Newspace parameters
Level: | \( N \) | \(=\) | \( 725 = 5^{2} \cdot 29 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 725.r (of order \(14\), degree \(6\), not minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(5.78915414654\) |
Analytic rank: | \(0\) |
Dimension: | \(12\) |
Relative dimension: | \(2\) over \(\Q(\zeta_{14})\) |
Coefficient field: | \(\Q(\zeta_{28})\) |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
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Defining polynomial: | \( x^{12} - x^{10} + x^{8} - x^{6} + x^{4} - x^{2} + 1 \) |
Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
Coefficient ring index: | \( 1 \) |
Twist minimal: | no (minimal twist has level 29) |
Sato-Tate group: | $\mathrm{SU}(2)[C_{14}]$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{28}\). We also show the integral \(q\)-expansion of the trace form.
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/725\mathbb{Z}\right)^\times\).
\(n\) | \(176\) | \(552\) |
\(\chi(n)\) | \(-1 + \zeta_{28}^{2} - \zeta_{28}^{4} + \zeta_{28}^{6} - \zeta_{28}^{8} + \zeta_{28}^{10}\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
24.1 |
|
−1.40881 | + | 1.12349i | 0.433884 | − | 0.0990311i | 0.277479 | − | 1.21572i | 0 | −0.500000 | + | 0.626980i | 3.94740 | − | 0.900969i | −0.588735 | − | 1.22252i | −2.52446 | + | 1.21572i | 0 | ||||||||||||||||||||||||||||||||||||||||
24.2 | 1.40881 | − | 1.12349i | −0.433884 | + | 0.0990311i | 0.277479 | − | 1.21572i | 0 | −0.500000 | + | 0.626980i | −3.94740 | + | 0.900969i | 0.588735 | + | 1.22252i | −2.52446 | + | 1.21572i | 0 | |||||||||||||||||||||||||||||||||||||||||
49.1 | −0.193096 | − | 0.400969i | 0.974928 | − | 0.777479i | 1.12349 | − | 1.40881i | 0 | −0.500000 | − | 0.240787i | 0.279032 | − | 0.222521i | −1.64960 | − | 0.376510i | −0.321552 | + | 1.40881i | 0 | |||||||||||||||||||||||||||||||||||||||||
49.2 | 0.193096 | + | 0.400969i | −0.974928 | + | 0.777479i | 1.12349 | − | 1.40881i | 0 | −0.500000 | − | 0.240787i | −0.279032 | + | 0.222521i | 1.64960 | + | 0.376510i | −0.321552 | + | 1.40881i | 0 | |||||||||||||||||||||||||||||||||||||||||
74.1 | −0.193096 | + | 0.400969i | 0.974928 | + | 0.777479i | 1.12349 | + | 1.40881i | 0 | −0.500000 | + | 0.240787i | 0.279032 | + | 0.222521i | −1.64960 | + | 0.376510i | −0.321552 | − | 1.40881i | 0 | |||||||||||||||||||||||||||||||||||||||||
74.2 | 0.193096 | − | 0.400969i | −0.974928 | − | 0.777479i | 1.12349 | + | 1.40881i | 0 | −0.500000 | + | 0.240787i | −0.279032 | − | 0.222521i | 1.64960 | − | 0.376510i | −0.321552 | − | 1.40881i | 0 | |||||||||||||||||||||||||||||||||||||||||
199.1 | −1.21572 | − | 0.277479i | 0.781831 | + | 1.62349i | −0.400969 | − | 0.193096i | 0 | −0.500000 | − | 2.19064i | −0.300257 | − | 0.623490i | 2.38374 | + | 1.90097i | −0.153989 | + | 0.193096i | 0 | |||||||||||||||||||||||||||||||||||||||||
