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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [80,11,Mod(17,80)] 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("80.17"); S:= CuspForms(chi, 11); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(80, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([0, 0, 1])) N = Newforms(chi, 11, names="a")
 
Level: \( N \) \(=\) \( 80 = 2^{4} \cdot 5 \)
Weight: \( k \) \(=\) \( 11 \)
Character orbit: \([\chi]\) \(=\) 80.p (of order \(4\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,366] 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: \(50.8285802139\)
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: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 10)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 17.1
Root \(1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 80.17
Dual form 80.11.p.b.33.1

$q$-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)
 
\(f(q)\) \(=\) \(q+(183.000 - 183.000i) q^{3} +(-1875.00 - 2500.00i) q^{5} +(8407.00 + 8407.00i) q^{7} -7929.00i q^{9} +173398. q^{11} +(-232623. + 232623. i) q^{13} +(-800625. - 114375. i) q^{15} +(1.88003e6 + 1.88003e6i) q^{17} +1.10156e6i q^{19} +3.07696e6 q^{21} +(5.22826e6 - 5.22826e6i) q^{23} +(-2.73438e6 + 9.37500e6i) q^{25} +(9.35496e6 + 9.35496e6i) q^{27} -2.47908e7i q^{29} +1.00660e7 q^{31} +(3.17318e7 - 3.17318e7i) q^{33} +(5.25438e6 - 3.67806e7i) q^{35} +(5.63879e7 + 5.63879e7i) q^{37} +8.51400e7i q^{39} -1.53004e8 q^{41} +(-5.93725e7 + 5.93725e7i) q^{43} +(-1.98225e7 + 1.48669e7i) q^{45} +(-1.72339e8 - 1.72339e8i) q^{47} -1.41120e8i q^{49} +6.88092e8 q^{51} +(1.96386e8 - 1.96386e8i) q^{53} +(-3.25121e8 - 4.33495e8i) q^{55} +(2.01585e8 + 2.01585e8i) q^{57} -6.94069e8i q^{59} +9.06186e8 q^{61} +(6.66591e7 - 6.66591e7i) q^{63} +(1.01773e9 + 1.45389e8i) q^{65} +(9.62074e8 + 9.62074e8i) q^{67} -1.91354e9i q^{69} +3.12088e9 q^{71} +(-6.36339e8 + 6.36339e8i) q^{73} +(1.21523e9 + 2.21602e9i) q^{75} +(1.45776e9 + 1.45776e9i) q^{77} +1.96800e9i q^{79} +3.89211e9 q^{81} +(5.18382e9 - 5.18382e9i) q^{83} +(1.17502e9 - 8.22514e9i) q^{85} +(-4.53672e9 - 4.53672e9i) q^{87} +7.77138e9i q^{89} -3.91132e9 q^{91} +(1.84208e9 - 1.84208e9i) q^{93} +(2.75390e9 - 2.06542e9i) q^{95} +(-6.40361e8 - 6.40361e8i) q^{97} -1.37487e9i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 366 q^{3} - 3750 q^{5} + 16814 q^{7} + 346796 q^{11} - 465246 q^{13} - 1601250 q^{15} + 3760066 q^{17} + 6153924 q^{21} + 10456526 q^{23} - 5468750 q^{25} + 18709920 q^{27} + 20131996 q^{31} + 63463668 q^{33}+ \cdots - 1280722494 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/80\mathbb{Z}\right)^\times\).

