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Properties

Label 325.6.b.g
Level $325$
Weight $6$
Character orbit 325.b
Analytic conductor $52.125$
Analytic rank $0$
Dimension $12$
Inner twists $2$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [325,6,Mod(274,325)]
 
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([1, 0]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("325.274");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 325 = 5^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 325.b (of order \(2\), degree \(1\), not minimal)

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: no
Analytic conductor: \(52.1247414392\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} + 326x^{10} + 38809x^{8} + 2034064x^{6} + 43897824x^{4} + 281822976x^{2} + 377913600 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{14}\cdot 3^{2} \)
Twist minimal: no (minimal twist has level 65)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + ( - \beta_{7} + 6 \beta_{3} - \beta_1) q^{3} + (\beta_{5} - \beta_{4} - 22) q^{4} + ( - \beta_{8} - \beta_{5} + 2 \beta_{4} + \cdots + 52) q^{6}+ \cdots + (2 \beta_{8} + 2 \beta_{6} - 4 \beta_{5} + \cdots - 83) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + ( - \beta_{7} + 6 \beta_{3} - \beta_1) q^{3} + (\beta_{5} - \beta_{4} - 22) q^{4} + ( - \beta_{8} - \beta_{5} + 2 \beta_{4} + \cdots + 52) q^{6}+ \cdots + ( - 1296 \beta_{8} - 325 \beta_{6} + \cdots + 4260) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 268 q^{4} + 636 q^{6} - 1036 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 268 q^{4} + 636 q^{6} - 1036 q^{9} - 340 q^{11} + 2880 q^{14} + 7012 q^{16} - 2436 q^{19} - 792 q^{21} - 25236 q^{24} - 16728 q^{29} + 5724 q^{31} + 42968 q^{34} - 4276 q^{36} - 12844 q^{39} + 4496 q^{41} + 146964 q^{44} - 79772 q^{46} - 171804 q^{49} - 110056 q^{51} - 91008 q^{54} - 377600 q^{56} + 57748 q^{59} + 1944 q^{61} - 28044 q^{64} - 316080 q^{66} - 306176 q^{69} - 148348 q^{71} - 452224 q^{74} - 186844 q^{76} - 429016 q^{79} + 232012 q^{81} + 470536 q^{84} - 609332 q^{86} - 188056 q^{89} + 74360 q^{91} + 251840 q^{94} - 771716 q^{96} + 64540 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} + 326x^{10} + 38809x^{8} + 2034064x^{6} + 43897824x^{4} + 281822976x^{2} + 377913600 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{10} - 502\nu^{8} + 454465\nu^{6} + 86698324\nu^{4} + 3394374192\nu^{2} + 10823025600 ) / 3687690240 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 1031 \nu^{11} - 336070 \nu^{9} - 40030151 \nu^{7} - 2080759244 \nu^{5} + \cdots - 168362017344 \nu ) / 132756848640 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 1471 \nu^{10} - 492758 \nu^{8} - 59320255 \nu^{6} - 3021732364 \nu^{4} - 56999366352 \nu^{2} - 200791794240 ) / 3971358720 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 1471 \nu^{10} - 492758 \nu^{8} - 59320255 \nu^{6} - 3021732364 \nu^{4} - 53028007632 \nu^{2} + 13661576640 ) / 3971358720 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 419 \nu^{10} - 109774 \nu^{8} - 10517891 \nu^{6} - 533274476 \nu^{4} - 16062475536 \nu^{2} - 99708770880 ) / 860461056 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 42439 \nu^{11} - 15188390 \nu^{9} - 2004833479 \nu^{7} - 118454680876 \nu^{5} + \cdots - 23673189128256 \nu ) / 929297940480 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 18847 \nu^{10} - 5263958 \nu^{8} - 504881695 \nu^{6} - 19132385356 \nu^{4} + \cdots - 212984313408 ) / 10325532672 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 287341 \nu^{11} + 95412002 \nu^{9} + 11315684077 \nu^{7} + 559500062884 \nu^{5} + \cdots + 25601512183488 \nu ) / 929297940480 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 347027 \nu^{11} - 111972574 \nu^{9} - 13012038419 \nu^{7} - 646224335708 \nu^{5} + \cdots - 37240250976576 \nu ) / 929297940480 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 27421 \nu^{11} - 8375738 \nu^{9} - 914911405 \nu^{7} - 42550380124 \nu^{5} + \cdots - 3672075332160 \nu ) / 38720747520 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{5} - \beta_{4} - 54 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{11} + 4\beta_{10} + 2\beta_{9} - 4\beta_{7} + 2\beta_{3} - 89\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 4\beta_{8} - 12\beta_{6} - 111\beta_{5} + 107\beta_{4} - 88\beta_{2} + 4742 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 127\beta_{11} - 644\beta_{10} - 326\beta_{9} + 300\beta_{7} + 4642\beta_{3} + 8747\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -636\beta_{8} + 1948\beta_{6} + 11941\beta_{5} - 11353\beta_{4} + 20256\beta_{2} - 461922 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -14485\beta_{11} + 86228\beta_{10} + 40866\beta_{9} - 18652\beta_{7} - 1088646\beta_{3} - 899817\beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 89828\beta_{8} - 249692\beta_{6} - 1307847\beta_{5} + 1191043\beta_{4} - 3282728\beta_{2} + 47271622 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 1667159 \beta_{11} - 10831380 \beta_{10} - 4726326 \beta_{9} + 664092 \beta_{7} + 176717010 \beta_{3} + 95289555 \beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 12659900 \beta_{8} + 29736684 \beta_{6} + 145834013 \beta_{5} - 124901745 \beta_{4} + \cdots - 4992535666 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 196473613 \beta_{11} + 1318953348 \beta_{10} + 530122898 \beta_{9} + 65745172 \beta_{7} + \cdots - 10303273025 \beta_1 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/325\mathbb{Z}\right)^\times\).

