Properties

Label 14.7.d.a
Level $14$
Weight $7$
Character orbit 14.d
Analytic conductor $3.221$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 14 = 2 \cdot 7 \)
Weight: \( k \) \(=\) \( 7 \)
Character orbit: \([\chi]\) \(=\) 14.d (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(3.22075717068\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
Defining polynomial: \( x^{8} - 2x^{7} + 285x^{6} + 282x^{5} + 62091x^{4} + 29260x^{3} + 4838750x^{2} + 2401000x + 294122500 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{6}\cdot 3^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( - \beta_{3} + \beta_{2}) q^{2} + ( - \beta_{6} + 2 \beta_{3} - \beta_{2}) q^{3} - 32 \beta_1 q^{4} + (\beta_{7} - 3 \beta_{6} - 3 \beta_{4} + 6 \beta_{3} + 5 \beta_{2} - 28 \beta_1 - 28) q^{5} + (4 \beta_{5} + 4 \beta_{4} - \beta_{3} - 2 \beta_{2} + 56 \beta_1 - 28) q^{6} + (5 \beta_{7} - 3 \beta_{6} + 3 \beta_{5} + 5 \beta_{4} - 13 \beta_{3} - 6 \beta_{2} + \cdots + 74) q^{7}+ \cdots + (7 \beta_{7} + 7 \beta_{6} - 14 \beta_{5} - 7 \beta_{4} - 48 \beta_{3} + 55 \beta_{2} + \cdots + 189) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{3} + \beta_{2}) q^{2} + ( - \beta_{6} + 2 \beta_{3} - \beta_{2}) q^{3} - 32 \beta_1 q^{4} + (\beta_{7} - 3 \beta_{6} - 3 \beta_{4} + 6 \beta_{3} + 5 \beta_{2} - 28 \beta_1 - 28) q^{5} + (4 \beta_{5} + 4 \beta_{4} - \beta_{3} - 2 \beta_{2} + 56 \beta_1 - 28) q^{6} + (5 \beta_{7} - 3 \beta_{6} + 3 \beta_{5} + 5 \beta_{4} - 13 \beta_{3} - 6 \beta_{2} + \cdots + 74) q^{7}+ \cdots + ( - 9702 \beta_{7} - 13902 \beta_{6} + 4851 \beta_{5} + \cdots - 578241) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 128 q^{4} - 336 q^{5} + 652 q^{7} + 756 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 128 q^{4} - 336 q^{5} + 652 q^{7} + 756 q^{9} - 2016 q^{10} - 1356 q^{11} + 2064 q^{14} + 27144 q^{15} - 4096 q^{16} - 17304 q^{17} - 6816 q^{18} - 32004 q^{19} + 9756 q^{21} + 25248 q^{22} - 4128 q^{23} + 10752 q^{24} + 4664 q^{25} - 4704 q^{26} - 7552 q^{28} - 30312 q^{29} + 9648 q^{30} - 3108 q^{31} + 3276 q^{33} + 98028 q^{35} - 48384 q^{36} - 6124 q^{37} + 155568 q^{38} + 100764 q^{39} + 64512 q^{40} - 315936 q^{42} - 297376 q^{43} - 43392 q^{44} - 172116 q^{45} - 194064 q^{46} + 313908 q^{47} + 32432 q^{49} + 38784 q^{50} + 253692 q^{51} + 255360 q^{52} + 278484 q^{53} + 386064 q^{54} - 125952 q^{56} - 81288 q^{57} - 169824 q^{58} - 835464 q^{59} - 434304 q^{60} - 995316 q^{61} + 1216188 q^{63} + 262144 q^{64} + 8316 q^{65} + 1673280 q^{66} + 648808 q^{67} + 553728 q^{68} - 1572816 q^{70} + 190128 q^{71} - 218112 q^{72} - 1617084 q^{73} - 1158144 q^{74} - 2042208 q^{75} + 1456224 q^{77} + 1122432 q^{78} + 70096 q^{79} + 344064 q^{80} + 1177920 q^{81} + 66528 q^{82} - 1133568 q^{84} + 2190984 q^{85} - 573024 q^{86} - 2057076 q^{87} - 403968 q^{88} + 739116 q^{89} + 2233752 q^{91} + 264192 q^{92} - 23364 q^{93} + 3795120 q^{94} + 725640 q^{95} - 344064 q^{96} - 3532320 q^{98} - 4625928 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 2x^{7} + 285x^{6} + 282x^{5} + 62091x^{4} + 29260x^{3} + 4838750x^{2} + 2401000x + 294122500 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 10737582 \nu^{7} + 884171859 \nu^{6} - 3763944460 \nu^{5} + 189328804101 \nu^{4} - 195625686972 \nu^{3} + \cdots + 41\!\cdots\!50 ) / 33\!\cdots\!50 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 512915233 \nu^{7} + 35396730954 \nu^{6} - 150684878760 \nu^{5} + 14474316021931 \nu^{4} - 7831633340232 \nu^{3} + \cdots + 16\!