LMFDB - Embedded newform 1008.6.a.bg.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1008,6,Mod(1,1008)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1008, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))

N = Newforms(chi, 6, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1008.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 1008 = 2^{4} \cdot 3^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	1008.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	yes
Analytic conductor:	$161.666890371$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{106}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 106 $ x^2 - 106
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2\cdot 3 $
Twist minimal:	no (minimal twist has level 504)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-10.2956$ of defining polynomial
Character	$\chi$	$=$	1008.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-99.7738 q^{5} -49.0000 q^{7} +O(q^{10})$	$q-99.7738 q^{5} -49.0000 q^{7} -26.8689 q^{11} +10.9049 q^{13} +593.964 q^{17} -1427.10 q^{19} +1514.73 q^{23} +6829.81 q^{25} -2924.52 q^{29} -2049.00 q^{31} +4888.92 q^{35} +2476.43 q^{37} +20188.3 q^{41} -9587.24 q^{43} +20069.8 q^{47} +2401.00 q^{49} +6554.12 q^{53} +2680.81 q^{55} +18212.8 q^{59} -35155.6 q^{61} -1088.02 q^{65} +28813.0 q^{67} +17116.0 q^{71} +21872.2 q^{73} +1316.58 q^{77} +87602.0 q^{79} +71085.9 q^{83} -59262.0 q^{85} -3299.03 q^{89} -534.339 q^{91} +142387. q^{95} -117193. q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 76 q^{5} - 98 q^{7}+O(q^{10}) $ 2 * q - 76 * q^5 - 98 * q^7	$ 2 q - 76 q^{5} - 98 q^{7} + 564 q^{11} + 516 q^{13} + 76 q^{17} - 2360 q^{19} - 2036 q^{23} + 4270 q^{25} - 8320 q^{29} + 6280 q^{31} + 3724 q^{35} - 2460 q^{37} + 14308 q^{41} - 23128 q^{43} + 12712 q^{47} + 4802 q^{49} + 9896 q^{53} + 16728 q^{55} + 60888 q^{59} + 16172 q^{61} + 10920 q^{65} + 62568 q^{67} - 732 q^{71} + 1244 q^{73} - 27636 q^{77} - 11600 q^{79} + 132288 q^{83} - 71576 q^{85} - 93452 q^{89} - 25284 q^{91} + 120208 q^{95} - 272932 q^{97}+O(q^{100}) $ 2 * q - 76 * q^5 - 98 * q^7 + 564 * q^11 + 516 * q^13 + 76 * q^17 - 2360 * q^19 - 2036 * q^23 + 4270 * q^25 - 8320 * q^29 + 6280 * q^31 + 3724 * q^35 - 2460 * q^37 + 14308 * q^41 - 23128 * q^43 + 12712 * q^47 + 4802 * q^49 + 9896 * q^53 + 16728 * q^55 + 60888 * q^59 + 16172 * q^61 + 10920 * q^65 + 62568 * q^67 - 732 * q^71 + 1244 * q^73 - 27636 * q^77 - 11600 * q^79 + 132288 * q^83 - 71576 * q^85 - 93452 * q^89 - 25284 * q^91 + 120208 * q^95 - 272932 * q^97

Display 100 coefficients 10 coefficients

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
$5$	−99.7738	−1.78481	−0.892404	−	0.451238i	$-0.850983\pi$
$5$	−99.7738	−1.78481	−0.892404	+	0.451238i	$0.850983\pi$
$6$	0	0
$7$	−49.0000	−0.377964
$7$	−49.0000	−0.377964
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−26.8689	−0.0669527	−0.0334764	−	0.999440i	$-0.510658\pi$
$11$	−26.8689	−0.0669527	−0.0334764	+	0.999440i	$0.510658\pi$
$12$	0	0
$13$	10.9049	0.0178963	0.00894813	−	0.999960i	$-0.497152\pi$
$13$	10.9049	0.0178963	0.00894813	+	0.999960i	$0.497152\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	593.964	0.498469	0.249234	−	0.968443i	$-0.419821\pi$
$17$	593.964	0.498469	0.249234	+	0.968443i	$0.419821\pi$
$18$	0	0
$19$	−1427.10	−0.906920	−0.453460	−	0.891277i	$-0.649810\pi$
$19$	−1427.10	−0.906920	−0.453460	+	0.891277i	$0.649810\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	1514.73	0.597055	0.298527	−	0.954401i	$-0.403505\pi$
$23$	1514.73	0.597055	0.298527	+	0.954401i	$0.403505\pi$
$24$	0	0
$25$	6829.81	2.18554
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−2924.52	−0.645744	−0.322872	−	0.946443i	$-0.604648\pi$
$29$	−2924.52	−0.645744	−0.322872	+	0.946443i	$0.604648\pi$
$30$	0	0
$31$	−2049.00	−0.382946	−0.191473	−	0.981498i	$-0.561326\pi$
$31$	−2049.00	−0.382946	−0.191473	+	0.981498i	$0.561326\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	4888.92	0.674594
$36$	0	0
$37$	2476.43	0.297386	0.148693	−	0.988883i	$-0.452493\pi$
$37$	2476.43	0.297386	0.148693	+	0.988883i	$0.452493\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	20188.3	1.87560	0.937798	−	0.347181i	$-0.112861\pi$
$41$	20188.3	1.87560	0.937798	+	0.347181i	$0.112861\pi$
$42$	0	0
$43$	−9587.24	−0.790719	−0.395360	−	0.918526i	$-0.629380\pi$
$43$	−9587.24	−0.790719	−0.395360	+	0.918526i	$0.629380\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	20069.8	1.32525	0.662625	−	0.748951i	$-0.269442\pi$
$47$	20069.8	1.32525	0.662625	+	0.748951i	$0.269442\pi$
$48$	0	0
$49$	2401.00	0.142857
$50$	0	0
$51$	0	0
$52$	0	0
$53$	6554.12	0.320497	0.160249	−	0.987077i	$-0.448770\pi$
$53$	6554.12	0.320497	0.160249	+	0.987077i	$0.448770\pi$
