LMFDB - Embedded newform 208.6.a.j.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [208,6,Mod(1,208)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(208, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("208.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 208 = 2^{4} \cdot 13 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	208.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:	$33.3598345211$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.168897.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 100x + 256 $ x^3 - x^2 - 100*x + 256
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 13)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-10.6486$ of defining polynomial
Character	$\chi$	$=$	208.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-10.2870 q^{3} +92.1784 q^{5} -2.86088 q^{7} -137.178 q^{9} +O(q^{10})$	$q-10.2870 q^{3} +92.1784 q^{5} -2.86088 q^{7} -137.178 q^{9} -571.711 q^{11} +169.000 q^{13} -948.236 q^{15} -855.571 q^{17} +2355.90 q^{19} +29.4298 q^{21} -2509.66 q^{23} +5371.86 q^{25} +3910.88 q^{27} -5497.50 q^{29} +144.924 q^{31} +5881.17 q^{33} -263.712 q^{35} -515.975 q^{37} -1738.50 q^{39} -13928.9 q^{41} -7753.58 q^{43} -12644.9 q^{45} -8344.77 q^{47} -16798.8 q^{49} +8801.22 q^{51} -5976.21 q^{53} -52699.5 q^{55} -24235.1 q^{57} -2110.84 q^{59} -17394.2 q^{61} +392.452 q^{63} +15578.2 q^{65} +3121.05 q^{67} +25816.8 q^{69} -43323.3 q^{71} +6808.31 q^{73} -55260.1 q^{75} +1635.60 q^{77} +1280.80 q^{79} -6896.72 q^{81} -7148.44 q^{83} -78865.2 q^{85} +56552.6 q^{87} +107123. q^{89} -483.489 q^{91} -1490.82 q^{93} +217163. q^{95} +42456.2 q^{97} +78426.5 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 8 q^{3} + 56 q^{5} + 60 q^{7} - 191 q^{9}+O(q^{10}) $ 3 * q - 8 * q^3 + 56 * q^5 + 60 * q^7 - 191 * q^9	$ 3 q - 8 q^{3} + 56 q^{5} + 60 q^{7} - 191 q^{9} - 556 q^{11} + 507 q^{13} - 1972 q^{15} + 908 q^{17} - 148 q^{19} + 1390 q^{21} - 3624 q^{23} + 2023 q^{25} + 4276 q^{27} - 8758 q^{29} + 2608 q^{31} - 1828 q^{33} - 4348 q^{35} - 20632 q^{37} - 1352 q^{39} - 10998 q^{41} - 2032 q^{43} - 13918 q^{45} - 34260 q^{47} - 44571 q^{49} - 13468 q^{51} - 12570 q^{53} - 35312 q^{55} + 1716 q^{57} - 63948 q^{59} - 12754 q^{61} + 1648 q^{63} + 9464 q^{65} - 56132 q^{67} - 20112 q^{69} - 77580 q^{71} - 43026 q^{73} - 23548 q^{75} - 21052 q^{77} + 61872 q^{79} - 108221 q^{81} - 98092 q^{83} - 55226 q^{85} - 139072 q^{87} + 33694 q^{89} + 10140 q^{91} + 105392 q^{93} + 196560 q^{95} + 76334 q^{97} + 60332 q^{99}+O(q^{100}) $ 3 * q - 8 * q^3 + 56 * q^5 + 60 * q^7 - 191 * q^9 - 556 * q^11 + 507 * q^13 - 1972 * q^15 + 908 * q^17 - 148 * q^19 + 1390 * q^21 - 3624 * q^23 + 2023 * q^25 + 4276 * q^27 - 8758 * q^29 + 2608 * q^31 - 1828 * q^33 - 4348 * q^35 - 20632 * q^37 - 1352 * q^39 - 10998 * q^41 - 2032 * q^43 - 13918 * q^45 - 34260 * q^47 - 44571 * q^49 - 13468 * q^51 - 12570 * q^53 - 35312 * q^55 + 1716 * q^57 - 63948 * q^59 - 12754 * q^61 + 1648 * q^63 + 9464 * q^65 - 56132 * q^67 - 20112 * q^69 - 77580 * q^71 - 43026 * q^73 - 23548 * q^75 - 21052 * q^77 + 61872 * q^79 - 108221 * q^81 - 98092 * q^83 - 55226 * q^85 - 139072 * q^87 + 33694 * q^89 + 10140 * q^91 + 105392 * q^93 + 196560 * q^95 + 76334 * q^97 + 60332 * q^99

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
$3$	−10.2870	−0.659909	−0.329954	−	0.943997i	$-0.607033\pi$
$3$	−10.2870	−0.659909	−0.329954	+	0.943997i	$0.607033\pi$
$4$	0	0
$5$	92.1784	1.64894	0.824469	−	0.565907i	$-0.191474\pi$
$5$	92.1784	1.64894	0.824469	+	0.565907i	$0.191474\pi$
$6$	0	0
$7$	−2.86088	−0.0220676	−0.0110338	−	0.999939i	$-0.503512\pi$
$7$	−2.86088	−0.0220676	−0.0110338	+	0.999939i	$0.503512\pi$
$8$	0	0
$9$	−137.178	−0.564520
$10$	0	0
$11$	−571.711	−1.42461	−0.712304	−	0.701871i	$-0.752348\pi$
$11$	−571.711	−1.42461	−0.712304	+	0.701871i	$0.752348\pi$
$12$	0	0
$13$	169.000	0.277350
$13$	169.000	0.277350
$14$	0	0
$15$	−948.236	−1.08815
$16$	0	0
$17$	−855.571	−0.718015	−0.359008	−	0.933335i	$-0.616885\pi$
$17$	−855.571	−0.718015	−0.359008	+	0.933335i	$0.616885\pi$
$18$	0	0
$19$	2355.90	1.49718	0.748589	−	0.663034i	$-0.230732\pi$
$19$	2355.90	1.49718	0.748589	+	0.663034i	$0.230732\pi$
$20$	0	0
$21$	29.4298	0.0145626
$22$	0	0
$23$	−2509.66	−0.989227	−0.494613	−	0.869113i	$-0.664690\pi$
$23$	−2509.66	−0.989227	−0.494613	+	0.869113i	$0.664690\pi$
$24$	0	0
$25$	5371.86	1.71900
$26$	0	0
$27$	3910.88	1.03244
$28$	0	0
$29$	−5497.50	−1.21387	−0.606933	−	0.794753i	$-0.707600\pi$
$29$	−5497.50	−1.21387	−0.606933	+	0.794753i	$0.707600\pi$
$30$	0	0
$31$	144.924	0.0270854	0.0135427	−	0.999908i	$-0.495689\pi$
$31$	144.924	0.0270854	0.0135427	+	0.999908i	$0.495689\pi$
$32$	0	0
$33$	5881.17	0.940111
$34$	0	0
$35$	−263.712	−0.0363881
$36$	0	0
$37$	−515.975	−0.0619618	−0.0309809	−	0.999520i	$-0.509863\pi$
$37$	−515.975	−0.0619618	−0.0309809	+	0.999520i	$0.509863\pi$
$38$	0	0
$39$	−1738.50	−0.183026
$40$	0	0
$41$	−13928.9	−1.29407	−0.647033	−	0.762462i	$-0.723991\pi$
$41$	−13928.9	−1.29407	−0.647033	+	0.762462i	$0.723991\pi$
$42$	0	0
$43$	−7753.58	−0.639486	−0.319743	−	0.947504i	$-0.603597\pi$
