LMFDB - Embedded newform 1850.2.b.d.149.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1850,2,Mod(149,1850)] 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("1850.149"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1850, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 1850 = 2 \cdot 5^{2} \cdot 37 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1850.b (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [2,0,0,-2,0,0,0,0,6,0,-8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

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

Self dual:	no
Analytic conductor:	$14.7723243739$
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 370)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			149.2
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	1850.149
Dual form			1850.2.b.d.149.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{2} -1.00000 q^{4} -1.00000i q^{8} +3.00000 q^{9} -4.00000 q^{11} +2.00000i q^{13} +1.00000 q^{16} +2.00000i q^{17} +3.00000i q^{18} +4.00000 q^{19} -4.00000i q^{22} -2.00000 q^{26} +6.00000 q^{29} -4.00000 q^{31} +1.00000i q^{32} -2.00000 q^{34} -3.00000 q^{36} +1.00000i q^{37} +4.00000i q^{38} -6.00000 q^{41} +4.00000i q^{43} +4.00000 q^{44} +8.00000i q^{47} +7.00000 q^{49} -2.00000i q^{52} +10.0000i q^{53} +6.00000i q^{58} -4.00000 q^{59} +10.0000 q^{61} -4.00000i q^{62} -1.00000 q^{64} +8.00000i q^{67} -2.00000i q^{68} -3.00000i q^{72} +10.0000i q^{73} -1.00000 q^{74} -4.00000 q^{76} +4.00000 q^{79} +9.00000 q^{81} -6.00000i q^{82} -4.00000 q^{86} +4.00000i q^{88} -2.00000 q^{89} -8.00000 q^{94} -6.00000i q^{97} +7.00000i q^{98} -12.0000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{4} + 6 q^{9} - 8 q^{11} + 2 q^{16} + 8 q^{19} - 4 q^{26} + 12 q^{29} - 8 q^{31} - 4 q^{34} - 6 q^{36} - 12 q^{41} + 8 q^{44} + 14 q^{49} - 8 q^{59} + 20 q^{61} - 2 q^{64} - 2 q^{74} - 8 q^{76}+ \cdots - 24 q^{99}+O(q^{100}) $ 2 * q - 2 * q^4 + 6 * q^9 - 8 * q^11 + 2 * q^16 + 8 * q^19 - 4 * q^26 + 12 * q^29 - 8 * q^31 - 4 * q^34 - 6 * q^36 - 12 * q^41 + 8 * q^44 + 14 * q^49 - 8 * q^59 + 20 * q^61 - 2 * q^64 - 2 * q^74 - 8 * q^76 + 8 * q^79 + 18 * q^81 - 8 * q^86 - 4 * q^89 - 16 * q^94 - 24 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1850\mathbb{Z}\right)^\times$.

$n$	$1001$	$1777$
$\chi(n)$	$1$	$-1$

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
$3$	0	0	1.00000			$0$
$3$	0	0	−1.00000			$\pi$
$4$	−1.00000	−0.500000
$5$	0	0
$5$	0	0
$6$	0	0
$7$	0	0	1.00000			$0$
$7$	0	0	−1.00000			$\pi$
$8$	− 1.00000i	− 0.353553i
$9$	3.00000	1.00000
$10$	0	0
$11$	−4.00000	−1.20605	−0.603023	−	0.797724i	$-0.706037\pi$
$11$	−4.00000	−1.20605	−0.603023	+	0.797724i	$0.706037\pi$
$12$	0	0
$13$	2.00000i	0.554700i	0.960769	+	0.277350i	$0.0894562\pi$
$13$	2.00000i	0.554700i	−0.960769	+	0.277350i	$0.910544\pi$
$14$	0	0
$15$	0	0
$16$	1.00000	0.250000
$17$	2.00000i	0.485071i	0.970143	+	0.242536i	$0.0779791\pi$
$17$	2.00000i	0.485071i	−0.970143	+	0.242536i	$0.922021\pi$
$18$	3.00000i	0.707107i
$19$	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
$21$	0	0
$22$	− 4.00000i	− 0.852803i
$23$	0	0	1.00000			$0$
$23$	0	0	−1.00000			$\pi$
$24$	0	0
$25$	0	0
$26$	−2.00000	−0.392232
$27$	0	0
$28$	0	0
$29$	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
$31$	−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
$33$	0	0
$34$	−2.00000	−0.342997
$35$	0	0
$36$	−3.00000	−0.500000
$37$	1.00000i	0.164399i
$37$	1.00000i	0.164399i
$38$	4.00000i	0.648886i
$39$	0	0
$40$	0	0
$41$	−6.00000	−0.937043	−0.468521	−	0.883452i	$-0.655213\pi$
$41$	−6.00000	−0.937043	−0.468521	+	0.883452i	$0.655213\pi$
$42$	0	0
$43$	4.00000i	0.609994i	0.952353	+	0.304997i	$0.0986555\pi$
$43$	4.00000i	0.609994i	−0.952353	+	0.304997i	$0.901344\pi$
$44$	4.00000	0.603023
$45$	0	0
$46$	0	0
$47$	8.00000i	1.16692i	0.812142	+	0.583460i	$0.198301\pi$
$47$	8.00000i	1.16692i	−0.812142	+	0.583460i	$0.801699\pi$
$48$	0	0
$49$	7.00000	1.00000
$50$	0	0
$51$	0	0
$52$	− 2.00000i	− 0.277350i
$53$	10.0000i	1.37361i	0.726844	+	0.686803i	$0.240986\pi$
$53$	10.0000i	1.37361i	−0.726844	+	0.686803i	$0.759014\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	0	0
$58$	6.00000i	0.787839i
$59$	−4.00000	−0.520756	−0.260378	−	0.965507i	$-0.583847\pi$
$59$	−4.00000	−0.520756	−0.260378	+	0.965507i	$0.583847\pi$
$60$	0	0
$61$	10.0000	1.28037	0.640184	−	0.768221i	$-0.278858\pi$
$61$	10.0000	1.28037	0.640184	+	0.768221i	$0.278858\pi$
$62$	− 4.00000i	− 0.508001i
$63$	0	0
$64$	−1.00000	−0.125000
$65$	0	0
$66$	0	0
$67$	8.00000i	0.977356i	0.872464	+	0.488678i	$0.162521\pi$
$67$	8.00000i	0.977356i	−0.872464	+	0.488678i	$0.837479\pi$
