↑

Properties

Label 10.11.c.c.7.3
Level $10$
Weight $11$
Character 10.7
Analytic conductor $6.354$
Analytic rank $0$
Dimension $6$
Inner twists $2$

Related objects

Downloads

Learn more

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 = [6,-96,128] 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: \(6\)
Relative dimension: \(3\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{6} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 1148x^{3} + 68121x^{2} - 299628x + 658952 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 2^{8}\cdot 5^{6} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 7.3
Root \(-12.3957 - 12.3957i\) of defining polynomial
Character \(\chi\) \(=\) 10.7
Dual form 10.11.c.c.3.3

$q$-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+(-16.0000 - 16.0000i) q^{2} +(326.149 - 326.149i) q^{3} +512.000i q^{4} +(3124.94 + 19.4459i) q^{5} -10436.8 q^{6} +(-1507.13 - 1507.13i) q^{7} +(8192.00 - 8192.00i) q^{8} -153697. i q^{9} +(-49687.9 - 50310.2i) q^{10} -71462.1 q^{11} +(166988. + 166988. i) q^{12} +(-300816. + 300816. i) q^{13} +48228.2i q^{14} +(1.02554e6 - 1.01285e6i) q^{15} -262144. q^{16} +(1.11637e6 + 1.11637e6i) q^{17} +(-2.45915e6 + 2.45915e6i) q^{18} -684142. i q^{19} +(-9956.30 + 1.59997e6i) q^{20} -983097. q^{21} +(1.14339e6 + 1.14339e6i) q^{22} +(1.52597e6 - 1.52597e6i) q^{23} -5.34362e6i q^{24} +(9.76487e6 + 121534. i) q^{25} +9.62613e6 q^{26} +(-3.08693e7 - 3.08693e7i) q^{27} +(771651. - 771651. i) q^{28} +2.47039e7i q^{29} +(-3.26142e7 - 202952. i) q^{30} +4.13378e7 q^{31} +(4.19430e6 + 4.19430e6i) q^{32} +(-2.33073e7 + 2.33073e7i) q^{33} -3.57237e7i q^{34} +(-4.68038e6 - 4.73900e6i) q^{35} +7.86928e7 q^{36} +(2.53176e7 + 2.53176e7i) q^{37} +(-1.09463e7 + 1.09463e7i) q^{38} +1.96222e8i q^{39} +(2.57588e7 - 2.54402e7i) q^{40} -1.32462e8 q^{41} +(1.57296e7 + 1.57296e7i) q^{42} +(-2.46531e7 + 2.46531e7i) q^{43} -3.65886e7i q^{44} +(2.98877e6 - 4.80293e8i) q^{45} -4.88310e7 q^{46} +(5.19304e7 + 5.19304e7i) q^{47} +(-8.54979e7 + 8.54979e7i) q^{48} -2.77932e8i q^{49} +(-1.54293e8 - 1.58182e8i) q^{50} +7.28202e8 q^{51} +(-1.54018e8 - 1.54018e8i) q^{52} +(-3.03473e8 + 3.03473e8i) q^{53} +9.87817e8i q^{54} +(-2.23315e8 - 1.38964e6i) q^{55} -2.46928e7 q^{56} +(-2.23132e8 - 2.23132e8i) q^{57} +(3.95262e8 - 3.95262e8i) q^{58} +6.92655e7i q^{59} +(5.18580e8 + 5.25075e8i) q^{60} +6.38536e8 q^{61} +(-6.61404e8 - 6.61404e8i) q^{62} +(-2.31641e8 + 2.31641e8i) q^{63} -1.34218e8i q^{64} +(-9.45883e8 + 9.34184e8i) q^{65} +7.45832e8 q^{66} +(7.98844e8 + 