Newspace parameters
Level: | \( N \) | \(=\) | \( 225 = 3^{2} \cdot 5^{2} \) |
Weight: | \( k \) | \(=\) | \( 3 \) |
Character orbit: | \([\chi]\) | \(=\) | 225.r (of order \(20\), degree \(8\), minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(6.13080594811\) |
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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
28.1 | −1.29583 | − | 2.54321i | 0 | −2.43759 | + | 3.35505i | 4.45624 | + | 2.26758i | 0 | −3.40272 | + | 3.40272i | 0.414625 | + | 0.0656701i | 0 | −0.00760491 | − | 14.2715i | ||||||
28.2 | 0.259330 | + | 0.508965i | 0 | 2.15935 | − | 2.97209i | 3.73307 | + | 3.32629i | 0 | 1.66138 | − | 1.66138i | 4.32944 | + | 0.685716i | 0 | −0.724866 | + | 2.76261i | ||||||
28.3 | 0.395527 | + | 0.776265i | 0 | 1.90500 | − | 2.62200i | −1.22928 | − | 4.84653i | 0 | −5.60844 | + | 5.60844i | 6.23083 | + | 0.986866i | 0 | 3.27598 | − | 2.87118i | ||||||
28.4 | 1.69523 | + | 3.32707i | 0 | −5.84445 | + | 8.04419i | −2.26962 | + | 4.45520i | 0 | 1.68463 | − | 1.68463i | −21.9189 | − | 3.47161i | 0 | −18.6703 | + | 0.00137996i | ||||||
37.1 | −0.463000 | − | 2.92327i | 0 | −4.52691 | + | 1.47088i | −0.953911 | + | 4.90816i | 0 | 4.44588 | + | 4.44588i | 1.02103 | + | 2.00389i | 0 | 14.7895 | + | 0.516059i | ||||||
37.2 | −0.0933465 | − | 0.589367i | 0 | 3.46559 | − | 1.12604i | 3.31432 | − | 3.74370i | 0 | −7.64532 | − | 7.64532i | −2.07076 | − | 4.06409i | 0 | −2.51579 | − | 1.60389i | ||||||
37.3 | 0.312579 | + | 1.97355i | 0 | 0.00704800 | − | 0.00229003i | −4.93389 | − | 0.810386i | 0 | 3.91191 | + | 3.91191i | 3.63528 | + | 7.13464i | 0 | 0.0571038 | − | 9.99057i | ||||||
37.4 | 0.513943 | + | 3.24491i | 0 | −6.46108 | + | 2.09933i | 4.99960 | + | 0.0628765i | 0 | 7.51823 | + | 7.51823i | −4.16668 | − | 8.17758i | 0 | 2.36548 | + | 16.2556i | ||||||
73.1 | −0.463000 | + | 2.92327i | 0 | −4.52691 | − | 1.47088i | −0.953911 | − | 4.90816i | 0 | 4.44588 | − | 4.44588i | 1.02103 | − | 2.00389i | 0 | 14.7895 | − | 0.516059i | ||||||
73.2 | −0.0933465 | + | 0.589367i | 0 | 3.46559 | + | 1.12604i | 3.31432 | + | 3.74370i | 0 | −7.64532 | + | 7.64532i | −2.07076 | + | 4.06409i | 0 | −2.51579 | + | 1.60389i | ||||||
73.3 | 0.312579 | − | 1.97355i | 0 | 0.00704800 | + | 0.00229003i | −4.93389 | + | 0.810386i | 0 | 3.91191 | − | 3.91191i | 3.63528 | − | 7.13464i | 0 | 0.0571038 | + | 9.99057i | ||||||
73.4 | 0.513943 | − | 3.24491i | 0 | −6.46108 | − | 2.09933i | 4.99960 | − | 0.0628765i | 0 | 7.51823 | − | 7.51823i | −4.16668 | + | 8.17758i | 0 | 2.36548 | − | 16.2556i | ||||||
127.1 | −1.80600 | − | 0.286042i | 0 | −0.624420 | − | 0.202886i | 3.20727 | − | 3.83580i | 0 | 3.62927 | + | 3.62927i | 7.58652 | + | 3.86553i | 0 | −6.88953 | + | 6.01004i | ||||||
