LMFDB - Embedded newform 550.6.b.f.199.1

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

Display 100 coefficients 10 coefficients

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 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	− 4.00000i	− 0.707107i
$2$	− 4.00000i	− 0.707107i
$3$	− 1.00000i	− 0.0641500i	−0.999485	−	0.0320750i	$-0.989788\pi$
$3$	− 1.00000i	− 0.0641500i	0.999485	−	0.0320750i	$-0.0102115\pi$
$4$	−16.0000	−0.500000
$5$	0	0
$5$	0	0
$6$	−4.00000	−0.0453609
$7$	− 166.000i	− 1.28045i	−0.768187	−	0.640226i	$-0.778841\pi$
$7$	− 166.000i	− 1.28045i	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
$11$	−121.000	−0.301511
$12$	16.0000i	0.0320750i
$13$	− 692.000i	− 1.13566i	−0.823146	−	0.567829i	$-0.807783\pi$
$13$	− 692.000i	− 1.13566i	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	$-0.899780\pi$
$17$	− 738.000i	− 0.619347i	0.950843	−	0.309674i	$-0.100220\pi$
$18$	− 968.000i	− 0.704197i
$19$	−1424.00	−0.904953	−0.452476	−	0.891776i	$-0.649459\pi$
$19$	−1424.00	−0.904953	−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	$0.114026\pi$
$23$	1779.00i	0.701223i	−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	$-0.426824\pi$
$29$	2064.00	0.455737	0.227869	+	0.973692i	$0.426824\pi$
$30$	0	0
$31$	6245.00	1.16715	0.583577	−	0.812058i	$-0.301653\pi$
$31$	6245.00	1.16715	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	$-0.652273\pi$
$37$	− 14785.0i	− 1.77549i	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	$-0.420757\pi$
$41$	5304.00	0.492770	0.246385	+	0.969172i	$0.420757\pi$
$42$	664.000i	0.0580824i
$43$	− 17798.0i	− 1.46791i	−0.679197	−	0.733956i	$-0.737672\pi$
$43$	− 17798.0i	− 1.46791i	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	$-0.807969\pi$
$47$	− 17184.0i	− 1.13470i	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	$0.270550\pi$
$53$	30726.0i	1.50251i	−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	$-0.272967\pi$
$59$	34989.0	1.30858	0.654292	+	0.756242i	$0.272967\pi$
$60$	0	0
$61$	−45940.0	−1.58076	−0.790381	−	0.612616i	$-0.790117\pi$
$61$	−45940.0	−1.58076	−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	$0.112073\pi$
$67$	25343.0i	0.689717i	−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	$-0.449918\pi$
$71$	13311.0	0.313375	0.156688	+	0.987648i	$0.449918\pi$
$72$	15488.0i	0.352098i
$73$	53260.0i	1.16975i	0.811123	+	0.584876i	$0.198857\pi$
$73$	53260.0i	1.16975i	−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	$-0.745111\pi$
$79$	−77234.0	−1.39233	−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	$-0.855589\pi$
$83$	− 55014.0i	− 0.876553i	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	$-0.816953\pi$
$89$	−125415.	−1.67832	−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	$-0.840938\pi$
$97$	− 88807.0i	− 0.958336i	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	$-0.497699\pi$
$101$	1482.00	0.0144559	0.00722794	+	0.999974i	$0.497699\pi$
$102$	2952.00i	0.0280942i
$103$	117496.i	1.09126i	0.838025	+	0.545632i	$0.183710\pi$
$103$	117496.i	1.09126i	−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	$-0.891243\pi$
$107$	− 79362.0i	− 0.670121i	0.942197	−	0.335060i	$-0.108757\pi$
$108$	7760.00i	0.0640180i
$109$	−87842.0	−0.708167	−0.354084	−	0.935214i	$-0.615207\pi$
$109$	−87842.0	−0.708167	−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	$0.0556821\pi$
$113$	47247.0i	0.348079i	−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	$-0.771154\pi$
$127$	− 239416.i	− 1.31718i	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	$-0.580375\pi$
$131$	−98142.0	−0.499662	−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	$0.364460\pi$
$137$	400137.i	1.82141i	−0.413059	+	0.910704i	$0.635540\pi$
$138$	− 7116.00i	− 0.0318081i
