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Properties

Label 4235.2.a.bm
Level $4235$
Weight $2$
Character orbit 4235.a
Self dual yes
Analytic conductor $33.817$
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] = [4235,2,Mod(1,4235)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4235, 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("4235.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4235 = 5 \cdot 7 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4235.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: \(33.8166452560\)
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: \( x^{14} - x^{13} - 24 x^{12} + 22 x^{11} + 223 x^{10} - 190 x^{9} - 1003 x^{8} + 814 x^{7} + 2214 x^{6} + \cdots + 80 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 385)
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.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} - \beta_{10} q^{3} + (\beta_{2} + 2) q^{4} - q^{5} + (\beta_{13} + \beta_{12} + \cdots + \beta_{4}) q^{6}+ \cdots + ( - \beta_{10} + \beta_{7} - \beta_{3} + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{2} - \beta_{10} q^{3} + (\beta_{2} + 2) q^{4} - q^{5} + (\beta_{13} + \beta_{12} + \cdots + \beta_{4}) q^{6}+ \cdots - \beta_1 q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 14 q - q^{2} + 5 q^{3} + 21 q^{4} - 14 q^{5} + q^{6} + 14 q^{7} - 3 q^{8} + 17 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 14 q - q^{2} + 5 q^{3} + 21 q^{4} - 14 q^{5} + q^{6} + 14 q^{7} - 3 q^{8} + 17 q^{9} + q^{10} + 15 q^{12} + 10 q^{13} - q^{14} - 5 q^{15} + 31 q^{16} + 11 q^{17} - 21 q^{18} + 21 q^{19} - 21 q^{20} + 5 q^{21} + 8 q^{23} + 22 q^{24} + 14 q^{25} - 18 q^{26} + 26 q^{27} + 21 q^{28} - 4 q^{29} - q^{30} + 6 q^{31} - 42 q^{32} + 16 q^{34} - 14 q^{35} + 28 q^{36} + 48 q^{37} + 35 q^{38} + 6 q^{39} + 3 q^{40} + 13 q^{41} + q^{42} - 17 q^{45} - 31 q^{46} + 11 q^{47} + 59 q^{48} + 14 q^{49} - q^{50} - 33 q^{51} + 52 q^{52} + 31 q^{53} - 57 q^{54} - 3 q^{56} - 4 q^{57} + 52 q^{58} - 4 q^{59} - 15 q^{60} + 17 q^{61} - 27 q^{62} + 17 q^{63} + 43 q^{64} - 10 q^{65} + 45 q^{67} + 14 q^{68} + 20 q^{69} + q^{70} - 6 q^{71} - 14 q^{72} + 11 q^{73} - 33 q^{74} + 5 q^{75} - 5 q^{76} + 52 q^{78} + 30 q^{79} - 31 q^{80} - 6 q^{81} + 26 q^{82} + 23 q^{83} + 15 q^{84} - 11 q^{85} - 23 q^{86} + 23 q^{87} - 15 q^{89} + 21 q^{90} + 10 q^{91} + 44 q^{92} + 29 q^{93} + 49 q^{94} - 21 q^{95} - 52 q^{96} + 44 q^{97} - q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{14} - x^{13} - 24 x^{12} + 22 x^{11} + 223 x^{10} - 190 x^{9} - 1003 x^{8} + 814 x^{7} + 2214 x^{6} + \cdots + 80 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 4 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 118 \nu^{13} - 1145 \nu^{12} - 5251 \nu^{11} + 28052 \nu^{10} + 79766 \nu^{9} - 265077 \nu^{8} + \cdots - 642628 ) / 44172 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 1109 \nu^{13} - 1309 \nu^{12} - 25424 \nu^{11} + 25936 \nu^{10} + 228211 \nu^{9} - 195048 \nu^{8} + \cdots - 245720 ) / 44172 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 1225 \nu^{13} - 61 \nu^{12} + 30586 \nu^{11} + 4510 \nu^{10} - 297017 \nu^{9} - 63576 \nu^{8} + \cdots + 38188 ) / 44172 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 6095 \nu^{13} - 2461 \nu^{12} - 147914 \nu^{11} + 49678 \nu^{10} + 1394305 \nu^{9} - 421740 \nu^{8} + \cdots - 1823888 ) / 88344 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 6143 \nu^{13} + 3925 \nu^{12} + 150050 \nu^{11} - 84298 \nu^{10} - 1421761 \nu^{9} + \cdots + 1466888 ) / 88344 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 3160 \nu^{13} - 1901 \nu^{12} - 78043 \nu^{11} + 41102 \nu^{10} + 744938 \nu^{9} - 345285 \nu^{8} + \cdots - 924616 ) / 44172 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 1423 \nu^{13} + 1070 \nu^{12} + 34489 \nu^{11} - 22346 \nu^{10} - 324383 \nu^{9} + 180111 \nu^{8} + \cdots + 301768 ) / 14724 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 14633 \nu^{13} + 8881 \nu^{12} + 354848 \nu^{11} - 183754 \nu^{10} - 3340603 \nu^{9} + \cdots + 3634496 ) / 88344 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 8327 \nu^{13} - 4279 \nu^{12} - 203066 \nu^{11} + 91402 \nu^{10} + 1920085 \nu^{9} - 785700 \nu^{8} + \cdots - 2185436 ) / 44172 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 18829 \nu^{13} - 14159 \nu^{12} - 453226 \nu^{11} + 291086 \nu^{10} + 4233959 \nu^{9} + \cdots - 4488256 ) / 88344 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 14410 \nu^{13} + 10055 \nu^{12} + 346765 \nu^{11} - 209726 \nu^{10} - 3233906 \nu^{9} + \cdots + 3443356 ) / 44172 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{10} - \beta_{9} + \beta_{6} + 5\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -\beta_{9} - \beta_{8} - \beta_{4} + 7\beta_{2} + 22 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 2 \beta_{13} - 2 \beta_{12} - \beta_{11} + 9 \beta_{10} - 11 \beta_{9} - \beta_{8} + 3 \beta_{7} + \cdots + 4 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - \beta_{12} - \beta_{11} - \beta_{10} - 13 \beta_{9} - 10 \beta_{8} + 2 \beta_{7} + 2 \beta_{6} + \cdots + 136 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 27 \beta_{13} - 27 \beta_{12} - 17 \beta_{11} + 66 \beta_{10} - 93 \beta_{9} - 15 \beta_{8} + \cdots + 54 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 2 \beta_{13} - 19 \beta_{12} - 17 \beta_{11} - 12 \beta_{10} - 130 \beta_{9} - 80 \beta_{8} + \cdots + 890 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 263 \beta_{13} - 269 \beta_{12} - 187 \beta_{11} + 456 \beta_{10} - 728 \beta_{9} - 160 \beta_{8} + \cdots + 538 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 50 \beta_{13} - 247 \beta_{12} - 202 \beta_{11} - 96 \beta_{10} - 1170 \beta_{9} - 609 \beta_{8} + \cdots + 6014 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 2267 \beta_{13} - 2396 \beta_{12} - 1740 \beta_{11} + 3083 \beta_{10} - 5542 \beta_{9} - 1487 \beta_{8} + \cdots + 4794 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 753 \beta_{13} - 2718 \beta_{12} - 2066 \beta_{11} - 643 \beta_{10} - 9956 \beta_{9} - 4611 \beta_{8} + \cdots + 41460 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 18445 \beta_{13} - 20242 \beta_{12} - 14942 \beta_{11} + 20662 \beta_{10} - 41765 \beta_{9} + \cdots + 40450 \) 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
2.79116
2.66229
2.18029
1.76635
1.36811
0.788750
0.352336
0.295402
−0.665869
−1.49190
−1.76010
−2.21273
−2.48059
−2.59350
−2.79116 2.87978 5.79056 −1.00000 −8.03793 1.00000 −10.5801 5.29315 2.79116
1.2 −2.66229 −1.47492 5.08779 −1.00000 3.92666 1.00000 −8.22059 −0.824612 2.66229
1.3 −2.18029 −2.33174 2.75366 −1.00000 5.08387 1.00000 −1.64319 2.43702 2.18029
1.4 −1.76635 0.244400 1.11998 −1.00000 −0.431694 1.00000 1.55443 −2.94027 1.76635
1.5 −1.36811 3.24763 −0.128288 −1.00000 −4.44310 1.00000 2.91172 7.54712 1.36811
1.6 −0.788750 0.680638 −1.37787 −1.00000 −0.536853 1.00000 2.66430 −2.53673 0.788750
