Newspace parameters
Level: | \( N \) | \(=\) | \( 100 = 2^{2} \cdot 5^{2} \) |
Weight: | \( k \) | \(=\) | \( 4 \) |
Character orbit: | \([\chi]\) | \(=\) | 100.e (of order \(4\), degree \(2\), minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(5.90019100057\) |
Analytic rank: | \(0\) |
Dimension: | \(12\) |
Relative dimension: | \(6\) over \(\Q(i)\) |
Coefficient field: | \(\mathbb{Q}[x]/(x^{12} - \cdots)\) |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
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Defining polynomial: | \( x^{12} - 7x^{10} + 44x^{8} - 156x^{6} + 704x^{4} - 1792x^{2} + 4096 \) |
Coefficient ring: | \(\Z[a_1, \ldots, a_{9}]\) |
Coefficient ring index: | \( 2^{15} \) |
Twist minimal: | no (minimal twist has level 20) |
Sato-Tate group: | $\mathrm{SU}(2)[C_{4}]$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{11}\) 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 \( x^{12} - 7x^{10} + 44x^{8} - 156x^{6} + 704x^{4} - 1792x^{2} + 4096 \) :
\(\beta_{1}\) | \(=\) | \( 2\nu^{2} - 2 \) |
\(\beta_{2}\) | \(=\) | \( ( \nu^{11} - 23\nu^{9} + 92\nu^{7} - 668\nu^{5} + 1152\nu^{3} - 8704\nu ) / 1280 \) |
\(\beta_{3}\) | \(=\) | \( ( \nu^{11} - 8\nu^{9} + 27\nu^{7} - 128\nu^{5} + 412\nu^{3} - 1504\nu ) / 640 \) |
\(\beta_{4}\) | \(=\) | \( ( - 5 \nu^{11} + 32 \nu^{10} + 35 \nu^{9} - 256 \nu^{8} - 220 \nu^{7} + 864 \nu^{6} + 780 \nu^{5} - 4096 \nu^{4} - 3520 \nu^{3} + 13184 \nu^{2} + 8960 \nu - 48128 ) / 5120 \) |
\(\beta_{5}\) | \(=\) | \( ( - 13 \nu^{11} - 32 \nu^{10} + 99 \nu^{9} + 256 \nu^{8} - 436 \nu^{7} - 864 \nu^{6} + 1804 \nu^{5} + 4096 \nu^{4} - 6816 \nu^{3} - 13184 \nu^{2} + 20992 \nu + 48128 ) / 5120 \) |
\(\beta_{6}\) | \(=\) | \( ( - 13 \nu^{11} + 48 \nu^{10} + 99 \nu^{9} + 16 \nu^{8} - 436 \nu^{7} + 416 \nu^{6} + 1804 \nu^{5} + 576 \nu^{4} - 6816 \nu^{3} + 8576 \nu^{2} + 10752 \nu + 22528 ) / 5120 \) |
\(\beta_{7}\) | \(=\) | \( ( 21 \nu^{11} + 48 \nu^{10} - 163 \nu^{9} + 16 \nu^{8} + 652 \nu^{7} + 416 \nu^{6} - 2828 \nu^{5} + 576 \nu^{4} + 10112 \nu^{3} + 8576 \nu^{2} - 22784 \nu + 22528 ) / 5120 \) |
\(\beta_{8}\) | \(=\) | \( ( -17\nu^{10} + 31\nu^{8} - 324\nu^{6} + 636\nu^{4} - 4384\nu^{2} - 512 ) / 640 \) |
