Newspace parameters
Level: | \( N \) | \(=\) | \( 525 = 3 \cdot 5^{2} \cdot 7 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 525.i (of order \(3\), degree \(2\), minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(4.19214610612\) |
Analytic rank: | \(0\) |
Dimension: | \(8\) |
Relative dimension: | \(4\) over \(\Q(\zeta_{3})\) |
Coefficient field: | \(\mathbb{Q}[x]/(x^{8} - \cdots)\) |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
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Defining polynomial: | \( x^{8} - 2x^{7} + 8x^{6} + 21x^{4} - 4x^{3} + 28x^{2} + 12x + 9 \) |
Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
Coefficient ring index: | \( 1 \) |
Twist minimal: | no (minimal twist has level 105) |
Sato-Tate group: | $\mathrm{SU}(2)[C_{3}]$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.
Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 2x^{7} + 8x^{6} + 21x^{4} - 4x^{3} + 28x^{2} + 12x + 9 \) :
\(\beta_{1}\) | \(=\) | \( \nu \) |
\(\beta_{2}\) | \(=\) | \( ( 68\nu^{7} - 215\nu^{6} + 357\nu^{5} + 646\nu^{4} - 1444\nu^{3} + 1156\nu^{2} + 561\nu + 5468 ) / 4243 \) |
\(\beta_{3}\) | \(=\) | \( ( 84\nu^{7} - 16\nu^{6} + 441\nu^{5} + 798\nu^{4} + 3208\nu^{3} + 1428\nu^{2} + 693\nu + 2262 ) / 4243 \) |
\(\beta_{4}\) | \(=\) | \( ( -754\nu^{7} + 1760\nu^{6} - 6080\nu^{5} + 1323\nu^{4} - 13440\nu^{3} + 12640\nu^{2} - 16828\nu + 5760 ) / 12729 \) |
\(\beta_{5}\) | \(=\) | \( ( -815\nu^{7} + 3388\nu^{6} - 11704\nu^{5} + 15594\nu^{4} - 25872\nu^{3} + 24332\nu^{2} - 56579\nu + 11088 ) / 12729 \) |
\(\beta_{6}\) | \(=\) | \( ( 1052\nu^{7} - 2827\nu^{6} + 9766\nu^{5} - 6978\nu^{4} + 21588\nu^{3} - 20303\nu^{2} + 17165\nu - 9252 ) / 12729 \) |
\(\beta_{7}\) | \(=\) | \( ( -556\nu^{7} + 510\nu^{6} - 2919\nu^{5} - 5282\nu^{4} - 8909\nu^{3} - 9452\nu^{2} - 4587\nu - 13760 ) / 4243 \) |
\(\nu\) | \(=\) | \( \beta_1 \) |
\(\nu^{2}\) | \(=\) | \( \beta_{6} + 2\beta_{4} + \beta_{3} + \beta_{2} + \beta _1 - 2 \) |
\(\nu^{3}\) | \(=\) | \( \beta_{7} + 5\beta_{3} + 2\beta_{2} - 2 \) |
\(\nu^{4}\) | \(=\) | \( -8\beta_{6} - 2\beta_{5} - 9\beta_{4} - 10\beta_1 \) |
\(\nu^{5}\) | \(=\) | \( -8\beta_{7} - 20\beta_{6} - 8\beta_{5} - 18\beta_{4} - 33\beta_{3} - 20\beta_{2} - 33\beta _1 + 18 \) |
\(\nu^{6}\) | \(=\) | \( -20\beta_{7} - 83\beta_{3} - 61\beta_{2} + 58 \) |
\(\nu^{7}\) | \(=\) | \( 164\beta_{6} + 61\beta_{5} + 146\beta_{4} + 243\beta_1 \) |
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/525\mathbb{Z}\right)^\times\).
