# Properties

 Label 6525.2.a.j.1.1 Level $6525$ Weight $2$ Character 6525.1 Self dual yes Analytic conductor $52.102$ 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] = [6525,2,Mod(1,6525)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6525, 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("6525.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$6525 = 3^{2} \cdot 5^{2} \cdot 29$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 6525.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: $$52.1023873189$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 435) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 6525.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+1.00000 q^{2} -1.00000 q^{4} -4.00000 q^{7} -3.00000 q^{8} +O(q^{10})$$ $$q+1.00000 q^{2} -1.00000 q^{4} -4.00000 q^{7} -3.00000 q^{8} +4.00000 q^{11} -6.00000 q^{13} -4.00000 q^{14} -1.00000 q^{16} +6.00000 q^{17} -4.00000 q^{19} +4.00000 q^{22} -4.00000 q^{23} -6.00000 q^{26} +4.00000 q^{28} -1.00000 q^{29} -8.00000 q^{31} +5.00000 q^{32} +6.00000 q^{34} -2.00000 q^{37} -4.00000 q^{38} +6.00000 q^{41} -4.00000 q^{43} -4.00000 q^{44} -4.00000 q^{46} +9.00000 q^{49} +6.00000 q^{52} -10.0000 q^{53} +12.0000 q^{56} -1.00000 q^{58} +12.0000 q^{59} -10.0000 q^{61} -8.00000 q^{62} +7.00000 q^{64} -8.00000 q^{67} -6.00000 q^{68} +8.00000 q^{71} +2.00000 q^{73} -2.00000 q^{74} +4.00000 q^{76} -16.0000 q^{77} +6.00000 q^{82} +8.00000 q^{83} -4.00000 q^{86} -12.0000 q^{88} +6.00000 q^{89} +24.0000 q^{91} +4.00000 q^{92} +2.00000 q^{97} +9.00000 q^{98} +O(q^{100})$$

## 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 0.353553 0.935414i $$-0.384973\pi$$
0.353553 + 0.935414i $$0.384973\pi$$
$$3$$ 0 0
$$4$$ −1.00000 −0.500000
$$5$$ 0 0
$$6$$ 0 0
$$7$$ −4.00000 −1.51186 −0.755929 0.654654i $$-0.772814\pi$$
−0.755929 + 0.654654i $$0.772814\pi$$
$$8$$ −3.00000 −1.06066
$$9$$ 0 0
$$10$$ 0 0
$$11$$ 4.00000 1.20605 0.603023 0.797724i $$-0.293963\pi$$
0.603023 + 0.797724i $$0.293963\pi$$
$$12$$ 0 0
$$13$$ −6.00000 −1.66410 −0.832050 0.554700i $$-0.812833\pi$$
−0.832050 + 0.554700i $$0.812833\pi$$
$$14$$ −4.00000 −1.06904
$$15$$ 0 0
$$16$$ −1.00000 −0.250000
$$17$$ 6.00000 1.45521 0.727607 0.685994i $$-0.240633\pi$$
0.727607 + 0.685994i $$0.240633\pi$$
$$18$$ 0 0
$$19$$ −4.00000 −0.917663 −0.458831 0.888523i $$-0.651732\pi$$
−0.458831 + 0.888523i $$0.651732\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 4.00000 0.852803
$$23$$ −4.00000 −0.834058 −0.417029 0.908893i $$-0.636929\pi$$
−0.417029 + 0.908893i $$0.636929\pi$$
$$24$$ 0 0
$$25$$ 0 0
$$26$$ −6.00000 −1.17670
$$27$$ 0 0
$$28$$ 4.00000 0.755929
$$29$$ −1.00000 −0.185695
$$30$$ 0 0
$$31$$ −8.00000 −1.43684 −0.718421 0.695608i $$-0.755135\pi$$
−0.718421 + 0.695608i $$0.755135\pi$$
$$32$$ 5.00000 0.883883
$$33$$ 0 0
$$34$$ 6.00000 1.02899
$$35$$ 0 0
$$36$$ 0 0
$$37$$ −2.00000 −0.328798 −0.164399 0.986394i $$-0.552568\pi$$
−0.164399 + 0.986394i $$0.552568\pi$$
$$38$$ −4.00000 −0.648886
$$39$$ 0 0
$$40$$ 0 0
$$41$$ 6.00000 0.937043 0.468521 0.883452i $$-0.344787\pi$$
0.468521 + 0.883452i $$0.344787\pi$$
$$42$$ 0 0
$$43$$ −4.00000 −0.609994 −0.304997 0.952353i $$-0.598656\pi$$
−0.304997 + 0.952353i $$0.598656\pi$$
$$44$$ −4.00000 −0.603023
$$45$$ 0 0
$$46$$ −4.00000 −0.589768
$$47$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$48$$ 0 0
$$49$$ 9.00000 1.28571
$$50$$ 0 0
$$51$$ 0 0
$$52$$ 6.00000 0.832050
$$53$$ −10.0000 −1.37361 −0.686803 0.726844i $$-0.740986\pi$$
−0.686803 + 0.726844i $$0.740986\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 12.0000 1.60357
$$57$$ 0 0
$$58$$ −1.00000 −0.131306
$$59$$ 12.0000 1.56227 0.781133 0.624364i $$-0.214642\pi$$
0.781133 + 0.624364i $$0.214642\pi$$
$$60$$ 0 0
$$61$$ −10.0000 −1.28037 −0.640184 0.768221i $$-0.721142\pi$$
−0.640184 + 0.768221i $$0.721142\pi$$
$$62$$ −8.00000 −1.01600
$$63$$ 0 0
$$64$$ 7.00000 0.875000
$$65$$ 0 0
$$66$$ 0 0
$$67$$ −8.00000 −0.977356 −0.488678 0.872464i $$-0.662521\pi$$
−0.488678 + 0.872464i $$0.662521\pi$$
$$68$$ −6.00000 −0.727607
$$69$$ 0 0
$$70$$ 0 0
$$71$$ 8.00000 0.949425 0.474713 0.880141i $$-0.342552\pi$$
0.474713 + 0.880141i $$0.342552\pi$$
$$72$$ 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
