# Properties

 Label 7942.2.a.c.1.1 Level $7942$ Weight $2$ Character 7942.1 Self dual yes Analytic conductor $63.417$ Analytic rank $2$ Dimension $1$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7942,2,Mod(1,7942)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7942, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("7942.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$7942 = 2 \cdot 11 \cdot 19^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 7942.a (trivial)

## 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: yes Analytic conductor: $$63.4171892853$$ Analytic rank: $$2$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 418) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 7942.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-1.00000 q^{2} -2.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} +2.00000 q^{6} -3.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$$ $$q-1.00000 q^{2} -2.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} +2.00000 q^{6} -3.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} -1.00000 q^{10} -1.00000 q^{11} -2.00000 q^{12} -4.00000 q^{13} +3.00000 q^{14} -2.00000 q^{15} +1.00000 q^{16} -4.00000 q^{17} -1.00000 q^{18} +1.00000 q^{20} +6.00000 q^{21} +1.00000 q^{22} +4.00000 q^{23} +2.00000 q^{24} -4.00000 q^{25} +4.00000 q^{26} +4.00000 q^{27} -3.00000 q^{28} -10.0000 q^{29} +2.00000 q^{30} -4.00000 q^{31} -1.00000 q^{32} +2.00000 q^{33} +4.00000 q^{34} -3.00000 q^{35} +1.00000 q^{36} -3.00000 q^{37} +8.00000 q^{39} -1.00000 q^{40} -6.00000 q^{41} -6.00000 q^{42} -4.00000 q^{43} -1.00000 q^{44} +1.00000 q^{45} -4.00000 q^{46} +6.00000 q^{47} -2.00000 q^{48} +2.00000 q^{49} +4.00000 q^{50} +8.00000 q^{51} -4.00000 q^{52} +1.00000 q^{53} -4.00000 q^{54} -1.00000 q^{55} +3.00000 q^{56} +10.0000 q^{58} -12.0000 q^{59} -2.00000 q^{60} +12.0000 q^{61} +4.00000 q^{62} -3.00000 q^{63} +1.00000 q^{64} -4.00000 q^{65} -2.00000 q^{66} -12.0000 q^{67} -4.00000 q^{68} -8.00000 q^{69} +3.00000 q^{70} -14.0000 q^{71} -1.00000 q^{72} +6.00000 q^{73} +3.00000 q^{74} +8.00000 q^{75} +3.00000 q^{77} -8.00000 q^{78} -17.0000 q^{79} +1.00000 q^{80} -11.0000 q^{81} +6.00000 q^{82} -13.0000 q^{83} +6.00000 q^{84} -4.00000 q^{85} +4.00000 q^{86} +20.0000 q^{87} +1.00000 q^{88} +2.00000 q^{89} -1.00000 q^{90} +12.0000 q^{91} +4.00000 q^{92} +8.00000 q^{93} -6.00000 q^{94} +2.00000 q^{96} -5.00000 q^{97} -2.00000 q^{98} -1.00000 q^{99} +O(q^{100})$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −1.00000 −0.707107
$$3$$ −2.00000 −1.15470 −0.577350 0.816497i $$-0.695913\pi$$
−0.577350 + 0.816497i $$0.695913\pi$$
$$4$$ 1.00000 0.500000
$$5$$ 1.00000 0.447214 0.223607 0.974679i $$-0.428217\pi$$
0.223607 + 0.974679i $$0.428217\pi$$
$$6$$ 2.00000 0.816497
$$7$$ −3.00000 −1.13389 −0.566947 0.823754i $$-0.691875\pi$$
−0.566947 + 0.823754i $$0.691875\pi$$
$$8$$ −1.00000 −0.353553
$$9$$ 1.00000 0.333333
$$10$$ −1.00000 −0.316228
$$11$$ −1.00000 −0.301511
$$12$$ −2.00000 −0.577350
$$13$$ −4.00000 −1.10940 −0.554700 0.832050i $$-0.687167\pi$$
−0.554700 + 0.832050i $$0.687167\pi$$
$$14$$ 3.00000 0.801784
$$15$$ −2.00000 −0.516398
$$16$$ 1.00000 0.250000
$$17$$ −4.00000 −0.970143 −0.485071 0.874475i $$-0.661206\pi$$
−0.485071 + 0.874475i $$0.661206\pi$$
$$18$$ −1.00000 −0.235702
$$19$$ 0 0
$$20$$ 1.00000 0.223607
$$21$$ 6.00000 1.30931
$$22$$ 1.00000 0.213201
$$23$$ 4.00000 0.834058 0.417029 0.908893i $$-0.363071\pi$$
0.417029 + 0.908893i $$0.363071\pi$$
$$24$$ 2.00000 0.408248
$$25$$ −4.00000 −0.800000
$$26$$ 4.00000 0.784465
$$27$$ 4.00000 0.769800
$$28$$ −3.00000 −0.566947
$$29$$ −10.0000 −1.85695 −0.928477 0.371391i $$-0.878881\pi$$
−0.928477 + 0.371391i $$0.878881\pi$$
$$30$$ 2.00000 0.365148
$$31$$ −4.00000 −0.718421 −0.359211 0.933257i $$-0.616954\pi$$
−0.359211 + 0.933257i $$0.616954\pi$$
$$32$$ −1.00000 −0.176777
$$33$$ 2.00000 0.348155
$$34$$ 4.00000 0.685994
$$35$$ −3.00000 −0.507093
$$36$$ 1.00000 0.166667
$$37$$ −3.00000 −0.493197 −0.246598 0.969118i $$-0.579313\pi$$
−0.246598 + 0.969118i $$0.579313\pi$$
$$38$$ 0 0
$$39$$ 8.00000 1.28103
$$40$$ −1.00000 −0.158114
$$41$$ −6.00000 −0.937043 −0.468521 0.883452i $$-0.655213\pi$$
−0.468521 + 0.883452i $$0.655213\pi$$
$$42$$ −6.00000 −0.925820
$$43$$ −4.00000 −0.609994 −0.304997 0.952353i $$-0.598656\pi$$
−0.304997 + 0.952353i $$0.598656\pi$$
$$44$$ −1.00000 −0.150756
$$45$$ 1.00000 0.149071
$$46$$ −4.00000 −0.589768
$$47$$ 6.00000 0.875190 0.437595 0.899172i $$-0.355830\pi$$
0.437595 + 0.899172i $$0.355830\pi$$
$$48$$ −2.00000 −0.288675
$$49$$ 2.00000 0.285714
$$50$$ 4.00000 0.565685
$$51$$ 8.00000 1.12022
$$52$$ −4.00000 −0.554700
$$53$$ 1.00000 0.137361 0.0686803 0.997639i $$-0.478121\pi$$
0.0686803 + 0.997639i $$0.478121\pi$$
$$54$$ −4.00000 −0.544331
$$55$$ −1.00000 −0.134840
$$56$$ 3.00000 0.400892
$$57$$ 0 0
$$58$$ 10.0000 1.31306
$$59$$ −12.0000 −1.56227 −0.781133 0.624364i $$-0.785358\pi$$
−0.781133 + 0.624364i $$0.785358\pi$$
