# Properties

 Label 3648.2.a.b.1.1 Level $3648$ Weight $2$ Character 3648.1 Self dual yes Analytic conductor $29.129$ Analytic rank $1$ 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] = [3648,2,Mod(1,3648)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3648, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("3648.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$3648 = 2^{6} \cdot 3 \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3648.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: $$29.1294266574$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 1824) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 3648.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-1.00000 q^{3} -3.00000 q^{5} +3.00000 q^{7} +1.00000 q^{9} +O(q^{10})$$ $$q-1.00000 q^{3} -3.00000 q^{5} +3.00000 q^{7} +1.00000 q^{9} -3.00000 q^{11} +3.00000 q^{15} +1.00000 q^{17} -1.00000 q^{19} -3.00000 q^{21} +4.00000 q^{25} -1.00000 q^{27} +8.00000 q^{29} -2.00000 q^{31} +3.00000 q^{33} -9.00000 q^{35} +4.00000 q^{37} -12.0000 q^{41} +1.00000 q^{43} -3.00000 q^{45} +9.00000 q^{47} +2.00000 q^{49} -1.00000 q^{51} -6.00000 q^{53} +9.00000 q^{55} +1.00000 q^{57} +6.00000 q^{59} +1.00000 q^{61} +3.00000 q^{63} +10.0000 q^{67} -10.0000 q^{71} -11.0000 q^{73} -4.00000 q^{75} -9.00000 q^{77} +6.00000 q^{79} +1.00000 q^{81} -3.00000 q^{85} -8.00000 q^{87} -6.00000 q^{89} +2.00000 q^{93} +3.00000 q^{95} -8.00000 q^{97} -3.00000 q^{99} +O(q^{100})$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 0 0
$$3$$ −1.00000 −0.577350
$$4$$ 0 0
$$5$$ −3.00000 −1.34164 −0.670820 0.741620i $$-0.734058\pi$$
−0.670820 + 0.741620i $$0.734058\pi$$
$$6$$ 0 0
$$7$$ 3.00000 1.13389 0.566947 0.823754i $$-0.308125\pi$$
0.566947 + 0.823754i $$0.308125\pi$$
$$8$$ 0 0
$$9$$ 1.00000 0.333333
$$10$$ 0 0
$$11$$ −3.00000 −0.904534 −0.452267 0.891883i $$-0.649385\pi$$
−0.452267 + 0.891883i $$0.649385\pi$$
$$12$$ 0 0
$$13$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$14$$ 0 0
$$15$$ 3.00000 0.774597
$$16$$ 0 0
$$17$$ 1.00000 0.242536 0.121268 0.992620i $$-0.461304\pi$$
0.121268 + 0.992620i $$0.461304\pi$$
$$18$$ 0 0
$$19$$ −1.00000 −0.229416
$$20$$ 0 0
$$21$$ −3.00000 −0.654654
$$22$$ 0 0
$$23$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$24$$ 0 0
$$25$$ 4.00000 0.800000
$$26$$ 0 0
$$27$$ −1.00000 −0.192450
$$28$$ 0 0
$$29$$ 8.00000 1.48556 0.742781 0.669534i $$-0.233506\pi$$
0.742781 + 0.669534i $$0.233506\pi$$
$$30$$ 0 0
$$31$$ −2.00000 −0.359211 −0.179605 0.983739i $$-0.557482\pi$$
−0.179605 + 0.983739i $$0.557482\pi$$
$$32$$ 0 0
$$33$$ 3.00000 0.522233
$$34$$ 0 0
$$35$$ −9.00000 −1.52128
$$36$$ 0 0
$$37$$ 4.00000 0.657596 0.328798 0.944400i $$-0.393356\pi$$
0.328798 + 0.944400i $$0.393356\pi$$
$$38$$ 0 0
$$39$$ 0 0
$$40$$ 0 0
$$41$$ −12.0000 −1.87409 −0.937043 0.349215i $$-0.886448\pi$$
−0.937043 + 0.349215i $$0.886448\pi$$
$$42$$ 0 0
$$43$$ 1.00000 0.152499 0.0762493 0.997089i $$-0.475706\pi$$
0.0762493 + 0.997089i $$0.475706\pi$$
$$44$$ 0 0
$$45$$ −3.00000 −0.447214
$$46$$ 0 0
$$47$$ 9.00000 1.31278 0.656392 0.754420i $$-0.272082\pi$$
0.656392 + 0.754420i $$0.272082\pi$$
$$48$$ 0 0
$$49$$ 2.00000 0.285714
$$50$$ 0 0
$$51$$ −1.00000 −0.140028
$$52$$ 0 0
$$53$$ −6.00000 −0.824163 −0.412082 0.911147i $$-0.635198\pi$$
−0.412082 + 0.911147i $$0.635198\pi$$
$$54$$ 0 0
$$55$$ 9.00000 1.21356
$$56$$ 0 0
$$57$$ 1.00000 0.132453
$$58$$ 0 0
$$59$$ 6.00000 0.781133 0.390567 0.920575i $$-0.372279\pi$$
0.390567 + 0.920575i $$0.372279\pi$$
$$60$$ 0 0
$$61$$ 1.00000 0.128037 0.0640184 0.997949i $$-0.479608\pi$$
0.0640184 + 0.997949i $$0.479608\pi$$
$$62$$ 0 0
$$63$$ 3.00000 0.377964
$$64$$ 0 0
$$65$$ 0 0
$$66$$ 0 0
$$67$$ 10.0000 1.22169 0.610847 0.791748i $$-0.290829\pi$$
0.610847 + 0.791748i $$0.290829\pi$$
$$68$$ 0 0
$$69$$ 0 0
$$70$$ 0 0
$$71$$ −10.0000 −1.18678 −0.593391 0.804914i $$-0.702211\pi$$
−0.593391 + 0.804914i $$0.702211\pi$$
$$72$$ 0 0
$$73$$ −11.0000 −1.28745 −0.643726 0.765256i $$-0.722612\pi$$
−0.643726 + 0.765256i $$0.722612\pi$$
$$74$$ 0 0
