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Properties

Label 520.2.w.f
Level $520$
Weight $2$
Character orbit 520.w
Analytic conductor $4.152$
Analytic rank $0$
Dimension $20$
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] = [520,2,Mod(57,520)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(520, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 0, 1, 3]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("520.57");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 520 = 2^{3} \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 520.w (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: \(4.15222090511\)
Analytic rank: \(0\)
Dimension: \(20\)
Relative dimension: \(10\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} + 41 x^{18} + 640 x^{16} + 4888 x^{14} + 19956 x^{12} + 45364 x^{10} + 57952 x^{8} + 41120 x^{6} + \cdots + 256 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 2^{10} \)
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 \(1,\beta_1,\ldots,\beta_{19}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{3} q^{3} + \beta_{7} q^{5} - \beta_{11} q^{7} + ( - \beta_{10} - \beta_1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{3} q^{3} + \beta_{7} q^{5} - \beta_{11} q^{7} + ( - \beta_{10} - \beta_1) q^{9} + ( - \beta_{19} - \beta_{16} + \cdots - \beta_{2}) q^{11}+ \cdots + ( - \beta_{19} - 2 \beta_{18} + \cdots - 2) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q - 2 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 20 q - 2 q^{5} + 2 q^{11} + 4 q^{13} - 16 q^{15} - 6 q^{17} - 2 q^{19} + 6 q^{21} + 14 q^{23} + 14 q^{25} + 6 q^{27} + 10 q^{31} - 12 q^{33} - 16 q^{35} + 32 q^{39} + 28 q^{41} + 14 q^{45} - 16 q^{49} + 20 q^{53} - 48 q^{55} + 32 q^{57} - 14 q^{59} + 12 q^{61} + 8 q^{63} - 40 q^{65} + 36 q^{67} + 24 q^{69} - 48 q^{71} - 8 q^{73} - 22 q^{75} - 12 q^{77} - 4 q^{81} - 6 q^{85} - 28 q^{89} - 8 q^{91} - 2 q^{95} + 4 q^{97} - 42 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} + 41 x^{18} + 640 x^{16} + 4888 x^{14} + 19956 x^{12} + 45364 x^{10} + 57952 x^{8} + 41120 x^{6} + \cdots + 256 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 3104559 \nu^{19} - 126413091 \nu^{17} - 1951301900 \nu^{15} - 14624510328 \nu^{13} + \cdots - 2745652800 \nu ) / 10137344 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 3299297 \nu^{18} + 133715617 \nu^{16} + 2048450680 \nu^{14} + 15159010536 \nu^{12} + \cdots + 1695293568 ) / 2534336 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 31608011 \nu^{19} + 25157822 \nu^{18} - 1280837719 \nu^{17} + 1020024054 \nu^{16} + \cdots + 14370606976 ) / 40549376 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 17285399 \nu^{19} - 76099192 \nu^{18} + 700235459 \nu^{17} - 3088305976 \nu^{16} + \cdots - 46990545920 ) / 40549376 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 21543149 \nu^{18} - 873127633 \nu^{16} - 13376692876 \nu^{14} - 99009793096 \nu^{12} + \cdots - 11774221376 ) / 10137344 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 31608011 \nu^{19} + 25157822 \nu^{18} + 1280837719 \nu^{17} + 1020024054 \nu^{16} + \cdots + 14370606976 ) / 40549376 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 17285399 \nu^{19} - 76099192 \nu^{18} - 700235459 \nu^{17} - 3088305976 \nu^{16} + \cdots - 46990545920 ) / 40549376 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 36298169 \nu^{19} - 104954648 \nu^{18} + 1476178893 \nu^{17} - 4264150744 \nu^{16} + \cdots - 73566300160 ) / 40549376 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 47636669 \nu^{19} - 31790564 \nu^{18} - 1927849889 \nu^{17} - 1289592724 \nu^{16} + \cdots - 17778438912 ) / 40549376 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 40567983 \nu^{19} - 1640319835 \nu^{17} - 25033637156 \nu^{15} - 184075029080 \nu^{13} + \cdots - 14602033344 \nu ) / 20274688 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 24051233 \nu^{19} - 975422861 \nu^{17} - 14959747028 \nu^{15} - 110921090120 \nu^{13} + \cdots - 13964851392 \nu ) / 10137344 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 74898849 \nu^{19} + 95243662 \nu^{18} + 3051757317 \nu^{17} + 3863689158 \nu^{16} + \cdots + 57168615296 ) / 40549376 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 51302060 \nu^{19} + 56555035 \nu^{18} - 2080598212 \nu^{17} + 2295307959 \nu^{16} + \cdots + 35483604416 ) / 20274688 