# Properties

 Label 1734.2.b.b.577.1 Level $1734$ Weight $2$ Character 1734.577 Analytic conductor $13.846$ Analytic rank $1$ Dimension $2$ Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1734,2,Mod(577,1734)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1734, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1734.577");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1734 = 2 \cdot 3 \cdot 17^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1734.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: $$13.8460597105$$ Analytic rank: $$1$$ 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: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 102) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 577.1 Root $$1.00000i$$ of defining polynomial Character $$\chi$$ $$=$$ 1734.577 Dual form 1734.2.b.b.577.2

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-1.00000 q^{2} -1.00000i q^{3} +1.00000 q^{4} +2.00000i q^{5} +1.00000i q^{6} -1.00000 q^{8} -1.00000 q^{9} +O(q^{10})$$ $$q-1.00000 q^{2} -1.00000i q^{3} +1.00000 q^{4} +2.00000i q^{5} +1.00000i q^{6} -1.00000 q^{8} -1.00000 q^{9} -2.00000i q^{10} -4.00000i q^{11} -1.00000i q^{12} -2.00000 q^{13} +2.00000 q^{15} +1.00000 q^{16} +1.00000 q^{18} -4.00000 q^{19} +2.00000i q^{20} +4.00000i q^{22} +1.00000i q^{24} +1.00000 q^{25} +2.00000 q^{26} +1.00000i q^{27} +10.0000i q^{29} -2.00000 q^{30} -8.00000i q^{31} -1.00000 q^{32} -4.00000 q^{33} -1.00000 q^{36} +2.00000i q^{37} +4.00000 q^{38} +2.00000i q^{39} -2.00000i q^{40} +10.0000i q^{41} -12.0000 q^{43} -4.00000i q^{44} -2.00000i q^{45} -1.00000i q^{48} +7.00000 q^{49} -1.00000 q^{50} -2.00000 q^{52} -6.00000 q^{53} -1.00000i q^{54} +8.00000 q^{55} +4.00000i q^{57} -10.0000i q^{58} -12.0000 q^{59} +2.00000 q^{60} -10.0000i q^{61} +8.00000i q^{62} +1.00000 q^{64} -4.00000i q^{65} +4.00000 q^{66} -12.0000 q^{67} +1.00000 q^{72} -10.0000i q^{73} -2.00000i q^{74} -1.00000i q^{75} -4.00000 q^{76} -2.00000i q^{78} -8.00000i q^{79} +2.00000i q^{80} +1.00000 q^{81} -10.0000i q^{82} -4.00000 q^{83} +12.0000 q^{86} +10.0000 q^{87} +4.00000i q^{88} -6.00000 q^{89} +2.00000i q^{90} -8.00000 q^{93} -8.00000i q^{95} +1.00000i q^{96} +14.0000i q^{97} -7.00000 q^{98} +4.00000i q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{2} + 2 q^{4} - 2 q^{8} - 2 q^{9}+O(q^{10})$$ 2 * q - 2 * q^2 + 2 * q^4 - 2 * q^8 - 2 * q^9 $$2 q - 2 q^{2} + 2 q^{4} - 2 q^{8} - 2 q^{9} - 4 q^{13} + 4 q^{15} + 2 q^{16} + 2 q^{18} - 8 q^{19} + 2 q^{25} + 4 q^{26} - 4 q^{30} - 2 q^{32} - 8 q^{33} - 2 q^{36} + 8 q^{38} - 24 q^{43} + 14 q^{49} - 2 q^{50} - 4 q^{52} - 12 q^{53} + 16 q^{55} - 24 q^{59} + 4 q^{60} + 2 q^{64} + 8 q^{66} - 24 q^{67} + 2 q^{72} - 8 q^{76} + 2 q^{81} - 8 q^{83} + 24 q^{86} + 20 q^{87} - 12 q^{89} - 16 q^{93} - 14 q^{98}+O(q^{100})$$ 2 * q - 2 * q^2 + 2 * q^4 - 2 * q^8 - 2 * q^9 - 4 * q^13 + 4 * q^15 + 2 * q^16 + 2 * q^18 - 8 * q^19 + 2 * q^25 + 4 * q^26 - 4 * q^30 - 2 * q^32 - 8 * q^33 - 2 * q^36 + 8 * q^38 - 24 * q^43 + 14 * q^49 - 2 * q^50 - 4 * q^52 - 12 * q^53 + 16 * q^55 - 24 * q^59 + 4 * q^60 + 2 * q^64 + 8 * q^66 - 24 * q^67 + 2 * q^72 - 8 * q^76 + 2 * q^81 - 8 * q^83 + 24 * q^86 + 20 * q^87 - 12 * q^89 - 16 * q^93 - 14 * q^98

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1734\mathbb{Z}\right)^\times$$.

 $$n$$ $$1157$$ $$1159$$ $$\chi(n)$$ $$1$$ $$-1$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −1.00000 −0.707107
$$3$$ − 1.00000i − 0.577350i
$$4$$ 1.00000 0.500000
$$5$$ 2.00000i 0.894427i 0.894427 + 0.447214i $$0.147584\pi$$
−0.894427 + 0.447214i $$0.852416\pi$$
$$6$$ 1.00000i 0.408248i
$$7$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$8$$ −1.00000 −0.353553
$$9$$ −1.00000 −0.333333
$$10$$ − 2.00000i − 0.632456i
$$11$$ − 4.00000i − 1.20605i −0.797724 0.603023i $$-0.793963\pi$$
0.797724 0.603023i $$-0.206037\pi$$
$$12$$ − 1.00000i − 0.288675i
$$13$$ −2.00000 −0.554700 −0.277350 0.960769i $$-0.589456\pi$$
−0.277350 + 0.960769i $$0.589456\pi$$
$$14$$ 0 0
$$15$$ 2.00000 0.516398
$$16$$ 1.00000 0.250000
$$17$$ 0 0
$$18$$ 1.00000 0.235702
$$19$$ −4.00000 −0.917663 −0.458831 0.888523i $$-0.651732\pi$$
−0.458831 + 0.888523i $$0.651732\pi$$
$$20$$ 2.00000i 0.447214i
$$21$$ 0 0
$$22$$ 4.00000i 0.852803i
$$23$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$24$$ 1.00000i 0.204124i
$$25$$ 1.00000 0.200000
$$26$$ 2.00000 0.392232
$$27$$ 1.00000i 0.192450i
$$28$$ 0 0
$$29$$ 10.0000i 1.85695i 0.371391 + 0.928477i $$0.378881\pi$$
−0.371391 + 0.928477i $$0.621119\pi$$
$$30$$ −2.00000 −0.365148
$$31$$ − 8.00000i − 1.43684i −0.695608 0.718421i $$-0.744865\pi$$
0.695608 0.718421i $$-0.255135\pi$$
$$32$$ −1.00000 −0.176777
$$33$$ −4.00000 −0.696311
$$34$$ 0 0
$$35$$ 0 0
$$36$$ −1.00000 −0.166667
$$37$$ 2.00000i 0.328798i 0.986394 + 0.164399i $$0.0525685\pi$$
−0.986394 + 0.164399i $$0.947432\pi$$
$$38$$ 4.00000 0.648886
$$39$$ 2.00000i 0.320256i
$$40$$ − 2.00000i − 0.316228i
$$41$$ 10.0000i 1.56174i 0.624695 + 0.780869i $$0.285223\pi$$
−0.624695 + 0.780869i $$0.714777\pi$$
$$42$$ 0 0
