# Properties

 Label 45.2.b.a.19.2 Level $45$ Weight $2$ Character 45.19 Analytic conductor $0.359$ Analytic rank $0$ Dimension $2$ CM discriminant -15 Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [45,2,Mod(19,45)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(45, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("45.19");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$45 = 3^{2} \cdot 5$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 45.b (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: $$0.359326809096$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-5})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} + 5$$ x^2 + 5 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

## Embedding invariants

 Embedding label 19.2 Root $$2.23607i$$ of defining polynomial Character $$\chi$$ $$=$$ 45.19 Dual form 45.2.b.a.19.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+2.23607i q^{2} -3.00000 q^{4} -2.23607i q^{5} -2.23607i q^{8} +O(q^{10})$$ $$q+2.23607i q^{2} -3.00000 q^{4} -2.23607i q^{5} -2.23607i q^{8} +5.00000 q^{10} -1.00000 q^{16} -4.47214i q^{17} -4.00000 q^{19} +6.70820i q^{20} +8.94427i q^{23} -5.00000 q^{25} +8.00000 q^{31} -6.70820i q^{32} +10.0000 q^{34} -8.94427i q^{38} -5.00000 q^{40} -20.0000 q^{46} +8.94427i q^{47} +7.00000 q^{49} -11.1803i q^{50} -4.47214i q^{53} +2.00000 q^{61} +17.8885i q^{62} +13.0000 q^{64} +13.4164i q^{68} +12.0000 q^{76} -16.0000 q^{79} +2.23607i q^{80} -17.8885i q^{83} -10.0000 q^{85} -26.8328i q^{92} -20.0000 q^{94} +8.94427i q^{95} +15.6525i q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 6 q^{4}+O(q^{10})$$ 2 * q - 6 * q^4 $$2 q - 6 q^{4} + 10 q^{10} - 2 q^{16} - 8 q^{19} - 10 q^{25} + 16 q^{31} + 20 q^{34} - 10 q^{40} - 40 q^{46} + 14 q^{49} + 4 q^{61} + 26 q^{64} + 24 q^{76} - 32 q^{79} - 20 q^{85} - 40 q^{94}+O(q^{100})$$ 2 * q - 6 * q^4 + 10 * q^10 - 2 * q^16 - 8 * q^19 - 10 * q^25 + 16 * q^31 + 20 * q^34 - 10 * q^40 - 40 * q^46 + 14 * q^49 + 4 * q^61 + 26 * q^64 + 24 * q^76 - 32 * q^79 - 20 * q^85 - 40 * q^94

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/45\mathbb{Z}\right)^\times$$.

 $$n$$ $$11$$ $$37$$ $$\chi(n)$$ $$1$$ $$-1$$

## Coefficient data

For each $$n$$ we display the coefficients of the $$q$$-expansion $$a_n$$, the Satake parameters $$\alpha_p$$, and the Satake angles $$\theta_p = \textrm{Arg}(\alpha_p)$$.

Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 2.23607i 1.58114i 0.612372 + 0.790569i $$0.290215\pi$$
−0.612372 + 0.790569i $$0.709785\pi$$
$$3$$ 0 0
$$4$$ −3.00000 −1.50000
$$5$$ − 2.23607i − 1.00000i
$$6$$ 0 0
$$7$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$8$$ − 2.23607i − 0.790569i
$$9$$ 0 0
$$10$$ 5.00000 1.58114
$$11$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$12$$ 0 0
$$13$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ −1.00000 −0.250000
$$17$$ − 4.47214i − 1.08465i −0.840168 0.542326i $$-0.817544\pi$$
0.840168 0.542326i $$-0.182456\pi$$
$$18$$ 0 0
$$19$$ −4.00000 −0.917663 −0.458831 0.888523i $$-0.651732\pi$$
−0.458831 + 0.888523i $$0.651732\pi$$
$$20$$ 6.70820i 1.50000i
$$21$$ 0 0
$$22$$ 0 0
$$23$$ 8.94427i 1.86501i 0.361158 + 0.932505i $$0.382382\pi$$
−0.361158 + 0.932505i $$0.617618\pi$$
$$24$$ 0 0
$$25$$ −5.00000 −1.00000
$$26$$ 0 0
$$27$$ 0 0
$$28$$ 0 0
$$29$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$30$$ 0 0
$$31$$ 8.00000 1.43684 0.718421 0.695608i $$-0.244865\pi$$
0.718421 + 0.695608i $$0.244865\pi$$
$$32$$ − 6.70820i − 1.18585i
$$33$$ 0 0
$$34$$ 10.0000 1.71499
$$35$$ 0 0
$$36$$ 0 0
$$37$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$38$$ − 8.94427i − 1.45095i
$$39$$ 0 0
$$40$$ −5.00000 −0.790569
$$41$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$42$$ 0 0
$$43$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ −20.0000 −2.94884
