Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [3751,2,Mod(1,3751)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3751, 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("3751.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 3751 = 11^{2} \cdot 31 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 3751.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: | \(29.9518857982\) |
| Analytic rank: | \(1\) |
| Dimension: | \(16\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{16} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{16} - 28x^{14} + 322x^{12} - 1955x^{10} + 6683x^{8} - 12608x^{6} + 11746x^{4} - 3964x^{2} + 289 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| 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 \(1,\beta_1,\ldots,\beta_{15}\) 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
\( x^{16} - 28x^{14} + 322x^{12} - 1955x^{10} + 6683x^{8} - 12608x^{6} + 11746x^{4} - 3964x^{2} + 289 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -16\nu^{15} + 312\nu^{13} - 2194\nu^{11} + 6358\nu^{9} - 4384\nu^{7} - 10721\nu^{5} + 15231\nu^{3} - 3641\nu ) / 459 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 7\nu^{15} - 162\nu^{13} + 1489\nu^{11} - 7021\nu^{9} + 18493\nu^{7} - 27787\nu^{5} + 21396\nu^{3} - 4390\nu ) / 459 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 14\nu^{15} - 324\nu^{13} + 2978\nu^{11} - 13889\nu^{9} + 34691\nu^{7} - 44405\nu^{5} + 23973\nu^{3} - 2660\nu ) / 459 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -14\nu^{15} + 324\nu^{13} - 2978\nu^{11} + 13889\nu^{9} - 34691\nu^{7} + 44405\nu^{5} - 23514\nu^{3} + 365\nu ) / 459 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( \nu^{14} - 24\nu^{12} + 226\nu^{10} - 1051\nu^{8} + 2479\nu^{6} - 2692\nu^{4} + 1005\nu^{2} - 133 ) / 27 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( -2\nu^{14} + 45\nu^{12} - 392\nu^{10} + 1664\nu^{8} - 3557\nu^{6} + 3578\nu^{4} - 1374\nu^{2} + 119 ) / 27 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( - 20 \nu^{15} + 492 \nu^{13} - 4859 \nu^{11} + 24599 \nu^{9} - 67190 \nu^{7} + 94400 \nu^{5} + \cdots + 9206 \nu ) / 459 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( 13\nu^{15} - 296\nu^{13} + 2639\nu^{11} - 11645\nu^{9} + 26342\nu^{7} - 28346\nu^{5} + 11683\nu^{3} - 1722\nu ) / 153 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( 4\nu^{14} - 90\nu^{12} + 793\nu^{10} - 3463\nu^{8} + 7771\nu^{6} - 8290\nu^{4} + 3270\nu^{2} - 346 ) / 27 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( 80 \nu^{15} - 1815 \nu^{13} + 16070 \nu^{11} - 69938 \nu^{9} + 153398 \nu^{7} - 151313 \nu^{5} + \cdots + 661 \nu ) / 459 \)
|
| \(\beta_{12}\) | \(=\) |
\( ( 2\nu^{14} - 45\nu^{12} + 395\nu^{10} - 1706\nu^{8} + 3731\nu^{6} - 3737\nu^{4} + 1188\nu^{2} - 80 ) / 9 \)
|
| \(\beta_{13}\) | \(=\) |
\( ( -5\nu^{14} + 117\nu^{12} - 1079\nu^{10} + 4961\nu^{8} - 11759\nu^{6} + 13148\nu^{4} - 5127\nu^{2} + 446 ) / 27 \)
|
| \(\beta_{14}\) | \(=\) |
\( ( 7\nu^{14} - 159\nu^{12} + 1411\nu^{10} - 6169\nu^{8} + 13672\nu^{6} - 13903\nu^{4} + 4542\nu^{2} - 265 ) / 27 \)
|
| \(\beta_{15}\) | \(=\) |
