Properties

 Label 3234.2.a.a.1.1 Level $3234$ Weight $2$ Character 3234.1 Self dual yes Analytic conductor $25.824$ 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] = [3234,2,Mod(1,3234)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3234, 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("3234.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$3234 = 2 \cdot 3 \cdot 7^{2} \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3234.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: $$25.8236200137$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 462) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 3234.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -3.00000 q^{5} +1.00000 q^{6} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$$ $$q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -3.00000 q^{5} +1.00000 q^{6} -1.00000 q^{8} +1.00000 q^{9} +3.00000 q^{10} -1.00000 q^{11} -1.00000 q^{12} -6.00000 q^{13} +3.00000 q^{15} +1.00000 q^{16} +5.00000 q^{17} -1.00000 q^{18} -6.00000 q^{19} -3.00000 q^{20} +1.00000 q^{22} +5.00000 q^{23} +1.00000 q^{24} +4.00000 q^{25} +6.00000 q^{26} -1.00000 q^{27} -6.00000 q^{29} -3.00000 q^{30} -4.00000 q^{31} -1.00000 q^{32} +1.00000 q^{33} -5.00000 q^{34} +1.00000 q^{36} -2.00000 q^{37} +6.00000 q^{38} +6.00000 q^{39} +3.00000 q^{40} -5.00000 q^{41} -10.0000 q^{43} -1.00000 q^{44} -3.00000 q^{45} -5.00000 q^{46} -9.00000 q^{47} -1.00000 q^{48} -4.00000 q^{50} -5.00000 q^{51} -6.00000 q^{52} +2.00000 q^{53} +1.00000 q^{54} +3.00000 q^{55} +6.00000 q^{57} +6.00000 q^{58} +12.0000 q^{59} +3.00000 q^{60} +5.00000 q^{61} +4.00000 q^{62} +1.00000 q^{64} +18.0000 q^{65} -1.00000 q^{66} +5.00000 q^{67} +5.00000 q^{68} -5.00000 q^{69} +4.00000 q^{71} -1.00000 q^{72} -12.0000 q^{73} +2.00000 q^{74} -4.00000 q^{75} -6.00000 q^{76} -6.00000 q^{78} -1.00000 q^{79} -3.00000 q^{80} +1.00000 q^{81} +5.00000 q^{82} -1.00000 q^{83} -15.0000 q^{85} +10.0000 q^{86} +6.00000 q^{87} +1.00000 q^{88} -6.00000 q^{89} +3.00000 q^{90} +5.00000 q^{92} +4.00000 q^{93} +9.00000 q^{94} +18.0000 q^{95} +1.00000 q^{96} -9.00000 q^{97} -1.00000 q^{99} +O(q^{100})$$

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
$$3$$ −1.00000 −0.577350
$$4$$ 1.00000 0.500000
$$5$$ −3.00000 −1.34164 −0.670820 0.741620i $$-0.734058\pi$$
−0.670820 + 0.741620i $$0.734058\pi$$
$$6$$ 1.00000 0.408248
$$7$$ 0 0
$$8$$ −1.00000 −0.353553
$$9$$ 1.00000 0.333333
$$10$$ 3.00000 0.948683
$$11$$ −1.00000 −0.301511
$$12$$ −1.00000 −0.288675
$$13$$ −6.00000 −1.66410 −0.832050 0.554700i $$-0.812833\pi$$
−0.832050 + 0.554700i $$0.812833\pi$$
$$14$$ 0 0
$$15$$ 3.00000 0.774597
$$16$$ 1.00000 0.250000
$$17$$ 5.00000 1.21268 0.606339 0.795206i $$-0.292637\pi$$
0.606339 + 0.795206i $$0.292637\pi$$
$$18$$ −1.00000 −0.235702
$$19$$ −6.00000 −1.37649 −0.688247 0.725476i $$-0.741620\pi$$
−0.688247 + 0.725476i $$0.741620\pi$$
$$20$$ −3.00000 −0.670820
$$21$$ 0 0
$$22$$ 1.00000 0.213201
$$23$$ 5.00000 1.04257 0.521286 0.853382i $$-0.325452\pi$$
0.521286 + 0.853382i $$0.325452\pi$$
$$24$$ 1.00000 0.204124
$$25$$ 4.00000 0.800000
$$26$$ 6.00000 1.17670
$$27$$ −1.00000 −0.192450
$$28$$ 0 0
$$29$$ −6.00000 −1.11417 −0.557086 0.830455i $$-0.688081\pi$$
−0.557086 + 0.830455i $$0.688081\pi$$
$$30$$ −3.00000 −0.547723
$$31$$ −4.00000 −0.718421 −0.359211 0.933257i $$-0.616954\pi$$
−0.359211 + 0.933257i $$0.616954\pi$$
$$32$$ −1.00000 −0.176777
$$33$$ 1.00000 0.174078
$$34$$ −5.00000 −0.857493
$$35$$ 0 0
$$36$$ 1.00000 0.166667
$$37$$ −2.00000 −0.328798 −0.164399 0.986394i $$-0.552568\pi$$
−0.164399 + 0.986394i $$0.552568\pi$$
$$38$$ 6.00000 0.973329
$$39$$ 6.00000 0.960769
$$40$$ 3.00000 0.474342
$$41$$ −5.00000 −0.780869 −0.390434 0.920631i $$-0.627675\pi$$
−0.390434 + 0.920631i $$0.627675\pi$$
$$42$$ 0 0
$$43$$ −10.0000 −1.52499 −0.762493 0.646997i $$-0.776025\pi$$
−0.762493 + 0.646997i $$0.776025\pi$$
$$44$$ −1.00000 −0.150756
$$45$$ −3.00000 −0.447214
$$46$$ −5.00000 −0.737210
$$47$$ −9.00000 −1.31278 −0.656392 0.754420i $$-0.727918\pi$$
−0.656392 + 0.754420i $$0.727918\pi$$
$$48$$ −1.00000 −0.144338
$$49$$ 0 0
$$50$$ −4.00000 −0.565685
$$51$$ −5.00000 −0.700140
$$52$$ −6.00000 −0.832050
$$53$$ 2.00000 0.274721 0.137361 0.990521i $$-0.456138\pi$$
0.137361 + 0.990521i $$0.456138\pi$$
$$54$$ 1.00000 0.136083
$$55$$ 3.00000 0.404520
$$56$$ 0 0
$$57$$ 6.00000 0.794719
$$58$$ 6.00000 0.787839
$$59$$ 12.0000 1.56227 0.781133 0.624364i $$-0.214642\pi$$
0.781133 + 0.624364i $$0.214642\pi$$
$$60$$ 3.00000 0.387298
$$61$$ 5.00000 0.640184 0.320092 0.947386i $$-0.396286\pi$$
0.320092 + 0.947386i $$0.396286\pi$$
$$62$$ 4.00000 0.508001
$$63$$ 0 0
$$64$$ 1.00000 0.125000
$$65$$ 18.0000 2.23263
$$66$$ −1.00000 −0.123091
$$67$$ 5.00000 0.610847 0.305424 0.952217i $$-0.401202\pi$$
0.305424 + 0.952217i $$0.401202\pi$$
$$68$$ 5.00000 0.606339