$$47$$ 8.94427i 1.30466i 0.757937 + 0.652328i $$0.226208\pi$$
−0.757937 + 0.652328i $$0.773792\pi$$
$$48$$ 0 0
$$49$$ 7.00000 1.00000
$$50$$ − 11.1803i − 1.58114i
$$51$$ 0 0
$$52$$ 0 0
$$53$$ − 4.47214i − 0.614295i −0.951662 0.307148i $$-0.900625\pi$$
0.951662 0.307148i $$-0.0993745\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 0 0
$$57$$ 0 0
$$58$$ 0 0
$$59$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$60$$ 0 0
$$61$$ 2.00000 0.256074 0.128037 0.991769i $$-0.459132\pi$$
0.128037 + 0.991769i $$0.459132\pi$$
$$62$$ 17.8885i 2.27185i
$$63$$ 0 0
$$64$$ 13.0000 1.62500
$$65$$ 0 0
$$66$$ 0 0
$$67$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$68$$ 13.4164i 1.62698i
$$69$$ 0 0
$$70$$ 0 0
$$71$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$72$$ 0 0
$$73$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 12.0000 1.37649
$$77$$ 0 0
$$78$$ 0 0
$$79$$ −16.0000 −1.80014 −0.900070 0.435745i $$-0.856485\pi$$
−0.900070 + 0.435745i $$0.856485\pi$$
$$80$$ 2.23607i 0.250000i
$$81$$ 0 0
$$82$$ 0 0
$$83$$ − 17.8885i − 1.96352i −0.190117 0.981761i $$-0.560887\pi$$
0.190117 0.981761i $$-0.439113\pi$$
$$84$$ 0 0
$$85$$ −10.0000 −1.08465
$$86$$ 0 0
$$87$$ 0 0
$$88$$ 0 0
$$89$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ − 26.8328i − 2.79751i
$$93$$ 0 0
$$94$$ −20.0000 −2.06284
$$95$$ 8.94427i 0.917663i
$$96$$ 0 0
$$97$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$98$$ 15.6525i 1.58114i
$$99$$ 0 0
$$100$$ 15.0000 1.50000
$$101$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$102$$ 0 0
$$103$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 10.0000 0.971286
$$107$$ − 17.8885i − 1.72935i −0.502331 0.864675i $$-0.667524\pi$$
0.502331 0.864675i $$-0.332476\pi$$
$$108$$ 0 0
$$109$$ 14.0000 1.34096 0.670478 0.741929i $$-0.266089\pi$$
0.670478 + 0.741929i $$0.266089\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ 0 0
$$113$$ − 4.47214i − 0.420703i −0.977626 0.210352i $$-0.932539\pi$$
0.977626 0.210352i $$-0.0674609\pi$$
$$114$$ 0 0
$$115$$ 20.0000 1.86501
$$116$$ 0 0
$$117$$ 0 0
$$118$$ 0 0
$$119$$ 0 0
$$120$$ 0 0
$$121$$ −11.0000 −1.00000
$$122$$ 4.47214i 0.404888i
$$123$$ 0 0
$$124$$ −24.0000 −2.15526
$$125$$ 11.1803i 1.00000i
$$126$$ 0 0
$$127$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$128$$ 15.6525i 1.38350i
$$129$$ 0 0
$$130$$ 0 0
$$131$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$132$$ 0 0
$$133$$ 0 0
$$134$$ 0 0
$$135$$ 0 0
$$136$$ −10.0000 −0.857493
$$137$$ 22.3607i 1.91040i 0.295958 + 0.955201i $$0.404361\pi$$
−0.295958 + 0.955201i $$0.595639\pi$$
$$138$$ 0 0
$$139$$ −4.00000 −0.339276 −0.169638 0.985506i $$-0.554260\pi$$
−0.169638 + 0.985506i $$0.554260\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 0 0
$$143$$ 0 0
$$144$$ 0 0
$$145$$ 0 0
$$146$$ 0 0
$$147$$ 0 0
$$148$$ 0 0
$$149$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$150$$ 0 0
$$151$$ 8.00000 0.651031 0.325515 0.945537i $$-0.394462\pi$$
0.325515 + 0.945537i $$0.394462\pi$$
$$152$$ 8.94427i 0.725476i
$$153$$ 0 0
$$154$$ 0 0
$$155$$ − 17.8885i − 1.43684i
$$156$$ 0 0
$$157$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$158$$ − 35.7771i − 2.84627i
$$159$$ 0 0
$$160$$ −15.0000 −1.18585
$$161$$ 0 0
$$162$$ 0 0
$$163$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 40.0000 3.10460
$$167$$ 8.94427i 0.692129i 0.938211 + 0.346064i $$0.112482\pi$$
−0.938211 + 0.346064i $$0.887518\pi$$
$$168$$ 0 0
$$169$$ 13.0000 1.00000
$$170$$ − 22.3607i − 1.71499i
$$171$$ 0 0
$$172$$ 0 0
$$173$$ 22.3607i 1.70005i 0.526742 + 0.850026i $$0.323414\pi$$
−0.526742 + 0.850026i $$0.676586\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 0 0
$$177$$ 0 0
$$178$$ 0 0
$$179$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$180$$ 0 0
$$181$$ −22.0000 −1.63525 −0.817624 0.575753i $$-0.804709\pi$$
