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Properties

Label 31.7.e.a.6.11
Level $31$
Weight $7$
Character 31.6
Analytic conductor $7.132$
Analytic rank $0$
Dimension $30$
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] = [31,7,Mod(6,31)] 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("31.6"); S:= CuspForms(chi, 7); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(31, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([5])) N = Newforms(chi, 7, names="a")
 
Level: \( N \) \(=\) \( 31 \)
Weight: \( k \) \(=\) \( 7 \)
Character orbit: \([\chi]\) \(=\) 31.e (of order \(6\), degree \(2\), 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: \(7.13167659222\)
Analytic rank: \(0\)
Dimension: \(30\)
Relative dimension: \(15\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 6.11
Character \(\chi\) \(=\) 31.6
Dual form 31.7.e.a.26.11

$q$-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+7.74164 q^{2} +(-12.6461 + 7.30123i) q^{3} -4.06698 q^{4} +(-28.4834 + 49.3346i) q^{5} +(-97.9016 + 56.5235i) q^{6} +(304.101 + 526.719i) q^{7} -526.950 q^{8} +(-257.884 + 446.668i) q^{9} +(-220.508 + 381.931i) q^{10} +(246.818 + 142.501i) q^{11} +(51.4314 - 29.6939i) q^{12} +(-1805.65 - 1042.49i) q^{13} +(2354.24 + 4077.67i) q^{14} -831.854i q^{15} -3819.17 q^{16} +(6903.49 - 3985.73i) q^{17} +(-1996.45 + 3457.95i) q^{18} +(-876.667 - 1518.43i) q^{19} +(115.841 - 200.643i) q^{20} +(-7691.39 - 4440.62i) q^{21} +(1910.78 + 1103.19i) q^{22} +20863.0i q^{23} +(6663.87 - 3847.38i) q^{24} +(6189.90 + 10721.2i) q^{25} +(-13978.7 - 8070.61i) q^{26} -18176.7i q^{27} +(-1236.77 - 2142.15i) q^{28} -42366.3i q^{29} -6439.92i q^{30} +(27400.6 + 11692.3i) q^{31} +4158.14 q^{32} -4161.72 q^{33} +(53444.4 - 30856.1i) q^{34} -34647.3 q^{35} +(1048.81 - 1816.59i) q^{36} +(1149.72 - 663.789i) q^{37} +(-6786.84 - 11755.2i) q^{38} +30445.9 q^{39} +(15009.3 - 25996.9i) q^{40} +(-49093.1 + 85031.7i) q^{41} +(-59544.0 - 34377.7i) q^{42} +(-38647.7 + 22313.3i) q^{43} +(-1003.81 - 579.547i) q^{44} +(-14690.8 - 25445.2i) q^{45} +161514. i q^{46} +53024.0 q^{47} +(48297.6 - 27884.7i) q^{48} +(-126130. + 218464. i) q^{49} +(47920.0 + 82999.8i) q^{50} +(-58201.5 + 100808. i) q^{51} +(7343.55 + 4239.80i) q^{52} +(92507.3 + 53409.1i) q^{53} -140717. i q^{54} +(-14060.4 + 8117.80i) q^{55} +(-160246. - 277554. i) q^{56} +(22172.8 + 12801.5i) q^{57} -327985. i q^{58} +(59948.5 + 103834. i) q^{59} +3383.13i q^{60} +211458. i q^{61} +(212126. + 90517.6i) q^{62} -313691. q^{63} +276618. q^{64} +(102862. - 59387.5i) q^{65} -32218.5 q^{66} +(255366. - 442307. i) q^{67} +(-28076.4 + 16209.9i) q^{68} +(-152325. - 263835. i) q^{69} -268227. q^{70} +(47835.7 - 82853.8i) q^{71} +(135892. - 235372. i) q^{72} +(90952.7 + 52511.6i) q^{73} +(8900.69 - 5138.81i) q^{74} +(-156556. - 90387.7i) q^{75} +(3565.39 + 6175.43i) q^{76} +173338. i q^{77} +235702. q^{78} +(77196.3 - 44569.3i) q^{79} +(108783. - 188418. i) q^{80} +(-55285.4 - 95757.1i) q^{81} +(-380061. + 658285. i) q^{82} +(156492. + 90350.6i) q^{83} +(31280.7 + 18059.9i) q^{84} +454108. i q^{85} +(-299197. + 172741. i) q^{86} +(309326. + 535769. i) q^{87} +(-130061. - 75090.7i) q^{88} +343254. i q^{89} +(-113731. - 196988. i) q^{90} -1.26809e6i q^{91} -84849.3i q^{92} +(-431879. + 52196.3i) q^{93} +410493. q^{94} +99881.7 q^{95} +(-52584.3 + 30359.5i) q^{96} -284177. q^{97} +(-976457. + 1.69127e6i) q^{98} +(-127301. + 73497.3i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 30 q - 4 q^{2} - 3 q^{3} + 1024 q^{4} - 73 q^{5} - 1446 q^{6} + 723 q^{7} - 1496 q^{8} + 3478 q^{9} - 848 q^{10} + 957 q^{11} - 2382 q^{12} + 8817 q^{13} - 8654 q^{14} + 25288 q^{16} - 3123 q^{17} + 6422 q^{18}+ \cdots - 3308442 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(3\)
