Properties

Label 7.12.c.a
Level $7$
Weight $12$
Character orbit 7.c
Analytic conductor $5.378$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $2$

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Newspace parameters

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

Newform invariants

Self dual: no
Analytic conductor: \(5.37840226392\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(6\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
Defining polynomial: \( x^{12} - x^{11} + 1846 x^{10} + 9475 x^{9} + 2735534 x^{8} + 11305015 x^{7} + 1247863105 x^{6} + 8299724444 x^{5} + 456730004116 x^{4} + \cdots + 4089842896896 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 2^{20}\cdot 3^{3}\cdot 7^{6} \)
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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - 4 \beta_{2} - \beta_1 + 4) q^{2} + (\beta_{6} + \beta_{3} - 40 \beta_{2} + \beta_1) q^{3} + ( - \beta_{7} + \beta_{3} - 426 \beta_{2} + \beta_1) q^{4} + (\beta_{11} + \beta_{9} + \beta_{6} - \beta_{5} + 1466 \beta_{2} + 5 \beta_1 - 1466) q^{5} + ( - \beta_{9} + \beta_{8} + 8 \beta_{5} - 2 \beta_{4} - 53 \beta_{3} + 3164) q^{6} + ( - 2 \beta_{11} - \beta_{10} + \beta_{9} + 3 \beta_{8} + 5 \beta_{7} - 17 \beta_{6} + \cdots + 568) q^{7}+ \cdots + (26 \beta_{11} - \beta_{10} + 26 \beta_{9} + \beta_{8} - 29 \beta_{7} - 59 \beta_{6} + \cdots - 28783) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - 4 \beta_{2} - \beta_1 + 4) q^{2} + (\beta_{6} + \beta_{3} - 40 \beta_{2} + \beta_1) q^{3} + ( - \beta_{7} + \beta_{3} - 426 \beta_{2} + \beta_1) q^{4} + (\beta_{11} + \beta_{9} + \beta_{6} - \beta_{5} + 1466 \beta_{2} + 5 \beta_1 - 1466) q^{5} + ( - \beta_{9} + \beta_{8} + 8 \beta_{5} - 2 \beta_{4} - 53 \beta_{3} + 3164) q^{6} + ( - 2 \beta_{11} - \beta_{10} + \beta_{9} + 3 \beta_{8} + 5 \beta_{7} - 17 \beta_{6} + \cdots + 568) q^{7}+ \cdots + ( - 615584 \beta_{9} + 2468405 \beta_{8} + \cdots + 30784020661) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 22 q^{2} - 244 q^{3} - 2556 q^{4} - 8782 q^{5} + 38140 q^{6} - 504 q^{7} + 97008 q^{8} - 172348 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 22 q^{2} - 244 q^{3} - 2556 q^{4} - 8782 q^{5} + 38140 q^{6} - 504 q^{7} + 97008 q^{8} - 172348 q^{9} + 111546 q^{10} - 1001572 q^{11} - 173684 q^{12} + 3864504 q^{13} - 1302994 q^{14} - 1286512 q^{15} + 39120 q^{16} - 6704802 q^{17} + 3768052 q^{18} + 4192212 q^{19} - 17646776 q^{20} + 44745358 q^{21} - 8505684 q^{22} - 33871872 q^{23} - 15734760 q^{24} + 13695456 q^{25} + 29350300 q^{26} + 73859384 q^{27} + 181285692 q^{28} - 255125224 q^{29} - 336068498 q^{30} - 331783920 q^{31} + 163252640 q^{32} - 80899438 q^{33} + 1853334396 q^{34} + 1407354844 q^{35} - 2197352048 q^{36} - 833082774 q^{37} - 2086458338 q^{38} + 737605904 q^{39} - 1219023432 q^{40} + 3104076808 q^{41} + 5982095000 q^{42} - 1722177552 q^{43} - 5105122436 q^{44} - 7406493484 q^{45} + 3435559326 q^{46} - 1327587552 q^{47} + 14535793696 q^{48} + 11976558636 q^{49} - 12237094384 q^{50} - 13921261140 q^{51} - 17237001432 q^{52} + 6725755626 q^{53} - 1674595226 q^{54} + 26323921200 q^{55} + 28163516640 q^{56} - 16884487756 q^{57} - 20073189204 q^{58} - 26237179548 q^{59} + 13611677716 q^{60} - 14411013726 q^{61} + 46185665964 q^{62} + 45955779184 q^{63} - 46365999744 q^{64} - 16224702172 q^{65} - 40938633602 q^{66} - 4241860068 q^{67} - 