$139$	−205766.	−0.903310	−0.451655	−	0.892193i	$-0.649166\pi$
$139$	−205766.	−0.903310	−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	$-0.551748\pi$
$149$	−87726.0	−0.323715	−0.161857	+	0.986814i	$0.551748\pi$
$150$	0	0
$151$	−432778.	−1.54462	−0.772312	−	0.635243i	$-0.780900\pi$
$151$	−432778.	−1.54462	−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	$-0.982432\pi$
$157$	− 34075.0i	− 0.110328i	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	$-0.978861\pi$
$163$	− 45020.0i	− 0.132720i	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	$0.233476\pi$
$167$	482556.i	1.33893i	−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	$0.426219\pi$
$173$	766254.i	1.94651i	−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	$-0.615137\pi$
$179$	−303399.	−0.707753	−0.353876	+	0.935292i	$0.615137\pi$
$180$	0	0
$181$	−285181.	−0.647030	−0.323515	−	0.946223i	$-0.604865\pi$
$181$	−285181.	−0.647030	−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	$-0.224850\pi$
$191$	767067.	1.52142	0.760711	+	0.649090i	$0.224850\pi$
$192$	4096.00i	0.00801875i
$193$	− 411668.i	− 0.795525i	−0.917488	−	0.397763i	$-0.869787\pi$
$193$	− 411668.i	− 0.795525i	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	$-0.754545\pi$
$197$	− 759258.i	− 1.39387i	0.717132	−	0.696937i	$-0.245455\pi$
$198$	117128.i	0.212323i
$199$	46600.0	0.0834167	0.0417084	−	0.999130i	$-0.486720\pi$
$199$	46600.0	0.0834167	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	$-0.756274\pi$
$211$	−932428.	−1.44181	−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	$-0.963541\pi$
$223$	− 169745.i	− 0.228578i	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	$0.0407171\pi$
$227$	198078.i	0.255136i	−0.991830	+	0.127568i	$0.959283\pi$
$228$	− 22784.0i	− 0.0290264i
$229$	849997.	1.07110	0.535548	−	0.844505i	$-0.320105\pi$
$229$	849997.	1.07110	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	$0.0779515\pi$
$233$	401832.i	0.484903i	−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	$-0.660891\pi$
$239$	−855174.	−0.968411	−0.484206	+	0.874954i	$0.660891\pi$
$240$	0	0
$241$	1.12546e6	1.24821	0.624107	−	0.781339i	$-0.285463\pi$
$241$	1.12546e6	1.24821	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	$-0.704786\pi$
$251$	−1.19751e6	−1.19976	−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	$0.00567570\pi$
$257$	37758.0i	0.0356596i	−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	$0.0907903\pi$
$263$	631254.i	0.562749i	−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	$-0.349587\pi$
$269$	1.08034e6	0.910292	0.455146	+	0.890417i	$0.349587\pi$
$270$	0	0
$271$	−816100.	−0.675025	−0.337513	−	0.941321i	$-0.609586\pi$
$271$	−816100.	−0.675025	−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	$0.229863\pi$
$277$	1.68820e6i	1.32198i	−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	$-0.607743\pi$
$281$	−879042.	−0.664116	−0.332058	+	0.943259i	$0.607743\pi$
$282$	68736.0i	0.0514709i
$283$	− 1.54027e6i	− 1.14322i	−0.820525	−	0.571611i	$-0.806319\pi$
$283$	− 1.54027e6i	− 1.14322i	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	$-0.921124\pi$
$293$	− 720840.i	− 0.490535i	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	$-0.898129\pi$
$307$	− 1.03905e6i	− 0.629201i	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	$-0.619552\pi$
$311$	−1.25135e6	−0.733630	−0.366815	+	0.930294i	$0.619552\pi$
$312$	− 44288.0i	− 0.0257573i
$313$	1.44336e6i	0.832749i	0.909193	+	0.416375i	$0.136699\pi$
$313$	1.44336e6i	0.832749i	−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	$-0.809917\pi$
$317$	− 2.01208e6i	− 1.12460i	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	$-0.331112\pi$
$331$	2.01734e6	1.01207	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	$0.0201763\pi$
