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Properties

Label 1375.2.b.e
Level $1375$
Weight $2$
Character orbit 1375.b
Analytic conductor $10.979$
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] = [1375,2,Mod(749,1375)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1375, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1375.749");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1375 = 5^{3} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1375.b (of order \(2\), degree \(1\), not 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: \(10.9794302779\)
Analytic rank: \(0\)
Dimension: \(20\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} + 31 x^{18} + 393 x^{16} + 2674 x^{14} + 10768 x^{12} + 26658 x^{10} + 40807 x^{8} + 37898 x^{6} + \cdots + 625 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 5^{4} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{19}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{7} q^{2} + \beta_{18} q^{3} + (\beta_{6} - 2) q^{4} + ( - \beta_{12} - \beta_{9} - \beta_{8} + \cdots - 1) q^{6}+ \cdots + (\beta_{8} - 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{7} q^{2} + \beta_{18} q^{3} + (\beta_{6} - 2) q^{4} + ( - \beta_{12} - \beta_{9} - \beta_{8} + \cdots - 1) q^{6}+ \cdots + ( - \beta_{8} + 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q - 30 q^{4} + 2 q^{6} - 16 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 20 q - 30 q^{4} + 2 q^{6} - 16 q^{9} - 20 q^{11} - 12 q^{14} + 42 q^{16} + 2 q^{19} - 42 q^{21} + 36 q^{24} - 12 q^{26} - 2 q^{29} + 26 q^{31} - 50 q^{34} + 52 q^{36} - 4 q^{41} + 30 q^{44} - 36 q^{46} - 2 q^{49} - 42 q^{51} - 30 q^{54} + 40 q^{56} + 16 q^{59} + 50 q^{61} - 116 q^{64} - 2 q^{66} + 36 q^{69} - 52 q^{71} + 76 q^{74} + 22 q^{76} + 26 q^{79} + 4 q^{81} - 48 q^{84} + 98 q^{86} - 26 q^{89} - 60 q^{91} + 52 q^{94} - 106 q^{96} + 16 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} + 31 x^{18} + 393 x^{16} + 2674 x^{14} + 10768 x^{12} + 26658 x^{10} + 40807 x^{8} + 37898 x^{6} + \cdots + 625 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 21734 \nu^{18} + 741879 \nu^{16} + 10683387 \nu^{14} + 85006241 \nu^{12} + 406857137 \nu^{10} + \cdots + 194529550 ) / 6705725 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 614 \nu^{18} + 18084 \nu^{16} + 213902 \nu^{14} + 1326911 \nu^{12} + 4731977 \nu^{10} + \cdots + 474275 ) / 15275 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 277672 \nu^{18} - 8160782 \nu^{16} - 95510571 \nu^{14} - 575548928 \nu^{12} - 1919967171 \nu^{10} + \cdots + 85358700 ) / 6705725 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 317612 \nu^{18} - 10117422 \nu^{16} - 131148641 \nu^{14} - 899104313 \nu^{12} + \cdots - 383988450 ) / 6705725 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 364999 \nu^{18} + 11361644 \nu^{16} + 144247857 \nu^{14} + 976240676 \nu^{12} + 3854103082 \nu^{10} + \cdots + 425590900 ) / 6705725 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 6408 \nu^{18} - 188958 \nu^{16} - 2232274 \nu^{14} - 13745602 \nu^{12} - 47998744 \nu^{10} + \cdots - 1043555 ) / 103165 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 27061 \nu^{19} - 758791 \nu^{17} - 8272998 \nu^{15} - 44457689 \nu^{13} - 119572823 \nu^{11} + \cdots + 48340925 \nu ) / 2579125 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 