Properties

Label 4205.2.a.w
Level $4205$
Weight $2$
Character orbit 4205.a
Self dual yes
Analytic conductor $33.577$
Analytic rank $0$
Dimension $20$
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: \(20\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} - x^{19} - 32 x^{18} + 33 x^{17} + 430 x^{16} - 452 x^{15} - 3156 x^{14} + 3333 x^{13} + \cdots + 531 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
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 \(1,\beta_1,\ldots,\beta_{19}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} - \beta_{17} q^{3} + (\beta_{2} + 1) q^{4} - q^{5} + (\beta_{19} - \beta_{17} - \beta_{12} + \cdots + 1) q^{6}+ \cdots + (\beta_{18} + \beta_{15} + \beta_{13} + \cdots + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} - \beta_{17} q^{3} + (\beta_{2} + 1) q^{4} - q^{5} + (\beta_{19} - \beta_{17} - \beta_{12} + \cdots + 1) q^{6}+ \cdots + (2 \beta_{17} + \beta_{16} + 2 \beta_{15} + \cdots + 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q + q^{2} + 3 q^{3} + 25 q^{4} - 20 q^{5} - 10 q^{6} - q^{7} - 6 q^{8} + 17 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 20 q + q^{2} + 3 q^{3} + 25 q^{4} - 20 q^{5} - 10 q^{6} - q^{7} - 6 q^{8} + 17 q^{9} - q^{10} + 23 q^{11} + 8 q^{12} + 8 q^{13} - q^{14} - 3 q^{15} + 15 q^{16} - 11 q^{17} + 39 q^{18} + 31 q^{19} - 25 q^{20} - 5 q^{21} - 22 q^{22} + 12 q^{23} - 37 q^{24} + 20 q^{25} - 17 q^{26} + 15 q^{27} - 24 q^{28} + 10 q^{30} + 52 q^{31} - 26 q^{32} + 43 q^{33} - 20 q^{34} + q^{35} + 55 q^{36} + 26 q^{37} - 12 q^{38} + 26 q^{39} + 6 q^{40} + 33 q^{41} + 7 q^{42} + 41 q^{43} + 34 q^{44} - 17 q^{45} + 34 q^{46} - 13 q^{47} + 14 q^{48} + 13 q^{49} + q^{50} - 45 q^{51} + 13 q^{52} - 21 q^{53} - 6 q^{54} - 23 q^{55} + 49 q^{56} - 38 q^{57} - q^{59} - 8 q^{60} + 12 q^{61} + 30 q^{62} + 16 q^{63} - 8 q^{65} + 12 q^{66} - 23 q^{67} - 26 q^{68} + 64 q^{69} + q^{70} - 6 q^{71} + 66 q^{72} + 32 q^{73} - 8 q^{74} + 3 q^{75} + 81 q^{76} + 35 q^{77} + 45 q^{78} + 71 q^{79} - 15 q^{80} + 80 q^{81} + 41 q^{82} - 5 q^{83} - 17 q^{84} + 11 q^{85} + 3 q^{86} - 11 q^{88} + 14 q^{89} - 39 q^{90} + 9 q^{91} - 48 q^{92} + 8 q^{93} + 9 q^{94} - 31 q^{95} - 176 q^{96} + 32 q^{97} + 38 q^{98} + 66 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} - x^{19} - 32 x^{18} + 33 x^{17} + 430 x^{16} - 452 x^{15} - 3156 x^{14} + 3333 x^{13} + \cdots + 531 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 11253920 \nu^{19} + 11665513 \nu^{18} - 380990777 \nu^{17} - 296959483 \nu^{16} + \cdots + 2348893611 ) / 1031305953 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 508361884 \nu^{19} + 4007565761 \nu^{18} - 15507702548 \nu^{17} - 117046953529 \nu^{16} + \cdots - 1211626049019 ) / 39189626214 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 22919433 \nu^{19} + 20865337 \nu^{18} + 668338843 \nu^{17} - 602657994 \nu^{16} + \cdots + 5975831520 ) / 1031305953 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 31707664 \nu^{19} - 30086474 \nu^{18} - 989289036 \nu^{17} + 966742532 \nu^{16} + \cdots + 16122981897 ) / 1031305953 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 1877988503 \nu^{19} + 5198803839 \nu^{18} - 54723512069 \nu^{17} - 154525370181 \nu^{16} + \cdots - 2647357784829 ) / 39189626214 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 2756447151 \nu^{19} + 6340309734 \nu^{18} - 86299830161 \nu^{17} - 181388331832 \nu^{16} + \cdots - 2181408250884 ) / 39189626214 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 4357094992 \nu^{19} - 5465424687 \nu^{18} + 131153193532 \nu^{17} + 155332145058 \nu^{16} + \cdots + 2338825304826 ) / 19594813107 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 10432885375 \nu^{19} - 9361904684 \nu^{18} + 316035881491 \nu^{17} + 260941887230 \nu^{16} + \cdots + 3552521236314 ) / 39189626214 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 6288478924 \nu^{19} + 8574161485 \nu^{18} - 193039854354 \nu^{17} - 240670451935 \nu^{16} + \cdots - 2962917932652 ) / 19594813107 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 16694014200 \nu^{19} - 16826257773 \nu^{18} + 507592296560 \nu^{17} + 472044041961 \nu^{16} + \cdots + 6891288342879 ) / 39189626214 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 11530503357 \nu^{19} - 9492520546 \nu^{18} + 349590863396 \nu^{17} + 260209840888 \nu^{16} + \cdots + 3360412430883 ) / 19594813107 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 25112798797 \nu^{19} + 20277459947 \nu^{18} - 763182951539 \nu^{17} - 552259520301 \nu^{16} + \cdots - 6583383259335 ) / 39189626214 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 26693057171 \nu^{19} - 17971809648 \nu^{18} + 818489858063 \nu^{17} + 473983679188 \nu^{16} + \cdots + 4799142015660 ) / 39189626214 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( 912741916 \nu^{19} + 552097956 \nu^{18} - 27884285882 \nu^{17} - 14287228873 \nu^{16} + \cdots - 119508033240 ) / 1031305953 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( 40946250473 \nu^{19} + 32896886734 \nu^{18} - 1244517242615 \nu^{17} - 895138330866 \nu^{16} + \cdots - 10632889707090 ) / 39189626214 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( - 42819030374 \nu^{19} - 39125662749 \nu^{18} + 1302401254348 \nu^{17} + \cdots + 13113834202443 ) / 39189626214 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( - 56776695727 \nu^{19} - 53394324095 \nu^{18} + 1725660209731 \nu^{17} + \cdots + 19440234559269 ) / 39189626214 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{16} + \beta_{15} - \beta_{14} - \beta_{12} + \beta_{3} + 4\beta _1 - 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - \beta_{19} + \beta_{18} + \beta_{17} + \beta_{15} + \beta_{10} + 2 \beta_{9} - \beta_{6} + 2 \beta_{3} + \cdots + 13 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - \beta_{19} + \beta_{17} + 9 \beta_{16} + 9 \beta_{15} - 10 \beta_{14} + 2 \beta_{13} - 9 \beta_{12} + \cdots - 11 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 11 \beta_{19} + 11 \beta_{18} + 12 \beta_{17} - \beta_{16} + 11 \beta_{15} + \beta_{12} + 2 \beta_{11} + \cdots + 66 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 13 \beta_{19} - 2 \beta_{18} + 9 \beta_{17} + 70 \beta_{16} + 67 \beta_{15} - 84 \beta_{14} + \cdots - 93 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 92 \beta_{19} + 96 \beta_{18} + 113 \beta_{17} - 15 \beta_{16} + 