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Properties

Label 1368.2.e.e.379.11
Level $1368$
Weight $2$
Character 1368.379
Analytic conductor $10.924$
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] = [1368,2,Mod(379,1368)] 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("1368.379"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1368, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 1, 0, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 1368 = 2^{3} \cdot 3^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1368.e (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [12,0,0,4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(4)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(10.9235349965\)
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: \(\Z[a_1, \ldots, a_{29}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: no (minimal twist has level 152)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 379.11
Root \(-1.37364 + 0.336338i\) of defining polynomial
Character \(\chi\) \(=\) 1368.379
Dual form 1368.2.e.e.379.12

$q$-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)
 
\(f(q)\) \(=\) \(q+(1.37364 - 0.336338i) q^{2} +(1.77375 - 0.924013i) q^{4} +3.04222i q^{5} +2.23607i q^{7} +(2.12571 - 1.86584i) q^{8} +(1.02321 + 4.17890i) q^{10} +2.29240 q^{11} +3.09769 q^{13} +(0.752075 + 3.07154i) q^{14} +(2.29240 - 3.27794i) q^{16} -5.58480 q^{17} +(-4.29240 + 0.758478i) q^{19} +(2.81105 + 5.39615i) q^{20} +(3.14893 - 0.771022i) q^{22} +5.51401i q^{23} -4.25511 q^{25} +(4.25511 - 1.04187i) q^{26} +(2.06615 + 3.96623i) q^{28} +1.85457 q^{29} +4.25142 q^{31} +(2.04643 - 5.27372i) q^{32} +(-7.67149 + 1.87838i) q^{34} -6.80261 q^{35} -1.24312 q^{37} +(-5.64109 + 2.48557i) q^{38} +(5.67629 + 6.46688i) q^{40} +8.33272i q^{41} +4.87720 q^{43} +(4.06615 - 2.11821i) q^{44} +(1.85457 + 7.57424i) q^{46} -9.36238i q^{47} +2.00000 q^{49} +(-5.84497 + 1.43115i) q^{50} +(5.49455 - 2.86231i) q^{52} -11.4420 q^{53} +6.97399i q^{55} +(4.17214 + 4.75323i) q^{56} +(2.54751 - 0.623763i) q^{58} +2.49721i q^{59} +2.26613i q^{61} +(5.83991 - 1.42992i) q^{62} +(1.03730 - 7.93247i) q^{64} +9.42387i q^{65} -4.49015i q^{67} +(-9.90606 + 5.16043i) q^{68} +(-9.34431 + 2.28798i) q^{70} +14.6982 q^{71} +1.51021 q^{73} +(-1.70760 + 0.418110i) q^{74} +(-6.91282 + 5.31159i) q^{76} +5.12597i q^{77} +11.5314 q^{79} +(9.97222 + 6.97399i) q^{80} +(2.80261 + 11.4461i) q^{82} -2.00000 q^{83} -16.9902i q^{85} +(6.69951 - 1.64039i) q^{86} +(4.87298 - 4.27725i) q^{88} -5.29881i q^{89} +6.92665i q^{91} +(5.09501 + 9.78049i) q^{92} +(-3.14893 - 12.8605i) q^{94} +(-2.30746 - 13.0584i) q^{95} -1.17375i q^{97} +(2.74727 - 0.672676i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 4 q^{4} - 12 q^{17} - 24 q^{19} - 4 q^{20} - 44 q^{25} + 44 q^{26} - 20 q^{28} - 40 q^{35} - 4 q^{38} - 24 q^{43} + 4 q^{44} + 24 q^{49} - 4 q^{58} + 8 q^{62} - 8 q^{64} - 12 q^{68} + 4 q^{73} - 48 q^{74}+ \cdots - 8 q^{92}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1368\mathbb{Z}\right)^\times\).

