# Properties

 Label 475.2.b.a.324.1 Level $475$ Weight $2$ Character 475.324 Analytic conductor $3.793$ Analytic rank $0$ Dimension $2$ Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [475,2,Mod(324,475)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(475, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("475.324");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$475 = 5^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 475.b (of order $$2$$, degree $$1$$, not minimal)

## 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: no Analytic conductor: $$3.79289409601$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(i)$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} + 1$$ x^2 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 19) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 324.1 Root $$-1.00000i$$ of defining polynomial Character $$\chi$$ $$=$$ 475.324 Dual form 475.2.b.a.324.2

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-2.00000i q^{3} +2.00000 q^{4} +1.00000i q^{7} -1.00000 q^{9} +O(q^{10})$$ $$q-2.00000i q^{3} +2.00000 q^{4} +1.00000i q^{7} -1.00000 q^{9} +3.00000 q^{11} -4.00000i q^{12} -4.00000i q^{13} +4.00000 q^{16} +3.00000i q^{17} -1.00000 q^{19} +2.00000 q^{21} -4.00000i q^{27} +2.00000i q^{28} -6.00000 q^{29} -4.00000 q^{31} -6.00000i q^{33} -2.00000 q^{36} -2.00000i q^{37} -8.00000 q^{39} -6.00000 q^{41} -1.00000i q^{43} +6.00000 q^{44} +3.00000i q^{47} -8.00000i q^{48} +6.00000 q^{49} +6.00000 q^{51} -8.00000i q^{52} +12.0000i q^{53} +2.00000i q^{57} +6.00000 q^{59} -1.00000 q^{61} -1.00000i q^{63} +8.00000 q^{64} +4.00000i q^{67} +6.00000i q^{68} +6.00000 q^{71} -7.00000i q^{73} -2.00000 q^{76} +3.00000i q^{77} -8.00000 q^{79} -11.0000 q^{81} +12.0000i q^{83} +4.00000 q^{84} +12.0000i q^{87} -12.0000 q^{89} +4.00000 q^{91} +8.00000i q^{93} -8.00000i q^{97} -3.00000 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 4 q^{4} - 2 q^{9}+O(q^{10})$$ 2 * q + 4 * q^4 - 2 * q^9 $$2 q + 4 q^{4} - 2 q^{9} + 6 q^{11} + 8 q^{16} - 2 q^{19} + 4 q^{21} - 12 q^{29} - 8 q^{31} - 4 q^{36} - 16 q^{39} - 12 q^{41} + 12 q^{44} + 12 q^{49} + 12 q^{51} + 12 q^{59} - 2 q^{61} + 16 q^{64} + 12 q^{71} - 4 q^{76} - 16 q^{79} - 22 q^{81} + 8 q^{84} - 24 q^{89} + 8 q^{91} - 6 q^{99}+O(q^{100})$$ 2 * q + 4 * q^4 - 2 * q^9 + 6 * q^11 + 8 * q^16 - 2 * q^19 + 4 * q^21 - 12 * q^29 - 8 * q^31 - 4 * q^36 - 16 * q^39 - 12 * q^41 + 12 * q^44 + 12 * q^49 + 12 * q^51 + 12 * q^59 - 2 * q^61 + 16 * q^64 + 12 * q^71 - 4 * q^76 - 16 * q^79 - 22 * q^81 + 8 * q^84 - 24 * q^89 + 8 * q^91 - 6 * q^99

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/475\mathbb{Z}\right)^\times$$.

 $$n$$ $$77$$ $$401$$ $$\chi(n)$$ $$-1$$ $$1$$

## 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 1.00000 $$0$$
−1.00000 $$\pi$$
$$3$$ − 2.00000i − 1.15470i −0.816497 0.577350i $$-0.804087\pi$$
0.816497 0.577350i $$-0.195913\pi$$
$$4$$ 2.00000 1.00000
$$5$$ 0 0
$$6$$ 0 0
$$7$$ 1.00000i 0.377964i 0.981981 + 0.188982i $$0.0605189\pi$$
−0.981981 + 0.188982i $$0.939481\pi$$
$$8$$ 0 0
$$9$$ −1.00000 −0.333333
$$10$$ 0 0
$$11$$ 3.00000 0.904534 0.452267 0.891883i $$-0.350615\pi$$
0.452267 + 0.891883i $$0.350615\pi$$
$$12$$ − 4.00000i − 1.15470i
$$13$$ − 4.00000i − 1.10940i −0.832050 0.554700i $$-0.812833\pi$$
0.832050 0.554700i $$-0.187167\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 4.00000 1.00000
$$17$$ 3.00000i 0.727607i 0.931476 + 0.363803i $$0.118522\pi$$
−0.931476 + 0.363803i $$0.881478\pi$$
$$18$$ 0 0
$$19$$ −1.00000 −0.229416
$$20$$ 0 0
$$21$$ 2.00000 0.436436
$$22$$ 0 0
$$23$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$24$$ 0 0
$$25$$ 0 0
$$26$$ 0 0
$$27$$ − 4.00000i − 0.769800i
$$28$$ 2.00000i 0.377964i
$$29$$ −6.00000 −1.11417 −0.557086 0.830455i $$-0.688081\pi$$
−0.557086 + 0.830455i $$0.688081\pi$$
$$30$$ 0 0
$$31$$ −4.00000 −0.718421 −0.359211 0.933257i $$-0.616954\pi$$
−0.359211 + 0.933257i $$0.616954\pi$$
$$32$$ 0 0
$$33$$ − 6.00000i − 1.04447i
$$34$$ 0 0
$$35$$ 0 0
$$36$$ −2.00000 −0.333333
$$37$$ − 2.00000i − 0.328798i −0.986394 0.164399i $$-0.947432\pi$$
0.986394 0.164399i $$-0.0525685\pi$$
$$38$$ 0 0
$$39$$ −8.00000 −1.28103
$$40$$ 0 0
$$41$$ −6.00000 −0.937043 −0.468521 0.883452i $$-0.655213\pi$$
−0.468521 + 0.883452i $$0.655213\pi$$
$$42$$ 0 0
$$43$$ − 1.00000i − 0.152499i −0.997089 0.0762493i $$-0.975706\pi$$
0.997089 0.0762493i $$-0.0242945\pi$$
$$44$$ 6.00000 0.904534
$$45$$ 0 0
$$46$$ 0 0
$$47$$ 3.00000i 0.437595i 0.975770 + 0.218797i $$0.0702134\pi$$
−0.975770 + 0.218797i $$0.929787\pi$$
$$48$$ − 8.00000i − 1.15470i
$$49$$ 6.00000 0.857143
$$50$$ 0 0
$$51$$ 6.00000 0.840168
$$52$$ − 8.00000i − 1.10940i
$$53$$ 12.0000i 1.64833i 0.566352 + 0.824163i $$0.308354\pi$$
