LMFDB - Embedded newform 2850.2.d.r.799.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2850,2,Mod(799,2850)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2850, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("2850.799");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2850 = 2 \cdot 3 \cdot 5^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2850.d (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:	$22.7573645761$
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 570)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			799.2
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	2850.799
Dual form			2850.2.d.r.799.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} +1.00000 q^{6} +2.00000i q^{7} -1.00000i q^{8} -1.00000 q^{9} +O(q^{10})$	\(q+1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} +1.00000 q^{6} +2.00000i q^{7} -1.00000i q^{8} -1.00000 q^{9} +4.00000 q^{11} +1.00000i q^{12} -6.00000i q^{13} -2.00000 q^{14} +1.00000 q^{16} -4.00000i q^{17} -1.00000i q^{18} -1.00000 q^{19} +2.00000 q^{21} +4.00000i q^{22} +4.00000i q^{23} -1.00000 q^{24} +6.00000 q^{26} +1.00000i q^{27} -2.00000i q^{28} -6.00000 q^{29} -6.00000 q^{31} +1.00000i q^{32} -4.00000i q^{33} +4.00000 q^{34} +1.00000 q^{36} -10.0000i q^{37} -1.00000i q^{38} -6.00000 q^{39} +4.00000 q^{41} +2.00000i q^{42} +12.0000i q^{43} -4.00000 q^{44} -4.00000 q^{46} -4.00000i q^{47} -1.00000i q^{48} +3.00000 q^{49} -4.00000 q^{51} +6.00000i q^{52} -10.0000i q^{53} -1.00000 q^{54} +2.00000 q^{56} +1.00000i q^{57} -6.00000i q^{58} -10.0000 q^{59} +2.00000 q^{61} -6.00000i q^{62} -2.00000i q^{63} -1.00000 q^{64} +4.00000 q^{66} -12.0000i q^{67} +4.00000i q^{68} +4.00000 q^{69} +8.00000 q^{71} +1.00000i q^{72} -2.00000i q^{73} +10.0000 q^{74} +1.00000 q^{76} +8.00000i q^{77} -6.00000i q^{78} -10.0000 q^{79} +1.00000 q^{81} +4.00000i q^{82} +2.00000i q^{83} -2.00000 q^{84} -12.0000 q^{86} +6.00000i q^{87} -4.00000i q^{88} +8.00000 q^{89} +12.0000 q^{91} -4.00000i q^{92} +6.00000i q^{93} +4.00000 q^{94} +1.00000 q^{96} -2.00000i q^{97} +3.00000i q^{98} -4.00000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{4} + 2 q^{6} - 2 q^{9}+O(q^{10}) $ 2 * q - 2 * q^4 + 2 * q^6 - 2 * q^9	$ 2 q - 2 q^{4} + 2 q^{6} - 2 q^{9} + 8 q^{11} - 4 q^{14} + 2 q^{16} - 2 q^{19} + 4 q^{21} - 2 q^{24} + 12 q^{26} - 12 q^{29} - 12 q^{31} + 8 q^{34} + 2 q^{36} - 12 q^{39} + 8 q^{41} - 8 q^{44} - 8 q^{46} + 6 q^{49} - 8 q^{51} - 2 q^{54} + 4 q^{56} - 20 q^{59} + 4 q^{61} - 2 q^{64} + 8 q^{66} + 8 q^{69} + 16 q^{71} + 20 q^{74} + 2 q^{76} - 20 q^{79} + 2 q^{81} - 4 q^{84} - 24 q^{86} + 16 q^{89} + 24 q^{91} + 8 q^{94} + 2 q^{96} - 8 q^{99}+O(q^{100}) $ 2 * q - 2 * q^4 + 2 * q^6 - 2 * q^9 + 8 * q^11 - 4 * q^14 + 2 * q^16 - 2 * q^19 + 4 * q^21 - 2 * q^24 + 12 * q^26 - 12 * q^29 - 12 * q^31 + 8 * q^34 + 2 * q^36 - 12 * q^39 + 8 * q^41 - 8 * q^44 - 8 * q^46 + 6 * q^49 - 8 * q^51 - 2 * q^54 + 4 * q^56 - 20 * q^59 + 4 * q^61 - 2 * q^64 + 8 * q^66 + 8 * q^69 + 16 * q^71 + 20 * q^74 + 2 * q^76 - 20 * q^79 + 2 * q^81 - 4 * q^84 - 24 * q^86 + 16 * q^89 + 24 * q^91 + 8 * q^94 + 2 * q^96 - 8 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2850\mathbb{Z}\right)^\times$.

$n$	$1027$	$1351$	$1901$
$\chi(n)$	$-1$	$1$	$1$

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 $

$2$	1.00000i	0.707107i
$2$	1.00000i	0.707107i
$3$	− 1.00000i	− 0.577350i
$3$	− 1.00000i	− 0.577350i
$4$	−1.00000	−0.500000
$5$	0	0
$5$	0	0
$6$	1.00000	0.408248
$7$	2.00000i	0.755929i	0.925820	+	0.377964i	$0.123376\pi$
$7$	2.00000i	0.755929i	−0.925820	+	0.377964i	$0.876624\pi$
$8$	− 1.00000i	− 0.353553i
$9$	−1.00000	−0.333333
$10$	0	0
$11$	4.00000	1.20605	0.603023	−	0.797724i	$-0.293963\pi$
$11$	4.00000	1.20605	0.603023	+	0.797724i	$0.293963\pi$
$12$	1.00000i	0.288675i
$13$	− 6.00000i	− 1.66410i	−0.554700	−	0.832050i	$-0.687167\pi$
$13$	− 6.00000i	− 1.66410i	0.554700	−	0.832050i	$-0.312833\pi$
$14$	−2.00000	−0.534522
$15$	0	0
$16$	1.00000	0.250000
$17$	− 4.00000i	− 0.970143i	−0.874475	−	0.485071i	$-0.838794\pi$
$17$	− 4.00000i	− 0.970143i	0.874475	−	0.485071i	$-0.161206\pi$
$18$	− 1.00000i	− 0.235702i
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	2.00000	0.436436
$22$	4.00000i	0.852803i
$23$	4.00000i	0.834058i	0.908893	+	0.417029i	$0.136929\pi$
$23$	4.00000i	0.834058i	−0.908893	+	0.417029i	$0.863071\pi$
$24$	−1.00000	−0.204124
$25$	0	0
$26$	6.00000	1.17670
$27$	1.00000i	0.192450i
$28$	− 2.00000i	− 0.377964i
$29$	−6.00000	−1.11417	−0.557086	−	0.830455i	$-0.688081\pi$
$29$	−6.00000	−1.11417	−0.557086	+	0.830455i	$0.688081\pi$
$30$	0	0
$31$	−6.00000	−1.07763	−0.538816	−	0.842424i	$-0.681128\pi$
$31$	−6.00000	−1.07763	−0.538816	+	0.842424i	$0.681128\pi$
$32$	1.00000i	0.176777i
$33$	− 4.00000i	− 0.696311i
$34$	4.00000	0.685994
$35$	0	0
$36$	1.00000	0.166667
$37$	− 10.0000i	− 1.64399i	−0.569495	−	0.821995i	$-0.692861\pi$
$37$	− 10.0000i	− 1.64399i	0.569495	−	0.821995i	$-0.307139\pi$
$38$	− 1.00000i	− 0.162221i
$39$	−6.00000	−0.960769
$40$	0	0
$41$	4.00000	0.624695	0.312348	−	0.949968i	$-0.398885\pi$
$41$	4.00000	0.624695	0.312348	+	0.949968i	$0.398885\pi$
$42$	2.00000i	0.308607i
$43$	12.0000i	1.82998i	0.403473	+	0.914991i	$0.367803\pi$
$43$	12.0000i	1.82998i	−0.403473	+	0.914991i	$0.632197\pi$
$44$	−4.00000	−0.603023
$45$	0	0
$46$	−4.00000	−0.589768
$47$	− 4.00000i	− 0.583460i	−0.956501	−	0.291730i	$-0.905769\pi$
$47$	− 4.00000i	− 0.583460i	0.956501	−	0.291730i	$-0.0942309\pi$
$48$	− 1.00000i	− 0.144338i
$49$	3.00000	0.428571
$50$	0	0
$51$	−4.00000	−0.560112