$$76$$ 4.00000 0.458831
$$77$$ −16.0000 −1.82337
$$78$$ 0 0
$$79$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$80$$ 0 0
$$81$$ 0 0
$$82$$ 6.00000 0.662589
$$83$$ 8.00000 0.878114 0.439057 0.898459i $$-0.355313\pi$$
0.439057 + 0.898459i $$0.355313\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ −4.00000 −0.431331
$$87$$ 0 0
$$88$$ −12.0000 −1.27920
$$89$$ 6.00000 0.635999 0.317999 0.948091i $$-0.396989\pi$$
0.317999 + 0.948091i $$0.396989\pi$$
$$90$$ 0 0
$$91$$ 24.0000 2.51588
$$92$$ 4.00000 0.417029
$$93$$ 0 0
$$94$$ 0 0
$$95$$ 0 0
$$96$$ 0 0
$$97$$ 2.00000 0.203069 0.101535 0.994832i $$-0.467625\pi$$
0.101535 + 0.994832i $$0.467625\pi$$
$$98$$ 9.00000 0.909137
$$99$$ 0 0
$$100$$ 0 0
$$101$$ 2.00000 0.199007 0.0995037 0.995037i $$-0.468274\pi$$
0.0995037 + 0.995037i $$0.468274\pi$$
$$102$$ 0 0
$$103$$ 12.0000 1.18240 0.591198 0.806527i $$-0.298655\pi$$
0.591198 + 0.806527i $$0.298655\pi$$
$$104$$ 18.0000 1.76505
$$105$$ 0 0
$$106$$ −10.0000 −0.971286
$$107$$ 8.00000 0.773389 0.386695 0.922208i $$-0.373617\pi$$
0.386695 + 0.922208i $$0.373617\pi$$
$$108$$ 0 0
$$109$$ 14.0000 1.34096 0.670478 0.741929i $$-0.266089\pi$$
0.670478 + 0.741929i $$0.266089\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ 4.00000 0.377964
$$113$$ −2.00000 −0.188144 −0.0940721 0.995565i $$-0.529988\pi$$
−0.0940721 + 0.995565i $$0.529988\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 1.00000 0.0928477
$$117$$ 0 0
$$118$$ 12.0000 1.10469
$$119$$ −24.0000 −2.20008
$$120$$ 0 0
$$121$$ 5.00000 0.454545
$$122$$ −10.0000 −0.905357
$$123$$ 0 0
$$124$$ 8.00000 0.718421
$$125$$ 0 0
$$126$$ 0 0
$$127$$ 16.0000 1.41977 0.709885 0.704317i $$-0.248747\pi$$
0.709885 + 0.704317i $$0.248747\pi$$
$$128$$ −3.00000 −0.265165
$$129$$ 0 0
$$130$$ 0 0
$$131$$ −12.0000 −1.04844 −0.524222 0.851581i $$-0.675644\pi$$
−0.524222 + 0.851581i $$0.675644\pi$$
$$132$$ 0 0
$$133$$ 16.0000 1.38738
$$134$$ −8.00000 −0.691095
$$135$$ 0 0
$$136$$ −18.0000 −1.54349
$$137$$ −2.00000 −0.170872 −0.0854358 0.996344i $$-0.527228\pi$$
−0.0854358 + 0.996344i $$0.527228\pi$$
$$138$$ 0 0
$$139$$ 4.00000 0.339276 0.169638 0.985506i $$-0.445740\pi$$
0.169638 + 0.985506i $$0.445740\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 8.00000 0.671345
$$143$$ −24.0000 −2.00698
$$144$$ 0 0
$$145$$ 0 0
$$146$$ 2.00000 0.165521
$$147$$ 0 0
$$148$$ 2.00000 0.164399
$$149$$ −6.00000 −0.491539 −0.245770 0.969328i $$-0.579041\pi$$
−0.245770 + 0.969328i $$0.579041\pi$$
$$150$$ 0 0
$$151$$ 16.0000 1.30206 0.651031 0.759051i $$-0.274337\pi$$
0.651031 + 0.759051i $$0.274337\pi$$
$$152$$ 12.0000 0.973329
$$153$$ 0 0
$$154$$ −16.0000 −1.28932
$$155$$ 0 0
$$156$$ 0 0
$$157$$ −18.0000 −1.43656 −0.718278 0.695756i $$-0.755069\pi$$
−0.718278 + 0.695756i $$0.755069\pi$$
$$158$$ 0 0
$$159$$ 0 0
$$160$$ 0 0
$$161$$ 16.0000 1.26098
$$162$$ 0 0
$$163$$ 12.0000 0.939913 0.469956 0.882690i $$-0.344270\pi$$
0.469956 + 0.882690i $$0.344270\pi$$
$$164$$ −6.00000 −0.468521
$$165$$ 0 0
$$166$$ 8.00000 0.620920
$$167$$ 12.0000 0.928588 0.464294 0.885681i $$-0.346308\pi$$
0.464294 + 0.885681i $$0.346308\pi$$
$$168$$ 0 0
$$169$$ 23.0000 1.76923
$$170$$ 0 0
$$171$$ 0 0
$$172$$ 4.00000 0.304997
$$173$$ 6.00000 0.456172 0.228086 0.973641i $$-0.426753\pi$$
0.228086 + 0.973641i $$0.426753\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ −4.00000 −0.301511
$$177$$ 0 0
$$178$$ 6.00000 0.449719
$$179$$ 12.0000 0.896922 0.448461 0.893802i $$-0.351972\pi$$
0.448461 + 0.893802i $$0.351972\pi$$
$$180$$ 0 0
$$181$$ 22.0000 1.63525 0.817624 0.575753i $$-0.195291\pi$$
0.817624 + 0.575753i $$0.195291\pi$$
$$182$$ 24.0000 1.77900
$$183$$ 0 0
$$184$$ 12.0000 0.884652
$$185$$ 0 0
$$186$$ 0 0
$$187$$ 24.0000 1.75505
$$188$$ 0 0
$$189$$ 0 0
$$190$$ 0 0
$$191$$ 16.0000 1.15772 0.578860 0.815427i $$-0.303498\pi$$
0.578860 + 0.815427i $$0.303498\pi$$
$$192$$ 0 0
$$193$$ −22.0000 −1.58359 −0.791797 0.610784i $$-0.790854\pi$$
−0.791797 + 0.610784i $$0.790854\pi$$
$$194$$ 2.00000 0.143592
$$195$$ 0 0
$$196$$ −9.00000 −0.642857
$$197$$ 14.0000 0.997459 0.498729 0.866758i $$-0.333800\pi$$
0.498729 + 0.866758i $$0.333800\pi$$
$$198$$ 0 0
$$199$$ −24.0000 −1.70131 −0.850657 0.525720i $$-0.823796\pi$$
