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Properties

Label 152.2.b.c.75.11
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 75.11
Root \(1.37364 - 0.336338i\) of defining polynomial
Character \(\chi\) \(=\) 152.75
Dual form 152.2.b.c.75.12

$q$-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+(1.37364 - 0.336338i) q^{2} +2.97320i q^{3} +(1.77375 - 0.924013i) q^{4} -3.04222i q^{5} +(1.00000 + 4.08409i) q^{6} +2.23607i q^{7} +(2.12571 - 1.86584i) q^{8} -5.83991 q^{9} +(-1.02321 - 4.17890i) q^{10} -2.29240 q^{11} +(2.74727 + 5.27372i) q^{12} -3.09769 q^{13} +(0.752075 + 3.07154i) q^{14} +9.04512 q^{15} +(2.29240 - 3.27794i) q^{16} +5.58480 q^{17} +(-8.02191 + 1.96418i) q^{18} +(-4.29240 - 0.758478i) q^{19} +(-2.81105 - 5.39615i) q^{20} -6.64827 q^{21} +(-3.14893 + 0.771022i) q^{22} -5.51401i q^{23} +(5.54751 + 6.32016i) q^{24} -4.25511 q^{25} +(-4.25511 + 1.04187i) q^{26} -8.44361i q^{27} +(2.06615 + 3.96623i) q^{28} +1.85457 q^{29} +(12.4247 - 3.04222i) q^{30} -4.25142 q^{31} +(2.04643 - 5.27372i) q^{32} -6.81576i q^{33} +(7.67149 - 1.87838i) q^{34} +6.80261 q^{35} +(-10.3586 + 5.39615i) q^{36} +1.24312 q^{37} +(-6.15130 + 0.401826i) q^{38} -9.21006i q^{39} +(-5.67629 - 6.46688i) q^{40} +8.33272i q^{41} +(-9.13231 + 2.23607i) q^{42} +4.87720 q^{43} +(-4.06615 + 2.11821i) q^{44} +17.7663i q^{45} +(-1.85457 - 7.57424i) q^{46} +9.36238i q^{47} +(9.74597 + 6.81576i) q^{48} +2.00000 q^{49} +(-5.84497 + 1.43115i) q^{50} +16.6047i q^{51} +(-5.49455 + 2.86231i) q^{52} -11.4420 q^{53} +(-2.83991 - 11.5984i) q^{54} +6.97399i q^{55} +(4.17214 + 4.75323i) q^{56} +(2.25511 - 12.7622i) q^{57} +(2.54751 - 0.623763i) q^{58} +2.49721i q^{59} +(16.0438 - 8.35781i) q^{60} +2.26613i q^{61} +(-5.83991 + 1.42992i) q^{62} -13.0584i q^{63} +(1.03730 - 7.93247i) q^{64} +9.42387i q^{65} +(-2.29240 - 9.36238i) q^{66} +4.49015i q^{67} +(9.90606 - 5.16043i) q^{68} +16.3942 q^{69} +(9.34431 - 2.28798i) q^{70} +14.6982 q^{71} +(-12.4140 + 10.8963i) q^{72} +1.51021 q^{73} +(1.70760 - 0.418110i) q^{74} -12.6513i q^{75} +(-8.31450 + 2.62088i) q^{76} -5.12597i q^{77} +(-3.09769 - 12.6513i) q^{78} -11.5314 q^{79} +(-9.97222 - 6.97399i) q^{80} +7.58480 q^{81} +(2.80261 + 11.4461i) q^{82} +2.00000 q^{83} +(-11.7924 + 6.14309i) q^{84} -16.9902i q^{85} +(6.69951 - 1.64039i) q^{86} +5.51401i q^{87} +(-4.87298 + 4.27725i) q^{88} -5.29881i q^{89} +(5.97548 + 24.4044i) q^{90} -6.92665i q^{91} +(-5.09501 - 9.78049i) q^{92} -12.6403i q^{93} +(3.14893 + 12.8605i) q^{94} +(-2.30746 + 13.0584i) q^{95} +(15.6798 + 6.08444i) q^{96} +1.17375i q^{97} +(2.74727 - 0.672676i) q^{98} +13.3874 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.37364 − 0.336338i 0.971308 − 0.237827i
\(3\) 2.97320i 1.71658i 0.513168 + 0.858288i \(0.328472\pi\)
−0.513168 + 0.858288i \(0.671528\pi\)
\(4\) 1.77375 − 0.924013i 0.886877 − 0.462006i
\(5\) − 3.04222i − 1.36052i −0.732970 − 0.680261i \(-0.761866\pi\)
0.732970 − 0.680261i \(-0.238134\pi\)
\(6\) 1.00000 + 4.08409i 0.408248 + 1.66732i
\(7\) 2.23607i 0.845154i 0.906327 + 0.422577i \(0.138874\pi\)
−0.906327 + 0.422577i \(0.861126\pi\)
