↑

Properties

Label 608.3.d.b
Level $608$
Weight $3$
Character orbit 608.d
Analytic conductor $16.567$
Analytic rank $0$
Dimension $20$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [608,3,Mod(191,608)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(608, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 0]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("608.191");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 608 = 2^{5} \cdot 19 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 608.d (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: \(16.5668000731\)
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: \( x^{20} + 124 x^{18} + 6294 x^{16} + 169580 x^{14} + 2633777 x^{12} + 23965840 x^{10} + 123396288 x^{8} + \cdots + 6553600 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{32}\cdot 3^{4} \)
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.

\(f(q)\) \(=\) \( q - \beta_1 q^{3} + \beta_{4} q^{5} + \beta_{8} q^{7} + (\beta_{2} - 3) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{3} + \beta_{4} q^{5} + \beta_{8} q^{7} + (\beta_{2} - 3) q^{9} + ( - \beta_{14} - \beta_1) q^{11} + ( - \beta_{7} + \beta_{5} + \beta_{2} - 1) q^{13} + (\beta_{17} + \beta_{15} - 2 \beta_{9}) q^{15} + ( - \beta_{10} - \beta_{4}) q^{17} - \beta_{9} q^{19} + (\beta_{7} - \beta_{6} - \beta_{4} + \cdots + 3) q^{21}+ \cdots + (3 \beta_{19} - \beta_{18} + \cdots + 17 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q - 68 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 20 q - 68 q^{9} - 16 q^{13} + 8 q^{17} + 64 q^{21} + 196 q^{25} - 88 q^{29} - 184 q^{33} + 16 q^{37} - 16 q^{41} + 16 q^{45} + 52 q^{49} + 88 q^{53} - 208 q^{61} - 192 q^{65} + 248 q^{69} - 152 q^{73} + 312 q^{77} + 500 q^{81} - 488 q^{85} + 304 q^{89} - 240 q^{93} - 72 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} + 124 x^{18} + 6294 x^{16} + 169580 x^{14} + 2633777 x^{12} + 23965840 x^{10} + 123396288 x^{8} + \cdots + 6553600 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} + 12 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 25103 \nu^{18} + 3237084 \nu^{16} + 169735218 \nu^{14} + 4648358572 \nu^{12} + \cdots - 32987668480 ) / 419098890240 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 1032450871 \nu^{18} - 126244048878 \nu^{16} - 6286063245726 \nu^{14} + \cdots - 91\!\cdots\!20 ) / 26\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 1487983051 \nu^{18} - 166419434268 \nu^{16} - 7295301858306 \nu^{14} + \cdots - 32\!\cdots\!20 ) / 35\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 4165 \nu^{18} + 572124 \nu^{16} + 32475982 \nu^{14} + 983850588 \nu^{12} + 17133441109 \nu^{10} + \cdots + 257697169408 ) / 8609480704 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 12631322621 \nu^{18} + 1587344183028 \nu^{16} + 81921041128806 \nu^{14} + \cdots + 73\!\cdots\!80 ) / 21\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 466405 \nu^{19} - 61504812 \nu^{17} - 3346585806 \nu^{15} - 97217180732 \nu^{13} + \cdots - 169616414261248 \nu ) / 7185002987520 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 1893775 \nu^{19} - 223484148 \nu^{17} - 10593807114 \nu^{15} + \cdots + 256322160508928 \nu ) / 13862592184320 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 7287122581 \nu^{18} + 859893789528 \nu^{16} + 40826526594546 \nu^{14} + \cdots + 17\!\cdots\!00 ) / 52\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 660516649 \nu^{18} + 78536441997 \nu^{16} + 3769382618364 \nu^{14} + 94198473006356 \nu^{12} + \cdots - 15\!\cdots\!20 ) / 437574166318080 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 19577143477 \nu^{18} - 2358245586996 \nu^{16} - 115250281142502 \nu^{14} + \cdots - 42\!\cdots\!80 ) / 10\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 1965845 \nu^{19} + 243363132 \nu^{17} + 12321235086 \nu^{15} + 330652231612 \nu^{13} + \cdots + 140188967026688 \nu ) / 6705582243840 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 3626217871 \nu^{19} + 454955199624 \nu^{17} + 23415709916874 \nu^{15} + \cdots + 17\!