Newspace parameters
| Level: | \( N \) | \(=\) | \( 3822 = 2 \cdot 3 \cdot 7^{2} \cdot 13 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 3822.c (of order \(2\), degree \(1\), minimal) |
Newform invariants
| Self dual: | no |
| Analytic conductor: | \(30.5188236525\) |
| Analytic rank: | \(0\) |
| Dimension: | \(2\) |
| Coefficient field: | \(\Q(\sqrt{-1}) \) |
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| Defining polynomial: |
\( x^{2} + 1 \)
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| Coefficient ring: | \(\Z[a_1, a_2]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 78) |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of \(i = \sqrt{-1}\). We also show the integral \(q\)-expansion of the trace form.
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3822\mathbb{Z}\right)^\times\).
| \(n\) | \(1471\) | \(2549\) | \(3433\) |
| \(\chi(n)\) | \(-1\) | \(1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 883.1 |
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− | 1.00000i | −1.00000 | −1.00000 | 2.00000i | 1.00000i | 0 | 1.00000i | 1.00000 | 2.00000 | |||||||||||||||||||||||
| 883.2 | 1.00000i | −1.00000 | −1.00000 | − | 2.00000i | − | 1.00000i | 0 | − | 1.00000i | 1.00000 | 2.00000 | ||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 13.b | even | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 3822.2.c.d | 2 | |
| 7.b | odd | 2 | 1 | 78.2.b.a | ✓ | 2 | |
| 13.b | even | 2 | 1 | inner | 3822.2.c.d | 2 | |
| 21.c | even | 2 | 1 | 234.2.b.a | 2 | ||
| 28.d | even | 2 | 1 | 624.2.c.a | 2 | ||
| 35.c | odd | 2 | 1 | 1950.2.b.c | 2 | ||
| 35.f | even | 4 | 1 | 1950.2.f.d | 2 | ||
| 35.f | even | 4 | 1 | 1950.2.f.g | 2 | ||
| 56.e | even | 2 | 1 | 2496.2.c.m | 2 | ||
| 56.h | odd | 2 | 1 | 2496.2.c.f | 2 | ||
| 84.h | odd | 2 | 1 | 1872.2.c.b | 2 | ||
| 91.b | odd | 2 | 1 | 78.2.b.a | ✓ | 2 | |
| 91.i | even | 4 | 1 | 1014.2.a.b | 1 | ||
| 91.i | even | 4 | 1 | 1014.2.a.g | 1 | ||
| 91.n | odd | 6 | 2 | 1014.2.i.c | 4 | ||
| 91.t | odd | 6 | 2 | 1014.2.i.c | 4 | ||
| 91.bc | even | 12 | 2 | 1014.2.e.b | 2 | ||
| 91.bc | even | 12 | 2 | 1014.2.e.e | 2 | ||
| 273.g | even | 2 | 1 | 234.2.b.a | 2 | ||
| 273.o | odd | 4 | 1 | 3042.2.a.c | 1 | ||
| 273.o | odd | 4 | 1 | 3042.2.a.n | 1 | ||
| 364.h | even | 2 | 1 | 624.2.c.a | 2 | ||
| 364.p | odd | 4 | 1 | 8112.2.a.g | 1 | ||
| 364.p | odd | 4 | 1 | 8112.2.a.j | 1 | ||
| 455.h | odd | 2 | 1 | 1950.2.b.c | 2 | ||
| 455.s | even | 4 | 1 | 1950.2.f.d | 2 | ||
| 455.s | even | 4 | 1 | 1950.2.f.g | 2 | ||
| 728.b | even | 2 | 1 | 2496.2.c.m | 2 | ||
| 728.l | odd | 2 | 1 | 2496.2.c.f | 2 | ||
| 1092.d | odd | 2 | 1 | 1872.2.c.b | 2 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 78.2.b.a | ✓ | 2 | 7.b | odd | 2 | 1 | |
| 78.2.b.a | ✓ | 2 | 91.b | odd | 2 | 1 | |
| 234.2.b.a | 2 | 21.c | even | 2 | 1 | ||
| 234.2.b.a | 2 | 273.g | even | 2 | 1 | ||
| 624.2.c.a | 2 | 28.d | even | 2 | 1 | ||
| 624.2.c.a | 2 | 364.h | even | 2 | 1 | ||
| 1014.2.a.b | 1 | 91.i | even | 4 | 1 | ||
| 1014.2.a.g | 1 | 91.i | even | 4 | 1 | ||
| 1014.2.e.b | 2 | 91.bc | even | 12 | 2 | ||
| 1014.2.e.e | 2 | 91.bc | even | 12 | 2 | ||
| 1014.2.i.c | 4 | 91.n | odd | 6 | 2 | ||
