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Properties

Label 154.2.f.c.71.1
Level $154$
Weight $2$
Character 154.71
Analytic conductor $1.230$
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] = [154,2,Mod(15,154)] 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("154.15"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(154, base_ring=CyclotomicField(10)) chi = DirichletCharacter(H, H._module([0, 2])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 154 = 2 \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 154.f (of order \(5\), degree \(4\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,1,-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: \(1.22969619113\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{10})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} + x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

Embedding invariants

Embedding label 71.1
Root \(-0.309017 + 0.951057i\) of defining polynomial
Character \(\chi\) \(=\) 154.71
Dual form 154.2.f.c.141.1

$q$-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+(-0.309017 - 0.951057i) q^{2} +(-0.500000 - 0.363271i) q^{3} +(-0.809017 + 0.587785i) q^{4} +(0.618034 - 1.90211i) q^{5} +(-0.190983 + 0.587785i) q^{6} +(0.809017 - 0.587785i) q^{7} +(0.809017 + 0.587785i) q^{8} +(-0.809017 - 2.48990i) q^{9} -2.00000 q^{10} +(-3.04508 - 1.31433i) q^{11} +0.618034 q^{12} +(-0.381966 - 1.17557i) q^{13} +(-0.809017 - 0.587785i) q^{14} +(-1.00000 + 0.726543i) q^{15} +(0.309017 - 0.951057i) q^{16} +(0.263932 - 0.812299i) q^{17} +(-2.11803 + 1.53884i) q^{18} +(5.54508 + 4.02874i) q^{19} +(0.618034 + 1.90211i) q^{20} -0.618034 q^{21} +(-0.309017 + 3.30220i) q^{22} -0.472136 q^{23} +(-0.190983 - 0.587785i) q^{24} +(0.809017 + 0.587785i) q^{25} +(-1.00000 + 0.726543i) q^{26} +(-1.07295 + 3.30220i) q^{27} +(-0.309017 + 0.951057i) q^{28} +(6.47214 - 4.70228i) q^{29} +(1.00000 + 0.726543i) q^{30} +(1.38197 + 4.25325i) q^{31} -1.00000 q^{32} +(1.04508 + 1.76336i) q^{33} -0.854102 q^{34} +(-0.618034 - 1.90211i) q^{35} +(2.11803 + 1.53884i) q^{36} +(-4.61803 + 3.35520i) q^{37} +(2.11803 - 6.51864i) q^{38} +(-0.236068 + 0.726543i) q^{39} +(1.61803 - 1.17557i) q^{40} +(4.73607 + 3.44095i) q^{41} +(0.190983 + 0.587785i) q^{42} +1.85410 q^{43} +(3.23607 - 0.726543i) q^{44} -5.23607 q^{45} +(0.145898 + 0.449028i) q^{46} +(9.47214 + 6.88191i) q^{47} +(-0.500000 + 0.363271i) q^{48} +(0.309017 - 0.951057i) q^{49} +(0.309017 - 0.951057i) q^{50} +(-0.427051 + 0.310271i) q^{51} +(1.00000 + 0.726543i) q^{52} +(-4.00000 - 12.3107i) q^{53} +3.47214 q^{54} +(-4.38197 + 4.97980i) q^{55} +1.00000 q^{56} +(-1.30902 - 4.02874i) q^{57} +(-6.47214 - 4.70228i) q^{58} +(-8.78115 + 6.37988i) q^{59} +(0.381966 - 1.17557i) q^{60} +(-0.763932 + 2.35114i) q^{61} +(3.61803 - 2.62866i) q^{62} +(-2.11803 - 1.53884i) q^{63} +(0.309017 + 0.951057i) q^{64} -2.47214 q^{65} +(1.35410 - 1.53884i) q^{66} -10.0902 q^{67} +(0.263932 + 0.812299i) q^{68} +(0.236068 + 0.171513i) q^{69} +(-1.61803 + 1.17557i) q^{70} +(0.472136 - 1.45309i) q^{71} +(0.809017 - 2.48990i) q^{72} +(3.73607 - 2.71441i) q^{73} +(4.61803 + 3.35520i) q^{74} +(-0.190983 - 0.587785i) q^{75} -6.85410 q^{76} +(-3.23607 + 0.726543i) q^{77} +0.763932 q^{78} +(1.00000 + 3.07768i) q^{79} +(-1.61803 - 1.17557i) q^{80} +(-4.61803 + 3.35520i) q^{81} +(1.80902 - 5.56758i) q^{82} +(4.95492 - 15.2497i) q^{83} +(0.500000 - 0.363271i) q^{84} +(-1.38197 - 1.00406i) q^{85} +(-0.572949 - 1.76336i) q^{86} -4.94427 q^{87} +(-1.69098 - 2.85317i) q^{88} +0.326238 q^{89} +(1.61803 + 4.97980i) q^{90} +(-1.00000 - 0.726543i) q^{91} +(0.381966 - 0.277515i) q^{92} +(0.854102 - 2.62866i) q^{93} +(3.61803 - 11.1352i) q^{94} +(11.0902 - 8.05748i) q^{95} +(0.500000 + 0.363271i) q^{96} +(-3.04508 - 9.37181i) q^{97} -1.00000 q^{98} +(-0.809017 + 8.64527i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + q^{2} - 2 q^{3} - q^{4} - 2 q^{5} - 3 q^{6} + q^{7} + q^{8} - q^{9} - 8 q^{10} - q^{11} - 2 q^{12} - 6 q^{13} - q^{14} - 4 q^{15} - q^{16} + 10 q^{17} - 4 q^{18} + 11 q^{19} - 2 q^{20} + 2 q^{21}+ \cdots - q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/154\mathbb{Z}\right)^\times\).

\(n\) \(45\) \(57\)
\(\chi(n)\) \(1\) \(e\left(\frac{2}{5}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.309017 − 0.951057i −0.218508 − 0.672499i
\(3\) −0.500000 − 0.363271i −0.288675 − 0.209735i 0.434017 − 0.900905i \(-0.357096\pi\)
−0.722692 + 0.691170i \(0.757096\pi\)
\(4\) −0.809017 + 0.587785i −0.404508 + 0.293893i
\(5\) 0.618034 − 1.90211i 0.276393 − 0.850651i −0.712454 − 0.701719i \(-0.752416\pi\)
0.988847 − 0.148932i \(-0.0475836\pi\)
\(6\) −0.190983 + 0.587785i −0.0779685 + 0.239962i
\(7\) 0.809017 − 0.587785i 0.305780 − 0.222162i
\(8\) 0.809017 + 0.587785i 0.286031 + 0.207813i
\(9\) −0.809017 − 2.48990i −0.269672 − 0.829966i
\(10\) −2.00000 −0.632456
\(11\) −3.04508 − 1.31433i −0.918128 − 0.396285i
\(12\) 0.618034 0.178411
\(13\) −0.381966 − 1.17557i −0.105938 − 0.326045i 0.884011 − 0.467466i \(-0.154833\pi\)
−0.989950 + 0.141421i \(0.954833\pi\)
\(14\) −0.809017 − 0.587785i −0.216219 − 0.157092i
\(15\) −1.00000 + 0.726543i −0.258199 + 0.187592i
\(16\) 0.309017 − 0.951057i 0.0772542 − 0.237764i
\(17\) 0.263932 − 0.812299i 0.0640129 − 0.197012i −0.913935 − 0.405861i \(-0.866972\pi\)
0.977948 + 0.208849i \(0.0669718\pi\)
\(18\) −2.11803 + 1.53884i −0.499225 + 0.362708i
