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Properties

Label 845.2.b.g
Level $845$
Weight $2$
Character orbit 845.b
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(339,845)]
 
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([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("845.339");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 845 = 5 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 845.b (of order \(2\), degree \(1\), 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: \( x^{18} + 26x^{16} + 281x^{14} + 1632x^{12} + 5482x^{10} + 10620x^{8} + 11052x^{6} + 5165x^{4} + 760x^{2} + 29 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
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.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} - \beta_{15} q^{3} + (\beta_{2} - 1) q^{4} - \beta_{12} q^{5} + (\beta_{14} - \beta_{13} + \cdots - \beta_{5}) q^{6}+ \cdots + ( - \beta_{13} - \beta_{12} + \beta_{10} + \cdots - 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} - \beta_{15} q^{3} + (\beta_{2} - 1) q^{4} - \beta_{12} q^{5} + (\beta_{14} - \beta_{13} + \cdots - \beta_{5}) q^{6}+ \cdots + ( - \beta_{17} - \beta_{16} + \beta_{14} + \cdots + 4) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 18 q - 16 q^{4} - 16 q^{6} - 18 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 18 q - 16 q^{4} - 16 q^{6} - 18 q^{9} + 13 q^{10} - 22 q^{11} - 4 q^{14} - 8 q^{15} - 12 q^{16} + 28 q^{19} + 10 q^{20} - 26 q^{21} + 34 q^{24} - 8 q^{25} + 20 q^{29} + 31 q^{30} - 32 q^{31} + 18 q^{34} + 10 q^{35} + 32 q^{36} + 3 q^{40} - 52 q^{41} + 50 q^{44} - 5 q^{45} - 30 q^{46} - 44 q^{49} - 29 q^{50} - 40 q^{51} + 90 q^{54} - 20 q^{55} - 20 q^{56} + 76 q^{59} - 43 q^{60} + 8 q^{61} + 68 q^{64} + 8 q^{66} - 30 q^{69} - 46 q^{70} - 72 q^{71} - 30 q^{74} + 13 q^{75} - 4 q^{76} - 16 q^{79} - 73 q^{80} - 30 q^{81} + 78 q^{84} - 29 q^{85} + 30 q^{86} + 94 q^{89} - 70 q^{90} + 128 q^{94} + 41 q^{95} + 18 q^{96} + 76 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{18} + 26x^{16} + 281x^{14} + 1632x^{12} + 5482x^{10} + 10620x^{8} + 11052x^{6} + 5165x^{4} + 760x^{2} + 29 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} + 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} + 4\nu \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 51 \nu^{16} - 1009 \nu^{14} - 7704 \nu^{12} - 28208 \nu^{10} - 47962 \nu^{8} - 21998 \nu^{6} + \cdots + 1435 ) / 788 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 149 \nu^{16} + 3141 \nu^{14} + 26216 \nu^{12} + 109868 \nu^{10} + 239026 \nu^{8} + 242882 \nu^{6} + \cdots - 2207 ) / 788 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 49 \nu^{17} + 1066 \nu^{15} + 9256 \nu^{13} + 40830 \nu^{11} + 95532 \nu^{9} + 110442 \nu^{7} + \cdots - 780 \nu ) / 394 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 149 \nu^{17} + 3141 \nu^{15} + 26216 \nu^{13} + 109868 \nu^{11} + 239026 \nu^{9} + 242882 \nu^{7} + \cdots - 2207 \nu ) / 788 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 41 \nu^{16} - 900 \nu^{14} - 7978 \nu^{12} - 36749 \nu^{10} - 93934 \nu^{8} - 131091 \nu^{6} + \cdots - 855 ) / 197 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 18 \nu^{17} + 