LMFDB - Embedded newform 150.2.c.a.49.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [150,2,Mod(49,150)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("150.49"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(150, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 150 = 2 \cdot 3 \cdot 5^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	150.c (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)

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

Self dual:	no
Analytic conductor:	$1.19775603032$
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 30)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			49.2
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	150.49
Dual form			150.2.c.a.49.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+1.00000i q^{2} +1.00000i q^{3} -1.00000 q^{4} -1.00000 q^{6} +4.00000i q^{7} -1.00000i q^{8} -1.00000 q^{9} -1.00000i q^{12} +2.00000i q^{13} -4.00000 q^{14} +1.00000 q^{16} -6.00000i q^{17} -1.00000i q^{18} +4.00000 q^{19} -4.00000 q^{21} +1.00000 q^{24} -2.00000 q^{26} -1.00000i q^{27} -4.00000i q^{28} +6.00000 q^{29} +8.00000 q^{31} +1.00000i q^{32} +6.00000 q^{34} +1.00000 q^{36} -2.00000i q^{37} +4.00000i q^{38} -2.00000 q^{39} -6.00000 q^{41} -4.00000i q^{42} -4.00000i q^{43} +1.00000i q^{48} -9.00000 q^{49} +6.00000 q^{51} -2.00000i q^{52} -6.00000i q^{53} +1.00000 q^{54} +4.00000 q^{56} +4.00000i q^{57} +6.00000i q^{58} -10.0000 q^{61} +8.00000i q^{62} -4.00000i q^{63} -1.00000 q^{64} +4.00000i q^{67} +6.00000i q^{68} +1.00000i q^{72} +2.00000i q^{73} +2.00000 q^{74} -4.00000 q^{76} -2.00000i q^{78} -8.00000 q^{79} +1.00000 q^{81} -6.00000i q^{82} +12.0000i q^{83} +4.00000 q^{84} +4.00000 q^{86} +6.00000i q^{87} -18.0000 q^{89} -8.00000 q^{91} +8.00000i q^{93} -1.00000 q^{96} -2.00000i q^{97} -9.00000i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{4} - 2 q^{6} - 2 q^{9} - 8 q^{14} + 2 q^{16} + 8 q^{19} - 8 q^{21} + 2 q^{24} - 4 q^{26} + 12 q^{29} + 16 q^{31} + 12 q^{34} + 2 q^{36} - 4 q^{39} - 12 q^{41} - 18 q^{49} + 12 q^{51} + 2 q^{54}+ \cdots - 2 q^{96}+O(q^{100}) $ 2 * q - 2 * q^4 - 2 * q^6 - 2 * q^9 - 8 * q^14 + 2 * q^16 + 8 * q^19 - 8 * q^21 + 2 * q^24 - 4 * q^26 + 12 * q^29 + 16 * q^31 + 12 * q^34 + 2 * q^36 - 4 * q^39 - 12 * q^41 - 18 * q^49 + 12 * q^51 + 2 * q^54 + 8 * q^56 - 20 * q^61 - 2 * q^64 + 4 * q^74 - 8 * q^76 - 16 * q^79 + 2 * q^81 + 8 * q^84 + 8 * q^86 - 36 * q^89 - 16 * q^91 - 2 * q^96

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/150\mathbb{Z}\right)^\times$.

$n$	$101$	$127$
$\chi(n)$	$1$	$-1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

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

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

$2$	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
$7$	4.00000i	1.51186i	0.654654	+	0.755929i	$0.272814\pi$
$7$	4.00000i	1.51186i	−0.654654	+	0.755929i	$0.727186\pi$
$8$	− 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
$13$	2.00000i	0.554700i	0.960769	+	0.277350i	$0.0894562\pi$
$13$	2.00000i	0.554700i	−0.960769	+	0.277350i	$0.910544\pi$
$14$	−4.00000	−1.06904
$15$	0	0
$16$	1.00000	0.250000
$17$	− 6.00000i	− 1.45521i	−0.685994	−	0.727607i	$-0.740633\pi$
$17$	− 6.00000i	− 1.45521i	0.685994	−	0.727607i	$-0.259367\pi$
$18$	− 1.00000i	− 0.235702i
$19$	4.00000	0.917663	0.458831	−	0.888523i	$-0.348268\pi$
$19$	4.00000	0.917663	0.458831	+	0.888523i	$0.348268\pi$
$20$	0	0
$21$	−4.00000	−0.872872
$22$	0	0
$23$	0	0	1.00000			$0$
$23$	0	0	−1.00000			$\pi$
$24$	1.00000	0.204124
$25$	0	0
$26$	−2.00000	−0.392232
$27$	− 1.00000i	− 0.192450i
$28$	− 4.00000i	− 0.755929i
$29$	6.00000	1.11417	0.557086	−	0.830455i	$-0.311919\pi$
$29$	6.00000	1.11417	0.557086	+	0.830455i	$0.311919\pi$
$30$	0	0
$31$	8.00000	1.43684	0.718421	−	0.695608i	$-0.244865\pi$
$31$	8.00000	1.43684	0.718421	+	0.695608i	$0.244865\pi$
$32$	1.00000i	0.176777i
$33$	0	0
$34$	6.00000	1.02899
$35$	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	$-0.0525685\pi$
$38$	4.00000i	0.648886i
$39$	−2.00000	−0.320256
$40$	0	0
$41$	−6.00000	−0.937043	−0.468521	−	0.883452i	$-0.655213\pi$
$41$	−6.00000	−0.937043	−0.468521	+	0.883452i	$0.655213\pi$
$42$	− 4.00000i	− 0.617213i
$43$	− 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
$46$	0	0
$47$	0	0	1.00000			$0$
$47$	0	0	−1.00000			$\pi$
$48$	1.00000i	0.144338i
$49$	−9.00000	−1.28571
$50$	0	0
$51$	6.00000	0.840168
$52$	− 2.00000i	− 0.277350i
$53$	− 6.00000i	− 0.824163i	−0.911147	−	0.412082i	$-0.864802\pi$
$53$	− 6.00000i	− 0.824163i	0.911147	−	0.412082i	$-0.135198\pi$
$54$	1.00000	0.136083
$55$	0	0
$56$	4.00000	0.534522
$57$	4.00000i	0.529813i
$58$	6.00000i	0.787839i
$59$	0	0		−	1.00000i	$-0.5\pi$
$59$	0	0			1.00000i	$0.5\pi$
$60$	0	0
$61$	−10.0000	−1.28037	−0.640184	−	0.768221i	$-0.721142\pi$
$61$	−10.0000	−1.28037	−0.640184	+	0.768221i	$0.721142\pi$
$62$	8.00000i	1.01600i
$63$	− 4.00000i	− 0.503953i
$64$	−1.00000	−0.125000
$65$	0	0
$66$	0	0
$67$	4.00000i	0.488678i	0.969690	+	0.244339i	$0.0785709\pi$
$67$	4.00000i	0.488678i	−0.969690	+	0.244339i	$0.921429\pi$
$68$	6.00000i	0.727607i
$69$	0	0
$70$	0	0
$71$	0	0		−	1.00000i	$-0.5\pi$
$71$	0	0			1.00000i	$0.5\pi$
$72$	1.00000i	0.117851i
$73$	2.00000i	0.234082i	0.993127	+	0.117041i	$0.0373409\pi$
