Properties

Label 3626.2.a.bi
Level $3626$
Weight $2$
Character orbit 3626.a
Self dual yes
Analytic conductor $28.954$
Analytic rank $0$
Dimension $14$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [3626,2,Mod(1,3626)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3626, 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("3626.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3626 = 2 \cdot 7^{2} \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3626.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: \(28.9537557729\)
Analytic rank: \(0\)
Dimension: \(14\)
Coefficient field: \(\mathbb{Q}[x]/(x^{14} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{14} - 32x^{12} + 379x^{10} - 2028x^{8} + 4673x^{6} - 3288x^{4} + 352x^{2} - 8 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
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 \(1,\beta_1,\ldots,\beta_{13}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - q^{2} + \beta_1 q^{3} + q^{4} + \beta_{10} q^{5} - \beta_1 q^{6} - q^{8} + (\beta_{7} + \beta_{6} + 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{2} + \beta_1 q^{3} + q^{4} + \beta_{10} q^{5} - \beta_1 q^{6} - q^{8} + (\beta_{7} + \beta_{6} + 2) q^{9} - \beta_{10} q^{10} + (\beta_{8} - \beta_{2} + 1) q^{11} + \beta_1 q^{12} + ( - \beta_{12} + \beta_{9} + \beta_1) q^{13} + (\beta_{8} + \beta_{7} - \beta_{6} + \cdots + 1) q^{15}+ \cdots + ( - 2 \beta_{8} + 2 \beta_{7} + \cdots + 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 14 q - 14 q^{2} + 14 q^{4} - 14 q^{8} + 22 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 14 q - 14 q^{2} + 14 q^{4} - 14 q^{8} + 22 q^{9} + 16 q^{11} + 8 q^{15} + 14 q^{16} - 22 q^{18} - 16 q^{22} + 46 q^{25} - 8 q^{29} - 8 q^{30} - 14 q^{32} + 22 q^{36} + 14 q^{37} + 48 q^{39} + 20 q^{43} + 16 q^{44} - 46 q^{50} + 44 q^{51} + 24 q^{53} - 68 q^{57} + 8 q^{58} + 8 q^{60} + 14 q^{64} + 52 q^{67} - 16 q^{71} - 22 q^{72} - 14 q^{74} - 48 q^{78} + 48 q^{79} + 82 q^{81} - 24 q^{85} - 20 q^{86} - 16 q^{88} - 16 q^{93} - 24 q^{95} + 16 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{14} - 32x^{12} + 379x^{10} - 2028x^{8} + 4673x^{6} - 3288x^{4} + 352x^{2} - 8 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -89\nu^{12} + 2242\nu^{10} - 16515\nu^{8} + 16360\nu^{6} + 168433\nu^{4} - 275752\nu^{2} - 13280 ) / 86786 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 1660\nu^{12} - 53031\nu^{10} + 626898\nu^{8} - 3349965\nu^{6} + 7740820\nu^{4} - 5583120\nu^{2} + 469535 ) / 43393 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 1660 \nu^{13} + 53031 \nu^{11} - 626898 \nu^{9} + 3349965 \nu^{7} - 7740820 \nu^{5} + \cdots - 860072 \nu ) / 86786 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -2073\nu^{12} + 66012\nu^{10} - 774928\nu^{8} + 4072529\nu^{6} - 8980925\nu^{4} + 5364093\nu^{2} - 239041 ) / 43393 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 5357 \nu^{12} + 170192 \nu^{10} - 1996203 \nu^{8} + 10529374 \nu^{6} - 23629681 \nu^{4} + \cdots - 1000630 ) / 86786 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 5357 \nu^{12} - 170192 \nu^{10} + 1996203 \nu^{8} - 10529374 \nu^{6} + 23629681 \nu^{4} + \cdots + 566700 ) / 86786 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 6403 \nu^{12} + 204900 \nu^{10} - 2424191 \nu^{8} + 12922086 \nu^{6} - 29410293 \nu^{4} + \cdots - 1119374 ) / 