Properties

Label 3075.2.a.t
Level 3075
Weight 2
Character orbit 3075.a
Self dual yes
Analytic conductor 24.554
Analytic rank 1
Dimension 3
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 3075 = 3 \cdot 5^{2} \cdot 41 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 3075.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(24.5539986215\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.316.1
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 123)
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.

\(f(q)\) \(=\) \( q -\beta_{1} q^{2} + q^{3} + ( 1 + \beta_{2} ) q^{4} -\beta_{1} q^{6} + ( -1 + \beta_{1} + \beta_{2} ) q^{7} + ( -1 - \beta_{2} ) q^{8} + q^{9} +O(q^{10})\) \( q -\beta_{1} q^{2} + q^{3} + ( 1 + \beta_{2} ) q^{4} -\beta_{1} q^{6} + ( -1 + \beta_{1} + \beta_{2} ) q^{7} + ( -1 - \beta_{2} ) q^{8} + q^{9} + ( -1 - \beta_{1} ) q^{11} + ( 1 + \beta_{2} ) q^{12} + ( -3 + \beta_{1} - \beta_{2} ) q^{13} + ( -4 - 2 \beta_{2} ) q^{14} + ( -1 + 2 \beta_{1} - \beta_{2} ) q^{16} + ( -1 + \beta_{1} - 2 \beta_{2} ) q^{17} -\beta_{1} q^{18} + ( 1 - \beta_{1} + \beta_{2} ) q^{19} + ( -1 + \beta_{1} + \beta_{2} ) q^{21} + ( 3 + \beta_{1} + \beta_{2} ) q^{22} + ( 3 + \beta_{1} - \beta_{2} ) q^{23} + ( -1 - \beta_{2} ) q^{24} + ( -2 + 4 \beta_{1} ) q^{26} + q^{27} + ( 4 + 4 \beta_{1} ) q^{28} + ( -1 - 3 \beta_{1} ) q^{29} + ( -2 + 4 \beta_{1} + \beta_{2} ) q^{31} + ( -3 + 2 \beta_{1} + \beta_{2} ) q^{32} + ( -1 - \beta_{1} ) q^{33} + ( -1 + 3 \beta_{1} + \beta_{2} ) q^{34} + ( 1 + \beta_{2} ) q^{36} + ( -6 - 2 \beta_{1} + \beta_{2} ) q^{37} + ( 2 - 2 \beta_{1} ) q^{38} + ( -3 + \beta_{1} - \beta_{2} ) q^{39} + q^{41} + ( -4 - 2 \beta_{2} ) q^{42} + ( -4 + 2 \beta_{1} - 5 \beta_{2} ) q^{43} + ( -2 - 2 \beta_{1} - 2 \beta_{2} ) q^{44} + ( -2 - 2 \beta_{1} ) q^{46} + ( -1 - \beta_{1} - 2 \beta_{2} ) q^{47} + ( -1 + 2 \beta_{1} - \beta_{2} ) q^{48} + ( 3 + 2 \beta_{1} ) q^{49} + ( -1 + \beta_{1} - 2 \beta_{2} ) q^{51} + ( -6 - 2 \beta_{2} ) q^{52} + ( -6 + 4 \beta_{1} - 2 \beta_{2} ) q^{53} -\beta_{1} q^{54} + ( -4 - 4 \beta_{1} ) q^{56} + ( 1 - \beta_{1} + \beta_{2} ) q^{57} + ( 9 + \beta_{1} + 3 \beta_{2} ) q^{58} + ( -2 - 2 \beta_{1} - 2 \beta_{2} ) q^{59} + ( -2 - 2 \beta_{1} + \beta_{2} ) q^{61} + ( -13 + \beta_{1} - 5 \beta_{2} ) q^{62} + ( -1 + \beta_{1} + \beta_{2} ) q^{63} + ( -5 - 2 \beta_{1} - \beta_{2} ) q^{64} + ( 3 + \beta_{1} + \beta_{2} ) q^{66} + ( -2 - 6 \beta_{1} + 4 \beta_{2} ) q^{67} + ( -8 - 2 \beta_{1} ) q^{68} + ( 3 + \beta_{1} - \beta_{2} ) q^{69} + ( -11 + \beta_{1} ) q^{71} + ( -1 - \beta_{2} ) q^{72} + ( -2 + 2 \beta_{1} + 3 \beta_{2} ) q^{73} + ( 5 + 5 \beta_{1} + \beta_{2} ) q^{74} + 4 q^{76} + ( -3 - \beta_{1} - 3 \beta_{2} ) q^{77} + ( -2 + 4 \beta_{1} ) q^{78} + ( -8 + 4 \beta_{1} - 2 \beta_{2} ) q^{79} + q^{81} -\beta_{1} q^{82} + ( 5 - \beta_{1} + 3 \beta_{2} ) q^{83} + ( 4 + 4 \beta_{1} ) q^{84} + ( -1 + 9 \beta_{1} + 3 \beta_{2} ) q^{86} + ( -1 - 3 \beta_{1} ) q^{87} + ( 2 + 2 \beta_{1} + 2 \beta_{2} ) q^{88} + ( 6 - 4 \beta_{1} ) q^{89} + ( 2 - 6 \beta_{1} ) q^{91} + 4 \beta_{2} q^{92} + ( -2 + 4 \beta_{1} + \beta_{2} ) q^{93} + ( 5 + 3 \beta_{1} + 3 \beta_{2} ) q^{94} + ( -3 + 2 \beta_{1} + \beta_{2} ) q^{96} + ( 3 + 3 \beta_{1} - \beta_{2} ) q^{97} + ( -6 - 3 \beta_{1} - 2 \beta_{2} ) q^{98} + ( -1 - \beta_{1} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q - q^{2} + 3q^{3} + 3q^{4} - q^{6} - 2q^{7} - 3q^{8} + 3q^{9} + O(q^{10}) \) \( 3q - q^{2} + 3q^{3} + 3q^{4} - q^{6} - 2q^{7} - 3q^{8} + 3q^{9} - 4q^{11} + 3q^{12} - 8q^{13} - 12q^{14} - q^{16} - 2q^{17} - q^{18} + 2q^{19} - 2q^{21} + 10q^{22} + 10q^{23} - 3q^{24} - 2q^{26} + 3q^{27} + 16q^{28} - 6q^{29} - 2q^{31} - 7q^{32} - 4q^{33} + 3q^{36} - 20q^{37} + 4q^{38} - 8q^{39} + 3q^{41} - 12q^{42} - 10q^{43} - 8q^{44} - 8q^{46} - 4q^{47} - q^{48} + 11q^{49} - 2q^{51} - 18q^{52} - 14q^{53} - q^{54} - 16q^{56} + 2q^{57} + 28q^{58} - 8q^{59} - 8q^{61} - 38q^{62} - 2q^{63} - 17q^{64} + 10q^{66} - 12q^{67} - 26q^{68} + 10q^{69} - 32q^{71} - 3q^{72} - 4q^{73} + 20q^{74} + 12q^{76} - 10q^{77} - 2q^{78} - 20q^{79} + 3q^{81} - q^{82} + 14q^{83} + 16q^{84} + 6q^{86} - 6q^{87} + 8q^{88} + 14q^{89} - 2q^{93} + 18q^{94} - 7q^{96} + 12q^{97} - 21q^{98} - 4q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{3} - x^{2} - 4 x + 2\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - 3 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + 3\)

