# Properties

 Label 5.5.c.a Level $5$ Weight $5$ Character orbit 5.c Analytic conductor $0.517$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$5$$ Weight: $$k$$ $$=$$ $$5$$ Character orbit: $$[\chi]$$ $$=$$ 5.c (of order $$4$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$0.516849815419$$ 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.

 $$f(q)$$ $$=$$ $$q + ( -1 - i ) q^{2} + ( -6 + 6 i ) q^{3} -14 i q^{4} + ( 20 + 15 i ) q^{5} + 12 q^{6} + ( -26 - 26 i ) q^{7} + ( -30 + 30 i ) q^{8} + 9 i q^{9} +O(q^{10})$$ $$q + ( -1 - i ) q^{2} + ( -6 + 6 i ) q^{3} -14 i q^{4} + ( 20 + 15 i ) q^{5} + 12 q^{6} + ( -26 - 26 i ) q^{7} + ( -30 + 30 i ) q^{8} + 9 i q^{9} + ( -5 - 35 i ) q^{10} -8 q^{11} + ( 84 + 84 i ) q^{12} + ( 139 - 139 i ) q^{13} + 52 i q^{14} + ( -210 + 30 i ) q^{15} -164 q^{16} + ( -1 - i ) q^{17} + ( 9 - 9 i ) q^{18} + 180 i q^{19} + ( 210 - 280 i ) q^{20} + 312 q^{21} + ( 8 + 8 i ) q^{22} + ( -166 + 166 i ) q^{23} -360 i q^{24} + ( 175 + 600 i ) q^{25} -278 q^{26} + ( -540 - 540 i ) q^{27} + ( -364 + 364 i ) q^{28} -480 i q^{29} + ( 240 + 180 i ) q^{30} + 572 q^{31} + ( 644 + 644 i ) q^{32} + ( 48 - 48 i ) q^{33} + 2 i q^{34} + ( -130 - 910 i ) q^{35} + 126 q^{36} + ( -251 - 251 i ) q^{37} + ( 180 - 180 i ) q^{38} + 1668 i q^{39} + ( -1050 + 150 i ) q^{40} -1688 q^{41} + ( -312 - 312 i ) q^{42} + ( 1474 - 1474 i ) q^{43} + 112 i q^{44} + ( -135 + 180 i ) q^{45} + 332 q^{46} + ( 2474 + 2474 i ) q^{47} + ( 984 - 984 i ) q^{48} -1049 i q^{49} + ( 425 - 775 i ) q^{50} + 12 q^{51} + ( -1946 - 1946 i ) q^{52} + ( -3331 + 3331 i ) q^{53} + 1080 i q^{54} + ( -160 - 120 i ) q^{55} + 1560 q^{56} + ( -1080 - 1080 i ) q^{57} + ( -480 + 480 i ) q^{58} -3660 i q^{59} + ( 420 + 2940 i ) q^{60} + 1592 q^{61} + ( -572 - 572 i ) q^{62} + ( 234 - 234 i ) q^{63} + 1336 i q^{64} + ( 4865 - 695 i ) q^{65} -96 q^{66} + ( 874 + 874 i ) q^{67} + ( -14 + 14 i ) q^{68} -1992 i q^{69} + ( -780 + 1040 i ) q^{70} -6068 q^{71} + ( -270 - 270 i ) q^{72} + ( -791 + 791 i ) q^{73} + 502 i q^{74} + ( -4650 - 2550 i ) q^{75} + 2520 q^{76} + ( 208 + 208 i ) q^{77} + ( 1668 - 1668 i ) q^{78} + 9120 i q^{79} + ( -3280 - 2460 i ) q^{80} + 5751 q^{81} + ( 1688 + 1688 i ) q^{82} + ( 5654 - 5654 i ) q^{83} -4368 i q^{84} + ( -5 - 35 i ) q^{85} -2948 q^{86} + ( 2880 + 2880 i ) q^{87} + ( 240 - 240 i ) q^{88} + 2160 i q^{89} + ( 315 - 45 i ) q^{90} -7228 q^{91} + ( 2324 + 2324 i ) q^{92} + ( -3432 + 3432 i ) q^{93} -4948 i q^{94} + ( -2700 + 3600 i ) q^{95} -7728 q^{96} + ( -6551 - 6551 i ) q^{97} + ( -1049 + 1049 i ) q^{98} -72 i q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{2} - 12 q^{3} + 40 q^{5} + 24 q^{6} - 52 q^{7} - 60 q^{8} + O(q^{10})$$ $$2 q - 2 q^{2} - 12 q^{3} + 40 q^{5} + 24 q^{6} - 52 q^{7} - 60 q^{8} - 10 q^{10} - 16 q^{11} + 168 q^{12} + 278 q^{13} - 420 q^{15} - 328 q^{16} - 2 q^{17} + 18 q^{18} + 420 q^{20} + 624 q^{21} + 16 q^{22} - 332 q^{23} + 350 q^{25} - 556 q^{26} - 1080 q^{27} - 728 q^{28} + 480 q^{30} + 1144 q^{31} + 1288 q^{32} + 96 q^{33} - 260 q^{35} + 252 q^{36} - 502 q^{37} + 360 q^{38} - 2100 q^{40} - 3376 q^{41} - 624 q^{42} + 2948 q^{43} - 270 q^{45} + 664 q^{46} + 4948 q^{47} + 1968 q^{48} + 850 q^{50} + 24 q^{51} - 3892 q^{52} - 6662 q^{53} - 320 q^{55} + 3120 q^{56} - 2160 q^{57} - 960 q^{58} + 840 q^{60} + 3184 q^{61} - 1144 q^{62} + 468 q^{63} + 9730 q^{65} - 192 q^{66} + 1748 q^{67} - 28 q^{68} - 1560 q^{70} - 12136 q^{71} - 540 q^{72} - 1582 q^{73} - 9300 q^{75} + 5040 q^{76} + 416 q^{77} + 3336 q^{78} - 6560 q^{80} + 11502 q^{81} + 3376 q^{82} + 11308 q^{83} - 10 q^{85} - 5896 q^{86} + 5760 q^{87} + 480 q^{88} + 630 q^{90} - 14456 q^{91} + 4648 q^{92} - 6864 q^{93} - 5400 q^{95} - 15456 q^{96} - 13102 q^{97} - 2098 q^{98} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/5\mathbb{Z}\right)^\times$$.