−0.566352 + 0.824163i $$0.691646\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 0 0
$$57$$ 2.00000i 0.264906i
$$58$$ 0 0
$$59$$ 6.00000 0.781133 0.390567 0.920575i $$-0.372279\pi$$
0.390567 + 0.920575i $$0.372279\pi$$
$$60$$ 0 0
$$61$$ −1.00000 −0.128037 −0.0640184 0.997949i $$-0.520392\pi$$
−0.0640184 + 0.997949i $$0.520392\pi$$
$$62$$ 0 0
$$63$$ − 1.00000i − 0.125988i
$$64$$ 8.00000 1.00000
$$65$$ 0 0
$$66$$ 0 0
$$67$$ 4.00000i 0.488678i 0.969690 + 0.244339i $$0.0785709\pi$$
−0.969690 + 0.244339i $$0.921429\pi$$
$$68$$ 6.00000i 0.727607i
$$69$$ 0 0
$$70$$ 0 0
$$71$$ 6.00000 0.712069 0.356034 0.934473i $$-0.384129\pi$$
0.356034 + 0.934473i $$0.384129\pi$$
$$72$$ 0 0
$$73$$ − 7.00000i − 0.819288i −0.912245 0.409644i $$-0.865653\pi$$
0.912245 0.409644i $$-0.134347\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ −2.00000 −0.229416
$$77$$ 3.00000i 0.341882i
$$78$$ 0 0
$$79$$ −8.00000 −0.900070 −0.450035 0.893011i $$-0.648589\pi$$
−0.450035 + 0.893011i $$0.648589\pi$$
$$80$$ 0 0
$$81$$ −11.0000 −1.22222
$$82$$ 0 0
$$83$$ 12.0000i 1.31717i 0.752506 + 0.658586i $$0.228845\pi$$
−0.752506 + 0.658586i $$0.771155\pi$$
$$84$$ 4.00000 0.436436
$$85$$ 0 0
$$86$$ 0 0
$$87$$ 12.0000i 1.28654i
$$88$$ 0 0
$$89$$ −12.0000 −1.27200 −0.635999 0.771690i $$-0.719412\pi$$
−0.635999 + 0.771690i $$0.719412\pi$$
$$90$$ 0 0
$$91$$ 4.00000 0.419314
$$92$$ 0 0
$$93$$ 8.00000i 0.829561i
$$94$$ 0 0
$$95$$ 0 0
$$96$$ 0 0
$$97$$ − 8.00000i − 0.812277i −0.913812 0.406138i $$-0.866875\pi$$
0.913812 0.406138i $$-0.133125\pi$$
$$98$$ 0 0
$$99$$ −3.00000 −0.301511
$$100$$ 0 0
$$101$$ 6.00000 0.597022 0.298511 0.954406i $$-0.403510\pi$$
0.298511 + 0.954406i $$0.403510\pi$$
$$102$$ 0 0
$$103$$ 14.0000i 1.37946i 0.724066 + 0.689730i $$0.242271\pi$$
−0.724066 + 0.689730i $$0.757729\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ 18.0000i 1.74013i 0.492941 + 0.870063i $$0.335922\pi$$
−0.492941 + 0.870063i $$0.664078\pi$$
$$108$$ − 8.00000i − 0.769800i
$$109$$ 16.0000 1.53252 0.766261 0.642529i $$-0.222115\pi$$
0.766261 + 0.642529i $$0.222115\pi$$
$$110$$ 0 0
$$111$$ −4.00000 −0.379663
$$112$$ 4.00000i 0.377964i
$$113$$ 6.00000i 0.564433i 0.959351 + 0.282216i $$0.0910696\pi$$
−0.959351 + 0.282216i $$0.908930\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ −12.0000 −1.11417
$$117$$ 4.00000i 0.369800i
$$118$$ 0 0
$$119$$ −3.00000 −0.275010
$$120$$ 0 0
$$121$$ −2.00000 −0.181818
$$122$$ 0 0
$$123$$ 12.0000i 1.08200i
$$124$$ −8.00000 −0.718421
$$125$$ 0 0
$$126$$ 0 0
$$127$$ − 2.00000i − 0.177471i −0.996055 0.0887357i $$-0.971717\pi$$
0.996055 0.0887357i $$-0.0282826\pi$$
$$128$$ 0 0
$$129$$ −2.00000 −0.176090
$$130$$ 0 0
$$131$$ −15.0000 −1.31056 −0.655278 0.755388i $$-0.727449\pi$$
−0.655278 + 0.755388i $$0.727449\pi$$
$$132$$ − 12.0000i − 1.04447i
$$133$$ − 1.00000i − 0.0867110i
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ 3.00000i 0.256307i 0.991754 + 0.128154i $$0.0409051\pi$$
−0.991754 + 0.128154i $$0.959095\pi$$
$$138$$ 0 0
$$139$$ 13.0000 1.10265 0.551323 0.834292i $$-0.314123\pi$$
0.551323 + 0.834292i $$0.314123\pi$$
$$140$$ 0 0
$$141$$ 6.00000 0.505291
$$142$$ 0 0
$$143$$ − 12.0000i − 1.00349i
$$144$$ −4.00000 −0.333333
$$145$$ 0 0
$$146$$ 0 0
$$147$$ − 12.0000i − 0.989743i
$$148$$ − 4.00000i − 0.328798i
$$149$$ −21.0000 −1.72039 −0.860194 0.509968i $$-0.829657\pi$$
−0.860194 + 0.509968i $$0.829657\pi$$
$$150$$ 0 0
$$151$$ −10.0000 −0.813788 −0.406894 0.913475i $$-0.633388\pi$$
−0.406894 + 0.913475i $$0.633388\pi$$
$$152$$ 0 0
$$153$$ − 3.00000i − 0.242536i
$$154$$ 0 0
$$155$$ 0 0
$$156$$ −16.0000 −1.28103
$$157$$ − 14.0000i − 1.11732i −0.829396 0.558661i $$-0.811315\pi$$
0.829396 0.558661i $$-0.188685\pi$$
$$158$$ 0 0
$$159$$ 24.0000 1.90332
$$160$$ 0 0
$$161$$ 0 0
$$162$$ 0 0
$$163$$ 20.0000i 1.56652i 0.621694 + 0.783260i $$0.286445\pi$$
−0.621694 + 0.783260i $$0.713555\pi$$
$$164$$ −12.0000 −0.937043
$$165$$ 0 0
$$166$$ 0 0
$$167$$ 18.0000i 1.39288i 0.717614 + 0.696441i $$0.245234\pi$$
−0.717614 + 0.696441i $$0.754766\pi$$
$$168$$ 0 0
$$169$$ −3.00000 −0.230769
$$170$$ 0 0
$$171$$ 1.00000 0.0764719
$$172$$ − 2.00000i − 0.152499i
$$173$$ − 18.0000i − 1.36851i −0.729241 0.684257i $$-0.760127\pi$$
0.729241 0.684257i $$-0.239873\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 12.0000 0.904534
$$177$$ − 12.0000i − 0.901975i
$$178$$ 0 0
$$179$$ 18.0000 1.34538 0.672692 0.739923i $$-0.265138\pi$$
0.672692 + 0.739923i $$0.265138\pi$$
$$180$$ 0 0
$$181$$ 2.00000 0.148659 0.0743294 0.997234i $$-0.476318\pi$$
0.0743294 + 0.997234i $$0.476318\pi$$
$$182$$ 0 0
$$183$$ 2.00000i 0.147844i
$$184$$ 0 0
$$185$$ 0 0
$$186$$ 0 0
$$187$$ 9.00000i 0.658145i
$$188$$ 6.00000i 0.437595i
$$189$$ 4.00000 0.290957
$$190$$ 0 0
