LMFDB - Embedded newform 124.2.d.c.123.4

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			123.4
Root			$-0.0837246 - 1.41173i$ of defining polynomial
Character	$\chi$	$=$	124.123
Dual form			124.2.d.c.123.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.0837246 + 1.41173i) q^{2} +(-1.98598 - 0.236394i) q^{4} +3.80451 q^{5} +3.29625i q^{7} +(0.500000 - 2.78388i) q^{8} -3.00000 q^{9} +O(q^{10})$	\(q+(-0.0837246 + 1.41173i) q^{2} +(-1.98598 - 0.236394i) q^{4} +3.80451 q^{5} +3.29625i q^{7} +(0.500000 - 2.78388i) q^{8} -3.00000 q^{9} +(-0.318531 + 5.37095i) q^{10} +(-4.65343 - 0.275977i) q^{14} +(3.88824 + 0.938946i) q^{16} +(0.251174 - 4.23520i) q^{18} -7.99761i q^{19} +(-7.55569 - 0.899362i) q^{20} +9.47431 q^{25} +(0.779213 - 6.54629i) q^{28} -5.56776i q^{31} +(-1.65108 + 5.41054i) q^{32} +12.5406i q^{35} +(5.95794 + 0.709181i) q^{36} +(11.2905 + 0.669597i) q^{38} +(1.90226 - 10.5913i) q^{40} -12.0833 q^{41} -11.4135 q^{45} -11.1355i q^{47} -3.86529 q^{49} +(-0.793233 + 13.3752i) q^{50} +(9.17638 + 1.64813i) q^{56} +14.5901i q^{59} +(7.86020 + 0.466159i) q^{62} -9.88876i q^{63} +(-7.50000 - 2.78388i) q^{64} +11.1355i q^{67} +(-17.7040 - 1.04996i) q^{70} +12.6990i q^{71} +(-1.50000 + 8.35165i) q^{72} +(-1.89058 + 15.8831i) q^{76} +(14.7928 + 3.57223i) q^{80} +9.00000 q^{81} +(1.01167 - 17.0584i) q^{82} +(0.955594 - 16.1129i) q^{90} +(15.7204 + 0.932318i) q^{94} -30.4270i q^{95} +19.6924 q^{97} +(0.323619 - 5.45675i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−0.0837246	+	1.41173i	−0.0592022	+	0.998246i
$2$	−0.0837246	+	1.41173i	−0.0592022	+	0.998246i
$3$		0			0			−	1.00000i	$-0.5\pi$
$3$		0			0				1.00000i	$0.5\pi$
$4$	−1.98598	−	0.236394i	−0.992990	−	0.118197i
$5$	3.80451			1.70143			0.850715	−	0.525628i	$-0.176170\pi$
$5$	3.80451			1.70143			0.850715	+	0.525628i	$0.176170\pi$
$6$		0			0
$7$			3.29625i			1.24587i	0.782275	+	0.622933i	$0.214059\pi$
$7$			3.29625i			1.24587i	−0.782275	+	0.622933i	$0.785941\pi$
$8$	0.500000	−	2.78388i	0.176777	−	0.984251i
$9$	−3.00000			−1.00000
$10$	−0.318531	+	5.37095i	−0.100728	+	1.69845i
$11$		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			$\pi$
$14$	−4.65343	−	0.275977i	−1.24368	−	0.0737581i
$15$		0			0
$16$	3.88824	+	0.938946i	0.972059	+	0.234736i
$17$		0			0		1.00000			$0$
$17$		0			0		−1.00000			$\pi$
$18$	0.251174	−	4.23520i	0.0592022	−	0.998246i
$19$		−	7.99761i		−	1.83478i	−0.397992	−	0.917389i	$-0.630293\pi$
$19$		−	7.99761i		−	1.83478i	0.397992	−	0.917389i	$-0.369707\pi$
$20$	−7.55569	−	0.899362i	−1.68950	−	0.201103i
$21$		0			0
$22$		0			0
$23$		0			0			−	1.00000i	$-0.5\pi$
$23$		0			0				1.00000i	$0.5\pi$
$24$		0			0
$25$	9.47431			1.89486
$26$		0			0
$27$		0			0
$28$	0.779213	−	6.54629i	0.147257	−	1.23713i
$29$		0			0		1.00000			$0$
$29$		0			0		−1.00000			$\pi$
$30$		0			0
$31$		−	5.56776i		−	1.00000i
$31$		−	5.56776i		−	1.00000i
$32$	−1.65108	+	5.41054i	−0.291873	+	0.956457i
$33$		0			0
$34$		0			0
$35$			12.5406i			2.11975i
$36$	5.95794	+	0.709181i	0.992990	+	0.118197i
$37$		0			0		1.00000			$0$
$37$		0			0		−1.00000			$\pi$
$38$	11.2905	+	0.669597i	1.83156	+	0.108623i
$39$		0			0
$40$	1.90226	−	10.5913i	0.300773	−	1.67463i
$41$	−12.0833			−1.88710			−0.943550	−	0.331231i	$-0.892536\pi$
$41$	−12.0833			−1.88710			−0.943550	+	0.331231i	$0.892536\pi$
$42$		0			0
$43$		0			0			−	1.00000i	$-0.5\pi$
$43$		0			0				1.00000i	$0.5\pi$
$44$		0			0
$45$	−11.4135			−1.70143
$46$		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
$49$	−3.86529			−0.552184
$50$	−0.793233	+	13.3752i	−0.112180	+	1.89154i
$51$		0			0
$52$		0			0
$53$		0			0		1.00000			$0$
$53$		0			0		−1.00000			$\pi$
$54$		0			0
$55$		0			0
$56$	9.17638	+	1.64813i	1.22625	+	0.220240i
$57$		0			0
$58$		0			0
$59$			14.5901i			1.89947i	0.313053	+	0.949736i	$0.398648\pi$
$59$			14.5901i			1.89947i	−0.313053	+	0.949736i	$0.601352\pi$
$60$		0			0
$61$		0			0		1.00000			$0$
$61$		0			0		−1.00000			$\pi$
$62$	7.86020	+	0.466159i	0.998246	+	0.0592022i
