LMFDB - Embedded newform 207.6.a.i.1.6

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:	$10$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} - 2 x^{9} - 251 x^{8} + 296 x^{7} + 21169 x^{6} - 9042 x^{5} - 685949 x^{4} - 270764 x^{3} + \cdots + 5446216 $ x^10 - 2x^9 - 251x^8 + 296x^7 + 21169x^6 - 9042x^5 - 685949x^4 - 270764x^3 + 6230110x^2 + 11573432*x + 5446216
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{6}\cdot 3^{4} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$-1.72516$ 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)

$f(q)$	$=$	$q+2.72516 q^{2} -24.5735 q^{4} +3.82575 q^{5} -182.948 q^{7} -154.172 q^{8} +O(q^{10})$	\(q+2.72516 q^{2} -24.5735 q^{4} +3.82575 q^{5} -182.948 q^{7} -154.172 q^{8} +10.4258 q^{10} -699.885 q^{11} +813.091 q^{13} -498.562 q^{14} +366.210 q^{16} +1982.60 q^{17} +1654.54 q^{19} -94.0121 q^{20} -1907.30 q^{22} +529.000 q^{23} -3110.36 q^{25} +2215.80 q^{26} +4495.68 q^{28} -774.405 q^{29} -1248.48 q^{31} +5931.48 q^{32} +5402.89 q^{34} -699.913 q^{35} +7715.93 q^{37} +4508.89 q^{38} -589.822 q^{40} -7021.66 q^{41} -11547.8 q^{43} +17198.6 q^{44} +1441.61 q^{46} +20829.5 q^{47} +16663.0 q^{49} -8476.23 q^{50} -19980.5 q^{52} +25868.7 q^{53} -2677.58 q^{55} +28205.4 q^{56} -2110.38 q^{58} +12167.1 q^{59} +52562.8 q^{61} -3402.31 q^{62} +4445.47 q^{64} +3110.68 q^{65} -69504.4 q^{67} -48719.4 q^{68} -1907.37 q^{70} +41256.0 q^{71} -18589.7 q^{73} +21027.1 q^{74} -40657.9 q^{76} +128043. q^{77} +19551.6 q^{79} +1401.03 q^{80} -19135.1 q^{82} -123259. q^{83} +7584.91 q^{85} -31469.6 q^{86} +107902. q^{88} +46120.6 q^{89} -148753. q^{91} -12999.4 q^{92} +56763.6 q^{94} +6329.86 q^{95} -71712.9 q^{97} +45409.2 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q + 8 q^{2} + 192 q^{4} + 100 q^{5} + 20 q^{7} + 384 q^{8}+O(q^{10}) $ 10 * q + 8 * q^2 + 192 * q^4 + 100 * q^5 + 20 * q^7 + 384 * q^8	$ 10 q + 8 q^{2} + 192 q^{4} + 100 q^{5} + 20 q^{7} + 384 q^{8} - 250 q^{10} + 460 q^{11} + 464 q^{13} + 3676 q^{14} + 4612 q^{16} + 4756 q^{17} - 1780 q^{19} + 10314 q^{20} - 4214 q^{22} + 5290 q^{23} + 1330 q^{25} - 5152 q^{26} + 7072 q^{28} + 4048 q^{29} + 2816 q^{31} + 27436 q^{32} + 420 q^{34} + 9452 q^{35} + 2872 q^{37} + 31038 q^{38} + 2618 q^{40} + 34056 q^{41} + 7316 q^{43} + 33562 q^{44} + 4232 q^{46} + 49300 q^{47} + 45118 q^{49} + 44764 q^{50} - 25120 q^{52} + 86676 q^{53} - 2120 q^{55} + 290684 q^{56} - 87408 q^{58} + 67100 q^{59} - 40432 q^{61} + 230992 q^{62} + 136776 q^{64} + 184000 q^{65} - 50108 q^{67} + 270592 q^{68} + 117456 q^{70} + 238584 q^{71} - 13804 q^{73} + 150074 q^{74} - 197622 q^{76} + 116248 q^{77} - 9228 q^{79} + 313010 q^{80} - 68604 q^{82} + 155300 q^{83} + 80444 q^{85} - 80914 q^{86} - 237738 q^{88} + 213732 q^{89} - 264352 q^{91} + 101568 q^{92} + 140280 q^{94} - 123612 q^{95} + 42516 q^{97} - 151388 q^{98}+O(q^{100}) $ 10 * q + 8 * q^2 + 192 * q^4 + 100 * q^5 + 20 * q^7 + 384 * q^8 - 250 * q^10 + 460 * q^11 + 464 * q^13 + 3676 * q^14 + 4612 * q^16 + 4756 * q^17 - 1780 * q^19 + 10314 * q^20 - 4214 * q^22 + 5290 * q^23 + 1330 * q^25 - 5152 * q^26 + 7072 * q^28 + 4048 * q^29 + 2816 * q^31 + 27436 * q^32 + 420 * q^34 + 9452 * q^35 + 2872 * q^37 + 31038 * q^38 + 2618 * q^40 + 34056 * q^41 + 7316 * q^43 + 33562 * q^44 + 4232 * q^46 + 49300 * q^47 + 45118 * q^49 + 44764 * q^50 - 25120 * q^52 + 86676 * q^53 - 2120 * q^55 + 290684 * q^56 - 87408 * q^58 + 67100 * q^59 - 40432 * q^61 + 230992 * q^62 + 136776 * q^64 + 184000 * q^65 - 50108 * q^67 + 270592 * q^68 + 117456 * q^70 + 238584 * q^71 - 13804 * q^73 + 150074 * q^74 - 197622 * q^76 + 116248 * q^77 - 9228 * q^79 + 313010 * q^80 - 68604 * q^82 + 155300 * q^83 + 80444 * q^85 - 80914 * q^86 - 237738 * q^88 + 213732 * q^89 - 264352 * q^91 + 101568 * q^92 + 140280 * q^94 - 123612 * q^95 + 42516 * q^97 - 151388 * 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 $

$2$	2.72516	0.481744	0.240872	−	0.970557i	$-0.422567\pi$
$2$	2.72516	0.481744	0.240872	+	0.970557i	$0.422567\pi$
$3$	0	0
$3$	0	0
$4$	−24.5735	−0.767922
$5$	3.82575	0.0684371	0.0342185	−	0.999414i	$-0.489106\pi$
$5$	3.82575	0.0684371	0.0342185	+	0.999414i	$0.489106\pi$
$6$	0	0
$7$	−182.948	−1.41118	−0.705590	−	0.708620i	$-0.749318\pi$
$7$	−182.948	−1.41118	−0.705590	+	0.708620i	$0.749318\pi$
$8$	−154.172	−0.851687
$9$	0	0
$10$	10.4258	0.0329692
$11$	−699.885	−1.74399	−0.871997	−	0.489511i	$-0.837175\pi$
$11$	−699.885	−1.74399	−0.871997	+	0.489511i	$0.837175\pi$
$12$	0	0
$13$	813.091	1.33438	0.667192	−	0.744886i	$-0.267496\pi$
$13$	813.091	1.33438	0.667192	+	0.744886i	$0.267496\pi$
$14$	−498.562	−0.679828
$15$	0	0
$16$	366.210	0.357627
$17$	1982.60	1.66384	0.831921	−	0.554894i	$-0.187241\pi$
$17$	1982.60	1.66384	0.831921	+	0.554894i	$0.187241\pi$
$18$	0	0
$19$	1654.54	1.05146	0.525731	−	0.850651i	$-0.323792\pi$
$19$	1654.54	1.05146	0.525731	+	0.850651i	$0.323792\pi$
$20$	−94.0121	−0.0525544
$21$	0	0
$22$	−1907.30	−0.840159
$23$	529.000	0.208514
$23$	529.000	0.208514
$24$	0	0
$25$	−3110.36	−0.995316
$26$	2215.80	0.642832
$27$	0	0
$28$	4495.68	1.08368
$29$	−774.405	−0.170991	−0.0854955	−	0.996339i	$-0.527247\pi$
$29$	−774.405	−0.170991	−0.0854955	+	0.996339i	$0.527247\pi$
$30$	0	0
$31$	−1248.48	−0.233334	−0.116667	−	0.993171i	$-0.537221\pi$
$31$	−1248.48	−0.233334	−0.116667	+	0.993171i	$0.537221\pi$
$32$	5931.48	1.02397
$33$	0	0
$34$	5402.89	0.801546
$35$	−699.913	−0.0965770
$36$	0	0
$37$	7715.93	0.926582	0.463291	−	0.886206i	$-0.346668\pi$
