# Properties

 Label 1950.2.e.l.1249.2 Level $1950$ Weight $2$ Character 1950.1249 Analytic conductor $15.571$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1950,2,Mod(1249,1950)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1950, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1950.1249");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1950 = 2 \cdot 3 \cdot 5^{2} \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1950.e (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

 Self dual: no Analytic conductor: $$15.5708283941$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(i)$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} + 1$$ x^2 + 1 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 390) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 1249.2 Root $$1.00000i$$ of defining polynomial Character $$\chi$$ $$=$$ 1950.1249 Dual form 1950.2.e.l.1249.1

## $q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} +1.00000 q^{6} -1.00000i q^{8} -1.00000 q^{9} +O(q^{10})$$ $$q+1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} +1.00000 q^{6} -1.00000i q^{8} -1.00000 q^{9} +1.00000i q^{12} -1.00000i q^{13} +1.00000 q^{16} +6.00000i q^{17} -1.00000i q^{18} -4.00000i q^{23} -1.00000 q^{24} +1.00000 q^{26} +1.00000i q^{27} +10.0000 q^{29} +1.00000i q^{32} -6.00000 q^{34} +1.00000 q^{36} +6.00000i q^{37} -1.00000 q^{39} +2.00000 q^{41} -4.00000i q^{43} +4.00000 q^{46} -1.00000i q^{48} +7.00000 q^{49} +6.00000 q^{51} +1.00000i q^{52} -6.00000i q^{53} -1.00000 q^{54} +10.0000i q^{58} +6.00000 q^{61} -1.00000 q^{64} -4.00000i q^{67} -6.00000i q^{68} -4.00000 q^{69} +16.0000 q^{71} +1.00000i q^{72} -2.00000i q^{73} -6.00000 q^{74} -1.00000i q^{78} +1.00000 q^{81} +2.00000i q^{82} +4.00000i q^{83} +4.00000 q^{86} -10.0000i q^{87} +6.00000 q^{89} +4.00000i q^{92} +1.00000 q^{96} -14.0000i q^{97} +7.00000i q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{4} + 2 q^{6} - 2 q^{9}+O(q^{10})$$ 2 * q - 2 * q^4 + 2 * q^6 - 2 * q^9 $$2 q - 2 q^{4} + 2 q^{6} - 2 q^{9} + 2 q^{16} - 2 q^{24} + 2 q^{26} + 20 q^{29} - 12 q^{34} + 2 q^{36} - 2 q^{39} + 4 q^{41} + 8 q^{46} + 14 q^{49} + 12 q^{51} - 2 q^{54} + 12 q^{61} - 2 q^{64} - 8 q^{69} + 32 q^{71} - 12 q^{74} + 2 q^{81} + 8 q^{86} + 12 q^{89} + 2 q^{96}+O(q^{100})$$ 2 * q - 2 * q^4 + 2 * q^6 - 2 * q^9 + 2 * q^16 - 2 * q^24 + 2 * q^26 + 20 * q^29 - 12 * q^34 + 2 * q^36 - 2 * q^39 + 4 * q^41 + 8 * q^46 + 14 * q^49 + 12 * q^51 - 2 * q^54 + 12 * q^61 - 2 * q^64 - 8 * q^69 + 32 * q^71 - 12 * q^74 + 2 * q^81 + 8 * q^86 + 12 * q^89 + 2 * q^96

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1950\mathbb{Z}\right)^\times$$.

 $$n$$ $$301$$ $$1301$$ $$1327$$ $$\chi(n)$$ $$1$$ $$1$$ $$-1$$

## Coefficient data

For each $$n$$ we display the coefficients of the $$q$$-expansion $$a_n$$, the Satake parameters $$\alpha_p$$, and the Satake angles $$\theta_p = \textrm{Arg}(\alpha_p)$$.

Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 1.00000i 0.707107i
$$3$$ − 1.00000i − 0.577350i
$$4$$ −1.00000 −0.500000
$$5$$ 0 0
$$6$$ 1.00000 0.408248
$$7$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$8$$ − 1.00000i − 0.353553i
$$9$$ −1.00000 −0.333333
$$10$$ 0 0
$$11$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$12$$ 1.00000i 0.288675i
$$13$$ − 1.00000i − 0.277350i
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 1.00000 0.250000
$$17$$ 6.00000i 1.45521i 0.685994 + 0.727607i $$0.259367\pi$$
−0.685994 + 0.727607i $$0.740633\pi$$
$$18$$ − 1.00000i − 0.235702i
$$19$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 0 0
$$23$$ − 4.00000i − 0.834058i −0.908893 0.417029i $$-0.863071\pi$$
0.908893 0.417029i $$-0.136929\pi$$
$$24$$ −1.00000 −0.204124
$$25$$ 0 0
$$26$$ 1.00000 0.196116
$$27$$ 1.00000i 0.192450i
$$28$$ 0 0
$$29$$ 10.0000 1.85695 0.928477 0.371391i $$-0.121119\pi$$
0.928477 + 0.371391i $$0.121119\pi$$
$$30$$ 0 0
$$31$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$32$$ 1.00000i 0.176777i
$$33$$ 0 0
$$34$$ −6.00000 −1.02899
$$35$$ 0 0
$$36$$ 1.00000 0.166667
$$37$$ 6.00000i 0.986394i 0.869918 + 0.493197i $$0.164172\pi$$
−0.869918 + 0.493197i $$0.835828\pi$$
$$38$$ 0 0
$$39$$ −1.00000 −0.160128
$$40$$ 0 0
$$41$$ 2.00000 0.312348 0.156174 0.987730i $$-0.450084\pi$$
0.156174 + 0.987730i $$0.450084\pi$$
$$42$$ 0 0
$$43$$ − 4.00000i − 0.609994i −0.952353 0.304997i $$-0.901344\pi$$
0.952353 0.304997i $$-0.0986555\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 4.00000 0.589768
$$47$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$48$$ − 1.00000i − 0.144338i
$$49$$ 7.00000 1.00000
$$50$$ 0 0
$$51$$ 6.00000 0.840168