$$60$$ −2.00000 −0.258199
$$61$$ 12.0000 1.53644 0.768221 0.640184i $$-0.221142\pi$$
0.768221 + 0.640184i $$0.221142\pi$$
$$62$$ 4.00000 0.508001
$$63$$ −3.00000 −0.377964
$$64$$ 1.00000 0.125000
$$65$$ −4.00000 −0.496139
$$66$$ −2.00000 −0.246183
$$67$$ −12.0000 −1.46603 −0.733017 0.680211i $$-0.761888\pi$$
−0.733017 + 0.680211i $$0.761888\pi$$
$$68$$ −4.00000 −0.485071
$$69$$ −8.00000 −0.963087
$$70$$ 3.00000 0.358569
$$71$$ −14.0000 −1.66149 −0.830747 0.556650i $$-0.812086\pi$$
−0.830747 + 0.556650i $$0.812086\pi$$
$$72$$ −1.00000 −0.117851
$$73$$ 6.00000 0.702247 0.351123 0.936329i $$-0.385800\pi$$
0.351123 + 0.936329i $$0.385800\pi$$
$$74$$ 3.00000 0.348743
$$75$$ 8.00000 0.923760
$$76$$ 0 0
$$77$$ 3.00000 0.341882
$$78$$ −8.00000 −0.905822
$$79$$ −17.0000 −1.91265 −0.956325 0.292306i $$-0.905577\pi$$
−0.956325 + 0.292306i $$0.905577\pi$$
$$80$$ 1.00000 0.111803
$$81$$ −11.0000 −1.22222
$$82$$ 6.00000 0.662589
$$83$$ −13.0000 −1.42694 −0.713468 0.700688i $$-0.752876\pi$$
−0.713468 + 0.700688i $$0.752876\pi$$
$$84$$ 6.00000 0.654654
$$85$$ −4.00000 −0.433861
$$86$$ 4.00000 0.431331
$$87$$ 20.0000 2.14423
$$88$$ 1.00000 0.106600
$$89$$ 2.00000 0.212000 0.106000 0.994366i $$-0.466196\pi$$
0.106000 + 0.994366i $$0.466196\pi$$
$$90$$ −1.00000 −0.105409
$$91$$ 12.0000 1.25794
$$92$$ 4.00000 0.417029
$$93$$ 8.00000 0.829561
$$94$$ −6.00000 −0.618853
$$95$$ 0 0
$$96$$ 2.00000 0.204124
$$97$$ −5.00000 −0.507673 −0.253837 0.967247i $$-0.581693\pi$$
−0.253837 + 0.967247i $$0.581693\pi$$
$$98$$ −2.00000 −0.202031
$$99$$ −1.00000 −0.100504
$$100$$ −4.00000 −0.400000
$$101$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$102$$ −8.00000 −0.792118
$$103$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$104$$ 4.00000 0.392232
$$105$$ 6.00000 0.585540
$$106$$ −1.00000 −0.0971286
$$107$$ 7.00000 0.676716 0.338358 0.941018i $$-0.390129\pi$$
0.338358 + 0.941018i $$0.390129\pi$$
$$108$$ 4.00000 0.384900
$$109$$ 4.00000 0.383131 0.191565 0.981480i $$-0.438644\pi$$
0.191565 + 0.981480i $$0.438644\pi$$
$$110$$ 1.00000 0.0953463
$$111$$ 6.00000 0.569495
$$112$$ −3.00000 −0.283473
$$113$$ 6.00000 0.564433 0.282216 0.959351i $$-0.408930\pi$$
0.282216 + 0.959351i $$0.408930\pi$$
$$114$$ 0 0
$$115$$ 4.00000 0.373002
$$116$$ −10.0000 −0.928477
$$117$$ −4.00000 −0.369800
$$118$$ 12.0000 1.10469
$$119$$ 12.0000 1.10004
$$120$$ 2.00000 0.182574
$$121$$ 1.00000 0.0909091
$$122$$ −12.0000 −1.08643
$$123$$ 12.0000 1.08200
$$124$$ −4.00000 −0.359211
$$125$$ −9.00000 −0.804984
$$126$$ 3.00000 0.267261
$$127$$ 8.00000 0.709885 0.354943 0.934888i $$-0.384500\pi$$
0.354943 + 0.934888i $$0.384500\pi$$
$$128$$ −1.00000 −0.0883883
$$129$$ 8.00000 0.704361
$$130$$ 4.00000 0.350823
$$131$$ −16.0000 −1.39793 −0.698963 0.715158i $$-0.746355\pi$$
−0.698963 + 0.715158i $$0.746355\pi$$
$$132$$ 2.00000 0.174078
$$133$$ 0 0
$$134$$ 12.0000 1.03664
$$135$$ 4.00000 0.344265
$$136$$ 4.00000 0.342997
$$137$$ 1.00000 0.0854358 0.0427179 0.999087i $$-0.486398\pi$$
0.0427179 + 0.999087i $$0.486398\pi$$
$$138$$ 8.00000 0.681005
$$139$$ −17.0000 −1.44192 −0.720961 0.692976i $$-0.756299\pi$$
−0.720961 + 0.692976i $$0.756299\pi$$
$$140$$ −3.00000 −0.253546
$$141$$ −12.0000 −1.01058
$$142$$ 14.0000 1.17485
$$143$$ 4.00000 0.334497
$$144$$ 1.00000 0.0833333
$$145$$ −10.0000 −0.830455
$$146$$ −6.00000 −0.496564
$$147$$ −4.00000 −0.329914
$$148$$ −3.00000 −0.246598
$$149$$ −6.00000 −0.491539 −0.245770 0.969328i $$-0.579041\pi$$
−0.245770 + 0.969328i $$0.579041\pi$$
$$150$$ −8.00000 −0.653197
$$151$$ −15.0000 −1.22068 −0.610341 0.792139i $$-0.708968\pi$$
−0.610341 + 0.792139i $$0.708968\pi$$
$$152$$ 0 0
$$153$$ −4.00000 −0.323381
$$154$$ −3.00000 −0.241747
$$155$$ −4.00000 −0.321288
$$156$$ 8.00000 0.640513
$$157$$ 5.00000 0.399043 0.199522 0.979893i $$-0.436061\pi$$
0.199522 + 0.979893i $$0.436061\pi$$
$$158$$ 17.0000 1.35245
$$159$$ −2.00000 −0.158610
$$160$$ −1.00000 −0.0790569
$$161$$ −12.0000 −0.945732
$$162$$ 11.0000 0.864242
$$163$$ 4.00000 0.313304 0.156652 0.987654i $$-0.449930\pi$$
0.156652 + 0.987654i $$0.449930\pi$$
$$164$$ −6.00000 −0.468521
$$165$$ 2.00000 0.155700
$$166$$ 13.0000 1.00900
$$167$$ −17.0000 −1.31550 −0.657750 0.753237i $$-0.728492\pi$$
−0.657750 + 0.753237i $$0.728492\pi$$
$$168$$ −6.00000 −0.462910
$$169$$ 3.00000 0.230769
$$170$$ 4.00000 0.306786
$$171$$ 0 0
$$172$$ −4.00000 −0.304997
$$173$$ 4.00000 0.304114 0.152057 0.988372i $$-0.451410\pi$$
0.152057 + 0.988372i $$0.451410\pi$$
$$174$$ −20.0000 −1.51620
$$175$$ 12.0000 0.907115
$$176$$ −1.00000 −0.0753778
$$177$$ 24.0000 1.80395
$$178$$ −2.00000 −0.149906
$$179$$ 20.0000 1.49487 0.747435 0.664335i $$-0.231285\pi$$
0.747435 + 0.664335i $$0.231285\pi$$
$$180$$ 1.00000 0.0745356
$$181$$ 17.0000 1.26360 0.631800 0.775131i $$-0.282316\pi$$
0.631800 + 0.775131i $$0.282316\pi$$
$$182$$ −12.0000 −0.889499
$$183$$ −24.0000 −1.77413
$$184$$ −4.00000 −0.294884
$$185$$ −3.00000 −0.220564
$$186$$ −8.00000 −0.586588
$$187$$ 4.00000 0.292509
