# Properties

 Label 1183.2.c.a.337.2 Level $1183$ Weight $2$ Character 1183.337 Analytic conductor $9.446$ Analytic rank $0$ 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: $$0$$ 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: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 91) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

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

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-2.00000 q^{3} +2.00000 q^{4} +3.00000i q^{5} +1.00000i q^{7} +1.00000 q^{9} +O(q^{10})$$ $$q-2.00000 q^{3} +2.00000 q^{4} +3.00000i q^{5} +1.00000i q^{7} +1.00000 q^{9} -4.00000 q^{12} -6.00000i q^{15} +4.00000 q^{16} +6.00000 q^{17} +7.00000i q^{19} +6.00000i q^{20} -2.00000i q^{21} -3.00000 q^{23} -4.00000 q^{25} +4.00000 q^{27} +2.00000i q^{28} -9.00000 q^{29} -5.00000i q^{31} -3.00000 q^{35} +2.00000 q^{36} +2.00000i q^{37} +6.00000i q^{41} +1.00000 q^{43} +3.00000i q^{45} +3.00000i q^{47} -8.00000 q^{48} -1.00000 q^{49} -12.0000 q^{51} -9.00000 q^{53} -14.0000i q^{57} -12.0000i q^{60} -10.0000 q^{61} +1.00000i q^{63} +8.00000 q^{64} -14.0000i q^{67} +12.0000 q^{68} +6.00000 q^{69} +6.00000i q^{71} +11.0000i q^{73} +8.00000 q^{75} +14.0000i q^{76} -1.00000 q^{79} +12.0000i q^{80} -11.0000 q^{81} -3.00000i q^{83} -4.00000i q^{84} +18.0000i q^{85} +18.0000 q^{87} +15.0000i q^{89} -6.00000 q^{92} +10.0000i q^{93} -21.0000 q^{95} +1.00000i q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 4 q^{3} + 4 q^{4} + 2 q^{9}+O(q^{10})$$ 2 * q - 4 * q^3 + 4 * q^4 + 2 * q^9 $$2 q - 4 q^{3} + 4 q^{4} + 2 q^{9} - 8 q^{12} + 8 q^{16} + 12 q^{17} - 6 q^{23} - 8 q^{25} + 8 q^{27} - 18 q^{29} - 6 q^{35} + 4 q^{36} + 2 q^{43} - 16 q^{48} - 2 q^{49} - 24 q^{51} - 18 q^{53} - 20 q^{61} + 16 q^{64} + 24 q^{68} + 12 q^{69} + 16 q^{75} - 2 q^{79} - 22 q^{81} + 36 q^{87} - 12 q^{92} - 42 q^{95}+O(q^{100})$$ 2 * q - 4 * q^3 + 4 * q^4 + 2 * q^9 - 8 * q^12 + 8 * q^16 + 12 * q^17 - 6 * q^23 - 8 * q^25 + 8 * q^27 - 18 * q^29 - 6 * q^35 + 4 * q^36 + 2 * q^43 - 16 * q^48 - 2 * q^49 - 24 * q^51 - 18 * q^53 - 20 * q^61 + 16 * q^64 + 24 * q^68 + 12 * q^69 + 16 * q^75 - 2 * q^79 - 22 * q^81 + 36 * q^87 - 12 * q^92 - 42 * 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$3$$ −2.00000 −1.15470 −0.577350 0.816497i $$-0.695913\pi$$
−0.577350 + 0.816497i $$0.695913\pi$$
$$4$$ 2.00000 1.00000
$$5$$ 3.00000i 1.34164i 0.741620 + 0.670820i $$0.234058\pi$$
−0.741620 + 0.670820i $$0.765942\pi$$
$$6$$ 0 0
$$7$$ 1.00000i 0.377964i
$$8$$ 0 0
$$9$$ 1.00000 0.333333
$$10$$ 0 0
$$11$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$12$$ −4.00000 −1.15470
$$13$$ 0 0
$$14$$ 0 0
$$15$$ − 6.00000i − 1.54919i
$$16$$ 4.00000 1.00000
$$17$$ 6.00000 1.45521 0.727607 0.685994i $$-0.240633\pi$$
0.727607 + 0.685994i $$0.240633\pi$$
$$18$$ 0 0
$$19$$ 7.00000i 1.60591i 0.596040 + 0.802955i $$0.296740\pi$$
−0.596040 + 0.802955i $$0.703260\pi$$
$$20$$ 6.00000i 1.34164i
$$21$$ − 2.00000i − 0.436436i
$$22$$ 0 0
$$23$$ −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
$$27$$ 4.00000 0.769800
$$28$$ 2.00000i 0.377964i
$$29$$ −9.00000 −1.67126 −0.835629 0.549294i $$-0.814897\pi$$
−0.835629 + 0.549294i $$0.814897\pi$$
$$30$$ 0 0
$$31$$ − 5.00000i − 0.898027i −0.893525 0.449013i $$-0.851776\pi$$
0.893525 0.449013i $$-0.148224\pi$$
$$32$$ 0 0
$$33$$ 0 0
$$34$$ 0 0
$$35$$ −3.00000 −0.507093
$$36$$ 2.00000 0.333333
$$37$$ 2.00000i 0.328798i 0.986394 + 0.164399i $$0.0525685\pi$$
−0.986394 + 0.164399i $$0.947432\pi$$
$$38$$ 0 0
$$39$$ 0 0
$$40$$ 0 0
$$41$$ 6.00000i 0.937043i 0.883452 + 0.468521i $$0.155213\pi$$
−0.883452 + 0.468521i $$0.844787\pi$$
$$42$$ 0 0
$$43$$ 1.00000 0.152499 0.0762493 0.997089i $$-0.475706\pi$$
0.0762493 + 0.997089i $$0.475706\pi$$
$$44$$ 0 0
$$45$$ 3.00000i 0.447214i
$$46$$ 0 0
$$47$$ 3.00000i 0.437595i 0.975770 + 0.218797i $$0.0702134\pi$$
−0.975770 + 0.218797i $$0.929787\pi$$
$$48$$ −8.00000 −1.15470
$$49$$ −1.00000 −0.142857
$$50$$ 0 0
$$51$$ −12.0000 −1.68034
$$52$$ 0 0
$$53$$ −9.00000 −1.23625 −0.618123 0.786082i $$-0.712106\pi$$
−0.618123 + 0.786082i $$0.712106\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 0 0
$$57$$ − 14.0000i − 1.85435i
$$58$$ 0 0
$$59$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$60$$ − 12.0000i − 1.54919i
$$61$$ −10.0000 −1.28037 −0.640184 0.768221i $$-0.721142\pi$$
−0.640184 + 0.768221i $$0.721142\pi$$
$$62$$ 0 0
$$63$$ 1.00000i 0.125988i
$$64$$ 8.00000 1.00000
