Properties

Label 14.14.c.a
Level $14$
Weight $14$
Character orbit 14.c
Analytic conductor $15.012$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Newspace parameters

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

Newform invariants

Self dual: no
Analytic conductor: \(15.0123300533\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
Defining polynomial: \( x^{8} - 2 x^{7} + 692094 x^{6} + 445371928 x^{5} + 480078817147 x^{4} + 153633603309480 x^{3} + \cdots + 23\!\cdots\!25 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{14}\cdot 3\cdot 7^{4} \)
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.

\(f(q)\) \(=\) \( q + ( - 64 \beta_1 - 64) q^{2} + (\beta_{4} + 45 \beta_1) q^{3} + 4096 \beta_1 q^{4} + (\beta_{7} + 5 \beta_{4} - \beta_{3} - 5 \beta_{2} - 16103 \beta_1 - 16102) q^{5} + ( - 64 \beta_{2} + 2880) q^{6} + (7 \beta_{7} - 3 \beta_{6} - \beta_{5} - 25 \beta_{4} - 9 \beta_{3} - 10 \beta_{2} + \cdots + 11651) q^{7}+ \cdots + ( - 6 \beta_{7} + 11 \beta_{6} + 22 \beta_{5} + 484 \beta_{4} + \cdots - 996183) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - 64 \beta_1 - 64) q^{2} + (\beta_{4} + 45 \beta_1) q^{3} + 4096 \beta_1 q^{4} + (\beta_{7} + 5 \beta_{4} - \beta_{3} - 5 \beta_{2} - 16103 \beta_1 - 16102) q^{5} + ( - 64 \beta_{2} + 2880) q^{6} + (7 \beta_{7} - 3 \beta_{6} - \beta_{5} - 25 \beta_{4} - 9 \beta_{3} - 10 \beta_{2} + \cdots + 11651) q^{7}+ \cdots + ( - 30229148 \beta_{6} + 30229148 \beta_{5} + \cdots - 3231197379261) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 256 q^{2} - 182 q^{3} - 16384 q^{4} - 64400 q^{5} + 23296 q^{6} - 113736 q^{7} + 2097152 q^{8} - 3983752 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 256 q^{2} - 182 q^{3} - 16384 q^{4} - 64400 q^{5} + 23296 q^{6} - 113736 q^{7} + 2097152 q^{8} - 3983752 q^{9} - 4121600 q^{10} - 1008790 q^{11} - 745472 q^{12} + 53807264 q^{13} + 23506560 q^{14} - 88888916 q^{15} - 67108864 q^{16} - 165333028 q^{17} - 254960128 q^{18} - 423405794 q^{19} + 527564800 q^{20} + 231089600 q^{21} + 129125120 q^{22} + 286233866 q^{23} - 47710208 q^{24} - 472017432 q^{25} - 1721832448 q^{26} - 10405941644 q^{27} - 1038557184 q^{28} + 18200674816 q^{29} + 2844445312 q^{30} + 1507094246 q^{31} - 4294967296 q^{32} - 16888935028 q^{33} + 21162627584 q^{34} - 4300332526 q^{35} + 32634896384 q^{36} - 18959705336 q^{37} - 27097970816 q^{38} + 35759388756 q^{39} - 16882073600 q^{40} + 113651910624 q^{41} + 25531294208 q^{42} - 91557619424 q^{43} - 4132003840 q^{44} - 119804452768 q^{45} + 18318967424 q^{46} - 81351201078 q^{47} + 6106906624 q^{48} - 60954502168 q^{49} + 60418231296 q^{50} + 14996824142 q^{51} - 110197276672 q^{52} + 87497947440 q^{53} + 332990132608 q^{54} + 701637410348 q^{55} - 29815209984 q^{56} - 1807326928264 q^{57} - 582421594112 q^{58} - 194140265102 q^{59} + 182044499968 q^{60} + 175816313120 q^{61} - 192908063488 q^{62} + 3245184239332 q^{63} + 549755813888 q^{64} - 1689866774568 q^{65} - 1080891841792 q^{66} - 243815218758 q^{67} - 677204082688 q^{68} + 6044157952784 q^{69} + 3010938632576 q^{70} - 3275394679072 q^{71} - 1044316684288 q^{72} - 3492491920596 q^{73} - 1213421141504 q^{74} - 2370218127424 q^{75} + 3468540264448 q^{76} + 6381735922240 q^{77} - 4577201760768 q^{78} - 1016380081246 q^{79} - 1080452710400 q^{80} + 17492694092 q^{81} - 3636861139968 q^{82} + 7027495743296 q^{83} - 2580545830912 q^{84} + 569114840528 q^{85} + 2929843821568 q^{86} - 9706955821052 q^{87} - 264448245760 q^{88} - 8034124428036 q^{89} + 15334969954304 q^{90} - 11436513503632 q^{91} - 2344827830272 q^{92} - 3388371390552 q^{93} - 5206476868992 q^{94} + 1429435505438 q^{95} - 195421011968 q^{96} + 54351131725632 q^{97} - 461748942592 q^{98} - 25837834576328 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 2 x^{7} + 692094 x^{6} + 445371928 x^{5} + 480078817147 x^{4} + 153633603309480 x^{3} + \cdots + 23\!