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Properties

Label 3610.2.a.bi
Level $3610$
Weight $2$
Character orbit 3610.a
Self dual yes
Analytic conductor $28.826$
Analytic rank $0$
Dimension $9$
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] = [3610,2,Mod(1,3610)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3610, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("3610.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3610 = 2 \cdot 5 \cdot 19^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3610.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: \(28.8259951297\)
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: \( x^{9} - 24x^{7} - 6x^{6} + 183x^{5} + 78x^{4} - 455x^{3} - 168x^{2} + 228x - 8 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 190)
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.

\(f(q)\) \(=\) \( q - q^{2} + \beta_1 q^{3} + q^{4} + q^{5} - \beta_1 q^{6} - \beta_{7} q^{7} - q^{8} + (\beta_{8} + \beta_{4} + 2 \beta_{3} + \cdots + 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{2} + \beta_1 q^{3} + q^{4} + q^{5} - \beta_1 q^{6} - \beta_{7} q^{7} - q^{8} + (\beta_{8} + \beta_{4} + 2 \beta_{3} + \cdots + 2) q^{9}+ \cdots + ( - 2 \beta_{8} + \beta_{7} + \beta_{6} + \cdots + 6) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 9 q - 9 q^{2} + 9 q^{4} + 9 q^{5} - 9 q^{8} + 21 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 9 q - 9 q^{2} + 9 q^{4} + 9 q^{5} - 9 q^{8} + 21 q^{9} - 9 q^{10} + 12 q^{11} - 9 q^{13} + 9 q^{16} + 6 q^{17} - 21 q^{18} + 9 q^{20} - 6 q^{21} - 12 q^{22} + 18 q^{23} + 9 q^{25} + 9 q^{26} + 18 q^{27} + 6 q^{31} - 9 q^{32} - 6 q^{33} - 6 q^{34} + 21 q^{36} - 6 q^{37} + 24 q^{39} - 9 q^{40} + 6 q^{42} + 18 q^{43} + 12 q^{44} + 21 q^{45} - 18 q^{46} - 3 q^{47} + 39 q^{49} - 9 q^{50} + 48 q^{51} - 9 q^{52} - 18 q^{54} + 12 q^{55} - 21 q^{59} + 18 q^{61} - 6 q^{62} - 12 q^{63} + 9 q^{64} - 9 q^{65} + 6 q^{66} + 6 q^{68} + 30 q^{69} - 18 q^{71} - 21 q^{72} - 36 q^{73} + 6 q^{74} + 15 q^{77} - 24 q^{78} + 6 q^{79} + 9 q^{80} + 69 q^{81} - 6 q^{83} - 6 q^{84} + 6 q^{85} - 18 q^{86} - 24 q^{87} - 12 q^{88} + 18 q^{89} - 21 q^{90} + 60 q^{91} + 18 q^{92} + 3 q^{94} - 18 q^{97} - 39 q^{98} + 54 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{9} - 24x^{7} - 6x^{6} + 183x^{5} + 78x^{4} - 455x^{3} - 168x^{2} + 228x - 8 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 131 \nu^{8} + 360 \nu^{7} + 2204 \nu^{6} - 5902 \nu^{5} - 8957 \nu^{4} + 24062 \nu^{3} + \cdots + 6504 ) / 3876 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 70 \nu^{8} - 111 \nu^{7} - 1444 \nu^{6} + 1556 \nu^{5} + 9106 \nu^{4} - 4033 \nu^{3} - 17246 \nu^{2} + \cdots + 3744 ) / 1938 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 9\nu^{8} - 32\nu^{7} - 140\nu^{6} + 542\nu^{5} + 415\nu^{4} - 2466\nu^{3} + 933\nu^{2} + 2832\nu - 1580 ) / 204 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 111 \nu^{8} + 236 \nu^{7} + 1976 \nu^{6} - 3704 \nu^{5} - 9493 \nu^{4} + 14604 \nu^{3} + \cdots + 560 ) / 1938 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 134 \nu^{8} - 9 \nu^{7} + 3078 \nu^{6} + 842 \nu^{5} - 21732 \nu^{4} - 9371 \nu^{3} + 45620 \nu^{2} + \cdots - 11726 ) / 1938 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 557 \nu^{8} - 1580 \nu^{7} - 10032 \nu^{6} + 26298 \nu^{5} + 49091 \nu^{4} - 113178 \nu^{3} + \cdots - 10888 ) / 3876 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 291 \nu^{8} + 706 \nu^{7} + 5320 \nu^{6} - 11212 \nu^{5} - 26633 \nu^{4} + 43524 \nu^{3} + \cdots + 1084 ) / 1938 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{8} + \beta_{4} + 2\beta_{3} - \beta_{2} + 5 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{8} + \beta_{5} + 2\beta_{3} - 4\beta_{2} + 8\beta _1 + 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 13\beta_{8} + \beta_{7} + \beta_{6} - 3\beta_{5} + 9\beta_{4} + 25\beta_{3} - 12\beta_{2} + 3\beta _1 + 44 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 19\beta_{8} + 4\beta_{7} - 2\beta_{6} + 12\beta_{5} + 24\beta_{3} - 58\beta_{2} + 74\beta _1 + 36 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 151\beta_{8} + 13\beta_{7} + 16\beta_{6} - 53\beta_{5} + 86\beta_{4} + 280\beta_{3} - 147\beta_{2} + 57\beta _1 + 436 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 269 \beta_{8} + 68 \beta_{7} - 40 \beta_{6} + 113 \beta_{5} + 4 \beta_{4} + 268 \beta_{3} - 724 \beta_{2} + \cdots + 494 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 1732 \beta_{8} + 157 \beta_{7} + 181 \beta_{6} - 733 \beta_{5} + 856 \beta_{4} + 3051 \beta_{3} + \cdots + 4547 \) 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
−3.21584
−2.37131
−2.31980
−1.10917
0.0361439
0.576411
1.79490
3.17969
3.42897
−1.00000 −3.21584 1.00000 1.00000 3.21584 −0.0233951 −1.00000 7.34161 −1.00000
1.2 −1.00000 −2.37131 1.00000 1.00000 2.37131 3.15542 −1.00000 2.62313 −1.00000
1.3 −1.00000 −2.31980 1.00000 1.00000 2.31980 −4.91674 −1.00000 2.38147 −1.00000
1.4 −1.00000 −1.10917 1.00000 1.00000 1.10917 4.92913 −1.00000 −1.76973 −1.00000
1.5 −1.00000 0.0361439 1.00000 1.00000 −0.0361439 1.83741 −1.00000 −2.99869 −1.00000
1.6 −1.00000 0.576411 1.00000 1.00000 −0.576411 −4.86419 −1.00000 −2.66775 −1.00000
1.7 −1.00000 1.79490 1.00000 1.00000 −1.79490 1.36147 −1.00000 0.221676 −1.00000
1.8 −1.00000 3.17969 1.00000 1.00000 −3.17969 −3.34610 −1.00000 7.11045 −1.00000
1.9 −1.00000 3.42897 1.00000 1.00000 −3.42897 1.86700 −1.00000 8.75785 −1.00000
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.9
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(5\) \(-1\)
\(19\) \(1\)

