Newspace parameters
Level: | \( N \) | \(=\) | \( 98 = 2 \cdot 7^{2} \) |
Weight: | \( k \) | \(=\) | \( 7 \) |
Character orbit: | \([\chi]\) | \(=\) | 98.d (of order \(6\), degree \(2\), not minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(22.5453001947\) |
Analytic rank: | \(0\) |
Dimension: | \(8\) |
Relative dimension: | \(4\) over \(\Q(\zeta_{6})\) |
Coefficient field: | 8.0.44930433024.14 |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
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Defining polynomial: | \( x^{8} + 10x^{6} + 77x^{4} + 230x^{2} + 529 \) |
Coefficient ring: | \(\Z[a_1, \ldots, a_{9}]\) |
Coefficient ring index: | \( 2^{14}\cdot 3^{2} \) |
Twist minimal: | no (minimal twist has level 14) |
Sato-Tate group: | $\mathrm{SU}(2)[C_{6}]$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{7}\) 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 \( x^{8} + 10x^{6} + 77x^{4} + 230x^{2} + 529 \) :
\(\beta_{1}\) | \(=\) | \( ( 10\nu^{6} + 77\nu^{4} + 770\nu^{2} + 2300 ) / 1771 \) |
\(\beta_{2}\) | \(=\) | \( ( 32\nu^{7} - 462\nu^{5} - 1078\nu^{3} - 3266\nu ) / 1771 \) |
\(\beta_{3}\) | \(=\) | \( ( 64\nu^{7} + 2618\nu^{5} + 15554\nu^{3} + 74934\nu ) / 1771 \) |
\(\beta_{4}\) | \(=\) | \( ( 106\nu^{7} + 462\nu^{5} + 1078\nu^{3} + 3128\nu ) / 1771 \) |
\(\beta_{5}\) | \(=\) | \( ( -108\nu^{6} - 1540\nu^{4} - 8316\nu^{2} - 24840 ) / 1771 \) |
\(\beta_{6}\) | \(=\) | \( ( -4\nu^{6} + 620 ) / 77 \) |
\(\beta_{7}\) | \(=\) | \( ( -478\nu^{7} - 2618\nu^{5} - 15554\nu^{3} + 10488\nu ) / 1771 \) |
\(\nu\) | \(=\) | \( ( \beta_{7} + 3\beta_{4} + \beta_{3} + 3\beta_{2} ) / 48 \) |
\(\nu^{2}\) | \(=\) | \( ( \beta_{6} + \beta_{5} + 20\beta _1 - 20 ) / 4 \) |
\(\nu^{3}\) | \(=\) | \( ( -6\beta_{7} - 34\beta_{4} + 3\beta_{3} + 17\beta_{2} ) / 48 \) |
\(\nu^{4}\) | \(=\) | \( ( -5\beta_{5} - 54\beta_1 ) / 2 \) |
\(\nu^{5}\) | \(=\) | \( ( 7\beta_{7} + 101\beta_{4} - 14\beta_{3} - 202\beta_{2} ) / 48 \) |
\(\nu^{6}\) | \(=\) | \( ( -77\beta_{6} + 620 ) / 4 \) |
\(\nu^{7}\) | \(=\) | \( ( \beta_{7} + 619\beta_{4} + \beta_{3} + 619\beta_{2} ) / 48 \) |
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/98\mathbb{Z}\right)^\times\).
