# Properties

 Label 3234.2.a.r.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: yes 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)

 $$f(q)$$ $$=$$ $$q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +2.00000 q^{5} -1.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$$ $$q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +2.00000 q^{5} -1.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} +2.00000 q^{10} +1.00000 q^{11} -1.00000 q^{12} +4.00000 q^{13} -2.00000 q^{15} +1.00000 q^{16} +6.00000 q^{17} +1.00000 q^{18} +2.00000 q^{19} +2.00000 q^{20} +1.00000 q^{22} -1.00000 q^{24} -1.00000 q^{25} +4.00000 q^{26} -1.00000 q^{27} -6.00000 q^{29} -2.00000 q^{30} +2.00000 q^{31} +1.00000 q^{32} -1.00000 q^{33} +6.00000 q^{34} +1.00000 q^{36} +2.00000 q^{37} +2.00000 q^{38} -4.00000 q^{39} +2.00000 q^{40} -2.00000 q^{41} -12.0000 q^{43} +1.00000 q^{44} +2.00000 q^{45} -6.00000 q^{47} -1.00000 q^{48} -1.00000 q^{50} -6.00000 q^{51} +4.00000 q^{52} +10.0000 q^{53} -1.00000 q^{54} +2.00000 q^{55} -2.00000 q^{57} -6.00000 q^{58} +8.00000 q^{59} -2.00000 q^{60} +4.00000 q^{61} +2.00000 q^{62} +1.00000 q^{64} +8.00000 q^{65} -1.00000 q^{66} +6.00000 q^{68} -8.00000 q^{71} +1.00000 q^{72} +6.00000 q^{73} +2.00000 q^{74} +1.00000 q^{75} +2.00000 q^{76} -4.00000 q^{78} +12.0000 q^{79} +2.00000 q^{80} +1.00000 q^{81} -2.00000 q^{82} +6.00000 q^{83} +12.0000 q^{85} -12.0000 q^{86} +6.00000 q^{87} +1.00000 q^{88} -12.0000 q^{89} +2.00000 q^{90} -2.00000 q^{93} -6.00000 q^{94} +4.00000 q^{95} -1.00000 q^{96} +1.00000 q^{99} +O(q^{100})$$

## 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
$$5$$ 2.00000 0.894427 0.447214 0.894427i $$-0.352416\pi$$
0.447214 + 0.894427i $$0.352416\pi$$
$$6$$ −1.00000 −0.408248
$$7$$ 0 0
$$8$$ 1.00000 0.353553
$$9$$ 1.00000 0.333333
$$10$$ 2.00000 0.632456
$$11$$ 1.00000 0.301511
$$12$$ −1.00000 −0.288675
$$13$$ 4.00000 1.10940 0.554700 0.832050i $$-0.312833\pi$$
0.554700 + 0.832050i $$0.312833\pi$$
$$14$$ 0 0
$$15$$ −2.00000 −0.516398
$$16$$ 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
$$19$$ 2.00000 0.458831 0.229416 0.973329i $$-0.426318\pi$$
0.229416 + 0.973329i $$0.426318\pi$$
$$20$$ 2.00000 0.447214
$$21$$ 0 0
$$22$$ 1.00000 0.213201
$$23$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$24$$ −1.00000 −0.204124
$$25$$ −1.00000 −0.200000
$$26$$ 4.00000 0.784465
$$27$$ −1.00000 −0.192450
$$28$$ 0 0
$$29$$ −6.00000 −1.11417 −0.557086 0.830455i $$-0.688081\pi$$
−0.557086 + 0.830455i $$0.688081\pi$$
$$30$$ −2.00000 −0.365148
$$31$$ 2.00000 0.359211 0.179605 0.983739i $$-0.442518\pi$$
0.179605 + 0.983739i $$0.442518\pi$$
$$32$$ 1.00000 0.176777
$$33$$ −1.00000 −0.174078
$$34$$ 6.00000 1.02899
$$35$$ 0 0
$$36$$ 1.00000 0.166667
$$37$$ 2.00000 0.328798 0.164399 0.986394i $$-0.447432\pi$$
0.164399 + 0.986394i $$0.447432\pi$$
$$38$$ 2.00000 0.324443
$$39$$ −4.00000 −0.640513
$$40$$ 2.00000 0.316228
$$41$$ −2.00000 −0.312348 −0.156174 0.987730i $$-0.549916\pi$$
−0.156174 + 0.987730i $$0.549916\pi$$
$$42$$ 0 0
$$43$$ −12.0000 −1.82998 −0.914991 0.403473i $$-0.867803\pi$$
−0.914991 + 0.403473i $$0.867803\pi$$
$$44$$ 1.00000 0.150756
$$45$$ 2.00000 0.298142
$$46$$ 0 0
$$47$$ −6.00000 −0.875190 −0.437595 0.899172i $$-0.644170\pi$$
−0.437595 + 0.899172i $$0.644170\pi$$
$$48$$ −1.00000 −0.144338
$$49$$ 0 0
$$50$$ −1.00000 −0.141421
$$51$$ −6.00000 −0.840168
$$52$$ 4.00000 0.554700
$$53$$ 10.0000 1.37361 0.686803 0.726844i $$-0.259014\pi$$
0.686803 + 0.726844i $$0.259014\pi$$
$$54$$ −1.00000 −0.136083
$$55$$ 2.00000 0.269680
$$56$$ 0 0
$$57$$ −2.00000 −0.264906
$$58$$ −6.00000 −0.787839
$$59$$ 8.00000 1.04151 0.520756 0.853706i $$-0.325650\pi$$
0.520756 + 0.853706i $$0.325650\pi$$
$$60$$ −2.00000 −0.258199
$$61$$ 4.00000 0.512148 0.256074 0.966657i $$-0.417571\pi$$
0.256074 + 0.966657i $$0.417571\pi$$
$$62$$ 2.00000 0.254000
$$63$$ 0 0
$$64$$ 1.00000 0.125000
$$65$$ 8.00000 0.992278
$$66$$ −1.00000 −0.123091
$$67$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$68$$ 6.00000 0.727607
$$69$$ 0 0
$$70$$ 0 0
$$71$$ −8.00000 −0.949425 −0.474713 0.880141i $$-0.657448\pi$$
−0.474713 + 0.880141i $$0.657448\pi$$
$$72$$ 1.00000 0.117851
$$73$$ 6.00000 0.702247 0.351123 0.936329i $$-0.385800\pi$$
0.351123 + 0.936329i $$0.385800\pi$$
$$74$$ 2.00000 0.232495
$$75$$ 1.00000 0.115470
$$76$$ 2.00000 0.229416
$$77$$ 0 0
$$78$$ −4.00000 −0.452911
$$79$$ 12.0000 1.35011 0.675053 0.737769i $$-0.264121\pi$$
0.675053 + 0.737769i $$0.264121\pi$$
$$80$$ 2.00000 0.223607
$$81$$ 1.00000 0.111111
$$82$$ −2.00000 −0.220863
