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Properties

Label 2075.4.a.f
Level $2075$
Weight $4$
Character orbit 2075.a
Self dual yes
Analytic conductor $122.429$
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] = [2075,4,Mod(1,2075)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2075, 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("2075.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2075 = 5^{2} \cdot 83 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 2075.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: \(122.428963262\)
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: \( x^{20} - 5 x^{19} - 101 x^{18} + 508 x^{17} + 4106 x^{16} - 21051 x^{15} - 85533 x^{14} + 459851 x^{13} + \cdots + 7355648 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: no (minimal twist has level 415)
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.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} + ( - \beta_{5} - 1) q^{3} + (\beta_{2} + 3) q^{4} + ( - \beta_{4} + \beta_1 + 1) q^{6} + ( - \beta_{16} - 2) q^{7} + ( - \beta_{3} - \beta_{2} - 4 \beta_1) q^{8} + (\beta_{16} + \beta_{14} + \beta_{11} + \cdots + 12) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{2} + ( - \beta_{5} - 1) q^{3} + (\beta_{2} + 3) q^{4} + ( - \beta_{4} + \beta_1 + 1) q^{6} + ( - \beta_{16} - 2) q^{7} + ( - \beta_{3} - \beta_{2} - 4 \beta_1) q^{8} + (\beta_{16} + \beta_{14} + \beta_{11} + \cdots + 12) q^{9}+ \cdots + (25 \beta_{19} - 31 \beta_{18} + \cdots + 626) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q - 5 q^{2} - 12 q^{3} + 67 q^{4} + 17 q^{6} - 31 q^{7} - 36 q^{8} + 216 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 20 q - 5 q^{2} - 12 q^{3} + 67 q^{4} + 17 q^{6} - 31 q^{7} - 36 q^{8} + 216 q^{9} + 36 q^{11} - 44 q^{12} - 67 q^{13} + 89 q^{14} + 375 q^{16} - 425 q^{17} + 11 q^{18} - 251 q^{19} - 23 q^{21} - 447 q^{22} - 584 q^{23} + 152 q^{24} + 799 q^{26} - 477 q^{27} - 831 q^{28} + 252 q^{29} + 329 q^{31} - 1088 q^{32} - 1405 q^{33} + 189 q^{34} + 1126 q^{36} - 260 q^{37} - 2204 q^{38} + 40 q^{39} + 1830 q^{41} - 1782 q^{42} + 87 q^{43} + 86 q^{44} - 549 q^{46} - 1028 q^{47} - 2931 q^{48} + 1531 q^{49} + 981 q^{51} - 623 q^{52} - 1491 q^{53} + 595 q^{54} + 1930 q^{56} - 1184 q^{57} - 1075 q^{58} + 782 q^{59} - 686 q^{61} - 797 q^{62} - 1554 q^{63} + 714 q^{64} + 1688 q^{66} - 661 q^{67} - 3632 q^{68} + 1326 q^{69} + 298 q^{71} - 1378 q^{72} - 656 q^{73} + 348 q^{74} - 3262 q^{76} - 1231 q^{77} - 587 q^{78} - 748 q^{79} + 2356 q^{81} - 526 q^{82} + 1660 q^{83} + 1927 q^{84} + 2632 q^{86} - 4001 q^{87} - 2200 q^{88} + 3702 q^{89} + 112 q^{91} - 7797 q^{92} - 1542 q^{93} - 1773 q^{94} + 18 q^{96} - 4506 q^{97} - 1942 q^{98} + 10603 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} - 5 x^{19} - 101 x^{18} + 508 x^{17} + 4106 x^{16} - 21051 x^{15} - 85533 x^{14} + 459851 x^{13} + \cdots + 7355648 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 11 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} - \nu^{2} - 20\nu + 11 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 28\!\cdots\!53 \nu^{19} + \cdots + 26\!\cdots\!56 ) / 24\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 32\!\cdots\!97 \nu^{19} + \cdots - 11\!\cdots\!56 ) / 24\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 42\!\cdots\!79 \nu^{19} + \cdots - 14\!\cdots\!92 ) / 12\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 55\!\cdots\!07 \nu^{19} + \cdots + 18\!\cdots\!36 ) / 61\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 60\!\cdots\!21 \nu^{19} + \cdots + 28\!\cdots\!08 ) / 61\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 12\!\cdots\!81 \nu^{19} + \cdots + 18\!\cdots\!88 ) / 12\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 30\!\cdots\!13 \nu^{19} + \cdots + 47\!\cdots\!76 ) / 24\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 32\!\cdots\!41 \nu^{19} + \cdots - 94\!