LMFDB - Embedded newform 124.5.c.a.61.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [124,5,Mod(61,124)]

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([0, 1]))

N = Newforms(chi, 5, names="a")

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

chi := DirichletCharacter("124.61");

S:= CuspForms(chi, 5);

N := Newforms(S);

Level:	$ N $	$=$	$ 124 = 2^{2} \cdot 31 $
Weight:	$ k $	$=$	$ 5 $
Character orbit:	$[\chi]$	$=$	124.c (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:	$12.8178754224$
Analytic rank:	$0$
Dimension:	$10$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} + 478x^{8} + 69668x^{6} + 4198200x^{4} + 101304000x^{2} + 622080000 $ x^10 + 478x^8 + 69668x^6 + 4198200x^4 + 101304000x^2 + 622080000
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{9}\cdot 5^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			61.7
Root			$6.79929i$ of defining polynomial
Character	$\chi$	$=$	124.61
Dual form			124.5.c.a.61.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+6.79929i q^{3} -34.9624 q^{5} -12.5932 q^{7} +34.7697 q^{9} +O(q^{10})$	$q+6.79929i q^{3} -34.9624 q^{5} -12.5932 q^{7} +34.7697 q^{9} -8.83961i q^{11} -264.619i q^{13} -237.720i q^{15} -75.3007i q^{17} +427.955 q^{19} -85.6246i q^{21} -3.44118i q^{23} +597.372 q^{25} +787.151i q^{27} -1247.29i q^{29} +(-381.004 - 882.245i) q^{31} +60.1031 q^{33} +440.288 q^{35} -762.838i q^{37} +1799.22 q^{39} -938.638 q^{41} -2824.41i q^{43} -1215.63 q^{45} +152.443 q^{47} -2242.41 q^{49} +511.991 q^{51} +3342.17i q^{53} +309.054i q^{55} +2909.79i q^{57} -990.978 q^{59} -1649.09i q^{61} -437.861 q^{63} +9251.74i q^{65} +3816.55 q^{67} +23.3975 q^{69} -4175.61 q^{71} +6354.67i q^{73} +4061.70i q^{75} +111.319i q^{77} -6456.90i q^{79} -2535.72 q^{81} -5591.64i q^{83} +2632.70i q^{85} +8480.71 q^{87} +3479.25i q^{89} +3332.40i q^{91} +(5998.64 - 2590.55i) q^{93} -14962.3 q^{95} -16279.6 q^{97} -307.351i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q - 6 q^{5} - 2 q^{7} - 146 q^{9}+O(q^{10}) $ 10 * q - 6 * q^5 - 2 * q^7 - 146 * q^9	$ 10 q - 6 q^{5} - 2 q^{7} - 146 q^{9} - 310 q^{19} - 756 q^{25} + 734 q^{31} - 372 q^{33} - 666 q^{35} - 1924 q^{39} - 210 q^{41} + 650 q^{45} - 1992 q^{47} + 188 q^{49} + 1352 q^{51} + 5610 q^{59} + 3478 q^{63} - 5420 q^{67} + 10160 q^{69} - 1734 q^{71} + 11598 q^{81} - 13244 q^{87} + 3088 q^{93} + 4302 q^{95} + 2942 q^{97}+O(q^{100}) $ 10 * q - 6 * q^5 - 2 * q^7 - 146 * q^9 - 310 * q^19 - 756 * q^25 + 734 * q^31 - 372 * q^33 - 666 * q^35 - 1924 * q^39 - 210 * q^41 + 650 * q^45 - 1992 * q^47 + 188 * q^49 + 1352 * q^51 + 5610 * q^59 + 3478 * q^63 - 5420 * q^67 + 10160 * q^69 - 1734 * q^71 + 11598 * q^81 - 13244 * q^87 + 3088 * q^93 + 4302 * q^95 + 2942 * q^97

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			0
$2$		0			0
$3$			6.79929i			0.755476i	0.925912	+	0.377738i	$0.123298\pi$
$3$			6.79929i			0.755476i	−0.925912	+	0.377738i	$0.876702\pi$
$4$		0			0
$5$	−34.9624			−1.39850			−0.699249	−	0.714879i	$-0.746482\pi$
$5$	−34.9624			−1.39850			−0.699249	+	0.714879i	$0.746482\pi$
$6$		0			0
$7$	−12.5932			−0.257004			−0.128502	−	0.991709i	$-0.541017\pi$
$7$	−12.5932			−0.257004			−0.128502	+	0.991709i	$0.541017\pi$
$8$		0			0
$9$	34.7697			0.429255
$10$		0			0
$11$		−	8.83961i		−	0.0730546i	−0.999333	−	0.0365273i	$-0.988370\pi$
$11$		−	8.83961i		−	0.0730546i	0.999333	−	0.0365273i	$-0.0116296\pi$
$12$		0			0
$13$		−	264.619i		−	1.56579i	−0.622151	−	0.782897i	$-0.713741\pi$
$13$		−	264.619i		−	1.56579i	0.622151	−	0.782897i	$-0.286259\pi$
$14$		0			0
$15$		−	237.720i		−	1.05653i
$16$		0			0
$17$		−	75.3007i		−	0.260556i	−0.991477	−	0.130278i	$-0.958413\pi$
$17$		−	75.3007i		−	0.260556i	0.991477	−	0.130278i	$-0.0415870\pi$
$18$		0			0
$19$	427.955			1.18547			0.592735	−	0.805397i	$-0.298048\pi$
$19$	427.955			1.18547			0.592735	+	0.805397i	$0.298048\pi$
$20$		0			0
$21$		−	85.6246i		−	0.194160i
$22$		0			0
$23$		−	3.44118i		−	0.00650506i	−0.999995	−	0.00325253i	$-0.998965\pi$
$23$		−	3.44118i		−	0.00650506i	0.999995	−	0.00325253i	$-0.00103531\pi$
$24$		0			0
$25$	597.372			0.955795
$26$		0			0
$27$			787.151i			1.07977i
$28$		0			0
$29$		−	1247.29i		−	1.48311i	−0.670893	−	0.741554i	$-0.734089\pi$
$29$		−	1247.29i		−	1.48311i	0.670893	−	0.741554i	$-0.265911\pi$
$30$		0			0
$31$	−381.004	−	882.245i	−0.396466	−	0.918049i
$31$	−381.004	−	882.245i	−0.396466	−	0.918049i
$32$		0			0
$33$	60.1031			0.0551910
$34$		0			0
$35$	440.288			0.359419
$36$		0			0
$37$		−	762.838i		−	0.557223i	−0.960404	−	0.278611i	$-0.910126\pi$
$37$		−	762.838i		−	0.557223i	0.960404	−	0.278611i	$-0.0898742\pi$
$38$		0			0
$39$	1799.22			1.18292
$40$		0			0
$41$	−938.638			−0.558381			−0.279190	−	0.960236i	$-0.590066\pi$
$41$	−938.638			−0.558381			−0.279190	+	0.960236i	$0.590066\pi$
$42$		0			0
$43$		−	2824.41i		−	1.52753i	−0.645492	−	0.763767i	$-0.723348\pi$
$43$		−	2824.41i		−	1.52753i	0.645492	−	0.763767i	$-0.276652\pi$
$44$		0			0
$45$	−1215.63			−0.600313
$46$		0			0
$47$	152.443			0.0690101			0.0345050	−	0.999405i	$-0.489015\pi$
$47$	152.443			0.0690101			0.0345050	+	0.999405i	$0.489015\pi$
$48$		0			0
$49$	−2242.41			−0.933949
$50$		0			0
$51$	511.991			0.196844
$52$		0			0
$53$			3342.17i			1.18981i	0.803797	+	0.594904i	$0.202810\pi$
$53$			3342.17i			1.18981i	−0.803797	+	0.594904i	$0.797190\pi$
$54$		0			0
$55$			309.054i			0.102167i
$56$		0			0
$57$			2909.79i			0.895595i
