Properties

Label 7.4.c.a
Level 7
Weight 4
Character orbit 7.c
Analytic conductor 0.413
Analytic rank 0
Dimension 2
CM No
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 7 \)
Weight: \( k \) = \( 4 \)
Character orbit: \([\chi]\) = 7.c (of order \(3\) and degree \(2\))

Newform invariants

Self dual: No
Analytic conductor: \(0.41301337004\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q\) \( + ( -2 + 2 \zeta_{6} ) q^{2} \) \( -7 \zeta_{6} q^{3} \) \( + 4 \zeta_{6} q^{4} \) \( + ( -7 + 7 \zeta_{6} ) q^{5} \) \( + 14 q^{6} \) \( + ( 21 - 14 \zeta_{6} ) q^{7} \) \( -24 q^{8} \) \( + ( -22 + 22 \zeta_{6} ) q^{9} \) \(+O(q^{10})\) \( q\) \( + ( -2 + 2 \zeta_{6} ) q^{2} \) \( -7 \zeta_{6} q^{3} \) \( + 4 \zeta_{6} q^{4} \) \( + ( -7 + 7 \zeta_{6} ) q^{5} \) \( + 14 q^{6} \) \( + ( 21 - 14 \zeta_{6} ) q^{7} \) \( -24 q^{8} \) \( + ( -22 + 22 \zeta_{6} ) q^{9} \) \( -14 \zeta_{6} q^{10} \) \( + 5 \zeta_{6} q^{11} \) \( + ( 28 - 28 \zeta_{6} ) q^{12} \) \( -14 q^{13} \) \( + ( -14 + 42 \zeta_{6} ) q^{14} \) \( + 49 q^{15} \) \( + ( 16 - 16 \zeta_{6} ) q^{16} \) \( + 21 \zeta_{6} q^{17} \) \( -44 \zeta_{6} q^{18} \) \( + ( -49 + 49 \zeta_{6} ) q^{19} \) \( -28 q^{20} \) \( + ( -98 - 49 \zeta_{6} ) q^{21} \) \( -10 q^{22} \) \( + ( 159 - 159 \zeta_{6} ) q^{23} \) \( + 168 \zeta_{6} q^{24} \) \( + 76 \zeta_{6} q^{25} \) \( + ( 28 - 28 \zeta_{6} ) q^{26} \) \( -35 q^{27} \) \( + ( 56 + 28 \zeta_{6} ) q^{28} \) \( + 58 q^{29} \) \( + ( -98 + 98 \zeta_{6} ) q^{30} \) \( -147 \zeta_{6} q^{31} \) \( -160 \zeta_{6} q^{32} \) \( + ( 35 - 35 \zeta_{6} ) q^{33} \) \( -42 q^{34} \) \( + ( -49 + 147 \zeta_{6} ) q^{35} \) \( -88 q^{36} \) \( + ( -219 + 219 \zeta_{6} ) q^{37} \) \( -98 \zeta_{6} q^{38} \) \( + 98 \zeta_{6} q^{39} \) \( + ( 168 - 168 \zeta_{6} ) q^{40} \) \( + 350 q^{41} \) \( + ( 294 - 196 \zeta_{6} ) q^{42} \) \( -124 q^{43} \) \( + ( -20 + 20 \zeta_{6} ) q^{44} \) \( -154 \zeta_{6} q^{45} \) \( + 318 \zeta_{6} q^{46} \) \( + ( -525 + 525 \zeta_{6} ) q^{47} \) \( -112 q^{48} \) \( + ( 245 - 392 \zeta_{6} ) q^{49} \) \( -152 q^{50} \) \( + ( 147 - 147 \zeta_{6} ) q^{51} \) \( -56 \zeta_{6} q^{52} \) \( -303 \zeta_{6} q^{53} \) \( + ( 70 - 70 \zeta_{6} ) q^{54} \) \( -35 q^{55} \) \( + ( -504 + 336 \zeta_{6} ) q^{56} \) \( + 343 q^{57} \) \( + ( -116 + 116 \zeta_{6} ) q^{58} \) \( + 105 \zeta_{6} q^{59} \) \( + 196 \zeta_{6} q^{60} \) \( + ( 413 - 413 \zeta_{6} ) q^{61} \) \( + 294 q^{62} \) \( + ( -154 + 462 \zeta_{6} ) q^{63} \) \( + 448 q^{64} \) \( + ( 98 - 98 \zeta_{6} ) q^{65} \) \( + 70 \zeta_{6} q^{66} \) \( -415 \zeta_{6} q^{67} \) \( + ( -84 + 84 \zeta_{6} ) q^{68} \) \( -1113 q^{69} \) \( + ( -196 - 98 \zeta_{6} ) q^{70} \) \( -432 q^{71} \) \( + ( 528 - 528 \zeta_{6} ) q^{72} \) \( + 1113 \zeta_{6} q^{73} \) \( -438 \zeta_{6} q^{74} \) \( + ( 532 - 532 \zeta_{6} ) q^{75} \) \( -196 q^{76} \) \( + ( 70 + 35 \zeta_{6} ) q^{77} \) \( -196 q^{78} \) \( + ( 103 - 103 \zeta_{6} ) q^{79} \) \( + 112 \zeta_{6} q^{80} \) \( + 839 \zeta_{6} q^{81} \) \( + ( -700 + 700 \zeta_{6} ) q^{82} \) \( + 1092 q^{83} \) \( + ( 196 - 588 \zeta_{6} ) q^{84} \) \( -147 q^{85} \) \( + ( 248 - 248 \zeta_{6} ) q^{86} \) \( -406 \zeta_{6} q^{87} \) \( -120 \zeta_{6} q^{88} \) \( + ( 329 - 329 \zeta_{6} ) q^{89} \) \( + 308 q^{90} \) \( + ( -294 + 196 \zeta_{6} ) q^{91} \) \( + 636 q^{92} \) \( + ( -1029 + 1029 \zeta_{6} ) q^{93} \) \( -1050 \zeta_{6} q^{94} \) \( -343 \zeta_{6} q^{95} \) \( + ( -1120 + 1120 \zeta_{6} ) q^{96} \) \( -882 q^{97} \) \( + ( 294 + 490 \zeta_{6} ) q^{98} \) \( -110 q^{99} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(2q \) \(\mathstrut -\mathstrut 2q^{2} \) \(\mathstrut -\mathstrut 7q^{3} \) \(\mathstrut +\mathstrut 4q^{4} \) \(\mathstrut -\mathstrut 7q^{5} \) \(\mathstrut +\mathstrut 28q^{6} \) \(\mathstrut +\mathstrut 28q^{7} \) \(\mathstrut -\mathstrut 48q^{8} \) \(\mathstrut -\mathstrut 22q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(2q \) \(\mathstrut -\mathstrut 2q^{2} \) \(\mathstrut -\mathstrut 7q^{3} \) \(\mathstrut +\mathstrut 4q^{4} \) \(\mathstrut -\mathstrut 7q^{5} \) \(\mathstrut +\mathstrut 28q^{6} \) \(\mathstrut +\mathstrut 28q^{7} \) \(\mathstrut -\mathstrut 48q^{8} \) \(\mathstrut -\mathstrut 22q^{9} \) \(\mathstrut -\mathstrut 14q^{10} \) \(\mathstrut +\mathstrut 5q^{11} \) \(\mathstrut +\mathstrut 28q^{12} \) \(\mathstrut -\mathstrut 28q^{13} \) \(\mathstrut +\mathstrut 14q^{14} \) \(\mathstrut +\mathstrut 98q^{15} \) \(\mathstrut +\mathstrut 16q^{16} \) \(\mathstrut +\mathstrut 21q^{17} \) \(\mathstrut -\mathstrut 44q^{18} \) \(\mathstrut -\mathstrut 49q^{19} \) \(\mathstrut -\mathstrut 56q^{20} \) \(\mathstrut -\mathstrut 245q^{21} \) \(\mathstrut -\mathstrut 20q^{22} \) \(\mathstrut +\mathstrut 159q^{23} \) \(\mathstrut +\mathstrut 168q^{24} \) \(\mathstrut +\mathstrut 76q^{25} \) \(\mathstrut +\mathstrut 28q^{26} \) \(\mathstrut -\mathstrut 70q^{27} \) \(\mathstrut +\mathstrut 140q^{28} \) \(\mathstrut +\mathstrut 116q^{29} \) \(\mathstrut -\mathstrut 98q^{30} \) \(\mathstrut -\mathstrut 147q^{31} \) \(\mathstrut -\mathstrut 160q^{32} \) \(\mathstrut +\mathstrut 35q^{33} \) \(\mathstrut -\mathstrut 84q^{34} \) \(\mathstrut +\mathstrut 49q^{35} \) \(\mathstrut -\mathstrut 176q^{36} \) \(\mathstrut -\mathstrut 219q^{37} \) \(\mathstrut -\mathstrut 98q^{38} \) \(\mathstrut +\mathstrut 98q^{39} \) \(\mathstrut +\mathstrut 168q^{40} \) \(\mathstrut +\mathstrut 700q^{41} \) \(\mathstrut +\mathstrut 392q^{42} \) \(\mathstrut -\mathstrut 248q^{43} \) \(\mathstrut -\mathstrut 