$68$	− 2.00000i	− 0.242536i
$69$	0	0
$70$	0	0
$71$	0	0		−	1.00000i	$-0.5\pi$
$71$	0	0			1.00000i	$0.5\pi$
$72$	− 3.00000i	− 0.353553i
$73$	10.0000i	1.17041i	0.810885	+	0.585206i	$0.198986\pi$
$73$	10.0000i	1.17041i	−0.810885	+	0.585206i	$0.801014\pi$
$74$	−1.00000	−0.116248
$75$	0	0
$76$	−4.00000	−0.458831
$77$	0	0
$78$	0	0
$79$	4.00000	0.450035	0.225018	−	0.974355i	$-0.427756\pi$
$79$	4.00000	0.450035	0.225018	+	0.974355i	$0.427756\pi$
$80$	0	0
$81$	9.00000	1.00000
$82$	− 6.00000i	− 0.662589i
$83$	0	0	1.00000			$0$
$83$	0	0	−1.00000			$\pi$
$84$	0	0
$85$	0	0
$86$	−4.00000	−0.431331
$87$	0	0
$88$	4.00000i	0.426401i
$89$	−2.00000	−0.212000	−0.106000	−	0.994366i	$-0.533804\pi$
$89$	−2.00000	−0.212000	−0.106000	+	0.994366i	$0.533804\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	0	0
$94$	−8.00000	−0.825137
$95$	0	0
$96$	0	0
$97$	− 6.00000i	− 0.609208i	−0.952479	−	0.304604i	$-0.901476\pi$
$97$	− 6.00000i	− 0.609208i	0.952479	−	0.304604i	$-0.0985241\pi$
$98$	7.00000i	0.707107i
$99$	−12.0000	−1.20605
$100$	0	0
$101$	6.00000	0.597022	0.298511	−	0.954406i	$-0.403510\pi$
$101$	6.00000	0.597022	0.298511	+	0.954406i	$0.403510\pi$
$102$	0	0
$103$	0	0	1.00000			$0$
$103$	0	0	−1.00000			$\pi$
$104$	2.00000	0.196116
$105$	0	0
$106$	−10.0000	−0.971286
$107$	8.00000i	0.773389i	0.922208	+	0.386695i	$0.126383\pi$
$107$	8.00000i	0.773389i	−0.922208	+	0.386695i	$0.873617\pi$
$108$	0	0
$109$	−18.0000	−1.72409	−0.862044	−	0.506834i	$-0.830816\pi$
$109$	−18.0000	−1.72409	−0.862044	+	0.506834i	$0.830816\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	14.0000i	1.31701i	0.752577	+	0.658505i	$0.228811\pi$
$113$	14.0000i	1.31701i	−0.752577	+	0.658505i	$0.771189\pi$
$114$	0	0
$115$	0	0
$116$	−6.00000	−0.557086
$117$	6.00000i	0.554700i
$118$	− 4.00000i	− 0.368230i
$119$	0	0
$120$	0	0
$121$	5.00000	0.454545
$122$	10.0000i	0.905357i
$123$	0	0
$124$	4.00000	0.359211
$125$	0	0
$126$	0	0
$127$	− 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
$129$	0	0
$130$	0	0
$131$	−12.0000	−1.04844	−0.524222	−	0.851581i	$-0.675644\pi$
$131$	−12.0000	−1.04844	−0.524222	+	0.851581i	$0.675644\pi$
$132$	0	0
$133$	0	0
$134$	−8.00000	−0.691095
$135$	0	0
$136$	2.00000	0.171499
$137$	6.00000i	0.512615i	0.966595	+	0.256307i	$0.0825059\pi$
$137$	6.00000i	0.512615i	−0.966595	+	0.256307i	$0.917494\pi$
$138$	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
$143$	− 8.00000i	− 0.668994i
$144$	3.00000	0.250000
$145$	0	0
$146$	−10.0000	−0.827606
$147$	0	0
$148$	− 1.00000i	− 0.0821995i
$149$	10.0000	0.819232	0.409616	−	0.912258i	$-0.365663\pi$
$149$	10.0000	0.819232	0.409616	+	0.912258i	$0.365663\pi$
$150$	0	0
$151$	−24.0000	−1.95309	−0.976546	−	0.215308i	$-0.930924\pi$
$151$	−24.0000	−1.95309	−0.976546	+	0.215308i	$0.930924\pi$
$152$	− 4.00000i	− 0.324443i
$153$	6.00000i	0.485071i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	6.00000i	0.478852i	0.970915	+	0.239426i	$0.0769593\pi$
$157$	6.00000i	0.478852i	−0.970915	+	0.239426i	$0.923041\pi$
$158$	4.00000i	0.318223i
$159$	0	0
$160$	0	0
$161$	0	0
$162$	9.00000i	0.707107i
$163$	− 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
$166$	0	0
$167$	16.0000i	1.23812i	0.785345	+	0.619059i	$0.212486\pi$
$167$	16.0000i	1.23812i	−0.785345	+	0.619059i	$0.787514\pi$
$168$	0	0
$169$	9.00000	0.692308
$170$	0	0
$171$	12.0000	0.917663
$172$	− 4.00000i	− 0.304997i
$173$	− 14.0000i	− 1.06440i	−0.846619	−	0.532200i	$-0.821365\pi$
$173$	− 14.0000i	− 1.06440i	0.846619	−	0.532200i	$-0.178635\pi$
$174$	0	0
$175$	0	0
$176$	−4.00000	−0.301511
$177$	0	0
$178$	− 2.00000i	− 0.149906i
$179$	−4.00000	−0.298974	−0.149487	−	0.988764i	$-0.547762\pi$
$179$	−4.00000	−0.298974	−0.149487	+	0.988764i	$0.547762\pi$
$180$	0	0
$181$	−2.00000	−0.148659	−0.0743294	−	0.997234i	$-0.523682\pi$
$181$	−2.00000	−0.148659	−0.0743294	+	0.997234i	$0.523682\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	0	0
$186$	0	0
$187$	− 8.00000i	− 0.585018i
$188$	− 8.00000i	− 0.583460i
$189$	0	0
$190$	0	0
$191$	−12.0000	−0.868290	−0.434145	−	0.900843i	$-0.642949\pi$
$191$	−12.0000	−0.868290	−0.434145	+	0.900843i	$0.642949\pi$
$192$	0	0
$193$	14.0000i	1.00774i	0.863779	+	0.503871i	$0.168091\pi$
$193$	14.0000i	1.00774i	−0.863779	+	0.503871i	$0.831909\pi$
$194$	6.00000	0.430775
$195$	0	0
$196$	−7.00000	−0.500000
$197$	− 2.00000i	− 0.142494i	−0.997459	−	0.0712470i	$-0.977302\pi$
$197$	− 2.00000i	− 0.142494i	0.997459	−	0.0712470i	$-0.0226979\pi$
$198$	− 12.0000i	− 0.852803i
$199$	20.0000	1.41776	0.708881	−	0.705328i	$-0.249200\pi$
$199$	20.0000	1.41776	0.708881	+	0.705328i	$0.249200\pi$
$200$	0	0
