Newspace parameters
Level: | \( N \) | \(=\) | \( 847 = 7 \cdot 11^{2} \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 847.n (of order \(15\), degree \(8\), not minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(6.76332905120\) |
Analytic rank: | \(0\) |
Dimension: | \(40\) |
Relative dimension: | \(5\) over \(\Q(\zeta_{15})\) |
Twist minimal: | no (minimal twist has level 77) |
Sato-Tate group: | $\mathrm{SU}(2)[C_{15}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
9.1 | −1.63776 | − | 1.81892i | 2.66301 | + | 1.18565i | −0.417144 | + | 3.96886i | 0.324669 | + | 0.0690106i | −2.20477 | − | 6.78559i | 1.08484 | + | 2.41312i | 3.94192 | − | 2.86397i | 3.67845 | + | 4.08533i | −0.406206 | − | 0.703570i |
9.2 | −1.50246 | − | 1.66865i | −1.74404 | − | 0.776495i | −0.317954 | + | 3.02513i | −2.41117 | − | 0.512511i | 1.32465 | + | 4.07684i | −2.63136 | + | 0.275607i | 1.89247 | − | 1.37496i | 0.431324 | + | 0.479033i | 2.76749 | + | 4.79344i |
9.3 | −0.292556 | − | 0.324917i | −2.36017 | − | 1.05082i | 0.189075 | − | 1.79893i | 1.18922 | + | 0.252776i | 0.349055 | + | 1.07428i | 2.48973 | + | 0.895120i | −1.34725 | + | 0.978836i | 2.45880 | + | 2.73077i | −0.265782 | − | 0.460348i |
9.4 | 0.853189 | + | 0.947563i | −0.127266 | − | 0.0566625i | 0.0391138 | − | 0.372143i | −1.15917 | − | 0.246390i | −0.0548908 | − | 0.168936i | −1.05400 | + | 2.42674i | 2.44911 | − | 1.77938i | −1.99441 | − | 2.21501i | −0.755525 | − | 1.30861i |
9.5 | 1.49691 | + | 1.66249i | 1.21952 | + | 0.542967i | −0.314069 | + | 2.98816i | 0.847402 | + | 0.180121i | 0.922843 | + | 2.84022i | 2.23951 | − | 1.40876i | −1.81822 | + | 1.32101i | −0.814967 | − | 0.905113i | 0.969038 | + | 1.67842i |
81.1 | −0.275074 | + | 2.61716i | −1.48109 | − | 1.64492i | −4.81754 | − | 1.02400i | −1.07466 | − | 0.478467i | 4.71241 | − | 3.42377i | 2.62673 | + | 0.316648i | 2.37875 | − | 7.32103i | −0.198538 | + | 1.88897i | 1.54783 | − | 2.68093i |
81.2 | −0.0611782 | + | 0.582072i | −0.925174 | − | 1.02751i | 1.62123 | + | 0.344603i | −3.62146 | − | 1.61238i | 0.654686 | − | 0.475657i | −2.53895 | + | 0.744116i | −0.661490 | + | 2.03586i | 0.113756 | − | 1.08231i | 1.16007 | − | 2.00931i |
81.3 | 0.00794644 | − | 0.0756054i | −1.57585 | − | 1.75016i | 1.95064 | + | 0.414622i | 2.35518 | + | 1.04860i | −0.144844 | + | 0.105235i | 1.83494 | − | 1.90604i | 0.0938324 | − | 0.288786i | −0.266166 | + | 2.53240i | 0.0979948 | − | 0.169732i |
81.4 | 0.0704484 | − | 0.670272i | 1.43939 | + | 1.59861i | 1.51199 | + | 0.321384i | 1.54306 | + | 0.687016i | 1.17290 | − | 0.852164i | −1.55543 | − | 2.14024i | 0.738465 | − | 2.27276i | −0.170108 | + | 1.61847i | 0.569194 | − | 0.985873i |
