LMFDB - Embedded newform 207.6.a.d.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [207,6,Mod(1,207)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(207, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("207.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 207 = 3^{2} \cdot 23 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	207.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:	$33.1994507013$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\mathbb{Q}[x]/(x^{4} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 39x^{2} - 30x + 20 $ x^4 - 39x^2 - 30x + 20
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 69)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-1.23317$ of defining polynomial
Character	$\chi$	$=$	207.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.233171 q^{2} -31.9456 q^{4} -18.8848 q^{5} -97.3695 q^{7} -14.9102 q^{8} +O(q^{10})$	\(q+0.233171 q^{2} -31.9456 q^{4} -18.8848 q^{5} -97.3695 q^{7} -14.9102 q^{8} -4.40338 q^{10} -331.380 q^{11} -212.534 q^{13} -22.7037 q^{14} +1018.78 q^{16} -1973.45 q^{17} +1412.55 q^{19} +603.286 q^{20} -77.2681 q^{22} -529.000 q^{23} -2768.36 q^{25} -49.5566 q^{26} +3110.53 q^{28} +1780.57 q^{29} +9986.80 q^{31} +714.678 q^{32} -460.150 q^{34} +1838.80 q^{35} +3479.68 q^{37} +329.365 q^{38} +281.577 q^{40} -3575.89 q^{41} +3386.82 q^{43} +10586.2 q^{44} -123.347 q^{46} +23186.4 q^{47} -7326.19 q^{49} -645.501 q^{50} +6789.53 q^{52} +14679.4 q^{53} +6258.05 q^{55} +1451.80 q^{56} +415.177 q^{58} -27304.7 q^{59} +9718.64 q^{61} +2328.63 q^{62} -32434.4 q^{64} +4013.66 q^{65} +66186.2 q^{67} +63043.0 q^{68} +428.754 q^{70} -23434.6 q^{71} +5814.11 q^{73} +811.360 q^{74} -45124.8 q^{76} +32266.3 q^{77} +57792.8 q^{79} -19239.5 q^{80} -833.792 q^{82} -48918.6 q^{83} +37268.1 q^{85} +789.708 q^{86} +4940.96 q^{88} -93715.8 q^{89} +20694.3 q^{91} +16899.2 q^{92} +5406.39 q^{94} -26675.7 q^{95} +150381. q^{97} -1708.25 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{2} - 46 q^{4} + 122 q^{5} + 62 q^{7} - 72 q^{8}+O(q^{10}) $ 4 * q - 4 * q^2 - 46 * q^4 + 122 * q^5 + 62 * q^7 - 72 * q^8	$ 4 q - 4 q^{2} - 46 q^{4} + 122 q^{5} + 62 q^{7} - 72 q^{8} + 642 q^{10} - 32 q^{11} + 1364 q^{13} - 2754 q^{14} + 18 q^{16} - 278 q^{17} + 2862 q^{19} - 3830 q^{20} + 3176 q^{22} - 2116 q^{23} + 5944 q^{25} - 6996 q^{26} + 4738 q^{28} + 5180 q^{29} - 1788 q^{31} + 7352 q^{32} + 15818 q^{34} - 11768 q^{35} + 3348 q^{37} - 1050 q^{38} - 13462 q^{40} + 17664 q^{41} + 25398 q^{43} + 16848 q^{44} + 2116 q^{46} + 26040 q^{47} + 55720 q^{49} + 35256 q^{50} - 2752 q^{52} + 32006 q^{53} + 34904 q^{55} + 68542 q^{56} - 40804 q^{58} + 61136 q^{59} + 35844 q^{61} + 47524 q^{62} - 35142 q^{64} + 48036 q^{65} + 73458 q^{67} - 17910 q^{68} - 59104 q^{70} - 24432 q^{71} + 122512 q^{73} + 20828 q^{74} + 56834 q^{76} - 159496 q^{77} + 90170 q^{79} + 36546 q^{80} + 84144 q^{82} - 28592 q^{83} + 355124 q^{85} - 103778 q^{86} - 150776 q^{88} + 27926 q^{89} + 334180 q^{91} + 24334 q^{92} + 113632 q^{94} - 113392 q^{95} + 16580 q^{97} - 49688 q^{98}+O(q^{100}) $ 4 * q - 4 * q^2 - 46 * q^4 + 122 * q^5 + 62 * q^7 - 72 * q^8 + 642 * q^10 - 32 * q^11 + 1364 * q^13 - 2754 * q^14 + 18 * q^16 - 278 * q^17 + 2862 * q^19 - 3830 * q^20 + 3176 * q^22 - 2116 * q^23 + 5944 * q^25 - 6996 * q^26 + 4738 * q^28 + 5180 * q^29 - 1788 * q^31 + 7352 * q^32 + 15818 * q^34 - 11768 * q^35 + 3348 * q^37 - 1050 * q^38 - 13462 * q^40 + 17664 * q^41 + 25398 * q^43 + 16848 * q^44 + 2116 * q^46 + 26040 * q^47 + 55720 * q^49 + 35256 * q^50 - 2752 * q^52 + 32006 * q^53 + 34904 * q^55 + 68542 * q^56 - 40804 * q^58 + 61136 * q^59 + 35844 * q^61 + 47524 * q^62 - 35142 * q^64 + 48036 * q^65 + 73458 * q^67 - 17910 * q^68 - 59104 * q^70 - 24432 * q^71 + 122512 * q^73 + 20828 * q^74 + 56834 * q^76 - 159496 * q^77 + 90170 * q^79 + 36546 * q^80 + 84144 * q^82 - 28592 * q^83 + 355124 * q^85 - 103778 * q^86 - 150776 * q^88 + 27926 * q^89 + 334180 * q^91 + 24334 * q^92 + 113632 * q^94 - 113392 * q^95 + 16580 * q^97 - 49688 * q^98

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 $

$2$	0.233171	0.0412191	0.0206096	−	0.999788i	$-0.493439\pi$
$2$	0.233171	0.0412191	0.0206096	+	0.999788i	$0.493439\pi$
$3$	0	0
$3$	0	0
$4$	−31.9456	−0.998301
$5$	−18.8848	−0.337821	−0.168911	−	0.985631i	$-0.554025\pi$
$5$	−18.8848	−0.337821	−0.168911	+	0.985631i	$0.554025\pi$
$6$	0	0
$7$	−97.3695	−0.751065	−0.375533	−	0.926809i	$-0.622540\pi$
$7$	−97.3695	−0.751065	−0.375533	+	0.926809i	$0.622540\pi$
$8$	−14.9102	−0.0823682
$9$	0	0
$10$	−4.40338	−0.0139247
$11$	−331.380	−0.825743	−0.412872	−	0.910789i	$-0.635474\pi$
$11$	−331.380	−0.825743	−0.412872	+	0.910789i	$0.635474\pi$
$12$	0	0
$13$	−212.534	−0.348795	−0.174397	−	0.984675i	$-0.555798\pi$
$13$	−212.534	−0.348795	−0.174397	+	0.984675i	$0.555798\pi$
$14$	−22.7037	−0.0309583
$15$	0	0
$16$	1018.78	0.994906
$17$	−1973.45	−1.65616	−0.828081	−	0.560608i	$-0.810568\pi$
$17$	−1973.45	−1.65616	−0.828081	+	0.560608i	$0.810568\pi$
$18$	0	0
$19$	1412.55	0.897676	0.448838	−	0.893613i	$-0.351838\pi$
$19$	1412.55	0.897676	0.448838	+	0.893613i	$0.351838\pi$
$20$	603.286	0.337247
$21$	0	0
$22$	−77.2681	−0.0340364
$23$	−529.000	−0.208514
$23$	−529.000	−0.208514
$24$	0	0
$25$	−2768.36	−0.885877
$26$	−49.5566	−0.0143770
$27$	0	0
$28$	3110.53	0.749789
$29$	1780.57	0.393155	0.196578	−	0.980488i	$-0.437017\pi$
$29$	1780.57	0.393155	0.196578	+	0.980488i	$0.437017\pi$
$30$	0	0
$31$	9986.80	1.86647	0.933237	−	0.359261i	$-0.116971\pi$
$31$	9986.80	1.86647	0.933237	+	0.359261i	$0.116971\pi$
