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Properties

Label 10.11.c.a.3.1
Level $10$
Weight $11$
Character 10.3
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,-366] 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 3.1
Root \(-1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 10.3
Dual form 10.11.c.a.7.1

$q$-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} +(-183.000 - 183.000i) q^{3} -512.000i q^{4} +(-1875.00 + 2500.00i) q^{5} -5856.00 q^{6} +(-8407.00 + 8407.00i) q^{7} +(-8192.00 - 8192.00i) q^{8} +7929.00i q^{9} +(10000.0 + 70000.0i) q^{10} -173398. q^{11} +(-93696.0 + 93696.0i) q^{12} +(-232623. - 232623. i) q^{13} +269024. i q^{14} +(800625. - 114375. i) q^{15} -262144. q^{16} +(1.88003e6 - 1.88003e6i) q^{17} +(126864. + 126864. i) q^{18} +1.10156e6i q^{19} +(1.28000e6 + 960000. i) q^{20} +3.07696e6 q^{21} +(-2.77437e6 + 2.77437e6i) q^{22} +(-5.22826e6 - 5.22826e6i) q^{23} +2.99827e6i q^{24} +(-2.73438e6 - 9.37500e6i) q^{25} -7.44394e6 q^{26} +(-9.35496e6 + 9.35496e6i) q^{27} +(4.30438e6 + 4.30438e6i) q^{28} +2.47908e7i q^{29} +(1.09800e7 - 1.46400e7i) q^{30} -1.00660e7 q^{31} +(-4.19430e6 + 4.19430e6i) q^{32} +(3.17318e7 + 3.17318e7i) q^{33} -6.01611e7i q^{34} +(-5.25438e6 - 3.67806e7i) q^{35} +4.05965e6 q^{36} +(5.63879e7 - 5.63879e7i) q^{37} +(1.76250e7 + 1.76250e7i) q^{38} +8.51400e7i q^{39} +(3.58400e7 - 5.12000e6i) q^{40} -1.53004e8 q^{41} +(4.92314e7 - 4.92314e7i) q^{42} +(5.93725e7 + 5.93725e7i) q^{43} +8.87798e7i q^{44} +(-1.98225e7 - 1.48669e7i) q^{45} -1.67304e8 q^{46} +(1.72339e8 - 1.72339e8i) q^{47} +(4.79724e7 + 4.79724e7i) q^{48} +1.41120e8i q^{49} +(-1.93750e8 - 1.06250e8i) q^{50} -6.88092e8 q^{51} +(-1.19103e8 + 1.19103e8i) q^{52} +(1.96386e8 + 1.96386e8i) q^{53} +2.99359e8i q^{54} +(3.25121e8 - 4.33495e8i) q^{55} +1.37740e8 q^{56} +(2.01585e8 - 2.01585e8i) q^{57} +(3.96653e8 + 3.96653e8i) q^{58} -6.94069e8i q^{59} +(-5.85600e7 - 4.09920e8i) q^{60} +9.06186e8 q^{61} +(-1.61056e8 + 1.61056e8i) q^{62} +(-6.66591e7 - 6.66591e7i) q^{63} +1.34218e8i q^{64} +(1.01773e9 - 1.45389e8i) q^{65} +1.01542e9 q^{66} +(-9.62074e8 + 9.62074e8i) q^{67} +(-9.62577e8 - 9.62577e8i) q^{68} +1.91354e9i q^{69} +(-6.72560e8 - 5.04420e8i) q^{70} -3.12088e9 q^{71} +(6.49544e7 - 6.49544e7i) q^{72} +(-6.36339e8 - 6.36339e8i) q^{73} -1.80441e9i q^{74} +(-1.21523e9 + 2.21602e9i) q^{75} +5.63999e8 q^{76} +(1.45776e9 - 1.45776e9i) q^{77} +(1.36224e9 + 1.36224e9i) q^{78} +1.96800e9i