# Properties

 Label 7942.2.a.a.1.1 Level $7942$ Weight $2$ Character 7942.1 Self dual yes Analytic conductor $63.417$ Analytic rank $1$ 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: $$1$$ 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)

 $$f(q)$$ $$=$$ $$q-1.00000 q^{2} -3.00000 q^{3} +1.00000 q^{4} -2.00000 q^{5} +3.00000 q^{6} +1.00000 q^{7} -1.00000 q^{8} +6.00000 q^{9} +O(q^{10})$$ $$q-1.00000 q^{2} -3.00000 q^{3} +1.00000 q^{4} -2.00000 q^{5} +3.00000 q^{6} +1.00000 q^{7} -1.00000 q^{8} +6.00000 q^{9} +2.00000 q^{10} +1.00000 q^{11} -3.00000 q^{12} +7.00000 q^{13} -1.00000 q^{14} +6.00000 q^{15} +1.00000 q^{16} -3.00000 q^{17} -6.00000 q^{18} -2.00000 q^{20} -3.00000 q^{21} -1.00000 q^{22} +3.00000 q^{23} +3.00000 q^{24} -1.00000 q^{25} -7.00000 q^{26} -9.00000 q^{27} +1.00000 q^{28} -1.00000 q^{29} -6.00000 q^{30} -2.00000 q^{31} -1.00000 q^{32} -3.00000 q^{33} +3.00000 q^{34} -2.00000 q^{35} +6.00000 q^{36} +6.00000 q^{37} -21.0000 q^{39} +2.00000 q^{40} +2.00000 q^{41} +3.00000 q^{42} +4.00000 q^{43} +1.00000 q^{44} -12.0000 q^{45} -3.00000 q^{46} -3.00000 q^{48} -6.00000 q^{49} +1.00000 q^{50} +9.00000 q^{51} +7.00000 q^{52} -3.00000 q^{53} +9.00000 q^{54} -2.00000 q^{55} -1.00000 q^{56} +1.00000 q^{58} -7.00000 q^{59} +6.00000 q^{60} -12.0000 q^{61} +2.00000 q^{62} +6.00000 q^{63} +1.00000 q^{64} -14.0000 q^{65} +3.00000 q^{66} -15.0000 q^{67} -3.00000 q^{68} -9.00000 q^{69} +2.00000 q^{70} -6.00000 q^{71} -6.00000 q^{72} -9.00000 q^{73} -6.00000 q^{74} +3.00000 q^{75} +1.00000 q^{77} +21.0000 q^{78} +8.00000 q^{79} -2.00000 q^{80} +9.00000 q^{81} -2.00000 q^{82} +16.0000 q^{83} -3.00000 q^{84} +6.00000 q^{85} -4.00000 q^{86} +3.00000 q^{87} -1.00000 q^{88} +16.0000 q^{89} +12.0000 q^{90} +7.00000 q^{91} +3.00000 q^{92} +6.00000 q^{93} +3.00000 q^{96} -8.00000 q^{97} +6.00000 q^{98} +6.00000 q^{99} +O(q^{100})$$

## 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
$$3$$ −3.00000 −1.73205 −0.866025 0.500000i $$-0.833333\pi$$
−0.866025 + 0.500000i $$0.833333\pi$$
$$4$$ 1.00000 0.500000
$$5$$ −2.00000 −0.894427 −0.447214 0.894427i $$-0.647584\pi$$
−0.447214 + 0.894427i $$0.647584\pi$$
$$6$$ 3.00000 1.22474
$$7$$ 1.00000 0.377964 0.188982 0.981981i $$-0.439481\pi$$
0.188982 + 0.981981i $$0.439481\pi$$
$$8$$ −1.00000 −0.353553
$$9$$ 6.00000 2.00000
$$10$$ 2.00000 0.632456
$$11$$ 1.00000 0.301511
$$12$$ −3.00000 −0.866025
$$13$$ 7.00000 1.94145 0.970725 0.240192i $$-0.0772105\pi$$
0.970725 + 0.240192i $$0.0772105\pi$$
$$14$$ −1.00000 −0.267261
$$15$$ 6.00000 1.54919
$$16$$ 1.00000 0.250000
$$17$$ −3.00000 −0.727607 −0.363803 0.931476i $$-0.618522\pi$$
−0.363803 + 0.931476i $$0.618522\pi$$
$$18$$ −6.00000 −1.41421
$$19$$ 0 0
$$20$$ −2.00000 −0.447214
$$21$$ −3.00000 −0.654654
$$22$$ −1.00000 −0.213201
$$23$$ 3.00000 0.625543 0.312772 0.949828i $$-0.398743\pi$$
0.312772 + 0.949828i $$0.398743\pi$$
$$24$$ 3.00000 0.612372
$$25$$ −1.00000 −0.200000
$$26$$ −7.00000 −1.37281
$$27$$ −9.00000 −1.73205
$$28$$ 1.00000 0.188982
$$29$$ −1.00000 −0.185695 −0.0928477 0.995680i $$-0.529597\pi$$
−0.0928477 + 0.995680i $$0.529597\pi$$
$$30$$ −6.00000 −1.09545
$$31$$ −2.00000 −0.359211 −0.179605 0.983739i $$-0.557482\pi$$
−0.179605 + 0.983739i $$0.557482\pi$$
$$32$$ −1.00000 −0.176777
$$33$$ −3.00000 −0.522233
$$34$$ 3.00000 0.514496
$$35$$ −2.00000 −0.338062
$$36$$ 6.00000 1.00000
$$37$$ 6.00000 0.986394 0.493197 0.869918i $$-0.335828\pi$$
0.493197 + 0.869918i $$0.335828\pi$$
$$38$$ 0 0
$$39$$ −21.0000 −3.36269
$$40$$ 2.00000 0.316228
$$41$$ 2.00000 0.312348 0.156174 0.987730i $$-0.450084\pi$$
0.156174 + 0.987730i $$0.450084\pi$$
$$42$$ 3.00000 0.462910
$$43$$ 4.00000 0.609994 0.304997 0.952353i $$-0.401344\pi$$
0.304997 + 0.952353i $$0.401344\pi$$
$$44$$ 1.00000 0.150756
$$45$$ −12.0000 −1.78885
$$46$$ −3.00000 −0.442326
$$47$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$48$$ −3.00000 −0.433013
$$49$$ −6.00000 −0.857143
$$50$$ 1.00000 0.141421
$$51$$ 9.00000 1.26025
$$52$$ 7.00000 0.970725
$$53$$ −3.00000 −0.412082 −0.206041 0.978543i $$-0.566058\pi$$
−0.206041 + 0.978543i $$0.566058\pi$$
$$54$$ 9.00000 1.22474
$$55$$ −2.00000 −0.269680
$$56$$ −1.00000 −0.133631
$$57$$ 0 0
$$58$$ 1.00000 0.131306
$$59$$ −7.00000 −0.911322 −0.455661 0.890153i $$-0.650597\pi$$
−0.455661 + 0.890153i $$0.650597\pi$$
$$60$$ 6.00000 0.774597
$$61$$ −12.0000 −1.53644 −0.768221 0.640184i $$-0.778858\pi$$
−0.768221 + 0.640184i $$0.778858\pi$$
$$62$$ 2.00000 0.254000
$$63$$ 6.00000 0.755929
$$64$$ 1.00000 0.125000
$$65$$ −14.0000 −1.73649
$$66$$ 3.00000 0.369274
$$67$$ −15.0000 −1.83254 −0.916271 0.400559i $$-0.868816\pi$$
−0.916271 + 0.400559i $$0.868816\pi$$
$$68$$ −3.00000 −0.363803
$$69$$ −9.00000 −1.08347
$$70$$ 2.00000 0.239046
$$71$$ −6.00000 −0.712069 −0.356034 0.934473i $$-0.615871\pi$$
−0.356034 + 0.934473i $$0.615871\pi$$
$$72$$ −6.00000 −0.707107
$$73$$ −9.00000 −1.05337 −0.526685 0.850060i $$-0.676565\pi$$
