# Properties

 Label 9216.2.a.bi.1.1 Level $9216$ Weight $2$ Character 9216.1 Self dual yes Analytic conductor $73.590$ Analytic rank $1$ Dimension $4$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [9216,2,Mod(1,9216)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(9216, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("9216.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$9216 = 2^{10} \cdot 3^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 9216.a (trivial)

## Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

 Self dual: yes Analytic conductor: $$73.5901305028$$ Analytic rank: $$1$$ Dimension: $$4$$ Coefficient field: $$\Q(\zeta_{24})^+$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - 4x^{2} + 1$$ x^4 - 4*x^2 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 768) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Root $$-1.93185$$ of defining polynomial Character $$\chi$$ $$=$$ 9216.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-3.86370 q^{5} +2.44949 q^{7} +O(q^{10})$$ $$q-3.86370 q^{5} +2.44949 q^{7} -1.46410 q^{11} +4.24264 q^{13} +3.46410 q^{17} -7.46410 q^{19} -2.82843 q^{23} +9.92820 q^{25} -8.76268 q^{29} +7.34847 q^{31} -9.46410 q^{35} +0.656339 q^{37} -4.53590 q^{41} +3.46410 q^{43} +2.82843 q^{47} -1.00000 q^{49} +4.62158 q^{53} +5.65685 q^{55} +13.8564 q^{59} -0.656339 q^{61} -16.3923 q^{65} -14.9282 q^{67} -6.41473 q^{71} +4.00000 q^{73} -3.58630 q^{77} -2.44949 q^{79} +5.46410 q^{83} -13.3843 q^{85} +4.92820 q^{89} +10.3923 q^{91} +28.8391 q^{95} +1.07180 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q+O(q^{10})$$ 4 * q $$4 q + 8 q^{11} - 16 q^{19} + 12 q^{25} - 24 q^{35} - 32 q^{41} - 4 q^{49} - 24 q^{65} - 32 q^{67} + 16 q^{73} + 8 q^{83} - 8 q^{89} + 32 q^{97}+O(q^{100})$$ 4 * q + 8 * q^11 - 16 * q^19 + 12 * q^25 - 24 * q^35 - 32 * q^41 - 4 * q^49 - 24 * q^65 - 32 * q^67 + 16 * q^73 + 8 * q^83 - 8 * q^89 + 32 * q^97

## 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
$$3$$ 0 0
$$4$$ 0 0
$$5$$ −3.86370 −1.72790 −0.863950 0.503577i $$-0.832017\pi$$
−0.863950 + 0.503577i $$0.832017\pi$$
$$6$$ 0 0
$$7$$ 2.44949 0.925820 0.462910 0.886405i $$-0.346805\pi$$
0.462910 + 0.886405i $$0.346805\pi$$
$$8$$ 0 0
$$9$$ 0 0
$$10$$ 0 0
$$11$$ −1.46410 −0.441443 −0.220722 0.975337i $$-0.570841\pi$$
−0.220722 + 0.975337i $$0.570841\pi$$
$$12$$ 0 0
$$13$$ 4.24264 1.17670 0.588348 0.808608i $$-0.299778\pi$$
0.588348 + 0.808608i $$0.299778\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ 3.46410 0.840168 0.420084 0.907485i $$-0.362001\pi$$
0.420084 + 0.907485i $$0.362001\pi$$
$$18$$ 0 0
$$19$$ −7.46410 −1.71238 −0.856191 0.516659i $$-0.827175\pi$$
−0.856191 + 0.516659i $$0.827175\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 0 0
$$23$$ −2.82843 −0.589768 −0.294884 0.955533i $$-0.595281\pi$$
−0.294884 + 0.955533i $$0.595281\pi$$
$$24$$ 0 0
$$25$$ 9.92820 1.98564
$$26$$ 0 0
$$27$$ 0 0
$$28$$ 0 0
$$29$$ −8.76268 −1.62719 −0.813595 0.581432i $$-0.802492\pi$$
−0.813595 + 0.581432i $$0.802492\pi$$
$$30$$ 0 0
$$31$$ 7.34847 1.31982 0.659912 0.751343i $$-0.270594\pi$$
0.659912 + 0.751343i $$0.270594\pi$$
$$32$$ 0 0
$$33$$ 0 0
$$34$$ 0 0
$$35$$ −9.46410 −1.59973
$$36$$ 0 0
$$37$$ 0.656339 0.107901 0.0539507 0.998544i $$-0.482819\pi$$
0.0539507 + 0.998544i $$0.482819\pi$$
$$38$$ 0 0
$$39$$ 0 0
$$40$$ 0 0
$$41$$ −4.53590 −0.708388 −0.354194 0.935172i $$-0.615245\pi$$
−0.354194 + 0.935172i $$0.615245\pi$$
$$42$$ 0 0
$$43$$ 3.46410 0.528271 0.264135 0.964486i $$-0.414913\pi$$
0.264135 + 0.964486i $$0.414913\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ 2.82843 0.412568 0.206284 0.978492i $$-0.433863\pi$$
0.206284 + 0.978492i $$0.433863\pi$$
$$48$$ 0 0
$$49$$ −1.00000 −0.142857
$$50$$ 0 0
$$51$$ 0 0
$$52$$ 0 0
$$53$$ 4.62158 0.634823 0.317411 0.948288i $$-0.397186\pi$$
0.317411 + 0.948288i $$0.397186\pi$$
$$54$$ 0 0
$$55$$ 5.65685 0.762770
$$56$$ 0 0
$$57$$ 0 0
$$58$$ 0 0
$$59$$ 13.8564 1.80395 0.901975 0.431788i $$-0.142117\pi$$
0.901975 + 0.431788i $$0.142117\pi$$
$$60$$ 0 0
$$61$$ −0.656339 −0.0840356 −0.0420178 0.999117i $$-0.513379\pi$$
−0.0420178 + 0.999117i $$0.513379\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ 0 0
