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Properties

Label 770.2.m.e
Level $770$
Weight $2$
Character orbit 770.m
Analytic conductor $6.148$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [770,2,Mod(43,770)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(770, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([3, 0, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("770.43");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 770 = 2 \cdot 5 \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 770.m (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: \(6.14848095564\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} + 32x^{14} + 404x^{12} + 2600x^{10} + 9170x^{8} + 17648x^{6} + 17180x^{4} + 6904x^{2} + 529 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{10} q^{2} + ( - \beta_{15} + \beta_{7} - \beta_{3} - 1) q^{3} + \beta_{3} q^{4} + (\beta_{7} - \beta_{3} - 1) q^{5} + (\beta_{10} - \beta_{8} + \cdots + \beta_{4}) q^{6}+ \cdots + ( - 2 \beta_{15} + \beta_{14} + \cdots + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{10} q^{2} + ( - \beta_{15} + \beta_{7} - \beta_{3} - 1) q^{3} + \beta_{3} q^{4} + (\beta_{7} - \beta_{3} - 1) q^{5} + (\beta_{10} - \beta_{8} + \cdots + \beta_{4}) q^{6}+ \cdots + (5 \beta_{15} - \beta_{11} + \cdots + 3 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 8 q^{3} - 8 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 8 q^{3} - 8 q^{5} - 8 q^{11} + 8 q^{12} + 24 q^{15} - 16 q^{16} + 16 q^{20} - 8 q^{22} + 32 q^{23} + 16 q^{25} + 16 q^{26} - 32 q^{27} - 24 q^{33} - 48 q^{36} - 48 q^{37} - 8 q^{38} - 8 q^{42} + 72 q^{45} + 8 q^{48} - 16 q^{53} - 48 q^{55} + 16 q^{56} + 32 q^{58} - 56 q^{60} - 32 q^{66} + 48 q^{67} - 16 q^{70} - 48 q^{71} + 112 q^{75} + 8 q^{77} + 32 q^{78} + 8 q^{80} - 80 q^{81} - 16 q^{82} + 32 q^{86} - 8 q^{88} - 16 q^{91} - 32 q^{92} + 32 q^{93}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} + 32x^{14} + 404x^{12} + 2600x^{10} + 9170x^{8} + 17648x^{6} + 17180x^{4} + 6904x^{2} + 529 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 133 \nu^{14} - 3381 \nu^{12} - 29156 \nu^{10} - 88672 \nu^{8} + 38697 \nu^{6} + 580417 \nu^{4} + \cdots - 104150 ) / 52140 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 96 \nu^{15} - 12857 \nu^{14} - 12594 \nu^{13} - 396267 \nu^{12} - 306090 \nu^{11} + \cdots - 44689 ) / 2398440 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 7 \nu^{15} - 270 \nu^{13} - 4277 \nu^{11} - 35680 \nu^{9} - 165183 \nu^{7} - 404918 \nu^{5} + \cdots - 136832 \nu ) / 30360 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -29\nu^{14} - 837\nu^{12} - 9095\nu^{10} - 46994\nu^{8} - 119553\nu^{6} - 139743\nu^{4} - 59095\nu^{2} - 688 ) / 1580 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 3267 \nu^{15} - 37904 \nu^{14} + 125037 \nu^{13} - 1115040 \nu^{12} + 1853445 \nu^{11} + \cdots - 5847244 ) / 4796880 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 2362 \nu^{15} - 9154 \nu^{14} + 97503 \nu^{13} - 239016 \nu^{12} + 1552478 \nu^{11} + \cdots + 5518597 ) / 2398440 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 14119 \nu^{15} + 4554 \nu^{14} - 384441 \nu^{13} + 185196 \nu^{12} - 3770627 \nu^{11} + \cdots + 12567522 ) / 4796880 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 14119 \nu^{15} - 51244 \nu^{14} + 384441 \nu^{13} - 1510962 \nu^{12} + 3770627 \nu^{11} + \cdots - 15822896 ) / 4796880 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 14119 \nu^{15} + 51244 \nu^{14} + 384441 \nu^{13} + 1510962 \nu^{12} + 3770627 \nu^{11} + \cdots + 15822896 ) / 4796880 