\(8\) 2.12571 − 1.86584i 0.751552 − 0.659673i
\(9\) −5.83991 −1.94664
\(10\) −1.02321 − 4.17890i −0.323569 − 1.32149i
\(11\) −2.29240 −0.691185 −0.345593 − 0.938385i \(-0.612322\pi\)
−0.345593 + 0.938385i \(0.612322\pi\)
\(12\) 2.74727 + 5.27372i 0.793069 + 1.52239i
\(13\) −3.09769 −0.859146 −0.429573 − 0.903032i \(-0.641336\pi\)
−0.429573 + 0.903032i \(0.641336\pi\)
\(14\) 0.752075 + 3.07154i 0.201000 + 0.820905i
\(15\) 9.04512 2.33544
\(16\) 2.29240 − 3.27794i 0.573100 − 0.819485i
\(17\) 5.58480 1.35451 0.677257 − 0.735747i \(-0.263169\pi\)
0.677257 + 0.735747i \(0.263169\pi\)
\(18\) −8.02191 + 1.96418i −1.89078 + 0.462963i
\(19\) −4.29240 − 0.758478i −0.984744 − 0.174007i
\(20\) −2.81105 − 5.39615i −0.628570 − 1.20662i
\(21\) −6.64827 −1.45077
\(22\) −3.14893 + 0.771022i −0.671353 + 0.164382i
\(23\) − 5.51401i − 1.14975i −0.818241 − 0.574875i \(-0.805051\pi\)
0.818241 − 0.574875i \(-0.194949\pi\)
\(24\) 5.54751 + 6.32016i 1.13238 + 1.29010i
\(25\) −4.25511 −0.851021
\(26\) −4.25511 + 1.04187i −0.834495 + 0.204328i
\(27\) − 8.44361i − 1.62497i
\(28\) 2.06615 + 3.96623i 0.390467 + 0.749548i
\(29\) 1.85457 0.344385 0.172193 − 0.985063i \(-0.444915\pi\)
0.172193 + 0.985063i \(0.444915\pi\)
\(30\) 12.4247 − 3.04222i 2.26843 − 0.555431i
\(31\) −4.25142 −0.763578 −0.381789 − 0.924250i \(-0.624692\pi\)
−0.381789 + 0.924250i \(0.624692\pi\)
\(32\) 2.04643 − 5.27372i 0.361761 − 0.932271i
\(33\) − 6.81576i − 1.18647i
\(34\) 7.67149 − 1.87838i 1.31565 − 0.322140i
\(35\) 6.80261 1.14985
\(36\) −10.3586 + 5.39615i −1.72643 + 0.899358i
\(37\) 1.24312 0.204368 0.102184 − 0.994766i \(-0.467417\pi\)
0.102184 + 0.994766i \(0.467417\pi\)
\(38\) −6.15130 + 0.401826i −0.997873 + 0.0651847i
\(39\) − 9.21006i − 1.47479i
\(40\) −5.67629 − 6.46688i −0.897500 − 1.02250i
\(41\) 8.33272i 1.30135i 0.759355 + 0.650676i \(0.225514\pi\)
−0.759355 + 0.650676i \(0.774486\pi\)
\(42\) −9.13231 + 2.23607i −1.40915 + 0.345033i
\(43\) 4.87720 0.743767 0.371883 − 0.928279i \(-0.378712\pi\)
0.371883 + 0.928279i \(0.378712\pi\)
\(44\) −4.06615 + 2.11821i −0.612996 + 0.319332i
\(45\) 17.7663i 2.64844i
\(46\) −1.85457 − 7.57424i −0.273442 − 1.11676i
\(47\) 9.36238i 1.36564i 0.730585 + 0.682822i \(0.239247\pi\)
−0.730585 + 0.682822i \(0.760753\pi\)
\(48\) 9.74597 + 6.81576i 1.40671 + 0.983771i
\(49\) 2.00000 0.285714
\(50\) −5.84497 + 1.43115i −0.826603 + 0.202396i
\(51\) 16.6047i 2.32513i
\(52\) −5.49455 + 2.86231i −0.761956 + 0.396931i
\(53\) −11.4420 −1.57168 −0.785838 − 0.618432i \(-0.787768\pi\)
−0.785838 + 0.618432i \(0.787768\pi\)
\(54\) −2.83991 − 11.5984i −0.386463 − 1.57835i
\(55\) 6.97399i 0.940373i
\(56\) 4.17214 + 4.75323i 0.557526 + 0.635178i
\(57\) 2.25511 − 12.7622i 0.298696 − 1.69039i
\(58\) 2.54751 − 0.623763i 0.334504 − 0.0819041i
\(59\) 2.49721i 0.325110i 0.986700 + 0.162555i \(0.0519734\pi\)
−0.986700 + 0.162555i \(0.948027\pi\)
\(60\) 16.0438 − 8.35781i 2.07125 − 1.07899i
\(61\) 2.26613i 0.290149i 0.989421 + 0.145074i \(0.0463422\pi\)
−0.989421 + 0.145074i \(0.953658\pi\)
\(62\) −5.83991 + 1.42992i −0.741669 + 0.181599i
\(63\) − 13.0584i − 1.64521i
