Properties

 Label 384.2.d.a.193.2 Level $384$ Weight $2$ Character 384.193 Analytic conductor $3.066$ 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] = [384,2,Mod(193,384)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(384, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("384.193");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$384 = 2^{7} \cdot 3$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 384.d (of order $$2$$, degree $$1$$, 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: $$3.06625543762$$ 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: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

 Embedding label 193.2 Root $$1.00000i$$ of defining polynomial Character $$\chi$$ $$=$$ 384.193 Dual form 384.2.d.a.193.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+1.00000i q^{3} -4.00000 q^{7} -1.00000 q^{9} +O(q^{10})$$ $$q+1.00000i q^{3} -4.00000 q^{7} -1.00000 q^{9} +4.00000i q^{11} +4.00000i q^{13} -2.00000 q^{17} +4.00000i q^{19} -4.00000i q^{21} -8.00000 q^{23} +5.00000 q^{25} -1.00000i q^{27} -8.00000i q^{29} +4.00000 q^{31} -4.00000 q^{33} +4.00000i q^{37} -4.00000 q^{39} -6.00000 q^{41} +4.00000i q^{43} +8.00000 q^{47} +9.00000 q^{49} -2.00000i q^{51} +8.00000i q^{53} -4.00000 q^{57} -12.0000i q^{59} +12.0000i q^{61} +4.00000 q^{63} -12.0000i q^{67} -8.00000i q^{69} +8.00000 q^{71} +6.00000 q^{73} +5.00000i q^{75} -16.0000i q^{77} +4.00000 q^{79} +1.00000 q^{81} +4.00000i q^{83} +8.00000 q^{87} +6.00000 q^{89} -16.0000i q^{91} +4.00000i q^{93} -2.00000 q^{97} -4.00000i q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 8 q^{7} - 2 q^{9}+O(q^{10})$$ 2 * q - 8 * q^7 - 2 * q^9 $$2 q - 8 q^{7} - 2 q^{9} - 4 q^{17} - 16 q^{23} + 10 q^{25} + 8 q^{31} - 8 q^{33} - 8 q^{39} - 12 q^{41} + 16 q^{47} + 18 q^{49} - 8 q^{57} + 8 q^{63} + 16 q^{71} + 12 q^{73} + 8 q^{79} + 2 q^{81} + 16 q^{87} + 12 q^{89} - 4 q^{97}+O(q^{100})$$ 2 * q - 8 * q^7 - 2 * q^9 - 4 * q^17 - 16 * q^23 + 10 * q^25 + 8 * q^31 - 8 * q^33 - 8 * q^39 - 12 * q^41 + 16 * q^47 + 18 * q^49 - 8 * q^57 + 8 * q^63 + 16 * q^71 + 12 * q^73 + 8 * q^79 + 2 * q^81 + 16 * q^87 + 12 * q^89 - 4 * q^97

Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/384\mathbb{Z}\right)^\times$$.

 $$n$$ $$127$$ $$133$$ $$257$$ $$\chi(n)$$ $$1$$ $$-1$$ $$1$$

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
$$5$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$6$$ 0 0
$$7$$ −4.00000 −1.51186 −0.755929 0.654654i $$-0.772814\pi$$
−0.755929 + 0.654654i $$0.772814\pi$$
$$8$$ 0 0
$$9$$ −1.00000 −0.333333
$$10$$ 0 0
$$11$$ 4.00000i 1.20605i 0.797724 + 0.603023i $$0.206037\pi$$
−0.797724 + 0.603023i $$0.793963\pi$$
$$12$$ 0 0
$$13$$ 4.00000i 1.10940i 0.832050 + 0.554700i $$0.187167\pi$$
−0.832050 + 0.554700i $$0.812833\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 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 $$0.151732\pi$$
−0.888523 + 0.458831i $$0.848268\pi$$
$$20$$ 0 0
$$21$$ − 4.00000i − 0.872872i
$$22$$ 0 0
$$23$$ −8.00000 −1.66812 −0.834058 0.551677i $$-0.813988\pi$$
−0.834058 + 0.551677i $$0.813988\pi$$
$$24$$ 0 0
$$25$$ 5.00000 1.00000
$$26$$ 0 0
$$27$$ − 1.00000i − 0.192450i
$$28$$ 0 0
$$29$$ − 8.00000i − 1.48556i −0.669534 0.742781i $$-0.733506\pi$$
0.669534 0.742781i $$-0.266494\pi$$
$$30$$ 0 0
$$31$$ 4.00000 0.718421 0.359211 0.933257i $$-0.383046\pi$$
0.359211 + 0.933257i $$0.383046\pi$$
$$32$$ 0 0
$$33$$ −4.00000 −0.696311
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 0 0
$$37$$ 4.00000i 0.657596i 0.944400 + 0.328798i $$0.106644\pi$$
−0.944400 + 0.328798i $$0.893356\pi$$
$$38$$ 0 0
$$39$$ −4.00000 −0.640513
$$40$$ 0 0
$$41$$ −6.00000 −0.937043 −0.468521 0.883452i $$-0.655213\pi$$
−0.468521 + 0.883452i $$0.655213\pi$$
$$42$$ 0 0
$$43$$ 4.00000i 0.609994i 0.952353 + 0.304997i $$0.0986555\pi$$
−0.952353 + 0.304997i $$0.901344\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ 8.00000 1.16692 0.583460 0.812142i $$-0.301699\pi$$
0.583460 + 0.812142i $$0.301699\pi$$
$$48$$ 0 0
$$49$$ 9.00000 1.28571
$$50$$ 0 0
$$51$$ − 2.00000i − 0.280056i
$$52$$ 0 0
$$53$$ 8.00000i 1.09888i 0.835532 + 0.549442i $$0.185160\pi$$
−0.835532 + 0.549442i $$0.814840\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 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
