LMFDB - Embedded newform 2178.3.d.b.1693.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2178,3,Mod(1693,2178)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2178, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1])) N = Newforms(chi, 3, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2178.1693"); S:= CuspForms(chi, 3); N := Newforms(S);

Level:	$ N $	$=$	$ 2178 = 2 \cdot 3^{2} \cdot 11^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	2178.d (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [2,0,0,-4,-4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

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

Self dual:	no
Analytic conductor:	$59.3462015777$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{-2}) $
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 2 $ x^2 + 2
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1693.2
Root			$1.41421i$ of defining polynomial
Character	$\chi$	$=$	2178.1693
Dual form			2178.3.d.b.1693.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+1.41421i q^{2} -2.00000 q^{4} -2.00000 q^{5} -2.82843i q^{8} -2.82843i q^{10} -7.07107i q^{13} +4.00000 q^{16} -9.89949i q^{17} +5.65685i q^{19} +4.00000 q^{20} -20.0000 q^{23} -21.0000 q^{25} +10.0000 q^{26} +7.07107i q^{29} +12.0000 q^{31} +5.65685i q^{32} +14.0000 q^{34} +30.0000 q^{37} -8.00000 q^{38} +5.65685i q^{40} +21.2132i q^{41} -28.2843i q^{46} -44.0000 q^{47} +49.0000 q^{49} -29.6985i q^{50} +14.1421i q^{52} +56.0000 q^{53} -10.0000 q^{58} +80.0000 q^{59} +29.6985i q^{61} +16.9706i q^{62} -8.00000 q^{64} +14.1421i q^{65} +80.0000 q^{67} +19.7990i q^{68} -60.0000 q^{71} +77.7817i q^{73} +42.4264i q^{74} -11.3137i q^{76} -62.2254i q^{79} -8.00000 q^{80} -30.0000 q^{82} -39.5980i q^{83} +19.7990i q^{85} -80.0000 q^{89} +40.0000 q^{92} -62.2254i q^{94} -11.3137i q^{95} +69.2965i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 4 q^{4} - 4 q^{5} + 8 q^{16} + 8 q^{20} - 40 q^{23} - 42 q^{25} + 20 q^{26} + 24 q^{31} + 28 q^{34} + 60 q^{37} - 16 q^{38} - 88 q^{47} + 98 q^{49} + 112 q^{53} - 20 q^{58} + 160 q^{59} - 16 q^{64}+ \cdots + 80 q^{92}+O(q^{100}) $ 2 * q - 4 * q^4 - 4 * q^5 + 8 * q^16 + 8 * q^20 - 40 * q^23 - 42 * q^25 + 20 * q^26 + 24 * q^31 + 28 * q^34 + 60 * q^37 - 16 * q^38 - 88 * q^47 + 98 * q^49 + 112 * q^53 - 20 * q^58 + 160 * q^59 - 16 * q^64 + 160 * q^67 - 120 * q^71 - 16 * q^80 - 60 * q^82 - 160 * q^89 + 80 * q^92

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2178\mathbb{Z}\right)^\times$.

$n$	$1333$	$1937$
$\chi(n)$	$-1$	$1$

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 $

$2$	1.41421i	0.707107i
$2$	1.41421i	0.707107i
$3$	0	0
$3$	0	0
$4$	−2.00000	−0.500000
$5$	−2.00000	−0.400000	−0.200000	−	0.979796i	$-0.564094\pi$
$5$	−2.00000	−0.400000	−0.200000	+	0.979796i	$0.564094\pi$
$6$	0	0
$7$	0	0	1.00000			$0$
$7$	0	0	−1.00000			$\pi$
$8$	− 2.82843i	− 0.353553i
$9$	0	0
$10$	− 2.82843i	− 0.282843i
$11$	0	0
$11$	0	0
$12$	0	0
$13$	− 7.07107i	− 0.543928i	−0.962307	−	0.271964i	$-0.912327\pi$
$13$	− 7.07107i	− 0.543928i	0.962307	−	0.271964i	$-0.0876732\pi$
$14$	0	0
$15$	0	0
$16$	4.00000	0.250000
$17$	− 9.89949i	− 0.582323i	−0.956674	−	0.291162i	$-0.905958\pi$
$17$	− 9.89949i	− 0.582323i	0.956674	−	0.291162i	$-0.0940417\pi$
$18$	0	0
$19$	5.65685i	0.297729i	0.988858	+	0.148865i	$0.0475619\pi$
$19$	5.65685i	0.297729i	−0.988858	+	0.148865i	$0.952438\pi$
$20$	4.00000	0.200000
$21$	0	0
$22$	0	0
$23$	−20.0000	−0.869565	−0.434783	−	0.900535i	$-0.643175\pi$
$23$	−20.0000	−0.869565	−0.434783	+	0.900535i	$0.643175\pi$
$24$	0	0
$25$	−21.0000	−0.840000
$26$	10.0000	0.384615
$27$	0	0
$28$	0	0
$29$	7.07107i	0.243830i	0.992541	+	0.121915i	$0.0389035\pi$
$29$	7.07107i	0.243830i	−0.992541	+	0.121915i	$0.961096\pi$
$30$	0	0
$31$	12.0000	0.387097	0.193548	−	0.981091i	$-0.438000\pi$
$31$	12.0000	0.387097	0.193548	+	0.981091i	$0.438000\pi$
$32$	5.65685i	0.176777i
$33$	0	0
$34$	14.0000	0.411765
$35$	0	0
$36$	0	0
$37$	30.0000	0.810811	0.405405	−	0.914137i	$-0.367130\pi$
$37$	30.0000	0.810811	0.405405	+	0.914137i	$0.367130\pi$
$38$	−8.00000	−0.210526
$39$	0	0
$40$	5.65685i	0.141421i
$41$	21.2132i	0.517395i	0.965958	+	0.258698i	$0.0832933\pi$
$41$	21.2132i	0.517395i	−0.965958	+	0.258698i	$0.916707\pi$
$42$	0	0
$43$	0	0	1.00000			$0$
$43$	0	0	−1.00000			$\pi$
$44$	0	0
$45$	0	0
$46$	− 28.2843i	− 0.614875i
$47$	−44.0000	−0.936170	−0.468085	−	0.883683i	$-0.655056\pi$
$47$	−44.0000	−0.936170	−0.468085	+	0.883683i	$0.655056\pi$
$48$	0	0
$49$	49.0000	1.00000
$50$	− 29.6985i	− 0.593970i
$51$	0	0
$52$	14.1421i	0.271964i
$53$	56.0000	1.05660	0.528302	−	0.849057i	$-0.322829\pi$
$53$	56.0000	1.05660	0.528302	+	0.849057i	$0.322829\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	0	0
$58$	−10.0000	−0.172414
$59$	80.0000	1.35593	0.677966	−	0.735093i	$-0.262862\pi$
$59$	80.0000	1.35593	0.677966	+	0.735093i	$0.262862\pi$
