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Properties

Label 960.4.h.b
Level $960$
Weight $4$
Character orbit 960.h
Analytic conductor $56.642$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $4$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [960,4,Mod(191,960)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(960, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 1, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("960.191");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 960 = 2^{6} \cdot 3 \cdot 5 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 960.h (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: \(56.6418336055\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 4 x^{15} + 5 x^{14} + 28 x^{13} - 121 x^{12} + 208 x^{11} + 48 x^{10} - 1168 x^{9} + \cdots + 390625 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{34}\cdot 3^{8}\cdot 5^{6} \)
Twist minimal: no (minimal twist has level 240)
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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{3} q^{3} - \beta_{6} q^{5} + (\beta_{9} + \beta_{7}) q^{7} + ( - 2 \beta_{6} + \beta_{2} - 4) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{3} q^{3} - \beta_{6} q^{5} + (\beta_{9} + \beta_{7}) q^{7} + ( - 2 \beta_{6} + \beta_{2} - 4) q^{9} + ( - \beta_{11} - \beta_{10} + \cdots - 2 \beta_{3}) q^{11}+ \cdots + ( - 18 \beta_{14} + 27 \beta_{11} + \cdots - 9 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 72 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 72 q^{9} - 112 q^{13} + 216 q^{21} - 400 q^{25} - 864 q^{33} + 880 q^{37} - 600 q^{45} - 1376 q^{49} - 1296 q^{57} - 560 q^{61} - 5112 q^{69} + 1792 q^{73} - 2304 q^{81} + 1440 q^{85} - 1152 q^{93} + 4768 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 4 x^{15} + 5 x^{14} + 28 x^{13} - 121 x^{12} + 208 x^{11} + 48 x^{10} - 1168 x^{9} + \cdots + 390625 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 20203 \nu^{15} - 9765408 \nu^{14} + 3504240 \nu^{13} + 60952716 \nu^{12} + \cdots - 1119042500000 ) / 1414375000 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 986006 \nu^{15} - 7925625 \nu^{14} + 25332624 \nu^{13} - 14281272 \nu^{12} + \cdots - 256826453125 ) / 8627687500 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 7114423 \nu^{15} + 32905262 \nu^{14} - 12075120 \nu^{13} - 209186644 \nu^{12} + \cdots + 1578501843750 ) / 34510750000 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 3697142 \nu^{15} + 16299903 \nu^{14} - 3759600 \nu^{13} - 159752976 \nu^{12} + \cdots + 393322359375 ) / 8627687500 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 9843 \nu^{15} - 255048 \nu^{14} + 479040 \nu^{13} + 209496 \nu^{12} - 5759757 \nu^{11} + \cdots - 14735625000 ) / 22812500 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 1902902 \nu^{15} - 4859838 \nu^{14} - 5706720 \nu^{13} + 63146331 \nu^{12} + \cdots - 282372734375 ) / 4313843750 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 423298 \nu^{15} + 1532817 \nu^{14} + 558960 \nu^{13} - 15141144 \nu^{12} + \cdots + 108842578125 ) / 695781250 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 33389032 \nu^{15} + 208022995 \nu^{14} - 61121088 \nu^{13} - 1377631096 \nu^{12} + \cdots + 12859696359375 ) / 34510750000 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 2971954 \nu^{15} - 9586561 \nu^{14} + 1304400 \nu^{13} + 75978512 \nu^{12} + \cdots - 308009453125 ) / 2783125000 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 49706839 \nu^{15} - 102896771 \nu^{14} - 76691520 \nu^{13} + 1312731292 \nu^{12} + \cdots - 3240789609375 ) / 43138437500 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 9022256 \nu^{15} + 23739895 \nu^{14} + 2770656 \nu^{13} - 205906968 \nu^{12} + \cdots + 314079171875 ) / 6902150000 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 60254518 \nu^{15} - 92612967 \nu^{14} - 230929680 \nu^{13} + 1891408554 \nu^{12} + \cdots - 2684831953125 ) / 43138437500 