$63$		−	9.88876i		−	1.24587i
$64$	−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
$70$	−17.7040	−	1.04996i	−2.11604	−	0.125494i
$71$			12.6990i			1.50709i	0.657396	+	0.753545i	$0.271658\pi$
$71$			12.6990i			1.50709i	−0.657396	+	0.753545i	$0.728342\pi$
$72$	−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
$76$	−1.89058	+	15.8831i	−0.216865	+	1.82192i
$77$		0			0
$78$		0			0
$79$		0			0			−	1.00000i	$-0.5\pi$
$79$		0			0				1.00000i	$0.5\pi$
$80$	14.7928	+	3.57223i	1.65389	+	0.399388i
$81$	9.00000			1.00000
$82$	1.01167	−	17.0584i	0.111720	−	1.88379i
$83$		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			$\pi$
$90$	0.955594	−	16.1129i	0.100728	−	1.69845i
$91$		0			0
$92$		0			0
$93$		0			0
$94$	15.7204	+	0.932318i	1.62143	+	0.0961612i
$95$		−	30.4270i		−	3.12174i
$96$		0			0
$97$	19.6924			1.99946			0.999728	−	0.0233291i	$-0.00742657\pi$
$97$	19.6924			1.99946			0.999728	+	0.0233291i	$0.00742657\pi$
$98$	0.323619	−	5.45675i	0.0326905	−	0.551215i
$99$		0			0
$100$	−18.8158	−	2.23967i	−1.88158	−	0.223967i
$101$	−12.7531			−1.26898			−0.634492	−	0.772930i	$-0.718791\pi$
$101$	−12.7531			−1.26898			−0.634492	+	0.772930i	$0.718791\pi$
$102$		0			0
$103$		−	6.10646i		−	0.601688i	−0.953673	−	0.300844i	$-0.902732\pi$
$103$		−	6.10646i		−	0.601688i	0.953673	−	0.300844i	$-0.0972683\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$			5.18740i			0.501485i	0.968054	+	0.250743i	$0.0806748\pi$
$107$			5.18740i			0.501485i	−0.968054	+	0.250743i	$0.919325\pi$
$108$		0			0
$109$	5.14411			0.492716			0.246358	−	0.969179i	$-0.420766\pi$
$109$	5.14411			0.492716			0.246358	+	0.969179i	$0.420766\pi$
$110$		0			0
$111$		0			0
$112$	−3.09500	+	12.8166i	−0.292450	+	1.21106i
$113$	−10.7437			−1.01069			−0.505343	−	0.862919i	$-0.668634\pi$
$113$	−10.7437			−1.01069			−0.505343	+	0.862919i	$0.668634\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$		0			0
$118$	−20.5974	−	1.22155i	−1.89614	−	0.112453i
$119$		0			0
$120$		0			0
$121$	−11.0000			−1.00000
$122$		0			0
$123$		0			0
$124$	−1.31618	+	11.0575i	−0.118197	+	0.992990i
$125$	17.0226			1.52254
$126$	13.9603	+	0.827932i	1.24368	+	0.0737581i
$127$		0			0			−	1.00000i	$-0.5\pi$
$127$		0			0				1.00000i	$0.5\pi$
$128$	4.55803	−	10.3549i	0.402877	−	0.915254i
$129$		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
$133$	26.3622			2.28589
$134$	−15.7204	−	0.932318i	−1.35804	−	0.0805400i
$135$		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	$0.5\pi$
$140$	2.96452	−	24.9055i	0.250548	−	2.10489i
$141$		0			0
$142$	−17.9276	−	1.06322i	−1.50445	−	0.0892231i
$143$		0			0
$144$	−11.6647	−	2.81684i	−0.972059	−	0.234736i
$145$		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	$0.5\pi$
$152$	−22.2644	−	3.99881i	−1.80588	−	0.324346i
$153$		0			0
$154$		0			0
$155$		−	21.1826i		−	1.70143i
$156$		0			0
$157$	2.46492			0.196722			0.0983609	−	0.995151i	$-0.468640\pi$
$157$	2.46492			0.196722			0.0983609	+	0.995151i	$0.468640\pi$
$158$		0			0
$159$		0			0
$160$	−6.28156	+	20.5845i	−0.496601	+	1.62734i
$161$		0			0
$162$	−0.753521	+	12.7056i	−0.0592022	+	0.998246i
$163$		−	17.4003i		−	1.36290i	−0.731865	−	0.681449i	$-0.761350\pi$
$163$		−	17.4003i		−	1.36290i	0.731865	−	0.681449i	$-0.238650\pi$
$164$	23.9973	+	2.85642i	1.87387	+	0.223049i
$165$		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
$171$			23.9928i			1.83478i
$172$		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
$175$			31.2297i			2.36074i
$176$		0			0
$177$		0			0
$178$		0			0
$179$		0			0			−	1.00000i	$-0.5\pi$
$179$		0			0				1.00000i	$0.5\pi$
$180$	22.6671	+	2.69809i	1.68950	+	0.201103i
$181$		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
$188$	−2.63237	+	22.1149i	−0.191985	+	1.61290i
$189$		0			0
$190$	42.9548	+	2.54749i	3.11627	+	0.184814i
$191$		−	0.486044i		−	0.0351689i	−0.999845	−	0.0175844i	$-0.994402\pi$
$191$		−	0.486044i		−	0.0351689i	0.999845	−	0.0175844i	$-0.00559759\pi$
$192$		0			0
$193$	−27.3014			−1.96520			−0.982598	−	0.185744i	$-0.940530\pi$