$201$	0	0
$202$	6.00000i	0.422159i
$203$	0	0
$204$	0	0
$205$	0	0
$206$	0	0
$207$	0	0
$208$	2.00000i	0.138675i
$209$	−16.0000	−1.10674
$210$	0	0
$211$	28.0000	1.92760	0.963800	−	0.266627i	$-0.0859092\pi$
$211$	28.0000	1.92760	0.963800	+	0.266627i	$0.0859092\pi$
$212$	− 10.0000i	− 0.686803i
$213$	0	0
$214$	−8.00000	−0.546869
$215$	0	0
$216$	0	0
$217$	0	0
$218$	− 18.0000i	− 1.21911i
$219$	0	0
$220$	0	0
$221$	−4.00000	−0.269069
$222$	0	0
$223$	− 16.0000i	− 1.07144i	−0.844396	−	0.535720i	$-0.820040\pi$
$223$	− 16.0000i	− 1.07144i	0.844396	−	0.535720i	$-0.179960\pi$
$224$	0	0
$225$	0	0
$226$	−14.0000	−0.931266
$227$	− 20.0000i	− 1.32745i	−0.747978	−	0.663723i	$-0.768975\pi$
$227$	− 20.0000i	− 1.32745i	0.747978	−	0.663723i	$-0.231025\pi$
$228$	0	0
$229$	18.0000	1.18947	0.594737	−	0.803921i	$-0.297256\pi$
$229$	18.0000	1.18947	0.594737	+	0.803921i	$0.297256\pi$
$230$	0	0
$231$	0	0
$232$	− 6.00000i	− 0.393919i
$233$	− 22.0000i	− 1.44127i	−0.693316	−	0.720634i	$-0.743851\pi$
$233$	− 22.0000i	− 1.44127i	0.693316	−	0.720634i	$-0.256149\pi$
$234$	−6.00000	−0.392232
$235$	0	0
$236$	4.00000	0.260378
$237$	0	0
$238$	0	0
$239$	−12.0000	−0.776215	−0.388108	−	0.921614i	$-0.626871\pi$
$239$	−12.0000	−0.776215	−0.388108	+	0.921614i	$0.626871\pi$
$240$	0	0
$241$	2.00000	0.128831	0.0644157	−	0.997923i	$-0.479482\pi$
$241$	2.00000	0.128831	0.0644157	+	0.997923i	$0.479482\pi$
$242$	5.00000i	0.321412i
$243$	0	0
$244$	−10.0000	−0.640184
$245$	0	0
$246$	0	0
$247$	8.00000i	0.509028i
$248$	4.00000i	0.254000i
$249$	0	0
$250$	0	0
$251$	20.0000	1.26239	0.631194	−	0.775625i	$-0.282565\pi$
$251$	20.0000	1.26239	0.631194	+	0.775625i	$0.282565\pi$
$252$	0	0
$253$	0	0
$254$	8.00000	0.501965
$255$	0	0
$256$	1.00000	0.0625000
$257$	− 14.0000i	− 0.873296i	−0.899632	−	0.436648i	$-0.856166\pi$
$257$	− 14.0000i	− 0.873296i	0.899632	−	0.436648i	$-0.143834\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	18.0000	1.11417
$262$	− 12.0000i	− 0.741362i
$263$	− 32.0000i	− 1.97320i	−0.163144	−	0.986602i	$-0.552164\pi$
$263$	− 32.0000i	− 1.97320i	0.163144	−	0.986602i	$-0.447836\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	0	0
$268$	− 8.00000i	− 0.488678i
$269$	−14.0000	−0.853595	−0.426798	−	0.904347i	$-0.640358\pi$
$269$	−14.0000	−0.853595	−0.426798	+	0.904347i	$0.640358\pi$
$270$	0	0
$271$	−8.00000	−0.485965	−0.242983	−	0.970031i	$-0.578126\pi$
$271$	−8.00000	−0.485965	−0.242983	+	0.970031i	$0.578126\pi$
$272$	2.00000i	0.121268i
$273$	0	0
$274$	−6.00000	−0.362473
$275$	0	0
$276$	0	0
$277$	22.0000i	1.32185i	0.750451	+	0.660926i	$0.229836\pi$
$277$	22.0000i	1.32185i	−0.750451	+	0.660926i	$0.770164\pi$
$278$	4.00000i	0.239904i
$279$	−12.0000	−0.718421
$280$	0	0
$281$	2.00000	0.119310	0.0596550	−	0.998219i	$-0.481000\pi$
$281$	2.00000	0.119310	0.0596550	+	0.998219i	$0.481000\pi$
$282$	0	0
$283$	4.00000i	0.237775i	0.992908	+	0.118888i	$0.0379328\pi$
$283$	4.00000i	0.237775i	−0.992908	+	0.118888i	$0.962067\pi$
$284$	0	0
$285$	0	0
$286$	8.00000	0.473050
$287$	0	0
$288$	3.00000i	0.176777i
$289$	13.0000	0.764706
$290$	0	0
$291$	0	0
$292$	− 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	$0.823771\pi$
$294$	0	0
$295$	0	0
$296$	1.00000	0.0581238
$297$	0	0
$298$	10.0000i	0.579284i
$299$	0	0
$300$	0	0
$301$	0	0
$302$	− 24.0000i	− 1.38104i
$303$	0	0
$304$	4.00000	0.229416
$305$	0	0
$306$	−6.00000	−0.342997
$307$	32.0000i	1.82634i	0.407583	+	0.913168i	$0.366372\pi$
$307$	32.0000i	1.82634i	−0.407583	+	0.913168i	$0.633628\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	12.0000	0.680458	0.340229	−	0.940343i	$-0.389495\pi$
$311$	12.0000	0.680458	0.340229	+	0.940343i	$0.389495\pi$
$312$	0	0
$313$	− 10.0000i	− 0.565233i	−0.959233	−	0.282617i	$-0.908798\pi$
$313$	− 10.0000i	− 0.565233i	0.959233	−	0.282617i	$-0.0912024\pi$
$314$	−6.00000	−0.338600
$315$	0	0
$316$	−4.00000	−0.225018
$317$	− 2.00000i	− 0.112331i	−0.998421	−	0.0561656i	$-0.982113\pi$
$317$	− 2.00000i	− 0.112331i	0.998421	−	0.0561656i	$-0.0178875\pi$
$318$	0	0
$319$	−24.0000	−1.34374
$320$	0	0
$321$	0	0
$322$	0	0
$323$	8.00000i	0.445132i
$324$	−9.00000	−0.500000
$325$	0	0
$326$	4.00000	0.221540
$327$	0	0
$328$	6.00000i	0.331295i
$329$	0	0
$330$	0	0
$331$	20.0000	1.09930	0.549650	−	0.835395i	$-0.314761\pi$
$331$	20.0000	1.09930	0.549650	+	0.835395i	$0.314761\pi$
$332$	0	0
$333$	3.00000i	0.164399i
$334$	−16.0000	−0.875481
$335$	0	0
$336$	0	0
$337$	− 34.0000i	− 1.85210i	−0.377403	−	0.926049i	$-0.623183\pi$
$337$	− 34.0000i	− 1.85210i	0.377403	−	0.926049i	$-0.376817\pi$
$338$	9.00000i	0.489535i
$339$	0	0
$340$	0	0
$341$	16.0000	0.866449
$342$	12.0000i	0.648886i
