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Properties

Label 1407.4.a.a
Level $1407$
Weight $4$
Character orbit 1407.a
Self dual yes
Analytic conductor $83.016$
Analytic rank $1$
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] = [1407,4,Mod(1,1407)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1407, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1407.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1407 = 3 \cdot 7 \cdot 67 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1407.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: \(83.0156873781\)
Analytic rank: \(1\)
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: \( x^{18} - 3 x^{17} - 81 x^{16} + 229 x^{15} + 2672 x^{14} - 7162 x^{13} - 45982 x^{12} + 117906 x^{11} + \cdots + 154368 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{5}\cdot 3^{2} \)
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 \(1,\beta_1,\ldots,\beta_{17}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} + 3 q^{3} + (\beta_{2} + 2) q^{4} + ( - \beta_{6} - 2) q^{5} - 3 \beta_1 q^{6} + 7 q^{7} + ( - \beta_{3} - \beta_{2} + \beta_1 - 2) q^{8} + 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{2} + 3 q^{3} + (\beta_{2} + 2) q^{4} + ( - \beta_{6} - 2) q^{5} - 3 \beta_1 q^{6} + 7 q^{7} + ( - \beta_{3} - \beta_{2} + \beta_1 - 2) q^{8} + 9 q^{9} + (\beta_{16} + \beta_{12} - \beta_{10} + \cdots - 3) q^{10}+ \cdots + ( - 9 \beta_{12} + 18 \beta_{6} + \cdots - 18) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 18 q - 3 q^{2} + 54 q^{3} + 27 q^{4} - 29 q^{5} - 9 q^{6} + 126 q^{7} - 21 q^{8} + 162 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 18 q - 3 q^{2} + 54 q^{3} + 27 q^{4} - 29 q^{5} - 9 q^{6} + 126 q^{7} - 21 q^{8} + 162 q^{9} - 35 q^{10} - 42 q^{11} + 81 q^{12} - 198 q^{13} - 21 q^{14} - 87 q^{15} - 269 q^{16} - 275 q^{17} - 27 q^{18} - 427 q^{19} - 213 q^{20} + 378 q^{21} - 128 q^{22} - 194 q^{23} - 63 q^{24} - 167 q^{25} - 252 q^{26} + 486 q^{27} + 189 q^{28} - 189 q^{29} - 105 q^{30} - 847 q^{31} - 641 q^{32} - 126 q^{33} - 921 q^{34} - 203 q^{35} + 243 q^{36} - 1160 q^{37} - 823 q^{38} - 594 q^{39} + 11 q^{40} - 936 q^{41} - 63 q^{42} - 1551 q^{43} - 334 q^{44} - 261 q^{45} - 1492 q^{46} - 625 q^{47} - 807 q^{48} + 882 q^{49} - 80 q^{50} - 825 q^{51} - 248 q^{52} - 1110 q^{53} - 81 q^{54} - 2138 q^{55} - 147 q^{56} - 1281 q^{57} + 725 q^{58} - 1728 q^{59} - 639 q^{60} - 1264 q^{61} + 1909 q^{62} + 1134 q^{63} - 2065 q^{64} - 310 q^{65} - 384 q^{66} + 1206 q^{67} + 267 q^{68} - 582 q^{69} - 245 q^{70} + 996 q^{71} - 189 q^{72} - 2454 q^{73} + 1446 q^{74} - 501 q^{75} - 1675 q^{76} - 294 q^{77} - 756 q^{78} - 1514 q^{79} + 909 q^{80} + 1458 q^{81} - 3712 q^{82} + 324 q^{83} + 567 q^{84} - 2106 q^{85} + 1481 q^{86} - 567 q^{87} - 1048 q^{88} - 2055 q^{89} - 315 q^{90} - 1386 q^{91} + 2090 q^{92} - 2541 q^{93} + 435 q^{94} + 102 q^{95} - 1923 q^{96} - 4275 q^{97} - 147 q^{98} - 378 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{18} - 3 x^{17} - 81 x^{16} + 229 x^{15} + 2672 x^{14} - 7162 x^{13} - 45982 x^{12} + 117906 x^{11} + \cdots + 154368 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 10 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} - \nu^{2} - 15\nu + 8 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 10\!\cdots\!69 \nu^{17} + \cdots + 93\!\cdots\!60 ) / 10\!\cdots\!32 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 85741135522163 \nu^{17} - 350085946096595 \nu^{16} + \cdots - 87\!\cdots\!84 ) / 64\!\cdots\!52 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 537404593688467 \nu^{17} + \cdots - 39\!\cdots\!12 ) / 38\!\cdots\!12 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 13\!\cdots\!83 \nu^{17} + 849964248327636 \nu^{16} + \cdots - 80\!\cdots\!20 ) / 77\!\cdots\!24 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 30\!\cdots\!39 \nu^{17} + \cdots - 64\!