Properties

Label 7.14.c.a
Level $7$
Weight $14$
Character orbit 7.c
Analytic conductor $7.506$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $2$

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Newspace parameters

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

Newform invariants

Self dual: no
Analytic conductor: \(7.50616502663\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
Defining polynomial: \( x^{16} - x^{15} + 12266 x^{14} - 55087 x^{13} + 107030285 x^{12} - 640627660 x^{11} + 438282746176 x^{10} - 5859940491984 x^{9} + \cdots + 12\!\cdots\!76 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{34}\cdot 3^{7}\cdot 7^{11} \)
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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} + ( - \beta_{4} + 91 \beta_{2}) q^{3} + (\beta_{8} + \beta_{4} - 6 \beta_{3} - 4075 \beta_{2} - 6 \beta_1) q^{4} + (\beta_{8} - \beta_{7} + 6 \beta_{6} + \beta_{5} - 5864 \beta_{2} + 8 \beta_1 + 5864) q^{5} + ( - \beta_{11} - \beta_{9} + 24 \beta_{6} + \beta_{5} - 24 \beta_{4} + 205 \beta_{3} + \cdots - 1837) q^{6}+ \cdots + (\beta_{15} - 2 \beta_{14} - \beta_{13} + 2 \beta_{12} - 3 \beta_{11} - 3 \beta_{10} + \cdots - 364763) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{2} + ( - \beta_{4} + 91 \beta_{2}) q^{3} + (\beta_{8} + \beta_{4} - 6 \beta_{3} - 4075 \beta_{2} - 6 \beta_1) q^{4} + (\beta_{8} - \beta_{7} + 6 \beta_{6} + \beta_{5} - 5864 \beta_{2} + 8 \beta_1 + 5864) q^{5} + ( - \beta_{11} - \beta_{9} + 24 \beta_{6} + \beta_{5} - 24 \beta_{4} + 205 \beta_{3} + \cdots - 1837) q^{6}+ \cdots + (3888775 \beta_{15} + 10992685 \beta_{14} + \cdots + 116618821539) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 2 q^{2} + 728 q^{3} - 32588 q^{4} + 46928 q^{5} - 30212 q^{6} + 257992 q^{7} - 1093248 q^{8} - 2910808 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 2 q^{2} + 728 q^{3} - 32588 q^{4} + 46928 q^{5} - 30212 q^{6} + 257992 q^{7} - 1093248 q^{8} - 2910808 q^{9} + 879018 q^{10} + 3746000 q^{11} + 13786276 q^{12} - 63556528 q^{13} - 45497410 q^{14} + 179729504 q^{15} - 158020976 q^{16} + 108548496 q^{17} + 358872748 q^{18} + 617001728 q^{19} - 1844979416 q^{20} + 400815352 q^{21} + 977603772 q^{22} - 274288968 q^{23} - 690136272 q^{24} + 342571888 q^{25} - 1890370748 q^{26} + 1818768224 q^{27} - 2830532012 q^{28} - 1243557136 q^{29} + 8470546342 q^{30} + 1983953552 q^{31} - 18584336224 q^{32} - 3534705832 q^{33} + 35123983356 q^{34} - 15057013664 q^{35} + 2810399728 q^{36} + 3822948680 q^{37} + 48271561702 q^{38} + 5559167432 q^{39} + 20095902624 q^{40} - 186830117264 q^{41} - 160102538512 q^{42} + 147613206080 q^{43} + 146754773860 q^{44} + 167965018760 q^{45} + 33234006318 q^{46} + 136146541776 q^{47} - 918554738528 q^{48} - 279115649456 q^{49} + 446501048144 q^{50} + 374908120992 q^{51} + 920059547960 q^{52} - 365779470792 q^{53} + 142841959918 q^{54} - 335157159792 q^{55} - 2232488591424 q^{56} + 1368890666384 q^{57} + 1968765302052 q^{58} + 1211924963928 q^{59} - 1696452613604 q^{60} - 155263824184 q^{61} - 3267358992324 q^{62} - 4123328121080 q^{63} + 3671045348608 q^{64} + 2491203210104 q^{65} + 6785854381222 q^{66} - 1812819047992 q^{67} + 2300642770860 q^{68} - 9833331511584 q^{69} - 10764563120010 q^{70} + 6413635527072 q^{71} + 9182131417728 q^{72} + 7135229075576 q^{73} - 1240472589270 q^{74} + 3092701132528 q^{75} - 24098923346008 q^{76} - 8899938075392 q^{77} + 16977900019528 q^{78} + 7912090175960 q^{79} + 18767635478704 q^{80} - 2029193868592 q^{81} + 2118830761068 q^{82} - 16937841300576 q^{83} - 32302920082804 q^{84} + 9546866646768 q^{85} + 17052910238024 q^{86} + 16147535411672 q^{87} - 15433836932304 q^{88} + 3654589406888 q^{89} - 6354662390216 q^{90} - 14961004874704 q^{91} + 2966525661672 q^{92} + 10829807426472 q^{93} + 7135101410430 q^{94} - 16751540432680 q^{95} - 4043645970784 q^{96} + 20623672673744 q^{97} + 1374873929086 q^{98} + 1803523183088 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - x^{15} + 12266 x^{14} - 55087 x^{13} + 107030285 x^{12} - 640627660 x^{11} + 438282746176 x^{10} - 5859940491984 x^{9} + \cdots + 12\!\cdots\!76 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( 2\nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 41\!