\(n\) \(17\) \(21\) \(31\)
\(\chi(n)\) \(e\left(\frac{1}{4}\right)\) \(1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 183.000 − 183.000i 0.753086 − 0.753086i −0.221968 − 0.975054i \(-0.571248\pi\)
0.975054 + 0.221968i \(0.0712479\pi\)
\(4\) 0 0
\(5\) −1875.00 − 2500.00i −0.600000 − 0.800000i
\(6\) 0 0
\(7\) 8407.00 + 8407.00i 0.500208 + 0.500208i 0.911503 − 0.411294i \(-0.134923\pi\)
−0.411294 + 0.911503i \(0.634923\pi\)
\(8\) 0 0
\(9\) − 7929.00i − 0.134278i
\(10\) 0 0
\(11\) 173398. 1.07667 0.538333 − 0.842732i \(-0.319054\pi\)
0.538333 + 0.842732i \(0.319054\pi\)
\(12\) 0 0
\(13\) −232623. + 232623.i −0.626521 + 0.626521i −0.947191 − 0.320670i \(-0.896092\pi\)
0.320670 + 0.947191i \(0.396092\pi\)
\(14\) 0 0
\(15\) −800625. − 114375.i −1.05432 − 0.150617i
\(16\) 0 0
\(17\) 1.88003e6 + 1.88003e6i 1.32410 + 1.32410i 0.910424 + 0.413676i \(0.135755\pi\)
0.413676 + 0.910424i \(0.364245\pi\)
\(18\) 0 0
\(19\) 1.10156e6i 0.444877i 0.974947 + 0.222439i \(0.0714017\pi\)
−0.974947 + 0.222439i \(0.928598\pi\)
\(20\) 0 0
\(21\) 3.07696e6 0.753400
\(22\) 0 0
\(23\) 5.22826e6 − 5.22826e6i 0.812303 − 0.812303i −0.172675 − 0.984979i \(-0.555241\pi\)
0.984979 + 0.172675i \(0.0552412\pi\)
\(24\) 0 0
\(25\) −2.73438e6 + 9.37500e6i −0.280000 + 0.960000i
\(26\) 0 0
\(27\) 9.35496e6 + 9.35496e6i 0.651963 + 0.651963i
\(28\) 0 0
\(29\) − 2.47908e7i − 1.20865i −0.796737 − 0.604326i \(-0.793442\pi\)
0.796737 − 0.604326i \(-0.206558\pi\)
\(30\) 0 0
\(31\) 1.00660e7 0.351600 0.175800 − 0.984426i \(-0.443749\pi\)
0.175800 + 0.984426i \(0.443749\pi\)
\(32\) 0 0
\(33\) 3.17318e7 − 3.17318e7i 0.810822 − 0.810822i
\(34\) 0 0
\(35\) 5.25438e6 − 3.67806e7i 0.100042 − 0.700292i
\(36\) 0 0
\(37\) 5.63879e7 + 5.63879e7i 0.813163 + 0.813163i 0.985107 − 0.171944i \(-0.0550048\pi\)
−0.171944 + 0.985107i \(0.555005\pi\)
\(38\) 0 0
\(39\) 8.51400e7i 0.943649i
\(40\) 0 0
\(41\) −1.53004e8 −1.32063 −0.660317 − 0.750987i \(-0.729578\pi\)
−0.660317 + 0.750987i \(0.729578\pi\)
\(42\) 0 0
\(43\) −5.93725e7 + 5.93725e7i −0.403871 + 0.403871i −0.879595 − 0.475724i \(-0.842186\pi\)
0.475724 + 0.879595i \(0.342186\pi\)
\(44\) 0 0
\(45\) −1.98225e7 + 1.48669e7i −0.107423 + 0.0805670i
\(46\) 0 0
\(47\) −1.72339e8 − 1.72339e8i −0.751441 − 0.751441i 0.223307 − 0.974748i \(-0.428315\pi\)
−0.974748 + 0.223307i \(0.928315\pi\)
\(48\) 0 0
\(49\) − 1.41120e8i − 0.499583i
\(50\) 0 0
\(51\) 6.88092e8 1.99432
\(52\) 0 0
\(53\) 1.96386e8 − 1.96386e8i 0.469602 − 0.469602i −0.432183 − 0.901786i \(-0.642257\pi\)
