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Properties

Label 5.17.c.a
Level $5$
Weight $17$
Character orbit 5.c
Analytic conductor $8.116$
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] = [5,17,Mod(2,5)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(5, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([1]))
 
N = Newforms(chi, 17, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("5.2");
 
S:= CuspForms(chi, 17);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 5 \)
Weight: \( k \) \(=\) \( 17 \)
Character orbit: \([\chi]\) \(=\) 5.c (of order \(4\), 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: \(8.11622719283\)
Analytic rank: \(0\)
Dimension: \(14\)
Relative dimension: \(7\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{14} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{14} + 354217 x^{12} + 47647865256 x^{10} + \cdots + 84\!\cdots\!36 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 2^{34}\cdot 3^{10}\cdot 5^{18} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-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.

\(f(q)\) \(=\) \( q - \beta_{2} q^{2} + ( - \beta_{6} - 565 \beta_{4} + \cdots + 565) q^{3}+ \cdots + (15 \beta_{13} + 14 \beta_{12} + \cdots - 15) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{2} q^{2} + ( - \beta_{6} - 565 \beta_{4} + \cdots + 565) q^{3}+ \cdots + (653840295 \beta_{13} + \cdots - 2813954820) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 14 q - 2 q^{2} + 7908 q^{3} + 192880 q^{5} + 4417368 q^{6} - 386452 q^{7} - 10493340 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 14 q - 2 q^{2} + 7908 q^{3} + 192880 q^{5} + 4417368 q^{6} - 386452 q^{7} - 10493340 q^{8} + 398727830 q^{10} - 539141632 q^{11} + 1973309928 q^{12} - 681358582 q^{13} + 3514452060 q^{15} - 22953579016 q^{16} + 13124182498 q^{17} - 11293583022 q^{18} - 20186820060 q^{20} + 133652286048 q^{21} - 244784660624 q^{22} + 236373052228 q^{23} + 13144083650 q^{25} + 223473840788 q^{26} + 608106053880 q^{27} - 1494176423768 q^{28} + 2252131613760 q^{30} - 1105223074952 q^{31} - 2024535639992 q^{32} - 6385551058704 q^{33} + 6956678093980 q^{35} + 12514339524924 q^{36} - 9027827087002 q^{37} - 7824802789560 q^{38} + 31912538913900 q^{40} + 39543901484288 q^{41} - 70814500818384 q^{42} - 63017457929452 q^{43} + 126481294123110 q^{45} + 44357914058008 q^{46} - 2826187575452 q^{47} - 405160526368272 q^{48} + 461109216233650 q^{50} + 208649050954008 q^{51} - 389623094881012 q^{52} - 275142037498442 q^{53} + 330504874256560 q^{55} + 865011118900080 q^{56} - 644237427525840 q^{57} - 991296507673440 q^{58} + 16\!\cdots\!80 q^{60}+ \cdots + 48\!\cdots\!02 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{14} + 354217 x^{12} + 47647865256 x^{10} + \cdots + 84\!\cdots\!36 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 19395026881 \nu^{12} + \cdots - 26\!\cdots\!36 ) / 13\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 19395026881 \nu^{12} + \cdots - 26\!\cdots\!36 ) / 13\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 28\!\cdots\!93 \nu^{12} + \cdots + 13\!\cdots\!08 ) / 91\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 34\!\cdots\!99 \nu^{13} + \cdots + 47\!\cdots\!44 \nu ) / 15\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 24\!\cdots\!31 \nu^{13} + \cdots + 77\!\cdots\!36 \nu ) / 19\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 25\!\cdots\!65 \nu^{13} + \cdots - 19\!\cdots\!32 ) / 17\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 25\!\cdots\!65 \nu^{13} + \cdots - 19\!\cdots\!32 ) / 17\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 16\!\cdots\!23 \nu^{13} + \cdots + 49\!\cdots\!40 ) / 38\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 16\!\cdots\!53 \nu^{13} + \cdots - 25\!\cdots\!92 ) / 38\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 88\!\cdots\!97 \nu^{13} + \cdots - 31\!\cdots\!72 ) / 19\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 11\!\cdots\!47 \nu^{13} + \cdots - 49\!\cdots\!92 ) / 19\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 68\!\cdots\!07 \nu^{13} + \cdots - 54\!\cdots\!40 ) / 34\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 11\!\cdots\!97 \nu^{13} + \cdots - 80\!\cdots\!84 ) / 38\!\cdots\!00 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -\beta_{2} + \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 2\beta_{7} + 2\beta_{6} + \beta_{3} + 7\beta_{2} + 7\beta _1 - 101206 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - 2 \beta_{13} + 6 \beta_{12} - 3 \beta_{11} + 17 \beta_{9} - 163 \beta_{7} + 158 \beta_{6} + 27 \beta_{5} + \cdots - 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 475 \beta_{13} + 10 \beta_{11} - 1385 \beta_{10} + 1952 \beta_{9} + 22 \beta_{8} - 432342 \beta_{7} + \cdots + 8621376925 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 285278 \beta_{13} - 908514 \beta_{12} + 331607 \beta_{11} + 437920 \beta_{10} - 2135933 \beta_{9} + \cdots + 46329 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 74784975 \beta_{13} - 14462850 \beta_{11} + 166503525 \beta_{10} - 353950176 \beta_{9} + \cdots - 856970002462529 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 34242288078 \beta_{13} + 112080642834 \beta_{12} - 33898082967 \beta_{11} - 79215175200 \beta_{10} + \cdots + 344205111 ) / 2 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 9193799374575 \beta_{13} + 2657937153090 \beta_{11} - 16949649511365 \beta_{10} + 47171245639392 \beta_{9} + \cdots + 90\!\cdots\!13 ) / 2 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 39\!\cdots\!02 \beta_{13} + \cdots - 408918043980839 ) / 2 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 10\!\cdots\!75 \beta_{13} + \cdots - 98\!\cdots\!97 ) / 2 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 44\!\cdots\!50 \beta_{13} + \cdots + 71\!\cdots\!75 ) / 2 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( 11\!\cdots\!75 \beta_{13} + \cdots + 10\!\cdots\!01 ) / 2 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 50\!\cdots\!22 \beta_{13} + \cdots - 98\!\cdots\!39 ) / 2 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/5\mathbb{Z}\right)^\times\).

