# Properties

 Label 1183.2.c.b.337.1 Level $1183$ Weight $2$ Character 1183.337 Analytic conductor $9.446$ 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] = [1183,2,Mod(337,1183)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1183, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1]))

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

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

chi := DirichletCharacter("1183.337");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1183 = 7 \cdot 13^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1183.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: $$9.44630255912$$ 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: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 91) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 337.1 Root $$-1.00000i$$ of defining polynomial Character $$\chi$$ $$=$$ 1183.337 Dual form 1183.2.c.b.337.2

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-2.00000i q^{2} -2.00000 q^{4} -3.00000i q^{5} +1.00000i q^{7} -3.00000 q^{9} +O(q^{10})$$ $$q-2.00000i q^{2} -2.00000 q^{4} -3.00000i q^{5} +1.00000i q^{7} -3.00000 q^{9} -6.00000 q^{10} +6.00000i q^{11} +2.00000 q^{14} -4.00000 q^{16} -4.00000 q^{17} +6.00000i q^{18} +5.00000i q^{19} +6.00000i q^{20} +12.0000 q^{22} -3.00000 q^{23} -4.00000 q^{25} -2.00000i q^{28} -5.00000 q^{29} -3.00000i q^{31} +8.00000i q^{32} +8.00000i q^{34} +3.00000 q^{35} +6.00000 q^{36} +4.00000i q^{37} +10.0000 q^{38} -6.00000i q^{41} +1.00000 q^{43} -12.0000i q^{44} +9.00000i q^{45} +6.00000i q^{46} -7.00000i q^{47} -1.00000 q^{49} +8.00000i q^{50} -9.00000 q^{53} +18.0000 q^{55} +10.0000i q^{58} -8.00000i q^{59} -10.0000 q^{61} -6.00000 q^{62} -3.00000i q^{63} +8.00000 q^{64} -6.00000i q^{67} +8.00000 q^{68} -6.00000i q^{70} -8.00000i q^{71} +13.0000i q^{73} +8.00000 q^{74} -10.0000i q^{76} -6.00000 q^{77} +3.00000 q^{79} +12.0000i q^{80} +9.00000 q^{81} -12.0000 q^{82} +15.0000i q^{83} +12.0000i q^{85} -2.00000i q^{86} -3.00000i q^{89} +18.0000 q^{90} +6.00000 q^{92} -14.0000 q^{94} +15.0000 q^{95} +7.00000i q^{97} +2.00000i q^{98} -18.0000i q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 4 q^{4} - 6 q^{9}+O(q^{10})$$ 2 * q - 4 * q^4 - 6 * q^9 $$2 q - 4 q^{4} - 6 q^{9} - 12 q^{10} + 4 q^{14} - 8 q^{16} - 8 q^{17} + 24 q^{22} - 6 q^{23} - 8 q^{25} - 10 q^{29} + 6 q^{35} + 12 q^{36} + 20 q^{38} + 2 q^{43} - 2 q^{49} - 18 q^{53} + 36 q^{55} - 20 q^{61} - 12 q^{62} + 16 q^{64} + 16 q^{68} + 16 q^{74} - 12 q^{77} + 6 q^{79} + 18 q^{81} - 24 q^{82} + 36 q^{90} + 12 q^{92} - 28 q^{94} + 30 q^{95}+O(q^{100})$$ 2 * q - 4 * q^4 - 6 * q^9 - 12 * q^10 + 4 * q^14 - 8 * q^16 - 8 * q^17 + 24 * q^22 - 6 * q^23 - 8 * q^25 - 10 * q^29 + 6 * q^35 + 12 * q^36 + 20 * q^38 + 2 * q^43 - 2 * q^49 - 18 * q^53 + 36 * q^55 - 20 * q^61 - 12 * q^62 + 16 * q^64 + 16 * q^68 + 16 * q^74 - 12 * q^77 + 6 * q^79 + 18 * q^81 - 24 * q^82 + 36 * q^90 + 12 * q^92 - 28 * q^94 + 30 * q^95

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1183\mathbb{Z}\right)^\times$$.

 $$n$$ $$339$$ $$1016$$ $$\chi(n)$$ $$1$$ $$-1$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ − 2.00000i − 1.41421i −0.707107 0.707107i $$-0.750000\pi$$
0.707107 0.707107i $$-0.250000\pi$$
$$3$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$4$$ −2.00000 −1.00000
$$5$$ − 3.00000i − 1.34164i −0.741620 0.670820i $$-0.765942\pi$$
0.741620 0.670820i $$-0.234058\pi$$
$$6$$ 0 0
$$7$$ 1.00000i 0.377964i
$$8$$ 0 0
$$9$$ −3.00000 −1.00000
$$10$$ −6.00000 −1.89737
$$11$$ 6.00000i 1.80907i 0.426401 + 0.904534i $$0.359781\pi$$
−0.426401 + 0.904534i $$0.640219\pi$$
$$12$$ 0 0
$$13$$ 0 0
$$14$$ 2.00000 0.534522
$$15$$ 0 0
$$16$$ −4.00000 −1.00000
$$17$$ −4.00000 −0.970143 −0.485071 0.874475i $$-0.661206\pi$$
−0.485071 + 0.874475i $$0.661206\pi$$
$$18$$ 6.00000i 1.41421i
$$19$$ 5.00000i 1.14708i 0.819178 + 0.573539i $$0.194430\pi$$
−0.819178 + 0.573539i $$0.805570\pi$$
$$20$$ 6.00000i 1.34164i
$$21$$ 0 0
$$22$$ 12.0000 2.55841
$$23$$ −3.00000 −0.625543 −0.312772 0.949828i $$-0.601257\pi$$
−0.312772 + 0.949828i $$0.601257\pi$$
$$24$$ 0 0
$$25$$ −4.00000 −0.800000
$$26$$ 0 0
$$27$$ 0 0
$$28$$ − 2.00000i − 0.377964i
$$29$$ −5.00000 −0.928477 −0.464238 0.885710i $$-0.653672\pi$$
−0.464238 + 0.885710i $$0.653672\pi$$
$$30$$ 0 0
$$31$$ − 3.00000i − 0.538816i −0.963026 0.269408i $$-0.913172\pi$$
0.963026 0.269408i $$-0.0868280\pi$$
$$32$$ 8.00000i 1.41421i
$$33$$ 0 0
$$34$$ 8.00000i 1.37199i
$$35$$ 3.00000 0.507093
$$36$$ 6.00000 1.00000
$$37$$ 4.00000i 0.657596i 0.944400 + 0.328798i $$0.106644\pi$$
−0.944400 + 0.328798i $$0.893356\pi$$
$$38$$ 10.0000 1.62221
$$39$$ 0 0
$$40$$ 0 0
$$41$$ − 6.00000i − 0.937043i −0.883452 0.468521i $$-0.844787\pi$$
0.883452 0.468521i $$-0.155213\pi$$
$$42$$ 0 0