$$61$$ 12.0000i 1.53644i 0.640184 + 0.768221i $$0.278858\pi$$
−0.640184 + 0.768221i $$0.721142\pi$$
$$62$$ 0 0
$$63$$ 4.00000 0.503953
$$64$$ 0 0
$$65$$ 0 0
$$66$$ 0 0
$$67$$ − 12.0000i − 1.46603i −0.680211 0.733017i $$-0.738112\pi$$
0.680211 0.733017i $$-0.261888\pi$$
$$68$$ 0 0
$$69$$ − 8.00000i − 0.963087i
$$70$$ 0 0
$$71$$ 8.00000 0.949425 0.474713 0.880141i $$-0.342552\pi$$
0.474713 + 0.880141i $$0.342552\pi$$
$$72$$ 0 0
$$73$$ 6.00000 0.702247 0.351123 0.936329i $$-0.385800\pi$$
0.351123 + 0.936329i $$0.385800\pi$$
$$74$$ 0 0
$$75$$ 5.00000i 0.577350i
$$76$$ 0 0
$$77$$ − 16.0000i − 1.82337i
$$78$$ 0 0
$$79$$ 4.00000 0.450035 0.225018 0.974355i $$-0.427756\pi$$
0.225018 + 0.974355i $$0.427756\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ 0 0
$$83$$ 4.00000i 0.439057i 0.975606 + 0.219529i $$0.0704519\pi$$
−0.975606 + 0.219529i $$0.929548\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 0 0
$$87$$ 8.00000 0.857690
$$88$$ 0 0
$$89$$ 6.00000 0.635999 0.317999 0.948091i $$-0.396989\pi$$
0.317999 + 0.948091i $$0.396989\pi$$
$$90$$ 0 0
$$91$$ − 16.0000i − 1.67726i
$$92$$ 0 0
$$93$$ 4.00000i 0.414781i
$$94$$ 0 0
$$95$$ 0 0
$$96$$ 0 0
$$97$$ −2.00000 −0.203069 −0.101535 0.994832i $$-0.532375\pi$$
−0.101535 + 0.994832i $$0.532375\pi$$
$$98$$ 0 0
$$99$$ − 4.00000i − 0.402015i
$$100$$ 0 0
$$101$$ 8.00000i 0.796030i 0.917379 + 0.398015i $$0.130301\pi$$
−0.917379 + 0.398015i $$0.869699\pi$$
$$102$$ 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
$$106$$ 0 0
$$107$$ 4.00000i 0.386695i 0.981130 + 0.193347i $$0.0619344\pi$$
−0.981130 + 0.193347i $$0.938066\pi$$
$$108$$ 0 0
$$109$$ 4.00000i 0.383131i 0.981480 + 0.191565i $$0.0613564\pi$$
−0.981480 + 0.191565i $$0.938644\pi$$
$$110$$ 0 0
$$111$$ −4.00000 −0.379663
$$112$$ 0 0
$$113$$ −14.0000 −1.31701 −0.658505 0.752577i $$-0.728811\pi$$
−0.658505 + 0.752577i $$0.728811\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 0 0
$$117$$ − 4.00000i − 0.369800i
$$118$$ 0 0
$$119$$ 8.00000 0.733359
$$120$$ 0 0
$$121$$ −5.00000 −0.454545
$$122$$ 0 0
$$123$$ − 6.00000i − 0.541002i
$$124$$ 0 0
$$125$$ 0 0
$$126$$ 0 0
$$127$$ −4.00000 −0.354943 −0.177471 0.984126i $$-0.556792\pi$$
−0.177471 + 0.984126i $$0.556792\pi$$
$$128$$ 0 0
$$129$$ −4.00000 −0.352180
$$130$$ 0 0
$$131$$ − 12.0000i − 1.04844i −0.851581 0.524222i $$-0.824356\pi$$
0.851581 0.524222i $$-0.175644\pi$$
$$132$$ 0 0
$$133$$ − 16.0000i − 1.38738i
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ −6.00000 −0.512615 −0.256307 0.966595i $$-0.582506\pi$$
−0.256307 + 0.966595i $$0.582506\pi$$
$$138$$ 0 0
$$139$$ 20.0000i 1.69638i 0.529694 + 0.848189i $$0.322307\pi$$
−0.529694 + 0.848189i $$0.677693\pi$$
$$140$$ 0 0
$$141$$ 8.00000i 0.673722i
$$142$$ 0 0
$$143$$ −16.0000 −1.33799
$$144$$ 0 0
$$145$$ 0 0
$$146$$ 0 0
$$147$$ 9.00000i 0.742307i
$$148$$ 0 0
$$149$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$150$$ 0 0
$$151$$ 12.0000 0.976546 0.488273 0.872691i $$-0.337627\pi$$
0.488273 + 0.872691i $$0.337627\pi$$
$$152$$ 0 0
$$153$$ 2.00000 0.161690
$$154$$ 0 0
$$155$$ 0 0
$$156$$ 0 0
$$157$$ − 4.00000i − 0.319235i −0.987179 0.159617i $$-0.948974\pi$$
0.987179 0.159617i $$-0.0510260\pi$$
$$158$$ 0 0
$$159$$ −8.00000 −0.634441
$$160$$ 0 0
$$161$$ 32.0000 2.52195
$$162$$ 0 0
$$163$$ 20.0000i 1.56652i 0.621694 + 0.783260i $$0.286445\pi$$
−0.621694 + 0.783260i $$0.713555\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$168$$ 0 0
$$169$$ −3.00000 −0.230769
$$170$$ 0 0
$$171$$ − 4.00000i − 0.305888i
$$172$$ 0 0
$$173$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$174$$ 0 0
$$175$$ −20.0000 −1.51186
$$176$$ 0 0
$$177$$ 12.0000 0.901975
$$178$$ 0 0
$$179$$ 4.00000i 0.298974i 0.988764 + 0.149487i $$0.0477622\pi$$
−0.988764 + 0.149487i $$0.952238\pi$$
$$180$$ 0 0
$$181$$ − 4.00000i − 0.297318i −0.988889 0.148659i $$-0.952504\pi$$
0.988889 0.148659i $$-0.0474956\pi$$
$$182$$ 0 0
$$183$$ −12.0000 −0.887066
$$184$$ 0 0
$$185$$ 0 0
$$186$$ 0 0
$$187$$ − 8.00000i − 0.585018i
$$188$$ 0 0
$$189$$ 4.00000i 0.290957i
$$190$$ 0 0
$$191$$ −16.0000 −1.15772 −0.578860 0.815427i $$-0.696502\pi$$
−0.578860 + 0.815427i $$0.696502\pi$$
$$192$$ 0 0
$$193$$ −18.0000 −1.29567 −0.647834 0.761781i $$-0.724325\pi$$
