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

Related objects

Downloads

Learn more about

Newspace parameters

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

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: \( 1 \)
Twist minimal: yes
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})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 4q^{2} + 6q^{3} - 20q^{4} - 30q^{5} + 304q^{8} - 24q^{9} + O(q^{10}) \) \( 4q - 4q^{2} + 6q^{3} - 20q^{4} - 30q^{5} + 304q^{8} - 24q^{9} - 204q^{10} - 58q^{11} - 588q^{12} + 560q^{14} + 468q^{15} - 72q^{16} - 246q^{17} + 216q^{18} + 642q^{19} - 1050q^{21} - 1264q^{22} + 290q^{23} + 720q^{24} - 572q^{25} + 1008q^{26} - 28q^{28} - 2176q^{29} - 72q^{30} + 3618q^{31} + 1584q^{32} + 2070q^{33} - 2478q^{35} - 1632q^{36} - 270q^{37} - 6168q^{38} - 1428q^{39} - 1752q^{40} + 6048q^{42} + 2472q^{43} + 2412q^{44} + 1944q^{45} + 2384q^{46} - 1542q^{47} - 980q^{49} + 7568q^{50} - 4734q^{51} - 3192q^{52} - 4510q^{53} - 11016q^{54} - 1232q^{56} + 11052q^{57} - 904q^{58} + 2526q^{59} - 756q^{60} - 282q^{61} - 336q^{63} - 5472q^{64} + 5796q^{65} - 2556q^{66} - 1318q^{67} + 20412q^{68} + 3360q^{70} - 10408q^{71} - 1296q^{72} + 5214q^{73} + 4036q^{74} - 9636q^{75} - 8890q^{77} - 9072q^{78} - 8110q^{79} - 3144q^{80} + 9306q^{81} - 4032q^{82} + 588q^{84} - 15492q^{85} + 12928q^{86} + 5976q^{87} - 2912q^{88} + 33990q^{89} + 17640q^{91} - 20232q^{92} + 7446q^{93} + 27768q^{94} + 6558q^{95} - 37828q^{98} + 10368q^{99} + O(q^{100}) \)

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

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

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 Parity Ord Mult Type
1.a even 1 1 trivial
7.d odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 7.5.d.a 4
3.b odd 2 1 63.5.m.d 4
4.b odd 2 1 112.5.s.a 4
5.b even 2 1 175.5.i.a 4
5.c odd 4 2 175.5.j.a 8
7.b odd 2 1 49.5.d.b 4
7.c even 3 1 49.5.b.a 4
7.c even 3 1 49.5.d.b 4
7.d odd 6 1 inner 7.5.d.a 4
7.d odd 6 1 49.5.b.a 4
21.g even 6 1 63.5.m.d 4
21.g even 6 1 441.5.d.d 4
21.h odd 6 1 441.5.d.d 4
28.f even 6 1 112.5.s.a 4
28.f even 6 1 784.5.c.c 4
28.g odd 6 1 784.5.c.c 4
35.i odd 6 1 175.5.i.a 4
35.k even 12 2 175.5.j.a 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
7.5.d.a 4 1.a even 1 1 trivial
7.5.d.a 4 7.d odd 6 1 inner
49.5.b.a 4 7.c even 3 1
49.5.b.a 4 7.d odd 6 1
49.5.d.b 4 7.b odd 2 1
49.5.d.b 4 7.c even 3 1
63.5.m.d 4 3.b odd 2 1
63.5.m.d 4 21.g even 6 1
112.5.s.a 4 4.b odd 2 1
112.5.s.a 4 28.f even 6 1
175.5.i.a 4 5.b even 2 1
175.5.i.a 4 35.i odd 6 1
175.5.j.a 8 5.c odd 4 2
175.5.j.a 8 35.k even 12 2
441.5.d.d 4 21.g even 6 1
441.5.d.d 4 21.h odd 6 1
784.5.c.c 4 28.f even 6 1
784.5.c.c 4 28.g odd 6 1