−0.850657 + 0.525720i $$0.823796\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ 2.00000 0.140720
$$203$$ 4.00000 0.280745
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 12.0000 0.836080
$$207$$ 0 0
$$208$$ 6.00000 0.416025
$$209$$ −16.0000 −1.10674
$$210$$ 0 0
$$211$$ 20.0000 1.37686 0.688428 0.725304i $$-0.258301\pi$$
0.688428 + 0.725304i $$0.258301\pi$$
$$212$$ 10.0000 0.686803
$$213$$ 0 0
$$214$$ 8.00000 0.546869
$$215$$ 0 0
$$216$$ 0 0
$$217$$ 32.0000 2.17230
$$218$$ 14.0000 0.948200
$$219$$ 0 0
$$220$$ 0 0
$$221$$ −36.0000 −2.42162
$$222$$ 0 0
$$223$$ −28.0000 −1.87502 −0.937509 0.347960i $$-0.886874\pi$$
−0.937509 + 0.347960i $$0.886874\pi$$
$$224$$ −20.0000 −1.33631
$$225$$ 0 0
$$226$$ −2.00000 −0.133038
$$227$$ 24.0000 1.59294 0.796468 0.604681i $$-0.206699\pi$$
0.796468 + 0.604681i $$0.206699\pi$$
$$228$$ 0 0
$$229$$ −2.00000 −0.132164 −0.0660819 0.997814i $$-0.521050\pi$$
−0.0660819 + 0.997814i $$0.521050\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 3.00000 0.196960
$$233$$ −22.0000 −1.44127 −0.720634 0.693316i $$-0.756149\pi$$
−0.720634 + 0.693316i $$0.756149\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ −12.0000 −0.781133
$$237$$ 0 0
$$238$$ −24.0000 −1.55569
$$239$$ 8.00000 0.517477 0.258738 0.965947i $$-0.416693\pi$$
0.258738 + 0.965947i $$0.416693\pi$$
$$240$$ 0 0
$$241$$ 2.00000 0.128831 0.0644157 0.997923i $$-0.479482\pi$$
0.0644157 + 0.997923i $$0.479482\pi$$
$$242$$ 5.00000 0.321412
$$243$$ 0 0
$$244$$ 10.0000 0.640184
$$245$$ 0 0
$$246$$ 0 0
$$247$$ 24.0000 1.52708
$$248$$ 24.0000 1.52400
$$249$$ 0 0
$$250$$ 0 0
$$251$$ 20.0000 1.26239 0.631194 0.775625i $$-0.282565\pi$$
0.631194 + 0.775625i $$0.282565\pi$$
$$252$$ 0 0
$$253$$ −16.0000 −1.00591
$$254$$ 16.0000 1.00393
$$255$$ 0 0
$$256$$ −17.0000 −1.06250
$$257$$ −22.0000 −1.37232 −0.686161 0.727450i $$-0.740706\pi$$
−0.686161 + 0.727450i $$0.740706\pi$$
$$258$$ 0 0
$$259$$ 8.00000 0.497096
$$260$$ 0 0
$$261$$ 0 0
$$262$$ −12.0000 −0.741362
$$263$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 16.0000 0.981023
$$267$$ 0 0
$$268$$ 8.00000 0.488678
$$269$$ −6.00000 −0.365826 −0.182913 0.983129i $$-0.558553\pi$$
−0.182913 + 0.983129i $$0.558553\pi$$
$$270$$ 0 0
$$271$$ 16.0000 0.971931 0.485965 0.873978i $$-0.338468\pi$$
0.485965 + 0.873978i $$0.338468\pi$$
$$272$$ −6.00000 −0.363803
$$273$$ 0 0
$$274$$ −2.00000 −0.120824
$$275$$ 0 0
$$276$$ 0 0
$$277$$ 26.0000 1.56219 0.781094 0.624413i $$-0.214662\pi$$
0.781094 + 0.624413i $$0.214662\pi$$
$$278$$ 4.00000 0.239904
$$279$$ 0 0
$$280$$ 0 0
$$281$$ −10.0000 −0.596550 −0.298275 0.954480i $$-0.596411\pi$$
−0.298275 + 0.954480i $$0.596411\pi$$
$$282$$ 0 0
$$283$$ −8.00000 −0.475551 −0.237775 0.971320i $$-0.576418\pi$$
−0.237775 + 0.971320i $$0.576418\pi$$
$$284$$ −8.00000 −0.474713
$$285$$ 0 0
$$286$$ −24.0000 −1.41915
$$287$$ −24.0000 −1.41668
$$288$$ 0 0
$$289$$ 19.0000 1.11765
$$290$$ 0 0
$$291$$ 0 0
$$292$$ −2.00000 −0.117041
$$293$$ 18.0000 1.05157 0.525786 0.850617i $$-0.323771\pi$$
0.525786 + 0.850617i $$0.323771\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 6.00000 0.348743
$$297$$ 0 0
$$298$$ −6.00000 −0.347571
$$299$$ 24.0000 1.38796
$$300$$ 0 0
$$301$$ 16.0000 0.922225
$$302$$ 16.0000 0.920697
$$303$$ 0 0
$$304$$ 4.00000 0.229416
$$305$$ 0 0
$$306$$ 0 0
$$307$$ −12.0000 −0.684876 −0.342438 0.939540i $$-0.611253\pi$$
−0.342438 + 0.939540i $$0.611253\pi$$
$$308$$ 16.0000 0.911685
$$309$$ 0 0
$$310$$ 0 0
$$311$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$312$$ 0 0
$$313$$ 30.0000 1.69570 0.847850 0.530236i $$-0.177897\pi$$
0.847850 + 0.530236i $$0.177897\pi$$
$$314$$ −18.0000 −1.01580
$$315$$ 0 0
$$316$$ 0 0
$$317$$ 2.00000 0.112331 0.0561656 0.998421i $$-0.482113\pi$$
0.0561656 + 0.998421i $$0.482113\pi$$
$$318$$ 0 0
$$319$$ −4.00000 −0.223957
$$320$$ 0 0
$$321$$ 0 0
$$322$$ 16.0000 0.891645
$$323$$ −24.0000 −1.33540
$$324$$ 0 0
$$325$$ 0 0
$$326$$ 12.0000 0.664619
$$327$$ 0 0
$$328$$ −18.0000 −0.993884
$$329$$ 0 0
$$330$$ 0 0
$$331$$ 4.00000 0.219860 0.109930 0.993939i $$-0.464937\pi$$
0.109930 + 0.993939i $$0.464937\pi$$
$$332$$ −8.00000 −0.439057
$$333$$ 0 0
$$334$$ 12.0000 0.656611
$$335$$ 0 0
$$336$$ 0 0
