LMFDB - Embedded newform 200.8.c.h.49.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [200,8,Mod(49,200)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(200, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1])) N = Newforms(chi, 8, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("200.49"); S:= CuspForms(chi, 8); N := Newforms(S);

Level:	$ N $	$=$	$ 200 = 2^{3} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	200.c (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [4,0,0,0,0,0,0,0,-3044] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(9)] == traces)

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

Self dual:	no
Analytic conductor:	$62.4770050968$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{46})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 529 $ x^4 + 529
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{9} $
Twist minimal:	no (minimal twist has level 40)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			49.1
Root			$3.39116 - 3.39116i$ of defining polynomial
Character	$\chi$	$=$	200.49
Dual form			200.8.c.h.49.4

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-56.2586i q^{3} +1344.40i q^{7} -978.035 q^{9} +4381.83 q^{11} +3760.14i q^{13} +3848.55i q^{17} -38959.0 q^{19} +75633.9 q^{21} -59312.5i q^{23} -68014.8i q^{27} -207630. q^{29} -286080. q^{31} -246516. i q^{33} -453930. i q^{37} +211540. q^{39} +5982.93 q^{41} +216595. i q^{43} -406755. i q^{47} -983860. q^{49} +216514. q^{51} +1.38096e6i q^{53} +2.19178e6i q^{57} +1.88711e6 q^{59} -2.77012e6 q^{61} -1.31487e6i q^{63} +3.24935e6i q^{67} -3.33684e6 q^{69} -571258. q^{71} -2.07058e6i q^{73} +5.89092e6i q^{77} +1.92993e6 q^{79} -5.96538e6 q^{81} -6.83187e6i q^{83} +1.16810e7i q^{87} +3.55603e6 q^{89} -5.05512e6 q^{91} +1.60945e7i q^{93} -1.26173e7i q^{97} -4.28558e6 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 3044 q^{9} + 9280 q^{11} - 79440 q^{19} + 203568 q^{21} - 373880 q^{29} - 654256 q^{31} + 1527216 q^{39} - 325928 q^{41} - 821428 q^{49} - 3190320 q^{51} - 2241552 q^{59} - 3711704 q^{61} - 8039568 q^{69}+ \cdots - 8852032 q^{99}+O(q^{100}) $ 4 * q - 3044 * q^9 + 9280 * q^11 - 79440 * q^19 + 203568 * q^21 - 373880 * q^29 - 654256 * q^31 + 1527216 * q^39 - 325928 * q^41 - 821428 * q^49 - 3190320 * q^51 - 2241552 * q^59 - 3711704 * q^61 - 8039568 * q^69 - 2142224 * q^71 - 8680288 * q^79 - 23284204 * q^81 - 84520 * q^89 - 20682768 * q^91 - 8852032 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/200\mathbb{Z}\right)^\times$.

$n$	$101$	$151$	$177$
$\chi(n)$	$1$	$1$	$-1$

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	− 56.2586i	− 1.20300i	−0.798874	−	0.601499i	$-0.794570\pi$
$3$	− 56.2586i	− 1.20300i	0.798874	−	0.601499i	$-0.205430\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	1344.40i	1.48144i	0.671813	+	0.740720i	$0.265516\pi$
$7$	1344.40i	1.48144i	−0.671813	+	0.740720i	$0.734484\pi$
$8$	0	0
$9$	−978.035	−0.447204
$10$	0	0
$11$	4381.83	0.992615	0.496308	−	0.868147i	$-0.334689\pi$
$11$	4381.83	0.992615	0.496308	+	0.868147i	$0.334689\pi$
$12$	0	0
$13$	3760.14i	0.474682i	0.971426	+	0.237341i	$0.0762758\pi$
$13$	3760.14i	0.474682i	−0.971426	+	0.237341i	$0.923724\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	3848.55i	0.189988i	0.995478	+	0.0949939i	$0.0302831\pi$
$17$	3848.55i	0.189988i	−0.995478	+	0.0949939i	$0.969717\pi$
$18$	0	0
$19$	−38959.0	−1.30308	−0.651539	−	0.758615i	$-0.725876\pi$
$19$	−38959.0	−1.30308	−0.651539	+	0.758615i	$0.725876\pi$
$20$	0	0
$21$	75633.9	1.78217
$22$	0	0
$23$	− 59312.5i	− 1.01648i	−0.861215	−	0.508240i	$-0.830296\pi$
$23$	− 59312.5i	− 1.01648i	0.861215	−	0.508240i	$-0.169704\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	− 68014.8i	− 0.665013i
$28$	0	0
$29$	−207630.	−1.58087	−0.790437	−	0.612543i	$-0.790147\pi$
$29$	−207630.	−1.58087	−0.790437	+	0.612543i	$0.790147\pi$
$30$	0	0
$31$	−286080.	−1.72473	−0.862366	−	0.506286i	$-0.831018\pi$
$31$	−286080.	−1.72473	−0.862366	+	0.506286i	$0.831018\pi$
$32$	0	0
$33$	− 246516.i	− 1.19411i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	− 453930.i	− 1.47327i	−0.676290	−	0.736635i	$-0.736413\pi$
$37$	− 453930.i	− 1.47327i	0.676290	−	0.736635i	$-0.263587\pi$
$38$	0	0
$39$	211540.	0.571041
$40$	0	0
$41$	5982.93	0.0135572	0.00677860	−	0.999977i	$-0.497842\pi$
$41$	5982.93	0.0135572	0.00677860	+	0.999977i	$0.497842\pi$
$42$	0	0
$43$	216595.i	0.415440i	0.978188	+	0.207720i	$0.0666043\pi$
$43$	216595.i	0.415440i	−0.978188	+	0.207720i	$0.933396\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 406755.i	− 0.571466i	−0.958309	−	0.285733i	$-0.907763\pi$
$47$	− 406755.i	− 0.571466i	0.958309	−	0.285733i	$-0.0922370\pi$
$48$	0	0
$49$	−983860.	−1.19467
$50$	0	0
$51$	216514.	0.228555
$52$	0	0
$53$	1.38096e6i	1.27414i	0.770806	+	0.637070i	$0.219854\pi$
$53$	1.38096e6i	1.27414i	−0.770806	+	0.637070i	$0.780146\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	2.19178e6i	1.56760i
$58$	0	0
$59$	1.88711e6	1.19623	0.598116	−	0.801410i	$-0.295916\pi$
