# Properties

 Label 1638.2.a.s.1.1 Level $1638$ Weight $2$ Character 1638.1 Self dual yes Analytic conductor $13.079$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

# Learn more

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1638,2,Mod(1,1638)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1638, 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("1638.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1638 = 2 \cdot 3^{2} \cdot 7 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1638.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: $$13.0794958511$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 546) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 1638.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+1.00000 q^{2} +1.00000 q^{4} +2.00000 q^{5} -1.00000 q^{7} +1.00000 q^{8} +O(q^{10})$$ $$q+1.00000 q^{2} +1.00000 q^{4} +2.00000 q^{5} -1.00000 q^{7} +1.00000 q^{8} +2.00000 q^{10} +4.00000 q^{11} +1.00000 q^{13} -1.00000 q^{14} +1.00000 q^{16} +2.00000 q^{17} -4.00000 q^{19} +2.00000 q^{20} +4.00000 q^{22} +4.00000 q^{23} -1.00000 q^{25} +1.00000 q^{26} -1.00000 q^{28} +2.00000 q^{29} +1.00000 q^{32} +2.00000 q^{34} -2.00000 q^{35} -2.00000 q^{37} -4.00000 q^{38} +2.00000 q^{40} -2.00000 q^{41} +4.00000 q^{43} +4.00000 q^{44} +4.00000 q^{46} +12.0000 q^{47} +1.00000 q^{49} -1.00000 q^{50} +1.00000 q^{52} -6.00000 q^{53} +8.00000 q^{55} -1.00000 q^{56} +2.00000 q^{58} -10.0000 q^{61} +1.00000 q^{64} +2.00000 q^{65} +4.00000 q^{67} +2.00000 q^{68} -2.00000 q^{70} +8.00000 q^{71} -6.00000 q^{73} -2.00000 q^{74} -4.00000 q^{76} -4.00000 q^{77} +8.00000 q^{79} +2.00000 q^{80} -2.00000 q^{82} -8.00000 q^{83} +4.00000 q^{85} +4.00000 q^{86} +4.00000 q^{88} +6.00000 q^{89} -1.00000 q^{91} +4.00000 q^{92} +12.0000 q^{94} -8.00000 q^{95} +2.00000 q^{97} +1.00000 q^{98} +O(q^{100})$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 1.00000 0.707107
$$3$$ 0 0
$$4$$ 1.00000 0.500000
$$5$$ 2.00000 0.894427 0.447214 0.894427i $$-0.352416\pi$$
0.447214 + 0.894427i $$0.352416\pi$$
$$6$$ 0 0
$$7$$ −1.00000 −0.377964
$$8$$ 1.00000 0.353553
$$9$$ 0 0
$$10$$ 2.00000 0.632456
$$11$$ 4.00000 1.20605 0.603023 0.797724i $$-0.293963\pi$$
0.603023 + 0.797724i $$0.293963\pi$$
$$12$$ 0 0
$$13$$ 1.00000 0.277350
$$14$$ −1.00000 −0.267261
$$15$$ 0 0
$$16$$ 1.00000 0.250000
$$17$$ 2.00000 0.485071 0.242536 0.970143i $$-0.422021\pi$$
0.242536 + 0.970143i $$0.422021\pi$$
$$18$$ 0 0
$$19$$ −4.00000 −0.917663 −0.458831 0.888523i $$-0.651732\pi$$
−0.458831 + 0.888523i $$0.651732\pi$$
$$20$$ 2.00000 0.447214
$$21$$ 0 0
$$22$$ 4.00000 0.852803
$$23$$ 4.00000 0.834058 0.417029 0.908893i $$-0.363071\pi$$
0.417029 + 0.908893i $$0.363071\pi$$
$$24$$ 0 0
$$25$$ −1.00000 −0.200000
$$26$$ 1.00000 0.196116
$$27$$ 0 0
$$28$$ −1.00000 −0.188982
$$29$$ 2.00000 0.371391 0.185695 0.982607i $$-0.440546\pi$$
0.185695 + 0.982607i $$0.440546\pi$$
$$30$$ 0 0
$$31$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$32$$ 1.00000 0.176777
$$33$$ 0 0
$$34$$ 2.00000 0.342997
$$35$$ −2.00000 −0.338062
$$36$$ 0 0
$$37$$ −2.00000 −0.328798 −0.164399 0.986394i $$-0.552568\pi$$
−0.164399 + 0.986394i $$0.552568\pi$$
$$38$$ −4.00000 −0.648886
$$39$$ 0 0
$$40$$ 2.00000 0.316228
$$41$$ −2.00000 −0.312348 −0.156174 0.987730i $$-0.549916\pi$$
−0.156174 + 0.987730i $$0.549916\pi$$
$$42$$ 0 0
$$43$$ 4.00000 0.609994 0.304997 0.952353i $$-0.401344\pi$$
0.304997 + 0.952353i $$0.401344\pi$$
$$44$$ 4.00000 0.603023
$$45$$ 0 0
$$46$$ 4.00000 0.589768
$$47$$ 12.0000 1.75038 0.875190 0.483779i $$-0.160736\pi$$
0.875190 + 0.483779i $$0.160736\pi$$
$$48$$ 0 0
$$49$$ 1.00000 0.142857
$$50$$ −1.00000 −0.141421
$$51$$ 0 0
$$52$$ 1.00000 0.138675
$$53$$ −6.00000 −0.824163 −0.412082 0.911147i $$-0.635198\pi$$
−0.412082 + 0.911147i $$0.635198\pi$$
$$54$$ 0 0
$$55$$ 8.00000 1.07872
$$56$$ −1.00000 −0.133631
$$57$$ 0 0
$$58$$ 2.00000 0.262613
$$59$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$60$$ 0 0
$$61$$ −10.0000 −1.28037 −0.640184 0.768221i $$-0.721142\pi$$
−0.640184 + 0.768221i $$0.721142\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ 1.00000 0.125000
$$65$$ 2.00000 0.248069
$$66$$ 0 0
$$67$$ 4.00000 0.488678 0.244339 0.969690i $$-0.421429\pi$$
0.244339 + 0.969690i $$0.421429\pi$$
$$68$$ 2.00000 0.242536
$$69$$ 0 0
$$70$$ −2.00000 −0.239046
$$71$$ 8.00000 0.949425 0.474713 0.880141i $$-0.342552\pi$$