−0.526685 + 0.850060i $$0.676565\pi$$
$$74$$ −6.00000 −0.697486
$$75$$ 3.00000 0.346410
$$76$$ 0 0
$$77$$ 1.00000 0.113961
$$78$$ 21.0000 2.37778
$$79$$ 8.00000 0.900070 0.450035 0.893011i $$-0.351411\pi$$
0.450035 + 0.893011i $$0.351411\pi$$
$$80$$ −2.00000 −0.223607
$$81$$ 9.00000 1.00000
$$82$$ −2.00000 −0.220863
$$83$$ 16.0000 1.75623 0.878114 0.478451i $$-0.158802\pi$$
0.878114 + 0.478451i $$0.158802\pi$$
$$84$$ −3.00000 −0.327327
$$85$$ 6.00000 0.650791
$$86$$ −4.00000 −0.431331
$$87$$ 3.00000 0.321634
$$88$$ −1.00000 −0.106600
$$89$$ 16.0000 1.69600 0.847998 0.529999i $$-0.177808\pi$$
0.847998 + 0.529999i $$0.177808\pi$$
$$90$$ 12.0000 1.26491
$$91$$ 7.00000 0.733799
$$92$$ 3.00000 0.312772
$$93$$ 6.00000 0.622171
$$94$$ 0 0
$$95$$ 0 0
$$96$$ 3.00000 0.306186
$$97$$ −8.00000 −0.812277 −0.406138 0.913812i $$-0.633125\pi$$
−0.406138 + 0.913812i $$0.633125\pi$$
$$98$$ 6.00000 0.606092
$$99$$ 6.00000 0.603023
$$100$$ −1.00000 −0.100000
$$101$$ −2.00000 −0.199007 −0.0995037 0.995037i $$-0.531726\pi$$
−0.0995037 + 0.995037i $$0.531726\pi$$
$$102$$ −9.00000 −0.891133
$$103$$ 12.0000 1.18240 0.591198 0.806527i $$-0.298655\pi$$
0.591198 + 0.806527i $$0.298655\pi$$
$$104$$ −7.00000 −0.686406
$$105$$ 6.00000 0.585540
$$106$$ 3.00000 0.291386
$$107$$ −11.0000 −1.06341 −0.531705 0.846930i $$-0.678449\pi$$
−0.531705 + 0.846930i $$0.678449\pi$$
$$108$$ −9.00000 −0.866025
$$109$$ −11.0000 −1.05361 −0.526804 0.849987i $$-0.676610\pi$$
−0.526804 + 0.849987i $$0.676610\pi$$
$$110$$ 2.00000 0.190693
$$111$$ −18.0000 −1.70848
$$112$$ 1.00000 0.0944911
$$113$$ −20.0000 −1.88144 −0.940721 0.339182i $$-0.889850\pi$$
−0.940721 + 0.339182i $$0.889850\pi$$
$$114$$ 0 0
$$115$$ −6.00000 −0.559503
$$116$$ −1.00000 −0.0928477
$$117$$ 42.0000 3.88290
$$118$$ 7.00000 0.644402
$$119$$ −3.00000 −0.275010
$$120$$ −6.00000 −0.547723
$$121$$ 1.00000 0.0909091
$$122$$ 12.0000 1.08643
$$123$$ −6.00000 −0.541002
$$124$$ −2.00000 −0.179605
$$125$$ 12.0000 1.07331
$$126$$ −6.00000 −0.534522
$$127$$ 16.0000 1.41977 0.709885 0.704317i $$-0.248747\pi$$
0.709885 + 0.704317i $$0.248747\pi$$
$$128$$ −1.00000 −0.0883883
$$129$$ −12.0000 −1.05654
$$130$$ 14.0000 1.22788
$$131$$ −14.0000 −1.22319 −0.611593 0.791173i $$-0.709471\pi$$
−0.611593 + 0.791173i $$0.709471\pi$$
$$132$$ −3.00000 −0.261116
$$133$$ 0 0
$$134$$ 15.0000 1.29580
$$135$$ 18.0000 1.54919
$$136$$ 3.00000 0.257248
$$137$$ −17.0000 −1.45241 −0.726204 0.687479i $$-0.758717\pi$$
−0.726204 + 0.687479i $$0.758717\pi$$
$$138$$ 9.00000 0.766131
$$139$$ 10.0000 0.848189 0.424094 0.905618i $$-0.360592\pi$$
0.424094 + 0.905618i $$0.360592\pi$$
$$140$$ −2.00000 −0.169031
$$141$$ 0 0
$$142$$ 6.00000 0.503509
$$143$$ 7.00000 0.585369
$$144$$ 6.00000 0.500000
$$145$$ 2.00000 0.166091
$$146$$ 9.00000 0.744845
$$147$$ 18.0000 1.48461
$$148$$ 6.00000 0.493197
$$149$$ 14.0000 1.14692 0.573462 0.819232i $$-0.305600\pi$$
0.573462 + 0.819232i $$0.305600\pi$$
$$150$$ −3.00000 −0.244949
$$151$$ 12.0000 0.976546 0.488273 0.872691i $$-0.337627\pi$$
0.488273 + 0.872691i $$0.337627\pi$$
$$152$$ 0 0
$$153$$ −18.0000 −1.45521
$$154$$ −1.00000 −0.0805823
$$155$$ 4.00000 0.321288
$$156$$ −21.0000 −1.68135
$$157$$ 18.0000 1.43656 0.718278 0.695756i $$-0.244931\pi$$
0.718278 + 0.695756i $$0.244931\pi$$
$$158$$ −8.00000 −0.636446
$$159$$ 9.00000 0.713746
$$160$$ 2.00000 0.158114
$$161$$ 3.00000 0.236433
$$162$$ −9.00000 −0.707107
$$163$$ 14.0000 1.09656 0.548282 0.836293i $$-0.315282\pi$$
0.548282 + 0.836293i $$0.315282\pi$$
$$164$$ 2.00000 0.156174
$$165$$ 6.00000 0.467099
$$166$$ −16.0000 −1.24184
$$167$$ −18.0000 −1.39288 −0.696441 0.717614i $$-0.745234\pi$$
−0.696441 + 0.717614i $$0.745234\pi$$
$$168$$ 3.00000 0.231455
$$169$$ 36.0000 2.76923
$$170$$ −6.00000 −0.460179
$$171$$ 0 0
$$172$$ 4.00000 0.304997
$$173$$ 2.00000 0.152057 0.0760286 0.997106i $$-0.475776\pi$$
0.0760286 + 0.997106i $$0.475776\pi$$
$$174$$ −3.00000 −0.227429
$$175$$ −1.00000 −0.0755929
$$176$$ 1.00000 0.0753778
$$177$$ 21.0000 1.57846
$$178$$ −16.0000 −1.19925
$$179$$ 4.00000 0.298974 0.149487 0.988764i $$-0.452238\pi$$
0.149487 + 0.988764i $$0.452238\pi$$
$$180$$ −12.0000 −0.894427
$$181$$ 2.00000 0.148659 0.0743294 0.997234i $$-0.476318\pi$$
0.0743294 + 0.997234i $$0.476318\pi$$
$$182$$ −7.00000 −0.518875
$$183$$ 36.0000 2.66120
$$184$$ −3.00000 −0.221163
$$185$$ −12.0000 −0.882258
$$186$$ −6.00000 −0.439941
$$187$$ −3.00000 −0.219382
$$188$$ 0 0
$$189$$ −9.00000 −0.654654
$$190$$ 0 0
$$191$$ 3.00000 0.217072 0.108536 0.994092i $$-0.465384\pi$$
0.108536 + 0.994092i $$0.465384\pi$$
$$192$$ −3.00000 −0.216506
$$193$$ −4.00000 −0.287926 −0.143963 0.989583i $$-0.545985\pi$$
−0.143963 + 0.989583i $$0.545985\pi$$
$$194$$ 8.00000 0.574367
$$195$$ 42.0000 3.00768
$$196$$ −6.00000 −0.428571
$$197$$ −28.0000 −1.99492 −0.997459 0.0712470i $$-0.977302\pi$$
−0.997459 + 0.0712470i $$0.977302\pi$$
