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Properties

Label 10.11.c.b.7.1
Level $10$
Weight $11$
Character 10.7
Analytic conductor $6.354$
Analytic rank $0$
Dimension $2$
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] = [10,11,Mod(3,10)] 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("10.3"); S:= CuspForms(chi, 11); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(10, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([3])) N = Newforms(chi, 11, names="a")
 
Level: \( N \) \(=\) \( 10 = 2 \cdot 5 \)
Weight: \( k \) \(=\) \( 11 \)
Character orbit: \([\chi]\) \(=\) 10.c (of order \(4\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,32,114] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(6.35357252674\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 7.1
Root \(1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 10.7
Dual form 10.11.c.b.3.1

$q$-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+(16.0000 + 16.0000i) q^{2} +(57.0000 - 57.0000i) q^{3} +512.000i q^{4} +(2925.00 + 1100.00i) q^{5} +1824.00 q^{6} +(6953.00 + 6953.00i) q^{7} +(-8192.00 + 8192.00i) q^{8} +52551.0i q^{9} +(29200.0 + 64400.0i) q^{10} +75242.0 q^{11} +(29184.0 + 29184.0i) q^{12} +(109857. - 109857. i) q^{13} +222496. i q^{14} +(229425. - 104025. i) q^{15} -262144. q^{16} +(-1.52893e6 - 1.52893e6i) q^{17} +(-840816. + 840816. i) q^{18} -4.03868e6i q^{19} +(-563200. + 1.49760e6i) q^{20} +792642. q^{21} +(1.20387e6 + 1.20387e6i) q^{22} +(-712423. + 712423. i) q^{23} +933888. i q^{24} +(7.34562e6 + 6.43500e6i) q^{25} +3.51542e6 q^{26} +(6.36120e6 + 6.36120e6i) q^{27} +(-3.55994e6 + 3.55994e6i) q^{28} +446120. i q^{29} +(5.33520e6 + 2.00640e6i) q^{30} -2.90807e7 q^{31} +(-4.19430e6 - 4.19430e6i) q^{32} +(4.28879e6 - 4.28879e6i) q^{33} -4.89257e7i q^{34} +(1.26892e7 + 2.79858e7i) q^{35} -2.69061e7 q^{36} +(-911847. - 911847. i) q^{37} +(6.46189e7 - 6.46189e7i) q^{38} -1.25237e7i q^{39} +(-3.29728e7 + 1.49504e7i) q^{40} -1.63946e8 q^{41} +(1.26823e7 + 1.26823e7i) q^{42} +(1.18423e8 - 1.18423e8i) q^{43} +3.85239e7i q^{44} +(-5.78061e7 + 1.53712e8i) q^{45} -2.27975e7 q^{46} +(2.76320e8 + 2.76320e8i) q^{47} +(-1.49422e7 + 1.49422e7i) q^{48} -1.85787e8i q^{49} +(1.45700e7 + 2.20490e8i) q^{50} -1.74298e8 q^{51} +(5.62468e7 + 5.62468e7i) q^{52} +(3.08460e8 - 3.08460e8i) q^{53} +2.03558e8i q^{54} +(2.20083e8 + 8.27662e7i) q^{55} -1.13918e8 q^{56} +(-2.30205e8 - 2.30205e8i) q^{57} +(-7.13792e6 + 7.13792e6i) q^{58} -9.40888e8i q^{59} +(5.32608e7 + 1.17466e8i) q^{60} -1.35361e9 q^{61} +(-4.65291e8 - 4.65291e8i) q^{62} +(-3.65387e8 + 3.65387e8i) q^{63} -1.34218e8i