Properties

Label 4205.2.a.u
Level $4205$
Weight $2$
Character orbit 4205.a
Self dual yes
Analytic conductor $33.577$
Analytic rank $1$
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: \(1\)
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: \( x^{20} - 9 x^{19} + 10 x^{18} + 135 x^{17} - 416 x^{16} - 496 x^{15} + 3448 x^{14} - 1537 x^{13} + \cdots - 9 \) 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.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} + ( - \beta_{9} - 1) q^{3} + (\beta_{2} + 1) q^{4} + q^{5} + ( - \beta_{18} + \beta_{16} + \cdots + 2 \beta_1) q^{6}+ \cdots + ( - \beta_{19} + \beta_{9} + \beta_{7} + \cdots + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{2} + ( - \beta_{9} - 1) q^{3} + (\beta_{2} + 1) q^{4} + q^{5} + ( - \beta_{18} + \beta_{16} + \cdots + 2 \beta_1) q^{6}+ \cdots + (2 \beta_{19} + \beta_{18} - \beta_{17} + \cdots - 8) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q - 9 q^{2} - 13 q^{3} + 21 q^{4} + 20 q^{5} + 6 q^{6} + 3 q^{7} - 18 q^{8} + 25 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 20 q - 9 q^{2} - 13 q^{3} + 21 q^{4} + 20 q^{5} + 6 q^{6} + 3 q^{7} - 18 q^{8} + 25 q^{9} - 9 q^{10} - 13 q^{11} - 40 q^{12} - 9 q^{14} - 13 q^{15} + 39 q^{16} - 11 q^{17} - 9 q^{18} - 41 q^{19} + 21 q^{20} - 9 q^{21} - 32 q^{22} - 14 q^{23} - 5 q^{24} + 20 q^{25} + 39 q^{26} - 49 q^{27} - 12 q^{28} + 6 q^{30} - 12 q^{31} - 50 q^{32} + 17 q^{33} + 3 q^{35} + 59 q^{36} - 48 q^{37} + 76 q^{38} - 10 q^{39} - 18 q^{40} + 17 q^{41} - 53 q^{42} - 53 q^{43} - 22 q^{44} + 25 q^{45} - 14 q^{46} - 67 q^{47} - 100 q^{48} + 41 q^{49} - 9 q^{50} + 11 q^{51} - 13 q^{52} - 11 q^{53} + 6 q^{54} - 13 q^{55} - 3 q^{56} - 14 q^{57} - 25 q^{59} - 40 q^{60} - 20 q^{61} - 60 q^{62} - 4 q^{63} + 32 q^{64} + 24 q^{66} + 21 q^{67} - 68 q^{68} + 16 q^{69} - 9 q^{70} - 6 q^{71} + 8 q^{72} - 12 q^{73} - 24 q^{74} - 13 q^{75} - 111 q^{76} - 47 q^{77} + 7 q^{78} - 57 q^{79} + 39 q^{80} - 16 q^{81} - 25 q^{82} - 13 q^{83} + 159 q^{84} - 11 q^{85} - 9 q^{86} - 65 q^{88} + 6 q^{89} - 9 q^{90} - 83 q^{91} + 22 q^{92} + 72 q^{93} - 35 q^{94} - 41 q^{95} + 128 q^{96} - 26 q^{97} - 58 q^{98} - 50 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} - 9 x^{19} + 10 x^{18} + 135 x^{17} - 416 x^{16} - 496 x^{15} + 3448 x^{14} - 1537 x^{13} + \cdots - 9 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 10141252 \nu^{19} + 74341551 \nu^{18} + 46968008 \nu^{17} - 1482191367 \nu^{16} + \cdots + 288447375 ) / 223202976 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 1320726 \nu^{19} + 9201683 \nu^{18} + 7978485 \nu^{17} - 181276105 \nu^{16} + 181081980 \nu^{15} + \cdots + 39939972 ) / 6975093 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 1364125 \nu^{19} - 11984062 \nu^{18} + 10956580 \nu^{17} + 187063931 \nu^{16} - 525912125 \nu^{15} + \cdots + 58801599 ) / 6975093 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 46388371 \nu^{19} + 395252717 \nu^{18} - 289055713 \nu^{17} - 6296069179 \nu^{16} + \cdots - 443415951 ) / 223202976 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 56850039 \nu^{19} - 521791603 \nu^{18} + 642841941 \nu^{17} + 7721723273 \nu^{16} + \cdots + 3526834329 ) / 223202976 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 57788162 \nu^{19} + 463868767 \nu^{18} - 95491514 \nu^{17} - 8132605571 \nu^{16} + \cdots + 655823751 ) / 223202976 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 59225453 \nu^{19} + 521410356 \nu^{18} - 490329059 \nu^{17} - 8105411094 \nu^{16} + \cdots - 2935925760 ) / 223202976 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 76167763 \nu^{19} + 576648654 \nu^{18} + 121194107 \nu^{17} - 10568742252 \nu^{16} + \cdots + 775450626 ) / 223202976 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 79026557 \nu^{19} - 685412713 \nu^{18} + 597080231 \nu^{17} + 10646842139 \nu^{16} + \cdots + 4342840539 ) / 223202976 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 107691052 \nu^{19} + 909108349 \nu^{18} - 576274600 \nu^{17} - 14801126981 \nu^{16} + \cdots - 2689819107 ) / 223202976 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 42045781 \nu^{19} - 313127303 \nu^{18} - 88887513 \nu^{17} + 5715451305 \nu^{16} + \cdots - 332352035 ) / 74400992 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 146026351 \nu^{19} + 1178579760 \nu^{18} - 358134001 \nu^{17} - 20090044998 \nu^{16} + \cdots - 1349588556 ) / 223202976 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 77648892 \nu^{19} - 664296617 \nu^{18} + 478314360 \nu^{17} + 10696184881 \nu^{16} + \cdots + 2211553431 ) / 111601488 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( - 158811749 \nu^{19} + 1296054303 \nu^{18} - 536394647 \nu^{17} - 21640448577 \nu^{16} + \cdots - 2010107013 ) / 223202976 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( 268820746 \nu^{19} - 2196902355 \nu^{18} + 889037314 \nu^{17} + 36888855303 \nu^{16} + \cdots + 2806095333 ) / 223202976 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( - 9465781 \nu^{19} + 75517723 \nu^{18} - 17890495 \nu^{17} - 1293701666 \nu^{16} + \cdots - 105773820 ) / 6975093 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( 376046241 \nu^{19} - 3145512869 \nu^{18} + 1814188659 \nu^{17} + 51514416127 \nu^{16} + \cdots + 8080497903 ) / 223202976 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{18} + \beta_{16} + \beta_{14} + \beta_{10} - \beta_{8} - \beta_{7} + 2\beta_{5} + \beta_{3} + \beta_{2} + 6\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - \beta_{18} + 2 \beta_{14} - \beta_{13} + \beta_{11} + \beta_{10} + 2 \beta_{9} - 2 \beta_{8} + \cdots + 15 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{19} - 9 \beta_{18} + 8 \beta_{16} + 11 \beta_{14} + 10 \beta_{10} + 2 \beta_{9} - 10 \beta_{8} + \cdots + 1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( \beta_{19} - 12 \beta_{18} + 2 \beta_{17} + 3 \beta_{16} + \beta_{15} + 23 \beta_{14} - 9 \beta_{13} + \cdots + 87 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 14 \beta_{19} - 70 \beta_{18} + 57 \beta_{16} + \beta_{15} + 98 \beta_{14} - 2 \beta_{13} + 3 \beta_{12} + \cdots + 10 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 14 \beta_{19} - 113 \beta_{18} + 26 \beta_{17} + 43 \beta_{16} + 14 \beta_{15} + 211 \beta_{14} + \cdots + 527 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 136 \beta_{19} - 527 \beta_{18} + \beta_{17} + 397 \beta_{16} + 16 \beta_{15} + 808 \beta_{14} + \cdots + 84 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 137 \beta_{19} - 970 \beta_{18} + 245 \beta_{17} + 435 \beta_{16} + 144 \beta_{15} + 1784 \beta_{14} + \cdots + 3255 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 1144 \beta_{19} - 3930 \beta_{18} + 22 \beta_{17} + 2757 \beta_{16} + 186 \beta_{15} + 6408 \beta_{14} + \cdots + 690 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 1162 \beta_{19} - 7948 \beta_{18} + 2046 \beta_{17} + 3845 \beta_{16} + 1328 \beta_{15} + 14477 \beta_{14} + \cdots + 20385 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 8927 \beta_{19} - 29246 \beta_{18} + 330 \beta_{17} + 19240 \beta_{16} + 1928 \beta_{15} + 49707 \beta_{14} + \cdots + 5653 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 9157 \beta_{19} - 63417 \beta_{18} + 16114 \beta_{17} + 31806 \beta_{16} + 11633 \beta_{15} + \cdots + 129256 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 66575 \beta_{19} - 217799 \beta_{18} + 4086 \beta_{17} + 135389 \beta_{16} + 18793 \beta_{15} + \cdots + 46091 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( 69092 \beta_{19} - 497729 \beta_{18} + 122894 \beta_{17} + 253634 \beta_{16} + 98952 \beta_{15} + \cdots + 829388 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( 481864 \beta_{19} - 1624688 \beta_{18} + 44744 \beta_{17} + 961723 \beta_{16} + 175429 \beta_{15} + \cdots + 372757 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( 506771 \beta_{19} - 3864019 \beta_{18} + 919720 \beta_{17} + 1978958 \beta_{16} + 825353 \beta_{15} + \cdots + 5384058 \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( 3415055 \beta_{19} - 12140872 \beta_{18} + 449798 \beta_{17} + 6894638 \beta_{16} + 1582130 \beta_{15} + \cdots + 2985929 \) 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
2.72492
2.69869
2.60254
2.39174
2.16829
2.07024
1.27509
1.20439
0.916581
0.872206
0.311030
0.257741
0.0966090
−0.265964
−0.831921
−1.24663
−1.57484
−1.76395
−2.36893
−2.53784
−2.72492 −1.94113 5.42521 1.00000 5.28943 0.0233174 −9.33344 0.767983 −2.72492
1.2 −2.69869 −2.66237 5.28293 1.00000 7.18491 −2.24968 −8.85960 4.08822 −2.69869
1.3 −2.60254 0.925438 4.77323 1.00000 −2.40849 4.60735 −7.21744 −2.14357 −2.60254
1.4 −2.39174 −3.12787 3.72044 1.00000 7.48108 −4.23970 −4.11486 6.78359 −2.39174
1.5 −2.16829 2.50536 2.70147 1.00000 −5.43235 1.53488 −1.52100 3.27684 −2.16829
1.6 −2.07024 0.132219 2.28589 1.00000 −0.273725 2.07989 −0.591858 −2.98252 −2.07024
1.7 −1.27509 −1.59671 −0.374142 1.00000 2.03595 4.23027 3.02725 −0.450518 −1.27509
1.8 −1.20439 1.44885 −0.549441 1.00000 −1.74498 −3.45384 3.07053 −0.900842 −1.20439
1.9 −0.916581 −0.984878 −1.15988 1.00000 0.902720 −4.70152 2.89629 −2.03002 −0.916581
1.10 −0.872206 −2.29496 −1.23926 1.00000 2.00168 1.14877 2.82530 2.26684 −0.872206
1.11 −0.311030 −2.25319 −1.90326 1.00000 0.700810 3.56565 1.21403 2.07686 −0.311030
1.12 −0.257741 −3.41588 −1.93357 1.00000 0.880413 3.96307 1.01384 8.66824 −0.257741