$$191$$ 3.00000 0.217072 0.108536 0.994092i $$-0.465384\pi$$
0.108536 + 0.994092i $$0.465384\pi$$
$$192$$ − 16.0000i − 1.15470i
$$193$$ − 4.00000i − 0.287926i −0.989583 0.143963i $$-0.954015\pi$$
0.989583 0.143963i $$-0.0459847\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 12.0000 0.857143
$$197$$ − 18.0000i − 1.28245i −0.767354 0.641223i $$-0.778427\pi$$
0.767354 0.641223i $$-0.221573\pi$$
$$198$$ 0 0
$$199$$ −11.0000 −0.779769 −0.389885 0.920864i $$-0.627485\pi$$
−0.389885 + 0.920864i $$0.627485\pi$$
$$200$$ 0 0
$$201$$ 8.00000 0.564276
$$202$$ 0 0
$$203$$ − 6.00000i − 0.421117i
$$204$$ 12.0000 0.840168
$$205$$ 0 0
$$206$$ 0 0
$$207$$ 0 0
$$208$$ − 16.0000i − 1.10940i
$$209$$ −3.00000 −0.207514
$$210$$ 0 0
$$211$$ 14.0000 0.963800 0.481900 0.876226i $$-0.339947\pi$$
0.481900 + 0.876226i $$0.339947\pi$$
$$212$$ 24.0000i 1.64833i
$$213$$ − 12.0000i − 0.822226i
$$214$$ 0 0
$$215$$ 0 0
$$216$$ 0 0
$$217$$ − 4.00000i − 0.271538i
$$218$$ 0 0
$$219$$ −14.0000 −0.946032
$$220$$ 0 0
$$221$$ 12.0000 0.807207
$$222$$ 0 0
$$223$$ − 10.0000i − 0.669650i −0.942280 0.334825i $$-0.891323\pi$$
0.942280 0.334825i $$-0.108677\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ − 12.0000i − 0.796468i −0.917284 0.398234i $$-0.869623\pi$$
0.917284 0.398234i $$-0.130377\pi$$
$$228$$ 4.00000i 0.264906i
$$229$$ −5.00000 −0.330409 −0.165205 0.986259i $$-0.552828\pi$$
−0.165205 + 0.986259i $$0.552828\pi$$
$$230$$ 0 0
$$231$$ 6.00000 0.394771
$$232$$ 0 0
$$233$$ − 21.0000i − 1.37576i −0.725826 0.687878i $$-0.758542\pi$$
0.725826 0.687878i $$-0.241458\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ 12.0000 0.781133
$$237$$ 16.0000i 1.03931i
$$238$$ 0 0
$$239$$ −15.0000 −0.970269 −0.485135 0.874439i $$-0.661229\pi$$
−0.485135 + 0.874439i $$0.661229\pi$$
$$240$$ 0 0
$$241$$ −10.0000 −0.644157 −0.322078 0.946713i $$-0.604381\pi$$
−0.322078 + 0.946713i $$0.604381\pi$$
$$242$$ 0 0
$$243$$ 10.0000i 0.641500i
$$244$$ −2.00000 −0.128037
$$245$$ 0 0
$$246$$ 0 0
$$247$$ 4.00000i 0.254514i
$$248$$ 0 0
$$249$$ 24.0000 1.52094
$$250$$ 0 0
$$251$$ 21.0000 1.32551 0.662754 0.748837i $$-0.269387\pi$$
0.662754 + 0.748837i $$0.269387\pi$$
$$252$$ − 2.00000i − 0.125988i
$$253$$ 0 0
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 16.0000 1.00000
$$257$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$258$$ 0 0
$$259$$ 2.00000 0.124274
$$260$$ 0 0
$$261$$ 6.00000 0.371391
$$262$$ 0 0
$$263$$ 9.00000i 0.554964i 0.960731 + 0.277482i $$0.0894999\pi$$
−0.960731 + 0.277482i $$0.910500\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 0 0
$$267$$ 24.0000i 1.46878i
$$268$$ 8.00000i 0.488678i
$$269$$ −24.0000 −1.46331 −0.731653 0.681677i $$-0.761251\pi$$
−0.731653 + 0.681677i $$0.761251\pi$$
$$270$$ 0 0
$$271$$ −16.0000 −0.971931 −0.485965 0.873978i $$-0.661532\pi$$
−0.485965 + 0.873978i $$0.661532\pi$$
$$272$$ 12.0000i 0.727607i
$$273$$ − 8.00000i − 0.484182i
$$274$$ 0 0
$$275$$ 0 0
$$276$$ 0 0
$$277$$ 19.0000i 1.14160i 0.821089 + 0.570800i $$0.193367\pi$$
−0.821089 + 0.570800i $$0.806633\pi$$
$$278$$ 0 0
$$279$$ 4.00000 0.239474
$$280$$ 0 0
$$281$$ 6.00000 0.357930 0.178965 0.983855i $$-0.442725\pi$$
0.178965 + 0.983855i $$0.442725\pi$$
$$282$$ 0 0
$$283$$ − 13.0000i − 0.772770i −0.922338 0.386385i $$-0.873724\pi$$
0.922338 0.386385i $$-0.126276\pi$$
$$284$$ 12.0000 0.712069
$$285$$ 0 0
$$286$$ 0 0
$$287$$ − 6.00000i − 0.354169i
$$288$$ 0 0
$$289$$ 8.00000 0.470588
$$290$$ 0 0
$$291$$ −16.0000 −0.937937
$$292$$ − 14.0000i − 0.819288i
$$293$$ − 12.0000i − 0.701047i −0.936554 0.350524i $$-0.886004\pi$$
0.936554 0.350524i $$-0.113996\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 0 0
$$297$$ − 12.0000i − 0.696311i
$$298$$ 0 0
$$299$$ 0 0
$$300$$ 0 0
$$301$$ 1.00000 0.0576390
$$302$$ 0 0
$$303$$ − 12.0000i − 0.689382i
$$304$$ −4.00000 −0.229416
$$305$$ 0 0
$$306$$ 0 0
$$307$$ − 20.0000i − 1.14146i −0.821138 0.570730i $$-0.806660\pi$$
0.821138 0.570730i $$-0.193340\pi$$
$$308$$ 6.00000i 0.341882i
$$309$$ 28.0000 1.59286
$$310$$ 0 0
$$311$$ −3.00000 −0.170114 −0.0850572 0.996376i $$-0.527107\pi$$
−0.0850572 + 0.996376i $$0.527107\pi$$
$$312$$ 0 0
$$313$$ − 10.0000i − 0.565233i −0.959233 0.282617i $$-0.908798\pi$$
0.959233 0.282617i $$-0.0912024\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ −16.0000 −0.900070
$$317$$ − 6.00000i − 0.336994i −0.985702 0.168497i $$-0.946109\pi$$
0.985702 0.168497i $$-0.0538913\pi$$
$$318$$ 0 0
$$319$$ −18.0000 −1.00781
$$320$$ 0 0
$$321$$ 36.0000 2.00932
$$322$$ 0 0
$$323$$ − 3.00000i − 0.166924i
$$324$$ −22.0000 −1.22222
$$325$$ 0 0
$$326$$ 0 0
$$327$$ − 32.0000i − 1.76960i
$$328$$ 0 0
$$329$$ −3.00000 −0.165395
$$330$$ 0 0
$$331$$ −28.0000 −1.53902 −0.769510 0.638635i $$-0.779499\pi$$
