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Properties

Label 225.2.k.c
Level $225$
Weight $2$
Character orbit 225.k
Analytic conductor $1.797$
Analytic rank $0$
Dimension $16$
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] = [225,2,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, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("225.49");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 225 = 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
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: \(1.79663404548\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 12x^{14} + 102x^{12} - 406x^{10} + 1167x^{8} - 1842x^{6} + 2023x^{4} - 441x^{2} + 81 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + (\beta_{14} - \beta_{10}) q^{3} + (\beta_{11} + \beta_{9} + \beta_{6} + \cdots + 1) q^{4}+ \cdots + ( - \beta_{9} - 2 \beta_{6} + \cdots - \beta_{3}) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + (\beta_{14} - \beta_{10}) q^{3} + (\beta_{11} + \beta_{9} + \beta_{6} + \cdots + 1) q^{4}+ \cdots + ( - 3 \beta_{13} + 6 \beta_{11} + \cdots + 6) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 8 q^{4} + 16 q^{6} - 10 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 8 q^{4} + 16 q^{6} - 10 q^{9} + 2 q^{11} + 6 q^{14} - 8 q^{16} - 8 q^{19} - 30 q^{21} + 66 q^{24} - 40 q^{26} + 2 q^{29} + 8 q^{31} + 18 q^{34} - 28 q^{36} - 50 q^{39} + 10 q^{41} - 88 q^{44} - 6 q^{49} + 22 q^{51} - 52 q^{54} + 60 q^{56} + 34 q^{59} + 26 q^{61} - 76 q^{64} - 16 q^{66} + 54 q^{69} - 32 q^{71} + 80 q^{74} - 22 q^{76} - 14 q^{79} + 34 q^{81} - 54 q^{84} + 68 q^{86} + 36 q^{89} - 68 q^{91} + 6 q^{94} + 68 q^{96} + 34 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 12x^{14} + 102x^{12} - 406x^{10} + 1167x^{8} - 1842x^{6} + 2023x^{4} - 441x^{2} + 81 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 3457 \nu^{14} - 28693 \nu^{12} + 210280 \nu^{10} - 306752 \nu^{8} + 709118 \nu^{6} + \cdots - 7606359 ) / 3976425 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -26\nu^{14} + 74\nu^{12} + 10\nu^{10} - 10439\nu^{8} + 38576\nu^{6} - 94895\nu^{4} + 82467\nu^{2} - 18063 ) / 29025 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 5917 \nu^{15} - 90503 \nu^{13} + 784955 \nu^{11} - 3838362 \nu^{9} + 10468603 \nu^{7} + \cdots - 16863264 \nu ) / 3976425 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 7559 \nu^{14} + 69641 \nu^{12} - 572135 \nu^{10} + 1494574 \nu^{8} - 4803841 \nu^{6} + \cdots - 5812992 ) / 3976425 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 11836 \nu^{14} - 138574 \nu^{12} + 1170340 \nu^{10} - 4500371 \nu^{8} + 12811274 \nu^{6} + \cdots - 814212 ) / 3976425 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 3766 \nu^{15} - 41414 \nu^{13} + 332215 \nu^{11} - 1090526 \nu^{9} + 2480389 \nu^{7} + \cdots + 2721243 \nu ) / 1325475 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 11836 \nu^{15} + 138574 \nu^{13} - 1170340 \nu^{11} + 4500371 \nu^{9} - 12811274 \nu^{7} + \cdots + 4790637 \nu ) / 3976425 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 14896 \nu^{14} + 165889 \nu^{12} - 1314040 \nu^{10} + 4313456 \nu^{8} - 9114389 \nu^{6} + \cdots - 5535918 ) / 3976425 