↑

Properties

Label 152.2.o.c.27.7
Level $152$
Weight $2$
Character 152.27
Analytic conductor $1.214$
Analytic rank $0$
Dimension $28$
Inner twists $4$

Related objects

Downloads

Learn more

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 27.7
Character \(\chi\) \(=\) 152.27
Dual form 152.2.o.c.107.7

$q$-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+(-0.285573 - 1.38508i) q^{2} +(0.705625 + 0.407393i) q^{3} +(-1.83690 + 0.791084i) q^{4} +(3.59735 + 2.07693i) q^{5} +(0.362764 - 1.09369i) q^{6} +3.24695i q^{7} +(1.62028 + 2.31834i) q^{8} +(-1.16806 - 2.02314i) q^{9} +(1.84941 - 5.57573i) q^{10} -1.21833 q^{11} +(-1.61844 - 0.190130i) q^{12} +(-2.01425 - 3.48878i) q^{13} +(4.49729 - 0.927243i) q^{14} +(1.69225 + 2.93107i) q^{15} +(2.74837 - 2.90628i) q^{16} +(1.50282 - 2.60296i) q^{17} +(-2.46865 + 2.19562i) q^{18} +(-2.16355 - 3.78405i) q^{19} +(-8.25097 - 0.969300i) q^{20} +(-1.32279 + 2.29113i) q^{21} +(0.347922 + 1.68748i) q^{22} +(-3.01328 + 1.73972i) q^{23} +(0.198839 + 2.29597i) q^{24} +(6.12726 + 10.6127i) q^{25} +(-4.25703 + 3.78620i) q^{26} -4.34780i q^{27} +(-2.56861 - 5.96432i) q^{28} +(-2.35755 - 4.08339i) q^{29} +(3.57650 - 3.18094i) q^{30} +5.99801 q^{31} +(-4.81029 - 2.97676i) q^{32} +(-0.859683 - 0.496338i) q^{33} +(-4.03447 - 1.33819i) q^{34} +(-6.74369 + 11.6804i) q^{35} +(3.74608 + 2.79227i) q^{36} +1.34860 q^{37} +(-4.62336 + 4.07731i) q^{38} -3.28236i q^{39} +(1.01370 + 11.7051i) q^{40} +(2.97093 + 1.71527i) q^{41} +(3.55116 + 1.17788i) q^{42} +(-1.01615 + 1.76003i) q^{43} +(2.23794 - 0.963799i) q^{44} -9.70392i q^{45} +(3.27016 + 3.67682i) q^{46} +(-8.37318 + 4.83426i) q^{47} +(3.12332 - 0.931075i) q^{48} -3.54271 q^{49} +(12.9497 - 11.5175i) q^{50} +(2.12085 - 1.22448i) q^{51} +(6.45988 + 4.81509i) q^{52} +(1.22182 + 2.11626i) q^{53} +(-6.02205 + 1.24161i) q^{54} +(-4.38275 - 2.53038i) q^{55} +(-7.52753 + 5.26099i) q^{56} +(0.0149406 - 3.55154i) q^{57} +(-4.98257 + 4.43150i) q^{58} +(-9.35127 - 5.39896i) q^{59} +(-5.42721 - 4.04535i) q^{60} +(-0.300922 + 0.173737i) q^{61} +(-1.71287 - 8.30773i) q^{62} +(6.56905 - 3.79264i) q^{63} +(-2.74937 + 7.51272i) q^{64} -16.7338i q^{65} +(-0.441966 + 1.33247i) q^{66} +(8.31107 - 4.79840i) q^{67} +(-0.701363 + 5.97022i) q^{68} -2.83500 q^{69} +(18.1041 + 6.00494i) q^{70} +(-0.581533 + 1.00725i) q^{71} +(2.79773 - 5.98602i) q^{72} +(1.46675 - 2.54048i) q^{73} +(-0.385123 - 1.86791i) q^{74} +9.98482i q^{75} +(6.96772 + 5.23936i) q^{76} -3.95586i q^{77} +(-4.54634 + 