LMFDB - Embedded newform 531.6.a.a.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [531,6,Mod(1,531)]

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([0, 0]))

N = Newforms(chi, 6, names="a")

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

chi := DirichletCharacter("531.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 531 = 3^{2} \cdot 59 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	531.a (trivial)

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:	yes
Analytic conductor:	$85.1638083207$
Analytic rank:	$1$
Dimension:	$9$
Coefficient field:	$\mathbb{Q}[x]/(x^{9} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{9} - 171x^{7} - 44x^{6} + 9767x^{5} + 2200x^{4} - 215105x^{3} - 33724x^{2} + 1380292x + 1109072 $ x^9 - 171x^7 - 44x^6 + 9767x^5 + 2200x^4 - 215105x^3 - 33724x^2 + 1380292*x + 1109072
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 59)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$4.80409$ of defining polynomial
Character	$\chi$	$=$	531.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.80409 q^{2} -17.5289 q^{4} +42.9333 q^{5} +214.202 q^{7} +188.412 q^{8} +O(q^{10})$	\(q-3.80409 q^{2} -17.5289 q^{4} +42.9333 q^{5} +214.202 q^{7} +188.412 q^{8} -163.322 q^{10} +720.507 q^{11} -882.947 q^{13} -814.844 q^{14} -155.814 q^{16} -1584.31 q^{17} -1234.05 q^{19} -752.572 q^{20} -2740.87 q^{22} -1180.66 q^{23} -1281.73 q^{25} +3358.81 q^{26} -3754.72 q^{28} -6690.90 q^{29} +407.702 q^{31} -5436.47 q^{32} +6026.86 q^{34} +9196.40 q^{35} +5606.29 q^{37} +4694.45 q^{38} +8089.16 q^{40} +9137.16 q^{41} -19293.4 q^{43} -12629.7 q^{44} +4491.33 q^{46} +4662.43 q^{47} +29075.5 q^{49} +4875.83 q^{50} +15477.1 q^{52} -10048.8 q^{53} +30933.7 q^{55} +40358.3 q^{56} +25452.8 q^{58} +3481.00 q^{59} -34785.8 q^{61} -1550.93 q^{62} +25666.9 q^{64} -37907.8 q^{65} -41086.0 q^{67} +27771.2 q^{68} -34983.9 q^{70} -73673.3 q^{71} -32905.0 q^{73} -21326.8 q^{74} +21631.6 q^{76} +154334. q^{77} -36261.8 q^{79} -6689.62 q^{80} -34758.6 q^{82} +38330.1 q^{83} -68019.6 q^{85} +73393.8 q^{86} +135752. q^{88} +39785.2 q^{89} -189129. q^{91} +20695.6 q^{92} -17736.3 q^{94} -52981.9 q^{95} -53489.3 q^{97} -110606. q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 9 q + 9 q^{2} + 63 q^{4} + 72 q^{5} - 208 q^{7} + 327 q^{8}+O(q^{10}) $ 9 * q + 9 * q^2 + 63 * q^4 + 72 * q^5 - 208 * q^7 + 327 * q^8	$ 9 q + 9 q^{2} + 63 q^{4} + 72 q^{5} - 208 q^{7} + 327 q^{8} - 402 q^{10} + 1176 q^{11} - 1928 q^{13} + 89 q^{14} - 3533 q^{16} + 3031 q^{17} - 4516 q^{19} + 5540 q^{20} - 2075 q^{22} + 2192 q^{23} - 5611 q^{25} - 3155 q^{26} - 10477 q^{28} + 3096 q^{29} - 10148 q^{31} - 15377 q^{32} + 13101 q^{34} - 9947 q^{35} - 11554 q^{37} - 21836 q^{38} + 24298 q^{40} - 11294 q^{41} - 12526 q^{43} - 24051 q^{44} + 22376 q^{46} - 11716 q^{47} + 19137 q^{49} - 66081 q^{50} + 23609 q^{52} - 16552 q^{53} - 9668 q^{55} - 51401 q^{56} + 62124 q^{58} + 31329 q^{59} - 143662 q^{61} - 96300 q^{62} - 79981 q^{64} + 69180 q^{65} - 119488 q^{67} + 19761 q^{68} - 119574 q^{70} + 38295 q^{71} - 197332 q^{73} - 50783 q^{74} - 104806 q^{76} + 67070 q^{77} - 75556 q^{79} - 45772 q^{80} - 100205 q^{82} + 49800 q^{83} - 189532 q^{85} - 23879 q^{86} + 43035 q^{88} + 85950 q^{89} - 117668 q^{91} - 167480 q^{92} - 5734 q^{94} - 169216 q^{95} - 23462 q^{97} + 70298 q^{98}+O(q^{100}) $ 9 * q + 9 * q^2 + 63 * q^4 + 72 * q^5 - 208 * q^7 + 327 * q^8 - 402 * q^10 + 1176 * q^11 - 1928 * q^13 + 89 * q^14 - 3533 * q^16 + 3031 * q^17 - 4516 * q^19 + 5540 * q^20 - 2075 * q^22 + 2192 * q^23 - 5611 * q^25 - 3155 * q^26 - 10477 * q^28 + 3096 * q^29 - 10148 * q^31 - 15377 * q^32 + 13101 * q^34 - 9947 * q^35 - 11554 * q^37 - 21836 * q^38 + 24298 * q^40 - 11294 * q^41 - 12526 * q^43 - 24051 * q^44 + 22376 * q^46 - 11716 * q^47 + 19137 * q^49 - 66081 * q^50 + 23609 * q^52 - 16552 * q^53 - 9668 * q^55 - 51401 * q^56 + 62124 * q^58 + 31329 * q^59 - 143662 * q^61 - 96300 * q^62 - 79981 * q^64 + 69180 * q^65 - 119488 * q^67 + 19761 * q^68 - 119574 * q^70 + 38295 * q^71 - 197332 * q^73 - 50783 * q^74 - 104806 * q^76 + 67070 * q^77 - 75556 * q^79 - 45772 * q^80 - 100205 * q^82 + 49800 * q^83 - 189532 * q^85 - 23879 * q^86 + 43035 * q^88 + 85950 * q^89 - 117668 * q^91 - 167480 * q^92 - 5734 * q^94 - 169216 * q^95 - 23462 * q^97 + 70298 * q^98

Display 100 coefficients 10 coefficients

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 $

$2$	−3.80409	−0.672475	−0.336237	−	0.941777i	$-0.609154\pi$
$2$	−3.80409	−0.672475	−0.336237	+	0.941777i	$0.609154\pi$
$3$	0	0
$3$	0	0
$4$	−17.5289	−0.547777
$5$	42.9333	0.768014	0.384007	−	0.923330i	$-0.374544\pi$
$5$	42.9333	0.768014	0.384007	+	0.923330i	$0.374544\pi$
$6$	0	0
$7$	214.202	1.65226	0.826130	−	0.563479i	$-0.190538\pi$
$7$	214.202	1.65226	0.826130	+	0.563479i	$0.190538\pi$
$8$	188.412	1.04084
$9$	0	0
$10$	−163.322	−0.516470
$11$	720.507	1.79538	0.897690	−	0.440628i	$-0.145244\pi$
$11$	720.507	1.79538	0.897690	+	0.440628i	$0.145244\pi$
$12$	0	0
$13$	−882.947	−1.44903	−0.724513	−	0.689261i	$-0.757935\pi$
$13$	−882.947	−1.44903	−0.724513	+	0.689261i	$0.757935\pi$
$14$	−814.844	−1.11110
$15$	0	0
$16$	−155.814	−0.152162
$17$	−1584.31	−1.32959	−0.664795	−	0.747026i	$-0.731481\pi$
$17$	−1584.31	−1.32959	−0.664795	+	0.747026i	$0.731481\pi$
$18$	0	0
$19$	−1234.05	−0.784242	−0.392121	−	0.919914i	$-0.628258\pi$
$19$	−1234.05	−0.784242	−0.392121	+	0.919914i	$0.628258\pi$
$20$	−752.572	−0.420701
$21$	0	0
$22$	−2740.87	−1.20735
$23$	−1180.66	−0.465376	−0.232688	−	0.972551i	$-0.574752\pi$