q^{64} +(4.42174e8 - 2.00489e8i) q^{65} +1.37241e8 q^{66} +(8.53571e8 + 8.53571e8i) q^{67} +(7.82811e8 - 7.82811e8i) q^{68} +8.12162e7i q^{69} +(-2.44746e8 + 6.50801e8i) q^{70} +2.82701e9 q^{71} +(-4.30498e8 - 4.30498e8i) q^{72} +(-2.75330e9 + 2.75330e9i) q^{73} -2.91791e7i q^{74} +(7.85496e8 - 5.19056e7i) q^{75} +2.06780e9 q^{76} +(5.23158e8 + 5.23158e8i) q^{77} +(2.00379e8 - 2.00379e8i) q^{78} +3.32450e9i q^{79} +(-7.66771e8 - 2.88358e8i) q^{80} -2.37791e9 q^{81} +(-2.62313e9 - 2.62313e9i) q^{82} +(1.34634e9 - 1.34634e9i) q^{83} +4.05833e8i q^{84} +(-2.79029e9 - 6.15393e9i) q^{85} +3.78953e9 q^{86} +(2.54288e7 + 2.54288e7i) q^{87} +(-6.16382e8 + 6.16382e8i) q^{88} -2.66745e9i q^{89} +(-3.38428e9 + 1.53449e9i) q^{90} +1.52767e9 q^{91} +(-3.64761e8 - 3.64761e8i) q^{92} +(-1.65760e9 + 1.65760e9i) q^{93} +8.84225e9i q^{94} +(4.44255e9 - 1.18131e10i) q^{95} -4.78151e8 q^{96} +(-5.26563e8 - 5.26563e8i) q^{97} +(2.97259e9 - 2.97259e9i) q^{98} +3.95404e9i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 32 q^{2} + 114 q^{3} + 5850 q^{5} + 3648 q^{6} + 13906 q^{7} - 16384 q^{8} + 58400 q^{10} + 150484 q^{11} + 58368 q^{12} + 219714 q^{13} + 458850 q^{15} - 524288 q^{16} - 3057854 q^{17} - 1681632 q^{18}+ \cdots + 5945178592 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/10\mathbb{Z}\right)^\times\).

\(n\) \(7\)
\(\chi(n)\) \(e\left(\frac{1}{4}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 16.0000 + 16.0000i 0.500000 + 0.500000i
\(3\) 57.0000 − 57.0000i 0.234568 − 0.234568i −0.580028 − 0.814596i \(-0.696959\pi\)
0.814596 + 0.580028i \(0.196959\pi\)
\(4\) 512.000i 0.500000i
\(5\) 2925.00 + 1100.00i 0.936000 + 0.352000i
\(6\) 1824.00 0.234568
\(7\) 6953.00 + 6953.00i 0.413697 + 0.413697i 0.883024 − 0.469328i \(-0.155504\pi\)
−0.469328 + 0.883024i \(0.655504\pi\)
\(8\) −8192.00 + 8192.00i −0.250000 + 0.250000i
\(9\) 52551.0i 0.889956i
\(10\) 29200.0 + 64400.0i 0.292000 + 0.644000i
\(11\) 75242.0 0.467194 0.233597 − 0.972334i \(-0.424950\pi\)
0.233597 + 0.972334i \(0.424950\pi\)
\(12\) 29184.0 + 29184.0i 0.117284 + 0.117284i
\(13\) 109857. − 109857.i 0.295877 − 0.295877i −0.543520 − 0.839396i \(-0.682909\pi\)
0.839396 + 0.543520i \(0.182909\pi\)
\(14\) 222496.i 0.413697i
\(15\) 229425. − 104025.i 0.302123 − 0.136988i
\(16\) −262144. −0.250000
\(17\) −1.52893e6 − 1.52893e6i −1.07682 − 1.07682i −0.996793 − 0.0800247i \(-0.974500\pi\)
−0.0800247 − 0.996793i \(-0.525500\pi\)
\(18\) −840816. + 840816.i −0.444978 + 0.444978i
\(19\) − 4.03868e6i − 1.63107i −0.578711 − 0.815533i \(-0.696444\pi\)
0.578711 − 0.815533i \(-0.303556\pi\)
\(20\) −563200. + 1.49760e6i −0.176000 + 0.468000i