$$65$$ −16.3923 −2.03322
$$66$$ 0 0
$$67$$ −14.9282 −1.82377 −0.911885 0.410445i $$-0.865373\pi$$
−0.911885 + 0.410445i $$0.865373\pi$$
$$68$$ 0 0
$$69$$ 0 0
$$70$$ 0 0
$$71$$ −6.41473 −0.761288 −0.380644 0.924722i $$-0.624298\pi$$
−0.380644 + 0.924722i $$0.624298\pi$$
$$72$$ 0 0
$$73$$ 4.00000 0.468165 0.234082 0.972217i $$-0.424791\pi$$
0.234082 + 0.972217i $$0.424791\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ −3.58630 −0.408697
$$78$$ 0 0
$$79$$ −2.44949 −0.275589 −0.137795 0.990461i $$-0.544001\pi$$
−0.137795 + 0.990461i $$0.544001\pi$$
$$80$$ 0 0
$$81$$ 0 0
$$82$$ 0 0
$$83$$ 5.46410 0.599763 0.299882 0.953976i $$-0.403053\pi$$
0.299882 + 0.953976i $$0.403053\pi$$
$$84$$ 0 0
$$85$$ −13.3843 −1.45173
$$86$$ 0 0
$$87$$ 0 0
$$88$$ 0 0
$$89$$ 4.92820 0.522388 0.261194 0.965286i $$-0.415884\pi$$
0.261194 + 0.965286i $$0.415884\pi$$
$$90$$ 0 0
$$91$$ 10.3923 1.08941
$$92$$ 0 0
$$93$$ 0 0
$$94$$ 0 0
$$95$$ 28.8391 2.95883
$$96$$ 0 0
$$97$$ 1.07180 0.108824 0.0544122 0.998519i $$-0.482671\pi$$
0.0544122 + 0.998519i $$0.482671\pi$$
$$98$$ 0 0
$$99$$ 0 0
$$100$$ 0 0
$$101$$ 14.4195 1.43480 0.717399 0.696663i $$-0.245333\pi$$
0.717399 + 0.696663i $$0.245333\pi$$
$$102$$ 0 0
$$103$$ −3.20736 −0.316031 −0.158016 0.987437i $$-0.550510\pi$$
−0.158016 + 0.987437i $$0.550510\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ −4.00000 −0.386695 −0.193347 0.981130i $$-0.561934\pi$$
−0.193347 + 0.981130i $$0.561934\pi$$
$$108$$ 0 0
$$109$$ 1.41421 0.135457 0.0677285 0.997704i $$-0.478425\pi$$
0.0677285 + 0.997704i $$0.478425\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ 0 0
$$113$$ −18.0000 −1.69330 −0.846649 0.532152i $$-0.821383\pi$$
−0.846649 + 0.532152i $$0.821383\pi$$
$$114$$ 0 0
$$115$$ 10.9282 1.01906
$$116$$ 0 0
$$117$$ 0 0
$$118$$ 0 0
$$119$$ 8.48528 0.777844
$$120$$ 0 0
$$121$$ −8.85641 −0.805128
$$122$$ 0 0
$$123$$ 0 0
$$124$$ 0 0
$$125$$ −19.0411 −1.70309
$$126$$ 0 0
$$127$$ 14.5211 1.28854 0.644268 0.764799i $$-0.277162\pi$$
0.644268 + 0.764799i $$0.277162\pi$$
$$128$$ 0 0
$$129$$ 0 0
$$130$$ 0 0
$$131$$ −14.9282 −1.30428 −0.652142 0.758097i $$-0.726129\pi$$
−0.652142 + 0.758097i $$0.726129\pi$$
$$132$$ 0 0
$$133$$ −18.2832 −1.58536
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ −14.3923 −1.22962 −0.614809 0.788676i $$-0.710767\pi$$
−0.614809 + 0.788676i $$0.710767\pi$$
$$138$$ 0 0
$$139$$ −6.92820 −0.587643 −0.293821 0.955860i $$-0.594927\pi$$
−0.293821 + 0.955860i $$0.594927\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 0 0
$$143$$ −6.21166 −0.519445
$$144$$ 0 0
$$145$$ 33.8564 2.81162
$$146$$ 0 0
$$147$$ 0 0
$$148$$ 0 0
$$149$$ 5.37945 0.440702 0.220351 0.975421i $$-0.429280\pi$$
0.220351 + 0.975421i $$0.429280\pi$$
$$150$$ 0 0
$$151$$ 6.59059 0.536335 0.268167 0.963372i $$-0.413582\pi$$
0.268167 + 0.963372i $$0.413582\pi$$
$$152$$ 0 0
$$153$$ 0 0
$$154$$ 0 0
$$155$$ −28.3923 −2.28052
$$156$$ 0 0
$$157$$ 2.17209 0.173352 0.0866758 0.996237i $$-0.472376\pi$$
0.0866758 + 0.996237i $$0.472376\pi$$
$$158$$ 0 0
$$159$$ 0 0
$$160$$ 0 0
$$161$$ −6.92820 −0.546019
$$162$$ 0 0
$$163$$ −4.53590 −0.355279 −0.177639 0.984096i $$-0.556846\pi$$
−0.177639 + 0.984096i $$0.556846\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ 7.72741 0.597965 0.298982 0.954259i $$-0.403353\pi$$
0.298982 + 0.954259i $$0.403353\pi$$
$$168$$ 0 0
$$169$$ 5.00000 0.384615
$$170$$ 0 0
$$171$$ 0 0
$$172$$ 0 0
$$173$$ 17.2480 1.31134 0.655669 0.755048i $$-0.272387\pi$$
0.655669 + 0.755048i $$0.272387\pi$$
$$174$$ 0 0
$$175$$ 24.3190 1.83835
$$176$$ 0 0
$$177$$ 0 0
$$178$$ 0 0
$$179$$ −18.9282 −1.41476 −0.707380 0.706833i $$-0.750123\pi$$
−0.707380 + 0.706833i $$0.750123\pi$$
$$180$$ 0 0
$$181$$ 7.07107 0.525588 0.262794 0.964852i $$-0.415356\pi$$
0.262794 + 0.964852i $$0.415356\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ 0 0
$$185$$ −2.53590 −0.186443
$$186$$ 0 0
$$187$$ −5.07180 −0.370887
$$188$$ 0 0
$$189$$ 0 0
$$190$$ 0 0
$$191$$ 7.72741 0.559136 0.279568 0.960126i $$-0.409809\pi$$
0.279568 + 0.960126i $$0.409809\pi$$
$$192$$ 0 0
$$193$$ 6.00000 0.431889 0.215945 0.976406i $$-0.430717\pi$$
0.215945 + 0.976406i $$0.430717\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ 8.76268 0.624315 0.312158 0.950030i $$-0.398948\pi$$