95 \beta_{15} - 2 \beta_{14} + \cdots + 381 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 128 \beta_{19} - 27 \beta_{18} + 59 \beta_{17} + 523 \beta_{16} + 472 \beta_{15} - 660 \beta_{14} + \cdots - 729 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 702 \beta_{19} + 775 \beta_{18} + 968 \beta_{17} - 156 \beta_{16} + 759 \beta_{15} - 36 \beta_{14} + \cdots + 2428 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 1132 \beta_{19} - 257 \beta_{18} + 322 \beta_{17} + 3843 \beta_{16} + 3252 \beta_{15} - 5015 \beta_{14} + \cdots - 5562 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 5141 \beta_{19} + 6018 \beta_{18} + 7885 \beta_{17} - 1415 \beta_{16} + 5856 \beta_{15} + \cdots + 16534 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 9443 \beta_{19} - 2159 \beta_{18} + 1369 \beta_{17} + 27998 \beta_{16} + 22159 \beta_{15} + \cdots - 41972 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 36836 \beta_{19} + 45661 \beta_{18} + 62306 \beta_{17} - 12044 \beta_{16} + 44338 \beta_{15} + \cdots + 117218 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 75913 \beta_{19} - 17229 \beta_{18} + 2227 \beta_{17} + 203020 \beta_{16} + 149993 \beta_{15} + \cdots - 315174 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( - 260451 \beta_{19} + 341353 \beta_{18} + 482812 \beta_{17} - 99136 \beta_{16} + 331949 \beta_{15} + \cdots + 850521 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( - 595115 \beta_{19} - 134745 \beta_{18} - 42439 \beta_{17} + 1468509 \beta_{16} + 1010494 \beta_{15} + \cdots - 2361113 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( - 1824823 \beta_{19} + 2527205 \beta_{18} + 3692981 \beta_{17} - 800261 \beta_{16} + 2467851 \beta_{15} + \cdots + 6253098 \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( - 4583140 \beta_{19} - 1047699 \beta_{18} - 745621 \beta_{17} + 10611215 \beta_{16} + 6780518 \beta_{15} + \cdots - 17667741 \) 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
−2.74216
−2.59819
−2.23299
−1.97558
−1.92331
−1.88992
−0.917077
−0.570711
−0.524833
−0.377848
0.384837
0.635547
1.33815
1.42752
1.85217
1.85907
1.90835
2.24889
2.44141
2.65668
−2.74216 1.17856 5.51946 −1.00000 −3.23181 0.485403 −9.65092 −1.61099 2.74216
1.2 −2.59819 2.74783 4.75061 −1.00000 −7.13939 −3.75279 −7.14662 4.55056 2.59819
1.3 −2.23299 0.560889 2.98623 −1.00000 −1.25246 −0.536528 −2.20223 −2.68540 2.23299
1.4 −1.97558 −0.929678 1.90291 −1.00000 1.83665 −0.161567 0.191811 −2.13570 1.97558
1.5 −1.92331 −0.480809 1.69913 −1.00000 0.924746 3.15245 0.578675 −2.76882 1.92331
1.6 −1.88992 −2.27083 1.57181 −1.00000 4.29169 −2.62326 0.809245 2.15667 1.88992
1.7 −0.917077 3.20820 −1.15897 −1.00000 −2.94217 3.34120 2.89702 7.29258 0.917077
1.8 −0.570711 −0.0145096 −1.67429 −1.00000 0.00828081 3.74775 2.09696 −2.99979 0.570711
1.9 −0.524833 1.29665 −1.72455 −1.00000 −0.680526 0.270633 1.95477 −1.31869 0.524833
1.10 −0.377848 0.0409643 −1.85723 −1.00000 −0.0154783 −1.29703 1.45745 −2.99832 0.377848
1.11 0.384837 −3.05566 −1.85190 −1.00000 −1.17593 2.20210 −1.48235 6.33704 −0.384837
1.12 0.635547 1.04324 −1.59608 −1.00000 0.663025 −4.92609 −2.28548 −1.91166 −0.635547