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 135275282 \nu^{15} + 625900893 \nu^{14} - 529440720 \nu^{13} - 3756722196 \nu^{12} + \cdots + 25543301640625 ) / 43138437500 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 298263532 \nu^{15} + 619177013 \nu^{14} + 239344800 \nu^{13} - 6805589096 \nu^{12} + \cdots + 9343558515625 ) / 86276875000 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 167452678 \nu^{15} + 484531167 \nu^{14} + 286740240 \nu^{13} - 5277421884 \nu^{12} + \cdots + 25714414609375 ) / 43138437500 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( - 2 \beta_{15} + 2 \beta_{14} - \beta_{13} - 5 \beta_{12} - 11 \beta_{11} + 10 \beta_{10} - 15 \beta_{9} + \cdots + 61 ) / 240 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 5 \beta_{15} - 3 \beta_{14} - 7 \beta_{13} + 13 \beta_{12} - 24 \beta_{11} - 42 \beta_{10} - 30 \beta_{9} + \cdots + 83 ) / 240 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 8\beta_{15} - 11\beta_{13} - 10\beta_{12} + 32\beta_{6} - 4\beta_{5} + 2\beta_{4} - 39\beta_{2} - 619 ) / 120 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 9 \beta_{15} - 37 \beta_{14} + 9 \beta_{13} - 3 \beta_{12} + 88 \beta_{11} + 43 \beta_{10} + \cdots + 295 ) / 240 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 6 \beta_{15} + 18 \beta_{14} + 11 \beta_{13} - 37 \beta_{12} + 3 \beta_{11} + 130 \beta_{10} + \cdots - 1155 ) / 80 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 22\beta_{15} + 9\beta_{13} - 17\beta_{12} + 273\beta_{6} - 19\beta_{4} + 77\beta_{2} + 1527 ) / 30 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 14 \beta_{15} + 978 \beta_{14} - 336 \beta_{13} - 388 \beta_{12} - 297 \beta_{11} + 1554 \beta_{10} + \cdots + 12050 ) / 240 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 447 \beta_{15} + 1123 \beta_{14} + 115 \beta_{13} + 2219 \beta_{12} - 2128 \beta_{11} + 1495 \beta_{10} + \cdots + 25073 ) / 240 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 1208 \beta_{15} - 778 \beta_{13} + 1763 \beta_{12} + 2699 \beta_{6} + 212 \beta_{5} + 2335 \beta_{4} + \cdots - 23666 ) / 120 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 1065 \beta_{15} - 4601 \beta_{14} + 569 \beta_{13} - 791 \beta_{12} + 4832 \beta_{11} - 2096 \beta_{10} + \cdots - 8161 ) / 80 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 5566 \beta_{15} - 10546 \beta_{14} - 568 \beta_{13} + 560 \beta_{12} + 21523 \beta_{11} + \cdots - 727982 ) / 240 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( 768 \beta_{15} + 941 \beta_{13} + 2702 \beta_{12} + 2418 \beta_{6} + 560 \beta_{5} - 2071 \beta_{4} + \cdots - 116367 ) / 15 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 37606 \beta_{15} + 53010 \beta_{14} - 32261 \beta_{13} + 81161 \beta_{12} + 158781 \beta_{11} + \cdots + 238913 ) / 240 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( - 53059 \beta_{15} + 55325 \beta_{14} + 83849 \beta_{13} - 15859 \beta_{12} + 323704 \beta_{11} + \cdots + 4048283 ) / 240 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( - 30024 \beta_{15} + 7521 \beta_{13} + 20358 \beta_{12} - 146264 \beta_{6} - 6612 \beta_{5} + \cdots + 1053865 ) / 40 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/960\mathbb{Z}\right)^\times\).

\(n\) \(511\) \(577\) \(641\) \(901\)
\(\chi(n)\) \(-1\) \(1\) \(-1\) \(1\)

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
−2.22641 + 0.207615i
−2.22641 − 0.207615i
−2.20232 − 0.387018i
−2.20232 + 0.387018i
1.56591 − 1.59622i
1.56591 + 1.59622i
−0.641852 + 2.14197i
−0.641852 − 2.14197i
2.17592 + 0.515124i
2.17592 − 0.515124i
0.599418 − 2.15423i
0.599418 + 2.15423i
1.43633 + 1.71376i
1.43633 − 1.71376i
1.29300 − 1.82432i
1.29300 + 1.82432i
0 −4.57475 − 2.46408i 0 5.00000i 0 35.9395i 0 14.8566 + 22.5451i 0
191.2 0 −4.57475 + 2.46408i 0 − 5.00000i 0 − 35.9395i 0 14.8566 − 22.5451i 0
191.3 0 −4.14878 − 3.12852i 0 5.00000i 0 − 16.1728i 0 7.42476 + 25.9591i 0
191.4 0 −4.14878 + 3.12852i 0 − 5.00000i 0 16.1728i 0 7.42476 − 25.9591i 0
