Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [37,8,Mod(1,37)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(37, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0]))
N = Newforms(chi, 8, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("37.1");
S:= CuspForms(chi, 8);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 37 \) |
| Weight: | \( k \) | \(=\) | \( 8 \) |
| Character orbit: | \([\chi]\) | \(=\) | 37.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: | \(11.5582459429\) |
| Analytic rank: | \(1\) |
| Dimension: | \(10\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{10} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{10} - 4 x^{9} - 905 x^{8} + 4018 x^{7} + 291290 x^{6} - 1367036 x^{5} - 39566544 x^{4} + \cdots - 45399525376 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 2^{4}\cdot 3 \) |
| 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 \(1,\beta_1,\ldots,\beta_{9}\) 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
\( x^{10} - 4 x^{9} - 905 x^{8} + 4018 x^{7} + 291290 x^{6} - 1367036 x^{5} - 39566544 x^{4} + \cdots - 45399525376 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( - 6903850151 \nu^{9} - 44030216064 \nu^{8} + 5848661370847 \nu^{7} + 31740379054046 \nu^{6} + \cdots - 29\!\cdots\!24 ) / 52\!\cdots\!28 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 62575071425 \nu^{9} - 287016653328 \nu^{8} + 51295267891465 \nu^{7} + \cdots - 13\!\cdots\!64 ) / 31\!\cdots\!68 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 824744993 \nu^{9} + 5598640392 \nu^{8} - 698144499529 \nu^{7} - 3961595322746 \nu^{6} + \cdots + 34\!\cdots\!60 ) / 372548736817152 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 23822773327 \nu^{9} + 160382537808 \nu^{8} - 20325864531719 \nu^{7} - 115632112359166 \nu^{6} + \cdots + 10\!\cdots\!64 ) / 10\!\cdots\!56 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 169096231607 \nu^{9} - 1115891967696 \nu^{8} + 141291187862959 \nu^{7} + \cdots - 74\!\cdots\!88 ) / 31\!\cdots\!68 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 169096231607 \nu^{9} + 1115891967696 \nu^{8} - 141291187862959 \nu^{7} + \cdots + 74\!\cdots\!44 ) / 31\!\cdots\!68 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( 81089504555 \nu^{9} + 473877845808 \nu^{8} - 67919217844483 \nu^{7} - 344607611020502 \nu^{6} + \cdots + 32\!\cdots\!72 ) / 78\!\cdots\!92 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( 396026017655 \nu^{9} + 2579512909920 \nu^{8} - 331845919248943 \nu^{7} + \cdots + 17\!\cdots\!04 ) / 15\!\cdots\!84 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{7} + \beta_{6} - \beta _1 + 183 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{9} - 6\beta_{7} - 2\beta_{6} - 5\beta_{5} + 2\beta_{4} + \beta_{3} - 4\beta_{2} + 258\beta _1 - 214 \)
|
| \(\nu^{4}\) | \(=\) |
\( - 11 \beta_{9} + \beta_{8} + 374 \beta_{7} + 335 \beta_{6} - 14 \beta_{5} - 3 \beta_{4} + 3 \beta_{3} + \cdots + 46927 \)
|
| \(\nu^{5}\) | \(=\) |
\( 329 \beta_{9} + 123 \beta_{8} - 1888 \beta_{7} - 469 \beta_{6} - 1956 \beta_{5} + 1055 \beta_{4} + \cdots - 111261 \)
|
| \(\nu^{6}\) | \(=\) |
\( - 7185 \beta_{9} + 495 \beta_{8} + 128160 \beta_{7} + 102799 \beta_{6} - 4486 \beta_{5} + \cdots + 13564839 \)
|
| \(\nu^{7}\) | \(=\) |
\( 83577 \beta_{9} + 72597 \beta_{8} - 520180 \beta_{7} - 131975 \beta_{6} - 654390 \beta_{5} + \cdots - 41517167 \)
|
| \(\nu^{8}\) | \(=\) |
\( - 3247261 \beta_{9} + 23355 \beta_{8} + 43190140 \beta_{7} + 31563511 \beta_{6} - 902246 \beta_{5} + \cdots + 4109646639 \)
|
| \(\nu^{9}\) | \(=\) |
