Properties

Label 7.9.b.b
Level 7
Weight 9
Character orbit 7.b
Analytic conductor 2.852
Analytic rank 0
Dimension 4
CM No
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 7 \)
Weight: \( k \) = \( 9 \)
Character orbit: \([\chi]\) = 7.b (of order \(2\) and degree \(1\))

Newform invariants

Self dual: No
Analytic conductor: \(2.85165027043\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\mathbb{Q}[x]/(x^{4} + \cdots)\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{4}\cdot 3\cdot 7 \)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q\) \( + ( 8 + \beta_{3} ) q^{2} \) \( -\beta_{1} q^{3} \) \( + ( -8 + 16 \beta_{3} ) q^{4} \) \( + ( 2 \beta_{1} + \beta_{2} ) q^{5} \) \( + ( -17 \beta_{1} - \beta_{2} ) q^{6} \) \( + ( 357 + 16 \beta_{1} - \beta_{2} + 119 \beta_{3} ) q^{7} \) \( + ( 832 - 136 \beta_{3} ) q^{8} \) \( + ( -2031 - 426 \beta_{3} ) q^{9} \) \(+O(q^{10})\) \( q\) \( + ( 8 + \beta_{3} ) q^{2} \) \( -\beta_{1} q^{3} \) \( + ( -8 + 16 \beta_{3} ) q^{4} \) \( + ( 2 \beta_{1} + \beta_{2} ) q^{5} \) \( + ( -17 \beta_{1} - \beta_{2} ) q^{6} \) \( + ( 357 + 16 \beta_{1} - \beta_{2} + 119 \beta_{3} ) q^{7} \) \( + ( 832 - 136 \beta_{3} ) q^{8} \) \( + ( -2031 - 426 \beta_{3} ) q^{9} \) \( + ( 137 \beta_{1} + \beta_{2} ) q^{10} \) \( + ( -5542 - 68 \beta_{3} ) q^{11} \) \( + ( -136 \beta_{1} - 16 \beta_{2} ) q^{12} \) \( + ( -272 \beta_{1} + 17 \beta_{2} ) q^{13} \) \( + ( 24752 + 169 \beta_{1} + 17 \beta_{2} + 1309 \beta_{3} ) q^{14} \) \( + ( 18240 + 5610 \beta_{3} ) q^{15} \) \( + ( -16320 - 4352 \beta_{3} ) q^{16} \) \( + ( -630 \beta_{1} - 18 \beta_{2} ) q^{17} \) \( + ( -94632 - 5439 \beta_{3} ) q^{18} \) \( + ( 969 \beta_{1} + 102 \beta_{2} ) q^{19} \) \( + ( 1920 \beta_{1} - 120 \beta_{2} ) q^{20} \) \( + ( 136416 - 1428 \beta_{1} - 119 \beta_{2} + 2058 \beta_{3} ) q^{21} \) \( + ( -56848 - 6086 \beta_{3} ) q^{22} \) \( + ( 227018 - 5984 \beta_{3} ) q^{23} \) \( + ( 392 \beta_{1} + 136 \beta_{2} ) q^{24} \) \( + ( -513935 + 21030 \beta_{3} ) q^{25} \) \( + ( -2873 \beta_{1} - 289 \beta_{2} ) q^{26} \) \( + ( -696 \beta_{1} + 426 \beta_{2} ) q^{27} \) \( + ( 347480 + 528 \beta_{1} + 408 \beta_{2} + 4760 \beta_{3} ) q^{28} \) \( + ( -368254 - 23324 \beta_{3} ) q^{29} \) \( + ( 1178160 + 63120 \beta_{3} ) q^{30} \) \( + ( -2582 \beta_{1} - 526 \beta_{2} ) q^{31} \) \( + ( -1144320 - 16320 \beta_{3} ) q^{32} \) \( + ( 6154 \beta_{1} + 68 \beta_{2} ) q^{33} \) \( + ( -12564 \beta_{1} - 612 \beta_{2} ) q^{34} \) \( + ( 576240 + 15113 \beta_{1} - 476 \beta_{2} - 122010 \beta_{3} ) q^{35} \) \( + ( -1237896 - 29088 \beta_{3} ) q^{36} \) \( + ( 1678818 + 29716 \beta_{3} ) q^{37} \) \( + ( 26979 \beta_{1} + 867 \beta_{2} ) q^{38} \) \( + ( -2319072 - 34986 \beta_{3} ) q^{39} \) \( + ( -14792 \beta_{1} + 1784 \beta_{2} ) q^{40} \) \( + ( 5456 \beta_{1} - 782 \beta_{2} ) q^{41} \) \( + ( 1470000 - 36533 \beta_{1} - 1309 \beta_{2} + 152880 \beta_{3} ) q^{42} \) \( + ( 1437018 + 173656 \beta_{3} ) q^{43} \) \( + ( -155856 - 88128 \beta_{3} ) q^{44} \) \( + ( -55608 \beta_{1} + 951 \beta_{2} ) q^{45} \) \( + ( 715088 + 179146 \beta_{3} ) q^{46} \) \( + ( 16966 \beta_{1} - 3910 \beta_{2} ) q^{47} \) \( + ( 55488 \beta_{1} + 4352 \beta_{2} ) q^{48} \) \( + ( -298655 + 21182 \beta_{1} + 5236 \beta_{2} + 169932 \beta_{3} ) q^{49} \) \( + ( -241960 - 345695 \beta_{3} ) q^{50} \) \( + ( -5431968 - 354024 \beta_{3} ) q^{51} \) \( + ( -8976 \beta_{1} - 6936 \beta_{2} ) q^{52} \) \( + ( 1687394 - 379832 \beta_{3} ) q^{53} \) \( + ( 32046 \beta_{1} - 1122 \beta_{2} ) q^{54} \) \( + ( -19312 \beta_{1} - 5066 \beta_{2} ) q^{55} \) \( + ( -2680832 + 7736 \beta_{1} - 4232 \beta_{2} + 50456 \beta_{3} ) q^{56} \) \( + ( 8433360 + 898110 \beta_{3} ) q^{57} \) \( + ( -7237648 - 554846 \beta_{3} ) q^{58} \) \( + ( 67847 \beta_{1} + 9826 \beta_{2} ) q^{59} \) \( + ( 16369920 + 246960 \beta_{3} ) q^{60} \) \( + ( -67474 \beta_{1} + 13243 \beta_{2} ) q^{61} \) \( + ( -98072 \beta_{1} - 2056 \beta_{2} ) q^{62} \) \( + ( -10052763 - 49962 \beta_{1} - 8619 \beta_{2} - 393771 \beta_{3} ) q^{63} \) \( + ( -7979520 - 160768 \beta_{3} ) q^{64} \) \( + ( -9796080 + 2074170 \beta_{3} ) q^{65} \) \( + ( 111622 \beta_{1} + 6086 \beta_{2} ) q^{66} \) \( + ( 17506778 - 580704 \beta_{3} ) q^{67} \) \( + ( -115344 \beta_{1} - 7344 \beta_{2} ) q^{68} \) \( + ( -173162 \beta_{1} + 5984 \beta_{2} ) q^{69} \) \( + ( -17839920 + 207893 \beta_{1} + 15589 \beta_{2} - 399840 \beta_{3} ) q^{70} \) \( + ( 12475178 - 1287818 \beta_{3} ) q^{71} \) \( + ( 8970432 - 78216 \beta_{3} ) q^{72} \) \( + ( -49794 \beta_{1} - 29784 \beta_{2} ) q^{73} \) \( + ( 18898288 + 1916546 \beta_{3} ) q^{74} \) \( + ( 324665 \beta_{1} - 21030 \beta_{2} ) q^{75} \) \( + 299880 \beta_{1} q^{76} \) \( + ( -3467422 - 91460 \beta_{1} + 3842 \beta_{2} - 683774 \beta_{3} ) q^{77} \) \( + ( -24990000 - 2598960 \beta_{3} ) q^{78} \) \( + ( -20541814 - 1054374 \beta_{3} ) q^{79} \) \( + ( -559232 \beta_{1} + 14144 \beta_{2} ) q^{80} \) \( + ( -18855567 - 1064574 \beta_{3} ) q^{81} \) \( + ( 12206 \beta_{1} + 6238 \beta_{2} ) q^{82} \) \( + ( 