\cdots\!68 ) / 24\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 91\!\cdots\!27 \nu^{19} + \cdots + 15\!\cdots\!96 ) / 61\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 37\!\cdots\!49 \nu^{19} + \cdots + 84\!\cdots\!52 ) / 24\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 27\!\cdots\!57 \nu^{19} + \cdots - 45\!\cdots\!36 ) / 12\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 57\!\cdots\!77 \nu^{19} + \cdots - 51\!\cdots\!04 ) / 24\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( - 28\!\cdots\!07 \nu^{19} + \cdots + 59\!\cdots\!36 ) / 12\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( - 29\!\cdots\!13 \nu^{19} + \cdots + 60\!\cdots\!24 ) / 12\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( 12\!\cdots\!17 \nu^{19} + \cdots - 12\!\cdots\!16 ) / 49\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( 73\!\cdots\!63 \nu^{19} + \cdots - 73\!\cdots\!24 ) / 24\!\cdots\!00 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 11 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} + \beta_{2} + 20\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - \beta_{17} - \beta_{16} - 2 \beta_{14} + \beta_{13} - \beta_{12} - \beta_{10} - \beta_{9} + \cdots + 216 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 5 \beta_{19} + \beta_{18} + 4 \beta_{17} - 5 \beta_{16} - 3 \beta_{15} - 6 \beta_{14} + 3 \beta_{13} + \cdots + 43 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 3 \beta_{19} - \beta_{18} - 47 \beta_{17} - 38 \beta_{16} + 10 \beta_{15} - 73 \beta_{14} + \cdots + 4956 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 255 \beta_{19} + 40 \beta_{18} + 225 \beta_{17} - 211 \beta_{16} - 148 \beta_{15} - 261 \beta_{14} + \cdots + 2072 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 110 \beta_{19} - 135 \beta_{18} - 1680 \beta_{17} - 1101 \beta_{16} + 549 \beta_{15} - 2165 \beta_{14} + \cdots + 121319 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 9219 \beta_{19} + 1347 \beta_{18} + 8644 \beta_{17} - 6687 \beta_{16} - 5300 \beta_{15} - 8555 \beta_{14} + \cdots + 74618 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 2784 \beta_{19} - 7317 \beta_{18} - 54027 \beta_{17} - 29382 \beta_{16} + 21207 \beta_{15} + \cdots + 3069042 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 291569 \beta_{19} + 43770 \beta_{18} + 286713 \beta_{17} - 192609 \beta_{16} - 168605 \beta_{15} + \cdots + 2387165 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 56902 \beta_{19} - 297391 \beta_{18} - 1647824 \beta_{17} - 756573 \beta_{16} + 714978 \beta_{15} + \cdots + 79136346 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 8644171 \beta_{19} + 1387107 \beta_{18} + 8846533 \beta_{17} - 5318542 \beta_{16} - 5064991 \beta_{15} + \cdots + 71667540 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 869343 \beta_{19} - 10542106 \beta_{18} - 48735092 \beta_{17} - 19091338 \beta_{16} + \cdots + 2065420773 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 247634835 \beta_{19} + 42809208 \beta_{18} + 262133484 \beta_{17} - 143656800 \beta_{16} + \cdots + 2070541272 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( - 3236122 \beta_{19} - 346027960 \beta_{18} - 1413062332 \beta_{17} - 475097080 \beta_{16} + \cdots + 54344953208 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( 6960218030 \beta_{19} + 1288138982 \beta_{18} + 7578391008 \beta_{17} - 3830268022 \beta_{16} + \cdots + 58349533755 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( 483772692 \beta_{19} - 10827474710 \beta_{18} - 40413387812 \beta_{17} - 11692115850 \beta_{16} + \cdots + 1437980643154 \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( 193520202550 \beta_{19} + 37911326838 \beta_{18} + 215630293117 \beta_{17} - 101280339297 \beta_{16} + \cdots + 1616626592686 \) 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
5.19643
5.17298
5.15033
4.02753
3.19006
2.89887
1.88261
1.52630
1.38863
0.837885