$$75$$ −4.00000 −0.461880
$$76$$ 0 0
$$77$$ −9.00000 −1.02565
$$78$$ 0 0
$$79$$ 6.00000 0.675053 0.337526 0.941316i $$-0.390410\pi$$
0.337526 + 0.941316i $$0.390410\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ 0 0
$$83$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$84$$ 0 0
$$85$$ −3.00000 −0.325396
$$86$$ 0 0
$$87$$ −8.00000 −0.857690
$$88$$ 0 0
$$89$$ −6.00000 −0.635999 −0.317999 0.948091i $$-0.603011\pi$$
−0.317999 + 0.948091i $$0.603011\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ 0 0
$$93$$ 2.00000 0.207390
$$94$$ 0 0
$$95$$ 3.00000 0.307794
$$96$$ 0 0
$$97$$ −8.00000 −0.812277 −0.406138 0.913812i $$-0.633125\pi$$
−0.406138 + 0.913812i $$0.633125\pi$$
$$98$$ 0 0
$$99$$ −3.00000 −0.301511
$$100$$ 0 0
$$101$$ −14.0000 −1.39305 −0.696526 0.717532i $$-0.745272\pi$$
−0.696526 + 0.717532i $$0.745272\pi$$
$$102$$ 0 0
$$103$$ 10.0000 0.985329 0.492665 0.870219i $$-0.336023\pi$$
0.492665 + 0.870219i $$0.336023\pi$$
$$104$$ 0 0
$$105$$ 9.00000 0.878310
$$106$$ 0 0
$$107$$ −18.0000 −1.74013 −0.870063 0.492941i $$-0.835922\pi$$
−0.870063 + 0.492941i $$0.835922\pi$$
$$108$$ 0 0
$$109$$ 10.0000 0.957826 0.478913 0.877862i $$-0.341031\pi$$
0.478913 + 0.877862i $$0.341031\pi$$
$$110$$ 0 0
$$111$$ −4.00000 −0.379663
$$112$$ 0 0
$$113$$ −4.00000 −0.376288 −0.188144 0.982141i $$-0.560247\pi$$
−0.188144 + 0.982141i $$0.560247\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 0 0
$$117$$ 0 0
$$118$$ 0 0
$$119$$ 3.00000 0.275010
$$120$$ 0 0
$$121$$ −2.00000 −0.181818
$$122$$ 0 0
$$123$$ 12.0000 1.08200
$$124$$ 0 0
$$125$$ 3.00000 0.268328
$$126$$ 0 0
$$127$$ 8.00000 0.709885 0.354943 0.934888i $$-0.384500\pi$$
0.354943 + 0.934888i $$0.384500\pi$$
$$128$$ 0 0
$$129$$ −1.00000 −0.0880451
$$130$$ 0 0
$$131$$ −13.0000 −1.13582 −0.567908 0.823092i $$-0.692247\pi$$
−0.567908 + 0.823092i $$0.692247\pi$$
$$132$$ 0 0
$$133$$ −3.00000 −0.260133
$$134$$ 0 0
$$135$$ 3.00000 0.258199
$$136$$ 0 0
$$137$$ 1.00000 0.0854358 0.0427179 0.999087i $$-0.486398\pi$$
0.0427179 + 0.999087i $$0.486398\pi$$
$$138$$ 0 0
$$139$$ 9.00000 0.763370 0.381685 0.924292i $$-0.375344\pi$$
0.381685 + 0.924292i $$0.375344\pi$$
$$140$$ 0 0
$$141$$ −9.00000 −0.757937
$$142$$ 0 0
$$143$$ 0 0
$$144$$ 0 0
$$145$$ −24.0000 −1.99309
$$146$$ 0 0
$$147$$ −2.00000 −0.164957
$$148$$ 0 0
$$149$$ −5.00000 −0.409616 −0.204808 0.978802i $$-0.565657\pi$$
−0.204808 + 0.978802i $$0.565657\pi$$
$$150$$ 0 0
$$151$$ 18.0000 1.46482 0.732410 0.680864i $$-0.238396\pi$$
0.732410 + 0.680864i $$0.238396\pi$$
$$152$$ 0 0
$$153$$ 1.00000 0.0808452
$$154$$ 0 0
$$155$$ 6.00000 0.481932
$$156$$ 0 0
$$157$$ −2.00000 −0.159617 −0.0798087 0.996810i $$-0.525431\pi$$
−0.0798087 + 0.996810i $$0.525431\pi$$
$$158$$ 0 0
$$159$$ 6.00000 0.475831
$$160$$ 0 0
$$161$$ 0 0
$$162$$ 0 0
$$163$$ 4.00000 0.313304 0.156652 0.987654i $$-0.449930\pi$$
0.156652 + 0.987654i $$0.449930\pi$$
$$164$$ 0 0
$$165$$ −9.00000 −0.700649
$$166$$ 0 0
$$167$$ −4.00000 −0.309529 −0.154765 0.987951i $$-0.549462\pi$$
−0.154765 + 0.987951i $$0.549462\pi$$
$$168$$ 0 0
$$169$$ −13.0000 −1.00000
$$170$$ 0 0
$$171$$ −1.00000 −0.0764719
$$172$$ 0 0
$$173$$ −18.0000 −1.36851 −0.684257 0.729241i $$-0.739873\pi$$
−0.684257 + 0.729241i $$0.739873\pi$$
$$174$$ 0 0
$$175$$ 12.0000 0.907115
$$176$$ 0 0
$$177$$ −6.00000 −0.450988
$$178$$ 0 0
$$179$$ −24.0000 −1.79384 −0.896922 0.442189i $$-0.854202\pi$$
−0.896922 + 0.442189i $$0.854202\pi$$
$$180$$ 0 0
$$181$$ −22.0000 −1.63525 −0.817624 0.575753i $$-0.804709\pi$$
−0.817624 + 0.575753i $$0.804709\pi$$
$$182$$ 0 0
$$183$$ −1.00000 −0.0739221
$$184$$ 0 0
$$185$$ −12.0000 −0.882258
$$186$$ 0 0
$$187$$ −3.00000 −0.219382
$$188$$ 0 0
$$189$$ −3.00000 −0.218218
$$190$$ 0 0
$$191$$ −13.0000 −0.940647 −0.470323 0.882494i $$-0.655863\pi$$
−0.470323 + 0.882494i $$0.655863\pi$$
$$192$$ 0 0
$$193$$ −14.0000 −1.00774 −0.503871 0.863779i $$-0.668091\pi$$
−0.503871 + 0.863779i $$0.668091\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ −18.0000 −1.28245 −0.641223 0.767354i $$-0.721573\pi$$
