Properties

Label 7.13.b.b
Level $7$
Weight $13$
Character orbit 7.b
Analytic conductor $6.398$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 7 \)
Weight: \( k \) \(=\) \( 13 \)
Character orbit: \([\chi]\) \(=\) 7.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(6.39795672093\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: \(\mathbb{Q}[x]/(x^{6} - \cdots)\)
Defining polynomial: \( x^{6} - x^{5} + 238188x^{4} - 14589496x^{3} + 11212054600x^{2} - 101757597480x + 81251686776288 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{11}\cdot 3\cdot 7^{3} \)
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 \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{2} q^{2} + \beta_{3} q^{3} + ( - 26 \beta_{2} + 2 \beta_1 + 2714) q^{4} + (\beta_{4} + 8 \beta_{3}) q^{5} + (\beta_{5} - \beta_{4} - 26 \beta_{3}) q^{6} + ( - \beta_{5} - \beta_{4} + 7 \beta_{3} + 674 \beta_{2} + 25 \beta_1 - 52450) q^{7} + (2024 \beta_{2} - 172800) q^{8} + (3126 \beta_{2} + 48 \beta_1 - 280143) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{2} q^{2} + \beta_{3} q^{3} + ( - 26 \beta_{2} + 2 \beta_1 + 2714) q^{4} + (\beta_{4} + 8 \beta_{3}) q^{5} + (\beta_{5} - \beta_{4} - 26 \beta_{3}) q^{6} + ( - \beta_{5} - \beta_{4} + 7 \beta_{3} + 674 \beta_{2} + 25 \beta_1 - 52450) q^{7} + (2024 \beta_{2} - 172800) q^{8} + (3126 \beta_{2} + 48 \beta_1 - 280143) q^{9} + (25 \beta_{5} + \beta_{4} - 712 \beta_{3}) q^{10} + ( - 2222 \beta_{2} + 594 \beta_1 + 283932) q^{11} + ( - 26 \beta_{5} + 224 \beta_{4} + 2754 \beta_{3}) q^{12} + (2 \beta_{5} - 223 \beta_{4} + 2056 \beta_{3}) q^{13} + ( - 27 \beta_{5} - 223 \beta_{4} - 5348 \beta_{3} + \cdots + 4643190) q^{14}+ \cdots + (4615961790 \beta_{2} - 160053630 \beta_1 + 10982049804) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 16288 q^{4} - 314650 q^{7} - 1036800 q^{8} - 1680762 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q + 16288 q^{4} - 314650 q^{7} - 1036800 q^{8} - 1680762 q^{9} + 1704780 q^{11} + 27863136 q^{14} - 40526880 q^{15} + 15993088 q^{16} + 128356800 q^{18} - 32694816 q^{21} - 83177600 q^{22} - 266735700 q^{23} - 1115987130 q^{25} - 150488800 q^{28} + 2090185452 q^{29} + 3469961280 q^{30} - 666316800 q^{32} + 1813281120 q^{35} - 5110975584 q^{36} - 4824866900 q^{37} - 9660671328 q^{39} + 26387188800 q^{42} - 28724189300 q^{43} + 40254705216 q^{44} - 56843757056 q^{46} + 23216390022 q^{49} + 30589686720 q^{50} + 74552249088 q^{51} - 104521857300 q^{53} + 110766507264 q^{56} + 6945343200 q^{57} + 169979689600 q^{58} - 621243786240 q^{60} + 209517066150 q^{63} - 404530321408 q^{64} + 420221360160 q^{65} - 129746603700 q^{67} + 715652629440 q^{70} - 15017604660 q^{71} + 550229836800 q^{72} - 538689695616 q^{74} + 272839020300 q^{77} - 336774849600 q^{78} - 172527631668 q^{79} - 1648796648346 q^{81} - 343188486144 q^{84} + 62636728320 q^{85} + 1808205928704 q^{86} - 462937446400 q^{88} - 474685856736 q^{91} + 2013091790400 q^{92} - 2143380662400 q^{93} + 3657881967840 q^{95} - 2066720745600 q^{98} + 65572191564 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - x^{5} + 238188x^{4} - 14589496x^{3} + 11212054600x^{2} - 101757597480x + 81251686776288 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 316322973 \nu^{5} - 1880954567990 \nu^{4} - 229888629934646 \nu^{3} + \cdots + 67\!\cdots\!71 ) / 78\!\cdots\!45 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 829712739 \nu^{5} - 436403341435 \nu^{4} - 197574368898172 \nu^{3} + \cdots - 23\!\cdots\!88 ) / 15\!\cdots\!90 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 214467963392 \nu^{5} + 2208641196555 \nu^{4} + \cdots + 34\!\cdots\!96 ) / 10\!\cdots\!30 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 29099137708946 \nu^{5} + \cdots + 57\!\cdots\!28 ) / 98\!\cdots\!70 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 36542487068476 \nu^{5} + \cdots - 31\!\cdots\!08 ) / 98\!\cdots\!70 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{5} + 8\beta_{4} - 105\beta_{3} + 132\beta_{2} + 12\beta _1 + 108 ) / 672 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 43\beta_{5} - 160\beta_{4} + 5285\beta_{3} - 70276\beta_{2} + 7540\beta _1 - 17787180 ) / 224 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 4319\beta_{5} - 435680\beta_{4} + 6648369\beta_{3} - 1152820\beta_{2} - 1586684\beta _1 + 1607875428 ) / 224 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 6442253 \beta_{5} + 79955072 \beta_{4} - 1576024835 \beta_{3} + 8005050140 \beta_{2} - 1545103820 \beta _1 + 2564427752724 ) / 224 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 2655070297 \beta_{5} + 71240832448 \beta_{4} - 1193406700087 \beta_{3} - 1339548073876 \beta_{2} + 438107594404 \beta _1 - 621325934265084 ) / 224 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/7\mathbb{Z}\right)^\times\).