$43$	−7753.58	−0.639486	−0.319743	+	0.947504i	$0.603597\pi$
$44$	0	0
$45$	−12644.9	−0.930859
$46$	0	0
$47$	−8344.77	−0.551023	−0.275512	−	0.961298i	$-0.588847\pi$
$47$	−8344.77	−0.551023	−0.275512	+	0.961298i	$0.588847\pi$
$48$	0	0
$49$	−16798.8	−0.999513
$50$	0	0
$51$	8801.22	0.473825
$52$	0	0
$53$	−5976.21	−0.292238	−0.146119	−	0.989267i	$-0.546678\pi$
$53$	−5976.21	−0.292238	−0.146119	+	0.989267i	$0.546678\pi$
$54$	0	0
$55$	−52699.5	−2.34909
$56$	0	0
$57$	−24235.1	−0.988001
$58$	0	0
$59$	−2110.84	−0.0789451	−0.0394726	−	0.999221i	$-0.512568\pi$
$59$	−2110.84	−0.0789451	−0.0394726	+	0.999221i	$0.512568\pi$
$60$	0	0
$61$	−17394.2	−0.598522	−0.299261	−	0.954171i	$-0.596740\pi$
$61$	−17394.2	−0.598522	−0.299261	+	0.954171i	$0.596740\pi$
$62$	0	0
$63$	392.452	0.0124576
$64$	0	0
$65$	15578.2	0.457333
$66$	0	0
$67$	3121.05	0.0849402	0.0424701	−	0.999098i	$-0.486477\pi$
$67$	3121.05	0.0849402	0.0424701	+	0.999098i	$0.486477\pi$
$68$	0	0
$69$	25816.8	0.652800
$70$	0	0
$71$	−43323.3	−1.01994	−0.509971	−	0.860191i	$-0.670344\pi$
$71$	−43323.3	−1.01994	−0.509971	+	0.860191i	$0.670344\pi$
$72$	0	0
$73$	6808.31	0.149531	0.0747657	−	0.997201i	$-0.476179\pi$
$73$	6808.31	0.149531	0.0747657	+	0.997201i	$0.476179\pi$
$74$	0	0
$75$	−55260.1	−1.13438
$76$	0	0
$77$	1635.60	0.0314377
$78$	0	0
$79$	1280.80	0.0230895	0.0115447	−	0.999933i	$-0.496325\pi$
$79$	1280.80	0.0230895	0.0115447	+	0.999933i	$0.496325\pi$
$80$	0	0
$81$	−6896.72	−0.116797
$82$	0	0
$83$	−7148.44	−0.113898	−0.0569490	−	0.998377i	$-0.518137\pi$
$83$	−7148.44	−0.113898	−0.0569490	+	0.998377i	$0.518137\pi$
$84$	0	0
$85$	−78865.2	−1.18396
$86$	0	0
$87$	56552.6	0.801041
$88$	0	0
$89$	107123.	1.43353	0.716765	−	0.697315i	$-0.245622\pi$
$89$	107123.	1.43353	0.716765	+	0.697315i	$0.245622\pi$
$90$	0	0
$91$	−483.489	−0.00612045
$92$	0	0
$93$	−1490.82	−0.0178739
$94$	0	0
$95$	217163.	2.46875
$96$	0	0
$97$	42456.2	0.458155	0.229077	−	0.973408i	$-0.426429\pi$
$97$	42456.2	0.458155	0.229077	+	0.973408i	$0.426429\pi$
$98$	0	0
$99$	78426.5	0.804220
$100$	0	0
$101$	−87947.3	−0.857866	−0.428933	−	0.903336i	$-0.641110\pi$
$101$	−87947.3	−0.857866	−0.428933	+	0.903336i	$0.641110\pi$
$102$	0	0
$103$	−96885.8	−0.899844	−0.449922	−	0.893068i	$-0.648548\pi$
$103$	−96885.8	−0.899844	−0.449922	+	0.893068i	$0.648548\pi$
$104$	0	0
$105$	2712.79	0.0240128
$106$	0	0
$107$	−106509.	−0.899346	−0.449673	−	0.893193i	$-0.648460\pi$
$107$	−106509.	−0.899346	−0.449673	+	0.893193i	$0.648460\pi$
$108$	0	0
$109$	−142520.	−1.14897	−0.574485	−	0.818515i	$-0.694798\pi$
$109$	−142520.	−1.14897	−0.574485	+	0.818515i	$0.694798\pi$
$110$	0	0
$111$	5307.81	0.0408892
$112$	0	0
$113$	−209540.	−1.54373	−0.771864	−	0.635787i	$-0.780676\pi$
$113$	−209540.	−1.54373	−0.771864	+	0.635787i	$0.780676\pi$
$114$	0	0
$115$	−231337.	−1.63117
$116$	0	0
$117$	−23183.2	−0.156570
$118$	0	0
$119$	2447.69	0.0158449
$120$	0	0
$121$	165803.	1.02951
$122$	0	0
$123$	143286.	0.853966
$124$	0	0
$125$	207112.	1.18558
$126$	0	0
$127$	54468.2	0.299663	0.149832	−	0.988712i	$-0.452127\pi$
$127$	54468.2	0.299663	0.149832	+	0.988712i	$0.452127\pi$
$128$	0	0
$129$	79760.8	0.422003
$130$	0	0
$131$	−258252.	−1.31482	−0.657408	−	0.753535i	$-0.728347\pi$
$131$	−258252.	−1.31482	−0.657408	+	0.753535i	$0.728347\pi$
$132$	0	0
$133$	−6739.97	−0.0330391
$134$	0	0
$135$	360499.	1.70243
$136$	0	0
$137$	293754.	1.33716	0.668580	−	0.743641i	$-0.266903\pi$
$137$	293754.	1.33716	0.668580	+	0.743641i	$0.266903\pi$
$138$	0	0
$139$	−180400.	−0.791955	−0.395977	−	0.918260i	$-0.629594\pi$
$139$	−180400.	−0.791955	−0.395977	+	0.918260i	$0.629594\pi$
$140$	0	0
$141$	85842.4	0.363625
$142$	0	0
$143$	−96619.2	−0.395115
$144$	0	0
$145$	−506751.	−2.00159
$146$	0	0
$147$	172809.	0.659588
$148$	0	0
$149$	516009.	1.90411	0.952054	−	0.305930i	$-0.0989675\pi$
$149$	516009.	1.90411	0.952054	+	0.305930i	$0.0989675\pi$
$150$	0	0
$151$	333770.	1.19126	0.595628	−	0.803260i	$-0.296903\pi$
$151$	333770.	1.19126	0.595628	+	0.803260i	$0.296903\pi$
$152$	0	0
$153$	117366.	0.405334
$154$	0	0
$155$	13358.8	0.0446621
$156$	0	0
$157$	−265860.	−0.860804	−0.430402	−	0.902637i	$-0.641628\pi$
$157$	−265860.	−0.860804	−0.430402	+	0.902637i	$0.641628\pi$
$158$	0	0
$159$	61477.0	0.192850
$160$	0	0
$161$	7179.86	0.0218299
$162$	0	0
$163$	231417.	0.682223	0.341112	−	0.940023i	$-0.389197\pi$
$163$	231417.	0.682223	0.341112	+	0.940023i	$0.389197\pi$
$164$	0	0
$165$	542117.	1.55018
$166$	0	0
$167$	656226.	1.82080	0.910400	−	0.413730i	$-0.135774\pi$
$167$	656226.	1.82080	0.910400	+	0.413730i	$0.135774\pi$
$168$	0	0
$169$	28561.0	0.0769231
$170$	0	0
$171$	−323179.	−0.845187
$172$	0	0
$173$	45111.6	0.114597	0.0572985	−	0.998357i	$-0.481751\pi$
$173$	45111.6	0.114597	0.0572985	+	0.998357i	$0.481751\pi$
$174$	0	0
$175$	−15368.3	−0.0379341
$176$	0	0
$177$	21714.1	0.0520966
$178$	0	0
$179$	−521846.	−1.21733	−0.608667	−	0.793426i	$-0.708295\pi$
$179$	−521846.	−1.21733	−0.608667	+	0.793426i	$0.708295\pi$
$180$	0	0
$181$	−122780.	−0.278569	−0.139284	−	0.990252i	$-0.544480\pi$
$181$	−122780.	−0.278569	−0.139284	+	0.990252i	$0.544480\pi$