$$43$$ 1.00000 0.152499 0.0762493 0.997089i $$-0.475706\pi$$
0.0762493 + 0.997089i $$0.475706\pi$$
$$44$$ − 12.0000i − 1.80907i
$$45$$ 9.00000i 1.34164i
$$46$$ 6.00000i 0.884652i
$$47$$ − 7.00000i − 1.02105i −0.859861 0.510527i $$-0.829450\pi$$
0.859861 0.510527i $$-0.170550\pi$$
$$48$$ 0 0
$$49$$ −1.00000 −0.142857
$$50$$ 8.00000i 1.13137i
$$51$$ 0 0
$$52$$ 0 0
$$53$$ −9.00000 −1.23625 −0.618123 0.786082i $$-0.712106\pi$$
−0.618123 + 0.786082i $$0.712106\pi$$
$$54$$ 0 0
$$55$$ 18.0000 2.42712
$$56$$ 0 0
$$57$$ 0 0
$$58$$ 10.0000i 1.31306i
$$59$$ − 8.00000i − 1.04151i −0.853706 0.520756i $$-0.825650\pi$$
0.853706 0.520756i $$-0.174350\pi$$
$$60$$ 0 0
$$61$$ −10.0000 −1.28037 −0.640184 0.768221i $$-0.721142\pi$$
−0.640184 + 0.768221i $$0.721142\pi$$
$$62$$ −6.00000 −0.762001
$$63$$ − 3.00000i − 0.377964i
$$64$$ 8.00000 1.00000
$$65$$ 0 0
$$66$$ 0 0
$$67$$ − 6.00000i − 0.733017i −0.930415 0.366508i $$-0.880553\pi$$
0.930415 0.366508i $$-0.119447\pi$$
$$68$$ 8.00000 0.970143
$$69$$ 0 0
$$70$$ − 6.00000i − 0.717137i
$$71$$ − 8.00000i − 0.949425i −0.880141 0.474713i $$-0.842552\pi$$
0.880141 0.474713i $$-0.157448\pi$$
$$72$$ 0 0
$$73$$ 13.0000i 1.52153i 0.649025 + 0.760767i $$0.275177\pi$$
−0.649025 + 0.760767i $$0.724823\pi$$
$$74$$ 8.00000 0.929981
$$75$$ 0 0
$$76$$ − 10.0000i − 1.14708i
$$77$$ −6.00000 −0.683763
$$78$$ 0 0
$$79$$ 3.00000 0.337526 0.168763 0.985657i $$-0.446023\pi$$
0.168763 + 0.985657i $$0.446023\pi$$
$$80$$ 12.0000i 1.34164i
$$81$$ 9.00000 1.00000
$$82$$ −12.0000 −1.32518
$$83$$ 15.0000i 1.64646i 0.567705 + 0.823232i $$0.307831\pi$$
−0.567705 + 0.823232i $$0.692169\pi$$
$$84$$ 0 0
$$85$$ 12.0000i 1.30158i
$$86$$ − 2.00000i − 0.215666i
$$87$$ 0 0
$$88$$ 0 0
$$89$$ − 3.00000i − 0.317999i −0.987279 0.159000i $$-0.949173\pi$$
0.987279 0.159000i $$-0.0508269\pi$$
$$90$$ 18.0000 1.89737
$$91$$ 0 0
$$92$$ 6.00000 0.625543
$$93$$ 0 0
$$94$$ −14.0000 −1.44399
$$95$$ 15.0000 1.53897
$$96$$ 0 0
$$97$$ 7.00000i 0.710742i 0.934725 + 0.355371i $$0.115646\pi$$
−0.934725 + 0.355371i $$0.884354\pi$$
$$98$$ 2.00000i 0.202031i
$$99$$ − 18.0000i − 1.80907i
$$100$$ 8.00000 0.800000
$$101$$ 14.0000 1.39305 0.696526 0.717532i $$-0.254728\pi$$
0.696526 + 0.717532i $$0.254728\pi$$
$$102$$ 0 0
$$103$$ 4.00000 0.394132 0.197066 0.980390i $$-0.436859\pi$$
0.197066 + 0.980390i $$0.436859\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 18.0000i 1.74831i
$$107$$ −4.00000 −0.386695 −0.193347 0.981130i $$-0.561934\pi$$
−0.193347 + 0.981130i $$0.561934\pi$$
$$108$$ 0 0
$$109$$ − 2.00000i − 0.191565i −0.995402 0.0957826i $$-0.969465\pi$$
0.995402 0.0957826i $$-0.0305354\pi$$
$$110$$ − 36.0000i − 3.43247i
$$111$$ 0 0
$$112$$ − 4.00000i − 0.377964i
$$113$$ −3.00000 −0.282216 −0.141108 0.989994i $$-0.545067\pi$$
−0.141108 + 0.989994i $$0.545067\pi$$
$$114$$ 0 0
$$115$$ 9.00000i 0.839254i
$$116$$ 10.0000 0.928477
$$117$$ 0 0
$$118$$ −16.0000 −1.47292
$$119$$ − 4.00000i − 0.366679i
$$120$$ 0 0
$$121$$ −25.0000 −2.27273
$$122$$ 20.0000i 1.81071i
$$123$$ 0 0
$$124$$ 6.00000i 0.538816i
$$125$$ − 3.00000i − 0.268328i
$$126$$ −6.00000 −0.534522
$$127$$ 4.00000 0.354943 0.177471 0.984126i $$-0.443208\pi$$
0.177471 + 0.984126i $$0.443208\pi$$
$$128$$ 0 0
$$129$$ 0 0
$$130$$ 0 0
$$131$$ 8.00000 0.698963 0.349482 0.936943i $$-0.386358\pi$$
0.349482 + 0.936943i $$0.386358\pi$$
$$132$$ 0 0
$$133$$ −5.00000 −0.433555
$$134$$ −12.0000 −1.03664
$$135$$ 0 0
$$136$$ 0 0
$$137$$ − 4.00000i − 0.341743i −0.985293 0.170872i $$-0.945342\pi$$
0.985293 0.170872i $$-0.0546583\pi$$
$$138$$ 0 0
$$139$$ −18.0000 −1.52674 −0.763370 0.645961i $$-0.776457\pi$$
−0.763370 + 0.645961i $$0.776457\pi$$
$$140$$ −6.00000 −0.507093
$$141$$ 0 0
$$142$$ −16.0000 −1.34269
$$143$$ 0 0
$$144$$ 12.0000 1.00000
$$145$$ 15.0000i 1.24568i
$$146$$ 26.0000 2.15178
$$147$$ 0 0
$$148$$ − 8.00000i − 0.657596i
$$149$$ − 18.0000i − 1.47462i −0.675556 0.737309i $$-0.736096\pi$$
0.675556 0.737309i $$-0.263904\pi$$
$$150$$ 0 0
$$151$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$152$$ 0 0
$$153$$ 12.0000 0.970143
$$154$$ 12.0000i 0.966988i
$$155$$ −9.00000 −0.722897
$$156$$ 0 0
$$157$$ −8.00000 −0.638470 −0.319235 0.947676i $$-0.603426\pi$$
−0.319235 + 0.947676i $$0.603426\pi$$
$$158$$ − 6.00000i − 0.477334i
$$159$$ 0 0
$$160$$ 24.0000 1.89737
$$161$$ − 3.00000i − 0.236433i
$$162$$ − 18.0000i − 1.41421i
$$163$$ 4.00000i 0.313304i 0.987654 + 0.156652i $$0.0500701\pi$$
−0.987654 + 0.156652i $$0.949930\pi$$
$$164$$ 12.0000i 0.937043i
$$165$$ 0 0
$$166$$ 30.0000 2.32845
$$167$$ − 5.00000i − 0.386912i −0.981109 0.193456i $$-0.938030\pi$$
0.981109 0.193456i $$-0.0619696\pi$$
$$168$$ 0 0
$$169$$ 0 0
$$170$$ 24.0000 1.84072
$$171$$ − 15.0000i − 1.14708i
$$172$$ −2.00000 −0.152499
$$173$$ 8.00000 0.608229 0.304114 0.952636i $$-0.401639\pi$$
