LMFDB - Newform orbit 8040.2.a.bb

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8040,2,Mod(1,8040)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8040, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("8040.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8040 = 2^{3} \cdot 3 \cdot 5 \cdot 67 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8040.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:	$64.1997232251$
Analytic rank:	$0$
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} - 4x^{8} - 34x^{7} + 123x^{6} + 375x^{5} - 1146x^{4} - 1662x^{3} + 3086x^{2} + 3372x - 64 $ x^9 - 4x^8 - 34x^7 + 123x^6 + 375x^5 - 1146x^4 - 1662x^3 + 3086x^2 + 3372x - 64
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

Coefficients of the $q$-expansion are expressed in terms of a basis $1,\beta_1,\ldots,\beta_{8}$ for the coefficient ring described below. We also show the integral $q$-expansion of the trace form.

$f(q)$	$=$	$ q + q^{3} + q^{5} + ( - \beta_1 + 1) q^{7} + q^{9} + (\beta_{5} - \beta_{2} + \beta_1 - 1) q^{11} - \beta_{2} q^{13} + q^{15} + ( - \beta_{8} - \beta_{6} + 1) q^{17} + ( - \beta_{3} + 1) q^{19} + ( - \beta_1 + 1) q^{21}+ \cdots + (\beta_{5} - \beta_{2} + \beta_1 - 1) q^{99}+O(q^{100}) $ q + q^3 + q^5 + (-b1 + 1) * q^7 + q^9 + (b5 - b2 + b1 - 1) * q^11 - b2 * q^13 + q^15 + (-b8 - b6 + 1) * q^17 + (-b3 + 1) * q^19 + (-b1 + 1) * q^21 + (-b6 + 2) * q^23 + q^25 + q^27 + (b5 - b4 - b3 + b2 + 1) * q^29 + (b6 + b3 + 3) * q^31 + (b5 - b2 + b1 - 1) * q^33 + (-b1 + 1) * q^35 + (b8 - b7 - b2 + 2) * q^37 - b2 * q^39 + (b8 + b7 + b3 - b2 - 1) * q^41 + (-b7 - b6 - b5 - b1 + 2) * q^43 + q^45 + (b7 - b5 + b1) * q^47 + (2b6 - b5 + b4 + b2 - b1 + 4) q^49 + (-b8 - b6 + 1) * q^51 + (-b8 + b4 + b2 - 1) * q^53 + (b5 - b2 + b1 - 1) * q^55 + (-b3 + 1) * q^57 + (-b8 + b6 + b5 - b4 + 2b2 + 1) q^59 + (b7 + b6 + 2b5 + b4 - 1) q^61 + (-b1 + 1) * q^63 - b2 * q^65 - q^67 + (-b6 + 2) * q^69 + (-2b8 - 2b6 - b5 + b2) * q^71 + (b8 - b6 - 2b5 - b4 + b2 - 3b1 - 2) * q^73 + q^75 + (b8 - 2b7 + b4 - b2 + 2b1 - 1) * q^77 + (b7 + 2b6 - 2b5 + b3 + b1 + 1) * q^79 + q^81 + (b6 + b3 + b2 + 5) * q^83 + (-b8 - b6 + 1) * q^85 + (b5 - b4 - b3 + b2 + 1) * q^87 + (2b8 + b7 + b6 + b5 + b4 + b3 - b2) q^89 + (2b8 - b7 + b6 - b5 + b4 - b2 + 5) q^91 + (b6 + b3 + 3) * q^93 + (-b3 + 1) * q^95 + (-b8 - b4 + 2b2 - b1 + 2) q^97 + (b5 - b2 + b1 - 1) * q^99
$\operatorname{Tr}(f)(q)$	$=$	$ 9 q + 9 q^{3} + 9 q^{5} + 5 q^{7} + 9 q^{9} - 7 q^{11} - 3 q^{13} + 9 q^{15} + 7 q^{17} + 8 q^{19} + 5 q^{21} + 19 q^{23} + 9 q^{25} + 9 q^{27} + 14 q^{29} + 27 q^{31} - 7 q^{33} + 5 q^{35} + 15 q^{37}+ \cdots - 7 q^{99}+O(q^{100}) $ 9 * q + 9 * q^3 + 9 * q^5 + 5 * q^7 + 9 * q^9 - 7 * q^11 - 3 * q^13 + 9 * q^15 + 7 * q^17 + 8 * q^19 + 5 * q^21 + 19 * q^23 + 9 * q^25 + 9 * q^27 + 14 * q^29 + 27 * q^31 - 7 * q^33 + 5 * q^35 + 15 * q^37 - 3 * q^39 - 5 * q^41 + 11 * q^43 + 9 * q^45 + 6 * q^47 + 30 * q^49 + 7 * q^51 - 11 * q^53 - 7 * q^55 + 8 * q^57 + 14 * q^59 - 7 * q^61 + 5 * q^63 - 3 * q^65 - 9 * q^67 + 19 * q^69 - 2 * q^71 - 23 * q^73 + 9 * q^75 - 9 * q^77 + 13 * q^79 + 9 * q^81 + 48 * q^83 + 7 * q^85 + 14 * q^87 + 5 * q^89 + 41 * q^91 + 27 * q^93 + 8 * q^95 + 19 * q^97 - 7 * q^99

