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 \(3\) and degree \(2\))

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: \( 2^{2} \)
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.

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

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

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

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. orbit Parity Mult. Self Twist Proved
1.a Even 1 trivial yes
7.c Even 1 yes

Hecke kernels

There are no other newforms in \(S_{6}^{\mathrm{new}}(7, [\chi])\).