Properties

Label 1775.4.a.g
Level $1775$
Weight $4$
Character orbit 1775.a
Self dual yes
Analytic conductor $104.728$
Analytic rank $1$
Dimension $20$
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: \(1\)
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: \( x^{20} - 9 x^{19} - 89 x^{18} + 952 x^{17} + 2911 x^{16} - 41549 x^{15} - 37799 x^{14} + 974485 x^{13} - 43478 x^{12} - 13402754 x^{11} + 6726996 x^{10} + \cdots - 106929408 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{4} \)
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 \(1,\beta_1,\ldots,\beta_{19}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} + (\beta_{5} - 1) q^{3} + (\beta_{2} + \beta_1 + 5) q^{4} + ( - \beta_{10} - \beta_{5} + \beta_1) q^{6} + (\beta_{4} - 2) q^{7} + ( - \beta_{3} - 4 \beta_1 - 5) q^{8} + ( - \beta_{15} - \beta_{5} - \beta_{2} + 12) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{2} + (\beta_{5} - 1) q^{3} + (\beta_{2} + \beta_1 + 5) q^{4} + ( - \beta_{10} - \beta_{5} + \beta_1) q^{6} + (\beta_{4} - 2) q^{7} + ( - \beta_{3} - 4 \beta_1 - 5) q^{8} + ( - \beta_{15} - \beta_{5} - \beta_{2} + 12) q^{9} + ( - \beta_{18} - \beta_{5} - \beta_{4} - \beta_1 + 4) q^{11} + (\beta_{10} - \beta_{9} + \beta_{6} + 3 \beta_{5} - 2 \beta_{2} - 2 \beta_1 - 9) q^{12} + (\beta_{11} + 2 \beta_1 - 12) q^{13} + ( - \beta_{19} + 2 \beta_{18} + 2 \beta_{15} + \beta_{12} + \beta_{11} - \beta_{9} + \beta_{6} + \beta_{4} + \cdots + 6 \beta_1) q^{14}+ \cdots + ( - 17 \beta_{19} - \beta_{18} - \beta_{16} + 30 \beta_{15} - 8 \beta_{14} - 9 \beta_{13} + \cdots - 258) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q - 9 q^{2} - 14 q^{3} + 99 q^{4} + 9 q^{6} - 40 q^{7} - 132 q^{8} + 236 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 20 q - 9 q^{2} - 14 q^{3} + 99 q^{4} + 9 q^{6} - 40 q^{7} - 132 q^{8} + 236 q^{9} + 62 q^{11} - 160 q^{12} - 232 q^{13} + 56 q^{14} + 387 q^{16} - 314 q^{17} - 224 q^{18} + 54 q^{19} + 58 q^{21} + 123 q^{22} - 344 q^{23} + 419 q^{24} - 387 q^{26} - 326 q^{27} - 786 q^{28} + 748 q^{29} - 18 q^{31} - 345 q^{32} - 746 q^{33} - 142 q^{34} + 90 q^{36} - 858 q^{37} - 723 q^{38} + 210 q^{39} + 386 q^{41} + 237 q^{42} - 1210 q^{43} + 1105 q^{44} - 57 q^{46} - 320 q^{47} - 2280 q^{48} + 1576 q^{49} - 350 q^{51} - 780 q^{52} - 2066 q^{53} - 536 q^{54} + 938 q^{56} - 710 q^{57} - 1852 q^{58} + 1198 q^{59} + 284 q^{61} - 2444 q^{62} - 732 q^{63} + 1118 q^{64} - 219 q^{66} - 976 q^{67} - 2904 q^{68} + 1026 q^{69} + 1420 q^{71} - 3374 q^{72} - 4310 q^{73} + 955 q^{74} + 740 q^{76} - 5196 q^{77} - 61 q^{78} - 340 q^{79} + 2556 q^{81} - 1191 q^{82} + 354 q^{83} - 1955 q^{84} + 1248 q^{86} - 3392 q^{87} - 135 q^{88} - 1446 q^{89} - 240 q^{91} - 4114 q^{92} + 1066 q^{93} - 3985 q^{94} - 2145 q^{96} - 3282 q^{97} + 4708 q^{98} - 6506 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} - 9 x^{19} - 89 x^{18} + 952 x^{17} + 2911 x^{16} - 41549 x^{15} - 37799 x^{14} + 974485 x^{13} - 43478 x^{12} - 13402754 x^{11} + 6726996 x^{10} + \cdots - 106929408 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - \nu - 13 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} - 20\nu - 5 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 26\!\cdots\!31 \nu^{19} + \cdots + 61\!\cdots\!72 ) / 72\!\cdots\!88 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 37\!\cdots\!97 \nu^{19} + \cdots + 87\!\cdots\!96 ) / 60\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 55\!