# Properties

 Label 378.2.a.d.1.1 Level $378$ Weight $2$ Character 378.1 Self dual yes Analytic conductor $3.018$ Analytic rank $0$ 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] = [378,2,Mod(1,378)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(378, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("378.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$378 = 2 \cdot 3^{3} \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 378.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: $$3.01834519640$$ Analytic rank: $$0$$ 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 $$\chi$$ $$=$$ 378.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{7} -1.00000 q^{8} +O(q^{10})$$ $$q-1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{7} -1.00000 q^{8} +5.00000 q^{13} -1.00000 q^{14} +1.00000 q^{16} -3.00000 q^{17} +2.00000 q^{19} +9.00000 q^{23} -5.00000 q^{25} -5.00000 q^{26} +1.00000 q^{28} +3.00000 q^{29} +5.00000 q^{31} -1.00000 q^{32} +3.00000 q^{34} +2.00000 q^{37} -2.00000 q^{38} +6.00000 q^{41} -1.00000 q^{43} -9.00000 q^{46} +6.00000 q^{47} +1.00000 q^{49} +5.00000 q^{50} +5.00000 q^{52} -3.00000 q^{53} -1.00000 q^{56} -3.00000 q^{58} +3.00000 q^{59} -10.0000 q^{61} -5.00000 q^{62} +1.00000 q^{64} -13.0000 q^{67} -3.00000 q^{68} -9.00000 q^{71} +2.00000 q^{73} -2.00000 q^{74} +2.00000 q^{76} -10.0000 q^{79} -6.00000 q^{82} +12.0000 q^{83} +1.00000 q^{86} -15.0000 q^{89} +5.00000 q^{91} +9.00000 q^{92} -6.00000 q^{94} +8.00000 q^{97} -1.00000 q^{98} +O(q^{100})$$

## 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
$$5$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$6$$ 0 0
$$7$$ 1.00000 0.377964
$$8$$ −1.00000 −0.353553
$$9$$ 0 0
$$10$$ 0 0
$$11$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$12$$ 0 0
$$13$$ 5.00000 1.38675 0.693375 0.720577i $$-0.256123\pi$$
0.693375 + 0.720577i $$0.256123\pi$$
$$14$$ −1.00000 −0.267261
$$15$$ 0 0
$$16$$ 1.00000 0.250000
$$17$$ −3.00000 −0.727607 −0.363803 0.931476i $$-0.618522\pi$$
−0.363803 + 0.931476i $$0.618522\pi$$
$$18$$ 0 0
$$19$$ 2.00000 0.458831 0.229416 0.973329i $$-0.426318\pi$$
0.229416 + 0.973329i $$0.426318\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 0 0
$$23$$ 9.00000 1.87663 0.938315 0.345782i $$-0.112386\pi$$
0.938315 + 0.345782i $$0.112386\pi$$
$$24$$ 0 0
$$25$$ −5.00000 −1.00000
$$26$$ −5.00000 −0.980581
$$27$$ 0 0
$$28$$ 1.00000 0.188982
$$29$$ 3.00000 0.557086 0.278543 0.960424i $$-0.410149\pi$$
0.278543 + 0.960424i $$0.410149\pi$$
$$30$$ 0 0
$$31$$ 5.00000 0.898027 0.449013 0.893525i $$-0.351776\pi$$
0.449013 + 0.893525i $$0.351776\pi$$
$$32$$ −1.00000 −0.176777
$$33$$ 0 0
$$34$$ 3.00000 0.514496
$$35$$ 0 0
$$36$$ 0 0
$$37$$ 2.00000 0.328798 0.164399 0.986394i $$-0.447432\pi$$
0.164399 + 0.986394i $$0.447432\pi$$
$$38$$ −2.00000 −0.324443
$$39$$ 0 0
$$40$$ 0 0
$$41$$ 6.00000 0.937043 0.468521 0.883452i $$-0.344787\pi$$
0.468521 + 0.883452i $$0.344787\pi$$
$$42$$ 0 0
$$43$$ −1.00000 −0.152499 −0.0762493 0.997089i $$-0.524294\pi$$
−0.0762493 + 0.997089i $$0.524294\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ −9.00000 −1.32698
$$47$$ 6.00000 0.875190 0.437595 0.899172i $$-0.355830\pi$$
0.437595 + 0.899172i $$0.355830\pi$$
$$48$$ 0 0
$$49$$ 1.00000 0.142857
$$50$$ 5.00000 0.707107
$$51$$ 0 0
$$52$$ 5.00000 0.693375
$$53$$ −3.00000 −0.412082 −0.206041 0.978543i $$-0.566058\pi$$
−0.206041 + 0.978543i $$0.566058\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ −1.00000 −0.133631
$$57$$ 0 0
$$58$$ −3.00000 −0.393919
$$59$$ 3.00000 0.390567 0.195283 0.980747i $$-0.437437\pi$$
0.195283 + 0.980747i $$0.437437\pi$$
$$60$$ 0 0
$$61$$ −10.0000 −1.28037 −0.640184 0.768221i $$-0.721142\pi$$
−0.640184 + 0.768221i $$0.721142\pi$$
$$62$$ −5.00000 −0.635001
$$63$$ 0 0
$$64$$ 1.00000 0.125000
$$65$$ 0 0
$$66$$ 0 0
$$67$$ −13.0000 −1.58820 −0.794101 0.607785i $$-0.792058\pi$$
−0.794101 + 0.607785i $$0.792058\pi$$
$$68$$ −3.00000 −0.363803
$$69$$ 0 0
$$70$$ 0 0
$$71$$ −9.00000 −1.06810 −0.534052 0.845452i $$-0.679331\pi$$
−0.534052 + 0.845452i $$0.679331\pi$$
$$72$$ 0 0
$$73$$ 2.00000 0.234082 0.117041 0.993127i $$-0.462659\pi$$
0.117041 + 0.993127i $$0.462659\pi$$
$$74$$ −2.00000 −0.232495
$$75$$ 0 0
$$76$$ 2.00000 0.229416
$$77$$ 0 0
$$78$$ 0 0
