LMFDB - Embedded newform 4350.2.e.j.349.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [4350,2,Mod(349,4350)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(4350, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 0])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("4350.349"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 4350 = 2 \cdot 3 \cdot 5^{2} \cdot 29 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4350.e (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [2,0,0,-2,0,-2,0,0,-2,0,12,0,0,10,0,2,0,0,2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(19)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$34.7349248793$
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 174)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			349.1
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	4350.349
Dual form			4350.2.e.j.349.2

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} -1.00000 q^{6} +5.00000i q^{7} +1.00000i q^{8} -1.00000 q^{9} +6.00000 q^{11} +1.00000i q^{12} +4.00000i q^{13} +5.00000 q^{14} +1.00000 q^{16} +3.00000i q^{17} +1.00000i q^{18} +1.00000 q^{19} +5.00000 q^{21} -6.00000i q^{22} +1.00000 q^{24} +4.00000 q^{26} +1.00000i q^{27} -5.00000i q^{28} +1.00000 q^{29} -4.00000 q^{31} -1.00000i q^{32} -6.00000i q^{33} +3.00000 q^{34} +1.00000 q^{36} -1.00000i q^{37} -1.00000i q^{38} +4.00000 q^{39} -9.00000 q^{41} -5.00000i q^{42} +7.00000i q^{43} -6.00000 q^{44} -3.00000i q^{47} -1.00000i q^{48} -18.0000 q^{49} +3.00000 q^{51} -4.00000i q^{52} +6.00000i q^{53} +1.00000 q^{54} -5.00000 q^{56} -1.00000i q^{57} -1.00000i q^{58} -3.00000 q^{59} -10.0000 q^{61} +4.00000i q^{62} -5.00000i q^{63} -1.00000 q^{64} -6.00000 q^{66} -4.00000i q^{67} -3.00000i q^{68} +12.0000 q^{71} -1.00000i q^{72} -2.00000i q^{73} -1.00000 q^{74} -1.00000 q^{76} +30.0000i q^{77} -4.00000i q^{78} -14.0000 q^{79} +1.00000 q^{81} +9.00000i q^{82} -5.00000 q^{84} +7.00000 q^{86} -1.00000i q^{87} +6.00000i q^{88} +6.00000 q^{89} -20.0000 q^{91} +4.00000i q^{93} -3.00000 q^{94} -1.00000 q^{96} +8.00000i q^{97} +18.0000i q^{98} -6.00000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{4} - 2 q^{6} - 2 q^{9} + 12 q^{11} + 10 q^{14} + 2 q^{16} + 2 q^{19} + 10 q^{21} + 2 q^{24} + 8 q^{26} + 2 q^{29} - 8 q^{31} + 6 q^{34} + 2 q^{36} + 8 q^{39} - 18 q^{41} - 12 q^{44} - 36 q^{49}+ \cdots - 12 q^{99}+O(q^{100}) $ 2 * q - 2 * q^4 - 2 * q^6 - 2 * q^9 + 12 * q^11 + 10 * q^14 + 2 * q^16 + 2 * q^19 + 10 * q^21 + 2 * q^24 + 8 * q^26 + 2 * q^29 - 8 * q^31 + 6 * q^34 + 2 * q^36 + 8 * q^39 - 18 * q^41 - 12 * q^44 - 36 * q^49 + 6 * q^51 + 2 * q^54 - 10 * q^56 - 6 * q^59 - 20 * q^61 - 2 * q^64 - 12 * q^66 + 24 * q^71 - 2 * q^74 - 2 * q^76 - 28 * q^79 + 2 * q^81 - 10 * q^84 + 14 * q^86 + 12 * q^89 - 40 * q^91 - 6 * q^94 - 2 * q^96 - 12 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/4350\mathbb{Z}\right)^\times$.

$n$	$901$	$1451$	$4177$
$\chi(n)$	$1$	$1$	$-1$

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
$7$	5.00000i	1.88982i	0.327327	+	0.944911i	$0.393852\pi$
$7$	5.00000i	1.88982i	−0.327327	+	0.944911i	$0.606148\pi$
$8$	1.00000i	0.353553i
$9$	−1.00000	−0.333333
$10$	0	0
$11$	6.00000	1.80907	0.904534	−	0.426401i	$-0.140219\pi$
$11$	6.00000	1.80907	0.904534	+	0.426401i	$0.140219\pi$
$12$	1.00000i	0.288675i
$13$	4.00000i	1.10940i	0.832050	+	0.554700i	$0.187167\pi$
$13$	4.00000i	1.10940i	−0.832050	+	0.554700i	$0.812833\pi$
$14$	5.00000	1.33631
$15$	0	0
$16$	1.00000	0.250000
$17$	3.00000i	0.727607i	0.931476	+	0.363803i	$0.118522\pi$
$17$	3.00000i	0.727607i	−0.931476	+	0.363803i	$0.881478\pi$
$18$	1.00000i	0.235702i
$19$	1.00000	0.229416	0.114708	−	0.993399i	$-0.463407\pi$
$19$	1.00000	0.229416	0.114708	+	0.993399i	$0.463407\pi$
$20$	0	0
$21$	5.00000	1.09109
$22$	− 6.00000i	− 1.27920i
$23$	0	0	1.00000			$0$
$23$	0	0	−1.00000			$\pi$
$24$	1.00000	0.204124
$25$	0	0
$26$	4.00000	0.784465
$27$	1.00000i	0.192450i
$28$	− 5.00000i	− 0.944911i
$29$	1.00000	0.185695
$29$	1.00000	0.185695
$30$	0	0
$31$	−4.00000	−0.718421	−0.359211	−	0.933257i	$-0.616954\pi$
$31$	−4.00000	−0.718421	−0.359211	+	0.933257i	$0.616954\pi$
$32$	− 1.00000i	− 0.176777i
$33$	− 6.00000i	− 1.04447i
$34$	3.00000	0.514496
$35$	0	0
$36$	1.00000	0.166667
$37$	− 1.00000i	− 0.164399i	−0.996616	−	0.0821995i	$-0.973806\pi$
$37$	− 1.00000i	− 0.164399i	0.996616	−	0.0821995i	$-0.0261945\pi$
$38$	− 1.00000i	− 0.162221i
$39$	4.00000	0.640513
$40$	0	0
$41$	−9.00000	−1.40556	−0.702782	−	0.711405i	$-0.748059\pi$
$41$	−9.00000	−1.40556	−0.702782	+	0.711405i	$0.748059\pi$
$42$	− 5.00000i	− 0.771517i
$43$	7.00000i	1.06749i	0.845645	+	0.533745i	$0.179216\pi$
$43$	7.00000i	1.06749i	−0.845645	+	0.533745i	$0.820784\pi$
$44$	−6.00000	−0.904534
$45$	0	0
$46$	0	0
$47$	− 3.00000i	− 0.437595i	−0.975770	−	0.218797i	$-0.929787\pi$
$47$	− 3.00000i	− 0.437595i	0.975770	−	0.218797i	$-0.0702134\pi$
$48$	− 1.00000i	− 0.144338i
$49$	−18.0000	−2.57143
$50$	0	0
$51$	3.00000	0.420084
$52$	− 4.00000i	− 0.554700i
$53$	6.00000i	0.824163i	0.911147	+	0.412082i	$0.135198\pi$
$53$	6.00000i	0.824163i	−0.911147	+	0.412082i	$0.864802\pi$
$54$	1.00000	0.136083
$55$	0	0
$56$	−5.00000	−0.668153
$57$	− 1.00000i	− 0.132453i
$58$	− 1.00000i	− 0.131306i
$59$	−3.00000	−0.390567	−0.195283	−	0.980747i	$-0.562563\pi$
$59$	−3.00000	−0.390567	−0.195283	+	0.980747i	$0.562563\pi$
$60$	0	0
