# Properties

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

# Related objects

## Newspace parameters

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

## Newform invariants

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

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$i = \sqrt{-1}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( 1 + i ) q^{2} + 2 i q^{4} + ( -3 - 3 i ) q^{5} + ( 2 - 2 i ) q^{7} + ( -2 + 2 i ) q^{8} -9 q^{9} +O(q^{10})$$ $$q + ( 1 + i ) q^{2} + 2 i q^{4} + ( -3 - 3 i ) q^{5} + ( 2 - 2 i ) q^{7} + ( -2 + 2 i ) q^{8} -9 q^{9} -6 i q^{10} + ( 6 - 6 i ) q^{11} + 13 i q^{13} + 4 q^{14} -4 q^{16} + 6 i q^{17} + ( -9 - 9 i ) q^{18} + ( 26 + 26 i ) q^{19} + ( 6 - 6 i ) q^{20} + 12 q^{22} -24 i q^{23} -7 i q^{25} + ( -13 + 13 i ) q^{26} + ( 4 + 4 i ) q^{28} -48 q^{29} + ( -14 - 14 i ) q^{31} + ( -4 - 4 i ) q^{32} + ( -6 + 6 i ) q^{34} -12 q^{35} -18 i q^{36} + ( 37 - 37 i ) q^{37} + 52 i q^{38} + 12 q^{40} + ( -9 - 9 i ) q^{41} + 36 i q^{43} + ( 12 + 12 i ) q^{44} + ( 27 + 27 i ) q^{45} + ( 24 - 24 i ) q^{46} + ( 42 - 42 i ) q^{47} + 41 i q^{49} + ( 7 - 7 i ) q^{50} -26 q^{52} + 30 q^{53} -36 q^{55} + 8 i q^{56} + ( -48 - 48 i ) q^{58} + ( -54 + 54 i ) q^{59} -18 q^{61} -28 i q^{62} + ( -18 + 18 i ) q^{63} -8 i q^{64} + ( 39 - 39 i ) q^{65} + ( -22 - 22 i ) q^{67} -12 q^{68} + ( -12 - 12 i ) q^{70} + ( 6 + 6 i ) q^{71} + ( 18 - 18 i ) q^{72} + ( 17 - 17 i ) q^{73} + 74 q^{74} + ( -52 + 52 i ) q^{76} -24 i q^{77} -108 q^{79} + ( 12 + 12 i ) q^{80} + 81 q^{81} -18 i q^{82} + ( 78 + 78 i ) q^{83} + ( 18 - 18 i ) q^{85} + ( -36 + 36 i ) q^{86} + 24 i q^{88} + ( -9 + 9 i ) q^{89} + 54 i q^{90} + ( 26 + 26 i ) q^{91} + 48 q^{92} + 84 q^{94} -156 i q^{95} + ( -47 - 47 i ) q^{97} + ( -41 + 41 i ) q^{98} + ( -54 + 54 i ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 2q^{2} - 6q^{5} + 4q^{7} - 4q^{8} - 18q^{9} + O(q^{10})$$ $$2q + 2q^{2} - 6q^{5} + 4q^{7} - 4q^{8} - 18q^{9} + 12q^{11} + 8q^{14} - 8q^{16} - 18q^{18} + 52q^{19} + 12q^{20} + 24q^{22} - 26q^{26} + 8q^{28} - 96q^{29} - 28q^{31} - 8q^{32} - 12q^{34} - 24q^{35} + 74q^{37} + 24q^{40} - 18q^{41} + 24q^{44} + 54q^{45} + 48q^{46} + 84q^{47} + 14q^{50} - 52q^{52} + 60q^{53} - 72q^{55} - 96q^{58} - 108q^{59} - 36q^{61} - 36q^{63} + 78q^{65} - 44q^{67} - 24q^{68} - 24q^{70} + 12q^{71} + 36q^{72} + 34q^{73} + 148q^{74} - 104q^{76} - 216q^{79} + 24q^{80} + 162q^{81} + 156q^{83} + 36q^{85} - 72q^{86} - 18q^{89} + 52q^{91} + 96q^{92} + 168q^{94} - 94q^{97} - 82q^{98} - 108q^{99} + O(q^{100})$$