\cdots\!51 \nu^{19} + \cdots - 11\!\cdots\!00 ) / 24\!\cdots\!96 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 87\!\cdots\!61 \nu^{19} + \cdots + 12\!\cdots\!68 ) / 36\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 21\!\cdots\!83 \nu^{19} + \cdots - 19\!\cdots\!04 ) / 88\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 69\!\cdots\!83 \nu^{19} + \cdots + 13\!\cdots\!48 ) / 24\!\cdots\!96 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 27\!\cdots\!75 \nu^{19} + \cdots - 48\!\cdots\!72 ) / 60\!\cdots\!24 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 88\!\cdots\!03 \nu^{19} + \cdots + 10\!\cdots\!76 ) / 18\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 23\!\cdots\!69 \nu^{19} + \cdots - 21\!\cdots\!00 ) / 36\!\cdots\!44 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 25\!\cdots\!29 \nu^{19} + \cdots - 16\!\cdots\!72 ) / 36\!\cdots\!44 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 23\!\cdots\!59 \nu^{19} + \cdots + 29\!\cdots\!92 ) / 30\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 11\!\cdots\!38 \nu^{19} + \cdots - 22\!\cdots\!36 ) / 15\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( 29\!\cdots\!11 \nu^{19} + \cdots - 90\!\cdots\!32 ) / 36\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( - 20\!\cdots\!31 \nu^{19} + \cdots - 29\!\cdots\!52 ) / 24\!\cdots\!96 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( - 29\!\cdots\!51 \nu^{19} + \cdots - 73\!\cdots\!08 ) / 18\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( - 80\!\cdots\!46 \nu^{19} + \cdots - 59\!\cdots\!88 ) / 37\!\cdots\!40 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + \beta _1 + 13 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} + 20\beta _1 + 5 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{19} - \beta_{18} - \beta_{17} - \beta_{16} - \beta_{12} + \beta_{10} + \beta_{9} + \beta_{8} - \beta_{5} + \beta_{3} + 28 \beta_{2} + 30 \beta _1 + 268 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 2 \beta_{18} + 2 \beta_{17} - 2 \beta_{16} + 2 \beta_{15} - 2 \beta_{14} - \beta_{13} - 2 \beta_{10} - 2 \beta_{8} + \beta_{6} + 2 \beta_{5} + \beta_{4} + 35 \beta_{3} + 463 \beta _1 + 170 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 35 \beta_{19} - 43 \beta_{18} - 34 \beta_{17} - 42 \beta_{16} + 5 \beta_{15} - 4 \beta_{14} + 11 \beta_{13} - 39 \beta_{12} + 8 \beta_{11} + 41 \beta_{10} + 39 \beta_{9} + 35 \beta_{8} + 6 \beta_{7} - 55 \beta_{5} - 6 \beta_{4} + 39 \beta_{3} + 745 \beta_{2} + \cdots + 6307 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 19 \beta_{19} - 93 \beta_{18} + 106 \beta_{17} - 111 \beta_{16} + 108 \beta_{15} - 106 \beta_{14} - 28 \beta_{13} - 10 \beta_{12} + 18 \beta_{11} - 120 \beta_{10} - 46 \beta_{9} - 98 \beta_{8} + 2 \beta_{7} + 57 \beta_{6} + 80 \beta_{5} + \cdots + 4750 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 970 \beta_{19} - 1454 \beta_{18} - 905 \beta_{17} - 1455 \beta_{16} + 249 \beta_{15} - 194 \beta_{14} + 666 \beta_{13} - 1262 \beta_{12} + 430 \beta_{11} + 1352 \beta_{10} + 1264 \beta_{9} + 938 \beta_{8} + 314 \beta_{7} + \cdots + 158023 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 1417 \beta_{19} - 2997 \beta_{18} + 4431 \beta_{17} - 4438 \beta_{16} + 4483 \beta_{15} - 3932 \beta_{14} - 236 \beta_{13} - 680 \beta_{12} + 1222 \beta_{11} - 4910 \beta_{10} - 2980 \beta_{9} - 3662 \beta_{8} + \cdots + 127502 