LMFDB - Embedded newform 2850.2.d.e.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:	$1$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, a_2]$
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.e.799.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{2} +1.00000i q^{3} -1.00000 q^{4} -1.00000 q^{6} -2.00000i q^{7} -1.00000i q^{8} -1.00000 q^{9} +O(q^{10})$	\(q+1.00000i q^{2} +1.00000i q^{3} -1.00000 q^{4} -1.00000 q^{6} -2.00000i q^{7} -1.00000i q^{8} -1.00000 q^{9} -1.00000i q^{12} -6.00000i q^{13} +2.00000 q^{14} +1.00000 q^{16} +8.00000i q^{17} -1.00000i q^{18} -1.00000 q^{19} +2.00000 q^{21} +4.00000i q^{23} +1.00000 q^{24} +6.00000 q^{26} -1.00000i q^{27} +2.00000i q^{28} -2.00000 q^{29} -2.00000 q^{31} +1.00000i q^{32} -8.00000 q^{34} +1.00000 q^{36} -2.00000i q^{37} -1.00000i q^{38} +6.00000 q^{39} -12.0000 q^{41} +2.00000i q^{42} -4.00000i q^{43} -4.00000 q^{46} +12.0000i q^{47} +1.00000i q^{48} +3.00000 q^{49} -8.00000 q^{51} +6.00000i q^{52} -10.0000i q^{53} +1.00000 q^{54} -2.00000 q^{56} -1.00000i q^{57} -2.00000i q^{58} -6.00000 q^{59} -14.0000 q^{61} -2.00000i q^{62} +2.00000i q^{63} -1.00000 q^{64} -12.0000i q^{67} -8.00000i q^{68} -4.00000 q^{69} -8.00000 q^{71} +1.00000i q^{72} +10.0000i q^{73} +2.00000 q^{74} +1.00000 q^{76} +6.00000i q^{78} -14.0000 q^{79} +1.00000 q^{81} -12.0000i q^{82} -2.00000i q^{83} -2.00000 q^{84} +4.00000 q^{86} -2.00000i q^{87} -12.0000 q^{91} -4.00000i q^{92} -2.00000i q^{93} -12.0000 q^{94} -1.00000 q^{96} +2.00000i q^{97} +3.00000i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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} + 4 q^{14} + 2 q^{16} - 2 q^{19} + 4 q^{21} + 2 q^{24} + 12 q^{26} - 4 q^{29} - 4 q^{31} - 16 q^{34} + 2 q^{36} + 12 q^{39} - 24 q^{41} - 8 q^{46} + 6 q^{49} - 16 q^{51} + 2 q^{54} - 4 q^{56} - 12 q^{59} - 28 q^{61} - 2 q^{64} - 8 q^{69} - 16 q^{71} + 4 q^{74} + 2 q^{76} - 28 q^{79} + 2 q^{81} - 4 q^{84} + 8 q^{86} - 24 q^{91} - 24 q^{94} - 2 q^{96}+O(q^{100}) $ 2 * q - 2 * q^4 - 2 * q^6 - 2 * q^9 + 4 * q^14 + 2 * q^16 - 2 * q^19 + 4 * q^21 + 2 * q^24 + 12 * q^26 - 4 * q^29 - 4 * q^31 - 16 * q^34 + 2 * q^36 + 12 * q^39 - 24 * q^41 - 8 * q^46 + 6 * q^49 - 16 * q^51 + 2 * q^54 - 4 * q^56 - 12 * q^59 - 28 * q^61 - 2 * q^64 - 8 * q^69 - 16 * q^71 + 4 * q^74 + 2 * q^76 - 28 * q^79 + 2 * q^81 - 4 * q^84 + 8 * q^86 - 24 * q^91 - 24 * q^94 - 2 * q^96

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	$-0.876624\pi$
$7$	− 2.00000i	− 0.755929i	0.925820	−	0.377964i	$-0.123376\pi$
$8$	− 1.00000i	− 0.353553i
$9$	−1.00000	−0.333333
$10$	0	0
$11$	0	0		−	1.00000i	$-0.5\pi$
$11$	0	0			1.00000i	$0.5\pi$
$12$	− 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
$17$	8.00000i	1.94029i	0.242536	+	0.970143i	$0.422021\pi$
$17$	8.00000i	1.94029i	−0.242536	+	0.970143i	$0.577979\pi$
$18$	− 1.00000i	− 0.235702i
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	2.00000	0.436436
$22$	0	0
$23$	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
$29$	−2.00000	−0.371391	−0.185695	−	0.982607i	$-0.559454\pi$
$29$	−2.00000	−0.371391	−0.185695	+	0.982607i	$0.559454\pi$
$30$	0	0
$31$	−2.00000	−0.359211	−0.179605	−	0.983739i	$-0.557482\pi$
$31$	−2.00000	−0.359211	−0.179605	+	0.983739i	$0.557482\pi$
$32$	1.00000i	0.176777i
$33$	0	0
$34$	−8.00000	−1.37199
$35$	0	0
$36$	1.00000	0.166667
$37$	− 2.00000i	− 0.328798i	−0.986394	−	0.164399i	$-0.947432\pi$
$37$	− 2.00000i	− 0.328798i	0.986394	−	0.164399i	$-0.0525685\pi$
$38$	− 1.00000i	− 0.162221i
$39$	6.00000	0.960769
$40$	0	0
$41$	−12.0000	−1.87409	−0.937043	−	0.349215i	$-0.886448\pi$
$41$	−12.0000	−1.87409	−0.937043	+	0.349215i	$0.886448\pi$
$42$	2.00000i	0.308607i
$43$	− 4.00000i	− 0.609994i	−0.952353	−	0.304997i	$-0.901344\pi$
$43$	− 4.00000i	− 0.609994i	0.952353	−	0.304997i	$-0.0986555\pi$
$44$	0	0
$45$	0	0
$46$	−4.00000	−0.589768
$47$	12.0000i	1.75038i	0.483779	+	0.875190i	$0.339264\pi$
$47$	12.0000i	1.75038i	−0.483779	+	0.875190i	$0.660736\pi$
$48$	1.00000i	0.144338i
$49$	3.00000	0.428571
$50$	0	0
$51$	−8.00000	−1.12022
$52$	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
$58$	− 2.00000i	− 0.262613i
$59$	−6.00000	−0.781133	−0.390567	−	0.920575i	$-0.627721\pi$
$59$	−6.00000	−0.781133	−0.390567	+	0.920575i	$0.627721\pi$
$60$	0	0
$61$	−14.0000	−1.79252	−0.896258	−	0.443533i	$-0.853725\pi$
$61$	−14.0000	−1.79252	−0.896258	+	0.443533i	$0.853725\pi$
$62$	− 2.00000i	− 0.254000i
$63$	2.00000i	0.251976i
$64$	−1.00000	−0.125000
$65$	0	0
$66$	0	0
$67$	− 12.0000i	− 1.46603i	−0.680211	−	0.733017i	$-0.738112\pi$
$67$	− 12.0000i	− 1.46603i	0.680211	−	0.733017i	$-0.261888\pi$
$68$	− 8.00000i	− 0.970143i
$69$	−4.00000	−0.481543
$70$	0	0
$71$	−8.00000	−0.949425	−0.474713	−	0.880141i	$-0.657448\pi$
$71$	−8.00000	−0.949425	−0.474713	+	0.880141i	$0.657448\pi$
$72$	1.00000i	0.117851i
$73$	10.0000i	1.17041i	0.810885	+	0.585206i	$0.198986\pi$
$73$	10.0000i	1.17041i	−0.810885	+	0.585206i	$0.801014\pi$
$74$	2.00000	0.232495
$75$	0	0
$76$	1.00000	0.114708
$77$	0	0
$78$	6.00000i	0.679366i
$79$	−14.0000	−1.57512	−0.787562	−	0.616236i	$-0.788657\pi$
