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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [230,4,Mod(31,230)] 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("230.31"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(230, base_ring=CyclotomicField(22)) chi = DirichletCharacter(H, H._module([0, 6])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 230 = 2 \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 230.g (of order \(11\), degree \(10\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [70,-14,3] 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: \(13.5704393013\)
Analytic rank: \(0\)
Dimension: \(70\)
Relative dimension: \(7\) over \(\Q(\zeta_{11})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{11}]$

Embedding invariants

Embedding label 71.2
Character \(\chi\) \(=\) 230.71
Dual form 230.4.g.d.81.2

$q$-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+(1.68251 + 1.08128i) q^{2} +(-0.573704 - 3.99020i) q^{3} +(1.66166 + 3.63853i) q^{4} +(-4.79746 - 1.40866i) q^{5} +(3.34927 - 7.33387i) q^{6} +(8.19371 - 9.45604i) q^{7} +(-1.13852 + 7.91857i) q^{8} +(10.3138 - 3.02840i) q^{9} +(-6.54861 - 7.55750i) q^{10} +(-6.56166 + 4.21692i) q^{11} +(13.5651 - 8.71779i) q^{12} +(-25.0844 - 28.9490i) q^{13} +(24.0106 - 7.05015i) q^{14} +(-2.86852 + 19.9510i) q^{15} +(-10.4778 + 12.0920i) q^{16} +(21.3989 - 46.8570i) q^{17} +(20.6275 + 6.05679i) q^{18} +(-65.8189 - 144.123i) q^{19} +(-2.84630 - 19.7964i) q^{20} +(-42.4322 - 27.2695i) q^{21} -15.5997 q^{22} +(94.1802 + 57.4203i) q^{23} +32.2498 q^{24} +(21.0313 + 13.5160i) q^{25} +(-10.9027 - 75.8301i) q^{26} +(-63.2160 - 138.424i) q^{27} +(48.0212 + 14.1003i) q^{28} +(61.7716 - 135.261i) q^{29} +(-26.3989 + 30.4660i) q^{30} +(17.4487 - 121.359i) q^{31} +(-30.7038 + 9.01544i) q^{32} +(20.5908 + 23.7630i) q^{33} +(86.6693 - 55.6990i) q^{34} +(-52.6294 + 33.8229i) q^{35} +(28.1569 + 32.4948i) q^{36} +(-137.987 + 40.5167i) q^{37} +(45.0971 - 313.657i) q^{38} +(-101.121 + 116.700i) q^{39} +(16.6166 - 36.3853i) q^{40} +(33.2362 + 9.75902i) q^{41} +(-41.9065 - 91.7624i) q^{42} +(18.9644 + 131.900i) q^{43} +(-26.2466 - 16.8677i) q^{44} -53.7459 q^{45} +(96.3714 + 198.445i) q^{46} +173.619 q^{47} +(54.2606 + 34.8712i) q^{48} +(26.5341 + 184.549i) q^{49} +(20.7708 + 45.4816i) q^{50} +(-199.245 - 58.5036i) q^{51} +(63.6498 - 139.374i) q^{52} +(39.2138 - 45.2551i) q^{53} +(43.3137 - 301.253i) q^{54} +(37.4195 - 10.9874i) q^{55} +(65.5497 + 75.6483i) q^{56} +(-537.320 + 345.314i) q^{57} +(250.186 - 160.785i) q^{58} +(-499.479 - 576.429i) q^{59} +(-77.3587 + 22.7146i) q^{60} +(-54.0706 + 376.070i) q^{61} +(160.581 - 185.320i) q^{62} +(55.8714 - 122.341i) q^{63} +(-61.4076 - 18.0309i) q^{64} +(79.5623 + 174.217i) q^{65} +(8.94962 + 62.2459i) q^{66} +(437.805 + 281.360i) q^{67} +206.048 q^{68} +(175.087 - 408.740i) q^{69} -125.121 q^{70} +(945.531 + 607.656i) q^{71} +(12.2382 + 85.1182i) q^{72} +(508.163 + 1112.72i) q^{73} +(-275.975 - 81.0334i) q^{74} +(41.8658 - 91.6734i) q^{75} +(415.028 - 478.968i) q^{76} +(-13.8889 + 96.5995i) q^{77} +(-296.322 + 87.0081i) q^{78} +(-53.2137 - 61.4118i) q^{79} +(67.3003 - 43.2513i) q^{80} +(-271.916 + 174.750i) q^{81} +(45.3678 + 52.3573i) q^{82} +(146.488 - 43.0128i) q^{83} +(28.7130 - 199.704i) q^{84} +(-168.666 + 194.651i) q^{85} +(-110.714 + 242.429i) q^{86} +(-575.156 - 168.881i) q^{87} +(-25.9214 - 56.7600i) q^{88} +(-57.4186 - 399.355i) q^{89} +(-90.4279 - 58.1145i) q^{90} -479.277 q^{91} +(-52.4298 + 438.090i) q^{92} -494.256 q^{93} +(292.115 + 187.731i) q^{94} +(112.743 + 784.143i) q^{95} +(53.5883 + 117.342i) q^{96} +(-252.695 - 74.1979i) q^{97} +(-154.905 + 339.195i) q^{98} +(-54.9049 + 63.3637i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 70 q - 14 q^{2} + 3 q^{3} - 28 q^{4} - 35 q^{5} + 6 q^{6} - 78 q^{7} - 56 q^{8} - 24 q^{9} - 70 q^{10} - 15 q^{11} - 120 q^{12} - 270 q^{13} + 64 q^{14} + 15 q^{15} - 112 q^{16} + 114 q^{17} - 48 q^{18}+ \cdots + 11285 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/230\mathbb{Z}\right)^\times\).

\(n\) \(47\) \(51\)
\(\chi(n)\) \(1\) \(e\left(\frac{1}{11}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.68251 + 1.08128i 0.594856 + 0.382291i
\(3\) −0.573704 − 3.99020i −0.110409 − 0.767914i −0.967522 − 0.252785i \(-0.918653\pi\)
0.857113 − 0.515128i \(-0.172256\pi\)
\(4\) 1.66166 + 3.63853i 0.207708 + 0.454816i
\(5\) −4.79746 − 1.40866i −0.429098 − 0.125995i
\(6\) 3.34927 − 7.33387i 0.227889 − 0.499007i
\(7\) 8.19371 − 9.45604i 0.442419 − 0.510578i −0.490117 − 0.871657i \(-0.663046\pi\)
0.932535 + 0.361078i \(0.117591\pi\)
\(8\) −1.13852 + 7.91857i −0.0503159 + 0.349955i
\(9\) 10.3138 − 3.02840i 0.381992 − 0.112163i
\(10\) −6.54861 − 7.55750i −0.207085 − 0.238989i
\(11\) −6.56166 + 4.21692i −0.179856 + 0.115586i −0.627471 − 0.778640i \(-0.715910\pi\)
0.447615 + 0.894226i \(0.352274\pi\)
\(12\) 13.5651 − 8.71779i 0.326327 − 0.209717i
\(13\) −25.0844 − 28.9490i −0.535166 − 0.617615i 0.422196 − 0.906505i \(-0.361259\pi\)
−0.957362 + 0.288890i \(0.906714\pi\)
\(14\) 24.0106 − 7.05015i 0.458365 − 0.134588i
\(15\) −2.86852 + 19.9510i −0.0493766 + 0.343422i
\(16\) −10.4778 + 12.0920i −0.163715 + 0.188937i
\(17\) 21.3989 − 46.8570i 0.305293 − 0.668499i −0.693348 − 0.720602i \(-0.743865\pi\)
0.998642 + 0.0521036i \(0.0165926\pi\)
