LMFDB - Embedded newform 531.5.b.a.296.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [531,5,Mod(296,531)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(531, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("531.296");

S:= CuspForms(chi, 5);

N := Newforms(S);

Level:	$ N $	$=$	$ 531 = 3^{2} \cdot 59 $
Weight:	$ k $	$=$	$ 5 $
Character orbit:	$[\chi]$	$=$	531.b (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:	$54.8894503975$
Analytic rank:	$0$
Dimension:	$76$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			296.3
Character	$\chi$	$=$	531.296
Dual form			531.5.b.a.296.74

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-7.65883i q^{2} -42.6577 q^{4} -8.87825i q^{5} +63.4860 q^{7} +204.167i q^{8} +O(q^{10})$	\(q-7.65883i q^{2} -42.6577 q^{4} -8.87825i q^{5} +63.4860 q^{7} +204.167i q^{8} -67.9970 q^{10} +136.985i q^{11} -49.0811 q^{13} -486.229i q^{14} +881.155 q^{16} -56.7036i q^{17} +178.317 q^{19} +378.726i q^{20} +1049.15 q^{22} -697.846i q^{23} +546.177 q^{25} +375.904i q^{26} -2708.17 q^{28} -94.1549i q^{29} -1532.25 q^{31} -3481.95i q^{32} -434.283 q^{34} -563.645i q^{35} -348.572 q^{37} -1365.70i q^{38} +1812.64 q^{40} -2091.74i q^{41} +822.999 q^{43} -5843.47i q^{44} -5344.69 q^{46} -2945.38i q^{47} +1629.48 q^{49} -4183.07i q^{50} +2093.69 q^{52} -3515.14i q^{53} +1216.19 q^{55} +12961.7i q^{56} -721.116 q^{58} -453.188i q^{59} -2109.78 q^{61} +11735.3i q^{62} -12569.2 q^{64} +435.755i q^{65} +3662.39 q^{67} +2418.85i q^{68} -4316.86 q^{70} -3905.27i q^{71} -6272.66 q^{73} +2669.65i q^{74} -7606.59 q^{76} +8696.65i q^{77} +8343.92 q^{79} -7823.12i q^{80} -16020.3 q^{82} +9055.37i q^{83} -503.429 q^{85} -6303.21i q^{86} -27967.8 q^{88} +6893.28i q^{89} -3115.97 q^{91} +29768.5i q^{92} -22558.2 q^{94} -1583.14i q^{95} +9649.80 q^{97} -12479.9i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 76 q - 632 q^{4} - 48 q^{7}+O(q^{10}) $ 76 * q - 632 * q^4 - 48 * q^7	$ 76 q - 632 q^{4} - 48 q^{7} - 72 q^{10} + 448 q^{13} + 5696 q^{16} - 2240 q^{19} + 1008 q^{22} - 11180 q^{25} + 5376 q^{28} - 3440 q^{31} - 7112 q^{34} + 7016 q^{37} - 832 q^{40} - 608 q^{43} + 17728 q^{46} + 30628 q^{49} - 27056 q^{52} - 9312 q^{55} + 4616 q^{58} - 10056 q^{61} - 42864 q^{64} + 31160 q^{67} + 7544 q^{70} + 4488 q^{73} - 15632 q^{76} + 19208 q^{79} + 9768 q^{82} - 7568 q^{85} - 41736 q^{88} - 31240 q^{91} + 27360 q^{94} + 22216 q^{97}+O(q^{100}) $ 76 * q - 632 * q^4 - 48 * q^7 - 72 * q^10 + 448 * q^13 + 5696 * q^16 - 2240 * q^19 + 1008 * q^22 - 11180 * q^25 + 5376 * q^28 - 3440 * q^31 - 7112 * q^34 + 7016 * q^37 - 832 * q^40 - 608 * q^43 + 17728 * q^46 + 30628 * q^49 - 27056 * q^52 - 9312 * q^55 + 4616 * q^58 - 10056 * q^61 - 42864 * q^64 + 31160 * q^67 + 7544 * q^70 + 4488 * q^73 - 15632 * q^76 + 19208 * q^79 + 9768 * q^82 - 7568 * q^85 - 41736 * q^88 - 31240 * q^91 + 27360 * q^94 + 22216 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/531\mathbb{Z}\right)^\times$.

$n$	$119$	$415$
$\chi(n)$	$-1$	$1$

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 $

$2$	− 7.65883i	− 1.91471i	−0.288918	−	0.957354i	$-0.593296\pi$
$2$	− 7.65883i	− 1.91471i	0.288918	−	0.957354i	$-0.406704\pi$
$3$	0	0
$3$	0	0
$4$	−42.6577	−2.66611
$5$	− 8.87825i	− 0.355130i	−0.984109	−	0.177565i	$-0.943178\pi$
$5$	− 8.87825i	− 0.355130i	0.984109	−	0.177565i	$-0.0568220\pi$
$6$	0	0
$7$	63.4860	1.29563	0.647817	−	0.761796i	$-0.275682\pi$
$7$	63.4860	1.29563	0.647817	+	0.761796i	$0.275682\pi$
$8$	204.167i	3.19010i
$9$	0	0
$10$	−67.9970	−0.679970
$11$	136.985i	1.13211i	0.824368	+	0.566055i	$0.191531\pi$
$11$	136.985i	1.13211i	−0.824368	+	0.566055i	$0.808469\pi$
$12$	0	0
$13$	−49.0811	−0.290421	−0.145210	−	0.989401i	$-0.546386\pi$
$13$	−49.0811	−0.290421	−0.145210	+	0.989401i	$0.546386\pi$
$14$	− 486.229i	− 2.48076i
$15$	0	0
$16$	881.155	3.44201
$17$	− 56.7036i	− 0.196206i	−0.995176	−	0.0981031i	$-0.968722\pi$
$17$	− 56.7036i	− 0.196206i	0.995176	−	0.0981031i	$-0.0312775\pi$
$18$	0	0
$19$	178.317	0.493953	0.246976	−	0.969022i	$-0.420563\pi$
$19$	178.317	0.493953	0.246976	+	0.969022i	$0.420563\pi$
$20$	378.726i	0.946815i
$21$	0	0
$22$	1049.15	2.16766
$23$	− 697.846i	− 1.31918i	−0.751626	−	0.659590i	$-0.770730\pi$
$23$	− 697.846i	− 1.31918i	0.751626	−	0.659590i	$-0.229270\pi$
$24$	0	0
$25$	546.177	0.873883
$26$	375.904i	0.556071i
$27$	0	0
$28$	−2708.17	−3.45430
$29$	− 94.1549i	− 0.111956i	−0.998432	−	0.0559779i	$-0.982172\pi$
$29$	− 94.1549i	− 0.111956i	0.998432	−	0.0559779i	$-0.0178276\pi$
$30$	0	0
$31$	−1532.25	−1.59444	−0.797219	−	0.603691i	$-0.793696\pi$
$31$	−1532.25	−1.59444	−0.797219	+	0.603691i	$0.793696\pi$
$32$	− 3481.95i	− 3.40034i
$33$	0	0
$34$	−434.283	−0.375678
$35$	− 563.645i	− 0.460119i
$36$	0	0
$37$	−348.572	−0.254618	−0.127309	−	0.991863i	$-0.540634\pi$
$37$	−348.572	−0.254618	−0.127309	+	0.991863i	$0.540634\pi$
$38$	− 1365.70i	− 0.945775i
$39$	0	0
$40$	1812.64	1.13290
$41$	− 2091.74i	− 1.24434i	−0.782880	−	0.622172i	$-0.786250\pi$
$41$	− 2091.74i	− 1.24434i	0.782880	−	0.622172i	$-0.213750\pi$
$42$	0	0
$43$	822.999	0.445105	0.222552	−	0.974921i	$-0.428561\pi$
$43$	822.999	0.445105	0.222552	+	0.974921i	$0.428561\pi$
$44$	− 5843.47i	− 3.01832i
$45$	0	0
$46$	−5344.69	−2.52584
$47$	− 2945.38i	− 1.33335i	−0.745347	−	0.666677i	$-0.767716\pi$