41 \nu^{16} + 472 \nu^{15} + 900 \nu^{14} + 5141 \nu^{13} + 7978 \nu^{12} + 30015 \nu^{11} + \cdots + 1840 ) / 394 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 65 \nu^{17} - 19 \nu^{16} + 1595 \nu^{15} - 345 \nu^{14} + 16146 \nu^{13} - 2198 \nu^{12} + \cdots + 4351 ) / 788 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 113 \nu^{17} - 2591 \nu^{15} - 24208 \nu^{13} - 118788 \nu^{11} - 327774 \nu^{9} - 502366 \nu^{7} + \cdots - 4067 \nu ) / 788 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 18 \nu^{17} + 41 \nu^{16} - 472 \nu^{15} + 900 \nu^{14} - 5141 \nu^{13} + 7978 \nu^{12} + \cdots + 1840 ) / 394 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 65 \nu^{17} + 19 \nu^{16} + 1595 \nu^{15} + 345 \nu^{14} + 16146 \nu^{13} + 2198 \nu^{12} + \cdots - 4351 ) / 788 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 143 \nu^{16} - 3115 \nu^{14} - 27326 \nu^{12} - 124094 \nu^{10} - 311210 \nu^{8} - 424292 \nu^{6} + \cdots - 5471 ) / 394 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 137 \nu^{17} - 3089 \nu^{15} - 28436 \nu^{13} - 138320 \nu^{11} - 383394 \nu^{9} - 605702 \nu^{7} + \cdots - 13149 \nu ) / 788 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( 132 \nu^{17} + 209 \nu^{16} + 2936 \nu^{15} + 4583 \nu^{14} + 26406 \nu^{13} + 40332 \nu^{12} + \cdots + 4541 ) / 788 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( - 132 \nu^{17} + 209 \nu^{16} - 2936 \nu^{15} + 4583 \nu^{14} - 26406 \nu^{13} + 40332 \nu^{12} + \cdots + 4541 ) / 788 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} - 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} - 4\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{12} + \beta_{9} + \beta_{8} - 6\beta_{2} + 13 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{17} - \beta_{16} + \beta_{13} + \beta_{11} + \beta_{10} + \beta_{7} + \beta_{6} - 8\beta_{3} + 17\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( \beta_{17} + \beta_{16} + \beta_{14} - \beta_{13} - 10 \beta_{12} + \beta_{10} - 10 \beta_{9} + \cdots - 59 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 10 \beta_{17} + 10 \beta_{16} + 3 \beta_{15} - 9 \beta_{13} - 12 \beta_{11} - 9 \beta_{10} + \cdots - 79 \beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 13 \beta_{17} - 13 \beta_{16} - 16 \beta_{14} + 12 \beta_{13} + 74 \beta_{12} - 12 \beta_{10} + \cdots + 275 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 75 \beta_{17} - 75 \beta_{16} - 40 \beta_{15} + 62 \beta_{13} - \beta_{12} + 105 \beta_{11} + \cdots + 398 \beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 119 \beta_{17} + 119 \beta_{16} + 163 \beta_{14} - 101 \beta_{13} - 494 \beta_{12} + 101 \beta_{10} + \cdots - 1311 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 512 \beta_{17} + 512 \beta_{16} + 365 \beta_{15} - 393 \beta_{13} + 18 \beta_{12} - 807 \beta_{11} + \cdots - 2140 \beta_1 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 944 \beta_{17} - 944 \beta_{16} - 1373 \beta_{14} + 740 \beta_{13} + 3161 \beta_{12} - 740 \beta_{10} + \cdots + 6387 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 3365 \beta_{17} - 3365 \beta_{16} - 2853 \beta_{15} + 2421 \beta_{13} - 204 \beta_{12} + \cdots + 