$182$	0	0
$183$	178934.	0.394970
$184$	0	0
$185$	−47561.7	−0.102171
$186$	0	0
$187$	489140.	1.02289
$188$	0	0
$189$	−11188.6	−0.0227835
$190$	0	0
$191$	−350208.	−0.694613	−0.347306	−	0.937752i	$-0.612904\pi$
$191$	−350208.	−0.694613	−0.347306	+	0.937752i	$0.612904\pi$
$192$	0	0
$193$	876316.	1.69343	0.846716	−	0.532046i	$-0.178577\pi$
$193$	876316.	1.69343	0.846716	+	0.532046i	$0.178577\pi$
$194$	0	0
$195$	−160252.	−0.301798
$196$	0	0
$197$	306582.	0.562835	0.281418	−	0.959585i	$-0.409195\pi$
$197$	306582.	0.562835	0.281418	+	0.959585i	$0.409195\pi$
$198$	0	0
$199$	327849.	0.586869	0.293434	−	0.955979i	$-0.405202\pi$
$199$	327849.	0.586869	0.293434	+	0.955979i	$0.405202\pi$
$200$	0	0
$201$	−32106.1	−0.0560528
$202$	0	0
$203$	15727.7	0.0267871
$204$	0	0
$205$	−1.28394e6	−2.13384
$206$	0	0
$207$	344272.	0.558439
$208$	0	0
$209$	−1.34690e6	−2.13289
$210$	0	0
$211$	509483.	0.787814	0.393907	−	0.919150i	$-0.371123\pi$
$211$	509483.	0.787814	0.393907	+	0.919150i	$0.371123\pi$
$212$	0	0
$213$	445665.	0.673069
$214$	0	0
$215$	−714713.	−1.05447
$216$	0	0
$217$	−414.610	−0.000597709 0
$218$	0	0
$219$	−70036.8	−0.0986771
$220$	0	0
$221$	−144591.	−0.199142
$222$	0	0
$223$	15061.4	0.0202816	0.0101408	−	0.999949i	$-0.496772\pi$
$223$	15061.4	0.0202816	0.0101408	+	0.999949i	$0.496772\pi$
$224$	0	0
$225$	−736904.	−0.970408
$226$	0	0
$227$	−273713.	−0.352558	−0.176279	−	0.984340i	$-0.556406\pi$
$227$	−273713.	−0.352558	−0.176279	+	0.984340i	$0.556406\pi$
$228$	0	0
$229$	743056.	0.936339	0.468169	−	0.883639i	$-0.344914\pi$
$229$	743056.	0.936339	0.468169	+	0.883639i	$0.344914\pi$
$230$	0	0
$231$	−16825.4	−0.0207460
$232$	0	0
$233$	1.12122e6	1.35301	0.676506	−	0.736437i	$-0.263493\pi$
$233$	1.12122e6	1.35301	0.676506	+	0.736437i	$0.263493\pi$
$234$	0	0
$235$	−769208.	−0.908603
$236$	0	0
$237$	−13175.6	−0.0152369
$238$	0	0
$239$	−279838.	−0.316892	−0.158446	−	0.987368i	$-0.550648\pi$
$239$	−279838.	−0.316892	−0.158446	+	0.987368i	$0.550648\pi$
$240$	0	0
$241$	816889.	0.905983	0.452992	−	0.891515i	$-0.350357\pi$
$241$	816889.	0.905983	0.452992	+	0.891515i	$0.350357\pi$
$242$	0	0
$243$	−879398.	−0.955366
$244$	0	0
$245$	−1.54849e6	−1.64813
$246$	0	0
$247$	398148.	0.415242
$248$	0	0
$249$	73535.7	0.0751623
$250$	0	0
$251$	766697.	0.768139	0.384069	−	0.923304i	$-0.374522\pi$
$251$	766697.	0.768139	0.384069	+	0.923304i	$0.374522\pi$
$252$	0	0
$253$	1.43480e6	1.40926
$254$	0	0
$255$	811283.	0.781307
$256$	0	0
$257$	−1.82413e6	−1.72276	−0.861379	−	0.507963i	$-0.830399\pi$
$257$	−1.82413e6	−1.72276	−0.861379	+	0.507963i	$0.830399\pi$
$258$	0	0
$259$	1476.14	0.00136735
$260$	0	0
$261$	754139.	0.685252
$262$	0	0
$263$	294257.	0.262323	0.131162	−	0.991361i	$-0.458129\pi$
$263$	294257.	0.262323	0.131162	+	0.991361i	$0.458129\pi$
$264$	0	0
$265$	−550878.	−0.481882
$266$	0	0
$267$	−1.10197e6	−0.945999
$268$	0	0
$269$	289738.	0.244132	0.122066	−	0.992522i	$-0.461048\pi$
$269$	289738.	0.244132	0.122066	+	0.992522i	$0.461048\pi$
$270$	0	0
$271$	−1.63179e6	−1.34971	−0.674855	−	0.737950i	$-0.735794\pi$
$271$	−1.63179e6	−1.34971	−0.674855	+	0.737950i	$0.735794\pi$
$272$	0	0
$273$	4973.64	0.00403894
$274$	0	0
$275$	−3.07116e6	−2.44889
$276$	0	0
$277$	398868.	0.312341	0.156171	−	0.987730i	$-0.450085\pi$
$277$	398868.	0.312341	0.156171	+	0.987730i	$0.450085\pi$
$278$	0	0
$279$	−19880.4	−0.0152902
$280$	0	0
$281$	274611.	0.207468	0.103734	−	0.994605i	$-0.466921\pi$
$281$	274611.	0.207468	0.103734	+	0.994605i	$0.466921\pi$
$282$	0	0
$283$	794683.	0.589831	0.294916	−	0.955523i	$-0.404708\pi$
$283$	794683.	0.589831	0.294916	+	0.955523i	$0.404708\pi$
$284$	0	0
$285$	−2.23395e6	−1.62915
$286$	0	0
$287$	39848.9	0.0285570
$288$	0	0
$289$	−687856.	−0.484454
$290$	0	0
$291$	−436746.	−0.302340
$292$	0	0
$293$	−1.55323e6	−1.05698	−0.528490	−	0.848940i	$-0.677241\pi$
$293$	−1.55323e6	−1.05698	−0.528490	+	0.848940i	$0.677241\pi$
$294$	0	0
$295$	−194574.	−0.130176
$296$	0	0
$297$	−2.23590e6	−1.47082
$298$	0	0
$299$	−424133.	−0.274362
$300$	0	0
$301$	22182.1	0.0141119
$302$	0	0
$303$	904711.	0.566113
$304$	0	0
$305$	−1.60337e6	−0.986926
$306$	0	0
$307$	−2.07136e6	−1.25433	−0.627163	−	0.778888i	$-0.715784\pi$
$307$	−2.07136e6	−1.25433	−0.627163	+	0.778888i	$0.715784\pi$
$308$	0	0
$309$	996660.	0.593815
$310$	0	0
$311$	2.50827e6	1.47053	0.735263	−	0.677782i	$-0.237059\pi$
$311$	2.50827e6	1.47053	0.735263	+	0.677782i	$0.237059\pi$
$312$	0	0
$313$	832384.	0.480245	0.240123	−	0.970743i	$-0.422812\pi$
$313$	832384.	0.480245	0.240123	+	0.970743i	$0.422812\pi$
$314$	0	0
$315$	36175.6	0.0205418
$316$	0	0
$317$	−1.76537e6	−0.986708	−0.493354	−	0.869829i	$-0.664229\pi$
$317$	−1.76537e6	−0.986708	−0.493354	+	0.869829i	$0.664229\pi$
$318$	0	0
$319$	3.14299e6	1.72928
$320$	0	0
$321$	1.09565e6	0.593487
$322$	0	0
$323$	−2.01564e6	−1.07500
$324$	0	0
$325$	907845.	0.476764
$326$	0	0
$327$	1.46609e6	0.758215
$328$	0	0
$329$	23873.4	0.0121598
$330$	0	0
$331$	285068.	0.143014	0.0715070	−	0.997440i	$-0.477219\pi$
$331$	285068.	0.143014	0.0715070	+	0.997440i	$0.477219\pi$
$332$	0	0
$333$	70780.6	0.0349787