$$43$$ −12.0000 −1.82998 −0.914991 0.403473i $$-0.867803\pi$$
−0.914991 + 0.403473i $$0.867803\pi$$
$$44$$ − 4.00000i − 0.603023i
$$45$$ − 2.00000i − 0.298142i
$$46$$ 0 0
$$47$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$48$$ − 1.00000i − 0.144338i
$$49$$ 7.00000 1.00000
$$50$$ −1.00000 −0.141421
$$51$$ 0 0
$$52$$ −2.00000 −0.277350
$$53$$ −6.00000 −0.824163 −0.412082 0.911147i $$-0.635198\pi$$
−0.412082 + 0.911147i $$0.635198\pi$$
$$54$$ − 1.00000i − 0.136083i
$$55$$ 8.00000 1.07872
$$56$$ 0 0
$$57$$ 4.00000i 0.529813i
$$58$$ − 10.0000i − 1.31306i
$$59$$ −12.0000 −1.56227 −0.781133 0.624364i $$-0.785358\pi$$
−0.781133 + 0.624364i $$0.785358\pi$$
$$60$$ 2.00000 0.258199
$$61$$ − 10.0000i − 1.28037i −0.768221 0.640184i $$-0.778858\pi$$
0.768221 0.640184i $$-0.221142\pi$$
$$62$$ 8.00000i 1.01600i
$$63$$ 0 0
$$64$$ 1.00000 0.125000
$$65$$ − 4.00000i − 0.496139i
$$66$$ 4.00000 0.492366
$$67$$ −12.0000 −1.46603 −0.733017 0.680211i $$-0.761888\pi$$
−0.733017 + 0.680211i $$0.761888\pi$$
$$68$$ 0 0
$$69$$ 0 0
$$70$$ 0 0
$$71$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$72$$ 1.00000 0.117851
$$73$$ − 10.0000i − 1.17041i −0.810885 0.585206i $$-0.801014\pi$$
0.810885 0.585206i $$-0.198986\pi$$
$$74$$ − 2.00000i − 0.232495i
$$75$$ − 1.00000i − 0.115470i
$$76$$ −4.00000 −0.458831
$$77$$ 0 0
$$78$$ − 2.00000i − 0.226455i
$$79$$ − 8.00000i − 0.900070i −0.893011 0.450035i $$-0.851411\pi$$
0.893011 0.450035i $$-0.148589\pi$$
$$80$$ 2.00000i 0.223607i
$$81$$ 1.00000 0.111111
$$82$$ − 10.0000i − 1.10432i
$$83$$ −4.00000 −0.439057 −0.219529 0.975606i $$-0.570452\pi$$
−0.219529 + 0.975606i $$0.570452\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 12.0000 1.29399
$$87$$ 10.0000 1.07211
$$88$$ 4.00000i 0.426401i
$$89$$ −6.00000 −0.635999 −0.317999 0.948091i $$-0.603011\pi$$
−0.317999 + 0.948091i $$0.603011\pi$$
$$90$$ 2.00000i 0.210819i
$$91$$ 0 0
$$92$$ 0 0
$$93$$ −8.00000 −0.829561
$$94$$ 0 0
$$95$$ − 8.00000i − 0.820783i
$$96$$ 1.00000i 0.102062i
$$97$$ 14.0000i 1.42148i 0.703452 + 0.710742i $$0.251641\pi$$
−0.703452 + 0.710742i $$0.748359\pi$$
$$98$$ −7.00000 −0.707107
$$99$$ 4.00000i 0.402015i
$$100$$ 1.00000 0.100000
$$101$$ −10.0000 −0.995037 −0.497519 0.867453i $$-0.665755\pi$$
−0.497519 + 0.867453i $$0.665755\pi$$
$$102$$ 0 0
$$103$$ −8.00000 −0.788263 −0.394132 0.919054i $$-0.628955\pi$$
−0.394132 + 0.919054i $$0.628955\pi$$
$$104$$ 2.00000 0.196116
$$105$$ 0 0
$$106$$ 6.00000 0.582772
$$107$$ 4.00000i 0.386695i 0.981130 + 0.193347i $$0.0619344\pi$$
−0.981130 + 0.193347i $$0.938066\pi$$
$$108$$ 1.00000i 0.0962250i
$$109$$ − 10.0000i − 0.957826i −0.877862 0.478913i $$-0.841031\pi$$
0.877862 0.478913i $$-0.158969\pi$$
$$110$$ −8.00000 −0.762770
$$111$$ 2.00000 0.189832
$$112$$ 0 0
$$113$$ 2.00000i 0.188144i 0.995565 + 0.0940721i $$0.0299884\pi$$
−0.995565 + 0.0940721i $$0.970012\pi$$
$$114$$ − 4.00000i − 0.374634i
$$115$$ 0 0
$$116$$ 10.0000i 0.928477i
$$117$$ 2.00000 0.184900
$$118$$ 12.0000 1.10469
$$119$$ 0 0
$$120$$ −2.00000 −0.182574
$$121$$ −5.00000 −0.454545
$$122$$ 10.0000i 0.905357i
$$123$$ 10.0000 0.901670
$$124$$ − 8.00000i − 0.718421i
$$125$$ 12.0000i 1.07331i
$$126$$ 0 0
$$127$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$128$$ −1.00000 −0.0883883
$$129$$ 12.0000i 1.05654i
$$130$$ 4.00000i 0.350823i
$$131$$ 12.0000i 1.04844i 0.851581 + 0.524222i $$0.175644\pi$$
−0.851581 + 0.524222i $$0.824356\pi$$
$$132$$ −4.00000 −0.348155
$$133$$ 0 0
$$134$$ 12.0000 1.03664
$$135$$ −2.00000 −0.172133
$$136$$ 0 0
$$137$$ 10.0000 0.854358 0.427179 0.904167i $$-0.359507\pi$$
0.427179 + 0.904167i $$0.359507\pi$$
$$138$$ 0 0
$$139$$ 4.00000i 0.339276i 0.985506 + 0.169638i $$0.0542598\pi$$
−0.985506 + 0.169638i $$0.945740\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 0 0
$$143$$ 8.00000i 0.668994i
$$144$$ −1.00000 −0.0833333
$$145$$ −20.0000 −1.66091
$$146$$ 10.0000i 0.827606i
$$147$$ − 7.00000i − 0.577350i
$$148$$ 2.00000i 0.164399i
$$149$$ −10.0000 −0.819232 −0.409616 0.912258i $$-0.634337\pi$$
−0.409616 + 0.912258i $$0.634337\pi$$
$$150$$ 1.00000i 0.0816497i
$$151$$ −24.0000 −1.95309 −0.976546 0.215308i $$-0.930924\pi$$
−0.976546 + 0.215308i $$0.930924\pi$$
$$152$$ 4.00000 0.324443
$$153$$ 0 0
$$154$$ 0 0
$$155$$ 16.0000 1.28515
$$156$$ 2.00000i 0.160128i
$$157$$ −2.00000 −0.159617 −0.0798087 0.996810i $$-0.525431\pi$$
−0.0798087 + 0.996810i $$0.525431\pi$$
$$158$$ 8.00000i 0.636446i
$$159$$ 6.00000i 0.475831i
$$160$$ − 2.00000i − 0.158114i
$$161$$ 0 0
$$162$$ −1.00000 −0.0785674
$$163$$ 4.00000i 0.313304i 0.987654 + 0.156652i $$0.0500701\pi$$
−0.987654 + 0.156652i $$0.949930\pi$$
$$164$$ 10.0000i 0.780869i
$$165$$ − 8.00000i − 0.622799i
$$166$$ 4.00000 0.310460
$$167$$ − 16.0000i − 1.23812i −0.785345 0.619059i $$-0.787514\pi$$
0.785345 0.619059i $$-0.212486\pi$$
$$168$$ 0 0
$$169$$ −9.00000 −0.692308
$$170$$ 0 0
$$171$$ 4.00000 0.305888
$$172$$ −12.0000 −0.914991
$$173$$ − 6.00000i − 0.456172i −0.973641 0.228086i $$-0.926753\pi$$
0.973641 0.228086i $$-0.0732467\pi$$
$$174$$ −10.0000 −0.758098
$$175$$ 0 0
$$176$$ − 4.00000i − 0.301511i