$37$	7715.93	0.926582	0.463291	+	0.886206i	$0.346668\pi$
$38$	4508.89	0.506536
$39$	0	0
$40$	−589.822	−0.0582869
$41$	−7021.66	−0.652350	−0.326175	−	0.945309i	$-0.605760\pi$
$41$	−7021.66	−0.652350	−0.326175	+	0.945309i	$0.605760\pi$
$42$	0	0
$43$	−11547.8	−0.952419	−0.476209	−	0.879332i	$-0.657990\pi$
$43$	−11547.8	−0.952419	−0.476209	+	0.879332i	$0.657990\pi$
$44$	17198.6	1.33925
$45$	0	0
$46$	1441.61	0.100451
$47$	20829.5	1.37541	0.687707	−	0.725988i	$-0.258617\pi$
$47$	20829.5	1.37541	0.687707	+	0.725988i	$0.258617\pi$
$48$	0	0
$49$	16663.0	0.991430
$50$	−8476.23	−0.479488
$51$	0	0
$52$	−19980.5	−1.02470
$53$	25868.7	1.26499	0.632493	−	0.774566i	$-0.282032\pi$
$53$	25868.7	1.26499	0.632493	+	0.774566i	$0.282032\pi$
$54$	0	0
$55$	−2677.58	−0.119354
$56$	28205.4	1.20188
$57$	0	0
$58$	−2110.38	−0.0823740
$59$	12167.1	0.455050	0.227525	−	0.973772i	$-0.426937\pi$
$59$	12167.1	0.455050	0.227525	+	0.973772i	$0.426937\pi$
$60$	0	0
$61$	52562.8	1.80865	0.904324	−	0.426848i	$-0.140376\pi$
$61$	52562.8	1.80865	0.904324	+	0.426848i	$0.140376\pi$
$62$	−3402.31	−0.112407
$63$	0	0
$64$	4445.47	0.135665
$65$	3110.68	0.0913213
$66$	0	0
$67$	−69504.4	−1.89158	−0.945791	−	0.324776i	$-0.894711\pi$
$67$	−69504.4	−1.89158	−0.945791	+	0.324776i	$0.894711\pi$
$68$	−48719.4	−1.27770
$69$	0	0
$70$	−1907.37	−0.0465254
$71$	41256.0	0.971273	0.485637	−	0.874161i	$-0.338588\pi$
$71$	41256.0	0.971273	0.485637	+	0.874161i	$0.338588\pi$
$72$	0	0
$73$	−18589.7	−0.408286	−0.204143	−	0.978941i	$-0.565441\pi$
$73$	−18589.7	−0.408286	−0.204143	+	0.978941i	$0.565441\pi$
$74$	21027.1	0.446376
$75$	0	0
$76$	−40657.9	−0.807441
$77$	128043.	2.46109
$78$	0	0
$79$	19551.6	0.352465	0.176232	−	0.984349i	$-0.443609\pi$
$79$	19551.6	0.352465	0.176232	+	0.984349i	$0.443609\pi$
$80$	1401.03	0.0244750
$81$	0	0
$82$	−19135.1	−0.314266
$83$	−123259.	−1.96391	−0.981956	−	0.189107i	$-0.939441\pi$
$83$	−123259.	−1.96391	−0.981956	+	0.189107i	$0.939441\pi$
$84$	0	0
$85$	7584.91	0.113868
$86$	−31469.6	−0.458822
$87$	0	0
$88$	107902.	1.48534
$89$	46120.6	0.617192	0.308596	−	0.951193i	$-0.400141\pi$
$89$	46120.6	0.617192	0.308596	+	0.951193i	$0.400141\pi$
$90$	0	0
$91$	−148753.	−1.88306
$92$	−12999.4	−0.160123
$93$	0	0
$94$	56763.6	0.662598
$95$	6329.86	0.0719590
$96$	0	0
$97$	−71712.9	−0.773870	−0.386935	−	0.922107i	$-0.626466\pi$
$97$	−71712.9	−0.773870	−0.386935	+	0.922107i	$0.626466\pi$
$98$	45409.2	0.477616
$99$	0	0
$100$	76432.6	0.764326
$101$	143943.	1.40406	0.702032	−	0.712145i	$-0.252276\pi$
$101$	143943.	1.40406	0.702032	+	0.712145i	$0.252276\pi$
$102$	0	0
$103$	−43763.9	−0.406465	−0.203232	−	0.979131i	$-0.565145\pi$
$103$	−43763.9	−0.406465	−0.203232	+	0.979131i	$0.565145\pi$
$104$	−125356.	−1.13648
$105$	0	0
$106$	70496.4	0.609400
$107$	200418.	1.69230	0.846148	−	0.532948i	$-0.178916\pi$
$107$	200418.	1.69230	0.846148	+	0.532948i	$0.178916\pi$
$108$	0	0
$109$	5154.25	0.0415527	0.0207764	−	0.999784i	$-0.493386\pi$
$109$	5154.25	0.0415527	0.0207764	+	0.999784i	$0.493386\pi$
$110$	−7296.84	−0.0574980
$111$	0	0
$112$	−66997.5	−0.504677
$113$	210130.	1.54808	0.774039	−	0.633138i	$-0.218233\pi$
$113$	210130.	1.54808	0.774039	+	0.633138i	$0.218233\pi$
$114$	0	0
$115$	2023.82	0.0142701
$116$	19029.9	0.131308
$117$	0	0
$118$	33157.4	0.219218
$119$	−362712.	−2.34798
$120$	0	0
$121$	328788.	2.04151
$122$	143242.	0.871305
$123$	0	0
$124$	30679.6	0.179182
$125$	−23854.9	−0.136554
$126$	0	0
$127$	−88331.6	−0.485967	−0.242983	−	0.970030i	$-0.578126\pi$
$127$	−88331.6	−0.485967	−0.242983	+	0.970030i	$0.578126\pi$
$128$	−177693.	−0.958616
$129$	0	0
$130$	8477.09	0.0439935
$131$	−161037.	−0.819874	−0.409937	−	0.912114i	$-0.634449\pi$
$131$	−161037.	−0.819874	−0.409937	+	0.912114i	$0.634449\pi$
$132$	0	0
$133$	−302695.	−1.48380
$134$	−189410.	−0.911259
$135$	0	0
$136$	−305660.	−1.41707
$137$	−289207.	−1.31646	−0.658230	−	0.752817i	$-0.728694\pi$
$137$	−289207.	−1.31646	−0.658230	+	0.752817i	$0.728694\pi$
$138$	0	0
$139$	368755.	1.61883	0.809414	−	0.587239i	$-0.199785\pi$
$139$	368755.	1.61883	0.809414	+	0.587239i	$0.199785\pi$
$140$	17199.3	0.0741637
$141$	0	0
$142$	112429.	0.467905
$143$	−569070.	−2.32716
$144$	0	0
$145$	−2962.68	−0.0117021
$146$	−50659.7	−0.196689
$147$	0	0
$148$	−189608.	−0.711543
$149$	9724.99	0.0358859	0.0179429	−	0.999839i	$-0.494288\pi$
$149$	9724.99	0.0358859	0.0179429	+	0.999839i	$0.494288\pi$
$150$	0	0
$151$	−12113.8	−0.0432354	−0.0216177	−	0.999766i	$-0.506882\pi$
$151$	−12113.8	−0.0432354	−0.0216177	+	0.999766i	$0.506882\pi$
$152$	−255084.	−0.895516
$153$	0	0
$154$	348936.	1.18562
$155$	−4776.38	−0.0159687
$156$	0	0
$157$	62748.5	0.203167	0.101584	−	0.994827i	$-0.467609\pi$
$157$	62748.5	0.203167	0.101584	+	0.994827i	$0.467609\pi$
$158$	53281.3	0.169798
$159$	0	0
$160$	22692.3	0.0700776
$161$	−96779.5	−0.294251
$162$	0	0
$163$	259317.	0.764473	0.382236	−	0.924065i	$-0.375154\pi$
$163$	259317.	0.764473	0.382236	+	0.924065i	$0.375154\pi$
$164$	172547.	0.500954
$165$	0	0
$166$	−335899.	−0.946104
$167$	122367.	0.339526	0.169763	−	0.985485i	$-0.445700\pi$
$167$	122367.	0.339526	0.169763	+	0.985485i	$0.445700\pi$
$168$	0	0
$169$	289824.	0.780579
$170$	20670.1	0.0548555
$171$	0	0
$172$	283770.	0.731384
$173$	7975.29	0.0202596	0.0101298	−	0.999949i	$-0.496776\pi$
$173$	7975.29	0.0202596	0.0101298	+	0.999949i	$0.496776\pi$
$174$	0	0
$175$	569035.	1.40457
$176$	−256305.	−0.623700
$177$	0	0
$178$	125686.	0.297329
$179$	402149.	0.938111	0.469055	−	0.883169i	$-0.344594\pi$
$179$	402149.	0.938111	0.469055	+	0.883169i	$0.344594\pi$