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 24983 \beta_{19} - 45649 \beta_{18} - 22034 \beta_{17} - 47681 \beta_{16} + 8938 \beta_{15} - 6470 \beta_{14} + 28476 \beta_{13} - 38836 \beta_{12} + 16356 \beta_{11} + 41204 \beta_{10} + 38886 \beta_{9} + \cdots + 4100882 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 69483 \beta_{19} - 83541 \beta_{18} + 166308 \beta_{17} - 157494 \beta_{16} + 166273 \beta_{15} - 127022 \beta_{14} + 17784 \beta_{13} - 31021 \beta_{12} + 55130 \beta_{11} - 173501 \beta_{10} + \cdots + 3433780 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 621390 \beta_{19} - 1388546 \beta_{18} - 508028 \beta_{17} - 1523753 \beta_{16} + 287843 \beta_{15} - 182986 \beta_{14} + 1059023 \beta_{13} - 1171823 \beta_{12} + 542994 \beta_{11} + \cdots + 108890412 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 2834830 \beta_{19} - 2164390 \beta_{18} + 5833150 \beta_{17} - 5282559 \beta_{16} + 5772651 \beta_{15} - 3834304 \beta_{14} + 1323824 \beta_{13} - 1205213 \beta_{12} + \cdots + 94206407 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 15060849 \beta_{19} - 41538065 \beta_{18} - 11078236 \beta_{17} - 47936989 \beta_{16} + 8929736 \beta_{15} - 4665044 \beta_{14} + 36631751 \beta_{13} - 35025756 \beta_{12} + \cdots + 2940292791 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 104462521 \beta_{19} - 53876727 \beta_{18} + 195382401 \beta_{17} - 171725205 \beta_{16} + 191805852 \beta_{15} - 111468206 \beta_{14} + 62470391 \beta_{13} + \cdots + 2645569879 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( 354974509 \beta_{19} - 1230108155 \beta_{18} - 221980178 \beta_{17} - 1490726562 \beta_{16} + 274702047 \beta_{15} - 108876842 \beta_{14} + 1213748184 \beta_{13} + \cdots + 80450438654 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( - 3613574983 \beta_{19} - 1314650605 \beta_{18} + 6336456253 \beta_{17} - 5474745144 \beta_{16} + 6178902523 \beta_{15} - 3168209618 \beta_{14} + \cdots + 76037890307 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( 8057876413 \beta_{19} - 36188714255 \beta_{18} - 3742095207 \beta_{17} - 45939075752 \beta_{16} + 8476980777 \beta_{15} - 2293767126 \beta_{14} + \cdots + 2225207611939 \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( - 119849655518 \beta_{19} - 31946898784 \beta_{18} + 200806051868 \beta_{17} - 172254900889 \beta_{16} + 194617458349 \beta_{15} - 88774663886 \beta_{14} + \cdots + 2230810952036 \) Copy content Toggle raw display

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
5.47301
5.11115
4.83558
4.78115
4.51125
3.70061
3.07004
2.43215
1.73073
0.535784
0.316870
−0.543504
−1.18246
−2.45426
−2.71848
−2.93017
−3.38207
−4.11567
−4.85260
−5.31910
−5.47301 −5.20034 21.9538 0 28.4615 13.4776 −76.3693 0.0435585 0
1.2 −5.11115 6.25577 18.1239 0 −31.9742 −23.9501 −51.7448 12.1347 0
1.3 −4.83558 −0.759478 15.3828 0 3.67251 5.98500 −35.7002 −26.4232 0
1.4 −4.78115 −8.52287 14.8594 0 40.7491 0.872689 −32.7957 45.6393 0
1.5 −4.51125 −3.38341 12.3514 0 15.2634 −34.8869 −19.6302 −15.5525 0
1.6 −3.70061 9.73685 5.69455 0 −36.0323 6.77131 8.53158 67.8063 0
1.7 −3.07004 −8.38792 1.42512 0 25.7512 −7.32876 20.1851 43.3572 0
1.8 −2.43215 4.90197 −2.08462 0 −11.9223 −7.59452 24.5274 −2.97072 0
1.9 −1.73073 1.98196 −5.00459 0 −3.43023 20.0698 22.5074 −23.0718 0
1.10 −0.535784 −4.19830 −7.71294 0 2.24938 −22.0571 8.41874 −9.37425 0
1.11 −0.316870 6.27919 −7.89959 0 −1.98969 28.3648 5.03811 12.4282 0