Basis of coefficient ring in terms of a root $\nu$ of $ x^{9} - 4x^{8} - 34x^{7} + 123x^{6} + 375x^{5} - 1146x^{4} - 1662x^{3} + 3086x^{2} + 3372x - 64 $ x^9 - 4*x^8 - 34*x^7 + 123*x^6 + 375*x^5 - 1146*x^4 - 1662*x^3 + 3086*x^2 + 3372*x - 64 :

$\beta_{1}$	$=$	$ ( - 60429 \nu^{8} + 54882 \nu^{7} + 2230564 \nu^{6} - 586265 \nu^{5} - 24550545 \nu^{4} + \cdots - 3019562 ) / 1184894 $ (-60429v^8 + 54882v^7 + 2230564v^6 - 586265v^5 - 24550545v^4 - 5493692v^3 + 82771688v^2 + 63680678v - 3019562) / 1184894
$\beta_{2}$	$=$	$ ( 48698 \nu^{8} - 55375 \nu^{7} - 1801381 \nu^{6} + 782418 \nu^{5} + 20127849 \nu^{4} + \cdots - 1839938 ) / 592447 $ (48698v^8 - 55375v^7 - 1801381v^6 + 782418v^5 + 20127849v^4 + 3018303v^3 - 68893991v^2 - 54778076v - 1839938) / 592447
$\beta_{3}$	$=$	$ ( - 261557 \nu^{8} + 206018 \nu^{7} + 9819906 \nu^{6} - 1328383 \nu^{5} - 110744457 \nu^{4} + \cdots + 868204 ) / 2369788 $ (-261557v^8 + 206018v^7 + 9819906v^6 - 1328383v^5 - 110744457v^4 - 40421256v^3 + 383711454v^2 + 344075870v + 868204) / 2369788
$\beta_{4}$	$=$	$ ( 157825 \nu^{8} - 165632 \nu^{7} - 5833326 \nu^{6} + 2151101 \nu^{5} + 64806243 \nu^{4} + \cdots - 1845208 ) / 1184894 $ (157825v^8 - 165632v^7 - 5833326v^6 + 2151101v^5 + 64806243v^4 + 11530298v^3 - 220559670v^2 - 170867042v - 1845208) / 1184894
$\beta_{5}$	$=$	$ ( 193129 \nu^{8} - 195460 \nu^{7} - 7087766 \nu^{6} + 2388727 \nu^{5} + 77645393 \nu^{4} + \cdots - 708176 ) / 1184894 $ (193129v^8 - 195460v^7 - 7087766v^6 + 2388727v^5 + 77645393v^4 + 15286020v^3 - 259065034v^2 - 207187326v - 708176) / 1184894
$\beta_{6}$	$=$	$ ( - 529981 \nu^{8} + 507450 \nu^{7} + 19528990 \nu^{6} - 5598063 \nu^{5} - 215175981 \nu^{4} + \cdots - 12210920 ) / 2369788 $ (-529981v^8 + 507450v^7 + 19528990v^6 - 5598063v^5 - 215175981v^4 - 50447320v^3 + 726025258v^2 + 582457966v - 12210920) / 2369788
$\beta_{7}$	$=$	$ ( 589495 \nu^{8} - 544442 \nu^{7} - 21728250 \nu^{6} + 5811565 \nu^{5} + 238666899 \nu^{4} + \cdots + 15388292 ) / 2369788 $ (589495v^8 - 544442v^7 - 21728250v^6 + 5811565v^5 + 238666899v^4 + 55598296v^3 - 799336970v^2 - 628579186v + 15388292) / 2369788
$\beta_{8}$	$=$	$ ( 157825 \nu^{8} - 165632 \nu^{7} - 5833326 \nu^{6} + 2151101 \nu^{5} + 64806243 \nu^{4} + \cdots + 4671709 ) / 592447 $ (157825v^8 - 165632v^7 - 5833326v^6 + 2151101v^5 + 64806243v^4 + 11530298v^3 - 221152117v^2 - 172051936v + 4671709) / 592447

