Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [4840,2,Mod(1,4840)] 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("4840.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(4840, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 4840 = 2^{3} \cdot 5 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4840.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,-2,0,4,0,-7,0,-4,0,0,0,3] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(38.6475945783\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{22 +2 \sqrt{5}})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} - 3x^{2} + x + 1 \) 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 440)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.4
Root \(0.737640\) of defining polynomial
Character \(\chi\) \(=\) 4840.1

$q$-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)
 
\(f(q)\) \(=\) \(q+1.19353 q^{3} +1.00000 q^{5} -1.83785 q^{7} -1.57549 q^{9} -1.25546 q^{13} +1.19353 q^{15} +6.44899 q^{17} +1.25120 q^{19} -2.19353 q^{21} -4.35567 q^{23} +1.00000 q^{25} -5.46097 q^{27} -0.726479 q^{29} -11.0790 q^{31} -1.83785 q^{35} -11.5086 q^{37} -1.49843 q^{39} +7.92427 q^{41} +0.606873 q^{43} -1.57549 q^{45} -3.98194 q^{47} -3.62230 q^{49} +7.69704 q^{51} +6.83290 q^{53} +1.49334 q^{57} -1.41201 q^{59} -9.75208 q^{61} +2.89552 q^{63} -1.25546 q^{65} +12.5899 q^{67} -5.19862 q^{69} -6.79665 q^{71} -10.0652 q^{73} +1.19353 q^{75} +6.51977 q^{79} -1.79134 q^{81} +0.644326 q^{83} +6.44899 q^{85} -0.867073 q^{87} +9.39723 q^{89} +2.30735 q^{91} -13.2231 q^{93} +1.25120 q^{95} -4.66191 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 2 q^{3} + 4 q^{5} - 7 q^{7} - 4 q^{9} + 3 q^{13} - 2 q^{15} + 11 q^{17} + 2 q^{19} - 2 q^{21} - 11 q^{23} + 4 q^{25} + q^{27} + 4 q^{29} - 17 q^{31} - 7 q^{35} - 3 q^{37} + q^{39} + 13 q^{41} - 7 q^{43}+ \cdots + 2 q^{97}+O(q^{100}) \) 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
\(3\) 1.19353 0.689083 0.344542 − 0.938771i \(-0.388034\pi\)
0.344542 + 0.938771i \(0.388034\pi\)
\(4\) 0 0
\(5\) 1.00000 0.447214
\(6\) 0 0
\(7\) −1.83785 −0.694643 −0.347322 − 0.937746i \(-0.612909\pi\)
−0.347322 + 0.937746i \(0.612909\pi\)
\(8\) 0 0
\(9\) −1.57549 −0.525164
\(10\) 0 0
\(11\) 0 0
\(12\) 0 0
\(13\) −1.25546 −0.348202 −0.174101 − 0.984728i \(-0.555702\pi\)
−0.174101 + 0.984728i \(0.555702\pi\)
\(14\) 0 0
\(15\) 1.19353 0.308167
\(16\) 0 0
\(17\) 6.44899 1.56411 0.782055 − 0.623210i \(-0.214172\pi\)
0.782055 + 0.623210i \(0.214172\pi\)
\(18\) 0 0
\(19\) 1.25120 0.287045 0.143522 − 0.989647i \(-0.454157\pi\)
0.143522 + 0.989647i \(0.454157\pi\)
\(20\) 0 0
\(21\) −2.19353 −0.478667
\(22\) 0 0
\(23\) −4.35567 −0.908221 −0.454110 − 0.890945i \(-0.650043\pi\)
−0.454110 + 0.890945i \(0.650043\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) −5.46097 −1.05097
\(28\) 0 0
\(29\) −0.726479 −0.134904 −0.0674519 − 0.997723i \(-0.521487\pi\)
−0.0674519 + 0.997723i \(0.521487\pi\)
\(30\) 0 0