$52$	6.00000i	0.832050i
$53$	− 10.0000i	− 1.37361i	−0.726844	−	0.686803i	$-0.759014\pi$
$53$	− 10.0000i	− 1.37361i	0.726844	−	0.686803i	$-0.240986\pi$
$54$	−1.00000	−0.136083
$55$	0	0
$56$	2.00000	0.267261
$57$	1.00000i	0.132453i
$58$	− 6.00000i	− 0.787839i
$59$	−10.0000	−1.30189	−0.650945	−	0.759125i	$-0.725627\pi$
$59$	−10.0000	−1.30189	−0.650945	+	0.759125i	$0.725627\pi$
$60$	0	0
$61$	2.00000	0.256074	0.128037	−	0.991769i	$-0.459132\pi$
$61$	2.00000	0.256074	0.128037	+	0.991769i	$0.459132\pi$
$62$	− 6.00000i	− 0.762001i
$63$	− 2.00000i	− 0.251976i
$64$	−1.00000	−0.125000
$65$	0	0
$66$	4.00000	0.492366
$67$	− 12.0000i	− 1.46603i	−0.680211	−	0.733017i	$-0.738112\pi$
$67$	− 12.0000i	− 1.46603i	0.680211	−	0.733017i	$-0.261888\pi$
$68$	4.00000i	0.485071i
$69$	4.00000	0.481543
$70$	0	0
$71$	8.00000	0.949425	0.474713	−	0.880141i	$-0.342552\pi$
$71$	8.00000	0.949425	0.474713	+	0.880141i	$0.342552\pi$
$72$	1.00000i	0.117851i
$73$	− 2.00000i	− 0.234082i	−0.993127	−	0.117041i	$-0.962659\pi$
$73$	− 2.00000i	− 0.234082i	0.993127	−	0.117041i	$-0.0373409\pi$
$74$	10.0000	1.16248
$75$	0	0
$76$	1.00000	0.114708
$77$	8.00000i	0.911685i
$78$	− 6.00000i	− 0.679366i
$79$	−10.0000	−1.12509	−0.562544	−	0.826767i	$-0.690177\pi$
$79$	−10.0000	−1.12509	−0.562544	+	0.826767i	$0.690177\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	4.00000i	0.441726i
$83$	2.00000i	0.219529i	0.993958	+	0.109764i	$0.0350096\pi$
$83$	2.00000i	0.219529i	−0.993958	+	0.109764i	$0.964990\pi$
$84$	−2.00000	−0.218218
$85$	0	0
$86$	−12.0000	−1.29399
$87$	6.00000i	0.643268i
$88$	− 4.00000i	− 0.426401i
$89$	8.00000	0.847998	0.423999	−	0.905663i	$-0.360626\pi$
$89$	8.00000	0.847998	0.423999	+	0.905663i	$0.360626\pi$
$90$	0	0
$91$	12.0000	1.25794
$92$	− 4.00000i	− 0.417029i
$93$	6.00000i	0.622171i
$94$	4.00000	0.412568
$95$	0	0
$96$	1.00000	0.102062
$97$	− 2.00000i	− 0.203069i	−0.994832	−	0.101535i	$-0.967625\pi$
$97$	− 2.00000i	− 0.203069i	0.994832	−	0.101535i	$-0.0323753\pi$
$98$	3.00000i	0.303046i
$99$	−4.00000	−0.402015
$100$	0	0
$101$	−10.0000	−0.995037	−0.497519	−	0.867453i	$-0.665755\pi$
$101$	−10.0000	−0.995037	−0.497519	+	0.867453i	$0.665755\pi$
$102$	− 4.00000i	− 0.396059i
$103$	− 12.0000i	− 1.18240i	−0.806527	−	0.591198i	$-0.798655\pi$
$103$	− 12.0000i	− 1.18240i	0.806527	−	0.591198i	$-0.201345\pi$
$104$	−6.00000	−0.588348
$105$	0	0
$106$	10.0000	0.971286
$107$	− 8.00000i	− 0.773389i	−0.922208	−	0.386695i	$-0.873617\pi$
$107$	− 8.00000i	− 0.773389i	0.922208	−	0.386695i	$-0.126383\pi$
$108$	− 1.00000i	− 0.0962250i
$109$	16.0000	1.53252	0.766261	−	0.642529i	$-0.222115\pi$
$109$	16.0000	1.53252	0.766261	+	0.642529i	$0.222115\pi$
$110$	0	0
$111$	−10.0000	−0.949158
$112$	2.00000i	0.188982i
$113$	− 14.0000i	− 1.31701i	−0.752577	−	0.658505i	$-0.771189\pi$
$113$	− 14.0000i	− 1.31701i	0.752577	−	0.658505i	$-0.228811\pi$
$114$	−1.00000	−0.0936586
$115$	0	0
$116$	6.00000	0.557086
$117$	6.00000i	0.554700i
$118$	− 10.0000i	− 0.920575i
$119$	8.00000	0.733359
$120$	0	0
$121$	5.00000	0.454545
$122$	2.00000i	0.181071i
$123$	− 4.00000i	− 0.360668i
$124$	6.00000	0.538816
$125$	0	0
$126$	2.00000	0.178174
$127$	20.0000i	1.77471i	0.461084	+	0.887357i	$0.347461\pi$
$127$	20.0000i	1.77471i	−0.461084	+	0.887357i	$0.652539\pi$
$128$	− 1.00000i	− 0.0883883i
$129$	12.0000	1.05654
$130$	0	0
$131$	12.0000	1.04844	0.524222	−	0.851581i	$-0.324356\pi$
$131$	12.0000	1.04844	0.524222	+	0.851581i	$0.324356\pi$
$132$	4.00000i	0.348155i
$133$	− 2.00000i	− 0.173422i
$134$	12.0000	1.03664
$135$	0	0
$136$	−4.00000	−0.342997
$137$	0	0	1.00000			$0$
$137$	0	0	−1.00000			$\pi$
$138$	4.00000i	0.340503i
$139$	4.00000	0.339276	0.169638	−	0.985506i	$-0.445740\pi$
$139$	4.00000	0.339276	0.169638	+	0.985506i	$0.445740\pi$
$140$	0	0
$141$	−4.00000	−0.336861
$142$	8.00000i	0.671345i
$143$	− 24.0000i	− 2.00698i
$144$	−1.00000	−0.0833333
$145$	0	0
$146$	2.00000	0.165521
$147$	− 3.00000i	− 0.247436i
$148$	10.0000i	0.821995i
$149$	−14.0000	−1.14692	−0.573462	−	0.819232i	$-0.694400\pi$
$149$	−14.0000	−1.14692	−0.573462	+	0.819232i	$0.694400\pi$
$150$	0	0
$151$	−2.00000	−0.162758	−0.0813788	−	0.996683i	$-0.525932\pi$
$151$	−2.00000	−0.162758	−0.0813788	+	0.996683i	$0.525932\pi$
$152$	1.00000i	0.0811107i
$153$	4.00000i	0.323381i
$154$	−8.00000	−0.644658
$155$	0	0
$156$	6.00000	0.480384
$157$	− 12.0000i	− 0.957704i	−0.877896	−	0.478852i	$-0.841053\pi$
$157$	− 12.0000i	− 0.957704i	0.877896	−	0.478852i	$-0.158947\pi$
$158$	− 10.0000i	− 0.795557i
$159$	−10.0000	−0.793052
$160$	0	0
$161$	−8.00000	−0.630488
$162$	1.00000i	0.0785674i
$163$	− 16.0000i	− 1.25322i	−0.779334	−	0.626608i	$-0.784443\pi$
$163$	− 16.0000i	− 1.25322i	0.779334	−	0.626608i	$-0.215557\pi$
$164$	−4.00000	−0.312348
$165$	0	0
$166$	−2.00000	−0.155230
$167$	− 24.0000i	− 1.85718i	−0.371113	−	0.928588i	$-0.621024\pi$
$167$	− 24.0000i	− 1.85718i	0.371113	−	0.928588i	$-0.378976\pi$
$168$	− 2.00000i	− 0.154303i
$169$	−23.0000	−1.76923
$170$	0	0
$171$	1.00000	0.0764719
$172$	− 12.0000i	− 0.914991i
$173$	− 18.0000i	− 1.36851i	−0.729241	−	0.684257i	$-0.760127\pi$
$173$	− 18.0000i	− 1.36851i	0.729241	−	0.684257i	$-0.239873\pi$
$174$	−6.00000	−0.454859
$175$	0	0
$176$	4.00000	0.301511
$177$	10.0000i	0.751646i
$178$	8.00000i	0.599625i
$179$	2.00000	0.149487	0.0747435	−	0.997203i	$-0.476186\pi$
$179$	2.00000	0.149487	0.0747435	+	0.997203i	$0.476186\pi$
$180$	0	0
$181$	0	0		−	1.00000i	$-0.5\pi$
$181$	0	0			1.00000i	$0.5\pi$
$182$	12.0000i	0.889499i
$183$	− 2.00000i	− 0.147844i
$184$	4.00000	0.294884
$185$	0	0
$186$	−6.00000	−0.439941
$187$	− 16.0000i	− 1.17004i
