# Properties

 Label 37.2.a.a.1.1 Level $37$ Weight $2$ Character 37.1 Self dual yes Analytic conductor $0.295$ Analytic rank $1$ Dimension $1$ CM no Inner twists $1$

# Learn more

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [37,2,Mod(1,37)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(37, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("37.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$37$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 37.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: $$0.295446487479$$ Analytic rank: $$1$$ 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 $$\chi$$ $$=$$ 37.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-2.00000 q^{2} -3.00000 q^{3} +2.00000 q^{4} -2.00000 q^{5} +6.00000 q^{6} -1.00000 q^{7} +6.00000 q^{9} +O(q^{10})$$ $$q-2.00000 q^{2} -3.00000 q^{3} +2.00000 q^{4} -2.00000 q^{5} +6.00000 q^{6} -1.00000 q^{7} +6.00000 q^{9} +4.00000 q^{10} -5.00000 q^{11} -6.00000 q^{12} -2.00000 q^{13} +2.00000 q^{14} +6.00000 q^{15} -4.00000 q^{16} -12.0000 q^{18} -4.00000 q^{20} +3.00000 q^{21} +10.0000 q^{22} +2.00000 q^{23} -1.00000 q^{25} +4.00000 q^{26} -9.00000 q^{27} -2.00000 q^{28} +6.00000 q^{29} -12.0000 q^{30} -4.00000 q^{31} +8.00000 q^{32} +15.0000 q^{33} +2.00000 q^{35} +12.0000 q^{36} -1.00000 q^{37} +6.00000 q^{39} -9.00000 q^{41} -6.00000 q^{42} +2.00000 q^{43} -10.0000 q^{44} -12.0000 q^{45} -4.00000 q^{46} -9.00000 q^{47} +12.0000 q^{48} -6.00000 q^{49} +2.00000 q^{50} -4.00000 q^{52} +1.00000 q^{53} +18.0000 q^{54} +10.0000 q^{55} -12.0000 q^{58} +8.00000 q^{59} +12.0000 q^{60} -8.00000 q^{61} +8.00000 q^{62} -6.00000 q^{63} -8.00000 q^{64} +4.00000 q^{65} -30.0000 q^{66} +8.00000 q^{67} -6.00000 q^{69} -4.00000 q^{70} +9.00000 q^{71} -1.00000 q^{73} +2.00000 q^{74} +3.00000 q^{75} +5.00000 q^{77} -12.0000 q^{78} +4.00000 q^{79} +8.00000 q^{80} +9.00000 q^{81} +18.0000 q^{82} -15.0000 q^{83} +6.00000 q^{84} -4.00000 q^{86} -18.0000 q^{87} +4.00000 q^{89} +24.0000 q^{90} +2.00000 q^{91} +4.00000 q^{92} +12.0000 q^{93} +18.0000 q^{94} -24.0000 q^{96} +4.00000 q^{97} +12.0000 q^{98} -30.0000 q^{99} +O(q^{100})$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −2.00000 −1.41421 −0.707107 0.707107i $$-0.750000\pi$$
−0.707107 + 0.707107i $$0.750000\pi$$
$$3$$ −3.00000 −1.73205 −0.866025 0.500000i $$-0.833333\pi$$
−0.866025 + 0.500000i $$0.833333\pi$$
$$4$$ 2.00000 1.00000
$$5$$ −2.00000 −0.894427 −0.447214 0.894427i $$-0.647584\pi$$
−0.447214 + 0.894427i $$0.647584\pi$$
$$6$$ 6.00000 2.44949
$$7$$ −1.00000 −0.377964 −0.188982 0.981981i $$-0.560519\pi$$
−0.188982 + 0.981981i $$0.560519\pi$$
$$8$$ 0 0
$$9$$ 6.00000 2.00000
$$10$$ 4.00000 1.26491
$$11$$ −5.00000 −1.50756 −0.753778 0.657129i $$-0.771771\pi$$
−0.753778 + 0.657129i $$0.771771\pi$$
$$12$$ −6.00000 −1.73205
$$13$$ −2.00000 −0.554700 −0.277350 0.960769i $$-0.589456\pi$$
−0.277350 + 0.960769i $$0.589456\pi$$
$$14$$ 2.00000 0.534522
$$15$$ 6.00000 1.54919
$$16$$ −4.00000 −1.00000
$$17$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$18$$ −12.0000 −2.82843
$$19$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$20$$ −4.00000 −0.894427
$$21$$ 3.00000 0.654654
$$22$$ 10.0000 2.13201
$$23$$ 2.00000 0.417029 0.208514 0.978019i $$-0.433137\pi$$
0.208514 + 0.978019i $$0.433137\pi$$
$$24$$ 0 0
$$25$$ −1.00000 −0.200000
$$26$$ 4.00000 0.784465
$$27$$ −9.00000 −1.73205
$$28$$ −2.00000 −0.377964
$$29$$ 6.00000 1.11417 0.557086 0.830455i $$-0.311919\pi$$
0.557086 + 0.830455i $$0.311919\pi$$
$$30$$ −12.0000 −2.19089
$$31$$ −4.00000 −0.718421 −0.359211 0.933257i $$-0.616954\pi$$
−0.359211 + 0.933257i $$0.616954\pi$$
$$32$$ 8.00000 1.41421
$$33$$ 15.0000 2.61116
$$34$$ 0 0
$$35$$ 2.00000 0.338062
$$36$$ 12.0000 2.00000
$$37$$ −1.00000 −0.164399
$$38$$ 0 0
$$39$$ 6.00000 0.960769
$$40$$ 0 0
$$41$$ −9.00000 −1.40556 −0.702782 0.711405i $$-0.748059\pi$$
−0.702782 + 0.711405i $$0.748059\pi$$
$$42$$ −6.00000 −0.925820
$$43$$ 2.00000 0.304997 0.152499 0.988304i $$-0.451268\pi$$
0.152499 + 0.988304i $$0.451268\pi$$
$$44$$ −10.0000 −1.50756
$$45$$ −12.0000 −1.78885
$$46$$ −4.00000 −0.589768
$$47$$ −9.00000 −1.31278 −0.656392 0.754420i $$-0.727918\pi$$
−0.656392 + 0.754420i $$0.727918\pi$$
$$48$$ 12.0000 1.73205
$$49$$ −6.00000 −0.857143
$$50$$ 2.00000 0.282843
$$51$$ 0 0
$$52$$ −4.00000 −0.554700
$$53$$ 1.00000 0.137361 0.0686803 0.997639i $$-0.478121\pi$$
0.0686803 + 0.997639i $$0.478121\pi$$
$$54$$ 18.0000 2.44949
$$55$$ 10.0000 1.34840
$$56$$ 0 0
$$57$$ 0 0
$$58$$ −12.0000 −1.57568
$$59$$ 8.00000 1.04151 0.520756 0.853706i $$-0.325650\pi$$
0.520756 + 0.853706i $$0.325650\pi$$
$$60$$ 12.0000 1.54919
$$61$$ −8.00000 −1.02430 −0.512148 0.858898i $$-0.671150\pi$$
−0.512148 + 0.858898i $$0.671150\pi$$
$$62$$ 8.00000 1.01600
$$63$$ −6.00000 −0.755929
$$64$$ −8.00000 −1.00000
$$65$$ 4.00000 0.496139