$73$	2.00000i	0.234082i	−0.993127	+	0.117041i	$0.962659\pi$
$74$	2.00000	0.232495
$75$	0	0
$76$	−4.00000	−0.458831
$77$	0	0
$78$	− 2.00000i	− 0.226455i
$79$	−8.00000	−0.900070	−0.450035	−	0.893011i	$-0.648589\pi$
$79$	−8.00000	−0.900070	−0.450035	+	0.893011i	$0.648589\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	− 6.00000i	− 0.662589i
$83$	12.0000i	1.31717i	0.752506	+	0.658586i	$0.228845\pi$
$83$	12.0000i	1.31717i	−0.752506	+	0.658586i	$0.771155\pi$
$84$	4.00000	0.436436
$85$	0	0
$86$	4.00000	0.431331
$87$	6.00000i	0.643268i
$88$	0	0
$89$	−18.0000	−1.90800	−0.953998	−	0.299813i	$-0.903076\pi$
$89$	−18.0000	−1.90800	−0.953998	+	0.299813i	$0.903076\pi$
$90$	0	0
$91$	−8.00000	−0.838628
$92$	0	0
$93$	8.00000i	0.829561i
$94$	0	0
$95$	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	$-0.0323753\pi$
$98$	− 9.00000i	− 0.909137i
$99$	0	0
$100$	0	0
$101$	18.0000	1.79107	0.895533	−	0.444994i	$-0.146794\pi$
$101$	18.0000	1.79107	0.895533	+	0.444994i	$0.146794\pi$
$102$	6.00000i	0.594089i
$103$	− 4.00000i	− 0.394132i	−0.980390	−	0.197066i	$-0.936859\pi$
$103$	− 4.00000i	− 0.394132i	0.980390	−	0.197066i	$-0.0631413\pi$
$104$	2.00000	0.196116
$105$	0	0
$106$	6.00000	0.582772
$107$	12.0000i	1.16008i	0.814587	+	0.580042i	$0.196964\pi$
$107$	12.0000i	1.16008i	−0.814587	+	0.580042i	$0.803036\pi$
$108$	1.00000i	0.0962250i
$109$	10.0000	0.957826	0.478913	−	0.877862i	$-0.341031\pi$
$109$	10.0000	0.957826	0.478913	+	0.877862i	$0.341031\pi$
$110$	0	0
$111$	2.00000	0.189832
$112$	4.00000i	0.377964i
$113$	− 18.0000i	− 1.69330i	−0.532152	−	0.846649i	$-0.678617\pi$
$113$	− 18.0000i	− 1.69330i	0.532152	−	0.846649i	$-0.321383\pi$
$114$	−4.00000	−0.374634
$115$	0	0
$116$	−6.00000	−0.557086
$117$	− 2.00000i	− 0.184900i
$118$	0	0
$119$	24.0000	2.20008
$120$	0	0
$121$	−11.0000	−1.00000
$122$	− 10.0000i	− 0.905357i
$123$	− 6.00000i	− 0.541002i
$124$	−8.00000	−0.718421
$125$	0	0
$126$	4.00000	0.356348
$127$	− 20.0000i	− 1.77471i	−0.461084	−	0.887357i	$-0.652539\pi$
$127$	− 20.0000i	− 1.77471i	0.461084	−	0.887357i	$-0.347461\pi$
$128$	− 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
$133$	16.0000i	1.38738i
$134$	−4.00000	−0.345547
$135$	0	0
$136$	−6.00000	−0.514496
$137$	− 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
$139$	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
$141$	0	0
$142$	0	0
$143$	0	0
$144$	−1.00000	−0.0833333
$145$	0	0
$146$	−2.00000	−0.165521
$147$	− 9.00000i	− 0.742307i
$148$	2.00000i	0.164399i
$149$	6.00000	0.491539	0.245770	−	0.969328i	$-0.420959\pi$
$149$	6.00000	0.491539	0.245770	+	0.969328i	$0.420959\pi$
$150$	0	0
$151$	8.00000	0.651031	0.325515	−	0.945537i	$-0.394462\pi$
$151$	8.00000	0.651031	0.325515	+	0.945537i	$0.394462\pi$
$152$	− 4.00000i	− 0.324443i
$153$	6.00000i	0.485071i
$154$	0	0
$155$	0	0
$156$	2.00000	0.160128
$157$	− 2.00000i	− 0.159617i	−0.996810	−	0.0798087i	$-0.974569\pi$
$157$	− 2.00000i	− 0.159617i	0.996810	−	0.0798087i	$-0.0254309\pi$
$158$	− 8.00000i	− 0.636446i
$159$	6.00000	0.475831
$160$	0	0
$161$	0	0
$162$	1.00000i	0.0785674i
$163$	− 4.00000i	− 0.313304i	−0.987654	−	0.156652i	$-0.949930\pi$
$163$	− 4.00000i	− 0.313304i	0.987654	−	0.156652i	$-0.0500701\pi$
$164$	6.00000	0.468521
$165$	0	0
$166$	−12.0000	−0.931381
$167$	0	0	1.00000			$0$
$167$	0	0	−1.00000			$\pi$
$168$	4.00000i	0.308607i
$169$	9.00000	0.692308
$170$	0	0
$171$	−4.00000	−0.305888
$172$	4.00000i	0.304997i
$173$	18.0000i	1.36851i	0.729241	+	0.684257i	$0.239873\pi$
$173$	18.0000i	1.36851i	−0.729241	+	0.684257i	$0.760127\pi$
$174$	−6.00000	−0.454859
$175$	0	0
$176$	0	0
$177$	0	0
$178$	− 18.0000i	− 1.34916i
$179$	−24.0000	−1.79384	−0.896922	−	0.442189i	$-0.854202\pi$
$179$	−24.0000	−1.79384	−0.896922	+	0.442189i	$0.854202\pi$
$180$	0	0
$181$	14.0000	1.04061	0.520306	−	0.853980i	$-0.325818\pi$
$181$	14.0000	1.04061	0.520306	+	0.853980i	$0.325818\pi$
$182$	− 8.00000i	− 0.592999i
$183$	− 10.0000i	− 0.739221i
$184$	0	0
$185$	0	0
$186$	−8.00000	−0.586588
$187$	0	0
$188$	0	0
$189$	4.00000	0.290957
$190$	0	0
$191$	−24.0000	−1.73658	−0.868290	−	0.496058i	$-0.834780\pi$
$191$	−24.0000	−1.73658	−0.868290	+	0.496058i	$0.834780\pi$
$192$	− 1.00000i	− 0.0721688i
$193$	− 22.0000i	− 1.58359i	−0.610784	−	0.791797i	$-0.709146\pi$
$193$	− 22.0000i	− 1.58359i	0.610784	−	0.791797i	$-0.290854\pi$
$194$	2.00000	0.143592
$195$	0	0
$196$	9.00000	0.642857
$197$	6.00000i	0.427482i	0.976890	+	0.213741i	$0.0685649\pi$
$197$	6.00000i	0.427482i	−0.976890	+	0.213741i	$0.931435\pi$
$198$	0	0
$199$	−8.00000	−0.567105	−0.283552	−	0.958957i	$-0.591513\pi$
$199$	−8.00000	−0.567105	−0.283552	+	0.958957i	$0.591513\pi$
$200$	0	0
$201$	−4.00000	−0.282138
$202$	18.0000i	1.26648i
$203$	24.0000i	1.68447i
$204$	−6.00000	−0.420084
$205$	0	0
$206$	4.00000	0.278693
$207$	0	0
$208$	2.00000i	0.138675i
$209$	0	0
$210$	0	0
$211$	20.0000	1.37686	0.688428	−	0.725304i	$-0.258301\pi$
$211$	20.0000	1.37686	0.688428	+	0.725304i	$0.258301\pi$
$212$	6.00000i	0.412082i
$213$	0	0
$214$	−12.0000	−0.820303
$215$	0	0
$216$	−1.00000	−0.0680414
$217$	32.0000i	2.17230i
$218$	10.0000i	0.677285i
$219$	−2.00000	−0.135147
$220$	0	0
$221$	12.0000	0.807207
$222$	2.00000i	0.134231i
$223$	20.0000i	1.33930i	0.742677	+	0.669650i	$0.233556\pi$
$223$	20.0000i	1.33930i	−0.742677	+	0.669650i	$0.766444\pi$
