Newspace parameters
| Level: | \( N \) | \(=\) | \( 75 = 3 \cdot 5^{2} \) |
| Weight: | \( k \) | \(=\) | \( 18 \) |
| Character orbit: | \([\chi]\) | \(=\) | 75.b (of order \(2\), degree \(1\), not minimal) |
Newform invariants
| Self dual: | no |
| Analytic conductor: | \(137.416565508\) |
| Analytic rank: | \(0\) |
| Dimension: | \(10\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{10} - \cdots)\) |
|
|
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| Defining polynomial: |
\( x^{10} - 2 x^{9} + 2 x^{8} - 1889754 x^{7} + 1938226081 x^{6} - 95481875708 x^{5} + \cdots + 14\!\cdots\!52 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 2^{26}\cdot 3^{6}\cdot 5^{8} \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{9}\) 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
\( x^{10} - 2 x^{9} + 2 x^{8} - 1889754 x^{7} + 1938226081 x^{6} - 95481875708 x^{5} + \cdots + 14\!\cdots\!52 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( - 66\!\cdots\!61 \nu^{9} + \cdots + 36\!\cdots\!52 ) / 38\!\cdots\!00 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 17\!\cdots\!31 \nu^{9} + \cdots - 19\!\cdots\!92 ) / 38\!\cdots\!00 \)
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| \(\beta_{3}\) | \(=\) |
\( ( - 25\!\cdots\!21 \nu^{9} + \cdots + 26\!\cdots\!72 ) / 53\!\cdots\!00 \)
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| \(\beta_{4}\) | \(=\) |
\( ( 47\!\cdots\!29 \nu^{9} + \cdots + 18\!\cdots\!72 ) / 10\!\cdots\!00 \)
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| \(\beta_{5}\) | \(=\) |
\( ( - 14\!\cdots\!84 \nu^{9} + \cdots + 80\!\cdots\!88 ) / 47\!\cdots\!00 \)
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| \(\beta_{6}\) | \(=\) |
\( ( 50\!\cdots\!61 \nu^{9} + \cdots - 24\!\cdots\!52 ) / 10\!\cdots\!00 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 78\!\cdots\!53 \nu^{9} + \cdots + 20\!\cdots\!04 ) / 46\!\cdots\!00 \)
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| \(\beta_{8}\) | \(=\) |
\( ( 35\!\cdots\!12 \nu^{9} + \cdots - 21\!\cdots\!84 ) / 95\!\cdots\!00 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( - 16\!\cdots\!87 \nu^{9} + \cdots + 72\!\cdots\!84 ) / 15\!\cdots\!00 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{3} + \beta_{2} - \beta _1 + 1 ) / 4 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{5} - 101\beta_{2} - 181823\beta_1 ) / 8 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( - 4 \beta_{8} - 4 \beta_{6} - 59 \beta_{5} - 59 \beta_{4} - 68129 \beta_{3} + 68129 \beta_{2} + \cdots + 4521779 ) / 8 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( -104\beta_{7} + 752\beta_{6} + 45715\beta_{4} + 8107675\beta_{3} - 6188607225 ) / 8 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 52 \beta_{9} + 113274 \beta_{8} - 52 \beta_{7} - 113274 \beta_{6} + 2154331 \beta_{5} + \cdots + 182950996774 ) / 4 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( 5910008\beta_{9} - 50292920\beta_{8} - 2061911115\beta_{5} + 492531704207\beta_{2} + 242675370236485\beta_1 ) / 8 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( - 92357200 \beta_{9} + 10994867996 \beta_{8} - 92357200 \beta_{7} + 10994867996 \beta_{6} + \cdots - 22\!\cdots\!25 ) / 8 \)
|
| \(\nu^{8}\) | \(=\) |
\( ( 268318699896 \beta_{7} - 2736128495664 \beta_{6} - 94803300527225 \beta_{4} + \cdots + 10\!\cdots\!75 ) / 8 \)
|
| \(\nu^{9}\) | \(=\) |
\( ( 5039696976756 \beta_{9} - 259893461169886 \beta_{8} - 5039696976756 \beta_{7} + 259893461169886 \beta_{6} + \cdots - 60\!\cdots\!32 ) / 4 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/75\mathbb{Z}\right)^\times\).