1.13 −0.0966090 1.58632 −1.99067 1.00000 −0.153252 −0.369143 0.385534 −0.483603 −0.0966090
1.14 0.265964 2.09410 −1.92926 1.00000 0.556953 −1.89937 −1.04504 1.38524 0.265964
1.15 0.831921 1.64551 −1.30791 1.00000 1.36893 0.563531 −2.75192 −0.292303 0.831921
1.16 1.24663 −0.0306507 −0.445906 1.00000 −0.0382102 4.65745 −3.04915 −2.99906 1.24663
1.17 1.57484 −1.46895 0.480108 1.00000 −2.31336 −2.35510 −2.39358 −0.842178 1.57484
1.18 1.76395 1.77419 1.11151 1.00000 3.12957 −3.20279 −1.56726 0.147741 1.76395
1.19 2.36893 −2.20406 3.61185 1.00000 −5.22127 −2.24323 3.81837 1.85788 2.36893
1.20 2.53784 −3.13132 4.44065 1.00000 −7.94681 1.34020 6.19400 6.80518 2.53784
\(n\): 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.u 20
29.b even 2 1 4205.2.a.x yes 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4205.2.a.u 20 1.a even 1 1 trivial
4205.2.a.x yes 20 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(4205))\):

\( T_{2}^{20} + 9 T_{2}^{19} + 10 T_{2}^{18} - 135 T_{2}^{17} - 416 T_{2}^{16} + 496 T_{2}^{15} + 3448 T_{2}^{14} + \cdots - 9 \) Copy content Toggle raw display
\( T_{3}^{20} + 13 T_{3}^{19} + 42 T_{3}^{18} - 144 T_{3}^{17} - 1090 T_{3}^{16} - 552 T_{3}^{15} + \cdots + 601 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{20} + 9 T^{19} + \cdots - 9 \) Copy content Toggle raw display
$3$ \( T^{20} + 13 T^{19} + \cdots + 601 \) Copy content Toggle raw display
$5$ \( (T - 1)^{20} \) Copy content Toggle raw display
$7$ \( T^{20} - 3 T^{19} + \cdots - 152219 \) Copy content Toggle raw display
$11$ \( T^{20} + \cdots + 149796531 \) Copy content Toggle raw display
$13$ \( T^{20} - 130 T^{18} + \cdots + 2131 \) Copy content Toggle raw display
$17$ \( T^{20} + \cdots + 700902171 \) Copy content Toggle raw display
$19$ \( T^{20} + \cdots + 33964903981 \) Copy content Toggle raw display
$23$ \( T^{20} + \cdots - 639945882999 \) Copy content Toggle raw display
$29$ \( T^{20} \) Copy content Toggle raw display
$31$ \( T^{20} + \cdots + 724172800711 \) Copy content Toggle raw display
$37$ \( T^{20} + \cdots + 275881111201 \) Copy content Toggle raw display
$41$ \( T^{20} + \cdots + 39543219351 \) Copy content Toggle raw display
$43$ \( T^{20} + \cdots - 172143979829 \) Copy content Toggle raw display
$47$ \( T^{20} + \cdots - 404481336703299 \) Copy content Toggle raw display
$53$ \( T^{20} + \cdots + 100434099951216 \) Copy content Toggle raw display
$59$ \( T^{20} + \cdots + 13218632181 \) Copy content Toggle raw display
$61$ \( T^{20} + \cdots + 124700620099711 \) Copy content Toggle raw display
$67$ \( T^{20} + \cdots + 61\!\cdots\!41 \) Copy content Toggle raw display
$71$ \( T^{20} + \cdots + 975995868501 \) Copy content Toggle raw display
$73$ \( T^{20} + \cdots + 478341883499581 \) Copy content Toggle raw display
$79$ \( T^{20} + \cdots - 938007788741909 \) Copy content Toggle raw display
$83$ \( T^{20} + \cdots + 40024363458861 \) Copy content Toggle raw display
$89$ \( T^{20} + \cdots + 89102195181 \) Copy content Toggle raw display
$97$ \( T^{20} + \cdots + 29\!\cdots\!61 \) Copy content Toggle raw display
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