Properties

Label 14.12.c.a
Level $14$
Weight $12$
Character orbit 14.c
Analytic conductor $10.757$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 14 = 2 \cdot 7 \)
Weight: \( k \) \(=\) \( 12 \)
Character orbit: \([\chi]\) \(=\) 14.c (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(10.7568045278\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
Defining polynomial: \( x^{8} - x^{7} + 101803 x^{6} + 6576400 x^{5} + 8539617914 x^{4} + 333205096780 x^{3} + 195840371083183 x^{2} + \cdots + 33\!\cdots\!81 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{12}\cdot 3\cdot 7^{4} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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.

\(f(q)\) \(=\) \( q + (32 \beta_{2} - 32) q^{2} + ( - \beta_{3} + 67 \beta_{2}) q^{3} - 1024 \beta_{2} q^{4} + ( - \beta_{6} - 1877 \beta_{2} - \beta_1 + 1877) q^{5} + (32 \beta_{3} + 32 \beta_1 - 2144) q^{6} + (\beta_{5} - \beta_{4} - 23 \beta_{3} - 3040 \beta_{2} - 42 \beta_1 - 3742) q^{7} + 32768 q^{8} + (\beta_{7} + 9 \beta_{6} + \beta_{5} + 30903 \beta_{2} + 38 \beta_1 - 30904) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (32 \beta_{2} - 32) q^{2} + ( - \beta_{3} + 67 \beta_{2}) q^{3} - 1024 \beta_{2} q^{4} + ( - \beta_{6} - 1877 \beta_{2} - \beta_1 + 1877) q^{5} + (32 \beta_{3} + 32 \beta_1 - 2144) q^{6} + (\beta_{5} - \beta_{4} - 23 \beta_{3} - 3040 \beta_{2} - 42 \beta_1 - 3742) q^{7} + 32768 q^{8} + (\beta_{7} + 9 \beta_{6} + \beta_{5} + 30903 \beta_{2} + 38 \beta_1 - 30904) q^{9} + (32 \beta_{4} - 32 \beta_{3} + 60032 \beta_{2}) q^{10} + (4 \beta_{7} - 8 \beta_{5} - 43 \beta_{4} + 628 \beta_{3} + 52962 \beta_{2} + \cdots + 4) q^{11}+ \cdots + (1428012 \beta_{7} + 1697841 \beta_{6} + \cdots + 49170955320) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 128 q^{2} + 266 q^{3} - 4096 q^{4} + 7504 q^{5} - 17024 q^{6} - 42224 q^{7} + 262144 q^{8} - 123520 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 128 q^{2} + 266 q^{3} - 4096 q^{4} + 7504 q^{5} - 17024 q^{6} - 42224 q^{7} + 262144 q^{8} - 123520 q^{9} + 240128 q^{10} + 213026 q^{11} + 272384 q^{12} - 2609712 q^{13} + 1257536 q^{14} + 2275500 q^{15} - 4194304 q^{16} + 8854244 q^{17} - 3952640 q^{18} + 7232806 q^{19} - 15368192 q^{20} + 49275128 q^{21} - 13633664 q^{22} - 10649134 q^{23} + 8716288 q^{24} - 119407256 q^{25} + 41755392 q^{26} + 19012196 q^{27} + 2996224 q^{28} - 221414576 q^{29} - 36408000 q^{30} + 486231270 q^{31} - 134217728 q^{32} + 489871116 q^{33} - 566671616 q^{34} - 756166390 q^{35} + 252968960 q^{36} + 463131040 q^{37} + 231449792 q^{38} + 493924396 q^{39} + 245891072 q^{40} - 1617723408 q^{41} - 1609760768 q^{42} + 2926896352 q^{43} + 218138624 q^{44} + 2871035832 q^{45} - 340772288 q^{46} - 894091254 q^{47} - 557842432 q^{48} - 11608958872 q^{49} + 7642064384 q^{50} + 6950287158 q^{51} + 1336172544 q^{52} - 1448863512 q^{53} - 304195136 q^{54} - 25123348964 q^{55} - 1383596032 q^{56} + 18014730520 q^{57} + 