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Properties

Label 154.2.f.c.113.1
Level $154$
Weight $2$
Character 154.113
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 113.1
Root \(0.809017 + 0.587785i\) of defining polynomial
Character \(\chi\) \(=\) 154.113
Dual form 154.2.f.c.15.1

$q$-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+(0.809017 - 0.587785i) q^{2} +(-0.500000 - 1.53884i) q^{3} +(0.309017 - 0.951057i) q^{4} +(-1.61803 - 1.17557i) q^{5} +(-1.30902 - 0.951057i) q^{6} +(-0.309017 + 0.951057i) q^{7} +(-0.309017 - 0.951057i) q^{8} +(0.309017 - 0.224514i) q^{9} -2.00000 q^{10} +(2.54508 - 2.12663i) q^{11} -1.61803 q^{12} +(-2.61803 + 1.90211i) q^{13} +(0.309017 + 0.951057i) q^{14} +(-1.00000 + 3.07768i) q^{15} +(-0.809017 - 0.587785i) q^{16} +(4.73607 + 3.44095i) q^{17} +(0.118034 - 0.363271i) q^{18} +(-0.0450850 - 0.138757i) q^{19} +(-1.61803 + 1.17557i) q^{20} +1.61803 q^{21} +(0.809017 - 3.21644i) q^{22} +8.47214 q^{23} +(-1.30902 + 0.951057i) q^{24} +(-0.309017 - 0.951057i) q^{25} +(-1.00000 + 3.07768i) q^{26} +(-4.42705 - 3.21644i) q^{27} +(0.809017 + 0.587785i) q^{28} +(-2.47214 + 7.60845i) q^{29} +(1.00000 + 3.07768i) q^{30} +(3.61803 - 2.62866i) q^{31} -1.00000 q^{32} +(-4.54508 - 2.85317i) q^{33} +5.85410 q^{34} +(1.61803 - 1.17557i) q^{35} +(-0.118034 - 0.363271i) q^{36} +(-2.38197 + 7.33094i) q^{37} +(-0.118034 - 0.0857567i) q^{38} +(4.23607 + 3.07768i) q^{39} +(-0.618034 + 1.90211i) q^{40} +(0.263932 + 0.812299i) q^{41} +(1.30902 - 0.951057i) q^{42} -4.85410 q^{43} +(-1.23607 - 3.07768i) q^{44} -0.763932 q^{45} +(6.85410 - 4.97980i) q^{46} +(0.527864 + 1.62460i) q^{47} +(-0.500000 + 1.53884i) q^{48} +(-0.809017 - 0.587785i) q^{49} +(-0.809017 - 0.587785i) q^{50} +(2.92705 - 9.00854i) q^{51} +(1.00000 + 3.07768i) q^{52} +(-4.00000 + 2.90617i) q^{53} -5.47214 q^{54} +(-6.61803 + 0.449028i) q^{55} +1.00000 q^{56} +(-0.190983 + 0.138757i) q^{57} +(2.47214 + 7.60845i) q^{58} +(1.28115 - 3.94298i) q^{59} +(2.61803 + 1.90211i) q^{60} +(-5.23607 - 3.80423i) q^{61} +(1.38197 - 4.25325i) q^{62} +(0.118034 + 0.363271i) q^{63} +(-0.809017 + 0.587785i) q^{64} +6.47214 q^{65} +(-5.35410 + 0.363271i) q^{66} +1.09017 q^{67} +(4.73607 - 3.44095i) q^{68} +(-4.23607 - 13.0373i) q^{69} +(0.618034 - 1.90211i) q^{70} +(-8.47214 - 6.15537i) q^{71} +(-0.309017 - 0.224514i) q^{72} +(-0.736068 + 2.26538i) q^{73} +(2.38197 + 7.33094i) q^{74} +(-1.30902 + 0.951057i) q^{75} -0.145898 q^{76} +(1.23607 + 3.07768i) q^{77} +5.23607 q^{78} +(1.00000 - 0.726543i) q^{79} +(0.618034 + 1.90211i) q^{80} +(-2.38197 + 7.33094i) q^{81} +(0.690983 + 0.502029i) q^{82} +(10.5451 + 7.66145i) q^{83} +(0.500000 - 1.53884i) q^{84} +(-3.61803 - 11.1352i) q^{85} +(-3.92705 + 2.85317i) q^{86} +12.9443 q^{87} +(-2.80902 - 1.76336i) q^{88} -15.3262 q^{89} +(-0.618034 + 0.449028i) q^{90} +(-1.00000 - 3.07768i) q^{91} +(2.61803 - 8.05748i) q^{92} +(-5.85410 - 4.25325i) q^{93} +(1.38197 + 1.00406i) q^{94} +(-0.0901699 + 0.277515i) q^{95} +(0.500000 + 1.53884i) q^{96} +(2.54508 - 1.84911i) q^{97} -1.00000 q^{98} +(0.309017 - 1.22857i) 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 \(\left(\mathbb{Z}/154\mathbb{Z}\right)^\times\).

