# Properties

 Label 7098.2.a.p.1.1 Level $7098$ Weight $2$ Character 7098.1 Self dual yes Analytic conductor $56.678$ Analytic rank $0$ 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] = [7098,2,Mod(1,7098)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7098, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("7098.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$7098 = 2 \cdot 3 \cdot 7 \cdot 13^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 7098.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: $$56.6778153547$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 546) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 7098.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +3.00000 q^{5} -1.00000 q^{6} +1.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$$ $$q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +3.00000 q^{5} -1.00000 q^{6} +1.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} -3.00000 q^{10} -5.00000 q^{11} +1.00000 q^{12} -1.00000 q^{14} +3.00000 q^{15} +1.00000 q^{16} -3.00000 q^{17} -1.00000 q^{18} +1.00000 q^{19} +3.00000 q^{20} +1.00000 q^{21} +5.00000 q^{22} +1.00000 q^{23} -1.00000 q^{24} +4.00000 q^{25} +1.00000 q^{27} +1.00000 q^{28} +5.00000 q^{29} -3.00000 q^{30} -1.00000 q^{32} -5.00000 q^{33} +3.00000 q^{34} +3.00000 q^{35} +1.00000 q^{36} -7.00000 q^{37} -1.00000 q^{38} -3.00000 q^{40} -1.00000 q^{42} +1.00000 q^{43} -5.00000 q^{44} +3.00000 q^{45} -1.00000 q^{46} +8.00000 q^{47} +1.00000 q^{48} +1.00000 q^{49} -4.00000 q^{50} -3.00000 q^{51} +14.0000 q^{53} -1.00000 q^{54} -15.0000 q^{55} -1.00000 q^{56} +1.00000 q^{57} -5.00000 q^{58} +14.0000 q^{59} +3.00000 q^{60} -3.00000 q^{61} +1.00000 q^{63} +1.00000 q^{64} +5.00000 q^{66} -8.00000 q^{67} -3.00000 q^{68} +1.00000 q^{69} -3.00000 q^{70} +10.0000 q^{71} -1.00000 q^{72} +11.0000 q^{73} +7.00000 q^{74} +4.00000 q^{75} +1.00000 q^{76} -5.00000 q^{77} +3.00000 q^{80} +1.00000 q^{81} -6.00000 q^{83} +1.00000 q^{84} -9.00000 q^{85} -1.00000 q^{86} +5.00000 q^{87} +5.00000 q^{88} -16.0000 q^{89} -3.00000 q^{90} +1.00000 q^{92} -8.00000 q^{94} +3.00000 q^{95} -1.00000 q^{96} +2.00000 q^{97} -1.00000 q^{98} -5.00000 q^{99} +O(q^{100})$$

## 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
$$3$$ 1.00000 0.577350
$$4$$ 1.00000 0.500000
$$5$$ 3.00000 1.34164 0.670820 0.741620i $$-0.265942\pi$$
0.670820 + 0.741620i $$0.265942\pi$$
$$6$$ −1.00000 −0.408248
$$7$$ 1.00000 0.377964
$$8$$ −1.00000 −0.353553
$$9$$ 1.00000 0.333333
$$10$$ −3.00000 −0.948683
$$11$$ −5.00000 −1.50756 −0.753778 0.657129i $$-0.771771\pi$$
−0.753778 + 0.657129i $$0.771771\pi$$
$$12$$ 1.00000 0.288675
$$13$$ 0 0
$$14$$ −1.00000 −0.267261
$$15$$ 3.00000 0.774597
$$16$$ 1.00000 0.250000
$$17$$ −3.00000 −0.727607 −0.363803 0.931476i $$-0.618522\pi$$
−0.363803 + 0.931476i $$0.618522\pi$$
$$18$$ −1.00000 −0.235702
$$19$$ 1.00000 0.229416 0.114708 0.993399i $$-0.463407\pi$$
0.114708 + 0.993399i $$0.463407\pi$$
$$20$$ 3.00000 0.670820
$$21$$ 1.00000 0.218218
$$22$$ 5.00000 1.06600
$$23$$ 1.00000 0.208514 0.104257 0.994550i $$-0.466753\pi$$
0.104257 + 0.994550i $$0.466753\pi$$
$$24$$ −1.00000 −0.204124
$$25$$ 4.00000 0.800000
$$26$$ 0 0
$$27$$ 1.00000 0.192450
$$28$$ 1.00000 0.188982
$$29$$ 5.00000 0.928477 0.464238 0.885710i $$-0.346328\pi$$
0.464238 + 0.885710i $$0.346328\pi$$
$$30$$ −3.00000 −0.547723
$$31$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$32$$ −1.00000 −0.176777
$$33$$ −5.00000 −0.870388
$$34$$ 3.00000 0.514496
$$35$$ 3.00000 0.507093
$$36$$ 1.00000 0.166667
$$37$$ −7.00000 −1.15079 −0.575396 0.817875i $$-0.695152\pi$$
−0.575396 + 0.817875i $$0.695152\pi$$
$$38$$ −1.00000 −0.162221
$$39$$ 0 0
$$40$$ −3.00000 −0.474342
$$41$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$42$$ −1.00000 −0.154303
$$43$$ 1.00000 0.152499 0.0762493 0.997089i $$-0.475706\pi$$
0.0762493 + 0.997089i $$0.475706\pi$$
$$44$$ −5.00000 −0.753778
$$45$$ 3.00000 0.447214
$$46$$ −1.00000 −0.147442
$$47$$ 8.00000 1.16692 0.583460 0.812142i $$-0.301699\pi$$
0.583460 + 0.812142i $$0.301699\pi$$
$$48$$ 1.00000 0.144338
$$49$$ 1.00000 0.142857
$$50$$ −4.00000 −0.565685
$$51$$ −3.00000 −0.420084
$$52$$ 0 0
$$53$$ 14.0000 1.92305 0.961524 0.274721i $$-0.0885855\pi$$
0.961524 + 0.274721i $$0.0885855\pi$$
$$54$$ −1.00000 −0.136083
$$55$$ −15.0000 −2.02260
$$56$$ −1.00000 −0.133631
$$57$$ 1.00000 0.132453
$$58$$ −5.00000 −0.656532
$$59$$ 14.0000 1.82264 0.911322 0.411693i $$-0.135063\pi$$
0.911322 + 0.411693i $$0.135063\pi$$
$$60$$ 3.00000 0.387298
$$61$$ −3.00000 −0.384111 −0.192055 0.981384i $$-0.561515\pi$$
−0.192055 + 0.981384i $$0.561515\pi$$
$$62$$ 0 0
$$63$$ 1.00000 0.125988
$$64$$ 1.00000 0.125000
$$65$$ 0 0
$$66$$ 5.00000 0.615457
$$67$$ −8.00000 −0.977356 −0.488678 0.872464i $$-0.662521\pi$$
−0.488678 + 0.872464i $$0.662521\pi$$
$$68$$ −3.00000 −0.363803