\(n\) \(3\)
\(\chi(n)\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
6.1
−1.79648 + 93.6923i
−1.79648 93.6923i
−50.4579 + 435.079i
−50.4579 435.079i
52.7544 213.185i
52.7544 + 213.185i
−108.657 1048.40i 7710.33 23251.4i 113916.i −98544.2 64267.7i −392722. −567695. 2.52642e6i
6.2 −108.657 1048.40i 7710.33 23251.4i 113916.i −98544.2 + 64267.7i −392722. −567695. 2.52642e6i
6.3 17.4333 949.924i −3792.08 18388.3i 16560.3i −116360. 17366.5i −137515. −370914. 320568.i
6.4 17.4333 949.924i −3792.08 18388.3i 16560.3i −116360. + 17366.5i −137515. −370914. 320568.i
6.5 91.2237 658.189i 4225.75 20289.4i 60042.4i 57579.4 + 102596.i 11836.7 98227.8 1.85087e6i
6.6 91.2237 658.189i 4225.75 20289.4i 60042.4i 57579.4 102596.i 11836.7 98227.8 1.85087e6i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 6.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 7.13.b.b 6
3.b odd 2 1 63.13.d.d 6
4.b odd 2 1 112.13.c.b 6
7.b odd 2 1 inner 7.13.b.b 6
7.c even 3 2 49.13.d.b 12
7.d odd 6 2 49.13.d.b 12
21.c even 2 1 63.13.d.d 6
28.d even 2 1 112.13.c.b 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
7.13.b.b 6 1.a even 1 1 trivial
7.13.b.b 6 7.b odd 2 1 inner
49.13.d.b 12 7.c even 3 2
49.13.d.b 12 7.d odd 6 2
63.13.d.d 6 3.b odd 2 1
63.13.d.d 6 21.c even 2 1
112.13.c.b 6 4.b odd 2 1
112.13.c.b 6 28.d even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{3} - 10216T_{2} + 172800 \) acting on \(S_{13}^{\mathrm{new}}(7, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{3} - 10216 T + 172800)^{2} \) Copy content Toggle raw display
$3$ \( T^{6} + 2434704 T^{4} + \cdots + 42\!\cdots\!00 \) Copy content Toggle raw display
$5$ \( T^{6} + 1290415440 T^{4} + \cdots + 75\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( T^{6} + 314650 T^{5} + \cdots + 26\!\cdots\!01 \) Copy content Toggle raw display
$11$ \( (T^{3} - 852390 T^{2} + \cdots + 22\!\cdots\!48)^{2} \) Copy content Toggle raw display
$13$ \( T^{6} + 65493616180944 T^{4} + \cdots + 33\!\cdots\!00 \) Copy content Toggle raw display
$17$ \( T^{6} + \cdots + 24\!\cdots\!00 \) Copy content Toggle raw display
$19$ \( T^{6} + \cdots + 62\!\cdots\!00 \) Copy content Toggle raw display
$23$ \( (T^{3} + 133367850 T^{2} + \cdots - 23\!\cdots\!00)^{2} \) Copy content Toggle raw display
$29$ \( (T^{3} - 1045092726 T^{2} + \cdots + 41\!\cdots\!92)^{2} \) Copy content Toggle raw display
$31$ \( T^{6} + \cdots + 37\!\cdots\!00 \) Copy content Toggle raw display
$37$ \( (T^{3} + 2412433450 T^{2} + \cdots - 10\!\cdots\!00)^{2} \) Copy content Toggle raw display
$41$ \( T^{6} + \cdots + 49\!\cdots\!00 \) Copy content Toggle raw display
$43$ \( (T^{3} + 14362094650 T^{2} + \cdots + 36\!\cdots\!00)^{2} \) Copy content Toggle raw display
$47$ \( T^{6} + \cdots + 20\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( (T^{3} + 52260928650 T^{2} + \cdots + 49\!\cdots\!00)^{2} \) Copy content Toggle raw display
$59$ \( T^{6} + \cdots + 27\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( T^{6} + \cdots + 30\!\cdots\!00 \) Copy content Toggle raw display
$67$ \( (T^{3} + 64873301850 T^{2} + \cdots - 76\!\cdots\!00)^{2} \) Copy content Toggle raw display
$71$ \( (T^{3} + 7508802330 T^{2} + \cdots - 18\!\cdots\!12)^{2} \) Copy content Toggle raw display
$73$ \( T^{6} + \cdots + 76\!\cdots\!00 \) Copy content Toggle raw display
$79$ \( (T^{3} + 86263815834 T^{2} + \cdots - 66\!\cdots\!68)^{2} \) Copy content Toggle raw display
$83$ \( T^{6} + \cdots + 13\!\cdots\!00 \) Copy content Toggle raw display
$89$ \( T^{6} + \cdots + 34\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{6} + \cdots + 30\!\cdots\!00 \) Copy content Toggle raw display
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