$23$	−1180.66	−0.465376	−0.232688	+	0.972551i	$0.574752\pi$
$24$	0	0
$25$	−1281.73	−0.410155
$26$	3358.81	0.974433
$27$	0	0
$28$	−3754.72	−0.905071
$29$	−6690.90	−1.47737	−0.738685	−	0.674050i	$-0.764553\pi$
$29$	−6690.90	−1.47737	−0.738685	+	0.674050i	$0.764553\pi$
$30$	0	0
$31$	407.702	0.0761971	0.0380985	−	0.999274i	$-0.487870\pi$
$31$	407.702	0.0761971	0.0380985	+	0.999274i	$0.487870\pi$
$32$	−5436.47	−0.938516
$33$	0	0
$34$	6026.86	0.894116
$35$	9196.40	1.26896
$36$	0	0
$37$	5606.29	0.673241	0.336621	−	0.941640i	$-0.390716\pi$
$37$	5606.29	0.673241	0.336621	+	0.941640i	$0.390716\pi$
$38$	4694.45	0.527383
$39$	0	0
$40$	8089.16	0.799381
$41$	9137.16	0.848890	0.424445	−	0.905454i	$-0.360469\pi$
$41$	9137.16	0.848890	0.424445	+	0.905454i	$0.360469\pi$
$42$	0	0
$43$	−19293.4	−1.59125	−0.795623	−	0.605793i	$-0.792856\pi$
$43$	−19293.4	−1.59125	−0.795623	+	0.605793i	$0.792856\pi$
$44$	−12629.7	−0.983468
$45$	0	0
$46$	4491.33	0.312954
$47$	4662.43	0.307870	0.153935	−	0.988081i	$-0.450805\pi$
$47$	4662.43	0.307870	0.153935	+	0.988081i	$0.450805\pi$
$48$	0	0
$49$	29075.5	1.72996
$50$	4875.83	0.275819
$51$	0	0
$52$	15477.1	0.793744
$53$	−10048.8	−0.491388	−0.245694	−	0.969347i	$-0.579016\pi$
$53$	−10048.8	−0.491388	−0.245694	+	0.969347i	$0.579016\pi$
$54$	0	0
$55$	30933.7	1.37888
$56$	40358.3	1.71974
$57$	0	0
$58$	25452.8	0.993495
$59$	3481.00	0.130189
$59$	3481.00	0.130189
$60$	0	0
$61$	−34785.8	−1.19696	−0.598478	−	0.801139i	$-0.704227\pi$
$61$	−34785.8	−1.19696	−0.598478	+	0.801139i	$0.704227\pi$
$62$	−1550.93	−0.0512406
$63$	0	0
$64$	25666.9	0.783291
$65$	−37907.8	−1.11287
$66$	0	0
$67$	−41086.0	−1.11817	−0.559084	−	0.829111i	$-0.688847\pi$
$67$	−41086.0	−1.11817	−0.559084	+	0.829111i	$0.688847\pi$
$68$	27771.2	0.728319
$69$	0	0
$70$	−34983.9	−0.853343
$71$	−73673.3	−1.73446	−0.867230	−	0.497908i	$-0.834102\pi$
$71$	−73673.3	−1.73446	−0.867230	+	0.497908i	$0.834102\pi$
$72$	0	0
$73$	−32905.0	−0.722694	−0.361347	−	0.932431i	$-0.617683\pi$
$73$	−32905.0	−0.722694	−0.361347	+	0.932431i	$0.617683\pi$
$74$	−21326.8	−0.452738
$75$	0	0
$76$	21631.6	0.429590
$77$	154334.	2.96643
$78$	0	0
$79$	−36261.8	−0.653704	−0.326852	−	0.945076i	$-0.605988\pi$
$79$	−36261.8	−0.653704	−0.326852	+	0.945076i	$0.605988\pi$
$80$	−6689.62	−0.116863
$81$	0	0
$82$	−34758.6	−0.570858
$83$	38330.1	0.610724	0.305362	−	0.952236i	$-0.401222\pi$
$83$	38330.1	0.610724	0.305362	+	0.952236i	$0.401222\pi$
$84$	0	0
$85$	−68019.6	−1.02114
$86$	73393.8	1.07007
$87$	0	0
$88$	135752.	1.86871
$89$	39785.2	0.532410	0.266205	−	0.963916i	$-0.414230\pi$
$89$	39785.2	0.532410	0.266205	+	0.963916i	$0.414230\pi$
$90$	0	0
$91$	−189129.	−2.39417
$92$	20695.6	0.254923
$93$	0	0
$94$	−17736.3	−0.207035
$95$	−52981.9	−0.602308
$96$	0	0
$97$	−53489.3	−0.577215	−0.288608	−	0.957447i	$-0.593192\pi$
$97$	−53489.3	−0.577215	−0.288608	+	0.957447i	$0.593192\pi$
$98$	−110606.	−1.16336
$99$	0	0
$100$	22467.4	0.224674
$101$	−21000.3	−0.204844	−0.102422	−	0.994741i	$-0.532659\pi$
$101$	−21000.3	−0.204844	−0.102422	+	0.994741i	$0.532659\pi$
$102$	0	0
$103$	77051.2	0.715627	0.357813	−	0.933793i	$-0.383522\pi$
$103$	77051.2	0.715627	0.357813	+	0.933793i	$0.383522\pi$
$104$	−166358.	−1.50821
$105$	0	0
$106$	38226.6	0.330446
$107$	47055.7	0.397331	0.198666	−	0.980067i	$-0.436339\pi$
$107$	47055.7	0.397331	0.198666	+	0.980067i	$0.436339\pi$
$108$	0	0
$109$	−185284.	−1.49373	−0.746865	−	0.664976i	$-0.768442\pi$
$109$	−185284.	−1.49373	−0.746865	+	0.664976i	$0.768442\pi$
$110$	−117675.	−0.927260
$111$	0	0
$112$	−33375.8	−0.251412
$113$	104331.	0.768633	0.384317	−	0.923201i	$-0.374437\pi$
$113$	104331.	0.768633	0.384317	+	0.923201i	$0.374437\pi$
$114$	0	0
$115$	−50689.5	−0.357415
$116$	117284.	0.809270
$117$	0	0
$118$	−13242.0	−0.0875488
$119$	−339362.	−2.19683
$120$	0	0
$121$	358079.	2.22339
$122$	132329.	0.804923
$123$	0	0
$124$	−7146.55	−0.0417390
$125$	−189196.	−1.08302
$126$	0	0
$127$	95904.7	0.527631	0.263816	−	0.964573i	$-0.415019\pi$
$127$	95904.7	0.527631	0.263816	+	0.964573i	$0.415019\pi$
$128$	76327.7	0.411772
$129$	0	0
$130$	144205.	0.748378
$131$	−109990.	−0.559981	−0.279991	−	0.960003i	$-0.590331\pi$
$131$	−109990.	−0.559981	−0.279991	+	0.960003i	$0.590331\pi$
$132$	0	0
$133$	−264337.	−1.29577
$134$	156295.	0.751939
$135$	0	0
$136$	−298504.	−1.38389
$137$	304032.	1.38394	0.691970	−	0.721926i	$-0.256743\pi$
$137$	304032.	1.38394	0.691970	+	0.721926i	$0.256743\pi$
$138$	0	0
$139$	−282945.	−1.24212	−0.621061	−	0.783762i	$-0.713298\pi$
$139$	−282945.	−1.24212	−0.621061	+	0.783762i	$0.713298\pi$
$140$	−161203.	−0.695107
$141$	0	0
$142$	280260.	1.16638
$143$	−636169.	−2.60155
$144$	0	0
$145$	−287262.	−1.13464
$146$	125174.	0.485994
$147$	0	0
$148$	−98271.9	−0.368786
$149$	263138.	0.970998	0.485499	−	0.874237i	$-0.338638\pi$
$149$	263138.	0.970998	0.485499	+	0.874237i	$0.338638\pi$
$150$	0	0
$151$	−21740.0	−0.0775922	−0.0387961	−	0.999247i	$-0.512352\pi$
$151$	−21740.0	−0.0775922	−0.0387961	+	0.999247i	$0.512352\pi$
$152$	−232511.	−0.816271
$153$	0	0
$154$	−587101.	−1.99485
$155$	17504.0	0.0585204
$156$	0	0
$157$	342216.	1.10803	0.554014	−	0.832507i	$-0.313095\pi$
$157$	342216.	1.10803	0.554014	+	0.832507i	$0.313095\pi$
$158$	137943.	0.439600
$159$	0	0
$160$	−233405.	−0.720793
$161$	−252899.	−0.768923
$162$	0	0
$163$	18154.2	0.0535191	0.0267596	−	0.999642i	$-0.491481\pi$
$163$	18154.2	0.0535191	0.0267596	+	0.999642i	$0.491481\pi$
$164$	−160164.	−0.465003
$165$	0	0
$166$	−145811.	−0.410697
$167$	498885.	1.38423	0.692116	−	0.721786i	$-0.256679\pi$
$167$	498885.	1.38423	0.692116	+	0.721786i	$0.256679\pi$
$168$	0	0