191.5 0 −2.57835 − 4.51133i 0 − 5.00000i 0 10.9905i 0 −13.7042 + 23.2636i 0
191.6 0 −2.57835 + 4.51133i 0 5.00000i 0 − 10.9905i 0 −13.7042 − 23.2636i 0
191.7 0 −0.459783 − 5.17577i 0 5.00000i 0 − 6.48076i 0 −26.5772 + 4.75946i 0
191.8 0 −0.459783 + 5.17577i 0 − 5.00000i 0 6.48076i 0 −26.5772 − 4.75946i 0
191.9 0 0.459783 − 5.17577i 0 − 5.00000i 0 − 6.48076i 0 −26.5772 − 4.75946i 0
191.10 0 0.459783 + 5.17577i 0 5.00000i 0 6.48076i 0 −26.5772 + 4.75946i 0
191.11 0 2.57835 − 4.51133i 0 5.00000i 0 10.9905i 0 −13.7042 − 23.2636i 0
191.12 0 2.57835 + 4.51133i 0 − 5.00000i 0 − 10.9905i 0 −13.7042 + 23.2636i 0
191.13 0 4.14878 − 3.12852i 0 − 5.00000i 0 − 16.1728i 0 7.42476 − 25.9591i 0
191.14 0 4.14878 + 3.12852i 0 5.00000i 0 16.1728i 0 7.42476 + 25.9591i 0
191.15 0 4.57475 − 2.46408i 0 − 5.00000i 0 35.9395i 0 14.8566 − 22.5451i 0
191.16 0 4.57475 + 2.46408i 0 5.00000i 0 − 35.9395i 0 14.8566 + 22.5451i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 191.16
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 960.4.h.b 16
3.b odd 2 1 inner 960.4.h.b 16
4.b odd 2 1 inner 960.4.h.b 16
8.b even 2 1 240.4.h.b ✓ 16
8.d odd 2 1 240.4.h.b ✓ 16
12.b even 2 1 inner 960.4.h.b 16
24.f even 2 1 240.4.h.b ✓ 16
24.h odd 2 1 240.4.h.b ✓ 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
240.4.h.b ✓ 16 8.b even 2 1
240.4.h.b ✓ 16 8.d odd 2 1
240.4.h.b ✓ 16 24.f even 2 1
240.4.h.b ✓ 16 24.h odd 2 1
960.4.h.b 16 1.a even 1 1 trivial
960.4.h.b 16 3.b odd 2 1 inner
960.4.h.b 16 4.b odd 2 1 inner
960.4.h.b 16 12.b even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(960, [\chi])\):

\( T_{7}^{8} + 1716T_{7}^{6} + 595764T_{7}^{4} + 62877600T_{7}^{2} + 1713960000 \) Copy content Toggle raw display
\( T_{11}^{8} - 8040T_{11}^{6} + 15790032T_{11}^{4} - 821636352T_{11}^{2} + 1743897600 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} \) Copy content Toggle raw display
$3$ \( T^{16} + \cdots + 282429536481 \) Copy content Toggle raw display
$5$ \( (T^{2} + 25)^{8} \) Copy content Toggle raw display
$7$ \( (T^{8} + 1716 T^{6} + \cdots + 1713960000)^{2} \) Copy content Toggle raw display
$11$ \( (T^{8} - 8040 T^{6} + \cdots + 1743897600)^{2} \) Copy content Toggle raw display
$13$ \( (T^{4} + 28 T^{3} + \cdots + 4948288)^{4} \) Copy content Toggle raw display
$17$ \( (T^{8} + \cdots + 28441315641600)^{2} \) Copy content Toggle raw display
$19$ \( (T^{8} + 7368 T^{6} + \cdots + 74384925696)^{2} \) Copy content Toggle raw display
$23$ \( (T^{8} + \cdots + 386287577640000)^{2} \) Copy content Toggle raw display
$29$ \( (T^{8} + \cdots + 23\!\cdots\!00)^{2} \) Copy content Toggle raw display
$31$ \( (T^{8} + \cdots + 56\!\cdots\!24)^{2} \) Copy content Toggle raw display
$37$ \( (T^{4} - 220 T^{3} + \cdots - 2015172800)^{4} \) Copy content Toggle raw display
$41$ \( (T^{8} + \cdots + 21\!\cdots\!00)^{2} \) Copy content Toggle raw display
$43$ \( (T^{8} + \cdots + 57\!\cdots\!76)^{2} \) Copy content Toggle raw display
$47$ \( (T^{8} + \cdots + 29\!\cdots\!00)^{2} \) Copy content Toggle raw display
$53$ \( (T^{8} + \cdots + 12\!\cdots\!00)^{2} \) Copy content Toggle raw display
$59$ \( (T^{8} + \cdots + 10\!\cdots\!00)^{2} \) Copy content Toggle raw display
$61$ \( (T^{4} + 140 T^{3} + \cdots + 20881227520)^{4} \) Copy content Toggle raw display
$67$ \( (T^{8} + \cdots + 82\!\cdots\!84)^{2} \) Copy content Toggle raw display
$71$ \( (T^{8} + \cdots + 14\!\cdots\!00)^{2} \) Copy content Toggle raw display
$73$ \( (T^{4} - 448 T^{3} + \cdots + 2158531600)^{4} \) Copy content Toggle raw display
$79$ \( (T^{8} + \cdots + 96\!\cdots\!00)^{2} \) Copy content Toggle raw display
$83$ \( (T^{8} + \cdots + 60\!\cdots\!00)^{2} \) Copy content Toggle raw display
$89$ \( (T^{8} + \cdots + 61\!\cdots\!00)^{2} \) Copy content Toggle raw display
$97$ \( (T^{4} - 1192 T^{3} + \cdots - 110002742000)^{4} \) Copy content Toggle raw display
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