$79$	−14.0000	−1.57512	−0.787562	+	0.616236i	$0.788657\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	− 12.0000i	− 1.32518i
$83$	− 2.00000i	− 0.219529i	−0.993958	−	0.109764i	$-0.964990\pi$
$83$	− 2.00000i	− 0.219529i	0.993958	−	0.109764i	$-0.0350096\pi$
$84$	−2.00000	−0.218218
$85$	0	0
$86$	4.00000	0.431331
$87$	− 2.00000i	− 0.214423i
$88$	0	0
$89$	0	0		−	1.00000i	$-0.5\pi$
$89$	0	0			1.00000i	$0.5\pi$
$90$	0	0
$91$	−12.0000	−1.25794
$92$	− 4.00000i	− 0.417029i
$93$	− 2.00000i	− 0.207390i
$94$	−12.0000	−1.23771
$95$	0	0
$96$	−1.00000	−0.102062
$97$	2.00000i	0.203069i	0.994832	+	0.101535i	$0.0323753\pi$
$97$	2.00000i	0.203069i	−0.994832	+	0.101535i	$0.967625\pi$
$98$	3.00000i	0.303046i
$99$	0	0
$100$	0	0
$101$	−6.00000	−0.597022	−0.298511	−	0.954406i	$-0.596490\pi$
$101$	−6.00000	−0.597022	−0.298511	+	0.954406i	$0.596490\pi$
$102$	− 8.00000i	− 0.792118i
$103$	16.0000i	1.57653i	0.615338	+	0.788263i	$0.289020\pi$
$103$	16.0000i	1.57653i	−0.615338	+	0.788263i	$0.710980\pi$
$104$	−6.00000	−0.588348
$105$	0	0
$106$	10.0000	0.971286
$107$	− 16.0000i	− 1.54678i	−0.633932	−	0.773389i	$-0.718560\pi$
$107$	− 16.0000i	− 1.54678i	0.633932	−	0.773389i	$-0.281440\pi$
$108$	1.00000i	0.0962250i
$109$	12.0000	1.14939	0.574696	−	0.818367i	$-0.305120\pi$
$109$	12.0000	1.14939	0.574696	+	0.818367i	$0.305120\pi$
$110$	0	0
$111$	2.00000	0.189832
$112$	− 2.00000i	− 0.188982i
$113$	− 6.00000i	− 0.564433i	−0.959351	−	0.282216i	$-0.908930\pi$
$113$	− 6.00000i	− 0.564433i	0.959351	−	0.282216i	$-0.0910696\pi$
$114$	1.00000	0.0936586
$115$	0	0
$116$	2.00000	0.185695
$117$	6.00000i	0.554700i
$118$	− 6.00000i	− 0.552345i
$119$	16.0000	1.46672
$120$	0	0
$121$	−11.0000	−1.00000
$122$	− 14.0000i	− 1.26750i
$123$	− 12.0000i	− 1.08200i
$124$	2.00000	0.179605
$125$	0	0
$126$	−2.00000	−0.178174
$127$	− 8.00000i	− 0.709885i	−0.934888	−	0.354943i	$-0.884500\pi$
$127$	− 8.00000i	− 0.709885i	0.934888	−	0.354943i	$-0.115500\pi$
$128$	− 1.00000i	− 0.0883883i
$129$	4.00000	0.352180
$130$	0	0
$131$	0	0		−	1.00000i	$-0.5\pi$
$131$	0	0			1.00000i	$0.5\pi$
$132$	0	0
$133$	2.00000i	0.173422i
$134$	12.0000	1.03664
$135$	0	0
$136$	8.00000	0.685994
$137$	12.0000i	1.02523i	0.858619	+	0.512615i	$0.171323\pi$
$137$	12.0000i	1.02523i	−0.858619	+	0.512615i	$0.828677\pi$
$138$	− 4.00000i	− 0.340503i
$139$	12.0000	1.01783	0.508913	−	0.860818i	$-0.330047\pi$
$139$	12.0000	1.01783	0.508913	+	0.860818i	$0.330047\pi$
$140$	0	0
$141$	−12.0000	−1.01058
$142$	− 8.00000i	− 0.671345i
$143$	0	0
$144$	−1.00000	−0.0833333
$145$	0	0
$146$	−10.0000	−0.827606
$147$	3.00000i	0.247436i
$148$	2.00000i	0.164399i
$149$	−10.0000	−0.819232	−0.409616	−	0.912258i	$-0.634337\pi$
$149$	−10.0000	−0.819232	−0.409616	+	0.912258i	$0.634337\pi$
$150$	0	0
$151$	18.0000	1.46482	0.732410	−	0.680864i	$-0.238396\pi$
$151$	18.0000	1.46482	0.732410	+	0.680864i	$0.238396\pi$
$152$	1.00000i	0.0811107i
$153$	− 8.00000i	− 0.646762i
$154$	0	0
$155$	0	0
$156$	−6.00000	−0.480384
$157$	− 4.00000i	− 0.319235i	−0.987179	−	0.159617i	$-0.948974\pi$
$157$	− 4.00000i	− 0.319235i	0.987179	−	0.159617i	$-0.0510260\pi$
$158$	− 14.0000i	− 1.11378i
$159$	10.0000	0.793052
$160$	0	0
$161$	8.00000	0.630488
$162$	1.00000i	0.0785674i
$163$	− 8.00000i	− 0.626608i	−0.949653	−	0.313304i	$-0.898564\pi$
$163$	− 8.00000i	− 0.626608i	0.949653	−	0.313304i	$-0.101436\pi$
$164$	12.0000	0.937043
$165$	0	0
$166$	2.00000	0.155230
$167$	8.00000i	0.619059i	0.950890	+	0.309529i	$0.100171\pi$
$167$	8.00000i	0.619059i	−0.950890	+	0.309529i	$0.899829\pi$
$168$	− 2.00000i	− 0.154303i
$169$	−23.0000	−1.76923
$170$	0	0
$171$	1.00000	0.0764719
$172$	4.00000i	0.304997i
$173$	− 2.00000i	− 0.152057i	−0.997106	−	0.0760286i	$-0.975776\pi$
$173$	− 2.00000i	− 0.152057i	0.997106	−	0.0760286i	$-0.0242240\pi$
$174$	2.00000	0.151620
$175$	0	0
$176$	0	0
$177$	− 6.00000i	− 0.450988i
$178$	0	0
$179$	−10.0000	−0.747435	−0.373718	−	0.927543i	$-0.621917\pi$
$179$	−10.0000	−0.747435	−0.373718	+	0.927543i	$0.621917\pi$
$180$	0	0
$181$	−12.0000	−0.891953	−0.445976	−	0.895045i	$-0.647144\pi$
$181$	−12.0000	−0.891953	−0.445976	+	0.895045i	$0.647144\pi$
$182$	− 12.0000i	− 0.889499i
$183$	− 14.0000i	− 1.03491i
$184$	4.00000	0.294884
$185$	0	0
$186$	2.00000	0.146647
$187$	0	0
$188$	− 12.0000i	− 0.875190i
$189$	−2.00000	−0.145479
$190$	0	0
$191$	0	0		−	1.00000i	$-0.5\pi$
$191$	0	0			1.00000i	$0.5\pi$
$192$	− 1.00000i	− 0.0721688i
$193$	2.00000i	0.143963i	0.997406	+	0.0719816i	$0.0229323\pi$
$193$	2.00000i	0.143963i	−0.997406	+	0.0719816i	$0.977068\pi$
$194$	−2.00000	−0.143592
$195$	0	0
$196$	−3.00000	−0.214286
$197$	18.0000i	1.28245i	0.767354	+	0.641223i	$0.221573\pi$
$197$	18.0000i	1.28245i	−0.767354	+	0.641223i	$0.778427\pi$
$198$	0	0
$199$	−4.00000	−0.283552	−0.141776	−	0.989899i	$-0.545281\pi$
$199$	−4.00000	−0.283552	−0.141776	+	0.989899i	$0.545281\pi$
$200$	0	0
$201$	12.0000	0.846415
$202$	− 6.00000i	− 0.422159i
$203$	4.00000i	0.280745i
$204$	8.00000	0.560112
$205$	0	0
$206$	−16.0000	−1.11477
$207$	− 4.00000i	− 0.278019i
$208$	− 6.00000i	− 0.416025i
$209$	0	0
$210$	0	0
$211$	−16.0000	−1.10149	−0.550743	−	0.834675i	$-0.685655\pi$