−0.769510 + 0.638635i $$0.779499\pi$$
$$332$$ 24.0000i 1.31717i
$$333$$ 2.00000i 0.109599i
$$334$$ 0 0
$$335$$ 0 0
$$336$$ 8.00000 0.436436
$$337$$ − 32.0000i − 1.74315i −0.490261 0.871576i $$-0.663099\pi$$
0.490261 0.871576i $$-0.336901\pi$$
$$338$$ 0 0
$$339$$ 12.0000 0.651751
$$340$$ 0 0
$$341$$ −12.0000 −0.649836
$$342$$ 0 0
$$343$$ 13.0000i 0.701934i
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ − 21.0000i − 1.12734i −0.826000 0.563670i $$-0.809389\pi$$
0.826000 0.563670i $$-0.190611\pi$$
$$348$$ 24.0000i 1.28654i
$$349$$ −17.0000 −0.909989 −0.454995 0.890494i $$-0.650359\pi$$
−0.454995 + 0.890494i $$0.650359\pi$$
$$350$$ 0 0
$$351$$ −16.0000 −0.854017
$$352$$ 0 0
$$353$$ − 6.00000i − 0.319348i −0.987170 0.159674i $$-0.948956\pi$$
0.987170 0.159674i $$-0.0510443\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ −24.0000 −1.27200
$$357$$ 6.00000i 0.317554i
$$358$$ 0 0
$$359$$ −15.0000 −0.791670 −0.395835 0.918322i $$-0.629545\pi$$
−0.395835 + 0.918322i $$0.629545\pi$$
$$360$$ 0 0
$$361$$ 1.00000 0.0526316
$$362$$ 0 0
$$363$$ 4.00000i 0.209946i
$$364$$ 8.00000 0.419314
$$365$$ 0 0
$$366$$ 0 0
$$367$$ − 8.00000i − 0.417597i −0.977959 0.208798i $$-0.933045\pi$$
0.977959 0.208798i $$-0.0669552\pi$$
$$368$$ 0 0
$$369$$ 6.00000 0.312348
$$370$$ 0 0
$$371$$ −12.0000 −0.623009
$$372$$ 16.0000i 0.829561i
$$373$$ − 4.00000i − 0.207112i −0.994624 0.103556i $$-0.966978\pi$$
0.994624 0.103556i $$-0.0330221\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ 24.0000i 1.23606i
$$378$$ 0 0
$$379$$ 34.0000 1.74646 0.873231 0.487306i $$-0.162020\pi$$
0.873231 + 0.487306i $$0.162020\pi$$
$$380$$ 0 0
$$381$$ −4.00000 −0.204926
$$382$$ 0 0
$$383$$ 12.0000i 0.613171i 0.951843 + 0.306586i $$0.0991866\pi$$
−0.951843 + 0.306586i $$0.900813\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ 1.00000i 0.0508329i
$$388$$ − 16.0000i − 0.812277i
$$389$$ −15.0000 −0.760530 −0.380265 0.924878i $$-0.624167\pi$$
−0.380265 + 0.924878i $$0.624167\pi$$
$$390$$ 0 0
$$391$$ 0 0
$$392$$ 0 0
$$393$$ 30.0000i 1.51330i
$$394$$ 0 0
$$395$$ 0 0
$$396$$ −6.00000 −0.301511
$$397$$ 7.00000i 0.351320i 0.984451 + 0.175660i $$0.0562059\pi$$
−0.984451 + 0.175660i $$0.943794\pi$$
$$398$$ 0 0
$$399$$ −2.00000 −0.100125
$$400$$ 0 0
$$401$$ 12.0000 0.599251 0.299626 0.954057i $$-0.403138\pi$$
0.299626 + 0.954057i $$0.403138\pi$$
$$402$$ 0 0
$$403$$ 16.0000i 0.797017i
$$404$$ 12.0000 0.597022
$$405$$ 0 0
$$406$$ 0 0
$$407$$ − 6.00000i − 0.297409i
$$408$$ 0 0
$$409$$ 4.00000 0.197787 0.0988936 0.995098i $$-0.468470\pi$$
0.0988936 + 0.995098i $$0.468470\pi$$
$$410$$ 0 0
$$411$$ 6.00000 0.295958
$$412$$ 28.0000i 1.37946i
$$413$$ 6.00000i 0.295241i
$$414$$ 0 0
$$415$$ 0 0
$$416$$ 0 0
$$417$$ − 26.0000i − 1.27323i
$$418$$ 0 0
$$419$$ 12.0000 0.586238 0.293119 0.956076i $$-0.405307\pi$$
0.293119 + 0.956076i $$0.405307\pi$$
$$420$$ 0 0
$$421$$ 8.00000 0.389896 0.194948 0.980814i $$-0.437546\pi$$
0.194948 + 0.980814i $$0.437546\pi$$
$$422$$ 0 0
$$423$$ − 3.00000i − 0.145865i
$$424$$ 0 0
$$425$$ 0 0
$$426$$ 0 0
$$427$$ − 1.00000i − 0.0483934i
$$428$$ 36.0000i 1.74013i
$$429$$ −24.0000 −1.15873
$$430$$ 0 0
$$431$$ −24.0000 −1.15604 −0.578020 0.816023i $$-0.696174\pi$$
−0.578020 + 0.816023i $$0.696174\pi$$
$$432$$ − 16.0000i − 0.769800i
$$433$$ 2.00000i 0.0961139i 0.998845 + 0.0480569i $$0.0153029\pi$$
−0.998845 + 0.0480569i $$0.984697\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 32.0000 1.53252
$$437$$ 0 0
$$438$$ 0 0
$$439$$ 10.0000 0.477274 0.238637 0.971109i $$-0.423299\pi$$
0.238637 + 0.971109i $$0.423299\pi$$
$$440$$ 0 0
$$441$$ −6.00000 −0.285714
$$442$$ 0 0
$$443$$ − 3.00000i − 0.142534i −0.997457 0.0712672i $$-0.977296\pi$$
0.997457 0.0712672i $$-0.0227043\pi$$
$$444$$ −8.00000 −0.379663
$$445$$ 0 0
$$446$$ 0 0
$$447$$ 42.0000i 1.98653i
$$448$$ 8.00000i 0.377964i
$$449$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$450$$ 0 0
$$451$$ −18.0000 −0.847587
$$452$$ 12.0000i 0.564433i
$$453$$ 20.0000i 0.939682i
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 37.0000i 1.73079i 0.501093 + 0.865393i $$0.332931\pi$$
−0.501093 + 0.865393i $$0.667069\pi$$
$$458$$ 0 0
$$459$$ 12.0000 0.560112
$$460$$ 0 0
$$461$$ 9.00000 0.419172 0.209586 0.977790i $$-0.432788\pi$$
0.209586 + 0.977790i $$0.432788\pi$$
$$462$$ 0 0
$$463$$ − 31.0000i − 1.44069i −0.693615 0.720346i $$-0.743983\pi$$
0.693615 0.720346i $$-0.256017\pi$$
$$464$$ −24.0000 −1.11417
$$465$$ 0 0
$$466$$ 0 0
$$467$$ 27.0000i 1.24941i 0.780860 + 0.624705i $$0.214781\pi$$
−0.780860 + 0.624705i $$0.785219\pi$$
$$468$$ 8.00000i 0.369800i
$$469$$ −4.00000 −0.184703
$$470$$ 0 0
$$471$$ −28.0000 −1.29017
$$472$$ 0 0
$$473$$ − 3.00000i − 0.137940i
$$474$$ 0 0
$$475$$ 0 0
$$476$$ −6.00000 −0.275010