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
2.34292
0.470683
−1.81361
−2.34292 1.00000 3.48929 0 −2.34292 3.83221 −3.48929 1.00000 0
1.2 −0.470683 1.00000 −1.77846 0 −0.470683 −3.30777 1.77846 1.00000 0
1.3 1.81361 1.00000 1.28917 0 1.81361 −2.52444 −1.28917 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3075.2.a.t 3
3.b odd 2 1 9225.2.a.bx 3
5.b even 2 1 123.2.a.d 3
15.d odd 2 1 369.2.a.e 3
20.d odd 2 1 1968.2.a.w 3
35.c odd 2 1 6027.2.a.s 3
40.e odd 2 1 7872.2.a.bs 3
40.f even 2 1 7872.2.a.bx 3
60.h even 2 1 5904.2.a.bd 3
205.c even 2 1 5043.2.a.n 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
123.2.a.d 3 5.b even 2 1
369.2.a.e 3 15.d odd 2 1
1968.2.a.w 3 20.d odd 2 1
3075.2.a.t 3 1.a even 1 1 trivial
5043.2.a.n 3 205.c even 2 1
5904.2.a.bd 3 60.h even 2 1
6027.2.a.s 3 35.c odd 2 1
7872.2.a.bs 3 40.e odd 2 1
7872.2.a.bx 3 40.f even 2 1
9225.2.a.bx 3 3.b odd 2 1

