LMFDB - Embedded newform 207.6.a.e.1.2

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:	$1$
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} - 75x^{2} - 42x + 736 $ x^4 - 75x^2 - 42x + 736
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
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			1.2
Root			$-3.86863$ 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)

$f(q)$	$=$	$q-4.86863 q^{2} -8.29644 q^{4} -66.7344 q^{5} -185.299 q^{7} +196.188 q^{8} +O(q^{10})$	\(q-4.86863 q^{2} -8.29644 q^{4} -66.7344 q^{5} -185.299 q^{7} +196.188 q^{8} +324.905 q^{10} +563.729 q^{11} +767.333 q^{13} +902.153 q^{14} -689.683 q^{16} +1737.11 q^{17} -2283.22 q^{19} +553.658 q^{20} -2744.59 q^{22} +529.000 q^{23} +1328.48 q^{25} -3735.86 q^{26} +1537.32 q^{28} -3574.17 q^{29} +2830.44 q^{31} -2920.22 q^{32} -8457.37 q^{34} +12365.8 q^{35} +4495.35 q^{37} +11116.2 q^{38} -13092.5 q^{40} -16207.8 q^{41} +7058.96 q^{43} -4676.94 q^{44} -2575.51 q^{46} +9336.01 q^{47} +17528.8 q^{49} -6467.88 q^{50} -6366.13 q^{52} +20013.2 q^{53} -37620.1 q^{55} -36353.6 q^{56} +17401.3 q^{58} -11065.7 q^{59} -43813.3 q^{61} -13780.4 q^{62} +36287.3 q^{64} -51207.5 q^{65} +34523.5 q^{67} -14411.9 q^{68} -60204.7 q^{70} +58611.8 q^{71} -62694.7 q^{73} -21886.2 q^{74} +18942.6 q^{76} -104459. q^{77} -57510.8 q^{79} +46025.6 q^{80} +78909.8 q^{82} -55180.2 q^{83} -115925. q^{85} -34367.5 q^{86} +110597. q^{88} +81686.4 q^{89} -142186. q^{91} -4388.81 q^{92} -45453.6 q^{94} +152370. q^{95} -171497. q^{97} -85341.1 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{2} + 26 q^{4} - 22 q^{5} - 62 q^{7} - 72 q^{8}+O(q^{10}) $ 4 * q - 4 * q^2 + 26 * q^4 - 22 * q^5 - 62 * q^7 - 72 * q^8	$ 4 q - 4 q^{2} + 26 q^{4} - 22 q^{5} - 62 q^{7} - 72 q^{8} - 496 q^{10} + 1076 q^{11} - 396 q^{13} + 1806 q^{14} - 1982 q^{16} - 70 q^{17} - 6366 q^{19} + 5240 q^{20} - 6974 q^{22} + 2116 q^{23} + 1264 q^{25} - 2464 q^{26} - 6474 q^{28} - 3948 q^{29} + 3092 q^{31} + 3672 q^{32} + 11682 q^{34} - 1304 q^{35} - 17464 q^{37} + 12628 q^{38} - 14108 q^{40} - 18680 q^{41} - 25846 q^{43} - 20746 q^{44} - 2116 q^{46} - 18392 q^{47} + 7952 q^{49} - 69444 q^{50} + 8844 q^{52} + 26518 q^{53} - 40848 q^{55} - 54890 q^{56} + 568 q^{58} + 14520 q^{59} - 13688 q^{61} - 120136 q^{62} - 30190 q^{64} - 38324 q^{65} - 11098 q^{67} - 112138 q^{68} - 29596 q^{70} + 57496 q^{71} - 112272 q^{73} + 21226 q^{74} - 76240 q^{76} + 4792 q^{77} - 240754 q^{79} - 41200 q^{80} + 49976 q^{82} + 93268 q^{83} - 323204 q^{85} + 88224 q^{86} + 42382 q^{88} + 107582 q^{89} - 301532 q^{91} + 13754 q^{92} + 79360 q^{94} + 18640 q^{95} - 53076 q^{97} - 59664 q^{98}+O(q^{100}) $ 4 * q - 4 * q^2 + 26 * q^4 - 22 * q^5 - 62 * q^7 - 72 * q^8 - 496 * q^10 + 1076 * q^11 - 396 * q^13 + 1806 * q^14 - 1982 * q^16 - 70 * q^17 - 6366 * q^19 + 5240 * q^20 - 6974 * q^22 + 2116 * q^23 + 1264 * q^25 - 2464 * q^26 - 6474 * q^28 - 3948 * q^29 + 3092 * q^31 + 3672 * q^32 + 11682 * q^34 - 1304 * q^35 - 17464 * q^37 + 12628 * q^38 - 14108 * q^40 - 18680 * q^41 - 25846 * q^43 - 20746 * q^44 - 2116 * q^46 - 18392 * q^47 + 7952 * q^49 - 69444 * q^50 + 8844 * q^52 + 26518 * q^53 - 40848 * q^55 - 54890 * q^56 + 568 * q^58 + 14520 * q^59 - 13688 * q^61 - 120136 * q^62 - 30190 * q^64 - 38324 * q^65 - 11098 * q^67 - 112138 * q^68 - 29596 * q^70 + 57496 * q^71 - 112272 * q^73 + 21226 * q^74 - 76240 * q^76 + 4792 * q^77 - 240754 * q^79 - 41200 * q^80 + 49976 * q^82 + 93268 * q^83 - 323204 * q^85 + 88224 * q^86 + 42382 * q^88 + 107582 * q^89 - 301532 * q^91 + 13754 * q^92 + 79360 * q^94 + 18640 * q^95 - 53076 * q^97 - 59664 * 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 $

$2$	−4.86863	−0.860660	−0.430330	−	0.902672i	$-0.641603\pi$
$2$	−4.86863	−0.860660	−0.430330	+	0.902672i	$0.641603\pi$
$3$	0	0
$3$	0	0
$4$	−8.29644	−0.259264
$5$	−66.7344	−1.19378	−0.596891	−	0.802323i	$-0.703597\pi$
$5$	−66.7344	−1.19378	−0.596891	+	0.802323i	$0.703597\pi$
$6$	0	0
$7$	−185.299	−1.42932	−0.714658	−	0.699474i	$-0.753418\pi$
$7$	−185.299	−1.42932	−0.714658	+	0.699474i	$0.753418\pi$
$8$	196.188	1.08380
$9$	0	0
$10$	324.905	1.02744
$11$	563.729	1.40472	0.702359	−	0.711823i	$-0.252130\pi$
$11$	563.729	1.40472	0.702359	+	0.711823i	$0.252130\pi$
$12$	0	0
$13$	767.333	1.25929	0.629645	−	0.776883i	$-0.283200\pi$
$13$	767.333	1.25929	0.629645	+	0.776883i	$0.283200\pi$
$14$	902.153	1.23016
$15$	0	0
$16$	−689.683	−0.673519
$17$	1737.11	1.45783	0.728913	−	0.684606i	$-0.240026\pi$
$17$	1737.11	1.45783	0.728913	+	0.684606i	$0.240026\pi$
$18$	0	0
$19$	−2283.22	−1.45099	−0.725495	−	0.688227i	$-0.758389\pi$
$19$	−2283.22	−1.45099	−0.725495	+	0.688227i	$0.758389\pi$
$20$	553.658	0.309504
$21$	0	0
$22$	−2744.59	−1.20898
$23$	529.000	0.208514
$23$	529.000	0.208514
$24$	0	0
$25$	1328.48	0.425114
$26$	−3735.86	−1.08382
$27$	0	0
$28$	1537.32	0.370570
$29$	−3574.17	−0.789188	−0.394594	−	0.918856i	$-0.629115\pi$
$29$	−3574.17	−0.789188	−0.394594	+	0.918856i	$0.629115\pi$
$30$	0	0
$31$	2830.44	0.528993	0.264497	−	0.964387i	$-0.414794\pi$
$31$	2830.44	0.528993	0.264497	+	0.964387i	$0.414794\pi$
$32$	−2920.22	−0.504127
$33$	0	0
$34$	−8457.37	−1.25469
$35$	12365.8	1.70629
$36$	0	0
$37$	4495.35	0.539833	0.269916	−	0.962884i	$-0.413004\pi$
$37$	4495.35	0.539833	0.269916	+	0.962884i	$0.413004\pi$
$38$	11116.2	1.24881
$39$	0	0
$40$	−13092.5	−1.29382
$41$	−16207.8	−1.50579	−0.752895	−	0.658140i	$-0.771343\pi$
$41$	−16207.8	−1.50579	−0.752895	+	0.658140i	$0.771343\pi$