0.312158 + 0.950030i $$0.398948\pi$$
$$198$$ 0 0
$$199$$ 18.6622 1.32293 0.661463 0.749977i $$-0.269936\pi$$
0.661463 + 0.749977i $$0.269936\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ 0 0
$$203$$ −21.4641 −1.50648
$$204$$ 0 0
$$205$$ 17.5254 1.22402
$$206$$ 0 0
$$207$$ 0 0
$$208$$ 0 0
$$209$$ 10.9282 0.755920
$$210$$ 0 0
$$211$$ −6.92820 −0.476957 −0.238479 0.971148i $$-0.576649\pi$$
−0.238479 + 0.971148i $$0.576649\pi$$
$$212$$ 0 0
$$213$$ 0 0
$$214$$ 0 0
$$215$$ −13.3843 −0.912799
$$216$$ 0 0
$$217$$ 18.0000 1.22192
$$218$$ 0 0
$$219$$ 0 0
$$220$$ 0 0
$$221$$ 14.6969 0.988623
$$222$$ 0 0
$$223$$ −14.5211 −0.972403 −0.486201 0.873847i $$-0.661618\pi$$
−0.486201 + 0.873847i $$0.661618\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ 11.3205 0.751369 0.375684 0.926748i $$-0.377408\pi$$
0.375684 + 0.926748i $$0.377408\pi$$
$$228$$ 0 0
$$229$$ −16.8690 −1.11474 −0.557368 0.830265i $$-0.688189\pi$$
−0.557368 + 0.830265i $$0.688189\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ −12.9282 −0.846955 −0.423477 0.905907i $$-0.639191\pi$$
−0.423477 + 0.905907i $$0.639191\pi$$
$$234$$ 0 0
$$235$$ −10.9282 −0.712877
$$236$$ 0 0
$$237$$ 0 0
$$238$$ 0 0
$$239$$ −7.72741 −0.499844 −0.249922 0.968266i $$-0.580405\pi$$
−0.249922 + 0.968266i $$0.580405\pi$$
$$240$$ 0 0
$$241$$ −26.7846 −1.72535 −0.862674 0.505760i $$-0.831212\pi$$
−0.862674 + 0.505760i $$0.831212\pi$$
$$242$$ 0 0
$$243$$ 0 0
$$244$$ 0 0
$$245$$ 3.86370 0.246843
$$246$$ 0 0
$$247$$ −31.6675 −2.01495
$$248$$ 0 0
$$249$$ 0 0
$$250$$ 0 0
$$251$$ −6.53590 −0.412542 −0.206271 0.978495i $$-0.566133\pi$$
−0.206271 + 0.978495i $$0.566133\pi$$
$$252$$ 0 0
$$253$$ 4.14110 0.260349
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ 22.7846 1.42126 0.710632 0.703563i $$-0.248409\pi$$
0.710632 + 0.703563i $$0.248409\pi$$
$$258$$ 0 0
$$259$$ 1.60770 0.0998973
$$260$$ 0 0
$$261$$ 0 0
$$262$$ 0 0
$$263$$ −22.6274 −1.39527 −0.697633 0.716455i $$-0.745763\pi$$
−0.697633 + 0.716455i $$0.745763\pi$$
$$264$$ 0 0
$$265$$ −17.8564 −1.09691
$$266$$ 0 0
$$267$$ 0 0
$$268$$ 0 0
$$269$$ −10.2784 −0.626687 −0.313344 0.949640i $$-0.601449\pi$$
−0.313344 + 0.949640i $$0.601449\pi$$
$$270$$ 0 0
$$271$$ −3.20736 −0.194834 −0.0974168 0.995244i $$-0.531058\pi$$
−0.0974168 + 0.995244i $$0.531058\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 0 0
$$275$$ −14.5359 −0.876548
$$276$$ 0 0
$$277$$ 12.5249 0.752545 0.376273 0.926509i $$-0.377206\pi$$
0.376273 + 0.926509i $$0.377206\pi$$
$$278$$ 0 0
$$279$$ 0 0
$$280$$ 0 0
$$281$$ −0.928203 −0.0553720 −0.0276860 0.999617i $$-0.508814\pi$$
−0.0276860 + 0.999617i $$0.508814\pi$$
$$282$$ 0 0
$$283$$ −9.85641 −0.585903 −0.292951 0.956127i $$-0.594637\pi$$
−0.292951 + 0.956127i $$0.594637\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ −11.1106 −0.655840
$$288$$ 0 0
$$289$$ −5.00000 −0.294118
$$290$$ 0 0
$$291$$ 0 0
$$292$$ 0 0
$$293$$ −10.8332 −0.632884 −0.316442 0.948612i $$-0.602488\pi$$
−0.316442 + 0.948612i $$0.602488\pi$$
$$294$$ 0 0
$$295$$ −53.5370 −3.11705
$$296$$ 0 0
$$297$$ 0 0
$$298$$ 0 0
$$299$$ −12.0000 −0.693978
$$300$$ 0 0
$$301$$ 8.48528 0.489083
$$302$$ 0 0
$$303$$ 0 0
$$304$$ 0 0
$$305$$ 2.53590 0.145205
$$306$$ 0 0
$$307$$ −22.9282 −1.30858 −0.654291 0.756243i $$-0.727033\pi$$
−0.654291 + 0.756243i $$0.727033\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ 0 0
$$311$$ −32.4254 −1.83867 −0.919337 0.393471i $$-0.871274\pi$$
−0.919337 + 0.393471i $$0.871274\pi$$
$$312$$ 0 0
$$313$$ 3.85641 0.217977 0.108988 0.994043i $$-0.465239\pi$$
0.108988 + 0.994043i $$0.465239\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ −6.69213 −0.375867 −0.187934 0.982182i $$-0.560179\pi$$
−0.187934 + 0.982182i $$0.560179\pi$$
$$318$$ 0 0
$$319$$ 12.8295 0.718312
$$320$$ 0 0
$$321$$ 0 0
$$322$$ 0 0
$$323$$ −25.8564 −1.43869
$$324$$ 0 0
$$325$$ 42.1218 2.33650
$$326$$ 0 0
$$327$$ 0 0
$$328$$ 0 0
$$329$$ 6.92820 0.381964
$$330$$ 0 0
$$331$$ −1.07180 −0.0589113 −0.0294556 0.999566i $$-0.509377\pi$$
−0.0294556 + 0.999566i $$0.509377\pi$$
$$332$$ 0 0
$$333$$ 0 0
$$334$$ 0 0
$$335$$ 57.6781 3.15129
$$336$$ 0 0
$$337$$ 9.85641 0.536913 0.268456 0.963292i $$-0.413486\pi$$
