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Properties

Label 1775.4.a.e
Level $1775$
Weight $4$
Character orbit 1775.a
Self dual yes
Analytic conductor $104.728$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1775,4,Mod(1,1775)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1775, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1775.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1775 = 5^{2} \cdot 71 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1775.a (trivial)

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: yes
Analytic conductor: \(104.728390260\)
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: \( x^{16} - 5 x^{15} - 78 x^{14} + 345 x^{13} + 2559 x^{12} - 9370 x^{11} - 45148 x^{10} + 126461 x^{9} + \cdots + 2185728 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: no (minimal twist has level 355)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-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.

\(f(q)\) \(=\) \( q + ( - \beta_1 + 1) q^{2} + ( - \beta_{6} + 1) q^{3} + (\beta_{2} - \beta_1 + 4) q^{4} + (\beta_{15} - \beta_{8} - 2 \beta_{6} + \cdots - 1) q^{6}+ \cdots + ( - \beta_{12} - \beta_{11} - \beta_{8} + \cdots + 5) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_1 + 1) q^{2} + ( - \beta_{6} + 1) q^{3} + (\beta_{2} - \beta_1 + 4) q^{4} + (\beta_{15} - \beta_{8} - 2 \beta_{6} + \cdots - 1) q^{6}+ \cdots + (18 \beta_{15} - 15 \beta_{14} + \cdots + 256) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 11 q^{2} + 10 q^{3} + 59 q^{4} - 39 q^{6} + 86 q^{7} + 108 q^{8} + 56 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 11 q^{2} + 10 q^{3} + 59 q^{4} - 39 q^{6} + 86 q^{7} + 108 q^{8} + 56 q^{9} - 114 q^{11} + 128 q^{12} + 262 q^{13} - 140 q^{14} + 35 q^{16} + 468 q^{17} + 388 q^{18} + 6 q^{19} - 30 q^{21} - 143 q^{22} + 438 q^{23} - 157 q^{24} + 481 q^{26} + 10 q^{27} + 726 q^{28} - 688 q^{29} - 26 q^{31} + 335 q^{32} + 838 q^{33} + 184 q^{34} + 1090 q^{36} + 646 q^{37} + 645 q^{38} - 278 q^{39} - 238 q^{41} - 513 q^{42} + 682 q^{43} - 251 q^{44} - 151 q^{46} + 746 q^{47} + 408 q^{48} + 248 q^{49} - 1378 q^{51} + 2970 q^{52} + 2704 q^{53} - 1176 q^{54} - 64 q^{56} + 3002 q^{57} + 700 q^{58} - 358 q^{59} - 1732 q^{61} + 524 q^{62} + 3426 q^{63} - 1658 q^{64} - 1069 q^{66} + 1766 q^{67} + 4304 q^{68} - 1986 q^{69} - 1136 q^{71} + 4066 q^{72} + 2438 q^{73} - 1213 q^{74} + 300 q^{76} + 3572 q^{77} - 1885 q^{78} + 112 q^{79} - 24 q^{81} + 3535 q^{82} + 842 q^{83} + 327 q^{84} - 1344 q^{86} + 2056 q^{87} + 201 q^{88} - 1574 q^{89} + 3960 q^{91} + 3224 q^{92} + 1350 q^{93} + 3091 q^{94} + 5355 q^{96} + 4628 q^{97} - 160 q^{98} + 4854 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 5 x^{15} - 78 x^{14} + 345 x^{13} + 2559 x^{12} - 9370 x^{11} - 45148 x^{10} + 126461 x^{9} + \cdots + 2185728 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - \nu - 11 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 30352734513959 \nu^{15} + 632985351527815 \nu^{14} + \cdots + 50\!\cdots\!44 ) / 33\!\cdots\!84 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 96886422306585 \nu^{15} + 978333546646921 \nu^{14} + \cdots - 68\!\cdots\!20 ) / 33\!\cdots\!84 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 172927659610743 \nu^{15} + \cdots - 46\!\cdots\!56 ) / 33\!\cdots\!84 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 44543414012733 \nu^{15} + 5390736224481 \nu^{14} + \cdots + 57\!\cdots\!76 ) / 82\!\cdots\!96 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 98767895814067 \nu^{15} + 143009842114019 \nu^{14} + \cdots + 11\!\cdots\!08 ) / 16\!