$42$	0	0
$43$	7058.96	0.582196	0.291098	−	0.956693i	$-0.405979\pi$
$43$	7058.96	0.582196	0.291098	+	0.956693i	$0.405979\pi$
$44$	−4676.94	−0.364192
$45$	0	0
$46$	−2575.51	−0.179460
$47$	9336.01	0.616476	0.308238	−	0.951309i	$-0.400261\pi$
$47$	9336.01	0.616476	0.308238	+	0.951309i	$0.400261\pi$
$48$	0	0
$49$	17528.8	1.04294
$50$	−6467.88	−0.365879
$51$	0	0
$52$	−6366.13	−0.326488
$53$	20013.2	0.978647	0.489323	−	0.872102i	$-0.337244\pi$
$53$	20013.2	0.978647	0.489323	+	0.872102i	$0.337244\pi$
$54$	0	0
$55$	−37620.1	−1.67692
$56$	−36353.6	−1.54909
$57$	0	0
$58$	17401.3	0.679223
$59$	−11065.7	−0.413857	−0.206928	−	0.978356i	$-0.566347\pi$
$59$	−11065.7	−0.413857	−0.206928	+	0.978356i	$0.566347\pi$
$60$	0	0
$61$	−43813.3	−1.50758	−0.753792	−	0.657113i	$-0.771778\pi$
$61$	−43813.3	−1.50758	−0.753792	+	0.657113i	$0.771778\pi$
$62$	−13780.4	−0.455284
$63$	0	0
$64$	36287.3	1.10740
$65$	−51207.5	−1.50332
$66$	0	0
$67$	34523.5	0.939567	0.469783	−	0.882782i	$-0.344332\pi$
$67$	34523.5	0.939567	0.469783	+	0.882782i	$0.344332\pi$
$68$	−14411.9	−0.377961
$69$	0	0
$70$	−60204.7	−1.46854
$71$	58611.8	1.37987	0.689936	−	0.723870i	$-0.257639\pi$
$71$	58611.8	1.37987	0.689936	+	0.723870i	$0.257639\pi$
$72$	0	0
$73$	−62694.7	−1.37697	−0.688484	−	0.725252i	$-0.741723\pi$
$73$	−62694.7	−1.37697	−0.688484	+	0.725252i	$0.741723\pi$
$74$	−21886.2	−0.464613
$75$	0	0
$76$	18942.6	0.376189
$77$	−104459.	−2.00778
$78$	0	0
$79$	−57510.8	−1.03677	−0.518384	−	0.855148i	$-0.673466\pi$
$79$	−57510.8	−1.03677	−0.518384	+	0.855148i	$0.673466\pi$
$80$	46025.6	0.804034
$81$	0	0
$82$	78909.8	1.29597
$83$	−55180.2	−0.879201	−0.439601	−	0.898193i	$-0.644880\pi$
$83$	−55180.2	−0.879201	−0.439601	+	0.898193i	$0.644880\pi$
$84$	0	0
$85$	−115925.	−1.74033
$86$	−34367.5	−0.501073
$87$	0	0
$88$	110597.	1.52243
$89$	81686.4	1.09314	0.546568	−	0.837414i	$-0.315934\pi$
$89$	81686.4	1.09314	0.546568	+	0.837414i	$0.315934\pi$
$90$	0	0
$91$	−142186.	−1.79992
$92$	−4388.81	−0.0540602
$93$	0	0
$94$	−45453.6	−0.530577
$95$	152370.	1.73216
$96$	0	0
$97$	−171497.	−1.85066	−0.925332	−	0.379158i	$-0.876214\pi$
$97$	−171497.	−1.85066	−0.925332	+	0.379158i	$0.876214\pi$
$98$	−85341.1	−0.897621
$99$	0	0
$100$	−11021.7	−0.110217
$101$	43082.5	0.420240	0.210120	−	0.977676i	$-0.432614\pi$
$101$	43082.5	0.420240	0.210120	+	0.977676i	$0.432614\pi$
$102$	0	0
$103$	41385.5	0.384375	0.192187	−	0.981358i	$-0.438442\pi$
$103$	41385.5	0.384375	0.192187	+	0.981358i	$0.438442\pi$
$104$	150542.	1.36482
$105$	0	0
$106$	−97436.7	−0.842282
$107$	−165235.	−1.39522	−0.697611	−	0.716477i	$-0.745754\pi$
$107$	−165235.	−1.39522	−0.697611	+	0.716477i	$0.745754\pi$
$108$	0	0
$109$	136414.	1.09974	0.549872	−	0.835249i	$-0.314676\pi$
$109$	136414.	1.09974	0.549872	+	0.835249i	$0.314676\pi$
$110$	183159.	1.44326
$111$	0	0
$112$	127798.	0.962671
$113$	−74143.6	−0.546232	−0.273116	−	0.961981i	$-0.588054\pi$
$113$	−74143.6	−0.546232	−0.273116	+	0.961981i	$0.588054\pi$
$114$	0	0
$115$	−35302.5	−0.248921
$116$	29652.9	0.204608
$117$	0	0
$118$	53874.9	0.356190
$119$	−321886.	−2.08370
$120$	0	0
$121$	156740.	0.973230
$122$	213311.	1.29752
$123$	0	0
$124$	−23482.6	−0.137149
$125$	119890.	0.686288
$126$	0	0
$127$	−49454.2	−0.272078	−0.136039	−	0.990703i	$-0.543437\pi$
$127$	−49454.2	−0.272078	−0.136039	+	0.990703i	$0.543437\pi$
$128$	−83222.6	−0.448969
$129$	0	0
$130$	249311.	1.29385
$131$	−96091.5	−0.489223	−0.244612	−	0.969621i	$-0.578660\pi$
$131$	−96091.5	−0.489223	−0.244612	+	0.969621i	$0.578660\pi$
$132$	0	0
$133$	423079.	2.07392
$134$	−168082.	−0.808648
$135$	0	0
$136$	340802.	1.57999
$137$	257144.	1.17051	0.585255	−	0.810850i	$-0.300995\pi$
$137$	257144.	1.17051	0.585255	+	0.810850i	$0.300995\pi$
$138$	0	0
$139$	127180.	0.558319	0.279159	−	0.960245i	$-0.409944\pi$
$139$	127180.	0.558319	0.279159	+	0.960245i	$0.409944\pi$
$140$	−102592.	−0.442379
$141$	0	0
$142$	−285359.	−1.18760
$143$	432568.	1.76895
$144$	0	0
$145$	238520.	0.942118
$146$	305237.	1.18510
$147$	0	0
$148$	−37295.4	−0.139959
$149$	−623.566	−0.00230100	−0.00115050	−	0.999999i	$-0.500366\pi$
$149$	−623.566	−0.00230100	−0.00115050	+	0.999999i	$0.500366\pi$
$150$	0	0
$151$	−333623.	−1.19073	−0.595366	−	0.803455i	$-0.702993\pi$
$151$	−333623.	−1.19073	−0.595366	+	0.803455i	$0.702993\pi$
$152$	−447942.	−1.57258
$153$	0	0
$154$	508570.	1.72802
$155$	−188888.	−0.631502
$156$	0	0
$157$	−574134.	−1.85893	−0.929467	−	0.368904i	$-0.879733\pi$
$157$	−574134.	−1.85893	−0.929467	+	0.368904i	$0.879733\pi$
$158$	279999.	0.892305
$159$	0	0
$160$	194879.	0.601818
$161$	−98023.3	−0.298033
$162$	0	0
$163$	239894.	0.707214	0.353607	−	0.935394i	$-0.384955\pi$
$163$	239894.	0.707214	0.353607	+	0.935394i	$0.384955\pi$
$164$	134467.	0.390397
$165$	0	0
$166$	268652.	0.756694
$167$	−309931.	−0.859951	−0.429975	−	0.902841i	$-0.641478\pi$
$167$	−309931.	−0.859951	−0.429975	+	0.902841i	$0.641478\pi$
$168$	0	0
$169$	217508.	0.585811
$170$	564397.	1.49783
$171$	0	0
$172$	−58564.2	−0.150942
$173$	81138.3	0.206115	0.103058	−	0.994675i	$-0.467137\pi$
$173$	81138.3	0.206115	0.103058	+	0.994675i	$0.467137\pi$
$174$	0	0
$175$	−246166.	−0.607622
$176$	−388795.	−0.946103
$177$	0	0
$178$	−397701.	−0.940820
$179$	118862.	0.277275	0.138637	−	0.990343i	$-0.455728\pi$
$179$	118862.	0.277275	0.138637	+	0.990343i	$0.455728\pi$
$180$	0	0
$181$	−541106.	−1.22768	−0.613841	−	0.789430i	$-0.710376\pi$
$181$	−541106.	−1.22768	−0.613841	+	0.789430i	$0.710376\pi$
$182$	692252.	1.54912
$183$	0	0
