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Properties

Label 75.4.e.d
Level $75$
Weight $4$
Character orbit 75.e
Analytic conductor $4.425$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [75,4,Mod(32,75)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(75, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([2, 1]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("75.32");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 75 = 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 75.e (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: \(4.42514325043\)
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: \( x^{16} - 36x^{14} + 562x^{12} - 3672x^{10} + 16413x^{8} - 6588x^{6} + 43024x^{4} + 499896x^{2} + 532900 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{6}\cdot 3^{8}\cdot 5^{4} \)
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.

\(f(q)\) \(=\) \( q + \beta_{3} q^{2} - \beta_{8} q^{3} + (\beta_{5} + 6 \beta_1) q^{4} + ( - \beta_{6} + \beta_{4} - 6) q^{6} + ( - \beta_{12} - \beta_{9} + \cdots - \beta_{3}) q^{7}+ \cdots + ( - \beta_{15} + \beta_{11} + \cdots - 11 \beta_1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{3} q^{2} - \beta_{8} q^{3} + (\beta_{5} + 6 \beta_1) q^{4} + ( - \beta_{6} + \beta_{4} - 6) q^{6} + ( - \beta_{12} - \beta_{9} + \cdots - \beta_{3}) q^{7}+ \cdots + ( - 33 \beta_{15} - 27 \beta_{11} + \cdots + 510 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 84 q^{6}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 84 q^{6} - 232 q^{16} + 816 q^{21} - 1208 q^{31} + 252 q^{36} + 1872 q^{46} + 156 q^{51} - 1528 q^{61} - 3420 q^{66} + 1064 q^{76} + 6876 q^{81} - 10008 q^{91} - 8172 q^{96}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 36x^{14} + 562x^{12} - 3672x^{10} + 16413x^{8} - 6588x^{6} + 43024x^{4} + 499896x^{2} + 532900 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 166 \nu^{14} + 6345 \nu^{12} - 107207 \nu^{10} + 837000 \nu^{8} - 4351513 \nu^{6} + \cdots - 61944300 ) / 17721150 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 1476838 \nu^{14} - 56632185 \nu^{12} + 969443611 \nu^{10} - 7877375550 \nu^{8} + \cdots + 850813760250 ) / 135306887300 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 5782554 \nu^{14} - 226662200 \nu^{12} + 3896596638 \nu^{10} - 30968795985 \nu^{8} + \cdots + 1184712292450 ) / 405920661900 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 1236 \nu^{14} - 46710 \nu^{12} + 752616 \nu^{10} - 4977045 \nu^{8} + 16234140 \nu^{6} + \cdots + 507816635 ) / 65682955 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 21947581 \nu^{14} - 815488875 \nu^{12} + 13314880712 \nu^{10} - 97200412890 \nu^{8} + \cdots + 7590938337300 ) / 202960330950 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 602367 \nu^{15} - 721824 \nu^{14} - 24352340 \nu^{13} + 27278640 \nu^{12} + \cdots - 296564914840 ) / 76717691440 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 1529999985 \nu^{15} - 3594093388 \nu^{14} - 56466906750 \nu^{13} + 134283386850 \nu^{12} + \cdots - 11\!\cdots\!00 ) / 118528833274800 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 2076350345 \nu^{15} + 5205196818 \nu^{14} + 77769418120 \nu^{13} + \cdots + 17\!\cdots\!00 ) / 118528833274800 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 155639007 \nu^{15} - 5942535720 \nu^{13} + 100420254144 \nu^{11} - 788497463630 \nu^{9} + \cdots + 45288096872360 \nu ) / 3950961109160 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 801486 \nu^{15} - 28294900 \nu^{13} + 429690357 \nu^{11} - 2628310960 \nu^{9} + \cdots + 482698562980 \nu ) / 19179422860 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 5705218035 \nu^{15} - 6408693652 \nu^{14} - 211546219050 \nu^{13} + 238122751500 \nu^{12} + \cdots - 22\!