LMFDB - Embedded newform 360.8.a.l.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [360,8,Mod(1,360)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(360, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 8, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("360.1"); S:= CuspForms(chi, 8); N := Newforms(S);

Level:	$ N $	$=$	$ 360 = 2^{3} \cdot 3^{2} \cdot 5 $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	360.a (trivial)

Newform invariants

comment:select newform

sage:traces = [2,0,0,0,250,0,844] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

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

Self dual:	yes
Analytic conductor:	$112.458609174$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{46}) $
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 46 $ x^2 - 46
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	no (minimal twist has level 40)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$6.78233$ of defining polynomial
Character	$\chi$	$=$	360.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+125.000 q^{5} +1344.40 q^{7} -4381.83 q^{11} -3760.14 q^{13} -3848.55 q^{17} +38959.0 q^{19} -59312.5 q^{23} +15625.0 q^{25} -207630. q^{29} -286080. q^{31} +168050. q^{35} -453930. q^{37} -5982.93 q^{41} -216595. q^{43} +406755. q^{47} +983860. q^{49} +1.38096e6 q^{53} -547729. q^{55} +1.88711e6 q^{59} -2.77012e6 q^{61} -470018. q^{65} +3.24935e6 q^{67} +571258. q^{71} +2.07058e6 q^{73} -5.89092e6 q^{77} -1.92993e6 q^{79} -6.83187e6 q^{83} -481068. q^{85} +3.55603e6 q^{89} -5.05512e6 q^{91} +4.86988e6 q^{95} -1.26173e7 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 250 q^{5} + 844 q^{7} - 4640 q^{11} + 6804 q^{13} - 38516 q^{17} + 39720 q^{19} - 46244 q^{23} + 31250 q^{25} - 186940 q^{29} - 327128 q^{31} + 105500 q^{35} + 67060 q^{37} + 162964 q^{41} - 1123468 q^{43}+ \cdots - 9119804 q^{97}+O(q^{100}) $ 2 * q + 250 * q^5 + 844 * q^7 - 4640 * q^11 + 6804 * q^13 - 38516 * q^17 + 39720 * q^19 - 46244 * q^23 + 31250 * q^25 - 186940 * q^29 - 327128 * q^31 + 105500 * q^35 + 67060 * q^37 + 162964 * q^41 - 1123468 * q^43 + 777156 * q^47 + 410714 * q^49 - 144164 * q^53 - 580000 * q^55 - 1120776 * q^59 - 1855852 * q^61 + 850500 * q^65 + 6239676 * q^67 + 1071112 * q^71 + 3467044 * q^73 - 5761728 * q^77 + 4340144 * q^79 - 10960692 * q^83 - 4814500 * q^85 - 42260 * q^89 - 10341384 * q^91 + 4965000 * q^95 - 9119804 * q^97

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
$3$	0	0
$3$	0	0
$4$	0	0
$5$	125.000	0.447214
$5$	125.000	0.447214
$6$	0	0
$7$	1344.40	1.48144	0.740720	−	0.671813i	$-0.234484\pi$
$7$	1344.40	1.48144	0.740720	+	0.671813i	$0.234484\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−4381.83	−0.992615	−0.496308	−	0.868147i	$-0.665311\pi$
$11$	−4381.83	−0.992615	−0.496308	+	0.868147i	$0.665311\pi$
$12$	0	0
$13$	−3760.14	−0.474682	−0.237341	−	0.971426i	$-0.576276\pi$
$13$	−3760.14	−0.474682	−0.237341	+	0.971426i	$0.576276\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−3848.55	−0.189988	−0.0949939	−	0.995478i	$-0.530283\pi$
$17$	−3848.55	−0.189988	−0.0949939	+	0.995478i	$0.530283\pi$
$18$	0	0
$19$	38959.0	1.30308	0.651539	−	0.758615i	$-0.274124\pi$
$19$	38959.0	1.30308	0.651539	+	0.758615i	$0.274124\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−59312.5	−1.01648	−0.508240	−	0.861215i	$-0.669704\pi$
$23$	−59312.5	−1.01648	−0.508240	+	0.861215i	$0.669704\pi$
$24$	0	0
$25$	15625.0	0.200000
$26$	0	0
$27$	0	0
$28$	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
$33$	0	0
$34$	0	0
$35$	168050.	0.662521
$36$	0	0
$37$	−453930.	−1.47327	−0.736635	−	0.676290i	$-0.763587\pi$
$37$	−453930.	−1.47327	−0.736635	+	0.676290i	$0.763587\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−5982.93	−0.0135572	−0.00677860	−	0.999977i	$-0.502158\pi$
$41$	−5982.93	−0.0135572	−0.00677860	+	0.999977i	$0.502158\pi$
$42$	0	0
$43$	−216595.	−0.415440	−0.207720	−	0.978188i	$-0.566604\pi$
$43$	−216595.	−0.415440	−0.207720	+	0.978188i	$0.566604\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	406755.	0.571466	0.285733	−	0.958309i	$-0.407763\pi$
$47$	406755.	0.571466	0.285733	+	0.958309i	$0.407763\pi$