$59$	1.88711e6	1.19623	0.598116	+	0.801410i	$0.295916\pi$
$60$	0	0
$61$	−2.77012e6	−1.56258	−0.781292	−	0.624166i	$-0.785439\pi$
$61$	−2.77012e6	−1.56258	−0.781292	+	0.624166i	$0.785439\pi$
$62$	0	0
$63$	− 1.31487e6i	− 0.662506i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	3.24935e6i	1.31988i	0.751317	+	0.659941i	$0.229419\pi$
$67$	3.24935e6i	1.31988i	−0.751317	+	0.659941i	$0.770581\pi$
$68$	0	0
$69$	−3.33684e6	−1.22282
$70$	0	0
$71$	−571258.	−0.189421	−0.0947105	−	0.995505i	$-0.530193\pi$
$71$	−571258.	−0.189421	−0.0947105	+	0.995505i	$0.530193\pi$
$72$	0	0
$73$	− 2.07058e6i	− 0.622962i	−0.950252	−	0.311481i	$-0.899175\pi$
$73$	− 2.07058e6i	− 0.622962i	0.950252	−	0.311481i	$-0.100825\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	5.89092e6i	1.47050i
$78$	0	0
$79$	1.92993e6	0.440399	0.220199	−	0.975455i	$-0.429329\pi$
$79$	1.92993e6	0.440399	0.220199	+	0.975455i	$0.429329\pi$
$80$	0	0
$81$	−5.96538e6	−1.24721
$82$	0	0
$83$	− 6.83187e6i	− 1.31149i	−0.754980	−	0.655747i	$-0.772354\pi$
$83$	− 6.83187e6i	− 1.31149i	0.754980	−	0.655747i	$-0.227646\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	1.16810e7i	1.90179i
$88$	0	0
$89$	3.55603e6	0.534688	0.267344	−	0.963601i	$-0.413854\pi$
$89$	3.55603e6	0.534688	0.267344	+	0.963601i	$0.413854\pi$
$90$	0	0
$91$	−5.05512e6	−0.703213
$92$	0	0
$93$	1.60945e7i	2.07485i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	− 1.26173e7i	− 1.40367i	−0.712338	−	0.701836i	$-0.752364\pi$
$97$	− 1.26173e7i	− 1.40367i	0.712338	−	0.701836i	$-0.247636\pi$
$98$	0	0
$99$	−4.28558e6	−0.443901
$100$	0	0
$101$	−9.61157e6	−0.928260	−0.464130	−	0.885767i	$-0.653633\pi$
$101$	−9.61157e6	−0.928260	−0.464130	+	0.885767i	$0.653633\pi$
$102$	0	0
$103$	7.89879e6i	0.712247i	0.934439	+	0.356123i	$0.115902\pi$
$103$	7.89879e6i	0.712247i	−0.934439	+	0.356123i	$0.884098\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	2.12283e7i	1.67522i	0.546268	+	0.837610i	$0.316048\pi$
$107$	2.12283e7i	1.67522i	−0.546268	+	0.837610i	$0.683952\pi$
$108$	0	0
$109$	−2.42928e7	−1.79674	−0.898369	−	0.439241i	$-0.855247\pi$
$109$	−2.42928e7	−1.79674	−0.898369	+	0.439241i	$0.855247\pi$
$110$	0	0
$111$	−2.55375e7	−1.77234
$112$	0	0
$113$	− 2.00007e7i	− 1.30398i	−0.758228	−	0.651989i	$-0.773935\pi$
$113$	− 2.00007e7i	− 1.30398i	0.758228	−	0.651989i	$-0.226065\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	− 3.67755e6i	− 0.212279i
$118$	0	0
$119$	−5.17397e6	−0.281456
$120$	0	0
$121$	−286752.	−0.0147149
$122$	0	0
$123$	− 336591.i	− 0.0163093i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	− 3.36067e7i	− 1.45584i	−0.685664	−	0.727918i	$-0.740488\pi$
$127$	− 3.36067e7i	− 1.45584i	0.685664	−	0.727918i	$-0.259512\pi$
$128$	0	0
$129$	1.21853e7	0.499774
$130$	0	0
$131$	−5.39131e6	−0.209529	−0.104765	−	0.994497i	$-0.533409\pi$
$131$	−5.39131e6	−0.209529	−0.104765	+	0.994497i	$0.533409\pi$
$132$	0	0
$133$	− 5.23764e7i	− 1.93043i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	2.80031e7i	0.930432i	0.885197	+	0.465216i	$0.154023\pi$
$137$	2.80031e7i	0.930432i	−0.885197	+	0.465216i	$0.845977\pi$
$138$	0	0
$139$	−3.18588e7	−1.00618	−0.503092	−	0.864233i	$-0.667804\pi$
$139$	−3.18588e7	−1.00618	−0.503092	+	0.864233i	$0.667804\pi$
$140$	0	0
$141$	−2.28835e7	−0.687472
$142$	0	0
$143$	1.64763e7i	0.471176i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	5.53506e7i	1.43718i
$148$	0	0
$149$	−6.66511e7	−1.65065	−0.825325	−	0.564657i	$-0.809008\pi$
$149$	−6.66511e7	−1.65065	−0.825325	+	0.564657i	$0.809008\pi$
$150$	0	0
$151$	−2.37628e7	−0.561666	−0.280833	−	0.959757i	$-0.590611\pi$
$151$	−2.37628e7	−0.561666	−0.280833	+	0.959757i	$0.590611\pi$
$152$	0	0
$153$	− 3.76401e6i	− 0.0849632i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	1.19191e7i	0.245806i	0.992419	+	0.122903i	$0.0392204\pi$
$157$	1.19191e7i	0.245806i	−0.992419	+	0.122903i	$0.960780\pi$
$158$	0	0
$159$	7.76912e7	1.53279
$160$	0	0
$161$	7.97396e7	1.50586
$162$	0	0
$163$	6.13425e7i	1.10944i	0.832036	+	0.554721i	$0.187175\pi$
$163$	6.13425e7i	1.10944i	−0.832036	+	0.554721i	$0.812825\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	6.44216e7i	1.07034i	0.844743	+	0.535172i	$0.179753\pi$
$167$	6.44216e7i	1.07034i	−0.844743	+	0.535172i	$0.820247\pi$
$168$	0	0
$169$	4.86099e7	0.774677
$170$	0	0
$171$	3.81033e7	0.582742
$172$	0	0
$173$	1.67708e7i	0.246260i	0.992391	+	0.123130i	$0.0392931\pi$
$173$	1.67708e7i	0.246260i	−0.992391	+	0.123130i	$0.960707\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	− 1.06166e8i	− 1.43906i
$178$	0	0
$179$	6.75477e7	0.880288	0.440144	−	0.897927i	$-0.354927\pi$
$179$	6.75477e7	0.880288	0.440144	+	0.897927i	$0.354927\pi$
$180$	0	0
$181$	−7.02414e7	−0.880477	−0.440238	−	0.897881i	$-0.645106\pi$
$181$	−7.02414e7	−0.880477	−0.440238	+	0.897881i	$0.645106\pi$
$182$	0	0
$183$	1.55843e8i	1.87979i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	1.68637e7i	0.188585i
$188$	0	0
$189$	9.14388e7	0.985177
$190$	0	0
$191$	−4.03750e7	−0.419272	−0.209636	−	0.977779i	$-0.567228\pi$