6528332916 q^{68} + 46750854252 q^{69} + 55705143270 q^{70} - 37335334656 q^{71} - 30237166608 q^{72} + 6005568990 q^{73} + 21663581922 q^{74} + 17116276792 q^{75} - 28817353320 q^{76} - 51928077698 q^{77} + 42636498520 q^{78} + 11712395640 q^{79} + 41748525232 q^{80} - 12455008366 q^{81} + 52795921668 q^{82} - 100821781200 q^{83} - 155783007940 q^{84} + 138884613396 q^{85} + 110437384472 q^{86} + 119455310144 q^{87} - 105039956616 q^{88} - 48633519778 q^{89} - 361508123864 q^{90} - 160908361488 q^{91} + 242956324248 q^{92} + 266530114134 q^{93} + 368497095702 q^{94} - 72161225128 q^{95} + 124456168928 q^{96} - 401308415928 q^{97} - 582375436706 q^{98} + 367357472240 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - x^{11} + 1846 x^{10} + 9475 x^{9} + 2735534 x^{8} + 11305015 x^{7} + 1247863105 x^{6} + 8299724444 x^{5} + 456730004116 x^{4} + \cdots + 4089842896896 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( 2\nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 25\!\cdots\!01 \nu^{11} + \cdots + 15\!\cdots\!84 ) / 12\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 77\!\cdots\!61 \nu^{11} + \cdots + 46\!\cdots\!24 ) / 28\!\cdots\!70 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 13\!\cdots\!27 \nu^{11} + \cdots + 69\!\cdots\!32 ) / 28\!\cdots\!70 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 65\!\cdots\!69 \nu^{11} + \cdots - 13\!\cdots\!04 ) / 18\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 57\!\cdots\!71 \nu^{11} + \cdots - 34\!\cdots\!04 ) / 39\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 44\!\cdots\!19 \nu^{11} + \cdots - 26\!\cdots\!96 ) / 91\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 15\!\cdots\!51 \nu^{11} + \cdots - 99\!\cdots\!76 ) / 18\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 25\!\cdots\!37 \nu^{11} + \cdots - 22\!\cdots\!12 ) / 27\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 32\!\cdots\!81 \nu^{11} + \cdots - 19\!\cdots\!44 ) / 27\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 55\!\cdots\!89 \nu^{11} + \cdots + 32\!\cdots\!96 ) / 29\!\cdots\!80 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{7} + 9\beta_{3} - 2458\beta_{2} + 9\beta_1 ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 3\beta_{9} - 5\beta_{8} + 6\beta_{5} + 7\beta_{4} + 4515\beta_{3} - 21136 ) / 8 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 61 \beta_{11} + 57 \beta_{10} - 61 \beta_{9} - 57 \beta_{8} + 2825 \beta_{7} - 3154 \beta_{6} + 3154 \beta_{5} + 2825 \beta_{4} + 5550640 \beta_{2} - 35581 \beta _1 - 5550640 ) / 8 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 2911 \beta_{11} + 4623 \beta_{10} + 10791 \beta_{7} - 12584 \beta_{6} - 2878987 \beta_{3} + 21861490 \beta_{2} - 2878987 \beta_1 ) / 4 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 47639\beta_{9} + 66743\beta_{8} - 2933594\beta_{5} - 1965530\beta_{4} - 32458762\beta_{3} + 3540780858 ) / 4 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 8419569 \beta_{11} - 14086431 \beta_{10} - 8419569 \beta_{9} + 14086431 \beta_{8} - 44000517 \beta_{7} + 60346602 \beta_{6} - 60346602 \beta_{5} - 44000517 \beta_{4} + \cdots + 80439326472 ) / 8 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 113423139 \beta_{11} - 243070119 \beta_{10} - 5476828259 \beta_{7} + 8867314350 \beta_{6} + 112554916767 \beta_{3} - 9551328686408 \beta_{2} + 112554916767 \beta_1 ) / 8 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 2840135421 \beta_{9} - 5104667528 \beta_{8} + 29146539381 \beta_{5} + 19749647785 \beta_{4} + 2685142522545 \beta_{3} - 34892880338323 ) / 2 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 59715637874 \beta_{11} + 203313584598 \beta_{10} - 59715637874 \beta_{9} - 203313584598 \beta_{8} + 3834814145287 \beta_{7} - 6363463903544 \beta_{6} + \cdots - 66\!