$54$	0	0
$55$	2680.81	0.119498
$56$	0	0
$57$	0	0
$58$	0	0
$59$	18212.8	0.681156	0.340578	−	0.940216i	$-0.389377\pi$
$59$	18212.8	0.681156	0.340578	+	0.940216i	$0.389377\pi$
$60$	0	0
$61$	−35155.6	−1.20968	−0.604840	−	0.796347i	$-0.706763\pi$
$61$	−35155.6	−1.20968	−0.604840	+	0.796347i	$0.706763\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−1088.02	−0.0319414
$66$	0	0
$67$	28813.0	0.784156	0.392078	−	0.919932i	$-0.371756\pi$
$67$	28813.0	0.784156	0.392078	+	0.919932i	$0.371756\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	17116.0	0.402954	0.201477	−	0.979493i	$-0.435426\pi$
$71$	17116.0	0.402954	0.201477	+	0.979493i	$0.435426\pi$
$72$	0	0
$73$	21872.2	0.480380	0.240190	−	0.970726i	$-0.422790\pi$
$73$	21872.2	0.480380	0.240190	+	0.970726i	$0.422790\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	1316.58	0.0253058
$78$	0	0
$79$	87602.0	1.57923	0.789616	−	0.613601i	$-0.210280\pi$
$79$	87602.0	1.57923	0.789616	+	0.613601i	$0.210280\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	71085.9	1.13263	0.566315	−	0.824189i	$-0.308368\pi$
$83$	71085.9	1.13263	0.566315	+	0.824189i	$0.308368\pi$
$84$	0	0
$85$	−59262.0	−0.889671
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−3299.03	−0.0441480	−0.0220740	−	0.999756i	$-0.507027\pi$
$89$	−3299.03	−0.0441480	−0.0220740	+	0.999756i	$0.507027\pi$
$90$	0	0
$91$	−534.339	−0.00676415
$92$	0	0
$93$	0	0
$94$	0	0
$95$	142387.	1.61868
$96$	0	0
$97$	−117193.	−1.26465	−0.632326	−	0.774703i	$-0.717900\pi$
$97$	−117193.	−1.26465	−0.632326	+	0.774703i	$0.717900\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−87975.0	−0.858135	−0.429068	−	0.903272i	$-0.641158\pi$
$101$	−87975.0	−0.858135	−0.429068	+	0.903272i	$0.641158\pi$
$102$	0	0
$103$	−185294.	−1.72095	−0.860474	−	0.509494i	$-0.829833\pi$
$103$	−185294.	−1.72095	−0.860474	+	0.509494i	$0.829833\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−15850.6	−0.133840	−0.0669200	−	0.997758i	$-0.521317\pi$
$107$	−15850.6	−0.133840	−0.0669200	+	0.997758i	$0.521317\pi$
$108$	0	0
$109$	154617.	1.24649	0.623247	−	0.782025i	$-0.285813\pi$
$109$	154617.	1.24649	0.623247	+	0.782025i	$0.285813\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−43252.8	−0.318653	−0.159327	−	0.987226i	$-0.550932\pi$
$113$	−43252.8	−0.318653	−0.159327	+	0.987226i	$0.550932\pi$
$114$	0	0
$115$	−151130.	−1.06563
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−29104.2	−0.188403
$120$	0	0
$121$	−160329.	−0.995517
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−369643.	−2.11596
$126$	0	0
$127$	−68236.6	−0.375412	−0.187706	−	0.982225i	$-0.560105\pi$
$127$	−68236.6	−0.375412	−0.187706	+	0.982225i	$0.560105\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	59398.9	0.302413	0.151206	−	0.988502i	$-0.451684\pi$
$131$	59398.9	0.302413	0.151206	+	0.988502i	$0.451684\pi$
$132$	0	0
$133$	69927.7	0.342783
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−32603.1	−0.148408	−0.0742040	−	0.997243i	$-0.523642\pi$
$137$	−32603.1	−0.148408	−0.0742040	+	0.997243i	$0.523642\pi$
$138$	0	0
$139$	−126248.	−0.554225	−0.277113	−	0.960837i	$-0.589378\pi$
$139$	−126248.	−0.554225	−0.277113	+	0.960837i	$0.589378\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−293.002	−0.00119820
$144$	0	0
$145$	291791.	1.15253
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−522092.	−1.92655	−0.963277	−	0.268509i	$-0.913469\pi$
$149$	−522092.	−1.92655	−0.963277	+	0.268509i	$0.913469\pi$
$150$	0	0
$151$	−144406.	−0.515397	−0.257698	−	0.966225i	$-0.582964\pi$
$151$	−144406.	−0.515397	−0.257698	+	0.966225i	$0.582964\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	204436.	0.683484
$156$	0	0
$157$	131612.	0.426133	0.213067	−	0.977038i	$-0.431655\pi$
$157$	131612.	0.426133	0.213067	+	0.977038i	$0.431655\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−74221.5	−0.225666
$162$	0	0
$163$	464116.	1.36823	0.684113	−	0.729376i	$-0.260189\pi$
$163$	464116.	1.36823	0.684113	+	0.729376i	$0.260189\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−321971.	−0.893358	−0.446679	−	0.894694i	$-0.647393\pi$
$167$	−321971.	−0.893358	−0.446679	+	0.894694i	$0.647393\pi$
$168$	0	0
$169$	−371174.	−0.999680
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−296228.	−0.752508	−0.376254	−	0.926517i	$-0.622788\pi$
$173$	−296228.	−0.752508	−0.376254	+	0.926517i	$0.622788\pi$
$174$	0	0
$175$	−334661.	−0.826056
$176$	0	0
$177$	0	0
$178$	0	0
$179$	114025.	0.265991	0.132995	−	0.991117i	$-0.457540\pi$
$179$	114025.	0.265991	0.132995	+	0.991117i	$0.457540\pi$
$180$	0	0
$181$	−498098.	−1.13010	−0.565051	−	0.825056i	$-0.691144\pi$
$181$	−498098.	−1.13010	−0.565051	+	0.825056i	$0.691144\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−247082.	−0.530778