94873 \nu^{18} - 2691953 \nu^{16} - 30051424 \nu^{14} - 169902737 \nu^{12} - 517840829 \nu^{10} + \cdots + 62675655 ) / 1341145 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 756929 \nu^{18} - 22822849 \nu^{16} - 278101447 \nu^{14} - 1787898096 \nu^{12} + \cdots - 607277475 ) / 6705725 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 15671 \nu^{19} + 487151 \nu^{17} + 6145178 \nu^{15} + 40941929 \nu^{13} + 156779603 \nu^{11} + \cdots + 17077650 \nu ) / 713375 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 160713 \nu^{19} + 4662838 \nu^{17} + 54265744 \nu^{15} + 332291867 \nu^{13} + 1185029099 \nu^{11} + \cdots + 194137835 \nu ) / 6705725 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 74219 \nu^{18} - 2253139 \nu^{16} - 27711817 \nu^{14} - 180413931 \nu^{12} - 680764792 \nu^{10} + \cdots - 71724975 ) / 515825 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 172629 \nu^{19} + 4866854 \nu^{17} + 53419002 \nu^{15} + 289622611 \nu^{13} + 792579517 \nu^{11} + \cdots - 208519695 \nu ) / 6705725 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 895958 \nu^{19} + 26244473 \nu^{17} + 305735144 \nu^{15} + 1829060267 \nu^{13} + \cdots - 625144825 \nu ) / 33528625 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 70398 \nu^{19} + 2197238 \nu^{17} + 28045564 \nu^{15} + 191744202 \nu^{13} + 771233639 \nu^{11} + \cdots + 132548300 \nu ) / 2579125 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( - 2068659 \nu^{19} - 63781504 \nu^{17} - 800671937 \nu^{15} - 5354554641 \nu^{13} + \cdots - 4254025150 \nu ) / 33528625 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( - 2087068 \nu^{19} - 62873658 \nu^{17} - 763099649 \nu^{15} - 4855600857 \nu^{13} + \cdots + 253286175 \nu ) / 33528625 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( - 5919447 \nu^{19} - 179456182 \nu^{17} - 2203349746 \nu^{15} - 14315368128 \nu^{13} + \cdots - 5815938300 \nu ) / 33528625 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( - 8156346 \nu^{19} - 247861576 \nu^{17} - 3054163203 \nu^{15} - 19948896254 \nu^{13} + \cdots - 8538351025 \nu ) / 33528625 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -2\beta_{14} + \beta_{10} - 3\beta_{7} ) / 5 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -2\beta_{12} - 4\beta_{8} + 5\beta_{6} - 2\beta_{5} - 2\beta_{4} + 4\beta_{3} - 2\beta_{2} - 17 ) / 5 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 6 \beta_{19} - 9 \beta_{18} + 3 \beta_{17} - \beta_{16} + 3 \beta_{15} + 18 \beta_{14} + \cdots + 7 \beta_{7} ) / 5 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 24 \beta_{12} - 7 \beta_{9} + 33 \beta_{8} - 38 \beta_{6} + 20 \beta_{5} + 16 \beta_{4} - 35 \beta_{3} + \cdots + 101 ) / 5 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 80 \beta_{19} + 125 \beta_{18} - 25 \beta_{17} + 15 \beta_{16} - 5 \beta_{15} - 152 \beta_{14} + \cdots - 23 \beta_{7} ) / 5 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 257 \beta_{12} + 101 \beta_{9} - 264 \beta_{8} + 294 \beta_{6} - 200 \beta_{5} - 118 \beta_{4} + \cdots - 763 ) / 5 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 832 \beta_{19} - 1343 \beta_{18} + 216 \beta_{17} - 152 \beta_{16} - 119 \beta_{15} + \cdots + 123 \beta_{7} ) / 5 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 2625 \beta_{12} - 