LMFDB - Embedded newform 531.6.a.b.1.11

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:	$0$
Dimension:	$11$
Coefficient field:	$\mathbb{Q}[x]/(x^{11} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{11} - 5 x^{10} - 238 x^{9} + 1067 x^{8} + 20782 x^{7} - 79077 x^{6} - 813818 x^{5} + 2364885 x^{4} + \cdots - 14846072 $ x^11 - 5x^10 - 238x^9 + 1067x^8 + 20782x^7 - 79077x^6 - 813818x^5 + 2364885x^4 + 13849341x^3 - 23890558x^2 - 74443300x - 14846072
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 177)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.11
Root			$-9.21944$ 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)

$f(q)$	$=$	$q+10.2194 q^{2} +72.4370 q^{4} +99.2561 q^{5} +109.985 q^{7} +413.244 q^{8} +O(q^{10})$	\(q+10.2194 q^{2} +72.4370 q^{4} +99.2561 q^{5} +109.985 q^{7} +413.244 q^{8} +1014.34 q^{10} +193.795 q^{11} +577.502 q^{13} +1123.98 q^{14} +1905.14 q^{16} +332.271 q^{17} -2111.75 q^{19} +7189.82 q^{20} +1980.48 q^{22} -3186.50 q^{23} +6726.76 q^{25} +5901.75 q^{26} +7966.96 q^{28} -6111.58 q^{29} -2024.27 q^{31} +6245.66 q^{32} +3395.63 q^{34} +10916.6 q^{35} +3581.82 q^{37} -21580.9 q^{38} +41017.0 q^{40} -4115.68 q^{41} -23103.3 q^{43} +14038.0 q^{44} -32564.3 q^{46} +6685.33 q^{47} -4710.38 q^{49} +68743.8 q^{50} +41832.5 q^{52} -18280.2 q^{53} +19235.4 q^{55} +45450.5 q^{56} -62457.0 q^{58} -3481.00 q^{59} -14865.7 q^{61} -20686.9 q^{62} +2862.70 q^{64} +57320.5 q^{65} -10836.8 q^{67} +24068.8 q^{68} +111562. q^{70} +80021.9 q^{71} -22189.8 q^{73} +36604.2 q^{74} -152969. q^{76} +21314.5 q^{77} -57520.3 q^{79} +189097. q^{80} -42060.0 q^{82} +98908.2 q^{83} +32979.9 q^{85} -236103. q^{86} +80084.8 q^{88} +91785.6 q^{89} +63516.3 q^{91} -230821. q^{92} +68320.4 q^{94} -209604. q^{95} +49151.1 q^{97} -48137.5 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 11 q + 6 q^{2} + 150 q^{4} + 192 q^{5} - 371 q^{7} + 621 q^{8}+O(q^{10}) $ 11 * q + 6 * q^2 + 150 * q^4 + 192 * q^5 - 371 * q^7 + 621 * q^8	$ 11 q + 6 q^{2} + 150 q^{4} + 192 q^{5} - 371 q^{7} + 621 q^{8} - 399 q^{10} + 698 q^{11} - 1556 q^{13} + 1679 q^{14} - 2662 q^{16} + 4793 q^{17} - 3753 q^{19} + 11023 q^{20} - 9534 q^{22} + 7323 q^{23} + 7867 q^{25} + 4844 q^{26} + 3650 q^{28} + 15467 q^{29} - 5151 q^{31} + 15368 q^{32} + 8452 q^{34} + 23285 q^{35} + 8623 q^{37} - 15205 q^{38} + 41530 q^{40} + 6369 q^{41} - 20506 q^{43} + 55632 q^{44} - 45191 q^{46} + 47899 q^{47} - 10322 q^{49} + 102147 q^{50} - 292 q^{52} + 80048 q^{53} - 2114 q^{55} + 108126 q^{56} - 58294 q^{58} - 38291 q^{59} - 82527 q^{61} + 67438 q^{62} - 51411 q^{64} + 167646 q^{65} - 166976 q^{67} + 136533 q^{68} + 76140 q^{70} + 183560 q^{71} - 36809 q^{73} + 116686 q^{74} + 55580 q^{76} + 164885 q^{77} - 281518 q^{79} + 32683 q^{80} + 178815 q^{82} + 254691 q^{83} + 4763 q^{85} - 349324 q^{86} + 251285 q^{88} + 89687 q^{89} + 34897 q^{91} + 20240 q^{92} + 96548 q^{94} + 155113 q^{95} - 45828 q^{97} - 465864 q^{98}+O(q^{100}) $ 11 * q + 6 * q^2 + 150 * q^4 + 192 * q^5 - 371 * q^7 + 621 * q^8 - 399 * q^10 + 698 * q^11 - 1556 * q^13 + 1679 * q^14 - 2662 * q^16 + 4793 * q^17 - 3753 * q^19 + 11023 * q^20 - 9534 * q^22 + 7323 * q^23 + 7867 * q^25 + 4844 * q^26 + 3650 * q^28 + 15467 * q^29 - 5151 * q^31 + 15368 * q^32 + 8452 * q^34 + 23285 * q^35 + 8623 * q^37 - 15205 * q^38 + 41530 * q^40 + 6369 * q^41 - 20506 * q^43 + 55632 * q^44 - 45191 * q^46 + 47899 * q^47 - 10322 * q^49 + 102147 * q^50 - 292 * q^52 + 80048 * q^53 - 2114 * q^55 + 108126 * q^56 - 58294 * q^58 - 38291 * q^59 - 82527 * q^61 + 67438 * q^62 - 51411 * q^64 + 167646 * q^65 - 166976 * q^67 + 136533 * q^68 + 76140 * q^70 + 183560 * q^71 - 36809 * q^73 + 116686 * q^74 + 55580 * q^76 + 164885 * q^77 - 281518 * q^79 + 32683 * q^80 + 178815 * q^82 + 254691 * q^83 + 4763 * q^85 - 349324 * q^86 + 251285 * q^88 + 89687 * q^89 + 34897 * q^91 + 20240 * q^92 + 96548 * q^94 + 155113 * q^95 - 45828 * q^97 - 465864 * 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 $

$2$	10.2194	1.80656	0.903280	−	0.429052i	$-0.141152\pi$
$2$	10.2194	1.80656	0.903280	+	0.429052i	$0.141152\pi$
$3$	0	0
$3$	0	0
$4$	72.4370	2.26366
$5$	99.2561	1.77555	0.887773	−	0.460281i	$-0.152251\pi$
$5$	99.2561	1.77555	0.887773	+	0.460281i	$0.152251\pi$
$6$	0	0
$7$	109.985	0.848373	0.424187	−	0.905575i	$-0.360560\pi$
$7$	109.985	0.848373	0.424187	+	0.905575i	$0.360560\pi$
$8$	413.244	2.28287
$9$	0	0
$10$	1014.34	3.20763
$11$	193.795	0.482905	0.241453	−	0.970413i	$-0.422376\pi$
$11$	193.795	0.482905	0.241453	+	0.970413i	$0.422376\pi$
$12$	0	0
$13$	577.502	0.947752	0.473876	−	0.880591i	$-0.342854\pi$
$13$	577.502	0.947752	0.473876	+	0.880591i	$0.342854\pi$
$14$	1123.98	1.53264
$15$	0	0
$16$	1905.14	1.86049
$17$	332.271	0.278850	0.139425	−	0.990233i	$-0.455475\pi$
$17$	332.271	0.278850	0.139425	+	0.990233i	$0.455475\pi$
$18$	0	0
$19$	−2111.75	−1.34202	−0.671010	−	0.741448i	$-0.734139\pi$
$19$	−2111.75	−1.34202	−0.671010	+	0.741448i	$0.734139\pi$
$20$	7189.82	4.01923
$21$	0	0
$22$	1980.48	0.872397
$23$	−3186.50	−1.25601	−0.628007	−	0.778208i	$-0.716129\pi$
$23$	−3186.50	−1.25601	−0.628007	+	0.778208i	$0.716129\pi$
$24$	0	0
$25$	6726.76	2.15256
$26$	5901.75	1.71217
$27$	0	0
$28$	7966.96	1.92043
$29$	−6111.58	−1.34946	−0.674728	−	0.738067i	$-0.735739\pi$
$29$	−6111.58	−1.34946	−0.674728	+	0.738067i	$0.735739\pi$
$30$	0	0
$31$	−2024.27	−0.378324	−0.189162	−	0.981946i	$-0.560577\pi$
$31$	−2024.27	−0.378324	−0.189162	+	0.981946i	$0.560577\pi$
$32$	6245.66	1.07821
$33$	0	0
