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Properties

Label 4641.2.a.w
Level $4641$
Weight $2$
Character orbit 4641.a
Self dual yes
Analytic conductor $37.059$
Analytic rank $0$
Dimension $14$
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] = [4641,2,Mod(1,4641)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4641, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("4641.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4641 = 3 \cdot 7 \cdot 13 \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4641.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: \(37.0585715781\)
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: \( x^{14} - x^{13} - 22 x^{12} + 19 x^{11} + 187 x^{10} - 135 x^{9} - 776 x^{8} + 443 x^{7} + 1636 x^{6} + \cdots - 19 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
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.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} - q^{3} + (\beta_{2} + 1) q^{4} - \beta_{11} q^{5} + \beta_1 q^{6} + q^{7} + ( - \beta_{3} - \beta_1) q^{8} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{2} - q^{3} + (\beta_{2} + 1) q^{4} - \beta_{11} q^{5} + \beta_1 q^{6} + q^{7} + ( - \beta_{3} - \beta_1) q^{8} + q^{9} + (\beta_{13} - \beta_{9} + \beta_{2}) q^{10} - \beta_{8} q^{11} + ( - \beta_{2} - 1) q^{12} + q^{13} - \beta_1 q^{14} + \beta_{11} q^{15} + (\beta_{13} - \beta_{12} + \cdots + \beta_1) q^{16}+ \cdots - \beta_{8} q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 14 q - q^{2} - 14 q^{3} + 17 q^{4} - q^{5} + q^{6} + 14 q^{7} - 6 q^{8} + 14 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 14 q - q^{2} - 14 q^{3} + 17 q^{4} - q^{5} + q^{6} + 14 q^{7} - 6 q^{8} + 14 q^{9} + 11 q^{10} - 4 q^{11} - 17 q^{12} + 14 q^{13} - q^{14} + q^{15} + 19 q^{16} + 14 q^{17} - q^{18} + 6 q^{19} + q^{20} - 14 q^{21} + 12 q^{22} + 7 q^{23} + 6 q^{24} + 19 q^{25} - q^{26} - 14 q^{27} + 17 q^{28} - 4 q^{29} - 11 q^{30} + 31 q^{31} - 18 q^{32} + 4 q^{33} - q^{34} - q^{35} + 17 q^{36} + 2 q^{37} + 9 q^{38} - 14 q^{39} + 50 q^{40} + 4 q^{41} + q^{42} + 14 q^{43} - 8 q^{44} - q^{45} - 17 q^{46} - q^{47} - 19 q^{48} + 14 q^{49} - 3 q^{50} - 14 q^{51} + 17 q^{52} - 43 q^{53} + q^{54} + 23 q^{55} - 6 q^{56} - 6 q^{57} - 10 q^{58} + 11 q^{59} - q^{60} + 25 q^{61} - 3 q^{62} + 14 q^{63} + 36 q^{64} - q^{65} - 12 q^{66} + 11 q^{67} + 17 q^{68} - 7 q^{69} + 11 q^{70} + 20 q^{71} - 6 q^{72} + 14 q^{73} - 24 q^{74} - 19 q^{75} + 9 q^{76} - 4 q^{77} + q^{78} + 42 q^{79} + 13 q^{80} + 14 q^{81} + 2 q^{82} + 15 q^{83} - 17 q^{84} - q^{85} - 11 q^{86} + 4 q^{87} + 63 q^{88} + 21 q^{89} + 11 q^{90} + 14 q^{91} + 30 q^{92} - 31 q^{93} - 29 q^{94} + 16 q^{95} + 18 q^{96} + 15 q^{97} - q^{98} - 4 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{14} - x^{13} - 22 x^{12} + 19 x^{11} + 187 x^{10} - 135 x^{9} - 776 x^{8} + 443 x^{7} + 1636 x^{6} + \cdots - 19 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} - 5\nu \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 409 \nu^{13} + 411 \nu^{12} - 8939 \nu^{11} - 9083 \nu^{10} + 73903 \nu^{9} + 71367 \nu^{8} + \cdots - 10039 ) / 4502 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 987 \nu^{13} - 502 \nu^{12} + 21390 \nu^{11} + 11231 \nu^{10} - 173186 \nu^{9} - 88105 \nu^{8} + \cdots + 9003 ) / 9004 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 1015 \nu^{13} + 1176 \nu^{12} + 20560 \nu^{11} - 22391 \nu^{10} - 155174 \nu^{9} + 161763 \nu^{8} + \cdots + 60313 ) / 9004 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 870 \nu^{13} + 1243 \nu^{12} - 18266 \nu^{11} - 27114 \nu^{10} + 140724 \nu^{9} + 214753 \nu^{8} + \cdots + 39951 ) / 4502 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 1749 \nu^{13} - 178 \nu^{12} + 36426 \nu^{11} + 6601 \nu^{10} - 280482 \nu^{9} - 64279 \nu^{8} + \cdots + 16501 ) / 9004 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 2071 \nu^{13} + 1140 \nu^{12} - 42638 \nu^{11} - 27379 \nu^{10} + 320954 \nu^{9} + 229647 \nu^{8} + \cdots - 5549 ) / 9004 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 1048 \nu^{13} + 866 \nu^{12} - 21672 \nu^{11} - 19823 \nu^{10} + 162887 \nu^{9} + 161810 \nu^{8} + \cdots + 8647 ) / 4502 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 2179 \nu^{13} + 456 \nu^{12} - 45868 \nu^{11} - 14103 \nu^{10} + 358884 \nu^{9} + 137329 \nu^{8} + \cdots + 12637 ) / 9004 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 2205 \nu^{13} + 1792 \nu^{12} - 46062 \nu^{11} - 41087 \nu^{10} + 353092 \nu^{9} + 334285 \nu^{8} + \cdots - 23753 ) / 9004 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 2353 \nu^{13} + 1605 \nu^{12} - 49071 \nu^{11} - 37984 \nu^{10} + 376224 \nu^{9} + 318491 \nu^{8} + \cdots + 31432 ) / 4502 \) 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
\(\nu^{3}\)\(=\) \( \beta_{3} + 5\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{13} - \beta_{12} - \beta_{11} - \beta_{9} - \beta_{8} + \beta_{3} + 7\beta_{2} + \beta _1 + 14 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -\beta_{12} + 2\beta_{9} - \beta_{7} + \beta_{6} + \beta_{4} + 9\beta_{3} - \beta_{2} + 30\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 10 \beta_{13} - 10 \beta_{12} - 10 \beta_{11} + 2 \beta_{10} - 13 \beta_{9} - 9 \beta_{8} + \beta_{7} + \cdots + 77 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( \beta_{13} - 12 \beta_{12} - \beta_{11} - \beta_{10} + 25 \beta_{9} + \beta_{8} - 13 \beta_{7} + \cdots + 5 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 79 \beta_{13} - 81 \beta_{12} - 78 \beta_{11} + 28 \beta_{10} - 120 \beta_{9} - 64 \beta_{8} + \cdots + 467 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 17 \beta_{13} - 108 \beta_{12} - 17 \beta_{11} - 17 \beta_{10} + 229 \beta_{9} + 18 \beta_{8} + \cdots + 83 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 580 \beta_{13} - 620 \beta_{12} - 568 \beta_{11} + 282 \beta_{10} - 974 \beta_{9} - 421 \beta_{8} + \cdots + 3005 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 197 \beta_{13} - 885 \beta_{12} - 200 \beta_{11} - 186 \beta_{10} + 1868 \beta_{9} + 217 \beta_{8} + \cdots + 924 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 4146 \beta_{13} - 4657 \beta_{12} - 4057 \beta_{11} + 2493 \beta_{10} - 7429 \beta_{9} - 2663 \beta_{8} + \cdots + 20032 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 1938 \beta_{13} - 6985 \beta_{12} - 2001 \beta_{11} - 1665 \beta_{10} + 14393 \beta_{9} + 2214 \beta_{8} + \cdots + 8707 \) 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
2.71911
2.63791
1.86457
1.68003
1.64154
0.913038
0.170473
−0.207817
−0.724018
−1.08303
−1.67687
−1.98662
−2.31666
−2.63166
−2.71911 −1.00000 5.39357 −2.29773 2.71911 1.00000 −9.22750 1.00000 6.24777
1.2 −2.63791 −1.00000 4.95855 −0.496275 2.63791 1.00000 −7.80439 1.00000 1.30913
1.3 −1.86457 −1.00000 1.47663 3.72897 1.86457 1.00000 0.975863 1.00000 −6.95294