$334$	0	0
$335$	287693.	0.140061
$336$	0	0
$337$	1.96932e6	0.944584	0.472292	−	0.881442i	$-0.343427\pi$
$337$	1.96932e6	0.944584	0.472292	+	0.881442i	$0.343427\pi$
$338$	0	0
$339$	2.15553e6	1.01872
$340$	0	0
$341$	−82854.5	−0.0385860
$342$	0	0
$343$	96142.4	0.0441245
$344$	0	0
$345$	2.37975e6	1.07643
$346$	0	0
$347$	−1.89753e6	−0.845988	−0.422994	−	0.906133i	$-0.639021\pi$
$347$	−1.89753e6	−0.845988	−0.422994	+	0.906133i	$0.639021\pi$
$348$	0	0
$349$	−365808.	−0.160764	−0.0803822	−	0.996764i	$-0.525614\pi$
$349$	−365808.	−0.160764	−0.0803822	+	0.996764i	$0.525614\pi$
$350$	0	0
$351$	660939.	0.286348
$352$	0	0
$353$	1.56672e6	0.669196	0.334598	−	0.942361i	$-0.391399\pi$
$353$	1.56672e6	0.669196	0.334598	+	0.942361i	$0.391399\pi$
$354$	0	0
$355$	−3.99348e6	−1.68182
$356$	0	0
$357$	−25179.3	−0.0104562
$358$	0	0
$359$	4.03307e6	1.65158	0.825789	−	0.563979i	$-0.190730\pi$
$359$	4.03307e6	1.65158	0.825789	+	0.563979i	$0.190730\pi$
$360$	0	0
$361$	3.07418e6	1.24154
$362$	0	0
$363$	−1.70561e6	−0.679380
$364$	0	0
$365$	627579.	0.246568
$366$	0	0
$367$	−683844.	−0.265028	−0.132514	−	0.991181i	$-0.542305\pi$
$367$	−683844.	−0.265028	−0.132514	+	0.991181i	$0.542305\pi$
$368$	0	0
$369$	1.91074e6	0.730527
$370$	0	0
$371$	17097.2	0.00644899
$372$	0	0
$373$	−3.09158e6	−1.15056	−0.575279	−	0.817957i	$-0.695107\pi$
$373$	−3.09158e6	−1.15056	−0.575279	+	0.817957i	$0.695107\pi$
$374$	0	0
$375$	−2.13056e6	−0.782374
$376$	0	0
$377$	−929078.	−0.336666
$378$	0	0
$379$	5.24600e6	1.87599	0.937995	−	0.346649i	$-0.112681\pi$
$379$	5.24600e6	1.87599	0.937995	+	0.346649i	$0.112681\pi$
$380$	0	0
$381$	−560312.	−0.197751
$382$	0	0
$383$	−1.57009e6	−0.546926	−0.273463	−	0.961883i	$-0.588169\pi$
$383$	−1.57009e6	−0.546926	−0.273463	+	0.961883i	$0.588169\pi$
$384$	0	0
$385$	150767.	0.0518388
$386$	0	0
$387$	1.06362e6	0.361003
$388$	0	0
$389$	1.54482e6	0.517611	0.258805	−	0.965929i	$-0.416671\pi$
$389$	1.54482e6	0.517611	0.258805	+	0.965929i	$0.416671\pi$
$390$	0	0
$391$	2.14719e6	0.710280
$392$	0	0
$393$	2.65663e6	0.867659
$394$	0	0
$395$	118062.	0.0380731
$396$	0	0
$397$	5.06711e6	1.61356	0.806778	−	0.590855i	$-0.201210\pi$
$397$	5.06711e6	1.61356	0.806778	+	0.590855i	$0.201210\pi$
$398$	0	0
$399$	69333.8	0.0218028
$400$	0	0
$401$	−138120.	−0.0428939	−0.0214469	−	0.999770i	$-0.506827\pi$
$401$	−138120.	−0.0428939	−0.0214469	+	0.999770i	$0.506827\pi$
$402$	0	0
$403$	24492.1	0.00751213
$404$	0	0
$405$	−635729.	−0.192590
$406$	0	0
$407$	294989.	0.0882713
$408$	0	0
$409$	−4.13317e6	−1.22173	−0.610865	−	0.791735i	$-0.709178\pi$
$409$	−4.13317e6	−1.22173	−0.610865	+	0.791735i	$0.709178\pi$
$410$	0	0
$411$	−3.02184e6	−0.882403
$412$	0	0
$413$	6038.87	0.00174213
$414$	0	0
$415$	−658932.	−0.187811
$416$	0	0
$417$	1.85577e6	0.522618
$418$	0	0
$419$	2.96703e6	0.825633	0.412817	−	0.910814i	$-0.364545\pi$
$419$	2.96703e6	0.825633	0.412817	+	0.910814i	$0.364545\pi$
$420$	0	0
$421$	−2.98125e6	−0.819773	−0.409887	−	0.912137i	$-0.634432\pi$
$421$	−2.98125e6	−0.819773	−0.409887	+	0.912137i	$0.634432\pi$
$422$	0	0
$423$	1.14472e6	0.311064
$424$	0	0
$425$	−4.59601e6	−1.23426
$426$	0	0
$427$	49762.8	0.0132080
$428$	0	0
$429$	993918.	0.260740
$430$	0	0
$431$	3.34790e6	0.868119	0.434059	−	0.900884i	$-0.357081\pi$
$431$	3.34790e6	0.868119	0.434059	+	0.900884i	$0.357081\pi$
$432$	0	0
$433$	−3.76782e6	−0.965764	−0.482882	−	0.875686i	$-0.660410\pi$
$433$	−3.76782e6	−0.965764	−0.482882	+	0.875686i	$0.660410\pi$
$434$	0	0
$435$	5.21293e6	1.32087
$436$	0	0
$437$	−5.91252e6	−1.48105
$438$	0	0
$439$	−5.63075e6	−1.39446	−0.697228	−	0.716849i	$-0.745584\pi$
$439$	−5.63075e6	−1.39446	−0.697228	+	0.716849i	$0.745584\pi$
$440$	0	0
$441$	2.30444e6	0.564245
$442$	0	0
$443$	−3.69803e6	−0.895284	−0.447642	−	0.894213i	$-0.647736\pi$
$443$	−3.69803e6	−0.895284	−0.447642	+	0.894213i	$0.647736\pi$
$444$	0	0
$445$	9.87441e6	2.36380
$446$	0	0
$447$	−5.30816e6	−1.25654
$448$	0	0
$449$	−505514.	−0.118336	−0.0591681	−	0.998248i	$-0.518845\pi$
$449$	−505514.	−0.118336	−0.0591681	+	0.998248i	$0.518845\pi$
$450$	0	0
$451$	7.96330e6	1.84354
$452$	0	0
$453$	−3.43348e6	−0.786121
$454$	0	0
$455$	−44567.3	−0.0100922
$456$	0	0
$457$	783627.	0.175517	0.0877584	−	0.996142i	$-0.472030\pi$
$457$	783627.	0.175517	0.0877584	+	0.996142i	$0.472030\pi$
$458$	0	0
$459$	−3.34603e6	−0.741308
$460$	0	0
$461$	4.64856e6	1.01875	0.509373	−	0.860546i	$-0.329877\pi$
$461$	4.64856e6	1.01875	0.509373	+	0.860546i	$0.329877\pi$
$462$	0	0
$463$	−2.64872e6	−0.574226	−0.287113	−	0.957897i	$-0.592696\pi$
$463$	−2.64872e6	−0.574226	−0.287113	+	0.957897i	$0.592696\pi$
$464$	0	0
$465$	−137422.	−0.0294729
$466$	0	0
$467$	2.42172e6	0.513845	0.256923	−	0.966432i	$-0.417291\pi$
$467$	2.42172e6	0.513845	0.256923	+	0.966432i	$0.417291\pi$
$468$	0	0
$469$	−8928.96	−0.00187443
$470$	0	0
$471$	2.73490e6	0.568052
$472$	0	0
$473$	4.43281e6	0.911017
$474$	0	0
$475$	1.26556e7	2.57364
$476$	0	0
$477$	819807.	0.164974
$478$	0	0
$479$	−5.58315e6	−1.11184	−0.555918	−	0.831237i	$-0.687633\pi$
$479$	−5.58315e6	−1.11184	−0.555918	+	0.831237i	$0.687633\pi$
$480$	0	0
$481$	−87199.7	−0.0171851
