# Properties

 Label 9075.2.a.bf.1.2 Level $9075$ Weight $2$ Character 9075.1 Self dual yes Analytic conductor $72.464$ 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] = [9075,2,Mod(1,9075)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(9075, 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("9075.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$9075 = 3 \cdot 5^{2} \cdot 11^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 9075.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: $$72.4642398343$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{15})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - 15$$ x^2 - 15 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.2 Root $$3.87298$$ of defining polynomial Character $$\chi$$ $$=$$ 9075.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-1.00000 q^{3} -2.00000 q^{4} +3.87298 q^{7} +1.00000 q^{9} +O(q^{10})$$ $$q-1.00000 q^{3} -2.00000 q^{4} +3.87298 q^{7} +1.00000 q^{9} +2.00000 q^{12} -3.87298 q^{13} +4.00000 q^{16} -7.74597 q^{17} -3.87298 q^{19} -3.87298 q^{21} -6.00000 q^{23} -1.00000 q^{27} -7.74597 q^{28} -7.74597 q^{29} -5.00000 q^{31} -2.00000 q^{36} -2.00000 q^{37} +3.87298 q^{39} +7.74597 q^{41} +3.87298 q^{43} +12.0000 q^{47} -4.00000 q^{48} +8.00000 q^{49} +7.74597 q^{51} +7.74597 q^{52} -6.00000 q^{53} +3.87298 q^{57} -11.6190 q^{61} +3.87298 q^{63} -8.00000 q^{64} +7.00000 q^{67} +15.4919 q^{68} +6.00000 q^{69} +12.0000 q^{71} +7.74597 q^{76} -7.74597 q^{79} +1.00000 q^{81} +7.74597 q^{83} +7.74597 q^{84} +7.74597 q^{87} +6.00000 q^{89} -15.0000 q^{91} +12.0000 q^{92} +5.00000 q^{93} -7.00000 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{3} - 4 q^{4} + 2 q^{9}+O(q^{10})$$ 2 * q - 2 * q^3 - 4 * q^4 + 2 * q^9 $$2 q - 2 q^{3} - 4 q^{4} + 2 q^{9} + 4 q^{12} + 8 q^{16} - 12 q^{23} - 2 q^{27} - 10 q^{31} - 4 q^{36} - 4 q^{37} + 24 q^{47} - 8 q^{48} + 16 q^{49} - 12 q^{53} - 16 q^{64} + 14 q^{67} + 12 q^{69} + 24 q^{71} + 2 q^{81} + 12 q^{89} - 30 q^{91} + 24 q^{92} + 10 q^{93} - 14 q^{97}+O(q^{100})$$ 2 * q - 2 * q^3 - 4 * q^4 + 2 * q^9 + 4 * q^12 + 8 * q^16 - 12 * q^23 - 2 * q^27 - 10 * q^31 - 4 * q^36 - 4 * q^37 + 24 * q^47 - 8 * q^48 + 16 * q^49 - 12 * q^53 - 16 * q^64 + 14 * q^67 + 12 * q^69 + 24 * q^71 + 2 * q^81 + 12 * q^89 - 30 * q^91 + 24 * q^92 + 10 * q^93 - 14 * 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 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$3$$ −1.00000 −0.577350
$$4$$ −2.00000 −1.00000
$$5$$ 0 0
$$6$$ 0 0
$$7$$ 3.87298 1.46385 0.731925 0.681385i $$-0.238622\pi$$
0.731925 + 0.681385i $$0.238622\pi$$
$$8$$ 0 0
$$9$$ 1.00000 0.333333
$$10$$ 0 0
$$11$$ 0 0
$$12$$ 2.00000 0.577350
$$13$$ −3.87298 −1.07417 −0.537086 0.843527i $$-0.680475\pi$$
−0.537086 + 0.843527i $$0.680475\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 4.00000 1.00000
$$17$$ −7.74597 −1.87867 −0.939336 0.342997i $$-0.888558\pi$$
−0.939336 + 0.342997i $$0.888558\pi$$
$$18$$ 0 0
$$19$$ −3.87298 −0.888523 −0.444262 0.895897i $$-0.646534\pi$$
−0.444262 + 0.895897i $$0.646534\pi$$
$$20$$ 0 0
$$21$$ −3.87298 −0.845154
$$22$$ 0 0
$$23$$ −6.00000 −1.25109 −0.625543 0.780189i $$-0.715123\pi$$
−0.625543 + 0.780189i $$0.715123\pi$$
$$24$$ 0 0
$$25$$ 0 0
$$26$$ 0 0
$$27$$ −1.00000 −0.192450
$$28$$ −7.74597 −1.46385
$$29$$ −7.74597 −1.43839 −0.719195 0.694808i $$-0.755489\pi$$
−0.719195 + 0.694808i $$0.755489\pi$$
$$30$$ 0 0
$$31$$ −5.00000 −0.898027 −0.449013 0.893525i $$-0.648224\pi$$
−0.449013 + 0.893525i $$0.648224\pi$$
$$32$$ 0 0
$$33$$ 0 0
$$34$$ 0 0
$$35$$ 0 0
$$36$$ −2.00000 −0.333333
$$37$$ −2.00000 −0.328798 −0.164399 0.986394i $$-0.552568\pi$$
−0.164399 + 0.986394i $$0.552568\pi$$
$$38$$ 0 0
$$39$$ 3.87298 0.620174
$$40$$ 0 0
$$41$$ 7.74597 1.20972 0.604858 0.796333i $$-0.293230\pi$$
0.604858 + 0.796333i $$0.293230\pi$$
$$42$$ 0 0
$$43$$ 3.87298 0.590624 0.295312 0.955401i $$-0.404576\pi$$
0.295312 + 0.955401i $$0.404576\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ 12.0000 1.75038 0.875190 0.483779i $$-0.160736\pi$$
0.875190 + 0.483779i $$0.160736\pi$$
$$48$$ −4.00000 −0.577350
$$49$$ 8.00000 1.14286
$$50$$ 0 0
$$51$$ 7.74597 1.08465
$$52$$ 7.74597 1.07417