81.5 | 0.193255 | − | 1.83870i | 0.790915 | + | 0.878400i | −1.38718 | − | 0.294855i | −2.15843 | − | 0.960996i | 1.76796 | − | 1.28450i | 2.47553 | + | 0.933664i | 0.332410 | − | 1.02305i | 0.167545 | − | 1.59409i | −2.18411 | + | 3.78299i |
130.1 | −2.18822 | + | 0.465120i | −0.139539 | − | 1.32762i | 2.74486 | − | 1.22209i | −0.579690 | + | 0.643811i | 0.922843 | + | 2.84022i | −0.647767 | + | 2.56523i | −1.81822 | + | 1.32101i | 1.19133 | − | 0.253226i | 0.969038 | − | 1.67842i |
130.2 | −1.24721 | + | 0.265102i | 0.0145619 | + | 0.138547i | −0.341843 | + | 0.152198i | 0.792967 | − | 0.880679i | −0.0548908 | − | 0.168936i | 1.98226 | − | 1.75232i | 2.44911 | − | 1.77938i | 2.91546 | − | 0.619700i | −0.755525 | + | 1.30861i |
130.3 | 0.427664 | − | 0.0909028i | 0.270052 | + | 2.56938i | −1.65246 | + | 0.735722i | −0.813520 | + | 0.903506i | 0.349055 | + | 1.07428i | 1.62068 | + | 2.09127i | −1.34725 | + | 0.978836i | −3.59432 | + | 0.763996i | −0.265782 | + | 0.460348i |
130.4 | 2.19633 | − | 0.466843i | 0.199554 | + | 1.89863i | 2.77881 | − | 1.23721i | 1.64943 | − | 1.83188i | 1.32465 | + | 4.07684i | −0.551016 | − | 2.58774i | 1.89247 | − | 1.37496i | −0.630517 | + | 0.134020i | 2.76749 | − | 4.79344i |
130.5 | 2.39411 | − | 0.508884i | −0.304703 | − | 2.89905i | 3.64571 | − | 1.62317i | −0.222100 | + | 0.246667i | −2.20477 | − | 6.78559i | 2.63024 | + | 0.286049i | 3.94192 | − | 2.86397i | −5.37722 | + | 1.14296i | −0.406206 | + | 0.703570i |
366.1 | −0.275074 | − | 2.61716i | −1.48109 | + | 1.64492i | −4.81754 | + | 1.02400i | −1.07466 | + | 0.478467i | 4.71241 | + | 3.42377i | 2.62673 | − | 0.316648i | 2.37875 | + | 7.32103i | −0.198538 | − | 1.88897i | 1.54783 | + | 2.68093i |
366.2 | −0.0611782 | − | 0.582072i | −0.925174 | + | 1.02751i | 1.62123 | − | 0.344603i | −3.62146 | + | 1.61238i | 0.654686 | + | 0.475657i | −2.53895 | − | 0.744116i | −0.661490 | − | 2.03586i | 0.113756 | + | 1.08231i | 1.16007 | + | 2.00931i |
366.3 | 0.00794644 | + | 0.0756054i | −1.57585 | + | 1.75016i | 1.95064 | − | 0.414622i | 2.35518 | − | 1.04860i | −0.144844 | − | 0.105235i | 1.83494 | + | 1.90604i | 0.0938324 | + | 0.288786i | −0.266166 | − | 2.53240i | 0.0979948 | + | 0.169732i |
366.4 | 0.0704484 | + | 0.670272i | 1.43939 | − | 1.59861i | 1.51199 | − | 0.321384i | 1.54306 | − | 0.687016i | 1.17290 | + | 0.852164i | −1.55543 | + | 2.14024i | 0.738465 | + | 2.27276i | −0.170108 | − | 1.61847i | 0.569194 | + | 0.985873i |
366.5 | 0.193255 | + | 1.83870i | 0.790915 | − | 0.878400i | −1.38718 | + | 0.294855i | −2.15843 | + | 0.960996i | 1.76796 | + | 1.28450i | 2.47553 | − | 0.933664i | 0.332410 | + | 1.02305i | 0.167545 | + | 1.59409i | −2.18411 | − | 3.78299i |
See all 40 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
7.c | even | 3 | 1 | inner |
