# Properties

 Label 1550.2.b.e.249.2 Level $1550$ Weight $2$ Character 1550.249 Analytic conductor $12.377$ 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] = [1550,2,Mod(249,1550)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1550, 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("1550.249");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1550 = 2 \cdot 5^{2} \cdot 31$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1550.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: $$12.3768123133$$ 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: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 310) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 249.2 Root $$-1.00000i$$ of defining polynomial Character $$\chi$$ $$=$$ 1550.249 Dual form 1550.2.b.e.249.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+1.00000i q^{2} -2.00000i q^{3} -1.00000 q^{4} +2.00000 q^{6} -1.00000i q^{8} -1.00000 q^{9} +O(q^{10})$$ $$q+1.00000i q^{2} -2.00000i q^{3} -1.00000 q^{4} +2.00000 q^{6} -1.00000i q^{8} -1.00000 q^{9} +2.00000 q^{11} +2.00000i q^{12} +1.00000 q^{16} +2.00000i q^{17} -1.00000i q^{18} +4.00000 q^{19} +2.00000i q^{22} +4.00000i q^{23} -2.00000 q^{24} -4.00000i q^{27} +4.00000 q^{29} -1.00000 q^{31} +1.00000i q^{32} -4.00000i q^{33} -2.00000 q^{34} +1.00000 q^{36} -8.00000i q^{37} +4.00000i q^{38} +6.00000 q^{41} -2.00000i q^{43} -2.00000 q^{44} -4.00000 q^{46} -2.00000i q^{48} +7.00000 q^{49} +4.00000 q^{51} -8.00000i q^{53} +4.00000 q^{54} -8.00000i q^{57} +4.00000i q^{58} -8.00000 q^{59} -1.00000i q^{62} -1.00000 q^{64} +4.00000 q^{66} +4.00000i q^{67} -2.00000i q^{68} +8.00000 q^{69} +1.00000i q^{72} -6.00000i q^{73} +8.00000 q^{74} -4.00000 q^{76} +4.00000 q^{79} -11.0000 q^{81} +6.00000i q^{82} -6.00000i q^{83} +2.00000 q^{86} -8.00000i q^{87} -2.00000i q^{88} +6.00000 q^{89} -4.00000i q^{92} +2.00000i q^{93} +2.00000 q^{96} -2.00000i q^{97} +7.00000i q^{98} -2.00000 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{4} + 4 q^{6} - 2 q^{9}+O(q^{10})$$ 2 * q - 2 * q^4 + 4 * q^6 - 2 * q^9 $$2 q - 2 q^{4} + 4 q^{6} - 2 q^{9} + 4 q^{11} + 2 q^{16} + 8 q^{19} - 4 q^{24} + 8 q^{29} - 2 q^{31} - 4 q^{34} + 2 q^{36} + 12 q^{41} - 4 q^{44} - 8 q^{46} + 14 q^{49} + 8 q^{51} + 8 q^{54} - 16 q^{59} - 2 q^{64} + 8 q^{66} + 16 q^{69} + 16 q^{74} - 8 q^{76} + 8 q^{79} - 22 q^{81} + 4 q^{86} + 12 q^{89} + 4 q^{96} - 4 q^{99}+O(q^{100})$$ 2 * q - 2 * q^4 + 4 * q^6 - 2 * q^9 + 4 * q^11 + 2 * q^16 + 8 * q^19 - 4 * q^24 + 8 * q^29 - 2 * q^31 - 4 * q^34 + 2 * q^36 + 12 * q^41 - 4 * q^44 - 8 * q^46 + 14 * q^49 + 8 * q^51 + 8 * q^54 - 16 * q^59 - 2 * q^64 + 8 * q^66 + 16 * q^69 + 16 * q^74 - 8 * q^76 + 8 * q^79 - 22 * q^81 + 4 * q^86 + 12 * q^89 + 4 * q^96 - 4 * q^99

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1550\mathbb{Z}\right)^\times$$.

 $$n$$ $$251$$ $$1427$$ $$\chi(n)$$ $$1$$ $$-1$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 1.00000i 0.707107i
$$3$$ − 2.00000i − 1.15470i −0.816497 0.577350i $$-0.804087\pi$$
0.816497 0.577350i $$-0.195913\pi$$
$$4$$ −1.00000 −0.500000
$$5$$ 0 0
$$6$$ 2.00000 0.816497
$$7$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$8$$ − 1.00000i − 0.353553i
$$9$$ −1.00000 −0.333333
$$10$$ 0 0
$$11$$ 2.00000 0.603023 0.301511 0.953463i $$-0.402509\pi$$
0.301511 + 0.953463i $$0.402509\pi$$
$$12$$ 2.00000i 0.577350i
$$13$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 1.00000 0.250000
$$17$$ 2.00000i 0.485071i 0.970143 + 0.242536i $$0.0779791\pi$$
−0.970143 + 0.242536i $$0.922021\pi$$
$$18$$ − 1.00000i − 0.235702i
$$19$$ 4.00000 0.917663 0.458831 0.888523i $$-0.348268\pi$$
0.458831 + 0.888523i $$0.348268\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 2.00000i 0.426401i
$$23$$ 4.00000i 0.834058i 0.908893 + 0.417029i $$0.136929\pi$$
−0.908893 + 0.417029i $$0.863071\pi$$
$$24$$ −2.00000 −0.408248
$$25$$ 0 0
$$26$$ 0 0
$$27$$ − 4.00000i − 0.769800i
$$28$$ 0 0
$$29$$ 4.00000 0.742781 0.371391 0.928477i $$-0.378881\pi$$
0.371391 + 0.928477i $$0.378881\pi$$
$$30$$ 0 0
$$31$$ −1.00000 −0.179605
$$32$$ 1.00000i 0.176777i
$$33$$ − 4.00000i − 0.696311i
$$34$$ −2.00000 −0.342997
$$35$$ 0 0
$$36$$ 1.00000 0.166667
$$37$$ − 8.00000i − 1.31519i −0.753371 0.657596i $$-0.771573\pi$$
0.753371 0.657596i $$-0.228427\pi$$
$$38$$ 4.00000i 0.648886i
$$39$$ 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
$$43$$ − 2.00000i − 0.304997i −0.988304 0.152499i $$-0.951268\pi$$
0.988304 0.152499i $$-0.0487319\pi$$
$$44$$ −2.00000 −0.301511
$$45$$ 0 0
$$46$$ −4.00000 −0.589768
$$47$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$48$$ − 2.00000i − 0.288675i
$$49$$ 7.00000 1.00000
$$50$$ 0 0