$$66$$ −30.0000 −3.69274
$$67$$ 8.00000 0.977356 0.488678 0.872464i $$-0.337479\pi$$
0.488678 + 0.872464i $$0.337479\pi$$
$$68$$ 0 0
$$69$$ −6.00000 −0.722315
$$70$$ −4.00000 −0.478091
$$71$$ 9.00000 1.06810 0.534052 0.845452i $$-0.320669\pi$$
0.534052 + 0.845452i $$0.320669\pi$$
$$72$$ 0 0
$$73$$ −1.00000 −0.117041 −0.0585206 0.998286i $$-0.518638\pi$$
−0.0585206 + 0.998286i $$0.518638\pi$$
$$74$$ 2.00000 0.232495
$$75$$ 3.00000 0.346410
$$76$$ 0 0
$$77$$ 5.00000 0.569803
$$78$$ −12.0000 −1.35873
$$79$$ 4.00000 0.450035 0.225018 0.974355i $$-0.427756\pi$$
0.225018 + 0.974355i $$0.427756\pi$$
$$80$$ 8.00000 0.894427
$$81$$ 9.00000 1.00000
$$82$$ 18.0000 1.98777
$$83$$ −15.0000 −1.64646 −0.823232 0.567705i $$-0.807831\pi$$
−0.823232 + 0.567705i $$0.807831\pi$$
$$84$$ 6.00000 0.654654
$$85$$ 0 0
$$86$$ −4.00000 −0.431331
$$87$$ −18.0000 −1.92980
$$88$$ 0 0
$$89$$ 4.00000 0.423999 0.212000 0.977270i $$-0.432002\pi$$
0.212000 + 0.977270i $$0.432002\pi$$
$$90$$ 24.0000 2.52982
$$91$$ 2.00000 0.209657
$$92$$ 4.00000 0.417029
$$93$$ 12.0000 1.24434
$$94$$ 18.0000 1.85656
$$95$$ 0 0
$$96$$ −24.0000 −2.44949
$$97$$ 4.00000 0.406138 0.203069 0.979164i $$-0.434908\pi$$
0.203069 + 0.979164i $$0.434908\pi$$
$$98$$ 12.0000 1.21218
$$99$$ −30.0000 −3.01511
$$100$$ −2.00000 −0.200000
$$101$$ 3.00000 0.298511 0.149256 0.988799i $$-0.452312\pi$$
0.149256 + 0.988799i $$0.452312\pi$$
$$102$$ 0 0
$$103$$ 18.0000 1.77359 0.886796 0.462160i $$-0.152926\pi$$
0.886796 + 0.462160i $$0.152926\pi$$
$$104$$ 0 0
$$105$$ −6.00000 −0.585540
$$106$$ −2.00000 −0.194257
$$107$$ −12.0000 −1.16008 −0.580042 0.814587i $$-0.696964\pi$$
−0.580042 + 0.814587i $$0.696964\pi$$
$$108$$ −18.0000 −1.73205
$$109$$ −16.0000 −1.53252 −0.766261 0.642529i $$-0.777885\pi$$
−0.766261 + 0.642529i $$0.777885\pi$$
$$110$$ −20.0000 −1.90693
$$111$$ 3.00000 0.284747
$$112$$ 4.00000 0.377964
$$113$$ −18.0000 −1.69330 −0.846649 0.532152i $$-0.821383\pi$$
−0.846649 + 0.532152i $$0.821383\pi$$
$$114$$ 0 0
$$115$$ −4.00000 −0.373002
$$116$$ 12.0000 1.11417
$$117$$ −12.0000 −1.10940
$$118$$ −16.0000 −1.47292
$$119$$ 0 0
$$120$$ 0 0
$$121$$ 14.0000 1.27273
$$122$$ 16.0000 1.44857
$$123$$ 27.0000 2.43451
$$124$$ −8.00000 −0.718421
$$125$$ 12.0000 1.07331
$$126$$ 12.0000 1.06904
$$127$$ 1.00000 0.0887357 0.0443678 0.999015i $$-0.485873\pi$$
0.0443678 + 0.999015i $$0.485873\pi$$
$$128$$ 0 0
$$129$$ −6.00000 −0.528271
$$130$$ −8.00000 −0.701646
$$131$$ −12.0000 −1.04844 −0.524222 0.851581i $$-0.675644\pi$$
−0.524222 + 0.851581i $$0.675644\pi$$
$$132$$ 30.0000 2.61116
$$133$$ 0 0
$$134$$ −16.0000 −1.38219
$$135$$ 18.0000 1.54919
$$136$$ 0 0
$$137$$ −6.00000 −0.512615 −0.256307 0.966595i $$-0.582506\pi$$
−0.256307 + 0.966595i $$0.582506\pi$$
$$138$$ 12.0000 1.02151
$$139$$ 4.00000 0.339276 0.169638 0.985506i $$-0.445740\pi$$
0.169638 + 0.985506i $$0.445740\pi$$
$$140$$ 4.00000 0.338062
$$141$$ 27.0000 2.27381
$$142$$ −18.0000 −1.51053
$$143$$ 10.0000 0.836242
$$144$$ −24.0000 −2.00000
$$145$$ −12.0000 −0.996546
$$146$$ 2.00000 0.165521
$$147$$ 18.0000 1.48461
$$148$$ −2.00000 −0.164399
$$149$$ −5.00000 −0.409616 −0.204808 0.978802i $$-0.565657\pi$$
−0.204808 + 0.978802i $$0.565657\pi$$
$$150$$ −6.00000 −0.489898
$$151$$ 16.0000 1.30206 0.651031 0.759051i $$-0.274337\pi$$
0.651031 + 0.759051i $$0.274337\pi$$
$$152$$ 0 0
$$153$$ 0 0
$$154$$ −10.0000 −0.805823
$$155$$ 8.00000 0.642575
$$156$$ 12.0000 0.960769
$$157$$ 23.0000 1.83560 0.917800 0.397043i $$-0.129964\pi$$
0.917800 + 0.397043i $$0.129964\pi$$
$$158$$ −8.00000 −0.636446
$$159$$ −3.00000 −0.237915
$$160$$ −16.0000 −1.26491
$$161$$ −2.00000 −0.157622
$$162$$ −18.0000 −1.41421
$$163$$ −18.0000 −1.40987 −0.704934 0.709273i $$-0.749024\pi$$
−0.704934 + 0.709273i $$0.749024\pi$$
$$164$$ −18.0000 −1.40556
$$165$$ −30.0000 −2.33550
$$166$$ 30.0000 2.32845
$$167$$ −12.0000 −0.928588 −0.464294 0.885681i $$-0.653692\pi$$
−0.464294 + 0.885681i $$0.653692\pi$$
$$168$$ 0 0
$$169$$ −9.00000 −0.692308
$$170$$ 0 0
$$171$$ 0 0
$$172$$ 4.00000 0.304997
$$173$$ 9.00000 0.684257 0.342129 0.939653i $$-0.388852\pi$$
0.342129 + 0.939653i $$0.388852\pi$$
$$174$$ 36.0000 2.72915
$$175$$ 1.00000 0.0755929
$$176$$ 20.0000 1.50756
$$177$$ −24.0000 −1.80395
$$178$$ −8.00000 −0.599625
$$179$$ 18.0000 1.34538 0.672692 0.739923i $$-0.265138\pi$$
0.672692 + 0.739923i $$0.265138\pi$$
$$180$$ −24.0000 −1.78885
$$181$$ 5.00000 0.371647 0.185824 0.982583i $$-0.440505\pi$$
0.185824 + 0.982583i $$0.440505\pi$$
$$182$$ −4.00000 −0.296500
$$183$$ 24.0000 1.77413
$$184$$ 0 0
$$185$$ 2.00000 0.147043
$$186$$ −24.0000 −1.75977
$$187$$ 0 0
$$188$$ −18.0000 −1.31278
$$189$$ 9.00000 0.654654
$$190$$ 0 0
$$191$$ −4.00000 −0.289430 −0.144715 0.989473i $$-0.546227\pi$$
−0.144715 + 0.989473i $$0.546227\pi$$
$$192$$ 24.0000 1.73205
$$193$$ −26.0000 −1.87152 −0.935760 0.352636i $$-0.885285\pi$$
