# Properties

 Label 1175.4.a.a Level 1175 Weight 4 Character orbit 1175.a Self dual Yes Analytic conductor 69.327 Analytic rank 0 Dimension 3 CM No Inner twists 1

# Related objects

## Newspace parameters

 Level: $$N$$ = $$1175 = 5^{2} \cdot 47$$ Weight: $$k$$ = $$4$$ Character orbit: $$[\chi]$$ = 1175.a (trivial)

## Newform invariants

 Self dual: Yes Analytic conductor: $$69.3272442567$$ Analytic rank: $$0$$ Dimension: $$3$$ Coefficient field: 3.3.1101.1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( 2 + \beta_{2} ) q^{2} + ( 1 + \beta_{1} - \beta_{2} ) q^{3} + ( 2 + 2 \beta_{1} + 3 \beta_{2} ) q^{4} + ( -2 + 2 \beta_{2} ) q^{6} + ( 16 + \beta_{1} + 4 \beta_{2} ) q^{7} + ( 10 + 10 \beta_{1} + \beta_{2} ) q^{8} + ( -17 + 3 \beta_{1} - 6 \beta_{2} ) q^{9} +O(q^{10})$$ $$q + ( 2 + \beta_{2} ) q^{2} + ( 1 + \beta_{1} - \beta_{2} ) q^{3} + ( 2 + 2 \beta_{1} + 3 \beta_{2} ) q^{4} + ( -2 + 2 \beta_{2} ) q^{6} + ( 16 + \beta_{1} + 4 \beta_{2} ) q^{7} + ( 10 + 10 \beta_{1} + \beta_{2} ) q^{8} + ( -17 + 3 \beta_{1} - 6 \beta_{2} ) q^{9} + ( -4 + 12 \beta_{1} - 2 \beta_{2} ) q^{11} + ( -4 \beta_{1} + 8 \beta_{2} ) q^{12} + ( 28 + 4 \beta_{1} + 8 \beta_{2} ) q^{13} + ( 58 + 10 \beta_{1} + 22 \beta_{2} ) q^{14} + ( 30 + 6 \beta_{1} + 7 \beta_{2} ) q^{16} + ( 7 + 21 \beta_{1} + 3 \beta_{2} ) q^{17} + ( -64 - 6 \beta_{1} - 17 \beta_{2} ) q^{18} + ( -4 - 10 \beta_{1} + 2 \beta_{2} ) q^{19} + ( 5 + 8 \beta_{1} - \beta_{2} ) q^{21} + ( 4 + 20 \beta_{1} + 18 \beta_{2} ) q^{22} + ( -58 + 20 \beta_{1} - 34 \beta_{2} ) q^{23} + ( 56 + 8 \beta_{1} - 16 \beta_{2} ) q^{24} + ( 112 + 24 \beta_{1} + 44 \beta_{2} ) q^{26} + ( -5 - 32 \beta_{1} + 17 \beta_{2} ) q^{27} + ( 140 + 56 \beta_{1} + 68 \beta_{2} ) q^{28} + ( -40 - 68 \beta_{1} - 4 \beta_{2} ) q^{29} + ( -36 + 96 \beta_{1} - 8 \beta_{2} ) q^{31} + ( 34 - 54 \beta_{1} + 41 \beta_{2} ) q^{32} + ( 64 - 16 \beta_{2} ) q^{33} + ( 74 + 48 \beta_{1} + 52 \beta_{2} ) q^{34} + ( -106 - 70 \beta_{1} - 45 \beta_{2} ) q^{36} + ( 193 + 7 \beta_{1} - 3 \beta_{2} ) q^{37} + ( -16 - 16 \beta_{1} - 22 \beta_{2} ) q^{38} + ( 16 + 12 \beta_{1} ) q^{39} + ( -30 + 42 \beta_{1} + 44 \beta_{2} ) q^{41} + ( 20 + 14 \beta_{1} + 20 \beta_{2} ) q^{42} + ( 38 + 44 \beta_{1} - 92 \beta_{2} ) q^{43} + ( 188 - 20 \beta_{1} + 78 \beta_{2} ) q^{44} + ( -280 - 28 \beta_{1} - 52 \beta_{2} ) q^{46} -47 q^{47} + ( 32 + 16 \beta_{1} - 8 \beta_{2} ) q^{48} + ( 32 + 63 \beta_{1} + 129 \beta_{2} ) q^{49} + ( 100 + \beta_{1} - 16 \beta_{2} ) q^{51} + ( 312 + 104 \beta_{1} + 140 \beta_{2} ) q^{52} + ( -96 - 161 \beta_{1} + 10 \beta_{2} ) q^{53} + ( 28 - 30 \beta_{1} - 52 \beta_{2} ) q^{54} + ( 336 + 168 \beta_{1} + 144 \beta_{2} ) q^{56} + ( -62 - 8 \beta_{1} + 22 \beta_{2} ) q^{57} + ( -240 - 144 \beta_{1} - 180 \beta_{2} ) q^{58} + ( 132 - 29 \beta_{1} - 212 \beta_{2} ) q^{59} + ( 108 + 49 \beta_{1} + 106 \beta_{2} ) q^{61} + ( 72 + 176 \beta_{1} + 148 \beta_{2} ) q^{62} + ( -383 - 20 \beta_{1} - 125 \beta_{2} ) q^{63} + ( -34 - 74 \beta_{1} - 89 \beta_{2} ) q^{64} + ( 32 - 32 \beta_{1} + 48 \beta_{2} ) q^{66} + ( 326 - 150 \beta_{1} + 