$$83$$ 6.00000 0.658586 0.329293 0.944228i $$-0.393190\pi$$
0.329293 + 0.944228i $$0.393190\pi$$
$$84$$ 0 0
$$85$$ 12.0000 1.30158
$$86$$ −12.0000 −1.29399
$$87$$ 6.00000 0.643268
$$88$$ 1.00000 0.106600
$$89$$ −12.0000 −1.27200 −0.635999 0.771690i $$-0.719412\pi$$
−0.635999 + 0.771690i $$0.719412\pi$$
$$90$$ 2.00000 0.210819
$$91$$ 0 0
$$92$$ 0 0
$$93$$ −2.00000 −0.207390
$$94$$ −6.00000 −0.618853
$$95$$ 4.00000 0.410391
$$96$$ −1.00000 −0.102062
$$97$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$98$$ 0 0
$$99$$ 1.00000 0.100504
$$100$$ −1.00000 −0.100000
$$101$$ 12.0000 1.19404 0.597022 0.802225i $$-0.296350\pi$$
0.597022 + 0.802225i $$0.296350\pi$$
$$102$$ −6.00000 −0.594089
$$103$$ 6.00000 0.591198 0.295599 0.955312i $$-0.404481\pi$$
0.295599 + 0.955312i $$0.404481\pi$$
$$104$$ 4.00000 0.392232
$$105$$ 0 0
$$106$$ 10.0000 0.971286
$$107$$ −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 $$0.658969\pi$$
$$110$$ 2.00000 0.190693
$$111$$ −2.00000 −0.189832
$$112$$ 0 0
$$113$$ 14.0000 1.31701 0.658505 0.752577i $$-0.271189\pi$$
0.658505 + 0.752577i $$0.271189\pi$$
$$114$$ −2.00000 −0.187317
$$115$$ 0 0
$$116$$ −6.00000 −0.557086
$$117$$ 4.00000 0.369800
$$118$$ 8.00000 0.736460
$$119$$ 0 0
$$120$$ −2.00000 −0.182574
$$121$$ 1.00000 0.0909091
$$122$$ 4.00000 0.362143
$$123$$ 2.00000 0.180334
$$124$$ 2.00000 0.179605
$$125$$ −12.0000 −1.07331
$$126$$ 0 0
$$127$$ 4.00000 0.354943 0.177471 0.984126i $$-0.443208\pi$$
0.177471 + 0.984126i $$0.443208\pi$$
$$128$$ 1.00000 0.0883883
$$129$$ 12.0000 1.05654
$$130$$ 8.00000 0.701646
$$131$$ 2.00000 0.174741 0.0873704 0.996176i $$-0.472154\pi$$
0.0873704 + 0.996176i $$0.472154\pi$$
$$132$$ −1.00000 −0.0870388
$$133$$ 0 0
$$134$$ 0 0
$$135$$ −2.00000 −0.172133
$$136$$ 6.00000 0.514496
$$137$$ 10.0000 0.854358 0.427179 0.904167i $$-0.359507\pi$$
0.427179 + 0.904167i $$0.359507\pi$$
$$138$$ 0 0
$$139$$ −10.0000 −0.848189 −0.424094 0.905618i $$-0.639408\pi$$
−0.424094 + 0.905618i $$0.639408\pi$$
$$140$$ 0 0
$$141$$ 6.00000 0.505291
$$142$$ −8.00000 −0.671345
$$143$$ 4.00000 0.334497
$$144$$ 1.00000 0.0833333
$$145$$ −12.0000 −0.996546
$$146$$ 6.00000 0.496564
$$147$$ 0 0
$$148$$ 2.00000 0.164399
$$149$$ 10.0000 0.819232 0.409616 0.912258i $$-0.365663\pi$$
0.409616 + 0.912258i $$0.365663\pi$$
$$150$$ 1.00000 0.0816497
$$151$$ −16.0000 −1.30206 −0.651031 0.759051i $$-0.725663\pi$$
−0.651031 + 0.759051i $$0.725663\pi$$
$$152$$ 2.00000 0.162221
$$153$$ 6.00000 0.485071
$$154$$ 0 0
$$155$$ 4.00000 0.321288
$$156$$ −4.00000 −0.320256
$$157$$ 14.0000 1.11732 0.558661 0.829396i $$-0.311315\pi$$
0.558661 + 0.829396i $$0.311315\pi$$
$$158$$ 12.0000 0.954669
$$159$$ −10.0000 −0.793052
$$160$$ 2.00000 0.158114
$$161$$ 0 0
$$162$$ 1.00000 0.0785674
$$163$$ −12.0000 −0.939913 −0.469956 0.882690i $$-0.655730\pi$$
−0.469956 + 0.882690i $$0.655730\pi$$
$$164$$ −2.00000 −0.156174
$$165$$ −2.00000 −0.155700
$$166$$ 6.00000 0.465690
$$167$$ 12.0000 0.928588 0.464294 0.885681i $$-0.346308\pi$$
0.464294 + 0.885681i $$0.346308\pi$$
$$168$$ 0 0
$$169$$ 3.00000 0.230769
$$170$$ 12.0000 0.920358
$$171$$ 2.00000 0.152944
$$172$$ −12.0000 −0.914991
$$173$$ −24.0000 −1.82469 −0.912343 0.409426i $$-0.865729\pi$$
−0.912343 + 0.409426i $$0.865729\pi$$
$$174$$ 6.00000 0.454859
$$175$$ 0 0
$$176$$ 1.00000 0.0753778
$$177$$ −8.00000 −0.601317
$$178$$ −12.0000 −0.899438
$$179$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$180$$ 2.00000 0.149071
$$181$$ 18.0000 1.33793 0.668965 0.743294i $$-0.266738\pi$$
0.668965 + 0.743294i $$0.266738\pi$$
$$182$$ 0 0
$$183$$ −4.00000 −0.295689
$$184$$ 0 0
$$185$$ 4.00000 0.294086
$$186$$ −2.00000 −0.146647
$$187$$ 6.00000 0.438763
$$188$$ −6.00000 −0.437595
$$189$$ 0 0
$$190$$ 4.00000 0.290191
$$191$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$192$$ −1.00000 −0.0721688
$$193$$ −14.0000 −1.00774 −0.503871 0.863779i $$-0.668091\pi$$
−0.503871 + 0.863779i $$0.668091\pi$$
$$194$$ 0 0
$$195$$ −8.00000 −0.572892
$$196$$ 0 0
$$197$$ −18.0000 −1.28245 −0.641223 0.767354i $$-0.721573\pi$$
−0.641223 + 0.767354i $$0.721573\pi$$
$$198$$ 1.00000 0.0710669
$$199$$ 14.0000 0.992434 0.496217 0.868199i $$-0.334722\pi$$
0.496217 + 0.868199i $$0.334722\pi$$
$$200$$ −1.00000 −0.0707107
$$201$$ 0 0
$$202$$ 12.0000 0.844317
$$203$$ 0 0
$$204$$ −6.00000 −0.420084
$$205$$ −4.00000 −0.279372
$$206$$ 6.00000 0.418040
$$207$$ 0 0
$$208$$ 4.00000 0.277350
$$209$$ 2.00000 0.138343
$$210$$ 0 0
$$211$$ 20.0000 1.37686 0.688428 0.725304i $$-0.258301\pi$$
0.688428 + 0.725304i $$0.258301\pi$$
$$212$$ 10.0000 0.686803
$$213$$ 8.00000 0.548151