$$69$$ 1.00000 0.120386
$$70$$ −3.00000 −0.358569
$$71$$ 10.0000 1.18678 0.593391 0.804914i $$-0.297789\pi$$
0.593391 + 0.804914i $$0.297789\pi$$
$$72$$ −1.00000 −0.117851
$$73$$ 11.0000 1.28745 0.643726 0.765256i $$-0.277388\pi$$
0.643726 + 0.765256i $$0.277388\pi$$
$$74$$ 7.00000 0.813733
$$75$$ 4.00000 0.461880
$$76$$ 1.00000 0.114708
$$77$$ −5.00000 −0.569803
$$78$$ 0 0
$$79$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$80$$ 3.00000 0.335410
$$81$$ 1.00000 0.111111
$$82$$ 0 0
$$83$$ −6.00000 −0.658586 −0.329293 0.944228i $$-0.606810\pi$$
−0.329293 + 0.944228i $$0.606810\pi$$
$$84$$ 1.00000 0.109109
$$85$$ −9.00000 −0.976187
$$86$$ −1.00000 −0.107833
$$87$$ 5.00000 0.536056
$$88$$ 5.00000 0.533002
$$89$$ −16.0000 −1.69600 −0.847998 0.529999i $$-0.822192\pi$$
−0.847998 + 0.529999i $$0.822192\pi$$
$$90$$ −3.00000 −0.316228
$$91$$ 0 0
$$92$$ 1.00000 0.104257
$$93$$ 0 0
$$94$$ −8.00000 −0.825137
$$95$$ 3.00000 0.307794
$$96$$ −1.00000 −0.102062
$$97$$ 2.00000 0.203069 0.101535 0.994832i $$-0.467625\pi$$
0.101535 + 0.994832i $$0.467625\pi$$
$$98$$ −1.00000 −0.101015
$$99$$ −5.00000 −0.502519
$$100$$ 4.00000 0.400000
$$101$$ 18.0000 1.79107 0.895533 0.444994i $$-0.146794\pi$$
0.895533 + 0.444994i $$0.146794\pi$$
$$102$$ 3.00000 0.297044
$$103$$ 11.0000 1.08386 0.541931 0.840423i $$-0.317693\pi$$
0.541931 + 0.840423i $$0.317693\pi$$
$$104$$ 0 0
$$105$$ 3.00000 0.292770
$$106$$ −14.0000 −1.35980
$$107$$ 18.0000 1.74013 0.870063 0.492941i $$-0.164078\pi$$
0.870063 + 0.492941i $$0.164078\pi$$
$$108$$ 1.00000 0.0962250
$$109$$ −9.00000 −0.862044 −0.431022 0.902342i $$-0.641847\pi$$
−0.431022 + 0.902342i $$0.641847\pi$$
$$110$$ 15.0000 1.43019
$$111$$ −7.00000 −0.664411
$$112$$ 1.00000 0.0944911
$$113$$ −6.00000 −0.564433 −0.282216 0.959351i $$-0.591070\pi$$
−0.282216 + 0.959351i $$0.591070\pi$$
$$114$$ −1.00000 −0.0936586
$$115$$ 3.00000 0.279751
$$116$$ 5.00000 0.464238
$$117$$ 0 0
$$118$$ −14.0000 −1.28880
$$119$$ −3.00000 −0.275010
$$120$$ −3.00000 −0.273861
$$121$$ 14.0000 1.27273
$$122$$ 3.00000 0.271607
$$123$$ 0 0
$$124$$ 0 0
$$125$$ −3.00000 −0.268328
$$126$$ −1.00000 −0.0890871
$$127$$ 12.0000 1.06483 0.532414 0.846484i $$-0.321285\pi$$
0.532414 + 0.846484i $$0.321285\pi$$
$$128$$ −1.00000 −0.0883883
$$129$$ 1.00000 0.0880451
$$130$$ 0 0
$$131$$ 7.00000 0.611593 0.305796 0.952097i $$-0.401077\pi$$
0.305796 + 0.952097i $$0.401077\pi$$
$$132$$ −5.00000 −0.435194
$$133$$ 1.00000 0.0867110
$$134$$ 8.00000 0.691095
$$135$$ 3.00000 0.258199
$$136$$ 3.00000 0.257248
$$137$$ 3.00000 0.256307 0.128154 0.991754i $$-0.459095\pi$$
0.128154 + 0.991754i $$0.459095\pi$$
$$138$$ −1.00000 −0.0851257
$$139$$ 20.0000 1.69638 0.848189 0.529694i $$-0.177693\pi$$
0.848189 + 0.529694i $$0.177693\pi$$
$$140$$ 3.00000 0.253546
$$141$$ 8.00000 0.673722
$$142$$ −10.0000 −0.839181
$$143$$ 0 0
$$144$$ 1.00000 0.0833333
$$145$$ 15.0000 1.24568
$$146$$ −11.0000 −0.910366
$$147$$ 1.00000 0.0824786
$$148$$ −7.00000 −0.575396
$$149$$ 6.00000 0.491539 0.245770 0.969328i $$-0.420959\pi$$
0.245770 + 0.969328i $$0.420959\pi$$
$$150$$ −4.00000 −0.326599
$$151$$ 15.0000 1.22068 0.610341 0.792139i $$-0.291032\pi$$
0.610341 + 0.792139i $$0.291032\pi$$
$$152$$ −1.00000 −0.0811107
$$153$$ −3.00000 −0.242536
$$154$$ 5.00000 0.402911
$$155$$ 0 0
$$156$$ 0 0
$$157$$ 13.0000 1.03751 0.518756 0.854922i $$-0.326395\pi$$
0.518756 + 0.854922i $$0.326395\pi$$
$$158$$ 0 0
$$159$$ 14.0000 1.11027
$$160$$ −3.00000 −0.237171
$$161$$ 1.00000 0.0788110
$$162$$ −1.00000 −0.0785674
$$163$$ −4.00000 −0.313304 −0.156652 0.987654i $$-0.550070\pi$$
−0.156652 + 0.987654i $$0.550070\pi$$
$$164$$ 0 0
$$165$$ −15.0000 −1.16775
$$166$$ 6.00000 0.465690
$$167$$ −7.00000 −0.541676 −0.270838 0.962625i $$-0.587301\pi$$
−0.270838 + 0.962625i $$0.587301\pi$$
$$168$$ −1.00000 −0.0771517
$$169$$ 0 0
$$170$$ 9.00000 0.690268
$$171$$ 1.00000 0.0764719
$$172$$ 1.00000 0.0762493
$$173$$ 26.0000 1.97674 0.988372 0.152057i $$-0.0485898\pi$$
0.988372 + 0.152057i $$0.0485898\pi$$
$$174$$ −5.00000 −0.379049
$$175$$ 4.00000 0.302372
$$176$$ −5.00000 −0.376889
$$177$$ 14.0000 1.05230
$$178$$ 16.0000 1.19925
$$179$$ −10.0000 −0.747435 −0.373718 0.927543i $$-0.621917\pi$$
−0.373718 + 0.927543i $$0.621917\pi$$
$$180$$ 3.00000 0.223607
$$181$$ −2.00000 −0.148659 −0.0743294 0.997234i $$-0.523682\pi$$
−0.0743294 + 0.997234i $$0.523682\pi$$
$$182$$ 0 0
$$183$$ −3.00000 −0.221766
$$184$$ −1.00000 −0.0737210
$$185$$ −21.0000 −1.54395
$$186$$ 0 0
$$187$$ 15.0000 1.09691
$$188$$ 8.00000 0.583460
$$189$$ 1.00000 0.0727393
$$190$$ −3.00000 −0.217643
$$191$$ 17.0000 1.23008 0.615038 0.788497i $$-0.289140\pi$$
0.615038 + 0.788497i $$0.289140\pi$$
$$192$$ 1.00000 0.0721688
$$193$$ 16.0000 1.15171 0.575853 0.817554i $$-0.304670\pi$$
0.575853 + 0.817554i $$0.304670\pi$$
$$194$$ −2.00000 −0.143592
$$195$$ 0 0
$$196$$ 1.00000 0.0714286