20q^{44} \) \(\mathstrut -\mathstrut 154q^{45} \) \(\mathstrut +\mathstrut 318q^{46} \) \(\mathstrut -\mathstrut 525q^{47} \) \(\mathstrut -\mathstrut 224q^{48} \) \(\mathstrut +\mathstrut 98q^{49} \) \(\mathstrut -\mathstrut 304q^{50} \) \(\mathstrut +\mathstrut 147q^{51} \) \(\mathstrut -\mathstrut 56q^{52} \) \(\mathstrut -\mathstrut 303q^{53} \) \(\mathstrut +\mathstrut 70q^{54} \) \(\mathstrut -\mathstrut 70q^{55} \) \(\mathstrut -\mathstrut 672q^{56} \) \(\mathstrut +\mathstrut 686q^{57} \) \(\mathstrut -\mathstrut 116q^{58} \) \(\mathstrut +\mathstrut 105q^{59} \) \(\mathstrut +\mathstrut 196q^{60} \) \(\mathstrut +\mathstrut 413q^{61} \) \(\mathstrut +\mathstrut 588q^{62} \) \(\mathstrut +\mathstrut 154q^{63} \) \(\mathstrut +\mathstrut 896q^{64} \) \(\mathstrut +\mathstrut 98q^{65} \) \(\mathstrut +\mathstrut 70q^{66} \) \(\mathstrut -\mathstrut 415q^{67} \) \(\mathstrut -\mathstrut 84q^{68} \) \(\mathstrut -\mathstrut 2226q^{69} \) \(\mathstrut -\mathstrut 490q^{70} \) \(\mathstrut -\mathstrut 864q^{71} \) \(\mathstrut +\mathstrut 528q^{72} \) \(\mathstrut +\mathstrut 1113q^{73} \) \(\mathstrut -\mathstrut 438q^{74} \) \(\mathstrut +\mathstrut 532q^{75} \) \(\mathstrut -\mathstrut 392q^{76} \) \(\mathstrut +\mathstrut 175q^{77} \) \(\mathstrut -\mathstrut 392q^{78} \) \(\mathstrut +\mathstrut 103q^{79} \) \(\mathstrut +\mathstrut 112q^{80} \) \(\mathstrut +\mathstrut 839q^{81} \) \(\mathstrut -\mathstrut 700q^{82} \) \(\mathstrut +\mathstrut 2184q^{83} \) \(\mathstrut -\mathstrut 196q^{84} \) \(\mathstrut -\mathstrut 294q^{85} \) \(\mathstrut +\mathstrut 248q^{86} \) \(\mathstrut -\mathstrut 406q^{87} \) \(\mathstrut -\mathstrut 120q^{88} \) \(\mathstrut +\mathstrut 329q^{89} \) \(\mathstrut +\mathstrut 616q^{90} \) \(\mathstrut -\mathstrut 392q^{91} \) \(\mathstrut +\mathstrut 1272q^{92} \) \(\mathstrut -\mathstrut 1029q^{93} \) \(\mathstrut -\mathstrut 1050q^{94} \) \(\mathstrut -\mathstrut 343q^{95} \) \(\mathstrut -\mathstrut 1120q^{96} \) \(\mathstrut -\mathstrut 1764q^{97} \) \(\mathstrut +\mathstrut 1078q^{98} \) \(\mathstrut -\mathstrut 220q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Character Values

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

\(n\) \(3\)
\(\chi(n)\) \(-1 + \zeta_{6}\)

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
0.500000 + 0.866025i
0.500000 0.866025i
−1.00000 + 1.73205i −3.50000 6.06218i 2.00000 + 3.46410i −3.50000 + 6.06218i 14.0000 14.0000 12.1244i −24.0000 −11.0000 + 19.0526i −7.00000 12.1244i
4.1 −1.00000 1.73205i −3.50000 + 6.06218i 2.00000 3.46410i −3.50000 6.06218i 14.0000 14.0000 + 12.1244i −24.0000 −11.0000 19.0526i −7.00000 + 12.1244i
\(n\): 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 \(S_{4}^{\mathrm{new}}(7, [\chi])\).