$61$	−10.0000	−1.28037	−0.640184	−	0.768221i	$-0.721142\pi$
$61$	−10.0000	−1.28037	−0.640184	+	0.768221i	$0.721142\pi$
$62$	4.00000i	0.508001i
$63$	− 5.00000i	− 0.629941i
$64$	−1.00000	−0.125000
$65$	0	0
$66$	−6.00000	−0.738549
$67$	− 4.00000i	− 0.488678i	−0.969690	−	0.244339i	$-0.921429\pi$
$67$	− 4.00000i	− 0.488678i	0.969690	−	0.244339i	$-0.0785709\pi$
$68$	− 3.00000i	− 0.363803i
$69$	0	0
$70$	0	0
$71$	12.0000	1.42414	0.712069	−	0.702109i	$-0.247758\pi$
$71$	12.0000	1.42414	0.712069	+	0.702109i	$0.247758\pi$
$72$	− 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	$-0.0373409\pi$
$74$	−1.00000	−0.116248
$75$	0	0
$76$	−1.00000	−0.114708
$77$	30.0000i	3.41882i
$78$	− 4.00000i	− 0.452911i
$79$	−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
$82$	9.00000i	0.993884i
$83$	0	0	1.00000			$0$
$83$	0	0	−1.00000			$\pi$
$84$	−5.00000	−0.545545
$85$	0	0
$86$	7.00000	0.754829
$87$	− 1.00000i	− 0.107211i
$88$	6.00000i	0.639602i
$89$	6.00000	0.635999	0.317999	−	0.948091i	$-0.396989\pi$
$89$	6.00000	0.635999	0.317999	+	0.948091i	$0.396989\pi$
$90$	0	0
$91$	−20.0000	−2.09657
$92$	0	0
$93$	4.00000i	0.414781i
$94$	−3.00000	−0.309426
$95$	0	0
$96$	−1.00000	−0.102062
$97$	8.00000i	0.812277i	0.913812	+	0.406138i	$0.133125\pi$
$97$	8.00000i	0.812277i	−0.913812	+	0.406138i	$0.866875\pi$
$98$	18.0000i	1.81827i
$99$	−6.00000	−0.603023
$100$	0	0
$101$	−6.00000	−0.597022	−0.298511	−	0.954406i	$-0.596490\pi$
$101$	−6.00000	−0.597022	−0.298511	+	0.954406i	$0.596490\pi$
$102$	− 3.00000i	− 0.297044i
$103$	13.0000i	1.28093i	0.767988	+	0.640464i	$0.221258\pi$
$103$	13.0000i	1.28093i	−0.767988	+	0.640464i	$0.778742\pi$
$104$	−4.00000	−0.392232
$105$	0	0
$106$	6.00000	0.582772
$107$	− 9.00000i	− 0.870063i	−0.900415	−	0.435031i	$-0.856737\pi$
$107$	− 9.00000i	− 0.870063i	0.900415	−	0.435031i	$-0.143263\pi$
$108$	− 1.00000i	− 0.0962250i
$109$	−8.00000	−0.766261	−0.383131	−	0.923694i	$-0.625154\pi$
$109$	−8.00000	−0.766261	−0.383131	+	0.923694i	$0.625154\pi$
$110$	0	0
$111$	−1.00000	−0.0949158
$112$	5.00000i	0.472456i
$113$	3.00000i	0.282216i	0.989994	+	0.141108i	$0.0450665\pi$
$113$	3.00000i	0.282216i	−0.989994	+	0.141108i	$0.954933\pi$
$114$	−1.00000	−0.0936586
$115$	0	0
$116$	−1.00000	−0.0928477
$117$	− 4.00000i	− 0.369800i
$118$	3.00000i	0.276172i
$119$	−15.0000	−1.37505
$120$	0	0
$121$	25.0000	2.27273
$122$	10.0000i	0.905357i
$123$	9.00000i	0.811503i
$124$	4.00000	0.359211
$125$	0	0
$126$	−5.00000	−0.445435
$127$	− 22.0000i	− 1.95218i	−0.217357	−	0.976092i	$-0.569744\pi$
$127$	− 22.0000i	− 1.95218i	0.217357	−	0.976092i	$-0.430256\pi$
$128$	1.00000i	0.0883883i
$129$	7.00000	0.616316
$130$	0	0
$131$	12.0000	1.04844	0.524222	−	0.851581i	$-0.324356\pi$
$131$	12.0000	1.04844	0.524222	+	0.851581i	$0.324356\pi$
$132$	6.00000i	0.522233i
$133$	5.00000i	0.433555i
$134$	−4.00000	−0.345547
$135$	0	0
$136$	−3.00000	−0.257248
$137$	− 6.00000i	− 0.512615i	−0.966595	−	0.256307i	$-0.917494\pi$
$137$	− 6.00000i	− 0.512615i	0.966595	−	0.256307i	$-0.0825059\pi$
$138$	0	0
$139$	−14.0000	−1.18746	−0.593732	−	0.804663i	$-0.702346\pi$
$139$	−14.0000	−1.18746	−0.593732	+	0.804663i	$0.702346\pi$
$140$	0	0
$141$	−3.00000	−0.252646
$142$	− 12.0000i	− 1.00702i
$143$	24.0000i	2.00698i
$144$	−1.00000	−0.0833333
$145$	0	0
$146$	−2.00000	−0.165521
$147$	18.0000i	1.48461i
$148$	1.00000i	0.0821995i
$149$	−3.00000	−0.245770	−0.122885	−	0.992421i	$-0.539215\pi$
$149$	−3.00000	−0.245770	−0.122885	+	0.992421i	$0.539215\pi$
$150$	0	0
$151$	17.0000	1.38344	0.691720	−	0.722166i	$-0.256853\pi$
$151$	17.0000	1.38344	0.691720	+	0.722166i	$0.256853\pi$
$152$	1.00000i	0.0811107i
$153$	− 3.00000i	− 0.242536i
$154$	30.0000	2.41747
$155$	0	0
$156$	−4.00000	−0.320256
$157$	− 13.0000i	− 1.03751i	−0.854922	−	0.518756i	$-0.826395\pi$
$157$	− 13.0000i	− 1.03751i	0.854922	−	0.518756i	$-0.173605\pi$
$158$	14.0000i	1.11378i
$159$	6.00000	0.475831
$160$	0	0
$161$	0	0
$162$	− 1.00000i	− 0.0785674i
$163$	− 5.00000i	− 0.391630i	−0.980641	−	0.195815i	$-0.937265\pi$
$163$	− 5.00000i	− 0.391630i	0.980641	−	0.195815i	$-0.0627352\pi$
$164$	9.00000	0.702782
$165$	0	0
$166$	0	0
$167$	6.00000i	0.464294i	0.972681	+	0.232147i	$0.0745750\pi$
$167$	6.00000i	0.464294i	−0.972681	+	0.232147i	$0.925425\pi$
$168$	5.00000i	0.385758i
$169$	−3.00000	−0.230769
$170$	0	0
$171$	−1.00000	−0.0764719
$172$	− 7.00000i	− 0.533745i
$173$	21.0000i	1.59660i	0.602260	+	0.798300i	$0.294267\pi$
$173$	21.0000i	1.59660i	−0.602260	+	0.798300i	$0.705733\pi$
$174$	−1.00000	−0.0758098
$175$	0	0
$176$	6.00000	0.452267
$177$	3.00000i	0.225494i
$178$	− 6.00000i	− 0.449719i
$179$	0	0		−	1.00000i	$-0.5\pi$
$179$	0	0			1.00000i	$0.5\pi$
$180$	0	0
$181$	8.00000	0.594635	0.297318	−	0.954779i	$-0.403908\pi$
$181$	8.00000	0.594635	0.297318	+	0.954779i	$0.403908\pi$
$182$	20.0000i	1.48250i
$183$	10.0000i	0.739221i
$184$	0	0
$185$	0	0
$186$	4.00000	0.293294
$187$	18.0000i	1.31629i
$188$	3.00000i	0.218797i
$189$	−5.00000	−0.363696
$190$	0	0
$191$	−21.0000	−1.51951	−0.759753	−	0.650211i	$-0.774680\pi$
$191$	−21.0000	−1.51951	−0.759753	+	0.650211i	$0.774680\pi$
$192$	1.00000i	0.0721688i
$193$	4.00000i	0.287926i	0.989583	+	0.143963i	$0.0459847\pi$
$193$	4.00000i	0.287926i	−0.989583	+	0.143963i	$0.954015\pi$
$194$	8.00000	0.574367
$195$	0	0
$196$	18.0000	1.28571
$197$	9.00000i	0.641223i	0.947211	+	0.320612i	$0.103888\pi$