\(31\) −11.0790 −1.98985 −0.994924 − 0.100626i \(-0.967916\pi\)
−0.994924 + 0.100626i \(0.967916\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −1.83785 −0.310654
\(36\) 0 0
\(37\) −11.5086 −1.89200 −0.946001 − 0.324162i \(-0.894918\pi\)
−0.946001 + 0.324162i \(0.894918\pi\)
\(38\) 0 0
\(39\) −1.49843 −0.239940
\(40\) 0 0
\(41\) 7.92427 1.23756 0.618781 − 0.785563i \(-0.287627\pi\)
0.618781 + 0.785563i \(0.287627\pi\)
\(42\) 0 0
\(43\) 0.606873 0.0925473 0.0462736 − 0.998929i \(-0.485265\pi\)
0.0462736 + 0.998929i \(0.485265\pi\)
\(44\) 0 0
\(45\) −1.57549 −0.234861
\(46\) 0 0
\(47\) −3.98194 −0.580826 −0.290413 − 0.956901i \(-0.593793\pi\)
−0.290413 + 0.956901i \(0.593793\pi\)
\(48\) 0 0
\(49\) −3.62230 −0.517471
\(50\) 0 0
\(51\) 7.69704 1.07780
\(52\) 0 0
\(53\) 6.83290 0.938571 0.469285 − 0.883047i \(-0.344512\pi\)
0.469285 + 0.883047i \(0.344512\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 1.49334 0.197798
\(58\) 0 0
\(59\) −1.41201 −0.183828 −0.0919141 − 0.995767i \(-0.529299\pi\)
−0.0919141 + 0.995767i \(0.529299\pi\)
\(60\) 0 0
\(61\) −9.75208 −1.24863 −0.624313 − 0.781174i \(-0.714621\pi\)
−0.624313 + 0.781174i \(0.714621\pi\)
\(62\) 0 0
\(63\) 2.89552 0.364802
\(64\) 0 0
\(65\) −1.25546 −0.155721
\(66\) 0 0
\(67\) 12.5899 1.53811 0.769053 − 0.639186i \(-0.220728\pi\)
0.769053 + 0.639186i \(0.220728\pi\)
\(68\) 0 0
\(69\) −5.19862 −0.625840
\(70\) 0 0
\(71\) −6.79665 −0.806614 −0.403307 − 0.915065i \(-0.632139\pi\)
−0.403307 + 0.915065i \(0.632139\pi\)
\(72\) 0 0
\(73\) −10.0652 −1.17804 −0.589022 − 0.808117i \(-0.700487\pi\)
−0.589022 + 0.808117i \(0.700487\pi\)
\(74\) 0 0
\(75\) 1.19353 0.137817
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 6.51977 0.733531 0.366765 − 0.930313i \(-0.380465\pi\)
0.366765 + 0.930313i \(0.380465\pi\)
\(80\) 0 0
\(81\) −1.79134 −0.199038
\(82\) 0 0
\(83\) 0.644326 0.0707239 0.0353620 − 0.999375i \(-0.488742\pi\)
0.0353620 + 0.999375i \(0.488742\pi\)
\(84\) 0 0
\(85\) 6.44899 0.699491
\(86\) 0 0
\(87\) −0.867073 −0.0929600
\(88\) 0 0
\(89\) 9.39723 0.996104 0.498052 − 0.867147i \(-0.334049\pi\)
0.498052 + 0.867147i \(0.334049\pi\)
\(90\) 0 0
\(91\) 2.30735 0.241876
\(92\) 0 0
\(93\) −13.2231 −1.37117
\(94\) 0 0
\(95\) 1.25120 0.128370
\(96\) 0 0
\(97\) −4.66191 −0.473345 −0.236673 − 0.971589i \(-0.576057\pi\)
−0.236673 + 0.971589i \(0.576057\pi\)
\(98\) 0 0
\(99\) 0 0
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 4840.2.a.y.1.4 4
4.3 odd 2 9680.2.a.cu.1.1 4
11.3 even 5 440.2.y.a.361.1 ✓ 8
11.4 even 5 440.2.y.a.401.1 yes 8
11.10 odd 2 4840.2.a.z.1.4 4
44.3 odd 10 880.2.bo.d.801.2 8
44.15 odd 10 880.2.bo.d.401.2 8
44.43 even 2 9680.2.a.ct.1.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
440.2.y.a.361.1 ✓ 8 11.3 even 5
440.2.y.a.401.1 yes 8 11.4 even 5
880.2.bo.d.401.2 8 44.15 odd 10
880.2.bo.d.801.2 8 44.3 odd 10
4840.2.a.y.1.4 4 1.1 even 1 trivial
4840.2.a.z.1.4 4 11.10 odd 2
9680.2.a.ct.1.1 4 44.43 even 2
9680.2.a.cu.1.1 4 4.3 odd 2