0.304114 + 0.952636i $$0.401639\pi$$
$$174$$ 0 0
$$175$$ − 4.00000i − 0.302372i
$$176$$ − 24.0000i − 1.80907i
$$177$$ 0 0
$$178$$ −6.00000 −0.449719
$$179$$ −23.0000 −1.71910 −0.859550 0.511051i $$-0.829256\pi$$
−0.859550 + 0.511051i $$0.829256\pi$$
$$180$$ − 18.0000i − 1.34164i
$$181$$ −14.0000 −1.04061 −0.520306 0.853980i $$-0.674182\pi$$
−0.520306 + 0.853980i $$0.674182\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ 0 0
$$185$$ 12.0000 0.882258
$$186$$ 0 0
$$187$$ − 24.0000i − 1.75505i
$$188$$ 14.0000i 1.02105i
$$189$$ 0 0
$$190$$ − 30.0000i − 2.17643i
$$191$$ −8.00000 −0.578860 −0.289430 0.957199i $$-0.593466\pi$$
−0.289430 + 0.957199i $$0.593466\pi$$
$$192$$ 0 0
$$193$$ − 22.0000i − 1.58359i −0.610784 0.791797i $$-0.709146\pi$$
0.610784 0.791797i $$-0.290854\pi$$
$$194$$ 14.0000 1.00514
$$195$$ 0 0
$$196$$ 2.00000 0.142857
$$197$$ 2.00000i 0.142494i 0.997459 + 0.0712470i $$0.0226979\pi$$
−0.997459 + 0.0712470i $$0.977302\pi$$
$$198$$ −36.0000 −2.55841
$$199$$ −4.00000 −0.283552 −0.141776 0.989899i $$-0.545281\pi$$
−0.141776 + 0.989899i $$0.545281\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ − 28.0000i − 1.97007i
$$203$$ − 5.00000i − 0.350931i
$$204$$ 0 0
$$205$$ −18.0000 −1.25717
$$206$$ − 8.00000i − 0.557386i
$$207$$ 9.00000 0.625543
$$208$$ 0 0
$$209$$ −30.0000 −2.07514
$$210$$ 0 0
$$211$$ −5.00000 −0.344214 −0.172107 0.985078i $$-0.555058\pi$$
−0.172107 + 0.985078i $$0.555058\pi$$
$$212$$ 18.0000 1.23625
$$213$$ 0 0
$$214$$ 8.00000i 0.546869i
$$215$$ − 3.00000i − 0.204598i
$$216$$ 0 0
$$217$$ 3.00000 0.203653
$$218$$ −4.00000 −0.270914
$$219$$ 0 0
$$220$$ −36.0000 −2.42712
$$221$$ 0 0
$$222$$ 0 0
$$223$$ 15.0000i 1.00447i 0.864730 + 0.502237i $$0.167490\pi$$
−0.864730 + 0.502237i $$0.832510\pi$$
$$224$$ −8.00000 −0.534522
$$225$$ 12.0000 0.800000
$$226$$ 6.00000i 0.399114i
$$227$$ 20.0000i 1.32745i 0.747978 + 0.663723i $$0.231025\pi$$
−0.747978 + 0.663723i $$0.768975\pi$$
$$228$$ 0 0
$$229$$ − 14.0000i − 0.925146i −0.886581 0.462573i $$-0.846926\pi$$
0.886581 0.462573i $$-0.153074\pi$$
$$230$$ 18.0000 1.18688
$$231$$ 0 0
$$232$$ 0 0
$$233$$ −15.0000 −0.982683 −0.491341 0.870967i $$-0.663493\pi$$
−0.491341 + 0.870967i $$0.663493\pi$$
$$234$$ 0 0
$$235$$ −21.0000 −1.36989
$$236$$ 16.0000i 1.04151i
$$237$$ 0 0
$$238$$ −8.00000 −0.518563
$$239$$ − 4.00000i − 0.258738i −0.991596 0.129369i $$-0.958705\pi$$
0.991596 0.129369i $$-0.0412952\pi$$
$$240$$ 0 0
$$241$$ 17.0000i 1.09507i 0.836784 + 0.547533i $$0.184433\pi$$
−0.836784 + 0.547533i $$0.815567\pi$$
$$242$$ 50.0000i 3.21412i
$$243$$ 0 0
$$244$$ 20.0000 1.28037
$$245$$ 3.00000i 0.191663i
$$246$$ 0 0
$$247$$ 0 0
$$248$$ 0 0
$$249$$ 0 0
$$250$$ −6.00000 −0.379473
$$251$$ 26.0000 1.64111 0.820553 0.571571i $$-0.193666\pi$$
0.820553 + 0.571571i $$0.193666\pi$$
$$252$$ 6.00000i 0.377964i
$$253$$ − 18.0000i − 1.13165i
$$254$$ − 8.00000i − 0.501965i
$$255$$ 0 0
$$256$$ 16.0000 1.00000
$$257$$ 2.00000 0.124757 0.0623783 0.998053i $$-0.480131\pi$$
0.0623783 + 0.998053i $$0.480131\pi$$
$$258$$ 0 0
$$259$$ −4.00000 −0.248548
$$260$$ 0 0
$$261$$ 15.0000 0.928477
$$262$$ − 16.0000i − 0.988483i
$$263$$ −15.0000 −0.924940 −0.462470 0.886635i $$-0.653037\pi$$
−0.462470 + 0.886635i $$0.653037\pi$$
$$264$$ 0 0
$$265$$ 27.0000i 1.65860i
$$266$$ 10.0000i 0.613139i
$$267$$ 0 0
$$268$$ 12.0000i 0.733017i
$$269$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$270$$ 0 0
$$271$$ − 8.00000i − 0.485965i −0.970031 0.242983i $$-0.921874\pi$$
0.970031 0.242983i $$-0.0781258\pi$$
$$272$$ 16.0000 0.970143
$$273$$ 0 0
$$274$$ −8.00000 −0.483298
$$275$$ − 24.0000i − 1.44725i
$$276$$ 0 0
$$277$$ −1.00000 −0.0600842 −0.0300421 0.999549i $$-0.509564\pi$$
−0.0300421 + 0.999549i $$0.509564\pi$$
$$278$$ 36.0000i 2.15914i
$$279$$ 9.00000i 0.538816i
$$280$$ 0 0
$$281$$ 30.0000i 1.78965i 0.446417 + 0.894825i $$0.352700\pi$$
−0.446417 + 0.894825i $$0.647300\pi$$
$$282$$ 0 0
$$283$$ −16.0000 −0.951101 −0.475551 0.879688i $$-0.657751\pi$$
−0.475551 + 0.879688i $$0.657751\pi$$
$$284$$ 16.0000i 0.949425i
$$285$$ 0 0
$$286$$ 0 0
$$287$$ 6.00000 0.354169
$$288$$ − 24.0000i − 1.41421i
$$289$$ −1.00000 −0.0588235
$$290$$ 30.0000 1.76166
$$291$$ 0 0
$$292$$ − 26.0000i − 1.52153i
$$293$$ 19.0000i 1.10999i 0.831853 + 0.554996i $$0.187280\pi$$
−0.831853 + 0.554996i $$0.812720\pi$$
$$294$$ 0 0
$$295$$ −24.0000 −1.39733
$$296$$ 0 0
$$297$$ 0 0
$$298$$ −36.0000 −2.08542
$$299$$ 0 0
$$300$$ 0 0
$$301$$ 1.00000i 0.0576390i
$$302$$ 0 0
$$303$$ 0 0
$$304$$ − 20.0000i − 1.14708i
$$305$$ 30.0000i 1.71780i
$$306$$ − 24.0000i − 1.37199i
$$307$$ 33.0000i 1.88341i 0.336440 + 0.941705i $$0.390777\pi$$
−0.336440 + 0.941705i $$0.609223\pi$$
$$308$$ 12.0000 0.683763
$$309$$ 0 0
$$310$$ 18.0000i 1.02233i
$$311$$ 6.00000 0.340229 0.170114 0.985424i $$-0.445586\pi$$