$193$	−27.3014			−1.96520			−0.982598	+	0.185744i	$0.940530\pi$
$194$	−1.64873	+	27.8003i	−0.118372	+	1.99595i
$195$		0			0
$196$	7.67638	+	0.913729i	0.548313	+	0.0652663i
$197$		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	$0.5\pi$
$200$	4.73715	−	26.3754i	0.334967	−	1.86502i
$201$		0			0
$202$	1.06775	−	18.0040i	0.0751267	−	1.26676i
$203$		0			0
$204$		0			0
$205$	−45.9712			−3.21077
$206$	8.62070	+	0.511261i	0.600632	+	0.0356213i
$207$		0			0
$208$		0			0
$209$		0			0
$210$		0			0
$211$			10.8078i			0.744041i	0.928225	+	0.372021i	$0.121335\pi$
$211$			10.8078i			0.744041i	−0.928225	+	0.372021i	$0.878665\pi$
$212$		0			0
$213$		0			0
$214$	−7.32323	−	0.434313i	−0.500605	−	0.0296890i
$215$		0			0
$216$		0			0
$217$	18.3528			1.24587
$218$	−0.430688	+	7.26210i	−0.0291699	+	0.491852i
$219$		0			0
$220$		0			0
$221$		0			0
$222$		0			0
$223$		0			0			−	1.00000i	$-0.5\pi$
$223$		0			0				1.00000i	$0.5\pi$
$224$	−17.8345	−	5.44238i	−1.19162	−	0.363635i
$225$	−28.4229			−1.89486
$226$	0.899515	−	15.1673i	0.0598349	−	1.00891i
$227$			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
$233$	21.0319			1.37785			0.688924	−	0.724833i	$-0.258083\pi$
$233$	21.0319			1.37785			0.688924	+	0.724833i	$0.258083\pi$
$234$		0			0
$235$		−	42.3652i		−	2.76360i
$236$	3.44901	−	28.9757i	0.224511	−	1.88616i
$237$		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			$\pi$
$242$	0.920971	−	15.5291i	0.0592022	−	0.998246i
$243$		0			0
$244$		0			0
$245$	−14.7055			−0.939501
$246$		0			0
$247$		0			0
$248$	−15.5000	−	2.78388i	−0.984251	−	0.176777i
$249$		0			0
$250$	−1.42521	+	24.0313i	−0.0901380	+	1.51987i
$251$		0			0			−	1.00000i	$-0.5\pi$
$251$		0			0				1.00000i	$0.5\pi$
$252$	−2.33764	+	19.6389i	−0.147257	+	1.23713i
$253$		0			0
$254$		0			0
$255$		0			0
$256$	14.2368	+	7.30169i	0.889798	+	0.456355i
$257$	5.81390			0.362661			0.181331	−	0.983422i	$-0.441960\pi$
$257$	5.81390			0.362661			0.181331	+	0.983422i	$0.441960\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$		0			0
$262$	−15.7204	−	0.932318i	−0.971209	−	0.0575988i
$263$		0			0			−	1.00000i	$-0.5\pi$
$263$		0			0				1.00000i	$0.5\pi$
$264$		0			0
$265$		0			0
$266$	−2.20716	+	37.2163i	−0.135330	+	2.28188i
$267$		0			0
$268$	2.63237	−	22.1149i	0.160797	−	1.35089i
$269$		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
$280$	34.9116	+	6.27032i	2.08637	+	0.374723i
$281$	−28.6410			−1.70858			−0.854289	−	0.519799i	$-0.826007\pi$
$281$	−28.6410			−1.70858			−0.854289	+	0.519799i	$0.826007\pi$
$282$		0			0
$283$		−	33.4066i		−	1.98582i	−0.118888	−	0.992908i	$-0.537933\pi$
$283$		−	33.4066i		−	1.98582i	0.118888	−	0.992908i	$-0.462067\pi$
$284$	3.00195	−	25.2199i	0.178133	−	1.49653i
$285$		0			0
$286$		0			0
$287$		−	39.8297i		−	2.35107i
$288$	4.95325	−	16.2316i	0.291873	−	0.956457i
$289$	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
$295$			55.5083i			3.23182i
$296$		0			0
$297$		0			0
$298$	0.837246	−	14.1173i	0.0485004	−	0.817795i
$299$		0			0
$300$		0			0
$301$		0			0
$302$		0			0
$303$		0			0
$304$	7.50932	−	31.0966i	0.430689	−	1.78351i
$305$		0			0
$306$		0			0
$307$			27.7751i			1.58521i	0.609735	+	0.792605i	$0.291276\pi$
$307$			27.7751i			1.58521i	−0.609735	+	0.792605i	$0.708724\pi$
$308$		0			0
$309$		0			0
$310$	29.9042	+	1.77351i	1.69845	+	0.100728i
$311$		−	32.4765i		−	1.84157i	−0.390067	−	0.920786i	$-0.627548\pi$
$311$		−	32.4765i		−	1.84157i	0.390067	−	0.920786i	$-0.372452\pi$
$312$		0			0
$313$		0			0		1.00000			$0$
$313$		0			0		−1.00000			$\pi$
$314$	−0.206374	+	3.47981i	−0.0116464	+	0.196377i
$315$		−	37.6219i		−	2.11975i
$316$		0			0
$317$	35.5802			1.99838			0.999191	−	0.0402038i	$-0.0128007\pi$
$317$	35.5802			1.99838			0.999191	+	0.0402038i	$0.0128007\pi$
$318$		0			0
$319$		0			0
$320$	−28.5338	−	10.5913i	−1.59509	−	0.592072i
$321$		0			0
$322$		0			0
$323$		0			0
$324$	−17.8738	−	2.12754i	−0.992990	−	0.118197i
$325$		0			0
$326$	24.5646	+	1.45684i	1.36051	+	0.0806866i
$327$		0			0