$211$	−16.0000	−1.10149	−0.550743	+	0.834675i	$0.685655\pi$
$212$	10.0000i	0.686803i
$213$	− 8.00000i	− 0.548151i
$214$	16.0000	1.09374
$215$	0	0
$216$	−1.00000	−0.0680414
$217$	4.00000i	0.271538i
$218$	12.0000i	0.812743i
$219$	−10.0000	−0.675737
$220$	0	0
$221$	48.0000	3.22883
$222$	2.00000i	0.134231i
$223$	4.00000i	0.267860i	0.990991	+	0.133930i	$0.0427597\pi$
$223$	4.00000i	0.267860i	−0.990991	+	0.133930i	$0.957240\pi$
$224$	2.00000	0.133631
$225$	0	0
$226$	6.00000	0.399114
$227$	20.0000i	1.32745i	0.747978	+	0.663723i	$0.231025\pi$
$227$	20.0000i	1.32745i	−0.747978	+	0.663723i	$0.768975\pi$
$228$	1.00000i	0.0662266i
$229$	−6.00000	−0.396491	−0.198246	−	0.980152i	$-0.563524\pi$
$229$	−6.00000	−0.396491	−0.198246	+	0.980152i	$0.563524\pi$
$230$	0	0
$231$	0	0
$232$	2.00000i	0.131306i
$233$	− 24.0000i	− 1.57229i	−0.618041	−	0.786146i	$-0.712073\pi$
$233$	− 24.0000i	− 1.57229i	0.618041	−	0.786146i	$-0.287927\pi$
$234$	−6.00000	−0.392232
$235$	0	0
$236$	6.00000	0.390567
$237$	− 14.0000i	− 0.909398i
$238$	16.0000i	1.03713i
$239$	8.00000	0.517477	0.258738	−	0.965947i	$-0.416693\pi$
$239$	8.00000	0.517477	0.258738	+	0.965947i	$0.416693\pi$
$240$	0	0
$241$	−14.0000	−0.901819	−0.450910	−	0.892570i	$-0.648900\pi$
$241$	−14.0000	−0.901819	−0.450910	+	0.892570i	$0.648900\pi$
$242$	− 11.0000i	− 0.707107i
$243$	1.00000i	0.0641500i
$244$	14.0000	0.896258
$245$	0	0
$246$	12.0000	0.765092
$247$	6.00000i	0.381771i
$248$	2.00000i	0.127000i
$249$	2.00000	0.126745
$250$	0	0
$251$	−12.0000	−0.757433	−0.378717	−	0.925513i	$-0.623635\pi$
$251$	−12.0000	−0.757433	−0.378717	+	0.925513i	$0.623635\pi$
$252$	− 2.00000i	− 0.125988i
$253$	0	0
$254$	8.00000	0.501965
$255$	0	0
$256$	1.00000	0.0625000
$257$	− 30.0000i	− 1.87135i	−0.352865	−	0.935674i	$-0.614792\pi$
$257$	− 30.0000i	− 1.87135i	0.352865	−	0.935674i	$-0.385208\pi$
$258$	4.00000i	0.249029i
$259$	−4.00000	−0.248548
$260$	0	0
$261$	2.00000	0.123797
$262$	0	0
$263$	12.0000i	0.739952i	0.929041	+	0.369976i	$0.120634\pi$
$263$	12.0000i	0.739952i	−0.929041	+	0.369976i	$0.879366\pi$
$264$	0	0
$265$	0	0
$266$	−2.00000	−0.122628
$267$	0	0
$268$	12.0000i	0.733017i
$269$	30.0000	1.82913	0.914566	−	0.404436i	$-0.132532\pi$
$269$	30.0000	1.82913	0.914566	+	0.404436i	$0.132532\pi$
$270$	0	0
$271$	−4.00000	−0.242983	−0.121491	−	0.992592i	$-0.538768\pi$
$271$	−4.00000	−0.242983	−0.121491	+	0.992592i	$0.538768\pi$
$272$	8.00000i	0.485071i
$273$	− 12.0000i	− 0.726273i
$274$	−12.0000	−0.724947
$275$	0	0
$276$	4.00000	0.240772
$277$	4.00000i	0.240337i	0.992754	+	0.120168i	$0.0383434\pi$
$277$	4.00000i	0.240337i	−0.992754	+	0.120168i	$0.961657\pi$
$278$	12.0000i	0.719712i
$279$	2.00000	0.119737
$280$	0	0
$281$	−20.0000	−1.19310	−0.596550	−	0.802576i	$-0.703462\pi$
$281$	−20.0000	−1.19310	−0.596550	+	0.802576i	$0.703462\pi$
$282$	− 12.0000i	− 0.714590i
$283$	− 20.0000i	− 1.18888i	−0.804141	−	0.594438i	$-0.797374\pi$
$283$	− 20.0000i	− 1.18888i	0.804141	−	0.594438i	$-0.202626\pi$
$284$	8.00000	0.474713
$285$	0	0
$286$	0	0
$287$	24.0000i	1.41668i
$288$	− 1.00000i	− 0.0589256i
$289$	−47.0000	−2.76471
$290$	0	0
$291$	−2.00000	−0.117242
$292$	− 10.0000i	− 0.585206i
$293$	18.0000i	1.05157i	0.850617	+	0.525786i	$0.176229\pi$
$293$	18.0000i	1.05157i	−0.850617	+	0.525786i	$0.823771\pi$
$294$	−3.00000	−0.174964
$295$	0	0
$296$	−2.00000	−0.116248
$297$	0	0
$298$	− 10.0000i	− 0.579284i
$299$	24.0000	1.38796
$300$	0	0
$301$	−8.00000	−0.461112
$302$	18.0000i	1.03578i
$303$	− 6.00000i	− 0.344691i
$304$	−1.00000	−0.0573539
$305$	0	0
$306$	8.00000	0.457330
$307$	− 28.0000i	− 1.59804i	−0.601302	−	0.799022i	$-0.705351\pi$
$307$	− 28.0000i	− 1.59804i	0.601302	−	0.799022i	$-0.294649\pi$
$308$	0	0
$309$	−16.0000	−0.910208
$310$	0	0
$311$	24.0000	1.36092	0.680458	−	0.732787i	$-0.261781\pi$
$311$	24.0000	1.36092	0.680458	+	0.732787i	$0.261781\pi$
$312$	− 6.00000i	− 0.339683i
$313$	− 6.00000i	− 0.339140i	−0.985518	−	0.169570i	$-0.945762\pi$
$313$	− 6.00000i	− 0.339140i	0.985518	−	0.169570i	$-0.0542379\pi$
$314$	4.00000	0.225733
$315$	0	0
$316$	14.0000	0.787562
$317$	− 2.00000i	− 0.112331i	−0.998421	−	0.0561656i	$-0.982113\pi$
$317$	− 2.00000i	− 0.112331i	0.998421	−	0.0561656i	$-0.0178875\pi$
$318$	10.0000i	0.560772i
$319$	0	0
$320$	0	0
$321$	16.0000	0.893033
$322$	8.00000i	0.445823i
$323$	− 8.00000i	− 0.445132i
$324$	−1.00000	−0.0555556
$325$	0	0
$326$	8.00000	0.443079
$327$	12.0000i	0.663602i
$328$	12.0000i	0.662589i
$329$	24.0000	1.32316
$330$	0	0
$331$	20.0000	1.09930	0.549650	−	0.835395i	$-0.314761\pi$
$331$	20.0000	1.09930	0.549650	+	0.835395i	$0.314761\pi$
$332$	2.00000i	0.109764i
$333$	2.00000i	0.109599i
$334$	−8.00000	−0.437741
$335$	0	0
$336$	2.00000	0.109109
$337$	− 2.00000i	− 0.108947i	−0.998515	−	0.0544735i	$-0.982652\pi$
$337$	− 2.00000i	− 0.108947i	0.998515	−	0.0544735i	$-0.0173480\pi$
$338$	− 23.0000i	− 1.25104i
$339$	6.00000	0.325875
$340$	0	0
$341$	0	0
$342$	1.00000i	0.0540738i
$343$	− 20.0000i	− 1.07990i
$344$	−4.00000	−0.215666
$345$	0	0
$346$	2.00000	0.107521
$347$	− 10.0000i	− 0.536828i	−0.963304	−	0.268414i	$-0.913500\pi$
$347$	− 10.0000i	− 0.536828i	0.963304	−	0.268414i	$-0.0864995\pi$
