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Properties

Label 4805.2.a.ba
Level $4805$
Weight $2$
Character orbit 4805.a
Self dual yes
Analytic conductor $38.368$
Analytic rank $0$
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] = [4805,2,Mod(1,4805)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4805, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("4805.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4805 = 5 \cdot 31^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4805.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: \(38.3681181712\)
Analytic rank: \(0\)
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: \( x^{20} - x^{19} - 29 x^{18} + 29 x^{17} + 348 x^{16} - 341 x^{15} - 2245 x^{14} + 2101 x^{13} + \cdots - 29 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 155)
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.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + (\beta_{13} + 1) q^{3} + (\beta_{7} + \beta_{6} + 1) q^{4} + q^{5} + (\beta_{17} + \beta_{15} - \beta_{12} + \cdots + 1) q^{6}+ \cdots + (\beta_{13} + \beta_{8} - \beta_{6} + \cdots + 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + (\beta_{13} + 1) q^{3} + (\beta_{7} + \beta_{6} + 1) q^{4} + q^{5} + (\beta_{17} + \beta_{15} - \beta_{12} + \cdots + 1) q^{6}+ \cdots + (2 \beta_{19} - \beta_{16} + 2 \beta_{15} + \cdots - 2) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q + q^{2} + 12 q^{3} + 19 q^{4} + 20 q^{5} + 12 q^{6} - 3 q^{7} - 3 q^{8} + 24 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 20 q + q^{2} + 12 q^{3} + 19 q^{4} + 20 q^{5} + 12 q^{6} - 3 q^{7} - 3 q^{8} + 24 q^{9} + q^{10} + 19 q^{11} - 9 q^{12} + 23 q^{13} - 10 q^{14} + 12 q^{15} + 17 q^{16} + 6 q^{17} - 12 q^{18} + 9 q^{19} + 19 q^{20} - 6 q^{21} + 42 q^{22} + 33 q^{23} + 40 q^{24} + 20 q^{25} + 4 q^{26} + 63 q^{27} - 8 q^{28} + 4 q^{29} + 12 q^{30} - 30 q^{32} - 8 q^{33} - 19 q^{34} - 3 q^{35} + q^{36} + 38 q^{37} - 13 q^{38} + 15 q^{39} - 3 q^{40} + 9 q^{41} + 48 q^{42} + 64 q^{43} + 27 q^{44} + 24 q^{45} - 20 q^{46} - 8 q^{47} + 5 q^{48} - 3 q^{49} + q^{50} - 18 q^{51} - 5 q^{52} + 7 q^{53} + 16 q^{54} + 19 q^{55} - 5 q^{56} + 36 q^{57} + 49 q^{58} - 26 q^{59} - 9 q^{60} + 34 q^{61} - 5 q^{63} + q^{64} + 23 q^{65} + 35 q^{66} + 2 q^{67} + 28 q^{68} + 18 q^{69} - 10 q^{70} + 55 q^{71} - 14 q^{72} + 50 q^{73} - 30 q^{74} + 12 q^{75} - 89 q^{76} + 8 q^{77} + 69 q^{78} + 41 q^{79} + 17 q^{80} + 40 q^{81} - 24 q^{82} + 63 q^{83} - 7 q^{84} + 6 q^{85} + 7 q^{86} + 36 q^{87} + 124 q^{88} + 32 q^{89} - 12 q^{90} + 24 q^{91} + 55 q^{92} - 47 q^{94} + 9 q^{95} + 18 q^{96} + 13 q^{97} + 13 q^{98} - 17 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} - x^{19} - 29 x^{18} + 29 x^{17} + 348 x^{16} - 341 x^{15} - 2245 x^{14} + 2101 x^{13} + \cdots - 29 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 795064044 \nu^{19} + 1166013655 \nu^{18} + 22673367132 \nu^{17} - 33645993769 \nu^{16} + \cdots + 69447586330 ) / 2926835085 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 12210781028 \nu^{19} + 16969351464 \nu^{18} + 346898828554 \nu^{17} - 490540294735 \nu^{16} + \cdots + 957961782796 ) / 14634175425 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 13511991166 \nu^{19} + 18776551643 \nu^{18} + 384924316338 \nu^{17} - 542845086150 \nu^{16} + \cdots + 1173927494402 ) / 14634175425 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 13896238881 \nu^{19} + 