1094 \beta_{9} + 2200 \beta_{8} - 2446 \beta_{6} + 1997 \beta_{5} + 874 \beta_{4} + \cdots + 6469 ) / 5 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 8057 \beta_{19} + 13338 \beta_{18} - 1966 \beta_{17} + 1407 \beta_{16} + 2099 \beta_{15} + \cdots - 1048 \beta_{7} ) / 5 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 26072 \beta_{12} + 10842 \beta_{9} - 18984 \beta_{8} + 21353 \beta_{6} - 19728 \beta_{5} + \cdots - 57864 ) / 5 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 76153 \beta_{19} - 128497 \beta_{18} + 18334 \beta_{17} - 12813 \beta_{16} - 25541 \beta_{15} + \cdots + 10703 \beta_{7} ) / 5 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( 254367 \beta_{12} - 103981 \beta_{9} + 167929 \beta_{8} - 191569 \beta_{6} + 192575 \beta_{5} + \cdots + 530393 ) / 5 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( - 713982 \beta_{19} + 1221898 \beta_{18} - 172511 \beta_{17} + 117047 \beta_{16} + \cdots - 111394 \beta_{7} ) / 5 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( - 2452614 \beta_{12} + 984554 \beta_{9} - 1511643 \beta_{8} + 1746091 \beta_{6} - 1860781 \beta_{5} + \cdots - 4917833 ) / 5 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( 6680119 \beta_{19} - 11550811 \beta_{18} + 1627317 \beta_{17} - 1076479 \beta_{16} + \cdots + 1135932 \beta_{7} ) / 5 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( ( 23461430 \beta_{12} - 9275596 \beta_{9} + 13777440 \beta_{8} - 16071669 \beta_{6} + 17839648 \beta_{5} + \cdots + 45859501 ) / 5 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( ( - 62506118 \beta_{19} + 108888272 \beta_{18} - 15352954 \beta_{17} + 9960898 \beta_{16} + \cdots - 11325599 \beta_{7} ) / 5 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( ( - 223220081 \beta_{12} + 87219798 \beta_{9} - 126691737 \beta_{8} + 148882057 \beta_{6} + \cdots - 428949924 ) / 5 \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( ( 585317191 \beta_{19} - 1025082139 \beta_{18} + 144762293 \beta_{17} - 92598166 \beta_{16} + \cdots + 110953380 \beta_{7} ) / 5 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1375\mathbb{Z}\right)^\times\).

\(n\) \(376\) \(1002\)
\(\chi(n)\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
749.1
− 1.80140i
− 1.74415i
− 1.27214i
− 3.06579i
− 0.852114i
− 0.717511i
− 2.56677i
− 2.06930i
0.556478i
− 1.12897i
1.12897i
− 0.556478i
2.06930i
2.56677i
0.717511i
0.852114i
3.06579i
1.27214i
1.74415i
1.80140i
− 2.80140i − 2.04055i −5.84783 0 −5.71640 1.73795i 10.7793i −1.16386 0
749.2 − 2.74415i 0.316646i −5.53039 0 0.868926 − 3.43900i 9.68792i 2.89974 0
749.3 − 2.27214i 2.82911i −3.16261 0 6.42812 0.492146i 2.64160i −5.00386 0
749.4 − 2.06579i − 1.82693i −2.26748 0 −3.77406 − 0.942454i 0.552552i −0.337688 0
749.5 − 1.85211i 2.68354i −1.43033 0 4.97021 − 0.339766i − 1.05510i −4.20136 0
749.6 − 1.71751i − 2.96355i −0.949843 0 −5.08993 − 3.35270i − 1.80366i −5.78262 0
749.7 − 1.56677i 0.600986i −0.454776 0 0.941609 1.45884i − 2.42101i 2.63882 0
749.8 − 1.06930i 2.38252i 0.856591 0 2.54763 4.07209i − 3.05456i −2.67639 0
749.9 − 0.443522i − 0.232972i 1.80329 0 −0.103328 − 0.482429i − 1.68684i 2.94572 0
749.10 − 0.128973i − 0.564354i 1.98337 0 −0.0727866 − 4.97227i − 0.513748i 2.68150 0
749.11 0.128973i 0.564354i 1.98337 0 −0.0727866 4.97227i 0.513748i 2.68150 0
