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Properties

Label 225.4.k.c
Level $225$
Weight $4$
Character orbit 225.k
Analytic conductor $13.275$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [225,4,Mod(49,225)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(225, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([2, 3]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("225.49");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 225 = 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 225.k (of order \(6\), degree \(2\), 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: \(13.2754297513\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(6\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 23x^{10} + 198x^{8} - 719x^{6} + 886x^{4} + 585x^{2} + 81 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{6}\cdot 3^{2} \)
Twist minimal: no (minimal twist has level 45)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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.

\(f(q)\) \(=\) \( q + (\beta_{11} - \beta_{7}) q^{2} + (\beta_{11} + \beta_{8} + \beta_{5}) q^{3} + ( - \beta_{6} - 2 \beta_{4} + \cdots - \beta_1) q^{4}+ \cdots + (\beta_{6} + 4 \beta_{4} + 22 \beta_{3} + \cdots + 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{11} - \beta_{7}) q^{2} + (\beta_{11} + \beta_{8} + \beta_{5}) q^{3} + ( - \beta_{6} - 2 \beta_{4} + \cdots - \beta_1) q^{4}+ \cdots + ( - 80 \beta_{6} + 133 \beta_{4} + \cdots - 208) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 22 q^{4} + 168 q^{6} - 114 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 22 q^{4} + 168 q^{6} - 114 q^{9} - 28 q^{11} - 54 q^{14} + 26 q^{16} + 656 q^{19} - 288 q^{21} + 126 q^{24} + 1736 q^{26} - 670 q^{29} + 704 q^{31} - 104 q^{34} + 2172 q^{36} + 780 q^{39} - 374 q^{41} - 3928 q^{44} - 804 q^{46} + 860 q^{49} - 360 q^{51} - 1278 q^{54} - 1410 q^{56} - 596 q^{59} + 2878 q^{61} + 6276 q^{64} - 1932 q^{66} + 1746 q^{69} + 280 q^{71} - 640 q^{74} - 408 q^{76} - 764 q^{79} - 2502 q^{81} + 1818 q^{84} - 3160 q^{86} - 6876 q^{89} - 2840 q^{91} - 4154 q^{94} + 2310 q^{96} + 1524 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 23x^{10} + 198x^{8} - 719x^{6} + 886x^{4} + 585x^{2} + 81 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 17\nu^{10} - 260\nu^{8} + 1311\nu^{6} - 2275\nu^{4} - 1388\nu^{2} + 33786 ) / 7875 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -19\nu^{10} + 445\nu^{8} - 3627\nu^{6} + 9800\nu^{4} + 5566\nu^{2} - 24327 ) / 7875 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -18\nu^{10} + 415\nu^{8} - 3594\nu^{6} + 13350\nu^{4} - 18398\nu^{2} - 5994 ) / 1125 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -368\nu^{10} + 8665\nu^{8} - 77019\nu^{6} + 295225\nu^{4} - 411623\nu^{2} - 133794 ) / 7875 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 181\nu^{11} - 4055\nu^{9} + 33723\nu^{7} - 115325\nu^{5} + 124141\nu^{3} + 132273\nu ) / 23625 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 106\nu^{10} - 2455\nu^{8} + 21423\nu^{6} - 80150\nu^{4} + 111416\nu^{2} + 29448 ) / 1575 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -88\nu^{11} + 2015\nu^{9} - 17154\nu^{7} + 60725\nu^{5} - 67168\nu^{3} - 70929\nu ) / 6750 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( -316\nu^{11} + 7355\nu^{9} - 64053\nu^{7} + 233450\nu^{5} - 264751\nu^{3} - 277353\nu ) / 23625 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( -1528\nu^{11} + 35465\nu^{9} - 309924\nu^{7} + 1163225\nu^{5} - 1604758\nu^{3} - 514449\nu ) / 23625 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 