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Properties

Label 845.2.c.h
Level $845$
Weight $2$
Character orbit 845.c
Analytic conductor $6.747$
Analytic rank $0$
Dimension $18$
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] = [845,2,Mod(506,845)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(845, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("845.506");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 845 = 5 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 845.c (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: \(6.74735897080\)
Analytic rank: \(0\)
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: \( x^{18} + 34x^{16} + 407x^{14} + 2175x^{12} + 5555x^{10} + 6664x^{8} + 3544x^{6} + 681x^{4} + 47x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{3} \)
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 \(1,\beta_1,\ldots,\beta_{17}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{6} - \beta_1) q^{2} + (\beta_{11} + 1) q^{3} + ( - \beta_{10} + \beta_{9} - \beta_{8} - 2) q^{4} + \beta_{13} q^{5} + (\beta_{17} - \beta_{16} + \cdots - \beta_1) q^{6}+ \cdots + (\beta_{11} + \beta_{9} - 2 \beta_{5} + \cdots + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{6} - \beta_1) q^{2} + (\beta_{11} + 1) q^{3} + ( - \beta_{10} + \beta_{9} - \beta_{8} - 2) q^{4} + \beta_{13} q^{5} + (\beta_{17} - \beta_{16} + \cdots - \beta_1) q^{6}+ \cdots + (5 \beta_{17} - \beta_{16} + \cdots - 3 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 18 q + 14 q^{3} - 34 q^{4} + 32 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 18 q + 14 q^{3} - 34 q^{4} + 32 q^{9} - 6 q^{10} - 24 q^{12} - 4 q^{14} + 74 q^{16} + 2 q^{17} + 24 q^{22} - 28 q^{23} - 18 q^{25} + 44 q^{27} + 24 q^{29} + 4 q^{30} + 14 q^{35} - 6 q^{36} + 94 q^{38} + 24 q^{40} - 22 q^{42} - 78 q^{43} - 6 q^{48} - 32 q^{49} - 86 q^{51} - 16 q^{53} - 18 q^{55} + 58 q^{56} - 6 q^{61} + 20 q^{62} - 68 q^{64} - 98 q^{66} - 40 q^{68} + 26 q^{69} - 30 q^{74} - 14 q^{75} + 8 q^{77} + 78 q^{79} + 58 q^{81} + 8 q^{82} + 32 q^{87} - 84 q^{88} + 20 q^{90} - 54 q^{92} + 32 q^{94} + 8 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{18} + 34x^{16} + 407x^{14} + 2175x^{12} + 5555x^{10} + 6664x^{8} + 3544x^{6} + 681x^{4} + 47x^{2} + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 314 \nu^{16} + 11531 \nu^{14} + 155943 \nu^{12} + 1000554 \nu^{10} + 3262234 \nu^{8} + 5215959 \nu^{6} + \cdots + 26257 ) / 23348 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 30751 \nu^{16} + 1054474 \nu^{14} + 12808346 \nu^{12} + 70153600 \nu^{10} + 186147393 \nu^{8} + \cdots + 1292422 ) / 957268 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 35814 \nu^{17} - 1200088 \nu^{15} - 13998849 \nu^{13} - 71418339 \nu^{11} - 168476899 \nu^{9} + \cdots - 3665123 \nu ) / 478634 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 85080 \nu^{16} - 2881455 \nu^{14} - 34266505 \nu^{12} - 181181422 \nu^{10} - 456103704 \nu^{8} + \cdots - 3980585 ) / 957268 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 335154 \nu^{17} - 11364485 \nu^{15} - 135353204 \nu^{13} - 716151604 \nu^{11} + \cdots + 3349792 \nu ) / 957268 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 363580 \nu^{16} - 12350593 \nu^{14} - 147564147 \nu^{12} - 785115096 \nu^{10} - 1982532264 \nu^{8} + \cdots - 1655871 ) / 957268 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 200967 \nu^{16} - 6775530 \nu^{14} - 79894886 \nu^{12} - 415449506 \nu^{10} - 1010742096 \nu^{8} + \cdots - 3057070 ) / 478634 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 435208 \nu^{16} + 14750769 \nu^{14} + 175561845 \nu^{12} + 927951774 \nu^{10} + 2319486062 \nu^{8} + \cdots + 8028849 ) / 957268 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 491862 \nu^{16} + 16646116 \nu^{14} + 197617407 \nu^{12} + 1040169704 \nu^{10} + 2584684372 \nu^{8} + \cdots + 7961039 ) / 957268 