$58$		0			0
$59$	−990.978			−0.284682			−0.142341	−	0.989818i	$-0.545463\pi$
$59$	−990.978			−0.284682			−0.142341	+	0.989818i	$0.545463\pi$
$60$		0			0
$61$		−	1649.09i		−	0.443184i	−0.975139	−	0.221592i	$-0.928875\pi$
$61$		−	1649.09i		−	0.443184i	0.975139	−	0.221592i	$-0.0711253\pi$
$62$		0			0
$63$	−437.861			−0.110320
$64$		0			0
$65$			9251.74i			2.18976i
$66$		0			0
$67$	3816.55			0.850201			0.425100	−	0.905146i	$-0.360239\pi$
$67$	3816.55			0.850201			0.425100	+	0.905146i	$0.360239\pi$
$68$		0			0
$69$	23.3975			0.00491442
$70$		0			0
$71$	−4175.61			−0.828331			−0.414165	−	0.910202i	$-0.635926\pi$
$71$	−4175.61			−0.828331			−0.414165	+	0.910202i	$0.635926\pi$
$72$		0			0
$73$			6354.67i			1.19247i	0.802810	+	0.596235i	$0.203337\pi$
$73$			6354.67i			1.19247i	−0.802810	+	0.596235i	$0.796663\pi$
$74$		0			0
$75$			4061.70i			0.722080i
$76$		0			0
$77$			111.319i			0.0187753i
$78$		0			0
$79$		−	6456.90i		−	1.03459i	−0.855806	−	0.517297i	$-0.826938\pi$
$79$		−	6456.90i		−	1.03459i	0.855806	−	0.517297i	$-0.173062\pi$
$80$		0			0
$81$	−2535.72			−0.386484
$82$		0			0
$83$		−	5591.64i		−	0.811677i	−0.913945	−	0.405838i	$-0.866980\pi$
$83$		−	5591.64i		−	0.811677i	0.913945	−	0.405838i	$-0.133020\pi$
$84$		0			0
$85$			2632.70i			0.364387i
$86$		0			0
$87$	8480.71			1.12045
$88$		0			0
$89$			3479.25i			0.439244i	0.975585	+	0.219622i	$0.0704824\pi$
$89$			3479.25i			0.439244i	−0.975585	+	0.219622i	$0.929518\pi$
$90$		0			0
$91$			3332.40i			0.402415i
$92$		0			0
$93$	5998.64	−	2590.55i	0.693565	−	0.299521i
$94$		0			0
$95$	−14962.3			−1.65788
$96$		0			0
$97$	−16279.6			−1.73021			−0.865107	−	0.501587i	$-0.832750\pi$
$97$	−16279.6			−1.73021			−0.865107	+	0.501587i	$0.832750\pi$
$98$		0			0
$99$		−	307.351i		−	0.0313591i
$100$		0			0
$101$	8732.60			0.856053			0.428027	−	0.903766i	$-0.359209\pi$
$101$	8732.60			0.856053			0.428027	+	0.903766i	$0.359209\pi$
$102$		0			0
$103$	1195.64			0.112701			0.0563505	−	0.998411i	$-0.482054\pi$
$103$	1195.64			0.112701			0.0563505	+	0.998411i	$0.482054\pi$
$104$		0			0
$105$			2993.65i			0.271532i
$106$		0			0
$107$	−6977.93			−0.609480			−0.304740	−	0.952436i	$-0.598570\pi$
$107$	−6977.93			−0.609480			−0.304740	+	0.952436i	$0.598570\pi$
$108$		0			0
$109$	3233.70			0.272174			0.136087	−	0.990697i	$-0.456547\pi$
$109$	3233.70			0.272174			0.136087	+	0.990697i	$0.456547\pi$
$110$		0			0
$111$	5186.76			0.420969
$112$		0			0
$113$	−11454.2			−0.897033			−0.448516	−	0.893775i	$-0.648047\pi$
$113$	−11454.2			−0.897033			−0.448516	+	0.893775i	$0.648047\pi$
$114$		0			0
$115$			120.312i			0.00909730i
$116$		0			0
$117$		−	9200.73i		−	0.672126i
$118$		0			0
$119$			948.275i			0.0669639i
$120$		0			0
$121$	14562.9			0.994663
$122$		0			0
$123$		−	6382.07i		−	0.421844i
$124$		0			0
$125$	965.957			0.0618212
$126$		0			0
$127$		−	16122.9i		−	0.999621i	−0.866135	−	0.499810i	$-0.833403\pi$
$127$		−	16122.9i		−	0.999621i	0.866135	−	0.499810i	$-0.166597\pi$
$128$		0			0
$129$	19204.0			1.15402
$130$		0			0
$131$	29454.3			1.71635			0.858176	−	0.513356i	$-0.171598\pi$
$131$	29454.3			1.71635			0.858176	+	0.513356i	$0.171598\pi$
$132$		0			0
$133$	−5389.31			−0.304670
$134$		0			0
$135$		−	27520.7i		−	1.51005i
$136$		0			0
$137$		−	20164.9i		−	1.07437i	−0.843464	−	0.537185i	$-0.819488\pi$
$137$		−	20164.9i		−	1.07437i	0.843464	−	0.537185i	$-0.180512\pi$
$138$		0			0
$139$		−	24894.5i		−	1.28847i	−0.764829	−	0.644234i	$-0.777176\pi$
$139$		−	24894.5i		−	1.28847i	0.764829	−	0.644234i	$-0.222824\pi$
$140$		0			0
$141$			1036.51i			0.0521355i
$142$		0			0
$143$	−2339.13			−0.114389
$144$		0			0
$145$			43608.4i			2.07412i
$146$		0			0
$147$		−	15246.8i		−	0.705576i
$148$		0			0
$149$	11873.2			0.534806			0.267403	−	0.963585i	$-0.413834\pi$
$149$	11873.2			0.534806			0.267403	+	0.963585i	$0.413834\pi$
$150$		0			0
$151$		−	17188.1i		−	0.753833i	−0.926247	−	0.376916i	$-0.876984\pi$
$151$		−	17188.1i		−	0.753833i	0.926247	−	0.376916i	$-0.123016\pi$
$152$		0			0
$153$		−	2618.18i		−	0.111845i
$154$		0			0
$155$	13320.8	+	30845.4i	0.554457	+	1.28389i
$156$		0			0
$157$	−22249.4			−0.902648			−0.451324	−	0.892360i	$-0.649048\pi$
$157$	−22249.4			−0.902648			−0.451324	+	0.892360i	$0.649048\pi$
$158$		0			0
$159$	−22724.4			−0.898872
$160$		0			0
$161$			43.3353i			0.00167182i
$162$		0			0
$163$	45127.3			1.69849			0.849247	−	0.527995i	$-0.177056\pi$
$163$	45127.3			1.69849			0.849247	+	0.527995i	$0.177056\pi$
$164$		0			0
$165$	−2101.35			−0.0771845
$166$		0			0
$167$			26653.4i			0.955695i	0.878443	+	0.477847i	$0.158583\pi$
$167$			26653.4i			0.955695i	−0.878443	+	0.477847i	$0.841417\pi$
$168$		0			0
$169$	−41462.4			−1.45171
$170$		0			0
$171$	14879.9			0.508870
$172$		0			0
$173$	−54370.6			−1.81665			−0.908326	−	0.418262i	$-0.862639\pi$
$173$	−54370.6			−1.81665			−0.908326	+	0.418262i	$0.862639\pi$
$174$		0			0
$175$	−7522.81			−0.245643
$176$		0			0
$177$		−	6737.95i		−	0.215071i
$178$		0			0
$179$		−	10769.2i		−	0.336107i	−0.985778	−	0.168054i	$-0.946252\pi$
$179$		−	10769.2i		−	0.336107i	0.985778	−	0.168054i	$-0.0537482\pi$
$180$		0			0
$181$		−	3119.04i		−	0.0952057i	−0.998866	−	0.0476029i	$-0.984842\pi$
$181$		−	3119.04i		−	0.0952057i	0.998866	−	0.0476029i	$-0.0151582\pi$
$182$		0			0
$183$	11212.6			0.334815
$184$		0			0
$185$			26670.7i			0.779275i
$186$		0			0
$187$	−665.629			−0.0190348
$188$		0			0
$189$		−	9912.74i		−	0.277504i
$190$		0			0