q^{79} +(4.91520e8 - 6.55360e8i) q^{80} +3.89211e9 q^{81} +(-2.44806e9 + 2.44806e9i) q^{82} +(-5.18382e9 - 5.18382e9i) q^{83} -1.57540e9i q^{84} +(1.17502e9 + 8.22514e9i) q^{85} +1.89992e9 q^{86} +(4.53672e9 - 4.53672e9i) q^{87} +(1.42048e9 + 1.42048e9i) q^{88} -7.77138e9i q^{89} +(-5.55030e8 + 7.92900e7i) q^{90} +3.91132e9 q^{91} +(-2.67687e9 + 2.67687e9i) q^{92} +(1.84208e9 + 1.84208e9i) q^{93} -5.51486e9i q^{94} +(-2.75390e9 - 2.06542e9i) q^{95} +1.53512e9 q^{96} +(-6.40361e8 + 6.40361e8i) q^{97} +(2.25792e9 + 2.25792e9i) q^{98} -1.37487e9i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 32 q^{2} - 366 q^{3} - 3750 q^{5} - 11712 q^{6} - 16814 q^{7} - 16384 q^{8} + 20000 q^{10} - 346796 q^{11} - 187392 q^{12} - 465246 q^{13} + 1601250 q^{15} - 524288 q^{16} + 3760066 q^{17} + 253728 q^{18}+ \cdots + 4515838432 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(7\)
\(\chi(n)\) \(e\left(\frac{3}{4}\right)\)

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
\(3\) −183.000 − 183.000i −0.753086 − 0.753086i 0.221968 − 0.975054i \(-0.428752\pi\)
−0.975054 + 0.221968i \(0.928752\pi\)
\(4\) − 512.000i − 0.500000i
\(5\) −1875.00 + 2500.00i −0.600000 + 0.800000i
\(6\) −5856.00 −0.753086
\(7\) −8407.00 + 8407.00i −0.500208 + 0.500208i −0.911503 − 0.411294i \(-0.865077\pi\)
0.411294 + 0.911503i \(0.365077\pi\)
\(8\) −8192.00 − 8192.00i −0.250000 − 0.250000i
\(9\) 7929.00i 0.134278i
\(10\) 10000.0 + 70000.0i 0.100000 + 0.700000i
\(11\) −173398. −1.07667 −0.538333 − 0.842732i \(-0.680946\pi\)
−0.538333 + 0.842732i \(0.680946\pi\)
\(12\) −93696.0 + 93696.0i −0.376543 + 0.376543i
\(13\) −232623. − 232623.i −0.626521 − 0.626521i 0.320670 − 0.947191i \(-0.396092\pi\)
−0.947191 + 0.320670i \(0.896092\pi\)
\(14\) 269024.i 0.500208i
\(15\) 800625. − 114375.i 1.05432 − 0.150617i
\(16\) −262144. −0.250000
\(17\) 1.88003e6 − 1.88003e6i 1.32410 − 1.32410i 0.413676 − 0.910424i \(-0.364245\pi\)
0.910424 − 0.413676i \(-0.135755\pi\)
\(18\) 126864. + 126864.i 0.0671392 + 0.0671392i
\(19\) 1.10156e6i 0.444877i 0.974947 + 0.222439i \(0.0714017\pi\)
−0.974947 + 0.222439i \(0.928598\pi\)
\(20\) 1.28000e6 + 960000.i 0.400000 + 0.300000i
\(21\) 3.07696e6 0.753400
\(22\) −2.77437e6 + 2.77437e6i −0.538333 + 0.538333i
\(23\) −5.22826e6 − 5.22826e6i −0.812303 − 0.812303i 0.172675 − 0.984979i \(-0.444759\pi\)
−0.984979 + 0.172675i \(0.944759\pi\)
\(24\) 2.99827e6i 0.376543i
\(25\) −2.73438e6 − 9.37500e6i −0.280000 − 0.960000i
