Properties

Label 13.4.e.a
Level $13$
Weight $4$
Character orbit 13.e
Analytic conductor $0.767$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 13 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 13.e (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.767024830075\)
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: $\mathrm{SU}(2)[C_{6}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( - 2 \zeta_{6} - 2) q^{2} + ( - 7 \zeta_{6} + 7) q^{3} + 4 \zeta_{6} q^{4} + (16 \zeta_{6} - 8) q^{5} + (14 \zeta_{6} - 28) q^{6} + ( - 13 \zeta_{6} + 26) q^{7} + (16 \zeta_{6} - 8) q^{8} - 22 \zeta_{6} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + ( - 2 \zeta_{6} - 2) q^{2} + ( - 7 \zeta_{6} + 7) q^{3} + 4 \zeta_{6} q^{4} + (16 \zeta_{6} - 8) q^{5} + (14 \zeta_{6} - 28) q^{6} + ( - 13 \zeta_{6} + 26) q^{7} + (16 \zeta_{6} - 8) q^{8} - 22 \zeta_{6} q^{9} + ( - 48 \zeta_{6} + 48) q^{10} + ( - 13 \zeta_{6} - 13) q^{11} + 28 q^{12} + (52 \zeta_{6} - 39) q^{13} - 78 q^{14} + (56 \zeta_{6} + 56) q^{15} + ( - 80 \zeta_{6} + 80) q^{16} - 27 \zeta_{6} q^{17} + (88 \zeta_{6} - 44) q^{18} + (51 \zeta_{6} - 102) q^{19} + (32 \zeta_{6} - 64) q^{20} + ( - 182 \zeta_{6} + 91) q^{21} + 78 \zeta_{6} q^{22} + (57 \zeta_{6} - 57) q^{23} + (56 \zeta_{6} + 56) q^{24} - 67 q^{25} + ( - 130 \zeta_{6} + 182) q^{26} + 35 q^{27} + (52 \zeta_{6} + 52) q^{28} + ( - 69 \zeta_{6} + 69) q^{29} - 336 \zeta_{6} q^{30} + ( - 84 \zeta_{6} + 42) q^{31} + (96 \zeta_{6} - 192) q^{32} + (91 \zeta_{6} - 182) q^{33} + (108 \zeta_{6} - 54) q^{34} + 312 \zeta_{6} q^{35} + ( - 88 \zeta_{6} + 88) q^{36} + ( - 23 \zeta_{6} - 23) q^{37} + 306 q^{38} + (273 \zeta_{6} + 91) q^{39} - 192 q^{40} + ( - 227 \zeta_{6} - 227) q^{41} + (546 \zeta_{6} - 546) q^{42} + 85 \zeta_{6} q^{43} + ( - 104 \zeta_{6} + 52) q^{44} + ( - 176 \zeta_{6} + 352) q^{45} + ( - 114 \zeta_{6} + 228) q^{46} + ( - 396 \zeta_{6} + 198) q^{47} - 560 \zeta_{6} q^{48} + ( - 164 \zeta_{6} + 164) q^{49} + (134 \zeta_{6} + 134) q^{50} - 189 q^{51} + (52 \zeta_{6} - 208) q^{52} + 426 q^{53} + ( - 70 \zeta_{6} - 70) q^{54} + ( - 312 \zeta_{6} + 312) q^{55} + 312 \zeta_{6} q^{56} + (714 \zeta_{6} - 357) q^{57} + (138 \zeta_{6} - 276) q^{58} + (11 \zeta_{6} - 22) q^{59} + (448 \zeta_{6} - 224) q^{60} + 17 \zeta_{6} q^{61} + (252 \zeta_{6} - 252) q^{62} + ( - 