$169$	408302.	1.09968
$170$	258753.	0.686693
$171$	0	0
$172$	338191.	0.871648
$173$	−351740.	−0.893525	−0.446762	−	0.894653i	$-0.647423\pi$
$173$	−351740.	−0.893525	−0.446762	+	0.894653i	$0.647423\pi$
$174$	0	0
$175$	−274550.	−0.677682
$176$	−112265.	−0.273189
$177$	0	0
$178$	−151347.	−0.358032
$179$	82608.1	0.192704	0.0963519	−	0.995347i	$-0.469283\pi$
$179$	82608.1	0.192704	0.0963519	+	0.995347i	$0.469283\pi$
$180$	0	0
$181$	554779.	1.25870	0.629352	−	0.777120i	$-0.283320\pi$
$181$	554779.	1.25870	0.629352	+	0.777120i	$0.283320\pi$
$182$	719464.	1.61002
$183$	0	0
$184$	−222450.	−0.484383
$185$	240696.	0.517059
$186$	0	0
$187$	−1.14151e6	−2.38712
$188$	−81727.2	−0.168644
$189$	0	0
$190$	201548.	0.405037
$191$	−548299.	−1.08751	−0.543756	−	0.839243i	$-0.682998\pi$
$191$	−548299.	−1.08751	−0.543756	+	0.839243i	$0.682998\pi$
$192$	0	0
$193$	−575410.	−1.11195	−0.555973	−	0.831200i	$-0.687654\pi$
$193$	−575410.	−1.11195	−0.555973	+	0.831200i	$0.687654\pi$
$194$	203478.	0.388163
$195$	0	0
$196$	−509661.	−0.947636
$197$	−218239.	−0.400652	−0.200326	−	0.979729i	$-0.564200\pi$
$197$	−218239.	−0.400652	−0.200326	+	0.979729i	$0.564200\pi$
$198$	0	0
$199$	808112.	1.44657	0.723284	−	0.690551i	$-0.242632\pi$
$199$	808112.	1.44657	0.723284	+	0.690551i	$0.242632\pi$
$200$	−241495.	−0.426906
$201$	0	0
$202$	79887.1	0.137752
$203$	−1.43320e6	−2.44100
$204$	0	0
$205$	392288.	0.651960
$206$	−293110.	−0.481241
$207$	0	0
$208$	137576.	0.220487
$209$	−889144.	−1.40801
$210$	0	0
$211$	−396679.	−0.613385	−0.306693	−	0.951809i	$-0.599222\pi$
$211$	−396679.	−0.613385	−0.306693	+	0.951809i	$0.599222\pi$
$212$	176144.	0.269171
$213$	0	0
$214$	−179004.	−0.267195
$215$	−828328.	−1.22210
$216$	0	0
$217$	87330.5	0.125897
$218$	704838.	1.00450
$219$	0	0
$220$	−542233.	−0.755317
$221$	1.39886e6	1.92661
$222$	0	0
$223$	−905385.	−1.21919	−0.609595	−	0.792713i	$-0.708668\pi$
$223$	−905385.	−1.21919	−0.609595	+	0.792713i	$0.708668\pi$
$224$	−1.16450e6	−1.55067
$225$	0	0
$226$	−396886.	−0.516886
$227$	−440138.	−0.566923	−0.283461	−	0.958984i	$-0.591483\pi$
$227$	−440138.	−0.566923	−0.283461	+	0.958984i	$0.591483\pi$
$228$	0	0
$229$	−310876.	−0.391741	−0.195870	−	0.980630i	$-0.562753\pi$
$229$	−310876.	−0.391741	−0.195870	+	0.980630i	$0.562753\pi$
$230$	192827.	0.240353
$231$	0	0
$232$	−1.26065e6	−1.53771
$233$	378998.	0.457349	0.228674	−	0.973503i	$-0.426561\pi$
$233$	378998.	0.457349	0.228674	+	0.973503i	$0.426561\pi$
$234$	0	0
$235$	200174.	0.236449
$236$	−61018.0	−0.0713145
$237$	0	0
$238$	1.29097e6	1.47731
$239$	−585450.	−0.662971	−0.331486	−	0.943460i	$-0.607550\pi$
$239$	−585450.	−0.662971	−0.331486	+	0.943460i	$0.607550\pi$
$240$	0	0
$241$	−657704.	−0.729437	−0.364718	−	0.931118i	$-0.618835\pi$
$241$	−657704.	−0.729437	−0.364718	+	0.931118i	$0.618835\pi$
$242$	−1.36217e6	−1.49517
$243$	0	0
$244$	609757.	0.655665
$245$	1.24831e6	1.32864
$246$	0	0
$247$	1.08960e6	1.13639
$248$	76816.1	0.0793091
$249$	0	0
$250$	719717.	0.728303
$251$	−719453.	−0.720806	−0.360403	−	0.932797i	$-0.617361\pi$
$251$	−719453.	−0.720806	−0.360403	+	0.932797i	$0.617361\pi$
$252$	0	0
$253$	−850671.	−0.835527
$254$	−364830.	−0.354819
$255$	0	0
$256$	−1.11170e6	−1.06020
$257$	818573.	0.773081	0.386540	−	0.922273i	$-0.373670\pi$
$257$	818573.	0.773081	0.386540	+	0.922273i	$0.373670\pi$
$258$	0	0
$259$	1.20088e6	1.11237
$260$	664481.	0.609606
$261$	0	0
$262$	418411.	0.376573
$263$	−761730.	−0.679066	−0.339533	−	0.940594i	$-0.610269\pi$
$263$	−761730.	−0.679066	−0.339533	+	0.940594i	$0.610269\pi$
$264$	0	0
$265$	−431428.	−0.377393
$266$	1.00556e6	0.871374
$267$	0	0
$268$	720191.	0.612507
$269$	−402782.	−0.339382	−0.169691	−	0.985497i	$-0.554277\pi$
$269$	−402782.	−0.339382	−0.169691	+	0.985497i	$0.554277\pi$
$270$	0	0
$271$	119947.	0.0992127	0.0496063	−	0.998769i	$-0.484203\pi$
$271$	119947.	0.0992127	0.0496063	+	0.998769i	$0.484203\pi$
$272$	246858.	0.202314
$273$	0	0
$274$	−1.15656e6	−0.930665
$275$	−923498.	−0.736383
$276$	0	0
$277$	−1.07216e6	−0.839573	−0.419787	−	0.907623i	$-0.637895\pi$
$277$	−1.07216e6	−0.839573	−0.419787	+	0.907623i	$0.637895\pi$
$278$	1.07635e6	0.835297
$279$	0	0
$280$	1.73272e6	1.32079
$281$	1.53110e6	1.15675	0.578374	−	0.815772i	$-0.303687\pi$
$281$	1.53110e6	1.15675	0.578374	+	0.815772i	$0.303687\pi$
$282$	0	0
$283$	−2.48859e6	−1.84709	−0.923543	−	0.383496i	$-0.874720\pi$
$283$	−2.48859e6	−1.84709	−0.923543	+	0.383496i	$0.874720\pi$
$284$	1.29141e6	0.950098
$285$	0	0
$286$	2.42005e6	1.74948
$287$	1.95720e6	1.40259
$288$	0	0
$289$	1.09018e6	0.767809
$290$	1.09277e6	0.763018
$291$	0	0
$292$	576787.	0.395876
$293$	−1.74603e6	−1.18818	−0.594090	−	0.804399i	$-0.702488\pi$
$293$	−1.74603e6	−1.18818	−0.594090	+	0.804399i	$0.702488\pi$
$294$	0	0
$295$	149451.	0.0999869
$296$	1.05629e6	0.700738
$297$	0	0
$298$	−1.00100e6	−0.652972
$299$	1.04246e6	0.674342
$300$	0	0
$301$	−4.13268e6	−2.62915
$302$	82701.1	0.0521788
$303$	0	0
$304$	192283.	0.119332
$305$	−1.49347e6	−0.919278
$306$	0	0
$307$	2.18289e6	1.32186	0.660930	−	0.750447i	$-0.270162\pi$
$307$	2.18289e6	1.32186	0.660930	+	0.750447i	$0.270162\pi$
$308$	−2.70530e6	−1.62495
$309$	0	0
$310$	−66586.7	−0.0393535
$311$	614560.	0.360299	0.180150	−	0.983639i	$-0.442342\pi$
$311$	614560.	0.360299	0.180150	+	0.983639i	$0.442342\pi$
$312$	0	0
$313$	2.96005e6	1.70781	0.853903	−	0.520432i	$-0.174229\pi$
$313$	2.96005e6	1.70781	0.853903	+	0.520432i	$0.174229\pi$
$314$	−1.30182e6	−0.745121
$315$	0	0
$316$	635628.	0.358084
$317$	1.97631e6	1.10461	0.552304	−	0.833643i	$-0.313749\pi$
$317$	1.97631e6	1.10461	0.552304	+	0.833643i	$0.313749\pi$
