# Properties

 Label 5376.2.c.m.2689.1 Level $5376$ Weight $2$ Character 5376.2689 Analytic conductor $42.928$ Analytic rank $1$ Dimension $2$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5376,2,Mod(2689,5376)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5376, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1, 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("5376.2689");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$5376 = 2^{8} \cdot 3 \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 5376.c (of order $$2$$, degree $$1$$, not minimal)

## 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: no Analytic conductor: $$42.9275761266$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(i)$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} + 1$$ x^2 + 1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 672) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 2689.1 Root $$1.00000i$$ of defining polynomial Character $$\chi$$ $$=$$ 5376.2689 Dual form 5376.2.c.m.2689.2

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-1.00000i q^{3} +2.00000i q^{5} -1.00000 q^{7} -1.00000 q^{9} +O(q^{10})$$ $$q-1.00000i q^{3} +2.00000i q^{5} -1.00000 q^{7} -1.00000 q^{9} -4.00000i q^{11} -6.00000i q^{13} +2.00000 q^{15} -2.00000 q^{17} -4.00000i q^{19} +1.00000i q^{21} -4.00000 q^{23} +1.00000 q^{25} +1.00000i q^{27} -2.00000i q^{29} -8.00000 q^{31} -4.00000 q^{33} -2.00000i q^{35} +10.0000i q^{37} -6.00000 q^{39} +2.00000 q^{41} +8.00000i q^{43} -2.00000i q^{45} +1.00000 q^{49} +2.00000i q^{51} +10.0000i q^{53} +8.00000 q^{55} -4.00000 q^{57} -12.0000i q^{59} +10.0000i q^{61} +1.00000 q^{63} +12.0000 q^{65} +8.00000i q^{67} +4.00000i q^{69} +12.0000 q^{71} -2.00000 q^{73} -1.00000i q^{75} +4.00000i q^{77} +1.00000 q^{81} -12.0000i q^{83} -4.00000i q^{85} -2.00000 q^{87} -6.00000 q^{89} +6.00000i q^{91} +8.00000i q^{93} +8.00000 q^{95} +2.00000 q^{97} +4.00000i q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{7} - 2 q^{9}+O(q^{10})$$ 2 * q - 2 * q^7 - 2 * q^9 $$2 q - 2 q^{7} - 2 q^{9} + 4 q^{15} - 4 q^{17} - 8 q^{23} + 2 q^{25} - 16 q^{31} - 8 q^{33} - 12 q^{39} + 4 q^{41} + 2 q^{49} + 16 q^{55} - 8 q^{57} + 2 q^{63} + 24 q^{65} + 24 q^{71} - 4 q^{73} + 2 q^{81} - 4 q^{87} - 12 q^{89} + 16 q^{95} + 4 q^{97}+O(q^{100})$$ 2 * q - 2 * q^7 - 2 * q^9 + 4 * q^15 - 4 * q^17 - 8 * q^23 + 2 * q^25 - 16 * q^31 - 8 * q^33 - 12 * q^39 + 4 * q^41 + 2 * q^49 + 16 * q^55 - 8 * q^57 + 2 * q^63 + 24 * q^65 + 24 * q^71 - 4 * q^73 + 2 * q^81 - 4 * q^87 - 12 * q^89 + 16 * q^95 + 4 * q^97

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/5376\mathbb{Z}\right)^\times$$.

 $$n$$ $$1793$$ $$2815$$ $$4609$$ $$5125$$ $$\chi(n)$$ $$1$$ $$1$$ $$1$$ $$-1$$

## 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
$$3$$ − 1.00000i − 0.577350i
$$4$$ 0 0
$$5$$ 2.00000i 0.894427i 0.894427 + 0.447214i $$0.147584\pi$$
−0.894427 + 0.447214i $$0.852416\pi$$
$$6$$ 0 0
$$7$$ −1.00000 −0.377964
$$8$$ 0 0
$$9$$ −1.00000 −0.333333
$$10$$ 0 0
$$11$$ − 4.00000i − 1.20605i −0.797724 0.603023i $$-0.793963\pi$$
0.797724 0.603023i $$-0.206037\pi$$
$$12$$ 0 0
$$13$$ − 6.00000i − 1.66410i −0.554700 0.832050i $$-0.687167\pi$$
0.554700 0.832050i $$-0.312833\pi$$
$$14$$ 0 0
$$15$$ 2.00000 0.516398
$$16$$ 0 0
$$17$$ −2.00000 −0.485071 −0.242536 0.970143i $$-0.577979\pi$$
−0.242536 + 0.970143i $$0.577979\pi$$
$$18$$ 0 0
$$19$$ − 4.00000i − 0.917663i −0.888523 0.458831i $$-0.848268\pi$$
0.888523 0.458831i $$-0.151732\pi$$
$$20$$ 0 0
$$21$$ 1.00000i 0.218218i
$$22$$ 0 0
$$23$$ −4.00000 −0.834058 −0.417029 0.908893i $$-0.636929\pi$$
−0.417029 + 0.908893i $$0.636929\pi$$
$$24$$ 0 0
$$25$$ 1.00000 0.200000
$$26$$ 0 0
$$27$$ 1.00000i 0.192450i
$$28$$ 0 0
$$29$$ − 2.00000i − 0.371391i −0.982607 0.185695i $$-0.940546\pi$$
0.982607 0.185695i $$-0.0594537\pi$$
$$30$$ 0 0
$$31$$ −8.00000 −1.43684 −0.718421 0.695608i $$-0.755135\pi$$
−0.718421 + 0.695608i $$0.755135\pi$$
$$32$$ 0 0
$$33$$ −4.00000 −0.696311
$$34$$ 0 0
$$35$$ − 2.00000i − 0.338062i
$$36$$ 0 0
$$37$$ 10.0000i 1.64399i 0.569495 + 0.821995i $$0.307139\pi$$
−0.569495 + 0.821995i $$0.692861\pi$$
$$38$$ 0 0
$$39$$ −6.00000 −0.960769
$$40$$ 0 0
$$41$$ 2.00000 0.312348 0.156174 0.987730i $$-0.450084\pi$$
0.156174 + 0.987730i $$0.450084\pi$$
$$42$$ 0 0
$$43$$ 8.00000i 1.21999i 0.792406 + 0.609994i $$0.208828\pi$$
−0.792406 + 0.609994i $$0.791172\pi$$
$$44$$ 0 0
$$45$$ − 2.00000i − 0.298142i
