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Properties

Label 4025.2.a.bc
Level $4025$
Weight $2$
Character orbit 4025.a
Self dual yes
Analytic conductor $32.140$
Analytic rank $0$
Dimension $14$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [4025,2,Mod(1,4025)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4025, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("4025.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4025 = 5^{2} \cdot 7 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4025.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: \(32.1397868136\)
Analytic rank: \(0\)
Dimension: \(14\)
Coefficient field: \(\mathbb{Q}[x]/(x^{14} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{14} - 3 x^{13} - 18 x^{12} + 58 x^{11} + 111 x^{10} - 414 x^{9} - 244 x^{8} + 1330 x^{7} - 27 x^{6} + \cdots - 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 5 \)
Twist minimal: yes
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 \(1,\beta_1,\ldots,\beta_{13}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} - \beta_{7} q^{3} + (\beta_{2} + 1) q^{4} + ( - \beta_{11} + \beta_1 - 1) q^{6} + q^{7} + (\beta_{3} + \beta_1) q^{8} + ( - \beta_{7} + \beta_{5} + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} - \beta_{7} q^{3} + (\beta_{2} + 1) q^{4} + ( - \beta_{11} + \beta_1 - 1) q^{6} + q^{7} + (\beta_{3} + \beta_1) q^{8} + ( - \beta_{7} + \beta_{5} + 1) q^{9} + \beta_{8} q^{11} + ( - \beta_{12} - \beta_{11} - \beta_{10} + \cdots + 1) q^{12}+ \cdots + (2 \beta_{10} - \beta_{9} + \beta_{8} + \cdots + 2) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 14 q + 3 q^{2} + 4 q^{3} + 17 q^{4} - 4 q^{6} + 14 q^{7} + 3 q^{8} + 18 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 14 q + 3 q^{2} + 4 q^{3} + 17 q^{4} - 4 q^{6} + 14 q^{7} + 3 q^{8} + 18 q^{9} + 5 q^{11} + 17 q^{12} + 11 q^{13} + 3 q^{14} + 23 q^{16} + 3 q^{17} + 15 q^{18} - 2 q^{19} + 4 q^{21} + 23 q^{22} + 14 q^{23} - 12 q^{24} - 9 q^{26} + 25 q^{27} + 17 q^{28} + 7 q^{29} - 3 q^{31} + 24 q^{32} + 6 q^{33} - 14 q^{34} + 13 q^{36} + 22 q^{37} + 20 q^{38} - 10 q^{39} - 17 q^{41} - 4 q^{42} + 18 q^{43} + 28 q^{44} + 3 q^{46} + 30 q^{47} + 8 q^{48} + 14 q^{49} + 4 q^{51} + 8 q^{52} + 11 q^{53} + 20 q^{54} + 3 q^{56} + 18 q^{57} + 38 q^{58} - 22 q^{59} - 8 q^{61} - 22 q^{62} + 18 q^{63} + 29 q^{64} - 9 q^{66} + 39 q^{67} + q^{68} + 4 q^{69} - 5 q^{71} - 24 q^{72} + 18 q^{73} + 35 q^{74} - 41 q^{76} + 5 q^{77} - 22 q^{78} + 10 q^{79} + 2 q^{81} - 8 q^{82} + 24 q^{83} + 17 q^{84} - 26 q^{86} + 5 q^{87} + 58 q^{88} + 25 q^{89} + 11 q^{91} + 17 q^{92} + 47 q^{93} - 2 q^{94} - 117 q^{96} + 43 q^{97} + 3 q^{98} + 55 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{14} - 3 x^{13} - 18 x^{12} + 58 x^{11} + 111 x^{10} - 414 x^{9} - 244 x^{8} + 1330 x^{7} - 27 x^{6} + \cdots - 9 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} - 5\nu \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 4055 \nu^{13} + 7975 \nu^{12} + 81881 \nu^{11} - 152987 \nu^{10} - 615369 \nu^{9} + 1080275 \nu^{8} + \cdots + 142730 ) / 22937 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 21776 \nu^{13} + 33409 \nu^{12} + 446796 \nu^{11} - 619227 \nu^{10} - 3443457 \nu^{9} + \cdots - 316562 ) / 22937 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 25090 \nu^{13} - 41963 \nu^{12} - 504992 \nu^{11} + 782303 \nu^{10} + 3774769 \nu^{9} - 5321389 \nu^{8} + \cdots + 89726 ) / 22937 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 33730 \nu^{13} + 54713 \nu^{12} + 686159 \nu^{11} - 1017543 \nu^{10} - 5217903 \nu^{9} + \cdots - 320996 ) / 22937 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 34523 \nu^{13} + 57715 \nu^{12} + 699434 \nu^{11} - 1075557 \nu^{10} - 5289707 \nu^{9} + \cdots - 436758 ) / 22937 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 42027 \nu^{13} - 75160 \nu^{12} - 839833 \nu^{11} + 1403897 \nu^{10} + 6219514 \nu^{9} + \cdots + 131732 ) / 22937 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 45854 \nu^{13} + 78020 \nu^{12} + 926777 \nu^{11} - 1457654 \nu^{10} - 6982144 \nu^{9} + \cdots - 264833 ) / 22937 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 46477 \nu^{13} + 79019 \nu^{12} + 938797 \nu^{11} - 1473873 \nu^{10} - 7064153 \nu^{9} + \cdots - 326507 ) / 22937 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 52513 \nu^{13} - 88882 \nu^{12} - 1064311 \nu^{11} + 1659877 \nu^{10} + 8052400 \nu^{9} + \cdots + 362877 ) / 22937 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 82827 \nu^{13} - 143014 \nu^{12} - 1668584 \nu^{11} + 2672763 \nu^{10} + 12509311 \nu^{9} + \cdots + 291069 ) / 22937 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} + 5\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{11} - \beta_{8} - \beta_{7} + \beta_{5} + \beta_{3} + 7\beta_{2} + 16 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 2\beta_{12} + \beta_{11} + \beta_{10} + \beta_{7} + \beta_{6} + \beta_{4} + 10\beta_{3} + \beta_{2} + 29\beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 3 \beta_{12} + 14 \beta_{11} + \beta_{10} + \beta_{9} - 9 \beta_{8} - 12 \beta_{7} + \beta_{6} + \cdots + 98 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 28 \beta_{12} + 18 \beta_{11} + 16 \beta_{10} + 2 \beta_{9} - \beta_{8} + 9 \beta_{7} + 14 \beta_{6} + \cdots + 20 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 2 \beta_{13} + 47 \beta_{12} + 143 \beta_{11} + 16 \beta_{10} + 17 \beta_{9} - 68 \beta_{8} + \cdots + 643 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( \beta_{13} + 285 \beta_{12} + 215 \beta_{11} + 173 \beta_{10} + 34 \beta_{9} - 18 \beta_{8} + 58 \beta_{7} + \cdots + 244 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 32 \beta_{13} + 517 \beta_{12} + 1293 \beta_{11} + 181 \beta_{10} + 190 \beta_{9} - 500 \beta_{8} + \cdots + 4397 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 20 \beta_{13} + 2567 \beta_{12} + 2171 \beta_{11} + 1602 \beta_{10} + 393 \beta_{9} - 221 \beta_{8} + \cdots + 2489 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 343 \beta_{13} + 4943 \beta_{12} + 11007 \beta_{11} + 1792 \beta_{10} + 1799 \beta_{9} - 3687 \beta_{8} + \cdots + 30917 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 252 \beta_{13} + 21770 \beta_{12} + 20092 \beta_{11} + 13735 \beta_{10} + 3872 \beta_{9} - 2314 \beta_{8} + \cdots + 23306 \) 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
−2.60812
−2.33696
−2.04050
−1.51853
−0.689960
−0.256730
−0.0822464
0.744662
1.29908
1.31920
1.63325
2.30520
2.42992
2.80172
−2.60812 2.93022 4.80228 0 −7.64237 1.00000 −7.30867 5.58621 0
1.2 −2.33696 0.00757499 3.46136 0 −0.0177024 1.00000 −3.41514 −2.99994 0
1.3 −2.04050 −1.31355 2.16365 0 2.68030 1.00000 −0.333918 −1.27458 0
1.4 −1.51853 0.630051 0.305932 0 −0.956751 1.00000 2.57249 −2.60304 0
1.5 −0.689960 2.51200 −1.52395 0 −1.73318 1.00000 2.43139 3.31014 0
