# Properties

 Label 30.2.a.a.1.1 Level $30$ Weight $2$ Character 30.1 Self dual yes Analytic conductor $0.240$ 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] = [30,2,Mod(1,30)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(30, 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("30.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$30 = 2 \cdot 3 \cdot 5$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 30.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: $$0.239551206064$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 30.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{6} -4.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$$ $$q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{6} -4.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +1.00000 q^{10} +1.00000 q^{12} +2.00000 q^{13} +4.00000 q^{14} -1.00000 q^{15} +1.00000 q^{16} +6.00000 q^{17} -1.00000 q^{18} -4.00000 q^{19} -1.00000 q^{20} -4.00000 q^{21} -1.00000 q^{24} +1.00000 q^{25} -2.00000 q^{26} +1.00000 q^{27} -4.00000 q^{28} -6.00000 q^{29} +1.00000 q^{30} +8.00000 q^{31} -1.00000 q^{32} -6.00000 q^{34} +4.00000 q^{35} +1.00000 q^{36} +2.00000 q^{37} +4.00000 q^{38} +2.00000 q^{39} +1.00000 q^{40} -6.00000 q^{41} +4.00000 q^{42} -4.00000 q^{43} -1.00000 q^{45} +1.00000 q^{48} +9.00000 q^{49} -1.00000 q^{50} +6.00000 q^{51} +2.00000 q^{52} -6.00000 q^{53} -1.00000 q^{54} +4.00000 q^{56} -4.00000 q^{57} +6.00000 q^{58} -1.00000 q^{60} -10.0000 q^{61} -8.00000 q^{62} -4.00000 q^{63} +1.00000 q^{64} -2.00000 q^{65} -4.00000 q^{67} +6.00000 q^{68} -4.00000 q^{70} -1.00000 q^{72} +2.00000 q^{73} -2.00000 q^{74} +1.00000 q^{75} -4.00000 q^{76} -2.00000 q^{78} +8.00000 q^{79} -1.00000 q^{80} +1.00000 q^{81} +6.00000 q^{82} +12.0000 q^{83} -4.00000 q^{84} -6.00000 q^{85} +4.00000 q^{86} -6.00000 q^{87} +18.0000 q^{89} +1.00000 q^{90} -8.00000 q^{91} +8.00000 q^{93} +4.00000 q^{95} -1.00000 q^{96} +2.00000 q^{97} -9.00000 q^{98} +O(q^{100})$$

## 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
$$5$$ −1.00000 −0.447214
$$6$$ −1.00000 −0.408248
$$7$$ −4.00000 −1.51186 −0.755929 0.654654i $$-0.772814\pi$$
−0.755929 + 0.654654i $$0.772814\pi$$
$$8$$ −1.00000 −0.353553
$$9$$ 1.00000 0.333333
$$10$$ 1.00000 0.316228
$$11$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$12$$ 1.00000 0.288675
$$13$$ 2.00000 0.554700 0.277350 0.960769i $$-0.410544\pi$$
0.277350 + 0.960769i $$0.410544\pi$$
$$14$$ 4.00000 1.06904
$$15$$ −1.00000 −0.258199
$$16$$ 1.00000 0.250000
$$17$$ 6.00000 1.45521 0.727607 0.685994i $$-0.240633\pi$$
0.727607 + 0.685994i $$0.240633\pi$$
$$18$$ −1.00000 −0.235702
$$19$$ −4.00000 −0.917663 −0.458831 0.888523i $$-0.651732\pi$$
−0.458831 + 0.888523i $$0.651732\pi$$
$$20$$ −1.00000 −0.223607
$$21$$ −4.00000 −0.872872
$$22$$ 0 0
$$23$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$24$$ −1.00000 −0.204124
$$25$$ 1.00000 0.200000
$$26$$ −2.00000 −0.392232
$$27$$ 1.00000 0.192450
$$28$$ −4.00000 −0.755929
$$29$$ −6.00000 −1.11417 −0.557086 0.830455i $$-0.688081\pi$$
−0.557086 + 0.830455i $$0.688081\pi$$
$$30$$ 1.00000 0.182574
$$31$$ 8.00000 1.43684 0.718421 0.695608i $$-0.244865\pi$$
0.718421 + 0.695608i $$0.244865\pi$$
$$32$$ −1.00000 −0.176777
$$33$$ 0 0
$$34$$ −6.00000 −1.02899
$$35$$ 4.00000 0.676123
$$36$$ 1.00000 0.166667
$$37$$ 2.00000 0.328798 0.164399 0.986394i $$-0.447432\pi$$
0.164399 + 0.986394i $$0.447432\pi$$
$$38$$ 4.00000 0.648886
$$39$$ 2.00000 0.320256
$$40$$ 1.00000 0.158114
$$41$$ −6.00000 −0.937043 −0.468521 0.883452i $$-0.655213\pi$$
−0.468521 + 0.883452i $$0.655213\pi$$
$$42$$ 4.00000 0.617213
$$43$$ −4.00000 −0.609994 −0.304997 0.952353i $$-0.598656\pi$$
−0.304997 + 0.952353i $$0.598656\pi$$
$$44$$ 0 0
$$45$$ −1.00000 −0.149071
$$46$$ 0 0
$$47$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$48$$ 1.00000 0.144338
$$49$$ 9.00000 1.28571
$$50$$ −1.00000 −0.141421
$$51$$ 6.00000 0.840168
$$52$$ 2.00000 0.277350
$$53$$ −6.00000 −0.824163 −0.412082 0.911147i $$-0.635198\pi$$
−0.412082 + 0.911147i $$0.635198\pi$$
$$54$$ −1.00000 −0.136083
$$55$$ 0 0
$$56$$ 4.00000 0.534522
$$57$$ −4.00000 −0.529813
$$58$$ 6.00000 0.787839
$$59$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$60$$ −1.00000 −0.129099
$$61$$ −10.0000 −1.28037 −0.640184 0.768221i $$-0.721142\pi$$
−0.640184 + 0.768221i $$0.721142\pi$$
$$62$$ −8.00000 −1.01600
$$63$$ −4.00000 −0.503953
$$64$$ 1.00000 0.125000
$$65$$ −2.00000 −0.248069
$$66$$ 0 0
$$67$$ −4.00000 −0.488678 −0.244339 0.969690i $$-0.578571\pi$$
−0.244339 + 0.969690i $$0.578571\pi$$
$$68$$ 6.00000 0.727607
$$69$$ 0 0
$$70$$ −4.00000 −0.478091
$$71$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$72$$ −1.00000 −0.117851
$$73$$ 2.00000 0.234082 0.117041 0.993127i $$-0.462659\pi$$
0.117041 + 0.993127i $$0.462659\pi$$
$$74$$ −2.00000 −0.232495
$$75$$ 1.00000 0.115470
$$76$$ −4.00000 −0.458831
$$77$$ 0 0
$$78$$ −2.00000 −0.226455
$$79$$ 8.00000 0.900070 0.450035 0.893011i $$-0.351411\pi$$
