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Properties

Label 123.4.d.a.40.7
Level $123$
Weight $4$
Character 123.40
Analytic conductor $7.257$
Analytic rank $0$
Dimension $20$
Inner twists $2$

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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] = [123,4,Mod(40,123)] 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("123.40"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(123, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 123 = 3 \cdot 41 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 123.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(7.25723493071\)
Analytic rank: \(0\)
Dimension: \(20\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} + \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} + 116 x^{18} + 5618 x^{16} + 148400 x^{14} + 2342961 x^{12} + 22779360 x^{10} + 135500500 x^{8} + \cdots + 331822656 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{8}\cdot 3^{6} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 40.7
Root \(-1.38830i\) of defining polynomial
Character \(\chi\) \(=\) 123.40
Dual form 123.4.d.a.40.8

$q$-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)
 
\(f(q)\) \(=\) \(q-1.38830 q^{2} -3.00000i q^{3} -6.07262 q^{4} -6.03337 q^{5} +4.16491i q^{6} -11.7247i q^{7} +19.5370 q^{8} -9.00000 q^{9} +8.37614 q^{10} +38.6012i q^{11} +18.2179i q^{12} +38.3615i q^{13} +16.2774i q^{14} +18.1001i q^{15} +21.4576 q^{16} -45.7952i q^{17} +12.4947 q^{18} +143.335i q^{19} +36.6383 q^{20} -35.1741 q^{21} -53.5902i q^{22} +215.302 q^{23} -58.6111i q^{24} -88.5985 q^{25} -53.2574i q^{26} +27.0000i q^{27} +71.1996i q^{28} +237.414i q^{29} -25.1284i q^{30} -193.357 q^{31} -186.086 q^{32} +115.804 q^{33} +63.5775i q^{34} +70.7395i q^{35} +54.6536 q^{36} +82.2711 q^{37} -198.992i q^{38} +115.085 q^{39} -117.874 q^{40} +(251.182 + 76.3445i) q^{41} +48.8323 q^{42} -444.490 q^{43} -234.411i q^{44} +54.3003 q^{45} -298.904 q^{46} -176.989i q^{47} -64.3728i q^{48} +205.531 q^{49} +123.001 q^{50} -137.386 q^{51} -232.955i q^{52} -271.247i q^{53} -37.4842i q^{54} -232.896i q^{55} -229.066i q^{56} +430.005 q^{57} -329.602i q^{58} -44.3529 q^{59} -109.915i q^{60} -875.886 q^{61} +268.437 q^{62} +105.522i q^{63} +86.6827 q^{64} -231.449i q^{65} -160.771 q^{66} +559.049i q^{67} +278.097i q^{68} -645.905i q^{69} -98.2078i q^{70} -130.818i q^{71} -175.833 q^{72} +327.860 q^{73} -114.217 q^{74} +265.795i q^{75} -870.418i q^{76} +452.588 q^{77} -159.772 q^{78} +1171.50i q^{79} -129.462 q^{80} +81.0000 q^{81} +(-348.717 - 105.989i) q^{82} -595.303 q^{83} +213.599 q^{84} +276.299i q^{85} +617.087 q^{86} +712.241 q^{87} +754.154i q^{88} +1356.87i q^{89} -75.3853 q^{90} +449.777 q^{91} -1307.44 q^{92} +580.070i q^{93} +245.714i q^{94} -864.792i q^{95} +558.258i q^{96} -979.433i q^{97} -285.340 q^{98} -347.411i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q + 4 q^{2} + 72 q^{4} - 16 q^{5} + 132 q^{8} - 180 q^{9} - 140 q^{10} + 152 q^{16} - 36 q^{18} + 140 q^{20} + 84 q^{21} + 280 q^{23} + 736 q^{25} - 408 q^{31} + 1600 q^{32} - 12 q^{33} - 648 q^{36}+ \cdots + 2264 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/123\mathbb{Z}\right)^\times\).