12080 \beta_1 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 6940 \beta_{17} + 6940 \beta_{16} + 10458 \beta_{14} - 5070 \beta_{13} - 19874 \beta_{12} + 5070 \beta_{10} + \cdots - 31821 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 21744 \beta_{17} + 21744 \beta_{16} + 20598 \beta_{15} - 14804 \beta_{13} + 1870 \beta_{12} + \cdots - 70597 \beta_1 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( - 48728 \beta_{17} - 48728 \beta_{16} - 75040 \beta_{14} + 33532 \beta_{13} + 124172 \beta_{12} + \cdots + 162343 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( 139368 \beta_{17} - 139368 \beta_{16} - 142104 \beta_{15} + 90640 \beta_{13} - 15196 \beta_{12} + \cdots + 422680 \beta_1 \) 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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
339.1
− 2.51756i
− 2.15692i
− 2.14790i
− 1.94561i
− 1.76651i
− 1.57695i
− 0.887876i
− 0.395078i
− 0.242854i
0.242854i
0.395078i
0.887876i
1.57695i
1.76651i
1.94561i
2.14790i
2.15692i
2.51756i
− 2.51756i − 2.61964i −4.33809 −2.23296 − 0.117790i −6.59510 1.23067i 5.88628i −3.86253 −0.296543 + 5.62161i
339.2 − 2.15692i 1.75445i −2.65230 0.176470 + 2.22909i 3.78422 3.86123i 1.40696i −0.0781086 4.80798 − 0.380631i
339.3 − 2.14790i − 2.37435i −2.61347 −1.11435 + 1.93861i −5.09986 − 3.54879i 1.31766i −2.63754 4.16394 + 2.39351i
339.4 − 1.94561i 0.752344i −1.78540 0.809176 − 2.08452i 1.46377 1.49584i − 0.417520i 2.43398 −4.05567 − 1.57434i
339.5 − 1.76651i 0.691389i −1.12056 2.04107 − 0.913245i 1.22135 − 4.36221i − 1.55354i 2.52198 −1.61326 − 3.60558i
339.6 − 1.57695i − 2.97563i −0.486782 2.01071 + 0.978285i −4.69244 − 2.50257i − 2.38627i −5.85440 1.54271 − 3.17080i
339.7 − 0.887876i 1.26907i 1.21168 −1.33257 + 1.79562i 1.12678 2.28128i − 2.85157i 1.38947 1.59429 + 1.18316i
339.8 − 0.395078i 2.73651i 1.84391 1.15119 + 1.91697i 1.08113 − 0.629405i − 1.51865i −4.48846 0.757353 − 0.454808i
339.9 − 0.242854i − 1.19348i 1.94102 −1.50873 − 1.65037i −0.289840 4.78046i − 0.957093i 1.57562 −0.400799 + 0.366402i
339.10 0.242854i 1.19348i 1.94102 −1.50873 + 1.65037i −0.289840 − 4.78046i 0.957093i 1.57562 −0.400799 − 0.366402i
339.11 0.395078i − 2.73651i 1.84391 1.15119 − 1.91697i 1.08113 0.629405i 1.51865i −4.48846 0.757353 + 0.454808i
339.12 0.887876i − 1.26907i 1.21168 −1.33257 − 1.79562i 1.12678 − 2.28128i 2.85157i 1.38947 1.59429 − 1.18316i
339.13 1.57695i 2.97563i −0.486782 2.01071 − 0.978285i −4.69244 2.50257i 2.38627i −5.85440 1.54271 + 3.17080i
339.14 1.76651i − 0.691389i −1.12056 2.04107 + 0.913245i 1.22135 4.36221i 1.55354i 2.52198 −1.61326 + 3.60558i
339.15 1.94561i − 0.752344i −1.78540 0.809176 + 2.08452i 1.46377 − 1.49584i 0.417520i 2.43398 −4.05567 + 1.57434i
339.16 2.14790i 2.37435i −2.61347 −1.11435 − 1.93861i −5.09986 3.54879i − 1.31766i −2.63754 4.16394 − 2.39351i
339.17 2.15692i − 1.75445i −2.65230 0.176470 − 2.22909i 3.78422 − 3.86123i − 1.40696i −0.0781086 4.80798 + 0.380631i
339.18 2.51756i 2.61964i −4.33809 −2.23296 + 0.117790i −6.59510 − 1.23067i − 5.88628i −3.86253 −0.296543 − 5.62161i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 339.18
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 845.2.b.g ✓ 18