$32$	714.678	0.123377
$33$	0	0
$34$	−460.150	−0.0682656
$35$	1838.80	0.253726
$36$	0	0
$37$	3479.68	0.417864	0.208932	−	0.977930i	$-0.433001\pi$
$37$	3479.68	0.417864	0.208932	+	0.977930i	$0.433001\pi$
$38$	329.365	0.0370014
$39$	0	0
$40$	281.577	0.0278257
$41$	−3575.89	−0.332219	−0.166110	−	0.986107i	$-0.553121\pi$
$41$	−3575.89	−0.332219	−0.166110	+	0.986107i	$0.553121\pi$
$42$	0	0
$43$	3386.82	0.279333	0.139666	−	0.990199i	$-0.455397\pi$
$43$	3386.82	0.279333	0.139666	+	0.990199i	$0.455397\pi$
$44$	10586.2	0.824340
$45$	0	0
$46$	−123.347	−0.00859478
$47$	23186.4	1.53105	0.765525	−	0.643406i	$-0.222479\pi$
$47$	23186.4	1.53105	0.765525	+	0.643406i	$0.222479\pi$
$48$	0	0
$49$	−7326.19	−0.435901
$50$	−645.501	−0.0365151
$51$	0	0
$52$	6789.53	0.348202
$53$	14679.4	0.717827	0.358913	−	0.933371i	$-0.383147\pi$
$53$	14679.4	0.717827	0.358913	+	0.933371i	$0.383147\pi$
$54$	0	0
$55$	6258.05	0.278954
$56$	1451.80	0.0618639
$57$	0	0
$58$	415.177	0.0162055
$59$	−27304.7	−1.02119	−0.510597	−	0.859820i	$-0.670576\pi$
$59$	−27304.7	−1.02119	−0.510597	+	0.859820i	$0.670576\pi$
$60$	0	0
$61$	9718.64	0.334411	0.167206	−	0.985922i	$-0.446526\pi$
$61$	9718.64	0.334411	0.167206	+	0.985922i	$0.446526\pi$
$62$	2328.63	0.0769344
$63$	0	0
$64$	−32434.4	−0.989820
$65$	4013.66	0.117830
$66$	0	0
$67$	66186.2	1.80128	0.900639	−	0.434568i	$-0.143099\pi$
$67$	66186.2	1.80128	0.900639	+	0.434568i	$0.143099\pi$
$68$	63043.0	1.65335
$69$	0	0
$70$	428.754	0.0104584
$71$	−23434.6	−0.551710	−0.275855	−	0.961199i	$-0.588961\pi$
$71$	−23434.6	−0.551710	−0.275855	+	0.961199i	$0.588961\pi$
$72$	0	0
$73$	5814.11	0.127696	0.0638479	−	0.997960i	$-0.479663\pi$
$73$	5814.11	0.127696	0.0638479	+	0.997960i	$0.479663\pi$
$74$	811.360	0.0172240
$75$	0	0
$76$	−45124.8	−0.896151
$77$	32266.3	0.620187
$78$	0	0
$79$	57792.8	1.04185	0.520926	−	0.853602i	$-0.325587\pi$
$79$	57792.8	1.04185	0.520926	+	0.853602i	$0.325587\pi$
$80$	−19239.5	−0.336100
$81$	0	0
$82$	−833.792	−0.0136938
$83$	−48918.6	−0.779434	−0.389717	−	0.920935i	$-0.627427\pi$
$83$	−48918.6	−0.779434	−0.389717	+	0.920935i	$0.627427\pi$
$84$	0	0
$85$	37268.1	0.559487
$86$	789.708	0.0115138
$87$	0	0
$88$	4940.96	0.0680150
$89$	−93715.8	−1.25412	−0.627058	−	0.778972i	$-0.715741\pi$
$89$	−93715.8	−1.25412	−0.627058	+	0.778972i	$0.715741\pi$
$90$	0	0
$91$	20694.3	0.261968
$92$	16899.2	0.208160
$93$	0	0
$94$	5406.39	0.0631085
$95$	−26675.7	−0.303254
$96$	0	0
$97$	150381.	1.62280	0.811399	−	0.584493i	$-0.198707\pi$
$97$	150381.	1.62280	0.811399	+	0.584493i	$0.198707\pi$
$98$	−1708.25	−0.0179674
$99$	0	0
$100$	88437.2	0.884372
$101$	146090.	1.42501	0.712504	−	0.701668i	$-0.247561\pi$
$101$	146090.	1.42501	0.712504	+	0.701668i	$0.247561\pi$
$102$	0	0
$103$	−19608.3	−0.182115	−0.0910575	−	0.995846i	$-0.529025\pi$
$103$	−19608.3	−0.182115	−0.0910575	+	0.995846i	$0.529025\pi$
$104$	3168.93	0.0287296
$105$	0	0
$106$	3422.81	0.0295882
$107$	−9563.34	−0.0807514	−0.0403757	−	0.999185i	$-0.512855\pi$
$107$	−9563.34	−0.0807514	−0.0403757	+	0.999185i	$0.512855\pi$
$108$	0	0
$109$	50850.7	0.409950	0.204975	−	0.978767i	$-0.434289\pi$
$109$	50850.7	0.409950	0.204975	+	0.978767i	$0.434289\pi$
$110$	1459.19	0.0114982
$111$	0	0
$112$	−99198.4	−0.747239
$113$	−82996.1	−0.611451	−0.305725	−	0.952120i	$-0.598899\pi$
$113$	−82996.1	−0.611451	−0.305725	+	0.952120i	$0.598899\pi$
$114$	0	0
$115$	9990.05	0.0704406
$116$	−56881.5	−0.392487
$117$	0	0
$118$	−6366.66	−0.0420927
$119$	192153.	1.24389
$120$	0	0
$121$	−51238.1	−0.318148
$122$	2266.10	0.0137841
$123$	0	0
$124$	−319034.	−1.86330
$125$	111295.	0.637089
$126$	0	0
$127$	−89569.5	−0.492777	−0.246389	−	0.969171i	$-0.579244\pi$
$127$	−89569.5	−0.492777	−0.246389	+	0.969171i	$0.579244\pi$
$128$	−30432.4	−0.164177
$129$	0	0
$130$	935.867	0.00485686
$131$	−302834.	−1.54180	−0.770898	−	0.636959i	$-0.780192\pi$
$131$	−302834.	−1.54180	−0.770898	+	0.636959i	$0.780192\pi$
$132$	0	0
$133$	−137539.	−0.674213
$134$	15432.7	0.0742471
$135$	0	0
$136$	29424.6	0.136415
$137$	54160.9	0.246538	0.123269	−	0.992373i	$-0.460662\pi$
$137$	54160.9	0.246538	0.123269	+	0.992373i	$0.460662\pi$
$138$	0	0
$139$	−305755.	−1.34226	−0.671130	−	0.741340i	$-0.734191\pi$
$139$	−305755.	−1.34226	−0.671130	+	0.741340i	$0.734191\pi$
$140$	−58741.7	−0.253295
$141$	0	0
$142$	−5464.25	−0.0227410
$143$	70429.5	0.288015
$144$	0	0
$145$	−33625.7	−0.132816
$146$	1355.68	0.00526351
$147$	0	0
$148$	−111161.	−0.417154
$149$	52473.3	0.193630	0.0968151	−	0.995302i	$-0.469134\pi$
$149$	52473.3	0.193630	0.0968151	+	0.995302i	$0.469134\pi$
$150$	0	0
$151$	−128555.	−0.458824	−0.229412	−	0.973329i	$-0.573680\pi$
$151$	−128555.	−0.458824	−0.229412	+	0.973329i	$0.573680\pi$
$152$	−21061.4	−0.0739399
$153$	0	0
$154$	7523.56	0.0255636
$155$	−188599.	−0.630535
$156$	0	0
$157$	−88511.3	−0.286583	−0.143291	−	0.989681i	$-0.545769\pi$
$157$	−88511.3	−0.286583	−0.143291	+	0.989681i	$0.545769\pi$
$158$	13475.6	0.0429442
$159$	0	0
$160$	−13496.5	−0.0416795
$161$	51508.5	0.156608
$162$	0	0
$163$	−238414.	−0.702851	−0.351426	−	0.936216i	$-0.614303\pi$
$163$	−238414.	−0.702851	−0.351426	+	0.936216i	$0.614303\pi$
$164$	114234.	0.331655
$165$	0	0
$166$	−11406.4	−0.0321276
$167$	−549507.	−1.52469	−0.762345	−	0.647171i	$-0.775952\pi$
$167$	−549507.	−1.52469	−0.762345	+	0.647171i	$0.775952\pi$
$168$	0	0
$169$	−326122.	−0.878342
$170$	8689.83	0.0230616
$171$	0	0
$172$	−108194.	−0.278858
$173$	553345.	1.40566	0.702831	−	0.711357i	$-0.251919\pi$
$173$	553345.	1.40566	0.702831	+	0.711357i	$0.251919\pi$