\cdots\!92 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 122937728629653 \nu^{15} - 26294869142451 \nu^{14} + \cdots - 49\!\cdots\!88 ) / 16\!\cdots\!92 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 273600015796445 \nu^{15} + 42222468542195 \nu^{14} + \cdots - 15\!\cdots\!64 ) / 33\!\cdots\!84 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 327882373917895 \nu^{15} + \cdots + 19\!\cdots\!44 ) / 33\!\cdots\!84 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 339467980867921 \nu^{15} - 106065478424081 \nu^{14} + \cdots + 22\!\cdots\!16 ) / 33\!\cdots\!84 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 138346259177385 \nu^{15} - 344498228794023 \nu^{14} + \cdots - 15\!\cdots\!40 ) / 82\!\cdots\!96 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 103812779742570 \nu^{15} - 142732780049301 \nu^{14} + \cdots - 16\!\cdots\!00 ) / 41\!\cdots\!48 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 430012201430543 \nu^{15} - 411495015462881 \nu^{14} + \cdots - 38\!\cdots\!80 ) / 16\!\cdots\!92 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 974531193759105 \nu^{15} + \cdots + 15\!\cdots\!36 ) / 33\!\cdots\!84 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + \beta _1 + 11 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( - \beta_{15} - \beta_{13} - \beta_{12} + \beta_{11} + \beta_{8} + \beta_{7} - \beta_{6} + \beta_{4} + \cdots + 11 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - 2 \beta_{15} + 2 \beta_{14} - 2 \beta_{13} - 4 \beta_{12} + 3 \beta_{11} - 3 \beta_{10} + 3 \beta_{8} + \cdots + 208 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 35 \beta_{15} + 9 \beta_{14} - 36 \beta_{13} - 42 \beta_{12} + 36 \beta_{11} - 14 \beta_{10} + \cdots + 478 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 103 \beta_{15} + 111 \beta_{14} - 118 \beta_{13} - 207 \beta_{12} + 147 \beta_{11} - 145 \beta_{10} + \cdots + 4837 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 1055 \beta_{15} + 559 \beta_{14} - 1170 \beta_{13} - 1505 \beta_{12} + 1198 \beta_{11} - 754 \beta_{10} + \cdots + 16636 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 3919 \beta_{15} + 4638 \beta_{14} - 5023 \beta_{13} - 8066 \beta_{12} + 5685 \beta_{11} + \cdots + 127879 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 31224 \beta_{15} + 24967 \beta_{14} - 38337 \beta_{13} - 51469 \beta_{12} + 39319 \beta_{11} + \cdots + 548520 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 135112 \beta_{15} + 175161 \beta_{14} - 190105 \beta_{13} - 285814 \beta_{12} + 203053 \beta_{11} + \cdots + 3673184 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 937384 \beta_{15} + 977624 \beta_{14} - 1275768 \beta_{13} - 1725959 \beta_{12} + 1292561 \beta_{11} + \cdots + 17843962 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 4490497 \beta_{15} + 6303286 \beta_{14} - 6824211 \beta_{13} - 9742179 \beta_{12} + 7009346 \beta_{11} + \cdots + 111172304 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 28761180 \beta_{15} + 35856981 \beta_{14} - 42860693 \beta_{13} - 57375205 \beta_{12} + \cdots + 579803129 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 147353306 \beta_{15} + 220889246 \beta_{14} - 238484452 \beta_{13} - 326565331 \beta_{12} + \cdots + 3477137215 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 900926053 \beta_{15} + 1268958157 \beta_{14} - 1445994464 \beta_{13} - 1900895314 \beta_{12} + \cdots + 18893058727 \) Copy content Toggle raw display

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
5.78524
5.06498
4.40525
3.64239
3.64020
1.58518
1.27672
1.03917
−0.618890
−0.839561
−2.09434
−3.02007