$$198$$ −6.00000 −0.426401
$$199$$ 7.00000 0.496217 0.248108 0.968732i $$-0.420191\pi$$
0.248108 + 0.968732i $$0.420191\pi$$
$$200$$ 1.00000 0.0707107
$$201$$ 45.0000 3.17406
$$202$$ 2.00000 0.140720
$$203$$ −1.00000 −0.0701862
$$204$$ 9.00000 0.630126
$$205$$ −4.00000 −0.279372
$$206$$ −12.0000 −0.836080
$$207$$ 18.0000 1.25109
$$208$$ 7.00000 0.485363
$$209$$ 0 0
$$210$$ −6.00000 −0.414039
$$211$$ 23.0000 1.58339 0.791693 0.610920i $$-0.209200\pi$$
0.791693 + 0.610920i $$0.209200\pi$$
$$212$$ −3.00000 −0.206041
$$213$$ 18.0000 1.23334
$$214$$ 11.0000 0.751945
$$215$$ −8.00000 −0.545595
$$216$$ 9.00000 0.612372
$$217$$ −2.00000 −0.135769
$$218$$ 11.0000 0.745014
$$219$$ 27.0000 1.82449
$$220$$ −2.00000 −0.134840
$$221$$ −21.0000 −1.41261
$$222$$ 18.0000 1.20808
$$223$$ −2.00000 −0.133930 −0.0669650 0.997755i $$-0.521332\pi$$
−0.0669650 + 0.997755i $$0.521332\pi$$
$$224$$ −1.00000 −0.0668153
$$225$$ −6.00000 −0.400000
$$226$$ 20.0000 1.33038
$$227$$ −25.0000 −1.65931 −0.829654 0.558278i $$-0.811462\pi$$
−0.829654 + 0.558278i $$0.811462\pi$$
$$228$$ 0 0
$$229$$ 20.0000 1.32164 0.660819 0.750546i $$-0.270209\pi$$
0.660819 + 0.750546i $$0.270209\pi$$
$$230$$ 6.00000 0.395628
$$231$$ −3.00000 −0.197386
$$232$$ 1.00000 0.0656532
$$233$$ 6.00000 0.393073 0.196537 0.980497i $$-0.437031\pi$$
0.196537 + 0.980497i $$0.437031\pi$$
$$234$$ −42.0000 −2.74563
$$235$$ 0 0
$$236$$ −7.00000 −0.455661
$$237$$ −24.0000 −1.55897
$$238$$ 3.00000 0.194461
$$239$$ 13.0000 0.840900 0.420450 0.907316i $$-0.361872\pi$$
0.420450 + 0.907316i $$0.361872\pi$$
$$240$$ 6.00000 0.387298
$$241$$ 20.0000 1.28831 0.644157 0.764894i $$-0.277208\pi$$
0.644157 + 0.764894i $$0.277208\pi$$
$$242$$ −1.00000 −0.0642824
$$243$$ 0 0
$$244$$ −12.0000 −0.768221
$$245$$ 12.0000 0.766652
$$246$$ 6.00000 0.382546
$$247$$ 0 0
$$248$$ 2.00000 0.127000
$$249$$ −48.0000 −3.04188
$$250$$ −12.0000 −0.758947
$$251$$ −30.0000 −1.89358 −0.946792 0.321847i $$-0.895696\pi$$
−0.946792 + 0.321847i $$0.895696\pi$$
$$252$$ 6.00000 0.377964
$$253$$ 3.00000 0.188608
$$254$$ −16.0000 −1.00393
$$255$$ −18.0000 −1.12720
$$256$$ 1.00000 0.0625000
$$257$$ −18.0000 −1.12281 −0.561405 0.827541i $$-0.689739\pi$$
−0.561405 + 0.827541i $$0.689739\pi$$
$$258$$ 12.0000 0.747087
$$259$$ 6.00000 0.372822
$$260$$ −14.0000 −0.868243
$$261$$ −6.00000 −0.371391
$$262$$ 14.0000 0.864923
$$263$$ 12.0000 0.739952 0.369976 0.929041i $$-0.379366\pi$$
0.369976 + 0.929041i $$0.379366\pi$$
$$264$$ 3.00000 0.184637
$$265$$ 6.00000 0.368577
$$266$$ 0 0
$$267$$ −48.0000 −2.93755
$$268$$ −15.0000 −0.916271
$$269$$ −18.0000 −1.09748 −0.548740 0.835993i $$-0.684892\pi$$
−0.548740 + 0.835993i $$0.684892\pi$$
$$270$$ −18.0000 −1.09545
$$271$$ −3.00000 −0.182237 −0.0911185 0.995840i $$-0.529044\pi$$
−0.0911185 + 0.995840i $$0.529044\pi$$
$$272$$ −3.00000 −0.181902
$$273$$ −21.0000 −1.27098
$$274$$ 17.0000 1.02701
$$275$$ −1.00000 −0.0603023
$$276$$ −9.00000 −0.541736
$$277$$ −8.00000 −0.480673 −0.240337 0.970690i $$-0.577258\pi$$
−0.240337 + 0.970690i $$0.577258\pi$$
$$278$$ −10.0000 −0.599760
$$279$$ −12.0000 −0.718421
$$280$$ 2.00000 0.119523
$$281$$ 16.0000 0.954480 0.477240 0.878773i $$-0.341637\pi$$
0.477240 + 0.878773i $$0.341637\pi$$
$$282$$ 0 0
$$283$$ −22.0000 −1.30776 −0.653882 0.756596i $$-0.726861\pi$$
−0.653882 + 0.756596i $$0.726861\pi$$
$$284$$ −6.00000 −0.356034
$$285$$ 0 0
$$286$$ −7.00000 −0.413919
$$287$$ 2.00000 0.118056
$$288$$ −6.00000 −0.353553
$$289$$ −8.00000 −0.470588
$$290$$ −2.00000 −0.117444
$$291$$ 24.0000 1.40690
$$292$$ −9.00000 −0.526685
$$293$$ −15.0000 −0.876309 −0.438155 0.898900i $$-0.644368\pi$$
−0.438155 + 0.898900i $$0.644368\pi$$
$$294$$ −18.0000 −1.04978
$$295$$ 14.0000 0.815112
$$296$$ −6.00000 −0.348743
$$297$$ −9.00000 −0.522233
$$298$$ −14.0000 −0.810998
$$299$$ 21.0000 1.21446
$$300$$ 3.00000 0.173205
$$301$$ 4.00000 0.230556
$$302$$ −12.0000 −0.690522
$$303$$ 6.00000 0.344691
$$304$$ 0 0
$$305$$ 24.0000 1.37424
$$306$$ 18.0000 1.02899
$$307$$ −16.0000 −0.913168 −0.456584 0.889680i $$-0.650927\pi$$
−0.456584 + 0.889680i $$0.650927\pi$$
$$308$$ 1.00000 0.0569803
$$309$$ −36.0000 −2.04797
$$310$$ −4.00000 −0.227185
$$311$$ 11.0000 0.623753 0.311876 0.950123i $$-0.399043\pi$$
0.311876 + 0.950123i $$0.399043\pi$$
$$312$$ 21.0000 1.18889
$$313$$ −27.0000 −1.52613 −0.763065 0.646322i $$-0.776306\pi$$
−0.763065 + 0.646322i $$0.776306\pi$$
$$314$$ −18.0000 −1.01580
$$315$$ −12.0000 −0.676123
$$316$$ 8.00000 0.450035
$$317$$ −1.00000 −0.0561656 −0.0280828 0.999606i $$-0.508940\pi$$
−0.0280828 + 0.999606i $$0.508940\pi$$
$$318$$ −9.00000 −0.504695
$$319$$ −1.00000 −0.0559893
$$320$$ −2.00000 −0.111803
$$321$$ 33.0000 1.84188
$$322$$ −3.00000 −0.167183
$$323$$ 0 0
$$324$$ 9.00000 0.500000
$$325$$ −7.00000 −0.388290
$$326$$ −14.0000 −0.775388
$$327$$ 33.0000 1.82490
$$328$$ −2.00000 −0.110432
$$329$$ 0 0
$$330$$ −6.00000 −0.330289