−0.647834 + 0.761781i $$0.724325\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ 24.0000i 1.70993i 0.518686 + 0.854965i $$0.326421\pi$$
−0.518686 + 0.854965i $$0.673579\pi$$
$$198$$ 0 0
$$199$$ 4.00000 0.283552 0.141776 0.989899i $$-0.454719\pi$$
0.141776 + 0.989899i $$0.454719\pi$$
$$200$$ 0 0
$$201$$ 12.0000 0.846415
$$202$$ 0 0
$$203$$ 32.0000i 2.24596i
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 0 0
$$207$$ 8.00000 0.556038
$$208$$ 0 0
$$209$$ −16.0000 −1.10674
$$210$$ 0 0
$$211$$ 4.00000i 0.275371i 0.990476 + 0.137686i $$0.0439664\pi$$
−0.990476 + 0.137686i $$0.956034\pi$$
$$212$$ 0 0
$$213$$ 8.00000i 0.548151i
$$214$$ 0 0
$$215$$ 0 0
$$216$$ 0 0
$$217$$ −16.0000 −1.08615
$$218$$ 0 0
$$219$$ 6.00000i 0.405442i
$$220$$ 0 0
$$221$$ − 8.00000i − 0.538138i
$$222$$ 0 0
$$223$$ −20.0000 −1.33930 −0.669650 0.742677i $$-0.733556\pi$$
−0.669650 + 0.742677i $$0.733556\pi$$
$$224$$ 0 0
$$225$$ −5.00000 −0.333333
$$226$$ 0 0
$$227$$ 4.00000i 0.265489i 0.991150 + 0.132745i $$0.0423790\pi$$
−0.991150 + 0.132745i $$0.957621\pi$$
$$228$$ 0 0
$$229$$ − 4.00000i − 0.264327i −0.991228 0.132164i $$-0.957808\pi$$
0.991228 0.132164i $$-0.0421925\pi$$
$$230$$ 0 0
$$231$$ 16.0000 1.05272
$$232$$ 0 0
$$233$$ 22.0000 1.44127 0.720634 0.693316i $$-0.243851\pi$$
0.720634 + 0.693316i $$0.243851\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ 0 0
$$237$$ 4.00000i 0.259828i
$$238$$ 0 0
$$239$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$240$$ 0 0
$$241$$ −2.00000 −0.128831 −0.0644157 0.997923i $$-0.520518\pi$$
−0.0644157 + 0.997923i $$0.520518\pi$$
$$242$$ 0 0
$$243$$ 1.00000i 0.0641500i
$$244$$ 0 0
$$245$$ 0 0
$$246$$ 0 0
$$247$$ −16.0000 −1.01806
$$248$$ 0 0
$$249$$ −4.00000 −0.253490
$$250$$ 0 0
$$251$$ − 12.0000i − 0.757433i −0.925513 0.378717i $$-0.876365\pi$$
0.925513 0.378717i $$-0.123635\pi$$
$$252$$ 0 0
$$253$$ − 32.0000i − 2.01182i
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ 2.00000 0.124757 0.0623783 0.998053i $$-0.480131\pi$$
0.0623783 + 0.998053i $$0.480131\pi$$
$$258$$ 0 0
$$259$$ − 16.0000i − 0.994192i
$$260$$ 0 0
$$261$$ 8.00000i 0.495188i
$$262$$ 0 0
$$263$$ −16.0000 −0.986602 −0.493301 0.869859i $$-0.664210\pi$$
−0.493301 + 0.869859i $$0.664210\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 0 0
$$267$$ 6.00000i 0.367194i
$$268$$ 0 0
$$269$$ 24.0000i 1.46331i 0.681677 + 0.731653i $$0.261251\pi$$
−0.681677 + 0.731653i $$0.738749\pi$$
$$270$$ 0 0
$$271$$ 12.0000 0.728948 0.364474 0.931214i $$-0.381249\pi$$
0.364474 + 0.931214i $$0.381249\pi$$
$$272$$ 0 0
$$273$$ 16.0000 0.968364
$$274$$ 0 0
$$275$$ 20.0000i 1.20605i
$$276$$ 0 0
$$277$$ − 20.0000i − 1.20168i −0.799368 0.600842i $$-0.794832\pi$$
0.799368 0.600842i $$-0.205168\pi$$
$$278$$ 0 0
$$279$$ −4.00000 −0.239474
$$280$$ 0 0
$$281$$ 6.00000 0.357930 0.178965 0.983855i $$-0.442725\pi$$
0.178965 + 0.983855i $$0.442725\pi$$
$$282$$ 0 0
$$283$$ 4.00000i 0.237775i 0.992908 + 0.118888i $$0.0379328\pi$$
−0.992908 + 0.118888i $$0.962067\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ 24.0000 1.41668
$$288$$ 0 0
$$289$$ −13.0000 −0.764706
$$290$$ 0 0
$$291$$ − 2.00000i − 0.117242i
$$292$$ 0 0
$$293$$ − 24.0000i − 1.40209i −0.713115 0.701047i $$-0.752716\pi$$
0.713115 0.701047i $$-0.247284\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 0 0
$$297$$ 4.00000 0.232104
$$298$$ 0 0
$$299$$ − 32.0000i − 1.85061i
$$300$$ 0 0
$$301$$ − 16.0000i − 0.922225i
$$302$$ 0 0
$$303$$ −8.00000 −0.459588
$$304$$ 0 0
$$305$$ 0 0
$$306$$ 0 0
$$307$$ − 12.0000i − 0.684876i −0.939540 0.342438i $$-0.888747\pi$$
0.939540 0.342438i $$-0.111253\pi$$
$$308$$ 0 0
$$309$$ 4.00000i 0.227552i
$$310$$ 0 0
$$311$$ 32.0000 1.81455 0.907277 0.420534i $$-0.138157\pi$$
0.907277 + 0.420534i $$0.138157\pi$$
$$312$$ 0 0
$$313$$ −22.0000 −1.24351 −0.621757 0.783210i $$-0.713581\pi$$
−0.621757 + 0.783210i $$0.713581\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ 8.00000i 0.449325i 0.974437 + 0.224662i $$0.0721279\pi$$
−0.974437 + 0.224662i $$0.927872\pi$$
$$318$$ 0 0
$$319$$ 32.0000 1.79166
$$320$$ 0 0
$$321$$ −4.00000 −0.223258
$$322$$ 0 0
$$323$$ − 8.00000i − 0.445132i
$$324$$ 0 0
$$325$$ 20.0000i 1.10940i
$$326$$ 0 0
$$327$$ −4.00000 −0.221201
$$328$$ 0 0
$$329$$ −32.0000 −1.76422
$$330$$ 0 0
$$331$$ 4.00000i 0.219860i 0.993939 + 0.109930i $$0.0350627\pi$$