$$69$$ −5.00000 −0.601929
$$70$$ 0 0
$$71$$ 4.00000 0.474713 0.237356 0.971423i $$-0.423719\pi$$
0.237356 + 0.971423i $$0.423719\pi$$
$$72$$ −1.00000 −0.117851
$$73$$ −12.0000 −1.40449 −0.702247 0.711934i $$-0.747820\pi$$
−0.702247 + 0.711934i $$0.747820\pi$$
$$74$$ 2.00000 0.232495
$$75$$ −4.00000 −0.461880
$$76$$ −6.00000 −0.688247
$$77$$ 0 0
$$78$$ −6.00000 −0.679366
$$79$$ −1.00000 −0.112509 −0.0562544 0.998416i $$-0.517916\pi$$
−0.0562544 + 0.998416i $$0.517916\pi$$
$$80$$ −3.00000 −0.335410
$$81$$ 1.00000 0.111111
$$82$$ 5.00000 0.552158
$$83$$ −1.00000 −0.109764 −0.0548821 0.998493i $$-0.517478\pi$$
−0.0548821 + 0.998493i $$0.517478\pi$$
$$84$$ 0 0
$$85$$ −15.0000 −1.62698
$$86$$ 10.0000 1.07833
$$87$$ 6.00000 0.643268
$$88$$ 1.00000 0.106600
$$89$$ −6.00000 −0.635999 −0.317999 0.948091i $$-0.603011\pi$$
−0.317999 + 0.948091i $$0.603011\pi$$
$$90$$ 3.00000 0.316228
$$91$$ 0 0
$$92$$ 5.00000 0.521286
$$93$$ 4.00000 0.414781
$$94$$ 9.00000 0.928279
$$95$$ 18.0000 1.84676
$$96$$ 1.00000 0.102062
$$97$$ −9.00000 −0.913812 −0.456906 0.889515i $$-0.651042\pi$$
−0.456906 + 0.889515i $$0.651042\pi$$
$$98$$ 0 0
$$99$$ −1.00000 −0.100504
$$100$$ 4.00000 0.400000
$$101$$ 8.00000 0.796030 0.398015 0.917379i $$-0.369699\pi$$
0.398015 + 0.917379i $$0.369699\pi$$
$$102$$ 5.00000 0.495074
$$103$$ 2.00000 0.197066 0.0985329 0.995134i $$-0.468585\pi$$
0.0985329 + 0.995134i $$0.468585\pi$$
$$104$$ 6.00000 0.588348
$$105$$ 0 0
$$106$$ −2.00000 −0.194257
$$107$$ −13.0000 −1.25676 −0.628379 0.777908i $$-0.716281\pi$$
−0.628379 + 0.777908i $$0.716281\pi$$
$$108$$ −1.00000 −0.0962250
$$109$$ −11.0000 −1.05361 −0.526804 0.849987i $$-0.676610\pi$$
−0.526804 + 0.849987i $$0.676610\pi$$
$$110$$ −3.00000 −0.286039
$$111$$ 2.00000 0.189832
$$112$$ 0 0
$$113$$ 10.0000 0.940721 0.470360 0.882474i $$-0.344124\pi$$
0.470360 + 0.882474i $$0.344124\pi$$
$$114$$ −6.00000 −0.561951
$$115$$ −15.0000 −1.39876
$$116$$ −6.00000 −0.557086
$$117$$ −6.00000 −0.554700
$$118$$ −12.0000 −1.10469
$$119$$ 0 0
$$120$$ −3.00000 −0.273861
$$121$$ 1.00000 0.0909091
$$122$$ −5.00000 −0.452679
$$123$$ 5.00000 0.450835
$$124$$ −4.00000 −0.359211
$$125$$ 3.00000 0.268328
$$126$$ 0 0
$$127$$ 5.00000 0.443678 0.221839 0.975083i $$-0.428794\pi$$
0.221839 + 0.975083i $$0.428794\pi$$
$$128$$ −1.00000 −0.0883883
$$129$$ 10.0000 0.880451
$$130$$ −18.0000 −1.57870
$$131$$ 4.00000 0.349482 0.174741 0.984614i $$-0.444091\pi$$
0.174741 + 0.984614i $$0.444091\pi$$
$$132$$ 1.00000 0.0870388
$$133$$ 0 0
$$134$$ −5.00000 −0.431934
$$135$$ 3.00000 0.258199
$$136$$ −5.00000 −0.428746
$$137$$ −2.00000 −0.170872 −0.0854358 0.996344i $$-0.527228\pi$$
−0.0854358 + 0.996344i $$0.527228\pi$$
$$138$$ 5.00000 0.425628
$$139$$ −22.0000 −1.86602 −0.933008 0.359856i $$-0.882826\pi$$
−0.933008 + 0.359856i $$0.882826\pi$$
$$140$$ 0 0
$$141$$ 9.00000 0.757937
$$142$$ −4.00000 −0.335673
$$143$$ 6.00000 0.501745
$$144$$ 1.00000 0.0833333
$$145$$ 18.0000 1.49482
$$146$$ 12.0000 0.993127
$$147$$ 0 0
$$148$$ −2.00000 −0.164399
$$149$$ 18.0000 1.47462 0.737309 0.675556i $$-0.236096\pi$$
0.737309 + 0.675556i $$0.236096\pi$$
$$150$$ 4.00000 0.326599
$$151$$ 9.00000 0.732410 0.366205 0.930534i $$-0.380657\pi$$
0.366205 + 0.930534i $$0.380657\pi$$
$$152$$ 6.00000 0.486664
$$153$$ 5.00000 0.404226
$$154$$ 0 0
$$155$$ 12.0000 0.963863
$$156$$ 6.00000 0.480384
$$157$$ 24.0000 1.91541 0.957704 0.287754i $$-0.0929087\pi$$
0.957704 + 0.287754i $$0.0929087\pi$$
$$158$$ 1.00000 0.0795557
$$159$$ −2.00000 −0.158610
$$160$$ 3.00000 0.237171
$$161$$ 0 0
$$162$$ −1.00000 −0.0785674
$$163$$ −1.00000 −0.0783260 −0.0391630 0.999233i $$-0.512469\pi$$
−0.0391630 + 0.999233i $$0.512469\pi$$
$$164$$ −5.00000 −0.390434
$$165$$ −3.00000 −0.233550
$$166$$ 1.00000 0.0776151
$$167$$ 24.0000 1.85718 0.928588 0.371113i $$-0.121024\pi$$
0.928588 + 0.371113i $$0.121024\pi$$
$$168$$ 0 0
$$169$$ 23.0000 1.76923
$$170$$ 15.0000 1.15045
$$171$$ −6.00000 −0.458831
$$172$$ −10.0000 −0.762493
$$173$$ 12.0000 0.912343 0.456172 0.889892i $$-0.349220\pi$$
0.456172 + 0.889892i $$0.349220\pi$$
$$174$$ −6.00000 −0.454859
$$175$$ 0 0
$$176$$ −1.00000 −0.0753778
$$177$$ −12.0000 −0.901975
$$178$$ 6.00000 0.449719
$$179$$ −8.00000 −0.597948 −0.298974 0.954261i $$-0.596644\pi$$
−0.298974 + 0.954261i $$0.596644\pi$$
$$180$$ −3.00000 −0.223607
$$181$$ 22.0000 1.63525 0.817624 0.575753i $$-0.195291\pi$$
0.817624 + 0.575753i $$0.195291\pi$$
$$182$$ 0 0
$$183$$ −5.00000 −0.369611
$$184$$ −5.00000 −0.368605
$$185$$ 6.00000 0.441129
$$186$$ −4.00000 −0.293294
$$187$$ −5.00000 −0.365636
$$188$$ −9.00000 −0.656392
$$189$$ 0 0
$$190$$ −18.0000 −1.30586
$$191$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$192$$ −1.00000 −0.0721688
$$193$$ 6.00000 0.431889 0.215945 0.976406i $$-0.430717\pi$$
0.215945 + 0.976406i $$0.430717\pi$$
$$194$$ 9.00000 0.646162
$$195$$ −18.0000 −1.28901
