Properties

Label 4205.2.a.s
Level $4205$
Weight $2$
Character orbit 4205.a
Self dual yes
Analytic conductor $33.577$
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] = [4205,2,Mod(1,4205)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4205, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("4205.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4205 = 5 \cdot 29^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4205.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: \(33.5770940499\)
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: \( x^{18} - x^{17} - 29 x^{16} + 26 x^{15} + 347 x^{14} - 277 x^{13} - 2215 x^{12} + 1567 x^{11} + \cdots + 41 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 145)
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.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} - \beta_{3} q^{3} + (\beta_{2} + 1) q^{4} + q^{5} + ( - \beta_{12} + \beta_{11} - \beta_{6}) q^{6} + (\beta_{15} + 1) q^{7} + ( - \beta_{16} - \beta_{14} + \cdots - \beta_1) q^{8}+ \cdots + (\beta_{13} - \beta_{9} + \beta_{8} + \cdots + \beta_1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{2} - \beta_{3} q^{3} + (\beta_{2} + 1) q^{4} + q^{5} + ( - \beta_{12} + \beta_{11} - \beta_{6}) q^{6} + (\beta_{15} + 1) q^{7} + ( - \beta_{16} - \beta_{14} + \cdots - \beta_1) q^{8}+ \cdots + (\beta_{17} + \beta_{16} + \beta_{15} + \cdots + 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 18 q - q^{2} - q^{3} + 23 q^{4} + 18 q^{5} + q^{6} + 17 q^{7} - 6 q^{8} + 19 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 18 q - q^{2} - q^{3} + 23 q^{4} + 18 q^{5} + q^{6} + 17 q^{7} - 6 q^{8} + 19 q^{9} - q^{10} - 5 q^{11} - 7 q^{12} + 17 q^{13} - 13 q^{14} - q^{15} + 25 q^{16} + 14 q^{17} - 21 q^{18} + 4 q^{19} + 23 q^{20} - 16 q^{21} + 29 q^{22} + 28 q^{23} + 8 q^{24} + 18 q^{25} + 5 q^{26} - 7 q^{27} + 30 q^{28} + q^{30} + 7 q^{31} - 33 q^{32} + 7 q^{33} + 32 q^{34} + 17 q^{35} + 8 q^{36} + 34 q^{37} + 4 q^{38} + 27 q^{39} - 6 q^{40} - 14 q^{41} + 5 q^{42} - 3 q^{43} - 26 q^{44} + 19 q^{45} + 38 q^{46} + 17 q^{47} + 58 q^{48} + 25 q^{49} - q^{50} + 28 q^{51} + 70 q^{52} + 24 q^{53} - 38 q^{54} - 5 q^{55} - 67 q^{56} + 10 q^{57} + q^{59} - 7 q^{60} + 40 q^{61} + 15 q^{62} + 80 q^{63} + 40 q^{64} + 17 q^{65} + 96 q^{66} + 38 q^{67} + 72 q^{68} + 4 q^{69} - 13 q^{70} - 4 q^{71} - 88 q^{72} - 29 q^{73} - 8 q^{74} - q^{75} - 41 q^{76} - 18 q^{77} + 12 q^{78} - 29 q^{79} + 25 q^{80} + 42 q^{81} - 14 q^{82} + 33 q^{83} - 52 q^{84} + 14 q^{85} - 63 q^{86} + 20 q^{88} + 4 q^{89} - 21 q^{90} + 86 q^{91} + 33 q^{92} - 2 q^{93} - 7 q^{94} + 4 q^{95} - 41 q^{96} + 21 q^{97} + 26 q^{98} - 4 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{18} - x^{17} - 29 x^{16} + 26 x^{15} + 347 x^{14} - 277 x^{13} - 2215 x^{12} + 1567 x^{11} + \cdots + 41 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 29840 \nu^{17} - 79467 \nu^{16} - 733103 \nu^{15} + 1992300 \nu^{14} + 7037561 \nu^{13} + \cdots + 416901 ) / 65884 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 238502 \nu^{17} - 646673 \nu^{16} - 5843139 \nu^{15} + 16273726 \nu^{14} + 55740803 \nu^{13} + \cdots + 8557209 ) / 461188 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 11948 \nu^{17} - 29039 \nu^{16} - 302635 \nu^{15} + 738738 \nu^{14} + 3028665 \nu^{13} + \cdots + 197727 ) / 17738 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 437793 \nu^{17} - 1163823 \nu^{16} - 10784092 \nu^{15} + 29347017 \nu^{14} + 103648205 \nu^{13} + \cdots + 14033920 ) / 461188 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 470453 \nu^{17} - 1206984 \nu^{16} - 11756001 \nu^{15} + 30629767 \nu^{14} + 115378314 \nu^{13} + \cdots + 10737833 ) / 461188 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 493901 \nu^{17} - 1243565 \nu^{16} - 12402410 \nu^{15} + 31606001 \nu^{14} + 122553663 \nu^{13} + \cdots + 11959042 ) / 461188 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 736531 \nu^{17} + 1887136 \nu^{16} + 18397989 \nu^{15} - 47868877 \nu^{14} - 180429682 \nu^{13} + \cdots - 19288573 ) / 461188 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 749664 \nu^{17} - 1920719 \nu^{16} - 18764575 \nu^{15} + 48829984 \nu^{14} + 184522593 \nu^{13} + \cdots + 20249941 ) / 461188 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 836399 \nu^{17} + 2186039 \nu^{16} + 20763412 \nu^{15} - 55350085 \nu^{14} - 201797451 \nu^{13} + \cdots - 26019218 ) / 461188 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 926803 \nu^{17} + 2424063 \nu^{16} + 23032284 \nu^{15} - 61478669 \nu^{14} - 224076039 \nu^{13} + \cdots - 31489058 ) / 461188 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 1151055 \nu^{17} - 3014842 \nu^{16} - 28643183 \nu^{15} + 76581673 \nu^{14} + 279158892 \nu^{13} + \cdots + 36483943 ) / 461188 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 207099 \nu^{17} - 534156 \nu^{16} - 5174933 \nu^{15} + 13579877 \nu^{14} + 50745382 \nu^{13} + \cdots + 6535141 ) / 65884 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 1498205 \nu^{17} - 3838619 \nu^{16} - 37449726 \nu^{15} + 97428583 \nu^{14} + 367638783 \nu^{13} + \cdots + 38645688 ) / 461188 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( - 960073 \nu^{17} + 2473038 \nu^{16} + 23990266 \nu^{15} - 62844453 \nu^{14} - 235297994 \nu^{13} + \cdots - 27550128 ) / 230594 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( 4633386 \nu^{17} - 11901709 \nu^{16} - 115689811 \nu^{15} + 301959672 \nu^{14} + 1133854541 \nu^{13} + \cdots + 119961671 ) / 461188 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{16} + \beta_{14} + \beta_{7} + \beta_{2} + 5\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{15} - \beta_{13} - \beta_{11} + \beta_{10} + \beta_{9} - 2\beta_{8} - \beta_{3} + 7\beta_{2} + 16 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 9 \beta_{16} + 8 \beta_{14} - \beta_{11} + 2 \beta_{10} - 2 \beta_{8} + 9 \beta_{7} + \beta_{5} + \cdots + 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( \beta_{17} + 2 \beta_{16} + 9 \beta_{15} + \beta_{14} - 10 \beta_{13} - 9 \beta_{11} + 12 \beta_{10} + \cdots + 99 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( \beta_{17} + 67 \beta_{16} - 2 \beta_{15} + 57 \beta_{14} - 3 \beta_{13} - \beta_{12} - 11 \beta_{11} + \cdots + 29 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 14 \beta_{17} + 30 \beta_{16} + 63 \beta_{15} + 19 \beta_{14} - 81 \beta_{13} - \beta_{12} - 65 \beta_{11} + \cdots + 655 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 15 \beta_{17} + 475 \beta_{16} - 33 \beta_{15} + 400 \beta_{14} - 49 \beta_{13} - 19 \beta_{12} + \cdots + 317 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 142 \beta_{17} + 321 \beta_{16} + 401 \beta_{15} + 235 \beta_{14} - 622 \beta_{13} - 27 \beta_{12} + \cdots + 4492 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 165 \beta_{17} + 3337 \beta_{16} - 378 \beta_{15} + 2828 \beta_{14} - 557 \beta_{13} - 252 \beta_{12} + \cdots + 3099 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 1279 \beta_{17} + 3017 \beta_{16} + 2399 \beta_{15} + 2427 \beta_{14} - 4711 \beta_{13} - 441 \beta_{12} + \cdots + 31518 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 1623 \beta_{17} + 23560 \beta_{16} - 3729 \beta_{15} + 20255 \beta_{14} - 5478 \beta_{13} - 2847 \beta_{12} + \cdots + 28438 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 10889 \beta_{17} + 26593 \beta_{16} + 13438 \beta_{15} + 22799 \beta_{14} - 35638 \beta_{13} + \cdots + 224873 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 15105 \beta_{17} + 168028 \beta_{16} - 34035 \beta_{15} + 147045 \beta_{14} - 50015 \beta_{13} + \cdots + 250318 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( 89928 \beta_{17} + 225980 \beta_{16} + 68039 \beta_{15} + 202516 \beta_{14} - 270408 \beta_{13} + \cdots + 1626032 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( 135905 \beta_{17} + 1212365 \beta_{16} - 296893 \beta_{15} + 1080928 \beta_{14} - 437159 \beta_{13} + \cdots + 2140037 \) Copy content Toggle raw display

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
2.80103
2.62027
2.30810
2.03141
1.81057
1.33016
1.14331
0.589764
0.200159
0.117638
−0.370427
−1.46716
−1.51612
−1.56310
−1.57163
−2.43831
−2.47806
−2.54760
−2.80103 −0.708357 5.84576 1.00000 1.98413 1.81823 −10.7721 −2.49823 −2.80103
1.2 −2.62027 2.90669 4.86584 1.00000 −7.61632 2.62123 −7.50929 5.44882 −2.62027
1.3 −2.30810 −2.53573 3.32733 1.00000 5.85273 3.10066 −3.06362 3.42994 −2.30810
1.4 −2.03141 −2.41065 2.12661 1.00000 4.89701 4.27304 −0.257201 2.81123 −2.03141
1.5 −1.81057 1.85470 1.27815 1.00000 −3.35806 −2.49392 1.30696 0.439927 −1.81057
1.6 −1.33016 −2.65160 −0.230684 1.00000 3.52704 −1.42983 2.96716 4.03098 −1.33016
1.7 −1.14331 −0.291587 −0.692836 1.00000 0.333376 −2.48191 3.07875 −2.91498 −1.14331
1.8 −0.589764 2.80954 −1.65218 1.00000 −1.65696 0.145738 2.15392 4.89351 −0.589764
1.9 −0.200159 0.328234 −1.95994 1.00000 −0.0656991 4.42737 0.792618 −2.89226 −0.200159
1.10 −0.117638 1.90235 −1.98616 1.00000 −0.223789 2.81872 0.468926 0.618929 −0.117638
1.11 0.370427 0.193908 −1.86278 1.00000 0.0718287 −0.400866 −1.43088 −2.96240 0.370427
1.12 1.46716 −2.83912 0.152571 1.00000 −4.16545 4.97656 −2.71048 5.06058 1.46716
1.13 1.51612 3.04030 0.298613 1.00000 4.60945 2.32872 −2.57950 6.24343 1.51612
1.14 1.56310 −0.512231 0.443278 1.00000 −0.800667 −3.87624 −2.43331 −2.73762 1.56310