$$46$$ 0 0
$$47$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$48$$ 0 0
$$49$$ 1.00000 0.142857
$$50$$ 0 0
$$51$$ 2.00000i 0.280056i
$$52$$ 0 0
$$53$$ 10.0000i 1.37361i 0.726844 + 0.686803i $$0.240986\pi$$
−0.726844 + 0.686803i $$0.759014\pi$$
$$54$$ 0 0
$$55$$ 8.00000 1.07872
$$56$$ 0 0
$$57$$ −4.00000 −0.529813
$$58$$ 0 0
$$59$$ − 12.0000i − 1.56227i −0.624364 0.781133i $$-0.714642\pi$$
0.624364 0.781133i $$-0.285358\pi$$
$$60$$ 0 0
$$61$$ 10.0000i 1.28037i 0.768221 + 0.640184i $$0.221142\pi$$
−0.768221 + 0.640184i $$0.778858\pi$$
$$62$$ 0 0
$$63$$ 1.00000 0.125988
$$64$$ 0 0
$$65$$ 12.0000 1.48842
$$66$$ 0 0
$$67$$ 8.00000i 0.977356i 0.872464 + 0.488678i $$0.162521\pi$$
−0.872464 + 0.488678i $$0.837479\pi$$
$$68$$ 0 0
$$69$$ 4.00000i 0.481543i
$$70$$ 0 0
$$71$$ 12.0000 1.42414 0.712069 0.702109i $$-0.247758\pi$$
0.712069 + 0.702109i $$0.247758\pi$$
$$72$$ 0 0
$$73$$ −2.00000 −0.234082 −0.117041 0.993127i $$-0.537341\pi$$
−0.117041 + 0.993127i $$0.537341\pi$$
$$74$$ 0 0
$$75$$ − 1.00000i − 0.115470i
$$76$$ 0 0
$$77$$ 4.00000i 0.455842i
$$78$$ 0 0
$$79$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ 0 0
$$83$$ − 12.0000i − 1.31717i −0.752506 0.658586i $$-0.771155\pi$$
0.752506 0.658586i $$-0.228845\pi$$
$$84$$ 0 0
$$85$$ − 4.00000i − 0.433861i
$$86$$ 0 0
$$87$$ −2.00000 −0.214423
$$88$$ 0 0
$$89$$ −6.00000 −0.635999 −0.317999 0.948091i $$-0.603011\pi$$
−0.317999 + 0.948091i $$0.603011\pi$$
$$90$$ 0 0
$$91$$ 6.00000i 0.628971i
$$92$$ 0 0
$$93$$ 8.00000i 0.829561i
$$94$$ 0 0
$$95$$ 8.00000 0.820783
$$96$$ 0 0
$$97$$ 2.00000 0.203069 0.101535 0.994832i $$-0.467625\pi$$
0.101535 + 0.994832i $$0.467625\pi$$
$$98$$ 0 0
$$99$$ 4.00000i 0.402015i
$$100$$ 0 0
$$101$$ 10.0000i 0.995037i 0.867453 + 0.497519i $$0.165755\pi$$
−0.867453 + 0.497519i $$0.834245\pi$$
$$102$$ 0 0
$$103$$ −8.00000 −0.788263 −0.394132 0.919054i $$-0.628955\pi$$
−0.394132 + 0.919054i $$0.628955\pi$$
$$104$$ 0 0
$$105$$ −2.00000 −0.195180
$$106$$ 0 0
$$107$$ − 12.0000i − 1.16008i −0.814587 0.580042i $$-0.803036\pi$$
0.814587 0.580042i $$-0.196964\pi$$
$$108$$ 0 0
$$109$$ − 2.00000i − 0.191565i −0.995402 0.0957826i $$-0.969465\pi$$
0.995402 0.0957826i $$-0.0305354\pi$$
$$110$$ 0 0
$$111$$ 10.0000 0.949158
$$112$$ 0 0
$$113$$ −14.0000 −1.31701 −0.658505 0.752577i $$-0.728811\pi$$
−0.658505 + 0.752577i $$0.728811\pi$$
$$114$$ 0 0
$$115$$ − 8.00000i − 0.746004i
$$116$$ 0 0
$$117$$ 6.00000i 0.554700i
$$118$$ 0 0
$$119$$ 2.00000 0.183340
$$120$$ 0 0
$$121$$ −5.00000 −0.454545
$$122$$ 0 0
$$123$$ − 2.00000i − 0.180334i
$$124$$ 0 0
$$125$$ 12.0000i 1.07331i
$$126$$ 0 0
$$127$$ −8.00000 −0.709885 −0.354943 0.934888i $$-0.615500\pi$$
−0.354943 + 0.934888i $$0.615500\pi$$
$$128$$ 0 0
$$129$$ 8.00000 0.704361
$$130$$ 0 0
$$131$$ 20.0000i 1.74741i 0.486458 + 0.873704i $$0.338289\pi$$
−0.486458 + 0.873704i $$0.661711\pi$$
$$132$$ 0 0
$$133$$ 4.00000i 0.346844i
$$134$$ 0 0
$$135$$ −2.00000 −0.172133
$$136$$ 0 0
$$137$$ −18.0000 −1.53784 −0.768922 0.639343i $$-0.779207\pi$$
−0.768922 + 0.639343i $$0.779207\pi$$
$$138$$ 0 0
$$139$$ 20.0000i 1.69638i 0.529694 + 0.848189i $$0.322307\pi$$
−0.529694 + 0.848189i $$0.677693\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 0 0
$$143$$ −24.0000 −2.00698
$$144$$ 0 0
$$145$$ 4.00000 0.332182
$$146$$ 0 0
$$147$$ − 1.00000i − 0.0824786i
$$148$$ 0 0
$$149$$ − 6.00000i − 0.491539i −0.969328 0.245770i $$-0.920959\pi$$
0.969328 0.245770i $$-0.0790407\pi$$
$$150$$ 0 0
$$151$$ −16.0000 −1.30206 −0.651031 0.759051i $$-0.725663\pi$$
−0.651031 + 0.759051i $$0.725663\pi$$
$$152$$ 0 0
$$153$$ 2.00000 0.161690
$$154$$ 0 0
$$155$$ − 16.0000i − 1.28515i
$$156$$ 0 0
$$157$$ 2.00000i 0.159617i 0.996810 + 0.0798087i $$0.0254309\pi$$
−0.996810 + 0.0798087i $$0.974569\pi$$
$$158$$ 0 0
$$159$$ 10.0000 0.793052
$$160$$ 0 0
$$161$$ 4.00000 0.315244
$$162$$ 0 0
$$163$$ − 16.0000i − 1.25322i −0.779334 0.626608i $$-0.784443\pi$$
0.779334 0.626608i $$-0.215557\pi$$
$$164$$ 0 0
$$165$$ − 8.00000i − 0.622799i
$$166$$ 0 0
$$167$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$168$$ 0 0
$$169$$ −23.0000 −1.76923
$$170$$ 0 0
$$171$$ 4.00000i 0.305888i
$$172$$ 0 0
$$173$$ 6.00000i 0.456172i 0.973641 + 0.228086i $$0.0732467\pi$$
−0.973641 + 0.228086i $$0.926753\pi$$
$$174$$ 0 0
$$175$$ −1.00000 −0.0755929
$$176$$ 0 0
$$177$$ −12.0000 −0.901975
$$178$$ 0 0
$$179$$ 20.0000i 1.49487i 0.664335 + 0.747435i $$0.268715\pi$$