$34$	3395.63	0.503759
$35$	10916.6	1.50633
$36$	0	0
$37$	3581.82	0.430129	0.215065	−	0.976600i	$-0.431004\pi$
$37$	3581.82	0.430129	0.215065	+	0.976600i	$0.431004\pi$
$38$	−21580.9	−2.42444
$39$	0	0
$40$	41017.0	4.05335
$41$	−4115.68	−0.382369	−0.191184	−	0.981554i	$-0.561233\pi$
$41$	−4115.68	−0.382369	−0.191184	+	0.981554i	$0.561233\pi$
$42$	0	0
$43$	−23103.3	−1.90547	−0.952737	−	0.303795i	$-0.901746\pi$
$43$	−23103.3	−1.90547	−0.952737	+	0.303795i	$0.901746\pi$
$44$	14038.0	1.09313
$45$	0	0
$46$	−32564.3	−2.26906
$47$	6685.33	0.441447	0.220723	−	0.975336i	$-0.429158\pi$
$47$	6685.33	0.441447	0.220723	+	0.975336i	$0.429158\pi$
$48$	0	0
$49$	−4710.38	−0.280263
$50$	68743.8	3.88874
$51$	0	0
$52$	41832.5	2.14539
$53$	−18280.2	−0.893904	−0.446952	−	0.894558i	$-0.647491\pi$
$53$	−18280.2	−0.893904	−0.446952	+	0.894558i	$0.647491\pi$
$54$	0	0
$55$	19235.4	0.857420
$56$	45450.5	1.93673
$57$	0	0
$58$	−62457.0	−2.43787
$59$	−3481.00	−0.130189
$59$	−3481.00	−0.130189
$60$	0	0
$61$	−14865.7	−0.511516	−0.255758	−	0.966741i	$-0.582325\pi$
$61$	−14865.7	−0.511516	−0.255758	+	0.966741i	$0.582325\pi$
$62$	−20686.9	−0.683465
$63$	0	0
$64$	2862.70	0.0873628
$65$	57320.5	1.68278
$66$	0	0
$67$	−10836.8	−0.294926	−0.147463	−	0.989068i	$-0.547111\pi$
$67$	−10836.8	−0.294926	−0.147463	+	0.989068i	$0.547111\pi$
$68$	24068.8	0.631221
$69$	0	0
$70$	111562.	2.72127
$71$	80021.9	1.88392	0.941961	−	0.335722i	$-0.108980\pi$
$71$	80021.9	1.88392	0.941961	+	0.335722i	$0.108980\pi$
$72$	0	0
$73$	−22189.8	−0.487357	−0.243678	−	0.969856i	$-0.578354\pi$
$73$	−22189.8	−0.487357	−0.243678	+	0.969856i	$0.578354\pi$
$74$	36604.2	0.777054
$75$	0	0
$76$	−152969.	−3.03787
$77$	21314.5	0.409684
$78$	0	0
$79$	−57520.3	−1.03694	−0.518470	−	0.855096i	$-0.673498\pi$
$79$	−57520.3	−1.03694	−0.518470	+	0.855096i	$0.673498\pi$
$80$	189097.	3.30338
$81$	0	0
$82$	−42060.0	−0.690772
$83$	98908.2	1.57593	0.787966	−	0.615719i	$-0.211134\pi$
$83$	98908.2	1.57593	0.787966	+	0.615719i	$0.211134\pi$
$84$	0	0
$85$	32979.9	0.495111
$86$	−236103.	−3.44235
$87$	0	0
$88$	80084.8	1.10241
$89$	91785.6	1.22829	0.614143	−	0.789195i	$-0.289502\pi$
$89$	91785.6	1.22829	0.614143	+	0.789195i	$0.289502\pi$
$90$	0	0
$91$	63516.3	0.804048
$92$	−230821.	−2.84319
$93$	0	0
$94$	68320.4	0.797500
$95$	−209604.	−2.38282
$96$	0	0
$97$	49151.1	0.530400	0.265200	−	0.964193i	$-0.414562\pi$
$97$	49151.1	0.530400	0.265200	+	0.964193i	$0.414562\pi$
$98$	−48137.5	−0.506312
$99$	0	0
$100$	487267.	4.87267
$101$	24936.4	0.243237	0.121619	−	0.992577i	$-0.461191\pi$
$101$	24936.4	0.243237	0.121619	+	0.992577i	$0.461191\pi$
$102$	0	0
$103$	−134019.	−1.24472	−0.622361	−	0.782731i	$-0.713826\pi$
$103$	−134019.	−1.24472	−0.622361	+	0.782731i	$0.713826\pi$
$104$	238649.	2.16360
$105$	0	0
$106$	−186813.	−1.61489
$107$	−106814.	−0.901924	−0.450962	−	0.892543i	$-0.648919\pi$
$107$	−106814.	−0.901924	−0.450962	+	0.892543i	$0.648919\pi$
$108$	0	0
$109$	186806.	1.50600	0.752998	−	0.658023i	$-0.228607\pi$
$109$	186806.	1.50600	0.752998	+	0.658023i	$0.228607\pi$
$110$	196575.	1.54898
$111$	0	0
$112$	209536.	1.57839
$113$	129207.	0.951900	0.475950	−	0.879472i	$-0.342104\pi$
$113$	129207.	0.951900	0.475950	+	0.879472i	$0.342104\pi$
$114$	0	0
$115$	−316279.	−2.23011
$116$	−442705.	−3.05471
$117$	0	0
$118$	−35573.9	−0.235194
$119$	36544.7	0.236569
$120$	0	0
$121$	−123494.	−0.766803
$122$	−151919.	−0.924085
$123$	0	0
$124$	−146632.	−0.856396
$125$	357497.	2.04643
$126$	0	0
$127$	−221537.	−1.21881	−0.609406	−	0.792858i	$-0.708592\pi$
$127$	−221537.	−1.21881	−0.609406	+	0.792858i	$0.708592\pi$
$128$	−170606.	−0.920384
$129$	0	0
$130$	585784.	3.04004
$131$	−103256.	−0.525700	−0.262850	−	0.964837i	$-0.584662\pi$
$131$	−103256.	−0.525700	−0.262850	+	0.964837i	$0.584662\pi$
$132$	0	0
$133$	−232260.	−1.13853
$134$	−110746.	−0.532801
$135$	0	0
$136$	137309.	0.636579
$137$	395523.	1.80041	0.900204	−	0.435469i	$-0.143418\pi$
$137$	395523.	1.80041	0.900204	+	0.435469i	$0.143418\pi$
$138$	0	0
$139$	239860.	1.05298	0.526492	−	0.850180i	$-0.323507\pi$
$139$	239860.	1.05298	0.526492	+	0.850180i	$0.323507\pi$
$140$	790769.	3.40981
$141$	0	0
$142$	817779.	3.40342
$143$	111917.	0.457674
$144$	0	0
$145$	−606611.	−2.39602
$146$	−226768.	−0.880439
$147$	0	0
$148$	259456.	0.973666
$149$	265499.	0.979708	0.489854	−	0.871804i	$-0.337050\pi$
$149$	265499.	0.979708	0.489854	+	0.871804i	$0.337050\pi$
$150$	0	0
$151$	481345.	1.71796	0.858982	−	0.512005i	$-0.171097\pi$
$151$	481345.	1.71796	0.858982	+	0.512005i	$0.171097\pi$
$152$	−872669.	−3.06366
$153$	0	0
$154$	217823.	0.740118
$155$	−200921.	−0.671732
$156$	0	0
$157$	108460.	0.351173	0.175587	−	0.984464i	$-0.443818\pi$
$157$	108460.	0.351173	0.175587	+	0.984464i	$0.443818\pi$
$158$	−587826.	−1.87329
$159$	0	0
$160$	619920.	1.91441
$161$	−350466.	−1.06557
$162$	0	0
$163$	167294.	0.493187	0.246594	−	0.969119i	$-0.420689\pi$
$163$	167294.	0.493187	0.246594	+	0.969119i	$0.420689\pi$
$164$	−298128.	−0.865552
$165$	0	0
$166$	1.01079e6	2.84701
$167$	700984.	1.94499	0.972494	−	0.232930i	$-0.0748313\pi$
$167$	700984.	1.94499	0.972494	+	0.232930i	$0.0748313\pi$
$168$	0	0
$169$	−37784.8	−0.101765
$170$	337037.	0.894448
$171$	0	0
$172$	−1.67354e6	−4.31334
$173$	375116.	0.952906	0.476453	−	0.879200i	$-0.341922\pi$
$173$	375116.	0.952906	0.476453	+	0.879200i	$0.341922\pi$
$174$	0	0
$175$	739841.	1.82618
$176$	369207.	0.898439
$177$	0	0
$178$	937998.	2.21897
$179$	−252271.	−0.588485	−0.294242	−	0.955731i	$-0.595067\pi$
