Properties

Label 13.4.e.b
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: \(\Z[a_1, a_2]\)
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.

\(f(q)\) \(=\) \( q + (\zeta_{6} + 1) q^{2} + (2 \zeta_{6} - 2) q^{3} - 5 \zeta_{6} q^{4} + ( - 2 \zeta_{6} + 1) q^{5} + (2 \zeta_{6} - 4) q^{6} + (8 \zeta_{6} - 16) q^{7} + ( - 26 \zeta_{6} + 13) q^{8} + 23 \zeta_{6} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + (\zeta_{6} + 1) q^{2} + (2 \zeta_{6} - 2) q^{3} - 5 \zeta_{6} q^{4} + ( - 2 \zeta_{6} + 1) q^{5} + (2 \zeta_{6} - 4) q^{6} + (8 \zeta_{6} - 16) q^{7} + ( - 26 \zeta_{6} + 13) q^{8} + 23 \zeta_{6} q^{9} + ( - 3 \zeta_{6} + 3) q^{10} + (8 \zeta_{6} + 8) q^{11} + 10 q^{12} + (13 \zeta_{6} + 39) q^{13} - 24 q^{14} + (2 \zeta_{6} + 2) q^{15} + (\zeta_{6} - 1) q^{16} - 117 \zeta_{6} q^{17} + (46 \zeta_{6} - 23) q^{18} + (66 \zeta_{6} - 132) q^{19} + (5 \zeta_{6} - 10) q^{20} + ( - 32 \zeta_{6} + 16) q^{21} + 24 \zeta_{6} q^{22} + ( - 78 \zeta_{6} + 78) q^{23} + (26 \zeta_{6} + 26) q^{24} + 122 q^{25} + (65 \zeta_{6} + 26) q^{26} - 100 q^{27} + (40 \zeta_{6} + 40) q^{28} + ( - 141 \zeta_{6} + 141) q^{29} + 6 \zeta_{6} q^{30} + (180 \zeta_{6} - 90) q^{31} + (105 \zeta_{6} - 210) q^{32} + (16 \zeta_{6} - 32) q^{33} + ( - 234 \zeta_{6} + 117) q^{34} + 24 \zeta_{6} q^{35} + ( - 115 \zeta_{6} + 115) q^{36} + ( - 83 \zeta_{6} - 83) q^{37} - 198 q^{38} + (78 \zeta_{6} - 104) q^{39} - 39 q^{40} + (157 \zeta_{6} + 157) q^{41} + ( - 48 \zeta_{6} + 48) q^{42} - 104 \zeta_{6} q^{43} + ( - 80 \zeta_{6} + 40) q^{44} + ( - 23 \zeta_{6} + 46) q^{45} + ( - 78 \zeta_{6} + 156) q^{46} + (348 \zeta_{6} - 174) q^{47} - 2 \zeta_{6} q^{48} + (151 \zeta_{6} - 151) q^{49} + (122 \zeta_{6} + 122) q^{50} + 234 q^{51} + ( - 260 \zeta_{6} + 65) q^{52} + 93 q^{53} + ( - 100 \zeta_{6} - 100) q^{54} + ( - 24 \zeta_{6} + 24) q^{55} + 312 \zeta_{6} q^{56} + ( - 264 \zeta_{6} + 132) q^{57} + ( - 141 \zeta_{6} + 282) q^{58} + (164 \zeta_{6} - 328) q^{59} + ( - 20 \zeta_{6} + 10) q^{60} - 145 \zeta_{6} q^{61} + (270 \zeta_{6} - 270) q^{62} + ( - 184 \zeta_{6} - 184) q^{63} - 307 q^{64} + ( - 91 \zeta_{6} + 65) q^{65} - 48 q^{66} + ( - 454 \zeta_{6} - 454) q^{67} + (585 \zeta_{6} - 585) q^{68} + 156 \zeta_{6} q^{69} + (48 \zeta_{6} - 24) q^{70} + ( - 610 \zeta_{6} + 1220) q^{71} + ( - 299 \zeta_{6} + 598) q^{72} + (530 \zeta_{6} - 265) q^{73} - 249 \zeta_{6} q^{74} + (244 \zeta_{6} - 244) q^{75} + (330 \zeta_{6} + 330) q^{76} - 192 q^{77} + (52 \zeta_{6} - 182) q^{78} + 1276 q^{79} + (\zeta_{6} + 1) q^{80} + (421 \zeta_{6} - 421) q^{81} + 471 \zeta_{6} q^{82} + ( - 912 \zeta_{6} + 456) q^{83} + (80 \zeta_{6} - 160) q^{84} + (117 \zeta_{6} - 234) q^{85} + ( - 208 \zeta_{6} + 104) q^{86} + 282 \zeta_{6} q^{87} + ( - 312 \zeta_{6} + 312) q^{88} + ( - 564 \zeta_{6} - 564) q^{89} + 69 q^{90} + (208 \zeta_{6} - 728) q^{91} - 390 q^{92} + ( - 180 \zeta_{6} - 180) q^{93} + (522 \zeta_{6} - 522) q^{94} + 198 \zeta_{6} q^{95} + ( - 420 \zeta_{6} + 210) q^{96} + ( - 116 \zeta_{6} + 232) q^{97} + (151 \zeta_{6} - 302) q^{98} + (368 \zeta_{6} - 184) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 3 q^{2} - 2 q^{3} - 5 q^{4} - 6 q^{6} - 24 q^{7} + 23 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 3 q^{2} - 2 q^{3} - 5 q^{4} - 6 q^{6} - 24 q^{7} + 23 q^{9} + 3 q^{10} + 24 q^{11} + 20 q^{12} + 91 q^{13} - 48 q^{14} + 6 q^{15} - q^{16} - 117 q^{17} - 198 q^{19} - 15 q^{20} + 24 q^{22} + 78 q^{23} + 78 q^{24} + 244 q^{25} + 117 q^{26} - 200 q^{27} + 120 q^{28} + 141 q^{29} + 6 q^{30} - 315 q^{32} - 48 q^{33} + 24 q^{35} + 115 q^{36} - 249 q^{37} - 396 