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Properties

Label 152.2.b.c.75.4
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 75.4
Root \(-1.17117 + 0.792696i\) of defining polynomial
Character \(\chi\) \(=\) 152.75
Dual form 152.2.b.c.75.3

$q$-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+(-1.17117 + 0.792696i) q^{2} -1.26152i q^{3} +(0.743268 - 1.85676i) q^{4} +3.51876i q^{5} +(1.00000 + 1.47745i) q^{6} -2.23607i q^{7} +(0.601353 + 2.76376i) q^{8} +1.40857 q^{9} +(-2.78930 - 4.12105i) q^{10} +2.89511 q^{11} +(-2.34234 - 0.937646i) q^{12} +6.30281 q^{13} +(1.77252 + 2.61881i) q^{14} +4.43898 q^{15} +(-2.89511 - 2.76014i) q^{16} -4.79021 q^{17} +(-1.64967 + 1.11657i) q^{18} +(0.895107 + 4.26600i) q^{19} +(6.53348 + 2.61538i) q^{20} -2.82084 q^{21} +(-3.39066 + 2.29494i) q^{22} -0.524069i q^{23} +(3.48654 - 0.758618i) q^{24} -7.38164 q^{25} +(-7.38164 + 4.99621i) q^{26} -5.56149i q^{27} +(-4.15184 - 1.66200i) q^{28} -0.415427 q^{29} +(-5.19878 + 3.51876i) q^{30} -1.20271 q^{31} +(5.57860 + 0.937646i) q^{32} -3.65223i q^{33} +(5.61014 - 3.79718i) q^{34} +7.86818 q^{35} +(1.04695 - 2.61538i) q^{36} -5.88738 q^{37} +(-4.42996 - 4.28666i) q^{38} -7.95110i q^{39} +(-9.72500 + 2.11602i) q^{40} -4.87978i q^{41} +(3.30368 - 2.23607i) q^{42} -10.6853 q^{43} +(2.15184 - 5.37551i) q^{44} +4.95642i q^{45} +(0.415427 + 0.613773i) q^{46} -4.27737i q^{47} +(-3.48196 + 3.65223i) q^{48} +2.00000 q^{49} +(8.64514 - 5.85139i) q^{50} +6.04294i q^{51} +(4.68467 - 11.7028i) q^{52} -6.05711 q^{53} +(4.40857 + 6.51344i) q^{54} +10.1872i q^{55} +(6.17996 - 1.34467i) q^{56} +(5.38164 - 1.12919i) q^{57} +(0.486535 - 0.329307i) q^{58} +8.08453i q^{59} +(3.29935 - 8.24211i) q^{60} +8.38042i q^{61} +(1.40857 - 0.953380i) q^{62} -3.14966i q^{63} +(-7.27675 + 3.32399i) q^{64} +22.1780i q^{65} +(2.89511 + 4.27737i) q^{66} -9.79353i q^{67} +(-3.56041 + 8.89427i) q^{68} -0.661123 q^{69} +(-9.21495 + 6.23707i) q^{70} -10.2002 q^{71} +(0.847049 + 3.89295i) q^{72} +7.76328 q^{73} +(6.89511 - 4.66690i) q^{74} +9.31208i q^{75} +(8.58624 + 1.50879i) q^{76} -6.47365i q^{77} +(6.30281 + 9.31208i) q^{78} +7.33578 q^{79} +(9.71225 - 10.1872i) q^{80} -2.79021 q^{81} +(3.86818 + 5.71504i) q^{82} +2.00000 q^{83} +(-2.09664 + 5.23762i) q^{84} -16.8556i q^{85} +(12.5143 - 8.47020i) q^{86} +0.524069i q^{87} +(1.74098 + 8.00138i) q^{88} -12.1842i q^{89} +(-3.92893 - 5.80480i) q^{90} -14.0935i q^{91} +(-0.973070 - 0.389524i) q^{92} +1.51724i q^{93} +(3.39066 + 5.00952i) q^{94} +(-15.0110 + 3.14966i) q^{95} +(1.18286 - 7.03751i) q^{96} +2.19044i q^{97} +(-2.34234 + 1.58539i) q^{98} +4.07796 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.17117 + 0.792696i −0.828141 + 0.560520i
\(3\) − 1.26152i − 0.728338i −0.931333 − 0.364169i \(-0.881353\pi\)
0.931333 − 0.364169i \(-0.118647\pi\)
\(4\) 0.743268 − 1.85676i 0.371634 − 0.928379i
\(5\) 3.51876i 1.57364i 0.617185 + 0.786818i \(0.288273\pi\)
−0.617185 + 0.786818i \(0.711727\pi\)
\(6\) 1.00000 + 1.47745i 0.408248 + 0.603166i
\(7\) − 2.23607i − 0.845154i −0.906327 − 0.422577i \(-0.861126\pi\)
0.906327 − 0.422577i \(-0.138874\pi\)