$482$	0	0
$483$	−73858.9	−0.0144057
$484$	0	0
$485$	3.91355e6	0.755468
$486$	0	0
$487$	−7.25183e6	−1.38556	−0.692780	−	0.721149i	$-0.743614\pi$
$487$	−7.25183e6	−1.38556	−0.692780	+	0.721149i	$0.743614\pi$
$488$	0	0
$489$	−2.38058e6	−0.450205
$490$	0	0
$491$	−4.17410e6	−0.781373	−0.390687	−	0.920524i	$-0.627762\pi$
$491$	−4.17410e6	−0.781373	−0.390687	+	0.920524i	$0.627762\pi$
$492$	0	0
$493$	4.70350e6	0.871574
$494$	0	0
$495$	7.22923e6	1.32611
$496$	0	0
$497$	123943.	0.0225077
$498$	0	0
$499$	7.83551e6	1.40869	0.704345	−	0.709857i	$-0.251241\pi$
$499$	7.83551e6	1.40869	0.704345	+	0.709857i	$0.251241\pi$
$500$	0	0
$501$	−6.75057e6	−1.20156
$502$	0	0
$503$	3.72420e6	0.656315	0.328158	−	0.944623i	$-0.393572\pi$
$503$	3.72420e6	0.656315	0.328158	+	0.944623i	$0.393572\pi$
$504$	0	0
$505$	−8.10685e6	−1.41457
$506$	0	0
$507$	−293806.	−0.0507622
$508$	0	0
$509$	−6.76675e6	−1.15767	−0.578836	−	0.815444i	$-0.696493\pi$
$509$	−6.76675e6	−1.15767	−0.578836	+	0.815444i	$0.696493\pi$
$510$	0	0
$511$	−19477.8	−0.00329980
$512$	0	0
$513$	9.21366e6	1.54575
$514$	0	0
$515$	−8.93078e6	−1.48379
$516$	0	0
$517$	4.77080e6	0.784992
$518$	0	0
$519$	−464062.	−0.0756236
$520$	0	0
$521$	9.08380e6	1.46613	0.733066	−	0.680157i	$-0.238088\pi$
$521$	9.08380e6	1.46613	0.733066	+	0.680157i	$0.238088\pi$
$522$	0	0
$523$	1.09284e7	1.74703	0.873515	−	0.486796i	$-0.161835\pi$
$523$	1.09284e7	1.74703	0.873515	+	0.486796i	$0.161835\pi$
$524$	0	0
$525$	158093.	0.0250331
$526$	0	0
$527$	−123992.	−0.0194477
$528$	0	0
$529$	−137934.	−0.0214305
$530$	0	0
$531$	289562.	0.0445661
$532$	0	0
$533$	−2.35398e6	−0.358910
$534$	0	0
$535$	−9.81783e6	−1.48297
$536$	0	0
$537$	5.36821e6	0.803329
$538$	0	0
$539$	9.60408e6	1.42391
$540$	0	0
$541$	1.17853e6	0.173120	0.0865598	−	0.996247i	$-0.472413\pi$
$541$	1.17853e6	0.173120	0.0865598	+	0.996247i	$0.472413\pi$
$542$	0	0
$543$	1.26304e6	0.183830
$544$	0	0
$545$	−1.31372e7	−1.89458
$546$	0	0
$547$	−9.57702e6	−1.36855	−0.684277	−	0.729222i	$-0.739882\pi$
$547$	−9.57702e6	−1.36855	−0.684277	+	0.729222i	$0.739882\pi$
$548$	0	0
$549$	2.38611e6	0.337878
$550$	0	0
$551$	−1.29516e7	−1.81737
$552$	0	0
$553$	−3664.22	−0.000509529 0
$554$	0	0
$555$	489266.	0.0674237
$556$	0	0
$557$	2.46873e6	0.337159	0.168580	−	0.985688i	$-0.446082\pi$
$557$	2.46873e6	0.337159	0.168580	+	0.985688i	$0.446082\pi$
$558$	0	0
$559$	−1.31036e6	−0.177362
$560$	0	0
$561$	−5.03176e6	−0.675014
$562$	0	0
$563$	−7.81681e6	−1.03934	−0.519671	−	0.854367i	$-0.673945\pi$
$563$	−7.81681e6	−1.03934	−0.519671	+	0.854367i	$0.673945\pi$
$564$	0	0
$565$	−1.93151e7	−2.54551
$566$	0	0
$567$	19730.7	0.00257742
$568$	0	0
$569$	−9.64303e6	−1.24863	−0.624314	−	0.781174i	$-0.714621\pi$
$569$	−9.64303e6	−1.24863	−0.624314	+	0.781174i	$0.714621\pi$
$570$	0	0
$571$	−1.12378e7	−1.44242	−0.721211	−	0.692715i	$-0.756414\pi$
$571$	−1.12378e7	−1.44242	−0.721211	+	0.692715i	$0.756414\pi$
$572$	0	0
$573$	3.60258e6	0.458381
$574$	0	0
$575$	−1.34816e7	−1.70048
$576$	0	0
$577$	−5.44674e6	−0.681078	−0.340539	−	0.940230i	$-0.610610\pi$
$577$	−5.44674e6	−0.681078	−0.340539	+	0.940230i	$0.610610\pi$
$578$	0	0
$579$	−9.01463e6	−1.11751
$580$	0	0
$581$	20450.9	0.00251346
$582$	0	0
$583$	3.41667e6	0.416324
$584$	0	0
$585$	−2.13699e6	−0.258174
$586$	0	0
$587$	−9.96240e6	−1.19335	−0.596676	−	0.802482i	$-0.703512\pi$
$587$	−9.96240e6	−1.19335	−0.596676	+	0.802482i	$0.703512\pi$
$588$	0	0
$589$	341426.	0.0405516
$590$	0	0
$591$	−3.15380e6	−0.371420
$592$	0	0
$593$	−3.17929e6	−0.371273	−0.185636	−	0.982619i	$-0.559435\pi$
$593$	−3.17929e6	−0.371273	−0.185636	+	0.982619i	$0.559435\pi$
$594$	0	0
$595$	225624.	0.0261272
$596$	0	0
$597$	−3.37257e6	−0.387280
$598$	0	0
$599$	1.66017e7	1.89054	0.945271	−	0.326285i	$-0.105797\pi$
$599$	1.66017e7	1.89054	0.945271	+	0.326285i	$0.105797\pi$
$600$	0	0
$601$	1.05771e6	0.119449	0.0597244	−	0.998215i	$-0.480978\pi$
$601$	1.05771e6	0.119449	0.0597244	+	0.998215i	$0.480978\pi$
$602$	0	0
$603$	−428141.	−0.0479505
$604$	0	0
$605$	1.52835e7	1.69759
$606$	0	0
$607$	−2.42026e6	−0.266619	−0.133309	−	0.991074i	$-0.542560\pi$
$607$	−2.42026e6	−0.266619	−0.133309	+	0.991074i	$0.542560\pi$
$608$	0	0
$609$	−161790.	−0.0176770
$610$	0	0
$611$	−1.41027e6	−0.152826
$612$	0	0
$613$	110410.	0.0118675	0.00593373	−	0.999982i	$-0.498111\pi$
$613$	110410.	0.0118675	0.00593373	+	0.999982i	$0.498111\pi$
$614$	0	0
$615$	1.32079e7	1.40814
$616$	0	0
$617$	−1.06544e7	−1.12672	−0.563361	−	0.826211i	$-0.690492\pi$
$617$	−1.06544e7	−1.12672	−0.563361	+	0.826211i	$0.690492\pi$
$618$	0	0
$619$	1.30456e7	1.36848	0.684240	−	0.729256i	$-0.260134\pi$
$619$	1.30456e7	1.36848	0.684240	+	0.729256i	$0.260134\pi$
$620$	0	0
$621$	−9.81499e6	−1.02132
$622$	0	0
$623$	−306466.	−0.0316346
$624$	0	0
$625$	2.30421e6	0.235951
$626$	0	0
$627$	1.38555e7	1.40751
$628$	0	0
$629$	441453.	0.0444895
$630$	0	0
$631$	1.53156e6	0.153130	0.0765651	−	0.997065i	$-0.475605\pi$
$631$	1.53156e6	0.153130	0.0765651	+	0.997065i	$0.475605\pi$
$632$	0	0
$633$	−5.24103e6	−0.519885
$634$	0	0
$635$	5.02079e6	0.494126
$636$	0	0
$637$	−2.83900e6	−0.277215
$638$	0	0