749.12 0.443522i 0.232972i 1.80329 0 −0.103328 0.482429i 1.68684i 2.94572 0
749.13 1.06930i − 2.38252i 0.856591 0 2.54763 − 4.07209i 3.05456i −2.67639 0
749.14 1.56677i − 0.600986i −0.454776 0 0.941609 − 1.45884i 2.42101i 2.63882 0
749.15 1.71751i 2.96355i −0.949843 0 −5.08993 3.35270i 1.80366i −5.78262 0
749.16 1.85211i − 2.68354i −1.43033 0 4.97021 0.339766i 1.05510i −4.20136 0
749.17 2.06579i 1.82693i −2.26748 0 −3.77406 0.942454i − 0.552552i −0.337688 0
749.18 2.27214i − 2.82911i −3.16261 0 6.42812 − 0.492146i − 2.64160i −5.00386 0
749.19 2.74415i − 0.316646i −5.53039 0 0.868926 3.43900i − 9.68792i 2.89974 0
749.20 2.80140i 2.04055i −5.84783 0 −5.71640 − 1.73795i − 10.7793i −1.16386 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 749.20
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1375.2.b.e 20
5.b even 2 1 inner 1375.2.b.e 20
5.c odd 4 1 1375.2.a.d ✓ 10
5.c odd 4 1 1375.2.a.g yes 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1375.2.a.d ✓ 10 5.c odd 4 1
1375.2.a.g yes 10 5.c odd 4 1
1375.2.b.e 20 1.a even 1 1 trivial
1375.2.b.e 20 5.b even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{20} + 35 T_{2}^{18} + 517 T_{2}^{16} + 4206 T_{2}^{14} + 20610 T_{2}^{12} + 62358 T_{2}^{10} + \cdots + 121 \) acting on \(S_{2}^{\mathrm{new}}(1375, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{20} + 35 T^{18} + \cdots + 121 \) Copy content Toggle raw display
$3$ \( T^{20} + 38 T^{18} + \cdots + 25 \) Copy content Toggle raw display
$5$ \( T^{20} \) Copy content Toggle raw display
$7$ \( T^{20} + 71 T^{18} + \cdots + 2025 \) Copy content Toggle raw display
$11$ \( (T + 1)^{20} \) Copy content Toggle raw display
$13$ \( T^{20} + 153 T^{18} + \cdots + 31640625 \) Copy content Toggle raw display
$17$ \( T^{20} + 184 T^{18} + \cdots + 8826841 \) Copy content Toggle raw display
$19$ \( (T^{10} - T^{9} - 97 T^{8} + \cdots + 211)^{2} \) Copy content Toggle raw display
$23$ \( T^{20} + \cdots + 947353769041 \) Copy content Toggle raw display
$29$ \( (T^{10} + T^{9} + \cdots - 11979)^{2} \) Copy content Toggle raw display
$31$ \( (T^{10} - 13 T^{9} + \cdots - 14771)^{2} \) Copy content Toggle raw display
$37$ \( T^{20} + \cdots + 12481799295681 \) Copy content Toggle raw display
$41$ \( (T^{10} + 2 T^{9} + \cdots - 5915961)^{2} \) Copy content Toggle raw display
$43$ \( T^{20} + \cdots + 86\!\cdots\!21 \) Copy content Toggle raw display
$47$ \( T^{20} + \cdots + 105489193681 \) Copy content Toggle raw display
$53$ \( T^{20} + \cdots + 30359229121 \) Copy content Toggle raw display
$59$ \( (T^{10} - 8 T^{9} + \cdots + 32212521)^{2} \) Copy content Toggle raw display
$61$ \( (T^{10} - 25 T^{9} + \cdots + 134404405)^{2} \) Copy content Toggle raw display
$67$ \( T^{20} + \cdots + 19\!\cdots\!01 \) Copy content Toggle raw display
$71$ \( (T^{10} + 26 T^{9} + \cdots - 11430981)^{2} \) Copy content Toggle raw display
$73$ \( T^{20} + \cdots + 12\!\cdots\!01 \) Copy content Toggle raw display
$79$ \( (T^{10} - 13 T^{9} + \cdots + 37761019)^{2} \) Copy content Toggle raw display
$83$ \( T^{20} + \cdots + 40\!\cdots\!21 \) Copy content Toggle raw display
$89$ \( (T^{10} + 13 T^{9} + \cdots + 59942655)^{2} \) Copy content Toggle raw display
$97$ \( T^{20} + \cdots + 69\!\cdots\!25 \) Copy content Toggle raw display
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