−23.0000			−1.00000
$530$		0			0
$531$			34.3799i			1.49196i
$532$	4.10036	+	9.56134i	0.177773	+	0.414537i
$533$		0			0
$534$		0			0
$535$			78.5152i			3.39451i
$536$	31.0000	+	5.56776i	1.33900	+	0.240491i
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	46.4578			1.99738			0.998688	−	0.0512107i	$-0.0163080\pi$
$541$	46.4578			1.99738			0.998688	+	0.0512107i	$0.0163080\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−58.8879			−2.52248
$546$		0			0
$547$			26.4276i			1.12996i	0.825104	+	0.564981i	$0.191117\pi$
$547$			26.4276i			1.12996i	−0.825104	+	0.564981i	$0.808883\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		0			0
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$		0			0
$557$		0			0		1.00000			$0$
$557$		0			0		−1.00000			$\pi$
$558$	13.0014	−	19.7222i	0.550394	−	0.834905i
$559$		0			0
$560$	−59.7170	−	56.8202i	−2.52350	−	2.40109i
$561$		0			0
$562$	0.911055	+	0.600594i	0.0384306	+	0.0253345i
$563$		−	15.4361i		−	0.650553i	−0.945619	−	0.325277i	$-0.894543\pi$
$563$		−	15.4361i		−	0.650553i	0.945619	−	0.325277i	$-0.105457\pi$
$564$		0			0
$565$	−83.7215			−3.52219
$566$	−26.0028	+	39.4443i	−1.09298	+	1.65797i
$567$		−	47.0966i		−	1.97787i
$568$	9.03341	+	1.62245i	0.379033	+	0.0680764i
$569$		0			0		1.00000			$0$
$569$		0			0		−1.00000			$\pi$
$570$		0			0
$571$		0			0			−	1.00000i	$-0.5\pi$
$571$		0			0				1.00000i	$0.5\pi$
$572$		0			0
$573$		0			0
$574$	−9.64589	+	14.6321i	−0.402612	+	0.610731i
$575$		0			0
$576$	22.5000	+	8.35165i	0.937500	+	0.347985i
$577$	18.0000			0.749350			0.374675	−	0.927156i	$-0.377754\pi$
$577$	18.0000			0.749350			0.374675	+	0.927156i	$0.377754\pi$
$578$	−20.0725	−	13.2324i	−0.834905	−	0.550394i
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$	30.6991	+	20.2377i	1.26817	+	0.836013i
$587$		0			0			−	1.00000i	$-0.5\pi$
$587$		0			0				1.00000i	$0.5\pi$
$588$		0			0
$589$	5.53454			0.228047
$590$	35.1275	−	53.2858i	1.44618	−	2.19374i
$591$		0			0
$592$		0			0
$593$	−48.0276			−1.97226			−0.986129	−	0.165978i	$-0.946922\pi$
$593$	−48.0276			−1.97226			−0.986129	+	0.165978i	$0.946922\pi$
$594$		0			0
$595$		0			0
$596$	−7.88267	−	18.3811i	−0.322887	−	0.752918i
$597$		0			0
$598$		0			0
$599$		−	30.1409i		−	1.23153i	−0.787932	−	0.615763i	$-0.788848\pi$
$599$		−	30.1409i		−	1.23153i	0.787932	−	0.615763i	$-0.211152\pi$
$600$		0			0
$601$		0			0		1.00000			$0$
$601$		0			0		−1.00000			$\pi$
$602$		0			0
$603$		−	33.4066i		−	1.36042i
$604$		0			0
$605$	43.3180			1.76113
$606$		0			0
$607$		−	11.1355i		−	0.451977i	−0.974130	−	0.225989i	$-0.927439\pi$
$607$		−	11.1355i		−	0.451977i	0.974130	−	0.225989i	$-0.0725612\pi$
$608$	1.26778	+	5.47832i	0.0514152	+	0.222175i
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$		0			0		1.00000			$0$
$613$		0			0		−1.00000			$\pi$
$614$	−25.2129	+	38.2461i	−1.01751	+	1.54349i
$615$		0			0
$616$		0			0
$617$	−22.0000			−0.885687			−0.442843	−	0.896599i	$-0.646030\pi$
$617$	−22.0000			−0.885687			−0.442843	+	0.896599i	$0.646030\pi$