$$51$$ 4.00000 0.560112
$$52$$ 0 0
$$53$$ − 8.00000i − 1.09888i −0.835532 0.549442i $$-0.814840\pi$$
0.835532 0.549442i $$-0.185160\pi$$
$$54$$ 4.00000 0.544331
$$55$$ 0 0
$$56$$ 0 0
$$57$$ − 8.00000i − 1.05963i
$$58$$ 4.00000i 0.525226i
$$59$$ −8.00000 −1.04151 −0.520756 0.853706i $$-0.674350\pi$$
−0.520756 + 0.853706i $$0.674350\pi$$
$$60$$ 0 0
$$61$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$62$$ − 1.00000i − 0.127000i
$$63$$ 0 0
$$64$$ −1.00000 −0.125000
$$65$$ 0 0
$$66$$ 4.00000 0.492366
$$67$$ 4.00000i 0.488678i 0.969690 + 0.244339i $$0.0785709\pi$$
−0.969690 + 0.244339i $$0.921429\pi$$
$$68$$ − 2.00000i − 0.242536i
$$69$$ 8.00000 0.963087
$$70$$ 0 0
$$71$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$72$$ 1.00000i 0.117851i
$$73$$ − 6.00000i − 0.702247i −0.936329 0.351123i $$-0.885800\pi$$
0.936329 0.351123i $$-0.114200\pi$$
$$74$$ 8.00000 0.929981
$$75$$ 0 0
$$76$$ −4.00000 −0.458831
$$77$$ 0 0
$$78$$ 0 0
$$79$$ 4.00000 0.450035 0.225018 0.974355i $$-0.427756\pi$$
0.225018 + 0.974355i $$0.427756\pi$$
$$80$$ 0 0
$$81$$ −11.0000 −1.22222
$$82$$ 6.00000i 0.662589i
$$83$$ − 6.00000i − 0.658586i −0.944228 0.329293i $$-0.893190\pi$$
0.944228 0.329293i $$-0.106810\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 2.00000 0.215666
$$87$$ − 8.00000i − 0.857690i
$$88$$ − 2.00000i − 0.213201i
$$89$$ 6.00000 0.635999 0.317999 0.948091i $$-0.396989\pi$$
0.317999 + 0.948091i $$0.396989\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ − 4.00000i − 0.417029i
$$93$$ 2.00000i 0.207390i
$$94$$ 0 0
$$95$$ 0 0
$$96$$ 2.00000 0.204124
$$97$$ − 2.00000i − 0.203069i −0.994832 0.101535i $$-0.967625\pi$$
0.994832 0.101535i $$-0.0323753\pi$$
$$98$$ 7.00000i 0.707107i
$$99$$ −2.00000 −0.201008
$$100$$ 0 0
$$101$$ 2.00000 0.199007 0.0995037 0.995037i $$-0.468274\pi$$
0.0995037 + 0.995037i $$0.468274\pi$$
$$102$$ 4.00000i 0.396059i
$$103$$ − 8.00000i − 0.788263i −0.919054 0.394132i $$-0.871045\pi$$
0.919054 0.394132i $$-0.128955\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 8.00000 0.777029
$$107$$ − 8.00000i − 0.773389i −0.922208 0.386695i $$-0.873617\pi$$
0.922208 0.386695i $$-0.126383\pi$$
$$108$$ 4.00000i 0.384900i
$$109$$ 18.0000 1.72409 0.862044 0.506834i $$-0.169184\pi$$
0.862044 + 0.506834i $$0.169184\pi$$
$$110$$ 0 0
$$111$$ −16.0000 −1.51865
$$112$$ 0 0
$$113$$ − 14.0000i − 1.31701i −0.752577 0.658505i $$-0.771189\pi$$
0.752577 0.658505i $$-0.228811\pi$$
$$114$$ 8.00000 0.749269
$$115$$ 0 0
$$116$$ −4.00000 −0.371391
$$117$$ 0 0
$$118$$ − 8.00000i − 0.736460i
$$119$$ 0 0
$$120$$ 0 0
$$121$$ −7.00000 −0.636364
$$122$$ 0 0
$$123$$ − 12.0000i − 1.08200i
$$124$$ 1.00000 0.0898027
$$125$$ 0 0
$$126$$ 0 0
$$127$$ 16.0000i 1.41977i 0.704317 + 0.709885i $$0.251253\pi$$
−0.704317 + 0.709885i $$0.748747\pi$$
$$128$$ − 1.00000i − 0.0883883i
$$129$$ −4.00000 −0.352180
$$130$$ 0 0
$$131$$ 20.0000 1.74741 0.873704 0.486458i $$-0.161711\pi$$
0.873704 + 0.486458i $$0.161711\pi$$
$$132$$ 4.00000i 0.348155i
$$133$$ 0 0
$$134$$ −4.00000 −0.345547
$$135$$ 0 0
$$136$$ 2.00000 0.171499
$$137$$ 18.0000i 1.53784i 0.639343 + 0.768922i $$0.279207\pi$$
−0.639343 + 0.768922i $$0.720793\pi$$
$$138$$ 8.00000i 0.681005i
$$139$$ −2.00000 −0.169638 −0.0848189 0.996396i $$-0.527031\pi$$
−0.0848189 + 0.996396i $$0.527031\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 0 0
$$143$$ 0 0
$$144$$ −1.00000 −0.0833333
$$145$$ 0 0
$$146$$ 6.00000 0.496564
$$147$$ − 14.0000i − 1.15470i
$$148$$ 8.00000i 0.657596i
$$149$$ 14.0000 1.14692 0.573462 0.819232i $$-0.305600\pi$$
0.573462 + 0.819232i $$0.305600\pi$$
$$150$$ 0 0
$$151$$ 4.00000 0.325515 0.162758 0.986666i $$-0.447961\pi$$
0.162758 + 0.986666i $$0.447961\pi$$
$$152$$ − 4.00000i − 0.324443i
$$153$$ − 2.00000i − 0.161690i
$$154$$ 0 0
$$155$$ 0 0
$$156$$ 0 0
$$157$$ 6.00000i 0.478852i 0.970915 + 0.239426i $$0.0769593\pi$$
−0.970915 + 0.239426i $$0.923041\pi$$
$$158$$ 4.00000i 0.318223i
$$159$$ −16.0000 −1.26888
$$160$$ 0 0
$$161$$ 0 0
$$162$$ − 11.0000i − 0.864242i
$$163$$ − 24.0000i − 1.87983i −0.341415 0.939913i $$-0.610906\pi$$
0.341415 0.939913i $$-0.389094\pi$$
$$164$$ −6.00000 −0.468521
$$165$$ 0 0
$$166$$ 6.00000 0.465690
$$167$$ 12.0000i 0.928588i 0.885681 + 0.464294i $$0.153692\pi$$
−0.885681 + 0.464294i $$0.846308\pi$$
$$168$$ 0 0
$$169$$ 13.0000 1.00000
$$170$$ 0 0
$$171$$ −4.00000 −0.305888
$$172$$ 2.00000i 0.152499i
$$173$$ 2.00000i 0.152057i 0.997106 + 0.0760286i $$0.0242240\pi$$
−0.997106 + 0.0760286i $$0.975776\pi$$
$$174$$ 8.00000 0.606478
$$175$$ 0 0
$$176$$ 2.00000 0.150756
$$177$$ 16.0000i 1.20263i
$$178$$ 6.00000i 0.449719i
$$179$$ −14.0000 −1.04641 −0.523205 0.852207i $$-0.675264\pi$$
−0.523205 + 0.852207i $$0.675264\pi$$