$$477$$ − 12.0000i − 0.549442i
$$478$$ 0 0
$$479$$ 12.0000 0.548294 0.274147 0.961688i $$-0.411605\pi$$
0.274147 + 0.961688i $$0.411605\pi$$
$$480$$ 0 0
$$481$$ −8.00000 −0.364769
$$482$$ 0 0
$$483$$ 0 0
$$484$$ −4.00000 −0.181818
$$485$$ 0 0
$$486$$ 0 0
$$487$$ − 2.00000i − 0.0906287i −0.998973 0.0453143i $$-0.985571\pi$$
0.998973 0.0453143i $$-0.0144289\pi$$
$$488$$ 0 0
$$489$$ 40.0000 1.80886
$$490$$ 0 0
$$491$$ 12.0000 0.541552 0.270776 0.962642i $$-0.412720\pi$$
0.270776 + 0.962642i $$0.412720\pi$$
$$492$$ 24.0000i 1.08200i
$$493$$ − 18.0000i − 0.810679i
$$494$$ 0 0
$$495$$ 0 0
$$496$$ −16.0000 −0.718421
$$497$$ 6.00000i 0.269137i
$$498$$ 0 0
$$499$$ −5.00000 −0.223831 −0.111915 0.993718i $$-0.535699\pi$$
−0.111915 + 0.993718i $$0.535699\pi$$
$$500$$ 0 0
$$501$$ 36.0000 1.60836
$$502$$ 0 0
$$503$$ 12.0000i 0.535054i 0.963550 + 0.267527i $$0.0862064\pi$$
−0.963550 + 0.267527i $$0.913794\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 0 0
$$507$$ 6.00000i 0.266469i
$$508$$ − 4.00000i − 0.177471i
$$509$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$510$$ 0 0
$$511$$ 7.00000 0.309662
$$512$$ 0 0
$$513$$ 4.00000i 0.176604i
$$514$$ 0 0
$$515$$ 0 0
$$516$$ −4.00000 −0.176090
$$517$$ 9.00000i 0.395820i
$$518$$ 0 0
$$519$$ −36.0000 −1.58022
$$520$$ 0 0
$$521$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$522$$ 0 0
$$523$$ 38.0000i 1.66162i 0.556553 + 0.830812i $$0.312124\pi$$
−0.556553 + 0.830812i $$0.687876\pi$$
$$524$$ −30.0000 −1.31056
$$525$$ 0 0
$$526$$ 0 0
$$527$$ − 12.0000i − 0.522728i
$$528$$ − 24.0000i − 1.04447i
$$529$$ 23.0000 1.00000
$$530$$ 0 0
$$531$$ −6.00000 −0.260378
$$532$$ − 2.00000i − 0.0867110i
$$533$$ 24.0000i 1.03956i
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 0 0
$$537$$ − 36.0000i − 1.55351i
$$538$$ 0 0
$$539$$ 18.0000 0.775315
$$540$$ 0 0
$$541$$ −25.0000 −1.07483 −0.537417 0.843317i $$-0.680600\pi$$
−0.537417 + 0.843317i $$0.680600\pi$$
$$542$$ 0 0
$$543$$ − 4.00000i − 0.171656i
$$544$$ 0 0
$$545$$ 0 0
$$546$$ 0 0
$$547$$ 28.0000i 1.19719i 0.801050 + 0.598597i $$0.204275\pi$$
−0.801050 + 0.598597i $$0.795725\pi$$
$$548$$ 6.00000i 0.256307i
$$549$$ 1.00000 0.0426790
$$550$$ 0 0
$$551$$ 6.00000 0.255609
$$552$$ 0 0
$$553$$ − 8.00000i − 0.340195i
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 26.0000 1.10265
$$557$$ − 21.0000i − 0.889799i −0.895581 0.444899i $$-0.853239\pi$$
0.895581 0.444899i $$-0.146761\pi$$
$$558$$ 0 0
$$559$$ −4.00000 −0.169182
$$560$$ 0 0
$$561$$ 18.0000 0.759961
$$562$$ 0 0
$$563$$ 6.00000i 0.252870i 0.991975 + 0.126435i $$0.0403535\pi$$
−0.991975 + 0.126435i $$0.959647\pi$$
$$564$$ 12.0000 0.505291
$$565$$ 0 0
$$566$$ 0 0
$$567$$ − 11.0000i − 0.461957i
$$568$$ 0 0
$$569$$ 24.0000 1.00613 0.503066 0.864248i $$-0.332205\pi$$
0.503066 + 0.864248i $$0.332205\pi$$
$$570$$ 0 0
$$571$$ −4.00000 −0.167395 −0.0836974 0.996491i $$-0.526673\pi$$
−0.0836974 + 0.996491i $$0.526673\pi$$
$$572$$ − 24.0000i − 1.00349i
$$573$$ − 6.00000i − 0.250654i
$$574$$ 0 0
$$575$$ 0 0
$$576$$ −8.00000 −0.333333
$$577$$ − 11.0000i − 0.457936i −0.973434 0.228968i $$-0.926465\pi$$
0.973434 0.228968i $$-0.0735351\pi$$
$$578$$ 0 0
$$579$$ −8.00000 −0.332469
$$580$$ 0 0
$$581$$ −12.0000 −0.497844
$$582$$ 0 0
$$583$$ 36.0000i 1.49097i
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ − 45.0000i − 1.85735i −0.370896 0.928674i $$-0.620949\pi$$
0.370896 0.928674i $$-0.379051\pi$$
$$588$$ − 24.0000i − 0.989743i
$$589$$ 4.00000 0.164817
$$590$$ 0 0
$$591$$ −36.0000 −1.48084
$$592$$ − 8.00000i − 0.328798i
$$593$$ − 42.0000i − 1.72473i −0.506284 0.862367i $$-0.668981\pi$$
0.506284 0.862367i $$-0.331019\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ −42.0000 −1.72039
$$597$$ 22.0000i 0.900400i
$$598$$ 0 0
$$599$$ 36.0000 1.47092 0.735460 0.677568i $$-0.236966\pi$$
0.735460 + 0.677568i $$0.236966\pi$$
$$600$$ 0 0
$$601$$ 26.0000 1.06056 0.530281 0.847822i $$-0.322086\pi$$
0.530281 + 0.847822i $$0.322086\pi$$
$$602$$ 0 0
$$603$$ − 4.00000i − 0.162893i
$$604$$ −20.0000 −0.813788
$$605$$ 0 0
$$606$$ 0 0
$$607$$ − 32.0000i − 1.29884i −0.760430 0.649420i $$-0.775012\pi$$
0.760430 0.649420i $$-0.224988\pi$$
$$608$$ 0 0
$$609$$ −12.0000 −0.486265
$$610$$ 0 0
$$611$$ 12.0000 0.485468
$$612$$ − 6.00000i − 0.242536i
$$613$$ 29.0000i 1.17130i 0.810564 + 0.585649i $$0.199160\pi$$
−0.810564 + 0.585649i $$0.800840\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ − 9.00000i − 0.362326i −0.983453 0.181163i $$-0.942014\pi$$
0.983453 0.181163i $$-0.0579862\pi$$
$$618$$ 0 0
$$619$$ −44.0000 −1.76851 −0.884255 0.467005i $$-0.845333\pi$$
−0.884255 + 0.467005i $$0.845333\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 0 0
$$623$$ − 12.0000i − 0.480770i
$$624$$ −32.0000 −1.28103
