# Properties

 Label 550.6.b.f.199.1 Level $550$ Weight $6$ Character 550.199 Analytic conductor $88.211$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [550,6,Mod(199,550)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(550, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 6, names="a")

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

chi := DirichletCharacter("550.199");

S:= CuspForms(chi, 6);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$550 = 2 \cdot 5^{2} \cdot 11$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 550.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: $$88.2111008971$$ 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: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 22) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 199.1 Root $$-1.00000i$$ of defining polynomial Character $$\chi$$ $$=$$ 550.199 Dual form 550.6.b.f.199.2

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-4.00000i q^{2} -1.00000i q^{3} -16.0000 q^{4} -4.00000 q^{6} -166.000i q^{7} +64.0000i q^{8} +242.000 q^{9} +O(q^{10})$$ $$q-4.00000i q^{2} -1.00000i q^{3} -16.0000 q^{4} -4.00000 q^{6} -166.000i q^{7} +64.0000i q^{8} +242.000 q^{9} -121.000 q^{11} +16.0000i q^{12} -692.000i q^{13} -664.000 q^{14} +256.000 q^{16} -738.000i q^{17} -968.000i q^{18} -1424.00 q^{19} -166.000 q^{21} +484.000i q^{22} +1779.00i q^{23} +64.0000 q^{24} -2768.00 q^{26} -485.000i q^{27} +2656.00i q^{28} +2064.00 q^{29} +6245.00 q^{31} -1024.00i q^{32} +121.000i q^{33} -2952.00 q^{34} -3872.00 q^{36} -14785.0i q^{37} +5696.00i q^{38} -692.000 q^{39} +5304.00 q^{41} +664.000i q^{42} -17798.0i q^{43} +1936.00 q^{44} +7116.00 q^{46} -17184.0i q^{47} -256.000i q^{48} -10749.0 q^{49} -738.000 q^{51} +11072.0i q^{52} +30726.0i q^{53} -1940.00 q^{54} +10624.0 q^{56} +1424.00i q^{57} -8256.00i q^{58} +34989.0 q^{59} -45940.0 q^{61} -24980.0i q^{62} -40172.0i q^{63} -4096.00 q^{64} +484.000 q^{66} +25343.0i q^{67} +11808.0i q^{68} +1779.00 q^{69} +13311.0 q^{71} +15488.0i q^{72} +53260.0i q^{73} -59140.0 q^{74} +22784.0 q^{76} +20086.0i q^{77} +2768.00i q^{78} -77234.0 q^{79} +58321.0 q^{81} -21216.0i q^{82} -55014.0i q^{83} +2656.00 q^{84} -71192.0 q^{86} -2064.00i q^{87} -7744.00i q^{88} -125415. q^{89} -114872. q^{91} -28464.0i q^{92} -6245.00i q^{93} -68736.0 q^{94} -1024.00 q^{96} -88807.0i q^{97} +42996.0i q^{98} -29282.0 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 32 q^{4} - 8 q^{6} + 484 q^{9}+O(q^{10})$$ 2 * q - 32 * q^4 - 8 * q^6 + 484 * q^9 $$2 q - 32 q^{4} - 8 q^{6} + 484 q^{9} - 242 q^{11} - 1328 q^{14} + 512 q^{16} - 2848 q^{19} - 332 q^{21} + 128 q^{24} - 5536 q^{26} + 4128 q^{29} + 12490 q^{31} - 5904 q^{34} - 7744 q^{36} - 1384 q^{39} + 10608 q^{41} + 3872 q^{44} + 14232 q^{46} - 21498 q^{49} - 1476 q^{51} - 3880 q^{54} + 21248 q^{56} + 69978 q^{59} - 91880 q^{61} - 8192 q^{64} + 968 q^{66} + 3558 q^{69} + 26622 q^{71} - 118280 q^{74} + 45568 q^{76} - 154468 q^{79} + 116642 q^{81} + 5312 q^{84} - 142384 q^{86} - 250830 q^{89} - 229744 q^{91} - 137472 q^{94} - 2048 q^{96} - 58564 q^{99}+O(q^{100})$$ 2 * q - 32 * q^4 - 8 * q^6 + 484 * q^9 - 242 * q^11 - 1328 * q^14 + 512 * q^16 - 2848 * q^19 - 332 * q^21 + 128 * q^24 - 5536 * q^26 + 4128 * q^29 + 12490 * q^31 - 5904 * q^34 - 7744 * q^36 - 1384 * q^39 + 10608 * q^41 + 3872 * q^44 + 14232 * q^46 - 21498 * q^49 - 1476 * q^51 - 3880 * q^54 + 21248 * q^56 + 69978 * q^59 - 91880 * q^61 - 8192 * q^64 + 968 * q^66 + 3558 * q^69 + 26622 * q^71 - 118280 * q^74 + 45568 * q^76 - 154468 * q^79 + 116642 * q^81 + 5312 * q^84 - 142384 * q^86 - 250830 * q^89 - 229744 * q^91 - 137472 * q^94 - 2048 * q^96 - 58564 * q^99

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/550\mathbb{Z}\right)^\times$$.

 $$n$$ $$101$$ $$177$$ $$\chi(n)$$ $$1$$ $$-1$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ − 4.00000i − 0.707107i
$$3$$ − 1.00000i − 0.0641500i −0.999485 0.0320750i $$-0.989788\pi$$
0.999485 0.0320750i $$-0.0102115\pi$$
$$4$$ −16.0000 −0.500000
$$5$$ 0 0
$$6$$ −4.00000 −0.0453609
$$7$$ − 166.000i − 1.28045i −0.768187 0.640226i $$-0.778841\pi$$
0.768187 0.640226i $$-0.221159\pi$$
$$8$$ 64.0000i 0.353553i
$$9$$ 242.000 0.995885
$$10$$ 0 0
$$11$$ −121.000 −0.301511
$$12$$ 16.0000i 0.0320750i
$$13$$ − 692.000i − 1.13566i −0.823146 0.567829i $$-0.807783\pi$$
0.823146 0.567829i $$-0.192217\pi$$
$$14$$ −664.000 −0.905416
$$15$$ 0 0
$$16$$ 256.000 0.250000
$$17$$ − 738.000i − 0.619347i −0.950843 0.309674i $$-0.899780\pi$$
0.950843 0.309674i $$-0.100220\pi$$
$$18$$ − 968.000i − 0.704197i
$$19$$ −1424.00 −0.904953 −0.452476 0.891776i $$-0.649459\pi$$
−0.452476 + 0.891776i $$0.649459\pi$$
$$20$$ 0 0
$$21$$ −166.000 −0.0821410
$$22$$ 484.000i 0.213201i
$$23$$ 1779.00i 0.701223i 0.936521 + 0.350612i $$0.114026\pi$$
−0.936521 + 0.350612i $$0.885974\pi$$
$$24$$ 64.0000 0.0226805
$$25$$ 0 0
$$26$$ −2768.00 −0.803032
$$27$$ − 485.000i − 0.128036i
$$28$$ 2656.00i 0.640226i
$$29$$ 2064.00 0.455737 0.227869 0.973692i $$-0.426824\pi$$
0.227869 + 0.973692i $$0.426824\pi$$
$$30$$ 0 0
$$31$$ 6245.00 1.16715 0.583577 0.812058i $$-0.301653\pi$$
0.583577 + 0.812058i $$0.301653\pi$$
$$32$$ − 1024.00i − 0.176777i
$$33$$ 121.000i 0.0193420i
$$34$$ −2952.00 −0.437944
$$35$$ 0 0
$$36$$ −3872.00 −0.497942