0.268456 + 0.963292i $$0.413486\pi$$
$$338$$ 0 0
$$339$$ 0 0
$$340$$ 0 0
$$341$$ −10.7589 −0.582627
$$342$$ 0 0
$$343$$ −19.5959 −1.05808
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ 2.53590 0.136134 0.0680671 0.997681i $$-0.478317\pi$$
0.0680671 + 0.997681i $$0.478317\pi$$
$$348$$ 0 0
$$349$$ 27.4249 1.46802 0.734010 0.679139i $$-0.237647\pi$$
0.734010 + 0.679139i $$0.237647\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 0 0
$$353$$ 0.928203 0.0494033 0.0247016 0.999695i $$-0.492136\pi$$
0.0247016 + 0.999695i $$0.492136\pi$$
$$354$$ 0 0
$$355$$ 24.7846 1.31543
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 0 0
$$359$$ −18.8380 −0.994234 −0.497117 0.867684i $$-0.665608\pi$$
−0.497117 + 0.867684i $$0.665608\pi$$
$$360$$ 0 0
$$361$$ 36.7128 1.93225
$$362$$ 0 0
$$363$$ 0 0
$$364$$ 0 0
$$365$$ −15.4548 −0.808942
$$366$$ 0 0
$$367$$ −16.3886 −0.855476 −0.427738 0.903903i $$-0.640689\pi$$
−0.427738 + 0.903903i $$0.640689\pi$$
$$368$$ 0 0
$$369$$ 0 0
$$370$$ 0 0
$$371$$ 11.3205 0.587731
$$372$$ 0 0
$$373$$ −13.2827 −0.687753 −0.343877 0.939015i $$-0.611740\pi$$
−0.343877 + 0.939015i $$0.611740\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ −37.1769 −1.91471
$$378$$ 0 0
$$379$$ −13.3205 −0.684229 −0.342114 0.939658i $$-0.611143\pi$$
−0.342114 + 0.939658i $$0.611143\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ 0 0
$$383$$ −28.8391 −1.47361 −0.736804 0.676106i $$-0.763666\pi$$
−0.736804 + 0.676106i $$0.763666\pi$$
$$384$$ 0 0
$$385$$ 13.8564 0.706188
$$386$$ 0 0
$$387$$ 0 0
$$388$$ 0 0
$$389$$ −9.52056 −0.482711 −0.241356 0.970437i $$-0.577592\pi$$
−0.241356 + 0.970437i $$0.577592\pi$$
$$390$$ 0 0
$$391$$ −9.79796 −0.495504
$$392$$ 0 0
$$393$$ 0 0
$$394$$ 0 0
$$395$$ 9.46410 0.476191
$$396$$ 0 0
$$397$$ −23.0807 −1.15839 −0.579193 0.815190i $$-0.696632\pi$$
−0.579193 + 0.815190i $$0.696632\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ 0 0
$$401$$ 18.3923 0.918468 0.459234 0.888315i $$-0.348124\pi$$
0.459234 + 0.888315i $$0.348124\pi$$
$$402$$ 0 0
$$403$$ 31.1769 1.55303
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ −0.960947 −0.0476324
$$408$$ 0 0
$$409$$ −9.07180 −0.448571 −0.224286 0.974523i $$-0.572005\pi$$
−0.224286 + 0.974523i $$0.572005\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ 0 0
$$413$$ 33.9411 1.67013
$$414$$ 0 0
$$415$$ −21.1117 −1.03633
$$416$$ 0 0
$$417$$ 0 0
$$418$$ 0 0
$$419$$ 0.392305 0.0191653 0.00958267 0.999954i $$-0.496950\pi$$
0.00958267 + 0.999954i $$0.496950\pi$$
$$420$$ 0 0
$$421$$ 19.6975 0.959995 0.479998 0.877270i $$-0.340638\pi$$
0.479998 + 0.877270i $$0.340638\pi$$
$$422$$ 0 0
$$423$$ 0 0
$$424$$ 0 0
$$425$$ 34.3923 1.66827
$$426$$ 0 0
$$427$$ −1.60770 −0.0778018
$$428$$ 0 0
$$429$$ 0 0
$$430$$ 0 0
$$431$$ −4.34418 −0.209252 −0.104626 0.994512i $$-0.533364\pi$$
−0.104626 + 0.994512i $$0.533364\pi$$
$$432$$ 0 0
$$433$$ −29.8564 −1.43481 −0.717404 0.696658i $$-0.754670\pi$$
−0.717404 + 0.696658i $$0.754670\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ 21.1117 1.00991
$$438$$ 0 0
$$439$$ 5.83272 0.278381 0.139190 0.990266i $$-0.455550\pi$$
0.139190 + 0.990266i $$0.455550\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 0 0
$$443$$ −35.3205 −1.67813 −0.839064 0.544033i $$-0.816897\pi$$
−0.839064 + 0.544033i $$0.816897\pi$$
$$444$$ 0 0
$$445$$ −19.0411 −0.902635
$$446$$ 0 0
$$447$$ 0 0
$$448$$ 0 0
$$449$$ 6.39230 0.301672 0.150836 0.988559i $$-0.451804\pi$$
0.150836 + 0.988559i $$0.451804\pi$$
$$450$$ 0 0
$$451$$ 6.64102 0.312713
$$452$$ 0 0
$$453$$ 0 0
$$454$$ 0 0
$$455$$ −40.1528 −1.88239
$$456$$ 0 0
$$457$$ −20.7846 −0.972263 −0.486132 0.873886i $$-0.661592\pi$$
−0.486132 + 0.873886i $$0.661592\pi$$
$$458$$ 0 0
$$459$$ 0 0
$$460$$ 0 0
$$461$$ −31.1870 −1.45252 −0.726262 0.687418i $$-0.758744\pi$$
−0.726262 + 0.687418i $$0.758744\pi$$
$$462$$ 0 0
$$463$$ −29.2180 −1.35788 −0.678938 0.734196i $$-0.737560\pi$$
−0.678938 + 0.734196i $$0.737560\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ −30.2487 −1.39974 −0.699872 0.714269i $$-0.746760\pi$$
−0.699872 + 0.714269i $$0.746760\pi$$
$$468$$ 0 0
$$469$$ −36.5665 −1.68848
$$470$$ 0 0
$$471$$ 0 0
$$472$$ 0 0
$$473$$ −5.07180 −0.233201