$343$	0	0
$344$	4.00000	0.215666
$345$	0	0
$346$	14.0000	0.752645
$347$	− 4.00000i	− 0.214731i	−0.994220	−	0.107366i	$-0.965758\pi$
$347$	− 4.00000i	− 0.214731i	0.994220	−	0.107366i	$-0.0342415\pi$
$348$	0	0
$349$	18.0000	0.963518	0.481759	−	0.876304i	$-0.339998\pi$
$349$	18.0000	0.963518	0.481759	+	0.876304i	$0.339998\pi$
$350$	0	0
$351$	0	0
$352$	− 4.00000i	− 0.213201i
$353$	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
$356$	2.00000	0.106000
$357$	0	0
$358$	− 4.00000i	− 0.211407i
$359$	0	0		−	1.00000i	$-0.5\pi$
$359$	0	0			1.00000i	$0.5\pi$
$360$	0	0
$361$	−3.00000	−0.157895
$362$	− 2.00000i	− 0.105118i
$363$	0	0
$364$	0	0
$365$	0	0
$366$	0	0
$367$	− 8.00000i	− 0.417597i	−0.977959	−	0.208798i	$-0.933045\pi$
$367$	− 8.00000i	− 0.417597i	0.977959	−	0.208798i	$-0.0669552\pi$
$368$	0	0
$369$	−18.0000	−0.937043
$370$	0	0
$371$	0	0
$372$	0	0
$373$	− 14.0000i	− 0.724893i	−0.932005	−	0.362446i	$-0.881942\pi$
$373$	− 14.0000i	− 0.724893i	0.932005	−	0.362446i	$-0.118058\pi$
$374$	8.00000	0.413670
$375$	0	0
$376$	8.00000	0.412568
$377$	12.0000i	0.618031i
$378$	0	0
$379$	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
$381$	0	0
$382$	− 12.0000i	− 0.613973i
$383$	− 24.0000i	− 1.22634i	−0.789950	−	0.613171i	$-0.789894\pi$
$383$	− 24.0000i	− 1.22634i	0.789950	−	0.613171i	$-0.210106\pi$
$384$	0	0
$385$	0	0
$386$	−14.0000	−0.712581
$387$	12.0000i	0.609994i
$388$	6.00000i	0.304604i
$389$	14.0000	0.709828	0.354914	−	0.934899i	$-0.384510\pi$
$389$	14.0000	0.709828	0.354914	+	0.934899i	$0.384510\pi$
$390$	0	0
$391$	0	0
$392$	− 7.00000i	− 0.353553i
$393$	0	0
$394$	2.00000	0.100759
$395$	0	0
$396$	12.0000	0.603023
$397$	− 34.0000i	− 1.70641i	−0.521575	−	0.853206i	$-0.674655\pi$
$397$	− 34.0000i	− 1.70641i	0.521575	−	0.853206i	$-0.325345\pi$
$398$	20.0000i	1.00251i
$399$	0	0
$400$	0	0
$401$	−22.0000	−1.09863	−0.549314	−	0.835616i	$-0.685111\pi$
$401$	−22.0000	−1.09863	−0.549314	+	0.835616i	$0.685111\pi$
$402$	0	0
$403$	− 8.00000i	− 0.398508i
$404$	−6.00000	−0.298511
$405$	0	0
$406$	0	0
$407$	− 4.00000i	− 0.198273i
$408$	0	0
$409$	−2.00000	−0.0988936	−0.0494468	−	0.998777i	$-0.515746\pi$
$409$	−2.00000	−0.0988936	−0.0494468	+	0.998777i	$0.515746\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	0	0
$414$	0	0
$415$	0	0
$416$	−2.00000	−0.0980581
$417$	0	0
$418$	− 16.0000i	− 0.782586i
$419$	−28.0000	−1.36789	−0.683945	−	0.729534i	$-0.739737\pi$
$419$	−28.0000	−1.36789	−0.683945	+	0.729534i	$0.739737\pi$
$420$	0	0
$421$	−6.00000	−0.292422	−0.146211	−	0.989253i	$-0.546708\pi$
$421$	−6.00000	−0.292422	−0.146211	+	0.989253i	$0.546708\pi$
$422$	28.0000i	1.36302i
$423$	24.0000i	1.16692i
$424$	10.0000	0.485643
$425$	0	0
$426$	0	0
$427$	0	0
$428$	− 8.00000i	− 0.386695i
$429$	0	0
$430$	0	0
$431$	−20.0000	−0.963366	−0.481683	−	0.876346i	$-0.659974\pi$
$431$	−20.0000	−0.963366	−0.481683	+	0.876346i	$0.659974\pi$
$432$	0	0
$433$	− 14.0000i	− 0.672797i	−0.941720	−	0.336399i	$-0.890791\pi$
$433$	− 14.0000i	− 0.672797i	0.941720	−	0.336399i	$-0.109209\pi$
$434$	0	0
$435$	0	0
$436$	18.0000	0.862044
$437$	0	0
$438$	0	0
$439$	20.0000	0.954548	0.477274	−	0.878755i	$-0.341625\pi$
$439$	20.0000	0.954548	0.477274	+	0.878755i	$0.341625\pi$
$440$	0	0
$441$	21.0000	1.00000
$442$	− 4.00000i	− 0.190261i
$443$	− 24.0000i	− 1.14027i	−0.821549	−	0.570137i	$-0.806890\pi$
$443$	− 24.0000i	− 1.14027i	0.821549	−	0.570137i	$-0.193110\pi$
$444$	0	0
$445$	0	0
$446$	16.0000	0.757622
$447$	0	0
$448$	0	0
$449$	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
$451$	24.0000	1.13012
$452$	− 14.0000i	− 0.658505i
$453$	0	0
$454$	20.0000	0.938647
$455$	0	0
$456$	0	0
$457$	18.0000i	0.842004i	0.907060	+	0.421002i	$0.138322\pi$
$457$	18.0000i	0.842004i	−0.907060	+	0.421002i	$0.861678\pi$
$458$	18.0000i	0.841085i
$459$	0	0
$460$	0	0
$461$	10.0000	0.465746	0.232873	−	0.972507i	$-0.425187\pi$
$461$	10.0000	0.465746	0.232873	+	0.972507i	$0.425187\pi$
$462$	0	0
$463$	− 40.0000i	− 1.85896i	−0.368875	−	0.929479i	$-0.620257\pi$
$463$	− 40.0000i	− 1.85896i	0.368875	−	0.929479i	$-0.379743\pi$
$464$	6.00000	0.278543
$465$	0	0
$466$	22.0000	1.01913
$467$	− 28.0000i	− 1.29569i	−0.761774	−	0.647843i	$-0.775671\pi$
$467$	− 28.0000i	− 1.29569i	0.761774	−	0.647843i	$-0.224329\pi$
$468$	− 6.00000i	− 0.277350i
$469$	0	0
$470$	0	0
$471$	0	0
$472$	4.00000i	0.184115i
$473$	− 16.0000i	− 0.735681i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	30.0000i	1.37361i
$478$	− 12.0000i	− 0.548867i
$479$	−28.0000	−1.27935	−0.639676	−	0.768644i	$-0.720932\pi$
$479$	−28.0000	−1.27935	−0.639676	+	0.768644i	$0.720932\pi$
$480$	0	0
$481$	−2.00000	−0.0911922
