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Properties

Label 152.2.b.c.75.7
Level $152$
Weight $2$
Character 152.75
Analytic conductor $1.214$
Analytic rank $0$
Dimension $12$
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(75,152)] 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("152.75"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(152, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 1, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 152 = 2^{3} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 152.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [12] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(1.21372611072\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: 12.0.319794774016000000.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 2x^{10} + 2x^{8} + 8x^{4} - 32x^{2} + 64 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 75.7
Root \(0.491416 - 1.32609i\) of defining polynomial
Character \(\chi\) \(=\) 152.75
Dual form 152.2.b.c.75.8

$q$-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+(0.491416 - 1.32609i) q^{2} +0.754098i q^{3} +(-1.51702 - 1.30332i) q^{4} -2.08884i q^{5} +(1.00000 + 0.370575i) q^{6} -2.23607i q^{7} +(-2.47381 + 1.37123i) q^{8} +2.43134 q^{9} +(-2.76999 - 1.02649i) q^{10} -0.602705 q^{11} +(0.982832 - 1.14398i) q^{12} +1.29574 q^{13} +(-2.96522 - 1.09884i) q^{14} +1.57519 q^{15} +(0.602705 + 3.95433i) q^{16} +2.20541 q^{17} +(1.19480 - 3.22417i) q^{18} +(-2.60270 + 3.49656i) q^{19} +(-2.72243 + 3.16881i) q^{20} +1.68621 q^{21} +(-0.296179 + 0.799240i) q^{22} +6.19040i q^{23} +(-1.03404 - 1.86549i) q^{24} +0.636747 q^{25} +(0.636747 - 1.71826i) q^{26} +4.09576i q^{27} +(-2.91432 + 3.39216i) q^{28} -8.20902 q^{29} +(0.774073 - 2.08884i) q^{30} +4.94762 q^{31} +(5.53997 + 1.14398i) q^{32} -0.454498i q^{33} +(1.08377 - 2.92457i) q^{34} -4.67079 q^{35} +(-3.68839 - 3.16881i) q^{36} +6.91328 q^{37} +(3.35774 + 5.16968i) q^{38} +0.977114i q^{39} +(2.86428 + 5.16739i) q^{40} -6.53862i q^{41} +(0.828632 - 2.23607i) q^{42} -0.191885 q^{43} +(0.914316 + 0.785518i) q^{44} -5.07867i q^{45} +(8.20902 + 3.04206i) q^{46} +0.223348i q^{47} +(-2.98195 + 0.454498i) q^{48} +2.00000 q^{49} +(0.312907 - 0.844383i) q^{50} +1.66309i q^{51} +(-1.96566 - 1.68877i) q^{52} -4.83659 q^{53} +(5.43134 + 2.01272i) q^{54} +1.25895i q^{55} +(3.06617 + 5.53160i) q^{56} +(-2.63675 - 1.96269i) q^{57} +(-4.03404 + 10.8859i) q^{58} -5.60395i q^{59} +(-2.38960 - 2.05298i) q^{60} +11.7743i q^{61} +(2.43134 - 6.56098i) q^{62} -5.43663i q^{63} +(4.23945 - 6.78432i) q^{64} -2.70659i q^{65} +(-0.602705 - 0.223348i) q^{66} -6.23902i