\(n\) \(83\) \(88\)
\(\chi(n)\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.38830 −0.490839 −0.245419 − 0.969417i \(-0.578926\pi\)
−0.245419 + 0.969417i \(0.578926\pi\)
\(3\) − 3.00000i − 0.577350i
\(4\) −6.07262 −0.759077
\(5\) −6.03337 −0.539641 −0.269820 − 0.962911i \(-0.586964\pi\)
−0.269820 + 0.962911i \(0.586964\pi\)
\(6\) 4.16491i 0.283386i
\(7\) − 11.7247i − 0.633074i −0.948580 − 0.316537i \(-0.897480\pi\)
0.948580 − 0.316537i \(-0.102520\pi\)
\(8\) 19.5370 0.863424
\(9\) −9.00000 −0.333333
\(10\) 8.37614 0.264877
\(11\) 38.6012i 1.05806i 0.848602 + 0.529032i \(0.177445\pi\)
−0.848602 + 0.529032i \(0.822555\pi\)
\(12\) 18.2179i 0.438253i
\(13\) 38.3615i 0.818428i 0.912438 + 0.409214i \(0.134197\pi\)
−0.912438 + 0.409214i \(0.865803\pi\)
\(14\) 16.2774i 0.310738i
\(15\) 18.1001i 0.311562i
\(16\) 21.4576 0.335275
\(17\) − 45.7952i − 0.653351i −0.945137 − 0.326675i \(-0.894072\pi\)
0.945137 − 0.326675i \(-0.105928\pi\)
\(18\) 12.4947 0.163613
\(19\) 143.335i 1.73070i 0.501169 + 0.865349i \(0.332903\pi\)
−0.501169 + 0.865349i \(0.667097\pi\)
\(20\) 36.6383 0.409629
\(21\) −35.1741 −0.365506
\(22\) − 53.5902i − 0.519339i
\(23\) 215.302 1.95189 0.975945 − 0.218015i \(-0.0699583\pi\)
0.975945 + 0.218015i \(0.0699583\pi\)
\(24\) − 58.6111i − 0.498498i
\(25\) −88.5985 −0.708788
\(26\) − 53.2574i − 0.401717i
\(27\) 27.0000i 0.192450i
\(28\) 71.1996i 0.480552i
\(29\) 237.414i 1.52023i 0.649790 + 0.760114i \(0.274857\pi\)
−0.649790 + 0.760114i \(0.725143\pi\)
\(30\) − 25.1284i − 0.152927i
\(31\) −193.357 −1.12025 −0.560127 − 0.828407i \(-0.689248\pi\)
−0.560127 + 0.828407i \(0.689248\pi\)
\(32\) −186.086 −1.02799
\(33\) 115.804 0.610874
\(34\) 63.5775i 0.320690i
\(35\) 70.7395i 0.341633i
\(36\) 54.6536 0.253026
\(37\) 82.2711 0.365548 0.182774 − 0.983155i \(-0.441492\pi\)
0.182774 + 0.983155i \(0.441492\pi\)
\(38\) − 198.992i − 0.849494i
\(39\) 115.085 0.472520
\(40\) −117.874 −0.465939
\(41\) 251.182 + 76.3445i 0.956782 + 0.290805i
\(42\) 48.8323 0.179404
\(43\) −444.490 −1.57637 −0.788187 − 0.615435i \(-0.788980\pi\)
−0.788187 + 0.615435i \(0.788980\pi\)
\(44\) − 234.411i − 0.803153i
\(45\) 54.3003 0.179880
\(46\) −298.904 −0.958064
\(47\) − 176.989i − 0.549286i −0.961546 − 0.274643i \(-0.911440\pi\)
0.961546 − 0.274643i \(-0.0885596\pi\)
\(48\) − 64.3728i − 0.193571i
\(49\) 205.531 0.599217
\(50\) 123.001 0.347901
\(51\) −137.386 −0.377212
\(52\) − 232.955i − 0.621250i