$$37$$ − 14785.0i − 1.77549i −0.460340 0.887743i $$-0.652273\pi$$
0.460340 0.887743i $$-0.347727\pi$$
$$38$$ 5696.00i 0.639898i
$$39$$ −692.000 −0.0728525
$$40$$ 0 0
$$41$$ 5304.00 0.492770 0.246385 0.969172i $$-0.420757\pi$$
0.246385 + 0.969172i $$0.420757\pi$$
$$42$$ 664.000i 0.0580824i
$$43$$ − 17798.0i − 1.46791i −0.679197 0.733956i $$-0.737672\pi$$
0.679197 0.733956i $$-0.262328\pi$$
$$44$$ 1936.00 0.150756
$$45$$ 0 0
$$46$$ 7116.00 0.495840
$$47$$ − 17184.0i − 1.13470i −0.823478 0.567348i $$-0.807969\pi$$
0.823478 0.567348i $$-0.192031\pi$$
$$48$$ − 256.000i − 0.0160375i
$$49$$ −10749.0 −0.639555
$$50$$ 0 0
$$51$$ −738.000 −0.0397311
$$52$$ 11072.0i 0.567829i
$$53$$ 30726.0i 1.50251i 0.660014 + 0.751253i $$0.270550\pi$$
−0.660014 + 0.751253i $$0.729450\pi$$
$$54$$ −1940.00 −0.0905352
$$55$$ 0 0
$$56$$ 10624.0 0.452708
$$57$$ 1424.00i 0.0580528i
$$58$$ − 8256.00i − 0.322255i
$$59$$ 34989.0 1.30858 0.654292 0.756242i $$-0.272967\pi$$
0.654292 + 0.756242i $$0.272967\pi$$
$$60$$ 0 0
$$61$$ −45940.0 −1.58076 −0.790381 0.612616i $$-0.790117\pi$$
−0.790381 + 0.612616i $$0.790117\pi$$
$$62$$ − 24980.0i − 0.825303i
$$63$$ − 40172.0i − 1.27518i
$$64$$ −4096.00 −0.125000
$$65$$ 0 0
$$66$$ 484.000 0.0136768
$$67$$ 25343.0i 0.689717i 0.938655 + 0.344859i $$0.112073\pi$$
−0.938655 + 0.344859i $$0.887927\pi$$
$$68$$ 11808.0i 0.309674i
$$69$$ 1779.00 0.0449835
$$70$$ 0 0
$$71$$ 13311.0 0.313375 0.156688 0.987648i $$-0.449918\pi$$
0.156688 + 0.987648i $$0.449918\pi$$
$$72$$ 15488.0i 0.352098i
$$73$$ 53260.0i 1.16975i 0.811123 + 0.584876i $$0.198857\pi$$
−0.811123 + 0.584876i $$0.801143\pi$$
$$74$$ −59140.0 −1.25546
$$75$$ 0 0
$$76$$ 22784.0 0.452476
$$77$$ 20086.0i 0.386071i
$$78$$ 2768.00i 0.0515145i
$$79$$ −77234.0 −1.39233 −0.696163 0.717884i $$-0.745111\pi$$
−0.696163 + 0.717884i $$0.745111\pi$$
$$80$$ 0 0
$$81$$ 58321.0 0.987671
$$82$$ − 21216.0i − 0.348441i
$$83$$ − 55014.0i − 0.876553i −0.898840 0.438276i $$-0.855589\pi$$
0.898840 0.438276i $$-0.144411\pi$$
$$84$$ 2656.00 0.0410705
$$85$$ 0 0
$$86$$ −71192.0 −1.03797
$$87$$ − 2064.00i − 0.0292356i
$$88$$ − 7744.00i − 0.106600i
$$89$$ −125415. −1.67832 −0.839159 0.543886i $$-0.816953\pi$$
−0.839159 + 0.543886i $$0.816953\pi$$
$$90$$ 0 0
$$91$$ −114872. −1.45416
$$92$$ − 28464.0i − 0.350612i
$$93$$ − 6245.00i − 0.0748730i
$$94$$ −68736.0 −0.802351
$$95$$ 0 0
$$96$$ −1024.00 −0.0113402
$$97$$ − 88807.0i − 0.958336i −0.877723 0.479168i $$-0.840938\pi$$
0.877723 0.479168i $$-0.159062\pi$$
$$98$$ 42996.0i 0.452234i
$$99$$ −29282.0 −0.300271
$$100$$ 0 0
$$101$$ 1482.00 0.0144559 0.00722794 0.999974i $$-0.497699\pi$$
0.00722794 + 0.999974i $$0.497699\pi$$
$$102$$ 2952.00i 0.0280942i
$$103$$ 117496.i 1.09126i 0.838025 + 0.545632i $$0.183710\pi$$
−0.838025 + 0.545632i $$0.816290\pi$$
$$104$$ 44288.0 0.401516
$$105$$ 0 0
$$106$$ 122904. 1.06243
$$107$$ − 79362.0i − 0.670121i −0.942197 0.335060i $$-0.891243\pi$$
0.942197 0.335060i $$-0.108757\pi$$
$$108$$ 7760.00i 0.0640180i
$$109$$ −87842.0 −0.708167 −0.354084 0.935214i $$-0.615207\pi$$
−0.354084 + 0.935214i $$0.615207\pi$$
$$110$$ 0 0
$$111$$ −14785.0 −0.113897
$$112$$ − 42496.0i − 0.320113i
$$113$$ 47247.0i 0.348079i 0.984739 + 0.174040i $$0.0556821\pi$$
−0.984739 + 0.174040i $$0.944318\pi$$
$$114$$ 5696.00 0.0410495
$$115$$ 0 0
$$116$$ −33024.0 −0.227869
$$117$$ − 167464.i − 1.13098i
$$118$$ − 139956.i − 0.925308i
$$119$$ −122508. −0.793044
$$120$$ 0 0
$$121$$ 14641.0 0.0909091
$$122$$ 183760.i 1.11777i
$$123$$ − 5304.00i − 0.0316112i
$$124$$ −99920.0 −0.583577
$$125$$ 0 0
$$126$$ −160688. −0.901690
$$127$$ − 239416.i − 1.31718i −0.752504 0.658588i $$-0.771154\pi$$
0.752504 0.658588i $$-0.228846\pi$$
$$128$$ 16384.0i 0.0883883i
$$129$$ −17798.0 −0.0941666
$$130$$ 0 0
$$131$$ −98142.0 −0.499662 −0.249831 0.968289i $$-0.580375\pi$$
−0.249831 + 0.968289i $$0.580375\pi$$
$$132$$ − 1936.00i − 0.00967098i
$$133$$ 236384.i 1.15875i
$$134$$ 101372. 0.487704
$$135$$ 0 0
$$136$$ 47232.0 0.218972
$$137$$ 400137.i 1.82141i 0.413059 + 0.910704i $$0.364460\pi$$
−0.413059 + 0.910704i $$0.635540\pi$$
$$138$$ − 7116.00i − 0.0318081i
$$139$$ −205766. −0.903310 −0.451655 0.892193i $$-0.649166\pi$$
−0.451655 + 0.892193i $$0.649166\pi$$
$$140$$ 0 0
$$141$$ −17184.0 −0.0727908
$$142$$ − 53244.0i − 0.221590i
$$143$$ 83732.0i 0.342414i
$$144$$ 61952.0 0.248971
$$145$$ 0 0
$$146$$ 213040. 0.827140
$$147$$ 10749.0i 0.0410275i
$$148$$ 236560.i 0.887743i
$$149$$ −87726.0 −0.323715 −0.161857 0.986814i $$-0.551748\pi$$
−0.161857 + 0.986814i $$0.551748\pi$$
$$150$$ 0 0
$$151$$ −432778. −1.54462 −0.772312 0.635243i $$-0.780900\pi$$
−0.772312 + 0.635243i $$0.780900\pi$$
$$152$$ − 91136.0i − 0.319949i
$$153$$ − 178596.i − 0.616798i
$$154$$ 80344.0 0.272993
$$155$$ 0 0
$$156$$ 11072.0 0.0364263
$$157$$ − 34075.0i − 0.110328i −0.998477 0.0551641i $$-0.982432\pi$$
0.998477 0.0551641i $$-0.0175682\pi$$
$$158$$ 308936.i 0.984523i
$$159$$ 30726.0 0.0963858
$$160$$ 0 0
$$161$$ 295314. 0.897882
$$162$$ − 233284.i − 0.698389i
$$163$$ − 45020.0i − 0.132720i −0.997796 0.0663600i $$-0.978861\pi$$
0.997796 0.0663600i $$-0.0211386\pi$$
$$164$$ −84864.0 −0.246385
$$165$$ 0 0
$$166$$ −220056. −0.619816
$$167$$ 482556.i 1.33893i 0.742845 + 0.669463i $$0.233476\pi$$
−0.742845 + 0.669463i $$0.766524\pi$$