\(n\) \(343\) \(685\) \(1009\) \(1217\)
\(\chi(n)\) \(-1\) \(-1\) \(-1\) \(1\)

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
\(3\) 0 0
\(4\) 1.77375 − 0.924013i 0.886877 − 0.462006i
\(5\) 3.04222i 1.36052i 0.732970 + 0.680261i \(0.238134\pi\)
−0.732970 + 0.680261i \(0.761866\pi\)
\(6\) 0 0
\(7\) 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
\(9\) 0 0
\(10\) 1.02321 + 4.17890i 0.323569 + 1.32149i
\(11\) 2.29240 0.691185 0.345593 − 0.938385i \(-0.387678\pi\)
0.345593 + 0.938385i \(0.387678\pi\)
\(12\) 0 0
\(13\) 3.09769 0.859146 0.429573 − 0.903032i \(-0.358664\pi\)
0.429573 + 0.903032i \(0.358664\pi\)
\(14\) 0.752075 + 3.07154i 0.201000 + 0.820905i
\(15\) 0 0
\(16\) 2.29240 − 3.27794i 0.573100 − 0.819485i
\(17\) −5.58480 −1.35451 −0.677257 − 0.735747i \(-0.736831\pi\)
−0.677257 + 0.735747i \(0.736831\pi\)
\(18\) 0 0
\(19\) −4.29240 + 0.758478i −0.984744 + 0.174007i
\(20\) 2.81105 + 5.39615i 0.628570 + 1.20662i
\(21\) 0 0
\(22\) 3.14893 − 0.771022i 0.671353 − 0.164382i
\(23\) 5.51401i 1.14975i 0.818241 + 0.574875i \(0.194949\pi\)
−0.818241 + 0.574875i \(0.805051\pi\)
\(24\) 0 0
\(25\) −4.25511 −0.851021
\(26\) 4.25511 − 1.04187i 0.834495 − 0.204328i
\(27\) 0 0
\(28\) 2.06615 + 3.96623i 0.390467 + 0.749548i
\(29\) 1.85457 0.344385 0.172193 − 0.985063i \(-0.444915\pi\)
0.172193 + 0.985063i \(0.444915\pi\)
\(30\) 0 0
\(31\) 4.25142 0.763578 0.381789 − 0.924250i \(-0.375308\pi\)
0.381789 + 0.924250i \(0.375308\pi\)
\(32\) 2.04643 − 5.27372i 0.361761 − 0.932271i
\(33\) 0 0
\(34\) −7.67149 + 1.87838i −1.31565 + 0.322140i
\(35\) −6.80261 −1.14985
\(36\) 0 0
\(37\) −1.24312 −0.204368 −0.102184 − 0.994766i \(-0.532583\pi\)
−0.102184 + 0.994766i \(0.532583\pi\)
\(38\) −5.64109 + 2.48557i −0.915106 + 0.403213i
\(39\) 0 0
\(40\) 5.67629 + 6.46688i 0.897500 + 1.02250i
\(41\) 8.33272i 1.30135i 0.759355 + 0.650676i \(0.225514\pi\)
−0.759355 + 0.650676i \(0.774486\pi\)
\(42\) 0 0
\(43\) 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
\(45\) 0 0
\(46\) 1.85457 + 7.57424i 0.273442 + 1.11676i
\(47\) − 9.36238i − 1.36564i −0.730585 − 0.682822i \(-0.760753\pi\)
0.730585 − 0.682822i \(-0.239247\pi\)
\(48\) 0 0
\(49\) 2.00000 0.285714
\(50\) −5.84497 + 1.43115i −0.826603 + 0.202396i
\(51\) 0 0
\(52\) 5.49455 − 2.86231i 0.761956 − 0.396931i
\(53\) −11.4420 −1.57168 −0.785838 − 0.618432i \(-0.787768\pi\)
−0.785838 + 0.618432i \(0.787768\pi\)
\(54\) 0 0
\(55\) 6.97399i 0.940373i
\(56\) 4.17214 + 4.75323i 0.557526 + 0.635178i
\(57\) 0 0
\(58\) 2.54751 − 0.623763i 0.334504 − 0.0819041i
\(59\) 2.49721i 0.325110i 0.986700 + 0.162555i \(0.0519734\pi\)
−0.986700 + 0.162555i \(0.948027\pi\)
\(60\) 0 0
\(61\) 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