1.4 −1.68003 −1.00000 0.822491 0.771577 1.68003 1.00000 1.97825 1.00000 −1.29627
1.5 −1.64154 −1.00000 0.694642 −4.21904 1.64154 1.00000 2.14279 1.00000 6.92570
1.6 −0.913038 −1.00000 −1.16636 0.426318 0.913038 1.00000 2.89101 1.00000 −0.389245
1.7 −0.170473 −1.00000 −1.97094 −2.73568 0.170473 1.00000 0.676939 1.00000 0.466361
1.8 0.207817 −1.00000 −1.95681 3.43880 −0.207817 1.00000 −0.822295 1.00000 0.714642
1.9 0.724018 −1.00000 −1.47580 −0.789548 −0.724018 1.00000 −2.51654 1.00000 −0.571646
1.10 1.08303 −1.00000 −0.827049 1.51909 −1.08303 1.00000 −3.06177 1.00000 1.64522
1.11 1.67687 −1.00000 0.811885 −3.65237 −1.67687 1.00000 −1.99231 1.00000 −6.12454
1.12 1.98662 −1.00000 1.94664 −1.69822 −1.98662 1.00000 −0.106000 1.00000 −3.37371
1.13 2.31666 −1.00000 3.36690 2.44310 −2.31666 1.00000 3.16663 1.00000 5.65982
1.14 2.63166 −1.00000 4.92565 2.56101 −2.63166 1.00000 7.69933 1.00000 6.73971
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.14
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(1\)
\(7\) \(-1\)
\(13\) \(-1\)
\(17\) \(-1\)

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 4641.2.a.w ✓ 14
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4641.2.a.w ✓ 14 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(4641))\):

\( T_{2}^{14} + T_{2}^{13} - 22 T_{2}^{12} - 19 T_{2}^{11} + 187 T_{2}^{10} + 135 T_{2}^{9} - 776 T_{2}^{8} + \cdots - 19 \) Copy content Toggle raw display
\( T_{5}^{14} + T_{5}^{13} - 44 T_{5}^{12} - 33 T_{5}^{11} + 728 T_{5}^{10} + 395 T_{5}^{9} - 5657 T_{5}^{8} + \cdots - 2584 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{14} + T^{13} + \cdots - 19 \) Copy content Toggle raw display
$3$ \( (T + 1)^{14} \) Copy content Toggle raw display
$5$ \( T^{14} + T^{13} + \cdots - 2584 \) Copy content Toggle raw display
$7$ \( (T - 1)^{14} \) Copy content Toggle raw display
$11$ \( T^{14} + 4 T^{13} + \cdots + 203776 \) Copy content Toggle raw display
$13$ \( (T - 1)^{14} \) Copy content Toggle raw display
$17$ \( (T - 1)^{14} \) Copy content Toggle raw display
$19$ \( T^{14} - 6 T^{13} + \cdots - 686848 \) Copy content Toggle raw display
$23$ \( T^{14} - 7 T^{13} + \cdots + 41725952 \) Copy content Toggle raw display
$29$ \( T^{14} + \cdots - 822123688 \) Copy content Toggle raw display
$31$ \( T^{14} - 31 T^{13} + \cdots + 32768 \) Copy content Toggle raw display
$37$ \( T^{14} - 2 T^{13} + \cdots + 411616 \) Copy content Toggle raw display
$41$ \( T^{14} - 4 T^{13} + \cdots - 30152104 \) Copy content Toggle raw display
$43$ \( T^{14} + \cdots + 6256400384 \) Copy content Toggle raw display
$47$ \( T^{14} + \cdots - 92618445824 \) Copy content Toggle raw display
$53$ \( T^{14} + \cdots + 1291452416 \) Copy content Toggle raw display
$59$ \( T^{14} + \cdots + 104123008 \) Copy content Toggle raw display
$61$ \( T^{14} - 25 T^{13} + \cdots + 95612104 \) Copy content Toggle raw display
$67$ \( T^{14} + \cdots + 264585673216 \) Copy content Toggle raw display
$71$ \( T^{14} + \cdots + 1179287552 \) Copy content Toggle raw display
$73$ \( T^{14} + \cdots - 71429149696 \) Copy content Toggle raw display
$79$ \( T^{14} + \cdots + 12665179000832 \) Copy content Toggle raw display
$83$ \( T^{14} + \cdots + 565001157248 \) Copy content Toggle raw display
$89$ \( T^{14} + \cdots - 543686656 \) Copy content Toggle raw display
$97$ \( T^{14} + \cdots + 4204057158656 \) Copy content Toggle raw display
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