$60$	0	0
$61$	29.6985i	0.486860i	0.969918	+	0.243430i	$0.0782727\pi$
$61$	29.6985i	0.486860i	−0.969918	+	0.243430i	$0.921727\pi$
$62$	16.9706i	0.273719i
$63$	0	0
$64$	−8.00000	−0.125000
$65$	14.1421i	0.217571i
$66$	0	0
$67$	80.0000	1.19403	0.597015	−	0.802230i	$-0.296353\pi$
$67$	80.0000	1.19403	0.597015	+	0.802230i	$0.296353\pi$
$68$	19.7990i	0.291162i
$69$	0	0
$70$	0	0
$71$	−60.0000	−0.845070	−0.422535	−	0.906347i	$-0.638860\pi$
$71$	−60.0000	−0.845070	−0.422535	+	0.906347i	$0.638860\pi$
$72$	0	0
$73$	77.7817i	1.06550i	0.846272	+	0.532752i	$0.178842\pi$
$73$	77.7817i	1.06550i	−0.846272	+	0.532752i	$0.821158\pi$
$74$	42.4264i	0.573330i
$75$	0	0
$76$	− 11.3137i	− 0.148865i
$77$	0	0
$78$	0	0
$79$	− 62.2254i	− 0.787663i	−0.919183	−	0.393832i	$-0.871149\pi$
$79$	− 62.2254i	− 0.787663i	0.919183	−	0.393832i	$-0.128851\pi$
$80$	−8.00000	−0.100000
$81$	0	0
$82$	−30.0000	−0.365854
$83$	− 39.5980i	− 0.477084i	−0.971132	−	0.238542i	$-0.923331\pi$
$83$	− 39.5980i	− 0.477084i	0.971132	−	0.238542i	$-0.0766695\pi$
$84$	0	0
$85$	19.7990i	0.232929i
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−80.0000	−0.898876	−0.449438	−	0.893311i	$-0.648376\pi$
$89$	−80.0000	−0.898876	−0.449438	+	0.893311i	$0.648376\pi$
$90$	0	0
$91$	0	0
$92$	40.0000	0.434783
$93$	0	0
$94$	− 62.2254i	− 0.661972i
$95$	− 11.3137i	− 0.119092i
$96$	0	0
$97$	0	0		−	1.00000i	$-0.5\pi$
$97$	0	0			1.00000i	$0.5\pi$
$98$	69.2965i	0.707107i
$99$	0	0
$100$	42.0000	0.420000
$101$	162.635i	1.61024i	0.593110	+	0.805122i	$0.297900\pi$
$101$	162.635i	1.61024i	−0.593110	+	0.805122i	$0.702100\pi$
$102$	0	0
$103$	140.000	1.35922	0.679612	−	0.733572i	$-0.262148\pi$
$103$	140.000	1.35922	0.679612	+	0.733572i	$0.262148\pi$
$104$	−20.0000	−0.192308
$105$	0	0
$106$	79.1960i	0.747132i
$107$	152.735i	1.42743i	0.700436	+	0.713715i	$0.252989\pi$
$107$	152.735i	1.42743i	−0.700436	+	0.713715i	$0.747011\pi$
$108$	0	0
$109$	15.5563i	0.142719i	0.997451	+	0.0713594i	$0.0227337\pi$
$109$	15.5563i	0.142719i	−0.997451	+	0.0713594i	$0.977266\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−210.000	−1.85841	−0.929204	−	0.369568i	$-0.879506\pi$
$113$	−210.000	−1.85841	−0.929204	+	0.369568i	$0.879506\pi$
$114$	0	0
$115$	40.0000	0.347826
$116$	− 14.1421i	− 0.121915i
$117$	0	0
$118$	113.137i	0.958789i
$119$	0	0
$120$	0	0
$121$	0	0
$122$	−42.0000	−0.344262
$123$	0	0
$124$	−24.0000	−0.193548
$125$	92.0000	0.736000
$126$	0	0
$127$	169.706i	1.33626i	0.744042	+	0.668132i	$0.232906\pi$
$127$	169.706i	1.33626i	−0.744042	+	0.668132i	$0.767094\pi$
$128$	− 11.3137i	− 0.0883883i
$129$	0	0
$130$	−20.0000	−0.153846
$131$	56.5685i	0.431821i	0.976413	+	0.215910i	$0.0692719\pi$
$131$	56.5685i	0.431821i	−0.976413	+	0.215910i	$0.930728\pi$
$132$	0	0
$133$	0	0
$134$	113.137i	0.844307i
$135$	0	0
$136$	−28.0000	−0.205882
$137$	−126.000	−0.919708	−0.459854	−	0.887995i	$-0.652098\pi$
$137$	−126.000	−0.919708	−0.459854	+	0.887995i	$0.652098\pi$
$138$	0	0
$139$	79.1960i	0.569755i	0.958564	+	0.284878i	$0.0919529\pi$
$139$	79.1960i	0.569755i	−0.958564	+	0.284878i	$0.908047\pi$
$140$	0	0
$141$	0	0
$142$	− 84.8528i	− 0.597555i
$143$	0	0
$144$	0	0
$145$	− 14.1421i	− 0.0975320i
$146$	−110.000	−0.753425
$147$	0	0
$148$	−60.0000	−0.405405
$149$	− 63.6396i	− 0.427111i	−0.976931	−	0.213556i	$-0.931496\pi$
$149$	− 63.6396i	− 0.427111i	0.976931	−	0.213556i	$-0.0685045\pi$
$150$	0	0
$151$	− 118.794i	− 0.786715i	−0.919386	−	0.393357i	$-0.871314\pi$
$151$	− 118.794i	− 0.786715i	0.919386	−	0.393357i	$-0.128686\pi$
$152$	16.0000	0.105263
$153$	0	0
$154$	0	0
$155$	−24.0000	−0.154839
$156$	0	0
$157$	40.0000	0.254777	0.127389	−	0.991853i	$-0.459340\pi$
$157$	40.0000	0.254777	0.127389	+	0.991853i	$0.459340\pi$
$158$	88.0000	0.556962
$159$	0	0
$160$	− 11.3137i	− 0.0707107i
$161$	0	0
$162$	0	0
$163$	40.0000	0.245399	0.122699	−	0.992444i	$-0.460845\pi$
$163$	40.0000	0.245399	0.122699	+	0.992444i	$0.460845\pi$
$164$	− 42.4264i	− 0.258698i
$165$	0	0
$166$	56.0000	0.337349
$167$	265.872i	1.59205i	0.605265	+	0.796024i	$0.293067\pi$
$167$	265.872i	1.59205i	−0.605265	+	0.796024i	$0.706933\pi$
$168$	0	0
$169$	119.000	0.704142
$170$	−28.0000	−0.164706
$171$	0	0
$172$	0	0
$173$	244.659i	1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$173$	244.659i	1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	0	0
$178$	− 113.137i	− 0.635602i
$179$	−320.000	−1.78771	−0.893855	−	0.448357i	$-0.852009\pi$
$179$	−320.000	−1.78771	−0.893855	+	0.448357i	$0.852009\pi$
$180$	0	0
$181$	130.000	0.718232	0.359116	−	0.933293i	$-0.383078\pi$
$181$	130.000	0.718232	0.359116	+	0.933293i	$0.383078\pi$
$182$	0	0
$183$	0	0
$184$	56.5685i	0.307438i
$185$	−60.0000	−0.324324
$186$	0	0
$187$	0	0
$188$	88.0000	0.468085
$189$	0	0
$190$	16.0000	0.0842105
$191$	260.000	1.36126	0.680628	−	0.732629i	$-0.261707\pi$