$191$	−4529.12			−0.124150			−0.0620751	−	0.998071i	$-0.519772\pi$
$191$	−4529.12			−0.124150			−0.0620751	+	0.998071i	$0.519772\pi$
$192$		0			0
$193$	35052.1			0.941022			0.470511	−	0.882394i	$-0.344070\pi$
$193$	35052.1			0.941022			0.470511	+	0.882394i	$0.344070\pi$
$194$		0			0
$195$	−62905.2			−1.65431
$196$		0			0
$197$			26084.7i			0.672131i	0.941839	+	0.336065i	$0.109096\pi$
$197$			26084.7i			0.672131i	−0.941839	+	0.336065i	$0.890904\pi$
$198$		0			0
$199$		−	33124.0i		−	0.836445i	−0.908345	−	0.418222i	$-0.862653\pi$
$199$		−	33124.0i		−	0.836445i	0.908345	−	0.418222i	$-0.137347\pi$
$200$		0			0
$201$			25949.8i			0.642307i
$202$		0			0
$203$			15707.4i			0.381164i
$204$		0			0
$205$	32817.1			0.780894
$206$		0			0
$207$		−	119.649i		−	0.00279233i
$208$		0			0
$209$		−	3782.95i		−	0.0866041i
$210$		0			0
$211$	53548.5			1.20277			0.601385	−	0.798959i	$-0.294616\pi$
$211$	53548.5			1.20277			0.601385	+	0.798959i	$0.294616\pi$
$212$		0			0
$213$		−	28391.2i		−	0.625784i
$214$		0			0
$215$			98748.2i			2.13625i
$216$		0			0
$217$	4798.05	+	11110.3i	0.101893	+	0.235942i
$218$		0			0
$219$	−43207.2			−0.900882
$220$		0			0
$221$	−19926.0			−0.407977
$222$		0			0
$223$			63305.1i			1.27300i	0.771276	+	0.636500i	$0.219619\pi$
$223$			63305.1i			1.27300i	−0.771276	+	0.636500i	$0.780381\pi$
$224$		0			0
$225$	20770.4			0.410280
$226$		0			0
$227$	−39997.0			−0.776204			−0.388102	−	0.921617i	$-0.626869\pi$
$227$	−39997.0			−0.776204			−0.388102	+	0.921617i	$0.626869\pi$
$228$		0			0
$229$			34513.6i			0.658141i	0.944305	+	0.329071i	$0.106735\pi$
$229$			34513.6i			0.658141i	−0.944305	+	0.329071i	$0.893265\pi$
$230$		0			0
$231$	−756.888			−0.0141843
$232$		0			0
$233$	−80328.0			−1.47964			−0.739818	−	0.672807i	$-0.765089\pi$
$233$	−80328.0			−1.47964			−0.739818	+	0.672807i	$0.765089\pi$
$234$		0			0
$235$	−5329.79			−0.0965104
$236$		0			0
$237$	43902.3			0.781611
$238$		0			0
$239$			89644.0i			1.56937i	0.619895	+	0.784685i	$0.287175\pi$
$239$			89644.0i			1.56937i	−0.619895	+	0.784685i	$0.712825\pi$
$240$		0			0
$241$		−	70059.0i		−	1.20623i	−0.797654	−	0.603115i	$-0.793926\pi$
$241$		−	70059.0i		−	1.20623i	0.797654	−	0.603115i	$-0.206074\pi$
$242$		0			0
$243$			46518.2i			0.787789i
$244$		0			0
$245$	78400.2			1.30613
$246$		0			0
$247$		−	113245.i		−	1.85620i
$248$		0			0
$249$	38019.2			0.613202
$250$		0			0
$251$			118206.i			1.87625i	0.346296	+	0.938125i	$0.387439\pi$
$251$			118206.i			1.87625i	−0.346296	+	0.938125i	$0.612561\pi$
$252$		0			0
$253$	−30.4187			−0.000475225
$254$		0			0
$255$	−17900.5			−0.275286
$256$		0			0
$257$	−52283.9			−0.791593			−0.395797	−	0.918338i	$-0.629531\pi$
$257$	−52283.9			−0.791593			−0.395797	+	0.918338i	$0.629531\pi$
$258$		0			0
$259$			9606.56i			0.143208i
$260$		0			0
$261$		−	43368.0i		−	0.636632i
$262$		0			0
$263$		−	40464.4i		−	0.585008i	−0.956264	−	0.292504i	$-0.905512\pi$
$263$		−	40464.4i		−	0.585008i	0.956264	−	0.292504i	$-0.0944884\pi$
$264$		0			0
$265$		−	116850.i		−	1.66394i
$266$		0			0
$267$	−23656.4			−0.331838
$268$		0			0
$269$			27782.6i			0.383944i	0.981400	+	0.191972i	$0.0614883\pi$
$269$			27782.6i			0.383944i	−0.981400	+	0.191972i	$0.938512\pi$
$270$		0			0
$271$		−	52687.3i		−	0.717410i	−0.933451	−	0.358705i	$-0.883218\pi$
$271$		−	52687.3i		−	0.717410i	0.933451	−	0.358705i	$-0.116782\pi$
$272$		0			0
$273$	−22657.9			−0.304015
$274$		0			0
$275$		−	5280.53i		−	0.0698252i
$276$		0			0
$277$		−	110138.i		−	1.43541i	−0.696347	−	0.717705i	$-0.745193\pi$
$277$		−	110138.i		−	1.43541i	0.696347	−	0.717705i	$-0.254807\pi$
$278$		0			0
$279$	−13247.4	−	30675.4i	−0.170185	−	0.394078i
$280$		0			0
$281$	42393.5			0.536891			0.268446	−	0.963295i	$-0.413490\pi$
$281$	42393.5			0.536891			0.268446	+	0.963295i	$0.413490\pi$
$282$		0			0
$283$	−44458.7			−0.555117			−0.277558	−	0.960709i	$-0.589525\pi$
$283$	−44458.7			−0.555117			−0.277558	+	0.960709i	$0.589525\pi$
$284$		0			0
$285$		−	101733.i		−	1.25249i
$286$		0			0
$287$	11820.4			0.143506
$288$		0			0
$289$	77850.8			0.932111
$290$		0			0
$291$		−	110690.i		−	1.30714i
$292$		0			0
$293$	41560.8			0.484116			0.242058	−	0.970262i	$-0.422178\pi$
$293$	41560.8			0.484116			0.242058	+	0.970262i	$0.422178\pi$
$294$		0			0
$295$	34647.0			0.398127
$296$		0			0
$297$	6958.11			0.0788821
$298$		0			0
$299$	−910.602			−0.0101856
$300$		0			0
$301$			35568.3i			0.392582i
$302$		0			0
$303$			59375.4i			0.646728i
$304$		0			0
$305$			57656.1i			0.619792i
$306$		0			0
$307$	17272.9			0.183268			0.0916341	−	0.995793i	$-0.470791\pi$
$307$	17272.9			0.183268			0.0916341	+	0.995793i	$0.470791\pi$
$308$		0			0
$309$			8129.53i			0.0851429i
$310$		0			0
$311$	−83035.0			−0.858500			−0.429250	−	0.903186i	$-0.641222\pi$
$311$	−83035.0			−0.858500			−0.429250	+	0.903186i	$0.641222\pi$
$312$		0			0
$313$			82900.4i			0.846190i	0.906085	+	0.423095i	$0.139056\pi$
$313$			82900.4i			0.846190i	−0.906085	+	0.423095i	$0.860944\pi$
$314$		0			0
$315$	15308.7			0.154283
$316$		0			0
$317$	149791.			1.49062			0.745312	−	0.666716i	$-0.232301\pi$
$317$	149791.			1.49062			0.745312	+	0.666716i	$0.232301\pi$
$318$		0			0
$319$	−11025.6			−0.108348
$320$		0			0
$321$		−	47445.0i		−	0.460448i
$322$		0			0
$323$		−	32225.3i		−	0.308882i
$324$		0			0
$325$		−	158076.i		−	1.49658i
$326$		0			0
$327$			21986.9i			0.205621i
$328$		0			0
$329$	−1919.75			−0.0177358
$330$		0			0