\(63\) 0 0
\(64\) 1.03730 − 7.93247i 0.129662 − 0.991558i
\(65\) 9.42387i 1.16889i
\(66\) 0 0
\(67\) − 4.49015i − 0.548560i −0.961650 − 0.274280i \(-0.911561\pi\)
0.961650 − 0.274280i \(-0.0884394\pi\)
\(68\) −9.90606 + 5.16043i −1.20129 + 0.625794i
\(69\) 0 0
\(70\) −9.34431 + 2.28798i −1.11686 + 0.273466i
\(71\) 14.6982 1.74436 0.872180 − 0.489186i \(-0.162706\pi\)
0.872180 + 0.489186i \(0.162706\pi\)
\(72\) 0 0
\(73\) 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
\(75\) 0 0
\(76\) −6.91282 + 5.31159i −0.792955 + 0.609281i
\(77\) 5.12597i 0.584158i
\(78\) 0 0
\(79\) 11.5314 1.29738 0.648690 − 0.761053i \(-0.275317\pi\)
0.648690 + 0.761053i \(0.275317\pi\)
\(80\) 9.97222 + 6.97399i 1.11493 + 0.779716i
\(81\) 0 0
\(82\) 2.80261 + 11.4461i 0.309497 + 1.26401i
\(83\) −2.00000 −0.219529 −0.109764 − 0.993958i \(-0.535010\pi\)
−0.109764 + 0.993958i \(0.535010\pi\)
\(84\) 0 0
\(85\) − 16.9902i − 1.84285i
\(86\) 6.69951 − 1.64039i 0.722426 − 0.176888i
\(87\) 0 0
\(88\) 4.87298 − 4.27725i 0.519462 − 0.455956i
\(89\) − 5.29881i − 0.561673i −0.959756 − 0.280836i \(-0.909388\pi\)
0.959756 − 0.280836i \(-0.0906118\pi\)
\(90\) 0 0
\(91\) 6.92665i 0.726111i
\(92\) 5.09501 + 9.78049i 0.531192 + 1.01969i
\(93\) 0 0
\(94\) −3.14893 − 12.8605i −0.324787 − 1.32646i
\(95\) −2.30746 − 13.0584i −0.236740 − 1.33977i
\(96\) 0 0
\(97\) − 1.17375i − 0.119176i −0.998223 − 0.0595881i \(-0.981021\pi\)
0.998223 − 0.0595881i \(-0.0189787\pi\)
\(98\) 2.74727 − 0.672676i 0.277516 − 0.0679506i
\(99\) 0 0
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 1368.2.e.e.379.11 12
3.2 odd 2 152.2.b.c.75.2 yes 12
4.3 odd 2 5472.2.e.e.5167.10 12
8.3 odd 2 inner 1368.2.e.e.379.1 12
8.5 even 2 5472.2.e.e.5167.3 12
12.11 even 2 608.2.b.c.303.11 12
19.18 odd 2 inner 1368.2.e.e.379.2 12
24.5 odd 2 608.2.b.c.303.12 12
24.11 even 2 152.2.b.c.75.12 yes 12
57.56 even 2 152.2.b.c.75.11 yes 12
76.75 even 2 5472.2.e.e.5167.9 12
152.37 odd 2 5472.2.e.e.5167.4 12
152.75 even 2 inner 1368.2.e.e.379.12 12
228.227 odd 2 608.2.b.c.303.1 12
456.227 odd 2 152.2.b.c.75.1 ✓ 12
456.341 even 2 608.2.b.c.303.2 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
152.2.b.c.75.1 ✓ 12 456.227 odd 2
152.2.b.c.75.2 yes 12 3.2 odd 2
152.2.b.c.75.11 yes 12 57.56 even 2
152.2.b.c.75.12 yes 12 24.11 even 2
608.2.b.c.303.1 12 228.227 odd 2
608.2.b.c.303.2 12 456.341 even 2
608.2.b.c.303.11 12 12.11 even 2
608.2.b.c.303.12 12 24.5 odd 2
1368.2.e.e.379.1 12 8.3 odd 2 inner
1368.2.e.e.379.2 12 19.18 odd 2 inner
1368.2.e.e.379.11 12 1.1 even 1 trivial
1368.2.e.e.379.12 12 152.75 even 2 inner
5472.2.e.e.5167.3 12 8.5 even 2
5472.2.e.e.5167.4 12 152.37 odd 2
5472.2.e.e.5167.9 12 76.75 even 2
5472.2.e.e.5167.10 12 4.3 odd 2