$197$	9.00000i	0.641223i	−0.947211	+	0.320612i	$0.896112\pi$
$198$	6.00000i	0.426401i
$199$	16.0000	1.13421	0.567105	−	0.823646i	$-0.308063\pi$
$199$	16.0000	1.13421	0.567105	+	0.823646i	$0.308063\pi$
$200$	0	0
$201$	−4.00000	−0.282138
$202$	6.00000i	0.422159i
$203$	5.00000i	0.350931i
$204$	−3.00000	−0.210042
$205$	0	0
$206$	13.0000	0.905753
$207$	0	0
$208$	4.00000i	0.277350i
$209$	6.00000	0.415029
$210$	0	0
$211$	−25.0000	−1.72107	−0.860535	−	0.509390i	$-0.829871\pi$
$211$	−25.0000	−1.72107	−0.860535	+	0.509390i	$0.829871\pi$
$212$	− 6.00000i	− 0.412082i
$213$	− 12.0000i	− 0.822226i
$214$	−9.00000	−0.615227
$215$	0	0
$216$	−1.00000	−0.0680414
$217$	− 20.0000i	− 1.35769i
$218$	8.00000i	0.541828i
$219$	−2.00000	−0.135147
$220$	0	0
$221$	−12.0000	−0.807207
$222$	1.00000i	0.0671156i
$223$	16.0000i	1.07144i	0.844396	+	0.535720i	$0.179960\pi$
$223$	16.0000i	1.07144i	−0.844396	+	0.535720i	$0.820040\pi$
$224$	5.00000	0.334077
$225$	0	0
$226$	3.00000	0.199557
$227$	3.00000i	0.199117i	0.995032	+	0.0995585i	$0.0317430\pi$
$227$	3.00000i	0.199117i	−0.995032	+	0.0995585i	$0.968257\pi$
$228$	1.00000i	0.0662266i
$229$	7.00000	0.462573	0.231287	−	0.972886i	$-0.425707\pi$
$229$	7.00000	0.462573	0.231287	+	0.972886i	$0.425707\pi$
$230$	0	0
$231$	30.0000	1.97386
$232$	1.00000i	0.0656532i
$233$	− 24.0000i	− 1.57229i	−0.618041	−	0.786146i	$-0.712073\pi$
$233$	− 24.0000i	− 1.57229i	0.618041	−	0.786146i	$-0.287927\pi$
$234$	−4.00000	−0.261488
$235$	0	0
$236$	3.00000	0.195283
$237$	14.0000i	0.909398i
$238$	15.0000i	0.972306i
$239$	−6.00000	−0.388108	−0.194054	−	0.980991i	$-0.562164\pi$
$239$	−6.00000	−0.388108	−0.194054	+	0.980991i	$0.562164\pi$
$240$	0	0
$241$	17.0000	1.09507	0.547533	−	0.836784i	$-0.315567\pi$
$241$	17.0000	1.09507	0.547533	+	0.836784i	$0.315567\pi$
$242$	− 25.0000i	− 1.60706i
$243$	− 1.00000i	− 0.0641500i
$244$	10.0000	0.640184
$245$	0	0
$246$	9.00000	0.573819
$247$	4.00000i	0.254514i
$248$	− 4.00000i	− 0.254000i
$249$	0	0
$250$	0	0
$251$	0	0		−	1.00000i	$-0.5\pi$
$251$	0	0			1.00000i	$0.5\pi$
$252$	5.00000i	0.314970i
$253$	0	0
$254$	−22.0000	−1.38040
$255$	0	0
$256$	1.00000	0.0625000
$257$	0	0	1.00000			$0$
$257$	0	0	−1.00000			$\pi$
$258$	− 7.00000i	− 0.435801i
$259$	5.00000	0.310685
$260$	0	0
$261$	−1.00000	−0.0618984
$262$	− 12.0000i	− 0.741362i
$263$	3.00000i	0.184988i	0.995713	+	0.0924940i	$0.0294839\pi$
$263$	3.00000i	0.184988i	−0.995713	+	0.0924940i	$0.970516\pi$
$264$	6.00000	0.369274
$265$	0	0
$266$	5.00000	0.306570
$267$	− 6.00000i	− 0.367194i
$268$	4.00000i	0.244339i
$269$	−6.00000	−0.365826	−0.182913	−	0.983129i	$-0.558553\pi$
$269$	−6.00000	−0.365826	−0.182913	+	0.983129i	$0.558553\pi$
$270$	0	0
$271$	20.0000	1.21491	0.607457	−	0.794353i	$-0.292190\pi$
$271$	20.0000	1.21491	0.607457	+	0.794353i	$0.292190\pi$
$272$	3.00000i	0.181902i
$273$	20.0000i	1.21046i
$274$	−6.00000	−0.362473
$275$	0	0
$276$	0	0
$277$	14.0000i	0.841178i	0.907251	+	0.420589i	$0.138177\pi$
$277$	14.0000i	0.841178i	−0.907251	+	0.420589i	$0.861823\pi$
$278$	14.0000i	0.839664i
$279$	4.00000	0.239474
$280$	0	0
$281$	−6.00000	−0.357930	−0.178965	−	0.983855i	$-0.557275\pi$
$281$	−6.00000	−0.357930	−0.178965	+	0.983855i	$0.557275\pi$
$282$	3.00000i	0.178647i
$283$	− 14.0000i	− 0.832214i	−0.909316	−	0.416107i	$-0.863394\pi$
$283$	− 14.0000i	− 0.832214i	0.909316	−	0.416107i	$-0.136606\pi$
$284$	−12.0000	−0.712069
$285$	0	0
$286$	24.0000	1.41915
$287$	− 45.0000i	− 2.65627i
$288$	1.00000i	0.0589256i
$289$	8.00000	0.470588
$290$	0	0
$291$	8.00000	0.468968
$292$	2.00000i	0.117041i
$293$	6.00000i	0.350524i	0.984522	+	0.175262i	$0.0560772\pi$
$293$	6.00000i	0.350524i	−0.984522	+	0.175262i	$0.943923\pi$
$294$	18.0000	1.04978
$295$	0	0
$296$	1.00000	0.0581238
$297$	6.00000i	0.348155i
$298$	3.00000i	0.173785i
$299$	0	0
$300$	0	0
$301$	−35.0000	−2.01737
$302$	− 17.0000i	− 0.978240i
$303$	6.00000i	0.344691i
$304$	1.00000	0.0573539
$305$	0	0
$306$	−3.00000	−0.171499
$307$	20.0000i	1.14146i	0.821138	+	0.570730i	$0.193340\pi$
$307$	20.0000i	1.14146i	−0.821138	+	0.570730i	$0.806660\pi$
$308$	− 30.0000i	− 1.70941i
$309$	13.0000	0.739544
$310$	0	0
$311$	21.0000	1.19080	0.595400	−	0.803429i	$-0.296993\pi$
$311$	21.0000	1.19080	0.595400	+	0.803429i	$0.296993\pi$
$312$	4.00000i	0.226455i
$313$	− 11.0000i	− 0.621757i	−0.950450	−	0.310878i	$-0.899377\pi$
$313$	− 11.0000i	− 0.621757i	0.950450	−	0.310878i	$-0.100623\pi$
$314$	−13.0000	−0.733632
$315$	0	0
$316$	14.0000	0.787562
$317$	0	0	1.00000			$0$
$317$	0	0	−1.00000			$\pi$
$318$	− 6.00000i	− 0.336463i
$319$	6.00000	0.335936
$320$	0	0
$321$	−9.00000	−0.502331
$322$	0	0
$323$	3.00000i	0.166924i
$324$	−1.00000	−0.0555556
$325$	0	0
$326$	−5.00000	−0.276924
$327$	8.00000i	0.442401i
$328$	− 9.00000i	− 0.496942i
$329$	15.0000	0.826977
$330$	0	0
$331$	17.0000	0.934405	0.467202	−	0.884150i	$-0.345262\pi$
$331$	17.0000	0.934405	0.467202	+	0.884150i	$0.345262\pi$
$332$	0	0
$333$	1.00000i	0.0547997i
$334$	6.00000	0.328305
$335$	0	0
$336$	5.00000	0.272772
$337$	− 10.0000i	− 0.544735i	−0.962193	−	0.272367i	$-0.912193\pi$
$337$	− 10.0000i	− 0.544735i	0.962193	−	0.272367i	$-0.0878066\pi$
$338$	3.00000i	0.163178i
$339$	3.00000	0.162938
$340$	0	0
$341$	−24.0000	−1.29967
$342$	1.00000i	0.0540738i
$343$	− 55.0000i	− 2.96972i
$344$	−7.00000	−0.377415
$345$	0	0
$346$	21.0000	1.12897
$347$	− 3.00000i	− 0.161048i	−0.996753	−	0.0805242i	$-0.974341\pi$