\(8\) 0.601353 + 2.76376i 0.212611 + 0.977137i
\(9\) 1.40857 0.469524
\(10\) −2.78930 − 4.12105i −0.882055 − 1.30319i
\(11\) 2.89511 0.872907 0.436454 − 0.899727i \(-0.356234\pi\)
0.436454 + 0.899727i \(0.356234\pi\)
\(12\) −2.34234 − 0.937646i −0.676174 − 0.270675i
\(13\) 6.30281 1.74808 0.874042 − 0.485851i \(-0.161490\pi\)
0.874042 + 0.485851i \(0.161490\pi\)
\(14\) 1.77252 + 2.61881i 0.473726 + 0.699907i
\(15\) 4.43898 1.14614
\(16\) −2.89511 − 2.76014i −0.723777 − 0.690034i
\(17\) −4.79021 −1.16180 −0.580899 − 0.813976i \(-0.697299\pi\)
−0.580899 + 0.813976i \(0.697299\pi\)
\(18\) −1.64967 + 1.11657i −0.388832 + 0.263178i
\(19\) 0.895107 + 4.26600i 0.205352 + 0.978688i
\(20\) 6.53348 + 2.61538i 1.46093 + 0.584816i
\(21\) −2.82084 −0.615558
\(22\) −3.39066 + 2.29494i −0.722890 + 0.489282i
\(23\) − 0.524069i − 0.109276i −0.998506 − 0.0546380i \(-0.982600\pi\)
0.998506 − 0.0546380i \(-0.0174005\pi\)
\(24\) 3.48654 − 0.758618i 0.711686 − 0.154852i
\(25\) −7.38164 −1.47633
\(26\) −7.38164 + 4.99621i −1.44766 + 0.979836i
\(27\) − 5.56149i − 1.07031i
\(28\) −4.15184 − 1.66200i −0.784624 − 0.314088i
\(29\) −0.415427 −0.0771429 −0.0385715 − 0.999256i \(-0.512281\pi\)
−0.0385715 + 0.999256i \(0.512281\pi\)
\(30\) −5.19878 + 3.51876i −0.949164 + 0.642434i
\(31\) −1.20271 −0.216013 −0.108006 − 0.994150i \(-0.534447\pi\)
−0.108006 + 0.994150i \(0.534447\pi\)
\(32\) 5.57860 + 0.937646i 0.986167 + 0.165754i
\(33\) − 3.65223i − 0.635772i
\(34\) 5.61014 − 3.79718i 0.962132 − 0.651211i
\(35\) 7.86818 1.32996
\(36\) 1.04695 − 2.61538i 0.174491 − 0.435896i
\(37\) −5.88738 −0.967879 −0.483939 − 0.875101i \(-0.660795\pi\)
−0.483939 + 0.875101i \(0.660795\pi\)
\(38\) −4.42996 − 4.28666i −0.718635 − 0.695388i
\(39\) − 7.95110i − 1.27320i
\(40\) −9.72500 + 2.11602i −1.53766 + 0.334571i
\(41\) − 4.87978i − 0.762093i −0.924556 − 0.381047i \(-0.875564\pi\)
0.924556 − 0.381047i \(-0.124436\pi\)
\(42\) 3.30368 − 2.23607i 0.509769 − 0.345033i
\(43\) −10.6853 −1.62950 −0.814748 − 0.579815i \(-0.803125\pi\)
−0.814748 + 0.579815i \(0.803125\pi\)
\(44\) 2.15184 − 5.37551i 0.324402 − 0.810389i
\(45\) 4.95642i 0.738859i
\(46\) 0.415427 + 0.613773i 0.0612514 + 0.0904959i
\(47\) − 4.27737i − 0.623919i −0.950095 − 0.311960i \(-0.899015\pi\)
0.950095 − 0.311960i \(-0.100985\pi\)
\(48\) −3.48196 + 3.65223i −0.502578 + 0.527154i
\(49\) 2.00000 0.285714
\(50\) 8.64514 − 5.85139i 1.22261 − 0.827512i
\(51\) 6.04294i 0.846181i
\(52\) 4.68467 − 11.7028i 0.649647 − 1.62288i
\(53\) −6.05711 −0.832008 −0.416004 − 0.909363i \(-0.636570\pi\)
−0.416004 + 0.909363i \(0.636570\pi\)
\(54\) 4.40857 + 6.51344i 0.599931 + 0.886367i
\(55\) 10.1872i 1.37364i
\(56\) 6.17996 − 1.34467i 0.825832 − 0.179689i
\(57\) 5.38164 − 1.12919i 0.712816 − 0.149565i
\(58\) 0.486535 − 0.329307i 0.0638852 − 0.0432402i
\(59\) 8.08453i 1.05252i 0.850325 + 0.526258i \(0.176405\pi\)
−0.850325 + 0.526258i \(0.823595\pi\)
\(60\) 3.29935 − 8.24211i 0.425944 − 1.06405i
\(61\) 8.38042i 1.07300i 0.843900 + 0.536501i \(0.180254\pi\)
−0.843900 + 0.536501i \(0.819746\pi\)
\(62\) 1.40857 − 0.953380i 0.178889 − 0.121079i