q^{67} +(-3.34565 - 2.87436i) q^{68} -4.66817 q^{69} +(-2.29530 + 6.19388i) q^{70} -12.4867 q^{71} +(-6.01466 + 3.33392i) q^{72} -8.27349 q^{73} +(3.39730 - 9.16762i) q^{74} +0.480169i q^{75} +(8.50550 - 1.91219i) q^{76} +1.34769i q^{77} +(1.29574 + 0.480169i) q^{78} -15.4017 q^{79} +(8.25997 - 1.25895i) q^{80} +4.20541 q^{81} +(-8.67079 - 3.21318i) q^{82} +2.00000 q^{83} +(-2.55802 - 2.19768i) q^{84} -4.60675i q^{85} +(-0.0942954 + 0.254457i) q^{86} -6.19040i q^{87} +(1.49098 - 0.826448i) q^{88} -7.44762i q^{89} +(-6.73477 - 2.49574i) q^{90} -2.89736i q^{91} +(8.06808 - 9.39097i) q^{92} +3.73098i q^{93} +(0.296179 + 0.109757i) q^{94} +(7.30375 + 5.43663i) q^{95} +(-0.862674 + 4.17768i) q^{96} +17.6018i q^{97} +(0.982832 - 2.65218i) q^{98} -1.46538 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 4 q^{4} + 12 q^{6} - 8 q^{9} + 12 q^{17} - 24 q^{19} + 4 q^{20} + 32 q^{24} - 44 q^{25} - 44 q^{26} - 20 q^{28} + 32 q^{30} + 40 q^{35} - 52 q^{36} + 4 q^{38} - 20 q^{42} - 24 q^{43} - 4 q^{44} + 24 q^{49}+ \cdots + 64 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(39\) \(77\) \(97\)
\(\chi(n)\) \(-1\) \(-1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.491416 − 1.32609i 0.347483 − 0.937686i
\(3\) 0.754098i 0.435378i 0.976018 + 0.217689i \(0.0698519\pi\)
−0.976018 + 0.217689i \(0.930148\pi\)
\(4\) −1.51702 − 1.30332i −0.758510 − 0.651661i
\(5\) − 2.08884i − 0.934158i −0.884216 − 0.467079i \(-0.845306\pi\)
0.884216 − 0.467079i \(-0.154694\pi\)
\(6\) 1.00000 + 0.370575i 0.408248 + 0.151287i
\(7\) − 2.23607i − 0.845154i −0.906327 − 0.422577i \(-0.861126\pi\)
0.906327 − 0.422577i \(-0.138874\pi\)
\(8\) −2.47381 + 1.37123i −0.874623 + 0.484803i
\(9\) 2.43134 0.810446
\(10\) −2.76999 − 1.02649i −0.875947 − 0.324604i
\(11\) −0.602705 −0.181722 −0.0908612 − 0.995864i \(-0.528962\pi\)
−0.0908612 + 0.995864i \(0.528962\pi\)
\(12\) 0.982832 − 1.14398i 0.283719 − 0.330239i
\(13\) 1.29574 0.359373 0.179687 − 0.983724i \(-0.442492\pi\)
0.179687 + 0.983724i \(0.442492\pi\)
\(14\) −2.96522 − 1.09884i −0.792489 − 0.293677i
\(15\) 1.57519 0.406712
\(16\) 0.602705 + 3.95433i 0.150676 + 0.988583i
\(17\) 2.20541 0.534890 0.267445 − 0.963573i \(-0.413821\pi\)
0.267445 + 0.963573i \(0.413821\pi\)
\(18\) 1.19480 − 3.22417i 0.281616 − 0.759944i
\(19\) −2.60270 + 3.49656i −0.597101 + 0.802166i
\(20\) −2.72243 + 3.16881i −0.608754 + 0.708568i
\(21\) 1.68621 0.367962
\(22\) −0.296179 + 0.799240i −0.0631455 + 0.170399i
\(23\) 6.19040i 1.29079i 0.763850 + 0.645394i \(0.223307\pi\)
−0.763850 + 0.645394i \(0.776693\pi\)
\(24\) −1.03404 − 1.86549i −0.211073 − 0.380792i
\(25\) 0.636747 0.127349