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 584675 \nu^{16} - 19943268 \nu^{14} - 240060381 \nu^{12} - 1294904608 \nu^{10} - 3354455949 \nu^{8} + \cdots - 15114581 ) / 957268 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 584675 \nu^{17} + 19943268 \nu^{15} + 240060381 \nu^{13} + 1294904608 \nu^{11} + \cdots + 15114581 \nu ) / 957268 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 26257 \nu^{17} - 892424 \nu^{15} - 10675068 \nu^{13} - 56953032 \nu^{11} - 144857081 \nu^{9} + \cdots - 562820 \nu ) / 23348 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 1292422 \nu^{17} + 43911597 \nu^{15} + 524961280 \nu^{13} + 2798209504 \nu^{11} + \cdots + 41641804 \nu ) / 957268 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 1511745 \nu^{17} + 51340153 \nu^{15} + 613239633 \nu^{13} + 3263026086 \nu^{11} + \cdots + 32061737 \nu ) / 957268 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( 973390 \nu^{17} + 33059157 \nu^{15} + 394965388 \nu^{13} + 2103186829 \nu^{11} + \cdots + 25690001 \nu ) / 478634 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( - 3090823 \nu^{17} - 104771128 \nu^{15} - 1247295356 \nu^{13} - 6597016758 \nu^{11} + \cdots - 48749924 \nu ) / 957268 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -2\beta_{10} + \beta_{9} - \beta_{8} - \beta_{5} + 2\beta_{3} - 5 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{17} - 3\beta_{15} + 2\beta_{14} + 3\beta_{13} + 3\beta_{12} - \beta_{6} - 9\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 3\beta_{11} + 27\beta_{10} - 9\beta_{9} + 16\beta_{8} + 5\beta_{5} - 27\beta_{3} + 6\beta_{2} + 51 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 16 \beta_{17} - 6 \beta_{16} + 44 \beta_{15} - 27 \beta_{14} - 72 \beta_{13} - 64 \beta_{12} + \cdots + 103 \beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 64 \beta_{11} - 354 \beta_{10} + 103 \beta_{9} - 215 \beta_{8} + 12 \beta_{7} - 5 \beta_{5} + \cdots - 630 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 215 \beta_{17} + 154 \beta_{16} - 555 \beta_{15} + 370 \beta_{14} + 1365 \beta_{13} + \cdots - 1318 \beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 1091 \beta_{11} + 4741 \beta_{10} - 1318 \beta_{9} + 2863 \beta_{8} - 316 \beta_{7} - 466 \beta_{5} + \cdots + 8472 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 2863 \beta_{17} - 2926 \beta_{16} + 7063 \beta_{15} - 5265 \beta_{14} - 23241 \beta_{13} + \cdots + 17918 \beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 17415 \beta_{11} - 65509 \beta_{10} + 17918 \beta_{9} - 39149 \beta_{8} + 6110 \beta_{7} + \cdots - 119143 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 39149 \beta_{17} + 49692 \beta_{16} - 93499 \beta_{15} + 76701 \beta_{14} + 374983 \beta_{13} + \cdots - 252911 \beta_1 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 270663 \beta_{11} + 930631 \beta_{10} - 252911 \beta_{9} + 551214 \beta_{8} - 105008 \beta_{7} + \cdots + 1720394 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 551214 \beta_{17} - 800346 \beta_{16} + 1287006 \beta_{15} - 1132827 \beta_{14} - 5879102 \beta_{13} + \cdots + 3657346 \beta_1 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 4152038 \beta_{11} - 13497390 \beta_{10} + 3657346 \beta_{9} - 7943412 \beta_{8} + 1704192 \beta_{7} + \cdots - 25251857 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 7943412 \beta_{17} + 12535678 \beta_{16} - 18265698 \beta_{15} + 16867706 \beta_{14} + \cdots - 53720321 \beta_1 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( 63242434 \beta_{11} + 198554992 \beta_{10} - 53720321 \beta_{9} + 116353757 \beta_{8} - 26818108 \beta_{7} + \cdots + 374457783 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( 116353757 \beta_{17} - 193303392 \beta_{16} + 264892277 \beta_{15} - 252390368 \beta_{14} + \cdots + 796922229 \beta_1 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/845\mathbb{Z}\right)^\times\).