0.450035 + 0.893011i $$0.351411\pi$$
$$80$$ −1.00000 −0.111803
$$81$$ 1.00000 0.111111
$$82$$ 6.00000 0.662589
$$83$$ 12.0000 1.31717 0.658586 0.752506i $$-0.271155\pi$$
0.658586 + 0.752506i $$0.271155\pi$$
$$84$$ −4.00000 −0.436436
$$85$$ −6.00000 −0.650791
$$86$$ 4.00000 0.431331
$$87$$ −6.00000 −0.643268
$$88$$ 0 0
$$89$$ 18.0000 1.90800 0.953998 0.299813i $$-0.0969242\pi$$
0.953998 + 0.299813i $$0.0969242\pi$$
$$90$$ 1.00000 0.105409
$$91$$ −8.00000 −0.838628
$$92$$ 0 0
$$93$$ 8.00000 0.829561
$$94$$ 0 0
$$95$$ 4.00000 0.410391
$$96$$ −1.00000 −0.102062
$$97$$ 2.00000 0.203069 0.101535 0.994832i $$-0.467625\pi$$
0.101535 + 0.994832i $$0.467625\pi$$
$$98$$ −9.00000 −0.909137
$$99$$ 0 0
$$100$$ 1.00000 0.100000
$$101$$ 18.0000 1.79107 0.895533 0.444994i $$-0.146794\pi$$
0.895533 + 0.444994i $$0.146794\pi$$
$$102$$ −6.00000 −0.594089
$$103$$ −4.00000 −0.394132 −0.197066 0.980390i $$-0.563141\pi$$
−0.197066 + 0.980390i $$0.563141\pi$$
$$104$$ −2.00000 −0.196116
$$105$$ 4.00000 0.390360
$$106$$ 6.00000 0.582772
$$107$$ −12.0000 −1.16008 −0.580042 0.814587i $$-0.696964\pi$$
−0.580042 + 0.814587i $$0.696964\pi$$
$$108$$ 1.00000 0.0962250
$$109$$ −10.0000 −0.957826 −0.478913 0.877862i $$-0.658969\pi$$
−0.478913 + 0.877862i $$0.658969\pi$$
$$110$$ 0 0
$$111$$ 2.00000 0.189832
$$112$$ −4.00000 −0.377964
$$113$$ −18.0000 −1.69330 −0.846649 0.532152i $$-0.821383\pi$$
−0.846649 + 0.532152i $$0.821383\pi$$
$$114$$ 4.00000 0.374634
$$115$$ 0 0
$$116$$ −6.00000 −0.557086
$$117$$ 2.00000 0.184900
$$118$$ 0 0
$$119$$ −24.0000 −2.20008
$$120$$ 1.00000 0.0912871
$$121$$ −11.0000 −1.00000
$$122$$ 10.0000 0.905357
$$123$$ −6.00000 −0.541002
$$124$$ 8.00000 0.718421
$$125$$ −1.00000 −0.0894427
$$126$$ 4.00000 0.356348
$$127$$ 20.0000 1.77471 0.887357 0.461084i $$-0.152539\pi$$
0.887357 + 0.461084i $$0.152539\pi$$
$$128$$ −1.00000 −0.0883883
$$129$$ −4.00000 −0.352180
$$130$$ 2.00000 0.175412
$$131$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$132$$ 0 0
$$133$$ 16.0000 1.38738
$$134$$ 4.00000 0.345547
$$135$$ −1.00000 −0.0860663
$$136$$ −6.00000 −0.514496
$$137$$ 6.00000 0.512615 0.256307 0.966595i $$-0.417494\pi$$
0.256307 + 0.966595i $$0.417494\pi$$
$$138$$ 0 0
$$139$$ −4.00000 −0.339276 −0.169638 0.985506i $$-0.554260\pi$$
−0.169638 + 0.985506i $$0.554260\pi$$
$$140$$ 4.00000 0.338062
$$141$$ 0 0
$$142$$ 0 0
$$143$$ 0 0
$$144$$ 1.00000 0.0833333
$$145$$ 6.00000 0.498273
$$146$$ −2.00000 −0.165521
$$147$$ 9.00000 0.742307
$$148$$ 2.00000 0.164399
$$149$$ −6.00000 −0.491539 −0.245770 0.969328i $$-0.579041\pi$$
−0.245770 + 0.969328i $$0.579041\pi$$
$$150$$ −1.00000 −0.0816497
$$151$$ 8.00000 0.651031 0.325515 0.945537i $$-0.394462\pi$$
0.325515 + 0.945537i $$0.394462\pi$$
$$152$$ 4.00000 0.324443
$$153$$ 6.00000 0.485071
$$154$$ 0 0
$$155$$ −8.00000 −0.642575
$$156$$ 2.00000 0.160128
$$157$$ 2.00000 0.159617 0.0798087 0.996810i $$-0.474569\pi$$
0.0798087 + 0.996810i $$0.474569\pi$$
$$158$$ −8.00000 −0.636446
$$159$$ −6.00000 −0.475831
$$160$$ 1.00000 0.0790569
$$161$$ 0 0
$$162$$ −1.00000 −0.0785674
$$163$$ −4.00000 −0.313304 −0.156652 0.987654i $$-0.550070\pi$$
−0.156652 + 0.987654i $$0.550070\pi$$
$$164$$ −6.00000 −0.468521
$$165$$ 0 0
$$166$$ −12.0000 −0.931381
$$167$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$168$$ 4.00000 0.308607
$$169$$ −9.00000 −0.692308
$$170$$ 6.00000 0.460179
$$171$$ −4.00000 −0.305888
$$172$$ −4.00000 −0.304997
$$173$$ 18.0000 1.36851 0.684257 0.729241i $$-0.260127\pi$$
0.684257 + 0.729241i $$0.260127\pi$$
$$174$$ 6.00000 0.454859
$$175$$ −4.00000 −0.302372
$$176$$ 0 0
$$177$$ 0 0
$$178$$ −18.0000 −1.34916
$$179$$ 24.0000 1.79384 0.896922 0.442189i $$-0.145798\pi$$
0.896922 + 0.442189i $$0.145798\pi$$
$$180$$ −1.00000 −0.0745356
$$181$$ 14.0000 1.04061 0.520306 0.853980i $$-0.325818\pi$$
0.520306 + 0.853980i $$0.325818\pi$$
$$182$$ 8.00000 0.592999
$$183$$ −10.0000 −0.739221
$$184$$ 0 0
$$185$$ −2.00000 −0.147043
$$186$$ −8.00000 −0.586588
$$187$$ 0 0
$$188$$ 0 0
$$189$$ −4.00000 −0.290957
$$190$$ −4.00000 −0.290191
$$191$$ −24.0000 −1.73658 −0.868290 0.496058i $$-0.834780\pi$$
−0.868290 + 0.496058i $$0.834780\pi$$
$$192$$ 1.00000 0.0721688
$$193$$ −22.0000 −1.58359 −0.791797 0.610784i $$-0.790854\pi$$
−0.791797 + 0.610784i $$0.790854\pi$$
$$194$$ −2.00000 −0.143592
$$195$$ −2.00000 −0.143223
$$196$$ 9.00000 0.642857
$$197$$ −6.00000 −0.427482 −0.213741 0.976890i $$-0.568565\pi$$
−0.213741 + 0.976890i $$0.568565\pi$$
$$198$$ 0 0
$$199$$ 8.00000 0.567105 0.283552 0.958957i $$-0.408487\pi$$
0.283552 + 0.958957i $$0.408487\pi$$
$$200$$ −1.00000 −0.0707107
$$201$$ −4.00000 −0.282138
$$202$$ −18.0000 −1.26648
$$203$$ 24.0000 1.68447
$$204$$ 6.00000 0.420084
$$205$$ 6.00000 0.419058
$$206$$ 4.00000 0.278693
$$207$$ 0 0
$$208$$ 2.00000 0.138675
$$209$$ 0 0
$$210$$ −4.00000 −0.276026
$$211$$ 20.0000 1.37686 0.688428 0.725304i $$-0.258301\pi$$
0.688428 + 0.725304i $$0.258301\pi$$
$$212$$ −6.00000 −0.412082
$$213$$ 0 0
$$214$$ 12.0000 0.820303
$$215$$ 4.00000 0.272798