$$53$$ −6.00000 −0.824163 −0.412082 0.911147i $$-0.635198\pi$$
−0.412082 + 0.911147i $$0.635198\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 0 0
$$57$$ 3.87298 0.512989
$$58$$ 0 0
$$59$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$60$$ 0 0
$$61$$ −11.6190 −1.48765 −0.743827 0.668372i $$-0.766991\pi$$
−0.743827 + 0.668372i $$0.766991\pi$$
$$62$$ 0 0
$$63$$ 3.87298 0.487950
$$64$$ −8.00000 −1.00000
$$65$$ 0 0
$$66$$ 0 0
$$67$$ 7.00000 0.855186 0.427593 0.903971i $$-0.359362\pi$$
0.427593 + 0.903971i $$0.359362\pi$$
$$68$$ 15.4919 1.87867
$$69$$ 6.00000 0.722315
$$70$$ 0 0
$$71$$ 12.0000 1.42414 0.712069 0.702109i $$-0.247758\pi$$
0.712069 + 0.702109i $$0.247758\pi$$
$$72$$ 0 0
$$73$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 7.74597 0.888523
$$77$$ 0 0
$$78$$ 0 0
$$79$$ −7.74597 −0.871489 −0.435745 0.900070i $$-0.643515\pi$$
−0.435745 + 0.900070i $$0.643515\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ 0 0
$$83$$ 7.74597 0.850230 0.425115 0.905139i $$-0.360234\pi$$
0.425115 + 0.905139i $$0.360234\pi$$
$$84$$ 7.74597 0.845154
$$85$$ 0 0
$$86$$ 0 0
$$87$$ 7.74597 0.830455
$$88$$ 0 0
$$89$$ 6.00000 0.635999 0.317999 0.948091i $$-0.396989\pi$$
0.317999 + 0.948091i $$0.396989\pi$$
$$90$$ 0 0
$$91$$ −15.0000 −1.57243
$$92$$ 12.0000 1.25109
$$93$$ 5.00000 0.518476
$$94$$ 0 0
$$95$$ 0 0
$$96$$ 0 0
$$97$$ −7.00000 −0.710742 −0.355371 0.934725i $$-0.615646\pi$$
−0.355371 + 0.934725i $$0.615646\pi$$
$$98$$ 0 0
$$99$$ 0 0
$$100$$ 0 0
$$101$$ 15.4919 1.54150 0.770752 0.637135i $$-0.219880\pi$$
0.770752 + 0.637135i $$0.219880\pi$$
$$102$$ 0 0
$$103$$ −16.0000 −1.57653 −0.788263 0.615338i $$-0.789020\pi$$
−0.788263 + 0.615338i $$0.789020\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ 7.74597 0.748831 0.374415 0.927261i $$-0.377843\pi$$
0.374415 + 0.927261i $$0.377843\pi$$
$$108$$ 2.00000 0.192450
$$109$$ 3.87298 0.370965 0.185482 0.982648i $$-0.440615\pi$$
0.185482 + 0.982648i $$0.440615\pi$$
$$110$$ 0 0
$$111$$ 2.00000 0.189832
$$112$$ 15.4919 1.46385
$$113$$ −6.00000 −0.564433 −0.282216 0.959351i $$-0.591070\pi$$
−0.282216 + 0.959351i $$0.591070\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 15.4919 1.43839
$$117$$ −3.87298 −0.358057
$$118$$ 0 0
$$119$$ −30.0000 −2.75010
$$120$$ 0 0
$$121$$ 0 0
$$122$$ 0 0
$$123$$ −7.74597 −0.698430
$$124$$ 10.0000 0.898027
$$125$$ 0 0
$$126$$ 0 0
$$127$$ 7.74597 0.687343 0.343672 0.939090i $$-0.388329\pi$$
0.343672 + 0.939090i $$0.388329\pi$$
$$128$$ 0 0
$$129$$ −3.87298 −0.340997
$$130$$ 0 0
$$131$$ 7.74597 0.676768 0.338384 0.941008i $$-0.390120\pi$$
0.338384 + 0.941008i $$0.390120\pi$$
$$132$$ 0 0
$$133$$ −15.0000 −1.30066
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ −12.0000 −1.02523 −0.512615 0.858619i $$-0.671323\pi$$
−0.512615 + 0.858619i $$0.671323\pi$$
$$138$$ 0 0
$$139$$ 7.74597 0.657004 0.328502 0.944503i $$-0.393456\pi$$
0.328502 + 0.944503i $$0.393456\pi$$
$$140$$ 0 0
$$141$$ −12.0000 −1.01058
$$142$$ 0 0
$$143$$ 0 0
$$144$$ 4.00000 0.333333
$$145$$ 0 0
$$146$$ 0 0
$$147$$ −8.00000 −0.659829
$$148$$ 4.00000 0.328798
$$149$$ 7.74597 0.634574 0.317287 0.948330i $$-0.397228\pi$$
0.317287 + 0.948330i $$0.397228\pi$$
$$150$$ 0 0
$$151$$ −11.6190 −0.945537 −0.472768 0.881187i $$-0.656745\pi$$
−0.472768 + 0.881187i $$0.656745\pi$$
$$152$$ 0 0
$$153$$ −7.74597 −0.626224
$$154$$ 0 0
$$155$$ 0 0
$$156$$ −7.74597 −0.620174
$$157$$ 13.0000 1.03751 0.518756 0.854922i $$-0.326395\pi$$
0.518756 + 0.854922i $$0.326395\pi$$
$$158$$ 0 0
$$159$$ 6.00000 0.475831
$$160$$ 0 0
$$161$$ −23.2379 −1.83140
$$162$$ 0 0
$$163$$ 11.0000 0.861586 0.430793 0.902451i $$-0.358234\pi$$
0.430793 + 0.902451i $$0.358234\pi$$
$$164$$ −15.4919 −1.20972
$$165$$ 0 0
$$166$$ 0 0
$$167$$ −15.4919 −1.19880 −0.599401 0.800449i $$-0.704594\pi$$
−0.599401 + 0.800449i $$0.704594\pi$$
$$168$$ 0 0
$$169$$ 2.00000 0.153846
$$170$$ 0 0
$$171$$ −3.87298 −0.296174
$$172$$ −7.74597 −0.590624
$$173$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 0 0
$$177$$ 0 0
$$178$$ 0 0
$$179$$ 6.00000 0.448461 0.224231 0.974536i $$-0.428013\pi$$
0.224231 + 0.974536i $$0.428013\pi$$
$$180$$ 0 0
$$181$$ 5.00000 0.371647 0.185824 0.982583i $$-0.440505\pi$$
0.185824 + 0.982583i $$0.440505\pi$$
$$182$$ 0 0