7.98844e8i) q^{67} +(-5.71579e8 + 5.71579e8i) q^{68} -9.95386e8i q^{69} +(-937840. + 1.50710e8i) q^{70} -4.06392e8 q^{71} +(-1.25908e9 - 1.25908e9i) q^{72} +(-5.91419e8 + 5.91419e8i) q^{73} -8.10163e8i q^{74} +(3.22444e9 - 3.14516e9i) q^{75} +3.50281e8 q^{76} +(1.07703e8 + 1.07703e8i) q^{77} +(3.13955e9 - 3.13955e9i) q^{78} -6.97925e8i q^{79} +(-8.19184e8 - 5.09762e6i) q^{80} -1.10603e10 q^{81} +(2.11939e9 + 2.11939e9i) q^{82} +(-2.76663e9 + 2.76663e9i) q^{83} -5.03346e8i q^{84} +(3.46687e9 + 3.51028e9i) q^{85} +7.88901e8 q^{86} +(8.05714e9 + 8.05714e9i) q^{87} +(-5.85417e8 + 5.85417e8i) q^{88} -3.28276e9i q^{89} +(-7.73251e9 + 7.63687e9i) q^{90} +9.06739e8 q^{91} +(7.81297e8 + 7.81297e8i) q^{92} +(1.34823e10 - 1.34823e10i) q^{93} -1.66177e9i q^{94} +(1.33038e7 - 2.13790e9i) q^{95} +2.73593e9 q^{96} +(-3.71980e9 - 3.71980e9i) q^{97} +(-4.44692e9 + 4.44692e9i) q^{98} +1.09835e10i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 96 q^{2} + 128 q^{3} + 5460 q^{5} - 4096 q^{6} + 13512 q^{7} + 49152 q^{8} - 173280 q^{10} + 647832 q^{11} + 65536 q^{12} - 742902 q^{13} + 1577720 q^{15} - 1572864 q^{16} - 755118 q^{17} - 5683744 q^{18}+ \cdots + 12874047264 q^{98}+O(q^{100}) \) 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
\(3\) 326.149 − 326.149i 1.34218 − 1.34218i 0.448284 − 0.893891i \(-0.352035\pi\)
0.893891 − 0.448284i \(-0.147965\pi\)
\(4\) 512.000i 0.500000i
\(5\) 3124.94 + 19.4459i 0.999981 + 0.00622269i
\(6\) −10436.8 −1.34218
\(7\) −1507.13 − 1507.13i −0.0896728 − 0.0896728i 0.660847 − 0.750520i \(-0.270197\pi\)
−0.750520 + 0.660847i \(0.770197\pi\)
\(8\) 8192.00 − 8192.00i 0.250000 − 0.250000i
\(9\) − 153697.i − 2.60287i
\(10\) −49687.9 − 50310.2i −0.496879 − 0.503102i
\(11\) −71462.1 −0.443723 −0.221862 − 0.975078i \(-0.571213\pi\)
−0.221862 + 0.975078i \(0.571213\pi\)
\(12\) 166988. + 166988.i 0.671088 + 0.671088i
\(13\) −300816. + 300816.i −0.810186 + 0.810186i −0.984662 − 0.174475i \(-0.944177\pi\)
0.174475 + 0.984662i \(0.444177\pi\)
\(14\) 48228.2i 0.0896728i
\(15\) 1.02554e6 − 1.01285e6i 1.35050 − 1.33380i
\(16\) −262144. −0.250000
\(17\) 1.11637e6 + 1.11637e6i 0.786252 + 0.786252i 0.980878 − 0.194626i \(-0.0623492\pi\)
−0.194626 + 0.980878i \(0.562349\pi\)
\(18\) −2.45915e6 + 2.45915e6i −1.30143 + 1.30143i
\(19\) − 684142.i − 0.276299i −0.990411 − 0.138149i \(-0.955885\pi\)
0.990411 − 0.138149i \(-0.0441154\pi\)
\(20\) −9956.30 + 1.59997e6i −0.00311134 + 0.499990i
\(21\) −983097. −0.240713
\(22\) 1.14339e6 + 1.14339e6i 0.221862 + 0.221862i
\(23\) 1.52597e6 − 1.52597e6i 0.237087 − 0.237087i −0.578556 − 0.815643i \(-0.696384\pi\)