$191$	−4.03750e7	−0.419272	−0.209636	+	0.977779i	$0.567228\pi$
$192$	0	0
$193$	− 3.48873e7i	− 0.349315i	−0.984629	−	0.174657i	$-0.944118\pi$
$193$	− 3.48873e7i	− 0.349315i	0.984629	−	0.174657i	$-0.0558818\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 1.29401e8i	− 1.20588i	−0.797786	−	0.602941i	$-0.793995\pi$
$197$	− 1.29401e8i	− 1.20588i	0.797786	−	0.602941i	$-0.206005\pi$
$198$	0	0
$199$	1.20451e8	1.08349	0.541746	−	0.840542i	$-0.317763\pi$
$199$	1.20451e8	1.08349	0.541746	+	0.840542i	$0.317763\pi$
$200$	0	0
$201$	1.82804e8	1.58782
$202$	0	0
$203$	− 2.79137e8i	− 2.34197i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	5.80097e7i	0.454574i
$208$	0	0
$209$	−1.70712e8	−1.29346
$210$	0	0
$211$	8.00773e7	0.586841	0.293421	−	0.955983i	$-0.405206\pi$
$211$	8.00773e7	0.586841	0.293421	+	0.955983i	$0.405206\pi$
$212$	0	0
$213$	3.21382e7i	0.227873i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	− 3.84605e8i	− 2.55509i
$218$	0	0
$219$	−1.16488e8	−0.749421
$220$	0	0
$221$	−1.44711e7	−0.0901837
$222$	0	0
$223$	− 3.09401e6i	− 0.0186833i	−0.999956	−	0.00934167i	$-0.997026\pi$
$223$	− 3.09401e6i	− 0.0186833i	0.999956	−	0.00934167i	$-0.00297359\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	9.59329e7i	0.544349i	0.962248	+	0.272174i	$0.0877428\pi$
$227$	9.59329e7i	0.544349i	−0.962248	+	0.272174i	$0.912257\pi$
$228$	0	0
$229$	−1.03824e8	−0.571314	−0.285657	−	0.958332i	$-0.592212\pi$
$229$	−1.03824e8	−0.571314	−0.285657	+	0.958332i	$0.592212\pi$
$230$	0	0
$231$	3.31415e8	1.76901
$232$	0	0
$233$	1.36559e8i	0.707255i	0.935386	+	0.353628i	$0.115052\pi$
$233$	1.36559e8i	0.707255i	−0.935386	+	0.353628i	$0.884948\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	− 1.08575e8i	− 0.529799i
$238$	0	0
$239$	1.60216e8	0.759125	0.379562	−	0.925166i	$-0.376075\pi$
$239$	1.60216e8	0.759125	0.379562	+	0.925166i	$0.376075\pi$
$240$	0	0
$241$	1.93437e8	0.890183	0.445092	−	0.895485i	$-0.353171\pi$
$241$	1.93437e8	0.890183	0.445092	+	0.895485i	$0.353171\pi$
$242$	0	0
$243$	1.86856e8i	0.835381i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	− 1.46491e8i	− 0.618547i
$248$	0	0
$249$	−3.84352e8	−1.57773
$250$	0	0
$251$	4.20551e8	1.67865	0.839326	−	0.543628i	$-0.182950\pi$
$251$	4.20551e8	1.67865	0.839326	+	0.543628i	$0.182950\pi$
$252$	0	0
$253$	− 2.59897e8i	− 1.00897i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 3.74041e8i	− 1.37453i	−0.726408	−	0.687264i	$-0.758812\pi$
$257$	− 3.74041e8i	− 1.37453i	0.726408	−	0.687264i	$-0.241188\pi$
$258$	0	0
$259$	6.10262e8	2.18256
$260$	0	0
$261$	2.03069e8	0.706973
$262$	0	0
$263$	− 2.64114e8i	− 0.895253i	−0.894221	−	0.447626i	$-0.852269\pi$
$263$	− 2.64114e8i	− 0.895253i	0.894221	−	0.447626i	$-0.147731\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	− 2.00058e8i	− 0.643229i
$268$	0	0
$269$	−4.09451e8	−1.28253	−0.641267	−	0.767318i	$-0.721591\pi$
$269$	−4.09451e8	−1.28253	−0.641267	+	0.767318i	$0.721591\pi$
$270$	0	0
$271$	−4.20454e8	−1.28329	−0.641647	−	0.767000i	$-0.721749\pi$
$271$	−4.20454e8	−1.28329	−0.641647	+	0.767000i	$0.721749\pi$
$272$	0	0
$273$	2.84394e8i	0.845963i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	5.93342e8i	1.67736i	0.544626	+	0.838679i	$0.316672\pi$
$277$	5.93342e8i	1.67736i	−0.544626	+	0.838679i	$0.683328\pi$
$278$	0	0
$279$	2.79796e8	0.771307
$280$	0	0
$281$	−4.93331e8	−1.32638	−0.663188	−	0.748453i	$-0.730797\pi$
$281$	−4.93331e8	−1.32638	−0.663188	+	0.748453i	$0.730797\pi$
$282$	0	0
$283$	3.01738e8i	0.791365i	0.918387	+	0.395683i	$0.129492\pi$
$283$	3.01738e8i	0.791365i	−0.918387	+	0.395683i	$0.870508\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	8.04343e6i	0.0200842i
$288$	0	0
$289$	3.95527e8	0.963905
$290$	0	0
$291$	−7.09833e8	−1.68861
$292$	0	0
$293$	− 3.08855e8i	− 0.717328i	−0.933467	−	0.358664i	$-0.883232\pi$
$293$	− 3.08855e8i	− 0.717328i	0.933467	−	0.358664i	$-0.116768\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	− 2.98029e8i	− 0.660102i
$298$	0	0
$299$	2.23023e8	0.482504
$300$	0	0
$301$	−2.91189e8	−0.615450
$302$	0	0
$303$	5.40734e8i	1.11669i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	4.53878e8i	0.895272i	0.894216	+	0.447636i	$0.147734\pi$
$307$	4.53878e8i	0.895272i	−0.894216	+	0.447636i	$0.852266\pi$
$308$	0	0
$309$	4.44375e8	0.856831
$310$	0	0
$311$	4.85507e8	0.915239	0.457619	−	0.889148i	$-0.348702\pi$
$311$	4.85507e8	0.915239	0.457619	+	0.889148i	$0.348702\pi$
$312$	0	0
$313$	− 3.18061e8i	− 0.586281i	−0.956069	−	0.293141i	$-0.905300\pi$
$313$	− 3.18061e8i	− 0.586281i	0.956069	−	0.293141i	$-0.0947004\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	3.85248e8i	0.679255i	0.940560	+	0.339627i	$0.110301\pi$
$317$	3.85248e8i	0.679255i	−0.940560	+	0.339627i	$0.889699\pi$
$318$	0	0
$319$	−9.09800e8	−1.56920
$320$	0	0
$321$	1.19428e9	2.01529
$322$	0	0
$323$	− 1.49936e8i	− 0.247569i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	1.36668e9i	2.16147i
$328$	0	0
$329$	5.46840e8	0.846592
$330$	0	0
$331$	−7.93180e8	−1.20219	−0.601096	−	0.799177i	$-0.705269\pi$
$331$	−7.93180e8	−1.20219	−0.601096	+	0.799177i	$0.705269\pi$