$224$	−4.00000	−0.267261
$225$	0	0
$226$	18.0000	1.19734
$227$	12.0000i	0.796468i	0.917284	+	0.398234i	$0.130377\pi$
$227$	12.0000i	0.796468i	−0.917284	+	0.398234i	$0.869623\pi$
$228$	− 4.00000i	− 0.264906i
$229$	10.0000	0.660819	0.330409	−	0.943838i	$-0.392813\pi$
$229$	10.0000	0.660819	0.330409	+	0.943838i	$0.392813\pi$
$230$	0	0
$231$	0	0
$232$	− 6.00000i	− 0.393919i
$233$	− 18.0000i	− 1.17922i	−0.807688	−	0.589610i	$-0.799282\pi$
$233$	− 18.0000i	− 1.17922i	0.807688	−	0.589610i	$-0.200718\pi$
$234$	2.00000	0.130744
$235$	0	0
$236$	0	0
$237$	− 8.00000i	− 0.519656i
$238$	24.0000i	1.55569i
$239$	−24.0000	−1.55243	−0.776215	−	0.630468i	$-0.782863\pi$
$239$	−24.0000	−1.55243	−0.776215	+	0.630468i	$0.782863\pi$
$240$	0	0
$241$	2.00000	0.128831	0.0644157	−	0.997923i	$-0.479482\pi$
$241$	2.00000	0.128831	0.0644157	+	0.997923i	$0.479482\pi$
$242$	− 11.0000i	− 0.707107i
$243$	1.00000i	0.0641500i
$244$	10.0000	0.640184
$245$	0	0
$246$	6.00000	0.382546
$247$	8.00000i	0.509028i
$248$	− 8.00000i	− 0.508001i
$249$	−12.0000	−0.760469
$250$	0	0
$251$	−24.0000	−1.51487	−0.757433	−	0.652913i	$-0.773547\pi$
$251$	−24.0000	−1.51487	−0.757433	+	0.652913i	$0.773547\pi$
$252$	4.00000i	0.251976i
$253$	0	0
$254$	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	$0.810261\pi$
$258$	4.00000i	0.249029i
$259$	8.00000	0.497096
$260$	0	0
$261$	−6.00000	−0.371391
$262$	0	0
$263$	0	0	1.00000			$0$
$263$	0	0	−1.00000			$\pi$
$264$	0	0
$265$	0	0
$266$	−16.0000	−0.981023
$267$	− 18.0000i	− 1.10158i
$268$	− 4.00000i	− 0.244339i
$269$	6.00000	0.365826	0.182913	−	0.983129i	$-0.441447\pi$
$269$	6.00000	0.365826	0.182913	+	0.983129i	$0.441447\pi$
$270$	0	0
$271$	−16.0000	−0.971931	−0.485965	−	0.873978i	$-0.661532\pi$
$271$	−16.0000	−0.971931	−0.485965	+	0.873978i	$0.661532\pi$
$272$	− 6.00000i	− 0.363803i
$273$	− 8.00000i	− 0.484182i
$274$	6.00000	0.362473
$275$	0	0
$276$	0	0
$277$	− 2.00000i	− 0.120168i	−0.998193	−	0.0600842i	$-0.980863\pi$
$277$	− 2.00000i	− 0.120168i	0.998193	−	0.0600842i	$-0.0191369\pi$
$278$	4.00000i	0.239904i
$279$	−8.00000	−0.478947
$280$	0	0
$281$	18.0000	1.07379	0.536895	−	0.843649i	$-0.319597\pi$
$281$	18.0000	1.07379	0.536895	+	0.843649i	$0.319597\pi$
$282$	0	0
$283$	− 28.0000i	− 1.66443i	−0.554455	−	0.832214i	$-0.687073\pi$
$283$	− 28.0000i	− 1.66443i	0.554455	−	0.832214i	$-0.312927\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	− 24.0000i	− 1.41668i
$288$	− 1.00000i	− 0.0589256i
$289$	−19.0000	−1.11765
$290$	0	0
$291$	2.00000	0.117242
$292$	− 2.00000i	− 0.117041i
$293$	− 6.00000i	− 0.350524i	−0.984522	−	0.175262i	$-0.943923\pi$
$293$	− 6.00000i	− 0.350524i	0.984522	−	0.175262i	$-0.0560772\pi$
$294$	9.00000	0.524891
$295$	0	0
$296$	−2.00000	−0.116248
$297$	0	0
$298$	6.00000i	0.347571i
$299$	0	0
$300$	0	0
$301$	16.0000	0.922225
$302$	8.00000i	0.460348i
$303$	18.0000i	1.03407i
$304$	4.00000	0.229416
$305$	0	0
$306$	−6.00000	−0.342997
$307$	− 20.0000i	− 1.14146i	−0.821138	−	0.570730i	$-0.806660\pi$
$307$	− 20.0000i	− 1.14146i	0.821138	−	0.570730i	$-0.193340\pi$
$308$	0	0
$309$	4.00000	0.227552
$310$	0	0
$311$	0	0		−	1.00000i	$-0.5\pi$
$311$	0	0			1.00000i	$0.5\pi$
$312$	2.00000i	0.113228i
$313$	2.00000i	0.113047i	0.998401	+	0.0565233i	$0.0180015\pi$
$313$	2.00000i	0.113047i	−0.998401	+	0.0565233i	$0.981998\pi$
$314$	2.00000	0.112867
$315$	0	0
$316$	8.00000	0.450035
$317$	− 18.0000i	− 1.01098i	−0.862832	−	0.505490i	$-0.831312\pi$
$317$	− 18.0000i	− 1.01098i	0.862832	−	0.505490i	$-0.168688\pi$
$318$	6.00000i	0.336463i
$319$	0	0
$320$	0	0
$321$	−12.0000	−0.669775
$322$	0	0
$323$	− 24.0000i	− 1.33540i
$324$	−1.00000	−0.0555556
$325$	0	0
$326$	4.00000	0.221540
$327$	10.0000i	0.553001i
$328$	6.00000i	0.331295i
$329$	0	0
$330$	0	0
$331$	−28.0000	−1.53902	−0.769510	−	0.638635i	$-0.779499\pi$
$331$	−28.0000	−1.53902	−0.769510	+	0.638635i	$0.779499\pi$
$332$	− 12.0000i	− 0.658586i
$333$	2.00000i	0.109599i
$334$	0	0
$335$	0	0
$336$	−4.00000	−0.218218
$337$	− 26.0000i	− 1.41631i	−0.706057	−	0.708155i	$-0.749528\pi$
$337$	− 26.0000i	− 1.41631i	0.706057	−	0.708155i	$-0.250472\pi$
$338$	9.00000i	0.489535i
$339$	18.0000	0.977626
$340$	0	0
$341$	0	0
$342$	− 4.00000i	− 0.216295i
$343$	− 8.00000i	− 0.431959i
$344$	−4.00000	−0.215666
$345$	0	0
$346$	−18.0000	−0.967686
$347$	− 12.0000i	− 0.644194i	−0.946707	−	0.322097i	$-0.895612\pi$
$347$	− 12.0000i	− 0.644194i	0.946707	−	0.322097i	$-0.104388\pi$
$348$	− 6.00000i	− 0.321634i
$349$	10.0000	0.535288	0.267644	−	0.963518i	$-0.413755\pi$
$349$	10.0000	0.535288	0.267644	+	0.963518i	$0.413755\pi$
$350$	0	0
$351$	2.00000	0.106752
$352$	0	0
$353$	6.00000i	0.319348i	0.987170	+	0.159674i	$0.0510443\pi$
$353$	6.00000i	0.319348i	−0.987170	+	0.159674i	$0.948956\pi$
$354$	0	0
$355$	0	0
$356$	18.0000	0.953998
$357$	24.0000i	1.27021i
$358$	− 24.0000i	− 1.26844i
$359$	24.0000	1.26667	0.633336	−	0.773877i	$-0.281685\pi$
$359$	24.0000	1.26667	0.633336	+	0.773877i	$0.281685\pi$
$360$	0	0
$361$	−3.00000	−0.157895
$362$	14.0000i	0.735824i
$363$	− 11.0000i	− 0.577350i
$364$	8.00000	0.419314
$365$	0	0
$366$	10.0000	0.522708
$367$	28.0000i	1.46159i	0.682598	+	0.730794i	$0.260850\pi$
$367$	28.0000i	1.46159i	−0.682598	+	0.730794i	$0.739150\pi$
$368$	0	0
$369$	6.00000	0.312348
$370$	0	0
$371$	24.0000	1.24602
$372$	− 8.00000i	− 0.414781i
