Properties

Label 688.6.a.e
Level $688$
Weight $6$
Character orbit 688.a
Self dual yes
Analytic conductor $110.344$
Analytic rank $1$
Dimension $8$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 688 = 2^{4} \cdot 43 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 688.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(110.344068031\)
Analytic rank: \(1\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
Defining polynomial: \( x^{8} - 4x^{7} - 173x^{6} + 462x^{5} + 9118x^{4} - 14192x^{3} - 167688x^{2} + 106368x + 681984 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{6} \)
Twist minimal: no (minimal twist has level 43)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-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.

\(f(q)\) \(=\) \( q + (\beta_{4} + 3) q^{3} + (\beta_{7} - \beta_{6} + \beta_{4} - 27) q^{5} + (\beta_{7} + 2 \beta_{6} + \beta_{5} - \beta_{4} + 4 \beta_{3} + 6 \beta_{2} + 4 \beta_1 + 17) q^{7} + ( - 4 \beta_{7} + \beta_{6} - 5 \beta_{5} + 9 \beta_{4} + 2 \beta_{3} + 3 \beta_{2} + \cdots + 68) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{4} + 3) q^{3} + (\beta_{7} - \beta_{6} + \beta_{4} - 27) q^{5} + (\beta_{7} + 2 \beta_{6} + \beta_{5} - \beta_{4} + 4 \beta_{3} + 6 \beta_{2} + 4 \beta_1 + 17) q^{7} + ( - 4 \beta_{7} + \beta_{6} - 5 \beta_{5} + 9 \beta_{4} + 2 \beta_{3} + 3 \beta_{2} + \cdots + 68) q^{9}+ \cdots + (563 \beta_{7} + 304 \beta_{6} + 1398 \beta_{5} - 3541 \beta_{4} + \cdots - 30494) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 26 q^{3} - 212 q^{5} + 136 q^{7} + 546 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 26 q^{3} - 212 q^{5} + 136 q^{7} + 546 q^{9} + 532 q^{11} - 2492 q^{13} + 1780 q^{15} - 2534 q^{17} + 1678 q^{19} - 2256 q^{21} + 2488 q^{23} + 4378 q^{25} + 8960 q^{27} - 4360 q^{29} - 5704 q^{31} - 12852 q^{33} - 5640 q^{35} - 3772 q^{37} - 11120 q^{39} - 10698 q^{41} + 14792 q^{43} - 44912 q^{45} + 77864 q^{47} + 7188 q^{49} + 80246 q^{51} - 62352 q^{53} + 49552 q^{55} - 808 q^{57} + 26224 q^{59} - 82540 q^{61} + 61768 q^{63} - 5000 q^{65} - 27784 q^{67} - 93776 q^{69} + 9504 q^{71} + 14260 q^{73} - 167420 q^{75} - 218140 q^{77} - 160248 q^{79} + 161076 q^{81} + 77176 q^{83} + 141096 q^{85} - 268136 q^{87} - 265692 q^{89} - 401148 q^{91} - 123860 q^{93} - 135884 q^{95} + 144742 q^{97} - 239516 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 4x^{7} - 173x^{6} + 462x^{5} + 9118x^{4} - 14192x^{3} - 167688x^{2} + 106368x + 681984 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 12949 \nu^{7} + 681148 \nu^{6} - 2601727 \nu^{5} - 80176782 \nu^{4} + 378607946 \nu^{3} + 1837549088 \nu^{2} - 8768955576 \nu - 1162574592 ) / 547265856 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 36205 \nu^{7} - 1830508 \nu^{6} - 539057 \nu^{5} + 265985910 \nu^{4} - 188725898 \nu^{3} - 9341109104 \nu^{2} + 5760293592 \nu + 58885512576 ) / 1094531712 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 67489 \nu^{7} + 623356 \nu^{6} + 8964549 \nu^{5} - 78907518 \nu^{4} - 250466958 \nu^{3} + 2271218384 \nu^{2} - 114021816 \nu - 8596825856 ) / 364843904 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 204587 \nu^{7} + 1315940 \nu^{6} + 28380103 \nu^{5} - 147318570 \nu^{4} - 993160922 \nu^{3} + 4310420080 \nu^{2} + 8605716312 \nu - 37440050688 ) / 1094531712 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 309497 \nu^{7} + 2935676 \nu^{6} + 38047933 \nu^{5} - 372264846 \nu^{4} - 955734974 \nu^{3} + 11960213584 \nu^{2} + 2287797192 \nu - 76606929024 ) / 1094531712 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 73259 \nu^{7} - 190788 \nu^{6} - 10643911 \nu^{5} + 12835930 \nu^{4} + 347431274 \nu^{3} - 15275088 \nu^{2} - 785010872 \nu - 3291957568 ) / 182421952 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 34066 \nu^{7} + 259912 \nu^{6} + 4750721 \nu^{5} - 30187209 \nu^{4} - 176761426 \nu^{3} + 777094526 \nu^{2} + 1922056332 \nu - 2232499728 ) / 68408232 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{6} + \beta_{5} + \beta_{3} + 2\beta_{2} + 4 ) / 8 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -2\beta_{7} + \beta_{6} - \beta_{5} + 8\beta_{4} + \beta_{3} + 180 ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -14\beta_{7} + 33\beta_{6} + 15\beta_{5} + 32\beta_{4} + 65\beta_{3} + 64\beta_{2} + 4\beta _1 + 380 ) / 4 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -99\beta_{7} + 84\beta_{6} - 117\beta_{5} + 488\beta_{4} + 140\beta_{3} + 63\beta_{2} + 46\beta _1 + 6796 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 869 \beta_{7} + 1527 \beta_{6} - 276 \beta_{5} + 2768 \beta_{4} + 3711 \beta_{3} + 2515 \beta_{2} + 142 \beta _1 + 27000 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 10339 \beta_{7} + 11590 \beta_{6} - 16231 \beta_{5} + 55680 \beta_{4} + 22478 \beta_{3} + 10127 \beta_{2} + 6850 \beta _1 + 639708 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 102849 \beta_{7} + 166838 \beta_{6} - 94865 \beta_{5} + 386624 \beta_{4} + 421838 \beta_{3} + 234341 \beta_{2} + 20926 \beta _1 + 3616388 ) / 2 \) Copy content Toggle raw display

