Properties

Label 975.4.a.j
Level $975$
Weight $4$
Character orbit 975.a
Self dual yes
Analytic conductor $57.527$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 975 = 3 \cdot 5^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 975.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(57.5268622556\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{14}) \)
Defining polynomial: \( x^{2} - 14 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 39)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{14}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta - 1) q^{2} + 3 q^{3} + ( - 2 \beta + 7) q^{4} + (3 \beta - 3) q^{6} - 2 \beta q^{7} + (\beta - 27) q^{8} + 9 q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + (\beta - 1) q^{2} + 3 q^{3} + ( - 2 \beta + 7) q^{4} + (3 \beta - 3) q^{6} - 2 \beta q^{7} + (\beta - 27) q^{8} + 9 q^{9} + (12 \beta - 22) q^{11} + ( - 6 \beta + 21) q^{12} + 13 q^{13} + (2 \beta - 28) q^{14} + ( - 12 \beta - 15) q^{16} + (4 \beta - 82) q^{17} + (9 \beta - 9) q^{18} + ( - 2 \beta + 24) q^{19} - 6 \beta q^{21} + ( - 34 \beta + 190) q^{22} + (48 \beta - 4) q^{23} + (3 \beta - 81) q^{24} + (13 \beta - 13) q^{26} + 27 q^{27} + ( - 14 \beta + 56) q^{28} + (24 \beta + 202) q^{29} + (26 \beta + 20) q^{31} + ( - 11 \beta + 63) q^{32} + (36 \beta - 66) q^{33} + ( - 86 \beta + 138) q^{34} + ( - 18 \beta + 63) q^{36} + (28 \beta + 50) q^{37} + (26 \beta - 52) q^{38} + 39 q^{39} + ( - 94 \beta + 100) q^{41} + (6 \beta - 84) q^{42} + (52 \beta + 308) q^{43} + (128 \beta - 490) q^{44} + ( - 52 \beta + 676) q^{46} + (32 \beta + 162) q^{47} + ( - 36 \beta - 45) q^{48} - 287 q^{49} + (12 \beta - 246) q^{51} + ( - 26 \beta + 91) q^{52} + ( - 120 \beta + 82) q^{53} + (27 \beta - 27) q^{54} + (54 \beta - 28) q^{56} + ( - 6 \beta + 72) q^{57} + (178 \beta + 134) q^{58} + ( - 40 \beta + 70) q^{59} + ( - 136 \beta + 314) q^{61} + ( - 6 \beta + 344) q^{62} - 18 \beta q^{63} + (170 \beta - 97) q^{64} + ( - 102 \beta + 570) q^{66} + ( - 170 \beta + 236) q^{67} + (192 \beta - 686) q^{68} + (144 \beta - 12) q^{69} + (84 \beta + 214) q^{71} + (9 \beta - 243) q^{72} + (76 \beta + 450) q^{73} + (22 \beta + 342) q^{74} + ( - 62 \beta + 224) q^{76} + (44 \beta - 336) q^{77} + (39 \beta - 39) q^{78} + (88 \beta - 216) q^{79} + 81 q^{81} + (194 \beta - 1416) q^{82} + (64 \beta + 694) q^{83} + ( - 42 \beta + 168) q^{84} + (256 \beta + 420) q^{86} + (72 \beta + 606) q^{87} + ( - 346 \beta + 762) q^{88} + (190 \beta + 480) q^{89} - 26 \beta q^{91} + (344 \beta - 1372) q^{92} + (78 \beta + 60) q^{93} + (130 \beta + 286) q^{94} + ( - 33 \beta + 189) q^{96} + ( - 220 \beta + 266) q^{97} + ( - 287 \beta + 287) q^{98} + (108 \beta - 198) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} + 6 q^{3} + 14 q^{4} - 6 q^{6} - 54 q^{8} + 18 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{2} + 6 q^{3} + 14 q^{4} - 6 q^{6} - 54 q^{8} + 18 q^{9} - 44 q^{11} + 42 q^{12} + 26 q^{13} - 56 q^{14} - 30 q^{16} - 164 q^{17} - 18 q^{18} + 48 q^{19} + 380 q^{22} - 8 q^{23} - 162 q^{24} - 26 q^{26} + 54 q^{27} + 112 q^{28} + 404 q^{29} + 40 q^{31} + 126 q^{32} - 132 q^{33} + 276 q^{34} + 126 q^{36} + 100 q^{37} - 104 q^{38} + 78 q^{39} + 200 q^{41} - 168 q^{42} + 616 q^{43} - 980 q^{44} + 1352 q^{46} + 324 q^{47} - 90 q^{48} - 574 q^{49} - 492 q^{51} + 182 q^{52} + 164 q^{53} - 54 q^{54} - 56 q^{56} + 144 q^{57} + 268 q^{58} + 140 q^{59} + 628 q^{61} + 688 q^{62} - 194 q^{64} + 1140 q^{66} + 472 q^{67} - 1372 q^{68} - 24 q^{69} + 428 q^{71} - 486 q^{72} + 900 q^{73} + 684 q^{74} + 448 q^{76} - 672 q^{77} - 78 q^{78} - 432 q^{79} + 162 q^{81} - 2832 q^{82} + 1388 q^{83} + 336 q^{84} + 840 q^{86} + 1212 q^{87} + 1524 q^{88} + 960 q^{89} - 2744 q^{92} + 120 q^{93} + 572 q^{94} + 378 q^{96} + 532 q^{97} + 574 q^{98} - 396 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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
−3.74166
3.74166
−4.74166 3.00000 14.4833 0 −14.2250 7.48331 −30.7417 9.00000 0
1.2 2.74166 3.00000 −0.483315 0 8.22497 −7.48331 −23.2583 9.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(5\) \(1\)
\(13\) \(-1\)