$184$	103784.	0.225988
$185$	−299994.	−0.644442
$186$	0	0
$187$	979262.	2.04783
$188$	−77455.6	−0.159830
$189$	0	0
$190$	−741831.	−1.49081
$191$	−268445.	−0.532441	−0.266220	−	0.963912i	$-0.585775\pi$
$191$	−268445.	−0.532441	−0.266220	+	0.963912i	$0.585775\pi$
$192$	0	0
$193$	−602619.	−1.16453	−0.582264	−	0.813000i	$-0.697833\pi$
$193$	−602619.	−1.16453	−0.582264	+	0.813000i	$0.697833\pi$
$194$	834956.	1.59279
$195$	0	0
$196$	−145426.	−0.270398
$197$	−482982.	−0.886676	−0.443338	−	0.896354i	$-0.646206\pi$
$197$	−482982.	−0.886676	−0.443338	+	0.896354i	$0.646206\pi$
$198$	0	0
$199$	573505.	1.02661	0.513304	−	0.858207i	$-0.328422\pi$
$199$	573505.	1.02661	0.513304	+	0.858207i	$0.328422\pi$
$200$	260633.	0.460738
$201$	0	0
$202$	−209753.	−0.361684
$203$	662291.	1.12800
$204$	0	0
$205$	1.08162e6	1.79759
$206$	−201491.	−0.330816
$207$	0	0
$208$	−529217.	−0.848155
$209$	−1.28712e6	−2.03823
$210$	0	0
$211$	−1.23893e6	−1.91577	−0.957883	−	0.287160i	$-0.907289\pi$
$211$	−1.23893e6	−1.91577	−0.957883	+	0.287160i	$0.907289\pi$
$212$	−166038.	−0.253727
$213$	0	0
$214$	804469.	1.20081
$215$	−471075.	−0.695015
$216$	0	0
$217$	−524479.	−0.756099
$218$	−664148.	−0.946507
$219$	0	0
$220$	312113.	0.434766
$221$	1.33295e6	1.83583
$222$	0	0
$223$	−308404.	−0.415296	−0.207648	−	0.978204i	$-0.566581\pi$
$223$	−308404.	−0.415296	−0.207648	+	0.978204i	$0.566581\pi$
$224$	541114.	0.720558
$225$	0	0
$226$	360978.	0.470120
$227$	−764147.	−0.984266	−0.492133	−	0.870520i	$-0.663783\pi$
$227$	−764147.	−0.984266	−0.492133	+	0.870520i	$0.663783\pi$
$228$	0	0
$229$	1.01846e6	1.28338	0.641692	−	0.766962i	$-0.278233\pi$
$229$	1.01846e6	1.28338	0.641692	+	0.766962i	$0.278233\pi$
$230$	171875.	0.214236
$231$	0	0
$232$	−701211.	−0.855321
$233$	120752.	0.145715	0.0728577	−	0.997342i	$-0.476788\pi$
$233$	120752.	0.145715	0.0728577	+	0.997342i	$0.476788\pi$
$234$	0	0
$235$	−623033.	−0.735938
$236$	91806.1	0.107298
$237$	0	0
$238$	1.56714e6	1.79335
$239$	317622.	0.359680	0.179840	−	0.983696i	$-0.442442\pi$
$239$	317622.	0.359680	0.179840	+	0.983696i	$0.442442\pi$
$240$	0	0
$241$	−66852.0	−0.0741432	−0.0370716	−	0.999313i	$-0.511803\pi$
$241$	−66852.0	−0.0741432	−0.0370716	+	0.999313i	$0.511803\pi$
$242$	−763107.	−0.837620
$243$	0	0
$244$	363495.	0.390862
$245$	−1.16977e6	−1.24505
$246$	0	0
$247$	−1.75199e6	−1.82722
$248$	555300.	0.573322
$249$	0	0
$250$	−583698.	−0.590661
$251$	101729.	0.101920	0.0509600	−	0.998701i	$-0.483772\pi$
$251$	101729.	0.101920	0.0509600	+	0.998701i	$0.483772\pi$
$252$	0	0
$253$	298213.	0.292904
$254$	240774.	0.234167
$255$	0	0
$256$	−756014.	−0.720991
$257$	−2.07807e6	−1.96258	−0.981291	−	0.192533i	$-0.938330\pi$
$257$	−2.07807e6	−1.96258	−0.981291	+	0.192533i	$0.938330\pi$
$258$	0	0
$259$	−832985.	−0.771592
$260$	424840.	0.389755
$261$	0	0
$262$	467834.	0.421055
$263$	437706.	0.390205	0.195102	−	0.980783i	$-0.437496\pi$
$263$	437706.	0.390205	0.195102	+	0.980783i	$0.437496\pi$
$264$	0	0
$265$	−1.33557e6	−1.16829
$266$	−2.05982e6	−1.78494
$267$	0	0
$268$	−286422.	−0.243596
$269$	−1.43024e6	−1.20511	−0.602556	−	0.798076i	$-0.705851\pi$
$269$	−1.43024e6	−1.20511	−0.602556	+	0.798076i	$0.705851\pi$
$270$	0	0
$271$	442343.	0.365877	0.182939	−	0.983124i	$-0.441439\pi$
$271$	442343.	0.365877	0.182939	+	0.983124i	$0.441439\pi$
$272$	−1.19806e6	−0.981874
$273$	0	0
$274$	−1.25194e6	−1.00741
$275$	748903.	0.597165
$276$	0	0
$277$	−1.01001e6	−0.790909	−0.395455	−	0.918486i	$-0.629413\pi$
$277$	−1.01001e6	−0.790909	−0.395455	+	0.918486i	$0.629413\pi$
$278$	−619193.	−0.480523
$279$	0	0
$280$	2.42603e6	1.84928
$281$	1.44669e6	1.09298	0.546488	−	0.837467i	$-0.315964\pi$
$281$	1.44669e6	1.09298	0.546488	+	0.837467i	$0.315964\pi$
$282$	0	0
$283$	647359.	0.480484	0.240242	−	0.970713i	$-0.422773\pi$
$283$	647359.	0.480484	0.240242	+	0.970713i	$0.422773\pi$
$284$	−486269.	−0.357751
$285$	0	0
$286$	−2.10602e6	−1.52246
$287$	3.00329e6	2.15225
$288$	0	0
$289$	1.59771e6	1.12526
$290$	−1.16127e6	−0.810844
$291$	0	0
$292$	520143.	0.356998
$293$	1.04046e6	0.708040	0.354020	−	0.935238i	$-0.384815\pi$
$293$	1.04046e6	0.708040	0.354020	+	0.935238i	$0.384815\pi$
$294$	0	0
$295$	738465.	0.494054
$296$	881936.	0.585070
$297$	0	0
$298$	3035.91	0.00198038
$299$	405919.	0.262580
$300$	0	0
$301$	−1.30802e6	−0.832143
$302$	1.62429e6	1.02482
$303$	0	0
$304$	1.57470e6	0.977269
$305$	2.92386e6	1.79973
$306$	0	0
$307$	792650.	0.479994	0.239997	−	0.970774i	$-0.422854\pi$
$307$	792650.	0.479994	0.239997	+	0.970774i	$0.422854\pi$
$308$	866634.	0.520546
$309$	0	0
$310$	919625.	0.543509
$311$	2.27129e6	1.33159	0.665797	−	0.746133i	$-0.268092\pi$
$311$	2.27129e6	1.33159	0.665797	+	0.746133i	$0.268092\pi$
$312$	0	0
$313$	1.32722e6	0.765741	0.382870	−	0.923802i	$-0.374936\pi$
$313$	1.32722e6	0.765741	0.382870	+	0.923802i	$0.374936\pi$
$314$	2.79525e6	1.59991
$315$	0	0
$316$	477134.	0.268796
$317$	−1.54042e6	−0.860977	−0.430489	−	0.902596i	$-0.641659\pi$
$317$	−1.54042e6	−0.860977	−0.430489	+	0.902596i	$0.641659\pi$
$318$	0	0
$319$	−2.01487e6	−1.10859
$320$	−2.42161e6	−1.32199
$321$	0	0
$322$	477239.	0.256505
$323$	−3.96622e6	−2.11529
$324$	0	0
$325$	1.01939e6	0.535342
$326$	−1.16796e6	−0.608671
$327$	0	0
$328$	−3.17979e6	−1.63197
$329$	−1.72995e6	−0.881140
$330$	0	0
$331$	−2.90735e6	−1.45857	−0.729286	−	0.684209i	$-0.760148\pi$
$331$	−2.90735e6	−1.45857	−0.729286	+	0.684209i	$0.760148\pi$
$332$	457799.	0.227945
$333$	0	0
$334$	1.50894e6	0.740126
$335$	−2.30390e6	−1.12164