$$65$$ 0 0
$$66$$ 0 0
$$67$$ − 14.0000i − 1.71037i −0.518321 0.855186i $$-0.673443\pi$$
0.518321 0.855186i $$-0.326557\pi$$
$$68$$ 12.0000 1.45521
$$69$$ 6.00000 0.722315
$$70$$ 0 0
$$71$$ 6.00000i 0.712069i 0.934473 + 0.356034i $$0.115871\pi$$
−0.934473 + 0.356034i $$0.884129\pi$$
$$72$$ 0 0
$$73$$ 11.0000i 1.28745i 0.765256 + 0.643726i $$0.222612\pi$$
−0.765256 + 0.643726i $$0.777388\pi$$
$$74$$ 0 0
$$75$$ 8.00000 0.923760
$$76$$ 14.0000i 1.60591i
$$77$$ 0 0
$$78$$ 0 0
$$79$$ −1.00000 −0.112509 −0.0562544 0.998416i $$-0.517916\pi$$
−0.0562544 + 0.998416i $$0.517916\pi$$
$$80$$ 12.0000i 1.34164i
$$81$$ −11.0000 −1.22222
$$82$$ 0 0
$$83$$ − 3.00000i − 0.329293i −0.986353 0.164646i $$-0.947352\pi$$
0.986353 0.164646i $$-0.0526483\pi$$
$$84$$ − 4.00000i − 0.436436i
$$85$$ 18.0000i 1.95237i
$$86$$ 0 0
$$87$$ 18.0000 1.92980
$$88$$ 0 0
$$89$$ 15.0000i 1.59000i 0.606612 + 0.794998i $$0.292528\pi$$
−0.606612 + 0.794998i $$0.707472\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ −6.00000 −0.625543
$$93$$ 10.0000i 1.03695i
$$94$$ 0 0
$$95$$ −21.0000 −2.15455
$$96$$ 0 0
$$97$$ 1.00000i 0.101535i 0.998711 + 0.0507673i $$0.0161667\pi$$
−0.998711 + 0.0507673i $$0.983833\pi$$
$$98$$ 0 0
$$99$$ 0 0
$$100$$ −8.00000 −0.800000
$$101$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$102$$ 0 0
$$103$$ 4.00000 0.394132 0.197066 0.980390i $$-0.436859\pi$$
0.197066 + 0.980390i $$0.436859\pi$$
$$104$$ 0 0
$$105$$ 6.00000 0.585540
$$106$$ 0 0
$$107$$ −12.0000 −1.16008 −0.580042 0.814587i $$-0.696964\pi$$
−0.580042 + 0.814587i $$0.696964\pi$$
$$108$$ 8.00000 0.769800
$$109$$ 16.0000i 1.53252i 0.642529 + 0.766261i $$0.277885\pi$$
−0.642529 + 0.766261i $$0.722115\pi$$
$$110$$ 0 0
$$111$$ − 4.00000i − 0.379663i
$$112$$ 4.00000i 0.377964i
$$113$$ 9.00000 0.846649 0.423324 0.905978i $$-0.360863\pi$$
0.423324 + 0.905978i $$0.360863\pi$$
$$114$$ 0 0
$$115$$ − 9.00000i − 0.839254i
$$116$$ −18.0000 −1.67126
$$117$$ 0 0
$$118$$ 0 0
$$119$$ 6.00000i 0.550019i
$$120$$ 0 0
$$121$$ 11.0000 1.00000
$$122$$ 0 0
$$123$$ − 12.0000i − 1.08200i
$$124$$ − 10.0000i − 0.898027i
$$125$$ 3.00000i 0.268328i
$$126$$ 0 0
$$127$$ 16.0000 1.41977 0.709885 0.704317i $$-0.248747\pi$$
0.709885 + 0.704317i $$0.248747\pi$$
$$128$$ 0 0
$$129$$ −2.00000 −0.176090
$$130$$ 0 0
$$131$$ −12.0000 −1.04844 −0.524222 0.851581i $$-0.675644\pi$$
−0.524222 + 0.851581i $$0.675644\pi$$
$$132$$ 0 0
$$133$$ −7.00000 −0.606977
$$134$$ 0 0
$$135$$ 12.0000i 1.03280i
$$136$$ 0 0
$$137$$ 6.00000i 0.512615i 0.966595 + 0.256307i $$0.0825059\pi$$
−0.966595 + 0.256307i $$0.917494\pi$$
$$138$$ 0 0
$$139$$ −4.00000 −0.339276 −0.169638 0.985506i $$-0.554260\pi$$
−0.169638 + 0.985506i $$0.554260\pi$$
$$140$$ −6.00000 −0.507093
$$141$$ − 6.00000i − 0.505291i
$$142$$ 0 0
$$143$$ 0 0
$$144$$ 4.00000 0.333333
$$145$$ − 27.0000i − 2.24223i
$$146$$ 0 0
$$147$$ 2.00000 0.164957
$$148$$ 4.00000i 0.328798i
$$149$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$150$$ 0 0
$$151$$ − 10.0000i − 0.813788i −0.913475 0.406894i $$-0.866612\pi$$
0.913475 0.406894i $$-0.133388\pi$$
$$152$$ 0 0
$$153$$ 6.00000 0.485071
$$154$$ 0 0
$$155$$ 15.0000 1.20483
$$156$$ 0 0
$$157$$ 14.0000 1.11732 0.558661 0.829396i $$-0.311315\pi$$
0.558661 + 0.829396i $$0.311315\pi$$
$$158$$ 0 0
$$159$$ 18.0000 1.42749
$$160$$ 0 0
$$161$$ − 3.00000i − 0.236433i
$$162$$ 0 0
$$163$$ − 16.0000i − 1.25322i −0.779334 0.626608i $$-0.784443\pi$$
0.779334 0.626608i $$-0.215557\pi$$
$$164$$ 12.0000i 0.937043i
$$165$$ 0 0
$$166$$ 0 0
$$167$$ − 15.0000i − 1.16073i −0.814355 0.580367i $$-0.802909\pi$$
0.814355 0.580367i $$-0.197091\pi$$
$$168$$ 0 0
$$169$$ 0 0
$$170$$ 0 0
$$171$$ 7.00000i 0.535303i
$$172$$ 2.00000 0.152499
$$173$$ −6.00000 −0.456172 −0.228086 0.973641i $$-0.573247\pi$$
−0.228086 + 0.973641i $$0.573247\pi$$
$$174$$ 0 0
$$175$$ − 4.00000i − 0.302372i
$$176$$ 0 0
$$177$$ 0 0
$$178$$ 0 0
$$179$$ −15.0000 −1.12115 −0.560576 0.828103i $$-0.689420\pi$$
−0.560576 + 0.828103i $$0.689420\pi$$
$$180$$ 6.00000i 0.447214i
$$181$$ 16.0000 1.18927 0.594635 0.803996i $$-0.297296\pi$$
0.594635 + 0.803996i $$0.297296\pi$$
$$182$$ 0 0
$$183$$ 20.0000 1.47844
$$184$$ 0 0
$$185$$ −6.00000 −0.441129
$$186$$ 0 0
$$187$$ 0 0
$$188$$ 6.00000i 0.437595i
$$189$$ 4.00000i 0.290957i
$$190$$ 0 0
$$191$$ 24.0000 1.73658 0.868290 0.496058i $$-0.165220\pi$$
0.868290 + 0.496058i $$0.165220\pi$$
$$192$$ −16.0000 −1.15470
$$193$$ − 22.0000i − 1.58359i −0.610784 0.791797i $$-0.709146\pi$$
