Newspace parameters
Level: | \( N \) | \(=\) | \( 225 = 3^{2} \cdot 5^{2} \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 225.k (of order \(6\), degree \(2\), not minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(1.79663404548\) |
Analytic rank: | \(0\) |
Dimension: | \(16\) |
Relative dimension: | \(8\) over \(\Q(\zeta_{6})\) |
Coefficient field: | \(\mathbb{Q}[x]/(x^{16} - \cdots)\) |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
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|
Defining polynomial: |
\( x^{16} - 12x^{14} + 102x^{12} - 406x^{10} + 1167x^{8} - 1842x^{6} + 2023x^{4} - 441x^{2} + 81 \)
|
Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
Coefficient ring index: | \( 1 \) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{6}]$ |
$q$-expansion
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
\( x^{16} - 12x^{14} + 102x^{12} - 406x^{10} + 1167x^{8} - 1842x^{6} + 2023x^{4} - 441x^{2} + 81 \)
:
\(\beta_{1}\) | \(=\) |
\( \nu \)
|
\(\beta_{2}\) | \(=\) |
\( ( 3457 \nu^{14} - 28693 \nu^{12} + 210280 \nu^{10} - 306752 \nu^{8} + 709118 \nu^{6} - 1299110 \nu^{4} + 7466556 \nu^{2} - 7606359 ) / 3976425 \)
|
\(\beta_{3}\) | \(=\) |
\( ( -26\nu^{14} + 74\nu^{12} + 10\nu^{10} - 10439\nu^{8} + 38576\nu^{6} - 94895\nu^{4} + 82467\nu^{2} - 18063 ) / 29025 \)
|
\(\beta_{4}\) | \(=\) |
\( ( 5917 \nu^{15} - 90503 \nu^{13} + 784955 \nu^{11} - 3838362 \nu^{9} + 10468603 \nu^{7} - 20515435 \nu^{5} + 19722086 \nu^{3} - 16863264 \nu ) / 3976425 \)
|
\(\beta_{5}\) | \(=\) |
\( ( - 7559 \nu^{14} + 69641 \nu^{12} - 572135 \nu^{10} + 1494574 \nu^{8} - 4803841 \nu^{6} + 3626995 \nu^{4} - 7975497 \nu^{2} - 5812992 ) / 3976425 \)
|
\(\beta_{6}\) | \(=\) |
\( ( 11836 \nu^{14} - 138574 \nu^{12} + 1170340 \nu^{10} - 4500371 \nu^{8} + 12811274 \nu^{6} - 19088030 \nu^{4} + 21981813 \nu^{2} - 814212 ) / 3976425 \)
|
\(\beta_{7}\) | \(=\) |
\( ( 3766 \nu^{15} - 41414 \nu^{13} + 332215 \nu^{11} - 1090526 \nu^{9} + 2480389 \nu^{7} - 2137205 \nu^{5} + 467253 \nu^{3} + 2721243 \nu ) / 1325475 \)
|
\(\beta_{8}\) | \(=\) |
\( ( - 11836 \nu^{15} + 138574 \nu^{13} - 1170340 \nu^{11} + 4500371 \nu^{9} - 12811274 \nu^{7} + 19088030 \nu^{5} - 21981813 \nu^{3} + 4790637 \nu ) / 3976425 \)
|
\(\beta_{9}\) | \(=\) |
\( ( - 14896 \nu^{14} + 165889 \nu^{12} - 1314040 \nu^{10} + 4313456 \nu^{8} - 9114389 \nu^{6} + 8453480 \nu^{4} - 1848168 \nu^{2} - 5535918 ) / 3976425 \)
|
\(\beta_{10}\) | \(=\) |
\( ( 13477 \nu^{15} - 168383 \nu^{13} + 1451855 \nu^{11} - 6027522 \nu^{9} + 17277208 \nu^{7} - 24805735 \nu^{5} + 20660066 \nu^{3} + 11503071 \nu ) / 3976425 \)
|
\(\beta_{11}\) | \(=\) |
\( ( - 3323 \nu^{14} + 38066 \nu^{12} - 324695 \nu^{10} + 1252588 \nu^{8} - 3846136 \nu^{6} + 6436600 \nu^{4} - 8169474 \nu^{2} + 1468098 ) / 795285 \)
|
\(\beta_{12}\) | \(=\) |