18764609698 \nu^{18} + 395720593933 \nu^{17} - 542540976180 \nu^{16} + \cdots + 1126040163872 ) / 14634175425 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 3495068301 \nu^{19} - 4778555788 \nu^{18} - 99579234178 \nu^{17} + 137891958170 \nu^{16} + \cdots - 269961905322 ) / 2926835085 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 3495068301 \nu^{19} + 4778555788 \nu^{18} + 99579234178 \nu^{17} - 137891958170 \nu^{16} + \cdots + 261181400067 ) / 2926835085 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 20631709743 \nu^{19} - 24258975774 \nu^{18} - 591745692299 \nu^{17} + 702998068755 \nu^{16} + \cdots - 2198358647886 ) / 14634175425 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 33769156948 \nu^{19} + 46859477389 \nu^{18} + 963275866989 \nu^{17} - 1352389422155 \nu^{16} + \cdots + 3069563077321 ) / 14634175425 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 38828971168 \nu^{19} - 52725210049 \nu^{18} - 1107275554174 \nu^{17} + 1521760757805 \nu^{16} + \cdots - 3066552995436 ) / 14634175425 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 42726446656 \nu^{19} - 58851208303 \nu^{18} - 1216788622908 \nu^{17} + 1699025642995 \nu^{16} + \cdots - 3456216131192 ) / 14634175425 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 9309031218 \nu^{19} - 12804099519 \nu^{18} - 265183349534 \nu^{17} + 369541139500 \nu^{16} + \cdots - 726780105776 ) / 2926835085 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 50172091784 \nu^{19} - 66094444292 \nu^{18} - 1432240185462 \nu^{17} + 1909363903105 \nu^{16} + \cdots - 4354818348563 ) / 14634175425 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 68849148714 \nu^{19} + 91503361917 \nu^{18} + 1964219039377 \nu^{17} - 2642851929710 \nu^{16} + \cdots + 5799520690963 ) / 14634175425 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 83652745191 \nu^{19} + 113951918908 \nu^{18} + 2387930033938 \nu^{17} + \cdots + 7778085460012 ) / 14634175425 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( 99909987391 \nu^{19} - 137906435538 \nu^{18} - 2846807818288 \nu^{17} + 3981029408710 \nu^{16} + \cdots - 8179548984282 ) / 14634175425 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( 100763557607 \nu^{19} - 136445388586 \nu^{18} - 2874549223901 \nu^{17} + 3939544279025 \nu^{16} + \cdots - 8753799481904 ) / 14634175425 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( 26643606167 \nu^{19} - 36634552006 \nu^{18} - 759015071161 \nu^{17} + 1057317836260 \nu^{16} + \cdots - 2087763841854 ) / 2926835085 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( 27456127314 \nu^{19} - 37560222576 \nu^{18} - 782257578932 \nu^{17} + 1084084408772 \nu^{16} + \cdots - 2128946103784 ) / 2926835085 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{7} + \beta_{6} + 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{18} + \beta_{14} + \beta_{13} - 2\beta_{12} - \beta_{6} + \beta_{2} + 5\beta _1 + 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -\beta_{19} - \beta_{17} + \beta_{16} - \beta_{14} + \beta_{12} + 6\beta_{7} + 8\beta_{6} + \beta_{3} - \beta _1 + 15 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 10 \beta_{18} + \beta_{17} - \beta_{16} + 9 \beta_{14} + 8 \beta_{13} - 20 \beta_{12} - \beta_{11} + \cdots + 16 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 10 \beta_{19} - 12 \beta_{17} + 10 \beta_{16} - 12 \beta_{14} - \beta_{13} + 11 \beta_{12} + \cdots + 85 