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Properties

Label 370.2.h.c.117.1
Level $370$
Weight $2$
Character 370.117
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(117,370)] 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("370.117"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(370, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([1, 3])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 370 = 2 \cdot 5 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 370.h (of order \(4\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,4,0] 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 \(\Q(i)\)
Coefficient field: \(\Q(\zeta_{8})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 117.1
Root \(-0.707107 - 0.707107i\) of defining polynomial
Character \(\chi\) \(=\) 370.117
Dual form 370.2.h.c.253.1

$q$-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+1.00000 q^{2} +(-1.41421 - 1.41421i) q^{3} +1.00000 q^{4} +(-0.707107 + 2.12132i) q^{5} +(-1.41421 - 1.41421i) q^{6} +(2.70711 + 2.70711i) q^{7} +1.00000 q^{8} +1.00000i q^{9} +(-0.707107 + 2.12132i) q^{10} +3.82843i q^{11} +(-1.41421 - 1.41421i) q^{12} +2.24264 q^{13} +(2.70711 + 2.70711i) q^{14} +(4.00000 - 2.00000i) q^{15} +1.00000 q^{16} -1.82843i q^{17} +1.00000i q^{18} +(5.82843 - 5.82843i) q^{19} +(-0.707107 + 2.12132i) q^{20} -7.65685i q^{21} +3.82843i q^{22} -2.58579 q^{23} +(-1.41421 - 1.41421i) q^{24} +(-4.00000 - 3.00000i) q^{25} +2.24264 q^{26} +(-2.82843 + 2.82843i) q^{27} +(2.70711 + 2.70711i) q^{28} +(6.70711 + 6.70711i) q^{29} +(4.00000 - 2.00000i) q^{30} +(-4.12132 + 4.12132i) q^{31} +1.00000 q^{32} +(5.41421 - 5.41421i) q^{33} -1.82843i q^{34} +(-7.65685 + 3.82843i) q^{35} +1.00000i q^{36} +(3.53553 - 4.94975i) q^{37} +(5.82843 - 5.82843i) q^{38} +(-3.17157 - 3.17157i) q^{39} +(-0.707107 + 2.12132i) q^{40} +7.00000i q^{41} -7.65685i q^{42} -7.00000 q^{43} +3.82843i q^{44} +(-2.12132 - 0.707107i) q^{45} -2.58579 q^{46} +(-8.24264 - 8.24264i) q^{47} +(-1.41421 - 1.41421i) q^{48} +7.65685i q^{49} +(-4.00000 - 3.00000i) q^{50} +(-2.58579 + 2.58579i) q^{51} +2.24264 q^{52} +(2.12132 - 2.12132i) q^{53} +(-2.82843 + 2.82843i) q^{54} +(-8.12132 - 2.70711i) q^{55} +(2.70711 + 2.70711i) q^{56} -16.4853 q^{57} +(6.70711 + 6.70711i) q^{58} +(1.17157 - 1.17157i) q^{59} +(4.00000 - 2.00000i) q^{60} +(9.29289 - 9.29289i) q^{61} +(-4.12132 + 4.12132i) q^{62} +(-2.70711 + 2.70711i) q^{63} +1.00000 q^{64} +(-1.58579 + 4.75736i) q^{65} +(5.41421 - 5.41421i) q^{66} +(-1.24264 + 1.24264i) q^{67} -1.82843i q^{68} +(3.65685 + 3.65685i) q^{69} +(-7.65685 + 3.82843i) q^{70} -0.343146 q^{71} +1.00000i q^{72} +(-6.00000 - 6.00000i) q^{73} +(3.53553 - 4.94975i) q^{74} +(1.41421 + 9.89949i) q^{75} +(5.82843 - 5.82843i) q^{76} +(-10.3640 + 10.3640i) q^{77} +(-3.17157 - 3.17157i) q^{78} +(-9.65685 + 9.65685i) q^{79} +(-0.707107 + 2.12132i) q^{80} +11.0000 q^{81} +7.00000i q^{82} +(-0.828427 + 0.828427i) q^{83} -7.65685i q^{84} +(3.87868 + 1.29289i) q^{85} -7.00000 q^{86} -18.9706i q^{87} +3.82843i q^{88} +(-7.41421 - 7.41421i) q^{89} +(-2.12132 - 0.707107i) q^{90} +(6.07107 + 6.07107i) q^{91} -2.58579 q^{92} +11.6569 q^{93} +(-8.24264 - 8.24264i) q^{94} +(8.24264 + 16.4853i) q^{95} +(-1.41421 - 1.41421i) q^{96} -7.00000i q^{97} +7.65685i q^{98} -3.82843 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{2} + 4 q^{4} + 8 q^{7} + 4 q^{8} - 8 q^{13} + 8 q^{14} + 16 q^{15} + 4 q^{16} + 12 q^{19} - 16 q^{23} - 16 q^{25} - 8 q^{26} + 8 q^{28} + 24 q^{29} + 16 q^{30} - 8 q^{31} + 4 q^{32} + 16 q^{33}+ \cdots - 4 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(261\) \(297\)
