Newspace parameters
| Level: | \( N \) | \(=\) | \( 325 = 5^{2} \cdot 13 \) |
| Weight: | \( k \) | \(=\) | \( 3 \) |
| Character orbit: | \([\chi]\) | \(=\) | 325.g (of order \(4\), degree \(2\), not minimal) |
Newform invariants
| Self dual: | no |
| Analytic conductor: | \(8.85560859171\) |
| Analytic rank: | \(0\) |
| Dimension: | \(20\) |
| Relative dimension: | \(10\) over \(\Q(i)\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{20} - \cdots)\) |
|
|
|
| Defining polynomial: |
\( x^{20} - 6 x^{17} + 336 x^{16} - 90 x^{15} + 18 x^{14} - 654 x^{13} + 30550 x^{12} - 9690 x^{11} + \cdots + 46656 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{13}]\) |
| Coefficient ring index: | \( 2^{3} \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{4}]$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{19}\) 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
\( x^{20} - 6 x^{17} + 336 x^{16} - 90 x^{15} + 18 x^{14} - 654 x^{13} + 30550 x^{12} - 9690 x^{11} + \cdots + 46656 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( - 13\!\cdots\!64 \nu^{19} + \cdots + 18\!\cdots\!84 ) / 30\!\cdots\!00 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 13\!\cdots\!96 \nu^{19} + \cdots + 20\!\cdots\!24 ) / 35\!\cdots\!20 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 19\!\cdots\!77 \nu^{19} + \cdots + 11\!\cdots\!96 ) / 30\!\cdots\!00 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 52\!\cdots\!28 \nu^{19} + \cdots - 12\!\cdots\!08 ) / 33\!\cdots\!00 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 21\!\cdots\!73 \nu^{19} + \cdots - 16\!\cdots\!88 ) / 10\!\cdots\!60 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - 13\!\cdots\!45 \nu^{19} + \cdots - 42\!\cdots\!96 ) / 60\!\cdots\!00 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( - 31\!\cdots\!57 \nu^{19} + \cdots - 21\!\cdots\!36 ) / 95\!\cdots\!40 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( - 31\!\cdots\!61 \nu^{19} + \cdots + 75\!\cdots\!56 ) / 95\!\cdots\!40 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( 43\!\cdots\!41 \nu^{19} + \cdots + 43\!\cdots\!24 ) / 60\!\cdots\!00 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( - 27\!\cdots\!43 \nu^{19} + \cdots + 69\!\cdots\!80 ) / 33\!\cdots\!00 \)
|
| \(\beta_{12}\) | \(=\) |
\( ( 11\!\cdots\!55 \nu^{19} + \cdots + 20\!\cdots\!76 ) / 11\!\cdots\!00 \)
|
| \(\beta_{13}\) | \(=\) |
\( ( 31\!\cdots\!57 \nu^{19} + \cdots + 21\!\cdots\!36 ) / 15\!\cdots\!40 \)
|
| \(\beta_{14}\) | \(=\) |
\( ( 83\!\cdots\!96 \nu^{19} + \cdots + 95\!\cdots\!72 ) / 33\!\cdots\!60 \)
|
| \(\beta_{15}\) | \(=\) |
\( ( - 17\!\cdots\!62 \nu^{19} + \cdots + 40\!\cdots\!08 ) / 60\!\cdots\!60 \)
|
| \(\beta_{16}\) | \(=\) |
\( ( - 12\!\cdots\!81 \nu^{19} + \cdots - 86\!\cdots\!28 ) / 33\!\cdots\!00 \)
|
| \(\beta_{17}\) | \(=\) |
\( ( - 14\!\cdots\!09 \nu^{19} + \cdots - 50\!\cdots\!00 ) / 33\!\cdots\!00 \)
|