$$331$$ −17.0000 −0.934405 −0.467202 0.884150i $$-0.654738\pi$$
−0.467202 + 0.884150i $$0.654738\pi$$
$$332$$ 16.0000 0.878114
$$333$$ 36.0000 1.97279
$$334$$ 18.0000 0.984916
$$335$$ 30.0000 1.63908
$$336$$ −3.00000 −0.163663
$$337$$ −4.00000 −0.217894 −0.108947 0.994048i $$-0.534748\pi$$
−0.108947 + 0.994048i $$0.534748\pi$$
$$338$$ −36.0000 −1.95814
$$339$$ 60.0000 3.25875
$$340$$ 6.00000 0.325396
$$341$$ −2.00000 −0.108306
$$342$$ 0 0
$$343$$ −13.0000 −0.701934
$$344$$ −4.00000 −0.215666
$$345$$ 18.0000 0.969087
$$346$$ −2.00000 −0.107521
$$347$$ 6.00000 0.322097 0.161048 0.986947i $$-0.448512\pi$$
0.161048 + 0.986947i $$0.448512\pi$$
$$348$$ 3.00000 0.160817
$$349$$ 10.0000 0.535288 0.267644 0.963518i $$-0.413755\pi$$
0.267644 + 0.963518i $$0.413755\pi$$
$$350$$ 1.00000 0.0534522
$$351$$ −63.0000 −3.36269
$$352$$ −1.00000 −0.0533002
$$353$$ −31.0000 −1.64996 −0.824982 0.565159i $$-0.808815\pi$$
−0.824982 + 0.565159i $$0.808815\pi$$
$$354$$ −21.0000 −1.11614
$$355$$ 12.0000 0.636894
$$356$$ 16.0000 0.847998
$$357$$ 9.00000 0.476331
$$358$$ −4.00000 −0.211407
$$359$$ 27.0000 1.42501 0.712503 0.701669i $$-0.247562\pi$$
0.712503 + 0.701669i $$0.247562\pi$$
$$360$$ 12.0000 0.632456
$$361$$ 0 0
$$362$$ −2.00000 −0.105118
$$363$$ −3.00000 −0.157459
$$364$$ 7.00000 0.366900
$$365$$ 18.0000 0.942163
$$366$$ −36.0000 −1.88175
$$367$$ 16.0000 0.835193 0.417597 0.908633i $$-0.362873\pi$$
0.417597 + 0.908633i $$0.362873\pi$$
$$368$$ 3.00000 0.156386
$$369$$ 12.0000 0.624695
$$370$$ 12.0000 0.623850
$$371$$ −3.00000 −0.155752
$$372$$ 6.00000 0.311086
$$373$$ 13.0000 0.673114 0.336557 0.941663i $$-0.390737\pi$$
0.336557 + 0.941663i $$0.390737\pi$$
$$374$$ 3.00000 0.155126
$$375$$ −36.0000 −1.85903
$$376$$ 0 0
$$377$$ −7.00000 −0.360518
$$378$$ 9.00000 0.462910
$$379$$ 1.00000 0.0513665 0.0256833 0.999670i $$-0.491824\pi$$
0.0256833 + 0.999670i $$0.491824\pi$$
$$380$$ 0 0
$$381$$ −48.0000 −2.45911
$$382$$ −3.00000 −0.153493
$$383$$ −26.0000 −1.32854 −0.664269 0.747494i $$-0.731257\pi$$
−0.664269 + 0.747494i $$0.731257\pi$$
$$384$$ 3.00000 0.153093
$$385$$ −2.00000 −0.101929
$$386$$ 4.00000 0.203595
$$387$$ 24.0000 1.21999
$$388$$ −8.00000 −0.406138
$$389$$ −14.0000 −0.709828 −0.354914 0.934899i $$-0.615490\pi$$
−0.354914 + 0.934899i $$0.615490\pi$$
$$390$$ −42.0000 −2.12675
$$391$$ −9.00000 −0.455150
$$392$$ 6.00000 0.303046
$$393$$ 42.0000 2.11862
$$394$$ 28.0000 1.41062
$$395$$ −16.0000 −0.805047
$$396$$ 6.00000 0.301511
$$397$$ −30.0000 −1.50566 −0.752828 0.658217i $$-0.771311\pi$$
−0.752828 + 0.658217i $$0.771311\pi$$
$$398$$ −7.00000 −0.350878
$$399$$ 0 0
$$400$$ −1.00000 −0.0500000
$$401$$ 2.00000 0.0998752 0.0499376 0.998752i $$-0.484098\pi$$
0.0499376 + 0.998752i $$0.484098\pi$$
$$402$$ −45.0000 −2.24440
$$403$$ −14.0000 −0.697390
$$404$$ −2.00000 −0.0995037
$$405$$ −18.0000 −0.894427
$$406$$ 1.00000 0.0496292
$$407$$ 6.00000 0.297409
$$408$$ −9.00000 −0.445566
$$409$$ 14.0000 0.692255 0.346128 0.938187i $$-0.387496\pi$$
0.346128 + 0.938187i $$0.387496\pi$$
$$410$$ 4.00000 0.197546
$$411$$ 51.0000 2.51564
$$412$$ 12.0000 0.591198
$$413$$ −7.00000 −0.344447
$$414$$ −18.0000 −0.884652
$$415$$ −32.0000 −1.57082
$$416$$ −7.00000 −0.343203
$$417$$ −30.0000 −1.46911
$$418$$ 0 0
$$419$$ −6.00000 −0.293119 −0.146560 0.989202i $$-0.546820\pi$$
−0.146560 + 0.989202i $$0.546820\pi$$
$$420$$ 6.00000 0.292770
$$421$$ −3.00000 −0.146211 −0.0731055 0.997324i $$-0.523291\pi$$
−0.0731055 + 0.997324i $$0.523291\pi$$
$$422$$ −23.0000 −1.11962
$$423$$ 0 0
$$424$$ 3.00000 0.145693
$$425$$ 3.00000 0.145521
$$426$$ −18.0000 −0.872103
$$427$$ −12.0000 −0.580721
$$428$$ −11.0000 −0.531705
$$429$$ −21.0000 −1.01389
$$430$$ 8.00000 0.385794
$$431$$ 22.0000 1.05970 0.529851 0.848091i $$-0.322248\pi$$
0.529851 + 0.848091i $$0.322248\pi$$
$$432$$ −9.00000 −0.433013
$$433$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$434$$ 2.00000 0.0960031
$$435$$ −6.00000 −0.287678
$$436$$ −11.0000 −0.526804
$$437$$ 0 0
$$438$$ −27.0000 −1.29011
$$439$$ 22.0000 1.05000 0.525001 0.851101i $$-0.324065\pi$$
0.525001 + 0.851101i $$0.324065\pi$$
$$440$$ 2.00000 0.0953463
$$441$$ −36.0000 −1.71429
$$442$$ 21.0000 0.998868
$$443$$ −14.0000 −0.665160 −0.332580 0.943075i $$-0.607919\pi$$
−0.332580 + 0.943075i $$0.607919\pi$$
$$444$$ −18.0000 −0.854242
$$445$$ −32.0000 −1.51695
$$446$$ 2.00000 0.0947027
$$447$$ −42.0000 −1.98653
$$448$$ 1.00000 0.0472456
$$449$$ 32.0000 1.51017 0.755087 0.655625i $$-0.227595\pi$$
0.755087 + 0.655625i $$0.227595\pi$$
$$450$$ 6.00000 0.282843
$$451$$ 2.00000 0.0941763
$$452$$ −20.0000 −0.940721
$$453$$ −36.0000 −1.69143
$$454$$ 25.0000 1.17331
$$455$$ −14.0000 −0.656330
$$456$$ 0 0
$$457$$ 23.0000 1.07589 0.537947 0.842978i $$-0.319200\pi$$
0.537947 + 0.842978i $$0.319200\pi$$
$$458$$ −20.0000 −0.934539
$$459$$ 27.0000 1.26025
$$460$$ −6.00000 −0.279751
$$461$$ −40.0000 −1.86299 −0.931493 0.363760i $$-0.881493\pi$$