$328$	−6.04167	+	33.6386i	−0.333595	+	1.85738i
$329$	36.7055			2.02364
$330$		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
$335$			42.3652i			2.31466i
$336$		0			0
$337$		0			0		1.00000			$0$
$337$		0			0		−1.00000			$\pi$
$338$	−1.08842	+	18.3525i	−0.0592022	+	0.998246i
$339$		0			0
$340$		0			0
$341$		0			0
$342$	−33.8715	−	2.00879i	−1.83156	−	0.108623i
$343$			10.3328i			0.557920i
$344$		0			0
$345$		0			0
$346$	−1.17214	+	19.7643i	−0.0630149	+	1.06253i
$347$		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	$0.203272\pi$
$350$	−44.0880	−	2.61470i	−2.35660	−	0.139761i
$351$		0			0
$352$		0			0
$353$		0			0		1.00000			$0$
$353$		0			0		−1.00000			$\pi$
$354$		0			0
$355$			48.3134i			2.56421i
$356$		0			0
$357$		0			0
$358$		0			0
$359$		−	15.5092i		−	0.818543i	−0.912413	−	0.409272i	$-0.865783\pi$
$359$		−	15.5092i		−	0.818543i	0.912413	−	0.409272i	$-0.134217\pi$
$360$	−5.70677	+	31.7739i	−0.300773	+	1.67463i
$361$	−44.9618			−2.36641
$362$		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
$369$	36.2500			1.88710
$370$		0			0
$371$		0			0
$372$		0			0
$373$	17.6830			0.915589			0.457795	−	0.889058i	$-0.348640\pi$
$373$	17.6830			0.915589			0.457795	+	0.889058i	$0.348640\pi$
$374$		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	$-0.328287\pi$
$380$	−7.19275	+	60.4274i	−0.368980	+	3.09986i
$381$		0			0
$382$	0.686164	+	0.0406938i	0.0351072	+	0.00208208i
$383$		0			0			−	1.00000i	$-0.5\pi$
$383$		0			0				1.00000i	$0.5\pi$
$384$		0			0
$385$		0			0
$386$	2.28580	−	38.5423i	0.116344	−	1.96175i
$387$		0			0
$388$	−39.1086	−	4.65515i	−1.98544	−	0.236329i
$389$		0			0		1.00000			$0$
$389$		0			0		−1.00000			$\pi$
$390$		0			0
$391$		0			0
$392$	−1.93264	+	10.7605i	−0.0976132	+	0.543487i
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$	21.7017			1.08918			0.544590	−	0.838703i	$-0.316685\pi$
$397$	21.7017			1.08918			0.544590	+	0.838703i	$0.316685\pi$
$398$		0			0
$399$		0			0
$400$	36.8383	+	8.89586i	1.84192	+	0.444793i
$401$		0			0		1.00000			$0$
$401$		0			0		−1.00000			$\pi$
$402$		0			0
$403$		0			0
$404$	25.3275	+	3.01476i	1.26009	+	0.149990i
$405$	34.2406			1.70143
$406$		0			0
$407$		0			0
$408$		0			0
$409$		0			0		1.00000			$0$
$409$		0			0		−1.00000			$\pi$
$410$	3.84892	−	64.8990i	0.190085	−	3.20513i
$411$		0			0
$412$	−1.44353	+	12.1273i	−0.0711175	+	0.597470i
$413$	−48.0927			−2.36649
$414$		0			0
$415$		0			0
$416$		0			0
$417$		0			0
$418$		0			0
$419$			18.3724i			0.897551i	0.893644	+	0.448776i	$0.148140\pi$
$419$			18.3724i			0.897551i	−0.893644	+	0.448776i	$0.851860\pi$
$420$		0			0
$421$	−29.3108			−1.42852			−0.714260	−	0.699881i	$-0.753236\pi$
$421$	−29.3108			−1.42852			−0.714260	+	0.699881i	$0.753236\pi$
$422$	−15.2578	−	0.904880i	−0.742736	−	0.0440489i
$423$			33.4066i			1.62428i
$424$		0			0
$425$		0			0
$426$		0			0
$427$		0			0
$428$	1.22627	−	10.3021i	0.0592739	−	0.497970i
$429$		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			$\pi$
$434$	−1.53658	+	25.9092i	−0.0737581	+	1.24368i
$435$		0			0
$436$	−10.2161	−	1.21603i	−0.489262	−	0.0582374i
$437$		0			0
$438$		0			0
$439$		−	41.8792i		−	1.99879i	−0.0348463	−	0.999393i	$-0.511094\pi$
$439$		−	41.8792i		−	1.99879i	0.0348463	−	0.999393i	$-0.488906\pi$
$440$		0			0
$441$	11.5959			0.552184
$442$		0			0
$443$			33.3955i			1.58667i	0.608785	+	0.793335i	$0.291657\pi$
$443$			33.3955i			1.58667i	−0.608785	+	0.793335i	$0.708343\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$	9.17638	−	24.7219i	0.433543	−	1.16800i
$449$		0			0		1.00000			$0$
$449$		0			0		−1.00000			$\pi$
$450$	2.37970	−	40.1256i	0.112180	−	1.89154i
$451$		0			0
$452$	21.3369	+	2.53975i	1.00360	+	0.119460i
$453$		0			0
$454$	−15.7204	−	0.932318i	−0.737795	−	0.0437558i
$455$		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
$466$	−1.76089	+	29.6915i	−0.0815717	+	1.37543i
$467$		−	26.8030i		−	1.24030i	−0.784484	−	0.620148i	$-0.787072\pi$
$467$		−	26.8030i		−	1.24030i	0.784484	−	0.620148i	$-0.212928\pi$
$468$		0			0
$469$	−36.7055			−1.69490