$$177$$ 12.0000i 0.901975i
$$178$$ 6.00000 0.449719
$$179$$ 12.0000 0.896922 0.448461 0.893802i $$-0.351972\pi$$
0.448461 + 0.893802i $$0.351972\pi$$
$$180$$ − 2.00000i − 0.149071i
$$181$$ 14.0000i 1.04061i 0.853980 + 0.520306i $$0.174182\pi$$
−0.853980 + 0.520306i $$0.825818\pi$$
$$182$$ 0 0
$$183$$ −10.0000 −0.739221
$$184$$ 0 0
$$185$$ −4.00000 −0.294086
$$186$$ 8.00000 0.586588
$$187$$ 0 0
$$188$$ 0 0
$$189$$ 0 0
$$190$$ 8.00000i 0.580381i
$$191$$ −16.0000 −1.15772 −0.578860 0.815427i $$-0.696502\pi$$
−0.578860 + 0.815427i $$0.696502\pi$$
$$192$$ − 1.00000i − 0.0721688i
$$193$$ 18.0000i 1.29567i 0.761781 + 0.647834i $$0.224325\pi$$
−0.761781 + 0.647834i $$0.775675\pi$$
$$194$$ − 14.0000i − 1.00514i
$$195$$ −4.00000 −0.286446
$$196$$ 7.00000 0.500000
$$197$$ 14.0000i 0.997459i 0.866758 + 0.498729i $$0.166200\pi$$
−0.866758 + 0.498729i $$0.833800\pi$$
$$198$$ − 4.00000i − 0.284268i
$$199$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$200$$ −1.00000 −0.0707107
$$201$$ 12.0000i 0.846415i
$$202$$ 10.0000 0.703598
$$203$$ 0 0
$$204$$ 0 0
$$205$$ −20.0000 −1.39686
$$206$$ 8.00000 0.557386
$$207$$ 0 0
$$208$$ −2.00000 −0.138675
$$209$$ 16.0000i 1.10674i
$$210$$ 0 0
$$211$$ − 28.0000i − 1.92760i −0.266627 0.963800i $$-0.585909\pi$$
0.266627 0.963800i $$-0.414091\pi$$
$$212$$ −6.00000 −0.412082
$$213$$ 0 0
$$214$$ − 4.00000i − 0.273434i
$$215$$ − 24.0000i − 1.63679i
$$216$$ − 1.00000i − 0.0680414i
$$217$$ 0 0
$$218$$ 10.0000i 0.677285i
$$219$$ −10.0000 −0.675737
$$220$$ 8.00000 0.539360
$$221$$ 0 0
$$222$$ −2.00000 −0.134231
$$223$$ −16.0000 −1.07144 −0.535720 0.844396i $$-0.679960\pi$$
−0.535720 + 0.844396i $$0.679960\pi$$
$$224$$ 0 0
$$225$$ −1.00000 −0.0666667
$$226$$ − 2.00000i − 0.133038i
$$227$$ 4.00000i 0.265489i 0.991150 + 0.132745i $$0.0423790\pi$$
−0.991150 + 0.132745i $$0.957621\pi$$
$$228$$ 4.00000i 0.264906i
$$229$$ 26.0000 1.71813 0.859064 0.511868i $$-0.171046\pi$$
0.859064 + 0.511868i $$0.171046\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ − 10.0000i − 0.656532i
$$233$$ − 26.0000i − 1.70332i −0.524097 0.851658i $$-0.675597\pi$$
0.524097 0.851658i $$-0.324403\pi$$
$$234$$ −2.00000 −0.130744
$$235$$ 0 0
$$236$$ −12.0000 −0.781133
$$237$$ −8.00000 −0.519656
$$238$$ 0 0
$$239$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$240$$ 2.00000 0.129099
$$241$$ − 2.00000i − 0.128831i −0.997923 0.0644157i $$-0.979482\pi$$
0.997923 0.0644157i $$-0.0205183\pi$$
$$242$$ 5.00000 0.321412
$$243$$ − 1.00000i − 0.0641500i
$$244$$ − 10.0000i − 0.640184i
$$245$$ 14.0000i 0.894427i
$$246$$ −10.0000 −0.637577
$$247$$ 8.00000 0.509028
$$248$$ 8.00000i 0.508001i
$$249$$ 4.00000i 0.253490i
$$250$$ − 12.0000i − 0.758947i
$$251$$ 28.0000 1.76734 0.883672 0.468106i $$-0.155064\pi$$
0.883672 + 0.468106i $$0.155064\pi$$
$$252$$ 0 0
$$253$$ 0 0
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 1.00000 0.0625000
$$257$$ −2.00000 −0.124757 −0.0623783 0.998053i $$-0.519869\pi$$
−0.0623783 + 0.998053i $$0.519869\pi$$
$$258$$ − 12.0000i − 0.747087i
$$259$$ 0 0
$$260$$ − 4.00000i − 0.248069i
$$261$$ − 10.0000i − 0.618984i
$$262$$ − 12.0000i − 0.741362i
$$263$$ 8.00000 0.493301 0.246651 0.969104i $$-0.420670\pi$$
0.246651 + 0.969104i $$0.420670\pi$$
$$264$$ 4.00000 0.246183
$$265$$ − 12.0000i − 0.737154i
$$266$$ 0 0
$$267$$ 6.00000i 0.367194i
$$268$$ −12.0000 −0.733017
$$269$$ − 6.00000i − 0.365826i −0.983129 0.182913i $$-0.941447\pi$$
0.983129 0.182913i $$-0.0585527\pi$$
$$270$$ 2.00000 0.121716
$$271$$ 16.0000 0.971931 0.485965 0.873978i $$-0.338468\pi$$
0.485965 + 0.873978i $$0.338468\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ −10.0000 −0.604122
$$275$$ − 4.00000i − 0.241209i
$$276$$ 0 0
$$277$$ − 30.0000i − 1.80253i −0.433273 0.901263i $$-0.642641\pi$$
0.433273 0.901263i $$-0.357359\pi$$
$$278$$ − 4.00000i − 0.239904i
$$279$$ 8.00000i 0.478947i
$$280$$ 0 0
$$281$$ 6.00000 0.357930 0.178965 0.983855i $$-0.442725\pi$$
0.178965 + 0.983855i $$0.442725\pi$$
$$282$$ 0 0
$$283$$ 12.0000i 0.713326i 0.934233 + 0.356663i $$0.116086\pi$$
−0.934233 + 0.356663i $$0.883914\pi$$
$$284$$ 0 0
$$285$$ −8.00000 −0.473879
$$286$$ − 8.00000i − 0.473050i
$$287$$ 0 0
$$288$$ 1.00000 0.0589256
$$289$$ 0 0
$$290$$ 20.0000 1.17444
$$291$$ 14.0000 0.820695
$$292$$ − 10.0000i − 0.585206i
$$293$$ −26.0000 −1.51894 −0.759468 0.650545i $$-0.774541\pi$$
−0.759468 + 0.650545i $$0.774541\pi$$
$$294$$ 7.00000i 0.408248i
$$295$$ − 24.0000i − 1.39733i
$$296$$ − 2.00000i − 0.116248i
$$297$$ 4.00000 0.232104
$$298$$ 10.0000 0.579284
$$299$$ 0 0
$$300$$ − 1.00000i − 0.0577350i
$$301$$ 0 0
$$302$$ 24.0000 1.38104
$$303$$ 10.0000i 0.574485i
$$304$$ −4.00000 −0.229416
$$305$$ 20.0000 1.14520
$$306$$ 0 0
$$307$$ −12.0000 −0.684876 −0.342438 0.939540i $$-0.611253\pi$$
−0.342438 + 0.939540i $$0.611253\pi$$
$$308$$ 0 0
$$309$$ 8.00000i 0.455104i
$$310$$ −16.0000 −0.908739
$$311$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$312$$ − 2.00000i − 0.113228i
$$313$$ 10.0000i 0.565233i 0.959233 + 0.282617i $$0.0912024\pi$$
−0.959233 + 0.282617i $$0.908798\pi$$