−0.935760 + 0.352636i $$0.885285\pi$$
$$194$$ −8.00000 −0.574367
$$195$$ −12.0000 −0.859338
$$196$$ −12.0000 −0.857143
$$197$$ 3.00000 0.213741 0.106871 0.994273i $$-0.465917\pi$$
0.106871 + 0.994273i $$0.465917\pi$$
$$198$$ 60.0000 4.26401
$$199$$ 2.00000 0.141776 0.0708881 0.997484i $$-0.477417\pi$$
0.0708881 + 0.997484i $$0.477417\pi$$
$$200$$ 0 0
$$201$$ −24.0000 −1.69283
$$202$$ −6.00000 −0.422159
$$203$$ −6.00000 −0.421117
$$204$$ 0 0
$$205$$ 18.0000 1.25717
$$206$$ −36.0000 −2.50824
$$207$$ 12.0000 0.834058
$$208$$ 8.00000 0.554700
$$209$$ 0 0
$$210$$ 12.0000 0.828079
$$211$$ −13.0000 −0.894957 −0.447478 0.894295i $$-0.647678\pi$$
−0.447478 + 0.894295i $$0.647678\pi$$
$$212$$ 2.00000 0.137361
$$213$$ −27.0000 −1.85001
$$214$$ 24.0000 1.64061
$$215$$ −4.00000 −0.272798
$$216$$ 0 0
$$217$$ 4.00000 0.271538
$$218$$ 32.0000 2.16731
$$219$$ 3.00000 0.202721
$$220$$ 20.0000 1.34840
$$221$$ 0 0
$$222$$ −6.00000 −0.402694
$$223$$ −17.0000 −1.13840 −0.569202 0.822198i $$-0.692748\pi$$
−0.569202 + 0.822198i $$0.692748\pi$$
$$224$$ −8.00000 −0.534522
$$225$$ −6.00000 −0.400000
$$226$$ 36.0000 2.39468
$$227$$ −16.0000 −1.06196 −0.530979 0.847385i $$-0.678176\pi$$
−0.530979 + 0.847385i $$0.678176\pi$$
$$228$$ 0 0
$$229$$ 7.00000 0.462573 0.231287 0.972886i $$-0.425707\pi$$
0.231287 + 0.972886i $$0.425707\pi$$
$$230$$ 8.00000 0.527504
$$231$$ −15.0000 −0.986928
$$232$$ 0 0
$$233$$ 6.00000 0.393073 0.196537 0.980497i $$-0.437031\pi$$
0.196537 + 0.980497i $$0.437031\pi$$
$$234$$ 24.0000 1.56893
$$235$$ 18.0000 1.17419
$$236$$ 16.0000 1.04151
$$237$$ −12.0000 −0.779484
$$238$$ 0 0
$$239$$ −6.00000 −0.388108 −0.194054 0.980991i $$-0.562164\pi$$
−0.194054 + 0.980991i $$0.562164\pi$$
$$240$$ −24.0000 −1.54919
$$241$$ 14.0000 0.901819 0.450910 0.892570i $$-0.351100\pi$$
0.450910 + 0.892570i $$0.351100\pi$$
$$242$$ −28.0000 −1.79991
$$243$$ 0 0
$$244$$ −16.0000 −1.02430
$$245$$ 12.0000 0.766652
$$246$$ −54.0000 −3.44291
$$247$$ 0 0
$$248$$ 0 0
$$249$$ 45.0000 2.85176
$$250$$ −24.0000 −1.51789
$$251$$ −2.00000 −0.126239 −0.0631194 0.998006i $$-0.520105\pi$$
−0.0631194 + 0.998006i $$0.520105\pi$$
$$252$$ −12.0000 −0.755929
$$253$$ −10.0000 −0.628695
$$254$$ −2.00000 −0.125491
$$255$$ 0 0
$$256$$ 16.0000 1.00000
$$257$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$258$$ 12.0000 0.747087
$$259$$ 1.00000 0.0621370
$$260$$ 8.00000 0.496139
$$261$$ 36.0000 2.22834
$$262$$ 24.0000 1.48272
$$263$$ 19.0000 1.17159 0.585795 0.810459i $$-0.300782\pi$$
0.585795 + 0.810459i $$0.300782\pi$$
$$264$$ 0 0
$$265$$ −2.00000 −0.122859
$$266$$ 0 0
$$267$$ −12.0000 −0.734388
$$268$$ 16.0000 0.977356
$$269$$ −6.00000 −0.365826 −0.182913 0.983129i $$-0.558553\pi$$
−0.182913 + 0.983129i $$0.558553\pi$$
$$270$$ −36.0000 −2.19089
$$271$$ −31.0000 −1.88312 −0.941558 0.336851i $$-0.890638\pi$$
−0.941558 + 0.336851i $$0.890638\pi$$
$$272$$ 0 0
$$273$$ −6.00000 −0.363137
$$274$$ 12.0000 0.724947
$$275$$ 5.00000 0.301511
$$276$$ −12.0000 −0.722315
$$277$$ 12.0000 0.721010 0.360505 0.932757i $$-0.382604\pi$$
0.360505 + 0.932757i $$0.382604\pi$$
$$278$$ −8.00000 −0.479808
$$279$$ −24.0000 −1.43684
$$280$$ 0 0
$$281$$ 12.0000 0.715860 0.357930 0.933748i $$-0.383483\pi$$
0.357930 + 0.933748i $$0.383483\pi$$
$$282$$ −54.0000 −3.21565
$$283$$ 4.00000 0.237775 0.118888 0.992908i $$-0.462067\pi$$
0.118888 + 0.992908i $$0.462067\pi$$
$$284$$ 18.0000 1.06810
$$285$$ 0 0
$$286$$ −20.0000 −1.18262
$$287$$ 9.00000 0.531253
$$288$$ 48.0000 2.82843
$$289$$ −17.0000 −1.00000
$$290$$ 24.0000 1.40933
$$291$$ −12.0000 −0.703452
$$292$$ −2.00000 −0.117041
$$293$$ −2.00000 −0.116841 −0.0584206 0.998292i $$-0.518606\pi$$
−0.0584206 + 0.998292i $$0.518606\pi$$
$$294$$ −36.0000 −2.09956
$$295$$ −16.0000 −0.931556
$$296$$ 0 0
$$297$$ 45.0000 2.61116
$$298$$ 10.0000 0.579284
$$299$$ −4.00000 −0.231326
$$300$$ 6.00000 0.346410
$$301$$ −2.00000 −0.115278
$$302$$ −32.0000 −1.84139
$$303$$ −9.00000 −0.517036
$$304$$ 0 0
$$305$$ 16.0000 0.916157
$$306$$ 0 0
$$307$$ −17.0000 −0.970241 −0.485121 0.874447i $$-0.661224\pi$$
−0.485121 + 0.874447i $$0.661224\pi$$
$$308$$ 10.0000 0.569803
$$309$$ −54.0000 −3.07195
$$310$$ −16.0000 −0.908739
$$311$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$312$$ 0 0
$$313$$ 22.0000 1.24351 0.621757 0.783210i $$-0.286419\pi$$
0.621757 + 0.783210i $$0.286419\pi$$
$$314$$ −46.0000 −2.59593
$$315$$ 12.0000 0.676123
$$316$$ 8.00000 0.450035
$$317$$ 22.0000 1.23564 0.617822 0.786318i $$-0.288015\pi$$
0.617822 + 0.786318i $$0.288015\pi$$
$$318$$ 6.00000 0.336463
$$319$$ −30.0000 −1.67968
$$320$$ 16.0000 0.894427
$$321$$ 36.0000 2.00932
$$322$$ 4.00000 0.222911
$$323$$ 0 0
$$324$$ 18.0000 1.00000
$$325$$ 2.00000 0.110940
$$326$$ 36.0000 1.99386
$$327$$ 48.0000 2.65441
$$328$$ 0 0
$$329$$ 9.00000 0.496186
$$330$$ 60.0000 3.30289
$$331$$ −2.00000 −0.109930 −0.0549650 0.998488i $$-0.517505\pi$$