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 7788 \nu^{15} - 9706 \nu^{14} + 227205 \nu^{13} - 281451 \nu^{12} + 2511069 \nu^{11} + \cdots - 4272089 ) / 2398440 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 7788 \nu^{15} + 9706 \nu^{14} + 227205 \nu^{13} + 281451 \nu^{12} + 2511069 \nu^{11} + \cdots + 4272089 ) / 2398440 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 16825 \nu^{15} + 44022 \nu^{14} + 477105 \nu^{13} + 1270566 \nu^{12} + 5030831 \nu^{11} + \cdots + 1044384 ) / 4796880 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 40961 \nu^{15} + 33442 \nu^{14} - 1235013 \nu^{13} + 906936 \nu^{12} - 14300983 \nu^{11} + \cdots - 13463050 ) / 4796880 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 40961 \nu^{15} - 21206 \nu^{14} - 1235013 \nu^{13} - 595884 \nu^{12} - 14300983 \nu^{11} + \cdots + 23044850 ) / 4796880 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 40769 \nu^{15} - 44022 \nu^{14} - 1209825 \nu^{13} - 1270566 \nu^{12} - 13688803 \nu^{11} + \cdots - 1044384 ) / 4796880 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -2\beta_{15} + \beta_{14} + \beta_{13} + \beta_{11} - \beta_{10} - \beta_{9} + \beta_{8} + \beta_{4} - 2\beta_{2} + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{10} + \beta_{8} + \beta_{6} - \beta_{5} - \beta _1 - 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 12 \beta_{15} - 5 \beta_{14} - 5 \beta_{13} - 6 \beta_{12} + \beta_{11} + 11 \beta_{10} + 5 \beta_{9} + \cdots - 5 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{14} - \beta_{13} - 2 \beta_{11} + 12 \beta_{10} + \beta_{9} - 13 \beta_{8} - 2 \beta_{7} + \cdots + 28 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 94 \beta_{15} + 45 \beta_{14} + 45 \beta_{13} + 100 \beta_{12} - 19 \beta_{11} - 99 \beta_{10} + \cdots + 35 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 18 \beta_{14} + 18 \beta_{13} + 47 \beta_{11} - 140 \beta_{10} - 13 \beta_{9} + 144 \beta_{8} + \cdots - 242 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 828 \beta_{15} - 473 \beta_{14} - 473 \beta_{13} - 1246 \beta_{12} + 147 \beta_{11} + 925 \beta_{10} + \cdots - 291 ) / 2 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 246 \beta_{14} - 246 \beta_{13} - 692 \beta_{11} + 1568 \beta_{10} + 134 \beta_{9} - 1558 \beta_{8} + \cdots + 2317 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 7750 \beta_{15} + 5091 \beta_{14} + 5091 \beta_{13} + 14208 \beta_{12} - 925 \beta_{11} - 9031 \beta_{10} + \cdots + 2679 ) / 2 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 2984 \beta_{14} + 2984 \beta_{13} + 8686 \beta_{11} - 17149 \beta_{10} - 1334 \beta_{9} + \cdots - 23316 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 75340 \beta_{15} - 54743 \beta_{14} - 54743 \beta_{13} - 156442 \beta_{12} + 4671 \beta_{11} + \cdots - 26187 ) / 2 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 34097 \beta_{14} - 34097 \beta_{13} - 101434 \beta_{11} + 185196 \beta_{10} + 13461 \beta_{9} + \cdots + 240514 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( - 751954 \beta_{15} + 586403 \beta_{14} + 586403 \beta_{13} + 1695148 \beta_{12} - 11285 \beta_{11} + \cdots + 265017 ) / 2 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 377518 \beta_{14} + 377518 \beta_{13} + 1140739 \beta_{11} - 1986410 \beta_{10} - 138513 \beta_{9} + \cdots - 2513158 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( 7648508 \beta_{15} - 6262871 \beta_{14} - 6262871 \beta_{13} - 18218730 \beta_{12} - 156783 \beta_{11} + \cdots - 2736265 ) / 2 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/770\mathbb{Z}\right)^\times\).