$188$	4.00000i	0.291730i
$189$	−2.00000	−0.145479
$190$	0	0
$191$	−8.00000	−0.578860	−0.289430	−	0.957199i	$-0.593466\pi$
$191$	−8.00000	−0.578860	−0.289430	+	0.957199i	$0.593466\pi$
$192$	1.00000i	0.0721688i
$193$	14.0000i	1.00774i	0.863779	+	0.503871i	$0.168091\pi$
$193$	14.0000i	1.00774i	−0.863779	+	0.503871i	$0.831909\pi$
$194$	2.00000	0.143592
$195$	0	0
$196$	−3.00000	−0.214286
$197$	− 22.0000i	− 1.56744i	−0.621117	−	0.783718i	$-0.713321\pi$
$197$	− 22.0000i	− 1.56744i	0.621117	−	0.783718i	$-0.286679\pi$
$198$	− 4.00000i	− 0.284268i
$199$	4.00000	0.283552	0.141776	−	0.989899i	$-0.454719\pi$
$199$	4.00000	0.283552	0.141776	+	0.989899i	$0.454719\pi$
$200$	0	0
$201$	−12.0000	−0.846415
$202$	− 10.0000i	− 0.703598i
$203$	− 12.0000i	− 0.842235i
$204$	4.00000	0.280056
$205$	0	0
$206$	12.0000	0.836080
$207$	− 4.00000i	− 0.278019i
$208$	− 6.00000i	− 0.416025i
$209$	−4.00000	−0.276686
$210$	0	0
$211$	−8.00000	−0.550743	−0.275371	−	0.961338i	$-0.588801\pi$
$211$	−8.00000	−0.550743	−0.275371	+	0.961338i	$0.588801\pi$
$212$	10.0000i	0.686803i
$213$	− 8.00000i	− 0.548151i
$214$	8.00000	0.546869
$215$	0	0
$216$	1.00000	0.0680414
$217$	− 12.0000i	− 0.814613i
$218$	16.0000i	1.08366i
$219$	−2.00000	−0.135147
$220$	0	0
$221$	−24.0000	−1.61441
$222$	− 10.0000i	− 0.671156i
$223$	24.0000i	1.60716i	0.595198	+	0.803579i	$0.297074\pi$
$223$	24.0000i	1.60716i	−0.595198	+	0.803579i	$0.702926\pi$
$224$	−2.00000	−0.133631
$225$	0	0
$226$	14.0000	0.931266
$227$	− 20.0000i	− 1.32745i	−0.747978	−	0.663723i	$-0.768975\pi$
$227$	− 20.0000i	− 1.32745i	0.747978	−	0.663723i	$-0.231025\pi$
$228$	− 1.00000i	− 0.0662266i
$229$	26.0000	1.71813	0.859064	−	0.511868i	$-0.171046\pi$
$229$	26.0000	1.71813	0.859064	+	0.511868i	$0.171046\pi$
$230$	0	0
$231$	8.00000	0.526361
$232$	6.00000i	0.393919i
$233$	− 12.0000i	− 0.786146i	−0.919507	−	0.393073i	$-0.871412\pi$
$233$	− 12.0000i	− 0.786146i	0.919507	−	0.393073i	$-0.128588\pi$
$234$	−6.00000	−0.392232
$235$	0	0
$236$	10.0000	0.650945
$237$	10.0000i	0.649570i
$238$	8.00000i	0.518563i
$239$	0	0		−	1.00000i	$-0.5\pi$
$239$	0	0			1.00000i	$0.5\pi$
$240$	0	0
$241$	10.0000	0.644157	0.322078	−	0.946713i	$-0.395619\pi$
$241$	10.0000	0.644157	0.322078	+	0.946713i	$0.395619\pi$
$242$	5.00000i	0.321412i
$243$	− 1.00000i	− 0.0641500i
$244$	−2.00000	−0.128037
$245$	0	0
$246$	4.00000	0.255031
$247$	6.00000i	0.381771i
$248$	6.00000i	0.381000i
$249$	2.00000	0.126745
$250$	0	0
$251$	0	0		−	1.00000i	$-0.5\pi$
$251$	0	0			1.00000i	$0.5\pi$
$252$	2.00000i	0.125988i
$253$	16.0000i	1.00591i
$254$	−20.0000	−1.25491
$255$	0	0
$256$	1.00000	0.0625000
$257$	18.0000i	1.12281i	0.827541	+	0.561405i	$0.189739\pi$
$257$	18.0000i	1.12281i	−0.827541	+	0.561405i	$0.810261\pi$
$258$	12.0000i	0.747087i
$259$	20.0000	1.24274
$260$	0	0
$261$	6.00000	0.371391
$262$	12.0000i	0.741362i
$263$	12.0000i	0.739952i	0.929041	+	0.369976i	$0.120634\pi$
$263$	12.0000i	0.739952i	−0.929041	+	0.369976i	$0.879366\pi$
$264$	−4.00000	−0.246183
$265$	0	0
$266$	2.00000	0.122628
$267$	− 8.00000i	− 0.489592i
$268$	12.0000i	0.733017i
$269$	−14.0000	−0.853595	−0.426798	−	0.904347i	$-0.640358\pi$
$269$	−14.0000	−0.853595	−0.426798	+	0.904347i	$0.640358\pi$
$270$	0	0
$271$	20.0000	1.21491	0.607457	−	0.794353i	$-0.292190\pi$
$271$	20.0000	1.21491	0.607457	+	0.794353i	$0.292190\pi$
$272$	− 4.00000i	− 0.242536i
$273$	− 12.0000i	− 0.726273i
$274$	0	0
$275$	0	0
$276$	−4.00000	−0.240772
$277$	− 28.0000i	− 1.68236i	−0.540758	−	0.841178i	$-0.681862\pi$
$277$	− 28.0000i	− 1.68236i	0.540758	−	0.841178i	$-0.318138\pi$
$278$	4.00000i	0.239904i
$279$	6.00000	0.359211
$280$	0	0
$281$	4.00000	0.238620	0.119310	−	0.992857i	$-0.461932\pi$
$281$	4.00000	0.238620	0.119310	+	0.992857i	$0.461932\pi$
$282$	− 4.00000i	− 0.238197i
$283$	28.0000i	1.66443i	0.554455	+	0.832214i	$0.312927\pi$
$283$	28.0000i	1.66443i	−0.554455	+	0.832214i	$0.687073\pi$
$284$	−8.00000	−0.474713
$285$	0	0
$286$	24.0000	1.41915
$287$	8.00000i	0.472225i
$288$	− 1.00000i	− 0.0589256i
$289$	1.00000	0.0588235
$290$	0	0
$291$	−2.00000	−0.117242
$292$	2.00000i	0.117041i
$293$	2.00000i	0.116841i	0.998292	+	0.0584206i	$0.0186065\pi$
$293$	2.00000i	0.116841i	−0.998292	+	0.0584206i	$0.981394\pi$
$294$	3.00000	0.174964
$295$	0	0
$296$	−10.0000	−0.581238
$297$	4.00000i	0.232104i
$298$	− 14.0000i	− 0.810998i
$299$	24.0000	1.38796
$300$	0	0
$301$	−24.0000	−1.38334
$302$	− 2.00000i	− 0.115087i
$303$	10.0000i	0.574485i
$304$	−1.00000	−0.0573539
$305$	0	0
$306$	−4.00000	−0.228665
$307$	12.0000i	0.684876i	0.939540	+	0.342438i	$0.111253\pi$
$307$	12.0000i	0.684876i	−0.939540	+	0.342438i	$0.888747\pi$
$308$	− 8.00000i	− 0.455842i
$309$	−12.0000	−0.682656
$310$	0	0
$311$	16.0000	0.907277	0.453638	−	0.891186i	$-0.350126\pi$
$311$	16.0000	0.907277	0.453638	+	0.891186i	$0.350126\pi$
$312$	6.00000i	0.339683i
$313$	6.00000i	0.339140i	0.985518	+	0.169570i	$0.0542379\pi$
$313$	6.00000i	0.339140i	−0.985518	+	0.169570i	$0.945762\pi$
$314$	12.0000	0.677199
$315$	0	0
$316$	10.0000	0.562544
$317$	− 18.0000i	− 1.01098i	−0.862832	−	0.505490i	$-0.831312\pi$
$317$	− 18.0000i	− 1.01098i	0.862832	−	0.505490i	$-0.168688\pi$
$318$	− 10.0000i	− 0.560772i
$319$	−24.0000	−1.34374
$320$	0	0
$321$	−8.00000	−0.446516
$322$	− 8.00000i	− 0.445823i
$323$	4.00000i	0.222566i
$324$	−1.00000	−0.0555556
$325$	0	0
$326$	16.0000	0.886158
$327$	− 16.0000i	− 0.884802i
$328$	− 4.00000i	− 0.220863i
$329$	8.00000	0.441054
$330$	0	0
$331$	−12.0000	−0.659580	−0.329790	−	0.944054i	$-0.606978\pi$
$331$	−12.0000	−0.659580	−0.329790	+	0.944054i	$0.606978\pi$