$48$	0	0
$49$	983860.	1.19467
$50$	0	0
$51$	0	0
$52$	0	0
$53$	1.38096e6	1.27414	0.637070	−	0.770806i	$-0.280146\pi$
$53$	1.38096e6	1.27414	0.637070	+	0.770806i	$0.280146\pi$
$54$	0	0
$55$	−547729.	−0.443911
$56$	0	0
$57$	0	0
$58$	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
$63$	0	0
$64$	0	0
$65$	−470018.	−0.212284
$66$	0	0
$67$	3.24935e6	1.31988	0.659941	−	0.751317i	$-0.270581\pi$
$67$	3.24935e6	1.31988	0.659941	+	0.751317i	$0.270581\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	571258.	0.189421	0.0947105	−	0.995505i	$-0.469807\pi$
$71$	571258.	0.189421	0.0947105	+	0.995505i	$0.469807\pi$
$72$	0	0
$73$	2.07058e6	0.622962	0.311481	−	0.950252i	$-0.399175\pi$
$73$	2.07058e6	0.622962	0.311481	+	0.950252i	$0.399175\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−5.89092e6	−1.47050
$78$	0	0
$79$	−1.92993e6	−0.440399	−0.220199	−	0.975455i	$-0.570671\pi$
$79$	−1.92993e6	−0.440399	−0.220199	+	0.975455i	$0.570671\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−6.83187e6	−1.31149	−0.655747	−	0.754980i	$-0.727646\pi$
$83$	−6.83187e6	−1.31149	−0.655747	+	0.754980i	$0.727646\pi$
$84$	0	0
$85$	−481068.	−0.0849651
$86$	0	0
$87$	0	0
$88$	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
$93$	0	0
$94$	0	0
$95$	4.86988e6	0.582755
$96$	0	0
$97$	−1.26173e7	−1.40367	−0.701836	−	0.712338i	$-0.747636\pi$
$97$	−1.26173e7	−1.40367	−0.701836	+	0.712338i	$0.747636\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	9.61157e6	0.928260	0.464130	−	0.885767i	$-0.346367\pi$
$101$	9.61157e6	0.928260	0.464130	+	0.885767i	$0.346367\pi$
$102$	0	0
$103$	−7.89879e6	−0.712247	−0.356123	−	0.934439i	$-0.615902\pi$
$103$	−7.89879e6	−0.712247	−0.356123	+	0.934439i	$0.615902\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−2.12283e7	−1.67522	−0.837610	−	0.546268i	$-0.816048\pi$
$107$	−2.12283e7	−1.67522	−0.837610	+	0.546268i	$0.816048\pi$
$108$	0	0
$109$	2.42928e7	1.79674	0.898369	−	0.439241i	$-0.144753\pi$
$109$	2.42928e7	1.79674	0.898369	+	0.439241i	$0.144753\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−2.00007e7	−1.30398	−0.651989	−	0.758228i	$-0.726065\pi$
$113$	−2.00007e7	−1.30398	−0.651989	+	0.758228i	$0.726065\pi$
$114$	0	0
$115$	−7.41406e6	−0.454584
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−5.17397e6	−0.281456
$120$	0	0
$121$	−286752.	−0.0147149
$122$	0	0
$123$	0	0
$124$	0	0
$125$	1.95312e6	0.0894427
$126$	0	0
$127$	−3.36067e7	−1.45584	−0.727918	−	0.685664i	$-0.759512\pi$
$127$	−3.36067e7	−1.45584	−0.727918	+	0.685664i	$0.759512\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	5.39131e6	0.209529	0.104765	−	0.994497i	$-0.466591\pi$
$131$	5.39131e6	0.209529	0.104765	+	0.994497i	$0.466591\pi$
$132$	0	0
$133$	5.23764e7	1.93043
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−2.80031e7	−0.930432	−0.465216	−	0.885197i	$-0.654023\pi$
$137$	−2.80031e7	−0.930432	−0.465216	+	0.885197i	$0.654023\pi$
$138$	0	0
$139$	3.18588e7	1.00618	0.503092	−	0.864233i	$-0.332196\pi$
$139$	3.18588e7	1.00618	0.503092	+	0.864233i	$0.332196\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	1.64763e7	0.471176
$144$	0	0
$145$	−2.59538e7	−0.706989
$146$	0	0
$147$	0	0
$148$	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
$153$	0	0
$154$	0	0
$155$	−3.57600e7	−0.771324
$156$	0	0
$157$	1.19191e7	0.245806	0.122903	−	0.992419i	$-0.460780\pi$
$157$	1.19191e7	0.245806	0.122903	+	0.992419i	$0.460780\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−7.97396e7	−1.50586
$162$	0	0
$163$	−6.13425e7	−1.10944	−0.554721	−	0.832036i	$-0.687175\pi$
$163$	−6.13425e7	−1.10944	−0.554721	+	0.832036i	$0.687175\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−6.44216e7	−1.07034	−0.535172	−	0.844743i	$-0.679753\pi$
$167$	−6.44216e7	−1.07034	−0.535172	+	0.844743i	$0.679753\pi$
$168$	0	0
$169$	−4.86099e7	−0.774677
$170$	0	0
$171$	0	0
$172$	0	0
$173$	1.67708e7	0.246260	0.123130	−	0.992391i	$-0.460707\pi$
$173$	1.67708e7	0.246260	0.123130	+	0.992391i	$0.460707\pi$
$174$	0	0
$175$	2.10062e7	0.296288
$176$	0	0
$177$	0	0
$178$	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
$183$	0	0
$184$	0	0
$185$	−5.67412e7	−0.658867
$186$	0	0
$187$	1.68637e7	0.188585
