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Properties

Label 65.4.e.a
Level $65$
Weight $4$
Character orbit 65.e
Analytic conductor $3.835$
Analytic rank $0$
Dimension $14$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [65,4,Mod(16,65)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(65, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 2]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("65.16");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 65 = 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 65.e (of order \(3\), degree \(2\), minimal)

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: no
Analytic conductor: \(3.83512415037\)
Analytic rank: \(0\)
Dimension: \(14\)
Relative dimension: \(7\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{14} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{14} - 2 x^{13} + 45 x^{12} - 52 x^{11} + 1311 x^{10} - 1336 x^{9} + 20343 x^{8} - 11166 x^{7} + 216447 x^{6} - 107836 x^{5} + 1201133 x^{4} + 4392 x^{3} + \cdots + 1157776 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{4}\cdot 3 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{13}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} + (\beta_{11} - \beta_{6}) q^{3} + (\beta_{13} - 4 \beta_{6} - \beta_{5} - 4) q^{4} - 5 q^{5} + (\beta_{13} - \beta_{11} + \beta_{7} - 3 \beta_{6} - \beta_{5} - \beta_{4} + \beta_{3} - 2 \beta_{2} - 2 \beta_1 - 3) q^{6} + (2 \beta_{12} - \beta_{10} + \beta_{9} + \beta_{8} + \beta_{4} - \beta_{3} + \beta_{2} + \beta_1) q^{7} + (\beta_{9} + \beta_{8} - \beta_{7} + \beta_{5} - \beta_{2} + 2) q^{8} + ( - \beta_{13} + 2 \beta_{12} + 3 \beta_{11} + \beta_{9} - 3 \beta_{7} - 12 \beta_{6} + \cdots - 12) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{2} + (\beta_{11} - \beta_{6}) q^{3} + (\beta_{13} - 4 \beta_{6} - \beta_{5} - 4) q^{4} - 5 q^{5} + (\beta_{13} - \beta_{11} + \beta_{7} - 3 \beta_{6} - \beta_{5} - \beta_{4} + \beta_{3} - 2 \beta_{2} - 2 \beta_1 - 3) q^{6} + (2 \beta_{12} - \beta_{10} + \beta_{9} + \beta_{8} + \beta_{4} - \beta_{3} + \beta_{2} + \beta_1) q^{7} + (\beta_{9} + \beta_{8} - \beta_{7} + \beta_{5} - \beta_{2} + 2) q^{8} + ( - \beta_{13} + 2 \beta_{12} + 3 \beta_{11} + \beta_{9} - 3 \beta_{7} - 12 \beta_{6} + \cdots - 12) q^{9}+ \cdots + ( - 17 \beta_{12} + 29 \beta_{9} - 7 \beta_{8} + 79 \beta_{7} + 40 \beta_{5} + \cdots + 853) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 14 q - 2 q^{2} + 4 q^{3} - 30 q^{4} - 70 q^{5} - 23 q^{6} - 7 q^{7} + 42 q^{8} - 87 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 14 q - 2 q^{2} + 4 q^{3} - 30 q^{4} - 70 q^{5} - 23 q^{6} - 7 q^{7} + 42 q^{8} - 87 q^{9} + 10 q^{10} - 87 q^{11} - 158 q^{12} + 123 q^{13} + 132 q^{14} - 20 q^{15} + 134 q^{16} + 114 q^{17} + 414 q^{18} - 245 q^{19} + 150 q^{20} - 76 q^{21} - 338 q^{22} + 74 q^{23} - 334 q^{24} + 350 q^{25} + 243 q^{26} - 884 q^{27} - 230 q^{28} + 88 q^{29} + 115 q^{30} + 1000 q^{31} - 80 q^{32} + 194 q^{33} + 854 q^{34} + 35 q^{35} - 425 q^{36} - 633 q^{37} - 596 q^{38} + 970 q^{39} - 210 q^{40} - 162 q^{41} + 1439 q^{42} + 280 q^{43} + 440 q^{44} + 435 q^{45} + 11 q^{46} + 950 q^{47} - 2281 q^{48} - 1694 q^{49} - 50 q^{50} - 860 q^{51} - 956 q^{52} - 1206 q^{53} - 51 q^{54} + 435 q^{55} + 1277 q^{56} + 916 q^{57} + 1213 q^{58} - 1410 q^{59} + 790 q^{60} - 412 q^{61} + 56 q^{62} - 1241 q^{63} - 2358 q^{64} - 615 q^{65} + 4346 q^{66} - 1398 q^{67} + 493 q^{68} - 1080 q^{69} - 660 q^{70} + 584 q^{71} - 1545 q^{72} + 5076 q^{73} - 3840 q^{74} + 100 q^{75} - 3292 q^{76} - 5506 q^{77} + 1179 q^{78} + 928 q^{79} - 670 q^{80} + 473 q^{81} + 1583 q^{82} + 932 q^{83} + 3081 q^{84} - 570 q^{85} + 9858 q^{86} + 282 q^{87} - 3389 q^{88} - 443 q^{89} - 2070 q^{90} + 487 q^{91} + 6182 q^{92} + 2116 q^{93} - 2017 q^{94} + 1225 q^{95} + 954 q^{96} + 1870 q^{97} - 1364 q^{98} + 11378 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{14} - 2 x^{13} + 45 x^{12} - 52 x^{11} + 1311 x^{10} - 1336 x^{9} + 20343 x^{8} - 11166 x^{7} + 216447 x^{6} - 107836 x^{5} + 1201133 x^{4} + 4392 x^{3} + \cdots + 1157776 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 3326902201366 \nu^{13} + \cdots - 34\!