\(n\) \(27\) \(301\)
\(\chi(n)\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
274.1
− 10.7882i
− 9.61672i
− 8.51599i
− 5.93318i
− 2.75663i
− 1.34530i
1.34530i
2.75663i
5.93318i
8.51599i
9.61672i
10.7882i
− 10.7882i − 0.527294i −84.3852 0 −5.68855 231.710i 565.142i 242.722 0
274.2 − 9.61672i 28.9791i −60.4813 0 278.684 189.995i 273.896i −596.790 0
274.3 − 8.51599i 11.2297i −40.5221 0 95.6316 − 229.647i 72.5738i 116.895 0
274.4 − 5.93318i 7.05430i −3.20258 0 41.8544 − 185.746i − 170.860i 193.237 0
274.5 − 2.75663i − 23.9621i 24.4010 0 −66.0547 85.7300i − 155.477i −331.182 0
274.6 − 1.34530i − 19.6439i 30.1902 0 −26.4269 − 48.6530i − 83.6645i −142.882 0
274.7 1.34530i 19.6439i 30.1902 0 −26.4269 48.6530i 83.6645i −142.882 0
274.8 2.75663i 23.9621i 24.4010 0 −66.0547 − 85.7300i 155.477i −331.182 0
274.9 5.93318i − 7.05430i −3.20258 0 41.8544 185.746i 170.860i 193.237 0
274.10 8.51599i − 11.2297i −40.5221 0 95.6316 229.647i − 72.5738i 116.895 0
274.11 9.61672i − 28.9791i −60.4813 0 278.684 − 189.995i − 273.896i −596.790 0
274.12 10.7882i 0.527294i −84.3852 0 −5.68855 − 231.710i − 565.142i 242.722 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 274.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 325.6.b.g 12
5.b even 2 1 inner 325.6.b.g 12
5.c odd 4 1 65.6.a.d ✓ 6
5.c odd 4 1 325.6.a.g 6
15.e even 4 1 585.6.a.m 6
20.e even 4 1 1040.6.a.q 6
65.h odd 4 1 845.6.a.h 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
65.6.a.d ✓ 6 5.c odd 4 1
325.6.a.g 6 5.c odd 4 1
325.6.b.g 12 1.a even 1 1 trivial
325.6.b.g 12 5.b even 2 1 inner
585.6.a.m 6 15.e even 4 1
845.6.a.h 6 65.h odd 4 1
1040.6.a.q 6 20.e even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{12} + 326T_{2}^{10} + 38809T_{2}^{8} + 2034064T_{2}^{6} + 43897824T_{2}^{4} + 281822976T_{2}^{2} + 377913600 \) acting on \(S_{6}^{\mathrm{new}}(325, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} + \cdots + 377913600 \) Copy content Toggle raw display
$3$ \( T^{12} + \cdots + 324653806656 \) Copy content Toggle raw display
$5$ \( T^{12} \) Copy content Toggle raw display
$7$ \( T^{12} + \cdots + 61\!\cdots\!44 \) Copy content Toggle raw display
$11$ \( (T^{6} + \cdots - 57\!\cdots\!56)^{2} \) Copy content Toggle raw display
$13$ \( (T^{2} + 28561)^{6} \) Copy content Toggle raw display
$17$ \( T^{12} + \cdots + 36\!\cdots\!24 \) Copy content Toggle raw display
$19$ \( (T^{6} + \cdots - 60\!\cdots\!80)^{2} \) Copy content Toggle raw display
$23$ \( T^{12} + \cdots + 42\!\cdots\!56 \) Copy content Toggle raw display
$29$ \( (T^{6} + \cdots + 38\!\cdots\!40)^{2} \) Copy content Toggle raw display
$31$ \( (T^{6} + \cdots - 20\!\cdots\!64)^{2} \) Copy content Toggle raw display
$37$ \( T^{12} + \cdots + 38\!\cdots\!76 \) Copy content Toggle raw display
$41$ \( (T^{6} + \cdots - 11\!\cdots\!12)^{2} \) Copy content Toggle raw display
$43$ \( T^{12} + \cdots + 34\!\cdots\!16 \) Copy content Toggle raw display
$47$ \( T^{12} + \cdots + 65\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( T^{12} + \cdots + 74\!\cdots\!64 \) Copy content Toggle raw display
$59$ \( (T^{6} + \cdots + 41\!\cdots\!80)^{2} \) Copy content Toggle raw display
$61$ \( (T^{6} + \cdots + 91\!\cdots\!28)^{2} \) Copy content Toggle raw display
$67$ \( T^{12} + \cdots + 19\!\cdots\!36 \) Copy content Toggle raw display
$71$ \( (T^{6} + \cdots - 10\!\cdots\!12)^{2} \) Copy content Toggle raw display
$73$ \( T^{12} + \cdots + 21\!\cdots\!16 \) Copy content Toggle raw display
$79$ \( (T^{6} + \cdots + 66\!\cdots\!20)^{2} \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots + 28\!\cdots\!56 \) Copy content Toggle raw display
$89$ \( (T^{6} + \cdots - 51\!\cdots\!00)^{2} \) Copy content Toggle raw display
$97$ \( T^{12} + \cdots + 19\!\cdots\!00 \) Copy content Toggle raw display
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