$332$	− 2.00000i	− 0.109764i
$333$	10.0000i	0.547997i
$334$	24.0000	1.31322
$335$	0	0
$336$	2.00000	0.109109
$337$	34.0000i	1.85210i	0.377403	+	0.926049i	$0.376817\pi$
$337$	34.0000i	1.85210i	−0.377403	+	0.926049i	$0.623183\pi$
$338$	− 23.0000i	− 1.25104i
$339$	−14.0000	−0.760376
$340$	0	0
$341$	−24.0000	−1.29967
$342$	1.00000i	0.0540738i
$343$	20.0000i	1.07990i
$344$	12.0000	0.646997
$345$	0	0
$346$	18.0000	0.967686
$347$	2.00000i	0.107366i	0.998558	+	0.0536828i	$0.0170960\pi$
$347$	2.00000i	0.107366i	−0.998558	+	0.0536828i	$0.982904\pi$
$348$	− 6.00000i	− 0.321634i
$349$	−22.0000	−1.17763	−0.588817	−	0.808267i	$-0.700406\pi$
$349$	−22.0000	−1.17763	−0.588817	+	0.808267i	$0.700406\pi$
$350$	0	0
$351$	6.00000	0.320256
$352$	4.00000i	0.213201i
$353$	0	0	1.00000			$0$
$353$	0	0	−1.00000			$\pi$
$354$	−10.0000	−0.531494
$355$	0	0
$356$	−8.00000	−0.423999
$357$	− 8.00000i	− 0.423405i
$358$	2.00000i	0.105703i
$359$	0	0		−	1.00000i	$-0.5\pi$
$359$	0	0			1.00000i	$0.5\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	− 5.00000i	− 0.262432i
$364$	−12.0000	−0.628971
$365$	0	0
$366$	2.00000	0.104542
$367$	22.0000i	1.14839i	0.818718	+	0.574195i	$0.194685\pi$
$367$	22.0000i	1.14839i	−0.818718	+	0.574195i	$0.805315\pi$
$368$	4.00000i	0.208514i
$369$	−4.00000	−0.208232
$370$	0	0
$371$	20.0000	1.03835
$372$	− 6.00000i	− 0.311086i
$373$	14.0000i	0.724893i	0.932005	+	0.362446i	$0.118058\pi$
$373$	14.0000i	0.724893i	−0.932005	+	0.362446i	$0.881942\pi$
$374$	16.0000	0.827340
$375$	0	0
$376$	−4.00000	−0.206284
$377$	36.0000i	1.85409i
$378$	− 2.00000i	− 0.102869i
$379$	−8.00000	−0.410932	−0.205466	−	0.978664i	$-0.565871\pi$
$379$	−8.00000	−0.410932	−0.205466	+	0.978664i	$0.565871\pi$
$380$	0	0
$381$	20.0000	1.02463
$382$	− 8.00000i	− 0.409316i
$383$	− 16.0000i	− 0.817562i	−0.912633	−	0.408781i	$-0.865954\pi$
$383$	− 16.0000i	− 0.817562i	0.912633	−	0.408781i	$-0.134046\pi$
$384$	−1.00000	−0.0510310
$385$	0	0
$386$	−14.0000	−0.712581
$387$	− 12.0000i	− 0.609994i
$388$	2.00000i	0.101535i
$389$	10.0000	0.507020	0.253510	−	0.967333i	$-0.418415\pi$
$389$	10.0000	0.507020	0.253510	+	0.967333i	$0.418415\pi$
$390$	0	0
$391$	16.0000	0.809155
$392$	− 3.00000i	− 0.151523i
$393$	− 12.0000i	− 0.605320i
$394$	22.0000	1.10834
$395$	0	0
$396$	4.00000	0.201008
$397$	24.0000i	1.20453i	0.798298	+	0.602263i	$0.205734\pi$
$397$	24.0000i	1.20453i	−0.798298	+	0.602263i	$0.794266\pi$
$398$	4.00000i	0.200502i
$399$	−2.00000	−0.100125
$400$	0	0
$401$	−24.0000	−1.19850	−0.599251	−	0.800561i	$-0.704535\pi$
$401$	−24.0000	−1.19850	−0.599251	+	0.800561i	$0.704535\pi$
$402$	− 12.0000i	− 0.598506i
$403$	36.0000i	1.79329i
$404$	10.0000	0.497519
$405$	0	0
$406$	12.0000	0.595550
$407$	− 40.0000i	− 1.98273i
$408$	4.00000i	0.198030i
$409$	6.00000	0.296681	0.148340	−	0.988936i	$-0.452607\pi$
$409$	6.00000	0.296681	0.148340	+	0.988936i	$0.452607\pi$
$410$	0	0
$411$	0	0
$412$	12.0000i	0.591198i
$413$	− 20.0000i	− 0.984136i
$414$	4.00000	0.196589
$415$	0	0
$416$	6.00000	0.294174
$417$	− 4.00000i	− 0.195881i
$418$	− 4.00000i	− 0.195646i
$419$	0	0		−	1.00000i	$-0.5\pi$
$419$	0	0			1.00000i	$0.5\pi$
$420$	0	0
$421$	−8.00000	−0.389896	−0.194948	−	0.980814i	$-0.562454\pi$
$421$	−8.00000	−0.389896	−0.194948	+	0.980814i	$0.562454\pi$
$422$	− 8.00000i	− 0.389434i
$423$	4.00000i	0.194487i
$424$	−10.0000	−0.485643
$425$	0	0
$426$	8.00000	0.387601
$427$	4.00000i	0.193574i
$428$	8.00000i	0.386695i
$429$	−24.0000	−1.15873
$430$	0	0
$431$	−40.0000	−1.92673	−0.963366	−	0.268190i	$-0.913575\pi$
$431$	−40.0000	−1.92673	−0.963366	+	0.268190i	$0.913575\pi$
$432$	1.00000i	0.0481125i
$433$	30.0000i	1.44171i	0.693087	+	0.720854i	$0.256250\pi$
$433$	30.0000i	1.44171i	−0.693087	+	0.720854i	$0.743750\pi$
$434$	12.0000	0.576018
$435$	0	0
$436$	−16.0000	−0.766261
$437$	− 4.00000i	− 0.191346i
$438$	− 2.00000i	− 0.0955637i
$439$	38.0000	1.81364	0.906821	−	0.421517i	$-0.138502\pi$
$439$	38.0000	1.81364	0.906821	+	0.421517i	$0.138502\pi$
$440$	0	0
$441$	−3.00000	−0.142857
$442$	− 24.0000i	− 1.14156i
$443$	6.00000i	0.285069i	0.989790	+	0.142534i	$0.0455251\pi$
$443$	6.00000i	0.285069i	−0.989790	+	0.142534i	$0.954475\pi$
$444$	10.0000	0.474579
$445$	0	0
$446$	−24.0000	−1.13643
$447$	14.0000i	0.662177i
$448$	− 2.00000i	− 0.0944911i
$449$	−12.0000	−0.566315	−0.283158	−	0.959073i	$-0.591382\pi$
$449$	−12.0000	−0.566315	−0.283158	+	0.959073i	$0.591382\pi$
$450$	0	0
$451$	16.0000	0.753411
$452$	14.0000i	0.658505i
$453$	2.00000i	0.0939682i
$454$	20.0000	0.938647
$455$	0	0
$456$	1.00000	0.0468293
$457$	22.0000i	1.02912i	0.857455	+	0.514558i	$0.172044\pi$
$457$	22.0000i	1.02912i	−0.857455	+	0.514558i	$0.827956\pi$
$458$	26.0000i	1.21490i
$459$	4.00000	0.186704
$460$	0	0
$461$	18.0000	0.838344	0.419172	−	0.907907i	$-0.362320\pi$
$461$	18.0000	0.838344	0.419172	+	0.907907i	$0.362320\pi$
$462$	8.00000i	0.372194i
$463$	− 18.0000i	− 0.836531i	−0.908325	−	0.418265i	$-0.862638\pi$
$463$	− 18.0000i	− 0.836531i	0.908325	−	0.418265i	$-0.137362\pi$
$464$	−6.00000	−0.278543
$465$	0	0
$466$	12.0000	0.555889
$467$	22.0000i	1.01804i	0.860755	+	0.509019i	$0.169992\pi$
$467$	22.0000i	1.01804i	−0.860755	+	0.509019i	$0.830008\pi$
$468$	− 6.00000i	− 0.277350i
$469$	24.0000	1.10822
$470$	0	0
$471$	−12.0000	−0.552931
$472$	10.0000i	0.460287i
$473$	48.0000i	2.20704i
$474$	−10.0000	−0.459315
$475$	0	0
$476$	−8.00000	−0.366679
$477$	10.0000i	0.457869i
$478$	0	0
$479$	24.0000	1.09659	0.548294	−	0.836286i	$-0.315277\pi$
$479$	24.0000	1.09659	0.548294	+	0.836286i	$0.315277\pi$
$480$	0	0
$481$	−60.0000	−2.73576
$482$	10.0000i	0.455488i