−0.0549650 + 0.998488i $$0.517505\pi$$
$$332$$ −30.0000 −1.64646
$$333$$ −6.00000 −0.328798
$$334$$ 24.0000 1.31322
$$335$$ −16.0000 −0.874173
$$336$$ −12.0000 −0.654654
$$337$$ −25.0000 −1.36184 −0.680918 0.732359i $$-0.738419\pi$$
−0.680918 + 0.732359i $$0.738419\pi$$
$$338$$ 18.0000 0.979071
$$339$$ 54.0000 2.93288
$$340$$ 0 0
$$341$$ 20.0000 1.08306
$$342$$ 0 0
$$343$$ 13.0000 0.701934
$$344$$ 0 0
$$345$$ 12.0000 0.646058
$$346$$ −18.0000 −0.967686
$$347$$ −10.0000 −0.536828 −0.268414 0.963304i $$-0.586500\pi$$
−0.268414 + 0.963304i $$0.586500\pi$$
$$348$$ −36.0000 −1.92980
$$349$$ 6.00000 0.321173 0.160586 0.987022i $$-0.448662\pi$$
0.160586 + 0.987022i $$0.448662\pi$$
$$350$$ −2.00000 −0.106904
$$351$$ 18.0000 0.960769
$$352$$ −40.0000 −2.13201
$$353$$ 8.00000 0.425797 0.212899 0.977074i $$-0.431710\pi$$
0.212899 + 0.977074i $$0.431710\pi$$
$$354$$ 48.0000 2.55117
$$355$$ −18.0000 −0.955341
$$356$$ 8.00000 0.423999
$$357$$ 0 0
$$358$$ −36.0000 −1.90266
$$359$$ −15.0000 −0.791670 −0.395835 0.918322i $$-0.629545\pi$$
−0.395835 + 0.918322i $$0.629545\pi$$
$$360$$ 0 0
$$361$$ −19.0000 −1.00000
$$362$$ −10.0000 −0.525588
$$363$$ −42.0000 −2.20443
$$364$$ 4.00000 0.209657
$$365$$ 2.00000 0.104685
$$366$$ −48.0000 −2.50900
$$367$$ 8.00000 0.417597 0.208798 0.977959i $$-0.433045\pi$$
0.208798 + 0.977959i $$0.433045\pi$$
$$368$$ −8.00000 −0.417029
$$369$$ −54.0000 −2.81113
$$370$$ −4.00000 −0.207950
$$371$$ −1.00000 −0.0519174
$$372$$ 24.0000 1.24434
$$373$$ −19.0000 −0.983783 −0.491891 0.870657i $$-0.663694\pi$$
−0.491891 + 0.870657i $$0.663694\pi$$
$$374$$ 0 0
$$375$$ −36.0000 −1.85903
$$376$$ 0 0
$$377$$ −12.0000 −0.618031
$$378$$ −18.0000 −0.925820
$$379$$ 15.0000 0.770498 0.385249 0.922813i $$-0.374116\pi$$
0.385249 + 0.922813i $$0.374116\pi$$
$$380$$ 0 0
$$381$$ −3.00000 −0.153695
$$382$$ 8.00000 0.409316
$$383$$ 20.0000 1.02195 0.510976 0.859595i $$-0.329284\pi$$
0.510976 + 0.859595i $$0.329284\pi$$
$$384$$ 0 0
$$385$$ −10.0000 −0.509647
$$386$$ 52.0000 2.64673
$$387$$ 12.0000 0.609994
$$388$$ 8.00000 0.406138
$$389$$ 4.00000 0.202808 0.101404 0.994845i $$-0.467667\pi$$
0.101404 + 0.994845i $$0.467667\pi$$
$$390$$ 24.0000 1.21529
$$391$$ 0 0
$$392$$ 0 0
$$393$$ 36.0000 1.81596
$$394$$ −6.00000 −0.302276
$$395$$ −8.00000 −0.402524
$$396$$ −60.0000 −3.01511
$$397$$ −5.00000 −0.250943 −0.125471 0.992097i $$-0.540044\pi$$
−0.125471 + 0.992097i $$0.540044\pi$$
$$398$$ −4.00000 −0.200502
$$399$$ 0 0
$$400$$ 4.00000 0.200000
$$401$$ 18.0000 0.898877 0.449439 0.893311i $$-0.351624\pi$$
0.449439 + 0.893311i $$0.351624\pi$$
$$402$$ 48.0000 2.39402
$$403$$ 8.00000 0.398508
$$404$$ 6.00000 0.298511
$$405$$ −18.0000 −0.894427
$$406$$ 12.0000 0.595550
$$407$$ 5.00000 0.247841
$$408$$ 0 0
$$409$$ 20.0000 0.988936 0.494468 0.869196i $$-0.335363\pi$$
0.494468 + 0.869196i $$0.335363\pi$$
$$410$$ −36.0000 −1.77791
$$411$$ 18.0000 0.887875
$$412$$ 36.0000 1.77359
$$413$$ −8.00000 −0.393654
$$414$$ −24.0000 −1.17954
$$415$$ 30.0000 1.47264
$$416$$ −16.0000 −0.784465
$$417$$ −12.0000 −0.587643
$$418$$ 0 0
$$419$$ 7.00000 0.341972 0.170986 0.985273i $$-0.445305\pi$$
0.170986 + 0.985273i $$0.445305\pi$$
$$420$$ −12.0000 −0.585540
$$421$$ −24.0000 −1.16969 −0.584844 0.811146i $$-0.698844\pi$$
−0.584844 + 0.811146i $$0.698844\pi$$
$$422$$ 26.0000 1.26566
$$423$$ −54.0000 −2.62557
$$424$$ 0 0
$$425$$ 0 0
$$426$$ 54.0000 2.61631
$$427$$ 8.00000 0.387147
$$428$$ −24.0000 −1.16008
$$429$$ −30.0000 −1.44841
$$430$$ 8.00000 0.385794
$$431$$ −30.0000 −1.44505 −0.722525 0.691345i $$-0.757018\pi$$
−0.722525 + 0.691345i $$0.757018\pi$$
$$432$$ 36.0000 1.73205
$$433$$ 9.00000 0.432512 0.216256 0.976337i $$-0.430615\pi$$
0.216256 + 0.976337i $$0.430615\pi$$
$$434$$ −8.00000 −0.384012
$$435$$ 36.0000 1.72607
$$436$$ −32.0000 −1.53252
$$437$$ 0 0
$$438$$ −6.00000 −0.286691
$$439$$ 28.0000 1.33637 0.668184 0.743996i $$-0.267072\pi$$
0.668184 + 0.743996i $$0.267072\pi$$
$$440$$ 0 0
$$441$$ −36.0000 −1.71429
$$442$$ 0 0
$$443$$ 1.00000 0.0475114 0.0237557 0.999718i $$-0.492438\pi$$
0.0237557 + 0.999718i $$0.492438\pi$$
$$444$$ 6.00000 0.284747
$$445$$ −8.00000 −0.379236
$$446$$ 34.0000 1.60995
$$447$$ 15.0000 0.709476
$$448$$ 8.00000 0.377964
$$449$$ 36.0000 1.69895 0.849473 0.527633i $$-0.176920\pi$$
0.849473 + 0.527633i $$0.176920\pi$$
$$450$$ 12.0000 0.565685
$$451$$ 45.0000 2.11897
$$452$$ −36.0000 −1.69330
$$453$$ −48.0000 −2.25524
$$454$$ 32.0000 1.50183
$$455$$ −4.00000 −0.187523
$$456$$ 0 0
$$457$$ 18.0000 0.842004 0.421002 0.907060i $$-0.361678\pi$$
0.421002 + 0.907060i $$0.361678\pi$$
$$458$$ −14.0000 −0.654177
$$459$$ 0 0
$$460$$ −8.00000 −0.373002
$$461$$ 30.0000 1.39724 0.698620 0.715493i $$-0.253798\pi$$
0.698620 + 0.715493i $$0.253798\pi$$
$$462$$ 30.0000 1.39573
$$463$$ −22.0000 −1.02243 −0.511213 0.859454i $$-0.670804\pi$$
−0.511213 + 0.859454i $$0.670804\pi$$
