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Properties

Label 31.8.g.a.7.4
Level $31$
Weight $8$
Character 31.7
Analytic conductor $9.684$
Analytic rank $0$
Dimension $144$
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] = [31,8,Mod(7,31)] 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("31.7"); S:= CuspForms(chi, 8); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(31, base_ring=CyclotomicField(30)) chi = DirichletCharacter(H, H._module([28])) N = Newforms(chi, 8, names="a")
 
Level: \( N \) \(=\) \( 31 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 31.g (of order \(15\), degree \(8\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(9.68393579001\)
Analytic rank: \(0\)
Dimension: \(144\)
Relative dimension: \(18\) over \(\Q(\zeta_{15})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{15}]$

Embedding invariants

Embedding label 7.4
Character \(\chi\) \(=\) 31.7
Dual form 31.8.g.a.9.4

$q$-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+(-5.03785 - 15.5049i) q^{2} +(8.90236 - 1.89225i) q^{3} +(-111.468 + 80.9861i) q^{4} +(146.446 - 253.652i) q^{5} +(-74.1879 - 128.497i) q^{6} +(-709.897 - 316.067i) q^{7} +(129.014 + 93.7340i) q^{8} +(-1922.25 + 855.842i) q^{9} +(-4670.62 - 992.770i) q^{10} +(-334.291 - 3180.56i) q^{11} +(-839.079 + 931.892i) q^{12} +(8489.72 + 9428.79i) q^{13} +(-1324.23 + 12599.2i) q^{14} +(823.740 - 2535.21i) q^{15} +(-4646.46 + 14300.3i) q^{16} +(2069.78 - 19692.7i) q^{17} +(22953.8 + 25492.7i) q^{18} +(-26436.1 + 29360.3i) q^{19} +(4218.26 + 40134.1i) q^{20} +(-6917.84 - 1470.43i) q^{21} +(-47630.2 + 21206.3i) q^{22} +(-41439.7 - 30107.7i) q^{23} +(1325.89 + 590.326i) q^{24} +(-3830.32 - 6634.31i) q^{25} +(103423. - 179133. i) q^{26} +(-31596.1 + 22955.9i) q^{27} +(104728. - 22260.6i) q^{28} +(-21466.2 - 66066.1i) q^{29} -43458.1 q^{30} +(-88043.0 - 140574. i) q^{31} +265545. q^{32} +(-8994.41 - 27681.9i) q^{33} +(-315760. + 67116.9i) q^{34} +(-184132. + 133780. i) q^{35} +(144958. - 251074. i) q^{36} +(-95124.3 - 164760. i) q^{37} +(588409. + 261976. i) q^{38} +(93420.2 + 67873.7i) q^{39} +(42669.3 - 18997.6i) q^{40} +(20526.4 + 4363.03i) q^{41} +(12052.1 + 114668. i) q^{42} +(410436. - 455835. i) q^{43} +(294844. + 327457. i) q^{44} +(-64420.2 + 612917. i) q^{45} +(-258050. + 794197. i) q^{46} +(415920. - 1.28007e6i) q^{47} +(-14304.6 + 136099. i) q^{48} +(-147002. - 163262. i) q^{49} +(-83567.7 + 92811.3i) q^{50} +(-18837.6 - 179228. i) q^{51} +(-1.70993e6 - 363457. i) q^{52} +(-1.45022e6 + 645680. i) q^{53} +(515106. + 374246. i) q^{54} +(-855711. - 380987. i) q^{55} +(-61960.3 - 107318. i) q^{56} +(-179786. + 311399. i) q^{57} +(-916204. + 665661. i) q^{58} +(-159380. + 33877.2i) q^{59} +(113496. + 349306. i) q^{60} +943722. q^{61} +(-1.73604e6 + 2.07329e6i) q^{62} +1.63511e6 