$483$	8.00000i	0.364013i
$484$	−5.00000	−0.227273
$485$	0	0
$486$	1.00000	0.0453609
$487$	− 20.0000i	− 0.906287i	−0.891438	−	0.453143i	$-0.850303\pi$
$487$	− 20.0000i	− 0.906287i	0.891438	−	0.453143i	$-0.149697\pi$
$488$	− 2.00000i	− 0.0905357i
$489$	−16.0000	−0.723545
$490$	0	0
$491$	12.0000	0.541552	0.270776	−	0.962642i	$-0.412720\pi$
$491$	12.0000	0.541552	0.270776	+	0.962642i	$0.412720\pi$
$492$	4.00000i	0.180334i
$493$	24.0000i	1.08091i
$494$	−6.00000	−0.269953
$495$	0	0
$496$	−6.00000	−0.269408
$497$	16.0000i	0.717698i
$498$	2.00000i	0.0896221i
$499$	4.00000	0.179065	0.0895323	−	0.995984i	$-0.471463\pi$
$499$	4.00000	0.179065	0.0895323	+	0.995984i	$0.471463\pi$
$500$	0	0
$501$	−24.0000	−1.07224
$502$	0	0
$503$	− 8.00000i	− 0.356702i	−0.983967	−	0.178351i	$-0.942924\pi$
$503$	− 8.00000i	− 0.356702i	0.983967	−	0.178351i	$-0.0570763\pi$
$504$	−2.00000	−0.0890871
$505$	0	0
$506$	−16.0000	−0.711287
$507$	23.0000i	1.02147i
$508$	− 20.0000i	− 0.887357i
$509$	−38.0000	−1.68432	−0.842160	−	0.539227i	$-0.818716\pi$
$509$	−38.0000	−1.68432	−0.842160	+	0.539227i	$0.818716\pi$
$510$	0	0
$511$	4.00000	0.176950
$512$	1.00000i	0.0441942i
$513$	− 1.00000i	− 0.0441511i
$514$	−18.0000	−0.793946
$515$	0	0
$516$	−12.0000	−0.528271
$517$	− 16.0000i	− 0.703679i
$518$	20.0000i	0.878750i
$519$	−18.0000	−0.790112
$520$	0	0
$521$	40.0000	1.75243	0.876216	−	0.481919i	$-0.160060\pi$
$521$	40.0000	1.75243	0.876216	+	0.481919i	$0.160060\pi$
$522$	6.00000i	0.262613i
$523$	4.00000i	0.174908i	0.996169	+	0.0874539i	$0.0278730\pi$
$523$	4.00000i	0.174908i	−0.996169	+	0.0874539i	$0.972127\pi$
$524$	−12.0000	−0.524222
$525$	0	0
$526$	−12.0000	−0.523225
$527$	24.0000i	1.04546i
$528$	− 4.00000i	− 0.174078i
$529$	7.00000	0.304348
$530$	0	0
$531$	10.0000	0.433963
$532$	2.00000i	0.0867110i
$533$	− 24.0000i	− 1.03956i
$534$	8.00000	0.346194
$535$	0	0
$536$	−12.0000	−0.518321
$537$	− 2.00000i	− 0.0863064i
$538$	− 14.0000i	− 0.603583i
$539$	12.0000	0.516877
$540$	0	0
$541$	38.0000	1.63375	0.816874	−	0.576816i	$-0.195705\pi$
$541$	38.0000	1.63375	0.816874	+	0.576816i	$0.195705\pi$
$542$	20.0000i	0.859074i
$543$	0	0
$544$	4.00000	0.171499
$545$	0	0
$546$	12.0000	0.513553
$547$	4.00000i	0.171028i	0.996337	+	0.0855138i	$0.0272532\pi$
$547$	4.00000i	0.171028i	−0.996337	+	0.0855138i	$0.972747\pi$
$548$	0	0
$549$	−2.00000	−0.0853579
$550$	0	0
$551$	6.00000	0.255609
$552$	− 4.00000i	− 0.170251i
$553$	− 20.0000i	− 0.850487i
$554$	28.0000	1.18961
$555$	0	0
$556$	−4.00000	−0.169638
$557$	− 10.0000i	− 0.423714i	−0.977301	−	0.211857i	$-0.932049\pi$
$557$	− 10.0000i	− 0.423714i	0.977301	−	0.211857i	$-0.0679510\pi$
$558$	6.00000i	0.254000i
$559$	72.0000	3.04528
$560$	0	0
$561$	−16.0000	−0.675521
$562$	4.00000i	0.168730i
$563$	24.0000i	1.01148i	0.862686	+	0.505740i	$0.168780\pi$
$563$	24.0000i	1.01148i	−0.862686	+	0.505740i	$0.831220\pi$
$564$	4.00000	0.168430
$565$	0	0
$566$	−28.0000	−1.17693
$567$	2.00000i	0.0839921i
$568$	− 8.00000i	− 0.335673i
$569$	−12.0000	−0.503066	−0.251533	−	0.967849i	$-0.580935\pi$
$569$	−12.0000	−0.503066	−0.251533	+	0.967849i	$0.580935\pi$
$570$	0	0
$571$	−28.0000	−1.17176	−0.585882	−	0.810397i	$-0.699252\pi$
$571$	−28.0000	−1.17176	−0.585882	+	0.810397i	$0.699252\pi$
$572$	24.0000i	1.00349i
$573$	8.00000i	0.334205i
$574$	−8.00000	−0.333914
$575$	0	0
$576$	1.00000	0.0416667
$577$	− 10.0000i	− 0.416305i	−0.978096	−	0.208153i	$-0.933255\pi$
$577$	− 10.0000i	− 0.416305i	0.978096	−	0.208153i	$-0.0667451\pi$
$578$	1.00000i	0.0415945i
$579$	14.0000	0.581820
$580$	0	0
$581$	−4.00000	−0.165948
$582$	− 2.00000i	− 0.0829027i
$583$	− 40.0000i	− 1.65663i
$584$	−2.00000	−0.0827606
$585$	0	0
$586$	−2.00000	−0.0826192
$587$	− 22.0000i	− 0.908037i	−0.890992	−	0.454019i	$-0.849990\pi$
$587$	− 22.0000i	− 0.908037i	0.890992	−	0.454019i	$-0.150010\pi$
$588$	3.00000i	0.123718i
$589$	6.00000	0.247226
$590$	0	0
$591$	−22.0000	−0.904959
$592$	− 10.0000i	− 0.410997i
$593$	− 8.00000i	− 0.328521i	−0.986417	−	0.164260i	$-0.947476\pi$
$593$	− 8.00000i	− 0.328521i	0.986417	−	0.164260i	$-0.0525237\pi$
$594$	−4.00000	−0.164122
$595$	0	0
$596$	14.0000	0.573462
$597$	− 4.00000i	− 0.163709i
$598$	24.0000i	0.981433i
$599$	28.0000	1.14405	0.572024	−	0.820237i	$-0.306158\pi$
$599$	28.0000	1.14405	0.572024	+	0.820237i	$0.306158\pi$
$600$	0	0
$601$	−6.00000	−0.244745	−0.122373	−	0.992484i	$-0.539050\pi$
$601$	−6.00000	−0.244745	−0.122373	+	0.992484i	$0.539050\pi$
$602$	− 24.0000i	− 0.978167i
$603$	12.0000i	0.488678i
$604$	2.00000	0.0813788
$605$	0	0
$606$	−10.0000	−0.406222
$607$	28.0000i	1.13648i	0.822861	+	0.568242i	$0.192376\pi$
$607$	28.0000i	1.13648i	−0.822861	+	0.568242i	$0.807624\pi$
$608$	− 1.00000i	− 0.0405554i
$609$	−12.0000	−0.486265
$610$	0	0
$611$	−24.0000	−0.970936
$612$	− 4.00000i	− 0.161690i
$613$	− 16.0000i	− 0.646234i	−0.946359	−	0.323117i	$-0.895269\pi$
$613$	− 16.0000i	− 0.646234i	0.946359	−	0.323117i	$-0.104731\pi$
$614$	−12.0000	−0.484281
$615$	0	0
$616$	8.00000	0.322329
$617$	48.0000i	1.93241i	0.257780	+	0.966204i	$0.417009\pi$
$617$	48.0000i	1.93241i	−0.257780	+	0.966204i	$0.582991\pi$
$618$	− 12.0000i	− 0.482711i
$619$	−4.00000	−0.160774	−0.0803868	−	0.996764i	$-0.525616\pi$
$619$	−4.00000	−0.160774	−0.0803868	+	0.996764i	$0.525616\pi$
$620$	0	0
$621$	−4.00000	−0.160514
$622$	16.0000i	0.641542i
$623$	16.0000i	0.641026i
$624$	−6.00000	−0.240192
$625$	0	0
$626$	−6.00000	−0.239808
$627$	4.00000i	0.159745i
$628$	12.0000i	0.478852i
$629$	−40.0000	−1.59490
$630$	0	0
$631$	8.00000	0.318475	0.159237	−	0.987240i	$-0.449096\pi$