−0.641223 + 0.767354i $$0.721573\pi$$
$$198$$ 0 0
$$199$$ −5.00000 −0.354441 −0.177220 0.984171i $$-0.556711\pi$$
−0.177220 + 0.984171i $$0.556711\pi$$
$$200$$ 0 0
$$201$$ −10.0000 −0.705346
$$202$$ 0 0
$$203$$ 24.0000 1.68447
$$204$$ 0 0
$$205$$ 36.0000 2.51435
$$206$$ 0 0
$$207$$ 0 0
$$208$$ 0 0
$$209$$ 3.00000 0.207514
$$210$$ 0 0
$$211$$ −22.0000 −1.51454 −0.757271 0.653101i $$-0.773468\pi$$
−0.757271 + 0.653101i $$0.773468\pi$$
$$212$$ 0 0
$$213$$ 10.0000 0.685189
$$214$$ 0 0
$$215$$ −3.00000 −0.204598
$$216$$ 0 0
$$217$$ −6.00000 −0.407307
$$218$$ 0 0
$$219$$ 11.0000 0.743311
$$220$$ 0 0
$$221$$ 0 0
$$222$$ 0 0
$$223$$ −16.0000 −1.07144 −0.535720 0.844396i $$-0.679960\pi$$
−0.535720 + 0.844396i $$0.679960\pi$$
$$224$$ 0 0
$$225$$ 4.00000 0.266667
$$226$$ 0 0
$$227$$ 4.00000 0.265489 0.132745 0.991150i $$-0.457621\pi$$
0.132745 + 0.991150i $$0.457621\pi$$
$$228$$ 0 0
$$229$$ −21.0000 −1.38772 −0.693860 0.720110i $$-0.744091\pi$$
−0.693860 + 0.720110i $$0.744091\pi$$
$$230$$ 0 0
$$231$$ 9.00000 0.592157
$$232$$ 0 0
$$233$$ 15.0000 0.982683 0.491341 0.870967i $$-0.336507\pi$$
0.491341 + 0.870967i $$0.336507\pi$$
$$234$$ 0 0
$$235$$ −27.0000 −1.76129
$$236$$ 0 0
$$237$$ −6.00000 −0.389742
$$238$$ 0 0
$$239$$ −13.0000 −0.840900 −0.420450 0.907316i $$-0.638128\pi$$
−0.420450 + 0.907316i $$0.638128\pi$$
$$240$$ 0 0
$$241$$ −24.0000 −1.54598 −0.772988 0.634421i $$-0.781239\pi$$
−0.772988 + 0.634421i $$0.781239\pi$$
$$242$$ 0 0
$$243$$ −1.00000 −0.0641500
$$244$$ 0 0
$$245$$ −6.00000 −0.383326
$$246$$ 0 0
$$247$$ 0 0
$$248$$ 0 0
$$249$$ 0 0
$$250$$ 0 0
$$251$$ −13.0000 −0.820553 −0.410276 0.911961i $$-0.634568\pi$$
−0.410276 + 0.911961i $$0.634568\pi$$
$$252$$ 0 0
$$253$$ 0 0
$$254$$ 0 0
$$255$$ 3.00000 0.187867
$$256$$ 0 0
$$257$$ 28.0000 1.74659 0.873296 0.487190i $$-0.161978\pi$$
0.873296 + 0.487190i $$0.161978\pi$$
$$258$$ 0 0
$$259$$ 12.0000 0.745644
$$260$$ 0 0
$$261$$ 8.00000 0.495188
$$262$$ 0 0
$$263$$ −3.00000 −0.184988 −0.0924940 0.995713i $$-0.529484\pi$$
−0.0924940 + 0.995713i $$0.529484\pi$$
$$264$$ 0 0
$$265$$ 18.0000 1.10573
$$266$$ 0 0
$$267$$ 6.00000 0.367194
$$268$$ 0 0
$$269$$ 22.0000 1.34136 0.670682 0.741745i $$-0.266002\pi$$
0.670682 + 0.741745i $$0.266002\pi$$
$$270$$ 0 0
$$271$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 0 0
$$275$$ −12.0000 −0.723627
$$276$$ 0 0
$$277$$ 23.0000 1.38194 0.690968 0.722885i $$-0.257185\pi$$
0.690968 + 0.722885i $$0.257185\pi$$
$$278$$ 0 0
$$279$$ −2.00000 −0.119737
$$280$$ 0 0
$$281$$ 6.00000 0.357930 0.178965 0.983855i $$-0.442725\pi$$
0.178965 + 0.983855i $$0.442725\pi$$
$$282$$ 0 0
$$283$$ 13.0000 0.772770 0.386385 0.922338i $$-0.373724\pi$$
0.386385 + 0.922338i $$0.373724\pi$$
$$284$$ 0 0
$$285$$ −3.00000 −0.177705
$$286$$ 0 0
$$287$$ −36.0000 −2.12501
$$288$$ 0 0
$$289$$ −16.0000 −0.941176
$$290$$ 0 0
$$291$$ 8.00000 0.468968
$$292$$ 0 0
$$293$$ −16.0000 −0.934730 −0.467365 0.884064i $$-0.654797\pi$$
−0.467365 + 0.884064i $$0.654797\pi$$
$$294$$ 0 0
$$295$$ −18.0000 −1.04800
$$296$$ 0 0
$$297$$ 3.00000 0.174078
$$298$$ 0 0
$$299$$ 0 0
$$300$$ 0 0
$$301$$ 3.00000 0.172917
$$302$$ 0 0
$$303$$ 14.0000 0.804279
$$304$$ 0 0
$$305$$ −3.00000 −0.171780
$$306$$ 0 0
$$307$$ 6.00000 0.342438 0.171219 0.985233i $$-0.445229\pi$$
0.171219 + 0.985233i $$0.445229\pi$$
$$308$$ 0 0
$$309$$ −10.0000 −0.568880
$$310$$ 0 0
$$311$$ −7.00000 −0.396934 −0.198467 0.980108i $$-0.563596\pi$$
−0.198467 + 0.980108i $$0.563596\pi$$
$$312$$ 0 0
$$313$$ −26.0000 −1.46961 −0.734803 0.678280i $$-0.762726\pi$$
−0.734803 + 0.678280i $$0.762726\pi$$
$$314$$ 0 0
$$315$$ −9.00000 −0.507093
$$316$$ 0 0
$$317$$ −18.0000 −1.01098 −0.505490 0.862832i $$-0.668688\pi$$
−0.505490 + 0.862832i $$0.668688\pi$$
$$318$$ 0 0
$$319$$ −24.0000 −1.34374
$$320$$ 0 0
$$321$$ 18.0000 1.00466
$$322$$ 0 0
$$323$$ −1.00000 −0.0556415
$$324$$ 0 0
$$325$$ 0 0
$$326$$ 0 0
$$327$$ −10.0000 −0.553001
$$328$$ 0 0
$$329$$ 27.0000 1.48856
$$330$$ 0 0
$$331$$ 18.0000 0.989369 0.494685 0.869072i $$-0.335284\pi$$
0.494685 + 0.869072i $$0.335284\pi$$