$188$	0	0
$189$	0	0
$190$	0	0
$191$	4.03750e7	0.419272	0.209636	−	0.977779i	$-0.432772\pi$
$191$	4.03750e7	0.419272	0.209636	+	0.977779i	$0.432772\pi$
$192$	0	0
$193$	3.48873e7	0.349315	0.174657	−	0.984629i	$-0.444118\pi$
$193$	3.48873e7	0.349315	0.174657	+	0.984629i	$0.444118\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	1.29401e8	1.20588	0.602941	−	0.797786i	$-0.293995\pi$
$197$	1.29401e8	1.20588	0.602941	+	0.797786i	$0.293995\pi$
$198$	0	0
$199$	−1.20451e8	−1.08349	−0.541746	−	0.840542i	$-0.682237\pi$
$199$	−1.20451e8	−1.08349	−0.541746	+	0.840542i	$0.682237\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−2.79137e8	−2.34197
$204$	0	0
$205$	−747866.	−0.00606297
$206$	0	0
$207$	0	0
$208$	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
$213$	0	0
$214$	0	0
$215$	−2.70743e7	−0.185790
$216$	0	0
$217$	−3.84605e8	−2.55509
$218$	0	0
$219$	0	0
$220$	0	0
$221$	1.44711e7	0.0901837
$222$	0	0
$223$	3.09401e6	0.0186833	0.00934167	−	0.999956i	$-0.497026\pi$
$223$	3.09401e6	0.0186833	0.00934167	+	0.999956i	$0.497026\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−9.59329e7	−0.544349	−0.272174	−	0.962248i	$-0.587743\pi$
$227$	−9.59329e7	−0.544349	−0.272174	+	0.962248i	$0.587743\pi$
$228$	0	0
$229$	1.03824e8	0.571314	0.285657	−	0.958332i	$-0.407788\pi$
$229$	1.03824e8	0.571314	0.285657	+	0.958332i	$0.407788\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	1.36559e8	0.707255	0.353628	−	0.935386i	$-0.384948\pi$
$233$	1.36559e8	0.707255	0.353628	+	0.935386i	$0.384948\pi$
$234$	0	0
$235$	5.08443e7	0.255567
$236$	0	0
$237$	0	0
$238$	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
$243$	0	0
$244$	0	0
$245$	1.22982e8	0.534271
$246$	0	0
$247$	−1.46491e8	−0.618547
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−4.20551e8	−1.67865	−0.839326	−	0.543628i	$-0.817050\pi$
$251$	−4.20551e8	−1.67865	−0.839326	+	0.543628i	$0.817050\pi$
$252$	0	0
$253$	2.59897e8	1.00897
$254$	0	0
$255$	0	0
$256$	0	0
$257$	3.74041e8	1.37453	0.687264	−	0.726408i	$-0.258812\pi$
$257$	3.74041e8	1.37453	0.687264	+	0.726408i	$0.258812\pi$
$258$	0	0
$259$	−6.10262e8	−2.18256
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−2.64114e8	−0.895253	−0.447626	−	0.894221i	$-0.647731\pi$
$263$	−2.64114e8	−0.895253	−0.447626	+	0.894221i	$0.647731\pi$
$264$	0	0
$265$	1.72621e8	0.569812
$266$	0	0
$267$	0	0
$268$	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
$273$	0	0
$274$	0	0
$275$	−6.84661e7	−0.198523
$276$	0	0
$277$	5.93342e8	1.67736	0.838679	−	0.544626i	$-0.183328\pi$
$277$	5.93342e8	1.67736	0.838679	+	0.544626i	$0.183328\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	4.93331e8	1.32638	0.663188	−	0.748453i	$-0.269203\pi$
$281$	4.93331e8	1.32638	0.663188	+	0.748453i	$0.269203\pi$
$282$	0	0
$283$	−3.01738e8	−0.791365	−0.395683	−	0.918387i	$-0.629492\pi$
$283$	−3.01738e8	−0.791365	−0.395683	+	0.918387i	$0.629492\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−8.04343e6	−0.0200842
$288$	0	0
$289$	−3.95527e8	−0.963905
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−3.08855e8	−0.717328	−0.358664	−	0.933467i	$-0.616768\pi$
$293$	−3.08855e8	−0.717328	−0.358664	+	0.933467i	$0.616768\pi$
$294$	0	0
$295$	2.35889e8	0.534971
$296$	0	0
$297$	0	0
$298$	0	0
$299$	2.23023e8	0.482504
$300$	0	0
$301$	−2.91189e8	−0.615450
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−3.46264e8	−0.698809
$306$	0	0
$307$	4.53878e8	0.895272	0.447636	−	0.894216i	$-0.352266\pi$
$307$	4.53878e8	0.895272	0.447636	+	0.894216i	$0.352266\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−4.85507e8	−0.915239	−0.457619	−	0.889148i	$-0.651298\pi$
$311$	−4.85507e8	−0.915239	−0.457619	+	0.889148i	$0.651298\pi$
$312$	0	0
$313$	3.18061e8	0.586281	0.293141	−	0.956069i	$-0.405300\pi$
$313$	3.18061e8	0.586281	0.293141	+	0.956069i	$0.405300\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−3.85248e8	−0.679255	−0.339627	−	0.940560i	$-0.610301\pi$
$317$	−3.85248e8	−0.679255	−0.339627	+	0.940560i	$0.610301\pi$
$318$	0	0
$319$	9.09800e8	1.56920
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−1.49936e8	−0.247569
$324$	0	0
$325$	−5.87522e7	−0.0949363
$326$	0	0
$327$	0	0