0.170114 + 0.985424i $$0.445586\pi$$
$$312$$ 0 0
$$313$$ 22.0000 1.24351 0.621757 0.783210i $$-0.286419\pi$$
0.621757 + 0.783210i $$0.286419\pi$$
$$314$$ 16.0000i 0.902932i
$$315$$ −9.00000 −0.507093
$$316$$ −6.00000 −0.337526
$$317$$ − 24.0000i − 1.34797i −0.738743 0.673987i $$-0.764580\pi$$
0.738743 0.673987i $$-0.235420\pi$$
$$318$$ 0 0
$$319$$ − 30.0000i − 1.67968i
$$320$$ − 24.0000i − 1.34164i
$$321$$ 0 0
$$322$$ −6.00000 −0.334367
$$323$$ − 20.0000i − 1.11283i
$$324$$ −18.0000 −1.00000
$$325$$ 0 0
$$326$$ 8.00000 0.443079
$$327$$ 0 0
$$328$$ 0 0
$$329$$ 7.00000 0.385922
$$330$$ 0 0
$$331$$ 22.0000i 1.20923i 0.796518 + 0.604615i $$0.206673\pi$$
−0.796518 + 0.604615i $$0.793327\pi$$
$$332$$ − 30.0000i − 1.64646i
$$333$$ − 12.0000i − 0.657596i
$$334$$ −10.0000 −0.547176
$$335$$ −18.0000 −0.983445
$$336$$ 0 0
$$337$$ −17.0000 −0.926049 −0.463025 0.886345i $$-0.653236\pi$$
−0.463025 + 0.886345i $$0.653236\pi$$
$$338$$ 0 0
$$339$$ 0 0
$$340$$ − 24.0000i − 1.30158i
$$341$$ 18.0000 0.974755
$$342$$ −30.0000 −1.62221
$$343$$ − 1.00000i − 0.0539949i
$$344$$ 0 0
$$345$$ 0 0
$$346$$ − 16.0000i − 0.860165i
$$347$$ −32.0000 −1.71785 −0.858925 0.512101i $$-0.828867\pi$$
−0.858925 + 0.512101i $$0.828867\pi$$
$$348$$ 0 0
$$349$$ − 11.0000i − 0.588817i −0.955680 0.294408i $$-0.904877\pi$$
0.955680 0.294408i $$-0.0951225\pi$$
$$350$$ −8.00000 −0.427618
$$351$$ 0 0
$$352$$ −48.0000 −2.55841
$$353$$ − 10.0000i − 0.532246i −0.963939 0.266123i $$-0.914257\pi$$
0.963939 0.266123i $$-0.0857428\pi$$
$$354$$ 0 0
$$355$$ −24.0000 −1.27379
$$356$$ 6.00000i 0.317999i
$$357$$ 0 0
$$358$$ 46.0000i 2.43118i
$$359$$ − 20.0000i − 1.05556i −0.849381 0.527780i $$-0.823025\pi$$
0.849381 0.527780i $$-0.176975\pi$$
$$360$$ 0 0
$$361$$ −6.00000 −0.315789
$$362$$ 28.0000i 1.47165i
$$363$$ 0 0
$$364$$ 0 0
$$365$$ 39.0000 2.04135
$$366$$ 0 0
$$367$$ 14.0000 0.730794 0.365397 0.930852i $$-0.380933\pi$$
0.365397 + 0.930852i $$0.380933\pi$$
$$368$$ 12.0000 0.625543
$$369$$ 18.0000i 0.937043i
$$370$$ − 24.0000i − 1.24770i
$$371$$ − 9.00000i − 0.467257i
$$372$$ 0 0
$$373$$ 30.0000 1.55334 0.776671 0.629907i $$-0.216907\pi$$
0.776671 + 0.629907i $$0.216907\pi$$
$$374$$ −48.0000 −2.48202
$$375$$ 0 0
$$376$$ 0 0
$$377$$ 0 0
$$378$$ 0 0
$$379$$ − 6.00000i − 0.308199i −0.988055 0.154100i $$-0.950752\pi$$
0.988055 0.154100i $$-0.0492477\pi$$
$$380$$ −30.0000 −1.53897
$$381$$ 0 0
$$382$$ 16.0000i 0.818631i
$$383$$ − 36.0000i − 1.83951i −0.392488 0.919757i $$-0.628386\pi$$
0.392488 0.919757i $$-0.371614\pi$$
$$384$$ 0 0
$$385$$ 18.0000i 0.917365i
$$386$$ −44.0000 −2.23954
$$387$$ −3.00000 −0.152499
$$388$$ − 14.0000i − 0.710742i
$$389$$ −30.0000 −1.52106 −0.760530 0.649303i $$-0.775061\pi$$
−0.760530 + 0.649303i $$0.775061\pi$$
$$390$$ 0 0
$$391$$ 12.0000 0.606866
$$392$$ 0 0
$$393$$ 0 0
$$394$$ 4.00000 0.201517
$$395$$ − 9.00000i − 0.452839i
$$396$$ 36.0000i 1.80907i
$$397$$ 13.0000i 0.652451i 0.945292 + 0.326226i $$0.105777\pi$$
−0.945292 + 0.326226i $$0.894223\pi$$
$$398$$ 8.00000i 0.401004i
$$399$$ 0 0
$$400$$ 16.0000 0.800000
$$401$$ 32.0000i 1.59800i 0.601329 + 0.799002i $$0.294638\pi$$
−0.601329 + 0.799002i $$0.705362\pi$$
$$402$$ 0 0
$$403$$ 0 0
$$404$$ −28.0000 −1.39305
$$405$$ − 27.0000i − 1.34164i
$$406$$ −10.0000 −0.496292
$$407$$ −24.0000 −1.18964
$$408$$ 0 0
$$409$$ − 13.0000i − 0.642809i −0.946942 0.321404i $$-0.895845\pi$$
0.946942 0.321404i $$-0.104155\pi$$
$$410$$ 36.0000i 1.77791i
$$411$$ 0 0
$$412$$ −8.00000 −0.394132
$$413$$ 8.00000 0.393654
$$414$$ − 18.0000i − 0.884652i
$$415$$ 45.0000 2.20896
$$416$$ 0 0
$$417$$ 0 0
$$418$$ 60.0000i 2.93470i
$$419$$ −10.0000 −0.488532 −0.244266 0.969708i $$-0.578547\pi$$
−0.244266 + 0.969708i $$0.578547\pi$$
$$420$$ 0 0
$$421$$ − 12.0000i − 0.584844i −0.956289 0.292422i $$-0.905539\pi$$
0.956289 0.292422i $$-0.0944612\pi$$
$$422$$ 10.0000i 0.486792i
$$423$$ 21.0000i 1.02105i
$$424$$ 0 0
$$425$$ 16.0000 0.776114
$$426$$ 0 0
$$427$$ − 10.0000i − 0.483934i
$$428$$ 8.00000 0.386695
$$429$$ 0 0
$$430$$ −6.00000 −0.289346
$$431$$ 6.00000i 0.289010i 0.989504 + 0.144505i $$0.0461589\pi$$
−0.989504 + 0.144505i $$0.953841\pi$$
$$432$$ 0 0
$$433$$ −12.0000 −0.576683 −0.288342 0.957528i $$-0.593104\pi$$
−0.288342 + 0.957528i $$0.593104\pi$$
$$434$$ − 6.00000i − 0.288009i
$$435$$ 0 0
$$436$$ 4.00000i 0.191565i
$$437$$ − 15.0000i − 0.717547i
$$438$$ 0 0
$$439$$ 22.0000 1.05000 0.525001 0.851101i $$-0.324065\pi$$
0.525001 + 0.851101i $$0.324065\pi$$
$$440$$ 0 0
$$441$$ 3.00000 0.142857
$$442$$ 0 0
$$443$$ 19.0000 0.902717 0.451359 0.892343i $$-0.350940\pi$$
0.451359 + 0.892343i $$0.350940\pi$$
$$444$$ 0 0
$$445$$ −9.00000 −0.426641
$$446$$ 30.0000 1.42054
$$447$$ 0 0
$$448$$ 8.00000i 0.377964i
$$449$$ − 36.0000i − 1.69895i −0.527633 0.849473i $$-0.676920\pi$$