\(\chi(n)\) \(e\left(\frac{5}{6}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 7.74164 0.967705 0.483853 − 0.875149i \(-0.339237\pi\)
0.483853 + 0.875149i \(0.339237\pi\)
\(3\) −12.6461 + 7.30123i −0.468374 + 0.270416i −0.715559 − 0.698552i \(-0.753828\pi\)
0.247185 + 0.968968i \(0.420495\pi\)
\(4\) −4.06698 −0.0635465
\(5\) −28.4834 + 49.3346i −0.227867 + 0.394677i −0.957176 − 0.289508i \(-0.906508\pi\)
0.729309 + 0.684185i \(0.239842\pi\)
\(6\) −97.9016 + 56.5235i −0.453248 + 0.261683i
\(7\) 304.101 + 526.719i 0.886592 + 1.53562i 0.843878 + 0.536535i \(0.180267\pi\)
0.0427141 + 0.999087i \(0.486400\pi\)
\(8\) −526.950 −1.02920
\(9\) −257.884 + 446.668i −0.353750 + 0.612714i
\(10\) −220.508 + 381.931i −0.220508 + 0.381931i
\(11\) 246.818 + 142.501i 0.185438 + 0.107063i 0.589845 − 0.807516i \(-0.299189\pi\)
−0.404407 + 0.914579i \(0.632522\pi\)
\(12\) 51.4314 − 29.6939i 0.0297636 − 0.0171840i
\(13\) −1805.65 − 1042.49i −0.821872 − 0.474508i 0.0291898 − 0.999574i \(-0.490707\pi\)
−0.851061 + 0.525066i \(0.824041\pi\)
\(14\) 2354.24 + 4077.67i 0.857960 + 1.48603i
\(15\) − 831.854i − 0.246475i
\(16\) −3819.17 −0.932415
\(17\) 6903.49 − 3985.73i 1.40515 − 0.811263i 0.410233 − 0.911981i \(-0.365447\pi\)
0.994915 + 0.100718i \(0.0321141\pi\)
\(18\) −1996.45 + 3457.95i −0.342326 + 0.592926i
\(19\) −876.667 − 1518.43i −0.127813 − 0.221378i 0.795016 − 0.606588i \(-0.207462\pi\)
−0.922829 + 0.385210i \(0.874129\pi\)
\(20\) 115.841 − 200.643i 0.0144802 − 0.0250804i
\(21\) −7691.39 − 4440.62i −0.830514 − 0.479497i
\(22\) 1910.78 + 1103.19i 0.179450 + 0.103605i
\(23\) 20863.0i 1.71472i 0.514718 + 0.857359i \(0.327897\pi\)
−0.514718 + 0.857359i \(0.672103\pi\)
\(24\) 6663.87 − 3847.38i 0.482050 − 0.278312i
\(25\) 6189.90 + 10721.2i 0.396153 + 0.686158i
\(26\) −13978.7 − 8070.61i −0.795330 − 0.459184i
\(27\) − 18176.7i − 0.923471i
\(28\) −1236.77 − 2142.15i −0.0563399 − 0.0975835i
\(29\) − 42366.3i − 1.73711i −0.495595 − 0.868554i \(-0.665050\pi\)
0.495595 − 0.868554i \(-0.334950\pi\)
\(30\) − 6439.92i − 0.238516i
\(31\) 27400.6 + 11692.3i 0.919762 + 0.392477i
\(32\) 4158.14 0.126896
\(33\) −4161.72 −0.115806
\(34\) 53444.4 − 30856.1i 1.35977 − 0.785063i
\(35\) −34647.3 −0.808100
\(36\) 1048.81 − 1816.59i 0.0224796 − 0.0389358i
\(37\) 1149.72 − 663.789i 0.0226979 − 0.0131046i −0.488608 − 0.872503i \(-0.662495\pi\)
0.511306 + 0.859399i \(0.329162\pi\)
\(38\) −6786.84 − 11755.2i −0.123685 − 0.214229i
\(39\) 30445.9 0.513258
\(40\) 15009.3 − 25996.9i 0.234521 − 0.406201i
\(41\) −49093.1 + 85031.7i −0.712310 + 1.23376i 0.251679 + 0.967811i \(0.419017\pi\)
−0.963988 + 0.265945i \(0.914316\pi\)
\(42\) −59544.0 − 34377.7i −0.803692 − 0.464012i
\(43\) −38647.7 + 22313.3i −0.486092 + 0.280645i −0.722952 − 0.690899i \(-0.757215\pi\)
0.236860 + 0.971544i \(0.423882\pi\)
\(44\) −1003.81 − 579.547i −0.0117840 − 0.00680347i
