Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [251,2,Mod(1,251)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(251, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("251.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 251 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 251.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: | \(2.00424509073\) |
| Analytic rank: | \(0\) |
| Dimension: | \(17\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{17} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{17} - 2 x^{16} - 28 x^{15} + 54 x^{14} + 317 x^{13} - 582 x^{12} - 1867 x^{11} + 3178 x^{10} + \cdots + 256 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| 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 \(1,\beta_1,\ldots,\beta_{16}\) 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
\( x^{17} - 2 x^{16} - 28 x^{15} + 54 x^{14} + 317 x^{13} - 582 x^{12} - 1867 x^{11} + 3178 x^{10} + \cdots + 256 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 4 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 69 \nu^{16} - 212 \nu^{15} - 1752 \nu^{14} + 5638 \nu^{13} + 17157 \nu^{12} - 59424 \nu^{11} + \cdots + 27776 ) / 1216 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 179 \nu^{16} + 230 \nu^{15} + 4968 \nu^{14} - 5722 \nu^{13} - 55271 \nu^{12} + 54778 \nu^{11} + \cdots + 17152 ) / 1216 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - \nu^{16} - \nu^{15} + 31 \nu^{14} + 30 \nu^{13} - 391 \nu^{12} - 363 \nu^{11} + 2570 \nu^{10} + \cdots + 448 ) / 16 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 51 \nu^{16} - 11 \nu^{15} + 1518 \nu^{14} + 482 \nu^{13} - 18373 \nu^{12} - 7691 \nu^{11} + \cdots + 19360 ) / 608 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 42 \nu^{16} + 37 \nu^{15} - 1306 \nu^{14} - 1148 \nu^{13} + 16568 \nu^{12} + 14297 \nu^{11} + \cdots - 23776 ) / 608 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( 92 \nu^{16} - 61 \nu^{15} - 2640 \nu^{14} + 1412 \nu^{13} + 30590 \nu^{12} - 12029 \nu^{11} + \cdots - 16672 ) / 608 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( 55 \nu^{16} - 66 \nu^{15} - 1551 \nu^{14} + 1676 \nu^{13} + 17575 \nu^{12} - 16544 \nu^{11} + \cdots - 1184 ) / 304 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( - 81 \nu^{16} + 120 \nu^{15} + 2231 \nu^{14} - 3068 \nu^{13} - 24529 \nu^{12} + 30574 \nu^{11} + \cdots + 3424 ) / 304 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( - 273 \nu^{16} + 54 \nu^{15} + 8052 \nu^{14} - 670 \nu^{13} - 96501 \nu^{12} - 3222 \nu^{11} + \cdots + 63040 ) / 1216 \)
|
| \(\beta_{12}\) | \(=\) |
\( ( - 439 \nu^{16} + 428 \nu^{15} + 12452 \nu^{14} - 10522 \nu^{13} - 142215 \nu^{12} + 98824 \nu^{11} + \cdots + 48064 ) / 1216 \)
|
| \(\beta_{13}\) | \(=\) |
\( ( 425 \nu^{16} - 320 \nu^{15} - 12232 \nu^{14} + 7662 \nu^{13} + 142177 \nu^{12} - 68940 \nu^{11} + \cdots - 61824 ) / 1216 \)
|
| \(\beta_{14}\) | \(=\) |
\( ( 65 \nu^{16} + 169 \nu^{15} - 2194 \nu^{14} - 4842 \nu^{13} + 30039 \nu^{12} + 55745 \nu^{11} + \cdots - 56064 ) / 608 \)
|
| \(\beta_{15}\) | \(=\) |
\( ( 461 \nu^{16} - 310 \nu^{15} - 13308 \nu^{14} + 7286 \nu^{13} + 155249 \nu^{12} - 63482 \nu^{11} + \cdots - 67264 ) / 1216 \)
|
| \(\beta_{16}\) | \(=\) |
