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Properties

Label 1170.2.w.g
Level $1170$
Weight $2$
Character orbit 1170.w
Analytic conductor $9.342$
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] = [1170,2,Mod(307,1170)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1170, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 1, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1170.307");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1170 = 2 \cdot 3^{2} \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1170.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: \(9.34249703649\)
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: \( x^{14} - 2 x^{13} + 2 x^{12} + 4 x^{11} + 112 x^{10} - 208 x^{9} + 200 x^{8} + 392 x^{7} + 1708 x^{6} + \cdots + 8 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{3} \)
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.

\(f(q)\) \(=\) \( q - \beta_{3} q^{2} - q^{4} + \beta_{11} q^{5} - \beta_{4} q^{7} + \beta_{3} q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{3} q^{2} - q^{4} + \beta_{11} q^{5} - \beta_{4} q^{7} + \beta_{3} q^{8} + \beta_{8} q^{10} + ( - \beta_{8} - \beta_{6} + \cdots + \beta_{3}) q^{11}+ \cdots + ( - \beta_{13} - \beta_{12} - \beta_{11} + \cdots - 1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 14 q - 14 q^{4} - 2 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 14 q - 14 q^{4} - 2 q^{5} + 4 q^{13} + 14 q^{16} + 14 q^{17} - 12 q^{19} + 2 q^{20} - 10 q^{25} + 6 q^{26} + 12 q^{31} + 14 q^{34} + 12 q^{35} + 20 q^{37} - 12 q^{38} - 2 q^{41} - 8 q^{47} + 14 q^{49} - 4 q^{52} + 6 q^{53} + 32 q^{55} + 16 q^{58} - 16 q^{59} + 24 q^{61} - 12 q^{62} - 14 q^{64} - 4 q^{65} - 14 q^{68} - 4 q^{70} - 24 q^{71} + 12 q^{76} + 16 q^{77} - 2 q^{80} + 2 q^{82} + 8 q^{83} + 2 q^{85} + 42 q^{89} - 8 q^{91} + 24 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{14} - 2 x^{13} + 2 x^{12} + 4 x^{11} + 112 x^{10} - 208 x^{9} + 200 x^{8} + 392 x^{7} + 1708 x^{6} + \cdots + 8 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 18173568310 \nu^{13} - 247381310443 \nu^{12} + 346095423701 \nu^{11} - 165652901047 \nu^{10} + \cdots - 9508243655196 ) / 2076047884764 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 30484029866 \nu^{13} + 10381236581 \nu^{12} - 45509689146 \nu^{11} + 204629354406 \nu^{10} + \cdots + 2344039589764 ) / 2076047884764 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 33301966367 \nu^{13} + 43374794091 \nu^{12} - 20753562656 \nu^{11} - 179944777142 \nu^{10} + \cdots + 210465790228 ) / 2076047884764 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 26113858887 \nu^{13} + 77092578313 \nu^{12} - 77409445594 \nu^{11} - 92404812016 \nu^{10} + \cdots + 2337185084848 ) / 1038023942382 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 69121579871 \nu^{13} + 87277353993 \nu^{12} - 23610801077 \nu^{11} - 380635108712 \nu^{10} + \cdots + 1946663734060 ) / 2076047884764 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 140562324261 \nu^{13} - 390865208924 \nu^{12} + 457700449226 \nu^{11} + 421218178202 \nu^{10} + \cdots - 2108327844356 ) / 2076047884764 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 170027010778 \nu^{13} + 425303531290 \nu^{12} - 431226654221 \nu^{11} + \cdots + 4416863379900 ) / 2076047884764 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 193256149421 \nu^{13} + 471153901368 \nu^{12} - 477963565895 \nu^{11} + \cdots + 4683279110836 ) / 2076047884764 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 111059415017 \nu^{13} + 168404528172 \nu^{12} - 121098378647 \nu^{11} + \cdots + 884052444052 ) / 1038023942382 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 36161069743 \nu^{13} - 64793036727 \nu^{12} + 57742896001 \nu^{11} + 156155172550 \nu^{10} + \cdots - 879526710332 ) / 296578269252 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 391040121791 \nu^{13} + 645210923282 \nu^{12} - 536443819903 \nu^{11} + \cdots + 3513550138068 ) / 2076047884764 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 414269260434 \nu^{13} + 691061293360 \nu^{12} - 583180731577 \nu^{11} + \cdots + 3779965869004 ) / 2076047884764 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 594513374825 \nu^{13} - 977596935143 \nu^{12} + 776523597301 \nu^{11} + \cdots - 11214898462380 ) / 2076047884764 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{12} - \beta_{11} - \beta_{8} + \beta_{7} ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{13} - \beta_{12} - \beta_{9} + \beta_{8} - 4\beta_{3} + \beta_1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -4\beta_{12} + 4\beta_{11} + \beta_{9} - 5\beta_{8} + 4\beta_{7} - \beta_{6} - \beta_{5} + \beta_{4} + \beta_{3} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - 9 \beta_{13} - 8 \beta_{11} + \beta_{10} + \beta_{8} - 9 \beta_{7} + \beta_{6} - \beta_{5} + \cdots - 33 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 36 \beta_{12} + 49 \beta_{11} + 11 \beta_{10} - 15 \beta_{9} + 37 \beta_{8} - 36 \beta_{7} + \cdots + 11 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 82 \beta_{13} + 83 \beta_{12} - 14 \beta_{11} - 15 \beta_{10} + 92 \beta_{9} - 70 \beta_{8} + 3 \beta_{7} + \cdots - 15 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 36 \beta_{13} + 331 \beta_{12} - 357 \beta_{11} - 20 \beta_{10} - 174 \beta_{9} + 479 \beta_{8} + \cdots - 32 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 760 \beta_{13} - 64 \beta_{12} + 646 \beta_{11} - 178 \beta_{10} - 154 \beta_{8} + 780 \beta_{7} + \cdots + 2718 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 3206 \beta_{12} - 4670 \beta_{11} - 1112 \beta_{10} + 1856 \beta_{9} - 3480 \beta_{8} + 3082 \beta_{7} + \cdots - 1364 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 7126 \beta_{13} - 7400 \beta_{12} + 1582 \beta_{11} + 1856 \beta_{10} - 7696 \beta_{9} + 6128 \beta_{8} + \cdots + 1856 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 5572 \beta_{13} - 28938 \beta_{12} + 33940 \beta_{11} + 3214 \beta_{10} + 19062 \beta_{9} - 45420 \beta_{8} + \cdots + 6862 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 67360 \beta_{13} + 11578 \beta_{12} - 59020 \beta_{11} + 19918 \beta_{10} + 15848 \beta_{8} + \cdots - 240558 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 295598 \beta_{12} + 440918 \beta_{11} + 106340 \beta_{10} - 191808 \beta_{9} + 330382 \beta_{8} + \cdots + 151796 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1170\mathbb{Z}\right)^\times\).

\(n\) \(911\) \(937\) \(1081\)
\(\chi(n)\) \(1\) \(-\beta_{3}\) \(-\beta_{3}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
307.1
2.18338 + 2.18338i
1.85561 + 1.85561i
0.372382 + 0.372382i
0.287242 + 0.287242i
−0.202596 − 0.202596i
−1.29182 − 1.29182i
−2.20421 − 2.20421i
2.18338 − 2.18338i
1.85561 − 1.85561i
0.372382 − 0.372382i
0.287242 − 0.287242i
−0.202596 + 0.202596i
−1.29182 + 1.29182i
−2.20421 + 2.20421i
1.00000i 0 −1.00000 −2.18338 + 0.482541i 0 2.10714 − 1.00000i 0 −0.482541 − 2.18338i
307.2 1.00000i 0 −1.00000 −1.85561 − 1.24768i 0 −4.63041 − 1.00000i 0 1.24768 − 1.85561i
307.3 1.00000i 0 −1.00000 −0.372382 + 2.20484i 0 1.64209 − 1.00000i 0 −2.20484 − 0.372382i
307.4 1.00000i 0 −1.00000 −0.287242 − 2.21754i 0 −1.27394 − 1.00000i 0 2.21754 − 0.287242i
307.5 1.00000i 0 −1.00000 0.202596 + 2.22687i 0 −0.902312 − 1.00000i 0 −2.22687 + 0.202596i
307.6 1.00000i 0 −1.00000 1.29182 − 1.82516i 0 4.71554 − 1.00000i 0 1.82516 + 1.29182i