\(n\) \(211\) \(617\) \(661\)
\(\chi(n)\) \(-1\) \(-\beta_{3}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
43.1
0.914718i
− 1.72791i
− 3.26007i
1.24483i
2.32893i
− 0.313694i
− 1.84585i
2.65904i
− 0.914718i
1.72791i
3.26007i
− 1.24483i
− 2.32893i
0.313694i
1.84585i
− 2.65904i
−0.707107 − 0.707107i −2.14984 − 2.14984i 1.00000i −1.71500 − 1.43484i 3.04034i 0.707107 + 0.707107i 0.707107 − 0.707107i 6.24365i 0.198102 + 2.22728i
43.2 −0.707107 − 0.707107i −1.40684 − 1.40684i 1.00000i −0.177857 − 2.22898i 1.98957i 0.707107 + 0.707107i 0.707107 − 0.707107i 0.958399i −1.45037 + 1.70189i
43.3 −0.707107 − 0.707107i −0.668611 − 0.668611i 1.00000i −2.17741 + 0.508801i 0.945559i 0.707107 + 0.707107i 0.707107 − 0.707107i − 2.10592i 1.89944 + 1.17989i
43.4 −0.707107 − 0.707107i 2.22529 + 2.22529i 1.00000i 2.07027 − 0.844975i − 3.14704i 0.707107 + 0.707107i 0.707107 − 0.707107i 6.90387i −2.06139 − 0.866414i
43.5 0.707107 + 0.707107i −2.14984 − 2.14984i 1.00000i −1.71500 − 1.43484i − 3.04034i −0.707107 − 0.707107i −0.707107 + 0.707107i 6.24365i −0.198102 − 2.22728i
43.6 0.707107 + 0.707107i −1.40684 − 1.40684i 1.00000i −0.177857 − 2.22898i − 1.98957i −0.707107 − 0.707107i −0.707107 + 0.707107i 0.958399i 1.45037 − 1.70189i
43.7 0.707107 + 0.707107i −0.668611 − 0.668611i 1.00000i −2.17741 + 0.508801i − 0.945559i −0.707107 − 0.707107i −0.707107 + 0.707107i − 2.10592i −1.89944 − 1.17989i
43.8 0.707107 + 0.707107i 2.22529 + 2.22529i 1.00000i 2.07027 − 0.844975i 3.14704i −0.707107 − 0.707107i −0.707107 + 0.707107i 6.90387i 2.06139 + 0.866414i
197.1 −0.707107 + 0.707107i −2.14984 + 2.14984i − 1.00000i −1.71500 + 1.43484i − 3.04034i 0.707107 − 0.707107i 0.707107 + 0.707107i − 6.24365i 0.198102 − 2.22728i
197.2 −0.707107 + 0.707107i −1.40684 + 1.40684i − 1.00000i −0.177857 + 2.22898i − 1.98957i 0.707107 − 0.707107i 0.707107 + 0.707107i − 0.958399i −1.45037 − 1.70189i
197.3 −0.707107 + 0.707107i −0.668611 + 0.668611i − 1.00000i −2.17741 − 0.508801i − 0.945559i 0.707107 − 0.707107i 0.707107 + 0.707107i 2.10592i 1.89944 − 1.17989i
197.4 −0.707107 + 0.707107i 2.22529 − 2.22529i − 1.00000i 2.07027 + 0.844975i 3.14704i 0.707107 − 0.707107i 0.707107 + 0.707107i − 6.90387i −2.06139 + 0.866414i
197.5 0.707107 − 0.707107i −2.14984 + 2.14984i − 1.00000i −1.71500 + 1.43484i 3.04034i −0.707107 + 0.707107i −0.707107 − 0.707107i − 6.24365i −0.198102 + 2.22728i
197.6 0.707107 − 0.707107i −1.40684 + 1.40684i − 1.00000i −0.177857 + 2.22898i 1.98957i −0.707107 + 0.707107i −0.707107 − 0.707107i − 0.958399i 1.45037 + 1.70189i
197.7 0.707107 − 0.707107i −0.668611 + 0.668611i − 1.00000i −2.17741 − 0.508801i 0.945559i −0.707107 + 0.707107i −0.707107 − 0.707107i 2.10592i −1.89944 + 1.17989i
197.8 0.707107 − 0.707107i 2.22529 − 2.22529i − 1.00000i 2.07027 + 0.844975i − 3.14704i −0.707107 + 0.707107i −0.707107 − 0.707107i − 6.90387i 2.06139 − 0.866414i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 43.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.c odd 4 1 inner
