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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [230,2,Mod(7,230)] 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("230.7"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(230, base_ring=CyclotomicField(44)) chi = DirichletCharacter(H, H._module([11, 38])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 230 = 2 \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 230.l (of order \(44\), degree \(20\), 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: \(1.83655924649\)
Analytic rank: \(0\)
Dimension: \(240\)
Relative dimension: \(12\) over \(\Q(\zeta_{44})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{44}]$

Embedding invariants

Embedding label 157.1
Character \(\chi\) \(=\) 230.157
Dual form 230.2.l.a.63.1

$q$-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+(-0.349464 + 0.936950i) q^{2} +(-1.06204 + 1.94498i) q^{3} +(-0.755750 - 0.654861i) q^{4} +(-0.498115 + 2.17988i) q^{5} +(-1.45120 - 1.67478i) q^{6} +(-4.02819 - 3.01546i) q^{7} +(0.877679 - 0.479249i) q^{8} +(-1.03309 - 1.60752i) q^{9} +(-1.86837 - 1.22850i) q^{10} +(1.58353 + 0.723173i) q^{11} +(2.07632 - 0.774429i) q^{12} +(0.214540 + 0.286592i) q^{13} +(4.23304 - 2.72041i) q^{14} +(-3.71080 - 3.28394i) q^{15} +(0.142315 + 0.989821i) q^{16} +(-2.13874 - 0.152966i) q^{17} +(1.86719 - 0.406183i) q^{18} +(0.840634 - 0.970143i) q^{19} +(1.80397 - 1.32125i) q^{20} +(10.1431 - 4.63220i) q^{21} +(-1.23096 + 1.23096i) q^{22} +(-2.54267 + 4.06631i) q^{23} +2.21605i q^{24} +(-4.50376 - 2.17166i) q^{25} +(-0.343496 + 0.100860i) q^{26} +(-2.40743 + 0.172183i) q^{27} +(1.06959 + 4.91683i) q^{28} +(-1.98168 + 1.71713i) q^{29} +(4.37368 - 2.32922i) q^{30} +(-8.43162 - 2.47575i) q^{31} +(-0.977147 - 0.212565i) q^{32} +(-3.08832 + 2.31189i) q^{33} +(0.890734 - 1.95044i) q^{34} +(8.57985 - 7.27892i) q^{35} +(-0.271944 + 1.89141i) q^{36} +(-1.87018 + 8.59708i) q^{37} +(0.615204 + 1.12666i) q^{38} +(-0.785264 + 0.112904i) q^{39} +(0.607521 + 2.15196i) q^{40} +(7.16256 + 4.60310i) q^{41} +(0.795486 + 11.1224i) q^{42} +(4.08392 + 2.22999i) q^{43} +(-0.723173 - 1.58353i) q^{44} +(4.01880 - 1.45129i) q^{45} +(-2.92135 - 3.80338i) q^{46} +(-1.60762 - 1.60762i) q^{47} +(-2.07632 - 0.774429i) q^{48} +(5.16114 + 17.5772i) q^{49} +(3.60864 - 3.46088i) q^{50} +(2.56894 - 3.99735i) q^{51} +(0.0255393 - 0.357085i) q^{52} +(4.62918 - 6.18386i) q^{53} +(0.679983 - 2.31581i) q^{54} +(-2.36521 + 3.09168i) q^{55} +(-4.98061 - 0.716104i) q^{56} +(0.994122 + 2.66534i) q^{57} +(-0.916342 - 2.45681i) q^{58} +(-12.7331 - 1.83074i) q^{59} +(0.653915 + 4.91190i) q^{60} +(-2.29221 + 7.80653i) q^{61} +(5.26620 - 7.03482i) q^{62} +(-0.685936 + 9.59064i) q^{63} +(0.540641 - 0.841254i) q^{64} +(-0.731601 + 