$331$			7417.31i			0.0677003i	0.999427	+	0.0338501i	$0.0107769\pi$
$331$			7417.31i			0.0677003i	−0.999427	+	0.0338501i	$0.989223\pi$
$332$		0			0
$333$		−	26523.6i		−	0.239191i
$334$		0			0
$335$	−133436.			−1.18900
$336$		0			0
$337$			170386.i			1.50028i	0.661278	+	0.750141i	$0.270014\pi$
$337$			170386.i			1.50028i	−0.661278	+	0.750141i	$0.729986\pi$
$338$		0			0
$339$		−	77880.5i		−	0.677687i
$340$		0			0
$341$	−7798.71	+	3367.93i	−0.0670678	+	0.0289637i
$342$		0			0
$343$	58475.3			0.497032
$344$		0			0
$345$	−818.035			−0.00687280
$346$		0			0
$347$			15520.8i			0.128901i	0.997921	+	0.0644503i	$0.0205294\pi$
$347$			15520.8i			0.128901i	−0.997921	+	0.0644503i	$0.979471\pi$
$348$		0			0
$349$	−43857.3			−0.360073			−0.180037	−	0.983660i	$-0.557622\pi$
$349$	−43857.3			−0.360073			−0.180037	+	0.983660i	$0.557622\pi$
$350$		0			0
$351$	208295.			1.69070
$352$		0			0
$353$		−	20603.8i		−	0.165348i	−0.996577	−	0.0826738i	$-0.973654\pi$
$353$		−	20603.8i		−	0.165348i	0.996577	−	0.0826738i	$-0.0263460\pi$
$354$		0			0
$355$	145990.			1.15842
$356$		0			0
$357$	−6447.60			−0.0505896
$358$		0			0
$359$	57571.4			0.446702			0.223351	−	0.974738i	$-0.428300\pi$
$359$	57571.4			0.446702			0.223351	+	0.974738i	$0.428300\pi$
$360$		0			0
$361$	52824.3			0.405340
$362$		0			0
$363$			99017.1i			0.751444i
$364$		0			0
$365$		−	222175.i		−	1.66766i
$366$		0			0
$367$			79385.2i			0.589396i	0.955590	+	0.294698i	$0.0952190\pi$
$367$			79385.2i			0.589396i	−0.955590	+	0.294698i	$0.904781\pi$
$368$		0			0
$369$	−32636.2			−0.239688
$370$		0			0
$371$		−	42088.6i		−	0.305785i
$372$		0			0
$373$	−133979.			−0.962983			−0.481491	−	0.876451i	$-0.659905\pi$
$373$	−133979.			−0.962983			−0.481491	+	0.876451i	$0.659905\pi$
$374$		0			0
$375$			6567.82i			0.0467045i
$376$		0			0
$377$	−330058.			−2.32224
$378$		0			0
$379$	−198010.			−1.37850			−0.689251	−	0.724522i	$-0.742060\pi$
$379$	−198010.			−1.37850			−0.689251	+	0.724522i	$0.742060\pi$
$380$		0			0
$381$	109624.			0.755190
$382$		0			0
$383$			139818.i			0.953162i	0.879131	+	0.476581i	$0.158124\pi$
$383$			139818.i			0.953162i	−0.879131	+	0.476581i	$0.841876\pi$
$384$		0			0
$385$		−	3891.98i		−	0.0262572i
$386$		0			0
$387$		−	98203.8i		−	0.655702i
$388$		0			0
$389$		−	173967.i		−	1.14965i	−0.818275	−	0.574827i	$-0.805069\pi$
$389$		−	173967.i		−	1.14965i	0.818275	−	0.574827i	$-0.194931\pi$
$390$		0			0
$391$	−259.123			−0.00169493
$392$		0			0
$393$			200268.i			1.29666i
$394$		0			0
$395$			225749.i			1.44688i
$396$		0			0
$397$	−219406.			−1.39209			−0.696046	−	0.717997i	$-0.745059\pi$
$397$	−219406.			−1.39209			−0.696046	+	0.717997i	$0.745059\pi$
$398$		0			0
$399$		−	36643.5i		−	0.230171i
$400$		0			0
$401$			286205.i			1.77987i	0.456088	+	0.889935i	$0.349250\pi$
$401$			286205.i			1.77987i	−0.456088	+	0.889935i	$0.650750\pi$
$402$		0			0
$403$	−233459.	+	100821.i	−1.43748	+	0.620784i
$404$		0			0
$405$	88655.1			0.540497
$406$		0			0
$407$	−6743.19			−0.0407077
$408$		0			0
$409$			92236.3i			0.551385i	0.961246	+	0.275693i	$0.0889072\pi$
$409$			92236.3i			0.551385i	−0.961246	+	0.275693i	$0.911093\pi$
$410$		0			0
$411$	137107.			0.811661
$412$		0			0
$413$	12479.6			0.0731643
$414$		0			0
$415$			195497.i			1.13513i
$416$		0			0
$417$	169265.			0.973406
$418$		0			0
$419$	−172486.			−0.982484			−0.491242	−	0.871023i	$-0.663457\pi$
$419$	−172486.			−0.982484			−0.491242	+	0.871023i	$0.663457\pi$
$420$		0			0
$421$	114061.			0.643538			0.321769	−	0.946818i	$-0.395723\pi$
$421$	114061.			0.643538			0.321769	+	0.946818i	$0.395723\pi$
$422$		0			0
$423$	5300.41			0.0296230
$424$		0			0
$425$		−	44982.5i		−	0.249038i
$426$		0			0
$427$			20767.3i			0.113900i
$428$		0			0
$429$		−	15904.4i		−	0.0864179i
$430$		0			0
$431$	−14812.8			−0.0797410			−0.0398705	−	0.999205i	$-0.512695\pi$
$431$	−14812.8			−0.0797410			−0.0398705	+	0.999205i	$0.512695\pi$
$432$		0			0
$433$			95474.9i			0.509229i	0.967043	+	0.254615i	$0.0819486\pi$
$433$			95474.9i			0.509229i	−0.967043	+	0.254615i	$0.918051\pi$
$434$		0			0
$435$	−296506.			−1.56695
$436$		0			0
$437$		−	1472.67i		−	0.00771155i
$438$		0			0
$439$	167535.			0.869315			0.434657	−	0.900596i	$-0.356870\pi$
$439$	167535.			0.869315			0.434657	+	0.900596i	$0.356870\pi$
$440$		0			0
$441$	−77968.0			−0.400903
$442$		0			0
$443$	17277.7			0.0880396			0.0440198	−	0.999031i	$-0.485984\pi$
$443$	17277.7			0.0880396			0.0440198	+	0.999031i	$0.485984\pi$
$444$		0			0
$445$		−	121643.i		−	0.614281i
$446$		0			0
$447$			80729.6i			0.404034i
$448$		0			0
$449$			274474.i			1.36147i	0.732530	+	0.680735i	$0.238339\pi$
$449$			274474.i			1.36147i	−0.732530	+	0.680735i	$0.761661\pi$
$450$		0			0
$451$			8297.20i			0.0407923i
$452$		0			0
$453$	116867.			0.569503
$454$		0			0
$455$		−	116509.i		−	0.562776i
$456$		0			0
$457$		−	145805.i		−	0.698135i	−0.937098	−	0.349067i	$-0.886498\pi$
$457$		−	145805.i		−	0.698135i	0.937098	−	0.349067i	$-0.113502\pi$
$458$		0			0
$459$	59273.1			0.281340
$460$		0			0
$461$		−	400014.i		−	1.88223i	−0.338081	−	0.941117i	$-0.609778\pi$
$461$		−	400014.i		−	1.88223i	0.338081	−	0.941117i	$-0.390222\pi$
$462$		0			0
$463$			124804.i			0.582193i	0.956694	+	0.291097i	$0.0940201\pi$
$463$			124804.i			0.582193i	−0.956694	+	0.291097i	$0.905980\pi$
$464$		0			0
$465$	−209727.	+	90572.1i	−0.969948	+	0.418879i
$466$		0			0
$467$	206843.			0.948434			0.474217	−	0.880408i	$-0.342731\pi$