$336$	0	0
$337$	−2.54377e6	−1.22012	−0.610061	−	0.792355i	$-0.708855\pi$
$337$	−2.54377e6	−1.22012	−0.610061	+	0.792355i	$0.708855\pi$
$338$	−1.05896e6	−0.504185
$339$	0	0
$340$	961766.	0.451203
$341$	1.59560e6	0.743086
$342$	0	0
$343$	−133745.	−0.0613820
$344$	1.38489e6	0.630984
$345$	0	0
$346$	−395032.	−0.177395
$347$	−1.06956e6	−0.476851	−0.238426	−	0.971161i	$-0.576631\pi$
$347$	−1.06956e6	−0.476851	−0.238426	+	0.971161i	$0.576631\pi$
$348$	0	0
$349$	−3.46366e6	−1.52220	−0.761099	−	0.648635i	$-0.775340\pi$
$349$	−3.46366e6	−1.52220	−0.761099	+	0.648635i	$0.775340\pi$
$350$	1.19849e6	0.522956
$351$	0	0
$352$	−1.64621e6	−0.708156
$353$	1.80510e6	0.771018	0.385509	−	0.922704i	$-0.374026\pi$
$353$	1.80510e6	0.771018	0.385509	+	0.922704i	$0.374026\pi$
$354$	0	0
$355$	−3.91142e6	−1.64727
$356$	−677706.	−0.283411
$357$	0	0
$358$	−578695.	−0.238640
$359$	2.44266e6	1.00029	0.500146	−	0.865941i	$-0.333280\pi$
$359$	2.44266e6	1.00029	0.500146	+	0.865941i	$0.333280\pi$
$360$	0	0
$361$	2.73701e6	1.10537
$362$	2.63444e6	1.05662
$363$	0	0
$364$	1.17964e6	0.466655
$365$	4.18389e6	1.64380
$366$	0	0
$367$	−1.73440e6	−0.672179	−0.336090	−	0.941830i	$-0.609105\pi$
$367$	−1.73440e6	−0.672179	−0.336090	+	0.941830i	$0.609105\pi$
$368$	−364842.	−0.140438
$369$	0	0
$370$	1.46056e6	0.554646
$371$	−3.70842e6	−1.39880
$372$	0	0
$373$	670423.	0.249503	0.124752	−	0.992188i	$-0.460187\pi$
$373$	670423.	0.249503	0.124752	+	0.992188i	$0.460187\pi$
$374$	−4.76766e6	−1.76249
$375$	0	0
$376$	1.83162e6	0.668136
$377$	−2.74258e6	−0.993817
$378$	0	0
$379$	−1.58515e6	−0.566857	−0.283428	−	0.958993i	$-0.591472\pi$
$379$	−1.58515e6	−0.566857	−0.283428	+	0.958993i	$0.591472\pi$
$380$	−1.26412e6	−0.449087
$381$	0	0
$382$	1.30696e6	0.458251
$383$	−1.13803e6	−0.396421	−0.198210	−	0.980160i	$-0.563513\pi$
$383$	−1.13803e6	−0.396421	−0.198210	+	0.980160i	$0.563513\pi$
$384$	0	0
$385$	6.97098e6	2.39686
$386$	2.93393e6	1.00226
$387$	0	0
$388$	1.42282e6	0.479810
$389$	−5.06039e6	−1.69555	−0.847774	−	0.530358i	$-0.822058\pi$
$389$	−5.06039e6	−1.69555	−0.847774	+	0.530358i	$0.822058\pi$
$390$	0	0
$391$	918933.	0.303978
$392$	3.43894e6	1.13034
$393$	0	0
$394$	2.35146e6	0.763127
$395$	3.83795e6	1.23767
$396$	0	0
$397$	2.94086e6	0.936480	0.468240	−	0.883601i	$-0.344888\pi$
$397$	2.94086e6	0.936480	0.468240	+	0.883601i	$0.344888\pi$
$398$	−2.79218e6	−0.883560
$399$	0	0
$400$	−916231.	−0.286322
$401$	−3.32124e6	−1.03143	−0.515715	−	0.856760i	$-0.672474\pi$
$401$	−3.32124e6	−1.03143	−0.515715	+	0.856760i	$0.672474\pi$
$402$	0	0
$403$	2.17189e6	0.666156
$404$	−357431.	−0.108953
$405$	0	0
$406$	−3.22445e6	−0.970824
$407$	2.53416e6	0.758312
$408$	0	0
$409$	−1.11681e6	−0.330120	−0.165060	−	0.986283i	$-0.552782\pi$
$409$	−1.11681e6	−0.330120	−0.165060	+	0.986283i	$0.552782\pi$
$410$	−5.26600e6	−1.54711
$411$	0	0
$412$	−343352.	−0.0996544
$413$	2.05047e6	0.591532
$414$	0	0
$415$	3.68242e6	1.04957
$416$	−2.24078e6	−0.634843
$417$	0	0
$418$	6.26651e6	1.75422
$419$	−707320.	−0.196825	−0.0984126	−	0.995146i	$-0.531376\pi$
$419$	−707320.	−0.196825	−0.0984126	+	0.995146i	$0.531376\pi$
$420$	0	0
$421$	5.82295e6	1.60117	0.800586	−	0.599218i	$-0.204522\pi$
$421$	5.82295e6	1.60117	0.800586	+	0.599218i	$0.204522\pi$
$422$	6.03192e6	1.64882
$423$	0	0
$424$	3.92635e6	1.06066
$425$	2.30772e6	0.619742
$426$	0	0
$427$	8.11858e6	2.15482
$428$	1.37086e6	0.361730
$429$	0	0
$430$	2.29349e6	0.598172
$431$	3.42318e6	0.887640	0.443820	−	0.896116i	$-0.353623\pi$
$431$	3.42318e6	0.887640	0.443820	+	0.896116i	$0.353623\pi$
$432$	0	0
$433$	1.45426e6	0.372754	0.186377	−	0.982478i	$-0.440325\pi$
$433$	1.45426e6	0.372754	0.186377	+	0.982478i	$0.440325\pi$
$434$	2.55349e6	0.650744
$435$	0	0
$436$	−1.13175e6	−0.285124
$437$	−1.20783e6	−0.302552
$438$	0	0
$439$	−3.79123e6	−0.938899	−0.469450	−	0.882959i	$-0.655548\pi$
$439$	−3.79123e6	−0.938899	−0.469450	+	0.882959i	$0.655548\pi$
$440$	−7.38064e6	−1.81745
$441$	0	0
$442$	−6.48962e6	−1.58002
$443$	672461.	0.162801	0.0814006	−	0.996681i	$-0.474061\pi$
$443$	672461.	0.162801	0.0814006	+	0.996681i	$0.474061\pi$
$444$	0	0
$445$	−5.45129e6	−1.30497
$446$	1.50150e6	0.357429
$447$	0	0
$448$	−6.72401e6	−1.58283
$449$	−4.85755e6	−1.13711	−0.568554	−	0.822646i	$-0.692497\pi$
$449$	−4.85755e6	−1.13711	−0.568554	+	0.822646i	$0.692497\pi$
$450$	0	0
$451$	−9.13682e6	−2.11521
$452$	615127.	0.141618
$453$	0	0
$454$	3.72035e6	0.847118
$455$	9.48871e6	2.14872
$456$	0	0
$457$	698808.	0.156519	0.0782596	−	0.996933i	$-0.475064\pi$
$457$	698808.	0.156519	0.0782596	+	0.996933i	$0.475064\pi$
$458$	−4.95852e6	−1.10456
$459$	0	0
$460$	292885.	0.0645361
$461$	−422691.	−0.0926341	−0.0463170	−	0.998927i	$-0.514748\pi$
$461$	−422691.	−0.0926341	−0.0463170	+	0.998927i	$0.514748\pi$
$462$	0	0
$463$	4.13790e6	0.897073	0.448536	−	0.893765i	$-0.351945\pi$
$463$	4.13790e6	0.897073	0.448536	+	0.893765i	$0.351945\pi$
$464$	2.46505e6	0.531533
$465$	0	0
$466$	−587898.	−0.125411
$467$	−6.02577e6	−1.27856	−0.639279	−	0.768975i	$-0.720767\pi$
$467$	−6.02577e6	−1.27856	−0.639279	+	0.768975i	$0.720767\pi$
$468$	0	0
$469$	−6.39717e6	−1.34294
$470$	3.03332e6	0.633393
$471$	0	0
$472$	−2.17097e6	−0.448537
$473$	3.97934e6	0.817821
$474$	0	0
$475$	−3.03322e6	−0.616836
$476$	2.67050e6	0.540227
$477$	0	0
$478$	−1.54639e6	−0.309562
$479$	2.25772e6	0.449604	0.224802	−	0.974404i	$-0.427826\pi$
$479$	2.25772e6	0.449604	0.224802	+	0.974404i	$0.427826\pi$
$480$	0	0
$481$	3.44943e6	0.679806
$482$	325478.	0.0638122
$483$	0	0