$$197$$ 2.00000 0.142494 0.0712470 0.997459i $$-0.477302\pi$$
0.0712470 + 0.997459i $$0.477302\pi$$
$$198$$ 5.00000 0.355335
$$199$$ 5.00000 0.354441 0.177220 0.984171i $$-0.443289\pi$$
0.177220 + 0.984171i $$0.443289\pi$$
$$200$$ −4.00000 −0.282843
$$201$$ −8.00000 −0.564276
$$202$$ −18.0000 −1.26648
$$203$$ 5.00000 0.350931
$$204$$ −3.00000 −0.210042
$$205$$ 0 0
$$206$$ −11.0000 −0.766406
$$207$$ 1.00000 0.0695048
$$208$$ 0 0
$$209$$ −5.00000 −0.345857
$$210$$ −3.00000 −0.207020
$$211$$ −13.0000 −0.894957 −0.447478 0.894295i $$-0.647678\pi$$
−0.447478 + 0.894295i $$0.647678\pi$$
$$212$$ 14.0000 0.961524
$$213$$ 10.0000 0.685189
$$214$$ −18.0000 −1.23045
$$215$$ 3.00000 0.204598
$$216$$ −1.00000 −0.0680414
$$217$$ 0 0
$$218$$ 9.00000 0.609557
$$219$$ 11.0000 0.743311
$$220$$ −15.0000 −1.01130
$$221$$ 0 0
$$222$$ 7.00000 0.469809
$$223$$ −16.0000 −1.07144 −0.535720 0.844396i $$-0.679960\pi$$
−0.535720 + 0.844396i $$0.679960\pi$$
$$224$$ −1.00000 −0.0668153
$$225$$ 4.00000 0.266667
$$226$$ 6.00000 0.399114
$$227$$ −8.00000 −0.530979 −0.265489 0.964114i $$-0.585534\pi$$
−0.265489 + 0.964114i $$0.585534\pi$$
$$228$$ 1.00000 0.0662266
$$229$$ −26.0000 −1.71813 −0.859064 0.511868i $$-0.828954\pi$$
−0.859064 + 0.511868i $$0.828954\pi$$
$$230$$ −3.00000 −0.197814
$$231$$ −5.00000 −0.328976
$$232$$ −5.00000 −0.328266
$$233$$ −14.0000 −0.917170 −0.458585 0.888650i $$-0.651644\pi$$
−0.458585 + 0.888650i $$0.651644\pi$$
$$234$$ 0 0
$$235$$ 24.0000 1.56559
$$236$$ 14.0000 0.911322
$$237$$ 0 0
$$238$$ 3.00000 0.194461
$$239$$ −4.00000 −0.258738 −0.129369 0.991596i $$-0.541295\pi$$
−0.129369 + 0.991596i $$0.541295\pi$$
$$240$$ 3.00000 0.193649
$$241$$ 10.0000 0.644157 0.322078 0.946713i $$-0.395619\pi$$
0.322078 + 0.946713i $$0.395619\pi$$
$$242$$ −14.0000 −0.899954
$$243$$ 1.00000 0.0641500
$$244$$ −3.00000 −0.192055
$$245$$ 3.00000 0.191663
$$246$$ 0 0
$$247$$ 0 0
$$248$$ 0 0
$$249$$ −6.00000 −0.380235
$$250$$ 3.00000 0.189737
$$251$$ −17.0000 −1.07303 −0.536515 0.843891i $$-0.680260\pi$$
−0.536515 + 0.843891i $$0.680260\pi$$
$$252$$ 1.00000 0.0629941
$$253$$ −5.00000 −0.314347
$$254$$ −12.0000 −0.752947
$$255$$ −9.00000 −0.563602
$$256$$ 1.00000 0.0625000
$$257$$ −18.0000 −1.12281 −0.561405 0.827541i $$-0.689739\pi$$
−0.561405 + 0.827541i $$0.689739\pi$$
$$258$$ −1.00000 −0.0622573
$$259$$ −7.00000 −0.434959
$$260$$ 0 0
$$261$$ 5.00000 0.309492
$$262$$ −7.00000 −0.432461
$$263$$ 24.0000 1.47990 0.739952 0.672660i $$-0.234848\pi$$
0.739952 + 0.672660i $$0.234848\pi$$
$$264$$ 5.00000 0.307729
$$265$$ 42.0000 2.58004
$$266$$ −1.00000 −0.0613139
$$267$$ −16.0000 −0.979184
$$268$$ −8.00000 −0.488678
$$269$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$270$$ −3.00000 −0.182574
$$271$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$272$$ −3.00000 −0.181902
$$273$$ 0 0
$$274$$ −3.00000 −0.181237
$$275$$ −20.0000 −1.20605
$$276$$ 1.00000 0.0601929
$$277$$ −28.0000 −1.68236 −0.841178 0.540758i $$-0.818138\pi$$
−0.841178 + 0.540758i $$0.818138\pi$$
$$278$$ −20.0000 −1.19952
$$279$$ 0 0
$$280$$ −3.00000 −0.179284
$$281$$ −10.0000 −0.596550 −0.298275 0.954480i $$-0.596411\pi$$
−0.298275 + 0.954480i $$0.596411\pi$$
$$282$$ −8.00000 −0.476393
$$283$$ −14.0000 −0.832214 −0.416107 0.909316i $$-0.636606\pi$$
−0.416107 + 0.909316i $$0.636606\pi$$
$$284$$ 10.0000 0.593391
$$285$$ 3.00000 0.177705
$$286$$ 0 0
$$287$$ 0 0
$$288$$ −1.00000 −0.0589256
$$289$$ −8.00000 −0.470588
$$290$$ −15.0000 −0.880830
$$291$$ 2.00000 0.117242
$$292$$ 11.0000 0.643726
$$293$$ 6.00000 0.350524 0.175262 0.984522i $$-0.443923\pi$$
0.175262 + 0.984522i $$0.443923\pi$$
$$294$$ −1.00000 −0.0583212
$$295$$ 42.0000 2.44533
$$296$$ 7.00000 0.406867
$$297$$ −5.00000 −0.290129
$$298$$ −6.00000 −0.347571
$$299$$ 0 0
$$300$$ 4.00000 0.230940
$$301$$ 1.00000 0.0576390
$$302$$ −15.0000 −0.863153
$$303$$ 18.0000 1.03407
$$304$$ 1.00000 0.0573539
$$305$$ −9.00000 −0.515339
$$306$$ 3.00000 0.171499
$$307$$ 8.00000 0.456584 0.228292 0.973593i $$-0.426686\pi$$
0.228292 + 0.973593i $$0.426686\pi$$
$$308$$ −5.00000 −0.284901
$$309$$ 11.0000 0.625768
$$310$$ 0 0
$$311$$ 8.00000 0.453638 0.226819 0.973937i $$-0.427167\pi$$
0.226819 + 0.973937i $$0.427167\pi$$
$$312$$ 0 0
$$313$$ −6.00000 −0.339140 −0.169570 0.985518i $$-0.554238\pi$$
−0.169570 + 0.985518i $$0.554238\pi$$
$$314$$ −13.0000 −0.733632
$$315$$ 3.00000 0.169031
$$316$$ 0 0
$$317$$ −8.00000 −0.449325 −0.224662 0.974437i $$-0.572128\pi$$
−0.224662 + 0.974437i $$0.572128\pi$$
$$318$$ −14.0000 −0.785081
$$319$$ −25.0000 −1.39973
$$320$$ 3.00000 0.167705
$$321$$ 18.0000 1.00466
$$322$$ −1.00000 −0.0557278
$$323$$ −3.00000 −0.166924
$$324$$ 1.00000 0.0555556
$$325$$ 0 0
$$326$$ 4.00000 0.221540
$$327$$ −9.00000 −0.497701
$$328$$ 0 0
$$329$$ 8.00000 0.441054
$$330$$ 15.0000 0.825723
$$331$$ −20.0000 −1.09930 −0.549650 0.835395i $$-0.685239\pi$$