0.324916i) q^{65} +(-1.08687 - 3.70153i) q^{66} +(5.12275 + 1.91069i) q^{67} +(1.51618 + 1.51618i) q^{68} +(-5.20847 - 9.26400i) q^{69} +(3.82163 + 10.5826i) q^{70} +(5.18399 + 11.3514i) q^{71} +(-1.67713 - 0.915780i) q^{72} +(0.691111 + 9.66299i) q^{73} +(-7.40147 - 4.75664i) q^{74} +(9.00700 - 6.45333i) q^{75} +(-1.27062 + 0.182687i) q^{76} +(-4.19805 - 7.68815i) q^{77} +(0.168636 - 0.775209i) q^{78} +(1.85277 - 12.8863i) q^{79} +(-2.22858 - 0.182815i) q^{80} +(4.60329 - 10.0798i) q^{81} +(-6.81593 + 5.10234i) q^{82} +(1.94207 + 0.422472i) q^{83} +(-10.6991 - 3.14153i) q^{84} +(1.39879 - 4.58600i) q^{85} +(-3.51657 + 3.04713i) q^{86} +(-1.23517 - 5.67798i) q^{87} +(1.73641 - 0.124191i) q^{88} +(0.850880 - 0.249841i) q^{89} +(-0.0446459 + 4.27259i) q^{90} -1.80138i q^{91} +(4.58448 - 1.40802i) q^{92} +(13.7700 - 13.7700i) q^{93} +(2.06807 - 0.944454i) q^{94} +(1.69606 + 2.31572i) q^{95} +(1.45120 - 1.67478i) q^{96} +(6.11554 - 1.33036i) q^{97} +(-18.2726 - 1.30688i) q^{98} +(-0.473413 - 3.29266i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 240 q - 8 q^{3} - 8 q^{6} - 8 q^{12} + 16 q^{13} + 24 q^{16} - 72 q^{18} - 80 q^{23} - 8 q^{26} + 16 q^{27} - 44 q^{28} + 24 q^{31} - 44 q^{33} - 8 q^{35} - 32 q^{36} - 88 q^{37} - 24 q^{41} - 8 q^{46}+ \cdots + 24 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(47\) \(51\)
\(\chi(n)\) \(e\left(\frac{1}{4}\right)\) \(e\left(\frac{15}{22}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.349464 + 0.936950i −0.247108 + 0.662524i
\(3\) −1.06204 + 1.94498i −0.613168 + 1.12293i 0.368009 + 0.929822i \(0.380040\pi\)
−0.981177 + 0.193111i \(0.938142\pi\)
\(4\) −0.755750 − 0.654861i −0.377875 − 0.327430i
\(5\) −0.498115 + 2.17988i −0.222764 + 0.974872i
\(6\) −1.45120 − 1.67478i −0.592451 − 0.683725i
\(7\) −4.02819 − 3.01546i −1.52251 − 1.13974i −0.946683 − 0.322166i \(-0.895589\pi\)
−0.575828 − 0.817571i \(-0.695320\pi\)
\(8\) 0.877679 − 0.479249i 0.310306 − 0.169440i
\(9\) −1.03309 − 1.60752i −0.344364 − 0.535840i
\(10\) −1.86837 − 1.22850i −0.590829 − 0.388485i
\(11\) 1.58353 + 0.723173i 0.477452 + 0.218045i 0.639580 − 0.768725i \(-0.279108\pi\)
−0.162128 + 0.986770i \(0.551836\pi\)
\(12\) 2.07632 − 0.774429i 0.599383 − 0.223558i
\(13\) 0.214540 + 0.286592i 0.0595027 + 0.0794863i 0.829305 − 0.558796i \(-0.188736\pi\)
−0.769803 + 0.638282i \(0.779645\pi\)
\(14\) 4.23304 − 2.72041i 1.13133 − 0.727060i
\(15\) −3.71080 − 3.28394i −0.958125 − 0.847909i
\(16\) 0.142315 + 0.989821i 0.0355787 + 0.247455i
\(17\) −2.13874 − 0.152966i −0.518721 − 0.0370996i −0.190475 − 0.981692i \(-0.561003\pi\)
−0.328246 + 0.944592i \(0.606457\pi\)
\(18\) 1.86719 − 0.406183i 0.440102 − 0.0957383i
\(19\) 0.840634 − 0.970143i 0.192855 − 0.222566i −0.651084 − 0.759005i \(-0.725686\pi\)
