Properties

Label 7.7.b.b
Level 7
Weight 7
Character orbit 7.b
Analytic conductor 1.610
Analytic rank 0
Dimension 2
CM No
Inner twists 2

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Newspace parameters

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

Newform invariants

Self dual: No
Analytic conductor: \(1.61037858534\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-510}) \)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = 2\sqrt{-510}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q\) \( -8 q^{2} \) \( + \beta q^{3} \) \( -\beta q^{5} \) \( -8 \beta q^{6} \) \( + ( 133 + 7 \beta ) q^{7} \) \( + 512 q^{8} \) \( -1311 q^{9} \) \(+O(q^{10})\) \( q\) \( -8 q^{2} \) \( + \beta q^{3} \) \( -\beta q^{5} \) \( -8 \beta q^{6} \) \( + ( 133 + 7 \beta ) q^{7} \) \( + 512 q^{8} \) \( -1311 q^{9} \) \( + 8 \beta q^{10} \) \( + 874 q^{11} \) \( + 49 \beta q^{13} \) \( + ( -1064 - 56 \beta ) q^{14} \) \( + 2040 q^{15} \) \( -4096 q^{16} \) \( -132 \beta q^{17} \) \( + 10488 q^{18} \) \( + 69 \beta q^{19} \) \( + ( -14280 + 133 \beta ) q^{21} \) \( -6992 q^{22} \) \( + 4738 q^{23} \) \( + 512 \beta q^{24} \) \( + 13585 q^{25} \) \( -392 \beta q^{26} \) \( -582 \beta q^{27} \) \( + 11146 q^{29} \) \( -16320 q^{30} \) \( -608 \beta q^{31} \) \( + 874 \beta q^{33} \) \( + 1056 \beta q^{34} \) \( + ( 14280 - 133 \beta ) q^{35} \) \( + 3002 q^{37} \) \( -552 \beta q^{38} \) \( -99960 q^{39} \) \( -512 \beta q^{40} \) \( + 1274 \beta q^{41} \) \( + ( 114240 - 1064 \beta ) q^{42} \) \( + 31418 q^{43} \) \( + 1311 \beta q^{45} \) \( -37904 q^{46} \) \( -1604 \beta q^{47} \) \( -4096 \beta q^{48} \) \( + ( -82271 + 1862 \beta ) q^{49} \) \( -108680 q^{50} \) \( + 269280 q^{51} \) \( -76406 q^{53} \) \( + 4656 \beta q^{54} \) \( -874 \beta q^{55} \) \( + ( 68096 + 3584 \beta ) q^{56} \) \( -140760 q^{57} \) \( -89168 q^{58} \) \( + 2507 \beta q^{59} \) \( -6091 \beta q^{61} \) \( + 4864 \beta q^{62} \) \( + ( -174363 - 9177 \beta ) q^{63} \) \( + 262144 q^{64} \) \( + 99960 q^{65} \) \( -6992 \beta q^{66} \) \( + 495242 q^{67} \) \( + 4738 \beta q^{69} \) \( + ( -114240 + 1064 \beta ) q^{70} \) \( -184406 q^{71} \) \( -671232 q^{72} \) \( -1350 \beta q^{73} \) \( -24016 q^{74} \) \( + 13585 \beta q^{75} \) \( + ( 116242 + 6118 \beta ) q^{77} \) \( + 799680 q^{78} \) \( -534934 q^{79} \) \( + 4096 \beta q^{80} \) \( + 231561 q^{81} \) \( -10192 \beta q^{82} \) \( -15827 \beta q^{83} \) \( -269280 q^{85} \) \( -251344 q^{86} \) \( + 11146 \beta q^{87} \) \( + 447488 q^{88} \) \( -13938 \beta q^{89} \) \( -10488 \beta q^{90} \) \( + ( -699720 + 6517 \beta ) q^{91} \) \( + 1240320 q^{93} \) \( + 12832 \beta q^{94} \) \( + 140760 q^{95} \) \( + 18032 \beta q^{97} \) \( + ( 658168 - 14896 \beta ) q^{98} \) \( -1145814 q^{99} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(2q \) \(\mathstrut -\mathstrut 16q^{2} \) \(\mathstrut +\mathstrut 266q^{7} \) \(\mathstrut +\mathstrut 1024q^{8} \) \(\mathstrut -\mathstrut 2622q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(2q \) \(\mathstrut -\mathstrut 16q^{2} \) \(\mathstrut +\mathstrut 266q^{7} \) \(\mathstrut +\mathstrut 1024q^{8} \) \(\mathstrut -\mathstrut 2622q^{9} \) \(\mathstrut +\mathstrut 1748q^{11} \) \(\mathstrut -\mathstrut 2128q^{14} \) \(\mathstrut +\mathstrut 4080q^{15} \) \(\mathstrut -\mathstrut 8192q^{16} \) \(\mathstrut +\mathstrut 20976q^{18} \) \(\mathstrut -\mathstrut 28560q^{21} \) \(\mathstrut -\mathstrut 13984q^{22} \) \(\mathstrut +\mathstrut 9476q^{23} \) \(\mathstrut +\mathstrut 27170q^{25} \) \(\mathstrut +\mathstrut 22292q^{29} \) \(\mathstrut -\mathstrut 32640q^{30} \) \(\mathstrut +\mathstrut 28560q^{35} \) \(\mathstrut +\mathstrut 6004q^{37} \) \(\mathstrut -\mathstrut 199920q^{39} \) \(\mathstrut +\mathstrut 228480q^{42} \) \(\mathstrut +\mathstrut 62836q^{43} \) \(\mathstrut -\mathstrut 75808q^{46} \) \(\mathstrut -\mathstrut 164542q^{49} \) \(\mathstrut -\mathstrut 217360q^{50} \) \(\mathstrut +\mathstrut 538560q^{51} \) \(\mathstrut -\mathstrut 152812q^{53} \) \(\mathstrut +\mathstrut 136192q^{56} \) \(\mathstrut -\mathstrut 281520q^{57} \) \(\mathstrut -\mathstrut 178336q^{58} \) \(\mathstrut -\mathstrut 348726q^{63} \) \(\mathstrut +\mathstrut 524288q^{64} \) \(\mathstrut +\mathstrut 199920q^{65} \) \(\mathstrut +\mathstrut 990484q^{67} \) \(\mathstrut -\mathstrut 228480q^{70} \) \(\mathstrut -\mathstrut 368812q^{71} \) \(\mathstrut -\mathstrut 1342464q^{72} \) \(\mathstrut -\mathstrut 48032q^{74} \) \(\mathstrut +\mathstrut 232484q^{77} \) \(\mathstrut +\mathstrut 1599360q^{78} \) \(\mathstrut -\mathstrut 1069868q^{79} \) \(\mathstrut +\mathstrut 463122q^{81} \) \(\mathstrut -\mathstrut 538560q^{85} \) \(\mathstrut -\mathstrut 502688q^{86} \) \(\mathstrut +\mathstrut 894976q^{88} \) \(\mathstrut -\mathstrut 1399440q^{91} \) \(\mathstrut +\mathstrut 2480640q^{93} \) \(\mathstrut +\mathstrut 281520q^{95} \) \(\mathstrut +\mathstrut 1316336q^{98} \) \(\mathstrut -\mathstrut 2291628q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

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
22.5832i
22.5832i
−8.00000 45.1664i 0 45.1664i 361.331i 133.000 316.165i 512.000 −1311.00 361.331i
6.2 −8.00000 45.1664i 0 45.1664i 361.331i 133.000 + 316.165i 512.000 −1311.00 361.331i
\(n\): 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 \(T_{2} \) \(\mathstrut +\mathstrut 8 \) acting on \(S_{7}^{\mathrm{new}}(7, [\chi])\).