3542633216 q^{58} + 14386900738 q^{59} - 1165056000 q^{60} + 10854402216 q^{61} - 31118801280 q^{62} - 11833302628 q^{63} + 8589934592 q^{64} + 5495584080 q^{65} + 15675875712 q^{66} + 19629545546 q^{67} + 9066745856 q^{68} - 92767078992 q^{69} + 13390987456 q^{70} + 1474804928 q^{71} - 4047503360 q^{72} + 21420158732 q^{73} + 14820193280 q^{74} + 63738659600 q^{75} - 14812786688 q^{76} + 28657880944 q^{77} - 31611161344 q^{78} - 60246238086 q^{79} + 7868514304 q^{80} + 100835656724 q^{81} + 25883574528 q^{82} - 139893582304 q^{83} + 1054613504 q^{84} + 109585554592 q^{85} - 46830341632 q^{86} - 77636109684 q^{87} + 6980435968 q^{88} + 126354105612 q^{89} - 183746293248 q^{90} + 44156589120 q^{91} + 21809426432 q^{92} - 213975312128 q^{93} - 28610920128 q^{94} + 368499346550 q^{95} + 8925478912 q^{96} - 538182884176 q^{97} + 210290620288 q^{98} + 393415805736 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - x^{7} + 101803 x^{6} + 6576400 x^{5} + 8539617914 x^{4} + 333205096780 x^{3} + 195840371083183 x^{2} + \cdots + 33\!\cdots\!81 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( 2\nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 13\!\cdots\!87 \nu^{7} + \cdots - 12\!\cdots\!31 ) / 10\!\cdots\!76 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 3452459022039 \nu^{7} + 135854022621670 \nu^{6} + \cdots + 24\!\cdots\!17 ) / 28\!\cdots\!68 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 46\!\cdots\!87 \nu^{7} + \cdots + 25\!\cdots\!33 ) / 28\!\cdots\!52 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 70\!\cdots\!39 \nu^{7} + \cdots - 47\!\cdots\!29 ) / 13\!\cdots\!12 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 42\!\cdots\!85 \nu^{7} + \cdots + 14\!\cdots\!09 ) / 70\!\cdots\!38 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 19\!\cdots\!43 \nu^{7} + \cdots - 29\!\cdots\!31 ) / 46\!\cdots\!92 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{7} - 2\beta_{5} - 9\beta_{4} - 96\beta_{3} - 203553\beta_{2} + 1 ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 1130 \beta_{7} + 2466 \beta_{6} - 565 \beta_{5} - 2466 \beta_{4} - 263827 \beta_{3} + 1901 \beta_{2} - 263827 \beta _1 - 19903645 ) / 8 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 204169\beta_{7} + 1834902\beta_{6} + 204169\beta_{5} + 26899401561\beta_{2} - 32759007\beta _1 - 26899605730 ) / 8 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 64196897 \beta_{7} + 128393794 \beta_{5} + 311150016 \beta_{4} + 20229268705 \beta_{3} + 3370858960210 \beta_{2} - 64196897 ) / 8 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 9518249694 \beta_{7} - 40579191729 \beta_{6} + 4759124847 \beta_{5} + 40579191729 \beta_{4} + 964914130080 \beta_{3} - 35820066882 \beta_{2} + \cdots + 516233389180301 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 3093308816199 \beta_{7} - 16644032027820 \beta_{6} - 3093308816199 \beta_{5} + \cdots + 19\!\cdots\!02 ) / 4 \) Copy content Toggle raw display