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 80 \beta_{18} + 11 \beta_{17} - 11 \beta_{16} - 3 \beta_{15} + 68 \beta_{14} + 53 \beta_{13} + \cdots + 108 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 80 \beta_{19} - 2 \beta_{18} - 111 \beta_{17} + 82 \beta_{16} + 2 \beta_{15} - 112 \beta_{14} + \cdots + 517 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 2 \beta_{19} + 598 \beta_{18} + 90 \beta_{17} - 94 \beta_{16} - 53 \beta_{15} + 496 \beta_{14} + \cdots + 689 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 598 \beta_{19} - 41 \beta_{18} - 923 \beta_{17} + 634 \beta_{16} + 43 \beta_{15} - 952 \beta_{14} + \cdots + 3297 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 41 \beta_{19} + 4350 \beta_{18} + 677 \beta_{17} - 747 \beta_{16} - 625 \beta_{15} + 3608 \beta_{14} + \cdots + 4274 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 4350 \beta_{19} - 556 \beta_{18} - 7266 \beta_{17} + 4796 \beta_{16} + 593 \beta_{15} - 7732 \beta_{14} + \cdots + 21738 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 556 \beta_{19} + 31278 \beta_{18} + 5024 \beta_{17} - 5811 \beta_{16} - 6213 \beta_{15} + 26424 \beta_{14} + \cdots + 25981 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 31278 \beta_{19} - 6273 \beta_{18} - 55493 \beta_{17} + 35977 \beta_{16} + 6703 \beta_{15} + \cdots + 146829 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 6273 \beta_{19} + 223953 \beta_{18} + 37800 \beta_{17} - 45055 \beta_{16} - 56386 \beta_{15} + \cdots + 154682 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( - 223953 \beta_{19} - 63774 \beta_{18} - 416599 \beta_{17} + 269070 \beta_{16} + 67769 \beta_{15} + \cdots + 1009741 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( 63774 \beta_{19} + 1603000 \beta_{18} + 290563 \beta_{17} - 350257 \beta_{16} - 484587 \beta_{15} + \cdots + 896344 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( - 1603000 \beta_{19} - 606788 \beta_{18} - 3097313 \beta_{17} + 2010798 \beta_{16} + 638812 \beta_{15} + \cdots + 7039983 \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( 606788 \beta_{19} + 11494868 \beta_{18} + 2277767 \beta_{17} - 2733291 \beta_{16} - 4021332 \beta_{15} + \cdots + 4989095 \) 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
−2.76895
−2.51863
−2.13105
−1.97715
−1.72743
−1.41551
−0.849572
−0.373369
−0.366800
−0.255998
−0.193483
0.834123
0.901239
1.08413
1.60031
1.92695
2.07874
2.12859
2.40368
2.62017
−2.76895 −1.90286 5.66708 1.00000 5.26892 1.75923 −10.1540 0.620871 −2.76895
1.2 −2.51863 1.94255 4.34352 1.00000 −4.89257 −0.835075 −5.90247 0.773495 −2.51863
1.3 −2.13105 −2.36490 2.54138 1.00000 5.03972 0.0496471 −1.15370 2.59274 −2.13105
1.4 −1.97715 −2.08385 1.90914 1.00000 4.12010 2.05405 0.179648 1.34244 −1.97715
1.5 −1.72743 1.21396 0.984000 1.00000 −2.09703 −4.71756 1.75506 −1.52629 −1.72743
1.6 −1.41551 3.11854 0.00366226 1.00000 −4.41432 0.949846 2.82583 6.72530 −1.41551
1.7 −0.849572 2.50747 −1.27823 1.00000 −2.13028 −2.84870 2.78509 3.28740 −0.849572
1.8 −0.373369 0.909417 −1.86060 1.00000 −0.339548 3.37231 1.44143 −2.17296 −0.373369
1.9 −0.366800 −0.788968 −1.86546 1.00000 0.289393 −0.970765 1.41785 −2.37753 −0.366800
1.10 −0.255998 3.30138 −1.93447 1.00000 −0.845146 3.25602 1.00721 7.89912 −0.255998
1.11 −0.193483 −1.09054 −1.96256 1.00000 0.211001 1.47241 0.766688 −1.81072 −0.193483
1.12 0.834123 3.30299 −1.30424 1.00000 2.75510 −4.92182 −2.75614 7.90976 0.834123