$618$		0			0
$619$		0			0			−	1.00000i	$-0.5\pi$
$619$		0			0				1.00000i	$0.5\pi$
$620$	−40.3021	+	17.2835i	−1.61857	+	0.694120i
$621$		0			0
$622$	21.9135	−	33.2411i	0.878650	−	1.33285i
$623$		0			0
$624$		0			0
$625$	32.8760			1.31504
$626$		0			0
$627$		0			0
$628$	−17.9959	−	41.9635i	−0.718117	−	1.67453i
$629$		0			0
$630$	−48.1208	+	72.9957i	−1.91718	+	2.90822i
$631$		0			0			−	1.00000i	$-0.5\pi$
$631$		0			0				1.00000i	$0.5\pi$
$632$		0			0
$633$		0			0
$634$	19.5415	+	12.8823i	0.776092	+	0.511622i
$635$		0			0
$636$		0			0
$637$		0			0
$638$		0			0
$639$		−	9.73469i		−	0.385098i
$640$	−26.3398	−	35.9336i	−1.04117	−	1.42040i
$641$		0			0		1.00000			$0$
$641$		0			0		−1.00000			$\pi$
$642$		0			0
$643$		0			0			−	1.00000i	$-0.5\pi$
$643$		0			0				1.00000i	$0.5\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0			−	1.00000i	$-0.5\pi$
$647$		0			0				1.00000i	$0.5\pi$
$648$	4.50000	−	25.0549i	0.176777	−	0.984251i
$649$		0			0
$650$		0			0
$651$		0			0
$652$	13.7561	−	5.89925i	0.538729	−	0.231033i
$653$	46.0000			1.80012			0.900060	−	0.435767i	$-0.143523\pi$
$653$	46.0000			1.80012			0.900060	+	0.435767i	$0.143523\pi$
$654$		0			0
$655$		−	43.8518i		−	1.71343i
$656$	−6.52959	+	6.86247i	−0.254938	+	0.267935i
$657$		0			0
$658$	68.8035	+	45.3573i	2.68224	+	1.76821i
$659$			51.3356i			1.99975i	0.0158077	+	0.999875i	$0.494968\pi$
$659$			51.3356i			1.99975i	−0.0158077	+	0.999875i	$0.505032\pi$
$660$		0			0
$661$	−32.3023			−1.25641			−0.628207	−	0.778046i	$-0.716211\pi$
$661$	−32.3023			−1.25641			−0.628207	+	0.778046i	$0.716211\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−20.4844			−0.794353
$666$		0			0
$667$		0			0
$668$		0			0
$669$		0			0
$670$	−34.1331	+	51.7773i	−1.31868	+	2.00033i
$671$		0			0
$672$		0			0
$673$		0			0		1.00000			$0$
$673$		0			0		−1.00000			$\pi$
$674$		0			0
$675$		0			0
$676$	10.2475	+	23.8954i	0.394134	+	0.919053i
$677$		0			0		1.00000			$0$
$677$		0			0		−1.00000			$\pi$
$678$		0			0
$679$			53.6072i			2.05726i
$680$		0			0
$681$		0			0
$682$		0			0
$683$		−	36.8935i		−	1.41169i	−0.708366	−	0.705846i	$-0.750567\pi$
$683$		−	36.8935i		−	1.41169i	0.708366	−	0.705846i	$-0.249433\pi$
$684$	5.48141	−	2.35069i	0.209587	−	0.0898809i
$685$		0			0
$686$	54.5152	−	82.6955i	2.08140	−	3.15733i
$687$		0			0
$688$		0			0
$689$		0			0
$690$		0			0
$691$		−	49.3475i		−	1.87727i	−0.344915	−	0.938634i	$-0.612092\pi$
$691$		−	49.3475i		−	1.87727i	0.344915	−	0.938634i	$-0.387908\pi$
$692$	11.0357	+	25.7335i	0.419516	+	0.978240i
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$		0			0
$698$	−35.4220	−	23.3512i	−1.34074	−	0.883857i
$699$		0			0
$700$	101.072	−	43.3447i	3.82018	−	1.63827i
$701$	49.5975			1.87327			0.936636	−	0.350304i	$-0.113922\pi$
$701$	49.5975			1.87327			0.936636	+	0.350304i	$0.113922\pi$
$702$		0			0
$703$		0			0
$704$		0			0
$705$		0			0
$706$		0			0
$707$			37.0375i			1.39294i
$708$		0			0
$709$		0			0		1.00000			$0$
