# Properties

 Label 26.3.f.a Level 26 Weight 3 Character orbit 26.f Analytic conductor 0.708 Analytic rank 0 Dimension 4 CM no Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$26 = 2 \cdot 13$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 26.f (of order $$12$$, degree $$4$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$0.708448687337$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\zeta_{12})$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{12}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( \zeta_{12} + \zeta_{12}^{2} - \zeta_{12}^{3} ) q^{2} + ( -\zeta_{12} - \zeta_{12}^{3} ) q^{3} + 2 \zeta_{12} q^{4} + ( -1 - 2 \zeta_{12} + 2 \zeta_{12}^{2} + \zeta_{12}^{3} ) q^{5} + ( -1 + \zeta_{12} - \zeta_{12}^{2} - 2 \zeta_{12}^{3} ) q^{6} + ( -4 - 4 \zeta_{12} - 3 \zeta_{12}^{2} + 7 \zeta_{12}^{3} ) q^{7} + ( 2 + 2 \zeta_{12}^{3} ) q^{8} + ( 6 - 6 \zeta_{12}^{2} ) q^{9} +O(q^{10})$$ $$q + ( \zeta_{12} + \zeta_{12}^{2} - \zeta_{12}^{3} ) q^{2} + ( -\zeta_{12} - \zeta_{12}^{3} ) q^{3} + 2 \zeta_{12} q^{4} + ( -1 - 2 \zeta_{12} + 2 \zeta_{12}^{2} + \zeta_{12}^{3} ) q^{5} + ( -1 + \zeta_{12} - \zeta_{12}^{2} - 2 \zeta_{12}^{3} ) q^{6} + ( -4 - 4 \zeta_{12} - 3 \zeta_{12}^{2} + 7 \zeta_{12}^{3} ) q^{7} + ( 2 + 2 \zeta_{12}^{3} ) q^{8} + ( 6 - 6 \zeta_{12}^{2} ) q^{9} + ( -4 + 2 \zeta_{12}^{2} ) q^{10} + ( 1 + 4 \zeta_{12} - 5 \zeta_{12}^{2} - 5 \zeta_{12}^{3} ) q^{11} + ( 2 - 4 \zeta_{12}^{2} ) q^{12} + ( -9 + 8 \zeta_{12} + 12 \zeta_{12}^{2} - 2 \zeta_{12}^{3} ) q^{13} + ( -1 - 14 \zeta_{12} + 7 \zeta_{12}^{3} ) q^{14} + ( 3 \zeta_{12} + 3 \zeta_{12}^{2} - 3 \zeta_{12}^{3} ) q^{15} + 4 \zeta_{12}^{2} q^{16} + ( 13 - 6 \zeta_{12} + 13 \zeta_{12}^{2} ) q^{17} + ( 6 - 6 \zeta_{12}^{3} ) q^{18} + ( -12 + 12 \zeta_{12} - 5 \zeta_{12}^{2} - 17 \zeta_{12}^{3} ) q^{19} + ( -2 - 2 \zeta_{12} - 2 \zeta_{12}^{2} + 4 \zeta_{12}^{3} ) q^{20} + ( 10 + \zeta_{12} + \zeta_{12}^{2} + 10 \zeta_{12}^{3} ) q^{21} + ( 9 + \zeta_{12} - 9 \zeta_{12}^{2} - 2 \zeta_{12}^{3} ) q^{22} + ( 4 + 21 \zeta_{12} - 2 \zeta_{12}^{2} - 21 \zeta_{12}^{3} ) q^{23} + ( 4 - 2 \zeta_{12} - 2 \zeta_{12}^{2} - 2 \zeta_{12}^{3} ) q^{24} + 19 \zeta_{12}^{3} q^{25} + ( -4 + 5 \zeta_{12} + \zeta_{12}^{2} + 15 \zeta_{12}^{3} ) q^{26} + ( -30 \zeta_{12} + 15 \zeta_{12}^{3} ) q^{27} + ( -14 - 8 \zeta_{12} + 6 \zeta_{12}^{2} - 6 \zeta_{12}^{3} ) q^{28} + ( -16 \zeta_{12} - 27 \zeta_{12}^{2} - 16 \zeta_{12}^{3} ) q^{29} + 6 \zeta_{12} q^{30} + ( -37 - 10 \zeta_{12} + 10 \zeta_{12}^{2} + 37 \zeta_{12}^{3} ) q^{31} + ( -4 + 4 \zeta_{12} + 4 \zeta_{12}^{2} ) q^{32} + ( -6 - 6 \zeta_{12} - 3 \zeta_{12}^{2} + 9 \zeta_{12}^{3} ) q^{33} + ( -19 + 26 \zeta_{12} + 26 \zeta_{12}^{2} - 19 \zeta_{12}^{3} ) q^{34} + ( 21 + \zeta_{12} - 21 \zeta_{12}^{2} - 2 \zeta_{12}^{3} ) q^{35} + ( 12 \zeta_{12} - 12 \zeta_{12}^{3} ) q^{36} + ( 7 - 13 \zeta_{12} + 6 \zeta_{12}^{2} + 6 \zeta_{12}^{3} ) q^{37} + ( 17 - 34 \zeta_{12}^{2} + 7 \zeta_{12}^{3} ) q^{38} + ( 4 + 21 \zeta_{12} - 14 \zeta_{12}^{2} - 15 \zeta_{12}^{3} ) q^{39} + ( -8 \zeta_{12} + 4 \zeta_{12}^{3} ) q^{40} + ( -13 + 13 \zeta_{12} + 26 \zeta_{12}^{2} - 26 \zeta_{12}^{3} ) q^{41} + ( \zeta_{12} + 21 \zeta_{12}^{2} + \zeta_{12}^{3} ) q^{42} + ( 20 - 39 \zeta_{12} + 20 \zeta_{12}^{2} ) q^{43} + ( 10 + 2 \zeta_{12} - 2 \zeta_{12}^{2} - 10 \zeta_{12}^{3} ) q^{44} + ( 6 - 6 \zeta_{12} + 6 \zeta_{12}^{2} + 12 \zeta_{12}^{3} ) q^{45} + ( 23 + 23 \zeta_{12} - 19 \zeta_{12}^{2} - 4 \zeta_{12}^{3} ) q^{46} + ( 33 + 33 \zeta_{12}^{3} ) q^{47} + ( 4 \zeta_{12} - 8 \zeta_{12}^{3} ) q^{48} + ( 14 + 25 \zeta_{12} - 7 \zeta_{12}^{2} - 25 \zeta_{12}^{3} ) q^{49} + ( -19 \zeta_{12} + 19 \zeta_{12}^{2} + 19 \zeta_{12}^{3} ) q^{50} + ( -6 + 12 \zeta_{12}^{2} - 39 \zeta_{12}^{3} ) q^{51} + ( 4 - 18 \zeta_{12} + 12 \zeta_{12}^{2} + 24 \zeta_{12}^{3} ) q^{52} + ( -42 + 44 \zeta_{12} - 22 \zeta_{12}^{3} ) q^{53} + ( -30 - 15 \zeta_{12} + 15 \zeta_{12}^{2} - 15 \zeta_{12}^{3} ) q^{54} + ( 9 \zeta_{12} + 3 \zeta_{12}^{2} + 9 \zeta_{12}^{3} ) q^{55} + ( -14 - 2 \zeta_{12} - 14 \zeta_{12}^{2} ) q^{56} + ( -22 + 7 \zeta_{12} - 7 \zeta_{12}^{2} + 22 \zeta_{12}^{3} ) q^{57} + ( 11 - 11 \zeta_{12} - 43 \zeta_{12}^{2} - 32 \zeta_{12}^{3} ) q^{58} + ( -10 - 10 \zeta_{12} + 23 \zeta_{12}^{2} - 13 \zeta_{12}^{3} ) q^{59} + ( 6 + 6 \zeta_{12}^{3} ) q^{60} + ( 39 + 6 \zeta_{12} - 39 \zeta_{12}^{2} - 12 \zeta_{12}^{3} ) q^{61} + ( -20 - 64 \zeta_{12} + 10 \zeta_{12}^{2} + 64 \zeta_{12}^{3} ) q^{62} + ( -42 + 18 \zeta_{12} + 24 \zeta_{12}^{2} + 24 \zeta_{12}^{3} ) q^{63} + 8 \zeta_{12}^{3} q^{64} + ( -25 + 2 \zeta_{12} - 10 \zeta_{12}^{2} - 7 \zeta_{12}^{3} ) q^{65} + ( -3 - 18 \zeta_{12} + 9 \zeta_{12}^{3} ) q^{66} + ( 33 + 10 \zeta_{12} - 23 \zeta_{12}^{2} + 23 \zeta_{12}^{3} ) q^{67} + ( 26 \zeta_{12} - 12 \zeta_{12}^{2} + 26 \zeta_{12}^{3} ) q^{68} + ( -21 - 6 \zeta_{12} - 21 \zeta_{12}^{2} ) q^{69} + ( 22 + 2 \zeta_{12} - 2 \zeta_{12}^{2} - 22 \zeta_{12}^{3} ) q^{70} + ( -4 + 4 \zeta_{12} + 29 \zeta_{12}^{2} + 25 \zeta_{12}^{3} ) q^{71} + ( 12 + 12 \zeta_{12} - 12 \zeta_{12}^{2} ) q^{72} + ( -7 - 54 \zeta_{12} - 54 \zeta_{12}^{2} - 7 \zeta_{12}^{3} ) q^{73} + ( -19 + 7 \zeta_{12} + 19 \zeta_{12}^{2} - 14 \zeta_{12}^{3} ) q^{74} + ( 38 - 19 \zeta_{12}^{2} ) q^{75} + ( 34 - 24 \zeta_{12} - 10 \zeta_{12}^{2} - 10 \zeta_{12}^{3} ) q^{76} + ( -32 + 64 \zeta_{12}^{2} + 15 \zeta_{12}^{3} ) q^{77} + ( 35 + 5 \zeta_{12} - 25 \zeta_{12}^{2} + 2 \zeta_{12}^{3} ) q^{78} + ( 48 + 72 \zeta_{12} - 36 \zeta_{12}^{3} ) q^{79} + ( -8 - 4 \zeta_{12} + 4 \zeta_{12}^{2} - 4 \zeta_{12}^{3} ) q^{80} -9 \zeta_{12}^{2} q^{81} + ( -13 + 39 \zeta_{12} - 13 \zeta_{12}^{2} ) q^{82} + ( 25 - 46 \zeta_{12} + 46 \zeta_{12}^{2} - 25 \zeta_{12}^{3} ) q^{83} + ( -20 + 20 \zeta_{12} + 22 \zeta_{12}^{2} + 2 \zeta_{12}^{3} ) q^{84} + ( -33 - 33 \zeta_{12} + 45 \zeta_{12}^{2} - 12 \zeta_{12}^{3} ) q^{85} + ( -59 + 40 \zeta_{12} + 40 \zeta_{12}^{2} - 59 \zeta_{12}^{3} ) q^{86} + ( -48 - 27 \zeta_{12} + 48 \zeta_{12}^{2} + 54 \zeta_{12}^{3} ) q^{87} + ( 4 + 18 \zeta_{12} - 2 \zeta_{12}^{2} - 18 \zeta_{12}^{3} ) q^{88} + ( 13 + 43 \zeta_{12} - 56 \zeta_{12}^{2} - 56 \zeta_{12}^{3} ) q^{89} + ( -12 + 24 \zeta_{12}^{2} ) q^{90} + ( 22 - 86 \zeta_{12} - 25 \zeta_{12}^{2} - 37 \zeta_{12}^{3} ) q^{91} + ( 42 + 8 \zeta_{12} - 4 \zeta_{12}^{3} ) q^{92} + ( 64 + 47 \zeta_{12} - 17 \zeta_{12}^{2} + 17 \zeta_{12}^{3} ) q^{93} + 66 \zeta_{12}^{2} q^{94} + ( -7 + 51 \zeta_{12} - 7 \zeta_{12}^{2} ) q^{95} + ( 4 + 8 \zeta_{12} - 8 \zeta_{12}^{2} - 4 \zeta_{12}^{3} ) q^{96} + ( 71 - 71 \zeta_{12} - 2 \zeta_{12}^{2} + 69 \zeta_{12}^{3} ) q^{97} + ( 32 + 32 \zeta_{12} - 18 \zeta_{12}^{2} - 14 \zeta_{12}^{3} ) q^{98} + ( -24 - 6 \zeta_{12} - 6 \zeta_{12}^{2} - 24 \zeta_{12}^{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 2q^{2} - 6q^{6} - 22q^{7} + 8q^{8} + 12q^{9} + O(q^{10})$$ $$4q + 2q^{2} - 6q^{6} - 22q^{7} + 8q^{8} + 12q^{9} - 12q^{10} - 6q^{11} - 12q^{13} - 4q^{14} + 6q^{15} + 8q^{16} + 78q^{17} + 24q^{18} - 58q^{19} - 12q^{20} + 42q^{21} + 18q^{22} + 12q^{23} + 12q^{24} - 14q^{26} - 44q^{28} - 54q^{29} - 128q^{31} - 8q^{32} - 30q^{33} - 24q^{34} + 42q^{35} + 40q^{37} - 12q^{39} + 42q^{42} + 120q^{43} + 36q^{44} + 36q^{45} + 54q^{46} + 132q^{47} + 42q^{49} + 38q^{50} + 40q^{52} - 168q^{53} - 90q^{54} + 6q^{55} - 84q^{56} - 102q^{57} - 42q^{58} + 6q^{59} + 24q^{60} + 78q^{61} - 60q^{62} - 120q^{63} - 120q^{65} - 12q^{66} + 86q^{67} - 24q^{68} - 126q^{69} + 84q^{70} + 42q^{71} + 24q^{72} - 136q^{73} - 38q^{74} + 114q^{75} + 116q^{76} + 90q^{78} + 192q^{79} - 24q^{80} - 18q^{81} - 78q^{82} + 192q^{83} - 36q^{84} - 42q^{85} - 156q^{86} - 96q^{87} + 12q^{88} - 60q^{89} + 38q^{91} + 168q^{92} + 222q^{93} + 132q^{94} - 42q^{95} + 280q^{97} + 92q^{98} - 108q^{99} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/26\mathbb{Z}\right)^\times$$.