0.474713 + 0.880141i $$0.342552\pi$$
$$72$$ 0 0
$$73$$ −6.00000 −0.702247 −0.351123 0.936329i $$-0.614200\pi$$
−0.351123 + 0.936329i $$0.614200\pi$$
$$74$$ −2.00000 −0.232495
$$75$$ 0 0
$$76$$ −4.00000 −0.458831
$$77$$ −4.00000 −0.455842
$$78$$ 0 0
$$79$$ 8.00000 0.900070 0.450035 0.893011i $$-0.351411\pi$$
0.450035 + 0.893011i $$0.351411\pi$$
$$80$$ 2.00000 0.223607
$$81$$ 0 0
$$82$$ −2.00000 −0.220863
$$83$$ −8.00000 −0.878114 −0.439057 0.898459i $$-0.644687\pi$$
−0.439057 + 0.898459i $$0.644687\pi$$
$$84$$ 0 0
$$85$$ 4.00000 0.433861
$$86$$ 4.00000 0.431331
$$87$$ 0 0
$$88$$ 4.00000 0.426401
$$89$$ 6.00000 0.635999 0.317999 0.948091i $$-0.396989\pi$$
0.317999 + 0.948091i $$0.396989\pi$$
$$90$$ 0 0
$$91$$ −1.00000 −0.104828
$$92$$ 4.00000 0.417029
$$93$$ 0 0
$$94$$ 12.0000 1.23771
$$95$$ −8.00000 −0.820783
$$96$$ 0 0
$$97$$ 2.00000 0.203069 0.101535 0.994832i $$-0.467625\pi$$
0.101535 + 0.994832i $$0.467625\pi$$
$$98$$ 1.00000 0.101015
$$99$$ 0 0
$$100$$ −1.00000 −0.100000
$$101$$ 6.00000 0.597022 0.298511 0.954406i $$-0.403510\pi$$
0.298511 + 0.954406i $$0.403510\pi$$
$$102$$ 0 0
$$103$$ −8.00000 −0.788263 −0.394132 0.919054i $$-0.628955\pi$$
−0.394132 + 0.919054i $$0.628955\pi$$
$$104$$ 1.00000 0.0980581
$$105$$ 0 0
$$106$$ −6.00000 −0.582772
$$107$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$108$$ 0 0
$$109$$ 14.0000 1.34096 0.670478 0.741929i $$-0.266089\pi$$
0.670478 + 0.741929i $$0.266089\pi$$
$$110$$ 8.00000 0.762770
$$111$$ 0 0
$$112$$ −1.00000 −0.0944911
$$113$$ −10.0000 −0.940721 −0.470360 0.882474i $$-0.655876\pi$$
−0.470360 + 0.882474i $$0.655876\pi$$
$$114$$ 0 0
$$115$$ 8.00000 0.746004
$$116$$ 2.00000 0.185695
$$117$$ 0 0
$$118$$ 0 0
$$119$$ −2.00000 −0.183340
$$120$$ 0 0
$$121$$ 5.00000 0.454545
$$122$$ −10.0000 −0.905357
$$123$$ 0 0
$$124$$ 0 0
$$125$$ −12.0000 −1.07331
$$126$$ 0 0
$$127$$ −8.00000 −0.709885 −0.354943 0.934888i $$-0.615500\pi$$
−0.354943 + 0.934888i $$0.615500\pi$$
$$128$$ 1.00000 0.0883883
$$129$$ 0 0
$$130$$ 2.00000 0.175412
$$131$$ −12.0000 −1.04844 −0.524222 0.851581i $$-0.675644\pi$$
−0.524222 + 0.851581i $$0.675644\pi$$
$$132$$ 0 0
$$133$$ 4.00000 0.346844
$$134$$ 4.00000 0.345547
$$135$$ 0 0
$$136$$ 2.00000 0.171499
$$137$$ −22.0000 −1.87959 −0.939793 0.341743i $$-0.888983\pi$$
−0.939793 + 0.341743i $$0.888983\pi$$
$$138$$ 0 0
$$139$$ 4.00000 0.339276 0.169638 0.985506i $$-0.445740\pi$$
0.169638 + 0.985506i $$0.445740\pi$$
$$140$$ −2.00000 −0.169031
$$141$$ 0 0
$$142$$ 8.00000 0.671345
$$143$$ 4.00000 0.334497
$$144$$ 0 0
$$145$$ 4.00000 0.332182
$$146$$ −6.00000 −0.496564
$$147$$ 0 0
$$148$$ −2.00000 −0.164399
$$149$$ −18.0000 −1.47462 −0.737309 0.675556i $$-0.763904\pi$$
−0.737309 + 0.675556i $$0.763904\pi$$
$$150$$ 0 0
$$151$$ −16.0000 −1.30206 −0.651031 0.759051i $$-0.725663\pi$$
−0.651031 + 0.759051i $$0.725663\pi$$
$$152$$ −4.00000 −0.324443
$$153$$ 0 0
$$154$$ −4.00000 −0.322329
$$155$$ 0 0
$$156$$ 0 0
$$157$$ 22.0000 1.75579 0.877896 0.478852i $$-0.158947\pi$$
0.877896 + 0.478852i $$0.158947\pi$$
$$158$$ 8.00000 0.636446
$$159$$ 0 0
$$160$$ 2.00000 0.158114
$$161$$ −4.00000 −0.315244
$$162$$ 0 0
$$163$$ 12.0000 0.939913 0.469956 0.882690i $$-0.344270\pi$$
0.469956 + 0.882690i $$0.344270\pi$$
$$164$$ −2.00000 −0.156174
$$165$$ 0 0
$$166$$ −8.00000 −0.620920
$$167$$ −12.0000 −0.928588 −0.464294 0.885681i $$-0.653692\pi$$
−0.464294 + 0.885681i $$0.653692\pi$$
$$168$$ 0 0
$$169$$ 1.00000 0.0769231
$$170$$ 4.00000 0.306786
$$171$$ 0 0
$$172$$ 4.00000 0.304997
$$173$$ −2.00000 −0.152057 −0.0760286 0.997106i $$-0.524224\pi$$
−0.0760286 + 0.997106i $$0.524224\pi$$
$$174$$ 0 0
$$175$$ 1.00000 0.0755929
$$176$$ 4.00000 0.301511
$$177$$ 0 0
$$178$$ 6.00000 0.449719
$$179$$ 24.0000 1.79384 0.896922 0.442189i $$-0.145798\pi$$
0.896922 + 0.442189i $$0.145798\pi$$
$$180$$ 0 0
$$181$$ −26.0000 −1.93256 −0.966282 0.257485i $$-0.917106\pi$$
−0.966282 + 0.257485i $$0.917106\pi$$
$$182$$ −1.00000 −0.0741249
$$183$$ 0 0
$$184$$ 4.00000 0.294884
$$185$$ −4.00000 −0.294086
$$186$$ 0 0
$$187$$ 8.00000 0.585018
$$188$$ 12.0000 0.875190
$$189$$ 0 0
$$190$$ −8.00000 −0.580381
$$191$$ 12.0000 0.868290 0.434145 0.900843i $$-0.357051\pi$$
0.434145 + 0.900843i $$0.357051\pi$$
$$192$$ 0 0
$$193$$ 2.00000 0.143963 0.0719816 0.997406i $$-0.477068\pi$$
0.0719816 + 0.997406i $$0.477068\pi$$
$$194$$ 2.00000 0.143592
$$195$$ 0 0
$$196$$ 1.00000 0.0714286
