LMFDB - Embedded newform 207.6.a.f.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:	$5$
Coefficient field:	$\mathbb{Q}[x]/(x^{5} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 113x^{3} - 257x^{2} + 1404x + 2197 $ x^5 - 113x^3 - 257x^2 + 1404*x + 2197
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{5} $
Twist minimal:	no (minimal twist has level 69)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-5.17654$ 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)

$f(q)$	$=$	$q-9.27315 q^{2} +53.9914 q^{4} +37.4928 q^{5} +154.850 q^{7} -203.929 q^{8} +O(q^{10})$	\(q-9.27315 q^{2} +53.9914 q^{4} +37.4928 q^{5} +154.850 q^{7} -203.929 q^{8} -347.676 q^{10} -521.289 q^{11} -28.8511 q^{13} -1435.95 q^{14} +163.344 q^{16} +428.492 q^{17} -2563.67 q^{19} +2024.29 q^{20} +4834.00 q^{22} +529.000 q^{23} -1719.29 q^{25} +267.541 q^{26} +8360.58 q^{28} +2401.11 q^{29} -2130.21 q^{31} +5011.02 q^{32} -3973.47 q^{34} +5805.77 q^{35} +3651.58 q^{37} +23773.3 q^{38} -7645.87 q^{40} -16198.0 q^{41} +6503.63 q^{43} -28145.1 q^{44} -4905.50 q^{46} +20712.3 q^{47} +7171.64 q^{49} +15943.3 q^{50} -1557.71 q^{52} -34393.6 q^{53} -19544.6 q^{55} -31578.5 q^{56} -22265.9 q^{58} +18299.4 q^{59} -26509.8 q^{61} +19753.7 q^{62} -51695.0 q^{64} -1081.71 q^{65} -26313.7 q^{67} +23134.9 q^{68} -53837.8 q^{70} -39459.2 q^{71} -37484.5 q^{73} -33861.6 q^{74} -138416. q^{76} -80721.9 q^{77} +37368.8 q^{79} +6124.22 q^{80} +150206. q^{82} -76901.5 q^{83} +16065.3 q^{85} -60309.1 q^{86} +106306. q^{88} -62991.4 q^{89} -4467.60 q^{91} +28561.4 q^{92} -192068. q^{94} -96118.9 q^{95} +61484.7 q^{97} -66503.7 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - 8 q^{2} + 118 q^{4} - 94 q^{5} + 272 q^{7} - 258 q^{8}+O(q^{10}) $ 5 * q - 8 * q^2 + 118 * q^4 - 94 * q^5 + 272 * q^7 - 258 * q^8	$ 5 q - 8 q^{2} + 118 q^{4} - 94 q^{5} + 272 q^{7} - 258 q^{8} - 172 q^{10} - 1100 q^{11} - 978 q^{13} + 344 q^{14} + 1218 q^{16} - 2522 q^{17} + 2060 q^{19} - 7720 q^{20} - 2572 q^{22} + 2645 q^{23} + 12035 q^{25} - 9280 q^{26} + 8072 q^{28} - 1526 q^{29} - 7392 q^{31} + 5086 q^{32} - 15608 q^{34} - 6056 q^{35} - 8210 q^{37} + 14276 q^{38} - 37472 q^{40} - 21250 q^{41} - 4548 q^{43} + 4260 q^{44} - 4232 q^{46} - 536 q^{47} - 27979 q^{49} + 81872 q^{50} - 76380 q^{52} + 11482 q^{53} - 77064 q^{55} + 28624 q^{56} - 79680 q^{58} - 74676 q^{59} - 44618 q^{61} - 64880 q^{62} - 137382 q^{64} + 24388 q^{65} - 1412 q^{67} - 80196 q^{68} - 222304 q^{70} - 37912 q^{71} + 46546 q^{73} - 111604 q^{74} - 79548 q^{76} - 157008 q^{77} + 50544 q^{79} - 69424 q^{80} - 233720 q^{82} - 89588 q^{83} + 147892 q^{85} - 77428 q^{86} + 54484 q^{88} - 280410 q^{89} - 27416 q^{91} + 62422 q^{92} + 113632 q^{94} - 203120 q^{95} + 90074 q^{97} - 32976 q^{98}+O(q^{100}) $ 5 * q - 8 * q^2 + 118 * q^4 - 94 * q^5 + 272 * q^7 - 258 * q^8 - 172 * q^10 - 1100 * q^11 - 978 * q^13 + 344 * q^14 + 1218 * q^16 - 2522 * q^17 + 2060 * q^19 - 7720 * q^20 - 2572 * q^22 + 2645 * q^23 + 12035 * q^25 - 9280 * q^26 + 8072 * q^28 - 1526 * q^29 - 7392 * q^31 + 5086 * q^32 - 15608 * q^34 - 6056 * q^35 - 8210 * q^37 + 14276 * q^38 - 37472 * q^40 - 21250 * q^41 - 4548 * q^43 + 4260 * q^44 - 4232 * q^46 - 536 * q^47 - 27979 * q^49 + 81872 * q^50 - 76380 * q^52 + 11482 * q^53 - 77064 * q^55 + 28624 * q^56 - 79680 * q^58 - 74676 * q^59 - 44618 * q^61 - 64880 * q^62 - 137382 * q^64 + 24388 * q^65 - 1412 * q^67 - 80196 * q^68 - 222304 * q^70 - 37912 * q^71 + 46546 * q^73 - 111604 * q^74 - 79548 * q^76 - 157008 * q^77 + 50544 * q^79 - 69424 * q^80 - 233720 * q^82 - 89588 * q^83 + 147892 * q^85 - 77428 * q^86 + 54484 * q^88 - 280410 * q^89 - 27416 * q^91 + 62422 * q^92 + 113632 * q^94 - 203120 * q^95 + 90074 * q^97 - 32976 * 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 $