q^{38} - 130 q^{39} - 78 q^{40} + 471 q^{41} + 48 q^{42} - 104 q^{43} + 69 q^{45} + 234 q^{46} - 2 q^{48} - 151 q^{49} + 366 q^{50} + 468 q^{51} - 130 q^{52} + 186 q^{53} - 300 q^{54} + 24 q^{55} + 312 q^{56} + 423 q^{58} - 492 q^{59} - 145 q^{61} - 270 q^{62} - 552 q^{63} - 614 q^{64} + 39 q^{65} - 96 q^{66} - 1362 q^{67} - 585 q^{68} + 156 q^{69} + 1830 q^{71} + 897 q^{72} - 249 q^{74} - 244 q^{75} + 990 q^{76} - 384 q^{77} - 312 q^{78} + 2552 q^{79} + 3 q^{80} - 421 q^{81} + 471 q^{82} - 240 q^{84} - 351 q^{85} + 282 q^{87} + 312 q^{88} - 1692 q^{89} + 138 q^{90} - 1248 q^{91} - 780 q^{92} - 540 q^{93} - 522 q^{94} + 198 q^{95} + 348 q^{97} - 453 q^{98}+O(q^{100}) \) 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
1.50000 + 0.866025i −1.00000 + 1.73205i −2.50000 4.33013i 1.73205i −3.00000 + 1.73205i −12.0000 + 6.92820i 22.5167i 11.5000 + 19.9186i 1.50000 2.59808i
10.1 1.50000 0.866025i −1.00000 1.73205i −2.50000 + 4.33013i 1.73205i −3.00000 1.73205i −12.0000 6.92820i 22.5167i 11.5000 19.9186i 1.50000 + 2.59808i
\(n\): 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.b 2
3.b odd 2 1 117.4.q.a 2
4.b odd 2 1 208.4.w.b 2
13.b even 2 1 169.4.e.a 2
13.c even 3 1 169.4.b.d 2
13.c even 3 1 169.4.e.a 2
13.d odd 4 2 169.4.c.h 4
13.e even 6 1 inner 13.4.e.b 2
13.e even 6 1 169.4.b.d 2
13.f odd 12 2 169.4.a.i 2
13.f odd 12 2 169.4.c.h 4
39.h odd 6 1 117.4.q.a 2
39.k even 12 2 1521.4.a.o 2
52.i odd 6 1 208.4.w.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
13.4.e.b 2 1.a even 1 1 trivial
13.4.e.b 2 13.e even 6 1 inner
117.4.q.a 2 3.b odd 2 1
117.4.q.a 2 39.h odd 6 1
169.4.a.i 2 13.f odd 12 2
169.4.b.d 2 13.c even 3 1
169.4.b.d 2 13.e even 6 1
169.4.c.h 4 13.d odd 4 2
169.4.c.h 4 13.f odd 12 2
169.4.e.a 2 13.b even 2 1
169.4.e.a 2 13.c even 3 1
208.4.w.b 2 4.b odd 2 1
208.4.w.b 2 52.i odd 6 1
1521.4.a.o 2 39.k even 12 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} - 3T_{2} + 3 \) acting on \(S_{4}^{\mathrm{new}}(13, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} - 3T + 3 \) Copy content Toggle raw display
$3$ \( T^{2} + 2T + 4 \) Copy content Toggle raw display
$5$ \( T^{2} + 3 \) Copy content Toggle raw display
$7$ \( T^{2} + 24T + 192 \) Copy content Toggle raw display
$11$ \( T^{2} - 24T + 192 \) Copy content Toggle raw display
$13$ \( T^{2} - 91T + 2197 \) Copy content Toggle raw display
$17$ \( T^{2} + 117T + 13689 \) Copy content Toggle raw display
$19$ \( T^{2} + 198T + 13068 \) Copy content Toggle raw display
$23$ \( T^{2} - 78T + 6084 \) Copy content Toggle raw display
$29$ \( T^{2} - 141T + 19881 \) Copy content Toggle raw display
$31$ \( T^{2} + 24300 \) Copy content Toggle raw display
$37$ \( T^{2} + 249T + 20667 \) Copy content Toggle raw display
$41$ \( T^{2} - 471T + 73947 \) Copy content Toggle raw display
$43$ \( T^{2} + 104T + 10816 \) Copy content Toggle raw display
$47$ \( T^{2} + 90828 \) Copy content Toggle raw display
$53$ \( (T - 93)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} + 492T + 80688 \) Copy content Toggle raw display
$61$ \( T^{2} + 145T + 21025 \) Copy content Toggle raw display
$67$ \( T^{2} + 1362 T + 618348 \) Copy content Toggle raw display
$71$ \( T^{2} - 1830 T + 1116300 \) Copy content Toggle raw display
$73$ \( T^{2} + 210675 \) Copy content Toggle raw display
$79$ \( (T - 1276)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 623808 \) Copy content Toggle raw display
$89$ \( T^{2} + 1692 T + 954288 \) Copy content Toggle raw display
$97$ \( T^{2} - 348T + 40368 \) Copy content Toggle raw display
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