# Properties

 Label 1216.2.a.a.1.1 Level $1216$ Weight $2$ Character 1216.1 Self dual yes Analytic conductor $9.710$ 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] = [1216,2,Mod(1,1216)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1216, 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("1216.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1216 = 2^{6} \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1216.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: $$9.70980888579$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 608) Fricke sign: $$+1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 1216.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-3.00000 q^{3} -1.00000 q^{7} +6.00000 q^{9} +O(q^{10})$$ $$q-3.00000 q^{3} -1.00000 q^{7} +6.00000 q^{9} -2.00000 q^{11} +1.00000 q^{13} +3.00000 q^{17} +1.00000 q^{19} +3.00000 q^{21} +3.00000 q^{23} -5.00000 q^{25} -9.00000 q^{27} -3.00000 q^{29} +8.00000 q^{31} +6.00000 q^{33} +10.0000 q^{37} -3.00000 q^{39} -12.0000 q^{41} -8.00000 q^{43} -8.00000 q^{47} -6.00000 q^{49} -9.00000 q^{51} +9.00000 q^{53} -3.00000 q^{57} +5.00000 q^{59} -10.0000 q^{61} -6.00000 q^{63} -7.00000 q^{67} -9.00000 q^{69} -10.0000 q^{71} +1.00000 q^{73} +15.0000 q^{75} +2.00000 q^{77} -14.0000 q^{79} +9.00000 q^{81} -6.00000 q^{83} +9.00000 q^{87} -4.00000 q^{89} -1.00000 q^{91} -24.0000 q^{93} -6.00000 q^{97} -12.0000 q^{99} +O(q^{100})$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 0 0
$$3$$ −3.00000 −1.73205 −0.866025 0.500000i $$-0.833333\pi$$
−0.866025 + 0.500000i $$0.833333\pi$$
$$4$$ 0 0
$$5$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$6$$ 0 0
$$7$$ −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
$$10$$ 0 0
$$11$$ −2.00000 −0.603023 −0.301511 0.953463i $$-0.597491\pi$$
−0.301511 + 0.953463i $$0.597491\pi$$
$$12$$ 0 0
$$13$$ 1.00000 0.277350 0.138675 0.990338i $$-0.455716\pi$$
0.138675 + 0.990338i $$0.455716\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ 3.00000 0.727607 0.363803 0.931476i $$-0.381478\pi$$
0.363803 + 0.931476i $$0.381478\pi$$
$$18$$ 0 0
$$19$$ 1.00000 0.229416
$$20$$ 0 0
$$21$$ 3.00000 0.654654
$$22$$ 0 0
$$23$$ 3.00000 0.625543 0.312772 0.949828i $$-0.398743\pi$$
0.312772 + 0.949828i $$0.398743\pi$$
$$24$$ 0 0
$$25$$ −5.00000 −1.00000
$$26$$ 0 0
$$27$$ −9.00000 −1.73205
$$28$$ 0 0
$$29$$ −3.00000 −0.557086 −0.278543 0.960424i $$-0.589851\pi$$
−0.278543 + 0.960424i $$0.589851\pi$$
$$30$$ 0 0
$$31$$ 8.00000 1.43684 0.718421 0.695608i $$-0.244865\pi$$
0.718421 + 0.695608i $$0.244865\pi$$
$$32$$ 0 0
$$33$$ 6.00000 1.04447
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 0 0
$$37$$ 10.0000 1.64399 0.821995 0.569495i $$-0.192861\pi$$
0.821995 + 0.569495i $$0.192861\pi$$
$$38$$ 0 0
$$39$$ −3.00000 −0.480384
$$40$$ 0 0
$$41$$ −12.0000 −1.87409 −0.937043 0.349215i $$-0.886448\pi$$
−0.937043 + 0.349215i $$0.886448\pi$$
$$42$$ 0 0
$$43$$ −8.00000 −1.21999 −0.609994 0.792406i $$-0.708828\pi$$
−0.609994 + 0.792406i $$0.708828\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ −8.00000 −1.16692 −0.583460 0.812142i $$-0.698301\pi$$
−0.583460 + 0.812142i $$0.698301\pi$$
$$48$$ 0 0
$$49$$ −6.00000 −0.857143
$$50$$ 0 0
$$51$$ −9.00000 −1.26025
$$52$$ 0 0
$$53$$ 9.00000 1.23625 0.618123 0.786082i $$-0.287894\pi$$
0.618123 + 0.786082i $$0.287894\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 0 0
$$57$$ −3.00000 −0.397360
$$58$$ 0 0
$$59$$ 5.00000 0.650945 0.325472 0.945552i $$-0.394477\pi$$
0.325472 + 0.945552i $$0.394477\pi$$
$$60$$ 0 0
$$61$$ −10.0000 −1.28037 −0.640184 0.768221i $$-0.721142\pi$$
−0.640184 + 0.768221i $$0.721142\pi$$
$$62$$ 0 0
$$63$$ −6.00000 −0.755929
$$64$$ 0 0
$$65$$ 0 0
$$66$$ 0 0
$$67$$ −7.00000 −0.855186 −0.427593 0.903971i $$-0.640638\pi$$
−0.427593 + 0.903971i $$0.640638\pi$$
$$68$$ 0 0
$$69$$ −9.00000 −1.08347
$$70$$ 0 0
$$71$$ −10.0000 −1.18678 −0.593391 0.804914i $$-0.702211\pi$$
−0.593391 + 0.804914i $$0.702211\pi$$
$$72$$ 0 0
$$73$$ 1.00000 0.117041 0.0585206 0.998286i $$-0.481362\pi$$
0.0585206 + 0.998286i $$0.481362\pi$$
