Newspace parameters
Level: | \( N \) | \(=\) | \( 136 = 2^{3} \cdot 17 \) |
Weight: | \( k \) | \(=\) | \( 3 \) |
Character orbit: | \([\chi]\) | \(=\) | 136.p (of order \(8\), degree \(4\), minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(3.70573159530\) |
Analytic rank: | \(0\) |
Dimension: | \(128\) |
Relative dimension: | \(32\) over \(\Q(\zeta_{8})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{8}]$ |
$q$-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 | −1.99989 | − | 0.0206161i | 2.82466 | − | 1.17001i | 3.99915 | + | 0.0824601i | −6.65544 | + | 2.75677i | −5.67315 | + | 2.28167i | −9.71464 | − | 4.02394i | −7.99617 | − | 0.247358i | 0.245829 | − | 0.245829i | 13.3670 | − | 5.37604i |
19.2 | −1.97912 | − | 0.288256i | 1.21149 | − | 0.501817i | 3.83382 | + | 1.14099i | 6.29726 | − | 2.60841i | −2.54234 | + | 0.643934i | 2.88787 | + | 1.19620i | −7.25868 | − | 3.36327i | −5.14807 | + | 5.14807i | −13.2149 | + | 3.34713i |
19.3 | −1.93870 | + | 0.491361i | −4.34673 | + | 1.80048i | 3.51713 | − | 1.90520i | 0.750539 | − | 0.310883i | 7.54234 | − | 5.62640i | −11.2238 | − | 4.64906i | −5.88252 | + | 5.42180i | 9.28842 | − | 9.28842i | −1.30232 | + | 0.971496i |
19.4 | −1.87130 | + | 0.705866i | −2.10387 | + | 0.871452i | 3.00351 | − | 2.64177i | −1.18140 | + | 0.489353i | 3.32184 | − | 3.11580i | 7.56608 | + | 3.13397i | −3.75572 | + | 7.06361i | −2.69711 | + | 2.69711i | 1.86534 | − | 1.74964i |
19.5 | −1.79251 | − | 0.887078i | −0.958919 | + | 0.397197i | 2.42619 | + | 3.18019i | 0.484158 | − | 0.200545i | 2.07122 | + | 0.138655i | −3.06598 | − | 1.26997i | −1.52789 | − | 7.85274i | −5.60220 | + | 5.60220i | −1.04576 | − | 0.0700070i |
19.6 | −1.75586 | + | 0.957582i | 4.91989 | − | 2.03789i | 2.16607 | − | 3.36276i | 4.41938 | − | 1.83057i | −6.68719 | + | 8.28944i | 0.225959 | + | 0.0935954i | −0.583198 | + | 7.97871i | 13.6884 | − | 13.6884i | −6.00688 | + | 7.44614i |
19.7 | −1.68630 | − | 1.07536i | −3.95884 | + | 1.63981i | 1.68722 | + | 3.62675i | −7.46648 | + | 3.09272i | 8.43917 | + | 1.49196i | 2.68777 | + | 1.11331i | 1.05489 | − | 7.93015i | 6.61949 | − | 6.61949i | 15.9165 | + | 2.81387i |
19.8 | −1.66343 | − | 1.11041i | 4.47944 | − | 1.85545i | 1.53399 | + | 3.69417i | −4.45144 | + | 1.84385i | −9.51154 | − | 1.88761i | 12.7752 | + | 5.29167i | 1.55035 | − | 7.84834i | 10.2588 | − | 10.2588i | 9.45207 | + | 1.87580i |
19.9 | −1.51389 | + | 1.30695i | 1.83762 | − | 0.761166i | 0.583738 | − | 3.95718i | −4.01021 | + | 1.66108i | −1.78715 | + | 3.55401i | 4.10587 | + | 1.70071i | 4.28814 | + | 6.75366i | −3.56650 | + | 3.56650i | 3.90006 | − | 7.75587i |
19.10 | −1.11041 | − | 1.66343i | 4.47944 | − | 1.85545i | −1.53399 | + | 3.69417i | 4.45144 | − | 1.84385i | −8.06041 | − | 5.39093i | −12.7752 | − | 5.29167i | 7.84834 | − | 1.55035i | 10.2588 | − | 10.2588i | −8.01001 | − | 5.35723i |
19.11 | −1.07536 | − | 1.68630i | −3.95884 | + | 1.63981i | −1.68722 | + | 3.62675i | 7.46648 | − | 3.09272i | 7.02237 | + | 4.91242i | −2.68777 | − | 1.11331i | 7.93015 | − | 1.05489i | 6.61949 | − | 6.61949i | −13.2444 | − | 9.26495i |
