# Properties

 Label 5070.2.b.c.1351.2 Level $5070$ Weight $2$ Character 5070.1351 Analytic conductor $40.484$ 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] = [5070,2,Mod(1351,5070)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5070, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("5070.1351");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$5070 = 2 \cdot 3 \cdot 5 \cdot 13^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 5070.b (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: $$40.4841538248$$ 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 1351.2 Root $$-1.00000i$$ of defining polynomial Character $$\chi$$ $$=$$ 5070.1351 Dual form 5070.2.b.c.1351.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+1.00000i q^{2} -1.00000 q^{3} -1.00000 q^{4} +1.00000i q^{5} -1.00000i q^{6} -1.00000i q^{8} +1.00000 q^{9} +O(q^{10})$$ $$q+1.00000i q^{2} -1.00000 q^{3} -1.00000 q^{4} +1.00000i q^{5} -1.00000i q^{6} -1.00000i q^{8} +1.00000 q^{9} -1.00000 q^{10} +1.00000 q^{12} -1.00000i q^{15} +1.00000 q^{16} +6.00000 q^{17} +1.00000i q^{18} -1.00000i q^{20} +4.00000 q^{23} +1.00000i q^{24} -1.00000 q^{25} -1.00000 q^{27} -10.0000 q^{29} +1.00000 q^{30} +1.00000i q^{32} +6.00000i q^{34} -1.00000 q^{36} -6.00000i q^{37} +1.00000 q^{40} -2.00000i q^{41} +4.00000 q^{43} +1.00000i q^{45} +4.00000i q^{46} -1.00000 q^{48} +7.00000 q^{49} -1.00000i q^{50} -6.00000 q^{51} -6.00000 q^{53} -1.00000i q^{54} -10.0000i q^{58} +1.00000i q^{60} +6.00000 q^{61} -1.00000 q^{64} -4.00000i q^{67} -6.00000 q^{68} -4.00000 q^{69} -16.0000i q^{71} -1.00000i q^{72} -2.00000i q^{73} +6.00000 q^{74} +1.00000 q^{75} +1.00000i q^{80} +1.00000 q^{81} +2.00000 q^{82} -4.00000i q^{83} +6.00000i q^{85} +4.00000i q^{86} +10.0000 q^{87} -6.00000i q^{89} -1.00000 q^{90} -4.00000 q^{92} -1.00000i q^{96} -14.0000i q^{97} +7.00000i q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{3} - 2 q^{4} + 2 q^{9}+O(q^{10})$$ 2 * q - 2 * q^3 - 2 * q^4 + 2 * q^9 $$2 q - 2 q^{3} - 2 q^{4} + 2 q^{9} - 2 q^{10} + 2 q^{12} + 2 q^{16} + 12 q^{17} + 8 q^{23} - 2 q^{25} - 2 q^{27} - 20 q^{29} + 2 q^{30} - 2 q^{36} + 2 q^{40} + 8 q^{43} - 2 q^{48} + 14 q^{49} - 12 q^{51} - 12 q^{53} + 12 q^{61} - 2 q^{64} - 12 q^{68} - 8 q^{69} + 12 q^{74} + 2 q^{75} + 2 q^{81} + 4 q^{82} + 20 q^{87} - 2 q^{90} - 8 q^{92}+O(q^{100})$$ 2 * q - 2 * q^3 - 2 * q^4 + 2 * q^9 - 2 * q^10 + 2 * q^12 + 2 * q^16 + 12 * q^17 + 8 * q^23 - 2 * q^25 - 2 * q^27 - 20 * q^29 + 2 * q^30 - 2 * q^36 + 2 * q^40 + 8 * q^43 - 2 * q^48 + 14 * q^49 - 12 * q^51 - 12 * q^53 + 12 * q^61 - 2 * q^64 - 12 * q^68 - 8 * q^69 + 12 * q^74 + 2 * q^75 + 2 * q^81 + 4 * q^82 + 20 * q^87 - 2 * q^90 - 8 * q^92

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/5070\mathbb{Z}\right)^\times$$.

 $$n$$ $$1691$$ $$1861$$ $$4057$$ $$\chi(n)$$ $$1$$ $$-1$$ $$1$$

## 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
$$3$$ −1.00000 −0.577350
$$4$$ −1.00000 −0.500000
$$5$$ 1.00000i 0.447214i
$$6$$ − 1.00000i − 0.408248i
$$7$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$8$$ − 1.00000i − 0.353553i
$$9$$ 1.00000 0.333333
$$10$$ −1.00000 −0.316228
$$11$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$12$$ 1.00000 0.288675
$$13$$ 0 0
$$14$$ 0 0
$$15$$ − 1.00000i − 0.258199i
$$16$$ 1.00000 0.250000
$$17$$ 6.00000 1.45521 0.727607 0.685994i $$-0.240633\pi$$
0.727607 + 0.685994i $$0.240633\pi$$
$$18$$ 1.00000i 0.235702i
$$19$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$20$$ − 1.00000i − 0.223607i
$$21$$ 0 0
$$22$$ 0 0
$$23$$ 4.00000 0.834058 0.417029 0.908893i $$-0.363071\pi$$
0.417029 + 0.908893i $$0.363071\pi$$
$$24$$ 1.00000i 0.204124i
$$25$$ −1.00000 −0.200000
$$26$$ 0 0
$$27$$ −1.00000 −0.192450
$$28$$ 0 0
$$29$$ −10.0000 −1.85695 −0.928477 0.371391i $$-0.878881\pi$$
−0.928477 + 0.371391i $$0.878881\pi$$
$$30$$ 1.00000 0.182574
$$31$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$32$$ 1.00000i 0.176777i
$$33$$ 0 0
$$34$$ 6.00000i 1.02899i
$$35$$ 0 0
$$36$$ −1.00000 −0.166667
$$37$$ − 6.00000i − 0.986394i −0.869918 0.493197i $$-0.835828\pi$$
0.869918 0.493197i $$-0.164172\pi$$
$$38$$ 0 0
$$39$$ 0 0
$$40$$ 1.00000 0.158114
$$41$$ − 2.00000i − 0.312348i −0.987730 0.156174i $$-0.950084\pi$$
0.987730 0.156174i $$-0.0499160\pi$$
$$42$$ 0 0
$$43$$ 4.00000 0.609994 0.304997 0.952353i $$-0.401344\pi$$