$$52$$ 1.00000i 0.138675i
$$53$$ − 6.00000i − 0.824163i −0.911147 0.412082i $$-0.864802\pi$$
0.911147 0.412082i $$-0.135198\pi$$
$$54$$ −1.00000 −0.136083
$$55$$ 0 0
$$56$$ 0 0
$$57$$ 0 0
$$58$$ 10.0000i 1.31306i
$$59$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$60$$ 0 0
$$61$$ 6.00000 0.768221 0.384111 0.923287i $$-0.374508\pi$$
0.384111 + 0.923287i $$0.374508\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ −1.00000 −0.125000
$$65$$ 0 0
$$66$$ 0 0
$$67$$ − 4.00000i − 0.488678i −0.969690 0.244339i $$-0.921429\pi$$
0.969690 0.244339i $$-0.0785709\pi$$
$$68$$ − 6.00000i − 0.727607i
$$69$$ −4.00000 −0.481543
$$70$$ 0 0
$$71$$ 16.0000 1.89885 0.949425 0.313993i $$-0.101667\pi$$
0.949425 + 0.313993i $$0.101667\pi$$
$$72$$ 1.00000i 0.117851i
$$73$$ − 2.00000i − 0.234082i −0.993127 0.117041i $$-0.962659\pi$$
0.993127 0.117041i $$-0.0373409\pi$$
$$74$$ −6.00000 −0.697486
$$75$$ 0 0
$$76$$ 0 0
$$77$$ 0 0
$$78$$ − 1.00000i − 0.113228i
$$79$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ 2.00000i 0.220863i
$$83$$ 4.00000i 0.439057i 0.975606 + 0.219529i $$0.0704519\pi$$
−0.975606 + 0.219529i $$0.929548\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 4.00000 0.431331
$$87$$ − 10.0000i − 1.07211i
$$88$$ 0 0
$$89$$ 6.00000 0.635999 0.317999 0.948091i $$-0.396989\pi$$
0.317999 + 0.948091i $$0.396989\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ 4.00000i 0.417029i
$$93$$ 0 0
$$94$$ 0 0
$$95$$ 0 0
$$96$$ 1.00000 0.102062
$$97$$ − 14.0000i − 1.42148i −0.703452 0.710742i $$-0.748359\pi$$
0.703452 0.710742i $$-0.251641\pi$$
$$98$$ 7.00000i 0.707107i
$$99$$ 0 0
$$100$$ 0 0
$$101$$ 6.00000 0.597022 0.298511 0.954406i $$-0.403510\pi$$
0.298511 + 0.954406i $$0.403510\pi$$
$$102$$ 6.00000i 0.594089i
$$103$$ 12.0000i 1.18240i 0.806527 + 0.591198i $$0.201345\pi$$
−0.806527 + 0.591198i $$0.798655\pi$$
$$104$$ −1.00000 −0.0980581
$$105$$ 0 0
$$106$$ 6.00000 0.582772
$$107$$ 4.00000i 0.386695i 0.981130 + 0.193347i $$0.0619344\pi$$
−0.981130 + 0.193347i $$0.938066\pi$$
$$108$$ − 1.00000i − 0.0962250i
$$109$$ 14.0000 1.34096 0.670478 0.741929i $$-0.266089\pi$$
0.670478 + 0.741929i $$0.266089\pi$$
$$110$$ 0 0
$$111$$ 6.00000 0.569495
$$112$$ 0 0
$$113$$ 10.0000i 0.940721i 0.882474 + 0.470360i $$0.155876\pi$$
−0.882474 + 0.470360i $$0.844124\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ −10.0000 −0.928477
$$117$$ 1.00000i 0.0924500i
$$118$$ 0 0
$$119$$ 0 0
$$120$$ 0 0
$$121$$ −11.0000 −1.00000
$$122$$ 6.00000i 0.543214i
$$123$$ − 2.00000i − 0.180334i
$$124$$ 0 0
$$125$$ 0 0
$$126$$ 0 0
$$127$$ 12.0000i 1.06483i 0.846484 + 0.532414i $$0.178715\pi$$
−0.846484 + 0.532414i $$0.821285\pi$$
$$128$$ − 1.00000i − 0.0883883i
$$129$$ −4.00000 −0.352180
$$130$$ 0 0
$$131$$ 12.0000 1.04844 0.524222 0.851581i $$-0.324356\pi$$
0.524222 + 0.851581i $$0.324356\pi$$
$$132$$ 0 0
$$133$$ 0 0
$$134$$ 4.00000 0.345547
$$135$$ 0 0
$$136$$ 6.00000 0.514496
$$137$$ − 6.00000i − 0.512615i −0.966595 0.256307i $$-0.917494\pi$$
0.966595 0.256307i $$-0.0825059\pi$$
$$138$$ − 4.00000i − 0.340503i
$$139$$ −4.00000 −0.339276 −0.169638 0.985506i $$-0.554260\pi$$
−0.169638 + 0.985506i $$0.554260\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 16.0000i 1.34269i
$$143$$ 0 0
$$144$$ −1.00000 −0.0833333
$$145$$ 0 0
$$146$$ 2.00000 0.165521
$$147$$ − 7.00000i − 0.577350i
$$148$$ − 6.00000i − 0.493197i
$$149$$ 14.0000 1.14692 0.573462 0.819232i $$-0.305600\pi$$
0.573462 + 0.819232i $$0.305600\pi$$
$$150$$ 0 0
$$151$$ −16.0000 −1.30206 −0.651031 0.759051i $$-0.725663\pi$$
−0.651031 + 0.759051i $$0.725663\pi$$
$$152$$ 0 0
$$153$$ − 6.00000i − 0.485071i
$$154$$ 0 0
$$155$$ 0 0
$$156$$ 1.00000 0.0800641
$$157$$ 14.0000i 1.11732i 0.829396 + 0.558661i $$0.188685\pi$$
−0.829396 + 0.558661i $$0.811315\pi$$
$$158$$ 0 0
$$159$$ −6.00000 −0.475831
$$160$$ 0 0
$$161$$ 0 0
$$162$$ 1.00000i 0.0785674i
$$163$$ − 4.00000i − 0.313304i −0.987654 0.156652i $$-0.949930\pi$$
0.987654 0.156652i $$-0.0500701\pi$$
$$164$$ −2.00000 −0.156174
$$165$$ 0 0
$$166$$ −4.00000 −0.310460
$$167$$ − 24.0000i − 1.85718i −0.371113 0.928588i $$-0.621024\pi$$
0.371113 0.928588i $$-0.378976\pi$$
$$168$$ 0 0
$$169$$ −1.00000 −0.0769231
$$170$$ 0 0
$$171$$ 0 0
$$172$$ 4.00000i 0.304997i
$$173$$ − 14.0000i − 1.06440i −0.846619 0.532200i $$-0.821365\pi$$
0.846619 0.532200i $$-0.178635\pi$$
$$174$$ 10.0000 0.758098
$$175$$ 0 0
$$176$$ 0 0
$$177$$ 0 0
$$178$$ 6.00000i 0.449719i
$$179$$ 20.0000 1.49487 0.747435 0.664335i $$-0.231285\pi$$
0.747435 + 0.664335i $$0.231285\pi$$
$$180$$ 0 0
$$181$$ −10.0000 −0.743294 −0.371647 0.928374i $$-0.621207\pi$$
−0.371647 + 0.928374i $$0.621207\pi$$
$$182$$ 0 0