\(\beta_{9}\) | \(=\) | \( ( - 45 \nu^{11} - 12 \nu^{10} + 115 \nu^{9} - 44 \nu^{8} - 900 \nu^{7} - 144 \nu^{6} + 1740 \nu^{5} - 2224 \nu^{4} - 10720 \nu^{3} + 256 \nu^{2} + 2560 \nu - 28672 ) / 5120 \) |
\(\beta_{10}\) | \(=\) | \( ( - 45 \nu^{11} + 12 \nu^{10} + 115 \nu^{9} + 44 \nu^{8} - 900 \nu^{7} + 144 \nu^{6} + 1740 \nu^{5} + 2224 \nu^{4} - 10720 \nu^{3} - 256 \nu^{2} + 2560 \nu + 28672 ) / 5120 \) |
\(\beta_{11}\) | \(=\) | \( ( 3\nu^{11} - 7\nu^{9} + 50\nu^{7} - 156\nu^{5} + 824\nu^{3} - 448\nu ) / 256 \) |
\(\nu\) | \(=\) | \( ( \beta_{7} - \beta_{6} + \beta_{5} + \beta_{4} - 2\beta_{3} ) / 4 \) |
\(\nu^{2}\) | \(=\) | \( ( \beta _1 + 2 ) / 2 \) |
\(\nu^{3}\) | \(=\) | \( ( \beta_{11} + \beta_{10} + \beta_{9} - 3\beta_{5} - 3\beta_{4} - 2\beta_{3} - 2\beta_{2} ) / 2 \) |
\(\nu^{4}\) | \(=\) | \( ( 3\beta_{10} - 3\beta_{9} + \beta_{8} - \beta_{5} + \beta_{4} - \beta_{3} + \beta _1 - 12 ) / 2 \) |
\(\nu^{5}\) | \(=\) | \( ( - 3 \beta_{11} - \beta_{10} - \beta_{9} - 5 \beta_{7} + 5 \beta_{6} - 10 \beta_{5} - 10 \beta_{4} + 14 \beta_{3} - 8 \beta_{2} ) / 2 \) |
\(\nu^{6}\) | \(=\) | \( ( 7 \beta_{10} - 7 \beta_{9} - 11 \beta_{8} - 14 \beta_{7} - 14 \beta_{6} + 5 \beta_{5} - 5 \beta_{4} + 19 \beta_{3} - \beta _1 - 60 ) / 2 \) |
\(\nu^{7}\) | \(=\) | \( ( - 25 \beta_{11} - 19 \beta_{10} - 19 \beta_{9} + 3 \beta_{7} - 3 \beta_{6} - 12 \beta_{5} - 12 \beta_{4} - 74 \beta_{3} + 16 \beta_{2} ) / 2 \) |
\(\nu^{8}\) | \(=\) | \( ( - 35 \beta_{10} + 35 \beta_{9} - 41 \beta_{8} - 18 \beta_{7} - 18 \beta_{6} + 47 \beta_{5} - 47 \beta_{4} + 65 \beta_{3} + 9 \beta _1 - 348 ) / 2 \) |
\(\nu^{9}\) | \(=\) | \( ( 49 \beta_{11} + 3 \beta_{10} + 3 \beta_{9} - 47 \beta_{7} + 47 \beta_{6} - 80 \beta_{5} - 80 \beta_{4} - 358 \beta_{3} + 88 \beta_{2} ) / 2 \) |
\(\nu^{10}\) | \(=\) | \( ( - 85 \beta_{10} + 85 \beta_{9} + 97 \beta_{8} + 234 \beta_{7} + 234 \beta_{6} - 47 \beta_{5} + 47 \beta_{4} - 281 \beta_{3} - 185 \beta _1 - 516 ) / 2 \) |
\(\nu^{11}\) | \(=\) | \( ( 271 \beta_{11} - 3 \beta_{10} - 3 \beta_{9} - 345 \beta_{7} + 345 \beta_{6} + 392 \beta_{5} + 392 \beta_{4} + 1526 \beta_{3} + 72 \beta_{2} ) / 2 \) |
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/100\mathbb{Z}\right)^\times\).