$$625$$ 0 0
$$626$$ 0 0
$$627$$ 6.00000i 0.239617i
$$628$$ − 28.0000i − 1.11732i
$$629$$ 6.00000 0.239236
$$630$$ 0 0
$$631$$ 11.0000 0.437903 0.218952 0.975736i $$-0.429736\pi$$
0.218952 + 0.975736i $$0.429736\pi$$
$$632$$ 0 0
$$633$$ − 28.0000i − 1.11290i
$$634$$ 0 0
$$635$$ 0 0
$$636$$ 48.0000 1.90332
$$637$$ − 24.0000i − 0.950915i
$$638$$ 0 0
$$639$$ −6.00000 −0.237356
$$640$$ 0 0
$$641$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$642$$ 0 0
$$643$$ − 13.0000i − 0.512670i −0.966588 0.256335i $$-0.917485\pi$$
0.966588 0.256335i $$-0.0825150\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ − 27.0000i − 1.06148i −0.847535 0.530740i $$-0.821914\pi$$
0.847535 0.530740i $$-0.178086\pi$$
$$648$$ 0 0
$$649$$ 18.0000 0.706562
$$650$$ 0 0
$$651$$ −8.00000 −0.313545
$$652$$ 40.0000i 1.56652i
$$653$$ − 39.0000i − 1.52619i −0.646288 0.763094i $$-0.723679\pi$$
0.646288 0.763094i $$-0.276321\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ −24.0000 −0.937043
$$657$$ 7.00000i 0.273096i
$$658$$ 0 0
$$659$$ 30.0000 1.16863 0.584317 0.811525i $$-0.301362\pi$$
0.584317 + 0.811525i $$0.301362\pi$$
$$660$$ 0 0
$$661$$ 32.0000 1.24466 0.622328 0.782757i $$-0.286187\pi$$
0.622328 + 0.782757i $$0.286187\pi$$
$$662$$ 0 0
$$663$$ − 24.0000i − 0.932083i
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 0 0
$$668$$ 36.0000i 1.39288i
$$669$$ −20.0000 −0.773245
$$670$$ 0 0
$$671$$ −3.00000 −0.115814
$$672$$ 0 0
$$673$$ − 10.0000i − 0.385472i −0.981251 0.192736i $$-0.938264\pi$$
0.981251 0.192736i $$-0.0617360\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ −6.00000 −0.230769
$$677$$ 42.0000i 1.61419i 0.590421 + 0.807096i $$0.298962\pi$$
−0.590421 + 0.807096i $$0.701038\pi$$
$$678$$ 0 0
$$679$$ 8.00000 0.307012
$$680$$ 0 0
$$681$$ −24.0000 −0.919682
$$682$$ 0 0
$$683$$ 36.0000i 1.37750i 0.724998 + 0.688751i $$0.241841\pi$$
−0.724998 + 0.688751i $$0.758159\pi$$
$$684$$ 2.00000 0.0764719
$$685$$ 0 0
$$686$$ 0 0
$$687$$ 10.0000i 0.381524i
$$688$$ − 4.00000i − 0.152499i
$$689$$ 48.0000 1.82865
$$690$$ 0 0
$$691$$ 17.0000 0.646710 0.323355 0.946278i $$-0.395189\pi$$
0.323355 + 0.946278i $$0.395189\pi$$
$$692$$ − 36.0000i − 1.36851i
$$693$$ − 3.00000i − 0.113961i
$$694$$ 0 0
$$695$$ 0 0
$$696$$ 0 0
$$697$$ − 18.0000i − 0.681799i
$$698$$ 0 0
$$699$$ −42.0000 −1.58859
$$700$$ 0 0
$$701$$ 6.00000 0.226617 0.113308 0.993560i $$-0.463855\pi$$
0.113308 + 0.993560i $$0.463855\pi$$
$$702$$ 0 0
$$703$$ 2.00000i 0.0754314i
$$704$$ 24.0000 0.904534
$$705$$ 0 0
$$706$$ 0 0
$$707$$ 6.00000i 0.225653i
$$708$$ − 24.0000i − 0.901975i
$$709$$ −26.0000 −0.976450 −0.488225 0.872718i $$-0.662356\pi$$
−0.488225 + 0.872718i $$0.662356\pi$$
$$710$$ 0 0
$$711$$ 8.00000 0.300023
$$712$$ 0 0
$$713$$ 0 0
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 36.0000 1.34538
$$717$$ 30.0000i 1.12037i
$$718$$ 0 0
$$719$$ −15.0000 −0.559406 −0.279703 0.960087i $$-0.590236\pi$$
−0.279703 + 0.960087i $$0.590236\pi$$
$$720$$ 0 0
$$721$$ −14.0000 −0.521387
$$722$$ 0 0
$$723$$ 20.0000i 0.743808i
$$724$$ 4.00000 0.148659
$$725$$ 0 0
$$726$$ 0 0
$$727$$ 19.0000i 0.704671i 0.935874 + 0.352335i $$0.114612\pi$$
−0.935874 + 0.352335i $$0.885388\pi$$
$$728$$ 0 0
$$729$$ −13.0000 −0.481481
$$730$$ 0 0
$$731$$ 3.00000 0.110959
$$732$$ 4.00000i 0.147844i
$$733$$ − 22.0000i − 0.812589i −0.913742 0.406294i $$-0.866821\pi$$
0.913742 0.406294i $$-0.133179\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ 12.0000i 0.442026i
$$738$$ 0 0
$$739$$ −11.0000 −0.404642 −0.202321 0.979319i $$-0.564848\pi$$
−0.202321 + 0.979319i $$0.564848\pi$$
$$740$$ 0 0
$$741$$ 8.00000 0.293887
$$742$$ 0 0
$$743$$ − 24.0000i − 0.880475i −0.897881 0.440237i $$-0.854894\pi$$
0.897881 0.440237i $$-0.145106\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 0 0
$$747$$ − 12.0000i − 0.439057i
$$748$$ 18.0000i 0.658145i
$$749$$ −18.0000 −0.657706
$$750$$ 0 0
$$751$$ 32.0000 1.16770 0.583848 0.811863i $$-0.301546\pi$$
0.583848 + 0.811863i $$0.301546\pi$$
$$752$$ 12.0000i 0.437595i
$$753$$ − 42.0000i − 1.53057i
$$754$$ 0 0
$$755$$ 0 0
$$756$$ 8.00000 0.290957
$$757$$ 25.0000i 0.908640i 0.890838 + 0.454320i $$0.150118\pi$$
−0.890838 + 0.454320i $$0.849882\pi$$
$$758$$ 0 0
$$759$$ 0 0
$$760$$ 0 0
$$761$$ 33.0000 1.19625 0.598125 0.801403i $$-0.295913\pi$$
0.598125 + 0.801403i $$0.295913\pi$$
$$762$$ 0 0
$$763$$ 16.0000i 0.579239i
$$764$$ 6.00000 0.217072
$$765$$ 0 0
$$766$$ 0 0
$$767$$ − 24.0000i − 0.866590i
$$768$$ − 32.0000i − 1.15470i
$$769$$ −23.0000 −0.829401 −0.414701 0.909958i $$-0.636114\pi$$
−0.414701 + 0.909958i $$0.636114\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ − 8.00000i − 0.287926i
$$773$$ − 6.00000i − 0.215805i −0.994161 0.107903i $$-0.965587\pi$$