$337$	264122.i	0.126686i	−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	$-0.875476\pi$
$347$	− 1.71049e6i	− 0.762601i	0.924451	−	0.381300i	$-0.124524\pi$
$348$	33024.0i	0.0146178i
$349$	−218822.	−0.0961673	−0.0480836	−	0.998843i	$-0.515311\pi$
$349$	−218822.	−0.0961673	−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	$-0.711977\pi$
$353$	− 3.68192e6i	− 1.57267i	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	$-0.626151\pi$
$359$	−1.88528e6	−0.772042	−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	$-0.793596\pi$
$367$	− 3.11666e6i	− 1.20788i	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	$-0.916452\pi$
$373$	− 1.39441e6i	− 0.518943i	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	$-0.224336\pi$
$379$	4.26036e6	1.52352	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	$-0.988812\pi$
$383$	− 201765.i	− 0.0702828i	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	$-0.605865\pi$
$389$	−1.94882e6	−0.652977	−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	$-0.924892\pi$
$397$	− 1.46826e6i	− 0.467548i	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	$-0.386596\pi$
$401$	2.24618e6	0.697563	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	$-0.320589\pi$
$409$	3.61488e6	1.06853	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	$-0.322029\pi$
$419$	3.81239e6	1.06087	0.530435	+	0.847726i	$0.322029\pi$
$420$	0	0
$421$	1.97346e6	0.542655	0.271327	−	0.962487i	$-0.412537\pi$
$421$	1.97346e6	0.542655	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	$-0.587070\pi$
$431$	−2.08359e6	−0.540280	−0.270140	+	0.962821i	$0.587070\pi$
$432$	− 124160.i	− 0.0320090i
$433$	72691.0i	0.0186321i	0.999957	+	0.00931603i	$0.00296543\pi$
$433$	72691.0i	0.0186321i	−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	$-0.523449\pi$
$439$	−594392.	−0.147201	−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	$-0.813571\pi$
$443$	− 4.56651e6i	− 1.10554i	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	$-0.280103\pi$
$449$	5.44382e6	1.27435	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	$0.270276\pi$
$457$	6.70312e6i	1.50137i	−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	$-0.544087\pi$
$461$	−1.25994e6	−0.276120	−0.138060	+	0.990424i	$0.544087\pi$
$462$	− 80344.0i	− 0.0175125i
$463$	5.02308e6i	1.08897i	0.838769	+	0.544487i	$0.183276\pi$
$463$	5.02308e6i	1.08897i	−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	$-0.919565\pi$
$467$	− 2.35660e6i	− 0.500028i	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	$-0.266563\pi$
$479$	6.72258e6	1.33874	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	$0.0599553\pi$
$487$	1.96001e6i	0.374487i	−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	$-0.517277\pi$
$491$	−579624.	−0.108503	−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	$-0.539273\pi$
$499$	−1.36905e6	−0.246132	−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	$-0.948350\pi$
$503$	− 1.83343e6i	− 0.323105i	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	$-0.453198\pi$
$509$	1.71266e6	0.293006	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	$-0.520293\pi$
$521$	−789435.	−0.127415	−0.0637077	+	0.997969i	$0.520293\pi$
$522$	− 1.99795e6i	− 0.320929i
$523$	− 627392.i	− 0.100296i	−0.998742	−	0.0501481i	$-0.984031\pi$
$523$	− 627392.i	− 0.100296i	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	$-0.424265\pi$
$541$	3.20895e6	0.471379	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	$0.0787319\pi$
$547$	3.42658e6i	0.489658i	−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	$0.255106\pi$
$557$	1.05198e7i	1.43672i	−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	$-0.881464\pi$