$348$	2.00000i	0.107211i
$349$	−14.0000	−0.749403	−0.374701	−	0.927146i	$-0.622255\pi$
$349$	−14.0000	−0.749403	−0.374701	+	0.927146i	$0.622255\pi$
$350$	0	0
$351$	−6.00000	−0.320256
$352$	0	0
$353$	− 20.0000i	− 1.06449i	−0.846590	−	0.532246i	$-0.821348\pi$
$353$	− 20.0000i	− 1.06449i	0.846590	−	0.532246i	$-0.178652\pi$
$354$	6.00000	0.318896
$355$	0	0
$356$	0	0
$357$	16.0000i	0.846810i
$358$	− 10.0000i	− 0.528516i
$359$	0	0		−	1.00000i	$-0.5\pi$
$359$	0	0			1.00000i	$0.5\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	− 12.0000i	− 0.630706i
$363$	− 11.0000i	− 0.577350i
$364$	12.0000	0.628971
$365$	0	0
$366$	14.0000	0.731792
$367$	34.0000i	1.77479i	0.461014	+	0.887393i	$0.347486\pi$
$367$	34.0000i	1.77479i	−0.461014	+	0.887393i	$0.652514\pi$
$368$	4.00000i	0.208514i
$369$	12.0000	0.624695
$370$	0	0
$371$	−20.0000	−1.03835
$372$	2.00000i	0.103695i
$373$	14.0000i	0.724893i	0.932005	+	0.362446i	$0.118058\pi$
$373$	14.0000i	0.724893i	−0.932005	+	0.362446i	$0.881942\pi$
$374$	0	0
$375$	0	0
$376$	12.0000	0.618853
$377$	12.0000i	0.618031i
$378$	− 2.00000i	− 0.102869i
$379$	16.0000	0.821865	0.410932	−	0.911666i	$-0.365203\pi$
$379$	16.0000	0.821865	0.410932	+	0.911666i	$0.365203\pi$
$380$	0	0
$381$	8.00000	0.409852
$382$	0	0
$383$	0	0	1.00000			$0$
$383$	0	0	−1.00000			$\pi$
$384$	1.00000	0.0510310
$385$	0	0
$386$	−2.00000	−0.101797
$387$	4.00000i	0.203331i
$388$	− 2.00000i	− 0.101535i
$389$	−18.0000	−0.912636	−0.456318	−	0.889817i	$-0.650832\pi$
$389$	−18.0000	−0.912636	−0.456318	+	0.889817i	$0.650832\pi$
$390$	0	0
$391$	−32.0000	−1.61831
$392$	− 3.00000i	− 0.151523i
$393$	0	0
$394$	−18.0000	−0.906827
$395$	0	0
$396$	0	0
$397$	8.00000i	0.401508i	0.979642	+	0.200754i	$0.0643393\pi$
$397$	8.00000i	0.401508i	−0.979642	+	0.200754i	$0.935661\pi$
$398$	− 4.00000i	− 0.200502i
$399$	−2.00000	−0.100125
$400$	0	0
$401$	0	0		−	1.00000i	$-0.5\pi$
$401$	0	0			1.00000i	$0.5\pi$
$402$	12.0000i	0.598506i
$403$	12.0000i	0.597763i
$404$	6.00000	0.298511
$405$	0	0
$406$	−4.00000	−0.198517
$407$	0	0
$408$	8.00000i	0.396059i
$409$	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
$411$	−12.0000	−0.591916
$412$	− 16.0000i	− 0.788263i
$413$	12.0000i	0.590481i
$414$	4.00000	0.196589
$415$	0	0
$416$	6.00000	0.294174
$417$	12.0000i	0.587643i
$418$	0	0
$419$	28.0000	1.36789	0.683945	−	0.729534i	$-0.260263\pi$
$419$	28.0000	1.36789	0.683945	+	0.729534i	$0.260263\pi$
$420$	0	0
$421$	4.00000	0.194948	0.0974740	−	0.995238i	$-0.468924\pi$
$421$	4.00000	0.194948	0.0974740	+	0.995238i	$0.468924\pi$
$422$	− 16.0000i	− 0.778868i
$423$	− 12.0000i	− 0.583460i
$424$	−10.0000	−0.485643
$425$	0	0
$426$	8.00000	0.387601
$427$	28.0000i	1.35501i
$428$	16.0000i	0.773389i
$429$	0	0
$430$	0	0
$431$	24.0000	1.15604	0.578020	−	0.816023i	$-0.303826\pi$
$431$	24.0000	1.15604	0.578020	+	0.816023i	$0.303826\pi$
$432$	− 1.00000i	− 0.0481125i
$433$	34.0000i	1.63394i	0.576683	+	0.816968i	$0.304347\pi$
$433$	34.0000i	1.63394i	−0.576683	+	0.816968i	$0.695653\pi$
$434$	−4.00000	−0.192006
$435$	0	0
$436$	−12.0000	−0.574696
$437$	− 4.00000i	− 0.191346i
$438$	− 10.0000i	− 0.477818i
$439$	10.0000	0.477274	0.238637	−	0.971109i	$-0.423299\pi$
$439$	10.0000	0.477274	0.238637	+	0.971109i	$0.423299\pi$
$440$	0	0
$441$	−3.00000	−0.142857
$442$	48.0000i	2.28313i
$443$	34.0000i	1.61539i	0.589601	+	0.807694i	$0.299285\pi$
$443$	34.0000i	1.61539i	−0.589601	+	0.807694i	$0.700715\pi$
$444$	−2.00000	−0.0949158
$445$	0	0
$446$	−4.00000	−0.189405
$447$	− 10.0000i	− 0.472984i
$448$	2.00000i	0.0944911i
$449$	−36.0000	−1.69895	−0.849473	−	0.527633i	$-0.823080\pi$
$449$	−36.0000	−1.69895	−0.849473	+	0.527633i	$0.823080\pi$
$450$	0	0
$451$	0	0
$452$	6.00000i	0.282216i
$453$	18.0000i	0.845714i
$454$	−20.0000	−0.938647
$455$	0	0
$456$	−1.00000	−0.0468293
$457$	10.0000i	0.467780i	0.972263	+	0.233890i	$0.0751456\pi$
$457$	10.0000i	0.467780i	−0.972263	+	0.233890i	$0.924854\pi$
$458$	− 6.00000i	− 0.280362i
$459$	8.00000	0.373408
$460$	0	0
$461$	−18.0000	−0.838344	−0.419172	−	0.907907i	$-0.637680\pi$
$461$	−18.0000	−0.838344	−0.419172	+	0.907907i	$0.637680\pi$
$462$	0	0
$463$	− 22.0000i	− 1.02243i	−0.859454	−	0.511213i	$-0.829196\pi$
$463$	− 22.0000i	− 1.02243i	0.859454	−	0.511213i	$-0.170804\pi$
$464$	−2.00000	−0.0928477
$465$	0	0
$466$	24.0000	1.11178
$467$	− 22.0000i	− 1.01804i	−0.860755	−	0.509019i	$-0.830008\pi$
$467$	− 22.0000i	− 1.01804i	0.860755	−	0.509019i	$-0.169992\pi$
$468$	− 6.00000i	− 0.277350i
$469$	−24.0000	−1.10822
$470$	0	0
$471$	4.00000	0.184310
$472$	6.00000i	0.276172i
$473$	0	0
$474$	14.0000	0.643041
$475$	0	0
$476$	−16.0000	−0.733359
$477$	10.0000i	0.457869i
$478$	8.00000i	0.365911i
$479$	0	0		−	1.00000i	$-0.5\pi$
$479$	0	0			1.00000i	$0.5\pi$
$480$	0	0
$481$	−12.0000	−0.547153
$482$	− 14.0000i	− 0.637683i
$483$	8.00000i	0.364013i
$484$	11.0000	0.500000
$485$	0	0
$486$	−1.00000	−0.0453609
$487$	40.0000i	1.81257i	0.422664	+	0.906287i	$0.361095\pi$
$487$	40.0000i	1.81257i	−0.422664	+	0.906287i	$0.638905\pi$
$488$	14.0000i	0.633750i
$489$	8.00000	0.361773
$490$	0	0
$491$	−16.0000	−0.722070	−0.361035	−	0.932552i	$-0.617576\pi$