$470$	59.8084	+	3.54701i	2.75876	+	0.163612i
$471$		0			0
$472$	40.6172	+	7.29506i	1.86956	+	0.335782i
$473$		0			0
$474$		0			0
$475$		−	75.7718i		−	3.47665i
$476$		0			0
$477$		0			0
$478$		0			0
$479$			8.91667i			0.407413i	0.979032	+	0.203707i	$0.0652989\pi$
$479$			8.91667i			0.407413i	−0.979032	+	0.203707i	$0.934701\pi$
$480$		0			0
$481$		0			0
$482$		0			0
$483$		0			0
$484$	21.8458	+	2.60033i	0.992990	+	0.118197i
$485$	74.9198			3.40193
$486$		0			0
$487$		0			0			−	1.00000i	$-0.5\pi$
$487$		0			0				1.00000i	$0.5\pi$
$488$		0			0
$489$		0			0
$490$	1.23121	−	20.7603i	0.0556206	−	0.937853i
$491$		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
$496$	5.22783	−	21.6488i	0.234736	−	0.972059i
$497$	−41.8590			−1.87763
$498$		0			0
$499$		0			0			−	1.00000i	$-0.5\pi$
$499$		0			0				1.00000i	$0.5\pi$
$500$	−33.8065	−	4.02402i	−1.51187	−	0.179960i
$501$		0			0
$502$		0			0
$503$			39.0690i			1.74200i	0.491283	+	0.871000i	$0.336528\pi$
$503$			39.0690i			1.74200i	−0.491283	+	0.871000i	$0.663472\pi$
$504$	−27.5291	−	4.94438i	−1.22625	−	0.220240i
$505$	−48.5194			−2.15909
$506$		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
$514$	−0.486767	+	8.20768i	−0.0214704	+	0.362025i
$515$		−	23.2321i		−	1.02373i
$516$		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	$0.5\pi$
$524$	2.63237	−	22.1149i	0.114995	−	0.966096i
$525$		0			0
$526$		0			0
$527$		0			0
$528$		0			0
$529$	−23.0000			−1.00000
$530$		0			0
$531$		−	43.7704i		−	1.89947i
$532$	−52.3547	−	6.23184i	−2.26986	−	0.270185i
$533$		0			0
$534$		0			0
$535$			19.7355i			0.853241i
$536$	31.0000	+	5.56776i	1.33900	+	0.240491i
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	−25.2920			−1.08739			−0.543694	−	0.839284i	$-0.682975\pi$
$541$	−25.2920			−1.08739			−0.543694	+	0.839284i	$0.682975\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	19.5708			0.838321
$546$		0			0
$547$			20.2105i			0.864140i	0.901840	+	0.432070i	$0.142217\pi$
$547$			20.2105i			0.864140i	−0.901840	+	0.432070i	$0.857783\pi$
$548$		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			$\pi$
$558$	−23.5806	−	1.39848i	−0.998246	−	0.0592022i
$559$		0			0
$560$	−11.7750	+	48.7609i	−0.497584	+	2.06053i
$561$		0			0
$562$	2.39795	−	40.4334i	0.101152	−	1.70558i
$563$			46.5806i			1.96314i	0.191112	+	0.981568i	$0.438791\pi$
$563$			46.5806i			1.96314i	−0.191112	+	0.981568i	$0.561209\pi$
$564$		0			0
$565$	−40.8747			−1.71961
$566$	47.1612	+	2.79695i	1.98233	+	0.117565i
$567$			29.6663i			1.24587i
$568$	35.3524	+	6.34948i	1.48336	+	0.266418i
$569$		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
$574$	56.2289	+	3.33473i	2.34695	+	0.139189i
$575$		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	$0.377754\pi$
$578$	−1.42332	+	23.9995i	−0.0592022	+	0.998246i
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$	2.17684	−	36.7051i	0.0899244	−	1.51627i
$587$		0			0			−	1.00000i	$-0.5\pi$
$587$		0			0				1.00000i	$0.5\pi$
$588$		0			0
$589$	−44.5288			−1.83478
$590$	−78.3629	−	4.64741i	−3.22615	−	0.191331i
$591$		0			0
$592$		0			0
$593$	17.0132			0.698647			0.349324	−	0.937002i	$-0.386411\pi$
$593$	17.0132			0.698647			0.349324	+	0.937002i	$0.386411\pi$
$594$		0			0
$595$		0			0
$596$	19.8598	+	2.36394i	0.813489	+	0.0968306i
$597$		0			0
$598$		0			0
$599$			48.4717i			1.98050i	0.139300	+	0.990250i	$0.455515\pi$
$599$			48.4717i			1.98050i	−0.139300	+	0.990250i	$0.544485\pi$
$600$		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
$605$	−41.8496			−1.70143
$606$		0			0
$607$		−	11.1355i		−	0.451977i	−0.974130	−	0.225989i	$-0.927439\pi$
$607$		−	11.1355i		−	0.451977i	0.974130	−	0.225989i	$-0.0725612\pi$
$608$	43.2714	+	13.2047i	1.75489	+	0.535522i
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$		0			0		1.00000			$0$
$613$		0			0		−1.00000			$\pi$
$614$	−39.2111	−	2.32546i	−1.58243	−	0.0938480i
$615$		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	$0.5\pi$
$620$	−5.00744	+	42.0683i	−0.201103	+	1.68950i
$621$		0			0
