# Properties

 Label 1305.2.a.g.1.1 Level $1305$ Weight $2$ Character 1305.1 Self dual yes Analytic conductor $10.420$ Analytic rank $1$ Dimension $1$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1305,2,Mod(1,1305)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1305, 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("1305.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1305 = 3^{2} \cdot 5 \cdot 29$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1305.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: $$10.4204774638$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$+1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 1305.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+2.00000 q^{2} +2.00000 q^{4} -1.00000 q^{5} -2.00000 q^{7} +O(q^{10})$$ $$q+2.00000 q^{2} +2.00000 q^{4} -1.00000 q^{5} -2.00000 q^{7} -2.00000 q^{10} -3.00000 q^{11} -4.00000 q^{13} -4.00000 q^{14} -4.00000 q^{16} -2.00000 q^{17} -2.00000 q^{19} -2.00000 q^{20} -6.00000 q^{22} +5.00000 q^{23} +1.00000 q^{25} -8.00000 q^{26} -4.00000 q^{28} -1.00000 q^{29} +2.00000 q^{31} -8.00000 q^{32} -4.00000 q^{34} +2.00000 q^{35} -5.00000 q^{37} -4.00000 q^{38} -1.00000 q^{41} -1.00000 q^{43} -6.00000 q^{44} +10.0000 q^{46} +6.00000 q^{47} -3.00000 q^{49} +2.00000 q^{50} -8.00000 q^{52} +3.00000 q^{53} +3.00000 q^{55} -2.00000 q^{58} +4.00000 q^{59} +6.00000 q^{61} +4.00000 q^{62} -8.00000 q^{64} +4.00000 q^{65} +2.00000 q^{67} -4.00000 q^{68} +4.00000 q^{70} -12.0000 q^{71} +9.00000 q^{73} -10.0000 q^{74} -4.00000 q^{76} +6.00000 q^{77} -16.0000 q^{79} +4.00000 q^{80} -2.00000 q^{82} +7.00000 q^{83} +2.00000 q^{85} -2.00000 q^{86} -6.00000 q^{89} +8.00000 q^{91} +10.0000 q^{92} +12.0000 q^{94} +2.00000 q^{95} -13.0000 q^{97} -6.00000 q^{98} +O(q^{100})$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 2.00000 1.41421 0.707107 0.707107i $$-0.250000\pi$$
0.707107 + 0.707107i $$0.250000\pi$$
$$3$$ 0 0
$$4$$ 2.00000 1.00000
$$5$$ −1.00000 −0.447214
$$6$$ 0 0
$$7$$ −2.00000 −0.755929 −0.377964 0.925820i $$-0.623376\pi$$
−0.377964 + 0.925820i $$0.623376\pi$$
$$8$$ 0 0
$$9$$ 0 0
$$10$$ −2.00000 −0.632456
$$11$$ −3.00000 −0.904534 −0.452267 0.891883i $$-0.649385\pi$$
−0.452267 + 0.891883i $$0.649385\pi$$
$$12$$ 0 0
$$13$$ −4.00000 −1.10940 −0.554700 0.832050i $$-0.687167\pi$$
−0.554700 + 0.832050i $$0.687167\pi$$
$$14$$ −4.00000 −1.06904
$$15$$ 0 0
$$16$$ −4.00000 −1.00000
$$17$$ −2.00000 −0.485071 −0.242536 0.970143i $$-0.577979\pi$$
−0.242536 + 0.970143i $$0.577979\pi$$
$$18$$ 0 0
$$19$$ −2.00000 −0.458831 −0.229416 0.973329i $$-0.573682\pi$$
−0.229416 + 0.973329i $$0.573682\pi$$
$$20$$ −2.00000 −0.447214
$$21$$ 0 0
$$22$$ −6.00000 −1.27920
$$23$$ 5.00000 1.04257 0.521286 0.853382i $$-0.325452\pi$$
0.521286 + 0.853382i $$0.325452\pi$$
$$24$$ 0 0
$$25$$ 1.00000 0.200000
$$26$$ −8.00000 −1.56893
$$27$$ 0 0
$$28$$ −4.00000 −0.755929
$$29$$ −1.00000 −0.185695
$$30$$ 0 0
$$31$$ 2.00000 0.359211 0.179605 0.983739i $$-0.442518\pi$$
0.179605 + 0.983739i $$0.442518\pi$$
$$32$$ −8.00000 −1.41421
$$33$$ 0 0
$$34$$ −4.00000 −0.685994
$$35$$ 2.00000 0.338062
$$36$$ 0 0
$$37$$ −5.00000 −0.821995 −0.410997 0.911636i $$-0.634819\pi$$
−0.410997 + 0.911636i $$0.634819\pi$$
$$38$$ −4.00000 −0.648886
$$39$$ 0 0
$$40$$ 0 0
$$41$$ −1.00000 −0.156174 −0.0780869 0.996947i $$-0.524881\pi$$
−0.0780869 + 0.996947i $$0.524881\pi$$
$$42$$ 0 0
$$43$$ −1.00000 −0.152499 −0.0762493 0.997089i $$-0.524294\pi$$
−0.0762493 + 0.997089i $$0.524294\pi$$
$$44$$ −6.00000 −0.904534
$$45$$ 0 0
$$46$$ 10.0000 1.47442
$$47$$ 6.00000 0.875190 0.437595 0.899172i $$-0.355830\pi$$
0.437595 + 0.899172i $$0.355830\pi$$
$$48$$ 0 0
$$49$$ −3.00000 −0.428571
$$50$$ 2.00000 0.282843
$$51$$ 0 0
$$52$$ −8.00000 −1.10940
$$53$$ 3.00000 0.412082 0.206041 0.978543i $$-0.433942\pi$$
0.206041 + 0.978543i $$0.433942\pi$$
$$54$$ 0 0
$$55$$ 3.00000 0.404520
$$56$$ 0 0
$$57$$ 0 0
$$58$$ −2.00000 −0.262613
$$59$$ 4.00000 0.520756 0.260378 0.965507i $$-0.416153\pi$$
0.260378 + 0.965507i $$0.416153\pi$$
$$60$$ 0 0
$$61$$ 6.00000 0.768221 0.384111 0.923287i $$-0.374508\pi$$
0.384111 + 0.923287i $$0.374508\pi$$
$$62$$ 4.00000 0.508001
$$63$$ 0 0
$$64$$ −8.00000 −1.00000
$$65$$ 4.00000 0.496139
$$66$$ 0 0
$$67$$ 2.00000 0.244339 0.122169 0.992509i $$-0.461015\pi$$
0.122169 + 0.992509i $$0.461015\pi$$
$$68$$ −4.00000 −0.485071