$179$	−252271.	−0.588485	−0.294242	+	0.955731i	$0.595067\pi$
$180$	0	0
$181$	737343.	1.67291	0.836456	−	0.548033i	$-0.184623\pi$
$181$	737343.	1.67291	0.836456	+	0.548033i	$0.184623\pi$
$182$	649101.	1.45256
$183$	0	0
$184$	−1.31680e6	−2.86732
$185$	355517.	0.763715
$186$	0	0
$187$	64392.7	0.134658
$188$	484266.	0.999285
$189$	0	0
$190$	−2.14204e6	−4.30470
$191$	−984468.	−1.95262	−0.976311	−	0.216370i	$-0.930578\pi$
$191$	−984468.	−1.95262	−0.976311	+	0.216370i	$0.930578\pi$
$192$	0	0
$193$	3026.14	0.00584785	0.00292392	−	0.999996i	$-0.499069\pi$
$193$	3026.14	0.00584785	0.00292392	+	0.999996i	$0.499069\pi$
$194$	502297.	0.958200
$195$	0	0
$196$	−341206.	−0.634419
$197$	416621.	0.764848	0.382424	−	0.923987i	$-0.375089\pi$
$197$	416621.	0.764848	0.382424	+	0.923987i	$0.375089\pi$
$198$	0	0
$199$	−1.00316e6	−1.79572	−0.897861	−	0.440280i	$-0.854879\pi$
$199$	−1.00316e6	−1.79572	−0.897861	+	0.440280i	$0.854879\pi$
$200$	2.77980e6	4.91403
$201$	0	0
$202$	254836.	0.439423
$203$	−672180.	−1.14484
$204$	0	0
$205$	−408506.	−0.678913
$206$	−1.36960e6	−2.24866
$207$	0	0
$208$	1.10022e6	1.76328
$209$	−409248.	−0.648068
$210$	0	0
$211$	241371.	0.373231	0.186616	−	0.982433i	$-0.440248\pi$
$211$	241371.	0.373231	0.186616	+	0.982433i	$0.440248\pi$
$212$	−1.32416e6	−2.02349
$213$	0	0
$214$	−1.09158e6	−1.62938
$215$	−2.29314e6	−3.38326
$216$	0	0
$217$	−222639.	−0.320960
$218$	1.90905e6	2.72067
$219$	0	0
$220$	1.39335e6	1.94091
$221$	191887.	0.264281
$222$	0	0
$223$	227725.	0.306654	0.153327	−	0.988176i	$-0.451001\pi$
$223$	227725.	0.306654	0.153327	+	0.988176i	$0.451001\pi$
$224$	686927.	0.914725
$225$	0	0
$226$	1.32043e6	1.71966
$227$	−1.00306e6	−1.29200	−0.645998	−	0.763339i	$-0.723558\pi$
$227$	−1.00306e6	−1.29200	−0.645998	+	0.763339i	$0.723558\pi$
$228$	0	0
$229$	−682918.	−0.860558	−0.430279	−	0.902696i	$-0.641585\pi$
$229$	−682918.	−0.860558	−0.430279	+	0.902696i	$0.641585\pi$
$230$	−3.23220e6	−4.02883
$231$	0	0
$232$	−2.52558e6	−3.08064
$233$	−1.32745e6	−1.60187	−0.800935	−	0.598752i	$-0.795664\pi$
$233$	−1.32745e6	−1.60187	−0.800935	+	0.598752i	$0.795664\pi$
$234$	0	0
$235$	663560.	0.783809
$236$	−252153.	−0.294703
$237$	0	0
$238$	373467.	0.427376
$239$	1.27290e6	1.44145	0.720727	−	0.693220i	$-0.243808\pi$
$239$	1.27290e6	1.44145	0.720727	+	0.693220i	$0.243808\pi$
$240$	0	0
$241$	641199.	0.711132	0.355566	−	0.934651i	$-0.384288\pi$
$241$	641199.	0.711132	0.355566	+	0.934651i	$0.384288\pi$
$242$	−1.26204e6	−1.38527
$243$	0	0
$244$	−1.07682e6	−1.15790
$245$	−467534.	−0.497620
$246$	0	0
$247$	−1.21954e6	−1.27190
$248$	−836518.	−0.863666
$249$	0	0
$250$	3.65342e6	3.69700
$251$	1.79177e6	1.79514	0.897571	−	0.440870i	$-0.145330\pi$
$251$	1.79177e6	1.79514	0.897571	+	0.440870i	$0.145330\pi$
$252$	0	0
$253$	−617529.	−0.606536
$254$	−2.26398e6	−2.20186
$255$	0	0
$256$	−1.83510e6	−1.75009
$257$	511016.	0.482616	0.241308	−	0.970449i	$-0.422424\pi$
$257$	511016.	0.482616	0.241308	+	0.970449i	$0.422424\pi$
$258$	0	0
$259$	393945.	0.364910
$260$	4.15213e6	3.80923
$261$	0	0
$262$	−1.05522e6	−0.949708
$263$	−453523.	−0.404306	−0.202153	−	0.979354i	$-0.564794\pi$
$263$	−453523.	−0.404306	−0.202153	+	0.979354i	$0.564794\pi$
$264$	0	0
$265$	−1.81442e6	−1.58717
$266$	−2.37357e6	−2.05683
$267$	0	0
$268$	−784983.	−0.667611
$269$	−1.74732e6	−1.47229	−0.736144	−	0.676825i	$-0.763355\pi$
$269$	−1.74732e6	−1.47229	−0.736144	+	0.676825i	$0.763355\pi$
$270$	0	0
$271$	−223578.	−0.184929	−0.0924647	−	0.995716i	$-0.529475\pi$
$271$	−223578.	−0.184929	−0.0924647	+	0.995716i	$0.529475\pi$
$272$	633024.	0.518797
$273$	0	0
$274$	4.04203e6	3.25254
$275$	1.30362e6	1.03948
$276$	0	0
$277$	588786.	0.461061	0.230530	−	0.973065i	$-0.425954\pi$
$277$	588786.	0.461061	0.230530	+	0.973065i	$0.425954\pi$
$278$	2.45124e6	1.90228
$279$	0	0
$280$	4.51124e6	3.43875
$281$	−815660.	−0.616231	−0.308115	−	0.951349i	$-0.599698\pi$
$281$	−815660.	−0.616231	−0.308115	+	0.951349i	$0.599698\pi$
$282$	0	0
$283$	−1.08887e6	−0.808186	−0.404093	−	0.914718i	$-0.632413\pi$
$283$	−1.08887e6	−0.808186	−0.404093	+	0.914718i	$0.632413\pi$
$284$	5.79655e6	4.26455
$285$	0	0
$286$	1.14373e6	0.826816
$287$	−452662.	−0.324391
$288$	0	0
$289$	−1.30945e6	−0.922243
$290$	−6.19923e6	−4.32855
$291$	0	0
$292$	−1.60737e6	−1.10321
$293$	1.13966e6	0.775541	0.387770	−	0.921756i	$-0.373245\pi$
$293$	1.13966e6	0.775541	0.387770	+	0.921756i	$0.373245\pi$
$294$	0	0
$295$	−345510.	−0.231156
$296$	1.48017e6	0.981931
$297$	0	0
$298$	2.71325e6	1.76990
$299$	−1.84021e6	−1.19039
$300$	0	0
$301$	−2.54101e6	−1.61655
$302$	4.91908e6	3.10361
$303$	0	0
$304$	−4.02318e6	−2.49681
$305$	−1.47551e6	−0.908221
$306$	0	0
$307$	−1.27600e6	−0.772688	−0.386344	−	0.922355i	$-0.626262\pi$
$307$	−1.27600e6	−0.772688	−0.386344	+	0.922355i	$0.626262\pi$
$308$	1.54396e6	0.927384
$309$	0	0
$310$	−2.05330e6	−1.21352
$311$	1.30216e6	0.763422	0.381711	−	0.924282i	$-0.375335\pi$
$311$	1.30216e6	0.763422	0.381711	+	0.924282i	$0.375335\pi$
$312$	0	0
$313$	−2.37612e6	−1.37090	−0.685452	−	0.728118i	$-0.740395\pi$
$313$	−2.37612e6	−1.37090	−0.685452	+	0.728118i	$0.740395\pi$
$314$	1.10840e6	0.634416
$315$	0	0
$316$	−4.16660e6	−2.34728
$317$	−2.63675e6	−1.47374	−0.736869	−	0.676035i	$-0.763697\pi$
$317$	−2.63675e6	−1.47374	−0.736869	+	0.676035i	$0.763697\pi$
$318$	0	0
$319$	−1.18440e6	−0.651659
$320$	284141.	0.155117
$321$	0	0
$322$	−3.58157e6	−1.92501
$323$	−701675.	−0.374222
$324$	0	0
$325$	3.88472e6	2.04010
$326$	1.70965e6	0.890972
$327$	0	0
$328$	−1.70078e6	−0.872899
$329$	735284.	0.374512
$330$	0	0