$639$	5.94302e6	0.575778
$640$	0	0
$641$	8.09995e6	0.778641	0.389321	−	0.921102i	$-0.372710\pi$
$641$	8.09995e6	0.778641	0.389321	+	0.921102i	$0.372710\pi$
$642$	0	0
$643$	5.29175e6	0.504745	0.252373	−	0.967630i	$-0.418789\pi$
$643$	5.29175e6	0.504745	0.252373	+	0.967630i	$0.418789\pi$
$644$	0	0
$645$	7.35223e6	0.695856
$646$	0	0
$647$	−1.80626e7	−1.69637	−0.848185	−	0.529700i	$-0.822305\pi$
$647$	−1.80626e7	−1.69637	−0.848185	+	0.529700i	$0.822305\pi$
$648$	0	0
$649$	1.20679e6	0.112466
$650$	0	0
$651$	4265.07	0.000394434 0
$652$	0	0
$653$	−7.10212e6	−0.651786	−0.325893	−	0.945407i	$-0.605665\pi$
$653$	−7.10212e6	−0.651786	−0.325893	+	0.945407i	$0.605665\pi$
$654$	0	0
$655$	−2.38052e7	−2.16805
$656$	0	0
$657$	−933954.	−0.0844135
$658$	0	0
$659$	−1.41430e7	−1.26861	−0.634303	−	0.773084i	$-0.718713\pi$
$659$	−1.41430e7	−1.26861	−0.634303	+	0.773084i	$0.718713\pi$
$660$	0	0
$661$	−675807.	−0.0601615	−0.0300808	−	0.999547i	$-0.509576\pi$
$661$	−675807.	−0.0601615	−0.0300808	+	0.999547i	$0.509576\pi$
$662$	0	0
$663$	1.48741e6	0.131415
$664$	0	0
$665$	−621280.	−0.0544795
$666$	0	0
$667$	1.37969e7	1.20079
$668$	0	0
$669$	−154936.	−0.0133840
$670$	0	0
$671$	9.94447e6	0.852659
$672$	0	0
$673$	2.66714e6	0.226991	0.113496	−	0.993539i	$-0.463795\pi$
$673$	2.66714e6	0.226991	0.113496	+	0.993539i	$0.463795\pi$
$674$	0	0
$675$	2.10087e7	1.77476
$676$	0	0
$677$	−1.20319e7	−1.00893	−0.504465	−	0.863432i	$-0.668310\pi$
$677$	−1.20319e7	−1.00893	−0.504465	+	0.863432i	$0.668310\pi$
$678$	0	0
$679$	−121462.	−0.0101104
$680$	0	0
$681$	2.81568e6	0.232656
$682$	0	0
$683$	1.83758e7	1.50728	0.753640	−	0.657287i	$-0.228296\pi$
$683$	1.83758e7	1.50728	0.753640	+	0.657287i	$0.228296\pi$
$684$	0	0
$685$	2.70778e7	2.20489
$686$	0	0
$687$	−7.64379e6	−0.617898
$688$	0	0
$689$	−1.00998e6	−0.0810521
$690$	0	0
$691$	−1.33412e7	−1.06292	−0.531459	−	0.847084i	$-0.678356\pi$
$691$	−1.33412e7	−1.06292	−0.531459	+	0.847084i	$0.678356\pi$
$692$	0	0
$693$	−224369.	−0.0177472
$694$	0	0
$695$	−1.66290e7	−1.30588
$696$	0	0
$697$	1.19171e7	0.929159
$698$	0	0
$699$	−1.15340e7	−0.892865
$700$	0	0
$701$	2.51952e6	0.193652	0.0968262	−	0.995301i	$-0.469131\pi$
$701$	2.51952e6	0.193652	0.0968262	+	0.995301i	$0.469131\pi$
$702$	0	0
$703$	−1.21559e6	−0.0927679
$704$	0	0
$705$	7.91281e6	0.599595
$706$	0	0
$707$	251607.	0.0189310
$708$	0	0
$709$	2.51441e7	1.87854	0.939272	−	0.343174i	$-0.111502\pi$
$709$	2.51441e7	1.87854	0.939272	+	0.343174i	$0.111502\pi$
$710$	0	0
$711$	−175698.	−0.0130345
$712$	0	0
$713$	−363709.	−0.0267936
$714$	0	0
$715$	−8.90621e6	−0.651520
$716$	0	0
$717$	2.87868e6	0.209120
$718$	0	0
$719$	−415865.	−0.0300006	−0.0150003	−	0.999887i	$-0.504775\pi$
$719$	−415865.	−0.0300006	−0.0150003	+	0.999887i	$0.504775\pi$
$720$	0	0
$721$	277179.	0.0198574
$722$	0	0
$723$	−8.40330e6	−0.597866
$724$	0	0
$725$	−2.95318e7	−2.08663
$726$	0	0
$727$	1.53987e7	1.08056	0.540278	−	0.841486i	$-0.318319\pi$
$727$	1.53987e7	1.08056	0.540278	+	0.841486i	$0.318319\pi$
$728$	0	0
$729$	1.07222e7	0.747251
$730$	0	0
$731$	6.63374e6	0.459161
$732$	0	0
$733$	−1.82125e7	−1.25202	−0.626008	−	0.779817i	$-0.715312\pi$
$733$	−1.82125e7	−1.25202	−0.626008	+	0.779817i	$0.715312\pi$
$734$	0	0
$735$	1.59292e7	1.08762
$736$	0	0
$737$	−1.78434e6	−0.121007
$738$	0	0
$739$	1.88107e7	1.26705	0.633524	−	0.773723i	$-0.281608\pi$
$739$	1.88107e7	1.26705	0.633524	+	0.773723i	$0.281608\pi$
$740$	0	0
$741$	−4.09573e6	−0.274022
$742$	0	0
$743$	−2.61292e7	−1.73642	−0.868209	−	0.496199i	$-0.834729\pi$
$743$	−2.61292e7	−1.73642	−0.868209	+	0.496199i	$0.834729\pi$
$744$	0	0
$745$	4.75649e7	3.13976
$746$	0	0
$747$	980611.	0.0642977
$748$	0	0
$749$	304710.	0.0198464
$750$	0	0
$751$	1.90282e7	1.23111	0.615557	−	0.788092i	$-0.288931\pi$
$751$	1.90282e7	1.23111	0.615557	+	0.788092i	$0.288931\pi$
$752$	0	0
$753$	−7.88699e6	−0.506902
$754$	0	0
$755$	3.07664e7	1.96431
$756$	0	0
$757$	−1.91639e7	−1.21547	−0.607734	−	0.794140i	$-0.707921\pi$
$757$	−1.91639e7	−1.21547	−0.607734	+	0.794140i	$0.707921\pi$
$758$	0	0
$759$	−1.47598e7	−0.929983
$760$	0	0
$761$	−468091.	−0.0293001	−0.0146500	−	0.999893i	$-0.504663\pi$
$761$	−468091.	−0.0293001	−0.0146500	+	0.999893i	$0.504663\pi$
$762$	0	0
$763$	407732.	0.0253550
$764$	0	0
$765$	1.08186e7	0.668371
$766$	0	0
$767$	−356732.	−0.0218954
$768$	0	0
$769$	−1.92729e7	−1.17525	−0.587627	−	0.809132i	$-0.699938\pi$
$769$	−1.92729e7	−1.17525	−0.587627	+	0.809132i	$0.699938\pi$
$770$	0	0
$771$	1.87648e7	1.13686
$772$	0	0
$773$	2.42083e7	1.45719	0.728593	−	0.684947i	$-0.240175\pi$
$773$	2.42083e7	1.45719	0.728593	+	0.684947i	$0.240175\pi$
$774$	0	0
$775$	778509.	0.0465596
$776$	0	0
$777$	−15185.0	−0.000902326 0
$778$	0	0
$779$	−3.28151e7	−1.93745
$780$	0	0
$781$	2.47684e7	1.45302
$782$	0	0
$783$	−2.15001e7	−1.25324
$784$	0	0
$785$	−2.45066e7	−1.41941
$786$	0	0
$787$	1.48087e7	0.852276	0.426138	−	0.904658i	$-0.359874\pi$
$787$	1.48087e7	0.852276	0.426138	+	0.904658i	$0.359874\pi$
$788$	0	0
$789$	−3.02701e6	−0.173110
$790$	0	0
$791$	599470.	0.0340664
$792$	0	0
$793$	−2.93962e6	−0.166000
$794$	0	0
$795$	5.66686e6	0.317998