\(n\) | \(51\) | \(77\) |
\(\chi(n)\) | \(-1\) | \(\beta_{3}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
7.1 |
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−2.70884 | + | 0.813737i | 2.61822 | + | 2.61822i | 6.67566 | − | 4.40857i | 0 | −9.22289 | − | 4.96181i | 17.7783 | − | 17.7783i | −14.4959 | + | 17.3744i | − | 13.2899i | 0 | |||||||||||||||||||||||||||||||||||||||||
7.2 | −1.03109 | + | 2.63379i | −5.55970 | − | 5.55970i | −5.87372 | − | 5.43134i | 0 | 20.3756 | − | 8.91056i | 1.14202 | − | 1.14202i | 20.3613 | − | 9.86997i | 34.8205i | 0 | |||||||||||||||||||||||||||||||||||||||||||
7.3 | 0.510409 | + | 2.78199i | 4.02923 | + | 4.02923i | −7.47897 | + | 2.83991i | 0 | −9.15273 | + | 13.2658i | −14.4440 | + | 14.4440i | −11.7179 | − | 19.3569i | 5.46937i | 0 | |||||||||||||||||||||||||||||||||||||||||||
7.4 | 0.813737 | − | 2.70884i | −2.61822 | − | 2.61822i | −6.67566 | − | 4.40857i | 0 | −9.22289 | + | 4.96181i | −17.7783 | + | 17.7783i | −17.3744 | + | 14.4959i | − | 13.2899i | 0 | ||||||||||||||||||||||||||||||||||||||||||
7.5 | 2.63379 | − | 1.03109i | 5.55970 | + | 5.55970i | 5.87372 | − | 5.43134i | 0 | 20.3756 | + | 8.91056i | −1.14202 | + | 1.14202i | 9.86997 | − | 20.3613i | 34.8205i | 0 | |||||||||||||||||||||||||||||||||||||||||||
7.6 | 2.78199 | + | 0.510409i | −4.02923 | − | 4.02923i | 7.47897 | + | 2.83991i | 0 | −9.15273 | − | 13.2658i | 14.4440 | − | 14.4440i | 19.3569 | + | 11.7179i | 5.46937i | 0 | |||||||||||||||||||||||||||||||||||||||||||
43.1 | −2.70884 | − | 0.813737i | 2.61822 | − | 2.61822i | 6.67566 | + | 4.40857i | 0 | −9.22289 | + | 4.96181i | 17.7783 | + | 17.7783i | −14.4959 | − | 17.3744i | 13.2899i | 0 | |||||||||||||||||||||||||||||||||||||||||||
43.2 | −1.03109 | − | 2.63379i | −5.55970 | + | 5.55970i | −5.87372 | + | 5.43134i | 0 | 20.3756 | + | 8.91056i | 1.14202 | + | 1.14202i | 20.3613 | + | 9.86997i | − | 34.8205i | 0 | ||||||||||||||||||||||||||||||||||||||||||
43.3 | 0.510409 | − | 2.78199i | 4.02923 | − | 4.02923i | −7.47897 | − | 2.83991i | 0 | −9.15273 | − | 13.2658i | −14.4440 | − | 14.4440i | −11.7179 | + | 19.3569i | − | 5.46937i | 0 | ||||||||||||||||||||||||||||||||||||||||||
43.4 | 0.813737 | + | 2.70884i | −2.61822 | + | 2.61822i | −6.67566 | + | 4.40857i | 0 | −9.22289 | − | 4.96181i | −17.7783 | − | 17.7783i | −17.3744 | − | 14.4959i | 13.2899i | 0 | |||||||||||||||||||||||||||||||||||||||||||
43.5 | 2.63379 | + | 1.03109i | 5.55970 | − | 5.55970i | 5.87372 | + | 5.43134i | 0 | 20.3756 | − | 8.91056i | −1.14202 | − | 1.14202i | 9.86997 | + | 20.3613i | − | 34.8205i | 0 | ||||||||||||||||||||||||||||||||||||||||||
43.6 | 2.78199 | − | 0.510409i | −4.02923 | + | 4.02923i | 7.47897 | − | 2.83991i | 0 | −9.15273 | + | 13.2658i | 14.4440 | + | 14.4440i | 19.3569 | − | 11.7179i | − | 5.46937i | 0 | ||||||||||||||||||||||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
4.b | odd | 2 | 1 | inner |
5.c | odd | 4 | 1 | inner |
20.e | even | 4 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 100.4.e.e | 12 | |
4.b | odd | 2 | 1 | inner | 100.4.e.e | 12 | |