$$183$$ − 6.00000i − 0.443533i
$$184$$ −4.00000 −0.294884
$$185$$ 0 0
$$186$$ 0 0
$$187$$ 0 0
$$188$$ 0 0
$$189$$ 0 0
$$190$$ 0 0
$$191$$ 24.0000 1.73658 0.868290 0.496058i $$-0.165220\pi$$
0.868290 + 0.496058i $$0.165220\pi$$
$$192$$ 1.00000i 0.0721688i
$$193$$ 14.0000i 1.00774i 0.863779 + 0.503871i $$0.168091\pi$$
−0.863779 + 0.503871i $$0.831909\pi$$
$$194$$ 14.0000 1.00514
$$195$$ 0 0
$$196$$ −7.00000 −0.500000
$$197$$ 22.0000i 1.56744i 0.621117 + 0.783718i $$0.286679\pi$$
−0.621117 + 0.783718i $$0.713321\pi$$
$$198$$ 0 0
$$199$$ 16.0000 1.13421 0.567105 0.823646i $$-0.308063\pi$$
0.567105 + 0.823646i $$0.308063\pi$$
$$200$$ 0 0
$$201$$ −4.00000 −0.282138
$$202$$ 6.00000i 0.422159i
$$203$$ 0 0
$$204$$ −6.00000 −0.420084
$$205$$ 0 0
$$206$$ −12.0000 −0.836080
$$207$$ 4.00000i 0.278019i
$$208$$ − 1.00000i − 0.0693375i
$$209$$ 0 0
$$210$$ 0 0
$$211$$ −12.0000 −0.826114 −0.413057 0.910705i $$-0.635539\pi$$
−0.413057 + 0.910705i $$0.635539\pi$$
$$212$$ 6.00000i 0.412082i
$$213$$ − 16.0000i − 1.09630i
$$214$$ −4.00000 −0.273434
$$215$$ 0 0
$$216$$ 1.00000 0.0680414
$$217$$ 0 0
$$218$$ 14.0000i 0.948200i
$$219$$ −2.00000 −0.135147
$$220$$ 0 0
$$221$$ 6.00000 0.403604
$$222$$ 6.00000i 0.402694i
$$223$$ 16.0000i 1.07144i 0.844396 + 0.535720i $$0.179960\pi$$
−0.844396 + 0.535720i $$0.820040\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ −10.0000 −0.665190
$$227$$ − 20.0000i − 1.32745i −0.747978 0.663723i $$-0.768975\pi$$
0.747978 0.663723i $$-0.231025\pi$$
$$228$$ 0 0
$$229$$ −2.00000 −0.132164 −0.0660819 0.997814i $$-0.521050\pi$$
−0.0660819 + 0.997814i $$0.521050\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ − 10.0000i − 0.656532i
$$233$$ 18.0000i 1.17922i 0.807688 + 0.589610i $$0.200718\pi$$
−0.807688 + 0.589610i $$0.799282\pi$$
$$234$$ −1.00000 −0.0653720
$$235$$ 0 0
$$236$$ 0 0
$$237$$ 0 0
$$238$$ 0 0
$$239$$ −16.0000 −1.03495 −0.517477 0.855697i $$-0.673129\pi$$
−0.517477 + 0.855697i $$0.673129\pi$$
$$240$$ 0 0
$$241$$ −14.0000 −0.901819 −0.450910 0.892570i $$-0.648900\pi$$
−0.450910 + 0.892570i $$0.648900\pi$$
$$242$$ − 11.0000i − 0.707107i
$$243$$ − 1.00000i − 0.0641500i
$$244$$ −6.00000 −0.384111
$$245$$ 0 0
$$246$$ 2.00000 0.127515
$$247$$ 0 0
$$248$$ 0 0
$$249$$ 4.00000 0.253490
$$250$$ 0 0
$$251$$ −4.00000 −0.252478 −0.126239 0.992000i $$-0.540291\pi$$
−0.126239 + 0.992000i $$0.540291\pi$$
$$252$$ 0 0
$$253$$ 0 0
$$254$$ −12.0000 −0.752947
$$255$$ 0 0
$$256$$ 1.00000 0.0625000
$$257$$ 6.00000i 0.374270i 0.982334 + 0.187135i $$0.0599201\pi$$
−0.982334 + 0.187135i $$0.940080\pi$$
$$258$$ − 4.00000i − 0.249029i
$$259$$ 0 0
$$260$$ 0 0
$$261$$ −10.0000 −0.618984
$$262$$ 12.0000i 0.741362i
$$263$$ − 28.0000i − 1.72655i −0.504730 0.863277i $$-0.668408\pi$$
0.504730 0.863277i $$-0.331592\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 0 0
$$267$$ − 6.00000i − 0.367194i
$$268$$ 4.00000i 0.244339i
$$269$$ −14.0000 −0.853595 −0.426798 0.904347i $$-0.640358\pi$$
−0.426798 + 0.904347i $$0.640358\pi$$
$$270$$ 0 0
$$271$$ −24.0000 −1.45790 −0.728948 0.684569i $$-0.759990\pi$$
−0.728948 + 0.684569i $$0.759990\pi$$
$$272$$ 6.00000i 0.363803i
$$273$$ 0 0
$$274$$ 6.00000 0.362473
$$275$$ 0 0
$$276$$ 4.00000 0.240772
$$277$$ − 2.00000i − 0.120168i −0.998193 0.0600842i $$-0.980863\pi$$
0.998193 0.0600842i $$-0.0191369\pi$$
$$278$$ − 4.00000i − 0.239904i
$$279$$ 0 0
$$280$$ 0 0
$$281$$ −6.00000 −0.357930 −0.178965 0.983855i $$-0.557275\pi$$
−0.178965 + 0.983855i $$0.557275\pi$$
$$282$$ 0 0
$$283$$ − 12.0000i − 0.713326i −0.934233 0.356663i $$-0.883914\pi$$
0.934233 0.356663i $$-0.116086\pi$$
$$284$$ −16.0000 −0.949425
$$285$$ 0 0
$$286$$ 0 0
$$287$$ 0 0
$$288$$ − 1.00000i − 0.0589256i
$$289$$ −19.0000 −1.11765
$$290$$ 0 0
$$291$$ −14.0000 −0.820695
$$292$$ 2.00000i 0.117041i
$$293$$ 26.0000i 1.51894i 0.650545 + 0.759468i $$0.274541\pi$$
−0.650545 + 0.759468i $$0.725459\pi$$
$$294$$ 7.00000 0.408248
$$295$$ 0 0
$$296$$ 6.00000 0.348743
$$297$$ 0 0
$$298$$ 14.0000i 0.810998i
$$299$$ −4.00000 −0.231326
$$300$$ 0 0
$$301$$ 0 0
$$302$$ − 16.0000i − 0.920697i
$$303$$ − 6.00000i − 0.344691i
$$304$$ 0 0
$$305$$ 0 0
$$306$$ 6.00000 0.342997
$$307$$ 4.00000i 0.228292i 0.993464 + 0.114146i $$0.0364132\pi$$
−0.993464 + 0.114146i $$0.963587\pi$$
$$308$$ 0 0
$$309$$ 12.0000 0.682656
$$310$$ 0 0
$$311$$ −16.0000 −0.907277 −0.453638 0.891186i $$-0.649874\pi$$
−0.453638 + 0.891186i $$0.649874\pi$$
$$312$$ 1.00000i 0.0566139i
$$313$$ 26.0000i 1.46961i 0.678280 + 0.734803i $$0.262726\pi$$
−0.678280 + 0.734803i $$0.737274\pi$$
$$314$$ −14.0000 −0.790066
$$315$$ 0 0
$$316$$ 0 0
$$317$$ − 2.00000i − 0.112331i −0.998421 0.0561656i $$-0.982113\pi$$