$347$	− 3.00000i	− 0.161048i	0.996753	−	0.0805242i	$-0.0256594\pi$
$348$	1.00000i	0.0536056i
$349$	−2.00000	−0.107058	−0.0535288	−	0.998566i	$-0.517047\pi$
$349$	−2.00000	−0.107058	−0.0535288	+	0.998566i	$0.517047\pi$
$350$	0	0
$351$	−4.00000	−0.213504
$352$	− 6.00000i	− 0.319801i
$353$	30.0000i	1.59674i	0.602168	+	0.798369i	$0.294304\pi$
$353$	30.0000i	1.59674i	−0.602168	+	0.798369i	$0.705696\pi$
$354$	3.00000	0.159448
$355$	0	0
$356$	−6.00000	−0.317999
$357$	15.0000i	0.793884i
$358$	0	0
$359$	−9.00000	−0.475002	−0.237501	−	0.971387i	$-0.576328\pi$
$359$	−9.00000	−0.475002	−0.237501	+	0.971387i	$0.576328\pi$
$360$	0	0
$361$	−18.0000	−0.947368
$362$	− 8.00000i	− 0.420471i
$363$	− 25.0000i	− 1.31216i
$364$	20.0000	1.04828
$365$	0	0
$366$	10.0000	0.522708
$367$	26.0000i	1.35719i	0.734513	+	0.678594i	$0.237411\pi$
$367$	26.0000i	1.35719i	−0.734513	+	0.678594i	$0.762589\pi$
$368$	0	0
$369$	9.00000	0.468521
$370$	0	0
$371$	−30.0000	−1.55752
$372$	− 4.00000i	− 0.207390i
$373$	− 20.0000i	− 1.03556i	−0.855514	−	0.517780i	$-0.826758\pi$
$373$	− 20.0000i	− 1.03556i	0.855514	−	0.517780i	$-0.173242\pi$
$374$	18.0000	0.930758
$375$	0	0
$376$	3.00000	0.154713
$377$	4.00000i	0.206010i
$378$	5.00000i	0.257172i
$379$	28.0000	1.43826	0.719132	−	0.694874i	$-0.244540\pi$
$379$	28.0000	1.43826	0.719132	+	0.694874i	$0.244540\pi$
$380$	0	0
$381$	−22.0000	−1.12709
$382$	21.0000i	1.07445i
$383$	6.00000i	0.306586i	0.988181	+	0.153293i	$0.0489878\pi$
$383$	6.00000i	0.306586i	−0.988181	+	0.153293i	$0.951012\pi$
$384$	1.00000	0.0510310
$385$	0	0
$386$	4.00000	0.203595
$387$	− 7.00000i	− 0.355830i
$388$	− 8.00000i	− 0.406138i
$389$	12.0000	0.608424	0.304212	−	0.952604i	$-0.401607\pi$
$389$	12.0000	0.608424	0.304212	+	0.952604i	$0.401607\pi$
$390$	0	0
$391$	0	0
$392$	− 18.0000i	− 0.909137i
$393$	− 12.0000i	− 0.605320i
$394$	9.00000	0.453413
$395$	0	0
$396$	6.00000	0.301511
$397$	26.0000i	1.30490i	0.757831	+	0.652451i	$0.226259\pi$
$397$	26.0000i	1.30490i	−0.757831	+	0.652451i	$0.773741\pi$
$398$	− 16.0000i	− 0.802008i
$399$	5.00000	0.250313
$400$	0	0
$401$	30.0000	1.49813	0.749064	−	0.662497i	$-0.230503\pi$
$401$	30.0000	1.49813	0.749064	+	0.662497i	$0.230503\pi$
$402$	4.00000i	0.199502i
$403$	− 16.0000i	− 0.797017i
$404$	6.00000	0.298511
$405$	0	0
$406$	5.00000	0.248146
$407$	− 6.00000i	− 0.297409i
$408$	3.00000i	0.148522i
$409$	−8.00000	−0.395575	−0.197787	−	0.980245i	$-0.563376\pi$
$409$	−8.00000	−0.395575	−0.197787	+	0.980245i	$0.563376\pi$
$410$	0	0
$411$	−6.00000	−0.295958
$412$	− 13.0000i	− 0.640464i
$413$	− 15.0000i	− 0.738102i
$414$	0	0
$415$	0	0
$416$	4.00000	0.196116
$417$	14.0000i	0.685583i
$418$	− 6.00000i	− 0.293470i
$419$	−21.0000	−1.02592	−0.512959	−	0.858413i	$-0.671451\pi$
$419$	−21.0000	−1.02592	−0.512959	+	0.858413i	$0.671451\pi$
$420$	0	0
$421$	14.0000	0.682318	0.341159	−	0.940006i	$-0.389181\pi$
$421$	14.0000	0.682318	0.341159	+	0.940006i	$0.389181\pi$
$422$	25.0000i	1.21698i
$423$	3.00000i	0.145865i
$424$	−6.00000	−0.291386
$425$	0	0
$426$	−12.0000	−0.581402
$427$	− 50.0000i	− 2.41967i
$428$	9.00000i	0.435031i
$429$	24.0000	1.15873
$430$	0	0
$431$	36.0000	1.73406	0.867029	−	0.498257i	$-0.166026\pi$
$431$	36.0000	1.73406	0.867029	+	0.498257i	$0.166026\pi$
$432$	1.00000i	0.0481125i
$433$	− 26.0000i	− 1.24948i	−0.780833	−	0.624740i	$-0.785205\pi$
$433$	− 26.0000i	− 1.24948i	0.780833	−	0.624740i	$-0.214795\pi$
$434$	−20.0000	−0.960031
$435$	0	0
$436$	8.00000	0.383131
$437$	0	0
$438$	2.00000i	0.0955637i
$439$	1.00000	0.0477274	0.0238637	−	0.999715i	$-0.492403\pi$
$439$	1.00000	0.0477274	0.0238637	+	0.999715i	$0.492403\pi$
$440$	0	0
$441$	18.0000	0.857143
$442$	12.0000i	0.570782i
$443$	− 12.0000i	− 0.570137i	−0.958507	−	0.285069i	$-0.907984\pi$
$443$	− 12.0000i	− 0.570137i	0.958507	−	0.285069i	$-0.0920164\pi$
$444$	1.00000	0.0474579
$445$	0	0
$446$	16.0000	0.757622
$447$	3.00000i	0.141895i
$448$	− 5.00000i	− 0.236228i
$449$	−15.0000	−0.707894	−0.353947	−	0.935266i	$-0.615161\pi$
$449$	−15.0000	−0.707894	−0.353947	+	0.935266i	$0.615161\pi$
$450$	0	0
$451$	−54.0000	−2.54276
$452$	− 3.00000i	− 0.141108i
$453$	− 17.0000i	− 0.798730i
$454$	3.00000	0.140797
$455$	0	0
$456$	1.00000	0.0468293
$457$	11.0000i	0.514558i	0.966337	+	0.257279i	$0.0828260\pi$
$457$	11.0000i	0.514558i	−0.966337	+	0.257279i	$0.917174\pi$
$458$	− 7.00000i	− 0.327089i
$459$	−3.00000	−0.140028
$460$	0	0
$461$	12.0000	0.558896	0.279448	−	0.960161i	$-0.409849\pi$
$461$	12.0000	0.558896	0.279448	+	0.960161i	$0.409849\pi$
$462$	− 30.0000i	− 1.39573i
$463$	16.0000i	0.743583i	0.928316	+	0.371792i	$0.121256\pi$
$463$	16.0000i	0.743583i	−0.928316	+	0.371792i	$0.878744\pi$
$464$	1.00000	0.0464238
$465$	0	0
$466$	−24.0000	−1.11178
$467$	− 12.0000i	− 0.555294i	−0.960683	−	0.277647i	$-0.910445\pi$
$467$	− 12.0000i	− 0.555294i	0.960683	−	0.277647i	$-0.0895545\pi$
$468$	4.00000i	0.184900i
$469$	20.0000	0.923514
$470$	0	0
$471$	−13.0000	−0.599008
$472$	− 3.00000i	− 0.138086i
$473$	42.0000i	1.93116i
$474$	14.0000	0.643041
$475$	0	0
$476$	15.0000	0.687524
$477$	− 6.00000i	− 0.274721i
$478$	6.00000i	0.274434i
$479$	−24.0000	−1.09659	−0.548294	−	0.836286i	$-0.684723\pi$
$479$	−24.0000	−1.09659	−0.548294	+	0.836286i	$0.684723\pi$
$480$	0	0
$481$	4.00000	0.182384
$482$	− 17.0000i	− 0.774329i
$483$	0	0
$484$	−25.0000	−1.13636
$485$	0	0
$486$	−1.00000	−0.0453609