\cdots\!00 ) / 13\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 14893 \nu^{7} - 236741 \nu^{6} + 3320715 \nu^{5} + 8132426 \nu^{4} + 331237078 \nu^{3} + 640160500 \nu^{2} + 15842998500 \nu + 846958238000 ) / 106489495000 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 10421298761 \nu^{7} - 383100933507 \nu^{6} + 14696779653705 \nu^{5} - 107198920144948 \nu^{4} + \cdots - 61\!\cdots\!00 ) / 52\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 4954059529 \nu^{7} + 53601861423 \nu^{6} - 2878465568645 \nu^{5} + 14314185841272 \nu^{4} - 544906891545834 \nu^{3} + \cdots + 11\!\cdots\!00 ) / 75\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 98330981563 \nu^{7} + 465055977756 \nu^{6} - 28111572790890 \nu^{5} + 2750456475509 \nu^{4} + \cdots - 20\!\cdots\!00 ) / 52\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 112677227274 \nu^{7} + 39166570437 \nu^{6} + 18385233372395 \nu^{5} + 149484497566193 \nu^{4} + \cdots + 11\!\cdots\!00 ) / 52\!\cdots\!00 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( -2\beta_{7} - 2\beta_{6} - 2\beta_{5} - 4\beta_{4} + \beta_{2} + 6\beta_1 ) / 12 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{7} + 3\beta_{6} - 2\beta_{5} - 3\beta_{4} + 54\beta_{3} - 53\beta_{2} + 849\beta _1 - 849 ) / 6 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 286\beta_{7} + 434\beta_{6} - 143\beta_{5} + 217\beta_{4} + 477\beta_{3} - 143\beta_{2} - 1911 ) / 6 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 427\beta_{7} + 1137\beta_{6} + 427\beta_{5} + 2274\beta_{4} + 15064\beta_{2} - 139071\beta_1 ) / 6 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 23747 \beta_{7} - 45681 \beta_{6} + 47494 \beta_{5} + 45681 \beta_{4} - 132588 \beta_{3} + 108841 \beta_{2} - 644823 \beta _1 + 644823 ) / 6 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 260622 \beta_{7} - 658058 \beta_{6} + 130311 \beta_{5} - 329029 \beta_{4} - 3745254 \beta_{3} + 130311 \beta_{2} + 25540707 ) / 6 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 4421299 \beta_{7} - 9613689 \beta_{6} - 4421299 \beta_{5} - 19227378 \beta_{4} - 25496068 \beta_{2} + 180033087 \beta_1 ) / 6 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/14\mathbb{Z}\right)^\times\).

\(n\) \(3\)
\(\chi(n)\) \(1 - \beta_{1}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
3.1
4.65421 8.06134i
−4.86132 + 8.42006i
−6.30576 + 10.9219i
7.51287 13.0127i
4.65421 + 8.06134i
−4.86132 8.42006i
−6.30576 10.9219i
7.51287 + 13.0127i
−2.82843 4.89898i −12.7609 7.36750i −16.0000 + 27.7128i −106.741 + 61.6269i 83.3537i −309.691 + 147.446i 181.019 −255.940 443.301i 603.818 + 348.614i
3.2 −2.82843 4.89898i 27.6101 + 15.9407i −16.0000 + 27.7128i 111.836 64.5687i 180.349i 298.743 168.528i 181.019 143.713 + 248.917i −632.642 365.256i
3.3 2.82843 + 4.89898i −36.7384 21.2109i −16.0000 + 27.7128i −162.347 + 93.7310i 239.974i 141.244 312.569i −181.019 535.305 + 927.175i −918.372 530.222i
3.4 2.82843 + 4.89898i 21.8891 + 12.6377i −16.0000 + 27.7128i −10.7486 + 6.20573i 142.979i 195.705 + 281.689i −181.019 −45.0775 78.0766i −60.8035 35.1049i
5.1 −2.82843 + 4.89898i −12.7609 + 7.36750i −16.0000 27.7128i −106.741 61.6269i 83.3537i −309.691 147.446i 181.019 −255.940 + 443.301i 603.818 348.614i
5.2 −2.82843 + 4.89898i 27.6101 15.9407i −16.0000 27.7128i 111.836 + 64.5687i 180.349i 298.743 + 168.528i 181.019 143.713 248.917i −632.642 + 365.256i