$$332$$ 0 0
$$333$$ 4.00000 0.219199
$$334$$ 0 0
$$335$$ −30.0000 −1.63908
$$336$$ 0 0
$$337$$ −12.0000 −0.653682 −0.326841 0.945079i $$-0.605984\pi$$
−0.326841 + 0.945079i $$0.605984\pi$$
$$338$$ 0 0
$$339$$ 4.00000 0.217250
$$340$$ 0 0
$$341$$ 6.00000 0.324918
$$342$$ 0 0
$$343$$ −15.0000 −0.809924
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ 23.0000 1.23470 0.617352 0.786687i $$-0.288205\pi$$
0.617352 + 0.786687i $$0.288205\pi$$
$$348$$ 0 0
$$349$$ −1.00000 −0.0535288 −0.0267644 0.999642i $$-0.508520\pi$$
−0.0267644 + 0.999642i $$0.508520\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 0 0
$$353$$ 18.0000 0.958043 0.479022 0.877803i $$-0.340992\pi$$
0.479022 + 0.877803i $$0.340992\pi$$
$$354$$ 0 0
$$355$$ 30.0000 1.59223
$$356$$ 0 0
$$357$$ −3.00000 −0.158777
$$358$$ 0 0
$$359$$ 19.0000 1.00278 0.501391 0.865221i $$-0.332822\pi$$
0.501391 + 0.865221i $$0.332822\pi$$
$$360$$ 0 0
$$361$$ 1.00000 0.0526316
$$362$$ 0 0
$$363$$ 2.00000 0.104973
$$364$$ 0 0
$$365$$ 33.0000 1.72730
$$366$$ 0 0
$$367$$ −4.00000 −0.208798 −0.104399 0.994535i $$-0.533292\pi$$
−0.104399 + 0.994535i $$0.533292\pi$$
$$368$$ 0 0
$$369$$ −12.0000 −0.624695
$$370$$ 0 0
$$371$$ −18.0000 −0.934513
$$372$$ 0 0
$$373$$ −16.0000 −0.828449 −0.414224 0.910175i $$-0.635947\pi$$
−0.414224 + 0.910175i $$0.635947\pi$$
$$374$$ 0 0
$$375$$ −3.00000 −0.154919
$$376$$ 0 0
$$377$$ 0 0
$$378$$ 0 0
$$379$$ −6.00000 −0.308199 −0.154100 0.988055i $$-0.549248\pi$$
−0.154100 + 0.988055i $$0.549248\pi$$
$$380$$ 0 0
$$381$$ −8.00000 −0.409852
$$382$$ 0 0
$$383$$ −14.0000 −0.715367 −0.357683 0.933843i $$-0.616433\pi$$
−0.357683 + 0.933843i $$0.616433\pi$$
$$384$$ 0 0
$$385$$ 27.0000 1.37605
$$386$$ 0 0
$$387$$ 1.00000 0.0508329
$$388$$ 0 0
$$389$$ −23.0000 −1.16615 −0.583073 0.812420i $$-0.698150\pi$$
−0.583073 + 0.812420i $$0.698150\pi$$
$$390$$ 0 0
$$391$$ 0 0
$$392$$ 0 0
$$393$$ 13.0000 0.655763
$$394$$ 0 0
$$395$$ −18.0000 −0.905678
$$396$$ 0 0
$$397$$ −17.0000 −0.853206 −0.426603 0.904439i $$-0.640290\pi$$
−0.426603 + 0.904439i $$0.640290\pi$$
$$398$$ 0 0
$$399$$ 3.00000 0.150188
$$400$$ 0 0
$$401$$ 10.0000 0.499376 0.249688 0.968326i $$-0.419672\pi$$
0.249688 + 0.968326i $$0.419672\pi$$
$$402$$ 0 0
$$403$$ 0 0
$$404$$ 0 0
$$405$$ −3.00000 −0.149071
$$406$$ 0 0
$$407$$ −12.0000 −0.594818
$$408$$ 0 0
$$409$$ 14.0000 0.692255 0.346128 0.938187i $$-0.387496\pi$$
0.346128 + 0.938187i $$0.387496\pi$$
$$410$$ 0 0
$$411$$ −1.00000 −0.0493264
$$412$$ 0 0
$$413$$ 18.0000 0.885722
$$414$$ 0 0
$$415$$ 0 0
$$416$$ 0 0
$$417$$ −9.00000 −0.440732
$$418$$ 0 0
$$419$$ 4.00000 0.195413 0.0977064 0.995215i $$-0.468849\pi$$
0.0977064 + 0.995215i $$0.468849\pi$$
$$420$$ 0 0
$$421$$ 38.0000 1.85201 0.926003 0.377515i $$-0.123221\pi$$
0.926003 + 0.377515i $$0.123221\pi$$
$$422$$ 0 0
$$423$$ 9.00000 0.437595
$$424$$ 0 0
$$425$$ 4.00000 0.194029
$$426$$ 0 0
$$427$$ 3.00000 0.145180
$$428$$ 0 0
$$429$$ 0 0
$$430$$ 0 0
$$431$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$432$$ 0 0
$$433$$ −4.00000 −0.192228 −0.0961139 0.995370i $$-0.530641\pi$$
−0.0961139 + 0.995370i $$0.530641\pi$$
$$434$$ 0 0
$$435$$ 24.0000 1.15071
$$436$$ 0 0
$$437$$ 0 0
$$438$$ 0 0
$$439$$ −20.0000 −0.954548 −0.477274 0.878755i $$-0.658375\pi$$
−0.477274 + 0.878755i $$0.658375\pi$$
$$440$$ 0 0
$$441$$ 2.00000 0.0952381
$$442$$ 0 0
$$443$$ 15.0000 0.712672 0.356336 0.934358i $$-0.384026\pi$$
0.356336 + 0.934358i $$0.384026\pi$$
$$444$$ 0 0
$$445$$ 18.0000 0.853282
$$446$$ 0 0
$$447$$ 5.00000 0.236492
$$448$$ 0 0
$$449$$ −20.0000 −0.943858 −0.471929 0.881636i $$-0.656442\pi$$
−0.471929 + 0.881636i $$0.656442\pi$$
$$450$$ 0 0
$$451$$ 36.0000 1.69517
$$452$$ 0 0
$$453$$ −18.0000 −0.845714
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 39.0000 1.82434 0.912172 0.409809i $$-0.134405\pi$$
0.912172 + 0.409809i $$0.134405\pi$$
$$458$$ 0 0
$$459$$ −1.00000 −0.0466760
$$460$$ 0 0
$$461$$ 3.00000 0.139724 0.0698620 0.997557i $$-0.477744\pi$$
0.0698620 + 0.997557i $$0.477744\pi$$
$$462$$ 0 0
$$463$$ −27.0000 −1.25480 −0.627398 0.778699i $$-0.715880\pi$$
