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Properties

Label 370.2.n.c.359.1
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 359.1
Root \(-0.866025 + 0.500000i\) of defining polynomial
Character \(\chi\) \(=\) 370.359
Dual form 370.2.n.c.269.1

$q$-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)
 
\(f(q)\) \(=\) \(q+(-0.866025 + 0.500000i) q^{2} +(0.500000 - 0.866025i) q^{4} +(1.86603 + 1.23205i) q^{5} +(2.36603 + 1.36603i) q^{7} +1.00000i q^{8} +(-1.50000 - 2.59808i) q^{9} +(-2.23205 - 0.133975i) q^{10} +1.26795 q^{11} +(1.26795 + 0.732051i) q^{13} -2.73205 q^{14} +(-0.500000 - 0.866025i) q^{16} +(-1.50000 + 0.866025i) q^{17} +(2.59808 + 1.50000i) q^{18} +(0.633975 - 1.09808i) q^{19} +(2.00000 - 1.00000i) q^{20} +(-1.09808 + 0.633975i) q^{22} +1.46410i q^{23} +(1.96410 + 4.59808i) q^{25} -1.46410 q^{26} +(2.36603 - 1.36603i) q^{28} +1.73205 q^{29} +7.66025 q^{31} +(0.866025 + 0.500000i) q^{32} +(0.866025 - 1.50000i) q^{34} +(2.73205 + 5.46410i) q^{35} -3.00000 q^{36} +(2.59808 + 5.50000i) q^{37} +1.26795i q^{38} +(-1.23205 + 1.86603i) q^{40} +(-2.96410 + 5.13397i) q^{41} -8.73205i q^{43} +(0.633975 - 1.09808i) q^{44} +(0.401924 - 6.69615i) q^{45} +(-0.732051 - 1.26795i) q^{46} +6.73205i q^{47} +(0.232051 + 0.401924i) q^{49} +(-4.00000 - 3.00000i) q^{50} +(1.26795 - 0.732051i) q^{52} +(-7.26795 + 4.19615i) q^{53} +(2.36603 + 1.56218i) q^{55} +(-1.36603 + 2.36603i) q^{56} +(-1.50000 + 0.866025i) q^{58} +(-4.73205 - 8.19615i) q^{59} +(-3.86603 + 6.69615i) q^{61} +(-6.63397 + 3.83013i) q^{62} -8.19615i q^{63} -1.00000 q^{64} +(1.46410 + 2.92820i) q^{65} +(-7.56218 - 4.36603i) q^{67} +1.73205i q^{68} +(-5.09808 - 3.36603i) q^{70} +(4.56218 - 7.90192i) q^{71} +(2.59808 - 1.50000i) q^{72} -11.8564i q^{73} +(-5.00000 - 3.46410i) q^{74} +(-0.633975 - 1.09808i) q^{76} +(3.00000 + 1.73205i) q^{77} +(7.83013 - 13.5622i) q^{79} +(0.133975 - 2.23205i) q^{80} +(-4.50000 + 7.79423i) q^{81} -5.92820i q^{82} +(-2.19615 + 1.26795i) q^{83} +(-3.86603 - 0.232051i) q^{85} +(4.36603 + 7.56218i) q^{86} +1.26795i q^{88} +(-3.23205 - 5.59808i) q^{89} +(3.00000 + 6.00000i) q^{90} +(2.00000 + 3.46410i) q^{91} +(1.26795 + 0.732051i) q^{92} +(-3.36603 - 5.83013i) q^{94} +(2.53590 - 1.26795i) q^{95} -7.19615i q^{97} +(-0.401924 - 0.232051i) q^{98} +(-1.90192 - 3.29423i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{4} + 4 q^{5} + 6 q^{7} - 6 q^{9} - 2 q^{10} + 12 q^{11} + 12 q^{13} - 4 q^{14} - 2 q^{16} - 6 q^{17} + 6 q^{19} + 8 q^{20} + 6 q^{22} - 6 q^{25} + 8 q^{26} + 6 q^{28} - 4 q^{31} + 4 q^{35} - 12 q^{36}+ \cdots - 18 q^{99}+O(q^{100}) \) 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