$186$	0	0
$187$	−15959.2	−0.0333738
$188$	0	0
$189$	0	0
$190$	0	0
$191$	525214.	1.04173	0.520863	−	0.853640i	$-0.325610\pi$
$191$	525214.	1.04173	0.520863	+	0.853640i	$0.325610\pi$
$192$	0	0
$193$	−230768.	−0.445946	−0.222973	−	0.974825i	$-0.571576\pi$
$193$	−230768.	−0.445946	−0.222973	+	0.974825i	$0.571576\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−896175.	−1.64523	−0.822617	−	0.568596i	$-0.807487\pi$
$197$	−896175.	−1.64523	−0.822617	+	0.568596i	$0.807487\pi$
$198$	0	0
$199$	−904772.	−1.61959	−0.809797	−	0.586710i	$-0.800423\pi$
$199$	−904772.	−1.61959	−0.809797	+	0.586710i	$0.800423\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	143302.	0.244068
$204$	0	0
$205$	−2.01426e6	−3.34758
$206$	0	0
$207$	0	0
$208$	0	0
$209$	38344.5	0.0607208
$210$	0	0
$211$	652886.	1.00956	0.504779	−	0.863249i	$-0.331574\pi$
$211$	652886.	1.00956	0.504779	+	0.863249i	$0.331574\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	956555.	1.41128
$216$	0	0
$217$	100401.	0.144740
$218$	0	0
$219$	0	0
$220$	0	0
$221$	6477.10	0.00892073
$222$	0	0
$223$	1.35439e6	1.82382	0.911910	−	0.410391i	$-0.134608\pi$
$223$	1.35439e6	1.82382	0.911910	+	0.410391i	$0.134608\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−51188.1	−0.0659333	−0.0329666	−	0.999456i	$-0.510496\pi$
$227$	−51188.1	−0.0659333	−0.0329666	+	0.999456i	$0.510496\pi$
$228$	0	0
$229$	727407.	0.916619	0.458309	−	0.888793i	$-0.348455\pi$
$229$	727407.	0.916619	0.458309	+	0.888793i	$0.348455\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	1.44907e6	1.74863	0.874317	−	0.485355i	$-0.161310\pi$
$233$	1.44907e6	1.74863	0.874317	+	0.485355i	$0.161310\pi$
$234$	0	0
$235$	−2.00244e6	−2.36532
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−522404.	−0.591578	−0.295789	−	0.955253i	$-0.595583\pi$
$239$	−522404.	−0.591578	−0.295789	+	0.955253i	$0.595583\pi$
$240$	0	0
$241$	437082.	0.484753	0.242377	−	0.970182i	$-0.422073\pi$
$241$	437082.	0.484753	0.242377	+	0.970182i	$0.422073\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−239557.	−0.254973
$246$	0	0
$247$	−15562.3	−0.0162305
$248$	0	0
$249$	0	0
$250$	0	0
$251$	528351.	0.529344	0.264672	−	0.964338i	$-0.414736\pi$
$251$	528351.	0.529344	0.264672	+	0.964338i	$0.414736\pi$
$252$	0	0
$253$	−40699.0	−0.0399744
$254$	0	0
$255$	0	0
$256$	0	0
$257$	1.79360e6	1.69392	0.846962	−	0.531653i	$-0.178429\pi$
$257$	1.79360e6	1.69392	0.846962	+	0.531653i	$0.178429\pi$
$258$	0	0
$259$	−121345.	−0.112402
$260$	0	0
$261$	0	0
$262$	0	0
$263$	2.22568e6	1.98415	0.992074	−	0.125654i	$-0.0401028\pi$
$263$	2.22568e6	1.98415	0.992074	+	0.125654i	$0.0401028\pi$
$264$	0	0
$265$	−653929.	−0.572026
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−520438.	−0.438519	−0.219259	−	0.975667i	$-0.570364\pi$
$269$	−520438.	−0.438519	−0.219259	+	0.975667i	$0.570364\pi$
$270$	0	0
$271$	−2.15215e6	−1.78012	−0.890061	−	0.455841i	$-0.849339\pi$
$271$	−2.15215e6	−1.78012	−0.890061	+	0.455841i	$0.849339\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−183509.	−0.146328
$276$	0	0
$277$	−656005.	−0.513698	−0.256849	−	0.966452i	$-0.582684\pi$
$277$	−656005.	−0.513698	−0.256849	+	0.966452i	$0.582684\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	1.05359e6	0.795985	0.397992	−	0.917389i	$-0.369707\pi$
$281$	1.05359e6	0.795985	0.397992	+	0.917389i	$0.369707\pi$
$282$	0	0
$283$	−1.98071e6	−1.47013	−0.735064	−	0.677997i	$-0.762848\pi$
$283$	−1.98071e6	−1.47013	−0.735064	+	0.677997i	$0.762848\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−989225.	−0.708909
$288$	0	0
$289$	−1.06706e6	−0.751529
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−1.66438e6	−1.13262	−0.566310	−	0.824192i	$-0.691630\pi$
$293$	−1.66438e6	−1.13262	−0.566310	+	0.824192i	$0.691630\pi$
$294$	0	0
$295$	−1.81716e6	−1.21573
$296$	0	0
$297$	0	0
$298$	0	0
$299$	16517.9	0.0106851
$300$	0	0
$301$	469775.	0.298864
$302$	0	0
$303$	0	0
$304$	0	0
$305$	3.50761e6	2.15905
$306$	0	0
$307$	−2.45524e6	−1.48678	−0.743392	−	0.668856i	$-0.766784\pi$
$307$	−2.45524e6	−1.48678	−0.743392	+	0.668856i	$0.766784\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−1.91960e6	−1.12541	−0.562704	−	0.826658i	$-0.690239\pi$
$311$	−1.91960e6	−1.12541	−0.562704	+	0.826658i	$0.690239\pi$
$312$	0	0
$313$	−1.62377e6	−0.936836	−0.468418	−	0.883507i	$-0.655176\pi$
$313$	−1.62377e6	−0.936836	−0.468418	+	0.883507i	$0.655176\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	1.22066e6	0.682255	0.341128	−	0.940017i	$-0.389191\pi$
$317$	1.22066e6	0.682255	0.341128	+	0.940017i	$0.389191\pi$
$318$	0	0
$319$	78578.8	0.0432343
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−847643.	−0.452071
$324$	0	0
$325$	74478.2	0.0391130
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−983419.	−0.500898