$$474$$ 0 0
$$475$$ −74.1051 −3.40018
$$476$$ 0 0
$$477$$ 0 0
$$478$$ 0 0
$$479$$ 29.0421 1.32697 0.663485 0.748190i $$-0.269077\pi$$
0.663485 + 0.748190i $$0.269077\pi$$
$$480$$ 0 0
$$481$$ 2.78461 0.126967
$$482$$ 0 0
$$483$$ 0 0
$$484$$ 0 0
$$485$$ −4.14110 −0.188038
$$486$$ 0 0
$$487$$ 11.4896 0.520642 0.260321 0.965522i $$-0.416172\pi$$
0.260321 + 0.965522i $$0.416172\pi$$
$$488$$ 0 0
$$489$$ 0 0
$$490$$ 0 0
$$491$$ 5.85641 0.264296 0.132148 0.991230i $$-0.457813\pi$$
0.132148 + 0.991230i $$0.457813\pi$$
$$492$$ 0 0
$$493$$ −30.3548 −1.36711
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ −15.7128 −0.704816
$$498$$ 0 0
$$499$$ −4.00000 −0.179065 −0.0895323 0.995984i $$-0.528537\pi$$
−0.0895323 + 0.995984i $$0.528537\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ 0 0
$$503$$ −23.3853 −1.04270 −0.521349 0.853343i $$-0.674571\pi$$
−0.521349 + 0.853343i $$0.674571\pi$$
$$504$$ 0 0
$$505$$ −55.7128 −2.47919
$$506$$ 0 0
$$507$$ 0 0
$$508$$ 0 0
$$509$$ 7.24693 0.321215 0.160607 0.987018i $$-0.448655\pi$$
0.160607 + 0.987018i $$0.448655\pi$$
$$510$$ 0 0
$$511$$ 9.79796 0.433436
$$512$$ 0 0
$$513$$ 0 0
$$514$$ 0 0
$$515$$ 12.3923 0.546070
$$516$$ 0 0
$$517$$ −4.14110 −0.182126
$$518$$ 0 0
$$519$$ 0 0
$$520$$ 0 0
$$521$$ 21.3205 0.934068 0.467034 0.884239i $$-0.345322\pi$$
0.467034 + 0.884239i $$0.345322\pi$$
$$522$$ 0 0
$$523$$ −31.4641 −1.37583 −0.687915 0.725792i $$-0.741474\pi$$
−0.687915 + 0.725792i $$0.741474\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ 25.4558 1.10887
$$528$$ 0 0
$$529$$ −15.0000 −0.652174
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 0 0
$$533$$ −19.2442 −0.833558
$$534$$ 0 0
$$535$$ 15.4548 0.668170
$$536$$ 0 0
$$537$$ 0 0
$$538$$ 0 0
$$539$$ 1.46410 0.0630633
$$540$$ 0 0
$$541$$ −28.1827 −1.21167 −0.605835 0.795590i $$-0.707161\pi$$
−0.605835 + 0.795590i $$0.707161\pi$$
$$542$$ 0 0
$$543$$ 0 0
$$544$$ 0 0
$$545$$ −5.46410 −0.234056
$$546$$ 0 0
$$547$$ −23.1769 −0.990973 −0.495487 0.868616i $$-0.665010\pi$$
−0.495487 + 0.868616i $$0.665010\pi$$
$$548$$ 0 0
$$549$$ 0 0
$$550$$ 0 0
$$551$$ 65.4056 2.78637
$$552$$ 0 0
$$553$$ −6.00000 −0.255146
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ 28.0069 1.18669 0.593345 0.804949i $$-0.297807\pi$$
0.593345 + 0.804949i $$0.297807\pi$$
$$558$$ 0 0
$$559$$ 14.6969 0.621614
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 0 0
$$563$$ 1.46410 0.0617045 0.0308523 0.999524i $$-0.490178\pi$$
0.0308523 + 0.999524i $$0.490178\pi$$
$$564$$ 0 0
$$565$$ 69.5467 2.92585
$$566$$ 0 0
$$567$$ 0 0
$$568$$ 0 0
$$569$$ 3.46410 0.145223 0.0726113 0.997360i $$-0.476867\pi$$
0.0726113 + 0.997360i $$0.476867\pi$$
$$570$$ 0 0
$$571$$ 31.7128 1.32714 0.663570 0.748114i $$-0.269041\pi$$
0.663570 + 0.748114i $$0.269041\pi$$
$$572$$ 0 0
$$573$$ 0 0
$$574$$ 0 0
$$575$$ −28.0812 −1.17107
$$576$$ 0 0
$$577$$ −21.8564 −0.909894 −0.454947 0.890518i $$-0.650342\pi$$
−0.454947 + 0.890518i $$0.650342\pi$$
$$578$$ 0 0
$$579$$ 0 0
$$580$$ 0 0
$$581$$ 13.3843 0.555273
$$582$$ 0 0
$$583$$ −6.76646 −0.280238
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ −42.9282 −1.77184 −0.885918 0.463841i $$-0.846471\pi$$
−0.885918 + 0.463841i $$0.846471\pi$$
$$588$$ 0 0
$$589$$ −54.8497 −2.26004
$$590$$ 0 0
$$591$$ 0 0
$$592$$ 0 0
$$593$$ −37.7128 −1.54868 −0.774340 0.632770i $$-0.781918\pi$$
−0.774340 + 0.632770i $$0.781918\pi$$
$$594$$ 0 0
$$595$$ −32.7846 −1.34404
$$596$$ 0 0
$$597$$ 0 0
$$598$$ 0 0
$$599$$ −6.96953 −0.284767 −0.142384 0.989812i $$-0.545477\pi$$
−0.142384 + 0.989812i $$0.545477\pi$$
$$600$$ 0 0
$$601$$ 19.7128 0.804102 0.402051 0.915617i $$-0.368297\pi$$
0.402051 + 0.915617i $$0.368297\pi$$
$$602$$ 0 0
$$603$$ 0 0
$$604$$ 0 0
$$605$$ 34.2185 1.39118
$$606$$ 0 0
$$607$$ −31.8434 −1.29248 −0.646241 0.763133i $$-0.723660\pi$$
−0.646241 + 0.763133i $$0.723660\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ 12.0000 0.485468
$$612$$ 0 0
$$613$$ −6.11012 −0.246785 −0.123393 0.992358i $$-0.539377\pi$$
−0.123393 + 0.992358i $$0.539377\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ 39.8564 1.60456 0.802279 0.596949i $$-0.203621\pi$$
0.802279 + 0.596949i $$0.203621\pi$$
$$618$$ 0 0