$$196$$ 0 0
$$197$$ −24.0000 −1.70993 −0.854965 0.518686i $$-0.826421\pi$$
−0.854965 + 0.518686i $$0.826421\pi$$
$$198$$ 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
$$201$$ −5.00000 −0.352673
$$202$$ −8.00000 −0.562878
$$203$$ 0 0
$$204$$ −5.00000 −0.350070
$$205$$ 15.0000 1.04765
$$206$$ −2.00000 −0.139347
$$207$$ 5.00000 0.347524
$$208$$ −6.00000 −0.416025
$$209$$ 6.00000 0.415029
$$210$$ 0 0
$$211$$ 14.0000 0.963800 0.481900 0.876226i $$-0.339947\pi$$
0.481900 + 0.876226i $$0.339947\pi$$
$$212$$ 2.00000 0.137361
$$213$$ −4.00000 −0.274075
$$214$$ 13.0000 0.888662
$$215$$ 30.0000 2.04598
$$216$$ 1.00000 0.0680414
$$217$$ 0 0
$$218$$ 11.0000 0.745014
$$219$$ 12.0000 0.810885
$$220$$ 3.00000 0.202260
$$221$$ −30.0000 −2.01802
$$222$$ −2.00000 −0.134231
$$223$$ −16.0000 −1.07144 −0.535720 0.844396i $$-0.679960\pi$$
−0.535720 + 0.844396i $$0.679960\pi$$
$$224$$ 0 0
$$225$$ 4.00000 0.266667
$$226$$ −10.0000 −0.665190
$$227$$ −3.00000 −0.199117 −0.0995585 0.995032i $$-0.531743\pi$$
−0.0995585 + 0.995032i $$0.531743\pi$$
$$228$$ 6.00000 0.397360
$$229$$ −26.0000 −1.71813 −0.859064 0.511868i $$-0.828954\pi$$
−0.859064 + 0.511868i $$0.828954\pi$$
$$230$$ 15.0000 0.989071
$$231$$ 0 0
$$232$$ 6.00000 0.393919
$$233$$ −29.0000 −1.89985 −0.949927 0.312473i $$-0.898843\pi$$
−0.949927 + 0.312473i $$0.898843\pi$$
$$234$$ 6.00000 0.392232
$$235$$ 27.0000 1.76129
$$236$$ 12.0000 0.781133
$$237$$ 1.00000 0.0649570
$$238$$ 0 0
$$239$$ 26.0000 1.68180 0.840900 0.541190i $$-0.182026\pi$$
0.840900 + 0.541190i $$0.182026\pi$$
$$240$$ 3.00000 0.193649
$$241$$ 26.0000 1.67481 0.837404 0.546585i $$-0.184072\pi$$
0.837404 + 0.546585i $$0.184072\pi$$
$$242$$ −1.00000 −0.0642824
$$243$$ −1.00000 −0.0641500
$$244$$ 5.00000 0.320092
$$245$$ 0 0
$$246$$ −5.00000 −0.318788
$$247$$ 36.0000 2.29063
$$248$$ 4.00000 0.254000
$$249$$ 1.00000 0.0633724
$$250$$ −3.00000 −0.189737
$$251$$ −12.0000 −0.757433 −0.378717 0.925513i $$-0.623635\pi$$
−0.378717 + 0.925513i $$0.623635\pi$$
$$252$$ 0 0
$$253$$ −5.00000 −0.314347
$$254$$ −5.00000 −0.313728
$$255$$ 15.0000 0.939336
$$256$$ 1.00000 0.0625000
$$257$$ 14.0000 0.873296 0.436648 0.899632i $$-0.356166\pi$$
0.436648 + 0.899632i $$0.356166\pi$$
$$258$$ −10.0000 −0.622573
$$259$$ 0 0
$$260$$ 18.0000 1.11631
$$261$$ −6.00000 −0.371391
$$262$$ −4.00000 −0.247121
$$263$$ 24.0000 1.47990 0.739952 0.672660i $$-0.234848\pi$$
0.739952 + 0.672660i $$0.234848\pi$$
$$264$$ −1.00000 −0.0615457
$$265$$ −6.00000 −0.368577
$$266$$ 0 0
$$267$$ 6.00000 0.367194
$$268$$ 5.00000 0.305424
$$269$$ 9.00000 0.548740 0.274370 0.961624i $$-0.411531\pi$$
0.274370 + 0.961624i $$0.411531\pi$$
$$270$$ −3.00000 −0.182574
$$271$$ 24.0000 1.45790 0.728948 0.684569i $$-0.240010\pi$$
0.728948 + 0.684569i $$0.240010\pi$$
$$272$$ 5.00000 0.303170
$$273$$ 0 0
$$274$$ 2.00000 0.120824
$$275$$ −4.00000 −0.241209
$$276$$ −5.00000 −0.300965
$$277$$ 18.0000 1.08152 0.540758 0.841178i $$-0.318138\pi$$
0.540758 + 0.841178i $$0.318138\pi$$
$$278$$ 22.0000 1.31947
$$279$$ −4.00000 −0.239474
$$280$$ 0 0
$$281$$ 11.0000 0.656205 0.328102 0.944642i $$-0.393591\pi$$
0.328102 + 0.944642i $$0.393591\pi$$
$$282$$ −9.00000 −0.535942
$$283$$ 6.00000 0.356663 0.178331 0.983970i $$-0.442930\pi$$
0.178331 + 0.983970i $$0.442930\pi$$
$$284$$ 4.00000 0.237356
$$285$$ −18.0000 −1.06623
$$286$$ −6.00000 −0.354787
$$287$$ 0 0
$$288$$ −1.00000 −0.0589256
$$289$$ 8.00000 0.470588
$$290$$ −18.0000 −1.05700
$$291$$ 9.00000 0.527589
$$292$$ −12.0000 −0.702247
$$293$$ −8.00000 −0.467365 −0.233682 0.972313i $$-0.575078\pi$$
−0.233682 + 0.972313i $$0.575078\pi$$
$$294$$ 0 0
$$295$$ −36.0000 −2.09600
$$296$$ 2.00000 0.116248
$$297$$ 1.00000 0.0580259
$$298$$ −18.0000 −1.04271
$$299$$ −30.0000 −1.73494
$$300$$ −4.00000 −0.230940
$$301$$ 0 0
$$302$$ −9.00000 −0.517892
$$303$$ −8.00000 −0.459588
$$304$$ −6.00000 −0.344124
$$305$$ −15.0000 −0.858898
$$306$$ −5.00000 −0.285831
$$307$$ −18.0000 −1.02731 −0.513657 0.857996i $$-0.671710\pi$$
−0.513657 + 0.857996i $$0.671710\pi$$
$$308$$ 0 0
$$309$$ −2.00000 −0.113776
$$310$$ −12.0000 −0.681554
$$311$$ 21.0000 1.19080 0.595400 0.803429i $$-0.296993\pi$$
0.595400 + 0.803429i $$0.296993\pi$$
$$312$$ −6.00000 −0.339683
$$313$$ 6.00000 0.339140 0.169570 0.985518i $$-0.445762\pi$$
0.169570 + 0.985518i $$0.445762\pi$$
$$314$$ −24.0000 −1.35440
$$315$$ 0 0
$$316$$ −1.00000 −0.0562544
$$317$$ 9.00000 0.505490 0.252745 0.967533i $$-0.418667\pi$$
0.252745 + 0.967533i $$0.418667\pi$$
$$318$$ 2.00000 0.112154
$$319$$ 6.00000 0.335936
$$320$$ −3.00000 −0.167705
$$321$$ 13.0000 0.725589
$$322$$ 0 0
$$323$$ −30.0000 −1.66924
$$324$$ 1.00000 0.0555556
$$325$$ −24.0000 −1.33128
$$326$$ 1.00000 0.0553849
$$327$$ 11.0000 0.608301
$$328$$ 5.00000 0.276079
$$329$$ 0 0
$$330$$ 3.00000 0.165145
$$331$$ 35.0000 1.92377 0.961887 0.273447i $$-0.0881639\pi$$