\cdots\!32 ) / 56\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 52\!\cdots\!09 \nu^{13} + \cdots - 26\!\cdots\!84 ) / 30\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 11\!\cdots\!27 \nu^{13} + \cdots - 84\!\cdots\!28 ) / 30\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 27\!\cdots\!33 \nu^{13} + \cdots - 68\!\cdots\!76 ) / 56\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 80\!\cdots\!83 \nu^{13} + \cdots - 24\!\cdots\!72 ) / 15\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 21\!\cdots\!61 \nu^{13} + \cdots - 31\!\cdots\!24 ) / 37\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 65\!\cdots\!50 \nu^{13} + \cdots - 18\!\cdots\!20 ) / 10\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 24\!\cdots\!43 \nu^{13} + \cdots + 32\!\cdots\!92 ) / 33\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 32\!\cdots\!23 \nu^{13} + \cdots - 30\!\cdots\!96 ) / 30\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 12\!\cdots\!32 \nu^{13} + \cdots - 41\!\cdots\!12 ) / 30\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 15\!\cdots\!52 \nu^{13} + \cdots - 36\!\cdots\!60 ) / 26\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 88\!\cdots\!19 \nu^{13} + \cdots - 29\!\cdots\!68 ) / 15\!\cdots\!20 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{13} - 12\beta_{6} - \beta_{5} - 12 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{9} - \beta_{8} + \beta_{7} - \beta_{5} + 17\beta_{2} - 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -25\beta_{13} - \beta_{12} + 8\beta_{11} + \beta_{10} - \beta_{9} + 200\beta_{6} - \beta_1 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 34 \beta_{13} - 2 \beta_{12} + 36 \beta_{11} - 27 \beta_{10} - \beta_{9} + 27 \beta_{8} - 36 \beta_{7} + 80 \beta_{6} + 34 \beta_{5} + 6 \beta_{4} - 6 \beta_{3} - 322 \beta_{2} - 322 \beta _1 + 80 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -23\beta_{12} + 9\beta_{9} - 20\beta_{8} - 307\beta_{7} + 590\beta_{5} + 6\beta_{4} - 55\beta_{2} + 3758 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 1000 \beta_{13} + 20 \beta_{12} - 1050 \beta_{11} + 630 \beta_{10} + 928 \beta_{9} - 2572 \beta_{6} + 258 \beta_{3} + 6477 \beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 13717 \beta_{13} + 824 \beta_{12} - 8848 \beta_{11} - 260 \beta_{10} + 412 \beta_{9} + 260 \beta_{8} + 8848 \beta_{7} - 75872 \beta_{6} - 13717 \beta_{5} - 360 \beta_{4} + 360 \beta_{3} + 2340 \beta_{2} + \cdots - 75872 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 204 \beta_{12} - 22313 \beta_{9} - 14237 \beta_{8} + 28673 \beta_{7} - 27617 \beta_{5} - 7872 \beta_{4} + 135469 \beta_{2} - 76778 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 317765 \beta_{13} - 6773 \beta_{12} + 230140 \beta_{11} + 857 \beta_{10} - 26513 \beta_{9} + 1604200 \beta_{6} - 13824 \beta_{3} - 83517 \beta_1 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 736038 \beta_{13} + 2270 \beta_{12} + 755784 \beta_{11} - 319479 \beta_{10} + 1135 \beta_{9} + 319479 \beta_{8} - 755784 \beta_{7} + 2178976 \beta_{6} + 736038 \beta_{5} + 210678 \beta_{4} + \cdots + 2178976 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 106531 \beta_{12} + 643321 \beta_{9} + 97080 \beta_{8} - 5715939 \beta_{7} + 7367150 \beta_{5} + 439710 \beta_{4} - 2663275 \beta_{2} + 34980918 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 19166580 \beta_{13} - 129568 \beta_{12} - 19461158 \beta_{11} + 7172990 \beta_{10} + 12213120 \beta_{9} - 59610892 \beta_{6} + 5299266 \beta_{3} + 63856489 \beta_1 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/65\mathbb{Z}\right)^\times\).