$$180$$ 0 0
$$181$$ −24.0000 −1.78391 −0.891953 0.452128i $$-0.850665\pi$$
−0.891953 + 0.452128i $$0.850665\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ 4.00000 0.294884
$$185$$ 0 0
$$186$$ −2.00000 −0.146647
$$187$$ 4.00000i 0.292509i
$$188$$ 0 0
$$189$$ 0 0
$$190$$ 0 0
$$191$$ −8.00000 −0.578860 −0.289430 0.957199i $$-0.593466\pi$$
−0.289430 + 0.957199i $$0.593466\pi$$
$$192$$ 2.00000i 0.144338i
$$193$$ − 2.00000i − 0.143963i −0.997406 0.0719816i $$-0.977068\pi$$
0.997406 0.0719816i $$-0.0229323\pi$$
$$194$$ 2.00000 0.143592
$$195$$ 0 0
$$196$$ −7.00000 −0.500000
$$197$$ 8.00000i 0.569976i 0.958531 + 0.284988i $$0.0919897\pi$$
−0.958531 + 0.284988i $$0.908010\pi$$
$$198$$ − 2.00000i − 0.142134i
$$199$$ −16.0000 −1.13421 −0.567105 0.823646i $$-0.691937\pi$$
−0.567105 + 0.823646i $$0.691937\pi$$
$$200$$ 0 0
$$201$$ 8.00000 0.564276
$$202$$ 2.00000i 0.140720i
$$203$$ 0 0
$$204$$ −4.00000 −0.280056
$$205$$ 0 0
$$206$$ 8.00000 0.557386
$$207$$ − 4.00000i − 0.278019i
$$208$$ 0 0
$$209$$ 8.00000 0.553372
$$210$$ 0 0
$$211$$ −16.0000 −1.10149 −0.550743 0.834675i $$-0.685655\pi$$
−0.550743 + 0.834675i $$0.685655\pi$$
$$212$$ 8.00000i 0.549442i
$$213$$ 0 0
$$214$$ 8.00000 0.546869
$$215$$ 0 0
$$216$$ −4.00000 −0.272166
$$217$$ 0 0
$$218$$ 18.0000i 1.21911i
$$219$$ −12.0000 −0.810885
$$220$$ 0 0
$$221$$ 0 0
$$222$$ − 16.0000i − 1.07385i
$$223$$ 8.00000i 0.535720i 0.963458 + 0.267860i $$0.0863164\pi$$
−0.963458 + 0.267860i $$0.913684\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 14.0000 0.931266
$$227$$ 8.00000i 0.530979i 0.964114 + 0.265489i $$0.0855335\pi$$
−0.964114 + 0.265489i $$0.914466\pi$$
$$228$$ 8.00000i 0.529813i
$$229$$ −4.00000 −0.264327 −0.132164 0.991228i $$-0.542192\pi$$
−0.132164 + 0.991228i $$0.542192\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ − 4.00000i − 0.262613i
$$233$$ 26.0000i 1.70332i 0.524097 + 0.851658i $$0.324403\pi$$
−0.524097 + 0.851658i $$0.675597\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ 8.00000 0.520756
$$237$$ − 8.00000i − 0.519656i
$$238$$ 0 0
$$239$$ −12.0000 −0.776215 −0.388108 0.921614i $$-0.626871\pi$$
−0.388108 + 0.921614i $$0.626871\pi$$
$$240$$ 0 0
$$241$$ −2.00000 −0.128831 −0.0644157 0.997923i $$-0.520518\pi$$
−0.0644157 + 0.997923i $$0.520518\pi$$
$$242$$ − 7.00000i − 0.449977i
$$243$$ 10.0000i 0.641500i
$$244$$ 0 0
$$245$$ 0 0
$$246$$ 12.0000 0.765092
$$247$$ 0 0
$$248$$ 1.00000i 0.0635001i
$$249$$ −12.0000 −0.760469
$$250$$ 0 0
$$251$$ 14.0000 0.883672 0.441836 0.897096i $$-0.354327\pi$$
0.441836 + 0.897096i $$0.354327\pi$$
$$252$$ 0 0
$$253$$ 8.00000i 0.502956i
$$254$$ −16.0000 −1.00393
$$255$$ 0 0
$$256$$ 1.00000 0.0625000
$$257$$ 10.0000i 0.623783i 0.950118 + 0.311891i $$0.100963\pi$$
−0.950118 + 0.311891i $$0.899037\pi$$
$$258$$ − 4.00000i − 0.249029i
$$259$$ 0 0
$$260$$ 0 0
$$261$$ −4.00000 −0.247594
$$262$$ 20.0000i 1.23560i
$$263$$ 24.0000i 1.47990i 0.672660 + 0.739952i $$0.265152\pi$$
−0.672660 + 0.739952i $$0.734848\pi$$
$$264$$ −4.00000 −0.246183
$$265$$ 0 0
$$266$$ 0 0
$$267$$ − 12.0000i − 0.734388i
$$268$$ − 4.00000i − 0.244339i
$$269$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$270$$ 0 0
$$271$$ −20.0000 −1.21491 −0.607457 0.794353i $$-0.707810\pi$$
−0.607457 + 0.794353i $$0.707810\pi$$
$$272$$ 2.00000i 0.121268i
$$273$$ 0 0
$$274$$ −18.0000 −1.08742
$$275$$ 0 0
$$276$$ −8.00000 −0.481543
$$277$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$278$$ − 2.00000i − 0.119952i
$$279$$ 1.00000 0.0598684
$$280$$ 0 0
$$281$$ 10.0000 0.596550 0.298275 0.954480i $$-0.403589\pi$$
0.298275 + 0.954480i $$0.403589\pi$$
$$282$$ 0 0
$$283$$ 24.0000i 1.42665i 0.700832 + 0.713326i $$0.252812\pi$$
−0.700832 + 0.713326i $$0.747188\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ 0 0
$$288$$ − 1.00000i − 0.0589256i
$$289$$ 13.0000 0.764706
$$290$$ 0 0
$$291$$ −4.00000 −0.234484
$$292$$ 6.00000i 0.351123i
$$293$$ 18.0000i 1.05157i 0.850617 + 0.525786i $$0.176229\pi$$
−0.850617 + 0.525786i $$0.823771\pi$$
$$294$$ 14.0000 0.816497
$$295$$ 0 0
$$296$$ −8.00000 −0.464991
$$297$$ − 8.00000i − 0.464207i
$$298$$ 14.0000i 0.810998i
$$299$$ 0 0
$$300$$ 0 0
$$301$$ 0 0
$$302$$ 4.00000i 0.230174i
$$303$$ − 4.00000i − 0.229794i
$$304$$ 4.00000 0.229416
$$305$$ 0 0
$$306$$ 2.00000 0.114332
$$307$$ − 16.0000i − 0.913168i −0.889680 0.456584i $$-0.849073\pi$$
0.889680 0.456584i $$-0.150927\pi$$
$$308$$ 0 0
$$309$$ −16.0000 −0.910208
$$310$$ 0 0
$$311$$ 8.00000 0.453638 0.226819 0.973937i $$-0.427167\pi$$
0.226819 + 0.973937i $$0.427167\pi$$
$$312$$ 0 0
$$313$$ 22.0000i 1.24351i 0.783210 + 0.621757i $$0.213581\pi$$
−0.783210 + 0.621757i $$0.786419\pi$$
$$314$$ −6.00000 −0.338600