$487$	23.0000i	1.04223i	0.853487	+	0.521115i	$0.174484\pi$
$487$	23.0000i	1.04223i	−0.853487	+	0.521115i	$0.825516\pi$
$488$	− 10.0000i	− 0.452679i
$489$	−5.00000	−0.226108
$490$	0	0
$491$	−30.0000	−1.35388	−0.676941	−	0.736038i	$-0.736695\pi$
$491$	−30.0000	−1.35388	−0.676941	+	0.736038i	$0.736695\pi$
$492$	− 9.00000i	− 0.405751i
$493$	3.00000i	0.135113i
$494$	4.00000	0.179969
$495$	0	0
$496$	−4.00000	−0.179605
$497$	60.0000i	2.69137i
$498$	0	0
$499$	22.0000	0.984855	0.492428	−	0.870353i	$-0.336110\pi$
$499$	22.0000	0.984855	0.492428	+	0.870353i	$0.336110\pi$
$500$	0	0
$501$	6.00000	0.268060
$502$	0	0
$503$	33.0000i	1.47140i	0.677309	+	0.735699i	$0.263146\pi$
$503$	33.0000i	1.47140i	−0.677309	+	0.735699i	$0.736854\pi$
$504$	5.00000	0.222718
$505$	0	0
$506$	0	0
$507$	3.00000i	0.133235i
$508$	22.0000i	0.976092i
$509$	21.0000	0.930809	0.465404	−	0.885098i	$-0.345909\pi$
$509$	21.0000	0.930809	0.465404	+	0.885098i	$0.345909\pi$
$510$	0	0
$511$	10.0000	0.442374
$512$	− 1.00000i	− 0.0441942i
$513$	1.00000i	0.0441511i
$514$	0	0
$515$	0	0
$516$	−7.00000	−0.308158
$517$	− 18.0000i	− 0.791639i
$518$	− 5.00000i	− 0.219687i
$519$	21.0000	0.921798
$520$	0	0
$521$	6.00000	0.262865	0.131432	−	0.991325i	$-0.458042\pi$
$521$	6.00000	0.262865	0.131432	+	0.991325i	$0.458042\pi$
$522$	1.00000i	0.0437688i
$523$	− 2.00000i	− 0.0874539i	−0.999044	−	0.0437269i	$-0.986077\pi$
$523$	− 2.00000i	− 0.0874539i	0.999044	−	0.0437269i	$-0.0139232\pi$
$524$	−12.0000	−0.524222
$525$	0	0
$526$	3.00000	0.130806
$527$	− 12.0000i	− 0.522728i
$528$	− 6.00000i	− 0.261116i
$529$	23.0000	1.00000
$530$	0	0
$531$	3.00000	0.130189
$532$	− 5.00000i	− 0.216777i
$533$	− 36.0000i	− 1.55933i
$534$	−6.00000	−0.259645
$535$	0	0
$536$	4.00000	0.172774
$537$	0	0
$538$	6.00000i	0.258678i
$539$	−108.000	−4.65189
$540$	0	0
$541$	−1.00000	−0.0429934	−0.0214967	−	0.999769i	$-0.506843\pi$
$541$	−1.00000	−0.0429934	−0.0214967	+	0.999769i	$0.506843\pi$
$542$	− 20.0000i	− 0.859074i
$543$	− 8.00000i	− 0.343313i
$544$	3.00000	0.128624
$545$	0	0
$546$	20.0000	0.855921
$547$	26.0000i	1.11168i	0.831289	+	0.555840i	$0.187603\pi$
$547$	26.0000i	1.11168i	−0.831289	+	0.555840i	$0.812397\pi$
$548$	6.00000i	0.256307i
$549$	10.0000	0.426790
$550$	0	0
$551$	1.00000	0.0426014
$552$	0	0
$553$	− 70.0000i	− 2.97670i
$554$	14.0000	0.594803
$555$	0	0
$556$	14.0000	0.593732
$557$	45.0000i	1.90671i	0.301849	+	0.953356i	$0.402396\pi$
$557$	45.0000i	1.90671i	−0.301849	+	0.953356i	$0.597604\pi$
$558$	− 4.00000i	− 0.169334i
$559$	−28.0000	−1.18427
$560$	0	0
$561$	18.0000	0.759961
$562$	6.00000i	0.253095i
$563$	30.0000i	1.26435i	0.774826	+	0.632175i	$0.217837\pi$
$563$	30.0000i	1.26435i	−0.774826	+	0.632175i	$0.782163\pi$
$564$	3.00000	0.126323
$565$	0	0
$566$	−14.0000	−0.588464
$567$	5.00000i	0.209980i
$568$	12.0000i	0.503509i
$569$	−33.0000	−1.38343	−0.691716	−	0.722170i	$-0.743145\pi$
$569$	−33.0000	−1.38343	−0.691716	+	0.722170i	$0.743145\pi$
$570$	0	0
$571$	−22.0000	−0.920671	−0.460336	−	0.887745i	$-0.652271\pi$
$571$	−22.0000	−0.920671	−0.460336	+	0.887745i	$0.652271\pi$
$572$	− 24.0000i	− 1.00349i
$573$	21.0000i	0.877288i
$574$	−45.0000	−1.87826
$575$	0	0
$576$	1.00000	0.0416667
$577$	− 16.0000i	− 0.666089i	−0.942911	−	0.333044i	$-0.891924\pi$
$577$	− 16.0000i	− 0.666089i	0.942911	−	0.333044i	$-0.108076\pi$
$578$	− 8.00000i	− 0.332756i
$579$	4.00000	0.166234
$580$	0	0
$581$	0	0
$582$	− 8.00000i	− 0.331611i
$583$	36.0000i	1.49097i
$584$	2.00000	0.0827606
$585$	0	0
$586$	6.00000	0.247858
$587$	21.0000i	0.866763i	0.901211	+	0.433381i	$0.142680\pi$
$587$	21.0000i	0.866763i	−0.901211	+	0.433381i	$0.857320\pi$
$588$	− 18.0000i	− 0.742307i
$589$	−4.00000	−0.164817
$590$	0	0
$591$	9.00000	0.370211
$592$	− 1.00000i	− 0.0410997i
$593$	24.0000i	0.985562i	0.870153	+	0.492781i	$0.164020\pi$
$593$	24.0000i	0.985562i	−0.870153	+	0.492781i	$0.835980\pi$
$594$	6.00000	0.246183
$595$	0	0
$596$	3.00000	0.122885
$597$	− 16.0000i	− 0.654836i
$598$	0	0
$599$	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
$601$	32.0000	1.30531	0.652654	−	0.757656i	$-0.273656\pi$
$601$	32.0000	1.30531	0.652654	+	0.757656i	$0.273656\pi$
$602$	35.0000i	1.42649i
$603$	4.00000i	0.162893i
$604$	−17.0000	−0.691720
$605$	0	0
$606$	6.00000	0.243733
$607$	8.00000i	0.324710i	0.986732	+	0.162355i	$0.0519090\pi$
$607$	8.00000i	0.324710i	−0.986732	+	0.162355i	$0.948091\pi$
$608$	− 1.00000i	− 0.0405554i
$609$	5.00000	0.202610
$610$	0	0
$611$	12.0000	0.485468
$612$	3.00000i	0.121268i
$613$	34.0000i	1.37325i	0.727013	+	0.686624i	$0.240908\pi$
$613$	34.0000i	1.37325i	−0.727013	+	0.686624i	$0.759092\pi$
$614$	20.0000	0.807134
$615$	0	0
$616$	−30.0000	−1.20873
$617$	33.0000i	1.32853i	0.747497	+	0.664265i	$0.231255\pi$
$617$	33.0000i	1.32853i	−0.747497	+	0.664265i	$0.768745\pi$
$618$	− 13.0000i	− 0.522937i
$619$	−17.0000	−0.683288	−0.341644	−	0.939829i	$-0.610984\pi$
$619$	−17.0000	−0.683288	−0.341644	+	0.939829i	$0.610984\pi$
$620$	0	0
$621$	0	0
$622$	− 21.0000i	− 0.842023i
$623$	30.0000i	1.20192i
$624$	4.00000	0.160128
$625$	0	0
$626$	−11.0000	−0.439648
$627$	− 6.00000i	− 0.239617i
$628$	13.0000i	0.518756i
$629$	3.00000	0.119618
$630$	0	0
$631$	−13.0000	−0.517522	−0.258761	−	0.965941i	$-0.583314\pi$
$631$	−13.0000	−0.517522	−0.258761	+	0.965941i	$0.583314\pi$
$632$	− 14.0000i	− 0.556890i
$633$	25.0000i	0.993661i
$634$	0	0
$635$	0	0
$636$	−6.00000	−0.237915