Display $\nu^j$ in terms of $\beta_i$

$\nu$	$=$	$ ( \beta_{4} - \beta_{2} + \beta _1 + 1 ) / 2 $ (b4 - b2 + b1 + 1) / 2
$\nu^{2}$	$=$	$ -\beta_{8} + \beta_{4} + \beta_{2} - \beta _1 + 10 $ -b8 + b4 + b2 - b1 + 10
$\nu^{3}$	$=$	$ -2\beta_{7} + \beta_{6} + 3\beta_{5} + 8\beta_{4} - \beta_{3} - 9\beta_{2} + 4\beta _1 + 15 $ -2b7 + b6 + 3b5 + 8b4 - b3 - 9b2 + 4*b1 + 15
$\nu^{4}$	$=$	$ -19\beta_{8} - 9\beta_{7} - 5\beta_{6} + \beta_{5} + 26\beta_{4} + 10\beta_{2} - 34\beta _1 + 168 $ -19b8 - 9b7 - 5b6 + b5 + 26b4 + 10b2 - 34b1 + 168
$\nu^{5}$	$=$	$ -13\beta_{8} - 59\beta_{7} + 26\beta_{6} + 75\beta_{5} + 168\beta_{4} - 23\beta_{3} - 166\beta_{2} - 9\beta _1 + 396 $ -13b8 - 59b7 + 26b6 + 75b5 + 168b4 - 23b3 - 166b2 - 9b1 + 396
$\nu^{6}$	$=$	$ - 384 \beta_{8} - 311 \beta_{7} - 132 \beta_{6} + 105 \beta_{5} + 661 \beta_{4} - 15 \beta_{3} + \cdots + 3365 $ -384b8 - 311b7 - 132b6 + 105b5 + 661b4 - 15b3 + 9b2 - 835b1 + 3365
$\nu^{7}$	$=$	$ - 618 \beta_{8} - 1558 \beta_{7} + 488 \beta_{6} + 1701 \beta_{5} + 3882 \beta_{4} - 508 \beta_{3} + \cdots + 10628 $ -618b8 - 1558b7 + 488b6 + 1701b5 + 3882b4 - 508b3 - 3268b2 - 1560b1 + 10628
$\nu^{8}$	$=$	$ - 8260 \beta_{8} - 8484 \beta_{7} - 2741 \beta_{6} + 4014 \beta_{5} + 16901 \beta_{4} - 701 \beta_{3} + \cdots + 74577 $ -8260b8 - 8484b7 - 2741b6 + 4014b5 + 16901b4 - 701b3 - 3427b2 - 19564b1 + 74577

Display $\beta_i$ in terms of $\nu^j$

Embeddings

For each embedding $\iota_m$ of the coefficient field, the values $\iota_m(a_n)$ are shown below.

For more information on an embedded modular form you can click on its label.

comment: embeddings in the coefficient field

gp: mfembed(f)

Label

$\iota_m(\nu)$

$ a_{2} $

$ a_{3} $

$ a_{4} $

$ a_{5} $

$ a_{6} $

$ a_{7} $

$ a_{8} $

$ a_{9} $

$ a_{10} $

1.1

3.11607
2.94981
3.03344
−1.81912
−1.02526
−4.02322
0.0186643
5.01663
−3.26701

1.00000

−5.24081

1.00000

1.2

1.00000

−2.42687

1.00000

1.3

1.00000

−1.95262

1.00000

1.4

1.00000

0.227701

1.00000

1.5

1.00000

0.487682

1.00000

1.6

1.00000

1.81240

1.00000

1.7

1.00000

2.52099

1.00000

1.8

1.00000

4.56825

1.00000

1.9

1.00000

5.00327

1.00000

Atkin-Lehner signs

$ p $	Sign
$2$	$ -1 $
$3$	$ -1 $
$5$	$ -1 $
$67$	$ +1 $

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	8040.2.a.bb	✓	9

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
8040.2.a.bb	✓	9	1.a	even	1	1	trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{2}^{\mathrm{new}}(\Gamma_0(8040))$:

$ T_{7}^{9} - 5T_{7}^{8} - 34T_{7}^{7} + 186T_{7}^{6} + 175T_{7}^{5} - 1430T_{7}^{4} + 264T_{7}^{3} + 2924T_{7}^{2} - 1928T_{7} + 288 $

$ T_{11}^{9} + 7 T_{11}^{8} - 48 T_{11}^{7} - 352 T_{11}^{6} + 553 T_{11}^{5} + 3848 T_{11}^{4} + \cdots + 2752 $

$ T_{13}^{9} + 3 T_{13}^{8} - 65 T_{13}^{7} - 107 T_{13}^{6} + 1471 T_{13}^{5} + 560 T_{13}^{4} + \cdots - 1296 $