\(n\) \(2\)
\(\chi(n)\) \(\beta_{4}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2.1
− 330.301i
− 205.093i
− 162.000i
38.1160i
88.8004i
233.161i
336.317i
330.301i
205.093i
162.000i
− 38.1160i
− 88.8004i
− 233.161i
− 336.317i
−330.301 − 330.301i 4939.05 − 4939.05i 152661.i 188004. + 342407.i −3.26275e6 5.35472e6 + 5.35472e6i 2.87776e7 − 2.87776e7i − 5.74179e6i 5.09995e7 − 1.75195e8i
2.2 −205.093 − 205.093i −8389.67 + 8389.67i 18590.3i −65048.3 + 385171.i 3.44133e6 −4.20104e6 − 4.20104e6i −9.62824e6 + 9.62824e6i − 9.77264e7i 9.23368e7 − 6.56549e7i
2.3 −162.000 − 162.000i 380.189 − 380.189i − 13048.1i −70298.4 − 384247.i −123181. −65928.6 − 65928.6i −1.27306e7 + 1.27306e7i 4.27576e7i −5.08597e7 + 7.36363e7i
2.4 38.1160 + 38.1160i 7207.82 − 7207.82i − 62630.3i −287350. + 264609.i 549467. −3.21528e6 − 3.21528e6i 4.88519e6 − 4.88519e6i − 6.08585e7i −2.10385e7 − 866777.i
2.5 88.8004 + 88.8004i −1526.86 + 1526.86i − 49765.0i 383800. + 72701.2i −271171. 2.52190e6 + 2.52190e6i 1.02388e7 − 1.02388e7i 3.83841e7i 2.76257e7 + 4.05375e7i
2.6 233.161 + 233.161i −4708.07 + 4708.07i 43191.7i −387973. − 45440.9i −2.19547e6 −216284. − 216284.i 5.20981e6 − 5.20981e6i − 1.28506e6i −7.98650e7 − 1.01055e8i
2.7 336.317 + 336.317i 6051.53 − 6051.53i 160682.i 335305. − 200395.i 4.07046e6 −371312. − 371312.i −3.19992e7 + 3.19992e7i − 3.01954e7i 1.80165e8 + 4.53727e7i
3.1 −330.301 + 330.301i 4939.05 + 4939.05i − 152661.i 188004. − 342407.i −3.26275e6 5.35472e6 − 5.35472e6i 2.87776e7 + 2.87776e7i 5.74179e6i 5.09995e7 + 1.75195e8i
3.2 −205.093 + 205.093i −8389.67 − 8389.67i − 18590.3i −65048.3 − 385171.i 3.44133e6 −4.20104e6 + 4.20104e6i −9.62824e6 − 9.62824e6i 9.77264e7i 9.23368e7 + 6.56549e7i
3.3 −162.000 + 162.000i 380.189 + 380.189i 13048.1i −70298.4 + 384247.i −123181. −65928.6 + 65928.6i −1.27306e7 − 1.27306e7i − 4.27576e7i −5.08597e7 − 7.36363e7i
3.4 38.1160 − 38.1160i 7207.82 + 7207.82i 62630.3i −287350. − 264609.i 549467. −3.21528e6 + 3.21528e6i 4.88519e6 + 4.88519e6i 6.08585e7i −2.10385e7 + 866777.i
3.5 88.8004 − 88.8004i −1526.86 − 1526.86i 49765.0i 383800. − 72701.2i −271171. 2.52190e6 − 2.52190e6i 1.02388e7 + 1.02388e7i − 3.83841e7i 2.76257e7 − 4.05375e7i
3.6 233.161 − 233.161i −4708.07 − 4708.07i − 43191.7i −387973. + 45440.9i −2.19547e6 −216284. + 216284.i 5.20981e6 + 5.20981e6i 1.28506e6i −7.98650e7 + 1.01055e8i
3.7 336.317 − 336.317i 6051.53 + 6051.53i − 160682.i 335305. + 200395.i 4.07046e6 −371312. + 371312.i −3.19992e7 − 3.19992e7i 3.01954e7i 1.80165e8 − 4.53727e7i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 2.7
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.c odd 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 5.17.c.a ✓ 14
3.b odd 2 1 45.17.g.a 14
4.b odd 2 1 80.17.p.c 14
5.b even 2 1 25.17.c.b 14
5.c odd 4 1 inner 5.17.c.a ✓ 14
5.c odd 4 1 25.17.c.b 14
15.e even 4 1 45.17.g.a 14
20.e even 4 1 80.17.p.c 14
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
5.17.c.a ✓ 14 1.a even 1 1 trivial
5.17.c.a ✓ 14 5.c odd 4 1 inner
25.17.c.b 14 5.b even 2 1
25.17.c.b 14 5.c odd 4 1
45.17.g.a 14 3.b odd 2 1
45.17.g.a 14 15.e even 4 1
80.17.p.c 14 4.b odd 2 1
80.17.p.c 14 20.e even 4 1