$191$	260.000	1.36126	0.680628	+	0.732629i	$0.261707\pi$
$192$	0	0
$193$	148.492i	0.769391i	0.923044	+	0.384695i	$0.125694\pi$
$193$	148.492i	0.769391i	−0.923044	+	0.384695i	$0.874306\pi$
$194$	0	0
$195$	0	0
$196$	−98.0000	−0.500000
$197$	222.032i	1.12706i	0.826094	+	0.563532i	$0.190558\pi$
$197$	222.032i	1.12706i	−0.826094	+	0.563532i	$0.809442\pi$
$198$	0	0
$199$	−140.000	−0.703518	−0.351759	−	0.936091i	$-0.614416\pi$
$199$	−140.000	−0.703518	−0.351759	+	0.936091i	$0.614416\pi$
$200$	59.3970i	0.296985i
$201$	0	0
$202$	−230.000	−1.13861
$203$	0	0
$204$	0	0
$205$	− 42.4264i	− 0.206958i
$206$	197.990i	0.961116i
$207$	0	0
$208$	− 28.2843i	− 0.135982i
$209$	0	0
$210$	0	0
$211$	118.794i	0.563004i	0.959561	+	0.281502i	$0.0908327\pi$
$211$	118.794i	0.563004i	−0.959561	+	0.281502i	$0.909167\pi$
$212$	−112.000	−0.528302
$213$	0	0
$214$	−216.000	−1.00935
$215$	0	0
$216$	0	0
$217$	0	0
$218$	−22.0000	−0.100917
$219$	0	0
$220$	0	0
$221$	−70.0000	−0.316742
$222$	0	0
$223$	−140.000	−0.627803	−0.313901	−	0.949456i	$-0.601636\pi$
$223$	−140.000	−0.627803	−0.313901	+	0.949456i	$0.601636\pi$
$224$	0	0
$225$	0	0
$226$	− 296.985i	− 1.31409i
$227$	124.451i	0.548241i	0.961695	+	0.274121i	$0.0883868\pi$
$227$	124.451i	0.548241i	−0.961695	+	0.274121i	$0.911613\pi$
$228$	0	0
$229$	190.000	0.829694	0.414847	−	0.909891i	$-0.363835\pi$
$229$	190.000	0.829694	0.414847	+	0.909891i	$0.363835\pi$
$230$	56.5685i	0.245950i
$231$	0	0
$232$	20.0000	0.0862069
$233$	306.884i	1.31710i	0.752537	+	0.658550i	$0.228830\pi$
$233$	306.884i	1.31710i	−0.752537	+	0.658550i	$0.771170\pi$
$234$	0	0
$235$	88.0000	0.374468
$236$	−160.000	−0.677966
$237$	0	0
$238$	0	0
$239$	254.558i	1.06510i	0.846399	+	0.532549i	$0.178766\pi$
$239$	254.558i	1.06510i	−0.846399	+	0.532549i	$0.821234\pi$
$240$	0	0
$241$	100.409i	0.416636i	0.978061	+	0.208318i	$0.0667988\pi$
$241$	100.409i	0.416636i	−0.978061	+	0.208318i	$0.933201\pi$
$242$	0	0
$243$	0	0
$244$	− 59.3970i	− 0.243430i
$245$	−98.0000	−0.400000
$246$	0	0
$247$	40.0000	0.161943
$248$	− 33.9411i	− 0.136859i
$249$	0	0
$250$	130.108i	0.520431i
$251$	160.000	0.637450	0.318725	−	0.947847i	$-0.396745\pi$
$251$	160.000	0.637450	0.318725	+	0.947847i	$0.396745\pi$
$252$	0	0
$253$	0	0
$254$	−240.000	−0.944882
$255$	0	0
$256$	16.0000	0.0625000
$257$	240.000	0.933852	0.466926	−	0.884296i	$-0.345361\pi$
$257$	240.000	0.933852	0.466926	+	0.884296i	$0.345361\pi$
$258$	0	0
$259$	0	0
$260$	− 28.2843i	− 0.108786i
$261$	0	0
$262$	−80.0000	−0.305344
$263$	67.8823i	0.258107i	0.991638	+	0.129054i	$0.0411940\pi$
$263$	67.8823i	0.258107i	−0.991638	+	0.129054i	$0.958806\pi$
$264$	0	0
$265$	−112.000	−0.422642
$266$	0	0
$267$	0	0
$268$	−160.000	−0.597015
$269$	−280.000	−1.04089	−0.520446	−	0.853895i	$-0.674234\pi$
$269$	−280.000	−1.04089	−0.520446	+	0.853895i	$0.674234\pi$
$270$	0	0
$271$	− 401.637i	− 1.48205i	−0.671475	−	0.741027i	$-0.734339\pi$
$271$	− 401.637i	− 1.48205i	0.671475	−	0.741027i	$-0.265661\pi$
$272$	− 39.5980i	− 0.145581i
$273$	0	0
$274$	− 178.191i	− 0.650332i
$275$	0	0
$276$	0	0
$277$	346.482i	1.25084i	0.780289	+	0.625419i	$0.215072\pi$
$277$	346.482i	1.25084i	−0.780289	+	0.625419i	$0.784928\pi$
$278$	−112.000	−0.402878
$279$	0	0
$280$	0	0
$281$	− 190.919i	− 0.679426i	−0.940529	−	0.339713i	$-0.889670\pi$
$281$	− 190.919i	− 0.679426i	0.940529	−	0.339713i	$-0.110330\pi$
$282$	0	0
$283$	− 367.696i	− 1.29928i	−0.760243	−	0.649639i	$-0.774920\pi$
$283$	− 367.696i	− 1.29928i	0.760243	−	0.649639i	$-0.225080\pi$
$284$	120.000	0.422535
$285$	0	0
$286$	0	0
$287$	0	0
$288$	0	0
$289$	191.000	0.660900
$290$	20.0000	0.0689655
$291$	0	0
$292$	− 155.563i	− 0.532752i
$293$	− 287.085i	− 0.979813i	−0.871775	−	0.489907i	$-0.837031\pi$
$293$	− 287.085i	− 0.979813i	0.871775	−	0.489907i	$-0.162969\pi$
$294$	0	0
$295$	−160.000	−0.542373
$296$	− 84.8528i	− 0.286665i
$297$	0	0
$298$	90.0000	0.302013
$299$	141.421i	0.472981i
$300$	0	0
$301$	0	0
$302$	168.000	0.556291
$303$	0	0
$304$	22.6274i	0.0744323i
$305$	− 59.3970i	− 0.194744i
$306$	0	0
$307$	509.117i	1.65836i	0.558981	+	0.829181i	$0.311193\pi$
$307$	509.117i	1.65836i	−0.558981	+	0.829181i	$0.688807\pi$
$308$	0	0
$309$	0	0
$310$	− 33.9411i	− 0.109488i
$311$	−140.000	−0.450161	−0.225080	−	0.974340i	$-0.572264\pi$
$311$	−140.000	−0.450161	−0.225080	+	0.974340i	$0.572264\pi$
$312$	0	0
$313$	350.000	1.11821	0.559105	−	0.829097i	$-0.311145\pi$
$313$	350.000	1.11821	0.559105	+	0.829097i	$0.311145\pi$
$314$	56.5685i	0.180155i
$315$	0	0
$316$	124.451i	0.393832i
$317$	50.0000	0.157729	0.0788644	−	0.996885i	$-0.474871\pi$
$317$	50.0000	0.157729	0.0788644	+	0.996885i	$0.474871\pi$
$318$	0	0
$319$	0	0
$320$	16.0000	0.0500000
$321$	0	0
$322$	0	0
$323$	56.0000	0.173375
$324$	0	0
$325$	148.492i	0.456900i
$326$	56.5685i	0.173523i
$327$	0	0
$328$	60.0000	0.182927
$329$	0	0
$330$	0	0
$331$	−280.000	−0.845921	−0.422961	−	0.906148i	$-0.639009\pi$