$$79$$ −10.0000 −1.12509 −0.562544 0.826767i $$-0.690177\pi$$
−0.562544 + 0.826767i $$0.690177\pi$$
$$80$$ 0 0
$$81$$ 0 0
$$82$$ −6.00000 −0.662589
$$83$$ 12.0000 1.31717 0.658586 0.752506i $$-0.271155\pi$$
0.658586 + 0.752506i $$0.271155\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 1.00000 0.107833
$$87$$ 0 0
$$88$$ 0 0
$$89$$ −15.0000 −1.59000 −0.794998 0.606612i $$-0.792528\pi$$
−0.794998 + 0.606612i $$0.792528\pi$$
$$90$$ 0 0
$$91$$ 5.00000 0.524142
$$92$$ 9.00000 0.938315
$$93$$ 0 0
$$94$$ −6.00000 −0.618853
$$95$$ 0 0
$$96$$ 0 0
$$97$$ 8.00000 0.812277 0.406138 0.913812i $$-0.366875\pi$$
0.406138 + 0.913812i $$0.366875\pi$$
$$98$$ −1.00000 −0.101015
$$99$$ 0 0
$$100$$ −5.00000 −0.500000
$$101$$ −18.0000 −1.79107 −0.895533 0.444994i $$-0.853206\pi$$
−0.895533 + 0.444994i $$0.853206\pi$$
$$102$$ 0 0
$$103$$ −13.0000 −1.28093 −0.640464 0.767988i $$-0.721258\pi$$
−0.640464 + 0.767988i $$0.721258\pi$$
$$104$$ −5.00000 −0.490290
$$105$$ 0 0
$$106$$ 3.00000 0.291386
$$107$$ 12.0000 1.16008 0.580042 0.814587i $$-0.303036\pi$$
0.580042 + 0.814587i $$0.303036\pi$$
$$108$$ 0 0
$$109$$ −16.0000 −1.53252 −0.766261 0.642529i $$-0.777885\pi$$
−0.766261 + 0.642529i $$0.777885\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ 1.00000 0.0944911
$$113$$ −12.0000 −1.12887 −0.564433 0.825479i $$-0.690905\pi$$
−0.564433 + 0.825479i $$0.690905\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 3.00000 0.278543
$$117$$ 0 0
$$118$$ −3.00000 −0.276172
$$119$$ −3.00000 −0.275010
$$120$$ 0 0
$$121$$ −11.0000 −1.00000
$$122$$ 10.0000 0.905357
$$123$$ 0 0
$$124$$ 5.00000 0.449013
$$125$$ 0 0
$$126$$ 0 0
$$127$$ 20.0000 1.77471 0.887357 0.461084i $$-0.152539\pi$$
0.887357 + 0.461084i $$0.152539\pi$$
$$128$$ −1.00000 −0.0883883
$$129$$ 0 0
$$130$$ 0 0
$$131$$ 9.00000 0.786334 0.393167 0.919467i $$-0.371379\pi$$
0.393167 + 0.919467i $$0.371379\pi$$
$$132$$ 0 0
$$133$$ 2.00000 0.173422
$$134$$ 13.0000 1.12303
$$135$$ 0 0
$$136$$ 3.00000 0.257248
$$137$$ −18.0000 −1.53784 −0.768922 0.639343i $$-0.779207\pi$$
−0.768922 + 0.639343i $$0.779207\pi$$
$$138$$ 0 0
$$139$$ 14.0000 1.18746 0.593732 0.804663i $$-0.297654\pi$$
0.593732 + 0.804663i $$0.297654\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 9.00000 0.755263
$$143$$ 0 0
$$144$$ 0 0
$$145$$ 0 0
$$146$$ −2.00000 −0.165521
$$147$$ 0 0
$$148$$ 2.00000 0.164399
$$149$$ −9.00000 −0.737309 −0.368654 0.929567i $$-0.620181\pi$$
−0.368654 + 0.929567i $$0.620181\pi$$
$$150$$ 0 0
$$151$$ −10.0000 −0.813788 −0.406894 0.913475i $$-0.633388\pi$$
−0.406894 + 0.913475i $$0.633388\pi$$
$$152$$ −2.00000 −0.162221
$$153$$ 0 0
$$154$$ 0 0
$$155$$ 0 0
$$156$$ 0 0
$$157$$ 23.0000 1.83560 0.917800 0.397043i $$-0.129964\pi$$
0.917800 + 0.397043i $$0.129964\pi$$
$$158$$ 10.0000 0.795557
$$159$$ 0 0
$$160$$ 0 0
$$161$$ 9.00000 0.709299
$$162$$ 0 0
$$163$$ 11.0000 0.861586 0.430793 0.902451i $$-0.358234\pi$$
0.430793 + 0.902451i $$0.358234\pi$$
$$164$$ 6.00000 0.468521
$$165$$ 0 0
$$166$$ −12.0000 −0.931381
$$167$$ −12.0000 −0.928588 −0.464294 0.885681i $$-0.653692\pi$$
−0.464294 + 0.885681i $$0.653692\pi$$
$$168$$ 0 0
$$169$$ 12.0000 0.923077
$$170$$ 0 0
$$171$$ 0 0
$$172$$ −1.00000 −0.0762493
$$173$$ 24.0000 1.82469 0.912343 0.409426i $$-0.134271\pi$$
0.912343 + 0.409426i $$0.134271\pi$$
$$174$$ 0 0
$$175$$ −5.00000 −0.377964
$$176$$ 0 0
$$177$$ 0 0
$$178$$ 15.0000 1.12430
$$179$$ 24.0000 1.79384 0.896922 0.442189i $$-0.145798\pi$$
0.896922 + 0.442189i $$0.145798\pi$$
$$180$$ 0 0
$$181$$ −7.00000 −0.520306 −0.260153 0.965567i $$-0.583773\pi$$
−0.260153 + 0.965567i $$0.583773\pi$$
$$182$$ −5.00000 −0.370625
$$183$$ 0 0
$$184$$ −9.00000 −0.663489
$$185$$ 0 0
$$186$$ 0 0
$$187$$ 0 0
$$188$$ 6.00000 0.437595
$$189$$ 0 0
$$190$$ 0 0
$$191$$ −12.0000 −0.868290 −0.434145 0.900843i $$-0.642949\pi$$
−0.434145 + 0.900843i $$0.642949\pi$$
$$192$$ 0 0
$$193$$ 5.00000 0.359908 0.179954 0.983675i $$-0.442405\pi$$
0.179954 + 0.983675i $$0.442405\pi$$
$$194$$ −8.00000 −0.574367
$$195$$ 0 0
$$196$$ 1.00000 0.0714286
$$197$$ −18.0000 −1.28245 −0.641223 0.767354i $$-0.721573\pi$$
−0.641223 + 0.767354i $$0.721573\pi$$
$$198$$ 0 0
$$199$$ 11.0000 0.779769 0.389885 0.920864i $$-0.372515\pi$$
0.389885 + 0.920864i $$0.372515\pi$$
$$200$$ 5.00000 0.353553