\cdots\!25 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 27\!\cdots\!26 \nu^{7} + \cdots - 92\!\cdots\!05 ) / 90\!\cdots\!05 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 14\!\cdots\!70 \nu^{7} + \cdots - 42\!\cdots\!05 ) / 25\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 16\!\cdots\!30 \nu^{7} + \cdots - 84\!\cdots\!45 ) / 25\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 99\!\cdots\!39 \nu^{7} + \cdots - 33\!\cdots\!05 ) / 19\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 17\!\cdots\!37 \nu^{7} + \cdots - 14\!\cdots\!70 ) / 23\!\cdots\!70 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 20\!\cdots\!67 \nu^{7} + \cdots + 21\!\cdots\!15 ) / 23\!\cdots\!70 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 10\!\cdots\!81 \nu^{7} + \cdots - 35\!\cdots\!95 ) / 10\!\cdots\!40 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( 3\beta_{7} + \beta_{6} + 2\beta_{5} - 9\beta_{4} - 4\beta_{3} + 8\beta_{2} + 171\beta _1 + 174 ) / 336 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -87\beta_{7} + 386\beta_{6} + 193\beta_{5} - 5751\beta_{4} + 193\beta_{3} + 193\beta_{2} + 29070783\beta_1 ) / 84 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 813235\beta_{6} - 813235\beta_{5} + 2971559\beta_{3} - 16649389\beta_{2} - 56301080754 ) / 336 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 26739030 \beta_{7} - 47369614 \beta_{6} - 94739228 \beta_{5} + 1122208434 \beta_{4} + 74108644 \beta_{3} - 1074838820 \beta_{2} - 5071528550799 \beta _1 - 5071555289829 ) / 21 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 859127137203 \beta_{7} - 1474522292642 \beta_{6} - 737261146321 \beta_{5} + 17805005133849 \beta_{4} - 737261146321 \beta_{3} + \cdots - 64\!\cdots\!31 \beta_1 ) / 336 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 176934179918233 \beta_{6} + 176934179918233 \beta_{5} - 502838085390899 \beta_{3} + \cdots + 17\!\cdots\!76 ) / 84 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 69\!\cdots\!09 \beta_{7} + \cdots + 63\!\cdots\!34 ) / 336 \) 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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
9.1
−3.53342 + 6.12007i
−258.759 + 448.183i
481.229 833.513i
−217.937 + 377.477i
−3.53342 6.12007i
−258.759 448.183i
481.229 + 833.513i
−217.937 377.477i
−32.0000 + 55.4256i −855.669 1482.06i −2048.00 3547.24i 12545.8 21729.9i 109526. 48813.9 307419.i 262144. −667176. + 1.15558e6i 802928. + 1.39071e6i
9.2 −32.0000 + 55.4256i −679.656 1177.20i −2048.00 3547.24i −29011.3 + 50249.0i 86996.0 −173759. + 258257.i 262144. −126703. + 219456.i −1.85672e6 3.21593e6i
9.3 −32.0000 + 55.4256i 301.970 + 523.027i −2048.00 3547.24i 2521.96 4368.16i −38652.2 302038. + 75247.5i 262144. 614790. 1.06485e6i 161406. + 279563.i
9.4 −32.0000 + 55.4256i 1142.35 + 1978.62i −2048.00 3547.24i −18256.5 + 31621.1i −146221. −233961. 205308.i 262144. −1.81279e6 + 3.13984e6i −1.16841e6 2.02375e6i
11.1 −32.0000 55.4256i −855.669 + 1482.06i −2048.00 + 3547.24i 12545.8 + 21729.9i 109526. 48813.9 + 307419.i 262144. −667176. 1.15558e6i 802928. 1.39071e6i
11.2 −32.0000 55.4256i −679.656 + 1177.20i −2048.00 + 3547.24i −29011.3 50249.0i 86996.0 −173759. 258257.i 262144. −126703. 219456.i −1.85672e6 + 3.21593e6i