286 \zeta_{6} - 286) q^{63} - 64 q^{64} + ( - 208 \zeta_{6} - 520) q^{65} + 546 q^{66} + (95 \zeta_{6} + 95) q^{67} + ( - 108 \zeta_{6} + 108) q^{68} + 399 \zeta_{6} q^{69} + ( - 1248 \zeta_{6} + 624) q^{70} + ( - 337 \zeta_{6} + 674) q^{71} + ( - 176 \zeta_{6} + 352) q^{72} + (1160 \zeta_{6} - 580) q^{73} + 138 \zeta_{6} q^{74} + (469 \zeta_{6} - 469) q^{75} + ( - 204 \zeta_{6} - 204) q^{76} - 507 q^{77} + ( - 1274 \zeta_{6} + 364) q^{78} - 1244 q^{79} + (640 \zeta_{6} + 640) q^{80} + ( - 839 \zeta_{6} + 839) q^{81} + 1362 \zeta_{6} q^{82} + ( - 492 \zeta_{6} + 246) q^{83} + ( - 364 \zeta_{6} + 728) q^{84} + ( - 216 \zeta_{6} + 432) q^{85} + ( - 340 \zeta_{6} + 170) q^{86} - 483 \zeta_{6} q^{87} + ( - 312 \zeta_{6} + 312) q^{88} + (177 \zeta_{6} + 177) q^{89} - 1056 q^{90} + (1183 \zeta_{6} - 338) q^{91} - 228 q^{92} + ( - 294 \zeta_{6} - 294) q^{93} + (1188 \zeta_{6} - 1188) q^{94} - 1224 \zeta_{6} q^{95} + (1344 \zeta_{6} - 672) q^{96} + ( - 713 \zeta_{6} + 1426) q^{97} + (328 \zeta_{6} - 656) q^{98} + (572 \zeta_{6} - 286) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 6 q^{2} + 7 q^{3} + 4 q^{4} - 42 q^{6} + 39 q^{7} - 22 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 6 q^{2} + 7 q^{3} + 4 q^{4} - 42 q^{6} + 39 q^{7} - 22 q^{9} + 48 q^{10} - 39 q^{11} + 56 q^{12} - 26 q^{13} - 156 q^{14} + 168 q^{15} + 80 q^{16} - 27 q^{17} - 153 q^{19} - 96 q^{20} + 78 q^{22} - 57 q^{23} + 168 q^{24} - 134 q^{25} + 234 q^{26} + 70 q^{27} + 156 q^{28} + 69 q^{29} - 336 q^{30} - 288 q^{32} - 273 q^{33} + 312 q^{35} + 88 q^{36} - 69 q^{37} + 612 q^{38} + 455 q^{39} - 384 q^{40} - 681 q^{41} - 546 q^{42} + 85 q^{43} + 528 q^{45} + 342 q^{46} - 560 q^{48} + 164 q^{49} + 402 q^{50} - 378 q^{51} - 364 q^{52} + 852 q^{53} - 210 q^{54} + 312 q^{55} + 312 q^{56} - 414 q^{58} - 33 q^{59} + 17 q^{61} - 252 q^{62} - 858 q^{63} - 128 q^{64} - 1248 q^{65} + 1092 q^{66} + 285 q^{67} + 108 q^{68} + 399 q^{69} + 1011 q^{71} + 528 q^{72} + 138 q^{74} - 469 q^{75} - 612 q^{76} - 1014 q^{77} - 546 q^{78} - 2488 q^{79} + 1920 q^{80} + 839 q^{81} + 1362 q^{82} + 1092 q^{84} + 648 q^{85} - 483 q^{87} + 312 q^{88} + 531 q^{89} - 2112 q^{90} + 507 q^{91} - 456 q^{92} - 882 q^{93} - 1188 q^{94} - 1224 q^{95} + 2139 q^{97} - 984 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/13\mathbb{Z}\right)^\times\).