$$214$$ −12.0000 −0.820303
$$215$$ −24.0000 −1.63679
$$216$$ −1.00000 −0.0680414
$$217$$ 0 0
$$218$$ −10.0000 −0.677285
$$219$$ −6.00000 −0.405442
$$220$$ 2.00000 0.134840
$$221$$ 24.0000 1.61441
$$222$$ −2.00000 −0.134231
$$223$$ −14.0000 −0.937509 −0.468755 0.883328i $$-0.655297\pi$$
−0.468755 + 0.883328i $$0.655297\pi$$
$$224$$ 0 0
$$225$$ −1.00000 −0.0666667
$$226$$ 14.0000 0.931266
$$227$$ −6.00000 −0.398234 −0.199117 0.979976i $$-0.563807\pi$$
−0.199117 + 0.979976i $$0.563807\pi$$
$$228$$ −2.00000 −0.132453
$$229$$ −6.00000 −0.396491 −0.198246 0.980152i $$-0.563524\pi$$
−0.198246 + 0.980152i $$0.563524\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ −6.00000 −0.393919
$$233$$ 26.0000 1.70332 0.851658 0.524097i $$-0.175597\pi$$
0.851658 + 0.524097i $$0.175597\pi$$
$$234$$ 4.00000 0.261488
$$235$$ −12.0000 −0.782794
$$236$$ 8.00000 0.520756
$$237$$ −12.0000 −0.779484
$$238$$ 0 0
$$239$$ −16.0000 −1.03495 −0.517477 0.855697i $$-0.673129\pi$$
−0.517477 + 0.855697i $$0.673129\pi$$
$$240$$ −2.00000 −0.129099
$$241$$ 22.0000 1.41714 0.708572 0.705638i $$-0.249340\pi$$
0.708572 + 0.705638i $$0.249340\pi$$
$$242$$ 1.00000 0.0642824
$$243$$ −1.00000 −0.0641500
$$244$$ 4.00000 0.256074
$$245$$ 0 0
$$246$$ 2.00000 0.127515
$$247$$ 8.00000 0.509028
$$248$$ 2.00000 0.127000
$$249$$ −6.00000 −0.380235
$$250$$ −12.0000 −0.758947
$$251$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$252$$ 0 0
$$253$$ 0 0
$$254$$ 4.00000 0.250982
$$255$$ −12.0000 −0.751469
$$256$$ 1.00000 0.0625000
$$257$$ 20.0000 1.24757 0.623783 0.781598i $$-0.285595\pi$$
0.623783 + 0.781598i $$0.285595\pi$$
$$258$$ 12.0000 0.747087
$$259$$ 0 0
$$260$$ 8.00000 0.496139
$$261$$ −6.00000 −0.371391
$$262$$ 2.00000 0.123560
$$263$$ 4.00000 0.246651 0.123325 0.992366i $$-0.460644\pi$$
0.123325 + 0.992366i $$0.460644\pi$$
$$264$$ −1.00000 −0.0615457
$$265$$ 20.0000 1.22859
$$266$$ 0 0
$$267$$ 12.0000 0.734388
$$268$$ 0 0
$$269$$ −6.00000 −0.365826 −0.182913 0.983129i $$-0.558553\pi$$
−0.182913 + 0.983129i $$0.558553\pi$$
$$270$$ −2.00000 −0.121716
$$271$$ −28.0000 −1.70088 −0.850439 0.526073i $$-0.823664\pi$$
−0.850439 + 0.526073i $$0.823664\pi$$
$$272$$ 6.00000 0.363803
$$273$$ 0 0
$$274$$ 10.0000 0.604122
$$275$$ −1.00000 −0.0603023
$$276$$ 0 0
$$277$$ 10.0000 0.600842 0.300421 0.953807i $$-0.402873\pi$$
0.300421 + 0.953807i $$0.402873\pi$$
$$278$$ −10.0000 −0.599760
$$279$$ 2.00000 0.119737
$$280$$ 0 0
$$281$$ 2.00000 0.119310 0.0596550 0.998219i $$-0.481000\pi$$
0.0596550 + 0.998219i $$0.481000\pi$$
$$282$$ 6.00000 0.357295
$$283$$ 14.0000 0.832214 0.416107 0.909316i $$-0.363394\pi$$
0.416107 + 0.909316i $$0.363394\pi$$
$$284$$ −8.00000 −0.474713
$$285$$ −4.00000 −0.236940
$$286$$ 4.00000 0.236525
$$287$$ 0 0
$$288$$ 1.00000 0.0589256
$$289$$ 19.0000 1.11765
$$290$$ −12.0000 −0.704664
$$291$$ 0 0
$$292$$ 6.00000 0.351123
$$293$$ 8.00000 0.467365 0.233682 0.972313i $$-0.424922\pi$$
0.233682 + 0.972313i $$0.424922\pi$$
$$294$$ 0 0
$$295$$ 16.0000 0.931556
$$296$$ 2.00000 0.116248
$$297$$ −1.00000 −0.0580259
$$298$$ 10.0000 0.579284
$$299$$ 0 0
$$300$$ 1.00000 0.0577350
$$301$$ 0 0
$$302$$ −16.0000 −0.920697
$$303$$ −12.0000 −0.689382
$$304$$ 2.00000 0.114708
$$305$$ 8.00000 0.458079
$$306$$ 6.00000 0.342997
$$307$$ −26.0000 −1.48390 −0.741949 0.670456i $$-0.766098\pi$$
−0.741949 + 0.670456i $$0.766098\pi$$
$$308$$ 0 0
$$309$$ −6.00000 −0.341328
$$310$$ 4.00000 0.227185
$$311$$ −34.0000 −1.92796 −0.963982 0.265969i $$-0.914308\pi$$
−0.963982 + 0.265969i $$0.914308\pi$$
$$312$$ −4.00000 −0.226455
$$313$$ −20.0000 −1.13047 −0.565233 0.824931i $$-0.691214\pi$$
−0.565233 + 0.824931i $$0.691214\pi$$
$$314$$ 14.0000 0.790066
$$315$$ 0 0
$$316$$ 12.0000 0.675053
$$317$$ 18.0000 1.01098 0.505490 0.862832i $$-0.331312\pi$$
0.505490 + 0.862832i $$0.331312\pi$$
$$318$$ −10.0000 −0.560772
$$319$$ −6.00000 −0.335936
$$320$$ 2.00000 0.111803
$$321$$ 12.0000 0.669775
$$322$$ 0 0
$$323$$ 12.0000 0.667698
$$324$$ 1.00000 0.0555556
$$325$$ −4.00000 −0.221880
$$326$$ −12.0000 −0.664619
$$327$$ 10.0000 0.553001
$$328$$ −2.00000 −0.110432
$$329$$ 0 0
$$330$$ −2.00000 −0.110096
$$331$$ −24.0000 −1.31916 −0.659580 0.751635i $$-0.729266\pi$$
−0.659580 + 0.751635i $$0.729266\pi$$
$$332$$ 6.00000 0.329293
$$333$$ 2.00000 0.109599
$$334$$ 12.0000 0.656611
$$335$$ 0 0
$$336$$ 0 0
$$337$$ −26.0000 −1.41631 −0.708155 0.706057i $$-0.750472\pi$$
−0.708155 + 0.706057i $$0.750472\pi$$
$$338$$ 3.00000 0.163178
$$339$$ −14.0000 −0.760376
$$340$$ 12.0000 0.650791
$$341$$ 2.00000 0.108306
$$342$$ 2.00000 0.108148
$$343$$ 0 0
$$344$$ −12.0000 −0.646997
