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Properties

Label 3201.2.a.t
Level $3201$
Weight $2$
Character orbit 3201.a
Self dual yes
Analytic conductor $25.560$
Analytic rank $0$
Dimension $17$
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] = [3201,2,Mod(1,3201)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3201, 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("3201.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3201 = 3 \cdot 11 \cdot 97 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3201.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: \(25.5601136870\)
Analytic rank: \(0\)
Dimension: \(17\)
Coefficient field: \(\mathbb{Q}[x]/(x^{17} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{17} - 3 x^{16} - 22 x^{15} + 71 x^{14} + 181 x^{13} - 662 x^{12} - 663 x^{11} + 3095 x^{10} + \cdots - 54 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
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 \(1,\beta_1,\ldots,\beta_{16}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} - q^{3} + (\beta_{2} + 1) q^{4} + ( - \beta_{5} - 1) q^{5} + \beta_1 q^{6} + (\beta_{6} + 1) q^{7} + ( - \beta_{3} - \beta_1) q^{8} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{2} - q^{3} + (\beta_{2} + 1) q^{4} + ( - \beta_{5} - 1) q^{5} + \beta_1 q^{6} + (\beta_{6} + 1) q^{7} + ( - \beta_{3} - \beta_1) q^{8} + q^{9} + ( - \beta_{13} - \beta_{10} + \beta_1 + 1) q^{10} - q^{11} + ( - \beta_{2} - 1) q^{12} + \beta_{16} q^{13} + (\beta_{14} + \beta_{12} + 1) q^{14} + (\beta_{5} + 1) q^{15} + (\beta_{14} - \beta_{11} + \cdots + \beta_{2}) q^{16}+ \cdots - q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 17 q - 3 q^{2} - 17 q^{3} + 19 q^{4} - 10 q^{5} + 3 q^{6} + 9 q^{7} + 17 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 17 q - 3 q^{2} - 17 q^{3} + 19 q^{4} - 10 q^{5} + 3 q^{6} + 9 q^{7} + 17 q^{9} + 17 q^{10} - 17 q^{11} - 19 q^{12} - 2 q^{13} + 10 q^{14} + 10 q^{15} + 15 q^{16} - 2 q^{17} - 3 q^{18} + 25 q^{19} - 32 q^{20} - 9 q^{21} + 3 q^{22} - 3 q^{23} + 35 q^{25} - 7 q^{26} - 17 q^{27} + q^{28} + 4 q^{29} - 17 q^{30} + 44 q^{31} - 3 q^{32} + 17 q^{33} + 22 q^{34} + 4 q^{35} + 19 q^{36} + 13 q^{37} - 2 q^{38} + 2 q^{39} + 41 q^{40} + 5 q^{41} - 10 q^{42} + 27 q^{43} - 19 q^{44} - 10 q^{45} - 29 q^{46} - 10 q^{47} - 15 q^{48} + 12 q^{49} - 2 q^{50} + 2 q^{51} + 32 q^{52} - 45 q^{53} + 3 q^{54} + 10 q^{55} + 35 q^{56} - 25 q^{57} + 17 q^{58} + q^{59} + 32 q^{60} + 22 q^{61} + 23 q^{62} + 9 q^{63} + 24 q^{64} + 23 q^{65} - 3 q^{66} + 7 q^{67} - 10 q^{68} + 3 q^{69} + 40 q^{70} - 12 q^{71} - 21 q^{73} + 45 q^{74} - 35 q^{75} + 34 q^{76} - 9 q^{77} + 7 q^{78} + 24 q^{79} - 50 q^{80} + 17 q^{81} - 6 q^{82} - 26 q^{83} - q^{84} - 43 q^{85} + 9 q^{86} - 4 q^{87} - 40 q^{89} + 17 q^{90} + 38 q^{91} + 47 q^{92} - 44 q^{93} + 52 q^{94} - 12 q^{95} + 3 q^{96} + 17 q^{97} + 15 q^{98} - 17 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{17} - 3 x^{16} - 22 x^{15} + 71 x^{14} + 181 x^{13} - 662 x^{12} - 663 x^{11} + 3095 x^{10} + \cdots - 54 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} - 5\nu \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 3868993 \nu^{16} + 17888814 \nu^{15} - 120311788 \nu^{14} - 431403421 \nu^{13} + \cdots + 2553275862 ) / 246693672 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 4154231 \nu^{16} + 19263396 \nu^{15} + 94634558 \nu^{14} - 451993675 \nu^{13} + \cdots + 134881938 ) / 246693672 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 2251682 \nu^{16} + 704319 \nu^{15} + 61599767 \nu^{14} - 17386345 \nu^{13} - 680902028 \nu^{12} + \cdots - 116916606 ) / 123346836 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 722987 \nu^{16} + 384156 \nu^{15} + 18581333 \nu^{14} - 9874372 \nu^{13} - 192932606 \nu^{12} + \cdots - 274721256 ) / 30836709 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 2469213 \nu^{16} + 3579016 \nu^{15} + 65485350 \nu^{14} - 86488837 \nu^{13} - 708784696 \nu^{12} + \cdots - 611714586 ) / 82231224 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 12112723 \nu^{16} + 26840814 \nu^{15} + 275268640 \nu^{14} - 631769033 \nu^{13} + \cdots + 300249558 ) / 246693672 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 6169951 \nu^{16} - 9709773 \nu^{15} - 154524667 \nu^{14} + 221922488 \nu^{13} + 1549650424 \nu^{12} + \cdots + 459832320 ) / 123346836 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 3311506 \nu^{16} - 6563400 \nu^{15} - 78240037 \nu^{14} + 150856094 \nu^{13} + 724526131 \nu^{12} + \cdots - 483686766 ) / 61673418 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 16388747 \nu^{16} + 30876030 \nu^{15} + 410299844 \nu^{14} - 725268001 \nu^{13} + \cdots - 2316595842 ) / 246693672 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 19140605 \nu^{16} + 16178070 \nu^{15} + 466092608 \nu^{14} - 368048563 \nu^{13} + \cdots - 448642494 ) / 246693672 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 28490201 \nu^{16} - 55001556 \nu^{15} - 695265998 \nu^{14} + 1271963173 \nu^{13} + \cdots + 2313083826 ) / 246693672 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 23152049 \nu^{16} + 35548374 \nu^{15} + 544739882 \nu^{14} - 824956999 \nu^{13} + \cdots + 114422634 ) / 123346836 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( - 50659655 \nu^{16} + 55015212 \nu^{15} + 1240624370 \nu^{14} - 1266731371 \nu^{13} + \cdots - 3211160310 ) / 246693672 \) 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
\(\nu^{3}\)\(=\) \( \beta_{3} + 5\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{14} - \beta_{11} - \beta_{10} + \beta_{8} - \beta_{6} + 7\beta_{2} + 14 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{14} + \beta_{13} + \beta_{12} + \beta_{11} - \beta_{10} - \beta_{7} - \beta_{6} + \beta_{5} + \cdots + 28 \beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( \beta_{16} + 11 \beta_{14} - 2 \beta_{13} - \beta_{12} - 10 \beta_{11} - 11 \beta_{10} + \beta_{9} + \cdots + 78 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 2 \beta_{16} - \beta_{15} + 12 \beta_{14} + 15 \beta_{13} + 11 \beta_{12} + 17 \beta_{11} + \cdots + 169 \beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 13 \beta_{16} + 94 \beta_{14} - 25 \beta_{13} - 16 \beta_{12} - 80 \beta_{11} - 94 \beta_{10} + \cdots + 478 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 30 \beta_{16} - 15 \beta_{15} + 107 \beta_{14} + 158 \beta_{13} + 89 \beta_{12} + 190 \beta_{11} + \cdots - 3 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 123 \beta_{16} - 2 \beta_{15} + 738 \beta_{14} - 221 \beta_{13} - 182 \beta_{12} - 594 \beta_{11} + \cdots + 3098 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 309 \beta_{16} - 152 \beta_{15} + 865 \beta_{14} + 1444 \beta_{13} + 639 \beta_{12} + 1792 \beta_{11} + \cdots - 49 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 1043 \beta_{16} - 44 \beta_{15} + 5595 \beta_{14} - 1707 \beta_{13} - 1786 \beta_{12} - 4258 \beta_{11} + \cdots + 20811 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 2750 \beta_{16} - 1320 \beta_{15} + 6728 \beta_{14} + 12260 \beta_{13} + 4302 \beta_{12} + 15489 \beta_{11} + \cdots - 492 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 8439 \beta_{16} - 612 \beta_{15} + 41786 \beta_{14} - 12300 \beta_{13} - 16108 \beta_{12} + \cdots + 143252 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 22819 \beta_{16} - 10650 \beta_{15} + 51550 \beta_{14} + 99671 \beta_{13} + 27734 \beta_{12} + \cdots - 3673 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( 66741 \beta_{16} - 6900 \beta_{15} + 310227 \beta_{14} - 84851 \beta_{13} - 137624 \beta_{12} + \cdots + 1003283 \) 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
2.76004
2.44987
2.08308
2.03494
1.70642
1.45373
1.21807
0.860499
0.307641
−0.187677
−0.407673
−0.927557
−1.34659
−1.81311
−2.12570
−2.38654
−2.67946
−2.76004 −1.00000 5.61780 −4.35547 2.76004 −1.40363 −9.98526 1.00000 12.0212
1.2 −2.44987 −1.00000 4.00188 −0.847477 2.44987 −3.08344 −4.90436 1.00000 2.07621
1.3 −2.08308 −1.00000 2.33921 −3.02334 2.08308 4.49625 −0.706607 1.00000 6.29785
1.4 −2.03494 −1.00000 2.14099 1.82975 2.03494 2.92375 −0.286914 1.00000 −3.72344