\cdots\!99 \nu^{15} + \cdots + 97\!\cdots\!28 ) / 54\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 95\!\cdots\!31 \nu^{15} + \cdots + 49\!\cdots\!68 ) / 24\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 43\!\cdots\!57 \nu^{15} + \cdots + 89\!\cdots\!28 ) / 86\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 14\!\cdots\!97 \nu^{15} + \cdots - 13\!\cdots\!24 ) / 12\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 32\!\cdots\!67 \nu^{15} + \cdots + 81\!\cdots\!44 ) / 17\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 11\!\cdots\!79 \nu^{15} + \cdots - 97\!\cdots\!64 ) / 27\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 31\!\cdots\!95 \nu^{15} + \cdots - 24\!\cdots\!44 ) / 34\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 15\!\cdots\!87 \nu^{15} + \cdots - 25\!\cdots\!40 ) / 60\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 12\!\cdots\!47 \nu^{15} + \cdots + 18\!\cdots\!72 ) / 28\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 35\!\cdots\!73 \nu^{15} + \cdots + 10\!\cdots\!92 ) / 31\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 57\!\cdots\!99 \nu^{15} + \cdots + 31\!\cdots\!16 ) / 38\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 14\!\cdots\!73 \nu^{15} + \cdots - 47\!\cdots\!04 ) / 73\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 18\!\cdots\!37 \nu^{15} + \cdots - 18\!\cdots\!60 ) / 57\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 27\!\cdots\!21 \nu^{15} + \cdots - 11\!\cdots\!96 ) / 34\!\cdots\!80 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{8} + \beta_{4} - 6\beta_{3} - 12267\beta_{2} - 6\beta_1 ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( \beta_{14} + \beta_{13} + 2 \beta_{12} - 2 \beta_{11} + 2 \beta_{10} - \beta_{9} - \beta_{8} + 39 \beta_{6} - 23 \beta_{5} - 41 \beta_{4} + 20716 \beta_{3} + \beta _1 + 69411 ) / 8 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 13 \beta_{15} + 26 \beta_{14} + 8 \beta_{13} - 11 \beta_{12} - 14 \beta_{11} - 9 \beta_{10} - 5 \beta_{9} - 15125 \beta_{8} + 721 \beta_{7} - 27491 \beta_{6} - 15104 \beta_{5} - 5 \beta_{4} + 13 \beta_{3} + 127041607 \beta_{2} + \cdots - 127041615 ) / 8 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 3869 \beta_{15} + 4973 \beta_{14} - 4145 \beta_{13} - 3317 \beta_{12} + 552 \beta_{11} - 15129 \beta_{10} - 3366 \beta_{9} - 135815 \beta_{8} + 7235 \beta_{7} - 552 \beta_{6} + 3869 \beta_{5} + 200717 \beta_{4} + \cdots + 276 ) / 4 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 63872 \beta_{15} - 361243 \beta_{14} - 233499 \beta_{13} - 339254 \beta_{12} - 179810 \beta_{11} - 466998 \beta_{10} - 6667885 \beta_{9} + 233499 \beta_{8} - 127744 \beta_{7} + \cdots + 780761183467 ) / 8 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 59473441 \beta_{15} - 118946882 \beta_{14} + 5488584 \beta_{13} - 75939193 \beta_{12} + 201794270 \beta_{11} + 136832245 \beta_{10} + 64962025 \beta_{9} + \cdots + 4956412440471 ) / 8 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 1029809010 \beta_{15} - 131765570 \beta_{14} + 805298150 \beta_{13} + 1478830730 \beta_{12} + 449021720 \beta_{11} + 873075648 \beta_{10} + 26612782350 \beta_{9} + \cdots + 224510860 ) / 4 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 42528388464 \beta_{15} + 289403197201 \beta_{14} + 374459974129 \beta_{13} + 833976725186 \beta_{12} - 1634182209362 \beta_{11} + 748919948258 \beta_{10} + \cdots - 67\!\cdots\!09 ) / 8 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 13683162391681 \beta_{15} + 27366324783362 \beta_{14} + 3082784083016 \beta_{13} + 4434810142633 \beta_{12} + 10435605495658 \beta_{11} + \cdots - 34\!\cdots\!31 ) / 8 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 13\!\cdots\!05 \beta_{15} + \cdots + 152105770167396 ) / 4 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( - 20\!\cdots\!32 \beta_{15} + \cdots + 23\!\cdots\!27 ) / 8 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 20\!\cdots\!65 \beta_{15} + \cdots + 60\!\cdots\!71 ) / 8 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( 50\!\cdots\!94 \beta_{15} + \cdots + 69\!\cdots\!92 ) / 4 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( - 14\!\cdots\!64 \beta_{15} + \cdots - 50\!\cdots\!73 ) / 8 \) Copy content Toggle raw display