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 33953090 \nu^{19} - 65056235 \nu^{18} - 1378508306 \nu^{17} - 2642087399 \nu^{16} + \cdots - 43968453568 ) / 20274688 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 63102019 \nu^{19} - 13362129 \nu^{18} + 2561400135 \nu^{17} - 542855621 \nu^{16} + \cdots - 9374307136 ) / 20274688 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( 63102019 \nu^{19} + 13362129 \nu^{18} + 2561400135 \nu^{17} + 542855621 \nu^{16} + \cdots + 9374307136 ) / 20274688 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( - 135313015 \nu^{19} + 10441494 \nu^{18} - 5492289747 \nu^{17} + 419468462 \nu^{16} + \cdots - 789507712 ) / 40549376 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( - 68255829 \nu^{19} + 2173335 \nu^{18} - 2771839889 \nu^{17} + 88282691 \nu^{16} + \cdots + 799124160 ) / 20274688 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( - 68255829 \nu^{19} - 2173335 \nu^{18} - 2771839889 \nu^{17} - 88282691 \nu^{16} + \cdots - 799124160 ) / 20274688 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{19} - \beta_{18} - \beta_{17} - 2 \beta_{15} + \beta_{13} + \beta_{12} + \beta_{11} + \cdots + 2 \beta_1 ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( - 2 \beta_{19} + 2 \beta_{18} - \beta_{17} - \beta_{13} - \beta_{12} + \beta_{11} + \beta_{10} + \cdots - 8 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - 3 \beta_{19} + 6 \beta_{18} + 6 \beta_{17} + 4 \beta_{16} + 10 \beta_{15} - \beta_{14} + \cdots - 8 \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 26 \beta_{19} - 27 \beta_{18} + 17 \beta_{17} + 6 \beta_{16} - 6 \beta_{15} - \beta_{14} + 17 \beta_{13} + \cdots + 82 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 33 \beta_{19} - 82 \beta_{18} - 76 \beta_{17} - 58 \beta_{16} - 136 \beta_{15} + 27 \beta_{14} + \cdots + 132 \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 313 \beta_{19} + 350 \beta_{18} - 270 \beta_{17} - 102 \beta_{16} + 102 \beta_{15} + 33 \beta_{14} + \cdots - 1048 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 461 \beta_{19} + 1160 \beta_{18} + 1040 \beta_{17} + 746 \beta_{16} + 1908 \beta_{15} + \cdots - 2180 \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 3785 \beta_{19} - 4546 \beta_{18} + 4074 \beta_{17} + 1430 \beta_{16} - 1430 \beta_{15} - 693 \beta_{14} + \cdots + 14200 ) / 2 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 7035 \beta_{19} - 16618 \beta_{18} - 14826 \beta_{17} - 9362 \beta_{16} - 27016 \beta_{15} + \cdots + 34648 \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 46497 \beta_{19} + 59570 \beta_{18} - 59974 \beta_{17} - 19054 \beta_{16} + 19054 \beta_{15} + \cdots - 196896 ) / 2 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 109315 \beta_{19} + 239658 \beta_{18} + 215174 \beta_{17} + 117478 \beta_{16} + 385076 \beta_{15} + \cdots - 536120 \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( 581141 \beta_{19} - 789894 \beta_{18} + 873986 \beta_{17} + 250438 \beta_{16} - 250438 \beta_{15} + \cdots + 2765320 ) / 2 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 1689707 \beta_{19} - 3469346 \beta_{18} - 3144222 \beta_{17} - 1487406 \beta_{16} - 5517068 \beta_{15} + \cdots + 8153320 \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( - 7389589 \beta_{19} + 10609086 \beta_{18} - 12691042 \beta_{17} - 3296110 \beta_{16} + 3296110 \beta_{15} + \cdots - 39173800 ) / 2 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( - 25842979 \beta_{19} + 50333994 \beta_{18} + 46037406 \beta_{17} + 19075766 \beta_{16} + \cdots - 122579512 \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( ( 95560565 \beta_{19} - 144292750 \beta_{18} + 184137458 \beta_{17} + 43739630 \beta_{16} + \cdots + 558466248 ) / 2 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( ( 391207875 \beta_{19} - 731240762 \beta_{18} - 674147342 \beta_{17} - 248214550 \beta_{16} + \cdots + 1828313752 \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( ( - 1255970197 \beta_{19} + 1985430478 \beta_{18} - 2672420306 \beta_{17} - 586919758 \beta_{16} + \cdots - 8000196328 ) / 2 \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( ( - 5871544243 \beta_{19} + 10632575258 \beta_{18} + 9866373182 \beta_{17} + 3278142470 \beta_{16} + \cdots - 27115332760 \beta_1 ) / 2 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/520\mathbb{Z}\right)^\times\).