$$197$$ 6.00000 0.427482 0.213741 0.976890i $$-0.431435\pi$$
0.213741 + 0.976890i $$0.431435\pi$$
$$198$$ 0 0
$$199$$ −8.00000 −0.567105 −0.283552 0.958957i $$-0.591513\pi$$
−0.283552 + 0.958957i $$0.591513\pi$$
$$200$$ −1.00000 −0.0707107
$$201$$ 0 0
$$202$$ 6.00000 0.422159
$$203$$ −2.00000 −0.140372
$$204$$ 0 0
$$205$$ −4.00000 −0.279372
$$206$$ −8.00000 −0.557386
$$207$$ 0 0
$$208$$ 1.00000 0.0693375
$$209$$ −16.0000 −1.10674
$$210$$ 0 0
$$211$$ 4.00000 0.275371 0.137686 0.990476i $$-0.456034\pi$$
0.137686 + 0.990476i $$0.456034\pi$$
$$212$$ −6.00000 −0.412082
$$213$$ 0 0
$$214$$ 0 0
$$215$$ 8.00000 0.545595
$$216$$ 0 0
$$217$$ 0 0
$$218$$ 14.0000 0.948200
$$219$$ 0 0
$$220$$ 8.00000 0.539360
$$221$$ 2.00000 0.134535
$$222$$ 0 0
$$223$$ −24.0000 −1.60716 −0.803579 0.595198i $$-0.797074\pi$$
−0.803579 + 0.595198i $$0.797074\pi$$
$$224$$ −1.00000 −0.0668153
$$225$$ 0 0
$$226$$ −10.0000 −0.665190
$$227$$ 8.00000 0.530979 0.265489 0.964114i $$-0.414466\pi$$
0.265489 + 0.964114i $$0.414466\pi$$
$$228$$ 0 0
$$229$$ −26.0000 −1.71813 −0.859064 0.511868i $$-0.828954\pi$$
−0.859064 + 0.511868i $$0.828954\pi$$
$$230$$ 8.00000 0.527504
$$231$$ 0 0
$$232$$ 2.00000 0.131306
$$233$$ −26.0000 −1.70332 −0.851658 0.524097i $$-0.824403\pi$$
−0.851658 + 0.524097i $$0.824403\pi$$
$$234$$ 0 0
$$235$$ 24.0000 1.56559
$$236$$ 0 0
$$237$$ 0 0
$$238$$ −2.00000 −0.129641
$$239$$ 24.0000 1.55243 0.776215 0.630468i $$-0.217137\pi$$
0.776215 + 0.630468i $$0.217137\pi$$
$$240$$ 0 0
$$241$$ 2.00000 0.128831 0.0644157 0.997923i $$-0.479482\pi$$
0.0644157 + 0.997923i $$0.479482\pi$$
$$242$$ 5.00000 0.321412
$$243$$ 0 0
$$244$$ −10.0000 −0.640184
$$245$$ 2.00000 0.127775
$$246$$ 0 0
$$247$$ −4.00000 −0.254514
$$248$$ 0 0
$$249$$ 0 0
$$250$$ −12.0000 −0.758947
$$251$$ 4.00000 0.252478 0.126239 0.992000i $$-0.459709\pi$$
0.126239 + 0.992000i $$0.459709\pi$$
$$252$$ 0 0
$$253$$ 16.0000 1.00591
$$254$$ −8.00000 −0.501965
$$255$$ 0 0
$$256$$ 1.00000 0.0625000
$$257$$ 18.0000 1.12281 0.561405 0.827541i $$-0.310261\pi$$
0.561405 + 0.827541i $$0.310261\pi$$
$$258$$ 0 0
$$259$$ 2.00000 0.124274
$$260$$ 2.00000 0.124035
$$261$$ 0 0
$$262$$ −12.0000 −0.741362
$$263$$ 20.0000 1.23325 0.616626 0.787256i $$-0.288499\pi$$
0.616626 + 0.787256i $$0.288499\pi$$
$$264$$ 0 0
$$265$$ −12.0000 −0.737154
$$266$$ 4.00000 0.245256
$$267$$ 0 0
$$268$$ 4.00000 0.244339
$$269$$ 6.00000 0.365826 0.182913 0.983129i $$-0.441447\pi$$
0.182913 + 0.983129i $$0.441447\pi$$
$$270$$ 0 0
$$271$$ −24.0000 −1.45790 −0.728948 0.684569i $$-0.759990\pi$$
−0.728948 + 0.684569i $$0.759990\pi$$
$$272$$ 2.00000 0.121268
$$273$$ 0 0
$$274$$ −22.0000 −1.32907
$$275$$ −4.00000 −0.241209
$$276$$ 0 0
$$277$$ 22.0000 1.32185 0.660926 0.750451i $$-0.270164\pi$$
0.660926 + 0.750451i $$0.270164\pi$$
$$278$$ 4.00000 0.239904
$$279$$ 0 0
$$280$$ −2.00000 −0.119523
$$281$$ −22.0000 −1.31241 −0.656205 0.754583i $$-0.727839\pi$$
−0.656205 + 0.754583i $$0.727839\pi$$
$$282$$ 0 0
$$283$$ −28.0000 −1.66443 −0.832214 0.554455i $$-0.812927\pi$$
−0.832214 + 0.554455i $$0.812927\pi$$
$$284$$ 8.00000 0.474713
$$285$$ 0 0
$$286$$ 4.00000 0.236525
$$287$$ 2.00000 0.118056
$$288$$ 0 0
$$289$$ −13.0000 −0.764706
$$290$$ 4.00000 0.234888
$$291$$ 0 0
$$292$$ −6.00000 −0.351123
$$293$$ −14.0000 −0.817889 −0.408944 0.912559i $$-0.634103\pi$$
−0.408944 + 0.912559i $$0.634103\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ −2.00000 −0.116248
$$297$$ 0 0
$$298$$ −18.0000 −1.04271
$$299$$ 4.00000 0.231326
$$300$$ 0 0
$$301$$ −4.00000 −0.230556
$$302$$ −16.0000 −0.920697
$$303$$ 0 0
$$304$$ −4.00000 −0.229416
$$305$$ −20.0000 −1.14520
$$306$$ 0 0
$$307$$ −12.0000 −0.684876 −0.342438 0.939540i $$-0.611253\pi$$
−0.342438 + 0.939540i $$0.611253\pi$$
$$308$$ −4.00000 −0.227921
$$309$$ 0 0
$$310$$ 0 0
$$311$$ −24.0000 −1.36092 −0.680458 0.732787i $$-0.738219\pi$$
−0.680458 + 0.732787i $$0.738219\pi$$
$$312$$ 0 0
$$313$$ −6.00000 −0.339140 −0.169570 0.985518i $$-0.554238\pi$$
−0.169570 + 0.985518i $$0.554238\pi$$
$$314$$ 22.0000 1.24153
$$315$$ 0 0
$$316$$ 8.00000 0.450035
$$317$$ −2.00000 −0.112331 −0.0561656 0.998421i $$-0.517887\pi$$
−0.0561656 + 0.998421i $$0.517887\pi$$
$$318$$ 0 0
$$319$$ 8.00000 0.447914
$$320$$ 2.00000 0.111803
$$321$$ 0 0
$$322$$ −4.00000 −0.222911
$$323$$ −8.00000 −0.445132
$$324$$ 0 0
$$325$$ −1.00000 −0.0554700
$$326$$ 12.0000 0.664619
$$327$$ 0 0
$$328$$ −2.00000 −0.110432