$637$	− 72.0000i	− 2.85274i
$638$	− 6.00000i	− 0.237542i
$639$	−12.0000	−0.474713
$640$	0	0
$641$	−27.0000	−1.06644	−0.533218	−	0.845978i	$-0.679017\pi$
$641$	−27.0000	−1.06644	−0.533218	+	0.845978i	$0.679017\pi$
$642$	9.00000i	0.355202i
$643$	40.0000i	1.57745i	0.614749	+	0.788723i	$0.289257\pi$
$643$	40.0000i	1.57745i	−0.614749	+	0.788723i	$0.710743\pi$
$644$	0	0
$645$	0	0
$646$	3.00000	0.118033
$647$	− 6.00000i	− 0.235884i	−0.993020	−	0.117942i	$-0.962370\pi$
$647$	− 6.00000i	− 0.235884i	0.993020	−	0.117942i	$-0.0376297\pi$
$648$	1.00000i	0.0392837i
$649$	−18.0000	−0.706562
$650$	0	0
$651$	−20.0000	−0.783862
$652$	5.00000i	0.195815i
$653$	− 42.0000i	− 1.64359i	−0.569785	−	0.821794i	$-0.692974\pi$
$653$	− 42.0000i	− 1.64359i	0.569785	−	0.821794i	$-0.307026\pi$
$654$	8.00000	0.312825
$655$	0	0
$656$	−9.00000	−0.351391
$657$	2.00000i	0.0780274i
$658$	− 15.0000i	− 0.584761i
$659$	0	0		−	1.00000i	$-0.5\pi$
$659$	0	0			1.00000i	$0.5\pi$
$660$	0	0
$661$	32.0000	1.24466	0.622328	−	0.782757i	$-0.286187\pi$
$661$	32.0000	1.24466	0.622328	+	0.782757i	$0.286187\pi$
$662$	− 17.0000i	− 0.660724i
$663$	12.0000i	0.466041i
$664$	0	0
$665$	0	0
$666$	1.00000	0.0387492
$667$	0	0
$668$	− 6.00000i	− 0.232147i
$669$	16.0000	0.618596
$670$	0	0
$671$	−60.0000	−2.31627
$672$	− 5.00000i	− 0.192879i
$673$	13.0000i	0.501113i	0.968102	+	0.250557i	$0.0806136\pi$
$673$	13.0000i	0.501113i	−0.968102	+	0.250557i	$0.919386\pi$
$674$	−10.0000	−0.385186
$675$	0	0
$676$	3.00000	0.115385
$677$	− 18.0000i	− 0.691796i	−0.938272	−	0.345898i	$-0.887574\pi$
$677$	− 18.0000i	− 0.691796i	0.938272	−	0.345898i	$-0.112426\pi$
$678$	− 3.00000i	− 0.115214i
$679$	−40.0000	−1.53506
$680$	0	0
$681$	3.00000	0.114960
$682$	24.0000i	0.919007i
$683$	9.00000i	0.344375i	0.985064	+	0.172188i	$0.0550836\pi$
$683$	9.00000i	0.344375i	−0.985064	+	0.172188i	$0.944916\pi$
$684$	1.00000	0.0382360
$685$	0	0
$686$	−55.0000	−2.09991
$687$	− 7.00000i	− 0.267067i
$688$	7.00000i	0.266872i
$689$	−24.0000	−0.914327
$690$	0	0
$691$	−52.0000	−1.97817	−0.989087	−	0.147335i	$-0.952930\pi$
$691$	−52.0000	−1.97817	−0.989087	+	0.147335i	$0.952930\pi$
$692$	− 21.0000i	− 0.798300i
$693$	− 30.0000i	− 1.13961i
$694$	−3.00000	−0.113878
$695$	0	0
$696$	1.00000	0.0379049
$697$	− 27.0000i	− 1.02270i
$698$	2.00000i	0.0757011i
$699$	−24.0000	−0.907763
$700$	0	0
$701$	45.0000	1.69963	0.849813	−	0.527084i	$-0.176715\pi$
$701$	45.0000	1.69963	0.849813	+	0.527084i	$0.176715\pi$
$702$	4.00000i	0.150970i
$703$	− 1.00000i	− 0.0377157i
$704$	−6.00000	−0.226134
$705$	0	0
$706$	30.0000	1.12906
$707$	− 30.0000i	− 1.12827i
$708$	− 3.00000i	− 0.112747i
$709$	−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
$711$	14.0000	0.525041
$712$	6.00000i	0.224860i
$713$	0	0
$714$	15.0000	0.561361
$715$	0	0
$716$	0	0
$717$	6.00000i	0.224074i
$718$	9.00000i	0.335877i
$719$	48.0000	1.79010	0.895049	−	0.445968i	$-0.147140\pi$
$719$	48.0000	1.79010	0.895049	+	0.445968i	$0.147140\pi$
$720$	0	0
$721$	−65.0000	−2.42073
$722$	18.0000i	0.669891i
$723$	− 17.0000i	− 0.632237i
$724$	−8.00000	−0.297318
$725$	0	0
$726$	−25.0000	−0.927837
$727$	− 10.0000i	− 0.370879i	−0.982656	−	0.185440i	$-0.940629\pi$
$727$	− 10.0000i	− 0.370879i	0.982656	−	0.185440i	$-0.0593710\pi$
$728$	− 20.0000i	− 0.741249i
$729$	−1.00000	−0.0370370
$730$	0	0
$731$	−21.0000	−0.776713
$732$	− 10.0000i	− 0.369611i
$733$	− 50.0000i	− 1.84679i	−0.383849	−	0.923396i	$-0.625402\pi$
$733$	− 50.0000i	− 1.84679i	0.383849	−	0.923396i	$-0.374598\pi$
$734$	26.0000	0.959678
$735$	0	0
$736$	0	0
$737$	− 24.0000i	− 0.884051i
$738$	− 9.00000i	− 0.331295i
$739$	−8.00000	−0.294285	−0.147142	−	0.989115i	$-0.547008\pi$
$739$	−8.00000	−0.294285	−0.147142	+	0.989115i	$0.547008\pi$
$740$	0	0
$741$	4.00000	0.146944
$742$	30.0000i	1.10133i
$743$	21.0000i	0.770415i	0.922830	+	0.385208i	$0.125870\pi$
$743$	21.0000i	0.770415i	−0.922830	+	0.385208i	$0.874130\pi$
$744$	−4.00000	−0.146647
$745$	0	0
$746$	−20.0000	−0.732252
$747$	0	0
$748$	− 18.0000i	− 0.658145i
$749$	45.0000	1.64426
$750$	0	0
$751$	−10.0000	−0.364905	−0.182453	−	0.983215i	$-0.558404\pi$
$751$	−10.0000	−0.364905	−0.182453	+	0.983215i	$0.558404\pi$
$752$	− 3.00000i	− 0.109399i
$753$	0	0
$754$	4.00000	0.145671
$755$	0	0
$756$	5.00000	0.181848
$757$	− 7.00000i	− 0.254419i	−0.991876	−	0.127210i	$-0.959398\pi$
$757$	− 7.00000i	− 0.254419i	0.991876	−	0.127210i	$-0.0406021\pi$
$758$	− 28.0000i	− 1.01701i
$759$	0	0
$760$	0	0
$761$	−36.0000	−1.30500	−0.652499	−	0.757789i	$-0.726280\pi$
$761$	−36.0000	−1.30500	−0.652499	+	0.757789i	$0.726280\pi$
$762$	22.0000i	0.796976i
$763$	− 40.0000i	− 1.44810i
$764$	21.0000	0.759753
$765$	0	0
$766$	6.00000	0.216789
$767$	− 12.0000i	− 0.433295i
$768$	− 1.00000i	− 0.0360844i
$769$	22.0000	0.793340	0.396670	−	0.917961i	$-0.370166\pi$
$769$	22.0000	0.793340	0.396670	+	0.917961i	$0.370166\pi$
$770$	0	0
$771$	0	0
$772$	− 4.00000i	− 0.143963i
$773$	6.00000i	0.215805i	0.994161	+	0.107903i	$0.0344134\pi$
$773$	6.00000i	0.215805i	−0.994161	+	0.107903i	$0.965587\pi$
$774$	−7.00000	−0.251610
$775$	0	0
$776$	−8.00000	−0.287183
$777$	− 5.00000i	− 0.179374i
$778$	− 12.0000i	− 0.430221i
$779$	−9.00000	−0.322458
$780$	0	0
$781$	72.0000	2.57636
$782$	0	0
$783$	1.00000i	0.0357371i
$784$	−18.0000	−0.642857
$785$	0	0
$786$	−12.0000	−0.428026
$787$	32.0000i	1.14068i	0.821410	+	0.570338i	$0.193188\pi$