Character values

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

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

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2.1
42.7191 73.9916i
24.0187 41.6016i
20.8043 36.0341i
12.1857 21.1063i
−6.80614 + 11.7886i
−18.3288 + 31.7465i
−33.6943 + 58.3603i
−40.3985 + 69.9723i
42.7191 + 73.9916i
24.0187 + 41.6016i
20.8043 + 36.0341i
12.1857 + 21.1063i
−6.80614 11.7886i
−18.3288 31.7465i
−33.6943 58.3603i
−40.3985 69.9723i
−85.4382 + 147.983i 275.475 + 477.137i −10503.4 18192.4i 18457.8 31969.8i −94144.4 −296175. + 95755.5i 2.18974e6 645388. 1.11785e6i 3.15400e6 + 5.46289e6i
2.2 −48.0374 + 83.2032i 45.1151 + 78.1417i −519.184 899.253i −28325.9 + 49061.9i −8668.85 52177.1 306866.i −687284. 793091. 1.37367e6i −2.72141e6 4.71362e6i
2.3 −41.6086 + 72.0682i −951.376 1647.83i 633.448 + 1097.16i 8420.78 14585.2i 158342. 92983.2 + 297057.i −787143. −1.01307e6 + 1.75469e6i 700754. + 1.21374e6i
2.4 −24.3715 + 42.2127i 1099.13 + 1903.74i 2908.06 + 5036.91i 12992.6 22503.8i −107149. 282613. + 130456.i −682798. −1.61900e6 + 2.80418e6i 633297. + 1.09690e6i
2.5 13.6123 23.5772i −273.794 474.225i 3725.41 + 6452.60i 16862.5 29206.7i −14907.8 −174823. 257538.i 425869. 647235. 1.12104e6i −459074. 795140.i
2.6 36.6577 63.4930i 306.948 + 531.649i 1408.43 + 2439.47i −16706.4 + 28936.3i 45008.0 −98806.7 + 295172.i 807118. 608728. 1.05435e6i 1.22484e6 + 2.12148e6i
2.7 67.3886 116.721i −948.121 1642.19i −4986.46 8636.79i −7164.19 + 12408.7i −255570. 311248. 3719.03i −240026. −1.00071e6 + 1.73328e6i 965570. + 1.67242e6i
2.8 80.7971 139.945i 810.627 + 1404.05i −8960.34 15519.8i 18926.8 32782.3i 261985. −40219.7 308661.i −1.57210e6 −517071. + 895593.i −3.05847e6 5.29742e6i
4.1 −85.4382 147.983i 275.475 477.137i −10503.4 + 18192.4i 18457.8 + 31969.8i −94144.4 −296175. 95755.5i 2.18974e6 645388. + 1.11785e6i 3.15400e6 5.46289e6i
4.2 −48.0374 83.2032i 45.1151 78.1417i −519.184 + 899.253i −28325.9 49061.9i −8668.85 52177.1 + 306866.i −687284. 793091. + 1.37367e6i −2.72141e6 + 4.71362e6i
4.3 −41.6086 72.0682i −951.376 + 1647.83i 633.448 1097.16i 8420.78 + 14585.2i 158342. 92983.2 297057.i −787143. −1.01307e6 1.75469e6i 700754. 1.21374e6i
4.4 −24.3715 42.2127i 1099.13 1903.74i 2908.06 5036.91i 12992.6 + 22503.8i −107149. 282613. 130456.i −682798. −1.61900e6 2.80418e6i 633297. 1.09690e6i
4.5 13.6123 + 23.5772i −273.794 + 474.225i 3725.41 6452.60i 16862.5 + 29206.7i −14907.8 −174823. + 257538.i 425869. 647235. + 1.12104e6i −459074. + 795140.i
4.6 36.6577 + 63.4930i 306.948 531.649i 1408.43 2439.47i −16706.4 28936.3i 45008.0 −98806.7 295172.i 807118. 608728. + 1.05435e6i 1.22484e6 2.12148e6i
4.7 67.3886 + 116.721i −948.121 + 1642.19i −4986.46 + 8636.79i −7164.19 12408.7i −255570. 311248. + 3719.03i −240026. −1.00071e6 1.73328e6i 965570. 1.67242e6i
4.8 80.7971 + 139.945i 810.627 1404.05i −8960.34 + 15519.8i 18926.8 + 32782.3i 261985. −40219.7 + 308661.i −1.57210e6 −517071. 895593.i −3.05847e6 + 5.29742e6i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 4.8
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 7.14.c.a 16
3.b odd 2 1 63.14.e.c 16
7.b odd 2 1 49.14.c.g 16
7.c even 3 1 inner 7.14.c.a 16
7.c even 3 1 49.14.a.e 8
7.d odd 6 1 49.14.a.f 8
7.d odd 6 1 49.14.c.g 16
21.h odd 6 1 63.14.e.c 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
7.14.c.a 16 1.a even 1 1 trivial
7.14.c.a 16 7.c even 3 1 inner
49.14.a.e 8 7.c even 3 1
49.14.a.f 8 7.d odd 6 1
49.14.c.g 16 7.b odd 2 1
49.14.c.g 16 7.d odd 6 1
63.14.e.c 16 3.b odd 2 1
63.14.e.c 16 21.h odd 6 1