$$201$$ 0 0
$$202$$ 18.0000 1.26648
$$203$$ 3.00000 0.210559
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 13.0000 0.905753
$$207$$ 0 0
$$208$$ 5.00000 0.346688
$$209$$ 0 0
$$210$$ 0 0
$$211$$ −13.0000 −0.894957 −0.447478 0.894295i $$-0.647678\pi$$
−0.447478 + 0.894295i $$0.647678\pi$$
$$212$$ −3.00000 −0.206041
$$213$$ 0 0
$$214$$ −12.0000 −0.820303
$$215$$ 0 0
$$216$$ 0 0
$$217$$ 5.00000 0.339422
$$218$$ 16.0000 1.08366
$$219$$ 0 0
$$220$$ 0 0
$$221$$ −15.0000 −1.00901
$$222$$ 0 0
$$223$$ 8.00000 0.535720 0.267860 0.963458i $$-0.413684\pi$$
0.267860 + 0.963458i $$0.413684\pi$$
$$224$$ −1.00000 −0.0668153
$$225$$ 0 0
$$226$$ 12.0000 0.798228
$$227$$ −9.00000 −0.597351 −0.298675 0.954355i $$-0.596545\pi$$
−0.298675 + 0.954355i $$0.596545\pi$$
$$228$$ 0 0
$$229$$ 14.0000 0.925146 0.462573 0.886581i $$-0.346926\pi$$
0.462573 + 0.886581i $$0.346926\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ −3.00000 −0.196960
$$233$$ 12.0000 0.786146 0.393073 0.919507i $$-0.371412\pi$$
0.393073 + 0.919507i $$0.371412\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ 3.00000 0.195283
$$237$$ 0 0
$$238$$ 3.00000 0.194461
$$239$$ −12.0000 −0.776215 −0.388108 0.921614i $$-0.626871\pi$$
−0.388108 + 0.921614i $$0.626871\pi$$
$$240$$ 0 0
$$241$$ −10.0000 −0.644157 −0.322078 0.946713i $$-0.604381\pi$$
−0.322078 + 0.946713i $$0.604381\pi$$
$$242$$ 11.0000 0.707107
$$243$$ 0 0
$$244$$ −10.0000 −0.640184
$$245$$ 0 0
$$246$$ 0 0
$$247$$ 10.0000 0.636285
$$248$$ −5.00000 −0.317500
$$249$$ 0 0
$$250$$ 0 0
$$251$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$252$$ 0 0
$$253$$ 0 0
$$254$$ −20.0000 −1.25491
$$255$$ 0 0
$$256$$ 1.00000 0.0625000
$$257$$ −18.0000 −1.12281 −0.561405 0.827541i $$-0.689739\pi$$
−0.561405 + 0.827541i $$0.689739\pi$$
$$258$$ 0 0
$$259$$ 2.00000 0.124274
$$260$$ 0 0
$$261$$ 0 0
$$262$$ −9.00000 −0.556022
$$263$$ −9.00000 −0.554964 −0.277482 0.960731i $$-0.589500\pi$$
−0.277482 + 0.960731i $$0.589500\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ −2.00000 −0.122628
$$267$$ 0 0
$$268$$ −13.0000 −0.794101
$$269$$ 24.0000 1.46331 0.731653 0.681677i $$-0.238749\pi$$
0.731653 + 0.681677i $$0.238749\pi$$
$$270$$ 0 0
$$271$$ −7.00000 −0.425220 −0.212610 0.977137i $$-0.568196\pi$$
−0.212610 + 0.977137i $$0.568196\pi$$
$$272$$ −3.00000 −0.181902
$$273$$ 0 0
$$274$$ 18.0000 1.08742
$$275$$ 0 0
$$276$$ 0 0
$$277$$ 8.00000 0.480673 0.240337 0.970690i $$-0.422742\pi$$
0.240337 + 0.970690i $$0.422742\pi$$
$$278$$ −14.0000 −0.839664
$$279$$ 0 0
$$280$$ 0 0
$$281$$ −6.00000 −0.357930 −0.178965 0.983855i $$-0.557275\pi$$
−0.178965 + 0.983855i $$0.557275\pi$$
$$282$$ 0 0
$$283$$ −4.00000 −0.237775 −0.118888 0.992908i $$-0.537933\pi$$
−0.118888 + 0.992908i $$0.537933\pi$$
$$284$$ −9.00000 −0.534052
$$285$$ 0 0
$$286$$ 0 0
$$287$$ 6.00000 0.354169
$$288$$ 0 0
$$289$$ −8.00000 −0.470588
$$290$$ 0 0
$$291$$ 0 0
$$292$$ 2.00000 0.117041
$$293$$ 6.00000 0.350524 0.175262 0.984522i $$-0.443923\pi$$
0.175262 + 0.984522i $$0.443923\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ −2.00000 −0.116248
$$297$$ 0 0
$$298$$ 9.00000 0.521356
$$299$$ 45.0000 2.60242
$$300$$ 0 0
$$301$$ −1.00000 −0.0576390
$$302$$ 10.0000 0.575435
$$303$$ 0 0
$$304$$ 2.00000 0.114708
$$305$$ 0 0
$$306$$ 0 0
$$307$$ 2.00000 0.114146 0.0570730 0.998370i $$-0.481823\pi$$
0.0570730 + 0.998370i $$0.481823\pi$$
$$308$$ 0 0
$$309$$ 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
$$313$$ −10.0000 −0.565233 −0.282617 0.959233i $$-0.591202\pi$$
−0.282617 + 0.959233i $$0.591202\pi$$
$$314$$ −23.0000 −1.29797
$$315$$ 0 0
$$316$$ −10.0000 −0.562544
$$317$$ −30.0000 −1.68497 −0.842484 0.538721i $$-0.818908\pi$$
−0.842484 + 0.538721i $$0.818908\pi$$
$$318$$ 0 0
$$319$$ 0 0
$$320$$ 0 0
$$321$$ 0 0
$$322$$ −9.00000 −0.501550
$$323$$ −6.00000 −0.333849
$$324$$ 0 0
$$325$$ −25.0000 −1.38675
$$326$$ −11.0000 −0.609234
$$327$$ 0 0
$$328$$ −6.00000 −0.331295
$$329$$ 6.00000 0.330791
$$330$$ 0 0
$$331$$ −19.0000 −1.04433 −0.522167 0.852843i $$-0.674876\pi$$
−0.522167 + 0.852843i $$0.674876\pi$$
$$332$$ 12.0000 0.658586
$$333$$ 0 0
$$334$$ 12.0000 0.656611
$$335$$ 0 0
$$336$$ 0 0
$$337$$ −13.0000 −0.708155 −0.354078 0.935216i $$-0.615205\pi$$
−0.354078 + 0.935216i $$0.615205\pi$$
