Newspace parameters
Level: | \( N \) | \(=\) | \( 861 = 3 \cdot 7 \cdot 41 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 861.z (of order \(10\), degree \(4\), minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(6.87511961403\) |
Analytic rank: | \(0\) |
Dimension: | \(72\) |
Relative dimension: | \(18\) over \(\Q(\zeta_{10})\) |
Twist minimal: | yes |
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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
64.1 | −0.752183 | − | 2.31498i | 1.00000i | −3.17533 | + | 2.30701i | −2.87489 | + | 2.08873i | 2.31498 | − | 0.752183i | 0.951057 | + | 0.309017i | 3.79063 | + | 2.75405i | −1.00000 | 6.99782 | + | 5.08421i | ||||
64.2 | −0.716198 | − | 2.20423i | 1.00000i | −2.72766 | + | 1.98176i | 1.33599 | − | 0.970654i | 2.20423 | − | 0.716198i | 0.951057 | + | 0.309017i | 2.57174 | + | 1.86848i | −1.00000 | −3.09638 | − | 2.24965i | ||||
64.3 | −0.636263 | − | 1.95822i | − | 1.00000i | −1.81175 | + | 1.31631i | −2.33402 | + | 1.69577i | −1.95822 | + | 0.636263i | −0.951057 | − | 0.309017i | 0.398855 | + | 0.289785i | −1.00000 | 4.80573 | + | 3.49156i | |||
64.4 | −0.592083 | − | 1.82224i | − | 1.00000i | −1.35198 | + | 0.982267i | 1.10944 | − | 0.806055i | −1.82224 | + | 0.592083i | −0.951057 | − | 0.309017i | −0.509774 | − | 0.370373i | −1.00000 | −2.12571 | − | 1.54442i | |||
64.5 | −0.424654 | − | 1.30695i | − | 1.00000i | 0.0902446 | − | 0.0655665i | −1.42837 | + | 1.03777i | −1.30695 | + | 0.424654i | −0.951057 | − | 0.309017i | −2.34753 | − | 1.70558i | −1.00000 | 1.96288 | + | 1.42612i | |||
64.6 | −0.358744 | − | 1.10410i | 1.00000i | 0.527693 | − | 0.383392i | −1.55355 | + | 1.12872i | 1.10410 | − | 0.358744i | 0.951057 | + | 0.309017i | −2.49102 | − | 1.80983i | −1.00000 | 1.80355 | + | 1.31035i | ||||
64.7 | −0.328363 | − | 1.01060i | 1.00000i | 0.704551 | − | 0.511886i | 1.51935 | − | 1.10387i | 1.01060 | − | 0.328363i | 0.951057 | + | 0.309017i | −2.46799 | − | 1.79310i | −1.00000 | −1.61446 | − | 1.17298i | ||||
64.8 | −0.204093 | − | 0.628132i | − | 1.00000i | 1.26514 | − | 0.919176i | 2.96102 | − | 2.15131i | −0.628132 | + | 0.204093i | −0.951057 | − | 0.309017i | −1.90421 | − | 1.38349i | −1.00000 | −1.95563 | − | 1.42085i | |||
64.9 | −0.0429603 | − | 0.132218i | 1.00000i | 1.60240 | − | 1.16421i | −1.31978 | + | 0.958875i | 0.132218 | − | 0.0429603i | 0.951057 | + | 0.309017i | −0.447712 | − | 0.325282i | −1.00000 | 0.183479 | + | 0.133305i | ||||
64.10 | 0.0790778 | + | 0.243376i | − | 1.00000i | 1.56506 | − | 1.13708i | −1.80571 | + | 1.31192i | 0.243376 | − | 0.0790778i | −0.951057 | − | 0.309017i | 0.814556 | + | 0.591810i | −1.00000 | −0.462083 | − | 0.335723i | |||
64.11 | 0.170649 | + | 0.525203i | − | 1.00000i | 1.37132 | − | 0.996320i | 1.90227 | − | 1.38208i | 0.525203 | − | 0.170649i | −0.951057 | − | 0.309017i | 1.65081 | + | 1.19939i | −1.00000 | 1.05050 | + | 0.763230i | |||
