Newspace parameters
Level: | \( N \) | \(=\) | \( 273 = 3 \cdot 7 \cdot 13 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 273.bv (of order \(12\), degree \(4\), minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(2.17991597518\) |
Analytic rank: | \(0\) |
Dimension: | \(128\) |
Relative dimension: | \(32\) over \(\Q(\zeta_{12})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{12}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
2.1 | −1.83790 | − | 1.83790i | 1.29065 | + | 1.15509i | 4.75574i | 2.20098 | − | 0.589752i | −0.249144 | − | 4.49501i | −2.64509 | − | 0.0592679i | 5.06476 | − | 5.06476i | 0.331543 | + | 2.98162i | −5.12909 | − | 2.96128i | ||
2.2 | −1.81876 | − | 1.81876i | −0.267828 | − | 1.71122i | 4.61580i | 2.15482 | − | 0.577382i | −2.62519 | + | 3.59942i | 1.30976 | − | 2.29881i | 4.75753 | − | 4.75753i | −2.85654 | + | 0.916623i | −4.96923 | − | 2.86898i | ||
2.3 | −1.73689 | − | 1.73689i | 1.00077 | − | 1.41367i | 4.03355i | −2.06624 | + | 0.553646i | −4.19360 | + | 0.717172i | −1.93609 | + | 1.80321i | 3.53204 | − | 3.53204i | −0.996937 | − | 2.82951i | 4.55044 | + | 2.62720i | ||
2.4 | −1.52398 | − | 1.52398i | −1.72942 | − | 0.0953716i | 2.64502i | −2.04381 | + | 0.547638i | 2.49026 | + | 2.78095i | −2.26018 | − | 1.37535i | 0.982996 | − | 0.982996i | 2.98181 | + | 0.329876i | 3.94931 | + | 2.28014i | ||
2.5 | −1.48334 | − | 1.48334i | 1.73149 | − | 0.0439971i | 2.40062i | −0.600039 | + | 0.160780i | −2.63366 | − | 2.50313i | 2.54988 | + | 0.705761i | 0.594253 | − | 0.594253i | 2.99613 | − | 0.152361i | 1.12856 | + | 0.651572i | ||
2.6 | −1.44872 | − | 1.44872i | −0.783380 | + | 1.54477i | 2.19760i | 0.629445 | − | 0.168659i | 3.37285 | − | 1.10304i | −0.719902 | + | 2.54593i | 0.286268 | − | 0.286268i | −1.77263 | − | 2.42029i | −1.15623 | − | 0.667551i | ||
2.7 | −1.32954 | − | 1.32954i | −1.62246 | − | 0.606325i | 1.53537i | 3.58803 | − | 0.961410i | 1.35099 | + | 2.96326i | 0.951973 | + | 2.46855i | −0.617751 | + | 0.617751i | 2.26474 | + | 1.96747i | −6.04868 | − | 3.49220i | ||
2.8 | −1.21487 | − | 1.21487i | 0.917597 | + | 1.46902i | 0.951801i | −2.79534 | + | 0.749010i | 0.669904 | − | 2.89942i | −0.107896 | − | 2.64355i | −1.27342 | + | 1.27342i | −1.31603 | + | 2.69593i | 4.30591 | + | 2.48602i | ||
2.9 | −0.879582 | − | 0.879582i | 1.39839 | − | 1.02202i | − | 0.452670i | 2.08749 | − | 0.559342i | −2.12894 | − | 0.331049i | 0.755530 | − | 2.53558i | −2.15733 | + | 2.15733i | 0.910969 | − | 2.85835i | −2.32811 | − | 1.34414i | |
2.10 | −0.824043 | − | 0.824043i | −0.772097 | − | 1.55044i | − | 0.641906i | −0.114643 | + | 0.0307184i | −0.641388 | + | 1.91387i | −2.54084 | − | 0.737668i | −2.17704 | + | 2.17704i | −1.80773 | + | 2.39418i | 0.119784 | + | 0.0691572i | |
2.11 | −0.748547 | − | 0.748547i | −1.33611 | + | 1.10219i | − | 0.879356i | −2.02207 | + | 0.541811i | 1.82518 | + | 0.175098i | 2.61787 | + | 0.383108i | −2.15533 | + | 2.15533i | 0.570358 | − | 2.94528i | 1.91918 | + | 1.10804i | |
2.12 | −0.583259 | − | 0.583259i | −0.386481 | + | 1.68838i | − | 1.31962i | 3.27322 | − | 0.877056i | 1.21018 | − | 0.759346i | −0.829143 | − | 2.51247i | −1.93620 | + | 1.93620i | −2.70127 | − | 1.30505i | −2.42069 | − | 1.39758i | |
