Newspace parameters
Level: | \( N \) | \(=\) | \( 350 = 2 \cdot 5^{2} \cdot 7 \) |
Weight: | \( k \) | \(=\) | \( 7 \) |
Character orbit: | \([\chi]\) | \(=\) | 350.d (of order \(2\), degree \(1\), not minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(80.5189292669\) |
Analytic rank: | \(0\) |
Dimension: | \(8\) |
Coefficient field: | 8.0.179721732096.20 |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
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Defining polynomial: | \( x^{8} - 4x^{7} + 28x^{6} - 40x^{5} + 258x^{4} - 32x^{3} - 620x^{2} + 7480x + 11698 \) |
Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
Coefficient ring index: | \( 2^{17} \) |
Twist minimal: | no (minimal twist has level 14) |
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 \( x^{8} - 4x^{7} + 28x^{6} - 40x^{5} + 258x^{4} - 32x^{3} - 620x^{2} + 7480x + 11698 \) :
\(\beta_{1}\) | \(=\) | \( ( 68088 \nu^{7} - 1192852 \nu^{6} + 3754608 \nu^{5} - 14542140 \nu^{4} + 2822728 \nu^{3} + 65626512 \nu^{2} - 319126368 \nu + 2280456368 ) / 502161073 \) |
\(\beta_{2}\) | \(=\) | \( ( 88482 \nu^{7} - 496428 \nu^{6} + 3438200 \nu^{5} - 8169695 \nu^{4} + 40056132 \nu^{3} - 61256098 \nu^{2} + 110856524 \nu + 629909935 ) / 297167111 \) |
\(\beta_{3}\) | \(=\) | \( ( 330528 \nu^{7} - 1756044 \nu^{6} + 12796256 \nu^{5} - 35079836 \nu^{4} + 160737304 \nu^{3} - 212573944 \nu^{2} + 330272096 \nu + 2243651672 ) / 197941847 \) |
\(\beta_{4}\) | \(=\) | \( ( 168752282 \nu^{7} - 537771504 \nu^{6} + 5539642408 \nu^{5} - 5130738670 \nu^{4} + 19457898344 \nu^{3} + 81862631972 \nu^{2} + \cdots + 864763629972 ) / 68796067001 \) |
\(\beta_{5}\) | \(=\) | \( ( 231831374 \nu^{7} - 656249376 \nu^{6} + 5414235240 \nu^{5} - 2422416074 \nu^{4} + 40448137544 \nu^{3} + 57693609052 \nu^{2} + \cdots + 1337919721068 ) / 68796067001 \) |
\(\beta_{6}\) | \(=\) | \( ( - 173141866 \nu^{7} + 1283513888 \nu^{6} - 8336224680 \nu^{5} + 28368672042 \nu^{4} - 101446760380 \nu^{3} + \cdots - 1378152307096 ) / 39670724767 \) |
\(\beta_{7}\) | \(=\) | \( ( 369202382 \nu^{7} - 2207604480 \nu^{6} + 13578418280 \nu^{5} - 40765776686 \nu^{4} + 148677452196 \nu^{3} - 325911446316 \nu^{2} + \cdots + 2171463328472 ) / 39670724767 \) |
\(\nu\) | \(=\) | \( ( \beta_{7} + 3\beta_{6} - 3\beta_{5} + \beta_{4} + 4\beta_{3} + 16\beta_{2} + 16 ) / 32 \) |
\(\nu^{2}\) | \(=\) | \( ( -\beta_{7} + \beta_{6} + 13\beta_{3} - 32\beta_{2} + 11\beta _1 - 40 ) / 8 \) |
\(\nu^{3}\) | \(=\) | \( ( -15\beta_{7} - 9\beta_{6} + 21\beta_{5} - 27\beta_{4} + 44\beta_{3} + 64\beta_{2} + 24\beta _1 - 304 ) / 16 \) |
\(\nu^{4}\) | \(=\) | \( ( -23\beta_{7} - 21\beta_{6} + 31\beta_{5} - 21\beta_{4} - 218\beta_{3} + 1464\beta_{2} - 18\beta _1 - 360 ) / 8 \) |
\(\nu^{5}\) | \(=\) | \( ( -160\beta_{7} - 284\beta_{6} - 187\beta_{5} + 469\beta_{4} - 1262\beta_{3} + 6528\beta_{2} - 790\beta _1 + 688 ) / 16 \) |
\(\nu^{6}\) | \(=\) | \( ( 60\beta_{7} - 30\beta_{6} - 216\beta_{5} + 438\beta_{4} + 953\beta_{3} - 7868\beta_{2} - 2098\beta _1 + 9772 ) / 4 \) |
\(\nu^{7}\) | \(=\) | \( ( 2024 \beta_{7} + 2516 \beta_{6} + 129 \beta_{5} - 169 \beta_{4} + 6436 \beta_{3} - 47364 \beta_{2} - 3836 \beta _1 + 21100 ) / 4 \) |
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/350\mathbb{Z}\right)^\times\).