$331$	−280.000	−0.845921	−0.422961	+	0.906148i	$0.639009\pi$
$332$	79.1960i	0.238542i
$333$	0	0
$334$	−376.000	−1.12575
$335$	−160.000	−0.477612
$336$	0	0
$337$	− 318.198i	− 0.944208i	−0.881543	−	0.472104i	$-0.843495\pi$
$337$	− 318.198i	− 0.944208i	0.881543	−	0.472104i	$-0.156505\pi$
$338$	168.291i	0.497904i
$339$	0	0
$340$	− 39.5980i	− 0.116465i
$341$	0	0
$342$	0	0
$343$	0	0
$344$	0	0
$345$	0	0
$346$	−346.000	−1.00000
$347$	− 356.382i	− 1.02704i	−0.858079	−	0.513518i	$-0.828342\pi$
$347$	− 356.382i	− 1.02704i	0.858079	−	0.513518i	$-0.171658\pi$
$348$	0	0
$349$	− 606.698i	− 1.73839i	−0.494471	−	0.869194i	$-0.664638\pi$
$349$	− 606.698i	− 1.73839i	0.494471	−	0.869194i	$-0.335362\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−256.000	−0.725212	−0.362606	−	0.931942i	$-0.618113\pi$
$353$	−256.000	−0.725212	−0.362606	+	0.931942i	$0.618113\pi$
$354$	0	0
$355$	120.000	0.338028
$356$	160.000	0.449438
$357$	0	0
$358$	− 452.548i	− 1.26410i
$359$	395.980i	1.10301i	0.834172	+	0.551504i	$0.185946\pi$
$359$	395.980i	1.10301i	−0.834172	+	0.551504i	$0.814054\pi$
$360$	0	0
$361$	329.000	0.911357
$362$	183.848i	0.507867i
$363$	0	0
$364$	0	0
$365$	− 155.563i	− 0.426201i
$366$	0	0
$367$	100.000	0.272480	0.136240	−	0.990676i	$-0.456498\pi$
$367$	100.000	0.272480	0.136240	+	0.990676i	$0.456498\pi$
$368$	−80.0000	−0.217391
$369$	0	0
$370$	− 84.8528i	− 0.229332i
$371$	0	0
$372$	0	0
$373$	− 106.066i	− 0.284359i	−0.989841	−	0.142180i	$-0.954589\pi$
$373$	− 106.066i	− 0.284359i	0.989841	−	0.142180i	$-0.0454111\pi$
$374$	0	0
$375$	0	0
$376$	124.451i	0.330986i
$377$	50.0000	0.132626
$378$	0	0
$379$	−8.00000	−0.0211082	−0.0105541	−	0.999944i	$-0.503360\pi$
$379$	−8.00000	−0.0211082	−0.0105541	+	0.999944i	$0.503360\pi$
$380$	22.6274i	0.0595458i
$381$	0	0
$382$	367.696i	0.962554i
$383$	116.000	0.302872	0.151436	−	0.988467i	$-0.451610\pi$
$383$	116.000	0.302872	0.151436	+	0.988467i	$0.451610\pi$
$384$	0	0
$385$	0	0
$386$	−210.000	−0.544041
$387$	0	0
$388$	0	0
$389$	30.0000	0.0771208	0.0385604	−	0.999256i	$-0.487723\pi$
$389$	30.0000	0.0771208	0.0385604	+	0.999256i	$0.487723\pi$
$390$	0	0
$391$	197.990i	0.506368i
$392$	− 138.593i	− 0.353553i
$393$	0	0
$394$	−314.000	−0.796954
$395$	124.451i	0.315065i
$396$	0	0
$397$	−280.000	−0.705290	−0.352645	−	0.935757i	$-0.614718\pi$
$397$	−280.000	−0.705290	−0.352645	+	0.935757i	$0.614718\pi$
$398$	− 197.990i	− 0.497462i
$399$	0	0
$400$	−84.0000	−0.210000
$401$	350.000	0.872818	0.436409	−	0.899748i	$-0.356250\pi$
$401$	350.000	0.872818	0.436409	+	0.899748i	$0.356250\pi$
$402$	0	0
$403$	− 84.8528i	− 0.210553i
$404$	− 325.269i	− 0.805122i
$405$	0	0
$406$	0	0
$407$	0	0
$408$	0	0
$409$	521.845i	1.27590i	0.770076	+	0.637952i	$0.220218\pi$
$409$	521.845i	1.27590i	−0.770076	+	0.637952i	$0.779782\pi$
$410$	60.0000	0.146341
$411$	0	0
$412$	−280.000	−0.679612
$413$	0	0
$414$	0	0
$415$	79.1960i	0.190834i
$416$	40.0000	0.0961538
$417$	0	0
$418$	0	0
$419$	160.000	0.381862	0.190931	−	0.981604i	$-0.438849\pi$
$419$	160.000	0.381862	0.190931	+	0.981604i	$0.438849\pi$
$420$	0	0
$421$	8.00000	0.0190024	0.00950119	−	0.999955i	$-0.496976\pi$
$421$	8.00000	0.0190024	0.00950119	+	0.999955i	$0.496976\pi$
$422$	−168.000	−0.398104
$423$	0	0
$424$	− 158.392i	− 0.373566i
$425$	207.889i	0.489152i
$426$	0	0
$427$	0	0
$428$	− 305.470i	− 0.713715i
$429$	0	0
$430$	0	0
$431$	− 424.264i	− 0.984371i	−0.870490	−	0.492186i	$-0.836198\pi$
$431$	− 424.264i	− 0.984371i	0.870490	−	0.492186i	$-0.163802\pi$
$432$	0	0
$433$	−80.0000	−0.184758	−0.0923788	−	0.995724i	$-0.529447\pi$
$433$	−80.0000	−0.184758	−0.0923788	+	0.995724i	$0.529447\pi$
$434$	0	0
$435$	0	0
$436$	− 31.1127i	− 0.0713594i
$437$	− 113.137i	− 0.258895i
$438$	0	0
$439$	192.333i	0.438116i	0.975712	+	0.219058i	$0.0702984\pi$
$439$	192.333i	0.438116i	−0.975712	+	0.219058i	$0.929702\pi$
$440$	0	0
$441$	0	0
$442$	− 98.9949i	− 0.223970i
$443$	336.000	0.758465	0.379233	−	0.925301i	$-0.376188\pi$
$443$	336.000	0.758465	0.379233	+	0.925301i	$0.376188\pi$
$444$	0	0
$445$	160.000	0.359551
$446$	− 197.990i	− 0.443924i
$447$	0	0
$448$	0	0
$449$	0	0		−	1.00000i	$-0.5\pi$
$449$	0	0			1.00000i	$0.5\pi$
$450$	0	0
$451$	0	0
$452$	420.000	0.929204
$453$	0	0
$454$	−176.000	−0.387665
$455$	0	0
$456$	0	0
$457$	91.9239i	0.201146i	0.994930	+	0.100573i	$0.0320677\pi$
$457$	91.9239i	0.201146i	−0.994930	+	0.100573i	$0.967932\pi$
$458$	268.701i	0.586682i
$459$	0	0
$460$	−80.0000	−0.173913
$461$	− 346.482i	− 0.751589i	−0.926703	−	0.375794i	$-0.877370\pi$
$461$	− 346.482i	− 0.751589i	0.926703	−	0.375794i	$-0.122630\pi$
$462$	0	0
$463$	−260.000	−0.561555	−0.280778	−	0.959773i	$-0.590592\pi$
$463$	−260.000	−0.561555	−0.280778	+	0.959773i	$0.590592\pi$
$464$	28.2843i	0.0609575i
$465$	0	0
$466$	−434.000	−0.931330
$467$	40.0000	0.0856531	0.0428266	−	0.999083i	$-0.486364\pi$