\cdots\!96 \nu ) / 42\!\cdots\!68 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 77339351927 \nu^{19} - 9661629702564 \nu^{17} - 494708103710106 \nu^{15} + \cdots - 39\!\cdots\!32 \nu ) / 84\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( 10232384963 \nu^{19} + 1242569240964 \nu^{17} + 61426179484098 \nu^{15} + \cdots + 11\!\cdots\!60 \nu ) / 98\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( 1821319438 \nu^{19} + 229012248099 \nu^{17} + 11821970657178 \nu^{15} + \cdots + 15\!\cdots\!20 \nu ) / 17\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( 6396460927 \nu^{19} + 786617024004 \nu^{17} + 39503185646826 \nu^{15} + \cdots + 43\!\cdots\!92 \nu ) / 27\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( - 460077190321 \nu^{19} - 56255483874636 \nu^{17} + \cdots - 22\!\cdots\!52 \nu ) / 16\!\cdots\!20 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} - 12 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{19} - \beta_{16} - \beta_{14} - \beta_{13} + 4\beta_{9} - 22\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 3\beta_{12} - 2\beta_{11} + \beta_{10} + 4\beta_{7} - 5\beta_{5} - 7\beta_{4} - \beta_{3} - 36\beta_{2} + 268 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 43 \beta_{19} + 6 \beta_{18} + 2 \beta_{17} + 39 \beta_{16} + 14 \beta_{15} + 45 \beta_{14} + \cdots + 584 \beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 129 \beta_{12} + 78 \beta_{11} - 3 \beta_{10} - 180 \beta_{7} + 16 \beta_{6} + 239 \beta_{5} + \cdots - 7088 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 1513 \beta_{19} - 328 \beta_{18} - 188 \beta_{17} - 1253 \beta_{16} - 884 \beta_{15} + \cdots - 16754 \beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 4491 \beta_{12} - 2626 \beta_{11} - 1031 \beta_{10} + 6580 \beta_{7} - 1216 \beta_{6} - 8877 \beta_{5} + \cdots + 202804 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 50291 \beta_{19} + 13114 \beta_{18} + 10294 \beta_{17} + 38815 \beta_{16} + 40378 \beta_{15} + \cdots + 500004 \beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 149401 \beta_{12} + 86286 \beta_{11} + 63909 \beta_{10} - 229092 \beta_{7} + 62448 \beta_{6} + \cdots - 6046280 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 1640577 \beta_{19} - 468276 \beta_{18} - 451976 \beta_{17} - 1201453 \beta_{16} + \cdots - 15282934 \beta_1 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 4931379 \beta_{12} - 2825730 \beta_{11} - 2844319 \beta_{10} + 7874948 \beta_{7} - 2719968 \beta_{6} + \cdots + 184806428 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 53223547 \beta_{19} + 15881294 \beta_{18} + 17745386 \beta_{17} + 37502071 \beta_{16} + \cdots + 475032544 \beta_1 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 163009361 \beta_{12} + 92589710 \beta_{11} + 111697389 \beta_{10} - 269544852 \beta_{7} + \cdots - 5747432160 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 1725866713 \beta_{19} - 524717152 \beta_{18} - 654214292 \beta_{17} - 1182317109 \beta_{16} + \cdots - 14956529338 \beta_1 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( 5404388507 \beta_{12} - 3037680962 \beta_{11} - 4120082167 \beta_{10} + 9200959316 \beta_{7} + \cdots + 181113098884 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( 56049877571 \beta_{19} + 17100490914 \beta_{18} + 23226065886 \beta_{17} + 37622356751 \beta_{16} + \cdots + 475785535196 \beta_1 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( - 179621167689 \beta_{12} + 99797809294 \beta_{11} + 146639895925 \beta_{10} - 313237137476 \beta_{7} + \cdots - 5767089628664 \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( - 1824456615025 \beta_{19} - 553461696204 \beta_{18} - 805418866144 \beta_{17} + \cdots - 15262436723134 \beta_1 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/608\mathbb{Z}\right)^\times\).