0.843939 + 0.536439i \(0.180231\pi\)
\(20\) 1.80397 − 1.32125i 0.403380 − 0.295440i
\(21\) 10.1431 − 4.63220i 2.21340 − 1.01083i
\(22\) −1.23096 + 1.23096i −0.262442 + 0.262442i
\(23\) −2.54267 + 4.06631i −0.530182 + 0.847884i
\(24\) 2.21605i 0.452349i
\(25\) −4.50376 − 2.17166i −0.900753 − 0.434332i
\(26\) −0.343496 + 0.100860i −0.0673651 + 0.0197802i
\(27\) −2.40743 + 0.172183i −0.463310 + 0.0331366i
\(28\) 1.06959 + 4.91683i 0.202134 + 0.929194i
\(29\) −1.98168 + 1.71713i −0.367988 + 0.318863i −0.819151 − 0.573578i \(-0.805555\pi\)
0.451163 + 0.892442i \(0.351009\pi\)
\(30\) 4.37368 − 2.32922i 0.798521 − 0.425255i
\(31\) −8.43162 − 2.47575i −1.51436 − 0.444658i −0.584141 − 0.811652i \(-0.698569\pi\)
−0.930223 + 0.366994i \(0.880387\pi\)
\(32\) −0.977147 − 0.212565i −0.172737 − 0.0375766i
\(33\) −3.08832 + 2.31189i −0.537608 + 0.402449i
\(34\) 0.890734 − 1.95044i 0.152760 − 0.334497i
\(35\) 8.57985 − 7.27892i 1.45026 − 1.23036i
\(36\) −0.271944 + 1.89141i −0.0453241 + 0.315236i
\(37\) −1.87018 + 8.59708i −0.307456 + 1.41335i 0.521124 + 0.853481i \(0.325513\pi\)
−0.828580 + 0.559871i \(0.810851\pi\)
\(38\) 0.615204 + 1.12666i 0.0997992 + 0.182769i
\(39\) −0.785264 + 0.112904i −0.125743 + 0.0180791i
\(40\) 0.607521 + 2.15196i 0.0960575 + 0.340254i
\(41\) 7.16256 + 4.60310i 1.11860 + 0.718884i 0.963151 − 0.268960i \(-0.0866799\pi\)
0.155453 + 0.987843i \(0.450316\pi\)
\(42\) 0.795486 + 11.1224i 0.122746 + 1.71622i
\(43\) 4.08392 + 2.22999i 0.622792 + 0.340070i 0.759467 − 0.650546i \(-0.225460\pi\)
−0.136675 + 0.990616i \(0.543642\pi\)
\(44\) −0.723173 − 1.58353i −0.109022 − 0.238726i
\(45\) 4.01880 − 1.45129i 0.599088 − 0.216345i
\(46\) −2.92135 − 3.80338i −0.430730 − 0.560778i
\(47\) −1.60762 − 1.60762i −0.234496 − 0.234496i 0.580071 − 0.814566i \(-0.303025\pi\)
−0.814566 + 0.580071i \(0.803025\pi\)
\(48\) −2.07632 − 0.774429i −0.299692 − 0.111779i
\(49\) 5.16114 + 17.5772i 0.737306 + 2.51103i
\(50\) 3.60864 − 3.46088i 0.510339 − 0.489443i
\(51\) 2.56894 − 3.99735i 0.359723 − 0.559740i
\(52\) 0.0255393 − 0.357085i 0.00354166 − 0.0495188i
\(53\) 4.62918 − 6.18386i 0.635867 − 0.849419i −0.360659 − 0.932698i \(-0.617448\pi\)
0.996526 + 0.0832788i \(0.0265392\pi\)
\(54\) 0.679983 − 2.31581i 0.0925340 − 0.315142i
\(55\) −2.36521 + 3.09168i −0.318925 + 0.416882i
\(56\) −4.98061 − 0.716104i −0.665562 − 0.0956934i
\(57\) 0.994122 + 2.66534i 0.131675 + 0.353033i
\(58\) −0.916342 − 2.45681i −0.120322 − 0.322595i
\(59\) −12.7331 − 1.83074i −1.65771 − 0.238342i −0.751060 − 0.660234i \(-0.770457\pi\)
−0.906646 + 0.421891i \(0.861366\pi\)
\(60\) 0.653915 + 4.91190i 0.0844201 + 0.634123i
\(61\) −2.29221 + 7.80653i −0.293487 + 0.999524i 0.672320 + 0.740261i \(0.265298\pi\)
−0.965807 + 0.259263i \(0.916520\pi\)