$467$	40.0000	0.0856531	0.0428266	+	0.999083i	$0.486364\pi$
$468$	0	0
$469$	0	0
$470$	124.451i	0.264789i
$471$	0	0
$472$	− 226.274i	− 0.479394i
$473$	0	0
$474$	0	0
$475$	− 118.794i	− 0.250093i
$476$	0	0
$477$	0	0
$478$	−360.000	−0.753138
$479$	707.107i	1.47621i	0.674683	+	0.738107i	$0.264280\pi$
$479$	707.107i	1.47621i	−0.674683	+	0.738107i	$0.735720\pi$
$480$	0	0
$481$	− 212.132i	− 0.441023i
$482$	−142.000	−0.294606
$483$	0	0
$484$	0	0
$485$	0	0
$486$	0	0
$487$	300.000	0.616016	0.308008	−	0.951384i	$-0.400338\pi$
$487$	300.000	0.616016	0.308008	+	0.951384i	$0.400338\pi$
$488$	84.0000	0.172131
$489$	0	0
$490$	− 138.593i	− 0.282843i
$491$	− 537.401i	− 1.09450i	−0.836968	−	0.547252i	$-0.815674\pi$
$491$	− 537.401i	− 1.09450i	0.836968	−	0.547252i	$-0.184326\pi$
$492$	0	0
$493$	70.0000	0.141988
$494$	56.5685i	0.114511i
$495$	0	0
$496$	48.0000	0.0967742
$497$	0	0
$498$	0	0
$499$	320.000	0.641283	0.320641	−	0.947201i	$-0.396102\pi$
$499$	320.000	0.641283	0.320641	+	0.947201i	$0.396102\pi$
$500$	−184.000	−0.368000
$501$	0	0
$502$	226.274i	0.450745i
$503$	− 463.862i	− 0.922191i	−0.887351	−	0.461095i	$-0.847457\pi$
$503$	− 463.862i	− 0.922191i	0.887351	−	0.461095i	$-0.152543\pi$
$504$	0	0
$505$	− 325.269i	− 0.644097i
$506$	0	0
$507$	0	0
$508$	− 339.411i	− 0.668132i
$509$	280.000	0.550098	0.275049	−	0.961430i	$-0.411306\pi$
$509$	280.000	0.550098	0.275049	+	0.961430i	$0.411306\pi$
$510$	0	0
$511$	0	0
$512$	22.6274i	0.0441942i
$513$	0	0
$514$	339.411i	0.660333i
$515$	−280.000	−0.543689
$516$	0	0
$517$	0	0
$518$	0	0
$519$	0	0
$520$	40.0000	0.0769231
$521$	−690.000	−1.32438	−0.662188	−	0.749338i	$-0.730372\pi$
$521$	−690.000	−1.32438	−0.662188	+	0.749338i	$0.730372\pi$
$522$	0	0
$523$	905.097i	1.73059i	0.501266	+	0.865293i	$0.332868\pi$
$523$	905.097i	1.73059i	−0.501266	+	0.865293i	$0.667132\pi$
$524$	− 113.137i	− 0.215910i
$525$	0	0
$526$	−96.0000	−0.182510
$527$	− 118.794i	− 0.225415i
$528$	0	0
$529$	−129.000	−0.243856
$530$	− 158.392i	− 0.298853i
$531$	0	0
$532$	0	0
$533$	150.000	0.281426
$534$	0	0
$535$	− 305.470i	− 0.570972i
$536$	− 226.274i	− 0.422153i
$537$	0	0
$538$	− 395.980i	− 0.736022i
$539$	0	0
$540$	0	0
$541$	323.855i	0.598623i	0.954155	+	0.299311i	$0.0967569\pi$
$541$	323.855i	0.598623i	−0.954155	+	0.299311i	$0.903243\pi$
$542$	568.000	1.04797
$543$	0	0
$544$	56.0000	0.102941
$545$	− 31.1127i	− 0.0570875i
$546$	0	0
$547$	− 593.970i	− 1.08587i	−0.839775	−	0.542934i	$-0.817313\pi$
$547$	− 593.970i	− 1.08587i	0.839775	−	0.542934i	$-0.182687\pi$
$548$	252.000	0.459854
$549$	0	0
$550$	0	0
$551$	−40.0000	−0.0725953
$552$	0	0
$553$	0	0
$554$	−490.000	−0.884477
$555$	0	0
$556$	− 158.392i	− 0.284878i
$557$	− 60.8112i	− 0.109176i	−0.998509	−	0.0545881i	$-0.982615\pi$
$557$	− 60.8112i	− 0.109176i	0.998509	−	0.0545881i	$-0.0173846\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	0	0
$562$	270.000	0.480427
$563$	− 746.705i	− 1.32630i	−0.748488	−	0.663148i	$-0.769220\pi$
$563$	− 746.705i	− 1.32630i	0.748488	−	0.663148i	$-0.230780\pi$
$564$	0	0
$565$	420.000	0.743363
$566$	520.000	0.918728
$567$	0	0
$568$	169.706i	0.298778i
$569$	− 247.487i	− 0.434951i	−0.976066	−	0.217476i	$-0.930218\pi$
$569$	− 247.487i	− 0.434951i	0.976066	−	0.217476i	$-0.0697823\pi$
$570$	0	0
$571$	712.764i	1.24827i	0.781315	+	0.624136i	$0.214549\pi$
$571$	712.764i	1.24827i	−0.781315	+	0.624136i	$0.785451\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	420.000	0.730435
$576$	0	0
$577$	320.000	0.554593	0.277296	−	0.960784i	$-0.410562\pi$
$577$	320.000	0.554593	0.277296	+	0.960784i	$0.410562\pi$
$578$	270.115i	0.467327i
$579$	0	0
$580$	28.2843i	0.0487660i
$581$	0	0
$582$	0	0
$583$	0	0
$584$	220.000	0.376712
$585$	0	0
$586$	406.000	0.692833
$587$	−680.000	−1.15843	−0.579216	−	0.815174i	$-0.696641\pi$
$587$	−680.000	−1.15843	−0.579216	+	0.815174i	$0.696641\pi$
$588$	0	0
$589$	67.8823i	0.115250i
$590$	− 226.274i	− 0.383516i
$591$	0	0
$592$	120.000	0.202703
$593$	− 89.0955i	− 0.150245i	−0.997174	−	0.0751226i	$-0.976065\pi$
$593$	− 89.0955i	− 0.150245i	0.997174	−	0.0751226i	$-0.0239348\pi$
$594$	0	0
$595$	0	0
$596$	127.279i	0.213556i
$597$	0	0
$598$	−200.000	−0.334448
$599$	660.000	1.10184	0.550918	−	0.834559i	$-0.314278\pi$
$599$	660.000	1.10184	0.550918	+	0.834559i	$0.314278\pi$
$600$	0	0
$601$	963.079i	1.60246i	0.598355	+	0.801231i	$0.295821\pi$
$601$	963.079i	1.60246i	−0.598355	+	0.801231i	$0.704179\pi$
$602$	0	0
$603$	0	0
$604$	237.588i	0.393357i
$605$	0	0
$606$	0	0
$607$	− 509.117i	− 0.838743i	−0.907815	−	0.419371i	$-0.862251\pi$
$607$	− 509.117i	− 0.838743i	0.907815	−	0.419371i	$-0.137749\pi$
$608$	−32.0000	−0.0526316
$609$	0	0
$610$	84.0000	0.137705
$611$	311.127i	0.509209i
$612$	0	0
$613$	742.462i	1.21119i	0.795771	+	0.605597i	$0.207066\pi$
$613$	742.462i	1.21119i	−0.795771	+	0.605597i	$0.792934\pi$
$614$	−720.000	−1.17264
$615$	0	0
$616$	0	0
