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Properties

Label 1840.2.m.a.1839.4
Level $1840$
Weight $2$
Character 1840.1839
Analytic conductor $14.692$
Analytic rank $0$
Dimension $4$
Inner twists $4$

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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(1839,1840)] 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("1840.1839"); 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([1, 0, 1, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 1840 = 2^{4} \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1840.m (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,-4,0,0,0,0,0,-8] 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: no
Analytic conductor: \(14.6924739719\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{6}, \sqrt{-14})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 4x^{2} + 25 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1839.4
Root \(1.22474 - 1.87083i\) of defining polynomial
Character \(\chi\) \(=\) 1840.1839
Dual form 1840.2.m.a.1839.3

$q$-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)
 
\(f(q)\) \(=\) \(q-1.00000 q^{3} +(1.22474 + 1.87083i) q^{5} -2.00000 q^{9} +4.89898 q^{11} -4.58258i q^{13} +(-1.22474 - 1.87083i) q^{15} -2.44949 q^{17} +2.44949 q^{19} +(3.00000 - 3.74166i) q^{23} +(-2.00000 + 4.58258i) q^{25} +5.00000 q^{27} -3.00000 q^{29} -4.58258i q^{31} -4.89898 q^{33} +4.89898 q^{37} +4.58258i q^{39} -3.00000 q^{41} -11.2250i q^{43} +(-2.44949 - 3.74166i) q^{45} +9.00000 q^{47} +7.00000 q^{49} +2.44949 q^{51} -4.89898 q^{53} +(6.00000 + 9.16515i) q^{55} -2.44949 q^{57} +9.16515i q^{59} +11.2250i q^{61} +(8.57321 - 5.61249i) q^{65} +(-3.00000 + 3.74166i) q^{69} +13.7477i q^{71} -4.58258i q^{73} +(2.00000 - 4.58258i) q^{75} +7.34847 q^{79} +1.00000 q^{81} +3.74166i q^{83} +(-3.00000 - 4.58258i) q^{85} +3.00000 q^{87} -3.74166i q^{89} +4.58258i q^{93} +(3.00000 + 4.58258i) q^{95} +9.79796 q^{97} -9.79796 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{3} - 8 q^{9} + 12 q^{23} - 8 q^{25} + 20 q^{27} - 12 q^{29} - 12 q^{41} + 36 q^{47} + 28 q^{49} + 24 q^{55} - 12 q^{69} + 8 q^{75} + 4 q^{81} - 12 q^{85} + 12 q^{87} + 12 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1840\mathbb{Z}\right)^\times\).

\(n\) \(737\) \(1151\) \(1201\) \(1381\)
\(\chi(n)\) \(-1\) \(-1\) \(-1\) \(1\)

