# Properties

 Label 345.2.a.d.1.1 Level $345$ Weight $2$ Character 345.1 Self dual yes Analytic conductor $2.755$ 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] = [345,2,Mod(1,345)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(345, 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("345.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$345 = 3 \cdot 5 \cdot 23$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 345.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: $$2.75483886973$$ 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 $$\chi$$ $$=$$ 345.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+1.00000 q^{3} -2.00000 q^{4} -1.00000 q^{5} -3.00000 q^{7} +1.00000 q^{9} +O(q^{10})$$ $$q+1.00000 q^{3} -2.00000 q^{4} -1.00000 q^{5} -3.00000 q^{7} +1.00000 q^{9} -4.00000 q^{11} -2.00000 q^{12} -1.00000 q^{15} +4.00000 q^{16} -3.00000 q^{17} -8.00000 q^{19} +2.00000 q^{20} -3.00000 q^{21} +1.00000 q^{23} +1.00000 q^{25} +1.00000 q^{27} +6.00000 q^{28} +9.00000 q^{29} -5.00000 q^{31} -4.00000 q^{33} +3.00000 q^{35} -2.00000 q^{36} -9.00000 q^{37} +7.00000 q^{41} +4.00000 q^{43} +8.00000 q^{44} -1.00000 q^{45} -2.00000 q^{47} +4.00000 q^{48} +2.00000 q^{49} -3.00000 q^{51} +13.0000 q^{53} +4.00000 q^{55} -8.00000 q^{57} -3.00000 q^{59} +2.00000 q^{60} -14.0000 q^{61} -3.00000 q^{63} -8.00000 q^{64} +13.0000 q^{67} +6.00000 q^{68} +1.00000 q^{69} -13.0000 q^{71} -4.00000 q^{73} +1.00000 q^{75} +16.0000 q^{76} +12.0000 q^{77} -4.00000 q^{80} +1.00000 q^{81} -1.00000 q^{83} +6.00000 q^{84} +3.00000 q^{85} +9.00000 q^{87} -8.00000 q^{89} -2.00000 q^{92} -5.00000 q^{93} +8.00000 q^{95} +10.0000 q^{97} -4.00000 q^{99} +O(q^{100})$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$3$$ 1.00000 0.577350
$$4$$ −2.00000 −1.00000
$$5$$ −1.00000 −0.447214
$$6$$ 0 0
$$7$$ −3.00000 −1.13389 −0.566947 0.823754i $$-0.691875\pi$$
−0.566947 + 0.823754i $$0.691875\pi$$
$$8$$ 0 0
$$9$$ 1.00000 0.333333
$$10$$ 0 0
$$11$$ −4.00000 −1.20605 −0.603023 0.797724i $$-0.706037\pi$$
−0.603023 + 0.797724i $$0.706037\pi$$
$$12$$ −2.00000 −0.577350
$$13$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$14$$ 0 0
$$15$$ −1.00000 −0.258199
$$16$$ 4.00000 1.00000
$$17$$ −3.00000 −0.727607 −0.363803 0.931476i $$-0.618522\pi$$
−0.363803 + 0.931476i $$0.618522\pi$$
$$18$$ 0 0
$$19$$ −8.00000 −1.83533 −0.917663 0.397360i $$-0.869927\pi$$
−0.917663 + 0.397360i $$0.869927\pi$$
$$20$$ 2.00000 0.447214
$$21$$ −3.00000 −0.654654
$$22$$ 0 0
$$23$$ 1.00000 0.208514
$$24$$ 0 0
$$25$$ 1.00000 0.200000
$$26$$ 0 0
$$27$$ 1.00000 0.192450
$$28$$ 6.00000 1.13389
$$29$$ 9.00000 1.67126 0.835629 0.549294i $$-0.185103\pi$$
0.835629 + 0.549294i $$0.185103\pi$$
$$30$$ 0 0
$$31$$ −5.00000 −0.898027 −0.449013 0.893525i $$-0.648224\pi$$
−0.449013 + 0.893525i $$0.648224\pi$$
$$32$$ 0 0
$$33$$ −4.00000 −0.696311
$$34$$ 0 0
$$35$$ 3.00000 0.507093
$$36$$ −2.00000 −0.333333
$$37$$ −9.00000 −1.47959 −0.739795 0.672832i $$-0.765078\pi$$
−0.739795 + 0.672832i $$0.765078\pi$$
$$38$$ 0 0
$$39$$ 0 0
$$40$$ 0 0
$$41$$ 7.00000 1.09322 0.546608 0.837389i $$-0.315919\pi$$
0.546608 + 0.837389i $$0.315919\pi$$
$$42$$ 0 0
$$43$$ 4.00000 0.609994 0.304997 0.952353i $$-0.401344\pi$$
0.304997 + 0.952353i $$0.401344\pi$$
$$44$$ 8.00000 1.20605
$$45$$ −1.00000 −0.149071
$$46$$ 0 0
$$47$$ −2.00000 −0.291730 −0.145865 0.989305i $$-0.546597\pi$$
−0.145865 + 0.989305i $$0.546597\pi$$
$$48$$ 4.00000 0.577350
$$49$$ 2.00000 0.285714
$$50$$ 0 0
$$51$$ −3.00000 −0.420084
$$52$$ 0 0
$$53$$ 13.0000 1.78569 0.892844 0.450367i $$-0.148707\pi$$
0.892844 + 0.450367i $$0.148707\pi$$
$$54$$ 0 0
$$55$$ 4.00000 0.539360
$$56$$ 0 0
$$57$$ −8.00000 −1.05963
$$58$$ 0 0
$$59$$ −3.00000 −0.390567 −0.195283 0.980747i $$-0.562563\pi$$
−0.195283 + 0.980747i $$0.562563\pi$$
$$60$$ 2.00000 0.258199
$$61$$ −14.0000 −1.79252 −0.896258 0.443533i $$-0.853725\pi$$
−0.896258 + 0.443533i $$0.853725\pi$$
$$62$$ 0 0
$$63$$ −3.00000 −0.377964
$$64$$ −8.00000 −1.00000
$$65$$ 0 0
$$66$$ 0 0
$$67$$ 13.0000 1.58820 0.794101 0.607785i $$-0.207942\pi$$
0.794101 + 0.607785i $$0.207942\pi$$
$$68$$ 6.00000 0.727607
$$69$$ 1.00000 0.120386
$$70$$ 0 0
$$71$$ −13.0000 −1.54282 −0.771408 0.636341i $$-0.780447\pi$$
−0.771408 + 0.636341i $$0.780447\pi$$
$$72$$ 0 0
$$73$$ −4.00000 −0.468165 −0.234082 0.972217i $$-0.575209\pi$$
