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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 33.1
Root \(-1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 80.33
Dual form 80.11.p.b.17.1

$q$-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 \(\left(\mathbb{Z}/80\mathbb{Z}\right)^\times\).

\(n\) \(17\) \(21\) \(31\)
\(\chi(n)\) \(e\left(\frac{3}{4}\right)\) \(1\) \(1\)

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 0.975054 − 0.221968i \(-0.0712479\pi\)
−0.221968 + 0.975054i \(0.571248\pi\)
\(4\) 0 0
\(5\) −1875.00 + 2500.00i −0.600000 + 0.800000i
\(6\) 0 0
\(7\) 8407.00 − 8407.00i 0.500208 − 0.500208i −0.411294 − 0.911503i \(-0.634923\pi\)
0.911503 + 0.411294i \(0.134923\pi\)
\(8\) 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 0.320670 − 0.947191i \(-0.396092\pi\)
−0.947191 + 0.320670i \(0.896092\pi\)
\(14\) 0 0
\(15\) −800625. + 114375.i −1.05432 + 0.150617i
\(16\) 0 0
\(17\) 1.88003e6 − 1.88003e6i 1.32410 − 1.32410i 0.413676 − 0.910424i \(-0.364245\pi\)
0.910424 − 0.413676i \(-0.135755\pi\)
\(18\) 0 0
\(19\) − 1.10156e6i − 0.444877i −0.974947 − 0.222439i \(-0.928598\pi\)
0.974947 − 0.222439i \(-0.0714017\pi\)
\(20\) 0 0
\(21\) 3.07696e6 0.753400
\(22\) 0 0
\(23\) 5.22826e6 + 5.22826e6i 0.812303 + 0.812303i 0.984979 − 0.172675i \(-0.0552412\pi\)
−0.172675 + 0.984979i \(0.555241\pi\)
\(24\) 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 \(0.206558\pi\)
−0.796737 + 0.604326i \(0.793442\pi\)
\(30\) 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 −0.171944 − 0.985107i \(-0.555005\pi\)
0.985107 + 0.171944i \(0.0550048\pi\)
\(38\) 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 0.475724 − 0.879595i \(-0.342186\pi\)
−0.879595 + 0.475724i \(0.842186\pi\)
\(44\) 0 0
\(45\) −1.98225e7 − 1.48669e7i −0.107423 − 0.0805670i
\(46\) 0 0
\(47\) −1.72339e8 + 1.72339e8i −0.751441 + 0.751441i −0.974748 − 0.223307i \(-0.928315\pi\)
0.223307 + 0.974748i \(0.428315\pi\)
\(48\) 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 0.901786 − 0.432183i \(-0.142257\pi\)
−0.432183 + 0.901786i \(0.642257\pi\)
\(54\) 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 \(0.161331\pi\)
−0.874284 + 0.485414i \(0.838669\pi\)
\(60\) 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 −0.254493 − 0.967075i \(-0.581909\pi\)
0.967075 + 0.254493i \(0.0819087\pi\)
\(68\) 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 0.536772 − 0.843727i \(-0.319643\pi\)
−0.843727 + 0.536772i \(0.819643\pi\)
\(74\) 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 \(-0.896389\pi\)
0.947490 − 0.319786i \(-0.103611\pi\)
\(80\) 0 0
\(81\) 3.89211e9 1.11625
\(82\) 0 0
\(83\) 5.18382e9 + 5.18382e9i 1.31601 + 1.31601i 0.916904 + 0.399107i \(0.130680\pi\)
0.399107 + 0.916904i \(0.369320\pi\)
\(84\) 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 \(-0.755026\pi\)
0.718183 − 0.695854i \(-0.244974\pi\)
\(90\) 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 −0.743408 − 0.668838i \(-0.766792\pi\)
0.668838 + 0.743408i \(0.266792\pi\)
\(98\) 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.33.1 2
4.3 odd 2 10.11.c.a.3.1 ✓ 2
5.2 odd 4 inner 80.11.p.b.17.1 2
12.11 even 2 90.11.g.b.73.1 2
20.3 even 4 50.11.c.c.7.1 2
20.7 even 4 10.11.c.a.7.1 yes 2
20.19 odd 2 50.11.c.c.43.1 2
60.47 odd 4 90.11.g.b.37.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
10.11.c.a.3.1 ✓ 2 4.3 odd 2
10.11.c.a.7.1 yes 2 20.7 even 4
50.11.c.c.7.1 2 20.3 even 4
50.11.c.c.43.1 2 20.19 odd 2
80.11.p.b.17.1 2 5.2 odd 4 inner
80.11.p.b.33.1 2 1.1 even 1 trivial
90.11.g.b.37.1 2 60.47 odd 4
90.11.g.b.73.1 2 12.11 even 2