$796$	0	0
$797$	−2.90994e7	−1.62270	−0.811351	−	0.584560i	$-0.801267\pi$
$797$	−2.90994e7	−1.62270	−0.811351	+	0.584560i	$0.801267\pi$
$798$	0	0
$799$	7.13954e6	0.395643
$800$	0	0
$801$	−1.46949e7	−0.809257
$802$	0	0
$803$	−3.89239e6	−0.213023
$804$	0	0
$805$	661828.	0.0359961
$806$	0	0
$807$	−2.98052e6	−0.161105
$808$	0	0
$809$	3.37792e7	1.81459	0.907295	−	0.420496i	$-0.138144\pi$
$809$	3.37792e7	1.81459	0.907295	+	0.420496i	$0.138144\pi$
$810$	0	0
$811$	2.42986e7	1.29727	0.648633	−	0.761101i	$-0.275341\pi$
$811$	2.42986e7	1.29727	0.648633	+	0.761101i	$0.275341\pi$
$812$	0	0
$813$	1.67861e7	0.890686
$814$	0	0
$815$	2.13317e7	1.12494
$816$	0	0
$817$	−1.82667e7	−0.957425
$818$	0	0
$819$	66324.3	0.00345512
$820$	0	0
$821$	−7.42442e6	−0.384419	−0.192209	−	0.981354i	$-0.561565\pi$
$821$	−7.42442e6	−0.384419	−0.192209	+	0.981354i	$0.561565\pi$
$822$	0	0
$823$	2.98540e7	1.53640	0.768198	−	0.640212i	$-0.221153\pi$
$823$	2.98540e7	1.53640	0.768198	+	0.640212i	$0.221153\pi$
$824$	0	0
$825$	3.15929e7	1.61605
$826$	0	0
$827$	−2.47154e7	−1.25662	−0.628311	−	0.777962i	$-0.716253\pi$
$827$	−2.47154e7	−1.25662	−0.628311	+	0.777962i	$0.716253\pi$
$828$	0	0
$829$	−1.14800e7	−0.580169	−0.290084	−	0.957001i	$-0.593683\pi$
$829$	−1.14800e7	−0.580169	−0.290084	+	0.957001i	$0.593683\pi$
$830$	0	0
$831$	−4.10314e6	−0.206117
$832$	0	0
$833$	1.43726e7	0.717665
$834$	0	0
$835$	6.04899e7	3.00239
$836$	0	0
$837$	566779.	0.0279640
$838$	0	0
$839$	−2.41188e7	−1.18291	−0.591454	−	0.806339i	$-0.701446\pi$
$839$	−2.41188e7	−1.18291	−0.591454	+	0.806339i	$0.701446\pi$
$840$	0	0
$841$	9.71140e6	0.473469
$842$	0	0
$843$	−2.82491e6	−0.136910
$844$	0	0
$845$	2.63271e6	0.126841
$846$	0	0
$847$	−474343.	−0.0227187
$848$	0	0
$849$	−8.17487e6	−0.389235
$850$	0	0
$851$	1.29492e6	0.0612943
$852$	0	0
$853$	−2.02681e6	−0.0953764	−0.0476882	−	0.998862i	$-0.515185\pi$
$853$	−2.02681e6	−0.0953764	−0.0476882	+	0.998862i	$0.515185\pi$
$854$	0	0
$855$	−2.97901e7	−1.39366
$856$	0	0
$857$	1.52859e7	0.710950	0.355475	−	0.934686i	$-0.384319\pi$
$857$	1.52859e7	0.710950	0.355475	+	0.934686i	$0.384319\pi$
$858$	0	0
$859$	−1.78567e7	−0.825693	−0.412846	−	0.910801i	$-0.635465\pi$
$859$	−1.78567e7	−0.825693	−0.412846	+	0.910801i	$0.635465\pi$
$860$	0	0
$861$	−409924.	−0.0188450
$862$	0	0
$863$	1.12315e7	0.513347	0.256673	−	0.966498i	$-0.417374\pi$
$863$	1.12315e7	0.513347	0.256673	+	0.966498i	$0.417374\pi$
$864$	0	0
$865$	4.15832e6	0.188963
$866$	0	0
$867$	7.07595e6	0.319696
$868$	0	0
$869$	−732249.	−0.0328934
$870$	0	0
$871$	527457.	0.0235582
$872$	0	0
$873$	−5.82408e6	−0.258638
$874$	0	0
$875$	−592524.	−0.0261629
$876$	0	0
$877$	1.75233e6	0.0769336	0.0384668	−	0.999260i	$-0.487753\pi$
$877$	1.75233e6	0.0769336	0.0384668	+	0.999260i	$0.487753\pi$
$878$	0	0
$879$	1.59780e7	0.697510
$880$	0	0
$881$	−1.04009e7	−0.451474	−0.225737	−	0.974188i	$-0.572479\pi$
$881$	−1.04009e7	−0.451474	−0.225737	+	0.974188i	$0.572479\pi$
$882$	0	0
$883$	−4.25645e7	−1.83715	−0.918577	−	0.395241i	$-0.870661\pi$
$883$	−4.25645e7	−1.83715	−0.918577	+	0.395241i	$0.870661\pi$
$884$	0	0
$885$	2.00157e6	0.0859040
$886$	0	0
$887$	−201500.	−0.00859935	−0.00429968	−	0.999991i	$-0.501369\pi$
$887$	−201500.	−0.00859935	−0.00429968	+	0.999991i	$0.501369\pi$
$888$	0	0
$889$	−155827.	−0.00661285
$890$	0	0
$891$	3.94294e6	0.166389
$892$	0	0
$893$	−1.96595e7	−0.824980
$894$	0	0
$895$	−4.81029e7	−2.00731
$896$	0	0
$897$	4.36304e6	0.181054
$898$	0	0
$899$	−796718.	−0.0328780
$900$	0	0
$901$	5.11307e6	0.209831
$902$	0	0
$903$	−228186.	−0.00931259
$904$	0	0
$905$	−1.13177e7	−0.459342
$906$	0	0
$907$	−1.47930e7	−0.597088	−0.298544	−	0.954396i	$-0.596501\pi$
$907$	−1.47930e7	−0.597088	−0.298544	+	0.954396i	$0.596501\pi$
$908$	0	0
$909$	1.20645e7	0.484283
$910$	0	0
$911$	3.50070e7	1.39752	0.698762	−	0.715354i	$-0.253735\pi$
$911$	3.50070e7	1.39752	0.698762	+	0.715354i	$0.253735\pi$
$912$	0	0
$913$	4.08684e6	0.162260
$914$	0	0
$915$	1.64938e7	0.651281
$916$	0	0
$917$	738828.	0.0290148
$918$	0	0
$919$	−4.24526e7	−1.65812	−0.829060	−	0.559160i	$-0.811124\pi$
$919$	−4.24526e7	−1.65812	−0.829060	+	0.559160i	$0.811124\pi$
$920$	0	0
$921$	2.13080e7	0.827741
$922$	0	0
$923$	−7.32164e6	−0.282881
$924$	0	0
$925$	−2.77175e6	−0.106512
$926$	0	0
$927$	1.32906e7	0.507980
$928$	0	0
$929$	−3.95487e7	−1.50347	−0.751733	−	0.659468i	$-0.770782\pi$
$929$	−3.95487e7	−1.50347	−0.751733	+	0.659468i	$0.770782\pi$
$930$	0	0
$931$	−3.95764e7	−1.49645
$932$	0	0
$933$	−2.58024e7	−0.970413
$934$	0	0
$935$	4.50881e7	1.68668
$936$	0	0
$937$	−4.62850e7	−1.72223	−0.861116	−	0.508409i	$-0.830234\pi$
$937$	−4.62850e7	−1.72223	−0.861116	+	0.508409i	$0.830234\pi$
$938$	0	0
$939$	−8.56270e6	−0.316918
$940$	0	0
$941$	5.74465e6	0.211490	0.105745	−	0.994393i	$-0.466277\pi$
$941$	5.74465e6	0.211490	0.105745	+	0.994393i	$0.466277\pi$
$942$	0	0
$943$	3.49568e7	1.28013
$944$	0	0
$945$	−1.03135e6	−0.0375686
$946$	0	0
$947$	−5.15966e7	−1.86959	−0.934795	−	0.355187i	$-0.884417\pi$
$947$	−5.15966e7	−1.86959	−0.934795	+	0.355187i	$0.884417\pi$
$948$	0	0
$949$	1.15060e6	0.0414725
$950$	0	0