−0.549650 + 0.835395i $$0.685239\pi$$
$$332$$ −6.00000 −0.329293
$$333$$ −7.00000 −0.383598
$$334$$ 7.00000 0.383023
$$335$$ −24.0000 −1.31126
$$336$$ 1.00000 0.0545545
$$337$$ −33.0000 −1.79762 −0.898812 0.438334i $$-0.855569\pi$$
−0.898812 + 0.438334i $$0.855569\pi$$
$$338$$ 0 0
$$339$$ −6.00000 −0.325875
$$340$$ −9.00000 −0.488094
$$341$$ 0 0
$$342$$ −1.00000 −0.0540738
$$343$$ 1.00000 0.0539949
$$344$$ −1.00000 −0.0539164
$$345$$ 3.00000 0.161515
$$346$$ −26.0000 −1.39777
$$347$$ 28.0000 1.50312 0.751559 0.659665i $$-0.229302\pi$$
0.751559 + 0.659665i $$0.229302\pi$$
$$348$$ 5.00000 0.268028
$$349$$ −16.0000 −0.856460 −0.428230 0.903670i $$-0.640863\pi$$
−0.428230 + 0.903670i $$0.640863\pi$$
$$350$$ −4.00000 −0.213809
$$351$$ 0 0
$$352$$ 5.00000 0.266501
$$353$$ 24.0000 1.27739 0.638696 0.769460i $$-0.279474\pi$$
0.638696 + 0.769460i $$0.279474\pi$$
$$354$$ −14.0000 −0.744092
$$355$$ 30.0000 1.59223
$$356$$ −16.0000 −0.847998
$$357$$ −3.00000 −0.158777
$$358$$ 10.0000 0.528516
$$359$$ 34.0000 1.79445 0.897226 0.441572i $$-0.145579\pi$$
0.897226 + 0.441572i $$0.145579\pi$$
$$360$$ −3.00000 −0.158114
$$361$$ −18.0000 −0.947368
$$362$$ 2.00000 0.105118
$$363$$ 14.0000 0.734809
$$364$$ 0 0
$$365$$ 33.0000 1.72730
$$366$$ 3.00000 0.156813
$$367$$ −32.0000 −1.67039 −0.835193 0.549957i $$-0.814644\pi$$
−0.835193 + 0.549957i $$0.814644\pi$$
$$368$$ 1.00000 0.0521286
$$369$$ 0 0
$$370$$ 21.0000 1.09174
$$371$$ 14.0000 0.726844
$$372$$ 0 0
$$373$$ −26.0000 −1.34623 −0.673114 0.739538i $$-0.735044\pi$$
−0.673114 + 0.739538i $$0.735044\pi$$
$$374$$ −15.0000 −0.775632
$$375$$ −3.00000 −0.154919
$$376$$ −8.00000 −0.412568
$$377$$ 0 0
$$378$$ −1.00000 −0.0514344
$$379$$ −4.00000 −0.205466 −0.102733 0.994709i $$-0.532759\pi$$
−0.102733 + 0.994709i $$0.532759\pi$$
$$380$$ 3.00000 0.153897
$$381$$ 12.0000 0.614779
$$382$$ −17.0000 −0.869796
$$383$$ −31.0000 −1.58403 −0.792013 0.610504i $$-0.790967\pi$$
−0.792013 + 0.610504i $$0.790967\pi$$
$$384$$ −1.00000 −0.0510310
$$385$$ −15.0000 −0.764471
$$386$$ −16.0000 −0.814379
$$387$$ 1.00000 0.0508329
$$388$$ 2.00000 0.101535
$$389$$ 30.0000 1.52106 0.760530 0.649303i $$-0.224939\pi$$
0.760530 + 0.649303i $$0.224939\pi$$
$$390$$ 0 0
$$391$$ −3.00000 −0.151717
$$392$$ −1.00000 −0.0505076
$$393$$ 7.00000 0.353103
$$394$$ −2.00000 −0.100759
$$395$$ 0 0
$$396$$ −5.00000 −0.251259
$$397$$ −12.0000 −0.602263 −0.301131 0.953583i $$-0.597364\pi$$
−0.301131 + 0.953583i $$0.597364\pi$$
$$398$$ −5.00000 −0.250627
$$399$$ 1.00000 0.0500626
$$400$$ 4.00000 0.200000
$$401$$ −30.0000 −1.49813 −0.749064 0.662497i $$-0.769497\pi$$
−0.749064 + 0.662497i $$0.769497\pi$$
$$402$$ 8.00000 0.399004
$$403$$ 0 0
$$404$$ 18.0000 0.895533
$$405$$ 3.00000 0.149071
$$406$$ −5.00000 −0.248146
$$407$$ 35.0000 1.73489
$$408$$ 3.00000 0.148522
$$409$$ −19.0000 −0.939490 −0.469745 0.882802i $$-0.655654\pi$$
−0.469745 + 0.882802i $$0.655654\pi$$
$$410$$ 0 0
$$411$$ 3.00000 0.147979
$$412$$ 11.0000 0.541931
$$413$$ 14.0000 0.688895
$$414$$ −1.00000 −0.0491473
$$415$$ −18.0000 −0.883585
$$416$$ 0 0
$$417$$ 20.0000 0.979404
$$418$$ 5.00000 0.244558
$$419$$ −35.0000 −1.70986 −0.854931 0.518742i $$-0.826401\pi$$
−0.854931 + 0.518742i $$0.826401\pi$$
$$420$$ 3.00000 0.146385
$$421$$ 30.0000 1.46211 0.731055 0.682318i $$-0.239028\pi$$
0.731055 + 0.682318i $$0.239028\pi$$
$$422$$ 13.0000 0.632830
$$423$$ 8.00000 0.388973
$$424$$ −14.0000 −0.679900
$$425$$ −12.0000 −0.582086
$$426$$ −10.0000 −0.484502
$$427$$ −3.00000 −0.145180
$$428$$ 18.0000 0.870063
$$429$$ 0 0
$$430$$ −3.00000 −0.144673
$$431$$ 30.0000 1.44505 0.722525 0.691345i $$-0.242982\pi$$
0.722525 + 0.691345i $$0.242982\pi$$
$$432$$ 1.00000 0.0481125
$$433$$ −4.00000 −0.192228 −0.0961139 0.995370i $$-0.530641\pi$$
−0.0961139 + 0.995370i $$0.530641\pi$$
$$434$$ 0 0
$$435$$ 15.0000 0.719195
$$436$$ −9.00000 −0.431022
$$437$$ 1.00000 0.0478365
$$438$$ −11.0000 −0.525600
$$439$$ 15.0000 0.715911 0.357955 0.933739i $$-0.383474\pi$$
0.357955 + 0.933739i $$0.383474\pi$$
$$440$$ 15.0000 0.715097
$$441$$ 1.00000 0.0476190
$$442$$ 0 0
$$443$$ −6.00000 −0.285069 −0.142534 0.989790i $$-0.545525\pi$$
−0.142534 + 0.989790i $$0.545525\pi$$
$$444$$ −7.00000 −0.332205
$$445$$ −48.0000 −2.27542
$$446$$ 16.0000 0.757622
$$447$$ 6.00000 0.283790
$$448$$ 1.00000 0.0472456
$$449$$ −21.0000 −0.991051 −0.495526 0.868593i $$-0.665025\pi$$
−0.495526 + 0.868593i $$0.665025\pi$$
$$450$$ −4.00000 −0.188562
$$451$$ 0 0
$$452$$ −6.00000 −0.282216
$$453$$ 15.0000 0.704761
$$454$$ 8.00000 0.375459
$$455$$ 0 0
$$456$$ −1.00000 −0.0468293
$$457$$ −8.00000 −0.374224 −0.187112 0.982339i $$-0.559913\pi$$
−0.187112 + 0.982339i $$0.559913\pi$$
$$458$$ 26.0000 1.21490
$$459$$ −3.00000 −0.140028
$$460$$ 3.00000 0.139876
$$461$$ −15.0000 −0.698620 −0.349310 0.937007i $$-0.613584\pi$$
−0.349310 + 0.937007i $$0.613584\pi$$
$$462$$ 5.00000 0.232621