−0.627398 + 0.778699i $$0.715880\pi$$
$$464$$ 0 0
$$465$$ −6.00000 −0.278243
$$466$$ 0 0
$$467$$ 15.0000 0.694117 0.347059 0.937843i $$-0.387180\pi$$
0.347059 + 0.937843i $$0.387180\pi$$
$$468$$ 0 0
$$469$$ 30.0000 1.38527
$$470$$ 0 0
$$471$$ 2.00000 0.0921551
$$472$$ 0 0
$$473$$ −3.00000 −0.137940
$$474$$ 0 0
$$475$$ −4.00000 −0.183533
$$476$$ 0 0
$$477$$ −6.00000 −0.274721
$$478$$ 0 0
$$479$$ 32.0000 1.46212 0.731059 0.682315i $$-0.239027\pi$$
0.731059 + 0.682315i $$0.239027\pi$$
$$480$$ 0 0
$$481$$ 0 0
$$482$$ 0 0
$$483$$ 0 0
$$484$$ 0 0
$$485$$ 24.0000 1.08978
$$486$$ 0 0
$$487$$ 34.0000 1.54069 0.770344 0.637629i $$-0.220085\pi$$
0.770344 + 0.637629i $$0.220085\pi$$
$$488$$ 0 0
$$489$$ −4.00000 −0.180886
$$490$$ 0 0
$$491$$ 20.0000 0.902587 0.451294 0.892375i $$-0.350963\pi$$
0.451294 + 0.892375i $$0.350963\pi$$
$$492$$ 0 0
$$493$$ 8.00000 0.360302
$$494$$ 0 0
$$495$$ 9.00000 0.404520
$$496$$ 0 0
$$497$$ −30.0000 −1.34568
$$498$$ 0 0
$$499$$ 31.0000 1.38775 0.693875 0.720095i $$-0.255902\pi$$
0.693875 + 0.720095i $$0.255902\pi$$
$$500$$ 0 0
$$501$$ 4.00000 0.178707
$$502$$ 0 0
$$503$$ 24.0000 1.07011 0.535054 0.844818i $$-0.320291\pi$$
0.535054 + 0.844818i $$0.320291\pi$$
$$504$$ 0 0
$$505$$ 42.0000 1.86898
$$506$$ 0 0
$$507$$ 13.0000 0.577350
$$508$$ 0 0
$$509$$ 24.0000 1.06378 0.531891 0.846813i $$-0.321482\pi$$
0.531891 + 0.846813i $$0.321482\pi$$
$$510$$ 0 0
$$511$$ −33.0000 −1.45983
$$512$$ 0 0
$$513$$ 1.00000 0.0441511
$$514$$ 0 0
$$515$$ −30.0000 −1.32196
$$516$$ 0 0
$$517$$ −27.0000 −1.18746
$$518$$ 0 0
$$519$$ 18.0000 0.790112
$$520$$ 0 0
$$521$$ 26.0000 1.13908 0.569540 0.821963i $$-0.307121\pi$$
0.569540 + 0.821963i $$0.307121\pi$$
$$522$$ 0 0
$$523$$ −14.0000 −0.612177 −0.306089 0.952003i $$-0.599020\pi$$
−0.306089 + 0.952003i $$0.599020\pi$$
$$524$$ 0 0
$$525$$ −12.0000 −0.523723
$$526$$ 0 0
$$527$$ −2.00000 −0.0871214
$$528$$ 0 0
$$529$$ −23.0000 −1.00000
$$530$$ 0 0
$$531$$ 6.00000 0.260378
$$532$$ 0 0
$$533$$ 0 0
$$534$$ 0 0
$$535$$ 54.0000 2.33462
$$536$$ 0 0
$$537$$ 24.0000 1.03568
$$538$$ 0 0
$$539$$ −6.00000 −0.258438
$$540$$ 0 0
$$541$$ 5.00000 0.214967 0.107483 0.994207i $$-0.465721\pi$$
0.107483 + 0.994207i $$0.465721\pi$$
$$542$$ 0 0
$$543$$ 22.0000 0.944110
$$544$$ 0 0
$$545$$ −30.0000 −1.28506
$$546$$ 0 0
$$547$$ −8.00000 −0.342055 −0.171028 0.985266i $$-0.554709\pi$$
−0.171028 + 0.985266i $$0.554709\pi$$
$$548$$ 0 0
$$549$$ 1.00000 0.0426790
$$550$$ 0 0
$$551$$ −8.00000 −0.340811
$$552$$ 0 0
$$553$$ 18.0000 0.765438
$$554$$ 0 0
$$555$$ 12.0000 0.509372
$$556$$ 0 0
$$557$$ 3.00000 0.127114 0.0635570 0.997978i $$-0.479756\pi$$
0.0635570 + 0.997978i $$0.479756\pi$$
$$558$$ 0 0
$$559$$ 0 0
$$560$$ 0 0
$$561$$ 3.00000 0.126660
$$562$$ 0 0
$$563$$ 24.0000 1.01148 0.505740 0.862686i $$-0.331220\pi$$
0.505740 + 0.862686i $$0.331220\pi$$
$$564$$ 0 0
$$565$$ 12.0000 0.504844
$$566$$ 0 0
$$567$$ 3.00000 0.125988
$$568$$ 0 0
$$569$$ −12.0000 −0.503066 −0.251533 0.967849i $$-0.580935\pi$$
−0.251533 + 0.967849i $$0.580935\pi$$
$$570$$ 0 0
$$571$$ −32.0000 −1.33916 −0.669579 0.742741i $$-0.733526\pi$$
−0.669579 + 0.742741i $$0.733526\pi$$
$$572$$ 0 0
$$573$$ 13.0000 0.543083
$$574$$ 0 0
$$575$$ 0 0
$$576$$ 0 0
$$577$$ −33.0000 −1.37381 −0.686904 0.726748i $$-0.741031\pi$$
−0.686904 + 0.726748i $$0.741031\pi$$
$$578$$ 0 0
$$579$$ 14.0000 0.581820
$$580$$ 0 0
$$581$$ 0 0
$$582$$ 0 0
$$583$$ 18.0000 0.745484
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ −13.0000 −0.536567 −0.268284 0.963340i $$-0.586456\pi$$
−0.268284 + 0.963340i $$0.586456\pi$$
$$588$$ 0 0
$$589$$ 2.00000 0.0824086
$$590$$ 0 0
$$591$$ 18.0000 0.740421
$$592$$ 0 0
$$593$$ −42.0000 −1.72473 −0.862367 0.506284i $$-0.831019\pi$$
−0.862367 + 0.506284i $$0.831019\pi$$
$$594$$ 0 0
$$595$$ −9.00000 −0.368964
$$596$$ 0 0
$$597$$ 5.00000 0.204636
$$598$$ 0 0
$$599$$ 42.0000 1.71607 0.858037 0.513588i $$-0.171684\pi$$
0.858037 + 0.513588i $$0.171684\pi$$
$$600$$ 0 0
$$601$$ −2.00000 −0.0815817 −0.0407909 0.999168i $$-0.512988\pi$$