11.b odd 2 1 inner
55.e even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 770.2.m.e ✓ 16
5.c odd 4 1 inner 770.2.m.e ✓ 16
11.b odd 2 1 inner 770.2.m.e ✓ 16
55.e even 4 1 inner 770.2.m.e ✓ 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
770.2.m.e ✓ 16 1.a even 1 1 trivial
770.2.m.e ✓ 16 5.c odd 4 1 inner
770.2.m.e ✓ 16 11.b odd 2 1 inner
770.2.m.e ✓ 16 55.e even 4 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(770, [\chi])\):

\( T_{3}^{8} + 4T_{3}^{7} + 8T_{3}^{6} + 8T_{3}^{5} + 100T_{3}^{4} + 392T_{3}^{3} + 800T_{3}^{2} + 720T_{3} + 324 \) Copy content Toggle raw display
\( T_{17}^{16} + 1040T_{17}^{12} + 276576T_{17}^{8} + 21971200T_{17}^{4} + 303595776 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} + 1)^{4} \) Copy content Toggle raw display
$3$ \( (T^{8} + 4 T^{7} + \cdots + 324)^{2} \) Copy content Toggle raw display
$5$ \( (T^{8} + 4 T^{7} + \cdots + 625)^{2} \) Copy content Toggle raw display
$7$ \( (T^{4} + 1)^{4} \) Copy content Toggle raw display
$11$ \( (T^{8} + 4 T^{7} + \cdots + 14641)^{2} \) Copy content Toggle raw display
$13$ \( T^{16} + \cdots + 126247696 \) Copy content Toggle raw display
$17$ \( T^{16} + \cdots + 303595776 \) Copy content Toggle raw display
$19$ \( (T^{8} - 40 T^{6} + \cdots + 324)^{2} \) Copy content Toggle raw display
$23$ \( (T^{8} - 16 T^{7} + \cdots + 1336336)^{2} \) Copy content Toggle raw display
$29$ \( (T^{8} - 176 T^{6} + \cdots + 1763584)^{2} \) Copy content Toggle raw display
$31$ \( (T^{4} - 28 T^{2} + \cdots + 132)^{4} \) Copy content Toggle raw display
$37$ \( (T^{8} + 24 T^{7} + \cdots + 484416)^{2} \) Copy content Toggle raw display
$41$ \( (T^{8} + 120 T^{6} + \cdots + 126736)^{2} \) Copy content Toggle raw display
$43$ \( T^{16} + \cdots + 6553600000000 \) Copy content Toggle raw display
$47$ \( (T^{8} + 160 T^{5} + \cdots + 1936)^{2} \) Copy content Toggle raw display
$53$ \( (T^{8} + 8 T^{7} + \cdots + 10086976)^{2} \) Copy content Toggle raw display
$59$ \( (T^{8} + 480 T^{6} + \cdots + 142229476)^{2} \) Copy content Toggle raw display
$61$ \( (T^{8} + 360 T^{6} + \cdots + 1295044)^{2} \) Copy content Toggle raw display
$67$ \( (T^{8} - 24 T^{7} + \cdots + 6718464)^{2} \) Copy content Toggle raw display
$71$ \( (T^{4} + 12 T^{3} + \cdots + 4444)^{4} \) Copy content Toggle raw display
$73$ \( T^{16} + \cdots + 218889236736 \) Copy content Toggle raw display
$79$ \( (T^{8} - 536 T^{6} + \cdots + 255872016)^{2} \) Copy content Toggle raw display
$83$ \( T^{16} + \cdots + 218794862224656 \) Copy content Toggle raw display
$89$ \( (T^{8} + 384 T^{6} + \cdots + 4596736)^{2} \) Copy content Toggle raw display
$97$ \( (T^{8} - 64 T^{5} + \cdots + 278784)^{2} \) Copy content Toggle raw display
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