Newspace parameters
| Level: | \( N \) | \(=\) | \( 225 = 3^{2} \cdot 5^{2} \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 225.e (of order \(3\), degree \(2\), minimal) |
Newform invariants
| Self dual: | no |
| Analytic conductor: | \(13.2754297513\) |
| Analytic rank: | \(0\) |
| Dimension: | \(14\) |
| Relative dimension: | \(7\) over \(\Q(\zeta_{3})\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{14} - \cdots)\) |
| Defining polynomial: |
\( x^{14} - 2 x^{13} + 48 x^{12} - 60 x^{11} + 1605 x^{10} - 1800 x^{9} + 23232 x^{8} - 2346 x^{7} + 209529 x^{6} - 55412 x^{5} + 765088 x^{4} + 276096 x^{3} + 1572480 x^{2} + \cdots + 82944 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{4}]\) |
| Coefficient ring index: | \( 2^{2}\cdot 3^{6} \) |
| Twist minimal: | no (minimal twist has level 45) |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{3}]$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{13}\) 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
\( x^{14} - 2 x^{13} + 48 x^{12} - 60 x^{11} + 1605 x^{10} - 1800 x^{9} + 23232 x^{8} - 2346 x^{7} + 209529 x^{6} - 55412 x^{5} + 765088 x^{4} + 276096 x^{3} + 1572480 x^{2} + \cdots + 82944 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( - 20061126179 \nu^{13} + 306116042626 \nu^{12} - 1952416155384 \nu^{11} + 13978725434868 \nu^{10} - 67214635202751 \nu^{9} + \cdots + 81\!\cdots\!72 ) / 34\!\cdots\!00 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 102404815617 \nu^{13} + 5266027591948 \nu^{12} - 8326481101332 \nu^{11} + 268246879929564 \nu^{10} + \cdots + 63\!\cdots\!56 ) / 68\!\cdots\!00 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 2939472898789 \nu^{13} + 5818762419041 \nu^{12} - 140176351013994 \nu^{11} + 170511125461188 \nu^{10} + \cdots + 93\!\cdots\!52 ) / 10\!\cdots\!00 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 376371909409 \nu^{13} + 779547299216 \nu^{12} - 17633627499780 \nu^{11} + 23331293206188 \nu^{10} + \cdots - 63\!\cdots\!28 ) / 81\!\cdots\!68 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 2181076654577 \nu^{13} + 4927484464048 \nu^{12} - 102520174110372 \nu^{11} + 135970911492204 \nu^{10} + \cdots - 22\!\cdots\!24 ) / 40\!\cdots\!40 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - 11125627233907 \nu^{13} + 27262271328908 \nu^{12} - 529205531611572 \nu^{11} + 877253848876644 \nu^{10} + \cdots + 11\!\cdots\!76 ) / 20\!\cdots\!00 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( 11877650203069 \nu^{13} - 84950703264836 \nu^{12} + 708621055239924 \nu^{11} + \cdots - 58\!\cdots\!92 ) / 20\!\cdots\!00 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( 10816153989491 \nu^{13} - 13376805115054 \nu^{12} + 497974097019636 \nu^{11} - 253186185774972 \nu^{10} + \cdots - 56\!\cdots\!88 ) / 10\!\cdots\!00 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( 4781648360579 \nu^{13} - 8884383807886 \nu^{12} + 231881123650104 \nu^{11} - 261650594664228 \nu^{10} + \cdots + 41\!\cdots\!28 ) / 40\!\cdots\!40 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( - 6918798848131 \nu^{13} + 13573388130414 \nu^{12} - 327948078905676 \nu^{11} + 388609132067052 \nu^{10} + \cdots + 26\!\cdots\!08 ) / 34\!\cdots\!00 \)
