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Properties

Label 370.2.e.d.211.1
Level $370$
Weight $2$
Character 370.211
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(121,370)] 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("370.121"); 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([0, 4])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 370 = 2 \cdot 5 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 370.e (of order \(3\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,-2,2] 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: no
Analytic conductor: \(2.95446487479\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
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: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2}\cdot 3 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 211.1
Root \(0.866025 - 0.500000i\) of defining polynomial
Character \(\chi\) \(=\) 370.211
Dual form 370.2.e.d.121.1

$q$-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+(-0.500000 - 0.866025i) q^{2} +(0.500000 - 0.866025i) q^{3} +(-0.500000 + 0.866025i) q^{4} +(-0.500000 + 0.866025i) q^{5} -1.00000 q^{6} +(-1.73205 + 3.00000i) q^{7} +1.00000 q^{8} +(1.00000 + 1.73205i) q^{9} +1.00000 q^{10} -5.46410 q^{11} +(0.500000 + 0.866025i) q^{12} +(-2.23205 + 3.86603i) q^{13} +3.46410 q^{14} +(0.500000 + 0.866025i) q^{15} +(-0.500000 - 0.866025i) q^{16} +(0.732051 + 1.26795i) q^{17} +(1.00000 - 1.73205i) q^{18} +(1.00000 - 1.73205i) q^{19} +(-0.500000 - 0.866025i) q^{20} +(1.73205 + 3.00000i) q^{21} +(2.73205 + 4.73205i) q^{22} -5.46410 q^{23} +(0.500000 - 0.866025i) q^{24} +(-0.500000 - 0.866025i) q^{25} +4.46410 q^{26} +5.00000 q^{27} +(-1.73205 - 3.00000i) q^{28} -2.00000 q^{29} +(0.500000 - 0.866025i) q^{30} +8.46410 q^{31} +(-0.500000 + 0.866025i) q^{32} +(-2.73205 + 4.73205i) q^{33} +(0.732051 - 1.26795i) q^{34} +(-1.73205 - 3.00000i) q^{35} -2.00000 q^{36} +(4.69615 + 3.86603i) q^{37} -2.00000 q^{38} +(2.23205 + 3.86603i) q^{39} +(-0.500000 + 0.866025i) q^{40} +(5.96410 - 10.3301i) q^{41} +(1.73205 - 3.00000i) q^{42} -9.92820 q^{43} +(2.73205 - 4.73205i) q^{44} -2.00000 q^{45} +(2.73205 + 4.73205i) q^{46} +3.46410 q^{47} -1.00000 q^{48} +(-2.50000 - 4.33013i) q^{49} +(-0.500000 + 0.866025i) q^{50} +1.46410 q^{51} +(-2.23205 - 3.86603i) q^{52} +(2.76795 + 4.79423i) q^{53} +(-2.50000 - 4.33013i) q^{54} +(2.73205 - 4.73205i) q^{55} +(-1.73205 + 3.00000i) q^{56} +(-1.00000 - 1.73205i) q^{57} +(1.00000 + 1.73205i) q^{58} +(2.73205 + 4.73205i) q^{59} -1.00000 q^{60} +(-4.46410 + 7.73205i) q^{61} +(-4.23205 - 7.33013i) q^{62} -6.92820 q^{63} +1.00000 q^{64} +(-2.23205 - 3.86603i) q^{65} +5.46410 q^{66} +(-7.46410 + 12.9282i) q^{67} -1.46410 q^{68} +(-2.73205 + 4.73205i) q^{69} +(-1.73205 + 3.00000i) q^{70} +(-0.535898 + 0.928203i) q^{71} +(1.00000 + 1.73205i) q^{72} +2.53590 q^{73} +(1.00000 - 6.00000i) q^{74} -1.00000 q^{75} +(1.00000 + 1.73205i) q^{76} +(9.46410 - 16.3923i) q^{77} +(2.23205 - 3.86603i) q^{78} +(7.46410 - 12.9282i) q^{79} +1.00000 q^{80} +(-0.500000 + 0.866025i) q^{81} -11.9282 q^{82} +(-4.92820 - 8.53590i) q^{83} -3.46410 q^{84} -1.46410 q^{85} +(4.96410 + 8.59808i) q^{86} +(-1.00000 + 1.73205i) q^{87} -5.46410 q^{88} +(1.00000 + 1.73205i) q^{89} +(1.00000 + 