$$168$$ − 10624.0i − 0.0290412i
$$169$$ −107571. −0.289720
$$170$$ 0 0
$$171$$ −344608. −0.901229
$$172$$ 284768.i 0.733956i
$$173$$ 766254.i 1.94651i 0.229719 + 0.973257i $$0.426219\pi$$
−0.229719 + 0.973257i $$0.573781\pi$$
$$174$$ −8256.00 −0.0206727
$$175$$ 0 0
$$176$$ −30976.0 −0.0753778
$$177$$ − 34989.0i − 0.0839457i
$$178$$ 501660.i 1.18675i
$$179$$ −303399. −0.707753 −0.353876 0.935292i $$-0.615137\pi$$
−0.353876 + 0.935292i $$0.615137\pi$$
$$180$$ 0 0
$$181$$ −285181. −0.647030 −0.323515 0.946223i $$-0.604865\pi$$
−0.323515 + 0.946223i $$0.604865\pi$$
$$182$$ 459488.i 1.02824i
$$183$$ 45940.0i 0.101406i
$$184$$ −113856. −0.247920
$$185$$ 0 0
$$186$$ −24980.0 −0.0529432
$$187$$ 89298.0i 0.186740i
$$188$$ 274944.i 0.567348i
$$189$$ −80510.0 −0.163944
$$190$$ 0 0
$$191$$ 767067. 1.52142 0.760711 0.649090i $$-0.224850\pi$$
0.760711 + 0.649090i $$0.224850\pi$$
$$192$$ 4096.00i 0.00801875i
$$193$$ − 411668.i − 0.795525i −0.917488 0.397763i $$-0.869787\pi$$
0.917488 0.397763i $$-0.130213\pi$$
$$194$$ −355228. −0.677646
$$195$$ 0 0
$$196$$ 171984. 0.319777
$$197$$ − 759258.i − 1.39387i −0.717132 0.696937i $$-0.754545\pi$$
0.717132 0.696937i $$-0.245455\pi$$
$$198$$ 117128.i 0.212323i
$$199$$ 46600.0 0.0834167 0.0417084 0.999130i $$-0.486720\pi$$
0.0417084 + 0.999130i $$0.486720\pi$$
$$200$$ 0 0
$$201$$ 25343.0 0.0442454
$$202$$ − 5928.00i − 0.0102219i
$$203$$ − 342624.i − 0.583549i
$$204$$ 11808.0 0.0198656
$$205$$ 0 0
$$206$$ 469984. 0.771641
$$207$$ 430518.i 0.698338i
$$208$$ − 177152.i − 0.283915i
$$209$$ 172304. 0.272854
$$210$$ 0 0
$$211$$ −932428. −1.44181 −0.720907 0.693032i $$-0.756274\pi$$
−0.720907 + 0.693032i $$0.756274\pi$$
$$212$$ − 491616.i − 0.751253i
$$213$$ − 13311.0i − 0.0201030i
$$214$$ −317448. −0.473847
$$215$$ 0 0
$$216$$ 31040.0 0.0452676
$$217$$ − 1.03667e6i − 1.49448i
$$218$$ 351368.i 0.500750i
$$219$$ 53260.0 0.0750397
$$220$$ 0 0
$$221$$ −510696. −0.703367
$$222$$ 59140.0i 0.0805376i
$$223$$ − 169745.i − 0.228578i −0.993448 0.114289i $$-0.963541\pi$$
0.993448 0.114289i $$-0.0364590\pi$$
$$224$$ −169984. −0.226354
$$225$$ 0 0
$$226$$ 188988. 0.246129
$$227$$ 198078.i 0.255136i 0.991830 + 0.127568i $$0.0407171\pi$$
−0.991830 + 0.127568i $$0.959283\pi$$
$$228$$ − 22784.0i − 0.0290264i
$$229$$ 849997. 1.07110 0.535548 0.844505i $$-0.320105\pi$$
0.535548 + 0.844505i $$0.320105\pi$$
$$230$$ 0 0
$$231$$ 20086.0 0.0247664
$$232$$ 132096.i 0.161128i
$$233$$ 401832.i 0.484903i 0.970164 + 0.242451i $$0.0779515\pi$$
−0.970164 + 0.242451i $$0.922048\pi$$
$$234$$ −669856. −0.799727
$$235$$ 0 0
$$236$$ −559824. −0.654292
$$237$$ 77234.0i 0.0893177i
$$238$$ 490032.i 0.560766i
$$239$$ −855174. −0.968411 −0.484206 0.874954i $$-0.660891\pi$$
−0.484206 + 0.874954i $$0.660891\pi$$
$$240$$ 0 0
$$241$$ 1.12546e6 1.24821 0.624107 0.781339i $$-0.285463\pi$$
0.624107 + 0.781339i $$0.285463\pi$$
$$242$$ − 58564.0i − 0.0642824i
$$243$$ − 176176.i − 0.191395i
$$244$$ 735040. 0.790381
$$245$$ 0 0
$$246$$ −21216.0 −0.0223525
$$247$$ 985408.i 1.02772i
$$248$$ 399680.i 0.412651i
$$249$$ −55014.0 −0.0562309
$$250$$ 0 0
$$251$$ −1.19751e6 −1.19976 −0.599882 0.800088i $$-0.704786\pi$$
−0.599882 + 0.800088i $$0.704786\pi$$
$$252$$ 642752.i 0.637591i
$$253$$ − 215259.i − 0.211427i
$$254$$ −957664. −0.931384
$$255$$ 0 0
$$256$$ 65536.0 0.0625000
$$257$$ 37758.0i 0.0356596i 0.999841 + 0.0178298i $$0.00567570\pi$$
−0.999841 + 0.0178298i $$0.994324\pi$$
$$258$$ 71192.0i 0.0665858i
$$259$$ −2.45431e6 −2.27342
$$260$$ 0 0
$$261$$ 499488. 0.453862
$$262$$ 392568.i 0.353315i
$$263$$ 631254.i 0.562749i 0.959598 + 0.281375i $$0.0907903\pi$$
−0.959598 + 0.281375i $$0.909210\pi$$
$$264$$ −7744.00 −0.00683842
$$265$$ 0 0
$$266$$ 945536. 0.819359
$$267$$ 125415.i 0.107664i
$$268$$ − 405488.i − 0.344859i
$$269$$ 1.08034e6 0.910292 0.455146 0.890417i $$-0.349587\pi$$
0.455146 + 0.890417i $$0.349587\pi$$
$$270$$ 0 0
$$271$$ −816100. −0.675025 −0.337513 0.941321i $$-0.609586\pi$$
−0.337513 + 0.941321i $$0.609586\pi$$
$$272$$ − 188928.i − 0.154837i
$$273$$ 114872.i 0.0932841i
$$274$$ 1.60055e6 1.28793
$$275$$ 0 0
$$276$$ −28464.0 −0.0224917
$$277$$ 1.68820e6i 1.32198i 0.750396 + 0.660989i $$0.229863\pi$$
−0.750396 + 0.660989i $$0.770137\pi$$
$$278$$ 823064.i 0.638736i
$$279$$ 1.51129e6 1.16235
$$280$$ 0 0
$$281$$ −879042. −0.664116 −0.332058 0.943259i $$-0.607743\pi$$
−0.332058 + 0.943259i $$0.607743\pi$$
$$282$$ 68736.0i 0.0514709i
$$283$$ − 1.54027e6i − 1.14322i −0.820525 0.571611i $$-0.806319\pi$$
0.820525 0.571611i $$-0.193681\pi$$
$$284$$ −212976. −0.156688
$$285$$ 0 0
$$286$$ 334928. 0.242123
$$287$$ − 880464.i − 0.630967i
$$288$$ − 247808.i − 0.176049i
$$289$$ 875213. 0.616409
$$290$$ 0 0
$$291$$ −88807.0 −0.0614773
$$292$$ − 852160.i − 0.584876i
$$293$$ − 720840.i − 0.490535i −0.969455 0.245267i $$-0.921124\pi$$
0.969455 0.245267i $$-0.0788758\pi$$
$$294$$ 42996.0 0.0290108
$$295$$ 0 0
$$296$$ 946240. 0.627729
$$297$$ 58685.0i 0.0386043i
$$298$$ 350904.i 0.228901i
$$299$$ 1.23107e6 0.796350
$$300$$ 0 0
$$301$$ −2.95447e6 −1.87959
$$302$$ 1.73111e6i 1.09221i
$$303$$ − 1482.00i 0 0.000927346i
$$304$$ −364544. −0.226238
$$305$$ 0 0
$$306$$ −714384. −0.436142
$$307$$ − 1.03905e6i − 0.629201i −0.949224 0.314601i $$-0.898129\pi$$
0.949224 0.314601i $$-0.101871\pi$$
