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Properties

Label 152.2.o.c.27.3
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 27.3
Character \(\chi\) \(=\) 152.27
Dual form 152.2.o.c.107.3

$q$-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+(-1.34230 + 0.445227i) q^{2} +(0.705625 + 0.407393i) q^{3} +(1.60355 - 1.19526i) q^{4} +(-3.59735 - 2.07693i) q^{5} +(-1.12854 - 0.232681i) q^{6} -3.24695i q^{7} +(-1.62028 + 2.31834i) q^{8} +(-1.16806 - 2.02314i) q^{9} +(5.75343 + 1.18623i) q^{10} -1.21833 q^{11} +(1.61844 - 0.190130i) q^{12} +(2.01425 + 3.48878i) q^{13} +(1.44563 + 4.35839i) q^{14} +(-1.69225 - 2.93107i) q^{15} +(1.14272 - 3.83330i) 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} +(1.63536 - 0.542432i) q^{22} +(3.01328 - 1.73972i) q^{23} +(-2.08779 + 0.975785i) q^{24} +(6.12726 + 10.6127i) q^{25} +(-4.25703 - 3.78620i) q^{26} -4.34780i q^{27} +(-3.88094 - 5.20664i) q^{28} +(2.35755 + 4.08339i) q^{29} +(3.57650 + 3.18094i) q^{30} -5.99801 q^{31} +(0.172808 + 5.65421i) q^{32} +(-0.859683 - 0.496338i) q^{33} +(-0.858329 + 4.16305i) q^{34} +(-6.74369 + 11.6804i) q^{35} +(-4.29122 - 1.84807i) q^{36} -1.34860 q^{37} +(4.58890 + 4.11607i) q^{38} +3.28236i q^{39} +(10.6437 - 4.97465i) q^{40} +(2.97093 + 1.71527i) q^{41} +(-0.755505 + 3.66433i) q^{42} +(-1.01615 + 1.76003i) q^{43} +(-1.95365 + 1.45621i) q^{44} +9.70392i q^{45} +(-3.27016 + 3.67682i) q^{46} +(8.37318 - 4.83426i) q^{47} +(2.36799 - 2.23934i) q^{48} -3.54271 q^{49} +(-12.9497 - 11.5175i) q^{50} +(2.12085 - 1.22448i) q^{51} +(7.39993 + 3.18688i) q^{52} +(-1.22182 - 2.11626i) q^{53} +(1.93576 + 5.83606i) 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} +(-6.21698 - 2.67743i) q^{60} +(0.300922 - 0.173737i) q^{61} +(8.05114 - 2.67048i) q^{62} +(-6.56905 + 3.79264i) q^{63} +(-2.74937 - 7.51272i) q^{64} -16.7338i q^{65} +(1.37494 + 0.283482i) q^{66} +(8.31107 - 4.79840i) q^{67} +(-0.701363 - 5.97022i) q^{68} +2.83500 q^{69} +(3.85164 - 18.6811i) q^{70} +(0.581533 - 1.00725i) q^{71} +(6.58292 + 0.570103i) q^{72} +(1.46675 - 2.54048i) q^{73} +(1.81022 - 0.600431i) q^{74} +9.98482i q^{75} +(-7.99227 - 3.48191i) q^{76} +3.95586i q^{77} +(-1.46140 - 4.40592i) q^{78} +(2.57780 - 4.46489i) q^{79} +(-12.0723 + 11.4163i) q^{80} +(-1.73292 + 3.00151i) q^{81} +(-4.75157 - 0.979669i) q^{82} +11.7265 q^{83} +(-0.617343 - 5.25501i) q^{84} +(-10.8123 + 6.24249i) q^{85} +(0.580372 - 2.81491i) q^{86} +3.84179i q^{87} +(1.97404 - 2.82449i) q^{88} +(-12.1643 + 7.02306i) q^{89} +(-4.32044 - 13.0256i) q^{90} +(11.3279 - 6.54017i) q^{91} +(2.75253 - 6.39137i) 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} +(4.75539 - 1.57731i) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.34230 + 0.445227i −0.949150 + 0.314823i