$318$	0	0
$319$	−4.82084e6	−2.65244
$320$	1.10196e6	0.601578
$321$	0	0
$322$	962052.	0.517081
$323$	1.95512e6	1.04272
$324$	0	0
$325$	1.13170e6	0.594325
$326$	−69060.4	−0.0359903
$327$	0	0
$328$	1.72155e6	0.883560
$329$	998703.	0.508682
$330$	0	0
$331$	2.33004e6	1.16894	0.584471	−	0.811415i	$-0.301302\pi$
$331$	2.33004e6	1.16894	0.584471	+	0.811415i	$0.301302\pi$
$332$	−671884.	−0.334541
$333$	0	0
$334$	−1.89780e6	−0.930861
$335$	−1.76396e6	−0.858768
$336$	0	0
$337$	3.84206e6	1.84285	0.921423	−	0.388562i	$-0.127028\pi$
$337$	3.84206e6	1.84285	0.921423	+	0.388562i	$0.127028\pi$
$338$	−1.55322e6	−0.739504
$339$	0	0
$340$	1.19231e6	0.559359
$341$	293752.	0.136803
$342$	0	0
$343$	2.62794e6	1.20609
$344$	−3.63511e6	−1.65623
$345$	0	0
$346$	1.33805e6	0.600873
$347$	−1.55090e6	−0.691447	−0.345723	−	0.938336i	$-0.612366\pi$
$347$	−1.55090e6	−0.691447	−0.345723	+	0.938336i	$0.612366\pi$
$348$	0	0
$349$	−1.17603e6	−0.516837	−0.258418	−	0.966033i	$-0.583201\pi$
$349$	−1.17603e6	−0.516837	−0.258418	+	0.966033i	$0.583201\pi$
$350$	1.04441e6	0.455724
$351$	0	0
$352$	−3.91701e6	−1.68499
$353$	−700737.	−0.299308	−0.149654	−	0.988738i	$-0.547816\pi$
$353$	−700737.	−0.299308	−0.149654	+	0.988738i	$0.547816\pi$
$354$	0	0
$355$	−3.16304e6	−1.33209
$356$	−697389.	−0.291642
$357$	0	0
$358$	−314249.	−0.129588
$359$	−2.69702e6	−1.10445	−0.552227	−	0.833694i	$-0.686222\pi$
$359$	−2.69702e6	−1.10445	−0.552227	+	0.833694i	$0.686222\pi$
$360$	0	0
$361$	−953212.	−0.384965
$362$	−2.11043e6	−0.846447
$363$	0	0
$364$	3.31522e6	1.31147
$365$	−1.41272e6	−0.555039
$366$	0	0
$367$	−3.95579e6	−1.53309	−0.766546	−	0.642189i	$-0.778026\pi$
$367$	−3.95579e6	−1.53309	−0.766546	+	0.642189i	$0.778026\pi$
$368$	183963.	0.0708128
$369$	0	0
$370$	−915631.	−0.347709
$371$	−2.15247e6	−0.811901
$372$	0	0
$373$	−989342.	−0.368192	−0.184096	−	0.982908i	$-0.558936\pi$
$373$	−989342.	−0.368192	−0.184096	+	0.982908i	$0.558936\pi$
$374$	4.34239e6	1.60528
$375$	0	0
$376$	878460.	0.320444
$377$	5.90771e6	2.14075
$378$	0	0
$379$	4.39320e6	1.57102	0.785512	−	0.618846i	$-0.212399\pi$
$379$	4.39320e6	1.57102	0.785512	+	0.618846i	$0.212399\pi$
$380$	928714.	0.329931
$381$	0	0
$382$	2.08578e6	0.731325
$383$	1.04054e6	0.362460	0.181230	−	0.983441i	$-0.441992\pi$
$383$	1.04054e6	0.362460	0.181230	+	0.983441i	$0.441992\pi$
$384$	0	0
$385$	6.62607e6	2.27826
$386$	2.18891e6	0.747756
$387$	0	0
$388$	937608.	0.316186
$389$	5.44528e6	1.82451	0.912255	−	0.409622i	$-0.134339\pi$
$389$	5.44528e6	1.82451	0.912255	+	0.409622i	$0.134339\pi$
$390$	0	0
$391$	1.87053e6	0.618759
$392$	5.47819e6	1.80062
$393$	0	0
$394$	830202.	0.269428
$395$	−1.55684e6	−0.502054
$396$	0	0
$397$	2.31060e6	0.735781	0.367890	−	0.929869i	$-0.380080\pi$
$397$	2.31060e6	0.735781	0.367890	+	0.929869i	$0.380080\pi$
$398$	−3.07413e6	−0.972781
$399$	0	0
$400$	199713.	0.0624102
$401$	4.86056e6	1.50947	0.754736	−	0.656029i	$-0.227765\pi$
$401$	4.86056e6	1.50947	0.754736	+	0.656029i	$0.227765\pi$
$402$	0	0
$403$	−359979.	−0.110411
$404$	368112.	0.112209
$405$	0	0
$406$	5.45204e6	1.64151
$407$	4.03937e6	1.20872
$408$	0	0
$409$	−3.64662e6	−1.07791	−0.538954	−	0.842335i	$-0.681180\pi$
$409$	−3.64662e6	−1.07791	−0.538954	+	0.842335i	$0.681180\pi$
$410$	−1.49230e6	−0.438427
$411$	0	0
$412$	−1.35062e6	−0.392004
$413$	745637.	0.215106
$414$	0	0
$415$	1.64564e6	0.469045
$416$	4.80011e6	1.35993
$417$	0	0
$418$	3.38238e6	0.946852
$419$	2.49044e6	0.693012	0.346506	−	0.938048i	$-0.387368\pi$
$419$	2.49044e6	0.693012	0.346506	+	0.938048i	$0.387368\pi$
$420$	0	0
$421$	581843.	0.159993	0.0799965	−	0.996795i	$-0.474509\pi$
$421$	581843.	0.159993	0.0799965	+	0.996795i	$0.474509\pi$
$422$	1.50900e6	0.412486
$423$	0	0
$424$	−1.89332e6	−0.511457
$425$	2.03066e6	0.545338
$426$	0	0
$427$	−7.45120e6	−1.97768
$428$	−824833.	−0.217649
$429$	0	0
$430$	3.15104e6	0.821831
$431$	55946.1	0.0145070	0.00725348	−	0.999974i	$-0.497691\pi$
$431$	55946.1	0.0145070	0.00725348	+	0.999974i	$0.497691\pi$
$432$	0	0
$433$	−1.43091e6	−0.366768	−0.183384	−	0.983041i	$-0.558705\pi$
$433$	−1.43091e6	−0.366768	−0.183384	+	0.983041i	$0.558705\pi$
$434$	−332213.	−0.0846628
$435$	0	0
$436$	3.24782e6	0.818231
$437$	1.45699e6	0.364967
$438$	0	0
$439$	−834782.	−0.206734	−0.103367	−	0.994643i	$-0.532962\pi$
$439$	−834782.	−0.206734	−0.103367	+	0.994643i	$0.532962\pi$
$440$	5.82830e6	1.43519
$441$	0	0
$442$	−5.32140e6	−1.29560
$443$	−88044.4	−0.0213153	−0.0106577	−	0.999943i	$-0.503393\pi$
$443$	−88044.4	−0.0213153	−0.0106577	+	0.999943i	$0.503393\pi$
$444$	0	0
$445$	1.70811e6	0.408898
$446$	3.44417e6	0.819874
$447$	0	0
$448$	5.49790e6	1.29420
$449$	−3.10559e6	−0.726990	−0.363495	−	0.931596i	$-0.618417\pi$
$449$	−3.10559e6	−0.726990	−0.363495	+	0.931596i	$0.618417\pi$
$450$	0	0
$451$	6.58339e6	1.52408
$452$	−1.82881e6	−0.421040
$453$	0	0
$454$	1.67432e6	0.381241
$455$	−8.11993e6	−1.83875
$456$	0	0
$457$	−3.63850e6	−0.814953	−0.407476	−	0.913216i	$-0.633591\pi$
$457$	−3.63850e6	−0.814953	−0.407476	+	0.913216i	$0.633591\pi$
$458$	1.18260e6	0.263436
$459$	0	0
$460$	888530.	0.195784
$461$	−8.07826e6	−1.77038	−0.885188	−	0.465234i	$-0.845970\pi$
$461$	−8.07826e6	−1.77038	−0.885188	+	0.465234i	$0.845970\pi$
$462$	0	0
$463$	4.70035e6	1.01901	0.509504	−	0.860468i	$-0.329829\pi$
$463$	4.70035e6	1.01901	0.509504	+	0.860468i	$0.329829\pi$
$464$	1.04254e6	0.224800
$465$	0	0
$466$	−1.44174e6	−0.307556
$467$	−3.34198e6	−0.709106	−0.354553	−	0.935036i	$-0.615367\pi$
$467$	−3.34198e6	−0.709106	−0.354553	+	0.935036i	$0.615367\pi$
$468$	0	0
$469$	−8.80070e6	−1.84750