$180$	0	0
$181$	−797542.	−1.80949	−0.904747	−	0.425950i	$-0.859940\pi$
$181$	−797542.	−1.80949	−0.904747	+	0.425950i	$0.859940\pi$
$182$	−405376.	−0.907152
$183$	0	0
$184$	−81556.8	−0.177589
$185$	29519.2	0.0634126
$186$	0	0
$187$	−1.38759e6	−2.90173
$188$	−511853.	−1.05621
$189$	0	0
$190$	17249.9	0.0346658
$191$	544255.	1.07949	0.539746	−	0.841828i	$-0.318520\pi$
$191$	544255.	1.07949	0.539746	+	0.841828i	$0.318520\pi$
$192$	0	0
$193$	−509182.	−0.983966	−0.491983	−	0.870605i	$-0.663728\pi$
$193$	−509182.	−0.983966	−0.491983	+	0.870605i	$0.663728\pi$
$194$	−195429.	−0.372808
$195$	0	0
$196$	−409468.	−0.761342
$197$	491389.	0.902110	0.451055	−	0.892496i	$-0.351048\pi$
$197$	491389.	0.902110	0.451055	+	0.892496i	$0.351048\pi$
$198$	0	0
$199$	−205588.	−0.368014	−0.184007	−	0.982925i	$-0.558907\pi$
$199$	−205588.	−0.368014	−0.184007	+	0.982925i	$0.558907\pi$
$200$	479530.	0.847698
$201$	0	0
$202$	392267.	0.676400
$203$	141676.	0.241299
$204$	0	0
$205$	−26863.1	−0.0446449
$206$	−119264.	−0.195812
$207$	0	0
$208$	297762.	0.477212
$209$	−1.15799e6	−1.83374
$210$	0	0
$211$	326555.	0.504952	0.252476	−	0.967603i	$-0.418755\pi$
$211$	326555.	0.504952	0.252476	+	0.967603i	$0.418755\pi$
$212$	−635686.	−0.971411
$213$	0	0
$214$	546169.	0.815254
$215$	−44178.9	−0.0651807
$216$	0	0
$217$	228407.	0.329277
$218$	14046.2	0.0200178
$219$	0	0
$220$	65797.6	0.0916545
$221$	1.61203e6	2.22020
$222$	0	0
$223$	864259.	1.16381	0.581904	−	0.813257i	$-0.302308\pi$
$223$	864259.	1.16381	0.581904	+	0.813257i	$0.302308\pi$
$224$	−1.08515e6	−1.44501
$225$	0	0
$226$	572639.	0.745778
$227$	632266.	0.814395	0.407198	−	0.913340i	$-0.366506\pi$
$227$	632266.	0.814395	0.407198	+	0.913340i	$0.366506\pi$
$228$	0	0
$229$	−1.08510e6	−1.36735	−0.683677	−	0.729785i	$-0.739620\pi$
$229$	−1.08510e6	−1.36735	−0.683677	+	0.729785i	$0.739620\pi$
$230$	5515.23	0.00687455
$231$	0	0
$232$	119391.	0.145631
$233$	222522.	0.268523	0.134262	−	0.990946i	$-0.457134\pi$
$233$	222522.	0.268523	0.134262	+	0.990946i	$0.457134\pi$
$234$	0	0
$235$	79688.3	0.0941293
$236$	−298990.	−0.349443
$237$	0	0
$238$	−988447.	−1.13113
$239$	479116.	0.542558	0.271279	−	0.962501i	$-0.412553\pi$
$239$	479116.	0.542558	0.271279	+	0.962501i	$0.412553\pi$
$240$	0	0
$241$	583602.	0.647253	0.323627	−	0.946185i	$-0.395098\pi$
$241$	583602.	0.647253	0.323627	+	0.946185i	$0.395098\pi$
$242$	895999.	0.983488
$243$	0	0
$244$	−1.29165e6	−1.38890
$245$	63748.3	0.0678506
$246$	0	0
$247$	1.34529e6	1.40305
$248$	192481.	0.198728
$249$	0	0
$250$	−65008.4	−0.0657839
$251$	782506.	0.783977	0.391988	−	0.919970i	$-0.371787\pi$
$251$	782506.	0.783977	0.391988	+	0.919970i	$0.371787\pi$
$252$	0	0
$253$	−370239.	−0.363648
$254$	−240717.	−0.234112
$255$	0	0
$256$	−626495.	−0.597473
$257$	866304.	0.818159	0.409079	−	0.912499i	$-0.365850\pi$
$257$	866304.	0.818159	0.409079	+	0.912499i	$0.365850\pi$
$258$	0	0
$259$	−1.41161e6	−1.30758
$260$	−76440.4	−0.0701277
$261$	0	0
$262$	−438851.	−0.394969
$263$	−1.52956e6	−1.36357	−0.681785	−	0.731553i	$-0.738796\pi$
$263$	−1.52956e6	−1.36357	−0.681785	+	0.731553i	$0.738796\pi$
$264$	0	0
$265$	98967.3	0.0865719
$266$	−824892.	−0.714814
$267$	0	0
$268$	1.70797e6	1.45259
$269$	905547.	0.763010	0.381505	−	0.924367i	$-0.375406\pi$
$269$	905547.	0.763010	0.381505	+	0.924367i	$0.375406\pi$
$270$	0	0
$271$	−710314.	−0.587526	−0.293763	−	0.955878i	$-0.594908\pi$
$271$	−710314.	−0.587526	−0.293763	+	0.955878i	$0.594908\pi$
$272$	726047.	0.595035
$273$	0	0
$274$	−788135.	−0.634197
$275$	2.17690e6	1.73583
$276$	0	0
$277$	1.67464e6	1.31136	0.655680	−	0.755039i	$-0.272382\pi$
$277$	1.67464e6	1.31136	0.655680	+	0.755039i	$0.272382\pi$
$278$	1.00491e6	0.779861
$279$	0	0
$280$	107907.	0.0822534
$281$	1.93320e6	1.46053	0.730265	−	0.683164i	$-0.239397\pi$
$281$	1.93320e6	1.46053	0.730265	+	0.683164i	$0.239397\pi$
$282$	0	0
$283$	1.32361e6	0.982409	0.491205	−	0.871044i	$-0.336557\pi$
$283$	1.32361e6	0.982409	0.491205	+	0.871044i	$0.336557\pi$
$284$	−1.01381e6	−0.745863
$285$	0	0
$286$	−1.55081e6	−1.12109
$287$	1.28460e6	0.920583
$288$	0	0
$289$	2.51083e6	1.76837
$290$	−8073.77	−0.00563743
$291$	0	0
$292$	456813.	0.313532
$293$	2.01599e6	1.37189	0.685945	−	0.727654i	$-0.259389\pi$
$293$	2.01599e6	1.37189	0.685945	+	0.727654i	$0.259389\pi$
$294$	0	0
$295$	46548.4	0.0311423
$296$	−1.18958e6	−0.789158
$297$	0	0
$298$	26502.1	0.0172878
$299$	430125.	0.278238
$300$	0	0
$301$	2.11265e6	1.34403
$302$	−33012.1	−0.0208284
$303$	0	0
$304$	605910.	0.376032
$305$	201092.	0.123778
$306$	0	0
$307$	−1.39881e6	−0.847054	−0.423527	−	0.905883i	$-0.639208\pi$
$307$	−1.39881e6	−0.847054	−0.423527	+	0.905883i	$0.639208\pi$
$308$	−3.14646e6	−1.88993
$309$	0	0
$310$	−13016.4	−0.00769283
$311$	−754025.	−0.442063	−0.221032	−	0.975267i	$-0.570942\pi$
$311$	−754025.	−0.442063	−0.221032	+	0.975267i	$0.570942\pi$
$312$	0	0
$313$	−32854.0	−0.0189552	−0.00947759	−	0.999955i	$-0.503017\pi$
$313$	−32854.0	−0.0189552	−0.00947759	+	0.999955i	$0.503017\pi$
$314$	170999.	0.0978747
$315$	0	0
$316$	−480453.	−0.270665
$317$	−1.67382e6	−0.935534	−0.467767	−	0.883852i	$-0.654941\pi$
$317$	−1.67382e6	−0.935534	−0.467767	+	0.883852i	$0.654941\pi$
$318$	0	0
$319$	541995.	0.298207
$320$	17007.3	0.00928452
$321$	0	0
$322$	−263739.	−0.141754
$323$	3.28029e6	1.74947
$324$	0	0
$325$	−2.52901e6	−1.32813
$326$	706680.	0.368280
$327$	0	0
$328$	1.08254e6	0.555597
$329$	−3.81071e6	−1.94096
$330$	0	0
$331$	775873.	0.389243	0.194621	−	0.980878i	$-0.437652\pi$