\(n\) | \(3\) |
\(\chi(n)\) | \(\beta_{1}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
19.1 |
|
−2.82843 | + | 4.89898i | −25.9408 | + | 14.9769i | −16.0000 | − | 27.7128i | 85.1967 | + | 49.1883i | − | 169.445i | 0 | 181.019 | 84.1173 | − | 145.695i | −481.945 | + | 278.251i | |||||||||||||||||||||||||||||
19.2 | −2.82843 | + | 4.89898i | 25.9408 | − | 14.9769i | −16.0000 | − | 27.7128i | −85.1967 | − | 49.1883i | 169.445i | 0 | 181.019 | 84.1173 | − | 145.695i | 481.945 | − | 278.251i | |||||||||||||||||||||||||||||||
19.3 | 2.82843 | − | 4.89898i | −3.32777 | + | 1.92129i | −16.0000 | − | 27.7128i | −177.318 | − | 102.374i | 21.7369i | 0 | −181.019 | −357.117 | + | 618.545i | −1003.06 | + | 579.117i | |||||||||||||||||||||||||||||||
19.4 | 2.82843 | − | 4.89898i | 3.32777 | − | 1.92129i | −16.0000 | − | 27.7128i | 177.318 | + | 102.374i | − | 21.7369i | 0 | −181.019 | −357.117 | + | 618.545i | 1003.06 | − | 579.117i | ||||||||||||||||||||||||||||||
31.1 | −2.82843 | − | 4.89898i | −25.9408 | − | 14.9769i | −16.0000 | + | 27.7128i | 85.1967 | − | 49.1883i | 169.445i | 0 | 181.019 | 84.1173 | + | 145.695i | −481.945 | − | 278.251i | |||||||||||||||||||||||||||||||
31.2 | −2.82843 | − | 4.89898i | 25.9408 | + | 14.9769i | −16.0000 | + | 27.7128i | −85.1967 | + | 49.1883i | − | 169.445i | 0 | 181.019 | 84.1173 | + | 145.695i | 481.945 | + | 278.251i | ||||||||||||||||||||||||||||||
31.3 | 2.82843 | + | 4.89898i | −3.32777 | − | 1.92129i | −16.0000 | + | 27.7128i | −177.318 | + | 102.374i | − | 21.7369i | 0 | −181.019 | −357.117 | − | 618.545i | −1003.06 | − | 579.117i | ||||||||||||||||||||||||||||||
31.4 | 2.82843 | + | 4.89898i | 3.32777 | + | 1.92129i | −16.0000 | + | 27.7128i | 177.318 | − | 102.374i | 21.7369i | 0 | −181.019 | −357.117 | − | 618.545i | 1003.06 | + | 579.117i | |||||||||||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
7.b | odd | 2 | 1 | inner |
7.c | even | 3 | 1 | inner |
7.d | odd | 6 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 98.7.d.b | 8 | |
7.b | odd | 2 | 1 | inner | 98.7.d.b | 8 | |
7.c | even | 3 | 1 | 14.7.b.a | ✓ | 4 | |
7.c | even | 3 | 1 | inner | 98.7.d.b | 8 | |
7.d | odd | 6 | 1 | 14.7.b.a | ✓ | 4 | |
7.d | odd | 6 | 1 | inner | 98.7.d.b | 8 | |
21.g | even | 6 | 1 | 126.7.c.a | 4 | ||
21.h | odd | 6 | 1 | 126.7.c.a | 4 | ||
28.f | even | 6 | 1 | 112.7.c.c | 4 | ||
28.g | odd | 6 | 1 | 112.7.c.c | 4 | ||
35.i | odd | 6 | 1 | 350.7.b.a | 4 | ||
35.j | even | 6 | 1 | 350.7.b.a | 4 | ||
35.k | even | 12 | 2 | 350.7.d.a | 8 | ||
35.l | odd | 12 | 2 | 350.7.d.a | 8 | ||
56.j | odd | 6 | 1 | 448.7.c.h | 4 | ||