| \(\beta_{18}\) | \(=\) |
\( ( 14\!\cdots\!85 \nu^{19} + \cdots + 10\!\cdots\!56 ) / 30\!\cdots\!00 \)
|
| \(\beta_{19}\) | \(=\) |
\( ( - 62\!\cdots\!33 \nu^{19} + \cdots - 95\!\cdots\!88 ) / 60\!\cdots\!60 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{13} + 6\beta_{8} \)
|
| \(\nu^{3}\) | \(=\) |
\( -\beta_{14} - \beta_{8} - 11\beta_{6} + 1 \)
|
| \(\nu^{4}\) | \(=\) |
\( - \beta_{18} - \beta_{17} + \beta_{14} + \beta_{12} + 2 \beta_{11} - \beta_{10} + \beta_{9} + \beta_{7} + \cdots - 62 \)
|
| \(\nu^{5}\) | \(=\) |
\( \beta_{17} - \beta_{16} + 2 \beta_{15} + 3 \beta_{13} - 3 \beta_{11} - 18 \beta_{9} + 24 \beta_{8} + \cdots + 24 \)
|
| \(\nu^{6}\) | \(=\) |
\( - 20 \beta_{19} + 22 \beta_{16} - 20 \beta_{15} - 25 \beta_{14} - 187 \beta_{13} - 2 \beta_{12} + \cdots + 35 \beta_1 \)
|
| \(\nu^{7}\) | \(=\) |
\( 47 \beta_{19} - 25 \beta_{18} - 27 \beta_{17} - 27 \beta_{16} + 5 \beta_{15} + 281 \beta_{14} + \cdots - 491 \)
|
| \(\nu^{8}\) | \(=\) |
\( - 10 \beta_{19} + 318 \beta_{18} + 379 \beta_{17} + 10 \beta_{15} - 484 \beta_{14} - 378 \beta_{12} + \cdots + 9734 \)
|
| \(\nu^{9}\) | \(=\) |
\( - 175 \beta_{19} - 175 \beta_{18} - 526 \beta_{17} + 526 \beta_{16} - 843 \beta_{15} - 2116 \beta_{13} + \cdots - 9316 \)
|
| \(\nu^{10}\) | \(=\) |
\( 5386 \beta_{19} + 279 \beta_{18} - 6048 \beta_{16} + 5386 \beta_{15} + 8514 \beta_{14} + \cdots - 17046 \beta_1 \)
|
| \(\nu^{11}\) | \(=\) |
\( - 13958 \beta_{19} + 8560 \beta_{18} + 9260 \beta_{17} + 9260 \beta_{16} - 4114 \beta_{15} + \cdots + 167833 \)
|
| \(\nu^{12}\) | \(=\) |
\( 7976 \beta_{19} - 70507 \beta_{18} - 93721 \beta_{17} - 7976 \beta_{15} + 142921 \beta_{14} + \cdots - 1907522 \)
|
| \(\nu^{13}\) | \(=\) |
\( 82176 \beta_{19} + 82176 \beta_{18} + 156629 \beta_{17} - 156629 \beta_{16} + 224806 \beta_{15} + \cdots + 2912892 \)
|
| \(\nu^{14}\) | \(=\) |
\( - 1300588 \beta_{19} - 94932 \beta_{18} + 1436330 \beta_{16} - 1300588 \beta_{15} + \cdots + 5659951 \beta_1 \)
|
| \(\nu^{15}\) | \(=\) |
\( 3584131 \beta_{19} - 2339225 \beta_{18} - 2598471 \beta_{17} - 2598471 \beta_{16} + 1509469 \beta_{15} + \cdots - 49215691 \)
|
| \(\nu^{16}\) | \(=\) |
\( - 2863034 \beta_{19} + 15721638 \beta_{18} + 21945611 \beta_{17} + 2863034 \beta_{15} + \cdots + 414743434 \)
|
| \(\nu^{17}\) | \(=\) |
\( - 26411879 \beta_{19} - 26411879 \beta_{18} - 42610094 \beta_{17} + 42610094 \beta_{16} + \cdots - 815603744 \)
|
| \(\nu^{18}\) | \(=\) |
\( 308186666 \beta_{19} + 20577123 \beta_{18} - 335505876 \beta_{16} + 308186666 \beta_{15} + \cdots - 1624246206 \beta_1 \)
|
| \(\nu^{19}\) | \(=\) |
\( - 901860190 \beta_{19} + 596387684 \beta_{18} + 692961460 \beta_{17} + 692961460 \beta_{16} + \cdots + 13328434457 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/325\mathbb{Z}\right)^\times\).
| \(n\) | \(27\) | \(301\) |
| \(\chi(n)\) | \(-1\) | \(\beta_{8}\) |
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.
