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Properties

Label 3211.2.a.m
Level $3211$
Weight $2$
Character orbit 3211.a
Self dual yes
Analytic conductor $25.640$
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] = [3211,2,Mod(1,3211)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3211, 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("3211.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3211 = 13^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3211.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: \(25.6399640890\)
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: \( x^{20} - 6 x^{19} - 11 x^{18} + 126 x^{17} - 50 x^{16} - 1020 x^{15} + 1272 x^{14} + 3908 x^{13} + \cdots + 1 \) 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 247)
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.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} - \beta_{11} q^{3} + (\beta_{2} + 1) q^{4} + (\beta_{13} - 1) q^{5} + (\beta_{18} - \beta_{16} + \beta_{11} + \cdots - 1) q^{6}+ \cdots + ( - \beta_{12} + \beta_{8} + \cdots + \beta_{3}) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{2} - \beta_{11} q^{3} + (\beta_{2} + 1) q^{4} + (\beta_{13} - 1) q^{5} + (\beta_{18} - \beta_{16} + \beta_{11} + \cdots - 1) q^{6}+ \cdots + (\beta_{19} + \beta_{18} + 2 \beta_{17} + \cdots + 3) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q - 6 q^{2} + 18 q^{4} - 16 q^{5} - 16 q^{6} - 4 q^{7} - 12 q^{8} + 20 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 20 q - 6 q^{2} + 18 q^{4} - 16 q^{5} - 16 q^{6} - 4 q^{7} - 12 q^{8} + 20 q^{9} + 12 q^{10} - 20 q^{11} + 2 q^{12} - 8 q^{14} - 6 q^{15} + 30 q^{16} - 12 q^{17} - 18 q^{18} + 20 q^{19} - 32 q^{20} - 14 q^{21} + 20 q^{22} - 14 q^{23} - 40 q^{24} + 18 q^{25} + 24 q^{27} + 20 q^{28} - 6 q^{29} + 20 q^{30} - 16 q^{31} - 60 q^{32} - 16 q^{33} + 30 q^{34} - 8 q^{35} + 20 q^{36} - 46 q^{37} - 6 q^{38} + 16 q^{40} - 32 q^{41} + 14 q^{42} - 52 q^{44} - 44 q^{45} - 14 q^{46} + 12 q^{47} + 18 q^{48} + 20 q^{49} - 58 q^{50} + 4 q^{51} + 4 q^{53} + 10 q^{55} - 14 q^{56} - 36 q^{58} - 32 q^{59} + 80 q^{60} - 18 q^{61} + 2 q^{62} - 80 q^{63} + 38 q^{64} - 50 q^{66} - 6 q^{67} - 22 q^{68} - 2 q^{69} + 6 q^{70} - 44 q^{71} - 24 q^{72} + 32 q^{73} - 12 q^{74} - 2 q^{75} + 18 q^{76} - 22 q^{77} - 16 q^{79} - 76 q^{80} + 4 q^{81} - 52 q^{82} - 8 q^{83} - 80 q^{84} - 20 q^{85} - 2 q^{86} - 48 q^{87} + 72 q^{88} - 84 q^{89} + 22 q^{90} - 80 q^{92} - 52 q^{93} + 10 q^{94} - 16 q^{95} - 48 q^{96} + 2 q^{97} + 46 q^{98} + 24 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} - 6 x^{19} - 11 x^{18} + 126 x^{17} - 50 x^{16} - 1020 x^{15} + 1272 x^{14} + 3908 x^{13} + \cdots + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} - \nu^{2} - 5\nu + 3 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 5584646611 \nu^{19} + 51921105877 \nu^{18} - 29691518881 \nu^{17} - 990847723305 \nu^{16} + \cdots - 484570995186 ) / 71367422325 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 9227955421 \nu^{19} - 56029832047 \nu^{18} - 91019720759 \nu^{17} + 1141052812905 \nu^{16} + \cdots + 268873298571 ) / 71367422325 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 3411796199 \nu^{19} - 22389120018 \nu^{18} - 28569713246 \nu^{17} + 464363756170 \nu^{16} + \cdots - 103020424851 ) / 23789140775 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 11098691 \nu^{19} - 53575912 \nu^{18} - 184912064 \nu^{17} + 1181537080 \nu^{16} + \cdots - 12228359 ) / 44081175 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 6399055264 \nu^{19} + 30074891398 \nu^{18} + 106878563331 \nu^{17} - 651322859895 \nu^{16} + \cdots - 124206554314 ) / 23789140775 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 12228359 \nu^{19} + 62271463 \nu^{18} + 188087861 \nu^{17} - 1355861170 \nu^{16} + \cdots + 72998741 ) / 44081175 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 4819007407 \nu^{19} + 33252691459 \nu^{18} + 33377719748 \nu^{17} - 685184876625 \nu^{16} + \cdots + 174165206778 ) / 14273484465 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 27474672069 \nu^{19} - 161123578058 \nu^{18} - 313638799876 \nu^{17} + 3371231595595 \nu^{16} + \cdots - 295602606281 ) / 71367422325 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 44906059097 \nu^{19} - 266618558129 \nu^{18} - 526159681363 \nu^{17} + 5698864398585 \nu^{16} + \cdots - 1041554083503 ) / 71367422325 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 45488179896 \nu^{19} - 273010224922 \nu^{18} - 493061297909 \nu^{17} + 5701376512355 \nu^{16} + \cdots - 277937989054 ) / 71367422325 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 49680512656 \nu^{19} - 275108655692 \nu^{18} - 673972937374 \nu^{17} + 5962859296805 \nu^{16} + \cdots - 1434808329619 ) / 71367422325 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 55463051718 \nu^{19} + 374291343301 \nu^{18} + 421607549672 \nu^{17} - 7727830075340 \nu^{16} + \cdots + 1075440466582 ) / 71367422325 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( - 56369709621 \nu^{19} + 353515719122 \nu^{18} + 552885587134 \nu^{17} - 7390344157180 \nu^{16} + \cdots + 986641657229 ) / 71367422325 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( 58281189878 \nu^{19} - 348004683896 \nu^{18} - 642536337262 \nu^{17} + 7285975470240 \nu^{16} + \cdots - 506858731797 ) / 71367422325 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( - 80119927334 \nu^{19} + 503221890063 \nu^{18} + 775947301911 \nu^{17} - 10483610256770 \nu^{16} + \cdots + 1112034746791 ) / 71367422325 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( - 82399735936 \nu^{19} + 476450116227 \nu^{18} + 996713046369 \nu^{17} - 10100519253055 \nu^{16} + \cdots + 958344304289 ) / 71367422325 \) 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
\(\nu^{3}\)\(=\) \( \beta_{3} + \beta_{2} + 5\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{19} - \beta_{18} + \beta_{17} + \beta_{16} - \beta_{13} + \beta_{12} - \beta_{11} - \beta_{8} + \cdots + 17 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{19} + \beta_{18} + 2 \beta_{17} + \beta_{16} - \beta_{15} + \beta_{12} - \beta_{11} - \beta_{8} + \cdots + 5 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 10 \beta_{19} - 8 \beta_{18} + 14 \beta_{17} + 12 \beta_{16} - \beta_{14} - 10 \beta_{13} + 12 \beta_{12} + \cdots + 109 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 15 \beta_{19} + 10 \beta_{18} + 28 \beta_{17} + 16 \beta_{16} - 13 \beta_{15} - 3 \beta_{13} + 14 \beta_{12} + \cdots + 72 