| 1014.2.i.c | 4 | 91.t | odd | 6 | 2 | ||
| 1872.2.c.b | 2 | 84.h | odd | 2 | 1 | ||
| 1872.2.c.b | 2 | 1092.d | odd | 2 | 1 | ||
| 1950.2.b.c | 2 | 35.c | odd | 2 | 1 | ||
| 1950.2.b.c | 2 | 455.h | odd | 2 | 1 | ||
| 1950.2.f.d | 2 | 35.f | even | 4 | 1 | ||
| 1950.2.f.d | 2 | 455.s | even | 4 | 1 | ||
| 1950.2.f.g | 2 | 35.f | even | 4 | 1 | ||
| 1950.2.f.g | 2 | 455.s | even | 4 | 1 | ||
| 2496.2.c.f | 2 | 56.h | odd | 2 | 1 | ||
| 2496.2.c.f | 2 | 728.l | odd | 2 | 1 | ||
| 2496.2.c.m | 2 | 56.e | even | 2 | 1 | ||
| 2496.2.c.m | 2 | 728.b | even | 2 | 1 | ||
| 3042.2.a.c | 1 | 273.o | odd | 4 | 1 | ||
| 3042.2.a.n | 1 | 273.o | odd | 4 | 1 | ||
| 3822.2.c.d | 2 | 1.a | even | 1 | 1 | trivial | |
| 3822.2.c.d | 2 | 13.b | even | 2 | 1 | inner | |
| 8112.2.a.g | 1 | 364.p | odd | 4 | 1 | ||
| 8112.2.a.j | 1 | 364.p | odd | 4 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(3822, [\chi])\):
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\( T_{5}^{2} + 4 \)
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\( T_{11} \)
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\( T_{17} - 2 \)
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\( T_{19}^{2} + 36 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{2} + 1 \)
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| $3$ |
\( (T + 1)^{2} \)
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| $5$ |
\( T^{2} + 4 \)
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| $7$ |
\( T^{2} \)
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| $11$ |
\( T^{2} \)
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| $13$ |
\( T^{2} - 6T + 13 \)
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| $17$ |
\( (T - 2)^{2} \)
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| $19$ |
\( T^{2} + 36 \)
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| $23$ |
\( (T - 4)^{2} \)
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| $29$ |
\( (T + 10)^{2} \)
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| $31$ |
\( T^{2} + 100 \)
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| $37$ |
\( T^{2} + 64 \)
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| $41$ |
\( T^{2} + 100 \)
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| $43$ |
\( (T - 4)^{2} \)
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| $47$ |
\( T^{2} + 144 \)
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| $53$ |
\( (T + 6)^{2} \)
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| $59$ |
\( T^{2} + 16 \)
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| $61$ |
\( (T + 2)^{2} \)
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| $67$ |
\( T^{2} + 4 \)
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| $71$ |
\( T^{2} \)
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| $73$ |
\( T^{2} + 16 \)
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| $79$ |
\( T^{2} \)
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| $83$ |
\( T^{2} + 16 \)
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| $89$ |
\( T^{2} + 36 \)
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| $97$ |
\( T^{2} + 144 \)
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