−0.664335 + 0.747435i $$0.731285\pi$$
$$180$$ 0 0
$$181$$ − 2.00000i − 0.148659i −0.997234 0.0743294i $$-0.976318\pi$$
0.997234 0.0743294i $$-0.0236816\pi$$
$$182$$ 0 0
$$183$$ 10.0000 0.739221
$$184$$ 0 0
$$185$$ −20.0000 −1.47043
$$186$$ 0 0
$$187$$ 8.00000i 0.585018i
$$188$$ 0 0
$$189$$ − 1.00000i − 0.0727393i
$$190$$ 0 0
$$191$$ −12.0000 −0.868290 −0.434145 0.900843i $$-0.642949\pi$$
−0.434145 + 0.900843i $$0.642949\pi$$
$$192$$ 0 0
$$193$$ −14.0000 −1.00774 −0.503871 0.863779i $$-0.668091\pi$$
−0.503871 + 0.863779i $$0.668091\pi$$
$$194$$ 0 0
$$195$$ − 12.0000i − 0.859338i
$$196$$ 0 0
$$197$$ − 6.00000i − 0.427482i −0.976890 0.213741i $$-0.931435\pi$$
0.976890 0.213741i $$-0.0685649\pi$$
$$198$$ 0 0
$$199$$ 16.0000 1.13421 0.567105 0.823646i $$-0.308063\pi$$
0.567105 + 0.823646i $$0.308063\pi$$
$$200$$ 0 0
$$201$$ 8.00000 0.564276
$$202$$ 0 0
$$203$$ 2.00000i 0.140372i
$$204$$ 0 0
$$205$$ 4.00000i 0.279372i
$$206$$ 0 0
$$207$$ 4.00000 0.278019
$$208$$ 0 0
$$209$$ −16.0000 −1.10674
$$210$$ 0 0
$$211$$ 16.0000i 1.10149i 0.834675 + 0.550743i $$0.185655\pi$$
−0.834675 + 0.550743i $$0.814345\pi$$
$$212$$ 0 0
$$213$$ − 12.0000i − 0.822226i
$$214$$ 0 0
$$215$$ −16.0000 −1.09119
$$216$$ 0 0
$$217$$ 8.00000 0.543075
$$218$$ 0 0
$$219$$ 2.00000i 0.135147i
$$220$$ 0 0
$$221$$ 12.0000i 0.807207i
$$222$$ 0 0
$$223$$ 8.00000 0.535720 0.267860 0.963458i $$-0.413684\pi$$
0.267860 + 0.963458i $$0.413684\pi$$
$$224$$ 0 0
$$225$$ −1.00000 −0.0666667
$$226$$ 0 0
$$227$$ − 20.0000i − 1.32745i −0.747978 0.663723i $$-0.768975\pi$$
0.747978 0.663723i $$-0.231025\pi$$
$$228$$ 0 0
$$229$$ − 18.0000i − 1.18947i −0.803921 0.594737i $$-0.797256\pi$$
0.803921 0.594737i $$-0.202744\pi$$
$$230$$ 0 0
$$231$$ 4.00000 0.263181
$$232$$ 0 0
$$233$$ 14.0000 0.917170 0.458585 0.888650i $$-0.348356\pi$$
0.458585 + 0.888650i $$0.348356\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ 0 0
$$237$$ 0 0
$$238$$ 0 0
$$239$$ −12.0000 −0.776215 −0.388108 0.921614i $$-0.626871\pi$$
−0.388108 + 0.921614i $$0.626871\pi$$
$$240$$ 0 0
$$241$$ −30.0000 −1.93247 −0.966235 0.257663i $$-0.917048\pi$$
−0.966235 + 0.257663i $$0.917048\pi$$
$$242$$ 0 0
$$243$$ − 1.00000i − 0.0641500i
$$244$$ 0 0
$$245$$ 2.00000i 0.127775i
$$246$$ 0 0
$$247$$ −24.0000 −1.52708
$$248$$ 0 0
$$249$$ −12.0000 −0.760469
$$250$$ 0 0
$$251$$ − 4.00000i − 0.252478i −0.992000 0.126239i $$-0.959709\pi$$
0.992000 0.126239i $$-0.0402906\pi$$
$$252$$ 0 0
$$253$$ 16.0000i 1.00591i
$$254$$ 0 0
$$255$$ −4.00000 −0.250490
$$256$$ 0 0
$$257$$ −18.0000 −1.12281 −0.561405 0.827541i $$-0.689739\pi$$
−0.561405 + 0.827541i $$0.689739\pi$$
$$258$$ 0 0
$$259$$ − 10.0000i − 0.621370i
$$260$$ 0 0
$$261$$ 2.00000i 0.123797i
$$262$$ 0 0
$$263$$ −12.0000 −0.739952 −0.369976 0.929041i $$-0.620634\pi$$
−0.369976 + 0.929041i $$0.620634\pi$$
$$264$$ 0 0
$$265$$ −20.0000 −1.22859
$$266$$ 0 0
$$267$$ 6.00000i 0.367194i
$$268$$ 0 0
$$269$$ − 2.00000i − 0.121942i −0.998140 0.0609711i $$-0.980580\pi$$
0.998140 0.0609711i $$-0.0194197\pi$$
$$270$$ 0 0
$$271$$ 32.0000 1.94386 0.971931 0.235267i $$-0.0755965\pi$$
0.971931 + 0.235267i $$0.0755965\pi$$
$$272$$ 0 0
$$273$$ 6.00000 0.363137
$$274$$ 0 0
$$275$$ − 4.00000i − 0.241209i
$$276$$ 0 0
$$277$$ 18.0000i 1.08152i 0.841178 + 0.540758i $$0.181862\pi$$
−0.841178 + 0.540758i $$0.818138\pi$$
$$278$$ 0 0
$$279$$ 8.00000 0.478947
$$280$$ 0 0
$$281$$ 30.0000 1.78965 0.894825 0.446417i $$-0.147300\pi$$
0.894825 + 0.446417i $$0.147300\pi$$
$$282$$ 0 0
$$283$$ − 12.0000i − 0.713326i −0.934233 0.356663i $$-0.883914\pi$$
0.934233 0.356663i $$-0.116086\pi$$
$$284$$ 0 0
$$285$$ − 8.00000i − 0.473879i
$$286$$ 0 0
$$287$$ −2.00000 −0.118056
$$288$$ 0 0
$$289$$ −13.0000 −0.764706
$$290$$ 0 0
$$291$$ − 2.00000i − 0.117242i
$$292$$ 0 0
$$293$$ − 6.00000i − 0.350524i −0.984522 0.175262i $$-0.943923\pi$$
0.984522 0.175262i $$-0.0560772\pi$$
$$294$$ 0 0
$$295$$ 24.0000 1.39733
$$296$$ 0 0
$$297$$ 4.00000 0.232104
$$298$$ 0 0
$$299$$ 24.0000i 1.38796i
$$300$$ 0 0
$$301$$ − 8.00000i − 0.461112i
$$302$$ 0 0
$$303$$ 10.0000 0.574485
$$304$$ 0 0
$$305$$ −20.0000 −1.14520
$$306$$ 0 0
$$307$$ − 4.00000i − 0.228292i −0.993464 0.114146i $$-0.963587\pi$$
0.993464 0.114146i $$-0.0364132\pi$$
$$308$$ 0 0
$$309$$ 8.00000i 0.455104i
$$310$$ 0 0
$$311$$ 8.00000 0.453638 0.226819 0.973937i $$-0.427167\pi$$
0.226819 + 0.973937i $$0.427167\pi$$
$$312$$ 0 0
$$313$$ 14.0000 0.791327 0.395663 0.918396i $$-0.370515\pi$$