199.2 | 1.21572 | + | 0.277479i | −0.781831 | − | 1.62349i | −0.400969 | − | 0.193096i | 0 | −0.500000 | − | 2.19064i | 0.300257 | + | 0.623490i | −2.38374 | − | 1.90097i | −0.153989 | + | 0.193096i | 0 | |||||||||||||||||||||||||||||||||||||||||
574.1 | −1.40881 | − | 1.12349i | 0.433884 | + | 0.0990311i | 0.277479 | + | 1.21572i | 0 | −0.500000 | − | 0.626980i | 3.94740 | + | 0.900969i | −0.588735 | + | 1.22252i | −2.52446 | − | 1.21572i | 0 | |||||||||||||||||||||||||||||||||||||||||
574.2 | 1.40881 | + | 1.12349i | −0.433884 | − | 0.0990311i | 0.277479 | + | 1.21572i | 0 | −0.500000 | − | 0.626980i | −3.94740 | − | 0.900969i | 0.588735 | − | 1.22252i | −2.52446 | − | 1.21572i | 0 | |||||||||||||||||||||||||||||||||||||||||
674.1 | −1.21572 | + | 0.277479i | 0.781831 | − | 1.62349i | −0.400969 | + | 0.193096i | 0 | −0.500000 | + | 2.19064i | −0.300257 | + | 0.623490i | 2.38374 | − | 1.90097i | −0.153989 | − | 0.193096i | 0 | |||||||||||||||||||||||||||||||||||||||||
674.2 | 1.21572 | − | 0.277479i | −0.781831 | + | 1.62349i | −0.400969 | + | 0.193096i | 0 | −0.500000 | + | 2.19064i | 0.300257 | − | 0.623490i | −2.38374 | + | 1.90097i | −0.153989 | − | 0.193096i | 0 | |||||||||||||||||||||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
5.b | even | 2 | 1 | inner |
29.d | even | 7 | 1 | inner |
145.n | even | 14 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 725.2.r.b | 12 | |
5.b | even | 2 | 1 | inner | 725.2.r.b | 12 | |
5.c | odd | 4 | 1 | 29.2.d.a | ✓ | 6 | |
5.c | odd | 4 | 1 | 725.2.l.b | 6 | ||
15.e | even | 4 | 1 | 261.2.k.a | 6 | ||
20.e | even | 4 | 1 | 464.2.u.f | 6 | ||
29.d | even | 7 | 1 | inner | 725.2.r.b | 12 | |
145.e | even | 4 | 1 | 841.2.e.d | 12 | ||
145.h | odd | 4 | 1 | 841.2.d.d | 6 | ||
145.j | even | 4 | 1 | 841.2.e.d | 12 | ||
145.n | even | 14 | 1 | inner | 725.2.r.b | 12 | |
145.o | even | 28 | 1 | 841.2.b.c | 6 | ||
145.o | even | 28 | 2 | 841.2.e.b | 12 | ||
145.o | even | 28 | 2 | 841.2.e.c | 12 | ||
145.o | even | 28 | 1 | 841.2.e.d | 12 | ||
145.p | odd | 28 | 1 | 29.2.d.a | ✓ | 6 | |
145.p | odd | 28 | 1 | 725.2.l.b | 6 | ||
145.p | odd | 28 | 1 | 841.2.a.e | 3 | ||
145.p | odd | 28 | 2 | 841.2.d.b | 6 | ||
145.p | odd | 28 | 2 | 841.2.d.e | 6 | ||
145.q | odd | 28 | 1 | 841.2.a.f | 3 | ||
145.q | odd | 28 | 2 | 841.2.d.a | 6 | ||
145.q | odd | 28 | 2 | 841.2.d.c | 6 | ||
145.q | odd | 28 | 1 | 841.2.d.d | 6 | ||
145.t | even | 28 | 1 | 841.2.b.c | 6 | ||
145.t | even | 28 | 2 | 841.2.e.b | 12 | ||
145.t | even | 28 | 2 | 841.2.e.c | 12 | ||
145.t | even | 28 | 1 | 841.2.e.d | 12 | ||
435.bg | even | 28 | 1 | 7569.2.a.p | 3 | ||
435.bj | even | 28 | 1 | 261.2.k.a | 6 | ||
435.bj | even | 28 | 1 | 7569.2.a.r | 3 | ||
580.bi | even | 28 | 1 | 464.2.u.f | 6 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
29.2.d.a | ✓ | 6 | 5.c | odd | 4 | 1 | |
29.2.d.a | ✓ | 6 | 145.p | odd | 28 | 1 | |
261.2.k.a | 6 | 15.e | even | 4 | 1 | ||
261.2.k.a | 6 | 435.bj | even | 28 | 1 | ||
464.2.u.f | 6 | 20.e | even | 4 | 1 | ||