−0.234082 + 0.972217i $$0.575209\pi$$
$$74$$ 0 0
$$75$$ 1.00000 0.115470
$$76$$ 16.0000 1.83533
$$77$$ 12.0000 1.36753
$$78$$ 0 0
$$79$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$80$$ −4.00000 −0.447214
$$81$$ 1.00000 0.111111
$$82$$ 0 0
$$83$$ −1.00000 −0.109764 −0.0548821 0.998493i $$-0.517478\pi$$
−0.0548821 + 0.998493i $$0.517478\pi$$
$$84$$ 6.00000 0.654654
$$85$$ 3.00000 0.325396
$$86$$ 0 0
$$87$$ 9.00000 0.964901
$$88$$ 0 0
$$89$$ −8.00000 −0.847998 −0.423999 0.905663i $$-0.639374\pi$$
−0.423999 + 0.905663i $$0.639374\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ −2.00000 −0.208514
$$93$$ −5.00000 −0.518476
$$94$$ 0 0
$$95$$ 8.00000 0.820783
$$96$$ 0 0
$$97$$ 10.0000 1.01535 0.507673 0.861550i $$-0.330506\pi$$
0.507673 + 0.861550i $$0.330506\pi$$
$$98$$ 0 0
$$99$$ −4.00000 −0.402015
$$100$$ −2.00000 −0.200000
$$101$$ 9.00000 0.895533 0.447767 0.894150i $$-0.352219\pi$$
0.447767 + 0.894150i $$0.352219\pi$$
$$102$$ 0 0
$$103$$ 8.00000 0.788263 0.394132 0.919054i $$-0.371045\pi$$
0.394132 + 0.919054i $$0.371045\pi$$
$$104$$ 0 0
$$105$$ 3.00000 0.292770
$$106$$ 0 0
$$107$$ 11.0000 1.06341 0.531705 0.846930i $$-0.321551\pi$$
0.531705 + 0.846930i $$0.321551\pi$$
$$108$$ −2.00000 −0.192450
$$109$$ −16.0000 −1.53252 −0.766261 0.642529i $$-0.777885\pi$$
−0.766261 + 0.642529i $$0.777885\pi$$
$$110$$ 0 0
$$111$$ −9.00000 −0.854242
$$112$$ −12.0000 −1.13389
$$113$$ 1.00000 0.0940721 0.0470360 0.998893i $$-0.485022\pi$$
0.0470360 + 0.998893i $$0.485022\pi$$
$$114$$ 0 0
$$115$$ −1.00000 −0.0932505
$$116$$ −18.0000 −1.67126
$$117$$ 0 0
$$118$$ 0 0
$$119$$ 9.00000 0.825029
$$120$$ 0 0
$$121$$ 5.00000 0.454545
$$122$$ 0 0
$$123$$ 7.00000 0.631169
$$124$$ 10.0000 0.898027
$$125$$ −1.00000 −0.0894427
$$126$$ 0 0
$$127$$ −18.0000 −1.59724 −0.798621 0.601834i $$-0.794437\pi$$
−0.798621 + 0.601834i $$0.794437\pi$$
$$128$$ 0 0
$$129$$ 4.00000 0.352180
$$130$$ 0 0
$$131$$ −4.00000 −0.349482 −0.174741 0.984614i $$-0.555909\pi$$
−0.174741 + 0.984614i $$0.555909\pi$$
$$132$$ 8.00000 0.696311
$$133$$ 24.0000 2.08106
$$134$$ 0 0
$$135$$ −1.00000 −0.0860663
$$136$$ 0 0
$$137$$ 2.00000 0.170872 0.0854358 0.996344i $$-0.472772\pi$$
0.0854358 + 0.996344i $$0.472772\pi$$
$$138$$ 0 0
$$139$$ −3.00000 −0.254457 −0.127228 0.991873i $$-0.540608\pi$$
−0.127228 + 0.991873i $$0.540608\pi$$
$$140$$ −6.00000 −0.507093
$$141$$ −2.00000 −0.168430
$$142$$ 0 0
$$143$$ 0 0
$$144$$ 4.00000 0.333333
$$145$$ −9.00000 −0.747409
$$146$$ 0 0
$$147$$ 2.00000 0.164957
$$148$$ 18.0000 1.47959
$$149$$ −14.0000 −1.14692 −0.573462 0.819232i $$-0.694400\pi$$
−0.573462 + 0.819232i $$0.694400\pi$$
$$150$$ 0 0
$$151$$ −8.00000 −0.651031 −0.325515 0.945537i $$-0.605538\pi$$
−0.325515 + 0.945537i $$0.605538\pi$$
$$152$$ 0 0
$$153$$ −3.00000 −0.242536
$$154$$ 0 0
$$155$$ 5.00000 0.401610
$$156$$ 0 0
$$157$$ −3.00000 −0.239426 −0.119713 0.992809i $$-0.538197\pi$$
−0.119713 + 0.992809i $$0.538197\pi$$
$$158$$ 0 0
$$159$$ 13.0000 1.03097
$$160$$ 0 0
$$161$$ −3.00000 −0.236433
$$162$$ 0 0
$$163$$ −2.00000 −0.156652 −0.0783260 0.996928i $$-0.524958\pi$$
−0.0783260 + 0.996928i $$0.524958\pi$$
$$164$$ −14.0000 −1.09322
$$165$$ 4.00000 0.311400
$$166$$ 0 0
$$167$$ −22.0000 −1.70241 −0.851206 0.524832i $$-0.824128\pi$$
−0.851206 + 0.524832i $$0.824128\pi$$
$$168$$ 0 0
$$169$$ −13.0000 −1.00000
$$170$$ 0 0
$$171$$ −8.00000 −0.611775
$$172$$ −8.00000 −0.609994
$$173$$ −24.0000 −1.82469 −0.912343 0.409426i $$-0.865729\pi$$
−0.912343 + 0.409426i $$0.865729\pi$$
$$174$$ 0 0
$$175$$ −3.00000 −0.226779
$$176$$ −16.0000 −1.20605
$$177$$ −3.00000 −0.225494
$$178$$ 0 0
$$179$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$180$$ 2.00000 0.149071
$$181$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$182$$ 0 0
$$183$$ −14.0000 −1.03491
$$184$$ 0 0
$$185$$ 9.00000 0.661693
$$186$$ 0 0
$$187$$ 12.0000 0.877527
$$188$$ 4.00000 0.291730
$$189$$ −3.00000 −0.218218
$$190$$ 0 0
$$191$$ −2.00000 −0.144715 −0.0723575 0.997379i $$-0.523052\pi$$
−0.0723575 + 0.997379i $$0.523052\pi$$
$$192$$ −8.00000 −0.577350
$$193$$ 4.00000 0.287926 0.143963 0.989583i $$-0.454015\pi$$
0.143963 + 0.989583i $$0.454015\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ −4.00000 −0.285714
$$197$$ 2.00000 0.142494 0.0712470 0.997459i $$-0.477302\pi$$
0.0712470 + 0.997459i $$0.477302\pi$$
$$198$$ 0 0