1.73205i) q^{90} +(-7.73205 - 13.3923i) q^{91} +(2.73205 - 4.73205i) q^{92} +(4.23205 - 7.33013i) q^{93} +(-1.73205 - 3.00000i) q^{94} +(1.00000 + 1.73205i) q^{95} +(0.500000 + 0.866025i) q^{96} -2.00000 q^{97} +(-2.50000 + 4.33013i) q^{98} +(-5.46410 - 9.46410i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 2 q^{2} + 2 q^{3} - 2 q^{4} - 2 q^{5} - 4 q^{6} + 4 q^{8} + 4 q^{9} + 4 q^{10} - 8 q^{11} + 2 q^{12} - 2 q^{13} + 2 q^{15} - 2 q^{16} - 4 q^{17} + 4 q^{18} + 4 q^{19} - 2 q^{20} + 4 q^{22} - 8 q^{23}+ \cdots - 8 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.500000 − 0.866025i −0.353553 − 0.612372i
\(3\) 0.500000 − 0.866025i 0.288675 − 0.500000i −0.684819 − 0.728714i \(-0.740119\pi\)
0.973494 + 0.228714i \(0.0734519\pi\)
\(4\) −0.500000 + 0.866025i −0.250000 + 0.433013i
\(5\) −0.500000 + 0.866025i −0.223607 + 0.387298i
\(6\) −1.00000 −0.408248
\(7\) −1.73205 + 3.00000i −0.654654 + 1.13389i 0.327327 + 0.944911i \(0.393852\pi\)
−0.981981 + 0.188982i \(0.939481\pi\)
\(8\) 1.00000 0.353553
\(9\) 1.00000 + 1.73205i 0.333333 + 0.577350i
\(10\) 1.00000 0.316228
\(11\) −5.46410 −1.64749 −0.823744 − 0.566961i \(-0.808119\pi\)
−0.823744 + 0.566961i \(0.808119\pi\)
\(12\) 0.500000 + 0.866025i 0.144338 + 0.250000i
\(13\) −2.23205 + 3.86603i −0.619060 + 1.07224i 0.370598 + 0.928793i \(0.379153\pi\)
−0.989658 + 0.143449i \(0.954181\pi\)
\(14\) 3.46410 0.925820
\(15\) 0.500000 + 0.866025i 0.129099 + 0.223607i
\(16\) −0.500000 − 0.866025i −0.125000 − 0.216506i
\(17\) 0.732051 + 1.26795i 0.177548 + 0.307523i 0.941040 − 0.338295i \(-0.109850\pi\)
−0.763492 + 0.645817i \(0.776517\pi\)
\(18\) 1.00000 − 1.73205i 0.235702 − 0.408248i
\(19\) 1.00000 − 1.73205i 0.229416 − 0.397360i −0.728219 − 0.685344i \(-0.759652\pi\)
0.957635 + 0.287984i \(0.0929851\pi\)
\(20\) −0.500000 − 0.866025i −0.111803 − 0.193649i
\(21\) 1.73205 + 3.00000i 0.377964 + 0.654654i
\(22\) 2.73205 + 4.73205i 0.582475 + 1.00888i
\(23\) −5.46410 −1.13934 −0.569672 − 0.821872i \(-0.692930\pi\)
−0.569672 + 0.821872i \(0.692930\pi\)
\(24\) 0.500000 − 0.866025i 0.102062 − 0.176777i
\(25\) −0.500000 − 0.866025i −0.100000 − 0.173205i
\(26\) 4.46410 0.875482
\(27\) 5.00000 0.962250
\(28\) −1.73205 − 3.00000i −0.327327 − 0.566947i
\(29\) −2.00000 −0.371391 −0.185695 − 0.982607i \(-0.559454\pi\)
−0.185695 + 0.982607i \(0.559454\pi\)
\(30\) 0.500000 − 0.866025i 0.0912871 − 0.158114i
\(31\) 8.46410 1.52020 0.760099 − 0.649808i \(-0.225151\pi\)
0.760099 + 0.649808i \(0.225151\pi\)
\(32\) −0.500000 + 0.866025i −0.0883883 + 0.153093i
\(33\) −2.73205 + 4.73205i −0.475589 + 0.823744i
\(34\) 0.732051 − 1.26795i 0.125546 − 0.217451i
\(35\) −1.73205 − 3.00000i −0.292770 − 0.507093i
\(36\) −2.00000 −0.333333
\(37\) 4.69615 + 3.86603i 0.772043 + 0.635571i
\(38\) −2.00000 −0.324443
\(39\) 2.23205 + 3.86603i 0.357414 + 0.619060i
\(40\) −0.500000 + 0.866025i −0.0790569 + 0.136931i
\(41\) 5.96410 − 10.3301i 0.931436 − 1.61329i 0.150567 − 0.988600i \(-0.451890\pi\)
0.780869 − 0.624695i \(-0.214777\pi\)
\(42\) 1.73205 − 3.00000i 0.267261 − 0.462910i
\(43\) −9.92820 −1.51404 −0.757018 − 0.653394i \(-0.773345\pi\)
−0.757018 + 0.653394i \(0.773345\pi\)
\(44\) 2.73205 − 4.73205i 0.411872 − 0.713384i
\(45\) −2.00000 −0.298142
\(46\) 2.73205 + 4.73205i 0.402819 + 0.697703i