−0.993939 + 0.109930i $$0.964937\pi$$
$$332$$ 0 0
$$333$$ − 4.00000i − 0.219199i
$$334$$ 0 0
$$335$$ 0 0
$$336$$ 0 0
$$337$$ 2.00000 0.108947 0.0544735 0.998515i $$-0.482652\pi$$
0.0544735 + 0.998515i $$0.482652\pi$$
$$338$$ 0 0
$$339$$ − 14.0000i − 0.760376i
$$340$$ 0 0
$$341$$ 16.0000i 0.866449i
$$342$$ 0 0
$$343$$ −8.00000 −0.431959
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ − 12.0000i − 0.644194i −0.946707 0.322097i $$-0.895612\pi$$
0.946707 0.322097i $$-0.104388\pi$$
$$348$$ 0 0
$$349$$ 28.0000i 1.49881i 0.662114 + 0.749403i $$0.269659\pi$$
−0.662114 + 0.749403i $$0.730341\pi$$
$$350$$ 0 0
$$351$$ 4.00000 0.213504
$$352$$ 0 0
$$353$$ 34.0000 1.80964 0.904819 0.425797i $$-0.140006\pi$$
0.904819 + 0.425797i $$0.140006\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 0 0
$$357$$ 8.00000i 0.423405i
$$358$$ 0 0
$$359$$ −24.0000 −1.26667 −0.633336 0.773877i $$-0.718315\pi$$
−0.633336 + 0.773877i $$0.718315\pi$$
$$360$$ 0 0
$$361$$ 3.00000 0.157895
$$362$$ 0 0
$$363$$ − 5.00000i − 0.262432i
$$364$$ 0 0
$$365$$ 0 0
$$366$$ 0 0
$$367$$ 36.0000 1.87918 0.939592 0.342296i $$-0.111204\pi$$
0.939592 + 0.342296i $$0.111204\pi$$
$$368$$ 0 0
$$369$$ 6.00000 0.312348
$$370$$ 0 0
$$371$$ − 32.0000i − 1.66136i
$$372$$ 0 0
$$373$$ 4.00000i 0.207112i 0.994624 + 0.103556i $$0.0330221\pi$$
−0.994624 + 0.103556i $$0.966978\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ 32.0000 1.64808
$$378$$ 0 0
$$379$$ 4.00000i 0.205466i 0.994709 + 0.102733i $$0.0327588\pi$$
−0.994709 + 0.102733i $$0.967241\pi$$
$$380$$ 0 0
$$381$$ − 4.00000i − 0.204926i
$$382$$ 0 0
$$383$$ 16.0000 0.817562 0.408781 0.912633i $$-0.365954\pi$$
0.408781 + 0.912633i $$0.365954\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ − 4.00000i − 0.203331i
$$388$$ 0 0
$$389$$ − 32.0000i − 1.62246i −0.584724 0.811232i $$-0.698797\pi$$
0.584724 0.811232i $$-0.301203\pi$$
$$390$$ 0 0
$$391$$ 16.0000 0.809155
$$392$$ 0 0
$$393$$ 12.0000 0.605320
$$394$$ 0 0
$$395$$ 0 0
$$396$$ 0 0
$$397$$ − 20.0000i − 1.00377i −0.864934 0.501886i $$-0.832640\pi$$
0.864934 0.501886i $$-0.167360\pi$$
$$398$$ 0 0
$$399$$ 16.0000 0.801002
$$400$$ 0 0
$$401$$ −34.0000 −1.69788 −0.848939 0.528490i $$-0.822758\pi$$
−0.848939 + 0.528490i $$0.822758\pi$$
$$402$$ 0 0
$$403$$ 16.0000i 0.797017i
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ −16.0000 −0.793091
$$408$$ 0 0
$$409$$ 26.0000 1.28562 0.642809 0.766027i $$-0.277769\pi$$
0.642809 + 0.766027i $$0.277769\pi$$
$$410$$ 0 0
$$411$$ − 6.00000i − 0.295958i
$$412$$ 0 0
$$413$$ 48.0000i 2.36193i
$$414$$ 0 0
$$415$$ 0 0
$$416$$ 0 0
$$417$$ −20.0000 −0.979404
$$418$$ 0 0
$$419$$ 20.0000i 0.977064i 0.872546 + 0.488532i $$0.162467\pi$$
−0.872546 + 0.488532i $$0.837533\pi$$
$$420$$ 0 0
$$421$$ − 20.0000i − 0.974740i −0.873195 0.487370i $$-0.837956\pi$$
0.873195 0.487370i $$-0.162044\pi$$
$$422$$ 0 0
$$423$$ −8.00000 −0.388973
$$424$$ 0 0
$$425$$ −10.0000 −0.485071
$$426$$ 0 0
$$427$$ − 48.0000i − 2.32288i
$$428$$ 0 0
$$429$$ − 16.0000i − 0.772487i
$$430$$ 0 0
$$431$$ 24.0000 1.15604 0.578020 0.816023i $$-0.303826\pi$$
0.578020 + 0.816023i $$0.303826\pi$$
$$432$$ 0 0
$$433$$ 14.0000 0.672797 0.336399 0.941720i $$-0.390791\pi$$
0.336399 + 0.941720i $$0.390791\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ − 32.0000i − 1.53077i
$$438$$ 0 0
$$439$$ −12.0000 −0.572729 −0.286364 0.958121i $$-0.592447\pi$$
−0.286364 + 0.958121i $$0.592447\pi$$
$$440$$ 0 0
$$441$$ −9.00000 −0.428571
$$442$$ 0 0
$$443$$ 36.0000i 1.71041i 0.518289 + 0.855206i $$0.326569\pi$$
−0.518289 + 0.855206i $$0.673431\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ 0 0
$$447$$ 0 0
$$448$$ 0 0
$$449$$ 14.0000 0.660701 0.330350 0.943858i $$-0.392833\pi$$
0.330350 + 0.943858i $$0.392833\pi$$
$$450$$ 0 0
$$451$$ − 24.0000i − 1.13012i
$$452$$ 0 0
$$453$$ 12.0000i 0.563809i
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 10.0000 0.467780 0.233890 0.972263i $$-0.424854\pi$$
0.233890 + 0.972263i $$0.424854\pi$$
$$458$$ 0 0
$$459$$ 2.00000i 0.0933520i
$$460$$ 0 0
$$461$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$462$$ 0 0
$$463$$ −20.0000 −0.929479 −0.464739 0.885448i $$-0.653852\pi$$
−0.464739 + 0.885448i $$0.653852\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ − 12.0000i − 0.555294i −0.960683 0.277647i $$-0.910445\pi$$