$$188$$ 6.00000 0.437595
$$189$$ −12.0000 −0.872872
$$190$$ 0 0
$$191$$ 18.0000 1.30243 0.651217 0.758891i $$-0.274259\pi$$
0.651217 + 0.758891i $$0.274259\pi$$
$$192$$ −2.00000 −0.144338
$$193$$ 10.0000 0.719816 0.359908 0.932988i $$-0.382808\pi$$
0.359908 + 0.932988i $$0.382808\pi$$
$$194$$ 5.00000 0.358979
$$195$$ 8.00000 0.572892
$$196$$ 2.00000 0.142857
$$197$$ −12.0000 −0.854965 −0.427482 0.904024i $$-0.640599\pi$$
−0.427482 + 0.904024i $$0.640599\pi$$
$$198$$ 1.00000 0.0710669
$$199$$ −10.0000 −0.708881 −0.354441 0.935079i $$-0.615329\pi$$
−0.354441 + 0.935079i $$0.615329\pi$$
$$200$$ 4.00000 0.282843
$$201$$ 24.0000 1.69283
$$202$$ 0 0
$$203$$ 30.0000 2.10559
$$204$$ 8.00000 0.560112
$$205$$ −6.00000 −0.419058
$$206$$ 0 0
$$207$$ 4.00000 0.278019
$$208$$ −4.00000 −0.277350
$$209$$ 0 0
$$210$$ −6.00000 −0.414039
$$211$$ 23.0000 1.58339 0.791693 0.610920i $$-0.209200\pi$$
0.791693 + 0.610920i $$0.209200\pi$$
$$212$$ 1.00000 0.0686803
$$213$$ 28.0000 1.91853
$$214$$ −7.00000 −0.478510
$$215$$ −4.00000 −0.272798
$$216$$ −4.00000 −0.272166
$$217$$ 12.0000 0.814613
$$218$$ −4.00000 −0.270914
$$219$$ −12.0000 −0.810885
$$220$$ −1.00000 −0.0674200
$$221$$ 16.0000 1.07628
$$222$$ −6.00000 −0.402694
$$223$$ −16.0000 −1.07144 −0.535720 0.844396i $$-0.679960\pi$$
−0.535720 + 0.844396i $$0.679960\pi$$
$$224$$ 3.00000 0.200446
$$225$$ −4.00000 −0.266667
$$226$$ −6.00000 −0.399114
$$227$$ 3.00000 0.199117 0.0995585 0.995032i $$-0.468257\pi$$
0.0995585 + 0.995032i $$0.468257\pi$$
$$228$$ 0 0
$$229$$ −13.0000 −0.859064 −0.429532 0.903052i $$-0.641321\pi$$
−0.429532 + 0.903052i $$0.641321\pi$$
$$230$$ −4.00000 −0.263752
$$231$$ −6.00000 −0.394771
$$232$$ 10.0000 0.656532
$$233$$ −16.0000 −1.04819 −0.524097 0.851658i $$-0.675597\pi$$
−0.524097 + 0.851658i $$0.675597\pi$$
$$234$$ 4.00000 0.261488
$$235$$ 6.00000 0.391397
$$236$$ −12.0000 −0.781133
$$237$$ 34.0000 2.20854
$$238$$ −12.0000 −0.777844
$$239$$ −5.00000 −0.323423 −0.161712 0.986838i $$-0.551701\pi$$
−0.161712 + 0.986838i $$0.551701\pi$$
$$240$$ −2.00000 −0.129099
$$241$$ 10.0000 0.644157 0.322078 0.946713i $$-0.395619\pi$$
0.322078 + 0.946713i $$0.395619\pi$$
$$242$$ −1.00000 −0.0642824
$$243$$ 10.0000 0.641500
$$244$$ 12.0000 0.768221
$$245$$ 2.00000 0.127775
$$246$$ −12.0000 −0.765092
$$247$$ 0 0
$$248$$ 4.00000 0.254000
$$249$$ 26.0000 1.64768
$$250$$ 9.00000 0.569210
$$251$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$252$$ −3.00000 −0.188982
$$253$$ −4.00000 −0.251478
$$254$$ −8.00000 −0.501965
$$255$$ 8.00000 0.500979
$$256$$ 1.00000 0.0625000
$$257$$ 1.00000 0.0623783 0.0311891 0.999514i $$-0.490071\pi$$
0.0311891 + 0.999514i $$0.490071\pi$$
$$258$$ −8.00000 −0.498058
$$259$$ 9.00000 0.559233
$$260$$ −4.00000 −0.248069
$$261$$ −10.0000 −0.618984
$$262$$ 16.0000 0.988483
$$263$$ −23.0000 −1.41824 −0.709120 0.705087i $$-0.750908\pi$$
−0.709120 + 0.705087i $$0.750908\pi$$
$$264$$ −2.00000 −0.123091
$$265$$ 1.00000 0.0614295
$$266$$ 0 0
$$267$$ −4.00000 −0.244796
$$268$$ −12.0000 −0.733017
$$269$$ −15.0000 −0.914566 −0.457283 0.889321i $$-0.651177\pi$$
−0.457283 + 0.889321i $$0.651177\pi$$
$$270$$ −4.00000 −0.243432
$$271$$ −21.0000 −1.27566 −0.637830 0.770178i $$-0.720168\pi$$
−0.637830 + 0.770178i $$0.720168\pi$$
$$272$$ −4.00000 −0.242536
$$273$$ −24.0000 −1.45255
$$274$$ −1.00000 −0.0604122
$$275$$ 4.00000 0.241209
$$276$$ −8.00000 −0.481543
$$277$$ −4.00000 −0.240337 −0.120168 0.992754i $$-0.538343\pi$$
−0.120168 + 0.992754i $$0.538343\pi$$
$$278$$ 17.0000 1.01959
$$279$$ −4.00000 −0.239474
$$280$$ 3.00000 0.179284
$$281$$ −4.00000 −0.238620 −0.119310 0.992857i $$-0.538068\pi$$
−0.119310 + 0.992857i $$0.538068\pi$$
$$282$$ 12.0000 0.714590
$$283$$ 13.0000 0.772770 0.386385 0.922338i $$-0.373724\pi$$
0.386385 + 0.922338i $$0.373724\pi$$
$$284$$ −14.0000 −0.830747
$$285$$ 0 0
$$286$$ −4.00000 −0.236525
$$287$$ 18.0000 1.06251
$$288$$ −1.00000 −0.0589256
$$289$$ −1.00000 −0.0588235
$$290$$ 10.0000 0.587220
$$291$$ 10.0000 0.586210
$$292$$ 6.00000 0.351123
$$293$$ −30.0000 −1.75262 −0.876309 0.481749i $$-0.840002\pi$$
−0.876309 + 0.481749i $$0.840002\pi$$
$$294$$ 4.00000 0.233285
$$295$$ −12.0000 −0.698667
$$296$$ 3.00000 0.174371
$$297$$ −4.00000 −0.232104
$$298$$ 6.00000 0.347571
$$299$$ −16.0000 −0.925304
$$300$$ 8.00000 0.461880
$$301$$ 12.0000 0.691669
$$302$$ 15.0000 0.863153
$$303$$ 0 0
$$304$$ 0 0
$$305$$ 12.0000 0.687118
$$306$$ 4.00000 0.228665
$$307$$ −29.0000 −1.65512 −0.827559 0.561379i $$-0.810271\pi$$
−0.827559 + 0.561379i $$0.810271\pi$$
$$308$$ 3.00000 0.170941
$$309$$ 0 0
$$310$$ 4.00000 0.227185
$$311$$ −18.0000 −1.02069 −0.510343 0.859971i $$-0.670482\pi$$
−0.510343 + 0.859971i $$0.670482\pi$$
$$312$$ −8.00000 −0.452911
$$313$$ −9.00000 −0.508710 −0.254355 0.967111i $$-0.581863\pi$$
−0.254355 + 0.967111i $$0.581863\pi$$
$$314$$ −5.00000 −0.282166
$$315$$ −3.00000 −0.169031
$$316$$ −17.0000 −0.956325
$$317$$ 18.0000 1.01098 0.505490 0.862832i $$-0.331312\pi$$
0.505490 + 0.862832i $$0.331312\pi$$