0.304997 + 0.952353i $$0.401344\pi$$
$$44$$ 0 0
$$45$$ 1.00000i 0.149071i
$$46$$ 4.00000i 0.589768i
$$47$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$48$$ −1.00000 −0.144338
$$49$$ 7.00000 1.00000
$$50$$ − 1.00000i − 0.141421i
$$51$$ −6.00000 −0.840168
$$52$$ 0 0
$$53$$ −6.00000 −0.824163 −0.412082 0.911147i $$-0.635198\pi$$
−0.412082 + 0.911147i $$0.635198\pi$$
$$54$$ − 1.00000i − 0.136083i
$$55$$ 0 0
$$56$$ 0 0
$$57$$ 0 0
$$58$$ − 10.0000i − 1.31306i
$$59$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$60$$ 1.00000i 0.129099i
$$61$$ 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 $$-0.0785709\pi$$
$$68$$ −6.00000 −0.727607
$$69$$ −4.00000 −0.481543
$$70$$ 0 0
$$71$$ − 16.0000i − 1.89885i −0.313993 0.949425i $$-0.601667\pi$$
0.313993 0.949425i $$-0.398333\pi$$
$$72$$ − 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
$$75$$ 1.00000 0.115470
$$76$$ 0 0
$$77$$ 0 0
$$78$$ 0 0
$$79$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$80$$ 1.00000i 0.111803i
$$81$$ 1.00000 0.111111
$$82$$ 2.00000 0.220863
$$83$$ − 4.00000i − 0.439057i −0.975606 0.219529i $$-0.929548\pi$$
0.975606 0.219529i $$-0.0704519\pi$$
$$84$$ 0 0
$$85$$ 6.00000i 0.650791i
$$86$$ 4.00000i 0.431331i
$$87$$ 10.0000 1.07211
$$88$$ 0 0
$$89$$ − 6.00000i − 0.635999i −0.948091 0.317999i $$-0.896989\pi$$
0.948091 0.317999i $$-0.103011\pi$$
$$90$$ −1.00000 −0.105409
$$91$$ 0 0
$$92$$ −4.00000 −0.417029
$$93$$ 0 0
$$94$$ 0 0
$$95$$ 0 0
$$96$$ − 1.00000i − 0.102062i
$$97$$ − 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
$$100$$ 1.00000 0.100000
$$101$$ −6.00000 −0.597022 −0.298511 0.954406i $$-0.596490\pi$$
−0.298511 + 0.954406i $$0.596490\pi$$
$$102$$ − 6.00000i − 0.594089i
$$103$$ −12.0000 −1.18240 −0.591198 0.806527i $$-0.701345\pi$$
−0.591198 + 0.806527i $$0.701345\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ − 6.00000i − 0.582772i
$$107$$ −4.00000 −0.386695 −0.193347 0.981130i $$-0.561934\pi$$
−0.193347 + 0.981130i $$0.561934\pi$$
$$108$$ 1.00000 0.0962250
$$109$$ 14.0000i 1.34096i 0.741929 + 0.670478i $$0.233911\pi$$
−0.741929 + 0.670478i $$0.766089\pi$$
$$110$$ 0 0
$$111$$ 6.00000i 0.569495i
$$112$$ 0 0
$$113$$ 10.0000 0.940721 0.470360 0.882474i $$-0.344124\pi$$
0.470360 + 0.882474i $$0.344124\pi$$
$$114$$ 0 0
$$115$$ 4.00000i 0.373002i
$$116$$ 10.0000 0.928477
$$117$$ 0 0
$$118$$ 0 0
$$119$$ 0 0
$$120$$ −1.00000 −0.0912871
$$121$$ 11.0000 1.00000
$$122$$ 6.00000i 0.543214i
$$123$$ 2.00000i 0.180334i
$$124$$ 0 0
$$125$$ − 1.00000i − 0.0894427i
$$126$$ 0 0
$$127$$ 12.0000 1.06483 0.532414 0.846484i $$-0.321285\pi$$
0.532414 + 0.846484i $$0.321285\pi$$
$$128$$ − 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
$$135$$ − 1.00000i − 0.0860663i
$$136$$ − 6.00000i − 0.514496i
$$137$$ 6.00000i 0.512615i 0.966595 + 0.256307i $$0.0825059\pi$$
−0.966595 + 0.256307i $$0.917494\pi$$
$$138$$ − 4.00000i − 0.340503i
$$139$$ 4.00000 0.339276 0.169638 0.985506i $$-0.445740\pi$$
0.169638 + 0.985506i $$0.445740\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 16.0000 1.34269
$$143$$ 0 0
$$144$$ 1.00000 0.0833333
$$145$$ − 10.0000i − 0.830455i
$$146$$ 2.00000 0.165521
$$147$$ −7.00000 −0.577350
$$148$$ 6.00000i 0.493197i
$$149$$ 14.0000i 1.14692i 0.819232 + 0.573462i $$0.194400\pi$$
−0.819232 + 0.573462i $$0.805600\pi$$
$$150$$ 1.00000i 0.0816497i
$$151$$ − 16.0000i − 1.30206i −0.759051 0.651031i $$-0.774337\pi$$
0.759051 0.651031i $$-0.225663\pi$$
$$152$$ 0 0
$$153$$ 6.00000 0.485071
$$154$$ 0 0
$$155$$ 0 0
$$156$$ 0 0
$$157$$ −14.0000 −1.11732 −0.558661 0.829396i $$-0.688685\pi$$
−0.558661 + 0.829396i $$0.688685\pi$$
$$158$$ 0 0
$$159$$ 6.00000 0.475831
$$160$$ −1.00000 −0.0790569
$$161$$ 0 0
$$162$$ 1.00000i 0.0785674i
$$163$$ − 4.00000i − 0.313304i −0.987654 0.156652i $$-0.949930\pi$$
0.987654 0.156652i $$-0.0500701\pi$$
$$164$$ 2.00000i 0.156174i
$$165$$ 0 0
$$166$$ 4.00000 0.310460
$$167$$ 24.0000i 1.85718i 0.371113 + 0.928588i $$0.378976\pi$$
−0.371113 + 0.928588i $$0.621024\pi$$
$$168$$ 0 0
$$169$$ 0 0
$$170$$ −6.00000 −0.460179
$$171$$ 0 0
$$172$$ −4.00000 −0.304997
$$173$$ 14.0000 1.06440 0.532200 0.846619i $$-0.321365\pi$$
0.532200 + 0.846619i $$0.321365\pi$$
$$174$$ 10.0000i 0.758098i
$$175$$ 0 0
$$176$$ 0 0
$$177$$ 0 0
$$178$$ 6.00000 0.449719
$$179$$ 20.0000 1.49487 0.747435 0.664335i $$-0.231285\pi$$
0.747435 + 0.664335i $$0.231285\pi$$