1.12 0.543504 −10.2939 −7.70460 0 −5.59480 26.1847 −8.53552 78.9650 0
1.13 1.18246 3.11873 −6.60178 0 3.68778 −22.9558 −17.2661 −17.2735 0
1.14 2.45426 −1.11646 −1.97660 0 −2.74008 26.9047 −24.4852 −25.7535 0
1.15 2.71848 −8.68292 −0.609867 0 −23.6043 −33.0011 −23.4058 48.3931 0
1.16 2.93017 8.68161 0.585880 0 25.4386 −7.51412 −21.7246 48.3704 0
1.17 3.38207 −6.44036 3.43842 0 −21.7818 3.63998 −15.4276 14.4782 0
1.18 4.11567 6.53750 8.93876 0 26.9062 −22.2039 3.86362 15.7389 0
1.19 4.85260 0.286450 15.5478 0 1.39003 26.3499 36.6263 −26.9179 0
1.20 5.31910 −4.79403 20.2928 0 −25.4999 −17.1283 65.3868 −4.01726 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.20
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.g 20
5.b even 2 1 355.4.a.d 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
355.4.a.d 20 5.b even 2 1
1775.4.a.g 20 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{20} + 9 T_{2}^{19} - 89 T_{2}^{18} - 952 T_{2}^{17} + 2911 T_{2}^{16} + 41549 T_{2}^{15} - 37799 T_{2}^{14} - 974485 T_{2}^{13} - 43478 T_{2}^{12} + 13402754 T_{2}^{11} + 6726996 T_{2}^{10} + \cdots - 106929408 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1775))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{20} + 9 T^{19} - 89 T^{18} + \cdots - 106929408 \) Copy content Toggle raw display
$3$ \( T^{20} + 14 T^{19} + \cdots - 2328215452160 \) Copy content Toggle raw display
$5$ \( T^{20} \) Copy content Toggle raw display
$7$ \( T^{20} + 40 T^{19} + \cdots + 40\!\cdots\!12 \) Copy content Toggle raw display
$11$ \( T^{20} - 62 T^{19} + \cdots + 12\!\cdots\!76 \) Copy content Toggle raw display
$13$ \( T^{20} + 232 T^{19} + \cdots + 17\!\cdots\!72 \) Copy content Toggle raw display
$17$ \( T^{20} + 314 T^{19} + \cdots + 44\!\cdots\!84 \) Copy content Toggle raw display
$19$ \( T^{20} - 54 T^{19} + \cdots + 21\!\cdots\!00 \) Copy content Toggle raw display
$23$ \( T^{20} + 344 T^{19} + \cdots + 73\!\cdots\!40 \) Copy content Toggle raw display
$29$ \( T^{20} - 748 T^{19} + \cdots - 11\!\cdots\!60 \) Copy content Toggle raw display
$31$ \( T^{20} + 18 T^{19} + \cdots - 13\!\cdots\!16 \) Copy content Toggle raw display
$37$ \( T^{20} + 858 T^{19} + \cdots + 23\!\cdots\!88 \) Copy content Toggle raw display
$41$ \( T^{20} - 386 T^{19} + \cdots + 31\!\cdots\!28 \) Copy content Toggle raw display
$43$ \( T^{20} + 1210 T^{19} + \cdots - 37\!\cdots\!60 \) Copy content Toggle raw display
$47$ \( T^{20} + 320 T^{19} + \cdots - 69\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( T^{20} + 2066 T^{19} + \cdots - 31\!\cdots\!48 \) Copy content Toggle raw display
$59$ \( T^{20} - 1198 T^{19} + \cdots + 74\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( T^{20} - 284 T^{19} + \cdots + 81\!\cdots\!60 \) Copy content Toggle raw display
$67$ \( T^{20} + 976 T^{19} + \cdots + 45\!\cdots\!00 \) Copy content Toggle raw display
$71$ \( (T - 71)^{20} \) Copy content Toggle raw display
$73$ \( T^{20} + 4310 T^{19} + \cdots - 24\!\cdots\!56 \) Copy content Toggle raw display
$79$ \( T^{20} + 340 T^{19} + \cdots + 25\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{20} - 354 T^{19} + \cdots - 42\!\cdots\!12 \) Copy content Toggle raw display
$89$ \( T^{20} + 1446 T^{19} + \cdots + 48\!\cdots\!80 \) Copy content Toggle raw display
$97$ \( T^{20} + 3282 T^{19} + \cdots - 43\!\cdots\!40 \) Copy content Toggle raw display
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