# 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.

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

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

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

## 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 $$T_{2}^{3}$$ $$\mathstrut -\mathstrut 5 T_{2}^{2}$$ $$\mathstrut -\mathstrut 2 T_{2}$$ $$\mathstrut +\mathstrut 12$$ acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(1175))$$.