$491$	−16.0000	−0.722070	−0.361035	+	0.932552i	$0.617576\pi$
$492$	12.0000i	0.541002i
$493$	− 16.0000i	− 0.720604i
$494$	−6.00000	−0.269953
$495$	0	0
$496$	−2.00000	−0.0898027
$497$	16.0000i	0.717698i
$498$	2.00000i	0.0896221i
$499$	−28.0000	−1.25345	−0.626726	−	0.779240i	$-0.715605\pi$
$499$	−28.0000	−1.25345	−0.626726	+	0.779240i	$0.715605\pi$
$500$	0	0
$501$	−8.00000	−0.357414
$502$	− 12.0000i	− 0.535586i
$503$	16.0000i	0.713405i	0.934218	+	0.356702i	$0.116099\pi$
$503$	16.0000i	0.713405i	−0.934218	+	0.356702i	$0.883901\pi$
$504$	2.00000	0.0890871
$505$	0	0
$506$	0	0
$507$	− 23.0000i	− 1.02147i
$508$	8.00000i	0.354943i
$509$	−2.00000	−0.0886484	−0.0443242	−	0.999017i	$-0.514113\pi$
$509$	−2.00000	−0.0886484	−0.0443242	+	0.999017i	$0.514113\pi$
$510$	0	0
$511$	20.0000	0.884748
$512$	1.00000i	0.0441942i
$513$	1.00000i	0.0441511i
$514$	30.0000	1.32324
$515$	0	0
$516$	−4.00000	−0.176090
$517$	0	0
$518$	− 4.00000i	− 0.175750i
$519$	2.00000	0.0877903
$520$	0	0
$521$	0	0		−	1.00000i	$-0.5\pi$
$521$	0	0			1.00000i	$0.5\pi$
$522$	2.00000i	0.0875376i
$523$	20.0000i	0.874539i	0.899331	+	0.437269i	$0.144054\pi$
$523$	20.0000i	0.874539i	−0.899331	+	0.437269i	$0.855946\pi$
$524$	0	0
$525$	0	0
$526$	−12.0000	−0.523225
$527$	− 16.0000i	− 0.696971i
$528$	0	0
$529$	7.00000	0.304348
$530$	0	0
$531$	6.00000	0.260378
$532$	− 2.00000i	− 0.0867110i
$533$	72.0000i	3.11867i
$534$	0	0
$535$	0	0
$536$	−12.0000	−0.518321
$537$	− 10.0000i	− 0.431532i
$538$	30.0000i	1.29339i
$539$	0	0
$540$	0	0
$541$	−18.0000	−0.773880	−0.386940	−	0.922105i	$-0.626468\pi$
$541$	−18.0000	−0.773880	−0.386940	+	0.922105i	$0.626468\pi$
$542$	− 4.00000i	− 0.171815i
$543$	− 12.0000i	− 0.514969i
$544$	−8.00000	−0.342997
$545$	0	0
$546$	12.0000	0.513553
$547$	− 36.0000i	− 1.53925i	−0.638497	−	0.769624i	$-0.720443\pi$
$547$	− 36.0000i	− 1.53925i	0.638497	−	0.769624i	$-0.279557\pi$
$548$	− 12.0000i	− 0.512615i
$549$	14.0000	0.597505
$550$	0	0
$551$	2.00000	0.0852029
$552$	4.00000i	0.170251i
$553$	28.0000i	1.19068i
$554$	−4.00000	−0.169944
$555$	0	0
$556$	−12.0000	−0.508913
$557$	− 2.00000i	− 0.0847427i	−0.999102	−	0.0423714i	$-0.986509\pi$
$557$	− 2.00000i	− 0.0847427i	0.999102	−	0.0423714i	$-0.0134913\pi$
$558$	2.00000i	0.0846668i
$559$	−24.0000	−1.01509
$560$	0	0
$561$	0	0
$562$	− 20.0000i	− 0.843649i
$563$	− 24.0000i	− 1.01148i	−0.862686	−	0.505740i	$-0.831220\pi$
$563$	− 24.0000i	− 1.01148i	0.862686	−	0.505740i	$-0.168780\pi$
$564$	12.0000	0.505291
$565$	0	0
$566$	20.0000	0.840663
$567$	− 2.00000i	− 0.0839921i
$568$	8.00000i	0.335673i
$569$	−4.00000	−0.167689	−0.0838444	−	0.996479i	$-0.526720\pi$
$569$	−4.00000	−0.167689	−0.0838444	+	0.996479i	$0.526720\pi$
$570$	0	0
$571$	12.0000	0.502184	0.251092	−	0.967963i	$-0.419210\pi$
$571$	12.0000	0.502184	0.251092	+	0.967963i	$0.419210\pi$
$572$	0	0
$573$	0	0
$574$	−24.0000	−1.00174
$575$	0	0
$576$	1.00000	0.0416667
$577$	− 30.0000i	− 1.24892i	−0.781058	−	0.624458i	$-0.785320\pi$
$577$	− 30.0000i	− 1.24892i	0.781058	−	0.624458i	$-0.214680\pi$
$578$	− 47.0000i	− 1.95494i
$579$	−2.00000	−0.0831172
$580$	0	0
$581$	−4.00000	−0.165948
$582$	− 2.00000i	− 0.0829027i
$583$	0	0
$584$	10.0000	0.413803
$585$	0	0
$586$	−18.0000	−0.743573
$587$	− 26.0000i	− 1.07313i	−0.843857	−	0.536567i	$-0.819721\pi$
$587$	− 26.0000i	− 1.07313i	0.843857	−	0.536567i	$-0.180279\pi$
$588$	− 3.00000i	− 0.123718i
$589$	2.00000	0.0824086
$590$	0	0
$591$	−18.0000	−0.740421
$592$	− 2.00000i	− 0.0821995i
$593$	− 36.0000i	− 1.47834i	−0.673517	−	0.739171i	$-0.735217\pi$
$593$	− 36.0000i	− 1.47834i	0.673517	−	0.739171i	$-0.264783\pi$
$594$	0	0
$595$	0	0
$596$	10.0000	0.409616
$597$	− 4.00000i	− 0.163709i
$598$	24.0000i	0.981433i
$599$	36.0000	1.47092	0.735460	−	0.677568i	$-0.236966\pi$
$599$	36.0000	1.47092	0.735460	+	0.677568i	$0.236966\pi$
$600$	0	0
$601$	−14.0000	−0.571072	−0.285536	−	0.958368i	$-0.592172\pi$
$601$	−14.0000	−0.571072	−0.285536	+	0.958368i	$0.592172\pi$
$602$	− 8.00000i	− 0.326056i
$603$	12.0000i	0.488678i
$604$	−18.0000	−0.732410
$605$	0	0
$606$	6.00000	0.243733
$607$	24.0000i	0.974130i	0.873366	+	0.487065i	$0.161933\pi$
$607$	24.0000i	0.974130i	−0.873366	+	0.487065i	$0.838067\pi$
$608$	− 1.00000i	− 0.0405554i
$609$	−4.00000	−0.162088
$610$	0	0
$611$	72.0000	2.91281
$612$	8.00000i	0.323381i
$613$	16.0000i	0.646234i	0.946359	+	0.323117i	$0.104731\pi$
$613$	16.0000i	0.646234i	−0.946359	+	0.323117i	$0.895269\pi$
$614$	28.0000	1.12999
$615$	0	0
$616$	0	0
$617$	20.0000i	0.805170i	0.915383	+	0.402585i	$0.131888\pi$
$617$	20.0000i	0.805170i	−0.915383	+	0.402585i	$0.868112\pi$
$618$	− 16.0000i	− 0.643614i
$619$	−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
$622$	24.0000i	0.962312i
$623$	0	0
$624$	6.00000	0.240192
$625$	0	0
$626$	6.00000	0.239808
$627$	0	0
$628$	4.00000i	0.159617i
$629$	16.0000	0.637962
$630$	0	0
$631$	16.0000	0.636950	0.318475	−	0.947931i	$-0.396829\pi$
$631$	16.0000	0.636950	0.318475	+	0.947931i	$0.396829\pi$
$632$	14.0000i	0.556890i
$633$	− 16.0000i	− 0.635943i
$634$	2.00000	0.0794301
$635$	0	0
$636$	−10.0000	−0.396526
$637$	− 18.0000i	− 0.713186i
$638$	0	0