$$180$$ − 1.00000i − 0.0745356i
$$181$$ 10.0000 0.743294 0.371647 0.928374i $$-0.378793\pi$$
0.371647 + 0.928374i $$0.378793\pi$$
$$182$$ 0 0
$$183$$ −6.00000 −0.443533
$$184$$ − 4.00000i − 0.294884i
$$185$$ 6.00000 0.441129
$$186$$ 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 $$0.165220\pi$$
$$192$$ 1.00000 0.0721688
$$193$$ 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 $$0.308063\pi$$
$$200$$ 1.00000i 0.0707107i
$$201$$ 4.00000i 0.282138i
$$202$$ − 6.00000i − 0.422159i
$$203$$ 0 0
$$204$$ 6.00000 0.420084
$$205$$ 2.00000 0.139686
$$206$$ − 12.0000i − 0.836080i
$$207$$ 4.00000 0.278019
$$208$$ 0 0
$$209$$ 0 0
$$210$$ 0 0
$$211$$ −12.0000 −0.826114 −0.413057 0.910705i $$-0.635539\pi$$
−0.413057 + 0.910705i $$0.635539\pi$$
$$212$$ 6.00000 0.412082
$$213$$ 16.0000i 1.09630i
$$214$$ − 4.00000i − 0.273434i
$$215$$ 4.00000i 0.272798i
$$216$$ 1.00000i 0.0680414i
$$217$$ 0 0
$$218$$ −14.0000 −0.948200
$$219$$ 2.00000i 0.135147i
$$220$$ 0 0
$$221$$ 0 0
$$222$$ −6.00000 −0.402694
$$223$$ − 16.0000i − 1.07144i −0.844396 0.535720i $$-0.820040\pi$$
0.844396 0.535720i $$-0.179960\pi$$
$$224$$ 0 0
$$225$$ −1.00000 −0.0666667
$$226$$ 10.0000i 0.665190i
$$227$$ − 20.0000i − 1.32745i −0.747978 0.663723i $$-0.768975\pi$$
0.747978 0.663723i $$-0.231025\pi$$
$$228$$ 0 0
$$229$$ 2.00000i 0.132164i 0.997814 + 0.0660819i $$0.0210498\pi$$
−0.997814 + 0.0660819i $$0.978950\pi$$
$$230$$ −4.00000 −0.263752
$$231$$ 0 0
$$232$$ 10.0000i 0.656532i
$$233$$ −18.0000 −1.17922 −0.589610 0.807688i $$-0.700718\pi$$
−0.589610 + 0.807688i $$0.700718\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ 0 0
$$237$$ 0 0
$$238$$ 0 0
$$239$$ − 16.0000i − 1.03495i −0.855697 0.517477i $$-0.826871\pi$$
0.855697 0.517477i $$-0.173129\pi$$
$$240$$ − 1.00000i − 0.0645497i
$$241$$ − 14.0000i − 0.901819i −0.892570 0.450910i $$-0.851100\pi$$
0.892570 0.450910i $$-0.148900\pi$$
$$242$$ 11.0000i 0.707107i
$$243$$ −1.00000 −0.0641500
$$244$$ −6.00000 −0.384111
$$245$$ 7.00000i 0.447214i
$$246$$ −2.00000 −0.127515
$$247$$ 0 0
$$248$$ 0 0
$$249$$ 4.00000i 0.253490i
$$250$$ 1.00000 0.0632456
$$251$$ 4.00000 0.252478 0.126239 0.992000i $$-0.459709\pi$$
0.126239 + 0.992000i $$0.459709\pi$$
$$252$$ 0 0
$$253$$ 0 0
$$254$$ 12.0000i 0.752947i
$$255$$ − 6.00000i − 0.375735i
$$256$$ 1.00000 0.0625000
$$257$$ 6.00000 0.374270 0.187135 0.982334i $$-0.440080\pi$$
0.187135 + 0.982334i $$0.440080\pi$$
$$258$$ − 4.00000i − 0.249029i
$$259$$ 0 0
$$260$$ 0 0
$$261$$ −10.0000 −0.618984
$$262$$ 12.0000i 0.741362i
$$263$$ −28.0000 −1.72655 −0.863277 0.504730i $$-0.831592\pi$$
−0.863277 + 0.504730i $$0.831592\pi$$
$$264$$ 0 0
$$265$$ − 6.00000i − 0.368577i
$$266$$ 0 0
$$267$$ 6.00000i 0.367194i
$$268$$ 4.00000i 0.244339i
$$269$$ 14.0000 0.853595 0.426798 0.904347i $$-0.359642\pi$$
0.426798 + 0.904347i $$0.359642\pi$$
$$270$$ 1.00000 0.0608581
$$271$$ − 24.0000i − 1.45790i −0.684569 0.728948i $$-0.740010\pi$$
0.684569 0.728948i $$-0.259990\pi$$
$$272$$ 6.00000 0.363803
$$273$$ 0 0
$$274$$ −6.00000 −0.362473
$$275$$ 0 0
$$276$$ 4.00000 0.240772
$$277$$ −2.00000 −0.120168 −0.0600842 0.998193i $$-0.519137\pi$$
−0.0600842 + 0.998193i $$0.519137\pi$$
$$278$$ 4.00000i 0.239904i
$$279$$ 0 0
$$280$$ 0 0
$$281$$ − 6.00000i − 0.357930i −0.983855 0.178965i $$-0.942725\pi$$
0.983855 0.178965i $$-0.0572749\pi$$
$$282$$ 0 0
$$283$$ 12.0000 0.713326 0.356663 0.934233i $$-0.383914\pi$$
0.356663 + 0.934233i $$0.383914\pi$$
$$284$$ 16.0000i 0.949425i
$$285$$ 0 0
$$286$$ 0 0
$$287$$ 0 0
$$288$$ 1.00000i 0.0589256i
$$289$$ 19.0000 1.11765
$$290$$ 10.0000 0.587220
$$291$$ 14.0000i 0.820695i
$$292$$ 2.00000i 0.117041i
$$293$$ 26.0000i 1.51894i 0.650545 + 0.759468i $$0.274541\pi$$
−0.650545 + 0.759468i $$0.725459\pi$$
$$294$$ − 7.00000i − 0.408248i
$$295$$ 0 0
$$296$$ −6.00000 −0.348743
$$297$$ 0 0
$$298$$ −14.0000 −0.810998
$$299$$ 0 0
$$300$$ −1.00000 −0.0577350
$$301$$ 0 0
$$302$$ 16.0000 0.920697
$$303$$ 6.00000 0.344691
$$304$$ 0 0
$$305$$ 6.00000i 0.343559i
$$306$$ 6.00000i 0.342997i
$$307$$ − 4.00000i − 0.228292i −0.993464 0.114146i $$-0.963587\pi$$
0.993464 0.114146i $$-0.0364132\pi$$
$$308$$ 0 0
$$309$$ 12.0000 0.682656
$$310$$ 0 0
$$311$$ 16.0000 0.907277 0.453638 0.891186i $$-0.350126\pi$$
0.453638 + 0.891186i $$0.350126\pi$$
$$312$$ 0 0
$$313$$ 26.0000 1.46961 0.734803 0.678280i $$-0.237274\pi$$