$$314$$ 2.00000 0.112867
$$315$$ 0 0
$$316$$ − 8.00000i − 0.450035i
$$317$$ 6.00000i 0.336994i 0.985702 + 0.168497i $$0.0538913\pi$$
−0.985702 + 0.168497i $$0.946109\pi$$
$$318$$ − 6.00000i − 0.336463i
$$319$$ 40.0000 2.23957
$$320$$ 2.00000i 0.111803i
$$321$$ 4.00000 0.223258
$$322$$ 0 0
$$323$$ 0 0
$$324$$ 1.00000 0.0555556
$$325$$ −2.00000 −0.110940
$$326$$ − 4.00000i − 0.221540i
$$327$$ −10.0000 −0.553001
$$328$$ − 10.0000i − 0.552158i
$$329$$ 0 0
$$330$$ 8.00000i 0.440386i
$$331$$ 20.0000 1.09930 0.549650 0.835395i $$-0.314761\pi$$
0.549650 + 0.835395i $$0.314761\pi$$
$$332$$ −4.00000 −0.219529
$$333$$ − 2.00000i − 0.109599i
$$334$$ 16.0000i 0.875481i
$$335$$ − 24.0000i − 1.31126i
$$336$$ 0 0
$$337$$ 14.0000i 0.762629i 0.924445 + 0.381314i $$0.124528\pi$$
−0.924445 + 0.381314i $$0.875472\pi$$
$$338$$ 9.00000 0.489535
$$339$$ 2.00000 0.108625
$$340$$ 0 0
$$341$$ −32.0000 −1.73290
$$342$$ −4.00000 −0.216295
$$343$$ 0 0
$$344$$ 12.0000 0.646997
$$345$$ 0 0
$$346$$ 6.00000i 0.322562i
$$347$$ 28.0000i 1.50312i 0.659665 + 0.751559i $$0.270698\pi$$
−0.659665 + 0.751559i $$0.729302\pi$$
$$348$$ 10.0000 0.536056
$$349$$ −14.0000 −0.749403 −0.374701 0.927146i $$-0.622255\pi$$
−0.374701 + 0.927146i $$0.622255\pi$$
$$350$$ 0 0
$$351$$ − 2.00000i − 0.106752i
$$352$$ 4.00000i 0.213201i
$$353$$ −30.0000 −1.59674 −0.798369 0.602168i $$-0.794304\pi$$
−0.798369 + 0.602168i $$0.794304\pi$$
$$354$$ − 12.0000i − 0.637793i
$$355$$ 0 0
$$356$$ −6.00000 −0.317999
$$357$$ 0 0
$$358$$ −12.0000 −0.634220
$$359$$ 24.0000 1.26667 0.633336 0.773877i $$-0.281685\pi$$
0.633336 + 0.773877i $$0.281685\pi$$
$$360$$ 2.00000i 0.105409i
$$361$$ −3.00000 −0.157895
$$362$$ − 14.0000i − 0.735824i
$$363$$ 5.00000i 0.262432i
$$364$$ 0 0
$$365$$ 20.0000 1.04685
$$366$$ 10.0000 0.522708
$$367$$ 24.0000i 1.25279i 0.779506 + 0.626395i $$0.215470\pi$$
−0.779506 + 0.626395i $$0.784530\pi$$
$$368$$ 0 0
$$369$$ − 10.0000i − 0.520579i
$$370$$ 4.00000 0.207950
$$371$$ 0 0
$$372$$ −8.00000 −0.414781
$$373$$ 6.00000 0.310668 0.155334 0.987862i $$-0.450355\pi$$
0.155334 + 0.987862i $$0.450355\pi$$
$$374$$ 0 0
$$375$$ 12.0000 0.619677
$$376$$ 0 0
$$377$$ − 20.0000i − 1.03005i
$$378$$ 0 0
$$379$$ 4.00000i 0.205466i 0.994709 + 0.102733i $$0.0327588\pi$$
−0.994709 + 0.102733i $$0.967241\pi$$
$$380$$ − 8.00000i − 0.410391i
$$381$$ 0 0
$$382$$ 16.0000 0.818631
$$383$$ 16.0000 0.817562 0.408781 0.912633i $$-0.365954\pi$$
0.408781 + 0.912633i $$0.365954\pi$$
$$384$$ 1.00000i 0.0510310i
$$385$$ 0 0
$$386$$ − 18.0000i − 0.916176i
$$387$$ 12.0000 0.609994
$$388$$ 14.0000i 0.710742i
$$389$$ 26.0000 1.31825 0.659126 0.752032i $$-0.270926\pi$$
0.659126 + 0.752032i $$0.270926\pi$$
$$390$$ 4.00000 0.202548
$$391$$ 0 0
$$392$$ −7.00000 −0.353553
$$393$$ 12.0000 0.605320
$$394$$ − 14.0000i − 0.705310i
$$395$$ 16.0000 0.805047
$$396$$ 4.00000i 0.201008i
$$397$$ − 26.0000i − 1.30490i −0.757831 0.652451i $$-0.773741\pi$$
0.757831 0.652451i $$-0.226259\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ 1.00000 0.0500000
$$401$$ − 14.0000i − 0.699127i −0.936913 0.349563i $$-0.886330\pi$$
0.936913 0.349563i $$-0.113670\pi$$
$$402$$ − 12.0000i − 0.598506i
$$403$$ 16.0000i 0.797017i
$$404$$ −10.0000 −0.497519
$$405$$ 2.00000i 0.0993808i
$$406$$ 0 0
$$407$$ 8.00000 0.396545
$$408$$ 0 0
$$409$$ 26.0000 1.28562 0.642809 0.766027i $$-0.277769\pi$$
0.642809 + 0.766027i $$0.277769\pi$$
$$410$$ 20.0000 0.987730
$$411$$ − 10.0000i − 0.493264i
$$412$$ −8.00000 −0.394132
$$413$$ 0 0
$$414$$ 0 0
$$415$$ − 8.00000i − 0.392705i
$$416$$ 2.00000 0.0980581
$$417$$ 4.00000 0.195881
$$418$$ − 16.0000i − 0.782586i
$$419$$ 4.00000i 0.195413i 0.995215 + 0.0977064i $$0.0311506\pi$$
−0.995215 + 0.0977064i $$0.968849\pi$$
$$420$$ 0 0
$$421$$ 22.0000 1.07221 0.536107 0.844150i $$-0.319894\pi$$
0.536107 + 0.844150i $$0.319894\pi$$
$$422$$ 28.0000i 1.36302i
$$423$$ 0 0
$$424$$ 6.00000 0.291386
$$425$$ 0 0
$$426$$ 0 0
$$427$$ 0 0
$$428$$ 4.00000i 0.193347i
$$429$$ 8.00000 0.386244
$$430$$ 24.0000i 1.15738i
$$431$$ − 8.00000i − 0.385346i −0.981263 0.192673i $$-0.938284\pi$$
0.981263 0.192673i $$-0.0617157\pi$$
$$432$$ 1.00000i 0.0481125i
$$433$$ 14.0000 0.672797 0.336399 0.941720i $$-0.390791\pi$$
0.336399 + 0.941720i $$0.390791\pi$$
$$434$$ 0 0
$$435$$ 20.0000i 0.958927i
$$436$$ − 10.0000i − 0.478913i
$$437$$ 0 0
$$438$$ 10.0000 0.477818
$$439$$ − 16.0000i − 0.763638i −0.924237 0.381819i $$-0.875298\pi$$
0.924237 0.381819i $$-0.124702\pi$$
$$440$$ −8.00000 −0.381385
$$441$$ −7.00000 −0.333333
$$442$$ 0 0
$$443$$ −4.00000 −0.190046 −0.0950229 0.995475i $$-0.530292\pi$$
−0.0950229 + 0.995475i $$0.530292\pi$$
$$444$$ 2.00000 0.0949158
$$445$$ − 12.0000i − 0.568855i
$$446$$ 16.0000 0.757622
$$447$$ 10.0000i 0.472984i
$$448$$ 0 0
$$449$$ 2.00000i 0.0943858i 0.998886 + 0.0471929i $$0.0150276\pi$$
−0.998886 + 0.0471929i $$0.984972\pi$$
$$450$$ 1.00000 0.0471405
$$451$$ 40.0000 1.88353
$$452$$ 2.00000i 0.0940721i
$$453$$ 24.0000i 1.12762i
$$454$$ − 4.00000i − 0.187729i
$$455$$ 0 0
$$456$$ − 4.00000i − 0.187317i