| Label | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 99.1 |
|
−2.50536 | + | 2.50536i | 5.52663i | − | 8.55366i | 0 | −13.8462 | − | 13.8462i | −7.45363 | − | 7.45363i | 11.4085 | + | 11.4085i | −21.5436 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 99.2 | −2.28875 | + | 2.28875i | − | 3.77733i | − | 6.47674i | 0 | 8.64537 | + | 8.64537i | −4.50772 | − | 4.50772i | 5.66863 | + | 5.66863i | −5.26826 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 99.3 | −1.67324 | + | 1.67324i | 0.982234i | − | 1.59947i | 0 | −1.64351 | − | 1.64351i | 6.39745 | + | 6.39745i | −4.01666 | − | 4.01666i | 8.03522 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 99.4 | −1.01991 | + | 1.01991i | 1.47834i | 1.91956i | 0 | −1.50777 | − | 1.50777i | −2.86721 | − | 2.86721i | −6.03743 | − | 6.03743i | 6.81452 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 99.5 | −0.375392 | + | 0.375392i | − | 5.24224i | 3.71816i | 0 | 1.96790 | + | 1.96790i | 7.11069 | + | 7.11069i | −2.89734 | − | 2.89734i | −18.4811 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 99.6 | 0.177531 | − | 0.177531i | − | 1.31502i | 3.93697i | 0 | −0.233457 | − | 0.233457i | −4.93361 | − | 4.93361i | 1.40906 | + | 1.40906i | 7.27073 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 99.7 | 0.962413 | − | 0.962413i | 3.84448i | 2.14752i | 0 | 3.69997 | + | 3.69997i | −0.488874 | − | 0.488874i | 5.91646 | + | 5.91646i | −5.78000 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 99.8 | 1.88633 | − | 1.88633i | − | 0.793970i | − | 3.11650i | 0 | −1.49769 | − | 1.49769i | 8.42514 | + | 8.42514i | 1.66658 | + | 1.66658i | 8.36961 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 99.9 | 2.03566 | − | 2.03566i | − | 4.42402i | − | 4.28784i | 0 | −9.00581 | − | 9.00581i | −7.36000 | − | 7.36000i | −0.585953 | − | 0.585953i | −10.5719 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 99.10 | 2.80071 | − | 2.80071i | 3.72091i | − | 11.6880i | 0 | 10.4212 | + | 10.4212i | −4.32223 | − | 4.32223i | −21.5319 | − | 21.5319i | −4.84515 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 174.1 | −2.50536 | − | 2.50536i | − | 5.52663i | 8.55366i | 0 | −13.8462 | + | 13.8462i | −7.45363 | + | 7.45363i | 11.4085 | − | 11.4085i | −21.5436 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 174.2 | −2.28875 | − | 2.28875i | 3.77733i | 6.47674i | 0 | 8.64537 | − | 8.64537i | −4.50772 | + | 4.50772i | 5.66863 | − | 5.66863i | −5.26826 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 174.3 | −1.67324 | − | 1.67324i | − | 0.982234i | 1.59947i | 0 | −1.64351 | + | 1.64351i | 6.39745 | − | 6.39745i | −4.01666 | + | 4.01666i | 8.03522 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 174.4 | −1.01991 | − | 1.01991i | − | 1.47834i | − | 1.91956i | 0 | −1.50777 | + | 1.50777i | −2.86721 | + | 2.86721i | −6.03743 | + | 6.03743i | 6.81452 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 174.5 | −0.375392 | − | 0.375392i | 5.24224i | − | 3.71816i | 0 | 1.96790 | − | 1.96790i | 7.11069 | − | 7.11069i | −2.89734 | + | 2.89734i | −18.4811 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 174.6 | 0.177531 | + | 0.177531i | 1.31502i | − | 3.93697i | 0 | −0.233457 | + | 0.233457i | −4.93361 | + | 4.93361i | 1.40906 | − | 1.40906i | 