−0.931493 + 0.363760i $$0.881493\pi$$
$$462$$ 3.00000 0.139573
$$463$$ −32.0000 −1.48717 −0.743583 0.668644i $$-0.766875\pi$$
−0.743583 + 0.668644i $$0.766875\pi$$
$$464$$ −1.00000 −0.0464238
$$465$$ −12.0000 −0.556487
$$466$$ −6.00000 −0.277945
$$467$$ −12.0000 −0.555294 −0.277647 0.960683i $$-0.589555\pi$$
−0.277647 + 0.960683i $$0.589555\pi$$
$$468$$ 42.0000 1.94145
$$469$$ −15.0000 −0.692636
$$470$$ 0 0
$$471$$ −54.0000 −2.48819
$$472$$ 7.00000 0.322201
$$473$$ 4.00000 0.183920
$$474$$ 24.0000 1.10236
$$475$$ 0 0
$$476$$ −3.00000 −0.137505
$$477$$ −18.0000 −0.824163
$$478$$ −13.0000 −0.594606
$$479$$ 28.0000 1.27935 0.639676 0.768644i $$-0.279068\pi$$
0.639676 + 0.768644i $$0.279068\pi$$
$$480$$ −6.00000 −0.273861
$$481$$ 42.0000 1.91504
$$482$$ −20.0000 −0.910975
$$483$$ −9.00000 −0.409514
$$484$$ 1.00000 0.0454545
$$485$$ 16.0000 0.726523
$$486$$ 0 0
$$487$$ 20.0000 0.906287 0.453143 0.891438i $$-0.350303\pi$$
0.453143 + 0.891438i $$0.350303\pi$$
$$488$$ 12.0000 0.543214
$$489$$ −42.0000 −1.89931
$$490$$ −12.0000 −0.542105
$$491$$ −6.00000 −0.270776 −0.135388 0.990793i $$-0.543228\pi$$
−0.135388 + 0.990793i $$0.543228\pi$$
$$492$$ −6.00000 −0.270501
$$493$$ 3.00000 0.135113
$$494$$ 0 0
$$495$$ −12.0000 −0.539360
$$496$$ −2.00000 −0.0898027
$$497$$ −6.00000 −0.269137
$$498$$ 48.0000 2.15093
$$499$$ −20.0000 −0.895323 −0.447661 0.894203i $$-0.647743\pi$$
−0.447661 + 0.894203i $$0.647743\pi$$
$$500$$ 12.0000 0.536656
$$501$$ 54.0000 2.41254
$$502$$ 30.0000 1.33897
$$503$$ −23.0000 −1.02552 −0.512760 0.858532i $$-0.671377\pi$$
−0.512760 + 0.858532i $$0.671377\pi$$
$$504$$ −6.00000 −0.267261
$$505$$ 4.00000 0.177998
$$506$$ −3.00000 −0.133366
$$507$$ −108.000 −4.79645
$$508$$ 16.0000 0.709885
$$509$$ 18.0000 0.797836 0.398918 0.916987i $$-0.369386\pi$$
0.398918 + 0.916987i $$0.369386\pi$$
$$510$$ 18.0000 0.797053
$$511$$ −9.00000 −0.398137
$$512$$ −1.00000 −0.0441942
$$513$$ 0 0
$$514$$ 18.0000 0.793946
$$515$$ −24.0000 −1.05757
$$516$$ −12.0000 −0.528271
$$517$$ 0 0
$$518$$ −6.00000 −0.263625
$$519$$ −6.00000 −0.263371
$$520$$ 14.0000 0.613941
$$521$$ 4.00000 0.175243 0.0876216 0.996154i $$-0.472073\pi$$
0.0876216 + 0.996154i $$0.472073\pi$$
$$522$$ 6.00000 0.262613
$$523$$ −11.0000 −0.480996 −0.240498 0.970650i $$-0.577311\pi$$
−0.240498 + 0.970650i $$0.577311\pi$$
$$524$$ −14.0000 −0.611593
$$525$$ 3.00000 0.130931
$$526$$ −12.0000 −0.523225
$$527$$ 6.00000 0.261364
$$528$$ −3.00000 −0.130558
$$529$$ −14.0000 −0.608696
$$530$$ −6.00000 −0.260623
$$531$$ −42.0000 −1.82264
$$532$$ 0 0
$$533$$ 14.0000 0.606407
$$534$$ 48.0000 2.07716
$$535$$ 22.0000 0.951143
$$536$$ 15.0000 0.647901
$$537$$ −12.0000 −0.517838
$$538$$ 18.0000 0.776035
$$539$$ −6.00000 −0.258438
$$540$$ 18.0000 0.774597
$$541$$ 8.00000 0.343947 0.171973 0.985102i $$-0.444986\pi$$
0.171973 + 0.985102i $$0.444986\pi$$
$$542$$ 3.00000 0.128861
$$543$$ −6.00000 −0.257485
$$544$$ 3.00000 0.128624
$$545$$ 22.0000 0.942376
$$546$$ 21.0000 0.898717
$$547$$ −8.00000 −0.342055 −0.171028 0.985266i $$-0.554709\pi$$
−0.171028 + 0.985266i $$0.554709\pi$$
$$548$$ −17.0000 −0.726204
$$549$$ −72.0000 −3.07289
$$550$$ 1.00000 0.0426401
$$551$$ 0 0
$$552$$ 9.00000 0.383065
$$553$$ 8.00000 0.340195
$$554$$ 8.00000 0.339887
$$555$$ 36.0000 1.52811
$$556$$ 10.0000 0.424094
$$557$$ −30.0000 −1.27114 −0.635570 0.772043i $$-0.719235\pi$$
−0.635570 + 0.772043i $$0.719235\pi$$
$$558$$ 12.0000 0.508001
$$559$$ 28.0000 1.18427
$$560$$ −2.00000 −0.0845154
$$561$$ 9.00000 0.379980
$$562$$ −16.0000 −0.674919
$$563$$ 8.00000 0.337160 0.168580 0.985688i $$-0.446082\pi$$
0.168580 + 0.985688i $$0.446082\pi$$
$$564$$ 0 0
$$565$$ 40.0000 1.68281
$$566$$ 22.0000 0.924729
$$567$$ 9.00000 0.377964
$$568$$ 6.00000 0.251754
$$569$$ −4.00000 −0.167689 −0.0838444 0.996479i $$-0.526720\pi$$
−0.0838444 + 0.996479i $$0.526720\pi$$
$$570$$ 0 0
$$571$$ 2.00000 0.0836974 0.0418487 0.999124i $$-0.486675\pi$$
0.0418487 + 0.999124i $$0.486675\pi$$
$$572$$ 7.00000 0.292685
$$573$$ −9.00000 −0.375980
$$574$$ −2.00000 −0.0834784
$$575$$ −3.00000 −0.125109
$$576$$ 6.00000 0.250000
$$577$$ −17.0000 −0.707719 −0.353860 0.935299i $$-0.615131\pi$$
−0.353860 + 0.935299i $$0.615131\pi$$
$$578$$ 8.00000 0.332756
$$579$$ 12.0000 0.498703
$$580$$ 2.00000 0.0830455
$$581$$ 16.0000 0.663792
$$582$$ −24.0000 −0.994832
$$583$$ −3.00000 −0.124247
$$584$$ 9.00000 0.372423
$$585$$ −84.0000 −3.47297
$$586$$ 15.0000 0.619644
$$587$$ 20.0000 0.825488 0.412744 0.910847i $$-0.364570\pi$$
0.412744 + 0.910847i $$0.364570\pi$$
$$588$$ 18.0000 0.742307
$$589$$ 0 0
$$590$$ −14.0000 −0.576371
$$591$$ 84.0000 3.45530
$$592$$ 6.00000 0.246598
$$593$$ 26.0000 1.06769 0.533846 0.845582i $$-0.320746\pi$$
0.533846 + 0.845582i $$0.320746\pi$$
$$594$$ 9.00000 0.369274
$$595$$ 6.00000 0.245976
$$596$$ 14.0000 0.573462
$$597$$ −21.0000 −0.859473
$$598$$ −21.0000 −0.858754
$$599$$ 4.00000 0.163436 0.0817178 0.996656i $$-0.473959\pi$$