$$74$$ 0 0
$$75$$ 15.0000 1.73205
$$76$$ 0 0
$$77$$ 2.00000 0.227921
$$78$$ 0 0
$$79$$ −14.0000 −1.57512 −0.787562 0.616236i $$-0.788657\pi$$
−0.787562 + 0.616236i $$0.788657\pi$$
$$80$$ 0 0
$$81$$ 9.00000 1.00000
$$82$$ 0 0
$$83$$ −6.00000 −0.658586 −0.329293 0.944228i $$-0.606810\pi$$
−0.329293 + 0.944228i $$0.606810\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 0 0
$$87$$ 9.00000 0.964901
$$88$$ 0 0
$$89$$ −4.00000 −0.423999 −0.212000 0.977270i $$-0.567998\pi$$
−0.212000 + 0.977270i $$0.567998\pi$$
$$90$$ 0 0
$$91$$ −1.00000 −0.104828
$$92$$ 0 0
$$93$$ −24.0000 −2.48868
$$94$$ 0 0
$$95$$ 0 0
$$96$$ 0 0
$$97$$ −6.00000 −0.609208 −0.304604 0.952479i $$-0.598524\pi$$
−0.304604 + 0.952479i $$0.598524\pi$$
$$98$$ 0 0
$$99$$ −12.0000 −1.20605
$$100$$ 0 0
$$101$$ −10.0000 −0.995037 −0.497519 0.867453i $$-0.665755\pi$$
−0.497519 + 0.867453i $$0.665755\pi$$
$$102$$ 0 0
$$103$$ 14.0000 1.37946 0.689730 0.724066i $$-0.257729\pi$$
0.689730 + 0.724066i $$0.257729\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ −5.00000 −0.483368 −0.241684 0.970355i $$-0.577700\pi$$
−0.241684 + 0.970355i $$0.577700\pi$$
$$108$$ 0 0
$$109$$ −9.00000 −0.862044 −0.431022 0.902342i $$-0.641847\pi$$
−0.431022 + 0.902342i $$0.641847\pi$$
$$110$$ 0 0
$$111$$ −30.0000 −2.84747
$$112$$ 0 0
$$113$$ 2.00000 0.188144 0.0940721 0.995565i $$-0.470012\pi$$
0.0940721 + 0.995565i $$0.470012\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 0 0
$$117$$ 6.00000 0.554700
$$118$$ 0 0
$$119$$ −3.00000 −0.275010
$$120$$ 0 0
$$121$$ −7.00000 −0.636364
$$122$$ 0 0
$$123$$ 36.0000 3.24601
$$124$$ 0 0
$$125$$ 0 0
$$126$$ 0 0
$$127$$ 18.0000 1.59724 0.798621 0.601834i $$-0.205563\pi$$
0.798621 + 0.601834i $$0.205563\pi$$
$$128$$ 0 0
$$129$$ 24.0000 2.11308
$$130$$ 0 0
$$131$$ 8.00000 0.698963 0.349482 0.936943i $$-0.386358\pi$$
0.349482 + 0.936943i $$0.386358\pi$$
$$132$$ 0 0
$$133$$ −1.00000 −0.0867110
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ −17.0000 −1.45241 −0.726204 0.687479i $$-0.758717\pi$$
−0.726204 + 0.687479i $$0.758717\pi$$
$$138$$ 0 0
$$139$$ 16.0000 1.35710 0.678551 0.734553i $$-0.262608\pi$$
0.678551 + 0.734553i $$0.262608\pi$$
$$140$$ 0 0
$$141$$ 24.0000 2.02116
$$142$$ 0 0
$$143$$ −2.00000 −0.167248
$$144$$ 0 0
$$145$$ 0 0
$$146$$ 0 0
$$147$$ 18.0000 1.48461
$$148$$ 0 0
$$149$$ −4.00000 −0.327693 −0.163846 0.986486i $$-0.552390\pi$$
−0.163846 + 0.986486i $$0.552390\pi$$
$$150$$ 0 0
$$151$$ 10.0000 0.813788 0.406894 0.913475i $$-0.366612\pi$$
0.406894 + 0.913475i $$0.366612\pi$$
$$152$$ 0 0
$$153$$ 18.0000 1.45521
$$154$$ 0 0
$$155$$ 0 0
$$156$$ 0 0
$$157$$ −2.00000 −0.159617 −0.0798087 0.996810i $$-0.525431\pi$$
−0.0798087 + 0.996810i $$0.525431\pi$$
$$158$$ 0 0
$$159$$ −27.0000 −2.14124
$$160$$ 0 0
$$161$$ −3.00000 −0.236433
$$162$$ 0 0
$$163$$ −8.00000 −0.626608 −0.313304 0.949653i $$-0.601436\pi$$
−0.313304 + 0.949653i $$0.601436\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ −16.0000 −1.23812 −0.619059 0.785345i $$-0.712486\pi$$
−0.619059 + 0.785345i $$0.712486\pi$$
$$168$$ 0 0
$$169$$ −12.0000 −0.923077
$$170$$ 0 0
$$171$$ 6.00000 0.458831
$$172$$ 0 0
$$173$$ 6.00000 0.456172 0.228086 0.973641i $$-0.426753\pi$$
0.228086 + 0.973641i $$0.426753\pi$$
$$174$$ 0 0
$$175$$ 5.00000 0.377964
$$176$$ 0 0
$$177$$ −15.0000 −1.12747
$$178$$ 0 0
$$179$$ −24.0000 −1.79384 −0.896922 0.442189i $$-0.854202\pi$$
−0.896922 + 0.442189i $$0.854202\pi$$
$$180$$ 0 0
$$181$$ 18.0000 1.33793 0.668965 0.743294i $$-0.266738\pi$$
0.668965 + 0.743294i $$0.266738\pi$$
$$182$$ 0 0
$$183$$ 30.0000 2.21766
$$184$$ 0 0
$$185$$ 0 0
$$186$$ 0 0
$$187$$ −6.00000 −0.438763
$$188$$ 0 0
$$189$$ 9.00000 0.654654
$$190$$ 0 0
$$191$$ −21.0000 −1.51951 −0.759753 0.650211i $$-0.774680\pi$$
−0.759753 + 0.650211i $$0.774680\pi$$
$$192$$ 0 0
$$193$$ 10.0000 0.719816 0.359908 0.932988i $$-0.382808\pi$$
0.359908 + 0.932988i $$0.382808\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ 4.00000 0.284988 0.142494 0.989796i $$-0.454488\pi$$
0.142494 + 0.989796i $$0.454488\pi$$