$332$	0	0
$333$	4.43959e8i	0.658852i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	9.20184e8i	1.30969i	0.755761	+	0.654847i	$0.227267\pi$
$337$	9.20184e8i	1.30969i	−0.755761	+	0.654847i	$0.772733\pi$
$338$	0	0
$339$	−1.12521e9	−1.56868
$340$	0	0
$341$	−1.25355e9	−1.71200
$342$	0	0
$343$	− 2.15530e8i	− 0.288388i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 3.26003e8i	− 0.418859i	−0.977824	−	0.209430i	$-0.932839\pi$
$347$	− 3.26003e8i	− 0.418859i	0.977824	−	0.209430i	$-0.0671607\pi$
$348$	0	0
$349$	−4.40162e8	−0.554273	−0.277136	−	0.960831i	$-0.589385\pi$
$349$	−4.40162e8	−0.554273	−0.277136	+	0.960831i	$0.589385\pi$
$350$	0	0
$351$	2.55745e8	0.315669
$352$	0	0
$353$	− 2.14414e8i	− 0.259443i	−0.991550	−	0.129721i	$-0.958592\pi$
$353$	− 2.14414e8i	− 0.259443i	0.991550	−	0.129721i	$-0.0414083\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	2.91081e8i	0.338590i
$358$	0	0
$359$	−6.01055e8	−0.685621	−0.342810	−	0.939405i	$-0.611379\pi$
$359$	−6.01055e8	−0.685621	−0.342810	+	0.939405i	$0.611379\pi$
$360$	0	0
$361$	6.23935e8	0.698014
$362$	0	0
$363$	1.61323e7i	0.0177020i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	− 2.28125e8i	− 0.240903i	−0.992719	−	0.120452i	$-0.961566\pi$
$367$	− 2.28125e8i	− 0.240903i	0.992719	−	0.120452i	$-0.0384343\pi$
$368$	0	0
$369$	−5.85151e6	−0.00606283
$370$	0	0
$371$	−1.85656e9	−1.88756
$372$	0	0
$373$	− 1.09122e9i	− 1.08876i	−0.838838	−	0.544381i	$-0.816765\pi$
$373$	− 1.09122e9i	− 1.08876i	0.838838	−	0.544381i	$-0.183235\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 7.80719e8i	− 0.750412i
$378$	0	0
$379$	2.92161e8	0.275667	0.137834	−	0.990455i	$-0.455986\pi$
$379$	2.92161e8	0.275667	0.137834	+	0.990455i	$0.455986\pi$
$380$	0	0
$381$	−1.89067e9	−1.75137
$382$	0	0
$383$	− 6.58727e8i	− 0.599114i	−0.954078	−	0.299557i	$-0.903161\pi$
$383$	− 6.58727e8i	− 0.599114i	0.954078	−	0.299557i	$-0.0968389\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	− 2.11837e8i	− 0.185786i
$388$	0	0
$389$	1.79238e8	0.154385	0.0771927	−	0.997016i	$-0.475404\pi$
$389$	1.79238e8	0.154385	0.0771927	+	0.997016i	$0.475404\pi$
$390$	0	0
$391$	2.28267e8	0.193119
$392$	0	0
$393$	3.03308e8i	0.252063i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	2.98717e8i	0.239603i	0.992798	+	0.119802i	$0.0382259\pi$
$397$	2.98717e8i	0.239603i	−0.992798	+	0.119802i	$0.961774\pi$
$398$	0	0
$399$	−2.94663e9	−2.32231
$400$	0	0
$401$	−1.70861e9	−1.32323	−0.661617	−	0.749842i	$-0.730129\pi$
$401$	−1.70861e9	−1.32323	−0.661617	+	0.749842i	$0.730129\pi$
$402$	0	0
$403$	− 1.07570e9i	− 0.818698i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 1.98904e9i	− 1.46239i
$408$	0	0
$409$	9.66030e8	0.698166	0.349083	−	0.937092i	$-0.386493\pi$
$409$	9.66030e8	0.698166	0.349083	+	0.937092i	$0.386493\pi$
$410$	0	0
$411$	1.57542e9	1.11931
$412$	0	0
$413$	2.53703e9i	1.77215i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	1.79233e9i	1.21044i
$418$	0	0
$419$	−1.38808e7	−0.00921862	−0.00460931	−	0.999989i	$-0.501467\pi$
$419$	−1.38808e7	−0.00921862	−0.00460931	+	0.999989i	$0.501467\pi$
$420$	0	0
$421$	−3.27466e8	−0.213884	−0.106942	−	0.994265i	$-0.534106\pi$
$421$	−3.27466e8	−0.213884	−0.106942	+	0.994265i	$0.534106\pi$
$422$	0	0
$423$	3.97820e8i	0.255562i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	− 3.72413e9i	− 2.31488i
$428$	0	0
$429$	9.26934e8	0.566824
$430$	0	0
$431$	2.79809e7	0.0168341	0.00841707	−	0.999965i	$-0.497321\pi$
$431$	2.79809e7	0.0168341	0.00841707	+	0.999965i	$0.497321\pi$
$432$	0	0
$433$	− 1.37019e9i	− 0.811098i	−0.914073	−	0.405549i	$-0.867080\pi$
$433$	− 1.37019e9i	− 0.811098i	0.914073	−	0.405549i	$-0.132920\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	2.31076e9i	1.32455i
$438$	0	0
$439$	2.52455e9	1.42416	0.712079	−	0.702099i	$-0.247754\pi$
$439$	2.52455e9	1.42416	0.712079	+	0.702099i	$0.247754\pi$
$440$	0	0
$441$	9.62249e8	0.534260
$442$	0	0
$443$	1.36683e9i	0.746969i	0.927636	+	0.373485i	$0.121837\pi$
$443$	1.36683e9i	0.746969i	−0.927636	+	0.373485i	$0.878163\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	3.74970e9i	1.98573i
$448$	0	0
$449$	2.85383e8	0.148787	0.0743937	−	0.997229i	$-0.476298\pi$
$449$	2.85383e8	0.148787	0.0743937	+	0.997229i	$0.476298\pi$
$450$	0	0
$451$	2.62162e7	0.0134571
$452$	0	0
$453$	1.33686e9i	0.675683i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	2.55438e9i	1.25193i	0.779852	+	0.625964i	$0.215294\pi$
$457$	2.55438e9i	1.25193i	−0.779852	+	0.625964i	$0.784706\pi$
$458$	0	0
$459$	2.61758e8	0.126344
$460$	0	0
$461$	2.81572e9	1.33855	0.669277	−	0.743013i	$-0.266604\pi$
$461$	2.81572e9	1.33855	0.669277	+	0.743013i	$0.266604\pi$
$462$	0	0
$463$	− 4.29378e8i	− 0.201051i	−0.994934	−	0.100526i	$-0.967948\pi$
$463$	− 4.29378e8i	− 0.201051i	0.994934	−	0.100526i	$-0.0320524\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	1.02740e9i	0.466800i	0.972381	+	0.233400i	$0.0749852\pi$
$467$	1.02740e9i	0.466800i	−0.972381	+	0.233400i	$0.925015\pi$
$468$	0	0
$469$	−4.36842e9	−1.95533
$470$	0	0
$471$	6.70550e8	0.295704