$328$	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
$333$	0	0
$334$	0	0
$335$	4.06169e8	0.590269
$336$	0	0
$337$	9.20184e8	1.30969	0.654847	−	0.755761i	$-0.272733\pi$
$337$	9.20184e8	1.30969	0.654847	+	0.755761i	$0.272733\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	1.25355e9	1.71200
$342$	0	0
$343$	2.15530e8	0.288388
$344$	0	0
$345$	0	0
$346$	0	0
$347$	3.26003e8	0.418859	0.209430	−	0.977824i	$-0.432839\pi$
$347$	3.26003e8	0.418859	0.209430	+	0.977824i	$0.432839\pi$
$348$	0	0
$349$	4.40162e8	0.554273	0.277136	−	0.960831i	$-0.410615\pi$
$349$	4.40162e8	0.554273	0.277136	+	0.960831i	$0.410615\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−2.14414e8	−0.259443	−0.129721	−	0.991550i	$-0.541408\pi$
$353$	−2.14414e8	−0.259443	−0.129721	+	0.991550i	$0.541408\pi$
$354$	0	0
$355$	7.14073e7	0.0847117
$356$	0	0
$357$	0	0
$358$	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
$363$	0	0
$364$	0	0
$365$	2.58822e8	0.278597
$366$	0	0
$367$	−2.28125e8	−0.240903	−0.120452	−	0.992719i	$-0.538434\pi$
$367$	−2.28125e8	−0.240903	−0.120452	+	0.992719i	$0.538434\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	1.85656e9	1.88756
$372$	0	0
$373$	1.09122e9	1.08876	0.544381	−	0.838838i	$-0.316765\pi$
$373$	1.09122e9	1.08876	0.544381	+	0.838838i	$0.316765\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	7.80719e8	0.750412
$378$	0	0
$379$	−2.92161e8	−0.275667	−0.137834	−	0.990455i	$-0.544014\pi$
$379$	−2.92161e8	−0.275667	−0.137834	+	0.990455i	$0.544014\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−6.58727e8	−0.599114	−0.299557	−	0.954078i	$-0.596839\pi$
$383$	−6.58727e8	−0.599114	−0.299557	+	0.954078i	$0.596839\pi$
$384$	0	0
$385$	−7.36365e8	−0.657628
$386$	0	0
$387$	0	0
$388$	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
$393$	0	0
$394$	0	0
$395$	−2.41241e8	−0.196952
$396$	0	0
$397$	2.98717e8	0.239603	0.119802	−	0.992798i	$-0.461774\pi$
$397$	2.98717e8	0.239603	0.119802	+	0.992798i	$0.461774\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	1.70861e9	1.32323	0.661617	−	0.749842i	$-0.269871\pi$
$401$	1.70861e9	1.32323	0.661617	+	0.749842i	$0.269871\pi$
$402$	0	0
$403$	1.07570e9	0.818698
$404$	0	0
$405$	0	0
$406$	0	0
$407$	1.98904e9	1.46239
$408$	0	0
$409$	−9.66030e8	−0.698166	−0.349083	−	0.937092i	$-0.613507\pi$
$409$	−9.66030e8	−0.698166	−0.349083	+	0.937092i	$0.613507\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	2.53703e9	1.77215
$414$	0	0
$415$	−8.53984e8	−0.586518
$416$	0	0
$417$	0	0
$418$	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
$423$	0	0
$424$	0	0
$425$	−6.01335e7	−0.0379975
$426$	0	0
$427$	−3.72413e9	−2.31488
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−2.79809e7	−0.0168341	−0.00841707	−	0.999965i	$-0.502679\pi$
$431$	−2.79809e7	−0.0168341	−0.00841707	+	0.999965i	$0.502679\pi$
$432$	0	0
$433$	1.37019e9	0.811098	0.405549	−	0.914073i	$-0.367080\pi$
$433$	1.37019e9	0.811098	0.405549	+	0.914073i	$0.367080\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−2.31076e9	−1.32455
$438$	0	0
$439$	−2.52455e9	−1.42416	−0.712079	−	0.702099i	$-0.752246\pi$
$439$	−2.52455e9	−1.42416	−0.712079	+	0.702099i	$0.752246\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	1.36683e9	0.746969	0.373485	−	0.927636i	$-0.378163\pi$
$443$	1.36683e9	0.746969	0.373485	+	0.927636i	$0.378163\pi$
$444$	0	0
$445$	4.44504e8	0.239120
$446$	0	0
$447$	0	0
$448$	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
$453$	0	0
$454$	0	0
$455$	−6.31890e8	−0.314486
$456$	0	0
$457$	2.55438e9	1.25193	0.625964	−	0.779852i	$-0.284706\pi$
$457$	2.55438e9	1.25193	0.625964	+	0.779852i	$0.284706\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−2.81572e9	−1.33855	−0.669277	−	0.743013i	$-0.733396\pi$
$461$	−2.81572e9	−1.33855	−0.669277	+	0.743013i	$0.733396\pi$
$462$	0	0
$463$	4.29378e8	0.201051	0.100526	−	0.994934i	$-0.467948\pi$
$463$	4.29378e8	0.201051	0.100526	+	0.994934i	$0.467948\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−1.02740e9	−0.466800	−0.233400	−	0.972381i	$-0.574985\pi$
$467$	−1.02740e9	−0.466800	−0.233400	+	0.972381i	$0.574985\pi$
$468$	0	0
$469$	4.36842e9	1.95533
$470$	0	0
$471$	0	0
$472$	0	0
$473$	9.49081e8	0.412372
