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Properties

Label 370.2.n.c.359.2
Level $370$
Weight $2$
Character 370.359
Analytic conductor $2.954$
Analytic rank $0$
Dimension $4$
Inner twists $2$

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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] = [370,2,Mod(269,370)] 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("370.269"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(370, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([3, 4])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 370 = 2 \cdot 5 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 370.n (of order \(6\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,0,2,4,0,6] 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: no
Analytic conductor: \(2.95446487479\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{12})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 359.2
Root \(0.866025 - 0.500000i\) of defining polynomial
Character \(\chi\) \(=\) 370.359
Dual form 370.2.n.c.269.2

$q$-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)
 
\(f(q)\) \(=\) \(q+(0.866025 - 0.500000i) q^{2} +(0.500000 - 0.866025i) q^{4} +(0.133975 + 2.23205i) q^{5} +(0.633975 + 0.366025i) q^{7} -1.00000i q^{8} +(-1.50000 - 2.59808i) q^{9} +(1.23205 + 1.86603i) q^{10} +4.73205 q^{11} +(4.73205 + 2.73205i) q^{13} +0.732051 q^{14} +(-0.500000 - 0.866025i) q^{16} +(-1.50000 + 0.866025i) q^{17} +(-2.59808 - 1.50000i) q^{18} +(2.36603 - 4.09808i) q^{19} +(2.00000 + 1.00000i) q^{20} +(4.09808 - 2.36603i) q^{22} +5.46410i q^{23} +(-4.96410 + 0.598076i) q^{25} +5.46410 q^{26} +(0.633975 - 0.366025i) q^{28} -1.73205 q^{29} -9.66025 q^{31} +(-0.866025 - 0.500000i) q^{32} +(-0.866025 + 1.50000i) q^{34} +(-0.732051 + 1.46410i) q^{35} -3.00000 q^{36} +(-2.59808 - 5.50000i) q^{37} -4.73205i q^{38} +(2.23205 - 0.133975i) q^{40} +(3.96410 - 6.86603i) q^{41} +5.26795i q^{43} +(2.36603 - 4.09808i) q^{44} +(5.59808 - 3.69615i) q^{45} +(2.73205 + 4.73205i) q^{46} -3.26795i q^{47} +(-3.23205 - 5.59808i) q^{49} +(-4.00000 + 3.00000i) q^{50} +(4.73205 - 2.73205i) q^{52} +(-10.7321 + 6.19615i) q^{53} +(0.633975 + 10.5622i) q^{55} +(0.366025 - 0.633975i) q^{56} +(-1.50000 + 0.866025i) q^{58} +(-1.26795 - 2.19615i) q^{59} +(-2.13397 + 3.69615i) q^{61} +(-8.36603 + 4.83013i) q^{62} -2.19615i q^{63} -1.00000 q^{64} +(-5.46410 + 10.9282i) q^{65} +(4.56218 + 2.63397i) q^{67} +1.73205i q^{68} +(0.0980762 + 1.63397i) q^{70} +(-7.56218 + 13.0981i) q^{71} +(-2.59808 + 1.50000i) q^{72} -15.8564i q^{73} +(-5.00000 - 3.46410i) q^{74} +(-2.36603 - 4.09808i) q^{76} +(3.00000 + 1.73205i) q^{77} +(-0.830127 + 1.43782i) q^{79} +(1.86603 - 1.23205i) q^{80} +(-4.50000 + 7.79423i) q^{81} -7.92820i q^{82} +(8.19615 - 4.73205i) q^{83} +(-2.13397 - 3.23205i) q^{85} +(2.63397 + 4.56218i) q^{86} -4.73205i q^{88} +(0.232051 + 0.401924i) q^{89} +(3.00000 - 6.00000i) q^{90} +(2.00000 + 3.46410i) q^{91} +(4.73205 + 2.73205i) q^{92} +(-1.63397 - 2.83013i) q^{94} +(9.46410 + 4.73205i) q^{95} -3.19615i q^{97} +(-5.59808 - 3.23205i) q^{98} +(-7.09808 - 12.2942i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{4} + 4 q^{5} + 6 q^{7} - 6 q^{9} - 2 q^{10} + 12 q^{11} + 12 q^{13} - 4 q^{14} - 2 q^{16} - 6 q^{17} + 6 q^{19} + 8 q^{20} + 6 q^{22} - 6 q^{25} + 8 q^{26} + 6 q^{28} - 4 q^{31} + 4 q^{35} - 12 q^{36}+ \cdots - 18 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/370\mathbb{Z}\right)^\times\).