\(n\) | \(127\) | \(176\) | \(451\) |
\(\chi(n)\) | \(1\) | \(1\) | \(-1 + \beta_{4}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
151.1 |
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−1.25829 | − | 2.17942i | 0.500000 | − | 0.866025i | −2.16659 | + | 3.75264i | 0 | −2.51658 | 2.29673 | + | 1.31340i | 5.87162 | −0.500000 | − | 0.866025i | 0 | ||||||||||||||||||||||||||||||||
151.2 | −0.776205 | − | 1.34443i | 0.500000 | − | 0.866025i | −0.204988 | + | 0.355049i | 0 | −1.55241 | −2.60214 | + | 0.478401i | −2.46837 | −0.500000 | − | 0.866025i | 0 | |||||||||||||||||||||||||||||||||
151.3 | 0.143668 | + | 0.248840i | 0.500000 | − | 0.866025i | 0.958719 | − | 1.66055i | 0 | 0.287336 | 2.39939 | − | 1.11487i | 1.12562 | −0.500000 | − | 0.866025i | 0 | |||||||||||||||||||||||||||||||||
151.4 | 0.890827 | + | 1.54296i | 0.500000 | − | 0.866025i | −0.587145 | + | 1.01696i | 0 | 1.78165 | −1.09398 | − | 2.40898i | 1.47113 | −0.500000 | − | 0.866025i | 0 | |||||||||||||||||||||||||||||||||
226.1 | −1.25829 | + | 2.17942i | 0.500000 | + | 0.866025i | −2.16659 | − | 3.75264i | 0 | −2.51658 | 2.29673 | − | 1.31340i | 5.87162 | −0.500000 | + | 0.866025i | 0 | |||||||||||||||||||||||||||||||||
226.2 | −0.776205 | + | 1.34443i | 0.500000 | + | 0.866025i | −0.204988 | − | 0.355049i | 0 | −1.55241 | −2.60214 | − | 0.478401i | −2.46837 | −0.500000 | + | 0.866025i | 0 | |||||||||||||||||||||||||||||||||
226.3 | 0.143668 | − | 0.248840i | 0.500000 | + | 0.866025i | 0.958719 | + | 1.66055i | 0 | 0.287336 | 2.39939 | + | 1.11487i | 1.12562 | −0.500000 | + | 0.866025i | 0 | |||||||||||||||||||||||||||||||||
226.4 | 0.890827 | − | 1.54296i | 0.500000 | + | 0.866025i | −0.587145 | − | 1.01696i | 0 | 1.78165 | −1.09398 | + | 2.40898i | 1.47113 | −0.500000 | + | 0.866025i | 0 | |||||||||||||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
7.c | even | 3 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 525.2.i.h | 8 | |
5.b | even | 2 | 1 | 525.2.i.k | 8 | ||
5.c | odd | 4 | 2 | 105.2.q.a | ✓ | 16 | |
7.c | even | 3 | 1 | inner | 525.2.i.h | 8 | |
7.c | even | 3 | 1 | 3675.2.a.bz | 4 | ||
7.d | odd | 6 | 1 | 3675.2.a.cb | 4 | ||
15.e | even | 4 | 2 | 315.2.bf.b | 16 | ||
20.e | even | 4 | 2 | 1680.2.di.d | 16 | ||
35.f | even | 4 | 2 | 735.2.q.g | 16 | ||
35.i | odd | 6 | 1 | 3675.2.a.bn | 4 | ||
35.j | even | 6 | 1 | 525.2.i.k | 8 | ||
35.j | even | 6 | 1 | 3675.2.a.bp | 4 | ||
35.k | even | 12 | 2 | 735.2.d.e | 8 | ||
35.k | even | 12 | 2 | 735.2.q.g | 16 | ||
35.l | odd | 12 | 2 | 105.2.q.a | ✓ | 16 | |
35.l | odd | 12 | 2 | 735.2.d.d | 8 | ||
105.w | odd | 12 | 2 | 2205.2.d.o | 8 | ||
105.x | even | 12 | 2 | 315.2.bf.b | 16 | ||
