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: \(x^{2} - x + 1\)
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.

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

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 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

$p$ $F_p(T)$
$2$ \( 1 + 2 T - 4 T^{2} + 16 T^{3} + 64 T^{4} \)
$3$ \( 1 + 7 T + 22 T^{2} + 189 T^{3} + 729 T^{4} \)
$5$ \( 1 + 7 T - 76 T^{2} + 875 T^{3} + 15625 T^{4} \)
$7$ \( 1 - 28 T + 343 T^{2} \)
$11$ \( 1 - 5 T - 1306 T^{2} - 6655 T^{3} + 1771561 T^{4} \)
$13$ \( ( 1 + 14 T + 2197 T^{2} )^{2} \)
$17$ \( 1 - 21 T - 4472 T^{2} - 103173 T^{3} + 24137569 T^{4} \)
$19$ \( 1 + 49 T - 4458 T^{2} + 336091 T^{3} + 47045881 T^{4} \)
$23$ \( 1 - 159 T + 13114 T^{2} - 1934553 T^{3} + 148035889 T^{4} \)
$29$ \( ( 1 - 58 T + 24389 T^{2} )^{2} \)
$31$ \( 1 + 147 T - 8182 T^{2} + 4379277 T^{3} + 887503681 T^{4} \)
$37$ \( 1 + 219 T - 2692 T^{2} + 11093007 T^{3} + 2565726409 T^{4} \)
$41$ \( ( 1 - 350 T + 68921 T^{2} )^{2} \)
$43$ \( ( 1 + 124 T + 79507 T^{2} )^{2} \)
$47$ \( 1 + 525 T + 171802 T^{2} + 54507075 T^{3} + 10779215329 T^{4} \)
$53$ \( 1 + 303 T - 57068 T^{2} + 45109731 T^{3} + 22164361129 T^{4} \)
$59$ \( 1 - 105 T - 194354 T^{2} - 21564795 T^{3} + 42180533641 T^{4} \)
$61$ \( 1 - 413 T - 56412 T^{2} - 93743153 T^{3} + 51520374361 T^{4} \)
$67$ \( 1 + 415 T - 128538 T^{2} + 124816645 T^{3} + 90458382169 T^{4} \)
$71$ \( ( 1 + 432 T + 357911 T^{2} )^{2} \)
$73$ \( 1 - 1113 T + 849752 T^{2} - 432975921 T^{3} + 151334226289 T^{4} \)
$79$ \( 1 - 103 T - 482430 T^{2} - 50783017 T^{3} + 243087455521 T^{4} \)
$83$ \( ( 1 - 1092 T + 571787 T^{2} )^{2} \)
$89$ \( 1 - 329 T - 596728 T^{2} - 231934801 T^{3} + 496981290961 T^{4} \)
$97$ \( ( 1 + 882 T + 912673 T^{2} )^{2} \)
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