$617$	830.000	1.34522	0.672609	−	0.739998i	$-0.265173\pi$
$617$	830.000	1.34522	0.672609	+	0.739998i	$0.265173\pi$
$618$	0	0
$619$	−840.000	−1.35703	−0.678514	−	0.734588i	$-0.737376\pi$
$619$	−840.000	−1.35703	−0.678514	+	0.734588i	$0.737376\pi$
$620$	48.0000	0.0774194
$621$	0	0
$622$	− 197.990i	− 0.318312i
$623$	0	0
$624$	0	0
$625$	341.000	0.545600
$626$	494.975i	0.790694i
$627$	0	0
$628$	−80.0000	−0.127389
$629$	− 296.985i	− 0.472154i
$630$	0	0
$631$	700.000	1.10935	0.554675	−	0.832067i	$-0.312843\pi$
$631$	700.000	1.10935	0.554675	+	0.832067i	$0.312843\pi$
$632$	−176.000	−0.278481
$633$	0	0
$634$	70.7107i	0.111531i
$635$	− 339.411i	− 0.534506i
$636$	0	0
$637$	− 346.482i	− 0.543928i
$638$	0	0
$639$	0	0
$640$	22.6274i	0.0353553i
$641$	880.000	1.37285	0.686427	−	0.727198i	$-0.259178\pi$
$641$	880.000	1.37285	0.686427	+	0.727198i	$0.259178\pi$
$642$	0	0
$643$	−280.000	−0.435459	−0.217729	−	0.976009i	$-0.569865\pi$
$643$	−280.000	−0.435459	−0.217729	+	0.976009i	$0.569865\pi$
$644$	0	0
$645$	0	0
$646$	79.1960i	0.122594i
$647$	−556.000	−0.859351	−0.429675	−	0.902983i	$-0.641372\pi$
$647$	−556.000	−0.859351	−0.429675	+	0.902983i	$0.641372\pi$
$648$	0	0
$649$	0	0
$650$	−210.000	−0.323077
$651$	0	0
$652$	−80.0000	−0.122699
$653$	−840.000	−1.28637	−0.643185	−	0.765711i	$-0.722388\pi$
$653$	−840.000	−1.28637	−0.643185	+	0.765711i	$0.722388\pi$
$654$	0	0
$655$	− 113.137i	− 0.172728i
$656$	84.8528i	0.129349i
$657$	0	0
$658$	0	0
$659$	− 933.381i	− 1.41636i	−0.706032	−	0.708180i	$-0.749517\pi$
$659$	− 933.381i	− 1.41636i	0.706032	−	0.708180i	$-0.250483\pi$
$660$	0	0
$661$	840.000	1.27080	0.635401	−	0.772182i	$-0.280835\pi$
$661$	840.000	1.27080	0.635401	+	0.772182i	$0.280835\pi$
$662$	− 395.980i	− 0.598157i
$663$	0	0
$664$	−112.000	−0.168675
$665$	0	0
$666$	0	0
$667$	− 141.421i	− 0.212026i
$668$	− 531.744i	− 0.796024i
$669$	0	0
$670$	− 226.274i	− 0.337723i
$671$	0	0
$672$	0	0
$673$	374.767i	0.556860i	0.960457	+	0.278430i	$0.0898140\pi$
$673$	374.767i	0.556860i	−0.960457	+	0.278430i	$0.910186\pi$
$674$	450.000	0.667656
$675$	0	0
$676$	−238.000	−0.352071
$677$	980.050i	1.44764i	0.689991	+	0.723818i	$0.257615\pi$
$677$	980.050i	1.44764i	−0.689991	+	0.723818i	$0.742385\pi$
$678$	0	0
$679$	0	0
$680$	56.0000	0.0823529
$681$	0	0
$682$	0	0
$683$	616.000	0.901903	0.450952	−	0.892548i	$-0.351085\pi$
$683$	616.000	0.901903	0.450952	+	0.892548i	$0.351085\pi$
$684$	0	0
$685$	252.000	0.367883
$686$	0	0
$687$	0	0
$688$	0	0
$689$	− 395.980i	− 0.574717i
$690$	0	0
$691$	−832.000	−1.20405	−0.602026	−	0.798476i	$-0.705640\pi$
$691$	−832.000	−1.20405	−0.602026	+	0.798476i	$0.705640\pi$
$692$	− 489.318i	− 0.707107i
$693$	0	0
$694$	504.000	0.726225
$695$	− 158.392i	− 0.227902i
$696$	0	0
$697$	210.000	0.301291
$698$	858.000	1.22923
$699$	0	0
$700$	0	0
$701$	− 742.462i	− 1.05915i	−0.848264	−	0.529574i	$-0.822352\pi$
$701$	− 742.462i	− 1.05915i	0.848264	−	0.529574i	$-0.177648\pi$
$702$	0	0
$703$	169.706i	0.241402i
$704$	0	0
$705$	0	0
$706$	− 362.039i	− 0.512803i
$707$	0	0
$708$	0	0
$709$	1118.00	1.57687	0.788434	−	0.615119i	$-0.210892\pi$
$709$	1118.00	1.57687	0.788434	+	0.615119i	$0.210892\pi$
$710$	169.706i	0.239022i
$711$	0	0
$712$	226.274i	0.317801i
$713$	−240.000	−0.336606
$714$	0	0
$715$	0	0
$716$	640.000	0.893855
$717$	0	0
$718$	−560.000	−0.779944
$719$	−380.000	−0.528512	−0.264256	−	0.964453i	$-0.585126\pi$
$719$	−380.000	−0.528512	−0.264256	+	0.964453i	$0.585126\pi$
$720$	0	0
$721$	0	0
$722$	465.276i	0.644427i
$723$	0	0
$724$	−260.000	−0.359116
$725$	− 148.492i	− 0.204817i
$726$	0	0
$727$	500.000	0.687758	0.343879	−	0.939014i	$-0.388259\pi$
$727$	500.000	0.687758	0.343879	+	0.939014i	$0.388259\pi$
$728$	0	0
$729$	0	0
$730$	220.000	0.301370
$731$	0	0
$732$	0	0
$733$	445.477i	0.607745i	0.952713	+	0.303873i	$0.0982797\pi$
$733$	445.477i	0.607745i	−0.952713	+	0.303873i	$0.901720\pi$
$734$	141.421i	0.192672i
$735$	0	0
$736$	− 113.137i	− 0.153719i
$737$	0	0
$738$	0	0
$739$	− 1193.60i	− 1.61515i	−0.589765	−	0.807575i	$-0.700779\pi$
$739$	− 1193.60i	− 1.61515i	0.589765	−	0.807575i	$-0.299221\pi$
$740$	120.000	0.162162
$741$	0	0
$742$	0	0
$743$	− 379.009i	− 0.510107i	−0.966927	−	0.255053i	$-0.917907\pi$
$743$	− 379.009i	− 0.510107i	0.966927	−	0.255053i	$-0.0820930\pi$
$744$	0	0
$745$	127.279i	0.170845i
$746$	150.000	0.201072
$747$	0	0
$748$	0	0
$749$	0	0
$750$	0	0
$751$	180.000	0.239680	0.119840	−	0.992793i	$-0.461762\pi$
$751$	180.000	0.239680	0.119840	+	0.992793i	$0.461762\pi$
$752$	−176.000	−0.234043
$753$	0	0
$754$	70.7107i	0.0937807i
$755$	237.588i	0.314686i
$756$	0	0
$757$	1240.00	1.63804	0.819022	−	0.573761i	$-0.194516\pi$
$757$	1240.00	1.63804	0.819022	+	0.573761i	$0.194516\pi$
$758$	− 11.3137i	− 0.0149257i
$759$	0	0
$760$	−32.0000	−0.0421053
$761$	− 1378.86i	− 1.81190i	−0.423381	−	0.905952i	$-0.639157\pi$
$761$	− 1378.86i	− 1.81190i	0.423381	−	0.905952i	$-0.360843\pi$
