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Properties

Label 380.2.a.c.1.1
Level $380$
Weight $2$
Character 380.1
Self dual yes
Analytic conductor $3.034$
Analytic rank $1$
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] = [380,2,Mod(1,380)] 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("380.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(380, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 380 = 2^{2} \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 380.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,-4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(3.03431527681\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{8})^+\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-1.41421\) of defining polynomial
Character \(\chi\) \(=\) 380.1

$q$-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)
 
\(f(q)\) \(=\) \(q-3.41421 q^{3} +1.00000 q^{5} +0.828427 q^{7} +8.65685 q^{9} -2.00000 q^{11} -6.24264 q^{13} -3.41421 q^{15} +0.828427 q^{17} -1.00000 q^{19} -2.82843 q^{21} -6.00000 q^{23} +1.00000 q^{25} -19.3137 q^{27} -6.48528 q^{29} -6.82843 q^{31} +6.82843 q^{33} +0.828427 q^{35} -1.75736 q^{37} +21.3137 q^{39} +3.65685 q^{41} +4.82843 q^{43} +8.65685 q^{45} -4.82843 q^{47} -6.31371 q^{49} -2.82843 q^{51} +9.07107 q^{53} -2.00000 q^{55} +3.41421 q^{57} +13.6569 q^{59} -13.6569 q^{61} +7.17157 q^{63} -6.24264 q^{65} -3.41421 q^{67} +20.4853 q^{69} +5.17157 q^{71} -2.48528 q^{73} -3.41421 q^{75} -1.65685 q^{77} +1.65685 q^{79} +39.9706 q^{81} -13.3137 q^{83} +0.828427 q^{85} +22.1421 q^{87} -6.48528 q^{89} -5.17157 q^{91} +23.3137 q^{93} -1.00000 q^{95} -10.2426 q^{97} -17.3137 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 4 q^{3} + 2 q^{5} - 4 q^{7} + 6 q^{9} - 4 q^{11} - 4 q^{13} - 4 q^{15} - 4 q^{17} - 2 q^{19} - 12 q^{23} + 2 q^{25} - 16 q^{27} + 4 q^{29} - 8 q^{31} + 8 q^{33} - 4 q^{35} - 12 q^{37} + 20 q^{39}+ \cdots - 12 q^{99}+O(q^{100}) \) 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
\(3\) −3.41421 −1.97120 −0.985599 − 0.169102i \(-0.945913\pi\)
−0.985599 + 0.169102i \(0.945913\pi\)
\(4\) 0 0
\(5\) 1.00000 0.447214
\(6\) 0 0
\(7\) 0.828427 0.313116 0.156558 − 0.987669i \(-0.449960\pi\)
0.156558 + 0.987669i \(0.449960\pi\)
\(8\) 0 0
\(9\) 8.65685 2.88562
\(10\) 0 0
\(11\) −2.00000 −0.603023 −0.301511 − 0.953463i \(-0.597491\pi\)
−0.301511 + 0.953463i \(0.597491\pi\)
\(12\) 0 0
\(13\) −6.24264 −1.73140 −0.865699 − 0.500566i \(-0.833125\pi\)
−0.865699 + 0.500566i \(0.833125\pi\)
\(14\) 0 0
\(15\) −3.41421 −0.881546
\(16\) 0 0
\(17\) 0.828427 0.200923 0.100462 − 0.994941i \(-0.467968\pi\)
0.100462 + 0.994941i \(0.467968\pi\)
\(18\) 0 0
\(19\) −1.00000 −0.229416
\(20\) 0 0
\(21\) −2.82843 −0.617213
\(22\) 0 0
\(23\) −6.00000 −1.25109 −0.625543 − 0.780189i \(-0.715123\pi\)
−0.625543 + 0.780189i \(0.715123\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) −19.3137 −3.71692
\(28\) 0 0
\(29\) −6.48528 −1.20429 −0.602143 − 0.798388i \(-0.705686\pi\)
−0.602143 + 0.798388i \(0.705686\pi\)
\(30\) 0 0
