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Properties

Label 152.2.o.c.27.4
Level $152$
Weight $2$
Character 152.27
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 27.4
Character \(\chi\) \(=\) 152.27
Dual form 152.2.o.c.107.4

$q$-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+(-0.861662 + 1.12140i) q^{2} +(2.07624 + 1.19872i) q^{3} +(-0.515076 - 1.93254i) q^{4} +(0.418341 + 0.241529i) q^{5} +(-3.13326 + 1.29541i) q^{6} +1.56686i q^{7} +(2.61097 + 1.08759i) q^{8} +(1.37385 + 2.37958i) q^{9} +(-0.631319 + 0.261011i) q^{10} +1.24363 q^{11} +(1.24714 - 4.62984i) q^{12} +(-1.80015 - 3.11796i) q^{13} +(-1.75708 - 1.35010i) q^{14} +(0.579050 + 1.00294i) q^{15} +(-3.46939 + 1.99081i) 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} +(-1.07159 + 1.39460i) q^{22} +(5.52533 - 3.19005i) q^{23} +(4.11729 + 5.38791i) q^{24} +(-2.38333 - 4.12804i) q^{25} +(5.04760 + 0.667934i) q^{26} -0.604878i q^{27} +(3.02801 - 0.807052i) q^{28} +(0.695157 + 1.20405i) q^{29} +(-1.62365 - 0.214852i) q^{30} +4.86016 q^{31} +(0.756953 - 5.60598i) q^{32} +(2.58207 + 1.49076i) q^{33} +(-1.46506 - 3.54360i) q^{34} +(-0.378442 + 0.655481i) q^{35} +(3.89098 - 3.88068i) q^{36} -10.9720 q^{37} +(2.61752 - 5.58109i) q^{38} -8.63150i q^{39} +(0.829590 + 1.08561i) q^{40} +(1.10008 + 0.635132i) q^{41} +(-2.02972 - 4.90938i) q^{42} +(4.78175 - 8.28223i) q^{43} +(-0.640562 - 2.40335i) q^{44} +1.32730i q^{45} +(-1.18364 + 8.94485i) q^{46} +(7.26763 - 4.19597i) q^{47} +(-9.58971 - 0.0254286i) q^{48} +4.54495 q^{49} +(6.68281 + 0.884316i) q^{50} +(-5.62954 + 3.25022i) q^{51} +(-5.09835 + 5.08485i) q^{52} +(-1.50024 - 2.59849i) q^{53} +(0.678311 + 0.521201i) 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} +(1.63997 - 1.63563i) q^{60} +(-9.42313 + 5.44044i) q^{61} +(-4.18781 + 5.45018i) q^{62} +(-3.72846 + 2.15263i) q^{63} +(5.63431 + 5.67931i) q^{64} -1.73916i q^{65} +(-3.89660 + 1.61100i) q^{66} +(-3.22460 + 1.86172i) q^{67} +(5.23618 + 1.41047i) q^{68} +15.2959 q^{69} +(-0.408967 - 0.989188i) q^{70} +(-6.62670 + 11.4778i) q^{71} +(0.999080 + 7.70718i) q^{72} +(-0.494526 + 0.856544i) q^{73} +(9.45417 - 12.3040i) q^{74} -11.4277i q^{75} +(4.00322 + 7.74431i) q^{76} +1.94859i q^{77} +(9.67937 + 7.43744i) q^{78} +(-5.81311 + 10.0686i) q^{79} +(-1.93223 - 0.00512359i) q^{80} +(4.84663 - 8.39460i) q^{81} +(-1.66013 + 0.686361i) q^{82} -12.1949 q^{83} +(7.25431 + 1.95410i) q^{84} +(-1.13429 + 0.654884i) q^{85} +(5.16744 + 12.4987i) q^{86} +3.33319i q^{87} +(3.24707 + 1.35255i) q^{88} +(-11.1625 + 6.44469i) q^{89} +(-1.48843 - 1.14368i) q^{90} +(4.88540 - 2.82059i) q^{91} +(-9.01085 - 9.03478i) 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} +(-3.91621 + 5.09671i) 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 \(\left(\mathbb{Z}/152\mathbb{Z}\right)^\times\).