0.395663 + 0.918396i $$0.370515\pi$$
$$314$$ 0 0
$$315$$ 2.00000i 0.112687i
$$316$$ 0 0
$$317$$ − 2.00000i − 0.112331i −0.998421 0.0561656i $$-0.982113\pi$$
0.998421 0.0561656i $$-0.0178875\pi$$
$$318$$ 0 0
$$319$$ −8.00000 −0.447914
$$320$$ 0 0
$$321$$ −12.0000 −0.669775
$$322$$ 0 0
$$323$$ 8.00000i 0.445132i
$$324$$ 0 0
$$325$$ − 6.00000i − 0.332820i
$$326$$ 0 0
$$327$$ −2.00000 −0.110600
$$328$$ 0 0
$$329$$ 0 0
$$330$$ 0 0
$$331$$ − 8.00000i − 0.439720i −0.975531 0.219860i $$-0.929440\pi$$
0.975531 0.219860i $$-0.0705600\pi$$
$$332$$ 0 0
$$333$$ − 10.0000i − 0.547997i
$$334$$ 0 0
$$335$$ −16.0000 −0.874173
$$336$$ 0 0
$$337$$ −14.0000 −0.762629 −0.381314 0.924445i $$-0.624528\pi$$
−0.381314 + 0.924445i $$0.624528\pi$$
$$338$$ 0 0
$$339$$ 14.0000i 0.760376i
$$340$$ 0 0
$$341$$ 32.0000i 1.73290i
$$342$$ 0 0
$$343$$ −1.00000 −0.0539949
$$344$$ 0 0
$$345$$ −8.00000 −0.430706
$$346$$ 0 0
$$347$$ − 28.0000i − 1.50312i −0.659665 0.751559i $$-0.729302\pi$$
0.659665 0.751559i $$-0.270698\pi$$
$$348$$ 0 0
$$349$$ 18.0000i 0.963518i 0.876304 + 0.481759i $$0.160002\pi$$
−0.876304 + 0.481759i $$0.839998\pi$$
$$350$$ 0 0
$$351$$ 6.00000 0.320256
$$352$$ 0 0
$$353$$ 6.00000 0.319348 0.159674 0.987170i $$-0.448956\pi$$
0.159674 + 0.987170i $$0.448956\pi$$
$$354$$ 0 0
$$355$$ 24.0000i 1.27379i
$$356$$ 0 0
$$357$$ − 2.00000i − 0.105851i
$$358$$ 0 0
$$359$$ −12.0000 −0.633336 −0.316668 0.948536i $$-0.602564\pi$$
−0.316668 + 0.948536i $$0.602564\pi$$
$$360$$ 0 0
$$361$$ 3.00000 0.157895
$$362$$ 0 0
$$363$$ 5.00000i 0.262432i
$$364$$ 0 0
$$365$$ − 4.00000i − 0.209370i
$$366$$ 0 0
$$367$$ 24.0000 1.25279 0.626395 0.779506i $$-0.284530\pi$$
0.626395 + 0.779506i $$0.284530\pi$$
$$368$$ 0 0
$$369$$ −2.00000 −0.104116
$$370$$ 0 0
$$371$$ − 10.0000i − 0.519174i
$$372$$ 0 0
$$373$$ 34.0000i 1.76045i 0.474554 + 0.880227i $$0.342610\pi$$
−0.474554 + 0.880227i $$0.657390\pi$$
$$374$$ 0 0
$$375$$ 12.0000 0.619677
$$376$$ 0 0
$$377$$ −12.0000 −0.618031
$$378$$ 0 0
$$379$$ − 8.00000i − 0.410932i −0.978664 0.205466i $$-0.934129\pi$$
0.978664 0.205466i $$-0.0658711\pi$$
$$380$$ 0 0
$$381$$ 8.00000i 0.409852i
$$382$$ 0 0
$$383$$ 8.00000 0.408781 0.204390 0.978889i $$-0.434479\pi$$
0.204390 + 0.978889i $$0.434479\pi$$
$$384$$ 0 0
$$385$$ −8.00000 −0.407718
$$386$$ 0 0
$$387$$ − 8.00000i − 0.406663i
$$388$$ 0 0
$$389$$ − 38.0000i − 1.92668i −0.268290 0.963338i $$-0.586458\pi$$
0.268290 0.963338i $$-0.413542\pi$$
$$390$$ 0 0
$$391$$ 8.00000 0.404577
$$392$$ 0 0
$$393$$ 20.0000 1.00887
$$394$$ 0 0
$$395$$ 0 0
$$396$$ 0 0
$$397$$ 34.0000i 1.70641i 0.521575 + 0.853206i $$0.325345\pi$$
−0.521575 + 0.853206i $$0.674655\pi$$
$$398$$ 0 0
$$399$$ 4.00000 0.200250
$$400$$ 0 0
$$401$$ 10.0000 0.499376 0.249688 0.968326i $$-0.419672\pi$$
0.249688 + 0.968326i $$0.419672\pi$$
$$402$$ 0 0
$$403$$ 48.0000i 2.39105i
$$404$$ 0 0
$$405$$ 2.00000i 0.0993808i
$$406$$ 0 0
$$407$$ 40.0000 1.98273
$$408$$ 0 0
$$409$$ 22.0000 1.08783 0.543915 0.839140i $$-0.316941\pi$$
0.543915 + 0.839140i $$0.316941\pi$$
$$410$$ 0 0
$$411$$ 18.0000i 0.887875i
$$412$$ 0 0
$$413$$ 12.0000i 0.590481i
$$414$$ 0 0
$$415$$ 24.0000 1.17811
$$416$$ 0 0
$$417$$ 20.0000 0.979404
$$418$$ 0 0
$$419$$ − 12.0000i − 0.586238i −0.956076 0.293119i $$-0.905307\pi$$
0.956076 0.293119i $$-0.0946933\pi$$
$$420$$ 0 0
$$421$$ 34.0000i 1.65706i 0.559946 + 0.828529i $$0.310822\pi$$
−0.559946 + 0.828529i $$0.689178\pi$$
$$422$$ 0 0
$$423$$ 0 0
$$424$$ 0 0
$$425$$ −2.00000 −0.0970143
$$426$$ 0 0
$$427$$ − 10.0000i − 0.483934i
$$428$$ 0 0
$$429$$ 24.0000i 1.15873i
$$430$$ 0 0
$$431$$ −36.0000 −1.73406 −0.867029 0.498257i $$-0.833974\pi$$
−0.867029 + 0.498257i $$0.833974\pi$$
$$432$$ 0 0
$$433$$ −6.00000 −0.288342 −0.144171 0.989553i $$-0.546051\pi$$
−0.144171 + 0.989553i $$0.546051\pi$$
$$434$$ 0 0
$$435$$ − 4.00000i − 0.191785i
$$436$$ 0 0
$$437$$ 16.0000i 0.765384i
$$438$$ 0 0
$$439$$ 8.00000 0.381819 0.190910 0.981608i $$-0.438856\pi$$
0.190910 + 0.981608i $$0.438856\pi$$
$$440$$ 0 0
$$441$$ −1.00000 −0.0476190
$$442$$ 0 0
$$443$$ 28.0000i 1.33032i 0.746701 + 0.665160i $$0.231637\pi$$
−0.746701 + 0.665160i $$0.768363\pi$$
$$444$$ 0 0
$$445$$ − 12.0000i − 0.568855i
$$446$$ 0 0
$$447$$ −6.00000 −0.283790
$$448$$ 0 0
$$449$$ 18.0000 0.849473 0.424736 0.905317i $$-0.360367\pi$$
0.424736 + 0.905317i $$0.360367\pi$$
$$450$$ 0 0
$$451$$ − 8.00000i − 0.376705i
$$452$$ 0 0
$$453$$ 16.0000i 0.751746i
$$454$$ 0 0
$$455$$ −12.0000 −0.562569