\( ( 10\nu^{14} - 225\nu^{12} + 1978\nu^{10} - 8590\nu^{8} + 19099\nu^{6} - 20131\nu^{4} + 7725\nu^{2} - 622 ) / 27 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( -\beta_{14} + \beta_{12} + \beta_{6} + 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{5} + \beta_{4} + 5\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{15} - 7\beta_{14} + 7\beta_{12} - 2\beta_{10} + \beta_{7} + 7\beta_{6} + 21 \)
|
| \(\nu^{5}\) | \(=\) |
\( 2\beta_{8} + 7\beta_{5} + 11\beta_{4} + \beta_{2} + 26\beta_1 \)
|
| \(\nu^{6}\) | \(=\) |
\( 11\beta_{15} - 45\beta_{14} + \beta_{13} + 44\beta_{12} - 20\beta_{10} + 11\beta_{7} + 48\beta_{6} + 118 \)
|
| \(\nu^{7}\) | \(=\) |
\( -2\beta_{11} + 5\beta_{9} + 26\beta_{8} + 44\beta_{5} + 94\beta_{4} - \beta_{3} + 13\beta_{2} + 141\beta_1 \)
|
| \(\nu^{8}\) | \(=\) |
\( 92\beta_{15} - 284\beta_{14} + 15\beta_{13} + 272\beta_{12} - 157\beta_{10} + 95\beta_{7} + 329\beta_{6} + 692 \)
|
| \(\nu^{9}\) | \(=\) |
\( -30\beta_{11} + 75\beta_{9} + 244\beta_{8} + 272\beta_{5} + 733\beta_{4} - 21\beta_{3} + 122\beta_{2} + 792\beta_1 \)
|
| \(\nu^{10}\) | \(=\) |
\( 703 \beta_{15} - 1799 \beta_{14} + 152 \beta_{13} + 1692 \beta_{12} - 1144 \beta_{10} + 754 \beta_{7} + \cdots + 4192 \)
|
| \(\nu^{11}\) | \(=\) |
\( - 310 \beta_{11} + 769 \beta_{9} + 2011 \beta_{8} + 1692 \beta_{5} + 5473 \beta_{4} - 263 \beta_{3} + \cdots + 4588 \beta_1 \)
|
| \(\nu^{12}\) | \(=\) |
\( 5163 \beta_{15} - 11529 \beta_{14} + 1317 \beta_{13} + 10674 \beta_{12} - 8094 \beta_{10} + 5736 \beta_{7} + \cdots + 26071 \)
|
| \(\nu^{13}\) | \(=\) |
\( - 2745 \beta_{11} + 6738 \beta_{9} + 15516 \beta_{8} + 10674 \beta_{5} + 39894 \beta_{4} + \cdots + 27334 \beta_1 \)
|
| \(\nu^{14}\) | \(=\) |
\( 37149 \beta_{15} - 74890 \beta_{14} + 10542 \beta_{13} + 68419 \beta_{12} - 56523 \beta_{10} + \cdots + 165727 \)
|
| \(\nu^{15}\) | \(=\) |
\( - 22392 \beta_{11} + 54375 \beta_{9} + 115299 \beta_{8} + 68419 \beta_{5} + 286549 \beta_{4} + \cdots + 167081 \beta_1 \)
|
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 |
|
−2.63537 | 0.271050 | 4.94515 | −1.28967 | −0.714317 | 3.77376 | −7.76156 | −2.92653 | 3.39876 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −2.30742 | −1.71505 | 3.32418 | −0.550514 | 3.95733 | −3.99146 | −3.05543 | −0.0586147 | 1.27027 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −2.28820 | 1.78652 | 3.23585 | −4.21400 | −4.08790 | 0.107652 | −2.82786 | 0.191641 | 9.64247 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −2.16177 | −2.40476 | 2.67324 | −2.05134 | 5.19852 | −1.15020 | −1.45538 | 2.78286 | 4.43451 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | −1.70371 | 0.602944 | 0.902636 | 2.13088 | −1.02724 | 0.429301 | 1.86959 | −2.63646 | −3.63041 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | −1.53178 | 1.35626 | 0.346364 | 0.457920 | −2.07750 | 1.60482 | 2.53301 | −1.16055 | −0.701435 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | −0.688150 | −2.16234 | −1.52645 | 3.01306 | 1.48802 | 1.44396 | 2.42673 | 1.67574 | −2.07344 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | −0.314704 | 2.26537 | −1.90096 | −1.49633 | −0.712923 | −2.15584 | 1.22765 | 2.13192 | 0.470902 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | 0.314704 | 2.26537 | −1.90096 | −1.49633 | 0.712923 | 2.15584 | −1.22765 | 2.13192 | −0.470902 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.10 | 0.688150 | −2.16234 | −1.52645 | 3.01306 | −1.48802 | −1.44396 | −2.42673 | 1.67574 | 2.07344 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.11 | 1.53178 | 1.35626 | 0.346364 | 0.457920 | 2.07750 | −1.60482 | −2.53301 | −1.16055 | 0.701435 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.12 | 1.70371 | 0.602944 | 0.902636 | 2.13088 | 1.02724 | −0.429301 | −1.86959 | −2.63646 | 3.63041 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.13 | 2.16177 | −2.40476 | 2.67324 | −2.05134 | −5.19852 | 1.15020 | 1.45538 | 2.78286 | −4.43451 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.14 | 2.28820 | 1.78652 | 3.23585 | −4.21400 | 4.08790 | −0.107652 | 2.82786 | 0.191641 | −9.64247 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.15 | 2.30742 | −1.71505 | 3.32418 | −0.550514 | −3.95733 | 3.99146 | 3.05543 | −0.0586147 | −1.27027 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.16 | 2.63537 | 0.271050 | 4.94515 | −1.28967 | 0.714317 | −3.77376 | 7.76156 | −2.92653 | −3.39876 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(11\) | \(1\) |
| \(31\) | \(1\) |
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 11.b | odd | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 3751.2.a.n | ✓ | 16 |
| 11.b | odd | 2 | 1 | inner | 3751.2.a.n | ✓ | 16 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 3751.2.a.n | ✓ | 16 | 1.a | even | 1 | 1 | trivial |
| 3751.2.a.n | ✓ | 16 | 11.b | odd | 2 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(3751))\):
|
\( T_{2}^{16} - 28T_{2}^{14} + 322T_{2}^{12} - 1955T_{2}^{10} + 6683T_{2}^{8} - 12608T_{2}^{6} + 11746T_{2}^{4} - 3964T_{2}^{2} + 289 \)
|
|
\( T_{3}^{8} - 12T_{3}^{6} + 3T_{3}^{5} + 46T_{3}^{4} - 24T_{3}^{3} - 54T_{3}^{2} + 45T_{3} - 8 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{16} - 28 T^{14} + \cdots + 289 \)
|
| $3$ |
\( (T^{8} - 12 T^{6} + 3 T^{5} + \cdots - 8)^{2} \)
|
| $5$ |
\( (T^{8} + 4 T^{7} - 12 T^{6} + \cdots - 27)^{2} \)
|
| $7$ |
\( T^{16} - 41 T^{14} + \cdots + 16 \)
|
| $11$ |
\( T^{16} \)
|
| $13$ |
\( T^{16} - 99 T^{14} + \cdots + 301401 \)
|
| $17$ |
\( T^{16} - 82 T^{14} + \cdots + 169 \)
|
| $19$ |
\( T^{16} - 121 T^{14} + \cdots + 47032164 \)
|
| $23$ |
\( (T^{8} + 20 T^{7} + \cdots + 8112)^{2} \)
|
| $29$ |
\( T^{16} + \cdots + 17797427649 \)
|
| $31$ |
\( (T + 1)^{16} \)
|
| $37$ |
\( (T^{8} + 7 T^{7} + \cdots - 55431)^{2} \)
|
| $41$ |
\( T^{16} + \cdots + 169767896841 \)
|
| $43$ |
\( T^{16} + \cdots + 4099328676 \)
|
| $47$ |
\( (T^{8} + 15 T^{7} + \cdots - 185688)^{2} \)
|
| $53$ |
\( (T^{8} + 34 T^{7} + \cdots - 112359)^{2} \)
|
| $59$ |
\( (T^{8} + 3 T^{7} + \cdots + 775384)^{2} \)
|
| $61$ |
\( T^{16} + \cdots + 1498881106944 \)
|
| $67$ |
\( (T^{8} - 22 T^{7} + \cdots + 34882)^{2} \)
|
| $71$ |
\( (T^{8} + 10 T^{7} + \cdots - 37662)^{2} \)
|
| $73$ |
\( T^{16} + \cdots + 498277868544 \)
|
| $79$ |
\( T^{16} + \cdots + 3504181826916 \)
|
| $83$ |
\( T^{16} + \cdots + 61978492060164 \)
|
| $89$ |
\( (T^{8} + 45 T^{7} + \cdots + 10418857)^{2} \)
|
| $97$ |
\( (T^{8} - 11 T^{7} + \cdots - 428777)^{2} \)
|
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