$$318$$ 2.00000 0.112154
$$319$$ 10.0000 0.559893
$$320$$ 1.00000 0.0559017
$$321$$ −14.0000 −0.781404
$$322$$ 12.0000 0.668734
$$323$$ 0 0
$$324$$ −11.0000 −0.611111
$$325$$ 16.0000 0.887520
$$326$$ −4.00000 −0.221540
$$327$$ −8.00000 −0.442401
$$328$$ 6.00000 0.331295
$$329$$ −18.0000 −0.992372
$$330$$ −2.00000 −0.110096
$$331$$ −20.0000 −1.09930 −0.549650 0.835395i $$-0.685239\pi$$
−0.549650 + 0.835395i $$0.685239\pi$$
$$332$$ −13.0000 −0.713468
$$333$$ −3.00000 −0.164399
$$334$$ 17.0000 0.930199
$$335$$ −12.0000 −0.655630
$$336$$ 6.00000 0.327327
$$337$$ −28.0000 −1.52526 −0.762629 0.646837i $$-0.776092\pi$$
−0.762629 + 0.646837i $$0.776092\pi$$
$$338$$ −3.00000 −0.163178
$$339$$ −12.0000 −0.651751
$$340$$ −4.00000 −0.216930
$$341$$ 4.00000 0.216612
$$342$$ 0 0
$$343$$ 15.0000 0.809924
$$344$$ 4.00000 0.215666
$$345$$ −8.00000 −0.430706
$$346$$ −4.00000 −0.215041
$$347$$ 13.0000 0.697877 0.348938 0.937146i $$-0.386542\pi$$
0.348938 + 0.937146i $$0.386542\pi$$
$$348$$ 20.0000 1.07211
$$349$$ −26.0000 −1.39175 −0.695874 0.718164i $$-0.744983\pi$$
−0.695874 + 0.718164i $$0.744983\pi$$
$$350$$ −12.0000 −0.641427
$$351$$ −16.0000 −0.854017
$$352$$ 1.00000 0.0533002
$$353$$ 19.0000 1.01127 0.505634 0.862748i $$-0.331259\pi$$
0.505634 + 0.862748i $$0.331259\pi$$
$$354$$ −24.0000 −1.27559
$$355$$ −14.0000 −0.743043
$$356$$ 2.00000 0.106000
$$357$$ −24.0000 −1.27021
$$358$$ −20.0000 −1.05703
$$359$$ 7.00000 0.369446 0.184723 0.982791i $$-0.440861\pi$$
0.184723 + 0.982791i $$0.440861\pi$$
$$360$$ −1.00000 −0.0527046
$$361$$ 0 0
$$362$$ −17.0000 −0.893500
$$363$$ −2.00000 −0.104973
$$364$$ 12.0000 0.628971
$$365$$ 6.00000 0.314054
$$366$$ 24.0000 1.25450
$$367$$ 16.0000 0.835193 0.417597 0.908633i $$-0.362873\pi$$
0.417597 + 0.908633i $$0.362873\pi$$
$$368$$ 4.00000 0.208514
$$369$$ −6.00000 −0.312348
$$370$$ 3.00000 0.155963
$$371$$ −3.00000 −0.155752
$$372$$ 8.00000 0.414781
$$373$$ 28.0000 1.44979 0.724893 0.688862i $$-0.241889\pi$$
0.724893 + 0.688862i $$0.241889\pi$$
$$374$$ −4.00000 −0.206835
$$375$$ 18.0000 0.929516
$$376$$ −6.00000 −0.309426
$$377$$ 40.0000 2.06010
$$378$$ 12.0000 0.617213
$$379$$ −4.00000 −0.205466 −0.102733 0.994709i $$-0.532759\pi$$
−0.102733 + 0.994709i $$0.532759\pi$$
$$380$$ 0 0
$$381$$ −16.0000 −0.819705
$$382$$ −18.0000 −0.920960
$$383$$ −8.00000 −0.408781 −0.204390 0.978889i $$-0.565521\pi$$
−0.204390 + 0.978889i $$0.565521\pi$$
$$384$$ 2.00000 0.102062
$$385$$ 3.00000 0.152894
$$386$$ −10.0000 −0.508987
$$387$$ −4.00000 −0.203331
$$388$$ −5.00000 −0.253837
$$389$$ 21.0000 1.06474 0.532371 0.846511i $$-0.321301\pi$$
0.532371 + 0.846511i $$0.321301\pi$$
$$390$$ −8.00000 −0.405096
$$391$$ −16.0000 −0.809155
$$392$$ −2.00000 −0.101015
$$393$$ 32.0000 1.61419
$$394$$ 12.0000 0.604551
$$395$$ −17.0000 −0.855363
$$396$$ −1.00000 −0.0502519
$$397$$ 25.0000 1.25471 0.627357 0.778732i $$-0.284137\pi$$
0.627357 + 0.778732i $$0.284137\pi$$
$$398$$ 10.0000 0.501255
$$399$$ 0 0
$$400$$ −4.00000 −0.200000
$$401$$ −10.0000 −0.499376 −0.249688 0.968326i $$-0.580328\pi$$
−0.249688 + 0.968326i $$0.580328\pi$$
$$402$$ −24.0000 −1.19701
$$403$$ 16.0000 0.797017
$$404$$ 0 0
$$405$$ −11.0000 −0.546594
$$406$$ −30.0000 −1.48888
$$407$$ 3.00000 0.148704
$$408$$ −8.00000 −0.396059
$$409$$ 20.0000 0.988936 0.494468 0.869196i $$-0.335363\pi$$
0.494468 + 0.869196i $$0.335363\pi$$
$$410$$ 6.00000 0.296319
$$411$$ −2.00000 −0.0986527
$$412$$ 0 0
$$413$$ 36.0000 1.77144
$$414$$ −4.00000 −0.196589
$$415$$ −13.0000 −0.638145
$$416$$ 4.00000 0.196116
$$417$$ 34.0000 1.66499
$$418$$ 0 0
$$419$$ −38.0000 −1.85642 −0.928211 0.372055i $$-0.878653\pi$$
−0.928211 + 0.372055i $$0.878653\pi$$
$$420$$ 6.00000 0.292770
$$421$$ 3.00000 0.146211 0.0731055 0.997324i $$-0.476709\pi$$
0.0731055 + 0.997324i $$0.476709\pi$$
$$422$$ −23.0000 −1.11962
$$423$$ 6.00000 0.291730
$$424$$ −1.00000 −0.0485643
$$425$$ 16.0000 0.776114
$$426$$ −28.0000 −1.35660
$$427$$ −36.0000 −1.74216
$$428$$ 7.00000 0.338358
$$429$$ −8.00000 −0.386244
$$430$$ 4.00000 0.192897
$$431$$ −13.0000 −0.626188 −0.313094 0.949722i $$-0.601365\pi$$
−0.313094 + 0.949722i $$0.601365\pi$$
$$432$$ 4.00000 0.192450
$$433$$ 18.0000 0.865025 0.432512 0.901628i $$-0.357627\pi$$
0.432512 + 0.901628i $$0.357627\pi$$
$$434$$ −12.0000 −0.576018
$$435$$ 20.0000 0.958927
$$436$$ 4.00000 0.191565
$$437$$ 0 0
$$438$$ 12.0000 0.573382
$$439$$ −1.00000 −0.0477274 −0.0238637 0.999715i $$-0.507597\pi$$
−0.0238637 + 0.999715i $$0.507597\pi$$
$$440$$ 1.00000 0.0476731
$$441$$ 2.00000 0.0952381
$$442$$ −16.0000 −0.761042
$$443$$ 24.0000 1.14027 0.570137 0.821549i $$-0.306890\pi$$
0.570137 + 0.821549i $$0.306890\pi$$
$$444$$ 6.00000 0.284747
$$445$$ 2.00000 0.0948091
$$446$$ 16.0000 0.757622
$$447$$ 12.0000 0.567581
$$448$$ −3.00000 −0.141737
$$449$$ 33.0000 1.55737 0.778683 0.627417i $$-0.215888\pi$$
0.778683 + 0.627417i $$0.215888\pi$$
$$450$$ 4.00000 0.188562
$$451$$ 6.00000 0.282529
$$452$$ 6.00000 0.282216
$$453$$ 30.0000 1.40952