$373$	26.0000i	1.34623i	0.739538	+	0.673114i	$0.235044\pi$
$373$	26.0000i	1.34623i	−0.739538	+	0.673114i	$0.764956\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	12.0000i	0.618031i
$378$	4.00000i	0.205738i
$379$	4.00000	0.205466	0.102733	−	0.994709i	$-0.467241\pi$
$379$	4.00000	0.205466	0.102733	+	0.994709i	$0.467241\pi$
$380$	0	0
$381$	20.0000	1.02463
$382$	− 24.0000i	− 1.22795i
$383$	0	0	1.00000			$0$
$383$	0	0	−1.00000			$\pi$
$384$	1.00000	0.0510310
$385$	0	0
$386$	22.0000	1.11977
$387$	4.00000i	0.203331i
$388$	2.00000i	0.101535i
$389$	6.00000	0.304212	0.152106	−	0.988364i	$-0.451394\pi$
$389$	6.00000	0.304212	0.152106	+	0.988364i	$0.451394\pi$
$390$	0	0
$391$	0	0
$392$	9.00000i	0.454569i
$393$	0	0
$394$	−6.00000	−0.302276
$395$	0	0
$396$	0	0
$397$	22.0000i	1.10415i	0.833795	+	0.552074i	$0.186163\pi$
$397$	22.0000i	1.10415i	−0.833795	+	0.552074i	$0.813837\pi$
$398$	− 8.00000i	− 0.401004i
$399$	−16.0000	−0.801002
$400$	0	0
$401$	−6.00000	−0.299626	−0.149813	−	0.988714i	$-0.547867\pi$
$401$	−6.00000	−0.299626	−0.149813	+	0.988714i	$0.547867\pi$
$402$	− 4.00000i	− 0.199502i
$403$	16.0000i	0.797017i
$404$	−18.0000	−0.895533
$405$	0	0
$406$	−24.0000	−1.19110
$407$	0	0
$408$	− 6.00000i	− 0.297044i
$409$	−26.0000	−1.28562	−0.642809	−	0.766027i	$-0.722231\pi$
$409$	−26.0000	−1.28562	−0.642809	+	0.766027i	$0.722231\pi$
$410$	0	0
$411$	6.00000	0.295958
$412$	4.00000i	0.197066i
$413$	0	0
$414$	0	0
$415$	0	0
$416$	−2.00000	−0.0980581
$417$	4.00000i	0.195881i
$418$	0	0
$419$	0	0		−	1.00000i	$-0.5\pi$
$419$	0	0			1.00000i	$0.5\pi$
$420$	0	0
$421$	−10.0000	−0.487370	−0.243685	−	0.969854i	$-0.578356\pi$
$421$	−10.0000	−0.487370	−0.243685	+	0.969854i	$0.578356\pi$
$422$	20.0000i	0.973585i
$423$	0	0
$424$	−6.00000	−0.291386
$425$	0	0
$426$	0	0
$427$	− 40.0000i	− 1.93574i
$428$	− 12.0000i	− 0.580042i
$429$	0	0
$430$	0	0
$431$	0	0		−	1.00000i	$-0.5\pi$
$431$	0	0			1.00000i	$0.5\pi$
$432$	− 1.00000i	− 0.0481125i
$433$	26.0000i	1.24948i	0.780833	+	0.624740i	$0.214795\pi$
$433$	26.0000i	1.24948i	−0.780833	+	0.624740i	$0.785205\pi$
$434$	−32.0000	−1.53605
$435$	0	0
$436$	−10.0000	−0.478913
$437$	0	0
$438$	− 2.00000i	− 0.0955637i
$439$	−8.00000	−0.381819	−0.190910	−	0.981608i	$-0.561144\pi$
$439$	−8.00000	−0.381819	−0.190910	+	0.981608i	$0.561144\pi$
$440$	0	0
$441$	9.00000	0.428571
$442$	12.0000i	0.570782i
$443$	12.0000i	0.570137i	0.958507	+	0.285069i	$0.0920164\pi$
$443$	12.0000i	0.570137i	−0.958507	+	0.285069i	$0.907984\pi$
$444$	−2.00000	−0.0949158
$445$	0	0
$446$	−20.0000	−0.947027
$447$	6.00000i	0.283790i
$448$	− 4.00000i	− 0.188982i
$449$	6.00000	0.283158	0.141579	−	0.989927i	$-0.454782\pi$
$449$	6.00000	0.283158	0.141579	+	0.989927i	$0.454782\pi$
$450$	0	0
$451$	0	0
$452$	18.0000i	0.846649i
$453$	8.00000i	0.375873i
$454$	−12.0000	−0.563188
$455$	0	0
$456$	4.00000	0.187317
$457$	− 26.0000i	− 1.21623i	−0.793849	−	0.608114i	$-0.791926\pi$
$457$	− 26.0000i	− 1.21623i	0.793849	−	0.608114i	$-0.208074\pi$
$458$	10.0000i	0.467269i
$459$	−6.00000	−0.280056
$460$	0	0
$461$	−30.0000	−1.39724	−0.698620	−	0.715493i	$-0.746202\pi$
$461$	−30.0000	−1.39724	−0.698620	+	0.715493i	$0.746202\pi$
$462$	0	0
$463$	− 4.00000i	− 0.185896i	−0.995671	−	0.0929479i	$-0.970371\pi$
$463$	− 4.00000i	− 0.185896i	0.995671	−	0.0929479i	$-0.0296290\pi$
$464$	6.00000	0.278543
$465$	0	0
$466$	18.0000	0.833834
$467$	36.0000i	1.66588i	0.553362	+	0.832941i	$0.313345\pi$
$467$	36.0000i	1.66588i	−0.553362	+	0.832941i	$0.686655\pi$
$468$	2.00000i	0.0924500i
$469$	−16.0000	−0.738811
$470$	0	0
$471$	2.00000	0.0921551
$472$	0	0
$473$	0	0
$474$	8.00000	0.367452
$475$	0	0
$476$	−24.0000	−1.10004
$477$	6.00000i	0.274721i
$478$	− 24.0000i	− 1.09773i
$479$	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
$481$	4.00000	0.182384
$482$	2.00000i	0.0910975i
$483$	0	0
$484$	11.0000	0.500000
$485$	0	0
$486$	−1.00000	−0.0453609
$487$	28.0000i	1.26880i	0.773004	+	0.634401i	$0.218753\pi$
$487$	28.0000i	1.26880i	−0.773004	+	0.634401i	$0.781247\pi$
$488$	10.0000i	0.452679i
$489$	4.00000	0.180886
$490$	0	0
$491$	24.0000	1.08310	0.541552	−	0.840667i	$-0.317837\pi$
$491$	24.0000	1.08310	0.541552	+	0.840667i	$0.317837\pi$
$492$	6.00000i	0.270501i
$493$	− 36.0000i	− 1.62136i
$494$	−8.00000	−0.359937
$495$	0	0
$496$	8.00000	0.359211
$497$	0	0
$498$	− 12.0000i	− 0.537733i
$499$	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
$501$	0	0
$502$	− 24.0000i	− 1.07117i
$503$	24.0000i	1.07011i	0.844818	+	0.535054i	$0.179709\pi$
$503$	24.0000i	1.07011i	−0.844818	+	0.535054i	$0.820291\pi$
$504$	−4.00000	−0.178174
$505$	0	0
$506$	0	0
$507$	9.00000i	0.399704i
$508$	20.0000i	0.887357i
$509$	6.00000	0.265945	0.132973	−	0.991120i	$-0.457548\pi$
$509$	6.00000	0.265945	0.132973	+	0.991120i	$0.457548\pi$
$510$	0	0
$511$	−8.00000	−0.353899
$512$	1.00000i	0.0441942i
$513$	− 4.00000i	− 0.176604i
$514$	−18.0000	−0.793946
$515$	0	0
$516$	−4.00000	−0.176090
$517$	0	0
$518$	8.00000i	0.351500i
$519$	−18.0000	−0.790112
$520$	0	0
$521$	18.0000	0.788594	0.394297	−	0.918983i	$-0.370988\pi$
$521$	18.0000	0.788594	0.394297	+	0.918983i	$0.370988\pi$
$522$	− 6.00000i	− 0.262613i
$523$	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
$526$	0	0
$527$	− 48.0000i	− 2.09091i
$528$	0	0
$529$	23.0000	1.00000
$530$	0	0
$531$	0	0