q^{63} +(-743030. - 2.28681e6i) q^{64} +(3.63492e6 - 772625. i) q^{65} +(-383893. + 278915. i) q^{66} +(457794. - 792922. i) q^{67} +(1.36412e6 + 2.36272e6i) q^{68} +(-425883. - 189615. i) q^{69} +(3.00188e6 + 2.18099e6i) q^{70} +(-1.12845e6 + 502418. i) q^{71} +(-328219. - 69765.0i) q^{72} +(-183897. - 1.74967e6i) q^{73} +(-2.07537e6 + 2.30493e6i) q^{74} +(-46652.7 - 51813.0i) q^{75} +(569000. - 5.41368e6i) q^{76} +(-767958. + 2.36353e6i) q^{77} +(581739. - 1.79041e6i) q^{78} +(125192. - 1.19112e6i) q^{79} +(2.94685e6 + 3.27281e6i) q^{80} +(2.84137e6 - 3.15566e6i) q^{81} +(-35760.7 - 340240. i) q^{82} +(2.54224e6 + 540370. i) q^{83} +(890200. - 396343. i) q^{84} +(-4.69197e6 - 3.40891e6i) q^{85} +(-9.13539e6 - 4.06734e6i) q^{86} +(-316113. - 547524. i) q^{87} +(254999. - 441671. i) q^{88} +(7.30544e6 - 5.30771e6i) q^{89} +(9.82776e6 - 2.08895e6i) q^{90} +(-3.04670e6 - 9.37679e6i) q^{91} +7.05750e6 q^{92} +(-1.04979e6 - 1.08484e6i) q^{93} -2.19427e7 q^{94} +(3.57582e6 + 1.10053e7i) q^{95} +(2.36398e6 - 502479. i) q^{96} +(-7.18610e6 + 5.22100e6i) q^{97} +(-1.79079e6 + 3.10174e6i) q^{98} +(3.36465e6 + 5.82774e6i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 144 q - 6 q^{2} - 8 q^{3} - 2362 q^{4} - 75 q^{5} - 1637 q^{6} + 5518 q^{7} + 3401 q^{8} + 14542 q^{9} - 5312 q^{10} + 1361 q^{11} - 18281 q^{12} + 22457 q^{13} + 28470 q^{14} - 86918 q^{15} - 155730 q^{16}+ \cdots - 102755966 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/31\mathbb{Z}\right)^\times\).

\(n\) \(3\)
\(\chi(n)\) \(e\left(\frac{14}{15}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −5.03785 − 15.5049i −0.445287 − 1.37045i −0.882169 − 0.470933i \(-0.843917\pi\)
0.436882 − 0.899519i \(-0.356083\pi\)
\(3\) 8.90236 − 1.89225i 0.190362 − 0.0404627i −0.111744 − 0.993737i \(-0.535644\pi\)
0.302106 + 0.953274i \(0.402310\pi\)
\(4\) −111.468 + 80.9861i −0.870842 + 0.632704i
\(5\) 146.446 − 253.652i 0.523941 − 0.907492i −0.475671 − 0.879623i \(-0.657795\pi\)
0.999612 − 0.0278688i \(-0.00887206\pi\)
\(6\) −74.1879 − 128.497i −0.140218 − 0.242865i
\(7\) −709.897 − 316.067i −0.782262 − 0.348286i −0.0235594 − 0.999722i \(-0.507500\pi\)
−0.758703 + 0.651437i \(0.774167\pi\)
\(8\) 129.014 + 93.7340i 0.0890884 + 0.0647265i
\(9\) −1922.25 + 855.842i −0.878945 + 0.391332i
\(10\) −4670.62 − 992.770i −1.47698 − 0.313942i
\(11\) −334.291 − 3180.56i −0.0757268 − 0.720493i −0.964846 − 0.262817i \(-0.915349\pi\)
0.889119 − 0.457676i \(-0.151318\pi\)
\(12\) −839.079 + 931.892i −0.140174 + 0.155679i
\(13\) 8489.72 + 9428.79i 1.07175 + 1.19029i 0.980917 + 0.194428i \(0.0622850\pi\)
0.0908289 + 0.995867i \(0.471048\pi\)
\(14\) −1324.23 + 12599.2i −0.128978 + 1.22714i
\(15\) 823.740 − 2535.21i 0.0630189 − 0.193952i
\(16\) −4646.46 + 14300.3i −0.283597 + 0.872823i