7.27073 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 174.7 | 0.962413 | + | 0.962413i | − | 3.84448i | − | 2.14752i | 0 | 3.69997 | − | 3.69997i | −0.488874 | + | 0.488874i | 5.91646 | − | 5.91646i | −5.78000 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 174.8 | 1.88633 | + | 1.88633i | 0.793970i | 3.11650i | 0 | −1.49769 | + | 1.49769i | 8.42514 | − | 8.42514i | 1.66658 | − | 1.66658i | 8.36961 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 174.9 | 2.03566 | + | 2.03566i | 4.42402i | 4.28784i | 0 | −9.00581 | + | 9.00581i | −7.36000 | + | 7.36000i | −0.585953 | + | 0.585953i | −10.5719 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 174.10 | 2.80071 | + | 2.80071i | − | 3.72091i | 11.6880i | 0 | 10.4212 | − | 10.4212i | −4.32223 | + | 4.32223i | −21.5319 | + | 21.5319i | −4.84515 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 65.g | odd | 4 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 325.3.g.e | 20 | |
| 5.b | even | 2 | 1 | 325.3.g.f | 20 | ||
| 5.c | odd | 4 | 1 | 325.3.j.c | ✓ | 20 | |
| 5.c | odd | 4 | 1 | 325.3.j.d | yes | 20 | |
| 13.d | odd | 4 | 1 | 325.3.g.f | 20 | ||
| 65.f | even | 4 | 1 | 325.3.j.c | ✓ | 20 | |
| 65.g | odd | 4 | 1 | inner | 325.3.g.e | 20 | |
| 65.k | even | 4 | 1 | 325.3.j.d | yes | 20 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 325.3.g.e | 20 | 1.a | even | 1 | 1 | trivial | |
| 325.3.g.e | 20 | 65.g | odd | 4 | 1 | inner | |
| 325.3.g.f | 20 | 5.b | even | 2 | 1 | ||
| 325.3.g.f | 20 | 13.d | odd | 4 | 1 | ||
| 325.3.j.c | ✓ | 20 | 5.c | odd | 4 | 1 | |
| 325.3.j.c | ✓ | 20 | 65.f | even | 4 | 1 | |
| 325.3.j.d | yes | 20 | 5.c | odd | 4 | 1 | |
| 325.3.j.d | yes | 20 | 65.k | even | 4 | 1 | |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{20} + 6 T_{2}^{17} + 336 T_{2}^{16} + 90 T_{2}^{15} + 18 T_{2}^{14} + 654 T_{2}^{13} + \cdots + 46656 \)
acting on \(S_{3}^{\mathrm{new}}(325, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{20} + 6 T^{17} + \cdots + 46656 \)
|
| $3$ |
\( T^{20} + \cdots + 110250000 \)
|
| $5$ |
\( T^{20} \)
|
| $7$ |
\( T^{20} + \cdots + 82\!\cdots\!76 \)
|
| $11$ |
\( T^{20} + \cdots + 29\!\cdots\!36 \)
|
| $13$ |
\( T^{20} + \cdots + 19\!\cdots\!01 \)
|
| $17$ |
\( (T^{10} - 8 T^{9} + \cdots - 29492667360)^{2} \)
|
| $19$ |
\( T^{20} + \cdots + 35\!\cdots\!00 \)
|
| $23$ |
\( (T^{10} + 14 T^{9} + \cdots + 61510337640)^{2} \)
|
| $29$ |
\( (T^{10} + \cdots + 40470047745000)^{2} \)
|
| $31$ |
\( T^{20} + \cdots + 11\!\cdots\!00 \)
|
| $37$ |
\( T^{20} + \cdots + 12\!\cdots\!00 \)
|
| $41$ |
\( T^{20} + \cdots + 40\!\cdots\!76 \)
|
| $43$ |
\( (T^{10} + \cdots + 9131165078400)^{2} \)
|
| $47$ |
\( T^{20} + \cdots + 16\!\cdots\!36 \)
|
| $53$ |
\( T^{20} + \cdots + 52\!\cdots\!00 \)
|
| $59$ |
\( T^{20} + \cdots + 15\!\cdots\!24 \)
|
| $61$ |
\( (T^{10} + \cdots - 13329562266414)^{2} \)
|
| $67$ |
\( T^{20} + \cdots + 12\!\cdots\!44 \)
|
| $71$ |
\( T^{20} + \cdots + 17\!\cdots\!04 \)
|
| $73$ |
\( T^{20} + \cdots + 10\!\cdots\!00 \)
|
| $79$ |
\( (T^{10} + \cdots - 69\!\cdots\!12)^{2} \)
|
| $83$ |
\( T^{20} + \cdots + 75\!\cdots\!76 \)
|
| $89$ |
\( T^{20} + \cdots + 58\!\cdots\!00 \)
|
| $97$ |
\( T^{20} + \cdots + 25\!\cdots\!24 \)
|
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