−0.817624 + 0.575753i $$0.804709\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ 20.0000 1.47442
$$185$$ 0 0
$$186$$ 0 0
$$187$$ 0 0
$$188$$ − 26.8328i − 1.95698i
$$189$$ 0 0
$$190$$ −20.0000 −1.45095
$$191$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$192$$ 0 0
$$193$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ −21.0000 −1.50000
$$197$$ − 4.47214i − 0.318626i −0.987228 0.159313i $$-0.949072\pi$$
0.987228 0.159313i $$-0.0509280\pi$$
$$198$$ 0 0
$$199$$ −16.0000 −1.13421 −0.567105 0.823646i $$-0.691937\pi$$
−0.567105 + 0.823646i $$0.691937\pi$$
$$200$$ 11.1803i 0.790569i
$$201$$ 0 0
$$202$$ 0 0
$$203$$ 0 0
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 0 0
$$207$$ 0 0
$$208$$ 0 0
$$209$$ 0 0
$$210$$ 0 0
$$211$$ −28.0000 −1.92760 −0.963800 0.266627i $$-0.914091\pi$$
−0.963800 + 0.266627i $$0.914091\pi$$
$$212$$ 13.4164i 0.921443i
$$213$$ 0 0
$$214$$ 40.0000 2.73434
$$215$$ 0 0
$$216$$ 0 0
$$217$$ 0 0
$$218$$ 31.3050i 2.12024i
$$219$$ 0 0
$$220$$ 0 0
$$221$$ 0 0
$$222$$ 0 0
$$223$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 10.0000 0.665190
$$227$$ − 17.8885i − 1.18730i −0.804722 0.593652i $$-0.797686\pi$$
0.804722 0.593652i $$-0.202314\pi$$
$$228$$ 0 0
$$229$$ 26.0000 1.71813 0.859064 0.511868i $$-0.171046\pi$$
0.859064 + 0.511868i $$0.171046\pi$$
$$230$$ 44.7214i 2.94884i
$$231$$ 0 0
$$232$$ 0 0
$$233$$ 22.3607i 1.46490i 0.680823 + 0.732448i $$0.261622\pi$$
−0.680823 + 0.732448i $$0.738378\pi$$
$$234$$ 0 0
$$235$$ 20.0000 1.30466
$$236$$ 0 0
$$237$$ 0 0
$$238$$ 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 $$-0.479482\pi$$
0.0644157 + 0.997923i $$0.479482\pi$$
$$242$$ − 24.5967i − 1.58114i
$$243$$ 0 0
$$244$$ −6.00000 −0.384111
$$245$$ − 15.6525i − 1.00000i
$$246$$ 0 0
$$247$$ 0 0
$$248$$ − 17.8885i − 1.13592i
$$249$$ 0 0
$$250$$ −25.0000 −1.58114
$$251$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$252$$ 0 0
$$253$$ 0 0
$$254$$ 0 0
$$255$$ 0 0
$$256$$ −9.00000 −0.562500
$$257$$ − 31.3050i − 1.95275i −0.216085 0.976375i $$-0.569329\pi$$
0.216085 0.976375i $$-0.430671\pi$$
$$258$$ 0 0
$$259$$ 0 0
$$260$$ 0 0
$$261$$ 0 0
$$262$$ 0 0
$$263$$ 8.94427i 0.551527i 0.961225 + 0.275764i $$0.0889307\pi$$
−0.961225 + 0.275764i $$0.911069\pi$$
$$264$$ 0 0
$$265$$ −10.0000 −0.614295
$$266$$ 0 0
$$267$$ 0 0
$$268$$ 0 0
$$269$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$270$$ 0 0
$$271$$ 32.0000 1.94386 0.971931 0.235267i $$-0.0755965\pi$$
0.971931 + 0.235267i $$0.0755965\pi$$
$$272$$ 4.47214i 0.271163i
$$273$$ 0 0
$$274$$ −50.0000 −3.02061
$$275$$ 0 0
$$276$$ 0 0
$$277$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$278$$ − 8.94427i − 0.536442i
$$279$$ 0 0
$$280$$ 0 0
$$281$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$282$$ 0 0
$$283$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ 0 0
$$288$$ 0 0
$$289$$ −3.00000 −0.176471
$$290$$ 0 0
$$291$$ 0 0
$$292$$ 0 0
$$293$$ − 31.3050i − 1.82885i −0.404750 0.914427i $$-0.632641\pi$$
0.404750 0.914427i $$-0.367359\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 0 0
$$297$$ 0 0
$$298$$ 0 0
$$299$$ 0 0
$$300$$ 0 0
$$301$$ 0 0
$$302$$ 17.8885i 1.02937i
$$303$$ 0 0
$$304$$ 4.00000 0.229416
$$305$$ − 4.47214i − 0.256074i
$$306$$ 0 0
$$307$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ 40.0000 2.27185
$$311$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$312$$ 0 0
$$313$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 48.0000 2.70021
$$317$$ 22.3607i 1.25590i 0.778253 + 0.627950i $$0.216106\pi$$
−0.778253 + 0.627950i $$0.783894\pi$$
$$318$$ 0 0
$$319$$ 0 0
$$320$$ − 29.0689i − 1.62500i
$$321$$ 0 0
$$322$$ 0 0
$$323$$ 17.8885i 0.995345i
$$324$$ 0 0
$$325$$ 0 0
$$326$$ 0 0
$$327$$ 0 0
$$328$$ 0 0
$$329$$ 0 0
$$330$$ 0 0
$$331$$ −28.0000 −1.53902 −0.769510 0.638635i $$-0.779499\pi$$
−0.769510 + 0.638635i $$0.779499\pi$$
