Properties

Label 7.6.c.a
Level 7
Weight 6
Character orbit 7.c
Analytic conductor 1.123
Analytic rank 0
Dimension 4
CM no
Inner twists 2

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Newspace parameters

Level: \( N \) \(=\) \( 7 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 7.c (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(1.12268673869\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{37})\)
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 2^{2} \)
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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( -\beta_{1} - \beta_{2} ) q^{2} + ( 4 - 4 \beta_{1} - \beta_{2} - \beta_{3} ) q^{3} + ( -6 + 6 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} ) q^{4} + ( 19 \beta_{1} + 10 \beta_{2} ) q^{5} + ( -41 + 5 \beta_{3} ) q^{6} + ( -14 - 56 \beta_{1} - 14 \beta_{2} - 21 \beta_{3} ) q^{7} + ( 48 + 24 \beta_{3} ) q^{8} + ( 190 \beta_{1} - 8 \beta_{2} ) q^{9} +O(q^{10})\) \( q + ( -\beta_{1} - \beta_{2} ) q^{2} + ( 4 - 4 \beta_{1} - \beta_{2} - \beta_{3} ) q^{3} + ( -6 + 6 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} ) q^{4} + ( 19 \beta_{1} + 10 \beta_{2} ) q^{5} + ( -41 + 5 \beta_{3} ) q^{6} + ( -14 - 56 \beta_{1} - 14 \beta_{2} - 21 \beta_{3} ) q^{7} + ( 48 + 24 \beta_{3} ) q^{8} + ( 190 \beta_{1} - 8 \beta_{2} ) q^{9} + ( 389 - 389 \beta_{1} - 29 \beta_{2} - 29 \beta_{3} ) q^{10} + ( -212 + 212 \beta_{1} + 23 \beta_{2} + 23 \beta_{3} ) q^{11} + ( 98 \beta_{1} + 14 \beta_{2} ) q^{12} + ( -462 - 28 \beta_{3} ) q^{13} + ( -574 - 189 \beta_{1} + 63 \beta_{2} + 70 \beta_{3} ) q^{14} + ( 446 - 59 \beta_{3} ) q^{15} + ( 1032 \beta_{1} + 40 \beta_{2} ) q^{16} + ( 1173 - 1173 \beta_{1} + 132 \beta_{2} + 132 \beta_{3} ) q^{17} + ( -106 + 106 \beta_{1} - 182 \beta_{2} - 182 \beta_{3} ) q^{18} + ( 180 \beta_{1} - 277 \beta_{2} ) q^{19} + ( -854 + 98 \beta_{3} ) q^{20} + ( -21 - 721 \beta_{1} - 70 \beta_{2} + 42 \beta_{3} ) q^{21} + ( 1063 - 235 \beta_{3} ) q^{22} + ( 6 \beta_{1} + 69 \beta_{2} ) q^{23} + ( -696 + 696 \beta_{1} + 48 \beta_{2} + 48 \beta_{3} ) q^{24} + ( -936 + 936 \beta_{1} + 380 \beta_{2} + 380 \beta_{3} ) q^{25} + ( -574 \beta_{1} + 434 \beta_{2} ) q^{26} + ( 1436 - 401 \beta_{3} ) q^{27} + ( -98 + 1470 \beta_{1} + 98 \beta_{2} - 98 \beta_{3} ) q^{28} + ( -3526 + 700 \beta_{3} ) q^{29} + ( -2629 \beta_{1} - 505 \beta_{2} ) q^{30} + ( -1774 + 1774 \beta_{1} - 715 \beta_{2} - 715 \beta_{3} ) q^{31} + ( 4048 - 4048 \beta_{1} - 304 \beta_{2} - 304 \beta_{3} ) q^{32} + ( 1699 \beta_{1} + 304 \beta_{2} ) q^{33} + ( 3711 + 1041 \beta_{3} ) q^{34} + ( 6244 + 1260 \beta_{1} - 567 \beta_{2} - 826 \beta_{3} ) q^{35} + ( -548 + 332 \beta_{3} ) q^{36} + ( -5545 \beta_{1} + 790 \beta_{2} ) q^{37} + ( -10069 + 10069 \beta_{1} + 97 \beta_{2} + 97 \beta_{3} ) q^{38} + ( -812 + 812 \beta_{1} + 350 \beta_{2} + 350 \beta_{3} ) q^{39} + ( -7968 \beta_{1} + 24 \beta_{2} ) q^{40} + ( 1750 - 868 \beta_{3} ) q^{41} + ( -3311 + 4886 \beta_{1} + 854 \beta_{2} + 791 \beta_{3} ) q^{42} + ( -6340 - 1344 \beta_{3} ) q^{43} + ( -2974 \beta_{1} - 562 \beta_{2} ) q^{44} + ( -650 + 650 \beta_{1} + 1748 \beta_{2} + 1748 \beta_{3} ) q^{45} + ( 2559 - 2559 \beta_{1} - 75 \beta_{2} - 75 \beta_{3} ) q^{46} + ( 11478 \beta_{1} - 1635 \beta_{2} ) q^{47} + ( 5608 - 1192 \beta_{3} ) q^{48} + ( 6125 - 9800 \beta_{1} - 392 \beta_{2} + 2156 \beta_{3} ) q^{49} + ( 14996 - 1316 \beta_{3} ) q^{50} + ( 192 \beta_{1} - 645 \beta_{2} ) q^{51} + ( 700 - 700 \beta_{1} - 756 \beta_{2} - 756 \beta_{3} ) q^{52} + ( -1521 + 1521 \beta_{1} - 1818 \beta_{2} - 1818 \beta_{3} ) q^{53} + ( -16273 \beta_{1} - 1837 \beta_{2} ) q^{54} + ( -12538 + 2557 \beta_{3} ) q^{55} + ( -19320 + 9744 \beta_{1} + 672 \beta_{2} - 1344 \beta_{3} ) q^{56} + ( -9529 + 928 \beta_{3} ) q^{57} + ( 29426 \beta_{1} + 4226 \beta_{2} ) q^{58} + ( 32904 - 32904 \beta_{1} + 531 \beta_{2} + 531 \beta_{3} ) q^{59} + ( -7042 + 7042 \beta_{1} + 1246 \beta_{2} + 1246 \beta_{3} ) q^{60} + ( 21243 \beta_{1} + 4154 \beta_{2} ) q^{61} + ( -24681 - 1059 \beta_{3} ) q^{62} + ( 6496 - 15372 \beta_{1} + 1890 \beta_{2} - 2212 \beta_{3} ) q^{63} + ( 17728 + 3072 \beta_{3} ) q^{64} + ( 1582 \beta_{1} - 4088 \beta_{2} ) q^{65} + ( 12947 - 12947 \beta_{1} - 2003 \beta_{2} - 2003 \beta_{3} ) q^{66} + ( -21156 + 21156 \beta_{1} + 919 \beta_{2} + 919 \beta_{3} ) q^{67} + ( -2730 \beta_{1} + 1554 \beta_{2} ) q^{68} + ( 2577 - 282 \beta_{3} ) q^{69} + ( -19719 - 17087 \beta_{1} - 7763 \beta_{2} - 693 \beta_{3} ) q^{70} + ( -1104 + 2184 \beta_{3} ) q^{71} + ( 16224 \beta_{1} - 4944 \beta_{2} ) q^{72} + ( 25253 - 25253 \beta_{1} - 7372 \beta_{2} - 7372 \beta_{3} ) q^{73} + ( 23685 - 23685 \beta_{1} + 4755 \beta_{2} + 4755 \beta_{3} ) q^{74} + ( 17804 \beta_{1} + 2456 \beta_{2} ) q^{75} + ( 19418 - 1302 \beta_{3} ) q^{76} + ( 8883 + 14903 \beta_{1} + 4130 \beta_{2} - 126 \beta_{3} ) q^{77} + ( 13762 - 1162 \beta_{3} ) q^{78} + ( -4502 \beta_{1} + 5193 \beta_{2} ) q^{79} + ( -34408 + 34408 \beta_{1} + 11080 \beta_{2} + 11080 \beta_{3} ) q^{80} + ( -25589 + 25589 \beta_{1} - 4984 \beta_{2} - 4984 \beta_{3} ) q^{81} + ( -33866 \beta_{1} - 2618 \beta_{2} ) q^{82} + ( -52164 + 4536 \beta_{3} ) q^{83} + ( 12740 - 3234 \beta_{1} - 294 \beta_{2} - 2156 \beta_{3} ) q^{84} + ( -26553 - 9222 \beta_{3} ) q^{85} + ( -43388 \beta_{1} + 4996 \beta_{2} ) q^{86} + ( -40004 + 40004 \beta_{1} + 6326 \beta_{2} + 6326 \beta_{3} ) q^{87} + ( 10248 - 10248 \beta_{1} - 3984 \beta_{2} - 3984 \beta_{3} ) q^{88} + ( 13333 \beta_{1} - 9356 \beta_{2} ) q^{89} + ( 65326 - 2398 \beta_{3} ) q^{90} + ( 28224 + 11368 \beta_{1} + 4900 \beta_{2} + 10094 \beta_{3} ) q^{91} + ( -5142 + 426 \beta_{3} ) q^{92} + ( -19359 \beta_{1} - 1086 \beta_{2} ) q^{93} + ( -49017 + 49017 \beta_{1} - 9843 \beta_{2} - 9843 \beta_{3} ) q^{94} + ( 99070 - 99070 \beta_{1} - 3463 \beta_{2} - 3463 \beta_{3} ) q^{95} + ( -27440 \beta_{1} - 5264 \beta_{2} ) q^{96} + ( 104566 + 196 \beta_{3} ) q^{97} + ( -24304 + 97951 \beta_{1} + 6223 \beta_{2} + 10192 \beta_{3} ) q^{98} + ( -33472 + 2674 \beta_{3} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 2q^{2} + 8q^{3} - 12q^{4} + 38q^{5} - 164q^{6} - 168q^{7} + 192q^{8} + 380q^{9} + O(q^{10}) \) \( 4q - 2q^{2} + 8q^{3} - 12q^{4} + 38q^{5} - 164q^{6} - 168q^{7} + 192q^{8} + 380q^{9} + 778q^{10} - 424q^{11} + 196q^{12} - 1848q^{13} - 2674q^{14} + 1784q^{15} + 2064q^{16} + 2346q^{17} - 212q^{18} + 360q^{19} - 3416q^{20} - 1526q^{21} + 4252q^{22} + 12q^{23} - 1392q^{24} - 1872q^{25} - 1148q^{26} + 5744q^{27} + 2548q^{28} - 14104q^{29} - 5258q^{30} - 3548q^{31} + 8096q^{32} + 3398q^{33} + 14844q^{34} + 27496q^{35} - 2192q^{36} - 11090q^{37} - 20138q^{38} - 1624q^{39} - 15936q^{40} + 7000q^{41} - 3472q^{42} - 25360q^{43} - 5948q^{44} - 1300q^{45} + 5118q^{46} + 22956q^{47} + 22432q^{48} + 4900q^{49} + 59984q^{50} + 384q^{51} + 1400q^{52} - 3042q^{53} - 32546q^{54} - 50152q^{55} - 57792q^{56} - 38116q^{57} + 58852q^{58} + 65808q^{59} - 14084q^{60} + 42486q^{61} - 98724q^{62} - 4760q^{63} + 70912q^{64} + 3164q^{65} + 25894q^{66} - 42312q^{67} - 5460q^{68} + 10308q^{69} - 113050q^{70} - 4416q^{71} + 32448q^{72} + 50506q^{73} + 47370q^{74} + 35608q^{75} + 77672q^{76} + 65338q^{77} + 55048q^{78} - 9004q^{79} - 68816q^{80} - 51178q^{81} - 67732q^{82} - 208656q^{83} + 44492q^{84} - 106212q^{85} - 86776q^{86} - 80008q^{87} + 20496q^{88} + 26666q^{89} + 261304q^{90} + 135632q^{91} - 20568q^{92} - 38718q^{93} - 98034q^{94} + 198140q^{95} - 54880q^{96} + 418264q^{97} + 98686q^{98} - 133888q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - x^{3} + 10 x^{2} + 9 x + 81\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( -\nu^{3} + 10 \nu^{2} - 10 \nu + 81 \)\()/90\)
\(\beta_{2}\)\(=\)\((\)\( \nu^{3} - 10 \nu^{2} + 190 \nu - 81 \)\()/90\)
\(\beta_{3}\)\(=\)\((\)\( \nu^{3} + 14 \)\()/5\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{2} + \beta_{1}\)\()/2\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{3} + \beta_{2} + 19 \beta_{1} - 19\)\()/2\)
\(\nu^{3}\)\(=\)\(5 \beta_{3} - 14\)