$470$	−761479.	−0.159006
$471$	0	0
$472$	655864.	0.135506
$473$	−1.39010e7	−2.85689
$474$	0	0
$475$	1.58173e6	0.321660
$476$	5.94864e6	1.20337
$477$	0	0
$478$	2.22710e6	0.445832
$479$	−95433.5	−0.0190048	−0.00950238	−	0.999955i	$-0.503025\pi$
$479$	−95433.5	−0.0190048	−0.00950238	+	0.999955i	$0.503025\pi$
$480$	0	0
$481$	−4.95005e6	−0.975544
$482$	2.50197e6	0.490528
$483$	0	0
$484$	−6.27672e6	−1.21792
$485$	−2.29647e6	−0.443309
$486$	0	0
$487$	−2.24933e6	−0.429764	−0.214882	−	0.976640i	$-0.568937\pi$
$487$	−2.24933e6	−0.429764	−0.214882	+	0.976640i	$0.568937\pi$
$488$	−6.55409e6	−1.24584
$489$	0	0
$490$	−4.74868e6	−0.893475
$491$	3.37521e6	0.631825	0.315912	−	0.948788i	$-0.397689\pi$
$491$	3.37521e6	0.631825	0.315912	+	0.948788i	$0.397689\pi$
$492$	0	0
$493$	1.06005e7	1.96430
$494$	−4.14495e6	−0.764191
$495$	0	0
$496$	−63525.8	−0.0115943
$497$	−1.57810e7	−2.86578
$498$	0	0
$499$	−1.87656e6	−0.337374	−0.168687	−	0.985670i	$-0.553953\pi$
$499$	−1.87656e6	−0.337374	−0.168687	+	0.985670i	$0.553953\pi$
$500$	3.31639e6	0.593253
$501$	0	0
$502$	2.73687e6	0.484724
$503$	−3.41948e6	−0.602614	−0.301307	−	0.953527i	$-0.597423\pi$
$503$	−3.41948e6	−0.602614	−0.301307	+	0.953527i	$0.597423\pi$
$504$	0	0
$505$	−901612.	−0.157323
$506$	3.23603e6	0.561871
$507$	0	0
$508$	−1.68110e6	−0.289025
$509$	−5.72399e6	−0.979274	−0.489637	−	0.871926i	$-0.662871\pi$
$509$	−5.72399e6	−0.979274	−0.489637	+	0.871926i	$0.662871\pi$
$510$	0	0
$511$	−7.04832e6	−1.19408
$512$	1.78652e6	0.301184
$513$	0	0
$514$	−3.11393e6	−0.519877
$515$	3.30806e6	0.549611
$516$	0	0
$517$	3.35931e6	0.552744
$518$	−4.56825e6	−0.748041
$519$	0	0
$520$	−7.14230e6	−1.15832
$521$	5.68622e6	0.917761	0.458881	−	0.888498i	$-0.348251\pi$
$521$	5.68622e6	0.917761	0.458881	+	0.888498i	$0.348251\pi$
$522$	0	0
$523$	5.68927e6	0.909499	0.454750	−	0.890619i	$-0.349729\pi$
$523$	5.68927e6	0.909499	0.454750	+	0.890619i	$0.349729\pi$
$524$	1.92799e6	0.306745
$525$	0	0
$526$	2.89769e6	0.456655
$527$	−645925.	−0.101311
$528$	0	0
$529$	−5.04239e6	−0.783425
$530$	1.64119e6	0.253787
$531$	0	0
$532$	4.63353e6	0.709794
$533$	−8.06763e6	−1.23006
$534$	0	0
$535$	2.02025e6	0.305156
$536$	−7.74111e6	−1.16383
$537$	0	0
$538$	1.53222e6	0.228226
$539$	2.09491e7	3.10594
$540$	0	0
$541$	−9.48284e6	−1.39298	−0.696491	−	0.717566i	$-0.745256\pi$
$541$	−9.48284e6	−1.39298	−0.696491	+	0.717566i	$0.745256\pi$
$542$	−456291.	−0.0667180
$543$	0	0
$544$	8.61304e6	1.24784
$545$	−7.95485e6	−1.14720
$546$	0	0
$547$	1.99761e6	0.285458	0.142729	−	0.989762i	$-0.454412\pi$
$547$	1.99761e6	0.285458	0.142729	+	0.989762i	$0.454412\pi$
$548$	−5.32933e6	−0.758091
$549$	0	0
$550$	3.51307e6	0.495199
$551$	8.25693e6	1.15862
$552$	0	0
$553$	−7.76734e6	−1.08009
$554$	4.07858e6	0.564592
$555$	0	0
$556$	4.95970e6	0.680407
$557$	−4.43912e6	−0.606260	−0.303130	−	0.952949i	$-0.598032\pi$
$557$	−4.43912e6	−0.606260	−0.303130	+	0.952949i	$0.598032\pi$
$558$	0	0
$559$	1.70350e7	2.30576
$560$	−1.43293e6	−0.193088
$561$	0	0
$562$	−5.82446e6	−0.777884
$563$	−1.47709e6	−0.196397	−0.0981985	−	0.995167i	$-0.531308\pi$
$563$	−1.47709e6	−0.196397	−0.0981985	+	0.995167i	$0.531308\pi$
$564$	0	0
$565$	4.47929e6	0.590321
$566$	9.46682e6	1.24212
$567$	0	0
$568$	−1.38810e7	−1.80530
$569$	−9.55992e6	−1.23787	−0.618933	−	0.785444i	$-0.712435\pi$
$569$	−9.55992e6	−1.23787	−0.618933	+	0.785444i	$0.712435\pi$
$570$	0	0
$571$	−6.14632e6	−0.788905	−0.394453	−	0.918916i	$-0.629066\pi$
$571$	−6.14632e6	−0.788905	−0.394453	+	0.918916i	$0.629066\pi$
$572$	1.11513e7	1.42507
$573$	0	0
$574$	−7.44537e6	−0.943205
$575$	1.51329e6	0.190876
$576$	0	0
$577$	−5.55332e6	−0.694405	−0.347202	−	0.937790i	$-0.612868\pi$
$577$	−5.55332e6	−0.694405	−0.347202	+	0.937790i	$0.612868\pi$
$578$	−4.14714e6	−0.516332
$579$	0	0
$580$	5.03539e6	0.621531
$581$	8.21040e6	1.00908
$582$	0	0
$583$	−7.24023e6	−0.882228
$584$	−6.19971e6	−0.752210
$585$	0	0
$586$	6.64205e6	0.799021
$587$	−8.34996e6	−1.00021	−0.500103	−	0.865966i	$-0.666704\pi$
$587$	−8.34996e6	−1.00021	−0.500103	+	0.865966i	$0.666704\pi$
$588$	0	0
$589$	−503125.	−0.0597569
$590$	−568525.	−0.0672387
$591$	0	0
$592$	−873540.	−0.102442
$593$	−6.20626e6	−0.724758	−0.362379	−	0.932031i	$-0.618035\pi$
$593$	−6.20626e6	−0.724758	−0.362379	+	0.932031i	$0.618035\pi$
$594$	0	0
$595$	−1.45699e7	−1.68719
$596$	−4.61252e6	−0.531891
$597$	0	0
$598$	−3.96560e6	−0.453478
$599$	7.14419e6	0.813554	0.406777	−	0.913528i	$-0.366653\pi$
$599$	7.14419e6	0.813554	0.406777	+	0.913528i	$0.366653\pi$
$600$	0	0
$601$	−9.11301e6	−1.02914	−0.514571	−	0.857448i	$-0.672049\pi$
$601$	−9.11301e6	−1.02914	−0.514571	+	0.857448i	$0.672049\pi$
$602$	1.57211e7	1.76804
$603$	0	0
$604$	381078.	0.0425032
$605$	1.53735e7	1.70759
$606$	0	0
$607$	4.34064e6	0.478169	0.239085	−	0.970999i	$-0.423153\pi$
$607$	4.34064e6	0.478169	0.239085	+	0.970999i	$0.423153\pi$
$608$	6.70889e6	0.736023
$609$	0	0
$610$	5.68130e6	0.618192
$611$	−4.11668e6	−0.446112
$612$	0	0
$613$	6.47088e6	0.695524	0.347762	−	0.937583i	$-0.386942\pi$
$613$	6.47088e6	0.695524	0.347762	+	0.937583i	$0.386942\pi$
$614$	−8.30391e6	−0.888918
$615$	0	0
$616$	2.90784e7	3.08759
$617$	−416530.	−0.0440488	−0.0220244	−	0.999757i	$-0.507011\pi$
$617$	−416530.	−0.0440488	−0.0220244	+	0.999757i	$0.507011\pi$
$618$	0	0
$619$	1.02724e7	1.07757	0.538785	−	0.842443i	$-0.318883\pi$
$619$	1.02724e7	1.07757	0.538785	+	0.842443i	$0.318883\pi$
$620$	−306825.	−0.0320562
$621$	0	0
$622$	−2.33784e6	−0.242292
$623$	8.52207e6	0.879680
$624$	0	0
$625$	−4.11737e6	−0.421618
$626$	−1.12603e7	−1.14846
$627$	0	0
$628$	−5.99866e6	−0.606953