307.7 1.00000i 0 −1.00000 2.20421 + 0.376123i 0 −1.65811 − 1.00000i 0 −0.376123 + 2.20421i
343.1 − 1.00000i 0 −1.00000 −2.18338 − 0.482541i 0 2.10714 1.00000i 0 −0.482541 + 2.18338i
343.2 − 1.00000i 0 −1.00000 −1.85561 + 1.24768i 0 −4.63041 1.00000i 0 1.24768 + 1.85561i
343.3 − 1.00000i 0 −1.00000 −0.372382 − 2.20484i 0 1.64209 1.00000i 0 −2.20484 + 0.372382i
343.4 − 1.00000i 0 −1.00000 −0.287242 + 2.21754i 0 −1.27394 1.00000i 0 2.21754 + 0.287242i
343.5 − 1.00000i 0 −1.00000 0.202596 − 2.22687i 0 −0.902312 1.00000i 0 −2.22687 − 0.202596i
343.6 − 1.00000i 0 −1.00000 1.29182 + 1.82516i 0 4.71554 1.00000i 0 1.82516 − 1.29182i
343.7 − 1.00000i 0 −1.00000 2.20421 − 0.376123i 0 −1.65811 1.00000i 0 −0.376123 − 2.20421i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 307.7
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1170.2.w.g yes 14
3.b odd 2 1 1170.2.w.h yes 14
5.c odd 4 1 1170.2.m.h yes 14
13.d odd 4 1 1170.2.m.h yes 14
15.e even 4 1 1170.2.m.g ✓ 14
39.f even 4 1 1170.2.m.g ✓ 14
65.f even 4 1 inner 1170.2.w.g yes 14
195.u odd 4 1 1170.2.w.h yes 14
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1170.2.m.g ✓ 14 15.e even 4 1
1170.2.m.g ✓ 14 39.f even 4 1
1170.2.m.h yes 14 5.c odd 4 1
1170.2.m.h yes 14 13.d odd 4 1
1170.2.w.g yes 14 1.a even 1 1 trivial
1170.2.w.g yes 14 65.f even 4 1 inner
1170.2.w.h yes 14 3.b odd 2 1
1170.2.w.h yes 14 195.u odd 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1170, [\chi])\):

\( T_{7}^{7} - 28T_{7}^{5} - 4T_{7}^{4} + 144T_{7}^{3} + 64T_{7}^{2} - 204T_{7} - 144 \) Copy content Toggle raw display
\( T_{11}^{14} + 88 T_{11}^{11} + 1052 T_{11}^{10} + 2736 T_{11}^{9} + 3872 T_{11}^{8} + 7424 T_{11}^{7} + \cdots + 1229312 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 1)^{7} \) Copy content Toggle raw display
$3$ \( T^{14} \) Copy content Toggle raw display
$5$ \( T^{14} + 2 T^{13} + \cdots + 78125 \) Copy content Toggle raw display
$7$ \( (T^{7} - 28 T^{5} + \cdots - 144)^{2} \) Copy content Toggle raw display
$11$ \( T^{14} + 88 T^{11} + \cdots + 1229312 \) Copy content Toggle raw display
$13$ \( T^{14} - 4 T^{13} + \cdots + 62748517 \) Copy content Toggle raw display
$17$ \( T^{14} + \cdots + 221509152 \) Copy content Toggle raw display
$19$ \( T^{14} + 12 T^{13} + \cdots + 415872 \) Copy content Toggle raw display
$23$ \( T^{14} + 176 T^{11} + \cdots + 123008 \) Copy content Toggle raw display
$29$ \( T^{14} + 140 T^{12} + \cdots + 451584 \) Copy content Toggle raw display
$31$ \( T^{14} + \cdots + 966592512 \) Copy content Toggle raw display
$37$ \( (T^{7} - 10 T^{6} + \cdots + 37584)^{2} \) Copy content Toggle raw display
$41$ \( T^{14} + 2 T^{13} + \cdots + 19668992 \) Copy content Toggle raw display
$43$ \( T^{14} + \cdots + 5960572928 \) Copy content Toggle raw display
$47$ \( (T^{7} + 4 T^{6} + \cdots + 27392)^{2} \) Copy content Toggle raw display
$53$ \( T^{14} + \cdots + 87408822272 \) Copy content Toggle raw display
$59$ \( T^{14} + \cdots + 131609088 \) Copy content Toggle raw display
$61$ \( (T^{7} - 12 T^{6} + \cdots + 1125888)^{2} \) Copy content Toggle raw display
$67$ \( T^{14} + \cdots + 1103302656 \) Copy content Toggle raw display
$71$ \( T^{14} + \cdots + 411041792 \) Copy content Toggle raw display
$73$ \( T^{14} + \cdots + 140499028224 \) Copy content Toggle raw display
$79$ \( T^{14} + \cdots + 2453815296 \) Copy content Toggle raw display
$83$ \( (T^{7} - 4 T^{6} + \cdots + 41472)^{2} \) Copy content Toggle raw display
$89$ \( T^{14} + \cdots + 341754368 \) Copy content Toggle raw display
$97$ \( T^{14} + \cdots + 760704929856 \) Copy content Toggle raw display
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