\(19\) 5.54508 + 4.02874i 1.27213 + 0.924256i 0.999285 − 0.0378018i \(-0.0120356\pi\)
0.272844 + 0.962058i \(0.412036\pi\)
\(20\) 0.618034 + 1.90211i 0.138197 + 0.425325i
\(21\) −0.618034 −0.134866
\(22\) −0.309017 + 3.30220i −0.0658826 + 0.704031i
\(23\) −0.472136 −0.0984472 −0.0492236 − 0.998788i \(-0.515675\pi\)
−0.0492236 + 0.998788i \(0.515675\pi\)
\(24\) −0.190983 − 0.587785i −0.0389842 − 0.119981i
\(25\) 0.809017 + 0.587785i 0.161803 + 0.117557i
\(26\) −1.00000 + 0.726543i −0.196116 + 0.142487i
\(27\) −1.07295 + 3.30220i −0.206489 + 0.635508i
\(28\) −0.309017 + 0.951057i −0.0583987 + 0.179733i
\(29\) 6.47214 − 4.70228i 1.20185 − 0.873192i 0.207380 − 0.978260i \(-0.433506\pi\)
0.994465 + 0.105069i \(0.0335062\pi\)
\(30\) 1.00000 + 0.726543i 0.182574 + 0.132648i
\(31\) 1.38197 + 4.25325i 0.248208 + 0.763907i 0.995092 + 0.0989523i \(0.0315491\pi\)
−0.746884 + 0.664955i \(0.768451\pi\)
\(32\) −1.00000 −0.176777
\(33\) 1.04508 + 1.76336i 0.181926 + 0.306961i
\(34\) −0.854102 −0.146477
\(35\) −0.618034 − 1.90211i −0.104467 − 0.321516i
\(36\) 2.11803 + 1.53884i 0.353006 + 0.256474i
\(37\) −4.61803 + 3.35520i −0.759200 + 0.551591i −0.898665 − 0.438636i \(-0.855462\pi\)
0.139465 + 0.990227i \(0.455462\pi\)
\(38\) 2.11803 − 6.51864i 0.343590 − 1.05746i
\(39\) −0.236068 + 0.726543i −0.0378011 + 0.116340i
\(40\) 1.61803 − 1.17557i 0.255834 − 0.185874i
\(41\) 4.73607 + 3.44095i 0.739650 + 0.537387i 0.892601 − 0.450847i \(-0.148878\pi\)
−0.152952 + 0.988234i \(0.548878\pi\)
\(42\) 0.190983 + 0.587785i 0.0294693 + 0.0906972i
\(43\) 1.85410 0.282748 0.141374 − 0.989956i \(-0.454848\pi\)
0.141374 + 0.989956i \(0.454848\pi\)
\(44\) 3.23607 − 0.726543i 0.487856 − 0.109530i
\(45\) −5.23607 −0.780547
\(46\) 0.145898 + 0.449028i 0.0215115 + 0.0662056i
\(47\) 9.47214 + 6.88191i 1.38165 + 1.00383i 0.996724 + 0.0808837i \(0.0257742\pi\)
0.384929 + 0.922946i \(0.374226\pi\)
\(48\) −0.500000 + 0.363271i −0.0721688 + 0.0524337i
\(49\) 0.309017 − 0.951057i 0.0441453 − 0.135865i
\(50\) 0.309017 − 0.951057i 0.0437016 − 0.134500i
\(51\) −0.427051 + 0.310271i −0.0597991 + 0.0434466i
\(52\) 1.00000 + 0.726543i 0.138675 + 0.100753i
\(53\) −4.00000 − 12.3107i −0.549442 − 1.69101i −0.710187 − 0.704013i \(-0.751390\pi\)
0.160744 − 0.986996i \(-0.448610\pi\)
\(54\) 3.47214 0.472498
\(55\) −4.38197 + 4.97980i −0.590864 + 0.671476i
\(56\) 1.00000 0.133631
\(57\) −1.30902 − 4.02874i −0.173384 − 0.533620i
\(58\) −6.47214 − 4.70228i −0.849833 − 0.617440i
\(59\) −8.78115 + 6.37988i −1.14321 + 0.830590i −0.987563 − 0.157223i \(-0.949746\pi\)
−0.155646 + 0.987813i \(0.549746\pi\)
\(60\) 0.381966 − 1.17557i 0.0493116 − 0.151765i