|
| \(\beta_{12}\) | \(=\) |
\( ( - 2499552103127 \nu^{13} + 3909850857688 \nu^{12} - 117122204938752 \nu^{11} + 101188546669524 \nu^{10} + \cdots + 88\!\cdots\!96 ) / 10\!\cdots\!60 \)
|
| \(\beta_{13}\) | \(=\) |
\( ( - 47810889109763 \nu^{13} + 95353418254572 \nu^{12} + \cdots + 18\!\cdots\!84 ) / 68\!\cdots\!00 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{11} - \beta_{10} + \beta_{7} - 13\beta_{4} - \beta_{2} + \beta_1 \)
|
| \(\nu^{3}\) | \(=\) |
\( -\beta_{12} - \beta_{11} - \beta_{10} - 2\beta_{9} - \beta_{8} + \beta_{5} + \beta_{4} - \beta_{3} - 21\beta_{2} - 1 \)
|
| \(\nu^{4}\) | \(=\) |
\( 3 \beta_{13} - 3 \beta_{12} - 30 \beta_{11} + 24 \beta_{10} + 3 \beta_{8} + 9 \beta_{7} - 27 \beta_{5} + 294 \beta_{4} - 25 \beta _1 - 291 \)
|
| \(\nu^{5}\) | \(=\) |
\( - 30 \beta_{13} + 66 \beta_{12} + 80 \beta_{11} + 4 \beta_{10} + 78 \beta_{9} + 26 \beta_{7} - 78 \beta_{6} + 36 \beta_{5} - 62 \beta_{4} - 36 \beta_{3} + 529 \beta_{2} - 529 \beta _1 + 24 \)
|
| \(\nu^{6}\) | \(=\) |
\( 130 \beta_{12} + 130 \beta_{11} + 382 \beta_{10} + 8 \beta_{9} - 122 \beta_{8} - 1029 \beta_{7} + 851 \beta_{5} - 106 \beta_{4} - 146 \beta_{3} + 585 \beta_{2} + 7591 \)
|
| \(\nu^{7}\) | \(=\) |
\( 819 \beta_{13} - 771 \beta_{12} - 1515 \beta_{11} + 687 \beta_{10} + 819 \beta_{8} - 939 \beta_{7} + 2490 \beta_{6} - 2658 \beta_{5} + 3033 \beta_{4} + 1914 \beta_{3} + 14197 \beta _1 - 3600 \)
|
| \(\nu^{8}\) | \(=\) |
\( - 4053 \beta_{13} + 840 \beta_{12} + 19675 \beta_{11} - 30877 \beta_{10} - 480 \beta_{9} + 18262 \beta_{7} + 480 \beta_{6} - 4365 \beta_{5} - 202939 \beta_{4} + 4365 \beta_{3} - 13585 \beta_{2} + \cdots - 3045 \)
|
| \(\nu^{9}\) | \(=\) |
\( - 34738 \beta_{12} - 34738 \beta_{11} - 29248 \beta_{10} - 74966 \beta_{9} - 22408 \beta_{8} + 9672 \beta_{7} + 48166 \beta_{5} + 14278 \beta_{4} - 19768 \beta_{3} - 391569 \beta_{2} + \cdots + 132320 \)
|
| \(\nu^{10}\) | \(=\) |
\( 125772 \beta_{13} - 166692 \beta_{12} - 681957 \beta_{11} + 522429 \beta_{10} + 125772 \beta_{8} + 385404 \beta_{7} - 18264 \beta_{6} - 547401 \beta_{5} + 5722041 \beta_{4} + 32136 \beta_{3} + \cdots - 5637885 \)
|
| \(\nu^{11}\) | \(=\) |
\( - 622773 \beta_{13} + 1551186 \beta_{12} + 2507858 \beta_{11} + 304264 \beta_{10} + 2204082 \beta_{9} + 579293 \beta_{7} - 2204082 \beta_{6} + 1029213 \beta_{5} - 5182736 \beta_{4} + \cdots + 376317 \)
|
| \(\nu^{12}\) | \(=\) |
\( 3988855 \beta_{12} + 3988855 \beta_{11} + 11399653 \beta_{10} + 566912 \beta_{9} - 3783527 \beta_{8} - 26396937 \beta_{7} + 19434914 \beta_{5} - 2321479 \beta_{4} + \cdots + 161007274 \)
|
| \(\nu^{13}\) | \(=\) |
\( 17562210 \beta_{13} - 14227458 \beta_{12} - 44588544 \beta_{11} + 17387382 \beta_{10} + 17562210 \beta_{8} - 25965810 \beta_{7} + 64059678 \beta_{6} - 74597040 \beta_{5} + \cdots - 170775582 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/225\mathbb{Z}\right)^\times\).