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 3610.2.a.bi 9
19.b odd 2 1 3610.2.a.bj 9
19.e even 9 2 190.2.k.d ✓ 18
95.p even 18 2 950.2.l.i 18
95.q odd 36 4 950.2.u.g 36
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
190.2.k.d ✓ 18 19.e even 9 2
950.2.l.i 18 95.p even 18 2
950.2.u.g 36 95.q odd 36 4
3610.2.a.bi 9 1.a even 1 1 trivial
3610.2.a.bj 9 19.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(3610))\):

\( T_{3}^{9} - 24T_{3}^{7} - 6T_{3}^{6} + 183T_{3}^{5} + 78T_{3}^{4} - 455T_{3}^{3} - 168T_{3}^{2} + 228T_{3} - 8 \) Copy content Toggle raw display
\( T_{7}^{9} - 51T_{7}^{7} + 35T_{7}^{6} + 804T_{7}^{5} - 1236T_{7}^{4} - 3540T_{7}^{3} + 9372T_{7}^{2} - 5592T_{7} - 136 \) Copy content Toggle raw display
\( T_{13}^{9} + 9 T_{13}^{8} - 27 T_{13}^{7} - 415 T_{13}^{6} - 486 T_{13}^{5} + 3894 T_{13}^{4} + \cdots - 13544 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T + 1)^{9} \) Copy content Toggle raw display
$3$ \( T^{9} - 24 T^{7} + \cdots - 8 \) Copy content Toggle raw display
$5$ \( (T - 1)^{9} \) Copy content Toggle raw display
$7$ \( T^{9} - 51 T^{7} + \cdots - 136 \) Copy content Toggle raw display
$11$ \( T^{9} - 12 T^{8} + \cdots + 71271 \) Copy content Toggle raw display
$13$ \( T^{9} + 9 T^{8} + \cdots - 13544 \) Copy content Toggle raw display
$17$ \( T^{9} - 6 T^{8} + \cdots - 14328 \) Copy content Toggle raw display
$19$ \( T^{9} \) Copy content Toggle raw display
$23$ \( T^{9} - 18 T^{8} + \cdots - 11016 \) Copy content Toggle raw display
$29$ \( T^{9} - 114 T^{7} + \cdots - 9792 \) Copy content Toggle raw display
$31$ \( T^{9} - 6 T^{8} + \cdots - 23104 \) Copy content Toggle raw display
$37$ \( T^{9} + 6 T^{8} + \cdots + 25992 \) Copy content Toggle raw display
$41$ \( T^{9} - 225 T^{7} + \cdots + 3749517 \) Copy content Toggle raw display
$43$ \( T^{9} - 18 T^{8} + \cdots + 21176 \) Copy content Toggle raw display
$47$ \( T^{9} + 3 T^{8} + \cdots - 1368 \) Copy content Toggle raw display
$53$ \( T^{9} - 93 T^{7} + \cdots + 155592 \) Copy content Toggle raw display
$59$ \( T^{9} + 21 T^{8} + \cdots + 3345957 \) Copy content Toggle raw display
$61$ \( T^{9} - 18 T^{8} + \cdots + 420444224 \) Copy content Toggle raw display
$67$ \( T^{9} - 204 T^{7} + \cdots - 511552 \) Copy content Toggle raw display
$71$ \( T^{9} + 18 T^{8} + \cdots + 13042368 \) Copy content Toggle raw display
$73$ \( T^{9} + 36 T^{8} + \cdots - 187272 \) Copy content Toggle raw display
$79$ \( T^{9} - 6 T^{8} + \cdots - 5796352 \) Copy content Toggle raw display
$83$ \( T^{9} + 6 T^{8} + \cdots - 7573752 \) Copy content Toggle raw display
$89$ \( T^{9} - 18 T^{8} + \cdots + 12500559 \) Copy content Toggle raw display
$97$ \( T^{9} + 18 T^{8} + \cdots - 58248 \) Copy content Toggle raw display
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