$$337$$ −30.0000 −1.63420 −0.817102 0.576493i $$-0.804421\pi$$
−0.817102 + 0.576493i $$0.804421\pi$$
$$338$$ 23.0000 1.25104
$$339$$ 0 0
$$340$$ 0 0
$$341$$ −32.0000 −1.73290
$$342$$ 0 0
$$343$$ −8.00000 −0.431959
$$344$$ 12.0000 0.646997
$$345$$ 0 0
$$346$$ 6.00000 0.322562
$$347$$ 16.0000 0.858925 0.429463 0.903085i $$-0.358703\pi$$
0.429463 + 0.903085i $$0.358703\pi$$
$$348$$ 0 0
$$349$$ −34.0000 −1.81998 −0.909989 0.414632i $$-0.863910\pi$$
−0.909989 + 0.414632i $$0.863910\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 20.0000 1.06600
$$353$$ 18.0000 0.958043 0.479022 0.877803i $$-0.340992\pi$$
0.479022 + 0.877803i $$0.340992\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ −6.00000 −0.317999
$$357$$ 0 0
$$358$$ 12.0000 0.634220
$$359$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$360$$ 0 0
$$361$$ −3.00000 −0.157895
$$362$$ 22.0000 1.15629
$$363$$ 0 0
$$364$$ −24.0000 −1.25794
$$365$$ 0 0
$$366$$ 0 0
$$367$$ −8.00000 −0.417597 −0.208798 0.977959i $$-0.566955\pi$$
−0.208798 + 0.977959i $$0.566955\pi$$
$$368$$ 4.00000 0.208514
$$369$$ 0 0
$$370$$ 0 0
$$371$$ 40.0000 2.07670
$$372$$ 0 0
$$373$$ 2.00000 0.103556 0.0517780 0.998659i $$-0.483511\pi$$
0.0517780 + 0.998659i $$0.483511\pi$$
$$374$$ 24.0000 1.24101
$$375$$ 0 0
$$376$$ 0 0
$$377$$ 6.00000 0.309016
$$378$$ 0 0
$$379$$ −20.0000 −1.02733 −0.513665 0.857991i $$-0.671713\pi$$
−0.513665 + 0.857991i $$0.671713\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ 16.0000 0.818631
$$383$$ 20.0000 1.02195 0.510976 0.859595i $$-0.329284\pi$$
0.510976 + 0.859595i $$0.329284\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ −22.0000 −1.11977
$$387$$ 0 0
$$388$$ −2.00000 −0.101535
$$389$$ 2.00000 0.101404 0.0507020 0.998714i $$-0.483854\pi$$
0.0507020 + 0.998714i $$0.483854\pi$$
$$390$$ 0 0
$$391$$ −24.0000 −1.21373
$$392$$ −27.0000 −1.36371
$$393$$ 0 0
$$394$$ 14.0000 0.705310
$$395$$ 0 0
$$396$$ 0 0
$$397$$ 10.0000 0.501886 0.250943 0.968002i $$-0.419259\pi$$
0.250943 + 0.968002i $$0.419259\pi$$
$$398$$ −24.0000 −1.20301
$$399$$ 0 0
$$400$$ 0 0
$$401$$ −2.00000 −0.0998752 −0.0499376 0.998752i $$-0.515902\pi$$
−0.0499376 + 0.998752i $$0.515902\pi$$
$$402$$ 0 0
$$403$$ 48.0000 2.39105
$$404$$ −2.00000 −0.0995037
$$405$$ 0 0
$$406$$ 4.00000 0.198517
$$407$$ −8.00000 −0.396545
$$408$$ 0 0
$$409$$ −22.0000 −1.08783 −0.543915 0.839140i $$-0.683059\pi$$
−0.543915 + 0.839140i $$0.683059\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ −12.0000 −0.591198
$$413$$ −48.0000 −2.36193
$$414$$ 0 0
$$415$$ 0 0
$$416$$ −30.0000 −1.47087
$$417$$ 0 0
$$418$$ −16.0000 −0.782586
$$419$$ 4.00000 0.195413 0.0977064 0.995215i $$-0.468849\pi$$
0.0977064 + 0.995215i $$0.468849\pi$$
$$420$$ 0 0
$$421$$ −18.0000 −0.877266 −0.438633 0.898666i $$-0.644537\pi$$
−0.438633 + 0.898666i $$0.644537\pi$$
$$422$$ 20.0000 0.973585
$$423$$ 0 0
$$424$$ 30.0000 1.45693
$$425$$ 0 0
$$426$$ 0 0
$$427$$ 40.0000 1.93574
$$428$$ −8.00000 −0.386695
$$429$$ 0 0
$$430$$ 0 0
$$431$$ −24.0000 −1.15604 −0.578020 0.816023i $$-0.696174\pi$$
−0.578020 + 0.816023i $$0.696174\pi$$
$$432$$ 0 0
$$433$$ 34.0000 1.63394 0.816968 0.576683i $$-0.195653\pi$$
0.816968 + 0.576683i $$0.195653\pi$$
$$434$$ 32.0000 1.53605
$$435$$ 0 0
$$436$$ −14.0000 −0.670478
$$437$$ 16.0000 0.765384
$$438$$ 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
$$442$$ −36.0000 −1.71235
$$443$$ 20.0000 0.950229 0.475114 0.879924i $$-0.342407\pi$$
0.475114 + 0.879924i $$0.342407\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ −28.0000 −1.32584
$$447$$ 0 0
$$448$$ −28.0000 −1.32288
$$449$$ −26.0000 −1.22702 −0.613508 0.789689i $$-0.710242\pi$$
−0.613508 + 0.789689i $$0.710242\pi$$
$$450$$ 0 0
$$451$$ 24.0000 1.13012
$$452$$ 2.00000 0.0940721
$$453$$ 0 0
$$454$$ 24.0000 1.12638
$$455$$ 0 0
$$456$$ 0 0
$$457$$ −10.0000 −0.467780 −0.233890 0.972263i $$-0.575146\pi$$
−0.233890 + 0.972263i $$0.575146\pi$$
$$458$$ −2.00000 −0.0934539
$$459$$ 0 0
$$460$$ 0 0
$$461$$ −6.00000 −0.279448 −0.139724 0.990190i $$-0.544622\pi$$
−0.139724 + 0.990190i $$0.544622\pi$$
$$462$$ 0 0
$$463$$ −36.0000 −1.67306 −0.836531 0.547920i $$-0.815420\pi$$
−0.836531 + 0.547920i $$0.815420\pi$$
$$464$$ 1.00000 0.0464238
$$465$$ 0 0
$$466$$ −22.0000 −1.01913