0.0817178 + 0.996656i $$0.473959\pi$$
$$600$$ −3.00000 −0.122474
$$601$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$602$$ −4.00000 −0.163028
$$603$$ −90.0000 −3.66508
$$604$$ 12.0000 0.488273
$$605$$ −2.00000 −0.0813116
$$606$$ −6.00000 −0.243733
$$607$$ 14.0000 0.568242 0.284121 0.958788i $$-0.408298\pi$$
0.284121 + 0.958788i $$0.408298\pi$$
$$608$$ 0 0
$$609$$ 3.00000 0.121566
$$610$$ −24.0000 −0.971732
$$611$$ 0 0
$$612$$ −18.0000 −0.727607
$$613$$ 28.0000 1.13091 0.565455 0.824779i $$-0.308701\pi$$
0.565455 + 0.824779i $$0.308701\pi$$
$$614$$ 16.0000 0.645707
$$615$$ 12.0000 0.483887
$$616$$ −1.00000 −0.0402911
$$617$$ 18.0000 0.724653 0.362326 0.932051i $$-0.381983\pi$$
0.362326 + 0.932051i $$0.381983\pi$$
$$618$$ 36.0000 1.44813
$$619$$ −20.0000 −0.803868 −0.401934 0.915669i $$-0.631662\pi$$
−0.401934 + 0.915669i $$0.631662\pi$$
$$620$$ 4.00000 0.160644
$$621$$ −27.0000 −1.08347
$$622$$ −11.0000 −0.441060
$$623$$ 16.0000 0.641026
$$624$$ −21.0000 −0.840673
$$625$$ −19.0000 −0.760000
$$626$$ 27.0000 1.07914
$$627$$ 0 0
$$628$$ 18.0000 0.718278
$$629$$ −18.0000 −0.717707
$$630$$ 12.0000 0.478091
$$631$$ 8.00000 0.318475 0.159237 0.987240i $$-0.449096\pi$$
0.159237 + 0.987240i $$0.449096\pi$$
$$632$$ −8.00000 −0.318223
$$633$$ −69.0000 −2.74250
$$634$$ 1.00000 0.0397151
$$635$$ −32.0000 −1.26988
$$636$$ 9.00000 0.356873
$$637$$ −42.0000 −1.66410
$$638$$ 1.00000 0.0395904
$$639$$ −36.0000 −1.42414
$$640$$ 2.00000 0.0790569
$$641$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$642$$ −33.0000 −1.30241
$$643$$ −34.0000 −1.34083 −0.670415 0.741987i $$-0.733884\pi$$
−0.670415 + 0.741987i $$0.733884\pi$$
$$644$$ 3.00000 0.118217
$$645$$ 24.0000 0.944999
$$646$$ 0 0
$$647$$ 3.00000 0.117942 0.0589711 0.998260i $$-0.481218\pi$$
0.0589711 + 0.998260i $$0.481218\pi$$
$$648$$ −9.00000 −0.353553
$$649$$ −7.00000 −0.274774
$$650$$ 7.00000 0.274563
$$651$$ 6.00000 0.235159
$$652$$ 14.0000 0.548282
$$653$$ 12.0000 0.469596 0.234798 0.972044i $$-0.424557\pi$$
0.234798 + 0.972044i $$0.424557\pi$$
$$654$$ −33.0000 −1.29040
$$655$$ 28.0000 1.09405
$$656$$ 2.00000 0.0780869
$$657$$ −54.0000 −2.10674
$$658$$ 0 0
$$659$$ −27.0000 −1.05177 −0.525885 0.850555i $$-0.676266\pi$$
−0.525885 + 0.850555i $$0.676266\pi$$
$$660$$ 6.00000 0.233550
$$661$$ −25.0000 −0.972387 −0.486194 0.873851i $$-0.661615\pi$$
−0.486194 + 0.873851i $$0.661615\pi$$
$$662$$ 17.0000 0.660724
$$663$$ 63.0000 2.44672
$$664$$ −16.0000 −0.620920
$$665$$ 0 0
$$666$$ −36.0000 −1.39497
$$667$$ −3.00000 −0.116160
$$668$$ −18.0000 −0.696441
$$669$$ 6.00000 0.231973
$$670$$ −30.0000 −1.15900
$$671$$ −12.0000 −0.463255
$$672$$ 3.00000 0.115728
$$673$$ −8.00000 −0.308377 −0.154189 0.988041i $$-0.549276\pi$$
−0.154189 + 0.988041i $$0.549276\pi$$
$$674$$ 4.00000 0.154074
$$675$$ 9.00000 0.346410
$$676$$ 36.0000 1.38462
$$677$$ −27.0000 −1.03769 −0.518847 0.854867i $$-0.673639\pi$$
−0.518847 + 0.854867i $$0.673639\pi$$
$$678$$ −60.0000 −2.30429
$$679$$ −8.00000 −0.307012
$$680$$ −6.00000 −0.230089
$$681$$ 75.0000 2.87401
$$682$$ 2.00000 0.0765840
$$683$$ 16.0000 0.612223 0.306111 0.951996i $$-0.400972\pi$$
0.306111 + 0.951996i $$0.400972\pi$$
$$684$$ 0 0
$$685$$ 34.0000 1.29907
$$686$$ 13.0000 0.496342
$$687$$ −60.0000 −2.28914
$$688$$ 4.00000 0.152499
$$689$$ −21.0000 −0.800036
$$690$$ −18.0000 −0.685248
$$691$$ 10.0000 0.380418 0.190209 0.981744i $$-0.439083\pi$$
0.190209 + 0.981744i $$0.439083\pi$$
$$692$$ 2.00000 0.0760286
$$693$$ 6.00000 0.227921
$$694$$ −6.00000 −0.227757
$$695$$ −20.0000 −0.758643
$$696$$ −3.00000 −0.113715
$$697$$ −6.00000 −0.227266
$$698$$ −10.0000 −0.378506
$$699$$ −18.0000 −0.680823
$$700$$ −1.00000 −0.0377964
$$701$$ −24.0000 −0.906467 −0.453234 0.891392i $$-0.649730\pi$$
−0.453234 + 0.891392i $$0.649730\pi$$
$$702$$ 63.0000 2.37778
$$703$$ 0 0
$$704$$ 1.00000 0.0376889
$$705$$ 0 0
$$706$$ 31.0000 1.16670
$$707$$ −2.00000 −0.0752177
$$708$$ 21.0000 0.789228
$$709$$ −50.0000 −1.87779 −0.938895 0.344204i $$-0.888149\pi$$
−0.938895 + 0.344204i $$0.888149\pi$$
$$710$$ −12.0000 −0.450352
$$711$$ 48.0000 1.80014
$$712$$ −16.0000 −0.599625
$$713$$ −6.00000 −0.224702
$$714$$ −9.00000 −0.336817
$$715$$ −14.0000 −0.523570
$$716$$ 4.00000 0.149487
$$717$$ −39.0000 −1.45648
$$718$$ −27.0000 −1.00763
$$719$$ −25.0000 −0.932343 −0.466171 0.884694i $$-0.654367\pi$$
−0.466171 + 0.884694i $$0.654367\pi$$
$$720$$ −12.0000 −0.447214
$$721$$ 12.0000 0.446903
$$722$$ 0 0
$$723$$ −60.0000 −2.23142
$$724$$ 2.00000 0.0743294
$$725$$ 1.00000 0.0371391
$$726$$ 3.00000 0.111340
$$727$$ 23.0000 0.853023 0.426511 0.904482i $$-0.359742\pi$$
0.426511 + 0.904482i $$0.359742\pi$$
$$728$$ −7.00000 −0.259437
$$729$$ −27.0000 −1.00000
$$730$$ −18.0000 −0.666210
$$731$$ −12.0000 −0.443836
$$732$$ 36.0000 1.33060
$$733$$ −6.00000 −0.221615 −0.110808 0.993842i $$-0.535344\pi$$
−0.110808 + 0.993842i $$0.535344\pi$$
$$734$$ −16.0000 −0.590571