$$308$$ − 321376.i − 0.193035i
$$309$$ 117496. 0.0700046
$$310$$ 0 0
$$311$$ −1.25135e6 −0.733630 −0.366815 0.930294i $$-0.619552\pi$$
−0.366815 + 0.930294i $$0.619552\pi$$
$$312$$ − 44288.0i − 0.0257573i
$$313$$ 1.44336e6i 0.832749i 0.909193 + 0.416375i $$0.136699\pi$$
−0.909193 + 0.416375i $$0.863301\pi$$
$$314$$ −136300. −0.0780139
$$315$$ 0 0
$$316$$ 1.23574e6 0.696163
$$317$$ − 2.01208e6i − 1.12460i −0.826934 0.562298i $$-0.809917\pi$$
0.826934 0.562298i $$-0.190083\pi$$
$$318$$ − 122904.i − 0.0681551i
$$319$$ −249744. −0.137410
$$320$$ 0 0
$$321$$ −79362.0 −0.0429883
$$322$$ − 1.18126e6i − 0.634899i
$$323$$ 1.05091e6i 0.560480i
$$324$$ −933136. −0.493836
$$325$$ 0 0
$$326$$ −180080. −0.0938472
$$327$$ 87842.0i 0.0454290i
$$328$$ 339456.i 0.174220i
$$329$$ −2.85254e6 −1.45292
$$330$$ 0 0
$$331$$ 2.01734e6 1.01207 0.506033 0.862514i $$-0.331112\pi$$
0.506033 + 0.862514i $$0.331112\pi$$
$$332$$ 880224.i 0.438276i
$$333$$ − 3.57797e6i − 1.76818i
$$334$$ 1.93022e6 0.946764
$$335$$ 0 0
$$336$$ −42496.0 −0.0205352
$$337$$ 264122.i 0.126686i 0.997992 + 0.0633432i $$0.0201763\pi$$
−0.997992 + 0.0633432i $$0.979824\pi$$
$$338$$ 430284.i 0.204863i
$$339$$ 47247.0 0.0223293
$$340$$ 0 0
$$341$$ −755645. −0.351910
$$342$$ 1.37843e6i 0.637265i
$$343$$ − 1.00563e6i − 0.461532i
$$344$$ 1.13907e6 0.518985
$$345$$ 0 0
$$346$$ 3.06502e6 1.37639
$$347$$ − 1.71049e6i − 0.762601i −0.924451 0.381300i $$-0.875476\pi$$
0.924451 0.381300i $$-0.124524\pi$$
$$348$$ 33024.0i 0.0146178i
$$349$$ −218822. −0.0961673 −0.0480836 0.998843i $$-0.515311\pi$$
−0.0480836 + 0.998843i $$0.515311\pi$$
$$350$$ 0 0
$$351$$ −335620. −0.145405
$$352$$ 123904.i 0.0533002i
$$353$$ − 3.68192e6i − 1.57267i −0.617802 0.786334i $$-0.711977\pi$$
0.617802 0.786334i $$-0.288023\pi$$
$$354$$ −139956. −0.0593586
$$355$$ 0 0
$$356$$ 2.00664e6 0.839159
$$357$$ 122508.i 0.0508738i
$$358$$ 1.21360e6i 0.500457i
$$359$$ −1.88528e6 −0.772042 −0.386021 0.922490i $$-0.626151\pi$$
−0.386021 + 0.922490i $$0.626151\pi$$
$$360$$ 0 0
$$361$$ −448323. −0.181060
$$362$$ 1.14072e6i 0.457519i
$$363$$ − 14641.0i − 0.00583182i
$$364$$ 1.83795e6 0.727078
$$365$$ 0 0
$$366$$ 183760. 0.0717048
$$367$$ − 3.11666e6i − 1.20788i −0.797029 0.603940i $$-0.793596\pi$$
0.797029 0.603940i $$-0.206404\pi$$
$$368$$ 455424.i 0.175306i
$$369$$ 1.28357e6 0.490742
$$370$$ 0 0
$$371$$ 5.10052e6 1.92389
$$372$$ 99920.0i 0.0374365i
$$373$$ − 1.39441e6i − 0.518943i −0.965751 0.259471i $$-0.916452\pi$$
0.965751 0.259471i $$-0.0835484\pi$$
$$374$$ 357192. 0.132045
$$375$$ 0 0
$$376$$ 1.09978e6 0.401176
$$377$$ − 1.42829e6i − 0.517562i
$$378$$ 322040.i 0.115926i
$$379$$ 4.26036e6 1.52352 0.761759 0.647860i $$-0.224336\pi$$
0.761759 + 0.647860i $$0.224336\pi$$
$$380$$ 0 0
$$381$$ −239416. −0.0844969
$$382$$ − 3.06827e6i − 1.07581i
$$383$$ − 201765.i − 0.0702828i −0.999382 0.0351414i $$-0.988812\pi$$
0.999382 0.0351414i $$-0.0111882\pi$$
$$384$$ 16384.0 0.00567012
$$385$$ 0 0
$$386$$ −1.64667e6 −0.562521
$$387$$ − 4.30712e6i − 1.46187i
$$388$$ 1.42091e6i 0.479168i
$$389$$ −1.94882e6 −0.652977 −0.326489 0.945201i $$-0.605865\pi$$
−0.326489 + 0.945201i $$0.605865\pi$$
$$390$$ 0 0
$$391$$ 1.31290e6 0.434301
$$392$$ − 687936.i − 0.226117i
$$393$$ 98142.0i 0.0320534i
$$394$$ −3.03703e6 −0.985618
$$395$$ 0 0
$$396$$ 468512. 0.150135
$$397$$ − 1.46826e6i − 0.467548i −0.972291 0.233774i $$-0.924892\pi$$
0.972291 0.233774i $$-0.0751076\pi$$
$$398$$ − 186400.i − 0.0589845i
$$399$$ 236384. 0.0743337
$$400$$ 0 0
$$401$$ 2.24618e6 0.697563 0.348781 0.937204i $$-0.386596\pi$$
0.348781 + 0.937204i $$0.386596\pi$$
$$402$$ − 101372.i − 0.0312862i
$$403$$ − 4.32154e6i − 1.32549i
$$404$$ −23712.0 −0.00722794
$$405$$ 0 0
$$406$$ −1.37050e6 −0.412632
$$407$$ 1.78898e6i 0.535329i
$$408$$ − 47232.0i − 0.0140471i
$$409$$ 3.61488e6 1.06853 0.534263 0.845318i $$-0.320589\pi$$
0.534263 + 0.845318i $$0.320589\pi$$
$$410$$ 0 0
$$411$$ 400137. 0.116843
$$412$$ − 1.87994e6i − 0.545632i
$$413$$ − 5.80817e6i − 1.67558i
$$414$$ 1.72207e6 0.493799
$$415$$ 0 0
$$416$$ −708608. −0.200758
$$417$$ 205766.i 0.0579473i
$$418$$ − 689216.i − 0.192937i
$$419$$ 3.81239e6 1.06087 0.530435 0.847726i $$-0.322029\pi$$
0.530435 + 0.847726i $$0.322029\pi$$
$$420$$ 0 0
$$421$$ 1.97346e6 0.542655 0.271327 0.962487i $$-0.412537\pi$$
0.271327 + 0.962487i $$0.412537\pi$$
$$422$$ 3.72971e6i 1.01952i
$$423$$ − 4.15853e6i − 1.13003i
$$424$$ −1.96646e6 −0.531216
$$425$$ 0 0
$$426$$ −53244.0 −0.0142150
$$427$$ 7.62604e6i 2.02409i
$$428$$ 1.26979e6i 0.335060i
$$429$$ 83732.0 0.0219659
$$430$$ 0 0
$$431$$ −2.08359e6 −0.540280 −0.270140 0.962821i $$-0.587070\pi$$
−0.270140 + 0.962821i $$0.587070\pi$$
$$432$$ − 124160.i − 0.0320090i
$$433$$ 72691.0i 0.0186321i 0.999957 + 0.00931603i $$0.00296543\pi$$
−0.999957 + 0.00931603i $$0.997035\pi$$
$$434$$ −4.14668e6 −1.05676
$$435$$ 0 0
$$436$$ 1.40547e6 0.354084
$$437$$ − 2.53330e6i − 0.634574i
$$438$$ − 213040.i − 0.0530611i
$$439$$ −594392. −0.147201 −0.0736007 0.997288i $$-0.523449\pi$$
−0.0736007 + 0.997288i $$0.523449\pi$$
$$440$$ 0 0
$$441$$ −2.60126e6 −0.636923
$$442$$ 2.04278e6i 0.497355i
$$443$$ − 4.56651e6i − 1.10554i −0.833334 0.552770i $$-0.813571\pi$$
0.833334 0.552770i $$-0.186429\pi$$
$$444$$ 236560. 0.0569487
$$445$$ 0 0
$$446$$ −678980. −0.161629
$$447$$ 87726.0i 0.0207663i