$622$	45.8481	+	2.71908i	1.83834	+	0.109025i
$623$		0			0
$624$		0			0
$625$	17.3910			0.695639
$626$		0			0
$627$		0			0
$628$	−4.89528	−	0.582691i	−0.195343	−	0.0232519i
$629$		0			0
$630$	53.1121	+	3.14988i	2.11604	+	0.125494i
$631$		0			0			−	1.00000i	$-0.5\pi$
$631$		0			0				1.00000i	$0.5\pi$
$632$		0			0
$633$		0			0
$634$	−2.97894	+	50.2297i	−0.118309	+	1.99488i
$635$		0			0
$636$		0			0
$637$		0			0
$638$		0			0
$639$		−	38.0969i		−	1.50709i
$640$	17.3411	−	39.3954i	0.685467	−	1.55724i
$641$		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
$652$	−4.11333	+	34.5567i	−0.161090	+	1.35335i
$653$	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
$655$			42.3652i			1.65535i
$656$	−46.9828	−	11.3456i	−1.83437	−	0.442971i
$657$		0			0
$658$	−3.07316	+	51.8184i	−0.119804	+	2.02009i
$659$		−	24.9649i		−	0.972495i	−0.873821	−	0.486248i	$-0.838365\pi$
$659$		−	24.9649i		−	0.972495i	0.873821	−	0.486248i	$-0.161635\pi$
$660$		0			0
$661$	50.7982			1.97582			0.987911	−	0.155020i	$-0.0495442\pi$
$661$	50.7982			1.97582			0.987911	+	0.155020i	$0.0495442\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	100.295			3.88928
$666$		0			0
$667$		0			0
$668$		0			0
$669$		0			0
$670$	−59.8084	−	3.54701i	−2.31060	−	0.137033i
$671$		0			0
$672$		0			0
$673$		0			0		1.00000			$0$
$673$		0			0		−1.00000			$\pi$
$674$		0			0
$675$		0			0
$676$	−25.8177	−	3.07312i	−0.992990	−	0.118197i
$677$		0			0		1.00000			$0$
$677$		0			0		−1.00000			$\pi$
$678$		0			0
$679$			64.9110i			2.49106i
$680$		0			0
$681$		0			0
$682$		0			0
$683$		−	13.6180i		−	0.521079i	−0.965463	−	0.260540i	$-0.916100\pi$
$683$		−	13.6180i		−	0.521079i	0.965463	−	0.260540i	$-0.0839005\pi$
$684$	5.67175	−	47.6493i	0.216865	−	1.82192i
$685$		0			0
$686$	−14.5872	−	0.865111i	−0.556941	−	0.0330301i
$687$		0			0
$688$		0			0
$689$		0			0
$690$		0			0
$691$			8.96970i			0.341223i	0.985338	+	0.170612i	$0.0545744\pi$
$691$			8.96970i			0.341223i	−0.985338	+	0.170612i	$0.945426\pi$
$692$	−27.8037	−	3.30951i	−1.05694	−	0.125809i
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$		0			0
$698$	−2.51174	+	42.3520i	−0.0950707	+	1.60305i
$699$		0			0
$700$	7.38250	−	62.0216i	0.279032	−	2.34420i
$701$	−8.73435			−0.329892			−0.164946	−	0.986303i	$-0.552745\pi$
$701$	−8.73435			−0.329892			−0.164946	+	0.986303i	$0.552745\pi$
$702$		0			0
$703$		0			0
$704$		0			0
$705$		0			0
$706$		0			0
$707$		−	42.0375i		−	1.58098i
$708$		0			0
$709$		0			0		1.00000			$0$
$709$		0			0		−1.00000			$\pi$
$710$	−68.2056	−	4.04502i	−2.55971	−	0.151807i
$711$		0			0
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$	21.8948	+	1.29850i	0.817107	+	0.0484596i
$719$		0			0			−	1.00000i	$-0.5\pi$
$719$		0			0				1.00000i	$0.5\pi$
$720$	−44.3785	−	10.7167i	−1.65389	−	0.399388i
$721$	20.1284			0.749623
$722$	3.76441	−	63.4740i	0.140097	−	2.36226i
$723$		0			0
$724$		0			0
$725$		0			0
$726$		0			0
$727$		−	51.2819i		−	1.90194i	−0.309281	−	0.950971i	$-0.600088\pi$
$727$		−	51.2819i		−	1.90194i	0.309281	−	0.950971i	$-0.399912\pi$
$728$		0			0
$729$	−27.0000			−1.00000
$730$		0			0
$731$		0			0
$732$		0			0
$733$	52.1378			1.92575			0.962877	−	0.269939i	$-0.0870035\pi$
$733$	52.1378			1.92575			0.962877	+	0.269939i	$0.0870035\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$	−3.03502	+	51.1753i	−0.111720	+	1.88379i
$739$		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
$745$	−38.0451			−1.39387
$746$	−1.48050	+	24.9636i	−0.0542049	+	0.913983i
$747$		0			0
$748$		0			0
$749$	−17.0990			−0.624783
$750$		0			0
$751$		−	45.6615i		−	1.66621i	−0.553113	−	0.833106i	$-0.686560\pi$
$751$		−	45.6615i		−	1.66621i	0.553113	−	0.833106i	$-0.313440\pi$
$752$	10.4557	−	43.2976i	0.381279	−	1.57890i
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$		0			0		1.00000			$0$
$757$		0			0		−1.00000			$\pi$
$758$	47.1612	+	2.79695i	1.71297	+	0.101590i
$759$		0			0
$760$	−84.7052	−	15.2135i	−3.07258	−	0.551852i
$761$		0			0		1.00000			$0$
$761$		0			0		−1.00000			$\pi$
