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Properties

Label 152.2.o.c.27.13
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 27.13
Character \(\chi\) \(=\) 152.27
Dual form 152.2.o.c.107.13

$q$-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+(1.37418 - 0.334110i) q^{2} +(1.27630 + 0.736872i) q^{3} +(1.77674 - 0.918256i) q^{4} +(-2.34524 - 1.35403i) q^{5} +(2.00006 + 0.586169i) q^{6} +3.01597i q^{7} +(2.13476 - 1.85548i) q^{8} +(-0.414040 - 0.717138i) q^{9} +(-3.67518 - 1.07711i) q^{10} -3.11370 q^{11} +(2.94429 + 0.137261i) q^{12} +(0.218909 + 0.379162i) q^{13} +(1.00767 + 4.14449i) q^{14} +(-1.99549 - 3.45629i) q^{15} +(2.31361 - 3.26300i) q^{16} +(-3.22274 + 5.58194i) q^{17} +(-0.808569 - 0.847142i) q^{18} +(3.58731 - 2.47613i) q^{19} +(-5.41023 - 0.252221i) q^{20} +(-2.22238 + 3.84928i) q^{21} +(-4.27879 + 1.04032i) q^{22} +(-1.81421 + 1.04744i) q^{23} +(4.09184 - 0.795097i) q^{24} +(1.16678 + 2.02092i) q^{25} +(0.427502 + 0.447896i) q^{26} -5.64161i q^{27} +(2.76943 + 5.35860i) q^{28} +(-1.58443 - 2.74431i) q^{29} +(-3.89694 - 4.08285i) q^{30} +9.77556 q^{31} +(2.08912 - 5.25696i) q^{32} +(-3.97402 - 2.29440i) q^{33} +(-2.56363 + 8.74735i) q^{34} +(4.08371 - 7.07319i) q^{35} +(-1.39416 - 0.893974i) q^{36} -8.36258 q^{37} +(4.10230 - 4.60121i) q^{38} +0.645232i q^{39} +(-7.51890 + 1.46102i) q^{40} +(4.66208 + 2.69165i) q^{41} +(-1.76787 + 6.03213i) q^{42} +(-2.43347 + 4.21489i) q^{43} +(-5.53225 + 2.85918i) q^{44} +2.24248i q^{45} +(-2.14310 + 2.04551i) q^{46} +(-1.22865 + 0.709359i) q^{47} +(5.35728 - 2.45973i) q^{48} -2.09609 q^{49} +(2.27857 + 2.38727i) q^{50} +(-8.22635 + 4.74949i) q^{51} +(0.737112 + 0.472657i) q^{52} +(1.63792 + 2.83695i) q^{53} +(-1.88492 - 7.75258i) q^{54} +(7.30239 + 4.21604i) q^{55} +(5.59606 + 6.43838i) q^{56} +(6.40307 - 0.516904i) q^{57} +(-3.09420 - 3.24181i) q^{58} +(-1.70599 - 0.984956i) q^{59} +(-6.71922 - 4.30856i) q^{60} +(9.56035 - 5.51967i) q^{61} +(13.4334 - 3.26612i) q^{62} +(2.16287 - 1.24873i) q^{63} +(1.11442 - 7.92200i) q^{64} -1.18563i q^{65} +(-6.22760 - 1.82516i) q^{66} +(4.49875 - 2.59735i) q^{67} +(-0.600315 + 12.8770i) q^{68} -3.08731 q^{69} +(3.24852 - 11.0842i) q^{70} +(4.40927 - 7.63707i) q^{71} +(-2.21451 - 0.762678i) q^{72} +(-6.90160 + 11.9539i) q^{73} +(-11.4917 + 2.79403i) q^{74} +3.43906i q^{75} +(4.09999 - 7.69351i) q^{76} -9.39085i q^{77} +(0.215579 + 0.886664i) q^{78} +(-0.939415 + 1.62711i) q^{79} +(-9.84418 + 4.51984i) q^{80} +(2.91502 - 5.04897i) q^{81} +(7.30584 + 2.14116i) q^{82} +3.79468 q^{83} +(-0.413974 + 8.87989i) q^{84} +(15.1162 - 8.72734i) q^{85} +(-1.93578 + 6.60506i) q^{86} -4.67009i q^{87} +(-6.64702 + 5.77740i) q^{88} +(10.7709 - 6.21857i) q^{89} +(0.749238 + 3.08158i) q^{90} +(-1.14354 + 0.660224i) q^{91} +(-2.26157 + 3.52694i) q^{92} +(12.4765 + 7.20333i) q^{93} +(-1.45138 + 1.38529i) q^{94} +(-11.7659 + 0.949828i) q^{95} +(6.54004 - 5.17004i) q^{96} +(-16.1008 - 9.29578i) q^{97} +(-2.88040 + 0.700325i) q^{98} +(1.28920 + 2.23296i) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.37418 − 0.334110i 0.971692 − 0.236252i
\(3\) 1.27630 + 0.736872i 0.736872 + 0.425433i 0.820931 − 0.571028i \(-0.193455\pi\)
−0.0840591 + 0.996461i \(0.526788\pi\)
\(4\) 1.77674 − 0.918256i 0.888370 − 0.459128i
\(5\) −2.34524 − 1.35403i −1.04882 − 0.605539i −0.126504 − 0.991966i \(-0.540376\pi\)
−0.922320 + 0.386427i \(0.873709\pi\)
\(6\) 2.00006 + 0.586169i 0.816522 + 0.239303i
\(7\) 3.01597i 1.13993i 0.821669 + 0.569965i \(0.193043\pi\)
−0.821669 + 0.569965i \(0.806957\pi\)
\(8\) 2.13476 − 1.85548i 0.754752 − 0.656010i
\(9\) −0.414040 − 0.717138i −0.138013 − 0.239046i
\(10\) −3.67518 − 1.07711i −1.16219 − 0.340611i
\(11\) −3.11370 −0.938817 −0.469409 − 0.882981i \(-0.655533\pi\)
−0.469409 + 0.882981i \(0.655533\pi\)
\(12\) 2.94429 + 0.137261i 0.849943 + 0.0396237i
\(13\) 0.218909 + 0.379162i 0.0607144 + 0.105160i 0.894785 − 0.446497i \(-0.147329\pi\)
−0.834071 + 0.551658i \(0.813995\pi\)
\(14\) 1.00767 + 4.14449i 0.269311 + 1.10766i
\(15\) −1.99549 − 3.45629i −0.515233 − 0.892409i
\(16\) 2.31361 − 3.26300i 0.578403 − 0.815751i
\(17\) −3.22274 + 5.58194i −0.781629 + 1.35382i 0.149364 + 0.988782i \(0.452277\pi\)
−0.930993 + 0.365038i \(0.881056\pi\)
\(18\) −0.808569 − 0.847142i −0.190582 − 0.199673i
\(19\) 3.58731 − 2.47613i 0.822984 − 0.568064i
\(20\) −5.41023 − 0.252221i −1.20976 − 0.0563984i
\(21\) −2.22238 + 3.84928i −0.484964 + 0.839982i
\(22\) −4.27879 + 1.04032i −0.912241 + 0.221797i
\(23\) −1.81421 + 1.04744i −0.378290 + 0.218406i −0.677074 − 0.735915i \(-0.736752\pi\)
0.298784 + 0.954321i \(0.403419\pi\)
\(24\) 4.09184 − 0.795097i 0.835244 − 0.162298i
\(25\) 1.16678 + 2.02092i 0.233355 + 0.404183i
\(26\) 0.427502 + 0.447896i 0.0838401 + 0.0878397i
\(27\) − 5.64161i − 1.08573i
\(28\) 2.76943 + 5.35860i 0.523374 + 1.01268i
\(29\) −1.58443 − 2.74431i −0.294221 − 0.509606i 0.680582 − 0.732672i \(-0.261727\pi\)
−0.974803 + 0.223066i \(0.928394\pi\)