$$463$$ 11.0000 0.511213 0.255607 0.966781i $$-0.417725\pi$$
0.255607 + 0.966781i $$0.417725\pi$$
$$464$$ 5.00000 0.232119
$$465$$ 0 0
$$466$$ 14.0000 0.648537
$$467$$ 7.00000 0.323921 0.161961 0.986797i $$-0.448218\pi$$
0.161961 + 0.986797i $$0.448218\pi$$
$$468$$ 0 0
$$469$$ −8.00000 −0.369406
$$470$$ −24.0000 −1.10704
$$471$$ 13.0000 0.599008
$$472$$ −14.0000 −0.644402
$$473$$ −5.00000 −0.229900
$$474$$ 0 0
$$475$$ 4.00000 0.183533
$$476$$ −3.00000 −0.137505
$$477$$ 14.0000 0.641016
$$478$$ 4.00000 0.182956
$$479$$ −21.0000 −0.959514 −0.479757 0.877401i $$-0.659275\pi$$
−0.479757 + 0.877401i $$0.659275\pi$$
$$480$$ −3.00000 −0.136931
$$481$$ 0 0
$$482$$ −10.0000 −0.455488
$$483$$ 1.00000 0.0455016
$$484$$ 14.0000 0.636364
$$485$$ 6.00000 0.272446
$$486$$ −1.00000 −0.0453609
$$487$$ 32.0000 1.45006 0.725029 0.688718i $$-0.241826\pi$$
0.725029 + 0.688718i $$0.241826\pi$$
$$488$$ 3.00000 0.135804
$$489$$ −4.00000 −0.180886
$$490$$ −3.00000 −0.135526
$$491$$ 28.0000 1.26362 0.631811 0.775122i $$-0.282312\pi$$
0.631811 + 0.775122i $$0.282312\pi$$
$$492$$ 0 0
$$493$$ −15.0000 −0.675566
$$494$$ 0 0
$$495$$ −15.0000 −0.674200
$$496$$ 0 0
$$497$$ 10.0000 0.448561
$$498$$ 6.00000 0.268866
$$499$$ −14.0000 −0.626726 −0.313363 0.949633i $$-0.601456\pi$$
−0.313363 + 0.949633i $$0.601456\pi$$
$$500$$ −3.00000 −0.134164
$$501$$ −7.00000 −0.312737
$$502$$ 17.0000 0.758747
$$503$$ −6.00000 −0.267527 −0.133763 0.991013i $$-0.542706\pi$$
−0.133763 + 0.991013i $$0.542706\pi$$
$$504$$ −1.00000 −0.0445435
$$505$$ 54.0000 2.40297
$$506$$ 5.00000 0.222277
$$507$$ 0 0
$$508$$ 12.0000 0.532414
$$509$$ 1.00000 0.0443242 0.0221621 0.999754i $$-0.492945\pi$$
0.0221621 + 0.999754i $$0.492945\pi$$
$$510$$ 9.00000 0.398527
$$511$$ 11.0000 0.486611
$$512$$ −1.00000 −0.0441942
$$513$$ 1.00000 0.0441511
$$514$$ 18.0000 0.793946
$$515$$ 33.0000 1.45415
$$516$$ 1.00000 0.0440225
$$517$$ −40.0000 −1.75920
$$518$$ 7.00000 0.307562
$$519$$ 26.0000 1.14127
$$520$$ 0 0
$$521$$ 27.0000 1.18289 0.591446 0.806345i $$-0.298557\pi$$
0.591446 + 0.806345i $$0.298557\pi$$
$$522$$ −5.00000 −0.218844
$$523$$ −16.0000 −0.699631 −0.349816 0.936819i $$-0.613756\pi$$
−0.349816 + 0.936819i $$0.613756\pi$$
$$524$$ 7.00000 0.305796
$$525$$ 4.00000 0.174574
$$526$$ −24.0000 −1.04645
$$527$$ 0 0
$$528$$ −5.00000 −0.217597
$$529$$ −22.0000 −0.956522
$$530$$ −42.0000 −1.82436
$$531$$ 14.0000 0.607548
$$532$$ 1.00000 0.0433555
$$533$$ 0 0
$$534$$ 16.0000 0.692388
$$535$$ 54.0000 2.33462
$$536$$ 8.00000 0.345547
$$537$$ −10.0000 −0.431532
$$538$$ 0 0
$$539$$ −5.00000 −0.215365
$$540$$ 3.00000 0.129099
$$541$$ 15.0000 0.644900 0.322450 0.946586i $$-0.395494\pi$$
0.322450 + 0.946586i $$0.395494\pi$$
$$542$$ 0 0
$$543$$ −2.00000 −0.0858282
$$544$$ 3.00000 0.128624
$$545$$ −27.0000 −1.15655
$$546$$ 0 0
$$547$$ 28.0000 1.19719 0.598597 0.801050i $$-0.295725\pi$$
0.598597 + 0.801050i $$0.295725\pi$$
$$548$$ 3.00000 0.128154
$$549$$ −3.00000 −0.128037
$$550$$ 20.0000 0.852803
$$551$$ 5.00000 0.213007
$$552$$ −1.00000 −0.0425628
$$553$$ 0 0
$$554$$ 28.0000 1.18961
$$555$$ −21.0000 −0.891400
$$556$$ 20.0000 0.848189
$$557$$ −22.0000 −0.932170 −0.466085 0.884740i $$-0.654336\pi$$
−0.466085 + 0.884740i $$0.654336\pi$$
$$558$$ 0 0
$$559$$ 0 0
$$560$$ 3.00000 0.126773
$$561$$ 15.0000 0.633300
$$562$$ 10.0000 0.421825
$$563$$ −9.00000 −0.379305 −0.189652 0.981851i $$-0.560736\pi$$
−0.189652 + 0.981851i $$0.560736\pi$$
$$564$$ 8.00000 0.336861
$$565$$ −18.0000 −0.757266
$$566$$ 14.0000 0.588464
$$567$$ 1.00000 0.0419961
$$568$$ −10.0000 −0.419591
$$569$$ 30.0000 1.25767 0.628833 0.777541i $$-0.283533\pi$$
0.628833 + 0.777541i $$0.283533\pi$$
$$570$$ −3.00000 −0.125656
$$571$$ −32.0000 −1.33916 −0.669579 0.742741i $$-0.733526\pi$$
−0.669579 + 0.742741i $$0.733526\pi$$
$$572$$ 0 0
$$573$$ 17.0000 0.710185
$$574$$ 0 0
$$575$$ 4.00000 0.166812
$$576$$ 1.00000 0.0416667
$$577$$ −38.0000 −1.58196 −0.790980 0.611842i $$-0.790429\pi$$
−0.790980 + 0.611842i $$0.790429\pi$$
$$578$$ 8.00000 0.332756
$$579$$ 16.0000 0.664937
$$580$$ 15.0000 0.622841
$$581$$ −6.00000 −0.248922
$$582$$ −2.00000 −0.0829027
$$583$$ −70.0000 −2.89910
$$584$$ −11.0000 −0.455183
$$585$$ 0 0
$$586$$ −6.00000 −0.247858
$$587$$ −8.00000 −0.330195 −0.165098 0.986277i $$-0.552794\pi$$
−0.165098 + 0.986277i $$0.552794\pi$$
$$588$$ 1.00000 0.0412393
$$589$$ 0 0
$$590$$ −42.0000 −1.72911
$$591$$ 2.00000 0.0822690
$$592$$ −7.00000 −0.287698
$$593$$ −24.0000 −0.985562 −0.492781 0.870153i $$-0.664020\pi$$
−0.492781 + 0.870153i $$0.664020\pi$$
$$594$$ 5.00000 0.205152
$$595$$ −9.00000 −0.368964
$$596$$ 6.00000 0.245770
$$597$$ 5.00000 0.204636
$$598$$ 0 0
$$599$$ −45.0000 −1.83865 −0.919325 0.393499i $$-0.871265\pi$$
−0.919325 + 0.393499i $$0.871265\pi$$
$$600$$ −4.00000 −0.163299
$$601$$ −48.0000 −1.95796 −0.978980 0.203954i $$-0.934621\pi$$
−0.978980 + 0.203954i $$0.934621\pi$$