$$329$$ −12.0000 −0.661581
$$330$$ 0 0
$$331$$ −12.0000 −0.659580 −0.329790 0.944054i $$-0.606978\pi$$
−0.329790 + 0.944054i $$0.606978\pi$$
$$332$$ −8.00000 −0.439057
$$333$$ 0 0
$$334$$ −12.0000 −0.656611
$$335$$ 8.00000 0.437087
$$336$$ 0 0
$$337$$ 18.0000 0.980522 0.490261 0.871576i $$-0.336901\pi$$
0.490261 + 0.871576i $$0.336901\pi$$
$$338$$ 1.00000 0.0543928
$$339$$ 0 0
$$340$$ 4.00000 0.216930
$$341$$ 0 0
$$342$$ 0 0
$$343$$ −1.00000 −0.0539949
$$344$$ 4.00000 0.215666
$$345$$ 0 0
$$346$$ −2.00000 −0.107521
$$347$$ −16.0000 −0.858925 −0.429463 0.903085i $$-0.641297\pi$$
−0.429463 + 0.903085i $$0.641297\pi$$
$$348$$ 0 0
$$349$$ −34.0000 −1.81998 −0.909989 0.414632i $$-0.863910\pi$$
−0.909989 + 0.414632i $$0.863910\pi$$
$$350$$ 1.00000 0.0534522
$$351$$ 0 0
$$352$$ 4.00000 0.213201
$$353$$ 14.0000 0.745145 0.372572 0.928003i $$-0.378476\pi$$
0.372572 + 0.928003i $$0.378476\pi$$
$$354$$ 0 0
$$355$$ 16.0000 0.849192
$$356$$ 6.00000 0.317999
$$357$$ 0 0
$$358$$ 24.0000 1.26844
$$359$$ −24.0000 −1.26667 −0.633336 0.773877i $$-0.718315\pi$$
−0.633336 + 0.773877i $$0.718315\pi$$
$$360$$ 0 0
$$361$$ −3.00000 −0.157895
$$362$$ −26.0000 −1.36653
$$363$$ 0 0
$$364$$ −1.00000 −0.0524142
$$365$$ −12.0000 −0.628109
$$366$$ 0 0
$$367$$ −24.0000 −1.25279 −0.626395 0.779506i $$-0.715470\pi$$
−0.626395 + 0.779506i $$0.715470\pi$$
$$368$$ 4.00000 0.208514
$$369$$ 0 0
$$370$$ −4.00000 −0.207950
$$371$$ 6.00000 0.311504
$$372$$ 0 0
$$373$$ −26.0000 −1.34623 −0.673114 0.739538i $$-0.735044\pi$$
−0.673114 + 0.739538i $$0.735044\pi$$
$$374$$ 8.00000 0.413670
$$375$$ 0 0
$$376$$ 12.0000 0.618853
$$377$$ 2.00000 0.103005
$$378$$ 0 0
$$379$$ 28.0000 1.43826 0.719132 0.694874i $$-0.244540\pi$$
0.719132 + 0.694874i $$0.244540\pi$$
$$380$$ −8.00000 −0.410391
$$381$$ 0 0
$$382$$ 12.0000 0.613973
$$383$$ 12.0000 0.613171 0.306586 0.951843i $$-0.400813\pi$$
0.306586 + 0.951843i $$0.400813\pi$$
$$384$$ 0 0
$$385$$ −8.00000 −0.407718
$$386$$ 2.00000 0.101797
$$387$$ 0 0
$$388$$ 2.00000 0.101535
$$389$$ −14.0000 −0.709828 −0.354914 0.934899i $$-0.615490\pi$$
−0.354914 + 0.934899i $$0.615490\pi$$
$$390$$ 0 0
$$391$$ 8.00000 0.404577
$$392$$ 1.00000 0.0505076
$$393$$ 0 0
$$394$$ 6.00000 0.302276
$$395$$ 16.0000 0.805047
$$396$$ 0 0
$$397$$ 14.0000 0.702640 0.351320 0.936255i $$-0.385733\pi$$
0.351320 + 0.936255i $$0.385733\pi$$
$$398$$ −8.00000 −0.401004
$$399$$ 0 0
$$400$$ −1.00000 −0.0500000
$$401$$ 10.0000 0.499376 0.249688 0.968326i $$-0.419672\pi$$
0.249688 + 0.968326i $$0.419672\pi$$
$$402$$ 0 0
$$403$$ 0 0
$$404$$ 6.00000 0.298511
$$405$$ 0 0
$$406$$ −2.00000 −0.0992583
$$407$$ −8.00000 −0.396545
$$408$$ 0 0
$$409$$ 26.0000 1.28562 0.642809 0.766027i $$-0.277769\pi$$
0.642809 + 0.766027i $$0.277769\pi$$
$$410$$ −4.00000 −0.197546
$$411$$ 0 0
$$412$$ −8.00000 −0.394132
$$413$$ 0 0
$$414$$ 0 0
$$415$$ −16.0000 −0.785409
$$416$$ 1.00000 0.0490290
$$417$$ 0 0
$$418$$ −16.0000 −0.782586
$$419$$ −20.0000 −0.977064 −0.488532 0.872546i $$-0.662467\pi$$
−0.488532 + 0.872546i $$0.662467\pi$$
$$420$$ 0 0
$$421$$ −26.0000 −1.26716 −0.633581 0.773676i $$-0.718416\pi$$
−0.633581 + 0.773676i $$0.718416\pi$$
$$422$$ 4.00000 0.194717
$$423$$ 0 0
$$424$$ −6.00000 −0.291386
$$425$$ −2.00000 −0.0970143
$$426$$ 0 0
$$427$$ 10.0000 0.483934
$$428$$ 0 0
$$429$$ 0 0
$$430$$ 8.00000 0.385794
$$431$$ 40.0000 1.92673 0.963366 0.268190i $$-0.0864254\pi$$
0.963366 + 0.268190i $$0.0864254\pi$$
$$432$$ 0 0
$$433$$ −30.0000 −1.44171 −0.720854 0.693087i $$-0.756250\pi$$
−0.720854 + 0.693087i $$0.756250\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 14.0000 0.670478
$$437$$ −16.0000 −0.765384
$$438$$ 0 0
$$439$$ −32.0000 −1.52728 −0.763638 0.645644i $$-0.776589\pi$$
−0.763638 + 0.645644i $$0.776589\pi$$
$$440$$ 8.00000 0.381385
$$441$$ 0 0
$$442$$ 2.00000 0.0951303
$$443$$ 24.0000 1.14027 0.570137 0.821549i $$-0.306890\pi$$
0.570137 + 0.821549i $$0.306890\pi$$
$$444$$ 0 0
$$445$$ 12.0000 0.568855
$$446$$ −24.0000 −1.13643
$$447$$ 0 0
$$448$$ −1.00000 −0.0472456
$$449$$ 18.0000 0.849473 0.424736 0.905317i $$-0.360367\pi$$
0.424736 + 0.905317i $$0.360367\pi$$
$$450$$ 0 0
$$451$$ −8.00000 −0.376705
$$452$$ −10.0000 −0.470360
$$453$$ 0 0
$$454$$ 8.00000 0.375459
$$455$$ −2.00000 −0.0937614
$$456$$ 0 0
$$457$$ 18.0000 0.842004 0.421002 0.907060i $$-0.361678\pi$$
0.421002 + 0.907060i $$0.361678\pi$$
$$458$$ −26.0000 −1.21490
$$459$$ 0 0
