Newspace parameters
Level: | \( N \) | \(=\) | \( 225 = 3^{2} \cdot 5^{2} \) |
Weight: | \( k \) | \(=\) | \( 4 \) |
Character orbit: | \([\chi]\) | \(=\) | 225.m (of order \(10\), degree \(4\), minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(13.2754297513\) |
Analytic rank: | \(0\) |
Dimension: | \(24\) |
Relative dimension: | \(6\) over \(\Q(\zeta_{10})\) |
Twist minimal: | no (minimal twist has level 25) |
Sato-Tate group: | $\mathrm{SU}(2)[C_{10}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
19.1 | −3.47870 | + | 1.13030i | 0 | 4.35165 | − | 3.16166i | 11.1800 | − | 0.0808329i | 0 | − | 8.69522i | 5.63517 | − | 7.75614i | 0 | −38.8007 | + | 12.9180i | |||||||
19.2 | −3.33666 | + | 1.08415i | 0 | 3.48577 | − | 2.53256i | −10.7075 | − | 3.21691i | 0 | 25.7483i | 7.61219 | − | 10.4773i | 0 | 39.2150 | − | 0.874818i | ||||||||
19.3 | −0.331937 | + | 0.107853i | 0 | −6.37359 | + | 4.63068i | −10.0178 | + | 4.96433i | 0 | − | 14.4107i | 3.25738 | − | 4.48340i | 0 | 2.78985 | − | 2.72829i | |||||||
19.4 | 1.48841 | − | 0.483614i | 0 | −4.49065 | + | 3.26265i | 5.33345 | + | 9.82621i | 0 | − | 1.18261i | −12.4652 | + | 17.1569i | 0 | 12.6905 | + | 12.0461i | |||||||
19.5 | 2.51328 | − | 0.816614i | 0 | −0.822422 | + | 0.597525i | 5.31827 | − | 9.83443i | 0 | 26.3705i | −14.0054 | + | 19.2767i | 0 | 5.33537 | − | 29.0596i | ||||||||
19.6 | 4.95462 | − | 1.60985i | 0 | 15.4845 | − | 11.2501i | −11.0057 | − | 1.96864i | 0 | − | 24.5811i | 34.1117 | − | 46.9507i | 0 | −57.6981 | + | 7.96365i | |||||||
64.1 | −2.81526 | − | 3.87487i | 0 | −4.61680 | + | 14.2091i | −11.1717 | + | 0.439337i | 0 | − | 14.5499i | 31.6143 | − | 10.2721i | 0 | 33.1536 | + | 42.0521i | |||||||
64.2 | −1.81265 | − | 2.49489i | 0 | −0.466674 | + | 1.43628i | 10.8670 | + | 2.62823i | 0 | 5.10302i | −19.0341 | + | 6.18456i | 0 | −13.1409 | − | 31.8762i | ||||||||
64.3 | 0.217515 | + | 0.299383i | 0 | 2.42982 | − | 7.47821i | −7.78705 | − | 8.02258i | 0 | − | 0.707538i | 5.58294 | − | 1.81401i | 0 | 0.708030 | − | 4.07634i | |||||||
64.4 | 0.442260 | + | 0.608718i | 0 | 2.29719 | − | 7.07003i | −1.00498 | + | 11.1351i | 0 | 18.3105i | 11.0443 | − | 3.58852i | 0 | −7.22259 | + | 4.31284i | ||||||||
64.5 | 2.23725 | + | 3.07931i | 0 | −2.00472 | + | 6.16988i | 3.48187 | + | 10.6243i | 0 | 4.54748i | 5.47554 | − | 1.77911i | 0 | −24.9258 | + | 34.4910i | ||||||||
64.6 | 2.42187 | + | 3.33341i | 0 | −2.77407 | + | 8.53772i | 8.01402 | − | 7.79587i | 0 | − | 26.4674i | −3.82887 | + | 1.24408i | 0 | 45.3957 | + | 7.83348i | |||||||