$ T_{17}^{9} - 7 T_{17}^{8} - 82 T_{17}^{7} + 519 T_{17}^{6} + 1990 T_{17}^{5} - 10273 T_{17}^{4} + \cdots + 18240 $

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{9} $ T^9
$3$	$ (T - 1)^{9} $ (T - 1)^9
$5$	$ (T - 1)^{9} $ (T - 1)^9
$7$	$ T^{9} - 5 T^{8} + \cdots + 288 $ T^9 - 5T^8 - 34T^7 + 186T^6 + 175T^5 - 1430T^4 + 264T^3 + 2924T^2 - 1928T + 288
$11$	$ T^{9} + 7 T^{8} + \cdots + 2752 $ T^9 + 7T^8 - 48T^7 - 352T^6 + 553T^5 + 3848T^4 - 4508T^3 - 8736T^2 + 8448T + 2752
$13$	$ T^{9} + 3 T^{8} + \cdots - 1296 $ T^9 + 3T^8 - 65T^7 - 107T^6 + 1471T^5 + 560T^4 - 12164T^3 + 5268T^2 + 19872T - 1296
$17$	$ T^{9} - 7 T^{8} + \cdots + 18240 $ T^9 - 7T^8 - 82T^7 + 519T^6 + 1990T^5 - 10273T^4 - 14700T^3 + 44128T^2 + 62992T + 18240
$19$	$ T^{9} - 8 T^{8} + \cdots - 163136 $ T^9 - 8T^8 - 74T^7 + 678T^6 + 908T^5 - 14045T^4 + 4332T^3 + 85408T^2 - 34224T - 163136
$23$	$ T^{9} - 19 T^{8} + \cdots - 8640 $ T^9 - 19T^8 + 98T^7 + 126T^6 - 2367T^5 + 6073T^4 - 1160T^3 - 16020T^2 + 22164T - 8640
$29$	$ T^{9} - 14 T^{8} + \cdots - 4648552 $ T^9 - 14T^8 - 96T^7 + 1724T^6 + 2728T^5 - 72745T^4 - 12656T^3 + 1139152T^2 - 441380T - 4648552
$31$	$ T^{9} - 27 T^{8} + \cdots - 2255200 $ T^9 - 27T^8 + 220T^7 + 256T^6 - 13313T^5 + 64080T^4 + 5420T^3 - 890572T^2 + 2560232T - 2255200
$37$	$ T^{9} - 15 T^{8} + \cdots - 1611936 $ T^9 - 15T^8 - 36T^7 + 1318T^6 - 1615T^5 - 39931T^4 + 82570T^3 + 472580T^2 - 874392T - 1611936
$41$	$ T^{9} + 5 T^{8} + \cdots - 40896 $ T^9 + 5T^8 - 212T^7 - 740T^6 + 13645T^5 + 15708T^4 - 299768T^3 + 448080T^2 - 132240T - 40896
$43$	$ T^{9} - 11 T^{8} + \cdots - 1108992 $ T^9 - 11T^8 - 68T^7 + 1050T^6 + 215T^5 - 29534T^4 + 32088T^3 + 311040T^2 - 328448T - 1108992
$47$	$ T^{9} - 6 T^{8} + \cdots - 2160432 $ T^9 - 6T^8 - 196T^7 + 1344T^6 + 9704T^5 - 80181T^4 - 47980T^3 + 1137040T^2 - 1133172T - 2160432
$53$	$ T^{9} + 11 T^{8} + \cdots - 1057920 $ T^9 + 11T^8 - 175T^7 - 1763T^6 + 8571T^5 + 66440T^4 - 202856T^3 - 643056T^2 + 1815664T - 1057920
$59$	$ T^{9} - 14 T^{8} + \cdots + 742272 $ T^9 - 14T^8 - 250T^7 + 3332T^6 + 19442T^5 - 218623T^4 - 449902T^3 + 2636652T^2 - 2628248T + 742272
$61$	$ T^{9} + 7 T^{8} + \cdots + 314122240 $ T^9 + 7T^8 - 453T^7 - 2743T^6 + 69955T^5 + 335858T^4 - 4452008T^3 - 16206516T^2 + 93998352T + 314122240
$67$	$ (T + 1)^{9} $ (T + 1)^9
$71$	$ T^{9} + 2 T^{8} + \cdots - 1509888 $ T^9 + 2T^8 - 386T^7 - 1749T^6 + 37883T^5 + 265954T^4 + 45420T^3 - 2491172T^2 - 3891888T - 1509888
$73$	$ T^{9} + 23 T^{8} + \cdots + 94324848 $ T^9 + 23T^8 - 236T^7 - 7167T^6 + 12032T^5 + 694985T^4 + 79260T^3 - 22940112T^2 - 6740720T + 94324848
$79$	$ T^{9} + \cdots + 2154480320 $ T^9 - 13T^8 - 570T^7 + 5254T^6 + 132889T^5 - 588716T^4 - 14995172T^3 - 97068T^2 + 660029352T + 2154480320
$83$	$ T^{9} - 48 T^{8} + \cdots - 1350144 $ T^9 - 48T^8 + 844T^7 - 6011T^6 + 3313T^5 + 176822T^4 - 861152T^3 + 1260480T^2 + 307968T - 1350144
$89$	$ T^{9} - 5 T^{8} + \cdots - 51857200 $ T^9 - 5T^8 - 460T^7 + 2811T^6 + 47842T^5 - 293023T^4 - 1149744T^3 + 7244728T^2 + 7918688T - 51857200
$97$	$ T^{9} - 19 T^{8} + \cdots - 886976 $ T^9 - 19T^8 - 101T^7 + 2039T^6 + 4899T^5 - 61292T^4 - 88492T^3 + 582352T^2 + 402816T - 886976
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Overview	Random
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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:	\( N \)	\(=\)	\( 8040 = 2^{3} \cdot 3 \cdot 5 \cdot 67 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8040.a (trivial)