−0.0407909 + 0.999168i $$0.512988\pi$$
$$602$$ 0 0
$$603$$ 10.0000 0.407231
$$604$$ 0 0
$$605$$ 6.00000 0.243935
$$606$$ 0 0
$$607$$ 4.00000 0.162355 0.0811775 0.996700i $$-0.474132\pi$$
0.0811775 + 0.996700i $$0.474132\pi$$
$$608$$ 0 0
$$609$$ −24.0000 −0.972529
$$610$$ 0 0
$$611$$ 0 0
$$612$$ 0 0
$$613$$ −9.00000 −0.363507 −0.181753 0.983344i $$-0.558177\pi$$
−0.181753 + 0.983344i $$0.558177\pi$$
$$614$$ 0 0
$$615$$ −36.0000 −1.45166
$$616$$ 0 0
$$617$$ 17.0000 0.684394 0.342197 0.939628i $$-0.388829\pi$$
0.342197 + 0.939628i $$0.388829\pi$$
$$618$$ 0 0
$$619$$ −28.0000 −1.12542 −0.562708 0.826656i $$-0.690240\pi$$
−0.562708 + 0.826656i $$0.690240\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 0 0
$$623$$ −18.0000 −0.721155
$$624$$ 0 0
$$625$$ −29.0000 −1.16000
$$626$$ 0 0
$$627$$ −3.00000 −0.119808
$$628$$ 0 0
$$629$$ 4.00000 0.159490
$$630$$ 0 0
$$631$$ −37.0000 −1.47295 −0.736473 0.676467i $$-0.763510\pi$$
−0.736473 + 0.676467i $$0.763510\pi$$
$$632$$ 0 0
$$633$$ 22.0000 0.874421
$$634$$ 0 0
$$635$$ −24.0000 −0.952411
$$636$$ 0 0
$$637$$ 0 0
$$638$$ 0 0
$$639$$ −10.0000 −0.395594
$$640$$ 0 0
$$641$$ 16.0000 0.631962 0.315981 0.948766i $$-0.397666\pi$$
0.315981 + 0.948766i $$0.397666\pi$$
$$642$$ 0 0
$$643$$ 25.0000 0.985904 0.492952 0.870057i $$-0.335918\pi$$
0.492952 + 0.870057i $$0.335918\pi$$
$$644$$ 0 0
$$645$$ 3.00000 0.118125
$$646$$ 0 0
$$647$$ −45.0000 −1.76913 −0.884566 0.466415i $$-0.845546\pi$$
−0.884566 + 0.466415i $$0.845546\pi$$
$$648$$ 0 0
$$649$$ −18.0000 −0.706562
$$650$$ 0 0
$$651$$ 6.00000 0.235159
$$652$$ 0 0
$$653$$ −21.0000 −0.821794 −0.410897 0.911682i $$-0.634784\pi$$
−0.410897 + 0.911682i $$0.634784\pi$$
$$654$$ 0 0
$$655$$ 39.0000 1.52386
$$656$$ 0 0
$$657$$ −11.0000 −0.429151
$$658$$ 0 0
$$659$$ 10.0000 0.389545 0.194772 0.980848i $$-0.437603\pi$$
0.194772 + 0.980848i $$0.437603\pi$$
$$660$$ 0 0
$$661$$ 40.0000 1.55582 0.777910 0.628376i $$-0.216280\pi$$
0.777910 + 0.628376i $$0.216280\pi$$
$$662$$ 0 0
$$663$$ 0 0
$$664$$ 0 0
$$665$$ 9.00000 0.349005
$$666$$ 0 0
$$667$$ 0 0
$$668$$ 0 0
$$669$$ 16.0000 0.618596
$$670$$ 0 0
$$671$$ −3.00000 −0.115814
$$672$$ 0 0
$$673$$ 34.0000 1.31060 0.655302 0.755367i $$-0.272541\pi$$
0.655302 + 0.755367i $$0.272541\pi$$
$$674$$ 0 0
$$675$$ −4.00000 −0.153960
$$676$$ 0 0
$$677$$ −12.0000 −0.461197 −0.230599 0.973049i $$-0.574068\pi$$
−0.230599 + 0.973049i $$0.574068\pi$$
$$678$$ 0 0
$$679$$ −24.0000 −0.921035
$$680$$ 0 0
$$681$$ −4.00000 −0.153280
$$682$$ 0 0
$$683$$ 20.0000 0.765279 0.382639 0.923898i $$-0.375015\pi$$
0.382639 + 0.923898i $$0.375015\pi$$
$$684$$ 0 0
$$685$$ −3.00000 −0.114624
$$686$$ 0 0
$$687$$ 21.0000 0.801200
$$688$$ 0 0
$$689$$ 0 0
$$690$$ 0 0
$$691$$ −41.0000 −1.55971 −0.779857 0.625958i $$-0.784708\pi$$
−0.779857 + 0.625958i $$0.784708\pi$$
$$692$$ 0 0
$$693$$ −9.00000 −0.341882
$$694$$ 0 0
$$695$$ −27.0000 −1.02417
$$696$$ 0 0
$$697$$ −12.0000 −0.454532
$$698$$ 0 0
$$699$$ −15.0000 −0.567352
$$700$$ 0 0
$$701$$ 42.0000 1.58632 0.793159 0.609015i $$-0.208435\pi$$
0.793159 + 0.609015i $$0.208435\pi$$
$$702$$ 0 0
$$703$$ −4.00000 −0.150863
$$704$$ 0 0
$$705$$ 27.0000 1.01688
$$706$$ 0 0
$$707$$ −42.0000 −1.57957
$$708$$ 0 0
$$709$$ −46.0000 −1.72757 −0.863783 0.503864i $$-0.831911\pi$$
−0.863783 + 0.503864i $$0.831911\pi$$
$$710$$ 0 0
$$711$$ 6.00000 0.225018
$$712$$ 0 0
$$713$$ 0 0
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 0 0
$$717$$ 13.0000 0.485494
$$718$$ 0 0
$$719$$ −1.00000 −0.0372937 −0.0186469 0.999826i $$-0.505936\pi$$
−0.0186469 + 0.999826i $$0.505936\pi$$
$$720$$ 0 0
$$721$$ 30.0000 1.11726
$$722$$ 0 0
$$723$$ 24.0000 0.892570
$$724$$ 0 0
$$725$$ 32.0000 1.18845
$$726$$ 0 0
$$727$$ 37.0000 1.37225 0.686127 0.727482i $$-0.259309\pi$$
0.686127 + 0.727482i $$0.259309\pi$$
$$728$$ 0 0
$$729$$ 1.00000 0.0370370
$$730$$ 0 0
$$731$$ 1.00000 0.0369863
$$732$$ 0 0
$$733$$ −50.0000 −1.84679 −0.923396 0.383849i $$-0.874598\pi$$
−0.923396 + 0.383849i $$0.874598\pi$$
$$734$$ 0 0
$$735$$ 6.00000 0.221313
$$736$$ 0 0
$$737$$ −30.0000 −1.10506