\( ( 19253 \nu^{15} - 245932 \nu^{13} + 2129695 \nu^{11} - 9130758 \nu^{9} + 26781707 \nu^{7} - 44578415 \nu^{5} + 47402299 \nu^{3} - 10338741 \nu ) / 3976425 \)
|
\(\beta_{13}\) | \(=\) |
\( ( 572 \nu^{14} - 6758 \nu^{12} + 56330 \nu^{10} - 215542 \nu^{8} + 579583 \nu^{6} - 869185 \nu^{4} + 809301 \nu^{2} - 212679 ) / 92475 \)
|
\(\beta_{14}\) | \(=\) |
\( ( 43568 \nu^{15} - 518857 \nu^{13} + 4432420 \nu^{11} - 17623398 \nu^{9} + 51780332 \nu^{7} - 81131165 \nu^{5} + 89205019 \nu^{3} - 6887691 \nu ) / 3976425 \)
|
\(\beta_{15}\) | \(=\) |
\( ( - 49089 \nu^{15} + 557311 \nu^{13} - 4656460 \nu^{11} + 17097154 \nu^{9} - 47712236 \nu^{7} + 67106795 \nu^{5} - 70902062 \nu^{3} - 564507 \nu ) / 3976425 \)
|
\(\nu\) | \(=\) |
\( \beta_1 \)
|
\(\nu^{2}\) | \(=\) |
\( \beta_{11} + \beta_{9} + 3\beta_{6} + \beta_{5} - \beta_{3} + 1 \)
|
\(\nu^{3}\) | \(=\) |
\( -\beta_{15} - \beta_{14} - 3\beta_{12} + 2\beta_{10} - 4\beta_{8} - 3\beta_{7} + 2\beta_{4} + 4\beta_1 \)
|
\(\nu^{4}\) | \(=\) |
\( -6\beta_{13} - 2\beta_{11} + 15\beta_{6} + 8\beta_{5} - 7\beta_{3} - 8\beta_{2} - 15 \)
|
\(\nu^{5}\) | \(=\) |
\( 8\beta_{15} + 11\beta_{14} - 27\beta_{12} + 8\beta_{10} - 22\beta_{8} + 11\beta_{4} \)
|
\(\nu^{6}\) | \(=\) |
\( -57\beta_{13} - 57\beta_{11} - 49\beta_{9} + 19\beta_{5} - 38\beta_{2} - 139 \)
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\(\nu^{7}\) | \(=\) |
\( 144\beta_{15} + 144\beta_{14} - 87\beta_{10} + 204\beta_{7} - 57\beta_{4} - 139\beta_1 \)
|
\(\nu^{8}\) | \(=\) |
\( -144\beta_{13} - 253\beta_{11} - 343\beta_{9} - 588\beta_{6} - 253\beta_{5} + 343\beta_{3} + 144\beta_{2} - 343 \)
|
\(\nu^{9}\) | \(=\) |
\( 631 \beta_{15} + 397 \beta_{14} + 1461 \beta_{12} - 1028 \beta_{10} + 931 \beta_{8} + 1461 \beta_{7} - 1028 \beta_{4} - 931 \beta_1 \)
|
\(\nu^{10}\) | \(=\) |
\( 1725\beta_{13} + 1028\beta_{11} - 3984\beta_{6} - 2753\beta_{5} + 2392\beta_{3} + 2753\beta_{2} + 3984 \)
|
\(\nu^{11}\) | \(=\) |
\( -2753\beta_{15} - 4448\beta_{14} + 10260\beta_{12} - 2753\beta_{10} + 6376\beta_{8} - 4448\beta_{4} \)
|
\(\nu^{12}\) | \(=\) |
\( 19083\beta_{13} + 19083\beta_{11} + 16636\beta_{9} - 7201\beta_{5} + 11882\beta_{2} + 44023 \)
|
\(\nu^{13}\) | \(=\) |
\( -50121\beta_{15} - 50121\beta_{14} + 31038\beta_{10} - 71511\beta_{7} + 19083\beta_{4} + 44023\beta_1 \)
|
\(\nu^{14}\) | \(=\) |
\( 50121 \beta_{13} + 82189 \beta_{11} + 115534 \beta_{9} + 189318 \beta_{6} + 82189 \beta_{5} - 115534 \beta_{3} - 50121 \beta_{2} + 115534 \)
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\(\nu^{15}\) | \(=\) |
\( - 215776 \beta_{15} - 132310 \beta_{14} - 496965 \beta_{12} + 348086 \beta_{10} - 304852 \beta_{8} - 496965 \beta_{7} + 348086 \beta_{4} + 304852 \beta_1 \)
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Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/225\mathbb{Z}\right)^\times\).