$331$	775873.	0.389243	0.194621	+	0.980878i	$0.437652\pi$
$332$	3.02890e6	1.50813
$333$	0	0
$334$	333469.	0.163565
$335$	−265906.	−0.129454
$336$	0	0
$337$	−780786.	−0.374505	−0.187252	−	0.982312i	$-0.559958\pi$
$337$	−780786.	−0.374505	−0.187252	+	0.982312i	$0.559958\pi$
$338$	789815.	0.376040
$339$	0	0
$340$	−186388.	−0.0874421
$341$	873794.	0.406933
$342$	0	0
$343$	26349.8	0.0120932
$344$	1.78034e6	0.811162
$345$	0	0
$346$	21733.9	0.00975995
$347$	−2.41616e6	−1.07722	−0.538608	−	0.842556i	$-0.681050\pi$
$347$	−2.41616e6	−1.07722	−0.538608	+	0.842556i	$0.681050\pi$
$348$	0	0
$349$	−3.82175e6	−1.67957	−0.839786	−	0.542917i	$-0.817320\pi$
$349$	−3.82175e6	−1.67957	−0.839786	+	0.542917i	$0.817320\pi$
$350$	1.55071e6	0.676644
$351$	0	0
$352$	−4.15135e6	−1.78580
$353$	3.79377e6	1.62045	0.810223	−	0.586122i	$-0.199346\pi$
$353$	3.79377e6	1.62045	0.810223	+	0.586122i	$0.199346\pi$
$354$	0	0
$355$	157835.	0.0664711
$356$	−1.13335e6	−0.473955
$357$	0	0
$358$	1.09592e6	0.451930
$359$	−199027.	−0.0815034	−0.0407517	−	0.999169i	$-0.512975\pi$
$359$	−199027.	−0.0815034	−0.0407517	+	0.999169i	$0.512975\pi$
$360$	0	0
$361$	261409.	0.105573
$362$	−2.17343e6	−0.871713
$363$	0	0
$364$	3.65539e6	1.44604
$365$	−71119.3	−0.0279419
$366$	0	0
$367$	2.62662e6	1.01796	0.508981	−	0.860778i	$-0.330022\pi$
$367$	2.62662e6	1.01796	0.508981	+	0.860778i	$0.330022\pi$
$368$	193725.	0.0745705
$369$	0	0
$370$	80444.5	0.0305486
$371$	−4.73264e6	−1.78512
$372$	0	0
$373$	309885.	0.115326	0.0576632	−	0.998336i	$-0.481635\pi$
$373$	309885.	0.115326	0.0576632	+	0.998336i	$0.481635\pi$
$374$	−3.78140e6	−1.39789
$375$	0	0
$376$	−3.21131e6	−1.17142
$377$	−629662.	−0.228168
$378$	0	0
$379$	−4.04023e6	−1.44480	−0.722400	−	0.691475i	$-0.756961\pi$
$379$	−4.04023e6	−1.44480	−0.722400	+	0.691475i	$0.756961\pi$
$380$	−155547.	−0.0552589
$381$	0	0
$382$	1.48318e6	0.520039
$383$	−2.28525e6	−0.796045	−0.398022	−	0.917376i	$-0.630303\pi$
$383$	−2.28525e6	−0.796045	−0.398022	+	0.917376i	$0.630303\pi$
$384$	0	0
$385$	489859.	0.168430
$386$	−1.38760e6	−0.474020
$387$	0	0
$388$	1.76224e6	0.594272
$389$	−1.34208e6	−0.449681	−0.224841	−	0.974396i	$-0.572186\pi$
$389$	−1.34208e6	−0.449681	−0.224841	+	0.974396i	$0.572186\pi$
$390$	0	0
$391$	1.04879e6	0.346935
$392$	−2.56896e6	−0.844388
$393$	0	0
$394$	1.33911e6	0.434587
$395$	74799.7	0.0241216
$396$	0	0
$397$	361451.	0.115099	0.0575497	−	0.998343i	$-0.481671\pi$
$397$	361451.	0.115099	0.0575497	+	0.998343i	$0.481671\pi$
$398$	−560258.	−0.177289
$399$	0	0
$400$	−1.13905e6	−0.355952
$401$	−2.00740e6	−0.623410	−0.311705	−	0.950179i	$-0.600900\pi$
$401$	−2.00740e6	−0.623410	−0.311705	+	0.950179i	$0.600900\pi$
$402$	0	0
$403$	−1.01513e6	−0.311357
$404$	−3.53718e6	−1.07821
$405$	0	0
$406$	386089.	0.116245
$407$	−5.40026e6	−1.61595
$408$	0	0
$409$	−671787.	−0.198575	−0.0992873	−	0.995059i	$-0.531656\pi$
$409$	−671787.	−0.198575	−0.0992873	+	0.995059i	$0.531656\pi$
$410$	−73206.2	−0.0215074
$411$	0	0
$412$	1.07543e6	0.312134
$413$	−2.22596e6	−0.642157
$414$	0	0
$415$	−471557.	−0.134404
$416$	4.82283e6	1.36637
$417$	0	0
$418$	−3.15570e6	−0.883396
$419$	4.30066e6	1.19674	0.598371	−	0.801219i	$-0.295815\pi$
$419$	4.30066e6	1.19674	0.598371	+	0.801219i	$0.295815\pi$
$420$	0	0
$421$	5.86137e6	1.61173	0.805867	−	0.592096i	$-0.201699\pi$
$421$	5.86137e6	1.61173	0.805867	+	0.592096i	$0.201699\pi$
$422$	889913.	0.243258
$423$	0	0
$424$	−3.98823e6	−1.07737
$425$	−6.16659e6	−1.65605
$426$	0	0
$427$	−9.61626e6	−2.55233
$428$	−4.92496e6	−1.29955
$429$	0	0
$430$	−120395.	−0.0314004
$431$	6.39806e6	1.65903	0.829516	−	0.558482i	$-0.188616\pi$
$431$	6.39806e6	1.65903	0.829516	+	0.558482i	$0.188616\pi$
$432$	0	0
$433$	−1.32484e6	−0.339581	−0.169791	−	0.985480i	$-0.554309\pi$
$433$	−1.32484e6	−0.339581	−0.169791	+	0.985480i	$0.554309\pi$
$434$	622446.	0.158627
$435$	0	0
$436$	−126658.	−0.0319093
$437$	875253.	0.219245
$438$	0	0
$439$	4.02534e6	0.996877	0.498438	−	0.866925i	$-0.333907\pi$
$439$	4.02534e6	0.996877	0.498438	+	0.866925i	$0.333907\pi$
$440$	412808.	0.101652
$441$	0	0
$442$	4.39304e6	1.06957
$443$	4.88498e6	1.18264	0.591321	−	0.806436i	$-0.298606\pi$
$443$	4.88498e6	1.18264	0.591321	+	0.806436i	$0.298606\pi$
$444$	0	0
$445$	176446.	0.0422388
$446$	2.35524e6	0.560658
$447$	0	0
$448$	−813290.	−0.191448
$449$	2.00441e6	0.469213	0.234606	−	0.972090i	$-0.424620\pi$
$449$	2.00441e6	0.469213	0.234606	+	0.972090i	$0.424620\pi$
$450$	0	0
$451$	4.91436e6	1.13769
$452$	−5.16365e6	−1.18880
$453$	0	0
$454$	1.72302e6	0.392330
$455$	−569093.	−0.128871
$456$	0	0
$457$	−7.66973e6	−1.71787	−0.858934	−	0.512087i	$-0.828873\pi$
$457$	−7.66973e6	−1.71787	−0.858934	+	0.512087i	$0.828873\pi$
$458$	−2.95707e6	−0.658715
$459$	0	0
$460$	−49732.4	−0.0109583
$461$	−1.15815e6	−0.253812	−0.126906	−	0.991915i	$-0.540505\pi$
$461$	−1.15815e6	−0.253812	−0.126906	+	0.991915i	$0.540505\pi$
$462$	0	0
$463$	4.48451e6	0.972215	0.486107	−	0.873899i	$-0.338416\pi$
$463$	4.48451e6	0.972215	0.486107	+	0.873899i	$0.338416\pi$
$464$	−283595.	−0.0611511
$465$	0	0
$466$	606406.	0.129360
$467$	−679028.	−0.144077	−0.0720386	−	0.997402i	$-0.522950\pi$
$467$	−679028.	−0.144077	−0.0720386	+	0.997402i	$0.522950\pi$
$468$	0	0
$469$	1.27157e7	2.66936
$470$	217163.	0.0453462
$471$	0	0
$472$	−1.87583e6	−0.387560
$473$	8.08213e6	1.66101
$474$	0	0
$475$	−5.14623e6	−1.04654
$476$	8.91311e6	1.80307
$477$	0	0
$478$	1.30567e6	0.261374
$479$	−5.55589e6	−1.10641	−0.553204	−	0.833046i	$-0.686595\pi$
$479$	−5.55589e6	−1.10641	−0.553204	+	0.833046i	$0.686595\pi$