$787$	32.0000i	1.14068i	−0.821410	+	0.570338i	$0.806812\pi$
$788$	− 9.00000i	− 0.320612i
$789$	3.00000	0.106803
$790$	0	0
$791$	−15.0000	−0.533339
$792$	− 6.00000i	− 0.213201i
$793$	− 40.0000i	− 1.42044i
$794$	26.0000	0.922705
$795$	0	0
$796$	−16.0000	−0.567105
$797$	12.0000i	0.425062i	0.977154	+	0.212531i	$0.0681706\pi$
$797$	12.0000i	0.425062i	−0.977154	+	0.212531i	$0.931829\pi$
$798$	− 5.00000i	− 0.176998i
$799$	9.00000	0.318397
$800$	0	0
$801$	−6.00000	−0.212000
$802$	− 30.0000i	− 1.05934i
$803$	− 12.0000i	− 0.423471i
$804$	4.00000	0.141069
$805$	0	0
$806$	−16.0000	−0.563576
$807$	6.00000i	0.211210i
$808$	− 6.00000i	− 0.211079i
$809$	18.0000	0.632846	0.316423	−	0.948618i	$-0.397518\pi$
$809$	18.0000	0.632846	0.316423	+	0.948618i	$0.397518\pi$
$810$	0	0
$811$	20.0000	0.702295	0.351147	−	0.936320i	$-0.385792\pi$
$811$	20.0000	0.702295	0.351147	+	0.936320i	$0.385792\pi$
$812$	− 5.00000i	− 0.175466i
$813$	− 20.0000i	− 0.701431i
$814$	−6.00000	−0.210300
$815$	0	0
$816$	3.00000	0.105021
$817$	7.00000i	0.244899i
$818$	8.00000i	0.279713i
$819$	20.0000	0.698857
$820$	0	0
$821$	54.0000	1.88461	0.942306	−	0.334751i	$-0.108652\pi$
$821$	54.0000	1.88461	0.942306	+	0.334751i	$0.108652\pi$
$822$	6.00000i	0.209274i
$823$	− 26.0000i	− 0.906303i	−0.891434	−	0.453152i	$-0.850300\pi$
$823$	− 26.0000i	− 0.906303i	0.891434	−	0.453152i	$-0.149700\pi$
$824$	−13.0000	−0.452876
$825$	0	0
$826$	−15.0000	−0.521917
$827$	− 12.0000i	− 0.417281i	−0.977992	−	0.208640i	$-0.933096\pi$
$827$	− 12.0000i	− 0.417281i	0.977992	−	0.208640i	$-0.0669038\pi$
$828$	0	0
$829$	49.0000	1.70184	0.850920	−	0.525295i	$-0.176045\pi$
$829$	49.0000	1.70184	0.850920	+	0.525295i	$0.176045\pi$
$830$	0	0
$831$	14.0000	0.485655
$832$	− 4.00000i	− 0.138675i
$833$	− 54.0000i	− 1.87099i
$834$	14.0000	0.484780
$835$	0	0
$836$	−6.00000	−0.207514
$837$	− 4.00000i	− 0.138260i
$838$	21.0000i	0.725433i
$839$	3.00000	0.103572	0.0517858	−	0.998658i	$-0.483509\pi$
$839$	3.00000	0.103572	0.0517858	+	0.998658i	$0.483509\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	− 14.0000i	− 0.482472i
$843$	6.00000i	0.206651i
$844$	25.0000	0.860535
$845$	0	0
$846$	3.00000	0.103142
$847$	125.000i	4.29505i
$848$	6.00000i	0.206041i
$849$	−14.0000	−0.480479
$850$	0	0
$851$	0	0
$852$	12.0000i	0.411113i
$853$	19.0000i	0.650548i	0.945620	+	0.325274i	$0.105456\pi$
$853$	19.0000i	0.650548i	−0.945620	+	0.325274i	$0.894544\pi$
$854$	−50.0000	−1.71096
$855$	0	0
$856$	9.00000	0.307614
$857$	− 18.0000i	− 0.614868i	−0.951569	−	0.307434i	$-0.900530\pi$
$857$	− 18.0000i	− 0.614868i	0.951569	−	0.307434i	$-0.0994704\pi$
$858$	− 24.0000i	− 0.819346i
$859$	−5.00000	−0.170598	−0.0852989	−	0.996355i	$-0.527185\pi$
$859$	−5.00000	−0.170598	−0.0852989	+	0.996355i	$0.527185\pi$
$860$	0	0
$861$	−45.0000	−1.53360
$862$	− 36.0000i	− 1.22616i
$863$	36.0000i	1.22545i	0.790295	+	0.612727i	$0.209928\pi$
$863$	36.0000i	1.22545i	−0.790295	+	0.612727i	$0.790072\pi$
$864$	1.00000	0.0340207
$865$	0	0
$866$	−26.0000	−0.883516
$867$	− 8.00000i	− 0.271694i
$868$	20.0000i	0.678844i
$869$	−84.0000	−2.84950
$870$	0	0
$871$	16.0000	0.542139
$872$	− 8.00000i	− 0.270914i
$873$	− 8.00000i	− 0.270759i
$874$	0	0
$875$	0	0
$876$	2.00000	0.0675737
$877$	− 16.0000i	− 0.540282i	−0.962821	−	0.270141i	$-0.912930\pi$
$877$	− 16.0000i	− 0.540282i	0.962821	−	0.270141i	$-0.0870703\pi$
$878$	− 1.00000i	− 0.0337484i
$879$	6.00000	0.202375
$880$	0	0
$881$	−42.0000	−1.41502	−0.707508	−	0.706705i	$-0.750181\pi$
$881$	−42.0000	−1.41502	−0.707508	+	0.706705i	$0.750181\pi$
$882$	− 18.0000i	− 0.606092i
$883$	− 56.0000i	− 1.88455i	−0.334840	−	0.942275i	$-0.608682\pi$
$883$	− 56.0000i	− 1.88455i	0.334840	−	0.942275i	$-0.391318\pi$
$884$	12.0000	0.403604
$885$	0	0
$886$	−12.0000	−0.403148
$887$	48.0000i	1.61168i	0.592132	+	0.805841i	$0.298286\pi$
$887$	48.0000i	1.61168i	−0.592132	+	0.805841i	$0.701714\pi$
$888$	− 1.00000i	− 0.0335578i
$889$	110.000	3.68928
$890$	0	0
$891$	6.00000	0.201008
$892$	− 16.0000i	− 0.535720i
$893$	− 3.00000i	− 0.100391i
$894$	3.00000	0.100335
$895$	0	0
$896$	−5.00000	−0.167038
$897$	0	0
$898$	15.0000i	0.500556i
$899$	−4.00000	−0.133407
$900$	0	0
$901$	−18.0000	−0.599667
$902$	54.0000i	1.79800i
$903$	35.0000i	1.16473i
$904$	−3.00000	−0.0997785
$905$	0	0
$906$	−17.0000	−0.564787
$907$	− 28.0000i	− 0.929725i	−0.885383	−	0.464862i	$-0.846104\pi$
$907$	− 28.0000i	− 0.929725i	0.885383	−	0.464862i	$-0.153896\pi$
$908$	− 3.00000i	− 0.0995585i
$909$	6.00000	0.199007
$910$	0	0
$911$	−24.0000	−0.795155	−0.397578	−	0.917568i	$-0.630149\pi$
$911$	−24.0000	−0.795155	−0.397578	+	0.917568i	$0.630149\pi$
$912$	− 1.00000i	− 0.0331133i
$913$	0	0
$914$	11.0000	0.363848
$915$	0	0
$916$	−7.00000	−0.231287
$917$	60.0000i	1.98137i
$918$	3.00000i	0.0990148i
$919$	−29.0000	−0.956622	−0.478311	−	0.878191i	$-0.658751\pi$
$919$	−29.0000	−0.956622	−0.478311	+	0.878191i	$0.658751\pi$
$920$	0	0
$921$	20.0000	0.659022
$922$	− 12.0000i	− 0.395199i
$923$	48.0000i	1.57994i
$924$	−30.0000	−0.986928
$925$	0	0
$926$	16.0000	0.525793
$927$	− 13.0000i	− 0.426976i
$928$	− 1.00000i	− 0.0328266i
$929$	−18.0000	−0.590561	−0.295280	−	0.955411i	$-0.595413\pi$
$929$	−18.0000	−0.590561	−0.295280	+	0.955411i	$0.595413\pi$
$930$	0	0
$931$	−18.0000	−0.589926
$932$	24.0000i	0.786146i
$933$	− 21.0000i	− 0.687509i
$934$	−12.0000	−0.392652
$935$	0	0
$936$	4.00000	0.130744