288 \beta_{2} ) q^{67} + ( 500 + 32 \beta_{1} + 198 \beta_{2} ) q^{68} + ( 178 + 10 \beta_{1} - 98 \beta_{2} ) q^{69} + ( 233 - 135 \beta_{1} - 185 \beta_{2} ) q^{71} + ( -110 - 182 \beta_{1} - 155 \beta_{2} ) q^{72} + ( 600 + 162 \beta_{1} + 38 \beta_{2} ) q^{73} + ( 382 + 8 \beta_{1} + 204 \beta_{2} ) q^{74} + ( -164 + 4 \beta_{1} - 86 \beta_{2} ) q^{76} + ( 64 + 160 \beta_{1} + 64 \beta_{2} ) q^{77} + ( 56 + 24 \beta_{1} + 40 \beta_{2} ) q^{78} + ( 345 - 7 \beta_{1} + 223 \beta_{2} ) q^{79} + ( 226 - 120 \beta_{1} + 267 \beta_{2} ) q^{81} + ( 288 + 172 \beta_{1} + 98 \beta_{2} ) q^{82} + ( -340 + 96 \beta_{1} - 212 \beta_{2} ) q^{83} + ( 148 + 4 \beta_{1} + 76 \beta_{2} ) q^{84} + ( -388 - 96 \beta_{1} + 34 \beta_{2} ) q^{86} + ( -364 - 32 \beta_{1} + 92 \beta_{2} ) q^{87} + ( 772 - 44 \beta_{1} + 82 \beta_{2} ) q^{88} + ( 192 + 181 \beta_{1} - 78 \beta_{2} ) q^{89} + ( 716 + 152 \beta_{1} + 260 \beta_{2} ) q^{91} + ( -464 - 320 \beta_{1} - 116 \beta_{2} ) q^{92} + ( 476 - 20 \beta_{1} - 92 \beta_{2} ) q^{93} + ( -94 - 47 \beta_{2} ) q^{94} + ( -400 - 48 \beta_{1} + 184 \beta_{2} ) q^{96} + ( 860 - 259 \beta_{1} + 78 \beta_{2} ) q^{97} + ( 964 + 384 \beta_{1} + 287 \beta_{2} ) q^{98} + ( 236 - 228 \beta_{1} - 74 \beta_{2} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3q + 5q^{2} + 5q^{3} + 5q^{4} - 8q^{6} + 45q^{7} + 39q^{8} - 42q^{9} + O(q^{10})$$ $$3q + 5q^{2} + 5q^{3} + 5q^{4} - 8q^{6} + 45q^{7} + 39q^{8} - 42q^{9} + 2q^{11} - 12q^{12} + 80q^{13} + 162q^{14} + 89q^{16} + 39q^{17} - 181q^{18} - 24q^{19} + 24q^{21} + 14q^{22} - 120q^{23} + 192q^{24} + 316q^{26} - 64q^{27} + 408q^{28} - 184q^{29} - 4q^{31} + 7q^{32} + 208q^{33} + 218q^{34} - 343q^{36} + 589q^{37} - 42q^{38} + 60q^{39} - 92q^{41} + 54q^{42} + 250q^{43} + 466q^{44} - 816q^{46} - 141q^{47} + 120q^{48} + 30q^{49} + 317q^{51} + 900q^{52} - 459q^{53} + 106q^{54} + 1032q^{56} - 216q^{57} - 684q^{58} + 579q^{59} + 267q^{61} + 244q^{62} - 1044q^{63} - 87q^{64} + 16q^{66} + 540q^{67} + 1334q^{68} + 642q^{69} + 749q^{71} - 357q^{72} + 1924q^{73} + 950q^{74} - 402q^{76} + 288q^{77} + 152q^{78} + 805q^{79} + 291q^{81} + 938q^{82} - 712q^{83} + 372q^{84} - 1294q^{86} - 1216q^{87} + 2190q^{88} + 835q^{89} + 2040q^{91} - 1596q^{92} + 1500q^{93} - 235q^{94} - 1432q^{96} + 2243q^{97} + 2989q^{98} + 554q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{3} - x^{2} - 9 x + 12$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2} + \nu - 7$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{2} - \beta_{1} + 7$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 1.43163 −3.11903 2.68740
−1.51882 5.95044 −5.69320 0 −9.03763 3.35636 20.7975 8.40778 0
1.2 1.60930 −1.72833 −5.41015 0 −2.78140 11.3182 −21.5810 −24.0129 0
1.3 4.90952 0.777884 16.1033 0 3.81903 30.3255 39.7835 −26.3949 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Atkin-Lehner signs

$$p$$ Sign
$$5$$ $$1$$
$$47$$ $$1$$

## Hecke kernels

This newform can be constructed as the kernel of the linear operator $$T_{2}^{3} - 5 T_{2}^{2} - 2 T_{2} + 12$$ acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(1175))$$.