$$315$$ 0 0
$$316$$ −4.00000 −0.225018
$$317$$ 2.00000i 0.112331i 0.998421 + 0.0561656i $$0.0178875\pi$$
−0.998421 + 0.0561656i $$0.982113\pi$$
$$318$$ − 16.0000i − 0.897235i
$$319$$ 8.00000 0.447914
$$320$$ 0 0
$$321$$ −16.0000 −0.893033
$$322$$ 0 0
$$323$$ 8.00000i 0.445132i
$$324$$ 11.0000 0.611111
$$325$$ 0 0
$$326$$ 24.0000 1.32924
$$327$$ − 36.0000i − 1.99080i
$$328$$ − 6.00000i − 0.331295i
$$329$$ 0 0
$$330$$ 0 0
$$331$$ −30.0000 −1.64895 −0.824475 0.565899i $$-0.808529\pi$$
−0.824475 + 0.565899i $$0.808529\pi$$
$$332$$ 6.00000i 0.329293i
$$333$$ 8.00000i 0.438397i
$$334$$ −12.0000 −0.656611
$$335$$ 0 0
$$336$$ 0 0
$$337$$ 14.0000i 0.762629i 0.924445 + 0.381314i $$0.124528\pi$$
−0.924445 + 0.381314i $$0.875472\pi$$
$$338$$ 13.0000i 0.707107i
$$339$$ −28.0000 −1.52075
$$340$$ 0 0
$$341$$ −2.00000 −0.108306
$$342$$ − 4.00000i − 0.216295i
$$343$$ 0 0
$$344$$ −2.00000 −0.107833
$$345$$ 0 0
$$346$$ −2.00000 −0.107521
$$347$$ − 22.0000i − 1.18102i −0.807030 0.590511i $$-0.798926\pi$$
0.807030 0.590511i $$-0.201074\pi$$
$$348$$ 8.00000i 0.428845i
$$349$$ −10.0000 −0.535288 −0.267644 0.963518i $$-0.586245\pi$$
−0.267644 + 0.963518i $$0.586245\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 2.00000i 0.106600i
$$353$$ − 30.0000i − 1.59674i −0.602168 0.798369i $$-0.705696\pi$$
0.602168 0.798369i $$-0.294304\pi$$
$$354$$ −16.0000 −0.850390
$$355$$ 0 0
$$356$$ −6.00000 −0.317999
$$357$$ 0 0
$$358$$ − 14.0000i − 0.739923i
$$359$$ 8.00000 0.422224 0.211112 0.977462i $$-0.432292\pi$$
0.211112 + 0.977462i $$0.432292\pi$$
$$360$$ 0 0
$$361$$ −3.00000 −0.157895
$$362$$ − 24.0000i − 1.26141i
$$363$$ 14.0000i 0.734809i
$$364$$ 0 0
$$365$$ 0 0
$$366$$ 0 0
$$367$$ 4.00000i 0.208798i 0.994535 + 0.104399i $$0.0332919\pi$$
−0.994535 + 0.104399i $$0.966708\pi$$
$$368$$ 4.00000i 0.208514i
$$369$$ −6.00000 −0.312348
$$370$$ 0 0
$$371$$ 0 0
$$372$$ − 2.00000i − 0.103695i
$$373$$ 26.0000i 1.34623i 0.739538 + 0.673114i $$0.235044\pi$$
−0.739538 + 0.673114i $$0.764956\pi$$
$$374$$ −4.00000 −0.206835
$$375$$ 0 0
$$376$$ 0 0
$$377$$ 0 0
$$378$$ 0 0
$$379$$ −16.0000 −0.821865 −0.410932 0.911666i $$-0.634797\pi$$
−0.410932 + 0.911666i $$0.634797\pi$$
$$380$$ 0 0
$$381$$ 32.0000 1.63941
$$382$$ − 8.00000i − 0.409316i
$$383$$ − 16.0000i − 0.817562i −0.912633 0.408781i $$-0.865954\pi$$
0.912633 0.408781i $$-0.134046\pi$$
$$384$$ −2.00000 −0.102062
$$385$$ 0 0
$$386$$ 2.00000 0.101797
$$387$$ 2.00000i 0.101666i
$$388$$ 2.00000i 0.101535i
$$389$$ 36.0000 1.82527 0.912636 0.408773i $$-0.134043\pi$$
0.912636 + 0.408773i $$0.134043\pi$$
$$390$$ 0 0
$$391$$ −8.00000 −0.404577
$$392$$ − 7.00000i − 0.353553i
$$393$$ − 40.0000i − 2.01773i
$$394$$ −8.00000 −0.403034
$$395$$ 0 0
$$396$$ 2.00000 0.100504
$$397$$ − 14.0000i − 0.702640i −0.936255 0.351320i $$-0.885733\pi$$
0.936255 0.351320i $$-0.114267\pi$$
$$398$$ − 16.0000i − 0.802008i
$$399$$ 0 0
$$400$$ 0 0
$$401$$ 38.0000 1.89763 0.948815 0.315833i $$-0.102284\pi$$
0.948815 + 0.315833i $$0.102284\pi$$
$$402$$ 8.00000i 0.399004i
$$403$$ 0 0
$$404$$ −2.00000 −0.0995037
$$405$$ 0 0
$$406$$ 0 0
$$407$$ − 16.0000i − 0.793091i
$$408$$ − 4.00000i − 0.198030i
$$409$$ −6.00000 −0.296681 −0.148340 0.988936i $$-0.547393\pi$$
−0.148340 + 0.988936i $$0.547393\pi$$
$$410$$ 0 0
$$411$$ 36.0000 1.77575
$$412$$ 8.00000i 0.394132i
$$413$$ 0 0
$$414$$ 4.00000 0.196589
$$415$$ 0 0
$$416$$ 0 0
$$417$$ 4.00000i 0.195881i
$$418$$ 8.00000i 0.391293i
$$419$$ 24.0000 1.17248 0.586238 0.810139i $$-0.300608\pi$$
0.586238 + 0.810139i $$0.300608\pi$$
$$420$$ 0 0
$$421$$ −22.0000 −1.07221 −0.536107 0.844150i $$-0.680106\pi$$
−0.536107 + 0.844150i $$0.680106\pi$$
$$422$$ − 16.0000i − 0.778868i
$$423$$ 0 0
$$424$$ −8.00000 −0.388514
$$425$$ 0 0
$$426$$ 0 0
$$427$$ 0 0
$$428$$ 8.00000i 0.386695i
$$429$$ 0 0
$$430$$ 0 0
$$431$$ −24.0000 −1.15604 −0.578020 0.816023i $$-0.696174\pi$$
−0.578020 + 0.816023i $$0.696174\pi$$
$$432$$ − 4.00000i − 0.192450i
$$433$$ − 26.0000i − 1.24948i −0.780833 0.624740i $$-0.785205\pi$$
0.780833 0.624740i $$-0.214795\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ −18.0000 −0.862044
$$437$$ 16.0000i 0.765384i
$$438$$ − 12.0000i − 0.573382i
$$439$$ −40.0000 −1.90910 −0.954548 0.298057i $$-0.903661\pi$$
−0.954548 + 0.298057i $$0.903661\pi$$
$$440$$ 0 0
$$441$$ −7.00000 −0.333333
$$442$$ 0 0
$$443$$ 28.0000i 1.33032i 0.746701 + 0.665160i $$0.231637\pi$$
−0.746701 + 0.665160i $$0.768363\pi$$
$$444$$ 16.0000 0.759326
$$445$$ 0 0
$$446$$ −8.00000 −0.378811
$$447$$ − 28.0000i − 1.32435i
$$448$$ 0 0
$$449$$ −6.00000 −0.283158 −0.141579 0.989927i $$-0.545218\pi$$
−0.141579 + 0.989927i $$0.545218\pi$$
$$450$$ 0 0
$$451$$ 12.0000 0.565058
$$452$$ 14.0000i 0.658505i