$$464$$ −24.0000 −1.11417
$$465$$ −24.0000 −1.11297
$$466$$ −12.0000 −0.555889
$$467$$ −2.00000 −0.0925490 −0.0462745 0.998929i $$-0.514735\pi$$
−0.0462745 + 0.998929i $$0.514735\pi$$
$$468$$ −24.0000 −1.10940
$$469$$ −8.00000 −0.369406
$$470$$ −36.0000 −1.66056
$$471$$ −69.0000 −3.17935
$$472$$ 0 0
$$473$$ −10.0000 −0.459800
$$474$$ 24.0000 1.10236
$$475$$ 0 0
$$476$$ 0 0
$$477$$ 6.00000 0.274721
$$478$$ 12.0000 0.548867
$$479$$ 14.0000 0.639676 0.319838 0.947472i $$-0.396371\pi$$
0.319838 + 0.947472i $$0.396371\pi$$
$$480$$ 48.0000 2.19089
$$481$$ 2.00000 0.0911922
$$482$$ −28.0000 −1.27537
$$483$$ 6.00000 0.273009
$$484$$ 28.0000 1.27273
$$485$$ −8.00000 −0.363261
$$486$$ 0 0
$$487$$ −24.0000 −1.08754 −0.543772 0.839233i $$-0.683004\pi$$
−0.543772 + 0.839233i $$0.683004\pi$$
$$488$$ 0 0
$$489$$ 54.0000 2.44196
$$490$$ −24.0000 −1.08421
$$491$$ −28.0000 −1.26362 −0.631811 0.775122i $$-0.717688\pi$$
−0.631811 + 0.775122i $$0.717688\pi$$
$$492$$ 54.0000 2.43451
$$493$$ 0 0
$$494$$ 0 0
$$495$$ 60.0000 2.69680
$$496$$ 16.0000 0.718421
$$497$$ −9.00000 −0.403705
$$498$$ −90.0000 −4.03300
$$499$$ 12.0000 0.537194 0.268597 0.963253i $$-0.413440\pi$$
0.268597 + 0.963253i $$0.413440\pi$$
$$500$$ 24.0000 1.07331
$$501$$ 36.0000 1.60836
$$502$$ 4.00000 0.178529
$$503$$ 16.0000 0.713405 0.356702 0.934218i $$-0.383901\pi$$
0.356702 + 0.934218i $$0.383901\pi$$
$$504$$ 0 0
$$505$$ −6.00000 −0.266996
$$506$$ 20.0000 0.889108
$$507$$ 27.0000 1.19911
$$508$$ 2.00000 0.0887357
$$509$$ −31.0000 −1.37405 −0.687025 0.726633i $$-0.741084\pi$$
−0.687025 + 0.726633i $$0.741084\pi$$
$$510$$ 0 0
$$511$$ 1.00000 0.0442374
$$512$$ −32.0000 −1.41421
$$513$$ 0 0
$$514$$ 0 0
$$515$$ −36.0000 −1.58635
$$516$$ −12.0000 −0.528271
$$517$$ 45.0000 1.97910
$$518$$ −2.00000 −0.0878750
$$519$$ −27.0000 −1.18517
$$520$$ 0 0
$$521$$ −33.0000 −1.44576 −0.722878 0.690976i $$-0.757181\pi$$
−0.722878 + 0.690976i $$0.757181\pi$$
$$522$$ −72.0000 −3.15135
$$523$$ −22.0000 −0.961993 −0.480996 0.876723i $$-0.659725\pi$$
−0.480996 + 0.876723i $$0.659725\pi$$
$$524$$ −24.0000 −1.04844
$$525$$ −3.00000 −0.130931
$$526$$ −38.0000 −1.65688
$$527$$ 0 0
$$528$$ −60.0000 −2.61116
$$529$$ −19.0000 −0.826087
$$530$$ 4.00000 0.173749
$$531$$ 48.0000 2.08302
$$532$$ 0 0
$$533$$ 18.0000 0.779667
$$534$$ 24.0000 1.03858
$$535$$ 24.0000 1.03761
$$536$$ 0 0
$$537$$ −54.0000 −2.33027
$$538$$ 12.0000 0.517357
$$539$$ 30.0000 1.29219
$$540$$ 36.0000 1.54919
$$541$$ 20.0000 0.859867 0.429934 0.902861i $$-0.358537\pi$$
0.429934 + 0.902861i $$0.358537\pi$$
$$542$$ 62.0000 2.66313
$$543$$ −15.0000 −0.643712
$$544$$ 0 0
$$545$$ 32.0000 1.37073
$$546$$ 12.0000 0.513553
$$547$$ 8.00000 0.342055 0.171028 0.985266i $$-0.445291\pi$$
0.171028 + 0.985266i $$0.445291\pi$$
$$548$$ −12.0000 −0.512615
$$549$$ −48.0000 −2.04859
$$550$$ −10.0000 −0.426401
$$551$$ 0 0
$$552$$ 0 0
$$553$$ −4.00000 −0.170097
$$554$$ −24.0000 −1.01966
$$555$$ −6.00000 −0.254686
$$556$$ 8.00000 0.339276
$$557$$ −18.0000 −0.762684 −0.381342 0.924434i $$-0.624538\pi$$
−0.381342 + 0.924434i $$0.624538\pi$$
$$558$$ 48.0000 2.03200
$$559$$ −4.00000 −0.169182
$$560$$ −8.00000 −0.338062
$$561$$ 0 0
$$562$$ −24.0000 −1.01238
$$563$$ −30.0000 −1.26435 −0.632175 0.774826i $$-0.717837\pi$$
−0.632175 + 0.774826i $$0.717837\pi$$
$$564$$ 54.0000 2.27381
$$565$$ 36.0000 1.51453
$$566$$ −8.00000 −0.336265
$$567$$ −9.00000 −0.377964
$$568$$ 0 0
$$569$$ −24.0000 −1.00613 −0.503066 0.864248i $$-0.667795\pi$$
−0.503066 + 0.864248i $$0.667795\pi$$
$$570$$ 0 0
$$571$$ 7.00000 0.292941 0.146470 0.989215i $$-0.453209\pi$$
0.146470 + 0.989215i $$0.453209\pi$$
$$572$$ 20.0000 0.836242
$$573$$ 12.0000 0.501307
$$574$$ −18.0000 −0.751305
$$575$$ −2.00000 −0.0834058
$$576$$ −48.0000 −2.00000
$$577$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$578$$ 34.0000 1.41421
$$579$$ 78.0000 3.24157
$$580$$ −24.0000 −0.996546
$$581$$ 15.0000 0.622305
$$582$$ 24.0000 0.994832
$$583$$ −5.00000 −0.207079
$$584$$ 0 0
$$585$$ 24.0000 0.992278
$$586$$ 4.00000 0.165238
$$587$$ −32.0000 −1.32078 −0.660391 0.750922i $$-0.729609\pi$$
−0.660391 + 0.750922i $$0.729609\pi$$
$$588$$ 36.0000 1.48461
$$589$$ 0 0
$$590$$ 32.0000 1.31742
$$591$$ −9.00000 −0.370211
$$592$$ 4.00000 0.164399
$$593$$ −5.00000 −0.205325 −0.102663 0.994716i $$-0.532736\pi$$
−0.102663 + 0.994716i $$0.532736\pi$$
$$594$$ −90.0000 −3.69274
$$595$$ 0 0
$$596$$ −10.0000 −0.409616
$$597$$ −6.00000 −0.245564
$$598$$ 8.00000 0.327144
$$599$$ 1.00000 0.0408589 0.0204294 0.999791i $$-0.493497\pi$$
0.0204294 + 0.999791i $$0.493497\pi$$
$$600$$ 0 0
$$601$$ −22.0000 −0.897399 −0.448699 0.893683i $$-0.648113\pi$$
−0.448699 + 0.893683i $$0.648113\pi$$
$$602$$ 4.00000 0.163028
$$603$$ 48.0000 1.95471
$$604$$ 32.0000 1.30206
$$605$$ −28.0000 −1.13836
$$606$$ 18.0000 0.731200
$$607$$ −32.0000 −1.29884 −0.649420 0.760430i $$-0.724988\pi$$