0.998421 0.0561656i $$-0.0178875\pi$$
$$318$$ − 6.00000i − 0.336463i
$$319$$ 0 0
$$320$$ 0 0
$$321$$ 4.00000 0.223258
$$322$$ 0 0
$$323$$ 0 0
$$324$$ −1.00000 −0.0555556
$$325$$ 0 0
$$326$$ 4.00000 0.221540
$$327$$ − 14.0000i − 0.774202i
$$328$$ − 2.00000i − 0.110432i
$$329$$ 0 0
$$330$$ 0 0
$$331$$ 8.00000 0.439720 0.219860 0.975531i $$-0.429440\pi$$
0.219860 + 0.975531i $$0.429440\pi$$
$$332$$ − 4.00000i − 0.219529i
$$333$$ − 6.00000i − 0.328798i
$$334$$ 24.0000 1.31322
$$335$$ 0 0
$$336$$ 0 0
$$337$$ − 18.0000i − 0.980522i −0.871576 0.490261i $$-0.836901\pi$$
0.871576 0.490261i $$-0.163099\pi$$
$$338$$ − 1.00000i − 0.0543928i
$$339$$ 10.0000 0.543125
$$340$$ 0 0
$$341$$ 0 0
$$342$$ 0 0
$$343$$ 0 0
$$344$$ −4.00000 −0.215666
$$345$$ 0 0
$$346$$ 14.0000 0.752645
$$347$$ − 12.0000i − 0.644194i −0.946707 0.322097i $$-0.895612\pi$$
0.946707 0.322097i $$-0.104388\pi$$
$$348$$ 10.0000i 0.536056i
$$349$$ −34.0000 −1.81998 −0.909989 0.414632i $$-0.863910\pi$$
−0.909989 + 0.414632i $$0.863910\pi$$
$$350$$ 0 0
$$351$$ 1.00000 0.0533761
$$352$$ 0 0
$$353$$ − 26.0000i − 1.38384i −0.721974 0.691920i $$-0.756765\pi$$
0.721974 0.691920i $$-0.243235\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ −6.00000 −0.317999
$$357$$ 0 0
$$358$$ 20.0000i 1.05703i
$$359$$ −24.0000 −1.26667 −0.633336 0.773877i $$-0.718315\pi$$
−0.633336 + 0.773877i $$0.718315\pi$$
$$360$$ 0 0
$$361$$ −19.0000 −1.00000
$$362$$ − 10.0000i − 0.525588i
$$363$$ 11.0000i 0.577350i
$$364$$ 0 0
$$365$$ 0 0
$$366$$ 6.00000 0.313625
$$367$$ − 20.0000i − 1.04399i −0.852948 0.521996i $$-0.825188\pi$$
0.852948 0.521996i $$-0.174812\pi$$
$$368$$ − 4.00000i − 0.208514i
$$369$$ −2.00000 −0.104116
$$370$$ 0 0
$$371$$ 0 0
$$372$$ 0 0
$$373$$ 10.0000i 0.517780i 0.965907 + 0.258890i $$0.0833568\pi$$
−0.965907 + 0.258890i $$0.916643\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ − 10.0000i − 0.515026i
$$378$$ 0 0
$$379$$ −24.0000 −1.23280 −0.616399 0.787434i $$-0.711409\pi$$
−0.616399 + 0.787434i $$0.711409\pi$$
$$380$$ 0 0
$$381$$ 12.0000 0.614779
$$382$$ 24.0000i 1.22795i
$$383$$ − 8.00000i − 0.408781i −0.978889 0.204390i $$-0.934479\pi$$
0.978889 0.204390i $$-0.0655212\pi$$
$$384$$ −1.00000 −0.0510310
$$385$$ 0 0
$$386$$ −14.0000 −0.712581
$$387$$ 4.00000i 0.203331i
$$388$$ 14.0000i 0.710742i
$$389$$ 18.0000 0.912636 0.456318 0.889817i $$-0.349168\pi$$
0.456318 + 0.889817i $$0.349168\pi$$
$$390$$ 0 0
$$391$$ 24.0000 1.21373
$$392$$ − 7.00000i − 0.353553i
$$393$$ − 12.0000i − 0.605320i
$$394$$ −22.0000 −1.10834
$$395$$ 0 0
$$396$$ 0 0
$$397$$ 14.0000i 0.702640i 0.936255 + 0.351320i $$0.114267\pi$$
−0.936255 + 0.351320i $$0.885733\pi$$
$$398$$ 16.0000i 0.802008i
$$399$$ 0 0
$$400$$ 0 0
$$401$$ 18.0000 0.898877 0.449439 0.893311i $$-0.351624\pi$$
0.449439 + 0.893311i $$0.351624\pi$$
$$402$$ − 4.00000i − 0.199502i
$$403$$ 0 0
$$404$$ −6.00000 −0.298511
$$405$$ 0 0
$$406$$ 0 0
$$407$$ 0 0
$$408$$ − 6.00000i − 0.297044i
$$409$$ −2.00000 −0.0988936 −0.0494468 0.998777i $$-0.515746\pi$$
−0.0494468 + 0.998777i $$0.515746\pi$$
$$410$$ 0 0
$$411$$ −6.00000 −0.295958
$$412$$ − 12.0000i − 0.591198i
$$413$$ 0 0
$$414$$ −4.00000 −0.196589
$$415$$ 0 0
$$416$$ 1.00000 0.0490290
$$417$$ 4.00000i 0.195881i
$$418$$ 0 0
$$419$$ −12.0000 −0.586238 −0.293119 0.956076i $$-0.594693\pi$$
−0.293119 + 0.956076i $$0.594693\pi$$
$$420$$ 0 0
$$421$$ −22.0000 −1.07221 −0.536107 0.844150i $$-0.680106\pi$$
−0.536107 + 0.844150i $$0.680106\pi$$
$$422$$ − 12.0000i − 0.584151i
$$423$$ 0 0
$$424$$ −6.00000 −0.291386
$$425$$ 0 0
$$426$$ 16.0000 0.775203
$$427$$ 0 0
$$428$$ − 4.00000i − 0.193347i
$$429$$ 0 0
$$430$$ 0 0
$$431$$ −24.0000 −1.15604 −0.578020 0.816023i $$-0.696174\pi$$
−0.578020 + 0.816023i $$0.696174\pi$$
$$432$$ 1.00000i 0.0481125i
$$433$$ 2.00000i 0.0961139i 0.998845 + 0.0480569i $$0.0153029\pi$$
−0.998845 + 0.0480569i $$0.984697\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ −14.0000 −0.670478
$$437$$ 0 0
$$438$$ − 2.00000i − 0.0955637i
$$439$$ 24.0000 1.14546 0.572729 0.819745i $$-0.305885\pi$$
0.572729 + 0.819745i $$0.305885\pi$$
$$440$$ 0 0
$$441$$ −7.00000 −0.333333
$$442$$ 6.00000i 0.285391i
$$443$$ − 4.00000i − 0.190046i −0.995475 0.0950229i $$-0.969708\pi$$
0.995475 0.0950229i $$-0.0302924\pi$$
$$444$$ −6.00000 −0.284747
$$445$$ 0 0
$$446$$ −16.0000 −0.757622
$$447$$ − 14.0000i − 0.662177i
$$448$$ 0 0
$$449$$ 14.0000 0.660701 0.330350 0.943858i $$-0.392833\pi$$
0.330350 + 0.943858i $$0.392833\pi$$
$$450$$ 0 0
$$451$$ 0 0
$$452$$ − 10.0000i − 0.470360i
$$453$$ 16.0000i 0.751746i
$$454$$ 20.0000 0.938647
$$455$$ 0 0
$$456$$ 0 0