\(n\) | \(101\) | \(127\) |
\(\chi(n)\) | \(-1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
349.1 |
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− | 5.65685i | −29.9539 | −32.0000 | 0 | 169.445i | −281.627 | − | 195.794i | 181.019i | 168.235 | 0 | |||||||||||||||||||||||||||||||||||||||
349.2 | − | 5.65685i | −3.84257 | −32.0000 | 0 | 21.7369i | −340.444 | − | 41.7939i | 181.019i | −714.235 | 0 | ||||||||||||||||||||||||||||||||||||||||
349.3 | − | 5.65685i | 3.84257 | −32.0000 | 0 | − | 21.7369i | 340.444 | − | 41.7939i | 181.019i | −714.235 | 0 | |||||||||||||||||||||||||||||||||||||||
349.4 | − | 5.65685i | 29.9539 | −32.0000 | 0 | − | 169.445i | 281.627 | − | 195.794i | 181.019i | 168.235 | 0 | |||||||||||||||||||||||||||||||||||||||
349.5 | 5.65685i | −29.9539 | −32.0000 | 0 | − | 169.445i | −281.627 | + | 195.794i | − | 181.019i | 168.235 | 0 | |||||||||||||||||||||||||||||||||||||||
349.6 | 5.65685i | −3.84257 | −32.0000 | 0 | − | 21.7369i | −340.444 | + | 41.7939i | − | 181.019i | −714.235 | 0 | |||||||||||||||||||||||||||||||||||||||
349.7 | 5.65685i | 3.84257 | −32.0000 | 0 | 21.7369i | 340.444 | + | 41.7939i | − | 181.019i | −714.235 | 0 | ||||||||||||||||||||||||||||||||||||||||
349.8 | 5.65685i | 29.9539 | −32.0000 | 0 | 169.445i | 281.627 | + | 195.794i | − | 181.019i | 168.235 | 0 | ||||||||||||||||||||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
5.b | even | 2 | 1 | inner |
7.b | odd | 2 | 1 | inner |
35.c | odd | 2 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 350.7.d.a | 8 | |
5.b | even | 2 | 1 | inner | 350.7.d.a | 8 | |
5.c | odd | 4 | 1 | 14.7.b.a | ✓ | 4 | |
5.c | odd | 4 | 1 | 350.7.b.a | 4 | ||
7.b | odd | 2 | 1 | inner | 350.7.d.a | 8 | |
15.e | even | 4 | 1 | 126.7.c.a | 4 | ||
20.e | even | 4 | 1 | 112.7.c.c | 4 | ||
35.c | odd | 2 | 1 | inner | 350.7.d.a | 8 | |
35.f | even | 4 | 1 | 14.7.b.a | ✓ | 4 | |
35.f | even | 4 | 1 | 350.7.b.a | 4 | ||
35.k | even | 12 | 2 | 98.7.d.b | 8 | ||
35.l | odd | 12 | 2 | 98.7.d.b | 8 | ||
40.i | odd | 4 | 1 | 448.7.c.h | 4 | ||
40.k | even | 4 | 1 | 448.7.c.e | 4 | ||
105.k | odd | 4 | 1 | 126.7.c.a | 4 | ||
140.j | odd | 4 | 1 | 112.7.c.c | 4 | ||
280.s | even | 4 | 1 | 448.7.c.h | 4 | ||
280.y | odd | 4 | 1 | 448.7.c.e | 4 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
14.7.b.a | ✓ | 4 | 5.c | odd | 4 | 1 | |
14.7.b.a | ✓ | 4 | 35.f | even | 4 | 1 | |