$$199$$ −18.0000 −1.27599 −0.637993 0.770042i $$-0.720235\pi$$
−0.637993 + 0.770042i $$0.720235\pi$$
$$200$$ 0 0
$$201$$ 13.0000 0.916949
$$202$$ 0 0
$$203$$ −27.0000 −1.89503
$$204$$ 6.00000 0.420084
$$205$$ −7.00000 −0.488901
$$206$$ 0 0
$$207$$ 1.00000 0.0695048
$$208$$ 0 0
$$209$$ 32.0000 2.21349
$$210$$ 0 0
$$211$$ 7.00000 0.481900 0.240950 0.970538i $$-0.422541\pi$$
0.240950 + 0.970538i $$0.422541\pi$$
$$212$$ −26.0000 −1.78569
$$213$$ −13.0000 −0.890745
$$214$$ 0 0
$$215$$ −4.00000 −0.272798
$$216$$ 0 0
$$217$$ 15.0000 1.01827
$$218$$ 0 0
$$219$$ −4.00000 −0.270295
$$220$$ −8.00000 −0.539360
$$221$$ 0 0
$$222$$ 0 0
$$223$$ 10.0000 0.669650 0.334825 0.942280i $$-0.391323\pi$$
0.334825 + 0.942280i $$0.391323\pi$$
$$224$$ 0 0
$$225$$ 1.00000 0.0666667
$$226$$ 0 0
$$227$$ −4.00000 −0.265489 −0.132745 0.991150i $$-0.542379\pi$$
−0.132745 + 0.991150i $$0.542379\pi$$
$$228$$ 16.0000 1.05963
$$229$$ 4.00000 0.264327 0.132164 0.991228i $$-0.457808\pi$$
0.132164 + 0.991228i $$0.457808\pi$$
$$230$$ 0 0
$$231$$ 12.0000 0.789542
$$232$$ 0 0
$$233$$ −8.00000 −0.524097 −0.262049 0.965055i $$-0.584398\pi$$
−0.262049 + 0.965055i $$0.584398\pi$$
$$234$$ 0 0
$$235$$ 2.00000 0.130466
$$236$$ 6.00000 0.390567
$$237$$ 0 0
$$238$$ 0 0
$$239$$ 13.0000 0.840900 0.420450 0.907316i $$-0.361872\pi$$
0.420450 + 0.907316i $$0.361872\pi$$
$$240$$ −4.00000 −0.258199
$$241$$ 20.0000 1.28831 0.644157 0.764894i $$-0.277208\pi$$
0.644157 + 0.764894i $$0.277208\pi$$
$$242$$ 0 0
$$243$$ 1.00000 0.0641500
$$244$$ 28.0000 1.79252
$$245$$ −2.00000 −0.127775
$$246$$ 0 0
$$247$$ 0 0
$$248$$ 0 0
$$249$$ −1.00000 −0.0633724
$$250$$ 0 0
$$251$$ 18.0000 1.13615 0.568075 0.822977i $$-0.307688\pi$$
0.568075 + 0.822977i $$0.307688\pi$$
$$252$$ 6.00000 0.377964
$$253$$ −4.00000 −0.251478
$$254$$ 0 0
$$255$$ 3.00000 0.187867
$$256$$ 16.0000 1.00000
$$257$$ 18.0000 1.12281 0.561405 0.827541i $$-0.310261\pi$$
0.561405 + 0.827541i $$0.310261\pi$$
$$258$$ 0 0
$$259$$ 27.0000 1.67770
$$260$$ 0 0
$$261$$ 9.00000 0.557086
$$262$$ 0 0
$$263$$ −23.0000 −1.41824 −0.709120 0.705087i $$-0.750908\pi$$
−0.709120 + 0.705087i $$0.750908\pi$$
$$264$$ 0 0
$$265$$ −13.0000 −0.798584
$$266$$ 0 0
$$267$$ −8.00000 −0.489592
$$268$$ −26.0000 −1.58820
$$269$$ −17.0000 −1.03651 −0.518254 0.855227i $$-0.673418\pi$$
−0.518254 + 0.855227i $$0.673418\pi$$
$$270$$ 0 0
$$271$$ −23.0000 −1.39715 −0.698575 0.715537i $$-0.746182\pi$$
−0.698575 + 0.715537i $$0.746182\pi$$
$$272$$ −12.0000 −0.727607
$$273$$ 0 0
$$274$$ 0 0
$$275$$ −4.00000 −0.241209
$$276$$ −2.00000 −0.120386
$$277$$ 18.0000 1.08152 0.540758 0.841178i $$-0.318138\pi$$
0.540758 + 0.841178i $$0.318138\pi$$
$$278$$ 0 0
$$279$$ −5.00000 −0.299342
$$280$$ 0 0
$$281$$ 24.0000 1.43172 0.715860 0.698244i $$-0.246035\pi$$
0.715860 + 0.698244i $$0.246035\pi$$
$$282$$ 0 0
$$283$$ −5.00000 −0.297219 −0.148610 0.988896i $$-0.547480\pi$$
−0.148610 + 0.988896i $$0.547480\pi$$
$$284$$ 26.0000 1.54282
$$285$$ 8.00000 0.473879
$$286$$ 0 0
$$287$$ −21.0000 −1.23959
$$288$$ 0 0
$$289$$ −8.00000 −0.470588
$$290$$ 0 0
$$291$$ 10.0000 0.586210
$$292$$ 8.00000 0.468165
$$293$$ 19.0000 1.10999 0.554996 0.831853i $$-0.312720\pi$$
0.554996 + 0.831853i $$0.312720\pi$$
$$294$$ 0 0
$$295$$ 3.00000 0.174667
$$296$$ 0 0
$$297$$ −4.00000 −0.232104
$$298$$ 0 0
$$299$$ 0 0
$$300$$ −2.00000 −0.115470
$$301$$ −12.0000 −0.691669
$$302$$ 0 0
$$303$$ 9.00000 0.517036
$$304$$ −32.0000 −1.83533
$$305$$ 14.0000 0.801638
$$306$$ 0 0
$$307$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$308$$ −24.0000 −1.36753
$$309$$ 8.00000 0.455104
$$310$$ 0 0
$$311$$ −8.00000 −0.453638 −0.226819 0.973937i $$-0.572833\pi$$
−0.226819 + 0.973937i $$0.572833\pi$$
$$312$$ 0 0
$$313$$ 25.0000 1.41308 0.706542 0.707671i $$-0.250254\pi$$
0.706542 + 0.707671i $$0.250254\pi$$
$$314$$ 0 0
$$315$$ 3.00000 0.169031
$$316$$ 0 0
$$317$$ 30.0000 1.68497 0.842484 0.538721i $$-0.181092\pi$$
0.842484 + 0.538721i $$0.181092\pi$$
$$318$$ 0 0
$$319$$ −36.0000 −2.01561
$$320$$ 8.00000 0.447214
$$321$$ 11.0000 0.613960
$$322$$ 0 0
$$323$$ 24.0000 1.33540
$$324$$ −2.00000 −0.111111
$$325$$ 0 0
$$326$$ 0 0
$$327$$ −16.0000 −0.884802
$$328$$ 0 0
$$329$$ 6.00000 0.330791
$$330$$ 0 0
$$331$$ 7.00000 0.384755 0.192377 0.981321i $$-0.438380\pi$$
0.192377 + 0.981321i $$0.438380\pi$$
$$332$$ 2.00000 0.109764
$$333$$ −9.00000 −0.493197
$$334$$ 0 0
$$335$$ −13.0000 −0.710266