$762$	0	0
$763$	0	0
$764$	−520.000	−0.680628
$765$	0	0
$766$	164.049i	0.214163i
$767$	− 565.685i	− 0.737530i
$768$	0	0
$769$	12.7279i	0.0165513i	0.999966	+	0.00827563i	$0.00263425\pi$
$769$	12.7279i	0.0165513i	−0.999966	+	0.00827563i	$0.997366\pi$
$770$	0	0
$771$	0	0
$772$	− 296.985i	− 0.384695i
$773$	360.000	0.465718	0.232859	−	0.972511i	$-0.425192\pi$
$773$	360.000	0.465718	0.232859	+	0.972511i	$0.425192\pi$
$774$	0	0
$775$	−252.000	−0.325161
$776$	0	0
$777$	0	0
$778$	42.4264i	0.0545327i
$779$	−120.000	−0.154044
$780$	0	0
$781$	0	0
$782$	−280.000	−0.358056
$783$	0	0
$784$	196.000	0.250000
$785$	−80.0000	−0.101911
$786$	0	0
$787$	− 169.706i	− 0.215636i	−0.994171	−	0.107818i	$-0.965614\pi$
$787$	− 169.706i	− 0.215636i	0.994171	−	0.107818i	$-0.0343864\pi$
$788$	− 444.063i	− 0.563532i
$789$	0	0
$790$	−176.000	−0.222785
$791$	0	0
$792$	0	0
$793$	210.000	0.264817
$794$	− 395.980i	− 0.498715i
$795$	0	0
$796$	280.000	0.351759
$797$	1170.00	1.46801	0.734003	−	0.679147i	$-0.237650\pi$
$797$	1170.00	1.46801	0.734003	+	0.679147i	$0.237650\pi$
$798$	0	0
$799$	435.578i	0.545154i
$800$	− 118.794i	− 0.148492i
$801$	0	0
$802$	494.975i	0.617175i
$803$	0	0
$804$	0	0
$805$	0	0
$806$	120.000	0.148883
$807$	0	0
$808$	460.000	0.569307
$809$	813.173i	1.00516i	0.864531	+	0.502579i	$0.167615\pi$
$809$	813.173i	1.00516i	−0.864531	+	0.502579i	$0.832385\pi$
$810$	0	0
$811$	− 135.765i	− 0.167404i	−0.996491	−	0.0837019i	$-0.973326\pi$
$811$	− 135.765i	− 0.167404i	0.996491	−	0.0837019i	$-0.0266744\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−80.0000	−0.0981595
$816$	0	0
$817$	0	0
$818$	−738.000	−0.902200
$819$	0	0
$820$	84.8528i	0.103479i
$821$	954.594i	1.16272i	0.813646	+	0.581361i	$0.197479\pi$
$821$	954.594i	1.16272i	−0.813646	+	0.581361i	$0.802521\pi$
$822$	0	0
$823$	−340.000	−0.413123	−0.206561	−	0.978434i	$-0.566227\pi$
$823$	−340.000	−0.413123	−0.206561	+	0.978434i	$0.566227\pi$
$824$	− 395.980i	− 0.480558i
$825$	0	0
$826$	0	0
$827$	− 181.019i	− 0.218887i	−0.993993	−	0.109443i	$-0.965093\pi$
$827$	− 181.019i	− 0.218887i	0.993993	−	0.109443i	$-0.0349068\pi$
$828$	0	0
$829$	−280.000	−0.337756	−0.168878	−	0.985637i	$-0.554014\pi$
$829$	−280.000	−0.337756	−0.168878	+	0.985637i	$0.554014\pi$
$830$	−112.000	−0.134940
$831$	0	0
$832$	56.5685i	0.0679910i
$833$	− 485.075i	− 0.582323i
$834$	0	0
$835$	− 531.744i	− 0.636820i
$836$	0	0
$837$	0	0
$838$	226.274i	0.270017i
$839$	1660.00	1.97855	0.989273	−	0.146079i	$-0.0466653\pi$
$839$	1660.00	1.97855	0.989273	+	0.146079i	$0.0466653\pi$
$840$	0	0
$841$	791.000	0.940547
$842$	11.3137i	0.0134367i
$843$	0	0
$844$	− 237.588i	− 0.281502i
$845$	−238.000	−0.281657
$846$	0	0
$847$	0	0
$848$	224.000	0.264151
$849$	0	0
$850$	−294.000	−0.345882
$851$	−600.000	−0.705053
$852$	0	0
$853$	63.6396i	0.0746068i	0.999304	+	0.0373034i	$0.0118768\pi$
$853$	63.6396i	0.0746068i	−0.999304	+	0.0373034i	$0.988123\pi$
$854$	0	0
$855$	0	0
$856$	432.000	0.504673
$857$	− 89.0955i	− 0.103962i	−0.998648	−	0.0519810i	$-0.983446\pi$
$857$	− 89.0955i	− 0.103962i	0.998648	−	0.0519810i	$-0.0165535\pi$
$858$	0	0
$859$	400.000	0.465658	0.232829	−	0.972518i	$-0.425202\pi$
$859$	400.000	0.465658	0.232829	+	0.972518i	$0.425202\pi$
$860$	0	0
$861$	0	0
$862$	600.000	0.696056
$863$	524.000	0.607184	0.303592	−	0.952802i	$-0.401814\pi$
$863$	524.000	0.607184	0.303592	+	0.952802i	$0.401814\pi$
$864$	0	0
$865$	− 489.318i	− 0.565685i
$866$	− 113.137i	− 0.130643i
$867$	0	0
$868$	0	0
$869$	0	0
$870$	0	0
$871$	− 565.685i	− 0.649467i
$872$	44.0000	0.0504587
$873$	0	0
$874$	160.000	0.183066
$875$	0	0
$876$	0	0
$877$	742.462i	0.846593i	0.905991	+	0.423297i	$0.139127\pi$
$877$	742.462i	0.846593i	−0.905991	+	0.423297i	$0.860873\pi$
$878$	−272.000	−0.309795
$879$	0	0
$880$	0	0
$881$	−190.000	−0.215664	−0.107832	−	0.994169i	$-0.534391\pi$
$881$	−190.000	−0.215664	−0.107832	+	0.994169i	$0.534391\pi$
$882$	0	0
$883$	−1280.00	−1.44960	−0.724802	−	0.688957i	$-0.758069\pi$
$883$	−1280.00	−1.44960	−0.724802	+	0.688957i	$0.758069\pi$
$884$	140.000	0.158371
$885$	0	0
$886$	475.176i	0.536316i
$887$	− 1227.54i	− 1.38392i	−0.721936	−	0.691960i	$-0.756747\pi$
$887$	− 1227.54i	− 1.38392i	0.721936	−	0.691960i	$-0.243253\pi$
$888$	0	0
$889$	0	0
$890$	226.274i	0.254241i
$891$	0	0
$892$	280.000	0.313901
$893$	− 248.902i	− 0.278725i
$894$	0	0
$895$	640.000	0.715084
$896$	0	0
$897$	0	0
$898$	0	0
$899$	84.8528i	0.0943858i
$900$	0	0
$901$	− 554.372i	− 0.615285i
$902$	0	0
$903$	0	0
$904$	593.970i	0.657046i
$905$	−260.000	−0.287293
$906$	0	0
$907$	−840.000	−0.926130	−0.463065	−	0.886324i	$-0.653250\pi$
$907$	−840.000	−0.926130	−0.463065	+	0.886324i	$0.653250\pi$
$908$	− 248.902i	− 0.274121i
$909$	0	0
$910$	0	0
$911$	700.000	0.768386	0.384193	−	0.923253i	$-0.374480\pi$
$911$	700.000	0.768386	0.384193	+	0.923253i	$0.374480\pi$
$912$	0	0
$913$	0	0