$$216$$ −1.00000 −0.0680414
$$217$$ −32.0000 −2.17230
$$218$$ 10.0000 0.677285
$$219$$ 2.00000 0.135147
$$220$$ 0 0
$$221$$ 12.0000 0.807207
$$222$$ −2.00000 −0.134231
$$223$$ 20.0000 1.33930 0.669650 0.742677i $$-0.266444\pi$$
0.669650 + 0.742677i $$0.266444\pi$$
$$224$$ 4.00000 0.267261
$$225$$ 1.00000 0.0666667
$$226$$ 18.0000 1.19734
$$227$$ −12.0000 −0.796468 −0.398234 0.917284i $$-0.630377\pi$$
−0.398234 + 0.917284i $$0.630377\pi$$
$$228$$ −4.00000 −0.264906
$$229$$ −10.0000 −0.660819 −0.330409 0.943838i $$-0.607187\pi$$
−0.330409 + 0.943838i $$0.607187\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 6.00000 0.393919
$$233$$ −18.0000 −1.17922 −0.589610 0.807688i $$-0.700718\pi$$
−0.589610 + 0.807688i $$0.700718\pi$$
$$234$$ −2.00000 −0.130744
$$235$$ 0 0
$$236$$ 0 0
$$237$$ 8.00000 0.519656
$$238$$ 24.0000 1.55569
$$239$$ 24.0000 1.55243 0.776215 0.630468i $$-0.217137\pi$$
0.776215 + 0.630468i $$0.217137\pi$$
$$240$$ −1.00000 −0.0645497
$$241$$ 2.00000 0.128831 0.0644157 0.997923i $$-0.479482\pi$$
0.0644157 + 0.997923i $$0.479482\pi$$
$$242$$ 11.0000 0.707107
$$243$$ 1.00000 0.0641500
$$244$$ −10.0000 −0.640184
$$245$$ −9.00000 −0.574989
$$246$$ 6.00000 0.382546
$$247$$ −8.00000 −0.509028
$$248$$ −8.00000 −0.508001
$$249$$ 12.0000 0.760469
$$250$$ 1.00000 0.0632456
$$251$$ −24.0000 −1.51487 −0.757433 0.652913i $$-0.773547\pi$$
−0.757433 + 0.652913i $$0.773547\pi$$
$$252$$ −4.00000 −0.251976
$$253$$ 0 0
$$254$$ −20.0000 −1.25491
$$255$$ −6.00000 −0.375735
$$256$$ 1.00000 0.0625000
$$257$$ −18.0000 −1.12281 −0.561405 0.827541i $$-0.689739\pi$$
−0.561405 + 0.827541i $$0.689739\pi$$
$$258$$ 4.00000 0.249029
$$259$$ −8.00000 −0.497096
$$260$$ −2.00000 −0.124035
$$261$$ −6.00000 −0.371391
$$262$$ 0 0
$$263$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$264$$ 0 0
$$265$$ 6.00000 0.368577
$$266$$ −16.0000 −0.981023
$$267$$ 18.0000 1.10158
$$268$$ −4.00000 −0.244339
$$269$$ −6.00000 −0.365826 −0.182913 0.983129i $$-0.558553\pi$$
−0.182913 + 0.983129i $$0.558553\pi$$
$$270$$ 1.00000 0.0608581
$$271$$ −16.0000 −0.971931 −0.485965 0.873978i $$-0.661532\pi$$
−0.485965 + 0.873978i $$0.661532\pi$$
$$272$$ 6.00000 0.363803
$$273$$ −8.00000 −0.484182
$$274$$ −6.00000 −0.362473
$$275$$ 0 0
$$276$$ 0 0
$$277$$ 2.00000 0.120168 0.0600842 0.998193i $$-0.480863\pi$$
0.0600842 + 0.998193i $$0.480863\pi$$
$$278$$ 4.00000 0.239904
$$279$$ 8.00000 0.478947
$$280$$ −4.00000 −0.239046
$$281$$ 18.0000 1.07379 0.536895 0.843649i $$-0.319597\pi$$
0.536895 + 0.843649i $$0.319597\pi$$
$$282$$ 0 0
$$283$$ −28.0000 −1.66443 −0.832214 0.554455i $$-0.812927\pi$$
−0.832214 + 0.554455i $$0.812927\pi$$
$$284$$ 0 0
$$285$$ 4.00000 0.236940
$$286$$ 0 0
$$287$$ 24.0000 1.41668
$$288$$ −1.00000 −0.0589256
$$289$$ 19.0000 1.11765
$$290$$ −6.00000 −0.352332
$$291$$ 2.00000 0.117242
$$292$$ 2.00000 0.117041
$$293$$ −6.00000 −0.350524 −0.175262 0.984522i $$-0.556077\pi$$
−0.175262 + 0.984522i $$0.556077\pi$$
$$294$$ −9.00000 −0.524891
$$295$$ 0 0
$$296$$ −2.00000 −0.116248
$$297$$ 0 0
$$298$$ 6.00000 0.347571
$$299$$ 0 0
$$300$$ 1.00000 0.0577350
$$301$$ 16.0000 0.922225
$$302$$ −8.00000 −0.460348
$$303$$ 18.0000 1.03407
$$304$$ −4.00000 −0.229416
$$305$$ 10.0000 0.572598
$$306$$ −6.00000 −0.342997
$$307$$ 20.0000 1.14146 0.570730 0.821138i $$-0.306660\pi$$
0.570730 + 0.821138i $$0.306660\pi$$
$$308$$ 0 0
$$309$$ −4.00000 −0.227552
$$310$$ 8.00000 0.454369
$$311$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$312$$ −2.00000 −0.113228
$$313$$ 2.00000 0.113047 0.0565233 0.998401i $$-0.481998\pi$$
0.0565233 + 0.998401i $$0.481998\pi$$
$$314$$ −2.00000 −0.112867
$$315$$ 4.00000 0.225374
$$316$$ 8.00000 0.450035
$$317$$ 18.0000 1.01098 0.505490 0.862832i $$-0.331312\pi$$
0.505490 + 0.862832i $$0.331312\pi$$
$$318$$ 6.00000 0.336463
$$319$$ 0 0
$$320$$ −1.00000 −0.0559017
$$321$$ −12.0000 −0.669775
$$322$$ 0 0
$$323$$ −24.0000 −1.33540
$$324$$ 1.00000 0.0555556
$$325$$ 2.00000 0.110940
$$326$$ 4.00000 0.221540
$$327$$ −10.0000 −0.553001
$$328$$ 6.00000 0.331295
$$329$$ 0 0
$$330$$ 0 0
$$331$$ −28.0000 −1.53902 −0.769510 0.638635i $$-0.779499\pi$$
−0.769510 + 0.638635i $$0.779499\pi$$
$$332$$ 12.0000 0.658586
$$333$$ 2.00000 0.109599
$$334$$ 0 0
$$335$$ 4.00000 0.218543
$$336$$ −4.00000 −0.218218
$$337$$ 26.0000 1.41631 0.708155 0.706057i $$-0.249528\pi$$
0.708155 + 0.706057i $$0.249528\pi$$
$$338$$ 9.00000 0.489535
$$339$$ −18.0000 −0.977626
$$340$$ −6.00000 −0.325396
$$341$$ 0 0
$$342$$ 4.00000 0.216295
$$343$$ −8.00000 −0.431959
$$344$$ 4.00000 0.215666
$$345$$ 0 0
$$346$$ −18.0000 −0.967686
$$347$$ 12.0000 0.644194 0.322097 0.946707i $$-0.395612\pi$$
0.322097 + 0.946707i $$0.395612\pi$$
$$348$$ −6.00000 −0.321634
$$349$$ −10.0000 −0.535288 −0.267644 0.963518i $$-0.586245\pi$$
−0.267644 + 0.963518i $$0.586245\pi$$
$$350$$ 4.00000 0.213809
$$351$$ 2.00000 0.106752
$$352$$ 0 0
$$353$$ 6.00000 0.319348 0.159674 0.987170i $$-0.448956\pi$$
0.159674 + 0.987170i $$0.448956\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 18.0000 0.953998