\(n\) \(45\) \(57\)
\(\chi(n)\) \(1\) \(e\left(\frac{4}{5}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.809017 − 0.587785i 0.572061 − 0.415627i
\(3\) −0.500000 − 1.53884i −0.288675 − 0.888451i −0.985273 − 0.170989i \(-0.945304\pi\)
0.696598 − 0.717462i \(-0.254696\pi\)
\(4\) 0.309017 − 0.951057i 0.154508 − 0.475528i
\(5\) −1.61803 − 1.17557i −0.723607 − 0.525731i 0.163928 − 0.986472i \(-0.447584\pi\)
−0.887535 + 0.460741i \(0.847584\pi\)
\(6\) −1.30902 − 0.951057i −0.534404 − 0.388267i
\(7\) −0.309017 + 0.951057i −0.116797 + 0.359466i
\(8\) −0.309017 − 0.951057i −0.109254 − 0.336249i
\(9\) 0.309017 − 0.224514i 0.103006 − 0.0748380i
\(10\) −2.00000 −0.632456
\(11\) 2.54508 − 2.12663i 0.767372 − 0.641202i
\(12\) −1.61803 −0.467086
\(13\) −2.61803 + 1.90211i −0.726112 + 0.527551i −0.888331 − 0.459204i \(-0.848135\pi\)
0.162219 + 0.986755i \(0.448135\pi\)
\(14\) 0.309017 + 0.951057i 0.0825883 + 0.254181i
\(15\) −1.00000 + 3.07768i −0.258199 + 0.794654i
\(16\) −0.809017 − 0.587785i −0.202254 − 0.146946i
\(17\) 4.73607 + 3.44095i 1.14867 + 0.834554i 0.988303 − 0.152504i \(-0.0487337\pi\)
0.160362 + 0.987058i \(0.448734\pi\)
\(18\) 0.118034 − 0.363271i 0.0278209 − 0.0856239i
\(19\) −0.0450850 − 0.138757i −0.0103432 − 0.0318331i 0.945752 − 0.324890i \(-0.105327\pi\)
−0.956095 + 0.293057i \(0.905327\pi\)
\(20\) −1.61803 + 1.17557i −0.361803 + 0.262866i
\(21\) 1.61803 0.353084
\(22\) 0.809017 − 3.21644i 0.172483 − 0.685747i
\(23\) 8.47214 1.76656 0.883281 − 0.468844i \(-0.155329\pi\)
0.883281 + 0.468844i \(0.155329\pi\)
\(24\) −1.30902 + 0.951057i −0.267202 + 0.194134i
\(25\) −0.309017 − 0.951057i −0.0618034 − 0.190211i
\(26\) −1.00000 + 3.07768i −0.196116 + 0.603583i
\(27\) −4.42705 − 3.21644i −0.851986 − 0.619004i
\(28\) 0.809017 + 0.587785i 0.152890 + 0.111081i
\(29\) −2.47214 + 7.60845i −0.459064 + 1.41285i 0.407233 + 0.913324i \(0.366494\pi\)
−0.866297 + 0.499530i \(0.833506\pi\)
\(30\) 1.00000 + 3.07768i 0.182574 + 0.561906i
\(31\) 3.61803 − 2.62866i 0.649818 − 0.472120i −0.213391 − 0.976967i \(-0.568451\pi\)
0.863209 + 0.504846i \(0.168451\pi\)
\(32\) −1.00000 −0.176777
\(33\) −4.54508 − 2.85317i −0.791198 − 0.496673i
\(34\) 5.85410 1.00397