Hecke kernels

This newform subspace is the entire newspace \(S_{5}^{\mathrm{new}}(7, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + 4 T + 2 T^{2} - 72 T^{3} - 316 T^{4} - 1152 T^{5} + 512 T^{6} + 16384 T^{7} + 65536 T^{8} \)
$3$ \( 1 - 6 T + 111 T^{2} - 594 T^{3} + 4212 T^{4} - 48114 T^{5} + 728271 T^{6} - 3188646 T^{7} + 43046721 T^{8} \)
$5$ \( 1 + 30 T + 1361 T^{2} + 31830 T^{3} + 922596 T^{4} + 19893750 T^{5} + 531640625 T^{6} + 7324218750 T^{7} + 152587890625 T^{8} \)
$7$ \( 1 + 490 T^{2} + 5764801 T^{4} \)
$11$ \( 1 + 58 T - 20401 T^{2} - 319986 T^{3} + 301164020 T^{4} - 4684915026 T^{5} - 4373135531281 T^{6} + 182028845849818 T^{7} + 45949729863572161 T^{8} \)
$13$ \( 1 - 58972 T^{2} + 1740248838 T^{4} - 48105272078812 T^{6} + 665416609183179841 T^{8} \)
$17$ \( 1 + 246 T + 115961 T^{2} + 23564094 T^{3} + 3884560692 T^{4} + 1968096694974 T^{5} + 808915808615801 T^{6} + 143325070358521206 T^{7} + 48661191875666868481 T^{8} \)
$19$ \( 1 - 642 T + 342023 T^{2} - 131375670 T^{3} + 42796461732 T^{4} - 17121008690070 T^{5} + 5808769181971943 T^{6} - 1420948178040475362 T^{7} + \)\(28\!\cdots\!81\)\( T^{8} \)
$23$ \( 1 - 290 T - 459625 T^{2} + 4627530 T^{3} + 193791262244 T^{4} + 1294972622730 T^{5} - 35993686609779625 T^{6} - 6355241085285893090 T^{7} + \)\(61\!\cdots\!61\)\( T^{8} \)
$29$ \( ( 1 + 1088 T + 1602698 T^{2} + 769521728 T^{3} + 500246412961 T^{4} )^{2} \)
$31$ \( 1 - 3618 T + 7245671 T^{2} - 10428389334 T^{3} + 11484731993796 T^{4} - 9630836546125014 T^{5} + 6179767856146167911 T^{6} - \)\(28\!\cdots\!98\)\( T^{7} + \)\(72\!\cdots\!81\)\( T^{8} \)
$37$ \( 1 + 270 T - 3455695 T^{2} - 59326290 T^{3} + 8801876883204 T^{4} - 111187018992690 T^{5} - 12138057686517530095 T^{6} + \)\(17\!\cdots\!70\)\( T^{7} + \)\(12\!\cdots\!41\)\( T^{8} \)
$41$ \( 1 - 7249372 T^{2} + 28218527830086 T^{4} - 57885693378083362012 T^{6} + \)\(63\!\cdots\!41\)\( T^{8} \)
$43$ \( ( 1 - 1236 T + 4524526 T^{2} - 4225638036 T^{3} + 11688200277601 T^{4} )^{2} \)
$47$ \( 1 + 1542 T + 8442143 T^{2} + 11795613810 T^{3} + 38571981640692 T^{4} + 57558832591994610 T^{5} + \)\(20\!\cdots\!23\)\( T^{6} + \)\(17\!\cdots\!22\)\( T^{7} + \)\(56\!\cdots\!21\)\( T^{8} \)
$53$ \( 1 + 4510 T + 2017313 T^{2} + 11463630750 T^{3} + 112971660447908 T^{4} + 90453560623890750 T^{5} + \)\(12\!\cdots\!93\)\( T^{6} + \)\(22\!\cdots\!10\)\( T^{7} + \)\(38\!\cdots\!21\)\( T^{8} \)
$59$ \( 1 - 2526 T + 13587671 T^{2} - 28949927754 T^{3} + 10291335854532 T^{4} - 350796725519137194 T^{5} + \)\(19\!\cdots\!91\)\( T^{6} - \)\(44\!\cdots\!06\)\( T^{7} + \)\(21\!\cdots\!41\)\( T^{8} \)
$61$ \( 1 + 282 T + 23923217 T^{2} + 6738871938 T^{3} + 379712413586628 T^{4} + 93305349372909858 T^{5} + \)\(45\!\cdots\!77\)\( T^{6} + \)\(74\!\cdots\!22\)\( T^{7} + \)\(36\!\cdots\!61\)\( T^{8} \)
$67$ \( 1 + 1318 T - 38954849 T^{2} + 513665458 T^{3} + 1214763993160084 T^{4} + 10350934797678418 T^{5} - \)\(15\!\cdots\!09\)\( T^{6} + \)\(10\!\cdots\!98\)\( T^{7} + \)\(16\!\cdots\!81\)\( T^{8} \)
$71$ \( ( 1 + 5204 T + 56347598 T^{2} + 132242387924 T^{3} + 645753531245761 T^{4} )^{2} \)
$73$ \( 1 - 5214 T + 63240953 T^{2} - 282489415494 T^{3} + 2386249153485972 T^{4} - 8022202501147746054 T^{5} + \)\(51\!\cdots\!93\)\( T^{6} - \)\(11\!\cdots\!94\)\( T^{7} + \)\(65\!\cdots\!61\)\( T^{8} \)
$79$ \( 1 + 8110 T - 25860665 T^{2} + 111371410330 T^{3} + 4317626189098564 T^{4} + 4337925453437736730 T^{5} - \)\(39\!\cdots\!65\)\( T^{6} + \)\(47\!\cdots\!10\)\( T^{7} + \)\(23\!\cdots\!21\)\( T^{8} \)
$83$ \( 1 - 166506148 T^{2} + 11414799577931910 T^{4} - \)\(37\!\cdots\!68\)\( T^{6} + \)\(50\!\cdots\!81\)\( T^{8} \)
$89$ \( 1 - 33990 T + 597537041 T^{2} - 7220507290590 T^{3} + 65352518353788900 T^{4} - \)\(45\!\cdots\!90\)\( T^{5} + \)\(23\!\cdots\!21\)\( T^{6} - \)\(83\!\cdots\!90\)\( T^{7} + \)\(15\!\cdots\!61\)\( T^{8} \)
$97$ \( 1 - 137047516 T^{2} + 19765432645146054 T^{4} - \)\(10\!\cdots\!76\)\( T^{6} + \)\(61\!\cdots\!21\)\( T^{8} \)
show more
show less