\(f(q)\)	\(=\)	\( q + q^{3} + q^{5} + ( - \beta_1 + 1) q^{7} + q^{9} + (\beta_{5} - \beta_{2} + \beta_1 - 1) q^{11} - \beta_{2} q^{13} + q^{15} + ( - \beta_{8} - \beta_{6} + 1) q^{17} + ( - \beta_{3} + 1) q^{19} + ( - \beta_1 + 1) q^{21}+ \cdots + (\beta_{5} - \beta_{2} + \beta_1 - 1) q^{99}+O(q^{100}) \) q + q^3 + q^5 + (-b1 + 1) * q^7 + q^9 + (b5 - b2 + b1 - 1) * q^11 - b2 * q^13 + q^15 + (-b8 - b6 + 1) * q^17 + (-b3 + 1) * q^19 + (-b1 + 1) * q^21 + (-b6 + 2) * q^23 + q^25 + q^27 + (b5 - b4 - b3 + b2 + 1) * q^29 + (b6 + b3 + 3) * q^31 + (b5 - b2 + b1 - 1) * q^33 + (-b1 + 1) * q^35 + (b8 - b7 - b2 + 2) * q^37 - b2 * q^39 + (b8 + b7 + b3 - b2 - 1) * q^41 + (-b7 - b6 - b5 - b1 + 2) * q^43 + q^45 + (b7 - b5 + b1) * q^47 + (2b6 - b5 + b4 + b2 - b1 + 4) q^49 + (-b8 - b6 + 1) * q^51 + (-b8 + b4 + b2 - 1) * q^53 + (b5 - b2 + b1 - 1) * q^55 + (-b3 + 1) * q^57 + (-b8 + b6 + b5 - b4 + 2b2 + 1) q^59 + (b7 + b6 + 2b5 + b4 - 1) q^61 + (-b1 + 1) * q^63 - b2 * q^65 - q^67 + (-b6 + 2) * q^69 + (-2b8 - 2b6 - b5 + b2) * q^71 + (b8 - b6 - 2b5 - b4 + b2 - 3b1 - 2) * q^73 + q^75 + (b8 - 2b7 + b4 - b2 + 2b1 - 1) * q^77 + (b7 + 2b6 - 2b5 + b3 + b1 + 1) * q^79 + q^81 + (b6 + b3 + b2 + 5) * q^83 + (-b8 - b6 + 1) * q^85 + (b5 - b4 - b3 + b2 + 1) * q^87 + (2b8 + b7 + b6 + b5 + b4 + b3 - b2) q^89 + (2b8 - b7 + b6 - b5 + b4 - b2 + 5) q^91 + (b6 + b3 + 3) * q^93 + (-b3 + 1) * q^95 + (-b8 - b4 + 2b2 - b1 + 2) q^97 + (b5 - b2 + b1 - 1) * q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 9 q + 9 q^{3} + 9 q^{5} + 5 q^{7} + 9 q^{9} - 7 q^{11} - 3 q^{13} + 9 q^{15} + 7 q^{17} + 8 q^{19} + 5 q^{21} + 19 q^{23} + 9 q^{25} + 9 q^{27} + 14 q^{29} + 27 q^{31} - 7 q^{33} + 5 q^{35} + 15 q^{37}+ \cdots - 7 q^{99}+O(q^{100}) \) 9 * q + 9 * q^3 + 9 * q^5 + 5 * q^7 + 9 * q^9 - 7 * q^11 - 3 * q^13 + 9 * q^15 + 7 * q^17 + 8 * q^19 + 5 * q^21 + 19 * q^23 + 9 * q^25 + 9 * q^27 + 14 * q^29 + 27 * q^31 - 7 * q^33 + 5 * q^35 + 15 * q^37 - 3 * q^39 - 5 * q^41 + 11 * q^43 + 9 * q^45 + 6 * q^47 + 30 * q^49 + 7 * q^51 - 11 * q^53 - 7 * q^55 + 8 * q^57 + 14 * q^59 - 7 * q^61 + 5 * q^63 - 3 * q^65 - 9 * q^67 + 19 * q^69 - 2 * q^71 - 23 * q^73 + 9 * q^75 - 9 * q^77 + 13 * q^79 + 9 * q^81 + 48 * q^83 + 7 * q^85 + 14 * q^87 + 5 * q^89 + 41 * q^91 + 27 * q^93 + 8 * q^95 + 19 * q^97 - 7 * q^99

