Newspace parameters
Level: | \( N \) | \(=\) | \( 3525 = 3 \cdot 5^{2} \cdot 47 \) |
Weight: | \( k \) | \(=\) | \( 1 \) |
Character orbit: | \([\chi]\) | \(=\) | 3525.l (of order \(4\), degree \(2\), minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(1.75920416953\) |
Analytic rank: | \(0\) |
Dimension: | \(32\) |
Relative dimension: | \(16\) over \(\Q(i)\) |
Coefficient field: | \(\Q(\zeta_{120})\) |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
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Defining polynomial: | \( x^{32} + x^{28} - x^{20} - x^{16} - x^{12} + x^{4} + 1 \) |
Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
Coefficient ring index: | \( 1 \) |
Twist minimal: | yes |
Projective image: | \(D_{30}\) |
Projective field: | Galois closure of \(\mathbb{Q}[x]/(x^{30} - \cdots)\) |
$q$-expansion
The \(q\)-expansion and trace form are shown below.
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3525\mathbb{Z}\right)^\times\).
\(n\) | \(1552\) | \(2026\) | \(2351\) |
\(\chi(n)\) | \(\zeta_{120}^{30}\) | \(-1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
1268.1 |
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−1.38331 | − | 1.38331i | −0.777146 | − | 0.629320i | 2.82709i | 0 | 0.204489 | + | 1.94558i | −1.05097 | + | 1.05097i | 2.52743 | − | 2.52743i | 0.207912 | + | 0.978148i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
1268.2 | −1.38331 | − | 1.38331i | 0.629320 | + | 0.777146i | 2.82709i | 0 | 0.204489 | − | 1.94558i | 1.05097 | − | 1.05097i | 2.52743 | − | 2.52743i | −0.207912 | + | 0.978148i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
1268.3 | −1.29195 | − | 1.29195i | 0.544639 | − | 0.838671i | 2.33826i | 0 | −1.78716 | + | 0.379874i | 1.40647 | − | 1.40647i | 1.72896 | − | 1.72896i | −0.406737 | − | 0.913545i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
1268.4 | −1.29195 | − | 1.29195i | 0.838671 | − | 0.544639i | 2.33826i | 0 | −1.78716 | − | 0.379874i | −1.40647 | + | 1.40647i | 1.72896 | − | 1.72896i | 0.406737 | − | 0.913545i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
1268.5 | −0.946294 | − | 0.946294i | −0.933580 | + | 0.358368i | 0.790943i | 0 | 1.22256 | + | 0.544320i | 0.294032 | − | 0.294032i | −0.197829 | + | 0.197829i | 0.743145 | − | 0.669131i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
1268.6 | −0.946294 | − | 0.946294i | −0.358368 | + | 0.933580i | 0.790943i | 0 | 1.22256 | − | 0.544320i | −0.294032 | + | 0.294032i | −0.197829 | + | 0.197829i | −0.743145 | − | 0.669131i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
1268.7 | −0.147826 | − | 0.147826i | −0.0523360 | − | 0.998630i | − | 0.956295i | 0 | −0.139886 | + | 0.155360i | −0.575212 | + | 0.575212i | −0.289190 | + | 0.289190i | −0.994522 | + | 0.104528i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
