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Properties

Label 152.2.o.c.107.10
Level $152$
Weight $2$
Character 152.107
Analytic conductor $1.214$
Analytic rank $0$
Dimension $28$
Inner twists $4$

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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] = [152,2,Mod(27,152)] 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("152.27"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(152, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([3, 3, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 152 = 2^{3} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 152.o (of order \(6\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [28,-3,-6] 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: \(1.21372611072\)
Analytic rank: \(0\)
Dimension: \(28\)
Relative dimension: \(14\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 107.10
Character \(\chi\) \(=\) 152.107
Dual form 152.2.o.c.27.10

$q$-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+(0.540330 + 1.30692i) q^{2} +(2.07624 - 1.19872i) q^{3} +(-1.41609 + 1.41234i) q^{4} +(-0.418341 + 0.241529i) q^{5} +(2.68848 + 2.06578i) q^{6} +1.56686i q^{7} +(-2.61097 - 1.08759i) q^{8} +(1.37385 - 2.37958i) q^{9} +(-0.541701 - 0.416233i) q^{10} +1.24363 q^{11} +(-1.24714 + 4.62984i) q^{12} +(1.80015 - 3.11796i) q^{13} +(-2.04776 + 0.846621i) q^{14} +(-0.579050 + 1.00294i) q^{15} +(0.0106066 - 3.99999i) q^{16} +(-1.35570 - 2.34815i) q^{17} +(3.85225 + 0.509757i) q^{18} +(-4.25703 - 0.936862i) q^{19} +(0.251286 - 0.932864i) q^{20} +(1.87822 + 3.25318i) q^{21} +(0.671968 + 1.62532i) q^{22} +(-5.52533 - 3.19005i) q^{23} +(-6.72471 + 0.871723i) q^{24} +(-2.38333 + 4.12804i) q^{25} +(5.04760 + 0.667934i) q^{26} +0.604878i q^{27} +(-2.21293 - 2.21881i) q^{28} +(-0.695157 + 1.20405i) q^{29} +(-1.62365 - 0.214852i) q^{30} -4.86016 q^{31} +(5.23340 - 2.14745i) q^{32} +(2.58207 - 1.49076i) q^{33} +(2.33632 - 3.04058i) q^{34} +(-0.378442 - 0.655481i) q^{35} +(1.41527 + 5.31003i) q^{36} +10.9720 q^{37} +(-1.07579 - 6.06982i) q^{38} -8.63150i q^{39} +(1.35496 - 0.175643i) q^{40} +(1.10008 - 0.635132i) q^{41} +(-3.23679 + 4.21248i) q^{42} +(4.78175 + 8.28223i) q^{43} +(-1.76108 + 1.75642i) q^{44} +1.32730i q^{45} +(1.18364 - 8.94485i) q^{46} +(-7.26763 - 4.19597i) q^{47} +(-4.77283 - 8.31765i) q^{48} +4.54495 q^{49} +(-6.68281 - 0.884316i) q^{50} +(-5.62954 - 3.25022i) q^{51} +(1.85443 + 6.95772i) q^{52} +(1.50024 - 2.59849i) q^{53} +(-0.790529 + 0.326834i) q^{54} +(-0.520259 + 0.300372i) q^{55} +(1.70410 - 4.09102i) q^{56} +(-9.96165 + 3.15782i) q^{57} +(-1.94921 - 0.257933i) q^{58} +(4.43723 - 2.56184i) q^{59} +(-0.596510 - 2.23807i) q^{60} +(9.42313 + 5.44044i) q^{61} +(-2.62609 - 6.35185i) q^{62} +(3.72846 + 2.15263i) q^{63} +(5.63431 + 5.67931i) q^{64} +1.73916i q^{65} +(3.34347 + 2.56906i) q^{66} +(-3.22460 - 1.86172i) q^{67} +(5.23618 + 1.41047i) q^{68} -15.2959 q^{69} +(0.652179 - 0.848770i) q^{70} +(6.62670 + 11.4778i) q^{71} +(-6.17507 + 4.71882i) q^{72} +(-0.494526 - 0.856544i) q^{73} +(5.92851 + 14.3396i) q^{74} +11.4277i q^{75} +(7.35149 - 4.68568i) q^{76} +1.94859i q^{77} +(11.2807 - 4.66386i) q^{78} +(5.81311 + 10.0686i) q^{79} +(0.961676 + 1.67592i) q^{80} +(4.84663 + 8.39460i) q^{81} +(1.42447 + 1.09454i) q^{82} -12.1949 q^{83} +(-7.25431 - 1.95410i) q^{84} +(1.13429 + 0.654884i) q^{85} +(-8.24050 + 10.7245i) q^{86} +3.33319i q^{87} +(-3.24707 - 1.35255i) q^{88} +(-11.1625 - 6.44469i) q^{89} +(-1.73467 + 0.717179i) q^{90} +(4.88540 + 2.82059i) q^{91} +(12.3298 - 3.28624i) q^{92} +(-10.0909 + 5.82596i) q^{93} +(1.55688 - 11.7654i) q^{94} +(2.00717 - 0.636268i) q^{95} +(8.29161 - 10.7320i) q^{96} +(2.42717 - 1.40133i) q^{97} +(2.45577 + 5.93989i) q^{98} +(1.70855 - 2.95930i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 28 q - 3 q^{2} - 6 q^{3} + q^{4} - 3 q^{6} + 8 q^{9} + 6 q^{10} - 16 q^{11} + 13 q^{16} - 22 q^{17} + 4 q^{19} - 40 q^{20} - 21 q^{22} - 11 q^{24} + 16 q^{25} + 36 q^{26} - 10 q^{28} + 4 q^{30} - 3 q^{32}+ \cdots - 28 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/152\mathbb{Z}\right)^\times\).

\(n\) \(39\) \(77\) \(97\)
\(\chi(n)\) \(-1\) \(-1\) \(e\left(\frac{5}{6}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.540330 + 1.30692i 0.382071 + 0.924133i
\(3\) 2.07624 − 1.19872i 1.19872 − 0.692080i 0.238449 − 0.971155i \(-0.423361\pi\)
0.960269 + 0.279075i \(0.0900278\pi\)
\(4\) −1.41609 + 1.41234i −0.708044 + 0.706169i
\(5\) −0.418341 + 0.241529i −0.187088 + 0.108015i −0.590618 − 0.806951i \(-0.701116\pi\)
0.403531 + 0.914966i \(0.367783\pi\)
\(6\) 2.68848 + 2.06578i 1.09757 + 0.843351i
\(7\) 1.56686i 0.592217i 0.955154 + 0.296109i \(0.0956891\pi\)
−0.955154 + 0.296109i \(0.904311\pi\)
\(8\) −2.61097 − 1.08759i −0.923117 − 0.384520i
\(9\) 1.37385 − 2.37958i 0.457950 − 0.793192i
\(10\) −0.541701 − 0.416233i −0.171301 − 0.131624i
\(11\) 1.24363 0.374967 0.187484 − 0.982268i \(-0.439967\pi\)
0.187484 + 0.982268i \(0.439967\pi\)
\(12\) −1.24714 + 4.62984i −0.360019 + 1.33652i
\(13\) 1.80015 − 3.11796i 0.499273 − 0.864766i −0.500727 − 0.865605i \(-0.666934\pi\)
1.00000 0.000839577i \(0.000267246\pi\)
\(14\) −2.04776 + 0.846621i −0.547288 + 0.226269i
\(15\) −0.579050 + 1.00294i −0.149510 + 0.258959i
\(16\) 0.0106066 − 3.99999i 0.00265164 − 0.999996i
\(17\) −1.35570 − 2.34815i −0.328807 − 0.569510i 0.653469 − 0.756954i \(-0.273313\pi\)
−0.982275 + 0.187444i \(0.939980\pi\)
\(18\) 3.85225 + 0.509757i 0.907984 + 0.120151i
\(19\) −4.25703 − 0.936862i −0.976629 − 0.214931i