$$456$$ 0 0
$$457$$ 6.00000 0.280668 0.140334 0.990104i $$-0.455182\pi$$
0.140334 + 0.990104i $$0.455182\pi$$
$$458$$ 0 0
$$459$$ − 2.00000i − 0.0933520i
$$460$$ 0 0
$$461$$ − 26.0000i − 1.21094i −0.795868 0.605470i $$-0.792985\pi$$
0.795868 0.605470i $$-0.207015\pi$$
$$462$$ 0 0
$$463$$ −16.0000 −0.743583 −0.371792 0.928316i $$-0.621256\pi$$
−0.371792 + 0.928316i $$0.621256\pi$$
$$464$$ 0 0
$$465$$ −16.0000 −0.741982
$$466$$ 0 0
$$467$$ − 36.0000i − 1.66588i −0.553362 0.832941i $$-0.686655\pi$$
0.553362 0.832941i $$-0.313345\pi$$
$$468$$ 0 0
$$469$$ − 8.00000i − 0.369406i
$$470$$ 0 0
$$471$$ 2.00000 0.0921551
$$472$$ 0 0
$$473$$ 32.0000 1.47136
$$474$$ 0 0
$$475$$ − 4.00000i − 0.183533i
$$476$$ 0 0
$$477$$ − 10.0000i − 0.457869i
$$478$$ 0 0
$$479$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$480$$ 0 0
$$481$$ 60.0000 2.73576
$$482$$ 0 0
$$483$$ − 4.00000i − 0.182006i
$$484$$ 0 0
$$485$$ 4.00000i 0.181631i
$$486$$ 0 0
$$487$$ −32.0000 −1.45006 −0.725029 0.688718i $$-0.758174\pi$$
−0.725029 + 0.688718i $$0.758174\pi$$
$$488$$ 0 0
$$489$$ −16.0000 −0.723545
$$490$$ 0 0
$$491$$ − 36.0000i − 1.62466i −0.583200 0.812329i $$-0.698200\pi$$
0.583200 0.812329i $$-0.301800\pi$$
$$492$$ 0 0
$$493$$ 4.00000i 0.180151i
$$494$$ 0 0
$$495$$ −8.00000 −0.359573
$$496$$ 0 0
$$497$$ −12.0000 −0.538274
$$498$$ 0 0
$$499$$ 16.0000i 0.716258i 0.933672 + 0.358129i $$0.116585\pi$$
−0.933672 + 0.358129i $$0.883415\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ 0 0
$$503$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$504$$ 0 0
$$505$$ −20.0000 −0.889988
$$506$$ 0 0
$$507$$ 23.0000i 1.02147i
$$508$$ 0 0
$$509$$ − 18.0000i − 0.797836i −0.916987 0.398918i $$-0.869386\pi$$
0.916987 0.398918i $$-0.130614\pi$$
$$510$$ 0 0
$$511$$ 2.00000 0.0884748
$$512$$ 0 0
$$513$$ 4.00000 0.176604
$$514$$ 0 0
$$515$$ − 16.0000i − 0.705044i
$$516$$ 0 0
$$517$$ 0 0
$$518$$ 0 0
$$519$$ 6.00000 0.263371
$$520$$ 0 0
$$521$$ −38.0000 −1.66481 −0.832405 0.554168i $$-0.813037\pi$$
−0.832405 + 0.554168i $$0.813037\pi$$
$$522$$ 0 0
$$523$$ 28.0000i 1.22435i 0.790721 + 0.612177i $$0.209706\pi$$
−0.790721 + 0.612177i $$0.790294\pi$$
$$524$$ 0 0
$$525$$ 1.00000i 0.0436436i
$$526$$ 0 0
$$527$$ 16.0000 0.696971
$$528$$ 0 0
$$529$$ −7.00000 −0.304348
$$530$$ 0 0
$$531$$ 12.0000i 0.520756i
$$532$$ 0 0
$$533$$ − 12.0000i − 0.519778i
$$534$$ 0 0
$$535$$ 24.0000 1.03761
$$536$$ 0 0
$$537$$ 20.0000 0.863064
$$538$$ 0 0
$$539$$ − 4.00000i − 0.172292i
$$540$$ 0 0
$$541$$ − 26.0000i − 1.11783i −0.829226 0.558914i $$-0.811218\pi$$
0.829226 0.558914i $$-0.188782\pi$$
$$542$$ 0 0
$$543$$ −2.00000 −0.0858282
$$544$$ 0 0
$$545$$ 4.00000 0.171341
$$546$$ 0 0
$$547$$ 32.0000i 1.36822i 0.729378 + 0.684111i $$0.239809\pi$$
−0.729378 + 0.684111i $$0.760191\pi$$
$$548$$ 0 0
$$549$$ − 10.0000i − 0.426790i
$$550$$ 0 0
$$551$$ −8.00000 −0.340811
$$552$$ 0 0
$$553$$ 0 0
$$554$$ 0 0
$$555$$ 20.0000i 0.848953i
$$556$$ 0 0
$$557$$ 30.0000i 1.27114i 0.772043 + 0.635570i $$0.219235\pi$$
−0.772043 + 0.635570i $$0.780765\pi$$
$$558$$ 0 0
$$559$$ 48.0000 2.03018
$$560$$ 0 0
$$561$$ 8.00000 0.337760
$$562$$ 0 0
$$563$$ 20.0000i 0.842900i 0.906852 + 0.421450i $$0.138479\pi$$
−0.906852 + 0.421450i $$0.861521\pi$$
$$564$$ 0 0
$$565$$ − 28.0000i − 1.17797i
$$566$$ 0 0
$$567$$ −1.00000 −0.0419961
$$568$$ 0 0
$$569$$ −18.0000 −0.754599 −0.377300 0.926091i $$-0.623147\pi$$
−0.377300 + 0.926091i $$0.623147\pi$$
$$570$$ 0 0
$$571$$ 16.0000i 0.669579i 0.942293 + 0.334790i $$0.108665\pi$$
−0.942293 + 0.334790i $$0.891335\pi$$
$$572$$ 0 0
$$573$$ 12.0000i 0.501307i
$$574$$ 0 0
$$575$$ −4.00000 −0.166812
$$576$$ 0 0
$$577$$ 2.00000 0.0832611 0.0416305 0.999133i $$-0.486745\pi$$
0.0416305 + 0.999133i $$0.486745\pi$$
$$578$$ 0 0
$$579$$ 14.0000i 0.581820i
$$580$$ 0 0
$$581$$ 12.0000i 0.497844i
$$582$$ 0 0
$$583$$ 40.0000 1.65663
$$584$$ 0 0
$$585$$ −12.0000 −0.496139
$$586$$ 0 0
$$587$$ 12.0000i 0.495293i 0.968850 + 0.247647i $$0.0796572\pi$$
−0.968850 + 0.247647i $$0.920343\pi$$
$$588$$ 0 0
$$589$$ 32.0000i 1.31854i
$$590$$ 0 0
$$591$$ −6.00000 −0.246807
$$592$$ 0 0
$$593$$ −42.0000 −1.72473 −0.862367 0.506284i $$-0.831019\pi$$
−0.862367 + 0.506284i $$0.831019\pi$$
$$594$$ 0 0
$$595$$ 4.00000i 0.163984i
$$596$$ 0 0
$$597$$ − 16.0000i − 0.654836i
$$598$$ 0 0
$$599$$ −36.0000 −1.47092 −0.735460 0.677568i $$-0.763034\pi$$
−0.735460 + 0.677568i $$0.763034\pi$$
$$600$$ 0 0