$$198$$ 0 0
$$199$$ 11.0000 0.779769 0.389885 0.920864i $$-0.372515\pi$$
0.389885 + 0.920864i $$0.372515\pi$$
$$200$$ 0 0
$$201$$ 21.0000 1.48123
$$202$$ 0 0
$$203$$ 3.00000 0.210559
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 0 0
$$207$$ 18.0000 1.25109
$$208$$ 0 0
$$209$$ −2.00000 −0.138343
$$210$$ 0 0
$$211$$ 1.00000 0.0688428 0.0344214 0.999407i $$-0.489041\pi$$
0.0344214 + 0.999407i $$0.489041\pi$$
$$212$$ 0 0
$$213$$ 30.0000 2.05557
$$214$$ 0 0
$$215$$ 0 0
$$216$$ 0 0
$$217$$ −8.00000 −0.543075
$$218$$ 0 0
$$219$$ −3.00000 −0.202721
$$220$$ 0 0
$$221$$ 3.00000 0.201802
$$222$$ 0 0
$$223$$ 2.00000 0.133930 0.0669650 0.997755i $$-0.478668\pi$$
0.0669650 + 0.997755i $$0.478668\pi$$
$$224$$ 0 0
$$225$$ −30.0000 −2.00000
$$226$$ 0 0
$$227$$ 21.0000 1.39382 0.696909 0.717159i $$-0.254558\pi$$
0.696909 + 0.717159i $$0.254558\pi$$
$$228$$ 0 0
$$229$$ −22.0000 −1.45380 −0.726900 0.686743i $$-0.759040\pi$$
−0.726900 + 0.686743i $$0.759040\pi$$
$$230$$ 0 0
$$231$$ −6.00000 −0.394771
$$232$$ 0 0
$$233$$ 26.0000 1.70332 0.851658 0.524097i $$-0.175597\pi$$
0.851658 + 0.524097i $$0.175597\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ 0 0
$$237$$ 42.0000 2.72819
$$238$$ 0 0
$$239$$ 27.0000 1.74648 0.873242 0.487286i $$-0.162013\pi$$
0.873242 + 0.487286i $$0.162013\pi$$
$$240$$ 0 0
$$241$$ 20.0000 1.28831 0.644157 0.764894i $$-0.277208\pi$$
0.644157 + 0.764894i $$0.277208\pi$$
$$242$$ 0 0
$$243$$ 0 0
$$244$$ 0 0
$$245$$ 0 0
$$246$$ 0 0
$$247$$ 1.00000 0.0636285
$$248$$ 0 0
$$249$$ 18.0000 1.14070
$$250$$ 0 0
$$251$$ 2.00000 0.126239 0.0631194 0.998006i $$-0.479895\pi$$
0.0631194 + 0.998006i $$0.479895\pi$$
$$252$$ 0 0
$$253$$ −6.00000 −0.377217
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ −12.0000 −0.748539 −0.374270 0.927320i $$-0.622107\pi$$
−0.374270 + 0.927320i $$0.622107\pi$$
$$258$$ 0 0
$$259$$ −10.0000 −0.621370
$$260$$ 0 0
$$261$$ −18.0000 −1.11417
$$262$$ 0 0
$$263$$ −16.0000 −0.986602 −0.493301 0.869859i $$-0.664210\pi$$
−0.493301 + 0.869859i $$0.664210\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 0 0
$$267$$ 12.0000 0.734388
$$268$$ 0 0
$$269$$ −22.0000 −1.34136 −0.670682 0.741745i $$-0.733998\pi$$
−0.670682 + 0.741745i $$0.733998\pi$$
$$270$$ 0 0
$$271$$ 3.00000 0.182237 0.0911185 0.995840i $$-0.470956\pi$$
0.0911185 + 0.995840i $$0.470956\pi$$
$$272$$ 0 0
$$273$$ 3.00000 0.181568
$$274$$ 0 0
$$275$$ 10.0000 0.603023
$$276$$ 0 0
$$277$$ −12.0000 −0.721010 −0.360505 0.932757i $$-0.617396\pi$$
−0.360505 + 0.932757i $$0.617396\pi$$
$$278$$ 0 0
$$279$$ 48.0000 2.87368
$$280$$ 0 0
$$281$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$282$$ 0 0
$$283$$ −6.00000 −0.356663 −0.178331 0.983970i $$-0.557070\pi$$
−0.178331 + 0.983970i $$0.557070\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ 12.0000 0.708338
$$288$$ 0 0
$$289$$ −8.00000 −0.470588
$$290$$ 0 0
$$291$$ 18.0000 1.05518
$$292$$ 0 0
$$293$$ −1.00000 −0.0584206 −0.0292103 0.999573i $$-0.509299\pi$$
−0.0292103 + 0.999573i $$0.509299\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 0 0
$$297$$ 18.0000 1.04447
$$298$$ 0 0
$$299$$ 3.00000 0.173494
$$300$$ 0 0
$$301$$ 8.00000 0.461112
$$302$$ 0 0
$$303$$ 30.0000 1.72345
$$304$$ 0 0
$$305$$ 0 0
$$306$$ 0 0
$$307$$ −12.0000 −0.684876 −0.342438 0.939540i $$-0.611253\pi$$
−0.342438 + 0.939540i $$0.611253\pi$$
$$308$$ 0 0
$$309$$ −42.0000 −2.38930
$$310$$ 0 0
$$311$$ −5.00000 −0.283524 −0.141762 0.989901i $$-0.545277\pi$$
−0.141762 + 0.989901i $$0.545277\pi$$
$$312$$ 0 0
$$313$$ −3.00000 −0.169570 −0.0847850 0.996399i $$-0.527020\pi$$
−0.0847850 + 0.996399i $$0.527020\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ −21.0000 −1.17948 −0.589739 0.807594i $$-0.700769\pi$$
−0.589739 + 0.807594i $$0.700769\pi$$
$$318$$ 0 0
$$319$$ 6.00000 0.335936
$$320$$ 0 0
$$321$$ 15.0000 0.837218
$$322$$ 0 0
$$323$$ 3.00000 0.166924
$$324$$ 0 0
$$325$$ −5.00000 −0.277350
$$326$$ 0 0
$$327$$ 27.0000 1.49310
$$328$$ 0 0
$$329$$ 8.00000 0.441054
$$330$$ 0 0
$$331$$ −13.0000 −0.714545 −0.357272 0.934000i $$-0.616293\pi$$
