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Properties

Label 1840.2.a.n.1.1
Level $1840$
Weight $2$
Character 1840.1
Self dual yes
Analytic conductor $14.692$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [1840,2,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, 2); 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, 2, names="a")
 
Level: \( N \) \(=\) \( 1840 = 2^{4} \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1840.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,1,0,-2,0,-1,0,5] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(9)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(14.6924739719\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{21}) \)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 5 \) 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 230)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-1.79129\) 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)
 
\(f(q)\) \(=\) \(q-1.79129 q^{3} -1.00000 q^{5} -2.79129 q^{7} +0.208712 q^{9} -3.79129 q^{11} +1.20871 q^{13} +1.79129 q^{15} -3.79129 q^{17} -1.20871 q^{19} +5.00000 q^{21} -1.00000 q^{23} +1.00000 q^{25} +5.00000 q^{27} -1.58258 q^{29} -10.3739 q^{31} +6.79129 q^{33} +2.79129 q^{35} -4.00000 q^{37} -2.16515 q^{39} -2.20871 q^{41} +7.16515 q^{43} -0.208712 q^{45} +13.5826 q^{47} +0.791288 q^{49} +6.79129 q^{51} +6.00000 q^{53} +3.79129 q^{55} +2.16515 q^{57} +4.41742 q^{59} -3.37386 q^{61} -0.582576 q^{63} -1.20871 q^{65} +7.16515 q^{67} +1.79129 q^{69} +5.37386 q^{71} -14.7477 q^{73} -1.79129 q^{75} +10.5826 q^{77} -8.00000 q^{79} -9.58258 q^{81} +6.00000 q^{83} +3.79129 q^{85} +2.83485 q^{87} -3.16515 q^{89} -3.37386 q^{91} +18.5826 q^{93} +1.20871 q^{95} +14.9564 q^{97} -0.791288 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + q^{3} - 2 q^{5} - q^{7} + 5 q^{9} - 3 q^{11} + 7 q^{13} - q^{15} - 3 q^{17} - 7 q^{19} + 10 q^{21} - 2 q^{23} + 2 q^{25} + 10 q^{27} + 6 q^{29} - 7 q^{31} + 9 q^{33} + q^{35} - 8 q^{37} + 14 q^{39}+ \cdots + 3 q^{99}+O(q^{100}) \) 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
\(3\) −1.79129 −1.03420 −0.517100 − 0.855925i \(-0.672989\pi\)
−0.517100 + 0.855925i \(0.672989\pi\)
\(4\) 0 0
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) −2.79129 −1.05501 −0.527504 − 0.849553i \(-0.676872\pi\)
−0.527504 + 0.849553i \(0.676872\pi\)
\(8\) 0 0
\(9\) 0.208712 0.0695707
\(10\) 0 0
\(11\) −3.79129 −1.14312 −0.571558 − 0.820562i \(-0.693661\pi\)
−0.571558 + 0.820562i \(0.693661\pi\)
\(12\) 0 0
\(13\) 1.20871 0.335236 0.167618 − 0.985852i \(-0.446392\pi\)
0.167618 + 0.985852i \(0.446392\pi\)
\(14\) 0 0
\(15\) 1.79129 0.462509
\(16\) 0 0
\(17\) −3.79129 −0.919522 −0.459761 − 0.888043i \(-0.652065\pi\)
−0.459761 + 0.888043i \(0.652065\pi\)
\(18\) 0 0
\(19\) −1.20871 −0.277298 −0.138649 − 0.990342i \(-0.544276\pi\)
−0.138649 + 0.990342i \(0.544276\pi\)
\(20\) 0 0
\(21\) 5.00000 1.09109
\(22\) 0 0
\(23\) −1.00000 −0.208514
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) 5.00000 0.962250
\(28\) 0 0
\(29\) −1.58258 −0.293877 −0.146938 − 0.989146i \(-0.546942\pi\)