\(5\) 1.86603 + 1.23205i 0.834512 + 0.550990i
\(6\) 0 0
\(7\) 2.36603 + 1.36603i 0.894274 + 0.516309i 0.875338 − 0.483512i \(-0.160639\pi\)
0.0189356 + 0.999821i \(0.493972\pi\)
\(8\) 1.00000i 0.353553i
\(9\) −1.50000 − 2.59808i −0.500000 − 0.866025i
\(10\) −2.23205 − 0.133975i −0.705836 − 0.0423665i
\(11\) 1.26795 0.382301 0.191151 − 0.981561i \(-0.438778\pi\)
0.191151 + 0.981561i \(0.438778\pi\)
\(12\) 0 0
\(13\) 1.26795 + 0.732051i 0.351666 + 0.203034i 0.665419 − 0.746470i \(-0.268253\pi\)
−0.313753 + 0.949505i \(0.601586\pi\)
\(14\) −2.73205 −0.730171
\(15\) 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
\(19\) 0.633975 − 1.09808i 0.145444 − 0.251916i −0.784095 − 0.620641i \(-0.786872\pi\)
0.929538 + 0.368725i \(0.120206\pi\)
\(20\) 2.00000 − 1.00000i 0.447214 − 0.223607i
\(21\) 0 0
\(22\) −1.09808 + 0.633975i −0.234111 + 0.135164i
\(23\) 1.46410i 0.305286i 0.988281 + 0.152643i \(0.0487785\pi\)
−0.988281 + 0.152643i \(0.951221\pi\)
\(24\) 0 0
\(25\) 1.96410 + 4.59808i 0.392820 + 0.919615i
\(26\) −1.46410 −0.287134
\(27\) 0 0
\(28\) 2.36603 − 1.36603i 0.447137 − 0.258155i
\(29\) 1.73205 0.321634 0.160817 − 0.986984i \(-0.448587\pi\)
0.160817 + 0.986984i \(0.448587\pi\)
\(30\) 0 0
\(31\) 7.66025 1.37582 0.687911 − 0.725795i \(-0.258528\pi\)
0.687911 + 0.725795i \(0.258528\pi\)
\(32\) 0.866025 + 0.500000i 0.153093 + 0.0883883i
\(33\) 0 0
\(34\) 0.866025 − 1.50000i 0.148522 − 0.257248i
\(35\) 2.73205 + 5.46410i 0.461801 + 0.923602i
\(36\) −3.00000 −0.500000
\(37\) 2.59808 + 5.50000i 0.427121 + 0.904194i
\(38\) 1.26795i 0.205689i
\(39\) 0 0
\(40\) −1.23205 + 1.86603i −0.194804 + 0.295045i
\(41\) −2.96410 + 5.13397i −0.462915 + 0.801792i −0.999105 − 0.0423053i \(-0.986530\pi\)
0.536190 + 0.844097i \(0.319863\pi\)
\(42\) 0 0
\(43\) − 8.73205i − 1.33163i −0.746119 − 0.665813i \(-0.768085\pi\)
0.746119 − 0.665813i \(-0.231915\pi\)
\(44\) 0.633975 − 1.09808i 0.0955753 − 0.165541i
\(45\) 0.401924 − 6.69615i 0.0599153 − 0.998203i
\(46\) −0.732051 − 1.26795i −0.107935 − 0.186949i
\(47\) 6.73205i 0.981971i 0.871168 + 0.490985i \(0.163363\pi\)
−0.871168 + 0.490985i \(0.836637\pi\)
\(48\) 0 0
\(49\) 0.232051 + 0.401924i 0.0331501 + 0.0574177i
\(50\) −4.00000 − 3.00000i −0.565685 − 0.424264i
\(51\) 0 0
\(52\) 1.26795 − 0.732051i 0.175833 − 0.101517i
\(53\) −7.26795 + 4.19615i −0.998330 + 0.576386i −0.907754 − 0.419504i \(-0.862204\pi\)
−0.0905760 + 0.995890i \(0.528871\pi\)
\(54\) 0 0
\(55\) 2.36603 + 1.56218i 0.319035 + 0.210644i
\(56\) −1.36603 + 2.36603i −0.182543 + 0.316173i
\(57\) 0 0
\(58\) −1.50000 + 0.866025i −0.196960 + 0.113715i