\(26\) 0.636747 − 1.71826i 0.124876 − 0.336979i
\(27\) 4.09576i 0.788229i
\(28\) −2.91432 + 3.39216i −0.550754 + 0.641058i
\(29\) −8.20902 −1.52438 −0.762188 − 0.647355i \(-0.775875\pi\)
−0.762188 + 0.647355i \(0.775875\pi\)
\(30\) 0.774073 − 2.08884i 0.141326 − 0.381368i
\(31\) 4.94762 0.888618 0.444309 − 0.895874i \(-0.353449\pi\)
0.444309 + 0.895874i \(0.353449\pi\)
\(32\) 5.53997 + 1.14398i 0.979338 + 0.202229i
\(33\) − 0.454498i − 0.0791180i
\(34\) 1.08377 − 2.92457i 0.185866 − 0.501559i
\(35\) −4.67079 −0.789507
\(36\) −3.68839 − 3.16881i −0.614731 − 0.528136i
\(37\) 6.91328 1.13654 0.568268 − 0.822843i \(-0.307614\pi\)
0.568268 + 0.822843i \(0.307614\pi\)
\(38\) 3.35774 + 5.16968i 0.544697 + 0.838633i
\(39\) 0.977114i 0.156463i
\(40\) 2.86428 + 5.16739i 0.452883 + 0.817036i
\(41\) − 6.53862i − 1.02116i −0.859830 − 0.510580i \(-0.829431\pi\)
0.859830 − 0.510580i \(-0.170569\pi\)
\(42\) 0.828632 − 2.23607i 0.127861 − 0.345033i
\(43\) −0.191885 −0.0292622 −0.0146311 − 0.999893i \(-0.504657\pi\)
−0.0146311 + 0.999893i \(0.504657\pi\)
\(44\) 0.914316 + 0.785518i 0.137838 + 0.118421i
\(45\) − 5.07867i − 0.757084i
\(46\) 8.20902 + 3.04206i 1.21035 + 0.448527i
\(47\) 0.223348i 0.0325786i 0.999867 + 0.0162893i \(0.00518528\pi\)
−0.999867 + 0.0162893i \(0.994815\pi\)
\(48\) −2.98195 + 0.454498i −0.430408 + 0.0656012i
\(49\) 2.00000 0.285714
\(50\) 0.312907 − 0.844383i 0.0442518 − 0.119414i
\(51\) 1.66309i 0.232880i
\(52\) −1.96566 − 1.68877i −0.272588 − 0.234190i
\(53\) −4.83659 −0.664357 −0.332178 − 0.943217i \(-0.607784\pi\)
−0.332178 + 0.943217i \(0.607784\pi\)
\(54\) 5.43134 + 2.01272i 0.739111 + 0.273897i
\(55\) 1.25895i 0.169757i
\(56\) 3.06617 + 5.53160i 0.409734 + 0.739192i
\(57\) −2.63675 − 1.96269i −0.349246 − 0.259965i
\(58\) −4.03404 + 10.8859i −0.529696 + 1.42939i
\(59\) − 5.60395i − 0.729573i −0.931091 − 0.364786i \(-0.881142\pi\)
0.931091 − 0.364786i \(-0.118858\pi\)
\(60\) −2.38960 − 2.05298i −0.308495 − 0.265038i
\(61\) 11.7743i 1.50754i 0.657138 + 0.753770i \(0.271767\pi\)
−0.657138 + 0.753770i \(0.728233\pi\)
\(62\) 2.43134 − 6.56098i 0.308780 − 0.833245i
\(63\) − 5.43663i − 0.684952i
\(64\) 4.23945 − 6.78432i 0.529931 − 0.848040i
\(65\) − 2.70659i − 0.335711i
\(66\) −0.602705 − 0.223348i −0.0741878 − 0.0274922i
\(67\) − 6.23902i − 0.762218i −0.924530 − 0.381109i \(-0.875542\pi\)
0.924530 − 0.381109i \(-0.124458\pi\)
\(68\) −3.34565 − 2.87436i −0.405720 − 0.348567i
\(69\) −4.66817 −0.561981
\(70\) −2.29530 + 6.19388i −0.274341 + 0.740310i
\(71\) −12.4867 −1.48190 −0.740950 − 0.671560i \(-0.765624\pi\)
−0.740950 + 0.671560i \(0.765624\pi\)