$480$	0	0
$481$	6.27375e6	1.23642
$482$	1.59041e6	0.311810
$483$	0	0
$484$	−8.07948e6	−1.56772
$485$	−274356.	−0.0529614
$486$	0	0
$487$	−436634.	−0.0834248	−0.0417124	−	0.999130i	$-0.513281\pi$
$487$	−436634.	−0.0834248	−0.0417124	+	0.999130i	$0.513281\pi$
$488$	−8.10370e6	−1.54040
$489$	0	0
$490$	173724.	0.0326866
$491$	−2.14241e6	−0.401050	−0.200525	−	0.979689i	$-0.564265\pi$
$491$	−2.14241e6	−0.401050	−0.200525	+	0.979689i	$0.564265\pi$
$492$	0	0
$493$	−1.53533e6	−0.284502
$494$	3.66613e6	0.675913
$495$	0	0
$496$	−457207.	−0.0834466
$497$	−7.54771e6	−1.37064
$498$	0	0
$499$	−1.18374e6	−0.212816	−0.106408	−	0.994323i	$-0.533935\pi$
$499$	−1.18374e6	−0.212816	−0.106408	+	0.994323i	$0.533935\pi$
$500$	586200.	0.104863
$501$	0	0
$502$	2.13245e6	0.377676
$503$	−1.39777e6	−0.246330	−0.123165	−	0.992386i	$-0.539304\pi$
$503$	−1.39777e6	−0.246330	−0.123165	+	0.992386i	$0.539304\pi$
$504$	0	0
$505$	550689.	0.0960900
$506$	−1.00896e6	−0.175185
$507$	0	0
$508$	2.17062e6	0.373185
$509$	9.10872e6	1.55834	0.779171	−	0.626812i	$-0.215640\pi$
$509$	9.10872e6	1.55834	0.779171	+	0.626812i	$0.215640\pi$
$510$	0	0
$511$	3.40094e6	0.576165
$512$	3.97887e6	0.670787
$513$	0	0
$514$	2.36082e6	0.394143
$515$	−167430.	−0.0278173
$516$	0	0
$517$	−1.45782e7	−2.39871
$518$	−3.84687e6	−0.629917
$519$	0	0
$520$	−479579.	−0.0777771
$521$	1.45675e6	0.235121	0.117561	−	0.993066i	$-0.462493\pi$
$521$	1.45675e6	0.235121	0.117561	+	0.993066i	$0.462493\pi$
$522$	0	0
$523$	−1.64908e6	−0.263626	−0.131813	−	0.991275i	$-0.542080\pi$
$523$	−1.64908e6	−0.263626	−0.131813	+	0.991275i	$0.542080\pi$
$524$	3.95724e6	0.629599
$525$	0	0
$526$	−4.16829e6	−0.656892
$527$	−2.47524e6	−0.388231
$528$	0	0
$529$	279841.	0.0434783
$530$	269701.	0.0417055
$531$	0	0
$532$	7.43828e6	1.13945
$533$	−5.70925e6	−0.870484
$534$	0	0
$535$	766747.	0.115816
$536$	1.07156e7	1.61103
$537$	0	0
$538$	2.46776e6	0.367576
$539$	−1.16622e7	−1.72905
$540$	0	0
$541$	3.31982e6	0.487665	0.243833	−	0.969817i	$-0.421595\pi$
$541$	3.31982e6	0.487665	0.243833	+	0.969817i	$0.421595\pi$
$542$	−1.93572e6	−0.283038
$543$	0	0
$544$	1.17597e7	1.70373
$545$	19718.9	0.00284375
$546$	0	0
$547$	1.29056e7	1.84421	0.922104	−	0.386941i	$-0.126468\pi$
$547$	1.29056e7	1.84421	0.922104	+	0.386941i	$0.126468\pi$
$548$	7.10683e6	1.01094
$549$	0	0
$550$	5.93239e6	0.836224
$551$	−1.28129e6	−0.179791
$552$	0	0
$553$	−3.57693e6	−0.497391
$554$	4.56365e6	0.631740
$555$	0	0
$556$	−9.06160e6	−1.24313
$557$	−1.19202e7	−1.62797	−0.813985	−	0.580885i	$-0.802706\pi$
$557$	−1.19202e7	−1.62797	−0.813985	+	0.580885i	$0.802706\pi$
$558$	0	0
$559$	−9.38940e6	−1.27089
$560$	−256315.	−0.0345386
$561$	0	0
$562$	5.26827e6	0.703602
$563$	2.60845e6	0.346826	0.173413	−	0.984849i	$-0.444520\pi$
$563$	2.60845e6	0.346826	0.173413	+	0.984849i	$0.444520\pi$
$564$	0	0
$565$	803906.	0.105946
$566$	3.60703e6	0.473270
$567$	0	0
$568$	−6.36051e6	−0.827221
$569$	4.74193e6	0.614008	0.307004	−	0.951708i	$-0.400673\pi$
$569$	4.74193e6	0.614008	0.307004	+	0.951708i	$0.400673\pi$
$570$	0	0
$571$	−4.33777e6	−0.556770	−0.278385	−	0.960470i	$-0.589799\pi$
$571$	−4.33777e6	−0.556770	−0.278385	+	0.960470i	$0.589799\pi$
$572$	1.39841e7	1.78708
$573$	0	0
$574$	3.50073e6	0.443486
$575$	−1.64538e6	−0.207538
$576$	0	0
$577$	−9.92005e6	−1.24044	−0.620218	−	0.784430i	$-0.712956\pi$
$577$	−9.92005e6	−1.24044	−0.620218	+	0.784430i	$0.712956\pi$
$578$	6.84241e6	0.851901
$579$	0	0
$580$	72803.5	0.00898633
$581$	2.25499e7	2.77144
$582$	0	0
$583$	−1.81051e7	−2.20613
$584$	2.86600e6	0.347731
$585$	0	0
$586$	5.49389e6	0.660900
$587$	−1.64436e6	−0.196971	−0.0984854	−	0.995138i	$-0.531400\pi$
$587$	−1.64436e6	−0.196971	−0.0984854	+	0.995138i	$0.531400\pi$
$588$	0	0
$589$	−2.06567e6	−0.245342
$590$	126852.	0.0150026
$591$	0	0
$592$	2.82565e6	0.331371
$593$	7.19901e6	0.840691	0.420345	−	0.907364i	$-0.361909\pi$
$593$	7.19901e6	0.840691	0.420345	+	0.907364i	$0.361909\pi$
$594$	0	0
$595$	−1.38764e6	−0.160689
$596$	−238977.	−0.0275576
$597$	0	0
$598$	1.17216e6	0.134040
$599$	−7.35774e6	−0.837872	−0.418936	−	0.908016i	$-0.637597\pi$
$599$	−7.35774e6	−0.837872	−0.418936	+	0.908016i	$0.637597\pi$
$600$	0	0
$601$	−2.69213e6	−0.304026	−0.152013	−	0.988379i	$-0.548576\pi$
$601$	−2.69213e6	−0.304026	−0.152013	+	0.988379i	$0.548576\pi$
$602$	5.75729e6	0.647481
$603$	0	0
$604$	297679.	0.0332014
$605$	1.25786e6	0.139715
$606$	0	0
$607$	9.26825e6	1.02100	0.510500	−	0.859878i	$-0.329460\pi$
$607$	9.26825e6	1.02100	0.510500	+	0.859878i	$0.329460\pi$
$608$	9.81387e6	1.07667
$609$	0	0
$610$	548007.	0.0596296
$611$	1.69362e7	1.83533
$612$	0	0
$613$	−6.12094e6	−0.657911	−0.328955	−	0.944345i	$-0.606697\pi$
$613$	−6.12094e6	−0.657911	−0.328955	+	0.944345i	$0.606697\pi$
$614$	−3.81196e6	−0.408064
$615$	0	0
$616$	−1.97405e7	−2.09608
$617$	−6.62760e6	−0.700880	−0.350440	−	0.936585i	$-0.613968\pi$
$617$	−6.62760e6	−0.700880	−0.350440	+	0.936585i	$0.613968\pi$
$618$	0	0
$619$	5.81014e6	0.609481	0.304741	−	0.952435i	$-0.401430\pi$
$619$	5.81014e6	0.609481	0.304741	+	0.952435i	$0.401430\pi$
$620$	117372.	0.0122627
$621$	0	0
$622$	−2.05484e6	−0.212961
$623$	−8.43768e6	−0.870969
$624$	0	0
$625$	9.62862e6	0.985971
$626$	−89532.4	−0.00913155
$627$	0	0
$628$	−1.54195e6	−0.156017
$629$	1.52976e7	1.54169
$630$	0	0
$631$	−657568.	−0.0657456	−0.0328728	−	0.999460i	$-0.510466\pi$
$631$	−657568.	−0.0657456	−0.0328728	+	0.999460i	$0.510466\pi$
$632$	−3.01431e6	−0.300189
$633$	0	0
$634$	−4.56141e6	−0.450688
$635$	−337934.	−0.0332581
$636$	0	0
