Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [1840,4,Mod(1,1840)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1840.1"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1840, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 1840 = 2^{4} \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1840.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [5,0,-4,0,-25,0,3] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(108.563514411\)
Analytic rank: \(0\)
Dimension: \(5\)
Coefficient field: \(\mathbb{Q}[x]/(x^{5} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{5} - x^{4} - 27x^{3} + 7x^{2} + 168x + 92 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 115)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-2.49214\) of defining polynomial
Character \(\chi\) \(=\) 1840.1

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-9.02447 q^{3} -5.00000 q^{5} -4.33445 q^{7} +54.4411 q^{9} -48.1836 q^{11} +53.0174 q^{13} +45.1224 q^{15} -61.0120 q^{17} -12.0115 q^{19} +39.1161 q^{21} +23.0000 q^{23} +25.0000 q^{25} -247.642 q^{27} +198.213 q^{29} +11.9661 q^{31} +434.832 q^{33} +21.6723 q^{35} -230.018 q^{37} -478.454 q^{39} -393.486 q^{41} +167.496 q^{43} -272.206 q^{45} -531.002 q^{47} -324.213 q^{49} +550.602 q^{51} +373.570 q^{53} +240.918 q^{55} +108.398 q^{57} +825.880 q^{59} -632.806 q^{61} -235.972 q^{63} -265.087 q^{65} +38.7503 q^{67} -207.563 q^{69} -1109.37 q^{71} -446.171 q^{73} -225.612 q^{75} +208.850 q^{77} -642.608 q^{79} +764.926 q^{81} +292.575 q^{83} +305.060 q^{85} -1788.77 q^{87} -273.272 q^{89} -229.801 q^{91} -107.988 q^{93} +60.0577 q^{95} -959.139 q^{97} -2623.17 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q - 4 q^{3} - 25 q^{5} + 3 q^{7} + 77 q^{9} - 23 q^{11} + 132 q^{13} + 20 q^{15} + 23 q^{17} + 161 q^{19} - 60 q^{21} + 115 q^{23} + 125 q^{25} - 577 q^{27} + 401 q^{29} - 32 q^{31} + 189 q^{33} - 15 q^{35}+ \cdots - 2021 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\). You can download additional coefficients here.



Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −9.02447 −1.73676 −0.868380 − 0.495899i \(-0.834839\pi\)
−0.868380 + 0.495899i \(0.834839\pi\)
\(4\) 0 0
\(5\) −5.00000 −0.447214
\(6\) 0 0
\(7\) −4.33445 −0.234038 −0.117019 − 0.993130i \(-0.537334\pi\)
−0.117019 + 0.993130i \(0.537334\pi\)
\(8\) 0 0
\(9\) 54.4411 2.01634
\(10\) 0 0
\(11\) −48.1836 −1.32072 −0.660360 − 0.750949i \(-0.729596\pi\)
−0.660360 + 0.750949i \(0.729596\pi\)
\(12\) 0 0
\(13\) 53.0174 1.13111 0.565553 − 0.824712i \(-0.308663\pi\)
0.565553 + 0.824712i \(0.308663\pi\)
\(14\) 0 0
\(15\) 45.1224 0.776703
\(16\) 0 0
\(17\) −61.0120 −0.870447 −0.435223 − 0.900323i \(-0.643331\pi\)
−0.435223 + 0.900323i \(0.643331\pi\)
\(18\) 0 0
\(19\) −12.0115 −0.145034 −0.0725168 − 0.997367i \(-0.523103\pi\)
−0.0725168 + 0.997367i \(0.523103\pi\)
\(20\) 0 0
\(21\) 39.1161 0.406469
\(22\) 0 0
\(23\) 23.0000 0.208514
\(24\) 0 0
\(25\) 25.0000 0.200000
\(26\) 0 0
\(27\) −247.642 −1.76514