$472$	0	0
$473$	9.49081e8i	0.412372i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	− 1.35063e9i	− 0.569800i
$478$	0	0
$479$	1.23612e9	0.513908	0.256954	−	0.966424i	$-0.417281\pi$
$479$	1.23612e9	0.513908	0.256954	+	0.966424i	$0.417281\pi$
$480$	0	0
$481$	1.70684e9	0.699334
$482$	0	0
$483$	− 4.48604e9i	− 1.81154i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	− 3.53667e9i	− 1.38753i	−0.720200	−	0.693767i	$-0.755950\pi$
$487$	− 3.53667e9i	− 1.38753i	0.720200	−	0.693767i	$-0.244050\pi$
$488$	0	0
$489$	3.45104e9	1.33466
$490$	0	0
$491$	−4.64544e8	−0.177109	−0.0885546	−	0.996071i	$-0.528225\pi$
$491$	−4.64544e8	−0.177109	−0.0885546	+	0.996071i	$0.528225\pi$
$492$	0	0
$493$	− 7.99074e8i	− 0.300347i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 7.67998e8i	− 0.280616i
$498$	0	0
$499$	9.22783e8	0.332467	0.166233	−	0.986086i	$-0.446840\pi$
$499$	9.22783e8	0.332467	0.166233	+	0.986086i	$0.446840\pi$
$500$	0	0
$501$	3.62427e9	1.28762
$502$	0	0
$503$	2.71743e9i	0.952072i	0.879426	+	0.476036i	$0.157927\pi$
$503$	2.71743e9i	0.952072i	−0.879426	+	0.476036i	$0.842073\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	− 2.73472e9i	− 0.931935i
$508$	0	0
$509$	3.75201e9	1.26111	0.630553	−	0.776146i	$-0.282828\pi$
$509$	3.75201e9	1.26111	0.630553	+	0.776146i	$0.282828\pi$
$510$	0	0
$511$	2.78368e9	0.922881
$512$	0	0
$513$	2.64979e9i	0.866564i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	− 1.78233e9i	− 0.567245i
$518$	0	0
$519$	9.43504e8	0.296250
$520$	0	0
$521$	−5.80618e9	−1.79870	−0.899349	−	0.437231i	$-0.855959\pi$
$521$	−5.80618e9	−1.79870	−0.899349	+	0.437231i	$0.855959\pi$
$522$	0	0
$523$	− 3.82763e9i	− 1.16997i	−0.811045	−	0.584983i	$-0.801101\pi$
$523$	− 3.82763e9i	− 1.16997i	0.811045	−	0.584983i	$-0.198899\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 1.10099e9i	− 0.327678i
$528$	0	0
$529$	−1.13149e8	−0.0332319
$530$	0	0
$531$	−1.84566e9	−0.534959
$532$	0	0
$533$	2.24966e7i	0.00643536i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	− 3.80014e9i	− 1.05898i
$538$	0	0
$539$	−4.31111e9	−1.18585
$540$	0	0
$541$	1.27166e9	0.345286	0.172643	−	0.984984i	$-0.444769\pi$
$541$	1.27166e9	0.345286	0.172643	+	0.984984i	$0.444769\pi$
$542$	0	0
$543$	3.95168e9i	1.05921i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	− 1.67963e8i	− 0.0438792i	−0.999759	−	0.0219396i	$-0.993016\pi$
$547$	− 1.67963e8i	− 0.0438792i	0.999759	−	0.0219396i	$-0.00698415\pi$
$548$	0	0
$549$	2.70927e9	0.698793
$550$	0	0
$551$	8.08907e9	2.06000
$552$	0	0
$553$	2.59459e9i	0.652425i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 1.43415e9i	− 0.351644i	−0.984422	−	0.175822i	$-0.943742\pi$
$557$	− 1.43415e9i	− 0.351644i	0.984422	−	0.175822i	$-0.0562582\pi$
$558$	0	0
$559$	−8.14427e8	−0.197202
$560$	0	0
$561$	9.48727e8	0.226867
$562$	0	0
$563$	− 3.27203e9i	− 0.772748i	−0.922342	−	0.386374i	$-0.873727\pi$
$563$	− 3.27203e9i	− 0.772748i	0.922342	−	0.386374i	$-0.126273\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	− 8.01984e9i	− 1.84767i
$568$	0	0
$569$	−4.11773e9	−0.937054	−0.468527	−	0.883449i	$-0.655215\pi$
$569$	−4.11773e9	−0.937054	−0.468527	+	0.883449i	$0.655215\pi$
$570$	0	0
$571$	5.65009e9	1.27007	0.635037	−	0.772481i	$-0.280985\pi$
$571$	5.65009e9	1.27007	0.635037	+	0.772481i	$0.280985\pi$
$572$	0	0
$573$	2.27144e9i	0.504384i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	3.08235e9i	0.667986i	0.942576	+	0.333993i	$0.108396\pi$
$577$	3.08235e9i	0.667986i	−0.942576	+	0.333993i	$0.891604\pi$
$578$	0	0
$579$	−1.96271e9	−0.420225
$580$	0	0
$581$	9.18475e9	1.94290
$582$	0	0
$583$	6.05115e9i	1.26473i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 3.66057e9i	− 0.746990i	−0.927632	−	0.373495i	$-0.878159\pi$
$587$	− 3.66057e9i	− 0.746990i	0.927632	−	0.373495i	$-0.121841\pi$
$588$	0	0
$589$	1.11454e10	2.24746
$590$	0	0
$591$	−7.27991e9	−1.45067
$592$	0	0
$593$	− 6.95365e9i	− 1.36937i	−0.728838	−	0.684686i	$-0.759939\pi$
$593$	− 6.95365e9i	− 1.36937i	0.728838	−	0.684686i	$-0.240061\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	− 6.77643e9i	− 1.30344i
$598$	0	0
$599$	−4.50491e9	−0.856431	−0.428216	−	0.903677i	$-0.640858\pi$
$599$	−4.50491e9	−0.856431	−0.428216	+	0.903677i	$0.640858\pi$
$600$	0	0
$601$	−9.17562e9	−1.72415	−0.862075	−	0.506781i	$-0.830835\pi$
$601$	−9.17562e9	−1.72415	−0.862075	+	0.506781i	$0.830835\pi$
$602$	0	0
$603$	− 3.17798e9i	− 0.590256i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	4.68250e9i	0.849801i	0.905240	+	0.424900i	$0.139691\pi$
$607$	4.68250e9i	0.849801i	−0.905240	+	0.424900i	$0.860309\pi$
$608$	0	0
$609$	−1.57039e10	−2.81739
$610$	0	0
$611$	1.52945e9	0.271264
$612$	0	0
$613$	− 7.11746e9i	− 1.24800i	−0.781426	−	0.623998i	$-0.785507\pi$
$613$	− 7.11746e9i	− 1.24800i	0.781426	−	0.623998i	$-0.214493\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	5.75577e9i	0.986518i	0.869882	+	0.493259i	$0.164195\pi$
$617$	5.75577e9i	0.986518i	−0.869882	+	0.493259i	$0.835805\pi$
$618$	0	0
$619$	3.50085e9	0.593276	0.296638	−	0.954990i	$-0.404135\pi$