\(47\) 3.46410 0.505291 0.252646 − 0.967559i \(-0.418699\pi\)
0.252646 + 0.967559i \(0.418699\pi\)
\(48\) −1.00000 −0.144338
\(49\) −2.50000 − 4.33013i −0.357143 − 0.618590i
\(50\) −0.500000 + 0.866025i −0.0707107 + 0.122474i
\(51\) 1.46410 0.205015
\(52\) −2.23205 − 3.86603i −0.309530 − 0.536121i
\(53\) 2.76795 + 4.79423i 0.380207 + 0.658538i 0.991092 − 0.133182i \(-0.0425194\pi\)
−0.610885 + 0.791720i \(0.709186\pi\)
\(54\) −2.50000 − 4.33013i −0.340207 − 0.589256i
\(55\) 2.73205 − 4.73205i 0.368390 − 0.638070i
\(56\) −1.73205 + 3.00000i −0.231455 + 0.400892i
\(57\) −1.00000 − 1.73205i −0.132453 − 0.229416i
\(58\) 1.00000 + 1.73205i 0.131306 + 0.227429i
\(59\) 2.73205 + 4.73205i 0.355683 + 0.616061i 0.987235 − 0.159273i \(-0.0509150\pi\)
−0.631552 + 0.775334i \(0.717582\pi\)
\(60\) −1.00000 −0.129099
\(61\) −4.46410 + 7.73205i −0.571570 + 0.989988i 0.424835 + 0.905271i \(0.360332\pi\)
−0.996405 + 0.0847171i \(0.973001\pi\)
\(62\) −4.23205 − 7.33013i −0.537471 − 0.930927i
\(63\) −6.92820 −0.872872
\(64\) 1.00000 0.125000
\(65\) −2.23205 − 3.86603i −0.276852 − 0.479521i
\(66\) 5.46410 0.672584
\(67\) −7.46410 + 12.9282i −0.911885 + 1.57943i −0.100486 + 0.994938i \(0.532040\pi\)
−0.811399 + 0.584493i \(0.801293\pi\)
\(68\) −1.46410 −0.177548
\(69\) −2.73205 + 4.73205i −0.328900 + 0.569672i
\(70\) −1.73205 + 3.00000i −0.207020 + 0.358569i
\(71\) −0.535898 + 0.928203i −0.0635994 + 0.110157i −0.896072 − 0.443909i \(-0.853591\pi\)
0.832472 + 0.554066i \(0.186925\pi\)
\(72\) 1.00000 + 1.73205i 0.117851 + 0.204124i
\(73\) 2.53590 0.296804 0.148402 − 0.988927i \(-0.452587\pi\)
0.148402 + 0.988927i \(0.452587\pi\)
\(74\) 1.00000 − 6.00000i 0.116248 − 0.697486i
\(75\) −1.00000 −0.115470
\(76\) 1.00000 + 1.73205i 0.114708 + 0.198680i
\(77\) 9.46410 − 16.3923i 1.07853 − 1.86808i
\(78\) 2.23205 − 3.86603i 0.252730 − 0.437741i
\(79\) 7.46410 − 12.9282i 0.839777 − 1.45454i −0.0503039 − 0.998734i \(-0.516019\pi\)
0.890081 − 0.455803i \(-0.150648\pi\)
\(80\) 1.00000 0.111803
\(81\) −0.500000 + 0.866025i −0.0555556 + 0.0962250i
\(82\) −11.9282 −1.31725
\(83\) −4.92820 − 8.53590i −0.540941 − 0.936937i −0.998850 − 0.0479379i \(-0.984735\pi\)
0.457910 − 0.888999i \(-0.348598\pi\)
\(84\) −3.46410 −0.377964
\(85\) −1.46410 −0.158804
\(86\) 4.96410 + 8.59808i 0.535293 + 0.927154i
\(87\) −1.00000 + 1.73205i −0.107211 + 0.185695i
\(88\) −5.46410 −0.582475
\(89\) 1.00000 + 1.73205i 0.106000 + 0.183597i 0.914146 − 0.405385i \(-0.132862\pi\)
−0.808146 + 0.588982i \(0.799529\pi\)
\(90\) 1.00000 + 1.73205i 0.105409 + 0.182574i
\(91\) −7.73205 − 13.3923i −0.810539 − 1.40390i
\(92\) 2.73205 − 4.73205i 0.284836 − 0.493350i
\(93\) 4.23205 − 7.33013i 0.438843 − 0.760099i
\(94\) −1.73205 − 3.00000i −0.178647 − 0.309426i
\(95\) 1.00000 + 1.73205i 0.102598 + 0.177705i
\(96\) 0.500000 + 0.866025i 0.0510310 + 0.0883883i
\(97\) −2.00000 −0.203069 −0.101535 − 0.994832i \(-0.532375\pi\)
−0.101535 + 0.994832i \(0.532375\pi\)
\(98\) −2.50000 + 4.33013i −0.252538 + 0.437409i
\(99\) −5.46410 − 9.46410i −0.549163 − 0.951178i
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.e.d.211.1 yes 4
37.10 even 3 inner 370.2.e.d.121.1 ✓ 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
370.2.e.d.121.1 ✓ 4 37.10 even 3 inner
370.2.e.d.211.1 yes 4 1.1 even 1 trivial