\(n\) \(2\)
\(\chi(n)\) \(\zeta_{6}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
4.1
0.500000 + 0.866025i
0.500000 0.866025i
−3.00000 1.73205i 3.50000 6.06218i 2.00000 + 3.46410i 13.8564i −21.0000 + 12.1244i 19.5000 11.2583i 13.8564i −11.0000 19.0526i 24.0000 41.5692i
10.1 −3.00000 + 1.73205i 3.50000 + 6.06218i 2.00000 3.46410i 13.8564i −21.0000 12.1244i 19.5000 + 11.2583i 13.8564i −11.0000 + 19.0526i 24.0000 + 41.5692i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.e even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 13.4.e.a 2
3.b odd 2 1 117.4.q.c 2
4.b odd 2 1 208.4.w.a 2
13.b even 2 1 169.4.e.b 2
13.c even 3 1 169.4.b.b 2
13.c even 3 1 169.4.e.b 2
13.d odd 4 2 169.4.c.i 4
13.e even 6 1 inner 13.4.e.a 2
13.e even 6 1 169.4.b.b 2
13.f odd 12 2 169.4.a.h 2
13.f odd 12 2 169.4.c.i 4
39.h odd 6 1 117.4.q.c 2
39.k even 12 2 1521.4.a.q 2
52.i odd 6 1 208.4.w.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
13.4.e.a 2 1.a even 1 1 trivial
13.4.e.a 2 13.e even 6 1 inner
117.4.q.c 2 3.b odd 2 1
117.4.q.c 2 39.h odd 6 1
169.4.a.h 2 13.f odd 12 2
169.4.b.b 2 13.c even 3 1
169.4.b.b 2 13.e even 6 1
169.4.c.i 4 13.d odd 4 2
169.4.c.i 4 13.f odd 12 2
169.4.e.b 2 13.b even 2 1
169.4.e.b 2 13.c even 3 1
208.4.w.a 2 4.b odd 2 1
208.4.w.a 2 52.i odd 6 1
1521.4.a.q 2 39.k even 12 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} + 6T_{2} + 12 \) acting on \(S_{4}^{\mathrm{new}}(13, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + 6T + 12 \) Copy content Toggle raw display
$3$ \( T^{2} - 7T + 49 \) Copy content Toggle raw display
$5$ \( T^{2} + 192 \) Copy content Toggle raw display
$7$ \( T^{2} - 39T + 507 \) Copy content Toggle raw display
$11$ \( T^{2} + 39T + 507 \) Copy content Toggle raw display
$13$ \( T^{2} + 26T + 2197 \) Copy content Toggle raw display
$17$ \( T^{2} + 27T + 729 \) Copy content Toggle raw display
$19$ \( T^{2} + 153T + 7803 \) Copy content Toggle raw display
$23$ \( T^{2} + 57T + 3249 \) Copy content Toggle raw display
$29$ \( T^{2} - 69T + 4761 \) Copy content Toggle raw display
$31$ \( T^{2} + 5292 \) Copy content Toggle raw display
$37$ \( T^{2} + 69T + 1587 \) Copy content Toggle raw display
$41$ \( T^{2} + 681T + 154587 \) Copy content Toggle raw display
$43$ \( T^{2} - 85T + 7225 \) Copy content Toggle raw display
$47$ \( T^{2} + 117612 \) Copy content Toggle raw display
$53$ \( (T - 426)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} + 33T + 363 \) Copy content Toggle raw display
$61$ \( T^{2} - 17T + 289 \) Copy content Toggle raw display
$67$ \( T^{2} - 285T + 27075 \) Copy content Toggle raw display
$71$ \( T^{2} - 1011 T + 340707 \) Copy content Toggle raw display
$73$ \( T^{2} + 1009200 \) Copy content Toggle raw display
$79$ \( (T + 1244)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 181548 \) Copy content Toggle raw display
$89$ \( T^{2} - 531T + 93987 \) Copy content Toggle raw display
$97$ \( T^{2} - 2139 T + 1525107 \) Copy content Toggle raw display
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