0.961887 + 0.273447i $$0.0881639\pi$$
$$332$$ −1.00000 −0.0548821
$$333$$ −2.00000 −0.109599
$$334$$ −24.0000 −1.31322
$$335$$ −15.0000 −0.819538
$$336$$ 0 0
$$337$$ −2.00000 −0.108947 −0.0544735 0.998515i $$-0.517348\pi$$
−0.0544735 + 0.998515i $$0.517348\pi$$
$$338$$ −23.0000 −1.25104
$$339$$ −10.0000 −0.543125
$$340$$ −15.0000 −0.813489
$$341$$ 4.00000 0.216612
$$342$$ 6.00000 0.324443
$$343$$ 0 0
$$344$$ 10.0000 0.539164
$$345$$ 15.0000 0.807573
$$346$$ −12.0000 −0.645124
$$347$$ 7.00000 0.375780 0.187890 0.982190i $$-0.439835\pi$$
0.187890 + 0.982190i $$0.439835\pi$$
$$348$$ 6.00000 0.321634
$$349$$ 19.0000 1.01705 0.508523 0.861048i $$-0.330192\pi$$
0.508523 + 0.861048i $$0.330192\pi$$
$$350$$ 0 0
$$351$$ 6.00000 0.320256
$$352$$ 1.00000 0.0533002
$$353$$ −24.0000 −1.27739 −0.638696 0.769460i $$-0.720526\pi$$
−0.638696 + 0.769460i $$0.720526\pi$$
$$354$$ 12.0000 0.637793
$$355$$ −12.0000 −0.636894
$$356$$ −6.00000 −0.317999
$$357$$ 0 0
$$358$$ 8.00000 0.422813
$$359$$ −10.0000 −0.527780 −0.263890 0.964553i $$-0.585006\pi$$
−0.263890 + 0.964553i $$0.585006\pi$$
$$360$$ 3.00000 0.158114
$$361$$ 17.0000 0.894737
$$362$$ −22.0000 −1.15629
$$363$$ −1.00000 −0.0524864
$$364$$ 0 0
$$365$$ 36.0000 1.88433
$$366$$ 5.00000 0.261354
$$367$$ 14.0000 0.730794 0.365397 0.930852i $$-0.380933\pi$$
0.365397 + 0.930852i $$0.380933\pi$$
$$368$$ 5.00000 0.260643
$$369$$ −5.00000 −0.260290
$$370$$ −6.00000 −0.311925
$$371$$ 0 0
$$372$$ 4.00000 0.207390
$$373$$ −7.00000 −0.362446 −0.181223 0.983442i $$-0.558006\pi$$
−0.181223 + 0.983442i $$0.558006\pi$$
$$374$$ 5.00000 0.258544
$$375$$ −3.00000 −0.154919
$$376$$ 9.00000 0.464140
$$377$$ 36.0000 1.85409
$$378$$ 0 0
$$379$$ 13.0000 0.667765 0.333883 0.942615i $$-0.391641\pi$$
0.333883 + 0.942615i $$0.391641\pi$$
$$380$$ 18.0000 0.923381
$$381$$ −5.00000 −0.256158
$$382$$ 0 0
$$383$$ −8.00000 −0.408781 −0.204390 0.978889i $$-0.565521\pi$$
−0.204390 + 0.978889i $$0.565521\pi$$
$$384$$ 1.00000 0.0510310
$$385$$ 0 0
$$386$$ −6.00000 −0.305392
$$387$$ −10.0000 −0.508329
$$388$$ −9.00000 −0.456906
$$389$$ −21.0000 −1.06474 −0.532371 0.846511i $$-0.678699\pi$$
−0.532371 + 0.846511i $$0.678699\pi$$
$$390$$ 18.0000 0.911465
$$391$$ 25.0000 1.26430
$$392$$ 0 0
$$393$$ −4.00000 −0.201773
$$394$$ 24.0000 1.20910
$$395$$ 3.00000 0.150946
$$396$$ −1.00000 −0.0502519
$$397$$ −26.0000 −1.30490 −0.652451 0.757831i $$-0.726259\pi$$
−0.652451 + 0.757831i $$0.726259\pi$$
$$398$$ 10.0000 0.501255
$$399$$ 0 0
$$400$$ 4.00000 0.200000
$$401$$ −6.00000 −0.299626 −0.149813 0.988714i $$-0.547867\pi$$
−0.149813 + 0.988714i $$0.547867\pi$$
$$402$$ 5.00000 0.249377
$$403$$ 24.0000 1.19553
$$404$$ 8.00000 0.398015
$$405$$ −3.00000 −0.149071
$$406$$ 0 0
$$407$$ 2.00000 0.0991363
$$408$$ 5.00000 0.247537
$$409$$ −10.0000 −0.494468 −0.247234 0.968956i $$-0.579522\pi$$
−0.247234 + 0.968956i $$0.579522\pi$$
$$410$$ −15.0000 −0.740797
$$411$$ 2.00000 0.0986527
$$412$$ 2.00000 0.0985329
$$413$$ 0 0
$$414$$ −5.00000 −0.245737
$$415$$ 3.00000 0.147264
$$416$$ 6.00000 0.294174
$$417$$ 22.0000 1.07734
$$418$$ −6.00000 −0.293470
$$419$$ 20.0000 0.977064 0.488532 0.872546i $$-0.337533\pi$$
0.488532 + 0.872546i $$0.337533\pi$$
$$420$$ 0 0
$$421$$ 8.00000 0.389896 0.194948 0.980814i $$-0.437546\pi$$
0.194948 + 0.980814i $$0.437546\pi$$
$$422$$ −14.0000 −0.681509
$$423$$ −9.00000 −0.437595
$$424$$ −2.00000 −0.0971286
$$425$$ 20.0000 0.970143
$$426$$ 4.00000 0.193801
$$427$$ 0 0
$$428$$ −13.0000 −0.628379
$$429$$ −6.00000 −0.289683
$$430$$ −30.0000 −1.44673
$$431$$ 22.0000 1.05970 0.529851 0.848091i $$-0.322248\pi$$
0.529851 + 0.848091i $$0.322248\pi$$
$$432$$ −1.00000 −0.0481125
$$433$$ −11.0000 −0.528626 −0.264313 0.964437i $$-0.585145\pi$$
−0.264313 + 0.964437i $$0.585145\pi$$
$$434$$ 0 0
$$435$$ −18.0000 −0.863034
$$436$$ −11.0000 −0.526804
$$437$$ −30.0000 −1.43509
$$438$$ −12.0000 −0.573382
$$439$$ 5.00000 0.238637 0.119318 0.992856i $$-0.461929\pi$$
0.119318 + 0.992856i $$0.461929\pi$$
$$440$$ −3.00000 −0.143019
$$441$$ 0 0
$$442$$ 30.0000 1.42695
$$443$$ 2.00000 0.0950229 0.0475114 0.998871i $$-0.484871\pi$$
0.0475114 + 0.998871i $$0.484871\pi$$
$$444$$ 2.00000 0.0949158
$$445$$ 18.0000 0.853282
$$446$$ 16.0000 0.757622
$$447$$ −18.0000 −0.851371
$$448$$ 0 0
$$449$$ 2.00000 0.0943858 0.0471929 0.998886i $$-0.484972\pi$$
0.0471929 + 0.998886i $$0.484972\pi$$
$$450$$ −4.00000 −0.188562
$$451$$ 5.00000 0.235441
$$452$$ 10.0000 0.470360
$$453$$ −9.00000 −0.422857
$$454$$ 3.00000 0.140797
$$455$$ 0 0
$$456$$ −6.00000 −0.280976
$$457$$ −26.0000 −1.21623 −0.608114 0.793849i $$-0.708074\pi$$
−0.608114 + 0.793849i $$0.708074\pi$$
$$458$$ 26.0000 1.21490
$$459$$ −5.00000 −0.233380
$$460$$ −15.0000 −0.699379
$$461$$ 10.0000 0.465746 0.232873 0.972507i $$-0.425187\pi$$
0.232873 + 0.972507i $$0.425187\pi$$
$$462$$ 0 0
$$463$$ 28.0000 1.30127 0.650635 0.759390i $$-0.274503\pi$$