\(n\) \(171\) \(677\)
\(\chi(n)\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
506.1
0.987478i
3.88295i
3.03152i
1.03381i
1.76052i
− 0.271062i
− 0.199774i
2.07331i
− 0.421015i
0.421015i
− 2.07331i
0.199774i
0.271062i
− 1.76052i
− 1.03381i
− 3.03152i
− 3.88295i
− 0.987478i
− 2.78942i −1.81462 −5.78084 − 1.00000i 5.06173i 0.269601i 10.5463i 0.292842 −2.78942
506.2 − 2.63597i 1.98944 −4.94835 − 1.00000i − 5.24412i − 3.28231i 7.77176i 0.957886 −2.63597
506.3 − 2.58648i 0.884825 −4.68989 1.00000i − 2.28858i − 0.858383i 6.95735i −2.21708 2.58648
506.4 − 2.28079i 3.21428 −3.20199 1.00000i − 7.33108i − 2.30369i 2.74149i 7.33158 2.28079
506.5 − 2.20556i −0.0130567 −2.86449 − 1.00000i 0.0287972i 4.60897i 1.90669i −2.99983 −2.20556
506.6 − 1.53088i 2.88726 −0.343581 − 1.00000i − 4.42003i 3.86493i − 2.53577i 5.33625 −1.53088
506.7 − 1.04721i −2.75868 0.903360 1.00000i 2.88890i − 3.42366i − 3.04042i 4.61031 1.04721
506.8 − 0.271374i −0.319618 1.92636 1.00000i 0.0867358i 3.38151i − 1.06551i −2.89784 0.271374
506.9 − 0.0240266i 2.93017 1.99942 − 1.00000i − 0.0704021i − 1.66541i − 0.0960927i 5.58589 −0.0240266
506.10 0.0240266i 2.93017 1.99942 1.00000i 0.0704021i 1.66541i 0.0960927i 5.58589 −0.0240266
506.11 0.271374i −0.319618 1.92636 − 1.00000i − 0.0867358i − 3.38151i 1.06551i −2.89784 0.271374
506.12 1.04721i −2.75868 0.903360 − 1.00000i − 2.88890i 3.42366i 3.04042i 4.61031 1.04721
506.13 1.53088i 2.88726 −0.343581 1.00000i 4.42003i − 3.86493i 2.53577i 5.33625 −1.53088
506.14 2.20556i −0.0130567 −2.86449 1.00000i − 0.0287972i − 4.60897i − 1.90669i −2.99983 −2.20556
506.15 2.28079i 3.21428 −3.20199 − 1.00000i 7.33108i 2.30369i − 2.74149i 7.33158 2.28079
506.16 2.58648i 0.884825 −4.68989 − 1.00000i 2.28858i 0.858383i − 6.95735i −2.21708 2.58648
506.17 2.63597i 1.98944 −4.94835 1.00000i 5.24412i 3.28231i − 7.77176i 0.957886 −2.63597
506.18 2.78942i −1.81462 −5.78084 1.00000i − 5.06173i − 0.269601i − 10.5463i 0.292842 −2.78942
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 506.18
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 845.2.c.h 18
13.b even 2 1 inner 845.2.c.h 18
13.c even 3 2 845.2.m.j 36
13.d odd 4 1 845.2.a.n ✓ 9
13.d odd 4 1 845.2.a.o yes 9
13.e even 6 2 845.2.m.j 36
13.f odd 12 2 845.2.e.o 18
13.f odd 12 2 845.2.e.p 18
39.f even 4 1 7605.2.a.cp 9
39.f even 4 1 7605.2.a.cs 9
65.g odd 4 1 4225.2.a.bs 9
65.g odd 4 1 4225.2.a.bt 9
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
845.2.a.n ✓ 9 13.d odd 4 1
845.2.a.o yes 9 13.d odd 4 1