\(\chi(n)\) \(e\left(\frac{3}{4}\right)\) \(e\left(\frac{1}{4}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.707107
\(3\) −1.41421 − 1.41421i −0.816497 − 0.816497i 0.169102 − 0.985599i \(-0.445913\pi\)
−0.985599 + 0.169102i \(0.945913\pi\)
\(4\) 1.00000 0.500000
\(5\) −0.707107 + 2.12132i −0.316228 + 0.948683i
\(6\) −1.41421 − 1.41421i −0.577350 − 0.577350i
\(7\) 2.70711 + 2.70711i 1.02319 + 1.02319i 0.999725 + 0.0234655i \(0.00747000\pi\)
0.0234655 + 0.999725i \(0.492530\pi\)
\(8\) 1.00000 0.353553
\(9\) 1.00000i 0.333333i
\(10\) −0.707107 + 2.12132i −0.223607 + 0.670820i
\(11\) 3.82843i 1.15431i 0.816633 + 0.577157i \(0.195838\pi\)
−0.816633 + 0.577157i \(0.804162\pi\)
\(12\) −1.41421 − 1.41421i −0.408248 − 0.408248i
\(13\) 2.24264 0.621997 0.310998 − 0.950410i \(-0.399337\pi\)
0.310998 + 0.950410i \(0.399337\pi\)
\(14\) 2.70711 + 2.70711i 0.723505 + 0.723505i
\(15\) 4.00000 − 2.00000i 1.03280 − 0.516398i
\(16\) 1.00000 0.250000
\(17\) − 1.82843i − 0.443459i −0.975108 − 0.221729i \(-0.928830\pi\)
0.975108 − 0.221729i \(-0.0711701\pi\)
\(18\) 1.00000i 0.235702i
\(19\) 5.82843 − 5.82843i 1.33713 − 1.33713i 0.438308 − 0.898825i \(-0.355578\pi\)
0.898825 − 0.438308i \(-0.144422\pi\)
\(20\) −0.707107 + 2.12132i −0.158114 + 0.474342i
\(21\) − 7.65685i − 1.67086i
\(22\) 3.82843i 0.816223i
\(23\) −2.58579 −0.539174 −0.269587 − 0.962976i \(-0.586887\pi\)
−0.269587 + 0.962976i \(0.586887\pi\)
\(24\) −1.41421 − 1.41421i −0.288675 − 0.288675i
\(25\) −4.00000 − 3.00000i −0.800000 − 0.600000i
\(26\) 2.24264 0.439818
\(27\) −2.82843 + 2.82843i −0.544331 + 0.544331i
\(28\) 2.70711 + 2.70711i 0.511595 + 0.511595i
\(29\) 6.70711 + 6.70711i 1.24548 + 1.24548i 0.957696 + 0.287783i \(0.0929181\pi\)
0.287783 + 0.957696i \(0.407082\pi\)
\(30\) 4.00000 − 2.00000i 0.730297 − 0.365148i
\(31\) −4.12132 + 4.12132i −0.740211 + 0.740211i −0.972619 − 0.232408i \(-0.925340\pi\)
0.232408 + 0.972619i \(0.425340\pi\)
\(32\) 1.00000 0.176777
\(33\) 5.41421 − 5.41421i 0.942494 − 0.942494i
\(34\) − 1.82843i − 0.313573i
\(35\) −7.65685 + 3.82843i −1.29424 + 0.647122i
\(36\) 1.00000i 0.166667i
\(37\) 3.53553 − 4.94975i 0.581238 − 0.813733i
\(38\) 5.82843 − 5.82843i 0.945496 − 0.945496i
\(39\) −3.17157 − 3.17157i −0.507858 − 0.507858i
\(40\) −0.707107 + 2.12132i −0.111803 + 0.335410i
\(41\) 7.00000i 1.09322i 0.837389 + 0.546608i \(0.184081\pi\)
−0.837389 + 0.546608i \(0.815919\pi\)
\(42\) − 7.65685i − 1.18148i
\(43\) −7.00000 −1.06749 −0.533745 − 0.845645i \(-0.679216\pi\)
−0.533745 + 0.845645i \(0.679216\pi\)
\(44\) 3.82843i 0.577157i
\(45\) −2.12132 − 0.707107i −0.316228 − 0.105409i
\(46\) −2.58579 −0.381253
\(47\) −8.24264 − 8.24264i −1.20231 − 1.20231i −0.973461 − 0.228851i \(-0.926503\pi\)
−0.228851 − 0.973461i \(-0.573497\pi\)
\(48\) −1.41421 − 1.41421i −0.204124 − 0.204124i