1268.8 | −0.147826 | − | 0.147826i | 0.998630 | + | 0.0523360i | − | 0.956295i | 0 | −0.139886 | − | 0.155360i | 0.575212 | − | 0.575212i | −0.289190 | + | 0.289190i | 0.994522 | + | 0.104528i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
1268.9 | 0.147826 | + | 0.147826i | −0.998630 | − | 0.0523360i | − | 0.956295i | 0 | −0.139886 | − | 0.155360i | −0.575212 | + | 0.575212i | 0.289190 | − | 0.289190i | 0.994522 | + | 0.104528i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
1268.10 | 0.147826 | + | 0.147826i | 0.0523360 | + | 0.998630i | − | 0.956295i | 0 | −0.139886 | + | 0.155360i | 0.575212 | − | 0.575212i | 0.289190 | − | 0.289190i | −0.994522 | + | 0.104528i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
1268.11 | 0.946294 | + | 0.946294i | 0.358368 | − | 0.933580i | 0.790943i | 0 | 1.22256 | − | 0.544320i | 0.294032 | − | 0.294032i | 0.197829 | − | 0.197829i | −0.743145 | − | 0.669131i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
1268.12 | 0.946294 | + | 0.946294i | 0.933580 | − | 0.358368i | 0.790943i | 0 | 1.22256 | + | 0.544320i | −0.294032 | + | 0.294032i | 0.197829 | − | 0.197829i | 0.743145 | − | 0.669131i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
1268.13 | 1.29195 | + | 1.29195i | −0.838671 | + | 0.544639i | 2.33826i | 0 | −1.78716 | − | 0.379874i | 1.40647 | − | 1.40647i | −1.72896 | + | 1.72896i | 0.406737 | − | 0.913545i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
1268.14 | 1.29195 | + | 1.29195i | −0.544639 | + | 0.838671i | 2.33826i | 0 | −1.78716 | + | 0.379874i | −1.40647 | + | 1.40647i | −1.72896 | + | 1.72896i | −0.406737 | − | 0.913545i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
1268.15 | 1.38331 | + | 1.38331i | −0.629320 | − | 0.777146i | 2.82709i | 0 | 0.204489 | − | 1.94558i | −1.05097 | + | 1.05097i | −2.52743 | + | 2.52743i | −0.207912 | + | 0.978148i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
1268.16 | 1.38331 | + | 1.38331i | 0.777146 | + | 0.629320i | 2.82709i | 0 | 0.204489 | + | 1.94558i | 1.05097 | − | 1.05097i | −2.52743 | + | 2.52743i | 0.207912 | + | 0.978148i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
1832.1 | −1.38331 | + | 1.38331i | −0.777146 | + | 0.629320i | − | 2.82709i | 0 | 0.204489 | − | 1.94558i | −1.05097 | − | 1.05097i | 2.52743 | + | 2.52743i | 0.207912 | − | 0.978148i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
1832.2 | −1.38331 | + | 1.38331i | 0.629320 | − | 0.777146i | − | 2.82709i | 0 | 0.204489 | + | 1.94558i | 1.05097 | + | 1.05097i | 2.52743 | + | 2.52743i | −0.207912 | − | 0.978148i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
1832.3 | −1.29195 | + | 1.29195i | 0.544639 | + | 0.838671i | − | 2.33826i | 0 | −1.78716 | − | 0.379874i | 1.40647 | + | 1.40647i | 1.72896 | + | 1.72896i | −0.406737 | + | 0.913545i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
1832.4 | −1.29195 | + | 1.29195i | 0.838671 | + | 0.544639i | − | 2.33826i | 0 | −1.78716 | + | 0.379874i | −1.40647 | − | 1.40647i | 1.72896 | + | 1.72896i | 0.406737 | + | 0.913545i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