$619$	3.50085e9	0.593276	0.296638	+	0.954990i	$0.404135\pi$
$620$	0	0
$621$	−4.03413e9	−0.675972
$622$	0	0
$623$	4.78072e9i	0.792109i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	9.60402e9i	1.55602i
$628$	0	0
$629$	1.74697e9	0.279903
$630$	0	0
$631$	7.67576e9	1.21624	0.608119	−	0.793846i	$-0.291924\pi$
$631$	7.67576e9	1.21624	0.608119	+	0.793846i	$0.291924\pi$
$632$	0	0
$633$	− 4.50504e9i	− 0.705969i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	− 3.69945e9i	− 0.567087i
$638$	0	0
$639$	5.58710e8	0.0847098
$640$	0	0
$641$	3.44253e9	0.516268	0.258134	−	0.966109i	$-0.416892\pi$
$641$	3.44253e9	0.516268	0.258134	+	0.966109i	$0.416892\pi$
$642$	0	0
$643$	− 2.94778e9i	− 0.437277i	−0.975806	−	0.218638i	$-0.929839\pi$
$643$	− 2.94778e9i	− 0.437277i	0.975806	−	0.218638i	$-0.0701615\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	9.86842e9i	1.43246i	0.697864	+	0.716230i	$0.254134\pi$
$647$	9.86842e9i	1.43246i	−0.697864	+	0.716230i	$0.745866\pi$
$648$	0	0
$649$	8.26900e9	1.18740
$650$	0	0
$651$	−2.16374e10	−3.07377
$652$	0	0
$653$	− 2.25724e9i	− 0.317236i	−0.987340	−	0.158618i	$-0.949296\pi$
$653$	− 2.25724e9i	− 0.317236i	0.987340	−	0.158618i	$-0.0507038\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	2.02510e9i	0.278591i
$658$	0	0
$659$	−5.14817e9	−0.700735	−0.350368	−	0.936612i	$-0.613943\pi$
$659$	−5.14817e9	−0.700735	−0.350368	+	0.936612i	$0.613943\pi$
$660$	0	0
$661$	3.55066e9	0.478194	0.239097	−	0.970996i	$-0.423149\pi$
$661$	3.55066e9	0.478194	0.239097	+	0.970996i	$0.423149\pi$
$662$	0	0
$663$	8.14123e8i	0.108491i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	1.23151e10i	1.60693i
$668$	0	0
$669$	−1.74065e8	−0.0224760
$670$	0	0
$671$	−1.21382e10	−1.55104
$672$	0	0
$673$	7.61740e9i	0.963284i	0.876368	+	0.481642i	$0.159959\pi$
$673$	7.61740e9i	0.963284i	−0.876368	+	0.481642i	$0.840041\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	5.23904e9i	0.648921i	0.945899	+	0.324460i	$0.105183\pi$
$677$	5.23904e9i	0.648921i	−0.945899	+	0.324460i	$0.894817\pi$
$678$	0	0
$679$	1.69627e10	2.07946
$680$	0	0
$681$	5.39706e9	0.654850
$682$	0	0
$683$	8.98470e9i	1.07902i	0.841978	+	0.539512i	$0.181391\pi$
$683$	8.98470e9i	1.07902i	−0.841978	+	0.539512i	$0.818609\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	5.84102e9i	0.687290i
$688$	0	0
$689$	−5.19262e9	−0.604810
$690$	0	0
$691$	−4.25054e9	−0.490085	−0.245042	−	0.969512i	$-0.578802\pi$
$691$	−4.25054e9	−0.490085	−0.245042	+	0.969512i	$0.578802\pi$
$692$	0	0
$693$	− 5.76152e9i	− 0.657614i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	2.30256e7i	0.00257570i
$698$	0	0
$699$	7.68265e9	0.850826
$700$	0	0
$701$	−3.84334e9	−0.421402	−0.210701	−	0.977551i	$-0.567575\pi$
$701$	−3.84334e9	−0.421402	−0.210701	+	0.977551i	$0.567575\pi$
$702$	0	0
$703$	1.76847e10i	1.91979i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 1.29218e10i	− 1.37516i
$708$	0	0
$709$	−1.20064e10	−1.26518	−0.632588	−	0.774489i	$-0.718007\pi$
$709$	−1.20064e10	−1.26518	−0.632588	+	0.774489i	$0.718007\pi$
$710$	0	0
$711$	−1.88754e9	−0.196948
$712$	0	0
$713$	1.69681e10i	1.75316i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	− 9.01353e9i	− 0.913225i
$718$	0	0
$719$	−6.87813e9	−0.690112	−0.345056	−	0.938582i	$-0.612140\pi$
$719$	−6.87813e9	−0.690112	−0.345056	+	0.938582i	$0.612140\pi$
$720$	0	0
$721$	−1.06191e10	−1.05515
$722$	0	0
$723$	− 1.08825e10i	− 1.07089i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	1.20128e10i	1.15951i	0.814791	+	0.579754i	$0.196851\pi$
$727$	1.20128e10i	1.15951i	−0.814791	+	0.579754i	$0.803149\pi$
$728$	0	0
$729$	−2.53403e9	−0.242251
$730$	0	0
$731$	−8.33575e8	−0.0789285
$732$	0	0
$733$	− 3.09361e9i	− 0.290136i	−0.989422	−	0.145068i	$-0.953660\pi$
$733$	− 3.09361e9i	− 0.290136i	0.989422	−	0.145068i	$-0.0463401\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	1.42381e10i	1.31014i
$738$	0	0
$739$	2.02192e10	1.84293	0.921465	−	0.388461i	$-0.126993\pi$
$739$	2.02192e10	1.84293	0.921465	+	0.388461i	$0.126993\pi$
$740$	0	0
$741$	−8.24141e9	−0.744111
$742$	0	0
$743$	− 2.20781e9i	− 0.197470i	−0.995114	−	0.0987351i	$-0.968520\pi$
$743$	− 2.20781e9i	− 0.197470i	0.995114	−	0.0987351i	$-0.0314796\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	6.68181e9i	0.586505i
$748$	0	0
$749$	−2.85393e10	−2.48174
$750$	0	0
$751$	1.46597e10	1.26295	0.631476	−	0.775396i	$-0.282450\pi$
$751$	1.46597e10	1.26295	0.631476	+	0.775396i	$0.282450\pi$
$752$	0	0
$753$	− 2.36596e10i	− 2.01942i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	8.01162e9i	0.671251i	0.941995	+	0.335626i	$0.108948\pi$
$757$	8.01162e9i	0.671251i	−0.941995	+	0.335626i	$0.891052\pi$
$758$	0	0
$759$	−1.46215e10	−1.21379
$760$	0	0
$761$	−1.63492e10	−1.34478	−0.672390	−	0.740197i	$-0.734732\pi$
$761$	−1.63492e10	−1.34478	−0.672390	+	0.740197i	$0.734732\pi$
$762$	0	0
$763$	− 3.26592e10i	− 2.66176i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	7.09580e9i	0.567829i
$768$	0	0
$769$	8.77857e9	0.696116	0.348058	−	0.937473i	$-0.386841\pi$