0.610784 0.791797i $$-0.290854\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ −2.00000 −0.142857
$$197$$ − 6.00000i − 0.427482i −0.976890 0.213741i $$-0.931435\pi$$
0.976890 0.213741i $$-0.0685649\pi$$
$$198$$ 0 0
$$199$$ −2.00000 −0.141776 −0.0708881 0.997484i $$-0.522583\pi$$
−0.0708881 + 0.997484i $$0.522583\pi$$
$$200$$ 0 0
$$201$$ 28.0000i 1.97497i
$$202$$ 0 0
$$203$$ − 9.00000i − 0.631676i
$$204$$ −24.0000 −1.68034
$$205$$ −18.0000 −1.25717
$$206$$ 0 0
$$207$$ −3.00000 −0.208514
$$208$$ 0 0
$$209$$ 0 0
$$210$$ 0 0
$$211$$ 23.0000 1.58339 0.791693 0.610920i $$-0.209200\pi$$
0.791693 + 0.610920i $$0.209200\pi$$
$$212$$ −18.0000 −1.23625
$$213$$ − 12.0000i − 0.822226i
$$214$$ 0 0
$$215$$ 3.00000i 0.204598i
$$216$$ 0 0
$$217$$ 5.00000 0.339422
$$218$$ 0 0
$$219$$ − 22.0000i − 1.48662i
$$220$$ 0 0
$$221$$ 0 0
$$222$$ 0 0
$$223$$ 1.00000i 0.0669650i 0.999439 + 0.0334825i $$0.0106598\pi$$
−0.999439 + 0.0334825i $$0.989340\pi$$
$$224$$ 0 0
$$225$$ −4.00000 −0.266667
$$226$$ 0 0
$$227$$ − 24.0000i − 1.59294i −0.604681 0.796468i $$-0.706699\pi$$
0.604681 0.796468i $$-0.293301\pi$$
$$228$$ − 28.0000i − 1.85435i
$$229$$ 14.0000i 0.925146i 0.886581 + 0.462573i $$0.153074\pi$$
−0.886581 + 0.462573i $$0.846926\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ 9.00000 0.589610 0.294805 0.955557i $$-0.404745\pi$$
0.294805 + 0.955557i $$0.404745\pi$$
$$234$$ 0 0
$$235$$ −9.00000 −0.587095
$$236$$ 0 0
$$237$$ 2.00000 0.129914
$$238$$ 0 0
$$239$$ 12.0000i 0.776215i 0.921614 + 0.388108i $$0.126871\pi$$
−0.921614 + 0.388108i $$0.873129\pi$$
$$240$$ − 24.0000i − 1.54919i
$$241$$ − 1.00000i − 0.0644157i −0.999481 0.0322078i $$-0.989746\pi$$
0.999481 0.0322078i $$-0.0102538\pi$$
$$242$$ 0 0
$$243$$ 10.0000 0.641500
$$244$$ −20.0000 −1.28037
$$245$$ − 3.00000i − 0.191663i
$$246$$ 0 0
$$247$$ 0 0
$$248$$ 0 0
$$249$$ 6.00000i 0.380235i
$$250$$ 0 0
$$251$$ −12.0000 −0.757433 −0.378717 0.925513i $$-0.623635\pi$$
−0.378717 + 0.925513i $$0.623635\pi$$
$$252$$ 2.00000i 0.125988i
$$253$$ 0 0
$$254$$ 0 0
$$255$$ − 36.0000i − 2.25441i
$$256$$ 16.0000 1.00000
$$257$$ 24.0000 1.49708 0.748539 0.663090i $$-0.230755\pi$$
0.748539 + 0.663090i $$0.230755\pi$$
$$258$$ 0 0
$$259$$ −2.00000 −0.124274
$$260$$ 0 0
$$261$$ −9.00000 −0.557086
$$262$$ 0 0
$$263$$ 9.00000 0.554964 0.277482 0.960731i $$-0.410500\pi$$
0.277482 + 0.960731i $$0.410500\pi$$
$$264$$ 0 0
$$265$$ − 27.0000i − 1.65860i
$$266$$ 0 0
$$267$$ − 30.0000i − 1.83597i
$$268$$ − 28.0000i − 1.71037i
$$269$$ −24.0000 −1.46331 −0.731653 0.681677i $$-0.761251\pi$$
−0.731653 + 0.681677i $$0.761251\pi$$
$$270$$ 0 0
$$271$$ − 16.0000i − 0.971931i −0.873978 0.485965i $$-0.838468\pi$$
0.873978 0.485965i $$-0.161532\pi$$
$$272$$ 24.0000 1.45521
$$273$$ 0 0
$$274$$ 0 0
$$275$$ 0 0
$$276$$ 12.0000 0.722315
$$277$$ 19.0000 1.14160 0.570800 0.821089i $$-0.306633\pi$$
0.570800 + 0.821089i $$0.306633\pi$$
$$278$$ 0 0
$$279$$ − 5.00000i − 0.299342i
$$280$$ 0 0
$$281$$ 12.0000i 0.715860i 0.933748 + 0.357930i $$0.116517\pi$$
−0.933748 + 0.357930i $$0.883483\pi$$
$$282$$ 0 0
$$283$$ 4.00000 0.237775 0.118888 0.992908i $$-0.462067\pi$$
0.118888 + 0.992908i $$0.462067\pi$$
$$284$$ 12.0000i 0.712069i
$$285$$ 42.0000 2.48787
$$286$$ 0 0
$$287$$ −6.00000 −0.354169
$$288$$ 0 0
$$289$$ 19.0000 1.11765
$$290$$ 0 0
$$291$$ − 2.00000i − 0.117242i
$$292$$ 22.0000i 1.28745i
$$293$$ 9.00000i 0.525786i 0.964825 + 0.262893i $$0.0846766\pi$$
−0.964825 + 0.262893i $$0.915323\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 0 0
$$297$$ 0 0
$$298$$ 0 0
$$299$$ 0 0
$$300$$ 16.0000 0.923760
$$301$$ 1.00000i 0.0576390i
$$302$$ 0 0
$$303$$ 0 0
$$304$$ 28.0000i 1.60591i
$$305$$ − 30.0000i − 1.71780i
$$306$$ 0 0
$$307$$ 11.0000i 0.627803i 0.949456 + 0.313902i $$0.101636\pi$$
−0.949456 + 0.313902i $$0.898364\pi$$
$$308$$ 0 0
$$309$$ −8.00000 −0.455104
$$310$$ 0 0
$$311$$ 12.0000 0.680458 0.340229 0.940343i $$-0.389495\pi$$
0.340229 + 0.940343i $$0.389495\pi$$
$$312$$ 0 0
$$313$$ 8.00000 0.452187 0.226093 0.974106i $$-0.427405\pi$$
0.226093 + 0.974106i $$0.427405\pi$$
$$314$$ 0 0
$$315$$ −3.00000 −0.169031
$$316$$ −2.00000 −0.112509
$$317$$ − 12.0000i − 0.673987i −0.941507 0.336994i $$-0.890590\pi$$
0.941507 0.336994i $$-0.109410\pi$$
$$318$$ 0 0
$$319$$ 0 0
$$320$$ 24.0000i 1.34164i
$$321$$ 24.0000 1.33955
$$322$$ 0 0
$$323$$ 42.0000i 2.33694i
$$324$$ −22.0000 −1.22222
$$325$$ 0 0
$$326$$ 0 0
$$327$$ − 32.0000i − 1.76960i