$762$		0			0
$763$			16.9563i			0.613858i
$764$	−0.114898	+	0.965273i	−0.00415685	+	0.0349224i
$765$		0			0
$766$		0			0
$767$		0			0
$768$		0			0
$769$	−41.1798			−1.48498			−0.742491	−	0.669856i	$-0.766356\pi$
$769$	−41.1798			−1.48498			−0.742491	+	0.669856i	$0.766356\pi$
$770$		0			0
$771$		0			0
$772$	54.2200	+	6.45387i	1.95142	+	0.232280i
$773$		0			0		1.00000			$0$
$773$		0			0		−1.00000			$\pi$
$774$		0			0
$775$		−	52.7507i		−	1.89486i
$776$	9.84618	−	54.8212i	0.353457	−	1.96797i
$777$		0			0
$778$		0			0
$779$			96.6378i			3.46241i
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$	−15.0291	−	3.62929i	−0.536755	−	0.129618i
$785$	9.37781			0.334708
$786$		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
$791$		−	35.4141i		−	1.25918i
$792$		0			0
$793$		0			0
$794$	−1.81697	+	30.6371i	−0.0644818	+	1.08727i
$795$		0			0
$796$		0			0
$797$		0			0		1.00000			$0$
$797$		0			0		−1.00000			$\pi$
$798$		0			0
$799$		0			0
$800$	−15.6429	+	51.2611i	−0.553059	+	1.81235i
$801$		0			0
$802$		0			0
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$	−6.37656	+	35.5032i	−0.224327	+	1.24900i
$809$		0			0		1.00000			$0$
$809$		0			0		−1.00000			$\pi$
$810$	−2.86678	+	48.3386i	−0.100728	+	1.69845i
$811$			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
$815$		−	66.1997i		−	2.31888i
$816$		0			0
$817$		0			0
$818$		0			0
$819$		0			0
$820$	91.2979	+	10.8673i	3.18826	+	0.379502i
$821$		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	$0.5\pi$
$824$	−16.9997	−	3.05323i	−0.592212	−	0.106364i
$825$		0			0
$826$	4.02654	−	67.8941i	0.140101	−	2.36234i
$827$		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
$838$	−25.9369	−	1.53822i	−0.895977	−	0.0531370i
$839$		−	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
$842$	2.45403	−	41.3790i	0.0845715	−	1.42601i
$843$		0			0
$844$	2.55490	−	21.4641i	0.0879433	−	0.738825i
$845$	49.4587			1.70143
$846$	−47.1612	−	2.79695i	−1.62143	−	0.0961612i
$847$		−	36.2588i		−	1.24587i
$848$		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
$855$			91.2810i			3.12174i
$856$	14.4411	+	2.59370i	0.493587	+	0.0886509i
$857$	−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
$862$	15.7204	+	0.932318i	0.535439	+	0.0317549i
$863$		0			0			−	1.00000i	$-0.5\pi$
$863$		0			0				1.00000i	$0.5\pi$
$864$		0			0
$865$	53.2632			1.81100
$866$		0			0
$867$		0			0
$868$	−36.4482	−	4.33847i	−1.23713	−	0.147257i
$869$		0			0
$870$		0			0
$871$		0			0
$872$	2.57205	−	14.3206i	0.0871007	−	0.484956i
$873$	−59.0771			−1.99946
$874$		0			0
$875$			56.1107i			1.89689i
$876$		0			0
$877$	16.3434			0.551876			0.275938	−	0.961175i	$-0.411011\pi$
$877$	16.3434			0.551876			0.275938	+	0.961175i	$0.411011\pi$
$878$	59.1223	+	3.50632i	1.99528	+	0.118333i
$879$		0			0
$880$		0			0
$881$		0			0		1.00000			$0$
$881$		0			0		−1.00000			$\pi$
$882$	−0.970858	+	16.3703i	−0.0326905	+	0.551215i
$883$		0			0			−	1.00000i	$-0.5\pi$
$883$		0			0				1.00000i	$0.5\pi$
$884$		0			0
$885$		0			0
$886$	−47.1456	−	2.79603i	−1.58389	−	0.0939344i
$887$		−	4.26834i		−	0.143317i	−0.997429	−	0.0716584i	$-0.977171\pi$
$887$		−	4.26834i		−	0.143317i	0.997429	−	0.0716584i	$-0.0228292\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−89.0576			−2.98020
$894$		0			0
$895$		0			0
$896$	34.1324	+	15.0244i	1.14028	+	0.501931i
$897$		0			0
$898$		0			0
$899$		0			0
$900$	56.4474	+	6.71900i	1.88158	+	0.223967i
$901$		0			0
$902$		0			0
$903$		0			0
$904$	−5.37187	+	29.9093i	−0.178666	+	0.994769i
$905$		0			0
$906$		0			0
$907$			59.7656i			1.98448i	0.124324	+	0.992242i	$0.460324\pi$
$907$			59.7656i			1.98448i	−0.124324	+	0.992242i	$0.539676\pi$
$908$	2.63237	−	22.1149i	0.0873582	−	0.733910i
$909$	38.2594			1.26898
$910$		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
$917$	−36.7055			−1.21212
$918$		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
$927$			18.3194i			0.601688i
$928$		0			0
$929$		0			0		1.00000			$0$
$929$		0			0		−1.00000			$\pi$
$930$		0			0
$931$			30.9130i			1.01313i