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

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.861662 + 1.12140i −0.609287 + 0.792950i
\(3\) 2.07624 + 1.19872i 1.19872 + 0.692080i 0.960269 − 0.279075i \(-0.0900278\pi\)
0.238449 + 0.971155i \(0.423361\pi\)
\(4\) −0.515076 − 1.93254i −0.257538 − 0.966268i
\(5\) 0.418341 + 0.241529i 0.187088 + 0.108015i 0.590618 − 0.806951i \(-0.298884\pi\)
−0.403531 + 0.914966i \(0.632217\pi\)
\(6\) −3.13326 + 1.29541i −1.27915 + 0.528847i
\(7\) 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
\(10\) −0.631319 + 0.261011i −0.199641 + 0.0825388i
\(11\) 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 \(-0.333066\pi\)
−1.00000 0.000839577i \(0.999733\pi\)
\(14\) −1.75708 − 1.35010i −0.469598 − 0.360830i
\(15\) 0.579050 + 1.00294i 0.149510 + 0.258959i
\(16\) −3.46939 + 1.99081i −0.867348 + 0.497702i
\(17\) −1.35570 + 2.34815i −0.328807 + 0.569510i −0.982275 − 0.187444i \(-0.939980\pi\)
0.653469 + 0.756954i \(0.273313\pi\)
\(18\) −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
\(22\) −1.07159 + 1.39460i −0.228463 + 0.297330i
\(23\) 5.52533 − 3.19005i 1.15211 − 0.665171i 0.202710 − 0.979239i \(-0.435025\pi\)
0.949401 + 0.314068i \(0.101692\pi\)
\(24\) 4.11729 + 5.38791i 0.840438 + 1.09980i
\(25\) −2.38333 − 4.12804i −0.476665 − 0.825609i
\(26\) 5.04760 + 0.667934i 0.989916 + 0.130993i
\(27\) − 0.604878i − 0.116409i
\(28\) 3.02801 − 0.807052i 0.572241 − 0.152519i
\(29\) 0.695157 + 1.20405i 0.129087 + 0.223586i 0.923323 − 0.384024i \(-0.125462\pi\)
−0.794236 + 0.607610i \(0.792129\pi\)
\(30\) −1.62365 − 0.214852i −0.296436 − 0.0392265i
\(31\) 4.86016 0.872910 0.436455 − 0.899726i \(-0.356234\pi\)
0.436455 + 0.899726i \(0.356234\pi\)
\(32\) 0.756953 − 5.60598i 0.133812 − 0.991007i
\(33\) 2.58207 + 1.49076i 0.449480 + 0.259507i
\(34\) −1.46506 − 3.54360i −0.251255 − 0.607722i
\(35\) −0.378442 + 0.655481i −0.0639684 + 0.110797i
\(36\) 3.89098 − 3.88068i 0.648497 − 0.646779i
\(37\) −10.9720 −1.80379 −0.901894 − 0.431957i \(-0.857823\pi\)
−0.901894 + 0.431957i \(0.857823\pi\)
\(38\) 2.61752 − 5.58109i 0.424618 − 0.905372i
\(39\) − 8.63150i − 1.38215i
\(40\) 0.829590 + 1.08561i 0.131170 + 0.171649i
\(41\) 1.10008 + 0.635132i 0.171804 + 0.0991909i 0.583436 − 0.812159i \(-0.301708\pi\)
−0.411632 + 0.911350i \(0.635041\pi\)
\(42\) −2.02972 − 4.90938i −0.313193 − 0.757534i
\(43\) 4.78175 − 8.28223i 0.729210 − 1.26303i −0.228008 − 0.973659i \(-0.573221\pi\)
0.957218 − 0.289369i \(-0.0934455\pi\)
\(44\) −0.640562 − 2.40335i −0.0965684 − 0.362319i
\(45\) 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 \(-0.457014\pi\)
0.925458 + 0.378850i \(0.123680\pi\)
\(48\) −9.58971 − 0.0254286i −1.38416 − 0.00367030i
\(49\) 4.54495 0.649279
\(50\) 6.68281 + 0.884316i 0.945092 + 0.125061i
\(51\) −5.62954 + 3.25022i −0.788293 + 0.455121i
\(52\) −5.09835 + 5.08485i −0.707014 + 0.705142i
\(53\) −1.50024 − 2.59849i −0.206074 − 0.356930i 0.744400 − 0.667733i \(-0.232735\pi\)