$174$	0	0
$175$	269554.	0.665351
$176$	−337605.	−0.821537
$177$	0	0
$178$	−21851.8	−0.0516936
$179$	639248.	1.49120	0.745602	−	0.666392i	$-0.232162\pi$
$179$	639248.	1.49120	0.745602	+	0.666392i	$0.232162\pi$
$180$	0	0
$181$	371569.	0.843030	0.421515	−	0.906822i	$-0.361499\pi$
$181$	371569.	0.843030	0.421515	+	0.906822i	$0.361499\pi$
$182$	4825.30	0.0107981
$183$	0	0
$184$	7887.52	0.0171750
$185$	−65713.1	−0.141163
$186$	0	0
$187$	653961.	1.36757
$188$	−740705.	−1.52845
$189$	0	0
$190$	−6219.98	−0.0124999
$191$	−476400.	−0.944905	−0.472453	−	0.881356i	$-0.656631\pi$
$191$	−476400.	−0.944905	−0.472453	+	0.881356i	$0.656631\pi$
$192$	0	0
$193$	8459.49	0.0163475	0.00817374	−	0.999967i	$-0.497398\pi$
$193$	8459.49	0.0163475	0.00817374	+	0.999967i	$0.497398\pi$
$194$	35064.5	0.0668903
$195$	0	0
$196$	234040.	0.435160
$197$	−218549.	−0.401220	−0.200610	−	0.979671i	$-0.564292\pi$
$197$	−218549.	−0.401220	−0.200610	+	0.979671i	$0.564292\pi$
$198$	0	0
$199$	261692.	0.468444	0.234222	−	0.972183i	$-0.424746\pi$
$199$	261692.	0.468444	0.234222	+	0.972183i	$0.424746\pi$
$200$	41277.0	0.0729681
$201$	0	0
$202$	34063.9	0.0587376
$203$	−173373.	−0.295285
$204$	0	0
$205$	67529.9	0.112231
$206$	−4572.07	−0.00750662
$207$	0	0
$208$	−216526.	−0.347018
$209$	−468091.	−0.741250
$210$	0	0
$211$	603463.	0.933135	0.466568	−	0.884486i	$-0.345490\pi$
$211$	603463.	0.933135	0.466568	+	0.884486i	$0.345490\pi$
$212$	−468944.	−0.716607
$213$	0	0
$214$	−2229.89	−0.00332850
$215$	−63959.5	−0.0943645
$216$	0	0
$217$	−972409.	−1.40184
$218$	11856.9	0.0168978
$219$	0	0
$220$	−199917.	−0.278480
$221$	419424.	0.577661
$222$	0	0
$223$	200587.	0.270110	0.135055	−	0.990838i	$-0.456879\pi$
$223$	200587.	0.270110	0.135055	+	0.990838i	$0.456879\pi$
$224$	−69587.8	−0.0926645
$225$	0	0
$226$	−19352.2	−0.0252035
$227$	1.38611e6	1.78539	0.892695	−	0.450661i	$-0.148812\pi$
$227$	1.38611e6	1.78539	0.892695	+	0.450661i	$0.148812\pi$
$228$	0	0
$229$	−176784.	−0.222769	−0.111385	−	0.993777i	$-0.535529\pi$
$229$	−176784.	−0.222769	−0.111385	+	0.993777i	$0.535529\pi$
$230$	2329.39	0.00290350
$231$	0	0
$232$	−26548.7	−0.0323835
$233$	623251.	0.752096	0.376048	−	0.926600i	$-0.377283\pi$
$233$	623251.	0.752096	0.376048	+	0.926600i	$0.377283\pi$
$234$	0	0
$235$	−437871.	−0.517221
$236$	872267.	1.01946
$237$	0	0
$238$	44804.5	0.0512719
$239$	96056.9	0.108776	0.0543881	−	0.998520i	$-0.482679\pi$
$239$	96056.9	0.108776	0.0543881	+	0.998520i	$0.482679\pi$
$240$	0	0
$241$	1.11624e6	1.23798	0.618991	−	0.785398i	$-0.287542\pi$
$241$	1.11624e6	1.23798	0.618991	+	0.785398i	$0.287542\pi$
$242$	−11947.2	−0.0131138
$243$	0	0
$244$	−310468.	−0.333843
$245$	138353.	0.147257
$246$	0	0
$247$	−300215.	−0.313105
$248$	−148905.	−0.153738
$249$	0	0
$250$	25950.7	0.0262603
$251$	−117749.	−0.117970	−0.0589849	−	0.998259i	$-0.518786\pi$
$251$	−117749.	−0.117970	−0.0589849	+	0.998259i	$0.518786\pi$
$252$	0	0
$253$	175300.	0.172179
$254$	−20885.0	−0.0203118
$255$	0	0
$256$	1.03081e6	0.983053
$257$	423854.	0.400298	0.200149	−	0.979766i	$-0.435857\pi$
$257$	423854.	0.400298	0.200149	+	0.979766i	$0.435857\pi$
$258$	0	0
$259$	−338815.	−0.313843
$260$	−128219.	−0.117630
$261$	0	0
$262$	−70612.0	−0.0635515
$263$	1.60498e6	1.43080	0.715402	−	0.698713i	$-0.246244\pi$
$263$	1.60498e6	1.43080	0.715402	+	0.698713i	$0.246244\pi$
$264$	0	0
$265$	−277218.	−0.242497
$266$	−32070.1	−0.0277905
$267$	0	0
$268$	−2.11436e6	−1.79822
$269$	−935362.	−0.788132	−0.394066	−	0.919082i	$-0.628932\pi$
$269$	−935362.	−0.788132	−0.394066	+	0.919082i	$0.628932\pi$
$270$	0	0
$271$	1.21171e6	1.00225	0.501124	−	0.865376i	$-0.332920\pi$
$271$	1.21171e6	1.00225	0.501124	+	0.865376i	$0.332920\pi$
$272$	−2.01051e6	−1.64773
$273$	0	0
$274$	12628.7	0.0101621
$275$	917382.	0.731507
$276$	0	0
$277$	85343.7	0.0668301	0.0334151	−	0.999442i	$-0.489362\pi$
$277$	85343.7	0.0668301	0.0334151	+	0.999442i	$0.489362\pi$
$278$	−71293.1	−0.0553268
$279$	0	0
$280$	−27417.0	−0.0208989
$281$	233077.	0.176089	0.0880447	−	0.996117i	$-0.471938\pi$
$281$	233077.	0.176089	0.0880447	+	0.996117i	$0.471938\pi$
$282$	0	0
$283$	1.99517e6	1.48086	0.740431	−	0.672133i	$-0.234622\pi$
$283$	1.99517e6	1.48086	0.740431	+	0.672133i	$0.234622\pi$
$284$	748632.	0.550773
$285$	0	0
$286$	16422.1	0.0118717
$287$	348183.	0.249518
$288$	0	0
$289$	2.47463e6	1.74288
$290$	−7840.52	−0.00547457
$291$	0	0
$292$	−185736.	−0.127479
$293$	2.15191e6	1.46438	0.732192	−	0.681098i	$-0.238497\pi$
$293$	2.15191e6	1.46438	0.732192	+	0.681098i	$0.238497\pi$
$294$	0	0
$295$	515644.	0.344981
$296$	−51882.9	−0.0344187
$297$	0	0
$298$	12235.2	0.00798126
$299$	112430.	0.0727287
$300$	0	0
$301$	−329773.	−0.209797
$302$	−29975.2	−0.0189123
$303$	0	0
$304$	1.43908e6	0.893103
$305$	−183534.	−0.112971
$306$	0	0
$307$	2.34688e6	1.42116	0.710582	−	0.703614i	$-0.248432\pi$
$307$	2.34688e6	1.42116	0.710582	+	0.703614i	$0.248432\pi$
$308$	−1.03077e6	−0.619133
$309$	0	0
$310$	−43975.6	−0.0259901
$311$	−380693.	−0.223190	−0.111595	−	0.993754i	$-0.535596\pi$
$311$	−380693.	−0.223190	−0.111595	+	0.993754i	$0.535596\pi$
$312$	0	0
$313$	−2.90408e6	−1.67551	−0.837756	−	0.546045i	$-0.816133\pi$
$313$	−2.90408e6	−1.67551	−0.837756	+	0.546045i	$0.816133\pi$
$314$	−20638.2	−0.0118127
$315$	0	0
$316$	−1.84623e6	−1.04008
$317$	1.94767e6	1.08860	0.544298	−	0.838892i	$-0.316796\pi$
$317$	1.94767e6	1.08860	0.544298	+	0.838892i	$0.316796\pi$
$318$	0	0
$319$	−590046.	−0.324645
$320$	612517.	0.334382
$321$	0	0
$322$	12010.3	0.00645524
$323$	−2.78759e6	−1.48670
$324$	0	0