\(3\) 0.705625 + 0.407393i 0.407393 + 0.235208i 0.689669 − 0.724125i \(-0.257756\pi\)
−0.282276 + 0.959333i \(0.591089\pi\)
\(4\) 1.60355 − 1.19526i 0.801773 − 0.597628i
\(5\) −3.59735 − 2.07693i −1.60878 − 0.928831i −0.989643 − 0.143549i \(-0.954149\pi\)
−0.619139 − 0.785282i \(-0.712518\pi\)
\(6\) −1.12854 − 0.232681i −0.460726 − 0.0949916i
\(7\) − 3.24695i − 1.22723i −0.789604 − 0.613617i \(-0.789714\pi\)
0.789604 − 0.613617i \(-0.210286\pi\)
\(8\) −1.62028 + 2.31834i −0.572857 + 0.819656i
\(9\) −1.16806 − 2.02314i −0.389354 − 0.674381i
\(10\) 5.75343 + 1.18623i 1.81939 + 0.375119i
\(11\) −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 \(0.0220146\pi\)
−0.438957 + 0.898508i \(0.644652\pi\)
\(14\) 1.44563 + 4.35839i 0.386361 + 1.16483i
\(15\) −1.69225 − 2.93107i −0.436938 − 0.756798i
\(16\) 1.14272 − 3.83330i 0.285681 − 0.958325i
\(17\) 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
\(22\) 1.63536 − 0.542432i 0.348661 − 0.115647i
\(23\) 3.01328 − 1.73972i 0.628313 − 0.362756i −0.151786 − 0.988413i \(-0.548502\pi\)
0.780098 + 0.625657i \(0.215169\pi\)
\(24\) −2.08779 + 0.975785i −0.426168 + 0.199181i
\(25\) 6.12726 + 10.6127i 1.22545 + 2.12255i
\(26\) −4.25703 − 3.78620i −0.834872 − 0.742535i
\(27\) − 4.34780i − 0.836734i
\(28\) −3.88094 − 5.20664i −0.733429 − 0.983963i
\(29\) 2.35755 + 4.08339i 0.437785 + 0.758266i 0.997518 − 0.0704067i \(-0.0224297\pi\)
−0.559733 + 0.828673i \(0.689096\pi\)
\(30\) 3.57650 + 3.18094i 0.652977 + 0.580757i
\(31\) −5.99801 −1.07728 −0.538638 − 0.842538i \(-0.681061\pi\)
−0.538638 + 0.842538i \(0.681061\pi\)
\(32\) 0.172808 + 5.65421i 0.0305484 + 0.999533i
\(33\) −0.859683 − 0.496338i −0.149652 − 0.0864014i
\(34\) −0.858329 + 4.16305i −0.147202 + 0.713957i
\(35\) −6.74369 + 11.6804i −1.13989 + 1.97435i
\(36\) −4.29122 − 1.84807i −0.715203 − 0.308012i
\(37\) −1.34860 −0.221708 −0.110854 − 0.993837i \(-0.535359\pi\)
−0.110854 + 0.993837i \(0.535359\pi\)
\(38\) 4.58890 + 4.11607i 0.744417 + 0.667715i
\(39\) 3.28236i 0.525599i
\(40\) 10.6437 − 4.97465i 1.68292 − 0.786560i
\(41\) 2.97093 + 1.71527i 0.463981 + 0.267880i 0.713717 − 0.700434i \(-0.247010\pi\)
−0.249736 + 0.968314i \(0.580344\pi\)
\(42\) −0.755505 + 3.66433i −0.116577 + 0.565419i
\(43\) −1.01615 + 1.76003i −0.154962 + 0.268402i −0.933045 − 0.359759i \(-0.882859\pi\)
0.778083 + 0.628161i \(0.216192\pi\)
\(44\) −1.95365 + 1.45621i −0.294523 + 0.219533i
\(45\) 9.70392i 1.44658i
\(46\) −3.27016 + 3.67682i −0.482159 + 0.542118i
\(47\) 8.37318 − 4.83426i 1.22135 − 0.705149i 0.256147 − 0.966638i \(-0.417547\pi\)
0.965207 + 0.261489i \(0.0842134\pi\)
\(48\) 2.36799 − 2.23934i 0.341790 − 0.323220i
\(49\) −3.54271 −0.506102
\(50\) −12.9497 − 11.5175i −1.83136 − 1.62882i