$$345$$ 0 0
$$346$$ −24.0000 −1.29025
$$347$$ 20.0000 1.07366 0.536828 0.843692i $$-0.319622\pi$$
0.536828 + 0.843692i $$0.319622\pi$$
$$348$$ 6.00000 0.321634
$$349$$ −16.0000 −0.856460 −0.428230 0.903670i $$-0.640863\pi$$
−0.428230 + 0.903670i $$0.640863\pi$$
$$350$$ 0 0
$$351$$ −4.00000 −0.213504
$$352$$ 1.00000 0.0533002
$$353$$ −12.0000 −0.638696 −0.319348 0.947638i $$-0.603464\pi$$
−0.319348 + 0.947638i $$0.603464\pi$$
$$354$$ −8.00000 −0.425195
$$355$$ −16.0000 −0.849192
$$356$$ −12.0000 −0.635999
$$357$$ 0 0
$$358$$ 0 0
$$359$$ 20.0000 1.05556 0.527780 0.849381i $$-0.323025\pi$$
0.527780 + 0.849381i $$0.323025\pi$$
$$360$$ 2.00000 0.105409
$$361$$ −15.0000 −0.789474
$$362$$ 18.0000 0.946059
$$363$$ −1.00000 −0.0524864
$$364$$ 0 0
$$365$$ 12.0000 0.628109
$$366$$ −4.00000 −0.209083
$$367$$ −22.0000 −1.14839 −0.574195 0.818718i $$-0.694685\pi$$
−0.574195 + 0.818718i $$0.694685\pi$$
$$368$$ 0 0
$$369$$ −2.00000 −0.104116
$$370$$ 4.00000 0.207950
$$371$$ 0 0
$$372$$ −2.00000 −0.103695
$$373$$ 6.00000 0.310668 0.155334 0.987862i $$-0.450355\pi$$
0.155334 + 0.987862i $$0.450355\pi$$
$$374$$ 6.00000 0.310253
$$375$$ 12.0000 0.619677
$$376$$ −6.00000 −0.309426
$$377$$ −24.0000 −1.23606
$$378$$ 0 0
$$379$$ −20.0000 −1.02733 −0.513665 0.857991i $$-0.671713\pi$$
−0.513665 + 0.857991i $$0.671713\pi$$
$$380$$ 4.00000 0.205196
$$381$$ −4.00000 −0.204926
$$382$$ 0 0
$$383$$ 6.00000 0.306586 0.153293 0.988181i $$-0.451012\pi$$
0.153293 + 0.988181i $$0.451012\pi$$
$$384$$ −1.00000 −0.0510310
$$385$$ 0 0
$$386$$ −14.0000 −0.712581
$$387$$ −12.0000 −0.609994
$$388$$ 0 0
$$389$$ 22.0000 1.11544 0.557722 0.830028i $$-0.311675\pi$$
0.557722 + 0.830028i $$0.311675\pi$$
$$390$$ −8.00000 −0.405096
$$391$$ 0 0
$$392$$ 0 0
$$393$$ −2.00000 −0.100887
$$394$$ −18.0000 −0.906827
$$395$$ 24.0000 1.20757
$$396$$ 1.00000 0.0502519
$$397$$ −34.0000 −1.70641 −0.853206 0.521575i $$-0.825345\pi$$
−0.853206 + 0.521575i $$0.825345\pi$$
$$398$$ 14.0000 0.701757
$$399$$ 0 0
$$400$$ −1.00000 −0.0500000
$$401$$ −30.0000 −1.49813 −0.749064 0.662497i $$-0.769497\pi$$
−0.749064 + 0.662497i $$0.769497\pi$$
$$402$$ 0 0
$$403$$ 8.00000 0.398508
$$404$$ 12.0000 0.597022
$$405$$ 2.00000 0.0993808
$$406$$ 0 0
$$407$$ 2.00000 0.0991363
$$408$$ −6.00000 −0.297044
$$409$$ 2.00000 0.0988936 0.0494468 0.998777i $$-0.484254\pi$$
0.0494468 + 0.998777i $$0.484254\pi$$
$$410$$ −4.00000 −0.197546
$$411$$ −10.0000 −0.493264
$$412$$ 6.00000 0.295599
$$413$$ 0 0
$$414$$ 0 0
$$415$$ 12.0000 0.589057
$$416$$ 4.00000 0.196116
$$417$$ 10.0000 0.489702
$$418$$ 2.00000 0.0978232
$$419$$ 12.0000 0.586238 0.293119 0.956076i $$-0.405307\pi$$
0.293119 + 0.956076i $$0.405307\pi$$
$$420$$ 0 0
$$421$$ 14.0000 0.682318 0.341159 0.940006i $$-0.389181\pi$$
0.341159 + 0.940006i $$0.389181\pi$$
$$422$$ 20.0000 0.973585
$$423$$ −6.00000 −0.291730
$$424$$ 10.0000 0.485643
$$425$$ −6.00000 −0.291043
$$426$$ 8.00000 0.387601
$$427$$ 0 0
$$428$$ −12.0000 −0.580042
$$429$$ −4.00000 −0.193122
$$430$$ −24.0000 −1.15738
$$431$$ 12.0000 0.578020 0.289010 0.957326i $$-0.406674\pi$$
0.289010 + 0.957326i $$0.406674\pi$$
$$432$$ −1.00000 −0.0481125
$$433$$ 8.00000 0.384455 0.192228 0.981350i $$-0.438429\pi$$
0.192228 + 0.981350i $$0.438429\pi$$
$$434$$ 0 0
$$435$$ 12.0000 0.575356
$$436$$ −10.0000 −0.478913
$$437$$ 0 0
$$438$$ −6.00000 −0.286691
$$439$$ 28.0000 1.33637 0.668184 0.743996i $$-0.267072\pi$$
0.668184 + 0.743996i $$0.267072\pi$$
$$440$$ 2.00000 0.0953463
$$441$$ 0 0
$$442$$ 24.0000 1.14156
$$443$$ 24.0000 1.14027 0.570137 0.821549i $$-0.306890\pi$$
0.570137 + 0.821549i $$0.306890\pi$$
$$444$$ −2.00000 −0.0949158
$$445$$ −24.0000 −1.13771
$$446$$ −14.0000 −0.662919
$$447$$ −10.0000 −0.472984
$$448$$ 0 0
$$449$$ −18.0000 −0.849473 −0.424736 0.905317i $$-0.639633\pi$$
−0.424736 + 0.905317i $$0.639633\pi$$
$$450$$ −1.00000 −0.0471405
$$451$$ −2.00000 −0.0941763
$$452$$ 14.0000 0.658505
$$453$$ 16.0000 0.751746
$$454$$ −6.00000 −0.281594
$$455$$ 0 0
$$456$$ −2.00000 −0.0936586
$$457$$ −6.00000 −0.280668 −0.140334 0.990104i $$-0.544818\pi$$
−0.140334 + 0.990104i $$0.544818\pi$$
$$458$$ −6.00000 −0.280362
$$459$$ −6.00000 −0.280056
$$460$$ 0 0
$$461$$ 12.0000 0.558896 0.279448 0.960161i $$-0.409849\pi$$
0.279448 + 0.960161i $$0.409849\pi$$
$$462$$ 0 0
$$463$$ 16.0000 0.743583 0.371792 0.928316i $$-0.378744\pi$$
0.371792 + 0.928316i $$0.378744\pi$$
$$464$$ −6.00000 −0.278543
$$465$$ −4.00000 −0.185496
$$466$$ 26.0000 1.20443
$$467$$ 28.0000 1.29569 0.647843 0.761774i $$-0.275671\pi$$
0.647843 + 0.761774i $$0.275671\pi$$