$$69$$ 0 0
$$70$$ 4.00000 0.478091
$$71$$ −12.0000 −1.42414 −0.712069 0.702109i $$-0.752242\pi$$
−0.712069 + 0.702109i $$0.752242\pi$$
$$72$$ 0 0
$$73$$ 9.00000 1.05337 0.526685 0.850060i $$-0.323435\pi$$
0.526685 + 0.850060i $$0.323435\pi$$
$$74$$ −10.0000 −1.16248
$$75$$ 0 0
$$76$$ −4.00000 −0.458831
$$77$$ 6.00000 0.683763
$$78$$ 0 0
$$79$$ −16.0000 −1.80014 −0.900070 0.435745i $$-0.856485\pi$$
−0.900070 + 0.435745i $$0.856485\pi$$
$$80$$ 4.00000 0.447214
$$81$$ 0 0
$$82$$ −2.00000 −0.220863
$$83$$ 7.00000 0.768350 0.384175 0.923260i $$-0.374486\pi$$
0.384175 + 0.923260i $$0.374486\pi$$
$$84$$ 0 0
$$85$$ 2.00000 0.216930
$$86$$ −2.00000 −0.215666
$$87$$ 0 0
$$88$$ 0 0
$$89$$ −6.00000 −0.635999 −0.317999 0.948091i $$-0.603011\pi$$
−0.317999 + 0.948091i $$0.603011\pi$$
$$90$$ 0 0
$$91$$ 8.00000 0.838628
$$92$$ 10.0000 1.04257
$$93$$ 0 0
$$94$$ 12.0000 1.23771
$$95$$ 2.00000 0.205196
$$96$$ 0 0
$$97$$ −13.0000 −1.31995 −0.659975 0.751288i $$-0.729433\pi$$
−0.659975 + 0.751288i $$0.729433\pi$$
$$98$$ −6.00000 −0.606092
$$99$$ 0 0
$$100$$ 2.00000 0.200000
$$101$$ 1.00000 0.0995037 0.0497519 0.998762i $$-0.484157\pi$$
0.0497519 + 0.998762i $$0.484157\pi$$
$$102$$ 0 0
$$103$$ −12.0000 −1.18240 −0.591198 0.806527i $$-0.701345\pi$$
−0.591198 + 0.806527i $$0.701345\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 6.00000 0.582772
$$107$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$108$$ 0 0
$$109$$ −5.00000 −0.478913 −0.239457 0.970907i $$-0.576969\pi$$
−0.239457 + 0.970907i $$0.576969\pi$$
$$110$$ 6.00000 0.572078
$$111$$ 0 0
$$112$$ 8.00000 0.755929
$$113$$ 8.00000 0.752577 0.376288 0.926503i $$-0.377200\pi$$
0.376288 + 0.926503i $$0.377200\pi$$
$$114$$ 0 0
$$115$$ −5.00000 −0.466252
$$116$$ −2.00000 −0.185695
$$117$$ 0 0
$$118$$ 8.00000 0.736460
$$119$$ 4.00000 0.366679
$$120$$ 0 0
$$121$$ −2.00000 −0.181818
$$122$$ 12.0000 1.08643
$$123$$ 0 0
$$124$$ 4.00000 0.359211
$$125$$ −1.00000 −0.0894427
$$126$$ 0 0
$$127$$ −7.00000 −0.621150 −0.310575 0.950549i $$-0.600522\pi$$
−0.310575 + 0.950549i $$0.600522\pi$$
$$128$$ 0 0
$$129$$ 0 0
$$130$$ 8.00000 0.701646
$$131$$ 8.00000 0.698963 0.349482 0.936943i $$-0.386358\pi$$
0.349482 + 0.936943i $$0.386358\pi$$
$$132$$ 0 0
$$133$$ 4.00000 0.346844
$$134$$ 4.00000 0.345547
$$135$$ 0 0
$$136$$ 0 0
$$137$$ 12.0000 1.02523 0.512615 0.858619i $$-0.328677\pi$$
0.512615 + 0.858619i $$0.328677\pi$$
$$138$$ 0 0
$$139$$ −15.0000 −1.27228 −0.636142 0.771572i $$-0.719471\pi$$
−0.636142 + 0.771572i $$0.719471\pi$$
$$140$$ 4.00000 0.338062
$$141$$ 0 0
$$142$$ −24.0000 −2.01404
$$143$$ 12.0000 1.00349
$$144$$ 0 0
$$145$$ 1.00000 0.0830455
$$146$$ 18.0000 1.48969
$$147$$ 0 0
$$148$$ −10.0000 −0.821995
$$149$$ 20.0000 1.63846 0.819232 0.573462i $$-0.194400\pi$$
0.819232 + 0.573462i $$0.194400\pi$$
$$150$$ 0 0
$$151$$ −7.00000 −0.569652 −0.284826 0.958579i $$-0.591936\pi$$
−0.284826 + 0.958579i $$0.591936\pi$$
$$152$$ 0 0
$$153$$ 0 0
$$154$$ 12.0000 0.966988
$$155$$ −2.00000 −0.160644
$$156$$ 0 0
$$157$$ −2.00000 −0.159617 −0.0798087 0.996810i $$-0.525431\pi$$
−0.0798087 + 0.996810i $$0.525431\pi$$
$$158$$ −32.0000 −2.54578
$$159$$ 0 0
$$160$$ 8.00000 0.632456
$$161$$ −10.0000 −0.788110
$$162$$ 0 0
$$163$$ 1.00000 0.0783260 0.0391630 0.999233i $$-0.487531\pi$$
0.0391630 + 0.999233i $$0.487531\pi$$
$$164$$ −2.00000 −0.156174
$$165$$ 0 0
$$166$$ 14.0000 1.08661
$$167$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$168$$ 0 0
$$169$$ 3.00000 0.230769
$$170$$ 4.00000 0.306786
$$171$$ 0 0
$$172$$ −2.00000 −0.152499
$$173$$ 19.0000 1.44454 0.722272 0.691609i $$-0.243098\pi$$
0.722272 + 0.691609i $$0.243098\pi$$
$$174$$ 0 0
$$175$$ −2.00000 −0.151186
$$176$$ 12.0000 0.904534
$$177$$ 0 0
$$178$$ −12.0000 −0.899438
$$179$$ −24.0000 −1.79384 −0.896922 0.442189i $$-0.854202\pi$$
−0.896922 + 0.442189i $$0.854202\pi$$
$$180$$ 0 0
$$181$$ 21.0000 1.56092 0.780459 0.625207i $$-0.214986\pi$$
0.780459 + 0.625207i $$0.214986\pi$$
$$182$$ 16.0000 1.18600
$$183$$ 0 0
$$184$$ 0 0
$$185$$ 5.00000 0.367607
$$186$$ 0 0
$$187$$ 6.00000 0.438763
$$188$$ 12.0000 0.875190
$$189$$ 0 0
$$190$$ 4.00000 0.290191
$$191$$ 5.00000 0.361787 0.180894 0.983503i $$-0.442101\pi$$
0.180894 + 0.983503i $$0.442101\pi$$
$$192$$ 0 0
$$193$$ −10.0000 −0.719816 −0.359908 0.932988i $$-0.617192\pi$$
−0.359908 + 0.932988i $$0.617192\pi$$
$$194$$ −26.0000 −1.86669