109.1 | −2.81526 | + | 3.87487i | 0 | −4.61680 | − | 14.2091i | −11.1717 | − | 0.439337i | 0 | 14.5499i | 31.6143 | + | 10.2721i | 0 | 33.1536 | − | 42.0521i | ||||||||
109.2 | −1.81265 | + | 2.49489i | 0 | −0.466674 | − | 1.43628i | 10.8670 | − | 2.62823i | 0 | − | 5.10302i | −19.0341 | − | 6.18456i | 0 | −13.1409 | + | 31.8762i | |||||||
109.3 | 0.217515 | − | 0.299383i | 0 | 2.42982 | + | 7.47821i | −7.78705 | + | 8.02258i | 0 | 0.707538i | 5.58294 | + | 1.81401i | 0 | 0.708030 | + | 4.07634i | ||||||||
109.4 | 0.442260 | − | 0.608718i | 0 | 2.29719 | + | 7.07003i | −1.00498 | − | 11.1351i | 0 | − | 18.3105i | 11.0443 | + | 3.58852i | 0 | −7.22259 | − | 4.31284i | |||||||
109.5 | 2.23725 | − | 3.07931i | 0 | −2.00472 | − | 6.16988i | 3.48187 | − | 10.6243i | 0 | − | 4.54748i | 5.47554 | + | 1.77911i | 0 | −24.9258 | − | 34.4910i | |||||||
109.6 | 2.42187 | − | 3.33341i | 0 | −2.77407 | − | 8.53772i | 8.01402 | + | 7.79587i | 0 | 26.4674i | −3.82887 | − | 1.24408i | 0 | 45.3957 | − | 7.83348i | ||||||||
154.1 | −3.47870 | − | 1.13030i | 0 | 4.35165 | + | 3.16166i | 11.1800 | + | 0.0808329i | 0 | 8.69522i | 5.63517 | + | 7.75614i | 0 | −38.8007 | − | 12.9180i | ||||||||
154.2 | −3.33666 | − | 1.08415i | 0 | 3.48577 | + | 2.53256i | −10.7075 | + | 3.21691i | 0 | − | 25.7483i | 7.61219 | + | 10.4773i | 0 | 39.2150 | + | 0.874818i | |||||||
See all 24 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
25.e | even | 10 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 225.4.m.a | 24 | |
3.b | odd | 2 | 1 | 25.4.e.a | ✓ | 24 | |
15.d | odd | 2 | 1 | 125.4.e.a | 24 | ||
15.e | even | 4 | 2 | 125.4.d.b | 48 | ||
25.e | even | 10 | 1 | inner | 225.4.m.a | 24 | |
75.h | odd | 10 | 1 | 25.4.e.a | ✓ | 24 | |
75.j | odd | 10 | 1 | 125.4.e.a | 24 | ||
75.l | even | 20 | 2 | 125.4.d.b | 48 | ||
75.l | even | 20 | 2 | 625.4.a.g | 24 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
25.4.e.a | ✓ | 24 | 3.b | odd | 2 | 1 | |
25.4.e.a | ✓ | 24 | 75.h | odd | 10 | 1 | |
125.4.d.b | 48 | 15.e | even | 4 | 2 | ||
125.4.d.b | 48 | 75.l | even | 20 | 2 | ||
125.4.e.a | 24 | 15.d | odd | 2 | 1 | ||
125.4.e.a | 24 | 75.j | odd | 10 | 1 | ||
225.4.m.a | 24 | 1.a | even | 1 | 1 | trivial | |
225.4.m.a | 24 | 25.e | even | 10 | 1 | inner | |
625.4.a.g | 24 | 75.l | even | 20 | 2 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{24} - 5 T_{2}^{23} - 18 T_{2}^{22} + 95 T_{2}^{21} + 571 T_{2}^{20} - 2445 T_{2}^{19} - 11935 T_{2}^{18} + 72075 T_{2}^{17} + 190000 T_{2}^{16} - 1217055 T_{2}^{15} - 720663 T_{2}^{14} + 13173050 T_{2}^{13} + \cdots + 38738176 \)
acting on \(S_{4}^{\mathrm{new}}(225, [\chi])\).