2.13 | −0.554014 | − | 0.554014i | 0.671048 | + | 1.59678i | − | 1.38614i | −1.68690 | + | 0.452004i | 0.512866 | − | 1.25641i | −1.01706 | + | 2.44246i | −1.87597 | + | 1.87597i | −2.09939 | + | 2.14303i | 1.18499 | + | 0.684151i | |
2.14 | −0.397102 | − | 0.397102i | 1.57374 | − | 0.723421i | − | 1.68462i | −3.29155 | + | 0.881968i | −0.912208 | − | 0.337664i | −2.64283 | + | 0.124223i | −1.46317 | + | 1.46317i | 1.95332 | − | 2.27696i | 1.65731 | + | 0.956850i | |
2.15 | −0.363959 | − | 0.363959i | −1.62227 | − | 0.606835i | − | 1.73507i | 0.585566 | − | 0.156902i | 0.369576 | + | 0.811302i | 2.22038 | − | 1.43871i | −1.35941 | + | 1.35941i | 2.26350 | + | 1.96890i | −0.270228 | − | 0.156016i | |
2.16 | −0.340464 | − | 0.340464i | 1.62034 | + | 0.611966i | − | 1.76817i | 3.37589 | − | 0.904567i | −0.343315 | − | 0.760019i | −0.304456 | + | 2.62818i | −1.28292 | + | 1.28292i | 2.25100 | + | 1.98318i | −1.45734 | − | 0.841396i | |
2.17 | 0.340464 | + | 0.340464i | −1.34015 | − | 1.09727i | − | 1.76817i | −3.37589 | + | 0.904567i | −0.0826902 | − | 0.829853i | −0.304456 | + | 2.62818i | 1.28292 | − | 1.28292i | 0.591989 | + | 2.94101i | −1.45734 | − | 0.841396i | |
2.18 | 0.363959 | + | 0.363959i | 1.33667 | + | 1.10151i | − | 1.73507i | −0.585566 | + | 0.156902i | 0.0855890 | + | 0.887396i | 2.22038 | − | 1.43871i | 1.35941 | − | 1.35941i | 0.573364 | + | 2.94470i | −0.270228 | − | 0.156016i | |
2.19 | 0.397102 | + | 0.397102i | −0.160369 | − | 1.72461i | − | 1.68462i | 3.29155 | − | 0.881968i | 0.621163 | − | 0.748529i | −2.64283 | + | 0.124223i | 1.46317 | − | 1.46317i | −2.94856 | + | 0.553150i | 1.65731 | + | 0.956850i | |
2.20 | 0.554014 | + | 0.554014i | −1.71837 | + | 0.217243i | − | 1.38614i | 1.68690 | − | 0.452004i | −1.07236 | − | 0.831647i | −1.01706 | + | 2.44246i | 1.87597 | − | 1.87597i | 2.90561 | − | 0.746610i | 1.18499 | + | 0.684151i | |
See next 80 embeddings (of 128 total) |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
3.b | odd | 2 | 1 | inner |
91.x | odd | 12 | 1 | inner |
273.bv | even | 12 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 273.2.bv.b | ✓ | 128 |
3.b | odd | 2 | 1 | inner | 273.2.bv.b | ✓ | 128 |
7.c | even | 3 | 1 | 273.2.bw.b | yes | 128 | |
13.f | odd | 12 | 1 | 273.2.bw.b | yes | 128 | |
21.h | odd | 6 | 1 | 273.2.bw.b | yes | 128 | |
39.k | even | 12 | 1 | 273.2.bw.b | yes | 128 | |
91.x | odd | 12 | 1 | inner | 273.2.bv.b | ✓ | 128 |
273.bv | even | 12 | 1 | inner | 273.2.bv.b | ✓ | 128 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
273.2.bv.b | ✓ | 128 | 1.a | even | 1 | 1 | trivial |
273.2.bv.b | ✓ | 128 | 3.b | odd | 2 | 1 | inner |
273.2.bv.b | ✓ | 128 | 91.x | odd | 12 | 1 | inner |
273.2.bv.b | ✓ | 128 | 273.bv | even | 12 | 1 | inner |
273.2.bw.b | yes | 128 | 7.c | even | 3 | 1 | |
273.2.bw.b | yes | 128 | 13.f | odd | 12 | 1 | |
273.2.bw.b | yes | 128 | 21.h | odd | 6 | 1 | |
273.2.bw.b | yes | 128 | 39.k | even | 12 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{128} + 452 T_{2}^{124} + 94644 T_{2}^{120} + 12196318 T_{2}^{116} + 1083883810 T_{2}^{112} + 70551504126 T_{2}^{108} + 3487095853283 T_{2}^{104} + 133909455267320 T_{2}^{100} + \cdots + 269517889 \)
acting on \(S_{2}^{\mathrm{new}}(273, [\chi])\).