5.3 2.82843 4.89898i −36.7384 + 21.2109i −16.0000 27.7128i −162.347 93.7310i 239.974i 141.244 + 312.569i −181.019 535.305 927.175i −918.372 + 530.222i
5.4 2.82843 4.89898i 21.8891 12.6377i −16.0000 27.7128i −10.7486 6.20573i 142.979i 195.705 281.689i −181.019 −45.0775 + 78.0766i −60.8035 + 35.1049i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 5.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.d odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 14.7.d.a 8
3.b odd 2 1 126.7.n.c 8
4.b odd 2 1 112.7.s.c 8
7.b odd 2 1 98.7.d.c 8
7.c even 3 1 98.7.b.c 8
7.c even 3 1 98.7.d.c 8
7.d odd 6 1 inner 14.7.d.a 8
7.d odd 6 1 98.7.b.c 8
21.g even 6 1 126.7.n.c 8
28.f even 6 1 112.7.s.c 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
14.7.d.a 8 1.a even 1 1 trivial
14.7.d.a 8 7.d odd 6 1 inner
98.7.b.c 8 7.c even 3 1
98.7.b.c 8 7.d odd 6 1
98.7.d.c 8 7.b odd 2 1
98.7.d.c 8 7.c even 3 1
112.7.s.c 8 4.b odd 2 1
112.7.s.c 8 28.f even 6 1
126.7.n.c 8 3.b odd 2 1
126.7.n.c 8 21.g even 6 1

Hecke kernels

This newform subspace is the entire newspace \(S_{7}^{\mathrm{new}}(14, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} + 32 T^{2} + 1024)^{2} \) Copy content Toggle raw display
$3$ \( T^{8} - 1836 T^{6} + \cdots + 253716712209 \) Copy content Toggle raw display
$5$ \( T^{8} + 336 T^{7} + \cdots + 13\!\cdots\!25 \) Copy content Toggle raw display
$7$ \( T^{8} - 652 T^{7} + \cdots + 19\!\cdots\!01 \) Copy content Toggle raw display
$11$ \( T^{8} + 1356 T^{7} + \cdots + 57\!\cdots\!09 \) Copy content Toggle raw display
$13$ \( T^{8} + 11814792 T^{6} + \cdots + 76\!\cdots\!64 \) Copy content Toggle raw display
$17$ \( T^{8} + 17304 T^{7} + \cdots + 12\!\cdots\!41 \) Copy content Toggle raw display
$19$ \( T^{8} + 32004 T^{7} + \cdots + 47\!\cdots\!25 \) Copy content Toggle raw display
$23$ \( T^{8} + 4128 T^{7} + \cdots + 11\!\cdots\!09 \) Copy content Toggle raw display
$29$ \( (T^{4} + 15156 T^{3} + \cdots - 14\!\cdots\!56)^{2} \) Copy content Toggle raw display
$31$ \( T^{8} + 3108 T^{7} + \cdots + 12\!\cdots\!21 \) Copy content Toggle raw display
$37$ \( T^{8} + 6124 T^{7} + \cdots + 23\!\cdots\!61 \) Copy content Toggle raw display
$41$ \( T^{8} + 25142408520 T^{6} + \cdots + 50\!\cdots\!84 \) Copy content Toggle raw display
$43$ \( (T^{4} + 148688 T^{3} + \cdots - 56\!\cdots\!48)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} - 313908 T^{7} + \cdots + 12\!\cdots\!41 \) Copy content Toggle raw display
$53$ \( T^{8} - 278484 T^{7} + \cdots + 50\!\cdots\!09 \) Copy content Toggle raw display
$59$ \( T^{8} + 835464 T^{7} + \cdots + 27\!\cdots\!49 \) Copy content Toggle raw display
$61$ \( T^{8} + 995316 T^{7} + \cdots + 35\!\cdots\!41 \) Copy content Toggle raw display
$67$ \( T^{8} - 648808 T^{7} + \cdots + 49\!\cdots\!69 \) Copy content Toggle raw display
$71$ \( (T^{4} - 95064 T^{3} + \cdots + 56\!\cdots\!56)^{2} \) Copy content Toggle raw display
$73$ \( T^{8} + 1617084 T^{7} + \cdots + 79\!\cdots\!89 \) Copy content Toggle raw display
$79$ \( T^{8} - 70096 T^{7} + \cdots + 83\!\cdots\!25 \) Copy content Toggle raw display
$83$ \( T^{8} + 1103755415712 T^{6} + \cdots + 34\!\cdots\!84 \) Copy content Toggle raw display
$89$ \( T^{8} - 739116 T^{7} + \cdots + 11\!\cdots\!01 \) Copy content Toggle raw display
$97$ \( T^{8} + 4399256051592 T^{6} + \cdots + 11\!\cdots\!36 \) Copy content Toggle raw display
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