$$457$$ − 6.00000i − 0.280668i −0.990104 0.140334i $$-0.955182\pi$$
0.990104 0.140334i $$-0.0448177\pi$$
$$458$$ − 2.00000i − 0.0934539i
$$459$$ −6.00000 −0.280056
$$460$$ 0 0
$$461$$ −14.0000 −0.652045 −0.326023 0.945362i $$-0.605709\pi$$
−0.326023 + 0.945362i $$0.605709\pi$$
$$462$$ 0 0
$$463$$ − 24.0000i − 1.11537i −0.830051 0.557687i $$-0.811689\pi$$
0.830051 0.557687i $$-0.188311\pi$$
$$464$$ 10.0000 0.464238
$$465$$ 0 0
$$466$$ −18.0000 −0.833834
$$467$$ 28.0000i 1.29569i 0.761774 + 0.647843i $$0.224329\pi$$
−0.761774 + 0.647843i $$0.775671\pi$$
$$468$$ − 1.00000i − 0.0462250i
$$469$$ 0 0
$$470$$ 0 0
$$471$$ 14.0000 0.645086
$$472$$ 0 0
$$473$$ 0 0
$$474$$ 0 0
$$475$$ 0 0
$$476$$ 0 0
$$477$$ 6.00000i 0.274721i
$$478$$ − 16.0000i − 0.731823i
$$479$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$480$$ 0 0
$$481$$ 6.00000 0.273576
$$482$$ − 14.0000i − 0.637683i
$$483$$ 0 0
$$484$$ 11.0000 0.500000
$$485$$ 0 0
$$486$$ 1.00000 0.0453609
$$487$$ − 8.00000i − 0.362515i −0.983436 0.181257i $$-0.941983\pi$$
0.983436 0.181257i $$-0.0580167\pi$$
$$488$$ − 6.00000i − 0.271607i
$$489$$ −4.00000 −0.180886
$$490$$ 0 0
$$491$$ 20.0000 0.902587 0.451294 0.892375i $$-0.350963\pi$$
0.451294 + 0.892375i $$0.350963\pi$$
$$492$$ 2.00000i 0.0901670i
$$493$$ 60.0000i 2.70226i
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ 0 0
$$498$$ 4.00000i 0.179244i
$$499$$ 40.0000 1.79065 0.895323 0.445418i $$-0.146945\pi$$
0.895323 + 0.445418i $$0.146945\pi$$
$$500$$ 0 0
$$501$$ −24.0000 −1.07224
$$502$$ − 4.00000i − 0.178529i
$$503$$ − 36.0000i − 1.60516i −0.596544 0.802580i $$-0.703460\pi$$
0.596544 0.802580i $$-0.296540\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 0 0
$$507$$ 1.00000i 0.0444116i
$$508$$ − 12.0000i − 0.532414i
$$509$$ 38.0000 1.68432 0.842160 0.539227i $$-0.181284\pi$$
0.842160 + 0.539227i $$0.181284\pi$$
$$510$$ 0 0
$$511$$ 0 0
$$512$$ 1.00000i 0.0441942i
$$513$$ 0 0
$$514$$ −6.00000 −0.264649
$$515$$ 0 0
$$516$$ 4.00000 0.176090
$$517$$ 0 0
$$518$$ 0 0
$$519$$ −14.0000 −0.614532
$$520$$ 0 0
$$521$$ 10.0000 0.438108 0.219054 0.975713i $$-0.429703\pi$$
0.219054 + 0.975713i $$0.429703\pi$$
$$522$$ − 10.0000i − 0.437688i
$$523$$ − 20.0000i − 0.874539i −0.899331 0.437269i $$-0.855946\pi$$
0.899331 0.437269i $$-0.144054\pi$$
$$524$$ −12.0000 −0.524222
$$525$$ 0 0
$$526$$ 28.0000 1.22086
$$527$$ 0 0
$$528$$ 0 0
$$529$$ 7.00000 0.304348
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 0 0
$$533$$ − 2.00000i − 0.0866296i
$$534$$ 6.00000 0.259645
$$535$$ 0 0
$$536$$ −4.00000 −0.172774
$$537$$ − 20.0000i − 0.863064i
$$538$$ − 14.0000i − 0.603583i
$$539$$ 0 0
$$540$$ 0 0
$$541$$ −22.0000 −0.945854 −0.472927 0.881102i $$-0.656803\pi$$
−0.472927 + 0.881102i $$0.656803\pi$$
$$542$$ − 24.0000i − 1.03089i
$$543$$ 10.0000i 0.429141i
$$544$$ −6.00000 −0.257248
$$545$$ 0 0
$$546$$ 0 0
$$547$$ − 28.0000i − 1.19719i −0.801050 0.598597i $$-0.795725\pi$$
0.801050 0.598597i $$-0.204275\pi$$
$$548$$ 6.00000i 0.256307i
$$549$$ −6.00000 −0.256074
$$550$$ 0 0
$$551$$ 0 0
$$552$$ 4.00000i 0.170251i
$$553$$ 0 0
$$554$$ 2.00000 0.0849719
$$555$$ 0 0
$$556$$ 4.00000 0.169638
$$557$$ 30.0000i 1.27114i 0.772043 + 0.635570i $$0.219235\pi$$
−0.772043 + 0.635570i $$0.780765\pi$$
$$558$$ 0 0
$$559$$ −4.00000 −0.169182
$$560$$ 0 0
$$561$$ 0 0
$$562$$ − 6.00000i − 0.253095i
$$563$$ − 28.0000i − 1.18006i −0.807382 0.590030i $$-0.799116\pi$$
0.807382 0.590030i $$-0.200884\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ 12.0000 0.504398
$$567$$ 0 0
$$568$$ − 16.0000i − 0.671345i
$$569$$ 6.00000 0.251533 0.125767 0.992060i $$-0.459861\pi$$
0.125767 + 0.992060i $$0.459861\pi$$
$$570$$ 0 0
$$571$$ −4.00000 −0.167395 −0.0836974 0.996491i $$-0.526673\pi$$
−0.0836974 + 0.996491i $$0.526673\pi$$
$$572$$ 0 0
$$573$$ − 24.0000i − 1.00261i
$$574$$ 0 0
$$575$$ 0 0
$$576$$ 1.00000 0.0416667
$$577$$ − 22.0000i − 0.915872i −0.888985 0.457936i $$-0.848589\pi$$
0.888985 0.457936i $$-0.151411\pi$$
$$578$$ − 19.0000i − 0.790296i
$$579$$ 14.0000 0.581820
$$580$$ 0 0
$$581$$ 0 0
$$582$$ − 14.0000i − 0.580319i
$$583$$ 0 0
$$584$$ −2.00000 −0.0827606
$$585$$ 0 0
$$586$$ −26.0000 −1.07405
$$587$$ 12.0000i 0.495293i 0.968850 + 0.247647i $$0.0796572\pi$$
−0.968850 + 0.247647i $$0.920343\pi$$
$$588$$ 7.00000i 0.288675i
$$589$$ 0 0
$$590$$ 0 0
$$591$$ 22.0000 0.904959
$$592$$ 6.00000i 0.246598i
$$593$$ 6.00000i 0.246390i 0.992382 + 0.123195i $$0.0393141\pi$$
−0.992382 + 0.123195i $$0.960686\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ −14.0000 −0.573462
$$597$$ − 16.0000i − 0.654836i
$$598$$ − 4.00000i − 0.163572i
$$599$$ −24.0000 −0.980613 −0.490307 0.871550i $$-0.663115\pi$$