$$602$$ −1.00000 −0.0407570
$$603$$ −8.00000 −0.325785
$$604$$ 15.0000 0.610341
$$605$$ 42.0000 1.70754
$$606$$ −18.0000 −0.731200
$$607$$ 23.0000 0.933541 0.466771 0.884378i $$-0.345417\pi$$
0.466771 + 0.884378i $$0.345417\pi$$
$$608$$ −1.00000 −0.0405554
$$609$$ 5.00000 0.202610
$$610$$ 9.00000 0.364399
$$611$$ 0 0
$$612$$ −3.00000 −0.121268
$$613$$ 39.0000 1.57520 0.787598 0.616190i $$-0.211325\pi$$
0.787598 + 0.616190i $$0.211325\pi$$
$$614$$ −8.00000 −0.322854
$$615$$ 0 0
$$616$$ 5.00000 0.201456
$$617$$ 17.0000 0.684394 0.342197 0.939628i $$-0.388829\pi$$
0.342197 + 0.939628i $$0.388829\pi$$
$$618$$ −11.0000 −0.442485
$$619$$ −11.0000 −0.442127 −0.221064 0.975259i $$-0.570953\pi$$
−0.221064 + 0.975259i $$0.570953\pi$$
$$620$$ 0 0
$$621$$ 1.00000 0.0401286
$$622$$ −8.00000 −0.320771
$$623$$ −16.0000 −0.641026
$$624$$ 0 0
$$625$$ −29.0000 −1.16000
$$626$$ 6.00000 0.239808
$$627$$ −5.00000 −0.199681
$$628$$ 13.0000 0.518756
$$629$$ 21.0000 0.837325
$$630$$ −3.00000 −0.119523
$$631$$ 35.0000 1.39333 0.696664 0.717398i $$-0.254667\pi$$
0.696664 + 0.717398i $$0.254667\pi$$
$$632$$ 0 0
$$633$$ −13.0000 −0.516704
$$634$$ 8.00000 0.317721
$$635$$ 36.0000 1.42862
$$636$$ 14.0000 0.555136
$$637$$ 0 0
$$638$$ 25.0000 0.989759
$$639$$ 10.0000 0.395594
$$640$$ −3.00000 −0.118585
$$641$$ 8.00000 0.315981 0.157991 0.987441i $$-0.449498\pi$$
0.157991 + 0.987441i $$0.449498\pi$$
$$642$$ −18.0000 −0.710403
$$643$$ 19.0000 0.749287 0.374643 0.927169i $$-0.377765\pi$$
0.374643 + 0.927169i $$0.377765\pi$$
$$644$$ 1.00000 0.0394055
$$645$$ 3.00000 0.118125
$$646$$ 3.00000 0.118033
$$647$$ 2.00000 0.0786281 0.0393141 0.999227i $$-0.487483\pi$$
0.0393141 + 0.999227i $$0.487483\pi$$
$$648$$ −1.00000 −0.0392837
$$649$$ −70.0000 −2.74774
$$650$$ 0 0
$$651$$ 0 0
$$652$$ −4.00000 −0.156652
$$653$$ −1.00000 −0.0391330 −0.0195665 0.999809i $$-0.506229\pi$$
−0.0195665 + 0.999809i $$0.506229\pi$$
$$654$$ 9.00000 0.351928
$$655$$ 21.0000 0.820538
$$656$$ 0 0
$$657$$ 11.0000 0.429151
$$658$$ −8.00000 −0.311872
$$659$$ −30.0000 −1.16863 −0.584317 0.811525i $$-0.698638\pi$$
−0.584317 + 0.811525i $$0.698638\pi$$
$$660$$ −15.0000 −0.583874
$$661$$ 50.0000 1.94477 0.972387 0.233373i $$-0.0749763\pi$$
0.972387 + 0.233373i $$0.0749763\pi$$
$$662$$ 20.0000 0.777322
$$663$$ 0 0
$$664$$ 6.00000 0.232845
$$665$$ 3.00000 0.116335
$$666$$ 7.00000 0.271244
$$667$$ 5.00000 0.193601
$$668$$ −7.00000 −0.270838
$$669$$ −16.0000 −0.618596
$$670$$ 24.0000 0.927201
$$671$$ 15.0000 0.579069
$$672$$ −1.00000 −0.0385758
$$673$$ 11.0000 0.424019 0.212009 0.977268i $$-0.431999\pi$$
0.212009 + 0.977268i $$0.431999\pi$$
$$674$$ 33.0000 1.27111
$$675$$ 4.00000 0.153960
$$676$$ 0 0
$$677$$ 8.00000 0.307465 0.153732 0.988113i $$-0.450871\pi$$
0.153732 + 0.988113i $$0.450871\pi$$
$$678$$ 6.00000 0.230429
$$679$$ 2.00000 0.0767530
$$680$$ 9.00000 0.345134
$$681$$ −8.00000 −0.306561
$$682$$ 0 0
$$683$$ 1.00000 0.0382639 0.0191320 0.999817i $$-0.493910\pi$$
0.0191320 + 0.999817i $$0.493910\pi$$
$$684$$ 1.00000 0.0382360
$$685$$ 9.00000 0.343872
$$686$$ −1.00000 −0.0381802
$$687$$ −26.0000 −0.991962
$$688$$ 1.00000 0.0381246
$$689$$ 0 0
$$690$$ −3.00000 −0.114208
$$691$$ 40.0000 1.52167 0.760836 0.648944i $$-0.224789\pi$$
0.760836 + 0.648944i $$0.224789\pi$$
$$692$$ 26.0000 0.988372
$$693$$ −5.00000 −0.189934
$$694$$ −28.0000 −1.06287
$$695$$ 60.0000 2.27593
$$696$$ −5.00000 −0.189525
$$697$$ 0 0
$$698$$ 16.0000 0.605609
$$699$$ −14.0000 −0.529529
$$700$$ 4.00000 0.151186
$$701$$ −2.00000 −0.0755390 −0.0377695 0.999286i $$-0.512025\pi$$
−0.0377695 + 0.999286i $$0.512025\pi$$
$$702$$ 0 0
$$703$$ −7.00000 −0.264010
$$704$$ −5.00000 −0.188445
$$705$$ 24.0000 0.903892
$$706$$ −24.0000 −0.903252
$$707$$ 18.0000 0.676960
$$708$$ 14.0000 0.526152
$$709$$ −26.0000 −0.976450 −0.488225 0.872718i $$-0.662356\pi$$
−0.488225 + 0.872718i $$0.662356\pi$$
$$710$$ −30.0000 −1.12588
$$711$$ 0 0
$$712$$ 16.0000 0.599625
$$713$$ 0 0
$$714$$ 3.00000 0.112272
$$715$$ 0 0
$$716$$ −10.0000 −0.373718
$$717$$ −4.00000 −0.149383
$$718$$ −34.0000 −1.26887
$$719$$ 20.0000 0.745874 0.372937 0.927857i $$-0.378351\pi$$
0.372937 + 0.927857i $$0.378351\pi$$
$$720$$ 3.00000 0.111803
$$721$$ 11.0000 0.409661
$$722$$ 18.0000 0.669891
$$723$$ 10.0000 0.371904
$$724$$ −2.00000 −0.0743294
$$725$$ 20.0000 0.742781
$$726$$ −14.0000 −0.519589
$$727$$ 7.00000 0.259616 0.129808 0.991539i $$-0.458564\pi$$
0.129808 + 0.991539i $$0.458564\pi$$
$$728$$ 0 0
$$729$$ 1.00000 0.0370370
$$730$$ −33.0000 −1.22138
$$731$$ −3.00000 −0.110959
$$732$$ −3.00000 −0.110883
$$733$$ −36.0000 −1.32969 −0.664845 0.746981i $$-0.731502\pi$$
−0.664845 + 0.746981i $$0.731502\pi$$
$$734$$ 32.0000 1.18114
$$735$$ 3.00000 0.110657
$$736$$ −1.00000 −0.0368605
$$737$$ 40.0000 1.47342
$$738$$ 0 0
$$739$$ 24.0000 0.882854 0.441427 0.897297i $$-0.354472\pi$$