\(21\) 792642. 0.194080
\(22\) 1.20387e6 + 1.20387e6i 0.233597 + 0.233597i
\(23\) −712423. + 712423.i −0.110688 + 0.110688i −0.760281 − 0.649594i \(-0.774939\pi\)
0.649594 + 0.760281i \(0.274939\pi\)
\(24\) 933888.i 0.117284i
\(25\) 7.34562e6 + 6.43500e6i 0.752192 + 0.658944i
\(26\) 3.51542e6 0.295877
\(27\) 6.36120e6 + 6.36120e6i 0.443323 + 0.443323i
\(28\) −3.55994e6 + 3.55994e6i −0.206848 + 0.206848i
\(29\) 446120.i 0.0217501i 0.999941 + 0.0108751i \(0.00346171\pi\)
−0.999941 + 0.0108751i \(0.996538\pi\)
\(30\) 5.33520e6 + 2.00640e6i 0.219556 + 0.0825679i
\(31\) −2.90807e7 −1.01577 −0.507886 − 0.861424i \(-0.669573\pi\)
−0.507886 + 0.861424i \(0.669573\pi\)
\(32\) −4.19430e6 − 4.19430e6i −0.125000 − 0.125000i
\(33\) 4.28879e6 − 4.28879e6i 0.109589 − 0.109589i
\(34\) − 4.89257e7i − 1.07682i
\(35\) 1.26892e7 + 2.79858e7i 0.241599 + 0.532841i
\(36\) −2.69061e7 −0.444978
\(37\) −911847. − 911847.i −0.0131496 − 0.0131496i 0.700501 − 0.713651i \(-0.252960\pi\)
−0.713651 + 0.700501i \(0.752960\pi\)
\(38\) 6.46189e7 − 6.46189e7i 0.815533 − 0.815533i
\(39\) − 1.25237e7i − 0.138806i
\(40\) −3.29728e7 + 1.49504e7i −0.322000 + 0.146000i
\(41\) −1.63946e8 −1.41508 −0.707540 − 0.706674i \(-0.750195\pi\)
−0.707540 + 0.706674i \(0.750195\pi\)
\(42\) 1.26823e7 + 1.26823e7i 0.0970400 + 0.0970400i
\(43\) 1.18423e8 − 1.18423e8i 0.805551 − 0.805551i −0.178406 − 0.983957i \(-0.557094\pi\)
0.983957 + 0.178406i \(0.0570941\pi\)
\(44\) 3.85239e7i 0.233597i
\(45\) −5.78061e7 + 1.53712e8i −0.313264 + 0.832999i
\(46\) −2.27975e7 −0.110688
\(47\) 2.76320e8 + 2.76320e8i 1.20482 + 1.20482i 0.972682 + 0.232142i \(0.0745733\pi\)
0.232142 + 0.972682i \(0.425427\pi\)
\(48\) −1.49422e7 + 1.49422e7i −0.0586420 + 0.0586420i
\(49\) − 1.85787e8i − 0.657710i
\(50\) 1.45700e7 + 2.20490e8i 0.0466240 + 0.705568i
\(51\) −1.74298e8 −0.505174
\(52\) 5.62468e7 + 5.62468e7i 0.147938 + 0.147938i
\(53\) 3.08460e8 − 3.08460e8i 0.737598 − 0.737598i −0.234515 − 0.972113i \(-0.575350\pi\)
0.972113 + 0.234515i \(0.0753501\pi\)
\(54\) 2.03558e8i 0.443323i
\(55\) 2.20083e8 + 8.27662e7i 0.437293 + 0.164452i
\(56\) −1.13918e8 −0.206848
\(57\) −2.30205e8 − 2.30205e8i −0.382596 − 0.382596i
\(58\) −7.13792e6 + 7.13792e6i −0.0108751 + 0.0108751i
\(59\) − 9.40888e8i − 1.31607i −0.752989 − 0.658034i \(-0.771388\pi\)
0.752989 − 0.658034i \(-0.228612\pi\)
\(60\) 5.32608e7 + 1.17466e8i 0.0684938 + 0.151062i
\(61\) −1.35361e9 −1.60267 −0.801336 − 0.598215i \(-0.795877\pi\)
−0.801336 + 0.598215i \(0.795877\pi\)
\(62\) −4.65291e8 − 4.65291e8i −0.507886 − 0.507886i
\(63\) −3.65387e8 + 3.65387e8i −0.368172 + 0.368172i