$$195$$ 0 0
$$196$$ −6.00000 −0.428571
$$197$$ −1.00000 −0.0712470 −0.0356235 0.999365i $$-0.511342\pi$$
−0.0356235 + 0.999365i $$0.511342\pi$$
$$198$$ 0 0
$$199$$ −25.0000 −1.77220 −0.886102 0.463491i $$-0.846597\pi$$
−0.886102 + 0.463491i $$0.846597\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ 2.00000 0.140720
$$203$$ 2.00000 0.140372
$$204$$ 0 0
$$205$$ 1.00000 0.0698430
$$206$$ −24.0000 −1.67216
$$207$$ 0 0
$$208$$ 16.0000 1.10940
$$209$$ 6.00000 0.415029
$$210$$ 0 0
$$211$$ 8.00000 0.550743 0.275371 0.961338i $$-0.411199\pi$$
0.275371 + 0.961338i $$0.411199\pi$$
$$212$$ 6.00000 0.412082
$$213$$ 0 0
$$214$$ 0 0
$$215$$ 1.00000 0.0681994
$$216$$ 0 0
$$217$$ −4.00000 −0.271538
$$218$$ −10.0000 −0.677285
$$219$$ 0 0
$$220$$ 6.00000 0.404520
$$221$$ 8.00000 0.538138
$$222$$ 0 0
$$223$$ −8.00000 −0.535720 −0.267860 0.963458i $$-0.586316\pi$$
−0.267860 + 0.963458i $$0.586316\pi$$
$$224$$ 16.0000 1.06904
$$225$$ 0 0
$$226$$ 16.0000 1.06430
$$227$$ 13.0000 0.862840 0.431420 0.902151i $$-0.358013\pi$$
0.431420 + 0.902151i $$0.358013\pi$$
$$228$$ 0 0
$$229$$ −6.00000 −0.396491 −0.198246 0.980152i $$-0.563524\pi$$
−0.198246 + 0.980152i $$0.563524\pi$$
$$230$$ −10.0000 −0.659380
$$231$$ 0 0
$$232$$ 0 0
$$233$$ −27.0000 −1.76883 −0.884414 0.466702i $$-0.845442\pi$$
−0.884414 + 0.466702i $$0.845442\pi$$
$$234$$ 0 0
$$235$$ −6.00000 −0.391397
$$236$$ 8.00000 0.520756
$$237$$ 0 0
$$238$$ 8.00000 0.518563
$$239$$ −12.0000 −0.776215 −0.388108 0.921614i $$-0.626871\pi$$
−0.388108 + 0.921614i $$0.626871\pi$$
$$240$$ 0 0
$$241$$ 25.0000 1.61039 0.805196 0.593009i $$-0.202060\pi$$
0.805196 + 0.593009i $$0.202060\pi$$
$$242$$ −4.00000 −0.257130
$$243$$ 0 0
$$244$$ 12.0000 0.768221
$$245$$ 3.00000 0.191663
$$246$$ 0 0
$$247$$ 8.00000 0.509028
$$248$$ 0 0
$$249$$ 0 0
$$250$$ −2.00000 −0.126491
$$251$$ 24.0000 1.51487 0.757433 0.652913i $$-0.226453\pi$$
0.757433 + 0.652913i $$0.226453\pi$$
$$252$$ 0 0
$$253$$ −15.0000 −0.943042
$$254$$ −14.0000 −0.878438
$$255$$ 0 0
$$256$$ 16.0000 1.00000
$$257$$ −27.0000 −1.68421 −0.842107 0.539311i $$-0.818685\pi$$
−0.842107 + 0.539311i $$0.818685\pi$$
$$258$$ 0 0
$$259$$ 10.0000 0.621370
$$260$$ 8.00000 0.496139
$$261$$ 0 0
$$262$$ 16.0000 0.988483
$$263$$ −6.00000 −0.369976 −0.184988 0.982741i $$-0.559225\pi$$
−0.184988 + 0.982741i $$0.559225\pi$$
$$264$$ 0 0
$$265$$ −3.00000 −0.184289
$$266$$ 8.00000 0.490511
$$267$$ 0 0
$$268$$ 4.00000 0.244339
$$269$$ −10.0000 −0.609711 −0.304855 0.952399i $$-0.598608\pi$$
−0.304855 + 0.952399i $$0.598608\pi$$
$$270$$ 0 0
$$271$$ −8.00000 −0.485965 −0.242983 0.970031i $$-0.578126\pi$$
−0.242983 + 0.970031i $$0.578126\pi$$
$$272$$ 8.00000 0.485071
$$273$$ 0 0
$$274$$ 24.0000 1.44989
$$275$$ −3.00000 −0.180907
$$276$$ 0 0
$$277$$ 6.00000 0.360505 0.180253 0.983620i $$-0.442309\pi$$
0.180253 + 0.983620i $$0.442309\pi$$
$$278$$ −30.0000 −1.79928
$$279$$ 0 0
$$280$$ 0 0
$$281$$ −8.00000 −0.477240 −0.238620 0.971113i $$-0.576695\pi$$
−0.238620 + 0.971113i $$0.576695\pi$$
$$282$$ 0 0
$$283$$ 16.0000 0.951101 0.475551 0.879688i $$-0.342249\pi$$
0.475551 + 0.879688i $$0.342249\pi$$
$$284$$ −24.0000 −1.42414
$$285$$ 0 0
$$286$$ 24.0000 1.41915
$$287$$ 2.00000 0.118056
$$288$$ 0 0
$$289$$ −13.0000 −0.764706
$$290$$ 2.00000 0.117444
$$291$$ 0 0
$$292$$ 18.0000 1.05337
$$293$$ −14.0000 −0.817889 −0.408944 0.912559i $$-0.634103\pi$$
−0.408944 + 0.912559i $$0.634103\pi$$
$$294$$ 0 0
$$295$$ −4.00000 −0.232889
$$296$$ 0 0
$$297$$ 0 0
$$298$$ 40.0000 2.31714
$$299$$ −20.0000 −1.15663
$$300$$ 0 0
$$301$$ 2.00000 0.115278
$$302$$ −14.0000 −0.805609
$$303$$ 0 0
$$304$$ 8.00000 0.458831
$$305$$ −6.00000 −0.343559
$$306$$ 0 0
$$307$$ −15.0000 −0.856095 −0.428048 0.903756i $$-0.640798\pi$$
−0.428048 + 0.903756i $$0.640798\pi$$
$$308$$ 12.0000 0.683763
$$309$$ 0 0
$$310$$ −4.00000 −0.227185
$$311$$ 5.00000 0.283524 0.141762 0.989901i $$-0.454723\pi$$
0.141762 + 0.989901i $$0.454723\pi$$
$$312$$ 0 0
$$313$$ 20.0000 1.13047 0.565233 0.824931i $$-0.308786\pi$$
0.565233 + 0.824931i $$0.308786\pi$$
$$314$$ −4.00000 −0.225733
$$315$$ 0 0
$$316$$ −32.0000 −1.80014
$$317$$ 28.0000 1.57264 0.786318 0.617822i $$-0.211985\pi$$
0.786318 + 0.617822i $$0.211985\pi$$
$$318$$ 0 0
$$319$$ 3.00000 0.167968
$$320$$ 8.00000 0.447214
$$321$$ 0 0
$$322$$ −20.0000 −1.11456
$$323$$ 4.00000 0.222566
$$324$$ 0 0