$$467$$ −20.0000 −0.925490 −0.462745 0.886492i $$-0.653135\pi$$
−0.462745 + 0.886492i $$0.653135\pi$$
$$468$$ 0 0
$$469$$ 32.0000 1.47762
$$470$$ 0 0
$$471$$ 0 0
$$472$$ −36.0000 −1.65703
$$473$$ −16.0000 −0.735681
$$474$$ 0 0
$$475$$ 0 0
$$476$$ 24.0000 1.10004
$$477$$ 0 0
$$478$$ 8.00000 0.365911
$$479$$ 16.0000 0.731059 0.365529 0.930800i $$-0.380888\pi$$
0.365529 + 0.930800i $$0.380888\pi$$
$$480$$ 0 0
$$481$$ 12.0000 0.547153
$$482$$ 2.00000 0.0910975
$$483$$ 0 0
$$484$$ −5.00000 −0.227273
$$485$$ 0 0
$$486$$ 0 0
$$487$$ 28.0000 1.26880 0.634401 0.773004i $$-0.281247\pi$$
0.634401 + 0.773004i $$0.281247\pi$$
$$488$$ 30.0000 1.35804
$$489$$ 0 0
$$490$$ 0 0
$$491$$ 44.0000 1.98569 0.992846 0.119401i $$-0.0380974\pi$$
0.992846 + 0.119401i $$0.0380974\pi$$
$$492$$ 0 0
$$493$$ −6.00000 −0.270226
$$494$$ 24.0000 1.07981
$$495$$ 0 0
$$496$$ 8.00000 0.359211
$$497$$ −32.0000 −1.43540
$$498$$ 0 0
$$499$$ −36.0000 −1.61158 −0.805791 0.592200i $$-0.798259\pi$$
−0.805791 + 0.592200i $$0.798259\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ 20.0000 0.892644
$$503$$ 24.0000 1.07011 0.535054 0.844818i $$-0.320291\pi$$
0.535054 + 0.844818i $$0.320291\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ −16.0000 −0.711287
$$507$$ 0 0
$$508$$ −16.0000 −0.709885
$$509$$ 2.00000 0.0886484 0.0443242 0.999017i $$-0.485887\pi$$
0.0443242 + 0.999017i $$0.485887\pi$$
$$510$$ 0 0
$$511$$ −8.00000 −0.353899
$$512$$ −11.0000 −0.486136
$$513$$ 0 0
$$514$$ −22.0000 −0.970378
$$515$$ 0 0
$$516$$ 0 0
$$517$$ 0 0
$$518$$ 8.00000 0.351500
$$519$$ 0 0
$$520$$ 0 0
$$521$$ 22.0000 0.963837 0.481919 0.876216i $$-0.339940\pi$$
0.481919 + 0.876216i $$0.339940\pi$$
$$522$$ 0 0
$$523$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$524$$ 12.0000 0.524222
$$525$$ 0 0
$$526$$ 0 0
$$527$$ −48.0000 −2.09091
$$528$$ 0 0
$$529$$ −7.00000 −0.304348
$$530$$ 0 0
$$531$$ 0 0
$$532$$ −16.0000 −0.693688
$$533$$ −36.0000 −1.55933
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 24.0000 1.03664
$$537$$ 0 0
$$538$$ −6.00000 −0.258678
$$539$$ 36.0000 1.55063
$$540$$ 0 0
$$541$$ −34.0000 −1.46177 −0.730887 0.682498i $$-0.760893\pi$$
−0.730887 + 0.682498i $$0.760893\pi$$
$$542$$ 16.0000 0.687259
$$543$$ 0 0
$$544$$ 30.0000 1.28624
$$545$$ 0 0
$$546$$ 0 0
$$547$$ −40.0000 −1.71028 −0.855138 0.518400i $$-0.826528\pi$$
−0.855138 + 0.518400i $$0.826528\pi$$
$$548$$ 2.00000 0.0854358
$$549$$ 0 0
$$550$$ 0 0
$$551$$ 4.00000 0.170406
$$552$$ 0 0
$$553$$ 0 0
$$554$$ 26.0000 1.10463
$$555$$ 0 0
$$556$$ −4.00000 −0.169638
$$557$$ 30.0000 1.27114 0.635570 0.772043i $$-0.280765\pi$$
0.635570 + 0.772043i $$0.280765\pi$$
$$558$$ 0 0
$$559$$ 24.0000 1.01509
$$560$$ 0 0
$$561$$ 0 0
$$562$$ −10.0000 −0.421825
$$563$$ −28.0000 −1.18006 −0.590030 0.807382i $$-0.700884\pi$$
−0.590030 + 0.807382i $$0.700884\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ −8.00000 −0.336265
$$567$$ 0 0
$$568$$ −24.0000 −1.00702
$$569$$ 30.0000 1.25767 0.628833 0.777541i $$-0.283533\pi$$
0.628833 + 0.777541i $$0.283533\pi$$
$$570$$ 0 0
$$571$$ 12.0000 0.502184 0.251092 0.967963i $$-0.419210\pi$$
0.251092 + 0.967963i $$0.419210\pi$$
$$572$$ 24.0000 1.00349
$$573$$ 0 0
$$574$$ −24.0000 −1.00174
$$575$$ 0 0
$$576$$ 0 0
$$577$$ 10.0000 0.416305 0.208153 0.978096i $$-0.433255\pi$$
0.208153 + 0.978096i $$0.433255\pi$$
$$578$$ 19.0000 0.790296
$$579$$ 0 0
$$580$$ 0 0
$$581$$ −32.0000 −1.32758
$$582$$ 0 0
$$583$$ −40.0000 −1.65663
$$584$$ −6.00000 −0.248282
$$585$$ 0 0
$$586$$ 18.0000 0.743573
$$587$$ −16.0000 −0.660391 −0.330195 0.943913i $$-0.607115\pi$$
−0.330195 + 0.943913i $$0.607115\pi$$
$$588$$ 0 0
$$589$$ 32.0000 1.31854
$$590$$ 0 0
$$591$$ 0 0
$$592$$ 2.00000 0.0821995
$$593$$ 34.0000 1.39621 0.698106 0.715994i $$-0.254026\pi$$
0.698106 + 0.715994i $$0.254026\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 6.00000 0.245770
$$597$$ 0 0
$$598$$ 24.0000 0.981433
$$599$$ −24.0000 −0.980613 −0.490307 0.871550i $$-0.663115\pi$$
−0.490307 + 0.871550i $$0.663115\pi$$
$$600$$ 0 0
$$601$$ −14.0000 −0.571072 −0.285536 0.958368i $$-0.592172\pi$$
−0.285536 + 0.958368i $$0.592172\pi$$
$$602$$ 16.0000 0.652111
$$603$$ 0 0
$$604$$ −16.0000 −0.651031
$$605$$ 0 0
$$606$$ 0 0