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Newspace parameters

Newform invariants

$q$-expansion

Embeddings

Atkin-Lehner signs

Inner twists

Twists

Hecke kernels

Hecke characteristic polynomials

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	8040.2.a.bb

Level	$8040$
Weight	$2$
Character orbit	8040.a
Self dual	yes
Analytic conductor	$64.200$
Analytic rank	$0$
Dimension	$9$
CM	no
Inner twists	$1$

Self dual:	yes
Analytic conductor:	\(64.1997232251\)
Analytic rank:	\(0\)
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:	\( x^{9} - 4x^{8} - 34x^{7} + 123x^{6} + 375x^{5} - 1146x^{4} - 1662x^{3} + 3086x^{2} + 3372x - 64 \) x^9 - 4x^8 - 34x^7 + 123x^6 + 375x^5 - 1146x^4 - 1662x^3 + 3086x^2 + 3372x - 64
Coefficient ring:	\(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index:	\( 2^{2} \)
Twist minimal:	yes
Fricke sign:	\(-1\)
Sato-Tate group:	$\mathrm{SU}(2)$

\(\beta_{1}\)	\(=\)	\( ( - 60429 \nu^{8} + 54882 \nu^{7} + 2230564 \nu^{6} - 586265 \nu^{5} - 24550545 \nu^{4} + \cdots - 3019562 ) / 1184894 \) (-60429v^8 + 54882v^7 + 2230564v^6 - 586265v^5 - 24550545v^4 - 5493692v^3 + 82771688v^2 + 63680678v - 3019562) / 1184894
\(\beta_{2}\)	\(=\)	\( ( 48698 \nu^{8} - 55375 \nu^{7} - 1801381 \nu^{6} + 782418 \nu^{5} + 20127849 \nu^{4} + \cdots - 1839938 ) / 592447 \) (48698v^8 - 55375v^7 - 1801381v^6 + 782418v^5 + 20127849v^4 + 3018303v^3 - 68893991v^2 - 54778076v - 1839938) / 592447
\(\beta_{3}\)	\(=\)	\( ( - 261557 \nu^{8} + 206018 \nu^{7} + 9819906 \nu^{6} - 1328383 \nu^{5} - 110744457 \nu^{4} + \cdots + 868204 ) / 2369788 \) (-261557v^8 + 206018v^7 + 9819906v^6 - 1328383v^5 - 110744457v^4 - 40421256v^3 + 383711454v^2 + 344075870v + 868204) / 2369788
\(\beta_{4}\)	\(=\)	\( ( 157825 \nu^{8} - 165632 \nu^{7} - 5833326 \nu^{6} + 2151101 \nu^{5} + 64806243 \nu^{4} + \cdots - 1845208 ) / 1184894 \) (157825v^8 - 165632v^7 - 5833326v^6 + 2151101v^5 + 64806243v^4 + 11530298v^3 - 220559670v^2 - 170867042v - 1845208) / 1184894
\(\beta_{5}\)	\(=\)	\( ( 193129 \nu^{8} - 195460 \nu^{7} - 7087766 \nu^{6} + 2388727 \nu^{5} + 77645393 \nu^{4} + \cdots - 708176 ) / 1184894 \) (193129v^8 - 195460v^7 - 7087766v^6 + 2388727v^5 + 77645393v^4 + 15286020v^3 - 259065034v^2 - 207187326v - 708176) / 1184894
\(\beta_{6}\)	\(=\)	\( ( - 529981 \nu^{8} + 507450 \nu^{7} + 19528990 \nu^{6} - 5598063 \nu^{5} - 215175981 \nu^{4} + \cdots - 12210920 ) / 2369788 \) (-529981v^8 + 507450v^7 + 19528990v^6 - 5598063v^5 - 215175981v^4 - 50447320v^3 + 726025258v^2 + 582457966v - 12210920) / 2369788
\(\beta_{7}\)	\(=\)	\( ( 589495 \nu^{8} - 544442 \nu^{7} - 21728250 \nu^{6} + 5811565 \nu^{5} + 238666899 \nu^{4} + \cdots + 15388292 ) / 2369788 \) (589495v^8 - 544442v^7 - 21728250v^6 + 5811565v^5 + 238666899v^4 + 55598296v^3 - 799336970v^2 - 628579186v + 15388292) / 2369788
\(\beta_{8}\)	\(=\)	\( ( 157825 \nu^{8} - 165632 \nu^{7} - 5833326 \nu^{6} + 2151101 \nu^{5} + 64806243 \nu^{4} + \cdots + 4671709 ) / 592447 \) (157825v^8 - 165632v^7 - 5833326v^6 + 2151101v^5 + 64806243v^4 + 11530298v^3 - 221152117v^2 - 172051936v + 4671709) / 592447