$532$	− 16.0000i	− 0.693688i
$533$	− 12.0000i	− 0.519778i
$534$	18.0000	0.778936
$535$	0	0
$536$	4.00000	0.172774
$537$	− 24.0000i	− 1.03568i
$538$	6.00000i	0.258678i
$539$	0	0
$540$	0	0
$541$	−10.0000	−0.429934	−0.214967	−	0.976621i	$-0.568964\pi$
$541$	−10.0000	−0.429934	−0.214967	+	0.976621i	$0.568964\pi$
$542$	− 16.0000i	− 0.687259i
$543$	14.0000i	0.600798i
$544$	6.00000	0.257248
$545$	0	0
$546$	8.00000	0.342368
$547$	28.0000i	1.19719i	0.801050	+	0.598597i	$0.204275\pi$
$547$	28.0000i	1.19719i	−0.801050	+	0.598597i	$0.795725\pi$
$548$	6.00000i	0.256307i
$549$	10.0000	0.426790
$550$	0	0
$551$	24.0000	1.02243
$552$	0	0
$553$	− 32.0000i	− 1.36078i
$554$	2.00000	0.0849719
$555$	0	0
$556$	−4.00000	−0.169638
$557$	− 18.0000i	− 0.762684i	−0.924434	−	0.381342i	$-0.875462\pi$
$557$	− 18.0000i	− 0.762684i	0.924434	−	0.381342i	$-0.124538\pi$
$558$	− 8.00000i	− 0.338667i
$559$	8.00000	0.338364
$560$	0	0
$561$	0	0
$562$	18.0000i	0.759284i
$563$	− 12.0000i	− 0.505740i	−0.967500	−	0.252870i	$-0.918626\pi$
$563$	− 12.0000i	− 0.505740i	0.967500	−	0.252870i	$-0.0813744\pi$
$564$	0	0
$565$	0	0
$566$	28.0000	1.17693
$567$	4.00000i	0.167984i
$568$	0	0
$569$	−18.0000	−0.754599	−0.377300	−	0.926091i	$-0.623147\pi$
$569$	−18.0000	−0.754599	−0.377300	+	0.926091i	$0.623147\pi$
$570$	0	0
$571$	20.0000	0.836974	0.418487	−	0.908223i	$-0.362561\pi$
$571$	20.0000	0.836974	0.418487	+	0.908223i	$0.362561\pi$
$572$	0	0
$573$	− 24.0000i	− 1.00261i
$574$	24.0000	1.00174
$575$	0	0
$576$	1.00000	0.0416667
$577$	− 2.00000i	− 0.0832611i	−0.999133	−	0.0416305i	$-0.986745\pi$
$577$	− 2.00000i	− 0.0832611i	0.999133	−	0.0416305i	$-0.0132552\pi$
$578$	− 19.0000i	− 0.790296i
$579$	22.0000	0.914289
$580$	0	0
$581$	−48.0000	−1.99138
$582$	2.00000i	0.0829027i
$583$	0	0
$584$	2.00000	0.0827606
$585$	0	0
$586$	6.00000	0.247858
$587$	12.0000i	0.495293i	0.968850	+	0.247647i	$0.0796572\pi$
$587$	12.0000i	0.495293i	−0.968850	+	0.247647i	$0.920343\pi$
$588$	9.00000i	0.371154i
$589$	32.0000	1.31854
$590$	0	0
$591$	−6.00000	−0.246807
$592$	− 2.00000i	− 0.0821995i
$593$	30.0000i	1.23195i	0.787765	+	0.615976i	$0.211238\pi$
$593$	30.0000i	1.23195i	−0.787765	+	0.615976i	$0.788762\pi$
$594$	0	0
$595$	0	0
$596$	−6.00000	−0.245770
$597$	− 8.00000i	− 0.327418i
$598$	0	0
$599$	24.0000	0.980613	0.490307	−	0.871550i	$-0.336885\pi$
$599$	24.0000	0.980613	0.490307	+	0.871550i	$0.336885\pi$
$600$	0	0
$601$	−22.0000	−0.897399	−0.448699	−	0.893683i	$-0.648113\pi$
$601$	−22.0000	−0.897399	−0.448699	+	0.893683i	$0.648113\pi$
$602$	16.0000i	0.652111i
$603$	− 4.00000i	− 0.162893i
$604$	−8.00000	−0.325515
$605$	0	0
$606$	−18.0000	−0.731200
$607$	4.00000i	0.162355i	0.996700	+	0.0811775i	$0.0258681\pi$
$607$	4.00000i	0.162355i	−0.996700	+	0.0811775i	$0.974132\pi$
$608$	4.00000i	0.162221i
$609$	−24.0000	−0.972529
$610$	0	0
$611$	0	0
$612$	− 6.00000i	− 0.242536i
$613$	2.00000i	0.0807792i	0.999184	+	0.0403896i	$0.0128599\pi$
$613$	2.00000i	0.0807792i	−0.999184	+	0.0403896i	$0.987140\pi$
$614$	20.0000	0.807134
$615$	0	0
$616$	0	0
$617$	− 30.0000i	− 1.20775i	−0.797077	−	0.603877i	$-0.793622\pi$
$617$	− 30.0000i	− 1.20775i	0.797077	−	0.603877i	$-0.206378\pi$
$618$	4.00000i	0.160904i
$619$	−44.0000	−1.76851	−0.884255	−	0.467005i	$-0.845333\pi$
$619$	−44.0000	−1.76851	−0.884255	+	0.467005i	$0.845333\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	− 72.0000i	− 2.88462i
$624$	−2.00000	−0.0800641
$625$	0	0
$626$	−2.00000	−0.0799361
$627$	0	0
$628$	2.00000i	0.0798087i
$629$	−12.0000	−0.478471
$630$	0	0
$631$	32.0000	1.27390	0.636950	−	0.770905i	$-0.280196\pi$
$631$	32.0000	1.27390	0.636950	+	0.770905i	$0.280196\pi$
$632$	8.00000i	0.318223i
$633$	20.0000i	0.794929i
$634$	18.0000	0.714871
$635$	0	0
$636$	−6.00000	−0.237915
$637$	− 18.0000i	− 0.713186i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−30.0000	−1.18493	−0.592464	−	0.805597i	$-0.701845\pi$
$641$	−30.0000	−1.18493	−0.592464	+	0.805597i	$0.701845\pi$
$642$	− 12.0000i	− 0.473602i
$643$	− 4.00000i	− 0.157745i	−0.996885	−	0.0788723i	$-0.974868\pi$
$643$	− 4.00000i	− 0.157745i	0.996885	−	0.0788723i	$-0.0251319\pi$
$644$	0	0
$645$	0	0
$646$	24.0000	0.944267
$647$	− 24.0000i	− 0.943537i	−0.881722	−	0.471769i	$-0.843616\pi$
$647$	− 24.0000i	− 0.943537i	0.881722	−	0.471769i	$-0.156384\pi$
$648$	− 1.00000i	− 0.0392837i
$649$	0	0
$650$	0	0
$651$	−32.0000	−1.25418
$652$	4.00000i	0.156652i
$653$	18.0000i	0.704394i	0.935926	+	0.352197i	$0.114565\pi$
$653$	18.0000i	0.704394i	−0.935926	+	0.352197i	$0.885435\pi$
$654$	−10.0000	−0.391031
$655$	0	0
$656$	−6.00000	−0.234261
$657$	− 2.00000i	− 0.0780274i
$658$	0	0
$659$	−48.0000	−1.86981	−0.934907	−	0.354892i	$-0.884518\pi$
$659$	−48.0000	−1.86981	−0.934907	+	0.354892i	$0.884518\pi$
$660$	0	0
$661$	14.0000	0.544537	0.272268	−	0.962221i	$-0.412226\pi$
$661$	14.0000	0.544537	0.272268	+	0.962221i	$0.412226\pi$
$662$	− 28.0000i	− 1.08825i
$663$	12.0000i	0.466041i
$664$	12.0000	0.465690
$665$	0	0
$666$	−2.00000	−0.0774984
$667$	0	0
$668$	0	0
$669$	−20.0000	−0.773245
$670$	0	0
$671$	0	0
$672$	− 4.00000i	− 0.154303i
$673$	26.0000i	1.00223i	0.865382	+	0.501113i	$0.167076\pi$
$673$	26.0000i	1.00223i	−0.865382	+	0.501113i	$0.832924\pi$
$674$	26.0000	1.00148
$675$	0	0
$676$	−9.00000	−0.346154
$677$	6.00000i	0.230599i	0.993331	+	0.115299i	$0.0367827\pi$