$$468$$ 4.00000 0.184900
$$469$$ 0 0
$$470$$ −12.0000 −0.553519
$$471$$ −14.0000 −0.645086
$$472$$ 8.00000 0.368230
$$473$$ −12.0000 −0.551761
$$474$$ −12.0000 −0.551178
$$475$$ −2.00000 −0.0917663
$$476$$ 0 0
$$477$$ 10.0000 0.457869
$$478$$ −16.0000 −0.731823
$$479$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$480$$ −2.00000 −0.0912871
$$481$$ 8.00000 0.364769
$$482$$ 22.0000 1.00207
$$483$$ 0 0
$$484$$ 1.00000 0.0454545
$$485$$ 0 0
$$486$$ −1.00000 −0.0453609
$$487$$ −16.0000 −0.725029 −0.362515 0.931978i $$-0.618082\pi$$
−0.362515 + 0.931978i $$0.618082\pi$$
$$488$$ 4.00000 0.181071
$$489$$ 12.0000 0.542659
$$490$$ 0 0
$$491$$ −20.0000 −0.902587 −0.451294 0.892375i $$-0.649037\pi$$
−0.451294 + 0.892375i $$0.649037\pi$$
$$492$$ 2.00000 0.0901670
$$493$$ −36.0000 −1.62136
$$494$$ 8.00000 0.359937
$$495$$ 2.00000 0.0898933
$$496$$ 2.00000 0.0898027
$$497$$ 0 0
$$498$$ −6.00000 −0.268866
$$499$$ −40.0000 −1.79065 −0.895323 0.445418i $$-0.853055\pi$$
−0.895323 + 0.445418i $$0.853055\pi$$
$$500$$ −12.0000 −0.536656
$$501$$ −12.0000 −0.536120
$$502$$ 0 0
$$503$$ −28.0000 −1.24846 −0.624229 0.781241i $$-0.714587\pi$$
−0.624229 + 0.781241i $$0.714587\pi$$
$$504$$ 0 0
$$505$$ 24.0000 1.06799
$$506$$ 0 0
$$507$$ −3.00000 −0.133235
$$508$$ 4.00000 0.177471
$$509$$ −30.0000 −1.32973 −0.664863 0.746965i $$-0.731510\pi$$
−0.664863 + 0.746965i $$0.731510\pi$$
$$510$$ −12.0000 −0.531369
$$511$$ 0 0
$$512$$ 1.00000 0.0441942
$$513$$ −2.00000 −0.0883022
$$514$$ 20.0000 0.882162
$$515$$ 12.0000 0.528783
$$516$$ 12.0000 0.528271
$$517$$ −6.00000 −0.263880
$$518$$ 0 0
$$519$$ 24.0000 1.05348
$$520$$ 8.00000 0.350823
$$521$$ 28.0000 1.22670 0.613351 0.789810i $$-0.289821\pi$$
0.613351 + 0.789810i $$0.289821\pi$$
$$522$$ −6.00000 −0.262613
$$523$$ −2.00000 −0.0874539 −0.0437269 0.999044i $$-0.513923\pi$$
−0.0437269 + 0.999044i $$0.513923\pi$$
$$524$$ 2.00000 0.0873704
$$525$$ 0 0
$$526$$ 4.00000 0.174408
$$527$$ 12.0000 0.522728
$$528$$ −1.00000 −0.0435194
$$529$$ −23.0000 −1.00000
$$530$$ 20.0000 0.868744
$$531$$ 8.00000 0.347170
$$532$$ 0 0
$$533$$ −8.00000 −0.346518
$$534$$ 12.0000 0.519291
$$535$$ −24.0000 −1.03761
$$536$$ 0 0
$$537$$ 0 0
$$538$$ −6.00000 −0.258678
$$539$$ 0 0
$$540$$ −2.00000 −0.0860663
$$541$$ −2.00000 −0.0859867 −0.0429934 0.999075i $$-0.513689\pi$$
−0.0429934 + 0.999075i $$0.513689\pi$$
$$542$$ −28.0000 −1.20270
$$543$$ −18.0000 −0.772454
$$544$$ 6.00000 0.257248
$$545$$ −20.0000 −0.856706
$$546$$ 0 0
$$547$$ −20.0000 −0.855138 −0.427569 0.903983i $$-0.640630\pi$$
−0.427569 + 0.903983i $$0.640630\pi$$
$$548$$ 10.0000 0.427179
$$549$$ 4.00000 0.170716
$$550$$ −1.00000 −0.0426401
$$551$$ −12.0000 −0.511217
$$552$$ 0 0
$$553$$ 0 0
$$554$$ 10.0000 0.424859
$$555$$ −4.00000 −0.169791
$$556$$ −10.0000 −0.424094
$$557$$ −26.0000 −1.10166 −0.550828 0.834619i $$-0.685688\pi$$
−0.550828 + 0.834619i $$0.685688\pi$$
$$558$$ 2.00000 0.0846668
$$559$$ −48.0000 −2.03018
$$560$$ 0 0
$$561$$ −6.00000 −0.253320
$$562$$ 2.00000 0.0843649
$$563$$ −14.0000 −0.590030 −0.295015 0.955493i $$-0.595325\pi$$
−0.295015 + 0.955493i $$0.595325\pi$$
$$564$$ 6.00000 0.252646
$$565$$ 28.0000 1.17797
$$566$$ 14.0000 0.588464
$$567$$ 0 0
$$568$$ −8.00000 −0.335673
$$569$$ 34.0000 1.42535 0.712677 0.701492i $$-0.247483\pi$$
0.712677 + 0.701492i $$0.247483\pi$$
$$570$$ −4.00000 −0.167542
$$571$$ 4.00000 0.167395 0.0836974 0.996491i $$-0.473327\pi$$
0.0836974 + 0.996491i $$0.473327\pi$$
$$572$$ 4.00000 0.167248
$$573$$ 0 0
$$574$$ 0 0
$$575$$ 0 0
$$576$$ 1.00000 0.0416667
$$577$$ −28.0000 −1.16566 −0.582828 0.812596i $$-0.698054\pi$$
−0.582828 + 0.812596i $$0.698054\pi$$
$$578$$ 19.0000 0.790296
$$579$$ 14.0000 0.581820
$$580$$ −12.0000 −0.498273
$$581$$ 0 0
$$582$$ 0 0
$$583$$ 10.0000 0.414158
$$584$$ 6.00000 0.248282
$$585$$ 8.00000 0.330759
$$586$$ 8.00000 0.330477
$$587$$ −24.0000 −0.990586 −0.495293 0.868726i $$-0.664939\pi$$
−0.495293 + 0.868726i $$0.664939\pi$$
$$588$$ 0 0
$$589$$ 4.00000 0.164817
$$590$$ 16.0000 0.658710
$$591$$ 18.0000 0.740421
$$592$$ 2.00000 0.0821995
$$593$$ 22.0000 0.903432 0.451716 0.892162i $$-0.350812\pi$$
0.451716 + 0.892162i $$0.350812\pi$$
$$594$$ −1.00000 −0.0410305
$$595$$ 0 0
$$596$$ 10.0000 0.409616
$$597$$ −14.0000 −0.572982
$$598$$ 0 0
$$599$$ −32.0000 −1.30748 −0.653742 0.756717i $$-0.726802\pi$$
−0.653742 + 0.756717i $$0.726802\pi$$
$$600$$ 1.00000 0.0408248
$$601$$ −46.0000 −1.87638 −0.938190 0.346122i $$-0.887498\pi$$
−0.938190 + 0.346122i $$0.887498\pi$$
$$602$$ 0 0
$$603$$ 0 0
$$604$$ −16.0000 −0.651031
$$605$$ 2.00000 0.0813116
$$606$$ −12.0000 −0.487467