$$332$$ 53.6656i 2.94528i
$$333$$ 0 0
$$334$$ −20.0000 −1.09435
$$335$$ 0 0
$$336$$ 0 0
$$337$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$338$$ 29.0689i 1.58114i
$$339$$ 0 0
$$340$$ 30.0000 1.62698
$$341$$ 0 0
$$342$$ 0 0
$$343$$ 0 0
$$344$$ 0 0
$$345$$ 0 0
$$346$$ −50.0000 −2.68802
$$347$$ 35.7771i 1.92061i 0.278944 + 0.960307i $$0.410016\pi$$
−0.278944 + 0.960307i $$0.589984\pi$$
$$348$$ 0 0
$$349$$ −34.0000 −1.81998 −0.909989 0.414632i $$-0.863910\pi$$
−0.909989 + 0.414632i $$0.863910\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 0 0
$$353$$ − 31.3050i − 1.66619i −0.553127 0.833097i $$-0.686565\pi$$
0.553127 0.833097i $$-0.313435\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 0 0
$$359$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$360$$ 0 0
$$361$$ −3.00000 −0.157895
$$362$$ − 49.1935i − 2.58555i
$$363$$ 0 0
$$364$$ 0 0
$$365$$ 0 0
$$366$$ 0 0
$$367$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$368$$ − 8.94427i − 0.466252i
$$369$$ 0 0
$$370$$ 0 0
$$371$$ 0 0
$$372$$ 0 0
$$373$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 20.0000 1.03142
$$377$$ 0 0
$$378$$ 0 0
$$379$$ −4.00000 −0.205466 −0.102733 0.994709i $$-0.532759\pi$$
−0.102733 + 0.994709i $$0.532759\pi$$
$$380$$ − 26.8328i − 1.37649i
$$381$$ 0 0
$$382$$ 0 0
$$383$$ 8.94427i 0.457031i 0.973540 + 0.228515i $$0.0733872\pi$$
−0.973540 + 0.228515i $$0.926613\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ 0 0
$$388$$ 0 0
$$389$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$390$$ 0 0
$$391$$ 40.0000 2.02289
$$392$$ − 15.6525i − 0.790569i
$$393$$ 0 0
$$394$$ 10.0000 0.503793
$$395$$ 35.7771i 1.80014i
$$396$$ 0 0
$$397$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$398$$ − 35.7771i − 1.79334i
$$399$$ 0 0
$$400$$ 5.00000 0.250000
$$401$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$402$$ 0 0
$$403$$ 0 0
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ 0 0
$$408$$ 0 0
$$409$$ 26.0000 1.28562 0.642809 0.766027i $$-0.277769\pi$$
0.642809 + 0.766027i $$0.277769\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ 0 0
$$413$$ 0 0
$$414$$ 0 0
$$415$$ −40.0000 −1.96352
$$416$$ 0 0
$$417$$ 0 0
$$418$$ 0 0
$$419$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$420$$ 0 0
$$421$$ 38.0000 1.85201 0.926003 0.377515i $$-0.123221\pi$$
0.926003 + 0.377515i $$0.123221\pi$$
$$422$$ − 62.6099i − 3.04780i
$$423$$ 0 0
$$424$$ −10.0000 −0.485643
$$425$$ 22.3607i 1.08465i
$$426$$ 0 0
$$427$$ 0 0
$$428$$ 53.6656i 2.59403i
$$429$$ 0 0
$$430$$ 0 0
$$431$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$432$$ 0 0
$$433$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ −42.0000 −2.01144
$$437$$ − 35.7771i − 1.71145i
$$438$$ 0 0
$$439$$ −16.0000 −0.763638 −0.381819 0.924237i $$-0.624702\pi$$
−0.381819 + 0.924237i $$0.624702\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 0 0
$$443$$ − 17.8885i − 0.849910i −0.905214 0.424955i $$-0.860290\pi$$
0.905214 0.424955i $$-0.139710\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ 0 0
$$447$$ 0 0
$$448$$ 0 0
$$449$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$450$$ 0 0
$$451$$ 0 0
$$452$$ 13.4164i 0.631055i
$$453$$ 0 0
$$454$$ 40.0000 1.87729
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$458$$ 58.1378i 2.71660i
$$459$$ 0 0
$$460$$ −60.0000 −2.79751
$$461$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$462$$ 0 0
$$463$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ −50.0000 −2.31621
$$467$$ 35.7771i 1.65557i 0.561048 + 0.827783i $$0.310398\pi$$
−0.561048 + 0.827783i $$0.689602\pi$$
$$468$$ 0 0
$$469$$ 0 0
$$470$$ 44.7214i 2.06284i
$$471$$ 0 0
$$472$$ 0 0
$$473$$ 0 0
$$474$$ 0 0
$$475$$ 20.0000 0.917663
$$476$$ 0 0
$$477$$ 0 0
$$478$$ 0 0
$$479$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$480$$ 0 0
$$481$$ 0 0
$$482$$ 4.47214i 0.203700i