$$325$$ −4.00000 −0.221880
$$326$$ 2.00000 0.110770
$$327$$ 0 0
$$328$$ 0 0
$$329$$ −12.0000 −0.661581
$$330$$ 0 0
$$331$$ −12.0000 −0.659580 −0.329790 0.944054i $$-0.606978\pi$$
−0.329790 + 0.944054i $$0.606978\pi$$
$$332$$ 14.0000 0.768350
$$333$$ 0 0
$$334$$ 0 0
$$335$$ −2.00000 −0.109272
$$336$$ 0 0
$$337$$ 14.0000 0.762629 0.381314 0.924445i $$-0.375472\pi$$
0.381314 + 0.924445i $$0.375472\pi$$
$$338$$ 6.00000 0.326357
$$339$$ 0 0
$$340$$ 4.00000 0.216930
$$341$$ −6.00000 −0.324918
$$342$$ 0 0
$$343$$ 20.0000 1.07990
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 38.0000 2.04289
$$347$$ −3.00000 −0.161048 −0.0805242 0.996753i $$-0.525659\pi$$
−0.0805242 + 0.996753i $$0.525659\pi$$
$$348$$ 0 0
$$349$$ 31.0000 1.65939 0.829696 0.558216i $$-0.188514\pi$$
0.829696 + 0.558216i $$0.188514\pi$$
$$350$$ −4.00000 −0.213809
$$351$$ 0 0
$$352$$ 24.0000 1.27920
$$353$$ −14.0000 −0.745145 −0.372572 0.928003i $$-0.621524\pi$$
−0.372572 + 0.928003i $$0.621524\pi$$
$$354$$ 0 0
$$355$$ 12.0000 0.636894
$$356$$ −12.0000 −0.635999
$$357$$ 0 0
$$358$$ −48.0000 −2.53688
$$359$$ −35.0000 −1.84723 −0.923615 0.383322i $$-0.874780\pi$$
−0.923615 + 0.383322i $$0.874780\pi$$
$$360$$ 0 0
$$361$$ −15.0000 −0.789474
$$362$$ 42.0000 2.20747
$$363$$ 0 0
$$364$$ 16.0000 0.838628
$$365$$ −9.00000 −0.471082
$$366$$ 0 0
$$367$$ −27.0000 −1.40939 −0.704694 0.709511i $$-0.748916\pi$$
−0.704694 + 0.709511i $$0.748916\pi$$
$$368$$ −20.0000 −1.04257
$$369$$ 0 0
$$370$$ 10.0000 0.519875
$$371$$ −6.00000 −0.311504
$$372$$ 0 0
$$373$$ 18.0000 0.932005 0.466002 0.884783i $$-0.345694\pi$$
0.466002 + 0.884783i $$0.345694\pi$$
$$374$$ 12.0000 0.620505
$$375$$ 0 0
$$376$$ 0 0
$$377$$ 4.00000 0.206010
$$378$$ 0 0
$$379$$ −6.00000 −0.308199 −0.154100 0.988055i $$-0.549248\pi$$
−0.154100 + 0.988055i $$0.549248\pi$$
$$380$$ 4.00000 0.205196
$$381$$ 0 0
$$382$$ 10.0000 0.511645
$$383$$ 17.0000 0.868659 0.434330 0.900754i $$-0.356985\pi$$
0.434330 + 0.900754i $$0.356985\pi$$
$$384$$ 0 0
$$385$$ −6.00000 −0.305788
$$386$$ −20.0000 −1.01797
$$387$$ 0 0
$$388$$ −26.0000 −1.31995
$$389$$ −5.00000 −0.253510 −0.126755 0.991934i $$-0.540456\pi$$
−0.126755 + 0.991934i $$0.540456\pi$$
$$390$$ 0 0
$$391$$ −10.0000 −0.505722
$$392$$ 0 0
$$393$$ 0 0
$$394$$ −2.00000 −0.100759
$$395$$ 16.0000 0.805047
$$396$$ 0 0
$$397$$ −30.0000 −1.50566 −0.752828 0.658217i $$-0.771311\pi$$
−0.752828 + 0.658217i $$0.771311\pi$$
$$398$$ −50.0000 −2.50627
$$399$$ 0 0
$$400$$ −4.00000 −0.200000
$$401$$ −8.00000 −0.399501 −0.199750 0.979847i $$-0.564013\pi$$
−0.199750 + 0.979847i $$0.564013\pi$$
$$402$$ 0 0
$$403$$ −8.00000 −0.398508
$$404$$ 2.00000 0.0995037
$$405$$ 0 0
$$406$$ 4.00000 0.198517
$$407$$ 15.0000 0.743522
$$408$$ 0 0
$$409$$ 14.0000 0.692255 0.346128 0.938187i $$-0.387496\pi$$
0.346128 + 0.938187i $$0.387496\pi$$
$$410$$ 2.00000 0.0987730
$$411$$ 0 0
$$412$$ −24.0000 −1.18240
$$413$$ −8.00000 −0.393654
$$414$$ 0 0
$$415$$ −7.00000 −0.343616
$$416$$ 32.0000 1.56893
$$417$$ 0 0
$$418$$ 12.0000 0.586939
$$419$$ 16.0000 0.781651 0.390826 0.920465i $$-0.372190\pi$$
0.390826 + 0.920465i $$0.372190\pi$$
$$420$$ 0 0
$$421$$ −4.00000 −0.194948 −0.0974740 0.995238i $$-0.531076\pi$$
−0.0974740 + 0.995238i $$0.531076\pi$$
$$422$$ 16.0000 0.778868
$$423$$ 0 0
$$424$$ 0 0
$$425$$ −2.00000 −0.0970143
$$426$$ 0 0
$$427$$ −12.0000 −0.580721
$$428$$ 0 0
$$429$$ 0 0
$$430$$ 2.00000 0.0964486
$$431$$ −12.0000 −0.578020 −0.289010 0.957326i $$-0.593326\pi$$
−0.289010 + 0.957326i $$0.593326\pi$$
$$432$$ 0 0
$$433$$ 17.0000 0.816968 0.408484 0.912766i $$-0.366058\pi$$
0.408484 + 0.912766i $$0.366058\pi$$
$$434$$ −8.00000 −0.384012
$$435$$ 0 0
$$436$$ −10.0000 −0.478913
$$437$$ −10.0000 −0.478365
$$438$$ 0 0
$$439$$ −24.0000 −1.14546 −0.572729 0.819745i $$-0.694115\pi$$
−0.572729 + 0.819745i $$0.694115\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 16.0000 0.761042
$$443$$ −12.0000 −0.570137 −0.285069 0.958507i $$-0.592016\pi$$
−0.285069 + 0.958507i $$0.592016\pi$$
$$444$$ 0 0
$$445$$ 6.00000 0.284427
$$446$$ −16.0000 −0.757622
$$447$$ 0 0
$$448$$ 16.0000 0.755929
$$449$$ 13.0000 0.613508 0.306754 0.951789i $$-0.400757\pi$$
0.306754 + 0.951789i $$0.400757\pi$$
$$450$$ 0 0
$$451$$ 3.00000 0.141264
$$452$$ 16.0000 0.752577
$$453$$ 0 0
$$454$$ 26.0000 1.22024
$$455$$ −8.00000 −0.375046
$$456$$ 0 0
$$457$$ 34.0000 1.59045 0.795226 0.606313i $$-0.207352\pi$$