−0.490307 + 0.871550i $$0.663115\pi$$
$$600$$ 0 0
$$601$$ 42.0000 1.71322 0.856608 0.515968i $$-0.172568\pi$$
0.856608 + 0.515968i $$0.172568\pi$$
$$602$$ 0 0
$$603$$ 4.00000i 0.162893i
$$604$$ 16.0000 0.651031
$$605$$ 0 0
$$606$$ 6.00000 0.243733
$$607$$ 36.0000i 1.46119i 0.682808 + 0.730597i $$0.260758\pi$$
−0.682808 + 0.730597i $$0.739242\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ 0 0
$$612$$ 6.00000i 0.242536i
$$613$$ − 22.0000i − 0.888572i −0.895885 0.444286i $$-0.853457\pi$$
0.895885 0.444286i $$-0.146543\pi$$
$$614$$ −4.00000 −0.161427
$$615$$ 0 0
$$616$$ 0 0
$$617$$ − 30.0000i − 1.20775i −0.797077 0.603877i $$-0.793622\pi$$
0.797077 0.603877i $$-0.206378\pi$$
$$618$$ 12.0000i 0.482711i
$$619$$ −40.0000 −1.60774 −0.803868 0.594808i $$-0.797228\pi$$
−0.803868 + 0.594808i $$0.797228\pi$$
$$620$$ 0 0
$$621$$ 4.00000 0.160514
$$622$$ − 16.0000i − 0.641542i
$$623$$ 0 0
$$624$$ −1.00000 −0.0400320
$$625$$ 0 0
$$626$$ −26.0000 −1.03917
$$627$$ 0 0
$$628$$ − 14.0000i − 0.558661i
$$629$$ −36.0000 −1.43541
$$630$$ 0 0
$$631$$ −48.0000 −1.91085 −0.955425 0.295234i $$-0.904602\pi$$
−0.955425 + 0.295234i $$0.904602\pi$$
$$632$$ 0 0
$$633$$ 12.0000i 0.476957i
$$634$$ 2.00000 0.0794301
$$635$$ 0 0
$$636$$ 6.00000 0.237915
$$637$$ − 7.00000i − 0.277350i
$$638$$ 0 0
$$639$$ −16.0000 −0.632950
$$640$$ 0 0
$$641$$ 2.00000 0.0789953 0.0394976 0.999220i $$-0.487424\pi$$
0.0394976 + 0.999220i $$0.487424\pi$$
$$642$$ 4.00000i 0.157867i
$$643$$ 4.00000i 0.157745i 0.996885 + 0.0788723i $$0.0251319\pi$$
−0.996885 + 0.0788723i $$0.974868\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ 36.0000i 1.41531i 0.706560 + 0.707653i $$0.250246\pi$$
−0.706560 + 0.707653i $$0.749754\pi$$
$$648$$ − 1.00000i − 0.0392837i
$$649$$ 0 0
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 4.00000i 0.156652i
$$653$$ − 6.00000i − 0.234798i −0.993085 0.117399i $$-0.962544\pi$$
0.993085 0.117399i $$-0.0374557\pi$$
$$654$$ 14.0000 0.547443
$$655$$ 0 0
$$656$$ 2.00000 0.0780869
$$657$$ 2.00000i 0.0780274i
$$658$$ 0 0
$$659$$ 20.0000 0.779089 0.389545 0.921008i $$-0.372632\pi$$
0.389545 + 0.921008i $$0.372632\pi$$
$$660$$ 0 0
$$661$$ −30.0000 −1.16686 −0.583432 0.812162i $$-0.698291\pi$$
−0.583432 + 0.812162i $$0.698291\pi$$
$$662$$ 8.00000i 0.310929i
$$663$$ − 6.00000i − 0.233021i
$$664$$ 4.00000 0.155230
$$665$$ 0 0
$$666$$ 6.00000 0.232495
$$667$$ − 40.0000i − 1.54881i
$$668$$ 24.0000i 0.928588i
$$669$$ 16.0000 0.618596
$$670$$ 0 0
$$671$$ 0 0
$$672$$ 0 0
$$673$$ 42.0000i 1.61898i 0.587133 + 0.809491i $$0.300257\pi$$
−0.587133 + 0.809491i $$0.699743\pi$$
$$674$$ 18.0000 0.693334
$$675$$ 0 0
$$676$$ 1.00000 0.0384615
$$677$$ 30.0000i 1.15299i 0.817099 + 0.576497i $$0.195581\pi$$
−0.817099 + 0.576497i $$0.804419\pi$$
$$678$$ 10.0000i 0.384048i
$$679$$ 0 0
$$680$$ 0 0
$$681$$ −20.0000 −0.766402
$$682$$ 0 0
$$683$$ 12.0000i 0.459167i 0.973289 + 0.229584i $$0.0737364\pi$$
−0.973289 + 0.229584i $$0.926264\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ 0 0
$$687$$ 2.00000i 0.0763048i
$$688$$ − 4.00000i − 0.152499i
$$689$$ −6.00000 −0.228582
$$690$$ 0 0
$$691$$ 32.0000 1.21734 0.608669 0.793424i $$-0.291704\pi$$
0.608669 + 0.793424i $$0.291704\pi$$
$$692$$ 14.0000i 0.532200i
$$693$$ 0 0
$$694$$ 12.0000 0.455514
$$695$$ 0 0
$$696$$ −10.0000 −0.379049
$$697$$ 12.0000i 0.454532i
$$698$$ − 34.0000i − 1.28692i
$$699$$ 18.0000 0.680823
$$700$$ 0 0
$$701$$ −18.0000 −0.679851 −0.339925 0.940452i $$-0.610402\pi$$
−0.339925 + 0.940452i $$0.610402\pi$$
$$702$$ 1.00000i 0.0377426i
$$703$$ 0 0
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 26.0000 0.978523
$$707$$ 0 0
$$708$$ 0 0
$$709$$ −2.00000 −0.0751116 −0.0375558 0.999295i $$-0.511957\pi$$
−0.0375558 + 0.999295i $$0.511957\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ − 6.00000i − 0.224860i
$$713$$ 0 0
$$714$$ 0 0
$$715$$ 0 0
$$716$$ −20.0000 −0.747435
$$717$$ 16.0000i 0.597531i
$$718$$ − 24.0000i − 0.895672i
$$719$$ −24.0000 −0.895049 −0.447524 0.894272i $$-0.647694\pi$$
−0.447524 + 0.894272i $$0.647694\pi$$
$$720$$ 0 0
$$721$$ 0 0
$$722$$ − 19.0000i − 0.707107i
$$723$$ 14.0000i 0.520666i
$$724$$ 10.0000 0.371647
$$725$$ 0 0
$$726$$ −11.0000 −0.408248
$$727$$ − 12.0000i − 0.445055i −0.974926 0.222528i $$-0.928569\pi$$
0.974926 0.222528i $$-0.0714308\pi$$
$$728$$ 0 0
$$729$$ −1.00000 −0.0370370
$$730$$ 0 0
$$731$$ 24.0000 0.887672
$$732$$ 6.00000i 0.221766i
$$733$$ 50.0000i 1.84679i 0.383849 + 0.923396i $$0.374598\pi$$
−0.383849 + 0.923396i $$0.625402\pi$$
$$734$$ 20.0000 0.738213
$$735$$ 0 0
$$736$$ 4.00000 0.147442
$$737$$ 0 0
$$738$$ − 2.00000i − 0.0736210i
$$739$$ 40.0000 1.47142 0.735712 0.677295i $$-0.236848\pi$$
