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Properties

Label 8025.2.a.bm
Level $8025$
Weight $2$
Character orbit 8025.a
Self dual yes
Analytic conductor $64.080$
Analytic rank $1$
Dimension $18$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [8025,2,Mod(1,8025)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8025, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("8025.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8025 = 3 \cdot 5^{2} \cdot 107 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8025.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: \(64.0799476221\)
Analytic rank: \(1\)
Dimension: \(18\)
Coefficient field: \(\mathbb{Q}[x]/(x^{18} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{18} - 4 x^{17} - 19 x^{16} + 87 x^{15} + 134 x^{14} - 763 x^{13} - 404 x^{12} + 3475 x^{11} + \cdots + 72 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2\cdot 3 \)
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 \(1,\beta_1,\ldots,\beta_{17}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} + q^{3} + (\beta_{2} + 1) q^{4} - \beta_1 q^{6} + ( - \beta_{4} - 1) q^{7} + ( - \beta_{3} - \beta_1 - 1) q^{8} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{2} + q^{3} + (\beta_{2} + 1) q^{4} - \beta_1 q^{6} + ( - \beta_{4} - 1) q^{7} + ( - \beta_{3} - \beta_1 - 1) q^{8} + q^{9} - \beta_{5} q^{11} + (\beta_{2} + 1) q^{12} - \beta_{14} q^{13} + (\beta_{16} + \beta_{13} - \beta_{10} + \cdots + 1) q^{14}+ \cdots - \beta_{5} q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 18 q - 4 q^{2} + 18 q^{3} + 18 q^{4} - 4 q^{6} - 12 q^{7} - 15 q^{8} + 18 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 18 q - 4 q^{2} + 18 q^{3} + 18 q^{4} - 4 q^{6} - 12 q^{7} - 15 q^{8} + 18 q^{9} - 3 q^{11} + 18 q^{12} - 7 q^{13} - 11 q^{14} + 14 q^{16} - 13 q^{17} - 4 q^{18} - q^{19} - 12 q^{21} - 15 q^{22} - 31 q^{23} - 15 q^{24} - 13 q^{26} + 18 q^{27} - 19 q^{28} - 15 q^{29} - 3 q^{31} - 34 q^{32} - 3 q^{33} + 17 q^{34} + 18 q^{36} - 9 q^{37} - 26 q^{38} - 7 q^{39} - 15 q^{41} - 11 q^{42} - 28 q^{43} + 4 q^{44} - 8 q^{46} - 37 q^{47} + 14 q^{48} + 26 q^{49} - 13 q^{51} + 5 q^{52} - 47 q^{53} - 4 q^{54} - 23 q^{56} - q^{57} - 32 q^{58} - 22 q^{59} - 8 q^{61} + 11 q^{62} - 12 q^{63} + 25 q^{64} - 15 q^{66} - 34 q^{67} - 60 q^{68} - 31 q^{69} - 23 q^{71} - 15 q^{72} - 38 q^{73} - 16 q^{74} - 14 q^{76} - 5 q^{77} - 13 q^{78} + 19 q^{79} + 18 q^{81} + 5 q^{82} - 31 q^{83} - 19 q^{84} + 4 q^{86} - 15 q^{87} - 102 q^{88} - 12 q^{89} - 5 q^{91} - 30 q^{92} - 3 q^{93} + 68 q^{94} - 34 q^{96} - 71 q^{98} - 3 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{18} - 4 x^{17} - 19 x^{16} + 87 x^{15} + 134 x^{14} - 763 x^{13} - 404 x^{12} + 3475 x^{11} + \cdots + 72 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} - 5\nu - 1 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 138405 \nu^{17} + 2212673 \nu^{16} - 10725857 \nu^{15} - 46893218 \nu^{14} + 187343271 \nu^{13} + \cdots - 78746844 ) / 42583944 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 227026 \nu^{17} + 2108595 \nu^{16} - 8459606 \nu^{15} - 55440757 \nu^{14} + 115938821 \nu^{13} + \cdots + 296654988 ) / 42583944 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 645361 \nu^{17} + 14449 \nu^{16} - 15419517 \nu^{15} - 10549688 \nu^{14} + 152326439 \nu^{13} + \cdots + 322944048 ) / 21291972 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 369453 \nu^{17} + 1242743 \nu^{16} + 8901946 \nu^{15} - 29967491 \nu^{14} - 90498930 \nu^{13} + \cdots + 200976054 ) / 10645986 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 874233 \nu^{17} + 5631053 \nu^{16} + 11028727 \nu^{15} - 121536050 \nu^{14} - 5389023 \nu^{13} + \cdots + 422034648 ) / 21291972 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 904136 \nu^{17} - 2343513 \nu^{16} - 19868500 \nu^{15} + 48859723 \nu^{14} + 178902637 \nu^{13} + \cdots - 84874932 ) / 21291972 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 502315 \nu^{17} - 2183924 \nu^{16} - 8936223 \nu^{15} + 47016790 \nu^{14} + 54554156 \nu^{13} + \cdots - 