\(59\) −4.73205 − 8.19615i −0.616061 − 1.06705i −0.990197 − 0.139675i \(-0.955394\pi\)
0.374137 − 0.927373i \(-0.377939\pi\)
\(60\) 0 0
\(61\) −3.86603 + 6.69615i −0.494994 + 0.857354i −0.999983 − 0.00577101i \(-0.998163\pi\)
0.504990 + 0.863125i \(0.331496\pi\)
\(62\) −6.63397 + 3.83013i −0.842516 + 0.486427i
\(63\) − 8.19615i − 1.03262i
\(64\) −1.00000 −0.125000
\(65\) 1.46410 + 2.92820i 0.181599 + 0.363199i
\(66\) 0 0
\(67\) −7.56218 − 4.36603i −0.923867 − 0.533395i −0.0390004 − 0.999239i \(-0.512417\pi\)
−0.884867 + 0.465844i \(0.845751\pi\)
\(68\) 1.73205i 0.210042i
\(69\) 0 0
\(70\) −5.09808 − 3.36603i −0.609337 − 0.402317i
\(71\) 4.56218 − 7.90192i 0.541431 − 0.937786i −0.457391 − 0.889266i \(-0.651216\pi\)
0.998822 − 0.0485203i \(-0.0154506\pi\)
\(72\) 2.59808 − 1.50000i 0.306186 − 0.176777i
\(73\) − 11.8564i − 1.38769i −0.720126 − 0.693844i \(-0.755916\pi\)
0.720126 − 0.693844i \(-0.244084\pi\)
\(74\) −5.00000 − 3.46410i −0.581238 − 0.402694i
\(75\) 0 0
\(76\) −0.633975 − 1.09808i −0.0727219 − 0.125958i
\(77\) 3.00000 + 1.73205i 0.341882 + 0.197386i
\(78\) 0 0
\(79\) 7.83013 − 13.5622i 0.880958 − 1.52586i 0.0306808 − 0.999529i \(-0.490232\pi\)
0.850277 − 0.526335i \(-0.176434\pi\)
\(80\) 0.133975 − 2.23205i 0.0149788 − 0.249551i
\(81\) −4.50000 + 7.79423i −0.500000 + 0.866025i
\(82\) − 5.92820i − 0.654661i
\(83\) −2.19615 + 1.26795i −0.241059 + 0.139176i −0.615663 − 0.788009i \(-0.711112\pi\)
0.374604 + 0.927185i \(0.377779\pi\)
\(84\) 0 0
\(85\) −3.86603 − 0.232051i −0.419329 − 0.0251694i
\(86\) 4.36603 + 7.56218i 0.470801 + 0.815451i
\(87\) 0 0
\(88\) 1.26795i 0.135164i
\(89\) −3.23205 − 5.59808i −0.342597 − 0.593395i 0.642317 − 0.766439i \(-0.277973\pi\)
−0.984914 + 0.173044i \(0.944640\pi\)
\(90\) 3.00000 + 6.00000i 0.316228 + 0.632456i
\(91\) 2.00000 + 3.46410i 0.209657 + 0.363137i
\(92\) 1.26795 + 0.732051i 0.132193 + 0.0763216i
\(93\) 0 0
\(94\) −3.36603 − 5.83013i −0.347179 − 0.601332i
\(95\) 2.53590 − 1.26795i 0.260178 − 0.130089i
\(96\) 0 0
\(97\) − 7.19615i − 0.730659i −0.930878 − 0.365329i \(-0.880956\pi\)
0.930878 − 0.365329i \(-0.119044\pi\)
\(98\) −0.401924 − 0.232051i −0.0406004 − 0.0234407i
\(99\) −1.90192 − 3.29423i −0.191151 − 0.331082i
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.359.1 yes 4
5.4 even 2 370.2.n.a.359.2 yes 4
37.10 even 3 370.2.n.a.269.2 ✓ 4
185.84 even 6 inner 370.2.n.c.269.1 yes 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
370.2.n.a.269.2 ✓ 4 37.10 even 3
370.2.n.a.359.2 yes 4 5.4 even 2
370.2.n.c.269.1 yes 4 185.84 even 6 inner
370.2.n.c.359.1 yes 4 1.1 even 1 trivial