$$357$$ −24.0000 −1.27021
$$358$$ −24.0000 −1.26844
$$359$$ −24.0000 −1.26667 −0.633336 0.773877i $$-0.718315\pi$$
−0.633336 + 0.773877i $$0.718315\pi$$
$$360$$ 1.00000 0.0527046
$$361$$ −3.00000 −0.157895
$$362$$ −14.0000 −0.735824
$$363$$ −11.0000 −0.577350
$$364$$ −8.00000 −0.419314
$$365$$ −2.00000 −0.104685
$$366$$ 10.0000 0.522708
$$367$$ −28.0000 −1.46159 −0.730794 0.682598i $$-0.760850\pi$$
−0.730794 + 0.682598i $$0.760850\pi$$
$$368$$ 0 0
$$369$$ −6.00000 −0.312348
$$370$$ 2.00000 0.103975
$$371$$ 24.0000 1.24602
$$372$$ 8.00000 0.414781
$$373$$ 26.0000 1.34623 0.673114 0.739538i $$-0.264956\pi$$
0.673114 + 0.739538i $$0.264956\pi$$
$$374$$ 0 0
$$375$$ −1.00000 −0.0516398
$$376$$ 0 0
$$377$$ −12.0000 −0.618031
$$378$$ 4.00000 0.205738
$$379$$ −4.00000 −0.205466 −0.102733 0.994709i $$-0.532759\pi$$
−0.102733 + 0.994709i $$0.532759\pi$$
$$380$$ 4.00000 0.205196
$$381$$ 20.0000 1.02463
$$382$$ 24.0000 1.22795
$$383$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$384$$ −1.00000 −0.0510310
$$385$$ 0 0
$$386$$ 22.0000 1.11977
$$387$$ −4.00000 −0.203331
$$388$$ 2.00000 0.101535
$$389$$ −6.00000 −0.304212 −0.152106 0.988364i $$-0.548606\pi$$
−0.152106 + 0.988364i $$0.548606\pi$$
$$390$$ 2.00000 0.101274
$$391$$ 0 0
$$392$$ −9.00000 −0.454569
$$393$$ 0 0
$$394$$ 6.00000 0.302276
$$395$$ −8.00000 −0.402524
$$396$$ 0 0
$$397$$ −22.0000 −1.10415 −0.552074 0.833795i $$-0.686163\pi$$
−0.552074 + 0.833795i $$0.686163\pi$$
$$398$$ −8.00000 −0.401004
$$399$$ 16.0000 0.801002
$$400$$ 1.00000 0.0500000
$$401$$ −6.00000 −0.299626 −0.149813 0.988714i $$-0.547867\pi$$
−0.149813 + 0.988714i $$0.547867\pi$$
$$402$$ 4.00000 0.199502
$$403$$ 16.0000 0.797017
$$404$$ 18.0000 0.895533
$$405$$ −1.00000 −0.0496904
$$406$$ −24.0000 −1.19110
$$407$$ 0 0
$$408$$ −6.00000 −0.297044
$$409$$ 26.0000 1.28562 0.642809 0.766027i $$-0.277769\pi$$
0.642809 + 0.766027i $$0.277769\pi$$
$$410$$ −6.00000 −0.296319
$$411$$ 6.00000 0.295958
$$412$$ −4.00000 −0.197066
$$413$$ 0 0
$$414$$ 0 0
$$415$$ −12.0000 −0.589057
$$416$$ −2.00000 −0.0980581
$$417$$ −4.00000 −0.195881
$$418$$ 0 0
$$419$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$420$$ 4.00000 0.195180
$$421$$ −10.0000 −0.487370 −0.243685 0.969854i $$-0.578356\pi$$
−0.243685 + 0.969854i $$0.578356\pi$$
$$422$$ −20.0000 −0.973585
$$423$$ 0 0
$$424$$ 6.00000 0.291386
$$425$$ 6.00000 0.291043
$$426$$ 0 0
$$427$$ 40.0000 1.93574
$$428$$ −12.0000 −0.580042
$$429$$ 0 0
$$430$$ −4.00000 −0.192897
$$431$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$432$$ 1.00000 0.0481125
$$433$$ 26.0000 1.24948 0.624740 0.780833i $$-0.285205\pi$$
0.624740 + 0.780833i $$0.285205\pi$$
$$434$$ 32.0000 1.53605
$$435$$ 6.00000 0.287678
$$436$$ −10.0000 −0.478913
$$437$$ 0 0
$$438$$ −2.00000 −0.0955637
$$439$$ 8.00000 0.381819 0.190910 0.981608i $$-0.438856\pi$$
0.190910 + 0.981608i $$0.438856\pi$$
$$440$$ 0 0
$$441$$ 9.00000 0.428571
$$442$$ −12.0000 −0.570782
$$443$$ 12.0000 0.570137 0.285069 0.958507i $$-0.407984\pi$$
0.285069 + 0.958507i $$0.407984\pi$$
$$444$$ 2.00000 0.0949158
$$445$$ −18.0000 −0.853282
$$446$$ −20.0000 −0.947027
$$447$$ −6.00000 −0.283790
$$448$$ −4.00000 −0.188982
$$449$$ −6.00000 −0.283158 −0.141579 0.989927i $$-0.545218\pi$$
−0.141579 + 0.989927i $$0.545218\pi$$
$$450$$ −1.00000 −0.0471405
$$451$$ 0 0
$$452$$ −18.0000 −0.846649
$$453$$ 8.00000 0.375873
$$454$$ 12.0000 0.563188
$$455$$ 8.00000 0.375046
$$456$$ 4.00000 0.187317
$$457$$ 26.0000 1.21623 0.608114 0.793849i $$-0.291926\pi$$
0.608114 + 0.793849i $$0.291926\pi$$
$$458$$ 10.0000 0.467269
$$459$$ 6.00000 0.280056
$$460$$ 0 0
$$461$$ −30.0000 −1.39724 −0.698620 0.715493i $$-0.746202\pi$$
−0.698620 + 0.715493i $$0.746202\pi$$
$$462$$ 0 0
$$463$$ −4.00000 −0.185896 −0.0929479 0.995671i $$-0.529629\pi$$
−0.0929479 + 0.995671i $$0.529629\pi$$
$$464$$ −6.00000 −0.278543
$$465$$ −8.00000 −0.370991
$$466$$ 18.0000 0.833834
$$467$$ −36.0000 −1.66588 −0.832941 0.553362i $$-0.813345\pi$$
−0.832941 + 0.553362i $$0.813345\pi$$
$$468$$ 2.00000 0.0924500
$$469$$ 16.0000 0.738811
$$470$$ 0 0
$$471$$ 2.00000 0.0921551
$$472$$ 0 0
$$473$$ 0 0
$$474$$ −8.00000 −0.367452
$$475$$ −4.00000 −0.183533
$$476$$ −24.0000 −1.10004
$$477$$ −6.00000 −0.274721
$$478$$ −24.0000 −1.09773
$$479$$ −24.0000 −1.09659 −0.548294 0.836286i $$-0.684723\pi$$
−0.548294 + 0.836286i $$0.684723\pi$$
$$480$$ 1.00000 0.0456435
$$481$$ 4.00000 0.182384
$$482$$ −2.00000 −0.0910975
$$483$$ 0 0
$$484$$ −11.0000 −0.500000
$$485$$ −2.00000 −0.0908153
$$486$$ −1.00000 −0.0453609
$$487$$ −28.0000 −1.26880 −0.634401 0.773004i $$-0.718753\pi$$
−0.634401 + 0.773004i $$0.718753\pi$$
$$488$$ 10.0000 0.452679
$$489$$ −4.00000 −0.180886
$$490$$ 9.00000 0.406579
$$491$$ 24.0000 1.08310 0.541552 0.840667i $$-0.317837\pi$$
0.541552 + 0.840667i $$0.317837\pi$$
$$492$$ −6.00000 −0.270501
$$493$$ −36.0000 −1.62136
$$494$$ 8.00000 0.359937
$$495$$ 0 0
$$496$$ 8.00000 0.359211
$$497$$ 0 0
$$498$$ −12.0000 −0.537733
