Newspace parameters
Level: | \( N \) | \(=\) | \( 225 = 3^{2} \cdot 5^{2} \) |
Weight: | \( k \) | \(=\) | \( 4 \) |
Character orbit: | \([\chi]\) | \(=\) | 225.k (of order \(6\), degree \(2\), not minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(13.2754297513\) |
Analytic rank: | \(0\) |
Dimension: | \(12\) |
Relative dimension: | \(6\) over \(\Q(\zeta_{6})\) |
Coefficient field: | \(\mathbb{Q}[x]/(x^{12} - \cdots)\) |
Defining polynomial: |
\( x^{12} - 23x^{10} + 198x^{8} - 719x^{6} + 886x^{4} + 585x^{2} + 81 \)
|
Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
Coefficient ring index: | \( 2^{6}\cdot 3^{2} \) |
Twist minimal: | no (minimal twist has level 45) |
Sato-Tate group: | $\mathrm{SU}(2)[C_{6}]$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{11}\) 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
\( x^{12} - 23x^{10} + 198x^{8} - 719x^{6} + 886x^{4} + 585x^{2} + 81 \)
:
\(\beta_{1}\) | \(=\) |
\( ( 17\nu^{10} - 260\nu^{8} + 1311\nu^{6} - 2275\nu^{4} - 1388\nu^{2} + 33786 ) / 7875 \)
|
\(\beta_{2}\) | \(=\) |
\( ( -19\nu^{10} + 445\nu^{8} - 3627\nu^{6} + 9800\nu^{4} + 5566\nu^{2} - 24327 ) / 7875 \)
|
\(\beta_{3}\) | \(=\) |
\( ( -18\nu^{10} + 415\nu^{8} - 3594\nu^{6} + 13350\nu^{4} - 18398\nu^{2} - 5994 ) / 1125 \)
|
\(\beta_{4}\) | \(=\) |
\( ( -368\nu^{10} + 8665\nu^{8} - 77019\nu^{6} + 295225\nu^{4} - 411623\nu^{2} - 133794 ) / 7875 \)
|
\(\beta_{5}\) | \(=\) |
\( ( 181\nu^{11} - 4055\nu^{9} + 33723\nu^{7} - 115325\nu^{5} + 124141\nu^{3} + 132273\nu ) / 23625 \)
|
\(\beta_{6}\) | \(=\) |
\( ( 106\nu^{10} - 2455\nu^{8} + 21423\nu^{6} - 80150\nu^{4} + 111416\nu^{2} + 29448 ) / 1575 \)
|
\(\beta_{7}\) | \(=\) |
\( ( -88\nu^{11} + 2015\nu^{9} - 17154\nu^{7} + 60725\nu^{5} - 67168\nu^{3} - 70929\nu ) / 6750 \)
|
\(\beta_{8}\) | \(=\) |
\( ( -316\nu^{11} + 7355\nu^{9} - 64053\nu^{7} + 233450\nu^{5} - 264751\nu^{3} - 277353\nu ) / 23625 \)
|
\(\beta_{9}\) | \(=\) |
\( ( -1528\nu^{11} + 35465\nu^{9} - 309924\nu^{7} + 1163225\nu^{5} - 1604758\nu^{3} - 514449\nu ) / 23625 \)
|
\(\beta_{10}\) | \(=\) |
\( ( 2818\nu^{11} - 65540\nu^{9} + 575244\nu^{7} - 2181725\nu^{5} + 3102073\nu^{3} + 800019\nu ) / 23625 \)
|
\(\beta_{11}\) | \(=\) |
\( ( -811\nu^{11} + 18830\nu^{9} - 164763\nu^{7} + 620450\nu^{5} - 860821\nu^{3} - 275688\nu ) / 6750 \)
|
\(\nu\) | \(=\) |
\( ( 4\beta_{11} + 4\beta_{10} + \beta_{9} - 2\beta_{8} - 2\beta_{7} + 2\beta_{5} ) / 6 \)