1.15 1.57163 −3.15664 0.470008 1.00000 −4.96105 1.09217 −2.40457 6.96437 1.57163
1.16 2.43831 1.39369 3.94538 1.00000 3.39825 −0.911770 4.74344 −1.05763 2.43831
1.17 2.47806 0.0240901 4.14079 1.00000 0.0596967 −2.90004 5.30501 −2.99942 2.47806
1.18 2.54760 −0.347586 4.49025 1.00000 −0.885510 3.89213 6.34414 −2.87918 2.54760
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.18
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(5\) \( -1 \)
\(29\) \( +1 \)

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 4205.2.a.s 18
29.b even 2 1 4205.2.a.t 18
29.d even 7 2 145.2.k.b 36
145.n even 14 2 725.2.l.e 36
145.p odd 28 4 725.2.r.d 72
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
145.2.k.b 36 29.d even 7 2
725.2.l.e 36 145.n even 14 2
725.2.r.d 72 145.p odd 28 4
4205.2.a.s 18 1.a even 1 1 trivial
4205.2.a.t 18 29.b even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4205))\):

\( T_{2}^{18} + T_{2}^{17} - 29 T_{2}^{16} - 26 T_{2}^{15} + 347 T_{2}^{14} + 277 T_{2}^{13} - 2215 T_{2}^{12} + \cdots + 41 \) Copy content Toggle raw display
\( T_{3}^{18} + T_{3}^{17} - 36 T_{3}^{16} - 32 T_{3}^{15} + 518 T_{3}^{14} + 394 T_{3}^{13} - 3781 T_{3}^{12} + \cdots - 1 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{18} + T^{17} + \cdots + 41 \) Copy content Toggle raw display
$3$ \( T^{18} + T^{17} + \cdots - 1 \) Copy content Toggle raw display
$5$ \( (T - 1)^{18} \) Copy content Toggle raw display
$7$ \( T^{18} - 17 T^{17} + \cdots - 205729 \) Copy content Toggle raw display
$11$ \( T^{18} + 5 T^{17} + \cdots + 17681728 \) Copy content Toggle raw display
$13$ \( T^{18} - 17 T^{17} + \cdots + 59082688 \) Copy content Toggle raw display
$17$ \( T^{18} - 14 T^{17} + \cdots - 3879616 \) Copy content Toggle raw display
$19$ \( T^{18} + \cdots + 575326144 \) Copy content Toggle raw display
$23$ \( T^{18} - 28 T^{17} + \cdots - 63413 \) Copy content Toggle raw display
$29$ \( T^{18} \) Copy content Toggle raw display
$31$ \( T^{18} - 7 T^{17} + \cdots - 80576 \) Copy content Toggle raw display
$37$ \( T^{18} + \cdots + 4756936947008 \) Copy content Toggle raw display
$41$ \( T^{18} + \cdots + 457112949881 \) Copy content Toggle raw display
$43$ \( T^{18} + \cdots - 28751525207 \) Copy content Toggle raw display
$47$ \( T^{18} + \cdots - 64893278323 \) Copy content Toggle raw display
$53$ \( T^{18} + \cdots + 1789774633472 \) Copy content Toggle raw display
$59$ \( T^{18} + \cdots + 600525977408 \) Copy content Toggle raw display
$61$ \( T^{18} + \cdots - 363101625152 \) Copy content Toggle raw display
$67$ \( T^{18} + \cdots + 388848546868928 \) Copy content Toggle raw display
$71$ \( T^{18} + \cdots + 806647744 \) Copy content Toggle raw display
$73$ \( T^{18} + \cdots - 46786424768 \) Copy content Toggle raw display
$79$ \( T^{18} + \cdots + 11470874176 \) Copy content Toggle raw display
$83$ \( T^{18} + \cdots - 772624375419197 \) Copy content Toggle raw display
$89$ \( T^{18} + \cdots + 28001490013 \) Copy content Toggle raw display
$97$ \( T^{18} + \cdots + 44674838248768 \) Copy content Toggle raw display
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