−0.649420 + 0.760430i $$0.724988\pi$$
$$608$$ 0 0
$$609$$ 18.0000 0.729397
$$610$$ −32.0000 −1.29564
$$611$$ 18.0000 0.728202
$$612$$ 0 0
$$613$$ 15.0000 0.605844 0.302922 0.953015i $$-0.402038\pi$$
0.302922 + 0.953015i $$0.402038\pi$$
$$614$$ 34.0000 1.37213
$$615$$ −54.0000 −2.17749
$$616$$ 0 0
$$617$$ 17.0000 0.684394 0.342197 0.939628i $$-0.388829\pi$$
0.342197 + 0.939628i $$0.388829\pi$$
$$618$$ 108.000 4.34440
$$619$$ −1.00000 −0.0401934 −0.0200967 0.999798i $$-0.506397\pi$$
−0.0200967 + 0.999798i $$0.506397\pi$$
$$620$$ 16.0000 0.642575
$$621$$ −18.0000 −0.722315
$$622$$ 0 0
$$623$$ −4.00000 −0.160257
$$624$$ −24.0000 −0.960769
$$625$$ −19.0000 −0.760000
$$626$$ −44.0000 −1.75859
$$627$$ 0 0
$$628$$ 46.0000 1.83560
$$629$$ 0 0
$$630$$ −24.0000 −0.956183
$$631$$ −28.0000 −1.11466 −0.557331 0.830290i $$-0.688175\pi$$
−0.557331 + 0.830290i $$0.688175\pi$$
$$632$$ 0 0
$$633$$ 39.0000 1.55011
$$634$$ −44.0000 −1.74746
$$635$$ −2.00000 −0.0793676
$$636$$ −6.00000 −0.237915
$$637$$ 12.0000 0.475457
$$638$$ 60.0000 2.37542
$$639$$ 54.0000 2.13621
$$640$$ 0 0
$$641$$ −1.00000 −0.0394976 −0.0197488 0.999805i $$-0.506287\pi$$
−0.0197488 + 0.999805i $$0.506287\pi$$
$$642$$ −72.0000 −2.84161
$$643$$ 14.0000 0.552106 0.276053 0.961142i $$-0.410973\pi$$
0.276053 + 0.961142i $$0.410973\pi$$
$$644$$ −4.00000 −0.157622
$$645$$ 12.0000 0.472500
$$646$$ 0 0
$$647$$ −8.00000 −0.314512 −0.157256 0.987558i $$-0.550265\pi$$
−0.157256 + 0.987558i $$0.550265\pi$$
$$648$$ 0 0
$$649$$ −40.0000 −1.57014
$$650$$ −4.00000 −0.156893
$$651$$ −12.0000 −0.470317
$$652$$ −36.0000 −1.40987
$$653$$ −24.0000 −0.939193 −0.469596 0.882881i $$-0.655601\pi$$
−0.469596 + 0.882881i $$0.655601\pi$$
$$654$$ −96.0000 −3.75390
$$655$$ 24.0000 0.937758
$$656$$ 36.0000 1.40556
$$657$$ −6.00000 −0.234082
$$658$$ −18.0000 −0.701713
$$659$$ −15.0000 −0.584317 −0.292159 0.956370i $$-0.594373\pi$$
−0.292159 + 0.956370i $$0.594373\pi$$
$$660$$ −60.0000 −2.33550
$$661$$ −28.0000 −1.08907 −0.544537 0.838737i $$-0.683295\pi$$
−0.544537 + 0.838737i $$0.683295\pi$$
$$662$$ 4.00000 0.155464
$$663$$ 0 0
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 12.0000 0.464991
$$667$$ 12.0000 0.464642
$$668$$ −24.0000 −0.928588
$$669$$ 51.0000 1.97177
$$670$$ 32.0000 1.23627
$$671$$ 40.0000 1.54418
$$672$$ 24.0000 0.925820
$$673$$ 27.0000 1.04077 0.520387 0.853931i $$-0.325788\pi$$
0.520387 + 0.853931i $$0.325788\pi$$
$$674$$ 50.0000 1.92593
$$675$$ 9.00000 0.346410
$$676$$ −18.0000 −0.692308
$$677$$ −11.0000 −0.422764 −0.211382 0.977403i $$-0.567796\pi$$
−0.211382 + 0.977403i $$0.567796\pi$$
$$678$$ −108.000 −4.14772
$$679$$ −4.00000 −0.153506
$$680$$ 0 0
$$681$$ 48.0000 1.83936
$$682$$ −40.0000 −1.53168
$$683$$ 18.0000 0.688751 0.344375 0.938832i $$-0.388091\pi$$
0.344375 + 0.938832i $$0.388091\pi$$
$$684$$ 0 0
$$685$$ 12.0000 0.458496
$$686$$ −26.0000 −0.992685
$$687$$ −21.0000 −0.801200
$$688$$ −8.00000 −0.304997
$$689$$ −2.00000 −0.0761939
$$690$$ −24.0000 −0.913664
$$691$$ −20.0000 −0.760836 −0.380418 0.924815i $$-0.624220\pi$$
−0.380418 + 0.924815i $$0.624220\pi$$
$$692$$ 18.0000 0.684257
$$693$$ 30.0000 1.13961
$$694$$ 20.0000 0.759190
$$695$$ −8.00000 −0.303457
$$696$$ 0 0
$$697$$ 0 0
$$698$$ −12.0000 −0.454207
$$699$$ −18.0000 −0.680823
$$700$$ 2.00000 0.0755929
$$701$$ −12.0000 −0.453234 −0.226617 0.973984i $$-0.572767\pi$$
−0.226617 + 0.973984i $$0.572767\pi$$
$$702$$ −36.0000 −1.35873
$$703$$ 0 0
$$704$$ 40.0000 1.50756
$$705$$ −54.0000 −2.03376
$$706$$ −16.0000 −0.602168
$$707$$ −3.00000 −0.112827
$$708$$ −48.0000 −1.80395
$$709$$ 40.0000 1.50223 0.751116 0.660171i $$-0.229516\pi$$
0.751116 + 0.660171i $$0.229516\pi$$
$$710$$ 36.0000 1.35106
$$711$$ 24.0000 0.900070
$$712$$ 0 0
$$713$$ −8.00000 −0.299602
$$714$$ 0 0
$$715$$ −20.0000 −0.747958
$$716$$ 36.0000 1.34538
$$717$$ 18.0000 0.672222
$$718$$ 30.0000 1.11959
$$719$$ 39.0000 1.45445 0.727227 0.686397i $$-0.240809\pi$$
0.727227 + 0.686397i $$0.240809\pi$$
$$720$$ 48.0000 1.78885
$$721$$ −18.0000 −0.670355
$$722$$ 38.0000 1.41421
$$723$$ −42.0000 −1.56200
$$724$$ 10.0000 0.371647
$$725$$ −6.00000 −0.222834
$$726$$ 84.0000 3.11753
$$727$$ 16.0000 0.593407 0.296704 0.954970i $$-0.404113\pi$$
0.296704 + 0.954970i $$0.404113\pi$$
$$728$$ 0 0
$$729$$ −27.0000 −1.00000
$$730$$ −4.00000 −0.148047
$$731$$ 0 0
$$732$$ 48.0000 1.77413
$$733$$ 7.00000 0.258551 0.129275 0.991609i $$-0.458735\pi$$
0.129275 + 0.991609i $$0.458735\pi$$
$$734$$ −16.0000 −0.590571
$$735$$ −36.0000 −1.32788
$$736$$ 16.0000 0.589768
$$737$$ −40.0000 −1.47342
$$738$$ 108.000 3.97553
$$739$$ −9.00000 −0.331070 −0.165535 0.986204i $$-0.552935\pi$$
−0.165535 + 0.986204i $$0.552935\pi$$
$$740$$ 4.00000 0.147043
$$741$$ 0 0
$$742$$ 2.00000 0.0734223
$$743$$ 21.0000 0.770415 0.385208 0.922830i $$-0.374130\pi$$
0.385208 + 0.922830i $$0.374130\pi$$