\(28\) 0 0
\(29\) 198.213 1.26922 0.634609 − 0.772834i \(-0.281161\pi\)
0.634609 + 0.772834i \(0.281161\pi\)
\(30\) 0 0
\(31\) 11.9661 0.0693284 0.0346642 − 0.999399i \(-0.488964\pi\)
0.0346642 + 0.999399i \(0.488964\pi\)
\(32\) 0 0
\(33\) 434.832 2.29377
\(34\) 0 0
\(35\) 21.6723 0.104665
\(36\) 0 0
\(37\) −230.018 −1.02202 −0.511010 − 0.859575i \(-0.670728\pi\)
−0.511010 + 0.859575i \(0.670728\pi\)
\(38\) 0 0
\(39\) −478.454 −1.96446
\(40\) 0 0
\(41\) −393.486 −1.49883 −0.749417 − 0.662098i \(-0.769666\pi\)
−0.749417 + 0.662098i \(0.769666\pi\)
\(42\) 0 0
\(43\) 167.496 0.594019 0.297010 − 0.954874i \(-0.404011\pi\)
0.297010 + 0.954874i \(0.404011\pi\)
\(44\) 0 0
\(45\) −272.206 −0.901734
\(46\) 0 0
\(47\) −531.002 −1.64797 −0.823984 − 0.566612i \(-0.808254\pi\)
−0.823984 + 0.566612i \(0.808254\pi\)
\(48\) 0 0
\(49\) −324.213 −0.945226
\(50\) 0 0
\(51\) 550.602 1.51176
\(52\) 0 0
\(53\) 373.570 0.968186 0.484093 − 0.875017i \(-0.339150\pi\)
0.484093 + 0.875017i \(0.339150\pi\)
\(54\) 0 0
\(55\) 240.918 0.590644
\(56\) 0 0
\(57\) 108.398 0.251889
\(58\) 0 0
\(59\) 825.880 1.82238 0.911190 − 0.411987i \(-0.135165\pi\)
0.911190 + 0.411987i \(0.135165\pi\)
\(60\) 0 0
\(61\) −632.806 −1.32824 −0.664119 − 0.747627i \(-0.731193\pi\)
−0.664119 + 0.747627i \(0.731193\pi\)
\(62\) 0 0
\(63\) −235.972 −0.471900
\(64\) 0 0
\(65\) −265.087 −0.505846
\(66\) 0 0
\(67\) 38.7503 0.0706583 0.0353292 − 0.999376i \(-0.488752\pi\)
0.0353292 + 0.999376i \(0.488752\pi\)
\(68\) 0 0
\(69\) −207.563 −0.362140
\(70\) 0 0
\(71\) −1109.37 −1.85434 −0.927168 − 0.374645i \(-0.877764\pi\)
−0.927168 + 0.374645i \(0.877764\pi\)
\(72\) 0 0
\(73\) −446.171 −0.715348 −0.357674 − 0.933847i \(-0.616430\pi\)
−0.357674 + 0.933847i \(0.616430\pi\)
\(74\) 0 0
\(75\) −225.612 −0.347352
\(76\) 0 0
\(77\) 208.850 0.309099
\(78\) 0 0
\(79\) −642.608 −0.915179 −0.457589 − 0.889164i \(-0.651287\pi\)
−0.457589 + 0.889164i \(0.651287\pi\)
\(80\) 0 0
\(81\) 764.926 1.04928
\(82\) 0 0
\(83\) 292.575 0.386919 0.193460 − 0.981108i \(-0.438029\pi\)
0.193460 + 0.981108i \(0.438029\pi\)
\(84\) 0 0
\(85\) 305.060 0.389276
\(86\) 0 0
\(87\) −1788.77 −2.20433
\(88\) 0 0
\(89\) −273.272 −0.325469 −0.162735 − 0.986670i \(-0.552031\pi\)
−0.162735 + 0.986670i \(0.552031\pi\)
\(90\) 0 0
\(91\) −229.801 −0.264722
\(92\) 0 0
\(93\) −107.988 −0.120407
\(94\) 0 0
\(95\) 60.0577 0.0648610
\(96\) 0 0
\(97\) −959.139 −1.00398 −0.501989 − 0.864874i \(-0.667398\pi\)
−0.501989 + 0.864874i \(0.667398\pi\)
\(98\) 0 0
\(99\) −2623.17 −2.66302
Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 1840.4.a.n.1.1 5
4.3 odd 2 115.4.a.e.1.2 ✓ 5
12.11 even 2 1035.4.a.k.1.4 5
20.3 even 4 575.4.b.i.24.7 10
20.7 even 4 575.4.b.i.24.4 10
20.19 odd 2 575.4.a.j.1.4 5
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
115.4.a.e.1.2 ✓ 5 4.3 odd 2
575.4.a.j.1.4 5 20.19 odd 2
575.4.b.i.24.4 10 20.7 even 4
575.4.b.i.24.7 10 20.3 even 4
1035.4.a.k.1.4 5 12.11 even 2
1840.4.a.n.1.1 5 1.1 even 1 trivial