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Properties

Label 4235.2.a.bp
Level $4235$
Weight $2$
Character orbit 4235.a
Self dual yes
Analytic conductor $33.817$
Analytic rank $0$
Dimension $18$
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: \(18\)
Coefficient field: \(\mathbb{Q}[x]/(x^{18} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{18} - 2 x^{17} - 28 x^{16} + 54 x^{15} + 317 x^{14} - 580 x^{13} - 1874 x^{12} + 3158 x^{11} + \cdots - 176 \) 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 \(1,\beta_1,\ldots,\beta_{17}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + \beta_{7} q^{3} + (\beta_{2} + 1) q^{4} + q^{5} + (\beta_{16} - \beta_{14} + \cdots - \beta_1) q^{6}+ \cdots + (\beta_{15} + 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + \beta_{7} q^{3} + (\beta_{2} + 1) q^{4} + q^{5} + (\beta_{16} - \beta_{14} + \cdots - \beta_1) q^{6}+ \cdots + \beta_1 q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 18 q + 2 q^{2} + 5 q^{3} + 24 q^{4} + 18 q^{5} - q^{6} + 18 q^{7} + 6 q^{8} + 37 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 18 q + 2 q^{2} + 5 q^{3} + 24 q^{4} + 18 q^{5} - q^{6} + 18 q^{7} + 6 q^{8} + 37 q^{9} + 2 q^{10} + 15 q^{12} + 8 q^{13} + 2 q^{14} + 5 q^{15} + 44 q^{16} - 5 q^{17} + 2 q^{18} + 15 q^{19} + 24 q^{20} + 5 q^{21} + 4 q^{23} + 8 q^{24} + 18 q^{25} + 14 q^{26} + 20 q^{27} + 24 q^{28} - 6 q^{29} - q^{30} + 22 q^{31} - 6 q^{32} + 44 q^{34} + 18 q^{35} + 83 q^{36} + 26 q^{37} - 11 q^{38} - 38 q^{39} + 6 q^{40} + 7 q^{41} - q^{42} + 10 q^{43} + 37 q^{45} - 40 q^{46} - q^{47} - 15 q^{48} + 18 q^{49} + 2 q^{50} - 11 q^{51} + 18 q^{52} + 23 q^{53} + 13 q^{54} + 6 q^{56} - 16 q^{57} + 2 q^{58} + 30 q^{59} + 15 q^{60} + 17 q^{61} + 57 q^{62} + 37 q^{63} + 64 q^{64} + 8 q^{65} + 29 q^{67} - 66 q^{68} + 54 q^{69} + 2 q^{70} - 2 q^{71} - 77 q^{72} + 3 q^{73} - 48 q^{74} + 5 q^{75} + 47 q^{76} + 10 q^{78} - 18 q^{79} + 44 q^{80} + 110 q^{81} + 56 q^{82} + 9 q^{83} + 15 q^{84} - 5 q^{85} + 25 q^{86} + 23 q^{87} + 59 q^{89} + 2 q^{90} + 8 q^{91} - 74 q^{92} + 33 q^{93} - 19 q^{94} + 15 q^{95} + 14 q^{96} + 30 q^{97} + 2 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{18} - 2 x^{17} - 28 x^{16} + 54 x^{15} + 317 x^{14} - 580 x^{13} - 1874 x^{12} + 3158 x^{11} + \cdots - 176 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} - 6\nu \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 816960 \nu^{17} + 2255867 \nu^{16} + 21055270 \nu^{15} - 60082628 \nu^{14} - 210456368 \nu^{13} + \cdots + 168084952 ) / 2227868 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 1319945 \nu^{17} - 3626264 \nu^{16} - 34314866 \nu^{15} + 96973182 \nu^{14} + 347707617 \nu^{13} + \cdots - 323928792 ) / 2227868 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 1404970 \nu^{17} + 3780947 \nu^{16} + 36540544 \nu^{15} - 100751078 \nu^{14} - 370573528 \nu^{13} + \cdots + 298271312 ) / 2227868 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 3494855 \nu^{17} + 9764098 \nu^{16} + 90446516 \nu^{15} - 260823938 \nu^{14} - 910154271 \nu^{13} + \cdots + 817140288 ) / 4455736 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 3681009 \nu^{17} - 10001908 \nu^{16} - 95815724 \nu^{15} + 267404218 \nu^{14} + 972933489 \nu^{13} + \cdots - 881675160 ) / 4455736 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 2019375 \nu^{17} + 5620788 \nu^{16} + 52264236 \nu^{15} - 150234226 \nu^{14} - 525947833 \nu^{13} + \cdots + 487365336 ) / 2227868 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 5620963 \nu^{17} + 15618254 \nu^{16} + 145277224 \nu^{15} - 416660350 \nu^{14} + \cdots + 1262317704 ) / 4455736 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 3688663 \nu^{17} - 10111498 \nu^{16} - 95822274 \nu^{15} + 270214862 \nu^{14} + 969897915 \nu^{13} + \cdots - 876816348 ) / 2227868 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 9070279 \nu^{17} + 