See all 32 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
47.b | odd | 2 | 1 | CM by \(\Q(\sqrt{-47}) \) |
3.b | odd | 2 | 1 | inner |
5.b | even | 2 | 1 | inner |
5.c | odd | 4 | 2 | inner |
15.d | odd | 2 | 1 | inner |
15.e | even | 4 | 2 | inner |
141.c | even | 2 | 1 | inner |
235.b | odd | 2 | 1 | inner |
235.e | even | 4 | 2 | inner |
705.g | even | 2 | 1 | inner |
705.l | odd | 4 | 2 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 3525.1.l.d | ✓ | 32 |
3.b | odd | 2 | 1 | inner | 3525.1.l.d | ✓ | 32 |
5.b | even | 2 | 1 | inner | 3525.1.l.d | ✓ | 32 |
5.c | odd | 4 | 2 | inner | 3525.1.l.d | ✓ | 32 |
15.d | odd | 2 | 1 | inner | 3525.1.l.d | ✓ | 32 |
15.e | even | 4 | 2 | inner | 3525.1.l.d | ✓ | 32 |
47.b | odd | 2 | 1 | CM | 3525.1.l.d | ✓ | 32 |
141.c | even | 2 | 1 | inner | 3525.1.l.d | ✓ | 32 |
235.b | odd | 2 | 1 | inner | 3525.1.l.d | ✓ | 32 |
235.e | even | 4 | 2 | inner | 3525.1.l.d | ✓ | 32 |
705.g | even | 2 | 1 | inner | 3525.1.l.d | ✓ | 32 |
705.l | odd | 4 | 2 | inner | 3525.1.l.d | ✓ | 32 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
3525.1.l.d | ✓ | 32 | 1.a | even | 1 | 1 | trivial |
3525.1.l.d | ✓ | 32 | 3.b | odd | 2 | 1 | inner |
3525.1.l.d | ✓ | 32 | 5.b | even | 2 | 1 | inner |
3525.1.l.d | ✓ | 32 | 5.c | odd | 4 | 2 | inner |
3525.1.l.d | ✓ | 32 | 15.d | odd | 2 | 1 | inner |
3525.1.l.d | ✓ | 32 | 15.e | even | 4 | 2 | inner |
3525.1.l.d | ✓ | 32 | 47.b | odd | 2 | 1 | CM |
3525.1.l.d | ✓ | 32 | 141.c | even | 2 | 1 | inner |
3525.1.l.d | ✓ | 32 | 235.b | odd | 2 | 1 | inner |
3525.1.l.d | ✓ | 32 | 235.e | even | 4 | 2 | inner |
3525.1.l.d | ✓ | 32 | 705.g | even | 2 | 1 | inner |
3525.1.l.d | ✓ | 32 | 705.l | odd | 4 | 2 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{16} + 29T_{2}^{12} + 246T_{2}^{8} + 524T_{2}^{4} + 1 \)
acting on \(S_{1}^{\mathrm{new}}(3525, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( (T^{16} + 29 T^{12} + 246 T^{8} + 524 T^{4} + \cdots + 1)^{2} \)
$3$
\( T^{32} + T^{28} - T^{20} - T^{16} - T^{12} + T^{4} + \cdots + 1 \)
$5$
\( T^{32} \)
$7$
\( (T^{16} + 21 T^{12} + 86 T^{8} + 36 T^{4} + \cdots + 1)^{2} \)
$11$
\( T^{32} \)
$13$
\( T^{32} \)
$17$
\( (T^{16} + 29 T^{12} + 246 T^{8} + 524 T^{4} + \cdots + 1)^{2} \)
$19$
\( T^{32} \)
$23$
\( T^{32} \)
$29$
\( T^{32} \)
$31$
\( T^{32} \)
$37$
\( (T^{8} + 15 T^{4} + 25)^{4} \)
$41$
\( T^{32} \)
$43$
\( T^{32} \)
$47$
\( (T^{4} + 1)^{8} \)
$53$
\( (T^{8} + 7 T^{4} + 1)^{4} \)
$59$
\( (T^{8} - 7 T^{6} + 14 T^{4} - 8 T^{2} + 1)^{4} \)
$61$
\( (T^{2} - T - 1)^{16} \)
$67$
\( T^{32} \)
$71$
\( (T^{8} + 7 T^{6} + 14 T^{4} + 8 T^{2} + 1)^{4} \)
$73$
\( T^{32} \)
$79$
\( (T^{4} + 3 T^{2} + 1)^{8} \)
$83$
\( (T^{4} + 1)^{8} \)
$89$
\( (T^{4} - 5 T^{2} + 5)^{8} \)
$97$
\( (T^{8} + 15 T^{4} + 25)^{4} \)
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