$$328$$ 0 0
$$329$$ −3.00000 −0.165395
$$330$$ 0 0
$$331$$ − 26.0000i − 1.42909i −0.699590 0.714545i $$-0.746634\pi$$
0.699590 0.714545i $$-0.253366\pi$$
$$332$$ − 6.00000i − 0.329293i
$$333$$ 2.00000i 0.109599i
$$334$$ 0 0
$$335$$ 42.0000 2.29471
$$336$$ − 8.00000i − 0.436436i
$$337$$ −5.00000 −0.272367 −0.136184 0.990684i $$-0.543484\pi$$
−0.136184 + 0.990684i $$0.543484\pi$$
$$338$$ 0 0
$$339$$ −18.0000 −0.977626
$$340$$ 36.0000i 1.95237i
$$341$$ 0 0
$$342$$ 0 0
$$343$$ − 1.00000i − 0.0539949i
$$344$$ 0 0
$$345$$ 18.0000i 0.969087i
$$346$$ 0 0
$$347$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$348$$ 36.0000 1.92980
$$349$$ − 1.00000i − 0.0535288i −0.999642 0.0267644i $$-0.991480\pi$$
0.999642 0.0267644i $$-0.00852039\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 0 0
$$353$$ 18.0000i 0.958043i 0.877803 + 0.479022i $$0.159008\pi$$
−0.877803 + 0.479022i $$0.840992\pi$$
$$354$$ 0 0
$$355$$ −18.0000 −0.955341
$$356$$ 30.0000i 1.59000i
$$357$$ − 12.0000i − 0.635107i
$$358$$ 0 0
$$359$$ 18.0000i 0.950004i 0.879985 + 0.475002i $$0.157553\pi$$
−0.879985 + 0.475002i $$0.842447\pi$$
$$360$$ 0 0
$$361$$ −30.0000 −1.57895
$$362$$ 0 0
$$363$$ −22.0000 −1.15470
$$364$$ 0 0
$$365$$ −33.0000 −1.72730
$$366$$ 0 0
$$367$$ −28.0000 −1.46159 −0.730794 0.682598i $$-0.760850\pi$$
−0.730794 + 0.682598i $$0.760850\pi$$
$$368$$ −12.0000 −0.625543
$$369$$ 6.00000i 0.312348i
$$370$$ 0 0
$$371$$ − 9.00000i − 0.467257i
$$372$$ 20.0000i 1.03695i
$$373$$ 14.0000 0.724893 0.362446 0.932005i $$-0.381942\pi$$
0.362446 + 0.932005i $$0.381942\pi$$
$$374$$ 0 0
$$375$$ − 6.00000i − 0.309839i
$$376$$ 0 0
$$377$$ 0 0
$$378$$ 0 0
$$379$$ 16.0000i 0.821865i 0.911666 + 0.410932i $$0.134797\pi$$
−0.911666 + 0.410932i $$0.865203\pi$$
$$380$$ −42.0000 −2.15455
$$381$$ −32.0000 −1.63941
$$382$$ 0 0
$$383$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ 1.00000 0.0508329
$$388$$ 2.00000i 0.101535i
$$389$$ 6.00000 0.304212 0.152106 0.988364i $$-0.451394\pi$$
0.152106 + 0.988364i $$0.451394\pi$$
$$390$$ 0 0
$$391$$ −18.0000 −0.910299
$$392$$ 0 0
$$393$$ 24.0000 1.21064
$$394$$ 0 0
$$395$$ − 3.00000i − 0.150946i
$$396$$ 0 0
$$397$$ 11.0000i 0.552074i 0.961147 + 0.276037i $$0.0890213\pi$$
−0.961147 + 0.276037i $$0.910979\pi$$
$$398$$ 0 0
$$399$$ 14.0000 0.700877
$$400$$ −16.0000 −0.800000
$$401$$ 12.0000i 0.599251i 0.954057 + 0.299626i $$0.0968618\pi$$
−0.954057 + 0.299626i $$0.903138\pi$$
$$402$$ 0 0
$$403$$ 0 0
$$404$$ 0 0
$$405$$ − 33.0000i − 1.63978i
$$406$$ 0 0
$$407$$ 0 0
$$408$$ 0 0
$$409$$ − 23.0000i − 1.13728i −0.822588 0.568638i $$-0.807470\pi$$
0.822588 0.568638i $$-0.192530\pi$$
$$410$$ 0 0
$$411$$ − 12.0000i − 0.591916i
$$412$$ 8.00000 0.394132
$$413$$ 0 0
$$414$$ 0 0
$$415$$ 9.00000 0.441793
$$416$$ 0 0
$$417$$ 8.00000 0.391762
$$418$$ 0 0
$$419$$ 12.0000 0.586238 0.293119 0.956076i $$-0.405307\pi$$
0.293119 + 0.956076i $$0.405307\pi$$
$$420$$ 12.0000 0.585540
$$421$$ 10.0000i 0.487370i 0.969854 + 0.243685i $$0.0783563\pi$$
−0.969854 + 0.243685i $$0.921644\pi$$
$$422$$ 0 0
$$423$$ 3.00000i 0.145865i
$$424$$ 0 0
$$425$$ −24.0000 −1.16417
$$426$$ 0 0
$$427$$ − 10.0000i − 0.483934i
$$428$$ −24.0000 −1.16008
$$429$$ 0 0
$$430$$ 0 0
$$431$$ 30.0000i 1.44505i 0.691345 + 0.722525i $$0.257018\pi$$
−0.691345 + 0.722525i $$0.742982\pi$$
$$432$$ 16.0000 0.769800
$$433$$ 16.0000 0.768911 0.384455 0.923144i $$-0.374389\pi$$
0.384455 + 0.923144i $$0.374389\pi$$
$$434$$ 0 0
$$435$$ 54.0000i 2.58910i
$$436$$ 32.0000i 1.53252i
$$437$$ − 21.0000i − 1.00457i
$$438$$ 0 0
$$439$$ 10.0000 0.477274 0.238637 0.971109i $$-0.423299\pi$$
0.238637 + 0.971109i $$0.423299\pi$$
$$440$$ 0 0
$$441$$ −1.00000 −0.0476190
$$442$$ 0 0
$$443$$ 39.0000 1.85295 0.926473 0.376361i $$-0.122825\pi$$
0.926473 + 0.376361i $$0.122825\pi$$
$$444$$ − 8.00000i − 0.379663i
$$445$$ −45.0000 −2.13320
$$446$$ 0 0
$$447$$ 0 0
$$448$$ 8.00000i 0.377964i
$$449$$ − 24.0000i − 1.13263i −0.824189 0.566315i $$-0.808369\pi$$
0.824189 0.566315i $$-0.191631\pi$$
$$450$$ 0 0
$$451$$ 0 0
$$452$$ 18.0000 0.846649
$$453$$ 20.0000i 0.939682i
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 10.0000i 0.467780i 0.972263 + 0.233890i $$0.0751456\pi$$
−0.972263 + 0.233890i $$0.924854\pi$$
$$458$$ 0 0
$$459$$ 24.0000 1.12022
$$460$$ − 18.0000i − 0.839254i
$$461$$ 30.0000i 1.39724i 0.715493 + 0.698620i $$0.246202\pi$$
−0.715493 + 0.698620i $$0.753798\pi$$
$$462$$ 0 0
$$463$$ − 4.00000i − 0.185896i −0.995671 0.0929479i $$-0.970371\pi$$