$709$		0			0		−1.00000			$\pi$
$710$	−9.94639	+	15.0879i	−0.373281	+	0.566239i
$711$		0			0
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$	−17.2711	+	26.1989i	−0.644551	+	0.977735i
$719$		0			0			−	1.00000i	$-0.5\pi$
$719$		0			0				1.00000i	$0.5\pi$
$720$	−32.5744	+	34.2351i	−1.21398	+	1.27587i
$721$	−71.7482			−2.67204
$722$	−21.2673	−	14.0200i	−0.791486	−	0.521770i
$723$		0			0
$724$		0			0
$725$		0			0
$726$		0			0
$727$			11.1972i			0.415280i	0.978205	+	0.207640i	$0.0665782\pi$
$727$			11.1972i			0.415280i	−0.978205	+	0.207640i	$0.933422\pi$
$728$		0			0
$729$	−27.0000			−1.00000
$730$		0			0
$731$		0			0
$732$		0			0
$733$	−13.4105			−0.495330			−0.247665	−	0.968846i	$-0.579663\pi$
$733$	−13.4105			−0.495330			−0.247665	+	0.968846i	$0.579663\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$	8.38842	+	5.52989i	0.308782	+	0.203558i
$739$		0			0			−	1.00000i	$-0.5\pi$
$739$		0			0				1.00000i	$0.5\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		0			0			−	1.00000i	$-0.5\pi$
$743$		0			0				1.00000i	$0.5\pi$
$744$		0			0
$745$	39.3800			1.44277
$746$	45.5548	+	30.0311i	1.66788	+	1.09951i
$747$		0			0
$748$		0			0
$749$	−104.334			−3.81227
$750$		0			0
$751$			49.0847i			1.79113i	0.444934	+	0.895563i	$0.353227\pi$
$751$			49.0847i			1.79113i	−0.444934	+	0.895563i	$0.646773\pi$
$752$	32.2690	+	30.7036i	1.17673	+	1.11965i
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$		0			0		1.00000			$0$
$757$		0			0		−1.00000			$\pi$
$758$	−26.0028	+	39.4443i	−0.944465	+	1.43268i
$759$		0			0
$760$	−10.8975	−	1.95725i	−0.395294	−	0.0709970i
$761$		0			0		1.00000			$0$
$761$		0			0		−1.00000			$\pi$
$762$		0			0
$763$		−	78.2524i		−	2.83293i
$764$	−44.4394	+	19.0577i	−1.60776	+	0.689485i
$765$		0			0
$766$		0			0
$767$		0			0
$768$		0			0
$769$	52.7639			1.90272			0.951358	−	0.308089i	$-0.0996893\pi$
$769$	52.7639			1.90272			0.951358	+	0.308089i	$0.0996893\pi$
$770$		0			0
$771$		0			0
$772$	14.2835	+	33.3068i	0.514075	+	1.19874i
$773$		0			0		1.00000			$0$
$773$		0			0		−1.00000			$\pi$
$774$		0			0
$775$		−	58.5054i		−	2.10157i
$776$	−5.12207	+	28.5185i	−0.183872	+	1.02375i
$777$		0			0
$778$		0			0
$779$			2.35400i			0.0843409i
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$	56.2038	−	59.0692i	2.00728	−	2.10961i
$785$	89.9036			3.20880
$786$		0			0
$787$		0			0			−	1.00000i	$-0.5\pi$
$787$		0			0				1.00000i	$0.5\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		−	111.252i		−	3.95567i
$792$		0			0
$793$		0			0
$794$	−21.3636	−	14.0835i	−0.758166	−	0.499805i
$795$		0			0
$796$		0			0
$797$		0			0		1.00000			$0$
$797$		0			0		−1.00000			$\pi$
$798$		0			0
$799$		0			0
$800$	57.9110	−	13.4016i	2.04746	−	0.473819i
$801$		0			0
$802$		0			0
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$	−3.53887	+	19.7036i	−0.124497	+	0.693170i
$809$		0			0		1.00000			$0$
$809$		0			0		−1.00000			$\pi$
$810$	41.8476	+	27.5872i	1.47038	+	0.969314i
$811$			55.6776i			1.95511i	0.210688	+	0.977553i	$0.432429\pi$