\(35\) 1.61803 − 1.17557i 0.273498 − 0.198708i
\(36\) −0.118034 − 0.363271i −0.0196723 − 0.0605452i
\(37\) −2.38197 + 7.33094i −0.391593 + 1.20520i 0.539991 + 0.841671i \(0.318428\pi\)
−0.931583 + 0.363528i \(0.881572\pi\)
\(38\) −0.118034 − 0.0857567i −0.0191476 − 0.0139116i
\(39\) 4.23607 + 3.07768i 0.678314 + 0.492824i
\(40\) −0.618034 + 1.90211i −0.0977198 + 0.300750i
\(41\) 0.263932 + 0.812299i 0.0412193 + 0.126860i 0.969549 − 0.244899i \(-0.0787548\pi\)
−0.928329 + 0.371759i \(0.878755\pi\)
\(42\) 1.30902 − 0.951057i 0.201986 − 0.146751i
\(43\) −4.85410 −0.740244 −0.370122 − 0.928983i \(-0.620684\pi\)
−0.370122 + 0.928983i \(0.620684\pi\)
\(44\) −1.23607 − 3.07768i −0.186344 − 0.463978i
\(45\) −0.763932 −0.113880
\(46\) 6.85410 − 4.97980i 1.01058 − 0.734231i
\(47\) 0.527864 + 1.62460i 0.0769969 + 0.236972i 0.982145 − 0.188123i \(-0.0602405\pi\)
−0.905149 + 0.425096i \(0.860241\pi\)
\(48\) −0.500000 + 1.53884i −0.0721688 + 0.222113i
\(49\) −0.809017 − 0.587785i −0.115574 − 0.0839693i
\(50\) −0.809017 − 0.587785i −0.114412 − 0.0831254i
\(51\) 2.92705 − 9.00854i 0.409869 − 1.26145i
\(52\) 1.00000 + 3.07768i 0.138675 + 0.426798i
\(53\) −4.00000 + 2.90617i −0.549442 + 0.399193i −0.827580 − 0.561348i \(-0.810283\pi\)
0.278138 + 0.960541i \(0.410283\pi\)
\(54\) −5.47214 −0.744663
\(55\) −6.61803 + 0.449028i −0.892376 + 0.0605469i
\(56\) 1.00000 0.133631
\(57\) −0.190983 + 0.138757i −0.0252963 + 0.0183789i
\(58\) 2.47214 + 7.60845i 0.324607 + 0.999039i
\(59\) 1.28115 − 3.94298i 0.166792 − 0.513333i −0.832372 − 0.554217i \(-0.813018\pi\)
0.999164 + 0.0408847i \(0.0130176\pi\)
\(60\) 2.61803 + 1.90211i 0.337987 + 0.245562i
\(61\) −5.23607 − 3.80423i −0.670410 − 0.487081i 0.199753 − 0.979846i \(-0.435986\pi\)
−0.870162 + 0.492765i \(0.835986\pi\)
\(62\) 1.38197 − 4.25325i 0.175510 − 0.540164i
\(63\) 0.118034 + 0.363271i 0.0148709 + 0.0457679i
\(64\) −0.809017 + 0.587785i −0.101127 + 0.0734732i
\(65\) 6.47214 0.802770
\(66\) −5.35410 + 0.363271i −0.659044 + 0.0447156i
\(67\) 1.09017 0.133185 0.0665927 − 0.997780i \(-0.478787\pi\)
0.0665927 + 0.997780i \(0.478787\pi\)
\(68\) 4.73607 − 3.44095i 0.574333 − 0.417277i
\(69\) −4.23607 − 13.0373i −0.509963 − 1.56950i