$$744$$ 0 0
$$745$$ 10.0000 0.366372
$$746$$ 38.0000 1.39128
$$747$$ −90.0000 −3.29293
$$748$$ 0 0
$$749$$ 12.0000 0.438470
$$750$$ 72.0000 2.62907
$$751$$ 25.0000 0.912263 0.456131 0.889912i $$-0.349235\pi$$
0.456131 + 0.889912i $$0.349235\pi$$
$$752$$ 36.0000 1.31278
$$753$$ 6.00000 0.218652
$$754$$ 24.0000 0.874028
$$755$$ −32.0000 −1.16460
$$756$$ 18.0000 0.654654
$$757$$ −50.0000 −1.81728 −0.908640 0.417579i $$-0.862879\pi$$
−0.908640 + 0.417579i $$0.862879\pi$$
$$758$$ −30.0000 −1.08965
$$759$$ 30.0000 1.08893
$$760$$ 0 0
$$761$$ −35.0000 −1.26875 −0.634375 0.773026i $$-0.718742\pi$$
−0.634375 + 0.773026i $$0.718742\pi$$
$$762$$ 6.00000 0.217357
$$763$$ 16.0000 0.579239
$$764$$ −8.00000 −0.289430
$$765$$ 0 0
$$766$$ −40.0000 −1.44526
$$767$$ −16.0000 −0.577727
$$768$$ −48.0000 −1.73205
$$769$$ 26.0000 0.937584 0.468792 0.883309i $$-0.344689\pi$$
0.468792 + 0.883309i $$0.344689\pi$$
$$770$$ 20.0000 0.720750
$$771$$ 0 0
$$772$$ −52.0000 −1.87152
$$773$$ −9.00000 −0.323708 −0.161854 0.986815i $$-0.551747\pi$$
−0.161854 + 0.986815i $$0.551747\pi$$
$$774$$ −24.0000 −0.862662
$$775$$ 4.00000 0.143684
$$776$$ 0 0
$$777$$ −3.00000 −0.107624
$$778$$ −8.00000 −0.286814
$$779$$ 0 0
$$780$$ −24.0000 −0.859338
$$781$$ −45.0000 −1.61023
$$782$$ 0 0
$$783$$ −54.0000 −1.92980
$$784$$ 24.0000 0.857143
$$785$$ −46.0000 −1.64181
$$786$$ −72.0000 −2.56815
$$787$$ −5.00000 −0.178231 −0.0891154 0.996021i $$-0.528404\pi$$
−0.0891154 + 0.996021i $$0.528404\pi$$
$$788$$ 6.00000 0.213741
$$789$$ −57.0000 −2.02925
$$790$$ 16.0000 0.569254
$$791$$ 18.0000 0.640006
$$792$$ 0 0
$$793$$ 16.0000 0.568177
$$794$$ 10.0000 0.354887
$$795$$ 6.00000 0.212798
$$796$$ 4.00000 0.141776
$$797$$ 52.0000 1.84193 0.920967 0.389640i $$-0.127401\pi$$
0.920967 + 0.389640i $$0.127401\pi$$
$$798$$ 0 0
$$799$$ 0 0
$$800$$ −8.00000 −0.282843
$$801$$ 24.0000 0.847998
$$802$$ −36.0000 −1.27120
$$803$$ 5.00000 0.176446
$$804$$ −48.0000 −1.69283
$$805$$ 4.00000 0.140981
$$806$$ −16.0000 −0.563576
$$807$$ 18.0000 0.633630
$$808$$ 0 0
$$809$$ 2.00000 0.0703163 0.0351581 0.999382i $$-0.488807\pi$$
0.0351581 + 0.999382i $$0.488807\pi$$
$$810$$ 36.0000 1.26491
$$811$$ 47.0000 1.65039 0.825197 0.564846i $$-0.191064\pi$$
0.825197 + 0.564846i $$0.191064\pi$$
$$812$$ −12.0000 −0.421117
$$813$$ 93.0000 3.26165
$$814$$ −10.0000 −0.350500
$$815$$ 36.0000 1.26102
$$816$$ 0 0
$$817$$ 0 0
$$818$$ −40.0000 −1.39857
$$819$$ 12.0000 0.419314
$$820$$ 36.0000 1.25717
$$821$$ −47.0000 −1.64031 −0.820156 0.572140i $$-0.806113\pi$$
−0.820156 + 0.572140i $$0.806113\pi$$
$$822$$ −36.0000 −1.25564
$$823$$ −16.0000 −0.557725 −0.278862 0.960331i $$-0.589957\pi$$
−0.278862 + 0.960331i $$0.589957\pi$$
$$824$$ 0 0
$$825$$ −15.0000 −0.522233
$$826$$ 16.0000 0.556711
$$827$$ 22.0000 0.765015 0.382507 0.923952i $$-0.375061\pi$$
0.382507 + 0.923952i $$0.375061\pi$$
$$828$$ 24.0000 0.834058
$$829$$ −4.00000 −0.138926 −0.0694629 0.997585i $$-0.522129\pi$$
−0.0694629 + 0.997585i $$0.522129\pi$$
$$830$$ −60.0000 −2.08263
$$831$$ −36.0000 −1.24883
$$832$$ 16.0000 0.554700
$$833$$ 0 0
$$834$$ 24.0000 0.831052
$$835$$ 24.0000 0.830554
$$836$$ 0 0
$$837$$ 36.0000 1.24434
$$838$$ −14.0000 −0.483622
$$839$$ 44.0000 1.51905 0.759524 0.650479i $$-0.225432\pi$$
0.759524 + 0.650479i $$0.225432\pi$$
$$840$$ 0 0
$$841$$ 7.00000 0.241379
$$842$$ 48.0000 1.65419
$$843$$ −36.0000 −1.23991
$$844$$ −26.0000 −0.894957
$$845$$ 18.0000 0.619219
$$846$$ 108.000 3.71312
$$847$$ −14.0000 −0.481046
$$848$$ −4.00000 −0.137361
$$849$$ −12.0000 −0.411839
$$850$$ 0 0
$$851$$ −2.00000 −0.0685591
$$852$$ −54.0000 −1.85001
$$853$$ 26.0000 0.890223 0.445112 0.895475i $$-0.353164\pi$$
0.445112 + 0.895475i $$0.353164\pi$$
$$854$$ −16.0000 −0.547509
$$855$$ 0 0
$$856$$ 0 0
$$857$$ −48.0000 −1.63965 −0.819824 0.572615i $$-0.805929\pi$$
−0.819824 + 0.572615i $$0.805929\pi$$
$$858$$ 60.0000 2.04837
$$859$$ −20.0000 −0.682391 −0.341196 0.939992i $$-0.610832\pi$$
−0.341196 + 0.939992i $$0.610832\pi$$
$$860$$ −8.00000 −0.272798
$$861$$ −27.0000 −0.920158
$$862$$ 60.0000 2.04361
$$863$$ −24.0000 −0.816970 −0.408485 0.912765i $$-0.633943\pi$$
−0.408485 + 0.912765i $$0.633943\pi$$
$$864$$ −72.0000 −2.44949
$$865$$ −18.0000 −0.612018
$$866$$ −18.0000 −0.611665
$$867$$ 51.0000 1.73205
$$868$$ 8.00000 0.271538
$$869$$ −20.0000 −0.678454
$$870$$ −72.0000 −2.44103
$$871$$ −16.0000 −0.542139
$$872$$ 0 0
$$873$$ 24.0000 0.812277
$$874$$ 0 0
$$875$$ −12.0000 −0.405674
$$876$$ 6.00000 0.202721
$$877$$ 50.0000 1.68838 0.844190 0.536044i $$-0.180082\pi$$
0.844190 + 0.536044i $$0.180082\pi$$
$$878$$ −56.0000 −1.88991
$$879$$ 6.00000 0.202375
$$880$$ −40.0000 −1.34840
$$881$$ −14.0000 −0.471672 −0.235836 0.971793i $$-0.575783\pi$$
−0.235836 + 0.971793i $$0.575783\pi$$
$$882$$ 72.0000 2.42437
$$883$$ 48.0000 1.61533 0.807664 0.589643i $$-0.200731\pi$$