0.994161 0.107903i $$-0.0344134\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ 0 0
$$777$$ − 4.00000i − 0.143499i
$$778$$ 0 0
$$779$$ 6.00000 0.214972
$$780$$ 0 0
$$781$$ 18.0000 0.644091
$$782$$ 0 0
$$783$$ 24.0000i 0.857690i
$$784$$ 24.0000 0.857143
$$785$$ 0 0
$$786$$ 0 0
$$787$$ 4.00000i 0.142585i 0.997455 + 0.0712923i $$0.0227123\pi$$
−0.997455 + 0.0712923i $$0.977288\pi$$
$$788$$ − 36.0000i − 1.28245i
$$789$$ 18.0000 0.640817
$$790$$ 0 0
$$791$$ −6.00000 −0.213335
$$792$$ 0 0
$$793$$ 4.00000i 0.142044i
$$794$$ 0 0
$$795$$ 0 0
$$796$$ −22.0000 −0.779769
$$797$$ 12.0000i 0.425062i 0.977154 + 0.212531i $$0.0681706\pi$$
−0.977154 + 0.212531i $$0.931829\pi$$
$$798$$ 0 0
$$799$$ −9.00000 −0.318397
$$800$$ 0 0
$$801$$ 12.0000 0.423999
$$802$$ 0 0
$$803$$ − 21.0000i − 0.741074i
$$804$$ 16.0000 0.564276
$$805$$ 0 0
$$806$$ 0 0
$$807$$ 48.0000i 1.68968i
$$808$$ 0 0
$$809$$ 9.00000 0.316423 0.158212 0.987405i $$-0.449427\pi$$
0.158212 + 0.987405i $$0.449427\pi$$
$$810$$ 0 0
$$811$$ −16.0000 −0.561836 −0.280918 0.959732i $$-0.590639\pi$$
−0.280918 + 0.959732i $$0.590639\pi$$
$$812$$ − 12.0000i − 0.421117i
$$813$$ 32.0000i 1.12229i
$$814$$ 0 0
$$815$$ 0 0
$$816$$ 24.0000 0.840168
$$817$$ 1.00000i 0.0349856i
$$818$$ 0 0
$$819$$ −4.00000 −0.139771
$$820$$ 0 0
$$821$$ 33.0000 1.15171 0.575854 0.817553i $$-0.304670\pi$$
0.575854 + 0.817553i $$0.304670\pi$$
$$822$$ 0 0
$$823$$ − 49.0000i − 1.70803i −0.520246 0.854016i $$-0.674160\pi$$
0.520246 0.854016i $$-0.325840\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ − 12.0000i − 0.417281i −0.977992 0.208640i $$-0.933096\pi$$
0.977992 0.208640i $$-0.0669038\pi$$
$$828$$ 0 0
$$829$$ 16.0000 0.555703 0.277851 0.960624i $$-0.410378\pi$$
0.277851 + 0.960624i $$0.410378\pi$$
$$830$$ 0 0
$$831$$ 38.0000 1.31821
$$832$$ − 32.0000i − 1.10940i
$$833$$ 18.0000i 0.623663i
$$834$$ 0 0
$$835$$ 0 0
$$836$$ −6.00000 −0.207514
$$837$$ 16.0000i 0.553041i
$$838$$ 0 0
$$839$$ −18.0000 −0.621429 −0.310715 0.950503i $$-0.600568\pi$$
−0.310715 + 0.950503i $$0.600568\pi$$
$$840$$ 0 0
$$841$$ 7.00000 0.241379
$$842$$ 0 0
$$843$$ − 12.0000i − 0.413302i
$$844$$ 28.0000 0.963800
$$845$$ 0 0
$$846$$ 0 0
$$847$$ − 2.00000i − 0.0687208i
$$848$$ 48.0000i 1.64833i
$$849$$ −26.0000 −0.892318
$$850$$ 0 0
$$851$$ 0 0
$$852$$ − 24.0000i − 0.822226i
$$853$$ 26.0000i 0.890223i 0.895475 + 0.445112i $$0.146836\pi$$
−0.895475 + 0.445112i $$0.853164\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ − 18.0000i − 0.614868i −0.951569 0.307434i $$-0.900530\pi$$
0.951569 0.307434i $$-0.0994704\pi$$
$$858$$ 0 0
$$859$$ 49.0000 1.67186 0.835929 0.548837i $$-0.184929\pi$$
0.835929 + 0.548837i $$0.184929\pi$$
$$860$$ 0 0
$$861$$ −12.0000 −0.408959
$$862$$ 0 0
$$863$$ 18.0000i 0.612727i 0.951915 + 0.306364i $$0.0991123\pi$$
−0.951915 + 0.306364i $$0.900888\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ 0 0
$$867$$ − 16.0000i − 0.543388i
$$868$$ − 8.00000i − 0.271538i
$$869$$ −24.0000 −0.814144
$$870$$ 0 0
$$871$$ 16.0000 0.542139
$$872$$ 0 0
$$873$$ 8.00000i 0.270759i
$$874$$ 0 0
$$875$$ 0 0
$$876$$ −28.0000 −0.946032
$$877$$ 22.0000i 0.742887i 0.928456 + 0.371444i $$0.121137\pi$$
−0.928456 + 0.371444i $$0.878863\pi$$
$$878$$ 0 0
$$879$$ −24.0000 −0.809500
$$880$$ 0 0
$$881$$ −27.0000 −0.909653 −0.454827 0.890580i $$-0.650299\pi$$
−0.454827 + 0.890580i $$0.650299\pi$$
$$882$$ 0 0
$$883$$ 47.0000i 1.58168i 0.612026 + 0.790838i $$0.290355\pi$$
−0.612026 + 0.790838i $$0.709645\pi$$
$$884$$ 24.0000 0.807207
$$885$$ 0 0
$$886$$ 0 0
$$887$$ − 18.0000i − 0.604381i −0.953248 0.302190i $$-0.902282\pi$$
0.953248 0.302190i $$-0.0977178\pi$$
$$888$$ 0 0
$$889$$ 2.00000 0.0670778
$$890$$ 0 0
$$891$$ −33.0000 −1.10554
$$892$$ − 20.0000i − 0.669650i
$$893$$ − 3.00000i − 0.100391i
$$894$$ 0 0
$$895$$ 0 0
$$896$$ 0 0
$$897$$ 0 0
$$898$$ 0 0
$$899$$ 24.0000 0.800445
$$900$$ 0 0
$$901$$ −36.0000 −1.19933
$$902$$ 0 0
$$903$$ − 2.00000i − 0.0665558i
$$904$$ 0 0
$$905$$ 0 0
$$906$$ 0 0
$$907$$ − 8.00000i − 0.265636i −0.991140 0.132818i $$-0.957597\pi$$
0.991140 0.132818i $$-0.0424025\pi$$
$$908$$ − 24.0000i − 0.796468i
$$909$$ −6.00000 −0.199007
$$910$$ 0 0
$$911$$ −6.00000 −0.198789 −0.0993944 0.995048i $$-0.531691\pi$$
−0.0993944 + 0.995048i $$0.531691\pi$$
$$912$$ 8.00000i 0.264906i
$$913$$ 36.0000i 1.19143i
$$914$$ 0 0
$$915$$ 0 0
$$916$$ −10.0000 −0.330409
$$917$$ − 15.0000i − 0.495344i
$$918$$ 0 0
$$919$$ −20.0000 −0.659739 −0.329870 0.944027i $$-0.607005\pi$$
−0.329870 + 0.944027i $$0.607005\pi$$
$$920$$ 0 0
$$921$$ −40.0000 −1.31804
$$922$$ 0 0
$$923$$ − 24.0000i − 0.789970i
$$924$$ 12.0000 0.394771
$$925$$ 0 0
$$926$$ 0 0
$$927$$ − 14.0000i − 0.459820i
$$928$$ 0 0
$$929$$ 18.0000 0.590561 0.295280 0.955411i $$-0.404587\pi$$