86786 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 12675 \nu^{13} + 403296 \nu^{11} - 4732969 \nu^{9} + 24912948 \nu^{7} - 55351751 \nu^{5} + \cdots - 457292 \nu ) / 173572 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 14165 \nu^{13} - 452950 \nu^{11} + 5361615 \nu^{9} - 28689902 \nu^{7} + 66209837 \nu^{5} + \cdots + 4461428 \nu ) / 173572 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 10100 \nu^{13} + 322061 \nu^{11} - 3793496 \nu^{9} + 20101495 \nu^{7} - 45299154 \nu^{5} + \cdots - 1433504 \nu ) / 86786 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 16503 \nu^{13} + 526961 \nu^{11} - 6217687 \nu^{9} + 33023581 \nu^{7} - 74709447 \nu^{5} + \cdots - 1858590 \nu ) / 86786 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 12357 \nu^{13} - 394937 \nu^{11} + 4667831 \nu^{9} - 24878523 \nu^{7} + 56790990 \nu^{5} + \cdots + 3029442 \nu ) / 43393 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{7} + \beta_{6} + 5 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{11} + \beta_{10} - \beta_{9} + 2\beta_{4} + 8\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 2\beta_{8} + 10\beta_{7} + 11\beta_{6} - 2\beta_{5} + 3\beta_{3} + \beta_{2} + 44 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 3\beta_{13} + 4\beta_{12} + 14\beta_{11} + 12\beta_{10} - 16\beta_{9} + 32\beta_{4} + 74\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 35\beta_{8} + 102\beta_{7} + 116\beta_{6} - 35\beta_{5} + 47\beta_{3} + 23\beta_{2} + 425 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 58\beta_{13} + 79\beta_{12} + 163\beta_{11} + 135\beta_{10} - 205\beta_{9} + 445\beta_{4} + 735\beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 477\beta_{8} + 1075\beta_{7} + 1215\beta_{6} - 454\beta_{5} + 589\beta_{3} + 377\beta_{2} + 4325 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 831\beta_{13} + 1168\beta_{12} + 1804\beta_{11} + 1547\beta_{10} - 2455\beta_{9} + 5757\beta_{4} + 7618\beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 6001\beta_{8} + 11620\beta_{7} + 12770\beta_{6} - 5347\beta_{5} + 6894\beta_{3} + 5301\beta_{2} + 45520 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 10648 \beta_{13} + 15499 \beta_{12} + 19664 \beta_{11} + 18018 \beta_{10} - 28712 \beta_{9} + \cdots + 81087 \beta_1 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 72877\beta_{8} + 127817\beta_{7} + 135274\beta_{6} - 60670\beta_{5} + 78688\beta_{3} + 68726\beta_{2} + 489893 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 129396 \beta_{13} + 194816 \beta_{12} + 213962 \beta_{11} + 211255 \beta_{10} - 332595 \beta_{9} + \cdots + 878296 \beta_1 \) Copy content Toggle raw display

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
−3.37101
−2.94589
−2.51188
−2.13577
−0.981300
−0.303902
−0.178025
0.178025
0.303902
0.981300
2.13577
2.51188
2.94589
3.37101
−1.00000 −3.37101 1.00000 −2.15229 3.37101 0 −1.00000 8.36369 2.15229
1.2 −1.00000 −2.94589 1.00000 3.60911 2.94589 0 −1.00000 5.67825 −3.60911
1.3 −1.00000 −2.51188 1.00000 −4.25891 2.51188 0 −1.00000 3.30954 4.25891
1.4 −1.00000 −2.13577 1.00000 0.474614 2.13577 0 −1.00000 1.56152 −0.474614
1.5 −1.00000 −0.981300 1.00000 3.28753 0.981300 0 −1.00000 −2.03705 −3.28753
1.6 −1.00000 −0.303902 1.00000 −1.19352 0.303902 0 −1.00000 −2.90764 1.19352
1.7 −1.00000 −0.178025 1.00000 −3.12188 0.178025 0 −1.00000 −2.96831 3.12188