$$738$$ 0 0
$$739$$ 45.0000 1.65535 0.827676 0.561206i $$-0.189663\pi$$
0.827676 + 0.561206i $$0.189663\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 0 0
$$743$$ −26.0000 −0.953847 −0.476924 0.878945i $$-0.658248\pi$$
−0.476924 + 0.878945i $$0.658248\pi$$
$$744$$ 0 0
$$745$$ 15.0000 0.549557
$$746$$ 0 0
$$747$$ 0 0
$$748$$ 0 0
$$749$$ −54.0000 −1.97312
$$750$$ 0 0
$$751$$ 28.0000 1.02173 0.510867 0.859660i $$-0.329324\pi$$
0.510867 + 0.859660i $$0.329324\pi$$
$$752$$ 0 0
$$753$$ 13.0000 0.473746
$$754$$ 0 0
$$755$$ −54.0000 −1.96526
$$756$$ 0 0
$$757$$ −23.0000 −0.835949 −0.417975 0.908459i $$-0.637260\pi$$
−0.417975 + 0.908459i $$0.637260\pi$$
$$758$$ 0 0
$$759$$ 0 0
$$760$$ 0 0
$$761$$ −47.0000 −1.70375 −0.851874 0.523746i $$-0.824534\pi$$
−0.851874 + 0.523746i $$0.824534\pi$$
$$762$$ 0 0
$$763$$ 30.0000 1.08607
$$764$$ 0 0
$$765$$ −3.00000 −0.108465
$$766$$ 0 0
$$767$$ 0 0
$$768$$ 0 0
$$769$$ 51.0000 1.83911 0.919554 0.392965i $$-0.128551\pi$$
0.919554 + 0.392965i $$0.128551\pi$$
$$770$$ 0 0
$$771$$ −28.0000 −1.00840
$$772$$ 0 0
$$773$$ 10.0000 0.359675 0.179838 0.983696i $$-0.442443\pi$$
0.179838 + 0.983696i $$0.442443\pi$$
$$774$$ 0 0
$$775$$ −8.00000 −0.287368
$$776$$ 0 0
$$777$$ −12.0000 −0.430498
$$778$$ 0 0
$$779$$ 12.0000 0.429945
$$780$$ 0 0
$$781$$ 30.0000 1.07348
$$782$$ 0 0
$$783$$ −8.00000 −0.285897
$$784$$ 0 0
$$785$$ 6.00000 0.214149
$$786$$ 0 0
$$787$$ −44.0000 −1.56843 −0.784215 0.620489i $$-0.786934\pi$$
−0.784215 + 0.620489i $$0.786934\pi$$
$$788$$ 0 0
$$789$$ 3.00000 0.106803
$$790$$ 0 0
$$791$$ −12.0000 −0.426671
$$792$$ 0 0
$$793$$ 0 0
$$794$$ 0 0
$$795$$ −18.0000 −0.638394
$$796$$ 0 0
$$797$$ −2.00000 −0.0708436 −0.0354218 0.999372i $$-0.511277\pi$$
−0.0354218 + 0.999372i $$0.511277\pi$$
$$798$$ 0 0
$$799$$ 9.00000 0.318397
$$800$$ 0 0
$$801$$ −6.00000 −0.212000
$$802$$ 0 0
$$803$$ 33.0000 1.16454
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ −22.0000 −0.774437
$$808$$ 0 0
$$809$$ 39.0000 1.37117 0.685583 0.727994i $$-0.259547\pi$$
0.685583 + 0.727994i $$0.259547\pi$$
$$810$$ 0 0
$$811$$ −28.0000 −0.983213 −0.491606 0.870817i $$-0.663590\pi$$
−0.491606 + 0.870817i $$0.663590\pi$$
$$812$$ 0 0
$$813$$ 0 0
$$814$$ 0 0
$$815$$ −12.0000 −0.420342
$$816$$ 0 0
$$817$$ −1.00000 −0.0349856
$$818$$ 0 0
$$819$$ 0 0
$$820$$ 0 0
$$821$$ −13.0000 −0.453703 −0.226852 0.973929i $$-0.572843\pi$$
−0.226852 + 0.973929i $$0.572843\pi$$
$$822$$ 0 0
$$823$$ 11.0000 0.383436 0.191718 0.981450i $$-0.438594\pi$$
0.191718 + 0.981450i $$0.438594\pi$$
$$824$$ 0 0
$$825$$ 12.0000 0.417786
$$826$$ 0 0
$$827$$ −2.00000 −0.0695468 −0.0347734 0.999395i $$-0.511071\pi$$
−0.0347734 + 0.999395i $$0.511071\pi$$
$$828$$ 0 0
$$829$$ 4.00000 0.138926 0.0694629 0.997585i $$-0.477871\pi$$
0.0694629 + 0.997585i $$0.477871\pi$$
$$830$$ 0 0
$$831$$ −23.0000 −0.797861
$$832$$ 0 0
$$833$$ 2.00000 0.0692959
$$834$$ 0 0
$$835$$ 12.0000 0.415277
$$836$$ 0 0
$$837$$ 2.00000 0.0691301
$$838$$ 0 0
$$839$$ −42.0000 −1.45000 −0.725001 0.688748i $$-0.758161\pi$$
−0.725001 + 0.688748i $$0.758161\pi$$
$$840$$ 0 0
$$841$$ 35.0000 1.20690
$$842$$ 0 0
$$843$$ −6.00000 −0.206651
$$844$$ 0 0
$$845$$ 39.0000 1.34164
$$846$$ 0 0
$$847$$ −6.00000 −0.206162
$$848$$ 0 0
$$849$$ −13.0000 −0.446159
$$850$$ 0 0
$$851$$ 0 0
$$852$$ 0 0
$$853$$ 46.0000 1.57501 0.787505 0.616308i $$-0.211372\pi$$
0.787505 + 0.616308i $$0.211372\pi$$
$$854$$ 0 0
$$855$$ 3.00000 0.102598
$$856$$ 0 0
$$857$$ 18.0000 0.614868 0.307434 0.951569i $$-0.400530\pi$$
0.307434 + 0.951569i $$0.400530\pi$$
$$858$$ 0 0
$$859$$ −7.00000 −0.238837 −0.119418 0.992844i $$-0.538103\pi$$
−0.119418 + 0.992844i $$0.538103\pi$$
$$860$$ 0 0
$$861$$ 36.0000 1.22688
$$862$$ 0 0
$$863$$ −8.00000 −0.272323 −0.136162 0.990687i $$-0.543477\pi$$
−0.136162 + 0.990687i $$0.543477\pi$$
$$864$$ 0 0
$$865$$ 54.0000 1.83606
$$866$$ 0 0
$$867$$ 16.0000 0.543388
$$868$$ 0 0
$$869$$ −18.0000 −0.610608
$$870$$ 0 0
$$871$$ 0 0
$$872$$ 0 0
$$873$$ −8.00000 −0.270759
$$874$$ 0 0
$$875$$ 9.00000 0.304256
$$876$$ 0 0