$330$	0	0
$331$	2.83639e6	1.42297	0.711487	−	0.702700i	$-0.248022\pi$
$331$	2.83639e6	1.42297	0.711487	+	0.702700i	$0.248022\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−2.87479e6	−1.39957
$336$	0	0
$337$	−1.42332e6	−0.682697	−0.341349	−	0.939937i	$-0.610884\pi$
$337$	−1.42332e6	−0.682697	−0.341349	+	0.939937i	$0.610884\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	55054.3	0.0256393
$342$	0	0
$343$	−117649.	−0.0539949
$344$	0	0
$345$	0	0
$346$	0	0
$347$	2.09903e6	0.935825	0.467912	−	0.883775i	$-0.345006\pi$
$347$	2.09903e6	0.935825	0.467912	+	0.883775i	$0.345006\pi$
$348$	0	0
$349$	−2.82234e6	−1.24035	−0.620177	−	0.784462i	$-0.712939\pi$
$349$	−2.82234e6	−1.24035	−0.620177	+	0.784462i	$0.712939\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	1.24004e6	0.529663	0.264831	−	0.964295i	$-0.414684\pi$
$353$	1.24004e6	0.529663	0.264831	+	0.964295i	$0.414684\pi$
$354$	0	0
$355$	−1.70773e6	−0.719196
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−234371.	−0.0959770	−0.0479885	−	0.998848i	$-0.515281\pi$
$359$	−234371.	−0.0959770	−0.0479885	+	0.998848i	$0.515281\pi$
$360$	0	0
$361$	−439499.	−0.177496
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−2.18227e6	−0.857386
$366$	0	0
$367$	2.45074e6	0.949800	0.474900	−	0.880040i	$-0.342484\pi$
$367$	2.45074e6	0.949800	0.474900	+	0.880040i	$0.342484\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−321152.	−0.121137
$372$	0	0
$373$	4924.89	0.00183284	0.000916420	−	1.00000i	$-0.499708\pi$
$373$	4924.89	0.00183284	0.000916420		1.00000i	$0.499708\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−31891.6	−0.0115564
$378$	0	0
$379$	540900.	0.193428	0.0967140	−	0.995312i	$-0.469167\pi$
$379$	540900.	0.193428	0.0967140	+	0.995312i	$0.469167\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−3.04844e6	−1.06189	−0.530947	−	0.847405i	$-0.678164\pi$
$383$	−3.04844e6	−1.06189	−0.530947	+	0.847405i	$0.678164\pi$
$384$	0	0
$385$	−131360.	−0.0451659
$386$	0	0
$387$	0	0
$388$	0	0
$389$	4.46106e6	1.49473	0.747366	−	0.664412i	$-0.231318\pi$
$389$	4.46106e6	1.49473	0.747366	+	0.664412i	$0.231318\pi$
$390$	0	0
$391$	899692.	0.297613
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−8.74038e6	−2.81863
$396$	0	0
$397$	−3.74992e6	−1.19411	−0.597057	−	0.802199i	$-0.703663\pi$
$397$	−3.74992e6	−1.19411	−0.597057	+	0.802199i	$0.703663\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−1.73486e6	−0.538770	−0.269385	−	0.963033i	$-0.586820\pi$
$401$	−1.73486e6	−0.538770	−0.269385	+	0.963033i	$0.586820\pi$
$402$	0	0
$403$	−22344.1	−0.00685330
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−66538.9	−0.0199108
$408$	0	0
$409$	−3.38651e6	−1.00102	−0.500512	−	0.865730i	$-0.666855\pi$
$409$	−3.38651e6	−1.00102	−0.500512	+	0.865730i	$0.666855\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−892427.	−0.257453
$414$	0	0
$415$	−7.09251e6	−2.02153
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−5.92994e6	−1.65012	−0.825060	−	0.565045i	$-0.808859\pi$
$419$	−5.92994e6	−1.65012	−0.825060	+	0.565045i	$0.808859\pi$
$420$	0	0
$421$	−3.61340e6	−0.993598	−0.496799	−	0.867866i	$-0.665491\pi$
$421$	−3.61340e6	−0.993598	−0.496799	+	0.867866i	$0.665491\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	4.05666e6	1.08942
$426$	0	0
$427$	1.72263e6	0.457216
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−46574.4	−0.0120769	−0.00603843	−	0.999982i	$-0.501922\pi$
$431$	−46574.4	−0.0120769	−0.00603843	+	0.999982i	$0.501922\pi$
$432$	0	0
$433$	3.55113e6	0.910222	0.455111	−	0.890435i	$-0.349600\pi$
$433$	3.55113e6	0.910222	0.455111	+	0.890435i	$0.349600\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−2.16166e6	−0.541481
$438$	0	0
$439$	991127.	0.245453	0.122726	−	0.992441i	$-0.460836\pi$
$439$	991127.	0.245453	0.122726	+	0.992441i	$0.460836\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−1.23482e6	−0.298948	−0.149474	−	0.988766i	$-0.547758\pi$
$443$	−1.23482e6	−0.298948	−0.149474	+	0.988766i	$0.547758\pi$
$444$	0	0
$445$	329157.	0.0787958
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−3.84247e6	−0.899485	−0.449743	−	0.893158i	$-0.648484\pi$
$449$	−3.84247e6	−0.899485	−0.449743	+	0.893158i	$0.648484\pi$
$450$	0	0
$451$	−542437.	−0.125576
$452$	0	0
$453$	0	0
$454$	0	0
$455$	53313.0	0.0120727
$456$	0	0
$457$	8.81800e6	1.97506	0.987528	−	0.157443i	$-0.0503250\pi$
$457$	8.81800e6	1.97506	0.987528	+	0.157443i	$0.0503250\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−2.76947e6	−0.606938	−0.303469	−	0.952841i	$-0.598145\pi$
$461$	−2.76947e6	−0.606938	−0.303469	+	0.952841i	$0.598145\pi$
$462$	0	0
$463$	−3.54086e6	−0.767638	−0.383819	−	0.923408i	$-0.625391\pi$