64.12 | 0.304564 | + | 0.937350i | 1.00000i | 0.832167 | − | 0.604605i | 2.63453 | − | 1.91410i | −0.937350 | + | 0.304564i | 0.951057 | + | 0.309017i | 2.41489 | + | 1.75452i | −1.00000 | 2.59656 | + | 1.88651i | ||||
64.13 | 0.348523 | + | 1.07264i | − | 1.00000i | 0.588940 | − | 0.427890i | 0.357062 | − | 0.259421i | 1.07264 | − | 0.348523i | −0.951057 | − | 0.309017i | 2.48912 | + | 1.80845i | −1.00000 | 0.402710 | + | 0.292586i | |||
64.14 | 0.459506 | + | 1.41421i | 1.00000i | −0.170822 | + | 0.124110i | 0.0702561 | − | 0.0510440i | −1.41421 | + | 0.459506i | 0.951057 | + | 0.309017i | 2.15199 | + | 1.56351i | −1.00000 | 0.104470 | + | 0.0759021i | ||||
64.15 | 0.598104 | + | 1.84077i | − | 1.00000i | −1.41269 | + | 1.02638i | −0.0719999 | + | 0.0523110i | 1.84077 | − | 0.598104i | −0.951057 | − | 0.309017i | 0.397448 | + | 0.288763i | −1.00000 | −0.139356 | − | 0.101248i | |||
64.16 | 0.660739 | + | 2.03355i | − | 1.00000i | −2.08070 | + | 1.51172i | −2.35668 | + | 1.71223i | 2.03355 | − | 0.660739i | −0.951057 | − | 0.309017i | −0.989274 | − | 0.718750i | −1.00000 | −5.03904 | − | 3.66108i | |||
64.17 | 0.665430 | + | 2.04798i | 1.00000i | −2.13340 | + | 1.55001i | 0.184706 | − | 0.134197i | −2.04798 | + | 0.665430i | 0.951057 | + | 0.309017i | −1.10978 | − | 0.806305i | −1.00000 | 0.397743 | + | 0.288977i | ||||
64.18 | 0.768948 | + | 2.36658i | 1.00000i | −3.39138 | + | 2.46399i | −3.56570 | + | 2.59063i | −2.36658 | + | 0.768948i | 0.951057 | + | 0.309017i | −4.41275 | − | 3.20605i | −1.00000 | −8.87278 | − | 6.44645i | ||||
127.1 | −2.17536 | + | 1.58049i | 1.00000i | 1.61621 | − | 4.97419i | −0.719358 | + | 2.21396i | −1.58049 | − | 2.17536i | 0.587785 | − | 0.809017i | 2.68400 | + | 8.26050i | −1.00000 | −1.93428 | − | 5.95310i | ||||
127.2 | −1.79621 | + | 1.30502i | − | 1.00000i | 0.905247 | − | 2.78606i | 0.572446 | − | 1.76181i | 1.30502 | + | 1.79621i | −0.587785 | + | 0.809017i | 0.637681 | + | 1.96258i | −1.00000 | 1.27096 | + | 3.91163i | |||
See all 72 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
41.f | even | 10 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 861.2.z.a | ✓ | 72 |
41.f | even | 10 | 1 | inner | 861.2.z.a | ✓ | 72 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
861.2.z.a | ✓ | 72 | 1.a | even | 1 | 1 | trivial |
861.2.z.a | ✓ | 72 | 41.f | even | 10 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{72} + 24 T_{2}^{70} + 350 T_{2}^{68} + 4038 T_{2}^{66} + 39947 T_{2}^{64} + 12 T_{2}^{63} + 336414 T_{2}^{62} + 532 T_{2}^{61} + 2462543 T_{2}^{60} + 9832 T_{2}^{59} + 16026179 T_{2}^{58} + 73236 T_{2}^{57} + 93604517 T_{2}^{56} + \cdots + 1 \)
acting on \(S_{2}^{\mathrm{new}}(861, [\chi])\).