\cdots\!36 ) / 15\!\cdots\!48 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 11\!\cdots\!99 \nu^{17} + \cdots - 27\!\cdots\!12 ) / 51\!\cdots\!16 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 39\!\cdots\!73 \nu^{17} + \cdots + 68\!\cdots\!72 ) / 15\!\cdots\!48 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 11\!\cdots\!65 \nu^{17} + \cdots - 10\!\cdots\!44 ) / 38\!\cdots\!12 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 728840649264811 \nu^{17} + \cdots + 95\!\cdots\!24 ) / 22\!\cdots\!64 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 10\!\cdots\!93 \nu^{17} + \cdots - 49\!\cdots\!60 ) / 30\!\cdots\!96 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 48\!\cdots\!79 \nu^{17} + \cdots - 52\!\cdots\!96 ) / 10\!\cdots\!32 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 27\!\cdots\!13 \nu^{17} - 265310450730978 \nu^{16} + \cdots - 12\!\cdots\!68 ) / 51\!\cdots\!16 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( 21\!\cdots\!95 \nu^{17} + \cdots - 86\!\cdots\!08 ) / 38\!\cdots\!12 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( - 93\!\cdots\!27 \nu^{17} + \cdots - 60\!\cdots\!24 ) / 15\!\cdots\!48 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 10 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} + \beta_{2} + 15\beta _1 + 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{16} + 2 \beta_{15} - \beta_{14} + \beta_{13} - \beta_{11} + \beta_{10} + \beta_{9} - \beta_{7} + \cdots + 160 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - \beta_{17} + \beta_{16} + \beta_{15} - 3 \beta_{14} + \beta_{13} + \beta_{11} + 4 \beta_{6} + \cdots + 105 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - \beta_{17} + 32 \beta_{16} + 61 \beta_{15} - 36 \beta_{14} + 34 \beta_{13} - 2 \beta_{12} + \cdots + 2933 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 36 \beta_{17} + 42 \beta_{16} + 58 \beta_{15} - 126 \beta_{14} + 52 \beta_{13} - 8 \beta_{12} + \cdots + 3642 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 44 \beta_{17} + 813 \beta_{16} + 1472 \beta_{15} - 981 \beta_{14} + 889 \beta_{13} - 54 \beta_{12} + \cdots + 57842 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 945 \beta_{17} + 1389 \beta_{16} + 2051 \beta_{15} - 3831 \beta_{14} + 1785 \beta_{13} + \cdots + 107411 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 1277 \beta_{17} + 19280 \beta_{16} + 33191 \beta_{15} - 24356 \beta_{14} + 21354 \beta_{13} + \cdots + 1196155 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 22148 \beta_{17} + 41562 \beta_{16} + 60688 \beta_{15} - 103170 \beta_{14} + 52128 \beta_{13} + \cdots + 2916872 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 31996 \beta_{17} + 447209 \beta_{16} + 732538 \beta_{15} - 583077 \beta_{14} + 496109 \beta_{13} + \cdots + 25584872 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 491485 \beta_{17} + 1169573 \beta_{16} + 1647489 \beta_{15} - 2620783 \beta_{14} + 1405029 \beta_{13} + \cdots + 75496657 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 750781 \beta_{17} + 10337112 \beta_{16} + 16099301 \beta_{15} - 13764924 \beta_{14} + \cdots + 560980245 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 10558164 \beta_{17} + 31563906 \beta_{16} + 42575414 \beta_{15} - 64500478 \beta_{14} + \cdots + 1895985814 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( - 16970612 \beta_{17} + 239988389 \beta_{16} + 354797668 \beta_{15} - 323515381 \beta_{14} + \cdots + 12528086454 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( - 221615841 \beta_{17} + 827066237 \beta_{16} + 1066350167 \beta_{15} - 1558645415 \beta_{14} + \cdots + 46679323679 \) 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
4.87556
4.80945
3.66748
3.45661
2.89995
2.20184
2.09177
1.32725
0.409543
0.0735288
−0.477756
−1.25427
−2.47592
−2.52654
−3.66607
−4.06577
−4.09471
−4.25195
