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Properties

Label 325.6.a.l
Level $325$
Weight $6$
Character orbit 325.a
Self dual yes
Analytic conductor $52.125$
Analytic rank $1$
Dimension $15$
CM no
Inner twists $1$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [325,6,Mod(1,325)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(325, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("325.1");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 325 = 5^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 325.a (trivial)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(52.1247414392\)
Analytic rank: \(1\)
Dimension: \(15\)
Coefficient field: \(\mathbb{Q}[x]/(x^{15} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{15} - 3 x^{14} - 356 x^{13} + 802 x^{12} + 49252 x^{11} - 78702 x^{10} - 3324132 x^{9} + \cdots + 3541906480 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{7}\cdot 5^{6} \)
Twist minimal: no (minimal twist has level 65)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{14}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_1 - 1) q^{2} + ( - \beta_{2} - 2) q^{3} + (\beta_{3} + \beta_{2} - \beta_1 + 16) q^{4} + ( - \beta_{6} + 2 \beta_{2} + \cdots - 13) q^{6}+ \cdots + ( - \beta_{10} - \beta_{9} + 5 \beta_{2} + \cdots + 63) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_1 - 1) q^{2} + ( - \beta_{2} - 2) q^{3} + (\beta_{3} + \beta_{2} - \beta_1 + 16) q^{4} + ( - \beta_{6} + 2 \beta_{2} + \cdots - 13) q^{6}+ \cdots + (9 \beta_{14} - 212 \beta_{13} + \cdots - 18288) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 15 q - 12 q^{2} - 36 q^{3} + 250 q^{4} - 202 q^{6} - 306 q^{7} - 576 q^{8} + 963 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 15 q - 12 q^{2} - 36 q^{3} + 250 q^{4} - 202 q^{6} - 306 q^{7} - 576 q^{8} + 963 q^{9} - 326 q^{11} - 2702 q^{12} + 2535 q^{13} + 1334 q^{14} + 6046 q^{16} - 2722 q^{17} - 2036 q^{18} - 2624 q^{19} + 3530 q^{21} - 7004 q^{22} - 9174 q^{23} + 2740 q^{24} - 2028 q^{26} - 12594 q^{27} - 14880 q^{28} - 3042 q^{29} - 11146 q^{31} - 22054 q^{32} - 18582 q^{33} + 23030 q^{34} + 25172 q^{36} - 10192 q^{37} - 23626 q^{38} - 6084 q^{39} - 7928 q^{41} - 13498 q^{42} - 45934 q^{43} + 22758 q^{44} + 22698 q^{46} - 38730 q^{47} - 103564 q^{48} + 70543 q^{49} + 48084 q^{51} + 42250 q^{52} - 85340 q^{53} + 2482 q^{54} + 33578 q^{56} + 11962 q^{57} - 102450 q^{58} - 62312 q^{59} - 29530 q^{61} - 8266 q^{62} - 209670 q^{63} + 15066 q^{64} - 1240 q^{66} - 43018 q^{67} - 281646 q^{68} + 199340 q^{69} - 22868 q^{71} + 54972 q^{72} - 165124 q^{73} - 258566 q^{74} - 139240 q^{76} - 86270 q^{77} - 34138 q^{78} + 129584 q^{79} + 57731 q^{81} - 95426 q^{82} - 375574 q^{83} - 15182 q^{84} - 241780 q^{86} - 373706 q^{87} - 881886 q^{88} - 167638 q^{89} - 51714 q^{91} - 672054 q^{92} - 538530 q^{93} + 104250 q^{94} - 46370 q^{96} - 265450 q^{97} - 1324762 q^{98} - 260488 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{15} - 3 x^{14} - 356 x^{13} + 802 x^{12} + 49252 x^{11} - 78702 x^{10} - 3324132 x^{9} + \cdots + 3541906480 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 34\!\cdots\!33 \nu^{14} + \cdots - 10\!\cdots\!20 ) / 40\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 34\!\cdots\!33 \nu^{14} + \cdots - 79\!\cdots\!80 ) / 40\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 47\!\cdots\!63 \nu^{14} + \cdots + 41\!\cdots\!40 ) / 40\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 13\!