\(n\) \(27\) \(41\)
\(\chi(n)\) \(1\) \(-1 - \beta_{6}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
16.1
2.45261 + 4.24804i
1.92689 + 3.33746i
1.48434 + 2.57096i
0.272699 + 0.472328i
−1.08655 − 1.88197i
−1.78789 − 3.09671i
−2.26209 − 3.91806i
2.45261 − 4.24804i
1.92689 − 3.33746i
1.48434 − 2.57096i
0.272699 − 0.472328i
−1.08655 + 1.88197i
−1.78789 + 3.09671i
−2.26209 + 3.91806i
−2.45261 − 4.24804i −3.39607 − 5.88216i −8.03055 + 13.9093i −5.00000 −16.6584 + 28.8532i 2.60104 − 4.50514i 39.5414 −9.56652 + 16.5697i 12.2630 + 21.2402i
16.2 −1.92689 − 3.33746i 4.40164 + 7.62386i −3.42577 + 5.93361i −5.00000 16.9629 − 29.3806i −17.9862 + 31.1531i −4.42588 −25.2488 + 43.7322i 9.63443 + 16.6873i
16.3 −1.48434 − 2.57096i 1.47367 + 2.55247i −0.406551 + 0.704167i −5.00000 4.37486 − 7.57749i 13.2000 − 22.8630i −21.3357 9.15660 − 15.8597i 7.42172 + 12.8548i
16.4 −0.272699 − 0.472328i −3.51122 − 6.08161i 3.85127 − 6.67060i −5.00000 −1.91501 + 3.31689i −11.4358 + 19.8074i −8.56412 −11.1573 + 19.3250i 1.36349 + 2.36164i
16.5 1.08655 + 1.88197i −1.99894 − 3.46226i 1.63880 − 2.83848i −5.00000 4.34391 − 7.52387i 10.9938 − 19.0418i 24.5074 5.50850 − 9.54100i −5.43277 − 9.40984i
16.6 1.78789 + 3.09671i 4.37841 + 7.58363i −2.39307 + 4.14492i −5.00000 −15.6562 + 27.1173i 11.1225 − 19.2648i 11.4920 −24.8410 + 43.0258i −8.93943 − 15.4835i
16.7 2.26209 + 3.91806i 0.652503 + 1.13017i −6.23413 + 10.7978i −5.00000 −2.95204 + 5.11309i −11.9953 + 20.7765i −20.2152 12.6485 − 21.9078i −11.3105 − 19.5903i
61.1 −2.45261 + 4.24804i −3.39607 + 5.88216i −8.03055 − 13.9093i −5.00000 −16.6584 − 28.8532i 2.60104 + 4.50514i 39.5414 −9.56652 − 16.5697i 12.2630 − 21.2402i
61.2 −1.92689 + 3.33746i 4.40164 − 7.62386i −3.42577 − 5.93361i −5.00000 16.9629 + 29.3806i −17.9862 − 31.1531i −4.42588 −25.2488 − 43.7322i 9.63443 − 16.6873i
61.3 −1.48434 + 2.57096i 1.47367 − 2.55247i −0.406551 − 0.704167i −5.00000 4.37486 + 7.57749i 13.2000 + 22.8630i −21.3357 9.15660 + 15.8597i 7.42172 − 12.8548i
61.4 −0.272699 + 0.472328i −3.51122 + 6.08161i 3.85127 + 6.67060i −5.00000 −1.91501 − 3.31689i −11.4358 − 19.8074i −8.56412 −11.1573 − 19.3250i 1.36349 − 2.36164i
61.5 1.08655 − 1.88197i −1.99894 + 3.46226i 1.63880 + 2.83848i −5.00000 4.34391 + 7.52387i 10.9938 + 19.0418i 24.5074 5.50850 + 9.54100i −5.43277 + 9.40984i
61.6 1.78789 − 3.09671i 4.37841 − 7.58363i −2.39307 − 4.14492i −5.00000 −15.6562 − 27.1173i 11.1225 + 19.2648i 11.4920 −24.8410 − 43.0258i −8.93943 + 15.4835i
61.7 2.26209 − 3.91806i 0.652503 − 1.13017i −6.23413 − 10.7978i −5.00000 −2.95204 − 5.11309i −11.9953 − 20.7765i −20.2152 12.6485 + 21.9078i −11.3105 + 19.5903i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 61.7
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 65.4.e.a ✓ 14
13.c even 3 1 inner 65.4.e.a ✓ 14
13.c even 3 1 845.4.a.k 7