0.960683 0.277647i $$-0.0895545\pi$$
$$468$$ 0 0
$$469$$ 48.0000i 2.21643i
$$470$$ 0 0
$$471$$ 4.00000 0.184310
$$472$$ 0 0
$$473$$ −16.0000 −0.735681
$$474$$ 0 0
$$475$$ 20.0000i 0.917663i
$$476$$ 0 0
$$477$$ − 8.00000i − 0.366295i
$$478$$ 0 0
$$479$$ 8.00000 0.365529 0.182765 0.983157i $$-0.441495\pi$$
0.182765 + 0.983157i $$0.441495\pi$$
$$480$$ 0 0
$$481$$ −16.0000 −0.729537
$$482$$ 0 0
$$483$$ 32.0000i 1.45605i
$$484$$ 0 0
$$485$$ 0 0
$$486$$ 0 0
$$487$$ −20.0000 −0.906287 −0.453143 0.891438i $$-0.649697\pi$$
−0.453143 + 0.891438i $$0.649697\pi$$
$$488$$ 0 0
$$489$$ −20.0000 −0.904431
$$490$$ 0 0
$$491$$ − 12.0000i − 0.541552i −0.962642 0.270776i $$-0.912720\pi$$
0.962642 0.270776i $$-0.0872803\pi$$
$$492$$ 0 0
$$493$$ 16.0000i 0.720604i
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ −32.0000 −1.43540
$$498$$ 0 0
$$499$$ 4.00000i 0.179065i 0.995984 + 0.0895323i $$0.0285372\pi$$
−0.995984 + 0.0895323i $$0.971463\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ 0 0
$$503$$ −24.0000 −1.07011 −0.535054 0.844818i $$-0.679709\pi$$
−0.535054 + 0.844818i $$0.679709\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 0 0
$$507$$ − 3.00000i − 0.133235i
$$508$$ 0 0
$$509$$ − 24.0000i − 1.06378i −0.846813 0.531891i $$-0.821482\pi$$
0.846813 0.531891i $$-0.178518\pi$$
$$510$$ 0 0
$$511$$ −24.0000 −1.06170
$$512$$ 0 0
$$513$$ 4.00000 0.176604
$$514$$ 0 0
$$515$$ 0 0
$$516$$ 0 0
$$517$$ 32.0000i 1.40736i
$$518$$ 0 0
$$519$$ 0 0
$$520$$ 0 0
$$521$$ −6.00000 −0.262865 −0.131432 0.991325i $$-0.541958\pi$$
−0.131432 + 0.991325i $$0.541958\pi$$
$$522$$ 0 0
$$523$$ 20.0000i 0.874539i 0.899331 + 0.437269i $$0.144054\pi$$
−0.899331 + 0.437269i $$0.855946\pi$$
$$524$$ 0 0
$$525$$ − 20.0000i − 0.872872i
$$526$$ 0 0
$$527$$ −8.00000 −0.348485
$$528$$ 0 0
$$529$$ 41.0000 1.78261
$$530$$ 0 0
$$531$$ 12.0000i 0.520756i
$$532$$ 0 0
$$533$$ − 24.0000i − 1.03956i
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 0 0
$$537$$ −4.00000 −0.172613
$$538$$ 0 0
$$539$$ 36.0000i 1.55063i
$$540$$ 0 0
$$541$$ − 28.0000i − 1.20381i −0.798566 0.601907i $$-0.794408\pi$$
0.798566 0.601907i $$-0.205592\pi$$
$$542$$ 0 0
$$543$$ 4.00000 0.171656
$$544$$ 0 0
$$545$$ 0 0
$$546$$ 0 0
$$547$$ 36.0000i 1.53925i 0.638497 + 0.769624i $$0.279557\pi$$
−0.638497 + 0.769624i $$0.720443\pi$$
$$548$$ 0 0
$$549$$ − 12.0000i − 0.512148i
$$550$$ 0 0
$$551$$ 32.0000 1.36325
$$552$$ 0 0
$$553$$ −16.0000 −0.680389
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ 16.0000i 0.677942i 0.940797 + 0.338971i $$0.110079\pi$$
−0.940797 + 0.338971i $$0.889921\pi$$
$$558$$ 0 0
$$559$$ −16.0000 −0.676728
$$560$$ 0 0
$$561$$ 8.00000 0.337760
$$562$$ 0 0
$$563$$ − 12.0000i − 0.505740i −0.967500 0.252870i $$-0.918626\pi$$
0.967500 0.252870i $$-0.0813744\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ 0 0
$$567$$ −4.00000 −0.167984
$$568$$ 0 0
$$569$$ 10.0000 0.419222 0.209611 0.977785i $$-0.432780\pi$$
0.209611 + 0.977785i $$0.432780\pi$$
$$570$$ 0 0
$$571$$ 4.00000i 0.167395i 0.996491 + 0.0836974i $$0.0266729\pi$$
−0.996491 + 0.0836974i $$0.973327\pi$$
$$572$$ 0 0
$$573$$ − 16.0000i − 0.668410i
$$574$$ 0 0
$$575$$ −40.0000 −1.66812
$$576$$ 0 0
$$577$$ 18.0000 0.749350 0.374675 0.927156i $$-0.377754\pi$$
0.374675 + 0.927156i $$0.377754\pi$$
$$578$$ 0 0
$$579$$ − 18.0000i − 0.748054i
$$580$$ 0 0
$$581$$ − 16.0000i − 0.663792i
$$582$$ 0 0
$$583$$ −32.0000 −1.32530
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ 4.00000i 0.165098i 0.996587 + 0.0825488i $$0.0263060\pi$$
−0.996587 + 0.0825488i $$0.973694\pi$$
$$588$$ 0 0
$$589$$ 16.0000i 0.659269i
$$590$$ 0 0
$$591$$ −24.0000 −0.987228
$$592$$ 0 0
$$593$$ 18.0000 0.739171 0.369586 0.929197i $$-0.379500\pi$$
0.369586 + 0.929197i $$0.379500\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ 4.00000i 0.163709i
$$598$$ 0 0
$$599$$ −24.0000 −0.980613 −0.490307 0.871550i $$-0.663115\pi$$
−0.490307 + 0.871550i $$0.663115\pi$$
$$600$$ 0 0
$$601$$ 22.0000 0.897399 0.448699 0.893683i $$-0.351887\pi$$
0.448699 + 0.893683i $$0.351887\pi$$
$$602$$ 0 0
$$603$$ 12.0000i 0.488678i
$$604$$ 0 0
$$605$$ 0 0
$$606$$ 0 0
$$607$$ 28.0000 1.13648 0.568242 0.822861i $$-0.307624\pi$$
0.568242 + 0.822861i $$0.307624\pi$$
$$608$$ 0 0
$$609$$ −32.0000 −1.29671