0.650635 + 0.759390i $$0.274503\pi$$
$$464$$ −6.00000 −0.278543
$$465$$ −12.0000 −0.556487
$$466$$ 29.0000 1.34340
$$467$$ −16.0000 −0.740392 −0.370196 0.928954i $$-0.620709\pi$$
−0.370196 + 0.928954i $$0.620709\pi$$
$$468$$ −6.00000 −0.277350
$$469$$ 0 0
$$470$$ −27.0000 −1.24542
$$471$$ −24.0000 −1.10586
$$472$$ −12.0000 −0.552345
$$473$$ 10.0000 0.459800
$$474$$ −1.00000 −0.0459315
$$475$$ −24.0000 −1.10120
$$476$$ 0 0
$$477$$ 2.00000 0.0915737
$$478$$ −26.0000 −1.18921
$$479$$ −12.0000 −0.548294 −0.274147 0.961688i $$-0.588395\pi$$
−0.274147 + 0.961688i $$0.588395\pi$$
$$480$$ −3.00000 −0.136931
$$481$$ 12.0000 0.547153
$$482$$ −26.0000 −1.18427
$$483$$ 0 0
$$484$$ 1.00000 0.0454545
$$485$$ 27.0000 1.22601
$$486$$ 1.00000 0.0453609
$$487$$ −4.00000 −0.181257 −0.0906287 0.995885i $$-0.528888\pi$$
−0.0906287 + 0.995885i $$0.528888\pi$$
$$488$$ −5.00000 −0.226339
$$489$$ 1.00000 0.0452216
$$490$$ 0 0
$$491$$ −3.00000 −0.135388 −0.0676941 0.997706i $$-0.521564\pi$$
−0.0676941 + 0.997706i $$0.521564\pi$$
$$492$$ 5.00000 0.225417
$$493$$ −30.0000 −1.35113
$$494$$ −36.0000 −1.61972
$$495$$ 3.00000 0.134840
$$496$$ −4.00000 −0.179605
$$497$$ 0 0
$$498$$ −1.00000 −0.0448111
$$499$$ 12.0000 0.537194 0.268597 0.963253i $$-0.413440\pi$$
0.268597 + 0.963253i $$0.413440\pi$$
$$500$$ 3.00000 0.134164
$$501$$ −24.0000 −1.07224
$$502$$ 12.0000 0.535586
$$503$$ 42.0000 1.87269 0.936344 0.351085i $$-0.114187\pi$$
0.936344 + 0.351085i $$0.114187\pi$$
$$504$$ 0 0
$$505$$ −24.0000 −1.06799
$$506$$ 5.00000 0.222277
$$507$$ −23.0000 −1.02147
$$508$$ 5.00000 0.221839
$$509$$ −2.00000 −0.0886484 −0.0443242 0.999017i $$-0.514113\pi$$
−0.0443242 + 0.999017i $$0.514113\pi$$
$$510$$ −15.0000 −0.664211
$$511$$ 0 0
$$512$$ −1.00000 −0.0441942
$$513$$ 6.00000 0.264906
$$514$$ −14.0000 −0.617514
$$515$$ −6.00000 −0.264392
$$516$$ 10.0000 0.440225
$$517$$ 9.00000 0.395820
$$518$$ 0 0
$$519$$ −12.0000 −0.526742
$$520$$ −18.0000 −0.789352
$$521$$ −2.00000 −0.0876216 −0.0438108 0.999040i $$-0.513950\pi$$
−0.0438108 + 0.999040i $$0.513950\pi$$
$$522$$ 6.00000 0.262613
$$523$$ −6.00000 −0.262362 −0.131181 0.991358i $$-0.541877\pi$$
−0.131181 + 0.991358i $$0.541877\pi$$
$$524$$ 4.00000 0.174741
$$525$$ 0 0
$$526$$ −24.0000 −1.04645
$$527$$ −20.0000 −0.871214
$$528$$ 1.00000 0.0435194
$$529$$ 2.00000 0.0869565
$$530$$ 6.00000 0.260623
$$531$$ 12.0000 0.520756
$$532$$ 0 0
$$533$$ 30.0000 1.29944
$$534$$ −6.00000 −0.259645
$$535$$ 39.0000 1.68612
$$536$$ −5.00000 −0.215967
$$537$$ 8.00000 0.345225
$$538$$ −9.00000 −0.388018
$$539$$ 0 0
$$540$$ 3.00000 0.129099
$$541$$ −39.0000 −1.67674 −0.838370 0.545101i $$-0.816491\pi$$
−0.838370 + 0.545101i $$0.816491\pi$$
$$542$$ −24.0000 −1.03089
$$543$$ −22.0000 −0.944110
$$544$$ −5.00000 −0.214373
$$545$$ 33.0000 1.41356
$$546$$ 0 0
$$547$$ −42.0000 −1.79579 −0.897895 0.440209i $$-0.854904\pi$$
−0.897895 + 0.440209i $$0.854904\pi$$
$$548$$ −2.00000 −0.0854358
$$549$$ 5.00000 0.213395
$$550$$ 4.00000 0.170561
$$551$$ 36.0000 1.53365
$$552$$ 5.00000 0.212814
$$553$$ 0 0
$$554$$ −18.0000 −0.764747
$$555$$ −6.00000 −0.254686
$$556$$ −22.0000 −0.933008
$$557$$ −32.0000 −1.35588 −0.677942 0.735116i $$-0.737128\pi$$
−0.677942 + 0.735116i $$0.737128\pi$$
$$558$$ 4.00000 0.169334
$$559$$ 60.0000 2.53773
$$560$$ 0 0
$$561$$ 5.00000 0.211100
$$562$$ −11.0000 −0.464007
$$563$$ 16.0000 0.674320 0.337160 0.941447i $$-0.390534\pi$$
0.337160 + 0.941447i $$0.390534\pi$$
$$564$$ 9.00000 0.378968
$$565$$ −30.0000 −1.26211
$$566$$ −6.00000 −0.252199
$$567$$ 0 0
$$568$$ −4.00000 −0.167836
$$569$$ −6.00000 −0.251533 −0.125767 0.992060i $$-0.540139\pi$$
−0.125767 + 0.992060i $$0.540139\pi$$
$$570$$ 18.0000 0.753937
$$571$$ −34.0000 −1.42286 −0.711428 0.702759i $$-0.751951\pi$$
−0.711428 + 0.702759i $$0.751951\pi$$
$$572$$ 6.00000 0.250873
$$573$$ 0 0
$$574$$ 0 0
$$575$$ 20.0000 0.834058
$$576$$ 1.00000 0.0416667
$$577$$ 7.00000 0.291414 0.145707 0.989328i $$-0.453454\pi$$
0.145707 + 0.989328i $$0.453454\pi$$
$$578$$ −8.00000 −0.332756
$$579$$ −6.00000 −0.249351
$$580$$ 18.0000 0.747409
$$581$$ 0 0
$$582$$ −9.00000 −0.373062
$$583$$ −2.00000 −0.0828315
$$584$$ 12.0000 0.496564
$$585$$ 18.0000 0.744208
$$586$$ 8.00000 0.330477
$$587$$ 24.0000 0.990586 0.495293 0.868726i $$-0.335061\pi$$
0.495293 + 0.868726i $$0.335061\pi$$
$$588$$ 0 0
$$589$$ 24.0000 0.988903
$$590$$ 36.0000 1.48210
$$591$$ 24.0000 0.987228
$$592$$ −2.00000 −0.0821995
$$593$$ −14.0000 −0.574911 −0.287456 0.957794i $$-0.592809\pi$$
−0.287456 + 0.957794i $$0.592809\pi$$
$$594$$ −1.00000 −0.0410305
$$595$$ 0 0
$$596$$ 18.0000 0.737309
$$597$$ 10.0000 0.409273
$$598$$ 30.0000 1.22679
$$599$$ −3.00000 −0.122577 −0.0612883 0.998120i $$-0.519521\pi$$
−0.0612883 + 0.998120i $$0.519521\pi$$
$$600$$ 4.00000 0.163299
$$601$$ 44.0000 1.79480 0.897399 0.441221i $$-0.145454\pi$$