\(n\) | \(101\) | \(127\) |
\(\chi(n)\) | \(-\beta_{6}\) | \(-1\) |
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.
Label | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
49.1 |
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−2.28087 | + | 1.31686i | −0.238330 | − | 1.71558i | 2.46825 | − | 4.27513i | 0 | 2.80278 | + | 3.59916i | 1.55662 | − | 0.898714i | 7.73393i | −2.88640 | + | 0.817746i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
49.2 | −1.41485 | + | 0.816862i | −1.36657 | − | 1.06419i | 0.334526 | − | 0.579416i | 0 | 2.80278 | + | 0.389365i | −0.437645 | + | 0.252674i | − | 2.17440i | 0.735010 | + | 2.90857i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
49.3 | −1.27588 | + | 0.736627i | −0.350156 | + | 1.69629i | 0.0852394 | − | 0.147639i | 0 | −0.802776 | − | 2.42219i | −3.34791 | + | 1.93291i | − | 2.69535i | −2.75478 | − | 1.18793i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
49.4 | −0.409850 | + | 0.236627i | 1.64411 | + | 0.544899i | −0.888015 | + | 1.53809i | 0 | −0.802776 | + | 0.165713i | −2.21967 | + | 1.28153i | − | 1.78702i | 2.40617 | + | 1.79175i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
49.5 | 0.409850 | − | 0.236627i | −1.64411 | − | 0.544899i | −0.888015 | + | 1.53809i | 0 | −0.802776 | + | 0.165713i | 2.21967 | − | 1.28153i | 1.78702i | 2.40617 | + | 1.79175i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
49.6 | 1.27588 | − | 0.736627i | 0.350156 | − | 1.69629i | 0.0852394 | − | 0.147639i | 0 | −0.802776 | − | 2.42219i | 3.34791 | − | 1.93291i | 2.69535i | −2.75478 | − | 1.18793i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
49.7 | 1.41485 | − | 0.816862i | 1.36657 | + | 1.06419i | 0.334526 | − | 0.579416i | 0 | 2.80278 | + | 0.389365i | 0.437645 | − | 0.252674i | 2.17440i | 0.735010 | + | 2.90857i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
49.8 | 2.28087 | − | 1.31686i | 0.238330 | + | 1.71558i | 2.46825 | − | 4.27513i | 0 | 2.80278 | + | 3.59916i | −1.55662 | + | 0.898714i | − | 7.73393i | −2.88640 | + | 0.817746i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
124.1 | −2.28087 | − | 1.31686i | −0.238330 | + | 1.71558i | 2.46825 | + | 4.27513i | 0 | 2.80278 | − | 3.59916i | 1.55662 | + | 0.898714i | − | 7.73393i | −2.88640 | − | 0.817746i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
124.2 | −1.41485 | − | 0.816862i | −1.36657 | + | 1.06419i | 0.334526 | + | 0.579416i | 0 | 2.80278 | − | 0.389365i | −0.437645 | − | 0.252674i | 2.17440i | 0.735010 | − | 2.90857i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