$47$	− 2945.38i	− 1.33335i	0.745347	−	0.666677i	$-0.232284\pi$
$48$	0	0
$49$	1629.48	0.678666
$50$	− 4183.07i	− 1.67323i
$51$	0	0
$52$	2093.69	0.774293
$53$	− 3515.14i	− 1.25139i	−0.780069	−	0.625693i	$-0.784816\pi$
$53$	− 3515.14i	− 1.25139i	0.780069	−	0.625693i	$-0.215184\pi$
$54$	0	0
$55$	1216.19	0.402046
$56$	12961.7i	4.13321i
$57$	0	0
$58$	−721.116	−0.214363
$59$	− 453.188i	− 0.130189i
$59$	− 453.188i	− 0.130189i
$60$	0	0
$61$	−2109.78	−0.566992	−0.283496	−	0.958973i	$-0.591494\pi$
$61$	−2109.78	−0.566992	−0.283496	+	0.958973i	$0.591494\pi$
$62$	11735.3i	3.05288i
$63$	0	0
$64$	−12569.2	−3.06865
$65$	435.755i	0.103137i
$66$	0	0
$67$	3662.39	0.815859	0.407929	−	0.913014i	$-0.366251\pi$
$67$	3662.39	0.815859	0.407929	+	0.913014i	$0.366251\pi$
$68$	2418.85i	0.523107i
$69$	0	0
$70$	−4316.86	−0.880992
$71$	− 3905.27i	− 0.774702i	−0.921932	−	0.387351i	$-0.873390\pi$
$71$	− 3905.27i	− 0.774702i	0.921932	−	0.387351i	$-0.126610\pi$
$72$	0	0
$73$	−6272.66	−1.17708	−0.588540	−	0.808468i	$-0.700297\pi$
$73$	−6272.66	−1.17708	−0.588540	+	0.808468i	$0.700297\pi$
$74$	2669.65i	0.487519i
$75$	0	0
$76$	−7606.59	−1.31693
$77$	8696.65i	1.46680i
$78$	0	0
$79$	8343.92	1.33695	0.668476	−	0.743734i	$-0.266947\pi$
$79$	8343.92	1.33695	0.668476	+	0.743734i	$0.266947\pi$
$80$	− 7823.12i	− 1.22236i
$81$	0	0
$82$	−16020.3	−2.38256
$83$	9055.37i	1.31447i	0.753686	+	0.657234i	$0.228274\pi$
$83$	9055.37i	1.31447i	−0.753686	+	0.657234i	$0.771726\pi$
$84$	0	0
$85$	−503.429	−0.0696788
$86$	− 6303.21i	− 0.852245i
$87$	0	0
$88$	−27967.8	−3.61155
$89$	6893.28i	0.870254i	0.900369	+	0.435127i	$0.143296\pi$
$89$	6893.28i	0.870254i	−0.900369	+	0.435127i	$0.856704\pi$
$90$	0	0
$91$	−3115.97	−0.376279
$92$	29768.5i	3.51707i
$93$	0	0
$94$	−22558.2	−2.55298
$95$	− 1583.14i	− 0.175417i
$96$	0	0
$97$	9649.80	1.02559	0.512797	−	0.858510i	$-0.328610\pi$
$97$	9649.80	1.02559	0.512797	+	0.858510i	$0.328610\pi$
$98$	− 12479.9i	− 1.29945i
$99$	0	0
$100$	−23298.6	−2.32986
$101$	− 14144.2i	− 1.38655i	−0.720673	−	0.693276i	$-0.756167\pi$
$101$	− 14144.2i	− 1.38655i	0.720673	−	0.693276i	$-0.243833\pi$
$102$	0	0
$103$	−3496.04	−0.329535	−0.164768	−	0.986332i	$-0.552687\pi$
$103$	−3496.04	−0.329535	−0.164768	+	0.986332i	$0.552687\pi$
$104$	− 10020.7i	− 0.926473i
$105$	0	0
$106$	−26921.9	−2.39604
$107$	− 10771.5i	− 0.940828i	−0.882446	−	0.470414i	$-0.844105\pi$
$107$	− 10771.5i	− 0.940828i	0.882446	−	0.470414i	$-0.155895\pi$
$108$	0	0
$109$	16179.0	1.36176	0.680879	−	0.732396i	$-0.261598\pi$
$109$	16179.0	1.36176	0.680879	+	0.732396i	$0.261598\pi$
$110$	− 9314.59i	− 0.769801i
$111$	0	0
$112$	55941.1	4.45959
$113$	− 1627.13i	− 0.127428i	−0.997968	−	0.0637139i	$-0.979705\pi$
$113$	− 1627.13i	− 0.127428i	0.997968	−	0.0637139i	$-0.0202945\pi$
$114$	0	0
$115$	−6195.66	−0.468481
$116$	4016.43i	0.298486i
$117$	0	0
$118$	−3470.89	−0.249274
$119$	− 3599.89i	− 0.254211i
$120$	0	0
$121$	−4123.96	−0.281672
$122$	16158.4i	1.08562i
$123$	0	0
$124$	65362.4	4.25094
$125$	− 10398.0i	− 0.665472i
$126$	0	0
$127$	−15549.0	−0.964043	−0.482021	−	0.876159i	$-0.660097\pi$
$127$	−15549.0	−0.964043	−0.482021	+	0.876159i	$0.660097\pi$
$128$	40554.1i	2.47523i
$129$	0	0
$130$	3337.37	0.197478
$131$	29979.0i	1.74693i	0.486891	+	0.873463i	$0.338131\pi$
$131$	29979.0i	1.74693i	−0.486891	+	0.873463i	$0.661869\pi$
$132$	0	0
$133$	11320.6	0.639982
$134$	− 28049.6i	− 1.56213i
$135$	0	0
$136$	11577.0	0.625919
$137$	− 20203.3i	− 1.07642i	−0.842812	−	0.538208i	$-0.819101\pi$
$137$	− 20203.3i	− 1.07642i	0.842812	−	0.538208i	$-0.180899\pi$
$138$	0	0
$139$	26676.4	1.38070	0.690348	−	0.723477i	$-0.257457\pi$
$139$	26676.4	1.38070	0.690348	+	0.723477i	$0.257457\pi$
$140$	24043.8i	1.22672i
$141$	0	0
$142$	−29909.8	−1.48333
$143$	− 6723.39i	− 0.328788i
$144$	0	0
$145$	−835.931	−0.0397589
$146$	48041.3i	2.25377i
$147$	0	0
$148$	14869.3	0.678838
$149$	− 767.855i	− 0.0345865i	−0.999850	−	0.0172933i	$-0.994495\pi$
$149$	− 767.855i	− 0.0345865i	0.999850	−	0.0172933i	$-0.00550489\pi$
$150$	0	0
$151$	−42048.1	−1.84414	−0.922068	−	0.387029i	$-0.873501\pi$
$151$	−42048.1	−1.84414	−0.922068	+	0.387029i	$0.873501\pi$
$152$	36406.4i	1.57576i
$153$	0	0
$154$	66606.2	2.80849
$155$	13603.7i	0.566233i
$156$	0	0
$157$	−1720.40	−0.0697961	−0.0348981	−	0.999391i	$-0.511111\pi$
$157$	−1720.40	−0.0697961	−0.0348981	+	0.999391i	$0.511111\pi$
$158$	− 63904.7i	− 2.55987i
$159$	0	0
$160$	−30913.7	−1.20756
$161$	− 44303.5i	− 1.70917i
$162$	0	0
$163$	−37367.9	−1.40645	−0.703224	−	0.710968i	$-0.748257\pi$
$163$	−37367.9	−1.40645	−0.703224	+	0.710968i	$0.748257\pi$
$164$	89229.0i	3.31755i
$165$	0	0
$166$	69353.6	2.51682
$167$	13657.8i	0.489719i	0.969559	+	0.244859i	$0.0787418\pi$
$167$	13657.8i	0.489719i	−0.969559	+	0.244859i	$0.921258\pi$
$168$	0	0
$169$	−26152.0	−0.915656
$170$	3855.68i	0.133414i
$171$	0	0
$172$	−35107.2	−1.18670
$173$	− 54026.4i	− 1.80515i	−0.430529	−	0.902577i	$-0.641673\pi$
$173$	− 54026.4i	− 1.80515i	0.430529	−	0.902577i	$-0.358327\pi$
$174$	0	0
$175$	34674.6	1.13223
$176$	120705.i	3.89674i
$177$	0	0
$178$	52794.5	1.66628
$179$	− 7138.88i	− 0.222804i	−0.993775	−	0.111402i	$-0.964466\pi$
$179$	− 7138.88i	− 0.222804i	0.993775	−	0.111402i	$-0.0355342\pi$
$180$	0	0
$181$	18365.5	0.560590	0.280295	−	0.959914i	$-0.409568\pi$
$181$	18365.5	0.560590	0.280295	+	0.959914i	$0.409568\pi$
$182$	23864.7i	0.720464i