$563$	− 5.47288e6i	− 0.727687i	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	$-0.223932\pi$
$569$	1.17787e7	1.52516	0.762580	+	0.646893i	$0.223932\pi$
$570$	0	0
$571$	−8.35628e6	−1.07256	−0.536281	−	0.844039i	$-0.680171\pi$
$571$	−8.35628e6	−1.07256	−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	$-0.669658\pi$
$577$	− 1.37758e7i	− 1.72258i	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	$-0.724612\pi$
$587$	− 1.27093e7i	− 1.52239i	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	$-0.981250\pi$
$593$	− 1.00825e6i	− 0.117742i	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	$-0.704205\pi$
$599$	−1.05100e7	−1.19684	−0.598421	+	0.801182i	$0.704205\pi$
$600$	0	0
$601$	−199390.	−0.0225173	−0.0112587	−	0.999937i	$-0.503584\pi$
$601$	−199390.	−0.0225173	−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	$0.000283854\pi$
$607$	16190.0i	0.00178351i	−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	$0.212625\pi$
$613$	1.15253e7i	1.23880i	−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	$0.355524\pi$
$617$	1.69974e7i	1.79750i	−0.438459	+	0.898751i	$0.644476\pi$
$618$	− 469984.i	− 0.0495008i
$619$	1.84875e7	1.93933	0.969663	−	0.244445i	$-0.0786058\pi$
$619$	1.84875e7	1.93933	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	$-0.572925\pi$
$631$	−4.54281e6	−0.454204	−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	$-0.153635\pi$
$641$	1.84286e7	1.77153	0.885764	+	0.464136i	$0.153635\pi$
$642$	317448.i	0.0303973i
$643$	− 9.66604e6i	− 0.921979i	−0.887406	−	0.460989i	$-0.847495\pi$
$643$	− 9.66604e6i	− 0.921979i	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	$-0.932008\pi$
$647$	− 4.51430e6i	− 0.423965i	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	$0.0792869\pi$
$653$	5.37235e6i	0.493039i	−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	$-0.646119\pi$
$659$	−9.87956e6	−0.886184	−0.443092	+	0.896476i	$0.646119\pi$
$660$	0	0
$661$	1.08052e7	0.961898	0.480949	−	0.876748i	$-0.340292\pi$
$661$	1.08052e7	0.961898	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	$-0.839903\pi$
$673$	− 1.13275e7i	− 0.964042i	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	$-0.831262\pi$
$677$	− 1.20595e7i	− 1.01125i	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	$0.0676308\pi$
$683$	5.14166e6i	0.421747i	−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	$-0.324881\pi$
$691$	1.31243e7	1.04563	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	$-0.455084\pi$
$701$	3.65956e6	0.281277	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	$-0.624753\pi$
$709$	−1.02252e7	−0.763935	−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	$-0.837017\pi$
$719$	−2.41683e7	−1.74351	−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	$0.200994\pi$
$727$	1.68246e7i	1.18062i	−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	$0.0554414\pi$
$733$	5.04168e6i	0.346590i	−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	$-0.432342\pi$
$739$	6.26375e6	0.421913	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	$-0.961409\pi$
$743$	− 3.63976e6i	− 0.241880i	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	$-0.707281\pi$
$751$	−1.87370e7	−1.21227	−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	$0.00493880\pi$
$757$	489242.i	0.0310302i	−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	$-0.347858\pi$
$761$	1.46969e7	0.919952	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	$-0.764264\pi$
$769$	−2.42072e7	−1.47615	−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	$-0.866543\pi$
$773$	− 1.35260e7i	− 0.814181i	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	$0.134085\pi$
$787$	1.42094e7i	0.817786i	−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	$-0.929003\pi$