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−2.08717
−6.09504
5.65705
2.58275
−5.06235
7.21373
10.9591
−9.16809
0 −25.6605 0 −61.4284 0 184.774 0 415.460 0
1.2 0 −11.1803 0 −63.3756 0 −223.489 0 −118.000 0
1.3 0 −7.84314 0 −107.102 0 25.5214 0 −181.485 0
1.4 0 −3.05838 0 27.7074 0 103.690 0 −233.646 0
1.5 0 9.19190 0 73.4416 0 4.24720 0 −158.509 0
1.6 0 11.2683 0 −9.80186 0 11.1041 0 −116.025 0
1.7 0 25.1057 0 −61.2927 0 166.001 0 387.294 0
1.8 0 28.1764 0 −10.1483 0 −135.849 0 550.912 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.8
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(43\) \(-1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 688.6.a.e 8
4.b odd 2 1 43.6.a.a 8
12.b even 2 1 387.6.a.c 8
20.d odd 2 1 1075.6.a.a 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
43.6.a.a 8 4.b odd 2 1
387.6.a.c 8 12.b even 2 1
688.6.a.e 8 1.a even 1 1 trivial
1075.6.a.a 8 20.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{8} - 26 T_{3}^{7} - 907 T_{3}^{6} + 22242 T_{3}^{5} + 184435 T_{3}^{4} - 3627880 T_{3}^{3} - 14906374 T_{3}^{2} + 156321960 T_{3} + 504223128 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(688))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( T^{8} - 26 T^{7} + \cdots + 504223128 \) Copy content Toggle raw display
$5$ \( T^{8} + 212 T^{7} + \cdots + 5172924974752 \) Copy content Toggle raw display
$7$ \( T^{8} + \cdots + 116222354316288 \) Copy content Toggle raw display
$11$ \( T^{8} - 532 T^{7} + \cdots + 16\!\cdots\!76 \) Copy content Toggle raw display
$13$ \( T^{8} + 2492 T^{7} + \cdots - 20\!\cdots\!44 \) Copy content Toggle raw display
$17$ \( T^{8} + 2534 T^{7} + \cdots + 37\!\cdots\!33 \) Copy content Toggle raw display
$19$ \( T^{8} - 1678 T^{7} + \cdots + 20\!\cdots\!68 \) Copy content Toggle raw display
$23$ \( T^{8} - 2488 T^{7} + \cdots - 18\!\cdots\!83 \) Copy content Toggle raw display
$29$ \( T^{8} + 4360 T^{7} + \cdots - 39\!\cdots\!36 \) Copy content Toggle raw display
$31$ \( T^{8} + 5704 T^{7} + \cdots + 10\!\cdots\!13 \) Copy content Toggle raw display
$37$ \( T^{8} + 3772 T^{7} + \cdots - 10\!\cdots\!04 \) Copy content Toggle raw display
$41$ \( T^{8} + 10698 T^{7} + \cdots + 17\!\cdots\!57 \) Copy content Toggle raw display
$43$ \( (T - 1849)^{8} \) Copy content Toggle raw display
$47$ \( T^{8} - 77864 T^{7} + \cdots - 19\!\cdots\!04 \) Copy content Toggle raw display
$53$ \( T^{8} + 62352 T^{7} + \cdots + 55\!\cdots\!84 \) Copy content Toggle raw display
$59$ \( T^{8} - 26224 T^{7} + \cdots - 68\!\cdots\!68 \) Copy content Toggle raw display
$61$ \( T^{8} + 82540 T^{7} + \cdots + 23\!\cdots\!84 \) Copy content Toggle raw display
$67$ \( T^{8} + 27784 T^{7} + \cdots + 69\!\cdots\!48 \) Copy content Toggle raw display
$71$ \( T^{8} - 9504 T^{7} + \cdots - 15\!\cdots\!64 \) Copy content Toggle raw display
$73$ \( T^{8} - 14260 T^{7} + \cdots + 80\!\cdots\!92 \) Copy content Toggle raw display
$79$ \( T^{8} + 160248 T^{7} + \cdots + 19\!\cdots\!12 \) Copy content Toggle raw display
$83$ \( T^{8} - 77176 T^{7} + \cdots - 19\!\cdots\!08 \) Copy content Toggle raw display
$89$ \( T^{8} + 265692 T^{7} + \cdots - 13\!\cdots\!64 \) Copy content Toggle raw display
$97$ \( T^{8} - 144742 T^{7} + \cdots + 38\!\cdots\!17 \) Copy content Toggle raw display
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