$$454$$ −3.00000 −0.140797
$$455$$ 12.0000 0.562569
$$456$$ 0 0
$$457$$ 38.0000 1.77757 0.888783 0.458329i $$-0.151552\pi$$
0.888783 + 0.458329i $$0.151552\pi$$
$$458$$ 13.0000 0.607450
$$459$$ −16.0000 −0.746816
$$460$$ 4.00000 0.186501
$$461$$ −2.00000 −0.0931493 −0.0465746 0.998915i $$-0.514831\pi$$
−0.0465746 + 0.998915i $$0.514831\pi$$
$$462$$ 6.00000 0.279145
$$463$$ 4.00000 0.185896 0.0929479 0.995671i $$-0.470371\pi$$
0.0929479 + 0.995671i $$0.470371\pi$$
$$464$$ −10.0000 −0.464238
$$465$$ 8.00000 0.370991
$$466$$ 16.0000 0.741186
$$467$$ 14.0000 0.647843 0.323921 0.946084i $$-0.394999\pi$$
0.323921 + 0.946084i $$0.394999\pi$$
$$468$$ −4.00000 −0.184900
$$469$$ 36.0000 1.66233
$$470$$ −6.00000 −0.276759
$$471$$ −10.0000 −0.460776
$$472$$ 12.0000 0.552345
$$473$$ 4.00000 0.183920
$$474$$ −34.0000 −1.56167
$$475$$ 0 0
$$476$$ 12.0000 0.550019
$$477$$ 1.00000 0.0457869
$$478$$ 5.00000 0.228695
$$479$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$480$$ 2.00000 0.0912871
$$481$$ 12.0000 0.547153
$$482$$ −10.0000 −0.455488
$$483$$ 24.0000 1.09204
$$484$$ 1.00000 0.0454545
$$485$$ −5.00000 −0.227038
$$486$$ −10.0000 −0.453609
$$487$$ −18.0000 −0.815658 −0.407829 0.913058i $$-0.633714\pi$$
−0.407829 + 0.913058i $$0.633714\pi$$
$$488$$ −12.0000 −0.543214
$$489$$ −8.00000 −0.361773
$$490$$ −2.00000 −0.0903508
$$491$$ 1.00000 0.0451294 0.0225647 0.999745i $$-0.492817\pi$$
0.0225647 + 0.999745i $$0.492817\pi$$
$$492$$ 12.0000 0.541002
$$493$$ 40.0000 1.80151
$$494$$ 0 0
$$495$$ −1.00000 −0.0449467
$$496$$ −4.00000 −0.179605
$$497$$ 42.0000 1.88396
$$498$$ −26.0000 −1.16509
$$499$$ −40.0000 −1.79065 −0.895323 0.445418i $$-0.853055\pi$$
−0.895323 + 0.445418i $$0.853055\pi$$
$$500$$ −9.00000 −0.402492
$$501$$ 34.0000 1.51901
$$502$$ 0 0
$$503$$ 8.00000 0.356702 0.178351 0.983967i $$-0.442924\pi$$
0.178351 + 0.983967i $$0.442924\pi$$
$$504$$ 3.00000 0.133631
$$505$$ 0 0
$$506$$ 4.00000 0.177822
$$507$$ −6.00000 −0.266469
$$508$$ 8.00000 0.354943
$$509$$ 13.0000 0.576215 0.288107 0.957598i $$-0.406974\pi$$
0.288107 + 0.957598i $$0.406974\pi$$
$$510$$ −8.00000 −0.354246
$$511$$ −18.0000 −0.796273
$$512$$ −1.00000 −0.0441942
$$513$$ 0 0
$$514$$ −1.00000 −0.0441081
$$515$$ 0 0
$$516$$ 8.00000 0.352180
$$517$$ −6.00000 −0.263880
$$518$$ −9.00000 −0.395437
$$519$$ −8.00000 −0.351161
$$520$$ 4.00000 0.175412
$$521$$ −25.0000 −1.09527 −0.547635 0.836717i $$-0.684472\pi$$
−0.547635 + 0.836717i $$0.684472\pi$$
$$522$$ 10.0000 0.437688
$$523$$ −9.00000 −0.393543 −0.196771 0.980449i $$-0.563046\pi$$
−0.196771 + 0.980449i $$0.563046\pi$$
$$524$$ −16.0000 −0.698963
$$525$$ −24.0000 −1.04745
$$526$$ 23.0000 1.00285
$$527$$ 16.0000 0.696971
$$528$$ 2.00000 0.0870388
$$529$$ −7.00000 −0.304348
$$530$$ −1.00000 −0.0434372
$$531$$ −12.0000 −0.520756
$$532$$ 0 0
$$533$$ 24.0000 1.03956
$$534$$ 4.00000 0.173097
$$535$$ 7.00000 0.302636
$$536$$ 12.0000 0.518321
$$537$$ −40.0000 −1.72613
$$538$$ 15.0000 0.646696
$$539$$ −2.00000 −0.0861461
$$540$$ 4.00000 0.172133
$$541$$ −22.0000 −0.945854 −0.472927 0.881102i $$-0.656803\pi$$
−0.472927 + 0.881102i $$0.656803\pi$$
$$542$$ 21.0000 0.902027
$$543$$ −34.0000 −1.45908
$$544$$ 4.00000 0.171499
$$545$$ 4.00000 0.171341
$$546$$ 24.0000 1.02711
$$547$$ −35.0000 −1.49649 −0.748246 0.663421i $$-0.769104\pi$$
−0.748246 + 0.663421i $$0.769104\pi$$
$$548$$ 1.00000 0.0427179
$$549$$ 12.0000 0.512148
$$550$$ −4.00000 −0.170561
$$551$$ 0 0
$$552$$ 8.00000 0.340503
$$553$$ 51.0000 2.16874
$$554$$ 4.00000 0.169944
$$555$$ 6.00000 0.254686
$$556$$ −17.0000 −0.720961
$$557$$ −12.0000 −0.508456 −0.254228 0.967144i $$-0.581821\pi$$
−0.254228 + 0.967144i $$0.581821\pi$$
$$558$$ 4.00000 0.169334
$$559$$ 16.0000 0.676728
$$560$$ −3.00000 −0.126773
$$561$$ −8.00000 −0.337760
$$562$$ 4.00000 0.168730
$$563$$ 33.0000 1.39078 0.695392 0.718631i $$-0.255231\pi$$
0.695392 + 0.718631i $$0.255231\pi$$
$$564$$ −12.0000 −0.505291
$$565$$ 6.00000 0.252422
$$566$$ −13.0000 −0.546431
$$567$$ 33.0000 1.38587
$$568$$ 14.0000 0.587427
$$569$$ 36.0000 1.50920 0.754599 0.656186i $$-0.227831\pi$$
0.754599 + 0.656186i $$0.227831\pi$$
$$570$$ 0 0
$$571$$ 19.0000 0.795125 0.397563 0.917575i $$-0.369856\pi$$
0.397563 + 0.917575i $$0.369856\pi$$
$$572$$ 4.00000 0.167248
$$573$$ −36.0000 −1.50392
$$574$$ −18.0000 −0.751305
$$575$$ −16.0000 −0.667246
$$576$$ 1.00000 0.0416667
$$577$$ 17.0000 0.707719 0.353860 0.935299i $$-0.384869\pi$$
0.353860 + 0.935299i $$0.384869\pi$$
$$578$$ 1.00000 0.0415945
$$579$$ −20.0000 −0.831172
$$580$$ −10.0000 −0.415227
$$581$$ 39.0000 1.61799
$$582$$ −10.0000 −0.414513
$$583$$ −1.00000 −0.0414158
$$584$$ −6.00000 −0.248282
$$585$$ −4.00000 −0.165380
$$586$$ 30.0000 1.23929
$$587$$ 24.0000 0.990586 0.495293 0.868726i $$-0.335061\pi$$
0.495293 + 0.868726i $$0.335061\pi$$
$$588$$ −4.00000 −0.164957
$$589$$ 0 0
$$590$$ 12.0000 0.494032
$$591$$ 24.0000 0.987228
$$592$$ −3.00000 −0.123299
$$593$$ −12.0000 −0.492781 −0.246390 0.969171i $$-0.579245\pi$$