\(\nu\)	\(=\)	\( ( \beta_{4} - \beta_{2} + \beta _1 + 1 ) / 2 \) (b4 - b2 + b1 + 1) / 2
\(\nu^{2}\)	\(=\)	\( -\beta_{8} + \beta_{4} + \beta_{2} - \beta _1 + 10 \) -b8 + b4 + b2 - b1 + 10
\(\nu^{3}\)	\(=\)	\( -2\beta_{7} + \beta_{6} + 3\beta_{5} + 8\beta_{4} - \beta_{3} - 9\beta_{2} + 4\beta _1 + 15 \) -2b7 + b6 + 3b5 + 8b4 - b3 - 9b2 + 4*b1 + 15
\(\nu^{4}\)	\(=\)	\( -19\beta_{8} - 9\beta_{7} - 5\beta_{6} + \beta_{5} + 26\beta_{4} + 10\beta_{2} - 34\beta _1 + 168 \) -19b8 - 9b7 - 5b6 + b5 + 26b4 + 10b2 - 34b1 + 168
\(\nu^{5}\)	\(=\)	\( -13\beta_{8} - 59\beta_{7} + 26\beta_{6} + 75\beta_{5} + 168\beta_{4} - 23\beta_{3} - 166\beta_{2} - 9\beta _1 + 396 \) -13b8 - 59b7 + 26b6 + 75b5 + 168b4 - 23b3 - 166b2 - 9b1 + 396
\(\nu^{6}\)	\(=\)	\( - 384 \beta_{8} - 311 \beta_{7} - 132 \beta_{6} + 105 \beta_{5} + 661 \beta_{4} - 15 \beta_{3} + \cdots + 3365 \) -384b8 - 311b7 - 132b6 + 105b5 + 661b4 - 15b3 + 9b2 - 835b1 + 3365
\(\nu^{7}\)	\(=\)	\( - 618 \beta_{8} - 1558 \beta_{7} + 488 \beta_{6} + 1701 \beta_{5} + 3882 \beta_{4} - 508 \beta_{3} + \cdots + 10628 \) -618b8 - 1558b7 + 488b6 + 1701b5 + 3882b4 - 508b3 - 3268b2 - 1560b1 + 10628
\(\nu^{8}\)	\(=\)	\( - 8260 \beta_{8} - 8484 \beta_{7} - 2741 \beta_{6} + 4014 \beta_{5} + 16901 \beta_{4} - 701 \beta_{3} + \cdots + 74577 \) -8260b8 - 8484b7 - 2741b6 + 4014b5 + 16901b4 - 701b3 - 3427b2 - 19564b1 + 74577