$325$	588371.	0.308989
$326$	−55591.2	−0.0289709
$327$	0	0
$328$	53317.4	0.0273643
$329$	−2.25765e6	−1.14992
$330$	0	0
$331$	2.77966e6	1.39451	0.697256	−	0.716822i	$-0.254404\pi$
$331$	2.77966e6	1.39451	0.697256	+	0.716822i	$0.254404\pi$
$332$	1.56274e6	0.778109
$333$	0	0
$334$	−128129.	−0.0628464
$335$	−1.24991e6	−0.608510
$336$	0	0
$337$	1.19218e6	0.571831	0.285916	−	0.958255i	$-0.407702\pi$
$337$	1.19218e6	0.571831	0.285916	+	0.958255i	$0.407702\pi$
$338$	−76042.1	−0.0362045
$339$	0	0
$340$	−1.19055e6	−0.558537
$341$	−3.30943e6	−1.54123
$342$	0	0
$343$	2.34984e6	1.07846
$344$	−50498.4	−0.0230081
$345$	0	0
$346$	129024.	0.0579402
$347$	−1.95342e6	−0.870909	−0.435455	−	0.900211i	$-0.643412\pi$
$347$	−1.95342e6	−0.870909	−0.435455	+	0.900211i	$0.643412\pi$
$348$	0	0
$349$	2.11470e6	0.929362	0.464681	−	0.885478i	$-0.346169\pi$
$349$	2.11470e6	0.929362	0.464681	+	0.885478i	$0.346169\pi$
$350$	62852.1	0.0274252
$351$	0	0
$352$	−236830.	−0.101878
$353$	3.72688e6	1.59188	0.795938	−	0.605379i	$-0.206978\pi$
$353$	3.72688e6	1.59188	0.795938	+	0.605379i	$0.206978\pi$
$354$	0	0
$355$	442557.	0.186380
$356$	2.99381e6	1.25199
$357$	0	0
$358$	149054.	0.0614661
$359$	−2.09511e6	−0.857968	−0.428984	−	0.903312i	$-0.641128\pi$
$359$	−2.09511e6	−0.857968	−0.428984	+	0.903312i	$0.641128\pi$
$360$	0	0
$361$	−480804.	−0.194178
$362$	86638.9	0.0347489
$363$	0	0
$364$	−661093.	−0.261522
$365$	−109798.	−0.0431383
$366$	0	0
$367$	−4.32739e6	−1.67711	−0.838554	−	0.544818i	$-0.816599\pi$
$367$	−4.32739e6	−1.67711	−0.838554	+	0.544818i	$0.816599\pi$
$368$	−538937.	−0.207452
$369$	0	0
$370$	−15322.4	−0.00581863
$371$	−1.42933e6	−0.539135
$372$	0	0
$373$	−2.20478e6	−0.820527	−0.410263	−	0.911967i	$-0.634563\pi$
$373$	−2.20478e6	−0.820527	−0.410263	+	0.911967i	$0.634563\pi$
$374$	152485.	0.0563698
$375$	0	0
$376$	−345715.	−0.126110
$377$	−378432.	−0.137130
$378$	0	0
$379$	−2.24027e6	−0.801130	−0.400565	−	0.916268i	$-0.631186\pi$
$379$	−2.24027e6	−0.801130	−0.400565	+	0.916268i	$0.631186\pi$
$380$	852172.	0.302739
$381$	0	0
$382$	−111082.	−0.0389482
$383$	2.14403e6	0.746849	0.373425	−	0.927661i	$-0.378183\pi$
$383$	2.14403e6	0.746849	0.373425	+	0.927661i	$0.378183\pi$
$384$	0	0
$385$	−609343.	−0.209512
$386$	1972.50	0.000673829 0
$387$	0	0
$388$	−4.80402e6	−1.62004
$389$	−1.52914e6	−0.512359	−0.256179	−	0.966629i	$-0.582464\pi$
$389$	−1.52914e6	−0.512359	−0.256179	+	0.966629i	$0.582464\pi$
$390$	0	0
$391$	1.04395e6	0.345334
$392$	109235.	0.0359044
$393$	0	0
$394$	−50959.1	−0.0165379
$395$	−1.09140e6	−0.351960
$396$	0	0
$397$	−3.37596e6	−1.07503	−0.537516	−	0.843254i	$-0.680637\pi$
$397$	−3.37596e6	−1.07503	−0.537516	+	0.843254i	$0.680637\pi$
$398$	61018.8	0.0193088
$399$	0	0
$400$	−2.82036e6	−0.881364
$401$	4.80377e6	1.49184	0.745919	−	0.666037i	$-0.232011\pi$
$401$	4.80377e6	1.49184	0.745919	+	0.666037i	$0.232011\pi$
$402$	0	0
$403$	−2.12253e6	−0.651016
$404$	−4.66694e6	−1.42259
$405$	0	0
$406$	−40425.5	−0.0121714
$407$	−1.15310e6	−0.345049
$408$	0	0
$409$	−6.03130e6	−1.78280	−0.891400	−	0.453218i	$-0.850276\pi$
$409$	−6.03130e6	−1.78280	−0.891400	+	0.453218i	$0.850276\pi$
$410$	15746.0	0.00462605
$411$	0	0
$412$	626398.	0.181806
$413$	2.65865e6	0.766983
$414$	0	0
$415$	923818.	0.263309
$416$	−151893.	−0.0430334
$417$	0	0
$418$	−109145.	−0.0305537
$419$	−5.07275e6	−1.41159	−0.705795	−	0.708416i	$-0.749410\pi$
$419$	−5.07275e6	−1.41159	−0.705795	+	0.708416i	$0.749410\pi$
$420$	0	0
$421$	6.74194e6	1.85387	0.926935	−	0.375221i	$-0.122433\pi$
$421$	6.74194e6	1.85387	0.926935	+	0.375221i	$0.122433\pi$
$422$	140710.	0.0384630
$423$	0	0
$424$	−218874.	−0.0591261
$425$	5.46322e6	1.46716
$426$	0	0
$427$	−946299.	−0.251165
$428$	305507.	0.0806142
$429$	0	0
$430$	−14913.5	−0.00388962
$431$	−3.35415e6	−0.869739	−0.434869	−	0.900493i	$-0.643205\pi$
$431$	−3.35415e6	−0.869739	−0.434869	+	0.900493i	$0.643205\pi$
$432$	0	0
$433$	−350372.	−0.0898069	−0.0449035	−	0.998991i	$-0.514298\pi$
$433$	−350372.	−0.0898069	−0.0449035	+	0.998991i	$0.514298\pi$
$434$	−226737.	−0.0577828
$435$	0	0
$436$	−1.62446e6	−0.409253
$437$	−747238.	−0.187178
$438$	0	0
$439$	−7.21424e6	−1.78661	−0.893304	−	0.449453i	$-0.851619\pi$
$439$	−7.21424e6	−1.78661	−0.893304	+	0.449453i	$0.851619\pi$
$440$	−93309.0	−0.0229769
$441$	0	0
$442$	97797.4	0.0238107
$443$	1.73598e6	0.420277	0.210139	−	0.977672i	$-0.432608\pi$
$443$	1.73598e6	0.420277	0.210139	+	0.977672i	$0.432608\pi$
$444$	0	0
$445$	1.76980e6	0.423667
$446$	46771.0	0.0111337
$447$	0	0
$448$	3.15812e6	0.743420
$449$	1.82067e6	0.426202	0.213101	−	0.977030i	$-0.431644\pi$
$449$	1.82067e6	0.426202	0.213101	+	0.977030i	$0.431644\pi$
$450$	0	0
$451$	1.18498e6	0.274328
$452$	2.65136e6	0.610412
$453$	0	0
$454$	323200.	0.0735922
$455$	−390808.	−0.0884982
$456$	0	0
$457$	6.35763e6	1.42398	0.711991	−	0.702188i	$-0.247794\pi$
$457$	6.35763e6	1.42398	0.711991	+	0.702188i	$0.247794\pi$
$458$	−41220.9	−0.00918235
$459$	0	0
$460$	−319139.	−0.0703209
$461$	1.59413e6	0.349358	0.174679	−	0.984625i	$-0.444111\pi$
$461$	1.59413e6	0.349358	0.174679	+	0.984625i	$0.444111\pi$
$462$	0	0
$463$	5.74960e6	1.24648	0.623239	−	0.782031i	$-0.285816\pi$
$463$	5.74960e6	1.24648	0.623239	+	0.782031i	$0.285816\pi$
$464$	1.81402e6	0.391153
$465$	0	0
$466$	145324.	0.0310007
$467$	−8.37476e6	−1.77697	−0.888485	−	0.458905i	$-0.848242\pi$
$467$	−8.37476e6	−1.77697	−0.888485	+	0.458905i	$0.848242\pi$
$468$	0	0
$469$	−6.44452e6	−1.35288
$470$	−102099.	−0.0213194
$471$	0	0
$472$	407120.	0.0841139
$473$	−1.12233e6	−0.230657
$474$	0	0
$475$	−3.91045e6	−0.795230