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(5\) \(1\)
\(41\) \(-1\)

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(3075))\):

\( T_{2}^{3} + T_{2}^{2} - 4 T_{2} - 2 \)
\( T_{7}^{3} + 2 T_{7}^{2} - 14 T_{7} - 32 \)
\( T_{13}^{3} + 8 T_{13}^{2} + 14 T_{13} - 4 \)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( 1 + T + 2 T^{2} + 2 T^{3} + 4 T^{4} + 4 T^{5} + 8 T^{6} \)
$3$ \( ( 1 - T )^{3} \)
$5$ \( \)
$7$ \( 1 + 2 T + 7 T^{2} - 4 T^{3} + 49 T^{4} + 98 T^{5} + 343 T^{6} \)
$11$ \( 1 + 4 T + 34 T^{2} + 84 T^{3} + 374 T^{4} + 484 T^{5} + 1331 T^{6} \)
$13$ \( 1 + 8 T + 53 T^{2} + 204 T^{3} + 689 T^{4} + 1352 T^{5} + 2197 T^{6} \)
$17$ \( 1 + 2 T + 28 T^{2} + 6 T^{3} + 476 T^{4} + 578 T^{5} + 4913 T^{6} \)
$19$ \( 1 - 2 T + 51 T^{2} - 68 T^{3} + 969 T^{4} - 722 T^{5} + 6859 T^{6} \)
$23$ \( 1 - 10 T + 95 T^{2} - 476 T^{3} + 2185 T^{4} - 5290 T^{5} + 12167 T^{6} \)
$29$ \( 1 + 6 T + 60 T^{2} + 262 T^{3} + 1740 T^{4} + 5046 T^{5} + 24389 T^{6} \)
$31$ \( 1 + 2 T + 2 T^{2} - 132 T^{3} + 62 T^{4} + 1922 T^{5} + 29791 T^{6} \)
$37$ \( 1 + 20 T + 228 T^{2} + 1646 T^{3} + 8436 T^{4} + 27380 T^{5} + 50653 T^{6} \)
$41$ \( ( 1 - T )^{3} \)
$43$ \( 1 + 10 T + 10 T^{2} - 296 T^{3} + 430 T^{4} + 18490 T^{5} + 79507 T^{6} \)
$47$ \( 1 + 4 T + 106 T^{2} + 384 T^{3} + 4982 T^{4} + 8836 T^{5} + 103823 T^{6} \)
$53$ \( 1 + 14 T + 159 T^{2} + 1452 T^{3} + 8427 T^{4} + 39326 T^{5} + 148877 T^{6} \)
$59$ \( 1 + 8 T + 137 T^{2} + 976 T^{3} + 8083 T^{4} + 27848 T^{5} + 205379 T^{6} \)
$61$ \( 1 + 8 T + 188 T^{2} + 930 T^{3} + 11468 T^{4} + 29768 T^{5} + 226981 T^{6} \)
$67$ \( 1 + 12 T + 77 T^{2} + 632 T^{3} + 5159 T^{4} + 53868 T^{5} + 300763 T^{6} \)
$71$ \( 1 + 32 T + 550 T^{2} + 5712 T^{3} + 39050 T^{4} + 161312 T^{5} + 357911 T^{6} \)
$73$ \( 1 + 4 T + 120 T^{2} + 130 T^{3} + 8760 T^{4} + 21316 T^{5} + 389017 T^{6} \)
$79$ \( 1 + 20 T + 305 T^{2} + 3192 T^{3} + 24095 T^{4} + 124820 T^{5} + 493039 T^{6} \)
$83$ \( 1 - 14 T + 259 T^{2} - 2028 T^{3} + 21497 T^{4} - 96446 T^{5} + 571787 T^{6} \)
$89$ \( 1 - 14 T + 263 T^{2} - 2308 T^{3} + 23407 T^{4} - 110894 T^{5} + 704969 T^{6} \)
$97$ \( 1 - 12 T + 305 T^{2} - 2180 T^{3} + 29585 T^{4} - 112908 T^{5} + 912673 T^{6} \)
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