0.995671 0.0929479i $$-0.0296290\pi$$
$$464$$ −36.0000 −1.67126
$$465$$ −30.0000 −1.39122
$$466$$ 0 0
$$467$$ 36.0000 1.66588 0.832941 0.553362i $$-0.186655\pi$$
0.832941 + 0.553362i $$0.186655\pi$$
$$468$$ 0 0
$$469$$ 14.0000 0.646460
$$470$$ 0 0
$$471$$ −28.0000 −1.29017
$$472$$ 0 0
$$473$$ 0 0
$$474$$ 0 0
$$475$$ − 28.0000i − 1.28473i
$$476$$ 12.0000i 0.550019i
$$477$$ −9.00000 −0.412082
$$478$$ 0 0
$$479$$ 9.00000i 0.411220i 0.978634 + 0.205610i $$0.0659179\pi$$
−0.978634 + 0.205610i $$0.934082\pi$$
$$480$$ 0 0
$$481$$ 0 0
$$482$$ 0 0
$$483$$ 6.00000i 0.273009i
$$484$$ 22.0000 1.00000
$$485$$ −3.00000 −0.136223
$$486$$ 0 0
$$487$$ − 2.00000i − 0.0906287i −0.998973 0.0453143i $$-0.985571\pi$$
0.998973 0.0453143i $$-0.0144289\pi$$
$$488$$ 0 0
$$489$$ 32.0000i 1.44709i
$$490$$ 0 0
$$491$$ 24.0000 1.08310 0.541552 0.840667i $$-0.317837\pi$$
0.541552 + 0.840667i $$0.317837\pi$$
$$492$$ − 24.0000i − 1.08200i
$$493$$ −54.0000 −2.43204
$$494$$ 0 0
$$495$$ 0 0
$$496$$ − 20.0000i − 0.898027i
$$497$$ −6.00000 −0.269137
$$498$$ 0 0
$$499$$ − 14.0000i − 0.626726i −0.949633 0.313363i $$-0.898544\pi$$
0.949633 0.313363i $$-0.101456\pi$$
$$500$$ 6.00000i 0.268328i
$$501$$ 30.0000i 1.34030i
$$502$$ 0 0
$$503$$ −6.00000 −0.267527 −0.133763 0.991013i $$-0.542706\pi$$
−0.133763 + 0.991013i $$0.542706\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 0 0
$$507$$ 0 0
$$508$$ 32.0000 1.41977
$$509$$ − 21.0000i − 0.930809i −0.885098 0.465404i $$-0.845909\pi$$
0.885098 0.465404i $$-0.154091\pi$$
$$510$$ 0 0
$$511$$ −11.0000 −0.486611
$$512$$ 0 0
$$513$$ 28.0000i 1.23623i
$$514$$ 0 0
$$515$$ 12.0000i 0.528783i
$$516$$ −4.00000 −0.176090
$$517$$ 0 0
$$518$$ 0 0
$$519$$ 12.0000 0.526742
$$520$$ 0 0
$$521$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$522$$ 0 0
$$523$$ 2.00000 0.0874539 0.0437269 0.999044i $$-0.486077\pi$$
0.0437269 + 0.999044i $$0.486077\pi$$
$$524$$ −24.0000 −1.04844
$$525$$ 8.00000i 0.349149i
$$526$$ 0 0
$$527$$ − 30.0000i − 1.30682i
$$528$$ 0 0
$$529$$ −14.0000 −0.608696
$$530$$ 0 0
$$531$$ 0 0
$$532$$ −14.0000 −0.606977
$$533$$ 0 0
$$534$$ 0 0
$$535$$ − 36.0000i − 1.55642i
$$536$$ 0 0
$$537$$ 30.0000 1.29460
$$538$$ 0 0
$$539$$ 0 0
$$540$$ 24.0000i 1.03280i
$$541$$ − 34.0000i − 1.46177i −0.682498 0.730887i $$-0.739107\pi$$
0.682498 0.730887i $$-0.260893\pi$$
$$542$$ 0 0
$$543$$ −32.0000 −1.37325
$$544$$ 0 0
$$545$$ −48.0000 −2.05609
$$546$$ 0 0
$$547$$ 17.0000 0.726868 0.363434 0.931620i $$-0.381604\pi$$
0.363434 + 0.931620i $$0.381604\pi$$
$$548$$ 12.0000i 0.512615i
$$549$$ −10.0000 −0.426790
$$550$$ 0 0
$$551$$ − 63.0000i − 2.68389i
$$552$$ 0 0
$$553$$ − 1.00000i − 0.0425243i
$$554$$ 0 0
$$555$$ 12.0000 0.509372
$$556$$ −8.00000 −0.339276
$$557$$ − 12.0000i − 0.508456i −0.967144 0.254228i $$-0.918179\pi$$
0.967144 0.254228i $$-0.0818214\pi$$
$$558$$ 0 0
$$559$$ 0 0
$$560$$ −12.0000 −0.507093
$$561$$ 0 0
$$562$$ 0 0
$$563$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$564$$ − 12.0000i − 0.505291i
$$565$$ 27.0000i 1.13590i
$$566$$ 0 0
$$567$$ − 11.0000i − 0.461957i
$$568$$ 0 0
$$569$$ −27.0000 −1.13190 −0.565949 0.824440i $$-0.691490\pi$$
−0.565949 + 0.824440i $$0.691490\pi$$
$$570$$ 0 0
$$571$$ −23.0000 −0.962520 −0.481260 0.876578i $$-0.659821\pi$$
−0.481260 + 0.876578i $$0.659821\pi$$
$$572$$ 0 0
$$573$$ −48.0000 −2.00523
$$574$$ 0 0
$$575$$ 12.0000 0.500435
$$576$$ 8.00000 0.333333
$$577$$ 34.0000i 1.41544i 0.706494 + 0.707719i $$0.250276\pi$$
−0.706494 + 0.707719i $$0.749724\pi$$
$$578$$ 0 0
$$579$$ 44.0000i 1.82858i
$$580$$ − 54.0000i − 2.24223i
$$581$$ 3.00000 0.124461
$$582$$ 0 0
$$583$$ 0 0
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ 9.00000i 0.371470i 0.982600 + 0.185735i $$0.0594666\pi$$
−0.982600 + 0.185735i $$0.940533\pi$$
$$588$$ 4.00000 0.164957
$$589$$ 35.0000 1.44215
$$590$$ 0 0
$$591$$ 12.0000i 0.493614i
$$592$$ 8.00000i 0.328798i
$$593$$ 9.00000i 0.369586i 0.982777 + 0.184793i $$0.0591614\pi$$
−0.982777 + 0.184793i $$0.940839\pi$$
$$594$$ 0 0
$$595$$ −18.0000 −0.737928
$$596$$ 0 0
$$597$$ 4.00000 0.163709
$$598$$ 0 0
$$599$$ 15.0000 0.612883 0.306442 0.951889i $$-0.400862\pi$$
0.306442 + 0.951889i $$0.400862\pi$$
$$600$$ 0 0
$$601$$ 26.0000 1.06056 0.530281 0.847822i $$-0.322086\pi$$
0.530281 + 0.847822i $$0.322086\pi$$
$$602$$ 0 0
$$603$$ − 14.0000i − 0.570124i
$$604$$ − 20.0000i − 0.813788i
$$605$$ 33.0000i 1.34164i
$$606$$ 0 0