0.734803 + 0.678280i $$0.237274\pi$$
$$314$$ − 14.0000i − 0.790066i
$$315$$ 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
$$320$$ − 1.00000i − 0.0559017i
$$321$$ 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
$$328$$ −2.00000 −0.110432
$$329$$ 0 0
$$330$$ 0 0
$$331$$ − 8.00000i − 0.439720i −0.975531 0.219860i $$-0.929440\pi$$
0.975531 0.219860i $$-0.0705600\pi$$
$$332$$ 4.00000i 0.219529i
$$333$$ − 6.00000i − 0.328798i
$$334$$ −24.0000 −1.31322
$$335$$ 4.00000 0.218543
$$336$$ 0 0
$$337$$ −18.0000 −0.980522 −0.490261 0.871576i $$-0.663099\pi$$
−0.490261 + 0.871576i $$0.663099\pi$$
$$338$$ 0 0
$$339$$ −10.0000 −0.543125
$$340$$ − 6.00000i − 0.325396i
$$341$$ 0 0
$$342$$ 0 0
$$343$$ 0 0
$$344$$ − 4.00000i − 0.215666i
$$345$$ − 4.00000i − 0.215353i
$$346$$ 14.0000i 0.752645i
$$347$$ 12.0000 0.644194 0.322097 0.946707i $$-0.395612\pi$$
0.322097 + 0.946707i $$0.395612\pi$$
$$348$$ −10.0000 −0.536056
$$349$$ 34.0000i 1.81998i 0.414632 + 0.909989i $$0.363910\pi$$
−0.414632 + 0.909989i $$0.636090\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 0 0
$$353$$ 26.0000i 1.38384i 0.721974 + 0.691920i $$0.243235\pi$$
−0.721974 + 0.691920i $$0.756765\pi$$
$$354$$ 0 0
$$355$$ 16.0000 0.849192
$$356$$ 6.00000i 0.317999i
$$357$$ 0 0
$$358$$ 20.0000i 1.05703i
$$359$$ 24.0000i 1.26667i 0.773877 + 0.633336i $$0.218315\pi$$
−0.773877 + 0.633336i $$0.781685\pi$$
$$360$$ 1.00000 0.0527046
$$361$$ 19.0000 1.00000
$$362$$ 10.0000i 0.525588i
$$363$$ −11.0000 −0.577350
$$364$$ 0 0
$$365$$ 2.00000 0.104685
$$366$$ − 6.00000i − 0.313625i
$$367$$ 20.0000 1.04399 0.521996 0.852948i $$-0.325188\pi$$
0.521996 + 0.852948i $$0.325188\pi$$
$$368$$ 4.00000 0.208514
$$369$$ − 2.00000i − 0.104116i
$$370$$ 6.00000i 0.311925i
$$371$$ 0 0
$$372$$ 0 0
$$373$$ 10.0000 0.517780 0.258890 0.965907i $$-0.416643\pi$$
0.258890 + 0.965907i $$0.416643\pi$$
$$374$$ 0 0
$$375$$ 1.00000i 0.0516398i
$$376$$ 0 0
$$377$$ 0 0
$$378$$ 0 0
$$379$$ − 24.0000i − 1.23280i −0.787434 0.616399i $$-0.788591\pi$$
0.787434 0.616399i $$-0.211409\pi$$
$$380$$ 0 0
$$381$$ −12.0000 −0.614779
$$382$$ 24.0000i 1.22795i
$$383$$ 8.00000i 0.408781i 0.978889 + 0.204390i $$0.0655212\pi$$
−0.978889 + 0.204390i $$0.934479\pi$$
$$384$$ 1.00000i 0.0510310i
$$385$$ 0 0
$$386$$ −14.0000 −0.712581
$$387$$ 4.00000 0.203331
$$388$$ 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
$$393$$ −12.0000 −0.605320
$$394$$ −22.0000 −1.10834
$$395$$ 0 0
$$396$$ 0 0
$$397$$ − 14.0000i − 0.702640i −0.936255 0.351320i $$-0.885733\pi$$
0.936255 0.351320i $$-0.114267\pi$$
$$398$$ 16.0000i 0.802008i
$$399$$ 0 0
$$400$$ −1.00000 −0.0500000
$$401$$ 18.0000i 0.898877i 0.893311 + 0.449439i $$0.148376\pi$$
−0.893311 + 0.449439i $$0.851624\pi$$
$$402$$ −4.00000 −0.199502
$$403$$ 0 0
$$404$$ 6.00000 0.298511
$$405$$ 1.00000i 0.0496904i
$$406$$ 0 0
$$407$$ 0 0
$$408$$ 6.00000i 0.297044i
$$409$$ − 2.00000i − 0.0988936i −0.998777 0.0494468i $$-0.984254\pi$$
0.998777 0.0494468i $$-0.0157458\pi$$
$$410$$ 2.00000i 0.0987730i
$$411$$ − 6.00000i − 0.295958i
$$412$$ 12.0000 0.591198
$$413$$ 0 0
$$414$$ 4.00000i 0.196589i
$$415$$ 4.00000 0.196352
$$416$$ 0 0
$$417$$ −4.00000 −0.195881
$$418$$ 0 0
$$419$$ 12.0000 0.586238 0.293119 0.956076i $$-0.405307\pi$$
0.293119 + 0.956076i $$0.405307\pi$$
$$420$$ 0 0
$$421$$ 22.0000i 1.07221i 0.844150 + 0.536107i $$0.180106\pi$$
−0.844150 + 0.536107i $$0.819894\pi$$
$$422$$ − 12.0000i − 0.584151i
$$423$$ 0 0
$$424$$ 6.00000i 0.291386i
$$425$$ −6.00000 −0.291043
$$426$$ −16.0000 −0.775203
$$427$$ 0 0
$$428$$ 4.00000 0.193347
$$429$$ 0 0
$$430$$ −4.00000 −0.192897
$$431$$ 24.0000i 1.15604i 0.816023 + 0.578020i $$0.196174\pi$$
−0.816023 + 0.578020i $$0.803826\pi$$
$$432$$ −1.00000 −0.0481125
$$433$$ −2.00000 −0.0961139 −0.0480569 0.998845i $$-0.515303\pi$$
−0.0480569 + 0.998845i $$0.515303\pi$$
$$434$$ 0 0
$$435$$ 10.0000i 0.479463i
$$436$$ − 14.0000i − 0.670478i
$$437$$ 0 0
$$438$$ −2.00000 −0.0955637
$$439$$ 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
$$442$$ 0 0
$$443$$ −4.00000 −0.190046 −0.0950229 0.995475i $$-0.530292\pi$$
−0.0950229 + 0.995475i $$0.530292\pi$$
$$444$$ − 6.00000i − 0.284747i
$$445$$ 6.00000 0.284427
$$446$$ 16.0000 0.757622
$$447$$ − 14.0000i − 0.662177i
$$448$$ 0 0
$$449$$ − 14.0000i − 0.660701i −0.943858 0.330350i $$-0.892833\pi$$
0.943858 0.330350i $$-0.107167\pi$$