$$448$$ 679936.i 0.160056i
$$449$$ 5.44382e6 1.27435 0.637174 0.770720i $$-0.280103\pi$$
0.637174 + 0.770720i $$0.280103\pi$$
$$450$$ 0 0
$$451$$ −641784. −0.148576
$$452$$ − 755952.i − 0.174040i
$$453$$ 432778.i 0.0990877i
$$454$$ 792312. 0.180408
$$455$$ 0 0
$$456$$ −91136.0 −0.0205247
$$457$$ 6.70312e6i 1.50137i 0.660662 + 0.750683i $$0.270276\pi$$
−0.660662 + 0.750683i $$0.729724\pi$$
$$458$$ − 3.39999e6i − 0.757380i
$$459$$ −357930. −0.0792988
$$460$$ 0 0
$$461$$ −1.25994e6 −0.276120 −0.138060 0.990424i $$-0.544087\pi$$
−0.138060 + 0.990424i $$0.544087\pi$$
$$462$$ − 80344.0i − 0.0175125i
$$463$$ 5.02308e6i 1.08897i 0.838769 + 0.544487i $$0.183276\pi$$
−0.838769 + 0.544487i $$0.816724\pi$$
$$464$$ 528384. 0.113934
$$465$$ 0 0
$$466$$ 1.60733e6 0.342878
$$467$$ − 2.35660e6i − 0.500028i −0.968242 0.250014i $$-0.919565\pi$$
0.968242 0.250014i $$-0.0804353\pi$$
$$468$$ 2.67942e6i 0.565492i
$$469$$ 4.20694e6 0.883149
$$470$$ 0 0
$$471$$ −34075.0 −0.00707756
$$472$$ 2.23930e6i 0.462654i
$$473$$ 2.15356e6i 0.442592i
$$474$$ 308936. 0.0631572
$$475$$ 0 0
$$476$$ 1.96013e6 0.396522
$$477$$ 7.43569e6i 1.49632i
$$478$$ 3.42070e6i 0.684770i
$$479$$ 6.72258e6 1.33874 0.669371 0.742928i $$-0.266563\pi$$
0.669371 + 0.742928i $$0.266563\pi$$
$$480$$ 0 0
$$481$$ −1.02312e7 −2.01634
$$482$$ − 4.50186e6i − 0.882620i
$$483$$ − 295314.i − 0.0575992i
$$484$$ −234256. −0.0454545
$$485$$ 0 0
$$486$$ −704704. −0.135337
$$487$$ 1.96001e6i 0.374487i 0.982314 + 0.187243i $$0.0599553\pi$$
−0.982314 + 0.187243i $$0.940045\pi$$
$$488$$ − 2.94016e6i − 0.558884i
$$489$$ −45020.0 −0.00851399
$$490$$ 0 0
$$491$$ −579624. −0.108503 −0.0542516 0.998527i $$-0.517277\pi$$
−0.0542516 + 0.998527i $$0.517277\pi$$
$$492$$ 84864.0i 0.0158056i
$$493$$ − 1.52323e6i − 0.282260i
$$494$$ 3.94163e6 0.726706
$$495$$ 0 0
$$496$$ 1.59872e6 0.291789
$$497$$ − 2.20963e6i − 0.401262i
$$498$$ 220056.i 0.0397612i
$$499$$ −1.36905e6 −0.246132 −0.123066 0.992398i $$-0.539273\pi$$
−0.123066 + 0.992398i $$0.539273\pi$$
$$500$$ 0 0
$$501$$ 482556. 0.0858921
$$502$$ 4.79005e6i 0.848361i
$$503$$ − 1.83343e6i − 0.323105i −0.986864 0.161552i $$-0.948350\pi$$
0.986864 0.161552i $$-0.0516501\pi$$
$$504$$ 2.57101e6 0.450845
$$505$$ 0 0
$$506$$ −861036. −0.149501
$$507$$ 107571.i 0.0185855i
$$508$$ 3.83066e6i 0.658588i
$$509$$ 1.71266e6 0.293006 0.146503 0.989210i $$-0.453198\pi$$
0.146503 + 0.989210i $$0.453198\pi$$
$$510$$ 0 0
$$511$$ 8.84116e6 1.49781
$$512$$ − 262144.i − 0.0441942i
$$513$$ 690640.i 0.115867i
$$514$$ 151032. 0.0252151
$$515$$ 0 0
$$516$$ 284768. 0.0470833
$$517$$ 2.07926e6i 0.342124i
$$518$$ 9.81724e6i 1.60755i
$$519$$ 766254. 0.124869
$$520$$ 0 0
$$521$$ −789435. −0.127415 −0.0637077 0.997969i $$-0.520293\pi$$
−0.0637077 + 0.997969i $$0.520293\pi$$
$$522$$ − 1.99795e6i − 0.320929i
$$523$$ − 627392.i − 0.100296i −0.998742 0.0501481i $$-0.984031\pi$$
0.998742 0.0501481i $$-0.0159693\pi$$
$$524$$ 1.57027e6 0.249831
$$525$$ 0 0
$$526$$ 2.52502e6 0.397924
$$527$$ − 4.60881e6i − 0.722873i
$$528$$ 30976.0i 0.00483549i
$$529$$ 3.27150e6 0.508286
$$530$$ 0 0
$$531$$ 8.46734e6 1.30320
$$532$$ − 3.78214e6i − 0.579374i
$$533$$ − 3.67037e6i − 0.559618i
$$534$$ 501660. 0.0761301
$$535$$ 0 0
$$536$$ −1.62195e6 −0.243852
$$537$$ 303399.i 0.0454024i
$$538$$ − 4.32137e6i − 0.643673i
$$539$$ 1.30063e6 0.192833
$$540$$ 0 0
$$541$$ 3.20895e6 0.471379 0.235689 0.971828i $$-0.424265\pi$$
0.235689 + 0.971828i $$0.424265\pi$$
$$542$$ 3.26440e6i 0.477315i
$$543$$ 285181.i 0.0415070i
$$544$$ −755712. −0.109486
$$545$$ 0 0
$$546$$ 459488. 0.0659618
$$547$$ 3.42658e6i 0.489658i 0.969566 + 0.244829i $$0.0787319\pi$$
−0.969566 + 0.244829i $$0.921268\pi$$
$$548$$ − 6.40219e6i − 0.910704i
$$549$$ −1.11175e7 −1.57426
$$550$$ 0 0
$$551$$ −2.93914e6 −0.412421
$$552$$ 113856.i 0.0159041i
$$553$$ 1.28208e7i 1.78280i
$$554$$ 6.75279e6 0.934779
$$555$$ 0 0
$$556$$ 3.29226e6 0.451655
$$557$$ 1.05198e7i 1.43672i 0.695674 + 0.718358i $$0.255106\pi$$
−0.695674 + 0.718358i $$0.744894\pi$$
$$558$$ − 6.04516e6i − 0.821906i
$$559$$ −1.23162e7 −1.66705
$$560$$ 0 0
$$561$$ 89298.0 0.0119794
$$562$$ 3.51617e6i 0.469601i
$$563$$ − 5.47288e6i − 0.727687i −0.931460 0.363844i $$-0.881464\pi$$
0.931460 0.363844i $$-0.118536\pi$$
$$564$$ 274944. 0.0363954
$$565$$ 0 0
$$566$$ −6.16107e6 −0.808379
$$567$$ − 9.68129e6i − 1.26466i
$$568$$ 851904.i 0.110795i
$$569$$ 1.17787e7 1.52516 0.762580 0.646893i $$-0.223932\pi$$
0.762580 + 0.646893i $$0.223932\pi$$
$$570$$ 0 0
$$571$$ −8.35628e6 −1.07256 −0.536281 0.844039i $$-0.680171\pi$$
−0.536281 + 0.844039i $$0.680171\pi$$
$$572$$ − 1.33971e6i − 0.171207i
$$573$$ − 767067.i − 0.0975993i
$$574$$ −3.52186e6 −0.446161
$$575$$ 0 0
$$576$$ −991232. −0.124486
$$577$$ − 1.37758e7i − 1.72258i −0.508117 0.861288i $$-0.669658\pi$$
0.508117 0.861288i $$-0.330342\pi$$
$$578$$ − 3.50085e6i − 0.435867i
$$579$$ −411668. −0.0510330
$$580$$ 0 0
$$581$$ −9.13232e6 −1.12238
$$582$$ 355228.i 0.0434710i
$$583$$ − 3.71785e6i − 0.453023i
$$584$$ −3.40864e6 −0.413570
$$585$$ 0 0
$$586$$ −2.88336e6 −0.346860
$$587$$ − 1.27093e7i − 1.52239i −0.648522 0.761196i $$-0.724612\pi$$
0.648522 0.761196i $$-0.275388\pi$$
$$588$$ − 171984.i − 0.0205137i
$$589$$ −8.89288e6 −1.05622
$$590$$ 0 0
$$591$$ −759258. −0.0894171