$$499$$ −4.00000 −0.179065 −0.0895323 0.995984i $$-0.528537\pi$$
−0.0895323 + 0.995984i $$0.528537\pi$$
$$500$$ −1.00000 −0.0447214
$$501$$ 0 0
$$502$$ 24.0000 1.07117
$$503$$ 24.0000 1.07011 0.535054 0.844818i $$-0.320291\pi$$
0.535054 + 0.844818i $$0.320291\pi$$
$$504$$ 4.00000 0.178174
$$505$$ −18.0000 −0.800989
$$506$$ 0 0
$$507$$ −9.00000 −0.399704
$$508$$ 20.0000 0.887357
$$509$$ −6.00000 −0.265945 −0.132973 0.991120i $$-0.542452\pi$$
−0.132973 + 0.991120i $$0.542452\pi$$
$$510$$ 6.00000 0.265684
$$511$$ −8.00000 −0.353899
$$512$$ −1.00000 −0.0441942
$$513$$ −4.00000 −0.176604
$$514$$ 18.0000 0.793946
$$515$$ 4.00000 0.176261
$$516$$ −4.00000 −0.176090
$$517$$ 0 0
$$518$$ 8.00000 0.351500
$$519$$ 18.0000 0.790112
$$520$$ 2.00000 0.0877058
$$521$$ 18.0000 0.788594 0.394297 0.918983i $$-0.370988\pi$$
0.394297 + 0.918983i $$0.370988\pi$$
$$522$$ 6.00000 0.262613
$$523$$ 20.0000 0.874539 0.437269 0.899331i $$-0.355946\pi$$
0.437269 + 0.899331i $$0.355946\pi$$
$$524$$ 0 0
$$525$$ −4.00000 −0.174574
$$526$$ 0 0
$$527$$ 48.0000 2.09091
$$528$$ 0 0
$$529$$ −23.0000 −1.00000
$$530$$ −6.00000 −0.260623
$$531$$ 0 0
$$532$$ 16.0000 0.693688
$$533$$ −12.0000 −0.519778
$$534$$ −18.0000 −0.778936
$$535$$ 12.0000 0.518805
$$536$$ 4.00000 0.172774
$$537$$ 24.0000 1.03568
$$538$$ 6.00000 0.258678
$$539$$ 0 0
$$540$$ −1.00000 −0.0430331
$$541$$ −10.0000 −0.429934 −0.214967 0.976621i $$-0.568964\pi$$
−0.214967 + 0.976621i $$0.568964\pi$$
$$542$$ 16.0000 0.687259
$$543$$ 14.0000 0.600798
$$544$$ −6.00000 −0.257248
$$545$$ 10.0000 0.428353
$$546$$ 8.00000 0.342368
$$547$$ −28.0000 −1.19719 −0.598597 0.801050i $$-0.704275\pi$$
−0.598597 + 0.801050i $$0.704275\pi$$
$$548$$ 6.00000 0.256307
$$549$$ −10.0000 −0.426790
$$550$$ 0 0
$$551$$ 24.0000 1.02243
$$552$$ 0 0
$$553$$ −32.0000 −1.36078
$$554$$ −2.00000 −0.0849719
$$555$$ −2.00000 −0.0848953
$$556$$ −4.00000 −0.169638
$$557$$ 18.0000 0.762684 0.381342 0.924434i $$-0.375462\pi$$
0.381342 + 0.924434i $$0.375462\pi$$
$$558$$ −8.00000 −0.338667
$$559$$ −8.00000 −0.338364
$$560$$ 4.00000 0.169031
$$561$$ 0 0
$$562$$ −18.0000 −0.759284
$$563$$ −12.0000 −0.505740 −0.252870 0.967500i $$-0.581374\pi$$
−0.252870 + 0.967500i $$0.581374\pi$$
$$564$$ 0 0
$$565$$ 18.0000 0.757266
$$566$$ 28.0000 1.17693
$$567$$ −4.00000 −0.167984
$$568$$ 0 0
$$569$$ 18.0000 0.754599 0.377300 0.926091i $$-0.376853\pi$$
0.377300 + 0.926091i $$0.376853\pi$$
$$570$$ −4.00000 −0.167542
$$571$$ 20.0000 0.836974 0.418487 0.908223i $$-0.362561\pi$$
0.418487 + 0.908223i $$0.362561\pi$$
$$572$$ 0 0
$$573$$ −24.0000 −1.00261
$$574$$ −24.0000 −1.00174
$$575$$ 0 0
$$576$$ 1.00000 0.0416667
$$577$$ 2.00000 0.0832611 0.0416305 0.999133i $$-0.486745\pi$$
0.0416305 + 0.999133i $$0.486745\pi$$
$$578$$ −19.0000 −0.790296
$$579$$ −22.0000 −0.914289
$$580$$ 6.00000 0.249136
$$581$$ −48.0000 −1.99138
$$582$$ −2.00000 −0.0829027
$$583$$ 0 0
$$584$$ −2.00000 −0.0827606
$$585$$ −2.00000 −0.0826898
$$586$$ 6.00000 0.247858
$$587$$ −12.0000 −0.495293 −0.247647 0.968850i $$-0.579657\pi$$
−0.247647 + 0.968850i $$0.579657\pi$$
$$588$$ 9.00000 0.371154
$$589$$ −32.0000 −1.31854
$$590$$ 0 0
$$591$$ −6.00000 −0.246807
$$592$$ 2.00000 0.0821995
$$593$$ 30.0000 1.23195 0.615976 0.787765i $$-0.288762\pi$$
0.615976 + 0.787765i $$0.288762\pi$$
$$594$$ 0 0
$$595$$ 24.0000 0.983904
$$596$$ −6.00000 −0.245770
$$597$$ 8.00000 0.327418
$$598$$ 0 0
$$599$$ −24.0000 −0.980613 −0.490307 0.871550i $$-0.663115\pi$$
−0.490307 + 0.871550i $$0.663115\pi$$
$$600$$ −1.00000 −0.0408248
$$601$$ −22.0000 −0.897399 −0.448699 0.893683i $$-0.648113\pi$$
−0.448699 + 0.893683i $$0.648113\pi$$
$$602$$ −16.0000 −0.652111
$$603$$ −4.00000 −0.162893
$$604$$ 8.00000 0.325515
$$605$$ 11.0000 0.447214
$$606$$ −18.0000 −0.731200
$$607$$ −4.00000 −0.162355 −0.0811775 0.996700i $$-0.525868\pi$$
−0.0811775 + 0.996700i $$0.525868\pi$$
$$608$$ 4.00000 0.162221
$$609$$ 24.0000 0.972529
$$610$$ −10.0000 −0.404888
$$611$$ 0 0
$$612$$ 6.00000 0.242536
$$613$$ 2.00000 0.0807792 0.0403896 0.999184i $$-0.487140\pi$$
0.0403896 + 0.999184i $$0.487140\pi$$
$$614$$ −20.0000 −0.807134
$$615$$ 6.00000 0.241943
$$616$$ 0 0
$$617$$ 30.0000 1.20775 0.603877 0.797077i $$-0.293622\pi$$
0.603877 + 0.797077i $$0.293622\pi$$
$$618$$ 4.00000 0.160904
$$619$$ 44.0000 1.76851 0.884255 0.467005i $$-0.154667\pi$$
0.884255 + 0.467005i $$0.154667\pi$$
$$620$$ −8.00000 −0.321288
$$621$$ 0 0
$$622$$ 0 0
$$623$$ −72.0000 −2.88462
$$624$$ 2.00000 0.0800641
$$625$$ 1.00000 0.0400000
$$626$$ −2.00000 −0.0799361
$$627$$ 0 0
$$628$$ 2.00000 0.0798087
$$629$$ 12.0000 0.478471
$$630$$ −4.00000 −0.159364
$$631$$ 32.0000 1.27390 0.636950 0.770905i $$-0.280196\pi$$
0.636950 + 0.770905i $$0.280196\pi$$
$$632$$ −8.00000 −0.318223
$$633$$ 20.0000 0.794929
$$634$$ −18.0000 −0.714871
$$635$$ −20.0000 −0.793676
$$636$$ −6.00000 −0.237915
$$637$$ 18.0000 0.713186
$$638$$ 0 0
$$639$$ 0 0
$$640$$ 1.00000 0.0395285
$$641$$ −30.0000 −1.18493 −0.592464 0.805597i $$-0.701845\pi$$