$$460$$ 8.00000 0.373002
$$461$$ 2.00000 0.0931493 0.0465746 0.998915i $$-0.485169\pi$$
0.0465746 + 0.998915i $$0.485169\pi$$
$$462$$ 0 0
$$463$$ 24.0000 1.11537 0.557687 0.830051i $$-0.311689\pi$$
0.557687 + 0.830051i $$0.311689\pi$$
$$464$$ 2.00000 0.0928477
$$465$$ 0 0
$$466$$ −26.0000 −1.20443
$$467$$ −4.00000 −0.185098 −0.0925490 0.995708i $$-0.529501\pi$$
−0.0925490 + 0.995708i $$0.529501\pi$$
$$468$$ 0 0
$$469$$ −4.00000 −0.184703
$$470$$ 24.0000 1.10704
$$471$$ 0 0
$$472$$ 0 0
$$473$$ 16.0000 0.735681
$$474$$ 0 0
$$475$$ 4.00000 0.183533
$$476$$ −2.00000 −0.0916698
$$477$$ 0 0
$$478$$ 24.0000 1.09773
$$479$$ −12.0000 −0.548294 −0.274147 0.961688i $$-0.588395\pi$$
−0.274147 + 0.961688i $$0.588395\pi$$
$$480$$ 0 0
$$481$$ −2.00000 −0.0911922
$$482$$ 2.00000 0.0910975
$$483$$ 0 0
$$484$$ 5.00000 0.227273
$$485$$ 4.00000 0.181631
$$486$$ 0 0
$$487$$ 32.0000 1.45006 0.725029 0.688718i $$-0.241826\pi$$
0.725029 + 0.688718i $$0.241826\pi$$
$$488$$ −10.0000 −0.452679
$$489$$ 0 0
$$490$$ 2.00000 0.0903508
$$491$$ 8.00000 0.361035 0.180517 0.983572i $$-0.442223\pi$$
0.180517 + 0.983572i $$0.442223\pi$$
$$492$$ 0 0
$$493$$ 4.00000 0.180151
$$494$$ −4.00000 −0.179969
$$495$$ 0 0
$$496$$ 0 0
$$497$$ −8.00000 −0.358849
$$498$$ 0 0
$$499$$ −20.0000 −0.895323 −0.447661 0.894203i $$-0.647743\pi$$
−0.447661 + 0.894203i $$0.647743\pi$$
$$500$$ −12.0000 −0.536656
$$501$$ 0 0
$$502$$ 4.00000 0.178529
$$503$$ 16.0000 0.713405 0.356702 0.934218i $$-0.383901\pi$$
0.356702 + 0.934218i $$0.383901\pi$$
$$504$$ 0 0
$$505$$ 12.0000 0.533993
$$506$$ 16.0000 0.711287
$$507$$ 0 0
$$508$$ −8.00000 −0.354943
$$509$$ −6.00000 −0.265945 −0.132973 0.991120i $$-0.542452\pi$$
−0.132973 + 0.991120i $$0.542452\pi$$
$$510$$ 0 0
$$511$$ 6.00000 0.265424
$$512$$ 1.00000 0.0441942
$$513$$ 0 0
$$514$$ 18.0000 0.793946
$$515$$ −16.0000 −0.705044
$$516$$ 0 0
$$517$$ 48.0000 2.11104
$$518$$ 2.00000 0.0878750
$$519$$ 0 0
$$520$$ 2.00000 0.0877058
$$521$$ 18.0000 0.788594 0.394297 0.918983i $$-0.370988\pi$$
0.394297 + 0.918983i $$0.370988\pi$$
$$522$$ 0 0
$$523$$ 36.0000 1.57417 0.787085 0.616844i $$-0.211589\pi$$
0.787085 + 0.616844i $$0.211589\pi$$
$$524$$ −12.0000 −0.524222
$$525$$ 0 0
$$526$$ 20.0000 0.872041
$$527$$ 0 0
$$528$$ 0 0
$$529$$ −7.00000 −0.304348
$$530$$ −12.0000 −0.521247
$$531$$ 0 0
$$532$$ 4.00000 0.173422
$$533$$ −2.00000 −0.0866296
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 4.00000 0.172774
$$537$$ 0 0
$$538$$ 6.00000 0.258678
$$539$$ 4.00000 0.172292
$$540$$ 0 0
$$541$$ −2.00000 −0.0859867 −0.0429934 0.999075i $$-0.513689\pi$$
−0.0429934 + 0.999075i $$0.513689\pi$$
$$542$$ −24.0000 −1.03089
$$543$$ 0 0
$$544$$ 2.00000 0.0857493
$$545$$ 28.0000 1.19939
$$546$$ 0 0
$$547$$ 20.0000 0.855138 0.427569 0.903983i $$-0.359370\pi$$
0.427569 + 0.903983i $$0.359370\pi$$
$$548$$ −22.0000 −0.939793
$$549$$ 0 0
$$550$$ −4.00000 −0.170561
$$551$$ −8.00000 −0.340811
$$552$$ 0 0
$$553$$ −8.00000 −0.340195
$$554$$ 22.0000 0.934690
$$555$$ 0 0
$$556$$ 4.00000 0.169638
$$557$$ 38.0000 1.61011 0.805056 0.593199i $$-0.202135\pi$$
0.805056 + 0.593199i $$0.202135\pi$$
$$558$$ 0 0
$$559$$ 4.00000 0.169182
$$560$$ −2.00000 −0.0845154
$$561$$ 0 0
$$562$$ −22.0000 −0.928014
$$563$$ 4.00000 0.168580 0.0842900 0.996441i $$-0.473138\pi$$
0.0842900 + 0.996441i $$0.473138\pi$$
$$564$$ 0 0
$$565$$ −20.0000 −0.841406
$$566$$ −28.0000 −1.17693
$$567$$ 0 0
$$568$$ 8.00000 0.335673
$$569$$ −26.0000 −1.08998 −0.544988 0.838444i $$-0.683466\pi$$
−0.544988 + 0.838444i $$0.683466\pi$$
$$570$$ 0 0
$$571$$ −28.0000 −1.17176 −0.585882 0.810397i $$-0.699252\pi$$
−0.585882 + 0.810397i $$0.699252\pi$$
$$572$$ 4.00000 0.167248
$$573$$ 0 0
$$574$$ 2.00000 0.0834784
$$575$$ −4.00000 −0.166812
$$576$$ 0 0
$$577$$ 18.0000 0.749350 0.374675 0.927156i $$-0.377754\pi$$
0.374675 + 0.927156i $$0.377754\pi$$
$$578$$ −13.0000 −0.540729
$$579$$ 0 0
$$580$$ 4.00000 0.166091
$$581$$ 8.00000 0.331896
$$582$$ 0 0
$$583$$ −24.0000 −0.993978
$$584$$ −6.00000 −0.248282
$$585$$ 0 0
$$586$$ −14.0000 −0.578335
$$587$$ 24.0000 0.990586 0.495293 0.868726i $$-0.335061\pi$$
0.495293 + 0.868726i $$0.335061\pi$$
$$588$$ 0 0
$$589$$ 0 0
$$590$$ 0 0
$$591$$ 0 0
$$592$$ −2.00000 −0.0821995
$$593$$ 22.0000 0.903432 0.451716 0.892162i $$-0.350812\pi$$
0.451716 + 0.892162i $$0.350812\pi$$
$$594$$ 0 0
$$595$$ −4.00000 −0.163984
$$596$$ −18.0000 −0.737309
$$597$$ 0 0