\(30\) −3.89694 − 4.08285i −0.711481 − 0.745422i
\(31\) 9.77556 1.75574 0.877871 − 0.478897i \(-0.158963\pi\)
0.877871 + 0.478897i \(0.158963\pi\)
\(32\) 2.08912 − 5.25696i 0.369307 − 0.929307i
\(33\) −3.97402 − 2.29440i −0.691788 − 0.399404i
\(34\) −2.56363 + 8.74735i −0.439660 + 1.50016i
\(35\) 4.08371 − 7.07319i 0.690272 − 1.19559i
\(36\) −1.39416 − 0.893974i −0.232360 − 0.148996i
\(37\) −8.36258 −1.37480 −0.687400 − 0.726279i \(-0.741248\pi\)
−0.687400 + 0.726279i \(0.741248\pi\)
\(38\) 4.10230 − 4.60121i 0.665481 − 0.746415i
\(39\) 0.645232i 0.103320i
\(40\) −7.51890 + 1.46102i −1.18884 + 0.231007i
\(41\) 4.66208 + 2.69165i 0.728094 + 0.420365i 0.817724 − 0.575610i \(-0.195235\pi\)
−0.0896306 + 0.995975i \(0.528569\pi\)
\(42\) −1.76787 + 6.03213i −0.272788 + 0.930778i
\(43\) −2.43347 + 4.21489i −0.371100 + 0.642764i −0.989735 − 0.142914i \(-0.954353\pi\)
0.618635 + 0.785679i \(0.287686\pi\)
\(44\) −5.53225 + 2.85918i −0.834017 + 0.431037i
\(45\) 2.24248i 0.334290i
\(46\) −2.14310 + 2.04551i −0.315982 + 0.301595i
\(47\) −1.22865 + 0.709359i −0.179216 + 0.103471i −0.586924 − 0.809642i \(-0.699661\pi\)
0.407708 + 0.913112i \(0.366328\pi\)
\(48\) 5.35728 − 2.45973i 0.773256 − 0.355032i
\(49\) −2.09609 −0.299441
\(50\) 2.27857 + 2.38727i 0.322238 + 0.337611i
\(51\) −8.22635 + 4.74949i −1.15192 + 0.665061i
\(52\) 0.737112 + 0.472657i 0.102219 + 0.0655458i
\(53\) 1.63792 + 2.83695i 0.224985 + 0.389686i 0.956315 − 0.292338i \(-0.0944333\pi\)
−0.731330 + 0.682024i \(0.761100\pi\)
\(54\) −1.88492 − 7.75258i −0.256505 − 1.05499i
\(55\) 7.30239 + 4.21604i 0.984655 + 0.568491i
\(56\) 5.59606 + 6.43838i 0.747805 + 0.860365i
\(57\) 6.40307 − 0.516904i 0.848107 − 0.0684655i
\(58\) −3.09420 − 3.24181i −0.406288 − 0.425670i
\(59\) −1.70599 − 0.984956i −0.222102 − 0.128230i 0.384821 − 0.922991i \(-0.374263\pi\)
−0.606923 + 0.794761i \(0.707596\pi\)
\(60\) −6.71922 − 4.30856i −0.867447 − 0.556232i
\(61\) 9.56035 − 5.51967i 1.22408 − 0.706722i 0.258293 − 0.966067i \(-0.416840\pi\)
0.965785 + 0.259345i \(0.0835067\pi\)
\(62\) 13.4334 − 3.26612i 1.70604 − 0.414797i
\(63\) 2.16287 − 1.24873i 0.272496 − 0.157326i
\(64\) 1.11442 − 7.92200i 0.139302 − 0.990250i
\(65\) − 1.18563i − 0.147060i
\(66\) −6.22760 − 1.82516i −0.766565 − 0.224661i
\(67\) 4.49875 − 2.59735i 0.549609 − 0.317317i −0.199355 − 0.979927i \(-0.563885\pi\)
0.748964 + 0.662610i \(0.230551\pi\)
\(68\) −0.600315 + 12.8770i −0.0727989 + 1.56156i
\(69\) −3.08731 −0.371668