$$735$$ −36.0000 −1.32788
$$736$$ −3.00000 −0.110581
$$737$$ −15.0000 −0.552532
$$738$$ −12.0000 −0.441726
$$739$$ 52.0000 1.91285 0.956425 0.291977i $$-0.0943129\pi$$
0.956425 + 0.291977i $$0.0943129\pi$$
$$740$$ −12.0000 −0.441129
$$741$$ 0 0
$$742$$ 3.00000 0.110133
$$743$$ −8.00000 −0.293492 −0.146746 0.989174i $$-0.546880\pi$$
−0.146746 + 0.989174i $$0.546880\pi$$
$$744$$ −6.00000 −0.219971
$$745$$ −28.0000 −1.02584
$$746$$ −13.0000 −0.475964
$$747$$ 96.0000 3.51246
$$748$$ −3.00000 −0.109691
$$749$$ −11.0000 −0.401931
$$750$$ 36.0000 1.31453
$$751$$ 34.0000 1.24068 0.620339 0.784334i $$-0.286995\pi$$
0.620339 + 0.784334i $$0.286995\pi$$
$$752$$ 0 0
$$753$$ 90.0000 3.27978
$$754$$ 7.00000 0.254925
$$755$$ −24.0000 −0.873449
$$756$$ −9.00000 −0.327327
$$757$$ −2.00000 −0.0726912 −0.0363456 0.999339i $$-0.511572\pi$$
−0.0363456 + 0.999339i $$0.511572\pi$$
$$758$$ −1.00000 −0.0363216
$$759$$ −9.00000 −0.326679
$$760$$ 0 0
$$761$$ −15.0000 −0.543750 −0.271875 0.962333i $$-0.587644\pi$$
−0.271875 + 0.962333i $$0.587644\pi$$
$$762$$ 48.0000 1.73886
$$763$$ −11.0000 −0.398227
$$764$$ 3.00000 0.108536
$$765$$ 36.0000 1.30158
$$766$$ 26.0000 0.939418
$$767$$ −49.0000 −1.76929
$$768$$ −3.00000 −0.108253
$$769$$ −37.0000 −1.33425 −0.667127 0.744944i $$-0.732476\pi$$
−0.667127 + 0.744944i $$0.732476\pi$$
$$770$$ 2.00000 0.0720750
$$771$$ 54.0000 1.94476
$$772$$ −4.00000 −0.143963
$$773$$ 19.0000 0.683383 0.341691 0.939812i $$-0.389000\pi$$
0.341691 + 0.939812i $$0.389000\pi$$
$$774$$ −24.0000 −0.862662
$$775$$ 2.00000 0.0718421
$$776$$ 8.00000 0.287183
$$777$$ −18.0000 −0.645746
$$778$$ 14.0000 0.501924
$$779$$ 0 0
$$780$$ 42.0000 1.50384
$$781$$ −6.00000 −0.214697
$$782$$ 9.00000 0.321839
$$783$$ 9.00000 0.321634
$$784$$ −6.00000 −0.214286
$$785$$ −36.0000 −1.28490
$$786$$ −42.0000 −1.49809
$$787$$ −41.0000 −1.46149 −0.730746 0.682649i $$-0.760828\pi$$
−0.730746 + 0.682649i $$0.760828\pi$$
$$788$$ −28.0000 −0.997459
$$789$$ −36.0000 −1.28163
$$790$$ 16.0000 0.569254
$$791$$ −20.0000 −0.711118
$$792$$ −6.00000 −0.213201
$$793$$ −84.0000 −2.98293
$$794$$ 30.0000 1.06466
$$795$$ −18.0000 −0.638394
$$796$$ 7.00000 0.248108
$$797$$ 33.0000 1.16892 0.584460 0.811423i $$-0.301306\pi$$
0.584460 + 0.811423i $$0.301306\pi$$
$$798$$ 0 0
$$799$$ 0 0
$$800$$ 1.00000 0.0353553
$$801$$ 96.0000 3.39199
$$802$$ −2.00000 −0.0706225
$$803$$ −9.00000 −0.317603
$$804$$ 45.0000 1.58703
$$805$$ −6.00000 −0.211472
$$806$$ 14.0000 0.493129
$$807$$ 54.0000 1.90089
$$808$$ 2.00000 0.0703598
$$809$$ −17.0000 −0.597688 −0.298844 0.954302i $$-0.596601\pi$$
−0.298844 + 0.954302i $$0.596601\pi$$
$$810$$ 18.0000 0.632456
$$811$$ 25.0000 0.877869 0.438934 0.898519i $$-0.355356\pi$$
0.438934 + 0.898519i $$0.355356\pi$$
$$812$$ −1.00000 −0.0350931
$$813$$ 9.00000 0.315644
$$814$$ −6.00000 −0.210300
$$815$$ −28.0000 −0.980797
$$816$$ 9.00000 0.315063
$$817$$ 0 0
$$818$$ −14.0000 −0.489499
$$819$$ 42.0000 1.46760
$$820$$ −4.00000 −0.139686
$$821$$ −8.00000 −0.279202 −0.139601 0.990208i $$-0.544582\pi$$
−0.139601 + 0.990208i $$0.544582\pi$$
$$822$$ −51.0000 −1.77883
$$823$$ 53.0000 1.84746 0.923732 0.383040i $$-0.125123\pi$$
0.923732 + 0.383040i $$0.125123\pi$$
$$824$$ −12.0000 −0.418040
$$825$$ 3.00000 0.104447
$$826$$ 7.00000 0.243561
$$827$$ −13.0000 −0.452054 −0.226027 0.974121i $$-0.572574\pi$$
−0.226027 + 0.974121i $$0.572574\pi$$
$$828$$ 18.0000 0.625543
$$829$$ −53.0000 −1.84077 −0.920383 0.391018i $$-0.872123\pi$$
−0.920383 + 0.391018i $$0.872123\pi$$
$$830$$ 32.0000 1.11074
$$831$$ 24.0000 0.832551
$$832$$ 7.00000 0.242681
$$833$$ 18.0000 0.623663
$$834$$ 30.0000 1.03882
$$835$$ 36.0000 1.24583
$$836$$ 0 0
$$837$$ 18.0000 0.622171
$$838$$ 6.00000 0.207267
$$839$$ −32.0000 −1.10476 −0.552381 0.833592i $$-0.686281\pi$$
−0.552381 + 0.833592i $$0.686281\pi$$
$$840$$ −6.00000 −0.207020
$$841$$ −28.0000 −0.965517
$$842$$ 3.00000 0.103387
$$843$$ −48.0000 −1.65321
$$844$$ 23.0000 0.791693
$$845$$ −72.0000 −2.47688
$$846$$ 0 0
$$847$$ 1.00000 0.0343604
$$848$$ −3.00000 −0.103020
$$849$$ 66.0000 2.26511
$$850$$ −3.00000 −0.102899
$$851$$ 18.0000 0.617032
$$852$$ 18.0000 0.616670
$$853$$ 30.0000 1.02718 0.513590 0.858036i $$-0.328315\pi$$
0.513590 + 0.858036i $$0.328315\pi$$
$$854$$ 12.0000 0.410632
$$855$$ 0 0
$$856$$ 11.0000 0.375972
$$857$$ 6.00000 0.204956 0.102478 0.994735i $$-0.467323\pi$$
0.102478 + 0.994735i $$0.467323\pi$$
$$858$$ 21.0000 0.716928
$$859$$ −14.0000 −0.477674 −0.238837 0.971060i $$-0.576766\pi$$
−0.238837 + 0.971060i $$0.576766\pi$$
$$860$$ −8.00000 −0.272798
$$861$$ −6.00000 −0.204479
$$862$$ −22.0000 −0.749323
$$863$$ 38.0000 1.29354 0.646768 0.762687i $$-0.276120\pi$$
0.646768 + 0.762687i $$0.276120\pi$$
$$864$$ 9.00000 0.306186
$$865$$ −4.00000 −0.136004
$$866$$ 0 0
$$867$$ 24.0000 0.815083
$$868$$ −2.00000 −0.0678844
$$869$$ 8.00000 0.271381
$$870$$ 6.00000 0.203419
$$871$$ −105.000 −3.55779