\(n\) \(261\) \(297\)
\(\chi(n)\) \(e\left(\frac{1}{3}\right)\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.866025 − 0.500000i 0.612372 − 0.353553i
\(3\) 0 0 0.500000 − 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(4\) 0.500000 − 0.866025i 0.250000 − 0.433013i
\(5\) 0.133975 + 2.23205i 0.0599153 + 0.998203i
\(6\) 0 0
\(7\) 0.633975 + 0.366025i 0.239620 + 0.138345i 0.615002 − 0.788526i \(-0.289155\pi\)
−0.375382 + 0.926870i \(0.622489\pi\)
\(8\) − 1.00000i − 0.353553i
\(9\) −1.50000 − 2.59808i −0.500000 − 0.866025i
\(10\) 1.23205 + 1.86603i 0.389609 + 0.590089i
\(11\) 4.73205 1.42677 0.713384 − 0.700774i \(-0.247162\pi\)
0.713384 + 0.700774i \(0.247162\pi\)
\(12\) 0 0
\(13\) 4.73205 + 2.73205i 1.31243 + 0.757735i 0.982499 − 0.186269i \(-0.0596395\pi\)
0.329936 + 0.944003i \(0.392973\pi\)
\(14\) 0.732051 0.195649
\(15\) 0 0
\(16\) −0.500000 − 0.866025i −0.125000 − 0.216506i
\(17\) −1.50000 + 0.866025i −0.363803 + 0.210042i −0.670748 − 0.741685i \(-0.734027\pi\)
0.306944 + 0.951727i \(0.400693\pi\)
\(18\) −2.59808 − 1.50000i −0.612372 − 0.353553i
\(19\) 2.36603 − 4.09808i 0.542803 − 0.940163i −0.455938 − 0.890011i \(-0.650696\pi\)
0.998742 − 0.0501517i \(-0.0159705\pi\)
\(20\) 2.00000 + 1.00000i 0.447214 + 0.223607i
\(21\) 0 0
\(22\) 4.09808 − 2.36603i 0.873713 − 0.504438i
\(23\) 5.46410i 1.13934i 0.821872 + 0.569672i \(0.192930\pi\)
−0.821872 + 0.569672i \(0.807070\pi\)
\(24\) 0 0
\(25\) −4.96410 + 0.598076i −0.992820 + 0.119615i
\(26\) 5.46410 1.07160
\(27\) 0 0
\(28\) 0.633975 − 0.366025i 0.119810 − 0.0691723i
\(29\) −1.73205 −0.321634 −0.160817 − 0.986984i \(-0.551413\pi\)
−0.160817 + 0.986984i \(0.551413\pi\)
\(30\) 0 0
\(31\) −9.66025 −1.73503 −0.867516 − 0.497409i \(-0.834285\pi\)
−0.867516 + 0.497409i \(0.834285\pi\)
\(32\) −0.866025 − 0.500000i −0.153093 − 0.0883883i
\(33\) 0 0
\(34\) −0.866025 + 1.50000i −0.148522 + 0.257248i
\(35\) −0.732051 + 1.46410i −0.123739 + 0.247478i
\(36\) −3.00000 −0.500000
\(37\) −2.59808 − 5.50000i −0.427121 − 0.904194i
\(38\) − 4.73205i − 0.767640i
\(39\) 0 0
\(40\) 2.23205 − 0.133975i 0.352918 − 0.0211832i
\(41\) 3.96410 − 6.86603i 0.619089 − 1.07229i −0.370564 − 0.928807i \(-0.620835\pi\)
0.989652 − 0.143486i \(-0.0458312\pi\)
\(42\) 0 0
\(43\) 5.26795i 0.803355i 0.915781 + 0.401677i \(0.131573\pi\)
−0.915781 + 0.401677i \(0.868427\pi\)
\(44\) 2.36603 − 4.09808i 0.356692 − 0.617808i
\(45\) 5.59808 − 3.69615i 0.834512 − 0.550990i
\(46\) 2.73205 + 4.73205i 0.402819 + 0.697703i
\(47\) − 3.26795i − 0.476679i −0.971182 − 0.238340i \(-0.923397\pi\)
0.971182 − 0.238340i \(-0.0766032\pi\)
\(48\) 0 0
\(49\) −3.23205 − 5.59808i −0.461722 − 0.799725i
\(50\) −4.00000 + 3.00000i −0.565685 + 0.424264i
\(51\) 0 0
\(52\) 4.73205 − 2.73205i 0.656217 − 0.378867i
\(53\) −10.7321 + 6.19615i −1.47416 + 0.851107i −0.999576 − 0.0291032i \(-0.990735\pi\)