$482$	2.00000i	0.0910975i
$483$	0	0
$484$	−5.00000	−0.227273
$485$	0	0
$486$	0	0
$487$	16.0000i	0.725029i	0.931978	+	0.362515i	$0.118082\pi$
$487$	16.0000i	0.725029i	−0.931978	+	0.362515i	$0.881918\pi$
$488$	− 10.0000i	− 0.452679i
$489$	0	0
$490$	0	0
$491$	20.0000	0.902587	0.451294	−	0.892375i	$-0.350963\pi$
$491$	20.0000	0.902587	0.451294	+	0.892375i	$0.350963\pi$
$492$	0	0
$493$	12.0000i	0.540453i
$494$	−8.00000	−0.359937
$495$	0	0
$496$	−4.00000	−0.179605
$497$	0	0
$498$	0	0
$499$	20.0000	0.895323	0.447661	−	0.894203i	$-0.352257\pi$
$499$	20.0000	0.895323	0.447661	+	0.894203i	$0.352257\pi$
$500$	0	0
$501$	0	0
$502$	20.0000i	0.892644i
$503$	24.0000i	1.07011i	0.844818	+	0.535054i	$0.179709\pi$
$503$	24.0000i	1.07011i	−0.844818	+	0.535054i	$0.820291\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	0	0
$508$	8.00000i	0.354943i
$509$	18.0000	0.797836	0.398918	−	0.916987i	$-0.369386\pi$
$509$	18.0000	0.797836	0.398918	+	0.916987i	$0.369386\pi$
$510$	0	0
$511$	0	0
$512$	1.00000i	0.0441942i
$513$	0	0
$514$	14.0000	0.617514
$515$	0	0
$516$	0	0
$517$	− 32.0000i	− 1.40736i
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−38.0000	−1.66481	−0.832405	−	0.554168i	$-0.813037\pi$
$521$	−38.0000	−1.66481	−0.832405	+	0.554168i	$0.813037\pi$
$522$	18.0000i	0.787839i
$523$	12.0000i	0.524723i	0.964970	+	0.262362i	$0.0845013\pi$
$523$	12.0000i	0.524723i	−0.964970	+	0.262362i	$0.915499\pi$
$524$	12.0000	0.524222
$525$	0	0
$526$	32.0000	1.39527
$527$	− 8.00000i	− 0.348485i
$528$	0	0
$529$	23.0000	1.00000
$530$	0	0
$531$	−12.0000	−0.520756
$532$	0	0
$533$	− 12.0000i	− 0.519778i
$534$	0	0
$535$	0	0
$536$	8.00000	0.345547
$537$	0	0
$538$	− 14.0000i	− 0.603583i
$539$	−28.0000	−1.20605
$540$	0	0
$541$	−6.00000	−0.257960	−0.128980	−	0.991647i	$-0.541170\pi$
$541$	−6.00000	−0.257960	−0.128980	+	0.991647i	$0.541170\pi$
$542$	− 8.00000i	− 0.343629i
$543$	0	0
$544$	−2.00000	−0.0857493
$545$	0	0
$546$	0	0
$547$	− 4.00000i	− 0.171028i	−0.996337	−	0.0855138i	$-0.972747\pi$
$547$	− 4.00000i	− 0.171028i	0.996337	−	0.0855138i	$-0.0272532\pi$
$548$	− 6.00000i	− 0.256307i
$549$	30.0000	1.28037
$550$	0	0
$551$	24.0000	1.02243
$552$	0	0
$553$	0	0
$554$	−22.0000	−0.934690
$555$	0	0
$556$	−4.00000	−0.169638
$557$	− 2.00000i	− 0.0847427i	−0.999102	−	0.0423714i	$-0.986509\pi$
$557$	− 2.00000i	− 0.0847427i	0.999102	−	0.0423714i	$-0.0134913\pi$
$558$	− 12.0000i	− 0.508001i
$559$	−8.00000	−0.338364
$560$	0	0
$561$	0	0
$562$	2.00000i	0.0843649i
$563$	36.0000i	1.51722i	0.651546	+	0.758610i	$0.274121\pi$
$563$	36.0000i	1.51722i	−0.651546	+	0.758610i	$0.725879\pi$
$564$	0	0
$565$	0	0
$566$	−4.00000	−0.168133
$567$	0	0
$568$	0	0
$569$	30.0000	1.25767	0.628833	−	0.777541i	$-0.283533\pi$
$569$	30.0000	1.25767	0.628833	+	0.777541i	$0.283533\pi$
$570$	0	0
$571$	12.0000	0.502184	0.251092	−	0.967963i	$-0.419210\pi$
$571$	12.0000	0.502184	0.251092	+	0.967963i	$0.419210\pi$
$572$	8.00000i	0.334497i
$573$	0	0
$574$	0	0
$575$	0	0
$576$	−3.00000	−0.125000
$577$	− 46.0000i	− 1.91501i	−0.288425	−	0.957503i	$-0.593132\pi$
$577$	− 46.0000i	− 1.91501i	0.288425	−	0.957503i	$-0.406868\pi$
$578$	13.0000i	0.540729i
$579$	0	0
$580$	0	0
$581$	0	0
$582$	0	0
$583$	− 40.0000i	− 1.65663i
$584$	10.0000	0.413803
$585$	0	0
$586$	−18.0000	−0.743573
$587$	12.0000i	0.495293i	0.968850	+	0.247647i	$0.0796572\pi$
$587$	12.0000i	0.495293i	−0.968850	+	0.247647i	$0.920343\pi$
$588$	0	0
$589$	−16.0000	−0.659269
$590$	0	0
$591$	0	0
$592$	1.00000i	0.0410997i
$593$	34.0000i	1.39621i	0.715994	+	0.698106i	$0.245974\pi$
$593$	34.0000i	1.39621i	−0.715994	+	0.698106i	$0.754026\pi$
$594$	0	0
$595$	0	0
$596$	−10.0000	−0.409616
$597$	0	0
$598$	0	0
$599$	−24.0000	−0.980613	−0.490307	−	0.871550i	$-0.663115\pi$
$599$	−24.0000	−0.980613	−0.490307	+	0.871550i	$0.663115\pi$
$600$	0	0
$601$	10.0000	0.407909	0.203954	−	0.978980i	$-0.434621\pi$
$601$	10.0000	0.407909	0.203954	+	0.978980i	$0.434621\pi$
$602$	0	0
$603$	24.0000i	0.977356i
$604$	24.0000	0.976546
$605$	0	0
$606$	0	0
$607$	− 24.0000i	− 0.974130i	−0.873366	−	0.487065i	$-0.838067\pi$
$607$	− 24.0000i	− 0.974130i	0.873366	−	0.487065i	$-0.161933\pi$
$608$	4.00000i	0.162221i
$609$	0	0
$610$	0	0
$611$	−16.0000	−0.647291
$612$	− 6.00000i	− 0.242536i
$613$	− 6.00000i	− 0.242338i	−0.992632	−	0.121169i	$-0.961336\pi$
$613$	− 6.00000i	− 0.242338i	0.992632	−	0.121169i	$-0.0386643\pi$
$614$	−32.0000	−1.29141
$615$	0	0
$616$	0	0
$617$	30.0000i	1.20775i	0.797077	+	0.603877i	$0.206378\pi$
$617$	30.0000i	1.20775i	−0.797077	+	0.603877i	$0.793622\pi$
$618$	0	0
$619$	−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
$621$	0	0