$484$	−1.30038e6	−0.252323
$485$	1.14448e7	2.20929
$486$	0	0
$487$	5.05901e6	0.966592	0.483296	−	0.875457i	$-0.339439\pi$
$487$	5.05901e6	0.966592	0.483296	+	0.875457i	$0.339439\pi$
$488$	−8.59567e6	−1.63392
$489$	0	0
$490$	5.69519e6	1.07156
$491$	5.75970e6	1.07819	0.539096	−	0.842244i	$-0.318766\pi$
$491$	5.75970e6	1.07819	0.539096	+	0.842244i	$0.318766\pi$
$492$	0	0
$493$	−6.20874e6	−1.15050
$494$	8.52981e6	1.57261
$495$	0	0
$496$	−1.95211e6	−0.356287
$497$	−1.08607e7	−1.97227
$498$	0	0
$499$	−956632.	−0.171986	−0.0859930	−	0.996296i	$-0.527406\pi$
$499$	−956632.	−0.171986	−0.0859930	+	0.996296i	$0.527406\pi$
$500$	−994657.	−0.177930
$501$	0	0
$502$	−495280.	−0.0877186
$503$	−4.84338e6	−0.853548	−0.426774	−	0.904358i	$-0.640350\pi$
$503$	−4.84338e6	−0.853548	−0.426774	+	0.904358i	$0.640350\pi$
$504$	0	0
$505$	−2.87509e6	−0.501675
$506$	−1.45189e6	−0.252091
$507$	0	0
$508$	410294.	0.0705400
$509$	−2.65927e6	−0.454955	−0.227478	−	0.973783i	$-0.573048\pi$
$509$	−2.65927e6	−0.454955	−0.227478	+	0.973783i	$0.573048\pi$
$510$	0	0
$511$	1.16173e7	1.96812
$512$	6.34388e6	1.06950
$513$	0	0
$514$	1.01174e7	1.68912
$515$	−2.76183e6	−0.458859
$516$	0	0
$517$	5.26298e6	0.865975
$518$	4.05549e6	0.664078
$519$	0	0
$520$	−1.00463e7	−1.62929
$521$	−1.09007e6	−0.175938	−0.0879691	−	0.996123i	$-0.528038\pi$
$521$	−1.09007e6	−0.175938	−0.0879691	+	0.996123i	$0.528038\pi$
$522$	0	0
$523$	−1.03436e7	−1.65354	−0.826772	−	0.562537i	$-0.809825\pi$
$523$	−1.03436e7	−1.65354	−0.826772	+	0.562537i	$0.809825\pi$
$524$	797217.	0.126838
$525$	0	0
$526$	−2.13103e6	−0.335834
$527$	4.91680e6	0.771181
$528$	0	0
$529$	279841.	0.0434783
$530$	6.50238e6	1.00550
$531$	0	0
$532$	−3.51005e6	−0.537693
$533$	−1.24368e7	−1.89623
$534$	0	0
$535$	1.10269e7	1.66559
$536$	6.77311e6	1.01830
$537$	0	0
$538$	6.96330e6	1.03719
$539$	9.88148e6	1.46504
$540$	0	0
$541$	−1.07591e7	−1.58045	−0.790227	−	0.612814i	$-0.790037\pi$
$541$	−1.07591e7	−1.58045	−0.790227	+	0.612814i	$0.790037\pi$
$542$	−2.15360e6	−0.314896
$543$	0	0
$544$	−5.07275e6	−0.734931
$545$	−9.10349e6	−1.31285
$546$	0	0
$547$	−1.31666e7	−1.88150	−0.940749	−	0.339103i	$-0.889876\pi$
$547$	−1.31666e7	−1.88150	−0.940749	+	0.339103i	$0.889876\pi$
$548$	−2.13338e6	−0.303470
$549$	0	0
$550$	−3.64613e6	−0.513956
$551$	8.16064e6	1.14510
$552$	0	0
$553$	1.06567e7	1.48187
$554$	4.91737e6	0.680704
$555$	0	0
$556$	−1.05514e6	−0.144752
$557$	−2.57299e6	−0.351399	−0.175699	−	0.984444i	$-0.556219\pi$
$557$	−2.57299e6	−0.351399	−0.175699	+	0.984444i	$0.556219\pi$
$558$	0	0
$559$	5.41658e6	0.733154
$560$	−8.52850e6	−1.14922
$561$	0	0
$562$	−7.04341e6	−0.940681
$563$	−5.77557e6	−0.767933	−0.383967	−	0.923347i	$-0.625442\pi$
$563$	−5.77557e6	−0.767933	−0.383967	+	0.923347i	$0.625442\pi$
$564$	0	0
$565$	4.94793e6	0.652082
$566$	−3.15175e6	−0.413533
$567$	0	0
$568$	1.14990e7	1.49550
$569$	−4.07120e6	−0.527160	−0.263580	−	0.964638i	$-0.584903\pi$
$569$	−4.07120e6	−0.527160	−0.263580	+	0.964638i	$0.584903\pi$
$570$	0	0
$571$	−5.03144e6	−0.645806	−0.322903	−	0.946432i	$-0.604659\pi$
$571$	−5.03144e6	−0.645806	−0.322903	+	0.946432i	$0.604659\pi$
$572$	−3.58878e6	−0.458623
$573$	0	0
$574$	−1.46219e7	−1.85236
$575$	702766.	0.0886424
$576$	0	0
$577$	−2.49388e6	−0.311843	−0.155921	−	0.987769i	$-0.549835\pi$
$577$	−2.49388e6	−0.311843	−0.155921	+	0.987769i	$0.549835\pi$
$578$	−7.77865e6	−0.968466
$579$	0	0
$580$	−1.97887e6	−0.244257
$581$	1.02249e7	1.25666
$582$	0	0
$583$	1.12820e7	1.37472
$584$	−1.23000e7	−1.49236
$585$	0	0
$586$	−5.06563e6	−0.609382
$587$	−5.36774e6	−0.642979	−0.321489	−	0.946913i	$-0.604183\pi$
$587$	−5.36774e6	−0.642979	−0.321489	+	0.946913i	$0.604183\pi$
$588$	0	0
$589$	−6.46253e6	−0.767564
$590$	−3.59531e6	−0.425213
$591$	0	0
$592$	−3.10037e6	−0.363587
$593$	−1.07373e7	−1.25389	−0.626943	−	0.779065i	$-0.715694\pi$
$593$	−1.07373e7	−1.25389	−0.626943	+	0.779065i	$0.715694\pi$
$594$	0	0
$595$	2.14809e7	2.48748
$596$	5173.38	0.000596566 0
$597$	0	0
$598$	−1.97627e6	−0.225992
$599$	3.52730e6	0.401675	0.200838	−	0.979625i	$-0.435634\pi$
$599$	3.52730e6	0.401675	0.200838	+	0.979625i	$0.435634\pi$
$600$	0	0
$601$	−2.73536e6	−0.308907	−0.154454	−	0.988000i	$-0.549362\pi$
$601$	−2.73536e6	−0.308907	−0.154454	+	0.988000i	$0.549362\pi$
$602$	6.36826e6	0.716192
$603$	0	0
$604$	2.76789e6	0.308714
$605$	−1.04599e7	−1.16182
$606$	0	0
$607$	3.19959e6	0.352470	0.176235	−	0.984348i	$-0.443608\pi$
$607$	3.19959e6	0.352470	0.176235	+	0.984348i	$0.443608\pi$
$608$	6.66751e6	0.731484
$609$	0	0
$610$	−1.42352e7	−1.54895
$611$	7.16383e6	0.776323
$612$	0	0
$613$	−1.31336e7	−1.41167	−0.705833	−	0.708378i	$-0.749427\pi$
$613$	−1.31336e7	−1.41167	−0.705833	+	0.708378i	$0.749427\pi$
$614$	−3.85912e6	−0.413112
$615$	0	0
$616$	−2.04936e7	−2.17603
$617$	−5.00019e6	−0.528778	−0.264389	−	0.964416i	$-0.585170\pi$
$617$	−5.00019e6	−0.528778	−0.264389	+	0.964416i	$0.585170\pi$
$618$	0	0
$619$	−8.11569e6	−0.851332	−0.425666	−	0.904880i	$-0.639960\pi$
$619$	−8.11569e6	−0.851332	−0.425666	+	0.904880i	$0.639960\pi$
$620$	1.56710e6	0.163726
$621$	0	0
$622$	−1.10581e7	−1.14605
$623$	−1.51364e7	−1.56244
$624$	0	0
$625$	−1.21523e7	−1.24439
$626$	−6.46174e6	−0.659043
$627$	0	0
$628$	4.76327e6	0.481954
$629$	7.80893e6	0.786983
$630$	0	0
$631$	−121821.	−0.0121801	−0.00609003	−	0.999981i	$-0.501939\pi$
$631$	−121821.	−0.0121801	−0.00609003	+	0.999981i	$0.501939\pi$
$632$	−1.12829e7	−1.12365
$633$	0	0
$634$	7.49975e6	0.741009
$635$	3.30030e6	0.324802