\(31\) −6.82843 −1.22642 −0.613211 − 0.789919i \(-0.710122\pi\)
−0.613211 + 0.789919i \(0.710122\pi\)
\(32\) 0 0
\(33\) 6.82843 1.18868
\(34\) 0 0
\(35\) 0.828427 0.140030
\(36\) 0 0
\(37\) −1.75736 −0.288908 −0.144454 − 0.989512i \(-0.546143\pi\)
−0.144454 + 0.989512i \(0.546143\pi\)
\(38\) 0 0
\(39\) 21.3137 3.41292
\(40\) 0 0
\(41\) 3.65685 0.571105 0.285552 − 0.958363i \(-0.407823\pi\)
0.285552 + 0.958363i \(0.407823\pi\)
\(42\) 0 0
\(43\) 4.82843 0.736328 0.368164 − 0.929761i \(-0.379986\pi\)
0.368164 + 0.929761i \(0.379986\pi\)
\(44\) 0 0
\(45\) 8.65685 1.29049
\(46\) 0 0
\(47\) −4.82843 −0.704298 −0.352149 − 0.935944i \(-0.614549\pi\)
−0.352149 + 0.935944i \(0.614549\pi\)
\(48\) 0 0
\(49\) −6.31371 −0.901958
\(50\) 0 0
\(51\) −2.82843 −0.396059
\(52\) 0 0
\(53\) 9.07107 1.24601 0.623003 − 0.782219i \(-0.285912\pi\)
0.623003 + 0.782219i \(0.285912\pi\)
\(54\) 0 0
\(55\) −2.00000 −0.269680
\(56\) 0 0
\(57\) 3.41421 0.452224
\(58\) 0 0
\(59\) 13.6569 1.77797 0.888985 − 0.457935i \(-0.151411\pi\)
0.888985 + 0.457935i \(0.151411\pi\)
\(60\) 0 0
\(61\) −13.6569 −1.74858 −0.874291 − 0.485403i \(-0.838673\pi\)
−0.874291 + 0.485403i \(0.838673\pi\)
\(62\) 0 0
\(63\) 7.17157 0.903533
\(64\) 0 0
\(65\) −6.24264 −0.774304
\(66\) 0 0
\(67\) −3.41421 −0.417113 −0.208556 − 0.978010i \(-0.566876\pi\)
−0.208556 + 0.978010i \(0.566876\pi\)
\(68\) 0 0
\(69\) 20.4853 2.46614
\(70\) 0 0
\(71\) 5.17157 0.613753 0.306876 − 0.951749i \(-0.400716\pi\)
0.306876 + 0.951749i \(0.400716\pi\)
\(72\) 0 0
\(73\) −2.48528 −0.290880 −0.145440 − 0.989367i \(-0.546460\pi\)
−0.145440 + 0.989367i \(0.546460\pi\)
\(74\) 0 0
\(75\) −3.41421 −0.394239
\(76\) 0 0
\(77\) −1.65685 −0.188816
\(78\) 0 0
\(79\) 1.65685 0.186411 0.0932053 − 0.995647i \(-0.470289\pi\)
0.0932053 + 0.995647i \(0.470289\pi\)
\(80\) 0 0
\(81\) 39.9706 4.44117
\(82\) 0 0
\(83\) −13.3137 −1.46137 −0.730685 − 0.682715i \(-0.760799\pi\)
−0.730685 + 0.682715i \(0.760799\pi\)
\(84\) 0 0
\(85\) 0.828427 0.0898555
\(86\) 0 0
\(87\) 22.1421 2.37389
\(88\) 0 0
\(89\) −6.48528 −0.687438 −0.343719 − 0.939072i \(-0.611687\pi\)
−0.343719 + 0.939072i \(0.611687\pi\)
\(90\) 0 0
\(91\) −5.17157 −0.542128
\(92\) 0 0
\(93\) 23.3137 2.41752
\(94\) 0 0
\(95\) −1.00000 −0.102598
\(96\) 0 0
\(97\) −10.2426 −1.03998 −0.519991 − 0.854172i \(-0.674065\pi\)
−0.519991 + 0.854172i \(0.674065\pi\)
\(98\) 0 0
\(99\) −17.3137 −1.74009
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 380.2.a.c.1.1 ✓ 2
3.2 odd 2 3420.2.a.g.1.2 2
4.3 odd 2 1520.2.a.o.1.2 2
5.2 odd 4 1900.2.c.d.1749.4 4
5.3 odd 4 1900.2.c.d.1749.1 4
5.4 even 2 1900.2.a.e.1.2 2
8.3 odd 2 6080.2.a.y.1.1 2
8.5 even 2 6080.2.a.bl.1.2 2
19.18 odd 2 7220.2.a.m.1.2 2
20.19 odd 2 7600.2.a.u.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
380.2.a.c.1.1 ✓ 2 1.1 even 1 trivial
1520.2.a.o.1.2 2 4.3 odd 2
1900.2.a.e.1.2 2 5.4 even 2
1900.2.c.d.1749.1 4 5.3 odd 4
1900.2.c.d.1749.4 4 5.2 odd 4
3420.2.a.g.1.2 2 3.2 odd 2
6080.2.a.y.1.1 2 8.3 odd 2
6080.2.a.bl.1.2 2 8.5 even 2
7220.2.a.m.1.2 2 19.18 odd 2
7600.2.a.u.1.1 2 20.19 odd 2