$$619$$ 12.0000 0.482321 0.241160 0.970485i $$-0.422472\pi$$
0.241160 + 0.970485i $$0.422472\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 0 0
$$623$$ 12.0716 0.483638
$$624$$ 0 0
$$625$$ 23.9282 0.957128
$$626$$ 0 0
$$627$$ 0 0
$$628$$ 0 0
$$629$$ 2.27362 0.0906553
$$630$$ 0 0
$$631$$ −23.5612 −0.937955 −0.468977 0.883210i $$-0.655377\pi$$
−0.468977 + 0.883210i $$0.655377\pi$$
$$632$$ 0 0
$$633$$ 0 0
$$634$$ 0 0
$$635$$ −56.1051 −2.22646
$$636$$ 0 0
$$637$$ −4.24264 −0.168100
$$638$$ 0 0
$$639$$ 0 0
$$640$$ 0 0
$$641$$ 25.6077 1.01144 0.505722 0.862697i $$-0.331226\pi$$
0.505722 + 0.862697i $$0.331226\pi$$
$$642$$ 0 0
$$643$$ 28.5359 1.12535 0.562673 0.826680i $$-0.309773\pi$$
0.562673 + 0.826680i $$0.309773\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ −40.9107 −1.60836 −0.804182 0.594383i $$-0.797396\pi$$
−0.804182 + 0.594383i $$0.797396\pi$$
$$648$$ 0 0
$$649$$ −20.2872 −0.796342
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 0 0
$$653$$ 24.9754 0.977362 0.488681 0.872463i $$-0.337478\pi$$
0.488681 + 0.872463i $$0.337478\pi$$
$$654$$ 0 0
$$655$$ 57.6781 2.25367
$$656$$ 0 0
$$657$$ 0 0
$$658$$ 0 0
$$659$$ −24.0000 −0.934907 −0.467454 0.884018i $$-0.654829\pi$$
−0.467454 + 0.884018i $$0.654829\pi$$
$$660$$ 0 0
$$661$$ 6.51626 0.253453 0.126727 0.991938i $$-0.459553\pi$$
0.126727 + 0.991938i $$0.459553\pi$$
$$662$$ 0 0
$$663$$ 0 0
$$664$$ 0 0
$$665$$ 70.6410 2.73934
$$666$$ 0 0
$$667$$ 24.7846 0.959664
$$668$$ 0 0
$$669$$ 0 0
$$670$$ 0 0
$$671$$ 0.960947 0.0370969
$$672$$ 0 0
$$673$$ 8.00000 0.308377 0.154189 0.988041i $$-0.450724\pi$$
0.154189 + 0.988041i $$0.450724\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ −3.30890 −0.127171 −0.0635857 0.997976i $$-0.520254\pi$$
−0.0635857 + 0.997976i $$0.520254\pi$$
$$678$$ 0 0
$$679$$ 2.62536 0.100752
$$680$$ 0 0
$$681$$ 0 0
$$682$$ 0 0
$$683$$ 25.1769 0.963368 0.481684 0.876345i $$-0.340025\pi$$
0.481684 + 0.876345i $$0.340025\pi$$
$$684$$ 0 0
$$685$$ 55.6076 2.12466
$$686$$ 0 0
$$687$$ 0 0
$$688$$ 0 0
$$689$$ 19.6077 0.746994
$$690$$ 0 0
$$691$$ −26.3923 −1.00401 −0.502005 0.864865i $$-0.667404\pi$$
−0.502005 + 0.864865i $$0.667404\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ 26.7685 1.01539
$$696$$ 0 0
$$697$$ −15.7128 −0.595165
$$698$$ 0 0
$$699$$ 0 0
$$700$$ 0 0
$$701$$ 14.9743 0.565573 0.282787 0.959183i $$-0.408741\pi$$
0.282787 + 0.959183i $$0.408741\pi$$
$$702$$ 0 0
$$703$$ −4.89898 −0.184769
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ 35.3205 1.32836
$$708$$ 0 0
$$709$$ −49.0913 −1.84366 −0.921832 0.387590i $$-0.873308\pi$$
−0.921832 + 0.387590i $$0.873308\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ 0 0
$$713$$ −20.7846 −0.778390
$$714$$ 0 0
$$715$$ 24.0000 0.897549
$$716$$ 0 0
$$717$$ 0 0
$$718$$ 0 0
$$719$$ 2.27362 0.0847919 0.0423959 0.999101i $$-0.486501\pi$$
0.0423959 + 0.999101i $$0.486501\pi$$
$$720$$ 0 0
$$721$$ −7.85641 −0.292588
$$722$$ 0 0
$$723$$ 0 0
$$724$$ 0 0
$$725$$ −86.9977 −3.23101
$$726$$ 0 0
$$727$$ −8.10634 −0.300648 −0.150324 0.988637i $$-0.548032\pi$$
−0.150324 + 0.988637i $$0.548032\pi$$
$$728$$ 0 0
$$729$$ 0 0
$$730$$ 0 0
$$731$$ 12.0000 0.443836
$$732$$ 0 0
$$733$$ 7.07107 0.261176 0.130588 0.991437i $$-0.458314\pi$$
0.130588 + 0.991437i $$0.458314\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ 21.8564 0.805091
$$738$$ 0 0
$$739$$ −36.0000 −1.32428 −0.662141 0.749380i $$-0.730352\pi$$
−0.662141 + 0.749380i $$0.730352\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 0 0
$$743$$ −27.3233 −1.00240 −0.501198 0.865333i $$-0.667107\pi$$
−0.501198 + 0.865333i $$0.667107\pi$$
$$744$$ 0 0
$$745$$ −20.7846 −0.761489
$$746$$ 0 0
$$747$$ 0 0
$$748$$ 0 0
$$749$$ −9.79796 −0.358010
$$750$$ 0 0
$$751$$ 46.1886 1.68545 0.842723 0.538348i $$-0.180951\pi$$
0.842723 + 0.538348i $$0.180951\pi$$
$$752$$ 0 0
$$753$$ 0 0
$$754$$ 0 0
$$755$$ −25.4641 −0.926734
$$756$$ 0 0
$$757$$ −33.6365 −1.22254 −0.611270 0.791422i $$-0.709341\pi$$
−0.611270 + 0.791422i $$0.709341\pi$$
$$758$$ 0 0
$$759$$ 0 0
$$760$$ 0 0
$$761$$ 52.2487 1.89401 0.947007 0.321212i $$-0.104090\pi$$
0.947007 + 0.321212i $$0.104090\pi$$
$$762$$ 0 0
$$763$$ 3.46410 0.125409
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ 58.7878 2.12270