$677$	6.00000i	0.230599i	−0.993331	+	0.115299i	$0.963217\pi$
$678$	18.0000i	0.691286i
$679$	8.00000	0.307012
$680$	0	0
$681$	−12.0000	−0.459841
$682$	0	0
$683$	− 12.0000i	− 0.459167i	−0.973289	−	0.229584i	$-0.926264\pi$
$683$	− 12.0000i	− 0.459167i	0.973289	−	0.229584i	$-0.0737364\pi$
$684$	4.00000	0.152944
$685$	0	0
$686$	8.00000	0.305441
$687$	10.0000i	0.381524i
$688$	− 4.00000i	− 0.152499i
$689$	12.0000	0.457164
$690$	0	0
$691$	44.0000	1.67384	0.836919	−	0.547326i	$-0.184354\pi$
$691$	44.0000	1.67384	0.836919	+	0.547326i	$0.184354\pi$
$692$	− 18.0000i	− 0.684257i
$693$	0	0
$694$	12.0000	0.455514
$695$	0	0
$696$	6.00000	0.227429
$697$	36.0000i	1.36360i
$698$	10.0000i	0.378506i
$699$	18.0000	0.680823
$700$	0	0
$701$	−6.00000	−0.226617	−0.113308	−	0.993560i	$-0.536145\pi$
$701$	−6.00000	−0.226617	−0.113308	+	0.993560i	$0.536145\pi$
$702$	2.00000i	0.0754851i
$703$	− 8.00000i	− 0.301726i
$704$	0	0
$705$	0	0
$706$	−6.00000	−0.225813
$707$	72.0000i	2.70784i
$708$	0	0
$709$	−38.0000	−1.42712	−0.713560	−	0.700594i	$-0.752918\pi$
$709$	−38.0000	−1.42712	−0.713560	+	0.700594i	$0.752918\pi$
$710$	0	0
$711$	8.00000	0.300023
$712$	18.0000i	0.674579i
$713$	0	0
$714$	−24.0000	−0.898177
$715$	0	0
$716$	24.0000	0.896922
$717$	− 24.0000i	− 0.896296i
$718$	24.0000i	0.895672i
$719$	0	0		−	1.00000i	$-0.5\pi$
$719$	0	0			1.00000i	$0.5\pi$
$720$	0	0
$721$	16.0000	0.595871
$722$	− 3.00000i	− 0.111648i
$723$	2.00000i	0.0743808i
$724$	−14.0000	−0.520306
$725$	0	0
$726$	11.0000	0.408248
$727$	28.0000i	1.03846i	0.854634	+	0.519231i	$0.173782\pi$
$727$	28.0000i	1.03846i	−0.854634	+	0.519231i	$0.826218\pi$
$728$	8.00000i	0.296500i
$729$	−1.00000	−0.0370370
$730$	0	0
$731$	−24.0000	−0.887672
$732$	10.0000i	0.369611i
$733$	− 22.0000i	− 0.812589i	−0.913742	−	0.406294i	$-0.866821\pi$
$733$	− 22.0000i	− 0.812589i	0.913742	−	0.406294i	$-0.133179\pi$
$734$	−28.0000	−1.03350
$735$	0	0
$736$	0	0
$737$	0	0
$738$	6.00000i	0.220863i
$739$	52.0000	1.91285	0.956425	−	0.291977i	$-0.0943129\pi$
$739$	52.0000	1.91285	0.956425	+	0.291977i	$0.0943129\pi$
$740$	0	0
$741$	−8.00000	−0.293887
$742$	24.0000i	0.881068i
$743$	− 24.0000i	− 0.880475i	−0.897881	−	0.440237i	$-0.854894\pi$
$743$	− 24.0000i	− 0.880475i	0.897881	−	0.440237i	$-0.145106\pi$
$744$	8.00000	0.293294
$745$	0	0
$746$	−26.0000	−0.951928
$747$	− 12.0000i	− 0.439057i
$748$	0	0
$749$	−48.0000	−1.75388
$750$	0	0
$751$	−40.0000	−1.45962	−0.729810	−	0.683650i	$-0.760392\pi$
$751$	−40.0000	−1.45962	−0.729810	+	0.683650i	$0.760392\pi$
$752$	0	0
$753$	− 24.0000i	− 0.874609i
$754$	−12.0000	−0.437014
$755$	0	0
$756$	−4.00000	−0.145479
$757$	− 2.00000i	− 0.0726912i	−0.999339	−	0.0363456i	$-0.988428\pi$
$757$	− 2.00000i	− 0.0726912i	0.999339	−	0.0363456i	$-0.0115717\pi$
$758$	4.00000i	0.145287i
$759$	0	0
$760$	0	0
$761$	18.0000	0.652499	0.326250	−	0.945284i	$-0.394215\pi$
$761$	18.0000	0.652499	0.326250	+	0.945284i	$0.394215\pi$
$762$	20.0000i	0.724524i
$763$	40.0000i	1.44810i
$764$	24.0000	0.868290
$765$	0	0
$766$	0	0
$767$	0	0
$768$	1.00000i	0.0360844i
$769$	−2.00000	−0.0721218	−0.0360609	−	0.999350i	$-0.511481\pi$
$769$	−2.00000	−0.0721218	−0.0360609	+	0.999350i	$0.511481\pi$
$770$	0	0
$771$	−18.0000	−0.648254
$772$	22.0000i	0.791797i
$773$	42.0000i	1.51064i	0.655359	+	0.755318i	$0.272517\pi$
$773$	42.0000i	1.51064i	−0.655359	+	0.755318i	$0.727483\pi$
$774$	−4.00000	−0.143777
$775$	0	0
$776$	−2.00000	−0.0717958
$777$	8.00000i	0.286998i
$778$	6.00000i	0.215110i
$779$	−24.0000	−0.859889
$780$	0	0
$781$	0	0
$782$	0	0
$783$	− 6.00000i	− 0.214423i
$784$	−9.00000	−0.321429
$785$	0	0
$786$	0	0
$787$	4.00000i	0.142585i	0.997455	+	0.0712923i	$0.0227123\pi$
$787$	4.00000i	0.142585i	−0.997455	+	0.0712923i	$0.977288\pi$
$788$	− 6.00000i	− 0.213741i
$789$	0	0
$790$	0	0
$791$	72.0000	2.56003
$792$	0	0
$793$	− 20.0000i	− 0.710221i
$794$	−22.0000	−0.780751
$795$	0	0
$796$	8.00000	0.283552
$797$	30.0000i	1.06265i	0.847167	+	0.531327i	$0.178307\pi$
$797$	30.0000i	1.06265i	−0.847167	+	0.531327i	$0.821693\pi$
$798$	− 16.0000i	− 0.566394i
$799$	0	0
$800$	0	0
$801$	18.0000	0.635999
$802$	− 6.00000i	− 0.211867i
$803$	0	0
$804$	4.00000	0.141069
$805$	0	0
$806$	−16.0000	−0.563576
$807$	6.00000i	0.211210i
$808$	− 18.0000i	− 0.633238i
$809$	54.0000	1.89854	0.949269	−	0.314464i	$-0.101825\pi$
$809$	54.0000	1.89854	0.949269	+	0.314464i	$0.101825\pi$
$810$	0	0
$811$	−4.00000	−0.140459	−0.0702295	−	0.997531i	$-0.522373\pi$
$811$	−4.00000	−0.140459	−0.0702295	+	0.997531i	$0.522373\pi$
$812$	− 24.0000i	− 0.842235i
$813$	− 16.0000i	− 0.561144i
$814$	0	0
$815$	0	0
$816$	6.00000	0.210042
$817$	− 16.0000i	− 0.559769i
$818$	− 26.0000i	− 0.909069i
$819$	8.00000	0.279543
$820$	0	0
$821$	18.0000	0.628204	0.314102	−	0.949389i	$-0.398297\pi$
$821$	18.0000	0.628204	0.314102	+	0.949389i	$0.398297\pi$
$822$	6.00000i	0.209274i
$823$	20.0000i	0.697156i	0.937280	+	0.348578i	$0.113335\pi$
$823$	20.0000i	0.697156i	−0.937280	+	0.348578i	$0.886665\pi$
$824$	−4.00000	−0.139347
$825$	0	0
$826$	0	0
$827$	− 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
$829$	−38.0000	−1.31979	−0.659897	−	0.751356i	$-0.729400\pi$
$829$	−38.0000	−1.31979	−0.659897	+	0.751356i	$0.729400\pi$
$830$	0	0
$831$	2.00000	0.0693792
$832$	− 2.00000i	− 0.0693375i
$833$	54.0000i	1.87099i