$467$	206843.			0.948434			0.474217	+	0.880408i	$0.342731\pi$
$468$		0			0
$469$	−48062.5			−0.218505
$470$		0			0
$471$		−	151280.i		−	0.681929i
$472$		0			0
$473$	−24966.7			−0.111593
$474$		0			0
$475$	255648.			1.13307
$476$		0			0
$477$			116206.i			0.510732i
$478$		0			0
$479$	446715.			1.94697			0.973485	−	0.228751i	$-0.0734643\pi$
$479$	446715.			1.94697			0.973485	+	0.228751i	$0.0734643\pi$
$480$		0			0
$481$	−201862.			−0.872497
$482$		0			0
$483$	−294.649			−0.00126302
$484$		0			0
$485$	569174.			2.41970
$486$		0			0
$487$		−	169429.i		−	0.714381i	−0.934031	−	0.357191i	$-0.883735\pi$
$487$		−	169429.i		−	0.714381i	0.934031	−	0.357191i	$-0.116265\pi$
$488$		0			0
$489$			306834.i			1.28317i
$490$		0			0
$491$		−	245875.i		−	1.01988i	−0.860209	−	0.509942i	$-0.829667\pi$
$491$		−	245875.i		−	1.01988i	0.860209	−	0.509942i	$-0.170333\pi$
$492$		0			0
$493$	−93922.1			−0.386433
$494$		0			0
$495$			10745.7i			0.0438556i
$496$		0			0
$497$	52584.2			0.212884
$498$		0			0
$499$		−	362998.i		−	1.45782i	−0.684611	−	0.728909i	$-0.740028\pi$
$499$		−	362998.i		−	1.45782i	0.684611	−	0.728909i	$-0.259972\pi$
$500$		0			0
$501$	−181224.			−0.722005
$502$		0			0
$503$	−340806.			−1.34701			−0.673505	−	0.739182i	$-0.735212\pi$
$503$	−340806.			−1.34701			−0.673505	+	0.739182i	$0.735212\pi$
$504$		0			0
$505$	−305313.			−1.19719
$506$		0			0
$507$		−	281915.i		−	1.09674i
$508$		0			0
$509$		−	77394.9i		−	0.298728i	−0.988782	−	0.149364i	$-0.952277\pi$
$509$		−	77394.9i		−	0.298728i	0.988782	−	0.149364i	$-0.0477227\pi$
$510$		0			0
$511$		−	80025.5i		−	0.306469i
$512$		0			0
$513$			336865.i			1.28003i
$514$		0			0
$515$	−41802.6			−0.157612
$516$		0			0
$517$		−	1347.54i		−	0.00504151i
$518$		0			0
$519$		−	369681.i		−	1.37244i
$520$		0			0
$521$	−431761.			−1.59062			−0.795312	−	0.606200i	$-0.792693\pi$
$521$	−431761.			−1.59062			−0.795312	+	0.606200i	$0.792693\pi$
$522$		0			0
$523$		−	16882.5i		−	0.0617211i	−0.999524	−	0.0308606i	$-0.990175\pi$
$523$		−	16882.5i		−	0.0617211i	0.999524	−	0.0308606i	$-0.00982478\pi$
$524$		0			0
$525$		−	51149.7i		−	0.185577i
$526$		0			0
$527$	−66433.7	+	28689.9i	−0.239203	+	0.103302i
$528$		0			0
$529$	279829.			0.999958
$530$		0			0
$531$	−34456.0			−0.122201
$532$		0			0
$533$			248382.i			0.874310i
$534$		0			0
$535$	243966.			0.852356
$536$		0			0
$537$	73222.9			0.253921
$538$		0			0
$539$			19822.0i			0.0682293i
$540$		0			0
$541$	380444.			1.29986			0.649929	−	0.759995i	$-0.274798\pi$
$541$	380444.			1.29986			0.649929	+	0.759995i	$0.274798\pi$
$542$		0			0
$543$	21207.2			0.0719257
$544$		0			0
$545$	−113058.			−0.380635
$546$		0			0
$547$	352052.			1.17661			0.588305	−	0.808639i	$-0.299795\pi$
$547$	352052.			1.17661			0.588305	+	0.808639i	$0.299795\pi$
$548$		0			0
$549$		−	57338.3i		−	0.190239i
$550$		0			0
$551$		−	533785.i		−	1.75818i
$552$		0			0
$553$			81312.9i			0.265894i
$554$		0			0
$555$	−181342.			−0.588724
$556$		0			0
$557$		−	229065.i		−	0.738326i	−0.929365	−	0.369163i	$-0.879644\pi$
$557$		−	229065.i		−	0.738326i	0.929365	−	0.369163i	$-0.120356\pi$
$558$		0			0
$559$	−747393.			−2.39180
$560$		0			0
$561$		−	4525.80i		−	0.0143804i
$562$		0			0
$563$	87970.9			0.277538			0.138769	−	0.990325i	$-0.455685\pi$
$563$	87970.9			0.277538			0.138769	+	0.990325i	$0.455685\pi$
$564$		0			0
$565$	400467.			1.25450
$566$		0			0
$567$	31932.8			0.0993279
$568$		0			0
$569$			208976.i			0.645463i	0.946491	+	0.322731i	$0.104601\pi$
$569$			208976.i			0.645463i	−0.946491	+	0.322731i	$0.895399\pi$
$570$		0			0
$571$		−	490410.i		−	1.50414i	−0.659085	−	0.752069i	$-0.729056\pi$
$571$		−	490410.i		−	1.50414i	0.659085	−	0.752069i	$-0.270944\pi$
$572$		0			0
$573$		−	30794.8i		−	0.0937925i
$574$		0			0
$575$		−	2055.66i		−	0.00621750i
$576$		0			0
$577$	274939.			0.825820			0.412910	−	0.910772i	$-0.364512\pi$
$577$	274939.			0.825820			0.412910	+	0.910772i	$0.364512\pi$
$578$		0			0
$579$			238330.i			0.710920i
$580$		0			0
$581$			70416.5i			0.208604i
$582$		0			0
$583$	29543.5			0.0869210
$584$		0			0
$585$			321680.i			0.939966i
$586$		0			0
$587$		−	564350.i		−	1.63784i	−0.573906	−	0.818921i	$-0.694572\pi$
$587$		−	564350.i		−	1.63784i	0.573906	−	0.818921i	$-0.305428\pi$
$588$		0			0
$589$	−163052.	−	377561.i	−0.469999	−	1.08832i
$590$		0			0
$591$	−177358.			−0.507779
$592$		0			0
$593$	−147849.			−0.420443			−0.210222	−	0.977654i	$-0.567419\pi$
$593$	−147849.			−0.420443			−0.210222	+	0.977654i	$0.567419\pi$
$594$		0			0
$595$		−	33154.0i		−	0.0936488i
$596$		0			0
$597$	225220.			0.631914
$598$		0			0
$599$	621873.			1.73320			0.866599	−	0.499005i	$-0.166301\pi$
$599$	621873.			1.73320			0.866599	+	0.499005i	$0.166301\pi$
$600$		0			0
$601$		−	242746.i		−	0.672053i	−0.941852	−	0.336027i	$-0.890917\pi$
$601$		−	242746.i		−	0.672053i	0.941852	−	0.336027i	$-0.109083\pi$
$602$		0			0
$603$	132700.			0.364953
$604$		0			0
$605$	−509153.			−1.39103
$606$		0			0
$607$	318345.			0.864015			0.432007	−	0.901870i	$-0.357805\pi$
$607$	318345.			0.864015			0.432007	+	0.901870i	$0.357805\pi$
$608$		0			0
$609$	−106799.			−0.287960
$610$		0			0
$611$		−	40339.4i		−	0.108056i
$612$		0			0
$613$			555808.i			1.47912i	0.673089	+	0.739561i	$0.264967\pi$
$613$			555808.i			1.47912i	−0.673089	+	0.739561i	$0.735033\pi$
$614$		0			0
$615$			223133.i			0.589947i
$616$		0			0
$617$	754481.			1.98188			0.990941	−	0.134295i	$-0.0428770\pi$