$636$	0	0
$637$	1.34504e7	1.31337
$638$	9.80964e6	0.954116
$639$	0	0
$640$	5.55381e6	0.535971
$641$	−4.06055e6	−0.390337	−0.195169	−	0.980770i	$-0.562525\pi$
$641$	−4.06055e6	−0.390337	−0.195169	+	0.980770i	$0.562525\pi$
$642$	0	0
$643$	1.62068e7	1.54586	0.772928	−	0.634494i	$-0.218792\pi$
$643$	1.62068e7	1.54586	0.772928	+	0.634494i	$0.218792\pi$
$644$	813244.	0.0772691
$645$	0	0
$646$	1.93101e7	1.82055
$647$	−4.33831e6	−0.407436	−0.203718	−	0.979030i	$-0.565303\pi$
$647$	−4.33831e6	−0.407436	−0.203718	+	0.979030i	$0.565303\pi$
$648$	0	0
$649$	−6.23807e6	−0.581351
$650$	−4.96302e6	−0.460747
$651$	0	0
$652$	−1.99027e6	−0.183355
$653$	−3.28611e6	−0.301578	−0.150789	−	0.988566i	$-0.548181\pi$
$653$	−3.28611e6	−0.301578	−0.150789	+	0.988566i	$0.548181\pi$
$654$	0	0
$655$	6.41261e6	0.584025
$656$	1.11783e7	1.01418
$657$	0	0
$658$	8.42251e6	0.758362
$659$	2.08664e7	1.87169	0.935847	−	0.352408i	$-0.114637\pi$
$659$	2.08664e7	1.87169	0.935847	+	0.352408i	$0.114637\pi$
$660$	0	0
$661$	−1.40139e7	−1.24754	−0.623769	−	0.781609i	$-0.714399\pi$
$661$	−1.40139e7	−1.24754	−0.623769	+	0.781609i	$0.714399\pi$
$662$	1.41548e7	1.25534
$663$	0	0
$664$	−1.08257e7	−0.952877
$665$	−2.82340e7	−2.47581
$666$	0	0
$667$	−1.89074e6	−0.164557
$668$	2.57132e6	0.222954
$669$	0	0
$670$	1.12169e7	0.965349
$671$	−2.46989e7	−2.11773
$672$	0	0
$673$	6.69255e6	0.569579	0.284790	−	0.958590i	$-0.408076\pi$
$673$	6.69255e6	0.569579	0.284790	+	0.958590i	$0.408076\pi$
$674$	1.23847e7	1.05011
$675$	0	0
$676$	−1.80454e6	−0.151880
$677$	1.24923e7	1.04754	0.523769	−	0.851860i	$-0.324525\pi$
$677$	1.24923e7	1.04754	0.523769	+	0.851860i	$0.324525\pi$
$678$	0	0
$679$	3.17783e7	2.64518
$680$	−2.27432e7	−1.88616
$681$	0	0
$682$	−7.76840e6	−0.639545
$683$	−975374.	−0.0800054	−0.0400027	−	0.999200i	$-0.512737\pi$
$683$	−975374.	−0.0800054	−0.0400027	+	0.999200i	$0.512737\pi$
$684$	0	0
$685$	−1.71603e7	−1.39733
$686$	651153.	0.0528290
$687$	0	0
$688$	−4.86845e6	−0.392120
$689$	1.53568e7	1.23240
$690$	0	0
$691$	5.56198e6	0.443133	0.221566	−	0.975145i	$-0.428883\pi$
$691$	5.56198e6	0.443133	0.221566	+	0.975145i	$0.428883\pi$
$692$	−673158.	−0.0534382
$693$	0	0
$694$	5.20731e6	0.410407
$695$	−8.48729e6	−0.666510
$696$	0	0
$697$	−2.81548e7	−2.19518
$698$	1.68633e7	1.31010
$699$	0	0
$700$	2.04230e6	0.157534
$701$	9.57353e6	0.735829	0.367915	−	0.929860i	$-0.380072\pi$
$701$	9.57353e6	0.735829	0.367915	+	0.929860i	$0.380072\pi$
$702$	0	0
$703$	−1.02639e7	−0.783292
$704$	2.04562e7	1.55559
$705$	0	0
$706$	−8.78836e6	−0.663584
$707$	−7.98316e6	−0.600656
$708$	0	0
$709$	1.85648e7	1.38700	0.693498	−	0.720458i	$-0.256068\pi$
$709$	1.85648e7	1.38700	0.693498	+	0.720458i	$0.256068\pi$
$710$	1.90433e7	1.41774
$711$	0	0
$712$	1.60259e7	1.18474
$713$	1.49730e6	0.110303
$714$	0	0
$715$	−2.88672e7	−2.11173
$716$	−986131.	−0.0718873
$717$	0	0
$718$	−1.18924e7	−0.860911
$719$	2.42441e7	1.74897	0.874487	−	0.485050i	$-0.161199\pi$
$719$	2.42441e7	1.74897	0.874487	+	0.485050i	$0.161199\pi$
$720$	0	0
$721$	−7.66869e6	−0.549393
$722$	−1.33255e7	−0.951350
$723$	0	0
$724$	4.48925e6	0.318293
$725$	−4.74822e6	−0.335495
$726$	0	0
$727$	1.23980e7	0.869990	0.434995	−	0.900433i	$-0.356750\pi$
$727$	1.23980e7	0.869990	0.434995	+	0.900433i	$0.356750\pi$
$728$	−2.78953e7	−1.95075
$729$	0	0
$730$	−2.03698e7	−1.41475
$731$	1.22622e7	0.848742
$732$	0	0
$733$	2.06634e7	1.42050	0.710252	−	0.703947i	$-0.248581\pi$
$733$	2.06634e7	1.42050	0.710252	+	0.703947i	$0.248581\pi$
$734$	8.44417e6	0.578518
$735$	0	0
$736$	−1.54480e6	−0.105118
$737$	1.94619e7	1.31983
$738$	0	0
$739$	1.86671e7	1.25738	0.628688	−	0.777658i	$-0.283592\pi$
$739$	1.86671e7	1.25738	0.628688	+	0.777658i	$0.283592\pi$
$740$	2.48889e6	0.167080
$741$	0	0
$742$	1.80549e7	1.20389
$743$	−8.53060e6	−0.566902	−0.283451	−	0.958987i	$-0.591479\pi$
$743$	−8.53060e6	−0.566902	−0.283451	+	0.958987i	$0.591479\pi$
$744$	0	0
$745$	41613.3	0.00274689
$746$	−3.26404e6	−0.214738
$747$	0	0
$748$	−8.12438e6	−0.530929
$749$	3.06179e7	1.99421
$750$	0	0
$751$	9.57195e6	0.619299	0.309650	−	0.950851i	$-0.399788\pi$
$751$	9.57195e6	0.619299	0.309650	+	0.950851i	$0.399788\pi$
$752$	−6.43889e6	−0.415208
$753$	0	0
$754$	1.33526e7	0.855339
$755$	2.22642e7	1.42147
$756$	0	0
$757$	3.09107e7	1.96051	0.980255	−	0.197738i	$-0.0633596\pi$
$757$	3.09107e7	1.96051	0.980255	+	0.197738i	$0.0633596\pi$
$758$	7.71752e6	0.487871
$759$	0	0
$760$	2.98932e7	1.87732
$761$	3.87764e6	0.242720	0.121360	−	0.992609i	$-0.461274\pi$
$761$	3.87764e6	0.242720	0.121360	+	0.992609i	$0.461274\pi$
$762$	0	0
$763$	−2.52774e7	−1.57188
$764$	2.22713e6	0.138043
$765$	0	0
$766$	5.54064e6	0.341184
$767$	−8.49110e6	−0.521165
$768$	0	0
$769$	−2.77315e7	−1.69105	−0.845527	−	0.533933i	$-0.820714\pi$
$769$	−2.77315e7	−1.69105	−0.845527	+	0.533933i	$0.820714\pi$
$770$	−3.39391e7	−2.06288
$771$	0	0
$772$	4.99959e6	0.301920
$773$	2.27867e7	1.37162	0.685810	−	0.727781i	$-0.259448\pi$
$773$	2.27867e7	1.37162	0.685810	+	0.727781i	$0.259448\pi$
$774$	0	0
$775$	3.76019e6	0.224882
$776$	−3.36458e7	−2.00575
$777$	0	0
$778$	2.46372e7	1.45929
$779$	3.70061e7	2.18489
$780$	0	0
$781$	3.30412e7	1.93833
$782$	−4.47395e6	−0.261622
$783$	0	0
$784$	−1.20893e7	−0.702443
$785$	3.83145e7	2.21916
$786$	0	0
$787$	1.70385e7	0.980604	0.490302	−	0.871553i	$-0.336886\pi$
$787$	1.70385e7	0.980604	0.490302	+	0.871553i	$0.336886\pi$
$788$	4.00703e6	0.229883
$789$	0	0
$790$	−1.86855e7	−1.06522
$791$	1.37387e7	0.780739
$792$	0	0
$793$	−3.36194e7	−1.89849