−0.592464 + 0.805597i $$0.701845\pi$$
$$642$$ 12.0000 0.473602
$$643$$ −4.00000 −0.157745 −0.0788723 0.996885i $$-0.525132\pi$$
−0.0788723 + 0.996885i $$0.525132\pi$$
$$644$$ 0 0
$$645$$ 4.00000 0.157500
$$646$$ 24.0000 0.944267
$$647$$ 24.0000 0.943537 0.471769 0.881722i $$-0.343616\pi$$
0.471769 + 0.881722i $$0.343616\pi$$
$$648$$ −1.00000 −0.0392837
$$649$$ 0 0
$$650$$ −2.00000 −0.0784465
$$651$$ −32.0000 −1.25418
$$652$$ −4.00000 −0.156652
$$653$$ 18.0000 0.704394 0.352197 0.935926i $$-0.385435\pi$$
0.352197 + 0.935926i $$0.385435\pi$$
$$654$$ 10.0000 0.391031
$$655$$ 0 0
$$656$$ −6.00000 −0.234261
$$657$$ 2.00000 0.0780274
$$658$$ 0 0
$$659$$ 48.0000 1.86981 0.934907 0.354892i $$-0.115482\pi$$
0.934907 + 0.354892i $$0.115482\pi$$
$$660$$ 0 0
$$661$$ 14.0000 0.544537 0.272268 0.962221i $$-0.412226\pi$$
0.272268 + 0.962221i $$0.412226\pi$$
$$662$$ 28.0000 1.08825
$$663$$ 12.0000 0.466041
$$664$$ −12.0000 −0.465690
$$665$$ −16.0000 −0.620453
$$666$$ −2.00000 −0.0774984
$$667$$ 0 0
$$668$$ 0 0
$$669$$ 20.0000 0.773245
$$670$$ −4.00000 −0.154533
$$671$$ 0 0
$$672$$ 4.00000 0.154303
$$673$$ 26.0000 1.00223 0.501113 0.865382i $$-0.332924\pi$$
0.501113 + 0.865382i $$0.332924\pi$$
$$674$$ −26.0000 −1.00148
$$675$$ 1.00000 0.0384900
$$676$$ −9.00000 −0.346154
$$677$$ −6.00000 −0.230599 −0.115299 0.993331i $$-0.536783\pi$$
−0.115299 + 0.993331i $$0.536783\pi$$
$$678$$ 18.0000 0.691286
$$679$$ −8.00000 −0.307012
$$680$$ 6.00000 0.230089
$$681$$ −12.0000 −0.459841
$$682$$ 0 0
$$683$$ −12.0000 −0.459167 −0.229584 0.973289i $$-0.573736\pi$$
−0.229584 + 0.973289i $$0.573736\pi$$
$$684$$ −4.00000 −0.152944
$$685$$ −6.00000 −0.229248
$$686$$ 8.00000 0.305441
$$687$$ −10.0000 −0.381524
$$688$$ −4.00000 −0.152499
$$689$$ −12.0000 −0.457164
$$690$$ 0 0
$$691$$ 44.0000 1.67384 0.836919 0.547326i $$-0.184354\pi$$
0.836919 + 0.547326i $$0.184354\pi$$
$$692$$ 18.0000 0.684257
$$693$$ 0 0
$$694$$ −12.0000 −0.455514
$$695$$ 4.00000 0.151729
$$696$$ 6.00000 0.227429
$$697$$ −36.0000 −1.36360
$$698$$ 10.0000 0.378506
$$699$$ −18.0000 −0.680823
$$700$$ −4.00000 −0.151186
$$701$$ −6.00000 −0.226617 −0.113308 0.993560i $$-0.536145\pi$$
−0.113308 + 0.993560i $$0.536145\pi$$
$$702$$ −2.00000 −0.0754851
$$703$$ −8.00000 −0.301726
$$704$$ 0 0
$$705$$ 0 0
$$706$$ −6.00000 −0.225813
$$707$$ −72.0000 −2.70784
$$708$$ 0 0
$$709$$ 38.0000 1.42712 0.713560 0.700594i $$-0.247082\pi$$
0.713560 + 0.700594i $$0.247082\pi$$
$$710$$ 0 0
$$711$$ 8.00000 0.300023
$$712$$ −18.0000 −0.674579
$$713$$ 0 0
$$714$$ 24.0000 0.898177
$$715$$ 0 0
$$716$$ 24.0000 0.896922
$$717$$ 24.0000 0.896296
$$718$$ 24.0000 0.895672
$$719$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$720$$ −1.00000 −0.0372678
$$721$$ 16.0000 0.595871
$$722$$ 3.00000 0.111648
$$723$$ 2.00000 0.0743808
$$724$$ 14.0000 0.520306
$$725$$ −6.00000 −0.222834
$$726$$ 11.0000 0.408248
$$727$$ −28.0000 −1.03846 −0.519231 0.854634i $$-0.673782\pi$$
−0.519231 + 0.854634i $$0.673782\pi$$
$$728$$ 8.00000 0.296500
$$729$$ 1.00000 0.0370370
$$730$$ 2.00000 0.0740233
$$731$$ −24.0000 −0.887672
$$732$$ −10.0000 −0.369611
$$733$$ −22.0000 −0.812589 −0.406294 0.913742i $$-0.633179\pi$$
−0.406294 + 0.913742i $$0.633179\pi$$
$$734$$ 28.0000 1.03350
$$735$$ −9.00000 −0.331970
$$736$$ 0 0
$$737$$ 0 0
$$738$$ 6.00000 0.220863
$$739$$ −52.0000 −1.91285 −0.956425 0.291977i $$-0.905687\pi$$
−0.956425 + 0.291977i $$0.905687\pi$$
$$740$$ −2.00000 −0.0735215
$$741$$ −8.00000 −0.293887
$$742$$ −24.0000 −0.881068
$$743$$ −24.0000 −0.880475 −0.440237 0.897881i $$-0.645106\pi$$
−0.440237 + 0.897881i $$0.645106\pi$$
$$744$$ −8.00000 −0.293294
$$745$$ 6.00000 0.219823
$$746$$ −26.0000 −0.951928
$$747$$ 12.0000 0.439057
$$748$$ 0 0
$$749$$ 48.0000 1.75388
$$750$$ 1.00000 0.0365148
$$751$$ −40.0000 −1.45962 −0.729810 0.683650i $$-0.760392\pi$$
−0.729810 + 0.683650i $$0.760392\pi$$
$$752$$ 0 0
$$753$$ −24.0000 −0.874609
$$754$$ 12.0000 0.437014
$$755$$ −8.00000 −0.291150
$$756$$ −4.00000 −0.145479
$$757$$ 2.00000 0.0726912 0.0363456 0.999339i $$-0.488428\pi$$
0.0363456 + 0.999339i $$0.488428\pi$$
$$758$$ 4.00000 0.145287
$$759$$ 0 0
$$760$$ −4.00000 −0.145095
$$761$$ 18.0000 0.652499 0.326250 0.945284i $$-0.394215\pi$$
0.326250 + 0.945284i $$0.394215\pi$$
$$762$$ −20.0000 −0.724524
$$763$$ 40.0000 1.44810
$$764$$ −24.0000 −0.868290
$$765$$ −6.00000 −0.216930
$$766$$ 0 0
$$767$$ 0 0
$$768$$ 1.00000 0.0360844
$$769$$ 2.00000 0.0721218 0.0360609 0.999350i $$-0.488519\pi$$
0.0360609 + 0.999350i $$0.488519\pi$$
$$770$$ 0 0
$$771$$ −18.0000 −0.648254
$$772$$ −22.0000 −0.791797
$$773$$ 42.0000 1.51064 0.755318 0.655359i $$-0.227483\pi$$
0.755318 + 0.655359i $$0.227483\pi$$
$$774$$ 4.00000 0.143777
$$775$$ 8.00000 0.287368
$$776$$ −2.00000 −0.0717958
$$777$$ −8.00000 −0.286998
$$778$$ 6.00000 0.215110
$$779$$ 24.0000 0.859889
$$780$$ −2.00000 −0.0716115
$$781$$ 0 0
$$782$$ 0 0
$$783$$ −6.00000 −0.214423
$$784$$ 9.00000 0.321429
$$785$$ −2.00000 −0.0713831
$$786$$ 0 0
$$787$$ −4.00000 −0.142585 −0.0712923 0.997455i $$-0.522712\pi$$