$932$	−41.7690	−	4.97182i	−1.36819	−	0.162857i
$933$		0			0
$934$	37.8387	+	2.24407i	1.23812	+	0.0734283i
$935$		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	$0.259347\pi$
$938$	3.07316	−	51.8184i	0.100342	−	1.69193i
$939$		0			0
$940$	−10.0149	+	84.1366i	−0.326649	+	2.74423i
$941$		0			0		1.00000			$0$
$941$		0			0		−1.00000			$\pi$
$942$		0			0
$943$		0			0
$944$	−13.6993	+	56.7298i	−0.445875	+	1.84640i
$945$		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
$950$	106.970	+	6.34397i	3.47055	+	0.205825i
$951$		0			0
$952$		0			0
$953$		0			0		1.00000			$0$
$953$		0			0		−1.00000			$\pi$
$954$		0			0
$955$		−	1.84916i		−	0.0598374i
$956$		0			0
$957$		0			0
$958$	−12.5880	−	0.746545i	−0.406699	−	0.0241198i
$959$		0			0
$960$		0			0
$961$	−31.0000			−1.00000
$962$		0			0
$963$		−	15.5622i		−	0.501485i
$964$		0			0
$965$	−103.868			−3.34364
$966$		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
$970$	−6.27263	+	105.767i	−0.201402	+	3.39597i
$971$			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
$977$	−57.7375			−1.84719			−0.923593	−	0.383375i	$-0.874762\pi$
$977$	−57.7375			−1.84719			−0.923593	+	0.383375i	$0.874762\pi$
$978$		0			0
$979$		0			0
$980$	29.2049	+	3.47629i	0.932916	+	0.111046i
$981$	−15.4323			−0.492716
$982$		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	$0.5\pi$
$992$	30.1246	+	9.19284i	0.956457	+	0.291873i
$993$		0			0
$994$	3.50463	−	59.0938i	0.111160	−	1.87434i
$995$		0			0
$996$		0			0
$997$	−40.5100			−1.28297			−0.641483	−	0.767137i	$-0.721680\pi$
$997$	−40.5100			−1.28297			−0.641483	+	0.767137i	$0.721680\pi$
$998$		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.123.4	yes	6
3.2	odd	2		1116.2.g.f.991.3		6
4.3	odd	2	inner	124.2.d.c.123.3	✓	6
8.3	odd	2		1984.2.h.f.1983.1		6
8.5	even	2		1984.2.h.f.1983.2		6
12.11	even	2		1116.2.g.f.991.4		6
31.30	odd	2	CM	124.2.d.c.123.4	yes	6
93.92	even	2		1116.2.g.f.991.3		6
124.123	even	2	inner	124.2.d.c.123.3	✓	6
248.61	odd	2		1984.2.h.f.1983.2		6
248.123	even	2		1984.2.h.f.1983.1		6
372.371	odd	2		1116.2.g.f.991.4		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
124.2.d.c.123.3	✓	6	4.3	odd	2	inner
124.2.d.c.123.3	✓	6	124.123	even	2	inner
124.2.d.c.123.4	yes	6	1.1	even	1	trivial
124.2.d.c.123.4	yes	6	31.30	odd	2	CM
1116.2.g.f.991.3		6	3.2	odd	2
1116.2.g.f.991.3		6	93.92	even	2
1116.2.g.f.991.4		6	12.11	even	2
1116.2.g.f.991.4		6	372.371	odd	2
1984.2.h.f.1983.1		6	8.3	odd	2
1984.2.h.f.1983.1		6	248.123	even	2
1984.2.h.f.1983.2		6	8.5	even	2
1984.2.h.f.1983.2		6	248.61	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q+(-0.0837246 + 1.41173i) q^{2} +(-1.98598 - 0.236394i) q^{4} +3.80451 q^{5} +3.29625i q^{7} +(0.500000 - 2.78388i) q^{8} -3.00000 q^{9} +O(q^{10})\)	\(q+(-0.0837246 + 1.41173i) q^{2} +(-1.98598 - 0.236394i) q^{4} +3.80451 q^{5} +3.29625i q^{7} +(0.500000 - 2.78388i) q^{8} -3.00000 q^{9} +(-0.318531 + 5.37095i) q^{10} +(-4.65343 - 0.275977i) q^{14} +(3.88824 + 0.938946i) q^{16} +(0.251174 - 4.23520i) q^{18} -7.99761i q^{19} +(-7.55569 - 0.899362i) q^{20} +9.47431 q^{25} +(0.779213 - 6.54629i) q^{28} -5.56776i q^{31} +(-1.65108 + 5.41054i) q^{32} +12.5406i q^{35} +(5.95794 + 0.709181i) q^{36} +(11.2905 + 0.669597i) q^{38} +(1.90226 - 10.5913i) q^{40} -12.0833 q^{41} -11.4135 q^{45} -11.1355i q^{47} -3.86529 q^{49} +(-0.793233 + 13.3752i) q^{50} +(9.17638 + 1.64813i) q^{56} +14.5901i q^{59} +(7.86020 + 0.466159i) q^{62} -9.88876i q^{63} +(-7.50000 - 2.78388i) q^{64} +11.1355i q^{67} +(-17.7040 - 1.04996i) q^{70} +12.6990i q^{71} +(-1.50000 + 8.35165i) q^{72} +(-1.89058 + 15.8831i) q^{76} +(14.7928 + 3.57223i) q^{80} +9.00000 q^{81} +(1.01167 - 17.0584i) q^{82} +(0.955594 - 16.1129i) q^{90} +(15.7204 + 0.932318i) q^{94} -30.4270i q^{95} +19.6924 q^{97} +(0.323619 - 5.45675i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 3 q^{8} - 18 q^{9}+O(q^{10}) \) 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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