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 13477 \nu^{15} - 168383 \nu^{13} + 1451855 \nu^{11} - 6027522 \nu^{9} + 17277208 \nu^{7} + \cdots + 11503071 \nu ) / 3976425 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 3323 \nu^{14} + 38066 \nu^{12} - 324695 \nu^{10} + 1252588 \nu^{8} - 3846136 \nu^{6} + \cdots + 1468098 ) / 795285 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 19253 \nu^{15} - 245932 \nu^{13} + 2129695 \nu^{11} - 9130758 \nu^{9} + 26781707 \nu^{7} + \cdots - 10338741 \nu ) / 3976425 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 572 \nu^{14} - 6758 \nu^{12} + 56330 \nu^{10} - 215542 \nu^{8} + 579583 \nu^{6} - 869185 \nu^{4} + \cdots - 212679 ) / 92475 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 43568 \nu^{15} - 518857 \nu^{13} + 4432420 \nu^{11} - 17623398 \nu^{9} + 51780332 \nu^{7} + \cdots - 6887691 \nu ) / 3976425 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 49089 \nu^{15} + 557311 \nu^{13} - 4656460 \nu^{11} + 17097154 \nu^{9} - 47712236 \nu^{7} + \cdots - 564507 \nu ) / 3976425 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{11} + \beta_{9} + 3\beta_{6} + \beta_{5} - \beta_{3} + 1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{15} - \beta_{14} - 3\beta_{12} + 2\beta_{10} - 4\beta_{8} - 3\beta_{7} + 2\beta_{4} + 4\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -6\beta_{13} - 2\beta_{11} + 15\beta_{6} + 8\beta_{5} - 7\beta_{3} - 8\beta_{2} - 15 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 8\beta_{15} + 11\beta_{14} - 27\beta_{12} + 8\beta_{10} - 22\beta_{8} + 11\beta_{4} \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -57\beta_{13} - 57\beta_{11} - 49\beta_{9} + 19\beta_{5} - 38\beta_{2} - 139 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 144\beta_{15} + 144\beta_{14} - 87\beta_{10} + 204\beta_{7} - 57\beta_{4} - 139\beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( -144\beta_{13} - 253\beta_{11} - 343\beta_{9} - 588\beta_{6} - 253\beta_{5} + 343\beta_{3} + 144\beta_{2} - 343 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 631 \beta_{15} + 397 \beta_{14} + 1461 \beta_{12} - 1028 \beta_{10} + 931 \beta_{8} + 1461 \beta_{7} + \cdots - 931 \beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 1725\beta_{13} + 1028\beta_{11} - 3984\beta_{6} - 2753\beta_{5} + 2392\beta_{3} + 2753\beta_{2} + 3984 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( -2753\beta_{15} - 4448\beta_{14} + 10260\beta_{12} - 2753\beta_{10} + 6376\beta_{8} - 4448\beta_{4} \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 19083\beta_{13} + 19083\beta_{11} + 16636\beta_{9} - 7201\beta_{5} + 11882\beta_{2} + 44023 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( -50121\beta_{15} - 50121\beta_{14} + 31038\beta_{10} - 71511\beta_{7} + 19083\beta_{4} + 44023\beta_1 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 50121 \beta_{13} + 82189 \beta_{11} + 115534 \beta_{9} + 189318 \beta_{6} + 82189 \beta_{5} + \cdots + 115534 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 215776 \beta_{15} - 132310 \beta_{14} - 496965 \beta_{12} + 348086 \beta_{10} - 304852 \beta_{8} + \cdots + 304852 \beta_1 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(101\) \(127\)