| \(n\) | \(101\) | \(127\) |
| \(\chi(n)\) | \(-1 + \beta_{4}\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 76.1 |
|
−2.65775 | − | 4.60336i | 5.19389 | + | 0.153351i | −10.1273 | + | 17.5410i | 0 | −13.0981 | − | 24.3169i | −6.71686 | − | 11.6339i | 65.1396 | 26.9530 | + | 1.59298i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||
| 76.2 | −2.13089 | − | 3.69081i | −4.39640 | − | 2.76977i | −5.08138 | + | 8.80120i | 0 | −0.854448 | + | 22.1284i | 15.3820 | + | 26.6423i | 9.21718 | 11.6567 | + | 24.3541i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 76.3 | −1.09722 | − | 1.90044i | −0.206141 | + | 5.19206i | 1.59221 | − | 2.75778i | 0 | 10.0934 | − | 5.30509i | 1.38302 | + | 2.39547i | −24.5436 | −26.9150 | − | 2.14059i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 76.4 | −0.112625 | − | 0.195072i | −2.06755 | − | 4.76710i | 3.97463 | − | 6.88426i | 0 | −0.697071 | + | 0.940215i | −15.5970 | − | 27.0148i | −3.59257 | −18.4505 | + | 19.7124i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 76.5 | 0.785104 | + | 1.35984i | 3.89934 | − | 3.43441i | 2.76722 | − | 4.79297i | 0 | 7.73164 | + | 2.60611i | 17.1199 | + | 29.6525i | 21.2519 | 3.40971 | − | 26.7838i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 76.6 | 1.52087 | + | 2.63422i | −4.01867 | + | 3.29398i | −0.626094 | + | 1.08443i | 0 | −14.7890 | − | 5.57636i | −6.85611 | − | 11.8751i | 20.5251 | 5.29939 | − | 26.4748i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 76.7 | 2.69252 | + | 4.66357i | 4.09553 | + | 3.19791i | −10.4993 | + | 18.1853i | 0 | −3.88642 | + | 27.7102i | 6.28510 | + | 10.8861i | −69.9976 | 6.54673 | + | 26.1943i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 151.1 | −2.65775 | + | 4.60336i | 5.19389 | − | 0.153351i | −10.1273 | − | 17.5410i | 0 | −13.0981 | + | 24.3169i | −6.71686 | + | 11.6339i | 65.1396 | 26.9530 | − | 1.59298i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 151.2 | −2.13089 | + | 3.69081i | −4.39640 | + | 2.76977i | −5.08138 | − | 8.80120i | 0 | −0.854448 | − | 22.1284i | 15.3820 | − | 26.6423i | 9.21718 | 11.6567 | − | 24.3541i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 151.3 | −1.09722 | + | 1.90044i | −0.206141 | − | 5.19206i | 1.59221 | + | 2.75778i | 0 | 10.0934 | + | 5.30509i | 1.38302 | − | 2.39547i | −24.5436 | −26.9150 | + | 2.14059i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 151.4 | −0.112625 | + | 0.195072i | −2.06755 | + | 4.76710i | 3.97463 | + | 6.88426i | 0 | −0.697071 | − | 0.940215i | −15.5970 | + | 27.0148i | −3.59257 | −18.4505 | − | 19.7124i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 151.5 | 0.785104 | − | 1.35984i | 3.89934 | + | 3.43441i | 2.76722 | + | 4.79297i | 0 | 7.73164 | − | 2.60611i | 17.1199 | − | 29.6525i | 21.2519 | 3.40971 | + | 26.7838i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 151.6 | 1.52087 | − | 2.63422i | −4.01867 | − | 3.29398i | −0.626094 | − | 1.08443i | 0 | −14.7890 | + | 5.57636i | −6.85611 | + | 11.8751i | 20.5251 | 5.29939 | + | 26.4748i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 151.7 | 2.69252 | − | 4.66357i | 4.09553 | − | 3.19791i | −10.4993 | − | 18.1853i | 0 | −3.88642 | − | 27.7102i | 6.28510 | − | 10.8861i | −69.9976 | 6.54673 | − | 26.1943i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 9.c | even | 3 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 225.4.e.d | 14 | |