|
\(\nu^{2}\) | \(=\) |
\( ( \beta_{6} + \beta_{4} + \beta_{3} + \beta_{2} - \beta _1 + 11 ) / 3 \)
|
\(\nu^{3}\) | \(=\) |
\( ( 26\beta_{11} + 20\beta_{10} - 7\beta_{9} - 10\beta_{8} - 4\beta_{7} + 13\beta_{5} ) / 6 \)
|
\(\nu^{4}\) | \(=\) |
\( ( 14\beta_{6} + 8\beta_{4} + 35\beta_{3} - \beta_{2} - 5\beta _1 + 79 ) / 3 \)
|
\(\nu^{5}\) | \(=\) |
\( ( 154\beta_{11} + 94\beta_{10} - 92\beta_{9} - 92\beta_{8} + 58\beta_{7} + 113\beta_{5} ) / 6 \)
|
\(\nu^{6}\) | \(=\) |
\( ( 151\beta_{6} + 46\beta_{4} + 505\beta_{3} - 26\beta_{2} + 2\beta _1 + 560 ) / 3 \)
|
\(\nu^{7}\) | \(=\) |
\( ( 692\beta_{11} + 350\beta_{10} - 565\beta_{9} - 868\beta_{8} + 1166\beta_{7} + 1102\beta_{5} ) / 6 \)
|
\(\nu^{8}\) | \(=\) |
\( ( 1430\beta_{6} + 242\beta_{4} + 5399\beta_{3} - 190\beta_{2} + 460\beta _1 + 3580 ) / 3 \)
|
\(\nu^{9}\) | \(=\) |
\( ( 262\beta_{11} - 218\beta_{10} - 947\beta_{9} - 7154\beta_{8} + 14116\beta_{7} + 11039\beta_{5} ) / 6 \)
|
\(\nu^{10}\) | \(=\) |
\( ( 12181\beta_{6} + 1306\beta_{4} + 48394\beta_{3} - 953\beta_{2} + 7520\beta _1 + 17075 ) / 3 \)
|
\(\nu^{11}\) | \(=\) |
\( ( -45694\beta_{11} - 26842\beta_{10} + 29504\beta_{9} - 48850\beta_{8} + 140162\beta_{7} + 104395\beta_{5} ) / 6 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/225\mathbb{Z}\right)^\times\).
\(n\) | \(101\) | \(127\) |
\(\chi(n)\) | \(\beta_{3}\) | \(-1\) |
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 |
|
−3.96084 | + | 2.28679i | −3.96084 | − | 3.36330i | 6.45882 | − | 11.1870i | 0 | 23.3794 | + | 4.26387i | −17.4197 | + | 10.0573i | 22.4912i | 4.37646 | + | 26.6429i | 0 | ||||||||||||||||||||||||||||||||||||||||||
49.2 | −3.24635 | + | 1.87428i | −3.24635 | − | 4.05724i | 3.02587 | − | 5.24096i | 0 | 18.1432 | + | 7.08665i | 27.1492 | − | 15.6746i | − | 7.30318i | −5.92239 | + | 26.3425i | 0 | ||||||||||||||||||||||||||||||||||||||||||
49.3 | −0.151541 | + | 0.0874923i | −0.151541 | − | 5.19394i | −3.98469 | + | 6.90169i | 0 | 0.477395 | + | 0.773837i | 7.32979 | − | 4.23186i | − | 2.79440i | −26.9541 | + | 1.57419i | 0 | ||||||||||||||||||||||||||||||||||||||||||
49.4 | 0.151541 | − | 0.0874923i | 0.151541 | + | 5.19394i | −3.98469 | + | 6.90169i | 0 | 0.477395 | + | 0.773837i | −7.32979 | + | 4.23186i | 2.79440i | −26.9541 | + | 1.57419i | 0 | |||||||||||||||||||||||||||||||||||||||||||