$463$	−3.54086e6	−0.767638	−0.383819	+	0.923408i	$0.625391\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	6.07387e6	1.28876	0.644382	−	0.764704i	$-0.277115\pi$
$467$	6.07387e6	1.28876	0.644382	+	0.764704i	$0.277115\pi$
$468$	0	0
$469$	−1.41184e6	−0.296383
$470$	0	0
$471$	0	0
$472$	0	0
$473$	257599.	0.0529408
$474$	0	0
$475$	−9.74678e6	−1.98211
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−3.31185e6	−0.659526	−0.329763	−	0.944064i	$-0.606969\pi$
$479$	−3.31185e6	−0.659526	−0.329763	+	0.944064i	$0.606969\pi$
$480$	0	0
$481$	27005.1	0.00532211
$482$	0	0
$483$	0	0
$484$	0	0
$485$	1.16927e7	2.25716
$486$	0	0
$487$	5.06263e6	0.967284	0.483642	−	0.875266i	$-0.339314\pi$
$487$	5.06263e6	0.967284	0.483642	+	0.875266i	$0.339314\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−6.68987e6	−1.25232	−0.626158	−	0.779696i	$-0.715373\pi$
$491$	−6.68987e6	−1.25232	−0.626158	+	0.779696i	$0.715373\pi$
$492$	0	0
$493$	−1.73706e6	−0.321883
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−838683.	−0.152302
$498$	0	0
$499$	−7.59272e6	−1.36504	−0.682521	−	0.730866i	$-0.739116\pi$
$499$	−7.59272e6	−1.36504	−0.682521	+	0.730866i	$0.739116\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	3.61812e6	0.637621	0.318811	−	0.947818i	$-0.396717\pi$
$503$	3.61812e6	0.637621	0.318811	+	0.947818i	$0.396717\pi$
$504$	0	0
$505$	8.77759e6	1.53161
$506$	0	0
$507$	0	0
$508$	0	0
$509$	6.94370e6	1.18795	0.593973	−	0.804485i	$-0.297559\pi$
$509$	6.94370e6	1.18795	0.593973	+	0.804485i	$0.297559\pi$
$510$	0	0
$511$	−1.07174e6	−0.181567
$512$	0	0
$513$	0	0
$514$	0	0
$515$	1.84875e7	3.07156
$516$	0	0
$517$	−539253.	−0.0887291
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−1.41143e6	−0.227807	−0.113903	−	0.993492i	$-0.536335\pi$
$521$	−1.41143e6	−0.227807	−0.113903	+	0.993492i	$0.536335\pi$
$522$	0	0
$523$	1.13103e7	1.80808	0.904042	−	0.427443i	$-0.140586\pi$
$523$	1.13103e7	1.80808	0.904042	+	0.427443i	$0.140586\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−1.21703e6	−0.190886
$528$	0	0
$529$	−4.14195e6	−0.643526
$530$	0	0
$531$	0	0
$532$	0	0
$533$	220151.	0.0335662
$534$	0	0
$535$	1.58147e6	0.238879
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−64512.2	−0.00956467
$540$	0	0
$541$	−190192.	−0.0279382	−0.0139691	−	0.999902i	$-0.504447\pi$
$541$	−190192.	−0.0279382	−0.0139691	+	0.999902i	$0.504447\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−1.54267e7	−2.22475
$546$	0	0
$547$	−1.06805e6	−0.152625	−0.0763123	−	0.997084i	$-0.524315\pi$
$547$	−1.06805e6	−0.152625	−0.0763123	+	0.997084i	$0.524315\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	4.17357e6	0.585638
$552$	0	0
$553$	−4.29250e6	−0.596894
$554$	0	0
$555$	0	0
$556$	0	0
$557$	8.25567e6	1.12749	0.563747	−	0.825947i	$-0.309359\pi$
$557$	8.25567e6	1.12749	0.563747	+	0.825947i	$0.309359\pi$
$558$	0	0
$559$	−104548.	−0.0141509
$560$	0	0
$561$	0	0
$562$	0	0
$563$	436297.	0.0580111	0.0290055	−	0.999579i	$-0.490766\pi$
$563$	436297.	0.0580111	0.0290055	+	0.999579i	$0.490766\pi$
$564$	0	0
$565$	4.31550e6	0.568735
$566$	0	0
$567$	0	0
$568$	0	0
$569$	1.17645e6	0.152332	0.0761660	−	0.997095i	$-0.475732\pi$
$569$	1.17645e6	0.152332	0.0761660	+	0.997095i	$0.475732\pi$
$570$	0	0
$571$	398202.	0.0511109	0.0255554	−	0.999673i	$-0.491865\pi$
$571$	398202.	0.0511109	0.0255554	+	0.999673i	$0.491865\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	1.03453e7	1.30489
$576$	0	0
$577$	−1.36513e7	−1.70700	−0.853502	−	0.521089i	$-0.825526\pi$
$577$	−1.36513e7	−1.70700	−0.853502	+	0.521089i	$0.825526\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−3.48321e6	−0.428094
$582$	0	0
$583$	−176102.	−0.0214582
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−423932.	−0.0507810	−0.0253905	−	0.999678i	$-0.508083\pi$
$587$	−423932.	−0.0507810	−0.0253905	+	0.999678i	$0.508083\pi$
$588$	0	0
$589$	2.92411e6	0.347301
$590$	0	0
$591$	0	0
$592$	0	0
$593$	1.05447e6	0.123140	0.0615698	−	0.998103i	$-0.480389\pi$
$593$	1.05447e6	0.123140	0.0615698	+	0.998103i	$0.480389\pi$
$594$	0	0
$595$	2.90384e6	0.336264
$596$	0	0
$597$	0	0
$598$	0	0
$599$	1.26516e7	1.44072	0.720359	−	0.693601i	$-0.243977\pi$
$599$	1.26516e7	1.44072	0.720359	+	0.693601i	$0.243977\pi$
$600$	0	0
$601$	4.90694e6	0.554147	0.277073	−	0.960849i	$-0.410635\pi$
$601$	4.90694e6	0.554147	0.277073	+	0.960849i	$0.410635\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	1.59966e7	1.77681
$606$	0	0
$607$	1.25925e7	1.38720	0.693600	−	0.720360i	$-0.256023\pi$
$607$	1.25925e7	1.38720	0.693600	+	0.720360i	$0.256023\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	218858.	0.0237170
$612$	0	0
$613$	−8.84587e6	−0.950801	−0.475400	−	0.879770i	$-0.657697\pi$
$613$	−8.84587e6	−0.950801	−0.475400	+	0.879770i	$0.657697\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−1.69655e7	−1.79413	−0.897066	−	0.441896i	$-0.854306\pi$