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 84 \beta_{19} - 52 \beta_{18} + 141 \beta_{17} + 111 \beta_{16} - 2 \beta_{15} - 13 \beta_{14} + \cdots + 739 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 157 \beta_{19} + 72 \beta_{18} + 289 \beta_{17} + 183 \beta_{16} - 123 \beta_{15} - 2 \beta_{14} + \cdots + 742 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 675 \beta_{19} - 326 \beta_{18} + 1253 \beta_{17} + 949 \beta_{16} - 46 \beta_{15} - 123 \beta_{14} + \cdots + 5171 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 1435 \beta_{19} + 435 \beta_{18} + 2656 \beta_{17} + 1820 \beta_{16} - 1043 \beta_{15} - 46 \beta_{14} + \cdots + 6739 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 5333 \beta_{19} - 2097 \beta_{18} + 10473 \beta_{17} + 7859 \beta_{16} - 663 \beta_{15} - 1043 \beta_{14} + \cdots + 36934 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 12282 \beta_{19} + 2184 \beta_{18} + 23008 \beta_{17} + 16764 \beta_{16} - 8456 \beta_{15} + \cdots + 57522 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 41781 \beta_{19} - 14252 \beta_{18} + 84641 \beta_{17} + 64047 \beta_{16} - 7749 \beta_{15} + \cdots + 267687 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 101379 \beta_{19} + 7404 \beta_{18} + 192446 \beta_{17} + 147235 \beta_{16} - 67312 \beta_{15} + \cdots + 474097 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( 325740 \beta_{19} - 103149 \beta_{18} + 670748 \beta_{17} + 516913 \beta_{16} - 80585 \beta_{15} + \cdots + 1961583 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( 818924 \beta_{19} - 13491 \beta_{18} + 1573509 \beta_{17} + 1252899 \beta_{16} - 532552 \beta_{15} + \cdots + 3824167 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( 2531957 \beta_{19} - 788669 \beta_{18} + 5252699 \beta_{17} + 4144025 \beta_{16} - 779329 \beta_{15} + \cdots + 14498180 \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( 6526077 \beta_{19} - 613858 \beta_{18} + 12666707 \beta_{17} + 10430435 \beta_{16} - 4211397 \beta_{15} + \cdots + 30415149 \) 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
2.79053
2.70963
2.48729
2.24282
1.72998
1.52695
1.30930
1.24486
0.792696
0.588593
0.0947618
0.0638809
−0.233288
−0.268816
−1.11499
−1.31425
−1.78785
−2.07025
−2.30256
−2.48929
−2.79053 2.20352 5.78708 0.210184 −6.14899 1.09591 −10.5680 1.85549 −0.586525
1.2 −2.70963 0.794731 5.34209 −3.51795 −2.15343 2.08838 −9.05582 −2.36840 9.53234
1.3 −2.48729 −2.76401 4.18660 −3.40389 6.87490 2.38542 −5.43870 4.63978 8.46646
1.4 −2.24282 2.77546 3.03022 1.18159 −6.22483 −3.65660 −2.31060 4.70315 −2.65009
1.5 −1.72998 −0.533809 0.992840 3.73502 0.923480 0.942617 1.74237 −2.71505 −6.46152
1.6 −1.52695 3.12306 0.331561 −4.41658 −4.76874 −1.01999 2.54761 6.75348 6.74387
1.7 −1.30930 −1.02071 −0.285737 −2.41231 1.33641 −3.92996 2.99271 −1.95815 3.15844
1.8 −1.24486 −0.546387 −0.450325 −2.14054 0.680174 −1.13458 3.05031 −2.70146 2.66467
1.9 −0.792696 −0.846838 −1.37163 −0.289760 0.671285 4.67588 2.67268 −2.28287 0.229692
1.10 −0.588593 −2.94379 −1.65356 1.89954 1.73269 −0.717912 2.15046 5.66588 −1.11806
1.11 −0.0947618 1.88046 −1.99102 0.275884 −0.178196 3.16320 0.378196 0.536138 −0.0261433
1.12 −0.0638809 3.09806 −1.99592 −2.16210 −0.197907 −3.19682 0.255263 6.59799 0.138117