\(49\) 7.65685i 1.09384i
\(50\) −4.00000 − 3.00000i −0.565685 − 0.424264i
\(51\) −2.58579 + 2.58579i −0.362083 + 0.362083i
\(52\) 2.24264 0.310998
\(53\) 2.12132 − 2.12132i 0.291386 − 0.291386i −0.546242 − 0.837628i \(-0.683942\pi\)
0.837628 + 0.546242i \(0.183942\pi\)
\(54\) −2.82843 + 2.82843i −0.384900 + 0.384900i
\(55\) −8.12132 − 2.70711i −1.09508 − 0.365026i
\(56\) 2.70711 + 2.70711i 0.361752 + 0.361752i
\(57\) −16.4853 −2.18353
\(58\) 6.70711 + 6.70711i 0.880686 + 0.880686i
\(59\) 1.17157 − 1.17157i 0.152526 − 0.152526i −0.626719 − 0.779245i \(-0.715603\pi\)
0.779245 + 0.626719i \(0.215603\pi\)
\(60\) 4.00000 − 2.00000i 0.516398 − 0.258199i
\(61\) 9.29289 − 9.29289i 1.18983 − 1.18983i 0.212720 − 0.977113i \(-0.431768\pi\)
0.977113 − 0.212720i \(-0.0682322\pi\)
\(62\) −4.12132 + 4.12132i −0.523408 + 0.523408i
\(63\) −2.70711 + 2.70711i −0.341063 + 0.341063i
\(64\) 1.00000 0.125000
\(65\) −1.58579 + 4.75736i −0.196693 + 0.590078i
\(66\) 5.41421 − 5.41421i 0.666444 − 0.666444i
\(67\) −1.24264 + 1.24264i −0.151813 + 0.151813i −0.778927 − 0.627114i \(-0.784236\pi\)
0.627114 + 0.778927i \(0.284236\pi\)
\(68\) − 1.82843i − 0.221729i
\(69\) 3.65685 + 3.65685i 0.440234 + 0.440234i
\(70\) −7.65685 + 3.82843i −0.915169 + 0.457585i
\(71\) −0.343146 −0.0407239 −0.0203620 − 0.999793i \(-0.506482\pi\)
−0.0203620 + 0.999793i \(0.506482\pi\)
\(72\) 1.00000i 0.117851i
\(73\) −6.00000 − 6.00000i −0.702247 − 0.702247i 0.262646 − 0.964892i \(-0.415405\pi\)
−0.964892 + 0.262646i \(0.915405\pi\)
\(74\) 3.53553 − 4.94975i 0.410997 − 0.575396i
\(75\) 1.41421 + 9.89949i 0.163299 + 1.14310i
\(76\) 5.82843 − 5.82843i 0.668566 − 0.668566i
\(77\) −10.3640 + 10.3640i −1.18108 + 1.18108i
\(78\) −3.17157 − 3.17157i −0.359110 − 0.359110i
\(79\) −9.65685 + 9.65685i −1.08648 + 1.08648i −0.0905930 + 0.995888i \(0.528876\pi\)
−0.995888 + 0.0905930i \(0.971124\pi\)
\(80\) −0.707107 + 2.12132i −0.0790569 + 0.237171i
\(81\) 11.0000 1.22222
\(82\) 7.00000i 0.773021i
\(83\) −0.828427 + 0.828427i −0.0909317 + 0.0909317i −0.751109 − 0.660178i \(-0.770481\pi\)
0.660178 + 0.751109i \(0.270481\pi\)
\(84\) − 7.65685i − 0.835431i
\(85\) 3.87868 + 1.29289i 0.420702 + 0.140234i
\(86\) −7.00000 −0.754829
\(87\) − 18.9706i − 2.03386i
\(88\) 3.82843i 0.408112i
\(89\) −7.41421 − 7.41421i −0.785905 − 0.785905i 0.194915 − 0.980820i \(-0.437557\pi\)
−0.980820 + 0.194915i \(0.937557\pi\)
\(90\) −2.12132 − 0.707107i −0.223607 − 0.0745356i
\(91\) 6.07107 + 6.07107i 0.636421 + 0.636421i
\(92\) −2.58579 −0.269587
\(93\) 11.6569 1.20876
\(94\) −8.24264 − 8.24264i −0.850163 − 0.850163i
\(95\) 8.24264 + 16.4853i 0.845677 + 1.69135i
\(96\) −1.41421 − 1.41421i −0.144338 − 0.144338i
\(97\) − 7.00000i − 0.710742i −0.934725 − 0.355371i \(-0.884354\pi\)
0.934725 − 0.355371i \(-0.115646\pi\)
\(98\) 7.65685i 0.773459i
\(99\) −3.82843 −0.384771
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.h.c.117.1 yes 4
5.3 odd 4 370.2.g.c.43.2 ✓ 4
37.31 odd 4 370.2.g.c.327.2 yes 4
185.68 even 4 inner 370.2.h.c.253.1 yes 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
370.2.g.c.43.2 ✓ 4 5.3 odd 4
370.2.g.c.327.2 yes 4 37.31 odd 4
370.2.h.c.117.1 yes 4 1.1 even 1 trivial
370.2.h.c.253.1 yes 4 185.68 even 4 inner