$$872$$ 11.0000 0.372507
$$873$$ −48.0000 −1.62455
$$874$$ 0 0
$$875$$ 12.0000 0.405674
$$876$$ 27.0000 0.912245
$$877$$ −27.0000 −0.911725 −0.455863 0.890050i $$-0.650669\pi$$
−0.455863 + 0.890050i $$0.650669\pi$$
$$878$$ −22.0000 −0.742464
$$879$$ 45.0000 1.51781
$$880$$ −2.00000 −0.0674200
$$881$$ −38.0000 −1.28025 −0.640126 0.768270i $$-0.721118\pi$$
−0.640126 + 0.768270i $$0.721118\pi$$
$$882$$ 36.0000 1.21218
$$883$$ 20.0000 0.673054 0.336527 0.941674i $$-0.390748\pi$$
0.336527 + 0.941674i $$0.390748\pi$$
$$884$$ −21.0000 −0.706306
$$885$$ −42.0000 −1.41181
$$886$$ 14.0000 0.470339
$$887$$ 20.0000 0.671534 0.335767 0.941945i $$-0.391004\pi$$
0.335767 + 0.941945i $$0.391004\pi$$
$$888$$ 18.0000 0.604040
$$889$$ 16.0000 0.536623
$$890$$ 32.0000 1.07264
$$891$$ 9.00000 0.301511
$$892$$ −2.00000 −0.0669650
$$893$$ 0 0
$$894$$ 42.0000 1.40469
$$895$$ −8.00000 −0.267411
$$896$$ −1.00000 −0.0334077
$$897$$ −63.0000 −2.10351
$$898$$ −32.0000 −1.06785
$$899$$ 2.00000 0.0667037
$$900$$ −6.00000 −0.200000
$$901$$ 9.00000 0.299833
$$902$$ −2.00000 −0.0665927
$$903$$ −12.0000 −0.399335
$$904$$ 20.0000 0.665190
$$905$$ −4.00000 −0.132964
$$906$$ 36.0000 1.19602
$$907$$ −1.00000 −0.0332045 −0.0166022 0.999862i $$-0.505285\pi$$
−0.0166022 + 0.999862i $$0.505285\pi$$
$$908$$ −25.0000 −0.829654
$$909$$ −12.0000 −0.398015
$$910$$ 14.0000 0.464095
$$911$$ 30.0000 0.993944 0.496972 0.867766i $$-0.334445\pi$$
0.496972 + 0.867766i $$0.334445\pi$$
$$912$$ 0 0
$$913$$ 16.0000 0.529523
$$914$$ −23.0000 −0.760772
$$915$$ −72.0000 −2.38025
$$916$$ 20.0000 0.660819
$$917$$ −14.0000 −0.462321
$$918$$ −27.0000 −0.891133
$$919$$ 39.0000 1.28649 0.643246 0.765660i $$-0.277587\pi$$
0.643246 + 0.765660i $$0.277587\pi$$
$$920$$ 6.00000 0.197814
$$921$$ 48.0000 1.58165
$$922$$ 40.0000 1.31733
$$923$$ −42.0000 −1.38245
$$924$$ −3.00000 −0.0986928
$$925$$ −6.00000 −0.197279
$$926$$ 32.0000 1.05159
$$927$$ 72.0000 2.36479
$$928$$ 1.00000 0.0328266
$$929$$ −51.0000 −1.67326 −0.836628 0.547772i $$-0.815476\pi$$
−0.836628 + 0.547772i $$0.815476\pi$$
$$930$$ 12.0000 0.393496
$$931$$ 0 0
$$932$$ 6.00000 0.196537
$$933$$ −33.0000 −1.08037
$$934$$ 12.0000 0.392652
$$935$$ 6.00000 0.196221
$$936$$ −42.0000 −1.37281
$$937$$ −41.0000 −1.33941 −0.669706 0.742627i $$-0.733580\pi$$
−0.669706 + 0.742627i $$0.733580\pi$$
$$938$$ 15.0000 0.489767
$$939$$ 81.0000 2.64334
$$940$$ 0 0
$$941$$ 15.0000 0.488986 0.244493 0.969651i $$-0.421378\pi$$
0.244493 + 0.969651i $$0.421378\pi$$
$$942$$ 54.0000 1.75942
$$943$$ 6.00000 0.195387
$$944$$ −7.00000 −0.227831
$$945$$ 18.0000 0.585540
$$946$$ −4.00000 −0.130051
$$947$$ −8.00000 −0.259965 −0.129983 0.991516i $$-0.541492\pi$$
−0.129983 + 0.991516i $$0.541492\pi$$
$$948$$ −24.0000 −0.779484
$$949$$ −63.0000 −2.04507
$$950$$ 0 0
$$951$$ 3.00000 0.0972817
$$952$$ 3.00000 0.0972306
$$953$$ −4.00000 −0.129573 −0.0647864 0.997899i $$-0.520637\pi$$
−0.0647864 + 0.997899i $$0.520637\pi$$
$$954$$ 18.0000 0.582772
$$955$$ −6.00000 −0.194155
$$956$$ 13.0000 0.420450
$$957$$ 3.00000 0.0969762
$$958$$ −28.0000 −0.904639
$$959$$ −17.0000 −0.548959
$$960$$ 6.00000 0.193649
$$961$$ −27.0000 −0.870968
$$962$$ −42.0000 −1.35413
$$963$$ −66.0000 −2.12682
$$964$$ 20.0000 0.644157
$$965$$ 8.00000 0.257529
$$966$$ 9.00000 0.289570
$$967$$ 56.0000 1.80084 0.900419 0.435023i $$-0.143260\pi$$
0.900419 + 0.435023i $$0.143260\pi$$
$$968$$ −1.00000 −0.0321412
$$969$$ 0 0
$$970$$ −16.0000 −0.513729
$$971$$ −28.0000 −0.898563 −0.449281 0.893390i $$-0.648320\pi$$
−0.449281 + 0.893390i $$0.648320\pi$$
$$972$$ 0 0
$$973$$ 10.0000 0.320585
$$974$$ −20.0000 −0.640841
$$975$$ 21.0000 0.672538
$$976$$ −12.0000 −0.384111
$$977$$ −12.0000 −0.383914 −0.191957 0.981403i $$-0.561483\pi$$
−0.191957 + 0.981403i $$0.561483\pi$$
$$978$$ 42.0000 1.34301
$$979$$ 16.0000 0.511362
$$980$$ 12.0000 0.383326
$$981$$ −66.0000 −2.10722
$$982$$ 6.00000 0.191468
$$983$$ −50.0000 −1.59475 −0.797376 0.603483i $$-0.793779\pi$$
−0.797376 + 0.603483i $$0.793779\pi$$
$$984$$ 6.00000 0.191273
$$985$$ 56.0000 1.78431
$$986$$ −3.00000 −0.0955395
$$987$$ 0 0
$$988$$ 0 0
$$989$$ 12.0000 0.381578
$$990$$ 12.0000 0.381385
$$991$$ 44.0000 1.39771 0.698853 0.715265i $$-0.253694\pi$$
0.698853 + 0.715265i $$0.253694\pi$$
$$992$$ 2.00000 0.0635001
$$993$$ 51.0000 1.61844
$$994$$ 6.00000 0.190308
$$995$$ −14.0000 −0.443830
$$996$$ −48.0000 −1.52094
$$997$$ −2.00000 −0.0633406 −0.0316703 0.999498i $$-0.510083\pi$$
−0.0316703 + 0.999498i $$0.510083\pi$$
$$998$$ 20.0000 0.633089
$$999$$ −54.0000 −1.70848
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.a.1.1 1
19.18 odd 2 418.2.a.c.1.1 1
57.56 even 2 3762.2.a.j.1.1 1
76.75 even 2 3344.2.a.a.1.1 1
209.208 even 2 4598.2.a.j.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
418.2.a.c.1.1 1 19.18 odd 2
3344.2.a.a.1.1 1 76.75 even 2
3762.2.a.j.1.1 1 57.56 even 2
4598.2.a.j.1.1 1 209.208 even 2
7942.2.a.a.1.1 1 1.1 even 1 trivial