$$610$$ 0 0
$$611$$ 32.0000i 1.29458i
$$612$$ 0 0
$$613$$ 36.0000i 1.45403i 0.686624 + 0.727013i $$0.259092\pi$$
−0.686624 + 0.727013i $$0.740908\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ −10.0000 −0.402585 −0.201292 0.979531i $$-0.564514\pi$$
−0.201292 + 0.979531i $$0.564514\pi$$
$$618$$ 0 0
$$619$$ − 28.0000i − 1.12542i −0.826656 0.562708i $$-0.809760\pi$$
0.826656 0.562708i $$-0.190240\pi$$
$$620$$ 0 0
$$621$$ 8.00000i 0.321029i
$$622$$ 0 0
$$623$$ −24.0000 −0.961540
$$624$$ 0 0
$$625$$ 25.0000 1.00000
$$626$$ 0 0
$$627$$ − 16.0000i − 0.638978i
$$628$$ 0 0
$$629$$ − 8.00000i − 0.318981i
$$630$$ 0 0
$$631$$ 44.0000 1.75161 0.875806 0.482663i $$-0.160330\pi$$
0.875806 + 0.482663i $$0.160330\pi$$
$$632$$ 0 0
$$633$$ −4.00000 −0.158986
$$634$$ 0 0
$$635$$ 0 0
$$636$$ 0 0
$$637$$ 36.0000i 1.42637i
$$638$$ 0 0
$$639$$ −8.00000 −0.316475
$$640$$ 0 0
$$641$$ −18.0000 −0.710957 −0.355479 0.934684i $$-0.615682\pi$$
−0.355479 + 0.934684i $$0.615682\pi$$
$$642$$ 0 0
$$643$$ 36.0000i 1.41970i 0.704352 + 0.709851i $$0.251238\pi$$
−0.704352 + 0.709851i $$0.748762\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ 24.0000 0.943537 0.471769 0.881722i $$-0.343616\pi$$
0.471769 + 0.881722i $$0.343616\pi$$
$$648$$ 0 0
$$649$$ 48.0000 1.88416
$$650$$ 0 0
$$651$$ − 16.0000i − 0.627089i
$$652$$ 0 0
$$653$$ 16.0000i 0.626128i 0.949732 + 0.313064i $$0.101356\pi$$
−0.949732 + 0.313064i $$0.898644\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 0 0
$$657$$ −6.00000 −0.234082
$$658$$ 0 0
$$659$$ 36.0000i 1.40236i 0.712984 + 0.701180i $$0.247343\pi$$
−0.712984 + 0.701180i $$0.752657\pi$$
$$660$$ 0 0
$$661$$ 4.00000i 0.155582i 0.996970 + 0.0777910i $$0.0247867\pi$$
−0.996970 + 0.0777910i $$0.975213\pi$$
$$662$$ 0 0
$$663$$ 8.00000 0.310694
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 64.0000i 2.47809i
$$668$$ 0 0
$$669$$ − 20.0000i − 0.773245i
$$670$$ 0 0
$$671$$ −48.0000 −1.85302
$$672$$ 0 0
$$673$$ −34.0000 −1.31060 −0.655302 0.755367i $$-0.727459\pi$$
−0.655302 + 0.755367i $$0.727459\pi$$
$$674$$ 0 0
$$675$$ − 5.00000i − 0.192450i
$$676$$ 0 0
$$677$$ 16.0000i 0.614930i 0.951559 + 0.307465i $$0.0994807\pi$$
−0.951559 + 0.307465i $$0.900519\pi$$
$$678$$ 0 0
$$679$$ 8.00000 0.307012
$$680$$ 0 0
$$681$$ −4.00000 −0.153280
$$682$$ 0 0
$$683$$ − 44.0000i − 1.68361i −0.539779 0.841807i $$-0.681492\pi$$
0.539779 0.841807i $$-0.318508\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ 0 0
$$687$$ 4.00000 0.152610
$$688$$ 0 0
$$689$$ −32.0000 −1.21910
$$690$$ 0 0
$$691$$ − 44.0000i − 1.67384i −0.547326 0.836919i $$-0.684354\pi$$
0.547326 0.836919i $$-0.315646\pi$$
$$692$$ 0 0
$$693$$ 16.0000i 0.607790i
$$694$$ 0 0
$$695$$ 0 0
$$696$$ 0 0
$$697$$ 12.0000 0.454532
$$698$$ 0 0
$$699$$ 22.0000i 0.832116i
$$700$$ 0 0
$$701$$ 8.00000i 0.302156i 0.988522 + 0.151078i $$0.0482744\pi$$
−0.988522 + 0.151078i $$0.951726\pi$$
$$702$$ 0 0
$$703$$ −16.0000 −0.603451
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ − 32.0000i − 1.20348i
$$708$$ 0 0
$$709$$ − 20.0000i − 0.751116i −0.926799 0.375558i $$-0.877451\pi$$
0.926799 0.375558i $$-0.122549\pi$$
$$710$$ 0 0
$$711$$ −4.00000 −0.150012
$$712$$ 0 0
$$713$$ −32.0000 −1.19841
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 0 0
$$717$$ 0 0
$$718$$ 0 0
$$719$$ 24.0000 0.895049 0.447524 0.894272i $$-0.352306\pi$$
0.447524 + 0.894272i $$0.352306\pi$$
$$720$$ 0 0
$$721$$ −16.0000 −0.595871
$$722$$ 0 0
$$723$$ − 2.00000i − 0.0743808i
$$724$$ 0 0
$$725$$ − 40.0000i − 1.48556i
$$726$$ 0 0
$$727$$ 28.0000 1.03846 0.519231 0.854634i $$-0.326218\pi$$
0.519231 + 0.854634i $$0.326218\pi$$
$$728$$ 0 0
$$729$$ −1.00000 −0.0370370
$$730$$ 0 0
$$731$$ − 8.00000i − 0.295891i
$$732$$ 0 0
$$733$$ − 28.0000i − 1.03420i −0.855924 0.517102i $$-0.827011\pi$$
0.855924 0.517102i $$-0.172989\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ 48.0000 1.76810
$$738$$ 0 0
$$739$$ − 28.0000i − 1.03000i −0.857191 0.514998i $$-0.827793\pi$$
0.857191 0.514998i $$-0.172207\pi$$
$$740$$ 0 0
$$741$$ − 16.0000i − 0.587775i
$$742$$ 0 0
$$743$$ 16.0000 0.586983 0.293492 0.955962i $$-0.405183\pi$$
0.293492 + 0.955962i $$0.405183\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 0 0
$$747$$ − 4.00000i − 0.146352i
$$748$$ 0 0
$$749$$ − 16.0000i − 0.584627i