0.441427 + 0.897297i $$0.354472\pi$$
$$740$$ −21.0000 −0.771975
$$741$$ 0 0
$$742$$ −14.0000 −0.513956
$$743$$ 4.00000 0.146746 0.0733729 0.997305i $$-0.476624\pi$$
0.0733729 + 0.997305i $$0.476624\pi$$
$$744$$ 0 0
$$745$$ 18.0000 0.659469
$$746$$ 26.0000 0.951928
$$747$$ −6.00000 −0.219529
$$748$$ 15.0000 0.548454
$$749$$ 18.0000 0.657706
$$750$$ 3.00000 0.109545
$$751$$ 48.0000 1.75154 0.875772 0.482724i $$-0.160353\pi$$
0.875772 + 0.482724i $$0.160353\pi$$
$$752$$ 8.00000 0.291730
$$753$$ −17.0000 −0.619514
$$754$$ 0 0
$$755$$ 45.0000 1.63772
$$756$$ 1.00000 0.0363696
$$757$$ 18.0000 0.654221 0.327111 0.944986i $$-0.393925\pi$$
0.327111 + 0.944986i $$0.393925\pi$$
$$758$$ 4.00000 0.145287
$$759$$ −5.00000 −0.181489
$$760$$ −3.00000 −0.108821
$$761$$ 30.0000 1.08750 0.543750 0.839248i $$-0.317004\pi$$
0.543750 + 0.839248i $$0.317004\pi$$
$$762$$ −12.0000 −0.434714
$$763$$ −9.00000 −0.325822
$$764$$ 17.0000 0.615038
$$765$$ −9.00000 −0.325396
$$766$$ 31.0000 1.12008
$$767$$ 0 0
$$768$$ 1.00000 0.0360844
$$769$$ 1.00000 0.0360609 0.0180305 0.999837i $$-0.494260\pi$$
0.0180305 + 0.999837i $$0.494260\pi$$
$$770$$ 15.0000 0.540562
$$771$$ −18.0000 −0.648254
$$772$$ 16.0000 0.575853
$$773$$ 9.00000 0.323708 0.161854 0.986815i $$-0.448253\pi$$
0.161854 + 0.986815i $$0.448253\pi$$
$$774$$ −1.00000 −0.0359443
$$775$$ 0 0
$$776$$ −2.00000 −0.0717958
$$777$$ −7.00000 −0.251124
$$778$$ −30.0000 −1.07555
$$779$$ 0 0
$$780$$ 0 0
$$781$$ −50.0000 −1.78914
$$782$$ 3.00000 0.107280
$$783$$ 5.00000 0.178685
$$784$$ 1.00000 0.0357143
$$785$$ 39.0000 1.39197
$$786$$ −7.00000 −0.249682
$$787$$ 13.0000 0.463400 0.231700 0.972787i $$-0.425571\pi$$
0.231700 + 0.972787i $$0.425571\pi$$
$$788$$ 2.00000 0.0712470
$$789$$ 24.0000 0.854423
$$790$$ 0 0
$$791$$ −6.00000 −0.213335
$$792$$ 5.00000 0.177667
$$793$$ 0 0
$$794$$ 12.0000 0.425864
$$795$$ 42.0000 1.48959
$$796$$ 5.00000 0.177220
$$797$$ −18.0000 −0.637593 −0.318796 0.947823i $$-0.603279\pi$$
−0.318796 + 0.947823i $$0.603279\pi$$
$$798$$ −1.00000 −0.0353996
$$799$$ −24.0000 −0.849059
$$800$$ −4.00000 −0.141421
$$801$$ −16.0000 −0.565332
$$802$$ 30.0000 1.05934
$$803$$ −55.0000 −1.94091
$$804$$ −8.00000 −0.282138
$$805$$ 3.00000 0.105736
$$806$$ 0 0
$$807$$ 0 0
$$808$$ −18.0000 −0.633238
$$809$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$810$$ −3.00000 −0.105409
$$811$$ −35.0000 −1.22902 −0.614508 0.788911i $$-0.710645\pi$$
−0.614508 + 0.788911i $$0.710645\pi$$
$$812$$ 5.00000 0.175466
$$813$$ 0 0
$$814$$ −35.0000 −1.22675
$$815$$ −12.0000 −0.420342
$$816$$ −3.00000 −0.105021
$$817$$ 1.00000 0.0349856
$$818$$ 19.0000 0.664319
$$819$$ 0 0
$$820$$ 0 0
$$821$$ 20.0000 0.698005 0.349002 0.937122i $$-0.386521\pi$$
0.349002 + 0.937122i $$0.386521\pi$$
$$822$$ −3.00000 −0.104637
$$823$$ −44.0000 −1.53374 −0.766872 0.641800i $$-0.778188\pi$$
−0.766872 + 0.641800i $$0.778188\pi$$
$$824$$ −11.0000 −0.383203
$$825$$ −20.0000 −0.696311
$$826$$ −14.0000 −0.487122
$$827$$ 23.0000 0.799788 0.399894 0.916561i $$-0.369047\pi$$
0.399894 + 0.916561i $$0.369047\pi$$
$$828$$ 1.00000 0.0347524
$$829$$ 25.0000 0.868286 0.434143 0.900844i $$-0.357051\pi$$
0.434143 + 0.900844i $$0.357051\pi$$
$$830$$ 18.0000 0.624789
$$831$$ −28.0000 −0.971309
$$832$$ 0 0
$$833$$ −3.00000 −0.103944
$$834$$ −20.0000 −0.692543
$$835$$ −21.0000 −0.726735
$$836$$ −5.00000 −0.172929
$$837$$ 0 0
$$838$$ 35.0000 1.20905
$$839$$ −36.0000 −1.24286 −0.621429 0.783470i $$-0.713448\pi$$
−0.621429 + 0.783470i $$0.713448\pi$$
$$840$$ −3.00000 −0.103510
$$841$$ −4.00000 −0.137931
$$842$$ −30.0000 −1.03387
$$843$$ −10.0000 −0.344418
$$844$$ −13.0000 −0.447478
$$845$$ 0 0
$$846$$ −8.00000 −0.275046
$$847$$ 14.0000 0.481046
$$848$$ 14.0000 0.480762
$$849$$ −14.0000 −0.480479
$$850$$ 12.0000 0.411597
$$851$$ −7.00000 −0.239957
$$852$$ 10.0000 0.342594
$$853$$ 16.0000 0.547830 0.273915 0.961754i $$-0.411681\pi$$
0.273915 + 0.961754i $$0.411681\pi$$
$$854$$ 3.00000 0.102658
$$855$$ 3.00000 0.102598
$$856$$ −18.0000 −0.615227
$$857$$ 42.0000 1.43469 0.717346 0.696717i $$-0.245357\pi$$
0.717346 + 0.696717i $$0.245357\pi$$
$$858$$ 0 0
$$859$$ −50.0000 −1.70598 −0.852989 0.521929i $$-0.825213\pi$$
−0.852989 + 0.521929i $$0.825213\pi$$
$$860$$ 3.00000 0.102299
$$861$$ 0 0
$$862$$ −30.0000 −1.02180
$$863$$ 24.0000 0.816970 0.408485 0.912765i $$-0.366057\pi$$
0.408485 + 0.912765i $$0.366057\pi$$
$$864$$ −1.00000 −0.0340207
$$865$$ 78.0000 2.65208
$$866$$ 4.00000 0.135926
$$867$$ −8.00000 −0.271694
$$868$$ 0 0
$$869$$ 0 0
$$870$$ −15.0000 −0.508548
$$871$$ 0 0
$$872$$ 9.00000 0.304778
$$873$$ 2.00000 0.0676897
$$874$$ −1.00000 −0.0338255
$$875$$ −3.00000 −0.101419
$$876$$ 11.0000 0.371656
$$877$$ −38.0000 −1.28317 −0.641584 0.767052i $$-0.721723\pi$$
−0.641584 + 0.767052i $$0.721723\pi$$
$$878$$ −15.0000 −0.506225
$$879$$ 6.00000 0.202375
$$880$$ −15.0000 −0.505650