$$877$$ −34.0000 −1.14810 −0.574049 0.818821i $$-0.694628\pi$$
−0.574049 + 0.818821i $$0.694628\pi$$
$$878$$ 0 0
$$879$$ 16.0000 0.539667
$$880$$ 0 0
$$881$$ −3.00000 −0.101073 −0.0505363 0.998722i $$-0.516093\pi$$
−0.0505363 + 0.998722i $$0.516093\pi$$
$$882$$ 0 0
$$883$$ 29.0000 0.975928 0.487964 0.872864i $$-0.337740\pi$$
0.487964 + 0.872864i $$0.337740\pi$$
$$884$$ 0 0
$$885$$ 18.0000 0.605063
$$886$$ 0 0
$$887$$ 30.0000 1.00730 0.503651 0.863907i $$-0.331990\pi$$
0.503651 + 0.863907i $$0.331990\pi$$
$$888$$ 0 0
$$889$$ 24.0000 0.804934
$$890$$ 0 0
$$891$$ −3.00000 −0.100504
$$892$$ 0 0
$$893$$ −9.00000 −0.301174
$$894$$ 0 0
$$895$$ 72.0000 2.40669
$$896$$ 0 0
$$897$$ 0 0
$$898$$ 0 0
$$899$$ −16.0000 −0.533630
$$900$$ 0 0
$$901$$ −6.00000 −0.199889
$$902$$ 0 0
$$903$$ −3.00000 −0.0998337
$$904$$ 0 0
$$905$$ 66.0000 2.19391
$$906$$ 0 0
$$907$$ −46.0000 −1.52740 −0.763702 0.645568i $$-0.776621\pi$$
−0.763702 + 0.645568i $$0.776621\pi$$
$$908$$ 0 0
$$909$$ −14.0000 −0.464351
$$910$$ 0 0
$$911$$ 18.0000 0.596367 0.298183 0.954509i $$-0.403619\pi$$
0.298183 + 0.954509i $$0.403619\pi$$
$$912$$ 0 0
$$913$$ 0 0
$$914$$ 0 0
$$915$$ 3.00000 0.0991769
$$916$$ 0 0
$$917$$ −39.0000 −1.28789
$$918$$ 0 0
$$919$$ 16.0000 0.527791 0.263896 0.964551i $$-0.414993\pi$$
0.263896 + 0.964551i $$0.414993\pi$$
$$920$$ 0 0
$$921$$ −6.00000 −0.197707
$$922$$ 0 0
$$923$$ 0 0
$$924$$ 0 0
$$925$$ 16.0000 0.526077
$$926$$ 0 0
$$927$$ 10.0000 0.328443
$$928$$ 0 0
$$929$$ 30.0000 0.984268 0.492134 0.870519i $$-0.336217\pi$$
0.492134 + 0.870519i $$0.336217\pi$$
$$930$$ 0 0
$$931$$ −2.00000 −0.0655474
$$932$$ 0 0
$$933$$ 7.00000 0.229170
$$934$$ 0 0
$$935$$ 9.00000 0.294331
$$936$$ 0 0
$$937$$ −43.0000 −1.40475 −0.702374 0.711808i $$-0.747877\pi$$
−0.702374 + 0.711808i $$0.747877\pi$$
$$938$$ 0 0
$$939$$ 26.0000 0.848478
$$940$$ 0 0
$$941$$ 6.00000 0.195594 0.0977972 0.995206i $$-0.468820\pi$$
0.0977972 + 0.995206i $$0.468820\pi$$
$$942$$ 0 0
$$943$$ 0 0
$$944$$ 0 0
$$945$$ 9.00000 0.292770
$$946$$ 0 0
$$947$$ −52.0000 −1.68977 −0.844886 0.534946i $$-0.820332\pi$$
−0.844886 + 0.534946i $$0.820332\pi$$
$$948$$ 0 0
$$949$$ 0 0
$$950$$ 0 0
$$951$$ 18.0000 0.583690
$$952$$ 0 0
$$953$$ 26.0000 0.842223 0.421111 0.907009i $$-0.361640\pi$$
0.421111 + 0.907009i $$0.361640\pi$$
$$954$$ 0 0
$$955$$ 39.0000 1.26201
$$956$$ 0 0
$$957$$ 24.0000 0.775810
$$958$$ 0 0
$$959$$ 3.00000 0.0968751
$$960$$ 0 0
$$961$$ −27.0000 −0.870968
$$962$$ 0 0
$$963$$ −18.0000 −0.580042
$$964$$ 0 0
$$965$$ 42.0000 1.35203
$$966$$ 0 0
$$967$$ −32.0000 −1.02905 −0.514525 0.857475i $$-0.672032\pi$$
−0.514525 + 0.857475i $$0.672032\pi$$
$$968$$ 0 0
$$969$$ 1.00000 0.0321246
$$970$$ 0 0
$$971$$ −60.0000 −1.92549 −0.962746 0.270408i $$-0.912841\pi$$
−0.962746 + 0.270408i $$0.912841\pi$$
$$972$$ 0 0
$$973$$ 27.0000 0.865580
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ −14.0000 −0.447900 −0.223950 0.974601i $$-0.571895\pi$$
−0.223950 + 0.974601i $$0.571895\pi$$
$$978$$ 0 0
$$979$$ 18.0000 0.575282
$$980$$ 0 0
$$981$$ 10.0000 0.319275
$$982$$ 0 0
$$983$$ 52.0000 1.65854 0.829271 0.558846i $$-0.188756\pi$$
0.829271 + 0.558846i $$0.188756\pi$$
$$984$$ 0 0
$$985$$ 54.0000 1.72058
$$986$$ 0 0
$$987$$ −27.0000 −0.859419
$$988$$ 0 0
$$989$$ 0 0
$$990$$ 0 0
$$991$$ 52.0000 1.65183 0.825917 0.563791i $$-0.190658\pi$$
0.825917 + 0.563791i $$0.190658\pi$$
$$992$$ 0 0
$$993$$ −18.0000 −0.571213
$$994$$ 0 0
$$995$$ 15.0000 0.475532
$$996$$ 0 0
$$997$$ −13.0000 −0.411714 −0.205857 0.978582i $$-0.565998\pi$$
−0.205857 + 0.978582i $$0.565998\pi$$
$$998$$ 0 0
$$999$$ −4.00000 −0.126554
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 3648.2.a.b.1.1 1
4.3 odd 2 3648.2.a.t.1.1 1
8.3 odd 2 1824.2.a.f.1.1 1
8.5 even 2 1824.2.a.l.1.1 yes 1
24.5 odd 2 5472.2.a.c.1.1 1
24.11 even 2 5472.2.a.b.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
1824.2.a.f.1.1 1 8.3 odd 2
1824.2.a.l.1.1 yes 1 8.5 even 2
3648.2.a.b.1.1 1 1.1 even 1 trivial
3648.2.a.t.1.1 1 4.3 odd 2
5472.2.a.b.1.1 1 24.11 even 2
5472.2.a.c.1.1 1 24.5 odd 2