$183$	0	0
$184$	142477.	4.20832
$185$	3094.71i	0.0904225i
$186$	0	0
$187$	7767.56	0.222127
$188$	125643.i	3.55486i
$189$	0	0
$190$	−12125.0	−0.335873
$191$	− 38429.0i	− 1.05340i	−0.850052	−	0.526698i	$-0.823430\pi$
$191$	− 38429.0i	− 1.05340i	0.850052	−	0.526698i	$-0.176570\pi$
$192$	0	0
$193$	10246.0	0.275068	0.137534	−	0.990497i	$-0.456082\pi$
$193$	10246.0	0.275068	0.137534	+	0.990497i	$0.456082\pi$
$194$	− 73906.2i	− 1.96371i
$195$	0	0
$196$	−69509.7	−1.80940
$197$	− 54974.7i	− 1.41654i	−0.705940	−	0.708272i	$-0.749475\pi$
$197$	− 54974.7i	− 1.41654i	0.705940	−	0.708272i	$-0.250525\pi$
$198$	0	0
$199$	312.943	0.00790240	0.00395120	−	0.999992i	$-0.498742\pi$
$199$	312.943	0.00790240	0.00395120	+	0.999992i	$0.498742\pi$
$200$	111511.i	2.78778i
$201$	0	0
$202$	−108328.	−2.65484
$203$	− 5977.52i	− 0.145054i
$204$	0	0
$205$	−18571.0	−0.441904
$206$	26775.6i	0.630963i
$207$	0	0
$208$	−43248.1	−0.999633
$209$	24426.8i	0.559209i
$210$	0	0
$211$	−36789.9	−0.826349	−0.413174	−	0.910652i	$-0.635580\pi$
$211$	−36789.9	−0.826349	−0.413174	+	0.910652i	$0.635580\pi$
$212$	149948.i	3.33633i
$213$	0	0
$214$	−82497.4	−1.80141
$215$	− 7306.79i	− 0.158070i
$216$	0	0
$217$	−97276.8	−2.06581
$218$	− 123912.i	− 2.60737i
$219$	0	0
$220$	−51879.9	−1.07190
$221$	2783.08i	0.0569824i
$222$	0	0
$223$	18588.2	0.373790	0.186895	−	0.982380i	$-0.440158\pi$
$223$	18588.2	0.373790	0.186895	+	0.982380i	$0.440158\pi$
$224$	− 221055.i	− 4.40560i
$225$	0	0
$226$	−12461.9	−0.243987
$227$	53064.9i	1.02981i	0.857248	+	0.514904i	$0.172172\pi$
$227$	53064.9i	1.02981i	−0.857248	+	0.514904i	$0.827828\pi$
$228$	0	0
$229$	−41369.1	−0.788869	−0.394435	−	0.918924i	$-0.629060\pi$
$229$	−41369.1	−0.788869	−0.394435	+	0.918924i	$0.629060\pi$
$230$	47451.5i	0.897003i
$231$	0	0
$232$	19223.3	0.357151
$233$	− 94834.2i	− 1.74684i	−0.486967	−	0.873420i	$-0.661897\pi$
$233$	− 94834.2i	− 1.74684i	0.486967	−	0.873420i	$-0.338103\pi$
$234$	0	0
$235$	−26149.8	−0.473514
$236$	19331.9i	0.347097i
$237$	0	0
$238$	−27570.9	−0.486741
$239$	64060.6i	1.12149i	0.827989	+	0.560745i	$0.189485\pi$
$239$	64060.6i	1.12149i	−0.827989	+	0.560745i	$0.810515\pi$
$240$	0	0
$241$	19607.2	0.337584	0.168792	−	0.985652i	$-0.446013\pi$
$241$	19607.2	0.337584	0.168792	+	0.985652i	$0.446013\pi$
$242$	31584.7i	0.539320i
$243$	0	0
$244$	89998.3	1.51166
$245$	− 14466.9i	− 0.241015i
$246$	0	0
$247$	−8752.00	−0.143454
$248$	− 312835.i	− 5.08642i
$249$	0	0
$250$	−79636.6	−1.27418
$251$	29107.9i	0.462022i	0.972951	+	0.231011i	$0.0742034\pi$
$251$	29107.9i	0.462022i	−0.972951	+	0.231011i	$0.925797\pi$
$252$	0	0
$253$	95594.6	1.49346
$254$	119087.i	1.84586i
$255$	0	0
$256$	109490.	1.67068
$257$	− 81206.5i	− 1.22949i	−0.788727	−	0.614744i	$-0.789259\pi$
$257$	− 81206.5i	− 1.22949i	0.788727	−	0.614744i	$-0.210741\pi$
$258$	0	0
$259$	−22129.5	−0.329891
$260$	− 18588.3i	− 0.274975i
$261$	0	0
$262$	229604.	3.34485
$263$	− 123679.i	− 1.78807i	−0.447995	−	0.894036i	$-0.647862\pi$
$263$	− 123679.i	− 1.78807i	0.447995	−	0.894036i	$-0.352138\pi$
$264$	0	0
$265$	−31208.3	−0.444405
$266$	− 86702.8i	− 1.22538i
$267$	0	0
$268$	−156229.	−2.17517
$269$	− 18442.4i	− 0.254867i	−0.991847	−	0.127433i	$-0.959326\pi$
$269$	− 18442.4i	− 0.254867i	0.991847	−	0.127433i	$-0.0406739\pi$
$270$	0	0
$271$	52364.6	0.713016	0.356508	−	0.934292i	$-0.383967\pi$
$271$	52364.6	0.713016	0.356508	+	0.934292i	$0.383967\pi$
$272$	− 49964.7i	− 0.675345i
$273$	0	0
$274$	−154733.	−2.06102
$275$	74818.1i	0.989331i
$276$	0	0
$277$	−5691.85	−0.0741812	−0.0370906	−	0.999312i	$-0.511809\pi$
$277$	−5691.85	−0.0741812	−0.0370906	+	0.999312i	$0.511809\pi$
$278$	− 204310.i	− 2.64363i
$279$	0	0
$280$	115078.	1.46783
$281$	70763.4i	0.896181i	0.893988	+	0.448091i	$0.147896\pi$
$281$	70763.4i	0.896181i	−0.893988	+	0.448091i	$0.852104\pi$
$282$	0	0
$283$	−145849.	−1.82108	−0.910541	−	0.413418i	$-0.864335\pi$
$283$	−145849.	−1.82108	−0.910541	+	0.413418i	$0.864335\pi$
$284$	166590.i	2.06544i
$285$	0	0
$286$	−51493.3	−0.629534
$287$	− 132797.i	− 1.61221i
$288$	0	0
$289$	80305.7	0.961503
$290$	6402.25i	0.0761267i
$291$	0	0
$292$	267577.	3.13822
$293$	− 21683.9i	− 0.252582i	−0.991993	−	0.126291i	$-0.959693\pi$
$293$	− 21683.9i	− 0.252582i	0.991993	−	0.126291i	$-0.0403074\pi$
$294$	0	0
$295$	−4023.51	−0.0462340
$296$	− 71166.8i	− 0.812258i
$297$	0	0
$298$	−5880.87	−0.0662231
$299$	34251.1i	0.383117i
$300$	0	0
$301$	52248.9	0.576693
$302$	322039.i	3.53098i
$303$	0	0
$304$	157125.	1.70019
$305$	18731.2i	0.201356i
$306$	0	0
$307$	111053.	1.17829	0.589147	−	0.808026i	$-0.299464\pi$
$307$	111053.	1.17829	0.589147	+	0.808026i	$0.299464\pi$
$308$	− 370979.i	− 3.91064i
$309$	0	0
$310$	104189.	1.08417
$311$	− 88436.3i	− 0.914345i	−0.889378	−	0.457172i	$-0.848862\pi$
$311$	− 88436.3i	− 0.914345i	0.889378	−	0.457172i	$-0.151138\pi$
$312$	0	0
$313$	99837.1	1.01907	0.509534	−	0.860450i	$-0.329818\pi$
$313$	99837.1	1.01907	0.509534	+	0.860450i	$0.329818\pi$
$314$	13176.3i	0.133639i
$315$	0	0
$316$	−355932.	−3.56446
$317$	9921.07i	0.0987279i	0.998781	+	0.0493639i	$0.0157194\pi$
$317$	9921.07i	0.0987279i	−0.998781	+	0.0493639i	$0.984281\pi$
$318$	0	0
$319$	12897.8	0.126746
$320$	111593.i	1.08977i
$321$	0	0
$322$	−339313.	−3.27257
$323$	− 10111.2i	− 0.0969166i
$324$	0	0
$325$	−26807.0	−0.253794
$326$	286195.i	2.69294i
$327$	0	0