0.897399 + 0.441221i $$0.145454\pi$$
$$602$$ 0 0
$$603$$ 5.00000 0.203616
$$604$$ 9.00000 0.366205
$$605$$ −3.00000 −0.121967
$$606$$ 8.00000 0.324978
$$607$$ −33.0000 −1.33943 −0.669714 0.742619i $$-0.733583\pi$$
−0.669714 + 0.742619i $$0.733583\pi$$
$$608$$ 6.00000 0.243332
$$609$$ 0 0
$$610$$ 15.0000 0.607332
$$611$$ 54.0000 2.18461
$$612$$ 5.00000 0.202113
$$613$$ −41.0000 −1.65597 −0.827987 0.560747i $$-0.810514\pi$$
−0.827987 + 0.560747i $$0.810514\pi$$
$$614$$ 18.0000 0.726421
$$615$$ −15.0000 −0.604858
$$616$$ 0 0
$$617$$ −6.00000 −0.241551 −0.120775 0.992680i $$-0.538538\pi$$
−0.120775 + 0.992680i $$0.538538\pi$$
$$618$$ 2.00000 0.0804518
$$619$$ 29.0000 1.16561 0.582804 0.812613i $$-0.301955\pi$$
0.582804 + 0.812613i $$0.301955\pi$$
$$620$$ 12.0000 0.481932
$$621$$ −5.00000 −0.200643
$$622$$ −21.0000 −0.842023
$$623$$ 0 0
$$624$$ 6.00000 0.240192
$$625$$ −29.0000 −1.16000
$$626$$ −6.00000 −0.239808
$$627$$ −6.00000 −0.239617
$$628$$ 24.0000 0.957704
$$629$$ −10.0000 −0.398726
$$630$$ 0 0
$$631$$ 28.0000 1.11466 0.557331 0.830290i $$-0.311825\pi$$
0.557331 + 0.830290i $$0.311825\pi$$
$$632$$ 1.00000 0.0397779
$$633$$ −14.0000 −0.556450
$$634$$ −9.00000 −0.357436
$$635$$ −15.0000 −0.595257
$$636$$ −2.00000 −0.0793052
$$637$$ 0 0
$$638$$ −6.00000 −0.237542
$$639$$ 4.00000 0.158238
$$640$$ 3.00000 0.118585
$$641$$ −40.0000 −1.57991 −0.789953 0.613168i $$-0.789895\pi$$
−0.789953 + 0.613168i $$0.789895\pi$$
$$642$$ −13.0000 −0.513069
$$643$$ −36.0000 −1.41970 −0.709851 0.704352i $$-0.751238\pi$$
−0.709851 + 0.704352i $$0.751238\pi$$
$$644$$ 0 0
$$645$$ −30.0000 −1.18125
$$646$$ 30.0000 1.18033
$$647$$ −17.0000 −0.668339 −0.334169 0.942513i $$-0.608456\pi$$
−0.334169 + 0.942513i $$0.608456\pi$$
$$648$$ −1.00000 −0.0392837
$$649$$ −12.0000 −0.471041
$$650$$ 24.0000 0.941357
$$651$$ 0 0
$$652$$ −1.00000 −0.0391630
$$653$$ −27.0000 −1.05659 −0.528296 0.849060i $$-0.677169\pi$$
−0.528296 + 0.849060i $$0.677169\pi$$
$$654$$ −11.0000 −0.430134
$$655$$ −12.0000 −0.468879
$$656$$ −5.00000 −0.195217
$$657$$ −12.0000 −0.468165
$$658$$ 0 0
$$659$$ 31.0000 1.20759 0.603794 0.797140i $$-0.293655\pi$$
0.603794 + 0.797140i $$0.293655\pi$$
$$660$$ −3.00000 −0.116775
$$661$$ −4.00000 −0.155582 −0.0777910 0.996970i $$-0.524787\pi$$
−0.0777910 + 0.996970i $$0.524787\pi$$
$$662$$ −35.0000 −1.36031
$$663$$ 30.0000 1.16510
$$664$$ 1.00000 0.0388075
$$665$$ 0 0
$$666$$ 2.00000 0.0774984
$$667$$ −30.0000 −1.16160
$$668$$ 24.0000 0.928588
$$669$$ 16.0000 0.618596
$$670$$ 15.0000 0.579501
$$671$$ −5.00000 −0.193023
$$672$$ 0 0
$$673$$ −26.0000 −1.00223 −0.501113 0.865382i $$-0.667076\pi$$
−0.501113 + 0.865382i $$0.667076\pi$$
$$674$$ 2.00000 0.0770371
$$675$$ −4.00000 −0.153960
$$676$$ 23.0000 0.884615
$$677$$ 18.0000 0.691796 0.345898 0.938272i $$-0.387574\pi$$
0.345898 + 0.938272i $$0.387574\pi$$
$$678$$ 10.0000 0.384048
$$679$$ 0 0
$$680$$ 15.0000 0.575224
$$681$$ 3.00000 0.114960
$$682$$ −4.00000 −0.153168
$$683$$ −12.0000 −0.459167 −0.229584 0.973289i $$-0.573736\pi$$
−0.229584 + 0.973289i $$0.573736\pi$$
$$684$$ −6.00000 −0.229416
$$685$$ 6.00000 0.229248
$$686$$ 0 0
$$687$$ 26.0000 0.991962
$$688$$ −10.0000 −0.381246
$$689$$ −12.0000 −0.457164
$$690$$ −15.0000 −0.571040
$$691$$ 27.0000 1.02713 0.513564 0.858051i $$-0.328325\pi$$
0.513564 + 0.858051i $$0.328325\pi$$
$$692$$ 12.0000 0.456172
$$693$$ 0 0
$$694$$ −7.00000 −0.265716
$$695$$ 66.0000 2.50352
$$696$$ −6.00000 −0.227429
$$697$$ −25.0000 −0.946943
$$698$$ −19.0000 −0.719161
$$699$$ 29.0000 1.09688
$$700$$ 0 0
$$701$$ −10.0000 −0.377695 −0.188847 0.982006i $$-0.560475\pi$$
−0.188847 + 0.982006i $$0.560475\pi$$
$$702$$ −6.00000 −0.226455
$$703$$ 12.0000 0.452589
$$704$$ −1.00000 −0.0376889
$$705$$ −27.0000 −1.01688
$$706$$ 24.0000 0.903252
$$707$$ 0 0
$$708$$ −12.0000 −0.450988
$$709$$ 20.0000 0.751116 0.375558 0.926799i $$-0.377451\pi$$
0.375558 + 0.926799i $$0.377451\pi$$
$$710$$ 12.0000 0.450352
$$711$$ −1.00000 −0.0375029
$$712$$ 6.00000 0.224860
$$713$$ −20.0000 −0.749006
$$714$$ 0 0
$$715$$ −18.0000 −0.673162
$$716$$ −8.00000 −0.298974
$$717$$ −26.0000 −0.970988
$$718$$ 10.0000 0.373197
$$719$$ −37.0000 −1.37987 −0.689934 0.723873i $$-0.742360\pi$$
−0.689934 + 0.723873i $$0.742360\pi$$
$$720$$ −3.00000 −0.111803
$$721$$ 0 0
$$722$$ −17.0000 −0.632674
$$723$$ −26.0000 −0.966950
$$724$$ 22.0000 0.817624
$$725$$ −24.0000 −0.891338
$$726$$ 1.00000 0.0371135
$$727$$ −22.0000 −0.815935 −0.407967 0.912996i $$-0.633762\pi$$
−0.407967 + 0.912996i $$0.633762\pi$$
$$728$$ 0 0
$$729$$ 1.00000 0.0370370
$$730$$ −36.0000 −1.33242
$$731$$ −50.0000 −1.84932
$$732$$ −5.00000 −0.184805
$$733$$ 15.0000 0.554038 0.277019 0.960864i $$-0.410654\pi$$
0.277019 + 0.960864i $$0.410654\pi$$
$$734$$ −14.0000 −0.516749
$$735$$ 0 0
$$736$$ −5.00000 −0.184302
$$737$$ −5.00000 −0.184177
$$738$$ 5.00000 0.184053