$476$	−6.13846e6	−1.24177
$477$	0	0
$478$	22397.6	0.00448366
$479$	5.39038e6	1.07345	0.536724	−	0.843758i	$-0.319662\pi$
$479$	5.39038e6	1.07345	0.536724	+	0.843758i	$0.319662\pi$
$480$	0	0
$481$	−739550.	−0.145749
$482$	260274.	0.0510286
$483$	0	0
$484$	1.63683e6	0.317608
$485$	−2.83992e6	−0.548216
$486$	0	0
$487$	3.80663e6	0.727307	0.363654	−	0.931534i	$-0.381529\pi$
$487$	3.80663e6	0.727307	0.363654	+	0.931534i	$0.381529\pi$
$488$	−144907.	−0.0275449
$489$	0	0
$490$	32259.9	0.00606979
$491$	4.50048e6	0.842472	0.421236	−	0.906951i	$-0.361596\pi$
$491$	4.50048e6	0.842472	0.421236	+	0.906951i	$0.361596\pi$
$492$	0	0
$493$	−3.51386e6	−0.651129
$494$	−70001.2	−0.0129059
$495$	0	0
$496$	1.01744e7	1.85697
$497$	2.28181e6	0.414371
$498$	0	0
$499$	8.52403e6	1.53248	0.766238	−	0.642557i	$-0.222126\pi$
$499$	8.52403e6	1.53248	0.766238	+	0.642557i	$0.222126\pi$
$500$	−3.55539e6	−0.636007
$501$	0	0
$502$	−27455.5	−0.00486261
$503$	6.88254e6	1.21291	0.606455	−	0.795118i	$-0.292591\pi$
$503$	6.88254e6	1.21291	0.606455	+	0.795118i	$0.292591\pi$
$504$	0	0
$505$	−2.75888e6	−0.481398
$506$	40874.8	0.00709708
$507$	0	0
$508$	2.86135e6	0.491940
$509$	−8.40311e6	−1.43762	−0.718812	−	0.695204i	$-0.755314\pi$
$509$	−8.40311e6	−1.43762	−0.718812	+	0.695204i	$0.755314\pi$
$510$	0	0
$511$	−566117.	−0.0959078
$512$	1.21419e6	0.204697
$513$	0	0
$514$	98830.2	0.0164999
$515$	370298.	0.0615224
$516$	0	0
$517$	−7.68353e6	−1.26425
$518$	−79001.6	−0.0129364
$519$	0	0
$520$	−59844.6	−0.00970547
$521$	−3.83335e6	−0.618705	−0.309353	−	0.950947i	$-0.600112\pi$
$521$	−3.83335e6	−0.618705	−0.309353	+	0.950947i	$0.600112\pi$
$522$	0	0
$523$	−5.98581e6	−0.956904	−0.478452	−	0.878114i	$-0.658802\pi$
$523$	−5.98581e6	−0.956904	−0.478452	+	0.878114i	$0.658802\pi$
$524$	9.67423e6	1.53918
$525$	0	0
$526$	374234.	0.0589765
$527$	−1.97084e7	−3.09118
$528$	0	0
$529$	279841.	0.0434783
$530$	−64639.1	−0.00999552
$531$	0	0
$532$	4.39378e6	0.673068
$533$	759998.	0.115876
$534$	0	0
$535$	180602.	0.0272795
$536$	−986853.	−0.148368
$537$	0	0
$538$	−218099.	−0.0324861
$539$	2.42775e6	0.359942
$540$	0	0
$541$	6.69946e6	0.984117	0.492058	−	0.870562i	$-0.336245\pi$
$541$	6.69946e6	0.984117	0.492058	+	0.870562i	$0.336245\pi$
$542$	282535.	0.0413118
$543$	0	0
$544$	−1.41038e6	−0.204333
$545$	−960304.	−0.138490
$546$	0	0
$547$	−5.18730e6	−0.741264	−0.370632	−	0.928780i	$-0.620859\pi$
$547$	−5.18730e6	−0.741264	−0.370632	+	0.928780i	$0.620859\pi$
$548$	−1.73020e6	−0.246119
$549$	0	0
$550$	213906.	0.0301521
$551$	2.51514e6	0.352926
$552$	0	0
$553$	−5.62726e6	−0.782499
$554$	19899.6	0.00275468
$555$	0	0
$556$	9.76754e6	1.33998
$557$	−166278.	−0.0227090	−0.0113545	−	0.999936i	$-0.503614\pi$
$557$	−166278.	−0.0227090	−0.0113545	+	0.999936i	$0.503614\pi$
$558$	0	0
$559$	−719815.	−0.0974297
$560$	1.87334e6	0.252433
$561$	0	0
$562$	54346.7	0.00725825
$563$	8.42768e6	1.12057	0.560283	−	0.828301i	$-0.310692\pi$
$563$	8.42768e6	1.12057	0.560283	+	0.828301i	$0.310692\pi$
$564$	0	0
$565$	1.56736e6	0.206561
$566$	465216.	0.0610398
$567$	0	0
$568$	349415.	0.0454434
$569$	1.48853e7	1.92743	0.963714	−	0.266935i	$-0.0860111\pi$
$569$	1.48853e7	1.92743	0.963714	+	0.266935i	$0.0860111\pi$
$570$	0	0
$571$	3.42577e6	0.439711	0.219856	−	0.975532i	$-0.429441\pi$
$571$	3.42577e6	0.439711	0.219856	+	0.975532i	$0.429441\pi$
$572$	−2.24992e6	−0.287525
$573$	0	0
$574$	81185.9	0.0102849
$575$	1.46446e6	0.184718
$576$	0	0
$577$	4.32818e6	0.541210	0.270605	−	0.962690i	$-0.412776\pi$
$577$	4.32818e6	0.541210	0.270605	+	0.962690i	$0.412776\pi$
$578$	577012.	0.0718398
$579$	0	0
$580$	1.07419e6	0.132591
$581$	4.76318e6	0.585406
$582$	0	0
$583$	−4.86448e6	−0.592741
$584$	−86689.8	−0.0105181
$585$	0	0
$586$	501762.	0.0603607
$587$	−596806.	−0.0714888	−0.0357444	−	0.999361i	$-0.511380\pi$
$587$	−596806.	−0.0714888	−0.0357444	+	0.999361i	$0.511380\pi$
$588$	0	0
$589$	1.41068e7	1.67549
$590$	120233.	0.0142198
$591$	0	0
$592$	3.54504e6	0.415736
$593$	8.58531e6	1.00258	0.501290	−	0.865279i	$-0.332859\pi$
$593$	8.58531e6	1.00258	0.501290	+	0.865279i	$0.332859\pi$
$594$	0	0
$595$	−3.62878e6	−0.420211
$596$	−1.67629e6	−0.193301
$597$	0	0
$598$	26215.5	0.00299781
$599$	6.67213e6	0.759797	0.379898	−	0.925028i	$-0.375959\pi$
$599$	6.67213e6	0.759797	0.379898	+	0.925028i	$0.375959\pi$
$600$	0	0
$601$	5.15927e6	0.582642	0.291321	−	0.956625i	$-0.405905\pi$
$601$	5.15927e6	0.582642	0.291321	+	0.956625i	$0.405905\pi$
$602$	−76893.4	−0.00864765
$603$	0	0
$604$	4.10676e6	0.458044
$605$	967620.	0.107477
$606$	0	0
$607$	−1.24988e7	−1.37689	−0.688443	−	0.725291i	$-0.741705\pi$
$607$	−1.24988e7	−1.37689	−0.688443	+	0.725291i	$0.741705\pi$
$608$	1.00952e6	0.110753
$609$	0	0
$610$	−42794.8	−0.00465658
$611$	−4.92790e6	−0.534022
$612$	0	0
$613$	185182.	0.0199044	0.00995219	−	0.999950i	$-0.496832\pi$
$613$	185182.	0.0199044	0.00995219	+	0.999950i	$0.496832\pi$
$614$	547222.	0.0585791
$615$	0	0
$616$	−481099.	−0.0510837
$617$	−8.27176e6	−0.874753	−0.437376	−	0.899279i	$-0.644092\pi$
$617$	−8.27176e6	−0.874753	−0.437376	+	0.899279i	$0.644092\pi$
$618$	0	0
$619$	1.38338e6	0.145116	0.0725581	−	0.997364i	$-0.476884\pi$
$619$	1.38338e6	0.145116	0.0725581	+	0.997364i	$0.476884\pi$
$620$	6.02490e6	0.629463
$621$	0	0
$622$	−88766.4	−0.00919968
$623$	9.12506e6	0.941923
$624$	0	0
$625$	6.54936e6	0.670654
$626$	−677145.	−0.0690631
$627$	0	0
$628$	2.82755e6	0.286096
$629$	−6.86697e6	−0.692051
$630$	0	0
$631$	−1.46459e7	−1.46434	−0.732170	−	0.681122i	$-0.761492\pi$
$631$	−1.46459e7	−1.46434	−0.732170	+	0.681122i	$0.761492\pi$