0.735712 + 0.677295i $$0.236848\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 0 0
$$743$$ − 48.0000i − 1.76095i −0.474093 0.880475i $$-0.657224\pi$$
0.474093 0.880475i $$-0.342776\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ −10.0000 −0.366126
$$747$$ − 4.00000i − 0.146352i
$$748$$ 0 0
$$749$$ 0 0
$$750$$ 0 0
$$751$$ 24.0000 0.875772 0.437886 0.899030i $$-0.355727\pi$$
0.437886 + 0.899030i $$0.355727\pi$$
$$752$$ 0 0
$$753$$ 4.00000i 0.145768i
$$754$$ 10.0000 0.364179
$$755$$ 0 0
$$756$$ 0 0
$$757$$ − 2.00000i − 0.0726912i −0.999339 0.0363456i $$-0.988428\pi$$
0.999339 0.0363456i $$-0.0115717\pi$$
$$758$$ − 24.0000i − 0.871719i
$$759$$ 0 0
$$760$$ 0 0
$$761$$ −22.0000 −0.797499 −0.398750 0.917060i $$-0.630556\pi$$
−0.398750 + 0.917060i $$0.630556\pi$$
$$762$$ 12.0000i 0.434714i
$$763$$ 0 0
$$764$$ −24.0000 −0.868290
$$765$$ 0 0
$$766$$ 8.00000 0.289052
$$767$$ 0 0
$$768$$ − 1.00000i − 0.0360844i
$$769$$ −34.0000 −1.22607 −0.613036 0.790055i $$-0.710052\pi$$
−0.613036 + 0.790055i $$0.710052\pi$$
$$770$$ 0 0
$$771$$ 6.00000 0.216085
$$772$$ − 14.0000i − 0.503871i
$$773$$ 26.0000i 0.935155i 0.883952 + 0.467578i $$0.154873\pi$$
−0.883952 + 0.467578i $$0.845127\pi$$
$$774$$ −4.00000 −0.143777
$$775$$ 0 0
$$776$$ −14.0000 −0.502571
$$777$$ 0 0
$$778$$ 18.0000i 0.645331i
$$779$$ 0 0
$$780$$ 0 0
$$781$$ 0 0
$$782$$ 24.0000i 0.858238i
$$783$$ 10.0000i 0.357371i
$$784$$ 7.00000 0.250000
$$785$$ 0 0
$$786$$ 12.0000 0.428026
$$787$$ 36.0000i 1.28326i 0.767014 + 0.641631i $$0.221742\pi$$
−0.767014 + 0.641631i $$0.778258\pi$$
$$788$$ − 22.0000i − 0.783718i
$$789$$ −28.0000 −0.996826
$$790$$ 0 0
$$791$$ 0 0
$$792$$ 0 0
$$793$$ − 6.00000i − 0.213066i
$$794$$ −14.0000 −0.496841
$$795$$ 0 0
$$796$$ −16.0000 −0.567105
$$797$$ − 18.0000i − 0.637593i −0.947823 0.318796i $$-0.896721\pi$$
0.947823 0.318796i $$-0.103279\pi$$
$$798$$ 0 0
$$799$$ 0 0
$$800$$ 0 0
$$801$$ −6.00000 −0.212000
$$802$$ 18.0000i 0.635602i
$$803$$ 0 0
$$804$$ 4.00000 0.141069
$$805$$ 0 0
$$806$$ 0 0
$$807$$ 14.0000i 0.492823i
$$808$$ − 6.00000i − 0.211079i
$$809$$ −26.0000 −0.914111 −0.457056 0.889438i $$-0.651096\pi$$
−0.457056 + 0.889438i $$0.651096\pi$$
$$810$$ 0 0
$$811$$ 32.0000 1.12367 0.561836 0.827249i $$-0.310095\pi$$
0.561836 + 0.827249i $$0.310095\pi$$
$$812$$ 0 0
$$813$$ 24.0000i 0.841717i
$$814$$ 0 0
$$815$$ 0 0
$$816$$ 6.00000 0.210042
$$817$$ 0 0
$$818$$ − 2.00000i − 0.0699284i
$$819$$ 0 0
$$820$$ 0 0
$$821$$ −22.0000 −0.767805 −0.383903 0.923374i $$-0.625420\pi$$
−0.383903 + 0.923374i $$0.625420\pi$$
$$822$$ − 6.00000i − 0.209274i
$$823$$ − 44.0000i − 1.53374i −0.641800 0.766872i $$-0.721812\pi$$
0.641800 0.766872i $$-0.278188\pi$$
$$824$$ 12.0000 0.418040
$$825$$ 0 0
$$826$$ 0 0
$$827$$ 12.0000i 0.417281i 0.977992 + 0.208640i $$0.0669038\pi$$
−0.977992 + 0.208640i $$0.933096\pi$$
$$828$$ − 4.00000i − 0.139010i
$$829$$ −54.0000 −1.87550 −0.937749 0.347314i $$-0.887094\pi$$
−0.937749 + 0.347314i $$0.887094\pi$$
$$830$$ 0 0
$$831$$ −2.00000 −0.0693792
$$832$$ 1.00000i 0.0346688i
$$833$$ 42.0000i 1.45521i
$$834$$ −4.00000 −0.138509
$$835$$ 0 0
$$836$$ 0 0
$$837$$ 0 0
$$838$$ − 12.0000i − 0.414533i
$$839$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$840$$ 0 0
$$841$$ 71.0000 2.44828
$$842$$ − 22.0000i − 0.758170i
$$843$$ 6.00000i 0.206651i
$$844$$ 12.0000 0.413057
$$845$$ 0 0
$$846$$ 0 0
$$847$$ 0 0
$$848$$ − 6.00000i − 0.206041i
$$849$$ −12.0000 −0.411839
$$850$$ 0 0
$$851$$ 24.0000 0.822709
$$852$$ 16.0000i 0.548151i
$$853$$ − 38.0000i − 1.30110i −0.759465 0.650548i $$-0.774539\pi$$
0.759465 0.650548i $$-0.225461\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 4.00000 0.136717
$$857$$ 38.0000i 1.29806i 0.760765 + 0.649028i $$0.224824\pi$$
−0.760765 + 0.649028i $$0.775176\pi$$
$$858$$ 0 0
$$859$$ 20.0000 0.682391 0.341196 0.939992i $$-0.389168\pi$$
0.341196 + 0.939992i $$0.389168\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ − 24.0000i − 0.817443i
$$863$$ 48.0000i 1.63394i 0.576681 + 0.816970i $$0.304348\pi$$
−0.576681 + 0.816970i $$0.695652\pi$$
$$864$$ −1.00000 −0.0340207
$$865$$ 0 0
$$866$$ −2.00000 −0.0679628
$$867$$ 19.0000i 0.645274i
$$868$$ 0 0
$$869$$ 0 0
$$870$$ 0 0
$$871$$ −4.00000 −0.135535
$$872$$ − 14.0000i − 0.474100i
$$873$$ 14.0000i 0.473828i
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 2.00000 0.0675737
$$877$$ 14.0000i 0.472746i 0.971662 + 0.236373i $$0.0759588\pi$$
−0.971662 + 0.236373i $$0.924041\pi$$
$$878$$ 24.0000i 0.809961i
$$879$$ 26.0000 0.876958
$$880$$ 0 0
$$881$$ 34.0000 1.14549 0.572745 0.819734i $$-0.305879\pi$$
0.572745 + 0.819734i $$0.305879\pi$$