$811$			55.6776i			1.95511i	−0.210688	+	0.977553i	$0.567571\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$			29.4713i			1.03234i
$816$		0			0
$817$		0			0
$818$		0			0
$819$		0			0
$820$	−7.35117	−	17.1417i	−0.256714	−	0.598613i
$821$		0			0		1.00000			$0$
$821$		0			0		−1.00000			$\pi$
$822$		0			0
$823$		0			0			−	1.00000i	$-0.5\pi$
$823$		0			0				1.00000i	$0.5\pi$
$824$	−38.1693	−	6.85541i	−1.32969	−	0.238819i
$825$		0			0
$826$	70.8080	+	46.6787i	2.46373	+	1.62416i
$827$		0			0			−	1.00000i	$-0.5\pi$
$827$		0			0				1.00000i	$0.5\pi$
$828$		0			0
$829$		0			0		1.00000			$0$
$829$		0			0		−1.00000			$\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		0			0
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$	−31.8119	+	48.2562i	−1.09892	+	1.66698i
$839$		−	55.6776i		−	1.92221i	−0.276191	−	0.961103i	$-0.589072\pi$
$839$		−	55.6776i		−	1.92221i	0.276191	−	0.961103i	$-0.410928\pi$
$840$		0			0
$841$	29.0000			1.00000
$842$	12.0641	+	7.95302i	0.415757	+	0.274079i
$843$		0			0
$844$	−32.9935	+	14.1492i	−1.13568	+	0.487036i
$845$	−51.1940			−1.76113
$846$	26.0028	−	39.4443i	0.893995	−	1.35612i
$847$			57.5626i			1.97787i
$848$		0			0
$849$		0			0
$850$		0			0
$851$		0			0
$852$		0			0
$853$	54.0000			1.84892			0.924462	−	0.381273i	$-0.124514\pi$
$853$	54.0000			1.84892			0.924462	+	0.381273i	$0.124514\pi$
$854$		0			0
$855$			11.7435i			0.401620i
$856$	−55.5045	−	9.96891i	−1.89711	−	0.340730i
$857$	−38.0000			−1.29806			−0.649028	−	0.760765i	$-0.724824\pi$
$857$	−38.0000			−1.29806			−0.649028	+	0.760765i	$0.724824\pi$
$858$		0			0
$859$		0			0			−	1.00000i	$-0.5\pi$
$859$		0			0				1.00000i	$0.5\pi$
$860$		0			0
$861$		0			0
$862$	−8.66761	+	13.1481i	−0.295220	+	0.447826i
$863$		0			0			−	1.00000i	$-0.5\pi$
$863$		0			0				1.00000i	$0.5\pi$
$864$		0			0
$865$	−55.1320			−1.87455
$866$		0			0
$867$		0			0
$868$	−22.9669	−	53.5549i	−0.779546	−	1.81777i
$869$		0			0
$870$		0			0
$871$		0			0
$872$	7.47687	−	41.6295i	0.253199	−	1.40975i
$873$	30.7324			1.04014
$874$		0			0
$875$			113.503i			3.83710i
$876$		0			0
$877$	−57.4735			−1.94074			−0.970371	−	0.241618i	$-0.922322\pi$
$877$	−57.4735			−1.94074			−0.970371	+	0.241618i	$0.922322\pi$
$878$	15.3145	−	23.2310i	0.516840	−	0.784007i
$879$		0			0
$880$		0			0
$881$		0			0		1.00000			$0$
$881$		0			0		−1.00000			$\pi$
$882$	−72.2038	−	47.5989i	−2.43123	−	1.60274i
$883$		0			0			−	1.00000i	$-0.5\pi$
$883$		0			0				1.00000i	$0.5\pi$
$884$		0			0
$885$		0			0
$886$	4.27776	−	6.48903i	0.143714	−	0.218003i
$887$			53.5864i			1.79926i	0.436657	+	0.899628i	$0.356162\pi$
$887$			53.5864i			1.79926i	−0.436657	+	0.899628i	$0.643838\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$		0			0
$892$		0			0
$893$	11.0691			0.370413
$894$		0			0
$895$		0			0
$896$	47.7499	−	35.0012i	1.59521	−	1.16931i
$897$		0			0
$898$		0			0
$899$		0			0
$900$	−24.8490	−	57.9438i	−0.828301	−	1.93146i
$901$		0			0
$902$		0			0
$903$		0			0
