Properties

Label 7.5.d.a
Level 7
Weight 5
Character orbit 7.d
Analytic conductor 0.724
Analytic rank 0
Dimension 4
CM No
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 7 \)
Weight: \( k \) = \( 5 \)
Character orbit: \([\chi]\) = 7.d (of order \(6\) and degree \(2\))

Newform invariants

Self dual: No
Analytic conductor: \(0.723589741587\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{22})\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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.

\(f(q)\) \(=\) \( q\) \( + ( -2 + \beta_{1} - 2 \beta_{2} ) q^{2} \) \( + ( 2 - \beta_{1} + \beta_{2} + \beta_{3} ) q^{3} \) \( + ( -4 \beta_{1} + 10 \beta_{2} - 4 \beta_{3} ) q^{4} \) \( + ( -5 + 2 \beta_{1} + 5 \beta_{2} + 4 \beta_{3} ) q^{5} \) \( + ( -24 + 6 \beta_{1} - 48 \beta_{2} + 3 \beta_{3} ) q^{6} \) \( + ( 21 + 42 \beta_{2} - 7 \beta_{3} ) q^{7} \) \( + ( 76 + 2 \beta_{3} ) q^{8} \) \( + ( -12 - 6 \beta_{1} - 12 \beta_{2} ) q^{9} \) \(+O(q^{10})\) \( q\) \( + ( -2 + \beta_{1} - 2 \beta_{2} ) q^{2} \) \( + ( 2 - \beta_{1} + \beta_{2} + \beta_{3} ) q^{3} \) \( + ( -4 \beta_{1} + 10 \beta_{2} - 4 \beta_{3} ) q^{4} \) \( + ( -5 + 2 \beta_{1} + 5 \beta_{2} + 4 \beta_{3} ) q^{5} \) \( + ( -24 + 6 \beta_{1} - 48 \beta_{2} + 3 \beta_{3} ) q^{6} \) \( + ( 21 + 42 \beta_{2} - 7 \beta_{3} ) q^{7} \) \( + ( 76 + 2 \beta_{3} ) q^{8} \) \( + ( -12 - 6 \beta_{1} - 12 \beta_{2} ) q^{9} \) \( + ( -68 - \beta_{1} - 34 \beta_{2} + \beta_{3} ) q^{10} \) \( + ( 17 \beta_{1} + 29 \beta_{2} + 17 \beta_{3} ) q^{11} \) \( + ( -98 - 14 \beta_{1} + 98 \beta_{2} - 28 \beta_{3} ) q^{12} \) \( + ( -70 - 28 \beta_{1} - 140 \beta_{2} - 14 \beta_{3} ) q^{13} \) \( + ( 196 + 7 \beta_{1} + 112 \beta_{2} + 42 \beta_{3} ) q^{14} \) \( + ( 117 - 9 \beta_{3} ) q^{15} \) \( + ( -36 + 16 \beta_{1} - 36 \beta_{2} ) q^{16} \) \( + ( -82 + 34 \beta_{1} - 41 \beta_{2} - 34 \beta_{3} ) q^{17} \) \( -108 \beta_{2} q^{18} \) \( + ( 107 + 37 \beta_{1} - 107 \beta_{2} + 74 \beta_{3} ) q^{19} \) \( + ( 126 + 252 \beta_{2} ) q^{20} \) \( + ( -308 - 56 \beta_{1} - 91 \beta_{2} - 70 \beta_{3} ) q^{21} \) \( + ( -316 - 5 \beta_{3} ) q^{22} \) \( + ( 145 - 41 \beta_{1} + 145 \beta_{2} ) q^{23} \) \( + ( 240 - 78 \beta_{1} + 120 \beta_{2} + 78 \beta_{3} ) q^{24} \) \( + ( -60 \beta_{1} + 286 \beta_{2} - 60 \beta_{3} ) q^{25} \) \( + ( 168 - 42 \beta_{1} - 168 \beta_{2} - 84 \beta_{3} ) q^{26} \) \( + ( 39 + 174 \beta_{1} + 78 \beta_{2} + 87 \beta_{3} ) q^{27} \) \( + ( -420 + 154 \beta_{1} - 826 \beta_{2} - 14 \beta_{3} ) q^{28} \) \( + ( -544 + 70 \beta_{3} ) q^{29} \) \( + ( -36 + 99 \beta_{1} - 36 \beta_{2} ) q^{30} \) \( + ( 1206 - 29 \beta_{1} + 603 \beta_{2} + 29 \beta_{3} ) q^{31} \) \( + ( -36 \beta_{1} - 792 \beta_{2} - 36 \beta_{3} ) q^{32} \) \( + ( 345 - 12 \beta_{1} - 345 \beta_{2} - 24 \beta_{3} ) q^{33} \) \( + ( 830 - 218 \beta_{1} + 1660 \beta_{2} - 109 \beta_{3} ) q^{34} \) \( + ( -623 - 91 \beta_{1} - 7 \beta_{2} + 70 \beta_{3} ) q^{35} \) \( + ( -408 - 12 \beta_{3} ) q^{36} \) \( + ( -135 - 104 \beta_{1} - 135 \beta_{2} ) q^{37} \) \( + ( -2056 + 181 \beta_{1} - 1028 \beta_{2} - 181 \beta_{3} ) q^{38} \) \( + ( 168 \beta_{1} + 714 \beta_{2} + 168 \beta_{3} ) q^{39} \) \( + ( -292 + 142 \beta_{1} + 292 \beta_{2} + 284 \beta_{3} ) q^{40} \) \( + ( -798 - 84 \beta_{1} - 1596 \beta_{2} - 42 \beta_{3} ) q^{41} \) \( + ( 1974 - 336 \beta_{1} + 924 \beta_{2} + 21 \beta_{3} ) q^{42} \) \( + ( 618 - 350 \beta_{3} ) q^{43} \) \( + ( 1206 - 54 \beta_{1} + 1206 \beta_{2} ) q^{44} \) \( + ( 648 + 54 \beta_{1} + 324 \beta_{2} - 54 \beta_{3} ) q^{45} \) \( + ( 227 \beta_{1} - 1192 \beta_{2} + 227 \beta_{3} ) q^{46} \) \( + ( -257 - 187 \beta_{1} + 257 \beta_{2} - 374 \beta_{3} ) q^{47} \) \( + ( -388 + 104 \beta_{1} - 776 \beta_{2} + 52 \beta_{3} ) q^{48} \) \( + ( -245 + 588 \beta_{1} + 294 \beta_{3} ) q^{49} \) \( + ( 1892 + 406 \beta_{3} ) q^{50} \) \( + ( -2367 + 225 \beta_{1} - 2367 \beta_{2} ) q^{51} \) \( + ( -1064 - 140 \beta_{1} - 532 \beta_{2} + 140 \beta_{3} ) q^{52} \) \( + ( -340 \beta_{1} + 2255 \beta_{2} - 340 \beta_{3} ) q^{53} \) \( + ( -1836 - 135 \beta_{1} + 1836 \beta_{2} - 270 \beta_{3} ) q^{54} \) \( + ( -893 - 286 \beta_{1} - 1786 \beta_{2} - 143 \beta_{3} ) q^{55} \) \( + ( 1288 - 84 \beta_{1} + 3192 \beta_{2} - 574 \beta_{3} ) q^{56} \) \( + ( 2763 + 432 \beta_{3} ) q^{57} \) \( + ( -452 - 404 \beta_{1} - 452 \beta_{2} ) q^{58} \) \( + ( 842 - 449 \beta_{1} + 421 \beta_{2} + 449 \beta_{3} ) q^{59} \) \( + ( -378 \beta_{1} + 378 \beta_{2} - 378 \beta_{3} ) q^{60} \) \( + ( -47 + 240 \beta_{1} + 47 \beta_{2} + 480 \beta_{3} ) q^{61} \) \( + ( -1844 + 1322 \beta_{1} - 3688 \beta_{2} + 661 \beta_{3} ) q^{62} \) \( + ( -672 - 210 \beta_{1} - 1176 \beta_{2} - 252 \beta_{3} ) q^{63} \) \( + ( -1368 - 976 \beta_{3} ) q^{64} \) \( + ( 2898 + 630 \beta_{1} + 2898 \beta_{2} ) q^{65} \) \( + ( -852 + 321 \beta_{1} - 426 \beta_{2} - 321 \beta_{3} ) q^{66} \) \( + ( 45 \beta_{1} + 659 \beta_{2} + 45 \beta_{3} ) q^{67} \) \( + ( 3402 + 504 \beta_{1} - 3402 \beta_{2} + 1008 \beta_{3} ) q^{68} \) \( + ( 1047 - 372 \beta_{1} + 2094 \beta_{2} - 186 \beta_{3} ) q^{69} \) \( + ( -308 - 301 \beta_{1} - 2296 \beta_{2} + 175 \beta_{3} ) q^{70} \) \( + ( -2602 + 238 \beta_{3} ) q^{71} \) \( + ( -648 - 432 \beta_{1} - 648 \beta_{2} ) q^{72} \) \( + ( 1738 + 272 \beta_{1} + 869 \beta_{2} - 272 \beta_{3} ) q^{73} \) \( + ( 73 \beta_{1} - 2018 \beta_{2} + 73 \beta_{3} ) q^{74} \) \( + ( -1606 - 346 \beta_{1} + 1606 \beta_{2} - 692 \beta_{3} ) q^{75} \) \( + ( 4326 - 1596 \beta_{1} + 8652 \beta_{2} - 798 \beta_{3} ) q^{76} \) \( + ( -1218 - 154 \beta_{1} + 2009 \beta_{2} + 560 \beta_{3} ) q^{77} \) \( + ( -2268 + 378 \beta_{3} ) q^{78} \) \( + ( -4055 + 351 \beta_{1} - 4055 \beta_{2} ) q^{79} \) \( + ( -1048 - 8 \beta_{1} - 524 \beta_{2} + 8 \beta_{3} ) q^{80} \) \( + ( 630 \beta_{1} - 4653 \beta_{2} + 630 \beta_{3} ) q^{81} \) \( + ( -672 - 714 \beta_{1} + 672 \beta_{2} - 1428 \beta_{3} ) q^{82} \) \( + ( 1932 - 168 \beta_{1} + 3864 \beta_{2} - 84 \beta_{3} ) q^{83} \) \( + ( -4018 + 1568 \beta_{1} - 8330 \beta_{2} + 1372 \beta_{3} ) q^{84} \) \( + ( -3873 + 264 \beta_{3} ) q^{85} \) \( + ( 6464 - 82 \beta_{1} + 6464 \beta_{2} ) q^{86} \) \( + ( 1992 + 474 \beta_{1} + 996 \beta_{2} - 474 \beta_{3} ) q^{87} \) \( + ( 1234 \beta_{1} + 1456 \beta_{2} + 1234 \beta_{3} ) q^{88} \) \( + ( 5665 - 376 \beta_{1} - 5665 \beta_{2} - 752 \beta_{3} ) q^{89} \) \( + ( 540 + 432 \beta_{1} + 1080 \beta_{2} + 216 \beta_{3} ) q^{90} \) \( + ( 2254 - 980 \beta_{1} - 4312 \beta_{2} - 1372 \beta_{3} ) q^{91} \) \( + ( -5058 - 990 \beta_{3} ) q^{92} \) \( + ( 3723 - 1896 \beta_{1} + 3723 \beta_{2} ) q^{93} \) \( + ( 9256 - 631 \beta_{1} + 4628 \beta_{2} + 631 \beta_{3} ) q^{94} \) \( + ( 87 \beta_{1} - 3279 \beta_{2} + 87 \beta_{3} ) q^{95} \) \( + ( 756 \beta_{1} + 1512 \beta_{3} ) q^{96} \) \( + ( -686 + 2548 \beta_{1} - 1372 \beta_{2} + 1274 \beta_{3} ) q^{97} \) \( + ( -5978 - 833 \beta_{1} + 6958 \beta_{2} - 1176 \beta_{3} ) q^{98} \) \( + ( 2592 - 378 \beta_{3} ) q^{99} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(4q \) \(\mathstrut -\mathstrut 4q^{2} \) \(\mathstrut +\mathstrut 6q^{3} \) \(\mathstrut -\mathstrut 20q^{4} \) \(\mathstrut -\mathstrut 30q^{5} \) \(\mathstrut +\mathstrut 304q^{8} \) \(\mathstrut -\mathstrut 24q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(4q \) \(\mathstrut -\mathstrut 4q^{2} \) \(\mathstrut +\mathstrut 6q^{3} \) \(\mathstrut -\mathstrut 20q^{4} \) \(\mathstrut -\mathstrut 30q^{5} \) \(\mathstrut +\mathstrut 304q^{8} \) \(\mathstrut -\mathstrut 24q^{9} \) \(\mathstrut -\mathstrut 204q^{10} \) \(\mathstrut -\mathstrut 58q^{11} \) \(\mathstrut -\mathstrut 588q^{12} \) \(\mathstrut +\mathstrut 560q^{14} \) \(\mathstrut +\mathstrut 468q^{15} \) \(\mathstrut -\mathstrut 72q^{16} \) \(\mathstrut -\mathstrut 246q^{17} \) \(\mathstrut +\mathstrut 216q^{18} \) \(\mathstrut +\mathstrut 642q^{19} \) \(\mathstrut -\mathstrut 1050q^{21} \) \(\mathstrut -\mathstrut 1264q^{22} \) \(\mathstrut +\mathstrut 290q^{23} \) \(\mathstrut +\mathstrut 720q^{24} \) \(\mathstrut -\mathstrut 572q^{25} \) \(\mathstrut +\mathstrut 1008q^{26} \) \(\mathstrut -\mathstrut 28q^{28} \) \(\mathstrut -\mathstrut 2176q^{29} \) \(\mathstrut -\mathstrut 72q^{30} \) \(\mathstrut +\mathstrut 3618q^{31} \) \(\mathstrut +\mathstrut 1584q^{32} \) \(\mathstrut +\mathstrut 2070q^{33} \) \(\mathstrut -\mathstrut 2478q^{35} \) \(\mathstrut -\mathstrut 1632q^{36} \) \(\mathstrut -\mathstrut 270q^{37} \) \(\mathstrut -\mathstrut 6168q^{38} \) \(\mathstrut -\mathstrut 1428q^{39} \) \(\mathstrut -\mathstrut 1752q^{40} \) \(\mathstrut +\mathstrut 6048q^{42} \) \(\mathstrut +\mathstrut 2472q^{43} \) \(\mathstrut +\mathstrut 2412q^{44} \) \(\mathstrut +\mathstrut 1944q^{45} \) \(\mathstrut +\mathstrut 2384q^{46} \) \(\mathstrut -\mathstrut 1542q^{47} \) \(\mathstrut -\mathstrut 980q^{49} \) \(\mathstrut +\mathstrut 7568q^{50} \) \(\mathstrut -\mathstrut 4734q^{51} \) \(\mathstrut -\mathstrut 3192q^{52} \) \(\mathstrut -\mathstrut 4510q^{53} \) \(\mathstrut -\mathstrut 11016q^{54} \) \(\mathstrut -\mathstrut 1232q^{56} \) \(\mathstrut +\mathstrut 11052q^{57} \) \(\mathstrut -\mathstrut 904q^{58} \) \(\mathstrut +\mathstrut 2526q^{59} \) \(\mathstrut -\mathstrut 756q^{60} \) \(\mathstrut -\mathstrut 282q^{61} \) \(\mathstrut -\mathstrut 336q^{63} \) \(\mathstrut -\mathstrut 5472q^{64} \) \(\mathstrut +\mathstrut 5796q^{65} \) \(\mathstrut -\mathstrut 2556q^{66} \) \(\mathstrut -\mathstrut 1318q^{67} \) \(\mathstrut +\mathstrut 20412q^{68} \) \(\mathstrut +\mathstrut 3360q^{70} \) \(\mathstrut -\mathstrut 10408q^{71} \) \(\mathstrut -\mathstrut 1296q^{72} \) \(\mathstrut +\mathstrut 5214q^{73} \) \(\mathstrut +\mathstrut 4036q^{74} \) \(\mathstrut -\mathstrut 9636q^{75} \) \(\mathstrut -\mathstrut 8890q^{77} \) \(\mathstrut -\mathstrut 9072q^{78} \) \(\mathstrut -\mathstrut 8110q^{79} \) \(\mathstrut -\mathstrut 3144q^{80} \) \(\mathstrut +\mathstrut 9306q^{81} \) \(\mathstrut -\mathstrut 4032q^{82} \) \(\mathstrut +\mathstrut 588q^{84} \) \(\mathstrut -\mathstrut 15492q^{85} \) \(\mathstrut +\mathstrut 12928q^{86} \) \(\mathstrut +\mathstrut 5976q^{87} \) \(\mathstrut -\mathstrut 2912q^{88} \) \(\mathstrut +\mathstrut 33990q^{89} \) \(\mathstrut +\mathstrut 17640q^{91} \) \(\mathstrut -\mathstrut 20232q^{92} \) \(\mathstrut +\mathstrut 7446q^{93} \) \(\mathstrut +\mathstrut 27768q^{94} \) \(\mathstrut +\mathstrut 6558q^{95} \) \(\mathstrut -\mathstrut 37828q^{98} \) \(\mathstrut +\mathstrut 10368q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4}\mathstrut +\mathstrut \) \(22\) \(x^{2}\mathstrut +\mathstrut \) \(484\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} \)\(/22\)
\(\beta_{3}\)\(=\)\( \nu^{3} \)\(/22\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(22\) \(\beta_{2}\)
\(\nu^{3}\)\(=\)\(22\) \(\beta_{3}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
3.1
−2.34521 4.06202i
2.34521 + 4.06202i
−2.34521 + 4.06202i
2.34521 4.06202i
−3.34521 5.79407i 8.53562 + 4.92804i −14.3808 + 24.9083i 6.57125 3.79391i 65.9413i −32.8329 + 36.3731i 85.3808 8.07125 + 13.9798i −43.9644 25.3828i
3.2 1.34521 + 2.32997i −5.53562 3.19599i 4.38083 7.58782i −21.5712 + 12.4542i 17.1971i 32.8329 + 36.3731i 66.6192 −20.0712 34.7644i −58.0356 33.5069i
5.1 −3.34521 + 5.79407i 8.53562 4.92804i −14.3808 24.9083i 6.57125 + 3.79391i 65.9413i −32.8329 36.3731i 85.3808 8.07125 13.9798i −43.9644 + 25.3828i
5.2 1.34521 2.32997i −5.53562 + 3.19599i 4.38083 + 7.58782i −21.5712 12.4542i 17.1971i 32.8329 36.3731i 66.6192 −20.0712 + 34.7644i −58.0356 + 33.5069i
\(n\): 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.d Odd 1 yes

Hecke kernels

There are no other newforms in \(S_{5}^{\mathrm{new}}(7, [\chi])\).