\cdots\!00 ) / 118528833274800 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 1529999985 \nu^{15} - 2219173136 \nu^{14} + 56466906750 \nu^{13} + \cdots - 642863561395300 ) / 29632208318700 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 658212131 \nu^{15} + 24539889600 \nu^{13} - 403007898822 \nu^{11} + \cdots - 282904140779460 \nu ) / 11852883327480 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 2076350345 \nu^{15} + 2926025931 \nu^{14} - 77769418120 \nu^{13} - 108048759270 \nu^{12} + \cdots + 10\!\cdots\!00 ) / 29632208318700 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 84726489 \nu^{15} - 3193068843 \nu^{13} + 52930433070 \nu^{11} - 397922684157 \nu^{9} + \cdots + 29738030133204 \nu ) / 395096110916 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{15} - 2\beta_{11} - 3\beta_{9} - \beta_{5} ) / 15 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{4} - 6\beta_{2} - 9\beta _1 + 14 ) / 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 4 \beta_{15} - 9 \beta_{13} - 15 \beta_{12} - 18 \beta_{11} - 9 \beta_{10} - 36 \beta_{9} + \cdots - 15 \beta_{3} ) / 15 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 4\beta_{14} + 16\beta_{8} - 18\beta_{5} - 9\beta_{4} - 36\beta_{3} - 76\beta_{2} - 252\beta _1 + 69 ) / 3 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 19 \beta_{15} - 150 \beta_{14} - 360 \beta_{13} - 225 \beta_{12} - 52 \beta_{11} + \cdots + 150 \beta_{2} ) / 15 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 54 \beta_{14} - 60 \beta_{12} + 216 \beta_{8} - 240 \beta_{7} - 405 \beta_{5} + 64 \beta_{4} + \cdots - 1166 ) / 3 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 1066 \beta_{15} - 4725 \beta_{14} - 8109 \beta_{13} - 1540 \beta_{12} + 4572 \beta_{11} + \cdots + 4725 \beta_{2} ) / 15 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 64 \beta_{14} - 1512 \beta_{12} + 256 \beta_{8} - 6048 \beta_{7} - 5208 \beta_{5} + 5121 \beta_{4} + \cdots - 54681 ) / 3 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 27269 \beta_{15} - 84600 \beta_{14} - 135000 \beta_{13} + 26325 \beta_{12} + 151108 \beta_{11} + \cdots + 84600 \beta_{2} ) / 15 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 14922 \beta_{14} - 22320 \beta_{12} - 59688 \beta_{8} - 89280 \beta_{7} - 23085 \beta_{5} + \cdots - 1274566 ) / 3 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 511286 \beta_{15} - 957825 \beta_{14} - 1512459 \beta_{13} + 1221110 \beta_{12} + \cdots + 957825 \beta_{2} ) / 15 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 145832 \beta_{14} - 56628 \beta_{12} - 583328 \beta_{8} - 226512 \beta_{7} + 305844 \beta_{5} + \cdots - 6858577 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( - 6773419 \beta_{15} - 1797900 \beta_{14} - 2693340 \beta_{13} + 27070875 \beta_{12} + \cdots + 1797900 \beta_{2} ) / 15 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( - 7976178 \beta_{14} + 1870440 \beta_{12} - 31904712 \beta_{8} + 7481760 \beta_{7} + 33427485 \beta_{5} + \cdots - 207545486 ) / 3 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( - 39995806 \beta_{15} + 239583825 \beta_{14} + 388141191 \beta_{13} + 409395810 \beta_{12} + \cdots - 239583825 \beta_{2} ) / 15 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/75\mathbb{Z}\right)^\times\).