$904$	10.6299	−	59.1850i	0.353547	−	1.96846i
$905$		0			0
$906$		0			0
$907$		−	36.3679i		−	1.20758i	−0.797144	−	0.603789i	$-0.793657\pi$
$907$		−	36.3679i		−	1.20758i	0.797144	−	0.603789i	$-0.206343\pi$
$908$	−20.4683	+	8.77777i	−0.679264	+	0.291301i
$909$	21.2332			0.704261
$910$		0			0
$911$		0			0			−	1.00000i	$-0.5\pi$
$911$		0			0				1.00000i	$0.5\pi$
$912$		0			0
$913$		0			0
$914$		0			0
$915$		0			0
$916$		0			0
$917$	58.2718			1.92430
$918$		0			0
$919$		−	55.6776i		−	1.83664i	−0.395843	−	0.918318i	$-0.629548\pi$
$919$		−	55.6776i		−	1.83664i	0.395843	−	0.918318i	$-0.370452\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		0			0
$924$		0			0
$925$		0			0
$926$		0			0
$927$			41.1325i			1.35097i
$928$		0			0
$929$		0			0		1.00000			$0$
$929$		0			0		−1.00000			$\pi$
$930$		0			0
$931$		−	20.2622i		−	0.664068i
$932$	6.81663	+	15.8952i	0.223286	+	0.520665i
$933$		0			0
$934$	−12.4242	+	18.8465i	−0.406531	+	0.616677i
$935$		0			0
$936$		0			0
$937$	42.0000			1.37208			0.686040	−	0.727564i	$-0.259347\pi$
$937$	42.0000			1.37208			0.686040	+	0.727564i	$0.259347\pi$
$938$	−68.8035	−	45.3573i	−2.24651	−	1.48097i
$939$		0			0
$940$	−80.6042	+	34.5669i	−2.62902	+	1.12745i
$941$		0			0		1.00000			$0$
$941$		0			0		−1.00000			$\pi$
$942$		0			0
$943$		0			0
$944$	33.2091	+	31.5982i	1.08086	+	1.02843i
$945$		0			0
$946$		0			0
$947$		0			0			−	1.00000i	$-0.5\pi$
$947$		0			0				1.00000i	$0.5\pi$
$948$		0			0
$949$		0			0
$950$	8.13025	−	12.3330i	0.263780	−	0.400134i
$951$		0			0
$952$		0			0
$953$		0			0		1.00000			$0$
$953$		0			0		−1.00000			$\pi$
$954$		0			0
$955$		−	95.2081i		−	3.08086i
$956$		0			0
$957$		0			0
$958$	25.4175	−	38.5564i	0.821202	−	1.24570i
$959$		0			0
$960$		0			0
$961$	−31.0000			−1.00000
$962$		0			0
$963$			59.8134i			1.92746i
$964$		0			0
$965$	−71.3572			−2.29707
$966$		0			0
$967$		0			0			−	1.00000i	$-0.5\pi$
$967$		0			0				1.00000i	$0.5\pi$
$968$	−5.50000	+	30.6227i	−0.176777	+	0.984251i
$969$		0			0
$970$	−47.6326	−	31.4008i	−1.52939	−	1.00822i
$971$			55.6776i			1.78678i	0.449281	+	0.893390i	$0.351680\pi$
$971$			55.6776i			1.78678i	−0.449281	+	0.893390i	$0.648320\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$		0			0
$976$		0			0
$977$	49.6242			1.58762			0.793809	−	0.608167i	$-0.208095\pi$
$977$	49.6242			1.58762			0.793809	+	0.608167i	$0.208095\pi$
$978$		0			0
$979$		0			0
$980$	63.2756	+	147.548i	2.02127	+	4.71325i
$981$	−44.8612			−1.43231
$982$		0			0
$983$		0			0			−	1.00000i	$-0.5\pi$
$983$		0			0				1.00000i	$0.5\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$		0			0			−	1.00000i	$-0.5\pi$
$991$		0			0				1.00000i	$0.5\pi$
$992$	−7.10107	−	30.6851i	−0.225459	−	0.974253i
$993$		0			0
$994$	−20.0494	−	13.2171i	−0.635927	−	0.419221i
$995$		0			0
$996$		0			0
$997$	62.2098			1.97020			0.985102	−	0.171972i	$-0.0550138\pi$
$997$	62.2098			1.97020			0.985102	+	0.171972i	$0.0550138\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	124.2.d.c.123.1	✓	6