Hecke kernels

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} + 2 T^{15} + \cdots + 83\!\cdots\!36 \) Copy content Toggle raw display
$3$ \( T^{16} - 728 T^{15} + \cdots + 46\!\cdots\!21 \) Copy content Toggle raw display
$5$ \( T^{16} - 46928 T^{15} + \cdots + 31\!\cdots\!25 \) Copy content Toggle raw display
$7$ \( T^{16} - 257992 T^{15} + \cdots + 77\!\cdots\!01 \) Copy content Toggle raw display
$11$ \( T^{16} - 3746000 T^{15} + \cdots + 43\!\cdots\!25 \) Copy content Toggle raw display
$13$ \( (T^{8} + 31778264 T^{7} + \cdots - 26\!\cdots\!64)^{2} \) Copy content Toggle raw display
$17$ \( T^{16} - 108548496 T^{15} + \cdots + 36\!\cdots\!61 \) Copy content Toggle raw display
$19$ \( T^{16} - 617001728 T^{15} + \cdots + 14\!\cdots\!29 \) Copy content Toggle raw display
$23$ \( T^{16} + 274288968 T^{15} + \cdots + 16\!\cdots\!61 \) Copy content Toggle raw display
$29$ \( (T^{8} + 621778568 T^{7} + \cdots - 85\!\cdots\!00)^{2} \) Copy content Toggle raw display
$31$ \( T^{16} - 1983953552 T^{15} + \cdots + 15\!\cdots\!41 \) Copy content Toggle raw display
$37$ \( T^{16} - 3822948680 T^{15} + \cdots + 53\!\cdots\!25 \) Copy content Toggle raw display
$41$ \( (T^{8} + 93415058632 T^{7} + \cdots + 45\!\cdots\!00)^{2} \) Copy content Toggle raw display
$43$ \( (T^{8} - 73806603040 T^{7} + \cdots - 51\!\cdots\!92)^{2} \) Copy content Toggle raw display
$47$ \( T^{16} - 136146541776 T^{15} + \cdots + 99\!\cdots\!25 \) Copy content Toggle raw display
$53$ \( T^{16} + 365779470792 T^{15} + \cdots + 68\!\cdots\!69 \) Copy content Toggle raw display
$59$ \( T^{16} - 1211924963928 T^{15} + \cdots + 20\!\cdots\!25 \) Copy content Toggle raw display
$61$ \( T^{16} + 155263824184 T^{15} + \cdots + 49\!\cdots\!09 \) Copy content Toggle raw display
$67$ \( T^{16} + 1812819047992 T^{15} + \cdots + 49\!\cdots\!25 \) Copy content Toggle raw display
$71$ \( (T^{8} - 3206817763536 T^{7} + \cdots + 83\!\cdots\!76)^{2} \) Copy content Toggle raw display
$73$ \( T^{16} - 7135229075576 T^{15} + \cdots + 11\!\cdots\!41 \) Copy content Toggle raw display
$79$ \( T^{16} - 7912090175960 T^{15} + \cdots + 26\!\cdots\!21 \) Copy content Toggle raw display
$83$ \( (T^{8} + 8468920650288 T^{7} + \cdots - 22\!\cdots\!24)^{2} \) Copy content Toggle raw display
$89$ \( T^{16} - 3654589406888 T^{15} + \cdots + 37\!\cdots\!09 \) Copy content Toggle raw display
$97$ \( (T^{8} - 10311836336872 T^{7} + \cdots + 40\!\cdots\!36)^{2} \) Copy content Toggle raw display
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