$$457$$ −10.0000 −0.467780 −0.233890 0.972263i $$-0.575146\pi$$
−0.233890 + 0.972263i $$0.575146\pi$$
$$458$$ −26.0000 −1.21490
$$459$$ 0 0
$$460$$ 0 0
$$461$$ −30.0000 −1.39724 −0.698620 0.715493i $$-0.746202\pi$$
−0.698620 + 0.715493i $$0.746202\pi$$
$$462$$ 0 0
$$463$$ 32.0000 1.48717 0.743583 0.668644i $$-0.233125\pi$$
0.743583 + 0.668644i $$0.233125\pi$$
$$464$$ 10.0000i 0.464238i
$$465$$ − 16.0000i − 0.741982i
$$466$$ 26.0000i 1.20443i
$$467$$ 12.0000 0.555294 0.277647 0.960683i $$-0.410445\pi$$
0.277647 + 0.960683i $$0.410445\pi$$
$$468$$ 2.00000 0.0924500
$$469$$ 0 0
$$470$$ 0 0
$$471$$ 2.00000i 0.0921551i
$$472$$ 12.0000 0.552345
$$473$$ 48.0000i 2.20704i
$$474$$ 8.00000 0.367452
$$475$$ −4.00000 −0.183533
$$476$$ 0 0
$$477$$ 6.00000 0.274721
$$478$$ 0 0
$$479$$ − 24.0000i − 1.09659i −0.836286 0.548294i $$-0.815277\pi$$
0.836286 0.548294i $$-0.184723\pi$$
$$480$$ −2.00000 −0.0912871
$$481$$ − 4.00000i − 0.182384i
$$482$$ 2.00000i 0.0910975i
$$483$$ 0 0
$$484$$ −5.00000 −0.227273
$$485$$ −28.0000 −1.27141
$$486$$ 1.00000i 0.0453609i
$$487$$ − 16.0000i − 0.725029i −0.931978 0.362515i $$-0.881918\pi$$
0.931978 0.362515i $$-0.118082\pi$$
$$488$$ 10.0000i 0.452679i
$$489$$ 4.00000 0.180886
$$490$$ − 14.0000i − 0.632456i
$$491$$ −12.0000 −0.541552 −0.270776 0.962642i $$-0.587280\pi$$
−0.270776 + 0.962642i $$0.587280\pi$$
$$492$$ 10.0000 0.450835
$$493$$ 0 0
$$494$$ −8.00000 −0.359937
$$495$$ −8.00000 −0.359573
$$496$$ − 8.00000i − 0.359211i
$$497$$ 0 0
$$498$$ − 4.00000i − 0.179244i
$$499$$ 4.00000i 0.179065i 0.995984 + 0.0895323i $$0.0285372\pi$$
−0.995984 + 0.0895323i $$0.971463\pi$$
$$500$$ 12.0000i 0.536656i
$$501$$ −16.0000 −0.714827
$$502$$ −28.0000 −1.24970
$$503$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$504$$ 0 0
$$505$$ − 20.0000i − 0.889988i
$$506$$ 0 0
$$507$$ 9.00000i 0.399704i
$$508$$ 0 0
$$509$$ 30.0000 1.32973 0.664863 0.746965i $$-0.268490\pi$$
0.664863 + 0.746965i $$0.268490\pi$$
$$510$$ 0 0
$$511$$ 0 0
$$512$$ −1.00000 −0.0441942
$$513$$ − 4.00000i − 0.176604i
$$514$$ 2.00000 0.0882162
$$515$$ − 16.0000i − 0.705044i
$$516$$ 12.0000i 0.528271i
$$517$$ 0 0
$$518$$ 0 0
$$519$$ −6.00000 −0.263371
$$520$$ 4.00000i 0.175412i
$$521$$ − 22.0000i − 0.963837i −0.876216 0.481919i $$-0.839940\pi$$
0.876216 0.481919i $$-0.160060\pi$$
$$522$$ 10.0000i 0.437688i
$$523$$ −4.00000 −0.174908 −0.0874539 0.996169i $$-0.527873\pi$$
−0.0874539 + 0.996169i $$0.527873\pi$$
$$524$$ 12.0000i 0.524222i
$$525$$ 0 0
$$526$$ −8.00000 −0.348817
$$527$$ 0 0
$$528$$ −4.00000 −0.174078
$$529$$ 23.0000 1.00000
$$530$$ 12.0000i 0.521247i
$$531$$ 12.0000 0.520756
$$532$$ 0 0
$$533$$ − 20.0000i − 0.866296i
$$534$$ − 6.00000i − 0.259645i
$$535$$ −8.00000 −0.345870
$$536$$ 12.0000 0.518321
$$537$$ − 12.0000i − 0.517838i
$$538$$ 6.00000i 0.258678i
$$539$$ − 28.0000i − 1.20605i
$$540$$ −2.00000 −0.0860663
$$541$$ 10.0000i 0.429934i 0.976621 + 0.214967i $$0.0689643\pi$$
−0.976621 + 0.214967i $$0.931036\pi$$
$$542$$ −16.0000 −0.687259
$$543$$ 14.0000 0.600798
$$544$$ 0 0
$$545$$ 20.0000 0.856706
$$546$$ 0 0
$$547$$ 12.0000i 0.513083i 0.966533 + 0.256541i $$0.0825830\pi$$
−0.966533 + 0.256541i $$0.917417\pi$$
$$548$$ 10.0000 0.427179
$$549$$ 10.0000i 0.426790i
$$550$$ 4.00000i 0.170561i
$$551$$ − 40.0000i − 1.70406i
$$552$$ 0 0
$$553$$ 0 0
$$554$$ 30.0000i 1.27458i
$$555$$ 4.00000i 0.169791i
$$556$$ 4.00000i 0.169638i
$$557$$ −34.0000 −1.44063 −0.720313 0.693649i $$-0.756002\pi$$
−0.720313 + 0.693649i $$0.756002\pi$$
$$558$$ − 8.00000i − 0.338667i
$$559$$ 24.0000 1.01509
$$560$$ 0 0
$$561$$ 0 0
$$562$$ −6.00000 −0.253095
$$563$$ −36.0000 −1.51722 −0.758610 0.651546i $$-0.774121\pi$$
−0.758610 + 0.651546i $$0.774121\pi$$
$$564$$ 0 0
$$565$$ −4.00000 −0.168281
$$566$$ − 12.0000i − 0.504398i
$$567$$ 0 0
$$568$$ 0 0
$$569$$ 6.00000 0.251533 0.125767 0.992060i $$-0.459861\pi$$
0.125767 + 0.992060i $$0.459861\pi$$
$$570$$ 8.00000 0.335083
$$571$$ 28.0000i 1.17176i 0.810397 + 0.585882i $$0.199252\pi$$
−0.810397 + 0.585882i $$0.800748\pi$$
$$572$$ 8.00000i 0.334497i
$$573$$ 16.0000i 0.668410i
$$574$$ 0 0
$$575$$ 0 0
$$576$$ −1.00000 −0.0416667
$$577$$ 2.00000 0.0832611 0.0416305 0.999133i $$-0.486745\pi$$
0.0416305 + 0.999133i $$0.486745\pi$$
$$578$$ 0 0
$$579$$ 18.0000 0.748054
$$580$$ −20.0000 −0.830455
$$581$$ 0 0
$$582$$ −14.0000 −0.580319
$$583$$ 24.0000i 0.993978i
$$584$$ 10.0000i 0.413803i
$$585$$ 4.00000i 0.165380i
$$586$$ 26.0000 1.07405
$$587$$ 20.0000 0.825488 0.412744 0.910847i $$-0.364570\pi$$
0.412744 + 0.910847i $$0.364570\pi$$
$$588$$ − 7.00000i − 0.288675i
$$589$$ 32.0000i 1.31854i
$$590$$ 24.0000i 0.988064i
$$591$$ 14.0000 0.575883
$$592$$ 2.00000i 0.0821995i
$$593$$ 14.0000 0.574911 0.287456 0.957794i $$-0.407191\pi$$
0.287456 + 0.957794i $$0.407191\pi$$
$$594$$ −4.00000 −0.164122
$$595$$ 0 0
$$596$$ −10.0000 −0.409616
$$597$$ 0 0
$$598$$ 0 0
$$599$$ −8.00000 −0.326871 −0.163436 0.986554i $$-0.552258\pi$$
−0.163436 + 0.986554i $$0.552258\pi$$
$$600$$ 1.00000i 0.0408248i
$$601$$ − 6.00000i − 0.244745i −0.992484 0.122373i $$-0.960950\pi$$