$$450$$ − 1.00000i − 0.0471405i
$$451$$ 0 0
$$452$$ −10.0000 −0.470360
$$453$$ 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 $$-0.0448177\pi$$
$$458$$ −2.00000 −0.0934539
$$459$$ −6.00000 −0.280056
$$460$$ − 4.00000i − 0.186501i
$$461$$ 14.0000i 0.652045i 0.945362 + 0.326023i $$0.105709\pi$$
−0.945362 + 0.326023i $$0.894291\pi$$
$$462$$ 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
$$466$$ − 18.0000i − 0.833834i
$$467$$ 28.0000 1.29569 0.647843 0.761774i $$-0.275671\pi$$
0.647843 + 0.761774i $$0.275671\pi$$
$$468$$ 0 0
$$469$$ 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
$$477$$ −6.00000 −0.274721
$$478$$ 16.0000 0.731823
$$479$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$480$$ 1.00000 0.0456435
$$481$$ 0 0
$$482$$ 14.0000 0.637683
$$483$$ 0 0
$$484$$ −11.0000 −0.500000
$$485$$ 14.0000 0.635707
$$486$$ − 1.00000i − 0.0453609i
$$487$$ − 8.00000i − 0.362515i −0.983436 0.181257i $$-0.941983\pi$$
0.983436 0.181257i $$-0.0580167\pi$$
$$488$$ − 6.00000i − 0.271607i
$$489$$ 4.00000i 0.180886i
$$490$$ −7.00000 −0.316228
$$491$$ −20.0000 −0.902587 −0.451294 0.892375i $$-0.649037\pi$$
−0.451294 + 0.892375i $$0.649037\pi$$
$$492$$ − 2.00000i − 0.0901670i
$$493$$ −60.0000 −2.70226
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ 0 0
$$498$$ −4.00000 −0.179244
$$499$$ 40.0000i 1.79065i 0.445418 + 0.895323i $$0.353055\pi$$
−0.445418 + 0.895323i $$0.646945\pi$$
$$500$$ 1.00000i 0.0447214i
$$501$$ − 24.0000i − 1.07224i
$$502$$ 4.00000i 0.178529i
$$503$$ −36.0000 −1.60516 −0.802580 0.596544i $$-0.796540\pi$$
−0.802580 + 0.596544i $$0.796540\pi$$
$$504$$ 0 0
$$505$$ − 6.00000i − 0.266996i
$$506$$ 0 0
$$507$$ 0 0
$$508$$ −12.0000 −0.532414
$$509$$ 38.0000i 1.68432i 0.539227 + 0.842160i $$0.318716\pi$$
−0.539227 + 0.842160i $$0.681284\pi$$
$$510$$ 6.00000 0.265684
$$511$$ 0 0
$$512$$ 1.00000i 0.0441942i
$$513$$ 0 0
$$514$$ 6.00000i 0.264649i
$$515$$ − 12.0000i − 0.528783i
$$516$$ 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
$$523$$ −20.0000 −0.874539 −0.437269 0.899331i $$-0.644054\pi$$
−0.437269 + 0.899331i $$0.644054\pi$$
$$524$$ −12.0000 −0.524222
$$525$$ 0 0
$$526$$ − 28.0000i − 1.22086i
$$527$$ 0 0
$$528$$ 0 0
$$529$$ −7.00000 −0.304348
$$530$$ 6.00000 0.260623
$$531$$ 0 0
$$532$$ 0 0
$$533$$ 0 0
$$534$$ −6.00000 −0.259645
$$535$$ − 4.00000i − 0.172935i
$$536$$ −4.00000 −0.172774
$$537$$ −20.0000 −0.863064
$$538$$ 14.0000i 0.603583i
$$539$$ 0 0
$$540$$ 1.00000i 0.0430331i
$$541$$ − 22.0000i − 0.945854i −0.881102 0.472927i $$-0.843197\pi$$
0.881102 0.472927i $$-0.156803\pi$$
$$542$$ 24.0000 1.03089
$$543$$ −10.0000 −0.429141
$$544$$ 6.00000i 0.257248i
$$545$$ −14.0000 −0.599694
$$546$$ 0 0
$$547$$ 28.0000 1.19719 0.598597 0.801050i $$-0.295725\pi$$
0.598597 + 0.801050i $$0.295725\pi$$
$$548$$ − 6.00000i − 0.256307i
$$549$$ 6.00000 0.256074
$$550$$ 0 0
$$551$$ 0 0
$$552$$ 4.00000i 0.170251i
$$553$$ 0 0
$$554$$ − 2.00000i − 0.0849719i
$$555$$ −6.00000 −0.254686
$$556$$ −4.00000 −0.169638
$$557$$ − 30.0000i − 1.27114i −0.772043 0.635570i $$-0.780765\pi$$
0.772043 0.635570i $$-0.219235\pi$$
$$558$$ 0 0
$$559$$ 0 0
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 6.00000 0.253095
$$563$$ 28.0000 1.18006 0.590030 0.807382i $$-0.299116\pi$$
0.590030 + 0.807382i $$0.299116\pi$$
$$564$$ 0 0
$$565$$ 10.0000i 0.420703i
$$566$$ 12.0000i 0.504398i
$$567$$ 0 0
$$568$$ −16.0000 −0.671345
$$569$$ 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 $$-0.473327\pi$$
0.0836974 + 0.996491i $$0.473327\pi$$
$$572$$ 0 0
$$573$$ −24.0000 −1.00261
$$574$$ 0 0
$$575$$ −4.00000 −0.166812
$$576$$ −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
$$579$$ − 14.0000i − 0.581820i
$$580$$ 10.0000i 0.415227i
$$581$$ 0 0
$$582$$ −14.0000 −0.580319
$$583$$ 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 $$0.920343\pi$$
$$588$$ 7.00000 0.288675
$$589$$ 0 0
$$590$$ 0 0
$$591$$ − 22.0000i − 0.904959i
$$592$$ − 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
$$596$$ − 14.0000i − 0.573462i
$$597$$ −16.0000 −0.654836
$$598$$ 0 0
$$599$$ 24.0000 0.980613 0.490307 0.871550i $$-0.336885\pi$$
0.490307 + 0.871550i $$0.336885\pi$$
$$600$$ − 1.00000i − 0.0408248i
$$601$$ 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
$$604$$ 16.0000i 0.651031i
$$605$$ 11.0000i 0.447214i
$$606$$ 6.00000i 0.243733i
$$607$$ −36.0000 −1.46119 −0.730597 0.682808i $$-0.760758\pi$$