$$483$$ 0 0
$$484$$ 33.0000 1.50000
$$485$$ 0 0
$$486$$ 0 0
$$487$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$488$$ − 4.47214i − 0.202444i
$$489$$ 0 0
$$490$$ 35.0000 1.58114
$$491$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$492$$ 0 0
$$493$$ 0 0
$$494$$ 0 0
$$495$$ 0 0
$$496$$ −8.00000 −0.359211
$$497$$ 0 0
$$498$$ 0 0
$$499$$ 44.0000 1.96971 0.984855 0.173379i $$-0.0554684\pi$$
0.984855 + 0.173379i $$0.0554684\pi$$
$$500$$ − 33.5410i − 1.50000i
$$501$$ 0 0
$$502$$ 0 0
$$503$$ − 44.7214i − 1.99403i −0.0772283 0.997013i $$-0.524607\pi$$
0.0772283 0.997013i $$-0.475393\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 0 0
$$507$$ 0 0
$$508$$ 0 0
$$509$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$510$$ 0 0
$$511$$ 0 0
$$512$$ 11.1803i 0.494106i
$$513$$ 0 0
$$514$$ 70.0000 3.08757
$$515$$ 0 0
$$516$$ 0 0
$$517$$ 0 0
$$518$$ 0 0
$$519$$ 0 0
$$520$$ 0 0
$$521$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$522$$ 0 0
$$523$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ −20.0000 −0.872041
$$527$$ − 35.7771i − 1.55847i
$$528$$ 0 0
$$529$$ −57.0000 −2.47826
$$530$$ − 22.3607i − 0.971286i
$$531$$ 0 0
$$532$$ 0 0
$$533$$ 0 0
$$534$$ 0 0
$$535$$ −40.0000 −1.72935
$$536$$ 0 0
$$537$$ 0 0
$$538$$ 0 0
$$539$$ 0 0
$$540$$ 0 0
$$541$$ 2.00000 0.0859867 0.0429934 0.999075i $$-0.486311\pi$$
0.0429934 + 0.999075i $$0.486311\pi$$
$$542$$ 71.5542i 3.07352i
$$543$$ 0 0
$$544$$ −30.0000 −1.28624
$$545$$ − 31.3050i − 1.34096i
$$546$$ 0 0
$$547$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$548$$ − 67.0820i − 2.86560i
$$549$$ 0 0
$$550$$ 0 0
$$551$$ 0 0
$$552$$ 0 0
$$553$$ 0 0
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 12.0000 0.508913
$$557$$ 22.3607i 0.947452i 0.880672 + 0.473726i $$0.157091\pi$$
−0.880672 + 0.473726i $$0.842909\pi$$
$$558$$ 0 0
$$559$$ 0 0
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 0 0
$$563$$ 35.7771i 1.50782i 0.656975 + 0.753912i $$0.271836\pi$$
−0.656975 + 0.753912i $$0.728164\pi$$
$$564$$ 0 0
$$565$$ −10.0000 −0.420703
$$566$$ 0 0
$$567$$ 0 0
$$568$$ 0 0
$$569$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$570$$ 0 0
$$571$$ −28.0000 −1.17176 −0.585882 0.810397i $$-0.699252\pi$$
−0.585882 + 0.810397i $$0.699252\pi$$
$$572$$ 0 0
$$573$$ 0 0
$$574$$ 0 0
$$575$$ − 44.7214i − 1.86501i
$$576$$ 0 0
$$577$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$578$$ − 6.70820i − 0.279024i
$$579$$ 0 0
$$580$$ 0 0
$$581$$ 0 0
$$582$$ 0 0
$$583$$ 0 0
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 70.0000 2.89167
$$587$$ − 17.8885i − 0.738339i −0.929362 0.369170i $$-0.879642\pi$$
0.929362 0.369170i $$-0.120358\pi$$
$$588$$ 0 0
$$589$$ −32.0000 −1.31854
$$590$$ 0 0
$$591$$ 0 0
$$592$$ 0 0
$$593$$ − 4.47214i − 0.183649i −0.995775 0.0918243i $$-0.970730\pi$$
0.995775 0.0918243i $$-0.0292698\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ 0 0
$$598$$ 0 0
$$599$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$600$$ 0 0
$$601$$ 38.0000 1.55005 0.775026 0.631929i $$-0.217737\pi$$
0.775026 + 0.631929i $$0.217737\pi$$
$$602$$ 0 0
$$603$$ 0 0
$$604$$ −24.0000 −0.976546
$$605$$ 24.5967i 1.00000i
$$606$$ 0 0
$$607$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$608$$ 26.8328i 1.08821i
$$609$$ 0 0
$$610$$ 10.0000 0.404888
$$611$$ 0 0
$$612$$ 0 0
$$613$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ 49.1935i 1.98046i 0.139459 + 0.990228i $$0.455464\pi$$
−0.139459 + 0.990228i $$0.544536\pi$$
$$618$$ 0 0
$$619$$ 44.0000 1.76851 0.884255 0.467005i $$-0.154667\pi$$
0.884255 + 0.467005i $$0.154667\pi$$
$$620$$ 53.6656i 2.15526i
$$621$$ 0 0
$$622$$ 0 0
$$623$$ 0 0
$$624$$ 0 0
$$625$$ 25.0000 1.00000
$$626$$ 0 0
$$627$$ 0 0
$$628$$ 0 0
$$629$$ 0 0
$$630$$ 0 0
$$631$$ 32.0000 1.27390 0.636950 0.770905i $$-0.280196\pi$$
0.636950 + 0.770905i $$0.280196\pi$$