127.2 | −0.287585 | − | 0.0455490i | 0 | −3.72360 | − | 1.20987i | −2.36408 | + | 4.40581i | 0 | −2.38950 | − | 2.38950i | 2.05348 | + | 1.04630i | 0 | 0.880552 | − | 1.15936i | ||||||
127.3 | 1.86717 | + | 0.295731i | 0 | −0.405347 | − | 0.131705i | −4.99561 | − | 0.209511i | 0 | −3.57009 | − | 3.57009i | −7.45551 | − | 3.79877i | 0 | −9.26571 | − | 1.86855i | ||||||
127.4 | 3.57427 | + | 0.566108i | 0 | 8.65068 | + | 2.81078i | 0.872190 | − | 4.92334i | 0 | −0.574149 | − | 0.574149i | 16.4311 | + | 8.37205i | 0 | 5.90458 | − | 17.1036i | ||||||
163.1 | −1.80600 | + | 0.286042i | 0 | −0.624420 | + | 0.202886i | 3.20727 | + | 3.83580i | 0 | 3.62927 | − | 3.62927i | 7.58652 | − | 3.86553i | 0 | −6.88953 | − | 6.01004i | ||||||
163.2 | −0.287585 | + | 0.0455490i | 0 | −3.72360 | + | 1.20987i | −2.36408 | − | 4.40581i | 0 | −2.38950 | + | 2.38950i | 2.05348 | − | 1.04630i | 0 | 0.880552 | + | 1.15936i | ||||||
163.3 | 1.86717 | − | 0.295731i | 0 | −0.405347 | + | 0.131705i | −4.99561 | + | 0.209511i | 0 | −3.57009 | + | 3.57009i | −7.45551 | + | 3.79877i | 0 | −9.26571 | + | 1.86855i | ||||||
163.4 | 3.57427 | − | 0.566108i | 0 | 8.65068 | − | 2.81078i | 0.872190 | + | 4.92334i | 0 | −0.574149 | + | 0.574149i | 16.4311 | − | 8.37205i | 0 | 5.90458 | + | 17.1036i | ||||||
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 | 225.3.r.a | 32 | |
3.b | odd | 2 | 1 | 25.3.f.a | ✓ | 32 | |
12.b | even | 2 | 1 | 400.3.bg.c | 32 | ||
15.d | odd | 2 | 1 | 125.3.f.c | 32 | ||
15.e | even | 4 | 1 | 125.3.f.a | 32 | ||
15.e | even | 4 | 1 | 125.3.f.b | 32 | ||
25.f | odd | 20 | 1 | inner | 225.3.r.a | 32 | |
75.h | odd | 10 | 1 | 125.3.f.b | 32 | ||
75.j | odd | 10 | 1 | 125.3.f.a | 32 | ||
75.l | even | 20 | 1 | 25.3.f.a | ✓ | 32 | |
75.l | even | 20 | 1 | 125.3.f.c | 32 | ||
300.u | odd | 20 | 1 | 400.3.bg.c | 32 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
25.3.f.a | ✓ | 32 | 3.b | odd | 2 | 1 | |
25.3.f.a | ✓ | 32 | 75.l | even | 20 | 1 | |
125.3.f.a | 32 | 15.e | even | 4 | 1 | ||
125.3.f.a | 32 | 75.j | odd | 10 | 1 | ||
125.3.f.b | 32 | 15.e | even | 4 | 1 | ||
125.3.f.b | 32 | 75.h | odd | 10 | 1 | ||
125.3.f.c | 32 | 15.d | odd | 2 | 1 | ||
125.3.f.c | 32 | 75.l | even | 20 | 1 | ||
225.3.r.a | 32 | 1.a | even | 1 | 1 | trivial | |
225.3.r.a | 32 | 25.f | odd | 20 | 1 | inner | |
400.3.bg.c | 32 | 12.b | even | 2 | 1 | ||
400.3.bg.c | 32 | 300.u | odd | 20 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{32} - 10 T_{2}^{31} + 55 T_{2}^{30} - 220 T_{2}^{29} + 612 T_{2}^{28} - 1380 T_{2}^{27} + 2695 T_{2}^{26} - 5200 T_{2}^{25} + 24863 T_{2}^{24} - 106160 T_{2}^{23} + 401985 T_{2}^{22} - 1356130 T_{2}^{21} + \cdots + 12952801 \)
acting on \(S_{3}^{\mathrm{new}}(225, [\chi])\).