$$881$$ −7.00000 −0.235836 −0.117918 0.993023i $$-0.537622\pi$$
−0.117918 + 0.993023i $$0.537622\pi$$
$$882$$ −1.00000 −0.0336718
$$883$$ 41.0000 1.37976 0.689880 0.723924i $$-0.257663\pi$$
0.689880 + 0.723924i $$0.257663\pi$$
$$884$$ 0 0
$$885$$ 42.0000 1.41181
$$886$$ 6.00000 0.201574
$$887$$ −2.00000 −0.0671534 −0.0335767 0.999436i $$-0.510690\pi$$
−0.0335767 + 0.999436i $$0.510690\pi$$
$$888$$ 7.00000 0.234905
$$889$$ 12.0000 0.402467
$$890$$ 48.0000 1.60896
$$891$$ −5.00000 −0.167506
$$892$$ −16.0000 −0.535720
$$893$$ 8.00000 0.267710
$$894$$ −6.00000 −0.200670
$$895$$ −30.0000 −1.00279
$$896$$ −1.00000 −0.0334077
$$897$$ 0 0
$$898$$ 21.0000 0.700779
$$899$$ 0 0
$$900$$ 4.00000 0.133333
$$901$$ −42.0000 −1.39922
$$902$$ 0 0
$$903$$ 1.00000 0.0332779
$$904$$ 6.00000 0.199557
$$905$$ −6.00000 −0.199447
$$906$$ −15.0000 −0.498342
$$907$$ 12.0000 0.398453 0.199227 0.979953i $$-0.436157\pi$$
0.199227 + 0.979953i $$0.436157\pi$$
$$908$$ −8.00000 −0.265489
$$909$$ 18.0000 0.597022
$$910$$ 0 0
$$911$$ 47.0000 1.55718 0.778590 0.627533i $$-0.215935\pi$$
0.778590 + 0.627533i $$0.215935\pi$$
$$912$$ 1.00000 0.0331133
$$913$$ 30.0000 0.992855
$$914$$ 8.00000 0.264616
$$915$$ −9.00000 −0.297531
$$916$$ −26.0000 −0.859064
$$917$$ 7.00000 0.231160
$$918$$ 3.00000 0.0990148
$$919$$ 10.0000 0.329870 0.164935 0.986304i $$-0.447259\pi$$
0.164935 + 0.986304i $$0.447259\pi$$
$$920$$ −3.00000 −0.0989071
$$921$$ 8.00000 0.263609
$$922$$ 15.0000 0.493999
$$923$$ 0 0
$$924$$ −5.00000 −0.164488
$$925$$ −28.0000 −0.920634
$$926$$ −11.0000 −0.361482
$$927$$ 11.0000 0.361287
$$928$$ −5.00000 −0.164133
$$929$$ −54.0000 −1.77168 −0.885841 0.463988i $$-0.846418\pi$$
−0.885841 + 0.463988i $$0.846418\pi$$
$$930$$ 0 0
$$931$$ 1.00000 0.0327737
$$932$$ −14.0000 −0.458585
$$933$$ 8.00000 0.261908
$$934$$ −7.00000 −0.229047
$$935$$ 45.0000 1.47166
$$936$$ 0 0
$$937$$ −32.0000 −1.04539 −0.522697 0.852518i $$-0.675074\pi$$
−0.522697 + 0.852518i $$0.675074\pi$$
$$938$$ 8.00000 0.261209
$$939$$ −6.00000 −0.195803
$$940$$ 24.0000 0.782794
$$941$$ 30.0000 0.977972 0.488986 0.872292i $$-0.337367\pi$$
0.488986 + 0.872292i $$0.337367\pi$$
$$942$$ −13.0000 −0.423563
$$943$$ 0 0
$$944$$ 14.0000 0.455661
$$945$$ 3.00000 0.0975900
$$946$$ 5.00000 0.162564
$$947$$ −57.0000 −1.85225 −0.926126 0.377215i $$-0.876882\pi$$
−0.926126 + 0.377215i $$0.876882\pi$$
$$948$$ 0 0
$$949$$ 0 0
$$950$$ −4.00000 −0.129777
$$951$$ −8.00000 −0.259418
$$952$$ 3.00000 0.0972306
$$953$$ −54.0000 −1.74923 −0.874616 0.484817i $$-0.838886\pi$$
−0.874616 + 0.484817i $$0.838886\pi$$
$$954$$ −14.0000 −0.453267
$$955$$ 51.0000 1.65032
$$956$$ −4.00000 −0.129369
$$957$$ −25.0000 −0.808135
$$958$$ 21.0000 0.678479
$$959$$ 3.00000 0.0968751
$$960$$ 3.00000 0.0968246
$$961$$ −31.0000 −1.00000
$$962$$ 0 0
$$963$$ 18.0000 0.580042
$$964$$ 10.0000 0.322078
$$965$$ 48.0000 1.54517
$$966$$ −1.00000 −0.0321745
$$967$$ 17.0000 0.546683 0.273342 0.961917i $$-0.411871\pi$$
0.273342 + 0.961917i $$0.411871\pi$$
$$968$$ −14.0000 −0.449977
$$969$$ −3.00000 −0.0963739
$$970$$ −6.00000 −0.192648
$$971$$ −28.0000 −0.898563 −0.449281 0.893390i $$-0.648320\pi$$
−0.449281 + 0.893390i $$0.648320\pi$$
$$972$$ 1.00000 0.0320750
$$973$$ 20.0000 0.641171
$$974$$ −32.0000 −1.02535
$$975$$ 0 0
$$976$$ −3.00000 −0.0960277
$$977$$ −3.00000 −0.0959785 −0.0479893 0.998848i $$-0.515281\pi$$
−0.0479893 + 0.998848i $$0.515281\pi$$
$$978$$ 4.00000 0.127906
$$979$$ 80.0000 2.55681
$$980$$ 3.00000 0.0958315
$$981$$ −9.00000 −0.287348
$$982$$ −28.0000 −0.893516
$$983$$ 11.0000 0.350846 0.175423 0.984493i $$-0.443871\pi$$
0.175423 + 0.984493i $$0.443871\pi$$
$$984$$ 0 0
$$985$$ 6.00000 0.191176
$$986$$ 15.0000 0.477697
$$987$$ 8.00000 0.254643
$$988$$ 0 0
$$989$$ 1.00000 0.0317982
$$990$$ 15.0000 0.476731
$$991$$ −28.0000 −0.889449 −0.444725 0.895667i $$-0.646698\pi$$
−0.444725 + 0.895667i $$0.646698\pi$$
$$992$$ 0 0
$$993$$ −20.0000 −0.634681
$$994$$ −10.0000 −0.317181
$$995$$ 15.0000 0.475532
$$996$$ −6.00000 −0.190117
$$997$$ 38.0000 1.20347 0.601736 0.798695i $$-0.294476\pi$$
0.601736 + 0.798695i $$0.294476\pi$$
$$998$$ 14.0000 0.443162
$$999$$ −7.00000 −0.221470
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 7098.2.a.p.1.1 1
13.5 odd 4 546.2.c.d.337.2 yes 2
13.8 odd 4 546.2.c.d.337.1 2
13.12 even 2 7098.2.a.x.1.1 1
39.5 even 4 1638.2.c.g.883.1 2
39.8 even 4 1638.2.c.g.883.2 2
52.31 even 4 4368.2.h.b.337.1 2
52.47 even 4 4368.2.h.b.337.2 2
91.34 even 4 3822.2.c.a.883.1 2
91.83 even 4 3822.2.c.a.883.2 2

By twisted newform
Twist Min Dim Char Parity Ord Type
546.2.c.d.337.1 2 13.8 odd 4
546.2.c.d.337.2 yes 2 13.5 odd 4
1638.2.c.g.883.1 2 39.5 even 4
1638.2.c.g.883.2 2 39.8 even 4
3822.2.c.a.883.1 2 91.34 even 4
3822.2.c.a.883.2 2 91.83 even 4
4368.2.h.b.337.1 2 52.31 even 4
4368.2.h.b.337.2 2 52.47 even 4
7098.2.a.p.1.1 1 1.1 even 1 trivial
7098.2.a.x.1.1 1 13.12 even 2