1.5 −1.70642 −1.00000 0.911872 1.67638 1.70642 −1.88863 1.85680 1.00000 −2.86060
1.6 −1.45373 −1.00000 0.113334 −2.86818 1.45373 1.27700 2.74271 1.00000 4.16956
1.7 −1.21807 −1.00000 −0.516295 −1.00665 1.21807 −0.856442 3.06503 1.00000 1.22617
1.8 −0.860499 −1.00000 −1.25954 −1.02546 0.860499 −2.10368 2.80483 1.00000 0.882406
1.9 −0.307641 −1.00000 −1.90536 3.24352 0.307641 0.497434 1.20145 1.00000 −0.997839
1.10 0.187677 −1.00000 −1.96478 −3.94913 −0.187677 2.55781 −0.744097 1.00000 −0.741160
1.11 0.407673 −1.00000 −1.83380 −1.15944 −0.407673 4.78856 −1.56294 1.00000 −0.472671
1.12 0.927557 −1.00000 −1.13964 2.43349 −0.927557 −3.10656 −2.91219 1.00000 2.25720
1.13 1.34659 −1.00000 −0.186709 3.61308 −1.34659 3.66792 −2.94459 1.00000 4.86532
1.14 1.81311 −1.00000 1.28737 −2.88104 −1.81311 −3.80494 −1.29208 1.00000 −5.22365
1.15 2.12570 −1.00000 2.51862 −4.05693 −2.12570 0.497891 1.10243 1.00000 −8.62384
1.16 2.38654 −1.00000 3.69555 1.78052 −2.38654 3.39776 4.04650 1.00000 4.24927
1.17 2.67946 −1.00000 5.17948 0.596376 −2.67946 1.14295 8.51928 1.00000 1.59796
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.17
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(1\)
\(11\) \(1\)
\(97\) \(-1\)

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 3201.2.a.t ✓ 17
3.b odd 2 1 9603.2.a.z 17
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3201.2.a.t ✓ 17 1.a even 1 1 trivial
9603.2.a.z 17 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(3201))\):

\( T_{2}^{17} + 3 T_{2}^{16} - 22 T_{2}^{15} - 71 T_{2}^{14} + 181 T_{2}^{13} + 662 T_{2}^{12} - 663 T_{2}^{11} + \cdots + 54 \) Copy content Toggle raw display
\( T_{5}^{17} + 10 T_{5}^{16} - 10 T_{5}^{15} - 374 T_{5}^{14} - 551 T_{5}^{13} + 5263 T_{5}^{12} + \cdots - 164248 \) Copy content Toggle raw display
\( T_{7}^{17} - 9 T_{7}^{16} - 25 T_{7}^{15} + 401 T_{7}^{14} - 81 T_{7}^{13} - 6990 T_{7}^{12} + \cdots + 126272 \) Copy content Toggle raw display
\( T_{17}^{17} + 2 T_{17}^{16} - 146 T_{17}^{15} - 309 T_{17}^{14} + 8198 T_{17}^{13} + 18287 T_{17}^{12} + \cdots + 63839232 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{17} + 3 T^{16} + \cdots + 54 \) Copy content Toggle raw display
$3$ \( (T + 1)^{17} \) Copy content Toggle raw display
$5$ \( T^{17} + 10 T^{16} + \cdots - 164248 \) Copy content Toggle raw display
$7$ \( T^{17} - 9 T^{16} + \cdots + 126272 \) Copy content Toggle raw display
$11$ \( (T + 1)^{17} \) Copy content Toggle raw display
$13$ \( T^{17} + 2 T^{16} + \cdots + 6912 \) Copy content Toggle raw display
$17$ \( T^{17} + 2 T^{16} + \cdots + 63839232 \) Copy content Toggle raw display
$19$ \( T^{17} - 25 T^{16} + \cdots + 10821312 \) Copy content Toggle raw display
$23$ \( T^{17} + 3 T^{16} + \cdots + 39716096 \) Copy content Toggle raw display
$29$ \( T^{17} - 4 T^{16} + \cdots - 4686336 \) Copy content Toggle raw display
$31$ \( T^{17} + \cdots - 1509705485568 \) Copy content Toggle raw display
$37$ \( T^{17} + \cdots - 90835686656 \) Copy content Toggle raw display
$41$ \( T^{17} + \cdots + 4872923136 \) Copy content Toggle raw display
$43$ \( T^{17} + \cdots - 75440570944 \) Copy content Toggle raw display
$47$ \( T^{17} + \cdots + 14369118336 \) Copy content Toggle raw display
$53$ \( T^{17} + \cdots + 6264396252032 \) Copy content Toggle raw display
$59$ \( T^{17} + \cdots + 154775505159072 \) Copy content Toggle raw display
$61$ \( T^{17} + \cdots + 17815413772864 \) Copy content Toggle raw display
$67$ \( T^{17} + \cdots - 26966098145776 \) Copy content Toggle raw display
$71$ \( T^{17} + \cdots - 27514153781408 \) Copy content Toggle raw display
$73$ \( T^{17} + \cdots + 467719374423488 \) Copy content Toggle raw display
$79$ \( T^{17} + \cdots + 2547922111296 \) Copy content Toggle raw display
$83$ \( T^{17} + \cdots + 1299510191224 \) Copy content Toggle raw display
$89$ \( T^{17} + \cdots + 1592011739136 \) Copy content Toggle raw display
$97$ \( (T - 1)^{17} \) Copy content Toggle raw display
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