−4.87556 3.00000 15.7711 −0.0559498 −14.6267 7.00000 −37.8885 9.00000 0.272787
1.2 −4.80945 3.00000 15.1309 −4.82711 −14.4284 7.00000 −34.2955 9.00000 23.2157
1.3 −3.66748 3.00000 5.45039 −9.56910 −11.0024 7.00000 9.35065 9.00000 35.0945
1.4 −3.45661 3.00000 3.94816 18.1196 −10.3698 7.00000 14.0056 9.00000 −62.6325
1.5 −2.89995 3.00000 0.409719 −17.8946 −8.69985 7.00000 22.0114 9.00000 51.8936
1.6 −2.20184 3.00000 −3.15189 −1.41870 −6.60553 7.00000 24.5547 9.00000 3.12374
1.7 −2.09177 3.00000 −3.62448 2.65299 −6.27532 7.00000 24.3158 9.00000 −5.54946
1.8 −1.32725 3.00000 −6.23840 12.4049 −3.98176 7.00000 18.8980 9.00000 −16.4644
1.9 −0.409543 3.00000 −7.83227 −16.5024 −1.22863 7.00000 6.48400 9.00000 6.75845
1.10 −0.0735288 3.00000 −7.99459 4.47838 −0.220586 7.00000 1.17606 9.00000 −0.329290
1.11 0.477756 3.00000 −7.77175 5.29671 1.43327 7.00000 −7.53505 9.00000 2.53053
1.12 1.25427 3.00000 −6.42681 −12.6198 3.76280 7.00000 −18.0951 9.00000 −15.8286
1.13 2.47592 3.00000 −1.86980 1.04456 7.42777 7.00000 −24.4369 9.00000 2.58625
1.14 2.52654 3.00000 −1.61657 16.9029 7.57963 7.00000 −24.2967 9.00000 42.7059
1.15 3.66607 3.00000 5.44010 −18.2124 10.9982 7.00000 −9.38479 9.00000 −66.7678
1.16 4.06577 3.00000 8.53047 −5.23853 12.1973 7.00000 2.15678 9.00000 −21.2987
1.17 4.09471 3.00000 8.76665 −5.29380 12.2841 7.00000 3.13922 9.00000 −21.6766
1.18 4.25195 3.00000 10.0791 1.73238 12.7559 7.00000 8.84030 9.00000 7.36600
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.18
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(7\) \(-1\)
\(67\) \(-1\)

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 1407.4.a.a ✓ 18
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1407.4.a.a ✓ 18 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{18} + 3 T_{2}^{17} - 81 T_{2}^{16} - 229 T_{2}^{15} + 2672 T_{2}^{14} + 7162 T_{2}^{13} + \cdots + 154368 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1407))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{18} + 3 T^{17} + \cdots + 154368 \) Copy content Toggle raw display
$3$ \( (T - 3)^{18} \) Copy content Toggle raw display
$5$ \( T^{18} + \cdots + 2985759304704 \) Copy content Toggle raw display
$7$ \( (T - 7)^{18} \) Copy content Toggle raw display
$11$ \( T^{18} + \cdots - 29\!\cdots\!92 \) Copy content Toggle raw display
$13$ \( T^{18} + \cdots - 25\!\cdots\!04 \) Copy content Toggle raw display
$17$ \( T^{18} + \cdots + 18\!\cdots\!32 \) Copy content Toggle raw display
$19$ \( T^{18} + \cdots + 13\!\cdots\!72 \) Copy content Toggle raw display
$23$ \( T^{18} + \cdots - 62\!\cdots\!44 \) Copy content Toggle raw display
$29$ \( T^{18} + \cdots + 61\!\cdots\!80 \) Copy content Toggle raw display
$31$ \( T^{18} + \cdots - 72\!\cdots\!40 \) Copy content Toggle raw display
$37$ \( T^{18} + \cdots - 26\!\cdots\!56 \) Copy content Toggle raw display
$41$ \( T^{18} + \cdots - 18\!\cdots\!16 \) Copy content Toggle raw display
$43$ \( T^{18} + \cdots + 89\!\cdots\!24 \) Copy content Toggle raw display
$47$ \( T^{18} + \cdots + 21\!\cdots\!60 \) Copy content Toggle raw display
$53$ \( T^{18} + \cdots - 12\!\cdots\!84 \) Copy content Toggle raw display
$59$ \( T^{18} + \cdots - 20\!\cdots\!28 \) Copy content Toggle raw display
$61$ \( T^{18} + \cdots + 70\!\cdots\!56 \) Copy content Toggle raw display
$67$ \( (T - 67)^{18} \) Copy content Toggle raw display
$71$ \( T^{18} + \cdots - 20\!\cdots\!24 \) Copy content Toggle raw display
$73$ \( T^{18} + \cdots - 23\!\cdots\!12 \) Copy content Toggle raw display
$79$ \( T^{18} + \cdots + 23\!\cdots\!12 \) Copy content Toggle raw display
$83$ \( T^{18} + \cdots + 57\!\cdots\!52 \) Copy content Toggle raw display
$89$ \( T^{18} + \cdots + 15\!\cdots\!40 \) Copy content Toggle raw display
$97$ \( T^{18} + \cdots - 19\!\cdots\!32 \) Copy content Toggle raw display
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