$$768$$ 0 0
$$769$$ −22.0000 −0.793340 −0.396670 0.917961i $$-0.629834\pi$$
−0.396670 + 0.917961i $$0.629834\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 0 0
$$773$$ −48.9155 −1.75937 −0.879684 0.475560i $$-0.842246\pi$$
−0.879684 + 0.475560i $$0.842246\pi$$
$$774$$ 0 0
$$775$$ 72.9571 2.62070
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 0 0
$$779$$ 33.8564 1.21303
$$780$$ 0 0
$$781$$ 9.39182 0.336066
$$782$$ 0 0
$$783$$ 0 0
$$784$$ 0 0
$$785$$ −8.39230 −0.299534
$$786$$ 0 0
$$787$$ 23.4641 0.836405 0.418202 0.908354i $$-0.362660\pi$$
0.418202 + 0.908354i $$0.362660\pi$$
$$788$$ 0 0
$$789$$ 0 0
$$790$$ 0 0
$$791$$ −44.0908 −1.56769
$$792$$ 0 0
$$793$$ −2.78461 −0.0988844
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ −21.9439 −0.777292 −0.388646 0.921387i $$-0.627057\pi$$
−0.388646 + 0.921387i $$0.627057\pi$$
$$798$$ 0 0
$$799$$ 9.79796 0.346627
$$800$$ 0 0
$$801$$ 0 0
$$802$$ 0 0
$$803$$ −5.85641 −0.206668
$$804$$ 0 0
$$805$$ 26.7685 0.943466
$$806$$ 0 0
$$807$$ 0 0
$$808$$ 0 0
$$809$$ 6.67949 0.234838 0.117419 0.993082i $$-0.462538\pi$$
0.117419 + 0.993082i $$0.462538\pi$$
$$810$$ 0 0
$$811$$ −54.3923 −1.90997 −0.954986 0.296651i $$-0.904130\pi$$
−0.954986 + 0.296651i $$0.904130\pi$$
$$812$$ 0 0
$$813$$ 0 0
$$814$$ 0 0
$$815$$ 17.5254 0.613887
$$816$$ 0 0
$$817$$ −25.8564 −0.904601
$$818$$ 0 0
$$819$$ 0 0
$$820$$ 0 0
$$821$$ 25.1784 0.878734 0.439367 0.898308i $$-0.355203\pi$$
0.439367 + 0.898308i $$0.355203\pi$$
$$822$$ 0 0
$$823$$ 9.97382 0.347666 0.173833 0.984775i $$-0.444385\pi$$
0.173833 + 0.984775i $$0.444385\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ 44.7846 1.55731 0.778657 0.627450i $$-0.215901\pi$$
0.778657 + 0.627450i $$0.215901\pi$$
$$828$$ 0 0
$$829$$ −1.61729 −0.0561706 −0.0280853 0.999606i $$-0.508941\pi$$
−0.0280853 + 0.999606i $$0.508941\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ 0 0
$$833$$ −3.46410 −0.120024
$$834$$ 0 0
$$835$$ −29.8564 −1.03322
$$836$$ 0 0
$$837$$ 0 0
$$838$$ 0 0
$$839$$ 26.0106 0.897987 0.448994 0.893535i $$-0.351783\pi$$
0.448994 + 0.893535i $$0.351783\pi$$
$$840$$ 0 0
$$841$$ 47.7846 1.64775
$$842$$ 0 0
$$843$$ 0 0
$$844$$ 0 0
$$845$$ −19.3185 −0.664577
$$846$$ 0 0
$$847$$ −21.6937 −0.745404
$$848$$ 0 0
$$849$$ 0 0
$$850$$ 0 0
$$851$$ −1.85641 −0.0636368
$$852$$ 0 0
$$853$$ 47.0208 1.60996 0.804980 0.593301i $$-0.202176\pi$$
0.804980 + 0.593301i $$0.202176\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ −43.1769 −1.47490 −0.737448 0.675404i $$-0.763969\pi$$
−0.737448 + 0.675404i $$0.763969\pi$$
$$858$$ 0 0
$$859$$ 32.2487 1.10031 0.550156 0.835062i $$-0.314568\pi$$
0.550156 + 0.835062i $$0.314568\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 0 0
$$863$$ 37.1213 1.26362 0.631812 0.775122i $$-0.282312\pi$$
0.631812 + 0.775122i $$0.282312\pi$$
$$864$$ 0 0
$$865$$ −66.6410 −2.26586
$$866$$ 0 0
$$867$$ 0 0
$$868$$ 0 0
$$869$$ 3.58630 0.121657
$$870$$ 0 0
$$871$$ −63.3350 −2.14602
$$872$$ 0 0
$$873$$ 0 0
$$874$$ 0 0
$$875$$ −46.6410 −1.57675
$$876$$ 0 0
$$877$$ −35.7071 −1.20574 −0.602871 0.797839i $$-0.705977\pi$$
−0.602871 + 0.797839i $$0.705977\pi$$
$$878$$ 0 0
$$879$$ 0 0
$$880$$ 0 0
$$881$$ 36.6410 1.23447 0.617234 0.786780i $$-0.288253\pi$$
0.617234 + 0.786780i $$0.288253\pi$$
$$882$$ 0 0
$$883$$ 5.60770 0.188714 0.0943570 0.995538i $$-0.469920\pi$$
0.0943570 + 0.995538i $$0.469920\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ 36.5665 1.22778 0.613891 0.789391i $$-0.289603\pi$$
0.613891 + 0.789391i $$0.289603\pi$$
$$888$$ 0 0
$$889$$ 35.5692 1.19295
$$890$$ 0 0
$$891$$ 0 0
$$892$$ 0 0
$$893$$ −21.1117 −0.706475
$$894$$ 0 0
$$895$$ 73.1330 2.44457
$$896$$ 0 0
$$897$$ 0 0
$$898$$ 0 0
$$899$$ −64.3923 −2.14760
$$900$$ 0 0
$$901$$ 16.0096 0.533358
$$902$$ 0 0
$$903$$ 0 0
$$904$$ 0 0
$$905$$ −27.3205 −0.908164
$$906$$ 0 0
$$907$$ 42.3923 1.40761 0.703807 0.710392i $$-0.251482\pi$$
0.703807 + 0.710392i $$0.251482\pi$$
$$908$$ 0 0
$$909$$ 0 0
$$910$$ 0 0
$$911$$ −22.0726 −0.731298 −0.365649 0.930753i $$-0.619153\pi$$
−0.365649 + 0.930753i $$0.619153\pi$$
$$912$$ 0 0
$$913$$ −8.00000 −0.264761
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 0 0
$$917$$ −36.5665 −1.20753