\(\chi(n)\) \(-\beta_{6}\) \(-1\)

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
−2.28087 + 1.31686i
−1.41485 + 0.816862i
−1.27588 + 0.736627i
−0.409850 + 0.236627i
0.409850 − 0.236627i
1.27588 − 0.736627i
1.41485 − 0.816862i
2.28087 − 1.31686i
−2.28087 − 1.31686i
−1.41485 − 0.816862i
−1.27588 − 0.736627i
−0.409850 − 0.236627i
0.409850 + 0.236627i
1.27588 + 0.736627i
1.41485 + 0.816862i
2.28087 + 1.31686i
−2.28087 + 1.31686i −0.238330 − 1.71558i 2.46825 − 4.27513i 0 2.80278 + 3.59916i 1.55662 − 0.898714i 7.73393i −2.88640 + 0.817746i 0
49.2 −1.41485 + 0.816862i −1.36657 − 1.06419i 0.334526 − 0.579416i 0 2.80278 + 0.389365i −0.437645 + 0.252674i − 2.17440i 0.735010 + 2.90857i 0
49.3 −1.27588 + 0.736627i −0.350156 + 1.69629i 0.0852394 − 0.147639i 0 −0.802776 − 2.42219i −3.34791 + 1.93291i − 2.69535i −2.75478 − 1.18793i 0
49.4 −0.409850 + 0.236627i 1.64411 + 0.544899i −0.888015 + 1.53809i 0 −0.802776 + 0.165713i −2.21967 + 1.28153i − 1.78702i 2.40617 + 1.79175i 0
49.5 0.409850 − 0.236627i −1.64411 − 0.544899i −0.888015 + 1.53809i 0 −0.802776 + 0.165713i 2.21967 − 1.28153i 1.78702i 2.40617 + 1.79175i 0
49.6 1.27588 − 0.736627i 0.350156 − 1.69629i 0.0852394 − 0.147639i 0 −0.802776 − 2.42219i 3.34791 − 1.93291i 2.69535i −2.75478 − 1.18793i 0
49.7 1.41485 − 0.816862i 1.36657 + 1.06419i 0.334526 − 0.579416i 0 2.80278 + 0.389365i 0.437645 − 0.252674i 2.17440i 0.735010 + 2.90857i 0
49.8 2.28087 − 1.31686i 0.238330 + 1.71558i 2.46825 − 4.27513i 0 2.80278 + 3.59916i −1.55662 + 0.898714i − 7.73393i −2.88640 + 0.817746i 0
124.1 −2.28087 − 1.31686i −0.238330 + 1.71558i 2.46825 + 4.27513i 0 2.80278 − 3.59916i 1.55662 + 0.898714i − 7.73393i −2.88640 − 0.817746i 0
124.2 −1.41485 − 0.816862i −1.36657 + 1.06419i 0.334526 + 0.579416i 0 2.80278 − 0.389365i −0.437645 − 0.252674i 2.17440i 0.735010 − 2.90857i 0
124.3 −1.27588 − 0.736627i −0.350156 − 1.69629i 0.0852394 + 0.147639i 0 −0.802776 + 2.42219i −3.34791 − 1.93291i 2.69535i −2.75478 + 1.18793i 0
124.4 −0.409850 − 0.236627i 1.64411 − 0.544899i −0.888015 − 1.53809i 0 −0.802776 − 0.165713i −2.21967 − 1.28153i 1.78702i 2.40617 − 1.79175i 0
124.5 0.409850 + 0.236627i −1.64411 + 0.544899i −0.888015 − 1.53809i 0 −0.802776 − 0.165713i 2.21967 + 1.28153i − 1.78702i 2.40617 − 1.79175i 0
124.6 1.27588 + 0.736627i 0.350156 + 1.69629i 0.0852394 + 0.147639i 0 −0.802776 + 2.42219i 3.34791 + 1.93291i − 2.69535i −2.75478 + 1.18793i 0
124.7 1.41485 + 0.816862i 1.36657 − 1.06419i 0.334526 + 0.579416i 0 2.80278 − 0.389365i 0.437645 + 0.252674i − 2.17440i 0.735010 − 2.90857i 0
124.8 2.28087 + 1.31686i 0.238330 − 1.71558i 2.46825 + 4.27513i 0 2.80278 − 3.59916i −1.55662 − 0.898714i 7.73393i −2.88640 − 0.817746i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 49.8
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 225.2.k.c 16
3.b odd 2 1 675.2.k.c 16
5.b even 2 1 inner 225.2.k.c 16
5.c odd 4 1 225.2.e.c ✓ 8
5.c odd 4 1 225.2.e.e yes 8
9.c even 3 1 inner 225.2.k.c 16
9.c even 3 1 2025.2.b.n 8
9.d odd 6 1 675.2.k.c 16
9.d odd 6 1 2025.2.b.o 8
15.d odd 2 1 675.2.k.c 16
15.e even 4 1 675.2.e.c 8
15.e even 4 1 675.2.e.e 8
45.h odd 6 1 675.2.k.c 16
45.h odd 6 1 2025.2.b.o 8
45.j even 6 1 inner 225.2.k.c 16
45.j even 6 1 2025.2.b.n 8
