Properties

Label 2352.4.a.cg
Level 2352
Weight 4
Character orbit 2352.a
Self dual yes
Analytic conductor 138.772
Analytic rank 0
Dimension 3
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 2352 = 2^{4} \cdot 3 \cdot 7^{2} \)
Weight: \( k \) = \( 4 \)
Character orbit: \([\chi]\) = 2352.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(138.772492334\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.57516.1
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 21)
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.

\(f(q)\) \(=\) \( q -3 q^{3} + ( -4 - \beta_{2} ) q^{5} + 9 q^{9} +O(q^{10})\) \( q -3 q^{3} + ( -4 - \beta_{2} ) q^{5} + 9 q^{9} + ( -11 + \beta_{1} + 2 \beta_{2} ) q^{11} + ( -20 - \beta_{1} + 2 \beta_{2} ) q^{13} + ( 12 + 3 \beta_{2} ) q^{15} + ( -17 - \beta_{1} - 3 \beta_{2} ) q^{17} + ( -67 - 2 \beta_{1} + \beta_{2} ) q^{19} + ( -73 + 7 \beta_{1} - 3 \beta_{2} ) q^{23} + ( 46 - 7 \beta_{1} + 8 \beta_{2} ) q^{25} -27 q^{27} + ( 21 - 5 \beta_{1} + 10 \beta_{2} ) q^{29} + ( -34 + 5 \beta_{1} - 7 \beta_{2} ) q^{31} + ( 33 - 3 \beta_{1} - 6 \beta_{2} ) q^{33} + ( 86 + 5 \beta_{1} - 4 \beta_{2} ) q^{37} + ( 60 + 3 \beta_{1} - 6 \beta_{2} ) q^{39} + ( -78 + 4 \beta_{1} + 10 \beta_{2} ) q^{41} + ( -125 + 18 \beta_{1} - 15 \beta_{2} ) q^{43} + ( -36 - 9 \beta_{2} ) q^{45} + ( -71 - 25 \beta_{1} - 3 \beta_{2} ) q^{47} + ( 51 + 3 \beta_{1} + 9 \beta_{2} ) q^{51} + ( 134 - 20 \beta_{1} + 9 \beta_{2} ) q^{53} + ( -339 + 11 \beta_{1} + 14 \beta_{2} ) q^{55} + ( 201 + 6 \beta_{1} - 3 \beta_{2} ) q^{57} + ( 368 + 10 \beta_{1} - 39 \beta_{2} ) q^{59} + ( 10 + 20 \beta_{1} - 40 \beta_{2} ) q^{61} + ( -157 + 17 \beta_{1} + \beta_{2} ) q^{65} + ( 199 - 16 \beta_{1} - 31 \beta_{2} ) q^{67} + ( 219 - 21 \beta_{1} + 9 \beta_{2} ) q^{69} + ( -99 - 33 \beta_{1} + 21 \beta_{2} ) q^{71} + ( -332 + 31 \beta_{1} - 8 \beta_{2} ) q^{73} + ( -138 + 21 \beta_{1} - 24 \beta_{2} ) q^{75} + ( -302 - 3 \beta_{1} - 45 \beta_{2} ) q^{79} + 81 q^{81} + ( 162 + 6 \beta_{1} - 33 \beta_{2} ) q^{83} + ( 606 - 18 \beta_{1} + 18 \beta_{2} ) q^{85} + ( -63 + 15 \beta_{1} - 30 \beta_{2} ) q^{87} + ( -580 - 48 \beta_{1} + 26 \beta_{2} ) q^{89} + ( 102 - 15 \beta_{1} + 21 \beta_{2} ) q^{93} + ( 259 + 13 \beta_{1} + 41 \beta_{2} ) q^{95} + ( -23 + \beta_{1} - 50 \beta_{2} ) q^{97} + ( -99 + 9 \beta_{1} + 18 \beta_{2} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q - 9q^{3} - 11q^{5} + 27q^{9} + O(q^{10}) \) \( 3q - 9q^{3} - 11q^{5} + 27q^{9} - 35q^{11} - 62q^{13} + 33q^{15} - 48q^{17} - 202q^{19} - 216q^{23} + 130q^{25} - 81q^{27} + 53q^{29} - 95q^{31} + 105q^{33} + 262q^{37} + 186q^{39} - 244q^{41} - 360q^{43} - 99q^{45} - 210q^{47} + 144q^{51} + 393q^{53} - 1031q^{55} + 606q^{57} + 1143q^{59} + 70q^{61} - 472q^{65} + 628q^{67} + 648q^{69} - 318q^{71} - 988q^{73} - 390q^{75} - 861q^{79} + 243q^{81} + 519q^{83} + 1800q^{85} - 159q^{87} - 1766q^{89} + 285q^{93} + 736q^{95} - 19q^{97} - 315q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{3} - x^{2} - 24 x + 6\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu^{2} + 2 \nu - 17 \)
\(\beta_{2}\)\(=\)\( \nu^{2} - 2 \nu - 16 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(-\beta_{2} + \beta_{1} + 1\)\()/4\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{2} + \beta_{1} + 33\)\()/2\)