\(61\) −0.763932 + 2.35114i −0.0978115 + 0.301033i −0.987976 − 0.154606i \(-0.950589\pi\)
0.890165 + 0.455639i \(0.150589\pi\)
\(62\) 3.61803 − 2.62866i 0.459491 − 0.333840i
\(63\) −2.11803 − 1.53884i −0.266847 − 0.193876i
\(64\) 0.309017 + 0.951057i 0.0386271 + 0.118882i
\(65\) −2.47214 −0.306631
\(66\) 1.35410 − 1.53884i 0.166678 − 0.189418i
\(67\) −10.0902 −1.23271 −0.616355 − 0.787468i \(-0.711391\pi\)
−0.616355 + 0.787468i \(0.711391\pi\)
\(68\) 0.263932 + 0.812299i 0.0320065 + 0.0985058i
\(69\) 0.236068 + 0.171513i 0.0284192 + 0.0206478i
\(70\) −1.61803 + 1.17557i −0.193392 + 0.140508i
\(71\) 0.472136 − 1.45309i 0.0560322 − 0.172449i −0.919124 − 0.393969i \(-0.871102\pi\)
0.975156 + 0.221520i \(0.0711017\pi\)
\(72\) 0.809017 − 2.48990i 0.0953436 − 0.293437i
\(73\) 3.73607 − 2.71441i 0.437274 − 0.317698i −0.347277 − 0.937763i \(-0.612894\pi\)
0.784551 + 0.620065i \(0.212894\pi\)
\(74\) 4.61803 + 3.35520i 0.536836 + 0.390034i
\(75\) −0.190983 − 0.587785i −0.0220528 − 0.0678716i
\(76\) −6.85410 −0.786219
\(77\) −3.23607 + 0.726543i −0.368784 + 0.0827972i
\(78\) 0.763932 0.0864983
\(79\) 1.00000 + 3.07768i 0.112509 + 0.346266i 0.991419 − 0.130720i \(-0.0417290\pi\)
−0.878911 + 0.476987i \(0.841729\pi\)
\(80\) −1.61803 − 1.17557i −0.180902 − 0.131433i
\(81\) −4.61803 + 3.35520i −0.513115 + 0.372800i
\(82\) 1.80902 − 5.56758i 0.199773 − 0.614837i
\(83\) 4.95492 − 15.2497i 0.543873 − 1.67387i −0.179783 − 0.983706i \(-0.557540\pi\)
0.723655 − 0.690161i \(-0.242460\pi\)
\(84\) 0.500000 − 0.363271i 0.0545545 − 0.0396361i
\(85\) −1.38197 − 1.00406i −0.149895 − 0.108905i
\(86\) −0.572949 − 1.76336i −0.0617827 − 0.190148i
\(87\) −4.94427 −0.530082
\(88\) −1.69098 − 2.85317i −0.180259 − 0.304149i
\(89\) 0.326238 0.0345812 0.0172906 − 0.999851i \(-0.494496\pi\)
0.0172906 + 0.999851i \(0.494496\pi\)
\(90\) 1.61803 + 4.97980i 0.170556 + 0.524917i
\(91\) −1.00000 − 0.726543i −0.104828 − 0.0761624i
\(92\) 0.381966 − 0.277515i 0.0398227 − 0.0289329i
\(93\) 0.854102 − 2.62866i 0.0885662 − 0.272579i
\(94\) 3.61803 − 11.1352i 0.373172 − 1.14850i
\(95\) 11.0902 − 8.05748i 1.13783 − 0.826680i
\(96\) 0.500000 + 0.363271i 0.0510310 + 0.0370762i
\(97\) −3.04508 − 9.37181i −0.309182 − 0.951563i −0.978084 − 0.208212i \(-0.933235\pi\)
0.668902 − 0.743351i \(-0.266765\pi\)
\(98\) −1.00000 −0.101015
\(99\) −0.809017 + 8.64527i −0.0813093 + 0.868882i
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 154.2.f.c.71.1 ✓ 4
11.3 even 5 1694.2.a.m.1.2 2
11.8 odd 10 1694.2.a.r.1.2 2
11.9 even 5 inner 154.2.f.c.141.1 yes 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
154.2.f.c.71.1 ✓ 4 1.1 even 1 trivial
154.2.f.c.141.1 yes 4 11.9 even 5 inner
1694.2.a.m.1.2 2 11.3 even 5
1694.2.a.r.1.2 2 11.8 odd 10