$$183$$ 11.6190 0.858898
$$184$$ 0 0
$$185$$ 0 0
$$186$$ 0 0
$$187$$ 0 0
$$188$$ −24.0000 −1.75038
$$189$$ −3.87298 −0.281718
$$190$$ 0 0
$$191$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$192$$ 8.00000 0.577350
$$193$$ 11.6190 0.836350 0.418175 0.908366i $$-0.362670\pi$$
0.418175 + 0.908366i $$0.362670\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ −16.0000 −1.14286
$$197$$ −15.4919 −1.10375 −0.551877 0.833925i $$-0.686088\pi$$
−0.551877 + 0.833925i $$0.686088\pi$$
$$198$$ 0 0
$$199$$ 25.0000 1.77220 0.886102 0.463491i $$-0.153403\pi$$
0.886102 + 0.463491i $$0.153403\pi$$
$$200$$ 0 0
$$201$$ −7.00000 −0.493742
$$202$$ 0 0
$$203$$ −30.0000 −2.10559
$$204$$ −15.4919 −1.08465
$$205$$ 0 0
$$206$$ 0 0
$$207$$ −6.00000 −0.417029
$$208$$ −15.4919 −1.07417
$$209$$ 0 0
$$210$$ 0 0
$$211$$ −19.3649 −1.33314 −0.666568 0.745444i $$-0.732237\pi$$
−0.666568 + 0.745444i $$0.732237\pi$$
$$212$$ 12.0000 0.824163
$$213$$ −12.0000 −0.822226
$$214$$ 0 0
$$215$$ 0 0
$$216$$ 0 0
$$217$$ −19.3649 −1.31458
$$218$$ 0 0
$$219$$ 0 0
$$220$$ 0 0
$$221$$ 30.0000 2.01802
$$222$$ 0 0
$$223$$ −19.0000 −1.27233 −0.636167 0.771551i $$-0.719481\pi$$
−0.636167 + 0.771551i $$0.719481\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ 15.4919 1.02824 0.514118 0.857720i $$-0.328119\pi$$
0.514118 + 0.857720i $$0.328119\pi$$
$$228$$ −7.74597 −0.512989
$$229$$ −5.00000 −0.330409 −0.165205 0.986259i $$-0.552828\pi$$
−0.165205 + 0.986259i $$0.552828\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ −15.4919 −1.01491 −0.507455 0.861678i $$-0.669414\pi$$
−0.507455 + 0.861678i $$0.669414\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ 0 0
$$237$$ 7.74597 0.503155
$$238$$ 0 0
$$239$$ 23.2379 1.50313 0.751567 0.659656i $$-0.229298\pi$$
0.751567 + 0.659656i $$0.229298\pi$$
$$240$$ 0 0
$$241$$ −27.1109 −1.74637 −0.873183 0.487393i $$-0.837948\pi$$
−0.873183 + 0.487393i $$0.837948\pi$$
$$242$$ 0 0
$$243$$ −1.00000 −0.0641500
$$244$$ 23.2379 1.48765
$$245$$ 0 0
$$246$$ 0 0
$$247$$ 15.0000 0.954427
$$248$$ 0 0
$$249$$ −7.74597 −0.490881
$$250$$ 0 0
$$251$$ 18.0000 1.13615 0.568075 0.822977i $$-0.307688\pi$$
0.568075 + 0.822977i $$0.307688\pi$$
$$252$$ −7.74597 −0.487950
$$253$$ 0 0
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 16.0000 1.00000
$$257$$ 12.0000 0.748539 0.374270 0.927320i $$-0.377893\pi$$
0.374270 + 0.927320i $$0.377893\pi$$
$$258$$ 0 0
$$259$$ −7.74597 −0.481311
$$260$$ 0 0
$$261$$ −7.74597 −0.479463
$$262$$ 0 0
$$263$$ 23.2379 1.43291 0.716455 0.697633i $$-0.245763\pi$$
0.716455 + 0.697633i $$0.245763\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 0 0
$$267$$ −6.00000 −0.367194
$$268$$ −14.0000 −0.855186
$$269$$ 30.0000 1.82913 0.914566 0.404436i $$-0.132532\pi$$
0.914566 + 0.404436i $$0.132532\pi$$
$$270$$ 0 0
$$271$$ 23.2379 1.41160 0.705801 0.708410i $$-0.250587\pi$$
0.705801 + 0.708410i $$0.250587\pi$$
$$272$$ −30.9839 −1.87867
$$273$$ 15.0000 0.907841
$$274$$ 0 0
$$275$$ 0 0
$$276$$ −12.0000 −0.722315
$$277$$ −27.1109 −1.62894 −0.814468 0.580209i $$-0.802971\pi$$
−0.814468 + 0.580209i $$0.802971\pi$$
$$278$$ 0 0
$$279$$ −5.00000 −0.299342
$$280$$ 0 0
$$281$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$282$$ 0 0
$$283$$ 3.87298 0.230225 0.115112 0.993352i $$-0.463277\pi$$
0.115112 + 0.993352i $$0.463277\pi$$
$$284$$ −24.0000 −1.42414
$$285$$ 0 0
$$286$$ 0 0
$$287$$ 30.0000 1.77084
$$288$$ 0 0
$$289$$ 43.0000 2.52941
$$290$$ 0 0
$$291$$ 7.00000 0.410347
$$292$$ 0 0
$$293$$ −23.2379 −1.35757 −0.678786 0.734336i $$-0.737494\pi$$
−0.678786 + 0.734336i $$0.737494\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 0 0
$$297$$ 0 0
$$298$$ 0 0
$$299$$ 23.2379 1.34388
$$300$$ 0 0
$$301$$ 15.0000 0.864586
$$302$$ 0 0
$$303$$ −15.4919 −0.889988
$$304$$ −15.4919 −0.888523
$$305$$ 0 0
$$306$$ 0 0
$$307$$ 11.6190 0.663129 0.331564 0.943433i $$-0.392424\pi$$
0.331564 + 0.943433i $$0.392424\pi$$
$$308$$ 0 0
$$309$$ 16.0000 0.910208
$$310$$ 0 0
$$311$$ 18.0000 1.02069 0.510343 0.859971i $$-0.329518\pi$$
0.510343 + 0.859971i $$0.329518\pi$$
$$312$$ 0 0
$$313$$ 1.00000 0.0565233 0.0282617 0.999601i $$-0.491003\pi$$
0.0282617 + 0.999601i $$0.491003\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 15.4919 0.871489
