Newspace parameters
Level: | \( N \) | \(=\) | \( 128 = 2^{7} \) |
Weight: | \( k \) | \(=\) | \( 8 \) |
Character orbit: | \([\chi]\) | \(=\) | 128.b (of order \(2\), degree \(1\), minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(39.9852832620\) |
Analytic rank: | \(0\) |
Dimension: | \(8\) |
Coefficient field: | 8.0.194307987513600.13 |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
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Defining polynomial: | \( x^{8} - 4x^{7} + 22x^{6} - 52x^{5} + 659x^{4} - 1236x^{3} + 4978x^{2} - 4368x + 59105 \) |
Coefficient ring: | \(\Z[a_1, \ldots, a_{13}]\) |
Coefficient ring index: | \( 2^{52} \) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-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 \( x^{8} - 4x^{7} + 22x^{6} - 52x^{5} + 659x^{4} - 1236x^{3} + 4978x^{2} - 4368x + 59105 \) :
\(\beta_{1}\) | \(=\) | \( ( -8\nu^{6} + 24\nu^{5} - 120\nu^{4} + 200\nu^{3} - 2792\nu^{2} + 2696\nu - 9760 ) / 225 \) |
\(\beta_{2}\) | \(=\) | \( ( 19\nu^{6} - 57\nu^{5} + 285\nu^{4} - 475\nu^{3} + 13831\nu^{2} - 13603\nu + 51980 ) / 450 \) |
\(\beta_{3}\) | \(=\) | \( ( 166 \nu^{7} - 581 \nu^{6} + 5289 \nu^{5} - 11770 \nu^{4} + 103509 \nu^{3} - 143784 \nu^{2} + 1398241 \nu - 675535 ) / 38550 \) |
\(\beta_{4}\) | \(=\) | \( 8\nu^{4} - 16\nu^{3} + 72\nu^{2} - 64\nu + 2184 \) |
\(\beta_{5}\) | \(=\) | \( ( 12\nu^{7} - 42\nu^{6} - 14\nu^{5} + 140\nu^{4} - 246\nu^{3} + 208\nu^{2} - 21886\nu + 10914 ) / 257 \) |
\(\beta_{6}\) | \(=\) | \( ( - 4412 \nu^{7} + 15442 \nu^{6} - 170298 \nu^{5} + 387140 \nu^{4} - 2097138 \nu^{3} + 2766288 \nu^{2} - 22329962 \nu + 10716470 ) / 19275 \) |
\(\beta_{7}\) | \(=\) | \( ( - 4928 \nu^{7} + 17248 \nu^{6} - 38112 \nu^{5} + 52160 \nu^{4} - 1741152 \nu^{3} + 2568192 \nu^{2} + 7715872 \nu - 4284640 ) / 11565 \) |
\(\nu\) | \(=\) | \( ( 3\beta_{7} + 4\beta_{6} + 16\beta_{5} + 336\beta_{3} + 4096 ) / 8192 \) |
\(\nu^{2}\) | \(=\) | \( ( 3\beta_{7} + 4\beta_{6} + 16\beta_{5} + 336\beta_{3} + 512\beta_{2} + 608\beta _1 - 28672 ) / 8192 \) |
\(\nu^{3}\) | \(=\) | \( ( -27\beta_{7} + 28\beta_{6} - 208\beta_{5} + 1072\beta_{3} + 384\beta_{2} + 456\beta _1 - 22528 ) / 4096 \) |
\(\nu^{4}\) | \(=\) | \( ( - 111 \beta_{7} + 108 \beta_{6} - 848 \beta_{5} + 1024 \beta_{4} + 3952 \beta_{3} - 3072 \beta_{2} - 3648 \beta _1 - 2035712 ) / 8192 \) |
\(\nu^{5}\) | \(=\) | \( ( - 75 \beta_{7} - 1956 \beta_{6} - 3856 \beta_{5} + 2560 \beta_{4} - 69584 \beta_{3} - 8960 \beta_{2} - 10640 \beta _1 - 5013504 ) / 8192 \) |
\(\nu^{6}\) | \(=\) | \( ( 27 \beta_{7} - 3068 \beta_{6} - 4720 \beta_{5} - 3840 \beta_{4} - 109232 \beta_{3} - 70144 \beta_{2} - 198496 \beta _1 + 7880704 ) / 4096 \) |
\(\nu^{7}\) | \(=\) | \( ( 5709 \beta_{7} - 16644 \beta_{6} + 168176 \beta_{5} - 35840 \beta_{4} - 240976 \beta_{3} - 458752 \beta_{2} - 1351168 \beta _1 + 72658944 ) / 8192 \) |
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/128\mathbb{Z}\right)^\times\).
