Newspace parameters
Level: | \( N \) | \(=\) | \( 825 = 3 \cdot 5^{2} \cdot 11 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 825.bs (of order \(10\), degree \(4\), not minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(6.58765816676\) |
Analytic rank: | \(0\) |
Dimension: | \(48\) |
Relative dimension: | \(12\) over \(\Q(\zeta_{10})\) |
Twist minimal: | no (minimal twist has level 165) |
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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
74.1 | −1.52016 | − | 2.09233i | −0.921840 | − | 1.46636i | −1.44890 | + | 4.45925i | 0 | −1.66676 | + | 4.15790i | 0.106178 | − | 0.326781i | 6.61341 | − | 2.14883i | −1.30042 | + | 2.70350i | 0 | ||||
74.2 | −1.52016 | − | 2.09233i | −0.116121 | − | 1.72815i | −1.44890 | + | 4.45925i | 0 | −3.43934 | + | 2.87004i | −0.106178 | + | 0.326781i | 6.61341 | − | 2.14883i | −2.97303 | + | 0.401348i | 0 | ||||
74.3 | −1.12347 | − | 1.54632i | −1.67101 | + | 0.455760i | −0.510897 | + | 1.57238i | 0 | 2.58208 | + | 2.07189i | 0.0258398 | − | 0.0795268i | −0.630238 | + | 0.204777i | 2.58457 | − | 1.52316i | 0 | ||||
74.4 | −1.12347 | − | 1.54632i | 1.61977 | − | 0.613479i | −0.510897 | + | 1.57238i | 0 | −2.76839 | − | 1.81546i | −0.0258398 | + | 0.0795268i | −0.630238 | + | 0.204777i | 2.24729 | − | 1.98739i | 0 | ||||
74.5 | −0.625329 | − | 0.860692i | −0.197973 | + | 1.72070i | 0.268280 | − | 0.825682i | 0 | 1.60479 | − | 0.905610i | −0.709430 | + | 2.18340i | −2.90203 | + | 0.942926i | −2.92161 | − | 0.681304i | 0 | ||||
74.6 | −0.625329 | − | 0.860692i | 1.17157 | + | 1.27571i | 0.268280 | − | 0.825682i | 0 | 0.365379 | − | 1.80609i | 0.709430 | − | 2.18340i | −2.90203 | + | 0.942926i | −0.254870 | + | 2.98915i | 0 | ||||
74.7 | 0.0958309 | + | 0.131900i | −0.923314 | + | 1.46543i | 0.609820 | − | 1.87683i | 0 | −0.281772 | + | 0.0186487i | −1.10960 | + | 3.41501i | 0.616109 | − | 0.200186i | −1.29498 | − | 2.70611i | 0 | ||||
74.8 | 0.0958309 | + | 0.131900i | 1.60834 | + | 0.642850i | 0.609820 | − | 1.87683i | 0 | 0.0693364 | + | 0.273744i | 1.10960 | − | 3.41501i | 0.616109 | − | 0.200186i | 2.17349 | + | 2.06784i | 0 | ||||
74.9 | 0.997022 | + | 1.37228i | −0.979464 | + | 1.42851i | −0.271073 | + | 0.834277i | 0 | −2.93687 | + | 0.0801565i | 1.24436 | − | 3.82974i | 1.81130 | − | 0.588527i | −1.08130 | − | 2.79836i | 0 | ||||
74.10 | 0.997022 | + | 1.37228i | 1.63206 | + | 0.579977i | −0.271073 | + | 0.834277i | 0 | 0.831309 | + | 2.81790i | −1.24436 | + | 3.82974i | 1.81130 | − | 0.588527i | 2.32725 | + | 1.89312i | 0 | ||||
74.11 | 1.05808 | + | 1.45632i | −1.44104 | − | 0.960940i | −0.383300 | + | 1.17968i | 0 | −0.125297 | − | 3.11536i | 0.0472713 | − | 0.145486i | 1.30046 | − | 0.422546i | 1.15319 | + | 2.76950i | 0 | ||||
74.12 | 1.05808 | + | 1.45632i | 0.600999 | − | 1.62444i | −0.383300 | + | 1.17968i | 0 | 3.00160 | − | 0.843534i | −0.0472713 | + | 0.145486i | 1.30046 | − | 0.422546i | −2.27760 | − | 1.95257i | 0 | ||||
149.1 | −2.04385 | + | 0.664086i | 0.200478 | + | 1.72041i | 2.11826 | − | 1.53901i | 0 | −1.55225 | − | 3.38312i | −1.60200 | + | 1.16392i | −0.781038 | + | 1.07501i | −2.91962 | + | 0.689809i | 0 | ||||
149.2 | −2.04385 | + | 0.664086i | 1.69816 | − | 0.340970i | 2.11826 | − | 1.53901i | 0 | −3.24434 | + | 1.82461i | 1.60200 | − | 1.16392i | −0.781038 | + | 1.07501i | 2.76748 | − | 1.15804i | 0 | ||||