$$336$$ −12.0000 −0.654654
$$337$$ 10.0000 0.544735 0.272367 0.962193i $$-0.412193\pi$$
0.272367 + 0.962193i $$0.412193\pi$$
$$338$$ 0 0
$$339$$ 1.00000 0.0543125
$$340$$ −6.00000 −0.325396
$$341$$ 20.0000 1.08306
$$342$$ 0 0
$$343$$ 15.0000 0.809924
$$344$$ 0 0
$$345$$ −1.00000 −0.0538382
$$346$$ 0 0
$$347$$ −16.0000 −0.858925 −0.429463 0.903085i $$-0.641297\pi$$
−0.429463 + 0.903085i $$0.641297\pi$$
$$348$$ −18.0000 −0.964901
$$349$$ −1.00000 −0.0535288 −0.0267644 0.999642i $$-0.508520\pi$$
−0.0267644 + 0.999642i $$0.508520\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 0 0
$$353$$ −6.00000 −0.319348 −0.159674 0.987170i $$-0.551044\pi$$
−0.159674 + 0.987170i $$0.551044\pi$$
$$354$$ 0 0
$$355$$ 13.0000 0.689968
$$356$$ 16.0000 0.847998
$$357$$ 9.00000 0.476331
$$358$$ 0 0
$$359$$ −2.00000 −0.105556 −0.0527780 0.998606i $$-0.516808\pi$$
−0.0527780 + 0.998606i $$0.516808\pi$$
$$360$$ 0 0
$$361$$ 45.0000 2.36842
$$362$$ 0 0
$$363$$ 5.00000 0.262432
$$364$$ 0 0
$$365$$ 4.00000 0.209370
$$366$$ 0 0
$$367$$ 31.0000 1.61819 0.809093 0.587680i $$-0.199959\pi$$
0.809093 + 0.587680i $$0.199959\pi$$
$$368$$ 4.00000 0.208514
$$369$$ 7.00000 0.364405
$$370$$ 0 0
$$371$$ −39.0000 −2.02478
$$372$$ 10.0000 0.518476
$$373$$ −6.00000 −0.310668 −0.155334 0.987862i $$-0.549645\pi$$
−0.155334 + 0.987862i $$0.549645\pi$$
$$374$$ 0 0
$$375$$ −1.00000 −0.0516398
$$376$$ 0 0
$$377$$ 0 0
$$378$$ 0 0
$$379$$ −8.00000 −0.410932 −0.205466 0.978664i $$-0.565871\pi$$
−0.205466 + 0.978664i $$0.565871\pi$$
$$380$$ −16.0000 −0.820783
$$381$$ −18.0000 −0.922168
$$382$$ 0 0
$$383$$ −23.0000 −1.17525 −0.587623 0.809135i $$-0.699936\pi$$
−0.587623 + 0.809135i $$0.699936\pi$$
$$384$$ 0 0
$$385$$ −12.0000 −0.611577
$$386$$ 0 0
$$387$$ 4.00000 0.203331
$$388$$ −20.0000 −1.01535
$$389$$ −30.0000 −1.52106 −0.760530 0.649303i $$-0.775061\pi$$
−0.760530 + 0.649303i $$0.775061\pi$$
$$390$$ 0 0
$$391$$ −3.00000 −0.151717
$$392$$ 0 0
$$393$$ −4.00000 −0.201773
$$394$$ 0 0
$$395$$ 0 0
$$396$$ 8.00000 0.402015
$$397$$ 8.00000 0.401508 0.200754 0.979642i $$-0.435661\pi$$
0.200754 + 0.979642i $$0.435661\pi$$
$$398$$ 0 0
$$399$$ 24.0000 1.20150
$$400$$ 4.00000 0.200000
$$401$$ 30.0000 1.49813 0.749064 0.662497i $$-0.230503\pi$$
0.749064 + 0.662497i $$0.230503\pi$$
$$402$$ 0 0
$$403$$ 0 0
$$404$$ −18.0000 −0.895533
$$405$$ −1.00000 −0.0496904
$$406$$ 0 0
$$407$$ 36.0000 1.78445
$$408$$ 0 0
$$409$$ 25.0000 1.23617 0.618085 0.786111i $$-0.287909\pi$$
0.618085 + 0.786111i $$0.287909\pi$$
$$410$$ 0 0
$$411$$ 2.00000 0.0986527
$$412$$ −16.0000 −0.788263
$$413$$ 9.00000 0.442861
$$414$$ 0 0
$$415$$ 1.00000 0.0490881
$$416$$ 0 0
$$417$$ −3.00000 −0.146911
$$418$$ 0 0
$$419$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$420$$ −6.00000 −0.292770
$$421$$ 4.00000 0.194948 0.0974740 0.995238i $$-0.468924\pi$$
0.0974740 + 0.995238i $$0.468924\pi$$
$$422$$ 0 0
$$423$$ −2.00000 −0.0972433
$$424$$ 0 0
$$425$$ −3.00000 −0.145521
$$426$$ 0 0
$$427$$ 42.0000 2.03252
$$428$$ −22.0000 −1.06341
$$429$$ 0 0
$$430$$ 0 0
$$431$$ 20.0000 0.963366 0.481683 0.876346i $$-0.340026\pi$$
0.481683 + 0.876346i $$0.340026\pi$$
$$432$$ 4.00000 0.192450
$$433$$ −37.0000 −1.77811 −0.889053 0.457804i $$-0.848636\pi$$
−0.889053 + 0.457804i $$0.848636\pi$$
$$434$$ 0 0
$$435$$ −9.00000 −0.431517
$$436$$ 32.0000 1.53252
$$437$$ −8.00000 −0.382692
$$438$$ 0 0
$$439$$ −24.0000 −1.14546 −0.572729 0.819745i $$-0.694115\pi$$
−0.572729 + 0.819745i $$0.694115\pi$$
$$440$$ 0 0
$$441$$ 2.00000 0.0952381
$$442$$ 0 0
$$443$$ 6.00000 0.285069 0.142534 0.989790i $$-0.454475\pi$$
0.142534 + 0.989790i $$0.454475\pi$$
$$444$$ 18.0000 0.854242
$$445$$ 8.00000 0.379236
$$446$$ 0 0
$$447$$ −14.0000 −0.662177
$$448$$ 24.0000 1.13389
$$449$$ −11.0000 −0.519122 −0.259561 0.965727i $$-0.583578\pi$$
−0.259561 + 0.965727i $$0.583578\pi$$
$$450$$ 0 0
$$451$$ −28.0000 −1.31847
$$452$$ −2.00000 −0.0940721
$$453$$ −8.00000 −0.375873
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ −35.0000 −1.63723 −0.818615 0.574342i $$-0.805258\pi$$
−0.818615 + 0.574342i $$0.805258\pi$$
$$458$$ 0 0
$$459$$ −3.00000 −0.140028
$$460$$ 2.00000 0.0932505
$$461$$ 26.0000 1.21094 0.605470 0.795868i $$-0.292985\pi$$
0.605470 + 0.795868i $$0.292985\pi$$
$$462$$ 0 0
$$463$$ −20.0000 −0.929479 −0.464739 0.885448i $$-0.653852\pi$$
−0.464739 + 0.885448i $$0.653852\pi$$