1.8 −1.00000 0.178025 1.00000 3.12188 −0.178025 0 −1.00000 −2.96831 −3.12188
1.9 −1.00000 0.303902 1.00000 1.19352 −0.303902 0 −1.00000 −2.90764 −1.19352
1.10 −1.00000 0.981300 1.00000 −3.28753 −0.981300 0 −1.00000 −2.03705 3.28753
1.11 −1.00000 2.13577 1.00000 −0.474614 −2.13577 0 −1.00000 1.56152 0.474614
1.12 −1.00000 2.51188 1.00000 4.25891 −2.51188 0 −1.00000 3.30954 −4.25891
1.13 −1.00000 2.94589 1.00000 −3.60911 −2.94589 0 −1.00000 5.67825 3.60911
1.14 −1.00000 3.37101 1.00000 2.15229 −3.37101 0 −1.00000 8.36369 −2.15229
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.14
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(7\) \(1\)
\(37\) \(-1\)

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3626.2.a.bi 14
7.b odd 2 1 inner 3626.2.a.bi 14
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3626.2.a.bi 14 1.a even 1 1 trivial
3626.2.a.bi 14 7.b odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(3626))\):

\( T_{3}^{14} - 32T_{3}^{12} + 379T_{3}^{10} - 2028T_{3}^{8} + 4673T_{3}^{6} - 3288T_{3}^{4} + 352T_{3}^{2} - 8 \) Copy content Toggle raw display
\( T_{5}^{14} - 58T_{5}^{12} + 1315T_{5}^{10} - 14722T_{5}^{8} + 83913T_{5}^{6} - 222610T_{5}^{4} + 210272T_{5}^{2} - 36992 \) Copy content Toggle raw display
\( T_{11}^{7} - 8T_{11}^{6} - 25T_{11}^{5} + 268T_{11}^{4} - 23T_{11}^{3} - 1918T_{11}^{2} + 1508T_{11} + 1680 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T + 1)^{14} \) Copy content Toggle raw display
$3$ \( T^{14} - 32 T^{12} + \cdots - 8 \) Copy content Toggle raw display
$5$ \( T^{14} - 58 T^{12} + \cdots - 36992 \) Copy content Toggle raw display
$7$ \( T^{14} \) Copy content Toggle raw display
$11$ \( (T^{7} - 8 T^{6} + \cdots + 1680)^{2} \) Copy content Toggle raw display
$13$ \( T^{14} - 130 T^{12} + \cdots - 24893568 \) Copy content Toggle raw display
$17$ \( T^{14} - 190 T^{12} + \cdots - 14450688 \) Copy content Toggle raw display
$19$ \( T^{14} + \cdots - 177020928 \) Copy content Toggle raw display
$23$ \( (T^{7} - 93 T^{5} + \cdots - 11424)^{2} \) Copy content Toggle raw display
$29$ \( (T^{7} + 4 T^{6} + \cdots + 109760)^{2} \) Copy content Toggle raw display
$31$ \( T^{14} + \cdots - 4302579848 \) Copy content Toggle raw display
$37$ \( (T - 1)^{14} \) Copy content Toggle raw display
$41$ \( T^{14} - 254 T^{12} + \cdots - 71760200 \) Copy content Toggle raw display
$43$ \( (T^{7} - 10 T^{6} + \cdots - 3584)^{2} \) Copy content Toggle raw display
$47$ \( T^{14} + \cdots - 3699376128 \) Copy content Toggle raw display
$53$ \( (T^{7} - 12 T^{6} + \cdots + 13824)^{2} \) Copy content Toggle raw display
$59$ \( T^{14} + \cdots - 1363673088 \) Copy content Toggle raw display
$61$ \( T^{14} + \cdots - 593004060800 \) Copy content Toggle raw display
$67$ \( (T^{7} - 26 T^{6} + \cdots + 17408)^{2} \) Copy content Toggle raw display
$71$ \( (T^{7} + 8 T^{6} + \cdots + 150528)^{2} \) Copy content Toggle raw display
$73$ \( T^{14} + \cdots - 14236085154312 \) Copy content Toggle raw display
$79$ \( (T^{7} - 24 T^{6} + \cdots + 648760)^{2} \) Copy content Toggle raw display
$83$ \( T^{14} + \cdots - 460621952 \) Copy content Toggle raw display
$89$ \( T^{14} + \cdots - 5602746368 \) Copy content Toggle raw display
$97$ \( T^{14} + \cdots - 384950476800 \) Copy content Toggle raw display
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