$834$	−4.00000	−0.138509
$835$	0	0
$836$	0	0
$837$	− 8.00000i	− 0.276520i
$838$	0	0
$839$	−24.0000	−0.828572	−0.414286	−	0.910147i	$-0.635969\pi$
$839$	−24.0000	−0.828572	−0.414286	+	0.910147i	$0.635969\pi$
$840$	0	0
$841$	7.00000	0.241379
$842$	− 10.0000i	− 0.344623i
$843$	18.0000i	0.619953i
$844$	−20.0000	−0.688428
$845$	0	0
$846$	0	0
$847$	− 44.0000i	− 1.51186i
$848$	− 6.00000i	− 0.206041i
$849$	28.0000	0.960958
$850$	0	0
$851$	0	0
$852$	0	0
$853$	− 46.0000i	− 1.57501i	−0.616308	−	0.787505i	$-0.711372\pi$
$853$	− 46.0000i	− 1.57501i	0.616308	−	0.787505i	$-0.288628\pi$
$854$	40.0000	1.36877
$855$	0	0
$856$	12.0000	0.410152
$857$	18.0000i	0.614868i	0.951569	+	0.307434i	$0.0994704\pi$
$857$	18.0000i	0.614868i	−0.951569	+	0.307434i	$0.900530\pi$
$858$	0	0
$859$	4.00000	0.136478	0.0682391	−	0.997669i	$-0.478262\pi$
$859$	4.00000	0.136478	0.0682391	+	0.997669i	$0.478262\pi$
$860$	0	0
$861$	24.0000	0.817918
$862$	0	0
$863$	24.0000i	0.816970i	0.912765	+	0.408485i	$0.133943\pi$
$863$	24.0000i	0.816970i	−0.912765	+	0.408485i	$0.866057\pi$
$864$	1.00000	0.0340207
$865$	0	0
$866$	−26.0000	−0.883516
$867$	− 19.0000i	− 0.645274i
$868$	− 32.0000i	− 1.08615i
$869$	0	0
$870$	0	0
$871$	−8.00000	−0.271070
$872$	− 10.0000i	− 0.338643i
$873$	2.00000i	0.0676897i
$874$	0	0
$875$	0	0
$876$	2.00000	0.0675737
$877$	− 2.00000i	− 0.0675352i	−0.999430	−	0.0337676i	$-0.989249\pi$
$877$	− 2.00000i	− 0.0675352i	0.999430	−	0.0337676i	$-0.0107506\pi$
$878$	− 8.00000i	− 0.269987i
$879$	6.00000	0.202375
$880$	0	0
$881$	−54.0000	−1.81931	−0.909653	−	0.415369i	$-0.863653\pi$
$881$	−54.0000	−1.81931	−0.909653	+	0.415369i	$0.863653\pi$
$882$	9.00000i	0.303046i
$883$	− 4.00000i	− 0.134611i	−0.997732	−	0.0673054i	$-0.978560\pi$
$883$	− 4.00000i	− 0.134611i	0.997732	−	0.0673054i	$-0.0214402\pi$
$884$	−12.0000	−0.403604
$885$	0	0
$886$	−12.0000	−0.403148
$887$	0	0	1.00000			$0$
$887$	0	0	−1.00000			$\pi$
$888$	− 2.00000i	− 0.0671156i
$889$	80.0000	2.68311
$890$	0	0
$891$	0	0
$892$	− 20.0000i	− 0.669650i
$893$	0	0
$894$	−6.00000	−0.200670
$895$	0	0
$896$	4.00000	0.133631
$897$	0	0
$898$	6.00000i	0.200223i
$899$	48.0000	1.60089
$900$	0	0
$901$	−36.0000	−1.19933
$902$	0	0
$903$	16.0000i	0.532447i
$904$	−18.0000	−0.598671
$905$	0	0
$906$	−8.00000	−0.265782
$907$	− 44.0000i	− 1.46100i	−0.682915	−	0.730498i	$-0.739288\pi$
$907$	− 44.0000i	− 1.46100i	0.682915	−	0.730498i	$-0.260712\pi$
$908$	− 12.0000i	− 0.398234i
$909$	−18.0000	−0.597022
$910$	0	0
$911$	−48.0000	−1.59031	−0.795155	−	0.606406i	$-0.792611\pi$
$911$	−48.0000	−1.59031	−0.795155	+	0.606406i	$0.792611\pi$
$912$	4.00000i	0.132453i
$913$	0	0
$914$	26.0000	0.860004
$915$	0	0
$916$	−10.0000	−0.330409
$917$	0	0
$918$	− 6.00000i	− 0.198030i
$919$	16.0000	0.527791	0.263896	−	0.964551i	$-0.414993\pi$
$919$	16.0000	0.527791	0.263896	+	0.964551i	$0.414993\pi$
$920$	0	0
$921$	20.0000	0.659022
$922$	− 30.0000i	− 0.987997i
$923$	0	0
$924$	0	0
$925$	0	0
$926$	4.00000	0.131448
$927$	4.00000i	0.131377i
$928$	6.00000i	0.196960i
$929$	6.00000	0.196854	0.0984268	−	0.995144i	$-0.468619\pi$
$929$	6.00000	0.196854	0.0984268	+	0.995144i	$0.468619\pi$
$930$	0	0
$931$	−36.0000	−1.17985
$932$	18.0000i	0.589610i
$933$	0	0
$934$	−36.0000	−1.17796
$935$	0	0
$936$	−2.00000	−0.0653720
$937$	− 26.0000i	− 0.849383i	−0.905338	−	0.424691i	$-0.860383\pi$
$937$	− 26.0000i	− 0.849383i	0.905338	−	0.424691i	$-0.139617\pi$
$938$	− 16.0000i	− 0.522419i
$939$	−2.00000	−0.0652675
$940$	0	0
$941$	18.0000	0.586783	0.293392	−	0.955992i	$-0.405216\pi$
$941$	18.0000	0.586783	0.293392	+	0.955992i	$0.405216\pi$
$942$	2.00000i	0.0651635i
$943$	0	0
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 36.0000i	− 1.16984i	−0.811090	−	0.584921i	$-0.801125\pi$
$947$	− 36.0000i	− 1.16984i	0.811090	−	0.584921i	$-0.198875\pi$
$948$	8.00000i	0.259828i
$949$	−4.00000	−0.129845
$950$	0	0
$951$	18.0000	0.583690
$952$	− 24.0000i	− 0.777844i
$953$	6.00000i	0.194359i	0.995267	+	0.0971795i	$0.0309821\pi$
$953$	6.00000i	0.194359i	−0.995267	+	0.0971795i	$0.969018\pi$
$954$	−6.00000	−0.194257
$955$	0	0
$956$	24.0000	0.776215
$957$	0	0
$958$	24.0000i	0.775405i
$959$	24.0000	0.775000
$960$	0	0
$961$	33.0000	1.06452
$962$	4.00000i	0.128965i
$963$	− 12.0000i	− 0.386695i
$964$	−2.00000	−0.0644157
$965$	0	0
$966$	0	0
$967$	4.00000i	0.128631i	0.997930	+	0.0643157i	$0.0204865\pi$
$967$	4.00000i	0.128631i	−0.997930	+	0.0643157i	$0.979514\pi$
$968$	11.0000i	0.353553i
$969$	24.0000	0.770991
$970$	0	0
$971$	24.0000	0.770197	0.385098	−	0.922876i	$-0.374168\pi$
$971$	24.0000	0.770197	0.385098	+	0.922876i	$0.374168\pi$
$972$	− 1.00000i	− 0.0320750i
$973$	16.0000i	0.512936i
$974$	−28.0000	−0.897178
$975$	0	0
$976$	−10.0000	−0.320092
$977$	42.0000i	1.34370i	0.740688	+	0.671850i	$0.234500\pi$
$977$	42.0000i	1.34370i	−0.740688	+	0.671850i	$0.765500\pi$
$978$	4.00000i	0.127906i
$979$	0	0
$980$	0	0
$981$	−10.0000	−0.319275
$982$	24.0000i	0.765871i
$983$	− 24.0000i	− 0.765481i	−0.923856	−	0.382741i	$-0.874980\pi$
$983$	− 24.0000i	− 0.765481i	0.923856	−	0.382741i	$-0.125020\pi$
$984$	−6.00000	−0.191273
$985$	0	0
$986$	36.0000	1.14647
$987$	0	0
$988$	− 8.00000i	− 0.254514i
$989$	0	0
$990$	0	0
$991$	−16.0000	−0.508257	−0.254128	−	0.967170i	$-0.581789\pi$
$991$	−16.0000	−0.508257	−0.254128	+	0.967170i	$0.581789\pi$