$$317$$ −18.0000 −1.01098 −0.505490 0.862832i $$-0.668688\pi$$
−0.505490 + 0.862832i $$0.668688\pi$$
$$318$$ 0 0
$$319$$ 0 0
$$320$$ 0 0
$$321$$ −7.74597 −0.432338
$$322$$ 0 0
$$323$$ 30.0000 1.66924
$$324$$ −2.00000 −0.111111
$$325$$ 0 0
$$326$$ 0 0
$$327$$ −3.87298 −0.214176
$$328$$ 0 0
$$329$$ 46.4758 2.56229
$$330$$ 0 0
$$331$$ 20.0000 1.09930 0.549650 0.835395i $$-0.314761\pi$$
0.549650 + 0.835395i $$0.314761\pi$$
$$332$$ −15.4919 −0.850230
$$333$$ −2.00000 −0.109599
$$334$$ 0 0
$$335$$ 0 0
$$336$$ −15.4919 −0.845154
$$337$$ −19.3649 −1.05487 −0.527437 0.849594i $$-0.676847\pi$$
−0.527437 + 0.849594i $$0.676847\pi$$
$$338$$ 0 0
$$339$$ 6.00000 0.325875
$$340$$ 0 0
$$341$$ 0 0
$$342$$ 0 0
$$343$$ 3.87298 0.209121
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ 15.4919 0.831651 0.415825 0.909445i $$-0.363493\pi$$
0.415825 + 0.909445i $$0.363493\pi$$
$$348$$ −15.4919 −0.830455
$$349$$ 15.4919 0.829264 0.414632 0.909989i $$-0.363910\pi$$
0.414632 + 0.909989i $$0.363910\pi$$
$$350$$ 0 0
$$351$$ 3.87298 0.206725
$$352$$ 0 0
$$353$$ −6.00000 −0.319348 −0.159674 0.987170i $$-0.551044\pi$$
−0.159674 + 0.987170i $$0.551044\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ −12.0000 −0.635999
$$357$$ 30.0000 1.58777
$$358$$ 0 0
$$359$$ 15.4919 0.817633 0.408816 0.912617i $$-0.365942\pi$$
0.408816 + 0.912617i $$0.365942\pi$$
$$360$$ 0 0
$$361$$ −4.00000 −0.210526
$$362$$ 0 0
$$363$$ 0 0
$$364$$ 30.0000 1.57243
$$365$$ 0 0
$$366$$ 0 0
$$367$$ −13.0000 −0.678594 −0.339297 0.940679i $$-0.610189\pi$$
−0.339297 + 0.940679i $$0.610189\pi$$
$$368$$ −24.0000 −1.25109
$$369$$ 7.74597 0.403239
$$370$$ 0 0
$$371$$ −23.2379 −1.20645
$$372$$ −10.0000 −0.518476
$$373$$ 3.87298 0.200535 0.100268 0.994960i $$-0.468030\pi$$
0.100268 + 0.994960i $$0.468030\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ 30.0000 1.54508
$$378$$ 0 0
$$379$$ 5.00000 0.256833 0.128416 0.991720i $$-0.459011\pi$$
0.128416 + 0.991720i $$0.459011\pi$$
$$380$$ 0 0
$$381$$ −7.74597 −0.396838
$$382$$ 0 0
$$383$$ 6.00000 0.306586 0.153293 0.988181i $$-0.451012\pi$$
0.153293 + 0.988181i $$0.451012\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ 3.87298 0.196875
$$388$$ 14.0000 0.710742
$$389$$ 30.0000 1.52106 0.760530 0.649303i $$-0.224939\pi$$
0.760530 + 0.649303i $$0.224939\pi$$
$$390$$ 0 0
$$391$$ 46.4758 2.35038
$$392$$ 0 0
$$393$$ −7.74597 −0.390732
$$394$$ 0 0
$$395$$ 0 0
$$396$$ 0 0
$$397$$ 23.0000 1.15434 0.577168 0.816625i $$-0.304158\pi$$
0.577168 + 0.816625i $$0.304158\pi$$
$$398$$ 0 0
$$399$$ 15.0000 0.750939
$$400$$ 0 0
$$401$$ −30.0000 −1.49813 −0.749064 0.662497i $$-0.769497\pi$$
−0.749064 + 0.662497i $$0.769497\pi$$
$$402$$ 0 0
$$403$$ 19.3649 0.964635
$$404$$ −30.9839 −1.54150
$$405$$ 0 0
$$406$$ 0 0
$$407$$ 0 0
$$408$$ 0 0
$$409$$ −27.1109 −1.34055 −0.670273 0.742114i $$-0.733823\pi$$
−0.670273 + 0.742114i $$0.733823\pi$$
$$410$$ 0 0
$$411$$ 12.0000 0.591916
$$412$$ 32.0000 1.57653
$$413$$ 0 0
$$414$$ 0 0
$$415$$ 0 0
$$416$$ 0 0
$$417$$ −7.74597 −0.379322
$$418$$ 0 0
$$419$$ 24.0000 1.17248 0.586238 0.810139i $$-0.300608\pi$$
0.586238 + 0.810139i $$0.300608\pi$$
$$420$$ 0 0
$$421$$ 10.0000 0.487370 0.243685 0.969854i $$-0.421644\pi$$
0.243685 + 0.969854i $$0.421644\pi$$
$$422$$ 0 0
$$423$$ 12.0000 0.583460
$$424$$ 0 0
$$425$$ 0 0
$$426$$ 0 0
$$427$$ −45.0000 −2.17770
$$428$$ −15.4919 −0.748831
$$429$$ 0 0
$$430$$ 0 0
$$431$$ 7.74597 0.373110 0.186555 0.982445i $$-0.440268\pi$$
0.186555 + 0.982445i $$0.440268\pi$$
$$432$$ −4.00000 −0.192450
$$433$$ 11.0000 0.528626 0.264313 0.964437i $$-0.414855\pi$$
0.264313 + 0.964437i $$0.414855\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ −7.74597 −0.370965
$$437$$ 23.2379 1.11162
$$438$$ 0 0
$$439$$ −34.8569 −1.66363 −0.831813 0.555055i $$-0.812697\pi$$
−0.831813 + 0.555055i $$0.812697\pi$$
$$440$$ 0 0
$$441$$ 8.00000 0.380952
$$442$$ 0 0
$$443$$ −6.00000 −0.285069 −0.142534 0.989790i $$-0.545525\pi$$
−0.142534 + 0.989790i $$0.545525\pi$$
$$444$$ −4.00000 −0.189832
$$445$$ 0 0
$$446$$ 0 0
$$447$$ −7.74597 −0.366372
$$448$$ −30.9839 −1.46385
$$449$$ −36.0000 −1.69895 −0.849473 0.527633i $$-0.823080\pi$$
−0.849473 + 0.527633i $$0.823080\pi$$
$$450$$ 0 0