0.295280 + 0.955411i $$0.404587\pi$$
$$930$$ 0 0
$$931$$ −6.00000 −0.196642
$$932$$ − 42.0000i − 1.37576i
$$933$$ 6.00000i 0.196431i
$$934$$ 0 0
$$935$$ 0 0
$$936$$ 0 0
$$937$$ 7.00000i 0.228680i 0.993442 + 0.114340i $$0.0364753\pi$$
−0.993442 + 0.114340i $$0.963525\pi$$
$$938$$ 0 0
$$939$$ −20.0000 −0.652675
$$940$$ 0 0
$$941$$ −18.0000 −0.586783 −0.293392 0.955992i $$-0.594784\pi$$
−0.293392 + 0.955992i $$0.594784\pi$$
$$942$$ 0 0
$$943$$ 0 0
$$944$$ 24.0000 0.781133
$$945$$ 0 0
$$946$$ 0 0
$$947$$ 36.0000i 1.16984i 0.811090 + 0.584921i $$0.198875\pi$$
−0.811090 + 0.584921i $$0.801125\pi$$
$$948$$ 32.0000i 1.03931i
$$949$$ −28.0000 −0.908918
$$950$$ 0 0
$$951$$ −12.0000 −0.389127
$$952$$ 0 0
$$953$$ − 48.0000i − 1.55487i −0.628962 0.777436i $$-0.716520\pi$$
0.628962 0.777436i $$-0.283480\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ −30.0000 −0.970269
$$957$$ 36.0000i 1.16371i
$$958$$ 0 0
$$959$$ −3.00000 −0.0968751
$$960$$ 0 0
$$961$$ −15.0000 −0.483871
$$962$$ 0 0
$$963$$ − 18.0000i − 0.580042i
$$964$$ −20.0000 −0.644157
$$965$$ 0 0
$$966$$ 0 0
$$967$$ 40.0000i 1.28631i 0.765735 + 0.643157i $$0.222376\pi$$
−0.765735 + 0.643157i $$0.777624\pi$$
$$968$$ 0 0
$$969$$ −6.00000 −0.192748
$$970$$ 0 0
$$971$$ 60.0000 1.92549 0.962746 0.270408i $$-0.0871586\pi$$
0.962746 + 0.270408i $$0.0871586\pi$$
$$972$$ 20.0000i 0.641500i
$$973$$ 13.0000i 0.416761i
$$974$$ 0 0
$$975$$ 0 0
$$976$$ −4.00000 −0.128037
$$977$$ − 24.0000i − 0.767828i −0.923369 0.383914i $$-0.874576\pi$$
0.923369 0.383914i $$-0.125424\pi$$
$$978$$ 0 0
$$979$$ −36.0000 −1.15056
$$980$$ 0 0
$$981$$ −16.0000 −0.510841
$$982$$ 0 0
$$983$$ − 36.0000i − 1.14822i −0.818778 0.574111i $$-0.805348\pi$$
0.818778 0.574111i $$-0.194652\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ 0 0
$$987$$ 6.00000i 0.190982i
$$988$$ 8.00000i 0.254514i
$$989$$ 0 0
$$990$$ 0 0
$$991$$ −34.0000 −1.08005 −0.540023 0.841650i $$-0.681584\pi$$
−0.540023 + 0.841650i $$0.681584\pi$$
$$992$$ 0 0
$$993$$ 56.0000i 1.77711i
$$994$$ 0 0
$$995$$ 0 0
$$996$$ 48.0000 1.52094
$$997$$ − 17.0000i − 0.538395i −0.963085 0.269198i $$-0.913241\pi$$
0.963085 0.269198i $$-0.0867585\pi$$
$$998$$ 0 0
$$999$$ −8.00000 −0.253109
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 475.2.b.a.324.1 2
5.2 odd 4 19.2.a.a.1.1 1
5.3 odd 4 475.2.a.b.1.1 1
5.4 even 2 inner 475.2.b.a.324.2 2
15.2 even 4 171.2.a.b.1.1 1
15.8 even 4 4275.2.a.i.1.1 1
20.3 even 4 7600.2.a.c.1.1 1
20.7 even 4 304.2.a.f.1.1 1
35.2 odd 12 931.2.f.c.704.1 2
35.12 even 12 931.2.f.b.704.1 2
35.17 even 12 931.2.f.b.324.1 2
35.27 even 4 931.2.a.a.1.1 1
35.32 odd 12 931.2.f.c.324.1 2
40.27 even 4 1216.2.a.b.1.1 1
40.37 odd 4 1216.2.a.o.1.1 1
55.32 even 4 2299.2.a.b.1.1 1
60.47 odd 4 2736.2.a.c.1.1 1
65.12 odd 4 3211.2.a.a.1.1 1
85.67 odd 4 5491.2.a.b.1.1 1
95.2 even 36 361.2.e.e.99.1 6
95.7 odd 12 361.2.c.c.68.1 2
95.12 even 12 361.2.c.a.68.1 2
95.17 odd 36 361.2.e.d.99.1 6
95.18 even 4 9025.2.a.d.1.1 1
95.22 even 36 361.2.e.e.28.1 6
95.27 even 12 361.2.c.a.292.1 2
95.32 even 36 361.2.e.e.245.1 6
95.37 even 4 361.2.a.b.1.1 1
95.42 odd 36 361.2.e.d.54.1 6
95.47 odd 36 361.2.e.d.62.1 6
95.52 even 36 361.2.e.e.234.1 6
95.62 odd 36 361.2.e.d.234.1 6
95.67 even 36 361.2.e.e.62.1 6
95.72 even 36 361.2.e.e.54.1 6
95.82 odd 36 361.2.e.d.245.1 6
95.87 odd 12 361.2.c.c.292.1 2
95.92 odd 36 361.2.e.d.28.1 6
105.62 odd 4 8379.2.a.j.1.1 1
285.227 odd 4 3249.2.a.d.1.1 1
380.227 odd 4 5776.2.a.c.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
19.2.a.a.1.1 1 5.2 odd 4
171.2.a.b.1.1 1 15.2 even 4
304.2.a.f.1.1 1 20.7 even 4
361.2.a.b.1.1 1 95.37 even 4
361.2.c.a.68.1 2 95.12 even 12
361.2.c.a.292.1 2 95.27 even 12
361.2.c.c.68.1 2 95.7 odd 12
361.2.c.c.292.1 2 95.87 odd 12
361.2.e.d.28.1 6 95.92 odd 36
361.2.e.d.54.1 6 95.42 odd 36
361.2.e.d.62.1 6 95.47 odd 36
361.2.e.d.99.1 6 95.17 odd 36
361.2.e.d.234.1 6 95.62 odd 36
361.2.e.d.245.1 6 95.82 odd 36
361.2.e.e.28.1 6 95.22 even 36
361.2.e.e.54.1 6 95.72 even 36
361.2.e.e.62.1 6 95.67 even 36
361.2.e.e.99.1 6 95.2 even 36
361.2.e.e.234.1 6 95.52 even 36
361.2.e.e.245.1 6 95.32 even 36
475.2.a.b.1.1 1 5.3 odd 4
475.2.b.a.324.1 2 1.1 even 1 trivial
475.2.b.a.324.2 2 5.4 even 2 inner
931.2.a.a.1.1 1 35.27 even 4
931.2.f.b.324.1 2 35.17 even 12
931.2.f.b.704.1 2 35.12 even 12
931.2.f.c.324.1 2 35.32 odd 12
931.2.f.c.704.1 2 35.2 odd 12
1216.2.a.b.1.1 1 40.27 even 4
1216.2.a.o.1.1 1 40.37 odd 4
2299.2.a.b.1.1 1 55.32 even 4
2736.2.a.c.1.1 1 60.47 odd 4
3211.2.a.a.1.1 1 65.12 odd 4
3249.2.a.d.1.1 1 285.227 odd 4
4275.2.a.i.1.1 1 15.8 even 4
5491.2.a.b.1.1 1 85.67 odd 4
5776.2.a.c.1.1 1 380.227 odd 4
7600.2.a.c.1.1 1 20.3 even 4
8379.2.a.j.1.1 1 105.62 odd 4
9025.2.a.d.1.1 1 95.18 even 4