$$632$$ 35.7771i 1.42314i
$$633$$ 0 0
$$634$$ −50.0000 −1.98575
$$635$$ 0 0
$$636$$ 0 0
$$637$$ 0 0
$$638$$ 0 0
$$639$$ 0 0
$$640$$ 35.0000 1.38350
$$641$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$642$$ 0 0
$$643$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ −40.0000 −1.57378
$$647$$ − 44.7214i − 1.75818i −0.476658 0.879089i $$-0.658152\pi$$
0.476658 0.879089i $$-0.341848\pi$$
$$648$$ 0 0
$$649$$ 0 0
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 0 0
$$653$$ 49.1935i 1.92509i 0.271122 + 0.962545i $$0.412605\pi$$
−0.271122 + 0.962545i $$0.587395\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 0 0
$$657$$ 0 0
$$658$$ 0 0
$$659$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$660$$ 0 0
$$661$$ −22.0000 −0.855701 −0.427850 0.903850i $$-0.640729\pi$$
−0.427850 + 0.903850i $$0.640729\pi$$
$$662$$ − 62.6099i − 2.43340i
$$663$$ 0 0
$$664$$ −40.0000 −1.55230
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 0 0
$$668$$ − 26.8328i − 1.03819i
$$669$$ 0 0
$$670$$ 0 0
$$671$$ 0 0
$$672$$ 0 0
$$673$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ −39.0000 −1.50000
$$677$$ − 31.3050i − 1.20315i −0.798817 0.601574i $$-0.794541\pi$$
0.798817 0.601574i $$-0.205459\pi$$
$$678$$ 0 0
$$679$$ 0 0
$$680$$ 22.3607i 0.857493i
$$681$$ 0 0
$$682$$ 0 0
$$683$$ 35.7771i 1.36897i 0.729026 + 0.684486i $$0.239973\pi$$
−0.729026 + 0.684486i $$0.760027\pi$$
$$684$$ 0 0
$$685$$ 50.0000 1.91040
$$686$$ 0 0
$$687$$ 0 0
$$688$$ 0 0
$$689$$ 0 0
$$690$$ 0 0
$$691$$ −52.0000 −1.97817 −0.989087 0.147335i $$-0.952930\pi$$
−0.989087 + 0.147335i $$0.952930\pi$$
$$692$$ − 67.0820i − 2.55008i
$$693$$ 0 0
$$694$$ −80.0000 −3.03676
$$695$$ 8.94427i 0.339276i
$$696$$ 0 0
$$697$$ 0 0
$$698$$ − 76.0263i − 2.87764i
$$699$$ 0 0
$$700$$ 0 0
$$701$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$702$$ 0 0
$$703$$ 0 0
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 70.0000 2.63448
$$707$$ 0 0
$$708$$ 0 0
$$709$$ 26.0000 0.976450 0.488225 0.872718i $$-0.337644\pi$$
0.488225 + 0.872718i $$0.337644\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ 0 0
$$713$$ 71.5542i 2.67972i
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 0 0
$$717$$ 0 0
$$718$$ 0 0
$$719$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$720$$ 0 0
$$721$$ 0 0
$$722$$ − 6.70820i − 0.249653i
$$723$$ 0 0
$$724$$ 66.0000 2.45287
$$725$$ 0 0
$$726$$ 0 0
$$727$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$728$$ 0 0
$$729$$ 0 0
$$730$$ 0 0
$$731$$ 0 0
$$732$$ 0 0
$$733$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 60.0000 2.21163
$$737$$ 0 0
$$738$$ 0 0
$$739$$ −4.00000 −0.147142 −0.0735712 0.997290i $$-0.523440\pi$$
−0.0735712 + 0.997290i $$0.523440\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 0 0
$$743$$ − 44.7214i − 1.64067i −0.571885 0.820334i $$-0.693788\pi$$
0.571885 0.820334i $$-0.306212\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 0 0
$$747$$ 0 0
$$748$$ 0 0
$$749$$ 0 0
$$750$$ 0 0
$$751$$ 8.00000 0.291924 0.145962 0.989290i $$-0.453372\pi$$
0.145962 + 0.989290i $$0.453372\pi$$
$$752$$ − 8.94427i − 0.326164i
$$753$$ 0 0
$$754$$ 0 0
$$755$$ − 17.8885i − 0.651031i
$$756$$ 0 0
$$757$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$758$$ − 8.94427i − 0.324871i
$$759$$ 0 0
$$760$$ 20.0000 0.725476
$$761$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$762$$ 0 0
$$763$$ 0 0
$$764$$ 0 0
$$765$$ 0 0
$$766$$ −20.0000 −0.722629
$$767$$ 0 0
$$768$$ 0 0
$$769$$ −46.0000 −1.65880 −0.829401 0.558653i $$-0.811318\pi$$
−0.829401 + 0.558653i $$0.811318\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 0 0
$$773$$ − 4.47214i − 0.160852i −0.996761 0.0804258i $$-0.974372\pi$$
0.996761 0.0804258i $$-0.0256280\pi$$
$$774$$ 0 0
$$775$$ −40.0000 −1.43684
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 0 0
$$779$$ 0 0