$769$	8.77857e9	0.696116	0.348058	+	0.937473i	$0.386841\pi$
$770$	0	0
$771$	−2.10430e10	−1.65355
$772$	0	0
$773$	8.03519e9i	0.625703i	0.949802	+	0.312851i	$0.101284\pi$
$773$	8.03519e9i	0.625703i	−0.949802	+	0.312851i	$0.898716\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	− 3.43325e10i	− 2.62562i
$778$	0	0
$779$	−2.33089e8	−0.0176661
$780$	0	0
$781$	−2.50316e9	−0.188022
$782$	0	0
$783$	1.41219e10i	1.05130i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	− 9.89014e9i	− 0.723254i	−0.932323	−	0.361627i	$-0.882221\pi$
$787$	− 9.89014e9i	− 0.723254i	0.932323	−	0.361627i	$-0.117779\pi$
$788$	0	0
$789$	−1.48587e10	−1.07699
$790$	0	0
$791$	2.68889e10	1.93177
$792$	0	0
$793$	− 1.04160e10i	− 0.741730i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 2.22521e10i	− 1.55692i	−0.627694	−	0.778460i	$-0.716001\pi$
$797$	− 2.22521e10i	− 1.55692i	0.627694	−	0.778460i	$-0.283999\pi$
$798$	0	0
$799$	1.56541e9	0.108571
$800$	0	0
$801$	−3.47792e9	−0.239115
$802$	0	0
$803$	− 9.07291e9i	− 0.618361i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	2.30351e10i	1.54288i
$808$	0	0
$809$	−2.40746e10	−1.59860	−0.799300	−	0.600932i	$-0.794796\pi$
$809$	−2.40746e10	−1.59860	−0.799300	+	0.600932i	$0.794796\pi$
$810$	0	0
$811$	7.05679e9	0.464552	0.232276	−	0.972650i	$-0.425383\pi$
$811$	7.05679e9	0.464552	0.232276	+	0.972650i	$0.425383\pi$
$812$	0	0
$813$	2.36542e10i	1.54380i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	− 8.43833e9i	− 0.541351i
$818$	0	0
$819$	4.94408e9	0.314479
$820$	0	0
$821$	4.39901e9	0.277430	0.138715	−	0.990332i	$-0.455703\pi$
$821$	4.39901e9	0.277430	0.138715	+	0.990332i	$0.455703\pi$
$822$	0	0
$823$	− 2.19932e10i	− 1.37528i	−0.726054	−	0.687638i	$-0.758648\pi$
$823$	− 2.19932e10i	− 1.37528i	0.726054	−	0.687638i	$-0.241352\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 5.85930e9i	− 0.360227i	−0.983646	−	0.180114i	$-0.942353\pi$
$827$	− 5.85930e9i	− 0.360227i	0.983646	−	0.180114i	$-0.0576466\pi$
$828$	0	0
$829$	1.44100e10	0.878464	0.439232	−	0.898374i	$-0.355251\pi$
$829$	1.44100e10	0.878464	0.439232	+	0.898374i	$0.355251\pi$
$830$	0	0
$831$	3.33806e10	2.01786
$832$	0	0
$833$	− 3.78643e9i	− 0.226972i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	1.94577e10i	1.14697i
$838$	0	0
$839$	2.46206e10	1.43923	0.719617	−	0.694371i	$-0.244317\pi$
$839$	2.46206e10	1.43923	0.719617	+	0.694371i	$0.244317\pi$
$840$	0	0
$841$	2.58604e10	1.49917
$842$	0	0
$843$	2.77541e10i	1.59563i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	− 3.85508e8i	− 0.0217992i
$848$	0	0
$849$	1.69753e10	0.952010
$850$	0	0
$851$	−2.69237e10	−1.49755
$852$	0	0
$853$	5.02066e9i	0.276974i	0.990364	+	0.138487i	$0.0442239\pi$
$853$	5.02066e9i	0.276974i	−0.990364	+	0.138487i	$0.955776\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	3.48488e9i	0.189127i	0.995519	+	0.0945637i	$0.0301456\pi$
$857$	3.48488e9i	0.189127i	−0.995519	+	0.0945637i	$0.969854\pi$
$858$	0	0
$859$	−3.27339e10	−1.76206	−0.881031	−	0.473058i	$-0.843150\pi$
$859$	−3.27339e10	−1.76206	−0.881031	+	0.473058i	$0.843150\pi$
$860$	0	0
$861$	4.52512e8	0.0241613
$862$	0	0
$863$	3.17065e10i	1.67923i	0.543179	+	0.839617i	$0.317220\pi$
$863$	3.17065e10i	1.67923i	−0.543179	+	0.839617i	$0.682780\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	− 2.22518e10i	− 1.15958i
$868$	0	0
$869$	8.45661e9	0.437147
$870$	0	0
$871$	−1.22180e10	−0.626524
$872$	0	0
$873$	1.23402e10i	0.627727i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	3.24939e9i	0.162668i	0.996687	+	0.0813342i	$0.0259181\pi$
$877$	3.24939e9i	0.162668i	−0.996687	+	0.0813342i	$0.974082\pi$
$878$	0	0
$879$	−1.73758e10	−0.862945
$880$	0	0
$881$	2.79940e10	1.37927	0.689636	−	0.724157i	$-0.257771\pi$
$881$	2.79940e10	1.37927	0.689636	+	0.724157i	$0.257771\pi$
$882$	0	0
$883$	− 1.41470e10i	− 0.691515i	−0.938324	−	0.345758i	$-0.887622\pi$
$883$	− 1.41470e10i	− 0.691515i	0.938324	−	0.345758i	$-0.112378\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	3.73596e9i	0.179750i	0.995953	+	0.0898751i	$0.0286468\pi$
$887$	3.73596e9i	0.179750i	−0.995953	+	0.0898751i	$0.971353\pi$
$888$	0	0
$889$	4.51807e10	2.15674
$890$	0	0
$891$	−2.61393e10	−1.23800
$892$	0	0
$893$	1.58468e10i	0.744665i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	− 1.25470e10i	− 0.580452i
$898$	0	0
$899$	5.93988e10	2.72659
$900$	0	0
$901$	−5.31471e9	−0.242071
$902$	0	0
$903$	1.63819e10i	0.740385i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	− 1.23717e10i	− 0.550557i	−0.961365	−	0.275278i	$-0.911230\pi$
$907$	− 1.23717e10i	− 0.550557i	0.961365	−	0.275278i	$-0.0887700\pi$
$908$	0	0
$909$	9.40045e9	0.415121
$910$	0	0
$911$	5.09259e9	0.223164	0.111582	−	0.993755i	$-0.464408\pi$
$911$	5.09259e9	0.223164	0.111582	+	0.993755i	$0.464408\pi$
$912$	0	0
$913$	− 2.99361e10i	− 1.30181i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 7.24806e9i	− 0.310405i
$918$	0	0
$919$	−7.85605e8	−0.0333887	−0.0166944	−	0.999861i	$-0.505314\pi$
$919$	−7.85605e8	−0.0333887	−0.0166944	+	0.999861i	$0.505314\pi$
$920$	0	0
$921$	2.55346e10	1.07701