−3.13614
−3.41658
−3.97068
−4.34286
−4.78524 3.70128 14.8985 0 −17.7115 33.4901 −33.0109 −13.3005 0
1.2 −4.06498 7.69622 8.52406 0 −31.2850 6.97708 −2.13031 32.2318 0
1.3 −3.40525 0.448998 3.59573 0 −1.52895 −1.13758 14.9977 −26.7984 0
1.4 −2.64239 −4.99876 −1.01778 0 13.2087 33.2218 23.8285 −2.01242 0
1.5 −2.64020 2.09237 −1.02936 0 −5.52427 −17.0725 23.8393 −22.6220 0
1.6 −0.585183 −5.50163 −7.65756 0 3.21947 −2.44966 9.16255 3.26798 0
1.7 −0.276715 8.81646 −7.92343 0 −2.43965 3.86738 4.40626 50.7300 0
1.8 −0.0391658 −4.52998 −7.99847 0 0.177420 12.9428 0.626592 −6.47931 0
1.9 1.61889 4.07210 −5.37920 0 6.59228 −7.09539 −21.6594 −10.4180 0
1.10 1.83956 0.908105 −4.61602 0 1.67051 4.96565 −23.2079 −26.1753 0
1.11 3.09434 −6.18356 1.57492 0 −19.1340 −8.29356 −19.8814 11.2364 0
1.12 4.02007 7.63082 8.16099 0 30.6765 30.3653 0.647179 31.2294 0
1.13 4.13614 −0.840068 9.10764 0 −3.47464 6.72356 4.58133 −26.2943 0
1.14 4.41658 −0.725222 11.5062 0 −3.20300 −34.1856 15.4853 −26.4741 0
1.15 4.97068 −9.61619 16.7077 0 −47.7990 26.3441 43.2831 65.4712 0
1.16 5.34286 7.02906 20.5461 0 37.5552 −2.66344 67.0321 22.4076 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.16
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(5\) \(1\)
\(71\) \(1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1775.4.a.e 16
5.b even 2 1 355.4.a.b ✓ 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
355.4.a.b ✓ 16 5.b even 2 1
1775.4.a.e 16 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{16} - 11 T_{2}^{15} - 33 T_{2}^{14} + 712 T_{2}^{13} - 509 T_{2}^{12} - 17399 T_{2}^{11} + \cdots + 52672 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1775))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} - 11 T^{15} + \cdots + 52672 \) Copy content Toggle raw display
$3$ \( T^{16} + \cdots - 211205984 \) Copy content Toggle raw display
$5$ \( T^{16} \) Copy content Toggle raw display
$7$ \( T^{16} + \cdots - 26\!\cdots\!04 \) Copy content Toggle raw display
$11$ \( T^{16} + \cdots - 27\!\cdots\!52 \) Copy content Toggle raw display
$13$ \( T^{16} + \cdots - 83\!\cdots\!16 \) Copy content Toggle raw display
$17$ \( T^{16} + \cdots + 13\!\cdots\!96 \) Copy content Toggle raw display
$19$ \( T^{16} + \cdots - 59\!\cdots\!44 \) Copy content Toggle raw display
$23$ \( T^{16} + \cdots + 11\!\cdots\!32 \) Copy content Toggle raw display
$29$ \( T^{16} + \cdots - 36\!\cdots\!44 \) Copy content Toggle raw display
$31$ \( T^{16} + \cdots - 80\!\cdots\!72 \) Copy content Toggle raw display
$37$ \( T^{16} + \cdots + 15\!\cdots\!68 \) Copy content Toggle raw display
$41$ \( T^{16} + \cdots - 14\!\cdots\!72 \) Copy content Toggle raw display
$43$ \( T^{16} + \cdots - 34\!\cdots\!84 \) Copy content Toggle raw display
$47$ \( T^{16} + \cdots + 54\!\cdots\!92 \) Copy content Toggle raw display
$53$ \( T^{16} + \cdots - 47\!\cdots\!64 \) Copy content Toggle raw display
$59$ \( T^{16} + \cdots + 15\!\cdots\!92 \) Copy content Toggle raw display
$61$ \( T^{16} + \cdots - 27\!\cdots\!64 \) Copy content Toggle raw display
$67$ \( T^{16} + \cdots + 14\!\cdots\!04 \) Copy content Toggle raw display
$71$ \( (T + 71)^{16} \) Copy content Toggle raw display
$73$ \( T^{16} + \cdots + 40\!\cdots\!12 \) Copy content Toggle raw display
$79$ \( T^{16} + \cdots - 51\!\cdots\!84 \) Copy content Toggle raw display
$83$ \( T^{16} + \cdots + 70\!\cdots\!96 \) Copy content Toggle raw display
$89$ \( T^{16} + \cdots + 20\!\cdots\!84 \) Copy content Toggle raw display
$97$ \( T^{16} + \cdots - 24\!\cdots\!12 \) Copy content Toggle raw display
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