$937$	− 49.0000i	− 1.60076i	−0.599493	−	0.800380i	$-0.704631\pi$
$937$	− 49.0000i	− 1.60076i	0.599493	−	0.800380i	$-0.295369\pi$
$938$	− 20.0000i	− 0.653023i
$939$	−11.0000	−0.358971
$940$	0	0
$941$	6.00000	0.195594	0.0977972	−	0.995206i	$-0.468820\pi$
$941$	6.00000	0.195594	0.0977972	+	0.995206i	$0.468820\pi$
$942$	13.0000i	0.423563i
$943$	0	0
$944$	−3.00000	−0.0976417
$945$	0	0
$946$	42.0000	1.36554
$947$	− 12.0000i	− 0.389948i	−0.980808	−	0.194974i	$-0.937538\pi$
$947$	− 12.0000i	− 0.389948i	0.980808	−	0.194974i	$-0.0624622\pi$
$948$	− 14.0000i	− 0.454699i
$949$	8.00000	0.259691
$950$	0	0
$951$	0	0
$952$	− 15.0000i	− 0.486153i
$953$	− 36.0000i	− 1.16615i	−0.812417	−	0.583077i	$-0.801849\pi$
$953$	− 36.0000i	− 1.16615i	0.812417	−	0.583077i	$-0.198151\pi$
$954$	−6.00000	−0.194257
$955$	0	0
$956$	6.00000	0.194054
$957$	− 6.00000i	− 0.193952i
$958$	24.0000i	0.775405i
$959$	30.0000	0.968751
$960$	0	0
$961$	−15.0000	−0.483871
$962$	− 4.00000i	− 0.128965i
$963$	9.00000i	0.290021i
$964$	−17.0000	−0.547533
$965$	0	0
$966$	0	0
$967$	− 4.00000i	− 0.128631i	−0.997930	−	0.0643157i	$-0.979514\pi$
$967$	− 4.00000i	− 0.128631i	0.997930	−	0.0643157i	$-0.0204865\pi$
$968$	25.0000i	0.803530i
$969$	3.00000	0.0963739
$970$	0	0
$971$	18.0000	0.577647	0.288824	−	0.957382i	$-0.406736\pi$
$971$	18.0000	0.577647	0.288824	+	0.957382i	$0.406736\pi$
$972$	1.00000i	0.0320750i
$973$	− 70.0000i	− 2.24410i
$974$	23.0000	0.736968
$975$	0	0
$976$	−10.0000	−0.320092
$977$	− 12.0000i	− 0.383914i	−0.981403	−	0.191957i	$-0.938517\pi$
$977$	− 12.0000i	− 0.383914i	0.981403	−	0.191957i	$-0.0614834\pi$
$978$	5.00000i	0.159882i
$979$	36.0000	1.15056
$980$	0	0
$981$	8.00000	0.255420
$982$	30.0000i	0.957338i
$983$	24.0000i	0.765481i	0.923856	+	0.382741i	$0.125020\pi$
$983$	24.0000i	0.765481i	−0.923856	+	0.382741i	$0.874980\pi$
$984$	−9.00000	−0.286910
$985$	0	0
$986$	3.00000	0.0955395
$987$	− 15.0000i	− 0.477455i
$988$	− 4.00000i	− 0.127257i
$989$	0	0
$990$	0	0
$991$	47.0000	1.49300	0.746502	−	0.665383i	$-0.231732\pi$
$991$	47.0000	1.49300	0.746502	+	0.665383i	$0.231732\pi$
$992$	4.00000i	0.127000i
$993$	− 17.0000i	− 0.539479i
$994$	60.0000	1.90308
$995$	0	0
$996$	0	0
$997$	− 19.0000i	− 0.601736i	−0.953666	−	0.300868i	$-0.902724\pi$
$997$	− 19.0000i	− 0.601736i	0.953666	−	0.300868i	$-0.0972764\pi$
$998$	− 22.0000i	− 0.696398i
$999$	1.00000	0.0316386

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4350.2.e.j.349.1		2
5.2	odd	4		4350.2.a.o.1.1		1
5.3	odd	4		174.2.a.b.1.1	✓	1
5.4	even	2	inner	4350.2.e.j.349.2		2
15.8	even	4		522.2.a.m.1.1		1
20.3	even	4		1392.2.a.b.1.1		1
35.13	even	4		8526.2.a.f.1.1		1
40.3	even	4		5568.2.a.bh.1.1		1
40.13	odd	4		5568.2.a.q.1.1		1
60.23	odd	4		4176.2.a.bd.1.1		1
145.28	odd	4		5046.2.a.i.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
174.2.a.b.1.1	✓	1	5.3	odd	4
522.2.a.m.1.1		1	15.8	even	4
1392.2.a.b.1.1		1	20.3	even	4
4176.2.a.bd.1.1		1	60.23	odd	4
4350.2.a.o.1.1		1	5.2	odd	4
4350.2.e.j.349.1		2	1.1	even	1	trivial
4350.2.e.j.349.2		2	5.4	even	2	inner
5046.2.a.i.1.1		1	145.28	odd	4
5568.2.a.q.1.1		1	40.13	odd	4
5568.2.a.bh.1.1		1	40.3	even	4
8526.2.a.f.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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 4350 = 2 \cdot 3 \cdot 5^{2} \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4350.e (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} -1.00000 q^{6} +5.00000i q^{7} +1.00000i q^{8} -1.00000 q^{9} +6.00000 q^{11} +1.00000i q^{12} +4.00000i q^{13} +5.00000 q^{14} +1.00000 q^{16} +3.00000i q^{17} +1.00000i q^{18} +1.00000 q^{19} +5.00000 q^{21} -6.00000i q^{22} +1.00000 q^{24} +4.00000 q^{26} +1.00000i q^{27} -5.00000i q^{28} +1.00000 q^{29} -4.00000 q^{31} -1.00000i q^{32} -6.00000i q^{33} +3.00000 q^{34} +1.00000 q^{36} -1.00000i q^{37} -1.00000i q^{38} +4.00000 q^{39} -9.00000 q^{41} -5.00000i q^{42} +7.00000i q^{43} -6.00000 q^{44} -3.00000i q^{47} -1.00000i q^{48} -18.0000 q^{49} +3.00000 q^{51} -4.00000i q^{52} +6.00000i q^{53} +1.00000 q^{54} -5.00000 q^{56} -1.00000i q^{57} -1.00000i q^{58} -3.00000 q^{59} -10.0000 q^{61} +4.00000i q^{62} -5.00000i q^{63} -1.00000 q^{64} -6.00000 q^{66} -4.00000i q^{67} -3.00000i q^{68} +12.0000 q^{71} -1.00000i q^{72} -2.00000i q^{73} -1.00000 q^{74} -1.00000 q^{76} +30.0000i q^{77} -4.00000i q^{78} -14.0000 q^{79} +1.00000 q^{81} +9.00000i q^{82} -5.00000 q^{84} +7.00000 q^{86} -1.00000i q^{87} +6.00000i q^{88} +6.00000 q^{89} -20.0000 q^{91} +4.00000i q^{93} -3.00000 q^{94} -1.00000 q^{96} +8.00000i q^{97} +18.0000i q^{98} -6.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{4} - 2 q^{6} - 2 q^{9} + 12 q^{11} + 10 q^{14} + 2 q^{16} + 2 q^{19} + 10 q^{21} + 2 q^{24} + 8 q^{26} + 2 q^{29} - 8 q^{31} + 6 q^{34} + 2 q^{36} + 8 q^{39} - 18 q^{41} - 12 q^{44} - 36 q^{49}+ \cdots - 12 q^{99}+O(q^{100}) \) 2 * q - 2 * q^4 - 2 * q^6 - 2 * q^9 + 12 * q^11 + 10 * q^14 + 2 * q^16 + 2 * q^19 + 10 * q^21 + 2 * q^24 + 8 * q^26 + 2 * q^29 - 8 * q^31 + 6 * q^34 + 2 * q^36 + 8 * q^39 - 18 * q^41 - 12 * q^44 - 36 * q^49 + 6 * q^51 + 2 * q^54 - 10 * q^56 - 6 * q^59 - 20 * q^61 - 2 * q^64 - 12 * q^66 + 24 * q^71 - 2 * q^74 - 2 * q^76 - 28 * q^79 + 2 * q^81 - 10 * q^84 + 14 * q^86 + 12 * q^89 - 40 * q^91 - 6 * q^94 - 2 * q^96 - 12 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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