0.937355i) q^{78} +(-2.57780 + 4.46489i) q^{79} +(15.9230 - 4.74671i) q^{80} +(-1.73292 + 3.00151i) q^{81} +(1.52736 - 4.60481i) q^{82} +11.7265 q^{83} +(0.617343 - 5.25501i) q^{84} +(10.8123 - 6.24249i) q^{85} +(2.72797 + 0.904836i) q^{86} -3.84179i q^{87} +(-1.97404 - 2.82449i) q^{88} +(-12.1643 + 7.02306i) q^{89} +(-13.4407 + 2.77118i) q^{90} +(11.3279 - 6.54017i) q^{91} +(4.15882 - 5.57944i) q^{92} +(4.23235 + 2.44355i) q^{93} +(9.08699 + 10.2170i) q^{94} +(0.0761686 - 18.1061i) q^{95} +(-2.18155 - 4.06016i) q^{96} +(12.1187 + 6.99673i) q^{97} +(1.01170 + 4.90694i) q^{98} +(1.42308 + 2.46485i) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.285573 − 1.38508i −0.201931 − 0.979400i
\(3\) 0.705625 + 0.407393i 0.407393 + 0.235208i 0.689669 − 0.724125i \(-0.257756\pi\)
−0.282276 + 0.959333i \(0.591089\pi\)
\(4\) −1.83690 + 0.791084i −0.918448 + 0.395542i
\(5\) 3.59735 + 2.07693i 1.60878 + 0.928831i 0.989643 + 0.143549i \(0.0458515\pi\)
0.619139 + 0.785282i \(0.287482\pi\)
\(6\) 0.362764 − 1.09369i 0.148098 − 0.446496i
\(7\) 3.24695i 1.22723i 0.789604 + 0.613617i \(0.210286\pi\)
−0.789604 + 0.613617i \(0.789714\pi\)
\(8\) 1.62028 + 2.31834i 0.572857 + 0.819656i
\(9\) −1.16806 − 2.02314i −0.389354 − 0.674381i
\(10\) 1.84941 − 5.57573i 0.584834 − 1.76320i
\(11\) −1.21833 −0.367340 −0.183670 − 0.982988i \(-0.558798\pi\)
−0.183670 + 0.982988i \(0.558798\pi\)
\(12\) −1.61844 − 0.190130i −0.467204 − 0.0548857i
\(13\) −2.01425 − 3.48878i −0.558652 − 0.967614i −0.997609 − 0.0691058i \(-0.977985\pi\)
0.438957 − 0.898508i \(-0.355348\pi\)
\(14\) 4.49729 − 0.927243i 1.20195 − 0.247816i
\(15\) 1.69225 + 2.93107i 0.436938 + 0.756798i
\(16\) 2.74837 − 2.90628i 0.687093 − 0.726569i
\(17\) 1.50282 − 2.60296i 0.364487 − 0.631310i −0.624207 − 0.781259i \(-0.714578\pi\)
0.988694 + 0.149949i \(0.0479110\pi\)
\(18\) −2.46865 + 2.19562i −0.581866 + 0.517511i
\(19\) −2.16355 − 3.78405i −0.496352 − 0.868121i
\(20\) −8.25097 − 0.969300i −1.84497 − 0.216742i
\(21\) −1.32279 + 2.29113i −0.288656 + 0.499966i
\(22\) 0.347922 + 1.68748i 0.0741772 + 0.359772i
\(23\) −3.01328 + 1.73972i −0.628313 + 0.362756i −0.780098 − 0.625657i \(-0.784831\pi\)
0.151786 + 0.988413i \(0.451498\pi\)
\(24\) 0.198839 + 2.29597i 0.0405878 + 0.468663i
\(25\) 6.12726 + 10.6127i 1.22545 + 2.12255i
\(26\) −4.25703 + 3.78620i −0.834872 + 0.742535i
\(27\) − 4.34780i − 0.836734i
\(28\) −2.56861 − 5.96432i −0.485422 − 1.12715i
\(29\) −2.35755 − 4.08339i −0.437785 − 0.758266i 0.559733 − 0.828673i \(-0.310904\pi\)