−0.246390 + 0.969171i $$0.579245\pi$$
$$594$$ 4.00000 0.164122
$$595$$ 12.0000 0.491952
$$596$$ −6.00000 −0.245770
$$597$$ 20.0000 0.818546
$$598$$ 16.0000 0.654289
$$599$$ −32.0000 −1.30748 −0.653742 0.756717i $$-0.726802\pi$$
−0.653742 + 0.756717i $$0.726802\pi$$
$$600$$ −8.00000 −0.326599
$$601$$ 10.0000 0.407909 0.203954 0.978980i $$-0.434621\pi$$
0.203954 + 0.978980i $$0.434621\pi$$
$$602$$ −12.0000 −0.489083
$$603$$ −12.0000 −0.488678
$$604$$ −15.0000 −0.610341
$$605$$ 1.00000 0.0406558
$$606$$ 0 0
$$607$$ 40.0000 1.62355 0.811775 0.583970i $$-0.198502\pi$$
0.811775 + 0.583970i $$0.198502\pi$$
$$608$$ 0 0
$$609$$ −60.0000 −2.43132
$$610$$ −12.0000 −0.485866
$$611$$ −24.0000 −0.970936
$$612$$ −4.00000 −0.161690
$$613$$ −16.0000 −0.646234 −0.323117 0.946359i $$-0.604731\pi$$
−0.323117 + 0.946359i $$0.604731\pi$$
$$614$$ 29.0000 1.17034
$$615$$ 12.0000 0.483887
$$616$$ −3.00000 −0.120873
$$617$$ 33.0000 1.32853 0.664265 0.747497i $$-0.268745\pi$$
0.664265 + 0.747497i $$0.268745\pi$$
$$618$$ 0 0
$$619$$ −32.0000 −1.28619 −0.643094 0.765787i $$-0.722350\pi$$
−0.643094 + 0.765787i $$0.722350\pi$$
$$620$$ −4.00000 −0.160644
$$621$$ 16.0000 0.642058
$$622$$ 18.0000 0.721734
$$623$$ −6.00000 −0.240385
$$624$$ 8.00000 0.320256
$$625$$ 11.0000 0.440000
$$626$$ 9.00000 0.359712
$$627$$ 0 0
$$628$$ 5.00000 0.199522
$$629$$ 12.0000 0.478471
$$630$$ 3.00000 0.119523
$$631$$ 40.0000 1.59237 0.796187 0.605050i $$-0.206847\pi$$
0.796187 + 0.605050i $$0.206847\pi$$
$$632$$ 17.0000 0.676224
$$633$$ −46.0000 −1.82834
$$634$$ −18.0000 −0.714871
$$635$$ 8.00000 0.317470
$$636$$ −2.00000 −0.0793052
$$637$$ −8.00000 −0.316972
$$638$$ −10.0000 −0.395904
$$639$$ −14.0000 −0.553831
$$640$$ −1.00000 −0.0395285
$$641$$ 41.0000 1.61940 0.809701 0.586842i $$-0.199629\pi$$
0.809701 + 0.586842i $$0.199629\pi$$
$$642$$ 14.0000 0.552536
$$643$$ −26.0000 −1.02534 −0.512670 0.858586i $$-0.671344\pi$$
−0.512670 + 0.858586i $$0.671344\pi$$
$$644$$ −12.0000 −0.472866
$$645$$ 8.00000 0.315000
$$646$$ 0 0
$$647$$ 42.0000 1.65119 0.825595 0.564263i $$-0.190840\pi$$
0.825595 + 0.564263i $$0.190840\pi$$
$$648$$ 11.0000 0.432121
$$649$$ 12.0000 0.471041
$$650$$ −16.0000 −0.627572
$$651$$ −24.0000 −0.940634
$$652$$ 4.00000 0.156652
$$653$$ 10.0000 0.391330 0.195665 0.980671i $$-0.437313\pi$$
0.195665 + 0.980671i $$0.437313\pi$$
$$654$$ 8.00000 0.312825
$$655$$ −16.0000 −0.625172
$$656$$ −6.00000 −0.234261
$$657$$ 6.00000 0.234082
$$658$$ 18.0000 0.701713
$$659$$ 31.0000 1.20759 0.603794 0.797140i $$-0.293655\pi$$
0.603794 + 0.797140i $$0.293655\pi$$
$$660$$ 2.00000 0.0778499
$$661$$ −5.00000 −0.194477 −0.0972387 0.995261i $$-0.531001\pi$$
−0.0972387 + 0.995261i $$0.531001\pi$$
$$662$$ 20.0000 0.777322
$$663$$ −32.0000 −1.24278
$$664$$ 13.0000 0.504498
$$665$$ 0 0
$$666$$ 3.00000 0.116248
$$667$$ −40.0000 −1.54881
$$668$$ −17.0000 −0.657750
$$669$$ 32.0000 1.23719
$$670$$ 12.0000 0.463600
$$671$$ −12.0000 −0.463255
$$672$$ −6.00000 −0.231455
$$673$$ 42.0000 1.61898 0.809491 0.587133i $$-0.199743\pi$$
0.809491 + 0.587133i $$0.199743\pi$$
$$674$$ 28.0000 1.07852
$$675$$ −16.0000 −0.615840
$$676$$ 3.00000 0.115385
$$677$$ 38.0000 1.46046 0.730229 0.683202i $$-0.239413\pi$$
0.730229 + 0.683202i $$0.239413\pi$$
$$678$$ 12.0000 0.460857
$$679$$ 15.0000 0.575647
$$680$$ 4.00000 0.153393
$$681$$ −6.00000 −0.229920
$$682$$ −4.00000 −0.153168
$$683$$ 2.00000 0.0765279 0.0382639 0.999268i $$-0.487817\pi$$
0.0382639 + 0.999268i $$0.487817\pi$$
$$684$$ 0 0
$$685$$ 1.00000 0.0382080
$$686$$ −15.0000 −0.572703
$$687$$ 26.0000 0.991962
$$688$$ −4.00000 −0.152499
$$689$$ −4.00000 −0.152388
$$690$$ 8.00000 0.304555
$$691$$ 16.0000 0.608669 0.304334 0.952565i $$-0.401566\pi$$
0.304334 + 0.952565i $$0.401566\pi$$
$$692$$ 4.00000 0.152057
$$693$$ 3.00000 0.113961
$$694$$ −13.0000 −0.493473
$$695$$ −17.0000 −0.644847
$$696$$ −20.0000 −0.758098
$$697$$ 24.0000 0.909065
$$698$$ 26.0000 0.984115
$$699$$ 32.0000 1.21035
$$700$$ 12.0000 0.453557
$$701$$ 12.0000 0.453234 0.226617 0.973984i $$-0.427233\pi$$
0.226617 + 0.973984i $$0.427233\pi$$
$$702$$ 16.0000 0.603881
$$703$$ 0 0
$$704$$ −1.00000 −0.0376889
$$705$$ −12.0000 −0.451946
$$706$$ −19.0000 −0.715074
$$707$$ 0 0
$$708$$ 24.0000 0.901975
$$709$$ 19.0000 0.713560 0.356780 0.934188i $$-0.383875\pi$$
0.356780 + 0.934188i $$0.383875\pi$$
$$710$$ 14.0000 0.525411
$$711$$ −17.0000 −0.637550
$$712$$ −2.00000 −0.0749532
$$713$$ −16.0000 −0.599205
$$714$$ 24.0000 0.898177
$$715$$ 4.00000 0.149592
$$716$$ 20.0000 0.747435
$$717$$ 10.0000 0.373457
$$718$$ −7.00000 −0.261238
$$719$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$720$$ 1.00000 0.0372678
$$721$$ 0 0
$$722$$ 0 0
$$723$$ −20.0000 −0.743808
$$724$$ 17.0000 0.631800
$$725$$ 40.0000 1.48556
$$726$$ 2.00000 0.0742270
$$727$$ 14.0000 0.519231 0.259616 0.965712i $$-0.416404\pi$$
0.259616 + 0.965712i $$0.416404\pi$$
$$728$$ −12.0000 −0.444750
$$729$$ 13.0000 0.481481
$$730$$ −6.00000 −0.222070
$$731$$ 16.0000 0.591781
