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 \(3\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.413013370040\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
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.

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

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 Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 7.4.c.a 2
3.b odd 2 1 63.4.e.b 2
4.b odd 2 1 112.4.i.c 2
5.b even 2 1 175.4.e.a 2
5.c odd 4 2 175.4.k.a 4
7.b odd 2 1 49.4.c.a 2
7.c even 3 1 inner 7.4.c.a 2
7.c even 3 1 49.4.a.d 1
7.d odd 6 1 49.4.a.c 1
7.d odd 6 1 49.4.c.a 2
8.b even 2 1 448.4.i.f 2
8.d odd 2 1 448.4.i.a 2
21.c even 2 1 441.4.e.k 2
21.g even 6 1 441.4.a.e 1
21.g even 6 1 441.4.e.k 2
21.h odd 6 1 63.4.e.b 2
21.h odd 6 1 441.4.a.d 1
28.f even 6 1 784.4.a.r 1
28.g odd 6 1 112.4.i.c 2
28.g odd 6 1 784.4.a.b 1
35.i odd 6 1 1225.4.a.d 1
35.j even 6 1 175.4.e.a 2
35.j even 6 1 1225.4.a.c 1
35.l odd 12 2 175.4.k.a 4
56.k odd 6 1 448.4.i.a 2
56.p even 6 1 448.4.i.f 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
7.4.c.a 2 1.a even 1 1 trivial
7.4.c.a 2 7.c even 3 1 inner
49.4.a.c 1 7.d odd 6 1
49.4.a.d 1 7.c even 3 1
49.4.c.a 2 7.b odd 2 1
49.4.c.a 2 7.d odd 6 1
63.4.e.b 2 3.b odd 2 1
63.4.e.b 2 21.h odd 6 1
112.4.i.c 2 4.b odd 2 1
112.4.i.c 2 28.g odd 6 1
175.4.e.a 2 5.b even 2 1
175.4.e.a 2 35.j even 6 1
175.4.k.a 4 5.c odd 4 2
175.4.k.a 4 35.l odd 12 2
441.4.a.d 1 21.h odd 6 1
441.4.a.e 1 21.g even 6 1
441.4.e.k 2 21.c even 2 1
441.4.e.k 2 21.g even 6 1
448.4.i.a 2 8.d odd 2 1
448.4.i.a 2 56.k odd 6 1
448.4.i.f 2 8.b even 2 1
448.4.i.f 2 56.p even 6 1
784.4.a.b 1 28.g odd 6 1
784.4.a.r 1 28.f even 6 1
1225.4.a.c 1 35.j even 6 1
1225.4.a.d 1 35.i odd 6 1

Hecke kernels

This newform subspace is the entire newspace \(S_{4}^{\mathrm{new}}(7, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + 2T + 4 \) Copy content Toggle raw display
$3$ \( T^{2} + 7T + 49 \) Copy content Toggle raw display
$5$ \( T^{2} + 7T + 49 \) Copy content Toggle raw display
$7$ \( T^{2} - 28T + 343 \) Copy content Toggle raw display
$11$ \( T^{2} - 5T + 25 \) Copy content Toggle raw display
$13$ \( (T + 14)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} - 21T + 441 \) Copy content Toggle raw display
$19$ \( T^{2} + 49T + 2401 \) Copy content Toggle raw display
$23$ \( T^{2} - 159T + 25281 \) Copy content Toggle raw display
$29$ \( (T - 58)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} + 147T + 21609 \) Copy content Toggle raw display
$37$ \( T^{2} + 219T + 47961 \) Copy content Toggle raw display
$41$ \( (T - 350)^{2} \) Copy content Toggle raw display
$43$ \( (T + 124)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} + 525T + 275625 \) Copy content Toggle raw display
$53$ \( T^{2} + 303T + 91809 \) Copy content Toggle raw display
$59$ \( T^{2} - 105T + 11025 \) Copy content Toggle raw display
$61$ \( T^{2} - 413T + 170569 \) Copy content Toggle raw display
$67$ \( T^{2} + 415T + 172225 \) Copy content Toggle raw display
$71$ \( (T + 432)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} - 1113 T + 1238769 \) Copy content Toggle raw display
$79$ \( T^{2} - 103T + 10609 \) Copy content Toggle raw display
$83$ \( (T - 1092)^{2} \) Copy content Toggle raw display
$89$ \( T^{2} - 329T + 108241 \) Copy content Toggle raw display
$97$ \( (T + 882)^{2} \) Copy content Toggle raw display
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