Properties

Label 627.4.a.g
Level $627$
Weight $4$
Character orbit 627.a
Self dual yes
Analytic conductor $36.994$
Analytic rank $0$
Dimension $12$
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] = [627,4,Mod(1,627)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(627, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("627.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 627 = 3 \cdot 11 \cdot 19 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 627.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: \(36.9941975736\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 6 x^{11} - 49 x^{10} + 299 x^{9} + 819 x^{8} - 5030 x^{7} - 5887 x^{6} + 35567 x^{5} + \cdots - 10032 \) 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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + 3 q^{3} + (\beta_{2} + \beta_1 + 3) q^{4} + ( - \beta_{4} + 4) q^{5} + 3 \beta_1 q^{6} + (\beta_{10} - 1) q^{7} + (\beta_{3} + \beta_{2} + 4 \beta_1 + 7) q^{8} + 9 q^{9} + ( - \beta_{7} + 7 \beta_1) q^{10}+ \cdots - 99 q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 6 q^{2} + 36 q^{3} + 38 q^{4} + 49 q^{5} + 18 q^{6} - 18 q^{7} + 105 q^{8} + 108 q^{9} + 38 q^{10} - 132 q^{11} + 114 q^{12} + 120 q^{13} + 37 q^{14} + 147 q^{15} + 254 q^{16} + 141 q^{17} + 54 q^{18}+ \cdots - 1188 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 6 x^{11} - 49 x^{10} + 299 x^{9} + 819 x^{8} - 5030 x^{7} - 5887 x^{6} + 35567 x^{5} + \cdots - 10032 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - \nu - 11 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} - \nu^{2} - 19\nu + 4 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 40499 \nu^{11} + 451503 \nu^{10} + 441637 \nu^{9} - 19158705 \nu^{8} + 31392696 \nu^{7} + \cdots - 1125801176 ) / 83950984 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 133285 \nu^{11} + 546897 \nu^{10} + 8723948 \nu^{9} - 33103661 \nu^{8} - 198047964 \nu^{7} + \cdots - 7081979704 ) / 83950984 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 182051 \nu^{11} + 1005316 \nu^{10} + 8291312 \nu^{9} - 43058907 \nu^{8} - 126970879 \nu^{7} + \cdots + 169520944 ) / 83950984 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 208509 \nu^{11} - 1542814 \nu^{10} - 7049504 \nu^{9} + 64561377 \nu^{8} + 45861395 \nu^{7} + \cdots - 406285968 ) / 83950984 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 527181 \nu^{11} + 4946019 \nu^{10} + 10683164 \nu^{9} - 205200553 \nu^{8} + 207971874 \nu^{7} + \cdots - 2834367680 ) / 167901968 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 304631 \nu^{11} - 3489200 \nu^{10} - 1950215 \nu^{9} + 142982941 \nu^{8} - 303550057 \nu^{7} + \cdots - 5173795208 ) / 83950984 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 336723 \nu^{11} + 2547519 \nu^{10} + 11553408 \nu^{9} - 111363341 \nu^{8} - 70575584 \nu^{7} + \cdots - 3037611384 ) / 83950984 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 947447 \nu^{11} + 6612671 \nu^{10} + 33999568 \nu^{9} - 281943307 \nu^{8} - 244837036 \nu^{7} + \cdots + 3639645200 ) / 167901968 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + \beta _1 + 11 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} + \beta_{2} + 20\beta _1 + 7 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -\beta_{9} - 2\beta_{8} + \beta_{6} + \beta_{4} + 2\beta_{3} + 28\beta_{2} + 37\beta _1 + 216 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -\beta_{11} - \beta_{9} - \beta_{8} + 2\beta_{7} + 4\beta_{6} + 3\beta_{4} + 37\beta_{3} + 41\beta_{2} + 479\beta _1 + 288 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 2 \beta_{11} - 8 \beta_{10} - 39 \beta_{9} - 74 \beta_{8} + 3 \beta_{7} + 50 \beta_{6} + 3 \beta_{5} + \cdots + 5113 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 44 \beta_{11} - 44 \beta_{10} - 71 \beta_{9} - 74 \beta_{8} + 82 \beta_{7} + 221 \beta_{6} + \cdots + 9516 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 135 \beta_{11} - 540 \beta_{10} - 1278 \beta_{9} - 2229 \beta_{8} + 114 \beta_{7} + 1911 \beta_{6} + \cdots + 131163 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 1633 \beta_{11} - 2992 \beta_{10} - 3323 \beta_{9} - 3415 \beta_{8} + 2307 \beta_{7} + 8985 \beta_{6} + \cdots + 300143 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 6377 \beta_{11} - 25068 \beta_{10} - 40427 \beta_{9} - 63983 \beta_{8} + 2429 \beta_{7} + \cdots + 3501013 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 57865 \beta_{11} - 140048 \beta_{10} - 131768 \beta_{9} - 131153 \beta_{8} + 53149 \beta_{7} + \cdots + 9367296 \) 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