45.k odd 12 1 225.2.e.c ✓ 8
45.k odd 12 1 225.2.e.e yes 8
45.k odd 12 1 2025.2.a.q 4
45.k odd 12 1 2025.2.a.y 4
45.l even 12 1 675.2.e.c 8
45.l even 12 1 675.2.e.e 8
45.l even 12 1 2025.2.a.p 4
45.l even 12 1 2025.2.a.z 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
225.2.e.c ✓ 8 5.c odd 4 1
225.2.e.c ✓ 8 45.k odd 12 1
225.2.e.e yes 8 5.c odd 4 1
225.2.e.e yes 8 45.k odd 12 1
225.2.k.c 16 1.a even 1 1 trivial
225.2.k.c 16 5.b even 2 1 inner
225.2.k.c 16 9.c even 3 1 inner
225.2.k.c 16 45.j even 6 1 inner
675.2.e.c 8 15.e even 4 1
675.2.e.c 8 45.l even 12 1
675.2.e.e 8 15.e even 4 1
675.2.e.e 8 45.l even 12 1
675.2.k.c 16 3.b odd 2 1
675.2.k.c 16 9.d odd 6 1
675.2.k.c 16 15.d odd 2 1
675.2.k.c 16 45.h odd 6 1
2025.2.a.p 4 45.l even 12 1
2025.2.a.q 4 45.k odd 12 1
2025.2.a.y 4 45.k odd 12 1
2025.2.a.z 4 45.l even 12 1
2025.2.b.n 8 9.c even 3 1
2025.2.b.n 8 45.j even 6 1
2025.2.b.o 8 9.d odd 6 1
2025.2.b.o 8 45.h odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{16} - 12T_{2}^{14} + 102T_{2}^{12} - 406T_{2}^{10} + 1167T_{2}^{8} - 1842T_{2}^{6} + 2023T_{2}^{4} - 441T_{2}^{2} + 81 \) acting on \(S_{2}^{\mathrm{new}}(225, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} - 12 T^{14} + \cdots + 81 \) Copy content Toggle raw display
$3$ \( T^{16} + 5 T^{14} + \cdots + 6561 \) Copy content Toggle raw display
$5$ \( T^{16} \) Copy content Toggle raw display
$7$ \( T^{16} - 25 T^{14} + \cdots + 6561 \) Copy content Toggle raw display
$11$ \( (T^{8} - T^{7} + 26 T^{6} + \cdots + 81)^{2} \) Copy content Toggle raw display
$13$ \( T^{16} + \cdots + 131079601 \) Copy content Toggle raw display
$17$ \( (T^{8} + 81 T^{6} + \cdots + 91809)^{2} \) Copy content Toggle raw display
$19$ \( (T^{4} + 2 T^{3} - 27 T^{2} + \cdots - 25)^{4} \) Copy content Toggle raw display
$23$ \( T^{16} + \cdots + 3486784401 \) Copy content Toggle raw display
$29$ \( (T^{8} - T^{7} + \cdots + 16641)^{2} \) Copy content Toggle raw display
$31$ \( (T^{8} - 4 T^{7} + \cdots + 59049)^{2} \) Copy content Toggle raw display
$37$ \( (T^{8} + 199 T^{6} + \cdots + 418609)^{2} \) Copy content Toggle raw display
$41$ \( (T^{8} - 5 T^{7} + \cdots + 42849)^{2} \) Copy content Toggle raw display
$43$ \( T^{16} + \cdots + 205144679041 \) Copy content Toggle raw display
$47$ \( T^{16} + \cdots + 21071715921 \) Copy content Toggle raw display
$53$ \( (T^{8} + 228 T^{6} + \cdots + 221841)^{2} \) Copy content Toggle raw display
$59$ \( (T^{8} - 17 T^{7} + \cdots + 5349969)^{2} \) Copy content Toggle raw display
$61$ \( (T^{8} - 13 T^{7} + 172 T^{6} + \cdots + 1)^{2} \) Copy content Toggle raw display
$67$ \( T^{16} + \cdots + 3486784401 \) Copy content Toggle raw display
$71$ \( (T^{4} + 8 T^{3} + \cdots + 381)^{4} \) Copy content Toggle raw display
$73$ \( (T^{8} + 196 T^{6} + \cdots + 12769)^{2} \) Copy content Toggle raw display
$79$ \( (T^{8} + 7 T^{7} + \cdots + 42849)^{2} \) Copy content Toggle raw display
$83$ \( T^{16} + \cdots + 282429536481 \) Copy content Toggle raw display
$89$ \( (T^{4} - 9 T^{3} + \cdots + 2025)^{4} \) Copy content Toggle raw display
$97$ \( T^{16} + \cdots + 824843587681 \) Copy content Toggle raw display
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