24718536 \nu^{16} + 236054868 \nu^{15} - 660782758 \nu^{14} + \cdots + 2180404840 ) / 4455736 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 5956650 \nu^{17} - 16368237 \nu^{16} - 154638528 \nu^{15} + 437355758 \nu^{14} + 1563734178 \nu^{13} + \cdots - 1411206752 ) / 2227868 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 11947639 \nu^{17} - 32671464 \nu^{16} - 310321848 \nu^{15} + 872715794 \nu^{14} + \cdots - 2766807104 ) / 4455736 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 7559885 \nu^{17} + 20794276 \nu^{16} + 196202226 \nu^{15} - 555649790 \nu^{14} + \cdots + 1807951924 ) / 2227868 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( 17825543 \nu^{17} - 48539622 \nu^{16} - 463327892 \nu^{15} + 1296748662 \nu^{14} + \cdots - 4151506792 ) / 4455736 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( - 23245789 \nu^{17} + 63356008 \nu^{16} + 604662944 \nu^{15} - 1693449166 \nu^{14} + \cdots + 5546657016 ) / 4455736 \) 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
\(\nu^{3}\)\(=\) \( \beta_{3} + 6\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -\beta_{13} + \beta_{11} + \beta_{8} - \beta_{7} - \beta_{5} + 7\beta_{2} + \beta _1 + 17 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - \beta_{16} - \beta_{15} + \beta_{14} - \beta_{13} + \beta_{12} + \beta_{11} + \beta_{9} + 3 \beta_{8} + \cdots + 1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - \beta_{15} - \beta_{14} - 11 \beta_{13} + \beta_{12} + 11 \beta_{11} + 13 \beta_{8} - 11 \beta_{7} + \cdots + 110 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 14 \beta_{16} - 14 \beta_{15} + 16 \beta_{14} - 14 \beta_{13} + 16 \beta_{12} + 15 \beta_{11} + \cdots + 14 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 2 \beta_{17} - 15 \beta_{15} - 11 \beta_{14} - 99 \beta_{13} + 19 \beta_{12} + 95 \beta_{11} + \cdots + 742 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 4 \beta_{17} - 142 \beta_{16} - 140 \beta_{15} + 176 \beta_{14} - 145 \beta_{13} + 182 \beta_{12} + \cdots + 142 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 42 \beta_{17} - 5 \beta_{16} - 162 \beta_{15} - 80 \beta_{14} - 832 \beta_{13} + 246 \beta_{12} + \cdots + 5103 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 84 \beta_{17} - 1270 \beta_{16} - 1235 \beta_{15} + 1663 \beta_{14} - 1337 \beta_{13} + 1795 \beta_{12} + \cdots + 1285 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 560 \beta_{17} - 120 \beta_{16} - 1541 \beta_{15} - 435 \beta_{14} - 6761 \beta_{13} + 2681 \beta_{12} + \cdots + 35540 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 1140 \beta_{17} - 10664 \beta_{16} - 10267 \beta_{15} + 14527 \beta_{14} - 11627 \beta_{13} + \cdots + 11055 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 6110 \beta_{17} - 1816 \beta_{16} - 13747 \beta_{15} - 1325 \beta_{14} - 53898 \beta_{13} + \cdots + 249958 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 12748 \beta_{17} - 86383 \beta_{16} - 82692 \beta_{15} + 121208 \beta_{14} - 97756 \beta_{13} + \cdots + 92746 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( - 59724 \beta_{17} - 22228 \beta_{16} - 118122 \beta_{15} + 7852 \beta_{14} - 424628 \beta_{13} + \cdots + 1772555 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( - 127964 \beta_{17} - 684398 \beta_{16} - 653941 \beta_{15} + 982463 \beta_{14} - 804613 \beta_{13} + \cdots + 767629 \) 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
−2.67006
−2.64654
−2.35862
−1.95691
−1.29163
−0.817277
−0.740902
−0.729335
−0.371973
0.238980
0.940613
1.11352
1.38970
1.97890
2.05277
2.52971
2.56132
2.77774
−2.67006 −3.09834 5.12921 1.00000 8.27276 1.00000 −8.35518 6.59974 −2.67006
1.2 −2.64654 2.16636 5.00419 1.00000 −5.73337 1.00000 −7.95071 1.69313 −2.64654
1.3 −2.35862 3.12567 3.56309 1.00000 −7.37227 1.00000 −3.68672 6.76983 −2.35862
1.4 −1.95691 0.270998 1.82949 1.00000 −0.530318 1.00000 0.333679 −2.92656 −1.95691
