Newspace parameters
Level: | \( N \) | \(=\) | \( 847 = 7 \cdot 11^{2} \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 847.r (of order \(30\), degree \(8\), not minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(6.76332905120\) |
Analytic rank: | \(0\) |
Dimension: | \(48\) |
Relative dimension: | \(6\) over \(\Q(\zeta_{30})\) |
Twist minimal: | no (minimal twist has level 77) |
Sato-Tate group: | $\mathrm{SU}(2)[C_{30}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
40.1 | −0.958667 | − | 2.15320i | −0.419789 | − | 1.97495i | −2.37898 | + | 2.64212i | −0.0295601 | + | 0.00310689i | −3.85003 | + | 2.79721i | 0.936082 | + | 2.47462i | 3.48644 | + | 1.13281i | −0.983572 | + | 0.437914i | 0.0350280 | + | 0.0606703i |
40.2 | −0.211548 | − | 0.475144i | 0.235625 | + | 1.10853i | 1.15725 | − | 1.28526i | 3.24614 | − | 0.341183i | 0.476865 | − | 0.346463i | 2.63196 | − | 0.269780i | −1.84481 | − | 0.599414i | 1.56732 | − | 0.697816i | −0.848825 | − | 1.47021i |
40.3 | 0.0110597 | + | 0.0248404i | −0.255965 | − | 1.20422i | 1.33777 | − | 1.48574i | −1.15068 | + | 0.120942i | 0.0270824 | − | 0.0196765i | −1.30554 | + | 2.30121i | 0.103422 | + | 0.0336040i | 1.35601 | − | 0.603736i | −0.0157304 | − | 0.0272459i |
40.4 | 0.608987 | + | 1.36781i | 0.125410 | + | 0.590009i | −0.161772 | + | 0.179666i | −2.35031 | + | 0.247028i | −0.730646 | + | 0.530845i | −0.609417 | − | 2.57461i | 2.50368 | + | 0.813494i | 2.40825 | − | 1.07222i | −1.76920 | − | 3.06434i |
40.5 | 0.729906 | + | 1.63940i | −0.600621 | − | 2.82570i | −0.816596 | + | 0.906921i | 2.00074 | − | 0.210287i | 4.19404 | − | 3.04715i | 2.34325 | − | 1.22849i | 1.33058 | + | 0.432332i | −4.88320 | + | 2.17414i | 1.80510 | + | 3.12652i |
40.6 | 1.07207 | + | 2.40791i | 0.332663 | + | 1.56506i | −3.31042 | + | 3.67659i | 2.13544 | − | 0.224443i | −3.41187 | + | 2.47887i | −0.295626 | + | 2.62918i | −7.38833 | − | 2.40061i | 0.401899 | − | 0.178937i | 2.82977 | + | 4.90131i |
94.1 | −1.53858 | − | 1.38534i | 1.27771 | + | 2.86979i | 0.238995 | + | 2.27389i | 0.246997 | + | 1.16203i | 2.00978 | − | 6.18547i | 2.24786 | + | 1.39540i | 0.348541 | − | 0.479726i | −4.59574 | + | 5.10408i | 1.22979 | − | 2.13006i |
94.2 | −1.04360 | − | 0.939665i | −0.0831894 | − | 0.186846i | −0.00291864 | − | 0.0277690i | 0.316039 | + | 1.48685i | −0.0887562 | + | 0.273164i | −2.50854 | − | 0.840980i | −1.67391 | + | 2.30394i | 1.97940 | − | 2.19835i | 1.06732 | − | 1.84865i |
94.3 | −0.830445 | − | 0.747736i | −0.634829 | − | 1.42585i | −0.0785273 | − | 0.747137i | −0.640220 | − | 3.01200i | −0.538968 | + | 1.65877i | 2.61986 | + | 0.369199i | −1.80712 | + | 2.48728i | 0.377352 | − | 0.419092i | −1.72051 | + | 2.98001i |
94.4 | 0.356247 | + | 0.320766i | 0.910648 | + | 2.04535i | −0.185036 | − | 1.76050i | −0.279700 | − | 1.31588i | −0.331663 | + | 1.02075i | −2.48234 | − | 0.915422i | 1.06233 | − | 1.46217i | −1.34678 | + | 1.49576i | 0.322449 | − | 0.558497i |