$617$	754481.			1.98188			0.990941	+	0.134295i	$0.0428770\pi$
$618$		0			0
$619$			556946.i			1.45356i	0.686872	+	0.726778i	$0.258983\pi$
$619$			556946.i			1.45356i	−0.686872	+	0.726778i	$0.741017\pi$
$620$		0			0
$621$	2708.73			0.00702396
$622$		0			0
$623$		−	43814.8i		−	0.112887i
$624$		0			0
$625$	−407129.			−1.04225
$626$		0			0
$627$	25721.4			0.0654273
$628$		0			0
$629$	−57442.3			−0.145188
$630$		0			0
$631$			263309.i			0.661312i	0.943751	+	0.330656i	$0.107270\pi$
$631$			263309.i			0.661312i	−0.943751	+	0.330656i	$0.892730\pi$
$632$		0			0
$633$			364092.i			0.908664i
$634$		0			0
$635$			563695.i			1.39797i
$636$		0			0
$637$			593386.i			1.46237i
$638$		0			0
$639$	−145185.			−0.355565
$640$		0			0
$641$		−	720715.i		−	1.75407i	−0.480424	−	0.877036i	$-0.659517\pi$
$641$		−	720715.i		−	1.75407i	0.480424	−	0.877036i	$-0.340483\pi$
$642$		0			0
$643$			586639.i			1.41889i	0.704760	+	0.709446i	$0.251055\pi$
$643$			586639.i			1.41889i	−0.704760	+	0.709446i	$0.748945\pi$
$644$		0			0
$645$	−671418.			−1.61389
$646$		0			0
$647$			224538.i			0.536390i	0.963365	+	0.268195i	$0.0864272\pi$
$647$			224538.i			0.536390i	−0.963365	+	0.268195i	$0.913573\pi$
$648$		0			0
$649$			8759.86i			0.0207973i
$650$		0			0
$651$	−75541.9	+	32623.3i	−0.178249	+	0.0769779i
$652$		0			0
$653$	−476782.			−1.11813			−0.559066	−	0.829123i	$-0.688840\pi$
$653$	−476782.			−1.11813			−0.559066	+	0.829123i	$0.688840\pi$
$654$		0			0
$655$	−1.02979e6			−2.40031
$656$		0			0
$657$			220950.i			0.511874i
$658$		0			0
$659$	796869.			1.83492			0.917458	−	0.397832i	$-0.130237\pi$
$659$	796869.			1.83492			0.917458	+	0.397832i	$0.130237\pi$
$660$		0			0
$661$	−46068.9			−0.105440			−0.0527199	−	0.998609i	$-0.516789\pi$
$661$	−46068.9			−0.105440			−0.0527199	+	0.998609i	$0.516789\pi$
$662$		0			0
$663$		−	135483.i		−	0.308217i
$664$		0			0
$665$	188423.			0.426080
$666$		0			0
$667$	−4292.16			−0.00964770
$668$		0			0
$669$	−430429.			−0.961722
$670$		0			0
$671$	−14577.3			−0.0323766
$672$		0			0
$673$			34147.7i			0.0753931i	0.999289	+	0.0376965i	$0.0120020\pi$
$673$			34147.7i			0.0753931i	−0.999289	+	0.0376965i	$0.987998\pi$
$674$		0			0
$675$			470222.i			1.03204i
$676$		0			0
$677$		−	895623.i		−	1.95411i	−0.212998	−	0.977053i	$-0.568323\pi$
$677$		−	895623.i		−	1.95411i	0.212998	−	0.977053i	$-0.431677\pi$
$678$		0			0
$679$	205012.			0.444671
$680$		0			0
$681$		−	271951.i		−	0.586403i
$682$		0			0
$683$	−411504.			−0.882131			−0.441065	−	0.897475i	$-0.645399\pi$
$683$	−411504.			−0.882131			−0.441065	+	0.897475i	$0.645399\pi$
$684$		0			0
$685$			705012.i			1.50250i
$686$		0			0
$687$	−234668.			−0.497210
$688$		0			0
$689$	884403.			1.86300
$690$		0			0
$691$	−540155.			−1.13126			−0.565630	−	0.824659i	$-0.691367\pi$
$691$	−540155.			−1.13126			−0.565630	+	0.824659i	$0.691367\pi$
$692$		0			0
$693$			3870.52i			0.00805940i
$694$		0			0
$695$			870371.i			1.80192i
$696$		0			0
$697$			70680.1i			0.145490i
$698$		0			0
$699$		−	546173.i		−	1.11783i
$700$		0			0
$701$	475956.			0.968569			0.484285	−	0.874911i	$-0.339080\pi$
$701$	475956.			0.968569			0.484285	+	0.874911i	$0.339080\pi$
$702$		0			0
$703$		−	326460.i		−	0.660571i
$704$		0			0
$705$		−	36238.8i		−	0.0729113i
$706$		0			0
$707$	−109971.			−0.220009
$708$		0			0
$709$			274721.i			0.546511i	0.961941	+	0.273256i	$0.0881005\pi$
$709$			274721.i			0.546511i	−0.961941	+	0.273256i	$0.911899\pi$
$710$		0			0
$711$		−	224505.i		−	0.444105i
$712$		0			0
$713$	−3035.96	+	1311.10i	−0.00597196	+	0.00257903i
$714$		0			0
$715$	81781.7			0.159972
$716$		0			0
$717$	−609515.			−1.18562
$718$		0			0
$719$		−	893268.i		−	1.72792i	−0.503560	−	0.863960i	$-0.667977\pi$
$719$		−	893268.i		−	1.72792i	0.503560	−	0.863960i	$-0.332023\pi$
$720$		0			0
$721$	−15057.0			−0.0289645
$722$		0			0
$723$	476352.			0.911278
$724$		0			0
$725$		−	745098.i		−	1.41755i
$726$		0			0
$727$	−22347.7			−0.0422829			−0.0211415	−	0.999776i	$-0.506730\pi$
$727$	−22347.7			−0.0422829			−0.0211415	+	0.999776i	$0.506730\pi$
$728$		0			0
$729$	−521684.			−0.981640
$730$		0			0
$731$	−212680.			−0.398008
$732$		0			0
$733$	243982.			0.454099			0.227049	−	0.973883i	$-0.427092\pi$
$733$	243982.			0.454099			0.227049	+	0.973883i	$0.427092\pi$
$734$		0			0
$735$			533065.i			0.986747i
$736$		0			0
$737$		−	33736.8i		−	0.0621111i
$738$		0			0
$739$			552771.i			1.01218i	0.862481	+	0.506089i	$0.168909\pi$
$739$			552771.i			1.01218i	−0.862481	+	0.506089i	$0.831091\pi$
$740$		0			0
$741$	769986.			1.40232
$742$		0			0
$743$			50422.4i			0.0913368i	0.998957	+	0.0456684i	$0.0145418\pi$
$743$			50422.4i			0.0913368i	−0.998957	+	0.0456684i	$0.985458\pi$
$744$		0			0
$745$	−415117.			−0.747925
$746$		0			0
$747$		−	194420.i		−	0.348417i
$748$		0			0
$749$	87874.4			0.156639
$750$		0			0
$751$	800704.			1.41969			0.709843	−	0.704360i	$-0.248766\pi$
$751$	800704.			1.41969			0.709843	+	0.704360i	$0.248766\pi$
$752$		0			0
$753$	−803714.			−1.41746
$754$		0			0
$755$			600939.i			1.05423i
$756$		0			0
$757$			728898.i			1.27196i	0.771704	+	0.635982i	$0.219405\pi$
$757$			728898.i			1.27196i	−0.771704	+	0.635982i	$0.780595\pi$
$758$		0			0
$759$		−	206.825i		−	0.000359021i
$760$		0			0
$761$		−	761497.i		−	1.31492i	−0.753490	−	0.657459i	$-0.771631\pi$
$761$		−	761497.i		−	1.31492i	0.753490	−	0.657459i	$-0.228369\pi$
$762$		0			0
$763$	−40722.6			−0.0699498
$764$		0			0
$765$			91538.0i			0.156415i
$766$		0			0