\(n\) | \(5\) | \(127\) |
\(\chi(n)\) | \(-1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
65.1 |
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0 | − | 77.7987i | 0 | − | 324.387i | 0 | 113.762 | 0 | −3865.64 | 0 | ||||||||||||||||||||||||||||||||||||||||
65.2 | 0 | − | 77.7987i | 0 | 324.387i | 0 | −113.762 | 0 | −3865.64 | 0 | ||||||||||||||||||||||||||||||||||||||||||
65.3 | 0 | − | 9.55816i | 0 | − | 172.387i | 0 | 1568.77 | 0 | 2095.64 | 0 | |||||||||||||||||||||||||||||||||||||||||
65.4 | 0 | − | 9.55816i | 0 | 172.387i | 0 | −1568.77 | 0 | 2095.64 | 0 | ||||||||||||||||||||||||||||||||||||||||||
65.5 | 0 | 9.55816i | 0 | − | 172.387i | 0 | −1568.77 | 0 | 2095.64 | 0 | ||||||||||||||||||||||||||||||||||||||||||
65.6 | 0 | 9.55816i | 0 | 172.387i | 0 | 1568.77 | 0 | 2095.64 | 0 | |||||||||||||||||||||||||||||||||||||||||||
65.7 | 0 | 77.7987i | 0 | − | 324.387i | 0 | −113.762 | 0 | −3865.64 | 0 | ||||||||||||||||||||||||||||||||||||||||||
65.8 | 0 | 77.7987i | 0 | 324.387i | 0 | 113.762 | 0 | −3865.64 | 0 | |||||||||||||||||||||||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
4.b | odd | 2 | 1 | inner |
8.b | even | 2 | 1 | inner |
8.d | odd | 2 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 128.8.b.g | ✓ | 8 |
4.b | odd | 2 | 1 | inner | 128.8.b.g | ✓ | 8 |
8.b | even | 2 | 1 | inner | 128.8.b.g | ✓ | 8 |
8.d | odd | 2 | 1 | inner | 128.8.b.g | ✓ | 8 |
16.e | even | 4 | 1 | 256.8.a.k | 4 | ||
16.e | even | 4 | 1 | 256.8.a.o | 4 | ||
16.f | odd | 4 | 1 | 256.8.a.k | 4 | ||
16.f | odd | 4 | 1 | 256.8.a.o | 4 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
128.8.b.g | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
128.8.b.g | ✓ | 8 | 4.b | odd | 2 | 1 | inner |
128.8.b.g | ✓ | 8 | 8.b | even | 2 | 1 | inner |
128.8.b.g | ✓ | 8 | 8.d | odd | 2 | 1 | inner |
256.8.a.k | 4 | 16.e | even | 4 | 1 | ||
256.8.a.k | 4 | 16.f | odd | 4 | 1 | ||
256.8.a.o | 4 | 16.e | even | 4 | 1 | ||
256.8.a.o | 4 | 16.f | odd | 4 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{4} + 6144T_{3}^{2} + 552960 \)
acting on \(S_{8}^{\mathrm{new}}(128, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{8} \)
$3$
\( (T^{4} + 6144 T^{2} + 552960)^{2} \)
$5$
\( (T^{4} + 134944 T^{2} + \cdots + 3127046400)^{2} \)
$7$
\( (T^{4} - 2473984 T^{2} + \cdots + 31850496000)^{2} \)
$11$
\( (T^{4} + 60872704 T^{2} + \cdots + 923324185866240)^{2} \)
$13$
\( (T^{4} + 232826144 T^{2} + \cdots + 82\!\cdots\!00)^{2} \)
$17$
\( (T^{2} - 8508 T - 30273148)^{4} \)
$19$
\( (T^{4} + 4381087744 T^{2} + \cdots + 47\!\cdots\!40)^{2} \)
$23$
\( (T^{4} - 2633089024 T^{2} + \cdots + 97\!\cdots\!60)^{2} \)
$29$
\( (T^{4} + 28662505760 T^{2} + \cdots + 17\!\cdots\!64)^{2} \)
$31$
\( (T^{4} - 110416101376 T^{2} + \cdots + 15\!\cdots\!60)^{2} \)
$37$
\( (T^{4} + 149733736480 T^{2} + \cdots + 25\!\cdots\!44)^{2} \)
$41$
\( (T^{2} - 584308 T + 61943042340)^{4} \)
$43$
\( (T^{4} + 1096961284096 T^{2} + \cdots + 28\!\cdots\!40)^{2} \)
$47$
\( (T^{4} - 646120407040 T^{2} + \cdots + 90\!\cdots\!00)^{2} \)
$53$
\( (T^{4} + 1975212319520 T^{2} + \cdots + 26\!\cdots\!84)^{2} \)
$59$
\( (T^{4} + 4388182677504 T^{2} + \cdots + 20\!\cdots\!40)^{2} \)
$61$
\( (T^{4} + 13929746948640 T^{2} + \cdots + 48\!\cdots\!84)^{2} \)
$67$
\( (T^{4} + 4003839350784 T^{2} + \cdots + 24\!\cdots\!40)^{2} \)
$71$
\( (T^{4} - 25329147559936 T^{2} + \cdots + 10\!\cdots\!60)^{2} \)
$73$
\( (T^{2} - 5184300 T + 5509556798436)^{4} \)
$79$
\( (T^{4} - 70685681975296 T^{2} + \cdots + 37\!\cdots\!00)^{2} \)
$83$
\( (T^{4} + 71887724222464 T^{2} + \cdots + 47\!\cdots\!40)^{2} \)
$89$
\( (T^{2} - 3521612 T - 10640851706460)^{4} \)
$97$
\( (T^{2} - 1967452 T - 708251541948)^{4} \)
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