−0.997518 + 0.0704067i \(0.977570\pi\)
\(30\) 3.57650 − 3.18094i 0.652977 − 0.580757i
\(31\) 5.99801 1.07728 0.538638 − 0.842538i \(-0.318939\pi\)
0.538638 + 0.842538i \(0.318939\pi\)
\(32\) −4.81029 − 2.97676i −0.850347 − 0.526222i
\(33\) −0.859683 − 0.496338i −0.149652 − 0.0864014i
\(34\) −4.03447 − 1.33819i −0.691906 − 0.229498i
\(35\) −6.74369 + 11.6804i −1.13989 + 1.97435i
\(36\) 3.74608 + 2.79227i 0.624347 + 0.465378i
\(37\) 1.34860 0.221708 0.110854 − 0.993837i \(-0.464641\pi\)
0.110854 + 0.993837i \(0.464641\pi\)
\(38\) −4.62336 + 4.07731i −0.750009 + 0.661428i
\(39\) − 3.28236i − 0.525599i
\(40\) 1.01370 + 11.7051i 0.160280 + 1.85073i
\(41\) 2.97093 + 1.71527i 0.463981 + 0.267880i 0.713717 − 0.700434i \(-0.247010\pi\)
−0.249736 + 0.968314i \(0.580344\pi\)
\(42\) 3.55116 + 1.17788i 0.547955 + 0.181751i
\(43\) −1.01615 + 1.76003i −0.154962 + 0.268402i −0.933045 − 0.359759i \(-0.882859\pi\)
0.778083 + 0.628161i \(0.216192\pi\)
\(44\) 2.23794 − 0.963799i 0.337382 − 0.145298i
\(45\) − 9.70392i − 1.44658i
\(46\) 3.27016 + 3.67682i 0.482159 + 0.542118i
\(47\) −8.37318 + 4.83426i −1.22135 + 0.705149i −0.965207 − 0.261489i \(-0.915787\pi\)
−0.256147 + 0.966638i \(0.582453\pi\)
\(48\) 3.12332 − 0.931075i 0.450812 − 0.134389i
\(49\) −3.54271 −0.506102
\(50\) 12.9497 − 11.5175i 1.83136 − 1.62882i
\(51\) 2.12085 − 1.22448i 0.296979 − 0.171461i
\(52\) 6.45988 + 4.81509i 0.895825 + 0.667733i
\(53\) 1.22182 + 2.11626i 0.167830 + 0.290690i 0.937657 − 0.347563i \(-0.112991\pi\)
−0.769827 + 0.638253i \(0.779657\pi\)
\(54\) −6.02205 + 1.24161i −0.819497 + 0.168962i
\(55\) −4.38275 − 2.53038i −0.590969 − 0.341196i
\(56\) −7.52753 + 5.26099i −1.00591 + 0.703029i
\(57\) 0.0149406 − 3.55154i 0.00197893 − 0.470413i
\(58\) −4.98257 + 4.43150i −0.654244 + 0.581884i
\(59\) −9.35127 − 5.39896i −1.21743 − 0.702885i −0.253064 − 0.967450i \(-0.581438\pi\)
−0.964368 + 0.264565i \(0.914772\pi\)
\(60\) −5.42721 − 4.04535i −0.700650 − 0.522253i
\(61\) −0.300922 + 0.173737i −0.0385291 + 0.0222448i −0.519141 − 0.854689i \(-0.673748\pi\)
0.480612 + 0.876934i \(0.340415\pi\)
\(62\) −1.71287 − 8.30773i −0.217535 − 1.05508i
\(63\) 6.56905 − 3.79264i 0.827623 − 0.477828i
\(64\) −2.74937 + 7.51272i −0.343671 + 0.939090i
\(65\) − 16.7338i − 2.07557i
\(66\) −0.441966 + 1.33247i −0.0544022 + 0.164016i
\(67\) 8.31107 − 4.79840i 1.01536 − 0.586218i 0.102603 − 0.994722i \(-0.467283\pi\)
0.912756 + 0.408505i \(0.133950\pi\)
\(68\) −0.701363 + 5.97022i −0.0850528 + 0.723995i