$$607$$ −32.0000 −1.29884 −0.649420 0.760430i $$-0.724988\pi$$
−0.649420 + 0.760430i $$0.724988\pi$$
$$608$$ −20.0000 −0.811107
$$609$$ 0 0
$$610$$ 0 0
$$611$$ 0 0
$$612$$ 0 0
$$613$$ −6.00000 −0.242338 −0.121169 0.992632i $$-0.538664\pi$$
−0.121169 + 0.992632i $$0.538664\pi$$
$$614$$ −12.0000 −0.484281
$$615$$ 0 0
$$616$$ 48.0000 1.93398
$$617$$ −18.0000 −0.724653 −0.362326 0.932051i $$-0.618017\pi$$
−0.362326 + 0.932051i $$0.618017\pi$$
$$618$$ 0 0
$$619$$ −20.0000 −0.803868 −0.401934 0.915669i $$-0.631662\pi$$
−0.401934 + 0.915669i $$0.631662\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 0 0
$$623$$ −24.0000 −0.961540
$$624$$ 0 0
$$625$$ 0 0
$$626$$ 30.0000 1.19904
$$627$$ 0 0
$$628$$ 18.0000 0.718278
$$629$$ −12.0000 −0.478471
$$630$$ 0 0
$$631$$ 24.0000 0.955425 0.477712 0.878516i $$-0.341466\pi$$
0.477712 + 0.878516i $$0.341466\pi$$
$$632$$ 0 0
$$633$$ 0 0
$$634$$ 2.00000 0.0794301
$$635$$ 0 0
$$636$$ 0 0
$$637$$ −54.0000 −2.13956
$$638$$ −4.00000 −0.158362
$$639$$ 0 0
$$640$$ 0 0
$$641$$ −34.0000 −1.34292 −0.671460 0.741041i $$-0.734332\pi$$
−0.671460 + 0.741041i $$0.734332\pi$$
$$642$$ 0 0
$$643$$ −16.0000 −0.630978 −0.315489 0.948929i $$-0.602169\pi$$
−0.315489 + 0.948929i $$0.602169\pi$$
$$644$$ −16.0000 −0.630488
$$645$$ 0 0
$$646$$ −24.0000 −0.944267
$$647$$ −28.0000 −1.10079 −0.550397 0.834903i $$-0.685524\pi$$
−0.550397 + 0.834903i $$0.685524\pi$$
$$648$$ 0 0
$$649$$ 48.0000 1.88416
$$650$$ 0 0
$$651$$ 0 0
$$652$$ −12.0000 −0.469956
$$653$$ 50.0000 1.95665 0.978326 0.207072i $$-0.0663936\pi$$
0.978326 + 0.207072i $$0.0663936\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ −6.00000 −0.234261
$$657$$ 0 0
$$658$$ 0 0
$$659$$ −12.0000 −0.467454 −0.233727 0.972302i $$-0.575092\pi$$
−0.233727 + 0.972302i $$0.575092\pi$$
$$660$$ 0 0
$$661$$ 22.0000 0.855701 0.427850 0.903850i $$-0.359271\pi$$
0.427850 + 0.903850i $$0.359271\pi$$
$$662$$ 4.00000 0.155464
$$663$$ 0 0
$$664$$ −24.0000 −0.931381
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 4.00000 0.154881
$$668$$ −12.0000 −0.464294
$$669$$ 0 0
$$670$$ 0 0
$$671$$ −40.0000 −1.54418
$$672$$ 0 0
$$673$$ 6.00000 0.231283 0.115642 0.993291i $$-0.463108\pi$$
0.115642 + 0.993291i $$0.463108\pi$$
$$674$$ −30.0000 −1.15556
$$675$$ 0 0
$$676$$ −23.0000 −0.884615
$$677$$ 50.0000 1.92166 0.960828 0.277145i $$-0.0893883\pi$$
0.960828 + 0.277145i $$0.0893883\pi$$
$$678$$ 0 0
$$679$$ −8.00000 −0.307012
$$680$$ 0 0
$$681$$ 0 0
$$682$$ −32.0000 −1.22534
$$683$$ 40.0000 1.53056 0.765279 0.643699i $$-0.222601\pi$$
0.765279 + 0.643699i $$0.222601\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ −8.00000 −0.305441
$$687$$ 0 0
$$688$$ 4.00000 0.152499
$$689$$ 60.0000 2.28582
$$690$$ 0 0
$$691$$ 44.0000 1.67384 0.836919 0.547326i $$-0.184354\pi$$
0.836919 + 0.547326i $$0.184354\pi$$
$$692$$ −6.00000 −0.228086
$$693$$ 0 0
$$694$$ 16.0000 0.607352
$$695$$ 0 0
$$696$$ 0 0
$$697$$ 36.0000 1.36360
$$698$$ −34.0000 −1.28692
$$699$$ 0 0
$$700$$ 0 0
$$701$$ −30.0000 −1.13308 −0.566542 0.824033i $$-0.691719\pi$$
−0.566542 + 0.824033i $$0.691719\pi$$
$$702$$ 0 0
$$703$$ 8.00000 0.301726
$$704$$ 28.0000 1.05529
$$705$$ 0 0
$$706$$ 18.0000 0.677439
$$707$$ −8.00000 −0.300871
$$708$$ 0 0
$$709$$ 22.0000 0.826227 0.413114 0.910679i $$-0.364441\pi$$
0.413114 + 0.910679i $$0.364441\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ −18.0000 −0.674579
$$713$$ 32.0000 1.19841
$$714$$ 0 0
$$715$$ 0 0
$$716$$ −12.0000 −0.448461
$$717$$ 0 0
$$718$$ 0 0
$$719$$ 24.0000 0.895049 0.447524 0.894272i $$-0.352306\pi$$
0.447524 + 0.894272i $$0.352306\pi$$
$$720$$ 0 0
$$721$$ −48.0000 −1.78761
$$722$$ −3.00000 −0.111648
$$723$$ 0 0
$$724$$ −22.0000 −0.817624
$$725$$ 0 0
$$726$$ 0 0
$$727$$ −16.0000 −0.593407 −0.296704 0.954970i $$-0.595887\pi$$
−0.296704 + 0.954970i $$0.595887\pi$$
$$728$$ −72.0000 −2.66850
$$729$$ 0 0
$$730$$ 0 0
$$731$$ −24.0000 −0.887672
$$732$$ 0 0
$$733$$ −2.00000 −0.0738717 −0.0369358 0.999318i $$-0.511760\pi$$
−0.0369358 + 0.999318i $$0.511760\pi$$
$$734$$ −8.00000 −0.295285
$$735$$ 0 0
$$736$$ −20.0000 −0.737210
$$737$$ −32.0000 −1.17874
$$738$$ 0 0
$$739$$ −36.0000 −1.32428 −0.662141 0.749380i $$-0.730352\pi$$
−0.662141 + 0.749380i $$0.730352\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 40.0000 1.46845