$632$	−861705.	−0.0858155
$633$	0	0
$634$	454139.	0.0448710
$635$	1.69150e6	0.166471
$636$	0	0
$637$	1.55706e6	0.152040
$638$	−137581.	−0.0133816
$639$	0	0
$640$	574710.	0.0554624
$641$	7.62817e6	0.733289	0.366645	−	0.930361i	$-0.380506\pi$
$641$	7.62817e6	0.733289	0.366645	+	0.930361i	$0.380506\pi$
$642$	0	0
$643$	−3.84764e6	−0.367001	−0.183500	−	0.983020i	$-0.558743\pi$
$643$	−3.84764e6	−0.367001	−0.183500	+	0.983020i	$0.558743\pi$
$644$	−1.64547e6	−0.156342
$645$	0	0
$646$	−649984.	−0.0612804
$647$	1.45701e6	0.136837	0.0684183	−	0.997657i	$-0.478205\pi$
$647$	1.45701e6	0.136837	0.0684183	+	0.997657i	$0.478205\pi$
$648$	0	0
$649$	9.04826e6	0.843244
$650$	137191.	0.0127363
$651$	0	0
$652$	7.61630e6	0.701657
$653$	−1.05115e7	−0.964675	−0.482337	−	0.875986i	$-0.660212\pi$
$653$	−1.05115e7	−0.964675	−0.482337	+	0.875986i	$0.660212\pi$
$654$	0	0
$655$	5.71896e6	0.520851
$656$	−3.64306e6	−0.330527
$657$	0	0
$658$	−526418.	−0.0473986
$659$	−226823.	−0.0203457	−0.0101729	−	0.999948i	$-0.503238\pi$
$659$	−226823.	−0.0203457	−0.0101729	+	0.999948i	$0.503238\pi$
$660$	0	0
$661$	−8.97104e6	−0.798618	−0.399309	−	0.916816i	$-0.630750\pi$
$661$	−8.97104e6	−0.798618	−0.399309	+	0.916816i	$0.630750\pi$
$662$	648136.	0.0574806
$663$	0	0
$664$	729389.	0.0642006
$665$	2.59740e6	0.227764
$666$	0	0
$667$	−941922.	−0.0819786
$668$	1.75543e7	1.52210
$669$	0	0
$670$	−291443.	−0.0250822
$671$	−3.22057e6	−0.276138
$672$	0	0
$673$	−7.79238e6	−0.663181	−0.331591	−	0.943423i	$-0.607585\pi$
$673$	−7.79238e6	−0.663181	−0.331591	+	0.943423i	$0.607585\pi$
$674$	277982.	0.0235704
$675$	0	0
$676$	1.04182e7	0.876850
$677$	2.15072e7	1.80348	0.901742	−	0.432276i	$-0.142289\pi$
$677$	2.15072e7	1.80348	0.901742	+	0.432276i	$0.142289\pi$
$678$	0	0
$679$	−1.46425e7	−1.21883
$680$	−555676.	−0.0460840
$681$	0	0
$682$	−771661.	−0.0635281
$683$	−2.29195e7	−1.87998	−0.939990	−	0.341201i	$-0.889166\pi$
$683$	−2.29195e7	−1.87998	−0.939990	+	0.341201i	$0.889166\pi$
$684$	0	0
$685$	−1.02282e6	−0.0832859
$686$	547912.	0.0444530
$687$	0	0
$688$	3.45044e6	0.277910
$689$	−3.11988e6	−0.250374
$690$	0	0
$691$	4.86536e6	0.387632	0.193816	−	0.981038i	$-0.437914\pi$
$691$	4.86536e6	0.387632	0.193816	+	0.981038i	$0.437914\pi$
$692$	−1.76770e7	−1.40327
$693$	0	0
$694$	−455481.	−0.0358981
$695$	5.77412e6	0.453444
$696$	0	0
$697$	7.05683e6	0.550209
$698$	493086.	0.0383075
$699$	0	0
$700$	−8.61108e6	−0.664221
$701$	−2.34422e7	−1.80179	−0.900894	−	0.434040i	$-0.857088\pi$
$701$	−2.34422e7	−1.80179	−0.900894	+	0.434040i	$0.857088\pi$
$702$	0	0
$703$	4.91522e6	0.375107
$704$	1.07481e7	0.817337
$705$	0	0
$706$	868999.	0.0656157
$707$	−1.42247e7	−1.07027
$708$	0	0
$709$	9.76246e6	0.729363	0.364681	−	0.931132i	$-0.381178\pi$
$709$	9.76246e6	0.729363	0.364681	+	0.931132i	$0.381178\pi$
$710$	103191.	0.00768240
$711$	0	0
$712$	1.39733e6	0.103299
$713$	−5.28301e6	−0.389187
$714$	0	0
$715$	−1.33005e6	−0.0972975
$716$	−2.04212e7	−1.48867
$717$	0	0
$718$	−488519.	−0.0353647
$719$	−1.53718e7	−1.10892	−0.554462	−	0.832209i	$-0.687076\pi$
$719$	−1.53718e7	−1.10892	−0.554462	+	0.832209i	$0.687076\pi$
$720$	0	0
$721$	1.90925e6	0.136780
$722$	−112109.	−0.00800385
$723$	0	0
$724$	−1.18700e7	−0.841597
$725$	−4.92927e6	−0.348287
$726$	0	0
$727$	−2.25494e7	−1.58233	−0.791167	−	0.611600i	$-0.790526\pi$
$727$	−2.25494e7	−1.58233	−0.791167	+	0.611600i	$0.790526\pi$
$728$	−308557.	−0.0215778
$729$	0	0
$730$	−25601.7	−0.00177812
$731$	−6.68372e6	−0.462620
$732$	0	0
$733$	7.32923e6	0.503847	0.251923	−	0.967747i	$-0.418937\pi$
$733$	7.32923e6	0.503847	0.251923	+	0.967747i	$0.418937\pi$
$734$	−1.00902e6	−0.0691289
$735$	0	0
$736$	−378065.	−0.0257260
$737$	−2.19328e7	−1.48739
$738$	0	0
$739$	−1.33090e7	−0.896466	−0.448233	−	0.893917i	$-0.647946\pi$
$739$	−1.33090e7	−0.896466	−0.448233	+	0.893917i	$0.647946\pi$
$740$	2.09925e6	0.140924
$741$	0	0
$742$	−333277.	−0.0222227
$743$	1.86284e7	1.23795	0.618974	−	0.785411i	$-0.287549\pi$
$743$	1.86284e7	1.23795	0.618974	+	0.785411i	$0.287549\pi$
$744$	0	0
$745$	−990948.	−0.0654124
$746$	−514089.	−0.0338214
$747$	0	0
$748$	−2.08912e7	−1.36524
$749$	931177.	0.0606496
$750$	0	0
$751$	6.85607e6	0.443584	0.221792	−	0.975094i	$-0.428809\pi$
$751$	6.85607e6	0.443584	0.221792	+	0.975094i	$0.428809\pi$
$752$	2.36220e7	1.52325
$753$	0	0
$754$	−88239.1	−0.00565240
$755$	2.42773e6	0.155000
$756$	0	0
$757$	4.10285e6	0.260223	0.130112	−	0.991499i	$-0.458466\pi$
$757$	4.10285e6	0.260223	0.130112	+	0.991499i	$0.458466\pi$
$758$	−522366.	−0.0330219
$759$	0	0
$760$	397741.	0.0249785
$761$	5.41460e6	0.338926	0.169463	−	0.985537i	$-0.445797\pi$
$761$	5.41460e6	0.338926	0.169463	+	0.985537i	$0.445797\pi$
$762$	0	0
$763$	−4.95130e6	−0.307899
$764$	1.52189e7	0.943300
$765$	0	0
$766$	499923.	0.0307845
$767$	5.80318e6	0.356187
$768$	0	0
$769$	1.64748e7	1.00463	0.502313	−	0.864686i	$-0.332483\pi$
$769$	1.64748e7	1.00463	0.502313	+	0.864686i	$0.332483\pi$
$770$	−142081.	−0.00863592
$771$	0	0
$772$	−270244.	−0.0163197
$773$	7.92947e6	0.477304	0.238652	−	0.971105i	$-0.423294\pi$
$773$	7.92947e6	0.477304	0.238652	+	0.971105i	$0.423294\pi$
$774$	0	0
$775$	−2.76471e7	−1.65347
$776$	−2.24222e6	−0.133667
$777$	0	0
$778$	−356551.	−0.0211190
$779$	−5.05112e6	−0.298225
$780$	0	0
$781$	7.76576e6	0.455571
$782$	243419.	0.0142344
$783$	0	0
$784$	−7.46380e6	−0.433680
$785$	1.67152e6	0.0968137
$786$	0	0
$787$	9.50104e6	0.546807	0.273404	−	0.961899i	$-0.411851\pi$
$787$	9.50104e6	0.546807	0.273404	+	0.961899i	$0.411851\pi$
$788$	6.98167e6	0.400538
$789$	0	0