$$918$$ 0 0
$$919$$ 0.582009 0.0191987 0.00959936 0.999954i $$-0.496944\pi$$
0.00959936 + 0.999954i $$0.496944\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ 0 0
$$923$$ −27.2154 −0.895805
$$924$$ 0 0
$$925$$ 6.51626 0.214253
$$926$$ 0 0
$$927$$ 0 0
$$928$$ 0 0
$$929$$ 45.3205 1.48692 0.743459 0.668782i $$-0.233184\pi$$
0.743459 + 0.668782i $$0.233184\pi$$
$$930$$ 0 0
$$931$$ 7.46410 0.244626
$$932$$ 0 0
$$933$$ 0 0
$$934$$ 0 0
$$935$$ 19.5959 0.640855
$$936$$ 0 0
$$937$$ 2.00000 0.0653372 0.0326686 0.999466i $$-0.489599\pi$$
0.0326686 + 0.999466i $$0.489599\pi$$
$$938$$ 0 0
$$939$$ 0 0
$$940$$ 0 0
$$941$$ 36.2891 1.18299 0.591495 0.806309i $$-0.298538\pi$$
0.591495 + 0.806309i $$0.298538\pi$$
$$942$$ 0 0
$$943$$ 12.8295 0.417785
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ 29.0718 0.944706 0.472353 0.881409i $$-0.343405\pi$$
0.472353 + 0.881409i $$0.343405\pi$$
$$948$$ 0 0
$$949$$ 16.9706 0.550888
$$950$$ 0 0
$$951$$ 0 0
$$952$$ 0 0
$$953$$ −46.3923 −1.50279 −0.751397 0.659850i $$-0.770620\pi$$
−0.751397 + 0.659850i $$0.770620\pi$$
$$954$$ 0 0
$$955$$ −29.8564 −0.966131
$$956$$ 0 0
$$957$$ 0 0
$$958$$ 0 0
$$959$$ −35.2538 −1.13840
$$960$$ 0 0
$$961$$ 23.0000 0.741935
$$962$$ 0 0
$$963$$ 0 0
$$964$$ 0 0
$$965$$ −23.1822 −0.746262
$$966$$ 0 0
$$967$$ −10.3800 −0.333797 −0.166899 0.985974i $$-0.553375\pi$$
−0.166899 + 0.985974i $$0.553375\pi$$
$$968$$ 0 0
$$969$$ 0 0
$$970$$ 0 0
$$971$$ 22.2487 0.713995 0.356998 0.934105i $$-0.383800\pi$$
0.356998 + 0.934105i $$0.383800\pi$$
$$972$$ 0 0
$$973$$ −16.9706 −0.544051
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ 35.1769 1.12541 0.562705 0.826658i $$-0.309761\pi$$
0.562705 + 0.826658i $$0.309761\pi$$
$$978$$ 0 0
$$979$$ −7.21539 −0.230605
$$980$$ 0 0
$$981$$ 0 0
$$982$$ 0 0
$$983$$ 23.7370 0.757093 0.378547 0.925582i $$-0.376424\pi$$
0.378547 + 0.925582i $$0.376424\pi$$
$$984$$ 0 0
$$985$$ −33.8564 −1.07875
$$986$$ 0 0
$$987$$ 0 0
$$988$$ 0 0
$$989$$ −9.79796 −0.311557
$$990$$ 0 0
$$991$$ 12.2474 0.389053 0.194527 0.980897i $$-0.437683\pi$$
0.194527 + 0.980897i $$0.437683\pi$$
$$992$$ 0 0
$$993$$ 0 0
$$994$$ 0 0
$$995$$ −72.1051 −2.28589
$$996$$ 0 0
$$997$$ 17.6269 0.558250 0.279125 0.960255i $$-0.409956\pi$$
0.279125 + 0.960255i $$0.409956\pi$$
$$998$$ 0 0
$$999$$ 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 9216.2.a.bi.1.1 4
3.2 odd 2 3072.2.a.l.1.4 4
4.3 odd 2 9216.2.a.bc.1.1 4
8.3 odd 2 inner 9216.2.a.bi.1.4 4
8.5 even 2 9216.2.a.bc.1.4 4
12.11 even 2 3072.2.a.r.1.4 4
24.5 odd 2 3072.2.a.r.1.1 4
24.11 even 2 3072.2.a.l.1.1 4
32.3 odd 8 2304.2.k.l.577.4 8
32.5 even 8 2304.2.k.e.1729.2 8
32.11 odd 8 2304.2.k.l.1729.3 8
32.13 even 8 2304.2.k.e.577.1 8
32.19 odd 8 2304.2.k.e.577.2 8
32.21 even 8 2304.2.k.l.1729.4 8
32.27 odd 8 2304.2.k.e.1729.1 8
32.29 even 8 2304.2.k.l.577.3 8
48.5 odd 4 3072.2.d.h.1537.8 8
48.11 even 4 3072.2.d.h.1537.4 8
48.29 odd 4 3072.2.d.h.1537.1 8
48.35 even 4 3072.2.d.h.1537.5 8
96.5 odd 8 768.2.j.f.193.4 yes 8
96.11 even 8 768.2.j.e.193.3 yes 8
96.29 odd 8 768.2.j.e.577.1 yes 8
96.35 even 8 768.2.j.e.577.3 yes 8
96.53 odd 8 768.2.j.e.193.1 8
96.59 even 8 768.2.j.f.193.2 yes 8
96.77 odd 8 768.2.j.f.577.4 yes 8
96.83 even 8 768.2.j.f.577.2 yes 8

By twisted newform
Twist Min Dim Char Parity Ord Type
768.2.j.e.193.1 8 96.53 odd 8
768.2.j.e.193.3 yes 8 96.11 even 8
768.2.j.e.577.1 yes 8 96.29 odd 8
768.2.j.e.577.3 yes 8 96.35 even 8
768.2.j.f.193.2 yes 8 96.59 even 8
768.2.j.f.193.4 yes 8 96.5 odd 8
768.2.j.f.577.2 yes 8 96.83 even 8
768.2.j.f.577.4 yes 8 96.77 odd 8
2304.2.k.e.577.1 8 32.13 even 8
2304.2.k.e.577.2 8 32.19 odd 8
2304.2.k.e.1729.1 8 32.27 odd 8
2304.2.k.e.1729.2 8 32.5 even 8
2304.2.k.l.577.3 8 32.29 even 8
2304.2.k.l.577.4 8 32.3 odd 8
2304.2.k.l.1729.3 8 32.11 odd 8
2304.2.k.l.1729.4 8 32.21 even 8
3072.2.a.l.1.1 4 24.11 even 2
3072.2.a.l.1.4 4 3.2 odd 2
3072.2.a.r.1.1 4 24.5 odd 2
3072.2.a.r.1.4 4 12.11 even 2
3072.2.d.h.1537.1 8 48.29 odd 4
3072.2.d.h.1537.4 8 48.11 even 4
3072.2.d.h.1537.5 8 48.35 even 4
3072.2.d.h.1537.8 8 48.5 odd 4
9216.2.a.bc.1.1 4 4.3 odd 2
9216.2.a.bc.1.4 4 8.5 even 2
9216.2.a.bi.1.1 4 1.1 even 1 trivial
9216.2.a.bi.1.4 4 8.3 odd 2 inner