0.277053
0.212679
−1.13420
−2.35792
−2.42598
−2.69285
−4.03318
−4.12687
−4.77137
−5.21898
−5.19643 4.66835 19.0029 0 −24.2588 −19.4864 −57.1757 −5.20647 0
1.2 −5.17298 −5.33695 18.7597 0 27.6079 −24.5134 −55.6596 1.48301 0
1.3 −5.15033 −7.34600 18.5259 0 37.8343 12.4401 −54.2120 26.9637 0
1.4 −4.02753 8.78887 8.22102 0 −35.3975 17.0997 −0.890184 50.2443 0
1.5 −3.19006 −0.0264944 2.17647 0 0.0845188 −7.10876 18.5774 −26.9993 0
1.6 −2.89887 −8.46043 0.403470 0 24.5257 −30.5247 22.0214 44.5789 0
1.7 −1.88261 3.23261 −4.45578 0 −6.08574 23.6314 23.4494 −16.5503 0
1.8 −1.52630 4.53506 −5.67041 0 −6.92186 −11.2127 20.8651 −6.43320 0
1.9 −1.38863 5.34900 −6.07172 0 −7.42775 15.4920 19.5403 1.61176 0
1.10 −0.837885 −5.91444 −7.29795 0 4.95562 −0.342901 12.8179 7.98055 0
1.11 −0.277053 −10.1425 −7.92324 0 2.81000 32.6746 4.41159 75.8695 0
1.12 −0.212679 −1.33214 −7.95477 0 0.283319 −34.7421 3.39324 −25.2254 0
1.13 1.13420 −1.45517 −6.71360 0 −1.65045 27.2256 −16.6881 −24.8825 0
1.14 2.35792 7.94020 −2.44022 0 18.7223 −11.7638 −24.6172 36.0468 0
1.15 2.42598 −7.15992 −2.11462 0 −17.3698 −16.2425 −24.5379 24.2645 0
1.16 2.69285 −2.70817 −0.748536 0 −7.29270 19.5739 −23.5585 −19.6658 0
1.17 4.03318 9.21280 8.26657 0 37.1569 −27.7232 1.07512 57.8757 0
1.18 4.12687 3.74680 9.03109 0 15.4626 1.36893 4.25516 −12.9615 0
1.19 4.77137 −8.97949 14.7660 0 −42.8445 −13.9148 32.2831 53.6312 0
1.20 5.21898 −0.612026 19.2377 0 −3.19415 17.0689 58.6494 −26.6254 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
\(5\) \(1\)
\(83\) \(-1\)

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 2075.4.a.f 20
5.b even 2 1 415.4.a.b ✓ 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
415.4.a.b ✓ 20 5.b even 2 1
2075.4.a.f 20 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{20} + 5 T_{2}^{19} - 101 T_{2}^{18} - 508 T_{2}^{17} + 4106 T_{2}^{16} + 21051 T_{2}^{15} + \cdots + 7355648 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(2075))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{20} + 5 T^{19} + \cdots + 7355648 \) Copy content Toggle raw display
$3$ \( T^{20} + \cdots + 96029869312 \) Copy content Toggle raw display
$5$ \( T^{20} \) Copy content Toggle raw display
$7$ \( T^{20} + \cdots - 32\!\cdots\!80 \) Copy content Toggle raw display
$11$ \( T^{20} + \cdots + 38\!\cdots\!56 \) Copy content Toggle raw display
$13$ \( T^{20} + \cdots - 10\!\cdots\!24 \) Copy content Toggle raw display
$17$ \( T^{20} + \cdots + 19\!\cdots\!20 \) Copy content Toggle raw display
$19$ \( T^{20} + \cdots - 12\!\cdots\!40 \) Copy content Toggle raw display
$23$ \( T^{20} + \cdots - 12\!\cdots\!00 \) Copy content Toggle raw display
$29$ \( T^{20} + \cdots + 38\!\cdots\!80 \) Copy content Toggle raw display
$31$ \( T^{20} + \cdots + 23\!\cdots\!00 \) Copy content Toggle raw display
$37$ \( T^{20} + \cdots + 12\!\cdots\!92 \) Copy content Toggle raw display
$41$ \( T^{20} + \cdots + 85\!\cdots\!96 \) Copy content Toggle raw display
$43$ \( T^{20} + \cdots + 17\!\cdots\!00 \) Copy content Toggle raw display
$47$ \( T^{20} + \cdots - 21\!\cdots\!40 \) Copy content Toggle raw display
$53$ \( T^{20} + \cdots - 13\!\cdots\!00 \) Copy content Toggle raw display
$59$ \( T^{20} + \cdots - 89\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( T^{20} + \cdots - 55\!\cdots\!00 \) Copy content Toggle raw display
$67$ \( T^{20} + \cdots + 11\!\cdots\!00 \) Copy content Toggle raw display
$71$ \( T^{20} + \cdots - 56\!\cdots\!80 \) Copy content Toggle raw display
$73$ \( T^{20} + \cdots - 60\!\cdots\!00 \) Copy content Toggle raw display
$79$ \( T^{20} + \cdots + 36\!\cdots\!32 \) Copy content Toggle raw display
$83$ \( (T - 83)^{20} \) Copy content Toggle raw display
$89$ \( T^{20} + \cdots + 30\!\cdots\!80 \) Copy content Toggle raw display
$97$ \( T^{20} + \cdots + 62\!\cdots\!64 \) Copy content Toggle raw display
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