$622$	12.0000i	0.481156i
$623$	0	0
$624$	0	0
$625$	0	0
$626$	10.0000	0.399680
$627$	0	0
$628$	− 6.00000i	− 0.239426i
$629$	−2.00000	−0.0797452
$630$	0	0
$631$	−36.0000	−1.43314	−0.716569	−	0.697517i	$-0.754288\pi$
$631$	−36.0000	−1.43314	−0.716569	+	0.697517i	$0.754288\pi$
$632$	− 4.00000i	− 0.159111i
$633$	0	0
$634$	2.00000	0.0794301
$635$	0	0
$636$	0	0
$637$	14.0000i	0.554700i
$638$	− 24.0000i	− 0.950169i
$639$	0	0
$640$	0	0
$641$	2.00000	0.0789953	0.0394976	−	0.999220i	$-0.487424\pi$
$641$	2.00000	0.0789953	0.0394976	+	0.999220i	$0.487424\pi$
$642$	0	0
$643$	− 44.0000i	− 1.73519i	−0.497271	−	0.867595i	$-0.665665\pi$
$643$	− 44.0000i	− 1.73519i	0.497271	−	0.867595i	$-0.334335\pi$
$644$	0	0
$645$	0	0
$646$	−8.00000	−0.314756
$647$	− 8.00000i	− 0.314512i	−0.987558	−	0.157256i	$-0.949735\pi$
$647$	− 8.00000i	− 0.314512i	0.987558	−	0.157256i	$-0.0502649\pi$
$648$	− 9.00000i	− 0.353553i
$649$	16.0000	0.628055
$650$	0	0
$651$	0	0
$652$	4.00000i	0.156652i
$653$	34.0000i	1.33052i	0.746611	+	0.665261i	$0.231680\pi$
$653$	34.0000i	1.33052i	−0.746611	+	0.665261i	$0.768320\pi$
$654$	0	0
$655$	0	0
$656$	−6.00000	−0.234261
$657$	30.0000i	1.17041i
$658$	0	0
$659$	−20.0000	−0.779089	−0.389545	−	0.921008i	$-0.627368\pi$
$659$	−20.0000	−0.779089	−0.389545	+	0.921008i	$0.627368\pi$
$660$	0	0
$661$	−22.0000	−0.855701	−0.427850	−	0.903850i	$-0.640729\pi$
$661$	−22.0000	−0.855701	−0.427850	+	0.903850i	$0.640729\pi$
$662$	20.0000i	0.777322i
$663$	0	0
$664$	0	0
$665$	0	0
$666$	−3.00000	−0.116248
$667$	0	0
$668$	− 16.0000i	− 0.619059i
$669$	0	0
$670$	0	0
$671$	−40.0000	−1.54418
$672$	0	0
$673$	10.0000i	0.385472i	0.981251	+	0.192736i	$0.0617360\pi$
$673$	10.0000i	0.385472i	−0.981251	+	0.192736i	$0.938264\pi$
$674$	34.0000	1.30963
$675$	0	0
$676$	−9.00000	−0.346154
$677$	− 2.00000i	− 0.0768662i	−0.999261	−	0.0384331i	$-0.987763\pi$
$677$	− 2.00000i	− 0.0768662i	0.999261	−	0.0384331i	$-0.0122367\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	0	0
$682$	16.0000i	0.612672i
$683$	36.0000i	1.37750i	0.724998	+	0.688751i	$0.241841\pi$
$683$	36.0000i	1.37750i	−0.724998	+	0.688751i	$0.758159\pi$
$684$	−12.0000	−0.458831
$685$	0	0
$686$	0	0
$687$	0	0
$688$	4.00000i	0.152499i
$689$	−20.0000	−0.761939
$690$	0	0
$691$	−20.0000	−0.760836	−0.380418	−	0.924815i	$-0.624220\pi$
$691$	−20.0000	−0.760836	−0.380418	+	0.924815i	$0.624220\pi$
$692$	14.0000i	0.532200i
$693$	0	0
$694$	4.00000	0.151838
$695$	0	0
$696$	0	0
$697$	− 12.0000i	− 0.454532i
$698$	18.0000i	0.681310i
$699$	0	0
$700$	0	0
$701$	50.0000	1.88847	0.944237	−	0.329267i	$-0.106802\pi$
$701$	50.0000	1.88847	0.944237	+	0.329267i	$0.106802\pi$
$702$	0	0
$703$	4.00000i	0.150863i
$704$	4.00000	0.150756
$705$	0	0
$706$	−6.00000	−0.225813
$707$	0	0
$708$	0	0
$709$	30.0000	1.12667	0.563337	−	0.826227i	$-0.309517\pi$
$709$	30.0000	1.12667	0.563337	+	0.826227i	$0.309517\pi$
$710$	0	0
$711$	12.0000	0.450035
$712$	2.00000i	0.0749532i
$713$	0	0
$714$	0	0
$715$	0	0
$716$	4.00000	0.149487
$717$	0	0
$718$	0	0
$719$	−32.0000	−1.19340	−0.596699	−	0.802465i	$-0.703521\pi$
$719$	−32.0000	−1.19340	−0.596699	+	0.802465i	$0.703521\pi$
$720$	0	0
$721$	0	0
$722$	− 3.00000i	− 0.111648i
$723$	0	0
$724$	2.00000	0.0743294
$725$	0	0
$726$	0	0
$727$	− 16.0000i	− 0.593407i	−0.954970	−	0.296704i	$-0.904113\pi$
$727$	− 16.0000i	− 0.593407i	0.954970	−	0.296704i	$-0.0958873\pi$
$728$	0	0
$729$	27.0000	1.00000
$730$	0	0
$731$	−8.00000	−0.295891
$732$	0	0
$733$	34.0000i	1.25582i	0.778287	+	0.627909i	$0.216089\pi$
$733$	34.0000i	1.25582i	−0.778287	+	0.627909i	$0.783911\pi$
$734$	8.00000	0.295285
$735$	0	0
$736$	0	0
$737$	− 32.0000i	− 1.17874i
$738$	− 18.0000i	− 0.662589i
$739$	−36.0000	−1.32428	−0.662141	−	0.749380i	$-0.730352\pi$
$739$	−36.0000	−1.32428	−0.662141	+	0.749380i	$0.730352\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	− 24.0000i	− 0.880475i	−0.897881	−	0.440237i	$-0.854894\pi$
$743$	− 24.0000i	− 0.880475i	0.897881	−	0.440237i	$-0.145106\pi$
$744$	0	0
$745$	0	0
$746$	14.0000	0.512576
$747$	0	0
$748$	8.00000i	0.292509i
$749$	0	0
$750$	0	0
$751$	−40.0000	−1.45962	−0.729810	−	0.683650i	$-0.760392\pi$
$751$	−40.0000	−1.45962	−0.729810	+	0.683650i	$0.760392\pi$
$752$	8.00000i	0.291730i
$753$	0	0
$754$	−12.0000	−0.437014
$755$	0	0
$756$	0	0
$757$	− 10.0000i	− 0.363456i	−0.983349	−	0.181728i	$-0.941831\pi$
$757$	− 10.0000i	− 0.363456i	0.983349	−	0.181728i	$-0.0581691\pi$
$758$	28.0000i	1.01701i
$759$	0	0
$760$	0	0
$761$	42.0000	1.52250	0.761249	−	0.648459i	$-0.224586\pi$
$761$	42.0000	1.52250	0.761249	+	0.648459i	$0.224586\pi$
$762$	0	0
$763$	0	0
$764$	12.0000	0.434145
$765$	0	0