\cdots\!18 ) / 4 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 15079829713775 \beta_{11} + 29232728119641 \beta_{10} + 133480919011635 \beta_{7} - 204263495750974 \beta_{6} + \cdots - 15\!\cdots\!11 \beta_1 ) / 8 \) 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
19.2454 33.3340i
11.5012 19.9207i
−0.427084 + 0.739732i
−1.81957 + 3.15159i
−10.4934 + 18.1751i
−17.5066 + 30.3223i
19.2454 + 33.3340i
11.5012 + 19.9207i
−0.427084 0.739732i
−1.81957 3.15159i
−10.4934 18.1751i
−17.5066 30.3223i
−36.4908 + 63.2040i −167.747 290.547i −1639.16 2839.11i −539.162 + 933.856i 24484.9 44465.4 388.777i 89790.9 32295.3 55937.0i −39348.9 68154.3i
2.2 −21.0025 + 36.3774i 205.214 + 355.441i 141.791 + 245.589i −41.2685 + 71.4792i −17240.0 −43319.9 + 10035.5i −97938.0 4347.82 7530.65i −1733.48 3002.48i
2.3 2.85417 4.94356i −366.163 634.213i 1007.71 + 1745.40i −4491.11 + 7778.83i −4180.36 −43014.2 11274.1i 23195.3 −179577. + 311037.i 25636.8 + 44404.2i
2.4 5.63915 9.76729i −23.0866 39.9871i 960.400 + 1663.46i 4304.93 7456.37i −520.754 44340.4 3354.68i 44761.3 87507.5 151567.i −48552.3 84095.1i
2.5 22.9868 39.8144i 311.801 + 540.056i −32.7899 56.7937i −5294.67 + 9170.64i 28669.3 23661.4 37649.3i 91139.2 −105867. + 183367.i 243416. + 421608.i
2.6 37.0132 64.1087i −82.0189 142.061i −1715.95 2972.11i 1670.28 2893.01i −12143.1 −26385.1 + 35793.2i −102445. 75119.3 130110.i −123645. 214159.i
4.1 −36.4908 63.2040i −167.747 + 290.547i −1639.16 + 2839.11i −539.162 933.856i 24484.9 44465.4 + 388.777i 89790.9 32295.3 + 55937.0i −39348.9 + 68154.3i
4.2 −21.0025 36.3774i 205.214 355.441i 141.791 245.589i −41.2685 71.4792i −17240.0 −43319.9 10035.5i −97938.0 4347.82 + 7530.65i −1733.48 + 3002.48i
4.3 2.85417 + 4.94356i −366.163 + 634.213i 1007.71 1745.40i −4491.11 7778.83i −4180.36 −43014.2 + 11274.1i 23195.3 −179577. 311037.i 25636.8 44404.2i
4.4 5.63915 + 9.76729i −23.0866 + 39.9871i 960.400 1663.46i 4304.93 + 7456.37i −520.754 44340.4 + 3354.68i 44761.3 87507.5 + 151567.i −48552.3 + 84095.1i
4.5 22.9868 + 39.8144i 311.801 540.056i −32.7899 + 56.7937i −5294.67 9170.64i 28669.3 23661.4 + 37649.3i 91139.2 −105867. 183367.i 243416. 421608.i
4.6 37.0132 + 64.1087i −82.0189 + 142.061i −1715.95 + 2972.11i 1670.28 + 2893.01i −12143.1 −26385.1 35793.2i −102445. 75119.3 + 130110.i −123645. + 214159.i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 4.6
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 7.12.c.a 12
3.b odd 2 1 63.12.e.b 12
4.b odd 2 1 112.12.i.c 12
7.b odd 2 1 49.12.c.i 12
7.c even 3 1 inner 7.12.c.a 12
7.c even 3 1 49.12.a.g 6
7.d odd 6 1 49.12.a.f 6
7.d odd 6 1 49.12.c.i 12
21.h odd 6 1 63.12.e.b 12
28.g odd 6 1 112.12.i.c 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
7.12.c.a 12 1.a even 1 1 trivial
7.12.c.a 12 7.c even 3 1 inner
49.12.a.f 6 7.d odd 6 1
49.12.a.g 6 7.c even 3 1
49.12.c.i 12 7.b odd 2 1
49.12.c.i 12 7.d odd 6 1
63.12.e.b 12 3.b odd 2 1
63.12.e.b 12 21.h odd 6 1
112.12.i.c 12 4.b odd 2 1
112.12.i.c 12 28.g odd 6 1