845.2.c.h 18 1.a even 1 1 trivial
845.2.c.h 18 13.b even 2 1 inner
845.2.e.o 18 13.f odd 12 2
845.2.e.p 18 13.f odd 12 2
845.2.m.j 36 13.c even 3 2
845.2.m.j 36 13.e even 6 2
4225.2.a.bs 9 65.g odd 4 1
4225.2.a.bt 9 65.g odd 4 1
7605.2.a.cp 9 39.f even 4 1
7605.2.a.cs 9 39.f even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{18} + 35 T_{2}^{16} + 507 T_{2}^{14} + 3912 T_{2}^{12} + 17195 T_{2}^{10} + 42488 T_{2}^{8} + \cdots + 1 \) acting on \(S_{2}^{\mathrm{new}}(845, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{18} + 35 T^{16} + \cdots + 1 \) Copy content Toggle raw display
$3$ \( (T^{9} - 7 T^{8} + 3 T^{7} + \cdots - 1)^{2} \) Copy content Toggle raw display
$5$ \( (T^{2} + 1)^{9} \) Copy content Toggle raw display
$7$ \( T^{18} + 79 T^{16} + \cdots + 361201 \) Copy content Toggle raw display
$11$ \( T^{18} + 105 T^{16} + \cdots + 79138816 \) Copy content Toggle raw display
$13$ \( T^{18} \) Copy content Toggle raw display
$17$ \( (T^{9} - T^{8} - 80 T^{7} + \cdots - 6656)^{2} \) Copy content Toggle raw display
$19$ \( T^{18} + \cdots + 116985856 \) Copy content Toggle raw display
$23$ \( (T^{9} + 14 T^{8} + \cdots + 43693)^{2} \) Copy content Toggle raw display
$29$ \( (T^{9} - 12 T^{8} - 22 T^{7} + \cdots + 1)^{2} \) Copy content Toggle raw display
$31$ \( T^{18} + \cdots + 116985856 \) Copy content Toggle raw display
$37$ \( T^{18} + \cdots + 100362568044544 \) Copy content Toggle raw display
$41$ \( T^{18} + \cdots + 986619064369 \) Copy content Toggle raw display
$43$ \( (T^{9} + 39 T^{8} + \cdots + 71513)^{2} \) Copy content Toggle raw display
$47$ \( T^{18} + \cdots + 136352329 \) Copy content Toggle raw display
$53$ \( (T^{9} + 8 T^{8} + \cdots - 4469312)^{2} \) Copy content Toggle raw display
$59$ \( T^{18} + \cdots + 74472226816 \) Copy content Toggle raw display
$61$ \( (T^{9} + 3 T^{8} + \cdots - 183247)^{2} \) Copy content Toggle raw display
$67$ \( T^{18} + \cdots + 13253305129 \) Copy content Toggle raw display
$71$ \( T^{18} + \cdots + 31208937558016 \) Copy content Toggle raw display
$73$ \( T^{18} + \cdots + 751369977856 \) Copy content Toggle raw display
$79$ \( (T^{9} - 39 T^{8} + \cdots - 10816)^{2} \) Copy content Toggle raw display
$83$ \( T^{18} + \cdots + 24\!\cdots\!21 \) Copy content Toggle raw display
$89$ \( T^{18} + 509 T^{16} + \cdots + 1100401 \) Copy content Toggle raw display
$97$ \( T^{18} + \cdots + 11\!\cdots\!76 \) Copy content Toggle raw display
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