3.2	odd	2		1116.2.g.f.991.6		6
4.3	odd	2	inner	124.2.d.c.123.2	yes	6
8.3	odd	2		1984.2.h.f.1983.6		6
8.5	even	2		1984.2.h.f.1983.5		6
12.11	even	2		1116.2.g.f.991.5		6
31.30	odd	2	CM	124.2.d.c.123.1	✓	6
93.92	even	2		1116.2.g.f.991.6		6
124.123	even	2	inner	124.2.d.c.123.2	yes	6
248.61	odd	2		1984.2.h.f.1983.5		6
248.123	even	2		1984.2.h.f.1983.6		6
372.371	odd	2		1116.2.g.f.991.5		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
124.2.d.c.123.1	✓	6	1.1	even	1	trivial
124.2.d.c.123.1	✓	6	31.30	odd	2	CM
124.2.d.c.123.2	yes	6	4.3	odd	2	inner
124.2.d.c.123.2	yes	6	124.123	even	2	inner
1116.2.g.f.991.5		6	12.11	even	2
1116.2.g.f.991.5		6	372.371	odd	2
1116.2.g.f.991.6		6	3.2	odd	2
1116.2.g.f.991.6		6	93.92	even	2
1984.2.h.f.1983.5		6	8.5	even	2
1984.2.h.f.1983.5		6	248.61	odd	2
1984.2.h.f.1983.6		6	8.3	odd	2
1984.2.h.f.1983.6		6	248.123	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 124 = 2^{2} \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	124.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(-1.18073 - 0.778374i) q^{2} +(0.788267 + 1.83811i) q^{4} -3.93800 q^{5} -5.23296i q^{7} +(0.500000 - 2.78388i) q^{8} -3.00000 q^{9} +O(q^{10})\)	\(q+(-1.18073 - 0.778374i) q^{2} +(0.788267 + 1.83811i) q^{4} -3.93800 q^{5} -5.23296i q^{7} +(0.500000 - 2.78388i) q^{8} -3.00000 q^{9} +(4.64974 + 3.06524i) q^{10} +(-4.07320 + 6.17874i) q^{14} +(-2.75727 + 2.89784i) q^{16} +(3.54220 + 2.33512i) q^{18} +0.994032i q^{19} +(-3.10420 - 7.23847i) q^{20} +10.5079 q^{25} +(9.61874 - 4.12497i) q^{28} -5.56776i q^{31} +(5.51120 - 1.27539i) q^{32} +20.6074i q^{35} +(-2.36480 - 5.51432i) q^{36} +(0.773729 - 1.17369i) q^{38} +(-1.96900 + 10.9629i) q^{40} +2.36814 q^{41} +11.8140 q^{45} -11.1355i q^{47} -20.3839 q^{49} +(-12.4070 - 8.17906i) q^{50} +(-14.5679 - 2.61648i) q^{56} -11.4600i q^{59} +(-4.33380 + 6.57405i) q^{62} +15.6989i q^{63} +(-7.50000 - 2.78388i) q^{64} +11.1355i q^{67} +(16.0403 - 24.3319i) q^{70} +3.24490i q^{71} +(-1.50000 + 8.35165i) q^{72} +(-1.82714 + 0.783563i) q^{76} +(10.8581 - 11.4117i) q^{80} +9.00000 q^{81} +(-2.79614 - 1.84330i) q^{82} +(-13.9492 - 9.19572i) q^{90} +(-8.66761 + 13.1481i) q^{94} -3.91450i q^{95} -10.2441 q^{97} +(24.0679 + 15.8663i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 3 q^{8} - 18 q^{9}+O(q^{10}) \) 6 * q + 3 * q^8 - 18 * q^9	\( 6 q + 3 q^{8} - 18 q^{9} + 9 q^{10} - 15 q^{14} - 21 q^{20} + 30 q^{25} + 27 q^{28} + 33 q^{38} - 42 q^{49} - 39 q^{50} - 45 q^{64} - 3 q^{70} - 9 q^{72} + 15 q^{76} + 51 q^{80} + 54 q^{81} + 21 q^{82} - 27 q^{90} + 57 q^{98}+O(q^{100}) \) 6 * q + 3 * q^8 - 18 * q^9 + 9 * q^10 - 15 * q^14 - 21 * q^20 + 30 * q^25 + 27 * q^28 + 33 * q^38 - 42 * q^49 - 39 * q^50 - 45 * q^64 - 3 * q^70 - 9 * q^72 + 15 * q^76 + 51 * q^80 + 54 * q^81 + 21 * q^82 - 27 * q^90 + 57 * q^98

Introduction

L-functions

Modular forms

Varieties

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Database

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