$637$	1.35485e7	1.32295
$638$	1.47702e6	0.143660
$639$	0	0
$640$	−679807.	−0.0656048
$641$	1.33378e6	0.128215	0.0641074	−	0.997943i	$-0.479580\pi$
$641$	1.33378e6	0.128215	0.0641074	+	0.997943i	$0.479580\pi$
$642$	0	0
$643$	1.02854e7	0.981052	0.490526	−	0.871426i	$-0.336805\pi$
$643$	1.02854e7	0.981052	0.490526	+	0.871426i	$0.336805\pi$
$644$	2.37821e6	0.225962
$645$	0	0
$646$	8.93930e6	0.842796
$647$	−1.00327e7	−0.942230	−0.471115	−	0.882072i	$-0.656148\pi$
$647$	−1.00327e7	−0.942230	−0.471115	+	0.882072i	$0.656148\pi$
$648$	0	0
$649$	−8.51560e6	−0.793604
$650$	−6.89194e6	−0.639821
$651$	0	0
$652$	−6.37233e6	−0.587056
$653$	1.15777e7	1.06252	0.531261	−	0.847208i	$-0.321718\pi$
$653$	1.15777e7	1.06252	0.531261	+	0.847208i	$0.321718\pi$
$654$	0	0
$655$	−616086.	−0.0561097
$656$	−2.57141e6	−0.233298
$657$	0	0
$658$	−1.03848e7	−0.935045
$659$	−1.68232e7	−1.50902	−0.754508	−	0.656291i	$-0.772124\pi$
$659$	−1.68232e7	−1.50902	−0.754508	+	0.656291i	$0.772124\pi$
$660$	0	0
$661$	461512.	0.0410846	0.0205423	−	0.999789i	$-0.493461\pi$
$661$	461512.	0.0410846	0.0205423	+	0.999789i	$0.493461\pi$
$662$	2.11438e6	0.187516
$663$	0	0
$664$	1.90030e7	1.67264
$665$	−1.15804e6	−0.101547
$666$	0	0
$667$	−409661.	−0.0356541
$668$	−3.00699e6	−0.260730
$669$	0	0
$670$	−724636.	−0.0623639
$671$	−3.67879e7	−3.15427
$672$	0	0
$673$	−1.09347e7	−0.930613	−0.465306	−	0.885150i	$-0.654056\pi$
$673$	−1.09347e7	−0.930613	−0.465306	+	0.885150i	$0.654056\pi$
$674$	−2.12776e6	−0.180416
$675$	0	0
$676$	−7.12199e6	−0.599424
$677$	−6.53146e6	−0.547695	−0.273847	−	0.961773i	$-0.588296\pi$
$677$	−6.53146e6	−0.547695	−0.273847	+	0.961773i	$0.588296\pi$
$678$	0	0
$679$	1.31197e7	1.09207
$680$	−1.16938e6	−0.0969802
$681$	0	0
$682$	2.38123e6	0.196038
$683$	1.88150e7	1.54330	0.771652	−	0.636045i	$-0.219431\pi$
$683$	1.88150e7	1.54330	0.771652	+	0.636045i	$0.219431\pi$
$684$	0	0
$685$	−1.10643e6	−0.0900946
$686$	71807.4	0.00582584
$687$	0	0
$688$	−4.22892e6	−0.340611
$689$	2.10336e7	1.68798
$690$	0	0
$691$	8.99229e6	0.716432	0.358216	−	0.933639i	$-0.383385\pi$
$691$	8.99229e6	0.716432	0.358216	+	0.933639i	$0.383385\pi$
$692$	−195981.	−0.0155578
$693$	0	0
$694$	−6.58443e6	−0.518943
$695$	1.41076e6	0.110788
$696$	0	0
$697$	−1.39211e7	−1.08541
$698$	−1.04149e7	−0.809124
$699$	0	0
$700$	−1.39832e7	−1.07860
$701$	−1.92528e6	−0.147979	−0.0739894	−	0.997259i	$-0.523573\pi$
$701$	−1.92528e6	−0.147979	−0.0739894	+	0.997259i	$0.523573\pi$
$702$	0	0
$703$	1.27663e7	0.974267
$704$	−3.11132e6	−0.236599
$705$	0	0
$706$	1.03386e7	0.780641
$707$	−2.63341e7	−1.98139
$708$	0	0
$709$	−464785.	−0.0347245	−0.0173623	−	0.999849i	$-0.505527\pi$
$709$	−464785.	−0.0347245	−0.0173623	+	0.999849i	$0.505527\pi$
$710$	430126.	0.0320221
$711$	0	0
$712$	−7.11050e6	−0.525654
$713$	−660447.	−0.0486535
$714$	0	0
$715$	−2.17712e6	−0.159264
$716$	−9.88221e6	−0.720396
$717$	0	0
$718$	−542380.	−0.0392638
$719$	−7.97792e6	−0.575529	−0.287765	−	0.957701i	$-0.592912\pi$
$719$	−7.97792e6	−0.575529	−0.287765	+	0.957701i	$0.592912\pi$
$720$	0	0
$721$	8.00652e6	0.573595
$722$	712381.	0.0508591
$723$	0	0
$724$	1.95984e7	1.38955
$725$	2.40868e6	0.170190
$726$	0	0
$727$	−1.78925e7	−1.25555	−0.627777	−	0.778394i	$-0.716035\pi$
$727$	−1.78925e7	−1.25555	−0.627777	+	0.778394i	$0.716035\pi$
$728$	2.29336e7	1.60377
$729$	0	0
$730$	−193811.	−0.0134608
$731$	−2.28946e7	−1.58467
$732$	0	0
$733$	−3.62180e6	−0.248980	−0.124490	−	0.992221i	$-0.539729\pi$
$733$	−3.62180e6	−0.248980	−0.124490	+	0.992221i	$0.539729\pi$
$734$	7.15794e6	0.490397
$735$	0	0
$736$	3.13775e6	0.213513
$737$	4.86451e7	3.29891
$738$	0	0
$739$	−1.37519e6	−0.0926300	−0.0463150	−	0.998927i	$-0.514748\pi$
$739$	−1.37519e6	−0.0926300	−0.0463150	+	0.998927i	$0.514748\pi$
$740$	−725391.	−0.0486959
$741$	0	0
$742$	−1.28972e7	−0.859973
$743$	−1.51506e7	−1.00683	−0.503416	−	0.864044i	$-0.667924\pi$
$743$	−1.51506e7	−1.00683	−0.503416	+	0.864044i	$0.667924\pi$
$744$	0	0
$745$	37205.3	0.00245592
$746$	844485.	0.0555578
$747$	0	0
$748$	3.40979e7	2.22830
$749$	−3.66660e7	−2.38813
$750$	0	0
$751$	−7.79348e6	−0.504233	−0.252117	−	0.967697i	$-0.581127\pi$
$751$	−7.79348e6	−0.504233	−0.252117	+	0.967697i	$0.581127\pi$
$752$	7.62797e6	0.491886
$753$	0	0
$754$	−1.71593e6	−0.109918
$755$	−46344.5	−0.00295890
$756$	0	0
$757$	2.94822e7	1.86991	0.934953	−	0.354773i	$-0.115442\pi$
$757$	2.94822e7	1.86991	0.934953	+	0.354773i	$0.115442\pi$
$758$	−1.10103e7	−0.696024
$759$	0	0
$760$	−975885.	−0.0612865
$761$	−1.81301e7	−1.13485	−0.567426	−	0.823424i	$-0.692061\pi$
$761$	−1.81301e7	−1.13485	−0.567426	+	0.823424i	$0.692061\pi$
$762$	0	0
$763$	−942960.	−0.0586384
$764$	−1.33743e7	−0.828966
$765$	0	0
$766$	−6.22768e6	−0.383490
$767$	9.89300e6	0.607211
$768$	0	0
$769$	−2.92322e6	−0.178257	−0.0891283	−	0.996020i	$-0.528408\pi$
$769$	−2.92322e6	−0.178257	−0.0891283	+	0.996020i	$0.528408\pi$
$770$	1.33494e6	0.0811401
$771$	0	0
$772$	1.25124e7	0.755610
$773$	−6.88834e6	−0.414635	−0.207317	−	0.978274i	$-0.566473\pi$
$773$	−6.88834e6	−0.414635	−0.207317	+	0.978274i	$0.566473\pi$
$774$	0	0
$775$	3.88323e6	0.232241
$776$	1.10561e7	0.659095
$777$	0	0
$778$	−3.65738e6	−0.216631
$779$	−1.16176e7	−0.685921
$780$	0	0
$781$	−2.88745e7	−1.69389
$782$	2.85813e6	0.167134
$783$	0	0
$784$	6.10215e6	0.354563
$785$	240060.	0.0139042
$786$	0	0
$787$	−2.76438e7	−1.59096	−0.795482	−	0.605977i	$-0.792782\pi$
$787$	−2.76438e7	−1.59096	−0.795482	+	0.605977i	$0.792782\pi$
$788$	−1.20752e7	−0.692751
$789$	0	0
$790$	203841.	0.0116205