$$750$$ 0 0
$$751$$ −44.0000 −1.60558 −0.802791 0.596260i $$-0.796653\pi$$
−0.802791 + 0.596260i $$0.796653\pi$$
$$752$$ 0 0
$$753$$ 12.0000 0.437304
$$754$$ 0 0
$$755$$ 0 0
$$756$$ 0 0
$$757$$ 12.0000i 0.436147i 0.975932 + 0.218074i $$0.0699773\pi$$
−0.975932 + 0.218074i $$0.930023\pi$$
$$758$$ 0 0
$$759$$ 32.0000 1.16153
$$760$$ 0 0
$$761$$ 10.0000 0.362500 0.181250 0.983437i $$-0.441986\pi$$
0.181250 + 0.983437i $$0.441986\pi$$
$$762$$ 0 0
$$763$$ − 16.0000i − 0.579239i
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ 48.0000 1.73318
$$768$$ 0 0
$$769$$ 46.0000 1.65880 0.829401 0.558653i $$-0.188682\pi$$
0.829401 + 0.558653i $$0.188682\pi$$
$$770$$ 0 0
$$771$$ 2.00000i 0.0720282i
$$772$$ 0 0
$$773$$ − 8.00000i − 0.287740i −0.989597 0.143870i $$-0.954045\pi$$
0.989597 0.143870i $$-0.0459547\pi$$
$$774$$ 0 0
$$775$$ 20.0000 0.718421
$$776$$ 0 0
$$777$$ 16.0000 0.573997
$$778$$ 0 0
$$779$$ − 24.0000i − 0.859889i
$$780$$ 0 0
$$781$$ 32.0000i 1.14505i
$$782$$ 0 0
$$783$$ −8.00000 −0.285897
$$784$$ 0 0
$$785$$ 0 0
$$786$$ 0 0
$$787$$ − 12.0000i − 0.427754i −0.976861 0.213877i $$-0.931391\pi$$
0.976861 0.213877i $$-0.0686091\pi$$
$$788$$ 0 0
$$789$$ − 16.0000i − 0.569615i
$$790$$ 0 0
$$791$$ 56.0000 1.99113
$$792$$ 0 0
$$793$$ −48.0000 −1.70453
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ − 16.0000i − 0.566749i −0.959009 0.283375i $$-0.908546\pi$$
0.959009 0.283375i $$-0.0914540\pi$$
$$798$$ 0 0
$$799$$ −16.0000 −0.566039
$$800$$ 0 0
$$801$$ −6.00000 −0.212000
$$802$$ 0 0
$$803$$ 24.0000i 0.846942i
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ −24.0000 −0.844840
$$808$$ 0 0
$$809$$ 26.0000 0.914111 0.457056 0.889438i $$-0.348904\pi$$
0.457056 + 0.889438i $$0.348904\pi$$
$$810$$ 0 0
$$811$$ − 44.0000i − 1.54505i −0.634985 0.772524i $$-0.718994\pi$$
0.634985 0.772524i $$-0.281006\pi$$
$$812$$ 0 0
$$813$$ 12.0000i 0.420858i
$$814$$ 0 0
$$815$$ 0 0
$$816$$ 0 0
$$817$$ −16.0000 −0.559769
$$818$$ 0 0
$$819$$ 16.0000i 0.559085i
$$820$$ 0 0
$$821$$ 24.0000i 0.837606i 0.908077 + 0.418803i $$0.137550\pi$$
−0.908077 + 0.418803i $$0.862450\pi$$
$$822$$ 0 0
$$823$$ −4.00000 −0.139431 −0.0697156 0.997567i $$-0.522209\pi$$
−0.0697156 + 0.997567i $$0.522209\pi$$
$$824$$ 0 0
$$825$$ −20.0000 −0.696311
$$826$$ 0 0
$$827$$ 4.00000i 0.139094i 0.997579 + 0.0695468i $$0.0221553\pi$$
−0.997579 + 0.0695468i $$0.977845\pi$$
$$828$$ 0 0
$$829$$ 4.00000i 0.138926i 0.997585 + 0.0694629i $$0.0221285\pi$$
−0.997585 + 0.0694629i $$0.977871\pi$$
$$830$$ 0 0
$$831$$ 20.0000 0.693792
$$832$$ 0 0
$$833$$ −18.0000 −0.623663
$$834$$ 0 0
$$835$$ 0 0
$$836$$ 0 0
$$837$$ − 4.00000i − 0.138260i
$$838$$ 0 0
$$839$$ 24.0000 0.828572 0.414286 0.910147i $$-0.364031\pi$$
0.414286 + 0.910147i $$0.364031\pi$$
$$840$$ 0 0
$$841$$ −35.0000 −1.20690
$$842$$ 0 0
$$843$$ 6.00000i 0.206651i
$$844$$ 0 0
$$845$$ 0 0
$$846$$ 0 0
$$847$$ 20.0000 0.687208
$$848$$ 0 0
$$849$$ −4.00000 −0.137280
$$850$$ 0 0
$$851$$ − 32.0000i − 1.09695i
$$852$$ 0 0
$$853$$ − 44.0000i − 1.50653i −0.657716 0.753266i $$-0.728477\pi$$
0.657716 0.753266i $$-0.271523\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ −22.0000 −0.751506 −0.375753 0.926720i $$-0.622616\pi$$
−0.375753 + 0.926720i $$0.622616\pi$$
$$858$$ 0 0
$$859$$ − 28.0000i − 0.955348i −0.878537 0.477674i $$-0.841480\pi$$
0.878537 0.477674i $$-0.158520\pi$$
$$860$$ 0 0
$$861$$ 24.0000i 0.817918i
$$862$$ 0 0
$$863$$ −16.0000 −0.544646 −0.272323 0.962206i $$-0.587792\pi$$
−0.272323 + 0.962206i $$0.587792\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ 0 0
$$867$$ − 13.0000i − 0.441503i
$$868$$ 0 0
$$869$$ 16.0000i 0.542763i
$$870$$ 0 0
$$871$$ 48.0000 1.62642
$$872$$ 0 0
$$873$$ 2.00000 0.0676897
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 0 0
$$877$$ − 20.0000i − 0.675352i −0.941262 0.337676i $$-0.890359\pi$$
0.941262 0.337676i $$-0.109641\pi$$
$$878$$ 0 0
$$879$$ 24.0000 0.809500
$$880$$ 0 0
$$881$$ −14.0000 −0.471672 −0.235836 0.971793i $$-0.575783\pi$$
−0.235836 + 0.971793i $$0.575783\pi$$
$$882$$ 0 0
$$883$$ − 28.0000i − 0.942275i −0.882060 0.471138i $$-0.843844\pi$$
0.882060 0.471138i $$-0.156156\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ −48.0000 −1.61168 −0.805841 0.592132i $$-0.798286\pi$$
−0.805841 + 0.592132i $$0.798286\pi$$