$$453$$ − 8.00000i − 0.375873i
$$454$$ −8.00000 −0.375459
$$455$$ 0 0
$$456$$ −8.00000 −0.374634
$$457$$ 2.00000i 0.0935561i 0.998905 + 0.0467780i $$0.0148953\pi$$
−0.998905 + 0.0467780i $$0.985105\pi$$
$$458$$ − 4.00000i − 0.186908i
$$459$$ 8.00000 0.373408
$$460$$ 0 0
$$461$$ −8.00000 −0.372597 −0.186299 0.982493i $$-0.559649\pi$$
−0.186299 + 0.982493i $$0.559649\pi$$
$$462$$ 0 0
$$463$$ − 36.0000i − 1.67306i −0.547920 0.836531i $$-0.684580\pi$$
0.547920 0.836531i $$-0.315420\pi$$
$$464$$ 4.00000 0.185695
$$465$$ 0 0
$$466$$ −26.0000 −1.20443
$$467$$ 8.00000i 0.370196i 0.982720 + 0.185098i $$0.0592602\pi$$
−0.982720 + 0.185098i $$0.940740\pi$$
$$468$$ 0 0
$$469$$ 0 0
$$470$$ 0 0
$$471$$ 12.0000 0.552931
$$472$$ 8.00000i 0.368230i
$$473$$ − 4.00000i − 0.183920i
$$474$$ 8.00000 0.367452
$$475$$ 0 0
$$476$$ 0 0
$$477$$ 8.00000i 0.366295i
$$478$$ − 12.0000i − 0.548867i
$$479$$ −24.0000 −1.09659 −0.548294 0.836286i $$-0.684723\pi$$
−0.548294 + 0.836286i $$0.684723\pi$$
$$480$$ 0 0
$$481$$ 0 0
$$482$$ − 2.00000i − 0.0910975i
$$483$$ 0 0
$$484$$ 7.00000 0.318182
$$485$$ 0 0
$$486$$ −10.0000 −0.453609
$$487$$ 32.0000i 1.45006i 0.688718 + 0.725029i $$0.258174\pi$$
−0.688718 + 0.725029i $$0.741826\pi$$
$$488$$ 0 0
$$489$$ −48.0000 −2.17064
$$490$$ 0 0
$$491$$ 2.00000 0.0902587 0.0451294 0.998981i $$-0.485630\pi$$
0.0451294 + 0.998981i $$0.485630\pi$$
$$492$$ 12.0000i 0.541002i
$$493$$ 8.00000i 0.360302i
$$494$$ 0 0
$$495$$ 0 0
$$496$$ −1.00000 −0.0449013
$$497$$ 0 0
$$498$$ − 12.0000i − 0.537733i
$$499$$ −10.0000 −0.447661 −0.223831 0.974628i $$-0.571856\pi$$
−0.223831 + 0.974628i $$0.571856\pi$$
$$500$$ 0 0
$$501$$ 24.0000 1.07224
$$502$$ 14.0000i 0.624851i
$$503$$ 16.0000i 0.713405i 0.934218 + 0.356702i $$0.116099\pi$$
−0.934218 + 0.356702i $$0.883901\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ −8.00000 −0.355643
$$507$$ − 26.0000i − 1.15470i
$$508$$ − 16.0000i − 0.709885i
$$509$$ −24.0000 −1.06378 −0.531891 0.846813i $$-0.678518\pi$$
−0.531891 + 0.846813i $$0.678518\pi$$
$$510$$ 0 0
$$511$$ 0 0
$$512$$ 1.00000i 0.0441942i
$$513$$ − 16.0000i − 0.706417i
$$514$$ −10.0000 −0.441081
$$515$$ 0 0
$$516$$ 4.00000 0.176090
$$517$$ 0 0
$$518$$ 0 0
$$519$$ 4.00000 0.175581
$$520$$ 0 0
$$521$$ −38.0000 −1.66481 −0.832405 0.554168i $$-0.813037\pi$$
−0.832405 + 0.554168i $$0.813037\pi$$
$$522$$ − 4.00000i − 0.175075i
$$523$$ − 38.0000i − 1.66162i −0.556553 0.830812i $$-0.687876\pi$$
0.556553 0.830812i $$-0.312124\pi$$
$$524$$ −20.0000 −0.873704
$$525$$ 0 0
$$526$$ −24.0000 −1.04645
$$527$$ − 2.00000i − 0.0871214i
$$528$$ − 4.00000i − 0.174078i
$$529$$ 7.00000 0.304348
$$530$$ 0 0
$$531$$ 8.00000 0.347170
$$532$$ 0 0
$$533$$ 0 0
$$534$$ 12.0000 0.519291
$$535$$ 0 0
$$536$$ 4.00000 0.172774
$$537$$ 28.0000i 1.20829i
$$538$$ 0 0
$$539$$ 14.0000 0.603023
$$540$$ 0 0
$$541$$ 38.0000 1.63375 0.816874 0.576816i $$-0.195705\pi$$
0.816874 + 0.576816i $$0.195705\pi$$
$$542$$ − 20.0000i − 0.859074i
$$543$$ 48.0000i 2.05988i
$$544$$ −2.00000 −0.0857493
$$545$$ 0 0
$$546$$ 0 0
$$547$$ − 32.0000i − 1.36822i −0.729378 0.684111i $$-0.760191\pi$$
0.729378 0.684111i $$-0.239809\pi$$
$$548$$ − 18.0000i − 0.768922i
$$549$$ 0 0
$$550$$ 0 0
$$551$$ 16.0000 0.681623
$$552$$ − 8.00000i − 0.340503i
$$553$$ 0 0
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 2.00000 0.0848189
$$557$$ 24.0000i 1.01691i 0.861088 + 0.508456i $$0.169784\pi$$
−0.861088 + 0.508456i $$0.830216\pi$$
$$558$$ 1.00000i 0.0423334i
$$559$$ 0 0
$$560$$ 0 0
$$561$$ 8.00000 0.337760
$$562$$ 10.0000i 0.421825i
$$563$$ − 12.0000i − 0.505740i −0.967500 0.252870i $$-0.918626\pi$$
0.967500 0.252870i $$-0.0813744\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ −24.0000 −1.00880
$$567$$ 0 0
$$568$$ 0 0
$$569$$ 42.0000 1.76073 0.880366 0.474295i $$-0.157297\pi$$
0.880366 + 0.474295i $$0.157297\pi$$
$$570$$ 0 0
$$571$$ −26.0000 −1.08807 −0.544033 0.839064i $$-0.683103\pi$$
−0.544033 + 0.839064i $$0.683103\pi$$
$$572$$ 0 0
$$573$$ 16.0000i 0.668410i
$$574$$ 0 0
$$575$$ 0 0
$$576$$ 1.00000 0.0416667
$$577$$ 30.0000i 1.24892i 0.781058 + 0.624458i $$0.214680\pi$$
−0.781058 + 0.624458i $$0.785320\pi$$
$$578$$ 13.0000i 0.540729i
$$579$$ −4.00000 −0.166234
$$580$$ 0 0
$$581$$ 0 0
$$582$$ − 4.00000i − 0.165805i
$$583$$ − 16.0000i − 0.662652i
$$584$$ −6.00000 −0.248282
$$585$$ 0 0
$$586$$ −18.0000 −0.743573
$$587$$ − 2.00000i − 0.0825488i −0.999148 0.0412744i $$-0.986858\pi$$
0.999148 0.0412744i $$-0.0131418\pi$$
$$588$$ 14.0000i 0.577350i
$$589$$ −4.00000 −0.164817
$$590$$ 0 0
$$591$$ 16.0000 0.658152
$$592$$ − 8.00000i − 0.328798i
$$593$$ 34.0000i 1.39621i 0.715994 + 0.698106i $$0.245974\pi$$
−0.715994 + 0.698106i $$0.754026\pi$$