$474$	0	0
$475$	6.08735e8	0.260616
$476$	0	0
$477$	0	0
$478$	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
$483$	0	0
$484$	0	0
$485$	−1.57716e9	−0.627741
$486$	0	0
$487$	−3.53667e9	−1.38753	−0.693767	−	0.720200i	$-0.744050\pi$
$487$	−3.53667e9	−1.38753	−0.693767	+	0.720200i	$0.744050\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	4.64544e8	0.177109	0.0885546	−	0.996071i	$-0.471775\pi$
$491$	4.64544e8	0.177109	0.0885546	+	0.996071i	$0.471775\pi$
$492$	0	0
$493$	7.99074e8	0.300347
$494$	0	0
$495$	0	0
$496$	0	0
$497$	7.67998e8	0.280616
$498$	0	0
$499$	−9.22783e8	−0.332467	−0.166233	−	0.986086i	$-0.553160\pi$
$499$	−9.22783e8	−0.332467	−0.166233	+	0.986086i	$0.553160\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	2.71743e9	0.952072	0.476036	−	0.879426i	$-0.342073\pi$
$503$	2.71743e9	0.952072	0.476036	+	0.879426i	$0.342073\pi$
$504$	0	0
$505$	1.20145e9	0.415130
$506$	0	0
$507$	0	0
$508$	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
$513$	0	0
$514$	0	0
$515$	−9.87349e8	−0.318526
$516$	0	0
$517$	−1.78233e9	−0.567245
$518$	0	0
$519$	0	0
$520$	0	0
$521$	5.80618e9	1.79870	0.899349	−	0.437231i	$-0.144041\pi$
$521$	5.80618e9	1.79870	0.899349	+	0.437231i	$0.144041\pi$
$522$	0	0
$523$	3.82763e9	1.16997	0.584983	−	0.811045i	$-0.301101\pi$
$523$	3.82763e9	1.16997	0.584983	+	0.811045i	$0.301101\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	1.10099e9	0.327678
$528$	0	0
$529$	1.13149e8	0.0332319
$530$	0	0
$531$	0	0
$532$	0	0
$533$	2.24966e7	0.00643536
$534$	0	0
$535$	−2.65354e9	−0.749181
$536$	0	0
$537$	0	0
$538$	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
$543$	0	0
$544$	0	0
$545$	3.03660e9	0.803526
$546$	0	0
$547$	−1.67963e8	−0.0438792	−0.0219396	−	0.999759i	$-0.506984\pi$
$547$	−1.67963e8	−0.0438792	−0.0219396	+	0.999759i	$0.506984\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−8.08907e9	−2.06000
$552$	0	0
$553$	−2.59459e9	−0.652425
$554$	0	0
$555$	0	0
$556$	0	0
$557$	1.43415e9	0.351644	0.175822	−	0.984422i	$-0.443742\pi$
$557$	1.43415e9	0.351644	0.175822	+	0.984422i	$0.443742\pi$
$558$	0	0
$559$	8.14427e8	0.197202
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−3.27203e9	−0.772748	−0.386374	−	0.922342i	$-0.626273\pi$
$563$	−3.27203e9	−0.772748	−0.386374	+	0.922342i	$0.626273\pi$
$564$	0	0
$565$	−2.50009e9	−0.583157
$566$	0	0
$567$	0	0
$568$	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
$573$	0	0
$574$	0	0
$575$	−9.26758e8	−0.203296
$576$	0	0
$577$	3.08235e9	0.667986	0.333993	−	0.942576i	$-0.391604\pi$
$577$	3.08235e9	0.667986	0.333993	+	0.942576i	$0.391604\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−9.18475e9	−1.94290
$582$	0	0
$583$	−6.05115e9	−1.26473
$584$	0	0
$585$	0	0
$586$	0	0
$587$	3.66057e9	0.746990	0.373495	−	0.927632i	$-0.378159\pi$
$587$	3.66057e9	0.746990	0.373495	+	0.927632i	$0.378159\pi$
$588$	0	0
$589$	−1.11454e10	−2.24746
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−6.95365e9	−1.36937	−0.684686	−	0.728838i	$-0.740061\pi$
$593$	−6.95365e9	−1.36937	−0.684686	+	0.728838i	$0.740061\pi$
$594$	0	0
$595$	−6.46747e8	−0.125871
$596$	0	0
$597$	0	0
$598$	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
$603$	0	0
$604$	0	0
$605$	−3.58440e7	−0.00658070
$606$	0	0
$607$	4.68250e9	0.849801	0.424900	−	0.905240i	$-0.360309\pi$
$607$	4.68250e9	0.849801	0.424900	+	0.905240i	$0.360309\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−1.52945e9	−0.271264
$612$	0	0
$613$	7.11746e9	1.24800	0.623998	−	0.781426i	$-0.285507\pi$
$613$	7.11746e9	1.24800	0.623998	+	0.781426i	$0.285507\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−5.75577e9	−0.986518	−0.493259	−	0.869882i	$-0.664195\pi$
$617$	−5.75577e9	−0.986518	−0.493259	+	0.869882i	$0.664195\pi$
$618$	0	0
$619$	−3.50085e9	−0.593276	−0.296638	−	0.954990i	$-0.595865\pi$
$619$	−3.50085e9	−0.593276	−0.296638	+	0.954990i	$0.595865\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	4.78072e9	0.792109
$624$	0	0
$625$	2.44141e8	0.0400000
$626$	0	0
$627$	0	0
$628$	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
$633$	0	0
$634$	0	0
$635$	−4.20084e9	−0.651070
$636$	0	0
$637$	−3.69945e9	−0.567087