\( 17596401 \beta_{9} + 32119609 \beta_{8} - 138258172 \beta_{7} - 46361323 \beta_{6} - 211558658 \beta_{5} + \cdots - 14473144979 \)
|
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 |
|
−19.8339 | −69.2191 | 265.384 | −437.776 | 1372.88 | 1289.75 | −2724.87 | 2604.28 | 8682.81 | ||||||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −18.8983 | 75.2961 | 229.145 | −439.542 | −1422.97 | 380.249 | −1911.48 | 3482.50 | 8306.59 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −12.5631 | −77.5936 | 29.8308 | 318.311 | 974.814 | −75.8943 | 1233.31 | 3833.76 | −3998.96 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −12.4614 | 49.0521 | 27.2868 | −16.1140 | −611.258 | −704.986 | 1255.03 | 219.106 | 200.803 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | −11.5549 | −22.2601 | 5.51464 | 118.339 | 257.212 | 925.681 | 1415.30 | −1691.49 | −1367.39 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 3.27803 | −7.54665 | −117.254 | 243.887 | −24.7382 | 767.318 | −803.953 | −2130.05 | 799.468 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 3.89746 | 51.1724 | −112.810 | −91.5007 | 199.442 | −1347.68 | −938.547 | 431.610 | −356.620 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 12.2020 | 8.53392 | 20.8894 | −415.456 | 104.131 | 362.670 | −1306.97 | −2114.17 | −5069.41 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | 15.7200 | −69.5612 | 119.119 | 316.724 | −1093.50 | −1289.70 | −139.607 | 2651.76 | 4978.91 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.10 | 16.2140 | −32.8739 | 134.894 | −220.871 | −533.017 | −808.413 | 111.778 | −1106.31 | −3581.21 | |||||||||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(37\) | \(-1\) |
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 | 37.8.a.a | ✓ | 10 |
| 3.b | odd | 2 | 1 | 333.8.a.c | 10 | ||
| 4.b | odd | 2 | 1 | 592.8.a.f | 10 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 37.8.a.a | ✓ | 10 | 1.a | even | 1 | 1 | trivial |
| 333.8.a.c | 10 | 3.b | odd | 2 | 1 | ||
| 592.8.a.f | 10 | 4.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{10} + 24 T_{2}^{9} - 653 T_{2}^{8} - 16962 T_{2}^{7} + 139726 T_{2}^{6} + 4135692 T_{2}^{5} + \cdots - 26941953024 \)
acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(37))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{10} + \cdots - 26941953024 \)
|
| $3$ |
\( T^{10} + \cdots + 33\!\cdots\!44 \)
|
| $5$ |
\( T^{10} + \cdots + 75\!\cdots\!00 \)
|
| $7$ |
\( T^{10} + \cdots - 94\!\cdots\!24 \)
|
| $11$ |
\( T^{10} + \cdots - 11\!\cdots\!20 \)
|
| $13$ |
\( T^{10} + \cdots - 11\!\cdots\!12 \)
|
| $17$ |
\( T^{10} + \cdots + 53\!\cdots\!60 \)
|
| $19$ |
\( T^{10} + \cdots + 50\!\cdots\!60 \)
|
| $23$ |
\( T^{10} + \cdots - 83\!\cdots\!12 \)
|
| $29$ |
\( T^{10} + \cdots - 13\!\cdots\!32 \)
|
| $31$ |
\( T^{10} + \cdots + 26\!\cdots\!76 \)
|
| $37$ |
\( (T - 50653)^{10} \)
|
| $41$ |
\( T^{10} + \cdots - 28\!\cdots\!98 \)
|
| $43$ |
\( T^{10} + \cdots + 30\!\cdots\!68 \)
|
| $47$ |
\( T^{10} + \cdots - 32\!\cdots\!64 \)
|
| $53$ |
\( T^{10} + \cdots + 44\!\cdots\!96 \)
|
| $59$ |
\( T^{10} + \cdots - 91\!\cdots\!52 \)
|
| $61$ |
\( T^{10} + \cdots - 19\!\cdots\!16 \)
|
| $67$ |
\( T^{10} + \cdots + 31\!\cdots\!12 \)
|
| $71$ |
\( T^{10} + \cdots + 38\!\cdots\!72 \)
|
| $73$ |
\( T^{10} + \cdots + 32\!\cdots\!06 \)
|
| $79$ |
\( T^{10} + \cdots - 64\!\cdots\!08 \)
|
| $83$ |
\( T^{10} + \cdots - 30\!\cdots\!60 \)
|
| $89$ |
\( T^{10} + \cdots + 74\!\cdots\!12 \)
|
| $97$ |
\( T^{10} + \cdots + 48\!\cdots\!24 \)
|
| show more | |
| show less |