$$594$$ 8.00000 0.328244
$$595$$ 0 0
$$596$$ −14.0000 −0.573462
$$597$$ 32.0000i 1.30967i
$$598$$ 0 0
$$599$$ 16.0000 0.653742 0.326871 0.945069i $$-0.394006\pi$$
0.326871 + 0.945069i $$0.394006\pi$$
$$600$$ 0 0
$$601$$ −14.0000 −0.571072 −0.285536 0.958368i $$-0.592172\pi$$
−0.285536 + 0.958368i $$0.592172\pi$$
$$602$$ 0 0
$$603$$ − 4.00000i − 0.162893i
$$604$$ −4.00000 −0.162758
$$605$$ 0 0
$$606$$ 4.00000 0.162489
$$607$$ − 24.0000i − 0.974130i −0.873366 0.487065i $$-0.838067\pi$$
0.873366 0.487065i $$-0.161933\pi$$
$$608$$ 4.00000i 0.162221i
$$609$$ 0 0
$$610$$ 0 0
$$611$$ 0 0
$$612$$ 2.00000i 0.0808452i
$$613$$ 44.0000i 1.77714i 0.458738 + 0.888572i $$0.348302\pi$$
−0.458738 + 0.888572i $$0.651698\pi$$
$$614$$ 16.0000 0.645707
$$615$$ 0 0
$$616$$ 0 0
$$617$$ − 38.0000i − 1.52982i −0.644136 0.764911i $$-0.722783\pi$$
0.644136 0.764911i $$-0.277217\pi$$
$$618$$ − 16.0000i − 0.643614i
$$619$$ 34.0000 1.36658 0.683288 0.730149i $$-0.260549\pi$$
0.683288 + 0.730149i $$0.260549\pi$$
$$620$$ 0 0
$$621$$ 16.0000 0.642058
$$622$$ 8.00000i 0.320771i
$$623$$ 0 0
$$624$$ 0 0
$$625$$ 0 0
$$626$$ −22.0000 −0.879297
$$627$$ − 16.0000i − 0.638978i
$$628$$ − 6.00000i − 0.239426i
$$629$$ 16.0000 0.637962
$$630$$ 0 0
$$631$$ −28.0000 −1.11466 −0.557331 0.830290i $$-0.688175\pi$$
−0.557331 + 0.830290i $$0.688175\pi$$
$$632$$ − 4.00000i − 0.159111i
$$633$$ 32.0000i 1.27189i
$$634$$ −2.00000 −0.0794301
$$635$$ 0 0
$$636$$ 16.0000 0.634441
$$637$$ 0 0
$$638$$ 8.00000i 0.316723i
$$639$$ 0 0
$$640$$ 0 0
$$641$$ 30.0000 1.18493 0.592464 0.805597i $$-0.298155\pi$$
0.592464 + 0.805597i $$0.298155\pi$$
$$642$$ − 16.0000i − 0.631470i
$$643$$ 2.00000i 0.0788723i 0.999222 + 0.0394362i $$0.0125562\pi$$
−0.999222 + 0.0394362i $$0.987444\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ −8.00000 −0.314756
$$647$$ 24.0000i 0.943537i 0.881722 + 0.471769i $$0.156384\pi$$
−0.881722 + 0.471769i $$0.843616\pi$$
$$648$$ 11.0000i 0.432121i
$$649$$ −16.0000 −0.628055
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 24.0000i 0.939913i
$$653$$ 6.00000i 0.234798i 0.993085 + 0.117399i $$0.0374557\pi$$
−0.993085 + 0.117399i $$0.962544\pi$$
$$654$$ 36.0000 1.40771
$$655$$ 0 0
$$656$$ 6.00000 0.234261
$$657$$ 6.00000i 0.234082i
$$658$$ 0 0
$$659$$ 36.0000 1.40236 0.701180 0.712984i $$-0.252657\pi$$
0.701180 + 0.712984i $$0.252657\pi$$
$$660$$ 0 0
$$661$$ −10.0000 −0.388955 −0.194477 0.980907i $$-0.562301\pi$$
−0.194477 + 0.980907i $$0.562301\pi$$
$$662$$ − 30.0000i − 1.16598i
$$663$$ 0 0
$$664$$ −6.00000 −0.232845
$$665$$ 0 0
$$666$$ −8.00000 −0.309994
$$667$$ 16.0000i 0.619522i
$$668$$ − 12.0000i − 0.464294i
$$669$$ 16.0000 0.618596
$$670$$ 0 0
$$671$$ 0 0
$$672$$ 0 0
$$673$$ − 34.0000i − 1.31060i −0.755367 0.655302i $$-0.772541\pi$$
0.755367 0.655302i $$-0.227459\pi$$
$$674$$ −14.0000 −0.539260
$$675$$ 0 0
$$676$$ −13.0000 −0.500000
$$677$$ − 48.0000i − 1.84479i −0.386248 0.922395i $$-0.626229\pi$$
0.386248 0.922395i $$-0.373771\pi$$
$$678$$ − 28.0000i − 1.07533i
$$679$$ 0 0
$$680$$ 0 0
$$681$$ 16.0000 0.613121
$$682$$ − 2.00000i − 0.0765840i
$$683$$ 48.0000i 1.83667i 0.395805 + 0.918334i $$0.370466\pi$$
−0.395805 + 0.918334i $$0.629534\pi$$
$$684$$ 4.00000 0.152944
$$685$$ 0 0
$$686$$ 0 0
$$687$$ 8.00000i 0.305219i
$$688$$ − 2.00000i − 0.0762493i
$$689$$ 0 0
$$690$$ 0 0
$$691$$ 28.0000 1.06517 0.532585 0.846376i $$-0.321221\pi$$
0.532585 + 0.846376i $$0.321221\pi$$
$$692$$ − 2.00000i − 0.0760286i
$$693$$ 0 0
$$694$$ 22.0000 0.835109
$$695$$ 0 0
$$696$$ −8.00000 −0.303239
$$697$$ 12.0000i 0.454532i
$$698$$ − 10.0000i − 0.378506i
$$699$$ 52.0000 1.96682
$$700$$ 0 0
$$701$$ −2.00000 −0.0755390 −0.0377695 0.999286i $$-0.512025\pi$$
−0.0377695 + 0.999286i $$0.512025\pi$$
$$702$$ 0 0
$$703$$ − 32.0000i − 1.20690i
$$704$$ −2.00000 −0.0753778
$$705$$ 0 0
$$706$$ 30.0000 1.12906
$$707$$ 0 0
$$708$$ − 16.0000i − 0.601317i
$$709$$ 16.0000 0.600893 0.300446 0.953799i $$-0.402864\pi$$
0.300446 + 0.953799i $$0.402864\pi$$
$$710$$ 0 0
$$711$$ −4.00000 −0.150012
$$712$$ − 6.00000i − 0.224860i
$$713$$ − 4.00000i − 0.149801i
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 14.0000 0.523205
$$717$$ 24.0000i 0.896296i
$$718$$ 8.00000i 0.298557i
$$719$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$720$$ 0 0
$$721$$ 0 0
$$722$$ − 3.00000i − 0.111648i
$$723$$ 4.00000i 0.148762i
$$724$$ 24.0000 0.891953
$$725$$ 0 0
$$726$$ −14.0000 −0.519589
$$727$$ 40.0000i 1.48352i 0.670667 + 0.741759i $$0.266008\pi$$
−0.670667 + 0.741759i $$0.733992\pi$$
$$728$$ 0 0
$$729$$ −13.0000 −0.481481
$$730$$ 0 0
$$731$$ 4.00000 0.147945
$$732$$ 0 0
$$733$$ − 2.00000i − 0.0738717i −0.999318 0.0369358i $$-0.988240\pi$$