0.527633 0.849473i $$-0.323080\pi$$
$$450$$ − 24.0000i − 1.13137i
$$451$$ 36.0000 1.69517
$$452$$ 6.00000 0.282216
$$453$$ 0 0
$$454$$ 40.0000 1.87729
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$458$$ −28.0000 −1.30835
$$459$$ 0 0
$$460$$ − 18.0000i − 0.839254i
$$461$$ − 22.0000i − 1.02464i −0.858794 0.512321i $$-0.828786\pi$$
0.858794 0.512321i $$-0.171214\pi$$
$$462$$ 0 0
$$463$$ 14.0000i 0.650635i 0.945605 + 0.325318i $$0.105471\pi$$
−0.945605 + 0.325318i $$0.894529\pi$$
$$464$$ 20.0000 0.928477
$$465$$ 0 0
$$466$$ 30.0000i 1.38972i
$$467$$ 22.0000 1.01804 0.509019 0.860755i $$-0.330008\pi$$
0.509019 + 0.860755i $$0.330008\pi$$
$$468$$ 0 0
$$469$$ 6.00000 0.277054
$$470$$ 42.0000i 1.93732i
$$471$$ 0 0
$$472$$ 0 0
$$473$$ 6.00000i 0.275880i
$$474$$ 0 0
$$475$$ − 20.0000i − 0.917663i
$$476$$ 8.00000i 0.366679i
$$477$$ 27.0000 1.23625
$$478$$ −8.00000 −0.365911
$$479$$ 11.0000i 0.502603i 0.967909 + 0.251301i $$0.0808585\pi$$
−0.967909 + 0.251301i $$0.919141\pi$$
$$480$$ 0 0
$$481$$ 0 0
$$482$$ 34.0000 1.54866
$$483$$ 0 0
$$484$$ 50.0000 2.27273
$$485$$ 21.0000 0.953561
$$486$$ 0 0
$$487$$ − 26.0000i − 1.17817i −0.808070 0.589086i $$-0.799488\pi$$
0.808070 0.589086i $$-0.200512\pi$$
$$488$$ 0 0
$$489$$ 0 0
$$490$$ 6.00000 0.271052
$$491$$ 12.0000 0.541552 0.270776 0.962642i $$-0.412720\pi$$
0.270776 + 0.962642i $$0.412720\pi$$
$$492$$ 0 0
$$493$$ 20.0000 0.900755
$$494$$ 0 0
$$495$$ −54.0000 −2.42712
$$496$$ 12.0000i 0.538816i
$$497$$ 8.00000 0.358849
$$498$$ 0 0
$$499$$ − 16.0000i − 0.716258i −0.933672 0.358129i $$-0.883415\pi$$
0.933672 0.358129i $$-0.116585\pi$$
$$500$$ 6.00000i 0.268328i
$$501$$ 0 0
$$502$$ − 52.0000i − 2.32087i
$$503$$ 2.00000 0.0891756 0.0445878 0.999005i $$-0.485803\pi$$
0.0445878 + 0.999005i $$0.485803\pi$$
$$504$$ 0 0
$$505$$ − 42.0000i − 1.86898i
$$506$$ −36.0000 −1.60040
$$507$$ 0 0
$$508$$ −8.00000 −0.354943
$$509$$ − 19.0000i − 0.842160i −0.907023 0.421080i $$-0.861651\pi$$
0.907023 0.421080i $$-0.138349\pi$$
$$510$$ 0 0
$$511$$ −13.0000 −0.575086
$$512$$ − 32.0000i − 1.41421i
$$513$$ 0 0
$$514$$ − 4.00000i − 0.176432i
$$515$$ − 12.0000i − 0.528783i
$$516$$ 0 0
$$517$$ 42.0000 1.84716
$$518$$ 8.00000i 0.351500i
$$519$$ 0 0
$$520$$ 0 0
$$521$$ 40.0000 1.75243 0.876216 0.481919i $$-0.160060\pi$$
0.876216 + 0.481919i $$0.160060\pi$$
$$522$$ − 30.0000i − 1.31306i
$$523$$ 10.0000 0.437269 0.218635 0.975807i $$-0.429840\pi$$
0.218635 + 0.975807i $$0.429840\pi$$
$$524$$ −16.0000 −0.698963
$$525$$ 0 0
$$526$$ 30.0000i 1.30806i
$$527$$ 12.0000i 0.522728i
$$528$$ 0 0
$$529$$ −14.0000 −0.608696
$$530$$ 54.0000 2.34561
$$531$$ 24.0000i 1.04151i
$$532$$ 10.0000 0.433555
$$533$$ 0 0
$$534$$ 0 0
$$535$$ 12.0000i 0.518805i
$$536$$ 0 0
$$537$$ 0 0
$$538$$ 0 0
$$539$$ − 6.00000i − 0.258438i
$$540$$ 0 0
$$541$$ 40.0000i 1.71973i 0.510518 + 0.859867i $$0.329454\pi$$
−0.510518 + 0.859867i $$0.670546\pi$$
$$542$$ −16.0000 −0.687259
$$543$$ 0 0
$$544$$ − 32.0000i − 1.37199i
$$545$$ −6.00000 −0.257012
$$546$$ 0 0
$$547$$ −7.00000 −0.299298 −0.149649 0.988739i $$-0.547814\pi$$
−0.149649 + 0.988739i $$0.547814\pi$$
$$548$$ 8.00000i 0.341743i
$$549$$ 30.0000 1.28037
$$550$$ −48.0000 −2.04673
$$551$$ − 25.0000i − 1.06504i
$$552$$ 0 0
$$553$$ 3.00000i 0.127573i
$$554$$ 2.00000i 0.0849719i
$$555$$ 0 0
$$556$$ 36.0000 1.52674
$$557$$ 12.0000i 0.508456i 0.967144 + 0.254228i $$0.0818214\pi$$
−0.967144 + 0.254228i $$0.918179\pi$$
$$558$$ 18.0000 0.762001
$$559$$ 0 0
$$560$$ −12.0000 −0.507093
$$561$$ 0 0
$$562$$ 60.0000 2.53095
$$563$$ −4.00000 −0.168580 −0.0842900 0.996441i $$-0.526862\pi$$
−0.0842900 + 0.996441i $$0.526862\pi$$
$$564$$ 0 0
$$565$$ 9.00000i 0.378633i
$$566$$ 32.0000i 1.34506i
$$567$$ 9.00000i 0.377964i
$$568$$ 0 0
$$569$$ −7.00000 −0.293455 −0.146728 0.989177i $$-0.546874\pi$$
−0.146728 + 0.989177i $$0.546874\pi$$
$$570$$ 0 0
$$571$$ 17.0000 0.711428 0.355714 0.934595i $$-0.384238\pi$$
0.355714 + 0.934595i $$0.384238\pi$$
$$572$$ 0 0
$$573$$ 0 0
$$574$$ − 12.0000i − 0.500870i
$$575$$ 12.0000 0.500435
$$576$$ −24.0000 −1.00000
$$577$$ 2.00000i 0.0832611i 0.999133 + 0.0416305i $$0.0132552\pi$$
−0.999133 + 0.0416305i $$0.986745\pi$$
$$578$$ 2.00000i 0.0831890i
$$579$$ 0 0
$$580$$ − 30.0000i − 1.24568i
$$581$$ −15.0000 −0.622305
$$582$$ 0 0
$$583$$ − 54.0000i − 2.23645i
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 38.0000 1.56977
$$587$$ 39.0000i 1.60970i 0.593477 + 0.804851i $$0.297755\pi$$
−0.593477 + 0.804851i $$0.702245\pi$$
$$588$$ 0 0
$$589$$ 15.0000 0.618064
$$590$$ 48.0000i 1.97613i
$$591$$ 0 0
$$592$$ − 16.0000i − 0.657596i
$$593$$ 27.0000i 1.10876i 0.832265 + 0.554379i $$0.187044\pi$$
−0.832265 + 0.554379i $$0.812956\pi$$
$$594$$ 0 0
$$595$$ −12.0000 −0.491952
$$596$$ 36.0000i 1.47462i
$$597$$ 0 0
$$598$$ 0 0