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 1.9
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( T^{9} \) T^9
$3$	\( (T - 1)^{9} \) (T - 1)^9
$5$	\( (T - 1)^{9} \) (T - 1)^9
$7$	\( T^{9} - 5 T^{8} + \cdots + 288 \) T^9 - 5T^8 - 34T^7 + 186T^6 + 175T^5 - 1430T^4 + 264T^3 + 2924T^2 - 1928T + 288
$11$	\( T^{9} + 7 T^{8} + \cdots + 2752 \) T^9 + 7T^8 - 48T^7 - 352T^6 + 553T^5 + 3848T^4 - 4508T^3 - 8736T^2 + 8448T + 2752
$13$	\( T^{9} + 3 T^{8} + \cdots - 1296 \) T^9 + 3T^8 - 65T^7 - 107T^6 + 1471T^5 + 560T^4 - 12164T^3 + 5268T^2 + 19872T - 1296
$17$	\( T^{9} - 7 T^{8} + \cdots + 18240 \) T^9 - 7T^8 - 82T^7 + 519T^6 + 1990T^5 - 10273T^4 - 14700T^3 + 44128T^2 + 62992T + 18240
$19$	\( T^{9} - 8 T^{8} + \cdots - 163136 \) T^9 - 8T^8 - 74T^7 + 678T^6 + 908T^5 - 14045T^4 + 4332T^3 + 85408T^2 - 34224T - 163136
$23$	\( T^{9} - 19 T^{8} + \cdots - 8640 \) T^9 - 19T^8 + 98T^7 + 126T^6 - 2367T^5 + 6073T^4 - 1160T^3 - 16020T^2 + 22164T - 8640
$29$	\( T^{9} - 14 T^{8} + \cdots - 4648552 \) T^9 - 14T^8 - 96T^7 + 1724T^6 + 2728T^5 - 72745T^4 - 12656T^3 + 1139152T^2 - 441380T - 4648552
$31$	\( T^{9} - 27 T^{8} + \cdots - 2255200 \) T^9 - 27T^8 + 220T^7 + 256T^6 - 13313T^5 + 64080T^4 + 5420T^3 - 890572T^2 + 2560232T - 2255200
$37$	\( T^{9} - 15 T^{8} + \cdots - 1611936 \) T^9 - 15T^8 - 36T^7 + 1318T^6 - 1615T^5 - 39931T^4 + 82570T^3 + 472580T^2 - 874392T - 1611936
$41$	\( T^{9} + 5 T^{8} + \cdots - 40896 \) T^9 + 5T^8 - 212T^7 - 740T^6 + 13645T^5 + 15708T^4 - 299768T^3 + 448080T^2 - 132240T - 40896
$43$	\( T^{9} - 11 T^{8} + \cdots - 1108992 \) T^9 - 11T^8 - 68T^7 + 1050T^6 + 215T^5 - 29534T^4 + 32088T^3 + 311040T^2 - 328448T - 1108992
$47$	\( T^{9} - 6 T^{8} + \cdots - 2160432 \) T^9 - 6T^8 - 196T^7 + 1344T^6 + 9704T^5 - 80181T^4 - 47980T^3 + 1137040T^2 - 1133172T - 2160432
$53$	\( T^{9} + 11 T^{8} + \cdots - 1057920 \) T^9 + 11T^8 - 175T^7 - 1763T^6 + 8571T^5 + 66440T^4 - 202856T^3 - 643056T^2 + 1815664T - 1057920
$59$	\( T^{9} - 14 T^{8} + \cdots + 742272 \) T^9 - 14T^8 - 250T^7 + 3332T^6 + 19442T^5 - 218623T^4 - 449902T^3 + 2636652T^2 - 2628248T + 742272
$61$	\( T^{9} + 7 T^{8} + \cdots + 314122240 \) T^9 + 7T^8 - 453T^7 - 2743T^6 + 69955T^5 + 335858T^4 - 4452008T^3 - 16206516T^2 + 93998352T + 314122240
$67$	\( (T + 1)^{9} \) (T + 1)^9
$71$	\( T^{9} + 2 T^{8} + \cdots - 1509888 \) T^9 + 2T^8 - 386T^7 - 1749T^6 + 37883T^5 + 265954T^4 + 45420T^3 - 2491172T^2 - 3891888T - 1509888
$73$	\( T^{9} + 23 T^{8} + \cdots + 94324848 \) T^9 + 23T^8 - 236T^7 - 7167T^6 + 12032T^5 + 694985T^4 + 79260T^3 - 22940112T^2 - 6740720T + 94324848
$79$	\( T^{9} + \cdots + 2154480320 \) T^9 - 13T^8 - 570T^7 + 5254T^6 + 132889T^5 - 588716T^4 - 14995172T^3 - 97068T^2 + 660029352T + 2154480320
$83$	\( T^{9} - 48 T^{8} + \cdots - 1350144 \) T^9 - 48T^8 + 844T^7 - 6011T^6 + 3313T^5 + 176822T^4 - 861152T^3 + 1260480T^2 + 307968T - 1350144
$89$	\( T^{9} - 5 T^{8} + \cdots - 51857200 \) T^9 - 5T^8 - 460T^7 + 2811T^6 + 47842T^5 - 293023T^4 - 1149744T^3 + 7244728T^2 + 7918688T - 51857200
$97$	\( T^{9} - 19 T^{8} + \cdots - 886976 \) T^9 - 19T^8 - 101T^7 + 2039T^6 + 4899T^5 - 61292T^4 - 88492T^3 + 582352T^2 + 402816T - 886976
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\( p \)	Sign
\(2\)	\( -1 \)
\(3\)	\( -1 \)
\(5\)	\( -1 \)
\(67\)	\( +1 \)