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

Downloads

Learn more about

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:

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 \(S_{4}^{\mathrm{new}}(\Gamma_0(1175))\).