\(20\) 0.251286 − 0.932864i 0.0561893 − 0.208595i
\(21\) 1.87822 + 3.25318i 0.409862 + 0.709901i
\(22\) 0.671968 + 1.62532i 0.143264 + 0.346520i
\(23\) −5.52533 − 3.19005i −1.15211 − 0.665171i −0.202710 − 0.979239i \(-0.564975\pi\)
−0.949401 + 0.314068i \(0.898308\pi\)
\(24\) −6.72471 + 0.871723i −1.37268 + 0.177940i
\(25\) −2.38333 + 4.12804i −0.476665 + 0.825609i
\(26\) 5.04760 + 0.667934i 0.989916 + 0.130993i
\(27\) 0.604878i 0.116409i
\(28\) −2.21293 − 2.21881i −0.418205 − 0.419316i
\(29\) −0.695157 + 1.20405i −0.129087 + 0.223586i −0.923323 − 0.384024i \(-0.874538\pi\)
0.794236 + 0.607610i \(0.207871\pi\)
\(30\) −1.62365 − 0.214852i −0.296436 − 0.0392265i
\(31\) −4.86016 −0.872910 −0.436455 − 0.899726i \(-0.643766\pi\)
−0.436455 + 0.899726i \(0.643766\pi\)
\(32\) 5.23340 − 2.14745i 0.925143 − 0.379619i
\(33\) 2.58207 − 1.49076i 0.449480 − 0.259507i
\(34\) 2.33632 − 3.04058i 0.400675 − 0.521454i
\(35\) −0.378442 − 0.655481i −0.0639684 − 0.110797i
\(36\) 1.41527 + 5.31003i 0.235879 + 0.885004i
\(37\) 10.9720 1.80379 0.901894 − 0.431957i \(-0.142177\pi\)
0.901894 + 0.431957i \(0.142177\pi\)
\(38\) −1.07579 − 6.06982i −0.174517 − 0.984654i
\(39\) − 8.63150i − 1.38215i
\(40\) 1.35496 − 0.175643i 0.214238 − 0.0277716i
\(41\) 1.10008 − 0.635132i 0.171804 − 0.0991909i −0.411632 − 0.911350i \(-0.635041\pi\)
0.583436 + 0.812159i \(0.301708\pi\)
\(42\) −3.23679 + 4.21248i −0.499447 + 0.649999i
\(43\) 4.78175 + 8.28223i 0.729210 + 1.26303i 0.957218 + 0.289369i \(0.0934455\pi\)
−0.228008 + 0.973659i \(0.573221\pi\)
\(44\) −1.76108 + 1.75642i −0.265493 + 0.264790i
\(45\) 1.32730i 0.197862i
\(46\) 1.18364 − 8.94485i 0.174519 − 1.31885i
\(47\) −7.26763 − 4.19597i −1.06009 − 0.612045i −0.134635 − 0.990895i \(-0.542986\pi\)
−0.925458 + 0.378850i \(0.876320\pi\)
\(48\) −4.77283 − 8.31765i −0.688899 − 1.20055i
\(49\) 4.54495 0.649279
\(50\) −6.68281 − 0.884316i −0.945092 − 0.125061i
\(51\) −5.62954 − 3.25022i −0.788293 − 0.455121i
\(52\) 1.85443 + 6.95772i 0.257164 + 0.964863i
\(53\) 1.50024 − 2.59849i 0.206074 − 0.356930i −0.744400 − 0.667733i \(-0.767265\pi\)
0.950474 + 0.310803i \(0.100598\pi\)
\(54\) −0.790529 + 0.326834i −0.107577 + 0.0444765i
\(55\) −0.520259 + 0.300372i −0.0701517 + 0.0405021i
\(56\) 1.70410 − 4.09102i 0.227719 − 0.546686i
\(57\) −9.96165 + 3.15782i −1.31945 + 0.418264i
\(58\) −1.94921 − 0.257933i −0.255944 − 0.0338682i
\(59\) 4.43723 − 2.56184i 0.577678 − 0.333523i −0.182532 − 0.983200i \(-0.558429\pi\)
0.760210 + 0.649677i \(0.225096\pi\)
\(60\) −0.596510 − 2.23807i −0.0770092 − 0.288934i
\(61\) 9.42313 + 5.44044i 1.20651 + 0.696578i 0.961995 − 0.273068i \(-0.0880385\pi\)
0.244513 + 0.969646i \(0.421372\pi\)
\(62\) −2.62609 − 6.35185i −0.333514 − 0.806685i
\(63\) 3.72846 + 2.15263i 0.469742 + 0.271206i