5.b even 2 1 inner 845.2.b.g ✓ 18
5.c odd 4 2 4225.2.a.ca 18
13.b even 2 1 845.2.b.h yes 18
13.c even 3 2 845.2.n.i 36
13.d odd 4 2 845.2.d.e 36
13.e even 6 2 845.2.n.h 36
13.f odd 12 4 845.2.l.g 72
65.d even 2 1 845.2.b.h yes 18
65.g odd 4 2 845.2.d.e 36
65.h odd 4 2 4225.2.a.cb 18
65.l even 6 2 845.2.n.h 36
65.n even 6 2 845.2.n.i 36
65.s odd 12 4 845.2.l.g 72
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
845.2.b.g ✓ 18 1.a even 1 1 trivial
845.2.b.g ✓ 18 5.b even 2 1 inner
845.2.b.h yes 18 13.b even 2 1
845.2.b.h yes 18 65.d even 2 1
845.2.d.e 36 13.d odd 4 2
845.2.d.e 36 65.g odd 4 2
845.2.l.g 72 13.f odd 12 4
845.2.l.g 72 65.s odd 12 4
845.2.n.h 36 13.e even 6 2
845.2.n.h 36 65.l even 6 2
845.2.n.i 36 13.c even 3 2
845.2.n.i 36 65.n even 6 2
4225.2.a.ca 18 5.c odd 4 2
4225.2.a.cb 18 65.h odd 4 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(845, [\chi])\):

\( T_{2}^{18} + 26 T_{2}^{16} + 281 T_{2}^{14} + 1632 T_{2}^{12} + 5482 T_{2}^{10} + 10620 T_{2}^{8} + \cdots + 29 \) Copy content Toggle raw display
\( T_{11}^{9} + 11 T_{11}^{8} - T_{11}^{7} - 335 T_{11}^{6} - 577 T_{11}^{5} + 3115 T_{11}^{4} + \cdots + 15336 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{18} + 26 T^{16} + \cdots + 29 \) Copy content Toggle raw display
$3$ \( T^{18} + 36 T^{16} + \cdots + 4901 \) Copy content Toggle raw display
$5$ \( T^{18} + 4 T^{16} + \cdots + 1953125 \) Copy content Toggle raw display
$7$ \( T^{18} + 85 T^{16} + \cdots + 3572829 \) Copy content Toggle raw display
$11$ \( (T^{9} + 11 T^{8} + \cdots + 15336)^{2} \) Copy content Toggle raw display
$13$ \( T^{18} \) Copy content Toggle raw display
$17$ \( T^{18} + 153 T^{16} + \cdots + 53009216 \) Copy content Toggle raw display
$19$ \( (T^{9} - 14 T^{8} + \cdots + 356504)^{2} \) Copy content Toggle raw display
$23$ \( T^{18} + \cdots + 1663159541 \) Copy content Toggle raw display
$29$ \( (T^{9} - 10 T^{8} + \cdots - 50679)^{2} \) Copy content Toggle raw display
$31$ \( (T^{9} + 16 T^{8} + \cdots + 4544)^{2} \) Copy content Toggle raw display
$37$ \( T^{18} + \cdots + 228661056 \) Copy content Toggle raw display
$41$ \( (T^{9} + 26 T^{8} + \cdots - 2460807)^{2} \) Copy content Toggle raw display
$43$ \( T^{18} + 293 T^{16} + \cdots + 21141 \) Copy content Toggle raw display
$47$ \( T^{18} + \cdots + 33109633009349 \) Copy content Toggle raw display
$53$ \( T^{18} + \cdots + 8958557504 \) Copy content Toggle raw display
$59$ \( (T^{9} - 38 T^{8} + \cdots - 20952)^{2} \) Copy content Toggle raw display
$61$ \( (T^{9} - 4 T^{8} + \cdots + 3297433)^{2} \) Copy content Toggle raw display
$67$ \( T^{18} + \cdots + 190023072011109 \) Copy content Toggle raw display
$71$ \( (T^{9} + 36 T^{8} + \cdots - 508248)^{2} \) Copy content Toggle raw display
$73$ \( T^{18} + \cdots + 158284670828544 \) Copy content Toggle raw display
$79$ \( (T^{9} + 8 T^{8} + \cdots - 53432392)^{2} \) Copy content Toggle raw display
$83$ \( T^{18} + \cdots + 269507038437029 \) Copy content Toggle raw display
$89$ \( (T^{9} - 47 T^{8} + \cdots + 21405033)^{2} \) Copy content Toggle raw display
$97$ \( T^{18} + \cdots + 294008165722944 \) Copy content Toggle raw display
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