$951$	1.81603e7	0.651137
$952$	0	0
$953$	−2.18333e7	−0.778729	−0.389365	−	0.921084i	$-0.627305\pi$
$953$	−2.18333e7	−0.778729	−0.389365	+	0.921084i	$0.627305\pi$
$954$	0	0
$955$	−3.22816e7	−1.14537
$956$	0	0
$957$	−3.23318e7	−1.14117
$958$	0	0
$959$	−840398.	−0.0295079
$960$	0	0
$961$	−2.86081e7	−0.999266
$962$	0	0
$963$	1.46107e7	0.507699
$964$	0	0
$965$	8.07774e7	2.79236
$966$	0	0
$967$	7.30509e6	0.251223	0.125611	−	0.992080i	$-0.459911\pi$
$967$	7.30509e6	0.251223	0.125611	+	0.992080i	$0.459911\pi$
$968$	0	0
$969$	2.07348e7	0.709400
$970$	0	0
$971$	1.62933e6	0.0554576	0.0277288	−	0.999615i	$-0.491173\pi$
$971$	1.62933e6	0.0554576	0.0277288	+	0.999615i	$0.491173\pi$
$972$	0	0
$973$	516105.	0.0174765
$974$	0	0
$975$	−9.33896e6	−0.314621
$976$	0	0
$977$	−2.55515e7	−0.856407	−0.428204	−	0.903682i	$-0.640853\pi$
$977$	−2.55515e7	−0.856407	−0.428204	+	0.903682i	$0.640853\pi$
$978$	0	0
$979$	−6.12433e7	−2.04222
$980$	0	0
$981$	1.95506e7	0.648617
$982$	0	0
$983$	8.48095e6	0.279937	0.139969	−	0.990156i	$-0.455300\pi$
$983$	8.48095e6	0.279937	0.139969	+	0.990156i	$0.455300\pi$
$984$	0	0
$985$	2.82603e7	0.928080
$986$	0	0
$987$	−245585.	−0.00802434
$988$	0	0
$989$	1.94589e7	0.632597
$990$	0	0
$991$	2.74483e7	0.887833	0.443916	−	0.896068i	$-0.353589\pi$
$991$	2.74483e7	0.887833	0.443916	+	0.896068i	$0.353589\pi$
$992$	0	0
$993$	−2.93248e6	−0.0943762
$994$	0	0
$995$	3.02206e7	0.967710
$996$	0	0
$997$	−2.26765e7	−0.722501	−0.361251	−	0.932469i	$-0.617650\pi$
$997$	−2.26765e7	−0.722501	−0.361251	+	0.932469i	$0.617650\pi$
$998$	0	0
$999$	−2.01792e6	−0.0639719

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	208.6.a.j.1.2		3
4.3	odd	2		13.6.a.b.1.1	✓	3
8.3	odd	2		832.6.a.s.1.2		3
8.5	even	2		832.6.a.t.1.2		3
12.11	even	2		117.6.a.d.1.3		3
20.3	even	4		325.6.b.c.274.5		6
20.7	even	4		325.6.b.c.274.2		6
20.19	odd	2		325.6.a.c.1.3		3
28.27	even	2		637.6.a.b.1.1		3
52.31	even	4		169.6.b.b.168.5		6
52.47	even	4		169.6.b.b.168.2		6
52.51	odd	2		169.6.a.b.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
13.6.a.b.1.1	✓	3	4.3	odd	2
117.6.a.d.1.3		3	12.11	even	2
169.6.a.b.1.3		3	52.51	odd	2
169.6.b.b.168.2		6	52.47	even	4
169.6.b.b.168.5		6	52.31	even	4
208.6.a.j.1.2		3	1.1	even	1	trivial
325.6.a.c.1.3		3	20.19	odd	2
325.6.b.c.274.2		6	20.7	even	4
325.6.b.c.274.5		6	20.3	even	4
637.6.a.b.1.1		3	28.27	even	2
832.6.a.s.1.2		3	8.3	odd	2
832.6.a.t.1.2		3	8.5	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 208 = 2^{4} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	208.a (trivial)

\(f(q)\)	\(=\)	\(q-10.2870 q^{3} +92.1784 q^{5} -2.86088 q^{7} -137.178 q^{9} +O(q^{10})\)	\(q-10.2870 q^{3} +92.1784 q^{5} -2.86088 q^{7} -137.178 q^{9} -571.711 q^{11} +169.000 q^{13} -948.236 q^{15} -855.571 q^{17} +2355.90 q^{19} +29.4298 q^{21} -2509.66 q^{23} +5371.86 q^{25} +3910.88 q^{27} -5497.50 q^{29} +144.924 q^{31} +5881.17 q^{33} -263.712 q^{35} -515.975 q^{37} -1738.50 q^{39} -13928.9 q^{41} -7753.58 q^{43} -12644.9 q^{45} -8344.77 q^{47} -16798.8 q^{49} +8801.22 q^{51} -5976.21 q^{53} -52699.5 q^{55} -24235.1 q^{57} -2110.84 q^{59} -17394.2 q^{61} +392.452 q^{63} +15578.2 q^{65} +3121.05 q^{67} +25816.8 q^{69} -43323.3 q^{71} +6808.31 q^{73} -55260.1 q^{75} +1635.60 q^{77} +1280.80 q^{79} -6896.72 q^{81} -7148.44 q^{83} -78865.2 q^{85} +56552.6 q^{87} +107123. q^{89} -483.489 q^{91} -1490.82 q^{93} +217163. q^{95} +42456.2 q^{97} +78426.5 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 8 q^{3} + 56 q^{5} + 60 q^{7} - 191 q^{9}+O(q^{10}) \) 3 * q - 8 * q^3 + 56 * q^5 + 60 * q^7 - 191 * q^9	\( 3 q - 8 q^{3} + 56 q^{5} + 60 q^{7} - 191 q^{9} - 556 q^{11} + 507 q^{13} - 1972 q^{15} + 908 q^{17} - 148 q^{19} + 1390 q^{21} - 3624 q^{23} + 2023 q^{25} + 4276 q^{27} - 8758 q^{29} + 2608 q^{31} - 1828 q^{33} - 4348 q^{35} - 20632 q^{37} - 1352 q^{39} - 10998 q^{41} - 2032 q^{43} - 13918 q^{45} - 34260 q^{47} - 44571 q^{49} - 13468 q^{51} - 12570 q^{53} - 35312 q^{55} + 1716 q^{57} - 63948 q^{59} - 12754 q^{61} + 1648 q^{63} + 9464 q^{65} - 56132 q^{67} - 20112 q^{69} - 77580 q^{71} - 43026 q^{73} - 23548 q^{75} - 21052 q^{77} + 61872 q^{79} - 108221 q^{81} - 98092 q^{83} - 55226 q^{85} - 139072 q^{87} + 33694 q^{89} + 10140 q^{91} + 105392 q^{93} + 196560 q^{95} + 76334 q^{97} + 60332 q^{99}+O(q^{100}) \) 3 * q - 8 * q^3 + 56 * q^5 + 60 * q^7 - 191 * q^9 - 556 * q^11 + 507 * q^13 - 1972 * q^15 + 908 * q^17 - 148 * q^19 + 1390 * q^21 - 3624 * q^23 + 2023 * q^25 + 4276 * q^27 - 8758 * q^29 + 2608 * q^31 - 1828 * q^33 - 4348 * q^35 - 20632 * q^37 - 1352 * q^39 - 10998 * q^41 - 2032 * q^43 - 13918 * q^45 - 34260 * q^47 - 44571 * q^49 - 13468 * q^51 - 12570 * q^53 - 35312 * q^55 + 1716 * q^57 - 63948 * q^59 - 12754 * q^61 + 1648 * q^63 + 9464 * q^65 - 56132 * q^67 - 20112 * q^69 - 77580 * q^71 - 43026 * q^73 - 23548 * q^75 - 21052 * q^77 + 61872 * q^79 - 108221 * q^81 - 98092 * q^83 - 55226 * q^85 - 139072 * q^87 + 33694 * q^89 + 10140 * q^91 + 105392 * q^93 + 196560 * q^95 + 76334 * q^97 + 60332 * q^99

Introduction

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