$331$	345030.	0.173096	0.0865480	−	0.996248i	$-0.472416\pi$
$331$	345030.	0.173096	0.0865480	+	0.996248i	$0.472416\pi$
$332$	7.16462e6	3.56737
$333$	0	0
$334$	7.16366e6	3.51374
$335$	−1.07561e6	−0.523654
$336$	0	0
$337$	1.21026e6	0.580503	0.290251	−	0.956950i	$-0.406261\pi$
$337$	1.21026e6	0.580503	0.290251	+	0.956950i	$0.406261\pi$
$338$	−386140.	−0.183845
$339$	0	0
$340$	2.38897e6	1.12076
$341$	−392294.	−0.182695
$342$	0	0
$343$	−2.36658e6	−1.08614
$344$	−9.54731e6	−4.34996
$345$	0	0
$346$	3.83348e6	1.72148
$347$	3.75698e6	1.67500	0.837500	−	0.546437i	$-0.184016\pi$
$347$	3.75698e6	1.67500	0.837500	+	0.546437i	$0.184016\pi$
$348$	0	0
$349$	402933.	0.177080	0.0885399	−	0.996073i	$-0.471780\pi$
$349$	402933.	0.177080	0.0885399	+	0.996073i	$0.471780\pi$
$350$	7.56076e6	3.29910
$351$	0	0
$352$	1.21038e6	0.520673
$353$	2.57154e6	1.09839	0.549196	−	0.835694i	$-0.314934\pi$
$353$	2.57154e6	1.09839	0.549196	+	0.835694i	$0.314934\pi$
$354$	0	0
$355$	7.94266e6	3.34499
$356$	6.64868e6	2.78042
$357$	0	0
$358$	−2.57807e6	−1.06313
$359$	−181007.	−0.0741241	−0.0370620	−	0.999313i	$-0.511800\pi$
$359$	−181007.	−0.0741241	−0.0370620	+	0.999313i	$0.511800\pi$
$360$	0	0
$361$	1.98340e6	0.801017
$362$	7.53524e6	3.02222
$363$	0	0
$364$	4.60093e6	1.82009
$365$	−2.20248e6	−0.865324
$366$	0	0
$367$	−395859.	−0.153418	−0.0767089	−	0.997054i	$-0.524441\pi$
$367$	−395859.	−0.153418	−0.0767089	+	0.997054i	$0.524441\pi$
$368$	−6.07073e6	−2.33680
$369$	0	0
$370$	3.63319e6	1.37970
$371$	−2.01054e6	−0.758365
$372$	0	0
$373$	−1.17285e6	−0.436486	−0.218243	−	0.975894i	$-0.570033\pi$
$373$	−1.17285e6	−0.436486	−0.218243	+	0.975894i	$0.570033\pi$
$374$	658057.	0.243268
$375$	0	0
$376$	2.76267e6	1.00777
$377$	−3.52945e6	−1.27895
$378$	0	0
$379$	−1.54397e6	−0.552130	−0.276065	−	0.961139i	$-0.589030\pi$
$379$	−1.54397e6	−0.552130	−0.276065	+	0.961139i	$0.589030\pi$
$380$	−1.51831e7	−5.39388
$381$	0	0
$382$	−1.00607e7	−3.52753
$383$	5.57141e6	1.94074	0.970371	−	0.241619i	$-0.0776784\pi$
$383$	5.57141e6	1.94074	0.970371	+	0.241619i	$0.0776784\pi$
$384$	0	0
$385$	2.11560e6	0.727412
$386$	30925.5	0.0105645
$387$	0	0
$388$	3.56036e6	1.20064
$389$	1.44734e6	0.484949	0.242474	−	0.970158i	$-0.422041\pi$
$389$	1.44734e6	0.484949	0.242474	+	0.970158i	$0.422041\pi$
$390$	0	0
$391$	−1.05878e6	−0.350239
$392$	−1.94654e6	−0.639805
$393$	0	0
$394$	4.25763e6	1.38174
$395$	−5.70924e6	−1.84113
$396$	0	0
$397$	5.20211e6	1.65654	0.828272	−	0.560326i	$-0.189324\pi$
$397$	5.20211e6	1.65654	0.828272	+	0.560326i	$0.189324\pi$
$398$	−1.02518e7	−3.24408
$399$	0	0
$400$	1.28154e7	4.00482
$401$	−1.93344e6	−0.600439	−0.300220	−	0.953870i	$-0.597060\pi$
$401$	−1.93344e6	−0.600439	−0.300220	+	0.953870i	$0.597060\pi$
$402$	0	0
$403$	−1.16902e6	−0.358558
$404$	1.80632e6	0.550606
$405$	0	0
$406$	−6.86931e6	−2.06823
$407$	694140.	0.207712
$408$	0	0
$409$	−4.68677e6	−1.38537	−0.692684	−	0.721241i	$-0.743572\pi$
$409$	−4.68677e6	−1.38537	−0.692684	+	0.721241i	$0.743572\pi$
$410$	−4.17471e6	−1.22650
$411$	0	0
$412$	−9.70791e6	−2.81762
$413$	−382857.	−0.110449
$414$	0	0
$415$	9.81724e6	2.79814
$416$	3.60688e6	1.02188
$417$	0	0
$418$	−4.18229e6	−1.17077
$419$	−5.57751e6	−1.55205	−0.776025	−	0.630703i	$-0.782767\pi$
$419$	−5.57751e6	−1.55205	−0.776025	+	0.630703i	$0.782767\pi$
$420$	0	0
$421$	−1.52289e6	−0.418759	−0.209379	−	0.977834i	$-0.567144\pi$
$421$	−1.52289e6	−0.418759	−0.209379	+	0.977834i	$0.567144\pi$
$422$	2.46667e6	0.674265
$423$	0	0
$424$	−7.55418e6	−2.04067
$425$	2.23511e6	0.600243
$426$	0	0
$427$	−1.63499e6	−0.433957
$428$	−7.73731e6	−2.04165
$429$	0	0
$430$	−2.34347e7	−6.11206
$431$	1.32186e6	0.342761	0.171380	−	0.985205i	$-0.445177\pi$
$431$	1.32186e6	0.342761	0.171380	+	0.985205i	$0.445177\pi$
$432$	0	0
$433$	4.13752e6	1.06052	0.530262	−	0.847834i	$-0.322094\pi$
$433$	4.13752e6	1.06052	0.530262	+	0.847834i	$0.322094\pi$
$434$	−2.27524e6	−0.579834
$435$	0	0
$436$	1.35316e7	3.40906
$437$	6.72910e6	1.68560
$438$	0	0
$439$	−916784.	−0.227042	−0.113521	−	0.993536i	$-0.536213\pi$
$439$	−916784.	−0.227042	−0.113521	+	0.993536i	$0.536213\pi$
$440$	7.94890e6	1.95738
$441$	0	0
$442$	1.96098e6	0.477439
$443$	−1.86032e6	−0.450378	−0.225189	−	0.974315i	$-0.572300\pi$
$443$	−1.86032e6	−0.450378	−0.225189	+	0.974315i	$0.572300\pi$
$444$	0	0
$445$	9.11027e6	2.18088
$446$	2.32722e6	0.553989
$447$	0	0
$448$	314854.	0.0741163
$449$	−2.38131e6	−0.557442	−0.278721	−	0.960372i	$-0.589910\pi$
$449$	−2.38131e6	−0.557442	−0.278721	+	0.960372i	$0.589910\pi$
$450$	0	0
$451$	−797601.	−0.184648
$452$	9.35940e6	2.15478
$453$	0	0
$454$	−1.02507e7	−2.33407
$455$	6.30438e6	1.42762
$456$	0	0
$457$	679422.	0.152177	0.0760885	−	0.997101i	$-0.475757\pi$
$457$	679422.	0.152177	0.0760885	+	0.997101i	$0.475757\pi$
$458$	−6.97905e6	−1.55465
$459$	0	0
$460$	−2.29104e7	−5.04821
$461$	278818.	0.0611039	0.0305520	−	0.999533i	$-0.490273\pi$
$461$	278818.	0.0611039	0.0305520	+	0.999533i	$0.490273\pi$
$462$	0	0
$463$	5.43255e6	1.17774	0.588872	−	0.808226i	$-0.299572\pi$
$463$	5.43255e6	1.17774	0.588872	+	0.808226i	$0.299572\pi$
$464$	−1.16434e7	−2.51065
$465$	0	0
$466$	−1.35658e7	−2.89387
$467$	373892.	0.0793330	0.0396665	−	0.999213i	$-0.487370\pi$
$467$	373892.	0.0793330	0.0396665	+	0.999213i	$0.487370\pi$
$468$	0	0
$469$	−1.19188e6	−0.250207
$470$	6.78121e6	1.41600
$471$	0	0
$472$	−1.43850e6	−0.297205
$473$	−4.47732e6	−0.920164
$474$	0	0
$475$	−1.42053e7	−2.88878
$476$	2.64719e6	0.535511
$477$	0	0
$478$	1.30084e7	2.60407
$479$	526098.	0.104768	0.0523839	−	0.998627i	$-0.483318\pi$