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−3.44253e9	−0.516268	−0.258134	−	0.966109i	$-0.583108\pi$
$641$	−3.44253e9	−0.516268	−0.258134	+	0.966109i	$0.583108\pi$
$642$	0	0
$643$	2.94778e9	0.437277	0.218638	−	0.975806i	$-0.429839\pi$
$643$	2.94778e9	0.437277	0.218638	+	0.975806i	$0.429839\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−9.86842e9	−1.43246	−0.716230	−	0.697864i	$-0.754134\pi$
$647$	−9.86842e9	−1.43246	−0.716230	+	0.697864i	$0.754134\pi$
$648$	0	0
$649$	−8.26900e9	−1.18740
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−2.25724e9	−0.317236	−0.158618	−	0.987340i	$-0.550704\pi$
$653$	−2.25724e9	−0.317236	−0.158618	+	0.987340i	$0.550704\pi$
$654$	0	0
$655$	6.73914e8	0.0937044
$656$	0	0
$657$	0	0
$658$	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
$663$	0	0
$664$	0	0
$665$	6.54705e9	0.863316
$666$	0	0
$667$	1.23151e10	1.60693
$668$	0	0
$669$	0	0
$670$	0	0
$671$	1.21382e10	1.55104
$672$	0	0
$673$	−7.61740e9	−0.963284	−0.481642	−	0.876368i	$-0.659959\pi$
$673$	−7.61740e9	−0.963284	−0.481642	+	0.876368i	$0.659959\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−5.23904e9	−0.648921	−0.324460	−	0.945899i	$-0.605183\pi$
$677$	−5.23904e9	−0.648921	−0.324460	+	0.945899i	$0.605183\pi$
$678$	0	0
$679$	−1.69627e10	−2.07946
$680$	0	0
$681$	0	0
$682$	0	0
$683$	8.98470e9	1.07902	0.539512	−	0.841978i	$-0.318609\pi$
$683$	8.98470e9	1.07902	0.539512	+	0.841978i	$0.318609\pi$
$684$	0	0
$685$	−3.50039e9	−0.416102
$686$	0	0
$687$	0	0
$688$	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
$693$	0	0
$694$	0	0
$695$	3.98235e9	0.449979
$696$	0	0
$697$	2.30256e7	0.00257570
$698$	0	0
$699$	0	0
$700$	0	0
$701$	3.84334e9	0.421402	0.210701	−	0.977551i	$-0.432425\pi$
$701$	3.84334e9	0.421402	0.210701	+	0.977551i	$0.432425\pi$
$702$	0	0
$703$	−1.76847e10	−1.91979
$704$	0	0
$705$	0	0
$706$	0	0
$707$	1.29218e10	1.37516
$708$	0	0
$709$	1.20064e10	1.26518	0.632588	−	0.774489i	$-0.281993\pi$
$709$	1.20064e10	1.26518	0.632588	+	0.774489i	$0.281993\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	1.69681e10	1.75316
$714$	0	0
$715$	2.05954e9	0.210716
$716$	0	0
$717$	0	0
$718$	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
$723$	0	0
$724$	0	0
$725$	−3.24422e9	−0.316175
$726$	0	0
$727$	1.20128e10	1.15951	0.579754	−	0.814791i	$-0.303149\pi$
$727$	1.20128e10	1.15951	0.579754	+	0.814791i	$0.303149\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	8.33575e8	0.0789285
$732$	0	0
$733$	3.09361e9	0.290136	0.145068	−	0.989422i	$-0.453660\pi$
$733$	3.09361e9	0.290136	0.145068	+	0.989422i	$0.453660\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−1.42381e10	−1.31014
$738$	0	0
$739$	−2.02192e10	−1.84293	−0.921465	−	0.388461i	$-0.873007\pi$
$739$	−2.02192e10	−1.84293	−0.921465	+	0.388461i	$0.873007\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−2.20781e9	−0.197470	−0.0987351	−	0.995114i	$-0.531480\pi$
$743$	−2.20781e9	−0.197470	−0.0987351	+	0.995114i	$0.531480\pi$
$744$	0	0
$745$	−8.33138e9	−0.738194
$746$	0	0
$747$	0	0
$748$	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
$753$	0	0
$754$	0	0
$755$	−2.97035e9	−0.251185
$756$	0	0
$757$	8.01162e9	0.671251	0.335626	−	0.941995i	$-0.391052\pi$
$757$	8.01162e9	0.671251	0.335626	+	0.941995i	$0.391052\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	1.63492e10	1.34478	0.672390	−	0.740197i	$-0.265268\pi$
$761$	1.63492e10	1.34478	0.672390	+	0.740197i	$0.265268\pi$
$762$	0	0
$763$	3.26592e10	2.66176
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−7.09580e9	−0.567829
$768$	0	0
$769$	−8.77857e9	−0.696116	−0.348058	−	0.937473i	$-0.613159\pi$
$769$	−8.77857e9	−0.696116	−0.348058	+	0.937473i	$0.613159\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	8.03519e9	0.625703	0.312851	−	0.949802i	$-0.398716\pi$
$773$	8.03519e9	0.625703	0.312851	+	0.949802i	$0.398716\pi$
$774$	0	0
$775$	−4.47000e9	−0.344946
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−2.33089e8	−0.0176661
$780$	0	0
$781$	−2.50316e9	−0.188022
$782$	0	0
$783$	0	0
$784$	0	0
$785$	1.48988e9	0.109928
$786$	0	0
$787$	−9.89014e9	−0.723254	−0.361627	−	0.932323i	$-0.617779\pi$
$787$	−9.89014e9	−0.723254	−0.361627	+	0.932323i	$0.617779\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−2.68889e10	−1.93177