$$732$$ −24.0000 −0.887066
$$733$$ −32.0000 −1.18195 −0.590973 0.806691i $$-0.701256\pi$$
−0.590973 + 0.806691i $$0.701256\pi$$
$$734$$ −16.0000 −0.590571
$$735$$ −4.00000 −0.147542
$$736$$ −4.00000 −0.147442
$$737$$ 12.0000 0.442026
$$738$$ 6.00000 0.220863
$$739$$ −4.00000 −0.147142 −0.0735712 0.997290i $$-0.523440\pi$$
−0.0735712 + 0.997290i $$0.523440\pi$$
$$740$$ −3.00000 −0.110282
$$741$$ 0 0
$$742$$ 3.00000 0.110133
$$743$$ −9.00000 −0.330178 −0.165089 0.986279i $$-0.552791\pi$$
−0.165089 + 0.986279i $$0.552791\pi$$
$$744$$ −8.00000 −0.293294
$$745$$ −6.00000 −0.219823
$$746$$ −28.0000 −1.02515
$$747$$ −13.0000 −0.475645
$$748$$ 4.00000 0.146254
$$749$$ −21.0000 −0.767323
$$750$$ −18.0000 −0.657267
$$751$$ −10.0000 −0.364905 −0.182453 0.983215i $$-0.558404\pi$$
−0.182453 + 0.983215i $$0.558404\pi$$
$$752$$ 6.00000 0.218797
$$753$$ 0 0
$$754$$ −40.0000 −1.45671
$$755$$ −15.0000 −0.545906
$$756$$ −12.0000 −0.436436
$$757$$ −22.0000 −0.799604 −0.399802 0.916602i $$-0.630921\pi$$
−0.399802 + 0.916602i $$0.630921\pi$$
$$758$$ 4.00000 0.145287
$$759$$ 8.00000 0.290382
$$760$$ 0 0
$$761$$ 22.0000 0.797499 0.398750 0.917060i $$-0.369444\pi$$
0.398750 + 0.917060i $$0.369444\pi$$
$$762$$ 16.0000 0.579619
$$763$$ −12.0000 −0.434429
$$764$$ 18.0000 0.651217
$$765$$ −4.00000 −0.144620
$$766$$ 8.00000 0.289052
$$767$$ 48.0000 1.73318
$$768$$ −2.00000 −0.0721688
$$769$$ −34.0000 −1.22607 −0.613036 0.790055i $$-0.710052\pi$$
−0.613036 + 0.790055i $$0.710052\pi$$
$$770$$ −3.00000 −0.108112
$$771$$ −2.00000 −0.0720282
$$772$$ 10.0000 0.359908
$$773$$ 26.0000 0.935155 0.467578 0.883952i $$-0.345127\pi$$
0.467578 + 0.883952i $$0.345127\pi$$
$$774$$ 4.00000 0.143777
$$775$$ 16.0000 0.574737
$$776$$ 5.00000 0.179490
$$777$$ −18.0000 −0.645746
$$778$$ −21.0000 −0.752886
$$779$$ 0 0
$$780$$ 8.00000 0.286446
$$781$$ 14.0000 0.500959
$$782$$ 16.0000 0.572159
$$783$$ −40.0000 −1.42948
$$784$$ 2.00000 0.0714286
$$785$$ 5.00000 0.178458
$$786$$ −32.0000 −1.14140
$$787$$ −7.00000 −0.249523 −0.124762 0.992187i $$-0.539817\pi$$
−0.124762 + 0.992187i $$0.539817\pi$$
$$788$$ −12.0000 −0.427482
$$789$$ 46.0000 1.63764
$$790$$ 17.0000 0.604833
$$791$$ −18.0000 −0.640006
$$792$$ 1.00000 0.0355335
$$793$$ −48.0000 −1.70453
$$794$$ −25.0000 −0.887217
$$795$$ −2.00000 −0.0709327
$$796$$ −10.0000 −0.354441
$$797$$ −37.0000 −1.31061 −0.655304 0.755366i $$-0.727459\pi$$
−0.655304 + 0.755366i $$0.727459\pi$$
$$798$$ 0 0
$$799$$ −24.0000 −0.849059
$$800$$ 4.00000 0.141421
$$801$$ 2.00000 0.0706665
$$802$$ 10.0000 0.353112
$$803$$ −6.00000 −0.211735
$$804$$ 24.0000 0.846415
$$805$$ −12.0000 −0.422944
$$806$$ −16.0000 −0.563576
$$807$$ 30.0000 1.05605
$$808$$ 0 0
$$809$$ −28.0000 −0.984428 −0.492214 0.870474i $$-0.663812\pi$$
−0.492214 + 0.870474i $$0.663812\pi$$
$$810$$ 11.0000 0.386501
$$811$$ −35.0000 −1.22902 −0.614508 0.788911i $$-0.710645\pi$$
−0.614508 + 0.788911i $$0.710645\pi$$
$$812$$ 30.0000 1.05279
$$813$$ 42.0000 1.47300
$$814$$ −3.00000 −0.105150
$$815$$ 4.00000 0.140114
$$816$$ 8.00000 0.280056
$$817$$ 0 0
$$818$$ −20.0000 −0.699284
$$819$$ 12.0000 0.419314
$$820$$ −6.00000 −0.209529
$$821$$ −52.0000 −1.81481 −0.907406 0.420255i $$-0.861941\pi$$
−0.907406 + 0.420255i $$0.861941\pi$$
$$822$$ 2.00000 0.0697580
$$823$$ 2.00000 0.0697156 0.0348578 0.999392i $$-0.488902\pi$$
0.0348578 + 0.999392i $$0.488902\pi$$
$$824$$ 0 0
$$825$$ −8.00000 −0.278524
$$826$$ −36.0000 −1.25260
$$827$$ 44.0000 1.53003 0.765015 0.644013i $$-0.222732\pi$$
0.765015 + 0.644013i $$0.222732\pi$$
$$828$$ 4.00000 0.139010
$$829$$ −46.0000 −1.59765 −0.798823 0.601566i $$-0.794544\pi$$
−0.798823 + 0.601566i $$0.794544\pi$$
$$830$$ 13.0000 0.451237
$$831$$ 8.00000 0.277517
$$832$$ −4.00000 −0.138675
$$833$$ −8.00000 −0.277184
$$834$$ −34.0000 −1.17732
$$835$$ −17.0000 −0.588309
$$836$$ 0 0
$$837$$ −16.0000 −0.553041
$$838$$ 38.0000 1.31269
$$839$$ 56.0000 1.93333 0.966667 0.256036i $$-0.0824164\pi$$
0.966667 + 0.256036i $$0.0824164\pi$$
$$840$$ −6.00000 −0.207020
$$841$$ 71.0000 2.44828
$$842$$ −3.00000 −0.103387
$$843$$ 8.00000 0.275535
$$844$$ 23.0000 0.791693
$$845$$ 3.00000 0.103203
$$846$$ −6.00000 −0.206284
$$847$$ −3.00000 −0.103081
$$848$$ 1.00000 0.0343401
$$849$$ −26.0000 −0.892318
$$850$$ −16.0000 −0.548795
$$851$$ −12.0000 −0.411355
$$852$$ 28.0000 0.959264
$$853$$ 42.0000 1.43805 0.719026 0.694983i $$-0.244588\pi$$
0.719026 + 0.694983i $$0.244588\pi$$
$$854$$ 36.0000 1.23189
$$855$$ 0 0
$$856$$ −7.00000 −0.239255
$$857$$ −48.0000 −1.63965 −0.819824 0.572615i $$-0.805929\pi$$
−0.819824 + 0.572615i $$0.805929\pi$$
$$858$$ 8.00000 0.273115
$$859$$ −6.00000 −0.204717 −0.102359 0.994748i $$-0.532639\pi$$
−0.102359 + 0.994748i $$0.532639\pi$$
$$860$$ −4.00000 −0.136399
$$861$$ −36.0000 −1.22688
$$862$$ 13.0000 0.442782
$$863$$ 6.00000 0.204242 0.102121 0.994772i $$-0.467437\pi$$
0.102121 + 0.994772i $$0.467437\pi$$
$$864$$ −4.00000 −0.136083
$$865$$ 4.00000 0.136004
$$866$$ −18.0000 −0.611665