0.815643 + 0.578556i \(0.196384\pi\)
\(24\) − 5.34362e6i − 0.671088i
\(25\) 9.76487e6 + 121534.i 0.999923 + 0.0124451i
\(26\) 9.62613e6 0.810186
\(27\) −3.08693e7 − 3.08693e7i −2.15133 − 2.15133i
\(28\) 771651. − 771651.i 0.0448364 − 0.0448364i
\(29\) 2.47039e7i 1.20441i 0.798340 + 0.602207i \(0.205712\pi\)
−0.798340 + 0.602207i \(0.794288\pi\)
\(30\) −3.26142e7 − 202952.i −1.34215 − 0.00835194i
\(31\) 4.13378e7 1.44390 0.721952 − 0.691943i \(-0.243245\pi\)
0.721952 + 0.691943i \(0.243245\pi\)
\(32\) 4.19430e6 + 4.19430e6i 0.125000 + 0.125000i
\(33\) −2.33073e7 + 2.33073e7i −0.595555 + 0.595555i
\(34\) − 3.57237e7i − 0.786252i
\(35\) −4.68038e6 − 4.73900e6i −0.0891130 − 0.0902290i
\(36\) 7.86928e7 1.30143
\(37\) 2.53176e7 + 2.53176e7i 0.365102 + 0.365102i 0.865687 − 0.500586i \(-0.166882\pi\)
−0.500586 + 0.865687i \(0.666882\pi\)
\(38\) −1.09463e7 + 1.09463e7i −0.138149 + 0.138149i
\(39\) 1.96222e8i 2.17482i
\(40\) 2.57588e7 − 2.54402e7i 0.251551 − 0.248439i
\(41\) −1.32462e8 −1.14333 −0.571664 − 0.820488i \(-0.693702\pi\)
−0.571664 + 0.820488i \(0.693702\pi\)
\(42\) 1.57296e7 + 1.57296e7i 0.120357 + 0.120357i
\(43\) −2.46531e7 + 2.46531e7i −0.167699 + 0.167699i −0.785967 − 0.618268i \(-0.787835\pi\)
0.618268 + 0.785967i \(0.287835\pi\)
\(44\) − 3.65886e7i − 0.221862i
\(45\) 2.98877e6 − 4.80293e8i 0.0161968 − 2.60282i
\(46\) −4.88310e7 −0.237087
\(47\) 5.19304e7 + 5.19304e7i 0.226429 + 0.226429i 0.811199 − 0.584770i \(-0.198815\pi\)
−0.584770 + 0.811199i \(0.698815\pi\)
\(48\) −8.54979e7 + 8.54979e7i −0.335544 + 0.335544i
\(49\) − 2.77932e8i − 0.983918i
\(50\) −1.54293e8 − 1.58182e8i −0.493739 − 0.506184i
\(51\) 7.28202e8 2.11058
\(52\) −1.54018e8 − 1.54018e8i −0.405093 − 0.405093i
\(53\) −3.03473e8 + 3.03473e8i −0.725672 + 0.725672i −0.969754 − 0.244083i \(-0.921513\pi\)
0.244083 + 0.969754i \(0.421513\pi\)
\(54\) 9.87817e8i 2.15133i
\(55\) −2.23315e8 − 1.38964e6i −0.443715 − 0.00276115i
\(56\) −2.46928e7 −0.0448364
\(57\) −2.23132e8 − 2.23132e8i −0.370841 − 0.370841i
\(58\) 3.95262e8 − 3.95262e8i 0.602207 − 0.602207i
\(59\) 6.92655e7i 0.0968851i 0.998826 + 0.0484426i \(0.0154258\pi\)
−0.998826 + 0.0484426i \(0.984574\pi\)
\(60\) 5.18580e8 + 5.25075e8i 0.666899 + 0.675251i
\(61\) 6.38536e8 0.756025 0.378012 − 0.925801i \(-0.376608\pi\)
0.378012 + 0.925801i \(0.376608\pi\)
\(62\) −6.61404e8 − 6.61404e8i −0.721952 − 0.721952i
\(63\) −2.31641e8 + 2.31641e8i −0.233407 + 0.233407i
\(64\) − 1.34218e8i − 0.125000i
\(65\) −9.45883e8 + 9.34184e8i −0.815212 + 0.805129i