49.5 | 3.24635 | − | 1.87428i | 3.24635 | + | 4.05724i | 3.02587 | − | 5.24096i | 0 | 18.1432 | + | 7.08665i | −27.1492 | + | 15.6746i | 7.30318i | −5.92239 | + | 26.3425i | 0 | |||||||||||||||||||||||||||||||||||||||||||
49.6 | 3.96084 | − | 2.28679i | 3.96084 | + | 3.36330i | 6.45882 | − | 11.1870i | 0 | 23.3794 | + | 4.26387i | 17.4197 | − | 10.0573i | − | 22.4912i | 4.37646 | + | 26.6429i | 0 | ||||||||||||||||||||||||||||||||||||||||||
124.1 | −3.96084 | − | 2.28679i | −3.96084 | + | 3.36330i | 6.45882 | + | 11.1870i | 0 | 23.3794 | − | 4.26387i | −17.4197 | − | 10.0573i | − | 22.4912i | 4.37646 | − | 26.6429i | 0 | ||||||||||||||||||||||||||||||||||||||||||
124.2 | −3.24635 | − | 1.87428i | −3.24635 | + | 4.05724i | 3.02587 | + | 5.24096i | 0 | 18.1432 | − | 7.08665i | 27.1492 | + | 15.6746i | 7.30318i | −5.92239 | − | 26.3425i | 0 | |||||||||||||||||||||||||||||||||||||||||||
124.3 | −0.151541 | − | 0.0874923i | −0.151541 | + | 5.19394i | −3.98469 | − | 6.90169i | 0 | 0.477395 | − | 0.773837i | 7.32979 | + | 4.23186i | 2.79440i | −26.9541 | − | 1.57419i | 0 | |||||||||||||||||||||||||||||||||||||||||||
124.4 | 0.151541 | + | 0.0874923i | 0.151541 | − | 5.19394i | −3.98469 | − | 6.90169i | 0 | 0.477395 | − | 0.773837i | −7.32979 | − | 4.23186i | − | 2.79440i | −26.9541 | − | 1.57419i | 0 | ||||||||||||||||||||||||||||||||||||||||||
124.5 | 3.24635 | + | 1.87428i | 3.24635 | − | 4.05724i | 3.02587 | + | 5.24096i | 0 | 18.1432 | − | 7.08665i | −27.1492 | − | 15.6746i | − | 7.30318i | −5.92239 | − | 26.3425i | 0 | ||||||||||||||||||||||||||||||||||||||||||
124.6 | 3.96084 | + | 2.28679i | 3.96084 | − | 3.36330i | 6.45882 | + | 11.1870i | 0 | 23.3794 | − | 4.26387i | 17.4197 | + | 10.0573i | 22.4912i | 4.37646 | − | 26.6429i | 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 | 225.4.k.c | 12 | |
5.b | even | 2 | 1 | inner | 225.4.k.c | 12 | |
5.c | odd | 4 | 1 | 45.4.e.b | ✓ | 6 | |
5.c | odd | 4 | 1 | 225.4.e.c | 6 | ||
9.c | even | 3 | 1 | inner | 225.4.k.c | 12 | |
15.e | even | 4 | 1 | 135.4.e.b | 6 | ||
45.j | even | 6 | 1 | inner | 225.4.k.c | 12 | |
45.k | odd | 12 | 1 | 45.4.e.b | ✓ | 6 | |
45.k | odd | 12 | 1 | 225.4.e.c | 6 | ||
45.k | odd | 12 | 1 | 405.4.a.h | 3 | ||
45.k | odd | 12 | 1 | 2025.4.a.s | 3 | ||
45.l | even | 12 | 1 | 135.4.e.b | 6 | ||
45.l | even | 12 | 1 | 405.4.a.j | 3 | ||