$$599$$ 11.0000 0.449448 0.224724 0.974422i $$-0.427852\pi$$
0.224724 + 0.974422i $$0.427852\pi$$
$$600$$ 0 0
$$601$$ −10.0000 −0.407909 −0.203954 0.978980i $$-0.565379\pi$$
−0.203954 + 0.978980i $$0.565379\pi$$
$$602$$ 2.00000 0.0815139
$$603$$ 18.0000i 0.733017i
$$604$$ 0 0
$$605$$ 75.0000i 3.04918i
$$606$$ 0 0
$$607$$ 2.00000 0.0811775 0.0405887 0.999176i $$-0.487077\pi$$
0.0405887 + 0.999176i $$0.487077\pi$$
$$608$$ −40.0000 −1.62221
$$609$$ 0 0
$$610$$ 60.0000 2.42933
$$611$$ 0 0
$$612$$ −24.0000 −0.970143
$$613$$ 8.00000i 0.323117i 0.986863 + 0.161558i $$0.0516520\pi$$
−0.986863 + 0.161558i $$0.948348\pi$$
$$614$$ 66.0000 2.66354
$$615$$ 0 0
$$616$$ 0 0
$$617$$ − 30.0000i − 1.20775i −0.797077 0.603877i $$-0.793622\pi$$
0.797077 0.603877i $$-0.206378\pi$$
$$618$$ 0 0
$$619$$ − 20.0000i − 0.803868i −0.915669 0.401934i $$-0.868338\pi$$
0.915669 0.401934i $$-0.131662\pi$$
$$620$$ 18.0000 0.722897
$$621$$ 0 0
$$622$$ − 12.0000i − 0.481156i
$$623$$ 3.00000 0.120192
$$624$$ 0 0
$$625$$ −29.0000 −1.16000
$$626$$ − 44.0000i − 1.75859i
$$627$$ 0 0
$$628$$ 16.0000 0.638470
$$629$$ − 16.0000i − 0.637962i
$$630$$ 18.0000i 0.717137i
$$631$$ − 22.0000i − 0.875806i −0.899022 0.437903i $$-0.855721\pi$$
0.899022 0.437903i $$-0.144279\pi$$
$$632$$ 0 0
$$633$$ 0 0
$$634$$ −48.0000 −1.90632
$$635$$ − 12.0000i − 0.476205i
$$636$$ 0 0
$$637$$ 0 0
$$638$$ −60.0000 −2.37542
$$639$$ 24.0000i 0.949425i
$$640$$ 0 0
$$641$$ −9.00000 −0.355479 −0.177739 0.984078i $$-0.556878\pi$$
−0.177739 + 0.984078i $$0.556878\pi$$
$$642$$ 0 0
$$643$$ 8.00000i 0.315489i 0.987480 + 0.157745i $$0.0504223\pi$$
−0.987480 + 0.157745i $$0.949578\pi$$
$$644$$ 6.00000i 0.236433i
$$645$$ 0 0
$$646$$ −40.0000 −1.57378
$$647$$ −18.0000 −0.707653 −0.353827 0.935311i $$-0.615120\pi$$
−0.353827 + 0.935311i $$0.615120\pi$$
$$648$$ 0 0
$$649$$ 48.0000 1.88416
$$650$$ 0 0
$$651$$ 0 0
$$652$$ − 8.00000i − 0.313304i
$$653$$ 18.0000 0.704394 0.352197 0.935926i $$-0.385435\pi$$
0.352197 + 0.935926i $$0.385435\pi$$
$$654$$ 0 0
$$655$$ − 24.0000i − 0.937758i
$$656$$ 24.0000i 0.937043i
$$657$$ − 39.0000i − 1.52153i
$$658$$ − 14.0000i − 0.545777i
$$659$$ 17.0000 0.662226 0.331113 0.943591i $$-0.392576\pi$$
0.331113 + 0.943591i $$0.392576\pi$$
$$660$$ 0 0
$$661$$ 33.0000i 1.28355i 0.766892 + 0.641776i $$0.221802\pi$$
−0.766892 + 0.641776i $$0.778198\pi$$
$$662$$ 44.0000 1.71011
$$663$$ 0 0
$$664$$ 0 0
$$665$$ 15.0000i 0.581675i
$$666$$ −24.0000 −0.929981
$$667$$ 15.0000 0.580802
$$668$$ 10.0000i 0.386912i
$$669$$ 0 0
$$670$$ 36.0000i 1.39080i
$$671$$ − 60.0000i − 2.31627i
$$672$$ 0 0
$$673$$ −1.00000 −0.0385472 −0.0192736 0.999814i $$-0.506135\pi$$
−0.0192736 + 0.999814i $$0.506135\pi$$
$$674$$ 34.0000i 1.30963i
$$675$$ 0 0
$$676$$ 0 0
$$677$$ 22.0000 0.845529 0.422764 0.906240i $$-0.361060\pi$$
0.422764 + 0.906240i $$0.361060\pi$$
$$678$$ 0 0
$$679$$ −7.00000 −0.268635
$$680$$ 0 0
$$681$$ 0 0
$$682$$ − 36.0000i − 1.37851i
$$683$$ 12.0000i 0.459167i 0.973289 + 0.229584i $$0.0737364\pi$$
−0.973289 + 0.229584i $$0.926264\pi$$
$$684$$ 30.0000i 1.14708i
$$685$$ −12.0000 −0.458496
$$686$$ −2.00000 −0.0763604
$$687$$ 0 0
$$688$$ −4.00000 −0.152499
$$689$$ 0 0
$$690$$ 0 0
$$691$$ 11.0000i 0.418460i 0.977866 + 0.209230i $$0.0670957\pi$$
−0.977866 + 0.209230i $$0.932904\pi$$
$$692$$ −16.0000 −0.608229
$$693$$ 18.0000 0.683763
$$694$$ 64.0000i 2.42941i
$$695$$ 54.0000i 2.04834i
$$696$$ 0 0
$$697$$ 24.0000i 0.909065i
$$698$$ −22.0000 −0.832712
$$699$$ 0 0
$$700$$ 8.00000i 0.302372i
$$701$$ 27.0000 1.01978 0.509888 0.860241i $$-0.329687\pi$$
0.509888 + 0.860241i $$0.329687\pi$$
$$702$$ 0 0
$$703$$ −20.0000 −0.754314
$$704$$ 48.0000i 1.80907i
$$705$$ 0 0
$$706$$ −20.0000 −0.752710
$$707$$ 14.0000i 0.526524i
$$708$$ 0 0
$$709$$ 10.0000i 0.375558i 0.982211 + 0.187779i $$0.0601289\pi$$
−0.982211 + 0.187779i $$0.939871\pi$$
$$710$$ 48.0000i 1.80141i
$$711$$ −9.00000 −0.337526
$$712$$ 0 0
$$713$$ 9.00000i 0.337053i
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 46.0000 1.71910
$$717$$ 0 0
$$718$$ −40.0000 −1.49279
$$719$$ −18.0000 −0.671287 −0.335643 0.941989i $$-0.608954\pi$$
−0.335643 + 0.941989i $$0.608954\pi$$
$$720$$ − 36.0000i − 1.34164i
$$721$$ 4.00000i 0.148968i
$$722$$ 12.0000i 0.446594i
$$723$$ 0 0
$$724$$ 28.0000 1.04061
$$725$$ 20.0000 0.742781
$$726$$ 0 0
$$727$$ −46.0000 −1.70605 −0.853023 0.521874i $$-0.825233\pi$$
−0.853023 + 0.521874i $$0.825233\pi$$
$$728$$ 0 0
$$729$$ −27.0000 −1.00000
$$730$$ − 78.0000i − 2.88691i
$$731$$ −4.00000 −0.147945
$$732$$ 0 0
$$733$$ 51.0000i 1.88373i 0.335994 + 0.941864i $$0.390928\pi$$
−0.335994 + 0.941864i $$0.609072\pi$$
$$734$$ − 28.0000i − 1.03350i
$$735$$ 0 0
$$736$$ − 24.0000i − 0.884652i
$$737$$ 36.0000 1.32608
$$738$$ 36.0000 1.32518
$$739$$ 26.0000i 0.956425i 0.878244 + 0.478213i $$0.158715\pi$$