$$867$$ 2.00000 0.0679236
$$868$$ 12.0000 0.407307
$$869$$ 17.0000 0.576686
$$870$$ −20.0000 −0.678064
$$871$$ 48.0000 1.62642
$$872$$ −4.00000 −0.135457
$$873$$ −5.00000 −0.169224
$$874$$ 0 0
$$875$$ 27.0000 0.912767
$$876$$ −12.0000 −0.405442
$$877$$ 12.0000 0.405211 0.202606 0.979260i $$-0.435059\pi$$
0.202606 + 0.979260i $$0.435059\pi$$
$$878$$ 1.00000 0.0337484
$$879$$ 60.0000 2.02375
$$880$$ −1.00000 −0.0337100
$$881$$ −6.00000 −0.202145 −0.101073 0.994879i $$-0.532227\pi$$
−0.101073 + 0.994879i $$0.532227\pi$$
$$882$$ −2.00000 −0.0673435
$$883$$ −26.0000 −0.874970 −0.437485 0.899226i $$-0.644131\pi$$
−0.437485 + 0.899226i $$0.644131\pi$$
$$884$$ 16.0000 0.538138
$$885$$ 24.0000 0.806751
$$886$$ −24.0000 −0.806296
$$887$$ −29.0000 −0.973725 −0.486862 0.873479i $$-0.661859\pi$$
−0.486862 + 0.873479i $$0.661859\pi$$
$$888$$ −6.00000 −0.201347
$$889$$ −24.0000 −0.804934
$$890$$ −2.00000 −0.0670402
$$891$$ 11.0000 0.368514
$$892$$ −16.0000 −0.535720
$$893$$ 0 0
$$894$$ −12.0000 −0.401340
$$895$$ 20.0000 0.668526
$$896$$ 3.00000 0.100223
$$897$$ 32.0000 1.06845
$$898$$ −33.0000 −1.10122
$$899$$ 40.0000 1.33407
$$900$$ −4.00000 −0.133333
$$901$$ −4.00000 −0.133259
$$902$$ −6.00000 −0.199778
$$903$$ −24.0000 −0.798670
$$904$$ −6.00000 −0.199557
$$905$$ 17.0000 0.565099
$$906$$ −30.0000 −0.996683
$$907$$ 2.00000 0.0664089 0.0332045 0.999449i $$-0.489429\pi$$
0.0332045 + 0.999449i $$0.489429\pi$$
$$908$$ 3.00000 0.0995585
$$909$$ 0 0
$$910$$ −12.0000 −0.397796
$$911$$ −30.0000 −0.993944 −0.496972 0.867766i $$-0.665555\pi$$
−0.496972 + 0.867766i $$0.665555\pi$$
$$912$$ 0 0
$$913$$ 13.0000 0.430237
$$914$$ −38.0000 −1.25693
$$915$$ −24.0000 −0.793416
$$916$$ −13.0000 −0.429532
$$917$$ 48.0000 1.58510
$$918$$ 16.0000 0.528079
$$919$$ 45.0000 1.48441 0.742207 0.670171i $$-0.233779\pi$$
0.742207 + 0.670171i $$0.233779\pi$$
$$920$$ −4.00000 −0.131876
$$921$$ 58.0000 1.91116
$$922$$ 2.00000 0.0658665
$$923$$ 56.0000 1.84326
$$924$$ −6.00000 −0.197386
$$925$$ 12.0000 0.394558
$$926$$ −4.00000 −0.131448
$$927$$ 0 0
$$928$$ 10.0000 0.328266
$$929$$ 14.0000 0.459325 0.229663 0.973270i $$-0.426238\pi$$
0.229663 + 0.973270i $$0.426238\pi$$
$$930$$ −8.00000 −0.262330
$$931$$ 0 0
$$932$$ −16.0000 −0.524097
$$933$$ 36.0000 1.17859
$$934$$ −14.0000 −0.458094
$$935$$ 4.00000 0.130814
$$936$$ 4.00000 0.130744
$$937$$ −38.0000 −1.24141 −0.620703 0.784046i $$-0.713153\pi$$
−0.620703 + 0.784046i $$0.713153\pi$$
$$938$$ −36.0000 −1.17544
$$939$$ 18.0000 0.587408
$$940$$ 6.00000 0.195698
$$941$$ 24.0000 0.782378 0.391189 0.920310i $$-0.372064\pi$$
0.391189 + 0.920310i $$0.372064\pi$$
$$942$$ 10.0000 0.325818
$$943$$ −24.0000 −0.781548
$$944$$ −12.0000 −0.390567
$$945$$ −12.0000 −0.390360
$$946$$ −4.00000 −0.130051
$$947$$ −30.0000 −0.974869 −0.487435 0.873160i $$-0.662067\pi$$
−0.487435 + 0.873160i $$0.662067\pi$$
$$948$$ 34.0000 1.10427
$$949$$ −24.0000 −0.779073
$$950$$ 0 0
$$951$$ −36.0000 −1.16738
$$952$$ −12.0000 −0.388922
$$953$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$954$$ −1.00000 −0.0323762
$$955$$ 18.0000 0.582466
$$956$$ −5.00000 −0.161712
$$957$$ −20.0000 −0.646508
$$958$$ 0 0
$$959$$ −3.00000 −0.0968751
$$960$$ −2.00000 −0.0645497
$$961$$ −15.0000 −0.483871
$$962$$ −12.0000 −0.386896
$$963$$ 7.00000 0.225572
$$964$$ 10.0000 0.322078
$$965$$ 10.0000 0.321911
$$966$$ −24.0000 −0.772187
$$967$$ −37.0000 −1.18984 −0.594920 0.803785i $$-0.702816\pi$$
−0.594920 + 0.803785i $$0.702816\pi$$
$$968$$ −1.00000 −0.0321412
$$969$$ 0 0
$$970$$ 5.00000 0.160540
$$971$$ 50.0000 1.60458 0.802288 0.596937i $$-0.203616\pi$$
0.802288 + 0.596937i $$0.203616\pi$$
$$972$$ 10.0000 0.320750
$$973$$ 51.0000 1.63498
$$974$$ 18.0000 0.576757
$$975$$ −32.0000 −1.02482
$$976$$ 12.0000 0.384111
$$977$$ −9.00000 −0.287936 −0.143968 0.989582i $$-0.545986\pi$$
−0.143968 + 0.989582i $$0.545986\pi$$
$$978$$ 8.00000 0.255812
$$979$$ −2.00000 −0.0639203
$$980$$ 2.00000 0.0638877
$$981$$ 4.00000 0.127710
$$982$$ −1.00000 −0.0319113
$$983$$ −38.0000 −1.21201 −0.606006 0.795460i $$-0.707229\pi$$
−0.606006 + 0.795460i $$0.707229\pi$$
$$984$$ −12.0000 −0.382546
$$985$$ −12.0000 −0.382352
$$986$$ −40.0000 −1.27386
$$987$$ 36.0000 1.14589
$$988$$ 0 0
$$989$$ −16.0000 −0.508770
$$990$$ 1.00000 0.0317821
$$991$$ −38.0000 −1.20711 −0.603555 0.797321i $$-0.706250\pi$$
−0.603555 + 0.797321i $$0.706250\pi$$
$$992$$ 4.00000 0.127000
$$993$$ 40.0000 1.26936
$$994$$ −42.0000 −1.33216
$$995$$ −10.0000 −0.317021
$$996$$ 26.0000 0.823842
$$997$$ 44.0000 1.39349 0.696747 0.717317i $$-0.254630\pi$$
0.696747 + 0.717317i $$0.254630\pi$$
$$998$$ 40.0000 1.26618
$$999$$ −12.0000 −0.379663
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 7942.2.a.c.1.1 1
19.8 odd 6 418.2.e.a.45.1 2
19.12 odd 6 418.2.e.a.353.1 yes 2
19.18 odd 2 7942.2.a.s.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
418.2.e.a.45.1 2 19.8 odd 6
418.2.e.a.353.1 yes 2 19.12 odd 6
7942.2.a.c.1.1 1 1.1 even 1 trivial
7942.2.a.s.1.1 1 19.18 odd 2