−0.357272 + 0.934000i $$0.616293\pi$$
$$332$$ 0 0
$$333$$ 60.0000 3.28798
$$334$$ 0 0
$$335$$ 0 0
$$336$$ 0 0
$$337$$ −4.00000 −0.217894 −0.108947 0.994048i $$-0.534748\pi$$
−0.108947 + 0.994048i $$0.534748\pi$$
$$338$$ 0 0
$$339$$ −6.00000 −0.325875
$$340$$ 0 0
$$341$$ −16.0000 −0.866449
$$342$$ 0 0
$$343$$ 13.0000 0.701934
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ −30.0000 −1.61048 −0.805242 0.592946i $$-0.797965\pi$$
−0.805242 + 0.592946i $$0.797965\pi$$
$$348$$ 0 0
$$349$$ −14.0000 −0.749403 −0.374701 0.927146i $$-0.622255\pi$$
−0.374701 + 0.927146i $$0.622255\pi$$
$$350$$ 0 0
$$351$$ −9.00000 −0.480384
$$352$$ 0 0
$$353$$ 9.00000 0.479022 0.239511 0.970894i $$-0.423013\pi$$
0.239511 + 0.970894i $$0.423013\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 0 0
$$357$$ 9.00000 0.476331
$$358$$ 0 0
$$359$$ −3.00000 −0.158334 −0.0791670 0.996861i $$-0.525226\pi$$
−0.0791670 + 0.996861i $$0.525226\pi$$
$$360$$ 0 0
$$361$$ 1.00000 0.0526316
$$362$$ 0 0
$$363$$ 21.0000 1.10221
$$364$$ 0 0
$$365$$ 0 0
$$366$$ 0 0
$$367$$ 20.0000 1.04399 0.521996 0.852948i $$-0.325188\pi$$
0.521996 + 0.852948i $$0.325188\pi$$
$$368$$ 0 0
$$369$$ −72.0000 −3.74817
$$370$$ 0 0
$$371$$ −9.00000 −0.467257
$$372$$ 0 0
$$373$$ −21.0000 −1.08734 −0.543669 0.839299i $$-0.682965\pi$$
−0.543669 + 0.839299i $$0.682965\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ −3.00000 −0.154508
$$378$$ 0 0
$$379$$ 5.00000 0.256833 0.128416 0.991720i $$-0.459011\pi$$
0.128416 + 0.991720i $$0.459011\pi$$
$$380$$ 0 0
$$381$$ −54.0000 −2.76650
$$382$$ 0 0
$$383$$ −38.0000 −1.94171 −0.970855 0.239669i $$-0.922961\pi$$
−0.970855 + 0.239669i $$0.922961\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ −48.0000 −2.43998
$$388$$ 0 0
$$389$$ 6.00000 0.304212 0.152106 0.988364i $$-0.451394\pi$$
0.152106 + 0.988364i $$0.451394\pi$$
$$390$$ 0 0
$$391$$ 9.00000 0.455150
$$392$$ 0 0
$$393$$ −24.0000 −1.21064
$$394$$ 0 0
$$395$$ 0 0
$$396$$ 0 0
$$397$$ 8.00000 0.401508 0.200754 0.979642i $$-0.435661\pi$$
0.200754 + 0.979642i $$0.435661\pi$$
$$398$$ 0 0
$$399$$ 3.00000 0.150188
$$400$$ 0 0
$$401$$ 8.00000 0.399501 0.199750 0.979847i $$-0.435987\pi$$
0.199750 + 0.979847i $$0.435987\pi$$
$$402$$ 0 0
$$403$$ 8.00000 0.398508
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ −20.0000 −0.991363
$$408$$ 0 0
$$409$$ 12.0000 0.593362 0.296681 0.954977i $$-0.404120\pi$$
0.296681 + 0.954977i $$0.404120\pi$$
$$410$$ 0 0
$$411$$ 51.0000 2.51564
$$412$$ 0 0
$$413$$ −5.00000 −0.246034
$$414$$ 0 0
$$415$$ 0 0
$$416$$ 0 0
$$417$$ −48.0000 −2.35057
$$418$$ 0 0
$$419$$ −28.0000 −1.36789 −0.683945 0.729534i $$-0.739737\pi$$
−0.683945 + 0.729534i $$0.739737\pi$$
$$420$$ 0 0
$$421$$ 37.0000 1.80327 0.901635 0.432498i $$-0.142368\pi$$
0.901635 + 0.432498i $$0.142368\pi$$
$$422$$ 0 0
$$423$$ −48.0000 −2.33384
$$424$$ 0 0
$$425$$ −15.0000 −0.727607
$$426$$ 0 0
$$427$$ 10.0000 0.483934
$$428$$ 0 0
$$429$$ 6.00000 0.289683
$$430$$ 0 0
$$431$$ −26.0000 −1.25238 −0.626188 0.779672i $$-0.715386\pi$$
−0.626188 + 0.779672i $$0.715386\pi$$
$$432$$ 0 0
$$433$$ 10.0000 0.480569 0.240285 0.970702i $$-0.422759\pi$$
0.240285 + 0.970702i $$0.422759\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ 3.00000 0.143509
$$438$$ 0 0
$$439$$ 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
$$443$$ −6.00000 −0.285069 −0.142534 0.989790i $$-0.545525\pi$$
−0.142534 + 0.989790i $$0.545525\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ 0 0
$$447$$ 12.0000 0.567581
$$448$$ 0 0
$$449$$ −6.00000 −0.283158 −0.141579 0.989927i $$-0.545218\pi$$
−0.141579 + 0.989927i $$0.545218\pi$$
$$450$$ 0 0
$$451$$ 24.0000 1.13012
$$452$$ 0 0
$$453$$ −30.0000 −1.40952
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 1.00000 0.0467780 0.0233890 0.999726i $$-0.492554\pi$$
0.0233890 + 0.999726i $$0.492554\pi$$
$$458$$ 0 0
$$459$$ −27.0000 −1.26025
$$460$$ 0 0
$$461$$ 20.0000 0.931493 0.465746 0.884918i $$-0.345786\pi$$
0.465746 + 0.884918i $$0.345786\pi$$
$$462$$ 0 0
$$463$$ −20.0000 −0.929479 −0.464739 0.885448i $$-0.653852\pi$$