\( ( - 613 \nu^{16} + 462 \nu^{15} + 17564 \nu^{14} - 10934 \nu^{13} - 203129 \nu^{12} + 96466 \nu^{11} + \cdots + 107392 ) / 1216 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( -\beta_{15} - \beta_{14} + \beta_{13} + \beta_{10} + \beta_{9} + \beta_{7} - \beta_{5} - \beta_{4} - \beta_{3} + 6\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( -\beta_{16} - \beta_{13} + \beta_{10} + \beta_{9} - \beta_{7} + 8\beta_{2} + 23 \)
|
| \(\nu^{5}\) | \(=\) |
\( - \beta_{16} - 10 \beta_{15} - 8 \beta_{14} + 9 \beta_{13} + 2 \beta_{12} + 9 \beta_{10} + 11 \beta_{9} + \cdots - 1 \)
|
| \(\nu^{6}\) | \(=\) |
\( - 13 \beta_{16} - 3 \beta_{15} + \beta_{14} - 12 \beta_{13} - 2 \beta_{12} + 12 \beta_{10} + 12 \beta_{9} + \cdots + 145 \)
|
| \(\nu^{7}\) | \(=\) |
\( - 12 \beta_{16} - 86 \beta_{15} - 56 \beta_{14} + 70 \beta_{13} + 26 \beta_{12} + 68 \beta_{10} + \cdots - 8 \)
|
| \(\nu^{8}\) | \(=\) |
\( - 124 \beta_{16} - 48 \beta_{15} + 14 \beta_{14} - 108 \beta_{13} - 28 \beta_{12} - 2 \beta_{11} + \cdots + 948 \)
|
| \(\nu^{9}\) | \(=\) |
\( - 108 \beta_{16} - 707 \beta_{15} - 387 \beta_{14} + 527 \beta_{13} + 248 \beta_{12} - 6 \beta_{11} + \cdots - 34 \)
|
| \(\nu^{10}\) | \(=\) |
\( - 1057 \beta_{16} - 552 \beta_{15} + 122 \beta_{14} - 857 \beta_{13} - 272 \beta_{12} - 46 \beta_{11} + \cdots + 6339 \)
|
| \(\nu^{11}\) | \(=\) |
\( - 889 \beta_{16} - 5684 \beta_{15} - 2706 \beta_{14} + 3923 \beta_{13} + 2110 \beta_{12} - 130 \beta_{11} + \cdots + 53 \)
|
| \(\nu^{12}\) | \(=\) |
\( - 8553 \beta_{16} - 5563 \beta_{15} + 815 \beta_{14} - 6328 \beta_{13} - 2270 \beta_{12} - 670 \beta_{11} + \cdots + 43149 \)
|
| \(\nu^{13}\) | \(=\) |
\( - 7116 \beta_{16} - 45112 \beta_{15} - 19254 \beta_{14} + 29056 \beta_{13} + 16986 \beta_{12} + \cdots + 3166 \)
|
| \(\nu^{14}\) | \(=\) |
\( - 67364 \beta_{16} - 52268 \beta_{15} + 4064 \beta_{14} - 44524 \beta_{13} - 17404 \beta_{12} + \cdots + 298240 \)
|
| \(\nu^{15}\) | \(=\) |
\( - 56728 \beta_{16} - 355337 \beta_{15} - 139449 \beta_{14} + 214733 \beta_{13} + 132652 \beta_{12} + \cdots + 46476 \)
|
| \(\nu^{16}\) | \(=\) |
\( - 522745 \beta_{16} - 470432 \beta_{15} + 8284 \beta_{14} - 301769 \beta_{13} - 125988 \beta_{12} + \cdots + 2089247 \)
|
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 |
|
−2.65791 | −2.62368 | 5.06447 | 1.15029 | 6.97351 | −2.51426 | −8.14509 | 3.88372 | −3.05737 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −2.51582 | −0.505139 | 4.32934 | −3.97764 | 1.27084 | 1.36760 | −5.86020 | −2.74483 | 10.0070 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −2.32547 | 3.27059 | 3.40783 | −2.03057 | −7.60566 | 1.64874 | −3.27388 | 7.69673 | 4.72203 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −2.27410 | 0.935470 | 3.17152 | 3.41593 | −2.12735 | 3.69332 | −2.66415 | −2.12490 | −7.76816 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | −1.09599 | −1.16074 | −0.798803 | 4.05107 | 1.27216 | −4.08218 | 3.06746 | −1.65268 | −4.43994 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | −0.932399 | −3.04458 | −1.13063 | −3.52221 | 2.83877 | −4.32039 | 2.91900 | 6.26949 | 3.28411 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | −0.787554 | 1.89108 | −1.37976 | −0.652808 | −1.48933 | 1.12991 | 2.66174 | 0.576182 | 0.514122 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | −0.622810 | −1.30185 | −1.61211 | −1.69081 | 0.810802 | 3.93205 | 2.24966 | −1.30520 | 1.05305 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | −0.139956 | 