43648704 ) / 10645986 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 1008876 \nu^{17} + 4644347 \nu^{16} + 18107756 \nu^{15} - 101378543 \nu^{14} + \cdots + 244550252 ) / 14194648 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 3073347 \nu^{17} - 6845476 \nu^{16} - 68191895 \nu^{15} + 136757401 \nu^{14} + 612674592 \nu^{13} + \cdots + 486067656 ) / 42583944 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 3226154 \nu^{17} + 5979349 \nu^{16} + 75637362 \nu^{15} - 122271863 \nu^{14} + \cdots + 169989348 ) / 42583944 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 5383793 \nu^{17} + 16981357 \nu^{16} + 109713177 \nu^{15} - 356212496 \nu^{14} + \cdots + 617038668 ) / 42583944 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 6527863 \nu^{17} - 21256990 \nu^{16} - 134484199 \nu^{15} + 454155495 \nu^{14} + 1096910006 \nu^{13} + \cdots - 919354752 ) / 42583944 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( 6876887 \nu^{17} - 22804575 \nu^{16} - 135837811 \nu^{15} + 476254156 \nu^{14} + 1027200223 \nu^{13} + \cdots - 143323860 ) / 42583944 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( 3620060 \nu^{17} - 12560789 \nu^{16} - 72757070 \nu^{15} + 268560351 \nu^{14} + 571919119 \nu^{13} + \cdots - 443119224 ) / 21291972 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} + 5\beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{17} - \beta_{15} + \beta_{14} - \beta_{13} - \beta_{11} - \beta_{9} + \beta_{5} + 2 \beta_{3} + \cdots + 15 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{16} - \beta_{15} + \beta_{12} - 2 \beta_{10} - \beta_{8} - \beta_{6} + \beta_{4} + 10 \beta_{3} + \cdots + 12 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 11 \beta_{17} + \beta_{16} - 11 \beta_{15} + 10 \beta_{14} - 10 \beta_{13} - \beta_{12} - 10 \beta_{11} + \cdots + 88 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( \beta_{17} + 11 \beta_{16} - 13 \beta_{15} - 2 \beta_{14} + 12 \beta_{12} - 2 \beta_{11} - 24 \beta_{10} + \cdots + 107 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 95 \beta_{17} + 15 \beta_{16} - 94 \beta_{15} + 76 \beta_{14} - 77 \beta_{13} - 15 \beta_{12} + \cdots + 561 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 24 \beta_{17} + 93 \beta_{16} - 125 \beta_{15} - 29 \beta_{14} - 2 \beta_{13} + 99 \beta_{12} + \cdots + 870 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 754 \beta_{17} + 153 \beta_{16} - 736 \beta_{15} + 517 \beta_{14} - 542 \beta_{13} - 149 \beta_{12} + \cdots + 3762 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 347 \beta_{17} + 724 \beta_{16} - 1079 \beta_{15} - 292 \beta_{14} - 36 \beta_{13} + 702 \beta_{12} + \cdots + 6811 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 5773 \beta_{17} + 1346 \beta_{16} - 5549 \beta_{15} + 3308 \beta_{14} - 3666 \beta_{13} - 1255 \beta_{12} + \cdots + 26077 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 4014 \beta_{17} + 5454 \beta_{16} - 8866 \beta_{15} - 2550 \beta_{14} - 434 \beta_{13} + 4579 \beta_{12} + \cdots + 52409 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 43545 \beta_{17} + 11029 \beta_{16} - 41132 \beta_{15} + 20282 \beta_{14} - 24323 \beta_{13} + \cdots + 184938 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 41032 \beta_{17} + 40484 \beta_{16} - 70998 \beta_{15} - 20830 \beta_{14} - 4420 \beta_{13} + \cdots + 400100 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( 326953 \beta_{17} + 86971 \beta_{16} - 302819 \beta_{15} + 119736 \beta_{14} - 159935 \beta_{13} + \cdots + 1333540 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( 388438 \beta_{17} + 298432 \beta_{16} - 560177 \beta_{15} - 164736 \beta_{14} - 41083 \beta_{13} + \cdots + 3044804 \) Copy content Toggle raw display

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
2.79383
2.57662
2.42703
1.91514
1.83302
1.71424
1.19867
0.812836
0.339806
0.271452
−0.395458
−0.805396
−0.963508
−1.36277
−1.64259
−2.04837
−2.17644
−2.48811
−2.79383 1.00000 5.80549 0 −2.79383 1.66812 −10.6319 1.00000 0
1.2 −2.57662 1.00000 4.63895 0 −2.57662 −4.64643 −6.79956 1.00000 0