$$607$$ −4.00000 −0.162355 −0.0811775 0.996700i $$-0.525868\pi$$
−0.0811775 + 0.996700i $$0.525868\pi$$
$$608$$ 0 0
$$609$$ 18.0000i 0.729397i
$$610$$ 0 0
$$611$$ 0 0
$$612$$ 12.0000 0.485071
$$613$$ 16.0000i 0.646234i 0.946359 + 0.323117i $$0.104731\pi$$
−0.946359 + 0.323117i $$0.895269\pi$$
$$614$$ 0 0
$$615$$ 36.0000 1.45166
$$616$$ 0 0
$$617$$ − 18.0000i − 0.724653i −0.932051 0.362326i $$-0.881983\pi$$
0.932051 0.362326i $$-0.118017\pi$$
$$618$$ 0 0
$$619$$ − 28.0000i − 1.12542i −0.826656 0.562708i $$-0.809760\pi$$
0.826656 0.562708i $$-0.190240\pi$$
$$620$$ 30.0000 1.20483
$$621$$ −12.0000 −0.481543
$$622$$ 0 0
$$623$$ −15.0000 −0.600962
$$624$$ 0 0
$$625$$ −29.0000 −1.16000
$$626$$ 0 0
$$627$$ 0 0
$$628$$ 28.0000 1.11732
$$629$$ 12.0000i 0.478471i
$$630$$ 0 0
$$631$$ 2.00000i 0.0796187i 0.999207 + 0.0398094i $$0.0126751\pi$$
−0.999207 + 0.0398094i $$0.987325\pi$$
$$632$$ 0 0
$$633$$ −46.0000 −1.82834
$$634$$ 0 0
$$635$$ 48.0000i 1.90482i
$$636$$ 36.0000 1.42749
$$637$$ 0 0
$$638$$ 0 0
$$639$$ 6.00000i 0.237356i
$$640$$ 0 0
$$641$$ 39.0000 1.54041 0.770204 0.637798i $$-0.220155\pi$$
0.770204 + 0.637798i $$0.220155\pi$$
$$642$$ 0 0
$$643$$ 4.00000i 0.157745i 0.996885 + 0.0788723i $$0.0251319\pi$$
−0.996885 + 0.0788723i $$0.974868\pi$$
$$644$$ − 6.00000i − 0.236433i
$$645$$ − 6.00000i − 0.236250i
$$646$$ 0 0
$$647$$ 48.0000 1.88707 0.943537 0.331266i $$-0.107476\pi$$
0.943537 + 0.331266i $$0.107476\pi$$
$$648$$ 0 0
$$649$$ 0 0
$$650$$ 0 0
$$651$$ −10.0000 −0.391931
$$652$$ − 32.0000i − 1.25322i
$$653$$ −6.00000 −0.234798 −0.117399 0.993085i $$-0.537456\pi$$
−0.117399 + 0.993085i $$0.537456\pi$$
$$654$$ 0 0
$$655$$ − 36.0000i − 1.40664i
$$656$$ 24.0000i 0.937043i
$$657$$ 11.0000i 0.429151i
$$658$$ 0 0
$$659$$ −3.00000 −0.116863 −0.0584317 0.998291i $$-0.518610\pi$$
−0.0584317 + 0.998291i $$0.518610\pi$$
$$660$$ 0 0
$$661$$ − 13.0000i − 0.505641i −0.967513 0.252821i $$-0.918642\pi$$
0.967513 0.252821i $$-0.0813583\pi$$
$$662$$ 0 0
$$663$$ 0 0
$$664$$ 0 0
$$665$$ − 21.0000i − 0.814345i
$$666$$ 0 0
$$667$$ 27.0000 1.04544
$$668$$ − 30.0000i − 1.16073i
$$669$$ − 2.00000i − 0.0773245i
$$670$$ 0 0
$$671$$ 0 0
$$672$$ 0 0
$$673$$ 19.0000 0.732396 0.366198 0.930537i $$-0.380659\pi$$
0.366198 + 0.930537i $$0.380659\pi$$
$$674$$ 0 0
$$675$$ −16.0000 −0.615840
$$676$$ 0 0
$$677$$ 12.0000 0.461197 0.230599 0.973049i $$-0.425932\pi$$
0.230599 + 0.973049i $$0.425932\pi$$
$$678$$ 0 0
$$679$$ −1.00000 −0.0383765
$$680$$ 0 0
$$681$$ 48.0000i 1.83936i
$$682$$ 0 0
$$683$$ 12.0000i 0.459167i 0.973289 + 0.229584i $$0.0737364\pi$$
−0.973289 + 0.229584i $$0.926264\pi$$
$$684$$ 14.0000i 0.535303i
$$685$$ −18.0000 −0.687745
$$686$$ 0 0
$$687$$ − 28.0000i − 1.06827i
$$688$$ 4.00000 0.152499
$$689$$ 0 0
$$690$$ 0 0
$$691$$ 37.0000i 1.40755i 0.710425 + 0.703773i $$0.248503\pi$$
−0.710425 + 0.703773i $$0.751497\pi$$
$$692$$ −12.0000 −0.456172
$$693$$ 0 0
$$694$$ 0 0
$$695$$ − 12.0000i − 0.455186i
$$696$$ 0 0
$$697$$ 36.0000i 1.36360i
$$698$$ 0 0
$$699$$ −18.0000 −0.680823
$$700$$ − 8.00000i − 0.302372i
$$701$$ −9.00000 −0.339925 −0.169963 0.985451i $$-0.554365\pi$$
−0.169963 + 0.985451i $$0.554365\pi$$
$$702$$ 0 0
$$703$$ −14.0000 −0.528020
$$704$$ 0 0
$$705$$ 18.0000 0.677919
$$706$$ 0 0
$$707$$ 0 0
$$708$$ 0 0
$$709$$ 8.00000i 0.300446i 0.988652 + 0.150223i $$0.0479992\pi$$
−0.988652 + 0.150223i $$0.952001\pi$$
$$710$$ 0 0
$$711$$ −1.00000 −0.0375029
$$712$$ 0 0
$$713$$ 15.0000i 0.561754i
$$714$$ 0 0
$$715$$ 0 0
$$716$$ −30.0000 −1.12115
$$717$$ − 24.0000i − 0.896296i
$$718$$ 0 0
$$719$$ −12.0000 −0.447524 −0.223762 0.974644i $$-0.571834\pi$$
−0.223762 + 0.974644i $$0.571834\pi$$
$$720$$ 12.0000i 0.447214i
$$721$$ 4.00000i 0.148968i
$$722$$ 0 0
$$723$$ 2.00000i 0.0743808i
$$724$$ 32.0000 1.18927
$$725$$ 36.0000 1.33701
$$726$$ 0 0
$$727$$ −26.0000 −0.964287 −0.482143 0.876092i $$-0.660142\pi$$
−0.482143 + 0.876092i $$0.660142\pi$$
$$728$$ 0 0
$$729$$ 13.0000 0.481481
$$730$$ 0 0
$$731$$ 6.00000 0.221918
$$732$$ 40.0000 1.47844
$$733$$ 49.0000i 1.80986i 0.425564 + 0.904928i $$0.360076\pi$$
−0.425564 + 0.904928i $$0.639924\pi$$
$$734$$ 0 0
$$735$$ 6.00000i 0.221313i
$$736$$ 0 0
$$737$$ 0 0
$$738$$ 0 0
$$739$$ 20.0000i 0.735712i 0.929883 + 0.367856i $$0.119908\pi$$
−0.929883 + 0.367856i $$0.880092\pi$$
$$740$$ −12.0000 −0.441129
$$741$$ 0 0
$$742$$ 0 0
$$743$$ − 48.0000i − 1.76095i −0.474093 0.880475i $$-0.657224\pi$$
0.474093 0.880475i $$-0.342776\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 0 0