\(64\) 1.03730 − 7.93247i 0.129662 − 0.991558i
\(65\) 9.42387i 1.16889i
\(66\) −2.29240 − 9.36238i −0.282175 − 1.15243i
\(67\) 4.49015i 0.548560i 0.961650 + 0.274280i \(0.0884394\pi\)
−0.961650 + 0.274280i \(0.911561\pi\)
\(68\) 9.90606 − 5.16043i 1.20129 − 0.625794i
\(69\) 16.3942 1.97363
\(70\) 9.34431 − 2.28798i 1.11686 − 0.273466i
\(71\) 14.6982 1.74436 0.872180 − 0.489186i \(-0.162706\pi\)
0.872180 + 0.489186i \(0.162706\pi\)
\(72\) −12.4140 + 10.8963i −1.46300 + 1.28414i
\(73\) 1.51021 0.176757 0.0883784 − 0.996087i \(-0.471832\pi\)
0.0883784 + 0.996087i \(0.471832\pi\)
\(74\) 1.70760 − 0.418110i 0.198504 − 0.0486043i
\(75\) − 12.6513i − 1.46084i
\(76\) −8.31450 + 2.62088i −0.953739 + 0.300636i
\(77\) − 5.12597i − 0.584158i
\(78\) −3.09769 − 12.6513i −0.350745 − 1.43247i
\(79\) −11.5314 −1.29738 −0.648690 − 0.761053i \(-0.724683\pi\)
−0.648690 + 0.761053i \(0.724683\pi\)
\(80\) −9.97222 − 6.97399i −1.11493 − 0.779716i
\(81\) 7.58480 0.842756
\(82\) 2.80261 + 11.4461i 0.309497 + 1.26401i
\(83\) 2.00000 0.219529 0.109764 − 0.993958i \(-0.464990\pi\)
0.109764 + 0.993958i \(0.464990\pi\)
\(84\) −11.7924 + 6.14309i −1.28666 + 0.670266i
\(85\) − 16.9902i − 1.84285i
\(86\) 6.69951 − 1.64039i 0.722426 − 0.176888i
\(87\) 5.51401i 0.591164i
\(88\) −4.87298 + 4.27725i −0.519462 + 0.455956i
\(89\) − 5.29881i − 0.561673i −0.959756 − 0.280836i \(-0.909388\pi\)
0.959756 − 0.280836i \(-0.0906118\pi\)
\(90\) 5.97548 + 24.4044i 0.629871 + 2.57245i
\(91\) − 6.92665i − 0.726111i
\(92\) −5.09501 − 9.78049i −0.531192 − 1.01969i
\(93\) − 12.6403i − 1.31074i
\(94\) 3.14893 + 12.8605i 0.324787 + 1.32646i
\(95\) −2.30746 + 13.0584i −0.236740 + 1.33977i
\(96\) 15.6798 + 6.08444i 1.60031 + 0.620991i
\(97\) 1.17375i 0.119176i 0.998223 + 0.0595881i \(0.0189787\pi\)
−0.998223 + 0.0595881i \(0.981021\pi\)
\(98\) 2.74727 − 0.672676i 0.277516 − 0.0679506i
\(99\) 13.3874 1.34549
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.75.11 yes 12
3.2 odd 2 1368.2.e.e.379.2 12
4.3 odd 2 608.2.b.c.303.1 12
8.3 odd 2 inner 152.2.b.c.75.1 ✓ 12
8.5 even 2 608.2.b.c.303.2 12
12.11 even 2 5472.2.e.e.5167.9 12
19.18 odd 2 inner 152.2.b.c.75.2 yes 12
24.5 odd 2 5472.2.e.e.5167.4 12
24.11 even 2 1368.2.e.e.379.12 12
57.56 even 2 1368.2.e.e.379.11 12
76.75 even 2 608.2.b.c.303.11 12
152.37 odd 2 608.2.b.c.303.12 12
152.75 even 2 inner 152.2.b.c.75.12 yes 12
228.227 odd 2 5472.2.e.e.5167.10 12
456.227 odd 2 1368.2.e.e.379.1 12
456.341 even 2 5472.2.e.e.5167.3 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
152.2.b.c.75.1 ✓ 12 8.3 odd 2 inner
152.2.b.c.75.2 yes 12 19.18 odd 2 inner
152.2.b.c.75.11 yes 12 1.1 even 1 trivial
152.2.b.c.75.12 yes 12 152.75 even 2 inner
608.2.b.c.303.1 12 4.3 odd 2
608.2.b.c.303.2 12 8.5 even 2
608.2.b.c.303.11 12 76.75 even 2
608.2.b.c.303.12 12 152.37 odd 2
1368.2.e.e.379.1 12 456.227 odd 2
1368.2.e.e.379.2 12 3.2 odd 2
1368.2.e.e.379.11 12 57.56 even 2
1368.2.e.e.379.12 12 24.11 even 2
5472.2.e.e.5167.3 12 456.341 even 2
5472.2.e.e.5167.4 12 24.5 odd 2
5472.2.e.e.5167.9 12 12.11 even 2
5472.2.e.e.5167.10 12 228.227 odd 2