$$451$$ 0 0
$$452$$ 12.0000 0.564433
$$453$$ 11.6190 0.545906
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ −15.4919 −0.724682 −0.362341 0.932046i $$-0.618022\pi$$
−0.362341 + 0.932046i $$0.618022\pi$$
$$458$$ 0 0
$$459$$ 7.74597 0.361551
$$460$$ 0 0
$$461$$ 7.74597 0.360766 0.180383 0.983596i $$-0.442266\pi$$
0.180383 + 0.983596i $$0.442266\pi$$
$$462$$ 0 0
$$463$$ −4.00000 −0.185896 −0.0929479 0.995671i $$-0.529629\pi$$
−0.0929479 + 0.995671i $$0.529629\pi$$
$$464$$ −30.9839 −1.43839
$$465$$ 0 0
$$466$$ 0 0
$$467$$ −42.0000 −1.94353 −0.971764 0.235954i $$-0.924178\pi$$
−0.971764 + 0.235954i $$0.924178\pi$$
$$468$$ 7.74597 0.358057
$$469$$ 27.1109 1.25186
$$470$$ 0 0
$$471$$ −13.0000 −0.599008
$$472$$ 0 0
$$473$$ 0 0
$$474$$ 0 0
$$475$$ 0 0
$$476$$ 60.0000 2.75010
$$477$$ −6.00000 −0.274721
$$478$$ 0 0
$$479$$ −30.9839 −1.41569 −0.707845 0.706368i $$-0.750332\pi$$
−0.707845 + 0.706368i $$0.750332\pi$$
$$480$$ 0 0
$$481$$ 7.74597 0.353186
$$482$$ 0 0
$$483$$ 23.2379 1.05736
$$484$$ 0 0
$$485$$ 0 0
$$486$$ 0 0
$$487$$ −23.0000 −1.04223 −0.521115 0.853487i $$-0.674484\pi$$
−0.521115 + 0.853487i $$0.674484\pi$$
$$488$$ 0 0
$$489$$ −11.0000 −0.497437
$$490$$ 0 0
$$491$$ 7.74597 0.349571 0.174785 0.984607i $$-0.444077\pi$$
0.174785 + 0.984607i $$0.444077\pi$$
$$492$$ 15.4919 0.698430
$$493$$ 60.0000 2.70226
$$494$$ 0 0
$$495$$ 0 0
$$496$$ −20.0000 −0.898027
$$497$$ 46.4758 2.08472
$$498$$ 0 0
$$499$$ 25.0000 1.11915 0.559577 0.828778i $$-0.310964\pi$$
0.559577 + 0.828778i $$0.310964\pi$$
$$500$$ 0 0
$$501$$ 15.4919 0.692129
$$502$$ 0 0
$$503$$ 7.74597 0.345376 0.172688 0.984977i $$-0.444755\pi$$
0.172688 + 0.984977i $$0.444755\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 0 0
$$507$$ −2.00000 −0.0888231
$$508$$ −15.4919 −0.687343
$$509$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$510$$ 0 0
$$511$$ 0 0
$$512$$ 0 0
$$513$$ 3.87298 0.170996
$$514$$ 0 0
$$515$$ 0 0
$$516$$ 7.74597 0.340997
$$517$$ 0 0
$$518$$ 0 0
$$519$$ 0 0
$$520$$ 0 0
$$521$$ 30.0000 1.31432 0.657162 0.753749i $$-0.271757\pi$$
0.657162 + 0.753749i $$0.271757\pi$$
$$522$$ 0 0
$$523$$ −11.6190 −0.508061 −0.254031 0.967196i $$-0.581756\pi$$
−0.254031 + 0.967196i $$0.581756\pi$$
$$524$$ −15.4919 −0.676768
$$525$$ 0 0
$$526$$ 0 0
$$527$$ 38.7298 1.68710
$$528$$ 0 0
$$529$$ 13.0000 0.565217
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 30.0000 1.30066
$$533$$ −30.0000 −1.29944
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 0 0
$$537$$ −6.00000 −0.258919
$$538$$ 0 0
$$539$$ 0 0
$$540$$ 0 0
$$541$$ 42.6028 1.83164 0.915819 0.401591i $$-0.131543\pi$$
0.915819 + 0.401591i $$0.131543\pi$$
$$542$$ 0 0
$$543$$ −5.00000 −0.214571
$$544$$ 0 0
$$545$$ 0 0
$$546$$ 0 0
$$547$$ 23.2379 0.993581 0.496790 0.867871i $$-0.334512\pi$$
0.496790 + 0.867871i $$0.334512\pi$$
$$548$$ 24.0000 1.02523
$$549$$ −11.6190 −0.495885
$$550$$ 0 0
$$551$$ 30.0000 1.27804
$$552$$ 0 0
$$553$$ −30.0000 −1.27573
$$554$$ 0 0
$$555$$ 0 0
$$556$$ −15.4919 −0.657004
$$557$$ −15.4919 −0.656414 −0.328207 0.944606i $$-0.606444\pi$$
−0.328207 + 0.944606i $$0.606444\pi$$
$$558$$ 0 0
$$559$$ −15.0000 −0.634432
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 0 0
$$563$$ 7.74597 0.326454 0.163227 0.986589i $$-0.447810\pi$$
0.163227 + 0.986589i $$0.447810\pi$$
$$564$$ 24.0000 1.01058
$$565$$ 0 0
$$566$$ 0 0
$$567$$ 3.87298 0.162650
$$568$$ 0 0
$$569$$ −23.2379 −0.974183 −0.487092 0.873351i $$-0.661942\pi$$
−0.487092 + 0.873351i $$0.661942\pi$$
$$570$$ 0 0
$$571$$ 19.3649 0.810397 0.405198 0.914229i $$-0.367202\pi$$
0.405198 + 0.914229i $$0.367202\pi$$
$$572$$ 0 0
$$573$$ 0 0
$$574$$ 0 0
$$575$$ 0 0
$$576$$ −8.00000 −0.333333
$$577$$ 37.0000 1.54033 0.770165 0.637845i $$-0.220174\pi$$
0.770165 + 0.637845i $$0.220174\pi$$
$$578$$ 0 0
$$579$$ −11.6190 −0.482867
$$580$$ 0 0
$$581$$ 30.0000 1.24461
$$582$$ 0 0
$$583$$ 0 0
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ 18.0000 0.742940 0.371470 0.928445i $$-0.378854\pi$$
0.371470 + 0.928445i $$0.378854\pi$$
$$588$$ 16.0000 0.659829
$$589$$ 19.3649 0.797917
$$590$$ 0 0
$$591$$ 15.4919 0.637253
$$592$$ −8.00000 −0.328798
$$593$$ −23.2379 −0.954266 −0.477133 0.878831i $$-0.658324\pi$$