\(26\) −7.44394e6 −0.626521
\(27\) −9.35496e6 + 9.35496e6i −0.651963 + 0.651963i
\(28\) 4.30438e6 + 4.30438e6i 0.250104 + 0.250104i
\(29\) 2.47908e7i 1.20865i 0.796737 + 0.604326i \(0.206558\pi\)
−0.796737 + 0.604326i \(0.793442\pi\)
\(30\) 1.09800e7 − 1.46400e7i 0.451852 − 0.602469i
\(31\) −1.00660e7 −0.351600 −0.175800 − 0.984426i \(-0.556251\pi\)
−0.175800 + 0.984426i \(0.556251\pi\)
\(32\) −4.19430e6 + 4.19430e6i −0.125000 + 0.125000i
\(33\) 3.17318e7 + 3.17318e7i 0.810822 + 0.810822i
\(34\) − 6.01611e7i − 1.32410i
\(35\) −5.25438e6 − 3.67806e7i −0.100042 − 0.700292i
\(36\) 4.05965e6 0.0671392
\(37\) 5.63879e7 − 5.63879e7i 0.813163 − 0.813163i −0.171944 − 0.985107i \(-0.555005\pi\)
0.985107 + 0.171944i \(0.0550048\pi\)
\(38\) 1.76250e7 + 1.76250e7i 0.222439 + 0.222439i
\(39\) 8.51400e7i 0.943649i
\(40\) 3.58400e7 − 5.12000e6i 0.350000 − 0.0500000i
\(41\) −1.53004e8 −1.32063 −0.660317 − 0.750987i \(-0.729578\pi\)
−0.660317 + 0.750987i \(0.729578\pi\)
\(42\) 4.92314e7 − 4.92314e7i 0.376700 − 0.376700i
\(43\) 5.93725e7 + 5.93725e7i 0.403871 + 0.403871i 0.879595 − 0.475724i \(-0.157814\pi\)
−0.475724 + 0.879595i \(0.657814\pi\)
\(44\) 8.87798e7i 0.538333i
\(45\) −1.98225e7 − 1.48669e7i −0.107423 − 0.0805670i
\(46\) −1.67304e8 −0.812303
\(47\) 1.72339e8 − 1.72339e8i 0.751441 − 0.751441i −0.223307 − 0.974748i \(-0.571685\pi\)
0.974748 + 0.223307i \(0.0716852\pi\)
\(48\) 4.79724e7 + 4.79724e7i 0.188272 + 0.188272i
\(49\) 1.41120e8i 0.499583i
\(50\) −1.93750e8 − 1.06250e8i −0.620000 − 0.340000i
\(51\) −6.88092e8 −1.99432
\(52\) −1.19103e8 + 1.19103e8i −0.313261 + 0.313261i
\(53\) 1.96386e8 + 1.96386e8i 0.469602 + 0.469602i 0.901786 − 0.432183i \(-0.142257\pi\)
−0.432183 + 0.901786i \(0.642257\pi\)
\(54\) 2.99359e8i 0.651963i
\(55\) 3.25121e8 − 4.33495e8i 0.645999 − 0.861332i
\(56\) 1.37740e8 0.250104
\(57\) 2.01585e8 − 2.01585e8i 0.335031 − 0.335031i
\(58\) 3.96653e8 + 3.96653e8i 0.604326 + 0.604326i
\(59\) − 6.94069e8i − 0.970829i −0.874284 − 0.485414i \(-0.838669\pi\)
0.874284 − 0.485414i \(-0.161331\pi\)
\(60\) −5.85600e7 − 4.09920e8i −0.0753086 − 0.527160i
\(61\) 9.06186e8 1.07292 0.536461 − 0.843925i \(-0.319761\pi\)
0.536461 + 0.843925i \(0.319761\pi\)
\(62\) −1.61056e8 + 1.61056e8i −0.175800 + 0.175800i
\(63\) −6.66591e7 − 6.66591e7i −0.0671671 − 0.0671671i
\(64\) 1.34218e8i 0.125000i
\(65\) 1.01773e9 − 1.45389e8i 0.877130 − 0.125304i
\(66\) 1.01542e9 0.810822
\(67\) −9.62074e8 + 9.62074e8i −0.712581 + 0.712581i −0.967075 − 0.254493i \(-0.918091\pi\)