$$338$$ −12.0000 −0.652714
$$339$$ 0 0
$$340$$ 0 0
$$341$$ 0 0
$$342$$ 0 0
$$343$$ 1.00000 0.0539949
$$344$$ 1.00000 0.0539164
$$345$$ 0 0
$$346$$ −24.0000 −1.29025
$$347$$ 30.0000 1.61048 0.805242 0.592946i $$-0.202035\pi$$
0.805242 + 0.592946i $$0.202035\pi$$
$$348$$ 0 0
$$349$$ −1.00000 −0.0535288 −0.0267644 0.999642i $$-0.508520\pi$$
−0.0267644 + 0.999642i $$0.508520\pi$$
$$350$$ 5.00000 0.267261
$$351$$ 0 0
$$352$$ 0 0
$$353$$ 27.0000 1.43706 0.718532 0.695493i $$-0.244814\pi$$
0.718532 + 0.695493i $$0.244814\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ −15.0000 −0.794998
$$357$$ 0 0
$$358$$ −24.0000 −1.26844
$$359$$ 3.00000 0.158334 0.0791670 0.996861i $$-0.474774\pi$$
0.0791670 + 0.996861i $$0.474774\pi$$
$$360$$ 0 0
$$361$$ −15.0000 −0.789474
$$362$$ 7.00000 0.367912
$$363$$ 0 0
$$364$$ 5.00000 0.262071
$$365$$ 0 0
$$366$$ 0 0
$$367$$ −19.0000 −0.991792 −0.495896 0.868382i $$-0.665160\pi$$
−0.495896 + 0.868382i $$0.665160\pi$$
$$368$$ 9.00000 0.469157
$$369$$ 0 0
$$370$$ 0 0
$$371$$ −3.00000 −0.155752
$$372$$ 0 0
$$373$$ 14.0000 0.724893 0.362446 0.932005i $$-0.381942\pi$$
0.362446 + 0.932005i $$0.381942\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ −6.00000 −0.309426
$$377$$ 15.0000 0.772539
$$378$$ 0 0
$$379$$ −16.0000 −0.821865 −0.410932 0.911666i $$-0.634797\pi$$
−0.410932 + 0.911666i $$0.634797\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ 12.0000 0.613973
$$383$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ −5.00000 −0.254493
$$387$$ 0 0
$$388$$ 8.00000 0.406138
$$389$$ −18.0000 −0.912636 −0.456318 0.889817i $$-0.650832\pi$$
−0.456318 + 0.889817i $$0.650832\pi$$
$$390$$ 0 0
$$391$$ −27.0000 −1.36545
$$392$$ −1.00000 −0.0505076
$$393$$ 0 0
$$394$$ 18.0000 0.906827
$$395$$ 0 0
$$396$$ 0 0
$$397$$ 2.00000 0.100377 0.0501886 0.998740i $$-0.484018\pi$$
0.0501886 + 0.998740i $$0.484018\pi$$
$$398$$ −11.0000 −0.551380
$$399$$ 0 0
$$400$$ −5.00000 −0.250000
$$401$$ 18.0000 0.898877 0.449439 0.893311i $$-0.351624\pi$$
0.449439 + 0.893311i $$0.351624\pi$$
$$402$$ 0 0
$$403$$ 25.0000 1.24534
$$404$$ −18.0000 −0.895533
$$405$$ 0 0
$$406$$ −3.00000 −0.148888
$$407$$ 0 0
$$408$$ 0 0
$$409$$ 14.0000 0.692255 0.346128 0.938187i $$-0.387496\pi$$
0.346128 + 0.938187i $$0.387496\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ −13.0000 −0.640464
$$413$$ 3.00000 0.147620
$$414$$ 0 0
$$415$$ 0 0
$$416$$ −5.00000 −0.245145
$$417$$ 0 0
$$418$$ 0 0
$$419$$ 15.0000 0.732798 0.366399 0.930458i $$-0.380591\pi$$
0.366399 + 0.930458i $$0.380591\pi$$
$$420$$ 0 0
$$421$$ −10.0000 −0.487370 −0.243685 0.969854i $$-0.578356\pi$$
−0.243685 + 0.969854i $$0.578356\pi$$
$$422$$ 13.0000 0.632830
$$423$$ 0 0
$$424$$ 3.00000 0.145693
$$425$$ 15.0000 0.727607
$$426$$ 0 0
$$427$$ −10.0000 −0.483934
$$428$$ 12.0000 0.580042
$$429$$ 0 0
$$430$$ 0 0
$$431$$ 24.0000 1.15604 0.578020 0.816023i $$-0.303826\pi$$
0.578020 + 0.816023i $$0.303826\pi$$
$$432$$ 0 0
$$433$$ −16.0000 −0.768911 −0.384455 0.923144i $$-0.625611\pi$$
−0.384455 + 0.923144i $$0.625611\pi$$
$$434$$ −5.00000 −0.240008
$$435$$ 0 0
$$436$$ −16.0000 −0.766261
$$437$$ 18.0000 0.861057
$$438$$ 0 0
$$439$$ 35.0000 1.67046 0.835229 0.549902i $$-0.185335\pi$$
0.835229 + 0.549902i $$0.185335\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 15.0000 0.713477
$$443$$ 6.00000 0.285069 0.142534 0.989790i $$-0.454475\pi$$
0.142534 + 0.989790i $$0.454475\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ −8.00000 −0.378811
$$447$$ 0 0
$$448$$ 1.00000 0.0472456
$$449$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$450$$ 0 0
$$451$$ 0 0
$$452$$ −12.0000 −0.564433
$$453$$ 0 0
$$454$$ 9.00000 0.422391
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 35.0000 1.63723 0.818615 0.574342i $$-0.194742\pi$$
0.818615 + 0.574342i $$0.194742\pi$$
$$458$$ −14.0000 −0.654177
$$459$$ 0 0
$$460$$ 0 0
$$461$$ 30.0000 1.39724 0.698620 0.715493i $$-0.253798\pi$$
0.698620 + 0.715493i $$0.253798\pi$$
$$462$$ 0 0
$$463$$ −22.0000 −1.02243 −0.511213 0.859454i $$-0.670804\pi$$
−0.511213 + 0.859454i $$0.670804\pi$$
$$464$$ 3.00000 0.139272
$$465$$ 0 0
$$466$$ −12.0000 −0.555889
$$467$$ −24.0000 −1.11059 −0.555294 0.831654i $$-0.687394\pi$$
−0.555294 + 0.831654i $$0.687394\pi$$
$$468$$ 0 0
