Newspace parameters
| Level: | \( N \) | = | \( 7 \) |
| Weight: | \( k \) | = | \( 10 \) |
| Character orbit: | \([\chi]\) | = | 7.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(3.60525085315\) |
| Analytic rank: | \(1\) |
| Dimension: | \(2\) |
| Coefficient field: | \(\Q(\sqrt{193}) \) |
| Coefficient ring: | \(\Z[a_1, a_2]\) |
| Coefficient ring index: | \( 2 \) |
| Twist minimal: | yes |
| Fricke sign: | \(1\) |
| Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{193}\). We also show the integral \(q\)-expansion of the trace form.
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1.1 |
|
−16.8924 | 109.817 | −226.645 | −2438.78 | −1855.08 | −2401.00 | 12477.5 | −7623.25 | 41197.0 | ||||||||||||||||||||||||
| 1.2 | 10.8924 | −195.817 | −393.355 | 200.782 | −2132.92 | −2401.00 | −9861.52 | 18661.3 | 2187.01 | |||||||||||||||||||||||||
Inner twists
This newform does not admit any (nontrivial) inner twists.
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 7.10.a.a | ✓ | 2 |
| 3.b | odd | 2 | 1 | 63.10.a.d | 2 | ||
| 4.b | odd | 2 | 1 | 112.10.a.e | 2 | ||
| 5.b | even | 2 | 1 | 175.10.a.b | 2 | ||
| 5.c | odd | 4 | 2 | 175.10.b.b | 4 | ||
| 7.b | odd | 2 | 1 | 49.10.a.b | 2 | ||
| 7.c | even | 3 | 2 | 49.10.c.c | 4 | ||
| 7.d | odd | 6 | 2 | 49.10.c.b | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 7.10.a.a | ✓ | 2 | 1.a | even | 1 | 1 | trivial |
| 49.10.a.b | 2 | 7.b | odd | 2 | 1 | ||
| 49.10.c.b | 4 | 7.d | odd | 6 | 2 | ||
| 49.10.c.c | 4 | 7.c | even | 3 | 2 | ||
| 63.10.a.d | 2 | 3.b | odd | 2 | 1 | ||
| 112.10.a.e | 2 | 4.b | odd | 2 | 1 | ||
| 175.10.a.b | 2 | 5.b | even | 2 | 1 | ||
| 175.10.b.b | 4 | 5.c | odd | 4 | 2 | ||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(7\) | \(1\) |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} + 6 T_{2} - 184 \) acting on \(S_{10}^{\mathrm{new}}(\Gamma_0(7))\).
Hecke Characteristic Polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ | \( 1 + 6 T + 840 T^{2} + 3072 T^{3} + 262144 T^{4} \) |
| $3$ | \( 1 + 86 T + 17862 T^{2} + 1692738 T^{3} + 387420489 T^{4} \) |
| $5$ | \( 1 + 2238 T + 3416586 T^{2} + 4371093750 T^{3} + 3814697265625 T^{4} \) |
| $7$ | \( ( 1 + 2401 T )^{2} \) |
| $11$ | \( 1 - 35316 T + 2892681078 T^{2} - 83273280655356 T^{3} + 5559917313492231481 T^{4} \) |
| $13$ | \( 1 + 26530 T - 1541163822 T^{2} + 281337368365690 T^{3} + \)\(11\!\cdots\!29\)\( T^{4} \) |
| $17$ | \( 1 + 463920 T + 273833245726 T^{2} + 55015287664488240 T^{3} + \)\(14\!\cdots\!09\)\( T^{4} \) |
| $19$ | \( 1 + 925426 T + 858791487510 T^{2} + 298623585404828854 T^{3} + \)\(10\!\cdots\!41\)\( T^{4} \) |
| $23$ | \( 1 - 778128 T + 3473691840430 T^{2} - 1401527318158881264 T^{3} + \)\(32\!\cdots\!69\)\( T^{4} \) |
| $29$ | \( 1 + 10003584 T + 52302031706070 T^{2} + \)\(14\!\cdots\!96\)\( T^{3} + \)\(21\!\cdots\!61\)\( T^{4} \) |
| $31$ | \( 1 - 2467260 T + 49371575832542 T^{2} - 65233422172137131460 T^{3} + \)\(69\!\cdots\!41\)\( T^{4} \) |
| $37$ | \( 1 - 30735552 T + 484209298874630 T^{2} - \)\(39\!\cdots\!04\)\( T^{3} + \)\(16\!\cdots\!29\)\( T^{4} \) |
| $41$ | \( 1 + 19103448 T + 602984827739166 T^{2} + \)\(62\!\cdots\!28\)\( T^{3} + \)\(10\!\cdots\!21\)\( T^{4} \) |
| $43$ | \( 1 - 4065100 T + 797231337676374 T^{2} - \)\(20\!\cdots\!00\)\( T^{3} + \)\(25\!\cdots\!49\)\( T^{4} \) |
| $47$ | \( 1 + 82195020 T + 3721245520696702 T^{2} + \)\(91\!\cdots\!40\)\( T^{3} + \)\(12\!\cdots\!89\)\( T^{4} \) |
| $53$ | \( 1 + 55189812 T + 3123778356606670 T^{2} + \)\(18\!\cdots\!96\)\( T^{3} + \)\(10\!\cdots\!89\)\( T^{4} \) |
| $59$ | \( 1 + 7069218 T + 16866494212382134 T^{2} + \)\(61\!\cdots\!02\)\( T^{3} + \)\(75\!\cdots\!21\)\( T^{4} \) |
| $61$ | \( 1 - 44316386 T + 21654336818123658 T^{2} - \)\(51\!\cdots\!26\)\( T^{3} + \)\(13\!\cdots\!81\)\( T^{4} \) |
| $67$ | \( 1 + 241921336 T + 59516583718815510 T^{2} + \)\(65\!\cdots\!92\)\( T^{3} + \)\(74\!\cdots\!09\)\( T^{4} \) |
| $71$ | \( 1 - 206493816 T + 58491352612128526 T^{2} - \)\(94\!\cdots\!96\)\( T^{3} + \)\(21\!\cdots\!61\)\( T^{4} \) |
| $73$ | \( 1 + 499153188 T + 178474458263805254 T^{2} + \)\(29\!\cdots\!44\)\( T^{3} + \)\(34\!\cdots\!69\)\( T^{4} \) |
| $79$ | \( 1 - 468535096 T + 239633073722978334 T^{2} - \)\(56\!\cdots\!24\)\( T^{3} + \)\(14\!\cdots\!61\)\( T^{4} \) |
| $83$ | \( 1 - 444023958 T + 333438010641681622 T^{2} - \)\(83\!\cdots\!74\)\( T^{3} + \)\(34\!\cdots\!09\)\( T^{4} \) |
| $89$ | \( 1 - 636267396 T + 801539802340191990 T^{2} - \)\(22\!\cdots\!64\)\( T^{3} + \)\(12\!\cdots\!81\)\( T^{4} \) |
| $97$ | \( 1 + 1632716064 T + 2180562419544849758 T^{2} + \)\(12\!\cdots\!88\)\( T^{3} + \)\(57\!\cdots\!89\)\( T^{4} \) |
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