$$780$$ 0 0
$$781$$ 0 0
$$782$$ 89.4427i 3.19847i
$$783$$ 0 0
$$784$$ −7.00000 −0.250000
$$785$$ 0 0
$$786$$ 0 0
$$787$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$788$$ 13.4164i 0.477940i
$$789$$ 0 0
$$790$$ −80.0000 −2.84627
$$791$$ 0 0
$$792$$ 0 0
$$793$$ 0 0
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 48.0000 1.70131
$$797$$ 49.1935i 1.74252i 0.490819 + 0.871262i $$0.336698\pi$$
−0.490819 + 0.871262i $$0.663302\pi$$
$$798$$ 0 0
$$799$$ 40.0000 1.41510
$$800$$ 33.5410i 1.18585i
$$801$$ 0 0
$$802$$ 0 0
$$803$$ 0 0
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ 0 0
$$808$$ 0 0
$$809$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$810$$ 0 0
$$811$$ −52.0000 −1.82597 −0.912983 0.407997i $$-0.866228\pi$$
−0.912983 + 0.407997i $$0.866228\pi$$
$$812$$ 0 0
$$813$$ 0 0
$$814$$ 0 0
$$815$$ 0 0
$$816$$ 0 0
$$817$$ 0 0
$$818$$ 58.1378i 2.03274i
$$819$$ 0 0
$$820$$ 0 0
$$821$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$822$$ 0 0
$$823$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ 35.7771i 1.24409i 0.782981 + 0.622046i $$0.213698\pi$$
−0.782981 + 0.622046i $$0.786302\pi$$
$$828$$ 0 0
$$829$$ −34.0000 −1.18087 −0.590434 0.807086i $$-0.701044\pi$$
−0.590434 + 0.807086i $$0.701044\pi$$
$$830$$ − 89.4427i − 3.10460i
$$831$$ 0 0
$$832$$ 0 0
$$833$$ − 31.3050i − 1.08465i
$$834$$ 0 0
$$835$$ 20.0000 0.692129
$$836$$ 0 0
$$837$$ 0 0
$$838$$ 0 0
$$839$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$840$$ 0 0
$$841$$ −29.0000 −1.00000
$$842$$ 84.9706i 2.92828i
$$843$$ 0 0
$$844$$ 84.0000 2.89140
$$845$$ − 29.0689i − 1.00000i
$$846$$ 0 0
$$847$$ 0 0
$$848$$ 4.47214i 0.153574i
$$849$$ 0 0
$$850$$ −50.0000 −1.71499
$$851$$ 0 0
$$852$$ 0 0
$$853$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ −40.0000 −1.36717
$$857$$ − 58.1378i − 1.98595i −0.118331 0.992974i $$-0.537755\pi$$
0.118331 0.992974i $$-0.462245\pi$$
$$858$$ 0 0
$$859$$ 44.0000 1.50126 0.750630 0.660722i $$-0.229750\pi$$
0.750630 + 0.660722i $$0.229750\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 0 0
$$863$$ − 44.7214i − 1.52233i −0.648557 0.761166i $$-0.724627\pi$$
0.648557 0.761166i $$-0.275373\pi$$
$$864$$ 0 0
$$865$$ 50.0000 1.70005
$$866$$ 0 0
$$867$$ 0 0
$$868$$ 0 0
$$869$$ 0 0
$$870$$ 0 0
$$871$$ 0 0
$$872$$ − 31.3050i − 1.06012i
$$873$$ 0 0
$$874$$ 80.0000 2.70604
$$875$$ 0 0
$$876$$ 0 0
$$877$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$878$$ − 35.7771i − 1.20742i
$$879$$ 0 0
$$880$$ 0 0
$$881$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$882$$ 0 0
$$883$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 40.0000 1.34383
$$887$$ 8.94427i 0.300319i 0.988662 + 0.150160i $$0.0479788\pi$$
−0.988662 + 0.150160i $$0.952021\pi$$
$$888$$ 0 0
$$889$$ 0 0
$$890$$ 0 0
$$891$$ 0 0
$$892$$ 0 0
$$893$$ − 35.7771i − 1.19723i
$$894$$ 0 0
$$895$$ 0 0
$$896$$ 0 0
$$897$$ 0 0
$$898$$ 0 0
$$899$$ 0 0
$$900$$ 0 0
$$901$$ −20.0000 −0.666297
$$902$$ 0 0
$$903$$ 0 0
$$904$$ −10.0000 −0.332595
$$905$$ 49.1935i 1.63525i
$$906$$ 0 0
$$907$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$908$$ 53.6656i 1.78096i
$$909$$ 0 0
$$910$$ 0 0
$$911$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$912$$ 0 0
$$913$$ 0 0
$$914$$ 0 0
$$915$$ 0 0
$$916$$ −78.0000 −2.57719
$$917$$ 0 0
$$918$$ 0 0
$$919$$ 56.0000 1.84727 0.923635 0.383274i $$-0.125203\pi$$
0.923635 + 0.383274i $$0.125203\pi$$
$$920$$ − 44.7214i − 1.47442i
$$921$$ 0 0
$$922$$ 0 0
$$923$$ 0 0
$$924$$ 0 0
$$925$$ 0 0
$$926$$ 0 0
$$927$$ 0 0
$$928$$ 0 0
$$929$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$930$$ 0 0
$$931$$ −28.0000 −0.917663
$$932$$ − 67.0820i − 2.19735i
$$933$$ 0 0
$$934$$ −80.0000 −2.61768
$$935$$ 0 0
$$936$$ 0 0
$$937$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$938$$ 0 0
$$939$$ 0 0
$$940$$ −60.0000 −1.95698