19.12 | −0.987360 | + | 1.73929i | −3.13949 | + | 1.30042i | −2.05024 | − | 3.43461i | 7.34303 | − | 3.04158i | 0.838003 | − | 6.74445i | 4.30870 | + | 1.78472i | 7.99809 | − | 0.174765i | 1.80133 | − | 1.80133i | −1.96003 | + | 15.7748i |
19.13 | −0.934433 | + | 1.76829i | 0.813063 | − | 0.336782i | −2.25367 | − | 3.30469i | −1.46394 | + | 0.606383i | −0.164227 | + | 1.75243i | −7.48231 | − | 3.09928i | 7.94954 | − | 0.897117i | −5.81631 | + | 5.81631i | 0.295694 | − | 3.15529i |
19.14 | −0.887078 | − | 1.79251i | −0.958919 | + | 0.397197i | −2.42619 | + | 3.18019i | −0.484158 | + | 0.200545i | 1.56262 | + | 1.36653i | 3.06598 | + | 1.26997i | 7.85274 | + | 1.52789i | −5.60220 | + | 5.60220i | 0.788964 | + | 0.689959i |
19.15 | −0.610091 | + | 1.90468i | −4.33611 | + | 1.79608i | −3.25558 | − | 2.32405i | −5.90990 | + | 2.44796i | −0.775521 | − | 9.35466i | 1.71003 | + | 0.708317i | 6.41276 | − | 4.78294i | 9.21202 | − | 9.21202i | −1.05700 | − | 12.7499i |
19.16 | −0.288256 | − | 1.97912i | 1.21149 | − | 0.501817i | −3.83382 | + | 1.14099i | −6.29726 | + | 2.60841i | −1.34237 | − | 2.25303i | −2.88787 | − | 1.19620i | 3.36327 | + | 7.25868i | −5.14807 | + | 5.14807i | 6.97758 | + | 11.7111i |
19.17 | −0.170951 | + | 1.99268i | 3.48144 | − | 1.44206i | −3.94155 | − | 0.681301i | 3.91706 | − | 1.62250i | 2.27841 | + | 7.18392i | 1.50905 | + | 0.625068i | 2.03143 | − | 7.73778i | 3.67692 | − | 3.67692i | 2.56350 | + | 8.08282i |
19.18 | −0.0206161 | − | 1.99989i | 2.82466 | − | 1.17001i | −3.99915 | + | 0.0824601i | 6.65544 | − | 2.75677i | −2.39814 | − | 5.62490i | 9.71464 | + | 4.02394i | 0.247358 | + | 7.99617i | 0.245829 | − | 0.245829i | −5.65046 | − | 13.2533i |
19.19 | 0.449222 | + | 1.94890i | 1.11155 | − | 0.460421i | −3.59640 | + | 1.75098i | −7.74186 | + | 3.20678i | 1.39665 | + | 1.95947i | 9.91980 | + | 4.10892i | −5.02805 | − | 6.22243i | −5.34040 | + | 5.34040i | −9.72751 | − | 13.6475i |
19.20 | 0.463170 | + | 1.94563i | −1.79304 | + | 0.742700i | −3.57095 | + | 1.80232i | 0.987744 | − | 0.409137i | −2.27550 | − | 3.14459i | −8.55035 | − | 3.54167i | −5.16059 | − | 6.11296i | −3.70059 | + | 3.70059i | 1.25352 | + | 1.73228i |
See next 80 embeddings (of 128 total) |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
8.d | odd | 2 | 1 | inner |
17.d | even | 8 | 1 | inner |
136.p | odd | 8 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 136.3.p.c | ✓ | 128 |
8.d | odd | 2 | 1 | inner | 136.3.p.c | ✓ | 128 |
17.d | even | 8 | 1 | inner | 136.3.p.c | ✓ | 128 |
136.p | odd | 8 | 1 | inner | 136.3.p.c | ✓ | 128 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
136.3.p.c | ✓ | 128 | 1.a | even | 1 | 1 | trivial |
136.3.p.c | ✓ | 128 | 8.d | odd | 2 | 1 | inner |
136.3.p.c | ✓ | 128 | 17.d | even | 8 | 1 | inner |
136.3.p.c | ✓ | 128 | 136.p | odd | 8 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{64} + 4 T_{3}^{63} + 24 T_{3}^{62} + 40 T_{3}^{61} + 128 T_{3}^{60} + 248 T_{3}^{59} - 5424 T_{3}^{58} - 1792 T_{3}^{57} + 992596 T_{3}^{56} + 5091712 T_{3}^{55} + 22737104 T_{3}^{54} + 29693376 T_{3}^{53} + \cdots + 25\!\cdots\!24 \)
acting on \(S_{3}^{\mathrm{new}}(136, [\chi])\).