−0.730597 + 0.682808i $$0.760758\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ −6.00000 −0.242933
$$611$$ 0 0
$$612$$ −6.00000 −0.242536
$$613$$ 22.0000i 0.888572i 0.895885 + 0.444286i $$0.146543\pi$$
−0.895885 + 0.444286i $$0.853457\pi$$
$$614$$ 4.00000 0.161427
$$615$$ −2.00000 −0.0806478
$$616$$ 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
$$619$$ 40.0000i 1.60774i 0.594808 + 0.803868i $$0.297228\pi$$
−0.594808 + 0.803868i $$0.702772\pi$$
$$620$$ 0 0
$$621$$ −4.00000 −0.160514
$$622$$ 16.0000i 0.641542i
$$623$$ 0 0
$$624$$ 0 0
$$625$$ 1.00000 0.0400000
$$626$$ 26.0000i 1.03917i
$$627$$ 0 0
$$628$$ 14.0000 0.558661
$$629$$ − 36.0000i − 1.43541i
$$630$$ 0 0
$$631$$ − 48.0000i − 1.91085i −0.295234 0.955425i $$-0.595398\pi$$
0.295234 0.955425i $$-0.404602\pi$$
$$632$$ 0 0
$$633$$ 12.0000 0.476957
$$634$$ 2.00000 0.0794301
$$635$$ 12.0000i 0.476205i
$$636$$ −6.00000 −0.237915
$$637$$ 0 0
$$638$$ 0 0
$$639$$ − 16.0000i − 0.632950i
$$640$$ 1.00000 0.0395285
$$641$$ −2.00000 −0.0789953 −0.0394976 0.999220i $$-0.512576\pi$$
−0.0394976 + 0.999220i $$0.512576\pi$$
$$642$$ 4.00000i 0.157867i
$$643$$ − 4.00000i − 0.157745i −0.996885 0.0788723i $$-0.974868\pi$$
0.996885 0.0788723i $$-0.0251319\pi$$
$$644$$ 0 0
$$645$$ − 4.00000i − 0.157500i
$$646$$ 0 0
$$647$$ 36.0000 1.41531 0.707653 0.706560i $$-0.249754\pi$$
0.707653 + 0.706560i $$0.249754\pi$$
$$648$$ − 1.00000i − 0.0392837i
$$649$$ 0 0
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 4.00000i 0.156652i
$$653$$ −6.00000 −0.234798 −0.117399 0.993085i $$-0.537456\pi$$
−0.117399 + 0.993085i $$0.537456\pi$$
$$654$$ 14.0000 0.547443
$$655$$ 12.0000i 0.468879i
$$656$$ − 2.00000i − 0.0780869i
$$657$$ − 2.00000i − 0.0780274i
$$658$$ 0 0
$$659$$ −20.0000 −0.779089 −0.389545 0.921008i $$-0.627368\pi$$
−0.389545 + 0.921008i $$0.627368\pi$$
$$660$$ 0 0
$$661$$ − 30.0000i − 1.16686i −0.812162 0.583432i $$-0.801709\pi$$
0.812162 0.583432i $$-0.198291\pi$$
$$662$$ 8.00000 0.310929
$$663$$ 0 0
$$664$$ −4.00000 −0.155230
$$665$$ 0 0
$$666$$ 6.00000 0.232495
$$667$$ −40.0000 −1.54881
$$668$$ − 24.0000i − 0.928588i
$$669$$ 16.0000i 0.618596i
$$670$$ 4.00000i 0.154533i
$$671$$ 0 0
$$672$$ 0 0
$$673$$ −42.0000 −1.61898 −0.809491 0.587133i $$-0.800257\pi$$
−0.809491 + 0.587133i $$0.800257\pi$$
$$674$$ − 18.0000i − 0.693334i
$$675$$ 1.00000 0.0384900
$$676$$ 0 0
$$677$$ −30.0000 −1.15299 −0.576497 0.817099i $$-0.695581\pi$$
−0.576497 + 0.817099i $$0.695581\pi$$
$$678$$ − 10.0000i − 0.384048i
$$679$$ 0 0
$$680$$ 6.00000 0.230089
$$681$$ 20.0000i 0.766402i
$$682$$ 0 0
$$683$$ 12.0000i 0.459167i 0.973289 + 0.229584i $$0.0737364\pi$$
−0.973289 + 0.229584i $$0.926264\pi$$
$$684$$ 0 0
$$685$$ −6.00000 −0.229248
$$686$$ 0 0
$$687$$ − 2.00000i − 0.0763048i
$$688$$ 4.00000 0.152499
$$689$$ 0 0
$$690$$ 4.00000 0.152277
$$691$$ − 32.0000i − 1.21734i −0.793424 0.608669i $$-0.791704\pi$$
0.793424 0.608669i $$-0.208296\pi$$
$$692$$ −14.0000 −0.532200
$$693$$ 0 0
$$694$$ 12.0000i 0.455514i
$$695$$ 4.00000i 0.151729i
$$696$$ − 10.0000i − 0.379049i
$$697$$ − 12.0000i − 0.454532i
$$698$$ −34.0000 −1.28692
$$699$$ 18.0000 0.680823
$$700$$ 0 0
$$701$$ 18.0000 0.679851 0.339925 0.940452i $$-0.389598\pi$$
0.339925 + 0.940452i $$0.389598\pi$$
$$702$$ 0 0
$$703$$ 0 0
$$704$$ 0 0
$$705$$ 0 0
$$706$$ −26.0000 −0.978523
$$707$$ 0 0
$$708$$ 0 0
$$709$$ 2.00000i 0.0751116i 0.999295 + 0.0375558i $$0.0119572\pi$$
−0.999295 + 0.0375558i $$0.988043\pi$$
$$710$$ 16.0000i 0.600469i
$$711$$ 0 0
$$712$$ −6.00000 −0.224860
$$713$$ 0 0
$$714$$ 0 0
$$715$$ 0 0
$$716$$ −20.0000 −0.747435
$$717$$ 16.0000i 0.597531i
$$718$$ −24.0000 −0.895672
$$719$$ −24.0000 −0.895049 −0.447524 0.894272i $$-0.647694\pi$$
−0.447524 + 0.894272i $$0.647694\pi$$
$$720$$ 1.00000i 0.0372678i
$$721$$ 0 0
$$722$$ 19.0000i 0.707107i
$$723$$ 14.0000i 0.520666i
$$724$$ −10.0000 −0.371647
$$725$$ 10.0000 0.371391
$$726$$ − 11.0000i − 0.408248i
$$727$$ −12.0000 −0.445055 −0.222528 0.974926i $$-0.571431\pi$$
−0.222528 + 0.974926i $$0.571431\pi$$
$$728$$ 0 0
$$729$$ 1.00000 0.0370370
$$730$$ 2.00000i 0.0740233i
$$731$$ 24.0000 0.887672
$$732$$ 6.00000 0.221766
$$733$$ − 50.0000i − 1.84679i −0.383849 0.923396i $$-0.625402\pi$$
0.383849 0.923396i $$-0.374598\pi$$
$$734$$ 20.0000i 0.738213i
$$735$$ − 7.00000i − 0.258199i
$$736$$ 4.00000i 0.147442i
$$737$$ 0 0