$766$	24.0000	0.867155
$767$	− 8.00000i	− 0.288863i
$768$	0	0
$769$	30.0000	1.08183	0.540914	−	0.841078i	$-0.318079\pi$
$769$	30.0000	1.08183	0.540914	+	0.841078i	$0.318079\pi$
$770$	0	0
$771$	0	0
$772$	− 14.0000i	− 0.503871i
$773$	− 22.0000i	− 0.791285i	−0.918405	−	0.395643i	$-0.870522\pi$
$773$	− 22.0000i	− 0.791285i	0.918405	−	0.395643i	$-0.129478\pi$
$774$	−12.0000	−0.431331
$775$	0	0
$776$	−6.00000	−0.215387
$777$	0	0
$778$	14.0000i	0.501924i
$779$	−24.0000	−0.859889
$780$	0	0
$781$	0	0
$782$	0	0
$783$	0	0
$784$	7.00000	0.250000
$785$	0	0
$786$	0	0
$787$	− 48.0000i	− 1.71102i	−0.517790	−	0.855508i	$-0.673245\pi$
$787$	− 48.0000i	− 1.71102i	0.517790	−	0.855508i	$-0.326755\pi$
$788$	2.00000i	0.0712470i
$789$	0	0
$790$	0	0
$791$	0	0
$792$	12.0000i	0.426401i
$793$	20.0000i	0.710221i
$794$	34.0000	1.20661
$795$	0	0
$796$	−20.0000	−0.708881
$797$	30.0000i	1.06265i	0.847167	+	0.531327i	$0.178307\pi$
$797$	30.0000i	1.06265i	−0.847167	+	0.531327i	$0.821693\pi$
$798$	0	0
$799$	−16.0000	−0.566039
$800$	0	0
$801$	−6.00000	−0.212000
$802$	− 22.0000i	− 0.776847i
$803$	− 40.0000i	− 1.41157i
$804$	0	0
$805$	0	0
$806$	8.00000	0.281788
$807$	0	0
$808$	− 6.00000i	− 0.211079i
$809$	−26.0000	−0.914111	−0.457056	−	0.889438i	$-0.651096\pi$
$809$	−26.0000	−0.914111	−0.457056	+	0.889438i	$0.651096\pi$
$810$	0	0
$811$	−20.0000	−0.702295	−0.351147	−	0.936320i	$-0.614208\pi$
$811$	−20.0000	−0.702295	−0.351147	+	0.936320i	$0.614208\pi$
$812$	0	0
$813$	0	0
$814$	4.00000	0.140200
$815$	0	0
$816$	0	0
$817$	16.0000i	0.559769i
$818$	− 2.00000i	− 0.0699284i
$819$	0	0
$820$	0	0
$821$	6.00000	0.209401	0.104701	−	0.994504i	$-0.466612\pi$
$821$	6.00000	0.209401	0.104701	+	0.994504i	$0.466612\pi$
$822$	0	0
$823$	8.00000i	0.278862i	0.990232	+	0.139431i	$0.0445274\pi$
$823$	8.00000i	0.278862i	−0.990232	+	0.139431i	$0.955473\pi$
$824$	0	0
$825$	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
$829$	14.0000	0.486240	0.243120	−	0.969996i	$-0.421829\pi$
$829$	14.0000	0.486240	0.243120	+	0.969996i	$0.421829\pi$
$830$	0	0
$831$	0	0
$832$	− 2.00000i	− 0.0693375i
$833$	14.0000i	0.485071i
$834$	0	0
$835$	0	0
$836$	16.0000	0.553372
$837$	0	0
$838$	− 28.0000i	− 0.967244i
$839$	−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
$842$	− 6.00000i	− 0.206774i
$843$	0	0
$844$	−28.0000	−0.963800
$845$	0	0
$846$	−24.0000	−0.825137
$847$	0	0
$848$	10.0000i	0.343401i
$849$	0	0
$850$	0	0
$851$	0	0
$852$	0	0
$853$	10.0000i	0.342393i	0.985237	+	0.171197i	$0.0547634\pi$
$853$	10.0000i	0.342393i	−0.985237	+	0.171197i	$0.945237\pi$
$854$	0	0
$855$	0	0
$856$	8.00000	0.273434
$857$	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
$859$	−20.0000	−0.682391	−0.341196	−	0.939992i	$-0.610832\pi$
$859$	−20.0000	−0.682391	−0.341196	+	0.939992i	$0.610832\pi$
$860$	0	0
$861$	0	0
$862$	− 20.0000i	− 0.681203i
$863$	24.0000i	0.816970i	0.912765	+	0.408485i	$0.133943\pi$
$863$	24.0000i	0.816970i	−0.912765	+	0.408485i	$0.866057\pi$
$864$	0	0
$865$	0	0
$866$	14.0000	0.475739
$867$	0	0
$868$	0	0
$869$	−16.0000	−0.542763
$870$	0	0
$871$	−16.0000	−0.542139
$872$	18.0000i	0.609557i
$873$	− 18.0000i	− 0.609208i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	− 42.0000i	− 1.41824i	−0.705088	−	0.709120i	$-0.749093\pi$
$877$	− 42.0000i	− 1.41824i	0.705088	−	0.709120i	$-0.250907\pi$
$878$	20.0000i	0.674967i
$879$	0	0
$880$	0	0
$881$	−30.0000	−1.01073	−0.505363	−	0.862907i	$-0.668641\pi$
$881$	−30.0000	−1.01073	−0.505363	+	0.862907i	$0.668641\pi$
$882$	21.0000i	0.707107i
$883$	20.0000i	0.673054i	0.941674	+	0.336527i	$0.109252\pi$
$883$	20.0000i	0.673054i	−0.941674	+	0.336527i	$0.890748\pi$
$884$	4.00000	0.134535
$885$	0	0
$886$	24.0000	0.806296
$887$	− 56.0000i	− 1.88030i	−0.340766	−	0.940148i	$-0.610687\pi$
$887$	− 56.0000i	− 1.88030i	0.340766	−	0.940148i	$-0.389313\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	−36.0000	−1.20605
$892$	16.0000i	0.535720i
$893$	32.0000i	1.07084i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	0	0
$898$	6.00000i	0.200223i
$899$	−24.0000	−0.800445
$900$	0	0
$901$	−20.0000	−0.666297
$902$	24.0000i	0.799113i
$903$	0	0
$904$	14.0000	0.465633
$905$	0	0
$906$	0	0
$907$	20.0000i	0.664089i	0.943264	+	0.332045i	$0.107738\pi$
$907$	20.0000i	0.664089i	−0.943264	+	0.332045i	$0.892262\pi$
$908$	20.0000i	0.663723i
$909$	18.0000	0.597022
$910$	0	0
$911$	−36.0000	−1.19273	−0.596367	−	0.802712i	$-0.703390\pi$
$911$	−36.0000	−1.19273	−0.596367	+	0.802712i	$0.703390\pi$
$912$	0	0
$913$	0	0
$914$	−18.0000	−0.595387
$915$	0	0
$916$	−18.0000	−0.594737
$917$	0	0
$918$	0	0
$919$	−20.0000	−0.659739	−0.329870	−	0.944027i	$-0.607005\pi$