0.999318 0.0369358i $$-0.0117597\pi$$
$$734$$ −4.00000 −0.147643
$$735$$ 0 0
$$736$$ −4.00000 −0.147442
$$737$$ 8.00000i 0.294684i
$$738$$ − 6.00000i − 0.220863i
$$739$$ −26.0000 −0.956425 −0.478213 0.878244i $$-0.658715\pi$$
−0.478213 + 0.878244i $$0.658715\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 0 0
$$743$$ − 24.0000i − 0.880475i −0.897881 0.440237i $$-0.854894\pi$$
0.897881 0.440237i $$-0.145106\pi$$
$$744$$ 2.00000 0.0733236
$$745$$ 0 0
$$746$$ −26.0000 −0.951928
$$747$$ 6.00000i 0.219529i
$$748$$ − 4.00000i − 0.146254i
$$749$$ 0 0
$$750$$ 0 0
$$751$$ 48.0000 1.75154 0.875772 0.482724i $$-0.160353\pi$$
0.875772 + 0.482724i $$0.160353\pi$$
$$752$$ 0 0
$$753$$ − 28.0000i − 1.02038i
$$754$$ 0 0
$$755$$ 0 0
$$756$$ 0 0
$$757$$ 12.0000i 0.436147i 0.975932 + 0.218074i $$0.0699773\pi$$
−0.975932 + 0.218074i $$0.930023\pi$$
$$758$$ − 16.0000i − 0.581146i
$$759$$ 16.0000 0.580763
$$760$$ 0 0
$$761$$ 22.0000 0.797499 0.398750 0.917060i $$-0.369444\pi$$
0.398750 + 0.917060i $$0.369444\pi$$
$$762$$ 32.0000i 1.15924i
$$763$$ 0 0
$$764$$ 8.00000 0.289430
$$765$$ 0 0
$$766$$ 16.0000 0.578103
$$767$$ 0 0
$$768$$ − 2.00000i − 0.0721688i
$$769$$ 2.00000 0.0721218 0.0360609 0.999350i $$-0.488519\pi$$
0.0360609 + 0.999350i $$0.488519\pi$$
$$770$$ 0 0
$$771$$ 20.0000 0.720282
$$772$$ 2.00000i 0.0719816i
$$773$$ 28.0000i 1.00709i 0.863969 + 0.503545i $$0.167971\pi$$
−0.863969 + 0.503545i $$0.832029\pi$$
$$774$$ −2.00000 −0.0718885
$$775$$ 0 0
$$776$$ −2.00000 −0.0717958
$$777$$ 0 0
$$778$$ 36.0000i 1.29066i
$$779$$ 24.0000 0.859889
$$780$$ 0 0
$$781$$ 0 0
$$782$$ − 8.00000i − 0.286079i
$$783$$ − 16.0000i − 0.571793i
$$784$$ 7.00000 0.250000
$$785$$ 0 0
$$786$$ 40.0000 1.42675
$$787$$ 22.0000i 0.784215i 0.919919 + 0.392108i $$0.128254\pi$$
−0.919919 + 0.392108i $$0.871746\pi$$
$$788$$ − 8.00000i − 0.284988i
$$789$$ 48.0000 1.70885
$$790$$ 0 0
$$791$$ 0 0
$$792$$ 2.00000i 0.0710669i
$$793$$ 0 0
$$794$$ 14.0000 0.496841
$$795$$ 0 0
$$796$$ 16.0000 0.567105
$$797$$ 24.0000i 0.850124i 0.905164 + 0.425062i $$0.139748\pi$$
−0.905164 + 0.425062i $$0.860252\pi$$
$$798$$ 0 0
$$799$$ 0 0
$$800$$ 0 0
$$801$$ −6.00000 −0.212000
$$802$$ 38.0000i 1.34183i
$$803$$ − 12.0000i − 0.423471i
$$804$$ −8.00000 −0.282138
$$805$$ 0 0
$$806$$ 0 0
$$807$$ 0 0
$$808$$ − 2.00000i − 0.0703598i
$$809$$ 10.0000 0.351581 0.175791 0.984428i $$-0.443752\pi$$
0.175791 + 0.984428i $$0.443752\pi$$
$$810$$ 0 0
$$811$$ 24.0000 0.842754 0.421377 0.906886i $$-0.361547\pi$$
0.421377 + 0.906886i $$0.361547\pi$$
$$812$$ 0 0
$$813$$ 40.0000i 1.40286i
$$814$$ 16.0000 0.560800
$$815$$ 0 0
$$816$$ 4.00000 0.140028
$$817$$ − 8.00000i − 0.279885i
$$818$$ − 6.00000i − 0.209785i
$$819$$ 0 0
$$820$$ 0 0
$$821$$ 20.0000 0.698005 0.349002 0.937122i $$-0.386521\pi$$
0.349002 + 0.937122i $$0.386521\pi$$
$$822$$ 36.0000i 1.25564i
$$823$$ − 16.0000i − 0.557725i −0.960331 0.278862i $$-0.910043\pi$$
0.960331 0.278862i $$-0.0899574\pi$$
$$824$$ −8.00000 −0.278693
$$825$$ 0 0
$$826$$ 0 0
$$827$$ − 14.0000i − 0.486828i −0.969923 0.243414i $$-0.921733\pi$$
0.969923 0.243414i $$-0.0782673\pi$$
$$828$$ 4.00000i 0.139010i
$$829$$ −32.0000 −1.11141 −0.555703 0.831381i $$-0.687551\pi$$
−0.555703 + 0.831381i $$0.687551\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ 0 0
$$833$$ 14.0000i 0.485071i
$$834$$ −4.00000 −0.138509
$$835$$ 0 0
$$836$$ −8.00000 −0.276686
$$837$$ 4.00000i 0.138260i
$$838$$ 24.0000i 0.829066i
$$839$$ −16.0000 −0.552381 −0.276191 0.961103i $$-0.589072\pi$$
−0.276191 + 0.961103i $$0.589072\pi$$
$$840$$ 0 0
$$841$$ −13.0000 −0.448276
$$842$$ − 22.0000i − 0.758170i
$$843$$ − 20.0000i − 0.688837i
$$844$$ 16.0000 0.550743
$$845$$ 0 0
$$846$$ 0 0
$$847$$ 0 0
$$848$$ − 8.00000i − 0.274721i
$$849$$ 48.0000 1.64736
$$850$$ 0 0
$$851$$ 32.0000 1.09695
$$852$$ 0 0
$$853$$ 14.0000i 0.479351i 0.970853 + 0.239675i $$0.0770410\pi$$
−0.970853 + 0.239675i $$0.922959\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ −8.00000 −0.273434
$$857$$ − 6.00000i − 0.204956i −0.994735 0.102478i $$-0.967323\pi$$
0.994735 0.102478i $$-0.0326771\pi$$
$$858$$ 0 0
$$859$$ −42.0000 −1.43302 −0.716511 0.697576i $$-0.754262\pi$$
−0.716511 + 0.697576i $$0.754262\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ − 24.0000i − 0.817443i
$$863$$ − 24.0000i − 0.816970i −0.912765 0.408485i $$-0.866057\pi$$
0.912765 0.408485i $$-0.133943\pi$$
$$864$$ 4.00000 0.136083
$$865$$ 0 0
$$866$$ 26.0000 0.883516
$$867$$ − 26.0000i − 0.883006i
$$868$$ 0 0
$$869$$ 8.00000 0.271381
$$870$$ 0 0
$$871$$ 0 0
$$872$$ − 18.0000i − 0.609557i
$$873$$ 2.00000i 0.0676897i
$$874$$ −16.0000 −0.541208
$$875$$ 0 0
$$876$$ 12.0000 0.405442
$$877$$ 34.0000i 1.14810i 0.818821 + 0.574049i $$0.194628\pi$$
