Newspace parameters
Level: | \( N \) | \(=\) | \( 400 = 2^{4} \cdot 5^{2} \) |
Weight: | \( k \) | \(=\) | \( 3 \) |
Character orbit: | \([\chi]\) | \(=\) | 400.bg (of order \(20\), degree \(8\), not minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(10.8992105744\) |
Analytic rank: | \(0\) |
Dimension: | \(32\) |
Relative dimension: | \(4\) over \(\Q(\zeta_{20})\) |
Twist minimal: | no (minimal twist has level 25) |
Sato-Tate group: | $\mathrm{SU}(2)[C_{20}]$ |
$q$-expansion
The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.
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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
17.1 | 0 | −0.838638 | + | 5.29495i | 0 | −3.73307 | + | 3.32629i | 0 | −1.66138 | − | 1.66138i | 0 | −18.7737 | − | 6.09994i | 0 | ||||||||||
17.2 | 0 | 0.0858318 | − | 0.541921i | 0 | 2.26962 | + | 4.45520i | 0 | −1.68463 | − | 1.68463i | 0 | 8.27320 | + | 2.68812i | 0 | ||||||||||
17.3 | 0 | 0.296456 | − | 1.87175i | 0 | 1.22928 | − | 4.84653i | 0 | 5.60844 | + | 5.60844i | 0 | 5.14394 | + | 1.67137i | 0 | ||||||||||
17.4 | 0 | 0.363254 | − | 2.29349i | 0 | −4.45624 | + | 2.26758i | 0 | 3.40272 | + | 3.40272i | 0 | 3.43135 | + | 1.11491i | 0 | ||||||||||
33.1 | 0 | −3.42034 | − | 0.541729i | 0 | 4.00059 | − | 2.99921i | 0 | 8.06323 | − | 8.06323i | 0 | 2.84577 | + | 0.924645i | 0 | ||||||||||
33.2 | 0 | −0.872241 | − | 0.138149i | 0 | −2.66494 | + | 4.23062i | 0 | −1.62783 | + | 1.62783i | 0 | −7.81779 | − | 2.54015i | 0 | ||||||||||
33.3 | 0 | 3.57679 | + | 0.566508i | 0 | −4.45026 | − | 2.27929i | 0 | −6.54971 | + | 6.54971i | 0 | 3.91299 | + | 1.27141i | 0 | ||||||||||
33.4 | 0 | 4.42692 | + | 0.701156i | 0 | 1.95091 | − | 4.60369i | 0 | 4.77540 | − | 4.77540i | 0 | 10.5465 | + | 3.42677i | 0 | ||||||||||
97.1 | 0 | −3.42034 | + | 0.541729i | 0 | 4.00059 | + | 2.99921i | 0 | 8.06323 | + | 8.06323i | 0 | 2.84577 | − | 0.924645i | 0 | ||||||||||
97.2 | 0 | −0.872241 | + | 0.138149i | 0 | −2.66494 | − | 4.23062i | 0 | −1.62783 | − | 1.62783i | 0 | −7.81779 | + | 2.54015i | 0 | ||||||||||
97.3 | 0 | 3.57679 | − | 0.566508i | 0 | −4.45026 | + | 2.27929i | 0 | −6.54971 | − | 6.54971i | 0 | 3.91299 | − | 1.27141i | 0 | ||||||||||
97.4 | 0 | 4.42692 | − | 0.701156i | 0 | 1.95091 | + | 4.60369i | 0 | 4.77540 | + | 4.77540i | 0 | 10.5465 | − | 3.42677i | 0 | ||||||||||
113.1 | 0 | −1.72787 | + | 3.39113i | 0 | 2.36408 | + | 4.40581i | 0 | 2.38950 | − | 2.38950i | 0 | −3.22416 | − | 4.43767i | 0 | ||||||||||