Hecke kernels

This newform subspace is the entire newspace \(S_{12}^{\mathrm{new}}(7, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} - 22 T^{11} + \cdots + 45\!\cdots\!44 \) Copy content Toggle raw display
$3$ \( T^{12} + 244 T^{11} + \cdots + 22\!\cdots\!09 \) Copy content Toggle raw display
$5$ \( T^{12} + 8782 T^{11} + \cdots + 59\!\cdots\!25 \) Copy content Toggle raw display
$7$ \( T^{12} + 504 T^{11} + \cdots + 59\!\cdots\!49 \) Copy content Toggle raw display
$11$ \( T^{12} + 1001572 T^{11} + \cdots + 14\!\cdots\!25 \) Copy content Toggle raw display
$13$ \( (T^{6} - 1932252 T^{5} + \cdots + 32\!\cdots\!44)^{2} \) Copy content Toggle raw display
$17$ \( T^{12} + 6704802 T^{11} + \cdots + 29\!\cdots\!69 \) Copy content Toggle raw display
$19$ \( T^{12} - 4192212 T^{11} + \cdots + 41\!\cdots\!21 \) Copy content Toggle raw display
$23$ \( T^{12} + 33871872 T^{11} + \cdots + 77\!\cdots\!09 \) Copy content Toggle raw display
$29$ \( (T^{6} + 127562612 T^{5} + \cdots + 28\!\cdots\!00)^{2} \) Copy content Toggle raw display
$31$ \( T^{12} + 331783920 T^{11} + \cdots + 18\!\cdots\!01 \) Copy content Toggle raw display
$37$ \( T^{12} + 833082774 T^{11} + \cdots + 14\!\cdots\!25 \) Copy content Toggle raw display
$41$ \( (T^{6} - 1552038404 T^{5} + \cdots - 95\!\cdots\!00)^{2} \) Copy content Toggle raw display
$43$ \( (T^{6} + 861088776 T^{5} + \cdots - 24\!\cdots\!68)^{2} \) Copy content Toggle raw display
$47$ \( T^{12} + 1327587552 T^{11} + \cdots + 59\!\cdots\!25 \) Copy content Toggle raw display
$53$ \( T^{12} - 6725755626 T^{11} + \cdots + 22\!\cdots\!01 \) Copy content Toggle raw display
$59$ \( T^{12} + 26237179548 T^{11} + \cdots + 24\!\cdots\!25 \) Copy content Toggle raw display
$61$ \( T^{12} + 14411013726 T^{11} + \cdots + 19\!\cdots\!69 \) Copy content Toggle raw display
$67$ \( T^{12} + 4241860068 T^{11} + \cdots + 93\!\cdots\!25 \) Copy content Toggle raw display
$71$ \( (T^{6} + 18667667328 T^{5} + \cdots + 10\!\cdots\!44)^{2} \) Copy content Toggle raw display
$73$ \( T^{12} - 6005568990 T^{11} + \cdots + 13\!\cdots\!89 \) Copy content Toggle raw display
$79$ \( T^{12} - 11712395640 T^{11} + \cdots + 20\!\cdots\!09 \) Copy content Toggle raw display
$83$ \( (T^{6} + 50410890600 T^{5} + \cdots - 24\!\cdots\!56)^{2} \) Copy content Toggle raw display
$89$ \( T^{12} + 48633519778 T^{11} + \cdots + 95\!\cdots\!41 \) Copy content Toggle raw display
$97$ \( (T^{6} + 200654207964 T^{5} + \cdots - 18\!\cdots\!64)^{2} \) Copy content Toggle raw display
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