1.6 −0.256730 −2.72326 −1.93409 0 0.699143 1.00000 1.01000 4.41613 0
1.7 −0.0822464 1.69790 −1.99324 0 −0.139646 1.00000 0.328429 −0.117136 0
1.8 0.744662 −1.85918 −1.44548 0 −1.38446 1.00000 −2.56572 0.456544 0
1.9 1.29908 −2.64602 −0.312383 0 −3.43741 1.00000 −3.00398 4.00145 0
1.10 1.31920 2.92477 −0.259709 0 3.85835 1.00000 −2.98101 5.55425 0
1.11 1.63325 −0.456743 0.667506 0 −0.745975 1.00000 −2.17630 −2.79139 0
1.12 2.30520 0.930494 3.31395 0 2.14497 1.00000 3.02891 −2.13418 0
1.13 2.42992 3.09771 3.90452 0 7.52721 1.00000 4.62785 6.59584 0
1.14 2.80172 −1.73197 5.84965 0 −4.85249 1.00000 10.7857 −0.000289796 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.14
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(5\) \(-1\)
\(7\) \(-1\)
\(23\) \(-1\)

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 4025.2.a.bc yes 14
5.b even 2 1 4025.2.a.z ✓ 14
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4025.2.a.z ✓ 14 5.b even 2 1
4025.2.a.bc yes 14 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4025))\):

\( T_{2}^{14} - 3 T_{2}^{13} - 18 T_{2}^{12} + 58 T_{2}^{11} + 111 T_{2}^{10} - 414 T_{2}^{9} - 244 T_{2}^{8} + \cdots - 9 \) Copy content Toggle raw display
\( T_{3}^{14} - 4 T_{3}^{13} - 22 T_{3}^{12} + 93 T_{3}^{11} + 182 T_{3}^{10} - 815 T_{3}^{9} - 712 T_{3}^{8} + \cdots + 7 \) Copy content Toggle raw display
\( T_{11}^{14} - 5 T_{11}^{13} - 78 T_{11}^{12} + 443 T_{11}^{11} + 1609 T_{11}^{10} - 12318 T_{11}^{9} + \cdots + 4800 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{14} - 3 T^{13} + \cdots - 9 \) Copy content Toggle raw display
$3$ \( T^{14} - 4 T^{13} + \cdots + 7 \) Copy content Toggle raw display
$5$ \( T^{14} \) Copy content Toggle raw display
$7$ \( (T - 1)^{14} \) Copy content Toggle raw display
$11$ \( T^{14} - 5 T^{13} + \cdots + 4800 \) Copy content Toggle raw display
$13$ \( T^{14} - 11 T^{13} + \cdots - 1625219 \) Copy content Toggle raw display
$17$ \( T^{14} - 3 T^{13} + \cdots - 4939200 \) Copy content Toggle raw display
$19$ \( T^{14} + 2 T^{13} + \cdots + 7294400 \) Copy content Toggle raw display
$23$ \( (T - 1)^{14} \) Copy content Toggle raw display
$29$ \( T^{14} + \cdots + 807142419 \) Copy content Toggle raw display
$31$ \( T^{14} + 3 T^{13} + \cdots - 253979 \) Copy content Toggle raw display
$37$ \( T^{14} + \cdots + 523289408 \) Copy content Toggle raw display
$41$ \( T^{14} + 17 T^{13} + \cdots + 35849475 \) Copy content Toggle raw display
$43$ \( T^{14} + \cdots - 253215808 \) Copy content Toggle raw display
$47$ \( T^{14} + \cdots + 1830657213 \) Copy content Toggle raw display
$53$ \( T^{14} + \cdots - 177216192 \) Copy content Toggle raw display
$59$ \( T^{14} + \cdots - 22715298117 \) Copy content Toggle raw display
$61$ \( T^{14} + \cdots - 23024244800 \) Copy content Toggle raw display
$67$ \( T^{14} + \cdots - 1461159760064 \) Copy content Toggle raw display
$71$ \( T^{14} + \cdots + 469402650987 \) Copy content Toggle raw display
$73$ \( T^{14} + \cdots + 197881165675 \) Copy content Toggle raw display
$79$ \( T^{14} + \cdots - 1971787774400 \) Copy content Toggle raw display
$83$ \( T^{14} + \cdots - 1693048609344 \) Copy content Toggle raw display
$89$ \( T^{14} + \cdots + 817868443200 \) Copy content Toggle raw display
$97$ \( T^{14} + \cdots - 390198905152 \) Copy content Toggle raw display
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