$$941$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$942$$ 0 0
$$943$$ 0 0
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ − 17.8885i − 0.581300i −0.956830 0.290650i $$-0.906129\pi$$
0.956830 0.290650i $$-0.0938715\pi$$
$$948$$ 0 0
$$949$$ 0 0
$$950$$ 44.7214i 1.45095i
$$951$$ 0 0
$$952$$ 0 0
$$953$$ − 58.1378i − 1.88327i −0.336640 0.941634i $$-0.609290\pi$$
0.336640 0.941634i $$-0.390710\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ 0 0
$$957$$ 0 0
$$958$$ 0 0
$$959$$ 0 0
$$960$$ 0 0
$$961$$ 33.0000 1.06452
$$962$$ 0 0
$$963$$ 0 0
$$964$$ −6.00000 −0.193247
$$965$$ 0 0
$$966$$ 0 0
$$967$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$968$$ 24.5967i 0.790569i
$$969$$ 0 0
$$970$$ 0 0
$$971$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$972$$ 0 0
$$973$$ 0 0
$$974$$ 0 0
$$975$$ 0 0
$$976$$ −2.00000 −0.0640184
$$977$$ − 4.47214i − 0.143076i −0.997438 0.0715382i $$-0.977209\pi$$
0.997438 0.0715382i $$-0.0227908\pi$$
$$978$$ 0 0
$$979$$ 0 0
$$980$$ 46.9574i 1.50000i
$$981$$ 0 0
$$982$$ 0 0
$$983$$ 62.6099i 1.99695i 0.0552438 + 0.998473i $$0.482406\pi$$
−0.0552438 + 0.998473i $$0.517594\pi$$
$$984$$ 0 0
$$985$$ −10.0000 −0.318626
$$986$$ 0 0
$$987$$ 0 0
$$988$$ 0 0
$$989$$ 0 0
$$990$$ 0 0
$$991$$ 32.0000 1.01651 0.508257 0.861206i $$-0.330290\pi$$
0.508257 + 0.861206i $$0.330290\pi$$
$$992$$ − 53.6656i − 1.70389i
$$993$$ 0 0
$$994$$ 0 0
$$995$$ 35.7771i 1.13421i
$$996$$ 0 0
$$997$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$998$$ 98.3870i 3.11439i
$$999$$ 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 45.2.b.a.19.2 yes 2
3.2 odd 2 inner 45.2.b.a.19.1 2
4.3 odd 2 720.2.f.d.289.1 2
5.2 odd 4 225.2.a.f.1.1 2
5.3 odd 4 225.2.a.f.1.2 2
5.4 even 2 inner 45.2.b.a.19.1 2
7.6 odd 2 2205.2.d.a.1324.2 2
8.3 odd 2 2880.2.f.j.1729.2 2
8.5 even 2 2880.2.f.k.1729.2 2
9.2 odd 6 405.2.j.c.109.1 4
9.4 even 3 405.2.j.c.379.1 4
9.5 odd 6 405.2.j.c.379.2 4
9.7 even 3 405.2.j.c.109.2 4
12.11 even 2 720.2.f.d.289.2 2
15.2 even 4 225.2.a.f.1.2 2
15.8 even 4 225.2.a.f.1.1 2
15.14 odd 2 CM 45.2.b.a.19.2 yes 2
20.3 even 4 3600.2.a.bs.1.1 2
20.7 even 4 3600.2.a.bs.1.2 2
20.19 odd 2 720.2.f.d.289.2 2
21.20 even 2 2205.2.d.a.1324.1 2
24.5 odd 2 2880.2.f.k.1729.1 2
24.11 even 2 2880.2.f.j.1729.1 2
35.34 odd 2 2205.2.d.a.1324.1 2
40.19 odd 2 2880.2.f.j.1729.1 2
40.29 even 2 2880.2.f.k.1729.1 2
45.4 even 6 405.2.j.c.379.2 4
45.14 odd 6 405.2.j.c.379.1 4
45.29 odd 6 405.2.j.c.109.2 4
45.34 even 6 405.2.j.c.109.1 4
60.23 odd 4 3600.2.a.bs.1.2 2
60.47 odd 4 3600.2.a.bs.1.1 2
60.59 even 2 720.2.f.d.289.1 2
105.104 even 2 2205.2.d.a.1324.2 2
120.29 odd 2 2880.2.f.k.1729.2 2
120.59 even 2 2880.2.f.j.1729.2 2

By twisted newform
Twist Min Dim Char Parity Ord Type
45.2.b.a.19.1 2 3.2 odd 2 inner
45.2.b.a.19.1 2 5.4 even 2 inner
45.2.b.a.19.2 yes 2 1.1 even 1 trivial
45.2.b.a.19.2 yes 2 15.14 odd 2 CM
225.2.a.f.1.1 2 5.2 odd 4
225.2.a.f.1.1 2 15.8 even 4
225.2.a.f.1.2 2 5.3 odd 4
225.2.a.f.1.2 2 15.2 even 4
405.2.j.c.109.1 4 9.2 odd 6
405.2.j.c.109.1 4 45.34 even 6
405.2.j.c.109.2 4 9.7 even 3
405.2.j.c.109.2 4 45.29 odd 6
405.2.j.c.379.1 4 9.4 even 3
405.2.j.c.379.1 4 45.14 odd 6
405.2.j.c.379.2 4 9.5 odd 6
405.2.j.c.379.2 4 45.4 even 6
720.2.f.d.289.1 2 4.3 odd 2
720.2.f.d.289.1 2 60.59 even 2
720.2.f.d.289.2 2 12.11 even 2
720.2.f.d.289.2 2 20.19 odd 2
2205.2.d.a.1324.1 2 21.20 even 2
2205.2.d.a.1324.1 2 35.34 odd 2
2205.2.d.a.1324.2 2 7.6 odd 2
2205.2.d.a.1324.2 2 105.104 even 2
2880.2.f.j.1729.1 2 24.11 even 2
2880.2.f.j.1729.1 2 40.19 odd 2
2880.2.f.j.1729.2 2 8.3 odd 2
2880.2.f.j.1729.2 2 120.59 even 2
2880.2.f.k.1729.1 2 24.5 odd 2
2880.2.f.k.1729.1 2 40.29 even 2
2880.2.f.k.1729.2 2 8.5 even 2
2880.2.f.k.1729.2 2 120.29 odd 2
3600.2.a.bs.1.1 2 20.3 even 4
3600.2.a.bs.1.1 2 60.47 odd 4
3600.2.a.bs.1.2 2 20.7 even 4
3600.2.a.bs.1.2 2 60.23 odd 4