$639$	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	$0.629247\pi$
$642$	16.0000i	0.631470i
$643$	28.0000i	1.10421i	0.833774	+	0.552106i	$0.186176\pi$
$643$	28.0000i	1.10421i	−0.833774	+	0.552106i	$0.813824\pi$
$644$	−8.00000	−0.315244
$645$	0	0
$646$	8.00000	0.314756
$647$	− 16.0000i	− 0.629025i	−0.949253	−	0.314512i	$-0.898159\pi$
$647$	− 16.0000i	− 0.629025i	0.949253	−	0.314512i	$-0.101841\pi$
$648$	− 1.00000i	− 0.0392837i
$649$	0	0
$650$	0	0
$651$	−4.00000	−0.156772
$652$	8.00000i	0.313304i
$653$	42.0000i	1.64359i	0.569785	+	0.821794i	$0.307026\pi$
$653$	42.0000i	1.64359i	−0.569785	+	0.821794i	$0.692974\pi$
$654$	−12.0000	−0.469237
$655$	0	0
$656$	−12.0000	−0.468521
$657$	− 10.0000i	− 0.390137i
$658$	24.0000i	0.935617i
$659$	6.00000	0.233727	0.116863	−	0.993148i	$-0.462716\pi$
$659$	6.00000	0.233727	0.116863	+	0.993148i	$0.462716\pi$
$660$	0	0
$661$	−4.00000	−0.155582	−0.0777910	−	0.996970i	$-0.524787\pi$
$661$	−4.00000	−0.155582	−0.0777910	+	0.996970i	$0.524787\pi$
$662$	20.0000i	0.777322i
$663$	48.0000i	1.86417i
$664$	−2.00000	−0.0776151
$665$	0	0
$666$	−2.00000	−0.0774984
$667$	− 8.00000i	− 0.309761i
$668$	− 8.00000i	− 0.309529i
$669$	−4.00000	−0.154649
$670$	0	0
$671$	0	0
$672$	2.00000i	0.0771517i
$673$	− 2.00000i	− 0.0770943i	−0.999257	−	0.0385472i	$-0.987727\pi$
$673$	− 2.00000i	− 0.0770943i	0.999257	−	0.0385472i	$-0.0122730\pi$
$674$	2.00000	0.0770371
$675$	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	$0.963217\pi$
$678$	6.00000i	0.230429i
$679$	4.00000	0.153506
$680$	0	0
$681$	−20.0000	−0.766402
$682$	0	0
$683$	36.0000i	1.37750i	0.724998	+	0.688751i	$0.241841\pi$
$683$	36.0000i	1.37750i	−0.724998	+	0.688751i	$0.758159\pi$
$684$	−1.00000	−0.0382360
$685$	0	0
$686$	20.0000	0.763604
$687$	− 6.00000i	− 0.228914i
$688$	− 4.00000i	− 0.152499i
$689$	−60.0000	−2.28582
$690$	0	0
$691$	28.0000	1.06517	0.532585	−	0.846376i	$-0.321221\pi$
$691$	28.0000	1.06517	0.532585	+	0.846376i	$0.321221\pi$
$692$	2.00000i	0.0760286i
$693$	0	0
$694$	10.0000	0.379595
$695$	0	0
$696$	−2.00000	−0.0758098
$697$	− 96.0000i	− 3.63626i
$698$	− 14.0000i	− 0.529908i
$699$	24.0000	0.907763
$700$	0	0
$701$	14.0000	0.528773	0.264386	−	0.964417i	$-0.414831\pi$
$701$	14.0000	0.528773	0.264386	+	0.964417i	$0.414831\pi$
$702$	− 6.00000i	− 0.226455i
$703$	2.00000i	0.0754314i
$704$	0	0
$705$	0	0
$706$	20.0000	0.752710
$707$	12.0000i	0.451306i
$708$	6.00000i	0.225494i
$709$	18.0000	0.676004	0.338002	−	0.941145i	$-0.390249\pi$
$709$	18.0000	0.676004	0.338002	+	0.941145i	$0.390249\pi$
$710$	0	0
$711$	14.0000	0.525041
$712$	0	0
$713$	− 8.00000i	− 0.299602i
$714$	−16.0000	−0.598785
$715$	0	0
$716$	10.0000	0.373718
$717$	8.00000i	0.298765i
$718$	0	0
$719$	8.00000	0.298350	0.149175	−	0.988811i	$-0.452338\pi$
$719$	8.00000	0.298350	0.149175	+	0.988811i	$0.452338\pi$
$720$	0	0
$721$	32.0000	1.19174
$722$	1.00000i	0.0372161i
$723$	− 14.0000i	− 0.520666i
$724$	12.0000	0.445976
$725$	0	0
$726$	11.0000	0.408248
$727$	42.0000i	1.55769i	0.627214	+	0.778847i	$0.284195\pi$
$727$	42.0000i	1.55769i	−0.627214	+	0.778847i	$0.715805\pi$
$728$	12.0000i	0.444750i
$729$	−1.00000	−0.0370370
$730$	0	0
$731$	32.0000	1.18356
$732$	14.0000i	0.517455i
$733$	− 8.00000i	− 0.295487i	−0.989026	−	0.147743i	$-0.952799\pi$
$733$	− 8.00000i	− 0.295487i	0.989026	−	0.147743i	$-0.0472010\pi$
$734$	−34.0000	−1.25496
$735$	0	0
$736$	−4.00000	−0.147442
$737$	0	0
$738$	12.0000i	0.441726i
$739$	−20.0000	−0.735712	−0.367856	−	0.929883i	$-0.619908\pi$
$739$	−20.0000	−0.735712	−0.367856	+	0.929883i	$0.619908\pi$
$740$	0	0
$741$	−6.00000	−0.220416
$742$	− 20.0000i	− 0.734223i
$743$	− 40.0000i	− 1.46746i	−0.679442	−	0.733729i	$-0.737778\pi$
$743$	− 40.0000i	− 1.46746i	0.679442	−	0.733729i	$-0.262222\pi$
$744$	−2.00000	−0.0733236
$745$	0	0
$746$	−14.0000	−0.512576
$747$	2.00000i	0.0731762i
$748$	0	0
$749$	−32.0000	−1.16925
$750$	0	0
$751$	−2.00000	−0.0729810	−0.0364905	−	0.999334i	$-0.511618\pi$
$751$	−2.00000	−0.0729810	−0.0364905	+	0.999334i	$0.511618\pi$
$752$	12.0000i	0.437595i
$753$	− 12.0000i	− 0.437304i
$754$	−12.0000	−0.437014
$755$	0	0
$756$	2.00000	0.0727393
$757$	40.0000i	1.45382i	0.686730	+	0.726912i	$0.259045\pi$
$757$	40.0000i	1.45382i	−0.686730	+	0.726912i	$0.740955\pi$
$758$	16.0000i	0.581146i
$759$	0	0
$760$	0	0
$761$	22.0000	0.797499	0.398750	−	0.917060i	$-0.369444\pi$
$761$	22.0000	0.797499	0.398750	+	0.917060i	$0.369444\pi$
$762$	8.00000i	0.289809i
$763$	− 24.0000i	− 0.868858i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	36.0000i	1.29988i
$768$	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
$771$	30.0000	1.08042
$772$	− 2.00000i	− 0.0719816i
$773$	− 10.0000i	− 0.359675i	−0.983696	−	0.179838i	$-0.942443\pi$
$773$	− 10.0000i	− 0.359675i	0.983696	−	0.179838i	$-0.0575572\pi$
$774$	−4.00000	−0.143777
$775$	0	0
$776$	2.00000	0.0717958
$777$	− 4.00000i	− 0.143499i
$778$	− 18.0000i	− 0.645331i
$779$	12.0000	0.429945
$780$	0	0
$781$	0	0
$782$	− 32.0000i	− 1.14432i
$783$	2.00000i	0.0714742i
$784$	3.00000	0.107143
$785$	0	0
$786$	0	0
$787$	− 20.0000i	− 0.712923i	−0.934310	−	0.356462i	$-0.883983\pi$