$$464$$ 36.0000 1.67126
$$465$$ 5.00000 0.231869
$$466$$ 0 0
$$467$$ −13.0000 −0.601568 −0.300784 0.953692i $$-0.597248\pi$$
−0.300784 + 0.953692i $$0.597248\pi$$
$$468$$ 0 0
$$469$$ −39.0000 −1.80085
$$470$$ 0 0
$$471$$ −3.00000 −0.138233
$$472$$ 0 0
$$473$$ −16.0000 −0.735681
$$474$$ 0 0
$$475$$ −8.00000 −0.367065
$$476$$ −18.0000 −0.825029
$$477$$ 13.0000 0.595229
$$478$$ 0 0
$$479$$ 6.00000 0.274147 0.137073 0.990561i $$-0.456230\pi$$
0.137073 + 0.990561i $$0.456230\pi$$
$$480$$ 0 0
$$481$$ 0 0
$$482$$ 0 0
$$483$$ −3.00000 −0.136505
$$484$$ −10.0000 −0.454545
$$485$$ −10.0000 −0.454077
$$486$$ 0 0
$$487$$ 2.00000 0.0906287 0.0453143 0.998973i $$-0.485571\pi$$
0.0453143 + 0.998973i $$0.485571\pi$$
$$488$$ 0 0
$$489$$ −2.00000 −0.0904431
$$490$$ 0 0
$$491$$ 3.00000 0.135388 0.0676941 0.997706i $$-0.478436\pi$$
0.0676941 + 0.997706i $$0.478436\pi$$
$$492$$ −14.0000 −0.631169
$$493$$ −27.0000 −1.21602
$$494$$ 0 0
$$495$$ 4.00000 0.179787
$$496$$ −20.0000 −0.898027
$$497$$ 39.0000 1.74939
$$498$$ 0 0
$$499$$ 21.0000 0.940089 0.470045 0.882643i $$-0.344238\pi$$
0.470045 + 0.882643i $$0.344238\pi$$
$$500$$ 2.00000 0.0894427
$$501$$ −22.0000 −0.982888
$$502$$ 0 0
$$503$$ 21.0000 0.936344 0.468172 0.883637i $$-0.344913\pi$$
0.468172 + 0.883637i $$0.344913\pi$$
$$504$$ 0 0
$$505$$ −9.00000 −0.400495
$$506$$ 0 0
$$507$$ −13.0000 −0.577350
$$508$$ 36.0000 1.59724
$$509$$ 30.0000 1.32973 0.664863 0.746965i $$-0.268490\pi$$
0.664863 + 0.746965i $$0.268490\pi$$
$$510$$ 0 0
$$511$$ 12.0000 0.530849
$$512$$ 0 0
$$513$$ −8.00000 −0.353209
$$514$$ 0 0
$$515$$ −8.00000 −0.352522
$$516$$ −8.00000 −0.352180
$$517$$ 8.00000 0.351840
$$518$$ 0 0
$$519$$ −24.0000 −1.05348
$$520$$ 0 0
$$521$$ −12.0000 −0.525730 −0.262865 0.964833i $$-0.584667\pi$$
−0.262865 + 0.964833i $$0.584667\pi$$
$$522$$ 0 0
$$523$$ −20.0000 −0.874539 −0.437269 0.899331i $$-0.644054\pi$$
−0.437269 + 0.899331i $$0.644054\pi$$
$$524$$ 8.00000 0.349482
$$525$$ −3.00000 −0.130931
$$526$$ 0 0
$$527$$ 15.0000 0.653410
$$528$$ −16.0000 −0.696311
$$529$$ 1.00000 0.0434783
$$530$$ 0 0
$$531$$ −3.00000 −0.130189
$$532$$ −48.0000 −2.08106
$$533$$ 0 0
$$534$$ 0 0
$$535$$ −11.0000 −0.475571
$$536$$ 0 0
$$537$$ 0 0
$$538$$ 0 0
$$539$$ −8.00000 −0.344584
$$540$$ 2.00000 0.0860663
$$541$$ −10.0000 −0.429934 −0.214967 0.976621i $$-0.568964\pi$$
−0.214967 + 0.976621i $$0.568964\pi$$
$$542$$ 0 0
$$543$$ 0 0
$$544$$ 0 0
$$545$$ 16.0000 0.685365
$$546$$ 0 0
$$547$$ 20.0000 0.855138 0.427569 0.903983i $$-0.359370\pi$$
0.427569 + 0.903983i $$0.359370\pi$$
$$548$$ −4.00000 −0.170872
$$549$$ −14.0000 −0.597505
$$550$$ 0 0
$$551$$ −72.0000 −3.06730
$$552$$ 0 0
$$553$$ 0 0
$$554$$ 0 0
$$555$$ 9.00000 0.382029
$$556$$ 6.00000 0.254457
$$557$$ 21.0000 0.889799 0.444899 0.895581i $$-0.353239\pi$$
0.444899 + 0.895581i $$0.353239\pi$$
$$558$$ 0 0
$$559$$ 0 0
$$560$$ 12.0000 0.507093
$$561$$ 12.0000 0.506640
$$562$$ 0 0
$$563$$ −11.0000 −0.463595 −0.231797 0.972764i $$-0.574461\pi$$
−0.231797 + 0.972764i $$0.574461\pi$$
$$564$$ 4.00000 0.168430
$$565$$ −1.00000 −0.0420703
$$566$$ 0 0
$$567$$ −3.00000 −0.125988
$$568$$ 0 0
$$569$$ −6.00000 −0.251533 −0.125767 0.992060i $$-0.540139\pi$$
−0.125767 + 0.992060i $$0.540139\pi$$
$$570$$ 0 0
$$571$$ −22.0000 −0.920671 −0.460336 0.887745i $$-0.652271\pi$$
−0.460336 + 0.887745i $$0.652271\pi$$
$$572$$ 0 0
$$573$$ −2.00000 −0.0835512
$$574$$ 0 0
$$575$$ 1.00000 0.0417029
$$576$$ −8.00000 −0.333333
$$577$$ −20.0000 −0.832611 −0.416305 0.909225i $$-0.636675\pi$$
−0.416305 + 0.909225i $$0.636675\pi$$
$$578$$ 0 0
$$579$$ 4.00000 0.166234
$$580$$ 18.0000 0.747409
$$581$$ 3.00000 0.124461
$$582$$ 0 0
$$583$$ −52.0000 −2.15362
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ 14.0000 0.577842 0.288921 0.957353i $$-0.406704\pi$$
0.288921 + 0.957353i $$0.406704\pi$$
$$588$$ −4.00000 −0.164957
$$589$$ 40.0000 1.64817
$$590$$ 0 0
$$591$$ 2.00000 0.0822690
$$592$$ −36.0000 −1.47959
$$593$$ 18.0000 0.739171 0.369586 0.929197i $$-0.379500\pi$$
0.369586 + 0.929197i $$0.379500\pi$$
$$594$$ 0 0
$$595$$ −9.00000 −0.368964
$$596$$ 28.0000 1.14692
$$597$$ −18.0000 −0.736691
$$598$$ 0 0
$$599$$ −4.00000 −0.163436 −0.0817178 0.996656i $$-0.526041\pi$$
−0.0817178 + 0.996656i $$0.526041\pi$$
$$600$$ 0 0
$$601$$ −41.0000 −1.67242 −0.836212 0.548406i $$-0.815235\pi$$
−0.836212 + 0.548406i $$0.815235\pi$$