98.7.d.b | 8 | 35.k | even | 12 | 2 | ||
98.7.d.b | 8 | 35.l | odd | 12 | 2 | ||
112.7.c.c | 4 | 20.e | even | 4 | 1 | ||
112.7.c.c | 4 | 140.j | odd | 4 | 1 | ||
126.7.c.a | 4 | 15.e | even | 4 | 1 | ||
126.7.c.a | 4 | 105.k | odd | 4 | 1 | ||
350.7.b.a | 4 | 5.c | odd | 4 | 1 | ||
350.7.b.a | 4 | 35.f | even | 4 | 1 | ||
350.7.d.a | 8 | 1.a | even | 1 | 1 | trivial | |
350.7.d.a | 8 | 5.b | even | 2 | 1 | inner | |
350.7.d.a | 8 | 7.b | odd | 2 | 1 | inner | |
350.7.d.a | 8 | 35.c | odd | 2 | 1 | inner | |
448.7.c.e | 4 | 40.k | even | 4 | 1 | ||
448.7.c.e | 4 | 280.y | odd | 4 | 1 | ||
448.7.c.h | 4 | 40.i | odd | 4 | 1 | ||
448.7.c.h | 4 | 280.s | even | 4 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{4} - 912T_{3}^{2} + 13248 \)
acting on \(S_{7}^{\mathrm{new}}(350, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( (T^{2} + 32)^{4} \)
$3$
\( (T^{4} - 912 T^{2} + 13248)^{2} \)
$5$
\( T^{8} \)
$7$
\( T^{8} - 310268 T^{6} + \cdots + 19\!\cdots\!01 \)
$11$
\( (T^{2} + 2220 T + 1227492)^{4} \)
$13$
\( (T^{4} - 15367056 T^{2} + \cdots + 26266196601792)^{2} \)
$17$
\( (T^{4} - 40214784 T^{2} + \cdots + 126735731638272)^{2} \)
$19$
\( (T^{4} + 65975568 T^{2} + \cdots + 551809934313408)^{2} \)
$23$
\( (T^{4} + 210343176 T^{2} + \cdots + 10\!\cdots\!36)^{2} \)
$29$
\( (T^{2} - 9132 T - 1357906716)^{4} \)
$31$
\( (T^{4} + 2274932736 T^{2} + \cdots + 86\!\cdots\!12)^{2} \)
$37$
\( (T^{4} + 3830319944 T^{2} + \cdots + 34\!\cdots\!96)^{2} \)
$41$
\( (T^{4} + 9071224896 T^{2} + \cdots + 28\!\cdots\!32)^{2} \)
$43$
\( (T^{4} + 21542362952 T^{2} + \cdots + 11\!\cdots\!76)^{2} \)
$47$
\( (T^{4} - 20120477952 T^{2} + \cdots + 12\!\cdots\!88)^{2} \)
$53$
\( (T^{4} + 15425248968 T^{2} + \cdots + 185366809042704)^{2} \)
$59$
\( (T^{4} + 180594109200 T^{2} + \cdots + 78\!\cdots\!00)^{2} \)
$61$
\( (T^{4} + 121774406928 T^{2} + \cdots + 82\!\cdots\!48)^{2} \)
$67$
\( (T^{4} + 23639424968 T^{2} + \cdots + 13\!\cdots\!44)^{2} \)
$71$
\( (T^{2} + 225804 T - 95443772508)^{4} \)
$73$
\( (T^{4} - 228009998400 T^{2} + \cdots + 26\!\cdots\!72)^{2} \)
$79$
\( (T^{2} + 1046452 T + 268612291876)^{4} \)
$83$
\( (T^{4} - 478841902608 T^{2} + \cdots + 21\!\cdots\!08)^{2} \)
$89$
\( (T^{4} + 258250641984 T^{2} + \cdots + 15\!\cdots\!52)^{2} \)
$97$
\( (T^{4} - 42611214336 T^{2} + \cdots + 39\!\cdots\!12)^{2} \)
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