124.3 | −1.27588 | − | 0.736627i | −0.350156 | − | 1.69629i | 0.0852394 | + | 0.147639i | 0 | −0.802776 | + | 2.42219i | −3.34791 | − | 1.93291i | 2.69535i | −2.75478 | + | 1.18793i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
124.4 | −0.409850 | − | 0.236627i | 1.64411 | − | 0.544899i | −0.888015 | − | 1.53809i | 0 | −0.802776 | − | 0.165713i | −2.21967 | − | 1.28153i | 1.78702i | 2.40617 | − | 1.79175i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
124.5 | 0.409850 | + | 0.236627i | −1.64411 | + | 0.544899i | −0.888015 | − | 1.53809i | 0 | −0.802776 | − | 0.165713i | 2.21967 | + | 1.28153i | − | 1.78702i | 2.40617 | − | 1.79175i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
124.6 | 1.27588 | + | 0.736627i | 0.350156 | + | 1.69629i | 0.0852394 | + | 0.147639i | 0 | −0.802776 | + | 2.42219i | 3.34791 | + | 1.93291i | − | 2.69535i | −2.75478 | + | 1.18793i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
124.7 | 1.41485 | + | 0.816862i | 1.36657 | − | 1.06419i | 0.334526 | + | 0.579416i | 0 | 2.80278 | − | 0.389365i | 0.437645 | + | 0.252674i | − | 2.17440i | 0.735010 | − | 2.90857i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
124.8 | 2.28087 | + | 1.31686i | 0.238330 | − | 1.71558i | 2.46825 | + | 4.27513i | 0 | 2.80278 | − | 3.59916i | −1.55662 | − | 0.898714i | 7.73393i | −2.88640 | − | 0.817746i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
5.b | even | 2 | 1 | inner |
9.c | even | 3 | 1 | inner |
45.j | even | 6 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 225.2.k.c | 16 | |
3.b | odd | 2 | 1 | 675.2.k.c | 16 | ||
5.b | even | 2 | 1 | inner | 225.2.k.c | 16 | |
5.c | odd | 4 | 1 | 225.2.e.c | ✓ | 8 | |
5.c | odd | 4 | 1 | 225.2.e.e | yes | 8 | |
9.c | even | 3 | 1 | inner | 225.2.k.c | 16 | |
9.c | even | 3 | 1 | 2025.2.b.n | 8 | ||
9.d | odd | 6 | 1 | 675.2.k.c | 16 | ||
9.d | odd | 6 | 1 | 2025.2.b.o | 8 | ||
15.d | odd | 2 | 1 | 675.2.k.c | 16 | ||
15.e | even | 4 | 1 | 675.2.e.c | 8 | ||
15.e | even | 4 | 1 | 675.2.e.e | 8 | ||
45.h | odd | 6 | 1 | 675.2.k.c | 16 | ||
45.h | odd | 6 | 1 | 2025.2.b.o | 8 | ||
45.j | even | 6 | 1 | inner | 225.2.k.c | 16 | |
45.j | even | 6 | 1 | 2025.2.b.n | 8 | ||
45.k | odd | 12 | 1 | 225.2.e.c | ✓ | 8 | |
45.k | odd | 12 | 1 | 225.2.e.e | yes | 8 | |
45.k | odd | 12 | 1 | 2025.2.a.q | 4 | ||
45.k | odd | 12 | 1 | 2025.2.a.y | 4 | ||
45.l | even | 12 | 1 | 675.2.e.c | 8 | ||
45.l | even | 12 | 1 | 675.2.e.e | 8 | ||
45.l | even | 12 | 1 | 2025.2.a.p | 4 | ||
45.l | even | 12 | 1 | 2025.2.a.z | 4 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
225.2.e.c | ✓ | 8 | 5.c | odd | 4 | 1 | |
225.2.e.c | ✓ | 8 | 45.k | odd | 12 | 1 | |
225.2.e.e | yes | 8 | 5.c | odd | 4 | 1 | |
225.2.e.e | yes | 8 | 45.k | odd | 12 | 1 | |