−0.477133 + 0.878831i $$0.658324\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ −15.4919 −0.634574
$$597$$ −25.0000 −1.02318
$$598$$ 0 0
$$599$$ 24.0000 0.980613 0.490307 0.871550i $$-0.336885\pi$$
0.490307 + 0.871550i $$0.336885\pi$$
$$600$$ 0 0
$$601$$ −27.1109 −1.10588 −0.552938 0.833222i $$-0.686493\pi$$
−0.552938 + 0.833222i $$0.686493\pi$$
$$602$$ 0 0
$$603$$ 7.00000 0.285062
$$604$$ 23.2379 0.945537
$$605$$ 0 0
$$606$$ 0 0
$$607$$ 38.7298 1.57200 0.785998 0.618229i $$-0.212150\pi$$
0.785998 + 0.618229i $$0.212150\pi$$
$$608$$ 0 0
$$609$$ 30.0000 1.21566
$$610$$ 0 0
$$611$$ −46.4758 −1.88021
$$612$$ 15.4919 0.626224
$$613$$ 46.4758 1.87714 0.938570 0.345089i $$-0.112151\pi$$
0.938570 + 0.345089i $$0.112151\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ 12.0000 0.483102 0.241551 0.970388i $$-0.422344\pi$$
0.241551 + 0.970388i $$0.422344\pi$$
$$618$$ 0 0
$$619$$ −1.00000 −0.0401934 −0.0200967 0.999798i $$-0.506397\pi$$
−0.0200967 + 0.999798i $$0.506397\pi$$
$$620$$ 0 0
$$621$$ 6.00000 0.240772
$$622$$ 0 0
$$623$$ 23.2379 0.931007
$$624$$ 15.4919 0.620174
$$625$$ 0 0
$$626$$ 0 0
$$627$$ 0 0
$$628$$ −26.0000 −1.03751
$$629$$ 15.4919 0.617704
$$630$$ 0 0
$$631$$ 7.00000 0.278666 0.139333 0.990246i $$-0.455504\pi$$
0.139333 + 0.990246i $$0.455504\pi$$
$$632$$ 0 0
$$633$$ 19.3649 0.769686
$$634$$ 0 0
$$635$$ 0 0
$$636$$ −12.0000 −0.475831
$$637$$ −30.9839 −1.22763
$$638$$ 0 0
$$639$$ 12.0000 0.474713
$$640$$ 0 0
$$641$$ 30.0000 1.18493 0.592464 0.805597i $$-0.298155\pi$$
0.592464 + 0.805597i $$0.298155\pi$$
$$642$$ 0 0
$$643$$ −16.0000 −0.630978 −0.315489 0.948929i $$-0.602169\pi$$
−0.315489 + 0.948929i $$0.602169\pi$$
$$644$$ 46.4758 1.83140
$$645$$ 0 0
$$646$$ 0 0
$$647$$ 12.0000 0.471769 0.235884 0.971781i $$-0.424201\pi$$
0.235884 + 0.971781i $$0.424201\pi$$
$$648$$ 0 0
$$649$$ 0 0
$$650$$ 0 0
$$651$$ 19.3649 0.758971
$$652$$ −22.0000 −0.861586
$$653$$ 24.0000 0.939193 0.469596 0.882881i $$-0.344399\pi$$
0.469596 + 0.882881i $$0.344399\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 30.9839 1.20972
$$657$$ 0 0
$$658$$ 0 0
$$659$$ −46.4758 −1.81044 −0.905220 0.424943i $$-0.860294\pi$$
−0.905220 + 0.424943i $$0.860294\pi$$
$$660$$ 0 0
$$661$$ −2.00000 −0.0777910 −0.0388955 0.999243i $$-0.512384\pi$$
−0.0388955 + 0.999243i $$0.512384\pi$$
$$662$$ 0 0
$$663$$ −30.0000 −1.16510
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 46.4758 1.79955
$$668$$ 30.9839 1.19880
$$669$$ 19.0000 0.734582
$$670$$ 0 0
$$671$$ 0 0
$$672$$ 0 0
$$673$$ 30.9839 1.19434 0.597170 0.802115i $$-0.296292\pi$$
0.597170 + 0.802115i $$0.296292\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ −4.00000 −0.153846
$$677$$ −23.2379 −0.893105 −0.446553 0.894757i $$-0.647348\pi$$
−0.446553 + 0.894757i $$0.647348\pi$$
$$678$$ 0 0
$$679$$ −27.1109 −1.04042
$$680$$ 0 0
$$681$$ −15.4919 −0.593652
$$682$$ 0 0
$$683$$ 6.00000 0.229584 0.114792 0.993390i $$-0.463380\pi$$
0.114792 + 0.993390i $$0.463380\pi$$
$$684$$ 7.74597 0.296174
$$685$$ 0 0
$$686$$ 0 0
$$687$$ 5.00000 0.190762
$$688$$ 15.4919 0.590624
$$689$$ 23.2379 0.885293
$$690$$ 0 0
$$691$$ −28.0000 −1.06517 −0.532585 0.846376i $$-0.678779\pi$$
−0.532585 + 0.846376i $$0.678779\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ 0 0
$$696$$ 0 0
$$697$$ −60.0000 −2.27266
$$698$$ 0 0
$$699$$ 15.4919 0.585959
$$700$$ 0 0
$$701$$ 30.9839 1.17024 0.585122 0.810945i $$-0.301047\pi$$
0.585122 + 0.810945i $$0.301047\pi$$
$$702$$ 0 0
$$703$$ 7.74597 0.292145
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ 60.0000 2.25653
$$708$$ 0 0
$$709$$ 19.0000 0.713560 0.356780 0.934188i $$-0.383875\pi$$
0.356780 + 0.934188i $$0.383875\pi$$
$$710$$ 0 0
$$711$$ −7.74597 −0.290496
$$712$$ 0 0
$$713$$ 30.0000 1.12351
$$714$$ 0 0
$$715$$ 0 0
$$716$$ −12.0000 −0.448461
$$717$$ −23.2379 −0.867835
$$718$$ 0 0
$$719$$ 6.00000 0.223762 0.111881 0.993722i $$-0.464312\pi$$
0.111881 + 0.993722i $$0.464312\pi$$
$$720$$ 0 0
$$721$$ −61.9677 −2.30780
$$722$$ 0 0
$$723$$ 27.1109 1.00826
$$724$$ −10.0000 −0.371647
$$725$$ 0 0
$$726$$ 0 0
$$727$$ −17.0000 −0.630495 −0.315248 0.949009i $$-0.602088\pi$$
−0.315248 + 0.949009i $$0.602088\pi$$
$$728$$ 0 0
$$729$$ 1.00000 0.0370370
$$730$$ 0 0