$629$	−8.88209e6	−0.895135
$630$	0	0
$631$	−1.64164e7	−1.64136	−0.820680	−	0.571388i	$-0.806405\pi$
$631$	−1.64164e7	−1.64136	−0.820680	+	0.571388i	$0.806405\pi$
$632$	−6.83217e6	−0.680402
$633$	0	0
$634$	−7.51808e6	−0.742821
$635$	4.11750e6	0.405228
$636$	0	0
$637$	−2.56721e7	−2.50676
$638$	1.83389e7	1.78370
$639$	0	0
$640$	3.27700e6	0.316247
$641$	845820.	0.0813080	0.0406540	−	0.999173i	$-0.487056\pi$
$641$	845820.	0.0813080	0.0406540	+	0.999173i	$0.487056\pi$
$642$	0	0
$643$	−6.06955e6	−0.578934	−0.289467	−	0.957188i	$-0.593478\pi$
$643$	−6.06955e6	−0.578934	−0.289467	+	0.957188i	$0.593478\pi$
$644$	4.43304e6	0.421199
$645$	0	0
$646$	−7.43746e6	−0.701203
$647$	6.25734e6	0.587664	0.293832	−	0.955857i	$-0.405069\pi$
$647$	6.25734e6	0.587664	0.293832	+	0.955857i	$0.405069\pi$
$648$	0	0
$649$	2.50808e6	0.233739
$650$	−4.30510e6	−0.399669
$651$	0	0
$652$	−318223.	−0.0293166
$653$	1.29415e6	0.118769	0.0593845	−	0.998235i	$-0.481086\pi$
$653$	1.29415e6	0.118769	0.0593845	+	0.998235i	$0.481086\pi$
$654$	0	0
$655$	−4.72221e6	−0.430073
$656$	−1.42370e6	−0.129169
$657$	0	0
$658$	−3.79916e6	−0.342076
$659$	1.68268e7	1.50934	0.754671	−	0.656104i	$-0.227797\pi$
$659$	1.68268e7	1.50934	0.754671	+	0.656104i	$0.227797\pi$
$660$	0	0
$661$	9.33549e6	0.831062	0.415531	−	0.909579i	$-0.363596\pi$
$661$	9.33549e6	0.831062	0.415531	+	0.909579i	$0.363596\pi$
$662$	−8.86368e6	−0.786084
$663$	0	0
$664$	7.22188e6	0.635667
$665$	−1.13488e7	−0.995170
$666$	0	0
$667$	7.89966e6	0.687533
$668$	−8.74489e6	−0.758251
$669$	0	0
$670$	6.71025e6	0.577500
$671$	−2.50634e7	−2.14899
$672$	0	0
$673$	1.36691e7	1.16333	0.581664	−	0.813429i	$-0.302402\pi$
$673$	1.36691e7	1.16333	0.581664	+	0.813429i	$0.302402\pi$
$674$	−1.46155e7	−1.23927
$675$	0	0
$676$	−7.15707e6	−0.602377
$677$	7.62952e6	0.639773	0.319886	−	0.947456i	$-0.396355\pi$
$677$	7.62952e6	0.639773	0.319886	+	0.947456i	$0.396355\pi$
$678$	0	0
$679$	−1.14575e7	−0.953710
$680$	−1.28157e7	−1.06285
$681$	0	0
$682$	−1.11746e6	−0.0919964
$683$	5.82854e6	0.478089	0.239044	−	0.971009i	$-0.423166\pi$
$683$	5.82854e6	0.478089	0.239044	+	0.971009i	$0.423166\pi$
$684$	0	0
$685$	1.30531e7	1.06289
$686$	−9.99693e6	−0.811067
$687$	0	0
$688$	3.00619e6	0.242128
$689$	8.87256e6	0.712034
$690$	0	0
$691$	1.53655e7	1.22420	0.612100	−	0.790781i	$-0.290325\pi$
$691$	1.53655e7	1.22420	0.612100	+	0.790781i	$0.290325\pi$
$692$	6.16561e6	0.489453
$693$	0	0
$694$	5.89975e6	0.464981
$695$	−1.21477e7	−0.953968
$696$	0	0
$697$	−1.44761e7	−1.12868
$698$	4.47371e6	0.347560
$699$	0	0
$700$	4.81255e6	0.371219
$701$	2.48878e6	0.191290	0.0956448	−	0.995416i	$-0.469509\pi$
$701$	2.48878e6	0.191290	0.0956448	+	0.995416i	$0.469509\pi$
$702$	0	0
$703$	−6.91845e6	−0.527984
$704$	1.84932e7	1.40630
$705$	0	0
$706$	2.66567e6	0.201277
$707$	−4.49831e6	−0.338455
$708$	0	0
$709$	−1.31587e7	−0.983099	−0.491550	−	0.870850i	$-0.663569\pi$
$709$	−1.31587e7	−0.983099	−0.491550	+	0.870850i	$0.663569\pi$
$710$	1.20325e7	0.895797
$711$	0	0
$712$	7.49602e6	0.554155
$713$	−481356.	−0.0354603
$714$	0	0
$715$	−2.73128e7	−1.99803
$716$	−1.44803e6	−0.105559
$717$	0	0
$718$	1.02597e7	0.742718
$719$	−1.67259e7	−1.20661	−0.603307	−	0.797509i	$-0.706151\pi$
$719$	−1.67259e7	−1.20661	−0.603307	+	0.797509i	$0.706151\pi$
$720$	0	0
$721$	1.65045e7	1.18240
$722$	3.62611e6	0.258879
$723$	0	0
$724$	−9.72465e6	−0.689490
$725$	8.57595e6	0.605951
$726$	0	0
$727$	−6.99729e6	−0.491014	−0.245507	−	0.969395i	$-0.578954\pi$
$727$	−6.99729e6	−0.491014	−0.245507	+	0.969395i	$0.578954\pi$
$728$	−3.56343e7	−2.49195
$729$	0	0
$730$	5.37411e6	0.373250
$731$	3.05667e7	2.11570
$732$	0	0
$733$	7.49643e6	0.515341	0.257670	−	0.966233i	$-0.417045\pi$
$733$	7.49643e6	0.515341	0.257670	+	0.966233i	$0.417045\pi$
$734$	1.50482e7	1.03097
$735$	0	0
$736$	6.41860e6	0.436763
$737$	−2.96027e7	−2.00753
$738$	0	0
$739$	9.42028e6	0.634531	0.317265	−	0.948337i	$-0.397235\pi$
$739$	9.42028e6	0.634531	0.317265	+	0.948337i	$0.397235\pi$
$740$	−4.21913e6	−0.283233
$741$	0	0
$742$	8.18821e6	0.545983
$743$	2.15995e7	1.43539	0.717697	−	0.696355i	$-0.245196\pi$
$743$	2.15995e7	1.43539	0.717697	+	0.696355i	$0.245196\pi$
$744$	0	0
$745$	1.12974e7	0.745740
$746$	3.76355e6	0.247600
$747$	0	0
$748$	2.00093e7	1.30761
$749$	1.00794e7	0.656495
$750$	0	0
$751$	8.83147e6	0.571391	0.285695	−	0.958321i	$-0.407775\pi$
$751$	8.83147e6	0.571391	0.285695	+	0.958321i	$0.407775\pi$
$752$	−726474.	−0.0468463
$753$	0	0
$754$	−2.24735e7	−1.43960
$755$	−933371.	−0.0595918
$756$	0	0
$757$	−8.40120e6	−0.532846	−0.266423	−	0.963856i	$-0.585842\pi$
$757$	−8.40120e6	−0.532846	−0.266423	+	0.963856i	$0.585842\pi$
$758$	−1.67121e7	−1.05647
$759$	0	0
$760$	−9.98246e6	−0.626908
$761$	−3.85408e6	−0.241246	−0.120623	−	0.992698i	$-0.538489\pi$
$761$	−3.85408e6	−0.241246	−0.120623	+	0.992698i	$0.538489\pi$
$762$	0	0
$763$	−3.96882e7	−2.46803
$764$	9.61107e6	0.595715
$765$	0	0
$766$	−3.95829e6	−0.243745
$767$	−3.07354e6	−0.188647
$768$	0	0
$769$	1.05834e7	0.645369	0.322685	−	0.946507i	$-0.395415\pi$
$769$	1.05834e7	0.645369	0.322685	+	0.946507i	$0.395415\pi$
$770$	−2.52062e7	−1.53207
$771$	0	0
$772$	1.00863e7	0.609099
$773$	2.39879e7	1.44392	0.721961	−	0.691933i	$-0.243241\pi$
$773$	2.39879e7	1.44392	0.721961	+	0.691933i	$0.243241\pi$
$774$	0	0
$775$	−522565.	−0.0312526
$776$	−1.00781e7	−0.600790
$777$	0	0
$778$	−2.07144e7	−1.22694
$779$	−1.12757e7	−0.665735
$780$	0	0
$781$	−5.30821e7	−3.11401
$782$	−7.11565e6	−0.416100
$783$	0	0
$784$	−4.53038e6	−0.263236
$785$	1.46924e7	0.850981
$786$	0	0
$787$	−2.62750e7	−1.51219	−0.756093	−	0.654464i	$-0.772894\pi$