1.5 −1.29163 −0.466181 −0.331697 1.00000 0.602132 1.00000 3.01169 −2.78268 −1.29163
1.6 −0.817277 −1.30837 −1.33206 1.00000 1.06930 1.00000 2.72321 −1.28816 −0.817277
1.7 −0.740902 3.32552 −1.45106 1.00000 −2.46389 1.00000 2.55690 8.05911 −0.740902
1.8 −0.729335 1.69348 −1.46807 1.00000 −1.23511 1.00000 2.52938 −0.132123 −0.729335
1.9 −0.371973 −1.74007 −1.86164 1.00000 0.647257 1.00000 1.43642 0.0278278 −0.371973
1.10 0.238980 −2.82984 −1.94289 1.00000 −0.676275 1.00000 −0.942271 5.00799 0.238980
1.11 0.940613 0.854982 −1.11525 1.00000 0.804208 1.00000 −2.93024 −2.26901 0.940613
1.12 1.11352 2.27506 −0.760083 1.00000 2.53332 1.00000 −3.07340 2.17591 1.11352
1.13 1.38970 −1.33908 −0.0687332 1.00000 −1.86092 1.00000 −2.87492 −1.20687 1.38970
1.14 1.97890 2.99507 1.91606 1.00000 5.92696 1.00000 −0.166100 5.97045 1.97890
1.15 2.05277 −3.41466 2.21386 1.00000 −7.00951 1.00000 0.439000 8.65991 2.05277
1.16 2.52971 3.17620 4.39941 1.00000 8.03485 1.00000 6.06980 7.08825 2.52971
1.17 2.56132 0.468192 4.56034 1.00000 1.19919 1.00000 6.55784 −2.78080 2.56132
1.18 2.77774 −1.15501 5.71583 1.00000 −3.20831 1.00000 10.3216 −1.66596 2.77774
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.18
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.bp 18
11.b odd 2 1 4235.2.a.bo 18
11.d odd 10 2 385.2.n.f ✓ 36
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
385.2.n.f ✓ 36 11.d odd 10 2
4235.2.a.bo 18 11.b odd 2 1
4235.2.a.bp 18 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(4235))\):

\( T_{2}^{18} - 2 T_{2}^{17} - 28 T_{2}^{16} + 54 T_{2}^{15} + 317 T_{2}^{14} - 580 T_{2}^{13} - 1874 T_{2}^{12} + \cdots - 176 \) Copy content Toggle raw display
\( T_{3}^{18} - 5 T_{3}^{17} - 33 T_{3}^{16} + 190 T_{3}^{15} + 384 T_{3}^{14} - 2832 T_{3}^{13} + \cdots + 4400 \) Copy content Toggle raw display
\( T_{13}^{18} - 8 T_{13}^{17} - 111 T_{13}^{16} + 986 T_{13}^{15} + 4291 T_{13}^{14} - 46981 T_{13}^{13} + \cdots + 156496 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{18} - 2 T^{17} + \cdots - 176 \) Copy content Toggle raw display
$3$ \( T^{18} - 5 T^{17} + \cdots + 4400 \) Copy content Toggle raw display
$5$ \( (T - 1)^{18} \) Copy content Toggle raw display
$7$ \( (T - 1)^{18} \) Copy content Toggle raw display
$11$ \( T^{18} \) Copy content Toggle raw display
$13$ \( T^{18} - 8 T^{17} + \cdots + 156496 \) Copy content Toggle raw display
$17$ \( T^{18} + \cdots - 122446139024 \) Copy content Toggle raw display
$19$ \( T^{18} - 15 T^{17} + \cdots - 23658496 \) Copy content Toggle raw display
$23$ \( T^{18} + \cdots - 53798725376 \) Copy content Toggle raw display
$29$ \( T^{18} + 6 T^{17} + \cdots + 2868305 \) Copy content Toggle raw display
$31$ \( T^{18} + \cdots + 1619504742400 \) Copy content Toggle raw display
$37$ \( T^{18} + \cdots + 281306189824 \) Copy content Toggle raw display
$41$ \( T^{18} + \cdots - 1359292436480 \) Copy content Toggle raw display
$43$ \( T^{18} + \cdots - 55605780736 \) Copy content Toggle raw display
$47$ \( T^{18} + \cdots + 311511344 \) Copy content Toggle raw display
$53$ \( T^{18} + \cdots - 4047305752576 \) Copy content Toggle raw display
$59$ \( T^{18} + \cdots - 12\!\cdots\!20 \) Copy content Toggle raw display
$61$ \( T^{18} + \cdots - 1920738463744 \) Copy content Toggle raw display
$67$ \( T^{18} + \cdots - 32144763136 \) Copy content Toggle raw display
$71$ \( T^{18} + \cdots + 34405546365455 \) Copy content Toggle raw display
$73$ \( T^{18} + \cdots + 3761805788656 \) Copy content Toggle raw display
$79$ \( T^{18} + \cdots + 16304004109 \) Copy content Toggle raw display
$83$ \( T^{18} + \cdots + 105847641630896 \) Copy content Toggle raw display
$89$ \( T^{18} + \cdots - 656130284154880 \) Copy content Toggle raw display
$97$ \( T^{18} + \cdots - 28\!\cdots\!20 \) Copy content Toggle raw display
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