$794$	−1.43180e7	−0.805992
$795$	0	0
$796$	−4.75804e6	−0.266162
$797$	2.04743e7	1.14173	0.570864	−	0.821045i	$-0.306608\pi$
$797$	2.04743e7	1.14173	0.570864	+	0.821045i	$0.306608\pi$
$798$	0	0
$799$	1.62177e7	0.898716
$800$	−3.87945e6	−0.214312
$801$	0	0
$802$	1.61699e7	0.887710
$803$	−3.53428e7	−1.93425
$804$	0	0
$805$	6.54152e6	0.355786
$806$	−1.05741e7	−0.573334
$807$	0	0
$808$	8.45230e6	0.455456
$809$	−8.90982e6	−0.478628	−0.239314	−	0.970942i	$-0.576922\pi$
$809$	−8.90982e6	−0.478628	−0.239314	+	0.970942i	$0.576922\pi$
$810$	0	0
$811$	4.16300e6	0.222256	0.111128	−	0.993806i	$-0.464554\pi$
$811$	4.16300e6	0.222256	0.111128	+	0.993806i	$0.464554\pi$
$812$	−5.49466e6	−0.292449
$813$	0	0
$814$	−1.23379e7	−0.652649
$815$	−1.60092e7	−0.844259
$816$	0	0
$817$	−1.61172e7	−0.844761
$818$	5.43735e6	0.284122
$819$	0	0
$820$	−8.97358e6	−0.466048
$821$	−8.55489e6	−0.442952	−0.221476	−	0.975166i	$-0.571087\pi$
$821$	−8.55489e6	−0.442952	−0.221476	+	0.975166i	$0.571087\pi$
$822$	0	0
$823$	2.62720e7	1.35205	0.676025	−	0.736878i	$-0.263701\pi$
$823$	2.62720e7	1.35205	0.676025	+	0.736878i	$0.263701\pi$
$824$	8.11935e6	0.416585
$825$	0	0
$826$	−9.98298e6	−0.509108
$827$	−777655.	−0.0395388	−0.0197694	−	0.999805i	$-0.506293\pi$
$827$	−777655.	−0.0395388	−0.0197694	+	0.999805i	$0.506293\pi$
$828$	0	0
$829$	−1.04189e6	−0.0526546	−0.0263273	−	0.999653i	$-0.508381\pi$
$829$	−1.04189e6	−0.0526546	−0.0263273	+	0.999653i	$0.508381\pi$
$830$	−1.79283e7	−0.903327
$831$	0	0
$832$	2.78445e7	1.39454
$833$	3.04495e7	1.52043
$834$	0	0
$835$	2.06830e7	1.02659
$836$	1.06785e7	0.528439
$837$	0	0
$838$	3.44368e6	0.169400
$839$	−2.90427e7	−1.42440	−0.712201	−	0.701976i	$-0.752301\pi$
$839$	−2.90427e7	−1.42440	−0.712201	+	0.701976i	$0.752301\pi$
$840$	0	0
$841$	−7.73644e6	−0.377182
$842$	−2.83498e7	−1.37806
$843$	0	0
$844$	1.02787e7	0.496688
$845$	−1.45152e7	−0.699331
$846$	0	0
$847$	−2.90437e7	−1.39105
$848$	−1.38027e7	−0.659137
$849$	0	0
$850$	−1.12354e7	−0.533388
$851$	2.37804e6	0.112563
$852$	0	0
$853$	−1.29014e7	−0.607103	−0.303552	−	0.952815i	$-0.598172\pi$
$853$	−1.29014e7	−0.607103	−0.303552	+	0.952815i	$0.598172\pi$
$854$	−3.95263e7	−1.85456
$855$	0	0
$856$	−3.24172e7	−1.51214
$857$	−2.64342e7	−1.22946	−0.614729	−	0.788738i	$-0.710735\pi$
$857$	−2.64342e7	−1.22946	−0.614729	+	0.788738i	$0.710735\pi$
$858$	0	0
$859$	3.34613e7	1.54725	0.773623	−	0.633646i	$-0.218442\pi$
$859$	3.34613e7	1.54725	0.773623	+	0.633646i	$0.218442\pi$
$860$	3.90825e6	0.180192
$861$	0	0
$862$	−1.66662e7	−0.763956
$863$	1.30469e7	0.596320	0.298160	−	0.954516i	$-0.403627\pi$
$863$	1.30469e7	0.596320	0.298160	+	0.954516i	$0.403627\pi$
$864$	0	0
$865$	−5.41471e6	−0.246057
$866$	−7.08025e6	−0.320814
$867$	0	0
$868$	4.35130e6	0.196029
$869$	−3.24205e7	−1.45637
$870$	0	0
$871$	2.64910e7	1.18319
$872$	2.67628e7	1.19190
$873$	0	0
$874$	5.88046e6	0.260395
$875$	−2.22154e7	−0.980923
$876$	0	0
$877$	−3.46134e7	−1.51966	−0.759828	−	0.650124i	$-0.774717\pi$
$877$	−3.46134e7	−1.51966	−0.759828	+	0.650124i	$0.774717\pi$
$878$	1.84581e7	0.808073
$879$	0	0
$880$	2.59460e7	1.12944
$881$	2.51488e7	1.09163	0.545817	−	0.837905i	$-0.316219\pi$
$881$	2.51488e7	1.09163	0.545817	+	0.837905i	$0.316219\pi$
$882$	0	0
$883$	8.32506e6	0.359323	0.179662	−	0.983728i	$-0.442500\pi$
$883$	8.32506e6	0.359323	0.179662	+	0.983728i	$0.442500\pi$
$884$	−1.10587e7	−0.475963
$885$	0	0
$886$	−3.27396e6	−0.140117
$887$	−3.90242e7	−1.66543	−0.832713	−	0.553705i	$-0.813214\pi$
$887$	−3.90242e7	−1.66543	−0.832713	+	0.553705i	$0.813214\pi$
$888$	0	0
$889$	9.16383e6	0.388886
$890$	2.65403e7	1.12313
$891$	0	0
$892$	2.55865e6	0.107671
$893$	−2.13162e7	−0.894501
$894$	0	0
$895$	−7.93219e6	−0.331006
$896$	1.54211e7	0.641719
$897$	0	0
$898$	2.36496e7	0.978664
$899$	−1.01165e7	−0.417475
$900$	0	0
$901$	3.47651e7	1.42670
$902$	4.44838e7	1.82048
$903$	0	0
$904$	−1.45461e7	−0.592006
$905$	3.61104e7	1.46558
$906$	0	0
$907$	−3.55617e7	−1.43537	−0.717685	−	0.696368i	$-0.754798\pi$
$907$	−3.55617e7	−1.43537	−0.717685	+	0.696368i	$0.754798\pi$
$908$	6.33970e6	0.255184
$909$	0	0
$910$	−4.61970e7	−1.84931
$911$	−1.54307e6	−0.0616013	−0.0308007	−	0.999526i	$-0.509806\pi$
$911$	−1.54307e6	−0.0616013	−0.0308007	+	0.999526i	$0.509806\pi$
$912$	0	0
$913$	−3.11067e7	−1.23503
$914$	−3.40224e6	−0.134710
$915$	0	0
$916$	−8.44962e6	−0.332735
$917$	1.78057e7	0.699254
$918$	0	0
$919$	5.41531e6	0.211512	0.105756	−	0.994392i	$-0.466274\pi$
$919$	5.41531e6	0.211512	0.105756	+	0.994392i	$0.466274\pi$
$920$	−6.92594e6	−0.269780
$921$	0	0
$922$	2.05793e6	0.0797265
$923$	4.49748e7	1.73766
$924$	0	0
$925$	5.97199e6	0.229490
$926$	−2.01459e7	−0.772075
$927$	0	0
$928$	1.04374e7	0.397851
$929$	2.37496e7	0.902851	0.451426	−	0.892309i	$-0.350916\pi$
$929$	2.37496e7	0.902851	0.451426	+	0.892309i	$0.350916\pi$
$930$	0	0
$931$	−4.00221e7	−1.51330
$932$	−1.00181e6	−0.0377787
$933$	0	0
$934$	2.93372e7	1.10040
$935$	−6.53505e7	−2.44467
$936$	0	0
$937$	−1.35178e7	−0.502987	−0.251493	−	0.967859i	$-0.580922\pi$
$937$	−1.35178e7	−0.502987	−0.251493	+	0.967859i	$0.580922\pi$
$938$	3.11455e7	1.15581
$939$	0	0
$940$	5.16895e6	0.190802
$941$	3.13260e7	1.15327	0.576635	−	0.817002i	$-0.304366\pi$
$941$	3.13260e7	1.15327	0.576635	+	0.817002i	$0.304366\pi$
$942$	0	0
$943$	−8.57393e6	−0.313979
$944$	7.63184e6	0.278740
$945$	0	0
$946$	−1.93739e7	−0.703866
$947$	2.57698e7	0.933761	0.466880	−	0.884320i	$-0.345378\pi$
$947$	2.57698e7	0.933761	0.466880	+	0.884320i	$0.345378\pi$