$$607$$ −8.00000 −0.324710 −0.162355 0.986732i $$-0.551909\pi$$
−0.162355 + 0.986732i $$0.551909\pi$$
$$608$$ 2.00000 0.0811107
$$609$$ 0 0
$$610$$ 8.00000 0.323911
$$611$$ −24.0000 −0.970936
$$612$$ 6.00000 0.242536
$$613$$ −6.00000 −0.242338 −0.121169 0.992632i $$-0.538664\pi$$
−0.121169 + 0.992632i $$0.538664\pi$$
$$614$$ −26.0000 −1.04927
$$615$$ 4.00000 0.161296
$$616$$ 0 0
$$617$$ −18.0000 −0.724653 −0.362326 0.932051i $$-0.618017\pi$$
−0.362326 + 0.932051i $$0.618017\pi$$
$$618$$ −6.00000 −0.241355
$$619$$ −4.00000 −0.160774 −0.0803868 0.996764i $$-0.525616\pi$$
−0.0803868 + 0.996764i $$0.525616\pi$$
$$620$$ 4.00000 0.160644
$$621$$ 0 0
$$622$$ −34.0000 −1.36328
$$623$$ 0 0
$$624$$ −4.00000 −0.160128
$$625$$ −19.0000 −0.760000
$$626$$ −20.0000 −0.799361
$$627$$ −2.00000 −0.0798723
$$628$$ 14.0000 0.558661
$$629$$ 12.0000 0.478471
$$630$$ 0 0
$$631$$ −40.0000 −1.59237 −0.796187 0.605050i $$-0.793153\pi$$
−0.796187 + 0.605050i $$0.793153\pi$$
$$632$$ 12.0000 0.477334
$$633$$ −20.0000 −0.794929
$$634$$ 18.0000 0.714871
$$635$$ 8.00000 0.317470
$$636$$ −10.0000 −0.396526
$$637$$ 0 0
$$638$$ −6.00000 −0.237542
$$639$$ −8.00000 −0.316475
$$640$$ 2.00000 0.0790569
$$641$$ 6.00000 0.236986 0.118493 0.992955i $$-0.462194\pi$$
0.118493 + 0.992955i $$0.462194\pi$$
$$642$$ 12.0000 0.473602
$$643$$ 40.0000 1.57745 0.788723 0.614749i $$-0.210743\pi$$
0.788723 + 0.614749i $$0.210743\pi$$
$$644$$ 0 0
$$645$$ 24.0000 0.944999
$$646$$ 12.0000 0.472134
$$647$$ −10.0000 −0.393141 −0.196570 0.980490i $$-0.562980\pi$$
−0.196570 + 0.980490i $$0.562980\pi$$
$$648$$ 1.00000 0.0392837
$$649$$ 8.00000 0.314027
$$650$$ −4.00000 −0.156893
$$651$$ 0 0
$$652$$ −12.0000 −0.469956
$$653$$ −18.0000 −0.704394 −0.352197 0.935926i $$-0.614565\pi$$
−0.352197 + 0.935926i $$0.614565\pi$$
$$654$$ 10.0000 0.391031
$$655$$ 4.00000 0.156293
$$656$$ −2.00000 −0.0780869
$$657$$ 6.00000 0.234082
$$658$$ 0 0
$$659$$ −12.0000 −0.467454 −0.233727 0.972302i $$-0.575092\pi$$
−0.233727 + 0.972302i $$0.575092\pi$$
$$660$$ −2.00000 −0.0778499
$$661$$ 18.0000 0.700119 0.350059 0.936727i $$-0.386161\pi$$
0.350059 + 0.936727i $$0.386161\pi$$
$$662$$ −24.0000 −0.932786
$$663$$ −24.0000 −0.932083
$$664$$ 6.00000 0.232845
$$665$$ 0 0
$$666$$ 2.00000 0.0774984
$$667$$ 0 0
$$668$$ 12.0000 0.464294
$$669$$ 14.0000 0.541271
$$670$$ 0 0
$$671$$ 4.00000 0.154418
$$672$$ 0 0
$$673$$ −18.0000 −0.693849 −0.346925 0.937893i $$-0.612774\pi$$
−0.346925 + 0.937893i $$0.612774\pi$$
$$674$$ −26.0000 −1.00148
$$675$$ 1.00000 0.0384900
$$676$$ 3.00000 0.115385
$$677$$ −48.0000 −1.84479 −0.922395 0.386248i $$-0.873771\pi$$
−0.922395 + 0.386248i $$0.873771\pi$$
$$678$$ −14.0000 −0.537667
$$679$$ 0 0
$$680$$ 12.0000 0.460179
$$681$$ 6.00000 0.229920
$$682$$ 2.00000 0.0765840
$$683$$ 16.0000 0.612223 0.306111 0.951996i $$-0.400972\pi$$
0.306111 + 0.951996i $$0.400972\pi$$
$$684$$ 2.00000 0.0764719
$$685$$ 20.0000 0.764161
$$686$$ 0 0
$$687$$ 6.00000 0.228914
$$688$$ −12.0000 −0.457496
$$689$$ 40.0000 1.52388
$$690$$ 0 0
$$691$$ 40.0000 1.52167 0.760836 0.648944i $$-0.224789\pi$$
0.760836 + 0.648944i $$0.224789\pi$$
$$692$$ −24.0000 −0.912343
$$693$$ 0 0
$$694$$ 20.0000 0.759190
$$695$$ −20.0000 −0.758643
$$696$$ 6.00000 0.227429
$$697$$ −12.0000 −0.454532
$$698$$ −16.0000 −0.605609
$$699$$ −26.0000 −0.983410
$$700$$ 0 0
$$701$$ 14.0000 0.528773 0.264386 0.964417i $$-0.414831\pi$$
0.264386 + 0.964417i $$0.414831\pi$$
$$702$$ −4.00000 −0.150970
$$703$$ 4.00000 0.150863
$$704$$ 1.00000 0.0376889
$$705$$ 12.0000 0.451946
$$706$$ −12.0000 −0.451626
$$707$$ 0 0
$$708$$ −8.00000 −0.300658
$$709$$ −38.0000 −1.42712 −0.713560 0.700594i $$-0.752918\pi$$
−0.713560 + 0.700594i $$0.752918\pi$$
$$710$$ −16.0000 −0.600469
$$711$$ 12.0000 0.450035
$$712$$ −12.0000 −0.449719
$$713$$ 0 0
$$714$$ 0 0
$$715$$ 8.00000 0.299183
$$716$$ 0 0
$$717$$ 16.0000 0.597531
$$718$$ 20.0000 0.746393
$$719$$ 46.0000 1.71551 0.857755 0.514058i $$-0.171858\pi$$
0.857755 + 0.514058i $$0.171858\pi$$
$$720$$ 2.00000 0.0745356
$$721$$ 0 0
$$722$$ −15.0000 −0.558242
$$723$$ −22.0000 −0.818189
$$724$$ 18.0000 0.668965
$$725$$ 6.00000 0.222834
$$726$$ −1.00000 −0.0371135
$$727$$ −30.0000 −1.11264 −0.556319 0.830969i $$-0.687787\pi$$
−0.556319 + 0.830969i $$0.687787\pi$$
$$728$$ 0 0
$$729$$ 1.00000 0.0370370
$$730$$ 12.0000 0.444140
$$731$$ −72.0000 −2.66302
$$732$$ −4.00000 −0.147844
$$733$$ 12.0000 0.443230 0.221615 0.975134i $$-0.428867\pi$$
0.221615 + 0.975134i $$0.428867\pi$$
$$734$$ −22.0000 −0.812035
$$735$$ 0 0
$$736$$ 0 0
$$737$$ 0 0
$$738$$ −2.00000 −0.0736210