$$738$$ 2.00000 0.0736210
$$739$$ − 40.0000i − 1.47142i −0.677295 0.735712i $$-0.736848\pi$$
0.677295 0.735712i $$-0.263152\pi$$
$$740$$ −6.00000 −0.220564
$$741$$ 0 0
$$742$$ 0 0
$$743$$ 48.0000i 1.76095i 0.474093 + 0.880475i $$0.342776\pi$$
−0.474093 + 0.880475i $$0.657224\pi$$
$$744$$ 0 0
$$745$$ −14.0000 −0.512920
$$746$$ 10.0000i 0.366126i
$$747$$ − 4.00000i − 0.146352i
$$748$$ 0 0
$$749$$ 0 0
$$750$$ −1.00000 −0.0365148
$$751$$ −24.0000 −0.875772 −0.437886 0.899030i $$-0.644273\pi$$
−0.437886 + 0.899030i $$0.644273\pi$$
$$752$$ 0 0
$$753$$ −4.00000 −0.145768
$$754$$ 0 0
$$755$$ 16.0000 0.582300
$$756$$ 0 0
$$757$$ 2.00000 0.0726912 0.0363456 0.999339i $$-0.488428\pi$$
0.0363456 + 0.999339i $$0.488428\pi$$
$$758$$ 24.0000 0.871719
$$759$$ 0 0
$$760$$ 0 0
$$761$$ − 22.0000i − 0.797499i −0.917060 0.398750i $$-0.869444\pi$$
0.917060 0.398750i $$-0.130556\pi$$
$$762$$ − 12.0000i − 0.434714i
$$763$$ 0 0
$$764$$ −24.0000 −0.868290
$$765$$ 6.00000i 0.216930i
$$766$$ −8.00000 −0.289052
$$767$$ 0 0
$$768$$ −1.00000 −0.0360844
$$769$$ − 34.0000i − 1.22607i −0.790055 0.613036i $$-0.789948\pi$$
0.790055 0.613036i $$-0.210052\pi$$
$$770$$ 0 0
$$771$$ −6.00000 −0.216085
$$772$$ − 14.0000i − 0.503871i
$$773$$ − 26.0000i − 0.935155i −0.883952 0.467578i $$-0.845127\pi$$
0.883952 0.467578i $$-0.154873\pi$$
$$774$$ 4.00000i 0.143777i
$$775$$ 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
$$783$$ 10.0000 0.357371
$$784$$ 7.00000 0.250000
$$785$$ − 14.0000i − 0.499681i
$$786$$ − 12.0000i − 0.428026i
$$787$$ − 36.0000i − 1.28326i −0.767014 0.641631i $$-0.778258\pi$$
0.767014 0.641631i $$-0.221742\pi$$
$$788$$ − 22.0000i − 0.783718i
$$789$$ 28.0000 0.996826
$$790$$ 0 0
$$791$$ 0 0
$$792$$ 0 0
$$793$$ 0 0
$$794$$ 14.0000 0.496841
$$795$$ 6.00000i 0.212798i
$$796$$ −16.0000 −0.567105
$$797$$ −18.0000 −0.637593 −0.318796 0.947823i $$-0.603279\pi$$
−0.318796 + 0.947823i $$0.603279\pi$$
$$798$$ 0 0
$$799$$ 0 0
$$800$$ − 1.00000i − 0.0353553i
$$801$$ − 6.00000i − 0.212000i
$$802$$ −18.0000 −0.635602
$$803$$ 0 0
$$804$$ − 4.00000i − 0.141069i
$$805$$ 0 0
$$806$$ 0 0
$$807$$ −14.0000 −0.492823
$$808$$ 6.00000i 0.211079i
$$809$$ 26.0000 0.914111 0.457056 0.889438i $$-0.348904\pi$$
0.457056 + 0.889438i $$0.348904\pi$$
$$810$$ −1.00000 −0.0351364
$$811$$ − 32.0000i − 1.12367i −0.827249 0.561836i $$-0.810095\pi$$
0.827249 0.561836i $$-0.189905\pi$$
$$812$$ 0 0
$$813$$ 24.0000i 0.841717i
$$814$$ 0 0
$$815$$ 4.00000 0.140114
$$816$$ −6.00000 −0.210042
$$817$$ 0 0
$$818$$ 2.00000 0.0699284
$$819$$ 0 0
$$820$$ −2.00000 −0.0698430
$$821$$ 22.0000i 0.767805i 0.923374 + 0.383903i $$0.125420\pi$$
−0.923374 + 0.383903i $$0.874580\pi$$
$$822$$ 6.00000 0.209274
$$823$$ 44.0000 1.53374 0.766872 0.641800i $$-0.221812\pi$$
0.766872 + 0.641800i $$0.221812\pi$$
$$824$$ 12.0000i 0.418040i
$$825$$ 0 0
$$826$$ 0 0
$$827$$ − 12.0000i − 0.417281i −0.977992 0.208640i $$-0.933096\pi$$
0.977992 0.208640i $$-0.0669038\pi$$
$$828$$ −4.00000 −0.139010
$$829$$ −54.0000 −1.87550 −0.937749 0.347314i $$-0.887094\pi$$
−0.937749 + 0.347314i $$0.887094\pi$$
$$830$$ 4.00000i 0.138842i
$$831$$ 2.00000 0.0693792
$$832$$ 0 0
$$833$$ 42.0000 1.45521
$$834$$ − 4.00000i − 0.138509i
$$835$$ −24.0000 −0.830554
$$836$$ 0 0
$$837$$ 0 0
$$838$$ 12.0000i 0.414533i
$$839$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$840$$ 0 0
$$841$$ 71.0000 2.44828
$$842$$ −22.0000 −0.758170
$$843$$ 6.00000i 0.206651i
$$844$$ 12.0000 0.413057
$$845$$ 0 0
$$846$$ 0 0
$$847$$ 0 0
$$848$$ −6.00000 −0.206041
$$849$$ −12.0000 −0.411839
$$850$$ − 6.00000i − 0.205798i
$$851$$ − 24.0000i − 0.822709i
$$852$$ − 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
$$856$$ 4.00000i 0.136717i
$$857$$ 38.0000 1.29806 0.649028 0.760765i $$-0.275176\pi$$
0.649028 + 0.760765i $$0.275176\pi$$
$$858$$ 0 0
$$859$$ −20.0000 −0.682391 −0.341196 0.939992i $$-0.610832\pi$$
−0.341196 + 0.939992i $$0.610832\pi$$
$$860$$ − 4.00000i − 0.136399i
$$861$$ 0 0
$$862$$ −24.0000 −0.817443
$$863$$ − 48.0000i − 1.63394i −0.576681 0.816970i $$-0.695652\pi$$
0.576681 0.816970i $$-0.304348\pi$$
$$864$$ − 1.00000i − 0.0340207i
$$865$$ 14.0000i 0.476014i
$$866$$ − 2.00000i − 0.0679628i
$$867$$ −19.0000 −0.645274
$$868$$ 0 0
$$869$$ 0 0
$$870$$ −10.0000 −0.339032
$$871$$ 0 0
$$872$$ 14.0000 0.474100
$$873$$ − 14.0000i − 0.473828i
$$874$$ 0 0
$$875$$ 0 0