\cdots\!15 \nu^{14} + \cdots - 64\!\cdots\!40 ) / 80\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 20\!\cdots\!99 \nu^{14} + \cdots - 73\!\cdots\!40 ) / 10\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 33\!\cdots\!05 \nu^{14} + \cdots - 13\!\cdots\!80 ) / 20\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 25\!\cdots\!01 \nu^{14} + \cdots + 98\!\cdots\!00 ) / 15\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 90\!\cdots\!95 \nu^{14} + \cdots - 33\!\cdots\!40 ) / 40\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 18\!\cdots\!87 \nu^{14} + \cdots + 12\!\cdots\!20 ) / 61\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 99\!\cdots\!13 \nu^{14} + \cdots - 67\!\cdots\!60 ) / 30\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 64\!\cdots\!77 \nu^{14} + \cdots - 34\!\cdots\!40 ) / 20\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 56\!\cdots\!51 \nu^{14} + \cdots + 33\!\cdots\!00 ) / 80\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 65\!\cdots\!21 \nu^{14} + \cdots - 38\!\cdots\!20 ) / 80\!\cdots\!00 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} + \beta_{2} + \beta _1 + 47 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{6} + \beta_{4} + 2\beta_{3} + \beta_{2} + 83\beta _1 + 37 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - 2 \beta_{12} - 2 \beta_{11} - 2 \beta_{10} - \beta_{8} + 2 \beta_{7} + 4 \beta_{6} + 2 \beta_{5} + \cdots + 3846 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 8 \beta_{13} - 2 \beta_{11} - 17 \beta_{10} - 11 \beta_{9} - 15 \beta_{8} + \beta_{7} + 134 \beta_{6} + \cdots + 7494 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 16 \beta_{14} + 34 \beta_{13} - 314 \beta_{12} - 350 \beta_{11} - 416 \beta_{10} - 80 \beta_{9} + \cdots + 363806 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 136 \beta_{14} + 1542 \beta_{13} - 196 \beta_{12} - 938 \beta_{11} - 3679 \beta_{10} - 2443 \beta_{9} + \cdots + 1123746 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 2688 \beta_{14} + 7708 \beta_{13} - 38682 \beta_{12} - 47262 \beta_{11} - 63044 \beta_{10} + \cdots + 36682106 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 21168 \beta_{14} + 216104 \beta_{13} - 53948 \beta_{12} - 196374 \beta_{11} - 590737 \beta_{10} + \cdots + 151050932 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 248256 \beta_{14} + 1228586 \beta_{13} - 4381850 \beta_{12} - 5885562 \beta_{11} - 8641014 \beta_{10} + \cdots + 3831763748 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 1441144 \beta_{14} + 26829658 \beta_{13} - 9550404 \beta_{12} - 31805410 \beta_{11} - 84548267 \beta_{10} + \cdots + 19323079826 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 5345888 \beta_{14} + 168903384 \beta_{13} - 477173538 \beta_{12} - 709778542 \beta_{11} + \cdots + 409918164010 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 111414240 \beta_{14} + 3136790332 \beta_{13} - 1393656140 \beta_{12} - 4569030326 \beta_{11} + \cdots + 2409721949492 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 3512806048 \beta_{14} + 21398072286 \beta_{13} - 50825997306 \beta_{12} - 84397501218 \beta_{11} + \cdots + 44657794158024 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−9.74364
−9.63516
−7.77693
−6.85882
−5.09884
−3.31587
−1.01498
0.556922
0.619549
4.30124
4.49260
5.87098
9.20857
10.4026
10.9918
−10.7436 10.0082 83.4258 0 −107.524 −239.875 −552.500 −142.837 0
1.2 −10.6352 −18.3028 81.1067 0 194.654 135.320 −522.258 91.9934 0