\(69\) −2.83500 −0.341294
\(70\) 18.1041 + 6.00494i 2.16386 + 0.717728i
\(71\) −0.581533 + 1.00725i −0.0690153 + 0.119538i −0.898468 − 0.439039i \(-0.855319\pi\)
0.829453 + 0.558577i \(0.188652\pi\)
\(72\) 2.79773 − 5.98602i 0.329716 − 0.705460i
\(73\) 1.46675 − 2.54048i 0.171670 − 0.297341i −0.767334 − 0.641248i \(-0.778417\pi\)
0.939004 + 0.343907i \(0.111750\pi\)
\(74\) −0.385123 − 1.86791i −0.0447696 − 0.217141i
\(75\) 9.98482i 1.15295i
\(76\) 6.96772 + 5.23936i 0.799252 + 0.600996i
\(77\) − 3.95586i − 0.450812i
\(78\) −4.54634 + 0.937355i −0.514771 + 0.106135i
\(79\) −2.57780 + 4.46489i −0.290026 + 0.502339i −0.973815 − 0.227340i \(-0.926997\pi\)
0.683790 + 0.729679i \(0.260330\pi\)
\(80\) 15.9230 − 4.74671i 1.78024 − 0.530698i
\(81\) −1.73292 + 3.00151i −0.192547 + 0.333501i
\(82\) 1.52736 − 4.60481i 0.168669 − 0.508516i
\(83\) 11.7265 1.28715 0.643574 − 0.765384i \(-0.277451\pi\)
0.643574 + 0.765384i \(0.277451\pi\)
\(84\) 0.617343 − 5.25501i 0.0673576 − 0.573368i
\(85\) 10.8123 − 6.24249i 1.17276 − 0.677093i
\(86\) 2.72797 + 0.904836i 0.294164 + 0.0975710i
\(87\) − 3.84179i − 0.411883i
\(88\) −1.97404 − 2.82449i −0.210433 − 0.301092i
\(89\) −12.1643 + 7.02306i −1.28941 + 0.744443i −0.978550 − 0.206011i \(-0.933952\pi\)
−0.310864 + 0.950455i \(0.600618\pi\)
\(90\) −13.4407 + 2.77118i −1.41678 + 0.292108i
\(91\) 11.3279 − 6.54017i 1.18749 − 0.685596i
\(92\) 4.15882 − 5.57944i 0.433587 − 0.581697i
\(93\) 4.23235 + 2.44355i 0.438874 + 0.253384i
\(94\) 9.08699 + 10.2170i 0.937252 + 1.05380i
\(95\) 0.0761686 − 18.1061i 0.00781474 − 1.85764i
\(96\) −2.18155 − 4.06016i −0.222653 − 0.414388i
\(97\) 12.1187 + 6.99673i 1.23047 + 0.710410i 0.967127 − 0.254293i \(-0.0818428\pi\)
0.263339 + 0.964703i \(0.415176\pi\)
\(98\) 1.01170 + 4.90694i 0.102198 + 0.495676i
\(99\) 1.42308 + 2.46485i 0.143025 + 0.247727i
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.27.7 yes 28
4.3 odd 2 608.2.s.c.559.6 28
8.3 odd 2 inner 152.2.o.c.27.3 ✓ 28
8.5 even 2 608.2.s.c.559.5 28
19.12 odd 6 inner 152.2.o.c.107.3 yes 28
76.31 even 6 608.2.s.c.335.5 28
152.69 odd 6 608.2.s.c.335.6 28
152.107 even 6 inner 152.2.o.c.107.7 yes 28
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
152.2.o.c.27.3 ✓ 28 8.3 odd 2 inner
152.2.o.c.27.7 yes 28 1.1 even 1 trivial
152.2.o.c.107.3 yes 28 19.12 odd 6 inner
152.2.o.c.107.7 yes 28 152.107 even 6 inner
608.2.s.c.335.5 28 76.31 even 6
608.2.s.c.335.6 28 152.69 odd 6
608.2.s.c.559.5 28 8.5 even 2
608.2.s.c.559.6 28 4.3 odd 2