 $$n$$ $$2$$ $$\chi(n)$$ $$i$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
2.1
 1.00000i − 1.00000i
−1.00000 1.00000i −6.00000 + 6.00000i 14.0000i 20.0000 + 15.0000i 12.0000 −26.0000 26.0000i −30.0000 + 30.0000i 9.00000i −5.00000 35.0000i
3.1 −1.00000 + 1.00000i −6.00000 6.00000i 14.0000i 20.0000 15.0000i 12.0000 −26.0000 + 26.0000i −30.0000 30.0000i 9.00000i −5.00000 + 35.0000i
 $$n$$: 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
5.c odd 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 5.5.c.a 2
3.b odd 2 1 45.5.g.b 2
4.b odd 2 1 80.5.p.d 2
5.b even 2 1 25.5.c.a 2
5.c odd 4 1 inner 5.5.c.a 2
5.c odd 4 1 25.5.c.a 2
8.b even 2 1 320.5.p.h 2
8.d odd 2 1 320.5.p.c 2
15.d odd 2 1 225.5.g.b 2
15.e even 4 1 45.5.g.b 2
15.e even 4 1 225.5.g.b 2
20.d odd 2 1 400.5.p.a 2
20.e even 4 1 80.5.p.d 2
20.e even 4 1 400.5.p.a 2
40.i odd 4 1 320.5.p.h 2
40.k even 4 1 320.5.p.c 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
5.5.c.a 2 1.a even 1 1 trivial
5.5.c.a 2 5.c odd 4 1 inner
25.5.c.a 2 5.b even 2 1
25.5.c.a 2 5.c odd 4 1
45.5.g.b 2 3.b odd 2 1
45.5.g.b 2 15.e even 4 1
80.5.p.d 2 4.b odd 2 1
80.5.p.d 2 20.e even 4 1
225.5.g.b 2 15.d odd 2 1
225.5.g.b 2 15.e even 4 1
320.5.p.c 2 8.d odd 2 1
320.5.p.c 2 40.k even 4 1
320.5.p.h 2 8.b even 2 1
320.5.p.h 2 40.i odd 4 1
400.5.p.a 2 20.d odd 2 1
400.5.p.a 2 20.e even 4 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{5}^{\mathrm{new}}(5, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$2 + 2 T + T^{2}$$
$3$ $$72 + 12 T + T^{2}$$
$5$ $$625 - 40 T + T^{2}$$
$7$ $$1352 + 52 T + T^{2}$$
$11$ $$( 8 + T )^{2}$$
$13$ $$38642 - 278 T + T^{2}$$
$17$ $$2 + 2 T + T^{2}$$
$19$ $$32400 + T^{2}$$
$23$ $$55112 + 332 T + T^{2}$$
$29$ $$230400 + T^{2}$$
$31$ $$( -572 + T )^{2}$$
$37$ $$126002 + 502 T + T^{2}$$
$41$ $$( 1688 + T )^{2}$$
$43$ $$4345352 - 2948 T + T^{2}$$
$47$ $$12241352 - 4948 T + T^{2}$$
$53$ $$22191122 + 6662 T + T^{2}$$
$59$ $$13395600 + T^{2}$$
$61$ $$( -1592 + T )^{2}$$
$67$ $$1527752 - 1748 T + T^{2}$$
$71$ $$( 6068 + T )^{2}$$
$73$ $$1251362 + 1582 T + T^{2}$$
$79$ $$83174400 + T^{2}$$
$83$ $$63935432 - 11308 T + T^{2}$$
$89$ $$4665600 + T^{2}$$
$97$ $$85831202 + 13102 T + T^{2}$$