1.13 1.33815 −1.75279 −0.209352 −1.00000 −2.34550 −0.122970 −2.95645 0.0722895 −1.33815
1.14 1.42752 −0.429988 0.0378047 −1.00000 −0.613816 4.41006 −2.80107 −2.81511 −1.42752
1.15 1.85217 3.37184 1.43054 −1.00000 6.24523 1.55791 −1.05473 8.36930 −1.85217
1.16 1.85907 −0.964666 1.45615 −1.00000 −1.79338 −3.77389 −1.01106 −2.06942 −1.85907
1.17 1.90835 2.03248 1.64180 −1.00000 3.87868 −5.06125 −0.683562 1.13097 −1.90835
1.18 2.24889 −1.76884 3.05749 −1.00000 −3.97791 0.442358 2.37817 0.128779 −2.24889
1.19 2.44141 2.41715 3.96048 −1.00000 5.90125 1.86399 4.78634 2.84260 −2.44141
1.20 2.65668 −3.23004 5.05796 −1.00000 −8.58118 −0.218471 8.12403 7.43313 −2.65668
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.20
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.w yes 20
29.b even 2 1 4205.2.a.v 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4205.2.a.v 20 29.b even 2 1
4205.2.a.w yes 20 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(4205))\):

\( T_{2}^{20} - T_{2}^{19} - 32 T_{2}^{18} + 33 T_{2}^{17} + 430 T_{2}^{16} - 452 T_{2}^{15} - 3156 T_{2}^{14} + \cdots + 531 \) Copy content Toggle raw display
\( T_{3}^{20} - 3 T_{3}^{19} - 34 T_{3}^{18} + 100 T_{3}^{17} + 456 T_{3}^{16} - 1302 T_{3}^{15} - 3122 T_{3}^{14} + \cdots + 1 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{20} - T^{19} + \cdots + 531 \) Copy content Toggle raw display
$3$ \( T^{20} - 3 T^{19} + \cdots + 1 \) Copy content Toggle raw display
$5$ \( (T + 1)^{20} \) Copy content Toggle raw display
$7$ \( T^{20} + T^{19} + \cdots + 181 \) Copy content Toggle raw display
$11$ \( T^{20} - 23 T^{19} + \cdots - 65890809 \) Copy content Toggle raw display
$13$ \( T^{20} - 8 T^{19} + \cdots + 98598751 \) Copy content Toggle raw display
$17$ \( T^{20} + \cdots + 1442168811 \) Copy content Toggle raw display
$19$ \( T^{20} - 31 T^{19} + \cdots - 10285259 \) Copy content Toggle raw display
$23$ \( T^{20} + \cdots - 207327555 \) Copy content Toggle raw display
$29$ \( T^{20} \) Copy content Toggle raw display
$31$ \( T^{20} + \cdots + 126740668195 \) Copy content Toggle raw display
$37$ \( T^{20} + \cdots - 2392991999 \) Copy content Toggle raw display
$41$ \( T^{20} + \cdots + 73\!\cdots\!71 \) Copy content Toggle raw display
$43$ \( T^{20} + \cdots + 714867558775 \) Copy content Toggle raw display
$47$ \( T^{20} + \cdots - 419379959979 \) Copy content Toggle raw display
$53$ \( T^{20} + \cdots - 383135553783504 \) Copy content Toggle raw display
$59$ \( T^{20} + \cdots - 15\!\cdots\!19 \) Copy content Toggle raw display
$61$ \( T^{20} + \cdots + 200711736974971 \) Copy content Toggle raw display
$67$ \( T^{20} + \cdots + 745138680901 \) Copy content Toggle raw display
$71$ \( T^{20} + \cdots + 11\!\cdots\!45 \) Copy content Toggle raw display
$73$ \( T^{20} + \cdots + 246641482597501 \) Copy content Toggle raw display
$79$ \( T^{20} + \cdots - 1105871247149 \) Copy content Toggle raw display
$83$ \( T^{20} + \cdots + 22\!\cdots\!21 \) Copy content Toggle raw display
$89$ \( T^{20} + \cdots + 46471302071541 \) Copy content Toggle raw display
$97$ \( T^{20} + \cdots - 18\!\cdots\!79 \) Copy content Toggle raw display
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