0.795226 + 0.606313i $$0.207352\pi$$
$$458$$ −12.0000 −0.560723
$$459$$ 0 0
$$460$$ −10.0000 −0.466252
$$461$$ −21.0000 −0.978068 −0.489034 0.872265i $$-0.662651\pi$$
−0.489034 + 0.872265i $$0.662651\pi$$
$$462$$ 0 0
$$463$$ −20.0000 −0.929479 −0.464739 0.885448i $$-0.653852\pi$$
−0.464739 + 0.885448i $$0.653852\pi$$
$$464$$ 4.00000 0.185695
$$465$$ 0 0
$$466$$ −54.0000 −2.50150
$$467$$ −36.0000 −1.66588 −0.832941 0.553362i $$-0.813345\pi$$
−0.832941 + 0.553362i $$0.813345\pi$$
$$468$$ 0 0
$$469$$ −4.00000 −0.184703
$$470$$ −12.0000 −0.553519
$$471$$ 0 0
$$472$$ 0 0
$$473$$ 3.00000 0.137940
$$474$$ 0 0
$$475$$ −2.00000 −0.0917663
$$476$$ 8.00000 0.366679
$$477$$ 0 0
$$478$$ −24.0000 −1.09773
$$479$$ 4.00000 0.182765 0.0913823 0.995816i $$-0.470871\pi$$
0.0913823 + 0.995816i $$0.470871\pi$$
$$480$$ 0 0
$$481$$ 20.0000 0.911922
$$482$$ 50.0000 2.27744
$$483$$ 0 0
$$484$$ −4.00000 −0.181818
$$485$$ 13.0000 0.590300
$$486$$ 0 0
$$487$$ −16.0000 −0.725029 −0.362515 0.931978i $$-0.618082\pi$$
−0.362515 + 0.931978i $$0.618082\pi$$
$$488$$ 0 0
$$489$$ 0 0
$$490$$ 6.00000 0.271052
$$491$$ 28.0000 1.26362 0.631811 0.775122i $$-0.282312\pi$$
0.631811 + 0.775122i $$0.282312\pi$$
$$492$$ 0 0
$$493$$ 2.00000 0.0900755
$$494$$ 16.0000 0.719874
$$495$$ 0 0
$$496$$ −8.00000 −0.359211
$$497$$ 24.0000 1.07655
$$498$$ 0 0
$$499$$ −36.0000 −1.61158 −0.805791 0.592200i $$-0.798259\pi$$
−0.805791 + 0.592200i $$0.798259\pi$$
$$500$$ −2.00000 −0.0894427
$$501$$ 0 0
$$502$$ 48.0000 2.14234
$$503$$ −14.0000 −0.624229 −0.312115 0.950044i $$-0.601037\pi$$
−0.312115 + 0.950044i $$0.601037\pi$$
$$504$$ 0 0
$$505$$ −1.00000 −0.0444994
$$506$$ −30.0000 −1.33366
$$507$$ 0 0
$$508$$ −14.0000 −0.621150
$$509$$ −14.0000 −0.620539 −0.310270 0.950649i $$-0.600419\pi$$
−0.310270 + 0.950649i $$0.600419\pi$$
$$510$$ 0 0
$$511$$ −18.0000 −0.796273
$$512$$ 32.0000 1.41421
$$513$$ 0 0
$$514$$ −54.0000 −2.38184
$$515$$ 12.0000 0.528783
$$516$$ 0 0
$$517$$ −18.0000 −0.791639
$$518$$ 20.0000 0.878750
$$519$$ 0 0
$$520$$ 0 0
$$521$$ −18.0000 −0.788594 −0.394297 0.918983i $$-0.629012\pi$$
−0.394297 + 0.918983i $$0.629012\pi$$
$$522$$ 0 0
$$523$$ 30.0000 1.31181 0.655904 0.754844i $$-0.272288\pi$$
0.655904 + 0.754844i $$0.272288\pi$$
$$524$$ 16.0000 0.698963
$$525$$ 0 0
$$526$$ −12.0000 −0.523225
$$527$$ −4.00000 −0.174243
$$528$$ 0 0
$$529$$ 2.00000 0.0869565
$$530$$ −6.00000 −0.260623
$$531$$ 0 0
$$532$$ 8.00000 0.346844
$$533$$ 4.00000 0.173259
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 0 0
$$537$$ 0 0
$$538$$ −20.0000 −0.862261
$$539$$ 9.00000 0.387657
$$540$$ 0 0
$$541$$ −38.0000 −1.63375 −0.816874 0.576816i $$-0.804295\pi$$
−0.816874 + 0.576816i $$0.804295\pi$$
$$542$$ −16.0000 −0.687259
$$543$$ 0 0
$$544$$ 16.0000 0.685994
$$545$$ 5.00000 0.214176
$$546$$ 0 0
$$547$$ 46.0000 1.96682 0.983409 0.181402i $$-0.0580636\pi$$
0.983409 + 0.181402i $$0.0580636\pi$$
$$548$$ 24.0000 1.02523
$$549$$ 0 0
$$550$$ −6.00000 −0.255841
$$551$$ 2.00000 0.0852029
$$552$$ 0 0
$$553$$ 32.0000 1.36078
$$554$$ 12.0000 0.509831
$$555$$ 0 0
$$556$$ −30.0000 −1.27228
$$557$$ −37.0000 −1.56774 −0.783870 0.620925i $$-0.786757\pi$$
−0.783870 + 0.620925i $$0.786757\pi$$
$$558$$ 0 0
$$559$$ 4.00000 0.169182
$$560$$ −8.00000 −0.338062
$$561$$ 0 0
$$562$$ −16.0000 −0.674919
$$563$$ 2.00000 0.0842900 0.0421450 0.999112i $$-0.486581\pi$$
0.0421450 + 0.999112i $$0.486581\pi$$
$$564$$ 0 0
$$565$$ −8.00000 −0.336563
$$566$$ 32.0000 1.34506
$$567$$ 0 0
$$568$$ 0 0
$$569$$ 38.0000 1.59304 0.796521 0.604610i $$-0.206671\pi$$
0.796521 + 0.604610i $$0.206671\pi$$
$$570$$ 0 0
$$571$$ −11.0000 −0.460336 −0.230168 0.973151i $$-0.573928\pi$$
−0.230168 + 0.973151i $$0.573928\pi$$
$$572$$ 24.0000 1.00349
$$573$$ 0 0
$$574$$ 4.00000 0.166957
$$575$$ 5.00000 0.208514
$$576$$ 0 0
$$577$$ 22.0000 0.915872 0.457936 0.888985i $$-0.348589\pi$$
0.457936 + 0.888985i $$0.348589\pi$$
$$578$$ −26.0000 −1.08146
$$579$$ 0 0
$$580$$ 2.00000 0.0830455
$$581$$ −14.0000 −0.580818
$$582$$ 0 0
$$583$$ −9.00000 −0.372742
$$584$$ 0 0
$$585$$ 0 0
$$586$$ −28.0000 −1.15667
$$587$$ −12.0000 −0.495293 −0.247647 0.968850i $$-0.579657\pi$$
−0.247647 + 0.968850i $$0.579657\pi$$
$$588$$ 0 0
$$589$$ −4.00000 −0.164817
$$590$$ −8.00000 −0.329355
$$591$$ 0 0
$$592$$ 20.0000 0.821995
$$593$$ −2.00000 −0.0821302 −0.0410651 0.999156i $$-0.513075\pi$$