$479$	526098.	0.104768	0.0523839	+	0.998627i	$0.483318\pi$
$480$	0	0
$481$	2.06851e6	0.407656
$482$	6.55270e6	1.28470
$483$	0	0
$484$	−8.94556e6	−1.73578
$485$	4.87854e6	0.941750
$486$	0	0
$487$	−5.03131e6	−0.961299	−0.480650	−	0.876913i	$-0.659599\pi$
$487$	−5.03131e6	−0.961299	−0.480650	+	0.876913i	$0.659599\pi$
$488$	−6.14315e6	−1.16773
$489$	0	0
$490$	−4.77793e6	−0.898980
$491$	−4.50265e6	−0.842878	−0.421439	−	0.906857i	$-0.638475\pi$
$491$	−4.50265e6	−0.842878	−0.421439	+	0.906857i	$0.638475\pi$
$492$	0	0
$493$	−2.03070e6	−0.376296
$494$	−1.24630e7	−2.29777
$495$	0	0
$496$	−3.85652e6	−0.703868
$497$	8.80118e6	1.59827
$498$	0	0
$499$	4.33681e6	0.779685	0.389842	−	0.920882i	$-0.372529\pi$
$499$	4.33681e6	0.779685	0.389842	+	0.920882i	$0.372529\pi$
$500$	2.58960e7	4.63242
$501$	0	0
$502$	1.83109e7	3.24303
$503$	4.73584e6	0.834597	0.417299	−	0.908769i	$-0.362977\pi$
$503$	4.73584e6	0.834597	0.417299	+	0.908769i	$0.362977\pi$
$504$	0	0
$505$	2.47509e6	0.431879
$506$	−6.31081e6	−1.09574
$507$	0	0
$508$	−1.60475e7	−2.75897
$509$	−5.24411e6	−0.897175	−0.448587	−	0.893739i	$-0.648073\pi$
$509$	−5.24411e6	−0.897175	−0.448587	+	0.893739i	$0.648073\pi$
$510$	0	0
$511$	−2.44054e6	−0.413460
$512$	−1.32944e7	−2.24126
$513$	0	0
$514$	5.22230e6	0.871875
$515$	−1.33022e7	−2.21006
$516$	0	0
$517$	1.29559e6	0.213177
$518$	4.02590e6	0.659232
$519$	0	0
$520$	2.36874e7	3.84157
$521$	−1.11744e7	−1.80356	−0.901781	−	0.432194i	$-0.857739\pi$
$521$	−1.11744e7	−1.80356	−0.901781	+	0.432194i	$0.857739\pi$
$522$	0	0
$523$	7.05297e6	1.12750	0.563752	−	0.825944i	$-0.309357\pi$
$523$	7.05297e6	1.12750	0.563752	+	0.825944i	$0.309357\pi$
$524$	−7.47957e6	−1.19000
$525$	0	0
$526$	−4.63476e6	−0.730403
$527$	−672607.	−0.105496
$528$	0	0
$529$	3.71744e6	0.577571
$530$	−1.85424e7	−2.86731
$531$	0	0
$532$	−1.68242e7	−2.57725
$533$	−2.37681e6	−0.362391
$534$	0	0
$535$	−1.06020e7	−1.60141
$536$	−4.47823e6	−0.673278
$537$	0	0
$538$	−1.78567e7	−2.65978
$539$	−912850.	−0.135340
$540$	0	0
$541$	−734520.	−0.107897	−0.0539486	−	0.998544i	$-0.517181\pi$
$541$	−734520.	−0.107897	−0.0539486	+	0.998544i	$0.517181\pi$
$542$	−2.28484e6	−0.334086
$543$	0	0
$544$	2.07525e6	0.300659
$545$	1.85416e7	2.67397
$546$	0	0
$547$	−5.09324e6	−0.727824	−0.363912	−	0.931433i	$-0.618559\pi$
$547$	−5.09324e6	−0.727824	−0.363912	+	0.931433i	$0.618559\pi$
$548$	2.86505e7	4.07551
$549$	0	0
$550$	1.33222e7	1.87789
$551$	1.29061e7	1.81100
$552$	0	0
$553$	−6.32635e6	−0.879712
$554$	6.01707e6	0.832934
$555$	0	0
$556$	1.73748e7	2.38359
$557$	3.78806e6	0.517343	0.258672	−	0.965965i	$-0.416715\pi$
$557$	3.78806e6	0.517343	0.258672	+	0.965965i	$0.416715\pi$
$558$	0	0
$559$	−1.33422e7	−1.80592
$560$	2.07977e7	2.80250
$561$	0	0
$562$	−8.33559e6	−1.11326
$563$	8.41523e6	1.11891	0.559455	−	0.828861i	$-0.311011\pi$
$563$	8.41523e6	1.11891	0.559455	+	0.828861i	$0.311011\pi$
$564$	0	0
$565$	1.28246e7	1.69014
$566$	−1.11277e7	−1.46004
$567$	0	0
$568$	3.30686e7	4.30075
$569$	4.34598e6	0.562739	0.281370	−	0.959599i	$-0.409211\pi$
$569$	4.34598e6	0.562739	0.281370	+	0.959599i	$0.409211\pi$
$570$	0	0
$571$	766497.	0.0983830	0.0491915	−	0.998789i	$-0.484336\pi$
$571$	766497.	0.0983830	0.0491915	+	0.998789i	$0.484336\pi$
$572$	8.10695e6	1.03602
$573$	0	0
$574$	−4.62595e6	−0.586032
$575$	−2.14348e7	−2.70365
$576$	0	0
$577$	460709.	0.0576086	0.0288043	−	0.999585i	$-0.490830\pi$
$577$	460709.	0.0576086	0.0288043	+	0.999585i	$0.490830\pi$
$578$	−1.33819e7	−1.66609
$579$	0	0
$580$	−4.39411e7	−5.42377
$581$	1.08784e7	1.33698
$582$	0	0
$583$	−3.54262e6	−0.431671
$584$	−9.16982e6	−1.11257
$585$	0	0
$586$	1.16467e7	1.40106
$587$	−279534.	−0.0334841	−0.0167421	−	0.999860i	$-0.505329\pi$
$587$	−279534.	−0.0334841	−0.0167421	+	0.999860i	$0.505329\pi$
$588$	0	0
$589$	4.27475e6	0.507718
$590$	−3.53092e6	−0.417598
$591$	0	0
$592$	6.82386e6	0.800251
$593$	−4.02162e6	−0.469639	−0.234819	−	0.972039i	$-0.575450\pi$
$593$	−4.02162e6	−0.469639	−0.234819	+	0.972039i	$0.575450\pi$
$594$	0	0
$595$	3.62729e6	0.420039
$596$	1.92319e7	2.21772
$597$	0	0
$598$	−1.88059e7	−2.15051
$599$	60227.1	0.00685843	0.00342922	−	0.999994i	$-0.498908\pi$
$599$	60227.1	0.00685843	0.00342922	+	0.999994i	$0.498908\pi$
$600$	0	0
$601$	4.60932e6	0.520536	0.260268	−	0.965536i	$-0.416189\pi$
$601$	4.60932e6	0.520536	0.260268	+	0.965536i	$0.416189\pi$
$602$	−2.59677e7	−2.92040
$603$	0	0
$604$	3.48672e7	3.88888
$605$	−1.22576e7	−1.36149
$606$	0	0
$607$	−6.23116e6	−0.686431	−0.343216	−	0.939257i	$-0.611516\pi$
$607$	−6.23116e6	−0.686431	−0.343216	+	0.939257i	$0.611516\pi$
$608$	−1.31893e7	−1.44698
$609$	0	0
$610$	−1.50789e7	−1.64076
$611$	3.86079e6	0.418382
$612$	0	0
$613$	5.27500e6	0.566984	0.283492	−	0.958975i	$-0.408507\pi$
$613$	5.27500e6	0.566984	0.283492	+	0.958975i	$0.408507\pi$
$614$	−1.30400e7	−1.39591
$615$	0	0
$616$	8.80810e6	0.935256
$617$	1.43161e7	1.51395	0.756975	−	0.653444i	$-0.226677\pi$
$617$	1.43161e7	1.51395	0.756975	+	0.653444i	$0.226677\pi$
$618$	0	0
$619$	1.67507e7	1.75714	0.878569	−	0.477615i	$-0.158499\pi$
$619$	1.67507e7	1.75714	0.878569	+	0.477615i	$0.158499\pi$
$620$	−1.45541e7	−1.52057
$621$	0	0
$622$	1.33074e7	1.37917
$623$	1.00950e7	1.04204
$624$	0	0
$625$	1.44626e7	1.48097
$626$	−2.42826e7	−2.47662
$627$	0	0
$628$	7.85654e6	0.794936
$629$	1.19014e6	0.119942
$630$	0	0
$631$	7.91210e6	0.791076	0.395538	−	0.918450i	$-0.370558\pi$
$631$	7.91210e6	0.791076	0.395538	+	0.918450i	$0.370558\pi$
$632$	−2.37699e7	−2.36720
$633$	0	0