\(n\) \(97\) \(191\) \(229\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
191.1
5.76701i
5.22378i
4.82918i
3.58592i
3.19537i
3.04083i
2.46116i
1.05616i
0.760762i
0.255387i
− 0.255387i
− 0.760762i
− 1.05616i
− 2.46116i
− 3.04083i
− 3.19537i
− 3.58592i
− 4.82918i
− 5.22378i
− 5.76701i
0 − 5.76701i 0 −6.90719 0 7.08805i 0 −24.2584 0
191.2 0 − 5.22378i 0 1.68508 0 − 3.72116i 0 −18.2879 0
191.3 0 − 4.82918i 0 9.30469 0 0.458734i 0 −14.3210 0
191.4 0 − 3.58592i 0 −5.60073 0 8.91584i 0 −3.85886 0
191.5 0 − 3.19537i 0 −1.24360 0 − 7.44432i 0 −1.21039 0
191.6 0 − 3.04083i 0 −0.295883 0 1.32785i 0 −0.246657 0
191.7 0 − 2.46116i 0 8.93826 0 6.32680i 0 2.94271 0
191.8 0 − 1.05616i 0 −9.08870 0 − 10.7326i 0 7.88453 0
191.9 0 − 0.760762i 0 3.86554 0 − 8.82674i 0 8.42124 0
191.10 0 − 0.255387i 0 −0.657472 0 − 5.46816i 0 8.93478 0
191.11 0 0.255387i 0 −0.657472 0 5.46816i 0 8.93478 0
191.12 0 0.760762i 0 3.86554 0 8.82674i 0 8.42124 0
191.13 0 1.05616i 0 −9.08870 0 10.7326i 0 7.88453 0
191.14 0 2.46116i 0 8.93826 0 − 6.32680i 0 2.94271 0
191.15 0 3.04083i 0 −0.295883 0 − 1.32785i 0 −0.246657 0
191.16 0 3.19537i 0 −1.24360 0 7.44432i 0 −1.21039 0
191.17 0 3.58592i 0 −5.60073 0 − 8.91584i 0 −3.85886 0
191.18 0 4.82918i 0 9.30469 0 − 0.458734i 0 −14.3210 0
191.19 0 5.22378i 0 1.68508 0 3.72116i 0 −18.2879 0
191.20 0 5.76701i 0 −6.90719 0 − 7.08805i 0 −24.2584 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 191.20
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 608.3.d.b ✓ 20
4.b odd 2 1 inner 608.3.d.b ✓ 20
8.b even 2 1 1216.3.d.f 20
8.d odd 2 1 1216.3.d.f 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
608.3.d.b ✓ 20 1.a even 1 1 trivial
608.3.d.b ✓ 20 4.b odd 2 1 inner
1216.3.d.f 20 8.b even 2 1
1216.3.d.f 20 8.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{20} + 124 T_{3}^{18} + 6294 T_{3}^{16} + 169580 T_{3}^{14} + 2633777 T_{3}^{12} + 23965840 T_{3}^{10} + \cdots + 6553600 \) acting on \(S_{3}^{\mathrm{new}}(608, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{20} \) Copy content Toggle raw display
$3$ \( T^{20} + 124 T^{18} + \cdots + 6553600 \) Copy content Toggle raw display
$5$ \( (T^{10} - 174 T^{8} + \cdots + 46080)^{2} \) Copy content Toggle raw display
$7$ \( T^{20} + \cdots + 12214186214400 \) Copy content Toggle raw display
$11$ \( T^{20} + \cdots + 41\!\cdots\!00 \) Copy content Toggle raw display
$13$ \( (T^{10} + 8 T^{9} + \cdots - 6220800)^{2} \) Copy content Toggle raw display
$17$ \( (T^{10} - 4 T^{9} + \cdots + 1676951236)^{2} \) Copy content Toggle raw display
$19$ \( (T^{2} + 19)^{10} \) Copy content Toggle raw display
$23$ \( T^{20} + \cdots + 26\!\cdots\!00 \) Copy content Toggle raw display
$29$ \( (T^{10} + \cdots - 47830231992960)^{2} \) Copy content Toggle raw display
$31$ \( T^{20} + \cdots + 46\!\cdots\!96 \) Copy content Toggle raw display
$37$ \( (T^{10} + \cdots + 389060288722944)^{2} \) Copy content Toggle raw display
$41$ \( (T^{10} + \cdots + 207062902374400)^{2} \) Copy content Toggle raw display
$43$ \( T^{20} + \cdots + 16\!\cdots\!00 \) Copy content Toggle raw display
$47$ \( T^{20} + \cdots + 47\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( (T^{10} + \cdots + 15\!\cdots\!20)^{2} \) Copy content Toggle raw display
$59$ \( T^{20} + \cdots + 29\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( (T^{10} + \cdots - 93\!\cdots\!96)^{2} \) Copy content Toggle raw display
$67$ \( T^{20} + \cdots + 33\!\cdots\!00 \) Copy content Toggle raw display
$71$ \( T^{20} + \cdots + 13\!\cdots\!56 \) Copy content Toggle raw display
$73$ \( (T^{10} + \cdots + 89\!\cdots\!00)^{2} \) Copy content Toggle raw display
$79$ \( T^{20} + \cdots + 44\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{20} + \cdots + 37\!\cdots\!00 \) Copy content Toggle raw display
$89$ \( (T^{10} + \cdots - 53\!\cdots\!00)^{2} \) Copy content Toggle raw display
$97$ \( (T^{10} + \cdots + 12\!\cdots\!00)^{2} \) Copy content Toggle raw display
show more
show less