0.807664 + 0.589643i $$0.200731\pi$$
$$884$$ 0 0
$$885$$ 48.0000 1.61350
$$886$$ −2.00000 −0.0671913
$$887$$ 25.0000 0.839418 0.419709 0.907659i $$-0.362132\pi$$
0.419709 + 0.907659i $$0.362132\pi$$
$$888$$ 0 0
$$889$$ −1.00000 −0.0335389
$$890$$ 16.0000 0.536321
$$891$$ −45.0000 −1.50756
$$892$$ −34.0000 −1.13840
$$893$$ 0 0
$$894$$ −30.0000 −1.00335
$$895$$ −36.0000 −1.20335
$$896$$ 0 0
$$897$$ 12.0000 0.400668
$$898$$ −72.0000 −2.40267
$$899$$ −24.0000 −0.800445
$$900$$ −12.0000 −0.400000
$$901$$ 0 0
$$902$$ −90.0000 −2.99667
$$903$$ 6.00000 0.199667
$$904$$ 0 0
$$905$$ −10.0000 −0.332411
$$906$$ 96.0000 3.18939
$$907$$ 52.0000 1.72663 0.863316 0.504664i $$-0.168384\pi$$
0.863316 + 0.504664i $$0.168384\pi$$
$$908$$ −32.0000 −1.06196
$$909$$ 18.0000 0.597022
$$910$$ 8.00000 0.265197
$$911$$ 26.0000 0.861418 0.430709 0.902491i $$-0.358263\pi$$
0.430709 + 0.902491i $$0.358263\pi$$
$$912$$ 0 0
$$913$$ 75.0000 2.48214
$$914$$ −36.0000 −1.19077
$$915$$ −48.0000 −1.58683
$$916$$ 14.0000 0.462573
$$917$$ 12.0000 0.396275
$$918$$ 0 0
$$919$$ −58.0000 −1.91324 −0.956622 0.291333i $$-0.905901\pi$$
−0.956622 + 0.291333i $$0.905901\pi$$
$$920$$ 0 0
$$921$$ 51.0000 1.68051
$$922$$ −60.0000 −1.97599
$$923$$ −18.0000 −0.592477
$$924$$ −30.0000 −0.986928
$$925$$ 1.00000 0.0328798
$$926$$ 44.0000 1.44593
$$927$$ 108.000 3.54719
$$928$$ 48.0000 1.57568
$$929$$ 18.0000 0.590561 0.295280 0.955411i $$-0.404587\pi$$
0.295280 + 0.955411i $$0.404587\pi$$
$$930$$ 48.0000 1.57398
$$931$$ 0 0
$$932$$ 12.0000 0.393073
$$933$$ 0 0
$$934$$ 4.00000 0.130884
$$935$$ 0 0
$$936$$ 0 0
$$937$$ 37.0000 1.20874 0.604369 0.796705i $$-0.293425\pi$$
0.604369 + 0.796705i $$0.293425\pi$$
$$938$$ 16.0000 0.522419
$$939$$ −66.0000 −2.15383
$$940$$ 36.0000 1.17419
$$941$$ −10.0000 −0.325991 −0.162995 0.986627i $$-0.552116\pi$$
−0.162995 + 0.986627i $$0.552116\pi$$
$$942$$ 138.000 4.49628
$$943$$ −18.0000 −0.586161
$$944$$ −32.0000 −1.04151
$$945$$ −18.0000 −0.585540
$$946$$ 20.0000 0.650256
$$947$$ 12.0000 0.389948 0.194974 0.980808i $$-0.437538\pi$$
0.194974 + 0.980808i $$0.437538\pi$$
$$948$$ −24.0000 −0.779484
$$949$$ 2.00000 0.0649227
$$950$$ 0 0
$$951$$ −66.0000 −2.14020
$$952$$ 0 0
$$953$$ 61.0000 1.97598 0.987992 0.154506i $$-0.0493785\pi$$
0.987992 + 0.154506i $$0.0493785\pi$$
$$954$$ −12.0000 −0.388514
$$955$$ 8.00000 0.258874
$$956$$ −12.0000 −0.388108
$$957$$ 90.0000 2.90929
$$958$$ −28.0000 −0.904639
$$959$$ 6.00000 0.193750
$$960$$ −48.0000 −1.54919
$$961$$ −15.0000 −0.483871
$$962$$ −4.00000 −0.128965
$$963$$ −72.0000 −2.32017
$$964$$ 28.0000 0.901819
$$965$$ 52.0000 1.67394
$$966$$ −12.0000 −0.386094
$$967$$ −14.0000 −0.450210 −0.225105 0.974335i $$-0.572272\pi$$
−0.225105 + 0.974335i $$0.572272\pi$$
$$968$$ 0 0
$$969$$ 0 0
$$970$$ 16.0000 0.513729
$$971$$ −8.00000 −0.256732 −0.128366 0.991727i $$-0.540973\pi$$
−0.128366 + 0.991727i $$0.540973\pi$$
$$972$$ 0 0
$$973$$ −4.00000 −0.128234
$$974$$ 48.0000 1.53802
$$975$$ −6.00000 −0.192154
$$976$$ 32.0000 1.02430
$$977$$ 28.0000 0.895799 0.447900 0.894084i $$-0.352172\pi$$
0.447900 + 0.894084i $$0.352172\pi$$
$$978$$ −108.000 −3.45346
$$979$$ −20.0000 −0.639203
$$980$$ 24.0000 0.766652
$$981$$ −96.0000 −3.06504
$$982$$ 56.0000 1.78703
$$983$$ 9.00000 0.287055 0.143528 0.989646i $$-0.454155\pi$$
0.143528 + 0.989646i $$0.454155\pi$$
$$984$$ 0 0
$$985$$ −6.00000 −0.191176
$$986$$ 0 0
$$987$$ −27.0000 −0.859419
$$988$$ 0 0
$$989$$ 4.00000 0.127193
$$990$$ −120.000 −3.81385
$$991$$ −18.0000 −0.571789 −0.285894 0.958261i $$-0.592291\pi$$
−0.285894 + 0.958261i $$0.592291\pi$$
$$992$$ −32.0000 −1.01600
$$993$$ 6.00000 0.190404
$$994$$ 18.0000 0.570925
$$995$$ −4.00000 −0.126809
$$996$$ 90.0000 2.85176
$$997$$ −42.0000 −1.33015 −0.665077 0.746775i $$-0.731601\pi$$
−0.665077 + 0.746775i $$0.731601\pi$$
$$998$$ −24.0000 −0.759707
$$999$$ 9.00000 0.284747
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 37.2.a.a.1.1 1
3.2 odd 2 333.2.a.d.1.1 1
4.3 odd 2 592.2.a.e.1.1 1
5.2 odd 4 925.2.b.b.149.1 2
5.3 odd 4 925.2.b.b.149.2 2
5.4 even 2 925.2.a.e.1.1 1
7.6 odd 2 1813.2.a.a.1.1 1
8.3 odd 2 2368.2.a.b.1.1 1
8.5 even 2 2368.2.a.q.1.1 1
11.10 odd 2 4477.2.a.b.1.1 1
12.11 even 2 5328.2.a.r.1.1 1
13.12 even 2 6253.2.a.c.1.1 1
15.14 odd 2 8325.2.a.e.1.1 1
37.6 odd 4 1369.2.b.c.1368.2 2
37.31 odd 4 1369.2.b.c.1368.1 2
37.36 even 2 1369.2.a.e.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
37.2.a.a.1.1 1 1.1 even 1 trivial
333.2.a.d.1.1 1 3.2 odd 2
592.2.a.e.1.1 1 4.3 odd 2
925.2.a.e.1.1 1 5.4 even 2
925.2.b.b.149.1 2 5.2 odd 4
925.2.b.b.149.2 2 5.3 odd 4
1369.2.a.e.1.1 1 37.36 even 2
1369.2.b.c.1368.1 2 37.31 odd 4
1369.2.b.c.1368.2 2 37.6 odd 4
1813.2.a.a.1.1 1 7.6 odd 2
2368.2.a.b.1.1 1 8.3 odd 2
2368.2.a.q.1.1 1 8.5 even 2
4477.2.a.b.1.1 1 11.10 odd 2
5328.2.a.r.1.1 1 12.11 even 2
6253.2.a.c.1.1 1 13.12 even 2
8325.2.a.e.1.1 1 15.14 odd 2