$$731$$ −30.0000 −1.10959
$$732$$ −23.2379 −0.858898
$$733$$ −30.9839 −1.14442 −0.572208 0.820109i $$-0.693913\pi$$
−0.572208 + 0.820109i $$0.693913\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ 0 0
$$738$$ 0 0
$$739$$ 7.74597 0.284940 0.142470 0.989799i $$-0.454496\pi$$
0.142470 + 0.989799i $$0.454496\pi$$
$$740$$ 0 0
$$741$$ −15.0000 −0.551039
$$742$$ 0 0
$$743$$ −23.2379 −0.852516 −0.426258 0.904602i $$-0.640168\pi$$
−0.426258 + 0.904602i $$0.640168\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 0 0
$$747$$ 7.74597 0.283410
$$748$$ 0 0
$$749$$ 30.0000 1.09618
$$750$$ 0 0
$$751$$ −8.00000 −0.291924 −0.145962 0.989290i $$-0.546628\pi$$
−0.145962 + 0.989290i $$0.546628\pi$$
$$752$$ 48.0000 1.75038
$$753$$ −18.0000 −0.655956
$$754$$ 0 0
$$755$$ 0 0
$$756$$ 7.74597 0.281718
$$757$$ −17.0000 −0.617876 −0.308938 0.951082i $$-0.599973\pi$$
−0.308938 + 0.951082i $$0.599973\pi$$
$$758$$ 0 0
$$759$$ 0 0
$$760$$ 0 0
$$761$$ −54.2218 −1.96554 −0.982769 0.184839i $$-0.940824\pi$$
−0.982769 + 0.184839i $$0.940824\pi$$
$$762$$ 0 0
$$763$$ 15.0000 0.543036
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ 0 0
$$768$$ −16.0000 −0.577350
$$769$$ 19.3649 0.698317 0.349158 0.937064i $$-0.386468\pi$$
0.349158 + 0.937064i $$0.386468\pi$$
$$770$$ 0 0
$$771$$ −12.0000 −0.432169
$$772$$ −23.2379 −0.836350
$$773$$ 6.00000 0.215805 0.107903 0.994161i $$-0.465587\pi$$
0.107903 + 0.994161i $$0.465587\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ 0 0
$$777$$ 7.74597 0.277885
$$778$$ 0 0
$$779$$ −30.0000 −1.07486
$$780$$ 0 0
$$781$$ 0 0
$$782$$ 0 0
$$783$$ 7.74597 0.276818
$$784$$ 32.0000 1.14286
$$785$$ 0 0
$$786$$ 0 0
$$787$$ −11.6190 −0.414171 −0.207085 0.978323i $$-0.566398\pi$$
−0.207085 + 0.978323i $$0.566398\pi$$
$$788$$ 30.9839 1.10375
$$789$$ −23.2379 −0.827291
$$790$$ 0 0
$$791$$ −23.2379 −0.826245
$$792$$ 0 0
$$793$$ 45.0000 1.59800
$$794$$ 0 0
$$795$$ 0 0
$$796$$ −50.0000 −1.77220
$$797$$ −12.0000 −0.425062 −0.212531 0.977154i $$-0.568171\pi$$
−0.212531 + 0.977154i $$0.568171\pi$$
$$798$$ 0 0
$$799$$ −92.9516 −3.28839
$$800$$ 0 0
$$801$$ 6.00000 0.212000
$$802$$ 0 0
$$803$$ 0 0
$$804$$ 14.0000 0.493742
$$805$$ 0 0
$$806$$ 0 0
$$807$$ −30.0000 −1.05605
$$808$$ 0 0
$$809$$ −15.4919 −0.544667 −0.272334 0.962203i $$-0.587795\pi$$
−0.272334 + 0.962203i $$0.587795\pi$$
$$810$$ 0 0
$$811$$ −3.87298 −0.135999 −0.0679994 0.997685i $$-0.521662\pi$$
−0.0679994 + 0.997685i $$0.521662\pi$$
$$812$$ 60.0000 2.10559
$$813$$ −23.2379 −0.814989
$$814$$ 0 0
$$815$$ 0 0
$$816$$ 30.9839 1.08465
$$817$$ −15.0000 −0.524784
$$818$$ 0 0
$$819$$ −15.0000 −0.524142
$$820$$ 0 0
$$821$$ 46.4758 1.62202 0.811008 0.585035i $$-0.198919\pi$$
0.811008 + 0.585035i $$0.198919\pi$$
$$822$$ 0 0
$$823$$ −31.0000 −1.08059 −0.540296 0.841475i $$-0.681688\pi$$
−0.540296 + 0.841475i $$0.681688\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ 23.2379 0.808061 0.404030 0.914746i $$-0.367609\pi$$
0.404030 + 0.914746i $$0.367609\pi$$
$$828$$ 12.0000 0.417029
$$829$$ 14.0000 0.486240 0.243120 0.969996i $$-0.421829\pi$$
0.243120 + 0.969996i $$0.421829\pi$$
$$830$$ 0 0
$$831$$ 27.1109 0.940466
$$832$$ 30.9839 1.07417
$$833$$ −61.9677 −2.14705
$$834$$ 0 0
$$835$$ 0 0
$$836$$ 0 0
$$837$$ 5.00000 0.172825
$$838$$ 0 0
$$839$$ −30.0000 −1.03572 −0.517858 0.855467i $$-0.673270\pi$$
−0.517858 + 0.855467i $$0.673270\pi$$
$$840$$ 0 0
$$841$$ 31.0000 1.06897
$$842$$ 0 0
$$843$$ 0 0
$$844$$ 38.7298 1.33314
$$845$$ 0 0
$$846$$ 0 0
$$847$$ 0 0
$$848$$ −24.0000 −0.824163
$$849$$ −3.87298 −0.132920
$$850$$ 0 0
$$851$$ 12.0000 0.411355
$$852$$ 24.0000 0.822226
$$853$$ −11.6190 −0.397825 −0.198913 0.980017i $$-0.563741\pi$$
−0.198913 + 0.980017i $$0.563741\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ −23.2379 −0.793792 −0.396896 0.917864i $$-0.629913\pi$$
−0.396896 + 0.917864i $$0.629913\pi$$
$$858$$ 0 0
$$859$$ 44.0000 1.50126 0.750630 0.660722i $$-0.229750\pi$$
0.750630 + 0.660722i $$0.229750\pi$$
$$860$$ 0 0
$$861$$ −30.0000 −1.02240
$$862$$ 0 0
$$863$$ 36.0000 1.22545 0.612727 0.790295i $$-0.290072\pi$$
0.612727 + 0.790295i $$0.290072\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ 0 0
$$867$$ −43.0000 −1.46036
$$868$$ 38.7298 1.31458
$$869$$ 0 0