$$592$$ − 3.78496e6i − 0.443871i
$$593$$ − 1.00825e6i − 0.117742i −0.998266 0.0588711i $$-0.981250\pi$$
0.998266 0.0588711i $$-0.0187501\pi$$
$$594$$ 234740. 0.0272974
$$595$$ 0 0
$$596$$ 1.40362e6 0.161857
$$597$$ − 46600.0i − 0.00535119i
$$598$$ − 4.92427e6i − 0.563105i
$$599$$ −1.05100e7 −1.19684 −0.598421 0.801182i $$-0.704205\pi$$
−0.598421 + 0.801182i $$0.704205\pi$$
$$600$$ 0 0
$$601$$ −199390. −0.0225173 −0.0112587 0.999937i $$-0.503584\pi$$
−0.0112587 + 0.999937i $$0.503584\pi$$
$$602$$ 1.18179e7i 1.32907i
$$603$$ 6.13301e6i 0.686879i
$$604$$ 6.92445e6 0.772312
$$605$$ 0 0
$$606$$ −5928.00 −0.000655732 0
$$607$$ 16190.0i 0.00178351i 1.00000 0.000891754i $$0.000283854\pi$$
−1.00000 0.000891754i $$0.999716\pi$$
$$608$$ 1.45818e6i 0.159975i
$$609$$ −342624. −0.0374347
$$610$$ 0 0
$$611$$ −1.18913e7 −1.28863
$$612$$ 2.85754e6i 0.308399i
$$613$$ 1.15253e7i 1.23880i 0.785074 + 0.619402i $$0.212625\pi$$
−0.785074 + 0.619402i $$0.787375\pi$$
$$614$$ −4.15619e6 −0.444913
$$615$$ 0 0
$$616$$ −1.28550e6 −0.136497
$$617$$ 1.69974e7i 1.79750i 0.438459 + 0.898751i $$0.355524\pi$$
−0.438459 + 0.898751i $$0.644476\pi$$
$$618$$ − 469984.i − 0.0495008i
$$619$$ 1.84875e7 1.93933 0.969663 0.244445i $$-0.0786058\pi$$
0.969663 + 0.244445i $$0.0786058\pi$$
$$620$$ 0 0
$$621$$ 862815. 0.0897819
$$622$$ 5.00539e6i 0.518755i
$$623$$ 2.08189e7i 2.14901i
$$624$$ −177152. −0.0182131
$$625$$ 0 0
$$626$$ 5.77344e6 0.588842
$$627$$ − 172304.i − 0.0175036i
$$628$$ 545200.i 0.0551641i
$$629$$ −1.09113e7 −1.09964
$$630$$ 0 0
$$631$$ −4.54281e6 −0.454204 −0.227102 0.973871i $$-0.572925\pi$$
−0.227102 + 0.973871i $$0.572925\pi$$
$$632$$ − 4.94298e6i − 0.492261i
$$633$$ 932428.i 0.0924924i
$$634$$ −8.04832e6 −0.795210
$$635$$ 0 0
$$636$$ −491616. −0.0481929
$$637$$ 7.43831e6i 0.726316i
$$638$$ 998976.i 0.0971635i
$$639$$ 3.22126e6 0.312086
$$640$$ 0 0
$$641$$ 1.84286e7 1.77153 0.885764 0.464136i $$-0.153635\pi$$
0.885764 + 0.464136i $$0.153635\pi$$
$$642$$ 317448.i 0.0303973i
$$643$$ − 9.66604e6i − 0.921979i −0.887406 0.460989i $$-0.847495\pi$$
0.887406 0.460989i $$-0.152505\pi$$
$$644$$ −4.72502e6 −0.448941
$$645$$ 0 0
$$646$$ 4.20365e6 0.396319
$$647$$ − 4.51430e6i − 0.423965i −0.977273 0.211982i $$-0.932008\pi$$
0.977273 0.211982i $$-0.0679920\pi$$
$$648$$ 3.73254e6i 0.349195i
$$649$$ −4.23367e6 −0.394553
$$650$$ 0 0
$$651$$ −1.03667e6 −0.0958712
$$652$$ 720320.i 0.0663600i
$$653$$ 5.37235e6i 0.493039i 0.969138 + 0.246519i $$0.0792869\pi$$
−0.969138 + 0.246519i $$0.920713\pi$$
$$654$$ 351368. 0.0321231
$$655$$ 0 0
$$656$$ 1.35782e6 0.123192
$$657$$ 1.28889e7i 1.16494i
$$658$$ 1.14102e7i 1.02737i
$$659$$ −9.87956e6 −0.886184 −0.443092 0.896476i $$-0.646119\pi$$
−0.443092 + 0.896476i $$0.646119\pi$$
$$660$$ 0 0
$$661$$ 1.08052e7 0.961898 0.480949 0.876748i $$-0.340292\pi$$
0.480949 + 0.876748i $$0.340292\pi$$
$$662$$ − 8.06935e6i − 0.715638i
$$663$$ 510696.i 0.0451210i
$$664$$ 3.52090e6 0.309908
$$665$$ 0 0
$$666$$ −1.43119e7 −1.25029
$$667$$ 3.67186e6i 0.319574i
$$668$$ − 7.72090e6i − 0.669463i
$$669$$ −169745. −0.0146633
$$670$$ 0 0
$$671$$ 5.55874e6 0.476618
$$672$$ 169984.i 0.0145206i
$$673$$ − 1.13275e7i − 0.964042i −0.876160 0.482021i $$-0.839903\pi$$
0.876160 0.482021i $$-0.160097\pi$$
$$674$$ 1.05649e6 0.0895808
$$675$$ 0 0
$$676$$ 1.72114e6 0.144860
$$677$$ − 1.20595e7i − 1.01125i −0.862754 0.505624i $$-0.831262\pi$$
0.862754 0.505624i $$-0.168738\pi$$
$$678$$ − 188988.i − 0.0157892i
$$679$$ −1.47420e7 −1.22710
$$680$$ 0 0
$$681$$ 198078. 0.0163670
$$682$$ 3.02258e6i 0.248838i
$$683$$ 5.14166e6i 0.421747i 0.977513 + 0.210873i $$0.0676308\pi$$
−0.977513 + 0.210873i $$0.932369\pi$$
$$684$$ 5.51373e6 0.450614
$$685$$ 0 0
$$686$$ −4.02251e6 −0.326353
$$687$$ − 849997.i − 0.0687109i
$$688$$ − 4.55629e6i − 0.366978i
$$689$$ 2.12624e7 1.70633
$$690$$ 0 0
$$691$$ 1.31243e7 1.04563 0.522817 0.852445i $$-0.324881\pi$$
0.522817 + 0.852445i $$0.324881\pi$$
$$692$$ − 1.22601e7i − 0.973257i
$$693$$ 4.86081e6i 0.384482i
$$694$$ −6.84197e6 −0.539240
$$695$$ 0 0
$$696$$ 132096. 0.0103363
$$697$$ − 3.91435e6i − 0.305195i
$$698$$ 875288.i 0.0680005i
$$699$$ 401832. 0.0311065
$$700$$ 0 0
$$701$$ 3.65956e6 0.281277 0.140638 0.990061i $$-0.455084\pi$$
0.140638 + 0.990061i $$0.455084\pi$$
$$702$$ 1.34248e6i 0.102817i
$$703$$ 2.10538e7i 1.60673i
$$704$$ 495616. 0.0376889
$$705$$ 0 0
$$706$$ −1.47277e7 −1.11204
$$707$$ − 246012.i − 0.0185101i
$$708$$ 559824.i 0.0419728i
$$709$$ −1.02252e7 −0.763935 −0.381968 0.924176i $$-0.624753\pi$$
−0.381968 + 0.924176i $$0.624753\pi$$
$$710$$ 0 0
$$711$$ −1.86906e7 −1.38660
$$712$$ − 8.02656e6i − 0.593375i
$$713$$ 1.11099e7i 0.818436i
$$714$$ 490032. 0.0359732
$$715$$ 0 0
$$716$$ 4.85438e6 0.353876
$$717$$ 855174.i 0.0621236i
$$718$$ 7.54114e6i 0.545916i
$$719$$ −2.41683e7 −1.74351 −0.871753 0.489945i $$-0.837017\pi$$
−0.871753 + 0.489945i $$0.837017\pi$$
$$720$$ 0 0
$$721$$ 1.95043e7 1.39731
$$722$$ 1.79329e6i 0.128029i
$$723$$ − 1.12546e6i − 0.0800730i
$$724$$ 4.56290e6 0.323515
$$725$$ 0 0
$$726$$ −58564.0 −0.00412372
$$727$$ 1.68246e7i 1.18062i 0.807177 + 0.590310i $$0.200994\pi$$
−0.807177 + 0.590310i $$0.799006\pi$$
$$728$$ − 7.35181e6i − 0.514121i
$$729$$ 1.39958e7 0.975393
$$730$$ 0 0
$$731$$ −1.31349e7 −0.909147
$$732$$ − 735040.i − 0.0507030i