$$469$$ −13.0000 −0.600284
$$470$$ 0 0
$$471$$ 0 0
$$472$$ −3.00000 −0.138086
$$473$$ 0 0
$$474$$ 0 0
$$475$$ −10.0000 −0.458831
$$476$$ −3.00000 −0.137505
$$477$$ 0 0
$$478$$ 12.0000 0.548867
$$479$$ −36.0000 −1.64488 −0.822441 0.568850i $$-0.807388\pi$$
−0.822441 + 0.568850i $$0.807388\pi$$
$$480$$ 0 0
$$481$$ 10.0000 0.455961
$$482$$ 10.0000 0.455488
$$483$$ 0 0
$$484$$ −11.0000 −0.500000
$$485$$ 0 0
$$486$$ 0 0
$$487$$ 38.0000 1.72194 0.860972 0.508652i $$-0.169856\pi$$
0.860972 + 0.508652i $$0.169856\pi$$
$$488$$ 10.0000 0.452679
$$489$$ 0 0
$$490$$ 0 0
$$491$$ −12.0000 −0.541552 −0.270776 0.962642i $$-0.587280\pi$$
−0.270776 + 0.962642i $$0.587280\pi$$
$$492$$ 0 0
$$493$$ −9.00000 −0.405340
$$494$$ −10.0000 −0.449921
$$495$$ 0 0
$$496$$ 5.00000 0.224507
$$497$$ −9.00000 −0.403705
$$498$$ 0 0
$$499$$ −4.00000 −0.179065 −0.0895323 0.995984i $$-0.528537\pi$$
−0.0895323 + 0.995984i $$0.528537\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ 0 0
$$503$$ −36.0000 −1.60516 −0.802580 0.596544i $$-0.796540\pi$$
−0.802580 + 0.596544i $$0.796540\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 0 0
$$507$$ 0 0
$$508$$ 20.0000 0.887357
$$509$$ 18.0000 0.797836 0.398918 0.916987i $$-0.369386\pi$$
0.398918 + 0.916987i $$0.369386\pi$$
$$510$$ 0 0
$$511$$ 2.00000 0.0884748
$$512$$ −1.00000 −0.0441942
$$513$$ 0 0
$$514$$ 18.0000 0.793946
$$515$$ 0 0
$$516$$ 0 0
$$517$$ 0 0
$$518$$ −2.00000 −0.0878750
$$519$$ 0 0
$$520$$ 0 0
$$521$$ −3.00000 −0.131432 −0.0657162 0.997838i $$-0.520933\pi$$
−0.0657162 + 0.997838i $$0.520933\pi$$
$$522$$ 0 0
$$523$$ 20.0000 0.874539 0.437269 0.899331i $$-0.355946\pi$$
0.437269 + 0.899331i $$0.355946\pi$$
$$524$$ 9.00000 0.393167
$$525$$ 0 0
$$526$$ 9.00000 0.392419
$$527$$ −15.0000 −0.653410
$$528$$ 0 0
$$529$$ 58.0000 2.52174
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 2.00000 0.0867110
$$533$$ 30.0000 1.29944
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 13.0000 0.561514
$$537$$ 0 0
$$538$$ −24.0000 −1.03471
$$539$$ 0 0
$$540$$ 0 0
$$541$$ −16.0000 −0.687894 −0.343947 0.938989i $$-0.611764\pi$$
−0.343947 + 0.938989i $$0.611764\pi$$
$$542$$ 7.00000 0.300676
$$543$$ 0 0
$$544$$ 3.00000 0.128624
$$545$$ 0 0
$$546$$ 0 0
$$547$$ 44.0000 1.88130 0.940652 0.339372i $$-0.110215\pi$$
0.940652 + 0.339372i $$0.110215\pi$$
$$548$$ −18.0000 −0.768922
$$549$$ 0 0
$$550$$ 0 0
$$551$$ 6.00000 0.255609
$$552$$ 0 0
$$553$$ −10.0000 −0.425243
$$554$$ −8.00000 −0.339887
$$555$$ 0 0
$$556$$ 14.0000 0.593732
$$557$$ −21.0000 −0.889799 −0.444899 0.895581i $$-0.646761\pi$$
−0.444899 + 0.895581i $$0.646761\pi$$
$$558$$ 0 0
$$559$$ −5.00000 −0.211477
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 6.00000 0.253095
$$563$$ 3.00000 0.126435 0.0632175 0.998000i $$-0.479864\pi$$
0.0632175 + 0.998000i $$0.479864\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ 4.00000 0.168133
$$567$$ 0 0
$$568$$ 9.00000 0.377632
$$569$$ 24.0000 1.00613 0.503066 0.864248i $$-0.332205\pi$$
0.503066 + 0.864248i $$0.332205\pi$$
$$570$$ 0 0
$$571$$ 5.00000 0.209243 0.104622 0.994512i $$-0.466637\pi$$
0.104622 + 0.994512i $$0.466637\pi$$
$$572$$ 0 0
$$573$$ 0 0
$$574$$ −6.00000 −0.250435
$$575$$ −45.0000 −1.87663
$$576$$ 0 0
$$577$$ −34.0000 −1.41544 −0.707719 0.706494i $$-0.750276\pi$$
−0.707719 + 0.706494i $$0.750276\pi$$
$$578$$ 8.00000 0.332756
$$579$$ 0 0
$$580$$ 0 0
$$581$$ 12.0000 0.497844
$$582$$ 0 0
$$583$$ 0 0
$$584$$ −2.00000 −0.0827606
$$585$$ 0 0
$$586$$ −6.00000 −0.247858
$$587$$ −33.0000 −1.36206 −0.681028 0.732257i $$-0.738467\pi$$
−0.681028 + 0.732257i $$0.738467\pi$$
$$588$$ 0 0
$$589$$ 10.0000 0.412043
$$590$$ 0 0
$$591$$ 0 0
$$592$$ 2.00000 0.0821995
$$593$$ −42.0000 −1.72473 −0.862367 0.506284i $$-0.831019\pi$$
−0.862367 + 0.506284i $$0.831019\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ −9.00000 −0.368654
$$597$$ 0 0
$$598$$ −45.0000 −1.84019
$$599$$ 3.00000 0.122577 0.0612883 0.998120i $$-0.480479\pi$$
0.0612883 + 0.998120i $$0.480479\pi$$
$$600$$ 0 0
$$601$$ 8.00000 0.326327 0.163163 0.986599i $$-0.447830\pi$$
0.163163 + 0.986599i $$0.447830\pi$$
$$602$$ 1.00000 0.0407570
$$603$$ 0 0
$$604$$ −10.0000 −0.406894
$$605$$ 0 0
$$606$$ 0 0
$$607$$ 5.00000 0.202944 0.101472 0.994838i $$-0.467645\pi$$
0.101472 + 0.994838i $$0.467645\pi$$