$797$	− 7.93333e6i	− 0.442395i	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	$-0.409274\pi$
$809$	1.04685e7	0.562359	0.281180	+	0.959655i	$0.409274\pi$
$810$	0	0
$811$	1.19147e7	0.636110	0.318055	−	0.948072i	$-0.396970\pi$
$811$	1.19147e7	0.636110	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	$-0.515343\pi$
$821$	−1.86112e6	−0.0963645	−0.0481822	+	0.998839i	$0.515343\pi$
$822$	− 1.60055e6i	− 0.0826208i
$823$	− 2.30153e7i	− 1.18445i	−0.805773	−	0.592225i	$-0.798250\pi$
$823$	− 2.30153e7i	− 1.18445i	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	$-0.859225\pi$
$827$	− 1.68351e7i	− 0.855959i	0.903788	−	0.427980i	$-0.140775\pi$
$828$	− 6.88829e6i	− 0.349169i
$829$	2.35299e7	1.18914	0.594570	−	0.804044i	$-0.297322\pi$
$829$	2.35299e7	1.18914	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	$-0.753552\pi$
$839$	−2.91549e7	−1.42990	−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	$0.0716828\pi$
$853$	9.49052e6i	0.446599i	−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	$-0.986557\pi$
$857$	− 1.81553e6i	− 0.0844405i	0.999108	−	0.0422203i	$-0.0134431\pi$
$858$	− 334928.i	− 0.0155322i
$859$	1.07812e7	0.498522	0.249261	−	0.968436i	$-0.419812\pi$
$859$	1.07812e7	0.498522	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	$0.224206\pi$
$863$	2.83355e7i	1.29510i	−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	$-0.798999\pi$
$877$	− 2.68919e7i	− 1.18065i	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	$-0.636917\pi$
$881$	−1.92132e7	−0.833989	−0.416995	+	0.908909i	$0.636917\pi$
$882$	1.04050e7i	0.450373i
$883$	− 1.15931e7i	− 0.500378i	−0.968197	−	0.250189i	$-0.919507\pi$
$883$	− 1.15931e7i	− 0.500378i	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	$0.0907857\pi$
$887$	1.31857e7i	0.562721i	−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	$0.0191675\pi$
$907$	2.98195e6i	0.120360i	−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	$-0.298343\pi$
$911$	2.96579e7	1.18398	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	$-0.713328\pi$
$919$	−3.18057e7	−1.24227	−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	$-0.353657\pi$
$929$	2.33444e7	0.887451	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	$0.126077\pi$
$937$	2.07372e7i	0.771616i	−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	$-0.334978\pi$
$941$	2.69193e7	0.991036	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	$0.0591016\pi$
$947$	1.01896e7i	0.369216i	−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	$-0.940664\pi$
$953$	− 1.03924e7i	− 0.370665i	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	$-0.955030\pi$
$967$	− 8.18877e6i	− 0.281613i	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	$-0.595282\pi$
$971$	−1.73274e7	−0.589775	−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	$-0.997658\pi$
$977$	− 438963.i	− 0.0147127i	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	$0.152388\pi$
$983$	2.79124e7i	0.921326i	−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	$-0.742535\pi$
$991$	−4.26846e7	−1.38066	−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	$-0.885455\pi$
$997$	− 2.21044e7i	− 0.704273i	0.935949	−	0.352137i	$-0.114545\pi$
$998$	5.47621e6i	0.174042i
$999$	−7.17072e6	−0.227326

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

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 550 = 2 \cdot 5^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	550.b (of order \(2\), degree \(1\), not minimal)

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

Introduction

L-functions

Modular forms

Varieties

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Representations

Groups

Database

Properties

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