$922$	0	0
$923$	− 2.14801e9i	− 0.0899147i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	− 7.72529e9i	− 0.318519i
$928$	0	0
$929$	9.71717e9	0.397635	0.198818	−	0.980036i	$-0.436290\pi$
$929$	9.71717e9	0.397635	0.198818	+	0.980036i	$0.436290\pi$
$930$	0	0
$931$	3.83302e10	1.55675
$932$	0	0
$933$	− 2.73140e10i	− 1.10103i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	− 1.27672e10i	− 0.506998i	−0.967336	−	0.253499i	$-0.918418\pi$
$937$	− 1.27672e10i	− 0.506998i	0.967336	−	0.253499i	$-0.0815815\pi$
$938$	0	0
$939$	−1.78937e10	−0.705295
$940$	0	0
$941$	−2.35922e10	−0.923008	−0.461504	−	0.887138i	$-0.652690\pi$
$941$	−2.35922e10	−0.923008	−0.461504	+	0.887138i	$0.652690\pi$
$942$	0	0
$943$	− 3.54862e8i	− 0.0137806i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 8.88895e9i	− 0.340115i	−0.985434	−	0.170057i	$-0.945605\pi$
$947$	− 8.88895e9i	− 0.340115i	0.985434	−	0.170057i	$-0.0543953\pi$
$948$	0	0
$949$	7.78566e9	0.295708
$950$	0	0
$951$	2.16735e10	0.817142
$952$	0	0
$953$	1.08612e10i	0.406491i	0.979128	+	0.203245i	$0.0651489\pi$
$953$	1.08612e10i	0.406491i	−0.979128	+	0.203245i	$0.934851\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	5.11841e10i	1.88774i
$958$	0	0
$959$	−3.76473e10	−1.37838
$960$	0	0
$961$	5.43292e10	1.97470
$962$	0	0
$963$	− 2.07620e10i	− 0.749165i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	− 2.24979e10i	− 0.800109i	−0.916491	−	0.400054i	$-0.868991\pi$
$967$	− 2.24979e10i	− 0.800109i	0.916491	−	0.400054i	$-0.131009\pi$
$968$	0	0
$969$	−8.43518e9	−0.297825
$970$	0	0
$971$	−4.57613e10	−1.60410	−0.802049	−	0.597258i	$-0.796257\pi$
$971$	−4.57613e10	−1.60410	−0.802049	+	0.597258i	$0.796257\pi$
$972$	0	0
$973$	− 4.28308e10i	− 1.49060i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	2.40739e10i	0.825879i	0.910759	+	0.412939i	$0.135498\pi$
$977$	2.40739e10i	0.825879i	−0.910759	+	0.412939i	$0.864502\pi$
$978$	0	0
$979$	1.55819e10	0.530740
$980$	0	0
$981$	2.37592e10	0.803508
$982$	0	0
$983$	1.95466e10i	0.656346i	0.944618	+	0.328173i	$0.106433\pi$
$983$	1.95466e10i	0.656346i	−0.944618	+	0.328173i	$0.893567\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	− 3.07645e10i	− 1.01845i
$988$	0	0
$989$	1.28468e10	0.422287
$990$	0	0
$991$	5.15969e10	1.68409	0.842046	−	0.539406i	$-0.181351\pi$
$991$	5.15969e10	1.68409	0.842046	+	0.539406i	$0.181351\pi$
$992$	0	0
$993$	4.46232e10i	1.44623i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	− 5.82401e9i	− 0.186118i	−0.995661	−	0.0930591i	$-0.970335\pi$
$997$	− 5.82401e9i	− 0.186118i	0.995661	−	0.0930591i	$-0.0296645\pi$
$998$	0	0
$999$	−3.08739e10	−0.979744

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	200.8.c.h.49.1		4
4.3	odd	2		400.8.c.q.49.4		4
5.2	odd	4		200.8.a.m.1.1		2
5.3	odd	4		40.8.a.b.1.2	✓	2
5.4	even	2	inner	200.8.c.h.49.4		4
15.8	even	4		360.8.a.l.1.2		2
20.3	even	4		80.8.a.h.1.1		2
20.7	even	4		400.8.a.z.1.2		2
20.19	odd	2		400.8.c.q.49.1		4
40.3	even	4		320.8.a.q.1.2		2
40.13	odd	4		320.8.a.o.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
40.8.a.b.1.2	✓	2	5.3	odd	4
80.8.a.h.1.1		2	20.3	even	4
200.8.a.m.1.1		2	5.2	odd	4
200.8.c.h.49.1		4	1.1	even	1	trivial
200.8.c.h.49.4		4	5.4	even	2	inner
320.8.a.o.1.1		2	40.13	odd	4
320.8.a.q.1.2		2	40.3	even	4
360.8.a.l.1.2		2	15.8	even	4
400.8.a.z.1.2		2	20.7	even	4
400.8.c.q.49.1		4	20.19	odd	2
400.8.c.q.49.4		4	4.3	odd	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:	\( N \)	\(=\)	\( 200 = 2^{3} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	200.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-56.2586i q^{3} +1344.40i q^{7} -978.035 q^{9} +4381.83 q^{11} +3760.14i q^{13} +3848.55i q^{17} -38959.0 q^{19} +75633.9 q^{21} -59312.5i q^{23} -68014.8i q^{27} -207630. q^{29} -286080. q^{31} -246516. i q^{33} -453930. i q^{37} +211540. q^{39} +5982.93 q^{41} +216595. i q^{43} -406755. i q^{47} -983860. q^{49} +216514. q^{51} +1.38096e6i q^{53} +2.19178e6i q^{57} +1.88711e6 q^{59} -2.77012e6 q^{61} -1.31487e6i q^{63} +3.24935e6i q^{67} -3.33684e6 q^{69} -571258. q^{71} -2.07058e6i q^{73} +5.89092e6i q^{77} +1.92993e6 q^{79} -5.96538e6 q^{81} -6.83187e6i q^{83} +1.16810e7i q^{87} +3.55603e6 q^{89} -5.05512e6 q^{91} +1.60945e7i q^{93} -1.26173e7i q^{97} -4.28558e6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 3044 q^{9} + 9280 q^{11} - 79440 q^{19} + 203568 q^{21} - 373880 q^{29} - 654256 q^{31} + 1527216 q^{39} - 325928 q^{41} - 821428 q^{49} - 3190320 q^{51} - 2241552 q^{59} - 3711704 q^{61} - 8039568 q^{69}+ \cdots - 8852032 q^{99}+O(q^{100}) \) 4 * q - 3044 * q^9 + 9280 * q^11 - 79440 * q^19 + 203568 * q^21 - 373880 * q^29 - 654256 * q^31 + 1527216 * q^39 - 325928 * q^41 - 821428 * q^49 - 3190320 * q^51 - 2241552 * q^59 - 3711704 * q^61 - 8039568 * q^69 - 2142224 * q^71 - 8680288 * q^79 - 23284204 * q^81 - 84520 * q^89 - 20682768 * q^91 - 8852032 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

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Groups

Database

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