\(18\) 20.6275 + 6.05679i 0.270109 + 0.0793111i
\(19\) −65.8189 − 144.123i −0.794731 − 1.74022i −0.662584 − 0.748988i \(-0.730540\pi\)
−0.132147 − 0.991230i \(-0.542187\pi\)
\(20\) −2.84630 − 19.7964i −0.0318226 − 0.221331i
\(21\) −42.4322 − 27.2695i −0.440927 − 0.283367i
\(22\) −15.5997 −0.151176
\(23\) 94.1802 + 57.4203i 0.853823 + 0.520563i
\(24\) 32.2498 0.274290
\(25\) 21.0313 + 13.5160i 0.168251 + 0.108128i
\(26\) −10.9027 − 75.8301i −0.0822385 − 0.571981i
\(27\) −63.2160 − 138.424i −0.450590 − 0.986655i
\(28\) 48.0212 + 14.1003i 0.324113 + 0.0951681i
\(29\) 61.7716 − 135.261i 0.395541 − 0.866114i −0.602162 − 0.798374i \(-0.705694\pi\)
0.997703 − 0.0677403i \(-0.0215789\pi\)
\(30\) −26.3989 + 30.4660i −0.160659 + 0.185410i
\(31\) 17.4487 − 121.359i 0.101093 − 0.703118i −0.874739 − 0.484595i \(-0.838967\pi\)
0.975832 − 0.218523i \(-0.0701240\pi\)
\(32\) −30.7038 + 9.01544i −0.169616 + 0.0498038i
\(33\) 20.5908 + 23.7630i 0.108618 + 0.125352i
\(34\) 86.6693 − 55.6990i 0.437166 − 0.280950i
\(35\) −52.6294 + 33.8229i −0.254171 + 0.163346i
\(36\) 28.1569 + 32.4948i 0.130356 + 0.150439i
\(37\) −137.987 + 40.5167i −0.613107 + 0.180025i −0.573520 − 0.819192i \(-0.694423\pi\)
−0.0395873 + 0.999216i \(0.512604\pi\)
\(38\) 45.0971 − 313.657i 0.192519 − 1.33900i
\(39\) −101.121 + 116.700i −0.415188 + 0.479152i
\(40\) 16.6166 − 36.3853i 0.0656829 − 0.143825i
\(41\) 33.2362 + 9.75902i 0.126600 + 0.0371732i 0.344419 − 0.938816i \(-0.388076\pi\)
−0.217819 + 0.975989i \(0.569894\pi\)
\(42\) −41.9065 − 91.7624i −0.153960 − 0.337125i
\(43\) 18.9644 + 131.900i 0.0672568 + 0.467782i 0.995419 + 0.0956061i \(0.0304789\pi\)
−0.928162 + 0.372175i \(0.878612\pi\)
\(44\) −26.2466 − 16.8677i −0.0899279 − 0.0577932i
\(45\) −53.7459 −0.178044
\(46\) 96.3714 + 198.445i 0.308895 + 0.636069i
\(47\) 173.619 0.538829 0.269414 − 0.963024i \(-0.413170\pi\)
0.269414 + 0.963024i \(0.413170\pi\)
\(48\) 54.2606 + 34.8712i 0.163163 + 0.104859i
\(49\) 26.5341 + 184.549i 0.0773589 + 0.538043i
\(50\) 20.7708 + 45.4816i 0.0587486 + 0.128641i
\(51\) −199.245 − 58.5036i −0.547057 − 0.160630i
\(52\) 63.6498 − 139.374i 0.169743 − 0.371686i
\(53\) 39.2138 − 45.2551i 0.101631 − 0.117288i −0.702655 − 0.711530i \(-0.748002\pi\)
0.804286 + 0.594242i \(0.202548\pi\)
\(54\) 43.3137 − 301.253i 0.109153 − 0.759174i
\(55\) 37.4195 − 10.9874i 0.0917391 − 0.0269370i
\(56\) 65.5497 + 75.6483i 0.156419 + 0.180517i
\(57\) −537.320 + 345.314i −1.24859 + 0.802421i
\(58\) 250.186 − 160.785i 0.566398 − 0.364002i
\(59\) −499.479 − 576.429i −1.10215 − 1.27194i −0.959358 − 0.282190i \(-0.908939\pi\)
−0.142787 − 0.989753i \(-0.545606\pi\)
\(60\) −77.3587 + 22.7146i −0.166449 + 0.0488740i