 $$n$$ $$15$$ $$\chi(n)$$ $$\zeta_{12}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
7.1
 0.866025 − 0.500000i −0.866025 − 0.500000i 0.866025 + 0.500000i −0.866025 + 0.500000i
1.36603 0.366025i −0.866025 + 1.50000i 1.73205 1.00000i −1.73205 1.73205i −0.633975 + 2.36603i −8.96410 2.40192i 2.00000 2.00000i 3.00000 + 5.19615i −3.00000 1.73205i
11.1 −0.366025 + 1.36603i 0.866025 + 1.50000i −1.73205 1.00000i 1.73205 + 1.73205i −2.36603 + 0.633975i −2.03590 7.59808i 2.00000 2.00000i 3.00000 5.19615i −3.00000 + 1.73205i
15.1 1.36603 + 0.366025i −0.866025 1.50000i 1.73205 + 1.00000i −1.73205 + 1.73205i −0.633975 2.36603i −8.96410 + 2.40192i 2.00000 + 2.00000i 3.00000 5.19615i −3.00000 + 1.73205i
19.1 −0.366025 1.36603i 0.866025 1.50000i −1.73205 + 1.00000i 1.73205 1.73205i −2.36603 0.633975i −2.03590 + 7.59808i 2.00000 + 2.00000i 3.00000 + 5.19615i −3.00000 1.73205i
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.f odd 12 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 26.3.f.a 4
3.b odd 2 1 234.3.bb.b 4
4.b odd 2 1 208.3.bd.c 4
13.b even 2 1 338.3.f.d 4
13.c even 3 1 338.3.d.d 4
13.c even 3 1 338.3.f.f 4
13.d odd 4 1 338.3.f.c 4
13.d odd 4 1 338.3.f.f 4
13.e even 6 1 338.3.d.e 4
13.e even 6 1 338.3.f.c 4
13.f odd 12 1 inner 26.3.f.a 4
13.f odd 12 1 338.3.d.d 4
13.f odd 12 1 338.3.d.e 4
13.f odd 12 1 338.3.f.d 4
39.k even 12 1 234.3.bb.b 4
52.l even 12 1 208.3.bd.c 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
26.3.f.a 4 1.a even 1 1 trivial
26.3.f.a 4 13.f odd 12 1 inner
208.3.bd.c 4 4.b odd 2 1
208.3.bd.c 4 52.l even 12 1
234.3.bb.b 4 3.b odd 2 1
234.3.bb.b 4 39.k even 12 1
338.3.d.d 4 13.c even 3 1
338.3.d.d 4 13.f odd 12 1
338.3.d.e 4 13.e even 6 1
338.3.d.e 4 13.f odd 12 1
338.3.f.c 4 13.d odd 4 1
338.3.f.c 4 13.e even 6 1
338.3.f.d 4 13.b even 2 1
338.3.f.d 4 13.f odd 12 1
338.3.f.f 4 13.c even 3 1
338.3.f.f 4 13.d odd 4 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{4} + 3 T_{3}^{2} + 9$$ acting on $$S_{3}^{\mathrm{new}}(26, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 - 2 T + 2 T^{2} - 4 T^{3} + 4 T^{4}$$
$3$ $$1 - 15 T^{2} + 144 T^{4} - 1215 T^{6} + 6561 T^{8}$$
$5$ $$1 + 686 T^{4} + 390625 T^{8}$$
$7$ $$1 + 22 T + 221 T^{2} + 1362 T^{3} + 8024 T^{4} + 66738 T^{5} + 530621 T^{6} + 2588278 T^{7} + 5764801 T^{8}$$
$11$ $$1 + 6 T + 45 T^{2} - 1710 T^{3} - 16024 T^{4} - 206910 T^{5} + 658845 T^{6} + 10629366 T^{7} + 214358881 T^{8}$$
$13$ $$1 + 12 T + 182 T^{2} + 2028 T^{3} + 28561 T^{4}$$
