Newspace parameters
Level: | \( N \) | \(=\) | \( 440 = 2^{3} \cdot 5 \cdot 11 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 440.bi (of order \(10\), degree \(4\), minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(3.51341768894\) |
Analytic rank: | \(0\) |
Dimension: | \(8\) |
Relative dimension: | \(2\) over \(\Q(\zeta_{10})\) |
Coefficient field: | \(\Q(\zeta_{20})\) |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
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Defining polynomial: | \( x^{8} - x^{6} + x^{4} - x^{2} + 1 \) |
Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
Coefficient ring index: | \( 1 \) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{10}]$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{20}\). We also show the integral \(q\)-expansion of the trace form.
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/440\mathbb{Z}\right)^\times\).
\(n\) | \(111\) | \(177\) | \(221\) | \(321\) |
\(\chi(n)\) | \(1\) | \(1\) | \(-1\) | \(-\zeta_{20}^{6}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
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141.1 |
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−0.642040 | − | 1.26007i | −0.363271 | − | 0.500000i | −1.17557 | + | 1.61803i | −0.951057 | + | 0.309017i | −0.396802 | + | 0.778768i | 0.618034 | + | 0.449028i | 2.79360 | + | 0.442463i | 0.809017 | − | 2.48990i | 1.00000 | + | 1.00000i | ||||||||||||||||||||||||
141.2 | 1.26007 | − | 0.642040i | 0.363271 | + | 0.500000i | 1.17557 | − | 1.61803i | 0.951057 | − | 0.309017i | 0.778768 | + | 0.396802i | 0.618034 | + | 0.449028i | 0.442463 | − | 2.79360i | 0.809017 | − | 2.48990i | 1.00000 | − | 1.00000i | |||||||||||||||||||||||||
181.1 | −0.642040 | + | 1.26007i | −0.363271 | + | 0.500000i | −1.17557 | − | 1.61803i | −0.951057 | − | 0.309017i | −0.396802 | − | 0.778768i | 0.618034 | − | 0.449028i | 2.79360 | − | 0.442463i | 0.809017 | + | 2.48990i | 1.00000 | − | 1.00000i | |||||||||||||||||||||||||
181.2 | 1.26007 | + | 0.642040i | 0.363271 | − | 0.500000i | 1.17557 | + | 1.61803i | 0.951057 | + | 0.309017i | 0.778768 | − | 0.396802i | 0.618034 | − | 0.449028i | 0.442463 | + | 2.79360i | 0.809017 | + | 2.48990i | 1.00000 | + | 1.00000i | |||||||||||||||||||||||||
301.1 | −1.39680 | + | 0.221232i | −1.53884 | − | 0.500000i | 1.90211 | − | 0.618034i | −0.587785 | − | 0.809017i | 2.26007 | + | 0.357960i | −1.61803 | − | 4.97980i | −2.52015 | + | 1.28408i | −0.309017 | − | 0.224514i | 1.00000 | + | 1.00000i | |||||||||||||||||||||||||
301.2 | −0.221232 | − | 1.39680i | 1.53884 | + | 0.500000i | −1.90211 | + | 0.618034i | 0.587785 | + | 0.809017i | 0.357960 | − | 2.26007i | −1.61803 | − | 4.97980i | 1.28408 | + | 2.52015i | −0.309017 | − | 0.224514i | 1.00000 | − | 1.00000i | |||||||||||||||||||||||||
421.1 | −1.39680 | − | 0.221232i | −1.53884 | + | 0.500000i | 1.90211 | + | 0.618034i | −0.587785 | + | 0.809017i | 2.26007 | − | 0.357960i | −1.61803 | + | 4.97980i | −2.52015 | − | 1.28408i | −0.309017 | + | 0.224514i | 1.00000 | − | 1.00000i | |||||||||||||||||||||||||
421.2 | −0.221232 | + | 1.39680i | 1.53884 | − | 0.500000i | −1.90211 | − | 0.618034i | 0.587785 | − | 0.809017i | 0.357960 | + | 2.26007i | −1.61803 | + | 4.97980i | 1.28408 | − | 2.52015i | −0.309017 | + | 0.224514i | 1.00000 | + | 1.00000i | |||||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
8.b | even | 2 | 1 | inner |
11.c | even | 5 | 1 | inner |
88.o | even | 10 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 440.2.bi.b | ✓ | 8 |
8.b | even | 2 | 1 | inner | 440.2.bi.b | ✓ | 8 |
11.c | even | 5 | 1 | inner | 440.2.bi.b | ✓ | 8 |
88.o | even | 10 | 1 | inner | 440.2.bi.b | ✓ | 8 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
440.2.bi.b | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
440.2.bi.b | ✓ | 8 | 8.b | even | 2 | 1 | inner |
440.2.bi.b | ✓ | 8 | 11.c | even | 5 | 1 | inner |
440.2.bi.b | ✓ | 8 | 88.o | even | 10 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{8} - 4T_{3}^{6} + 6T_{3}^{4} + T_{3}^{2} + 1 \)
acting on \(S_{2}^{\mathrm{new}}(440, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{8} + 2 T^{7} + 2 T^{6} - 4 T^{4} + \cdots + 16 \)
$3$
\( T^{8} - 4 T^{6} + 6 T^{4} + T^{2} + 1 \)
$5$
\( T^{8} - T^{6} + T^{4} - T^{2} + 1 \)
$7$
\( (T^{4} + 2 T^{3} + 24 T^{2} - 32 T + 16)^{2} \)
$11$
\( T^{8} - 41 T^{6} + 661 T^{4} + \cdots + 14641 \)
$13$
\( T^{8} + 16 T^{6} + 736 T^{4} + \cdots + 256 \)
$17$
\( (T^{4} - 4 T^{3} + 6 T^{2} + T + 1)^{2} \)
$19$
\( T^{8} + 25 T^{6} + 3750 T^{4} + \cdots + 390625 \)
$23$
\( (T^{2} + 2 T - 4)^{4} \)
$29$
\( T^{8} - 16 T^{6} + 256 T^{4} + \cdots + 65536 \)
$31$
\( (T^{4} + 2 T^{3} + 4 T^{2} + 8 T + 16)^{2} \)
$37$
\( T^{8} - 64 T^{6} + 4096 T^{4} + \cdots + 16777216 \)
$41$
\( (T^{4} - 6 T^{3} + 76 T^{2} - 781 T + 5041)^{2} \)
$43$
\( (T^{4} + 27 T^{2} + 81)^{2} \)
$47$
\( (T^{4} + 22 T^{3} + 244 T^{2} + 1368 T + 5776)^{2} \)
$53$
\( T^{8} + 20 T^{6} + 2400 T^{4} + \cdots + 160000 \)
$59$
\( T^{8} + T^{6} + 6 T^{4} - 4 T^{2} + 1 \)
$61$
\( T^{8} - 20 T^{6} + 400 T^{4} + \cdots + 160000 \)
$67$
\( (T^{4} + 183 T^{2} + 961)^{2} \)
$71$
\( (T^{4} - 18 T^{3} + 244 T^{2} - 1672 T + 5776)^{2} \)
$73$
\( (T^{4} - 12 T^{3} + 54 T^{2} + 27 T + 81)^{2} \)
$79$
\( (T^{4} + 24 T^{3} + 216 T^{2} - 216 T + 1296)^{2} \)
$83$
\( T^{8} - 31 T^{6} + 366 T^{4} + \cdots + 14641 \)
$89$
\( (T^{2} - 9 T - 81)^{4} \)
$97$
\( (T^{4} - 21 T^{3} + 166 T^{2} + 164 T + 1681)^{2} \)
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