$328$	427064.	3.96959
$329$	− 186990.i	− 1.72754i
$330$	0	0
$331$	196610.	1.79453	0.897263	−	0.441496i	$-0.145552\pi$
$331$	196610.	1.79453	0.897263	+	0.441496i	$0.145552\pi$
$332$	− 386281.i	− 3.50451i
$333$	0	0
$334$	104603.	0.937668
$335$	− 32515.6i	− 0.289736i
$336$	0	0
$337$	−189182.	−1.66579	−0.832895	−	0.553431i	$-0.813318\pi$
$337$	−189182.	−1.66579	−0.832895	+	0.553431i	$0.813318\pi$
$338$	200294.i	1.75321i
$339$	0	0
$340$	21475.1	0.185771
$341$	− 209896.i	− 1.80508i
$342$	0	0
$343$	−48980.9	−0.416331
$344$	168029.i	1.41993i
$345$	0	0
$346$	−413779.	−3.45634
$347$	− 128050.i	− 1.06346i	−0.846915	−	0.531728i	$-0.821543\pi$
$347$	− 128050.i	− 1.06346i	0.846915	−	0.531728i	$-0.178457\pi$
$348$	0	0
$349$	65219.4	0.535459	0.267729	−	0.963494i	$-0.413727\pi$
$349$	65219.4	0.535459	0.267729	+	0.963494i	$0.413727\pi$
$350$	− 265567.i	− 2.16789i
$351$	0	0
$352$	476976.	3.84956
$353$	142688.i	1.14509i	0.819874	+	0.572544i	$0.194043\pi$
$353$	142688.i	1.14509i	−0.819874	+	0.572544i	$0.805957\pi$
$354$	0	0
$355$	−34672.0	−0.275120
$356$	− 294051.i	− 2.32019i
$357$	0	0
$358$	−54675.5	−0.426605
$359$	47897.4i	0.371641i	0.982584	+	0.185820i	$0.0594942\pi$
$359$	47897.4i	0.371641i	−0.982584	+	0.185820i	$0.940506\pi$
$360$	0	0
$361$	−98524.1	−0.756011
$362$	− 140658.i	− 1.07337i
$363$	0	0
$364$	132920.	1.00320
$365$	55690.3i	0.418017i
$366$	0	0
$367$	10767.2	0.0799415	0.0399708	−	0.999201i	$-0.487274\pi$
$367$	10767.2	0.0799415	0.0399708	+	0.999201i	$0.487274\pi$
$368$	− 614911.i	− 4.54063i
$369$	0	0
$370$	23701.9	0.173133
$371$	− 223162.i	− 1.62134i
$372$	0	0
$373$	117383.	0.843702	0.421851	−	0.906665i	$-0.361381\pi$
$373$	117383.	0.843702	0.421851	+	0.906665i	$0.361381\pi$
$374$	− 59490.4i	− 0.425308i
$375$	0	0
$376$	601348.	4.25354
$377$	4621.23i	0.0325143i
$378$	0	0
$379$	−38343.5	−0.266940	−0.133470	−	0.991053i	$-0.542612\pi$
$379$	−38343.5	−0.266940	−0.133470	+	0.991053i	$0.542612\pi$
$380$	67533.2i	0.467681i
$381$	0	0
$382$	−294321.	−2.01695
$383$	− 66827.0i	− 0.455569i	−0.973712	−	0.227785i	$-0.926852\pi$
$383$	− 66827.0i	− 0.455569i	0.973712	−	0.227785i	$-0.0731482\pi$
$384$	0	0
$385$	77211.1	0.520905
$386$	− 78472.5i	− 0.526675i
$387$	0	0
$388$	−411638.	−2.73434
$389$	180199.i	1.19084i	0.803416	+	0.595418i	$0.203014\pi$
$389$	180199.i	1.19084i	−0.803416	+	0.595418i	$0.796986\pi$
$390$	0	0
$391$	−39570.4	−0.258831
$392$	332685.i	2.16502i
$393$	0	0
$394$	−421042.	−2.71227
$395$	− 74079.4i	− 0.474792i
$396$	0	0
$397$	−201272.	−1.27703	−0.638516	−	0.769609i	$-0.720451\pi$
$397$	−201272.	−1.27703	−0.638516	+	0.769609i	$0.720451\pi$
$398$	− 2396.78i	− 0.0151308i
$399$	0	0
$400$	481266.	3.00792
$401$	33419.8i	0.207834i	0.994586	+	0.103917i	$0.0331376\pi$
$401$	33419.8i	0.207834i	−0.994586	+	0.103917i	$0.966862\pi$
$402$	0	0
$403$	75204.8	0.463058
$404$	603359.i	3.69669i
$405$	0	0
$406$	−45780.8	−0.277735
$407$	− 47749.2i	− 0.288255i
$408$	0	0
$409$	86560.1	0.517453	0.258727	−	0.965951i	$-0.416697\pi$
$409$	86560.1	0.517453	0.258727	+	0.965951i	$0.416697\pi$
$410$	142232.i	0.846118i
$411$	0	0
$412$	149133.	0.878575
$413$	− 28771.1i	− 0.168677i
$414$	0	0
$415$	80395.9	0.466807
$416$	170898.i	0.987531i
$417$	0	0
$418$	187081.	1.07072
$419$	307989.i	1.75431i	0.480205	+	0.877156i	$0.340562\pi$
$419$	307989.i	1.75431i	−0.480205	+	0.877156i	$0.659438\pi$
$420$	0	0
$421$	−149578.	−0.843924	−0.421962	−	0.906613i	$-0.638658\pi$
$421$	−149578.	−0.843924	−0.421962	+	0.906613i	$0.638658\pi$
$422$	281767.i	1.58222i
$423$	0	0
$424$	717675.	3.99205
$425$	− 30970.2i	− 0.171461i
$426$	0	0
$427$	−133942.	−0.734614
$428$	459489.i	2.50835i
$429$	0	0
$430$	−55961.5	−0.302658
$431$	196908.i	1.06001i	0.847995	+	0.530004i	$0.177809\pi$
$431$	196908.i	1.06001i	−0.847995	+	0.530004i	$0.822191\pi$
$432$	0	0
$433$	91122.5	0.486015	0.243007	−	0.970024i	$-0.421866\pi$
$433$	91122.5	0.486015	0.243007	+	0.970024i	$0.421866\pi$
$434$	745026.i	3.95542i
$435$	0	0
$436$	−690160.	−3.63059
$437$	− 124438.i	− 0.651612i
$438$	0	0
$439$	−344596.	−1.78806	−0.894029	−	0.448009i	$-0.852133\pi$
$439$	−344596.	−1.78806	−0.894029	+	0.448009i	$0.852133\pi$
$440$	248306.i	1.28257i
$441$	0	0
$442$	21315.1	0.109105
$443$	− 157302.i	− 0.801544i	−0.916178	−	0.400772i	$-0.868742\pi$
$443$	− 157302.i	− 0.801544i	0.916178	−	0.400772i	$-0.131258\pi$
$444$	0	0
$445$	61200.3	0.309053
$446$	− 142364.i	− 0.715699i
$447$	0	0
$448$	−797968.	−3.97585
$449$	288643.i	1.43175i	0.698227	+	0.715876i	$0.253973\pi$
$449$	288643.i	1.43175i	−0.698227	+	0.715876i	$0.746027\pi$
$450$	0	0
$451$	286538.	1.40873
$452$	69409.5i	0.339736i
$453$	0	0
$454$	406415.	1.97178
$455$	27664.3i	0.133628i
$456$	0	0
$457$	372901.	1.78551	0.892753	−	0.450547i	$-0.148771\pi$
$457$	372901.	1.78551	0.892753	+	0.450547i	$0.148771\pi$
$458$	316839.i	1.51045i
$459$	0	0
$460$	264292.	1.24902
$461$	71310.9i	0.335548i	0.985826	+	0.167774i	$0.0536578\pi$
$461$	71310.9i	0.335548i	−0.985826	+	0.167774i	$0.946342\pi$
$462$	0	0
$463$	−58668.7	−0.273681	−0.136840	−	0.990593i	$-0.543695\pi$
$463$	−58668.7	−0.273681	−0.136840	+	0.990593i	$0.543695\pi$
$464$	− 82965.0i	− 0.385353i
$465$	0	0
$466$	−726319.	−3.34469
$467$	426796.i	1.95698i	0.206293	+	0.978490i	$0.433860\pi$
$467$	426796.i	1.95698i	−0.206293	+	0.978490i	$0.566140\pi$
$468$	0	0
$469$	232511.	1.05705
$470$	200277.i	0.906641i