$787$	−2.62750e7	−1.51219	−0.756093	+	0.654464i	$0.772894\pi$
$788$	3.82549e6	0.219468
$789$	0	0
$790$	5.92235e6	0.337619
$791$	2.23480e7	1.26998
$792$	0	0
$793$	3.07140e7	1.73442
$794$	−8.78974e6	−0.494794
$795$	0	0
$796$	−1.41653e7	−0.792397
$797$	−2.35177e7	−1.31144	−0.655722	−	0.755003i	$-0.727636\pi$
$797$	−2.35177e7	−1.31144	−0.655722	+	0.755003i	$0.727636\pi$
$798$	0	0
$799$	−7.38674e6	−0.409341
$800$	6.96810e6	0.384937
$801$	0	0
$802$	−1.84900e7	−1.01508
$803$	−2.37083e7	−1.29751
$804$	0	0
$805$	−1.08578e7	−0.590543
$806$	1.36939e6	0.0742490
$807$	0	0
$808$	−3.95672e6	−0.213210
$809$	−9.35084e6	−0.502318	−0.251159	−	0.967946i	$-0.580812\pi$
$809$	−9.35084e6	−0.502318	−0.251159	+	0.967946i	$0.580812\pi$
$810$	0	0
$811$	−1.09556e7	−0.584901	−0.292450	−	0.956281i	$-0.594471\pi$
$811$	−1.09556e7	−0.584901	−0.292450	+	0.956281i	$0.594471\pi$
$812$	2.51225e7	1.33713
$813$	0	0
$814$	−1.53661e7	−0.812837
$815$	779421.	0.0411034
$816$	0	0
$817$	2.38091e7	1.24792
$818$	1.38721e7	0.724866
$819$	0	0
$820$	−6.87637e6	−0.357129
$821$	1.17996e7	0.610954	0.305477	−	0.952199i	$-0.401184\pi$
$821$	1.17996e7	0.610954	0.305477	+	0.952199i	$0.401184\pi$
$822$	0	0
$823$	−2.24306e7	−1.15436	−0.577181	−	0.816616i	$-0.695847\pi$
$823$	−2.24306e7	−1.15436	−0.577181	+	0.816616i	$0.695847\pi$
$824$	1.45174e7	0.744854
$825$	0	0
$826$	−2.83647e6	−0.144653
$827$	2.22074e7	1.12910	0.564552	−	0.825398i	$-0.309049\pi$
$827$	2.22074e7	1.12910	0.564552	+	0.825398i	$0.309049\pi$
$828$	0	0
$829$	−2.00226e7	−1.01189	−0.505946	−	0.862565i	$-0.668856\pi$
$829$	−2.00226e7	−1.01189	−0.505946	+	0.862565i	$0.668856\pi$
$830$	−6.26016e6	−0.315421
$831$	0	0
$832$	−2.26625e7	−1.13501
$833$	−4.60646e7	−2.30014
$834$	0	0
$835$	2.14188e7	1.06311
$836$	1.55857e7	0.771277
$837$	0	0
$838$	−9.47386e6	−0.466033
$839$	1.18425e7	0.580818	0.290409	−	0.956903i	$-0.406209\pi$
$839$	1.18425e7	0.580818	0.290409	+	0.956903i	$0.406209\pi$
$840$	0	0
$841$	2.42570e7	1.18262
$842$	−2.21339e6	−0.107591
$843$	0	0
$844$	6.95334e6	0.335999
$845$	1.75297e7	0.844566
$846$	0	0
$847$	7.67012e7	3.67362
$848$	1.56575e6	0.0747708
$849$	0	0
$850$	−7.72483e6	−0.366726
$851$	−6.61910e6	−0.313311
$852$	0	0
$853$	−3.48707e7	−1.64092	−0.820461	−	0.571703i	$-0.806283\pi$
$853$	−3.48707e7	−1.64092	−0.820461	+	0.571703i	$0.806283\pi$
$854$	2.83451e7	1.32994
$855$	0	0
$856$	8.86588e6	0.413559
$857$	2.78040e7	1.29317	0.646584	−	0.762842i	$-0.276197\pi$
$857$	2.78040e7	1.29317	0.646584	+	0.762842i	$0.276197\pi$
$858$	0	0
$859$	4.10095e7	1.89628	0.948139	−	0.317856i	$-0.102963\pi$
$859$	4.10095e7	1.89628	0.948139	+	0.317856i	$0.102963\pi$
$860$	1.45197e7	0.669438
$861$	0	0
$862$	−212824.	−0.00975556
$863$	−6.62680e6	−0.302885	−0.151442	−	0.988466i	$-0.548392\pi$
$863$	−6.62680e6	−0.302885	−0.151442	+	0.988466i	$0.548392\pi$
$864$	0	0
$865$	−1.51014e7	−0.686239
$866$	5.44330e6	0.246642
$867$	0	0
$868$	−1.53081e6	−0.0689637
$869$	−2.61268e7	−1.17365
$870$	0	0
$871$	3.62767e7	1.62025
$872$	−3.49098e7	−1.55474
$873$	0	0
$874$	−5.54254e6	−0.245431
$875$	−4.05261e7	−1.78943
$876$	0	0
$877$	−1.20837e7	−0.530520	−0.265260	−	0.964177i	$-0.585458\pi$
$877$	−1.20837e7	−0.530520	−0.265260	+	0.964177i	$0.585458\pi$
$878$	3.17559e6	0.139023
$879$	0	0
$880$	−4.81992e6	−0.209813
$881$	3.09734e6	0.134446	0.0672232	−	0.997738i	$-0.478586\pi$
$881$	3.09734e6	0.134446	0.0672232	+	0.997738i	$0.478586\pi$
$882$	0	0
$883$	−3.46717e7	−1.49649	−0.748245	−	0.663422i	$-0.769103\pi$
$883$	−3.46717e7	−1.49649	−0.748245	+	0.663422i	$0.769103\pi$
$884$	−2.45205e7	−1.05535
$885$	0	0
$886$	334929.	0.0143340
$887$	−1.72660e7	−0.736856	−0.368428	−	0.929656i	$-0.620104\pi$
$887$	−1.72660e7	−0.736856	−0.368428	+	0.929656i	$0.620104\pi$
$888$	0	0
$889$	2.05430e7	0.871784
$890$	−6.49780e6	−0.274974
$891$	0	0
$892$	1.58704e7	0.667844
$893$	−5.75369e6	−0.241445
$894$	0	0
$895$	3.54664e6	0.147999
$896$	1.63496e7	0.680355
$897$	0	0
$898$	1.18140e7	0.488883
$899$	−2.72789e6	−0.112571
$900$	0	0
$901$	1.59204e7	0.653345
$902$	−2.50438e7	−1.02491
$903$	0	0
$904$	1.96573e7	0.800025
$905$	2.38185e7	0.966702
$906$	0	0
$907$	−9.07025e6	−0.366101	−0.183051	−	0.983104i	$-0.558597\pi$
$907$	−9.07025e6	−0.366101	−0.183051	+	0.983104i	$0.558597\pi$
$908$	7.71512e6	0.310547
$909$	0	0
$910$	3.08890e7	1.23652
$911$	−1.32947e7	−0.530743	−0.265371	−	0.964146i	$-0.585495\pi$
$911$	−1.32947e7	−0.530743	−0.265371	+	0.964146i	$0.585495\pi$
$912$	0	0
$913$	2.76171e7	1.09648
$914$	1.38412e7	0.548035
$915$	0	0
$916$	5.44931e6	0.214587
$917$	−2.35600e7	−0.925235
$918$	0	0
$919$	1.84320e7	0.719920	0.359960	−	0.932968i	$-0.382790\pi$
$919$	1.84320e7	0.719920	0.359960	+	0.932968i	$0.382790\pi$
$920$	−9.55053e6	−0.372013
$921$	0	0
$922$	3.07304e7	1.19053
$923$	6.50496e7	2.51328
$924$	0	0
$925$	−7.18576e6	−0.276133
$926$	−1.78806e7	−0.685257
$927$	0	0
$928$	3.63748e7	1.38654
$929$	−1.94185e7	−0.738204	−0.369102	−	0.929389i	$-0.620335\pi$
$929$	−1.94185e7	−0.738204	−0.369102	+	0.929389i	$0.620335\pi$
$930$	0	0
$931$	−3.58807e7	−1.35671
$932$	−6.64342e6	−0.250525
$933$	0	0
$934$	1.27132e7	0.476856
$935$	−4.90086e7	−1.83334
$936$	0	0
$937$	4.00416e7	1.48992	0.744959	−	0.667111i	$-0.232469\pi$
$937$	4.00416e7	1.48992	0.744959	+	0.667111i	$0.232469\pi$
$938$	3.34787e7	1.24240
$939$	0	0
$940$	−3.50882e6	−0.129521
$941$	3.79786e6	0.139818	0.0699092	−	0.997553i	$-0.477729\pi$
$941$	3.79786e6	0.139818	0.0699092	+	0.997553i	$0.477729\pi$
$942$	0	0
$943$	−1.07879e7	−0.395053
$944$	−542390.	−0.0198099
$945$	0	0
$946$	5.28807e7	1.92119
$947$	−2.18081e7	−0.790209	−0.395104	−	0.918636i	$-0.629292\pi$