\(66\) 7.45832e8 0.595555
\(67\) 7.98844e8 + 7.98844e8i 0.591682 + 0.591682i 0.938085 − 0.346404i \(-0.112597\pi\)
−0.346404 + 0.938085i \(0.612597\pi\)
\(68\) −5.71579e8 + 5.71579e8i −0.393126 + 0.393126i
\(69\) − 9.95386e8i − 0.636423i
\(70\) −937840. + 1.50710e8i −0.000558005 + 0.0896710i
\(71\) −4.06392e8 −0.225244 −0.112622 − 0.993638i \(-0.535925\pi\)
−0.112622 + 0.993638i \(0.535925\pi\)
\(72\) −1.25908e9 − 1.25908e9i −0.650717 − 0.650717i
\(73\) −5.91419e8 + 5.91419e8i −0.285286 + 0.285286i −0.835213 − 0.549927i \(-0.814656\pi\)
0.549927 + 0.835213i \(0.314656\pi\)
\(74\) − 8.10163e8i − 0.365102i
\(75\) 3.22444e9 − 3.14516e9i 1.35878 − 1.32537i
\(76\) 3.50281e8 0.138149
\(77\) 1.07703e8 + 1.07703e8i 0.0397899 + 0.0397899i
\(78\) 3.13955e9 − 3.13955e9i 1.08741 − 1.08741i
\(79\) − 6.97925e8i − 0.226816i −0.993549 − 0.113408i \(-0.963823\pi\)
0.993549 − 0.113408i \(-0.0361767\pi\)
\(80\) −8.19184e8 − 5.09762e6i −0.249995 − 0.00155567i
\(81\) −1.10603e10 −3.17206
\(82\) 2.11939e9 + 2.11939e9i 0.571664 + 0.571664i
\(83\) −2.76663e9 + 2.76663e9i −0.702362 + 0.702362i −0.964917 − 0.262555i \(-0.915435\pi\)
0.262555 + 0.964917i \(0.415435\pi\)
\(84\) − 5.03346e8i − 0.120357i
\(85\) 3.46687e9 + 3.51028e9i 0.781344 + 0.791129i
\(86\) 7.88901e8 0.167699
\(87\) 8.05714e9 + 8.05714e9i 1.61653 + 1.61653i
\(88\) −5.85417e8 + 5.85417e8i −0.110931 + 0.110931i
\(89\) − 3.28276e9i − 0.587880i −0.955824 − 0.293940i \(-0.905033\pi\)
0.955824 − 0.293940i \(-0.0949667\pi\)
\(90\) −7.73251e9 + 7.63687e9i −1.30951 + 1.29331i
\(91\) 9.06739e8 0.145303
\(92\) 7.81297e8 + 7.81297e8i 0.118543 + 0.118543i
\(93\) 1.34823e10 − 1.34823e10i 1.93797 − 1.93797i
\(94\) − 1.66177e9i − 0.226429i
\(95\) 1.33038e7 − 2.13790e9i 0.00171932 − 0.276293i
\(96\) 2.73593e9 0.335544
\(97\) −3.71980e9 − 3.71980e9i −0.433172 − 0.433172i 0.456534 − 0.889706i \(-0.349091\pi\)
−0.889706 + 0.456534i \(0.849091\pi\)
\(98\) −4.44692e9 + 4.44692e9i −0.491959 + 0.491959i
\(99\) 1.09835e10i 1.15495i
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.c.7.3 yes 6
3.2 odd 2 90.11.g.c.37.1 6
4.3 odd 2 80.11.p.c.17.1 6
5.2 odd 4 50.11.c.e.43.1 6
5.3 odd 4 inner 10.11.c.c.3.3 ✓ 6
5.4 even 2 50.11.c.e.7.1 6
15.8 even 4 90.11.g.c.73.1 6
20.3 even 4 80.11.p.c.33.1 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
10.11.c.c.3.3 ✓ 6 5.3 odd 4 inner
10.11.c.c.7.3 yes 6 1.1 even 1 trivial
50.11.c.e.7.1 6 5.4 even 2
50.11.c.e.43.1 6 5.2 odd 4
80.11.p.c.17.1 6 4.3 odd 2
80.11.p.c.33.1 6 20.3 even 4
90.11.g.c.37.1 6 3.2 odd 2
90.11.g.c.73.1 6 15.8 even 4