$$608$$ −2.00000 −0.0811107
$$609$$ 0 0
$$610$$ 0 0
$$611$$ 30.0000 1.21367
$$612$$ 0 0
$$613$$ 38.0000 1.53481 0.767403 0.641165i $$-0.221549\pi$$
0.767403 + 0.641165i $$0.221549\pi$$
$$614$$ −2.00000 −0.0807134
$$615$$ 0 0
$$616$$ 0 0
$$617$$ 42.0000 1.69086 0.845428 0.534089i $$-0.179345\pi$$
0.845428 + 0.534089i $$0.179345\pi$$
$$618$$ 0 0
$$619$$ −10.0000 −0.401934 −0.200967 0.979598i $$-0.564408\pi$$
−0.200967 + 0.979598i $$0.564408\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 24.0000 0.962312
$$623$$ −15.0000 −0.600962
$$624$$ 0 0
$$625$$ 25.0000 1.00000
$$626$$ 10.0000 0.399680
$$627$$ 0 0
$$628$$ 23.0000 0.917800
$$629$$ −6.00000 −0.239236
$$630$$ 0 0
$$631$$ −16.0000 −0.636950 −0.318475 0.947931i $$-0.603171\pi$$
−0.318475 + 0.947931i $$0.603171\pi$$
$$632$$ 10.0000 0.397779
$$633$$ 0 0
$$634$$ 30.0000 1.19145
$$635$$ 0 0
$$636$$ 0 0
$$637$$ 5.00000 0.198107
$$638$$ 0 0
$$639$$ 0 0
$$640$$ 0 0
$$641$$ −36.0000 −1.42191 −0.710957 0.703235i $$-0.751738\pi$$
−0.710957 + 0.703235i $$0.751738\pi$$
$$642$$ 0 0
$$643$$ −40.0000 −1.57745 −0.788723 0.614749i $$-0.789257\pi$$
−0.788723 + 0.614749i $$0.789257\pi$$
$$644$$ 9.00000 0.354650
$$645$$ 0 0
$$646$$ 6.00000 0.236067
$$647$$ −30.0000 −1.17942 −0.589711 0.807614i $$-0.700758\pi$$
−0.589711 + 0.807614i $$0.700758\pi$$
$$648$$ 0 0
$$649$$ 0 0
$$650$$ 25.0000 0.980581
$$651$$ 0 0
$$652$$ 11.0000 0.430793
$$653$$ 9.00000 0.352197 0.176099 0.984373i $$-0.443652\pi$$
0.176099 + 0.984373i $$0.443652\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 6.00000 0.234261
$$657$$ 0 0
$$658$$ −6.00000 −0.233904
$$659$$ 12.0000 0.467454 0.233727 0.972302i $$-0.424908\pi$$
0.233727 + 0.972302i $$0.424908\pi$$
$$660$$ 0 0
$$661$$ 14.0000 0.544537 0.272268 0.962221i $$-0.412226\pi$$
0.272268 + 0.962221i $$0.412226\pi$$
$$662$$ 19.0000 0.738456
$$663$$ 0 0
$$664$$ −12.0000 −0.465690
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 27.0000 1.04544
$$668$$ −12.0000 −0.464294
$$669$$ 0 0
$$670$$ 0 0
$$671$$ 0 0
$$672$$ 0 0
$$673$$ −1.00000 −0.0385472 −0.0192736 0.999814i $$-0.506135\pi$$
−0.0192736 + 0.999814i $$0.506135\pi$$
$$674$$ 13.0000 0.500741
$$675$$ 0 0
$$676$$ 12.0000 0.461538
$$677$$ −12.0000 −0.461197 −0.230599 0.973049i $$-0.574068\pi$$
−0.230599 + 0.973049i $$0.574068\pi$$
$$678$$ 0 0
$$679$$ 8.00000 0.307012
$$680$$ 0 0
$$681$$ 0 0
$$682$$ 0 0
$$683$$ −12.0000 −0.459167 −0.229584 0.973289i $$-0.573736\pi$$
−0.229584 + 0.973289i $$0.573736\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ −1.00000 −0.0381802
$$687$$ 0 0
$$688$$ −1.00000 −0.0381246
$$689$$ −15.0000 −0.571454
$$690$$ 0 0
$$691$$ −28.0000 −1.06517 −0.532585 0.846376i $$-0.678779\pi$$
−0.532585 + 0.846376i $$0.678779\pi$$
$$692$$ 24.0000 0.912343
$$693$$ 0 0
$$694$$ −30.0000 −1.13878
$$695$$ 0 0
$$696$$ 0 0
$$697$$ −18.0000 −0.681799
$$698$$ 1.00000 0.0378506
$$699$$ 0 0
$$700$$ −5.00000 −0.188982
$$701$$ 18.0000 0.679851 0.339925 0.940452i $$-0.389598\pi$$
0.339925 + 0.940452i $$0.389598\pi$$
$$702$$ 0 0
$$703$$ 4.00000 0.150863
$$704$$ 0 0
$$705$$ 0 0
$$706$$ −27.0000 −1.01616
$$707$$ −18.0000 −0.676960
$$708$$ 0 0
$$709$$ −28.0000 −1.05156 −0.525781 0.850620i $$-0.676227\pi$$
−0.525781 + 0.850620i $$0.676227\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ 15.0000 0.562149
$$713$$ 45.0000 1.68526
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 24.0000 0.896922
$$717$$ 0 0
$$718$$ −3.00000 −0.111959
$$719$$ −24.0000 −0.895049 −0.447524 0.894272i $$-0.647694\pi$$
−0.447524 + 0.894272i $$0.647694\pi$$
$$720$$ 0 0
$$721$$ −13.0000 −0.484145
$$722$$ 15.0000 0.558242
$$723$$ 0 0
$$724$$ −7.00000 −0.260153
$$725$$ −15.0000 −0.557086
$$726$$ 0 0
$$727$$ 17.0000 0.630495 0.315248 0.949009i $$-0.397912\pi$$
0.315248 + 0.949009i $$0.397912\pi$$
$$728$$ −5.00000 −0.185312
$$729$$ 0 0
$$730$$ 0 0
$$731$$ 3.00000 0.110959
$$732$$ 0 0
$$733$$ 23.0000 0.849524 0.424762 0.905305i $$-0.360358\pi$$
0.424762 + 0.905305i $$0.360358\pi$$
$$734$$ 19.0000 0.701303
$$735$$ 0 0
$$736$$ −9.00000 −0.331744
$$737$$ 0 0
$$738$$ 0 0
$$739$$ 20.0000 0.735712 0.367856 0.929883i $$-0.380092\pi$$
0.367856 + 0.929883i $$0.380092\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 3.00000 0.110133