$792$	0	0
$793$	1.04160e10	0.741730
$794$	0	0
$795$	0	0
$796$	0	0
$797$	2.22521e10	1.55692	0.778460	−	0.627694i	$-0.216001\pi$
$797$	2.22521e10	1.55692	0.778460	+	0.627694i	$0.216001\pi$
$798$	0	0
$799$	−1.56541e9	−0.108571
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−9.07291e9	−0.618361
$804$	0	0
$805$	−9.96744e9	−0.673439
$806$	0	0
$807$	0	0
$808$	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
$813$	0	0
$814$	0	0
$815$	−7.66781e9	−0.496158
$816$	0	0
$817$	−8.43833e9	−0.541351
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−4.39901e9	−0.277430	−0.138715	−	0.990332i	$-0.544297\pi$
$821$	−4.39901e9	−0.277430	−0.138715	+	0.990332i	$0.544297\pi$
$822$	0	0
$823$	2.19932e10	1.37528	0.687638	−	0.726054i	$-0.258648\pi$
$823$	2.19932e10	1.37528	0.687638	+	0.726054i	$0.258648\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	5.85930e9	0.360227	0.180114	−	0.983646i	$-0.442353\pi$
$827$	5.85930e9	0.360227	0.180114	+	0.983646i	$0.442353\pi$
$828$	0	0
$829$	−1.44100e10	−0.878464	−0.439232	−	0.898374i	$-0.644749\pi$
$829$	−1.44100e10	−0.878464	−0.439232	+	0.898374i	$0.644749\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−3.78643e9	−0.226972
$834$	0	0
$835$	−8.05270e9	−0.478673
$836$	0	0
$837$	0	0
$838$	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
$843$	0	0
$844$	0	0
$845$	−6.07623e9	−0.346446
$846$	0	0
$847$	−3.85508e8	−0.0217992
$848$	0	0
$849$	0	0
$850$	0	0
$851$	2.69237e10	1.49755
$852$	0	0
$853$	−5.02066e9	−0.276974	−0.138487	−	0.990364i	$-0.544224\pi$
$853$	−5.02066e9	−0.276974	−0.138487	+	0.990364i	$0.544224\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−3.48488e9	−0.189127	−0.0945637	−	0.995519i	$-0.530146\pi$
$857$	−3.48488e9	−0.189127	−0.0945637	+	0.995519i	$0.530146\pi$
$858$	0	0
$859$	3.27339e10	1.76206	0.881031	−	0.473058i	$-0.156850\pi$
$859$	3.27339e10	1.76206	0.881031	+	0.473058i	$0.156850\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	3.17065e10	1.67923	0.839617	−	0.543179i	$-0.182780\pi$
$863$	3.17065e10	1.67923	0.839617	+	0.543179i	$0.182780\pi$
$864$	0	0
$865$	2.09635e9	0.110131
$866$	0	0
$867$	0	0
$868$	0	0
$869$	8.45661e9	0.437147
$870$	0	0
$871$	−1.22180e10	−0.626524
$872$	0	0
$873$	0	0
$874$	0	0
$875$	2.62578e9	0.132504
$876$	0	0
$877$	3.24939e9	0.162668	0.0813342	−	0.996687i	$-0.474082\pi$
$877$	3.24939e9	0.162668	0.0813342	+	0.996687i	$0.474082\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−2.79940e10	−1.37927	−0.689636	−	0.724157i	$-0.742229\pi$
$881$	−2.79940e10	−1.37927	−0.689636	+	0.724157i	$0.742229\pi$
$882$	0	0
$883$	1.41470e10	0.691515	0.345758	−	0.938324i	$-0.387622\pi$
$883$	1.41470e10	0.691515	0.345758	+	0.938324i	$0.387622\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−3.73596e9	−0.179750	−0.0898751	−	0.995953i	$-0.528647\pi$
$887$	−3.73596e9	−0.179750	−0.0898751	+	0.995953i	$0.528647\pi$
$888$	0	0
$889$	−4.51807e10	−2.15674
$890$	0	0
$891$	0	0
$892$	0	0
$893$	1.58468e10	0.744665
$894$	0	0
$895$	8.44346e9	0.393677
$896$	0	0
$897$	0	0
$898$	0	0
$899$	5.93988e10	2.72659
$900$	0	0
$901$	−5.31471e9	−0.242071
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−8.78017e9	−0.393761
$906$	0	0
$907$	−1.23717e10	−0.550557	−0.275278	−	0.961365i	$-0.588770\pi$
$907$	−1.23717e10	−0.550557	−0.275278	+	0.961365i	$0.588770\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−5.09259e9	−0.223164	−0.111582	−	0.993755i	$-0.535592\pi$
$911$	−5.09259e9	−0.223164	−0.111582	+	0.993755i	$0.535592\pi$
$912$	0	0
$913$	2.99361e10	1.30181
$914$	0	0
$915$	0	0
$916$	0	0
$917$	7.24806e9	0.310405
$918$	0	0
$919$	7.85605e8	0.0333887	0.0166944	−	0.999861i	$-0.494686\pi$
$919$	7.85605e8	0.0333887	0.0166944	+	0.999861i	$0.494686\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−2.14801e9	−0.0899147
$924$	0	0
$925$	−7.09265e9	−0.294654
$926$	0	0
$927$	0	0
$928$	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
$933$	0	0
$934$	0	0
$935$	2.10796e9	0.0843376
$936$	0	0
$937$	−1.27672e10	−0.506998	−0.253499	−	0.967336i	$-0.581582\pi$
$937$	−1.27672e10	−0.506998	−0.253499	+	0.967336i	$0.581582\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	2.35922e10	0.923008	0.461504	−	0.887138i	$-0.347310\pi$