105.x | even | 12 | 2 | 2205.2.d.s | 8 | ||
140.w | even | 12 | 2 | 1680.2.di.d | 16 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
105.2.q.a | ✓ | 16 | 5.c | odd | 4 | 2 | |
105.2.q.a | ✓ | 16 | 35.l | odd | 12 | 2 | |
315.2.bf.b | 16 | 15.e | even | 4 | 2 | ||
315.2.bf.b | 16 | 105.x | even | 12 | 2 | ||
525.2.i.h | 8 | 1.a | even | 1 | 1 | trivial | |
525.2.i.h | 8 | 7.c | even | 3 | 1 | inner | |
525.2.i.k | 8 | 5.b | even | 2 | 1 | ||
525.2.i.k | 8 | 35.j | even | 6 | 1 | ||
735.2.d.d | 8 | 35.l | odd | 12 | 2 | ||
735.2.d.e | 8 | 35.k | even | 12 | 2 | ||
735.2.q.g | 16 | 35.f | even | 4 | 2 | ||
735.2.q.g | 16 | 35.k | even | 12 | 2 | ||
1680.2.di.d | 16 | 20.e | even | 4 | 2 | ||
1680.2.di.d | 16 | 140.w | even | 12 | 2 | ||
2205.2.d.o | 8 | 105.w | odd | 12 | 2 | ||
2205.2.d.s | 8 | 105.x | even | 12 | 2 | ||
3675.2.a.bn | 4 | 35.i | odd | 6 | 1 | ||
3675.2.a.bp | 4 | 35.j | even | 6 | 1 | ||
3675.2.a.bz | 4 | 7.c | even | 3 | 1 | ||
3675.2.a.cb | 4 | 7.d | odd | 6 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{8} + 2T_{2}^{7} + 8T_{2}^{6} + 4T_{2}^{5} + 26T_{2}^{4} + 16T_{2}^{3} + 44T_{2}^{2} - 12T_{2} + 4 \)
acting on \(S_{2}^{\mathrm{new}}(525, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{8} + 2 T^{7} + 8 T^{6} + 4 T^{5} + \cdots + 4 \)
$3$
\( (T^{2} - T + 1)^{4} \)
$5$
\( T^{8} \)
$7$
\( T^{8} - 2 T^{7} - 8 T^{6} + 14 T^{5} + \cdots + 2401 \)
$11$
\( T^{8} + 18 T^{6} - 28 T^{5} + \cdots + 900 \)
$13$
\( (T^{4} - 2 T^{3} - 28 T^{2} + 36 T + 127)^{2} \)
$17$
\( T^{8} + 2 T^{7} + 32 T^{6} + \cdots + 17956 \)
$19$
\( T^{8} - 12 T^{7} + 106 T^{6} + \cdots + 81 \)
$23$
\( T^{8} + 10 T^{7} + 132 T^{6} + \cdots + 2268036 \)
$29$
\( (T^{4} + 6 T^{3} - 38 T^{2} - 190 T - 22)^{2} \)
$31$
\( T^{8} - 8 T^{7} + 70 T^{6} + \cdots + 3721 \)
$37$
\( T^{8} + 24 T^{7} + 372 T^{6} + \cdots + 822649 \)
$41$
\( (T^{4} - 4 T^{3} - 50 T^{2} + 146 T - 10)^{2} \)
$43$
\( (T^{4} - 8 T^{3} - 32 T^{2} + 154 T - 49)^{2} \)
$47$
\( T^{8} - 10 T^{7} + 84 T^{6} + \cdots + 14400 \)
$53$
\( T^{8} + 20 T^{7} + 280 T^{6} + \cdots + 9216 \)
$59$
\( T^{8} + 2 T^{7} + 10 T^{6} + 38 T^{4} + \cdots + 100 \)
$61$
\( T^{8} - 8 T^{7} + 164 T^{6} + \cdots + 250000 \)
$67$
\( T^{8} + 6 T^{7} + 108 T^{6} + \cdots + 819025 \)
$71$
\( (T^{4} + 14 T^{3} - 90 T^{2} - 1334 T - 3202)^{2} \)
$73$
\( T^{8} + 12 T^{7} + 208 T^{6} + \cdots + 1929321 \)
$79$
\( T^{8} - 8 T^{7} + 218 T^{6} + \cdots + 50140561 \)
$83$
\( (T^{4} + 6 T^{3} - 56 T^{2} - 238 T + 362)^{2} \)
$89$
\( T^{8} + 8 T^{7} + 258 T^{6} + \cdots + 285156 \)
$97$
\( (T^{4} + 2 T^{3} - 8 T^{2} - 16 T - 4)^{2} \)
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