\(45\) −14690.8 − 25445.2i −0.161216 − 0.279234i
\(46\) 161514.i 1.65934i
\(47\) 53024.0 0.510715 0.255358 − 0.966847i \(-0.417807\pi\)
0.255358 + 0.966847i \(0.417807\pi\)
\(48\) 48297.6 − 27884.7i 0.436719 − 0.252140i
\(49\) −126130. + 218464.i −1.07209 + 1.85692i
\(50\) 47920.0 + 82999.8i 0.383360 + 0.663998i
\(51\) −58201.5 + 100808.i −0.438757 + 0.759949i
\(52\) 7343.55 + 4239.80i 0.0522271 + 0.0301533i
\(53\) 92507.3 + 53409.1i 0.621367 + 0.358746i 0.777401 − 0.629005i \(-0.216538\pi\)
−0.156034 + 0.987752i \(0.549871\pi\)
\(54\) − 140717.i − 0.893648i
\(55\) −14060.4 + 8117.80i −0.0845105 + 0.0487922i
\(56\) −160246. − 277554.i −0.912480 − 1.58046i
\(57\) 22172.8 + 12801.5i 0.119728 + 0.0691252i
\(58\) − 327985.i − 1.68101i
\(59\) 59948.5 + 103834.i 0.291892 + 0.505572i 0.974257 − 0.225440i \(-0.0723818\pi\)
−0.682365 + 0.731012i \(0.739048\pi\)
\(60\) 3383.13i 0.0156627i
\(61\) 211458.i 0.931613i 0.884887 + 0.465806i \(0.154236\pi\)
−0.884887 + 0.465806i \(0.845764\pi\)
\(62\) 212126. + 90517.6i 0.890058 + 0.379802i
\(63\) −313691. −1.25453
\(64\) 276618. 1.05521
\(65\) 102862. − 59387.5i 0.374555 − 0.216249i
\(66\) −32218.5 −0.112066
\(67\) 255366. − 442307.i 0.849060 − 1.47062i −0.0329876 − 0.999456i \(-0.510502\pi\)
0.882048 − 0.471160i \(-0.156164\pi\)
\(68\) −28076.4 + 16209.9i −0.0892923 + 0.0515529i
\(69\) −152325. − 263835.i −0.463687 − 0.803130i
\(70\) −268227. −0.782003
\(71\) 47835.7 − 82853.8i 0.133652 − 0.231493i −0.791429 − 0.611260i \(-0.790663\pi\)
0.925082 + 0.379768i \(0.123996\pi\)
\(72\) 135892. − 235372.i 0.364080 − 0.630605i
\(73\) 90952.7 + 52511.6i 0.233801 + 0.134985i 0.612325 − 0.790607i \(-0.290235\pi\)
−0.378523 + 0.925592i \(0.623568\pi\)
\(74\) 8900.69 − 5138.81i 0.0219649 − 0.0126814i
\(75\) −156556. − 90387.7i −0.371096 − 0.214252i
\(76\) 3565.39 + 6175.43i 0.00812205 + 0.0140678i
\(77\) 173338.i 0.379684i
\(78\) 235702. 0.496682
\(79\) 77196.3 − 44569.3i 0.156572 − 0.0903971i −0.419667 − 0.907678i \(-0.637853\pi\)
0.576239 + 0.817281i \(0.304520\pi\)
\(80\) 108783. − 188418.i 0.212467 − 0.368003i
\(81\) −55285.4 − 95757.1i −0.104029 − 0.180184i
\(82\) −380061. + 658285.i −0.689306 + 1.19391i
\(83\) 156492. + 90350.6i 0.273689 + 0.158014i 0.630563 − 0.776138i \(-0.282824\pi\)
−0.356874 + 0.934153i \(0.616157\pi\)
\(84\) 31280.7 + 18059.9i 0.0527763 + 0.0304704i
\(85\) 454108.i 0.739440i
\(86\) −299197. + 172741.i −0.470393 + 0.271582i
\(87\) 309326. + 535769.i 0.469741 + 0.813616i
\(88\) −130061. − 75090.7i −0.190853 − 0.110189i
\(89\) 343254.i 0.486907i 0.969913 + 0.243453i \(0.0782803\pi\)
−0.969913 + 0.243453i \(0.921720\pi\)
\(90\) −113731. − 196988.i −0.156010 − 0.270217i
\(91\) − 1.26809e6i − 1.68278i
\(92\) − 84849.3i − 0.108964i
\(93\) −431879. + 52196.3i −0.536925 + 0.0648919i
\(94\) 410493. 0.494222
\(95\) 99881.7 0.116497
\(96\) −52584.3 + 30359.5i −0.0594350 + 0.0343148i
\(97\) −284177. −0.311368 −0.155684 − 0.987807i \(-0.549758\pi\)
−0.155684 + 0.987807i \(0.549758\pi\)
\(98\) −976457. + 1.69127e6i −1.03747 + 1.79695i
\(99\) −127301. + 73497.3i −0.131198 + 0.0757471i
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 31.7.e.a.6.11 ✓ 30
31.26 odd 6 inner 31.7.e.a.26.11 yes 30
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
31.7.e.a.6.11 ✓ 30 1.1 even 1 trivial
31.7.e.a.26.11 yes 30 31.26 odd 6 inner