$$739$$ −10.0000 −0.367856 −0.183928 0.982940i $$-0.558881\pi$$
−0.183928 + 0.982940i $$0.558881\pi$$
$$740$$ 6.00000 0.220564
$$741$$ −36.0000 −1.32249
$$742$$ 0 0
$$743$$ 22.0000 0.807102 0.403551 0.914957i $$-0.367776\pi$$
0.403551 + 0.914957i $$0.367776\pi$$
$$744$$ −4.00000 −0.146647
$$745$$ −54.0000 −1.97841
$$746$$ 7.00000 0.256288
$$747$$ −1.00000 −0.0365881
$$748$$ −5.00000 −0.182818
$$749$$ 0 0
$$750$$ 3.00000 0.109545
$$751$$ −2.00000 −0.0729810 −0.0364905 0.999334i $$-0.511618\pi$$
−0.0364905 + 0.999334i $$0.511618\pi$$
$$752$$ −9.00000 −0.328196
$$753$$ 12.0000 0.437304
$$754$$ −36.0000 −1.31104
$$755$$ −27.0000 −0.982631
$$756$$ 0 0
$$757$$ −2.00000 −0.0726912 −0.0363456 0.999339i $$-0.511572\pi$$
−0.0363456 + 0.999339i $$0.511572\pi$$
$$758$$ −13.0000 −0.472181
$$759$$ 5.00000 0.181489
$$760$$ −18.0000 −0.652929
$$761$$ −47.0000 −1.70375 −0.851874 0.523746i $$-0.824534\pi$$
−0.851874 + 0.523746i $$0.824534\pi$$
$$762$$ 5.00000 0.181131
$$763$$ 0 0
$$764$$ 0 0
$$765$$ −15.0000 −0.542326
$$766$$ 8.00000 0.289052
$$767$$ −72.0000 −2.59977
$$768$$ −1.00000 −0.0360844
$$769$$ −46.0000 −1.65880 −0.829401 0.558653i $$-0.811318\pi$$
−0.829401 + 0.558653i $$0.811318\pi$$
$$770$$ 0 0
$$771$$ −14.0000 −0.504198
$$772$$ 6.00000 0.215945
$$773$$ −11.0000 −0.395643 −0.197821 0.980238i $$-0.563387\pi$$
−0.197821 + 0.980238i $$0.563387\pi$$
$$774$$ 10.0000 0.359443
$$775$$ −16.0000 −0.574737
$$776$$ 9.00000 0.323081
$$777$$ 0 0
$$778$$ 21.0000 0.752886
$$779$$ 30.0000 1.07486
$$780$$ −18.0000 −0.644503
$$781$$ −4.00000 −0.143131
$$782$$ −25.0000 −0.893998
$$783$$ 6.00000 0.214423
$$784$$ 0 0
$$785$$ −72.0000 −2.56979
$$786$$ 4.00000 0.142675
$$787$$ 50.0000 1.78231 0.891154 0.453701i $$-0.149897\pi$$
0.891154 + 0.453701i $$0.149897\pi$$
$$788$$ −24.0000 −0.854965
$$789$$ −24.0000 −0.854423
$$790$$ −3.00000 −0.106735
$$791$$ 0 0
$$792$$ 1.00000 0.0355335
$$793$$ −30.0000 −1.06533
$$794$$ 26.0000 0.922705
$$795$$ 6.00000 0.212798
$$796$$ −10.0000 −0.354441
$$797$$ 53.0000 1.87736 0.938678 0.344795i $$-0.112051\pi$$
0.938678 + 0.344795i $$0.112051\pi$$
$$798$$ 0 0
$$799$$ −45.0000 −1.59199
$$800$$ −4.00000 −0.141421
$$801$$ −6.00000 −0.212000
$$802$$ 6.00000 0.211867
$$803$$ 12.0000 0.423471
$$804$$ −5.00000 −0.176336
$$805$$ 0 0
$$806$$ −24.0000 −0.845364
$$807$$ −9.00000 −0.316815
$$808$$ −8.00000 −0.281439
$$809$$ 15.0000 0.527372 0.263686 0.964609i $$-0.415062\pi$$
0.263686 + 0.964609i $$0.415062\pi$$
$$810$$ 3.00000 0.105409
$$811$$ −42.0000 −1.47482 −0.737410 0.675446i $$-0.763951\pi$$
−0.737410 + 0.675446i $$0.763951\pi$$
$$812$$ 0 0
$$813$$ −24.0000 −0.841717
$$814$$ −2.00000 −0.0701000
$$815$$ 3.00000 0.105085
$$816$$ −5.00000 −0.175035
$$817$$ 60.0000 2.09913
$$818$$ 10.0000 0.349642
$$819$$ 0 0
$$820$$ 15.0000 0.523823
$$821$$ −8.00000 −0.279202 −0.139601 0.990208i $$-0.544582\pi$$
−0.139601 + 0.990208i $$0.544582\pi$$
$$822$$ −2.00000 −0.0697580
$$823$$ 10.0000 0.348578 0.174289 0.984695i $$-0.444237\pi$$
0.174289 + 0.984695i $$0.444237\pi$$
$$824$$ −2.00000 −0.0696733
$$825$$ 4.00000 0.139262
$$826$$ 0 0
$$827$$ −15.0000 −0.521601 −0.260801 0.965393i $$-0.583986\pi$$
−0.260801 + 0.965393i $$0.583986\pi$$
$$828$$ 5.00000 0.173762
$$829$$ 54.0000 1.87550 0.937749 0.347314i $$-0.112906\pi$$
0.937749 + 0.347314i $$0.112906\pi$$
$$830$$ −3.00000 −0.104132
$$831$$ −18.0000 −0.624413
$$832$$ −6.00000 −0.208013
$$833$$ 0 0
$$834$$ −22.0000 −0.761798
$$835$$ −72.0000 −2.49166
$$836$$ 6.00000 0.207514
$$837$$ 4.00000 0.138260
$$838$$ −20.0000 −0.690889
$$839$$ 9.00000 0.310715 0.155357 0.987858i $$-0.450347\pi$$
0.155357 + 0.987858i $$0.450347\pi$$
$$840$$ 0 0
$$841$$ 7.00000 0.241379
$$842$$ −8.00000 −0.275698
$$843$$ −11.0000 −0.378860
$$844$$ 14.0000 0.481900
$$845$$ −69.0000 −2.37367
$$846$$ 9.00000 0.309426
$$847$$ 0 0
$$848$$ 2.00000 0.0686803
$$849$$ −6.00000 −0.205919
$$850$$ −20.0000 −0.685994
$$851$$ −10.0000 −0.342796
$$852$$ −4.00000 −0.137038
$$853$$ −15.0000 −0.513590 −0.256795 0.966466i $$-0.582667\pi$$
−0.256795 + 0.966466i $$0.582667\pi$$
$$854$$ 0 0
$$855$$ 18.0000 0.615587
$$856$$ 13.0000 0.444331
$$857$$ −3.00000 −0.102478 −0.0512390 0.998686i $$-0.516317\pi$$
−0.0512390 + 0.998686i $$0.516317\pi$$
$$858$$ 6.00000 0.204837
$$859$$ 45.0000 1.53538 0.767690 0.640821i $$-0.221406\pi$$
0.767690 + 0.640821i $$0.221406\pi$$
$$860$$ 30.0000 1.02299
$$861$$ 0 0
$$862$$ −22.0000 −0.749323
$$863$$ −39.0000 −1.32758 −0.663788 0.747921i $$-0.731052\pi$$
−0.663788 + 0.747921i $$0.731052\pi$$
$$864$$ 1.00000 0.0340207
$$865$$ −36.0000 −1.22404
$$866$$ 11.0000 0.373795
$$867$$ −8.00000 −0.271694
$$868$$ 0 0
$$869$$ 1.00000 0.0339227
$$870$$ 18.0000 0.610257
$$871$$ −30.0000 −1.01651
$$872$$ 11.0000 0.372507
$$873$$ −9.00000 −0.304604
$$874$$ 30.0000 1.01477
$$875$$ 0 0
$$876$$ 12.0000 0.405442