$634$	−2.69461e7	−2.66240
$635$	−2.19889e7	−2.16406
$636$	0	0
$637$	−2.72025e6	−0.265620
$638$	−1.21039e7	−1.17726
$639$	0	0
$640$	−1.69337e7	−1.63418
$641$	1.24774e7	1.19945	0.599723	−	0.800208i	$-0.295278\pi$
$641$	1.24774e7	1.19945	0.599723	+	0.800208i	$0.295278\pi$
$642$	0	0
$643$	1.49984e6	0.143060	0.0715299	−	0.997438i	$-0.477212\pi$
$643$	1.49984e6	0.143060	0.0715299	+	0.997438i	$0.477212\pi$
$644$	−2.53867e7	−2.41208
$645$	0	0
$646$	−7.17073e6	−0.676055
$647$	6.33662e6	0.595109	0.297555	−	0.954705i	$-0.403829\pi$
$647$	6.33662e6	0.595109	0.297555	+	0.954705i	$0.403829\pi$
$648$	0	0
$649$	−674602.	−0.0628689
$650$	3.96997e7	3.68556
$651$	0	0
$652$	1.21183e7	1.11641
$653$	6.01496e6	0.552014	0.276007	−	0.961156i	$-0.410989\pi$
$653$	6.01496e6	0.552014	0.276007	+	0.961156i	$0.410989\pi$
$654$	0	0
$655$	−1.02488e7	−0.933404
$656$	−7.84095e6	−0.711393
$657$	0	0
$658$	7.51419e6	0.676578
$659$	−9.61791e6	−0.862714	−0.431357	−	0.902181i	$-0.641965\pi$
$659$	−9.61791e6	−0.862714	−0.431357	+	0.902181i	$0.641965\pi$
$660$	0	0
$661$	−8.42469e6	−0.749981	−0.374990	−	0.927029i	$-0.622354\pi$
$661$	−8.42469e6	−0.749981	−0.374990	+	0.927029i	$0.622354\pi$
$662$	3.52601e6	0.312708
$663$	0	0
$664$	4.08733e7	3.59765
$665$	−2.30532e7	−2.02152
$666$	0	0
$667$	1.94746e7	1.69494
$668$	5.07772e7	4.40278
$669$	0	0
$670$	−1.09922e7	−0.946012
$671$	−2.88090e6	−0.247014
$672$	0	0
$673$	4.81831e6	0.410069	0.205035	−	0.978755i	$-0.434269\pi$
$673$	4.81831e6	0.410069	0.205035	+	0.978755i	$0.434269\pi$
$674$	1.23682e7	1.04871
$675$	0	0
$676$	−2.73702e6	−0.230362
$677$	1.49503e7	1.25365	0.626826	−	0.779159i	$-0.284353\pi$
$677$	1.49503e7	1.25365	0.626826	+	0.779159i	$0.284353\pi$
$678$	0	0
$679$	5.40586e6	0.449977
$680$	1.36288e7	1.13028
$681$	0	0
$682$	−4.00903e6	−0.330049
$683$	−1.14545e6	−0.0939559	−0.0469780	−	0.998896i	$-0.514959\pi$
$683$	−1.14545e6	−0.0939559	−0.0469780	+	0.998896i	$0.514959\pi$
$684$	0	0
$685$	3.92581e7	3.19671
$686$	−2.41851e7	−1.96218
$687$	0	0
$688$	−4.40151e7	−3.54511
$689$	−1.05568e7	−0.847200
$690$	0	0
$691$	1.14659e7	0.913511	0.456755	−	0.889592i	$-0.349011\pi$
$691$	1.14659e7	0.913511	0.456755	+	0.889592i	$0.349011\pi$
$692$	2.71723e7	2.15705
$693$	0	0
$694$	3.83942e7	3.02599
$695$	2.38076e7	1.86962
$696$	0	0
$697$	−1.36752e6	−0.106624
$698$	4.11775e6	0.319905
$699$	0	0
$700$	5.35919e7	4.13384
$701$	5.89914e6	0.453413	0.226706	−	0.973963i	$-0.427204\pi$
$701$	5.89914e6	0.453413	0.226706	+	0.973963i	$0.427204\pi$
$702$	0	0
$703$	−7.56391e6	−0.577242
$704$	554779.	0.0421880
$705$	0	0
$706$	2.62798e7	1.98431
$707$	2.74262e6	0.206356
$708$	0	0
$709$	1.68552e6	0.125927	0.0629636	−	0.998016i	$-0.479945\pi$
$709$	1.68552e6	0.125927	0.0629636	+	0.998016i	$0.479945\pi$
$710$	8.11695e7	6.04293
$711$	0	0
$712$	3.79299e7	2.80402
$713$	6.45034e6	0.475180
$714$	0	0
$715$	1.11085e7	0.812622
$716$	−1.82738e7	−1.33213
$717$	0	0
$718$	−1.84979e6	−0.133910
$719$	−1.79183e7	−1.29263	−0.646314	−	0.763072i	$-0.723690\pi$
$719$	−1.79183e7	−1.29263	−0.646314	+	0.763072i	$0.723690\pi$
$720$	0	0
$721$	−1.47400e7	−1.05599
$722$	2.02692e7	1.44708
$723$	0	0
$724$	5.34110e7	3.78690
$725$	−4.11112e7	−2.90479
$726$	0	0
$727$	−8.60696e6	−0.603968	−0.301984	−	0.953313i	$-0.597649\pi$
$727$	−8.60696e6	−0.603968	−0.301984	+	0.953313i	$0.597649\pi$
$728$	2.62477e7	1.83554
$729$	0	0
$730$	−2.25081e7	−1.56326
$731$	−7.67657e6	−0.531342
$732$	0	0
$733$	−4.93390e6	−0.339180	−0.169590	−	0.985515i	$-0.554244\pi$
$733$	−4.93390e6	−0.339180	−0.169590	+	0.985515i	$0.554244\pi$
$734$	−4.04546e6	−0.277158
$735$	0	0
$736$	−1.99018e7	−1.35425
$737$	−2.10011e6	−0.142421
$738$	0	0
$739$	−6.04050e6	−0.406876	−0.203438	−	0.979088i	$-0.565212\pi$
$739$	−6.04050e6	−0.406876	−0.203438	+	0.979088i	$0.565212\pi$
$740$	2.57526e7	1.72879
$741$	0	0
$742$	−2.05466e7	−1.37003
$743$	7.90866e6	0.525570	0.262785	−	0.964854i	$-0.415359\pi$
$743$	7.90866e6	0.525570	0.262785	+	0.964854i	$0.415359\pi$
$744$	0	0
$745$	2.63524e7	1.73952
$746$	−1.19859e7	−0.788539
$747$	0	0
$748$	4.66442e6	0.304820
$749$	−1.17479e7	−0.765168
$750$	0	0
$751$	−6.20100e6	−0.401201	−0.200601	−	0.979673i	$-0.564289\pi$
$751$	−6.20100e6	−0.401201	−0.200601	+	0.979673i	$0.564289\pi$
$752$	1.27365e7	0.821307
$753$	0	0
$754$	−3.60690e7	−2.31050
$755$	4.77764e7	3.05033
$756$	0	0
$757$	6.28292e6	0.398494	0.199247	−	0.979949i	$-0.436150\pi$
$757$	6.28292e6	0.398494	0.199247	+	0.979949i	$0.436150\pi$
$758$	−1.57785e7	−0.997455
$759$	0	0
$760$	−8.66177e7	−5.43967
$761$	4.81278e6	0.301255	0.150627	−	0.988591i	$-0.451871\pi$
$761$	4.81278e6	0.301255	0.150627	+	0.988591i	$0.451871\pi$
$762$	0	0
$763$	2.05458e7	1.27765
$764$	−7.13120e7	−4.42007
$765$	0	0
$766$	5.69367e7	3.50607
$767$	−2.01028e6	−0.123387
$768$	0	0
$769$	−7.75895e6	−0.473137	−0.236569	−	0.971615i	$-0.576023\pi$
$769$	−7.75895e6	−0.473137	−0.236569	+	0.971615i	$0.576023\pi$
$770$	2.16202e7	1.31411
$771$	0	0
$772$	219205.	0.0132375
$773$	−1.00740e7	−0.606389	−0.303195	−	0.952929i	$-0.598053\pi$
$773$	−1.00740e7	−0.606389	−0.303195	+	0.952929i	$0.598053\pi$
$774$	0	0
$775$	−1.36168e7	−0.814367
$776$	2.03114e7	1.21084
$777$	0	0
$778$	1.47910e7	0.876089
$779$	8.69130e6	0.513146
$780$	0	0
$781$	1.55079e7	0.909756
$782$	−1.08202e7	−0.632728
$783$	0	0
$784$	−8.97393e6	−0.521426
$785$	1.07653e7	0.623525
$786$	0	0
$787$	−8.19431e6	−0.471602	−0.235801	−	0.971801i	$-0.575771\pi$
$787$	−8.19431e6	−0.471602	−0.235801	+	0.971801i	$0.575771\pi$
$788$	3.01788e7	1.73136
$789$	0	0
$790$	−5.83452e7	−3.32612