149.3 | −0.654275 | + | 0.212587i | −1.40962 | + | 1.00646i | −1.23515 | + | 0.897390i | 0 | 0.708322 | − | 0.958169i | −1.22699 | + | 0.891460i | 1.42608 | − | 1.96284i | 0.974079 | − | 2.83746i | 0 | ||||
149.4 | −0.654275 | + | 0.212587i | 0.521602 | − | 1.65165i | −1.23515 | + | 0.897390i | 0 | 0.00984690 | + | 1.19152i | 1.22699 | − | 0.891460i | 1.42608 | − | 1.96284i | −2.45586 | − | 1.72300i | 0 | ||||
149.5 | −0.431318 | + | 0.140144i | 0.703911 | + | 1.58256i | −1.45164 | + | 1.05468i | 0 | −0.525396 | − | 0.583940i | −3.74621 | + | 2.72178i | 1.01145 | − | 1.39214i | −2.00902 | + | 2.22797i | 0 | ||||
149.6 | −0.431318 | + | 0.140144i | 1.72263 | + | 0.180419i | −1.45164 | + | 1.05468i | 0 | −0.768285 | + | 0.163597i | 3.74621 | − | 2.72178i | 1.01145 | − | 1.39214i | 2.93490 | + | 0.621591i | 0 | ||||
149.7 | 0.124137 | − | 0.0403345i | −1.60787 | + | 0.644015i | −1.60425 | + | 1.16556i | 0 | −0.173620 | + | 0.144799i | 2.06924 | − | 1.50339i | −0.305576 | + | 0.420589i | 2.17049 | − | 2.07099i | 0 | ||||
149.8 | 0.124137 | − | 0.0403345i | 0.115636 | − | 1.72819i | −1.60425 | + | 1.16556i | 0 | −0.0553509 | − | 0.219196i | −2.06924 | + | 1.50339i | −0.305576 | + | 0.420589i | −2.97326 | − | 0.399681i | 0 | ||||
See all 48 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
11.d | odd | 10 | 1 | inner |
15.d | odd | 2 | 1 | inner |
165.r | even | 10 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 825.2.bs.h | 48 | |
3.b | odd | 2 | 1 | 825.2.bs.g | 48 | ||
5.b | even | 2 | 1 | 825.2.bs.g | 48 | ||
5.c | odd | 4 | 1 | 165.2.p.b | ✓ | 48 | |
5.c | odd | 4 | 1 | 825.2.bi.e | 48 | ||
11.d | odd | 10 | 1 | inner | 825.2.bs.h | 48 | |
15.d | odd | 2 | 1 | inner | 825.2.bs.h | 48 | |
15.e | even | 4 | 1 | 165.2.p.b | ✓ | 48 | |
15.e | even | 4 | 1 | 825.2.bi.e | 48 | ||
33.f | even | 10 | 1 | 825.2.bs.g | 48 | ||
55.h | odd | 10 | 1 | 825.2.bs.g | 48 | ||
55.l | even | 20 | 1 | 165.2.p.b | ✓ | 48 | |
55.l | even | 20 | 1 | 825.2.bi.e | 48 | ||
165.r | even | 10 | 1 | inner | 825.2.bs.h | 48 | |
165.u | odd | 20 | 1 | 165.2.p.b | ✓ | 48 | |
165.u | odd | 20 | 1 | 825.2.bi.e | 48 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
165.2.p.b | ✓ | 48 | 5.c | odd | 4 | 1 | |
165.2.p.b | ✓ | 48 | 15.e | even | 4 | 1 | |
165.2.p.b | ✓ | 48 | 55.l | even | 20 | 1 | |
165.2.p.b | ✓ | 48 | 165.u | odd | 20 | 1 | |
825.2.bi.e | 48 | 5.c | odd | 4 | 1 | ||
825.2.bi.e | 48 | 15.e | even | 4 | 1 | ||
825.2.bi.e | 48 | 55.l | even | 20 | 1 | ||
825.2.bi.e | 48 | 165.u | odd | 20 | 1 | ||
825.2.bs.g | 48 | 3.b | odd | 2 | 1 | ||
825.2.bs.g | 48 | 5.b | even | 2 | 1 | ||
825.2.bs.g | 48 | 33.f | even | 10 | 1 | ||
825.2.bs.g | 48 | 55.h | odd | 10 | 1 | ||
825.2.bs.h | 48 | 1.a | even | 1 | 1 | trivial | |
825.2.bs.h | 48 | 11.d | odd | 10 | 1 | inner | |
825.2.bs.h | 48 | 15.d | odd | 2 | 1 | inner | |
825.2.bs.h | 48 | 165.r | even | 10 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{24} - 7 T_{2}^{22} - 15 T_{2}^{21} + 43 T_{2}^{20} + 105 T_{2}^{19} - 128 T_{2}^{18} - 270 T_{2}^{17} + 886 T_{2}^{16} + 1425 T_{2}^{15} - 1264 T_{2}^{14} - 4970 T_{2}^{13} + 6374 T_{2}^{12} + 6395 T_{2}^{11} - 8398 T_{2}^{10} - 12920 T_{2}^{9} + \cdots + 1 \)
acting on \(S_{2}^{\mathrm{new}}(825, [\chi])\).