$$602$$ 0 0
$$603$$ 13.0000 0.529401
$$604$$ 16.0000 0.651031
$$605$$ −5.00000 −0.203279
$$606$$ 0 0
$$607$$ −10.0000 −0.405887 −0.202944 0.979190i $$-0.565051\pi$$
−0.202944 + 0.979190i $$0.565051\pi$$
$$608$$ 0 0
$$609$$ −27.0000 −1.09410
$$610$$ 0 0
$$611$$ 0 0
$$612$$ 6.00000 0.242536
$$613$$ 14.0000 0.565455 0.282727 0.959200i $$-0.408761\pi$$
0.282727 + 0.959200i $$0.408761\pi$$
$$614$$ 0 0
$$615$$ −7.00000 −0.282267
$$616$$ 0 0
$$617$$ −33.0000 −1.32853 −0.664265 0.747497i $$-0.731255\pi$$
−0.664265 + 0.747497i $$0.731255\pi$$
$$618$$ 0 0
$$619$$ −4.00000 −0.160774 −0.0803868 0.996764i $$-0.525616\pi$$
−0.0803868 + 0.996764i $$0.525616\pi$$
$$620$$ −10.0000 −0.401610
$$621$$ 1.00000 0.0401286
$$622$$ 0 0
$$623$$ 24.0000 0.961540
$$624$$ 0 0
$$625$$ 1.00000 0.0400000
$$626$$ 0 0
$$627$$ 32.0000 1.27796
$$628$$ 6.00000 0.239426
$$629$$ 27.0000 1.07656
$$630$$ 0 0
$$631$$ 38.0000 1.51276 0.756378 0.654135i $$-0.226967\pi$$
0.756378 + 0.654135i $$0.226967\pi$$
$$632$$ 0 0
$$633$$ 7.00000 0.278225
$$634$$ 0 0
$$635$$ 18.0000 0.714308
$$636$$ −26.0000 −1.03097
$$637$$ 0 0
$$638$$ 0 0
$$639$$ −13.0000 −0.514272
$$640$$ 0 0
$$641$$ −20.0000 −0.789953 −0.394976 0.918691i $$-0.629247\pi$$
−0.394976 + 0.918691i $$0.629247\pi$$
$$642$$ 0 0
$$643$$ 13.0000 0.512670 0.256335 0.966588i $$-0.417485\pi$$
0.256335 + 0.966588i $$0.417485\pi$$
$$644$$ 6.00000 0.236433
$$645$$ −4.00000 −0.157500
$$646$$ 0 0
$$647$$ 12.0000 0.471769 0.235884 0.971781i $$-0.424201\pi$$
0.235884 + 0.971781i $$0.424201\pi$$
$$648$$ 0 0
$$649$$ 12.0000 0.471041
$$650$$ 0 0
$$651$$ 15.0000 0.587896
$$652$$ 4.00000 0.156652
$$653$$ 16.0000 0.626128 0.313064 0.949732i $$-0.398644\pi$$
0.313064 + 0.949732i $$0.398644\pi$$
$$654$$ 0 0
$$655$$ 4.00000 0.156293
$$656$$ 28.0000 1.09322
$$657$$ −4.00000 −0.156055
$$658$$ 0 0
$$659$$ 12.0000 0.467454 0.233727 0.972302i $$-0.424908\pi$$
0.233727 + 0.972302i $$0.424908\pi$$
$$660$$ −8.00000 −0.311400
$$661$$ −14.0000 −0.544537 −0.272268 0.962221i $$-0.587774\pi$$
−0.272268 + 0.962221i $$0.587774\pi$$
$$662$$ 0 0
$$663$$ 0 0
$$664$$ 0 0
$$665$$ −24.0000 −0.930680
$$666$$ 0 0
$$667$$ 9.00000 0.348481
$$668$$ 44.0000 1.70241
$$669$$ 10.0000 0.386622
$$670$$ 0 0
$$671$$ 56.0000 2.16186
$$672$$ 0 0
$$673$$ −34.0000 −1.31060 −0.655302 0.755367i $$-0.727459\pi$$
−0.655302 + 0.755367i $$0.727459\pi$$
$$674$$ 0 0
$$675$$ 1.00000 0.0384900
$$676$$ 26.0000 1.00000
$$677$$ −45.0000 −1.72949 −0.864745 0.502211i $$-0.832520\pi$$
−0.864745 + 0.502211i $$0.832520\pi$$
$$678$$ 0 0
$$679$$ −30.0000 −1.15129
$$680$$ 0 0
$$681$$ −4.00000 −0.153280
$$682$$ 0 0
$$683$$ 26.0000 0.994862 0.497431 0.867503i $$-0.334277\pi$$
0.497431 + 0.867503i $$0.334277\pi$$
$$684$$ 16.0000 0.611775
$$685$$ −2.00000 −0.0764161
$$686$$ 0 0
$$687$$ 4.00000 0.152610
$$688$$ 16.0000 0.609994
$$689$$ 0 0
$$690$$ 0 0
$$691$$ 8.00000 0.304334 0.152167 0.988355i $$-0.451375\pi$$
0.152167 + 0.988355i $$0.451375\pi$$
$$692$$ 48.0000 1.82469
$$693$$ 12.0000 0.455842
$$694$$ 0 0
$$695$$ 3.00000 0.113796
$$696$$ 0 0
$$697$$ −21.0000 −0.795432
$$698$$ 0 0
$$699$$ −8.00000 −0.302588
$$700$$ 6.00000 0.226779
$$701$$ −6.00000 −0.226617 −0.113308 0.993560i $$-0.536145\pi$$
−0.113308 + 0.993560i $$0.536145\pi$$
$$702$$ 0 0
$$703$$ 72.0000 2.71553
$$704$$ 32.0000 1.20605
$$705$$ 2.00000 0.0753244
$$706$$ 0 0
$$707$$ −27.0000 −1.01544
$$708$$ 6.00000 0.225494
$$709$$ 22.0000 0.826227 0.413114 0.910679i $$-0.364441\pi$$
0.413114 + 0.910679i $$0.364441\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ 0 0
$$713$$ −5.00000 −0.187251
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 0 0
$$717$$ 13.0000 0.485494
$$718$$ 0 0
$$719$$ −7.00000 −0.261056 −0.130528 0.991445i $$-0.541667\pi$$
−0.130528 + 0.991445i $$0.541667\pi$$
$$720$$ −4.00000 −0.149071
$$721$$ −24.0000 −0.893807
$$722$$ 0 0
$$723$$ 20.0000 0.743808
$$724$$ 0 0
$$725$$ 9.00000 0.334252
$$726$$ 0 0
$$727$$ −37.0000 −1.37225 −0.686127 0.727482i $$-0.740691\pi$$
−0.686127 + 0.727482i $$0.740691\pi$$
$$728$$ 0 0
$$729$$ 1.00000 0.0370370
$$730$$ 0 0
$$731$$ −12.0000 −0.443836
$$732$$ 28.0000 1.03491
$$733$$ 51.0000 1.88373 0.941864 0.335994i $$-0.109072\pi$$
0.941864 + 0.335994i $$0.109072\pi$$
$$734$$ 0 0
$$735$$ −2.00000 −0.0737711
$$736$$ 0 0
$$737$$ −52.0000 −1.91544
$$738$$ 0 0
$$739$$ 27.0000 0.993211 0.496606 0.867976i $$-0.334580\pi$$