597295 \beta_{1} + 18428 \beta_{2} ) q^{83} \) \( + ( 4967424 - 390320 \beta_{1} - 4760 \beta_{2} + 2166192 \beta_{3} ) q^{84} \) \( + ( 27116640 + 2953800 \beta_{3} ) q^{85} \) \( + ( 43448848 + 2826266 \beta_{3} ) q^{86} \) \( + ( 578170 \beta_{1} + 23324 \beta_{2} ) q^{87} \) \( + ( -2909312 + 697136 \beta_{3} ) q^{88} \) \( + ( -801618 \beta_{1} + 46716 \beta_{2} ) q^{89} \) \( + ( -847383 \beta_{1} - 56559 \beta_{2} ) q^{90} \) \( + ( 51539376 - 180047 \beta_{1} - 44506 \beta_{2} - 1444422 \beta_{3} ) q^{91} \) \( + ( -19433040 + 3680160 \beta_{3} ) q^{92} \) \( + ( -22740000 - 3602640 \beta_{3} ) q^{93} \) \( + ( -114308 \beta_{1} + 20876 \beta_{2} ) q^{94} \) \( + ( -106218720 - 2146590 \beta_{3} ) q^{95} \) \( + ( 1291200 \beta_{1} + 16320 \beta_{2} ) q^{96} \) \( + ( 203798 \beta_{1} + 13006 \beta_{2} ) q^{97} \) \( + ( 28878248 + 899402 \beta_{1} + 15946 \beta_{2} + 1060801 \beta_{3} ) q^{98} \) \( + ( 16585914 + 2499000 \beta_{3} ) q^{99} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(4q \) \(\mathstrut +\mathstrut 32q^{2} \) \(\mathstrut -\mathstrut 32q^{4} \) \(\mathstrut +\mathstrut 1428q^{7} \) \(\mathstrut +\mathstrut 3328q^{8} \) \(\mathstrut -\mathstrut 8124q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(4q \) \(\mathstrut +\mathstrut 32q^{2} \) \(\mathstrut -\mathstrut 32q^{4} \) \(\mathstrut +\mathstrut 1428q^{7} \) \(\mathstrut +\mathstrut 3328q^{8} \) \(\mathstrut -\mathstrut 8124q^{9} \) \(\mathstrut -\mathstrut 22168q^{11} \) \(\mathstrut +\mathstrut 99008q^{14} \) \(\mathstrut +\mathstrut 72960q^{15} \) \(\mathstrut -\mathstrut 65280q^{16} \) \(\mathstrut -\mathstrut 378528q^{18} \) \(\mathstrut +\mathstrut 545664q^{21} \) \(\mathstrut -\mathstrut 227392q^{22} \) \(\mathstrut +\mathstrut 908072q^{23} \) \(\mathstrut -\mathstrut 2055740q^{25} \) \(\mathstrut +\mathstrut 1389920q^{28} \) \(\mathstrut -\mathstrut 1473016q^{29} \) \(\mathstrut +\mathstrut 4712640q^{30} \) \(\mathstrut -\mathstrut 4577280q^{32} \) \(\mathstrut +\mathstrut 2304960q^{35} \) \(\mathstrut -\mathstrut 4951584q^{36} \) \(\mathstrut +\mathstrut 6715272q^{37} \) \(\mathstrut -\mathstrut 9276288q^{39} \) \(\mathstrut +\mathstrut 5880000q^{42} \) \(\mathstrut +\mathstrut 5748072q^{43} \) \(\mathstrut -\mathstrut 623424q^{44} \) \(\mathstrut +\mathstrut 2860352q^{46} \) \(\mathstrut -\mathstrut 1194620q^{49} \) \(\mathstrut -\mathstrut 967840q^{50} \) \(\mathstrut -\mathstrut 21727872q^{51} \) \(\mathstrut +\mathstrut 6749576q^{53} \) \(\mathstrut -\mathstrut 10723328q^{56} \) \(\mathstrut +\mathstrut 33733440q^{57} \) \(\mathstrut -\mathstrut 28950592q^{58} \) \(\mathstrut +\mathstrut 65479680q^{60} \) \(\mathstrut -\mathstrut 40211052q^{63} \) \(\mathstrut -\mathstrut 31918080q^{64} \) \(\mathstrut -\mathstrut 39184320q^{65} \) \(\mathstrut +\mathstrut 70027112q^{67} \) \(\mathstrut -\mathstrut 71359680q^{70} \) \(\mathstrut +\mathstrut 49900712q^{71} \) \(\mathstrut +\mathstrut 35881728q^{72} \) \(\mathstrut +\mathstrut 75593152q^{74} \) \(\mathstrut -\mathstrut 13869688q^{77} \) \(\mathstrut -\mathstrut 99960000q^{78} \) \(\mathstrut -\mathstrut 82167256q^{79} \) \(\mathstrut -\mathstrut 75422268q^{81} \) \(\mathstrut +\mathstrut 19869696q^{84} \) \(\mathstrut +\mathstrut 108466560q^{85} \) \(\mathstrut +\mathstrut 173795392q^{86} \) \(\mathstrut -\mathstrut 11637248q^{88} \) \(\mathstrut +\mathstrut 206157504q^{91} \) \(\mathstrut -\mathstrut 77732160q^{92} \) \(\mathstrut -\mathstrut 90960000q^{93} \) \(\mathstrut -\mathstrut 424874880q^{95} \) \(\mathstrut +\mathstrut 115512992q^{98} \) \(\mathstrut +\mathstrut 66343656q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4}\mathstrut +\mathstrut \) \(1016\) \(x^{2}\mathstrut +\mathstrut \) \(51570\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( 4 \nu^{3} + 3506 \nu \)\()/201\)
\(\beta_{2}\)\(=\)\((\)\( 8 \nu^{3} + 15454 \nu \)\()/201\)
\(\beta_{3}\)\(=\)\((\)\( 2 \nu^{2} + 1016 \)\()/67\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{2}\mathstrut -\mathstrut \) \(2\) \(\beta_{1}\)\()/42\)
\(\nu^{2}\)\(=\)\((\)\(67\) \(\beta_{3}\mathstrut -\mathstrut \) \(1016\)\()/2\)
\(\nu^{3}\)\(=\)\((\)\(-\)\(1753\) \(\beta_{2}\mathstrut +\mathstrut \) \(7727\) \(\beta_{1}\)\()/84\)

Character Values

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

\(n\) \(3\)
\(\chi(n)\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
6.1
31.0228i
31.0228i
7.32010i
7.32010i
−5.56466 53.0420i −225.035 1090.79i 295.161i −1257.19 + 2045.55i 2676.79 3747.55 6069.88i
6.2 −5.56466 53.0420i −225.035 1090.79i 295.161i −1257.19 2045.55i 2676.79 3747.55 6069.88i
6.3 21.5647 119.877i 209.035 786.953i 2585.11i 1971.19 + 1370.84i −1012.79 −7809.55 16970.4i
6.4 21.5647 119.877i 209.035 786.953i 2585.11i 1971.19 1370.84i −1012.79 −7809.55 16970.4i
\(n\): 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.b Odd 1 yes

Hecke kernels

This newform can be constructed as the kernel of the linear operator \(T_{2}^{2} \) \(\mathstrut -\mathstrut 16 T_{2} \) \(\mathstrut -\mathstrut 120 \) acting on \(S_{9}^{\mathrm{new}}(7, [\chi])\).