11.3 −32.0000 55.4256i 301.970 523.027i −2048.00 + 3547.24i 2521.96 + 4368.16i −38652.2 302038. 75247.5i 262144. 614790. + 1.06485e6i 161406. 279563.i
11.4 −32.0000 55.4256i 1142.35 1978.62i −2048.00 + 3547.24i −18256.5 31621.1i −146221. −233961. + 205308.i 262144. −1.81279e6 3.13984e6i −1.16841e6 + 2.02375e6i
\(n\): 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 14.14.c.a 8
3.b odd 2 1 126.14.g.d 8
7.b odd 2 1 98.14.c.n 8
7.c even 3 1 inner 14.14.c.a 8
7.c even 3 1 98.14.a.l 4
7.d odd 6 1 98.14.a.k 4
7.d odd 6 1 98.14.c.n 8
21.h odd 6 1 126.14.g.d 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
14.14.c.a 8 1.a even 1 1 trivial
14.14.c.a 8 7.c even 3 1 inner
98.14.a.k 4 7.d odd 6 1
98.14.a.l 4 7.c even 3 1
98.14.c.n 8 7.b odd 2 1
98.14.c.n 8 7.d odd 6 1
126.14.g.d 8 3.b odd 2 1
126.14.g.d 8 21.h odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{8} + 182 T_{3}^{7} + 5197084 T_{3}^{6} + 4025617932 T_{3}^{5} + 23908528667637 T_{3}^{4} + \cdots + 10\!\cdots\!25 \) acting on \(S_{14}^{\mathrm{new}}(14, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 64 T + 4096)^{4} \) Copy content Toggle raw display
$3$ \( T^{8} + 182 T^{7} + \cdots + 10\!\cdots\!25 \) Copy content Toggle raw display
$5$ \( T^{8} + 64400 T^{7} + \cdots + 71\!\cdots\!25 \) Copy content Toggle raw display
$7$ \( T^{8} + 113736 T^{7} + \cdots + 88\!\cdots\!01 \) Copy content Toggle raw display
$11$ \( T^{8} + 1008790 T^{7} + \cdots + 29\!\cdots\!25 \) Copy content Toggle raw display
$13$ \( (T^{4} - 26903632 T^{3} + \cdots - 94\!\cdots\!80)^{2} \) Copy content Toggle raw display
$17$ \( T^{8} + 165333028 T^{7} + \cdots + 17\!\cdots\!25 \) Copy content Toggle raw display
$19$ \( T^{8} + 423405794 T^{7} + \cdots + 13\!\cdots\!25 \) Copy content Toggle raw display
$23$ \( T^{8} - 286233866 T^{7} + \cdots + 17\!\cdots\!81 \) Copy content Toggle raw display
$29$ \( (T^{4} - 9100337408 T^{3} + \cdots + 47\!\cdots\!28)^{2} \) Copy content Toggle raw display
$31$ \( T^{8} - 1507094246 T^{7} + \cdots + 34\!\cdots\!25 \) Copy content Toggle raw display
$37$ \( T^{8} + 18959705336 T^{7} + \cdots + 55\!\cdots\!61 \) Copy content Toggle raw display
$41$ \( (T^{4} - 56825955312 T^{3} + \cdots - 10\!\cdots\!08)^{2} \) Copy content Toggle raw display
$43$ \( (T^{4} + 45778809712 T^{3} + \cdots - 18\!\cdots\!00)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} + 81351201078 T^{7} + \cdots + 82\!\cdots\!25 \) Copy content Toggle raw display
$53$ \( T^{8} - 87497947440 T^{7} + \cdots + 64\!\cdots\!41 \) Copy content Toggle raw display
$59$ \( T^{8} + 194140265102 T^{7} + \cdots + 14\!\cdots\!25 \) Copy content Toggle raw display
$61$ \( T^{8} - 175816313120 T^{7} + \cdots + 12\!\cdots\!25 \) Copy content Toggle raw display
$67$ \( T^{8} + 243815218758 T^{7} + \cdots + 98\!\cdots\!25 \) Copy content Toggle raw display
$71$ \( (T^{4} + 1637697339536 T^{3} + \cdots - 79\!\cdots\!40)^{2} \) Copy content Toggle raw display
$73$ \( T^{8} + 3492491920596 T^{7} + \cdots + 21\!\cdots\!25 \) Copy content Toggle raw display
$79$ \( T^{8} + 1016380081246 T^{7} + \cdots + 33\!\cdots\!25 \) Copy content Toggle raw display
$83$ \( (T^{4} - 3513747871648 T^{3} + \cdots + 13\!\cdots\!00)^{2} \) Copy content Toggle raw display
$89$ \( T^{8} + 8034124428036 T^{7} + \cdots + 62\!\cdots\!25 \) Copy content Toggle raw display
$97$ \( (T^{4} - 27175565862816 T^{3} + \cdots - 27\!\cdots\!00)^{2} \) Copy content Toggle raw display
show more
show less