$471$	0	0
$472$	92525.8	0.415316
$473$	112739.i	0.503907i
$474$	0	0
$475$	97392.5	0.431657
$476$	153563.i	0.677754i
$477$	0	0
$478$	490629.	2.14732
$479$	82239.2i	0.358433i	0.983810	+	0.179217i	$0.0573563\pi$
$479$	82239.2i	0.358433i	−0.983810	+	0.179217i	$0.942644\pi$
$480$	0	0
$481$	17108.3	0.0739464
$482$	− 150169.i	− 0.646376i
$483$	0	0
$484$	175919.	0.750968
$485$	− 85673.4i	− 0.364219i
$486$	0	0
$487$	160029.	0.674747	0.337374	−	0.941371i	$-0.390461\pi$
$487$	160029.	0.674747	0.337374	+	0.941371i	$0.390461\pi$
$488$	− 430747.i	− 1.80877i
$489$	0	0
$490$	−110800.	−0.461473
$491$	− 385995.i	− 1.60110i	−0.599264	−	0.800551i	$-0.704540\pi$
$491$	− 385995.i	− 1.60110i	0.599264	−	0.800551i	$-0.295460\pi$
$492$	0	0
$493$	−5338.92	−0.0219664
$494$	67030.1i	0.274673i
$495$	0	0
$496$	−1.35015e6	−5.48807
$497$	− 247930.i	− 1.00373i
$498$	0	0
$499$	151782.	0.609564	0.304782	−	0.952422i	$-0.401416\pi$
$499$	151782.	0.609564	0.304782	+	0.952422i	$0.401416\pi$
$500$	443555.i	1.77422i
$501$	0	0
$502$	222932.	0.884637
$503$	− 429613.i	− 1.69801i	−0.528381	−	0.849007i	$-0.677201\pi$
$503$	− 429613.i	− 1.69801i	0.528381	−	0.849007i	$-0.322799\pi$
$504$	0	0
$505$	−125576.	−0.492406
$506$	− 732143.i	− 2.85953i
$507$	0	0
$508$	663286.	2.57024
$509$	72974.9i	0.281668i	0.990033	+	0.140834i	$0.0449784\pi$
$509$	72974.9i	0.281668i	−0.990033	+	0.140834i	$0.955022\pi$
$510$	0	0
$511$	−398226.	−1.52506
$512$	− 189699.i	− 0.723645i
$513$	0	0
$514$	−621946.	−2.35411
$515$	31038.7i	0.117028i
$516$	0	0
$517$	403473.	1.50950
$518$	169486.i	0.631646i
$519$	0	0
$520$	−88966.6	−0.329019
$521$	147677.i	0.544050i	0.962290	+	0.272025i	$0.0876933\pi$
$521$	147677.i	0.544050i	−0.962290	+	0.272025i	$0.912307\pi$
$522$	0	0
$523$	176987.	0.647050	0.323525	−	0.946220i	$-0.395132\pi$
$523$	176987.	0.647050	0.323525	+	0.946220i	$0.395132\pi$
$524$	− 1.27883e6i	− 4.65749i
$525$	0	0
$526$	−947238.	−3.42364
$527$	86884.4i	0.312839i
$528$	0	0
$529$	−207148.	−0.740236
$530$	239019.i	0.850905i
$531$	0	0
$532$	−482912.	−1.70626
$533$	102665.i	0.361384i
$534$	0	0
$535$	−95632.5	−0.334116
$536$	747738.i	2.60267i
$537$	0	0
$538$	−141247.	−0.487995
$539$	223214.i	0.768324i
$540$	0	0
$541$	537372.	1.83603	0.918016	−	0.396544i	$-0.129790\pi$
$541$	537372.	1.83603	0.918016	+	0.396544i	$0.129790\pi$
$542$	− 401052.i	− 1.36522i
$543$	0	0
$544$	−197439.	−0.667169
$545$	− 143642.i	− 0.483601i
$546$	0	0
$547$	364742.	1.21902	0.609511	−	0.792778i	$-0.291366\pi$
$547$	364742.	1.21902	0.609511	+	0.792778i	$0.291366\pi$
$548$	861824.i	2.86984i
$549$	0	0
$550$	573020.	1.89428
$551$	− 16789.4i	− 0.0553009i
$552$	0	0
$553$	529722.	1.73220
$554$	43592.9i	0.142035i
$555$	0	0
$556$	−1.13795e6	−3.68108
$557$	− 247324.i	− 0.797178i	−0.917130	−	0.398589i	$-0.869500\pi$
$557$	− 247324.i	− 0.797178i	0.917130	−	0.398589i	$-0.130500\pi$
$558$	0	0
$559$	−40393.7	−0.129268
$560$	− 496659.i	− 1.58373i
$561$	0	0
$562$	541965.	1.71593
$563$	115026.i	0.362893i	0.983401	+	0.181447i	$0.0580779\pi$
$563$	115026.i	0.362893i	−0.983401	+	0.181447i	$0.941922\pi$
$564$	0	0
$565$	−14446.0	−0.0452535
$566$	1.11703e6i	3.48684i
$567$	0	0
$568$	797326.	2.47138
$569$	− 6712.24i	− 0.0207321i	−0.999946	−	0.0103660i	$-0.996700\pi$
$569$	− 6712.24i	− 0.0207321i	0.999946	−	0.0103660i	$-0.00329967\pi$
$570$	0	0
$571$	598189.	1.83470	0.917352	−	0.398077i	$-0.130322\pi$
$571$	598189.	1.83470	0.917352	+	0.398077i	$0.130322\pi$
$572$	286804.i	0.876584i
$573$	0	0
$574$	−1.01707e6	−3.08692
$575$	− 381147.i	− 1.15281i
$576$	0	0
$577$	−538291.	−1.61683	−0.808417	−	0.588610i	$-0.799675\pi$
$577$	−538291.	−1.61683	−0.808417	+	0.588610i	$0.799675\pi$
$578$	− 615048.i	− 1.84100i
$579$	0	0
$580$	35658.9	0.106001
$581$	574890.i	1.70307i
$582$	0	0
$583$	481523.	1.41671
$584$	− 1.28067e6i	− 3.75501i
$585$	0	0
$586$	−166074.	−0.483621
$587$	24508.3i	0.0711274i	0.999367	+	0.0355637i	$0.0113227\pi$
$587$	24508.3i	0.0711274i	−0.999367	+	0.0355637i	$0.988677\pi$
$588$	0	0
$589$	−273227.	−0.787577
$590$	30815.4i	0.0885246i
$591$	0	0
$592$	−307146.	−0.876398
$593$	317.075i	0 0.000901680i	1.00000		0.000450840i	$0.000143507\pi$
$593$	317.075i	0 0.000901680i	−1.00000		0.000450840i	$0.999856\pi$
$594$	0	0
$595$	−31960.7	−0.0902781
$596$	32754.9i	0.0922113i
$597$	0	0
$598$	262323.	0.733558
$599$	3216.43i	0.00896439i	0.999990	+	0.00448219i	$0.00142673\pi$
$599$	3216.43i	0.00896439i	−0.999990	+	0.00448219i	$0.998573\pi$
$600$	0	0
$601$	280008.	0.775214	0.387607	−	0.921825i	$-0.373302\pi$
$601$	280008.	0.775214	0.387607	+	0.921825i	$0.373302\pi$
$602$	− 400166.i	− 1.10420i
$603$	0	0
$604$	1.79368e6	4.91666
$605$	36613.6i	0.100030i
$606$	0	0
$607$	555403.	1.50741	0.753704	−	0.657214i	$-0.228265\pi$
$607$	555403.	1.50741	0.753704	+	0.657214i	$0.228265\pi$
$608$	− 620891.i	− 1.67961i
$609$	0	0
$610$	143459.	0.385538
$611$	144563.i	0.387234i
$612$	0	0
$613$	−429205.	−1.14220	−0.571102	−	0.820879i	$-0.693484\pi$
$613$	−429205.	−1.14220	−0.571102	+	0.820879i	$0.693484\pi$
$614$	− 850537.i	− 2.25609i
$615$	0	0
$616$	−1.77557e6	−4.67924
$617$	− 643275.i	− 1.68976i	−0.534953	−	0.844882i	$-0.679671\pi$
$617$	− 643275.i	− 1.68976i	0.534953	−	0.844882i	$-0.320329\pi$
$618$	0	0
$619$	196343.	0.512431	0.256215	−	0.966620i	$-0.417524\pi$
$619$	196343.	0.512431	0.256215	+	0.966620i	$0.417524\pi$
$620$	− 580304.i	− 1.50964i