464.2.u.f | 6 | 580.bi | even | 28 | 1 | ||
725.2.l.b | 6 | 5.c | odd | 4 | 1 | ||
725.2.l.b | 6 | 145.p | odd | 28 | 1 | ||
725.2.r.b | 12 | 1.a | even | 1 | 1 | trivial | |
725.2.r.b | 12 | 5.b | even | 2 | 1 | inner | |
725.2.r.b | 12 | 29.d | even | 7 | 1 | inner | |
725.2.r.b | 12 | 145.n | even | 14 | 1 | inner | |
841.2.a.e | 3 | 145.p | odd | 28 | 1 | ||
841.2.a.f | 3 | 145.q | odd | 28 | 1 | ||
841.2.b.c | 6 | 145.o | even | 28 | 1 | ||
841.2.b.c | 6 | 145.t | even | 28 | 1 | ||
841.2.d.a | 6 | 145.q | odd | 28 | 2 | ||
841.2.d.b | 6 | 145.p | odd | 28 | 2 | ||
841.2.d.c | 6 | 145.q | odd | 28 | 2 | ||
841.2.d.d | 6 | 145.h | odd | 4 | 1 | ||
841.2.d.d | 6 | 145.q | odd | 28 | 1 | ||
841.2.d.e | 6 | 145.p | odd | 28 | 2 | ||
841.2.e.b | 12 | 145.o | even | 28 | 2 | ||
841.2.e.b | 12 | 145.t | even | 28 | 2 | ||
841.2.e.c | 12 | 145.o | even | 28 | 2 | ||
841.2.e.c | 12 | 145.t | even | 28 | 2 | ||
841.2.e.d | 12 | 145.e | even | 4 | 1 | ||
841.2.e.d | 12 | 145.j | even | 4 | 1 | ||
841.2.e.d | 12 | 145.o | even | 28 | 1 | ||
841.2.e.d | 12 | 145.t | even | 28 | 1 | ||
7569.2.a.p | 3 | 435.bg | even | 28 | 1 | ||
7569.2.a.r | 3 | 435.bj | even | 28 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{12} - 4T_{2}^{10} + 16T_{2}^{8} - 29T_{2}^{6} + 18T_{2}^{4} + 5T_{2}^{2} + 1 \)
acting on \(S_{2}^{\mathrm{new}}(725, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{12} - 4 T^{10} + 16 T^{8} - 29 T^{6} + \cdots + 1 \)
$3$
\( T^{12} + 3 T^{10} + 9 T^{8} - T^{6} + 25 T^{4} + \cdots + 1 \)
$5$
\( T^{12} \)
$7$
\( T^{12} - 29 T^{10} + 253 T^{8} + 139 T^{6} + \cdots + 1 \)
$11$
\( (T^{6} + 11 T^{5} + 79 T^{4} + 365 T^{3} + \cdots + 1681)^{2} \)
$13$
\( T^{12} - 25 T^{10} - 159 T^{8} + \cdots + 3418801 \)
$17$
\( (T^{6} + 24 T^{4} + 80 T^{2} + 64)^{2} \)
$19$
\( (T^{6} + T^{5} + T^{4} + 15 T^{3} + 29 T^{2} + \cdots + 169)^{2} \)
$23$
\( T^{12} - 49 T^{10} + 1029 T^{8} + \cdots + 5764801 \)
$29$
\( (T^{6} + 6 T^{5} - 13 T^{4} - 316 T^{3} + \cdots + 24389)^{2} \)
$31$
\( (T^{6} - 5 T^{5} - 3 T^{4} - 139 T^{3} + \cdots + 6889)^{2} \)
$37$
\( T^{12} - 37 T^{10} + 389 T^{8} + \cdots + 2825761 \)
$41$
\( (T^{3} - 10 T^{2} + 24 T - 8)^{4} \)
$43$
\( T^{12} - T^{10} + 1009 T^{8} + \cdots + 28561 \)
$47$
\( T^{12} - 9 T^{10} + \cdots + 3262808641 \)
$53$
\( T^{12} + 19 T^{10} + 613 T^{8} + 5795 T^{6} + \cdots + 1 \)
$59$
\( (T^{3} - 28 T^{2} + 252 T - 728)^{4} \)
$61$
\( (T^{6} - 3 T^{5} + 37 T^{4} - 13 T^{3} + \cdots + 169)^{2} \)
$67$
\( T^{12} - 109 T^{10} + 50913 T^{8} + \cdots + 28561 \)
$71$
\( (T^{6} - 21 T^{5} + 189 T^{4} + \cdots + 35721)^{2} \)
$73$
\( T^{12} - 121 T^{10} + \cdots + 30821664721 \)
$79$
\( (T^{6} - 9 T^{5} + 81 T^{4} + 27 T^{3} + \cdots + 729)^{2} \)
$83$
\( T^{12} - 149 T^{10} + \cdots + 815730721 \)
$89$
\( (T^{6} + 7 T^{5} + 119 T^{4} + 1337 T^{3} + \cdots + 8281)^{2} \)
$97$
\( T^{12} - 421 T^{10} + 50345 T^{8} + \cdots + 28561 \)
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