2.55238 | −1.98041 | 2.96478 | −0.357221 | 0.820250 | 0.557083 | 3.51462 | −0.414940 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.10 | 0.779516 | −3.14584 | −1.39236 | 1.66398 | −2.45223 | 2.07256 | −2.64439 | 6.89631 | 1.29710 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.11 | 1.37191 | 2.36731 | −0.117867 | −1.32016 | 3.24773 | 4.19189 | −2.90552 | 2.60414 | −1.81114 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.12 | 1.37907 | 2.46273 | −0.0981670 | 1.31548 | 3.39627 | −3.48956 | −2.89352 | 3.06502 | 1.81414 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.13 | 1.84638 | −0.508487 | 1.40912 | 3.51373 | −0.938859 | 0.924114 | −1.09099 | −2.74144 | 6.48767 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.14 | 2.18124 | 0.923498 | 2.75779 | 1.20531 | 2.01437 | −1.96022 | 1.65293 | −2.14715 | 2.62906 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.15 | 2.33247 | −0.826533 | 3.44041 | −1.37815 | −1.92786 | 4.67534 | 3.35971 | −2.31684 | −3.21448 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.16 | 2.64128 | 1.66988 | 4.97638 | −3.99830 | 4.41062 | −2.26241 | 7.86146 | −0.211508 | −10.5606 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.17 | 2.82015 | −2.95607 | 5.95323 | 2.29008 | −8.33655 | −2.82675 | 11.1487 | 5.73835 | 6.45836 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(251\) | \(-1\) |
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 | 251.2.a.b | ✓ | 17 |
| 3.b | odd | 2 | 1 | 2259.2.a.k | 17 | ||
| 4.b | odd | 2 | 1 | 4016.2.a.k | 17 | ||
| 5.b | even | 2 | 1 | 6275.2.a.e | 17 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 251.2.a.b | ✓ | 17 | 1.a | even | 1 | 1 | trivial |
| 2259.2.a.k | 17 | 3.b | odd | 2 | 1 | ||
| 4016.2.a.k | 17 | 4.b | odd | 2 | 1 | ||
| 6275.2.a.e | 17 | 5.b | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{17} - 2 T_{2}^{16} - 28 T_{2}^{15} + 54 T_{2}^{14} + 317 T_{2}^{13} - 582 T_{2}^{12} - 1867 T_{2}^{11} + \cdots + 256 \)
acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(251))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{17} - 2 T^{16} + \cdots + 256 \)
|
| $3$ |
\( T^{17} - 38 T^{15} + \cdots + 3164 \)
|
| $5$ |
\( T^{17} - 3 T^{16} + \cdots - 228857 \)
|
| $7$ |
\( T^{17} - 3 T^{16} + \cdots + 2209789 \)
|
| $11$ |
\( T^{17} + T^{16} + \cdots + 10657792 \)
|
| $13$ |
\( T^{17} - 22 T^{16} + \cdots - 504874 \)
|
| $17$ |
\( T^{17} + T^{16} + \cdots - 54097717 \)
|
| $19$ |
\( T^{17} + \cdots + 130088960 \)
|
| $23$ |
\( T^{17} + 2 T^{16} + \cdots + 201949 \)
|
| $29$ |
\( T^{17} + \cdots + 1937776640 \)
|
| $31$ |
\( T^{17} + \cdots + 10307640389 \)
|
| $37$ |
\( T^{17} + \cdots - 1861132288 \)
|
| $41$ |
\( T^{17} + \cdots - 11114425387 \)
|
| $43$ |
\( T^{17} - 9 T^{16} + \cdots - 11640832 \)
|
| $47$ |
\( T^{17} + \cdots - 62409392128 \)
|
| $53$ |
\( T^{17} + \cdots - 7243329708032 \)
|
| $59$ |
\( T^{17} + \cdots - 139809955840 \)
|
| $61$ |
\( T^{17} + \cdots - 3666674696192 \)
|
| $67$ |
\( T^{17} + \cdots - 1042048845953 \)
|
| $71$ |
\( T^{17} + \cdots - 12296978432 \)
|
| $73$ |
\( T^{17} + \cdots - 371103914897 \)
|
| $79$ |
\( T^{17} + \cdots - 1616495596055 \)
|
| $83$ |
\( T^{17} + \cdots + 295625646813184 \)
|
| $89$ |
\( T^{17} + \cdots - 91153496990 \)
|
| $97$ |
\( T^{17} + \cdots - 41770891288576 \)
|
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