| \(n\) | \(26\) | \(52\) |
| \(\chi(n)\) | \(1\) | \(-1\) |
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 |
|
− | 582.966i | − | 6561.00i | −208777. | 0 | −3.82484e6 | − | 8.47467e6i | 4.52994e7i | −4.30467e7 | 0 | |||||||||||||||||||||||||||||||||||||||||||||
| 49.2 | − | 519.013i | 6561.00i | −138303. | 0 | 3.40525e6 | − | 2.20319e7i | 3.75290e6i | −4.30467e7 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||
| 49.3 | − | 464.877i | 6561.00i | −85038.8 | 0 | 3.05006e6 | 1.02079e7i | − | 2.13998e7i | −4.30467e7 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||
| 49.4 | − | 289.266i | − | 6561.00i | 47397.4 | 0 | −1.89787e6 | 8.64842e6i | − | 5.16251e7i | −4.30467e7 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||
| 49.5 | − | 92.3407i | 6561.00i | 122545. | 0 | 605848. | 1.41672e7i | − | 2.34192e7i | −4.30467e7 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||
| 49.6 | 92.3407i | − | 6561.00i | 122545. | 0 | 605848. | − | 1.41672e7i | 2.34192e7i | −4.30467e7 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||
| 49.7 | 289.266i | 6561.00i | 47397.4 | 0 | −1.89787e6 | − | 8.64842e6i | 5.16251e7i | −4.30467e7 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||
| 49.8 | 464.877i | − | 6561.00i | −85038.8 | 0 | 3.05006e6 | − | 1.02079e7i | 2.13998e7i | −4.30467e7 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||
| 49.9 | 519.013i | − | 6561.00i | −138303. | 0 | 3.40525e6 | 2.20319e7i | − | 3.75290e6i | −4.30467e7 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||
| 49.10 | 582.966i | 6561.00i | −208777. | 0 | −3.82484e6 | 8.47467e6i | − | 4.52994e7i | −4.30467e7 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 5.b | even | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 75.18.b.g | 10 | |
| 5.b | even | 2 | 1 | inner | 75.18.b.g | 10 | |
| 5.c | odd | 4 | 1 | 75.18.a.g | ✓ | 5 | |
| 5.c | odd | 4 | 1 | 75.18.a.h | yes | 5 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 75.18.a.g | ✓ | 5 | 5.c | odd | 4 | 1 | |
| 75.18.a.h | yes | 5 | 5.c | odd | 4 | 1 | |
| 75.18.b.g | 10 | 1.a | even | 1 | 1 | trivial | |
| 75.18.b.g | 10 | 5.b | even | 2 | 1 | inner | |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{10} + 917536 T_{2}^{8} + 300017076288 T_{2}^{6} + \cdots + 14\!\cdots\!96 \)
acting on \(S_{18}^{\mathrm{new}}(75, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{10} + \cdots + 14\!\cdots\!96 \)
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| $3$ |
\( (T^{2} + 43046721)^{5} \)
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| $5$ |
\( T^{10} \)
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| $7$ |
\( T^{10} + \cdots + 54\!\cdots\!25 \)
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| $11$ |
\( (T^{5} + \cdots + 19\!\cdots\!96)^{2} \)
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| $13$ |
\( T^{10} + \cdots + 10\!\cdots\!69 \)
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| $17$ |
\( T^{10} + \cdots + 37\!\cdots\!64 \)
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| $19$ |
\( (T^{5} + \cdots - 12\!\cdots\!75)^{2} \)
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| $23$ |
\( T^{10} + \cdots + 85\!\cdots\!00 \)
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| $29$ |
\( (T^{5} + \cdots + 10\!\cdots\!00)^{2} \)
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| $31$ |
\( (T^{5} + \cdots + 75\!\cdots\!25)^{2} \)
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| $37$ |
\( T^{10} + \cdots + 46\!\cdots\!00 \)
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| $41$ |
\( (T^{5} + \cdots - 34\!\cdots\!00)^{2} \)
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| $43$ |
\( T^{10} + \cdots + 92\!\cdots\!41 \)
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| $47$ |
\( T^{10} + \cdots + 23\!\cdots\!04 \)
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| $53$ |
\( T^{10} + \cdots + 23\!\cdots\!00 \)
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| $59$ |
\( (T^{5} + \cdots + 10\!\cdots\!00)^{2} \)
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| $61$ |
\( (T^{5} + \cdots + 13\!\cdots\!97)^{2} \)
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| $67$ |
\( T^{10} + \cdots + 43\!\cdots\!69 \)
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| $71$ |
\( (T^{5} + \cdots - 10\!\cdots\!52)^{2} \)
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| $73$ |
\( T^{10} + \cdots + 61\!\cdots\!00 \)
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| $79$ |
\( (T^{5} + \cdots + 62\!\cdots\!00)^{2} \)
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| $83$ |
\( T^{10} + \cdots + 21\!\cdots\!84 \)
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| $89$ |
\( (T^{5} + \cdots + 49\!\cdots\!00)^{2} \)
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| $97$ |
\( T^{10} + \cdots + 15\!\cdots\!29 \)
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