Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1175,2,Mod(1,1175)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1175, 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("1175.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 1175 = 5^{2} \cdot 47 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1175.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: | \(9.38242223750\) |
| Analytic rank: | \(0\) |
| Dimension: | \(13\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{13} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{13} - 23 x^{11} - x^{10} + 200 x^{9} + 11 x^{8} - 816 x^{7} - 19 x^{6} + 1581 x^{5} - 102 x^{4} + \cdots - 117 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 1 \) |
| 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 \(1,\beta_1,\ldots,\beta_{12}\) 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
\( x^{13} - 23 x^{11} - x^{10} + 200 x^{9} + 11 x^{8} - 816 x^{7} - 19 x^{6} + 1581 x^{5} - 102 x^{4} + \cdots - 117 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 4 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 6 \nu^{12} + 193 \nu^{11} - 302 \nu^{10} - 4069 \nu^{9} + 4014 \nu^{8} + 31611 \nu^{7} - 21664 \nu^{6} + \cdots + 14016 ) / 1211 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 149 \nu^{12} - 266 \nu^{11} + 3463 \nu^{10} + 5868 \nu^{9} - 30135 \nu^{8} - 46902 \nu^{7} + \cdots - 34069 ) / 1211 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 149 \nu^{12} + 266 \nu^{11} - 3463 \nu^{10} - 5868 \nu^{9} + 30135 \nu^{8} + 46902 \nu^{7} + \cdots + 43757 ) / 1211 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 155 \nu^{12} - 286 \nu^{11} + 3765 \nu^{10} + 6477 \nu^{9} - 34322 \nu^{8} - 53082 \nu^{7} + \cdots - 54659 ) / 1211 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - 522 \nu^{12} - 183 \nu^{11} + 11742 \nu^{10} + 4889 \nu^{9} - 98195 \nu^{8} - 44783 \nu^{7} + \cdots - 76381 ) / 1211 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( 578 \nu^{12} + 312 \nu^{11} - 12946 \nu^{10} - 7805 \nu^{9} + 107460 \nu^{8} + 68555 \nu^{7} + \cdots + 81426 ) / 1211 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( 578 \nu^{12} + 312 \nu^{11} - 12946 \nu^{10} - 7805 \nu^{9} + 107460 \nu^{8} + 68555 \nu^{7} + \cdots + 81426 ) / 1211 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( 570 \nu^{12} + 516 \nu^{11} - 12947 \nu^{10} - 12010 \nu^{9} + 109720 \nu^{8} + 100278 \nu^{7} + \cdots + 115849 ) / 1211 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( 648 \nu^{12} + 430 \nu^{11} - 14451 \nu^{10} - 10585 \nu^{9} + 119690 \nu^{8} + 92215 \nu^{7} + \cdots + 102394 ) / 1211 \)
|
| \(\beta_{12}\) | \(=\) |
\( ( 842 \nu^{12} + 500 \nu^{11} - 18968 \nu^{10} - 12284 \nu^{9} + 159170 \nu^{8} + 106752 \nu^{7} + \cdots + 157505 ) / 1211 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{9} - \beta_{8} + 6\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{5} + \beta_{4} + 8\beta_{2} + \beta _1 + 24 \)
|
| \(\nu^{5}\) | \(=\) |
\( -\beta_{12} + 2\beta_{10} + 9\beta_{9} - 9\beta_{8} - \beta_{5} + \beta_{4} + 39\beta _1 + 3 \)
|
| \(\nu^{6}\) | \(=\) |
\( - 2 \beta_{12} + \beta_{11} + \beta_{10} + \beta_{9} - 2 \beta_{8} - 2 \beta_{7} + 11 \beta_{5} + \cdots + 157 \)
|
| \(\nu^{7}\) | \(=\) |
\( - 10 \beta_{12} + 24 \beta_{10} + 69 \beta_{9} - 70 \beta_{8} + \beta_{7} + \beta_{6} - 14 \beta_{5} + \cdots + 45 \)
|
| \(\nu^{8}\) | \(=\) |
\( - 31 \beta_{12} + 16 \beta_{11} + 18 \beta_{10} + 14 \beta_{9} - 32 \beta_{8} - 31 \beta_{7} + \cdots + 1064 \)
|
| \(\nu^{9}\) | \(=\) |
\( - 77 \beta_{12} - 3 \beta_{11} + 219 \beta_{10} + 507 \beta_{9} - 522 \beta_{8} + 15 \beta_{7} + \cdots + 489 \)
|
| \(\nu^{10}\) | \(=\) |
\( - 331 \beta_{12} + 177 \beta_{11} + 212 \beta_{10} + 138 \beta_{9} - 349 \beta_{8} - 328 \beta_{7} + \cdots + 7361 \)
|
| \(\nu^{11}\) | \(=\) |
\( - 551 \beta_{12} - 59 \beta_{11} + 1815 \beta_{10} + 3671 \beta_{9} - 3827 \beta_{8} + 152 \beta_{7} + \cdots + 4644 \)