Character values

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

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

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2.1
1.77069 3.06693i
−1.27069 + 2.20090i
1.77069 + 3.06693i
−1.27069 2.20090i
−3.54138 + 6.13385i 5.04138 + 8.73193i −9.08276 15.7318i 39.9138 69.1328i −71.4138 43.1587 + 122.247i −97.9863 70.6689 122.402i 282.700 + 489.651i
2.2 2.54138 4.40180i −1.04138 1.80373i 3.08276 + 5.33950i −20.9138 + 36.2238i −10.5862 −127.159 25.2522i 193.986 119.331 206.687i 106.300 + 184.117i
4.1 −3.54138 6.13385i 5.04138 8.73193i −9.08276 + 15.7318i 39.9138 + 69.1328i −71.4138 43.1587 122.247i −97.9863 70.6689 + 122.402i 282.700 489.651i
4.2 2.54138 + 4.40180i −1.04138 + 1.80373i 3.08276 5.33950i −20.9138 36.2238i −10.5862 −127.159 + 25.2522i 193.986 119.331 + 206.687i 106.300 184.117i
\(n\): e.g. 2-40 or 990-1000
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 7.6.c.a 4
3.b odd 2 1 63.6.e.d 4
4.b odd 2 1 112.6.i.c 4
7.b odd 2 1 49.6.c.f 4
7.c even 3 1 inner 7.6.c.a 4
7.c even 3 1 49.6.a.d 2
7.d odd 6 1 49.6.a.e 2
7.d odd 6 1 49.6.c.f 4
21.g even 6 1 441.6.a.m 2
21.h odd 6 1 63.6.e.d 4
21.h odd 6 1 441.6.a.n 2
28.f even 6 1 784.6.a.t 2
28.g odd 6 1 112.6.i.c 4
28.g odd 6 1 784.6.a.ba 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
7.6.c.a 4 1.a even 1 1 trivial
7.6.c.a 4 7.c even 3 1 inner
49.6.a.d 2 7.c even 3 1
49.6.a.e 2 7.d odd 6 1
49.6.c.f 4 7.b odd 2 1
49.6.c.f 4 7.d odd 6 1
63.6.e.d 4 3.b odd 2 1
63.6.e.d 4 21.h odd 6 1
112.6.i.c 4 4.b odd 2 1
112.6.i.c 4 28.g odd 6 1
441.6.a.m 2 21.g even 6 1
441.6.a.n 2 21.h odd 6 1
784.6.a.t 2 28.f even 6 1
784.6.a.ba 2 28.g odd 6 1