$$743$$ −3.00000 −0.110059 −0.0550297 0.998485i $$-0.517525\pi$$
−0.0550297 + 0.998485i $$0.517525\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ −14.0000 −0.512576
$$747$$ 0 0
$$748$$ 0 0
$$749$$ 12.0000 0.438470
$$750$$ 0 0
$$751$$ 14.0000 0.510867 0.255434 0.966827i $$-0.417782\pi$$
0.255434 + 0.966827i $$0.417782\pi$$
$$752$$ 6.00000 0.218797
$$753$$ 0 0
$$754$$ −15.0000 −0.546268
$$755$$ 0 0
$$756$$ 0 0
$$757$$ −16.0000 −0.581530 −0.290765 0.956795i $$-0.593910\pi$$
−0.290765 + 0.956795i $$0.593910\pi$$
$$758$$ 16.0000 0.581146
$$759$$ 0 0
$$760$$ 0 0
$$761$$ −9.00000 −0.326250 −0.163125 0.986605i $$-0.552157\pi$$
−0.163125 + 0.986605i $$0.552157\pi$$
$$762$$ 0 0
$$763$$ −16.0000 −0.579239
$$764$$ −12.0000 −0.434145
$$765$$ 0 0
$$766$$ 0 0
$$767$$ 15.0000 0.541619
$$768$$ 0 0
$$769$$ −40.0000 −1.44244 −0.721218 0.692708i $$-0.756418\pi$$
−0.721218 + 0.692708i $$0.756418\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 5.00000 0.179954
$$773$$ 24.0000 0.863220 0.431610 0.902060i $$-0.357946\pi$$
0.431610 + 0.902060i $$0.357946\pi$$
$$774$$ 0 0
$$775$$ −25.0000 −0.898027
$$776$$ −8.00000 −0.287183
$$777$$ 0 0
$$778$$ 18.0000 0.645331
$$779$$ 12.0000 0.429945
$$780$$ 0 0
$$781$$ 0 0
$$782$$ 27.0000 0.965518
$$783$$ 0 0
$$784$$ 1.00000 0.0357143
$$785$$ 0 0
$$786$$ 0 0
$$787$$ 14.0000 0.499046 0.249523 0.968369i $$-0.419726\pi$$
0.249523 + 0.968369i $$0.419726\pi$$
$$788$$ −18.0000 −0.641223
$$789$$ 0 0
$$790$$ 0 0
$$791$$ −12.0000 −0.426671
$$792$$ 0 0
$$793$$ −50.0000 −1.77555
$$794$$ −2.00000 −0.0709773
$$795$$ 0 0
$$796$$ 11.0000 0.389885
$$797$$ 6.00000 0.212531 0.106265 0.994338i $$-0.466111\pi$$
0.106265 + 0.994338i $$0.466111\pi$$
$$798$$ 0 0
$$799$$ −18.0000 −0.636794
$$800$$ 5.00000 0.176777
$$801$$ 0 0
$$802$$ −18.0000 −0.635602
$$803$$ 0 0
$$804$$ 0 0
$$805$$ 0 0
$$806$$ −25.0000 −0.880587
$$807$$ 0 0
$$808$$ 18.0000 0.633238
$$809$$ 24.0000 0.843795 0.421898 0.906644i $$-0.361364\pi$$
0.421898 + 0.906644i $$0.361364\pi$$
$$810$$ 0 0
$$811$$ 2.00000 0.0702295 0.0351147 0.999383i $$-0.488820\pi$$
0.0351147 + 0.999383i $$0.488820\pi$$
$$812$$ 3.00000 0.105279
$$813$$ 0 0
$$814$$ 0 0
$$815$$ 0 0
$$816$$ 0 0
$$817$$ −2.00000 −0.0699711
$$818$$ −14.0000 −0.489499
$$819$$ 0 0
$$820$$ 0 0
$$821$$ −39.0000 −1.36111 −0.680555 0.732697i $$-0.738261\pi$$
−0.680555 + 0.732697i $$0.738261\pi$$
$$822$$ 0 0
$$823$$ −22.0000 −0.766872 −0.383436 0.923567i $$-0.625259\pi$$
−0.383436 + 0.923567i $$0.625259\pi$$
$$824$$ 13.0000 0.452876
$$825$$ 0 0
$$826$$ −3.00000 −0.104383
$$827$$ 18.0000 0.625921 0.312961 0.949766i $$-0.398679\pi$$
0.312961 + 0.949766i $$0.398679\pi$$
$$828$$ 0 0
$$829$$ 2.00000 0.0694629 0.0347314 0.999397i $$-0.488942\pi$$
0.0347314 + 0.999397i $$0.488942\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ 5.00000 0.173344
$$833$$ −3.00000 −0.103944
$$834$$ 0 0
$$835$$ 0 0
$$836$$ 0 0
$$837$$ 0 0
$$838$$ −15.0000 −0.518166
$$839$$ −42.0000 −1.45000 −0.725001 0.688748i $$-0.758161\pi$$
−0.725001 + 0.688748i $$0.758161\pi$$
$$840$$ 0 0
$$841$$ −20.0000 −0.689655
$$842$$ 10.0000 0.344623
$$843$$ 0 0
$$844$$ −13.0000 −0.447478
$$845$$ 0 0
$$846$$ 0 0
$$847$$ −11.0000 −0.377964
$$848$$ −3.00000 −0.103020
$$849$$ 0 0
$$850$$ −15.0000 −0.514496
$$851$$ 18.0000 0.617032
$$852$$ 0 0
$$853$$ −37.0000 −1.26686 −0.633428 0.773802i $$-0.718353\pi$$
−0.633428 + 0.773802i $$0.718353\pi$$
$$854$$ 10.0000 0.342193
$$855$$ 0 0
$$856$$ −12.0000 −0.410152
$$857$$ −9.00000 −0.307434 −0.153717 0.988115i $$-0.549124\pi$$
−0.153717 + 0.988115i $$0.549124\pi$$
$$858$$ 0 0
$$859$$ 14.0000 0.477674 0.238837 0.971060i $$-0.423234\pi$$
0.238837 + 0.971060i $$0.423234\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ −24.0000 −0.817443
$$863$$ −33.0000 −1.12333 −0.561667 0.827364i $$-0.689840\pi$$
−0.561667 + 0.827364i $$0.689840\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ 16.0000 0.543702
$$867$$ 0 0
$$868$$ 5.00000 0.169711
$$869$$ 0 0
$$870$$ 0 0
$$871$$ −65.0000 −2.20244
$$872$$ 16.0000 0.541828
$$873$$ 0 0
$$874$$ −18.0000 −0.608859
$$875$$ 0 0
$$876$$ 0 0
$$877$$ 32.0000 1.08056 0.540282 0.841484i $$-0.318318\pi$$
0.540282 + 0.841484i $$0.318318\pi$$
$$878$$ −35.0000 −1.18119
$$879$$ 0 0
$$880$$ 0 0
$$881$$ 27.0000 0.909653 0.454827 0.890580i $$-0.349701\pi$$