$791$	1.42108e7	0.807567
$792$	0	0
$793$	−8.58494e6	−0.484791
$794$	5.31626e7	2.99265
$795$	0	0
$796$	−7.26662e7	−4.06490
$797$	1.92457e7	1.07322	0.536610	−	0.843830i	$-0.319705\pi$
$797$	1.92457e7	1.07322	0.536610	+	0.843830i	$0.319705\pi$
$798$	0	0
$799$	2.22134e6	0.123097
$800$	4.20131e7	2.32092
$801$	0	0
$802$	−1.97587e7	−1.08473
$803$	−4.30029e6	−0.235347
$804$	0	0
$805$	−3.47859e7	−1.89197
$806$	−1.19467e7	−0.647756
$807$	0	0
$808$	1.03048e7	0.555280
$809$	3.71300e6	0.199459	0.0997296	−	0.995015i	$-0.468202\pi$
$809$	3.71300e6	0.199459	0.0997296	+	0.995015i	$0.468202\pi$
$810$	0	0
$811$	−2.16439e7	−1.15553	−0.577767	−	0.816201i	$-0.696076\pi$
$811$	−2.16439e7	−1.15553	−0.577767	+	0.816201i	$0.696076\pi$
$812$	−4.86907e7	−2.59153
$813$	0	0
$814$	7.09372e6	0.375244
$815$	1.66050e7	0.875677
$816$	0	0
$817$	4.87885e7	2.55718
$818$	−4.78962e7	−2.50275
$819$	0	0
$820$	−2.95910e7	−1.53683
$821$	2.51582e7	1.30263	0.651316	−	0.758807i	$-0.274217\pi$
$821$	2.51582e7	1.30263	0.651316	+	0.758807i	$0.274217\pi$
$822$	0	0
$823$	−1.27945e7	−0.658451	−0.329225	−	0.944251i	$-0.606788\pi$
$823$	−1.27945e7	−0.658451	−0.329225	+	0.944251i	$0.606788\pi$
$824$	−5.53824e7	−2.84154
$825$	0	0
$826$	−3.91258e6	−0.199532
$827$	4.64226e6	0.236029	0.118015	−	0.993012i	$-0.462347\pi$
$827$	4.64226e6	0.236029	0.118015	+	0.993012i	$0.462347\pi$
$828$	0	0
$829$	−1.53561e7	−0.776059	−0.388029	−	0.921647i	$-0.626844\pi$
$829$	−1.53561e7	−0.776059	−0.388029	+	0.921647i	$0.626844\pi$
$830$	1.00327e8	5.05500
$831$	0	0
$832$	1.65322e6	0.0827983
$833$	−1.56512e6	−0.0781513
$834$	0	0
$835$	6.95769e7	3.45341
$836$	−2.96447e7	−1.46700
$837$	0	0
$838$	−5.69991e7	−2.80387
$839$	1.78634e6	0.0876112	0.0438056	−	0.999040i	$-0.486052\pi$
$839$	1.78634e6	0.0876112	0.0438056	+	0.999040i	$0.486052\pi$
$840$	0	0
$841$	1.68403e7	0.821031
$842$	−1.55631e7	−0.756513
$843$	0	0
$844$	1.74842e7	0.844868
$845$	−3.75037e6	−0.180689
$846$	0	0
$847$	−1.35825e7	−0.650535
$848$	−3.48263e7	−1.66310
$849$	0	0
$850$	2.28416e7	1.08437
$851$	−1.14135e7	−0.540248
$852$	0	0
$853$	−2.41934e7	−1.13848	−0.569239	−	0.822172i	$-0.692762\pi$
$853$	−2.41934e7	−1.13848	−0.569239	+	0.822172i	$0.692762\pi$
$854$	−1.67087e7	−0.783969
$855$	0	0
$856$	−4.41404e7	−2.05898
$857$	2.04166e6	0.0949578	0.0474789	−	0.998872i	$-0.484881\pi$
$857$	2.04166e6	0.0949578	0.0474789	+	0.998872i	$0.484881\pi$
$858$	0	0
$859$	4.39878e6	0.203399	0.101700	−	0.994815i	$-0.467572\pi$
$859$	4.39878e6	0.203399	0.101700	+	0.994815i	$0.467572\pi$
$860$	−1.66109e8	−7.65854
$861$	0	0
$862$	1.35086e7	0.619217
$863$	−2.14857e7	−0.982023	−0.491012	−	0.871153i	$-0.663373\pi$
$863$	−2.14857e7	−0.982023	−0.491012	+	0.871153i	$0.663373\pi$
$864$	0	0
$865$	3.72325e7	1.69193
$866$	4.22832e7	1.91590
$867$	0	0
$868$	−1.61273e7	−0.726544
$869$	−1.11472e7	−0.500743
$870$	0	0
$871$	−6.25825e6	−0.279516
$872$	7.71963e7	3.43800
$873$	0	0
$874$	6.87676e7	3.04513
$875$	3.93192e7	1.73614
$876$	0	0
$877$	3.80492e7	1.67050	0.835251	−	0.549869i	$-0.185322\pi$
$877$	3.80492e7	1.67050	0.835251	+	0.549869i	$0.185322\pi$
$878$	−9.36902e6	−0.410164
$879$	0	0
$880$	3.66461e7	1.59522
$881$	2.79654e7	1.21390	0.606949	−	0.794741i	$-0.292393\pi$
$881$	2.79654e7	1.21390	0.606949	+	0.794741i	$0.292393\pi$
$882$	0	0
$883$	−4.11323e7	−1.77534	−0.887669	−	0.460481i	$-0.847677\pi$
$883$	−4.11323e7	−1.77534	−0.887669	+	0.460481i	$0.847677\pi$
$884$	1.38997e7	0.598241
$885$	0	0
$886$	−1.90114e7	−0.813635
$887$	2.45600e7	1.04814	0.524070	−	0.851675i	$-0.324413\pi$
$887$	2.45600e7	1.04814	0.524070	+	0.851675i	$0.324413\pi$
$888$	0	0
$889$	−2.43656e7	−1.03401
$890$	9.31019e7	3.93989
$891$	0	0
$892$	1.64957e7	0.694159
$893$	−1.41178e7	−0.592430
$894$	0	0
$895$	−2.50394e7	−1.04488
$896$	−1.87640e7	−0.780829
$897$	0	0
$898$	−2.43356e7	−1.00705
$899$	1.23715e7	0.510532
$900$	0	0
$901$	−6.07399e6	−0.249265
$902$	−8.15104e6	−0.333577
$903$	0	0
$904$	5.33942e7	2.17307
$905$	7.31858e7	2.97033
$906$	0	0
$907$	2.64205e7	1.06640	0.533202	−	0.845988i	$-0.320988\pi$
$907$	2.64205e7	1.06640	0.533202	+	0.845988i	$0.320988\pi$
$908$	−7.26585e7	−2.92463
$909$	0	0
$910$	6.44272e7	2.57909
$911$	−4.37194e7	−1.74533	−0.872666	−	0.488317i	$-0.837611\pi$
$911$	−4.37194e7	−1.74533	−0.872666	+	0.488317i	$0.837611\pi$
$912$	0	0
$913$	1.91680e7	0.761025
$914$	6.94331e6	0.274917
$915$	0	0
$916$	−4.94686e7	−1.94801
$917$	−1.13566e7	−0.445990
$918$	0	0
$919$	−3.62009e7	−1.41394	−0.706969	−	0.707245i	$-0.749938\pi$
$919$	−3.62009e7	−1.41394	−0.706969	+	0.707245i	$0.749938\pi$
$920$	−1.30701e8	−5.09106
$921$	0	0
$922$	2.84937e6	0.110388
$923$	4.62128e7	1.78549
$924$	0	0
$925$	2.40940e7	0.925881
$926$	5.55176e7	2.12767
$927$	0	0
$928$	−3.81709e7	−1.45500
$929$	−2.73689e7	−1.04044	−0.520221	−	0.854032i	$-0.674150\pi$
$929$	−2.73689e7	−1.04044	−0.520221	+	0.854032i	$0.674150\pi$
$930$	0	0
$931$	9.94715e6	0.376118
$932$	−9.61563e7	−3.62608
$933$	0	0
$934$	3.82097e6	0.143320
$935$	6.39136e6	0.239092
$936$	0	0
$937$	4.36536e7	1.62432	0.812158	−	0.583437i	$-0.198292\pi$
$937$	4.36536e7	1.62432	0.812158	+	0.583437i	$0.198292\pi$
$938$	−1.21803e7	−0.452014
$939$	0	0
$940$	4.80663e7	1.77428
$941$	−9.38560e6	−0.345532	−0.172766	−	0.984963i	$-0.555270\pi$
$941$	−9.38560e6	−0.345532	−0.172766	+	0.984963i	$0.555270\pi$
$942$	0	0
$943$	1.31146e7	0.480260
$944$	−6.63179e6	−0.242215
$945$	0	0
$946$	−4.57557e7	−1.66233
$947$	−1.33282e7	−0.482942	−0.241471	−	0.970408i	$-0.577630\pi$
$947$	−1.33282e7	−0.482942	−0.241471	+	0.970408i	$0.577630\pi$