\(62\) 5.26620 − 7.03482i 0.668808 − 0.893423i
\(63\) −0.685936 + 9.59064i −0.0864198 + 1.20831i
\(64\) 0.540641 − 0.841254i 0.0675801 − 0.105157i
\(65\) −0.731601 + 0.324916i −0.0907440 + 0.0403009i
\(66\) −1.08687 − 3.70153i −0.133784 − 0.455627i
\(67\) 5.12275 + 1.91069i 0.625843 + 0.233427i 0.642320 − 0.766437i \(-0.277972\pi\)
−0.0164765 + 0.999864i \(0.505245\pi\)
\(68\) 1.51618 + 1.51618i 0.183864 + 0.183864i
\(69\) −5.20847 − 9.26400i −0.627026 − 1.11525i
\(70\) 3.82163 + 10.5826i 0.456772 + 1.26486i
\(71\) 5.18399 + 11.3514i 0.615227 + 1.34716i 0.918941 + 0.394396i \(0.129046\pi\)
−0.303714 + 0.952763i \(0.598227\pi\)
\(72\) −1.67713 − 0.915780i −0.197651 − 0.107926i
\(73\) 0.691111 + 9.66299i 0.0808884 + 1.13097i 0.862113 + 0.506717i \(0.169141\pi\)
−0.781224 + 0.624251i \(0.785404\pi\)
\(74\) −7.40147 − 4.75664i −0.860404 − 0.552948i
\(75\) 9.00700 − 6.45333i 1.04004 − 0.745167i
\(76\) −1.27062 + 0.182687i −0.145750 + 0.0209557i
\(77\) −4.19805 − 7.68815i −0.478412 − 0.876146i
\(78\) 0.168636 − 0.775209i 0.0190943 − 0.0877751i
\(79\) 1.85277 − 12.8863i 0.208453 − 1.44982i −0.569755 − 0.821814i \(-0.692962\pi\)
0.778208 − 0.628007i \(-0.216129\pi\)
\(80\) −2.22858 − 0.182815i −0.249163 − 0.0204394i
\(81\) 4.60329 − 10.0798i 0.511477 − 1.11998i
\(82\) −6.81593 + 5.10234i −0.752694 + 0.563460i
\(83\) 1.94207 + 0.422472i 0.213170 + 0.0463723i 0.317882 − 0.948130i \(-0.397028\pi\)
−0.104712 + 0.994503i \(0.533392\pi\)
\(84\) −10.6991 − 3.14153i −1.16737 − 0.342769i
\(85\) 1.39879 − 4.58600i 0.151720 − 0.497422i
\(86\) −3.51657 + 3.04713i −0.379202 + 0.328580i
\(87\) −1.23517 − 5.67798i −0.132424 − 0.608743i
\(88\) 1.73641 − 0.124191i 0.185102 − 0.0132388i
\(89\) 0.850880 − 0.249841i 0.0901931 − 0.0264831i −0.236325 − 0.971674i \(-0.575943\pi\)
0.326518 + 0.945191i \(0.394125\pi\)
\(90\) −0.0446459 + 4.27259i −0.00470609 + 0.450370i
\(91\) − 1.80138i − 0.188836i
\(92\) 4.58448 − 1.40802i 0.477965 − 0.146796i
\(93\) 13.7700 − 13.7700i 1.42788 − 1.42788i
\(94\) 2.06807 − 0.944454i 0.213305 − 0.0974130i
\(95\) 1.69606 + 2.31572i 0.174013 + 0.237588i
\(96\) 1.45120 − 1.67478i 0.148113 − 0.170931i
\(97\) 6.11554 − 1.33036i 0.620939 − 0.135077i 0.108919 − 0.994051i \(-0.465261\pi\)
0.512020 + 0.858973i \(0.328897\pi\)
\(98\) −18.2726 − 1.30688i −1.84581 − 0.132015i
\(99\) −0.473413 − 3.29266i −0.0475798 − 0.330925i
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 230.2.l.a.157.1 yes 240
5.3 odd 4 inner 230.2.l.a.203.7 yes 240
23.17 odd 22 inner 230.2.l.a.17.7 ✓ 240
115.63 even 44 inner 230.2.l.a.63.1 yes 240
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
230.2.l.a.17.7 ✓ 240 23.17 odd 22 inner
230.2.l.a.63.1 yes 240 115.63 even 44 inner
230.2.l.a.157.1 yes 240 1.1 even 1 trivial
230.2.l.a.203.7 yes 240 5.3 odd 4 inner