0.454827 + 0.890580i $$0.349701\pi$$
$$882$$ 0 0
$$883$$ −7.00000 −0.235569 −0.117784 0.993039i $$-0.537579\pi$$
−0.117784 + 0.993039i $$0.537579\pi$$
$$884$$ −15.0000 −0.504505
$$885$$ 0 0
$$886$$ −6.00000 −0.201574
$$887$$ −36.0000 −1.20876 −0.604381 0.796696i $$-0.706579\pi$$
−0.604381 + 0.796696i $$0.706579\pi$$
$$888$$ 0 0
$$889$$ 20.0000 0.670778
$$890$$ 0 0
$$891$$ 0 0
$$892$$ 8.00000 0.267860
$$893$$ 12.0000 0.401565
$$894$$ 0 0
$$895$$ 0 0
$$896$$ −1.00000 −0.0334077
$$897$$ 0 0
$$898$$ 0 0
$$899$$ 15.0000 0.500278
$$900$$ 0 0
$$901$$ 9.00000 0.299833
$$902$$ 0 0
$$903$$ 0 0
$$904$$ 12.0000 0.399114
$$905$$ 0 0
$$906$$ 0 0
$$907$$ −28.0000 −0.929725 −0.464862 0.885383i $$-0.653896\pi$$
−0.464862 + 0.885383i $$0.653896\pi$$
$$908$$ −9.00000 −0.298675
$$909$$ 0 0
$$910$$ 0 0
$$911$$ −24.0000 −0.795155 −0.397578 0.917568i $$-0.630149\pi$$
−0.397578 + 0.917568i $$0.630149\pi$$
$$912$$ 0 0
$$913$$ 0 0
$$914$$ −35.0000 −1.15770
$$915$$ 0 0
$$916$$ 14.0000 0.462573
$$917$$ 9.00000 0.297206
$$918$$ 0 0
$$919$$ 2.00000 0.0659739 0.0329870 0.999456i $$-0.489498\pi$$
0.0329870 + 0.999456i $$0.489498\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ −30.0000 −0.987997
$$923$$ −45.0000 −1.48119
$$924$$ 0 0
$$925$$ −10.0000 −0.328798
$$926$$ 22.0000 0.722965
$$927$$ 0 0
$$928$$ −3.00000 −0.0984798
$$929$$ −30.0000 −0.984268 −0.492134 0.870519i $$-0.663783\pi$$
−0.492134 + 0.870519i $$0.663783\pi$$
$$930$$ 0 0
$$931$$ 2.00000 0.0655474
$$932$$ 12.0000 0.393073
$$933$$ 0 0
$$934$$ 24.0000 0.785304
$$935$$ 0 0
$$936$$ 0 0
$$937$$ 56.0000 1.82944 0.914720 0.404088i $$-0.132411\pi$$
0.914720 + 0.404088i $$0.132411\pi$$
$$938$$ 13.0000 0.424465
$$939$$ 0 0
$$940$$ 0 0
$$941$$ 30.0000 0.977972 0.488986 0.872292i $$-0.337367\pi$$
0.488986 + 0.872292i $$0.337367\pi$$
$$942$$ 0 0
$$943$$ 54.0000 1.75848
$$944$$ 3.00000 0.0976417
$$945$$ 0 0
$$946$$ 0 0
$$947$$ 6.00000 0.194974 0.0974869 0.995237i $$-0.468920\pi$$
0.0974869 + 0.995237i $$0.468920\pi$$
$$948$$ 0 0
$$949$$ 10.0000 0.324614
$$950$$ 10.0000 0.324443
$$951$$ 0 0
$$952$$ 3.00000 0.0972306
$$953$$ 36.0000 1.16615 0.583077 0.812417i $$-0.301849\pi$$
0.583077 + 0.812417i $$0.301849\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ −12.0000 −0.388108
$$957$$ 0 0
$$958$$ 36.0000 1.16311
$$959$$ −18.0000 −0.581250
$$960$$ 0 0
$$961$$ −6.00000 −0.193548
$$962$$ −10.0000 −0.322413
$$963$$ 0 0
$$964$$ −10.0000 −0.322078
$$965$$ 0 0
$$966$$ 0 0
$$967$$ −4.00000 −0.128631 −0.0643157 0.997930i $$-0.520486\pi$$
−0.0643157 + 0.997930i $$0.520486\pi$$
$$968$$ 11.0000 0.353553
$$969$$ 0 0
$$970$$ 0 0
$$971$$ 57.0000 1.82922 0.914609 0.404341i $$-0.132499\pi$$
0.914609 + 0.404341i $$0.132499\pi$$
$$972$$ 0 0
$$973$$ 14.0000 0.448819
$$974$$ −38.0000 −1.21760
$$975$$ 0 0
$$976$$ −10.0000 −0.320092
$$977$$ −42.0000 −1.34370 −0.671850 0.740688i $$-0.734500\pi$$
−0.671850 + 0.740688i $$0.734500\pi$$
$$978$$ 0 0
$$979$$ 0 0
$$980$$ 0 0
$$981$$ 0 0
$$982$$ 12.0000 0.382935
$$983$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ 9.00000 0.286618
$$987$$ 0 0
$$988$$ 10.0000 0.318142
$$989$$ −9.00000 −0.286183
$$990$$ 0 0
$$991$$ 2.00000 0.0635321 0.0317660 0.999495i $$-0.489887\pi$$
0.0317660 + 0.999495i $$0.489887\pi$$
$$992$$ −5.00000 −0.158750
$$993$$ 0 0
$$994$$ 9.00000 0.285463
$$995$$ 0 0
$$996$$ 0 0
$$997$$ −19.0000 −0.601736 −0.300868 0.953666i $$-0.597276\pi$$
−0.300868 + 0.953666i $$0.597276\pi$$
$$998$$ 4.00000 0.126618
$$999$$ 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 378.2.a.d.1.1 1
3.2 odd 2 378.2.a.e.1.1 yes 1
4.3 odd 2 3024.2.a.o.1.1 1
5.4 even 2 9450.2.a.cl.1.1 1
7.6 odd 2 2646.2.a.f.1.1 1
9.2 odd 6 1134.2.f.e.757.1 2
9.4 even 3 1134.2.f.k.379.1 2
9.5 odd 6 1134.2.f.e.379.1 2
9.7 even 3 1134.2.f.k.757.1 2
12.11 even 2 3024.2.a.p.1.1 1
15.14 odd 2 9450.2.a.l.1.1 1
21.20 even 2 2646.2.a.y.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
378.2.a.d.1.1 1 1.1 even 1 trivial
378.2.a.e.1.1 yes 1 3.2 odd 2
1134.2.f.e.379.1 2 9.5 odd 6
1134.2.f.e.757.1 2 9.2 odd 6
1134.2.f.k.379.1 2 9.4 even 3
1134.2.f.k.757.1 2 9.7 even 3
2646.2.a.f.1.1 1 7.6 odd 2
2646.2.a.y.1.1 1 21.20 even 2
3024.2.a.o.1.1 1 4.3 odd 2
3024.2.a.p.1.1 1 12.11 even 2
9450.2.a.l.1.1 1 15.14 odd 2
9450.2.a.cl.1.1 1 5.4 even 2