2818\nu^{11} - 65540\nu^{9} + 575244\nu^{7} - 2181725\nu^{5} + 3102073\nu^{3} + 800019\nu ) / 23625 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( -811\nu^{11} + 18830\nu^{9} - 164763\nu^{7} + 620450\nu^{5} - 860821\nu^{3} - 275688\nu ) / 6750 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( 4\beta_{11} + 4\beta_{10} + \beta_{9} - 2\beta_{8} - 2\beta_{7} + 2\beta_{5} ) / 6 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{6} + \beta_{4} + \beta_{3} + \beta_{2} - \beta _1 + 11 ) / 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 26\beta_{11} + 20\beta_{10} - 7\beta_{9} - 10\beta_{8} - 4\beta_{7} + 13\beta_{5} ) / 6 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 14\beta_{6} + 8\beta_{4} + 35\beta_{3} - \beta_{2} - 5\beta _1 + 79 ) / 3 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 154\beta_{11} + 94\beta_{10} - 92\beta_{9} - 92\beta_{8} + 58\beta_{7} + 113\beta_{5} ) / 6 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 151\beta_{6} + 46\beta_{4} + 505\beta_{3} - 26\beta_{2} + 2\beta _1 + 560 ) / 3 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 692\beta_{11} + 350\beta_{10} - 565\beta_{9} - 868\beta_{8} + 1166\beta_{7} + 1102\beta_{5} ) / 6 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 1430\beta_{6} + 242\beta_{4} + 5399\beta_{3} - 190\beta_{2} + 460\beta _1 + 3580 ) / 3 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 262\beta_{11} - 218\beta_{10} - 947\beta_{9} - 7154\beta_{8} + 14116\beta_{7} + 11039\beta_{5} ) / 6 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 12181\beta_{6} + 1306\beta_{4} + 48394\beta_{3} - 953\beta_{2} + 7520\beta _1 + 17075 ) / 3 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( -45694\beta_{11} - 26842\beta_{10} + 29504\beta_{9} - 48850\beta_{8} + 140162\beta_{7} + 104395\beta_{5} ) / 6 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/225\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(127\)
\(\chi(n)\) \(\beta_{3}\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
49.1
−2.88506 − 0.500000i
−0.0378788 + 0.500000i
1.98116 − 0.500000i
−1.98116 + 0.500000i
0.0378788 − 0.500000i
2.88506 + 0.500000i
−2.88506 + 0.500000i
−0.0378788 − 0.500000i
1.98116 + 0.500000i
−1.98116 − 0.500000i
0.0378788 + 0.500000i
2.88506 − 0.500000i
−3.96084 + 2.28679i −3.96084 − 3.36330i 6.45882 − 11.1870i 0 23.3794 + 4.26387i −17.4197 + 10.0573i 22.4912i 4.37646 + 26.6429i 0
49.2 −3.24635 + 1.87428i −3.24635 − 4.05724i 3.02587 − 5.24096i 0 18.1432 + 7.08665i 27.1492 − 15.6746i − 7.30318i −5.92239 + 26.3425i 0
49.3 −0.151541 + 0.0874923i −0.151541 − 5.19394i −3.98469 + 6.90169i 0 0.477395 + 0.773837i 7.32979 − 4.23186i − 2.79440i −26.9541 + 1.57419i 0
49.4 0.151541 − 0.0874923i 0.151541 + 5.19394i −3.98469 + 6.90169i 0 0.477395 + 0.773837i −7.32979 + 4.23186i 2.79440i −26.9541 + 1.57419i 0
49.5 3.24635 − 1.87428i 3.24635 + 4.05724i 3.02587 − 5.24096i 0 18.1432 + 7.08665i −27.1492 + 15.6746i 7.30318i −5.92239 + 26.3425i 0
49.6 3.96084 − 2.28679i 3.96084 + 3.36330i 6.45882 − 11.1870i 0 23.3794 + 4.26387i 17.4197 − 10.0573i − 22.4912i 4.37646 + 26.6429i 0
124.1 −3.96084 − 2.28679i −3.96084 + 3.36330i 6.45882 + 11.1870i 0 23.3794 − 4.26387i −17.4197 − 10.0573i − 22.4912i 4.37646 − 26.6429i 0
124.2 −3.24635 − 1.87428i −3.24635 + 4.05724i 3.02587 + 5.24096i 0 18.1432 − 7.08665i 27.1492 + 15.6746i 7.30318i −5.92239 − 26.3425i 0
124.3 −0.151541 − 0.0874923i −0.151541 + 5.19394i −3.98469 − 6.90169i 0 0.477395 − 0.773837i 7.32979 + 4.23186i 2.79440i −26.9541 − 1.57419i 0
124.4 0.151541 + 0.0874923i 0.151541 − 5.19394i −3.98469 − 6.90169i 0 0.477395 − 0.773837i −7.32979 − 4.23186i − 2.79440i −26.9541 − 1.57419i 0