11.c | even | 5 | 1 | inner |
77.m | even | 15 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 847.2.n.i | 40 | |
7.c | even | 3 | 1 | inner | 847.2.n.i | 40 | |
11.b | odd | 2 | 1 | 847.2.n.h | 40 | ||
11.c | even | 5 | 2 | 77.2.m.b | ✓ | 40 | |
11.c | even | 5 | 1 | 847.2.e.i | 20 | ||
11.c | even | 5 | 1 | inner | 847.2.n.i | 40 | |
11.d | odd | 10 | 1 | 847.2.e.h | 20 | ||
11.d | odd | 10 | 1 | 847.2.n.h | 40 | ||
11.d | odd | 10 | 2 | 847.2.n.j | 40 | ||
33.h | odd | 10 | 2 | 693.2.by.b | 40 | ||
77.h | odd | 6 | 1 | 847.2.n.h | 40 | ||
77.j | odd | 10 | 2 | 539.2.q.h | 40 | ||
77.m | even | 15 | 2 | 77.2.m.b | ✓ | 40 | |
77.m | even | 15 | 2 | 539.2.f.h | 20 | ||
77.m | even | 15 | 1 | 847.2.e.i | 20 | ||
77.m | even | 15 | 1 | inner | 847.2.n.i | 40 | |
77.m | even | 15 | 1 | 5929.2.a.bw | 10 | ||
77.n | even | 30 | 1 | 5929.2.a.bz | 10 | ||
77.o | odd | 30 | 1 | 847.2.e.h | 20 | ||
77.o | odd | 30 | 1 | 847.2.n.h | 40 | ||
77.o | odd | 30 | 2 | 847.2.n.j | 40 | ||
77.o | odd | 30 | 1 | 5929.2.a.by | 10 | ||
77.p | odd | 30 | 2 | 539.2.f.g | 20 | ||
77.p | odd | 30 | 2 | 539.2.q.h | 40 | ||
77.p | odd | 30 | 1 | 5929.2.a.bx | 10 | ||
231.z | odd | 30 | 2 | 693.2.by.b | 40 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
77.2.m.b | ✓ | 40 | 11.c | even | 5 | 2 | |
77.2.m.b | ✓ | 40 | 77.m | even | 15 | 2 | |
539.2.f.g | 20 | 77.p | odd | 30 | 2 | ||
539.2.f.h | 20 | 77.m | even | 15 | 2 | ||
539.2.q.h | 40 | 77.j | odd | 10 | 2 | ||
539.2.q.h | 40 | 77.p | odd | 30 | 2 | ||
693.2.by.b | 40 | 33.h | odd | 10 | 2 | ||
693.2.by.b | 40 | 231.z | odd | 30 | 2 | ||
847.2.e.h | 20 | 11.d | odd | 10 | 1 | ||
847.2.e.h | 20 | 77.o | odd | 30 | 1 | ||
847.2.e.i | 20 | 11.c | even | 5 | 1 | ||
847.2.e.i | 20 | 77.m | even | 15 | 1 | ||
847.2.n.h | 40 | 11.b | odd | 2 | 1 | ||
847.2.n.h | 40 | 11.d | odd | 10 | 1 | ||
847.2.n.h | 40 | 77.h | odd | 6 | 1 | ||
847.2.n.h | 40 | 77.o | odd | 30 | 1 | ||
847.2.n.i | 40 | 1.a | even | 1 | 1 | trivial | |
847.2.n.i | 40 | 7.c | even | 3 | 1 | inner | |
847.2.n.i | 40 | 11.c | even | 5 | 1 | inner | |
847.2.n.i | 40 | 77.m | even | 15 | 1 | inner | |
847.2.n.j | 40 | 11.d | odd | 10 | 2 | ||
847.2.n.j | 40 | 77.o | odd | 30 | 2 | ||
5929.2.a.bw | 10 | 77.m | even | 15 | 1 | ||
5929.2.a.bx | 10 | 77.p | odd | 30 | 1 | ||
5929.2.a.by | 10 | 77.o | odd | 30 | 1 | ||
5929.2.a.bz | 10 | 77.n | even | 30 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{40} - 2 T_{2}^{39} - 9 T_{2}^{38} + 2 T_{2}^{37} + 66 T_{2}^{36} + 90 T_{2}^{35} - 137 T_{2}^{34} - 817 T_{2}^{33} - 1752 T_{2}^{32} + 4999 T_{2}^{31} + 16655 T_{2}^{30} - 25573 T_{2}^{29} - 76929 T_{2}^{28} + 79672 T_{2}^{27} + 271995 T_{2}^{26} + \cdots + 1 \)
acting on \(S_{2}^{\mathrm{new}}(847, [\chi])\).