1.3 −8.77693 −27.1524 45.0345 0 238.314 −203.885 −114.402 494.251 0
1.4 −7.85882 17.7271 29.7611 0 −139.314 223.370 17.5950 71.2505 0
1.5 −6.09884 5.72858 5.19588 0 −34.9377 −69.8784 163.474 −210.183 0
1.6 −4.31587 25.7397 −13.3733 0 −111.089 −123.574 195.825 419.532 0
1.7 −2.01498 1.68184 −27.9399 0 −3.38888 176.446 120.777 −240.171 0
1.8 −0.443078 −13.8115 −31.8037 0 6.11956 81.9449 28.2700 −52.2434 0
1.9 −0.380451 −19.9398 −31.8553 0 7.58611 −249.944 24.2938 154.594 0
1.10 3.30124 24.0437 −21.1018 0 79.3742 −91.7264 −175.302 335.101 0
1.11 3.49260 3.04304 −19.8017 0 10.6281 77.0767 −180.923 −233.740 0
1.12 4.87098 −21.8793 −8.27359 0 −106.574 33.1862 −196.172 235.705 0
1.13 8.20857 3.74903 35.3806 0 30.7741 83.0932 27.7498 −228.945 0
1.14 9.40255 0.823884 56.4080 0 7.74661 −107.754 229.497 −242.321 0
1.15 9.99183 −27.4593 67.8367 0 −274.369 −29.7993 358.074 511.014 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.15
Significant digits:
Format:

Atkin-Lehner signs

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

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 325.6.a.l 15
5.b even 2 1 325.6.a.m 15
5.c odd 4 2 65.6.b.a ✓ 30
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
65.6.b.a ✓ 30 5.c odd 4 2
325.6.a.l 15 1.a even 1 1 trivial
325.6.a.m 15 5.b even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{15} + 12 T_{2}^{14} - 293 T_{2}^{13} - 3644 T_{2}^{12} + 31381 T_{2}^{11} + 414186 T_{2}^{10} + \cdots + 3051840000 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(325))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{15} + \cdots + 3051840000 \) Copy content Toggle raw display
$3$ \( T^{15} + \cdots - 817583487836544 \) Copy content Toggle raw display
$5$ \( T^{15} \) Copy content Toggle raw display
$7$ \( T^{15} + \cdots - 28\!\cdots\!68 \) Copy content Toggle raw display
$11$ \( T^{15} + \cdots + 22\!\cdots\!96 \) Copy content Toggle raw display
$13$ \( (T - 169)^{15} \) Copy content Toggle raw display
$17$ \( T^{15} + \cdots - 22\!\cdots\!36 \) Copy content Toggle raw display
$19$ \( T^{15} + \cdots + 25\!\cdots\!80 \) Copy content Toggle raw display
$23$ \( T^{15} + \cdots - 24\!\cdots\!32 \) Copy content Toggle raw display
$29$ \( T^{15} + \cdots + 72\!\cdots\!40 \) Copy content Toggle raw display
$31$ \( T^{15} + \cdots - 53\!\cdots\!20 \) Copy content Toggle raw display
$37$ \( T^{15} + \cdots - 92\!\cdots\!16 \) Copy content Toggle raw display
$41$ \( T^{15} + \cdots + 31\!\cdots\!96 \) Copy content Toggle raw display
$43$ \( T^{15} + \cdots + 41\!\cdots\!60 \) Copy content Toggle raw display
$47$ \( T^{15} + \cdots + 45\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( T^{15} + \cdots - 32\!\cdots\!56 \) Copy content Toggle raw display
$59$ \( T^{15} + \cdots + 85\!\cdots\!20 \) Copy content Toggle raw display
$61$ \( T^{15} + \cdots + 34\!\cdots\!16 \) Copy content Toggle raw display
$67$ \( T^{15} + \cdots + 28\!\cdots\!56 \) Copy content Toggle raw display
$71$ \( T^{15} + \cdots + 17\!\cdots\!72 \) Copy content Toggle raw display
$73$ \( T^{15} + \cdots - 16\!\cdots\!28 \) Copy content Toggle raw display
$79$ \( T^{15} + \cdots - 58\!\cdots\!20 \) Copy content Toggle raw display
$83$ \( T^{15} + \cdots - 12\!\cdots\!92 \) Copy content Toggle raw display
$89$ \( T^{15} + \cdots + 17\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{15} + \cdots + 86\!\cdots\!00 \) Copy content Toggle raw display
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