$17$ $$1 - 78 T + 3077 T^{2} - 81822 T^{3} + 1602972 T^{4} - 23646558 T^{5} + 256994117 T^{6} - 1882730382 T^{7} + 6975757441 T^{8}$$
$19$ $$1 + 58 T + 1325 T^{2} + 12942 T^{3} + 74504 T^{4} + 4672062 T^{5} + 172675325 T^{6} + 2728661098 T^{7} + 16983563041 T^{8}$$
$23$ $$1 - 12 T + 677 T^{2} - 7548 T^{3} + 141192 T^{4} - 3992892 T^{5} + 189452357 T^{6} - 1776430668 T^{7} + 78310985281 T^{8}$$
$29$ $$1 + 54 T + 1273 T^{2} - 2106 T^{3} - 460188 T^{4} - 1771146 T^{5} + 900368713 T^{6} + 32120459334 T^{7} + 500246412961 T^{8}$$
$31$ $$1 + 128 T + 8192 T^{2} + 365952 T^{3} + 12745358 T^{4} + 351679872 T^{5} + 7565484032 T^{6} + 113600471168 T^{7} + 852891037441 T^{8}$$
$37$ $$1 - 40 T + 401 T^{2} + 52236 T^{3} - 3248140 T^{4} + 71511084 T^{5} + 751538561 T^{6} - 102629056360 T^{7} + 3512479453921 T^{8}$$
$41$ $$1 + 1521 T^{2} + 91572 T^{3} + 840356 T^{4} + 153932532 T^{5} + 4297982481 T^{6} + 7984925229121 T^{8}$$
$43$ $$1 - 120 T + 8177 T^{2} - 405240 T^{3} + 16860528 T^{4} - 749288760 T^{5} + 27955535777 T^{6} - 758563565880 T^{7} + 11688200277601 T^{8}$$
$47$ $$( 1 - 66 T + 2178 T^{2} - 145794 T^{3} + 4879681 T^{4} )^{2}$$
$53$ $$( 1 + 84 T + 5930 T^{2} + 235956 T^{3} + 7890481 T^{4} )^{2}$$
$59$ $$1 - 6 T + 1305 T^{2} - 208794 T^{3} - 5098528 T^{4} - 726811914 T^{5} + 15813156105 T^{6} - 253083201846 T^{7} + 146830437604321 T^{8}$$
$61$ $$1 - 78 T - 2771 T^{2} - 110214 T^{3} + 41926620 T^{4} - 410106294 T^{5} - 38366825411 T^{6} - 4018589200158 T^{7} + 191707312997281 T^{8}$$
$67$ $$1 - 86 T + 4985 T^{2} - 290202 T^{3} + 7709888 T^{4} - 1302716778 T^{5} + 100453338185 T^{6} - 7779420866534 T^{7} + 406067677556641 T^{8}$$
$71$ $$1 - 42 T + 3357 T^{2} - 375150 T^{3} + 3657944 T^{4} - 1891131150 T^{5} + 85307013117 T^{6} - 5380211924682 T^{7} + 645753531245761 T^{8}$$
$73$ $$1 + 136 T + 9248 T^{2} + 444312 T^{3} + 17094734 T^{4} + 2367738648 T^{5} + 262626932768 T^{6} + 20581454775304 T^{7} + 806460091894081 T^{8}$$
$79$ $$( 1 - 96 T + 10898 T^{2} - 599136 T^{3} + 38950081 T^{4} )^{2}$$
$83$ $$1 - 192 T + 18432 T^{2} - 1598016 T^{3} + 136488302 T^{4} - 11008732224 T^{5} + 874751772672 T^{6} - 62772551686848 T^{7} + 2252292232139041 T^{8}$$
$89$ $$1 + 60 T + 5661 T^{2} - 927072 T^{3} - 53599444 T^{4} - 7343337312 T^{5} + 355183826301 T^{6} + 29818877457660 T^{7} + 3936588805702081 T^{8}$$
$97$ $$1 - 280 T + 24089 T^{2} + 773796 T^{3} - 268596364 T^{4} + 7280646564 T^{5} + 2132581850009 T^{6} - 233232161380120 T^{7} + 7837433594376961 T^{8}$$