\(53\) − 271.247i − 0.702993i −0.936189 − 0.351497i \(-0.885673\pi\)
0.936189 − 0.351497i \(-0.114327\pi\)
\(54\) − 37.4842i − 0.0944620i
\(55\) − 232.896i − 0.570975i
\(56\) − 229.066i − 0.546611i
\(57\) 430.005 0.999219
\(58\) − 329.602i − 0.746187i
\(59\) −44.3529 −0.0978688 −0.0489344 − 0.998802i \(-0.515583\pi\)
−0.0489344 + 0.998802i \(0.515583\pi\)
\(60\) − 109.915i − 0.236499i
\(61\) −875.886 −1.83845 −0.919227 − 0.393729i \(-0.871185\pi\)
−0.919227 + 0.393729i \(0.871185\pi\)
\(62\) 268.437 0.549864
\(63\) 105.522i 0.211025i
\(64\) 86.6827 0.169302
\(65\) − 231.449i − 0.441657i
\(66\) −160.771 −0.299841
\(67\) 559.049i 1.01938i 0.860357 + 0.509692i \(0.170241\pi\)
−0.860357 + 0.509692i \(0.829759\pi\)
\(68\) 278.097i 0.495943i
\(69\) − 645.905i − 1.12692i
\(70\) − 98.2078i − 0.167687i
\(71\) − 130.818i − 0.218666i −0.994005 − 0.109333i \(-0.965129\pi\)
0.994005 − 0.109333i \(-0.0348714\pi\)
\(72\) −175.833 −0.287808
\(73\) 327.860 0.525659 0.262830 − 0.964842i \(-0.415344\pi\)
0.262830 + 0.964842i \(0.415344\pi\)
\(74\) −114.217 −0.179425
\(75\) 265.795i 0.409219i
\(76\) − 870.418i − 1.31373i
\(77\) 452.588 0.669834
\(78\) −159.772 −0.231931
\(79\) 1171.50i 1.66841i 0.551457 + 0.834203i \(0.314072\pi\)
−0.551457 + 0.834203i \(0.685928\pi\)
\(80\) −129.462 −0.180928
\(81\) 81.0000 0.111111
\(82\) −348.717 − 105.989i −0.469626 − 0.142739i
\(83\) −595.303 −0.787265 −0.393633 − 0.919268i \(-0.628782\pi\)
−0.393633 + 0.919268i \(0.628782\pi\)
\(84\) 213.599 0.277447
\(85\) 276.299i 0.352575i
\(86\) 617.087 0.773746
\(87\) 712.241 0.877704
\(88\) 754.154i 0.913558i
\(89\) 1356.87i 1.61604i 0.589156 + 0.808019i \(0.299460\pi\)
−0.589156 + 0.808019i \(0.700540\pi\)
\(90\) −75.3853 −0.0882923
\(91\) 449.777 0.518126
\(92\) −1307.44 −1.48164
\(93\) 580.070i 0.646779i
\(94\) 245.714i 0.269611i
\(95\) − 864.792i − 0.933956i
\(96\) 558.258i 0.593510i
\(97\) − 979.433i − 1.02522i −0.858621 − 0.512610i \(-0.828679\pi\)
0.858621 − 0.512610i \(-0.171321\pi\)
\(98\) −285.340 −0.294119
\(99\) − 347.411i − 0.352688i
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 123.4.d.a.40.7 ✓ 20
3.2 odd 2 369.4.d.c.163.13 20
41.40 even 2 inner 123.4.d.a.40.8 yes 20
123.122 odd 2 369.4.d.c.163.14 20
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
123.4.d.a.40.7 ✓ 20 1.1 even 1 trivial
123.4.d.a.40.8 yes 20 41.40 even 2 inner
369.4.d.c.163.13 20 3.2 odd 2
369.4.d.c.163.14 20 123.122 odd 2