−0.0712923 + 0.997455i $$0.522712\pi$$
$$788$$ −6.00000 −0.213741
$$789$$ 0 0
$$790$$ 8.00000 0.284627
$$791$$ 72.0000 2.56003
$$792$$ 0 0
$$793$$ −20.0000 −0.710221
$$794$$ 22.0000 0.780751
$$795$$ 6.00000 0.212798
$$796$$ 8.00000 0.283552
$$797$$ −30.0000 −1.06265 −0.531327 0.847167i $$-0.678307\pi$$
−0.531327 + 0.847167i $$0.678307\pi$$
$$798$$ −16.0000 −0.566394
$$799$$ 0 0
$$800$$ −1.00000 −0.0353553
$$801$$ 18.0000 0.635999
$$802$$ 6.00000 0.211867
$$803$$ 0 0
$$804$$ −4.00000 −0.141069
$$805$$ 0 0
$$806$$ −16.0000 −0.563576
$$807$$ −6.00000 −0.211210
$$808$$ −18.0000 −0.633238
$$809$$ −54.0000 −1.89854 −0.949269 0.314464i $$-0.898175\pi$$
−0.949269 + 0.314464i $$0.898175\pi$$
$$810$$ 1.00000 0.0351364
$$811$$ −4.00000 −0.140459 −0.0702295 0.997531i $$-0.522373\pi$$
−0.0702295 + 0.997531i $$0.522373\pi$$
$$812$$ 24.0000 0.842235
$$813$$ −16.0000 −0.561144
$$814$$ 0 0
$$815$$ 4.00000 0.140114
$$816$$ 6.00000 0.210042
$$817$$ 16.0000 0.559769
$$818$$ −26.0000 −0.909069
$$819$$ −8.00000 −0.279543
$$820$$ 6.00000 0.209529
$$821$$ 18.0000 0.628204 0.314102 0.949389i $$-0.398297\pi$$
0.314102 + 0.949389i $$0.398297\pi$$
$$822$$ −6.00000 −0.209274
$$823$$ 20.0000 0.697156 0.348578 0.937280i $$-0.386665\pi$$
0.348578 + 0.937280i $$0.386665\pi$$
$$824$$ 4.00000 0.139347
$$825$$ 0 0
$$826$$ 0 0
$$827$$ 12.0000 0.417281 0.208640 0.977992i $$-0.433096\pi$$
0.208640 + 0.977992i $$0.433096\pi$$
$$828$$ 0 0
$$829$$ 38.0000 1.31979 0.659897 0.751356i $$-0.270600\pi$$
0.659897 + 0.751356i $$0.270600\pi$$
$$830$$ 12.0000 0.416526
$$831$$ 2.00000 0.0693792
$$832$$ 2.00000 0.0693375
$$833$$ 54.0000 1.87099
$$834$$ 4.00000 0.138509
$$835$$ 0 0
$$836$$ 0 0
$$837$$ 8.00000 0.276520
$$838$$ 0 0
$$839$$ 24.0000 0.828572 0.414286 0.910147i $$-0.364031\pi$$
0.414286 + 0.910147i $$0.364031\pi$$
$$840$$ −4.00000 −0.138013
$$841$$ 7.00000 0.241379
$$842$$ 10.0000 0.344623
$$843$$ 18.0000 0.619953
$$844$$ 20.0000 0.688428
$$845$$ 9.00000 0.309609
$$846$$ 0 0
$$847$$ 44.0000 1.51186
$$848$$ −6.00000 −0.206041
$$849$$ −28.0000 −0.960958
$$850$$ −6.00000 −0.205798
$$851$$ 0 0
$$852$$ 0 0
$$853$$ −46.0000 −1.57501 −0.787505 0.616308i $$-0.788628\pi$$
−0.787505 + 0.616308i $$0.788628\pi$$
$$854$$ −40.0000 −1.36877
$$855$$ 4.00000 0.136797
$$856$$ 12.0000 0.410152
$$857$$ −18.0000 −0.614868 −0.307434 0.951569i $$-0.599470\pi$$
−0.307434 + 0.951569i $$0.599470\pi$$
$$858$$ 0 0
$$859$$ −4.00000 −0.136478 −0.0682391 0.997669i $$-0.521738\pi$$
−0.0682391 + 0.997669i $$0.521738\pi$$
$$860$$ 4.00000 0.136399
$$861$$ 24.0000 0.817918
$$862$$ 0 0
$$863$$ 24.0000 0.816970 0.408485 0.912765i $$-0.366057\pi$$
0.408485 + 0.912765i $$0.366057\pi$$
$$864$$ −1.00000 −0.0340207
$$865$$ −18.0000 −0.612018
$$866$$ −26.0000 −0.883516
$$867$$ 19.0000 0.645274
$$868$$ −32.0000 −1.08615
$$869$$ 0 0
$$870$$ −6.00000 −0.203419
$$871$$ −8.00000 −0.271070
$$872$$ 10.0000 0.338643
$$873$$ 2.00000 0.0676897
$$874$$ 0 0
$$875$$ 4.00000 0.135225
$$876$$ 2.00000 0.0675737
$$877$$ 2.00000 0.0675352 0.0337676 0.999430i $$-0.489249\pi$$
0.0337676 + 0.999430i $$0.489249\pi$$
$$878$$ −8.00000 −0.269987
$$879$$ −6.00000 −0.202375
$$880$$ 0 0
$$881$$ −54.0000 −1.81931 −0.909653 0.415369i $$-0.863653\pi$$
−0.909653 + 0.415369i $$0.863653\pi$$
$$882$$ −9.00000 −0.303046
$$883$$ −4.00000 −0.134611 −0.0673054 0.997732i $$-0.521440\pi$$
−0.0673054 + 0.997732i $$0.521440\pi$$
$$884$$ 12.0000 0.403604
$$885$$ 0 0
$$886$$ −12.0000 −0.403148
$$887$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$888$$ −2.00000 −0.0671156
$$889$$ −80.0000 −2.68311
$$890$$ 18.0000 0.603361
$$891$$ 0 0
$$892$$ 20.0000 0.669650
$$893$$ 0 0
$$894$$ 6.00000 0.200670
$$895$$ −24.0000 −0.802232
$$896$$ 4.00000 0.133631
$$897$$ 0 0
$$898$$ 6.00000 0.200223
$$899$$ −48.0000 −1.60089
$$900$$ 1.00000 0.0333333
$$901$$ −36.0000 −1.19933
$$902$$ 0 0
$$903$$ 16.0000 0.532447
$$904$$ 18.0000 0.598671
$$905$$ −14.0000 −0.465376
$$906$$ −8.00000 −0.265782
$$907$$ 44.0000 1.46100 0.730498 0.682915i $$-0.239288\pi$$
0.730498 + 0.682915i $$0.239288\pi$$
$$908$$ −12.0000 −0.398234
$$909$$ 18.0000 0.597022
$$910$$ −8.00000 −0.265197
$$911$$ −48.0000 −1.59031 −0.795155 0.606406i $$-0.792611\pi$$
−0.795155 + 0.606406i $$0.792611\pi$$
$$912$$ −4.00000 −0.132453
$$913$$ 0 0
$$914$$ −26.0000 −0.860004
$$915$$ 10.0000 0.330590
$$916$$ −10.0000 −0.330409
$$917$$ 0 0
$$918$$ −6.00000 −0.198030
$$919$$ −16.0000 −0.527791 −0.263896 0.964551i $$-0.585007\pi$$
−0.263896 + 0.964551i $$0.585007\pi$$
$$920$$ 0 0
$$921$$ 20.0000 0.659022
$$922$$ 30.0000 0.987997
$$923$$ 0 0
$$924$$ 0 0
$$925$$ 2.00000 0.0657596
$$926$$ 4.00000 0.131448
$$927$$ −4.00000 −0.131377
$$928$$ 6.00000 0.196960
$$929$$ −6.00000 −0.196854 −0.0984268 0.995144i $$-0.531381\pi$$
−0.0984268 + 0.995144i $$0.531381\pi$$
$$930$$ 8.00000 0.262330
$$931$$ −36.0000 −1.17985
$$932$$ −18.0000 −0.589610
$$933$$ 0 0
$$934$$ 36.0000 1.17796
$$935$$ 0 0
$$936$$ −2.00000 −0.0653720
$$937$$ 26.0000 0.849383 0.424691 0.905338i $$-0.360383\pi$$