$948$	0	0
$949$	−1.28147e7	−0.461893
$950$	−1.45170e8	−5.21876
$951$	0	0
$952$	1.51019e7	0.540057
$953$	−3.84674e7	−1.37202	−0.686010	−	0.727592i	$-0.740639\pi$
$953$	−3.84674e7	−1.37202	−0.686010	+	0.727592i	$0.740639\pi$
$954$	0	0
$955$	−9.77145e7	−3.46697
$956$	9.22053e7	3.26296
$957$	0	0
$958$	5.37643e6	0.189269
$959$	4.35015e7	1.52742
$960$	0	0
$961$	−2.45315e7	−0.856871
$962$	2.11390e7	0.736455
$963$	0	0
$964$	4.64466e7	1.60976
$965$	300363.	0.0103831
$966$	0	0
$967$	−4.34724e7	−1.49502	−0.747511	−	0.664249i	$-0.768751\pi$
$967$	−4.34724e7	−1.49502	−0.747511	+	0.664249i	$0.768751\pi$
$968$	−5.10333e7	−1.75051
$969$	0	0
$970$	4.98560e7	1.70133
$971$	1.32711e6	0.0451710	0.0225855	−	0.999745i	$-0.492810\pi$
$971$	1.32711e6	0.0451710	0.0225855	+	0.999745i	$0.492810\pi$
$972$	0	0
$973$	2.63810e7	0.893323
$974$	−5.14172e7	−1.73664
$975$	0	0
$976$	−2.83212e7	−0.951670
$977$	2.29185e7	0.768158	0.384079	−	0.923300i	$-0.374519\pi$
$977$	2.29185e7	0.768158	0.384079	+	0.923300i	$0.374519\pi$
$978$	0	0
$979$	1.77876e7	0.593146
$980$	−3.38668e7	−1.12644
$981$	0	0
$982$	−4.60146e7	−1.52271
$983$	−5.50750e7	−1.81790	−0.908952	−	0.416900i	$-0.863116\pi$
$983$	−5.50750e7	−1.81790	−0.908952	+	0.416900i	$0.863116\pi$
$984$	0	0
$985$	4.13521e7	1.35802
$986$	−2.07527e7	−0.679801
$987$	0	0
$988$	−8.83399e7	−2.87915
$989$	7.36187e7	2.39330
$990$	0	0
$991$	5.09544e7	1.64815	0.824077	−	0.566478i	$-0.191694\pi$
$991$	5.09544e7	1.64815	0.824077	+	0.566478i	$0.191694\pi$
$992$	−1.26429e7	−0.407913
$993$	0	0
$994$	8.99432e7	2.88737
$995$	−9.95700e7	−3.18839
$996$	0	0
$997$	−3.69215e7	−1.17636	−0.588181	−	0.808729i	$-0.700156\pi$
$997$	−3.69215e7	−1.17636	−0.588181	+	0.808729i	$0.700156\pi$
$998$	4.43198e7	1.40855
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	531.6.a.b.1.11		11
3.2	odd	2		177.6.a.a.1.1	✓	11

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
177.6.a.a.1.1	✓	11	3.2	odd	2
531.6.a.b.1.11		11	1.1	even	1	trivial

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)

\(f(q)\)	\(=\)	\(q+10.2194 q^{2} +72.4370 q^{4} +99.2561 q^{5} +109.985 q^{7} +413.244 q^{8} +O(q^{10})\)	\(q+10.2194 q^{2} +72.4370 q^{4} +99.2561 q^{5} +109.985 q^{7} +413.244 q^{8} +1014.34 q^{10} +193.795 q^{11} +577.502 q^{13} +1123.98 q^{14} +1905.14 q^{16} +332.271 q^{17} -2111.75 q^{19} +7189.82 q^{20} +1980.48 q^{22} -3186.50 q^{23} +6726.76 q^{25} +5901.75 q^{26} +7966.96 q^{28} -6111.58 q^{29} -2024.27 q^{31} +6245.66 q^{32} +3395.63 q^{34} +10916.6 q^{35} +3581.82 q^{37} -21580.9 q^{38} +41017.0 q^{40} -4115.68 q^{41} -23103.3 q^{43} +14038.0 q^{44} -32564.3 q^{46} +6685.33 q^{47} -4710.38 q^{49} +68743.8 q^{50} +41832.5 q^{52} -18280.2 q^{53} +19235.4 q^{55} +45450.5 q^{56} -62457.0 q^{58} -3481.00 q^{59} -14865.7 q^{61} -20686.9 q^{62} +2862.70 q^{64} +57320.5 q^{65} -10836.8 q^{67} +24068.8 q^{68} +111562. q^{70} +80021.9 q^{71} -22189.8 q^{73} +36604.2 q^{74} -152969. q^{76} +21314.5 q^{77} -57520.3 q^{79} +189097. q^{80} -42060.0 q^{82} +98908.2 q^{83} +32979.9 q^{85} -236103. q^{86} +80084.8 q^{88} +91785.6 q^{89} +63516.3 q^{91} -230821. q^{92} +68320.4 q^{94} -209604. q^{95} +49151.1 q^{97} -48137.5 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 11 q + 6 q^{2} + 150 q^{4} + 192 q^{5} - 371 q^{7} + 621 q^{8}+O(q^{10}) \) 11 * q + 6 * q^2 + 150 * q^4 + 192 * q^5 - 371 * q^7 + 621 * q^8	\( 11 q + 6 q^{2} + 150 q^{4} + 192 q^{5} - 371 q^{7} + 621 q^{8} - 399 q^{10} + 698 q^{11} - 1556 q^{13} + 1679 q^{14} - 2662 q^{16} + 4793 q^{17} - 3753 q^{19} + 11023 q^{20} - 9534 q^{22} + 7323 q^{23} + 7867 q^{25} + 4844 q^{26} + 3650 q^{28} + 15467 q^{29} - 5151 q^{31} + 15368 q^{32} + 8452 q^{34} + 23285 q^{35} + 8623 q^{37} - 15205 q^{38} + 41530 q^{40} + 6369 q^{41} - 20506 q^{43} + 55632 q^{44} - 45191 q^{46} + 47899 q^{47} - 10322 q^{49} + 102147 q^{50} - 292 q^{52} + 80048 q^{53} - 2114 q^{55} + 108126 q^{56} - 58294 q^{58} - 38291 q^{59} - 82527 q^{61} + 67438 q^{62} - 51411 q^{64} + 167646 q^{65} - 166976 q^{67} + 136533 q^{68} + 76140 q^{70} + 183560 q^{71} - 36809 q^{73} + 116686 q^{74} + 55580 q^{76} + 164885 q^{77} - 281518 q^{79} + 32683 q^{80} + 178815 q^{82} + 254691 q^{83} + 4763 q^{85} - 349324 q^{86} + 251285 q^{88} + 89687 q^{89} + 34897 q^{91} + 20240 q^{92} + 96548 q^{94} + 155113 q^{95} - 45828 q^{97} - 465864 q^{98}+O(q^{100}) \) 11 * q + 6 * q^2 + 150 * q^4 + 192 * q^5 - 371 * q^7 + 621 * q^8 - 399 * q^10 + 698 * q^11 - 1556 * q^13 + 1679 * q^14 - 2662 * q^16 + 4793 * q^17 - 3753 * q^19 + 11023 * q^20 - 9534 * q^22 + 7323 * q^23 + 7867 * q^25 + 4844 * q^26 + 3650 * q^28 + 15467 * q^29 - 5151 * q^31 + 15368 * q^32 + 8452 * q^34 + 23285 * q^35 + 8623 * q^37 - 15205 * q^38 + 41530 * q^40 + 6369 * q^41 - 20506 * q^43 + 55632 * q^44 - 45191 * q^46 + 47899 * q^47 - 10322 * q^49 + 102147 * q^50 - 292 * q^52 + 80048 * q^53 - 2114 * q^55 + 108126 * q^56 - 58294 * q^58 - 38291 * q^59 - 82527 * q^61 + 67438 * q^62 - 51411 * q^64 + 167646 * q^65 - 166976 * q^67 + 136533 * q^68 + 76140 * q^70 + 183560 * q^71 - 36809 * q^73 + 116686 * q^74 + 55580 * q^76 + 164885 * q^77 - 281518 * q^79 + 32683 * q^80 + 178815 * q^82 + 254691 * q^83 + 4763 * q^85 - 349324 * q^86 + 251285 * q^88 + 89687 * q^89 + 34897 * q^91 + 20240 * q^92 + 96548 * q^94 + 155113 * q^95 - 45828 * q^97 - 465864 * q^98

Introduction

L-functions

Modular forms

Varieties

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Groups

Database

Properties

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