\(72\) −6.01466 + 3.33392i −0.708835 + 0.392907i
\(73\) −8.27349 −0.968339 −0.484170 − 0.874974i \(-0.660878\pi\)
−0.484170 + 0.874974i \(0.660878\pi\)
\(74\) 3.39730 − 9.16762i 0.394928 − 1.06571i
\(75\) 0.480169i 0.0554452i
\(76\) 8.50550 − 1.91219i 0.975648 − 0.219343i
\(77\) 1.34769i 0.153583i
\(78\) 1.29574 + 0.480169i 0.146714 + 0.0543685i
\(79\) −15.4017 −1.73283 −0.866416 − 0.499323i \(-0.833582\pi\)
−0.866416 + 0.499323i \(0.833582\pi\)
\(80\) 8.25997 − 1.25895i 0.923493 − 0.140755i
\(81\) 4.20541 0.467268
\(82\) −8.67079 − 3.21318i −0.957528 − 0.354837i
\(83\) 2.00000 0.219529 0.109764 − 0.993958i \(-0.464990\pi\)
0.109764 + 0.993958i \(0.464990\pi\)
\(84\) −2.55802 − 2.19768i −0.279103 − 0.239786i
\(85\) − 4.60675i − 0.499672i
\(86\) −0.0942954 + 0.254457i −0.0101681 + 0.0274388i
\(87\) − 6.19040i − 0.663681i
\(88\) 1.49098 − 0.826448i 0.158939 − 0.0880996i
\(89\) − 7.44762i − 0.789446i −0.918800 − 0.394723i \(-0.870841\pi\)
0.918800 − 0.394723i \(-0.129159\pi\)
\(90\) −6.73477 − 2.49574i −0.709907 − 0.263074i
\(91\) − 2.89736i − 0.303726i
\(92\) 8.06808 − 9.39097i 0.841156 − 0.979076i
\(93\) 3.73098i 0.386885i
\(94\) 0.296179 + 0.109757i 0.0305485 + 0.0113205i
\(95\) 7.30375 + 5.43663i 0.749349 + 0.557787i
\(96\) −0.862674 + 4.17768i −0.0880463 + 0.426383i
\(97\) 17.6018i 1.78719i 0.448870 + 0.893597i \(0.351827\pi\)
−0.448870 + 0.893597i \(0.648173\pi\)
\(98\) 0.982832 − 2.65218i 0.0992810 − 0.267910i
\(99\) −1.46538 −0.147276
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.b.c.75.7 yes 12
3.2 odd 2 1368.2.e.e.379.6 12
4.3 odd 2 608.2.b.c.303.5 12
8.3 odd 2 inner 152.2.b.c.75.5 ✓ 12
8.5 even 2 608.2.b.c.303.6 12
12.11 even 2 5472.2.e.e.5167.8 12
19.18 odd 2 inner 152.2.b.c.75.6 yes 12
24.5 odd 2 5472.2.e.e.5167.5 12
24.11 even 2 1368.2.e.e.379.8 12
57.56 even 2 1368.2.e.e.379.7 12
76.75 even 2 608.2.b.c.303.7 12
152.37 odd 2 608.2.b.c.303.8 12
152.75 even 2 inner 152.2.b.c.75.8 yes 12
228.227 odd 2 5472.2.e.e.5167.7 12
456.227 odd 2 1368.2.e.e.379.5 12
456.341 even 2 5472.2.e.e.5167.6 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
152.2.b.c.75.5 ✓ 12 8.3 odd 2 inner
152.2.b.c.75.6 yes 12 19.18 odd 2 inner
152.2.b.c.75.7 yes 12 1.1 even 1 trivial
152.2.b.c.75.8 yes 12 152.75 even 2 inner
608.2.b.c.303.5 12 4.3 odd 2
608.2.b.c.303.6 12 8.5 even 2
608.2.b.c.303.7 12 76.75 even 2
608.2.b.c.303.8 12 152.37 odd 2
1368.2.e.e.379.5 12 456.227 odd 2
1368.2.e.e.379.6 12 3.2 odd 2
1368.2.e.e.379.7 12 57.56 even 2
1368.2.e.e.379.8 12 24.11 even 2
5472.2.e.e.5167.5 12 24.5 odd 2
5472.2.e.e.5167.6 12 456.341 even 2
5472.2.e.e.5167.7 12 228.227 odd 2
5472.2.e.e.5167.8 12 12.11 even 2