−0.464739 + 0.885448i $$0.653852\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ 6.00000 0.277647 0.138823 0.990317i $$-0.455668\pi$$
0.138823 + 0.990317i $$0.455668\pi$$
$$468$$ 0 0
$$469$$ 7.00000 0.323230
$$470$$ 0 0
$$471$$ 6.00000 0.276465
$$472$$ 0 0
$$473$$ 16.0000 0.735681
$$474$$ 0 0
$$475$$ −5.00000 −0.229416
$$476$$ 0 0
$$477$$ 54.0000 2.47249
$$478$$ 0 0
$$479$$ −28.0000 −1.27935 −0.639676 0.768644i $$-0.720932\pi$$
−0.639676 + 0.768644i $$0.720932\pi$$
$$480$$ 0 0
$$481$$ 10.0000 0.455961
$$482$$ 0 0
$$483$$ 9.00000 0.409514
$$484$$ 0 0
$$485$$ 0 0
$$486$$ 0 0
$$487$$ 18.0000 0.815658 0.407829 0.913058i $$-0.366286\pi$$
0.407829 + 0.913058i $$0.366286\pi$$
$$488$$ 0 0
$$489$$ 24.0000 1.08532
$$490$$ 0 0
$$491$$ −28.0000 −1.26362 −0.631811 0.775122i $$-0.717688\pi$$
−0.631811 + 0.775122i $$0.717688\pi$$
$$492$$ 0 0
$$493$$ −9.00000 −0.405340
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ 10.0000 0.448561
$$498$$ 0 0
$$499$$ −28.0000 −1.25345 −0.626726 0.779240i $$-0.715605\pi$$
−0.626726 + 0.779240i $$0.715605\pi$$
$$500$$ 0 0
$$501$$ 48.0000 2.14448
$$502$$ 0 0
$$503$$ 27.0000 1.20387 0.601935 0.798545i $$-0.294397\pi$$
0.601935 + 0.798545i $$0.294397\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 0 0
$$507$$ 36.0000 1.59882
$$508$$ 0 0
$$509$$ −34.0000 −1.50702 −0.753512 0.657434i $$-0.771642\pi$$
−0.753512 + 0.657434i $$0.771642\pi$$
$$510$$ 0 0
$$511$$ −1.00000 −0.0442374
$$512$$ 0 0
$$513$$ −9.00000 −0.397360
$$514$$ 0 0
$$515$$ 0 0
$$516$$ 0 0
$$517$$ 16.0000 0.703679
$$518$$ 0 0
$$519$$ −18.0000 −0.790112
$$520$$ 0 0
$$521$$ −16.0000 −0.700973 −0.350486 0.936568i $$-0.613984\pi$$
−0.350486 + 0.936568i $$0.613984\pi$$
$$522$$ 0 0
$$523$$ −1.00000 −0.0437269 −0.0218635 0.999761i $$-0.506960\pi$$
−0.0218635 + 0.999761i $$0.506960\pi$$
$$524$$ 0 0
$$525$$ −15.0000 −0.654654
$$526$$ 0 0
$$527$$ 24.0000 1.04546
$$528$$ 0 0
$$529$$ −14.0000 −0.608696
$$530$$ 0 0
$$531$$ 30.0000 1.30189
$$532$$ 0 0
$$533$$ −12.0000 −0.519778
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 0 0
$$537$$ 72.0000 3.10703
$$538$$ 0 0
$$539$$ 12.0000 0.516877
$$540$$ 0 0
$$541$$ −22.0000 −0.945854 −0.472927 0.881102i $$-0.656803\pi$$
−0.472927 + 0.881102i $$0.656803\pi$$
$$542$$ 0 0
$$543$$ −54.0000 −2.31736
$$544$$ 0 0
$$545$$ 0 0
$$546$$ 0 0
$$547$$ 28.0000 1.19719 0.598597 0.801050i $$-0.295725\pi$$
0.598597 + 0.801050i $$0.295725\pi$$
$$548$$ 0 0
$$549$$ −60.0000 −2.56074
$$550$$ 0 0
$$551$$ −3.00000 −0.127804
$$552$$ 0 0
$$553$$ 14.0000 0.595341
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ 32.0000 1.35588 0.677942 0.735116i $$-0.262872\pi$$
0.677942 + 0.735116i $$0.262872\pi$$
$$558$$ 0 0
$$559$$ −8.00000 −0.338364
$$560$$ 0 0
$$561$$ 18.0000 0.759961
$$562$$ 0 0
$$563$$ −4.00000 −0.168580 −0.0842900 0.996441i $$-0.526862\pi$$
−0.0842900 + 0.996441i $$0.526862\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ 0 0
$$567$$ −9.00000 −0.377964
$$568$$ 0 0
$$569$$ −36.0000 −1.50920 −0.754599 0.656186i $$-0.772169\pi$$
−0.754599 + 0.656186i $$0.772169\pi$$
$$570$$ 0 0
$$571$$ 32.0000 1.33916 0.669579 0.742741i $$-0.266474\pi$$
0.669579 + 0.742741i $$0.266474\pi$$
$$572$$ 0 0
$$573$$ 63.0000 2.63186
$$574$$ 0 0
$$575$$ −15.0000 −0.625543
$$576$$ 0 0
$$577$$ 43.0000 1.79011 0.895057 0.445952i $$-0.147135\pi$$
0.895057 + 0.445952i $$0.147135\pi$$
$$578$$ 0 0
$$579$$ −30.0000 −1.24676
$$580$$ 0 0
$$581$$ 6.00000 0.248922
$$582$$ 0 0
$$583$$ −18.0000 −0.745484
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ 4.00000 0.165098 0.0825488 0.996587i $$-0.473694\pi$$
0.0825488 + 0.996587i $$0.473694\pi$$
$$588$$ 0 0
$$589$$ 8.00000 0.329634
$$590$$ 0 0
$$591$$ −12.0000 −0.493614
$$592$$ 0 0
$$593$$ 34.0000 1.39621 0.698106 0.715994i $$-0.254026\pi$$
0.698106 + 0.715994i $$0.254026\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ −33.0000 −1.35060
$$598$$ 0 0
$$599$$ −40.0000 −1.63436 −0.817178 0.576386i $$-0.804463\pi$$
−0.817178 + 0.576386i $$0.804463\pi$$
$$600$$ 0 0
$$601$$ 40.0000 1.63163 0.815817 0.578310i $$-0.196288\pi$$