0.424691 + 0.905338i $$0.360383\pi$$
$$938$$ −16.0000 −0.522419
$$939$$ 2.00000 0.0652675
$$940$$ 0 0
$$941$$ 18.0000 0.586783 0.293392 0.955992i $$-0.405216\pi$$
0.293392 + 0.955992i $$0.405216\pi$$
$$942$$ −2.00000 −0.0651635
$$943$$ 0 0
$$944$$ 0 0
$$945$$ 4.00000 0.130120
$$946$$ 0 0
$$947$$ 36.0000 1.16984 0.584921 0.811090i $$-0.301125\pi$$
0.584921 + 0.811090i $$0.301125\pi$$
$$948$$ 8.00000 0.259828
$$949$$ 4.00000 0.129845
$$950$$ 4.00000 0.129777
$$951$$ 18.0000 0.583690
$$952$$ 24.0000 0.777844
$$953$$ 6.00000 0.194359 0.0971795 0.995267i $$-0.469018\pi$$
0.0971795 + 0.995267i $$0.469018\pi$$
$$954$$ 6.00000 0.194257
$$955$$ 24.0000 0.776622
$$956$$ 24.0000 0.776215
$$957$$ 0 0
$$958$$ 24.0000 0.775405
$$959$$ −24.0000 −0.775000
$$960$$ −1.00000 −0.0322749
$$961$$ 33.0000 1.06452
$$962$$ −4.00000 −0.128965
$$963$$ −12.0000 −0.386695
$$964$$ 2.00000 0.0644157
$$965$$ 22.0000 0.708205
$$966$$ 0 0
$$967$$ −4.00000 −0.128631 −0.0643157 0.997930i $$-0.520486\pi$$
−0.0643157 + 0.997930i $$0.520486\pi$$
$$968$$ 11.0000 0.353553
$$969$$ −24.0000 −0.770991
$$970$$ 2.00000 0.0642161
$$971$$ 24.0000 0.770197 0.385098 0.922876i $$-0.374168\pi$$
0.385098 + 0.922876i $$0.374168\pi$$
$$972$$ 1.00000 0.0320750
$$973$$ 16.0000 0.512936
$$974$$ 28.0000 0.897178
$$975$$ 2.00000 0.0640513
$$976$$ −10.0000 −0.320092
$$977$$ −42.0000 −1.34370 −0.671850 0.740688i $$-0.734500\pi$$
−0.671850 + 0.740688i $$0.734500\pi$$
$$978$$ 4.00000 0.127906
$$979$$ 0 0
$$980$$ −9.00000 −0.287494
$$981$$ −10.0000 −0.319275
$$982$$ −24.0000 −0.765871
$$983$$ −24.0000 −0.765481 −0.382741 0.923856i $$-0.625020\pi$$
−0.382741 + 0.923856i $$0.625020\pi$$
$$984$$ 6.00000 0.191273
$$985$$ 6.00000 0.191176
$$986$$ 36.0000 1.14647
$$987$$ 0 0
$$988$$ −8.00000 −0.254514
$$989$$ 0 0
$$990$$ 0 0
$$991$$ −16.0000 −0.508257 −0.254128 0.967170i $$-0.581789\pi$$
−0.254128 + 0.967170i $$0.581789\pi$$
$$992$$ −8.00000 −0.254000
$$993$$ −28.0000 −0.888553
$$994$$ 0 0
$$995$$ −8.00000 −0.253617
$$996$$ 12.0000 0.380235
$$997$$ 26.0000 0.823428 0.411714 0.911313i $$-0.364930\pi$$
0.411714 + 0.911313i $$0.364930\pi$$
$$998$$ 4.00000 0.126618
$$999$$ 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 30.2.a.a.1.1 1
3.2 odd 2 90.2.a.c.1.1 1
4.3 odd 2 240.2.a.b.1.1 1
5.2 odd 4 150.2.c.a.49.1 2
5.3 odd 4 150.2.c.a.49.2 2
5.4 even 2 150.2.a.b.1.1 1
7.2 even 3 1470.2.i.o.361.1 2
7.3 odd 6 1470.2.i.q.961.1 2
7.4 even 3 1470.2.i.o.961.1 2
7.5 odd 6 1470.2.i.q.361.1 2
7.6 odd 2 1470.2.a.d.1.1 1
8.3 odd 2 960.2.a.p.1.1 1
8.5 even 2 960.2.a.e.1.1 1
9.2 odd 6 810.2.e.b.271.1 2
9.4 even 3 810.2.e.l.541.1 2
9.5 odd 6 810.2.e.b.541.1 2
9.7 even 3 810.2.e.l.271.1 2
11.10 odd 2 3630.2.a.w.1.1 1
12.11 even 2 720.2.a.j.1.1 1
13.5 odd 4 5070.2.b.k.1351.2 2
13.8 odd 4 5070.2.b.k.1351.1 2
13.12 even 2 5070.2.a.w.1.1 1
15.2 even 4 450.2.c.b.199.2 2
15.8 even 4 450.2.c.b.199.1 2
15.14 odd 2 450.2.a.d.1.1 1
16.3 odd 4 3840.2.k.f.1921.1 2
16.5 even 4 3840.2.k.y.1921.1 2
16.11 odd 4 3840.2.k.f.1921.2 2
16.13 even 4 3840.2.k.y.1921.2 2
17.16 even 2 8670.2.a.g.1.1 1
20.3 even 4 1200.2.f.e.49.1 2
20.7 even 4 1200.2.f.e.49.2 2
20.19 odd 2 1200.2.a.k.1.1 1
21.20 even 2 4410.2.a.z.1.1 1
24.5 odd 2 2880.2.a.a.1.1 1
24.11 even 2 2880.2.a.q.1.1 1
35.34 odd 2 7350.2.a.ct.1.1 1
40.3 even 4 4800.2.f.w.3649.2 2
40.13 odd 4 4800.2.f.p.3649.1 2
40.19 odd 2 4800.2.a.d.1.1 1
40.27 even 4 4800.2.f.w.3649.1 2
40.29 even 2 4800.2.a.cq.1.1 1
40.37 odd 4 4800.2.f.p.3649.2 2
60.23 odd 4 3600.2.f.i.2449.1 2
60.47 odd 4 3600.2.f.i.2449.2 2
60.59 even 2 3600.2.a.f.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
30.2.a.a.1.1 1 1.1 even 1 trivial
90.2.a.c.1.1 1 3.2 odd 2
150.2.a.b.1.1 1 5.4 even 2
150.2.c.a.49.1 2 5.2 odd 4
150.2.c.a.49.2 2 5.3 odd 4
240.2.a.b.1.1 1 4.3 odd 2
450.2.a.d.1.1 1 15.14 odd 2
450.2.c.b.199.1 2 15.8 even 4
450.2.c.b.199.2 2 15.2 even 4
720.2.a.j.1.1 1 12.11 even 2
810.2.e.b.271.1 2 9.2 odd 6
810.2.e.b.541.1 2 9.5 odd 6
810.2.e.l.271.1 2 9.7 even 3
810.2.e.l.541.1 2 9.4 even 3
960.2.a.e.1.1 1 8.5 even 2
960.2.a.p.1.1 1 8.3 odd 2
1200.2.a.k.1.1 1 20.19 odd 2
1200.2.f.e.49.1 2 20.3 even 4
1200.2.f.e.49.2 2 20.7 even 4
1470.2.a.d.1.1 1 7.6 odd 2
1470.2.i.o.361.1 2 7.2 even 3
1470.2.i.o.961.1 2 7.4 even 3
1470.2.i.q.361.1 2 7.5 odd 6
1470.2.i.q.961.1 2 7.3 odd 6
2880.2.a.a.1.1 1 24.5 odd 2
2880.2.a.q.1.1 1 24.11 even 2
3600.2.a.f.1.1 1 60.59 even 2
3600.2.f.i.2449.1 2 60.23 odd 4
3600.2.f.i.2449.2 2 60.47 odd 4
3630.2.a.w.1.1 1 11.10 odd 2
3840.2.k.f.1921.1 2 16.3 odd 4
3840.2.k.f.1921.2 2 16.11 odd 4
3840.2.k.y.1921.1 2 16.5 even 4
3840.2.k.y.1921.2 2 16.13 even 4
4410.2.a.z.1.1 1 21.20 even 2
4800.2.a.d.1.1 1 40.19 odd 2
4800.2.a.cq.1.1 1 40.29 even 2
4800.2.f.p.3649.1 2 40.13 odd 4
4800.2.f.p.3649.2 2 40.37 odd 4
4800.2.f.w.3649.1 2 40.27 even 4
4800.2.f.w.3649.2 2 40.3 even 4
5070.2.a.w.1.1 1 13.12 even 2
5070.2.b.k.1351.1 2 13.8 odd 4
5070.2.b.k.1351.2 2 13.5 odd 4
7350.2.a.ct.1.1 1 35.34 odd 2
8670.2.a.g.1.1 1 17.16 even 2