$947$	−2.18081e7	−0.790209	−0.395104	+	0.918636i	$0.629292\pi$
$948$	0	0
$949$	2.90533e7	1.04720
$950$	−6.01704e6	−0.216309
$951$	0	0
$952$	−6.39401e7	−2.28655
$953$	5.08318e7	1.81302	0.906511	−	0.422182i	$-0.138736\pi$
$953$	5.08318e7	1.81302	0.906511	+	0.422182i	$0.138736\pi$
$954$	0	0
$955$	−2.35403e7	−0.835225
$956$	1.02623e7	0.363161
$957$	0	0
$958$	363038.	0.0127802
$959$	6.51242e7	2.28663
$960$	0	0
$961$	−2.84629e7	−0.994194
$962$	1.88305e7	0.656029
$963$	0	0
$964$	1.15288e7	0.399569
$965$	−2.47042e7	−0.853990
$966$	0	0
$967$	6.26036e6	0.215294	0.107647	−	0.994189i	$-0.465668\pi$
$967$	6.26036e6	0.215294	0.107647	+	0.994189i	$0.465668\pi$
$968$	6.74665e7	2.31419
$969$	0	0
$970$	8.73600e6	0.298115
$971$	−3.00670e7	−1.02339	−0.511697	−	0.859166i	$-0.670983\pi$
$971$	−3.00670e7	−1.02339	−0.511697	+	0.859166i	$0.670983\pi$
$972$	0	0
$973$	−6.06073e7	−2.05231
$974$	8.55665e6	0.289006
$975$	0	0
$976$	5.42013e6	0.182132
$977$	−6.71816e6	−0.225172	−0.112586	−	0.993642i	$-0.535913\pi$
$977$	−6.71816e6	−0.225172	−0.112586	+	0.993642i	$0.535913\pi$
$978$	0	0
$979$	2.86655e7	0.955878
$980$	−2.18814e7	−0.727797
$981$	0	0
$982$	−1.28396e7	−0.424886
$983$	3.04386e7	1.00471	0.502355	−	0.864661i	$-0.332467\pi$
$983$	3.04386e7	1.00471	0.502355	+	0.864661i	$0.332467\pi$
$984$	0	0
$985$	−9.36972e6	−0.307706
$986$	−4.03251e7	−1.32094
$987$	0	0
$988$	−1.90995e7	−0.622487
$989$	2.27789e7	0.740528
$990$	0	0
$991$	5.63685e7	1.82327	0.911637	−	0.410996i	$-0.134819\pi$
$991$	5.63685e7	1.82327	0.911637	+	0.410996i	$0.134819\pi$
$992$	−2.21646e6	−0.0715122
$993$	0	0
$994$	6.00323e7	1.92716
$995$	3.46949e7	1.11098
$996$	0	0
$997$	−1.21101e7	−0.385843	−0.192922	−	0.981214i	$-0.561796\pi$
$997$	−1.21101e7	−0.385843	−0.192922	+	0.981214i	$0.561796\pi$
$998$	7.13861e6	0.226875
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	531.6.a.a.1.3		9
3.2	odd	2		59.6.a.a.1.7	✓	9
12.11	even	2		944.6.a.e.1.5		9

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
59.6.a.a.1.7	✓	9	3.2	odd	2
531.6.a.a.1.3		9	1.1	even	1	trivial
944.6.a.e.1.5		9	12.11	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 531 = 3^{2} \cdot 59 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	531.a (trivial)

\(f(q)\)	\(=\)	\(q-3.80409 q^{2} -17.5289 q^{4} +42.9333 q^{5} +214.202 q^{7} +188.412 q^{8} +O(q^{10})\)	\(q-3.80409 q^{2} -17.5289 q^{4} +42.9333 q^{5} +214.202 q^{7} +188.412 q^{8} -163.322 q^{10} +720.507 q^{11} -882.947 q^{13} -814.844 q^{14} -155.814 q^{16} -1584.31 q^{17} -1234.05 q^{19} -752.572 q^{20} -2740.87 q^{22} -1180.66 q^{23} -1281.73 q^{25} +3358.81 q^{26} -3754.72 q^{28} -6690.90 q^{29} +407.702 q^{31} -5436.47 q^{32} +6026.86 q^{34} +9196.40 q^{35} +5606.29 q^{37} +4694.45 q^{38} +8089.16 q^{40} +9137.16 q^{41} -19293.4 q^{43} -12629.7 q^{44} +4491.33 q^{46} +4662.43 q^{47} +29075.5 q^{49} +4875.83 q^{50} +15477.1 q^{52} -10048.8 q^{53} +30933.7 q^{55} +40358.3 q^{56} +25452.8 q^{58} +3481.00 q^{59} -34785.8 q^{61} -1550.93 q^{62} +25666.9 q^{64} -37907.8 q^{65} -41086.0 q^{67} +27771.2 q^{68} -34983.9 q^{70} -73673.3 q^{71} -32905.0 q^{73} -21326.8 q^{74} +21631.6 q^{76} +154334. q^{77} -36261.8 q^{79} -6689.62 q^{80} -34758.6 q^{82} +38330.1 q^{83} -68019.6 q^{85} +73393.8 q^{86} +135752. q^{88} +39785.2 q^{89} -189129. q^{91} +20695.6 q^{92} -17736.3 q^{94} -52981.9 q^{95} -53489.3 q^{97} -110606. q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 9 q + 9 q^{2} + 63 q^{4} + 72 q^{5} - 208 q^{7} + 327 q^{8}+O(q^{10}) \) 9 * q + 9 * q^2 + 63 * q^4 + 72 * q^5 - 208 * q^7 + 327 * q^8	\( 9 q + 9 q^{2} + 63 q^{4} + 72 q^{5} - 208 q^{7} + 327 q^{8} - 402 q^{10} + 1176 q^{11} - 1928 q^{13} + 89 q^{14} - 3533 q^{16} + 3031 q^{17} - 4516 q^{19} + 5540 q^{20} - 2075 q^{22} + 2192 q^{23} - 5611 q^{25} - 3155 q^{26} - 10477 q^{28} + 3096 q^{29} - 10148 q^{31} - 15377 q^{32} + 13101 q^{34} - 9947 q^{35} - 11554 q^{37} - 21836 q^{38} + 24298 q^{40} - 11294 q^{41} - 12526 q^{43} - 24051 q^{44} + 22376 q^{46} - 11716 q^{47} + 19137 q^{49} - 66081 q^{50} + 23609 q^{52} - 16552 q^{53} - 9668 q^{55} - 51401 q^{56} + 62124 q^{58} + 31329 q^{59} - 143662 q^{61} - 96300 q^{62} - 79981 q^{64} + 69180 q^{65} - 119488 q^{67} + 19761 q^{68} - 119574 q^{70} + 38295 q^{71} - 197332 q^{73} - 50783 q^{74} - 104806 q^{76} + 67070 q^{77} - 75556 q^{79} - 45772 q^{80} - 100205 q^{82} + 49800 q^{83} - 189532 q^{85} - 23879 q^{86} + 43035 q^{88} + 85950 q^{89} - 117668 q^{91} - 167480 q^{92} - 5734 q^{94} - 169216 q^{95} - 23462 q^{97} + 70298 q^{98}+O(q^{100}) \) 9 * q + 9 * q^2 + 63 * q^4 + 72 * q^5 - 208 * q^7 + 327 * q^8 - 402 * q^10 + 1176 * q^11 - 1928 * q^13 + 89 * q^14 - 3533 * q^16 + 3031 * q^17 - 4516 * q^19 + 5540 * q^20 - 2075 * q^22 + 2192 * q^23 - 5611 * q^25 - 3155 * q^26 - 10477 * q^28 + 3096 * q^29 - 10148 * q^31 - 15377 * q^32 + 13101 * q^34 - 9947 * q^35 - 11554 * q^37 - 21836 * q^38 + 24298 * q^40 - 11294 * q^41 - 12526 * q^43 - 24051 * q^44 + 22376 * q^46 - 11716 * q^47 + 19137 * q^49 - 66081 * q^50 + 23609 * q^52 - 16552 * q^53 - 9668 * q^55 - 51401 * q^56 + 62124 * q^58 + 31329 * q^59 - 143662 * q^61 - 96300 * q^62 - 79981 * q^64 + 69180 * q^65 - 119488 * q^67 + 19761 * q^68 - 119574 * q^70 + 38295 * q^71 - 197332 * q^73 - 50783 * q^74 - 104806 * q^76 + 67070 * q^77 - 75556 * q^79 - 45772 * q^80 - 100205 * q^82 + 49800 * q^83 - 189532 * q^85 - 23879 * q^86 + 43035 * q^88 + 85950 * q^89 - 117668 * q^91 - 167480 * q^92 - 5734 * q^94 - 169216 * q^95 - 23462 * q^97 + 70298 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more