$787$	− 20.0000i	− 0.712923i	0.934310	−	0.356462i	$-0.116017\pi$
$788$	− 18.0000i	− 0.641223i
$789$	−12.0000	−0.427211
$790$	0	0
$791$	−12.0000	−0.426671
$792$	0	0
$793$	84.0000i	2.98293i
$794$	−8.00000	−0.283909
$795$	0	0
$796$	4.00000	0.141776
$797$	− 18.0000i	− 0.637593i	−0.947823	−	0.318796i	$-0.896721\pi$
$797$	− 18.0000i	− 0.637593i	0.947823	−	0.318796i	$-0.103279\pi$
$798$	− 2.00000i	− 0.0707992i
$799$	−96.0000	−3.39624
$800$	0	0
$801$	0	0
$802$	0	0
$803$	0	0
$804$	−12.0000	−0.423207
$805$	0	0
$806$	−12.0000	−0.422682
$807$	30.0000i	1.05605i
$808$	6.00000i	0.211079i
$809$	−38.0000	−1.33601	−0.668004	−	0.744157i	$-0.732851\pi$
$809$	−38.0000	−1.33601	−0.668004	+	0.744157i	$0.732851\pi$
$810$	0	0
$811$	16.0000	0.561836	0.280918	−	0.959732i	$-0.409361\pi$
$811$	16.0000	0.561836	0.280918	+	0.959732i	$0.409361\pi$
$812$	− 4.00000i	− 0.140372i
$813$	− 4.00000i	− 0.140286i
$814$	0	0
$815$	0	0
$816$	−8.00000	−0.280056
$817$	4.00000i	0.139942i
$818$	6.00000i	0.209785i
$819$	12.0000	0.419314
$820$	0	0
$821$	−38.0000	−1.32621	−0.663105	−	0.748527i	$-0.730762\pi$
$821$	−38.0000	−1.32621	−0.663105	+	0.748527i	$0.730762\pi$
$822$	− 12.0000i	− 0.418548i
$823$	− 14.0000i	− 0.488009i	−0.969774	−	0.244005i	$-0.921539\pi$
$823$	− 14.0000i	− 0.488009i	0.969774	−	0.244005i	$-0.0784612\pi$
$824$	16.0000	0.557386
$825$	0	0
$826$	−12.0000	−0.417533
$827$	32.0000i	1.11275i	0.830932	+	0.556375i	$0.187808\pi$
$827$	32.0000i	1.11275i	−0.830932	+	0.556375i	$0.812192\pi$
$828$	4.00000i	0.139010i
$829$	−28.0000	−0.972480	−0.486240	−	0.873825i	$-0.661632\pi$
$829$	−28.0000	−0.972480	−0.486240	+	0.873825i	$0.661632\pi$
$830$	0	0
$831$	−4.00000	−0.138758
$832$	6.00000i	0.208013i
$833$	24.0000i	0.831551i
$834$	−12.0000	−0.415526
$835$	0	0
$836$	0	0
$837$	2.00000i	0.0691301i
$838$	28.0000i	0.967244i
$839$	0	0		−	1.00000i	$-0.5\pi$
$839$	0	0			1.00000i	$0.5\pi$
$840$	0	0
$841$	−25.0000	−0.862069
$842$	4.00000i	0.137849i
$843$	− 20.0000i	− 0.688837i
$844$	16.0000	0.550743
$845$	0	0
$846$	12.0000	0.412568
$847$	22.0000i	0.755929i
$848$	− 10.0000i	− 0.343401i
$849$	20.0000	0.686398
$850$	0	0
$851$	8.00000	0.274236
$852$	8.00000i	0.274075i
$853$	28.0000i	0.958702i	0.877623	+	0.479351i	$0.159128\pi$
$853$	28.0000i	0.958702i	−0.877623	+	0.479351i	$0.840872\pi$
$854$	−28.0000	−0.958140
$855$	0	0
$856$	−16.0000	−0.546869
$857$	42.0000i	1.43469i	0.696717	+	0.717346i	$0.254643\pi$
$857$	42.0000i	1.43469i	−0.696717	+	0.717346i	$0.745357\pi$
$858$	0	0
$859$	28.0000	0.955348	0.477674	−	0.878537i	$-0.341480\pi$
$859$	28.0000	0.955348	0.477674	+	0.878537i	$0.341480\pi$
$860$	0	0
$861$	−24.0000	−0.817918
$862$	24.0000i	0.817443i
$863$	− 8.00000i	− 0.272323i	−0.990687	−	0.136162i	$-0.956523\pi$
$863$	− 8.00000i	− 0.272323i	0.990687	−	0.136162i	$-0.0434766\pi$
$864$	1.00000	0.0340207
$865$	0	0
$866$	−34.0000	−1.15537
$867$	− 47.0000i	− 1.59620i
$868$	− 4.00000i	− 0.135769i
$869$	0	0
$870$	0	0
$871$	−72.0000	−2.43963
$872$	− 12.0000i	− 0.406371i
$873$	− 2.00000i	− 0.0676897i
$874$	4.00000	0.135302
$875$	0	0
$876$	10.0000	0.337869
$877$	18.0000i	0.607817i	0.952701	+	0.303908i	$0.0982917\pi$
$877$	18.0000i	0.607817i	−0.952701	+	0.303908i	$0.901708\pi$
$878$	10.0000i	0.337484i
$879$	−18.0000	−0.607125
$880$	0	0
$881$	42.0000	1.41502	0.707508	−	0.706705i	$-0.249819\pi$
$881$	42.0000	1.41502	0.707508	+	0.706705i	$0.249819\pi$
$882$	− 3.00000i	− 0.101015i
$883$	52.0000i	1.74994i	0.484178	+	0.874970i	$0.339119\pi$
$883$	52.0000i	1.74994i	−0.484178	+	0.874970i	$0.660881\pi$
$884$	−48.0000	−1.61441
$885$	0	0
$886$	−34.0000	−1.14225
$887$	32.0000i	1.07445i	0.843437	+	0.537227i	$0.180528\pi$
$887$	32.0000i	1.07445i	−0.843437	+	0.537227i	$0.819472\pi$
$888$	− 2.00000i	− 0.0671156i
$889$	−16.0000	−0.536623
$890$	0	0
$891$	0	0
$892$	− 4.00000i	− 0.133930i
$893$	− 12.0000i	− 0.401565i
$894$	10.0000	0.334450
$895$	0	0
$896$	−2.00000	−0.0668153
$897$	24.0000i	0.801337i
$898$	− 36.0000i	− 1.20134i
$899$	4.00000	0.133407
$900$	0	0
$901$	80.0000	2.66519
$902$	0	0
$903$	− 8.00000i	− 0.266223i
$904$	−6.00000	−0.199557
$905$	0	0
$906$	−18.0000	−0.598010
$907$	− 36.0000i	− 1.19536i	−0.801735	−	0.597680i	$-0.796089\pi$
$907$	− 36.0000i	− 1.19536i	0.801735	−	0.597680i	$-0.203911\pi$
$908$	− 20.0000i	− 0.663723i
$909$	6.00000	0.199007
$910$	0	0
$911$	20.0000	0.662630	0.331315	−	0.943520i	$-0.392508\pi$
$911$	20.0000	0.662630	0.331315	+	0.943520i	$0.392508\pi$
$912$	− 1.00000i	− 0.0331133i
$913$	0	0
$914$	−10.0000	−0.330771
$915$	0	0
$916$	6.00000	0.198246
$917$	0	0
$918$	8.00000i	0.264039i
$919$	24.0000	0.791687	0.395843	−	0.918318i	$-0.370452\pi$
$919$	24.0000	0.791687	0.395843	+	0.918318i	$0.370452\pi$
$920$	0	0
$921$	28.0000	0.922631
$922$	− 18.0000i	− 0.592798i
$923$	48.0000i	1.57994i
$924$	0	0
$925$	0	0
$926$	22.0000	0.722965
$927$	− 16.0000i	− 0.525509i
$928$	− 2.00000i	− 0.0656532i
$929$	−34.0000	−1.11550	−0.557752	−	0.830008i	$-0.688336\pi$
$929$	−34.0000	−1.11550	−0.557752	+	0.830008i	$0.688336\pi$
$930$	0	0
$931$	−3.00000	−0.0983210
$932$	24.0000i	0.786146i
$933$	24.0000i	0.785725i
$934$	22.0000	0.719862
$935$	0	0
$936$	6.00000	0.196116