$$598$$ 4.00000 0.163572
$$599$$ 12.0000 0.490307 0.245153 0.969484i $$-0.421162\pi$$
0.245153 + 0.969484i $$0.421162\pi$$
$$600$$ 0 0
$$601$$ −22.0000 −0.897399 −0.448699 0.893683i $$-0.648113\pi$$
−0.448699 + 0.893683i $$0.648113\pi$$
$$602$$ −4.00000 −0.163028
$$603$$ 0 0
$$604$$ −16.0000 −0.651031
$$605$$ 10.0000 0.406558
$$606$$ 0 0
$$607$$ 8.00000 0.324710 0.162355 0.986732i $$-0.448091\pi$$
0.162355 + 0.986732i $$0.448091\pi$$
$$608$$ −4.00000 −0.162221
$$609$$ 0 0
$$610$$ −20.0000 −0.809776
$$611$$ 12.0000 0.485468
$$612$$ 0 0
$$613$$ 14.0000 0.565455 0.282727 0.959200i $$-0.408761\pi$$
0.282727 + 0.959200i $$0.408761\pi$$
$$614$$ −12.0000 −0.484281
$$615$$ 0 0
$$616$$ −4.00000 −0.161165
$$617$$ −22.0000 −0.885687 −0.442843 0.896599i $$-0.646030\pi$$
−0.442843 + 0.896599i $$0.646030\pi$$
$$618$$ 0 0
$$619$$ −28.0000 −1.12542 −0.562708 0.826656i $$-0.690240\pi$$
−0.562708 + 0.826656i $$0.690240\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ −24.0000 −0.962312
$$623$$ −6.00000 −0.240385
$$624$$ 0 0
$$625$$ −19.0000 −0.760000
$$626$$ −6.00000 −0.239808
$$627$$ 0 0
$$628$$ 22.0000 0.877896
$$629$$ −4.00000 −0.159490
$$630$$ 0 0
$$631$$ 40.0000 1.59237 0.796187 0.605050i $$-0.206847\pi$$
0.796187 + 0.605050i $$0.206847\pi$$
$$632$$ 8.00000 0.318223
$$633$$ 0 0
$$634$$ −2.00000 −0.0794301
$$635$$ −16.0000 −0.634941
$$636$$ 0 0
$$637$$ 1.00000 0.0396214
$$638$$ 8.00000 0.316723
$$639$$ 0 0
$$640$$ 2.00000 0.0790569
$$641$$ 22.0000 0.868948 0.434474 0.900684i $$-0.356934\pi$$
0.434474 + 0.900684i $$0.356934\pi$$
$$642$$ 0 0
$$643$$ 20.0000 0.788723 0.394362 0.918955i $$-0.370966\pi$$
0.394362 + 0.918955i $$0.370966\pi$$
$$644$$ −4.00000 −0.157622
$$645$$ 0 0
$$646$$ −8.00000 −0.314756
$$647$$ 48.0000 1.88707 0.943537 0.331266i $$-0.107476\pi$$
0.943537 + 0.331266i $$0.107476\pi$$
$$648$$ 0 0
$$649$$ 0 0
$$650$$ −1.00000 −0.0392232
$$651$$ 0 0
$$652$$ 12.0000 0.469956
$$653$$ −30.0000 −1.17399 −0.586995 0.809590i $$-0.699689\pi$$
−0.586995 + 0.809590i $$0.699689\pi$$
$$654$$ 0 0
$$655$$ −24.0000 −0.937758
$$656$$ −2.00000 −0.0780869
$$657$$ 0 0
$$658$$ −12.0000 −0.467809
$$659$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$660$$ 0 0
$$661$$ 22.0000 0.855701 0.427850 0.903850i $$-0.359271\pi$$
0.427850 + 0.903850i $$0.359271\pi$$
$$662$$ −12.0000 −0.466393
$$663$$ 0 0
$$664$$ −8.00000 −0.310460
$$665$$ 8.00000 0.310227
$$666$$ 0 0
$$667$$ 8.00000 0.309761
$$668$$ −12.0000 −0.464294
$$669$$ 0 0
$$670$$ 8.00000 0.309067
$$671$$ −40.0000 −1.54418
$$672$$ 0 0
$$673$$ 18.0000 0.693849 0.346925 0.937893i $$-0.387226\pi$$
0.346925 + 0.937893i $$0.387226\pi$$
$$674$$ 18.0000 0.693334
$$675$$ 0 0
$$676$$ 1.00000 0.0384615
$$677$$ −34.0000 −1.30673 −0.653363 0.757045i $$-0.726642\pi$$
−0.653363 + 0.757045i $$0.726642\pi$$
$$678$$ 0 0
$$679$$ −2.00000 −0.0767530
$$680$$ 4.00000 0.153393
$$681$$ 0 0
$$682$$ 0 0
$$683$$ 44.0000 1.68361 0.841807 0.539779i $$-0.181492\pi$$
0.841807 + 0.539779i $$0.181492\pi$$
$$684$$ 0 0
$$685$$ −44.0000 −1.68115
$$686$$ −1.00000 −0.0381802
$$687$$ 0 0
$$688$$ 4.00000 0.152499
$$689$$ −6.00000 −0.228582
$$690$$ 0 0
$$691$$ 28.0000 1.06517 0.532585 0.846376i $$-0.321221\pi$$
0.532585 + 0.846376i $$0.321221\pi$$
$$692$$ −2.00000 −0.0760286
$$693$$ 0 0
$$694$$ −16.0000 −0.607352
$$695$$ 8.00000 0.303457
$$696$$ 0 0
$$697$$ −4.00000 −0.151511
$$698$$ −34.0000 −1.28692
$$699$$ 0 0
$$700$$ 1.00000 0.0377964
$$701$$ 42.0000 1.58632 0.793159 0.609015i $$-0.208435\pi$$
0.793159 + 0.609015i $$0.208435\pi$$
$$702$$ 0 0
$$703$$ 8.00000 0.301726
$$704$$ 4.00000 0.150756
$$705$$ 0 0
$$706$$ 14.0000 0.526897
$$707$$ −6.00000 −0.225653
$$708$$ 0 0
$$709$$ 6.00000 0.225335 0.112667 0.993633i $$-0.464061\pi$$
0.112667 + 0.993633i $$0.464061\pi$$
$$710$$ 16.0000 0.600469
$$711$$ 0 0
$$712$$ 6.00000 0.224860
$$713$$ 0 0
$$714$$ 0 0
$$715$$ 8.00000 0.299183
$$716$$ 24.0000 0.896922
$$717$$ 0 0
$$718$$ −24.0000 −0.895672
$$719$$ −8.00000 −0.298350 −0.149175 0.988811i $$-0.547662\pi$$
−0.149175 + 0.988811i $$0.547662\pi$$
$$720$$ 0 0
$$721$$ 8.00000 0.297936
$$722$$ −3.00000 −0.111648
$$723$$ 0 0
$$724$$ −26.0000 −0.966282
$$725$$ −2.00000 −0.0742781
$$726$$ 0 0
$$727$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$728$$ −1.00000 −0.0370625
$$729$$ 0 0
$$730$$ −12.0000 −0.444140
$$731$$ 8.00000 0.295891
$$732$$ 0 0
$$733$$ 30.0000 1.10808 0.554038 0.832492i $$-0.313086\pi$$