$791$	−3.84430e7	−2.18462
$792$	0	0
$793$	4.27383e7	2.41343
$794$	985011.	0.0554485
$795$	0	0
$796$	5.05201e6	0.282606
$797$	−1.99716e7	−1.11370	−0.556850	−	0.830613i	$-0.687990\pi$
$797$	−1.99716e7	−1.11370	−0.556850	+	0.830613i	$0.687990\pi$
$798$	0	0
$799$	4.12964e7	2.28847
$800$	−1.84490e7	−1.01918
$801$	0	0
$802$	−5.47049e6	−0.300324
$803$	1.30106e7	0.712048
$804$	0	0
$805$	−370254.	−0.0201377
$806$	−2.76639e6	−0.149995
$807$	0	0
$808$	−2.21919e7	−1.19582
$809$	1.18906e7	0.638751	0.319376	−	0.947628i	$-0.396527\pi$
$809$	1.18906e7	0.638751	0.319376	+	0.947628i	$0.396527\pi$
$810$	0	0
$811$	2.06076e7	1.10021	0.550103	−	0.835097i	$-0.314588\pi$
$811$	2.06076e7	1.10021	0.550103	+	0.835097i	$0.314588\pi$
$812$	−3.48148e6	−0.185299
$813$	0	0
$814$	−1.47166e7	−0.778477
$815$	992082.	0.0523183
$816$	0	0
$817$	−1.91063e7	−1.00143
$818$	−1.83073e6	−0.0956621
$819$	0	0
$820$	660121.	0.0342838
$821$	−1.30392e7	−0.675136	−0.337568	−	0.941301i	$-0.609604\pi$
$821$	−1.30392e7	−0.675136	−0.337568	+	0.941301i	$0.609604\pi$
$822$	0	0
$823$	1.57128e7	0.808640	0.404320	−	0.914618i	$-0.367508\pi$
$823$	1.57128e7	0.808640	0.404320	+	0.914618i	$0.367508\pi$
$824$	6.74716e6	0.346181
$825$	0	0
$826$	−6.06608e6	−0.309356
$827$	1.57822e7	0.802422	0.401211	−	0.915986i	$-0.368589\pi$
$827$	1.57822e7	0.802422	0.401211	+	0.915986i	$0.368589\pi$
$828$	0	0
$829$	1.97429e7	0.997755	0.498877	−	0.866673i	$-0.333746\pi$
$829$	1.97429e7	0.997755	0.498877	+	0.866673i	$0.333746\pi$
$830$	−1.28507e6	−0.0647486
$831$	0	0
$832$	3.61457e6	0.181029
$833$	3.30359e7	1.64958
$834$	0	0
$835$	468145.	0.0232362
$836$	2.84559e7	1.40817
$837$	0	0
$838$	1.17200e7	0.576523
$839$	−3.20565e6	−0.157221	−0.0786106	−	0.996905i	$-0.525048\pi$
$839$	−3.20565e6	−0.157221	−0.0786106	+	0.996905i	$0.525048\pi$
$840$	0	0
$841$	−1.99114e7	−0.970762
$842$	1.59731e7	0.776444
$843$	0	0
$844$	−8.02460e6	−0.387764
$845$	1.10879e6	0.0534206
$846$	0	0
$847$	−6.01511e7	−2.88095
$848$	9.47340e6	0.452394
$849$	0	0
$850$	−1.68049e7	−0.797792
$851$	4.08173e6	0.193206
$852$	0	0
$853$	1.46500e7	0.689392	0.344696	−	0.938714i	$-0.387982\pi$
$853$	1.46500e7	0.689392	0.344696	+	0.938714i	$0.387982\pi$
$854$	−2.62058e7	−1.22957
$855$	0	0
$856$	−3.08987e7	−1.44131
$857$	−9.64538e6	−0.448608	−0.224304	−	0.974519i	$-0.572011\pi$
$857$	−9.64538e6	−0.448608	−0.224304	+	0.974519i	$0.572011\pi$
$858$	0	0
$859$	−4.37947e6	−0.202506	−0.101253	−	0.994861i	$-0.532285\pi$
$859$	−4.37947e6	−0.202506	−0.101253	+	0.994861i	$0.532285\pi$
$860$	1.08563e6	0.0500537
$861$	0	0
$862$	1.74357e7	0.799230
$863$	−3.56875e6	−0.163113	−0.0815567	−	0.996669i	$-0.525989\pi$
$863$	−3.56875e6	−0.163113	−0.0815567	+	0.996669i	$0.525989\pi$
$864$	0	0
$865$	30511.4	0.00138651
$866$	−3.61040e6	−0.163591
$867$	0	0
$868$	−5.61277e6	−0.252859
$869$	−1.36839e7	−0.614696
$870$	0	0
$871$	−5.65134e7	−2.52410
$872$	−794640.	−0.0353899
$873$	0	0
$874$	2.38520e6	0.105620
$875$	4.36421e6	0.192702
$876$	0	0
$877$	−1.78453e6	−0.0783476	−0.0391738	−	0.999232i	$-0.512473\pi$
$877$	−1.78453e6	−0.0783476	−0.0391738	+	0.999232i	$0.512473\pi$
$878$	1.09697e7	0.480240
$879$	0	0
$880$	−980559.	−0.0426842
$881$	−1.43263e7	−0.621863	−0.310931	−	0.950432i	$-0.600641\pi$
$881$	−1.43263e7	−0.621863	−0.310931	+	0.950432i	$0.600641\pi$
$882$	0	0
$883$	1.45225e7	0.626815	0.313408	−	0.949619i	$-0.398529\pi$
$883$	1.45225e7	0.626815	0.313408	+	0.949619i	$0.398529\pi$
$884$	−3.96133e7	−1.70494
$885$	0	0
$886$	1.33123e7	0.569731
$887$	−6.72663e6	−0.287071	−0.143535	−	0.989645i	$-0.545847\pi$
$887$	−6.72663e6	−0.287071	−0.143535	+	0.989645i	$0.545847\pi$
$888$	0	0
$889$	1.61601e7	0.685787
$890$	480843.	0.0203483
$891$	0	0
$892$	−2.12379e7	−0.893715
$893$	3.44632e7	1.44620
$894$	0	0
$895$	1.53852e6	0.0642015
$896$	3.25085e7	1.35278
$897$	0	0
$898$	5.46232e6	0.226041
$899$	966832.	0.0398980
$900$	0	0
$901$	5.12873e7	2.10474
$902$	1.33924e7	0.548077
$903$	0	0
$904$	−3.23962e7	−1.31848
$905$	−3.05119e6	−0.123836
$906$	0	0
$907$	−2.05444e6	−0.0829231	−0.0414616	−	0.999140i	$-0.513201\pi$
$907$	−2.05444e6	−0.0829231	−0.0414616	+	0.999140i	$0.513201\pi$
$908$	−1.55370e7	−0.625392
$909$	0	0
$910$	−1.55087e6	−0.0620828
$911$	8.41668e6	0.336004	0.168002	−	0.985787i	$-0.446268\pi$
$911$	8.41668e6	0.336004	0.168002	+	0.985787i	$0.446268\pi$
$912$	0	0
$913$	8.62669e7	3.42505
$914$	−2.09012e7	−0.827573
$915$	0	0
$916$	2.66647e7	1.05002
$917$	2.94614e7	1.15699
$918$	0	0
$919$	−2.36646e7	−0.924293	−0.462147	−	0.886804i	$-0.652921\pi$
$919$	−2.36646e7	−0.924293	−0.462147	+	0.886804i	$0.652921\pi$
$920$	−312016.	−0.0121537
$921$	0	0
$922$	−3.15613e6	−0.122272
$923$	3.35449e7	1.29605
$924$	0	0
$925$	−2.39994e7	−0.922243
$926$	1.22210e7	0.468359
$927$	0	0
$928$	−4.59337e6	−0.175090
$929$	−2.64464e7	−1.00537	−0.502687	−	0.864469i	$-0.667655\pi$
$929$	−2.64464e7	−1.00537	−0.502687	+	0.864469i	$0.667655\pi$
$930$	0	0
$931$	2.75696e7	1.04245
$932$	−5.46814e6	−0.206205
$933$	0	0
$934$	−1.85046e6	−0.0694084
$935$	−5.30857e6	−0.198586
$936$	0	0
$937$	−1.05897e7	−0.394036	−0.197018	−	0.980400i	$-0.563126\pi$
$937$	−1.05897e7	−0.394036	−0.197018	+	0.980400i	$0.563126\pi$
$938$	3.46522e7	1.28595
$939$	0	0
$940$	−1.95822e6	−0.0722840
$941$	1.57304e7	0.579115	0.289557	−	0.957161i	$-0.406492\pi$
$941$	1.57304e7	0.579115	0.289557	+	0.957161i	$0.406492\pi$
$942$	0	0
$943$	−3.71446e6	−0.136024
$944$	4.45574e6	0.162738
$945$	0	0
$946$	2.20251e7	0.800183
$947$	−3.96194e6	−0.143560	−0.0717799	−	0.997420i	$-0.522868\pi$