Hecke kernels

This newform subspace is the entire newspace \(S_{17}^{\mathrm{new}}(5, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{14} + \cdots + 10\!\cdots\!08 \) Copy content Toggle raw display
$3$ \( T^{14} + \cdots + 31\!\cdots\!08 \) Copy content Toggle raw display
$5$ \( T^{14} + \cdots + 19\!\cdots\!25 \) Copy content Toggle raw display
$7$ \( T^{14} + \cdots + 11\!\cdots\!28 \) Copy content Toggle raw display
$11$ \( (T^{7} + \cdots + 12\!\cdots\!92)^{2} \) Copy content Toggle raw display
$13$ \( T^{14} + \cdots + 27\!\cdots\!88 \) Copy content Toggle raw display
$17$ \( T^{14} + \cdots + 10\!\cdots\!68 \) Copy content Toggle raw display
$19$ \( T^{14} + \cdots + 10\!\cdots\!00 \) Copy content Toggle raw display
$23$ \( T^{14} + \cdots + 19\!\cdots\!68 \) Copy content Toggle raw display
$29$ \( T^{14} + \cdots + 47\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( (T^{7} + \cdots + 12\!\cdots\!32)^{2} \) Copy content Toggle raw display
$37$ \( T^{14} + \cdots + 39\!\cdots\!48 \) Copy content Toggle raw display
$41$ \( (T^{7} + \cdots - 13\!\cdots\!48)^{2} \) Copy content Toggle raw display
$43$ \( T^{14} + \cdots + 96\!\cdots\!28 \) Copy content Toggle raw display
$47$ \( T^{14} + \cdots + 57\!\cdots\!88 \) Copy content Toggle raw display
$53$ \( T^{14} + \cdots + 33\!\cdots\!08 \) Copy content Toggle raw display
$59$ \( T^{14} + \cdots + 72\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( (T^{7} + \cdots - 33\!\cdots\!08)^{2} \) Copy content Toggle raw display
$67$ \( T^{14} + \cdots + 52\!\cdots\!68 \) Copy content Toggle raw display
$71$ \( (T^{7} + \cdots - 10\!\cdots\!88)^{2} \) Copy content Toggle raw display
$73$ \( T^{14} + \cdots + 42\!\cdots\!68 \) Copy content Toggle raw display
$79$ \( T^{14} + \cdots + 12\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{14} + \cdots + 50\!\cdots\!48 \) Copy content Toggle raw display
$89$ \( T^{14} + \cdots + 54\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{14} + \cdots + 22\!\cdots\!88 \) Copy content Toggle raw display
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