\(64\) − 1.34218e8i − 0.125000i
\(65\) 4.42174e8 − 2.00489e8i 0.381089 − 0.172792i
\(66\) 1.37241e8 0.109589
\(67\) 8.53571e8 + 8.53571e8i 0.632216 + 0.632216i 0.948623 − 0.316407i \(-0.102477\pi\)
−0.316407 + 0.948623i \(0.602477\pi\)
\(68\) 7.82811e8 − 7.82811e8i 0.538409 − 0.538409i
\(69\) 8.12162e7i 0.0519275i
\(70\) −2.44746e8 + 6.50801e8i −0.145621 + 0.387220i
\(71\) 2.82701e9 1.56688 0.783441 − 0.621466i \(-0.213463\pi\)
0.783441 + 0.621466i \(0.213463\pi\)
\(72\) −4.30498e8 − 4.30498e8i −0.222489 − 0.222489i
\(73\) −2.75330e9 + 2.75330e9i −1.32812 + 1.32812i −0.421119 + 0.907005i \(0.638363\pi\)
−0.907005 + 0.421119i \(0.861637\pi\)
\(74\) − 2.91791e7i − 0.0131496i
\(75\) 7.85496e8 − 5.19056e7i 0.331007 − 0.0218730i
\(76\) 2.06780e9 0.815533
\(77\) 5.23158e8 + 5.23158e8i 0.193276 + 0.193276i
\(78\) 2.00379e8 − 2.00379e8i 0.0694032 − 0.0694032i
\(79\) 3.32450e9i 1.08042i 0.841532 + 0.540208i \(0.181654\pi\)
−0.841532 + 0.540208i \(0.818346\pi\)
\(80\) −7.66771e8 − 2.88358e8i −0.234000 − 0.0880000i
\(81\) −2.37791e9 −0.681977
\(82\) −2.62313e9 − 2.62313e9i −0.707540 − 0.707540i
\(83\) 1.34634e9 − 1.34634e9i 0.341794 − 0.341794i −0.515248 − 0.857041i \(-0.672300\pi\)
0.857041 + 0.515248i \(0.172300\pi\)
\(84\) 4.05833e8i 0.0970400i
\(85\) −2.79029e9 − 6.15393e9i −0.628861 − 1.38694i
\(86\) 3.78953e9 0.805551
\(87\) 2.54288e7 + 2.54288e7i 0.00510188 + 0.00510188i
\(88\) −6.16382e8 + 6.16382e8i −0.116798 + 0.116798i
\(89\) − 2.66745e9i − 0.477690i −0.971058 − 0.238845i \(-0.923231\pi\)
0.971058 − 0.238845i \(-0.0767688\pi\)
\(90\) −3.38428e9 + 1.53449e9i −0.573132 + 0.259867i
\(91\) 1.52767e9 0.244807
\(92\) −3.64761e8 − 3.64761e8i −0.0553438 − 0.0553438i
\(93\) −1.65760e9 + 1.65760e9i −0.238268 + 0.238268i
\(94\) 8.84225e9i 1.20482i
\(95\) 4.44255e9 − 1.18131e10i 0.574135 − 1.52668i
\(96\) −4.78151e8 −0.0586420
\(97\) −5.26563e8 − 5.26563e8i −0.0613185 − 0.0613185i 0.675783 − 0.737101i \(-0.263806\pi\)
−0.737101 + 0.675783i \(0.763806\pi\)
\(98\) 2.97259e9 − 2.97259e9i 0.328855 − 0.328855i
\(99\) 3.95404e9i 0.415782i
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 10.11.c.b.7.1 yes 2
3.2 odd 2 90.11.g.a.37.1 2
4.3 odd 2 80.11.p.a.17.1 2
5.2 odd 4 50.11.c.b.43.1 2
5.3 odd 4 inner 10.11.c.b.3.1 ✓ 2
5.4 even 2 50.11.c.b.7.1 2
15.8 even 4 90.11.g.a.73.1 2
20.3 even 4 80.11.p.a.33.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
10.11.c.b.3.1 ✓ 2 5.3 odd 4 inner
10.11.c.b.7.1 yes 2 1.1 even 1 trivial
50.11.c.b.7.1 2 5.4 even 2
50.11.c.b.43.1 2 5.2 odd 4
80.11.p.a.17.1 2 4.3 odd 2
80.11.p.a.33.1 2 20.3 even 4
90.11.g.a.37.1 2 3.2 odd 2
90.11.g.a.73.1 2 15.8 even 4