$621$	0	0
$622$	−677319.	−1.75070
$623$	437627.i	1.12753i
$624$	0	0
$625$	249044.	0.637553
$626$	− 764635.i	− 1.95122i
$627$	0	0
$628$	73388.5	0.186084
$629$	19765.3i	0.0499576i
$630$	0	0
$631$	−656278.	−1.64827	−0.824137	−	0.566391i	$-0.808339\pi$
$631$	−656278.	−1.64827	−0.824137	+	0.566391i	$0.808339\pi$
$632$	1.70355e6i	4.26502i
$633$	0	0
$634$	75983.8	0.189035
$635$	138048.i	0.342361i
$636$	0	0
$637$	−79976.6	−0.197099
$638$	− 98782.3i	− 0.242682i
$639$	0	0
$640$	360050.	0.879028
$641$	352122.i	0.856993i	0.903544	+	0.428496i	$0.140957\pi$
$641$	352122.i	0.856993i	−0.903544	+	0.428496i	$0.859043\pi$
$642$	0	0
$643$	333471.	0.806560	0.403280	−	0.915077i	$-0.367870\pi$
$643$	333471.	0.806560	0.403280	+	0.915077i	$0.367870\pi$
$644$	1.88988e6i	4.55684i
$645$	0	0
$646$	−77440.1	−0.185567
$647$	576523.i	1.37724i	0.725125	+	0.688618i	$0.241782\pi$
$647$	576523.i	1.37724i	−0.725125	+	0.688618i	$0.758218\pi$
$648$	0	0
$649$	62080.0	0.147388
$650$	205310.i	0.485941i
$651$	0	0
$652$	1.59403e6	3.74974
$653$	312773.i	0.733506i	0.930318	+	0.366753i	$0.119531\pi$
$653$	312773.i	0.733506i	−0.930318	+	0.366753i	$0.880469\pi$
$654$	0	0
$655$	266161.	0.620386
$656$	− 1.84315e6i	− 4.28305i
$657$	0	0
$658$	−1.43213e6	−3.30773
$659$	174086.i	0.400860i	0.979708	+	0.200430i	$0.0642340\pi$
$659$	174086.i	0.400860i	−0.979708	+	0.200430i	$0.935766\pi$
$660$	0	0
$661$	−311630.	−0.713241	−0.356620	−	0.934249i	$-0.616071\pi$
$661$	−311630.	−0.713241	−0.356620	+	0.934249i	$0.616071\pi$
$662$	− 1.50580e6i	− 3.43599i
$663$	0	0
$664$	−1.84881e6	−4.19329
$665$	− 100507.i	− 0.227277i
$666$	0	0
$667$	−65705.6	−0.147690
$668$	− 582609.i	− 1.30564i
$669$	0	0
$670$	−249032.	−0.554760
$671$	− 289009.i	− 0.641898i
$672$	0	0
$673$	−657508.	−1.45168	−0.725840	−	0.687864i	$-0.758549\pi$
$673$	−657508.	−1.45168	−0.725840	+	0.687864i	$0.758549\pi$
$674$	1.44891e6i	3.18950i
$675$	0	0
$676$	1.11559e6	2.44123
$677$	− 357902.i	− 0.780885i	−0.920627	−	0.390442i	$-0.872322\pi$
$677$	− 357902.i	− 0.780885i	0.920627	−	0.390442i	$-0.127678\pi$
$678$	0	0
$679$	612628.	1.32879
$680$	− 102783.i	− 0.222283i
$681$	0	0
$682$	−1.60756e6	−3.45620
$683$	− 231985.i	− 0.497300i	−0.968593	−	0.248650i	$-0.920013\pi$
$683$	− 231985.i	− 0.497300i	0.968593	−	0.248650i	$-0.0799869\pi$
$684$	0	0
$685$	−179370.	−0.382268
$686$	375137.i	0.797152i
$687$	0	0
$688$	725190.	1.53206
$689$	172527.i	0.363429i
$690$	0	0
$691$	34366.8	0.0719752	0.0359876	−	0.999352i	$-0.488542\pi$
$691$	34366.8	0.0719752	0.0359876	+	0.999352i	$0.488542\pi$
$692$	2.30464e6i	4.81273i
$693$	0	0
$694$	−980711.	−2.03621
$695$	− 236840.i	− 0.490327i
$696$	0	0
$697$	−118609.	−0.244148
$698$	− 499505.i	− 1.02525i
$699$	0	0
$700$	−1.47914e6	−3.01865
$701$	204222.i	0.415590i	0.978172	+	0.207795i	$0.0666288\pi$
$701$	204222.i	0.415590i	−0.978172	+	0.207795i	$0.933371\pi$
$702$	0	0
$703$	−62156.3	−0.125769
$704$	− 1.72179e6i	− 3.47405i
$705$	0	0
$706$	1.09283e6	2.19251
$707$	− 897960.i	− 1.79646i
$708$	0	0
$709$	−191874.	−0.381701	−0.190851	−	0.981619i	$-0.561125\pi$
$709$	−191874.	−0.381701	−0.190851	+	0.981619i	$0.561125\pi$
$710$	265547.i	0.526774i
$711$	0	0
$712$	−1.40738e6	−2.77620
$713$	1.06928e6i	2.10335i
$714$	0	0
$715$	−59692.0	−0.116763
$716$	304528.i	0.594020i
$717$	0	0
$718$	366838.	0.711583
$719$	− 637580.i	− 1.23332i	−0.787228	−	0.616662i	$-0.788485\pi$
$719$	− 637580.i	− 1.23332i	0.787228	−	0.616662i	$-0.211515\pi$
$720$	0	0
$721$	−221950.	−0.426957
$722$	754579.i	1.44754i
$723$	0	0
$724$	−783430.	−1.49459
$725$	− 51425.2i	− 0.0978362i
$726$	0	0
$727$	11669.2	0.0220786	0.0110393	−	0.999939i	$-0.496486\pi$
$727$	11669.2	0.0220786	0.0110393	+	0.999939i	$0.496486\pi$
$728$	− 636177.i	− 1.20037i
$729$	0	0
$730$	426523.	0.800380
$731$	− 46667.0i	− 0.0873323i
$732$	0	0
$733$	−304104.	−0.565997	−0.282999	−	0.959120i	$-0.591329\pi$
$733$	−304104.	−0.565997	−0.282999	+	0.959120i	$0.591329\pi$
$734$	− 82464.5i	− 0.153065i
$735$	0	0
$736$	−2.42987e6	−4.48567
$737$	501693.i	0.923641i
$738$	0	0
$739$	625858.	1.14601	0.573003	−	0.819553i	$-0.305778\pi$
$739$	625858.	1.14601	0.573003	+	0.819553i	$0.305778\pi$
$740$	− 132013.i	− 0.241076i
$741$	0	0
$742$	−1.70916e6	−3.10439
$743$	93152.6i	0.168740i	0.996435	+	0.0843699i	$0.0268877\pi$
$743$	93152.6i	0.168740i	−0.996435	+	0.0843699i	$0.973112\pi$
$744$	0	0
$745$	−6817.22	−0.0122827
$746$	− 899019.i	− 1.61544i
$747$	0	0
$748$	−331346.	−0.592214
$749$	− 683843.i	− 1.21897i
$750$	0	0
$751$	129073.	0.228853	0.114426	−	0.993432i	$-0.463497\pi$
$751$	129073.	0.228853	0.114426	+	0.993432i	$0.463497\pi$
$752$	− 2.59534e6i	− 4.58942i
$753$	0	0
$754$	35393.2	0.0622554
$755$	373314.i	0.654908i
$756$	0	0
$757$	61203.6	0.106803	0.0534017	−	0.998573i	$-0.482994\pi$
$757$	61203.6	0.106803	0.0534017	+	0.998573i	$0.482994\pi$
$758$	293667.i	0.511112i
$759$	0	0
$760$	323225.	0.559600
$761$	927429.i	1.60144i	0.599037	+	0.800721i	$0.295550\pi$
$761$	927429.i	1.60144i	−0.599037	+	0.800721i	$0.704450\pi$
$762$	0	0
$763$	1.02714e6	1.76434
$764$	1.63929e6i	2.80847i
$765$	0	0
$766$	−511816.	−0.872282
$767$	22243.0i	0.0378096i
$768$	0	0
$769$	534814.	0.904379	0.452189	−	0.891922i	$-0.350643\pi$
$769$	534814.	0.904379	0.452189	+	0.891922i	$0.350643\pi$
$770$	− 591347.i	− 0.997380i
$771$	0	0
$772$	−437071.	−0.733361