5.b | even | 2 | 1 | 20.4.e.b | ✓ | 12 | |
5.c | odd | 4 | 1 | 20.4.e.b | ✓ | 12 | |
5.c | odd | 4 | 1 | inner | 100.4.e.e | 12 | |
15.d | odd | 2 | 1 | 180.4.k.e | 12 | ||
15.e | even | 4 | 1 | 180.4.k.e | 12 | ||
20.d | odd | 2 | 1 | 20.4.e.b | ✓ | 12 | |
20.e | even | 4 | 1 | 20.4.e.b | ✓ | 12 | |
20.e | even | 4 | 1 | inner | 100.4.e.e | 12 | |
40.e | odd | 2 | 1 | 320.4.n.k | 12 | ||
40.f | even | 2 | 1 | 320.4.n.k | 12 | ||
40.i | odd | 4 | 1 | 320.4.n.k | 12 | ||
40.k | even | 4 | 1 | 320.4.n.k | 12 | ||
60.h | even | 2 | 1 | 180.4.k.e | 12 | ||
60.l | odd | 4 | 1 | 180.4.k.e | 12 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
20.4.e.b | ✓ | 12 | 5.b | even | 2 | 1 | |
20.4.e.b | ✓ | 12 | 5.c | odd | 4 | 1 | |
20.4.e.b | ✓ | 12 | 20.d | odd | 2 | 1 | |
20.4.e.b | ✓ | 12 | 20.e | even | 4 | 1 | |
100.4.e.e | 12 | 1.a | even | 1 | 1 | trivial | |
100.4.e.e | 12 | 4.b | odd | 2 | 1 | inner | |
100.4.e.e | 12 | 5.c | odd | 4 | 1 | inner | |
100.4.e.e | 12 | 20.e | even | 4 | 1 | inner | |
180.4.k.e | 12 | 15.d | odd | 2 | 1 | ||
180.4.k.e | 12 | 15.e | even | 4 | 1 | ||
180.4.k.e | 12 | 60.h | even | 2 | 1 | ||
180.4.k.e | 12 | 60.l | odd | 4 | 1 | ||
320.4.n.k | 12 | 40.e | odd | 2 | 1 | ||
320.4.n.k | 12 | 40.f | even | 2 | 1 | ||
320.4.n.k | 12 | 40.i | odd | 4 | 1 | ||
320.4.n.k | 12 | 40.k | even | 4 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{12} + 5064T_{3}^{8} + 4945680T_{3}^{4} + 757350400 \)
acting on \(S_{4}^{\mathrm{new}}(100, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{12} - 6 T^{11} + 18 T^{10} + \cdots + 262144 \)
$3$
\( T^{12} + 5064 T^{8} + \cdots + 757350400 \)
$5$
\( T^{12} \)
$7$
\( T^{12} + 573704 T^{8} + \cdots + 473344000000 \)
$11$
\( (T^{6} + 3000 T^{4} + 1778960 T^{2} + \cdots + 88064000)^{2} \)
$13$
\( (T^{6} + 58 T^{5} + 1682 T^{4} + \cdots + 7296200)^{2} \)
$17$
\( (T^{6} - 166 T^{5} + 13778 T^{4} + \cdots + 3024864200)^{2} \)
$19$
\( (T^{6} - 24160 T^{4} + \cdots - 148035584000)^{2} \)
$23$
\( T^{12} + 150061064 T^{8} + \cdots + 43\!\cdots\!00 \)
$29$
\( (T^{6} + 65648 T^{4} + \cdots + 234782887936)^{2} \)
$31$
\( (T^{6} + 75640 T^{4} + \cdots + 4998782336000)^{2} \)
$37$
\( (T^{6} + 254 T^{5} + \cdots + 6862252857800)^{2} \)
$41$
\( (T^{3} + 164 T^{2} - 18428 T - 1791008)^{4} \)
$43$
\( T^{12} + 14590421064 T^{8} + \cdots + 22\!\cdots\!00 \)
$47$
\( T^{12} + 61550198664 T^{8} + \cdots + 28\!\cdots\!00 \)
$53$
\( (T^{6} - 322 T^{5} + \cdots + 273645700473800)^{2} \)
$59$
\( (T^{6} - 688480 T^{4} + \cdots - 742151346176000)^{2} \)
$61$
\( (T^{3} + 224 T^{2} - 55468 T - 11698768)^{4} \)
$67$
\( T^{12} + 442279393224 T^{8} + \cdots + 16\!\cdots\!00 \)
$71$
\( (T^{6} + 715960 T^{4} + \cdots + 29\!\cdots\!00)^{2} \)
$73$
\( (T^{6} + 718 T^{5} + \cdots + 17037736128200)^{2} \)
$79$
\( (T^{6} - 2383360 T^{4} + \cdots - 42\!\cdots\!00)^{2} \)
$83$
\( T^{12} + 2795286470344 T^{8} + \cdots + 13\!\cdots\!00 \)
$89$
\( (T^{6} + 1431168 T^{4} + \cdots + 84\!\cdots\!16)^{2} \)
$97$
\( (T^{6} - 2386 T^{5} + \cdots + 60\!\cdots\!00)^{2} \)
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