$$743$$ −8.00000 −0.293492 −0.146746 0.989174i $$-0.546880\pi$$
−0.146746 + 0.989174i $$0.546880\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 2.00000 0.0732252
$$747$$ 0 0
$$748$$ −24.0000 −0.877527
$$749$$ −32.0000 −1.16925
$$750$$ 0 0
$$751$$ −8.00000 −0.291924 −0.145962 0.989290i $$-0.546628\pi$$
−0.145962 + 0.989290i $$0.546628\pi$$
$$752$$ 0 0
$$753$$ 0 0
$$754$$ 6.00000 0.218507
$$755$$ 0 0
$$756$$ 0 0
$$757$$ 14.0000 0.508839 0.254419 0.967094i $$-0.418116\pi$$
0.254419 + 0.967094i $$0.418116\pi$$
$$758$$ −20.0000 −0.726433
$$759$$ 0 0
$$760$$ 0 0
$$761$$ −26.0000 −0.942499 −0.471250 0.882000i $$-0.656197\pi$$
−0.471250 + 0.882000i $$0.656197\pi$$
$$762$$ 0 0
$$763$$ −56.0000 −2.02734
$$764$$ −16.0000 −0.578860
$$765$$ 0 0
$$766$$ 20.0000 0.722629
$$767$$ −72.0000 −2.59977
$$768$$ 0 0
$$769$$ 42.0000 1.51456 0.757279 0.653091i $$-0.226528\pi$$
0.757279 + 0.653091i $$0.226528\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 22.0000 0.791797
$$773$$ −38.0000 −1.36677 −0.683383 0.730061i $$-0.739492\pi$$
−0.683383 + 0.730061i $$0.739492\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ −6.00000 −0.215387
$$777$$ 0 0
$$778$$ 2.00000 0.0717035
$$779$$ −24.0000 −0.859889
$$780$$ 0 0
$$781$$ 32.0000 1.14505
$$782$$ −24.0000 −0.858238
$$783$$ 0 0
$$784$$ −9.00000 −0.321429
$$785$$ 0 0
$$786$$ 0 0
$$787$$ 48.0000 1.71102 0.855508 0.517790i $$-0.173245\pi$$
0.855508 + 0.517790i $$0.173245\pi$$
$$788$$ −14.0000 −0.498729
$$789$$ 0 0
$$790$$ 0 0
$$791$$ 8.00000 0.284447
$$792$$ 0 0
$$793$$ 60.0000 2.13066
$$794$$ 10.0000 0.354887
$$795$$ 0 0
$$796$$ 24.0000 0.850657
$$797$$ 50.0000 1.77109 0.885545 0.464553i $$-0.153785\pi$$
0.885545 + 0.464553i $$0.153785\pi$$
$$798$$ 0 0
$$799$$ 0 0
$$800$$ 0 0
$$801$$ 0 0
$$802$$ −2.00000 −0.0706225
$$803$$ 8.00000 0.282314
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 48.0000 1.69073
$$807$$ 0 0
$$808$$ −6.00000 −0.211079
$$809$$ 6.00000 0.210949 0.105474 0.994422i $$-0.466364\pi$$
0.105474 + 0.994422i $$0.466364\pi$$
$$810$$ 0 0
$$811$$ −20.0000 −0.702295 −0.351147 0.936320i $$-0.614208\pi$$
−0.351147 + 0.936320i $$0.614208\pi$$
$$812$$ −4.00000 −0.140372
$$813$$ 0 0
$$814$$ −8.00000 −0.280400
$$815$$ 0 0
$$816$$ 0 0
$$817$$ 16.0000 0.559769
$$818$$ −22.0000 −0.769212
$$819$$ 0 0
$$820$$ 0 0
$$821$$ 10.0000 0.349002 0.174501 0.984657i $$-0.444169\pi$$
0.174501 + 0.984657i $$0.444169\pi$$
$$822$$ 0 0
$$823$$ −16.0000 −0.557725 −0.278862 0.960331i $$-0.589957\pi$$
−0.278862 + 0.960331i $$0.589957\pi$$
$$824$$ −36.0000 −1.25412
$$825$$ 0 0
$$826$$ −48.0000 −1.67013
$$827$$ 20.0000 0.695468 0.347734 0.937593i $$-0.386951\pi$$
0.347734 + 0.937593i $$0.386951\pi$$
$$828$$ 0 0
$$829$$ −2.00000 −0.0694629 −0.0347314 0.999397i $$-0.511058\pi$$
−0.0347314 + 0.999397i $$0.511058\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ −42.0000 −1.45609
$$833$$ 54.0000 1.87099
$$834$$ 0 0
$$835$$ 0 0
$$836$$ 16.0000 0.553372
$$837$$ 0 0
$$838$$ 4.00000 0.138178
$$839$$ 24.0000 0.828572 0.414286 0.910147i $$-0.364031\pi$$
0.414286 + 0.910147i $$0.364031\pi$$
$$840$$ 0 0
$$841$$ 1.00000 0.0344828
$$842$$ −18.0000 −0.620321
$$843$$ 0 0
$$844$$ −20.0000 −0.688428
$$845$$ 0 0
$$846$$ 0 0
$$847$$ −20.0000 −0.687208
$$848$$ 10.0000 0.343401
$$849$$ 0 0
$$850$$ 0 0
$$851$$ 8.00000 0.274236
$$852$$ 0 0
$$853$$ 38.0000 1.30110 0.650548 0.759465i $$-0.274539\pi$$
0.650548 + 0.759465i $$0.274539\pi$$
$$854$$ 40.0000 1.36877
$$855$$ 0 0
$$856$$ −24.0000 −0.820303
$$857$$ 2.00000 0.0683187 0.0341593 0.999416i $$-0.489125\pi$$
0.0341593 + 0.999416i $$0.489125\pi$$
$$858$$ 0 0
$$859$$ 44.0000 1.50126 0.750630 0.660722i $$-0.229750\pi$$
0.750630 + 0.660722i $$0.229750\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ −24.0000 −0.817443
$$863$$ 4.00000 0.136162 0.0680808 0.997680i $$-0.478312\pi$$
0.0680808 + 0.997680i $$0.478312\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ 34.0000 1.15537
$$867$$ 0 0
$$868$$ −32.0000 −1.08615
$$869$$ 0 0
$$870$$ 0 0
$$871$$ 48.0000 1.62642
$$872$$ −42.0000 −1.42230
$$873$$ 0 0
$$874$$ 16.0000 0.541208
$$875$$ 0 0
$$876$$ 0 0
$$877$$ −14.0000 −0.472746 −0.236373 0.971662i $$-0.575959\pi$$