1.13 0.233288 −1.96908 −1.94558 1.83554 −0.459363 −3.52264 −0.920454 0.877295 0.428208
1.14 0.268816 0.756198 −1.92774 −3.33003 0.203278 2.89193 −1.05584 −2.42816 −0.895166
1.15 1.11499 −2.44874 −0.756803 −2.94951 −2.73032 −2.81477 −3.07380 2.99634 −3.28867
1.16 1.31425 0.475327 −0.272760 2.70049 0.624696 1.89627 −2.98696 −2.77406 3.54910
1.17 1.78785 2.39996 1.19640 −0.394379 4.29076 −4.52083 −1.43672 2.75981 −0.705090
1.18 2.07025 −0.286146 2.28593 0.477501 −0.592394 −0.790233 0.591954 −2.91812 0.988546
1.19 2.30256 −2.35741 3.30181 −0.231301 −5.42810 −1.95199 2.99749 2.55739 −0.532586
1.20 2.48929 −1.78984 4.19654 −3.06739 −4.45542 4.11672 5.46782 0.203530 −7.63560
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.20
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(13\) \(-1\)
\(19\) \(-1\)

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 3211.2.a.m 20
13.b even 2 1 3211.2.a.n 20
13.f odd 12 2 247.2.p.a ✓ 40
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
247.2.p.a ✓ 40 13.f odd 12 2
3211.2.a.m 20 1.a even 1 1 trivial
3211.2.a.n 20 13.b even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{20} + 6 T_{2}^{19} - 11 T_{2}^{18} - 126 T_{2}^{17} - 50 T_{2}^{16} + 1020 T_{2}^{15} + 1272 T_{2}^{14} + \cdots + 1 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(3211))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{20} + 6 T^{19} + \cdots + 1 \) Copy content Toggle raw display
$3$ \( T^{20} - 40 T^{18} + \cdots - 911 \) Copy content Toggle raw display
$5$ \( T^{20} + 16 T^{19} + \cdots + 541 \) Copy content Toggle raw display
$7$ \( T^{20} + 4 T^{19} + \cdots - 4534847 \) Copy content Toggle raw display
$11$ \( T^{20} + 20 T^{19} + \cdots - 2053199 \) Copy content Toggle raw display
$13$ \( T^{20} \) Copy content Toggle raw display
$17$ \( T^{20} + \cdots + 5093120869 \) Copy content Toggle raw display
$19$ \( (T - 1)^{20} \) Copy content Toggle raw display
$23$ \( T^{20} + \cdots - 13159647539 \) Copy content Toggle raw display
$29$ \( T^{20} + \cdots + 3625304437 \) Copy content Toggle raw display
$31$ \( T^{20} + \cdots - 4263058511783 \) Copy content Toggle raw display
$37$ \( T^{20} + \cdots + 169731533401 \) Copy content Toggle raw display
$41$ \( T^{20} + \cdots + 31827886857 \) Copy content Toggle raw display
$43$ \( T^{20} + \cdots - 24984291564227 \) Copy content Toggle raw display
$47$ \( T^{20} + \cdots + 1611747823009 \) Copy content Toggle raw display
$53$ \( T^{20} + \cdots - 21848293275899 \) Copy content Toggle raw display
$59$ \( T^{20} + \cdots + 89\!\cdots\!81 \) Copy content Toggle raw display
$61$ \( T^{20} + \cdots + 218385344700001 \) Copy content Toggle raw display
$67$ \( T^{20} + \cdots + 20279269051717 \) Copy content Toggle raw display
$71$ \( T^{20} + \cdots + 152661132638029 \) Copy content Toggle raw display
$73$ \( T^{20} + \cdots - 54212001058319 \) Copy content Toggle raw display
$79$ \( T^{20} + \cdots - 479213202993131 \) Copy content Toggle raw display
$83$ \( T^{20} + \cdots - 57653504935559 \) Copy content Toggle raw display
$89$ \( T^{20} + \cdots - 41913723841223 \) Copy content Toggle raw display
$97$ \( T^{20} + \cdots - 39\!\cdots\!47 \) Copy content Toggle raw display
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