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
\(3\) −1.00000 −0.577350 −0.288675 − 0.957427i \(-0.593215\pi\)
−0.288675 + 0.957427i \(0.593215\pi\)
\(4\) 0 0
\(5\) 1.22474 + 1.87083i 0.547723 + 0.836660i
\(6\) 0 0
\(7\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(8\) 0 0
\(9\) −2.00000 −0.666667
\(10\) 0 0
\(11\) 4.89898 1.47710 0.738549 − 0.674200i \(-0.235511\pi\)
0.738549 + 0.674200i \(0.235511\pi\)
\(12\) 0 0
\(13\) − 4.58258i − 1.27098i −0.772110 − 0.635489i \(-0.780799\pi\)
0.772110 − 0.635489i \(-0.219201\pi\)
\(14\) 0 0
\(15\) −1.22474 − 1.87083i −0.316228 − 0.483046i
\(16\) 0 0
\(17\) −2.44949 −0.594089 −0.297044 − 0.954864i \(-0.596001\pi\)
−0.297044 + 0.954864i \(0.596001\pi\)
\(18\) 0 0
\(19\) 2.44949 0.561951 0.280976 − 0.959715i \(-0.409342\pi\)
0.280976 + 0.959715i \(0.409342\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 3.00000 − 3.74166i 0.625543 − 0.780189i
\(24\) 0 0
\(25\) −2.00000 + 4.58258i −0.400000 + 0.916515i
\(26\) 0 0
\(27\) 5.00000 0.962250
\(28\) 0 0
\(29\) −3.00000 −0.557086 −0.278543 − 0.960424i \(-0.589851\pi\)
−0.278543 + 0.960424i \(0.589851\pi\)
\(30\) 0 0
\(31\) − 4.58258i − 0.823055i −0.911397 − 0.411527i \(-0.864995\pi\)
0.911397 − 0.411527i \(-0.135005\pi\)
\(32\) 0 0
\(33\) −4.89898 −0.852803
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 4.89898 0.805387 0.402694 − 0.915335i \(-0.368074\pi\)
0.402694 + 0.915335i \(0.368074\pi\)
\(38\) 0 0
\(39\) 4.58258i 0.733799i
\(40\) 0 0
\(41\) −3.00000 −0.468521 −0.234261 − 0.972174i \(-0.575267\pi\)
−0.234261 + 0.972174i \(0.575267\pi\)
\(42\) 0 0
\(43\) − 11.2250i − 1.71179i −0.517148 − 0.855896i \(-0.673006\pi\)
0.517148 − 0.855896i \(-0.326994\pi\)
\(44\) 0 0
\(45\) −2.44949 − 3.74166i −0.365148 − 0.557773i
\(46\) 0 0
\(47\) 9.00000 1.31278 0.656392 − 0.754420i \(-0.272082\pi\)
0.656392 + 0.754420i \(0.272082\pi\)
\(48\) 0 0
\(49\) 7.00000 1.00000
\(50\) 0 0
\(51\) 2.44949 0.342997
\(52\) 0 0
\(53\) −4.89898 −0.672927 −0.336463 − 0.941697i \(-0.609231\pi\)
−0.336463 + 0.941697i \(0.609231\pi\)
\(54\) 0 0
\(55\) 6.00000 + 9.16515i 0.809040 + 1.23583i
\(56\) 0 0
\(57\) −2.44949 −0.324443
\(58\) 0 0
\(59\) 9.16515i 1.19320i 0.802538 + 0.596601i \(0.203482\pi\)
−0.802538 + 0.596601i \(0.796518\pi\)
\(60\) 0 0
\(61\) 11.2250i 1.43721i 0.695418 + 0.718605i \(0.255219\pi\)
−0.695418 + 0.718605i \(0.744781\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 8.57321 − 5.61249i 1.06338 − 0.696143i
\(66\) 0 0
\(67\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(68\) 0 0
\(69\) −3.00000 + 3.74166i −0.361158 + 0.450443i
\(70\) 0 0
\(71\) 13.7477i 1.63156i 0.578366 + 0.815778i \(0.303691\pi\)
−0.578366 + 0.815778i \(0.696309\pi\)
\(72\) 0 0
\(73\) − 4.58258i − 0.536350i −0.963370 − 0.268175i \(-0.913579\pi\)
0.963370 − 0.268175i \(-0.0864205\pi\)
\(74\) 0 0
\(75\) 2.00000 − 4.58258i 0.230940 − 0.529150i
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 7.34847 0.826767 0.413384 − 0.910557i \(-0.364347\pi\)
0.413384 + 0.910557i \(0.364347\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 3.74166i 0.410700i 0.978689 + 0.205350i \(0.0658333\pi\)
−0.978689 + 0.205350i \(0.934167\pi\)
\(84\) 0 0
\(85\) −3.00000 − 4.58258i −0.325396 − 0.497050i
\(86\) 0 0
\(87\) 3.00000 0.321634
\(88\) 0 0
\(89\) − 3.74166i − 0.396615i −0.980140 − 0.198307i \(-0.936456\pi\)
0.980140 − 0.198307i \(-0.0635444\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 4.58258i 0.475191i
\(94\) 0 0
\(95\) 3.00000 + 4.58258i 0.307794 + 0.470162i
\(96\) 0 0
\(97\) 9.79796 0.994832 0.497416 − 0.867512i \(-0.334282\pi\)
0.497416 + 0.867512i \(0.334282\pi\)
\(98\) 0 0
\(99\) −9.79796 −0.984732
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.m.a.1839.4 yes 4
4.3 odd 2 1840.2.m.d.1839.4 yes 4
5.4 even 2 1840.2.m.d.1839.2 yes 4
20.19 odd 2 inner 1840.2.m.a.1839.2 yes 4
23.22 odd 2 inner 1840.2.m.a.1839.1 ✓ 4
92.91 even 2 1840.2.m.d.1839.1 yes 4
115.114 odd 2 1840.2.m.d.1839.3 yes 4
460.459 even 2 inner 1840.2.m.a.1839.3 yes 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1840.2.m.a.1839.1 ✓ 4 23.22 odd 2 inner
1840.2.m.a.1839.2 yes 4 20.19 odd 2 inner
1840.2.m.a.1839.3 yes 4 460.459 even 2 inner
1840.2.m.a.1839.4 yes 4 1.1 even 1 trivial
1840.2.m.d.1839.1 yes 4 92.91 even 2
1840.2.m.d.1839.2 yes 4 5.4 even 2
1840.2.m.d.1839.3 yes 4 115.114 odd 2
1840.2.m.d.1839.4 yes 4 4.3 odd 2