$$739$$ −4.00000 −0.147142 −0.0735712 0.997290i $$-0.523440\pi$$
−0.0735712 + 0.997290i $$0.523440\pi$$
$$740$$ 4.00000 0.147043
$$741$$ −8.00000 −0.293887
$$742$$ 0 0
$$743$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$744$$ −2.00000 −0.0733236
$$745$$ 20.0000 0.732743
$$746$$ 6.00000 0.219676
$$747$$ 6.00000 0.219529
$$748$$ 6.00000 0.219382
$$749$$ 0 0
$$750$$ 12.0000 0.438178
$$751$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$752$$ −6.00000 −0.218797
$$753$$ 0 0
$$754$$ −24.0000 −0.874028
$$755$$ −32.0000 −1.16460
$$756$$ 0 0
$$757$$ −30.0000 −1.09037 −0.545184 0.838316i $$-0.683540\pi$$
−0.545184 + 0.838316i $$0.683540\pi$$
$$758$$ −20.0000 −0.726433
$$759$$ 0 0
$$760$$ 4.00000 0.145095
$$761$$ 34.0000 1.23250 0.616250 0.787551i $$-0.288651\pi$$
0.616250 + 0.787551i $$0.288651\pi$$
$$762$$ −4.00000 −0.144905
$$763$$ 0 0
$$764$$ 0 0
$$765$$ 12.0000 0.433861
$$766$$ 6.00000 0.216789
$$767$$ 32.0000 1.15545
$$768$$ −1.00000 −0.0360844
$$769$$ −14.0000 −0.504853 −0.252426 0.967616i $$-0.581229\pi$$
−0.252426 + 0.967616i $$0.581229\pi$$
$$770$$ 0 0
$$771$$ −20.0000 −0.720282
$$772$$ −14.0000 −0.503871
$$773$$ −26.0000 −0.935155 −0.467578 0.883952i $$-0.654873\pi$$
−0.467578 + 0.883952i $$0.654873\pi$$
$$774$$ −12.0000 −0.431331
$$775$$ −2.00000 −0.0718421
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 22.0000 0.788738
$$779$$ −4.00000 −0.143315
$$780$$ −8.00000 −0.286446
$$781$$ −8.00000 −0.286263
$$782$$ 0 0
$$783$$ 6.00000 0.214423
$$784$$ 0 0
$$785$$ 28.0000 0.999363
$$786$$ −2.00000 −0.0713376
$$787$$ −38.0000 −1.35455 −0.677277 0.735728i $$-0.736840\pi$$
−0.677277 + 0.735728i $$0.736840\pi$$
$$788$$ −18.0000 −0.641223
$$789$$ −4.00000 −0.142404
$$790$$ 24.0000 0.853882
$$791$$ 0 0
$$792$$ 1.00000 0.0355335
$$793$$ 16.0000 0.568177
$$794$$ −34.0000 −1.20661
$$795$$ −20.0000 −0.709327
$$796$$ 14.0000 0.496217
$$797$$ 6.00000 0.212531 0.106265 0.994338i $$-0.466111\pi$$
0.106265 + 0.994338i $$0.466111\pi$$
$$798$$ 0 0
$$799$$ −36.0000 −1.27359
$$800$$ −1.00000 −0.0353553
$$801$$ −12.0000 −0.423999
$$802$$ −30.0000 −1.05934
$$803$$ 6.00000 0.211735
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 8.00000 0.281788
$$807$$ 6.00000 0.211210
$$808$$ 12.0000 0.422159
$$809$$ 14.0000 0.492214 0.246107 0.969243i $$-0.420849\pi$$
0.246107 + 0.969243i $$0.420849\pi$$
$$810$$ 2.00000 0.0702728
$$811$$ −34.0000 −1.19390 −0.596951 0.802278i $$-0.703621\pi$$
−0.596951 + 0.802278i $$0.703621\pi$$
$$812$$ 0 0
$$813$$ 28.0000 0.982003
$$814$$ 2.00000 0.0701000
$$815$$ −24.0000 −0.840683
$$816$$ −6.00000 −0.210042
$$817$$ −24.0000 −0.839654
$$818$$ 2.00000 0.0699284
$$819$$ 0 0
$$820$$ −4.00000 −0.139686
$$821$$ 42.0000 1.46581 0.732905 0.680331i $$-0.238164\pi$$
0.732905 + 0.680331i $$0.238164\pi$$
$$822$$ −10.0000 −0.348790
$$823$$ 40.0000 1.39431 0.697156 0.716919i $$-0.254448\pi$$
0.697156 + 0.716919i $$0.254448\pi$$
$$824$$ 6.00000 0.209020
$$825$$ 1.00000 0.0348155
$$826$$ 0 0
$$827$$ −36.0000 −1.25184 −0.625921 0.779886i $$-0.715277\pi$$
−0.625921 + 0.779886i $$0.715277\pi$$
$$828$$ 0 0
$$829$$ 10.0000 0.347314 0.173657 0.984806i $$-0.444442\pi$$
0.173657 + 0.984806i $$0.444442\pi$$
$$830$$ 12.0000 0.416526
$$831$$ −10.0000 −0.346896
$$832$$ 4.00000 0.138675
$$833$$ 0 0
$$834$$ 10.0000 0.346272
$$835$$ 24.0000 0.830554
$$836$$ 2.00000 0.0691714
$$837$$ −2.00000 −0.0691301
$$838$$ 12.0000 0.414533
$$839$$ −42.0000 −1.45000 −0.725001 0.688748i $$-0.758161\pi$$
−0.725001 + 0.688748i $$0.758161\pi$$
$$840$$ 0 0
$$841$$ 7.00000 0.241379
$$842$$ 14.0000 0.482472
$$843$$ −2.00000 −0.0688837
$$844$$ 20.0000 0.688428
$$845$$ 6.00000 0.206406
$$846$$ −6.00000 −0.206284
$$847$$ 0 0
$$848$$ 10.0000 0.343401
$$849$$ −14.0000 −0.480479
$$850$$ −6.00000 −0.205798
$$851$$ 0 0
$$852$$ 8.00000 0.274075
$$853$$ 36.0000 1.23262 0.616308 0.787505i $$-0.288628\pi$$
0.616308 + 0.787505i $$0.288628\pi$$
$$854$$ 0 0
$$855$$ 4.00000 0.136797
$$856$$ −12.0000 −0.410152
$$857$$ 42.0000 1.43469 0.717346 0.696717i $$-0.245357\pi$$
0.717346 + 0.696717i $$0.245357\pi$$
$$858$$ −4.00000 −0.136558
$$859$$ −8.00000 −0.272956 −0.136478 0.990643i $$-0.543578\pi$$
−0.136478 + 0.990643i $$0.543578\pi$$
$$860$$ −24.0000 −0.818393
$$861$$ 0 0
$$862$$ 12.0000 0.408722
$$863$$ 56.0000 1.90626 0.953131 0.302558i $$-0.0978405\pi$$
0.953131 + 0.302558i $$0.0978405\pi$$
$$864$$ −1.00000 −0.0340207
$$865$$ −48.0000 −1.63205
$$866$$ 8.00000 0.271851
$$867$$ −19.0000 −0.645274
$$868$$ 0 0
$$869$$ 12.0000 0.407072
$$870$$ 12.0000 0.406838
$$871$$ 0 0
$$872$$ −10.0000 −0.338643
$$873$$ 0 0