$$747$$ − 3.00000i − 0.109764i
$$748$$ 0 0
$$749$$ − 12.0000i − 0.438470i
$$750$$ 0 0
$$751$$ 13.0000 0.474377 0.237188 0.971464i $$-0.423774\pi$$
0.237188 + 0.971464i $$0.423774\pi$$
$$752$$ 12.0000i 0.437595i
$$753$$ 24.0000 0.874609
$$754$$ 0 0
$$755$$ 30.0000 1.09181
$$756$$ 8.00000i 0.290957i
$$757$$ −43.0000 −1.56286 −0.781431 0.623992i $$-0.785510\pi$$
−0.781431 + 0.623992i $$0.785510\pi$$
$$758$$ 0 0
$$759$$ 0 0
$$760$$ 0 0
$$761$$ − 27.0000i − 0.978749i −0.872074 0.489375i $$-0.837225\pi$$
0.872074 0.489375i $$-0.162775\pi$$
$$762$$ 0 0
$$763$$ −16.0000 −0.579239
$$764$$ 48.0000 1.73658
$$765$$ 18.0000i 0.650791i
$$766$$ 0 0
$$767$$ 0 0
$$768$$ −32.0000 −1.15470
$$769$$ − 5.00000i − 0.180305i −0.995928 0.0901523i $$-0.971265\pi$$
0.995928 0.0901523i $$-0.0287354\pi$$
$$770$$ 0 0
$$771$$ −48.0000 −1.72868
$$772$$ − 44.0000i − 1.58359i
$$773$$ − 30.0000i − 1.07903i −0.841978 0.539513i $$-0.818609\pi$$
0.841978 0.539513i $$-0.181391\pi$$
$$774$$ 0 0
$$775$$ 20.0000i 0.718421i
$$776$$ 0 0
$$777$$ 4.00000 0.143499
$$778$$ 0 0
$$779$$ −42.0000 −1.50481
$$780$$ 0 0
$$781$$ 0 0
$$782$$ 0 0
$$783$$ −36.0000 −1.28654
$$784$$ −4.00000 −0.142857
$$785$$ 42.0000i 1.49904i
$$786$$ 0 0
$$787$$ 5.00000i 0.178231i 0.996021 + 0.0891154i $$0.0284040\pi$$
−0.996021 + 0.0891154i $$0.971596\pi$$
$$788$$ − 12.0000i − 0.427482i
$$789$$ −18.0000 −0.640817
$$790$$ 0 0
$$791$$ 9.00000i 0.320003i
$$792$$ 0 0
$$793$$ 0 0
$$794$$ 0 0
$$795$$ 54.0000i 1.91518i
$$796$$ −4.00000 −0.141776
$$797$$ −54.0000 −1.91278 −0.956389 0.292096i $$-0.905647\pi$$
−0.956389 + 0.292096i $$0.905647\pi$$
$$798$$ 0 0
$$799$$ 18.0000i 0.636794i
$$800$$ 0 0
$$801$$ 15.0000i 0.529999i
$$802$$ 0 0
$$803$$ 0 0
$$804$$ 56.0000i 1.97497i
$$805$$ 9.00000 0.317208
$$806$$ 0 0
$$807$$ 48.0000 1.68968
$$808$$ 0 0
$$809$$ −3.00000 −0.105474 −0.0527372 0.998608i $$-0.516795\pi$$
−0.0527372 + 0.998608i $$0.516795\pi$$
$$810$$ 0 0
$$811$$ − 20.0000i − 0.702295i −0.936320 0.351147i $$-0.885792\pi$$
0.936320 0.351147i $$-0.114208\pi$$
$$812$$ − 18.0000i − 0.631676i
$$813$$ 32.0000i 1.12229i
$$814$$ 0 0
$$815$$ 48.0000 1.68137
$$816$$ −48.0000 −1.68034
$$817$$ 7.00000i 0.244899i
$$818$$ 0 0
$$819$$ 0 0
$$820$$ −36.0000 −1.25717
$$821$$ 54.0000i 1.88461i 0.334751 + 0.942306i $$0.391348\pi$$
−0.334751 + 0.942306i $$0.608652\pi$$
$$822$$ 0 0
$$823$$ −32.0000 −1.11545 −0.557725 0.830026i $$-0.688326\pi$$
−0.557725 + 0.830026i $$0.688326\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$828$$ −6.00000 −0.208514
$$829$$ −20.0000 −0.694629 −0.347314 0.937749i $$-0.612906\pi$$
−0.347314 + 0.937749i $$0.612906\pi$$
$$830$$ 0 0
$$831$$ −38.0000 −1.31821
$$832$$ 0 0
$$833$$ −6.00000 −0.207888
$$834$$ 0 0
$$835$$ 45.0000 1.55729
$$836$$ 0 0
$$837$$ − 20.0000i − 0.691301i
$$838$$ 0 0
$$839$$ − 12.0000i − 0.414286i −0.978311 0.207143i $$-0.933583\pi$$
0.978311 0.207143i $$-0.0664165\pi$$
$$840$$ 0 0
$$841$$ 52.0000 1.79310
$$842$$ 0 0
$$843$$ − 24.0000i − 0.826604i
$$844$$ 46.0000 1.58339
$$845$$ 0 0
$$846$$ 0 0
$$847$$ 11.0000i 0.377964i
$$848$$ −36.0000 −1.23625
$$849$$ −8.00000 −0.274559
$$850$$ 0 0
$$851$$ − 6.00000i − 0.205677i
$$852$$ − 24.0000i − 0.822226i
$$853$$ 17.0000i 0.582069i 0.956713 + 0.291034i $$0.0939994\pi$$
−0.956713 + 0.291034i $$0.906001\pi$$
$$854$$ 0 0
$$855$$ −21.0000 −0.718185
$$856$$ 0 0
$$857$$ 42.0000 1.43469 0.717346 0.696717i $$-0.245357\pi$$
0.717346 + 0.696717i $$0.245357\pi$$
$$858$$ 0 0
$$859$$ 50.0000 1.70598 0.852989 0.521929i $$-0.174787\pi$$
0.852989 + 0.521929i $$0.174787\pi$$
$$860$$ 6.00000i 0.204598i
$$861$$ 12.0000 0.408959
$$862$$ 0 0
$$863$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$864$$ 0 0
$$865$$ − 18.0000i − 0.612018i
$$866$$ 0 0
$$867$$ −38.0000 −1.29055
$$868$$ 10.0000 0.339422
$$869$$ 0 0
$$870$$ 0 0
$$871$$ 0 0
$$872$$ 0 0
$$873$$ 1.00000i 0.0338449i
$$874$$ 0 0
$$875$$ −3.00000 −0.101419
$$876$$ − 44.0000i − 1.48662i
$$877$$ 22.0000i 0.742887i 0.928456 + 0.371444i $$0.121137\pi$$
−0.928456 + 0.371444i $$0.878863\pi$$
$$878$$ 0 0
$$879$$ − 18.0000i − 0.607125i
$$880$$ 0 0
$$881$$ −6.00000 −0.202145 −0.101073 0.994879i $$-0.532227\pi$$
−0.101073 + 0.994879i $$0.532227\pi$$
$$882$$ 0 0
$$883$$ 16.0000 0.538443 0.269221 0.963078i $$-0.413234\pi$$
0.269221 + 0.963078i $$0.413234\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ −24.0000 −0.805841 −0.402921 0.915235i $$-0.632005\pi$$
−0.402921 + 0.915235i $$0.632005\pi$$