−0.950474 + 0.310803i \(0.899402\pi\)
\(54\) 0.678311 + 0.521201i 0.0923064 + 0.0709265i
\(55\) 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 0.760210 − 0.649677i \(-0.225096\pi\)
−0.182532 + 0.983200i \(0.558429\pi\)
\(60\) 1.63997 − 1.63563i 0.211719 − 0.211159i
\(61\) −9.42313 + 5.44044i −1.20651 + 0.696578i −0.961995 − 0.273068i \(-0.911961\pi\)
−0.244513 + 0.969646i \(0.578628\pi\)
\(62\) −4.18781 + 5.45018i −0.531853 + 0.692174i
\(63\) −3.72846 + 2.15263i −0.469742 + 0.271206i
\(64\) 5.63431 + 5.67931i 0.704289 + 0.709914i
\(65\) − 1.73916i − 0.215716i
\(66\) −3.89660 + 1.61100i −0.479639 + 0.198300i
\(67\) −3.22460 + 1.86172i −0.393947 + 0.227446i −0.683869 − 0.729605i \(-0.739704\pi\)
0.289922 + 0.957050i \(0.406371\pi\)
\(68\) 5.23618 + 1.41047i 0.634980 + 0.171045i
\(69\) 15.2959 1.84141
\(70\) −0.408967 − 0.989188i −0.0488809 − 0.118231i
\(71\) −6.62670 + 11.4778i −0.786444 + 1.36216i 0.141688 + 0.989911i \(0.454747\pi\)
−0.928132 + 0.372250i \(0.878586\pi\)
\(72\) 0.999080 + 7.70718i 0.117743 + 0.908300i
\(73\) −0.494526 + 0.856544i −0.0578799 + 0.100251i −0.893514 − 0.449036i \(-0.851767\pi\)
0.835634 + 0.549287i \(0.185101\pi\)
\(74\) 9.45417 − 12.3040i 1.09902 − 1.43031i
\(75\) − 11.4277i − 1.31956i
\(76\) 4.00322 + 7.74431i 0.459200 + 0.888333i
\(77\) 1.94859i 0.222062i
\(78\) 9.67937 + 7.43744i 1.09597 + 0.842124i
\(79\) −5.81311 + 10.0686i −0.654026 + 1.13281i 0.328111 + 0.944639i \(0.393588\pi\)
−0.982137 + 0.188168i \(0.939745\pi\)
\(80\) −1.93223 − 0.00512359i −0.216029 − 0.000572835i
\(81\) 4.84663 − 8.39460i 0.538514 − 0.932733i
\(82\) −1.66013 + 0.686361i −0.183331 + 0.0757959i
\(83\) −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
\(86\) 5.16744 + 12.4987i 0.557220 + 1.34777i
\(87\) 3.33319i 0.357355i
\(88\) 3.24707 + 1.35255i 0.346139 + 0.144182i
\(89\) −11.1625 + 6.44469i −1.18323 + 0.683136i −0.956759 − 0.290883i \(-0.906051\pi\)
−0.226467 + 0.974019i \(0.572718\pi\)
\(90\) −1.48843 − 1.14368i −0.156894 − 0.120555i
\(91\) 4.88540 − 2.82059i 0.512129 − 0.295678i
\(92\) −9.01085 − 9.03478i −0.939446 − 0.941941i
\(93\) 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 0.618134 − 0.786073i \(-0.287889\pi\)
−0.371692 + 0.928356i \(0.621222\pi\)
\(98\) −3.91621 + 5.09671i −0.395597 + 0.514845i
\(99\) 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.27.4 ✓ 28
4.3 odd 2 608.2.s.c.559.2 28
8.3 odd 2 inner 152.2.o.c.27.10 yes 28
8.5 even 2 608.2.s.c.559.1 28
19.12 odd 6 inner 152.2.o.c.107.10 yes 28
76.31 even 6 608.2.s.c.335.1 28
152.69 odd 6 608.2.s.c.335.2 28
152.107 even 6 inner 152.2.o.c.107.4 yes 28
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
152.2.o.c.27.4 ✓ 28 1.1 even 1 trivial
152.2.o.c.27.10 yes 28 8.3 odd 2 inner
152.2.o.c.107.4 yes 28 152.107 even 6 inner
152.2.o.c.107.10 yes 28 19.12 odd 6 inner
608.2.s.c.335.1 28 76.31 even 6
608.2.s.c.335.2 28 152.69 odd 6
608.2.s.c.559.1 28 8.5 even 2
608.2.s.c.559.2 28 4.3 odd 2