$992$	8.00000i	0.254000i
$993$	− 28.0000i	− 0.888553i
$994$	0	0
$995$	0	0
$996$	12.0000	0.380235
$997$	− 26.0000i	− 0.823428i	−0.911313	−	0.411714i	$-0.864930\pi$
$997$	− 26.0000i	− 0.823428i	0.911313	−	0.411714i	$-0.135070\pi$
$998$	4.00000i	0.126618i
$999$	−2.00000	−0.0632772

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	150.2.c.a.49.2		2
3.2	odd	2		450.2.c.b.199.1		2
4.3	odd	2		1200.2.f.e.49.1		2
5.2	odd	4		30.2.a.a.1.1	✓	1
5.3	odd	4		150.2.a.b.1.1		1
5.4	even	2	inner	150.2.c.a.49.1		2
8.3	odd	2		4800.2.f.w.3649.2		2
8.5	even	2		4800.2.f.p.3649.1		2
12.11	even	2		3600.2.f.i.2449.1		2
15.2	even	4		90.2.a.c.1.1		1
15.8	even	4		450.2.a.d.1.1		1
15.14	odd	2		450.2.c.b.199.2		2
20.3	even	4		1200.2.a.k.1.1		1
20.7	even	4		240.2.a.b.1.1		1
20.19	odd	2		1200.2.f.e.49.2		2
35.2	odd	12		1470.2.i.o.361.1		2
35.12	even	12		1470.2.i.q.361.1		2
35.13	even	4		7350.2.a.ct.1.1		1
35.17	even	12		1470.2.i.q.961.1		2
35.27	even	4		1470.2.a.d.1.1		1
35.32	odd	12		1470.2.i.o.961.1		2
40.3	even	4		4800.2.a.d.1.1		1
40.13	odd	4		4800.2.a.cq.1.1		1
40.19	odd	2		4800.2.f.w.3649.1		2
40.27	even	4		960.2.a.p.1.1		1
40.29	even	2		4800.2.f.p.3649.2		2
40.37	odd	4		960.2.a.e.1.1		1
45.2	even	12		810.2.e.b.271.1		2
45.7	odd	12		810.2.e.l.271.1		2
45.22	odd	12		810.2.e.l.541.1		2
45.32	even	12		810.2.e.b.541.1		2
55.32	even	4		3630.2.a.w.1.1		1
60.23	odd	4		3600.2.a.f.1.1		1
60.47	odd	4		720.2.a.j.1.1		1
60.59	even	2		3600.2.f.i.2449.2		2
65.12	odd	4		5070.2.a.w.1.1		1
65.47	even	4		5070.2.b.k.1351.1		2
65.57	even	4		5070.2.b.k.1351.2		2
80.27	even	4		3840.2.k.f.1921.2		2
80.37	odd	4		3840.2.k.y.1921.1		2
80.67	even	4		3840.2.k.f.1921.1		2
80.77	odd	4		3840.2.k.y.1921.2		2
85.67	odd	4		8670.2.a.g.1.1		1
105.62	odd	4		4410.2.a.z.1.1		1
120.77	even	4		2880.2.a.a.1.1		1
120.107	odd	4		2880.2.a.q.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
30.2.a.a.1.1	✓	1	5.2	odd	4
90.2.a.c.1.1		1	15.2	even	4
150.2.a.b.1.1		1	5.3	odd	4
150.2.c.a.49.1		2	5.4	even	2	inner
150.2.c.a.49.2		2	1.1	even	1	trivial
240.2.a.b.1.1		1	20.7	even	4
450.2.a.d.1.1		1	15.8	even	4
450.2.c.b.199.1		2	3.2	odd	2
450.2.c.b.199.2		2	15.14	odd	2
720.2.a.j.1.1		1	60.47	odd	4
810.2.e.b.271.1		2	45.2	even	12
810.2.e.b.541.1		2	45.32	even	12
810.2.e.l.271.1		2	45.7	odd	12
810.2.e.l.541.1		2	45.22	odd	12
960.2.a.e.1.1		1	40.37	odd	4
960.2.a.p.1.1		1	40.27	even	4
1200.2.a.k.1.1		1	20.3	even	4
1200.2.f.e.49.1		2	4.3	odd	2
1200.2.f.e.49.2		2	20.19	odd	2
1470.2.a.d.1.1		1	35.27	even	4
1470.2.i.o.361.1		2	35.2	odd	12
1470.2.i.o.961.1		2	35.32	odd	12
1470.2.i.q.361.1		2	35.12	even	12
1470.2.i.q.961.1		2	35.17	even	12
2880.2.a.a.1.1		1	120.77	even	4
2880.2.a.q.1.1		1	120.107	odd	4
3600.2.a.f.1.1		1	60.23	odd	4
3600.2.f.i.2449.1		2	12.11	even	2
3600.2.f.i.2449.2		2	60.59	even	2
3630.2.a.w.1.1		1	55.32	even	4
3840.2.k.f.1921.1		2	80.67	even	4
3840.2.k.f.1921.2		2	80.27	even	4
3840.2.k.y.1921.1		2	80.37	odd	4
3840.2.k.y.1921.2		2	80.77	odd	4
4410.2.a.z.1.1		1	105.62	odd	4
4800.2.a.d.1.1		1	40.3	even	4
4800.2.a.cq.1.1		1	40.13	odd	4
4800.2.f.p.3649.1		2	8.5	even	2
4800.2.f.p.3649.2		2	40.29	even	2
4800.2.f.w.3649.1		2	40.19	odd	2
4800.2.f.w.3649.2		2	8.3	odd	2
5070.2.a.w.1.1		1	65.12	odd	4
5070.2.b.k.1351.1		2	65.47	even	4
5070.2.b.k.1351.2		2	65.57	even	4
7350.2.a.ct.1.1		1	35.13	even	4
8670.2.a.g.1.1		1	85.67	odd	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:	\( N \)	\(=\)	\( 150 = 2 \cdot 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	150.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{2} +1.00000i q^{3} -1.00000 q^{4} -1.00000 q^{6} +4.00000i q^{7} -1.00000i q^{8} -1.00000 q^{9} -1.00000i q^{12} +2.00000i q^{13} -4.00000 q^{14} +1.00000 q^{16} -6.00000i q^{17} -1.00000i q^{18} +4.00000 q^{19} -4.00000 q^{21} +1.00000 q^{24} -2.00000 q^{26} -1.00000i q^{27} -4.00000i q^{28} +6.00000 q^{29} +8.00000 q^{31} +1.00000i q^{32} +6.00000 q^{34} +1.00000 q^{36} -2.00000i q^{37} +4.00000i q^{38} -2.00000 q^{39} -6.00000 q^{41} -4.00000i q^{42} -4.00000i q^{43} +1.00000i q^{48} -9.00000 q^{49} +6.00000 q^{51} -2.00000i q^{52} -6.00000i q^{53} +1.00000 q^{54} +4.00000 q^{56} +4.00000i q^{57} +6.00000i q^{58} -10.0000 q^{61} +8.00000i q^{62} -4.00000i q^{63} -1.00000 q^{64} +4.00000i q^{67} +6.00000i q^{68} +1.00000i q^{72} +2.00000i q^{73} +2.00000 q^{74} -4.00000 q^{76} -2.00000i q^{78} -8.00000 q^{79} +1.00000 q^{81} -6.00000i q^{82} +12.0000i q^{83} +4.00000 q^{84} +4.00000 q^{86} +6.00000i q^{87} -18.0000 q^{89} -8.00000 q^{91} +8.00000i q^{93} -1.00000 q^{96} -2.00000i q^{97} -9.00000i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{4} - 2 q^{6} - 2 q^{9} - 8 q^{14} + 2 q^{16} + 8 q^{19} - 8 q^{21} + 2 q^{24} - 4 q^{26} + 12 q^{29} + 16 q^{31} + 12 q^{34} + 2 q^{36} - 4 q^{39} - 12 q^{41} - 18 q^{49} + 12 q^{51} + 2 q^{54}+ \cdots - 2 q^{96}+O(q^{100}) \) 2 * q - 2 * q^4 - 2 * q^6 - 2 * q^9 - 8 * q^14 + 2 * q^16 + 8 * q^19 - 8 * q^21 + 2 * q^24 - 4 * q^26 + 12 * q^29 + 16 * q^31 + 12 * q^34 + 2 * q^36 - 4 * q^39 - 12 * q^41 - 18 * q^49 + 12 * q^51 + 2 * q^54 + 8 * q^56 - 20 * q^61 - 2 * q^64 + 4 * q^74 - 8 * q^76 - 16 * q^79 + 2 * q^81 + 8 * q^84 + 8 * q^86 - 36 * q^89 - 16 * q^91 - 2 * q^96

Introduction

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