−0.146938 + 0.989146i \(0.546942\pi\)
\(30\) 0 0
\(31\) −10.3739 −1.86320 −0.931600 − 0.363484i \(-0.881587\pi\)
−0.931600 + 0.363484i \(0.881587\pi\)
\(32\) 0 0
\(33\) 6.79129 1.18221
\(34\) 0 0
\(35\) 2.79129 0.471814
\(36\) 0 0
\(37\) −4.00000 −0.657596 −0.328798 − 0.944400i \(-0.606644\pi\)
−0.328798 + 0.944400i \(0.606644\pi\)
\(38\) 0 0
\(39\) −2.16515 −0.346702
\(40\) 0 0
\(41\) −2.20871 −0.344943 −0.172471 − 0.985015i \(-0.555175\pi\)
−0.172471 + 0.985015i \(0.555175\pi\)
\(42\) 0 0
\(43\) 7.16515 1.09268 0.546338 − 0.837565i \(-0.316022\pi\)
0.546338 + 0.837565i \(0.316022\pi\)
\(44\) 0 0
\(45\) −0.208712 −0.0311130
\(46\) 0 0
\(47\) 13.5826 1.98122 0.990611 − 0.136710i \(-0.0436528\pi\)
0.990611 + 0.136710i \(0.0436528\pi\)
\(48\) 0 0
\(49\) 0.791288 0.113041
\(50\) 0 0
\(51\) 6.79129 0.950971
\(52\) 0 0
\(53\) 6.00000 0.824163 0.412082 − 0.911147i \(-0.364802\pi\)
0.412082 + 0.911147i \(0.364802\pi\)
\(54\) 0 0
\(55\) 3.79129 0.511217
\(56\) 0 0
\(57\) 2.16515 0.286781
\(58\) 0 0
\(59\) 4.41742 0.575100 0.287550 − 0.957766i \(-0.407159\pi\)
0.287550 + 0.957766i \(0.407159\pi\)
\(60\) 0 0
\(61\) −3.37386 −0.431979 −0.215989 − 0.976396i \(-0.569298\pi\)
−0.215989 + 0.976396i \(0.569298\pi\)
\(62\) 0 0
\(63\) −0.582576 −0.0733976
\(64\) 0 0
\(65\) −1.20871 −0.149922
\(66\) 0 0
\(67\) 7.16515 0.875363 0.437681 − 0.899130i \(-0.355800\pi\)
0.437681 + 0.899130i \(0.355800\pi\)
\(68\) 0 0
\(69\) 1.79129 0.215646
\(70\) 0 0
\(71\) 5.37386 0.637760 0.318880 − 0.947795i \(-0.396693\pi\)
0.318880 + 0.947795i \(0.396693\pi\)
\(72\) 0 0
\(73\) −14.7477 −1.72609 −0.863045 − 0.505126i \(-0.831446\pi\)
−0.863045 + 0.505126i \(0.831446\pi\)
\(74\) 0 0
\(75\) −1.79129 −0.206840
\(76\) 0 0
\(77\) 10.5826 1.20600
\(78\) 0 0
\(79\) −8.00000 −0.900070 −0.450035 − 0.893011i \(-0.648589\pi\)
−0.450035 + 0.893011i \(0.648589\pi\)
\(80\) 0 0
\(81\) −9.58258 −1.06473
\(82\) 0 0
\(83\) 6.00000 0.658586 0.329293 − 0.944228i \(-0.393190\pi\)
0.329293 + 0.944228i \(0.393190\pi\)
\(84\) 0 0
\(85\) 3.79129 0.411223
\(86\) 0 0
\(87\) 2.83485 0.303928
\(88\) 0 0
\(89\) −3.16515 −0.335505 −0.167753 − 0.985829i \(-0.553651\pi\)
−0.167753 + 0.985829i \(0.553651\pi\)
\(90\) 0 0
\(91\) −3.37386 −0.353677
\(92\) 0 0
\(93\) 18.5826 1.92692
\(94\) 0 0
\(95\) 1.20871 0.124011
\(96\) 0 0
\(97\) 14.9564 1.51860 0.759298 − 0.650743i \(-0.225542\pi\)
0.759298 + 0.650743i \(0.225542\pi\)
\(98\) 0 0
\(99\) −0.791288 −0.0795274
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 1840.2.a.n.1.1 2
4.3 odd 2 230.2.a.a.1.2 ✓ 2
5.4 even 2 9200.2.a.bs.1.2 2
8.3 odd 2 7360.2.a.bq.1.1 2
8.5 even 2 7360.2.a.bk.1.2 2
12.11 even 2 2070.2.a.x.1.2 2
20.3 even 4 1150.2.b.g.599.4 4
20.7 even 4 1150.2.b.g.599.1 4
20.19 odd 2 1150.2.a.o.1.1 2
92.91 even 2 5290.2.a.e.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
230.2.a.a.1.2 ✓ 2 4.3 odd 2
1150.2.a.o.1.1 2 20.19 odd 2
1150.2.b.g.599.1 4 20.7 even 4
1150.2.b.g.599.4 4 20.3 even 4
1840.2.a.n.1.1 2 1.1 even 1 trivial
2070.2.a.x.1.2 2 12.11 even 2
5290.2.a.e.1.2 2 92.91 even 2
7360.2.a.bk.1.2 2 8.5 even 2
7360.2.a.bq.1.1 2 8.3 odd 2
9200.2.a.bs.1.2 2 5.4 even 2