| 5.b | even | 2 | 1 | 45.4.e.c | ✓ | 14 | |
| 5.c | odd | 4 | 2 | 225.4.k.d | 28 | ||
| 9.c | even | 3 | 1 | inner | 225.4.e.d | 14 | |
| 9.c | even | 3 | 1 | 2025.4.a.bb | 7 | ||
| 9.d | odd | 6 | 1 | 2025.4.a.ba | 7 | ||
| 15.d | odd | 2 | 1 | 135.4.e.c | 14 | ||
| 45.h | odd | 6 | 1 | 135.4.e.c | 14 | ||
| 45.h | odd | 6 | 1 | 405.4.a.n | 7 | ||
| 45.j | even | 6 | 1 | 45.4.e.c | ✓ | 14 | |
| 45.j | even | 6 | 1 | 405.4.a.m | 7 | ||
| 45.k | odd | 12 | 2 | 225.4.k.d | 28 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 45.4.e.c | ✓ | 14 | 5.b | even | 2 | 1 | |
| 45.4.e.c | ✓ | 14 | 45.j | even | 6 | 1 | |
| 135.4.e.c | 14 | 15.d | odd | 2 | 1 | ||
| 135.4.e.c | 14 | 45.h | odd | 6 | 1 | ||
| 225.4.e.d | 14 | 1.a | even | 1 | 1 | trivial | |
| 225.4.e.d | 14 | 9.c | even | 3 | 1 | inner | |
| 225.4.k.d | 28 | 5.c | odd | 4 | 2 | ||
| 225.4.k.d | 28 | 45.k | odd | 12 | 2 | ||
| 405.4.a.m | 7 | 45.j | even | 6 | 1 | ||
| 405.4.a.n | 7 | 45.h | odd | 6 | 1 | ||
| 2025.4.a.ba | 7 | 9.d | odd | 6 | 1 | ||
| 2025.4.a.bb | 7 | 9.c | even | 3 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{14} + 2 T_{2}^{13} + 48 T_{2}^{12} + 60 T_{2}^{11} + 1605 T_{2}^{10} + 1800 T_{2}^{9} + 23232 T_{2}^{8} + 2346 T_{2}^{7} + 209529 T_{2}^{6} + 55412 T_{2}^{5} + 765088 T_{2}^{4} - 276096 T_{2}^{3} + 1572480 T_{2}^{2} + \cdots + 82944 \)
acting on \(S_{4}^{\mathrm{new}}(225, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{14} + 2 T^{13} + 48 T^{12} + \cdots + 82944 \)
$3$
\( T^{14} - 5 T^{13} + \cdots + 10460353203 \)
$5$
\( T^{14} \)
$7$
\( T^{14} - 22 T^{13} + \cdots + 44\!\cdots\!36 \)
$11$
\( T^{14} - 23 T^{13} + \cdots + 40\!\cdots\!24 \)
$13$
\( T^{14} - 96 T^{13} + \cdots + 25\!\cdots\!76 \)
$17$
\( (T^{7} - 161 T^{6} + \cdots + 60588009792)^{2} \)
$19$
\( (T^{7} - 279 T^{6} + \cdots - 1377989598400)^{2} \)
$23$
\( T^{14} + 96 T^{13} + \cdots + 16\!\cdots\!44 \)
$29$
\( T^{14} + 296 T^{13} + \cdots + 14\!\cdots\!76 \)
$31$
\( T^{14} + 244 T^{13} + \cdots + 86\!\cdots\!16 \)
$37$
\( (T^{7} + 404 T^{6} + \cdots + 83646911884544)^{2} \)
$41$
\( T^{14} + 47 T^{13} + \cdots + 23\!\cdots\!21 \)
$43$
\( T^{14} - 525 T^{13} + \cdots + 17\!\cdots\!56 \)
$47$
\( T^{14} + 164 T^{13} + \cdots + 42\!\cdots\!84 \)
$53$
\( (T^{7} - 506 T^{6} + \cdots + 11\!\cdots\!92)^{2} \)
$59$
\( T^{14} + 85 T^{13} + \cdots + 35\!\cdots\!64 \)
$61$
\( T^{14} + 828 T^{13} + \cdots + 14\!\cdots\!16 \)
$67$
\( T^{14} - 1093 T^{13} + \cdots + 85\!\cdots\!09 \)
$71$
\( (T^{7} + 328 T^{6} + \cdots - 32\!\cdots\!28)^{2} \)
$73$
\( (T^{7} + 2085 T^{6} + \cdots - 82\!\cdots\!12)^{2} \)
$79$
\( T^{14} + 2110 T^{13} + \cdots + 26\!\cdots\!64 \)
$83$
\( T^{14} + 1290 T^{13} + \cdots + 43\!\cdots\!84 \)
$89$
\( (T^{7} - 3048 T^{6} + \cdots - 32\!\cdots\!50)^{2} \)
$97$
\( T^{14} - 1787 T^{13} + \cdots + 22\!\cdots\!76 \)
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