1.13 0.901239 −0.395231 −1.18777 1.00000 −0.356198 4.02554 −2.87294 −2.84379 0.901239
1.14 1.08413 1.99352 −0.824667 1.00000 2.16123 0.585164 −3.06230 0.974129 1.08413
1.15 1.60031 −2.12913 0.561000 1.00000 −3.40727 −3.72378 −2.30285 1.53318 1.60031
1.16 1.92695 −0.481087 1.71315 1.00000 −0.927033 −3.73279 −0.552745 −2.76855 1.92695
1.17 2.07874 −1.07111 2.32115 1.00000 −2.22655 −0.878521 0.667595 −1.85273 2.07874
1.18 2.12859 3.27171 2.53089 1.00000 6.96412 1.62381 1.13005 7.70408 2.12859
1.19 2.40368 1.70432 3.77770 1.00000 4.09665 0.756609 4.27303 −0.0952873 2.40368
1.20 2.62017 1.04180 4.86531 1.00000 2.72970 −0.275631 7.50762 −1.91465 2.62017
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.20
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(5\) \( -1 \)
\(31\) \( +1 \)

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 4805.2.a.ba 20
31.b odd 2 1 4805.2.a.z 20
31.g even 15 2 155.2.q.b ✓ 40
155.u even 30 2 775.2.bl.b 40
155.w odd 60 4 775.2.ck.b 80
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
155.2.q.b ✓ 40 31.g even 15 2
775.2.bl.b 40 155.u even 30 2
775.2.ck.b 80 155.w odd 60 4
4805.2.a.z 20 31.b odd 2 1
4805.2.a.ba 20 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4805))\):

\( T_{2}^{20} - T_{2}^{19} - 29 T_{2}^{18} + 29 T_{2}^{17} + 348 T_{2}^{16} - 341 T_{2}^{15} - 2245 T_{2}^{14} + \cdots - 29 \) Copy content Toggle raw display
\( T_{3}^{20} - 12 T_{3}^{19} + 30 T_{3}^{18} + 171 T_{3}^{17} - 892 T_{3}^{16} - 353 T_{3}^{15} + \cdots - 7409 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{20} - T^{19} + \cdots - 29 \) Copy content Toggle raw display
$3$ \( T^{20} - 12 T^{19} + \cdots - 7409 \) Copy content Toggle raw display
$5$ \( (T - 1)^{20} \) Copy content Toggle raw display
$7$ \( T^{20} + 3 T^{19} + \cdots - 1439 \) Copy content Toggle raw display
$11$ \( T^{20} - 19 T^{19} + \cdots - 1013855 \) Copy content Toggle raw display
$13$ \( T^{20} - 23 T^{19} + \cdots + 36950791 \) Copy content Toggle raw display
$17$ \( T^{20} + \cdots - 358459169 \) Copy content Toggle raw display
$19$ \( T^{20} + \cdots - 4687440179 \) Copy content Toggle raw display
$23$ \( T^{20} + \cdots + 874501351 \) Copy content Toggle raw display
$29$ \( T^{20} + \cdots - 967850969 \) Copy content Toggle raw display
$31$ \( T^{20} \) Copy content Toggle raw display
$37$ \( T^{20} + \cdots - 403640563509 \) Copy content Toggle raw display
$41$ \( T^{20} + \cdots - 69466967239619 \) Copy content Toggle raw display
$43$ \( T^{20} + \cdots + 539860642795 \) Copy content Toggle raw display
$47$ \( T^{20} + \cdots - 245924415419 \) Copy content Toggle raw display
$53$ \( T^{20} + \cdots + 293831973330631 \) Copy content Toggle raw display
$59$ \( T^{20} + \cdots + 100621390231 \) Copy content Toggle raw display
$61$ \( T^{20} + \cdots - 36\!\cdots\!49 \) Copy content Toggle raw display
$67$ \( T^{20} + \cdots + 1868221821451 \) Copy content Toggle raw display
$71$ \( T^{20} + \cdots - 10\!\cdots\!39 \) Copy content Toggle raw display
$73$ \( T^{20} + \cdots + 75\!\cdots\!36 \) Copy content Toggle raw display
$79$ \( T^{20} + \cdots - 20934879388109 \) Copy content Toggle raw display
$83$ \( T^{20} + \cdots + 453675261660496 \) Copy content Toggle raw display
$89$ \( T^{20} + \cdots + 3941028278911 \) Copy content Toggle raw display
$97$ \( T^{20} + \cdots + 110469595 \) Copy content Toggle raw display
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