$790$	−254483.	−0.0145075
$791$	8.08128e6	0.459239
$792$	0	0
$793$	−2.06554e6	−0.116641
$794$	−787175.	−0.0443119
$795$	0	0
$796$	−8.35991e6	−0.467648
$797$	5.90940e6	0.329532	0.164766	−	0.986333i	$-0.447313\pi$
$797$	5.90940e6	0.329532	0.164766	+	0.986333i	$0.447313\pi$
$798$	0	0
$799$	−4.57572e7	−2.53567
$800$	−1.97849e6	−0.109297
$801$	0	0
$802$	1.12010e6	0.0614923
$803$	−1.92668e6	−0.105444
$804$	0	0
$805$	−972726.	−0.0529055
$806$	−494912.	−0.0268343
$807$	0	0
$808$	−2.17824e6	−0.117375
$809$	−3.48663e7	−1.87299	−0.936494	−	0.350684i	$-0.885949\pi$
$809$	−3.48663e7	−1.87299	−0.936494	+	0.350684i	$0.885949\pi$
$810$	0	0
$811$	1.72646e7	0.921733	0.460867	−	0.887469i	$-0.347539\pi$
$811$	1.72646e7	0.921733	0.460867	+	0.887469i	$0.347539\pi$
$812$	5.53852e6	0.294784
$813$	0	0
$814$	−268869.	−0.0142226
$815$	4.50241e6	0.237438
$816$	0	0
$817$	4.78406e6	0.250750
$818$	−1.40632e6	−0.0734854
$819$	0	0
$820$	−2.15729e6	−0.112040
$821$	−2.61709e7	−1.35507	−0.677534	−	0.735492i	$-0.736951\pi$
$821$	−2.61709e7	−1.35507	−0.677534	+	0.735492i	$0.736951\pi$
$822$	0	0
$823$	−1.09360e7	−0.562807	−0.281403	−	0.959590i	$-0.590800\pi$
$823$	−1.09360e7	−0.562807	−0.281403	+	0.959590i	$0.590800\pi$
$824$	292364.	0.0150005
$825$	0	0
$826$	619919.	0.0316144
$827$	2.49552e7	1.26881	0.634405	−	0.773001i	$-0.281245\pi$
$827$	2.49552e7	1.26881	0.634405	+	0.773001i	$0.281245\pi$
$828$	0	0
$829$	1.11425e7	0.563115	0.281558	−	0.959544i	$-0.409149\pi$
$829$	1.11425e7	0.563115	0.281558	+	0.959544i	$0.409149\pi$
$830$	215407.	0.0108534
$831$	0	0
$832$	6.89342e6	0.345244
$833$	1.44578e7	0.721923
$834$	0	0
$835$	1.03773e7	0.515073
$836$	1.49535e7	0.739990
$837$	0	0
$838$	−1.18282e6	−0.0581845
$839$	−2.66589e7	−1.30749	−0.653744	−	0.756716i	$-0.726803\pi$
$839$	−2.66589e7	−1.30749	−0.653744	+	0.756716i	$0.726803\pi$
$840$	0	0
$841$	−1.73407e7	−0.845429
$842$	1.57202e6	0.0764149
$843$	0	0
$844$	−1.92780e7	−0.931550
$845$	6.15875e6	0.296723
$846$	0	0
$847$	4.98903e6	0.238950
$848$	1.49552e7	0.714170
$849$	0	0
$850$	1.27386e6	0.0604749
$851$	−1.84075e6	−0.0871307
$852$	0	0
$853$	1.97404e7	0.928929	0.464464	−	0.885592i	$-0.346247\pi$
$853$	1.97404e7	0.928929	0.464464	+	0.885592i	$0.346247\pi$
$854$	−220649.	−0.0103528
$855$	0	0
$856$	142592.	0.00665135
$857$	1.03672e7	0.482182	0.241091	−	0.970502i	$-0.422495\pi$
$857$	1.03672e7	0.482182	0.241091	+	0.970502i	$0.422495\pi$
$858$	0	0
$859$	−2.37753e7	−1.09937	−0.549685	−	0.835372i	$-0.685252\pi$
$859$	−2.37753e7	−1.09937	−0.549685	+	0.835372i	$0.685252\pi$
$860$	2.04323e6	0.0942042
$861$	0	0
$862$	−782088.	−0.0358499
$863$	−2.00019e7	−0.914207	−0.457103	−	0.889414i	$-0.651113\pi$
$863$	−2.00019e7	−0.914207	−0.457103	+	0.889414i	$0.651113\pi$
$864$	0	0
$865$	−1.04498e7	−0.474863
$866$	−81696.4	−0.00370176
$867$	0	0
$868$	3.10642e7	1.39946
$869$	−1.91514e7	−0.860302
$870$	0	0
$871$	−1.40668e7	−0.628276
$872$	−758196.	−0.0337668
$873$	0	0
$874$	−174234.	−0.00771533
$875$	−1.08367e7	−0.478496
$876$	0	0
$877$	2.39285e6	0.105055	0.0525275	−	0.998619i	$-0.483272\pi$
$877$	2.39285e6	0.105055	0.0525275	+	0.998619i	$0.483272\pi$
$878$	−1.68215e6	−0.0736424
$879$	0	0
$880$	6.37560e6	0.277533
$881$	2.76200e7	1.19890	0.599452	−	0.800411i	$-0.295385\pi$
$881$	2.76200e7	1.19890	0.599452	+	0.800411i	$0.295385\pi$
$882$	0	0
$883$	1.34991e7	0.582644	0.291322	−	0.956625i	$-0.405905\pi$
$883$	1.34991e7	0.582644	0.291322	+	0.956625i	$0.405905\pi$
$884$	−1.33988e7	−0.576679
$885$	0	0
$886$	404780.	0.0173235
$887$	3.89899e7	1.66396	0.831980	−	0.554806i	$-0.187208\pi$
$887$	3.89899e7	1.66396	0.831980	+	0.554806i	$0.187208\pi$
$888$	0	0
$889$	8.72133e6	0.370108
$890$	412666.	0.0174632
$891$	0	0
$892$	−6.40788e6	−0.269651
$893$	3.27520e7	1.37439
$894$	0	0
$895$	−1.20721e7	−0.503760
$896$	2.96319e6	0.123308
$897$	0	0
$898$	424526.	0.0175677
$899$	1.77822e7	0.733814
$900$	0	0
$901$	−2.89691e7	−1.18884
$902$	276302.	0.0113075
$903$	0	0
$904$	1.23749e6	0.0503641
$905$	−7.01700e6	−0.284793
$906$	0	0
$907$	2.05655e7	0.830083	0.415041	−	0.909803i	$-0.363767\pi$
$907$	2.05655e7	0.830083	0.415041	+	0.909803i	$0.363767\pi$
$908$	−4.42802e7	−1.78236
$909$	0	0
$910$	−91124.8	−0.00364782
$911$	−2.19394e7	−0.875846	−0.437923	−	0.899012i	$-0.644286\pi$
$911$	−2.19394e7	−0.875846	−0.437923	+	0.899012i	$0.644286\pi$
$912$	0	0
$913$	1.62107e7	0.643612
$914$	1.48241e6	0.0586953
$915$	0	0
$916$	5.64749e6	0.222391
$917$	2.94868e7	1.15799
$918$	0	0
$919$	1.41323e7	0.551980	0.275990	−	0.961161i	$-0.410994\pi$
$919$	1.41323e7	0.551980	0.275990	+	0.961161i	$0.410994\pi$
$920$	−148954.	−0.00580207
$921$	0	0
$922$	371704.	0.0144002
$923$	4.98064e6	0.192434
$924$	0	0
$925$	−9.63303e6	−0.370176
$926$	1.34064e6	0.0513788
$927$	0	0
$928$	1.27253e6	0.0485065
$929$	−3.09478e7	−1.17650	−0.588248	−	0.808681i	$-0.700182\pi$
$929$	−3.09478e7	−1.17650	−0.588248	+	0.808681i	$0.700182\pi$
$930$	0	0
$931$	−1.03486e7	−0.391298
$932$	−1.99102e7	−0.750819
$933$	0	0
$934$	−1.95275e6	−0.0732452
$935$	−1.23499e7	−0.461993
$936$	0	0
$937$	3.18975e7	1.18688	0.593441	−	0.804877i	$-0.297769\pi$
$937$	3.18975e7	1.18688	0.593441	+	0.804877i	$0.297769\pi$
$938$	−1.50267e6	−0.0557644
$939$	0	0
$940$	1.39881e7	0.516343
$941$	−3.73584e7	−1.37535	−0.687676	−	0.726018i	$-0.741369\pi$
$941$	−3.73584e7	−1.37535	−0.687676	+	0.726018i	$0.741369\pi$
$942$	0	0
$943$	1.89165e6	0.0692725
$944$	−2.78176e7	−1.01599
$945$	0	0
$946$	−261694.	−0.00950748
$947$	2.65744e7	0.962917	0.481459	−	0.876469i	$-0.340107\pi$
$947$	2.65744e7	0.962917	0.481459	+	0.876469i	$0.340107\pi$