0.815817 + 0.578310i $$0.196288\pi$$
$$602$$ 0 0
$$603$$ −42.0000 −1.71037
$$604$$ 0 0
$$605$$ 0 0
$$606$$ 0 0
$$607$$ −2.00000 −0.0811775 −0.0405887 0.999176i $$-0.512923\pi$$
−0.0405887 + 0.999176i $$0.512923\pi$$
$$608$$ 0 0
$$609$$ −9.00000 −0.364698
$$610$$ 0 0
$$611$$ −8.00000 −0.323645
$$612$$ 0 0
$$613$$ −6.00000 −0.242338 −0.121169 0.992632i $$-0.538664\pi$$
−0.121169 + 0.992632i $$0.538664\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ 10.0000 0.402585 0.201292 0.979531i $$-0.435486\pi$$
0.201292 + 0.979531i $$0.435486\pi$$
$$618$$ 0 0
$$619$$ 14.0000 0.562708 0.281354 0.959604i $$-0.409217\pi$$
0.281354 + 0.959604i $$0.409217\pi$$
$$620$$ 0 0
$$621$$ −27.0000 −1.08347
$$622$$ 0 0
$$623$$ 4.00000 0.160257
$$624$$ 0 0
$$625$$ 25.0000 1.00000
$$626$$ 0 0
$$627$$ 6.00000 0.239617
$$628$$ 0 0
$$629$$ 30.0000 1.19618
$$630$$ 0 0
$$631$$ −40.0000 −1.59237 −0.796187 0.605050i $$-0.793153\pi$$
−0.796187 + 0.605050i $$0.793153\pi$$
$$632$$ 0 0
$$633$$ −3.00000 −0.119239
$$634$$ 0 0
$$635$$ 0 0
$$636$$ 0 0
$$637$$ −6.00000 −0.237729
$$638$$ 0 0
$$639$$ −60.0000 −2.37356
$$640$$ 0 0
$$641$$ 46.0000 1.81689 0.908445 0.418004i $$-0.137270\pi$$
0.908445 + 0.418004i $$0.137270\pi$$
$$642$$ 0 0
$$643$$ −50.0000 −1.97181 −0.985904 0.167313i $$-0.946491\pi$$
−0.985904 + 0.167313i $$0.946491\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ 3.00000 0.117942 0.0589711 0.998260i $$-0.481218\pi$$
0.0589711 + 0.998260i $$0.481218\pi$$
$$648$$ 0 0
$$649$$ −10.0000 −0.392534
$$650$$ 0 0
$$651$$ 24.0000 0.940634
$$652$$ 0 0
$$653$$ 4.00000 0.156532 0.0782660 0.996933i $$-0.475062\pi$$
0.0782660 + 0.996933i $$0.475062\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 0 0
$$657$$ 6.00000 0.234082
$$658$$ 0 0
$$659$$ 31.0000 1.20759 0.603794 0.797140i $$-0.293655\pi$$
0.603794 + 0.797140i $$0.293655\pi$$
$$660$$ 0 0
$$661$$ 39.0000 1.51692 0.758462 0.651717i $$-0.225951\pi$$
0.758462 + 0.651717i $$0.225951\pi$$
$$662$$ 0 0
$$663$$ −9.00000 −0.349531
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ −9.00000 −0.348481
$$668$$ 0 0
$$669$$ −6.00000 −0.231973
$$670$$ 0 0
$$671$$ 20.0000 0.772091
$$672$$ 0 0
$$673$$ −32.0000 −1.23351 −0.616755 0.787155i $$-0.711553\pi$$
−0.616755 + 0.787155i $$0.711553\pi$$
$$674$$ 0 0
$$675$$ 45.0000 1.73205
$$676$$ 0 0
$$677$$ 11.0000 0.422764 0.211382 0.977403i $$-0.432204\pi$$
0.211382 + 0.977403i $$0.432204\pi$$
$$678$$ 0 0
$$679$$ 6.00000 0.230259
$$680$$ 0 0
$$681$$ −63.0000 −2.41417
$$682$$ 0 0
$$683$$ −36.0000 −1.37750 −0.688751 0.724998i $$-0.741841\pi$$
−0.688751 + 0.724998i $$0.741841\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ 0 0
$$687$$ 66.0000 2.51806
$$688$$ 0 0
$$689$$ 9.00000 0.342873
$$690$$ 0 0
$$691$$ −18.0000 −0.684752 −0.342376 0.939563i $$-0.611232\pi$$
−0.342376 + 0.939563i $$0.611232\pi$$
$$692$$ 0 0
$$693$$ 12.0000 0.455842
$$694$$ 0 0
$$695$$ 0 0
$$696$$ 0 0
$$697$$ −36.0000 −1.36360
$$698$$ 0 0
$$699$$ −78.0000 −2.95023
$$700$$ 0 0
$$701$$ −36.0000 −1.35970 −0.679851 0.733351i $$-0.737955\pi$$
−0.679851 + 0.733351i $$0.737955\pi$$
$$702$$ 0 0
$$703$$ 10.0000 0.377157
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ 10.0000 0.376089
$$708$$ 0 0
$$709$$ 22.0000 0.826227 0.413114 0.910679i $$-0.364441\pi$$
0.413114 + 0.910679i $$0.364441\pi$$
$$710$$ 0 0
$$711$$ −84.0000 −3.15025
$$712$$ 0 0
$$713$$ 24.0000 0.898807
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 0 0
$$717$$ −81.0000 −3.02500
$$718$$ 0 0
$$719$$ 15.0000 0.559406 0.279703 0.960087i $$-0.409764\pi$$
0.279703 + 0.960087i $$0.409764\pi$$
$$720$$ 0 0
$$721$$ −14.0000 −0.521387
$$722$$ 0 0
$$723$$ −60.0000 −2.23142
$$724$$ 0 0
$$725$$ 15.0000 0.557086
$$726$$ 0 0
$$727$$ 11.0000 0.407967 0.203984 0.978974i $$-0.434611\pi$$
0.203984 + 0.978974i $$0.434611\pi$$
$$728$$ 0 0
$$729$$ −27.0000 −1.00000
$$730$$ 0 0
$$731$$ −24.0000 −0.887672
$$732$$ 0 0
$$733$$ 24.0000 0.886460 0.443230 0.896408i $$-0.353832\pi$$
0.443230 + 0.896408i $$0.353832\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ 14.0000 0.515697