$$882$$ − 7.00000i − 0.235702i
$$883$$ 52.0000i 1.74994i 0.484178 + 0.874970i $$0.339119\pi$$
−0.484178 + 0.874970i $$0.660881\pi$$
$$884$$ −6.00000 −0.201802
$$885$$ 0 0
$$886$$ 4.00000 0.134383
$$887$$ 36.0000i 1.20876i 0.796696 + 0.604381i $$0.206579\pi$$
−0.796696 + 0.604381i $$0.793421\pi$$
$$888$$ − 6.00000i − 0.201347i
$$889$$ 0 0
$$890$$ 0 0
$$891$$ 0 0
$$892$$ − 16.0000i − 0.535720i
$$893$$ 0 0
$$894$$ 14.0000 0.468230
$$895$$ 0 0
$$896$$ 0 0
$$897$$ 4.00000i 0.133556i
$$898$$ 14.0000i 0.467186i
$$899$$ 0 0
$$900$$ 0 0
$$901$$ 36.0000 1.19933
$$902$$ 0 0
$$903$$ 0 0
$$904$$ 10.0000 0.332595
$$905$$ 0 0
$$906$$ −16.0000 −0.531564
$$907$$ − 28.0000i − 0.929725i −0.885383 0.464862i $$-0.846104\pi$$
0.885383 0.464862i $$-0.153896\pi$$
$$908$$ 20.0000i 0.663723i
$$909$$ −6.00000 −0.199007
$$910$$ 0 0
$$911$$ 40.0000 1.32526 0.662630 0.748947i $$-0.269440\pi$$
0.662630 + 0.748947i $$0.269440\pi$$
$$912$$ 0 0
$$913$$ 0 0
$$914$$ 6.00000 0.198462
$$915$$ 0 0
$$916$$ 2.00000 0.0660819
$$917$$ 0 0
$$918$$ − 6.00000i − 0.198030i
$$919$$ 16.0000 0.527791 0.263896 0.964551i $$-0.414993\pi$$
0.263896 + 0.964551i $$0.414993\pi$$
$$920$$ 0 0
$$921$$ 4.00000 0.131804
$$922$$ − 14.0000i − 0.461065i
$$923$$ − 16.0000i − 0.526646i
$$924$$ 0 0
$$925$$ 0 0
$$926$$ 24.0000 0.788689
$$927$$ − 12.0000i − 0.394132i
$$928$$ 10.0000i 0.328266i
$$929$$ 46.0000 1.50921 0.754606 0.656179i $$-0.227828\pi$$
0.754606 + 0.656179i $$0.227828\pi$$
$$930$$ 0 0
$$931$$ 0 0
$$932$$ − 18.0000i − 0.589610i
$$933$$ 16.0000i 0.523816i
$$934$$ −28.0000 −0.916188
$$935$$ 0 0
$$936$$ 1.00000 0.0326860
$$937$$ 22.0000i 0.718709i 0.933201 + 0.359354i $$0.117003\pi$$
−0.933201 + 0.359354i $$0.882997\pi$$
$$938$$ 0 0
$$939$$ 26.0000 0.848478
$$940$$ 0 0
$$941$$ 10.0000 0.325991 0.162995 0.986627i $$-0.447884\pi$$
0.162995 + 0.986627i $$0.447884\pi$$
$$942$$ 14.0000i 0.456145i
$$943$$ − 8.00000i − 0.260516i
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ 36.0000i 1.16984i 0.811090 + 0.584921i $$0.198875\pi$$
−0.811090 + 0.584921i $$0.801125\pi$$
$$948$$ 0 0
$$949$$ −2.00000 −0.0649227
$$950$$ 0 0
$$951$$ −2.00000 −0.0648544
$$952$$ 0 0
$$953$$ 10.0000i 0.323932i 0.986796 + 0.161966i $$0.0517835\pi$$
−0.986796 + 0.161966i $$0.948217\pi$$
$$954$$ −6.00000 −0.194257
$$955$$ 0 0
$$956$$ 16.0000 0.517477
$$957$$ 0 0
$$958$$ 0 0
$$959$$ 0 0
$$960$$ 0 0
$$961$$ −31.0000 −1.00000
$$962$$ 6.00000i 0.193448i
$$963$$ − 4.00000i − 0.128898i
$$964$$ 14.0000 0.450910
$$965$$ 0 0
$$966$$ 0 0
$$967$$ 32.0000i 1.02905i 0.857475 + 0.514525i $$0.172032\pi$$
−0.857475 + 0.514525i $$0.827968\pi$$
$$968$$ 11.0000i 0.353553i
$$969$$ 0 0
$$970$$ 0 0
$$971$$ 12.0000 0.385098 0.192549 0.981287i $$-0.438325\pi$$
0.192549 + 0.981287i $$0.438325\pi$$
$$972$$ 1.00000i 0.0320750i
$$973$$ 0 0
$$974$$ 8.00000 0.256337
$$975$$ 0 0
$$976$$ 6.00000 0.192055
$$977$$ 2.00000i 0.0639857i 0.999488 + 0.0319928i $$0.0101854\pi$$
−0.999488 + 0.0319928i $$0.989815\pi$$
$$978$$ − 4.00000i − 0.127906i
$$979$$ 0 0
$$980$$ 0 0
$$981$$ −14.0000 −0.446986
$$982$$ 20.0000i 0.638226i
$$983$$ 16.0000i 0.510321i 0.966899 + 0.255160i $$0.0821283\pi$$
−0.966899 + 0.255160i $$0.917872\pi$$
$$984$$ −2.00000 −0.0637577
$$985$$ 0 0
$$986$$ −60.0000 −1.91079
$$987$$ 0 0
$$988$$ 0 0
$$989$$ −16.0000 −0.508770
$$990$$ 0 0
$$991$$ −8.00000 −0.254128 −0.127064 0.991894i $$-0.540555\pi$$
−0.127064 + 0.991894i $$0.540555\pi$$
$$992$$ 0 0
$$993$$ − 8.00000i − 0.253872i
$$994$$ 0 0
$$995$$ 0 0
$$996$$ −4.00000 −0.126745
$$997$$ 22.0000i 0.696747i 0.937356 + 0.348373i $$0.113266\pi$$
−0.937356 + 0.348373i $$0.886734\pi$$
$$998$$ 40.0000i 1.26618i
$$999$$ −6.00000 −0.189832
Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000

## Twists

By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 1950.2.e.l.1249.2 2
3.2 odd 2 5850.2.e.p.5149.1 2
5.2 odd 4 390.2.a.a.1.1 1
5.3 odd 4 1950.2.a.y.1.1 1
5.4 even 2 inner 1950.2.e.l.1249.1 2
15.2 even 4 1170.2.a.m.1.1 1
15.8 even 4 5850.2.a.m.1.1 1
15.14 odd 2 5850.2.e.p.5149.2 2
20.7 even 4 3120.2.a.q.1.1 1
60.47 odd 4 9360.2.a.bn.1.1 1
65.12 odd 4 5070.2.a.s.1.1 1
65.47 even 4 5070.2.b.c.1351.1 2
65.57 even 4 5070.2.b.c.1351.2 2

By twisted newform
Twist Min Dim Char Parity Ord Type
390.2.a.a.1.1 1 5.2 odd 4
1170.2.a.m.1.1 1 15.2 even 4
1950.2.a.y.1.1 1 5.3 odd 4
1950.2.e.l.1249.1 2 5.4 even 2 inner
1950.2.e.l.1249.2 2 1.1 even 1 trivial
3120.2.a.q.1.1 1 20.7 even 4
5070.2.a.s.1.1 1 65.12 odd 4
5070.2.b.c.1351.1 2 65.47 even 4
5070.2.b.c.1351.2 2 65.57 even 4
5850.2.a.m.1.1 1 15.8 even 4
5850.2.e.p.5149.1 2 3.2 odd 2
5850.2.e.p.5149.2 2 15.14 odd 2
9360.2.a.bn.1.1 1 60.47 odd 4