0.554038 + 0.832492i $$0.313086\pi$$
$$734$$ −24.0000 −0.885856
$$735$$ 0 0
$$736$$ 4.00000 0.147442
$$737$$ 16.0000 0.589368
$$738$$ 0 0
$$739$$ 4.00000 0.147142 0.0735712 0.997290i $$-0.476560\pi$$
0.0735712 + 0.997290i $$0.476560\pi$$
$$740$$ −4.00000 −0.147043
$$741$$ 0 0
$$742$$ 6.00000 0.220267
$$743$$ −40.0000 −1.46746 −0.733729 0.679442i $$-0.762222\pi$$
−0.733729 + 0.679442i $$0.762222\pi$$
$$744$$ 0 0
$$745$$ −36.0000 −1.31894
$$746$$ −26.0000 −0.951928
$$747$$ 0 0
$$748$$ 8.00000 0.292509
$$749$$ 0 0
$$750$$ 0 0
$$751$$ −8.00000 −0.291924 −0.145962 0.989290i $$-0.546628\pi$$
−0.145962 + 0.989290i $$0.546628\pi$$
$$752$$ 12.0000 0.437595
$$753$$ 0 0
$$754$$ 2.00000 0.0728357
$$755$$ −32.0000 −1.16460
$$756$$ 0 0
$$757$$ 22.0000 0.799604 0.399802 0.916602i $$-0.369079\pi$$
0.399802 + 0.916602i $$0.369079\pi$$
$$758$$ 28.0000 1.01701
$$759$$ 0 0
$$760$$ −8.00000 −0.290191
$$761$$ 14.0000 0.507500 0.253750 0.967270i $$-0.418336\pi$$
0.253750 + 0.967270i $$0.418336\pi$$
$$762$$ 0 0
$$763$$ −14.0000 −0.506834
$$764$$ 12.0000 0.434145
$$765$$ 0 0
$$766$$ 12.0000 0.433578
$$767$$ 0 0
$$768$$ 0 0
$$769$$ 34.0000 1.22607 0.613036 0.790055i $$-0.289948\pi$$
0.613036 + 0.790055i $$0.289948\pi$$
$$770$$ −8.00000 −0.288300
$$771$$ 0 0
$$772$$ 2.00000 0.0719816
$$773$$ 34.0000 1.22290 0.611448 0.791285i $$-0.290588\pi$$
0.611448 + 0.791285i $$0.290588\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ 2.00000 0.0717958
$$777$$ 0 0
$$778$$ −14.0000 −0.501924
$$779$$ 8.00000 0.286630
$$780$$ 0 0
$$781$$ 32.0000 1.14505
$$782$$ 8.00000 0.286079
$$783$$ 0 0
$$784$$ 1.00000 0.0357143
$$785$$ 44.0000 1.57043
$$786$$ 0 0
$$787$$ 44.0000 1.56843 0.784215 0.620489i $$-0.213066\pi$$
0.784215 + 0.620489i $$0.213066\pi$$
$$788$$ 6.00000 0.213741
$$789$$ 0 0
$$790$$ 16.0000 0.569254
$$791$$ 10.0000 0.355559
$$792$$ 0 0
$$793$$ −10.0000 −0.355110
$$794$$ 14.0000 0.496841
$$795$$ 0 0
$$796$$ −8.00000 −0.283552
$$797$$ −42.0000 −1.48772 −0.743858 0.668338i $$-0.767006\pi$$
−0.743858 + 0.668338i $$0.767006\pi$$
$$798$$ 0 0
$$799$$ 24.0000 0.849059
$$800$$ −1.00000 −0.0353553
$$801$$ 0 0
$$802$$ 10.0000 0.353112
$$803$$ −24.0000 −0.846942
$$804$$ 0 0
$$805$$ −8.00000 −0.281963
$$806$$ 0 0
$$807$$ 0 0
$$808$$ 6.00000 0.211079
$$809$$ 30.0000 1.05474 0.527372 0.849635i $$-0.323177\pi$$
0.527372 + 0.849635i $$0.323177\pi$$
$$810$$ 0 0
$$811$$ −4.00000 −0.140459 −0.0702295 0.997531i $$-0.522373\pi$$
−0.0702295 + 0.997531i $$0.522373\pi$$
$$812$$ −2.00000 −0.0701862
$$813$$ 0 0
$$814$$ −8.00000 −0.280400
$$815$$ 24.0000 0.840683
$$816$$ 0 0
$$817$$ −16.0000 −0.559769
$$818$$ 26.0000 0.909069
$$819$$ 0 0
$$820$$ −4.00000 −0.139686
$$821$$ −34.0000 −1.18661 −0.593304 0.804978i $$-0.702177\pi$$
−0.593304 + 0.804978i $$0.702177\pi$$
$$822$$ 0 0
$$823$$ −8.00000 −0.278862 −0.139431 0.990232i $$-0.544527\pi$$
−0.139431 + 0.990232i $$0.544527\pi$$
$$824$$ −8.00000 −0.278693
$$825$$ 0 0
$$826$$ 0 0
$$827$$ −28.0000 −0.973655 −0.486828 0.873498i $$-0.661846\pi$$
−0.486828 + 0.873498i $$0.661846\pi$$
$$828$$ 0 0
$$829$$ 30.0000 1.04194 0.520972 0.853574i $$-0.325570\pi$$
0.520972 + 0.853574i $$0.325570\pi$$
$$830$$ −16.0000 −0.555368
$$831$$ 0 0
$$832$$ 1.00000 0.0346688
$$833$$ 2.00000 0.0692959
$$834$$ 0 0
$$835$$ −24.0000 −0.830554
$$836$$ −16.0000 −0.553372
$$837$$ 0 0
$$838$$ −20.0000 −0.690889
$$839$$ 12.0000 0.414286 0.207143 0.978311i $$-0.433583\pi$$
0.207143 + 0.978311i $$0.433583\pi$$
$$840$$ 0 0
$$841$$ −25.0000 −0.862069
$$842$$ −26.0000 −0.896019
$$843$$ 0 0
$$844$$ 4.00000 0.137686
$$845$$ 2.00000 0.0688021
$$846$$ 0 0
$$847$$ −5.00000 −0.171802
$$848$$ −6.00000 −0.206041
$$849$$ 0 0
$$850$$ −2.00000 −0.0685994
$$851$$ −8.00000 −0.274236
$$852$$ 0 0
$$853$$ −26.0000 −0.890223 −0.445112 0.895475i $$-0.646836\pi$$
−0.445112 + 0.895475i $$0.646836\pi$$
$$854$$ 10.0000 0.342193
$$855$$ 0 0
$$856$$ 0 0
$$857$$ −6.00000 −0.204956 −0.102478 0.994735i $$-0.532677\pi$$
−0.102478 + 0.994735i $$0.532677\pi$$
$$858$$ 0 0
$$859$$ 4.00000 0.136478 0.0682391 0.997669i $$-0.478262\pi$$
0.0682391 + 0.997669i $$0.478262\pi$$
$$860$$ 8.00000 0.272798
$$861$$ 0 0
$$862$$ 40.0000 1.36241
$$863$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$864$$ 0 0
$$865$$ −4.00000 −0.136004
$$866$$ −30.0000 −1.01944
$$867$$ 0 0
$$868$$ 0 0
$$869$$ 32.0000 1.08553
$$870$$ 0 0
$$871$$ 4.00000 0.135535
$$872$$ 14.0000 0.474100
$$873$$ 0 0