$$888$$ 0 0
$$889$$ 16.0000i 0.536623i
$$890$$ 0 0
$$891$$ 0 0
$$892$$ 2.00000i 0.0669650i
$$893$$ −21.0000 −0.702738
$$894$$ 0 0
$$895$$ − 45.0000i − 1.50418i
$$896$$ 0 0
$$897$$ 0 0
$$898$$ 0 0
$$899$$ 45.0000i 1.50083i
$$900$$ −8.00000 −0.266667
$$901$$ −54.0000 −1.79900
$$902$$ 0 0
$$903$$ − 2.00000i − 0.0665558i
$$904$$ 0 0
$$905$$ 48.0000i 1.59557i
$$906$$ 0 0
$$907$$ 37.0000 1.22856 0.614282 0.789086i $$-0.289446\pi$$
0.614282 + 0.789086i $$0.289446\pi$$
$$908$$ − 48.0000i − 1.59294i
$$909$$ 0 0
$$910$$ 0 0
$$911$$ −15.0000 −0.496972 −0.248486 0.968635i $$-0.579933\pi$$
−0.248486 + 0.968635i $$0.579933\pi$$
$$912$$ − 56.0000i − 1.85435i
$$913$$ 0 0
$$914$$ 0 0
$$915$$ 60.0000i 1.98354i
$$916$$ 28.0000i 0.925146i
$$917$$ − 12.0000i − 0.396275i
$$918$$ 0 0
$$919$$ −16.0000 −0.527791 −0.263896 0.964551i $$-0.585007\pi$$
−0.263896 + 0.964551i $$0.585007\pi$$
$$920$$ 0 0
$$921$$ − 22.0000i − 0.724925i
$$922$$ 0 0
$$923$$ 0 0
$$924$$ 0 0
$$925$$ − 8.00000i − 0.263038i
$$926$$ 0 0
$$927$$ 4.00000 0.131377
$$928$$ 0 0
$$929$$ 3.00000i 0.0984268i 0.998788 + 0.0492134i $$0.0156714\pi$$
−0.998788 + 0.0492134i $$0.984329\pi$$
$$930$$ 0 0
$$931$$ − 7.00000i − 0.229416i
$$932$$ 18.0000 0.589610
$$933$$ −24.0000 −0.785725
$$934$$ 0 0
$$935$$ 0 0
$$936$$ 0 0
$$937$$ 2.00000 0.0653372 0.0326686 0.999466i $$-0.489599\pi$$
0.0326686 + 0.999466i $$0.489599\pi$$
$$938$$ 0 0
$$939$$ −16.0000 −0.522140
$$940$$ −18.0000 −0.587095
$$941$$ − 15.0000i − 0.488986i −0.969651 0.244493i $$-0.921378\pi$$
0.969651 0.244493i $$-0.0786215\pi$$
$$942$$ 0 0
$$943$$ − 18.0000i − 0.586161i
$$944$$ 0 0
$$945$$ −12.0000 −0.390360
$$946$$ 0 0
$$947$$ 54.0000i 1.75476i 0.479792 + 0.877382i $$0.340712\pi$$
−0.479792 + 0.877382i $$0.659288\pi$$
$$948$$ 4.00000 0.129914
$$949$$ 0 0
$$950$$ 0 0
$$951$$ 24.0000i 0.778253i
$$952$$ 0 0
$$953$$ −21.0000 −0.680257 −0.340128 0.940379i $$-0.610471\pi$$
−0.340128 + 0.940379i $$0.610471\pi$$
$$954$$ 0 0
$$955$$ 72.0000i 2.32987i
$$956$$ 24.0000i 0.776215i
$$957$$ 0 0
$$958$$ 0 0
$$959$$ −6.00000 −0.193750
$$960$$ − 48.0000i − 1.54919i
$$961$$ 6.00000 0.193548
$$962$$ 0 0
$$963$$ −12.0000 −0.386695
$$964$$ − 2.00000i − 0.0644157i
$$965$$ 66.0000 2.12462
$$966$$ 0 0
$$967$$ − 32.0000i − 1.02905i −0.857475 0.514525i $$-0.827968\pi$$
0.857475 0.514525i $$-0.172032\pi$$
$$968$$ 0 0
$$969$$ − 84.0000i − 2.69847i
$$970$$ 0 0
$$971$$ −6.00000 −0.192549 −0.0962746 0.995355i $$-0.530693\pi$$
−0.0962746 + 0.995355i $$0.530693\pi$$
$$972$$ 20.0000 0.641500
$$973$$ − 4.00000i − 0.128234i
$$974$$ 0 0
$$975$$ 0 0
$$976$$ −40.0000 −1.28037
$$977$$ − 48.0000i − 1.53566i −0.640656 0.767828i $$-0.721338\pi$$
0.640656 0.767828i $$-0.278662\pi$$
$$978$$ 0 0
$$979$$ 0 0
$$980$$ − 6.00000i − 0.191663i
$$981$$ 16.0000i 0.510841i
$$982$$ 0 0
$$983$$ 57.0000i 1.81802i 0.416777 + 0.909009i $$0.363160\pi$$
−0.416777 + 0.909009i $$0.636840\pi$$
$$984$$ 0 0
$$985$$ 18.0000 0.573528
$$986$$ 0 0
$$987$$ 6.00000 0.190982
$$988$$ 0 0
$$989$$ −3.00000 −0.0953945
$$990$$ 0 0
$$991$$ −16.0000 −0.508257 −0.254128 0.967170i $$-0.581789\pi$$
−0.254128 + 0.967170i $$0.581789\pi$$
$$992$$ 0 0
$$993$$ 52.0000i 1.65017i
$$994$$ 0 0
$$995$$ − 6.00000i − 0.190213i
$$996$$ 12.0000i 0.380235i
$$997$$ 8.00000 0.253363 0.126681 0.991943i $$-0.459567\pi$$
0.126681 + 0.991943i $$0.459567\pi$$
$$998$$ 0 0
$$999$$ 8.00000i 0.253109i
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.a.337.2 2
13.5 odd 4 1183.2.a.a.1.1 1
13.8 odd 4 91.2.a.b.1.1 1
13.12 even 2 inner 1183.2.c.a.337.1 2
39.8 even 4 819.2.a.c.1.1 1
52.47 even 4 1456.2.a.k.1.1 1
65.34 odd 4 2275.2.a.d.1.1 1
91.34 even 4 637.2.a.b.1.1 1
91.47 even 12 637.2.e.b.508.1 2
91.60 odd 12 637.2.e.c.79.1 2
91.73 even 12 637.2.e.b.79.1 2
91.83 even 4 8281.2.a.h.1.1 1
91.86 odd 12 637.2.e.c.508.1 2
104.21 odd 4 5824.2.a.bd.1.1 1
104.99 even 4 5824.2.a.f.1.1 1
273.125 odd 4 5733.2.a.f.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
91.2.a.b.1.1 1 13.8 odd 4
637.2.a.b.1.1 1 91.34 even 4
637.2.e.b.79.1 2 91.73 even 12
637.2.e.b.508.1 2 91.47 even 12
637.2.e.c.79.1 2 91.60 odd 12
637.2.e.c.508.1 2 91.86 odd 12
819.2.a.c.1.1 1 39.8 even 4
1183.2.a.a.1.1 1 13.5 odd 4
1183.2.c.a.337.1 2 13.12 even 2 inner
1183.2.c.a.337.2 2 1.1 even 1 trivial
1456.2.a.k.1.1 1 52.47 even 4
2275.2.a.d.1.1 1 65.34 odd 4
5733.2.a.f.1.1 1 273.125 odd 4
5824.2.a.f.1.1 1 104.99 even 4
5824.2.a.bd.1.1 1 104.21 odd 4
8281.2.a.h.1.1 1 91.83 even 4