225.2.k.c | 16 | 1.a | even | 1 | 1 | trivial | |
225.2.k.c | 16 | 5.b | even | 2 | 1 | inner | |
225.2.k.c | 16 | 9.c | even | 3 | 1 | inner | |
225.2.k.c | 16 | 45.j | even | 6 | 1 | inner | |
675.2.e.c | 8 | 15.e | even | 4 | 1 | ||
675.2.e.c | 8 | 45.l | even | 12 | 1 | ||
675.2.e.e | 8 | 15.e | even | 4 | 1 | ||
675.2.e.e | 8 | 45.l | even | 12 | 1 | ||
675.2.k.c | 16 | 3.b | odd | 2 | 1 | ||
675.2.k.c | 16 | 9.d | odd | 6 | 1 | ||
675.2.k.c | 16 | 15.d | odd | 2 | 1 | ||
675.2.k.c | 16 | 45.h | odd | 6 | 1 | ||
2025.2.a.p | 4 | 45.l | even | 12 | 1 | ||
2025.2.a.q | 4 | 45.k | odd | 12 | 1 | ||
2025.2.a.y | 4 | 45.k | odd | 12 | 1 | ||
2025.2.a.z | 4 | 45.l | even | 12 | 1 | ||
2025.2.b.n | 8 | 9.c | even | 3 | 1 | ||
2025.2.b.n | 8 | 45.j | even | 6 | 1 | ||
2025.2.b.o | 8 | 9.d | odd | 6 | 1 | ||
2025.2.b.o | 8 | 45.h | odd | 6 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{16} - 12T_{2}^{14} + 102T_{2}^{12} - 406T_{2}^{10} + 1167T_{2}^{8} - 1842T_{2}^{6} + 2023T_{2}^{4} - 441T_{2}^{2} + 81 \)
acting on \(S_{2}^{\mathrm{new}}(225, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{16} - 12 T^{14} + 102 T^{12} + \cdots + 81 \)
$3$
\( T^{16} + 5 T^{14} + 4 T^{12} + 15 T^{10} + \cdots + 6561 \)
$5$
\( T^{16} \)
$7$
\( T^{16} - 25 T^{14} + 451 T^{12} + \cdots + 6561 \)
$11$
\( (T^{8} - T^{7} + 26 T^{6} + 107 T^{5} + \cdots + 81)^{2} \)
$13$
\( T^{16} - 64 T^{14} + \cdots + 131079601 \)
$17$
\( (T^{8} + 81 T^{6} + 2214 T^{4} + \cdots + 91809)^{2} \)
$19$
\( (T^{4} + 2 T^{3} - 27 T^{2} - 80 T - 25)^{4} \)
$23$
\( T^{16} - 111 T^{14} + \cdots + 3486784401 \)
$29$
\( (T^{8} - T^{7} + 41 T^{6} - 244 T^{5} + \cdots + 16641)^{2} \)
$31$
\( (T^{8} - 4 T^{7} + 58 T^{6} + 114 T^{5} + \cdots + 59049)^{2} \)
$37$
\( (T^{8} + 199 T^{6} + 9513 T^{4} + \cdots + 418609)^{2} \)
$41$
\( (T^{8} - 5 T^{7} + 50 T^{6} - 197 T^{5} + \cdots + 42849)^{2} \)
$43$
\( T^{16} - 196 T^{14} + \cdots + 205144679041 \)
$47$
\( T^{16} - 186 T^{14} + \cdots + 21071715921 \)
$53$
\( (T^{8} + 228 T^{6} + 13614 T^{4} + \cdots + 221841)^{2} \)
$59$
\( (T^{8} - 17 T^{7} + 287 T^{6} + \cdots + 5349969)^{2} \)
$61$
\( (T^{8} - 13 T^{7} + 172 T^{6} - 143 T^{5} + \cdots + 1)^{2} \)
$67$
\( T^{16} - 217 T^{14} + \cdots + 3486784401 \)
$71$
\( (T^{4} + 8 T^{3} - 40 T^{2} - 263 T + 381)^{4} \)
$73$
\( (T^{8} + 196 T^{6} + 8478 T^{4} + \cdots + 12769)^{2} \)
$79$
\( (T^{8} + 7 T^{7} + 82 T^{6} - 93 T^{5} + \cdots + 42849)^{2} \)
$83$
\( T^{16} - 324 T^{14} + \cdots + 282429536481 \)
$89$
\( (T^{4} - 9 T^{3} - 99 T^{2} + 405 T + 2025)^{4} \)
$97$
\( T^{16} - 199 T^{14} + \cdots + 824843587681 \)
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