$$733$$ 5.04168e6i 0.346590i 0.984870 + 0.173295i $$0.0554414\pi$$
−0.984870 + 0.173295i $$0.944559\pi$$
$$734$$ −1.24666e7 −0.854101
$$735$$ 0 0
$$736$$ 1.82170e6 0.123960
$$737$$ − 3.06650e6i − 0.207958i
$$738$$ − 5.13427e6i − 0.347007i
$$739$$ 6.26375e6 0.421913 0.210957 0.977495i $$-0.432342\pi$$
0.210957 + 0.977495i $$0.432342\pi$$
$$740$$ 0 0
$$741$$ 985408. 0.0659281
$$742$$ − 2.04021e7i − 1.36039i
$$743$$ − 3.63976e6i − 0.241880i −0.992660 0.120940i $$-0.961409\pi$$
0.992660 0.120940i $$-0.0385909\pi$$
$$744$$ 399680. 0.0264716
$$745$$ 0 0
$$746$$ −5.57766e6 −0.366948
$$747$$ − 1.33134e7i − 0.872945i
$$748$$ − 1.42877e6i − 0.0933701i
$$749$$ −1.31741e7 −0.858057
$$750$$ 0 0
$$751$$ −1.87370e7 −1.21227 −0.606135 0.795362i $$-0.707281\pi$$
−0.606135 + 0.795362i $$0.707281\pi$$
$$752$$ − 4.39910e6i − 0.283674i
$$753$$ 1.19751e6i 0.0769649i
$$754$$ −5.71315e6 −0.365972
$$755$$ 0 0
$$756$$ 1.28816e6 0.0819720
$$757$$ 489242.i 0.0310302i 0.999880 + 0.0155151i $$0.00493880\pi$$
−0.999880 + 0.0155151i $$0.995061\pi$$
$$758$$ − 1.70414e7i − 1.07729i
$$759$$ −215259. −0.0135630
$$760$$ 0 0
$$761$$ 1.46969e7 0.919952 0.459976 0.887931i $$-0.347858\pi$$
0.459976 + 0.887931i $$0.347858\pi$$
$$762$$ 957664.i 0.0597483i
$$763$$ 1.45818e7i 0.906774i
$$764$$ −1.22731e7 −0.760711
$$765$$ 0 0
$$766$$ −807060. −0.0496974
$$767$$ − 2.42124e7i − 1.48610i
$$768$$ − 65536.0i − 0.00400938i
$$769$$ −2.42072e7 −1.47615 −0.738073 0.674721i $$-0.764264\pi$$
−0.738073 + 0.674721i $$0.764264\pi$$
$$770$$ 0 0
$$771$$ 37758.0 0.00228756
$$772$$ 6.58669e6i 0.397763i
$$773$$ − 1.35260e7i − 0.814181i −0.913388 0.407091i $$-0.866543\pi$$
0.913388 0.407091i $$-0.133457\pi$$
$$774$$ −1.72285e7 −1.03370
$$775$$ 0 0
$$776$$ 5.68365e6 0.338823
$$777$$ 2.45431e6i 0.145840i
$$778$$ 7.79528e6i 0.461725i
$$779$$ −7.55290e6 −0.445933
$$780$$ 0 0
$$781$$ −1.61063e6 −0.0944862
$$782$$ − 5.25161e6i − 0.307097i
$$783$$ − 1.00104e6i − 0.0583508i
$$784$$ −2.75174e6 −0.159889
$$785$$ 0 0
$$786$$ 392568. 0.0226651
$$787$$ 1.42094e7i 0.817786i 0.912582 + 0.408893i $$0.134085\pi$$
−0.912582 + 0.408893i $$0.865915\pi$$
$$788$$ 1.21481e7i 0.696937i
$$789$$ 631254. 0.0361004
$$790$$ 0 0
$$791$$ 7.84300e6 0.445698
$$792$$ − 1.87405e6i − 0.106162i
$$793$$ 3.17905e7i 1.79521i
$$794$$ −5.87303e6 −0.330606
$$795$$ 0 0
$$796$$ −745600. −0.0417084
$$797$$ − 7.93333e6i − 0.442395i −0.975229 0.221197i $$-0.929003\pi$$
0.975229 0.221197i $$-0.0709965\pi$$
$$798$$ − 945536.i − 0.0525619i
$$799$$ −1.26818e7 −0.702771
$$800$$ 0 0
$$801$$ −3.03504e7 −1.67141
$$802$$ − 8.98471e6i − 0.493251i
$$803$$ − 6.44446e6i − 0.352694i
$$804$$ −405488. −0.0221227
$$805$$ 0 0
$$806$$ −1.72862e7 −0.937262
$$807$$ − 1.08034e6i − 0.0583952i
$$808$$ 94848.0i 0.00511093i
$$809$$ 1.04685e7 0.562359 0.281180 0.959655i $$-0.409274\pi$$
0.281180 + 0.959655i $$0.409274\pi$$
$$810$$ 0 0
$$811$$ 1.19147e7 0.636110 0.318055 0.948072i $$-0.396970\pi$$
0.318055 + 0.948072i $$0.396970\pi$$
$$812$$ 5.48198e6i 0.291775i
$$813$$ 816100.i 0.0433029i
$$814$$ 7.15594e6 0.378535
$$815$$ 0 0
$$816$$ −188928. −0.00993278
$$817$$ 2.53444e7i 1.32839i
$$818$$ − 1.44595e7i − 0.755562i
$$819$$ −2.77990e7 −1.44817
$$820$$ 0 0
$$821$$ −1.86112e6 −0.0963645 −0.0481822 0.998839i $$-0.515343\pi$$
−0.0481822 + 0.998839i $$0.515343\pi$$
$$822$$ − 1.60055e6i − 0.0826208i
$$823$$ − 2.30153e7i − 1.18445i −0.805773 0.592225i $$-0.798250\pi$$
0.805773 0.592225i $$-0.201750\pi$$
$$824$$ −7.51974e6 −0.385820
$$825$$ 0 0
$$826$$ −2.32327e7 −1.18481
$$827$$ − 1.68351e7i − 0.855959i −0.903788 0.427980i $$-0.859225\pi$$
0.903788 0.427980i $$-0.140775\pi$$
$$828$$ − 6.88829e6i − 0.349169i
$$829$$ 2.35299e7 1.18914 0.594570 0.804044i $$-0.297322\pi$$
0.594570 + 0.804044i $$0.297322\pi$$
$$830$$ 0 0
$$831$$ 1.68820e6 0.0848049
$$832$$ 2.83443e6i 0.141957i
$$833$$ 7.93276e6i 0.396106i
$$834$$ 823064. 0.0409750
$$835$$ 0 0
$$836$$ −2.75686e6 −0.136427
$$837$$ − 3.02882e6i − 0.149438i
$$838$$ − 1.52496e7i − 0.750148i
$$839$$ −2.91549e7 −1.42990 −0.714952 0.699173i $$-0.753552\pi$$
−0.714952 + 0.699173i $$0.753552\pi$$
$$840$$ 0 0
$$841$$ −1.62511e7 −0.792303
$$842$$ − 7.89385e6i − 0.383715i
$$843$$ 879042.i 0.0426030i
$$844$$ 1.49188e7 0.720907
$$845$$ 0 0
$$846$$ −1.66341e7 −0.799050
$$847$$ − 2.43041e6i − 0.116405i
$$848$$ 7.86586e6i 0.375627i
$$849$$ −1.54027e6 −0.0733377
$$850$$ 0 0
$$851$$ 2.63025e7 1.24501
$$852$$ 212976.i 0.0100515i
$$853$$ 9.49052e6i 0.446599i 0.974750 + 0.223299i $$0.0716828\pi$$
−0.974750 + 0.223299i $$0.928317\pi$$
$$854$$ 3.05042e7 1.43125
$$855$$ 0 0
$$856$$ 5.07917e6 0.236924
$$857$$ − 1.81553e6i − 0.0844405i −0.999108 0.0422203i $$-0.986557\pi$$
0.999108 0.0422203i $$-0.0134431\pi$$
$$858$$ − 334928.i − 0.0155322i
$$859$$ 1.07812e7 0.498522 0.249261 0.968436i $$-0.419812\pi$$
0.249261 + 0.968436i $$0.419812\pi$$
$$860$$ 0 0
$$861$$ −880464. −0.0404766
$$862$$ 8.33436e6i 0.382036i
$$863$$ 2.83355e7i 1.29510i 0.762023 + 0.647550i $$0.224206\pi$$
−0.762023 + 0.647550i $$0.775794\pi$$
$$864$$ −496640. −0.0226338
$$865$$ 0 0
$$866$$ 290764. 0.0131749
$$867$$ − 875213.i − 0.0395427i
$$868$$ 1.65867e7i 0.747242i
$$869$$ 9.34531e6 0.419802
$$870$$ 0 0
$$871$$ 1.75374e7 0.783283
$$872$$ − 5.62189e6i − 0.250375i
$$873$$ − 2.14913e7i − 0.954392i
$$874$$ −1.01332e7 −0.448712