$617$	−1.69655e7	−1.79413	−0.897066	+	0.441896i	$0.854306\pi$
$618$	0	0
$619$	1.15237e7	1.20884	0.604418	−	0.796668i	$-0.293406\pi$
$619$	1.15237e7	1.20884	0.604418	+	0.796668i	$0.293406\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	161653.	0.0166864
$624$	0	0
$625$	1.55375e7	1.59104
$626$	0	0
$627$	0	0
$628$	0	0
$629$	1.47091e6	0.148238
$630$	0	0
$631$	1.10729e7	1.10711	0.553553	−	0.832814i	$-0.313272\pi$
$631$	1.10729e7	1.10711	0.553553	+	0.832814i	$0.313272\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	6.80822e6	0.670038
$636$	0	0
$637$	26182.6	0.00255661
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−6.37605e6	−0.612924	−0.306462	−	0.951883i	$-0.599145\pi$
$641$	−6.37605e6	−0.612924	−0.306462	+	0.951883i	$0.599145\pi$
$642$	0	0
$643$	9.80768e6	0.935489	0.467745	−	0.883864i	$-0.345067\pi$
$643$	9.80768e6	0.935489	0.467745	+	0.883864i	$0.345067\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−1.05994e7	−0.995455	−0.497728	−	0.867333i	$-0.665832\pi$
$647$	−1.05994e7	−0.995455	−0.497728	+	0.867333i	$0.665832\pi$
$648$	0	0
$649$	−489358.	−0.0456052
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−1.16241e7	−1.06679	−0.533393	−	0.845868i	$-0.679083\pi$
$653$	−1.16241e7	−1.06679	−0.533393	+	0.845868i	$0.679083\pi$
$654$	0	0
$655$	−5.92645e6	−0.539748
$656$	0	0
$657$	0	0
$658$	0	0
$659$	1.23526e7	1.10801	0.554005	−	0.832514i	$-0.313099\pi$
$659$	1.23526e7	1.10801	0.554005	+	0.832514i	$0.313099\pi$
$660$	0	0
$661$	−677746.	−0.0603341	−0.0301671	−	0.999545i	$-0.509604\pi$
$661$	−677746.	−0.0603341	−0.0301671	+	0.999545i	$0.509604\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−6.97695e6	−0.611803
$666$	0	0
$667$	−4.42985e6	−0.385544
$668$	0	0
$669$	0	0
$670$	0	0
$671$	944594.	0.0809914
$672$	0	0
$673$	1.33598e7	1.13701	0.568504	−	0.822680i	$-0.307522\pi$
$673$	1.33598e7	1.13701	0.568504	+	0.822680i	$0.307522\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−5.11219e6	−0.428682	−0.214341	−	0.976759i	$-0.568760\pi$
$677$	−5.11219e6	−0.428682	−0.214341	+	0.976759i	$0.568760\pi$
$678$	0	0
$679$	5.74244e6	0.477993
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−3.13650e6	−0.257273	−0.128636	−	0.991692i	$-0.541060\pi$
$683$	−3.13650e6	−0.257273	−0.128636	+	0.991692i	$0.541060\pi$
$684$	0	0
$685$	3.25293e6	0.264880
$686$	0	0
$687$	0	0
$688$	0	0
$689$	71471.9	0.00573571
$690$	0	0
$691$	−6.46277e6	−0.514901	−0.257450	−	0.966291i	$-0.582882\pi$
$691$	−6.46277e6	−0.514901	−0.257450	+	0.966291i	$0.582882\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	1.25962e7	0.989185
$696$	0	0
$697$	1.19911e7	0.934926
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−9.52280e6	−0.731930	−0.365965	−	0.930629i	$-0.619261\pi$
$701$	−9.52280e6	−0.731930	−0.365965	+	0.930629i	$0.619261\pi$
$702$	0	0
$703$	−3.53410e6	−0.269706
$704$	0	0
$705$	0	0
$706$	0	0
$707$	4.31077e6	0.324345
$708$	0	0
$709$	6.51985e6	0.487105	0.243552	−	0.969888i	$-0.421687\pi$
$709$	6.51985e6	0.487105	0.243552	+	0.969888i	$0.421687\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−3.10367e6	−0.228640
$714$	0	0
$715$	29233.9	0.00213856
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−2.18789e7	−1.57835	−0.789175	−	0.614168i	$-0.789492\pi$
$719$	−2.18789e7	−1.57835	−0.789175	+	0.614168i	$0.789492\pi$
$720$	0	0
$721$	9.07940e6	0.650457
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−1.99739e7	−1.41130
$726$	0	0
$727$	−1.72693e7	−1.21182	−0.605911	−	0.795533i	$-0.707191\pi$
$727$	−1.72693e7	−1.21182	−0.605911	+	0.795533i	$0.707191\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−5.69448e6	−0.394149
$732$	0	0
$733$	−3.05461e6	−0.209988	−0.104994	−	0.994473i	$-0.533482\pi$
$733$	−3.05461e6	−0.209988	−0.104994	+	0.994473i	$0.533482\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−774175.	−0.0525013
$738$	0	0
$739$	1.41503e7	0.953136	0.476568	−	0.879138i	$-0.341881\pi$
$739$	1.41503e7	0.953136	0.476568	+	0.879138i	$0.341881\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	1.65710e7	1.10122	0.550612	−	0.834761i	$-0.314394\pi$
$743$	1.65710e7	1.10122	0.550612	+	0.834761i	$0.314394\pi$
$744$	0	0
$745$	5.20911e7	3.43853
$746$	0	0
$747$	0	0
$748$	0	0
$749$	776679.	0.0505868
$750$	0	0
$751$	2.51204e7	1.62527	0.812637	−	0.582771i	$-0.198031\pi$
$751$	2.51204e7	1.62527	0.812637	+	0.582771i	$0.198031\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	1.44079e7	0.919884
$756$	0	0
$757$	−8.12854e6	−0.515553	−0.257776	−	0.966205i	$-0.582990\pi$
$757$	−8.12854e6	−0.515553	−0.257776	+	0.966205i	$0.582990\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−1.77695e7	−1.11228	−0.556139	−	0.831089i	$-0.687718\pi$