0.254493 + 0.967075i \(0.418091\pi\)
\(68\) −9.62577e8 − 9.62577e8i −0.662050 − 0.662050i
\(69\) 1.91354e9i 1.22347i
\(70\) −6.72560e8 − 5.04420e8i −0.400167 − 0.300125i
\(71\) −3.12088e9 −1.72976 −0.864878 − 0.501982i \(-0.832605\pi\)
−0.864878 + 0.501982i \(0.832605\pi\)
\(72\) 6.49544e7 − 6.49544e7i 0.0335696 − 0.0335696i
\(73\) −6.36339e8 − 6.36339e8i −0.306955 − 0.306955i 0.536772 − 0.843727i \(-0.319643\pi\)
−0.843727 + 0.536772i \(0.819643\pi\)
\(74\) − 1.80441e9i − 0.813163i
\(75\) −1.21523e9 + 2.21602e9i −0.512099 + 0.933827i
\(76\) 5.63999e8 0.222439
\(77\) 1.45776e9 − 1.45776e9i 0.538557 − 0.538557i
\(78\) 1.36224e9 + 1.36224e9i 0.471825 + 0.471825i
\(79\) 1.96800e9i 0.639571i 0.947490 + 0.319786i \(0.103611\pi\)
−0.947490 + 0.319786i \(0.896389\pi\)
\(80\) 4.91520e8 − 6.55360e8i 0.150000 − 0.200000i
\(81\) 3.89211e9 1.11625
\(82\) −2.44806e9 + 2.44806e9i −0.660317 + 0.660317i
\(83\) −5.18382e9 − 5.18382e9i −1.31601 − 1.31601i −0.916904 − 0.399107i \(-0.869320\pi\)
−0.399107 − 0.916904i \(-0.630680\pi\)
\(84\) − 1.57540e9i − 0.376700i
\(85\) 1.17502e9 + 8.22514e9i 0.264820 + 1.85374i
\(86\) 1.89992e9 0.403871
\(87\) 4.53672e9 − 4.53672e9i 0.910219 − 0.910219i
\(88\) 1.42048e9 + 1.42048e9i 0.269166 + 0.269166i
\(89\) − 7.77138e9i − 1.39171i −0.718183 − 0.695854i \(-0.755026\pi\)
0.718183 − 0.695854i \(-0.244974\pi\)
\(90\) −5.55030e8 + 7.92900e7i −0.0939948 + 0.0134278i
\(91\) 3.91132e9 0.626782
\(92\) −2.67687e9 + 2.67687e9i −0.406152 + 0.406152i
\(93\) 1.84208e9 + 1.84208e9i 0.264785 + 0.264785i
\(94\) − 5.51486e9i − 0.751441i
\(95\) −2.75390e9 − 2.06542e9i −0.355902 − 0.266926i
\(96\) 1.53512e9 0.188272
\(97\) −6.40361e8 + 6.40361e8i −0.0745704 + 0.0745704i −0.743408 − 0.668838i \(-0.766792\pi\)
0.668838 + 0.743408i \(0.266792\pi\)
\(98\) 2.25792e9 + 2.25792e9i 0.249792 + 0.249792i
\(99\) − 1.37487e9i − 0.144573i
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.a.3.1 ✓ 2
3.2 odd 2 90.11.g.b.73.1 2
4.3 odd 2 80.11.p.b.33.1 2
5.2 odd 4 inner 10.11.c.a.7.1 yes 2
5.3 odd 4 50.11.c.c.7.1 2
5.4 even 2 50.11.c.c.43.1 2
15.2 even 4 90.11.g.b.37.1 2
20.7 even 4 80.11.p.b.17.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
10.11.c.a.3.1 ✓ 2 1.1 even 1 trivial
10.11.c.a.7.1 yes 2 5.2 odd 4 inner
50.11.c.c.7.1 2 5.3 odd 4
50.11.c.c.43.1 2 5.4 even 2
80.11.p.b.17.1 2 20.7 even 4
80.11.p.b.33.1 2 4.3 odd 2
90.11.g.b.37.1 2 15.2 even 4
90.11.g.b.73.1 2 3.2 odd 2