1.3 −2.42703 1.00000 3.89049 0 −2.42703 4.50494 −4.58829 1.00000 0
1.4 −1.91514 1.00000 1.66775 0 −1.91514 −3.13555 0.636297 1.00000 0
1.5 −1.83302 1.00000 1.35996 0 −1.83302 −2.16134 1.17320 1.00000 0
1.6 −1.71424 1.00000 0.938628 0 −1.71424 2.30306 1.81945 1.00000 0
1.7 −1.19867 1.00000 −0.563192 0 −1.19867 −4.56044 3.07242 1.00000 0
1.8 −0.812836 1.00000 −1.33930 0 −0.812836 1.82207 2.71430 1.00000 0
1.9 −0.339806 1.00000 −1.88453 0 −0.339806 1.11619 1.31999 1.00000 0
1.10 −0.271452 1.00000 −1.92631 0 −0.271452 −0.0530818 1.06581 1.00000 0
1.11 0.395458 1.00000 −1.84361 0 0.395458 3.15066 −1.51999 1.00000 0
1.12 0.805396 1.00000 −1.35134 0 0.805396 −4.76227 −2.69915 1.00000 0
1.13 0.963508 1.00000 −1.07165 0 0.963508 −2.23939 −2.95956 1.00000 0
1.14 1.36277 1.00000 −0.142868 0 1.36277 −0.825322 −2.92023 1.00000 0
1.15 1.64259 1.00000 0.698093 0 1.64259 2.65311 −2.13850 1.00000 0
1.16 2.04837 1.00000 2.19583 0 2.04837 −2.39715 0.401125 1.00000 0
1.17 2.17644 1.00000 2.73690 0 2.17644 −1.43481 1.60382 1.00000 0
1.18 2.48811 1.00000 4.19072 0 2.48811 −3.00238 5.45075 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.18
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(5\) \(-1\)
\(107\) \(1\)

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 8025.2.a.bm ✓ 18
5.b even 2 1 8025.2.a.bn yes 18
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8025.2.a.bm ✓ 18 1.a even 1 1 trivial
8025.2.a.bn yes 18 5.b even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8025))\):

\( T_{2}^{18} + 4 T_{2}^{17} - 19 T_{2}^{16} - 87 T_{2}^{15} + 134 T_{2}^{14} + 763 T_{2}^{13} - 404 T_{2}^{12} + \cdots + 72 \) Copy content Toggle raw display
\( T_{7}^{18} + 12 T_{7}^{17} - 4 T_{7}^{16} - 546 T_{7}^{15} - 1355 T_{7}^{14} + 8663 T_{7}^{13} + \cdots - 203852 \) Copy content Toggle raw display
\( T_{11}^{18} + 3 T_{11}^{17} - 120 T_{11}^{16} - 289 T_{11}^{15} + 5949 T_{11}^{14} + 10373 T_{11}^{13} + \cdots + 25995726 \) Copy content Toggle raw display
\( T_{13}^{18} + 7 T_{13}^{17} - 93 T_{13}^{16} - 671 T_{13}^{15} + 3354 T_{13}^{14} + 25321 T_{13}^{13} + \cdots - 36992 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{18} + 4 T^{17} + \cdots + 72 \) Copy content Toggle raw display
$3$ \( (T - 1)^{18} \) Copy content Toggle raw display
$5$ \( T^{18} \) Copy content Toggle raw display
$7$ \( T^{18} + 12 T^{17} + \cdots - 203852 \) Copy content Toggle raw display
$11$ \( T^{18} + 3 T^{17} + \cdots + 25995726 \) Copy content Toggle raw display
$13$ \( T^{18} + 7 T^{17} + \cdots - 36992 \) Copy content Toggle raw display
$17$ \( T^{18} + \cdots - 253067256 \) Copy content Toggle raw display
$19$ \( T^{18} + T^{17} + \cdots + 29333287 \) Copy content Toggle raw display
$23$ \( T^{18} + \cdots - 405364608 \) Copy content Toggle raw display
$29$ \( T^{18} + \cdots + 1797504228 \) Copy content Toggle raw display
$31$ \( T^{18} + \cdots + 25262211268 \) Copy content Toggle raw display
$37$ \( T^{18} + \cdots - 2882919173464 \) Copy content Toggle raw display
$41$ \( T^{18} + \cdots - 12325872744 \) Copy content Toggle raw display
$43$ \( T^{18} + \cdots + 15848764832716 \) Copy content Toggle raw display
$47$ \( T^{18} + \cdots - 135432906336 \) Copy content Toggle raw display
$53$ \( T^{18} + \cdots + 293027202288 \) Copy content Toggle raw display
$59$ \( T^{18} + \cdots + 5035851014832 \) Copy content Toggle raw display
$61$ \( T^{18} + \cdots + 5079710465063 \) Copy content Toggle raw display
$67$ \( T^{18} + \cdots - 286671987712 \) Copy content Toggle raw display
$71$ \( T^{18} + \cdots - 631314658584 \) Copy content Toggle raw display
$73$ \( T^{18} + \cdots + 17220424121360 \) Copy content Toggle raw display
$79$ \( T^{18} + \cdots - 111552118528 \) Copy content Toggle raw display
$83$ \( T^{18} + \cdots + 28\!\cdots\!74 \) Copy content Toggle raw display
$89$ \( T^{18} + \cdots + 45\!\cdots\!14 \) Copy content Toggle raw display
$97$ \( T^{18} + \cdots - 81\!\cdots\!96 \) Copy content Toggle raw display
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