$767$			262232.i			0.445754i
$768$		0			0
$769$	262427.			0.443767			0.221884	−	0.975073i	$-0.428779\pi$
$769$	262427.			0.443767			0.221884	+	0.975073i	$0.428779\pi$
$770$		0			0
$771$		−	355494.i		−	0.598030i
$772$		0			0
$773$			740606.i			1.23945i	0.784820	+	0.619724i	$0.212756\pi$
$773$			740606.i			1.23945i	−0.784820	+	0.619724i	$0.787244\pi$
$774$		0			0
$775$	−227601.	−	527028.i	−0.378940	−	0.877467i
$776$		0			0
$777$	−65317.7			−0.108190
$778$		0			0
$779$	−401695.			−0.661944
$780$		0			0
$781$			36910.8i			0.0605134i
$782$		0			0
$783$	981809.			1.60141
$784$		0			0
$785$	777892.			1.26235
$786$		0			0
$787$			417760.i			0.674493i	0.941416	+	0.337246i	$0.109496\pi$
$787$			417760.i			0.674493i	−0.941416	+	0.337246i	$0.890504\pi$
$788$		0			0
$789$	275129.			0.441959
$790$		0			0
$791$	144245.			0.230541
$792$		0			0
$793$	−436380.			−0.693935
$794$		0			0
$795$	794500.			1.25707
$796$		0			0
$797$		−	1.21326e6i		−	1.91001i	−0.296587	−	0.955006i	$-0.595848\pi$
$797$		−	1.21326e6i		−	1.91001i	0.296587	−	0.955006i	$-0.404152\pi$
$798$		0			0
$799$		−	11479.1i		−	0.0179810i
$800$		0			0
$801$			120972.i			0.188548i
$802$		0			0
$803$	56172.8			0.0871154
$804$		0			0
$805$		−	1515.11i		−	0.00233804i
$806$		0			0
$807$	−188902.			−0.290061
$808$		0			0
$809$		−	773978.i		−	1.18258i	−0.806458	−	0.591291i	$-0.798618\pi$
$809$		−	773978.i		−	1.18258i	0.806458	−	0.591291i	$-0.201382\pi$
$810$		0			0
$811$	674119.			1.02493			0.512466	−	0.858708i	$-0.328732\pi$
$811$	674119.			1.02493			0.512466	+	0.858708i	$0.328732\pi$
$812$		0			0
$813$	358236.			0.541986
$814$		0			0
$815$	−1.57776e6			−2.37534
$816$		0			0
$817$		−	1.20872e6i		−	1.81085i
$818$		0			0
$819$			115866.i			0.172739i
$820$		0			0
$821$		−	740519.i		−	1.09863i	−0.835617	−	0.549313i	$-0.814889\pi$
$821$		−	740519.i		−	1.09863i	0.835617	−	0.549313i	$-0.185111\pi$
$822$		0			0
$823$			824157.i			1.21677i	0.793640	+	0.608387i	$0.208183\pi$
$823$			824157.i			1.21677i	−0.793640	+	0.608387i	$0.791817\pi$
$824$		0			0
$825$	35903.9			0.0527513
$826$		0			0
$827$		−	896626.i		−	1.31099i	−0.755198	−	0.655496i	$-0.772460\pi$
$827$		−	896626.i		−	1.31099i	0.755198	−	0.655496i	$-0.227540\pi$
$828$		0			0
$829$			205371.i			0.298834i	0.988774	+	0.149417i	$0.0477396\pi$
$829$			205371.i			0.298834i	−0.988774	+	0.149417i	$0.952260\pi$
$830$		0			0
$831$	748857.			1.08442
$832$		0			0
$833$			168855.i			0.243346i
$834$		0			0
$835$		−	931867.i		−	1.33654i
$836$		0			0
$837$	694461.	−	299908.i	0.991281	−	0.428092i
$838$		0			0
$839$	−919336.			−1.30602			−0.653011	−	0.757349i	$-0.726494\pi$
$839$	−919336.			−1.30602			−0.653011	+	0.757349i	$0.726494\pi$
$840$		0			0
$841$	−848460.			−1.19961
$842$		0			0
$843$			288245.i			0.405608i
$844$		0			0
$845$	1.44963e6			2.03022
$846$		0			0
$847$	−183393.			−0.255632
$848$		0			0
$849$		−	302288.i		−	0.419377i
$850$		0			0
$851$	−2625.06			−0.00362477
$852$		0			0
$853$	−1.20914e6			−1.66180			−0.830898	−	0.556425i	$-0.812173\pi$
$853$	−1.20914e6			−1.66180			−0.830898	+	0.556425i	$0.812173\pi$
$854$		0			0
$855$	−520236.			−0.711653
$856$		0			0
$857$	886898.			1.20757			0.603785	−	0.797147i	$-0.293659\pi$
$857$	886898.			1.20757			0.603785	+	0.797147i	$0.293659\pi$
$858$		0			0
$859$		−	1.11896e6i		−	1.51645i	−0.651992	−	0.758226i	$-0.726067\pi$
$859$		−	1.11896e6i		−	1.51645i	0.651992	−	0.758226i	$-0.273933\pi$
$860$		0			0
$861$			80370.6i			0.108415i
$862$		0			0
$863$		−	458852.i		−	0.616100i	−0.951370	−	0.308050i	$-0.900324\pi$
$863$		−	458852.i		−	0.616100i	0.951370	−	0.308050i	$-0.0996764\pi$
$864$		0			0
$865$	1.90093e6			2.54058
$866$		0			0
$867$			529330.i			0.704187i
$868$		0			0
$869$	−57076.5			−0.0755819
$870$		0			0
$871$		−	1.00993e6i		−	1.33124i
$872$		0			0
$873$	−566036.			−0.742704
$874$		0			0
$875$	−12164.5			−0.0158883
$876$		0			0
$877$	−1.01499e6			−1.31966			−0.659832	−	0.751413i	$-0.729373\pi$
$877$	−1.01499e6			−1.31966			−0.659832	+	0.751413i	$0.729373\pi$
$878$		0			0
$879$			282584.i			0.365738i
$880$		0			0
$881$		−	488963.i		−	0.629976i	−0.949096	−	0.314988i	$-0.897999\pi$
$881$		−	488963.i		−	0.629976i	0.949096	−	0.314988i	$-0.102001\pi$
$882$		0			0
$883$			382828.i			0.491001i	0.969396	+	0.245500i	$0.0789523\pi$
$883$			382828.i			0.491001i	−0.969396	+	0.245500i	$0.921048\pi$
$884$		0			0
$885$			235575.i			0.300776i
$886$		0			0
$887$	−332289.			−0.422346			−0.211173	−	0.977449i	$-0.567728\pi$
$887$	−332289.			−0.422346			−0.211173	+	0.977449i	$0.567728\pi$
$888$		0			0
$889$			203038.i			0.256906i
$890$		0			0
$891$			22414.8i			0.0282345i
$892$		0			0
$893$	65238.8			0.0818094
$894$		0			0
$895$			376518.i			0.470045i
$896$		0			0
$897$		−	6191.44i		−	0.00769497i
$898$		0			0
$899$	−1.10042e6	+	475224.i	−1.36157	+	0.588002i
$900$		0			0
$901$	251668.			0.310012
$902$		0			0
$903$	−241839.			−0.296586
$904$		0			0
$905$			109049.i			0.133145i
$906$		0			0
$907$	1.01409e6			1.23271			0.616354	−	0.787469i	$-0.288609\pi$
$907$	1.01409e6			1.23271			0.616354	+	0.787469i	$0.288609\pi$
$908$		0			0
$909$	303630.			0.367466
$910$		0			0
$911$			928914.i			1.11928i	0.828736	+	0.559640i	$0.189061\pi$
$911$			928914.i			1.11928i	−0.828736	+	0.559640i	$0.810939\pi$
$912$		0			0
$913$	−49427.9			−0.0592967
$914$		0			0
$915$	−392020.			−0.468238
$916$		0			0
$917$	−370923.			−0.441109
$918$		0			0
$919$	−475355.			−0.562842			−0.281421	−	0.959584i	$-0.590806\pi$