\(17\) 2069.78 − 19692.7i 0.102177 − 0.972150i −0.816555 − 0.577268i \(-0.804119\pi\)
0.918732 − 0.394882i \(-0.129215\pi\)
\(18\) 22953.8 + 25492.7i 0.927684 + 1.03030i
\(19\) −26436.1 + 29360.3i −0.884219 + 0.982024i −0.999937 − 0.0112579i \(-0.996416\pi\)
0.115718 + 0.993282i \(0.463083\pi\)
\(20\) 4218.26 + 40134.1i 0.117904 + 1.12178i
\(21\) −6917.84 − 1470.43i −0.163006 − 0.0346479i
\(22\) −47630.2 + 21206.3i −0.953680 + 0.424606i
\(23\) −41439.7 − 30107.7i −0.710181 − 0.515977i 0.173051 − 0.984913i \(-0.444638\pi\)
−0.883232 + 0.468936i \(0.844638\pi\)
\(24\) 1325.89 + 590.326i 0.0195781 + 0.00871672i
\(25\) −3830.32 − 6634.31i −0.0490281 − 0.0849191i
\(26\) 103423. − 179133.i 1.15401 − 1.99880i
\(27\) −31596.1 + 22955.9i −0.308930 + 0.224451i
\(28\) 104728. − 22260.6i 0.901588 − 0.191639i
\(29\) −21466.2 − 66066.1i −0.163441 − 0.503020i 0.835477 − 0.549526i \(-0.185192\pi\)
−0.998918 + 0.0465054i \(0.985192\pi\)
\(30\) −43458.1 −0.293864
\(31\) −88043.0 − 140574.i −0.530798 − 0.847499i
\(32\) 265545. 1.43256
\(33\) −8994.41 − 27681.9i −0.0435686 − 0.134090i
\(34\) −315760. + 67116.9i −1.37778 + 0.292857i
\(35\) −184132. + 133780.i −0.725926 + 0.527416i
\(36\) 144958. − 251074.i 0.517825 − 0.896900i
\(37\) −95124.3 − 164760.i −0.308735 − 0.534744i 0.669351 − 0.742946i \(-0.266572\pi\)
−0.978086 + 0.208202i \(0.933239\pi\)
\(38\) 588409. + 261976.i 1.73955 + 0.774497i
\(39\) 93420.2 + 67873.7i 0.252182 + 0.183221i
\(40\) 42669.3 − 18997.6i 0.105416 − 0.0469342i
\(41\) 20526.4 + 4363.03i 0.0465125 + 0.00988654i 0.231109 − 0.972928i \(-0.425765\pi\)
−0.184597 + 0.982814i \(0.559098\pi\)
\(42\) 12052.1 + 114668.i 0.0251010 + 0.238820i
\(43\) 410436. − 455835.i 0.787237 − 0.874316i −0.207345 − 0.978268i \(-0.566482\pi\)
0.994582 + 0.103952i \(0.0331489\pi\)
\(44\) 294844. + 327457.i 0.521804 + 0.579522i
\(45\) −64420.2 + 612917.i −0.105385 + 1.00267i
\(46\) −258050. + 794197.i −0.390887 + 1.20303i
\(47\) 415920. − 1.28007e6i 0.584342 − 1.79842i −0.0175540 − 0.999846i \(-0.505588\pi\)
0.601896 − 0.798574i \(-0.294412\pi\)
\(48\) −14304.6 + 136099.i −0.0186694 + 0.177628i
\(49\) −147002. − 163262.i −0.178499 − 0.198243i
\(50\) −83567.7 + 92811.3i −0.0945461 + 0.105004i
\(51\) −18837.6 − 179228.i −0.0198852 − 0.189195i
\(52\) −1.70993e6 − 363457.i −1.68642 − 0.358461i
\(53\) −1.45022e6 + 645680.i −1.33804 + 0.595734i −0.945986 − 0.324209i \(-0.894902\pi\)
−0.392054 + 0.919942i \(0.628235\pi\)
\(54\) 515106. + 374246.i 0.445162 + 0.323429i
\(55\) −855711. − 380987.i −0.693518 − 0.308774i
\(56\) −61960.3 − 107318.i −0.0471472 − 0.0816613i
\(57\) −179786. + 311399.i −0.128586 + 0.222718i
\(58\) −916204. + 665661.i −0.616587 + 0.447977i
\(59\) −159380. + 33877.2i −0.101030 + 0.0214746i −0.258149 − 0.966105i \(-0.583113\pi\)
0.157119 + 0.987580i \(0.449779\pi\)