## Character values

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

 $$n$$ $$15$$ $$\chi(n)$$ $$i$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
5.1
 − 1.00000i 1.00000i
1.00000 1.00000i 0 2.00000i −3.00000 + 3.00000i 0 2.00000 + 2.00000i −2.00000 2.00000i −9.00000 6.00000i
21.1 1.00000 + 1.00000i 0 2.00000i −3.00000 3.00000i 0 2.00000 2.00000i −2.00000 + 2.00000i −9.00000 6.00000i
 $$n$$: 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.d odd 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 26.3.d.a 2
3.b odd 2 1 234.3.i.a 2
4.b odd 2 1 208.3.t.b 2
5.b even 2 1 650.3.k.b 2
5.c odd 4 1 650.3.f.b 2
5.c odd 4 1 650.3.f.e 2
13.b even 2 1 338.3.d.a 2
13.c even 3 2 338.3.f.b 4
13.d odd 4 1 inner 26.3.d.a 2
13.d odd 4 1 338.3.d.a 2
13.e even 6 2 338.3.f.g 4
13.f odd 12 2 338.3.f.b 4
13.f odd 12 2 338.3.f.g 4
39.f even 4 1 234.3.i.a 2
52.f even 4 1 208.3.t.b 2
65.f even 4 1 650.3.f.b 2
65.g odd 4 1 650.3.k.b 2
65.k even 4 1 650.3.f.e 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
26.3.d.a 2 1.a even 1 1 trivial
26.3.d.a 2 13.d odd 4 1 inner
208.3.t.b 2 4.b odd 2 1
208.3.t.b 2 52.f even 4 1
234.3.i.a 2 3.b odd 2 1
234.3.i.a 2 39.f even 4 1
338.3.d.a 2 13.b even 2 1
338.3.d.a 2 13.d odd 4 1
338.3.f.b 4 13.c even 3 2
338.3.f.b 4 13.f odd 12 2
338.3.f.g 4 13.e even 6 2
338.3.f.g 4 13.f odd 12 2
650.3.f.b 2 5.c odd 4 1
650.3.f.b 2 65.f even 4 1
650.3.f.e 2 5.c odd 4 1
650.3.f.e 2 65.k even 4 1
650.3.k.b 2 5.b even 2 1
650.3.k.b 2 65.g odd 4 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{3}^{\mathrm{new}}(26, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 - 2 T + 2 T^{2}$$
$3$ $$( 1 + 9 T^{2} )^{2}$$
$5$ $$1 + 6 T + 18 T^{2} + 150 T^{3} + 625 T^{4}$$
$7$ $$1 - 4 T + 8 T^{2} - 196 T^{3} + 2401 T^{4}$$
$11$ $$1 - 12 T + 72 T^{2} - 1452 T^{3} + 14641 T^{4}$$
$13$ $$1 + 169 T^{2}$$
$17$ $$1 - 542 T^{2} + 83521 T^{4}$$
$19$ $$1 - 52 T + 1352 T^{2} - 18772 T^{3} + 130321 T^{4}$$
$23$ $$1 - 482 T^{2} + 279841 T^{4}$$
$29$ $$( 1 + 48 T + 841 T^{2} )^{2}$$
$31$ $$1 + 28 T + 392 T^{2} + 26908 T^{3} + 923521 T^{4}$$
$37$ $$( 1 - 37 T )^{2}( 1 + 1369 T^{2} )$$
$41$ $$1 + 18 T + 162 T^{2} + 30258 T^{3} + 2825761 T^{4}$$
$43$ $$1 - 2402 T^{2} + 3418801 T^{4}$$
$47$ $$1 - 84 T + 3528 T^{2} - 185556 T^{3} + 4879681 T^{4}$$
$53$ $$( 1 - 30 T + 2809 T^{2} )^{2}$$
$59$ $$1 + 108 T + 5832 T^{2} + 375948 T^{3} + 12117361 T^{4}$$
$61$ $$( 1 + 18 T + 3721 T^{2} )^{2}$$
$67$ $$1 + 44 T + 968 T^{2} + 197516 T^{3} + 20151121 T^{4}$$
$71$ $$1 - 12 T + 72 T^{2} - 60492 T^{3} + 25411681 T^{4}$$
$73$ $$1 - 34 T + 578 T^{2} - 181186 T^{3} + 28398241 T^{4}$$
$79$ $$( 1 + 108 T + 6241 T^{2} )^{2}$$
$83$ $$1 - 156 T + 12168 T^{2} - 1074684 T^{3} + 47458321 T^{4}$$
$89$ $$1 + 18 T + 162 T^{2} + 142578 T^{3} + 62742241 T^{4}$$
$97$ $$1 + 94 T + 4418 T^{2} + 884446 T^{3} + 88529281 T^{4}$$