\(n\) \(26\) \(52\)
\(\chi(n)\) \(-1\) \(-\beta_{1}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
32.1
1.55042 − 1.22474i
−1.55042 + 1.22474i
0.0852547 + 1.22474i
−0.0852547 − 1.22474i
2.36424 + 1.22474i
−2.36424 − 1.22474i
−3.99991 − 1.22474i
3.99991 + 1.22474i
1.55042 + 1.22474i
−1.55042 − 1.22474i
0.0852547 − 1.22474i
−0.0852547 + 1.22474i
2.36424 − 1.22474i
−2.36424 + 1.22474i
−3.99991 + 1.22474i
3.99991 − 1.22474i
−3.39887 + 3.39887i 0.624963 + 5.15843i − 15.1047i 0 −19.6570 − 15.4087i −19.7241 − 19.7241i 24.1479 + 24.1479i −26.2188 + 6.44766i 0
32.2 −3.39887 + 3.39887i 5.15843 + 0.624963i − 15.1047i 0 −19.6570 + 15.4087i 19.7241 + 19.7241i 24.1479 + 24.1479i 26.2188 + 6.44766i 0
32.3 −1.39558 + 1.39558i −4.93508 − 1.62635i 4.10469i 0 9.15703 − 4.61761i −3.80245 − 3.80245i −16.8931 − 16.8931i 21.7100 + 16.0523i 0
32.4 −1.39558 + 1.39558i −1.62635 − 4.93508i 4.10469i 0 9.15703 + 4.61761i 3.80245 + 3.80245i −16.8931 − 16.8931i −21.7100 + 16.0523i 0
32.5 1.39558 − 1.39558i 1.62635 + 4.93508i 4.10469i 0 9.15703 + 4.61761i −3.80245 − 3.80245i 16.8931 + 16.8931i −21.7100 + 16.0523i 0
32.6 1.39558 − 1.39558i 4.93508 + 1.62635i 4.10469i 0 9.15703 − 4.61761i 3.80245 + 3.80245i 16.8931 + 16.8931i 21.7100 + 16.0523i 0
32.7 3.39887 − 3.39887i −5.15843 − 0.624963i − 15.1047i 0 −19.6570 + 15.4087i −19.7241 − 19.7241i −24.1479 − 24.1479i 26.2188 + 6.44766i 0
32.8 3.39887 − 3.39887i −0.624963 − 5.15843i − 15.1047i 0 −19.6570 − 15.4087i 19.7241 + 19.7241i −24.1479 − 24.1479i −26.2188 + 6.44766i 0
68.1 −3.39887 − 3.39887i 0.624963 − 5.15843i 15.1047i 0 −19.6570 + 15.4087i −19.7241 + 19.7241i 24.1479 − 24.1479i −26.2188 − 6.44766i 0
68.2 −3.39887 − 3.39887i 5.15843 − 0.624963i 15.1047i 0 −19.6570 − 15.4087i 19.7241 − 19.7241i 24.1479 − 24.1479i 26.2188 − 6.44766i 0
68.3 −1.39558 − 1.39558i −4.93508 + 1.62635i − 4.10469i 0 9.15703 + 4.61761i −3.80245 + 3.80245i −16.8931 + 16.8931i 21.7100 − 16.0523i 0
68.4 −1.39558 − 1.39558i −1.62635 + 4.93508i − 4.10469i 0 9.15703 − 4.61761i 3.80245 − 3.80245i −16.8931 + 16.8931i −21.7100 − 16.0523i 0
68.5 1.39558 + 1.39558i 1.62635 − 4.93508i − 4.10469i 0 9.15703 − 4.61761i −3.80245 + 3.80245i 16.8931 − 16.8931i −21.7100 − 16.0523i 0
68.6 1.39558 + 1.39558i 4.93508 − 1.62635i − 4.10469i 0 9.15703 + 4.61761i 3.80245 − 3.80245i 16.8931 − 16.8931i 21.7100 − 16.0523i 0
68.7 3.39887 + 3.39887i −5.15843 + 0.624963i 15.1047i 0 −19.6570 − 15.4087i −19.7241 + 19.7241i −24.1479 + 24.1479i 26.2188 − 6.44766i 0
68.8 3.39887 + 3.39887i −0.624963 + 5.15843i 15.1047i 0 −19.6570 + 15.4087i 19.7241 − 19.7241i −24.1479 + 24.1479i −26.2188 − 6.44766i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 32.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