$631$	8.00000	0.318475	0.159237	+	0.987240i	$0.449096\pi$
$632$	10.0000i	0.397779i
$633$	8.00000i	0.317971i
$634$	18.0000	0.714871
$635$	0	0
$636$	10.0000	0.396526
$637$	− 18.0000i	− 0.713186i
$638$	− 24.0000i	− 0.950169i
$639$	−8.00000	−0.316475
$640$	0	0
$641$	−20.0000	−0.789953	−0.394976	−	0.918691i	$-0.629247\pi$
$641$	−20.0000	−0.789953	−0.394976	+	0.918691i	$0.629247\pi$
$642$	− 8.00000i	− 0.315735i
$643$	− 20.0000i	− 0.788723i	−0.918955	−	0.394362i	$-0.870966\pi$
$643$	− 20.0000i	− 0.788723i	0.918955	−	0.394362i	$-0.129034\pi$
$644$	8.00000	0.315244
$645$	0	0
$646$	−4.00000	−0.157378
$647$	24.0000i	0.943537i	0.881722	+	0.471769i	$0.156384\pi$
$647$	24.0000i	0.943537i	−0.881722	+	0.471769i	$0.843616\pi$
$648$	− 1.00000i	− 0.0392837i
$649$	−40.0000	−1.57014
$650$	0	0
$651$	−12.0000	−0.470317
$652$	16.0000i	0.626608i
$653$	− 6.00000i	− 0.234798i	−0.993085	−	0.117399i	$-0.962544\pi$
$653$	− 6.00000i	− 0.234798i	0.993085	−	0.117399i	$-0.0374557\pi$
$654$	16.0000	0.625650
$655$	0	0
$656$	4.00000	0.156174
$657$	2.00000i	0.0780274i
$658$	8.00000i	0.311872i
$659$	−22.0000	−0.856998	−0.428499	−	0.903542i	$-0.640958\pi$
$659$	−22.0000	−0.856998	−0.428499	+	0.903542i	$0.640958\pi$
$660$	0	0
$661$	−8.00000	−0.311164	−0.155582	−	0.987823i	$-0.549725\pi$
$661$	−8.00000	−0.311164	−0.155582	+	0.987823i	$0.549725\pi$
$662$	− 12.0000i	− 0.466393i
$663$	24.0000i	0.932083i
$664$	2.00000	0.0776151
$665$	0	0
$666$	−10.0000	−0.387492
$667$	− 24.0000i	− 0.929284i
$668$	24.0000i	0.928588i
$669$	24.0000	0.927894
$670$	0	0
$671$	8.00000	0.308837
$672$	2.00000i	0.0771517i
$673$	26.0000i	1.00223i	0.865382	+	0.501113i	$0.167076\pi$
$673$	26.0000i	1.00223i	−0.865382	+	0.501113i	$0.832924\pi$
$674$	−34.0000	−1.30963
$675$	0	0
$676$	23.0000	0.884615
$677$	6.00000i	0.230599i	0.993331	+	0.115299i	$0.0367827\pi$
$677$	6.00000i	0.230599i	−0.993331	+	0.115299i	$0.963217\pi$
$678$	− 14.0000i	− 0.537667i
$679$	4.00000	0.153506
$680$	0	0
$681$	−20.0000	−0.766402
$682$	− 24.0000i	− 0.919007i
$683$	− 12.0000i	− 0.459167i	−0.973289	−	0.229584i	$-0.926264\pi$
$683$	− 12.0000i	− 0.459167i	0.973289	−	0.229584i	$-0.0737364\pi$
$684$	−1.00000	−0.0382360
$685$	0	0
$686$	−20.0000	−0.763604
$687$	− 26.0000i	− 0.991962i
$688$	12.0000i	0.457496i
$689$	−60.0000	−2.28582
$690$	0	0
$691$	4.00000	0.152167	0.0760836	−	0.997101i	$-0.475758\pi$
$691$	4.00000	0.152167	0.0760836	+	0.997101i	$0.475758\pi$
$692$	18.0000i	0.684257i
$693$	− 8.00000i	− 0.303895i
$694$	−2.00000	−0.0759190
$695$	0	0
$696$	6.00000	0.227429
$697$	− 16.0000i	− 0.606043i
$698$	− 22.0000i	− 0.832712i
$699$	−12.0000	−0.453882
$700$	0	0
$701$	10.0000	0.377695	0.188847	−	0.982006i	$-0.439525\pi$
$701$	10.0000	0.377695	0.188847	+	0.982006i	$0.439525\pi$
$702$	6.00000i	0.226455i
$703$	10.0000i	0.377157i
$704$	−4.00000	−0.150756
$705$	0	0
$706$	0	0
$707$	− 20.0000i	− 0.752177i
$708$	− 10.0000i	− 0.375823i
$709$	−14.0000	−0.525781	−0.262891	−	0.964826i	$-0.584676\pi$
$709$	−14.0000	−0.525781	−0.262891	+	0.964826i	$0.584676\pi$
$710$	0	0
$711$	10.0000	0.375029
$712$	− 8.00000i	− 0.299813i
$713$	− 24.0000i	− 0.898807i
$714$	8.00000	0.299392
$715$	0	0
$716$	−2.00000	−0.0747435
$717$	0	0
$718$	0	0
$719$	−40.0000	−1.49175	−0.745874	−	0.666087i	$-0.767968\pi$
$719$	−40.0000	−1.49175	−0.745874	+	0.666087i	$0.767968\pi$
$720$	0	0
$721$	24.0000	0.893807
$722$	1.00000i	0.0372161i
$723$	− 10.0000i	− 0.371904i
$724$	0	0
$725$	0	0
$726$	5.00000	0.185567
$727$	− 26.0000i	− 0.964287i	−0.876092	−	0.482143i	$-0.839858\pi$
$727$	− 26.0000i	− 0.964287i	0.876092	−	0.482143i	$-0.160142\pi$
$728$	− 12.0000i	− 0.444750i
$729$	−1.00000	−0.0370370
$730$	0	0
$731$	48.0000	1.77534
$732$	2.00000i	0.0739221i
$733$	− 8.00000i	− 0.295487i	−0.989026	−	0.147743i	$-0.952799\pi$
$733$	− 8.00000i	− 0.295487i	0.989026	−	0.147743i	$-0.0472010\pi$
$734$	−22.0000	−0.812035
$735$	0	0
$736$	−4.00000	−0.147442
$737$	− 48.0000i	− 1.76810i
$738$	− 4.00000i	− 0.147242i
$739$	20.0000	0.735712	0.367856	−	0.929883i	$-0.380092\pi$
$739$	20.0000	0.735712	0.367856	+	0.929883i	$0.380092\pi$
$740$	0	0
$741$	6.00000	0.220416
$742$	20.0000i	0.734223i
$743$	48.0000i	1.76095i	0.474093	+	0.880475i	$0.342776\pi$
$743$	48.0000i	1.76095i	−0.474093	+	0.880475i	$0.657224\pi$
$744$	6.00000	0.219971
$745$	0	0
$746$	−14.0000	−0.512576
$747$	− 2.00000i	− 0.0731762i
$748$	16.0000i	0.585018i
$749$	16.0000	0.584627
$750$	0	0
$751$	−38.0000	−1.38664	−0.693320	−	0.720630i	$-0.743853\pi$
$751$	−38.0000	−1.38664	−0.693320	+	0.720630i	$0.743853\pi$
$752$	− 4.00000i	− 0.145865i
$753$	0	0
$754$	−36.0000	−1.31104
$755$	0	0
$756$	2.00000	0.0727393
$757$	− 8.00000i	− 0.290765i	−0.989376	−	0.145382i	$-0.953559\pi$
$757$	− 8.00000i	− 0.290765i	0.989376	−	0.145382i	$-0.0464413\pi$
$758$	− 8.00000i	− 0.290573i
$759$	16.0000	0.580763
$760$	0	0
$761$	−34.0000	−1.23250	−0.616250	−	0.787551i	$-0.711349\pi$
$761$	−34.0000	−1.23250	−0.616250	+	0.787551i	$0.711349\pi$
$762$	20.0000i	0.724524i
$763$	32.0000i	1.15848i
$764$	8.00000	0.289430
$765$	0	0
$766$	16.0000	0.578103
$767$	60.0000i	2.16647i
$768$	− 1.00000i	− 0.0360844i
$769$	18.0000	0.649097	0.324548	−	0.945869i	$-0.394788\pi$
$769$	18.0000	0.649097	0.324548	+	0.945869i	$0.394788\pi$
$770$	0	0
$771$	18.0000	0.648254
$772$	− 14.0000i	− 0.503871i
$773$	6.00000i	0.215805i	0.994161	+	0.107903i	$0.0344134\pi$
$773$	6.00000i	0.215805i	−0.994161	+	0.107903i	$0.965587\pi$
$774$	12.0000	0.431331
$775$	0	0
$776$	−2.00000	−0.0717958
$777$	− 20.0000i	− 0.717496i
$778$	10.0000i	0.358517i
$779$	−4.00000	−0.143315
$780$	0	0
$781$	32.0000	1.14505
$782$	16.0000i	0.572159i