$919$	−20.0000	−0.659739	−0.329870	+	0.944027i	$0.607005\pi$
$920$	0	0
$921$	0	0
$922$	10.0000i	0.329332i
$923$	0	0
$924$	0	0
$925$	0	0
$926$	40.0000	1.31448
$927$	0	0
$928$	6.00000i	0.196960i
$929$	−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
$931$	28.0000	0.917663
$932$	22.0000i	0.720634i
$933$	0	0
$934$	28.0000	0.916188
$935$	0	0
$936$	6.00000	0.196116
$937$	14.0000i	0.457360i	0.973502	+	0.228680i	$0.0734410\pi$
$937$	14.0000i	0.457360i	−0.973502	+	0.228680i	$0.926559\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−42.0000	−1.36916	−0.684580	−	0.728937i	$-0.740015\pi$
$941$	−42.0000	−1.36916	−0.684580	+	0.728937i	$0.740015\pi$
$942$	0	0
$943$	0	0
$944$	−4.00000	−0.130189
$945$	0	0
$946$	16.0000	0.520205
$947$	4.00000i	0.129983i	0.997886	+	0.0649913i	$0.0207020\pi$
$947$	4.00000i	0.129983i	−0.997886	+	0.0649913i	$0.979298\pi$
$948$	0	0
$949$	−20.0000	−0.649227
$950$	0	0
$951$	0	0
$952$	0	0
$953$	− 6.00000i	− 0.194359i	−0.995267	−	0.0971795i	$-0.969018\pi$
$953$	− 6.00000i	− 0.194359i	0.995267	−	0.0971795i	$-0.0309821\pi$
$954$	−30.0000	−0.971286
$955$	0	0
$956$	12.0000	0.388108
$957$	0	0
$958$	− 28.0000i	− 0.904639i
$959$	0	0
$960$	0	0
$961$	−15.0000	−0.483871
$962$	− 2.00000i	− 0.0644826i
$963$	24.0000i	0.773389i
$964$	−2.00000	−0.0644157
$965$	0	0
$966$	0	0
$967$	8.00000i	0.257263i	0.991692	+	0.128631i	$0.0410584\pi$
$967$	8.00000i	0.257263i	−0.991692	+	0.128631i	$0.958942\pi$
$968$	− 5.00000i	− 0.160706i
$969$	0	0
$970$	0	0
$971$	−12.0000	−0.385098	−0.192549	−	0.981287i	$-0.561675\pi$
$971$	−12.0000	−0.385098	−0.192549	+	0.981287i	$0.561675\pi$
$972$	0	0
$973$	0	0
$974$	−16.0000	−0.512673
$975$	0	0
$976$	10.0000	0.320092
$977$	50.0000i	1.59964i	0.600239	+	0.799821i	$0.295072\pi$
$977$	50.0000i	1.59964i	−0.600239	+	0.799821i	$0.704928\pi$
$978$	0	0
$979$	8.00000	0.255681
$980$	0	0
$981$	−54.0000	−1.72409
$982$	20.0000i	0.638226i
$983$	24.0000i	0.765481i	0.923856	+	0.382741i	$0.125020\pi$
$983$	24.0000i	0.765481i	−0.923856	+	0.382741i	$0.874980\pi$
$984$	0	0
$985$	0	0
$986$	−12.0000	−0.382158
$987$	0	0
$988$	− 8.00000i	− 0.254514i
$989$	0	0
$990$	0	0
$991$	20.0000	0.635321	0.317660	−	0.948205i	$-0.397103\pi$
$991$	20.0000	0.635321	0.317660	+	0.948205i	$0.397103\pi$
$992$	− 4.00000i	− 0.127000i
$993$	0	0
$994$	0	0
$995$	0	0
$996$	0	0
$997$	− 10.0000i	− 0.316703i	−0.987383	−	0.158352i	$-0.949382\pi$
$997$	− 10.0000i	− 0.316703i	0.987383	−	0.158352i	$-0.0506179\pi$
$998$	20.0000i	0.633089i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1850.2.b.d.149.2		2
5.2	odd	4		370.2.a.b.1.1	✓	1
5.3	odd	4		1850.2.a.k.1.1		1
5.4	even	2	inner	1850.2.b.d.149.1		2
15.2	even	4		3330.2.a.w.1.1		1
20.7	even	4		2960.2.a.g.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
370.2.a.b.1.1	✓	1	5.2	odd	4
1850.2.a.k.1.1		1	5.3	odd	4
1850.2.b.d.149.1		2	5.4	even	2	inner
1850.2.b.d.149.2		2	1.1	even	1	trivial
2960.2.a.g.1.1		1	20.7	even	4
3330.2.a.w.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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 1850 = 2 \cdot 5^{2} \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1850.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{2} -1.00000 q^{4} -1.00000i q^{8} +3.00000 q^{9} -4.00000 q^{11} +2.00000i q^{13} +1.00000 q^{16} +2.00000i q^{17} +3.00000i q^{18} +4.00000 q^{19} -4.00000i q^{22} -2.00000 q^{26} +6.00000 q^{29} -4.00000 q^{31} +1.00000i q^{32} -2.00000 q^{34} -3.00000 q^{36} +1.00000i q^{37} +4.00000i q^{38} -6.00000 q^{41} +4.00000i q^{43} +4.00000 q^{44} +8.00000i q^{47} +7.00000 q^{49} -2.00000i q^{52} +10.0000i q^{53} +6.00000i q^{58} -4.00000 q^{59} +10.0000 q^{61} -4.00000i q^{62} -1.00000 q^{64} +8.00000i q^{67} -2.00000i q^{68} -3.00000i q^{72} +10.0000i q^{73} -1.00000 q^{74} -4.00000 q^{76} +4.00000 q^{79} +9.00000 q^{81} -6.00000i q^{82} -4.00000 q^{86} +4.00000i q^{88} -2.00000 q^{89} -8.00000 q^{94} -6.00000i q^{97} +7.00000i q^{98} -12.0000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{4} + 6 q^{9} - 8 q^{11} + 2 q^{16} + 8 q^{19} - 4 q^{26} + 12 q^{29} - 8 q^{31} - 4 q^{34} - 6 q^{36} - 12 q^{41} + 8 q^{44} + 14 q^{49} - 8 q^{59} + 20 q^{61} - 2 q^{64} - 2 q^{74} - 8 q^{76}+ \cdots - 24 q^{99}+O(q^{100}) \) 2 * q - 2 * q^4 + 6 * q^9 - 8 * q^11 + 2 * q^16 + 8 * q^19 - 4 * q^26 + 12 * q^29 - 8 * q^31 - 4 * q^34 - 6 * q^36 - 12 * q^41 + 8 * q^44 + 14 * q^49 - 8 * q^59 + 20 * q^61 - 2 * q^64 - 2 * q^74 - 8 * q^76 + 8 * q^79 + 18 * q^81 - 8 * q^86 - 4 * q^89 - 16 * q^94 - 24 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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