Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [74,18,Mod(1,74)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(74, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0]))
N = Newforms(chi, 18, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("74.1");
S:= CuspForms(chi, 18);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 74 = 2 \cdot 37 \) |
| Weight: | \( k \) | \(=\) | \( 18 \) |
| Character orbit: | \([\chi]\) | \(=\) | 74.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: | \(135.584344635\) |
| 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: |
\( x^{14} - 5 x^{13} - 1336450185 x^{12} + 1999539874104 x^{11} + \cdots - 10\!\cdots\!60 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{19}]\) |
| Coefficient ring index: | multiple of \( 2^{27}\cdot 3^{10}\cdot 7 \) |
| 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.
Basis of coefficient ring in terms of a root \(\nu\) of
\( x^{14} - 5 x^{13} - 1336450185 x^{12} + 1999539874104 x^{11} + \cdots - 10\!\cdots\!60 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 25\!\cdots\!87 \nu^{13} + \cdots + 30\!\cdots\!20 ) / 53\!\cdots\!00 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 43\!\cdots\!79 \nu^{13} + \cdots - 51\!\cdots\!40 ) / 26\!\cdots\!00 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 14\!\cdots\!44 \nu^{13} + \cdots - 87\!\cdots\!60 ) / 48\!\cdots\!00 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 19\!\cdots\!58 \nu^{13} + \cdots - 81\!\cdots\!20 ) / 48\!\cdots\!00 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 14\!\cdots\!40 \nu^{13} + \cdots + 30\!\cdots\!80 ) / 32\!\cdots\!20 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - 98\!\cdots\!09 \nu^{13} + \cdots + 38\!\cdots\!60 ) / 16\!\cdots\!00 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( - 93\!\cdots\!97 \nu^{13} + \cdots + 27\!\cdots\!80 ) / 96\!\cdots\!60 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( 16\!\cdots\!63 \nu^{13} + \cdots + 15\!\cdots\!80 ) / 16\!\cdots\!00 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( 18\!\cdots\!61 \nu^{13} + \cdots + 52\!\cdots\!20 ) / 13\!\cdots\!80 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( 89\!\cdots\!67 \nu^{13} + \cdots + 10\!\cdots\!20 ) / 48\!\cdots\!00 \)
|
| \(\beta_{12}\) | \(=\) |
\( ( 10\!\cdots\!02 \nu^{13} + \cdots - 30\!\cdots\!80 ) / 48\!\cdots\!00 \)
|
| \(\beta_{13}\) | \(=\) |
\( ( - 32\!\cdots\!29 \nu^{13} + \cdots + 42\!\cdots\!20 ) / 96\!\cdots\!60 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{3} + 34\beta_{2} - 2253\beta _1 + 190922276 \)
|
| \(\nu^{3}\) | \(=\) |
\( 474 \beta_{13} - 462 \beta_{12} + 275 \beta_{11} + 1202 \beta_{10} - 309 \beta_{9} - 493 \beta_{8} + \cdots - 427153887263 \)
|
| \(\nu^{4}\) | \(=\) |
\( 1346249 \beta_{13} + 6004886 \beta_{12} - 3158325 \beta_{11} + 8109385 \beta_{10} + \cdots + 60\!\cdots\!92 \)
|
| \(\nu^{5}\) | \(=\) |
\( 247469192713 \beta_{13} - 278119446743 \beta_{12} + 142117400388 \beta_{11} + 635866315565 \beta_{10} + \cdots - 33\!\cdots\!10 \)
|
| \(\nu^{6}\) | \(=\) |
\( 11\!\cdots\!90 \beta_{13} + \cdots + 21\!\cdots\!04 \)
|
| \(\nu^{7}\) | \(=\) |
\( 11\!\cdots\!32 \beta_{13} + \cdots - 17\!\cdots\!06 \)
|
| \(\nu^{8}\) | \(=\) |
\( 53\!\cdots\!13 \beta_{13} + \cdots + 81\!\cdots\!75 \)
|
| \(\nu^{9}\) | \(=\) |
\( 46\!\cdots\!48 \beta_{13} + \cdots - 83\!\cdots\!74 \)
|
| \(\nu^{10}\) | \(=\) |
\( 18\!\cdots\!06 \beta_{13} + \cdots + 31\!\cdots\!16 \)
|
| \(\nu^{11}\) | \(=\) |
\( 18\!\cdots\!92 \beta_{13} + \cdots - 37\!\cdots\!03 \)
|
| \(\nu^{12}\) | \(=\) |
\( 54\!\cdots\!92 \beta_{13} + \cdots + 12\!\cdots\!35 \)
|
| \(\nu^{13}\) | \(=\) |
\( 74\!\cdots\!75 \beta_{13} + \cdots - 16\!\cdots\!46 \)
|
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 |
|