\(61\) −54.0706 + 376.070i −0.113492 + 0.789357i 0.850985 + 0.525190i \(0.176006\pi\)
−0.964477 + 0.264166i \(0.914903\pi\)
\(62\) 160.581 − 185.320i 0.328932 − 0.379607i
\(63\) 55.8714 − 122.341i 0.111732 − 0.244660i
\(64\) −61.4076 − 18.0309i −0.119937 − 0.0352166i
\(65\) 79.5623 + 174.217i 0.151823 + 0.332446i
\(66\) 8.94962 + 62.2459i 0.0166912 + 0.116090i
\(67\) 437.805 + 281.360i 0.798304 + 0.513039i 0.875062 − 0.484011i \(-0.160820\pi\)
−0.0767578 + 0.997050i \(0.524457\pi\)
\(68\) 206.048 0.367456
\(69\) 175.087 − 408.740i 0.305478 − 0.713138i
\(70\) −125.121 −0.213641
\(71\) 945.531 + 607.656i 1.58048 + 1.01571i 0.975644 + 0.219359i \(0.0703967\pi\)
0.604834 + 0.796352i \(0.293240\pi\)
\(72\) 12.2382 + 85.1182i 0.0200317 + 0.139323i
\(73\) 508.163 + 1112.72i 0.814740 + 1.78403i 0.585521 + 0.810657i \(0.300890\pi\)
0.229219 + 0.973375i \(0.426383\pi\)
\(74\) −275.975 − 81.0334i −0.433532 − 0.127297i
\(75\) 41.8658 − 91.6734i 0.0644567 − 0.141140i
\(76\) 415.028 − 478.968i 0.626407 − 0.722913i
\(77\) −13.8889 + 96.5995i −0.0205557 + 0.142968i
\(78\) −296.322 + 87.0081i −0.430152 + 0.126304i
\(79\) −53.2137 − 61.4118i −0.0757849 − 0.0874604i 0.716591 − 0.697494i \(-0.245701\pi\)
−0.792376 + 0.610033i \(0.791156\pi\)
\(80\) 67.3003 − 43.2513i 0.0940550 − 0.0604455i
\(81\) −271.916 + 174.750i −0.372998 + 0.239711i
\(82\) 45.3678 + 52.3573i 0.0610980 + 0.0705109i
\(83\) 146.488 − 43.0128i 0.193725 − 0.0568828i −0.183431 − 0.983033i \(-0.558720\pi\)
0.377156 + 0.926150i \(0.376902\pi\)
\(84\) 28.7130 − 199.704i 0.0372958 − 0.259398i
\(85\) −168.666 + 194.651i −0.215228 + 0.248386i
\(86\) −110.714 + 242.429i −0.138820 + 0.303974i
\(87\) −575.156 − 168.881i −0.708773 − 0.208114i
\(88\) −25.9214 − 56.7600i −0.0314004 − 0.0687572i
\(89\) −57.4186 − 399.355i −0.0683861 − 0.475636i −0.995020 − 0.0996718i \(-0.968221\pi\)
0.926634 − 0.375964i \(-0.122688\pi\)
\(90\) −90.4279 − 58.1145i −0.105910 − 0.0680645i
\(91\) −479.277 −0.552108
\(92\) −52.4298 + 438.090i −0.0594151 + 0.496457i
\(93\) −494.256 −0.551096
\(94\) 292.115 + 187.731i 0.320525 + 0.205989i
\(95\) 112.743 + 784.143i 0.121760 + 0.846856i
\(96\) 53.5883 + 117.342i 0.0569722 + 0.124752i
\(97\) −252.695 − 74.1979i −0.264508 − 0.0776666i 0.146789 − 0.989168i \(-0.453106\pi\)
−0.411297 + 0.911501i \(0.634924\pi\)
\(98\) −154.905 + 339.195i −0.159671 + 0.349632i
\(99\) −54.9049 + 63.3637i −0.0557389 + 0.0643261i
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 230.4.g.d.71.2 ✓ 70
23.12 even 11 inner 230.4.g.d.81.2 yes 70
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
230.4.g.d.71.2 ✓ 70 1.1 even 1 trivial
230.4.g.d.81.2 yes 70 23.12 even 11 inner