$$738$$ 0 0
$$739$$ 32.0000 1.17714 0.588570 0.808447i $$-0.299691\pi$$
0.588570 + 0.808447i $$0.299691\pi$$
$$740$$ 0 0
$$741$$ −3.00000 −0.110208
$$742$$ 0 0
$$743$$ −40.0000 −1.46746 −0.733729 0.679442i $$-0.762222\pi$$
−0.733729 + 0.679442i $$0.762222\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 0 0
$$747$$ −36.0000 −1.31717
$$748$$ 0 0
$$749$$ 5.00000 0.182696
$$750$$ 0 0
$$751$$ 20.0000 0.729810 0.364905 0.931045i $$-0.381101\pi$$
0.364905 + 0.931045i $$0.381101\pi$$
$$752$$ 0 0
$$753$$ −6.00000 −0.218652
$$754$$ 0 0
$$755$$ 0 0
$$756$$ 0 0
$$757$$ −38.0000 −1.38113 −0.690567 0.723269i $$-0.742639\pi$$
−0.690567 + 0.723269i $$0.742639\pi$$
$$758$$ 0 0
$$759$$ 18.0000 0.653359
$$760$$ 0 0
$$761$$ −29.0000 −1.05125 −0.525625 0.850717i $$-0.676168\pi$$
−0.525625 + 0.850717i $$0.676168\pi$$
$$762$$ 0 0
$$763$$ 9.00000 0.325822
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ 5.00000 0.180540
$$768$$ 0 0
$$769$$ 13.0000 0.468792 0.234396 0.972141i $$-0.424689\pi$$
0.234396 + 0.972141i $$0.424689\pi$$
$$770$$ 0 0
$$771$$ 36.0000 1.29651
$$772$$ 0 0
$$773$$ 31.0000 1.11499 0.557496 0.830179i $$-0.311762\pi$$
0.557496 + 0.830179i $$0.311762\pi$$
$$774$$ 0 0
$$775$$ −40.0000 −1.43684
$$776$$ 0 0
$$777$$ 30.0000 1.07624
$$778$$ 0 0
$$779$$ −12.0000 −0.429945
$$780$$ 0 0
$$781$$ 20.0000 0.715656
$$782$$ 0 0
$$783$$ 27.0000 0.964901
$$784$$ 0 0
$$785$$ 0 0
$$786$$ 0 0
$$787$$ 53.0000 1.88925 0.944623 0.328158i $$-0.106428\pi$$
0.944623 + 0.328158i $$0.106428\pi$$
$$788$$ 0 0
$$789$$ 48.0000 1.70885
$$790$$ 0 0
$$791$$ −2.00000 −0.0711118
$$792$$ 0 0
$$793$$ −10.0000 −0.355110
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ 37.0000 1.31061 0.655304 0.755366i $$-0.272541\pi$$
0.655304 + 0.755366i $$0.272541\pi$$
$$798$$ 0 0
$$799$$ −24.0000 −0.849059
$$800$$ 0 0
$$801$$ −24.0000 −0.847998
$$802$$ 0 0
$$803$$ −2.00000 −0.0705785
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ 66.0000 2.32331
$$808$$ 0 0
$$809$$ −15.0000 −0.527372 −0.263686 0.964609i $$-0.584938\pi$$
−0.263686 + 0.964609i $$0.584938\pi$$
$$810$$ 0 0
$$811$$ 39.0000 1.36948 0.684738 0.728790i $$-0.259917\pi$$
0.684738 + 0.728790i $$0.259917\pi$$
$$812$$ 0 0
$$813$$ −9.00000 −0.315644
$$814$$ 0 0
$$815$$ 0 0
$$816$$ 0 0
$$817$$ −8.00000 −0.279885
$$818$$ 0 0
$$819$$ −6.00000 −0.209657
$$820$$ 0 0
$$821$$ 12.0000 0.418803 0.209401 0.977830i $$-0.432848\pi$$
0.209401 + 0.977830i $$0.432848\pi$$
$$822$$ 0 0
$$823$$ 41.0000 1.42917 0.714585 0.699549i $$-0.246616\pi$$
0.714585 + 0.699549i $$0.246616\pi$$
$$824$$ 0 0
$$825$$ −30.0000 −1.04447
$$826$$ 0 0
$$827$$ −51.0000 −1.77344 −0.886722 0.462303i $$-0.847023\pi$$
−0.886722 + 0.462303i $$0.847023\pi$$
$$828$$ 0 0
$$829$$ 23.0000 0.798823 0.399412 0.916772i $$-0.369214\pi$$
0.399412 + 0.916772i $$0.369214\pi$$
$$830$$ 0 0
$$831$$ 36.0000 1.24883
$$832$$ 0 0
$$833$$ −18.0000 −0.623663
$$834$$ 0 0
$$835$$ 0 0
$$836$$ 0 0
$$837$$ −72.0000 −2.48868
$$838$$ 0 0
$$839$$ −24.0000 −0.828572 −0.414286 0.910147i $$-0.635969\pi$$
−0.414286 + 0.910147i $$0.635969\pi$$
$$840$$ 0 0
$$841$$ −20.0000 −0.689655
$$842$$ 0 0
$$843$$ 0 0
$$844$$ 0 0
$$845$$ 0 0
$$846$$ 0 0
$$847$$ 7.00000 0.240523
$$848$$ 0 0
$$849$$ 18.0000 0.617758
$$850$$ 0 0
$$851$$ 30.0000 1.02839
$$852$$ 0 0
$$853$$ 2.00000 0.0684787 0.0342393 0.999414i $$-0.489099\pi$$
0.0342393 + 0.999414i $$0.489099\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ 20.0000 0.683187 0.341593 0.939848i $$-0.389033\pi$$
0.341593 + 0.939848i $$0.389033\pi$$
$$858$$ 0 0
$$859$$ 22.0000 0.750630 0.375315 0.926897i $$-0.377534\pi$$
0.375315 + 0.926897i $$0.377534\pi$$
$$860$$ 0 0
$$861$$ −36.0000 −1.22688
$$862$$ 0 0
$$863$$ 38.0000 1.29354 0.646768 0.762687i $$-0.276120\pi$$
0.646768 + 0.762687i $$0.276120\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ 0 0
$$867$$ 24.0000 0.815083
$$868$$ 0 0
$$869$$ 28.0000 0.949835
$$870$$ 0 0
$$871$$ −7.00000 −0.237186
$$872$$ 0 0
$$873$$ −36.0000 −1.21842
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 0 0