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 975.4.a.j 2
5.b even 2 1 39.4.a.b 2
15.d odd 2 1 117.4.a.c 2
20.d odd 2 1 624.4.a.r 2
35.c odd 2 1 1911.4.a.h 2
40.e odd 2 1 2496.4.a.s 2
40.f even 2 1 2496.4.a.bc 2
60.h even 2 1 1872.4.a.t 2
65.d even 2 1 507.4.a.f 2
65.g odd 4 2 507.4.b.f 4
195.e odd 2 1 1521.4.a.s 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
39.4.a.b 2 5.b even 2 1
117.4.a.c 2 15.d odd 2 1
507.4.a.f 2 65.d even 2 1
507.4.b.f 4 65.g odd 4 2
624.4.a.r 2 20.d odd 2 1
975.4.a.j 2 1.a even 1 1 trivial
1521.4.a.s 2 195.e odd 2 1
1872.4.a.t 2 60.h even 2 1
1911.4.a.h 2 35.c odd 2 1
2496.4.a.s 2 40.e odd 2 1
2496.4.a.bc 2 40.f even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(975))\):

\( T_{2}^{2} + 2T_{2} - 13 \) Copy content Toggle raw display
\( T_{7}^{2} - 56 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + 2T - 13 \) Copy content Toggle raw display
$3$ \( (T - 3)^{2} \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} - 56 \) Copy content Toggle raw display
$11$ \( T^{2} + 44T - 1532 \) Copy content Toggle raw display
$13$ \( (T - 13)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} + 164T + 6500 \) Copy content Toggle raw display
$19$ \( T^{2} - 48T + 520 \) Copy content Toggle raw display
$23$ \( T^{2} + 8T - 32240 \) Copy content Toggle raw display
$29$ \( T^{2} - 404T + 32740 \) Copy content Toggle raw display
$31$ \( T^{2} - 40T - 9064 \) Copy content Toggle raw display
$37$ \( T^{2} - 100T - 8476 \) Copy content Toggle raw display
$41$ \( T^{2} - 200T - 113704 \) Copy content Toggle raw display
$43$ \( T^{2} - 616T + 57008 \) Copy content Toggle raw display
$47$ \( T^{2} - 324T + 11908 \) Copy content Toggle raw display
$53$ \( T^{2} - 164T - 194876 \) Copy content Toggle raw display
$59$ \( T^{2} - 140T - 17500 \) Copy content Toggle raw display
$61$ \( T^{2} - 628T - 160348 \) Copy content Toggle raw display
$67$ \( T^{2} - 472T - 348904 \) Copy content Toggle raw display
$71$ \( T^{2} - 428T - 52988 \) Copy content Toggle raw display
$73$ \( T^{2} - 900T + 121636 \) Copy content Toggle raw display
$79$ \( T^{2} + 432T - 61760 \) Copy content Toggle raw display
$83$ \( T^{2} - 1388 T + 424292 \) Copy content Toggle raw display
$89$ \( T^{2} - 960T - 275000 \) Copy content Toggle raw display
$97$ \( T^{2} - 532T - 606844 \) Copy content Toggle raw display
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