113.2 | 0 | −1.61679 | + | 3.17313i | 0 | −0.872190 | − | 4.92334i | 0 | 0.574149 | − | 0.574149i | 0 | −2.16466 | − | 2.97940i | 0 | ||||||||||
113.3 | 0 | 0.665351 | − | 1.30583i | 0 | −3.20727 | − | 3.83580i | 0 | −3.62927 | + | 3.62927i | 0 | 4.02758 | + | 5.54349i | 0 | ||||||||||
113.4 | 0 | 2.19472 | − | 4.30737i | 0 | 4.99561 | − | 0.209511i | 0 | 3.57009 | − | 3.57009i | 0 | −8.44662 | − | 11.6258i | 0 | ||||||||||
177.1 | 0 | −1.72787 | − | 3.39113i | 0 | 2.36408 | − | 4.40581i | 0 | 2.38950 | + | 2.38950i | 0 | −3.22416 | + | 4.43767i | 0 | ||||||||||
177.2 | 0 | −1.61679 | − | 3.17313i | 0 | −0.872190 | + | 4.92334i | 0 | 0.574149 | + | 0.574149i | 0 | −2.16466 | + | 2.97940i | 0 | ||||||||||
177.3 | 0 | 0.665351 | + | 1.30583i | 0 | −3.20727 | + | 3.83580i | 0 | −3.62927 | − | 3.62927i | 0 | 4.02758 | − | 5.54349i | 0 | ||||||||||
177.4 | 0 | 2.19472 | + | 4.30737i | 0 | 4.99561 | + | 0.209511i | 0 | 3.57009 | + | 3.57009i | 0 | −8.44662 | + | 11.6258i | 0 | ||||||||||
See all 32 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
25.f | odd | 20 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 400.3.bg.c | 32 | |
4.b | odd | 2 | 1 | 25.3.f.a | ✓ | 32 | |
12.b | even | 2 | 1 | 225.3.r.a | 32 | ||
20.d | odd | 2 | 1 | 125.3.f.c | 32 | ||
20.e | even | 4 | 1 | 125.3.f.a | 32 | ||
20.e | even | 4 | 1 | 125.3.f.b | 32 | ||
25.f | odd | 20 | 1 | inner | 400.3.bg.c | 32 | |
100.h | odd | 10 | 1 | 125.3.f.b | 32 | ||
100.j | odd | 10 | 1 | 125.3.f.a | 32 | ||
100.l | even | 20 | 1 | 25.3.f.a | ✓ | 32 | |
100.l | even | 20 | 1 | 125.3.f.c | 32 | ||
300.u | odd | 20 | 1 | 225.3.r.a | 32 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
25.3.f.a | ✓ | 32 | 4.b | odd | 2 | 1 | |
25.3.f.a | ✓ | 32 | 100.l | even | 20 | 1 | |
125.3.f.a | 32 | 20.e | even | 4 | 1 | ||
125.3.f.a | 32 | 100.j | odd | 10 | 1 | ||
125.3.f.b | 32 | 20.e | even | 4 | 1 | ||
125.3.f.b | 32 | 100.h | odd | 10 | 1 | ||
125.3.f.c | 32 | 20.d | odd | 2 | 1 | ||
125.3.f.c | 32 | 100.l | even | 20 | 1 | ||
225.3.r.a | 32 | 12.b | even | 2 | 1 | ||
225.3.r.a | 32 | 300.u | odd | 20 | 1 | ||
400.3.bg.c | 32 | 1.a | even | 1 | 1 | trivial | |
400.3.bg.c | 32 | 25.f | odd | 20 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{32} - 10 T_{3}^{31} + 55 T_{3}^{30} - 310 T_{3}^{29} + 1192 T_{3}^{28} - 1960 T_{3}^{27} - 2380 T_{3}^{26} + 30930 T_{3}^{25} - 86732 T_{3}^{24} + 603820 T_{3}^{23} - 3704615 T_{3}^{22} - 6537080 T_{3}^{21} + \cdots + 40837943056 \)
acting on \(S_{3}^{\mathrm{new}}(400, [\chi])\).