$948$	0	0
$949$	−4.81078e7	−1.73400
$950$	1.47676e7	0.530886
$951$	0	0
$952$	−6.31503e7	−2.25831
$953$	−1.18796e7	−0.423710	−0.211855	−	0.977301i	$-0.567950\pi$
$953$	−1.18796e7	−0.423710	−0.211855	+	0.977301i	$0.567950\pi$
$954$	0	0
$955$	1.79145e7	0.635618
$956$	−2.63513e6	−0.0932520
$957$	0	0
$958$	−1.09920e7	−0.386957
$959$	−4.76485e7	−1.67303
$960$	0	0
$961$	−2.06177e7	−0.720166
$962$	−1.67940e7	−0.585082
$963$	0	0
$964$	554633.	0.0192226
$965$	4.02154e7	1.39019
$966$	0	0
$967$	−8.80618e6	−0.302846	−0.151423	−	0.988469i	$-0.548386\pi$
$967$	−8.80618e6	−0.302846	−0.151423	+	0.988469i	$0.548386\pi$
$968$	3.07505e7	1.05478
$969$	0	0
$970$	−5.57203e7	−1.90145
$971$	−1.00214e7	−0.341099	−0.170550	−	0.985349i	$-0.554554\pi$
$971$	−1.00214e7	−0.341099	−0.170550	+	0.985349i	$0.554554\pi$
$972$	0	0
$973$	−2.35664e7	−0.798014
$974$	−2.46305e7	−0.831908
$975$	0	0
$976$	3.02173e7	1.01539
$977$	−3.76716e7	−1.26264	−0.631318	−	0.775524i	$-0.717486\pi$
$977$	−3.76716e7	−1.26264	−0.631318	+	0.775524i	$0.717486\pi$
$978$	0	0
$979$	4.60490e7	1.53555
$980$	9.70494e6	0.322796
$981$	0	0
$982$	−2.80419e7	−0.927958
$983$	4.31076e7	1.42288	0.711442	−	0.702744i	$-0.248042\pi$
$983$	4.31076e7	1.42288	0.711442	+	0.702744i	$0.248042\pi$
$984$	0	0
$985$	3.22315e7	1.05850
$986$	3.02281e7	0.990189
$987$	0	0
$988$	1.45353e7	0.473731
$989$	3.73419e6	0.121396
$990$	0	0
$991$	3.25349e6	0.105236	0.0526181	−	0.998615i	$-0.483243\pi$
$991$	3.25349e6	0.105236	0.0526181	+	0.998615i	$0.483243\pi$
$992$	−8.26551e6	−0.266680
$993$	0	0
$994$	5.28768e7	1.69746
$995$	−3.82725e7	−1.22554
$996$	0	0
$997$	1.82210e7	0.580544	0.290272	−	0.956944i	$-0.406254\pi$
$997$	1.82210e7	0.580544	0.290272	+	0.956944i	$0.406254\pi$
$998$	4.65749e6	0.148022
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	207.6.a.e.1.2		4
3.2	odd	2		69.6.a.d.1.3	✓	4
12.11	even	2		1104.6.a.o.1.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
69.6.a.d.1.3	✓	4	3.2	odd	2
207.6.a.e.1.2		4	1.1	even	1	trivial
1104.6.a.o.1.4		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)

\(f(q)\)	\(=\)	\(q-4.86863 q^{2} -8.29644 q^{4} -66.7344 q^{5} -185.299 q^{7} +196.188 q^{8} +O(q^{10})\)	\(q-4.86863 q^{2} -8.29644 q^{4} -66.7344 q^{5} -185.299 q^{7} +196.188 q^{8} +324.905 q^{10} +563.729 q^{11} +767.333 q^{13} +902.153 q^{14} -689.683 q^{16} +1737.11 q^{17} -2283.22 q^{19} +553.658 q^{20} -2744.59 q^{22} +529.000 q^{23} +1328.48 q^{25} -3735.86 q^{26} +1537.32 q^{28} -3574.17 q^{29} +2830.44 q^{31} -2920.22 q^{32} -8457.37 q^{34} +12365.8 q^{35} +4495.35 q^{37} +11116.2 q^{38} -13092.5 q^{40} -16207.8 q^{41} +7058.96 q^{43} -4676.94 q^{44} -2575.51 q^{46} +9336.01 q^{47} +17528.8 q^{49} -6467.88 q^{50} -6366.13 q^{52} +20013.2 q^{53} -37620.1 q^{55} -36353.6 q^{56} +17401.3 q^{58} -11065.7 q^{59} -43813.3 q^{61} -13780.4 q^{62} +36287.3 q^{64} -51207.5 q^{65} +34523.5 q^{67} -14411.9 q^{68} -60204.7 q^{70} +58611.8 q^{71} -62694.7 q^{73} -21886.2 q^{74} +18942.6 q^{76} -104459. q^{77} -57510.8 q^{79} +46025.6 q^{80} +78909.8 q^{82} -55180.2 q^{83} -115925. q^{85} -34367.5 q^{86} +110597. q^{88} +81686.4 q^{89} -142186. q^{91} -4388.81 q^{92} -45453.6 q^{94} +152370. q^{95} -171497. q^{97} -85341.1 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{2} + 26 q^{4} - 22 q^{5} - 62 q^{7} - 72 q^{8}+O(q^{10}) \) 4 * q - 4 * q^2 + 26 * q^4 - 22 * q^5 - 62 * q^7 - 72 * q^8	\( 4 q - 4 q^{2} + 26 q^{4} - 22 q^{5} - 62 q^{7} - 72 q^{8} - 496 q^{10} + 1076 q^{11} - 396 q^{13} + 1806 q^{14} - 1982 q^{16} - 70 q^{17} - 6366 q^{19} + 5240 q^{20} - 6974 q^{22} + 2116 q^{23} + 1264 q^{25} - 2464 q^{26} - 6474 q^{28} - 3948 q^{29} + 3092 q^{31} + 3672 q^{32} + 11682 q^{34} - 1304 q^{35} - 17464 q^{37} + 12628 q^{38} - 14108 q^{40} - 18680 q^{41} - 25846 q^{43} - 20746 q^{44} - 2116 q^{46} - 18392 q^{47} + 7952 q^{49} - 69444 q^{50} + 8844 q^{52} + 26518 q^{53} - 40848 q^{55} - 54890 q^{56} + 568 q^{58} + 14520 q^{59} - 13688 q^{61} - 120136 q^{62} - 30190 q^{64} - 38324 q^{65} - 11098 q^{67} - 112138 q^{68} - 29596 q^{70} + 57496 q^{71} - 112272 q^{73} + 21226 q^{74} - 76240 q^{76} + 4792 q^{77} - 240754 q^{79} - 41200 q^{80} + 49976 q^{82} + 93268 q^{83} - 323204 q^{85} + 88224 q^{86} + 42382 q^{88} + 107582 q^{89} - 301532 q^{91} + 13754 q^{92} + 79360 q^{94} + 18640 q^{95} - 53076 q^{97} - 59664 q^{98}+O(q^{100}) \) 4 * q - 4 * q^2 + 26 * q^4 - 22 * q^5 - 62 * q^7 - 72 * q^8 - 496 * q^10 + 1076 * q^11 - 396 * q^13 + 1806 * q^14 - 1982 * q^16 - 70 * q^17 - 6366 * q^19 + 5240 * q^20 - 6974 * q^22 + 2116 * q^23 + 1264 * q^25 - 2464 * q^26 - 6474 * q^28 - 3948 * q^29 + 3092 * q^31 + 3672 * q^32 + 11682 * q^34 - 1304 * q^35 - 17464 * q^37 + 12628 * q^38 - 14108 * q^40 - 18680 * q^41 - 25846 * q^43 - 20746 * q^44 - 2116 * q^46 - 18392 * q^47 + 7952 * q^49 - 69444 * q^50 + 8844 * q^52 + 26518 * q^53 - 40848 * q^55 - 54890 * q^56 + 568 * q^58 + 14520 * q^59 - 13688 * q^61 - 120136 * q^62 - 30190 * q^64 - 38324 * q^65 - 11098 * q^67 - 112138 * q^68 - 29596 * q^70 + 57496 * q^71 - 112272 * q^73 + 21226 * q^74 - 76240 * q^76 + 4792 * q^77 - 240754 * q^79 - 41200 * q^80 + 49976 * q^82 + 93268 * q^83 - 323204 * q^85 + 88224 * q^86 + 42382 * q^88 + 107582 * q^89 - 301532 * q^91 + 13754 * q^92 + 79360 * q^94 + 18640 * q^95 - 53076 * q^97 - 59664 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

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Database

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