124.5 3.24635 + 1.87428i 3.24635 − 4.05724i 3.02587 + 5.24096i 0 18.1432 − 7.08665i −27.1492 − 15.6746i − 7.30318i −5.92239 − 26.3425i 0
124.6 3.96084 + 2.28679i 3.96084 − 3.36330i 6.45882 + 11.1870i 0 23.3794 − 4.26387i 17.4197 + 10.0573i 22.4912i 4.37646 − 26.6429i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 49.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
9.c even 3 1 inner
45.j even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 225.4.k.c 12
5.b even 2 1 inner 225.4.k.c 12
5.c odd 4 1 45.4.e.b ✓ 6
5.c odd 4 1 225.4.e.c 6
9.c even 3 1 inner 225.4.k.c 12
15.e even 4 1 135.4.e.b 6
45.j even 6 1 inner 225.4.k.c 12
45.k odd 12 1 45.4.e.b ✓ 6
45.k odd 12 1 225.4.e.c 6
45.k odd 12 1 405.4.a.h 3
45.k odd 12 1 2025.4.a.s 3
45.l even 12 1 135.4.e.b 6
45.l even 12 1 405.4.a.j 3
45.l even 12 1 2025.4.a.q 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
45.4.e.b ✓ 6 5.c odd 4 1
45.4.e.b ✓ 6 45.k odd 12 1
135.4.e.b 6 15.e even 4 1
135.4.e.b 6 45.l even 12 1
225.4.e.c 6 5.c odd 4 1
225.4.e.c 6 45.k odd 12 1
225.4.k.c 12 1.a even 1 1 trivial
225.4.k.c 12 5.b even 2 1 inner
225.4.k.c 12 9.c even 3 1 inner
225.4.k.c 12 45.j even 6 1 inner
405.4.a.h 3 45.k odd 12 1
405.4.a.j 3 45.l even 12 1
2025.4.a.q 3 45.l even 12 1
2025.4.a.s 3 45.k odd 12 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{12} - 35T_{2}^{10} + 930T_{2}^{8} - 10307T_{2}^{6} + 86710T_{2}^{4} - 2655T_{2}^{2} + 81 \) acting on \(S_{4}^{\mathrm{new}}(225, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} - 35 T^{10} + \cdots + 81 \) Copy content Toggle raw display
$3$ \( T^{12} + \cdots + 387420489 \) Copy content Toggle raw display
$5$ \( T^{12} \) Copy content Toggle raw display
$7$ \( T^{12} + \cdots + 811313702977761 \) Copy content Toggle raw display
$11$ \( (T^{6} + 14 T^{5} + \cdots + 1896428304)^{2} \) Copy content Toggle raw display
$13$ \( T^{12} + \cdots + 32\!\cdots\!16 \) Copy content Toggle raw display
$17$ \( (T^{6} + 9716 T^{4} + \cdots + 24437192976)^{2} \) Copy content Toggle raw display
$19$ \( (T^{3} - 164 T^{2} + \cdots - 57316)^{4} \) Copy content Toggle raw display
$23$ \( T^{12} + \cdots + 14\!\cdots\!61 \) Copy content Toggle raw display
$29$ \( (T^{6} + 335 T^{5} + \cdots + 11463342489)^{2} \) Copy content Toggle raw display
$31$ \( (T^{6} + \cdots + 97238137683600)^{2} \) Copy content Toggle raw display
$37$ \( (T^{6} + \cdots + 11124119360656)^{2} \) Copy content Toggle raw display
$41$ \( (T^{6} + \cdots + 52247959475625)^{2} \) Copy content Toggle raw display
$43$ \( T^{12} + \cdots + 56\!\cdots\!76 \) Copy content Toggle raw display
$47$ \( T^{12} + \cdots + 38\!\cdots\!21 \) Copy content Toggle raw display
$53$ \( (T^{6} + \cdots + 10565984287296)^{2} \) Copy content Toggle raw display
$59$ \( (T^{6} + \cdots + 16\!\cdots\!16)^{2} \) Copy content Toggle raw display
$61$ \( (T^{6} + \cdots + 30\!\cdots\!09)^{2} \) Copy content Toggle raw display
$67$ \( T^{12} + \cdots + 18\!\cdots\!61 \) Copy content Toggle raw display
$71$ \( (T^{3} - 70 T^{2} + \cdots + 223775052)^{4} \) Copy content Toggle raw display
$73$ \( (T^{6} + \cdots + 18\!\cdots\!36)^{2} \) Copy content Toggle raw display
$79$ \( (T^{6} + \cdots + 19\!\cdots\!56)^{2} \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots + 83\!\cdots\!81 \) Copy content Toggle raw display
$89$ \( (T^{3} + 1719 T^{2} + \cdots - 125506395)^{4} \) Copy content Toggle raw display
$97$ \( T^{12} + \cdots + 81\!\cdots\!16 \) Copy content Toggle raw display
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