$947$	−3.96194e6	−0.143560	−0.0717799	+	0.997420i	$0.522868\pi$
$948$	0	0
$949$	−1.51151e7	−0.544810
$950$	−1.40243e7	−0.504164
$951$	0	0
$952$	5.59199e7	1.99974
$953$	−1.04977e7	−0.374423	−0.187211	−	0.982320i	$-0.559945\pi$
$953$	−1.04977e7	−0.374423	−0.187211	+	0.982320i	$0.559945\pi$
$954$	0	0
$955$	2.08218e6	0.0738772
$956$	−1.17736e7	−0.416642
$957$	0	0
$958$	−1.51407e7	−0.533005
$959$	5.29098e7	1.85776
$960$	0	0
$961$	−2.70704e7	−0.945555
$962$	1.70970e7	0.595637
$963$	0	0
$964$	−1.43412e7	−0.497040
$965$	−1.94800e6	−0.0673397
$966$	0	0
$967$	−2.02069e7	−0.694917	−0.347459	−	0.937695i	$-0.612955\pi$
$967$	−2.02069e7	−0.694917	−0.347459	+	0.937695i	$0.612955\pi$
$968$	−5.06898e7	−1.73873
$969$	0	0
$970$	−747662.	−0.0255139
$971$	−2.37623e7	−0.808800	−0.404400	−	0.914582i	$-0.632520\pi$
$971$	−2.37623e7	−0.808800	−0.404400	+	0.914582i	$0.632520\pi$
$972$	0	0
$973$	−6.74629e7	−2.28446
$974$	−1.18990e6	−0.0401894
$975$	0	0
$976$	1.92490e7	0.646822
$977$	−1.39240e6	−0.0466688	−0.0233344	−	0.999728i	$-0.507428\pi$
$977$	−1.39240e6	−0.0466688	−0.0233344	+	0.999728i	$0.507428\pi$
$978$	0	0
$979$	−3.22791e7	−1.07638
$980$	−1.56652e6	−0.0521040
$981$	0	0
$982$	−5.83841e6	−0.193204
$983$	3.64073e7	1.20172	0.600862	−	0.799353i	$-0.294824\pi$
$983$	3.64073e7	1.20172	0.600862	+	0.799353i	$0.294824\pi$
$984$	0	0
$985$	1.87993e6	0.0617378
$986$	−4.18402e6	−0.137057
$987$	0	0
$988$	−3.30586e7	−1.07744
$989$	−6.10878e6	−0.198593
$990$	0	0
$991$	3.91635e6	0.126677	0.0633384	−	0.997992i	$-0.479825\pi$
$991$	3.91635e6	0.126677	0.0633384	+	0.997992i	$0.479825\pi$
$992$	−7.40534e6	−0.238927
$993$	0	0
$994$	−2.05687e7	−0.660299
$995$	−786526.	−0.0251858
$996$	0	0
$997$	4.91717e6	0.156667	0.0783334	−	0.996927i	$-0.475040\pi$
$997$	4.91717e6	0.156667	0.0783334	+	0.996927i	$0.475040\pi$
$998$	−3.22587e6	−0.102523
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	207.6.a.i.1.6	yes	10
3.2	odd	2		207.6.a.h.1.5	✓	10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
207.6.a.h.1.5	✓	10	3.2	odd	2
207.6.a.i.1.6	yes	10	1.1	even	1	trivial

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)

\(f(q)\)	\(=\)	\(q+2.72516 q^{2} -24.5735 q^{4} +3.82575 q^{5} -182.948 q^{7} -154.172 q^{8} +O(q^{10})\)	\(q+2.72516 q^{2} -24.5735 q^{4} +3.82575 q^{5} -182.948 q^{7} -154.172 q^{8} +10.4258 q^{10} -699.885 q^{11} +813.091 q^{13} -498.562 q^{14} +366.210 q^{16} +1982.60 q^{17} +1654.54 q^{19} -94.0121 q^{20} -1907.30 q^{22} +529.000 q^{23} -3110.36 q^{25} +2215.80 q^{26} +4495.68 q^{28} -774.405 q^{29} -1248.48 q^{31} +5931.48 q^{32} +5402.89 q^{34} -699.913 q^{35} +7715.93 q^{37} +4508.89 q^{38} -589.822 q^{40} -7021.66 q^{41} -11547.8 q^{43} +17198.6 q^{44} +1441.61 q^{46} +20829.5 q^{47} +16663.0 q^{49} -8476.23 q^{50} -19980.5 q^{52} +25868.7 q^{53} -2677.58 q^{55} +28205.4 q^{56} -2110.38 q^{58} +12167.1 q^{59} +52562.8 q^{61} -3402.31 q^{62} +4445.47 q^{64} +3110.68 q^{65} -69504.4 q^{67} -48719.4 q^{68} -1907.37 q^{70} +41256.0 q^{71} -18589.7 q^{73} +21027.1 q^{74} -40657.9 q^{76} +128043. q^{77} +19551.6 q^{79} +1401.03 q^{80} -19135.1 q^{82} -123259. q^{83} +7584.91 q^{85} -31469.6 q^{86} +107902. q^{88} +46120.6 q^{89} -148753. q^{91} -12999.4 q^{92} +56763.6 q^{94} +6329.86 q^{95} -71712.9 q^{97} +45409.2 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q + 8 q^{2} + 192 q^{4} + 100 q^{5} + 20 q^{7} + 384 q^{8}+O(q^{10}) \) 10 * q + 8 * q^2 + 192 * q^4 + 100 * q^5 + 20 * q^7 + 384 * q^8	\( 10 q + 8 q^{2} + 192 q^{4} + 100 q^{5} + 20 q^{7} + 384 q^{8} - 250 q^{10} + 460 q^{11} + 464 q^{13} + 3676 q^{14} + 4612 q^{16} + 4756 q^{17} - 1780 q^{19} + 10314 q^{20} - 4214 q^{22} + 5290 q^{23} + 1330 q^{25} - 5152 q^{26} + 7072 q^{28} + 4048 q^{29} + 2816 q^{31} + 27436 q^{32} + 420 q^{34} + 9452 q^{35} + 2872 q^{37} + 31038 q^{38} + 2618 q^{40} + 34056 q^{41} + 7316 q^{43} + 33562 q^{44} + 4232 q^{46} + 49300 q^{47} + 45118 q^{49} + 44764 q^{50} - 25120 q^{52} + 86676 q^{53} - 2120 q^{55} + 290684 q^{56} - 87408 q^{58} + 67100 q^{59} - 40432 q^{61} + 230992 q^{62} + 136776 q^{64} + 184000 q^{65} - 50108 q^{67} + 270592 q^{68} + 117456 q^{70} + 238584 q^{71} - 13804 q^{73} + 150074 q^{74} - 197622 q^{76} + 116248 q^{77} - 9228 q^{79} + 313010 q^{80} - 68604 q^{82} + 155300 q^{83} + 80444 q^{85} - 80914 q^{86} - 237738 q^{88} + 213732 q^{89} - 264352 q^{91} + 101568 q^{92} + 140280 q^{94} - 123612 q^{95} + 42516 q^{97} - 151388 q^{98}+O(q^{100}) \) 10 * q + 8 * q^2 + 192 * q^4 + 100 * q^5 + 20 * q^7 + 384 * q^8 - 250 * q^10 + 460 * q^11 + 464 * q^13 + 3676 * q^14 + 4612 * q^16 + 4756 * q^17 - 1780 * q^19 + 10314 * q^20 - 4214 * q^22 + 5290 * q^23 + 1330 * q^25 - 5152 * q^26 + 7072 * q^28 + 4048 * q^29 + 2816 * q^31 + 27436 * q^32 + 420 * q^34 + 9452 * q^35 + 2872 * q^37 + 31038 * q^38 + 2618 * q^40 + 34056 * q^41 + 7316 * q^43 + 33562 * q^44 + 4232 * q^46 + 49300 * q^47 + 45118 * q^49 + 44764 * q^50 - 25120 * q^52 + 86676 * q^53 - 2120 * q^55 + 290684 * q^56 - 87408 * q^58 + 67100 * q^59 - 40432 * q^61 + 230992 * q^62 + 136776 * q^64 + 184000 * q^65 - 50108 * q^67 + 270592 * q^68 + 117456 * q^70 + 238584 * q^71 - 13804 * q^73 + 150074 * q^74 - 197622 * q^76 + 116248 * q^77 - 9228 * q^79 + 313010 * q^80 - 68604 * q^82 + 155300 * q^83 + 80444 * q^85 - 80914 * q^86 - 237738 * q^88 + 213732 * q^89 - 264352 * q^91 + 101568 * q^92 + 140280 * q^94 - 123612 * q^95 + 42516 * q^97 - 151388 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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