\(70\) 0.618034 − 1.90211i 0.0738692 − 0.227346i
\(71\) −8.47214 − 6.15537i −1.00546 − 0.730508i −0.0422061 − 0.999109i \(-0.513439\pi\)
−0.963251 + 0.268601i \(0.913439\pi\)
\(72\) −0.309017 − 0.224514i −0.0364180 − 0.0264592i
\(73\) −0.736068 + 2.26538i −0.0861502 + 0.265143i −0.984846 − 0.173428i \(-0.944515\pi\)
0.898696 + 0.438572i \(0.144515\pi\)
\(74\) 2.38197 + 7.33094i 0.276898 + 0.852204i
\(75\) −1.30902 + 0.951057i −0.151152 + 0.109819i
\(76\) −0.145898 −0.0167357
\(77\) 1.23607 + 3.07768i 0.140863 + 0.350735i
\(78\) 5.23607 0.592868
\(79\) 1.00000 − 0.726543i 0.112509 − 0.0817424i −0.530108 − 0.847930i \(-0.677849\pi\)
0.642617 + 0.766188i \(0.277849\pi\)
\(80\) 0.618034 + 1.90211i 0.0690983 + 0.212663i
\(81\) −2.38197 + 7.33094i −0.264663 + 0.814549i
\(82\) 0.690983 + 0.502029i 0.0763063 + 0.0554398i
\(83\) 10.5451 + 7.66145i 1.15747 + 0.840954i 0.989456 − 0.144830i \(-0.0462637\pi\)
0.168017 + 0.985784i \(0.446264\pi\)
\(84\) 0.500000 − 1.53884i 0.0545545 − 0.167901i
\(85\) −3.61803 − 11.1352i −0.392431 − 1.20778i
\(86\) −3.92705 + 2.85317i −0.423465 + 0.307665i
\(87\) 12.9443 1.38777
\(88\) −2.80902 − 1.76336i −0.299442 − 0.187974i
\(89\) −15.3262 −1.62458 −0.812289 − 0.583255i \(-0.801779\pi\)
−0.812289 + 0.583255i \(0.801779\pi\)
\(90\) −0.618034 + 0.449028i −0.0651465 + 0.0473317i
\(91\) −1.00000 − 3.07768i −0.104828 − 0.322629i
\(92\) 2.61803 − 8.05748i 0.272949 − 0.840050i
\(93\) −5.85410 − 4.25325i −0.607042 − 0.441042i
\(94\) 1.38197 + 1.00406i 0.142539 + 0.103561i
\(95\) −0.0901699 + 0.277515i −0.00925124 + 0.0284724i
\(96\) 0.500000 + 1.53884i 0.0510310 + 0.157057i
\(97\) 2.54508 − 1.84911i 0.258414 − 0.187749i −0.451033 − 0.892507i \(-0.648944\pi\)
0.709448 + 0.704758i \(0.248944\pi\)
\(98\) −1.00000 −0.101015
\(99\) 0.309017 − 1.22857i 0.0310574 − 0.123476i
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.113.1 yes 4
11.2 odd 10 1694.2.a.r.1.1 2
11.4 even 5 inner 154.2.f.c.15.1 ✓ 4
11.9 even 5 1694.2.a.m.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
154.2.f.c.15.1 ✓ 4 11.4 even 5 inner
154.2.f.c.113.1 yes 4 1.1 even 1 trivial
1694.2.a.m.1.1 2 11.9 even 5
1694.2.a.r.1.1 2 11.2 odd 10