0.992484 0.122373i $$-0.0390503\pi$$
$$602$$ 0 0
$$603$$ 12.0000 0.488678
$$604$$ −24.0000 −0.976546
$$605$$ − 10.0000i − 0.406558i
$$606$$ − 10.0000i − 0.406222i
$$607$$ 24.0000i 0.974130i 0.873366 + 0.487065i $$0.161933\pi$$
−0.873366 + 0.487065i $$0.838067\pi$$
$$608$$ 4.00000 0.162221
$$609$$ 0 0
$$610$$ −20.0000 −0.809776
$$611$$ 0 0
$$612$$ 0 0
$$613$$ 38.0000 1.53481 0.767403 0.641165i $$-0.221549\pi$$
0.767403 + 0.641165i $$0.221549\pi$$
$$614$$ 12.0000 0.484281
$$615$$ 20.0000i 0.806478i
$$616$$ 0 0
$$617$$ 6.00000i 0.241551i 0.992680 + 0.120775i $$0.0385381\pi$$
−0.992680 + 0.120775i $$0.961462\pi$$
$$618$$ − 8.00000i − 0.321807i
$$619$$ − 20.0000i − 0.803868i −0.915669 0.401934i $$-0.868338\pi$$
0.915669 0.401934i $$-0.131662\pi$$
$$620$$ 16.0000 0.642575
$$621$$ 0 0
$$622$$ 0 0
$$623$$ 0 0
$$624$$ 2.00000i 0.0800641i
$$625$$ −19.0000 −0.760000
$$626$$ − 10.0000i − 0.399680i
$$627$$ 16.0000 0.638978
$$628$$ −2.00000 −0.0798087
$$629$$ 0 0
$$630$$ 0 0
$$631$$ 24.0000 0.955425 0.477712 0.878516i $$-0.341466\pi$$
0.477712 + 0.878516i $$0.341466\pi$$
$$632$$ 8.00000i 0.318223i
$$633$$ −28.0000 −1.11290
$$634$$ − 6.00000i − 0.238290i
$$635$$ 0 0
$$636$$ 6.00000i 0.237915i
$$637$$ −14.0000 −0.554700
$$638$$ −40.0000 −1.58362
$$639$$ 0 0
$$640$$ − 2.00000i − 0.0790569i
$$641$$ − 2.00000i − 0.0789953i −0.999220 0.0394976i $$-0.987424\pi$$
0.999220 0.0394976i $$-0.0125758\pi$$
$$642$$ −4.00000 −0.157867
$$643$$ 28.0000i 1.10421i 0.833774 + 0.552106i $$0.186176\pi$$
−0.833774 + 0.552106i $$0.813824\pi$$
$$644$$ 0 0
$$645$$ −24.0000 −0.944999
$$646$$ 0 0
$$647$$ −40.0000 −1.57256 −0.786281 0.617869i $$-0.787996\pi$$
−0.786281 + 0.617869i $$0.787996\pi$$
$$648$$ −1.00000 −0.0392837
$$649$$ 48.0000i 1.88416i
$$650$$ 2.00000 0.0784465
$$651$$ 0 0
$$652$$ 4.00000i 0.156652i
$$653$$ 6.00000i 0.234798i 0.993085 + 0.117399i $$0.0374557\pi$$
−0.993085 + 0.117399i $$0.962544\pi$$
$$654$$ 10.0000 0.391031
$$655$$ −24.0000 −0.937758
$$656$$ 10.0000i 0.390434i
$$657$$ 10.0000i 0.390137i
$$658$$ 0 0
$$659$$ 36.0000 1.40236 0.701180 0.712984i $$-0.252657\pi$$
0.701180 + 0.712984i $$0.252657\pi$$
$$660$$ − 8.00000i − 0.311400i
$$661$$ −6.00000 −0.233373 −0.116686 0.993169i $$-0.537227\pi$$
−0.116686 + 0.993169i $$0.537227\pi$$
$$662$$ −20.0000 −0.777322
$$663$$ 0 0
$$664$$ 4.00000 0.155230
$$665$$ 0 0
$$666$$ 2.00000i 0.0774984i
$$667$$ 0 0
$$668$$ − 16.0000i − 0.619059i
$$669$$ 16.0000i 0.618596i
$$670$$ 24.0000i 0.927201i
$$671$$ −40.0000 −1.54418
$$672$$ 0 0
$$673$$ − 46.0000i − 1.77317i −0.462566 0.886585i $$-0.653071\pi$$
0.462566 0.886585i $$-0.346929\pi$$
$$674$$ − 14.0000i − 0.539260i
$$675$$ 1.00000i 0.0384900i
$$676$$ −9.00000 −0.346154
$$677$$ − 46.0000i − 1.76792i −0.467559 0.883962i $$-0.654866\pi$$
0.467559 0.883962i $$-0.345134\pi$$
$$678$$ −2.00000 −0.0768095
$$679$$ 0 0
$$680$$ 0 0
$$681$$ 4.00000 0.153280
$$682$$ 32.0000 1.22534
$$683$$ 20.0000i 0.765279i 0.923898 + 0.382639i $$0.124985\pi$$
−0.923898 + 0.382639i $$0.875015\pi$$
$$684$$ 4.00000 0.152944
$$685$$ 20.0000i 0.764161i
$$686$$ 0 0
$$687$$ − 26.0000i − 0.991962i
$$688$$ −12.0000 −0.457496
$$689$$ 12.0000 0.457164
$$690$$ 0 0
$$691$$ − 28.0000i − 1.06517i −0.846376 0.532585i $$-0.821221\pi$$
0.846376 0.532585i $$-0.178779\pi$$
$$692$$ − 6.00000i − 0.228086i
$$693$$ 0 0
$$694$$ − 28.0000i − 1.06287i
$$695$$ −8.00000 −0.303457
$$696$$ −10.0000 −0.379049
$$697$$ 0 0
$$698$$ 14.0000 0.529908
$$699$$ −26.0000 −0.983410
$$700$$ 0 0
$$701$$ 46.0000 1.73740 0.868698 0.495342i $$-0.164957\pi$$
0.868698 + 0.495342i $$0.164957\pi$$
$$702$$ 2.00000i 0.0754851i
$$703$$ − 8.00000i − 0.301726i
$$704$$ − 4.00000i − 0.150756i
$$705$$ 0 0
$$706$$ 30.0000 1.12906
$$707$$ 0 0
$$708$$ 12.0000i 0.450988i
$$709$$ − 46.0000i − 1.72757i −0.503864 0.863783i $$-0.668089\pi$$
0.503864 0.863783i $$-0.331911\pi$$
$$710$$ 0 0
$$711$$ 8.00000i 0.300023i
$$712$$ 6.00000 0.224860
$$713$$ 0 0
$$714$$ 0 0
$$715$$ −16.0000 −0.598366
$$716$$ 12.0000 0.448461
$$717$$ 0 0
$$718$$ −24.0000 −0.895672
$$719$$ − 40.0000i − 1.49175i −0.666087 0.745874i $$-0.732032\pi$$
0.666087 0.745874i $$-0.267968\pi$$
$$720$$ − 2.00000i − 0.0745356i
$$721$$ 0 0
$$722$$ 3.00000 0.111648
$$723$$ −2.00000 −0.0743808
$$724$$ 14.0000i 0.520306i
$$725$$ 10.0000i 0.371391i
$$726$$ − 5.00000i − 0.185567i
$$727$$ −8.00000 −0.296704 −0.148352 0.988935i $$-0.547397\pi$$
−0.148352 + 0.988935i $$0.547397\pi$$
$$728$$ 0 0
$$729$$ −1.00000 −0.0370370
$$730$$ −20.0000 −0.740233
$$731$$ 0 0
$$732$$ −10.0000 −0.369611
$$733$$ −46.0000 −1.69905 −0.849524 0.527549i $$-0.823111\pi$$
−0.849524 + 0.527549i $$0.823111\pi$$
$$734$$ − 24.0000i − 0.885856i
$$735$$ 14.0000 0.516398
$$736$$ 0 0
$$737$$ 48.0000i 1.76810i
$$738$$ 10.0000i 0.368105i
$$739$$ −52.0000 −1.91285 −0.956425 0.291977i $$-0.905687\pi$$
−0.956425 + 0.291977i $$0.905687\pi$$
$$740$$ −4.00000 −0.147043
$$741$$ − 8.00000i − 0.293887i
$$742$$ 0 0
$$743$$ 16.0000i 0.586983i 0.955962 + 0.293492i $$0.0948173\pi$$
−0.955962 + 0.293492i $$0.905183\pi$$
$$744$$ 8.00000 0.293294
$$745$$ − 20.0000i − 0.732743i