0.496606 + 0.867976i $$0.334580\pi$$
$$740$$ −18.0000 −0.661693
$$741$$ 0 0
$$742$$ 0 0
$$743$$ −48.0000 −1.76095 −0.880475 0.474093i $$-0.842776\pi$$
−0.880475 + 0.474093i $$0.842776\pi$$
$$744$$ 0 0
$$745$$ 14.0000 0.512920
$$746$$ 0 0
$$747$$ −1.00000 −0.0365881
$$748$$ −24.0000 −0.877527
$$749$$ −33.0000 −1.20579
$$750$$ 0 0
$$751$$ −16.0000 −0.583848 −0.291924 0.956441i $$-0.594295\pi$$
−0.291924 + 0.956441i $$0.594295\pi$$
$$752$$ −8.00000 −0.291730
$$753$$ 18.0000 0.655956
$$754$$ 0 0
$$755$$ 8.00000 0.291150
$$756$$ 6.00000 0.218218
$$757$$ 45.0000 1.63555 0.817776 0.575536i $$-0.195207\pi$$
0.817776 + 0.575536i $$0.195207\pi$$
$$758$$ 0 0
$$759$$ −4.00000 −0.145191
$$760$$ 0 0
$$761$$ −41.0000 −1.48625 −0.743124 0.669153i $$-0.766657\pi$$
−0.743124 + 0.669153i $$0.766657\pi$$
$$762$$ 0 0
$$763$$ 48.0000 1.73772
$$764$$ 4.00000 0.144715
$$765$$ 3.00000 0.108465
$$766$$ 0 0
$$767$$ 0 0
$$768$$ 16.0000 0.577350
$$769$$ 6.00000 0.216366 0.108183 0.994131i $$-0.465497\pi$$
0.108183 + 0.994131i $$0.465497\pi$$
$$770$$ 0 0
$$771$$ 18.0000 0.648254
$$772$$ −8.00000 −0.287926
$$773$$ −46.0000 −1.65451 −0.827253 0.561830i $$-0.810097\pi$$
−0.827253 + 0.561830i $$0.810097\pi$$
$$774$$ 0 0
$$775$$ −5.00000 −0.179605
$$776$$ 0 0
$$777$$ 27.0000 0.968620
$$778$$ 0 0
$$779$$ −56.0000 −2.00641
$$780$$ 0 0
$$781$$ 52.0000 1.86071
$$782$$ 0 0
$$783$$ 9.00000 0.321634
$$784$$ 8.00000 0.285714
$$785$$ 3.00000 0.107075
$$786$$ 0 0
$$787$$ −27.0000 −0.962446 −0.481223 0.876598i $$-0.659807\pi$$
−0.481223 + 0.876598i $$0.659807\pi$$
$$788$$ −4.00000 −0.142494
$$789$$ −23.0000 −0.818822
$$790$$ 0 0
$$791$$ −3.00000 −0.106668
$$792$$ 0 0
$$793$$ 0 0
$$794$$ 0 0
$$795$$ −13.0000 −0.461062
$$796$$ 36.0000 1.27599
$$797$$ 27.0000 0.956389 0.478195 0.878254i $$-0.341291\pi$$
0.478195 + 0.878254i $$0.341291\pi$$
$$798$$ 0 0
$$799$$ 6.00000 0.212265
$$800$$ 0 0
$$801$$ −8.00000 −0.282666
$$802$$ 0 0
$$803$$ 16.0000 0.564628
$$804$$ −26.0000 −0.916949
$$805$$ 3.00000 0.105736
$$806$$ 0 0
$$807$$ −17.0000 −0.598428
$$808$$ 0 0
$$809$$ −39.0000 −1.37117 −0.685583 0.727994i $$-0.740453\pi$$
−0.685583 + 0.727994i $$0.740453\pi$$
$$810$$ 0 0
$$811$$ −25.0000 −0.877869 −0.438934 0.898519i $$-0.644644\pi$$
−0.438934 + 0.898519i $$0.644644\pi$$
$$812$$ 54.0000 1.89503
$$813$$ −23.0000 −0.806645
$$814$$ 0 0
$$815$$ 2.00000 0.0700569
$$816$$ −12.0000 −0.420084
$$817$$ −32.0000 −1.11954
$$818$$ 0 0
$$819$$ 0 0
$$820$$ 14.0000 0.488901
$$821$$ −10.0000 −0.349002 −0.174501 0.984657i $$-0.555831\pi$$
−0.174501 + 0.984657i $$0.555831\pi$$
$$822$$ 0 0
$$823$$ −16.0000 −0.557725 −0.278862 0.960331i $$-0.589957\pi$$
−0.278862 + 0.960331i $$0.589957\pi$$
$$824$$ 0 0
$$825$$ −4.00000 −0.139262
$$826$$ 0 0
$$827$$ 37.0000 1.28662 0.643308 0.765607i $$-0.277561\pi$$
0.643308 + 0.765607i $$0.277561\pi$$
$$828$$ −2.00000 −0.0695048
$$829$$ −27.0000 −0.937749 −0.468874 0.883265i $$-0.655340\pi$$
−0.468874 + 0.883265i $$0.655340\pi$$
$$830$$ 0 0
$$831$$ 18.0000 0.624413
$$832$$ 0 0
$$833$$ −6.00000 −0.207888
$$834$$ 0 0
$$835$$ 22.0000 0.761341
$$836$$ −64.0000 −2.21349
$$837$$ −5.00000 −0.172825
$$838$$ 0 0
$$839$$ −6.00000 −0.207143 −0.103572 0.994622i $$-0.533027\pi$$
−0.103572 + 0.994622i $$0.533027\pi$$
$$840$$ 0 0
$$841$$ 52.0000 1.79310
$$842$$ 0 0
$$843$$ 24.0000 0.826604
$$844$$ −14.0000 −0.481900
$$845$$ 13.0000 0.447214
$$846$$ 0 0
$$847$$ −15.0000 −0.515406
$$848$$ 52.0000 1.78569
$$849$$ −5.00000 −0.171600
$$850$$ 0 0
$$851$$ −9.00000 −0.308516
$$852$$ 26.0000 0.890745
$$853$$ −46.0000 −1.57501 −0.787505 0.616308i $$-0.788628\pi$$
−0.787505 + 0.616308i $$0.788628\pi$$
$$854$$ 0 0
$$855$$ 8.00000 0.273594
$$856$$ 0 0
$$857$$ −18.0000 −0.614868 −0.307434 0.951569i $$-0.599470\pi$$
−0.307434 + 0.951569i $$0.599470\pi$$
$$858$$ 0 0
$$859$$ 43.0000 1.46714 0.733571 0.679613i $$-0.237852\pi$$
0.733571 + 0.679613i $$0.237852\pi$$
$$860$$ 8.00000 0.272798
$$861$$ −21.0000 −0.715678
$$862$$ 0 0
$$863$$ 54.0000 1.83818 0.919091 0.394046i $$-0.128925\pi$$
0.919091 + 0.394046i $$0.128925\pi$$
$$864$$ 0 0
$$865$$ 24.0000 0.816024
$$866$$ 0 0
$$867$$ −8.00000 −0.271694
$$868$$ −30.0000 −1.01827
$$869$$ 0 0
$$870$$ 0 0
$$871$$ 0 0
$$872$$ 0 0
$$873$$ 10.0000 0.338449
$$874$$ 0 0
$$875$$ 3.00000 0.101419
$$876$$ 8.00000 0.270295
$$877$$ −22.0000 −0.742887 −0.371444 0.928456i $$-0.621137\pi$$