$$870$$ 0 0
$$871$$ −27.1109 −0.918617
$$872$$ 0 0
$$873$$ −7.00000 −0.236914
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 0 0
$$877$$ 19.3649 0.653907 0.326953 0.945040i $$-0.393978\pi$$
0.326953 + 0.945040i $$0.393978\pi$$
$$878$$ 0 0
$$879$$ 23.2379 0.783795
$$880$$ 0 0
$$881$$ −18.0000 −0.606435 −0.303218 0.952921i $$-0.598061\pi$$
−0.303218 + 0.952921i $$0.598061\pi$$
$$882$$ 0 0
$$883$$ −29.0000 −0.975928 −0.487964 0.872864i $$-0.662260\pi$$
−0.487964 + 0.872864i $$0.662260\pi$$
$$884$$ −60.0000 −2.01802
$$885$$ 0 0
$$886$$ 0 0
$$887$$ 7.74597 0.260084 0.130042 0.991508i $$-0.458489\pi$$
0.130042 + 0.991508i $$0.458489\pi$$
$$888$$ 0 0
$$889$$ 30.0000 1.00617
$$890$$ 0 0
$$891$$ 0 0
$$892$$ 38.0000 1.27233
$$893$$ −46.4758 −1.55525
$$894$$ 0 0
$$895$$ 0 0
$$896$$ 0 0
$$897$$ −23.2379 −0.775891
$$898$$ 0 0
$$899$$ 38.7298 1.29171
$$900$$ 0 0
$$901$$ 46.4758 1.54833
$$902$$ 0 0
$$903$$ −15.0000 −0.499169
$$904$$ 0 0
$$905$$ 0 0
$$906$$ 0 0
$$907$$ −28.0000 −0.929725 −0.464862 0.885383i $$-0.653896\pi$$
−0.464862 + 0.885383i $$0.653896\pi$$
$$908$$ −30.9839 −1.02824
$$909$$ 15.4919 0.513835
$$910$$ 0 0
$$911$$ 30.0000 0.993944 0.496972 0.867766i $$-0.334445\pi$$
0.496972 + 0.867766i $$0.334445\pi$$
$$912$$ 15.4919 0.512989
$$913$$ 0 0
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 10.0000 0.330409
$$917$$ 30.0000 0.990687
$$918$$ 0 0
$$919$$ 11.6190 0.383274 0.191637 0.981466i $$-0.438620\pi$$
0.191637 + 0.981466i $$0.438620\pi$$
$$920$$ 0 0
$$921$$ −11.6190 −0.382857
$$922$$ 0 0
$$923$$ −46.4758 −1.52977
$$924$$ 0 0
$$925$$ 0 0
$$926$$ 0 0
$$927$$ −16.0000 −0.525509
$$928$$ 0 0
$$929$$ 60.0000 1.96854 0.984268 0.176682i $$-0.0565363\pi$$
0.984268 + 0.176682i $$0.0565363\pi$$
$$930$$ 0 0
$$931$$ −30.9839 −1.01546
$$932$$ 30.9839 1.01491
$$933$$ −18.0000 −0.589294
$$934$$ 0 0
$$935$$ 0 0
$$936$$ 0 0
$$937$$ −27.1109 −0.885674 −0.442837 0.896602i $$-0.646028\pi$$
−0.442837 + 0.896602i $$0.646028\pi$$
$$938$$ 0 0
$$939$$ −1.00000 −0.0326338
$$940$$ 0 0
$$941$$ 30.9839 1.01005 0.505023 0.863106i $$-0.331484\pi$$
0.505023 + 0.863106i $$0.331484\pi$$
$$942$$ 0 0
$$943$$ −46.4758 −1.51346
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ 18.0000 0.584921 0.292461 0.956278i $$-0.405526\pi$$
0.292461 + 0.956278i $$0.405526\pi$$
$$948$$ −15.4919 −0.503155
$$949$$ 0 0
$$950$$ 0 0
$$951$$ 18.0000 0.583690
$$952$$ 0 0
$$953$$ 7.74597 0.250916 0.125458 0.992099i $$-0.459960\pi$$
0.125458 + 0.992099i $$0.459960\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ −46.4758 −1.50313
$$957$$ 0 0
$$958$$ 0 0
$$959$$ −46.4758 −1.50078
$$960$$ 0 0
$$961$$ −6.00000 −0.193548
$$962$$ 0 0
$$963$$ 7.74597 0.249610
$$964$$ 54.2218 1.74637
$$965$$ 0 0
$$966$$ 0 0
$$967$$ 23.2379 0.747280 0.373640 0.927574i $$-0.378109\pi$$
0.373640 + 0.927574i $$0.378109\pi$$
$$968$$ 0 0
$$969$$ −30.0000 −0.963739
$$970$$ 0 0
$$971$$ −30.0000 −0.962746 −0.481373 0.876516i $$-0.659862\pi$$
−0.481373 + 0.876516i $$0.659862\pi$$
$$972$$ 2.00000 0.0641500
$$973$$ 30.0000 0.961756
$$974$$ 0 0
$$975$$ 0 0
$$976$$ −46.4758 −1.48765
$$977$$ 42.0000 1.34370 0.671850 0.740688i $$-0.265500\pi$$
0.671850 + 0.740688i $$0.265500\pi$$
$$978$$ 0 0
$$979$$ 0 0
$$980$$ 0 0
$$981$$ 3.87298 0.123655
$$982$$ 0 0
$$983$$ 24.0000 0.765481 0.382741 0.923856i $$-0.374980\pi$$
0.382741 + 0.923856i $$0.374980\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ 0 0
$$987$$ −46.4758 −1.47934
$$988$$ −30.0000 −0.954427
$$989$$ −23.2379 −0.738922
$$990$$ 0 0
$$991$$ 55.0000 1.74713 0.873566 0.486705i $$-0.161801\pi$$
0.873566 + 0.486705i $$0.161801\pi$$
$$992$$ 0 0
$$993$$ −20.0000 −0.634681
$$994$$ 0 0
$$995$$ 0 0
$$996$$ 15.4919 0.490881
$$997$$ −46.4758 −1.47190 −0.735952 0.677034i $$-0.763265\pi$$
−0.735952 + 0.677034i $$0.763265\pi$$
$$998$$ 0 0
$$999$$ 2.00000 0.0632772
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 9075.2.a.bf.1.2 yes 2
5.4 even 2 9075.2.a.bm.1.1 yes 2
11.10 odd 2 inner 9075.2.a.bf.1.1 2
55.54 odd 2 9075.2.a.bm.1.2 yes 2

By twisted newform
Twist Min Dim Char Parity Ord Type
9075.2.a.bf.1.1 2 11.10 odd 2 inner
9075.2.a.bf.1.2 yes 2 1.1 even 1 trivial
9075.2.a.bm.1.1 yes 2 5.4 even 2
9075.2.a.bm.1.2 yes 2 55.54 odd 2