$$746$$ −6.00000 −0.219676
$$747$$ 4.00000 0.146352
$$748$$ 0 0
$$749$$ 0 0
$$750$$ −12.0000 −0.438178
$$751$$ − 8.00000i − 0.291924i −0.989290 0.145962i $$-0.953372\pi$$
0.989290 0.145962i $$-0.0466277\pi$$
$$752$$ 0 0
$$753$$ − 28.0000i − 1.02038i
$$754$$ 20.0000i 0.728357i
$$755$$ − 48.0000i − 1.74690i
$$756$$ 0 0
$$757$$ 10.0000 0.363456 0.181728 0.983349i $$-0.441831\pi$$
0.181728 + 0.983349i $$0.441831\pi$$
$$758$$ − 4.00000i − 0.145287i
$$759$$ 0 0
$$760$$ 8.00000i 0.290191i
$$761$$ −6.00000 −0.217500 −0.108750 0.994069i $$-0.534685\pi$$
−0.108750 + 0.994069i $$0.534685\pi$$
$$762$$ 0 0
$$763$$ 0 0
$$764$$ −16.0000 −0.578860
$$765$$ 0 0
$$766$$ −16.0000 −0.578103
$$767$$ 24.0000 0.866590
$$768$$ − 1.00000i − 0.0360844i
$$769$$ −30.0000 −1.08183 −0.540914 0.841078i $$-0.681921\pi$$
−0.540914 + 0.841078i $$0.681921\pi$$
$$770$$ 0 0
$$771$$ 2.00000i 0.0720282i
$$772$$ 18.0000i 0.647834i
$$773$$ −38.0000 −1.36677 −0.683383 0.730061i $$-0.739492\pi$$
−0.683383 + 0.730061i $$0.739492\pi$$
$$774$$ −12.0000 −0.431331
$$775$$ − 8.00000i − 0.287368i
$$776$$ − 14.0000i − 0.502571i
$$777$$ 0 0
$$778$$ −26.0000 −0.932145
$$779$$ − 40.0000i − 1.43315i
$$780$$ −4.00000 −0.143223
$$781$$ 0 0
$$782$$ 0 0
$$783$$ −10.0000 −0.357371
$$784$$ 7.00000 0.250000
$$785$$ − 4.00000i − 0.142766i
$$786$$ −12.0000 −0.428026
$$787$$ − 4.00000i − 0.142585i −0.997455 0.0712923i $$-0.977288\pi$$
0.997455 0.0712923i $$-0.0227123\pi$$
$$788$$ 14.0000i 0.498729i
$$789$$ − 8.00000i − 0.284808i
$$790$$ −16.0000 −0.569254
$$791$$ 0 0
$$792$$ − 4.00000i − 0.142134i
$$793$$ 20.0000i 0.710221i
$$794$$ 26.0000i 0.922705i
$$795$$ −12.0000 −0.425596
$$796$$ 0 0
$$797$$ −14.0000 −0.495905 −0.247953 0.968772i $$-0.579758\pi$$
−0.247953 + 0.968772i $$0.579758\pi$$
$$798$$ 0 0
$$799$$ 0 0
$$800$$ −1.00000 −0.0353553
$$801$$ 6.00000 0.212000
$$802$$ 14.0000i 0.494357i
$$803$$ −40.0000 −1.41157
$$804$$ 12.0000i 0.423207i
$$805$$ 0 0
$$806$$ − 16.0000i − 0.563576i
$$807$$ −6.00000 −0.211210
$$808$$ 10.0000 0.351799
$$809$$ − 22.0000i − 0.773479i −0.922189 0.386739i $$-0.873601\pi$$
0.922189 0.386739i $$-0.126399\pi$$
$$810$$ − 2.00000i − 0.0702728i
$$811$$ 20.0000i 0.702295i 0.936320 + 0.351147i $$0.114208\pi$$
−0.936320 + 0.351147i $$0.885792\pi$$
$$812$$ 0 0
$$813$$ − 16.0000i − 0.561144i
$$814$$ −8.00000 −0.280400
$$815$$ −8.00000 −0.280228
$$816$$ 0 0
$$817$$ 48.0000 1.67931
$$818$$ −26.0000 −0.909069
$$819$$ 0 0
$$820$$ −20.0000 −0.698430
$$821$$ 34.0000i 1.18661i 0.804978 + 0.593304i $$0.202177\pi$$
−0.804978 + 0.593304i $$0.797823\pi$$
$$822$$ 10.0000i 0.348790i
$$823$$ 32.0000i 1.11545i 0.830026 + 0.557725i $$0.188326\pi$$
−0.830026 + 0.557725i $$0.811674\pi$$
$$824$$ 8.00000 0.278693
$$825$$ −4.00000 −0.139262
$$826$$ 0 0
$$827$$ − 20.0000i − 0.695468i −0.937593 0.347734i $$-0.886951\pi$$
0.937593 0.347734i $$-0.113049\pi$$
$$828$$ 0 0
$$829$$ −34.0000 −1.18087 −0.590434 0.807086i $$-0.701044\pi$$
−0.590434 + 0.807086i $$0.701044\pi$$
$$830$$ 8.00000i 0.277684i
$$831$$ −30.0000 −1.04069
$$832$$ −2.00000 −0.0693375
$$833$$ 0 0
$$834$$ −4.00000 −0.138509
$$835$$ 32.0000 1.10741
$$836$$ 16.0000i 0.553372i
$$837$$ 8.00000 0.276520
$$838$$ − 4.00000i − 0.138178i
$$839$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$840$$ 0 0
$$841$$ −71.0000 −2.44828
$$842$$ −22.0000 −0.758170
$$843$$ − 6.00000i − 0.206651i
$$844$$ − 28.0000i − 0.963800i
$$845$$ − 18.0000i − 0.619219i
$$846$$ 0 0
$$847$$ 0 0
$$848$$ −6.00000 −0.206041
$$849$$ 12.0000 0.411839
$$850$$ 0 0
$$851$$ 0 0
$$852$$ 0 0
$$853$$ 2.00000i 0.0684787i 0.999414 + 0.0342393i $$0.0109009\pi$$
−0.999414 + 0.0342393i $$0.989099\pi$$
$$854$$ 0 0
$$855$$ 8.00000i 0.273594i
$$856$$ − 4.00000i − 0.136717i
$$857$$ − 6.00000i − 0.204956i −0.994735 0.102478i $$-0.967323\pi$$
0.994735 0.102478i $$-0.0326771\pi$$
$$858$$ −8.00000 −0.273115
$$859$$ 36.0000 1.22830 0.614152 0.789188i $$-0.289498\pi$$
0.614152 + 0.789188i $$0.289498\pi$$
$$860$$ − 24.0000i − 0.818393i
$$861$$ 0 0
$$862$$ 8.00000i 0.272481i
$$863$$ −32.0000 −1.08929 −0.544646 0.838666i $$-0.683336\pi$$
−0.544646 + 0.838666i $$0.683336\pi$$
$$864$$ − 1.00000i − 0.0340207i
$$865$$ 12.0000 0.408012
$$866$$ −14.0000 −0.475739
$$867$$ 0 0
$$868$$ 0 0
$$869$$ −32.0000 −1.08553
$$870$$ − 20.0000i − 0.678064i
$$871$$ 24.0000 0.813209
$$872$$ 10.0000i 0.338643i
$$873$$ − 14.0000i − 0.473828i
$$874$$ 0 0
$$875$$ 0 0
$$876$$ −10.0000 −0.337869
$$877$$ 22.0000i 0.742887i 0.928456 + 0.371444i $$0.121137\pi$$
−0.928456 + 0.371444i $$0.878863\pi$$
$$878$$ 16.0000i 0.539974i
$$879$$ 26.0000i 0.876958i
$$880$$ 8.00000 0.269680
$$881$$ − 2.00000i − 0.0673817i −0.999432 0.0336909i $$-0.989274\pi$$
0.999432 0.0336909i $$-0.0107262\pi$$
$$882$$ 7.00000 0.235702
$$883$$ 36.0000 1.21150 0.605748 0.795656i $$-0.292874\pi$$
0.605748 + 0.795656i $$0.292874\pi$$
$$884$$ 0 0
$$885$$ −24.0000 −0.806751
$$886$$ 4.00000 0.134383
$$887$$ 48.0000i 1.61168i 0.592132 + 0.805841i $$0.298286\pi$$
−0.592132 + 0.805841i $$0.701714\pi$$
$$888$$ −2.00000 −0.0671156
$$889$$ 0 0
$$890$$ 12.0000i 0.402241i