$$877$$ −3.00000 −0.101303 −0.0506514 0.998716i $$-0.516130\pi$$
−0.0506514 + 0.998716i $$0.516130\pi$$
$$878$$ −5.00000 −0.168742
$$879$$ 8.00000 0.269833
$$880$$ 3.00000 0.101130
$$881$$ 26.0000 0.875962 0.437981 0.898984i $$-0.355694\pi$$
0.437981 + 0.898984i $$0.355694\pi$$
$$882$$ 0 0
$$883$$ −13.0000 −0.437485 −0.218742 0.975783i $$-0.570195\pi$$
−0.218742 + 0.975783i $$0.570195\pi$$
$$884$$ −30.0000 −1.00901
$$885$$ 36.0000 1.21013
$$886$$ −2.00000 −0.0671913
$$887$$ 24.0000 0.805841 0.402921 0.915235i $$-0.367995\pi$$
0.402921 + 0.915235i $$0.367995\pi$$
$$888$$ −2.00000 −0.0671156
$$889$$ 0 0
$$890$$ −18.0000 −0.603361
$$891$$ −1.00000 −0.0335013
$$892$$ −16.0000 −0.535720
$$893$$ 54.0000 1.80704
$$894$$ 18.0000 0.602010
$$895$$ 24.0000 0.802232
$$896$$ 0 0
$$897$$ 30.0000 1.00167
$$898$$ −2.00000 −0.0667409
$$899$$ 24.0000 0.800445
$$900$$ 4.00000 0.133333
$$901$$ 10.0000 0.333148
$$902$$ −5.00000 −0.166482
$$903$$ 0 0
$$904$$ −10.0000 −0.332595
$$905$$ −66.0000 −2.19391
$$906$$ 9.00000 0.299005
$$907$$ 23.0000 0.763702 0.381851 0.924224i $$-0.375287\pi$$
0.381851 + 0.924224i $$0.375287\pi$$
$$908$$ −3.00000 −0.0995585
$$909$$ 8.00000 0.265343
$$910$$ 0 0
$$911$$ 51.0000 1.68971 0.844853 0.534999i $$-0.179688\pi$$
0.844853 + 0.534999i $$0.179688\pi$$
$$912$$ 6.00000 0.198680
$$913$$ 1.00000 0.0330952
$$914$$ 26.0000 0.860004
$$915$$ 15.0000 0.495885
$$916$$ −26.0000 −0.859064
$$917$$ 0 0
$$918$$ 5.00000 0.165025
$$919$$ −17.0000 −0.560778 −0.280389 0.959886i $$-0.590464\pi$$
−0.280389 + 0.959886i $$0.590464\pi$$
$$920$$ 15.0000 0.494535
$$921$$ 18.0000 0.593120
$$922$$ −10.0000 −0.329332
$$923$$ −24.0000 −0.789970
$$924$$ 0 0
$$925$$ −8.00000 −0.263038
$$926$$ −28.0000 −0.920137
$$927$$ 2.00000 0.0656886
$$928$$ 6.00000 0.196960
$$929$$ 26.0000 0.853032 0.426516 0.904480i $$-0.359741\pi$$
0.426516 + 0.904480i $$0.359741\pi$$
$$930$$ 12.0000 0.393496
$$931$$ 0 0
$$932$$ −29.0000 −0.949927
$$933$$ −21.0000 −0.687509
$$934$$ 16.0000 0.523536
$$935$$ 15.0000 0.490552
$$936$$ 6.00000 0.196116
$$937$$ −30.0000 −0.980057 −0.490029 0.871706i $$-0.663014\pi$$
−0.490029 + 0.871706i $$0.663014\pi$$
$$938$$ 0 0
$$939$$ −6.00000 −0.195803
$$940$$ 27.0000 0.880643
$$941$$ −6.00000 −0.195594 −0.0977972 0.995206i $$-0.531180\pi$$
−0.0977972 + 0.995206i $$0.531180\pi$$
$$942$$ 24.0000 0.781962
$$943$$ −25.0000 −0.814112
$$944$$ 12.0000 0.390567
$$945$$ 0 0
$$946$$ −10.0000 −0.325128
$$947$$ −30.0000 −0.974869 −0.487435 0.873160i $$-0.662067\pi$$
−0.487435 + 0.873160i $$0.662067\pi$$
$$948$$ 1.00000 0.0324785
$$949$$ 72.0000 2.33722
$$950$$ 24.0000 0.778663
$$951$$ −9.00000 −0.291845
$$952$$ 0 0
$$953$$ −21.0000 −0.680257 −0.340128 0.940379i $$-0.610471\pi$$
−0.340128 + 0.940379i $$0.610471\pi$$
$$954$$ −2.00000 −0.0647524
$$955$$ 0 0
$$956$$ 26.0000 0.840900
$$957$$ −6.00000 −0.193952
$$958$$ 12.0000 0.387702
$$959$$ 0 0
$$960$$ 3.00000 0.0968246
$$961$$ −15.0000 −0.483871
$$962$$ −12.0000 −0.386896
$$963$$ −13.0000 −0.418919
$$964$$ 26.0000 0.837404
$$965$$ −18.0000 −0.579441
$$966$$ 0 0
$$967$$ −21.0000 −0.675314 −0.337657 0.941269i $$-0.609634\pi$$
−0.337657 + 0.941269i $$0.609634\pi$$
$$968$$ −1.00000 −0.0321412
$$969$$ 30.0000 0.963739
$$970$$ −27.0000 −0.866918
$$971$$ 34.0000 1.09111 0.545556 0.838074i $$-0.316319\pi$$
0.545556 + 0.838074i $$0.316319\pi$$
$$972$$ −1.00000 −0.0320750
$$973$$ 0 0
$$974$$ 4.00000 0.128168
$$975$$ 24.0000 0.768615
$$976$$ 5.00000 0.160046
$$977$$ −28.0000 −0.895799 −0.447900 0.894084i $$-0.647828\pi$$
−0.447900 + 0.894084i $$0.647828\pi$$
$$978$$ −1.00000 −0.0319765
$$979$$ 6.00000 0.191761
$$980$$ 0 0
$$981$$ −11.0000 −0.351203
$$982$$ 3.00000 0.0957338
$$983$$ 59.0000 1.88181 0.940904 0.338674i $$-0.109978\pi$$
0.940904 + 0.338674i $$0.109978\pi$$
$$984$$ −5.00000 −0.159394
$$985$$ 72.0000 2.29411
$$986$$ 30.0000 0.955395
$$987$$ 0 0
$$988$$ 36.0000 1.14531
$$989$$ −50.0000 −1.58991
$$990$$ −3.00000 −0.0953463
$$991$$ 22.0000 0.698853 0.349427 0.936964i $$-0.386376\pi$$
0.349427 + 0.936964i $$0.386376\pi$$
$$992$$ 4.00000 0.127000
$$993$$ −35.0000 −1.11069
$$994$$ 0 0
$$995$$ 30.0000 0.951064
$$996$$ 1.00000 0.0316862
$$997$$ −14.0000 −0.443384 −0.221692 0.975117i $$-0.571158\pi$$
−0.221692 + 0.975117i $$0.571158\pi$$
$$998$$ −12.0000 −0.379853
$$999$$ 2.00000 0.0632772
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 3234.2.a.a.1.1 1
3.2 odd 2 9702.2.a.ce.1.1 1
7.3 odd 6 462.2.i.a.331.1 yes 2
7.5 odd 6 462.2.i.a.67.1 2
7.6 odd 2 3234.2.a.o.1.1 1
21.5 even 6 1386.2.k.j.991.1 2
21.17 even 6 1386.2.k.j.793.1 2
21.20 even 2 9702.2.a.be.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
462.2.i.a.67.1 2 7.5 odd 6
462.2.i.a.331.1 yes 2 7.3 odd 6
1386.2.k.j.793.1 2 21.17 even 6
1386.2.k.j.991.1 2 21.5 even 6
3234.2.a.a.1.1 1 1.1 even 1 trivial
3234.2.a.o.1.1 1 7.6 odd 2
9702.2.a.be.1.1 1 21.20 even 2
9702.2.a.ce.1.1 1 3.2 odd 2