\(64\) 5.63431 + 5.67931i 0.704289 + 0.709914i
\(65\) 1.73916i 0.215716i
\(66\) 3.34347 + 2.56906i 0.411553 + 0.316229i
\(67\) −3.22460 − 1.86172i −0.393947 − 0.227446i 0.289922 − 0.957050i \(-0.406371\pi\)
−0.683869 + 0.729605i \(0.739704\pi\)
\(68\) 5.23618 + 1.41047i 0.634980 + 0.171045i
\(69\) −15.2959 −1.84141
\(70\) 0.652179 − 0.848770i 0.0779503 − 0.101447i
\(71\) 6.62670 + 11.4778i 0.786444 + 1.36216i 0.928132 + 0.372250i \(0.121414\pi\)
−0.141688 + 0.989911i \(0.545253\pi\)
\(72\) −6.17507 + 4.71882i −0.727739 + 0.556118i
\(73\) −0.494526 − 0.856544i −0.0578799 − 0.100251i 0.835634 − 0.549287i \(-0.185101\pi\)
−0.893514 + 0.449036i \(0.851767\pi\)
\(74\) 5.92851 + 14.3396i 0.689175 + 1.66694i
\(75\) 11.4277i 1.31956i
\(76\) 7.35149 − 4.68568i 0.843274 − 0.537484i
\(77\) 1.94859i 0.222062i
\(78\) 11.2807 − 4.66386i 1.27729 − 0.528078i
\(79\) 5.81311 + 10.0686i 0.654026 + 1.13281i 0.982137 + 0.188168i \(0.0602548\pi\)
−0.328111 + 0.944639i \(0.606412\pi\)
\(80\) 0.961676 + 1.67592i 0.107519 + 0.187373i
\(81\) 4.84663 + 8.39460i 0.538514 + 0.932733i
\(82\) 1.42447 + 1.09454i 0.157307 + 0.120871i
\(83\) −12.1949 −1.33856 −0.669280 − 0.743010i \(-0.733397\pi\)
−0.669280 + 0.743010i \(0.733397\pi\)
\(84\) −7.25431 − 1.95410i −0.791510 − 0.213210i
\(85\) 1.13429 + 0.654884i 0.123031 + 0.0710322i
\(86\) −8.24050 + 10.7245i −0.888596 + 1.15645i
\(87\) 3.33319i 0.357355i
\(88\) −3.24707 − 1.35255i −0.346139 − 0.144182i
\(89\) −11.1625 − 6.44469i −1.18323 − 0.683136i −0.226467 − 0.974019i \(-0.572718\pi\)
−0.956759 + 0.290883i \(0.906051\pi\)
\(90\) −1.73467 + 0.717179i −0.182851 + 0.0755973i
\(91\) 4.88540 + 2.82059i 0.512129 + 0.295678i
\(92\) 12.3298 − 3.28624i 1.28547 − 0.342614i
\(93\) −10.0909 + 5.82596i −1.04637 + 0.604124i
\(94\) 1.55688 − 11.7654i 0.160580 − 1.21351i
\(95\) 2.00717 − 0.636268i 0.205931 − 0.0652798i
\(96\) 8.29161 − 10.7320i 0.846258 − 1.09533i
\(97\) 2.42717 − 1.40133i 0.246442 − 0.142283i −0.371692 − 0.928356i \(-0.621222\pi\)
0.618134 + 0.786073i \(0.287889\pi\)
\(98\) 2.45577 + 5.93989i 0.248070 + 0.600020i
\(99\) 1.70855 − 2.95930i 0.171716 − 0.297421i
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 152.2.o.c.107.10 yes 28
4.3 odd 2 608.2.s.c.335.1 28
8.3 odd 2 inner 152.2.o.c.107.4 yes 28
8.5 even 2 608.2.s.c.335.2 28
19.8 odd 6 inner 152.2.o.c.27.4 ✓ 28
76.27 even 6 608.2.s.c.559.2 28
152.27 even 6 inner 152.2.o.c.27.10 yes 28
152.141 odd 6 608.2.s.c.559.1 28
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
152.2.o.c.27.4 ✓ 28 19.8 odd 6 inner
152.2.o.c.27.10 yes 28 152.27 even 6 inner
152.2.o.c.107.4 yes 28 8.3 odd 2 inner
152.2.o.c.107.10 yes 28 1.1 even 1 trivial
608.2.s.c.335.1 28 4.3 odd 2
608.2.s.c.335.2 28 8.5 even 2
608.2.s.c.559.1 28 152.141 odd 6
608.2.s.c.559.2 28 76.27 even 6