$773$	− 547577.i	− 0.916402i	−0.888849	−	0.458201i	$-0.848494\pi$
$773$	− 547577.i	− 0.916402i	0.888849	−	0.458201i	$-0.151506\pi$
$774$	0	0
$775$	−836882.	−1.39335
$776$	1.97017e6i	3.27175i
$777$	0	0
$778$	1.38011e6	2.28010
$779$	− 372993.i	− 0.614647i
$780$	0	0
$781$	534965.	0.877047
$782$	303063.i	0.495586i
$783$	0	0
$784$	1.43582e6	2.33598
$785$	15274.2i	0.0247867i
$786$	0	0
$787$	1.20838e6	1.95098	0.975490	−	0.220043i	$-0.0706198\pi$
$787$	1.20838e6	1.95098	0.975490	+	0.220043i	$0.0706198\pi$
$788$	2.34509e6i	3.77666i
$789$	0	0
$790$	−567362.	−0.909088
$791$	− 103300.i	− 0.165100i
$792$	0	0
$793$	103550.	0.164666
$794$	1.54151e6i	2.44514i
$795$	0	0
$796$	−13349.4	−0.0210686
$797$	− 694506.i	− 1.09335i	−0.837345	−	0.546675i	$-0.815893\pi$
$797$	− 694506.i	− 1.09335i	0.837345	−	0.546675i	$-0.184107\pi$
$798$	0	0
$799$	−167014.	−0.261612
$800$	− 1.90176e6i	− 2.97150i
$801$	0	0
$802$	255957.	0.397940
$803$	− 859262.i	− 1.33258i
$804$	0	0
$805$	−393338.	−0.606979
$806$	− 575981.i	− 0.886621i
$807$	0	0
$808$	2.88778e6	4.42324
$809$	− 508173.i	− 0.776452i	−0.921564	−	0.388226i	$-0.873088\pi$
$809$	− 508173.i	− 0.776452i	0.921564	−	0.388226i	$-0.126912\pi$
$810$	0	0
$811$	−115454.	−0.175536	−0.0877681	−	0.996141i	$-0.527973\pi$
$811$	−115454.	−0.175536	−0.0877681	+	0.996141i	$0.527973\pi$
$812$	254987.i	0.386729i
$813$	0	0
$814$	−365703.	−0.551925
$815$	331762.i	0.499472i
$816$	0	0
$817$	146755.	0.219861
$818$	− 662949.i	− 0.990771i
$819$	0	0
$820$	792197.	1.17816
$821$	973501.i	1.44428i	0.691749	+	0.722138i	$0.256840\pi$
$821$	973501.i	1.44428i	−0.691749	+	0.722138i	$0.743160\pi$
$822$	0	0
$823$	254624.	0.375924	0.187962	−	0.982176i	$-0.439812\pi$
$823$	254624.	0.375924	0.187962	+	0.982176i	$0.439812\pi$
$824$	− 713775.i	− 1.05125i
$825$	0	0
$826$	−220353.	−0.322967
$827$	− 173362.i	− 0.253480i	−0.991936	−	0.126740i	$-0.959549\pi$
$827$	− 173362.i	− 0.253480i	0.991936	−	0.126740i	$-0.0404514\pi$
$828$	0	0
$829$	75748.0	0.110220	0.0551102	−	0.998480i	$-0.482449\pi$
$829$	75748.0	0.110220	0.0551102	+	0.998480i	$0.482449\pi$
$830$	− 615739.i	− 0.893800i
$831$	0	0
$832$	616910.	0.891201
$833$	− 92397.2i	− 0.133159i
$834$	0	0
$835$	121257.	0.173914
$836$	− 1.04199e6i	− 1.49091i
$837$	0	0
$838$	2.35883e6	3.35899
$839$	1.17811e6i	1.67365i	0.547474	+	0.836823i	$0.315589\pi$
$839$	1.17811e6i	1.67365i	−0.547474	+	0.836823i	$0.684411\pi$
$840$	0	0
$841$	698416.	0.987466
$842$	1.14559e6i	1.61587i
$843$	0	0
$844$	1.56937e6	2.20313
$845$	232184.i	0.325177i
$846$	0	0
$847$	−261814.	−0.364944
$848$	− 3.09739e6i	− 4.30729i
$849$	0	0
$850$	−237195.	−0.328298
$851$	243250.i	0.335887i
$852$	0	0
$853$	325449.	0.447285	0.223643	−	0.974671i	$-0.428205\pi$
$853$	325449.	0.447285	0.223643	+	0.974671i	$0.428205\pi$
$854$	1.02584e6i	1.40657i
$855$	0	0
$856$	2.19919e6	3.00134
$857$	837861.i	1.14080i	0.821366	+	0.570401i	$0.193212\pi$
$857$	837861.i	1.14080i	−0.821366	+	0.570401i	$0.806788\pi$
$858$	0	0
$859$	596355.	0.808199	0.404099	−	0.914715i	$-0.367585\pi$
$859$	596355.	0.808199	0.404099	+	0.914715i	$0.367585\pi$
$860$	311691.i	0.421432i
$861$	0	0
$862$	1.50809e6	2.02960
$863$	289501.i	0.388712i	0.980931	+	0.194356i	$0.0622617\pi$
$863$	289501.i	0.388712i	−0.980931	+	0.194356i	$0.937738\pi$
$864$	0	0
$865$	−479660.	−0.641064
$866$	− 697892.i	− 0.930577i
$867$	0	0
$868$	4.14960e6	5.50766
$869$	1.14299e6i	1.51358i
$870$	0	0
$871$	−179754.	−0.236942
$872$	3.30322e6i	4.34415i
$873$	0	0
$874$	−953048.	−1.24765
$875$	− 660128.i	− 0.862208i
$876$	0	0
$877$	−1.42801e6	−1.85666	−0.928329	−	0.371759i	$-0.878755\pi$
$877$	−1.42801e6	−1.85666	−0.928329	+	0.371759i	$0.878755\pi$
$878$	2.63920e6i	3.42361i
$879$	0	0
$880$	1.07165e6	1.38385
$881$	270761.i	0.348846i	0.984671	+	0.174423i	$0.0558060\pi$
$881$	270761.i	0.348846i	−0.984671	+	0.174423i	$0.944194\pi$
$882$	0	0
$883$	923108.	1.18394	0.591972	−	0.805958i	$-0.298350\pi$
$883$	923108.	1.18394	0.591972	+	0.805958i	$0.298350\pi$
$884$	− 118720.i	− 0.151921i
$885$	0	0
$886$	−1.20475e6	−1.53472
$887$	840536.i	1.06834i	0.845378	+	0.534169i	$0.179375\pi$
$887$	840536.i	1.06834i	−0.845378	+	0.534169i	$0.820625\pi$
$888$	0	0
$889$	−987147.	−1.24905
$890$	− 468723.i	− 0.591747i
$891$	0	0
$892$	−792931.	−0.996565
$893$	− 525211.i	− 0.658614i
$894$	0	0
$895$	−63380.8	−0.0791246
$896$	2.57462e6i	3.20699i
$897$	0	0
$898$	2.21067e6	2.74139
$899$	144269.i	0.178507i
$900$	0	0
$901$	−199321.	−0.245530
$902$	− 2.19455e6i	− 2.69732i
$903$	0	0
$904$	332205.	0.406508
$905$	− 163054.i	− 0.199083i
$906$	0	0
$907$	−1.44160e6	−1.75239	−0.876196	−	0.481955i	$-0.839927\pi$
$907$	−1.44160e6	−1.75239	−0.876196	+	0.481955i	$0.839927\pi$
$908$	− 2.26363e6i	− 2.74558i
$909$	0	0
$910$	211877.	0.255859
$911$	220395.i	0.265561i	0.991145	+	0.132780i	$0.0423905\pi$
$911$	220395.i	0.265561i	−0.991145	+	0.132780i	$0.957609\pi$
$912$	0	0
$913$	−1.24045e6	−1.48812
$914$	− 2.85599e6i	− 3.41872i
$915$	0	0
$916$	1.76471e6	2.10321
$917$	1.90325e6i	2.26337i
$918$	0	0
$919$	517625.	0.612893	0.306446	−	0.951888i	$-0.400860\pi$
$919$	517625.	0.612893	0.306446	+	0.951888i	$0.400860\pi$
$920$	− 1.26495e6i	− 1.49450i
$921$	0	0
$922$	546158.	0.642475
$923$	191675.i	0.224990i
$924$	0	0
$925$	−190382.	−0.222506
$926$	449334.i	0.524019i
$927$	0	0
$928$	−327843.	−0.380688
$929$	− 894023.i	− 1.03590i	−0.855411	−	0.517950i	$-0.826695\pi$