$$601$$ −10.0000 −0.407909 −0.203954 0.978980i $$-0.565379\pi$$
−0.203954 + 0.978980i $$0.565379\pi$$
$$602$$ 0 0
$$603$$ − 8.00000i − 0.325785i
$$604$$ 0 0
$$605$$ − 10.0000i − 0.406558i
$$606$$ 0 0
$$607$$ −8.00000 −0.324710 −0.162355 0.986732i $$-0.551909\pi$$
−0.162355 + 0.986732i $$0.551909\pi$$
$$608$$ 0 0
$$609$$ 2.00000 0.0810441
$$610$$ 0 0
$$611$$ 0 0
$$612$$ 0 0
$$613$$ 10.0000i 0.403896i 0.979396 + 0.201948i $$0.0647272\pi$$
−0.979396 + 0.201948i $$0.935273\pi$$
$$614$$ 0 0
$$615$$ 4.00000 0.161296
$$616$$ 0 0
$$617$$ −18.0000 −0.724653 −0.362326 0.932051i $$-0.618017\pi$$
−0.362326 + 0.932051i $$0.618017\pi$$
$$618$$ 0 0
$$619$$ − 20.0000i − 0.803868i −0.915669 0.401934i $$-0.868338\pi$$
0.915669 0.401934i $$-0.131662\pi$$
$$620$$ 0 0
$$621$$ − 4.00000i − 0.160514i
$$622$$ 0 0
$$623$$ 6.00000 0.240385
$$624$$ 0 0
$$625$$ −19.0000 −0.760000
$$626$$ 0 0
$$627$$ 16.0000i 0.638978i
$$628$$ 0 0
$$629$$ − 20.0000i − 0.797452i
$$630$$ 0 0
$$631$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$632$$ 0 0
$$633$$ 16.0000 0.635943
$$634$$ 0 0
$$635$$ − 16.0000i − 0.634941i
$$636$$ 0 0
$$637$$ − 6.00000i − 0.237729i
$$638$$ 0 0
$$639$$ −12.0000 −0.474713
$$640$$ 0 0
$$641$$ 26.0000 1.02694 0.513469 0.858108i $$-0.328360\pi$$
0.513469 + 0.858108i $$0.328360\pi$$
$$642$$ 0 0
$$643$$ − 28.0000i − 1.10421i −0.833774 0.552106i $$-0.813824\pi$$
0.833774 0.552106i $$-0.186176\pi$$
$$644$$ 0 0
$$645$$ 16.0000i 0.629999i
$$646$$ 0 0
$$647$$ −16.0000 −0.629025 −0.314512 0.949253i $$-0.601841\pi$$
−0.314512 + 0.949253i $$0.601841\pi$$
$$648$$ 0 0
$$649$$ −48.0000 −1.88416
$$650$$ 0 0
$$651$$ − 8.00000i − 0.313545i
$$652$$ 0 0
$$653$$ 14.0000i 0.547862i 0.961749 + 0.273931i $$0.0883240\pi$$
−0.961749 + 0.273931i $$0.911676\pi$$
$$654$$ 0 0
$$655$$ −40.0000 −1.56293
$$656$$ 0 0
$$657$$ 2.00000 0.0780274
$$658$$ 0 0
$$659$$ − 44.0000i − 1.71400i −0.515319 0.856998i $$-0.672327\pi$$
0.515319 0.856998i $$-0.327673\pi$$
$$660$$ 0 0
$$661$$ − 18.0000i − 0.700119i −0.936727 0.350059i $$-0.886161\pi$$
0.936727 0.350059i $$-0.113839\pi$$
$$662$$ 0 0
$$663$$ 12.0000 0.466041
$$664$$ 0 0
$$665$$ −8.00000 −0.310227
$$666$$ 0 0
$$667$$ 8.00000i 0.309761i
$$668$$ 0 0
$$669$$ − 8.00000i − 0.309298i
$$670$$ 0 0
$$671$$ 40.0000 1.54418
$$672$$ 0 0
$$673$$ 50.0000 1.92736 0.963679 0.267063i $$-0.0860531\pi$$
0.963679 + 0.267063i $$0.0860531\pi$$
$$674$$ 0 0
$$675$$ 1.00000i 0.0384900i
$$676$$ 0 0
$$677$$ 18.0000i 0.691796i 0.938272 + 0.345898i $$0.112426\pi$$
−0.938272 + 0.345898i $$0.887574\pi$$
$$678$$ 0 0
$$679$$ −2.00000 −0.0767530
$$680$$ 0 0
$$681$$ −20.0000 −0.766402
$$682$$ 0 0
$$683$$ 28.0000i 1.07139i 0.844411 + 0.535695i $$0.179950\pi$$
−0.844411 + 0.535695i $$0.820050\pi$$
$$684$$ 0 0
$$685$$ − 36.0000i − 1.37549i
$$686$$ 0 0
$$687$$ −18.0000 −0.686743
$$688$$ 0 0
$$689$$ 60.0000 2.28582
$$690$$ 0 0
$$691$$ − 12.0000i − 0.456502i −0.973602 0.228251i $$-0.926699\pi$$
0.973602 0.228251i $$-0.0733006\pi$$
$$692$$ 0 0
$$693$$ − 4.00000i − 0.151947i
$$694$$ 0 0
$$695$$ −40.0000 −1.51729
$$696$$ 0 0
$$697$$ −4.00000 −0.151511
$$698$$ 0 0
$$699$$ − 14.0000i − 0.529529i
$$700$$ 0 0
$$701$$ 30.0000i 1.13308i 0.824033 + 0.566542i $$0.191719\pi$$
−0.824033 + 0.566542i $$0.808281\pi$$
$$702$$ 0 0
$$703$$ 40.0000 1.50863
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ − 10.0000i − 0.376089i
$$708$$ 0 0
$$709$$ − 38.0000i − 1.42712i −0.700594 0.713560i $$-0.747082\pi$$
0.700594 0.713560i $$-0.252918\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ 0 0
$$713$$ 32.0000 1.19841
$$714$$ 0 0
$$715$$ − 48.0000i − 1.79510i
$$716$$ 0 0
$$717$$ 12.0000i 0.448148i
$$718$$ 0 0
$$719$$ −8.00000 −0.298350 −0.149175 0.988811i $$-0.547662\pi$$
−0.149175 + 0.988811i $$0.547662\pi$$
$$720$$ 0 0
$$721$$ 8.00000 0.297936
$$722$$ 0 0
$$723$$ 30.0000i 1.11571i
$$724$$ 0 0
$$725$$ − 2.00000i − 0.0742781i
$$726$$ 0 0
$$727$$ −40.0000 −1.48352 −0.741759 0.670667i $$-0.766008\pi$$
−0.741759 + 0.670667i $$0.766008\pi$$
$$728$$ 0 0
$$729$$ −1.00000 −0.0370370
$$730$$ 0 0
$$731$$ − 16.0000i − 0.591781i
$$732$$ 0 0
$$733$$ − 14.0000i − 0.517102i −0.965998 0.258551i $$-0.916755\pi$$
0.965998 0.258551i $$-0.0832450\pi$$
$$734$$ 0 0
$$735$$ 2.00000 0.0737711
$$736$$ 0 0
$$737$$ 32.0000 1.17874
$$738$$ 0 0
$$739$$ − 32.0000i − 1.17714i −0.808447 0.588570i $$-0.799691\pi$$
0.808447 0.588570i $$-0.200309\pi$$
$$740$$ 0 0
$$741$$ 24.0000i 0.881662i
$$742$$ 0 0