−0.474584 + 0.880210i \(0.657402\pi\)
\(54\) 0 0
\(55\) 0.633975 + 10.5622i 0.0854851 + 1.42420i
\(56\) 0.366025 − 0.633975i 0.0489122 − 0.0847184i
\(57\) 0 0
\(58\) −1.50000 + 0.866025i −0.196960 + 0.113715i
\(59\) −1.26795 − 2.19615i −0.165073 − 0.285915i 0.771608 − 0.636098i \(-0.219453\pi\)
−0.936681 + 0.350183i \(0.886119\pi\)
\(60\) 0 0
\(61\) −2.13397 + 3.69615i −0.273227 + 0.473244i −0.969686 − 0.244353i \(-0.921424\pi\)
0.696459 + 0.717597i \(0.254758\pi\)
\(62\) −8.36603 + 4.83013i −1.06249 + 0.613427i
\(63\) − 2.19615i − 0.276689i
\(64\) −1.00000 −0.125000
\(65\) −5.46410 + 10.9282i −0.677738 + 1.35548i
\(66\) 0 0
\(67\) 4.56218 + 2.63397i 0.557359 + 0.321791i 0.752085 − 0.659066i \(-0.229048\pi\)
−0.194726 + 0.980858i \(0.562382\pi\)
\(68\) 1.73205i 0.210042i
\(69\) 0 0
\(70\) 0.0980762 + 1.63397i 0.0117223 + 0.195297i
\(71\) −7.56218 + 13.0981i −0.897465 + 1.55446i −0.0667420 + 0.997770i \(0.521260\pi\)
−0.830723 + 0.556685i \(0.812073\pi\)
\(72\) −2.59808 + 1.50000i −0.306186 + 0.176777i
\(73\) − 15.8564i − 1.85585i −0.372764 − 0.927926i \(-0.621590\pi\)
0.372764 − 0.927926i \(-0.378410\pi\)
\(74\) −5.00000 − 3.46410i −0.581238 − 0.402694i
\(75\) 0 0
\(76\) −2.36603 − 4.09808i −0.271402 − 0.470082i
\(77\) 3.00000 + 1.73205i 0.341882 + 0.197386i
\(78\) 0 0
\(79\) −0.830127 + 1.43782i −0.0933966 + 0.161768i −0.908938 − 0.416931i \(-0.863106\pi\)
0.815542 + 0.578698i \(0.196439\pi\)
\(80\) 1.86603 − 1.23205i 0.208628 − 0.137747i
\(81\) −4.50000 + 7.79423i −0.500000 + 0.866025i
\(82\) − 7.92820i − 0.875524i
\(83\) 8.19615 − 4.73205i 0.899645 − 0.519410i 0.0225597 − 0.999745i \(-0.492818\pi\)
0.877085 + 0.480335i \(0.159485\pi\)
\(84\) 0 0
\(85\) −2.13397 − 3.23205i −0.231462 − 0.350565i
\(86\) 2.63397 + 4.56218i 0.284029 + 0.491952i
\(87\) 0 0
\(88\) − 4.73205i − 0.504438i
\(89\) 0.232051 + 0.401924i 0.0245973 + 0.0426038i 0.878062 − 0.478547i \(-0.158836\pi\)
−0.853465 + 0.521151i \(0.825503\pi\)
\(90\) 3.00000 − 6.00000i 0.316228 − 0.632456i
\(91\) 2.00000 + 3.46410i 0.209657 + 0.363137i
\(92\) 4.73205 + 2.73205i 0.493350 + 0.284836i
\(93\) 0 0
\(94\) −1.63397 − 2.83013i −0.168532 − 0.291905i
\(95\) 9.46410 + 4.73205i 0.970996 + 0.485498i
\(96\) 0 0
\(97\) − 3.19615i − 0.324520i −0.986748 − 0.162260i \(-0.948122\pi\)
0.986748 − 0.162260i \(-0.0518783\pi\)
\(98\) −5.59808 − 3.23205i −0.565491 − 0.326486i
\(99\) −7.09808 − 12.2942i −0.713384 − 1.23562i
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 370.2.n.c.359.2 yes 4
5.4 even 2 370.2.n.a.359.1 yes 4
37.10 even 3 370.2.n.a.269.1 ✓ 4
185.84 even 6 inner 370.2.n.c.269.2 yes 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
370.2.n.a.269.1 ✓ 4 37.10 even 3
370.2.n.a.359.1 yes 4 5.4 even 2
370.2.n.c.269.2 yes 4 185.84 even 6 inner
370.2.n.c.359.2 yes 4 1.1 even 1 trivial