$$877$$ −37.0000 −1.24940 −0.624701 0.780864i $$-0.714779\pi$$
−0.624701 + 0.780864i $$0.714779\pi$$
$$878$$ 0 0
$$879$$ 3.00000 0.101187
$$880$$ 0 0
$$881$$ −18.0000 −0.606435 −0.303218 0.952921i $$-0.598061\pi$$
−0.303218 + 0.952921i $$0.598061\pi$$
$$882$$ 0 0
$$883$$ −34.0000 −1.14419 −0.572096 0.820187i $$-0.693869\pi$$
−0.572096 + 0.820187i $$0.693869\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ −10.0000 −0.335767 −0.167884 0.985807i $$-0.553693\pi$$
−0.167884 + 0.985807i $$0.553693\pi$$
$$888$$ 0 0
$$889$$ −18.0000 −0.603701
$$890$$ 0 0
$$891$$ −18.0000 −0.603023
$$892$$ 0 0
$$893$$ −8.00000 −0.267710
$$894$$ 0 0
$$895$$ 0 0
$$896$$ 0 0
$$897$$ −9.00000 −0.300501
$$898$$ 0 0
$$899$$ −24.0000 −0.800445
$$900$$ 0 0
$$901$$ 27.0000 0.899500
$$902$$ 0 0
$$903$$ −24.0000 −0.798670
$$904$$ 0 0
$$905$$ 0 0
$$906$$ 0 0
$$907$$ −33.0000 −1.09575 −0.547874 0.836561i $$-0.684562\pi$$
−0.547874 + 0.836561i $$0.684562\pi$$
$$908$$ 0 0
$$909$$ −60.0000 −1.99007
$$910$$ 0 0
$$911$$ 24.0000 0.795155 0.397578 0.917568i $$-0.369851\pi$$
0.397578 + 0.917568i $$0.369851\pi$$
$$912$$ 0 0
$$913$$ 12.0000 0.397142
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 0 0
$$917$$ −8.00000 −0.264183
$$918$$ 0 0
$$919$$ 49.0000 1.61636 0.808180 0.588935i $$-0.200453\pi$$
0.808180 + 0.588935i $$0.200453\pi$$
$$920$$ 0 0
$$921$$ 36.0000 1.18624
$$922$$ 0 0
$$923$$ −10.0000 −0.329154
$$924$$ 0 0
$$925$$ −50.0000 −1.64399
$$926$$ 0 0
$$927$$ 84.0000 2.75892
$$928$$ 0 0
$$929$$ −39.0000 −1.27955 −0.639774 0.768563i $$-0.720972\pi$$
−0.639774 + 0.768563i $$0.720972\pi$$
$$930$$ 0 0
$$931$$ −6.00000 −0.196642
$$932$$ 0 0
$$933$$ 15.0000 0.491078
$$934$$ 0 0
$$935$$ 0 0
$$936$$ 0 0
$$937$$ −23.0000 −0.751377 −0.375689 0.926746i $$-0.622594\pi$$
−0.375689 + 0.926746i $$0.622594\pi$$
$$938$$ 0 0
$$939$$ 9.00000 0.293704
$$940$$ 0 0
$$941$$ 33.0000 1.07577 0.537885 0.843018i $$-0.319224\pi$$
0.537885 + 0.843018i $$0.319224\pi$$
$$942$$ 0 0
$$943$$ −36.0000 −1.17232
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ 16.0000 0.519930 0.259965 0.965618i $$-0.416289\pi$$
0.259965 + 0.965618i $$0.416289\pi$$
$$948$$ 0 0
$$949$$ 1.00000 0.0324614
$$950$$ 0 0
$$951$$ 63.0000 2.04291
$$952$$ 0 0
$$953$$ −2.00000 −0.0647864 −0.0323932 0.999475i $$-0.510313\pi$$
−0.0323932 + 0.999475i $$0.510313\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ 0 0
$$957$$ −18.0000 −0.581857
$$958$$ 0 0
$$959$$ 17.0000 0.548959
$$960$$ 0 0
$$961$$ 33.0000 1.06452
$$962$$ 0 0
$$963$$ −30.0000 −0.966736
$$964$$ 0 0
$$965$$ 0 0
$$966$$ 0 0
$$967$$ −8.00000 −0.257263 −0.128631 0.991692i $$-0.541058\pi$$
−0.128631 + 0.991692i $$0.541058\pi$$
$$968$$ 0 0
$$969$$ −9.00000 −0.289122
$$970$$ 0 0
$$971$$ −44.0000 −1.41203 −0.706014 0.708198i $$-0.749508\pi$$
−0.706014 + 0.708198i $$0.749508\pi$$
$$972$$ 0 0
$$973$$ −16.0000 −0.512936
$$974$$ 0 0
$$975$$ 15.0000 0.480384
$$976$$ 0 0
$$977$$ 16.0000 0.511885 0.255943 0.966692i $$-0.417614\pi$$
0.255943 + 0.966692i $$0.417614\pi$$
$$978$$ 0 0
$$979$$ 8.00000 0.255681
$$980$$ 0 0
$$981$$ −54.0000 −1.72409
$$982$$ 0 0
$$983$$ −14.0000 −0.446531 −0.223265 0.974758i $$-0.571672\pi$$
−0.223265 + 0.974758i $$0.571672\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ 0 0
$$987$$ −24.0000 −0.763928
$$988$$ 0 0
$$989$$ −24.0000 −0.763156
$$990$$ 0 0
$$991$$ −28.0000 −0.889449 −0.444725 0.895667i $$-0.646698\pi$$
−0.444725 + 0.895667i $$0.646698\pi$$
$$992$$ 0 0
$$993$$ 39.0000 1.23763
$$994$$ 0 0
$$995$$ 0 0
$$996$$ 0 0
$$997$$ 20.0000 0.633406 0.316703 0.948525i $$-0.397424\pi$$
0.316703 + 0.948525i $$0.397424\pi$$
$$998$$ 0 0
$$999$$ −90.0000 −2.84747
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 1216.2.a.a.1.1 1
4.3 odd 2 1216.2.a.r.1.1 1
8.3 odd 2 608.2.a.a.1.1 1
8.5 even 2 608.2.a.f.1.1 yes 1
24.5 odd 2 5472.2.a.i.1.1 1
24.11 even 2 5472.2.a.l.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
608.2.a.a.1.1 1 8.3 odd 2
608.2.a.f.1.1 yes 1 8.5 even 2
1216.2.a.a.1.1 1 1.1 even 1 trivial
1216.2.a.r.1.1 1 4.3 odd 2
5472.2.a.i.1.1 1 24.5 odd 2
5472.2.a.l.1.1 1 24.11 even 2