−0.236373 + 0.971662i $$0.575959\pi$$
$$878$$ −24.0000 −0.809961
$$879$$ 0 0
$$880$$ 0 0
$$881$$ −18.0000 −0.606435 −0.303218 0.952921i $$-0.598061\pi$$
−0.303218 + 0.952921i $$0.598061\pi$$
$$882$$ 0 0
$$883$$ 56.0000 1.88455 0.942275 0.334840i $$-0.108682\pi$$
0.942275 + 0.334840i $$0.108682\pi$$
$$884$$ 36.0000 1.21081
$$885$$ 0 0
$$886$$ 20.0000 0.671913
$$887$$ 8.00000 0.268614 0.134307 0.990940i $$-0.457119\pi$$
0.134307 + 0.990940i $$0.457119\pi$$
$$888$$ 0 0
$$889$$ −64.0000 −2.14649
$$890$$ 0 0
$$891$$ 0 0
$$892$$ 28.0000 0.937509
$$893$$ 0 0
$$894$$ 0 0
$$895$$ 0 0
$$896$$ 12.0000 0.400892
$$897$$ 0 0
$$898$$ −26.0000 −0.867631
$$899$$ 8.00000 0.266815
$$900$$ 0 0
$$901$$ −60.0000 −1.99889
$$902$$ 24.0000 0.799113
$$903$$ 0 0
$$904$$ 6.00000 0.199557
$$905$$ 0 0
$$906$$ 0 0
$$907$$ 44.0000 1.46100 0.730498 0.682915i $$-0.239288\pi$$
0.730498 + 0.682915i $$0.239288\pi$$
$$908$$ −24.0000 −0.796468
$$909$$ 0 0
$$910$$ 0 0
$$911$$ 56.0000 1.85536 0.927681 0.373373i $$-0.121799\pi$$
0.927681 + 0.373373i $$0.121799\pi$$
$$912$$ 0 0
$$913$$ 32.0000 1.05905
$$914$$ −10.0000 −0.330771
$$915$$ 0 0
$$916$$ 2.00000 0.0660819
$$917$$ 48.0000 1.58510
$$918$$ 0 0
$$919$$ 8.00000 0.263896 0.131948 0.991257i $$-0.457877\pi$$
0.131948 + 0.991257i $$0.457877\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ −6.00000 −0.197599
$$923$$ −48.0000 −1.57994
$$924$$ 0 0
$$925$$ 0 0
$$926$$ −36.0000 −1.18303
$$927$$ 0 0
$$928$$ −5.00000 −0.164133
$$929$$ −34.0000 −1.11550 −0.557752 0.830008i $$-0.688336\pi$$
−0.557752 + 0.830008i $$0.688336\pi$$
$$930$$ 0 0
$$931$$ −36.0000 −1.17985
$$932$$ 22.0000 0.720634
$$933$$ 0 0
$$934$$ −20.0000 −0.654420
$$935$$ 0 0
$$936$$ 0 0
$$937$$ −10.0000 −0.326686 −0.163343 0.986569i $$-0.552228\pi$$
−0.163343 + 0.986569i $$0.552228\pi$$
$$938$$ 32.0000 1.04484
$$939$$ 0 0
$$940$$ 0 0
$$941$$ 18.0000 0.586783 0.293392 0.955992i $$-0.405216\pi$$
0.293392 + 0.955992i $$0.405216\pi$$
$$942$$ 0 0
$$943$$ −24.0000 −0.781548
$$944$$ −12.0000 −0.390567
$$945$$ 0 0
$$946$$ −16.0000 −0.520205
$$947$$ 28.0000 0.909878 0.454939 0.890523i $$-0.349661\pi$$
0.454939 + 0.890523i $$0.349661\pi$$
$$948$$ 0 0
$$949$$ −12.0000 −0.389536
$$950$$ 0 0
$$951$$ 0 0
$$952$$ 72.0000 2.33353
$$953$$ −6.00000 −0.194359 −0.0971795 0.995267i $$-0.530982\pi$$
−0.0971795 + 0.995267i $$0.530982\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ −8.00000 −0.258738
$$957$$ 0 0
$$958$$ 16.0000 0.516937
$$959$$ 8.00000 0.258333
$$960$$ 0 0
$$961$$ 33.0000 1.06452
$$962$$ 12.0000 0.386896
$$963$$ 0 0
$$964$$ −2.00000 −0.0644157
$$965$$ 0 0
$$966$$ 0 0
$$967$$ −32.0000 −1.02905 −0.514525 0.857475i $$-0.672032\pi$$
−0.514525 + 0.857475i $$0.672032\pi$$
$$968$$ −15.0000 −0.482118
$$969$$ 0 0
$$970$$ 0 0
$$971$$ −20.0000 −0.641831 −0.320915 0.947108i $$-0.603990\pi$$
−0.320915 + 0.947108i $$0.603990\pi$$
$$972$$ 0 0
$$973$$ −16.0000 −0.512936
$$974$$ 28.0000 0.897178
$$975$$ 0 0
$$976$$ 10.0000 0.320092
$$977$$ −6.00000 −0.191957 −0.0959785 0.995383i $$-0.530598\pi$$
−0.0959785 + 0.995383i $$0.530598\pi$$
$$978$$ 0 0
$$979$$ 24.0000 0.767043
$$980$$ 0 0
$$981$$ 0 0
$$982$$ 44.0000 1.40410
$$983$$ 40.0000 1.27580 0.637901 0.770118i $$-0.279803\pi$$
0.637901 + 0.770118i $$0.279803\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ −6.00000 −0.191079
$$987$$ 0 0
$$988$$ −24.0000 −0.763542
$$989$$ 16.0000 0.508770
$$990$$ 0 0
$$991$$ −40.0000 −1.27064 −0.635321 0.772248i $$-0.719132\pi$$
−0.635321 + 0.772248i $$0.719132\pi$$
$$992$$ −40.0000 −1.27000
$$993$$ 0 0
$$994$$ −32.0000 −1.01498
$$995$$ 0 0
$$996$$ 0 0
$$997$$ −50.0000 −1.58352 −0.791758 0.610835i $$-0.790834\pi$$
−0.791758 + 0.610835i $$0.790834\pi$$
$$998$$ −36.0000 −1.13956
$$999$$ 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 6525.2.a.j.1.1 1
3.2 odd 2 2175.2.a.b.1.1 1
5.4 even 2 1305.2.a.b.1.1 1
15.2 even 4 2175.2.c.b.349.1 2
15.8 even 4 2175.2.c.b.349.2 2
15.14 odd 2 435.2.a.d.1.1 1
60.59 even 2 6960.2.a.l.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
435.2.a.d.1.1 1 15.14 odd 2
1305.2.a.b.1.1 1 5.4 even 2
2175.2.a.b.1.1 1 3.2 odd 2
2175.2.c.b.349.1 2 15.2 even 4
2175.2.c.b.349.2 2 15.8 even 4
6525.2.a.j.1.1 1 1.1 even 1 trivial
6960.2.a.l.1.1 1 60.59 even 2