$919$	−475355.			−0.562842			−0.281421	+	0.959584i	$0.590806\pi$
$920$		0			0
$921$			117443.i			0.138455i
$922$		0			0
$923$			1.10495e6i			1.29700i
$924$		0			0
$925$		−	455698.i		−	0.532591i
$926$		0			0
$927$	41572.2			0.0483775
$928$		0			0
$929$			1.28915e6i			1.49373i	0.664977	+	0.746864i	$0.268441\pi$
$929$			1.28915e6i			1.49373i	−0.664977	+	0.746864i	$0.731559\pi$
$930$		0			0
$931$	−959651.			−1.10717
$932$		0			0
$933$		−	564579.i		−	0.648576i
$934$		0			0
$935$	23272.0			0.0266202
$936$		0			0
$937$	539638.			0.614644			0.307322	−	0.951606i	$-0.400567\pi$
$937$	539638.			0.614644			0.307322	+	0.951606i	$0.400567\pi$
$938$		0			0
$939$	−563664.			−0.639277
$940$		0			0
$941$		−	490513.i		−	0.553951i	−0.960877	−	0.276975i	$-0.910668\pi$
$941$		−	490513.i		−	0.553951i	0.960877	−	0.276975i	$-0.0893320\pi$
$942$		0			0
$943$			3230.02i			0.00363230i
$944$		0			0
$945$			346573.i			0.388089i
$946$		0			0
$947$			1.16556e6i			1.29967i	0.760074	+	0.649837i	$0.225163\pi$
$947$			1.16556e6i			1.29967i	−0.760074	+	0.649837i	$0.774837\pi$
$948$		0			0
$949$	1.68157e6			1.86716
$950$		0			0
$951$			1.01847e6i			1.12613i
$952$		0			0
$953$		−	271838.i		−	0.299312i	−0.988738	−	0.149656i	$-0.952183\pi$
$953$		−	271838.i		−	0.299312i	0.988738	−	0.149656i	$-0.0478167\pi$
$954$		0			0
$955$	158349.			0.173624
$956$		0			0
$957$		−	74966.1i		−	0.0818543i
$958$		0			0
$959$			253940.i			0.276117i
$960$		0			0
$961$	−633193.	+	672278.i	−0.685629	+	0.727951i
$962$		0			0
$963$	−242621.			−0.261623
$964$		0			0
$965$	−1.22551e6			−1.31602
$966$		0			0
$967$		−	463205.i		−	0.495359i	−0.968842	−	0.247680i	$-0.920332\pi$
$967$		−	463205.i		−	0.495359i	0.968842	−	0.247680i	$-0.0796681\pi$
$968$		0			0
$969$	219109.			0.233353
$970$		0			0
$971$	1.38208e6			1.46587			0.732934	−	0.680300i	$-0.238151\pi$
$971$	1.38208e6			1.46587			0.732934	+	0.680300i	$0.238151\pi$
$972$		0			0
$973$			313501.i			0.331141i
$974$		0			0
$975$	1.07480e6			1.13063
$976$		0			0
$977$	62082.8			0.0650402			0.0325201	−	0.999471i	$-0.489647\pi$
$977$	62082.8			0.0650402			0.0325201	+	0.999471i	$0.489647\pi$
$978$		0			0
$979$	30755.2			0.0320888
$980$		0			0
$981$	112435.			0.116832
$982$		0			0
$983$			1.24381e6i			1.28720i	0.765363	+	0.643599i	$0.222560\pi$
$983$			1.24381e6i			1.28720i	−0.765363	+	0.643599i	$0.777440\pi$
$984$		0			0
$985$		−	911985.i		−	0.939973i
$986$		0			0
$987$		−	13052.9i		−	0.0133990i
$988$		0			0
$989$	−9719.29			−0.00993669
$990$		0			0
$991$		−	1.16663e6i		−	1.18792i	−0.804494	−	0.593961i	$-0.797563\pi$
$991$		−	1.16663e6i		−	1.18792i	0.804494	−	0.593961i	$-0.202437\pi$
$992$		0			0
$993$	−50432.4			−0.0511460
$994$		0			0
$995$			1.15810e6i			1.16977i
$996$		0			0
$997$	−802091.			−0.806926			−0.403463	−	0.914996i	$-0.632194\pi$
$997$	−802091.			−0.806926			−0.403463	+	0.914996i	$0.632194\pi$
$998$		0			0
$999$	600469.			0.601672

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	124.5.c.a.61.7	yes	10
3.2	odd	2		1116.5.h.b.433.10		10
4.3	odd	2		496.5.e.c.433.4		10
31.30	odd	2	inner	124.5.c.a.61.4	✓	10
93.92	even	2		1116.5.h.b.433.9		10
124.123	even	2		496.5.e.c.433.7		10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
124.5.c.a.61.4	✓	10	31.30	odd	2	inner
124.5.c.a.61.7	yes	10	1.1	even	1	trivial
496.5.e.c.433.4		10	4.3	odd	2
496.5.e.c.433.7		10	124.123	even	2
1116.5.h.b.433.9		10	93.92	even	2
1116.5.h.b.433.10		10	3.2	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:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	124.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+6.79929i q^{3} -34.9624 q^{5} -12.5932 q^{7} +34.7697 q^{9} +O(q^{10})\)	\(q+6.79929i q^{3} -34.9624 q^{5} -12.5932 q^{7} +34.7697 q^{9} -8.83961i q^{11} -264.619i q^{13} -237.720i q^{15} -75.3007i q^{17} +427.955 q^{19} -85.6246i q^{21} -3.44118i q^{23} +597.372 q^{25} +787.151i q^{27} -1247.29i q^{29} +(-381.004 - 882.245i) q^{31} +60.1031 q^{33} +440.288 q^{35} -762.838i q^{37} +1799.22 q^{39} -938.638 q^{41} -2824.41i q^{43} -1215.63 q^{45} +152.443 q^{47} -2242.41 q^{49} +511.991 q^{51} +3342.17i q^{53} +309.054i q^{55} +2909.79i q^{57} -990.978 q^{59} -1649.09i q^{61} -437.861 q^{63} +9251.74i q^{65} +3816.55 q^{67} +23.3975 q^{69} -4175.61 q^{71} +6354.67i q^{73} +4061.70i q^{75} +111.319i q^{77} -6456.90i q^{79} -2535.72 q^{81} -5591.64i q^{83} +2632.70i q^{85} +8480.71 q^{87} +3479.25i q^{89} +3332.40i q^{91} +(5998.64 - 2590.55i) q^{93} -14962.3 q^{95} -16279.6 q^{97} -307.351i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q - 6 q^{5} - 2 q^{7} - 146 q^{9}+O(q^{10}) \) 10 * q - 6 * q^5 - 2 * q^7 - 146 * q^9	\( 10 q - 6 q^{5} - 2 q^{7} - 146 q^{9} - 310 q^{19} - 756 q^{25} + 734 q^{31} - 372 q^{33} - 666 q^{35} - 1924 q^{39} - 210 q^{41} + 650 q^{45} - 1992 q^{47} + 188 q^{49} + 1352 q^{51} + 5610 q^{59} + 3478 q^{63} - 5420 q^{67} + 10160 q^{69} - 1734 q^{71} + 11598 q^{81} - 13244 q^{87} + 3088 q^{93} + 4302 q^{95} + 2942 q^{97}+O(q^{100}) \) 10 * q - 6 * q^5 - 2 * q^7 - 146 * q^9 - 310 * q^19 - 756 * q^25 + 734 * q^31 - 372 * q^33 - 666 * q^35 - 1924 * q^39 - 210 * q^41 + 650 * q^45 - 1992 * q^47 + 188 * q^49 + 1352 * q^51 + 5610 * q^59 + 3478 * q^63 - 5420 * q^67 + 10160 * q^69 - 1734 * q^71 + 11598 * q^81 - 13244 * q^87 + 3088 * q^93 + 4302 * q^95 + 2942 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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