$929$	− 894023.i	− 1.03590i	0.855411	−	0.517950i	$-0.173305\pi$
$930$	0	0
$931$	290563.	0.335229
$932$	4.04541e6i	4.65726i
$933$	0	0
$934$	3.26876e6	3.74705
$935$	− 68962.4i	− 0.0788840i
$936$	0	0
$937$	−622617.	−0.709156	−0.354578	−	0.935026i	$-0.615375\pi$
$937$	−622617.	−0.709156	−0.354578	+	0.935026i	$0.615375\pi$
$938$	− 1.78076e6i	− 2.02395i
$939$	0	0
$940$	1.11549e6	1.26244
$941$	− 1.32733e6i	− 1.49899i	−0.662010	−	0.749495i	$-0.730296\pi$
$941$	− 1.32733e6i	− 1.49899i	0.662010	−	0.749495i	$-0.269704\pi$
$942$	0	0
$943$	−1.45972e6	−1.64151
$944$	− 399329.i	− 0.448112i
$945$	0	0
$946$	863446.	0.964835
$947$	− 1.73543e6i	− 1.93511i	−0.252657	−	0.967556i	$-0.581305\pi$
$947$	− 1.73543e6i	− 1.93511i	0.252657	−	0.967556i	$-0.418695\pi$
$948$	0	0
$949$	307869.	0.341849
$950$	− 745913.i	− 0.826496i
$951$	0	0
$952$	734977.	0.810961
$953$	− 436680.i	− 0.480814i	−0.970672	−	0.240407i	$-0.922719\pi$
$953$	− 436680.i	− 0.480814i	0.970672	−	0.240407i	$-0.0772810\pi$
$954$	0	0
$955$	−341182.	−0.374093
$956$	− 2.73268e6i	− 2.99001i
$957$	0	0
$958$	629856.	0.686295
$959$	− 1.28262e6i	− 1.39464i
$960$	0	0
$961$	1.42428e6	1.54223
$962$	− 131030.i	− 0.141586i
$963$	0	0
$964$	−836400.	−0.900036
$965$	− 90966.7i	− 0.0976850i
$966$	0	0
$967$	1.14062e6	1.21979	0.609897	−	0.792481i	$-0.291211\pi$
$967$	1.14062e6	1.21979	0.609897	+	0.792481i	$0.291211\pi$
$968$	− 841976.i	− 0.898564i
$969$	0	0
$970$	−656158.	−0.697373
$971$	454074.i	0.481602i	0.970574	+	0.240801i	$0.0774102\pi$
$971$	454074.i	0.481602i	−0.970574	+	0.240801i	$0.922590\pi$
$972$	0	0
$973$	1.69358e6	1.78888
$974$	− 1.22564e6i	− 1.29194i
$975$	0	0
$976$	−1.85904e6	−1.95160
$977$	− 85711.3i	− 0.0897944i	−0.998992	−	0.0448972i	$-0.985704\pi$
$977$	− 85711.3i	− 0.0897944i	0.998992	−	0.0448972i	$-0.0142960\pi$
$978$	0	0
$979$	−944278.	−0.985223
$980$	617125.i	0.642571i
$981$	0	0
$982$	−2.95627e6	−3.06564
$983$	− 779306.i	− 0.806494i	−0.915091	−	0.403247i	$-0.867882\pi$
$983$	− 779306.i	− 0.806494i	0.915091	−	0.403247i	$-0.132118\pi$
$984$	0	0
$985$	−488079.	−0.503058
$986$	40889.9i	0.0420593i
$987$	0	0
$988$	373340.	0.382464
$989$	− 574326.i	− 0.587173i
$990$	0	0
$991$	−1.45956e6	−1.48619	−0.743093	−	0.669188i	$-0.766642\pi$
$991$	−1.45956e6	−1.48619	−0.743093	+	0.669188i	$0.766642\pi$
$992$	5.33524e6i	5.42164i
$993$	0	0
$994$	−1.89886e6	−1.92185
$995$	− 2778.39i	− 0.00280638i
$996$	0	0
$997$	−924174.	−0.929744	−0.464872	−	0.885378i	$-0.653900\pi$
$997$	−924174.	−0.929744	−0.464872	+	0.885378i	$0.653900\pi$
$998$	− 1.16247e6i	− 1.16714i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	531.5.b.a.296.3	✓	76
3.2	odd	2	inner	531.5.b.a.296.74	yes	76

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
531.5.b.a.296.3	✓	76	1.1	even	1	trivial
531.5.b.a.296.74	yes	76	3.2	odd	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 531 = 3^{2} \cdot 59 \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	531.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-7.65883i q^{2} -42.6577 q^{4} -8.87825i q^{5} +63.4860 q^{7} +204.167i q^{8} +O(q^{10})\)	\(q-7.65883i q^{2} -42.6577 q^{4} -8.87825i q^{5} +63.4860 q^{7} +204.167i q^{8} -67.9970 q^{10} +136.985i q^{11} -49.0811 q^{13} -486.229i q^{14} +881.155 q^{16} -56.7036i q^{17} +178.317 q^{19} +378.726i q^{20} +1049.15 q^{22} -697.846i q^{23} +546.177 q^{25} +375.904i q^{26} -2708.17 q^{28} -94.1549i q^{29} -1532.25 q^{31} -3481.95i q^{32} -434.283 q^{34} -563.645i q^{35} -348.572 q^{37} -1365.70i q^{38} +1812.64 q^{40} -2091.74i q^{41} +822.999 q^{43} -5843.47i q^{44} -5344.69 q^{46} -2945.38i q^{47} +1629.48 q^{49} -4183.07i q^{50} +2093.69 q^{52} -3515.14i q^{53} +1216.19 q^{55} +12961.7i q^{56} -721.116 q^{58} -453.188i q^{59} -2109.78 q^{61} +11735.3i q^{62} -12569.2 q^{64} +435.755i q^{65} +3662.39 q^{67} +2418.85i q^{68} -4316.86 q^{70} -3905.27i q^{71} -6272.66 q^{73} +2669.65i q^{74} -7606.59 q^{76} +8696.65i q^{77} +8343.92 q^{79} -7823.12i q^{80} -16020.3 q^{82} +9055.37i q^{83} -503.429 q^{85} -6303.21i q^{86} -27967.8 q^{88} +6893.28i q^{89} -3115.97 q^{91} +29768.5i q^{92} -22558.2 q^{94} -1583.14i q^{95} +9649.80 q^{97} -12479.9i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 76 q - 632 q^{4} - 48 q^{7}+O(q^{10}) \) 76 * q - 632 * q^4 - 48 * q^7	\( 76 q - 632 q^{4} - 48 q^{7} - 72 q^{10} + 448 q^{13} + 5696 q^{16} - 2240 q^{19} + 1008 q^{22} - 11180 q^{25} + 5376 q^{28} - 3440 q^{31} - 7112 q^{34} + 7016 q^{37} - 832 q^{40} - 608 q^{43} + 17728 q^{46} + 30628 q^{49} - 27056 q^{52} - 9312 q^{55} + 4616 q^{58} - 10056 q^{61} - 42864 q^{64} + 31160 q^{67} + 7544 q^{70} + 4488 q^{73} - 15632 q^{76} + 19208 q^{79} + 9768 q^{82} - 7568 q^{85} - 41736 q^{88} - 31240 q^{91} + 27360 q^{94} + 22216 q^{97}+O(q^{100}) \) 76 * q - 632 * q^4 - 48 * q^7 - 72 * q^10 + 448 * q^13 + 5696 * q^16 - 2240 * q^19 + 1008 * q^22 - 11180 * q^25 + 5376 * q^28 - 3440 * q^31 - 7112 * q^34 + 7016 * q^37 - 832 * q^40 - 608 * q^43 + 17728 * q^46 + 30628 * q^49 - 27056 * q^52 - 9312 * q^55 + 4616 * q^58 - 10056 * q^61 - 42864 * q^64 + 31160 * q^67 + 7544 * q^70 + 4488 * q^73 - 15632 * q^76 + 19208 * q^79 + 9768 * q^82 - 7568 * q^85 - 41736 * q^88 - 31240 * q^91 + 27360 * q^94 + 22216 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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