$$875$$ 0 0
$$876$$ −852160. −0.0375198
$$877$$ − 2.68919e7i − 1.18065i −0.807165 0.590326i $$-0.798999\pi$$
0.807165 0.590326i $$-0.201001\pi$$
$$878$$ 2.37757e6i 0.104087i
$$879$$ −720840. −0.0314678
$$880$$ 0 0
$$881$$ −1.92132e7 −0.833989 −0.416995 0.908909i $$-0.636917\pi$$
−0.416995 + 0.908909i $$0.636917\pi$$
$$882$$ 1.04050e7i 0.450373i
$$883$$ − 1.15931e7i − 0.500378i −0.968197 0.250189i $$-0.919507\pi$$
0.968197 0.250189i $$-0.0804927\pi$$
$$884$$ 8.17114e6 0.351683
$$885$$ 0 0
$$886$$ −1.82660e7 −0.781735
$$887$$ 1.31857e7i 0.562721i 0.959602 + 0.281361i $$0.0907857\pi$$
−0.959602 + 0.281361i $$0.909214\pi$$
$$888$$ − 946240.i − 0.0402688i
$$889$$ −3.97431e7 −1.68658
$$890$$ 0 0
$$891$$ −7.05684e6 −0.297794
$$892$$ 2.71592e6i 0.114289i
$$893$$ 2.44700e7i 1.02685i
$$894$$ 350904. 0.0146840
$$895$$ 0 0
$$896$$ 2.71974e6 0.113177
$$897$$ − 1.23107e6i − 0.0510859i
$$898$$ − 2.17753e7i − 0.901100i
$$899$$ 1.28897e7 0.531916
$$900$$ 0 0
$$901$$ 2.26758e7 0.930573
$$902$$ 2.56714e6i 0.105059i
$$903$$ 2.95447e6i 0.120576i
$$904$$ −3.02381e6 −0.123065
$$905$$ 0 0
$$906$$ 1.73111e6 0.0700656
$$907$$ 2.98195e6i 0.120360i 0.998188 + 0.0601800i $$0.0191675\pi$$
−0.998188 + 0.0601800i $$0.980833\pi$$
$$908$$ − 3.16925e6i − 0.127568i
$$909$$ 358644. 0.0143964
$$910$$ 0 0
$$911$$ 2.96579e7 1.18398 0.591989 0.805946i $$-0.298343\pi$$
0.591989 + 0.805946i $$0.298343\pi$$
$$912$$ 364544.i 0.0145132i
$$913$$ 6.65669e6i 0.264291i
$$914$$ 2.68125e7 1.06163
$$915$$ 0 0
$$916$$ −1.36000e7 −0.535548
$$917$$ 1.62916e7i 0.639793i
$$918$$ 1.43172e6i 0.0560727i
$$919$$ −3.18057e7 −1.24227 −0.621135 0.783704i $$-0.713328\pi$$
−0.621135 + 0.783704i $$0.713328\pi$$
$$920$$ 0 0
$$921$$ −1.03905e6 −0.0403633
$$922$$ 5.03976e6i 0.195246i
$$923$$ − 9.21121e6i − 0.355887i
$$924$$ −321376. −0.0123832
$$925$$ 0 0
$$926$$ 2.00923e7 0.770021
$$927$$ 2.84340e7i 1.08677i
$$928$$ − 2.11354e6i − 0.0805638i
$$929$$ 2.33444e7 0.887451 0.443725 0.896163i $$-0.353657\pi$$
0.443725 + 0.896163i $$0.353657\pi$$
$$930$$ 0 0
$$931$$ 1.53066e7 0.578767
$$932$$ − 6.42931e6i − 0.242451i
$$933$$ 1.25135e6i 0.0470624i
$$934$$ −9.42642e6 −0.353573
$$935$$ 0 0
$$936$$ 1.07177e7 0.399864
$$937$$ 2.07372e7i 0.771616i 0.922579 + 0.385808i $$0.126077\pi$$
−0.922579 + 0.385808i $$0.873923\pi$$
$$938$$ − 1.68278e7i − 0.624481i
$$939$$ 1.44336e6 0.0534209
$$940$$ 0 0
$$941$$ 2.69193e7 0.991036 0.495518 0.868598i $$-0.334978\pi$$
0.495518 + 0.868598i $$0.334978\pi$$
$$942$$ 136300.i 0.00500459i
$$943$$ 9.43582e6i 0.345542i
$$944$$ 8.95718e6 0.327146
$$945$$ 0 0
$$946$$ 8.61423e6 0.312960
$$947$$ 1.01896e7i 0.369216i 0.982812 + 0.184608i $$0.0591016\pi$$
−0.982812 + 0.184608i $$0.940898\pi$$
$$948$$ − 1.23574e6i − 0.0446589i
$$949$$ 3.68559e7 1.32844
$$950$$ 0 0
$$951$$ −2.01208e6 −0.0721429
$$952$$ − 7.84051e6i − 0.280383i
$$953$$ − 1.03924e7i − 0.370665i −0.982676 0.185333i $$-0.940664\pi$$
0.982676 0.185333i $$-0.0593362\pi$$
$$954$$ 2.97428e7 1.05806
$$955$$ 0 0
$$956$$ 1.36828e7 0.484206
$$957$$ 249744.i 0.00881486i
$$958$$ − 2.68903e7i − 0.946634i
$$959$$ 6.64227e7 2.33222
$$960$$ 0 0
$$961$$ 1.03709e7 0.362249
$$962$$ 4.09249e7i 1.42577i
$$963$$ − 1.92056e7i − 0.667363i
$$964$$ −1.80074e7 −0.624107
$$965$$ 0 0
$$966$$ −1.18126e6 −0.0407288
$$967$$ − 8.18877e6i − 0.281613i −0.990037 0.140806i $$-0.955030\pi$$
0.990037 0.140806i $$-0.0449695\pi$$
$$968$$ 937024.i 0.0321412i
$$969$$ 1.05091e6 0.0359548
$$970$$ 0 0
$$971$$ −1.73274e7 −0.589775 −0.294887 0.955532i $$-0.595282\pi$$
−0.294887 + 0.955532i $$0.595282\pi$$
$$972$$ 2.81882e6i 0.0956976i
$$973$$ 3.41572e7i 1.15664i
$$974$$ 7.84005e6 0.264802
$$975$$ 0 0
$$976$$ −1.17606e7 −0.395190
$$977$$ − 438963.i − 0.0147127i −0.999973 0.00735634i $$-0.997658\pi$$
0.999973 0.00735634i $$-0.00234162\pi$$
$$978$$ 180080.i 0.00602030i
$$979$$ 1.51752e7 0.506032
$$980$$ 0 0
$$981$$ −2.12578e7 −0.705253
$$982$$ 2.31850e6i 0.0767234i
$$983$$ 2.79124e7i 0.921326i 0.887575 + 0.460663i $$0.152388\pi$$
−0.887575 + 0.460663i $$0.847612\pi$$
$$984$$ 339456. 0.0111762
$$985$$ 0 0
$$986$$ −6.09293e6 −0.199588
$$987$$ 2.85254e6i 0.0932051i
$$988$$ − 1.57665e7i − 0.513859i
$$989$$ 3.16626e7 1.02933
$$990$$ 0 0
$$991$$ −4.26846e7 −1.38066 −0.690331 0.723494i $$-0.742535\pi$$
−0.690331 + 0.723494i $$0.742535\pi$$
$$992$$ − 6.39488e6i − 0.206326i
$$993$$ − 2.01734e6i − 0.0649240i
$$994$$ −8.83850e6 −0.283735
$$995$$ 0 0
$$996$$ 880224. 0.0281154
$$997$$ − 2.21044e7i − 0.704273i −0.935949 0.352137i $$-0.885455\pi$$
0.935949 0.352137i $$-0.114545\pi$$
$$998$$ 5.47621e6i 0.174042i
$$999$$ −7.17072e6 −0.227326
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 550.6.b.f.199.1 2
5.2 odd 4 550.6.a.f.1.1 1
5.3 odd 4 22.6.a.b.1.1 1
5.4 even 2 inner 550.6.b.f.199.2 2
15.8 even 4 198.6.a.i.1.1 1
20.3 even 4 176.6.a.b.1.1 1
35.13 even 4 1078.6.a.a.1.1 1
40.3 even 4 704.6.a.f.1.1 1
40.13 odd 4 704.6.a.e.1.1 1
55.43 even 4 242.6.a.d.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
22.6.a.b.1.1 1 5.3 odd 4
176.6.a.b.1.1 1 20.3 even 4
198.6.a.i.1.1 1 15.8 even 4
242.6.a.d.1.1 1 55.43 even 4
550.6.a.f.1.1 1 5.2 odd 4
550.6.b.f.199.1 2 1.1 even 1 trivial
550.6.b.f.199.2 2 5.4 even 2 inner
704.6.a.e.1.1 1 40.13 odd 4
704.6.a.f.1.1 1 40.3 even 4
1078.6.a.a.1.1 1 35.13 even 4