|
| \(\nu^{12}\) | \(=\) |
\( - 3043 \beta_{12} + 1679 \beta_{11} + 2112 \beta_{10} + 1193 \beta_{9} - 3268 \beta_{8} - 2977 \beta_{7} + \cdots + 51641 \)
|
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 |
|
−2.63214 | −2.36135 | 4.92818 | 0 | 6.21541 | 2.69509 | −7.70738 | 2.57596 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −2.43775 | −0.173250 | 3.94261 | 0 | 0.422339 | −2.26378 | −4.73559 | −2.96998 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −2.30741 | 2.74069 | 3.32413 | 0 | −6.32389 | −4.14817 | −3.05531 | 4.51138 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −1.51512 | −3.07956 | 0.295574 | 0 | 4.66590 | −2.65977 | 2.58240 | 6.48372 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | −1.41853 | 3.19052 | 0.0122215 | 0 | −4.52584 | 4.71614 | 2.81972 | 7.17942 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | −0.684957 | −1.97644 | −1.53083 | 0 | 1.35378 | 4.30619 | 2.41847 | 0.906331 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 0.452387 | −0.461453 | −1.79535 | 0 | −0.208755 | −1.59927 | −1.71697 | −2.78706 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 0.649794 | 1.04162 | −1.57777 | 0 | 0.676842 | 4.46925 | −2.32481 | −1.91502 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | 0.719439 | −2.00949 | −1.48241 | 0 | −1.44570 | −3.86564 | −2.50538 | 1.03803 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.10 | 1.63744 | 3.34240 | 0.681219 | 0 | 5.47299 | 0.857318 | −2.15943 | 8.17165 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.11 | 2.09045 | 0.807628 | 2.36999 | 0 | 1.68831 | 0.424369 | 0.773439 | −2.34774 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.12 | 2.70217 | 1.87663 | 5.30175 | 0 | 5.07098 | 0.754185 | 8.92190 | 0.521743 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.13 | 2.74421 | −2.93795 | 5.53069 | 0 | −8.06236 | −3.68590 | 9.68894 | 5.63156 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(5\) | \(1\) |
| \(47\) | \(-1\) |
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 | 1175.2.a.l | yes | 13 |
| 5.b | even | 2 | 1 | 1175.2.a.k | ✓ | 13 | |
| 5.c | odd | 4 | 2 | 1175.2.c.h | 26 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 1175.2.a.k | ✓ | 13 | 5.b | even | 2 | 1 | |
| 1175.2.a.l | yes | 13 | 1.a | even | 1 | 1 | trivial |
| 1175.2.c.h | 26 | 5.c | odd | 4 | 2 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1175))\):
|
\( T_{2}^{13} - 23 T_{2}^{11} - T_{2}^{10} + 200 T_{2}^{9} + 11 T_{2}^{8} - 816 T_{2}^{7} - 19 T_{2}^{6} + \cdots - 117 \)
|
|
\( T_{11}^{13} - 9 T_{11}^{12} - 66 T_{11}^{11} + 781 T_{11}^{10} + 612 T_{11}^{9} - 22396 T_{11}^{8} + \cdots - 162816 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{13} - 23 T^{11} + \cdots - 117 \)
|
| $3$ |
\( T^{13} - 33 T^{11} + \cdots + 313 \)
|
| $5$ |
\( T^{13} \)
|
| $7$ |
\( T^{13} - 65 T^{11} + \cdots - 38201 \)
|
| $11$ |
\( T^{13} - 9 T^{12} + \cdots - 162816 \)
|
| $13$ |
\( T^{13} + 2 T^{12} + \cdots + 2993152 \)
|
| $17$ |
\( T^{13} - 5 T^{12} + \cdots + 965253 \)
|
| $19$ |
\( T^{13} + \cdots + 137098240 \)
|
| $23$ |
\( T^{13} + 10 T^{12} + \cdots + 1575936 \)
|
| $29$ |
\( T^{13} + \cdots + 334586880 \)
|
| $31$ |
\( T^{13} + \cdots - 1721648128 \)
|
| $37$ |
\( T^{13} + \cdots + 121578796 \)
|
| $41$ |
\( T^{13} + \cdots + 11649862656 \)
|
| $43$ |
\( T^{13} + \cdots + 23218762752 \)
|
| $47$ |
\( (T - 1)^{13} \)
|
| $53$ |
\( T^{13} + \cdots + 4070026788 \)
|
| $59$ |
\( T^{13} + 25 T^{12} + \cdots + 2096085 \)
|
| $61$ |
\( T^{13} + \cdots + 12115114912 \)
|
| $67$ |
\( T^{13} + \cdots + 31152864256 \)
|
| $71$ |
\( T^{13} + \cdots + 1045573191 \)
|
| $73$ |
\( T^{13} + 3 T^{12} + \cdots + 93979648 \)
|
| $79$ |
\( T^{13} + \cdots + 23780854640 \)
|
| $83$ |
\( T^{13} + \cdots + 120218439213 \)
|
| $89$ |
\( T^{13} + \cdots - 130290180 \)
|
| $97$ |
\( T^{13} - 25 T^{12} + \cdots + 7116012 \)
|
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