$$743$$ −28.0000 −1.02722 −0.513610 0.858024i $$-0.671692\pi$$
−0.513610 + 0.858024i $$0.671692\pi$$
$$744$$ 0 0
$$745$$ 12.0000 0.439646
$$746$$ 0 0
$$747$$ 12.0000i 0.439057i
$$748$$ 0 0
$$749$$ 12.0000i 0.438470i
$$750$$ 0 0
$$751$$ −16.0000 −0.583848 −0.291924 0.956441i $$-0.594295\pi$$
−0.291924 + 0.956441i $$0.594295\pi$$
$$752$$ 0 0
$$753$$ −4.00000 −0.145768
$$754$$ 0 0
$$755$$ − 32.0000i − 1.16460i
$$756$$ 0 0
$$757$$ 26.0000i 0.944986i 0.881334 + 0.472493i $$0.156646\pi$$
−0.881334 + 0.472493i $$0.843354\pi$$
$$758$$ 0 0
$$759$$ 16.0000 0.580763
$$760$$ 0 0
$$761$$ −30.0000 −1.08750 −0.543750 0.839248i $$-0.682996\pi$$
−0.543750 + 0.839248i $$0.682996\pi$$
$$762$$ 0 0
$$763$$ 2.00000i 0.0724049i
$$764$$ 0 0
$$765$$ 4.00000i 0.144620i
$$766$$ 0 0
$$767$$ −72.0000 −2.59977
$$768$$ 0 0
$$769$$ 26.0000 0.937584 0.468792 0.883309i $$-0.344689\pi$$
0.468792 + 0.883309i $$0.344689\pi$$
$$770$$ 0 0
$$771$$ 18.0000i 0.648254i
$$772$$ 0 0
$$773$$ 42.0000i 1.51064i 0.655359 + 0.755318i $$0.272517\pi$$
−0.655359 + 0.755318i $$0.727483\pi$$
$$774$$ 0 0
$$775$$ −8.00000 −0.287368
$$776$$ 0 0
$$777$$ −10.0000 −0.358748
$$778$$ 0 0
$$779$$ − 8.00000i − 0.286630i
$$780$$ 0 0
$$781$$ − 48.0000i − 1.71758i
$$782$$ 0 0
$$783$$ 2.00000 0.0714742
$$784$$ 0 0
$$785$$ −4.00000 −0.142766
$$786$$ 0 0
$$787$$ 4.00000i 0.142585i 0.997455 + 0.0712923i $$0.0227123\pi$$
−0.997455 + 0.0712923i $$0.977288\pi$$
$$788$$ 0 0
$$789$$ 12.0000i 0.427211i
$$790$$ 0 0
$$791$$ 14.0000 0.497783
$$792$$ 0 0
$$793$$ 60.0000 2.13066
$$794$$ 0 0
$$795$$ 20.0000i 0.709327i
$$796$$ 0 0
$$797$$ − 42.0000i − 1.48772i −0.668338 0.743858i $$-0.732994\pi$$
0.668338 0.743858i $$-0.267006\pi$$
$$798$$ 0 0
$$799$$ 0 0
$$800$$ 0 0
$$801$$ 6.00000 0.212000
$$802$$ 0 0
$$803$$ 8.00000i 0.282314i
$$804$$ 0 0
$$805$$ 8.00000i 0.281963i
$$806$$ 0 0
$$807$$ −2.00000 −0.0704033
$$808$$ 0 0
$$809$$ −10.0000 −0.351581 −0.175791 0.984428i $$-0.556248\pi$$
−0.175791 + 0.984428i $$0.556248\pi$$
$$810$$ 0 0
$$811$$ − 4.00000i − 0.140459i −0.997531 0.0702295i $$-0.977627\pi$$
0.997531 0.0702295i $$-0.0223732\pi$$
$$812$$ 0 0
$$813$$ − 32.0000i − 1.12229i
$$814$$ 0 0
$$815$$ 32.0000 1.12091
$$816$$ 0 0
$$817$$ 32.0000 1.11954
$$818$$ 0 0
$$819$$ − 6.00000i − 0.209657i
$$820$$ 0 0
$$821$$ 10.0000i 0.349002i 0.984657 + 0.174501i $$0.0558313\pi$$
−0.984657 + 0.174501i $$0.944169\pi$$
$$822$$ 0 0
$$823$$ 32.0000 1.11545 0.557725 0.830026i $$-0.311674\pi$$
0.557725 + 0.830026i $$0.311674\pi$$
$$824$$ 0 0
$$825$$ −4.00000 −0.139262
$$826$$ 0 0
$$827$$ 4.00000i 0.139094i 0.997579 + 0.0695468i $$0.0221553\pi$$
−0.997579 + 0.0695468i $$0.977845\pi$$
$$828$$ 0 0
$$829$$ − 14.0000i − 0.486240i −0.969996 0.243120i $$-0.921829\pi$$
0.969996 0.243120i $$-0.0781709\pi$$
$$830$$ 0 0
$$831$$ 18.0000 0.624413
$$832$$ 0 0
$$833$$ −2.00000 −0.0692959
$$834$$ 0 0
$$835$$ 0 0
$$836$$ 0 0
$$837$$ − 8.00000i − 0.276520i
$$838$$ 0 0
$$839$$ −16.0000 −0.552381 −0.276191 0.961103i $$-0.589072\pi$$
−0.276191 + 0.961103i $$0.589072\pi$$
$$840$$ 0 0
$$841$$ 25.0000 0.862069
$$842$$ 0 0
$$843$$ − 30.0000i − 1.03325i
$$844$$ 0 0
$$845$$ − 46.0000i − 1.58245i
$$846$$ 0 0
$$847$$ 5.00000 0.171802
$$848$$ 0 0
$$849$$ −12.0000 −0.411839
$$850$$ 0 0
$$851$$ − 40.0000i − 1.37118i
$$852$$ 0 0
$$853$$ − 42.0000i − 1.43805i −0.694983 0.719026i $$-0.744588\pi$$
0.694983 0.719026i $$-0.255412\pi$$
$$854$$ 0 0
$$855$$ −8.00000 −0.273594
$$856$$ 0 0
$$857$$ 18.0000 0.614868 0.307434 0.951569i $$-0.400530\pi$$
0.307434 + 0.951569i $$0.400530\pi$$
$$858$$ 0 0
$$859$$ 12.0000i 0.409435i 0.978821 + 0.204717i $$0.0656275\pi$$
−0.978821 + 0.204717i $$0.934372\pi$$
$$860$$ 0 0
$$861$$ 2.00000i 0.0681598i
$$862$$ 0 0
$$863$$ −52.0000 −1.77010 −0.885050 0.465495i $$-0.845876\pi$$
−0.885050 + 0.465495i $$0.845876\pi$$
$$864$$ 0 0
$$865$$ −12.0000 −0.408012
$$866$$ 0 0
$$867$$ 13.0000i 0.441503i
$$868$$ 0 0
$$869$$ 0 0
$$870$$ 0 0
$$871$$ 48.0000 1.62642
$$872$$ 0 0
$$873$$ −2.00000 −0.0676897
$$874$$ 0 0
$$875$$ − 12.0000i − 0.405674i
$$876$$ 0 0
$$877$$ 14.0000i 0.472746i 0.971662 + 0.236373i $$0.0759588\pi$$
−0.971662 + 0.236373i $$0.924041\pi$$
$$878$$ 0 0
$$879$$ −6.00000 −0.202375
$$880$$ 0 0
$$881$$ −42.0000 −1.41502 −0.707508 0.706705i $$-0.750181\pi$$
−0.707508 + 0.706705i $$0.750181\pi$$
$$882$$ 0 0
$$883$$ − 16.0000i − 0.538443i −0.963078 0.269221i $$-0.913234\pi$$
0.963078 0.269221i $$-0.0867663\pi$$
$$884$$ 0 0
$$885$$ − 24.0000i − 0.806751i