$$874$$ −16.0000 −0.541208
$$875$$ 12.0000 0.405674
$$876$$ 0 0
$$877$$ −10.0000 −0.337676 −0.168838 0.985644i $$-0.554001\pi$$
−0.168838 + 0.985644i $$0.554001\pi$$
$$878$$ −32.0000 −1.07995
$$879$$ 0 0
$$880$$ 8.00000 0.269680
$$881$$ −22.0000 −0.741199 −0.370599 0.928793i $$-0.620848\pi$$
−0.370599 + 0.928793i $$0.620848\pi$$
$$882$$ 0 0
$$883$$ 20.0000 0.673054 0.336527 0.941674i $$-0.390748\pi$$
0.336527 + 0.941674i $$0.390748\pi$$
$$884$$ 2.00000 0.0672673
$$885$$ 0 0
$$886$$ 24.0000 0.806296
$$887$$ 16.0000 0.537227 0.268614 0.963248i $$-0.413434\pi$$
0.268614 + 0.963248i $$0.413434\pi$$
$$888$$ 0 0
$$889$$ 8.00000 0.268311
$$890$$ 12.0000 0.402241
$$891$$ 0 0
$$892$$ −24.0000 −0.803579
$$893$$ −48.0000 −1.60626
$$894$$ 0 0
$$895$$ 48.0000 1.60446
$$896$$ −1.00000 −0.0334077
$$897$$ 0 0
$$898$$ 18.0000 0.600668
$$899$$ 0 0
$$900$$ 0 0
$$901$$ −12.0000 −0.399778
$$902$$ −8.00000 −0.266371
$$903$$ 0 0
$$904$$ −10.0000 −0.332595
$$905$$ −52.0000 −1.72854
$$906$$ 0 0
$$907$$ −36.0000 −1.19536 −0.597680 0.801735i $$-0.703911\pi$$
−0.597680 + 0.801735i $$0.703911\pi$$
$$908$$ 8.00000 0.265489
$$909$$ 0 0
$$910$$ −2.00000 −0.0662994
$$911$$ 12.0000 0.397578 0.198789 0.980042i $$-0.436299\pi$$
0.198789 + 0.980042i $$0.436299\pi$$
$$912$$ 0 0
$$913$$ −32.0000 −1.05905
$$914$$ 18.0000 0.595387
$$915$$ 0 0
$$916$$ −26.0000 −0.859064
$$917$$ 12.0000 0.396275
$$918$$ 0 0
$$919$$ −40.0000 −1.31948 −0.659739 0.751495i $$-0.729333\pi$$
−0.659739 + 0.751495i $$0.729333\pi$$
$$920$$ 8.00000 0.263752
$$921$$ 0 0
$$922$$ 2.00000 0.0658665
$$923$$ 8.00000 0.263323
$$924$$ 0 0
$$925$$ 2.00000 0.0657596
$$926$$ 24.0000 0.788689
$$927$$ 0 0
$$928$$ 2.00000 0.0656532
$$929$$ 6.00000 0.196854 0.0984268 0.995144i $$-0.468619\pi$$
0.0984268 + 0.995144i $$0.468619\pi$$
$$930$$ 0 0
$$931$$ −4.00000 −0.131095
$$932$$ −26.0000 −0.851658
$$933$$ 0 0
$$934$$ −4.00000 −0.130884
$$935$$ 16.0000 0.523256
$$936$$ 0 0
$$937$$ 42.0000 1.37208 0.686040 0.727564i $$-0.259347\pi$$
0.686040 + 0.727564i $$0.259347\pi$$
$$938$$ −4.00000 −0.130605
$$939$$ 0 0
$$940$$ 24.0000 0.782794
$$941$$ 34.0000 1.10837 0.554184 0.832394i $$-0.313030\pi$$
0.554184 + 0.832394i $$0.313030\pi$$
$$942$$ 0 0
$$943$$ −8.00000 −0.260516
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 16.0000 0.520205
$$947$$ 20.0000 0.649913 0.324956 0.945729i $$-0.394650\pi$$
0.324956 + 0.945729i $$0.394650\pi$$
$$948$$ 0 0
$$949$$ −6.00000 −0.194768
$$950$$ 4.00000 0.129777
$$951$$ 0 0
$$952$$ −2.00000 −0.0648204
$$953$$ 46.0000 1.49009 0.745043 0.667016i $$-0.232429\pi$$
0.745043 + 0.667016i $$0.232429\pi$$
$$954$$ 0 0
$$955$$ 24.0000 0.776622
$$956$$ 24.0000 0.776215
$$957$$ 0 0
$$958$$ −12.0000 −0.387702
$$959$$ 22.0000 0.710417
$$960$$ 0 0
$$961$$ −31.0000 −1.00000
$$962$$ −2.00000 −0.0644826
$$963$$ 0 0
$$964$$ 2.00000 0.0644157
$$965$$ 4.00000 0.128765
$$966$$ 0 0
$$967$$ −24.0000 −0.771788 −0.385894 0.922543i $$-0.626107\pi$$
−0.385894 + 0.922543i $$0.626107\pi$$
$$968$$ 5.00000 0.160706
$$969$$ 0 0
$$970$$ 4.00000 0.128432
$$971$$ −12.0000 −0.385098 −0.192549 0.981287i $$-0.561675\pi$$
−0.192549 + 0.981287i $$0.561675\pi$$
$$972$$ 0 0
$$973$$ −4.00000 −0.128234
$$974$$ 32.0000 1.02535
$$975$$ 0 0
$$976$$ −10.0000 −0.320092
$$977$$ 42.0000 1.34370 0.671850 0.740688i $$-0.265500\pi$$
0.671850 + 0.740688i $$0.265500\pi$$
$$978$$ 0 0
$$979$$ 24.0000 0.767043
$$980$$ 2.00000 0.0638877
$$981$$ 0 0
$$982$$ 8.00000 0.255290
$$983$$ 4.00000 0.127580 0.0637901 0.997963i $$-0.479681\pi$$
0.0637901 + 0.997963i $$0.479681\pi$$
$$984$$ 0 0
$$985$$ 12.0000 0.382352
$$986$$ 4.00000 0.127386
$$987$$ 0 0
$$988$$ −4.00000 −0.127257
$$989$$ 16.0000 0.508770
$$990$$ 0 0
$$991$$ 24.0000 0.762385 0.381193 0.924496i $$-0.375513\pi$$
0.381193 + 0.924496i $$0.375513\pi$$
$$992$$ 0 0
$$993$$ 0 0
$$994$$ −8.00000 −0.253745
$$995$$ −16.0000 −0.507234
$$996$$ 0 0
$$997$$ 46.0000 1.45683 0.728417 0.685134i $$-0.240256\pi$$
0.728417 + 0.685134i $$0.240256\pi$$
$$998$$ −20.0000 −0.633089
$$999$$ 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 1638.2.a.s.1.1 1
3.2 odd 2 546.2.a.b.1.1 1
12.11 even 2 4368.2.a.d.1.1 1
21.20 even 2 3822.2.a.h.1.1 1
39.38 odd 2 7098.2.a.be.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
546.2.a.b.1.1 1 3.2 odd 2
1638.2.a.s.1.1 1 1.1 even 1 trivial
3822.2.a.h.1.1 1 21.20 even 2
4368.2.a.d.1.1 1 12.11 even 2
7098.2.a.be.1.1 1 39.38 odd 2