$$888$$ 0 0
$$889$$ 16.0000 0.536623
$$890$$ 0 0
$$891$$ 4.00000i 0.134005i
$$892$$ 0 0
$$893$$ 32.0000i 1.07084i
$$894$$ 0 0
$$895$$ 0 0
$$896$$ 0 0
$$897$$ 32.0000 1.06845
$$898$$ 0 0
$$899$$ − 32.0000i − 1.06726i
$$900$$ 0 0
$$901$$ − 16.0000i − 0.533037i
$$902$$ 0 0
$$903$$ 16.0000 0.532447
$$904$$ 0 0
$$905$$ 0 0
$$906$$ 0 0
$$907$$ 4.00000i 0.132818i 0.997792 + 0.0664089i $$0.0211542\pi$$
−0.997792 + 0.0664089i $$0.978846\pi$$
$$908$$ 0 0
$$909$$ − 8.00000i − 0.265343i
$$910$$ 0 0
$$911$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$912$$ 0 0
$$913$$ −16.0000 −0.529523
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 0 0
$$917$$ 48.0000i 1.58510i
$$918$$ 0 0
$$919$$ −36.0000 −1.18753 −0.593765 0.804638i $$-0.702359\pi$$
−0.593765 + 0.804638i $$0.702359\pi$$
$$920$$ 0 0
$$921$$ 12.0000 0.395413
$$922$$ 0 0
$$923$$ 32.0000i 1.05329i
$$924$$ 0 0
$$925$$ 20.0000i 0.657596i
$$926$$ 0 0
$$927$$ −4.00000 −0.131377
$$928$$ 0 0
$$929$$ 14.0000 0.459325 0.229663 0.973270i $$-0.426238\pi$$
0.229663 + 0.973270i $$0.426238\pi$$
$$930$$ 0 0
$$931$$ 36.0000i 1.17985i
$$932$$ 0 0
$$933$$ 32.0000i 1.04763i
$$934$$ 0 0
$$935$$ 0 0
$$936$$ 0 0
$$937$$ 22.0000 0.718709 0.359354 0.933201i $$-0.382997\pi$$
0.359354 + 0.933201i $$0.382997\pi$$
$$938$$ 0 0
$$939$$ − 22.0000i − 0.717943i
$$940$$ 0 0
$$941$$ − 32.0000i − 1.04317i −0.853199 0.521585i $$-0.825341\pi$$
0.853199 0.521585i $$-0.174659\pi$$
$$942$$ 0 0
$$943$$ 48.0000 1.56310
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ − 12.0000i − 0.389948i −0.980808 0.194974i $$-0.937538\pi$$
0.980808 0.194974i $$-0.0624622\pi$$
$$948$$ 0 0
$$949$$ 24.0000i 0.779073i
$$950$$ 0 0
$$951$$ −8.00000 −0.259418
$$952$$ 0 0
$$953$$ 42.0000 1.36051 0.680257 0.732974i $$-0.261868\pi$$
0.680257 + 0.732974i $$0.261868\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ 0 0
$$957$$ 32.0000i 1.03441i
$$958$$ 0 0
$$959$$ 24.0000 0.775000
$$960$$ 0 0
$$961$$ −15.0000 −0.483871
$$962$$ 0 0
$$963$$ − 4.00000i − 0.128898i
$$964$$ 0 0
$$965$$ 0 0
$$966$$ 0 0
$$967$$ −12.0000 −0.385894 −0.192947 0.981209i $$-0.561805\pi$$
−0.192947 + 0.981209i $$0.561805\pi$$
$$968$$ 0 0
$$969$$ 8.00000 0.256997
$$970$$ 0 0
$$971$$ 20.0000i 0.641831i 0.947108 + 0.320915i $$0.103990\pi$$
−0.947108 + 0.320915i $$0.896010\pi$$
$$972$$ 0 0
$$973$$ − 80.0000i − 2.56468i
$$974$$ 0 0
$$975$$ −20.0000 −0.640513
$$976$$ 0 0
$$977$$ 30.0000 0.959785 0.479893 0.877327i $$-0.340676\pi$$
0.479893 + 0.877327i $$0.340676\pi$$
$$978$$ 0 0
$$979$$ 24.0000i 0.767043i
$$980$$ 0 0
$$981$$ − 4.00000i − 0.127710i
$$982$$ 0 0
$$983$$ 16.0000 0.510321 0.255160 0.966899i $$-0.417872\pi$$
0.255160 + 0.966899i $$0.417872\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ 0 0
$$987$$ − 32.0000i − 1.01857i
$$988$$ 0 0
$$989$$ − 32.0000i − 1.01754i
$$990$$ 0 0
$$991$$ 20.0000 0.635321 0.317660 0.948205i $$-0.397103\pi$$
0.317660 + 0.948205i $$0.397103\pi$$
$$992$$ 0 0
$$993$$ −4.00000 −0.126936
$$994$$ 0 0
$$995$$ 0 0
$$996$$ 0 0
$$997$$ − 44.0000i − 1.39349i −0.717317 0.696747i $$-0.754630\pi$$
0.717317 0.696747i $$-0.245370\pi$$
$$998$$ 0 0
$$999$$ 4.00000 0.126554
Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000

Twists

By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 384.2.d.a.193.2 yes 2
3.2 odd 2 1152.2.d.a.577.1 2
4.3 odd 2 384.2.d.b.193.1 yes 2
8.3 odd 2 384.2.d.b.193.2 yes 2
8.5 even 2 inner 384.2.d.a.193.1 2
12.11 even 2 1152.2.d.f.577.2 2
16.3 odd 4 768.2.a.f.1.1 1
16.5 even 4 768.2.a.g.1.1 1
16.11 odd 4 768.2.a.b.1.1 1
16.13 even 4 768.2.a.c.1.1 1
24.5 odd 2 1152.2.d.a.577.2 2
24.11 even 2 1152.2.d.f.577.1 2
48.5 odd 4 2304.2.a.k.1.1 1
48.11 even 4 2304.2.a.f.1.1 1
48.29 odd 4 2304.2.a.j.1.1 1
48.35 even 4 2304.2.a.g.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
384.2.d.a.193.1 2 8.5 even 2 inner
384.2.d.a.193.2 yes 2 1.1 even 1 trivial
384.2.d.b.193.1 yes 2 4.3 odd 2
384.2.d.b.193.2 yes 2 8.3 odd 2
768.2.a.b.1.1 1 16.11 odd 4
768.2.a.c.1.1 1 16.13 even 4
768.2.a.f.1.1 1 16.3 odd 4
768.2.a.g.1.1 1 16.5 even 4
1152.2.d.a.577.1 2 3.2 odd 2
1152.2.d.a.577.2 2 24.5 odd 2
1152.2.d.f.577.1 2 24.11 even 2
1152.2.d.f.577.2 2 12.11 even 2
2304.2.a.f.1.1 1 48.11 even 4
2304.2.a.g.1.1 1 48.35 even 4
2304.2.a.j.1.1 1 48.29 odd 4
2304.2.a.k.1.1 1 48.5 odd 4