\(51\) 2.12085 − 1.22448i 0.296979 − 0.171461i
\(52\) 7.39993 + 3.18688i 1.02619 + 0.441941i
\(53\) −1.22182 − 2.11626i −0.167830 − 0.290690i 0.769827 − 0.638253i \(-0.220343\pi\)
−0.937657 + 0.347563i \(0.887009\pi\)
\(54\) 1.93576 + 5.83606i 0.263423 + 0.794187i
\(55\) 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 \(0.914772\pi\)
\(60\) −6.21698 − 2.67743i −0.802609 − 0.345654i
\(61\) 0.300922 − 0.173737i 0.0385291 − 0.0222448i −0.480612 − 0.876934i \(-0.659585\pi\)
0.519141 + 0.854689i \(0.326252\pi\)
\(62\) 8.05114 − 2.67048i 1.02250 − 0.339151i
\(63\) −6.56905 + 3.79264i −0.827623 + 0.477828i
\(64\) −2.74937 − 7.51272i −0.343671 − 0.939090i
\(65\) − 16.7338i − 2.07557i
\(66\) 1.37494 + 0.283482i 0.169243 + 0.0348942i
\(67\) 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
\(70\) 3.85164 − 18.6811i 0.460359 − 2.23282i
\(71\) 0.581533 − 1.00725i 0.0690153 − 0.119538i −0.829453 − 0.558577i \(-0.811348\pi\)
0.898468 + 0.439039i \(0.144681\pi\)
\(72\) 6.58292 + 0.570103i 0.775804 + 0.0671873i
\(73\) 1.46675 − 2.54048i 0.171670 − 0.297341i −0.767334 − 0.641248i \(-0.778417\pi\)
0.939004 + 0.343907i \(0.111750\pi\)
\(74\) 1.81022 − 0.600431i 0.210434 − 0.0697987i
\(75\) 9.98482i 1.15295i
\(76\) −7.99227 − 3.48191i −0.916776 − 0.399402i
\(77\) 3.95586i 0.450812i
\(78\) −1.46140 − 4.40592i −0.165470 − 0.498872i
\(79\) 2.57780 − 4.46489i 0.290026 − 0.502339i −0.683790 − 0.729679i \(-0.739670\pi\)
0.973815 + 0.227340i \(0.0730029\pi\)
\(80\) −12.0723 + 11.4163i −1.34972 + 1.27639i
\(81\) −1.73292 + 3.00151i −0.192547 + 0.333501i
\(82\) −4.75157 − 0.979669i −0.524723 − 0.108186i
\(83\) 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
\(86\) 0.580372 − 2.81491i 0.0625831 − 0.303539i
\(87\) 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 \(0.600618\pi\)
\(90\) −4.32044 − 13.0256i −0.455415 − 1.37302i
\(91\) 11.3279 − 6.54017i 1.18749 − 0.685596i
\(92\) 2.75253 − 6.39137i 0.286971 − 0.666346i
\(93\) −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 \(0.415176\pi\)
\(98\) 4.75539 − 1.57731i 0.480367 − 0.159332i
\(99\) 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.27.3 ✓ 28
4.3 odd 2 608.2.s.c.559.5 28
8.3 odd 2 inner 152.2.o.c.27.7 yes 28
8.5 even 2 608.2.s.c.559.6 28
19.12 odd 6 inner 152.2.o.c.107.7 yes 28
76.31 even 6 608.2.s.c.335.6 28
152.69 odd 6 608.2.s.c.335.5 28
152.107 even 6 inner 152.2.o.c.107.3 yes 28
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
152.2.o.c.27.3 ✓ 28 1.1 even 1 trivial
152.2.o.c.27.7 yes 28 8.3 odd 2 inner
152.2.o.c.107.3 yes 28 152.107 even 6 inner
152.2.o.c.107.7 yes 28 19.12 odd 6 inner
608.2.s.c.335.5 28 152.69 odd 6
608.2.s.c.335.6 28 76.31 even 6
608.2.s.c.559.5 28 4.3 odd 2
608.2.s.c.559.6 28 8.5 even 2