$$886$$ 0 0
$$887$$ −16.0000 −0.537227 −0.268614 0.963248i $$-0.586566\pi$$
−0.268614 + 0.963248i $$0.586566\pi$$
$$888$$ 0 0
$$889$$ 8.00000 0.268311
$$890$$ 0 0
$$891$$ − 4.00000i − 0.134005i
$$892$$ 0 0
$$893$$ 0 0
$$894$$ 0 0
$$895$$ −40.0000 −1.33705
$$896$$ 0 0
$$897$$ 24.0000 0.801337
$$898$$ 0 0
$$899$$ 16.0000i 0.533630i
$$900$$ 0 0
$$901$$ − 20.0000i − 0.666297i
$$902$$ 0 0
$$903$$ −8.00000 −0.266223
$$904$$ 0 0
$$905$$ 4.00000 0.132964
$$906$$ 0 0
$$907$$ 48.0000i 1.59381i 0.604102 + 0.796907i $$0.293532\pi$$
−0.604102 + 0.796907i $$0.706468\pi$$
$$908$$ 0 0
$$909$$ − 10.0000i − 0.331679i
$$910$$ 0 0
$$911$$ 12.0000 0.397578 0.198789 0.980042i $$-0.436299\pi$$
0.198789 + 0.980042i $$0.436299\pi$$
$$912$$ 0 0
$$913$$ −48.0000 −1.58857
$$914$$ 0 0
$$915$$ 20.0000i 0.661180i
$$916$$ 0 0
$$917$$ − 20.0000i − 0.660458i
$$918$$ 0 0
$$919$$ 8.00000 0.263896 0.131948 0.991257i $$-0.457877\pi$$
0.131948 + 0.991257i $$0.457877\pi$$
$$920$$ 0 0
$$921$$ −4.00000 −0.131804
$$922$$ 0 0
$$923$$ − 72.0000i − 2.36991i
$$924$$ 0 0
$$925$$ 10.0000i 0.328798i
$$926$$ 0 0
$$927$$ 8.00000 0.262754
$$928$$ 0 0
$$929$$ 22.0000 0.721797 0.360898 0.932605i $$-0.382470\pi$$
0.360898 + 0.932605i $$0.382470\pi$$
$$930$$ 0 0
$$931$$ − 4.00000i − 0.131095i
$$932$$ 0 0
$$933$$ − 8.00000i − 0.261908i
$$934$$ 0 0
$$935$$ −16.0000 −0.523256
$$936$$ 0 0
$$937$$ −10.0000 −0.326686 −0.163343 0.986569i $$-0.552228\pi$$
−0.163343 + 0.986569i $$0.552228\pi$$
$$938$$ 0 0
$$939$$ − 14.0000i − 0.456873i
$$940$$ 0 0
$$941$$ 6.00000i 0.195594i 0.995206 + 0.0977972i $$0.0311797\pi$$
−0.995206 + 0.0977972i $$0.968820\pi$$
$$942$$ 0 0
$$943$$ −8.00000 −0.260516
$$944$$ 0 0
$$945$$ 2.00000 0.0650600
$$946$$ 0 0
$$947$$ − 44.0000i − 1.42981i −0.699223 0.714904i $$-0.746470\pi$$
0.699223 0.714904i $$-0.253530\pi$$
$$948$$ 0 0
$$949$$ 12.0000i 0.389536i
$$950$$ 0 0
$$951$$ −2.00000 −0.0648544
$$952$$ 0 0
$$953$$ 54.0000 1.74923 0.874616 0.484817i $$-0.161114\pi$$
0.874616 + 0.484817i $$0.161114\pi$$
$$954$$ 0 0
$$955$$ − 24.0000i − 0.776622i
$$956$$ 0 0
$$957$$ 8.00000i 0.258603i
$$958$$ 0 0
$$959$$ 18.0000 0.581250
$$960$$ 0 0
$$961$$ 33.0000 1.06452
$$962$$ 0 0
$$963$$ 12.0000i 0.386695i
$$964$$ 0 0
$$965$$ − 28.0000i − 0.901352i
$$966$$ 0 0
$$967$$ 8.00000 0.257263 0.128631 0.991692i $$-0.458942\pi$$
0.128631 + 0.991692i $$0.458942\pi$$
$$968$$ 0 0
$$969$$ 8.00000 0.256997
$$970$$ 0 0
$$971$$ 28.0000i 0.898563i 0.893390 + 0.449281i $$0.148320\pi$$
−0.893390 + 0.449281i $$0.851680\pi$$
$$972$$ 0 0
$$973$$ − 20.0000i − 0.641171i
$$974$$ 0 0
$$975$$ −6.00000 −0.192154
$$976$$ 0 0
$$977$$ −22.0000 −0.703842 −0.351921 0.936030i $$-0.614471\pi$$
−0.351921 + 0.936030i $$0.614471\pi$$
$$978$$ 0 0
$$979$$ 24.0000i 0.767043i
$$980$$ 0 0
$$981$$ 2.00000i 0.0638551i
$$982$$ 0 0
$$983$$ 24.0000 0.765481 0.382741 0.923856i $$-0.374980\pi$$
0.382741 + 0.923856i $$0.374980\pi$$
$$984$$ 0 0
$$985$$ 12.0000 0.382352
$$986$$ 0 0
$$987$$ 0 0
$$988$$ 0 0
$$989$$ − 32.0000i − 1.01754i
$$990$$ 0 0
$$991$$ −8.00000 −0.254128 −0.127064 0.991894i $$-0.540555\pi$$
−0.127064 + 0.991894i $$0.540555\pi$$
$$992$$ 0 0
$$993$$ −8.00000 −0.253872
$$994$$ 0 0
$$995$$ 32.0000i 1.01447i
$$996$$ 0 0
$$997$$ 62.0000i 1.96356i 0.190022 + 0.981780i $$0.439144\pi$$
−0.190022 + 0.981780i $$0.560856\pi$$
$$998$$ 0 0
$$999$$ −10.0000 −0.316386
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 5376.2.c.m.2689.1 2
4.3 odd 2 5376.2.c.s.2689.2 2
8.3 odd 2 5376.2.c.s.2689.1 2
8.5 even 2 inner 5376.2.c.m.2689.2 2
16.3 odd 4 1344.2.a.h.1.1 1
16.5 even 4 672.2.a.b.1.1 1
16.11 odd 4 672.2.a.f.1.1 yes 1
16.13 even 4 1344.2.a.r.1.1 1
48.5 odd 4 2016.2.a.l.1.1 1
48.11 even 4 2016.2.a.k.1.1 1
48.29 odd 4 4032.2.a.n.1.1 1
48.35 even 4 4032.2.a.f.1.1 1
112.13 odd 4 9408.2.a.g.1.1 1
112.27 even 4 4704.2.a.m.1.1 1
112.69 odd 4 4704.2.a.be.1.1 1
112.83 even 4 9408.2.a.cb.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
672.2.a.b.1.1 1 16.5 even 4
672.2.a.f.1.1 yes 1 16.11 odd 4
1344.2.a.h.1.1 1 16.3 odd 4
1344.2.a.r.1.1 1 16.13 even 4
2016.2.a.k.1.1 1 48.11 even 4
2016.2.a.l.1.1 1 48.5 odd 4
4032.2.a.f.1.1 1 48.35 even 4
4032.2.a.n.1.1 1 48.29 odd 4
4704.2.a.m.1.1 1 112.27 even 4
4704.2.a.be.1.1 1 112.69 odd 4
5376.2.c.m.2689.1 2 1.1 even 1 trivial
5376.2.c.m.2689.2 2 8.5 even 2 inner
5376.2.c.s.2689.1 2 8.3 odd 2
5376.2.c.s.2689.2 2 4.3 odd 2
9408.2.a.g.1.1 1 112.13 odd 4
9408.2.a.cb.1.1 1 112.83 even 4