13.e even 6 1 845.4.a.h 7
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
65.4.e.a ✓ 14 1.a even 1 1 trivial
65.4.e.a ✓ 14 13.c even 3 1 inner
845.4.a.h 7 13.e even 6 1
845.4.a.k 7 13.c even 3 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{14} + 2 T_{2}^{13} + 45 T_{2}^{12} + 52 T_{2}^{11} + 1311 T_{2}^{10} + 1336 T_{2}^{9} + 20343 T_{2}^{8} + 11166 T_{2}^{7} + 216447 T_{2}^{6} + 107836 T_{2}^{5} + 1201133 T_{2}^{4} - 4392 T_{2}^{3} + \cdots + 1157776 \) acting on \(S_{4}^{\mathrm{new}}(65, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{14} + 2 T^{13} + 45 T^{12} + \cdots + 1157776 \) Copy content Toggle raw display
$3$ \( T^{14} - 4 T^{13} + \cdots + 3196771600 \) Copy content Toggle raw display
$5$ \( (T + 5)^{14} \) Copy content Toggle raw display
$7$ \( T^{14} + 7 T^{13} + \cdots + 17\!\cdots\!25 \) Copy content Toggle raw display
$11$ \( T^{14} + 87 T^{13} + \cdots + 59\!\cdots\!89 \) Copy content Toggle raw display
$13$ \( T^{14} - 123 T^{13} + \cdots + 24\!\cdots\!13 \) Copy content Toggle raw display
$17$ \( T^{14} - 114 T^{13} + \cdots + 41\!\cdots\!44 \) Copy content Toggle raw display
$19$ \( T^{14} + 245 T^{13} + \cdots + 65\!\cdots\!25 \) Copy content Toggle raw display
$23$ \( T^{14} - 74 T^{13} + \cdots + 95\!\cdots\!64 \) Copy content Toggle raw display
$29$ \( T^{14} - 88 T^{13} + \cdots + 86\!\cdots\!76 \) Copy content Toggle raw display
$31$ \( (T^{7} - 500 T^{6} + \cdots - 274963230720000)^{2} \) Copy content Toggle raw display
$37$ \( T^{14} + 633 T^{13} + \cdots + 70\!\cdots\!69 \) Copy content Toggle raw display
$41$ \( T^{14} + 162 T^{13} + \cdots + 67\!\cdots\!00 \) Copy content Toggle raw display
$43$ \( T^{14} - 280 T^{13} + \cdots + 24\!\cdots\!36 \) Copy content Toggle raw display
$47$ \( (T^{7} - 475 T^{6} + \cdots - 35\!\cdots\!00)^{2} \) Copy content Toggle raw display
$53$ \( (T^{7} + 603 T^{6} + \cdots - 88\!\cdots\!00)^{2} \) Copy content Toggle raw display
$59$ \( T^{14} + 1410 T^{13} + \cdots + 33\!\cdots\!24 \) Copy content Toggle raw display
$61$ \( T^{14} + 412 T^{13} + \cdots + 46\!\cdots\!00 \) Copy content Toggle raw display
$67$ \( T^{14} + 1398 T^{13} + \cdots + 13\!\cdots\!44 \) Copy content Toggle raw display
$71$ \( T^{14} - 584 T^{13} + \cdots + 85\!\cdots\!36 \) Copy content Toggle raw display
$73$ \( (T^{7} - 2538 T^{6} + \cdots - 16\!\cdots\!00)^{2} \) Copy content Toggle raw display
$79$ \( (T^{7} - 464 T^{6} + \cdots - 13\!\cdots\!00)^{2} \) Copy content Toggle raw display
$83$ \( (T^{7} - 466 T^{6} + \cdots - 21\!\cdots\!00)^{2} \) Copy content Toggle raw display
$89$ \( T^{14} + 443 T^{13} + \cdots + 11\!\cdots\!29 \) Copy content Toggle raw display
$97$ \( T^{14} - 1870 T^{13} + \cdots + 13\!\cdots\!04 \) Copy content Toggle raw display
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