$937$	2.00000i	0.0653372i	0.999466	+	0.0326686i	$0.0104006\pi$
$937$	2.00000i	0.0653372i	−0.999466	+	0.0326686i	$0.989599\pi$
$938$	− 24.0000i	− 0.783628i
$939$	6.00000	0.195803
$940$	0	0
$941$	30.0000	0.977972	0.488986	−	0.872292i	$-0.337367\pi$
$941$	30.0000	0.977972	0.488986	+	0.872292i	$0.337367\pi$
$942$	4.00000i	0.130327i
$943$	− 48.0000i	− 1.56310i
$944$	−6.00000	−0.195283
$945$	0	0
$946$	0	0
$947$	2.00000i	0.0649913i	0.999472	+	0.0324956i	$0.0103455\pi$
$947$	2.00000i	0.0649913i	−0.999472	+	0.0324956i	$0.989654\pi$
$948$	14.0000i	0.454699i
$949$	60.0000	1.94768
$950$	0	0
$951$	2.00000	0.0648544
$952$	− 16.0000i	− 0.518563i
$953$	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
$956$	−8.00000	−0.258738
$957$	0	0
$958$	0	0
$959$	24.0000	0.775000
$960$	0	0
$961$	−27.0000	−0.870968
$962$	− 12.0000i	− 0.386896i
$963$	16.0000i	0.515593i
$964$	14.0000	0.450910
$965$	0	0
$966$	−8.00000	−0.257396
$967$	− 34.0000i	− 1.09337i	−0.837340	−	0.546683i	$-0.815890\pi$
$967$	− 34.0000i	− 1.09337i	0.837340	−	0.546683i	$-0.184110\pi$
$968$	11.0000i	0.353553i
$969$	8.00000	0.256997
$970$	0	0
$971$	−6.00000	−0.192549	−0.0962746	−	0.995355i	$-0.530693\pi$
$971$	−6.00000	−0.192549	−0.0962746	+	0.995355i	$0.530693\pi$
$972$	− 1.00000i	− 0.0320750i
$973$	− 24.0000i	− 0.769405i
$974$	−40.0000	−1.28168
$975$	0	0
$976$	−14.0000	−0.448129
$977$	− 30.0000i	− 0.959785i	−0.877327	−	0.479893i	$-0.840676\pi$
$977$	− 30.0000i	− 0.959785i	0.877327	−	0.479893i	$-0.159324\pi$
$978$	8.00000i	0.255812i
$979$	0	0
$980$	0	0
$981$	−12.0000	−0.383131
$982$	− 16.0000i	− 0.510581i
$983$	0	0	1.00000			$0$
$983$	0	0	−1.00000			$\pi$
$984$	−12.0000	−0.382546
$985$	0	0
$986$	16.0000	0.509544
$987$	24.0000i	0.763928i
$988$	− 6.00000i	− 0.190885i
$989$	16.0000	0.508770
$990$	0	0
$991$	50.0000	1.58830	0.794151	−	0.607720i	$-0.207916\pi$
$991$	50.0000	1.58830	0.794151	+	0.607720i	$0.207916\pi$
$992$	− 2.00000i	− 0.0635001i
$993$	20.0000i	0.634681i
$994$	−16.0000	−0.507489
$995$	0	0
$996$	−2.00000	−0.0633724
$997$	44.0000i	1.39349i	0.717317	+	0.696747i	$0.245370\pi$
$997$	44.0000i	1.39349i	−0.717317	+	0.696747i	$0.754630\pi$
$998$	− 28.0000i	− 0.886325i
$999$	−2.00000	−0.0632772

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2850.2.d.e.799.2		2
5.2	odd	4		2850.2.a.n.1.1		1
5.3	odd	4		570.2.a.h.1.1	✓	1
5.4	even	2	inner	2850.2.d.e.799.1		2
15.2	even	4		8550.2.a.bh.1.1		1
15.8	even	4		1710.2.a.c.1.1		1
20.3	even	4		4560.2.a.bc.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
570.2.a.h.1.1	✓	1	5.3	odd	4
1710.2.a.c.1.1		1	15.8	even	4
2850.2.a.n.1.1		1	5.2	odd	4
2850.2.d.e.799.1		2	5.4	even	2	inner
2850.2.d.e.799.2		2	1.1	even	1	trivial
4560.2.a.bc.1.1		1	20.3	even	4
8550.2.a.bh.1.1		1	15.2	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)

\(f(q)\)	\(=\)	\(q+1.00000i q^{2} +1.00000i q^{3} -1.00000 q^{4} -1.00000 q^{6} -2.00000i q^{7} -1.00000i q^{8} -1.00000 q^{9} +O(q^{10})\)	\(q+1.00000i q^{2} +1.00000i q^{3} -1.00000 q^{4} -1.00000 q^{6} -2.00000i q^{7} -1.00000i q^{8} -1.00000 q^{9} -1.00000i q^{12} -6.00000i q^{13} +2.00000 q^{14} +1.00000 q^{16} +8.00000i q^{17} -1.00000i q^{18} -1.00000 q^{19} +2.00000 q^{21} +4.00000i q^{23} +1.00000 q^{24} +6.00000 q^{26} -1.00000i q^{27} +2.00000i q^{28} -2.00000 q^{29} -2.00000 q^{31} +1.00000i q^{32} -8.00000 q^{34} +1.00000 q^{36} -2.00000i q^{37} -1.00000i q^{38} +6.00000 q^{39} -12.0000 q^{41} +2.00000i q^{42} -4.00000i q^{43} -4.00000 q^{46} +12.0000i q^{47} +1.00000i q^{48} +3.00000 q^{49} -8.00000 q^{51} +6.00000i q^{52} -10.0000i q^{53} +1.00000 q^{54} -2.00000 q^{56} -1.00000i q^{57} -2.00000i q^{58} -6.00000 q^{59} -14.0000 q^{61} -2.00000i q^{62} +2.00000i q^{63} -1.00000 q^{64} -12.0000i q^{67} -8.00000i q^{68} -4.00000 q^{69} -8.00000 q^{71} +1.00000i q^{72} +10.0000i q^{73} +2.00000 q^{74} +1.00000 q^{76} +6.00000i q^{78} -14.0000 q^{79} +1.00000 q^{81} -12.0000i q^{82} -2.00000i q^{83} -2.00000 q^{84} +4.00000 q^{86} -2.00000i q^{87} -12.0000 q^{91} -4.00000i q^{92} -2.00000i q^{93} -12.0000 q^{94} -1.00000 q^{96} +2.00000i q^{97} +3.00000i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{4} - 2 q^{6} - 2 q^{9}+O(q^{10}) \) 2 * q - 2 * q^4 - 2 * q^6 - 2 * q^9	\( 2 q - 2 q^{4} - 2 q^{6} - 2 q^{9} + 4 q^{14} + 2 q^{16} - 2 q^{19} + 4 q^{21} + 2 q^{24} + 12 q^{26} - 4 q^{29} - 4 q^{31} - 16 q^{34} + 2 q^{36} + 12 q^{39} - 24 q^{41} - 8 q^{46} + 6 q^{49} - 16 q^{51} + 2 q^{54} - 4 q^{56} - 12 q^{59} - 28 q^{61} - 2 q^{64} - 8 q^{69} - 16 q^{71} + 4 q^{74} + 2 q^{76} - 28 q^{79} + 2 q^{81} - 4 q^{84} + 8 q^{86} - 24 q^{91} - 24 q^{94} - 2 q^{96}+O(q^{100}) \) 2 * q - 2 * q^4 - 2 * q^6 - 2 * q^9 + 4 * q^14 + 2 * q^16 - 2 * q^19 + 4 * q^21 + 2 * q^24 + 12 * q^26 - 4 * q^29 - 4 * q^31 - 16 * q^34 + 2 * q^36 + 12 * q^39 - 24 * q^41 - 8 * q^46 + 6 * q^49 - 16 * q^51 + 2 * q^54 - 4 * q^56 - 12 * q^59 - 28 * q^61 - 2 * q^64 - 8 * q^69 - 16 * q^71 + 4 * q^74 + 2 * q^76 - 28 * q^79 + 2 * q^81 - 4 * q^84 + 8 * q^86 - 24 * q^91 - 24 * q^94 - 2 * q^96

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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