$948$	0	0
$949$	−1.23570e6	−0.0445396
$950$	−911802.	−0.0327787
$951$	0	0
$952$	−2.86505e6	−0.102457
$953$	−2.55780e7	−0.912294	−0.456147	−	0.889905i	$-0.650771\pi$
$953$	−2.55780e7	−0.912294	−0.456147	+	0.889905i	$0.650771\pi$
$954$	0	0
$955$	8.99671e6	0.319209
$956$	−3.06860e6	−0.108591
$957$	0	0
$958$	1.25688e6	0.0442465
$959$	−5.27362e6	−0.185166
$960$	0	0
$961$	7.11069e7	2.48373
$962$	−172441.	−0.00600764
$963$	0	0
$964$	−3.56590e7	−1.23588
$965$	−159756.	−0.00552253
$966$	0	0
$967$	−2.86028e7	−0.983655	−0.491828	−	0.870693i	$-0.663671\pi$
$967$	−2.86028e7	−0.983655	−0.491828	+	0.870693i	$0.663671\pi$
$968$	763972.	0.0262053
$969$	0	0
$970$	−662185.	−0.0225970
$971$	5.12662e7	1.74495	0.872475	−	0.488659i	$-0.162514\pi$
$971$	5.12662e7	1.74495	0.872475	+	0.488659i	$0.162514\pi$
$972$	0	0
$973$	2.97712e7	1.00812
$974$	887593.	0.0299790
$975$	0	0
$976$	9.90119e6	0.332708
$977$	2.19414e7	0.735406	0.367703	−	0.929943i	$-0.380144\pi$
$977$	2.19414e7	0.735406	0.367703	+	0.929943i	$0.380144\pi$
$978$	0	0
$979$	3.10556e7	1.03558
$980$	−4.41979e6	−0.147006
$981$	0	0
$982$	1.04938e6	0.0347260
$983$	−2.48179e7	−0.819182	−0.409591	−	0.912269i	$-0.634329\pi$
$983$	−2.48179e7	−0.819182	−0.409591	+	0.912269i	$0.634329\pi$
$984$	0	0
$985$	4.12724e6	0.135541
$986$	−819329.	−0.0268390
$987$	0	0
$988$	9.59054e6	0.312573
$989$	−1.79163e6	−0.0582449
$990$	0	0
$991$	−2.00450e7	−0.648367	−0.324184	−	0.945994i	$-0.605090\pi$
$991$	−2.00450e7	−0.648367	−0.324184	+	0.945994i	$0.605090\pi$
$992$	7.13734e6	0.230281
$993$	0	0
$994$	532051.	0.0170800
$995$	−4.94199e6	−0.158250
$996$	0	0
$997$	4.12650e7	1.31475	0.657375	−	0.753563i	$-0.271667\pi$
$997$	4.12650e7	1.31475	0.657375	+	0.753563i	$0.271667\pi$
$998$	1.98755e6	0.0631673
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	207.6.a.d.1.3		4
3.2	odd	2		69.6.a.c.1.2	✓	4
12.11	even	2		1104.6.a.n.1.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
69.6.a.c.1.2	✓	4	3.2	odd	2
207.6.a.d.1.3		4	1.1	even	1	trivial
1104.6.a.n.1.3		4	12.11	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:	\( N \)	\(=\)	\( 207 = 3^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	207.a (trivial)

\(f(q)\)	\(=\)	\(q+0.233171 q^{2} -31.9456 q^{4} -18.8848 q^{5} -97.3695 q^{7} -14.9102 q^{8} +O(q^{10})\)	\(q+0.233171 q^{2} -31.9456 q^{4} -18.8848 q^{5} -97.3695 q^{7} -14.9102 q^{8} -4.40338 q^{10} -331.380 q^{11} -212.534 q^{13} -22.7037 q^{14} +1018.78 q^{16} -1973.45 q^{17} +1412.55 q^{19} +603.286 q^{20} -77.2681 q^{22} -529.000 q^{23} -2768.36 q^{25} -49.5566 q^{26} +3110.53 q^{28} +1780.57 q^{29} +9986.80 q^{31} +714.678 q^{32} -460.150 q^{34} +1838.80 q^{35} +3479.68 q^{37} +329.365 q^{38} +281.577 q^{40} -3575.89 q^{41} +3386.82 q^{43} +10586.2 q^{44} -123.347 q^{46} +23186.4 q^{47} -7326.19 q^{49} -645.501 q^{50} +6789.53 q^{52} +14679.4 q^{53} +6258.05 q^{55} +1451.80 q^{56} +415.177 q^{58} -27304.7 q^{59} +9718.64 q^{61} +2328.63 q^{62} -32434.4 q^{64} +4013.66 q^{65} +66186.2 q^{67} +63043.0 q^{68} +428.754 q^{70} -23434.6 q^{71} +5814.11 q^{73} +811.360 q^{74} -45124.8 q^{76} +32266.3 q^{77} +57792.8 q^{79} -19239.5 q^{80} -833.792 q^{82} -48918.6 q^{83} +37268.1 q^{85} +789.708 q^{86} +4940.96 q^{88} -93715.8 q^{89} +20694.3 q^{91} +16899.2 q^{92} +5406.39 q^{94} -26675.7 q^{95} +150381. q^{97} -1708.25 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{2} - 46 q^{4} + 122 q^{5} + 62 q^{7} - 72 q^{8}+O(q^{10}) \) 4 * q - 4 * q^2 - 46 * q^4 + 122 * q^5 + 62 * q^7 - 72 * q^8	\( 4 q - 4 q^{2} - 46 q^{4} + 122 q^{5} + 62 q^{7} - 72 q^{8} + 642 q^{10} - 32 q^{11} + 1364 q^{13} - 2754 q^{14} + 18 q^{16} - 278 q^{17} + 2862 q^{19} - 3830 q^{20} + 3176 q^{22} - 2116 q^{23} + 5944 q^{25} - 6996 q^{26} + 4738 q^{28} + 5180 q^{29} - 1788 q^{31} + 7352 q^{32} + 15818 q^{34} - 11768 q^{35} + 3348 q^{37} - 1050 q^{38} - 13462 q^{40} + 17664 q^{41} + 25398 q^{43} + 16848 q^{44} + 2116 q^{46} + 26040 q^{47} + 55720 q^{49} + 35256 q^{50} - 2752 q^{52} + 32006 q^{53} + 34904 q^{55} + 68542 q^{56} - 40804 q^{58} + 61136 q^{59} + 35844 q^{61} + 47524 q^{62} - 35142 q^{64} + 48036 q^{65} + 73458 q^{67} - 17910 q^{68} - 59104 q^{70} - 24432 q^{71} + 122512 q^{73} + 20828 q^{74} + 56834 q^{76} - 159496 q^{77} + 90170 q^{79} + 36546 q^{80} + 84144 q^{82} - 28592 q^{83} + 355124 q^{85} - 103778 q^{86} - 150776 q^{88} + 27926 q^{89} + 334180 q^{91} + 24334 q^{92} + 113632 q^{94} - 113392 q^{95} + 16580 q^{97} - 49688 q^{98}+O(q^{100}) \) 4 * q - 4 * q^2 - 46 * q^4 + 122 * q^5 + 62 * q^7 - 72 * q^8 + 642 * q^10 - 32 * q^11 + 1364 * q^13 - 2754 * q^14 + 18 * q^16 - 278 * q^17 + 2862 * q^19 - 3830 * q^20 + 3176 * q^22 - 2116 * q^23 + 5944 * q^25 - 6996 * q^26 + 4738 * q^28 + 5180 * q^29 - 1788 * q^31 + 7352 * q^32 + 15818 * q^34 - 11768 * q^35 + 3348 * q^37 - 1050 * q^38 - 13462 * q^40 + 17664 * q^41 + 25398 * q^43 + 16848 * q^44 + 2116 * q^46 + 26040 * q^47 + 55720 * q^49 + 35256 * q^50 - 2752 * q^52 + 32006 * q^53 + 34904 * q^55 + 68542 * q^56 - 40804 * q^58 + 61136 * q^59 + 35844 * q^61 + 47524 * q^62 - 35142 * q^64 + 48036 * q^65 + 73458 * q^67 - 17910 * q^68 - 59104 * q^70 - 24432 * q^71 + 122512 * q^73 + 20828 * q^74 + 56834 * q^76 - 159496 * q^77 + 90170 * q^79 + 36546 * q^80 + 84144 * q^82 - 28592 * q^83 + 355124 * q^85 - 103778 * q^86 - 150776 * q^88 + 27926 * q^89 + 334180 * q^91 + 24334 * q^92 + 113632 * q^94 - 113392 * q^95 + 16580 * q^97 - 49688 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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