\(63\) − 3.14966i − 0.396820i
\(64\) −7.27675 + 3.32399i −0.909594 + 0.415499i
\(65\) 22.1780i 2.75085i
\(66\) 2.89511 + 4.27737i 0.356363 + 0.526508i
\(67\) − 9.79353i − 1.19647i −0.801321 − 0.598235i \(-0.795869\pi\)
0.801321 − 0.598235i \(-0.204131\pi\)
\(68\) −3.56041 + 8.89427i −0.431763 + 1.07859i
\(69\) −0.661123 −0.0795899
\(70\) −9.21495 + 6.23707i −1.10140 + 0.745472i
\(71\) −10.2002 −1.21054 −0.605270 − 0.796020i \(-0.706935\pi\)
−0.605270 + 0.796020i \(0.706935\pi\)
\(72\) 0.847049 + 3.89295i 0.0998257 + 0.458789i
\(73\) 7.76328 0.908624 0.454312 − 0.890843i \(-0.349885\pi\)
0.454312 + 0.890843i \(0.349885\pi\)
\(74\) 6.89511 − 4.66690i 0.801540 − 0.542516i
\(75\) 9.31208i 1.07527i
\(76\) 8.58624 + 1.50879i 0.984910 + 0.173070i
\(77\) − 6.47365i − 0.737741i
\(78\) 6.30281 + 9.31208i 0.713652 + 1.05439i
\(79\) 7.33578 0.825340 0.412670 − 0.910881i \(-0.364596\pi\)
0.412670 + 0.910881i \(0.364596\pi\)
\(80\) 9.71225 − 10.1872i 1.08586 − 1.13896i
\(81\) −2.79021 −0.310024
\(82\) 3.86818 + 5.71504i 0.427169 + 0.631120i
\(83\) 2.00000 0.219529 0.109764 − 0.993958i \(-0.464990\pi\)
0.109764 + 0.993958i \(0.464990\pi\)
\(84\) −2.09664 + 5.23762i −0.228762 + 0.571471i
\(85\) − 16.8556i − 1.82825i
\(86\) 12.5143 − 8.47020i 1.34945 − 0.913366i
\(87\) 0.524069i 0.0561861i
\(88\) 1.74098 + 8.00138i 0.185589 + 0.852950i
\(89\) − 12.1842i − 1.29153i −0.763538 − 0.645763i \(-0.776539\pi\)
0.763538 − 0.645763i \(-0.223461\pi\)
\(90\) −3.92893 − 5.80480i −0.414146 − 0.611879i
\(91\) − 14.0935i − 1.47740i
\(92\) −0.973070 − 0.389524i −0.101450 − 0.0406107i
\(93\) 1.51724i 0.157330i
\(94\) 3.39066 + 5.00952i 0.349719 + 0.516693i
\(95\) −15.0110 + 3.14966i −1.54010 + 0.323148i
\(96\) 1.18286 − 7.03751i 0.120725 − 0.718263i
\(97\) 2.19044i 0.222406i 0.993798 + 0.111203i \(0.0354703\pi\)
−0.993798 + 0.111203i \(0.964530\pi\)
\(98\) −2.34234 + 1.58539i −0.236612 + 0.160149i
\(99\) 4.07796 0.409851
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.75.4 yes 12
3.2 odd 2 1368.2.e.e.379.9 12
4.3 odd 2 608.2.b.c.303.10 12
8.3 odd 2 inner 152.2.b.c.75.10 yes 12
8.5 even 2 608.2.b.c.303.9 12
12.11 even 2 5472.2.e.e.5167.2 12
19.18 odd 2 inner 152.2.b.c.75.9 yes 12
24.5 odd 2 5472.2.e.e.5167.11 12
24.11 even 2 1368.2.e.e.379.3 12
57.56 even 2 1368.2.e.e.379.4 12
76.75 even 2 608.2.b.c.303.4 12
152.37 odd 2 608.2.b.c.303.3 12
152.75 even 2 inner 152.2.b.c.75.3 ✓ 12
228.227 odd 2 5472.2.e.e.5167.1 12
456.227 odd 2 1368.2.e.e.379.10 12
456.341 even 2 5472.2.e.e.5167.12 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
152.2.b.c.75.3 ✓ 12 152.75 even 2 inner
152.2.b.c.75.4 yes 12 1.1 even 1 trivial
152.2.b.c.75.9 yes 12 19.18 odd 2 inner
152.2.b.c.75.10 yes 12 8.3 odd 2 inner
608.2.b.c.303.3 12 152.37 odd 2
608.2.b.c.303.4 12 76.75 even 2
608.2.b.c.303.9 12 8.5 even 2
608.2.b.c.303.10 12 4.3 odd 2
1368.2.e.e.379.3 12 24.11 even 2
1368.2.e.e.379.4 12 57.56 even 2
1368.2.e.e.379.9 12 3.2 odd 2
1368.2.e.e.379.10 12 456.227 odd 2
5472.2.e.e.5167.1 12 228.227 odd 2
5472.2.e.e.5167.2 12 12.11 even 2
5472.2.e.e.5167.11 12 24.5 odd 2
5472.2.e.e.5167.12 12 456.341 even 2