\(n\) \(41\) \(261\) \(391\) \(417\)
\(\chi(n)\) \(-\beta_{1}\) \(1\) \(1\) \(\beta_{1}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
57.1
0.681256i
− 1.43898i
− 3.36342i
1.34002i
0.591081i
0.654187i
− 0.535565i
− 2.54561i
1.79929i
3.81775i
− 0.681256i
1.43898i
3.36342i
− 1.34002i
− 0.591081i
− 0.654187i
0.535565i
2.54561i
− 1.79929i
− 3.81775i
0 −2.09540 + 2.09540i 0 2.20590 − 0.366044i 0 − 2.09184i 0 − 5.78137i 0
57.2 0 −2.00495 + 2.00495i 0 −1.29543 + 1.82260i 0 3.21560i 0 − 5.03968i 0
57.3 0 −1.27538 + 1.27538i 0 −2.21266 − 0.322673i 0 − 0.873016i 0 − 0.253208i 0
57.4 0 −0.847275 + 0.847275i 0 0.665310 + 2.13480i 0 − 3.46256i 0 1.56425i 0
57.5 0 0.0984267 − 0.0984267i 0 2.21134 + 0.331643i 0 1.39172i 0 2.98062i 0
57.6 0 0.200960 − 0.200960i 0 −1.91559 − 1.15348i 0 3.02649i 0 2.91923i 0
57.7 0 0.465613 − 0.465613i 0 0.646900 − 2.14045i 0 − 1.65752i 0 2.56641i 0
57.8 0 1.70333 − 1.70333i 0 −0.846581 + 2.06961i 0 3.32755i 0 − 2.80266i 0
57.9 0 1.75903 − 1.75903i 0 −2.07584 − 0.831197i 0 − 4.71535i 0 − 3.18840i 0
57.10 0 1.99564 − 1.99564i 0 1.61665 − 1.54481i 0 1.83893i 0 − 4.96519i 0
73.1 0 −2.09540 − 2.09540i 0 2.20590 + 0.366044i 0 2.09184i 0 5.78137i 0
73.2 0 −2.00495 − 2.00495i 0 −1.29543 − 1.82260i 0 − 3.21560i 0 5.03968i 0
73.3 0 −1.27538 − 1.27538i 0 −2.21266 + 0.322673i 0 0.873016i 0 0.253208i 0
73.4 0 −0.847275 − 0.847275i 0 0.665310 − 2.13480i 0 3.46256i 0 − 1.56425i 0
73.5 0 0.0984267 + 0.0984267i 0 2.21134 − 0.331643i 0 − 1.39172i 0 − 2.98062i 0
73.6 0 0.200960 + 0.200960i 0 −1.91559 + 1.15348i 0 − 3.02649i 0 − 2.91923i 0
73.7 0 0.465613 + 0.465613i 0 0.646900 + 2.14045i 0 1.65752i 0 − 2.56641i 0
73.8 0 1.70333 + 1.70333i 0 −0.846581 − 2.06961i 0 − 3.32755i 0 2.80266i 0
73.9 0 1.75903 + 1.75903i 0 −2.07584 + 0.831197i 0 4.71535i 0 3.18840i 0
73.10 0 1.99564 + 1.99564i 0 1.61665 + 1.54481i 0 − 1.83893i 0 4.96519i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 57.10
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
65.k even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 520.2.w.f ✓ 20
4.b odd 2 1 1040.2.bg.p 20
5.c odd 4 1 520.2.bh.f yes 20
13.d odd 4 1 520.2.bh.f yes 20
20.e even 4 1 1040.2.cd.p 20
52.f even 4 1 1040.2.cd.p 20
65.k even 4 1 inner 520.2.w.f ✓ 20
260.s odd 4 1 1040.2.bg.p 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
520.2.w.f ✓ 20 1.a even 1 1 trivial
520.2.w.f ✓ 20 65.k even 4 1 inner