94.5 | 0.973480 | + | 0.876525i | −1.11095 | − | 2.49522i | −0.0296902 | − | 0.282484i | 0.0850810 | + | 0.400275i | 1.10564 | − | 3.40282i | 0.0315105 | − | 2.64556i | 1.75864 | − | 2.42055i | −2.98455 | + | 3.31468i | −0.268026 | + | 0.464235i |
94.6 | 1.54298 | + | 1.38930i | 0.0760020 | + | 0.170703i | 0.241558 | + | 2.29827i | −0.533436 | − | 2.50962i | −0.119889 | + | 0.368981i | 0.538211 | + | 2.59043i | −0.379465 | + | 0.522289i | 1.98403 | − | 2.20349i | 2.66354 | − | 4.61339i |
215.1 | −0.430453 | + | 2.02512i | 3.12416 | + | 0.328363i | −2.08874 | − | 0.929967i | −0.882850 | + | 0.794922i | −2.00978 | + | 6.18547i | 0.632479 | + | 2.56904i | 0.348541 | − | 0.479726i | 6.71813 | + | 1.42798i | −1.22979 | − | 2.13006i |
215.2 | −0.291972 | + | 1.37362i | −0.203408 | − | 0.0213791i | 0.0255080 | + | 0.0113569i | −1.12963 | + | 1.01712i | 0.0887562 | − | 0.273164i | −0.0246392 | − | 2.64564i | −1.67391 | + | 2.30394i | −2.89352 | − | 0.615038i | −1.06732 | − | 1.84865i |
215.3 | −0.232336 | + | 1.09305i | −1.55224 | − | 0.163147i | 0.686304 | + | 0.305562i | 2.28836 | − | 2.06045i | 0.538968 | − | 1.65877i | −0.458454 | + | 2.60573i | −1.80712 | + | 2.48728i | −0.551621 | − | 0.117251i | 1.72051 | + | 2.98001i |
215.4 | 0.0996681 | − | 0.468902i | 2.22665 | + | 0.234030i | 1.61716 | + | 0.720004i | 0.999739 | − | 0.900169i | 0.331663 | − | 1.02075i | −0.103533 | − | 2.64372i | 1.06233 | − | 1.46217i | 1.96875 | + | 0.418472i | −0.322449 | − | 0.558497i |
215.5 | 0.272353 | − | 1.28132i | −2.71640 | − | 0.285505i | 0.259483 | + | 0.115529i | −0.304108 | + | 0.273820i | −1.10564 | + | 3.40282i | −2.52582 | − | 0.787556i | 1.75864 | − | 2.42055i | 4.36287 | + | 0.927357i | 0.268026 | + | 0.464235i |
215.6 | 0.431683 | − | 2.03091i | 0.185834 | + | 0.0195320i | −2.11114 | − | 0.939941i | 1.90668 | − | 1.71678i | 0.119889 | − | 0.368981i | 2.29733 | + | 1.31236i | −0.379465 | + | 0.522289i | −2.90029 | − | 0.616476i | −2.66354 | − | 4.61339i |
360.1 | −0.958667 | + | 2.15320i | −0.419789 | + | 1.97495i | −2.37898 | − | 2.64212i | −0.0295601 | − | 0.00310689i | −3.85003 | − | 2.79721i | 0.936082 | − | 2.47462i | 3.48644 | − | 1.13281i | −0.983572 | − | 0.437914i | 0.0350280 | − | 0.0606703i |
360.2 | −0.211548 | + | 0.475144i | 0.235625 | − | 1.10853i | 1.15725 | + | 1.28526i | 3.24614 | + | 0.341183i | 0.476865 | + | 0.346463i | 2.63196 | + | 0.269780i | −1.84481 | + | 0.599414i | 1.56732 | + | 0.697816i | −0.848825 | + | 1.47021i |
See all 48 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
7.d | odd | 6 | 1 | inner |
11.d | odd | 10 | 1 | inner |
77.n | even | 30 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 847.2.r.a | 48 | |
7.d | odd | 6 | 1 | inner | 847.2.r.a | 48 | |
11.b | odd | 2 | 1 | 847.2.r.d | 48 | ||
11.c | even | 5 | 1 | 77.2.n.a | ✓ | 48 | |
11.c | even | 5 | 1 | 847.2.i.b | 48 | ||
11.c | even | 5 | 1 | 847.2.r.c | 48 | ||