5.b even 2 1 inner
5.c odd 4 2 inner
15.d odd 2 1 inner
15.e even 4 2 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 75.4.e.d ✓ 16
3.b odd 2 1 inner 75.4.e.d ✓ 16
5.b even 2 1 inner 75.4.e.d ✓ 16
5.c odd 4 2 inner 75.4.e.d ✓ 16
15.d odd 2 1 inner 75.4.e.d ✓ 16
15.e even 4 2 inner 75.4.e.d ✓ 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
75.4.e.d ✓ 16 1.a even 1 1 trivial
75.4.e.d ✓ 16 3.b odd 2 1 inner
75.4.e.d ✓ 16 5.b even 2 1 inner
75.4.e.d ✓ 16 5.c odd 4 2 inner
75.4.e.d ✓ 16 15.d odd 2 1 inner
75.4.e.d ✓ 16 15.e even 4 2 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{8} + 549T_{2}^{4} + 8100 \) acting on \(S_{4}^{\mathrm{new}}(75, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{8} + 549 T^{4} + 8100)^{2} \) Copy content Toggle raw display
$3$ \( T^{16} + \cdots + 282429536481 \) Copy content Toggle raw display
$5$ \( T^{16} \) Copy content Toggle raw display
$7$ \( (T^{8} + 606249 T^{4} + 506250000)^{2} \) Copy content Toggle raw display
$11$ \( (T^{4} + 3915 T^{2} + 324000)^{4} \) Copy content Toggle raw display
$13$ \( (T^{8} + \cdots + 21767823360000)^{2} \) Copy content Toggle raw display
$17$ \( (T^{8} + \cdots + 539435927577600)^{2} \) Copy content Toggle raw display
$19$ \( (T^{4} + 9002 T^{2} + 2399401)^{4} \) Copy content Toggle raw display
$23$ \( (T^{8} + \cdots + 113427612057600)^{2} \) Copy content Toggle raw display
$29$ \( (T^{4} - 73440 T^{2} + 1327104000)^{4} \) Copy content Toggle raw display
$31$ \( (T^{2} + 151 T + 3394)^{8} \) Copy content Toggle raw display
$37$ \( (T^{8} + \cdots + 58027829760000)^{2} \) Copy content Toggle raw display
$41$ \( (T^{4} + 3915 T^{2} + 324000)^{4} \) Copy content Toggle raw display
$43$ \( (T^{8} + \cdots + 82\!\cdots\!00)^{2} \) Copy content Toggle raw display
$47$ \( (T^{8} + \cdots + 36\!\cdots\!00)^{2} \) Copy content Toggle raw display
$53$ \( (T^{8} + \cdots + 18\!\cdots\!00)^{2} \) Copy content Toggle raw display
$59$ \( (T^{4} - 166860 T^{2} + 6718464000)^{4} \) Copy content Toggle raw display
$61$ \( (T^{2} + 191 T - 103886)^{8} \) Copy content Toggle raw display
$67$ \( (T^{8} + \cdots + 30\!\cdots\!25)^{2} \) Copy content Toggle raw display
$71$ \( (T^{4} + 1697760 T^{2} + 613453824000)^{4} \) Copy content Toggle raw display
$73$ \( (T^{8} + \cdots + 57\!\cdots\!00)^{2} \) Copy content Toggle raw display
$79$ \( (T^{4} + 500852 T^{2} + 19593280576)^{4} \) Copy content Toggle raw display
$83$ \( (T^{8} + \cdots + 36\!\cdots\!00)^{2} \) Copy content Toggle raw display
$89$ \( (T^{4} - 370035 T^{2} + 16548624000)^{4} \) Copy content Toggle raw display
$97$ \( (T^{8} + \cdots + 19\!\cdots\!00)^{2} \) Copy content Toggle raw display
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