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
−4.55637
5.30829
0.248072
0 −3.00000 0 −17.8732 0 0 0 9.00000 0
1.2 0 −3.00000 0 −5.56140 0 0 0 9.00000 0
1.3 0 −3.00000 0 12.4346 0 0 0 9.00000 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 2352.4.a.cg 3
4.b odd 2 1 147.4.a.m 3
7.b odd 2 1 2352.4.a.ci 3
7.d odd 6 2 336.4.q.k 6
12.b even 2 1 441.4.a.t 3
28.d even 2 1 147.4.a.l 3
28.f even 6 2 21.4.e.b 6
28.g odd 6 2 147.4.e.n 6
84.h odd 2 1 441.4.a.s 3
84.j odd 6 2 63.4.e.c 6
84.n even 6 2 441.4.e.w 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.4.e.b 6 28.f even 6 2
63.4.e.c 6 84.j odd 6 2
147.4.a.l 3 28.d even 2 1
147.4.a.m 3 4.b odd 2 1
147.4.e.n 6 28.g odd 6 2
336.4.q.k 6 7.d odd 6 2
441.4.a.s 3 84.h odd 2 1
441.4.a.t 3 12.b even 2 1
441.4.e.w 6 84.n even 6 2
2352.4.a.cg 3 1.a even 1 1 trivial
2352.4.a.ci 3 7.b odd 2 1

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(1\)
\(7\) \(-1\)

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(2352))\):

\( T_{5}^{3} + 11 T_{5}^{2} - 192 T_{5} - 1236 \)
\( T_{11}^{3} + 35 T_{11}^{2} - 1368 T_{11} + 9564 \)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( \)
$3$ \( ( 1 + 3 T )^{3} \)
$5$ \( 1 + 11 T + 183 T^{2} + 1514 T^{3} + 22875 T^{4} + 171875 T^{5} + 1953125 T^{6} \)
$7$ \( \)
$11$ \( 1 + 35 T + 2625 T^{2} + 102734 T^{3} + 3493875 T^{4} + 62004635 T^{5} + 2357947691 T^{6} \)
$13$ \( 1 + 62 T + 7016 T^{2} + 253976 T^{3} + 15414152 T^{4} + 299262158 T^{5} + 10604499373 T^{6} \)
$17$ \( 1 + 48 T + 12339 T^{2} + 358752 T^{3} + 60621507 T^{4} + 1158603312 T^{5} + 118587876497 T^{6} \)
$19$ \( 1 + 202 T + 32858 T^{2} + 3004840 T^{3} + 225373022 T^{4} + 9503267962 T^{5} + 322687697779 T^{6} \)
$23$ \( 1 + 216 T + 35829 T^{2} + 3675600 T^{3} + 435931443 T^{4} + 31975752024 T^{5} + 1801152661463 T^{6} \)
$29$ \( 1 - 53 T + 52695 T^{2} - 3410210 T^{3} + 1285178355 T^{4} - 31525636013 T^{5} + 14507145975869 T^{6} \)
$31$ \( 1 + 95 T + 79372 T^{2} + 5648467 T^{3} + 2364571252 T^{4} + 84312849695 T^{5} + 26439622160671 T^{6} \)
$37$ \( 1 - 262 T + 166048 T^{2} - 26591324 T^{3} + 8410829344 T^{4} - 672220319158 T^{5} + 129961739795077 T^{6} \)
$41$ \( 1 + 244 T + 187983 T^{2} + 33933832 T^{3} + 12955976343 T^{4} + 1159025434804 T^{5} + 327381934393961 T^{6} \)
$43$ \( 1 + 360 T + 166158 T^{2} + 38975294 T^{3} + 13210724106 T^{4} + 2275690697640 T^{5} + 502592611936843 T^{6} \)
$47$ \( 1 + 210 T + 64953 T^{2} + 48724788 T^{3} + 6743615319 T^{4} + 2263635219090 T^{5} + 1119130473102767 T^{6} \)
$53$ \( 1 - 393 T + 365895 T^{2} - 83847930 T^{3} + 54473349915 T^{4} - 8710593923697 T^{5} + 3299763591802133 T^{6} \)
$59$ \( 1 - 1143 T + 749241 T^{2} - 369027450 T^{3} + 153878367339 T^{4} - 48212349951663 T^{5} + 8662995818654939 T^{6} \)
$61$ \( 1 - 70 T + 340043 T^{2} + 52853660 T^{3} + 77183300183 T^{4} - 3606426205270 T^{5} + 11694146092834141 T^{6} \)
$67$ \( 1 - 628 T + 597326 T^{2} - 405751330 T^{3} + 179653559738 T^{4} - 56807864002132 T^{5} + 27206534396294947 T^{6} \)
$71$ \( 1 + 318 T + 742929 T^{2} + 256167372 T^{3} + 265902461319 T^{4} + 40735890286878 T^{5} + 45848500718449031 T^{6} \)
$73$ \( 1 + 988 T + 1162696 T^{2} + 625490474 T^{3} + 452308509832 T^{4} + 149518215573532 T^{5} + 58871586708267913 T^{6} \)
$79$ \( 1 + 861 T + 1221216 T^{2} + 655056821 T^{3} + 602107115424 T^{4} + 209298299203581 T^{5} + 119851595982618319 T^{6} \)
$83$ \( 1 - 519 T + 1583745 T^{2} - 545598870 T^{3} + 905564802315 T^{4} - 169682053778511 T^{5} + 186940255267540403 T^{6} \)
$89$ \( 1 + 1766 T + 2392827 T^{2} + 2476945964 T^{3} + 1686868857363 T^{4} + 877668959837126 T^{5} + 350356403707485209 T^{6} \)
$97$ \( 1 + 19 T + 2168419 T^{2} - 10094878 T^{3} + 1979057473987 T^{4} + 15826468093651 T^{5} + 760231058654565217 T^{6} \)
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