\(70\) 3.24852 − 11.0842i 0.388273 − 1.32482i
\(71\) 4.40927 − 7.63707i 0.523284 − 0.906354i −0.476349 − 0.879256i \(-0.658040\pi\)
0.999633 − 0.0270976i \(-0.00862647\pi\)
\(72\) −2.21451 − 0.762678i −0.260983 − 0.0898825i
\(73\) −6.90160 + 11.9539i −0.807772 + 1.39910i 0.106632 + 0.994299i \(0.465993\pi\)
−0.914404 + 0.404803i \(0.867340\pi\)
\(74\) −11.4917 + 2.79403i −1.33588 + 0.324799i
\(75\) 3.43906i 0.397108i
\(76\) 4.09999 − 7.69351i 0.470301 − 0.882506i
\(77\) − 9.39085i − 1.07019i
\(78\) 0.215579 + 0.886664i 0.0244095 + 0.100395i
\(79\) −0.939415 + 1.62711i −0.105692 + 0.183065i −0.914021 − 0.405667i \(-0.867039\pi\)
0.808328 + 0.588732i \(0.200373\pi\)
\(80\) −9.84418 + 4.51984i −1.10061 + 0.505334i
\(81\) 2.91502 − 5.04897i 0.323891 − 0.560996i
\(82\) 7.30584 + 2.14116i 0.806795 + 0.236452i
\(83\) 3.79468 0.416520 0.208260 − 0.978073i \(-0.433220\pi\)
0.208260 + 0.978073i \(0.433220\pi\)
\(84\) −0.413974 + 8.87989i −0.0451683 + 0.968876i
\(85\) 15.1162 − 8.72734i 1.63958 − 0.946613i
\(86\) −1.93578 + 6.60506i −0.208741 + 0.712242i
\(87\) − 4.67009i − 0.500686i
\(88\) −6.64702 + 5.77740i −0.708575 + 0.615873i
\(89\) 10.7709 − 6.21857i 1.14171 − 0.659167i 0.194857 − 0.980832i \(-0.437576\pi\)
0.946854 + 0.321664i \(0.104242\pi\)
\(90\) 0.749238 + 3.08158i 0.0789766 + 0.324827i
\(91\) −1.14354 + 0.660224i −0.119876 + 0.0692102i
\(92\) −2.26157 + 3.52694i −0.235785 + 0.367708i
\(93\) 12.4765 + 7.20333i 1.29376 + 0.746951i
\(94\) −1.45138 + 1.38529i −0.149698 + 0.142882i
\(95\) −11.7659 + 0.949828i −1.20715 + 0.0974503i
\(96\) 6.54004 − 5.17004i 0.667490 − 0.527665i
\(97\) −16.1008 − 9.29578i −1.63479 − 0.943844i −0.982589 − 0.185794i \(-0.940514\pi\)
−0.652197 − 0.758050i \(-0.726152\pi\)
\(98\) −2.88040 + 0.700325i −0.290964 + 0.0707435i
\(99\) 1.28920 + 2.23296i 0.129569 + 0.224421i
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.27.13 yes 28
4.3 odd 2 608.2.s.c.559.3 28
8.3 odd 2 inner 152.2.o.c.27.9 ✓ 28
8.5 even 2 608.2.s.c.559.4 28
19.12 odd 6 inner 152.2.o.c.107.9 yes 28
76.31 even 6 608.2.s.c.335.4 28
152.69 odd 6 608.2.s.c.335.3 28
152.107 even 6 inner 152.2.o.c.107.13 yes 28
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
152.2.o.c.27.9 ✓ 28 8.3 odd 2 inner
152.2.o.c.27.13 yes 28 1.1 even 1 trivial
152.2.o.c.107.9 yes 28 19.12 odd 6 inner
152.2.o.c.107.13 yes 28 152.107 even 6 inner
608.2.s.c.335.3 28 152.69 odd 6
608.2.s.c.335.4 28 76.31 even 6
608.2.s.c.559.3 28 4.3 odd 2
608.2.s.c.559.4 28 8.5 even 2