$761$	−1.77695e7	−1.11228	−0.556139	+	0.831089i	$0.687718\pi$
$762$	0	0
$763$	−7.57622e6	−0.471130
$764$	0	0
$765$	0	0
$766$	0	0
$767$	198608.	0.0121901
$768$	0	0
$769$	2.76525e6	0.168624	0.0843119	−	0.996439i	$-0.473131\pi$
$769$	2.76525e6	0.168624	0.0843119	+	0.996439i	$0.473131\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	1.27256e7	0.765999	0.382999	−	0.923749i	$-0.374891\pi$
$773$	1.27256e7	0.765999	0.382999	+	0.923749i	$0.374891\pi$
$774$	0	0
$775$	−1.39943e7	−0.836943
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−2.88106e7	−1.70102
$780$	0	0
$781$	−459888.	−0.0269789
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−1.31314e7	−0.760566
$786$	0	0
$787$	8.99512e6	0.517691	0.258845	−	0.965919i	$-0.416658\pi$
$787$	8.99512e6	0.517691	0.258845	+	0.965919i	$0.416658\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	2.11939e6	0.120440
$792$	0	0
$793$	−383368.	−0.0216488
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−2.16172e7	−1.20546	−0.602731	−	0.797944i	$-0.705921\pi$
$797$	−2.16172e7	−1.20546	−0.602731	+	0.797944i	$0.705921\pi$
$798$	0	0
$799$	1.19207e7	0.660596
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−587682.	−0.0321628
$804$	0	0
$805$	7.40536e6	0.402770
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−2.63701e7	−1.41658	−0.708289	−	0.705923i	$-0.750533\pi$
$809$	−2.63701e7	−1.41658	−0.708289	+	0.705923i	$0.750533\pi$
$810$	0	0
$811$	3.10910e7	1.65990	0.829952	−	0.557835i	$-0.188368\pi$
$811$	3.10910e7	1.65990	0.829952	+	0.557835i	$0.188368\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−4.63066e7	−2.44202
$816$	0	0
$817$	1.36819e7	0.717119
$818$	0	0
$819$	0	0
$820$	0	0
$821$	2.17372e7	1.12550	0.562751	−	0.826627i	$-0.309743\pi$
$821$	2.17372e7	1.12550	0.562751	+	0.826627i	$0.309743\pi$
$822$	0	0
$823$	−1.33966e7	−0.689436	−0.344718	−	0.938706i	$-0.612026\pi$
$823$	−1.33966e7	−0.689436	−0.344718	+	0.938706i	$0.612026\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−1.67048e7	−0.849334	−0.424667	−	0.905350i	$-0.639609\pi$
$827$	−1.67048e7	−0.849334	−0.424667	+	0.905350i	$0.639609\pi$
$828$	0	0
$829$	−1.99150e7	−1.00646	−0.503228	−	0.864154i	$-0.667855\pi$
$829$	−1.99150e7	−1.00646	−0.503228	+	0.864154i	$0.667855\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	1.42611e6	0.0712098
$834$	0	0
$835$	3.21243e7	1.59447
$836$	0	0
$837$	0	0
$838$	0	0
$839$	3.56979e7	1.75081	0.875404	−	0.483393i	$-0.160596\pi$
$839$	3.56979e7	1.75081	0.875404	+	0.483393i	$0.160596\pi$
$840$	0	0
$841$	−1.19583e7	−0.583015
$842$	0	0
$843$	0	0
$844$	0	0
$845$	3.70334e7	1.78424
$846$	0	0
$847$	7.85612e6	0.376270
$848$	0	0
$849$	0	0
$850$	0	0
$851$	3.75111e6	0.177556
$852$	0	0
$853$	−2.85813e7	−1.34496	−0.672480	−	0.740115i	$-0.734771\pi$
$853$	−2.85813e7	−1.34496	−0.672480	+	0.740115i	$0.734771\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	2.35988e7	1.09759	0.548793	−	0.835958i	$-0.315087\pi$
$857$	2.35988e7	1.09759	0.548793	+	0.835958i	$0.315087\pi$
$858$	0	0
$859$	−1.57178e7	−0.726789	−0.363395	−	0.931635i	$-0.618382\pi$
$859$	−1.57178e7	−0.726789	−0.363395	+	0.931635i	$0.618382\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−1.57903e7	−0.721712	−0.360856	−	0.932622i	$-0.617515\pi$
$863$	−1.57903e7	−0.721712	−0.360856	+	0.932622i	$0.617515\pi$
$864$	0	0
$865$	2.95558e7	1.34308
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−2.35377e6	−0.105734
$870$	0	0
$871$	314203.	0.0140335
$872$	0	0
$873$	0	0
$874$	0	0
$875$	1.81125e7	0.799757
$876$	0	0
$877$	−2.98385e7	−1.31002	−0.655010	−	0.755620i	$-0.727336\pi$
$877$	−2.98385e7	−1.31002	−0.655010	+	0.755620i	$0.727336\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	2.42418e7	1.05227	0.526133	−	0.850402i	$-0.323641\pi$
$881$	2.42418e7	1.05227	0.526133	+	0.850402i	$0.323641\pi$
$882$	0	0
$883$	1.08649e7	0.468946	0.234473	−	0.972123i	$-0.424664\pi$
$883$	1.08649e7	0.468946	0.234473	+	0.972123i	$0.424664\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	620748.	0.0264915	0.0132457	−	0.999912i	$-0.495784\pi$
$887$	620748.	0.0264915	0.0132457	+	0.999912i	$0.495784\pi$
$888$	0	0
$889$	3.34359e6	0.141892
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−2.86415e7	−1.20190
$894$	0	0
$895$	−1.13767e7	−0.474743
$896$	0	0
$897$	0	0
$898$	0	0
$899$	5.99234e6	0.247285
$900$	0	0
$901$	3.89291e6	0.159758
$902$	0	0
$903$	0	0
$904$	0	0
$905$	4.96971e7	2.01702
$906$	0	0
$907$	9.31727e6	0.376071	0.188036	−	0.982162i	$-0.439788\pi$
$907$	9.31727e6	0.376071	0.188036	+	0.982162i	$0.439788\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	2.48642e7	0.992608	0.496304	−	0.868149i	$-0.334690\pi$
$911$	2.48642e7	0.992608	0.496304	+	0.868149i	$0.334690\pi$
$912$	0	0
$913$	−1.91000e6	−0.0758327
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−2.91054e6	−0.114301