1.7 −0.352336 2.82407 −1.87586 −1.00000 −0.995021 1.00000 1.36560 4.97538 0.352336
1.8 −0.295402 −2.62617 −1.91274 −1.00000 0.775775 1.00000 1.15583 3.89676 0.295402
1.9 0.665869 −0.570975 −1.55662 −1.00000 −0.380194 1.00000 −2.36824 −2.67399 −0.665869
1.10 1.49190 1.35098 0.225752 −1.00000 2.01553 1.00000 −2.64699 −1.17484 −1.49190
1.11 1.76010 −2.23537 1.09795 −1.00000 −3.93447 1.00000 −1.58770 1.99687 −1.76010
1.12 2.21273 −1.10851 2.89616 −1.00000 −2.45284 1.00000 1.98295 −1.77120 −2.21273
1.13 2.48059 2.43917 4.15331 −1.00000 6.05056 1.00000 5.34146 2.94954 −2.48059
1.14 2.59350 1.68101 4.72623 −1.00000 4.35970 1.00000 7.07047 −0.174198 −2.59350
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.14
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(5\) \(1\)
\(7\) \(-1\)
\(11\) \(1\)

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 4235.2.a.bm 14
11.b odd 2 1 4235.2.a.bn 14
11.c even 5 2 385.2.n.e ✓ 28
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
385.2.n.e ✓ 28 11.c even 5 2
4235.2.a.bm 14 1.a even 1 1 trivial
4235.2.a.bn 14 11.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4235))\):

\( T_{2}^{14} + T_{2}^{13} - 24 T_{2}^{12} - 22 T_{2}^{11} + 223 T_{2}^{10} + 190 T_{2}^{9} - 1003 T_{2}^{8} + \cdots + 80 \) Copy content Toggle raw display
\( T_{3}^{14} - 5 T_{3}^{13} - 17 T_{3}^{12} + 108 T_{3}^{11} + 86 T_{3}^{10} - 888 T_{3}^{9} - 15 T_{3}^{8} + \cdots + 311 \) Copy content Toggle raw display
\( T_{13}^{14} - 10 T_{13}^{13} - 51 T_{13}^{12} + 730 T_{13}^{11} + 207 T_{13}^{10} - 17885 T_{13}^{9} + \cdots + 922625 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{14} + T^{13} + \cdots + 80 \) Copy content Toggle raw display
$3$ \( T^{14} - 5 T^{13} + \cdots + 311 \) Copy content Toggle raw display
$5$ \( (T + 1)^{14} \) Copy content Toggle raw display
$7$ \( (T - 1)^{14} \) Copy content Toggle raw display
$11$ \( T^{14} \) Copy content Toggle raw display
$13$ \( T^{14} - 10 T^{13} + \cdots + 922625 \) Copy content Toggle raw display
$17$ \( T^{14} - 11 T^{13} + \cdots - 43651 \) Copy content Toggle raw display
$19$ \( T^{14} - 21 T^{13} + \cdots - 2560000 \) Copy content Toggle raw display
$23$ \( T^{14} - 8 T^{13} + \cdots - 228096 \) Copy content Toggle raw display
$29$ \( T^{14} + \cdots - 585442071 \) Copy content Toggle raw display
$31$ \( T^{14} - 6 T^{13} + \cdots + 90342400 \) Copy content Toggle raw display
$37$ \( T^{14} + \cdots - 1160892416 \) Copy content Toggle raw display
$41$ \( T^{14} + \cdots + 1694192896 \) Copy content Toggle raw display
$43$ \( T^{14} + \cdots + 98764325120 \) Copy content Toggle raw display
$47$ \( T^{14} + \cdots + 103261605229 \) Copy content Toggle raw display
$53$ \( T^{14} + \cdots - 5237653504 \) Copy content Toggle raw display
$59$ \( T^{14} + \cdots - 12868824320 \) Copy content Toggle raw display
$61$ \( T^{14} + \cdots + 2825042176 \) Copy content Toggle raw display
$67$ \( T^{14} + \cdots - 19129901824 \) Copy content Toggle raw display
$71$ \( T^{14} + \cdots + 2958698655 \) Copy content Toggle raw display
$73$ \( T^{14} + \cdots + 131908409 \) Copy content Toggle raw display
$79$ \( T^{14} + \cdots - 115624120151 \) Copy content Toggle raw display
$83$ \( T^{14} + \cdots + 6493968225 \) Copy content Toggle raw display
$89$ \( T^{14} + \cdots - 326827264 \) Copy content Toggle raw display
$97$ \( T^{14} + \cdots + 109290228784 \) Copy content Toggle raw display
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