$$874$$ 0 0
$$875$$ 0 0
$$876$$ −6.00000 −0.202721
$$877$$ −2.00000 −0.0675352 −0.0337676 0.999430i $$-0.510751\pi$$
−0.0337676 + 0.999430i $$0.510751\pi$$
$$878$$ 28.0000 0.944954
$$879$$ −8.00000 −0.269833
$$880$$ 2.00000 0.0674200
$$881$$ 24.0000 0.808581 0.404290 0.914631i $$-0.367519\pi$$
0.404290 + 0.914631i $$0.367519\pi$$
$$882$$ 0 0
$$883$$ −52.0000 −1.74994 −0.874970 0.484178i $$-0.839119\pi$$
−0.874970 + 0.484178i $$0.839119\pi$$
$$884$$ 24.0000 0.807207
$$885$$ −16.0000 −0.537834
$$886$$ 24.0000 0.806296
$$887$$ 40.0000 1.34307 0.671534 0.740973i $$-0.265636\pi$$
0.671534 + 0.740973i $$0.265636\pi$$
$$888$$ −2.00000 −0.0671156
$$889$$ 0 0
$$890$$ −24.0000 −0.804482
$$891$$ 1.00000 0.0335013
$$892$$ −14.0000 −0.468755
$$893$$ −12.0000 −0.401565
$$894$$ −10.0000 −0.334450
$$895$$ 0 0
$$896$$ 0 0
$$897$$ 0 0
$$898$$ −18.0000 −0.600668
$$899$$ −12.0000 −0.400222
$$900$$ −1.00000 −0.0333333
$$901$$ 60.0000 1.99889
$$902$$ −2.00000 −0.0665927
$$903$$ 0 0
$$904$$ 14.0000 0.465633
$$905$$ 36.0000 1.19668
$$906$$ 16.0000 0.531564
$$907$$ −8.00000 −0.265636 −0.132818 0.991140i $$-0.542403\pi$$
−0.132818 + 0.991140i $$0.542403\pi$$
$$908$$ −6.00000 −0.199117
$$909$$ 12.0000 0.398015
$$910$$ 0 0
$$911$$ 40.0000 1.32526 0.662630 0.748947i $$-0.269440\pi$$
0.662630 + 0.748947i $$0.269440\pi$$
$$912$$ −2.00000 −0.0662266
$$913$$ 6.00000 0.198571
$$914$$ −6.00000 −0.198462
$$915$$ −8.00000 −0.264472
$$916$$ −6.00000 −0.198246
$$917$$ 0 0
$$918$$ −6.00000 −0.198030
$$919$$ 40.0000 1.31948 0.659739 0.751495i $$-0.270667\pi$$
0.659739 + 0.751495i $$0.270667\pi$$
$$920$$ 0 0
$$921$$ 26.0000 0.856729
$$922$$ 12.0000 0.395199
$$923$$ −32.0000 −1.05329
$$924$$ 0 0
$$925$$ −2.00000 −0.0657596
$$926$$ 16.0000 0.525793
$$927$$ 6.00000 0.197066
$$928$$ −6.00000 −0.196960
$$929$$ −44.0000 −1.44359 −0.721797 0.692105i $$-0.756683\pi$$
−0.721797 + 0.692105i $$0.756683\pi$$
$$930$$ −4.00000 −0.131165
$$931$$ 0 0
$$932$$ 26.0000 0.851658
$$933$$ 34.0000 1.11311
$$934$$ 28.0000 0.916188
$$935$$ 12.0000 0.392442
$$936$$ 4.00000 0.130744
$$937$$ 38.0000 1.24141 0.620703 0.784046i $$-0.286847\pi$$
0.620703 + 0.784046i $$0.286847\pi$$
$$938$$ 0 0
$$939$$ 20.0000 0.652675
$$940$$ −12.0000 −0.391397
$$941$$ 48.0000 1.56476 0.782378 0.622804i $$-0.214007\pi$$
0.782378 + 0.622804i $$0.214007\pi$$
$$942$$ −14.0000 −0.456145
$$943$$ 0 0
$$944$$ 8.00000 0.260378
$$945$$ 0 0
$$946$$ −12.0000 −0.390154
$$947$$ 12.0000 0.389948 0.194974 0.980808i $$-0.437538\pi$$
0.194974 + 0.980808i $$0.437538\pi$$
$$948$$ −12.0000 −0.389742
$$949$$ 24.0000 0.779073
$$950$$ −2.00000 −0.0648886
$$951$$ −18.0000 −0.583690
$$952$$ 0 0
$$953$$ −2.00000 −0.0647864 −0.0323932 0.999475i $$-0.510313\pi$$
−0.0323932 + 0.999475i $$0.510313\pi$$
$$954$$ 10.0000 0.323762
$$955$$ 0 0
$$956$$ −16.0000 −0.517477
$$957$$ 6.00000 0.193952
$$958$$ 0 0
$$959$$ 0 0
$$960$$ −2.00000 −0.0645497
$$961$$ −27.0000 −0.870968
$$962$$ 8.00000 0.257930
$$963$$ −12.0000 −0.386695
$$964$$ 22.0000 0.708572
$$965$$ −28.0000 −0.901352
$$966$$ 0 0
$$967$$ 8.00000 0.257263 0.128631 0.991692i $$-0.458942\pi$$
0.128631 + 0.991692i $$0.458942\pi$$
$$968$$ 1.00000 0.0321412
$$969$$ −12.0000 −0.385496
$$970$$ 0 0
$$971$$ 60.0000 1.92549 0.962746 0.270408i $$-0.0871586\pi$$
0.962746 + 0.270408i $$0.0871586\pi$$
$$972$$ −1.00000 −0.0320750
$$973$$ 0 0
$$974$$ −16.0000 −0.512673
$$975$$ 4.00000 0.128103
$$976$$ 4.00000 0.128037
$$977$$ −2.00000 −0.0639857 −0.0319928 0.999488i $$-0.510185\pi$$
−0.0319928 + 0.999488i $$0.510185\pi$$
$$978$$ 12.0000 0.383718
$$979$$ −12.0000 −0.383522
$$980$$ 0 0
$$981$$ −10.0000 −0.319275
$$982$$ −20.0000 −0.638226
$$983$$ −38.0000 −1.21201 −0.606006 0.795460i $$-0.707229\pi$$
−0.606006 + 0.795460i $$0.707229\pi$$
$$984$$ 2.00000 0.0637577
$$985$$ −36.0000 −1.14706
$$986$$ −36.0000 −1.14647
$$987$$ 0 0
$$988$$ 8.00000 0.254514
$$989$$ 0 0
$$990$$ 2.00000 0.0635642
$$991$$ −40.0000 −1.27064 −0.635321 0.772248i $$-0.719132\pi$$
−0.635321 + 0.772248i $$0.719132\pi$$
$$992$$ 2.00000 0.0635001
$$993$$ 24.0000 0.761617
$$994$$ 0 0
$$995$$ 28.0000 0.887660
$$996$$ −6.00000 −0.190117
$$997$$ −8.00000 −0.253363 −0.126681 0.991943i $$-0.540433\pi$$
−0.126681 + 0.991943i $$0.540433\pi$$
$$998$$ −40.0000 −1.26618
$$999$$ −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.r.1.1 1
3.2 odd 2 9702.2.a.g.1.1 1
7.6 odd 2 3234.2.a.u.1.1 yes 1
21.20 even 2 9702.2.a.q.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
3234.2.a.r.1.1 1 1.1 even 1 trivial
3234.2.a.u.1.1 yes 1 7.6 odd 2
9702.2.a.g.1.1 1 3.2 odd 2
9702.2.a.q.1.1 1 21.20 even 2