$918$	0	0
$919$	4.09652e7	1.60003	0.800013	−	0.599983i	$-0.204826\pi$
$919$	4.09652e7	1.60003	0.800013	+	0.599983i	$0.204826\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	186648.	0.00721138
$924$	0	0
$925$	1.69135e7	0.649950
$926$	0	0
$927$	0	0
$928$	0	0
$929$	1.84805e6	0.0702544	0.0351272	−	0.999383i	$-0.488816\pi$
$929$	1.84805e6	0.0702544	0.0351272	+	0.999383i	$0.488816\pi$
$930$	0	0
$931$	−3.42646e6	−0.129560
$932$	0	0
$933$	0	0
$934$	0	0
$935$	1.59231e6	0.0595659
$936$	0	0
$937$	−1.60973e7	−0.598968	−0.299484	−	0.954101i	$-0.596815\pi$
$937$	−1.60973e7	−0.598968	−0.299484	+	0.954101i	$0.596815\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	3.47172e7	1.27812	0.639059	−	0.769158i	$-0.279324\pi$
$941$	3.47172e7	1.27812	0.639059	+	0.769158i	$0.279324\pi$
$942$	0	0
$943$	3.05797e7	1.11983
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−3.93018e7	−1.42409	−0.712046	−	0.702133i	$-0.752231\pi$
$947$	−3.93018e7	−1.42409	−0.712046	+	0.702133i	$0.752231\pi$
$948$	0	0
$949$	238513.	0.00859701
$950$	0	0
$951$	0	0
$952$	0	0
$953$	3.99314e7	1.42424	0.712118	−	0.702060i	$-0.247736\pi$
$953$	3.99314e7	1.42424	0.712118	+	0.702060i	$0.247736\pi$
$954$	0	0
$955$	−5.24026e7	−1.85928
$956$	0	0
$957$	0	0
$958$	0	0
$959$	1.59755e6	0.0560929
$960$	0	0
$961$	−2.44308e7	−0.853353
$962$	0	0
$963$	0	0
$964$	0	0
$965$	2.30246e7	0.795927
$966$	0	0
$967$	2.42085e7	0.832534	0.416267	−	0.909242i	$-0.363338\pi$
$967$	2.42085e7	0.832534	0.416267	+	0.909242i	$0.363338\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−4.00227e7	−1.36226	−0.681128	−	0.732164i	$-0.738510\pi$
$971$	−4.00227e7	−1.36226	−0.681128	+	0.732164i	$0.738510\pi$
$972$	0	0
$973$	6.18614e6	0.209477
$974$	0	0
$975$	0	0
$976$	0	0
$977$	6.69145e6	0.224277	0.112138	−	0.993693i	$-0.464230\pi$
$977$	6.69145e6	0.224277	0.112138	+	0.993693i	$0.464230\pi$
$978$	0	0
$979$	88641.4	0.00295583
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−3.41067e7	−1.12579	−0.562894	−	0.826529i	$-0.690312\pi$
$983$	−3.41067e7	−1.12579	−0.562894	+	0.826529i	$0.690312\pi$
$984$	0	0
$985$	8.94148e7	2.93642
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−1.45220e7	−0.472103
$990$	0	0
$991$	−2.75297e7	−0.890465	−0.445232	−	0.895415i	$-0.646879\pi$
$991$	−2.75297e7	−0.890465	−0.445232	+	0.895415i	$0.646879\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	9.02725e7	2.89066
$996$	0	0
$997$	3.40958e7	1.08633	0.543166	−	0.839625i	$-0.317226\pi$
$997$	3.40958e7	1.08633	0.543166	+	0.839625i	$0.317226\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1008.6.a.bg.1.1		2
3.2	odd	2		1008.6.a.bv.1.2		2
4.3	odd	2		504.6.a.k.1.1	✓	2
12.11	even	2		504.6.a.u.1.2	yes	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
504.6.a.k.1.1	✓	2	4.3	odd	2
504.6.a.u.1.2	yes	2	12.11	even	2
1008.6.a.bg.1.1		2	1.1	even	1	trivial
1008.6.a.bv.1.2		2	3.2	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 1008 = 2^{4} \cdot 3^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	1008.a (trivial)

\(f(q)\)	\(=\)	\(q-99.7738 q^{5} -49.0000 q^{7} +O(q^{10})\)	\(q-99.7738 q^{5} -49.0000 q^{7} -26.8689 q^{11} +10.9049 q^{13} +593.964 q^{17} -1427.10 q^{19} +1514.73 q^{23} +6829.81 q^{25} -2924.52 q^{29} -2049.00 q^{31} +4888.92 q^{35} +2476.43 q^{37} +20188.3 q^{41} -9587.24 q^{43} +20069.8 q^{47} +2401.00 q^{49} +6554.12 q^{53} +2680.81 q^{55} +18212.8 q^{59} -35155.6 q^{61} -1088.02 q^{65} +28813.0 q^{67} +17116.0 q^{71} +21872.2 q^{73} +1316.58 q^{77} +87602.0 q^{79} +71085.9 q^{83} -59262.0 q^{85} -3299.03 q^{89} -534.339 q^{91} +142387. q^{95} -117193. q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 76 q^{5} - 98 q^{7}+O(q^{10}) \) 2 * q - 76 * q^5 - 98 * q^7	\( 2 q - 76 q^{5} - 98 q^{7} + 564 q^{11} + 516 q^{13} + 76 q^{17} - 2360 q^{19} - 2036 q^{23} + 4270 q^{25} - 8320 q^{29} + 6280 q^{31} + 3724 q^{35} - 2460 q^{37} + 14308 q^{41} - 23128 q^{43} + 12712 q^{47} + 4802 q^{49} + 9896 q^{53} + 16728 q^{55} + 60888 q^{59} + 16172 q^{61} + 10920 q^{65} + 62568 q^{67} - 732 q^{71} + 1244 q^{73} - 27636 q^{77} - 11600 q^{79} + 132288 q^{83} - 71576 q^{85} - 93452 q^{89} - 25284 q^{91} + 120208 q^{95} - 272932 q^{97}+O(q^{100}) \) 2 * q - 76 * q^5 - 98 * q^7 + 564 * q^11 + 516 * q^13 + 76 * q^17 - 2360 * q^19 - 2036 * q^23 + 4270 * q^25 - 8320 * q^29 + 6280 * q^31 + 3724 * q^35 - 2460 * q^37 + 14308 * q^41 - 23128 * q^43 + 12712 * q^47 + 4802 * q^49 + 9896 * q^53 + 16728 * q^55 + 60888 * q^59 + 16172 * q^61 + 10920 * q^65 + 62568 * q^67 - 732 * q^71 + 1244 * q^73 - 27636 * q^77 - 11600 * q^79 + 132288 * q^83 - 71576 * q^85 - 93452 * q^89 - 25284 * q^91 + 120208 * q^95 - 272932 * q^97

Introduction

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