$$891$$ − 4.00000i − 0.134005i
$$892$$ −16.0000 −0.535720
$$893$$ 0 0
$$894$$ − 10.0000i − 0.334450i
$$895$$ 24.0000i 0.802232i
$$896$$ 0 0
$$897$$ 0 0
$$898$$ − 2.00000i − 0.0667409i
$$899$$ 80.0000 2.66815
$$900$$ −1.00000 −0.0333333
$$901$$ 0 0
$$902$$ −40.0000 −1.33185
$$903$$ 0 0
$$904$$ − 2.00000i − 0.0665190i
$$905$$ −28.0000 −0.930751
$$906$$ − 24.0000i − 0.797347i
$$907$$ 28.0000i 0.929725i 0.885383 + 0.464862i $$0.153896\pi$$
−0.885383 + 0.464862i $$0.846104\pi$$
$$908$$ 4.00000i 0.132745i
$$909$$ 10.0000 0.331679
$$910$$ 0 0
$$911$$ 8.00000i 0.265052i 0.991180 + 0.132526i $$0.0423088\pi$$
−0.991180 + 0.132526i $$0.957691\pi$$
$$912$$ 4.00000i 0.132453i
$$913$$ 16.0000i 0.529523i
$$914$$ 10.0000 0.330771
$$915$$ − 20.0000i − 0.661180i
$$916$$ 26.0000 0.859064
$$917$$ 0 0
$$918$$ 0 0
$$919$$ 24.0000 0.791687 0.395843 0.918318i $$-0.370452\pi$$
0.395843 + 0.918318i $$0.370452\pi$$
$$920$$ 0 0
$$921$$ 12.0000i 0.395413i
$$922$$ 30.0000 0.987997
$$923$$ 0 0
$$924$$ 0 0
$$925$$ 2.00000i 0.0657596i
$$926$$ −32.0000 −1.05159
$$927$$ 8.00000 0.262754
$$928$$ − 10.0000i − 0.328266i
$$929$$ 2.00000i 0.0656179i 0.999462 + 0.0328089i $$0.0104453\pi$$
−0.999462 + 0.0328089i $$0.989555\pi$$
$$930$$ 16.0000i 0.524661i
$$931$$ −28.0000 −0.917663
$$932$$ − 26.0000i − 0.851658i
$$933$$ 0 0
$$934$$ −12.0000 −0.392652
$$935$$ 0 0
$$936$$ −2.00000 −0.0653720
$$937$$ 22.0000 0.718709 0.359354 0.933201i $$-0.382997\pi$$
0.359354 + 0.933201i $$0.382997\pi$$
$$938$$ 0 0
$$939$$ 10.0000 0.326338
$$940$$ 0 0
$$941$$ − 10.0000i − 0.325991i −0.986627 0.162995i $$-0.947884\pi$$
0.986627 0.162995i $$-0.0521156\pi$$
$$942$$ − 2.00000i − 0.0651635i
$$943$$ 0 0
$$944$$ −12.0000 −0.390567
$$945$$ 0 0
$$946$$ − 48.0000i − 1.56061i
$$947$$ 28.0000i 0.909878i 0.890523 + 0.454939i $$0.150339\pi$$
−0.890523 + 0.454939i $$0.849661\pi$$
$$948$$ −8.00000 −0.259828
$$949$$ 20.0000i 0.649227i
$$950$$ 4.00000 0.129777
$$951$$ 6.00000 0.194563
$$952$$ 0 0
$$953$$ −6.00000 −0.194359 −0.0971795 0.995267i $$-0.530982\pi$$
−0.0971795 + 0.995267i $$0.530982\pi$$
$$954$$ −6.00000 −0.194257
$$955$$ − 32.0000i − 1.03550i
$$956$$ 0 0
$$957$$ − 40.0000i − 1.29302i
$$958$$ 24.0000i 0.775405i
$$959$$ 0 0
$$960$$ 2.00000 0.0645497
$$961$$ −33.0000 −1.06452
$$962$$ 4.00000i 0.128965i
$$963$$ − 4.00000i − 0.128898i
$$964$$ − 2.00000i − 0.0644157i
$$965$$ −36.0000 −1.15888
$$966$$ 0 0
$$967$$ 8.00000 0.257263 0.128631 0.991692i $$-0.458942\pi$$
0.128631 + 0.991692i $$0.458942\pi$$
$$968$$ 5.00000 0.160706
$$969$$ 0 0
$$970$$ 28.0000 0.899026
$$971$$ 4.00000 0.128366 0.0641831 0.997938i $$-0.479556\pi$$
0.0641831 + 0.997938i $$0.479556\pi$$
$$972$$ − 1.00000i − 0.0320750i
$$973$$ 0 0
$$974$$ 16.0000i 0.512673i
$$975$$ 2.00000i 0.0640513i
$$976$$ − 10.0000i − 0.320092i
$$977$$ 46.0000 1.47167 0.735835 0.677161i $$-0.236790\pi$$
0.735835 + 0.677161i $$0.236790\pi$$
$$978$$ −4.00000 −0.127906
$$979$$ 24.0000i 0.767043i
$$980$$ 14.0000i 0.447214i
$$981$$ 10.0000i 0.319275i
$$982$$ 12.0000 0.382935
$$983$$ − 48.0000i − 1.53096i −0.643458 0.765481i $$-0.722501\pi$$
0.643458 0.765481i $$-0.277499\pi$$
$$984$$ −10.0000 −0.318788
$$985$$ −28.0000 −0.892154
$$986$$ 0 0
$$987$$ 0 0
$$988$$ 8.00000 0.254514
$$989$$ 0 0
$$990$$ 8.00000 0.254257
$$991$$ 40.0000i 1.27064i 0.772248 + 0.635321i $$0.219132\pi$$
−0.772248 + 0.635321i $$0.780868\pi$$
$$992$$ 8.00000i 0.254000i
$$993$$ − 20.0000i − 0.634681i
$$994$$ 0 0
$$995$$ 0 0
$$996$$ 4.00000i 0.126745i
$$997$$ − 2.00000i − 0.0633406i −0.999498 0.0316703i $$-0.989917\pi$$
0.999498 0.0316703i $$-0.0100827\pi$$
$$998$$ − 4.00000i − 0.126618i
$$999$$ −2.00000 −0.0632772
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 1734.2.b.b.577.1 2
17.2 even 8 1734.2.f.e.829.1 4
17.4 even 4 1734.2.a.j.1.1 1
17.8 even 8 1734.2.f.e.1483.1 4
17.9 even 8 1734.2.f.e.1483.2 4
17.13 even 4 102.2.a.c.1.1 1
17.15 even 8 1734.2.f.e.829.2 4
17.16 even 2 inner 1734.2.b.b.577.2 2
51.38 odd 4 5202.2.a.c.1.1 1
51.47 odd 4 306.2.a.b.1.1 1
68.47 odd 4 816.2.a.b.1.1 1
85.13 odd 4 2550.2.d.m.2449.1 2
85.47 odd 4 2550.2.d.m.2449.2 2
85.64 even 4 2550.2.a.c.1.1 1
119.13 odd 4 4998.2.a.be.1.1 1
136.13 even 4 3264.2.a.m.1.1 1
136.115 odd 4 3264.2.a.bc.1.1 1
204.47 even 4 2448.2.a.p.1.1 1
255.149 odd 4 7650.2.a.ca.1.1 1
408.149 odd 4 9792.2.a.k.1.1 1
408.251 even 4 9792.2.a.l.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
102.2.a.c.1.1 1 17.13 even 4
306.2.a.b.1.1 1 51.47 odd 4
816.2.a.b.1.1 1 68.47 odd 4
1734.2.a.j.1.1 1 17.4 even 4
1734.2.b.b.577.1 2 1.1 even 1 trivial
1734.2.b.b.577.2 2 17.16 even 2 inner
1734.2.f.e.829.1 4 17.2 even 8
1734.2.f.e.829.2 4 17.15 even 8
1734.2.f.e.1483.1 4 17.8 even 8
1734.2.f.e.1483.2 4 17.9 even 8
2448.2.a.p.1.1 1 204.47 even 4
2550.2.a.c.1.1 1 85.64 even 4
2550.2.d.m.2449.1 2 85.13 odd 4
2550.2.d.m.2449.2 2 85.47 odd 4
3264.2.a.m.1.1 1 136.13 even 4
3264.2.a.bc.1.1 1 136.115 odd 4
4998.2.a.be.1.1 1 119.13 odd 4
5202.2.a.c.1.1 1 51.38 odd 4
7650.2.a.ca.1.1 1 255.149 odd 4
9792.2.a.k.1.1 1 408.149 odd 4
9792.2.a.l.1.1 1 408.251 even 4