$783$	− 6.00000i	− 0.214423i
$784$	3.00000	0.107143
$785$	0	0
$786$	12.0000	0.428026
$787$	− 4.00000i	− 0.142585i	−0.997455	−	0.0712923i	$-0.977288\pi$
$787$	− 4.00000i	− 0.142585i	0.997455	−	0.0712923i	$-0.0227123\pi$
$788$	22.0000i	0.783718i
$789$	12.0000	0.427211
$790$	0	0
$791$	28.0000	0.995565
$792$	4.00000i	0.142134i
$793$	− 12.0000i	− 0.426132i
$794$	−24.0000	−0.851728
$795$	0	0
$796$	−4.00000	−0.141776
$797$	− 2.00000i	− 0.0708436i	−0.999372	−	0.0354218i	$-0.988723\pi$
$797$	− 2.00000i	− 0.0708436i	0.999372	−	0.0354218i	$-0.0112775\pi$
$798$	− 2.00000i	− 0.0707992i
$799$	−16.0000	−0.566039
$800$	0	0
$801$	−8.00000	−0.282666
$802$	− 24.0000i	− 0.847469i
$803$	− 8.00000i	− 0.282314i
$804$	12.0000	0.423207
$805$	0	0
$806$	−36.0000	−1.26805
$807$	14.0000i	0.492823i
$808$	10.0000i	0.351799i
$809$	42.0000	1.47664	0.738321	−	0.674450i	$-0.235619\pi$
$809$	42.0000	1.47664	0.738321	+	0.674450i	$0.235619\pi$
$810$	0	0
$811$	−8.00000	−0.280918	−0.140459	−	0.990086i	$-0.544858\pi$
$811$	−8.00000	−0.280918	−0.140459	+	0.990086i	$0.544858\pi$
$812$	12.0000i	0.421117i
$813$	− 20.0000i	− 0.701431i
$814$	40.0000	1.40200
$815$	0	0
$816$	−4.00000	−0.140028
$817$	− 12.0000i	− 0.419827i
$818$	6.00000i	0.209785i
$819$	−12.0000	−0.419314
$820$	0	0
$821$	−42.0000	−1.46581	−0.732905	−	0.680331i	$-0.761836\pi$
$821$	−42.0000	−1.46581	−0.732905	+	0.680331i	$0.761836\pi$
$822$	0	0
$823$	− 26.0000i	− 0.906303i	−0.891434	−	0.453152i	$-0.850300\pi$
$823$	− 26.0000i	− 0.906303i	0.891434	−	0.453152i	$-0.149700\pi$
$824$	−12.0000	−0.418040
$825$	0	0
$826$	20.0000	0.695889
$827$	− 48.0000i	− 1.66912i	−0.550914	−	0.834562i	$-0.685721\pi$
$827$	− 48.0000i	− 1.66912i	0.550914	−	0.834562i	$-0.314279\pi$
$828$	4.00000i	0.139010i
$829$	40.0000	1.38926	0.694629	−	0.719368i	$-0.255569\pi$
$829$	40.0000	1.38926	0.694629	+	0.719368i	$0.255569\pi$
$830$	0	0
$831$	−28.0000	−0.971309
$832$	6.00000i	0.208013i
$833$	− 12.0000i	− 0.415775i
$834$	4.00000	0.138509
$835$	0	0
$836$	4.00000	0.138343
$837$	− 6.00000i	− 0.207390i
$838$	0	0
$839$	0	0		−	1.00000i	$-0.5\pi$
$839$	0	0			1.00000i	$0.5\pi$
$840$	0	0
$841$	7.00000	0.241379
$842$	− 8.00000i	− 0.275698i
$843$	− 4.00000i	− 0.137767i
$844$	8.00000	0.275371
$845$	0	0
$846$	−4.00000	−0.137523
$847$	10.0000i	0.343604i
$848$	− 10.0000i	− 0.343401i
$849$	28.0000	0.960958
$850$	0	0
$851$	40.0000	1.37118
$852$	8.00000i	0.274075i
$853$	− 36.0000i	− 1.23262i	−0.787505	−	0.616308i	$-0.788628\pi$
$853$	− 36.0000i	− 1.23262i	0.787505	−	0.616308i	$-0.211372\pi$
$854$	−4.00000	−0.136877
$855$	0	0
$856$	−8.00000	−0.273434
$857$	− 30.0000i	− 1.02478i	−0.858753	−	0.512390i	$-0.828760\pi$
$857$	− 30.0000i	− 1.02478i	0.858753	−	0.512390i	$-0.171240\pi$
$858$	− 24.0000i	− 0.819346i
$859$	−4.00000	−0.136478	−0.0682391	−	0.997669i	$-0.521738\pi$
$859$	−4.00000	−0.136478	−0.0682391	+	0.997669i	$0.521738\pi$
$860$	0	0
$861$	8.00000	0.272639
$862$	− 40.0000i	− 1.36241i
$863$	32.0000i	1.08929i	0.838666	+	0.544646i	$0.183336\pi$
$863$	32.0000i	1.08929i	−0.838666	+	0.544646i	$0.816664\pi$
$864$	−1.00000	−0.0340207
$865$	0	0
$866$	−30.0000	−1.01944
$867$	− 1.00000i	− 0.0339618i
$868$	12.0000i	0.407307i
$869$	−40.0000	−1.35691
$870$	0	0
$871$	−72.0000	−2.43963
$872$	− 16.0000i	− 0.541828i
$873$	2.00000i	0.0676897i
$874$	4.00000	0.135302
$875$	0	0
$876$	2.00000	0.0675737
$877$	34.0000i	1.14810i	0.818821	+	0.574049i	$0.194628\pi$
$877$	34.0000i	1.14810i	−0.818821	+	0.574049i	$0.805372\pi$
$878$	38.0000i	1.28244i
$879$	2.00000	0.0674583
$880$	0	0
$881$	18.0000	0.606435	0.303218	−	0.952921i	$-0.401939\pi$
$881$	18.0000	0.606435	0.303218	+	0.952921i	$0.401939\pi$
$882$	− 3.00000i	− 0.101015i
$883$	36.0000i	1.21150i	0.795656	+	0.605748i	$0.207126\pi$
$883$	36.0000i	1.21150i	−0.795656	+	0.605748i	$0.792874\pi$
$884$	24.0000	0.807207
$885$	0	0
$886$	−6.00000	−0.201574
$887$	− 48.0000i	− 1.61168i	−0.592132	−	0.805841i	$-0.701714\pi$
$887$	− 48.0000i	− 1.61168i	0.592132	−	0.805841i	$-0.298286\pi$
$888$	10.0000i	0.335578i
$889$	−40.0000	−1.34156
$890$	0	0
$891$	4.00000	0.134005
$892$	− 24.0000i	− 0.803579i
$893$	4.00000i	0.133855i
$894$	−14.0000	−0.468230
$895$	0	0
$896$	2.00000	0.0668153
$897$	− 24.0000i	− 0.801337i
$898$	− 12.0000i	− 0.400445i
$899$	36.0000	1.20067
$900$	0	0
$901$	−40.0000	−1.33259
$902$	16.0000i	0.532742i
$903$	24.0000i	0.798670i
$904$	−14.0000	−0.465633
$905$	0	0
$906$	−2.00000	−0.0664455
$907$	20.0000i	0.664089i	0.943264	+	0.332045i	$0.107738\pi$
$907$	20.0000i	0.664089i	−0.943264	+	0.332045i	$0.892262\pi$
$908$	20.0000i	0.663723i
$909$	10.0000	0.331679
$910$	0	0
$911$	12.0000	0.397578	0.198789	−	0.980042i	$-0.436299\pi$
$911$	12.0000	0.397578	0.198789	+	0.980042i	$0.436299\pi$
$912$	1.00000i	0.0331133i
$913$	8.00000i	0.264761i
$914$	−22.0000	−0.727695
$915$	0	0
$916$	−26.0000	−0.859064
$917$	24.0000i	0.792550i
$918$	4.00000i	0.132020i
$919$	−16.0000	−0.527791	−0.263896	−	0.964551i	$-0.585007\pi$
$919$	−16.0000	−0.527791	−0.263896	+	0.964551i	$0.585007\pi$
$920$	0	0
$921$	12.0000	0.395413
$922$	18.0000i	0.592798i
$923$	− 48.0000i	− 1.57994i
$924$	−8.00000	−0.263181
$925$	0	0
$926$	18.0000	0.591517
$927$	12.0000i	0.394132i
$928$	− 6.00000i	− 0.196960i
$929$	−42.0000	−1.37798	−0.688988	−	0.724773i	$-0.741945\pi$
$929$	−42.0000	−1.37798	−0.688988	+	0.724773i	$0.741945\pi$
$930$	0	0
$931$	−3.00000	−0.0983210
$932$	12.0000i	0.393073i
$933$	− 16.0000i	− 0.523816i
$934$	−22.0000	−0.719862
$935$	0	0
$936$	6.00000	0.196116
$937$	− 26.0000i	− 0.849383i	−0.905338	−	0.424691i	$-0.860383\pi$