\(60\) 113496. + 349306.i 0.0678348 + 0.208774i
\(61\) 943722. 0.532341 0.266170 − 0.963926i \(-0.414242\pi\)
0.266170 + 0.963926i \(0.414242\pi\)
\(62\) −1.73604e6 + 2.07329e6i −0.925099 + 1.10481i
\(63\) 1.63511e6 0.823861
\(64\) −743030. − 2.28681e6i −0.354304 − 1.09044i
\(65\) 3.63492e6 − 772625.i 1.64171 − 0.348957i
\(66\) −383893. + 278915.i −0.164364 + 0.119417i
\(67\) 457794. − 792922.i 0.185955 − 0.322084i −0.757943 − 0.652321i \(-0.773795\pi\)
0.943898 + 0.330237i \(0.107129\pi\)
\(68\) 1.36412e6 + 2.36272e6i 0.526103 + 0.911237i
\(69\) −425883. − 189615.i −0.156069 − 0.0694866i
\(70\) 3.00188e6 + 2.18099e6i 1.04604 + 0.759995i
\(71\) −1.12845e6 + 502418.i −0.374178 + 0.166595i −0.585205 − 0.810886i \(-0.698986\pi\)
0.211027 + 0.977480i \(0.432319\pi\)
\(72\) −328219. − 69765.0i −0.103633 − 0.0220279i
\(73\) −183897. − 1.74967e6i −0.0553281 − 0.526412i −0.986724 − 0.162405i \(-0.948075\pi\)
0.931396 − 0.364007i \(-0.118592\pi\)
\(74\) −2.07537e6 + 2.30493e6i −0.595366 + 0.661220i
\(75\) −46652.7 − 51813.0i −0.0127692 − 0.0141816i
\(76\) 569000. − 5.41368e6i 0.148684 − 1.41464i
\(77\) −767958. + 2.36353e6i −0.191699 + 0.589989i
\(78\) 581739. − 1.79041e6i 0.138802 − 0.427190i
\(79\) 125192. − 1.19112e6i 0.0285681 − 0.271807i −0.970908 − 0.239451i \(-0.923032\pi\)
0.999476 − 0.0323557i \(-0.0103009\pi\)
\(80\) 2.94685e6 + 3.27281e6i 0.643492 + 0.714670i
\(81\) 2.84137e6 − 3.15566e6i 0.594061 − 0.659771i
\(82\) −35760.7 − 340240.i −0.00716238 − 0.0681455i
\(83\) 2.54224e6 + 540370.i 0.488027 + 0.103733i 0.445352 − 0.895356i \(-0.353079\pi\)
0.0426749 + 0.999089i \(0.486412\pi\)
\(84\) 890200. − 396343.i 0.163874 − 0.0729614i
\(85\) −4.69197e6 − 3.40891e6i −0.828684 − 0.602074i
\(86\) −9.13539e6 − 4.06734e6i −1.54875 − 0.689550i
\(87\) −316113. − 547524.i −0.0514666 − 0.0891427i
\(88\) 254999. − 441671.i 0.0398886 − 0.0690890i
\(89\) 7.30544e6 − 5.30771e6i 1.09845 − 0.798073i 0.117646 − 0.993056i \(-0.462465\pi\)
0.980807 + 0.194983i \(0.0624651\pi\)
\(90\) 9.82776e6 − 2.08895e6i 1.42104 − 0.302051i
\(91\) −3.04670e6 − 9.37679e6i −0.423824 − 1.30440i
\(92\) 7.05750e6 0.944916
\(93\) −1.04979e6 − 1.08484e6i −0.135336 − 0.139854i
\(94\) −2.19427e7 −2.72485
\(95\) 3.57582e6 + 1.10053e7i 0.427901 + 1.31694i
\(96\) 2.36398e6 − 502479.i 0.272706 − 0.0579654i
\(97\) −7.18610e6 + 5.22100e6i −0.799451 + 0.580835i −0.910753 − 0.412951i \(-0.864498\pi\)
0.111302 + 0.993787i \(0.464498\pi\)
\(98\) −1.79079e6 + 3.10174e6i −0.192200 + 0.332900i
\(99\) 3.36465e6 + 5.82774e6i 0.348511 + 0.603639i
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 31.8.g.a.7.4 ✓ 144
31.9 even 15 inner 31.8.g.a.9.4 yes 144
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
31.8.g.a.7.4 ✓ 144 1.1 even 1 trivial
31.8.g.a.9.4 yes 144 31.9 even 15 inner