Hecke kernels

This newform subspace is the entire newspace \(S_{6}^{\mathrm{new}}(7, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + 2 T - 24 T^{2} - 72 T^{3} - 368 T^{4} - 2304 T^{5} - 24576 T^{6} + 65536 T^{7} + 1048576 T^{8} \)
$3$ \( 1 - 8 T - 401 T^{2} + 168 T^{3} + 141624 T^{4} + 40824 T^{5} - 23678649 T^{6} - 114791256 T^{7} + 3486784401 T^{8} \)
$5$ \( 1 - 38 T - 1467 T^{2} + 126882 T^{3} - 5804204 T^{4} + 396506250 T^{5} - 14326171875 T^{6} - 1159667968750 T^{7} + 95367431640625 T^{8} \)
$7$ \( 1 + 168 T + 11662 T^{2} + 2823576 T^{3} + 282475249 T^{4} \)
$11$ \( 1 + 424 T - 167697 T^{2} + 10757304 T^{3} + 65846956552 T^{4} + 1732474566504 T^{5} - 4349628293313897 T^{6} + 1771153223832236024 T^{7} + \)\(67\!\cdots\!01\)\( T^{8} \)
$13$ \( ( 1 + 924 T + 927022 T^{2} + 343074732 T^{3} + 137858491849 T^{4} )^{2} \)
$17$ \( 1 - 2346 T + 1932761 T^{2} - 1715491386 T^{3} + 2921236022964 T^{4} - 2435752452851802 T^{5} + 3896434387025709689 T^{6} - \)\(67\!\cdots\!78\)\( T^{7} + \)\(40\!\cdots\!01\)\( T^{8} \)
$19$ \( 1 - 360 T - 2016025 T^{2} + 1010366280 T^{3} - 1848262047576 T^{4} + 2501766935541720 T^{5} - 12360382852383261025 T^{6} - \)\(54\!\cdots\!40\)\( T^{7} + \)\(37\!\cdots\!01\)\( T^{8} \)
$23$ \( 1 - 12 T - 12696421 T^{2} + 2113452 T^{3} + 119775324752184 T^{4} + 13602901986036 T^{5} - \)\(52\!\cdots\!29\)\( T^{6} - \)\(31\!\cdots\!84\)\( T^{7} + \)\(17\!\cdots\!01\)\( T^{8} \)
$29$ \( ( 1 + 7052 T + 35324974 T^{2} + 144644622748 T^{3} + 420707233300201 T^{4} )^{2} \)
$31$ \( 1 + 3548 T - 28901749 T^{2} - 55945747452 T^{3} + 541403754912104 T^{4} - 1601679251611173252 T^{5} - \)\(23\!\cdots\!49\)\( T^{6} + \)\(83\!\cdots\!48\)\( T^{7} + \)\(67\!\cdots\!01\)\( T^{8} \)
$37$ \( 1 + 11090 T - 23355139 T^{2} + 84897554250 T^{3} + 8079277710681572 T^{4} + 5887132351317167250 T^{5} - \)\(11\!\cdots\!11\)\( T^{6} + \)\(36\!\cdots\!70\)\( T^{7} + \)\(23\!\cdots\!01\)\( T^{8} \)
$41$ \( ( 1 - 3500 T + 206898214 T^{2} - 405496703500 T^{3} + 13422659310152401 T^{4} )^{2} \)
$43$ \( ( 1 + 12680 T + 267378054 T^{2} + 1864067057240 T^{3} + 21611482313284249 T^{4} )^{2} \)
$47$ \( 1 - 22956 T + 35452763 T^{2} - 753763910004 T^{3} + 68138105036084328 T^{4} - \)\(17\!\cdots\!28\)\( T^{5} + \)\(18\!\cdots\!87\)\( T^{6} - \)\(27\!\cdots\!08\)\( T^{7} + \)\(27\!\cdots\!01\)\( T^{8} \)
$53$ \( 1 + 3042 T - 707161075 T^{2} - 364967439174 T^{3} + 334492868775797220 T^{4} - \)\(15\!\cdots\!82\)\( T^{5} - \)\(12\!\cdots\!75\)\( T^{6} + \)\(22\!\cdots\!94\)\( T^{7} + \)\(30\!\cdots\!01\)\( T^{8} \)
$59$ \( 1 - 65808 T + 1828603607 T^{2} - 70562013287472 T^{3} + 2653216336714668312 T^{4} - \)\(50\!\cdots\!28\)\( T^{5} + \)\(93\!\cdots\!07\)\( T^{6} - \)\(24\!\cdots\!92\)\( T^{7} + \)\(26\!\cdots\!01\)\( T^{8} \)
$61$ \( 1 - 42486 T + 303064037 T^{2} + 7953228077298 T^{3} + 18102385363935684 T^{4} + \)\(67\!\cdots\!98\)\( T^{5} + \)\(21\!\cdots\!37\)\( T^{6} - \)\(25\!\cdots\!86\)\( T^{7} + \)\(50\!\cdots\!01\)\( T^{8} \)
$67$ \( 1 + 42312 T - 1326272449 T^{2} + 17615652522648 T^{3} + 5473083141138317592 T^{4} + \)\(23\!\cdots\!36\)\( T^{5} - \)\(24\!\cdots\!01\)\( T^{6} + \)\(10\!\cdots\!16\)\( T^{7} + \)\(33\!\cdots\!01\)\( T^{8} \)
$71$ \( ( 1 + 2208 T + 3433192846 T^{2} + 3983738407008 T^{3} + 3255243551009881201 T^{4} )^{2} \)
$73$ \( 1 - 50506 T - 222184951 T^{2} + 69349899662694 T^{3} - 1895976700185808828 T^{4} + \)\(14\!\cdots\!42\)\( T^{5} - \)\(95\!\cdots\!99\)\( T^{6} - \)\(44\!\cdots\!42\)\( T^{7} + \)\(18\!\cdots\!01\)\( T^{8} \)
$79$ \( 1 + 9004 T - 5095520573 T^{2} - 8801591961836 T^{3} + 17079371584250245336 T^{4} - \)\(27\!\cdots\!64\)\( T^{5} - \)\(48\!\cdots\!73\)\( T^{6} + \)\(26\!\cdots\!96\)\( T^{7} + \)\(89\!\cdots\!01\)\( T^{8} \)
$83$ \( ( 1 + 104328 T + 9837878230 T^{2} + 410952232202904 T^{3} + 15516041187205853449 T^{4} )^{2} \)
$89$ \( 1 - 26666 T - 7396026999 T^{2} + 81625061802438 T^{3} + 30572703729886615780 T^{4} + \)\(45\!\cdots\!62\)\( T^{5} - \)\(23\!\cdots\!99\)\( T^{6} - \)\(46\!\cdots\!34\)\( T^{7} + \)\(97\!\cdots\!01\)\( T^{8} \)
$97$ \( ( 1 - 209132 T + 28107307478 T^{2} - 1795887642626924 T^{3} + 73742412689492826049 T^{4} )^{2} \)
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