56.k | odd | 6 | 1 | 448.7.c.e | 4 | ||
56.m | even | 6 | 1 | 448.7.c.e | 4 | ||
56.p | even | 6 | 1 | 448.7.c.h | 4 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
14.7.b.a | ✓ | 4 | 7.c | even | 3 | 1 | |
14.7.b.a | ✓ | 4 | 7.d | odd | 6 | 1 | |
98.7.d.b | 8 | 1.a | even | 1 | 1 | trivial | |
98.7.d.b | 8 | 7.b | odd | 2 | 1 | inner | |
98.7.d.b | 8 | 7.c | even | 3 | 1 | inner | |
98.7.d.b | 8 | 7.d | odd | 6 | 1 | inner | |
112.7.c.c | 4 | 28.f | even | 6 | 1 | ||
112.7.c.c | 4 | 28.g | odd | 6 | 1 | ||
126.7.c.a | 4 | 21.g | even | 6 | 1 | ||
126.7.c.a | 4 | 21.h | odd | 6 | 1 | ||
350.7.b.a | 4 | 35.i | odd | 6 | 1 | ||
350.7.b.a | 4 | 35.j | even | 6 | 1 | ||
350.7.d.a | 8 | 35.k | even | 12 | 2 | ||
350.7.d.a | 8 | 35.l | odd | 12 | 2 | ||
448.7.c.e | 4 | 56.k | odd | 6 | 1 | ||
448.7.c.e | 4 | 56.m | even | 6 | 1 | ||
448.7.c.h | 4 | 56.j | odd | 6 | 1 | ||
448.7.c.h | 4 | 56.p | even | 6 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{8} - 912T_{3}^{6} + 818496T_{3}^{4} - 12082176T_{3}^{2} + 175509504 \)
acting on \(S_{7}^{\mathrm{new}}(98, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( (T^{4} + 32 T^{2} + 1024)^{2} \)
$3$
\( T^{8} - 912 T^{6} + \cdots + 175509504 \)
$5$
\( T^{8} - 51600 T^{6} + \cdots + 16\!\cdots\!00 \)
$7$
\( T^{8} \)
$11$
\( (T^{4} - 2220 T^{3} + \cdots + 1506736610064)^{2} \)
$13$
\( (T^{4} + 15367056 T^{2} + \cdots + 26266196601792)^{2} \)
$17$
\( T^{8} - 40214784 T^{6} + \cdots + 16\!\cdots\!84 \)
$19$
\( T^{8} - 65975568 T^{6} + \cdots + 30\!\cdots\!64 \)
$23$
\( (T^{4} + 20292 T^{3} + \cdots + 10\!\cdots\!36)^{2} \)
$29$
\( (T^{2} + 9132 T - 1357906716)^{4} \)
$31$
\( T^{8} - 2274932736 T^{6} + \cdots + 75\!\cdots\!44 \)
$37$
\( (T^{4} - 11596 T^{3} + \cdots + 34\!\cdots\!96)^{2} \)
$41$
\( (T^{4} + 9071224896 T^{2} + \cdots + 28\!\cdots\!32)^{2} \)
$43$
\( (T^{2} + 22348 T - 10521464924)^{4} \)
$47$
\( T^{8} - 20120477952 T^{6} + \cdots + 14\!\cdots\!44 \)
$53$
\( (T^{4} + 124308 T^{3} + \cdots + 185366809042704)^{2} \)
$59$
\( T^{8} - 180594109200 T^{6} + \cdots + 62\!\cdots\!00 \)
$61$
\( T^{8} - 121774406928 T^{6} + \cdots + 67\!\cdots\!04 \)
$67$
\( (T^{4} - 217388 T^{3} + \cdots + 13\!\cdots\!44)^{2} \)
$71$
\( (T^{2} + 225804 T - 95443772508)^{4} \)
$73$
\( T^{8} - 228009998400 T^{6} + \cdots + 72\!\cdots\!84 \)
$79$
\( (T^{4} + 1046452 T^{3} + \cdots + 72\!\cdots\!76)^{2} \)
$83$
\( (T^{4} + 478841902608 T^{2} + \cdots + 21\!\cdots\!08)^{2} \)
$89$
\( T^{8} - 258250641984 T^{6} + \cdots + 22\!\cdots\!04 \)
$97$
\( (T^{4} + 42611214336 T^{2} + \cdots + 39\!\cdots\!12)^{2} \)
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