11.c | even | 5 | 1 | 847.2.r.d | 48 | ||
11.d | odd | 10 | 1 | 77.2.n.a | ✓ | 48 | |
11.d | odd | 10 | 1 | 847.2.i.b | 48 | ||
11.d | odd | 10 | 1 | inner | 847.2.r.a | 48 | |
11.d | odd | 10 | 1 | 847.2.r.c | 48 | ||
33.f | even | 10 | 1 | 693.2.cg.a | 48 | ||
33.h | odd | 10 | 1 | 693.2.cg.a | 48 | ||
77.i | even | 6 | 1 | 847.2.r.d | 48 | ||
77.j | odd | 10 | 1 | 539.2.s.d | 48 | ||
77.l | even | 10 | 1 | 539.2.s.d | 48 | ||
77.m | even | 15 | 1 | 539.2.m.a | 48 | ||
77.m | even | 15 | 1 | 539.2.s.d | 48 | ||
77.n | even | 30 | 1 | 77.2.n.a | ✓ | 48 | |
77.n | even | 30 | 1 | 539.2.m.a | 48 | ||
77.n | even | 30 | 1 | 847.2.i.b | 48 | ||
77.n | even | 30 | 1 | inner | 847.2.r.a | 48 | |
77.n | even | 30 | 1 | 847.2.r.c | 48 | ||
77.o | odd | 30 | 1 | 539.2.m.a | 48 | ||
77.o | odd | 30 | 1 | 539.2.s.d | 48 | ||
77.p | odd | 30 | 1 | 77.2.n.a | ✓ | 48 | |
77.p | odd | 30 | 1 | 539.2.m.a | 48 | ||
77.p | odd | 30 | 1 | 847.2.i.b | 48 | ||
77.p | odd | 30 | 1 | 847.2.r.c | 48 | ||
77.p | odd | 30 | 1 | 847.2.r.d | 48 | ||
231.bc | even | 30 | 1 | 693.2.cg.a | 48 | ||
231.bf | odd | 30 | 1 | 693.2.cg.a | 48 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
77.2.n.a | ✓ | 48 | 11.c | even | 5 | 1 | |
77.2.n.a | ✓ | 48 | 11.d | odd | 10 | 1 | |
77.2.n.a | ✓ | 48 | 77.n | even | 30 | 1 | |
77.2.n.a | ✓ | 48 | 77.p | odd | 30 | 1 | |
539.2.m.a | 48 | 77.m | even | 15 | 1 | ||
539.2.m.a | 48 | 77.n | even | 30 | 1 | ||
539.2.m.a | 48 | 77.o | odd | 30 | 1 | ||
539.2.m.a | 48 | 77.p | odd | 30 | 1 | ||
539.2.s.d | 48 | 77.j | odd | 10 | 1 | ||
539.2.s.d | 48 | 77.l | even | 10 | 1 | ||
539.2.s.d | 48 | 77.m | even | 15 | 1 | ||
539.2.s.d | 48 | 77.o | odd | 30 | 1 | ||
693.2.cg.a | 48 | 33.f | even | 10 | 1 | ||
693.2.cg.a | 48 | 33.h | odd | 10 | 1 | ||
693.2.cg.a | 48 | 231.bc | even | 30 | 1 | ||
693.2.cg.a | 48 | 231.bf | odd | 30 | 1 | ||
847.2.i.b | 48 | 11.c | even | 5 | 1 | ||
847.2.i.b | 48 | 11.d | odd | 10 | 1 | ||
847.2.i.b | 48 | 77.n | even | 30 | 1 | ||
847.2.i.b | 48 | 77.p | odd | 30 | 1 | ||
847.2.r.a | 48 | 1.a | even | 1 | 1 | trivial | |
847.2.r.a | 48 | 7.d | odd | 6 | 1 | inner | |
847.2.r.a | 48 | 11.d | odd | 10 | 1 | inner | |
847.2.r.a | 48 | 77.n | even | 30 | 1 | inner | |
847.2.r.c | 48 | 11.c | even | 5 | 1 | ||
847.2.r.c | 48 | 11.d | odd | 10 | 1 | ||
847.2.r.c | 48 | 77.n | even | 30 | 1 | ||
847.2.r.c | 48 | 77.p | odd | 30 | 1 | ||
847.2.r.d | 48 | 11.b | odd | 2 | 1 | ||
847.2.r.d | 48 | 11.c | even | 5 | 1 | ||
847.2.r.d | 48 | 77.i | even | 6 | 1 | ||
847.2.r.d | 48 | 77.p | odd | 30 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{48} + 5 T_{2}^{47} + 18 T_{2}^{46} + 65 T_{2}^{45} + 162 T_{2}^{44} + 360 T_{2}^{43} + 502 T_{2}^{42} - 50 T_{2}^{41} - 2375 T_{2}^{40} - 11210 T_{2}^{39} - 31718 T_{2}^{38} - 76060 T_{2}^{37} - 152303 T_{2}^{36} - 260405 T_{2}^{35} + \cdots + 1 \)
acting on \(S_{2}^{\mathrm{new}}(847, [\chi])\).