256.000 | −20021.5 | 65536.0 | −554011. | −5.12551e6 | 1.26616e7 | 1.67772e7 | 2.71722e8 | −1.41827e8 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | 256.000 | −19402.5 | 65536.0 | 452002. | −4.96703e6 | −2.20899e7 | 1.67772e7 | 2.47315e8 | 1.15713e8 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | 256.000 | −19317.6 | 65536.0 | 1.29187e6 | −4.94530e6 | −8.25640e6 | 1.67772e7 | 2.44029e8 | 3.30719e8 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | 256.000 | −9171.52 | 65536.0 | −636083. | −2.34791e6 | −1.18196e7 | 1.67772e7 | −4.50235e7 | −1.62837e8 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | 256.000 | −5823.71 | 65536.0 | 1.19281e6 | −1.49087e6 | 1.68109e7 | 1.67772e7 | −9.52245e7 | 3.05360e8 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 256.000 | −4756.78 | 65536.0 | 401290. | −1.21774e6 | −6.32728e6 | 1.67772e7 | −1.06513e8 | 1.02730e8 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 256.000 | −1901.42 | 65536.0 | −1.32899e6 | −486763. | 2.02145e7 | 1.67772e7 | −1.25525e8 | −3.40221e8 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 256.000 | 871.168 | 65536.0 | 323928. | 223019. | 1.49416e7 | 1.67772e7 | −1.28381e8 | 8.29257e7 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | 256.000 | 9364.09 | 65536.0 | −1.64194e6 | 2.39721e6 | −2.06556e7 | 1.67772e7 | −4.14540e7 | −4.20338e8 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.10 | 256.000 | 10661.5 | 65536.0 | −422211. | 2.72935e6 | −2.47188e7 | 1.67772e7 | −1.54721e7 | −1.08086e8 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.11 | 256.000 | 14721.2 | 65536.0 | 1.30106e6 | 3.76862e6 | 1.06371e7 | 1.67772e7 | 8.75733e7 | 3.33071e8 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.12 | 256.000 | 16056.1 | 65536.0 | 1.43142e6 | 4.11036e6 | 9.87024e6 | 1.67772e7 | 1.28659e8 | 3.66443e8 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.13 | 256.000 | 17852.4 | 65536.0 | −902149. | 4.57022e6 | 2.80074e7 | 1.67772e7 | 1.89568e8 | −2.30950e8 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.14 | 256.000 | 19707.5 | 65536.0 | 268386. | 5.04513e6 | 1.34319e6 | 1.67772e7 | 2.59246e8 | 6.87068e7 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(-1\) |
| \(37\) | \(1\) |
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 | 74.18.a.d | ✓ | 14 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 74.18.a.d | ✓ | 14 | 1.a | even | 1 | 1 | trivial |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{14} - 8839 T_{3}^{13} - 1300176519 T_{3}^{12} + 12027534181010 T_{3}^{11} + \cdots - 26\!\cdots\!00 \)
acting on \(S_{18}^{\mathrm{new}}(\Gamma_0(74))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T - 256)^{14} \)
|
| $3$ |
\( T^{14} + \cdots - 26\!\cdots\!00 \)
|
| $5$ |
\( T^{14} + \cdots + 13\!\cdots\!00 \)
|
| $7$ |
\( T^{14} + \cdots + 17\!\cdots\!20 \)
|
| $11$ |
\( T^{14} + \cdots + 21\!\cdots\!00 \)
|
| $13$ |
\( T^{14} + \cdots - 90\!\cdots\!56 \)
|
| $17$ |
\( T^{14} + \cdots + 24\!\cdots\!00 \)
|
| $19$ |
\( T^{14} + \cdots + 37\!\cdots\!00 \)
|
| $23$ |
\( T^{14} + \cdots - 15\!\cdots\!36 \)
|
| $29$ |
\( T^{14} + \cdots + 44\!\cdots\!60 \)
|
| $31$ |
\( T^{14} + \cdots - 17\!\cdots\!00 \)
|
| $37$ |
\( (T + 3512479453921)^{14} \)
|
| $41$ |
\( T^{14} + \cdots + 16\!\cdots\!60 \)
|
| $43$ |
\( T^{14} + \cdots - 18\!\cdots\!40 \)
|
| $47$ |
\( T^{14} + \cdots - 10\!\cdots\!28 \)
|
| $53$ |
\( T^{14} + \cdots + 73\!\cdots\!16 \)
|
| $59$ |
\( T^{14} + \cdots - 17\!\cdots\!00 \)
|
| $61$ |
\( T^{14} + \cdots - 13\!\cdots\!00 \)
|
| $67$ |
\( T^{14} + \cdots + 42\!\cdots\!00 \)
|
| $71$ |
\( T^{14} + \cdots + 16\!\cdots\!00 \)
|
| $73$ |
\( T^{14} + \cdots - 10\!\cdots\!92 \)
|
| $79$ |
\( T^{14} + \cdots + 13\!\cdots\!72 \)
|
| $83$ |
\( T^{14} + \cdots + 22\!\cdots\!00 \)
|
| $89$ |
\( T^{14} + \cdots + 90\!\cdots\!00 \)
|
| $97$ |
\( T^{14} + \cdots + 97\!\cdots\!40 \)
|
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