−0.0410651 + 0.999156i $$0.513075\pi$$
$$594$$ 0 0
$$595$$ −4.00000 −0.163984
$$596$$ 40.0000 1.63846
$$597$$ 0 0
$$598$$ −40.0000 −1.63572
$$599$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$600$$ 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
$$604$$ −14.0000 −0.569652
$$605$$ 2.00000 0.0813116
$$606$$ 0 0
$$607$$ 40.0000 1.62355 0.811775 0.583970i $$-0.198502\pi$$
0.811775 + 0.583970i $$0.198502\pi$$
$$608$$ 16.0000 0.648886
$$609$$ 0 0
$$610$$ −12.0000 −0.485866
$$611$$ −24.0000 −0.970936
$$612$$ 0 0
$$613$$ 12.0000 0.484675 0.242338 0.970192i $$-0.422086\pi$$
0.242338 + 0.970192i $$0.422086\pi$$
$$614$$ −30.0000 −1.21070
$$615$$ 0 0
$$616$$ 0 0
$$617$$ −18.0000 −0.724653 −0.362326 0.932051i $$-0.618017\pi$$
−0.362326 + 0.932051i $$0.618017\pi$$
$$618$$ 0 0
$$619$$ −16.0000 −0.643094 −0.321547 0.946894i $$-0.604203\pi$$
−0.321547 + 0.946894i $$0.604203\pi$$
$$620$$ −4.00000 −0.160644
$$621$$ 0 0
$$622$$ 10.0000 0.400963
$$623$$ 12.0000 0.480770
$$624$$ 0 0
$$625$$ 1.00000 0.0400000
$$626$$ 40.0000 1.59872
$$627$$ 0 0
$$628$$ −4.00000 −0.159617
$$629$$ 10.0000 0.398726
$$630$$ 0 0
$$631$$ −8.00000 −0.318475 −0.159237 0.987240i $$-0.550904\pi$$
−0.159237 + 0.987240i $$0.550904\pi$$
$$632$$ 0 0
$$633$$ 0 0
$$634$$ 56.0000 2.22404
$$635$$ 7.00000 0.277787
$$636$$ 0 0
$$637$$ 12.0000 0.475457
$$638$$ 6.00000 0.237542
$$639$$ 0 0
$$640$$ 0 0
$$641$$ 5.00000 0.197488 0.0987441 0.995113i $$-0.468517\pi$$
0.0987441 + 0.995113i $$0.468517\pi$$
$$642$$ 0 0
$$643$$ −8.00000 −0.315489 −0.157745 0.987480i $$-0.550422\pi$$
−0.157745 + 0.987480i $$0.550422\pi$$
$$644$$ −20.0000 −0.788110
$$645$$ 0 0
$$646$$ 8.00000 0.314756
$$647$$ 27.0000 1.06148 0.530740 0.847535i $$-0.321914\pi$$
0.530740 + 0.847535i $$0.321914\pi$$
$$648$$ 0 0
$$649$$ −12.0000 −0.471041
$$650$$ −8.00000 −0.313786
$$651$$ 0 0
$$652$$ 2.00000 0.0783260
$$653$$ 14.0000 0.547862 0.273931 0.961749i $$-0.411676\pi$$
0.273931 + 0.961749i $$0.411676\pi$$
$$654$$ 0 0
$$655$$ −8.00000 −0.312586
$$656$$ 4.00000 0.156174
$$657$$ 0 0
$$658$$ −24.0000 −0.935617
$$659$$ 1.00000 0.0389545 0.0194772 0.999810i $$-0.493800\pi$$
0.0194772 + 0.999810i $$0.493800\pi$$
$$660$$ 0 0
$$661$$ −13.0000 −0.505641 −0.252821 0.967513i $$-0.581358\pi$$
−0.252821 + 0.967513i $$0.581358\pi$$
$$662$$ −24.0000 −0.932786
$$663$$ 0 0
$$664$$ 0 0
$$665$$ −4.00000 −0.155113
$$666$$ 0 0
$$667$$ −5.00000 −0.193601
$$668$$ 0 0
$$669$$ 0 0
$$670$$ −4.00000 −0.154533
$$671$$ −18.0000 −0.694882
$$672$$ 0 0
$$673$$ 10.0000 0.385472 0.192736 0.981251i $$-0.438264\pi$$
0.192736 + 0.981251i $$0.438264\pi$$
$$674$$ 28.0000 1.07852
$$675$$ 0 0
$$676$$ 6.00000 0.230769
$$677$$ −6.00000 −0.230599 −0.115299 0.993331i $$-0.536783\pi$$
−0.115299 + 0.993331i $$0.536783\pi$$
$$678$$ 0 0
$$679$$ 26.0000 0.997788
$$680$$ 0 0
$$681$$ 0 0
$$682$$ −12.0000 −0.459504
$$683$$ 33.0000 1.26271 0.631355 0.775494i $$-0.282499\pi$$
0.631355 + 0.775494i $$0.282499\pi$$
$$684$$ 0 0
$$685$$ −12.0000 −0.458496
$$686$$ 40.0000 1.52721
$$687$$ 0 0
$$688$$ 4.00000 0.152499
$$689$$ −12.0000 −0.457164
$$690$$ 0 0
$$691$$ 28.0000 1.06517 0.532585 0.846376i $$-0.321221\pi$$
0.532585 + 0.846376i $$0.321221\pi$$
$$692$$ 38.0000 1.44454
$$693$$ 0 0
$$694$$ −6.00000 −0.227757
$$695$$ 15.0000 0.568982
$$696$$ 0 0
$$697$$ 2.00000 0.0757554
$$698$$ 62.0000 2.34673
$$699$$ 0 0
$$700$$ −4.00000 −0.151186
$$701$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$702$$ 0 0
$$703$$ 10.0000 0.377157
$$704$$ 24.0000 0.904534
$$705$$ 0 0
$$706$$ −28.0000 −1.05379
$$707$$ −2.00000 −0.0752177
$$708$$ 0 0
$$709$$ −21.0000 −0.788672 −0.394336 0.918966i $$-0.629025\pi$$
−0.394336 + 0.918966i $$0.629025\pi$$
$$710$$ 24.0000 0.900704
$$711$$ 0 0
$$712$$ 0 0
$$713$$ 10.0000 0.374503
$$714$$ 0 0
$$715$$ −12.0000 −0.448775
$$716$$ −48.0000 −1.79384
$$717$$ 0 0
$$718$$ −70.0000 −2.61238
$$719$$ −24.0000 −0.895049 −0.447524 0.894272i $$-0.647694\pi$$
−0.447524 + 0.894272i $$0.647694\pi$$
$$720$$ 0 0
$$721$$ 24.0000 0.893807
$$722$$ −30.0000 −1.11648
$$723$$ 0 0
$$724$$ 42.0000 1.56092
$$725$$ −1.00000 −0.0371391
$$726$$ 0 0
$$727$$ −8.00000 −0.296704 −0.148352 0.988935i $$-0.547397\pi$$
−0.148352 + 0.988935i $$0.547397\pi$$
$$728$$ 0 0
$$729$$ 0 0
$$730$$ −18.0000 −0.666210
$$731$$ 2.00000 0.0739727
$$732$$ 0 0
$$733$$ 42.0000 1.55131 0.775653 0.631160i $$-0.217421\pi$$