Character values

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

\(n\) \(3\)
\(\chi(n)\) \(-1 + \beta_{2}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
9.1
152.619 264.344i
64.1153 111.051i
−100.397 + 173.893i
−115.837 + 200.636i
152.619 + 264.344i
64.1153 + 111.051i
−100.397 173.893i
−115.837 200.636i
−16.0000 + 27.7128i −271.738 470.665i −512.000 886.810i 3667.11 6351.63i 17391.3 −28136.3 + 34433.6i 32768.0 −59110.0 + 102381.i 117348. + 203252.i
9.2 −16.0000 + 27.7128i −94.7305 164.078i −512.000 886.810i −3211.13 + 5561.84i 6062.75 −9575.58 43423.9i 32768.0 70625.8 122327.i −102756. 177979.i
9.3 −16.0000 + 27.7128i 234.294 + 405.809i −512.000 886.810i −3326.37 + 5761.45i −14994.8 12255.3 + 42745.0i 32768.0 −21214.1 + 36743.8i −106444. 184366.i
9.4 −16.0000 + 27.7128i 265.175 + 459.296i −512.000 886.810i 6622.39 11470.3i −16971.2 4344.64 44254.4i 32768.0 −52061.7 + 90173.6i 211917. + 367050.i
11.1 −16.0000 27.7128i −271.738 + 470.665i −512.000 + 886.810i 3667.11 + 6351.63i 17391.3 −28136.3 34433.6i 32768.0 −59110.0 102381.i 117348. 203252.i
11.2 −16.0000 27.7128i −94.7305 + 164.078i −512.000 + 886.810i −3211.13 5561.84i 6062.75 −9575.58 + 43423.9i 32768.0 70625.8 + 122327.i −102756. + 177979.i
11.3 −16.0000 27.7128i 234.294 405.809i −512.000 + 886.810i −3326.37 5761.45i −14994.8 12255.3 42745.0i 32768.0 −21214.1 36743.8i −106444. + 184366.i
11.4 −16.0000 27.7128i 265.175 459.296i −512.000 + 886.810i 6622.39 + 11470.3i −16971.2 4344.64 + 44254.4i 32768.0 −52061.7 90173.6i 211917. 367050.i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 11.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 14.12.c.a 8
3.b odd 2 1 126.12.g.e 8
4.b odd 2 1 112.12.i.a 8
7.b odd 2 1 98.12.c.l 8
7.c even 3 1 inner 14.12.c.a 8
7.c even 3 1 98.12.a.j 4
7.d odd 6 1 98.12.a.l 4
7.d odd 6 1 98.12.c.l 8
21.h odd 6 1 126.12.g.e 8
28.g odd 6 1 112.12.i.a 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
14.12.c.a 8 1.a even 1 1 trivial
14.12.c.a 8 7.c even 3 1 inner
98.12.a.j 4 7.c even 3 1
98.12.a.l 4 7.d odd 6 1
98.12.c.l 8 7.b odd 2 1
98.12.c.l 8 7.d odd 6 1
112.12.i.a 8 4.b odd 2 1
112.12.i.a 8 28.g odd 6 1
126.12.g.e 8 3.b odd 2 1
126.12.g.e 8 21.h odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{8} - 266 T_{3}^{7} + 451432 T_{3}^{6} - 57316476 T_{3}^{5} + 140415791337 T_{3}^{4} - 16569707709996 T_{3}^{3} + \cdots + 65\!\cdots\!25 \) acting on \(S_{12}^{\mathrm{new}}(14, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 32 T + 1024)^{4} \) Copy content Toggle raw display
$3$ \( T^{8} - 266 T^{7} + \cdots + 65\!\cdots\!25 \) Copy content Toggle raw display
$5$ \( T^{8} - 7504 T^{7} + \cdots + 17\!\cdots\!25 \) Copy content Toggle raw display
$7$ \( T^{8} + 42224 T^{7} + \cdots + 15\!\cdots\!01 \) Copy content Toggle raw display
$11$ \( T^{8} - 213026 T^{7} + \cdots + 84\!\cdots\!25 \) Copy content Toggle raw display
$13$ \( (T^{4} + 1304856 T^{3} + \cdots + 45\!\cdots\!00)^{2} \) Copy content Toggle raw display
$17$ \( T^{8} - 8854244 T^{7} + \cdots + 48\!\cdots\!41 \) Copy content Toggle raw display
$19$ \( T^{8} - 7232806 T^{7} + \cdots + 16\!\cdots\!25 \) Copy content Toggle raw display
$23$ \( T^{8} + 10649134 T^{7} + \cdots + 25\!\cdots\!09 \) Copy content Toggle raw display
$29$ \( (T^{4} + 110707288 T^{3} + \cdots - 37\!\cdots\!16)^{2} \) Copy content Toggle raw display
$31$ \( T^{8} - 486231270 T^{7} + \cdots + 52\!\cdots\!25 \) Copy content Toggle raw display
$37$ \( T^{8} - 463131040 T^{7} + \cdots + 67\!\cdots\!89 \) Copy content Toggle raw display
$41$ \( (T^{4} + 808861704 T^{3} + \cdots - 17\!\cdots\!96)^{2} \) Copy content Toggle raw display
$43$ \( (T^{4} - 1463448176 T^{3} + \cdots + 31\!\cdots\!00)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} + 894091254 T^{7} + \cdots + 24\!\cdots\!81 \) Copy content Toggle raw display
$53$ \( T^{8} + 1448863512 T^{7} + \cdots + 31\!\cdots\!01 \) Copy content Toggle raw display
$59$ \( T^{8} - 14386900738 T^{7} + \cdots + 16\!\cdots\!25 \) Copy content Toggle raw display
$61$ \( T^{8} - 10854402216 T^{7} + \cdots + 84\!\cdots\!81 \) Copy content Toggle raw display
$67$ \( T^{8} - 19629545546 T^{7} + \cdots + 34\!\cdots\!41 \) Copy content Toggle raw display
$71$ \( (T^{4} - 737402464 T^{3} + \cdots + 29\!\cdots\!96)^{2} \) Copy content Toggle raw display
$73$ \( T^{8} - 21420158732 T^{7} + \cdots + 40\!\cdots\!25 \) Copy content Toggle raw display
$79$ \( T^{8} + 60246238086 T^{7} + \cdots + 73\!\cdots\!25 \) Copy content Toggle raw display
$83$ \( (T^{4} + 69946791152 T^{3} + \cdots - 47\!\cdots\!84)^{2} \) Copy content Toggle raw display
$89$ \( T^{8} - 126354105612 T^{7} + \cdots + 59\!\cdots\!21 \) Copy content Toggle raw display
$97$ \( (T^{4} + 269091442088 T^{3} + \cdots - 18\!\cdots\!00)^{2} \) Copy content Toggle raw display
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