−0.878244 + 0.478213i $$0.841285\pi$$
$$740$$ −24.0000 −0.882258
$$741$$ 0 0
$$742$$ −18.0000 −0.660801
$$743$$ 36.0000i 1.32071i 0.750953 + 0.660356i $$0.229595\pi$$
−0.750953 + 0.660356i $$0.770405\pi$$
$$744$$ 0 0
$$745$$ −54.0000 −1.97841
$$746$$ − 60.0000i − 2.19676i
$$747$$ − 45.0000i − 1.64646i
$$748$$ 48.0000i 1.75505i
$$749$$ − 4.00000i − 0.146157i
$$750$$ 0 0
$$751$$ 17.0000 0.620339 0.310169 0.950681i $$-0.399614\pi$$
0.310169 + 0.950681i $$0.399614\pi$$
$$752$$ 28.0000i 1.02105i
$$753$$ 0 0
$$754$$ 0 0
$$755$$ 0 0
$$756$$ 0 0
$$757$$ −15.0000 −0.545184 −0.272592 0.962130i $$-0.587881\pi$$
−0.272592 + 0.962130i $$0.587881\pi$$
$$758$$ −12.0000 −0.435860
$$759$$ 0 0
$$760$$ 0 0
$$761$$ − 9.00000i − 0.326250i −0.986605 0.163125i $$-0.947843\pi$$
0.986605 0.163125i $$-0.0521573\pi$$
$$762$$ 0 0
$$763$$ 2.00000 0.0724049
$$764$$ 16.0000 0.578860
$$765$$ − 36.0000i − 1.30158i
$$766$$ −72.0000 −2.60147
$$767$$ 0 0
$$768$$ 0 0
$$769$$ − 35.0000i − 1.26213i −0.775729 0.631066i $$-0.782618\pi$$
0.775729 0.631066i $$-0.217382\pi$$
$$770$$ 36.0000 1.29735
$$771$$ 0 0
$$772$$ 44.0000i 1.58359i
$$773$$ 54.0000i 1.94225i 0.238581 + 0.971123i $$0.423318\pi$$
−0.238581 + 0.971123i $$0.576682\pi$$
$$774$$ 6.00000i 0.215666i
$$775$$ 12.0000i 0.431053i
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 60.0000i 2.15110i
$$779$$ 30.0000 1.07486
$$780$$ 0 0
$$781$$ 48.0000 1.71758
$$782$$ − 24.0000i − 0.858238i
$$783$$ 0 0
$$784$$ 4.00000 0.142857
$$785$$ 24.0000i 0.856597i
$$786$$ 0 0
$$787$$ − 37.0000i − 1.31891i −0.751745 0.659454i $$-0.770788\pi$$
0.751745 0.659454i $$-0.229212\pi$$
$$788$$ − 4.00000i − 0.142494i
$$789$$ 0 0
$$790$$ −18.0000 −0.640411
$$791$$ − 3.00000i − 0.106668i
$$792$$ 0 0
$$793$$ 0 0
$$794$$ 26.0000 0.922705
$$795$$ 0 0
$$796$$ 8.00000 0.283552
$$797$$ −18.0000 −0.637593 −0.318796 0.947823i $$-0.603279\pi$$
−0.318796 + 0.947823i $$0.603279\pi$$
$$798$$ 0 0
$$799$$ 28.0000i 0.990569i
$$800$$ − 32.0000i − 1.13137i
$$801$$ 9.00000i 0.317999i
$$802$$ 64.0000 2.25992
$$803$$ −78.0000 −2.75256
$$804$$ 0 0
$$805$$ −9.00000 −0.317208
$$806$$ 0 0
$$807$$ 0 0
$$808$$ 0 0
$$809$$ −31.0000 −1.08990 −0.544951 0.838468i $$-0.683452\pi$$
−0.544951 + 0.838468i $$0.683452\pi$$
$$810$$ −54.0000 −1.89737
$$811$$ − 52.0000i − 1.82597i −0.407997 0.912983i $$-0.633772\pi$$
0.407997 0.912983i $$-0.366228\pi$$
$$812$$ 10.0000i 0.350931i
$$813$$ 0 0
$$814$$ 48.0000i 1.68240i
$$815$$ 12.0000 0.420342
$$816$$ 0 0
$$817$$ 5.00000i 0.174928i
$$818$$ −26.0000 −0.909069
$$819$$ 0 0
$$820$$ 36.0000 1.25717
$$821$$ − 6.00000i − 0.209401i −0.994504 0.104701i $$-0.966612\pi$$
0.994504 0.104701i $$-0.0333885\pi$$
$$822$$ 0 0
$$823$$ 32.0000 1.11545 0.557725 0.830026i $$-0.311674\pi$$
0.557725 + 0.830026i $$0.311674\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ − 16.0000i − 0.556711i
$$827$$ − 4.00000i − 0.139094i −0.997579 0.0695468i $$-0.977845\pi$$
0.997579 0.0695468i $$-0.0221553\pi$$
$$828$$ −18.0000 −0.625543
$$829$$ 10.0000 0.347314 0.173657 0.984806i $$-0.444442\pi$$
0.173657 + 0.984806i $$0.444442\pi$$
$$830$$ − 90.0000i − 3.12395i
$$831$$ 0 0
$$832$$ 0 0
$$833$$ 4.00000 0.138592
$$834$$ 0 0
$$835$$ −15.0000 −0.519096
$$836$$ 60.0000 2.07514
$$837$$ 0 0
$$838$$ 20.0000i 0.690889i
$$839$$ 8.00000i 0.276191i 0.990419 + 0.138095i $$0.0440980\pi$$
−0.990419 + 0.138095i $$0.955902\pi$$
$$840$$ 0 0
$$841$$ −4.00000 −0.137931
$$842$$ −24.0000 −0.827095
$$843$$ 0 0
$$844$$ 10.0000 0.344214
$$845$$ 0 0
$$846$$ 42.0000 1.44399
$$847$$ − 25.0000i − 0.859010i
$$848$$ 36.0000 1.23625
$$849$$ 0 0
$$850$$ − 32.0000i − 1.09759i
$$851$$ − 12.0000i − 0.411355i
$$852$$ 0 0
$$853$$ − 45.0000i − 1.54077i −0.637579 0.770385i $$-0.720064\pi$$
0.637579 0.770385i $$-0.279936\pi$$
$$854$$ −20.0000 −0.684386
$$855$$ −45.0000 −1.53897
$$856$$ 0 0
$$857$$ 6.00000 0.204956 0.102478 0.994735i $$-0.467323\pi$$
0.102478 + 0.994735i $$0.467323\pi$$
$$858$$ 0 0
$$859$$ −2.00000 −0.0682391 −0.0341196 0.999418i $$-0.510863\pi$$
−0.0341196 + 0.999418i $$0.510863\pi$$
$$860$$ 6.00000i 0.204598i
$$861$$ 0 0
$$862$$ 12.0000 0.408722
$$863$$ − 32.0000i − 1.08929i −0.838666 0.544646i $$-0.816664\pi$$
0.838666 0.544646i $$-0.183336\pi$$
$$864$$ 0 0
$$865$$ − 24.0000i − 0.816024i
$$866$$ 24.0000i 0.815553i
$$867$$ 0 0
$$868$$ −6.00000 −0.203653
$$869$$ 18.0000i 0.610608i
$$870$$ 0 0
$$871$$ 0 0
$$872$$ 0 0
$$873$$ − 21.0000i − 0.710742i
$$874$$ −30.0000 −1.01477
$$875$$ 3.00000 0.101419
$$876$$ 0 0
$$877$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$878$$ − 44.0000i − 1.48493i
$$879$$ 0 0
$$880$$ −72.0000 −2.42712
$$881$$ 30.0000 1.01073 0.505363 0.862907i $$-0.331359\pi$$
0.505363 + 0.862907i $$0.331359\pi$$
$$882$$ − 6.00000i − 0.202031i
$$883$$ 4.00000 0.134611 0.0673054 0.997732i $$-0.478560\pi$$