−0.818821 + 0.574049i $$0.805372\pi$$
$$878$$ − 40.0000i − 1.34993i
$$879$$ 36.0000 1.21425
$$880$$ 0 0
$$881$$ −6.00000 −0.202145 −0.101073 0.994879i $$-0.532227\pi$$
−0.101073 + 0.994879i $$0.532227\pi$$
$$882$$ − 7.00000i − 0.235702i
$$883$$ 26.0000i 0.874970i 0.899226 + 0.437485i $$0.144131\pi$$
−0.899226 + 0.437485i $$0.855869\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ −28.0000 −0.940678
$$887$$ 48.0000i 1.61168i 0.592132 + 0.805841i $$0.298286\pi$$
−0.592132 + 0.805841i $$0.701714\pi$$
$$888$$ 16.0000i 0.536925i
$$889$$ 0 0
$$890$$ 0 0
$$891$$ −22.0000 −0.737028
$$892$$ − 8.00000i − 0.267860i
$$893$$ 0 0
$$894$$ 28.0000 0.936460
$$895$$ 0 0
$$896$$ 0 0
$$897$$ 0 0
$$898$$ − 6.00000i − 0.200223i
$$899$$ −4.00000 −0.133407
$$900$$ 0 0
$$901$$ 16.0000 0.533037
$$902$$ 12.0000i 0.399556i
$$903$$ 0 0
$$904$$ −14.0000 −0.465633
$$905$$ 0 0
$$906$$ 8.00000 0.265782
$$907$$ − 32.0000i − 1.06254i −0.847202 0.531271i $$-0.821714\pi$$
0.847202 0.531271i $$-0.178286\pi$$
$$908$$ − 8.00000i − 0.265489i
$$909$$ −2.00000 −0.0663358
$$910$$ 0 0
$$911$$ −24.0000 −0.795155 −0.397578 0.917568i $$-0.630149\pi$$
−0.397578 + 0.917568i $$0.630149\pi$$
$$912$$ − 8.00000i − 0.264906i
$$913$$ − 12.0000i − 0.397142i
$$914$$ −2.00000 −0.0661541
$$915$$ 0 0
$$916$$ 4.00000 0.132164
$$917$$ 0 0
$$918$$ 8.00000i 0.264039i
$$919$$ −40.0000 −1.31948 −0.659739 0.751495i $$-0.729333\pi$$
−0.659739 + 0.751495i $$0.729333\pi$$
$$920$$ 0 0
$$921$$ −32.0000 −1.05444
$$922$$ − 8.00000i − 0.263466i
$$923$$ 0 0
$$924$$ 0 0
$$925$$ 0 0
$$926$$ 36.0000 1.18303
$$927$$ 8.00000i 0.262754i
$$928$$ 4.00000i 0.131306i
$$929$$ −30.0000 −0.984268 −0.492134 0.870519i $$-0.663783\pi$$
−0.492134 + 0.870519i $$0.663783\pi$$
$$930$$ 0 0
$$931$$ 28.0000 0.917663
$$932$$ − 26.0000i − 0.851658i
$$933$$ − 16.0000i − 0.523816i
$$934$$ −8.00000 −0.261768
$$935$$ 0 0
$$936$$ 0 0
$$937$$ − 14.0000i − 0.457360i −0.973502 0.228680i $$-0.926559\pi$$
0.973502 0.228680i $$-0.0734410\pi$$
$$938$$ 0 0
$$939$$ 44.0000 1.43589
$$940$$ 0 0
$$941$$ −60.0000 −1.95594 −0.977972 0.208736i $$-0.933065\pi$$
−0.977972 + 0.208736i $$0.933065\pi$$
$$942$$ 12.0000i 0.390981i
$$943$$ 24.0000i 0.781548i
$$944$$ −8.00000 −0.260378
$$945$$ 0 0
$$946$$ 4.00000 0.130051
$$947$$ − 14.0000i − 0.454939i −0.973785 0.227469i $$-0.926955\pi$$
0.973785 0.227469i $$-0.0730452\pi$$
$$948$$ 8.00000i 0.259828i
$$949$$ 0 0
$$950$$ 0 0
$$951$$ 4.00000 0.129709
$$952$$ 0 0
$$953$$ − 18.0000i − 0.583077i −0.956559 0.291539i $$-0.905833\pi$$
0.956559 0.291539i $$-0.0941672\pi$$
$$954$$ −8.00000 −0.259010
$$955$$ 0 0
$$956$$ 12.0000 0.388108
$$957$$ − 16.0000i − 0.517207i
$$958$$ − 24.0000i − 0.775405i
$$959$$ 0 0
$$960$$ 0 0
$$961$$ 1.00000 0.0322581
$$962$$ 0 0
$$963$$ 8.00000i 0.257796i
$$964$$ 2.00000 0.0644157
$$965$$ 0 0
$$966$$ 0 0
$$967$$ − 8.00000i − 0.257263i −0.991692 0.128631i $$-0.958942\pi$$
0.991692 0.128631i $$-0.0410584\pi$$
$$968$$ 7.00000i 0.224989i
$$969$$ 16.0000 0.513994
$$970$$ 0 0
$$971$$ 36.0000 1.15529 0.577647 0.816286i $$-0.303971\pi$$
0.577647 + 0.816286i $$0.303971\pi$$
$$972$$ − 10.0000i − 0.320750i
$$973$$ 0 0
$$974$$ −32.0000 −1.02535
$$975$$ 0 0
$$976$$ 0 0
$$977$$ 34.0000i 1.08776i 0.839164 + 0.543878i $$0.183045\pi$$
−0.839164 + 0.543878i $$0.816955\pi$$
$$978$$ − 48.0000i − 1.53487i
$$979$$ 12.0000 0.383522
$$980$$ 0 0
$$981$$ −18.0000 −0.574696
$$982$$ 2.00000i 0.0638226i
$$983$$ − 8.00000i − 0.255160i −0.991828 0.127580i $$-0.959279\pi$$
0.991828 0.127580i $$-0.0407210\pi$$
$$984$$ −12.0000 −0.382546
$$985$$ 0 0
$$986$$ −8.00000 −0.254772
$$987$$ 0 0
$$988$$ 0 0
$$989$$ 8.00000 0.254385
$$990$$ 0 0
$$991$$ −40.0000 −1.27064 −0.635321 0.772248i $$-0.719132\pi$$
−0.635321 + 0.772248i $$0.719132\pi$$
$$992$$ − 1.00000i − 0.0317500i
$$993$$ 60.0000i 1.90404i
$$994$$ 0 0
$$995$$ 0 0
$$996$$ 12.0000 0.380235
$$997$$ − 10.0000i − 0.316703i −0.987383 0.158352i $$-0.949382\pi$$
0.987383 0.158352i $$-0.0506179\pi$$
$$998$$ − 10.0000i − 0.316544i
$$999$$ −32.0000 −1.01244
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 1550.2.b.e.249.2 2
5.2 odd 4 1550.2.a.a.1.1 1
5.3 odd 4 310.2.a.b.1.1 1
5.4 even 2 inner 1550.2.b.e.249.1 2
15.8 even 4 2790.2.a.h.1.1 1
20.3 even 4 2480.2.a.c.1.1 1
40.3 even 4 9920.2.a.bg.1.1 1
40.13 odd 4 9920.2.a.d.1.1 1
155.123 even 4 9610.2.a.a.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
310.2.a.b.1.1 1 5.3 odd 4
1550.2.a.a.1.1 1 5.2 odd 4
1550.2.b.e.249.1 2 5.4 even 2 inner
1550.2.b.e.249.2 2 1.1 even 1 trivial
2480.2.a.c.1.1 1 20.3 even 4
2790.2.a.h.1.1 1 15.8 even 4
9610.2.a.a.1.1 1 155.123 even 4
9920.2.a.d.1.1 1 40.13 odd 4
9920.2.a.bg.1.1 1 40.3 even 4