0.901786 + 0.432183i \(0.142257\pi\)
\(54\) 0 0
\(55\) −3.25121e8 − 4.33495e8i −0.645999 − 0.861332i
\(56\) 0 0
\(57\) 2.01585e8 + 2.01585e8i 0.335031 + 0.335031i
\(58\) 0 0
\(59\) − 6.94069e8i − 0.970829i −0.874284 − 0.485414i \(-0.838669\pi\)
0.874284 − 0.485414i \(-0.161331\pi\)
\(60\) 0 0
\(61\) 9.06186e8 1.07292 0.536461 − 0.843925i \(-0.319761\pi\)
0.536461 + 0.843925i \(0.319761\pi\)
\(62\) 0 0
\(63\) 6.66591e7 − 6.66591e7i 0.0671671 − 0.0671671i
\(64\) 0 0
\(65\) 1.01773e9 + 1.45389e8i 0.877130 + 0.125304i
\(66\) 0 0
\(67\) 9.62074e8 + 9.62074e8i 0.712581 + 0.712581i 0.967075 − 0.254493i \(-0.0819087\pi\)
−0.254493 + 0.967075i \(0.581909\pi\)
\(68\) 0 0
\(69\) − 1.91354e9i − 1.22347i
\(70\) 0 0
\(71\) 3.12088e9 1.72976 0.864878 − 0.501982i \(-0.167395\pi\)
0.864878 + 0.501982i \(0.167395\pi\)
\(72\) 0 0
\(73\) −6.36339e8 + 6.36339e8i −0.306955 + 0.306955i −0.843727 − 0.536772i \(-0.819643\pi\)
0.536772 + 0.843727i \(0.319643\pi\)
\(74\) 0 0
\(75\) 1.21523e9 + 2.21602e9i 0.512099 + 0.933827i
\(76\) 0 0
\(77\) 1.45776e9 + 1.45776e9i 0.538557 + 0.538557i
\(78\) 0 0
\(79\) 1.96800e9i 0.639571i 0.947490 + 0.319786i \(0.103611\pi\)
−0.947490 + 0.319786i \(0.896389\pi\)
\(80\) 0 0
\(81\) 3.89211e9 1.11625
\(82\) 0 0
\(83\) 5.18382e9 − 5.18382e9i 1.31601 − 1.31601i 0.399107 − 0.916904i \(-0.369320\pi\)
0.916904 − 0.399107i \(-0.130680\pi\)
\(84\) 0 0
\(85\) 1.17502e9 − 8.22514e9i 0.264820 − 1.85374i
\(86\) 0 0
\(87\) −4.53672e9 − 4.53672e9i −0.910219 − 0.910219i
\(88\) 0 0
\(89\) 7.77138e9i 1.39171i 0.718183 + 0.695854i \(0.244974\pi\)
−0.718183 + 0.695854i \(0.755026\pi\)
\(90\) 0 0
\(91\) −3.91132e9 −0.626782
\(92\) 0 0
\(93\) 1.84208e9 − 1.84208e9i 0.264785 − 0.264785i
\(94\) 0 0
\(95\) 2.75390e9 − 2.06542e9i 0.355902 − 0.266926i
\(96\) 0 0
\(97\) −6.40361e8 − 6.40361e8i −0.0745704 − 0.0745704i 0.668838 − 0.743408i \(-0.266792\pi\)
−0.743408 + 0.668838i \(0.766792\pi\)
\(98\) 0 0
\(99\) − 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 80.11.p.b.17.1 2
4.3 odd 2 10.11.c.a.7.1 yes 2
5.3 odd 4 inner 80.11.p.b.33.1 2
12.11 even 2 90.11.g.b.37.1 2
20.3 even 4 10.11.c.a.3.1 ✓ 2
20.7 even 4 50.11.c.c.43.1 2
20.19 odd 2 50.11.c.c.7.1 2
60.23 odd 4 90.11.g.b.73.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
10.11.c.a.3.1 ✓ 2 20.3 even 4
10.11.c.a.7.1 yes 2 4.3 odd 2
50.11.c.c.7.1 2 20.19 odd 2
50.11.c.c.43.1 2 20.7 even 4
80.11.p.b.17.1 2 1.1 even 1 trivial
80.11.p.b.33.1 2 5.3 odd 4 inner
90.11.g.b.37.1 2 12.11 even 2
90.11.g.b.73.1 2 60.23 odd 4