0.0673054 + 0.997732i $$0.478560\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ − 38.0000i − 1.27663i
$$887$$ −12.0000 −0.402921 −0.201460 0.979497i $$-0.564569\pi$$
−0.201460 + 0.979497i $$0.564569\pi$$
$$888$$ 0 0
$$889$$ 4.00000i 0.134156i
$$890$$ 18.0000i 0.603361i
$$891$$ 54.0000i 1.80907i
$$892$$ − 30.0000i − 1.00447i
$$893$$ 35.0000 1.17123
$$894$$ 0 0
$$895$$ 69.0000i 2.30642i
$$896$$ 0 0
$$897$$ 0 0
$$898$$ −72.0000 −2.40267
$$899$$ 15.0000i 0.500278i
$$900$$ −24.0000 −0.800000
$$901$$ 36.0000 1.19933
$$902$$ − 72.0000i − 2.39734i
$$903$$ 0 0
$$904$$ 0 0
$$905$$ 42.0000i 1.39613i
$$906$$ 0 0
$$907$$ −7.00000 −0.232431 −0.116216 0.993224i $$-0.537076\pi$$
−0.116216 + 0.993224i $$0.537076\pi$$
$$908$$ − 40.0000i − 1.32745i
$$909$$ −42.0000 −1.39305
$$910$$ 0 0
$$911$$ −15.0000 −0.496972 −0.248486 0.968635i $$-0.579933\pi$$
−0.248486 + 0.968635i $$0.579933\pi$$
$$912$$ 0 0
$$913$$ −90.0000 −2.97857
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 28.0000i 0.925146i
$$917$$ 8.00000i 0.264183i
$$918$$ 0 0
$$919$$ 8.00000 0.263896 0.131948 0.991257i $$-0.457877\pi$$
0.131948 + 0.991257i $$0.457877\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ −44.0000 −1.44906
$$923$$ 0 0
$$924$$ 0 0
$$925$$ − 16.0000i − 0.526077i
$$926$$ 28.0000 0.920137
$$927$$ −12.0000 −0.394132
$$928$$ − 40.0000i − 1.31306i
$$929$$ 5.00000i 0.164045i 0.996630 + 0.0820223i $$0.0261379\pi$$
−0.996630 + 0.0820223i $$0.973862\pi$$
$$930$$ 0 0
$$931$$ − 5.00000i − 0.163868i
$$932$$ 30.0000 0.982683
$$933$$ 0 0
$$934$$ − 44.0000i − 1.43972i
$$935$$ −72.0000 −2.35465
$$936$$ 0 0
$$937$$ −8.00000 −0.261349 −0.130674 0.991425i $$-0.541714\pi$$
−0.130674 + 0.991425i $$0.541714\pi$$
$$938$$ − 12.0000i − 0.391814i
$$939$$ 0 0
$$940$$ 42.0000 1.36989
$$941$$ 55.0000i 1.79295i 0.443096 + 0.896474i $$0.353880\pi$$
−0.443096 + 0.896474i $$0.646120\pi$$
$$942$$ 0 0
$$943$$ 18.0000i 0.586161i
$$944$$ 32.0000i 1.04151i
$$945$$ 0 0
$$946$$ 12.0000 0.390154
$$947$$ 18.0000i 0.584921i 0.956278 + 0.292461i $$0.0944741\pi$$
−0.956278 + 0.292461i $$0.905526\pi$$
$$948$$ 0 0
$$949$$ 0 0
$$950$$ −40.0000 −1.29777
$$951$$ 0 0
$$952$$ 0 0
$$953$$ 39.0000 1.26333 0.631667 0.775240i $$-0.282371\pi$$
0.631667 + 0.775240i $$0.282371\pi$$
$$954$$ − 54.0000i − 1.74831i
$$955$$ 24.0000i 0.776622i
$$956$$ 8.00000i 0.258738i
$$957$$ 0 0
$$958$$ 22.0000 0.710788
$$959$$ 4.00000 0.129167
$$960$$ 0 0
$$961$$ 22.0000 0.709677
$$962$$ 0 0
$$963$$ 12.0000 0.386695
$$964$$ − 34.0000i − 1.09507i
$$965$$ −66.0000 −2.12462
$$966$$ 0 0
$$967$$ 22.0000i 0.707472i 0.935345 + 0.353736i $$0.115089\pi$$
−0.935345 + 0.353736i $$0.884911\pi$$
$$968$$ 0 0
$$969$$ 0 0
$$970$$ − 42.0000i − 1.34854i
$$971$$ 38.0000 1.21948 0.609739 0.792602i $$-0.291274\pi$$
0.609739 + 0.792602i $$0.291274\pi$$
$$972$$ 0 0
$$973$$ − 18.0000i − 0.577054i
$$974$$ −52.0000 −1.66619
$$975$$ 0 0
$$976$$ 40.0000 1.28037
$$977$$ 10.0000i 0.319928i 0.987123 + 0.159964i $$0.0511379\pi$$
−0.987123 + 0.159964i $$0.948862\pi$$
$$978$$ 0 0
$$979$$ 18.0000 0.575282
$$980$$ − 6.00000i − 0.191663i
$$981$$ 6.00000i 0.191565i
$$982$$ − 24.0000i − 0.765871i
$$983$$ − 17.0000i − 0.542216i −0.962549 0.271108i $$-0.912610\pi$$
0.962549 0.271108i $$-0.0873900\pi$$
$$984$$ 0 0
$$985$$ 6.00000 0.191176
$$986$$ − 40.0000i − 1.27386i
$$987$$ 0 0
$$988$$ 0 0
$$989$$ −3.00000 −0.0953945
$$990$$ 108.000i 3.43247i
$$991$$ 4.00000 0.127064 0.0635321 0.997980i $$-0.479763\pi$$
0.0635321 + 0.997980i $$0.479763\pi$$
$$992$$ 24.0000 0.762001
$$993$$ 0 0
$$994$$ − 16.0000i − 0.507489i
$$995$$ 12.0000i 0.380426i
$$996$$ 0 0
$$997$$ −28.0000 −0.886769 −0.443384 0.896332i $$-0.646222\pi$$
−0.443384 + 0.896332i $$0.646222\pi$$
$$998$$ −32.0000 −1.01294
$$999$$ 0 0
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 1183.2.c.b.337.1 2
13.5 odd 4 91.2.a.a.1.1 1
13.8 odd 4 1183.2.a.b.1.1 1
13.12 even 2 inner 1183.2.c.b.337.2 2
39.5 even 4 819.2.a.f.1.1 1
52.31 even 4 1456.2.a.g.1.1 1
65.44 odd 4 2275.2.a.h.1.1 1
91.5 even 12 637.2.e.d.508.1 2
91.18 odd 12 637.2.e.e.79.1 2
91.31 even 12 637.2.e.d.79.1 2
91.34 even 4 8281.2.a.l.1.1 1
91.44 odd 12 637.2.e.e.508.1 2
91.83 even 4 637.2.a.a.1.1 1
104.5 odd 4 5824.2.a.s.1.1 1
104.83 even 4 5824.2.a.t.1.1 1
273.83 odd 4 5733.2.a.l.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
91.2.a.a.1.1 1 13.5 odd 4
637.2.a.a.1.1 1 91.83 even 4
637.2.e.d.79.1 2 91.31 even 12
637.2.e.d.508.1 2 91.5 even 12
637.2.e.e.79.1 2 91.18 odd 12
637.2.e.e.508.1 2 91.44 odd 12
819.2.a.f.1.1 1 39.5 even 4
1183.2.a.b.1.1 1 13.8 odd 4
1183.2.c.b.337.1 2 1.1 even 1 trivial
1183.2.c.b.337.2 2 13.12 even 2 inner
1456.2.a.g.1.1 1 52.31 even 4
2275.2.a.h.1.1 1 65.44 odd 4
5733.2.a.l.1.1 1 273.83 odd 4
5824.2.a.s.1.1 1 104.5 odd 4
5824.2.a.t.1.1 1 104.83 even 4
8281.2.a.l.1.1 1 91.34 even 4