−0.371444 + 0.928456i $$0.621137\pi$$
$$878$$ 0 0
$$879$$ 19.0000 0.640854
$$880$$ 16.0000 0.539360
$$881$$ 30.0000 1.01073 0.505363 0.862907i $$-0.331359\pi$$
0.505363 + 0.862907i $$0.331359\pi$$
$$882$$ 0 0
$$883$$ −34.0000 −1.14419 −0.572096 0.820187i $$-0.693869\pi$$
−0.572096 + 0.820187i $$0.693869\pi$$
$$884$$ 0 0
$$885$$ 3.00000 0.100844
$$886$$ 0 0
$$887$$ −46.0000 −1.54453 −0.772264 0.635301i $$-0.780876\pi$$
−0.772264 + 0.635301i $$0.780876\pi$$
$$888$$ 0 0
$$889$$ 54.0000 1.81110
$$890$$ 0 0
$$891$$ −4.00000 −0.134005
$$892$$ −20.0000 −0.669650
$$893$$ 16.0000 0.535420
$$894$$ 0 0
$$895$$ 0 0
$$896$$ 0 0
$$897$$ 0 0
$$898$$ 0 0
$$899$$ −45.0000 −1.50083
$$900$$ −2.00000 −0.0666667
$$901$$ −39.0000 −1.29928
$$902$$ 0 0
$$903$$ −12.0000 −0.399335
$$904$$ 0 0
$$905$$ 0 0
$$906$$ 0 0
$$907$$ 53.0000 1.75984 0.879918 0.475125i $$-0.157597\pi$$
0.879918 + 0.475125i $$0.157597\pi$$
$$908$$ 8.00000 0.265489
$$909$$ 9.00000 0.298511
$$910$$ 0 0
$$911$$ −40.0000 −1.32526 −0.662630 0.748947i $$-0.730560\pi$$
−0.662630 + 0.748947i $$0.730560\pi$$
$$912$$ −32.0000 −1.05963
$$913$$ 4.00000 0.132381
$$914$$ 0 0
$$915$$ 14.0000 0.462826
$$916$$ −8.00000 −0.264327
$$917$$ 12.0000 0.396275
$$918$$ 0 0
$$919$$ 24.0000 0.791687 0.395843 0.918318i $$-0.370452\pi$$
0.395843 + 0.918318i $$0.370452\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ 0 0
$$923$$ 0 0
$$924$$ −24.0000 −0.789542
$$925$$ −9.00000 −0.295918
$$926$$ 0 0
$$927$$ 8.00000 0.262754
$$928$$ 0 0
$$929$$ 57.0000 1.87011 0.935055 0.354504i $$-0.115350\pi$$
0.935055 + 0.354504i $$0.115350\pi$$
$$930$$ 0 0
$$931$$ −16.0000 −0.524379
$$932$$ 16.0000 0.524097
$$933$$ −8.00000 −0.261908
$$934$$ 0 0
$$935$$ −12.0000 −0.392442
$$936$$ 0 0
$$937$$ 10.0000 0.326686 0.163343 0.986569i $$-0.447772\pi$$
0.163343 + 0.986569i $$0.447772\pi$$
$$938$$ 0 0
$$939$$ 25.0000 0.815844
$$940$$ −4.00000 −0.130466
$$941$$ 10.0000 0.325991 0.162995 0.986627i $$-0.447884\pi$$
0.162995 + 0.986627i $$0.447884\pi$$
$$942$$ 0 0
$$943$$ 7.00000 0.227951
$$944$$ −12.0000 −0.390567
$$945$$ 3.00000 0.0975900
$$946$$ 0 0
$$947$$ −8.00000 −0.259965 −0.129983 0.991516i $$-0.541492\pi$$
−0.129983 + 0.991516i $$0.541492\pi$$
$$948$$ 0 0
$$949$$ 0 0
$$950$$ 0 0
$$951$$ 30.0000 0.972817
$$952$$ 0 0
$$953$$ −22.0000 −0.712650 −0.356325 0.934362i $$-0.615970\pi$$
−0.356325 + 0.934362i $$0.615970\pi$$
$$954$$ 0 0
$$955$$ 2.00000 0.0647185
$$956$$ −26.0000 −0.840900
$$957$$ −36.0000 −1.16371
$$958$$ 0 0
$$959$$ −6.00000 −0.193750
$$960$$ 8.00000 0.258199
$$961$$ −6.00000 −0.193548
$$962$$ 0 0
$$963$$ 11.0000 0.354470
$$964$$ −40.0000 −1.28831
$$965$$ −4.00000 −0.128765
$$966$$ 0 0
$$967$$ 10.0000 0.321578 0.160789 0.986989i $$-0.448596\pi$$
0.160789 + 0.986989i $$0.448596\pi$$
$$968$$ 0 0
$$969$$ 24.0000 0.770991
$$970$$ 0 0
$$971$$ −12.0000 −0.385098 −0.192549 0.981287i $$-0.561675\pi$$
−0.192549 + 0.981287i $$0.561675\pi$$
$$972$$ −2.00000 −0.0641500
$$973$$ 9.00000 0.288527
$$974$$ 0 0
$$975$$ 0 0
$$976$$ −56.0000 −1.79252
$$977$$ −25.0000 −0.799821 −0.399910 0.916554i $$-0.630959\pi$$
−0.399910 + 0.916554i $$0.630959\pi$$
$$978$$ 0 0
$$979$$ 32.0000 1.02272
$$980$$ 4.00000 0.127775
$$981$$ −16.0000 −0.510841
$$982$$ 0 0
$$983$$ 15.0000 0.478426 0.239213 0.970967i $$-0.423111\pi$$
0.239213 + 0.970967i $$0.423111\pi$$
$$984$$ 0 0
$$985$$ −2.00000 −0.0637253
$$986$$ 0 0
$$987$$ 6.00000 0.190982
$$988$$ 0 0
$$989$$ 4.00000 0.127193
$$990$$ 0 0
$$991$$ −33.0000 −1.04828 −0.524140 0.851632i $$-0.675613\pi$$
−0.524140 + 0.851632i $$0.675613\pi$$
$$992$$ 0 0
$$993$$ 7.00000 0.222138
$$994$$ 0 0
$$995$$ 18.0000 0.570638
$$996$$ 2.00000 0.0633724
$$997$$ 8.00000 0.253363 0.126681 0.991943i $$-0.459567\pi$$
0.126681 + 0.991943i $$0.459567\pi$$
$$998$$ 0 0
$$999$$ −9.00000 −0.284747
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 345.2.a.d.1.1 1
3.2 odd 2 1035.2.a.d.1.1 1
4.3 odd 2 5520.2.a.h.1.1 1
5.2 odd 4 1725.2.b.j.1174.1 2
5.3 odd 4 1725.2.b.j.1174.2 2
5.4 even 2 1725.2.a.k.1.1 1
15.14 odd 2 5175.2.a.o.1.1 1
23.22 odd 2 7935.2.a.i.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
345.2.a.d.1.1 1 1.1 even 1 trivial
1035.2.a.d.1.1 1 3.2 odd 2
1725.2.a.k.1.1 1 5.4 even 2
1725.2.b.j.1174.1 2 5.2 odd 4
1725.2.b.j.1174.2 2 5.3 odd 4
5175.2.a.o.1.1 1 15.14 odd 2
5520.2.a.h.1.1 1 4.3 odd 2
7935.2.a.i.1.1 1 23.22 odd 2