$937$	− 26.0000i	− 0.849383i	0.905338	−	0.424691i	$-0.139617\pi$
$938$	24.0000i	0.783628i
$939$	6.00000	0.195803
$940$	0	0
$941$	10.0000	0.325991	0.162995	−	0.986627i	$-0.447884\pi$
$941$	10.0000	0.325991	0.162995	+	0.986627i	$0.447884\pi$
$942$	− 12.0000i	− 0.390981i
$943$	16.0000i	0.521032i
$944$	−10.0000	−0.325472
$945$	0	0
$946$	−48.0000	−1.56061
$947$	38.0000i	1.23483i	0.786636	+	0.617417i	$0.211821\pi$
$947$	38.0000i	1.23483i	−0.786636	+	0.617417i	$0.788179\pi$
$948$	− 10.0000i	− 0.324785i
$949$	−12.0000	−0.389536
$950$	0	0
$951$	−18.0000	−0.583690
$952$	− 8.00000i	− 0.259281i
$953$	6.00000i	0.194359i	0.995267	+	0.0971795i	$0.0309821\pi$
$953$	6.00000i	0.194359i	−0.995267	+	0.0971795i	$0.969018\pi$
$954$	−10.0000	−0.323762
$955$	0	0
$956$	0	0
$957$	24.0000i	0.775810i
$958$	24.0000i	0.775405i
$959$	0	0
$960$	0	0
$961$	5.00000	0.161290
$962$	− 60.0000i	− 1.93448i
$963$	8.00000i	0.257796i
$964$	−10.0000	−0.322078
$965$	0	0
$966$	−8.00000	−0.257396
$967$	18.0000i	0.578841i	0.957202	+	0.289420i	$0.0934626\pi$
$967$	18.0000i	0.578841i	−0.957202	+	0.289420i	$0.906537\pi$
$968$	− 5.00000i	− 0.160706i
$969$	4.00000	0.128499
$970$	0	0
$971$	−42.0000	−1.34784	−0.673922	−	0.738802i	$-0.735392\pi$
$971$	−42.0000	−1.34784	−0.673922	+	0.738802i	$0.735392\pi$
$972$	1.00000i	0.0320750i
$973$	8.00000i	0.256468i
$974$	20.0000	0.640841
$975$	0	0
$976$	2.00000	0.0640184
$977$	42.0000i	1.34370i	0.740688	+	0.671850i	$0.234500\pi$
$977$	42.0000i	1.34370i	−0.740688	+	0.671850i	$0.765500\pi$
$978$	− 16.0000i	− 0.511624i
$979$	32.0000	1.02272
$980$	0	0
$981$	−16.0000	−0.510841
$982$	12.0000i	0.382935i
$983$	24.0000i	0.765481i	0.923856	+	0.382741i	$0.125020\pi$
$983$	24.0000i	0.765481i	−0.923856	+	0.382741i	$0.874980\pi$
$984$	−4.00000	−0.127515
$985$	0	0
$986$	−24.0000	−0.764316
$987$	− 8.00000i	− 0.254643i
$988$	− 6.00000i	− 0.190885i
$989$	−48.0000	−1.52631
$990$	0	0
$991$	−2.00000	−0.0635321	−0.0317660	−	0.999495i	$-0.510113\pi$
$991$	−2.00000	−0.0635321	−0.0317660	+	0.999495i	$0.510113\pi$
$992$	− 6.00000i	− 0.190500i
$993$	12.0000i	0.380808i
$994$	−16.0000	−0.507489
$995$	0	0
$996$	−2.00000	−0.0633724
$997$	52.0000i	1.64686i	0.567420	+	0.823428i	$0.307941\pi$
$997$	52.0000i	1.64686i	−0.567420	+	0.823428i	$0.692059\pi$
$998$	4.00000i	0.126618i
$999$	10.0000	0.316386

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2850.2.d.r.799.2		2
5.2	odd	4		570.2.a.a.1.1	✓	1
5.3	odd	4		2850.2.a.bb.1.1		1
5.4	even	2	inner	2850.2.d.r.799.1		2
15.2	even	4		1710.2.a.q.1.1		1
15.8	even	4		8550.2.a.n.1.1		1
20.7	even	4		4560.2.a.t.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
570.2.a.a.1.1	✓	1	5.2	odd	4
1710.2.a.q.1.1		1	15.2	even	4
2850.2.a.bb.1.1		1	5.3	odd	4
2850.2.d.r.799.1		2	5.4	even	2	inner
2850.2.d.r.799.2		2	1.1	even	1	trivial
4560.2.a.t.1.1		1	20.7	even	4
8550.2.a.n.1.1		1	15.8	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:	\( N \)	\(=\)	\( 2850 = 2 \cdot 3 \cdot 5^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2850.d (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} +1.00000 q^{6} +2.00000i q^{7} -1.00000i q^{8} -1.00000 q^{9} +O(q^{10})\)	\(q+1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} +1.00000 q^{6} +2.00000i q^{7} -1.00000i q^{8} -1.00000 q^{9} +4.00000 q^{11} +1.00000i q^{12} -6.00000i q^{13} -2.00000 q^{14} +1.00000 q^{16} -4.00000i q^{17} -1.00000i q^{18} -1.00000 q^{19} +2.00000 q^{21} +4.00000i q^{22} +4.00000i q^{23} -1.00000 q^{24} +6.00000 q^{26} +1.00000i q^{27} -2.00000i q^{28} -6.00000 q^{29} -6.00000 q^{31} +1.00000i q^{32} -4.00000i q^{33} +4.00000 q^{34} +1.00000 q^{36} -10.0000i q^{37} -1.00000i q^{38} -6.00000 q^{39} +4.00000 q^{41} +2.00000i q^{42} +12.0000i q^{43} -4.00000 q^{44} -4.00000 q^{46} -4.00000i q^{47} -1.00000i q^{48} +3.00000 q^{49} -4.00000 q^{51} +6.00000i q^{52} -10.0000i q^{53} -1.00000 q^{54} +2.00000 q^{56} +1.00000i q^{57} -6.00000i q^{58} -10.0000 q^{59} +2.00000 q^{61} -6.00000i q^{62} -2.00000i q^{63} -1.00000 q^{64} +4.00000 q^{66} -12.0000i q^{67} +4.00000i q^{68} +4.00000 q^{69} +8.00000 q^{71} +1.00000i q^{72} -2.00000i q^{73} +10.0000 q^{74} +1.00000 q^{76} +8.00000i q^{77} -6.00000i q^{78} -10.0000 q^{79} +1.00000 q^{81} +4.00000i q^{82} +2.00000i q^{83} -2.00000 q^{84} -12.0000 q^{86} +6.00000i q^{87} -4.00000i q^{88} +8.00000 q^{89} +12.0000 q^{91} -4.00000i q^{92} +6.00000i q^{93} +4.00000 q^{94} +1.00000 q^{96} -2.00000i q^{97} +3.00000i q^{98} -4.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{4} + 2 q^{6} - 2 q^{9}+O(q^{10}) \) 2 * q - 2 * q^4 + 2 * q^6 - 2 * q^9	\( 2 q - 2 q^{4} + 2 q^{6} - 2 q^{9} + 8 q^{11} - 4 q^{14} + 2 q^{16} - 2 q^{19} + 4 q^{21} - 2 q^{24} + 12 q^{26} - 12 q^{29} - 12 q^{31} + 8 q^{34} + 2 q^{36} - 12 q^{39} + 8 q^{41} - 8 q^{44} - 8 q^{46} + 6 q^{49} - 8 q^{51} - 2 q^{54} + 4 q^{56} - 20 q^{59} + 4 q^{61} - 2 q^{64} + 8 q^{66} + 8 q^{69} + 16 q^{71} + 20 q^{74} + 2 q^{76} - 20 q^{79} + 2 q^{81} - 4 q^{84} - 24 q^{86} + 16 q^{89} + 24 q^{91} + 8 q^{94} + 2 q^{96} - 8 q^{99}+O(q^{100}) \) 2 * q - 2 * q^4 + 2 * q^6 - 2 * q^9 + 8 * q^11 - 4 * q^14 + 2 * q^16 - 2 * q^19 + 4 * q^21 - 2 * q^24 + 12 * q^26 - 12 * q^29 - 12 * q^31 + 8 * q^34 + 2 * q^36 - 12 * q^39 + 8 * q^41 - 8 * q^44 - 8 * q^46 + 6 * q^49 - 8 * q^51 - 2 * q^54 + 4 * q^56 - 20 * q^59 + 4 * q^61 - 2 * q^64 + 8 * q^66 + 8 * q^69 + 16 * q^71 + 20 * q^74 + 2 * q^76 - 20 * q^79 + 2 * q^81 - 4 * q^84 - 24 * q^86 + 16 * q^89 + 24 * q^91 + 8 * q^94 + 2 * q^96 - 8 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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