0.775653 + 0.631160i $$0.217421\pi$$
$$734$$ −54.0000 −1.99318
$$735$$ 0 0
$$736$$ −40.0000 −1.47442
$$737$$ −6.00000 −0.221013
$$738$$ 0 0
$$739$$ −50.0000 −1.83928 −0.919640 0.392763i $$-0.871519\pi$$
−0.919640 + 0.392763i $$0.871519\pi$$
$$740$$ 10.0000 0.367607
$$741$$ 0 0
$$742$$ −12.0000 −0.440534
$$743$$ 12.0000 0.440237 0.220119 0.975473i $$-0.429356\pi$$
0.220119 + 0.975473i $$0.429356\pi$$
$$744$$ 0 0
$$745$$ −20.0000 −0.732743
$$746$$ 36.0000 1.31805
$$747$$ 0 0
$$748$$ 12.0000 0.438763
$$749$$ 0 0
$$750$$ 0 0
$$751$$ −8.00000 −0.291924 −0.145962 0.989290i $$-0.546628\pi$$
−0.145962 + 0.989290i $$0.546628\pi$$
$$752$$ −24.0000 −0.875190
$$753$$ 0 0
$$754$$ 8.00000 0.291343
$$755$$ 7.00000 0.254756
$$756$$ 0 0
$$757$$ −47.0000 −1.70824 −0.854122 0.520073i $$-0.825905\pi$$
−0.854122 + 0.520073i $$0.825905\pi$$
$$758$$ −12.0000 −0.435860
$$759$$ 0 0
$$760$$ 0 0
$$761$$ −10.0000 −0.362500 −0.181250 0.983437i $$-0.558014\pi$$
−0.181250 + 0.983437i $$0.558014\pi$$
$$762$$ 0 0
$$763$$ 10.0000 0.362024
$$764$$ 10.0000 0.361787
$$765$$ 0 0
$$766$$ 34.0000 1.22847
$$767$$ −16.0000 −0.577727
$$768$$ 0 0
$$769$$ 14.0000 0.504853 0.252426 0.967616i $$-0.418771\pi$$
0.252426 + 0.967616i $$0.418771\pi$$
$$770$$ −12.0000 −0.432450
$$771$$ 0 0
$$772$$ −20.0000 −0.719816
$$773$$ 46.0000 1.65451 0.827253 0.561830i $$-0.189903\pi$$
0.827253 + 0.561830i $$0.189903\pi$$
$$774$$ 0 0
$$775$$ 2.00000 0.0718421
$$776$$ 0 0
$$777$$ 0 0
$$778$$ −10.0000 −0.358517
$$779$$ 2.00000 0.0716574
$$780$$ 0 0
$$781$$ 36.0000 1.28818
$$782$$ −20.0000 −0.715199
$$783$$ 0 0
$$784$$ 12.0000 0.428571
$$785$$ 2.00000 0.0713831
$$786$$ 0 0
$$787$$ 42.0000 1.49714 0.748569 0.663057i $$-0.230741\pi$$
0.748569 + 0.663057i $$0.230741\pi$$
$$788$$ −2.00000 −0.0712470
$$789$$ 0 0
$$790$$ 32.0000 1.13851
$$791$$ −16.0000 −0.568895
$$792$$ 0 0
$$793$$ −24.0000 −0.852265
$$794$$ −60.0000 −2.12932
$$795$$ 0 0
$$796$$ −50.0000 −1.77220
$$797$$ 2.00000 0.0708436 0.0354218 0.999372i $$-0.488723\pi$$
0.0354218 + 0.999372i $$0.488723\pi$$
$$798$$ 0 0
$$799$$ −12.0000 −0.424529
$$800$$ −8.00000 −0.282843
$$801$$ 0 0
$$802$$ −16.0000 −0.564980
$$803$$ −27.0000 −0.952809
$$804$$ 0 0
$$805$$ 10.0000 0.352454
$$806$$ −16.0000 −0.563576
$$807$$ 0 0
$$808$$ 0 0
$$809$$ 19.0000 0.668004 0.334002 0.942572i $$-0.391601\pi$$
0.334002 + 0.942572i $$0.391601\pi$$
$$810$$ 0 0
$$811$$ −41.0000 −1.43970 −0.719852 0.694127i $$-0.755791\pi$$
−0.719852 + 0.694127i $$0.755791\pi$$
$$812$$ 4.00000 0.140372
$$813$$ 0 0
$$814$$ 30.0000 1.05150
$$815$$ −1.00000 −0.0350285
$$816$$ 0 0
$$817$$ 2.00000 0.0699711
$$818$$ 28.0000 0.978997
$$819$$ 0 0
$$820$$ 2.00000 0.0698430
$$821$$ −20.0000 −0.698005 −0.349002 0.937122i $$-0.613479\pi$$
−0.349002 + 0.937122i $$0.613479\pi$$
$$822$$ 0 0
$$823$$ −32.0000 −1.11545 −0.557725 0.830026i $$-0.688326\pi$$
−0.557725 + 0.830026i $$0.688326\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ −16.0000 −0.556711
$$827$$ −24.0000 −0.834562 −0.417281 0.908778i $$-0.637017\pi$$
−0.417281 + 0.908778i $$0.637017\pi$$
$$828$$ 0 0
$$829$$ 28.0000 0.972480 0.486240 0.873825i $$-0.338368\pi$$
0.486240 + 0.873825i $$0.338368\pi$$
$$830$$ −14.0000 −0.485947
$$831$$ 0 0
$$832$$ 32.0000 1.10940
$$833$$ 6.00000 0.207888
$$834$$ 0 0
$$835$$ 0 0
$$836$$ 12.0000 0.415029
$$837$$ 0 0
$$838$$ 32.0000 1.10542
$$839$$ −8.00000 −0.276191 −0.138095 0.990419i $$-0.544098\pi$$
−0.138095 + 0.990419i $$0.544098\pi$$
$$840$$ 0 0
$$841$$ 1.00000 0.0344828
$$842$$ −8.00000 −0.275698
$$843$$ 0 0
$$844$$ 16.0000 0.550743
$$845$$ −3.00000 −0.103203
$$846$$ 0 0
$$847$$ 4.00000 0.137442
$$848$$ −12.0000 −0.412082
$$849$$ 0 0
$$850$$ −4.00000 −0.137199
$$851$$ −25.0000 −0.856989
$$852$$ 0 0
$$853$$ 5.00000 0.171197 0.0855984 0.996330i $$-0.472720\pi$$
0.0855984 + 0.996330i $$0.472720\pi$$
$$854$$ −24.0000 −0.821263
$$855$$ 0 0
$$856$$ 0 0
$$857$$ 19.0000 0.649028 0.324514 0.945881i $$-0.394799\pi$$
0.324514 + 0.945881i $$0.394799\pi$$
$$858$$ 0 0
$$859$$ 30.0000 1.02359 0.511793 0.859109i $$-0.328981\pi$$
0.511793 + 0.859109i $$0.328981\pi$$
$$860$$ 2.00000 0.0681994
$$861$$ 0 0
$$862$$ −24.0000 −0.817443
$$863$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$864$$ 0 0
$$865$$ −19.0000 −0.646019
$$866$$ 34.0000 1.15537
$$867$$ 0 0
$$868$$ −8.00000 −0.271538
$$869$$ 48.0000 1.62829
$$870$$ 0 0