−4.90542
−3.07205
−2.82927
−2.70585
−0.886154
0.173827
0.929351
2.07111
2.58173
4.09251
5.01354
5.53667
−4.90542 3.00000 16.0632 10.5947 −14.7163 2.99167 −39.5534 9.00000 −51.9717
1.2 −3.07205 3.00000 1.43747 18.9795 −9.21614 9.05552 20.1604 9.00000 −58.3059
1.3 −2.82927 3.00000 0.00475762 −7.98443 −8.48780 19.3632 22.6207 9.00000 22.5901
1.4 −2.70585 3.00000 −0.678365 −4.71721 −8.11756 −33.7180 23.4824 9.00000 12.7641
1.5 −0.886154 3.00000 −7.21473 −7.23582 −2.65846 −8.56440 13.4826 9.00000 6.41205
1.6 0.173827 3.00000 −7.96978 21.5510 0.521482 −36.0850 −2.77599 9.00000 3.74615
1.7 0.929351 3.00000 −7.13631 7.66653 2.78805 27.2467 −14.0669 9.00000 7.12489
1.8 2.07111 3.00000 −3.71049 −21.5435 6.21334 −11.0349 −24.2538 9.00000 −44.6190
1.9 2.58173 3.00000 −1.33465 1.36032 7.74520 −0.566774 −24.0996 9.00000 3.51198
1.10 4.09251 3.00000 8.74866 19.7775 12.2775 17.0797 3.06389 9.00000 80.9397
1.11 5.01354 3.00000 17.1356 4.99216 15.0406 15.8989 45.8014 9.00000 25.0284
1.12 5.53667 3.00000 22.6547 5.55919 16.6100 −19.6665 81.1382 9.00000 30.7794
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.12
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \( -1 \)
\(11\) \( +1 \)
\(19\) \( -1 \)

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 627.4.a.g 12
3.b odd 2 1 1881.4.a.g 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
627.4.a.g 12 1.a even 1 1 trivial
1881.4.a.g 12 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{12} - 6 T_{2}^{11} - 49 T_{2}^{10} + 299 T_{2}^{9} + 819 T_{2}^{8} - 5030 T_{2}^{7} + \cdots - 10032 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(627))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} - 6 T^{11} + \cdots - 10032 \) Copy content Toggle raw display
$3$ \( (T - 3)^{12} \) Copy content Toggle raw display
$5$ \( T^{12} + \cdots + 145641974016 \) Copy content Toggle raw display
$7$ \( T^{12} + \cdots + 4974597007616 \) Copy content Toggle raw display
$11$ \( (T + 11)^{12} \) Copy content Toggle raw display
$13$ \( T^{12} + \cdots + 24\!\cdots\!00 \) Copy content Toggle raw display
$17$ \( T^{12} + \cdots + 24\!\cdots\!60 \) Copy content Toggle raw display
$19$ \( (T - 19)^{12} \) Copy content Toggle raw display
$23$ \( T^{12} + \cdots + 14\!\cdots\!84 \) Copy content Toggle raw display
$29$ \( T^{12} + \cdots - 17\!\cdots\!60 \) Copy content Toggle raw display
$31$ \( T^{12} + \cdots + 20\!\cdots\!32 \) Copy content Toggle raw display
$37$ \( T^{12} + \cdots + 22\!\cdots\!64 \) Copy content Toggle raw display
$41$ \( T^{12} + \cdots - 26\!\cdots\!84 \) Copy content Toggle raw display
$43$ \( T^{12} + \cdots + 16\!\cdots\!28 \) Copy content Toggle raw display
$47$ \( T^{12} + \cdots - 89\!\cdots\!84 \) Copy content Toggle raw display
$53$ \( T^{12} + \cdots + 34\!\cdots\!92 \) Copy content Toggle raw display
$59$ \( T^{12} + \cdots + 17\!\cdots\!76 \) Copy content Toggle raw display
$61$ \( T^{12} + \cdots - 98\!\cdots\!80 \) Copy content Toggle raw display
$67$ \( T^{12} + \cdots + 31\!\cdots\!40 \) Copy content Toggle raw display
$71$ \( T^{12} + \cdots - 28\!\cdots\!60 \) Copy content Toggle raw display
$73$ \( T^{12} + \cdots - 36\!\cdots\!64 \) Copy content Toggle raw display
$79$ \( T^{12} + \cdots - 53\!\cdots\!96 \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots + 75\!\cdots\!48 \) Copy content Toggle raw display
$89$ \( T^{12} + \cdots + 29\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{12} + \cdots - 30\!\cdots\!04 \) Copy content Toggle raw display
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