520.2.bh.f yes 20 5.c odd 4 1
520.2.bh.f yes 20 13.d odd 4 1
1040.2.bg.p 20 4.b odd 2 1
1040.2.bg.p 20 260.s odd 4 1
1040.2.cd.p 20 20.e even 4 1
1040.2.cd.p 20 52.f even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{20} - 2 T_{3}^{17} + 145 T_{3}^{16} - 22 T_{3}^{15} + 2 T_{3}^{14} - 76 T_{3}^{13} + 6340 T_{3}^{12} + \cdots + 64 \) acting on \(S_{2}^{\mathrm{new}}(520, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{20} \) Copy content Toggle raw display
$3$ \( T^{20} - 2 T^{17} + \cdots + 64 \) Copy content Toggle raw display
$5$ \( T^{20} + 2 T^{19} + \cdots + 9765625 \) Copy content Toggle raw display
$7$ \( T^{20} + 78 T^{18} + \cdots + 16777216 \) Copy content Toggle raw display
$11$ \( T^{20} - 2 T^{19} + \cdots + 891136 \) Copy content Toggle raw display
$13$ \( T^{20} + \cdots + 137858491849 \) Copy content Toggle raw display
$17$ \( T^{20} + \cdots + 2398648576 \) Copy content Toggle raw display
$19$ \( T^{20} + 2 T^{19} + \cdots + 14258176 \) Copy content Toggle raw display
$23$ \( T^{20} + \cdots + 13901353216 \) Copy content Toggle raw display
$29$ \( T^{20} + \cdots + 4294967296 \) Copy content Toggle raw display
$31$ \( T^{20} + \cdots + 5974671616 \) Copy content Toggle raw display
$37$ \( T^{20} + 290 T^{18} + \cdots + 67108864 \) Copy content Toggle raw display
$41$ \( T^{20} + \cdots + 62524002304 \) Copy content Toggle raw display
$43$ \( T^{20} + \cdots + 21066780736 \) Copy content Toggle raw display
$47$ \( T^{20} + \cdots + 100844013420544 \) Copy content Toggle raw display
$53$ \( T^{20} + \cdots + 415354834625536 \) Copy content Toggle raw display
$59$ \( T^{20} + \cdots + 17632205661184 \) Copy content Toggle raw display
$61$ \( (T^{10} - 6 T^{9} + \cdots - 238336)^{2} \) Copy content Toggle raw display
$67$ \( (T^{10} - 18 T^{9} + \cdots + 152065024)^{2} \) Copy content Toggle raw display
$71$ \( T^{20} + \cdots + 36\!\cdots\!64 \) Copy content Toggle raw display
$73$ \( (T^{10} + 4 T^{9} + \cdots + 67913728)^{2} \) Copy content Toggle raw display
$79$ \( T^{20} + \cdots + 17925996544 \) Copy content Toggle raw display
$83$ \( T^{20} + \cdots + 374034989056 \) Copy content Toggle raw display
$89$ \( T^{20} + \cdots + 11\!\cdots\!96 \) Copy content Toggle raw display
$97$ \( (T^{10} - 2 T^{9} + \cdots - 671028224)^{2} \) Copy content Toggle raw display
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