$$871$$ −8.00000 −0.271070
$$872$$ 0 0
$$873$$ 0 0
$$874$$ −20.0000 −0.676510
$$875$$ 2.00000 0.0676123
$$876$$ 0 0
$$877$$ 28.0000 0.945493 0.472746 0.881199i $$-0.343263\pi$$
0.472746 + 0.881199i $$0.343263\pi$$
$$878$$ −48.0000 −1.61992
$$879$$ 0 0
$$880$$ −12.0000 −0.404520
$$881$$ −3.00000 −0.101073 −0.0505363 0.998722i $$-0.516093\pi$$
−0.0505363 + 0.998722i $$0.516093\pi$$
$$882$$ 0 0
$$883$$ 22.0000 0.740359 0.370179 0.928960i $$-0.379296\pi$$
0.370179 + 0.928960i $$0.379296\pi$$
$$884$$ 16.0000 0.538138
$$885$$ 0 0
$$886$$ −24.0000 −0.806296
$$887$$ 4.00000 0.134307 0.0671534 0.997743i $$-0.478608\pi$$
0.0671534 + 0.997743i $$0.478608\pi$$
$$888$$ 0 0
$$889$$ 14.0000 0.469545
$$890$$ 12.0000 0.402241
$$891$$ 0 0
$$892$$ −16.0000 −0.535720
$$893$$ −12.0000 −0.401565
$$894$$ 0 0
$$895$$ 24.0000 0.802232
$$896$$ 0 0
$$897$$ 0 0
$$898$$ 26.0000 0.867631
$$899$$ −2.00000 −0.0667037
$$900$$ 0 0
$$901$$ −6.00000 −0.199889
$$902$$ 6.00000 0.199778
$$903$$ 0 0
$$904$$ 0 0
$$905$$ −21.0000 −0.698064
$$906$$ 0 0
$$907$$ −3.00000 −0.0996134 −0.0498067 0.998759i $$-0.515861\pi$$
−0.0498067 + 0.998759i $$0.515861\pi$$
$$908$$ 26.0000 0.862840
$$909$$ 0 0
$$910$$ −16.0000 −0.530395
$$911$$ −49.0000 −1.62344 −0.811721 0.584045i $$-0.801469\pi$$
−0.811721 + 0.584045i $$0.801469\pi$$
$$912$$ 0 0
$$913$$ −21.0000 −0.694999
$$914$$ 68.0000 2.24924
$$915$$ 0 0
$$916$$ −12.0000 −0.396491
$$917$$ −16.0000 −0.528367
$$918$$ 0 0
$$919$$ −32.0000 −1.05558 −0.527791 0.849374i $$-0.676980\pi$$
−0.527791 + 0.849374i $$0.676980\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ −42.0000 −1.38320
$$923$$ 48.0000 1.57994
$$924$$ 0 0
$$925$$ −5.00000 −0.164399
$$926$$ −40.0000 −1.31448
$$927$$ 0 0
$$928$$ 8.00000 0.262613
$$929$$ 38.0000 1.24674 0.623370 0.781927i $$-0.285763\pi$$
0.623370 + 0.781927i $$0.285763\pi$$
$$930$$ 0 0
$$931$$ 6.00000 0.196642
$$932$$ −54.0000 −1.76883
$$933$$ 0 0
$$934$$ −72.0000 −2.35591
$$935$$ −6.00000 −0.196221
$$936$$ 0 0
$$937$$ 36.0000 1.17607 0.588034 0.808836i $$-0.299902\pi$$
0.588034 + 0.808836i $$0.299902\pi$$
$$938$$ −8.00000 −0.261209
$$939$$ 0 0
$$940$$ −12.0000 −0.391397
$$941$$ 10.0000 0.325991 0.162995 0.986627i $$-0.447884\pi$$
0.162995 + 0.986627i $$0.447884\pi$$
$$942$$ 0 0
$$943$$ −5.00000 −0.162822
$$944$$ −16.0000 −0.520756
$$945$$ 0 0
$$946$$ 6.00000 0.195077
$$947$$ −24.0000 −0.779895 −0.389948 0.920837i $$-0.627507\pi$$
−0.389948 + 0.920837i $$0.627507\pi$$
$$948$$ 0 0
$$949$$ −36.0000 −1.16861
$$950$$ −4.00000 −0.129777
$$951$$ 0 0
$$952$$ 0 0
$$953$$ 54.0000 1.74923 0.874616 0.484817i $$-0.161114\pi$$
0.874616 + 0.484817i $$0.161114\pi$$
$$954$$ 0 0
$$955$$ −5.00000 −0.161796
$$956$$ −24.0000 −0.776215
$$957$$ 0 0
$$958$$ 8.00000 0.258468
$$959$$ −24.0000 −0.775000
$$960$$ 0 0
$$961$$ −27.0000 −0.870968
$$962$$ 40.0000 1.28965
$$963$$ 0 0
$$964$$ 50.0000 1.61039
$$965$$ 10.0000 0.321911
$$966$$ 0 0
$$967$$ −47.0000 −1.51142 −0.755709 0.654907i $$-0.772708\pi$$
−0.755709 + 0.654907i $$0.772708\pi$$
$$968$$ 0 0
$$969$$ 0 0
$$970$$ 26.0000 0.834810
$$971$$ −27.0000 −0.866471 −0.433236 0.901281i $$-0.642628\pi$$
−0.433236 + 0.901281i $$0.642628\pi$$
$$972$$ 0 0
$$973$$ 30.0000 0.961756
$$974$$ −32.0000 −1.02535
$$975$$ 0 0
$$976$$ −24.0000 −0.768221
$$977$$ −45.0000 −1.43968 −0.719839 0.694141i $$-0.755784\pi$$
−0.719839 + 0.694141i $$0.755784\pi$$
$$978$$ 0 0
$$979$$ 18.0000 0.575282
$$980$$ 6.00000 0.191663
$$981$$ 0 0
$$982$$ 56.0000 1.78703
$$983$$ −48.0000 −1.53096 −0.765481 0.643458i $$-0.777499\pi$$
−0.765481 + 0.643458i $$0.777499\pi$$
$$984$$ 0 0
$$985$$ 1.00000 0.0318626
$$986$$ 4.00000 0.127386
$$987$$ 0 0
$$988$$ 16.0000 0.509028
$$989$$ −5.00000 −0.158991
$$990$$ 0 0
$$991$$ 19.0000 0.603555 0.301777 0.953378i $$-0.402420\pi$$
0.301777 + 0.953378i $$0.402420\pi$$
$$992$$ −16.0000 −0.508001
$$993$$ 0 0
$$994$$ 48.0000 1.52247
$$995$$ 25.0000 0.792553
$$996$$ 0 0
$$997$$ 59.0000 1.86855 0.934274 0.356555i $$-0.116049\pi$$
0.934274 + 0.356555i $$0.116049\pi$$
$$998$$ −72.0000 −2.27912
$$999$$ 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 1305.2.a.g.1.1 yes 1
3.2 odd 2 1305.2.a.a.1.1 1
5.4 even 2 6525.2.a.a.1.1 1
15.14 odd 2 6525.2.a.m.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
1305.2.a.a.1.1 1 3.2 odd 2
1305.2.a.g.1.1 yes 1 1.1 even 1 trivial
6525.2.a.a.1.1 1 5.4 even 2
6525.2.a.m.1.1 1 15.14 odd 2