$$876$$ − 2.00000i − 0.0675737i
$$877$$ 14.0000i 0.472746i 0.971662 + 0.236373i $$0.0759588\pi$$
−0.971662 + 0.236373i $$0.924041\pi$$
$$878$$ 24.0000i 0.809961i
$$879$$ − 26.0000i − 0.876958i
$$880$$ 0 0
$$881$$ −34.0000 −1.14549 −0.572745 0.819734i $$-0.694121\pi$$
−0.572745 + 0.819734i $$0.694121\pi$$
$$882$$ 7.00000i 0.235702i
$$883$$ −52.0000 −1.74994 −0.874970 0.484178i $$-0.839119\pi$$
−0.874970 + 0.484178i $$0.839119\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ − 4.00000i − 0.134383i
$$887$$ −36.0000 −1.20876 −0.604381 0.796696i $$-0.706579\pi$$
−0.604381 + 0.796696i $$0.706579\pi$$
$$888$$ 6.00000 0.201347
$$889$$ 0 0
$$890$$ 6.00000i 0.201120i
$$891$$ 0 0
$$892$$ 16.0000i 0.535720i
$$893$$ 0 0
$$894$$ 14.0000 0.468230
$$895$$ 20.0000i 0.668526i
$$896$$ 0 0
$$897$$ 0 0
$$898$$ 14.0000 0.467186
$$899$$ 0 0
$$900$$ 1.00000 0.0333333
$$901$$ −36.0000 −1.19933
$$902$$ 0 0
$$903$$ 0 0
$$904$$ − 10.0000i − 0.332595i
$$905$$ 10.0000i 0.332411i
$$906$$ −16.0000 −0.531564
$$907$$ −28.0000 −0.929725 −0.464862 0.885383i $$-0.653896\pi$$
−0.464862 + 0.885383i $$0.653896\pi$$
$$908$$ 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
$$915$$ − 6.00000i − 0.198354i
$$916$$ − 2.00000i − 0.0660819i
$$917$$ 0 0
$$918$$ − 6.00000i − 0.198030i
$$919$$ −16.0000 −0.527791 −0.263896 0.964551i $$-0.585007\pi$$
−0.263896 + 0.964551i $$0.585007\pi$$
$$920$$ 4.00000 0.131876
$$921$$ 4.00000i 0.131804i
$$922$$ −14.0000 −0.461065
$$923$$ 0 0
$$924$$ 0 0
$$925$$ 6.00000i 0.197279i
$$926$$ 24.0000 0.788689
$$927$$ −12.0000 −0.394132
$$928$$ − 10.0000i − 0.328266i
$$929$$ 46.0000i 1.50921i 0.656179 + 0.754606i $$0.272172\pi$$
−0.656179 + 0.754606i $$0.727828\pi$$
$$930$$ 0 0
$$931$$ 0 0
$$932$$ 18.0000 0.589610
$$933$$ −16.0000 −0.523816
$$934$$ 28.0000i 0.916188i
$$935$$ 0 0
$$936$$ 0 0
$$937$$ −22.0000 −0.718709 −0.359354 0.933201i $$-0.617003\pi$$
−0.359354 + 0.933201i $$0.617003\pi$$
$$938$$ 0 0
$$939$$ −26.0000 −0.848478
$$940$$ 0 0
$$941$$ − 10.0000i − 0.325991i −0.986627 0.162995i $$-0.947884\pi$$
0.986627 0.162995i $$-0.0521156\pi$$
$$942$$ 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 $$-0.801125\pi$$
0.811090 0.584921i $$-0.198875\pi$$
$$948$$ 0 0
$$949$$ 0 0
$$950$$ 0 0
$$951$$ 2.00000i 0.0648544i
$$952$$ 0 0
$$953$$ −10.0000 −0.323932 −0.161966 0.986796i $$-0.551783\pi$$
−0.161966 + 0.986796i $$0.551783\pi$$
$$954$$ − 6.00000i − 0.194257i
$$955$$ 24.0000i 0.776622i
$$956$$ 16.0000i 0.517477i
$$957$$ 0 0
$$958$$ 0 0
$$959$$ 0 0
$$960$$ 1.00000i 0.0322749i
$$961$$ 31.0000 1.00000
$$962$$ 0 0
$$963$$ −4.00000 −0.128898
$$964$$ 14.0000i 0.450910i
$$965$$ −14.0000 −0.450676
$$966$$ 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
$$970$$ 14.0000i 0.449513i
$$971$$ 12.0000 0.385098 0.192549 0.981287i $$-0.438325\pi$$
0.192549 + 0.981287i $$0.438325\pi$$
$$972$$ 1.00000 0.0320750
$$973$$ 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 $$0.989815\pi$$
$$978$$ −4.00000 −0.127906
$$979$$ 0 0
$$980$$ − 7.00000i − 0.223607i
$$981$$ 14.0000i 0.446986i
$$982$$ − 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
$$985$$ −22.0000 −0.700978
$$986$$ − 60.0000i − 1.91079i
$$987$$ 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
$$995$$ 16.0000i 0.507234i
$$996$$ − 4.00000i − 0.126745i
$$997$$ −22.0000 −0.696747 −0.348373 0.937356i $$-0.613266\pi$$
−0.348373 + 0.937356i $$0.613266\pi$$
$$998$$ −40.0000 −1.26618
$$999$$ 6.00000i 0.189832i
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 5070.2.b.c.1351.2 2
13.5 odd 4 5070.2.a.s.1.1 1
13.8 odd 4 390.2.a.a.1.1 1
13.12 even 2 inner 5070.2.b.c.1351.1 2
39.8 even 4 1170.2.a.m.1.1 1
52.47 even 4 3120.2.a.q.1.1 1
65.8 even 4 1950.2.e.l.1249.2 2
65.34 odd 4 1950.2.a.y.1.1 1
65.47 even 4 1950.2.e.l.1249.1 2
156.47 odd 4 9360.2.a.bn.1.1 1
195.8 odd 4 5850.2.e.p.5149.1 2
195.47 odd 4 5850.2.e.p.5149.2 2
195.164 even 4 5850.2.a.m.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
390.2.a.a.1.1 1 13.8 odd 4
1170.2.a.m.1.1 1 39.8 even 4
1950.2.a.y.1.1 1 65.34 odd 4
1950.2.e.l.1249.1 2 65.47 even 4
1950.2.e.l.1249.2 2 65.8 even 4
3120.2.a.q.1.1 1 52.47 even 4
5070.2.a.s.1.1 1 13.5 odd 4
5070.2.b.c.1351.1 2 13.12 even 2 inner
5070.2.b.c.1351.2 2 1.1 even 1 trivial
5850.2.a.m.1.1 1 195.164 even 4
5850.2.e.p.5149.1 2 195.8 odd 4
5850.2.e.p.5149.2 2 195.47 odd 4
9360.2.a.bn.1.1 1 156.47 odd 4