$941$	2.35922e10	0.923008	0.461504	+	0.887138i	$0.347310\pi$
$942$	0	0
$943$	3.54862e8	0.0137806
$944$	0	0
$945$	0	0
$946$	0	0
$947$	8.88895e9	0.340115	0.170057	−	0.985434i	$-0.445605\pi$
$947$	8.88895e9	0.340115	0.170057	+	0.985434i	$0.445605\pi$
$948$	0	0
$949$	−7.78566e9	−0.295708
$950$	0	0
$951$	0	0
$952$	0	0
$953$	1.08612e10	0.406491	0.203245	−	0.979128i	$-0.434851\pi$
$953$	1.08612e10	0.406491	0.203245	+	0.979128i	$0.434851\pi$
$954$	0	0
$955$	5.04688e9	0.187504
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−3.76473e10	−1.37838
$960$	0	0
$961$	5.43292e10	1.97470
$962$	0	0
$963$	0	0
$964$	0	0
$965$	4.36092e9	0.156218
$966$	0	0
$967$	−2.24979e10	−0.800109	−0.400054	−	0.916491i	$-0.631009\pi$
$967$	−2.24979e10	−0.800109	−0.400054	+	0.916491i	$0.631009\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	4.57613e10	1.60410	0.802049	−	0.597258i	$-0.203743\pi$
$971$	4.57613e10	1.60410	0.802049	+	0.597258i	$0.203743\pi$
$972$	0	0
$973$	4.28308e10	1.49060
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−2.40739e10	−0.825879	−0.412939	−	0.910759i	$-0.635498\pi$
$977$	−2.40739e10	−0.825879	−0.412939	+	0.910759i	$0.635498\pi$
$978$	0	0
$979$	−1.55819e10	−0.530740
$980$	0	0
$981$	0	0
$982$	0	0
$983$	1.95466e10	0.656346	0.328173	−	0.944618i	$-0.393567\pi$
$983$	1.95466e10	0.656346	0.328173	+	0.944618i	$0.393567\pi$
$984$	0	0
$985$	1.61751e10	0.539287
$986$	0	0
$987$	0	0
$988$	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
$993$	0	0
$994$	0	0
$995$	−1.50564e10	−0.484553
$996$	0	0
$997$	−5.82401e9	−0.186118	−0.0930591	−	0.995661i	$-0.529665\pi$
$997$	−5.82401e9	−0.186118	−0.0930591	+	0.995661i	$0.529665\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	360.8.a.l.1.2		2
3.2	odd	2		40.8.a.b.1.2	✓	2
12.11	even	2		80.8.a.h.1.1		2
15.2	even	4		200.8.c.h.49.1		4
15.8	even	4		200.8.c.h.49.4		4
15.14	odd	2		200.8.a.m.1.1		2
24.5	odd	2		320.8.a.o.1.1		2
24.11	even	2		320.8.a.q.1.2		2
60.23	odd	4		400.8.c.q.49.1		4
60.47	odd	4		400.8.c.q.49.4		4
60.59	even	2		400.8.a.z.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
40.8.a.b.1.2	✓	2	3.2	odd	2
80.8.a.h.1.1		2	12.11	even	2
200.8.a.m.1.1		2	15.14	odd	2
200.8.c.h.49.1		4	15.2	even	4
200.8.c.h.49.4		4	15.8	even	4
320.8.a.o.1.1		2	24.5	odd	2
320.8.a.q.1.2		2	24.11	even	2
360.8.a.l.1.2		2	1.1	even	1	trivial
400.8.a.z.1.2		2	60.59	even	2
400.8.c.q.49.1		4	60.23	odd	4
400.8.c.q.49.4		4	60.47	odd	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 360 = 2^{3} \cdot 3^{2} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	360.a (trivial)

\(f(q)\)	\(=\)	\(q+125.000 q^{5} +1344.40 q^{7} -4381.83 q^{11} -3760.14 q^{13} -3848.55 q^{17} +38959.0 q^{19} -59312.5 q^{23} +15625.0 q^{25} -207630. q^{29} -286080. q^{31} +168050. q^{35} -453930. q^{37} -5982.93 q^{41} -216595. q^{43} +406755. q^{47} +983860. q^{49} +1.38096e6 q^{53} -547729. q^{55} +1.88711e6 q^{59} -2.77012e6 q^{61} -470018. q^{65} +3.24935e6 q^{67} +571258. q^{71} +2.07058e6 q^{73} -5.89092e6 q^{77} -1.92993e6 q^{79} -6.83187e6 q^{83} -481068. q^{85} +3.55603e6 q^{89} -5.05512e6 q^{91} +4.86988e6 q^{95} -1.26173e7 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 250 q^{5} + 844 q^{7} - 4640 q^{11} + 6804 q^{13} - 38516 q^{17} + 39720 q^{19} - 46244 q^{23} + 31250 q^{25} - 186940 q^{29} - 327128 q^{31} + 105500 q^{35} + 67060 q^{37} + 162964 q^{41} - 1123468 q^{43}+ \cdots - 9119804 q^{97}+O(q^{100}) \) 2 * q + 250 * q^5 + 844 * q^7 - 4640 * q^11 + 6804 * q^13 - 38516 * q^17 + 39720 * q^19 - 46244 * q^23 + 31250 * q^25 - 186940 * q^29 - 327128 * q^31 + 105500 * q^35 + 67060 * q^37 + 162964 * q^41 - 1123468 * q^43 + 777156 * q^47 + 410714 * q^49 - 144164 * q^53 - 580000 * q^55 - 1120776 * q^59 - 1855852 * q^61 + 850500 * q^65 + 6239676 * q^67 + 1071112 * q^71 + 3467044 * q^73 - 5761728 * q^77 + 4340144 * q^79 - 10960692 * q^83 - 4814500 * q^85 - 42260 * q^89 - 10341384 * q^91 + 4965000 * q^95 - 9119804 * q^97

Introduction

L-functions

Modular forms

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Database

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