$914$	−130.000	−0.142232
$915$	0	0
$916$	−380.000	−0.414847
$917$	0	0
$918$	0	0
$919$	475.176i	0.517057i	0.966004	+	0.258529i	$0.0832377\pi$
$919$	475.176i	0.517057i	−0.966004	+	0.258529i	$0.916762\pi$
$920$	− 113.137i	− 0.122975i
$921$	0	0
$922$	490.000	0.531453
$923$	424.264i	0.459658i
$924$	0	0
$925$	−630.000	−0.681081
$926$	− 367.696i	− 0.397079i
$927$	0	0
$928$	−40.0000	−0.0431034
$929$	−1010.00	−1.08719	−0.543595	−	0.839347i	$-0.682937\pi$
$929$	−1010.00	−1.08719	−0.543595	+	0.839347i	$0.682937\pi$
$930$	0	0
$931$	277.186i	0.297729i
$932$	− 613.769i	− 0.658550i
$933$	0	0
$934$	56.5685i	0.0605659i
$935$	0	0
$936$	0	0
$937$	148.492i	0.158476i	0.996856	+	0.0792382i	$0.0252488\pi$
$937$	148.492i	0.158476i	−0.996856	+	0.0792382i	$0.974751\pi$
$938$	0	0
$939$	0	0
$940$	−176.000	−0.187234
$941$	− 1647.56i	− 1.75086i	−0.483345	−	0.875430i	$-0.660578\pi$
$941$	− 1647.56i	− 1.75086i	0.483345	−	0.875430i	$-0.339422\pi$
$942$	0	0
$943$	− 424.264i	− 0.449909i
$944$	320.000	0.338983
$945$	0	0
$946$	0	0
$947$	−344.000	−0.363252	−0.181626	−	0.983368i	$-0.558136\pi$
$947$	−344.000	−0.363252	−0.181626	+	0.983368i	$0.558136\pi$
$948$	0	0
$949$	550.000	0.579557
$950$	168.000	0.176842
$951$	0	0
$952$	0	0
$953$	− 363.453i	− 0.381378i	−0.981651	−	0.190689i	$-0.938928\pi$
$953$	− 363.453i	− 0.381378i	0.981651	−	0.190689i	$-0.0610721\pi$
$954$	0	0
$955$	−520.000	−0.544503
$956$	− 509.117i	− 0.532549i
$957$	0	0
$958$	−1000.00	−1.04384
$959$	0	0
$960$	0	0
$961$	−817.000	−0.850156
$962$	300.000	0.311850
$963$	0	0
$964$	− 200.818i	− 0.208318i
$965$	− 296.985i	− 0.307756i
$966$	0	0
$967$	1187.94i	1.22848i	0.789120	+	0.614240i	$0.210537\pi$
$967$	1187.94i	1.22848i	−0.789120	+	0.614240i	$0.789463\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	800.000	0.823893	0.411946	−	0.911208i	$-0.364849\pi$
$971$	800.000	0.823893	0.411946	+	0.911208i	$0.364849\pi$
$972$	0	0
$973$	0	0
$974$	424.264i	0.435589i
$975$	0	0
$976$	118.794i	0.121715i
$977$	446.000	0.456499	0.228250	−	0.973603i	$-0.426700\pi$
$977$	446.000	0.456499	0.228250	+	0.973603i	$0.426700\pi$
$978$	0	0
$979$	0	0
$980$	196.000	0.200000
$981$	0	0
$982$	760.000	0.773931
$983$	84.0000	0.0854527	0.0427263	−	0.999087i	$-0.486396\pi$
$983$	84.0000	0.0854527	0.0427263	+	0.999087i	$0.486396\pi$
$984$	0	0
$985$	− 444.063i	− 0.450825i
$986$	98.9949i	0.100401i
$987$	0	0
$988$	−80.0000	−0.0809717
$989$	0	0
$990$	0	0
$991$	732.000	0.738648	0.369324	−	0.929301i	$-0.379589\pi$
$991$	732.000	0.738648	0.369324	+	0.929301i	$0.379589\pi$
$992$	67.8823i	0.0684297i
$993$	0	0
$994$	0	0
$995$	280.000	0.281407
$996$	0	0
$997$	1138.44i	1.14187i	0.820996	+	0.570934i	$0.193419\pi$
$997$	1138.44i	1.14187i	−0.820996	+	0.570934i	$0.806581\pi$
$998$	452.548i	0.453455i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2178.3.d.b.1693.2	yes	2
3.2	odd	2		2178.3.d.c.1693.1	yes	2
11.10	odd	2	inner	2178.3.d.b.1693.1	✓	2
33.32	even	2		2178.3.d.c.1693.2	yes	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2178.3.d.b.1693.1	✓	2	11.10	odd	2	inner
2178.3.d.b.1693.2	yes	2	1.1	even	1	trivial
2178.3.d.c.1693.1	yes	2	3.2	odd	2
2178.3.d.c.1693.2	yes	2	33.32	even	2

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:	\( N \)	\(=\)	\( 2178 = 2 \cdot 3^{2} \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	2178.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.41421i q^{2} -2.00000 q^{4} -2.00000 q^{5} -2.82843i q^{8} -2.82843i q^{10} -7.07107i q^{13} +4.00000 q^{16} -9.89949i q^{17} +5.65685i q^{19} +4.00000 q^{20} -20.0000 q^{23} -21.0000 q^{25} +10.0000 q^{26} +7.07107i q^{29} +12.0000 q^{31} +5.65685i q^{32} +14.0000 q^{34} +30.0000 q^{37} -8.00000 q^{38} +5.65685i q^{40} +21.2132i q^{41} -28.2843i q^{46} -44.0000 q^{47} +49.0000 q^{49} -29.6985i q^{50} +14.1421i q^{52} +56.0000 q^{53} -10.0000 q^{58} +80.0000 q^{59} +29.6985i q^{61} +16.9706i q^{62} -8.00000 q^{64} +14.1421i q^{65} +80.0000 q^{67} +19.7990i q^{68} -60.0000 q^{71} +77.7817i q^{73} +42.4264i q^{74} -11.3137i q^{76} -62.2254i q^{79} -8.00000 q^{80} -30.0000 q^{82} -39.5980i q^{83} +19.7990i q^{85} -80.0000 q^{89} +40.0000 q^{92} -62.2254i q^{94} -11.3137i q^{95} +69.2965i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 4 q^{4} - 4 q^{5} + 8 q^{16} + 8 q^{20} - 40 q^{23} - 42 q^{25} + 20 q^{26} + 24 q^{31} + 28 q^{34} + 60 q^{37} - 16 q^{38} - 88 q^{47} + 98 q^{49} + 112 q^{53} - 20 q^{58} + 160 q^{59} - 16 q^{64}+ \cdots + 80 q^{92}+O(q^{100}) \) 2 * q - 4 * q^4 - 4 * q^5 + 8 * q^16 + 8 * q^20 - 40 * q^23 - 42 * q^25 + 20 * q^26 + 24 * q^31 + 28 * q^34 + 60 * q^37 - 16 * q^38 - 88 * q^47 + 98 * q^49 + 112 * q^53 - 20 * q^58 + 160 * q^59 - 16 * q^64 + 160 * q^67 - 120 * q^71 - 16 * q^80 - 60 * q^82 - 160 * q^89 + 80 * q^92

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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