45.l | even | 12 | 1 | 2025.4.a.q | 3 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
45.4.e.b | ✓ | 6 | 5.c | odd | 4 | 1 | |
45.4.e.b | ✓ | 6 | 45.k | odd | 12 | 1 | |
135.4.e.b | 6 | 15.e | even | 4 | 1 | ||
135.4.e.b | 6 | 45.l | even | 12 | 1 | ||
225.4.e.c | 6 | 5.c | odd | 4 | 1 | ||
225.4.e.c | 6 | 45.k | odd | 12 | 1 | ||
225.4.k.c | 12 | 1.a | even | 1 | 1 | trivial | |
225.4.k.c | 12 | 5.b | even | 2 | 1 | inner | |
225.4.k.c | 12 | 9.c | even | 3 | 1 | inner | |
225.4.k.c | 12 | 45.j | even | 6 | 1 | inner | |
405.4.a.h | 3 | 45.k | odd | 12 | 1 | ||
405.4.a.j | 3 | 45.l | even | 12 | 1 | ||
2025.4.a.q | 3 | 45.l | even | 12 | 1 | ||
2025.4.a.s | 3 | 45.k | odd | 12 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{12} - 35T_{2}^{10} + 930T_{2}^{8} - 10307T_{2}^{6} + 86710T_{2}^{4} - 2655T_{2}^{2} + 81 \)
acting on \(S_{4}^{\mathrm{new}}(225, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{12} - 35 T^{10} + 930 T^{8} + \cdots + 81 \)
$3$
\( T^{12} + 57 T^{10} + \cdots + 387420489 \)
$5$
\( T^{12} \)
$7$
\( T^{12} + \cdots + 811313702977761 \)
$11$
\( (T^{6} + 14 T^{5} + 3012 T^{4} + \cdots + 1896428304)^{2} \)
$13$
\( T^{12} - 6504 T^{10} + \cdots + 32\!\cdots\!16 \)
$17$
\( (T^{6} + 9716 T^{4} + \cdots + 24437192976)^{2} \)
$19$
\( (T^{3} - 164 T^{2} + 7292 T - 57316)^{4} \)
$23$
\( T^{12} - 38907 T^{10} + \cdots + 14\!\cdots\!61 \)
$29$
\( (T^{6} + 335 T^{5} + \cdots + 11463342489)^{2} \)
$31$
\( (T^{6} - 352 T^{5} + \cdots + 97238137683600)^{2} \)
$37$
\( (T^{6} + 112188 T^{4} + \cdots + 11124119360656)^{2} \)
$41$
\( (T^{6} + 187 T^{5} + \cdots + 52247959475625)^{2} \)
$43$
\( T^{12} - 342492 T^{10} + \cdots + 56\!\cdots\!76 \)
$47$
\( T^{12} - 251663 T^{10} + \cdots + 38\!\cdots\!21 \)
$53$
\( (T^{6} + 320300 T^{4} + \cdots + 10565984287296)^{2} \)
$59$
\( (T^{6} + 298 T^{5} + \cdots + 16\!\cdots\!16)^{2} \)
$61$
\( (T^{6} - 1439 T^{5} + \cdots + 30\!\cdots\!09)^{2} \)
$67$
\( T^{12} - 1231447 T^{10} + \cdots + 18\!\cdots\!61 \)
$71$
\( (T^{3} - 70 T^{2} - 685460 T + 223775052)^{4} \)
$73$
\( (T^{6} + 881280 T^{4} + \cdots + 18\!\cdots\!36)^{2} \)
$79$
\( (T^{6} + 382 T^{5} + \cdots + 19\!\cdots\!56)^{2} \)
$83$
\( T^{12} - 3012507 T^{10} + \cdots + 83\!\cdots\!81 \)
$89$
\( (T^{3} + 1719 T^{2} + 238491 T - 125506395)^{4} \)
$97$
\( T^{12} - 214860 T^{10} + \cdots + 81\!\cdots\!16 \)
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