$2$	−9.27315	−1.63928	−0.819639	−	0.572881i	$-0.805826\pi$
$2$	−9.27315	−1.63928	−0.819639	+	0.572881i	$0.805826\pi$
$3$	0	0
$3$	0	0
$4$	53.9914	1.68723
$5$	37.4928	0.670691	0.335345	−	0.942095i	$-0.391147\pi$
$5$	37.4928	0.670691	0.335345	+	0.942095i	$0.391147\pi$
$6$	0	0
$7$	154.850	1.19445	0.597224	−	0.802075i	$-0.296270\pi$
$7$	154.850	1.19445	0.597224	+	0.802075i	$0.296270\pi$
$8$	−203.929	−1.12656
$9$	0	0
$10$	−347.676	−1.09945
$11$	−521.289	−1.29896	−0.649482	−	0.760377i	$-0.725014\pi$
$11$	−521.289	−1.29896	−0.649482	+	0.760377i	$0.725014\pi$
$12$	0	0
$13$	−28.8511	−0.0473483	−0.0236741	−	0.999720i	$-0.507536\pi$
$13$	−28.8511	−0.0473483	−0.0236741	+	0.999720i	$0.507536\pi$
$14$	−1435.95	−1.95803
$15$	0	0
$16$	163.344	0.159516
$17$	428.492	0.359601	0.179800	−	0.983703i	$-0.442455\pi$
$17$	428.492	0.359601	0.179800	+	0.983703i	$0.442455\pi$
$18$	0	0
$19$	−2563.67	−1.62921	−0.814606	−	0.580015i	$-0.803047\pi$
$19$	−2563.67	−1.62921	−0.814606	+	0.580015i	$0.803047\pi$
$20$	2024.29	1.13161
$21$	0	0
$22$	4834.00	2.12936
$23$	529.000	0.208514
$23$	529.000	0.208514
$24$	0	0
$25$	−1719.29	−0.550174
$26$	267.541	0.0776169
$27$	0	0
$28$	8360.58	2.01531
$29$	2401.11	0.530172	0.265086	−	0.964225i	$-0.414600\pi$
$29$	2401.11	0.530172	0.265086	+	0.964225i	$0.414600\pi$
$30$	0	0
$31$	−2130.21	−0.398124	−0.199062	−	0.979987i	$-0.563789\pi$
$31$	−2130.21	−0.398124	−0.199062	+	0.979987i	$0.563789\pi$
$32$	5011.02	0.865071
$33$	0	0
$34$	−3973.47	−0.589485
$35$	5805.77	0.801105
$36$	0	0
$37$	3651.58	0.438507	0.219253	−	0.975668i	$-0.429638\pi$
$37$	3651.58	0.438507	0.219253	+	0.975668i	$0.429638\pi$
$38$	23773.3	2.67073
$39$	0	0
$40$	−7645.87	−0.755574
$41$	−16198.0	−1.50488	−0.752440	−	0.658661i	$-0.771123\pi$
$41$	−16198.0	−1.50488	−0.752440	+	0.658661i	$0.771123\pi$
$42$	0	0
$43$	6503.63	0.536395	0.268197	−	0.963364i	$-0.413572\pi$
$43$	6503.63	0.536395	0.268197	+	0.963364i	$0.413572\pi$
$44$	−28145.1	−2.19165
$45$	0	0
$46$	−4905.50	−0.341813
$47$	20712.3	1.36768	0.683838	−	0.729633i	$-0.260309\pi$
$47$	20712.3	1.36768	0.683838	+	0.729633i	$0.260309\pi$
$48$	0	0
$49$	7171.64	0.426706
$50$	15943.3	0.901887
$51$	0	0
$52$	−1557.71	−0.0798874
$53$	−34393.6	−1.68185	−0.840926	−	0.541149i	$-0.817989\pi$
$53$	−34393.6	−1.68185	−0.840926	+	0.541149i	$0.817989\pi$
$54$	0	0
$55$	−19544.6	−0.871203
$56$	−31578.5	−1.34562
$57$	0	0
$58$	−22265.9	−0.869099
$59$	18299.4	0.684393	0.342197	−	0.939628i	$-0.388829\pi$
$59$	18299.4	0.684393	0.342197	+	0.939628i	$0.388829\pi$
$60$	0	0
$61$	−26509.8	−0.912183	−0.456092	−	0.889933i	$-0.650751\pi$
$61$	−26509.8	−0.912183	−0.456092	+	0.889933i	$0.650751\pi$
$62$	19753.7	0.652635
$63$	0	0
$64$	−51695.0	−1.57761
$65$	−1081.71	−0.0317560
$66$	0	0
$67$	−26313.7	−0.716136	−0.358068	−	0.933696i	$-0.616564\pi$
$67$	−26313.7	−0.716136	−0.358068	+	0.933696i	$0.616564\pi$
$68$	23134.9	0.606729
$69$	0	0
$70$	−53837.8	−1.31323
$71$	−39459.2	−0.928971	−0.464486	−	0.885581i	$-0.653761\pi$
$71$	−39459.2	−0.928971	−0.464486	+	0.885581i	$0.653761\pi$
$72$	0	0
$73$	−37484.5	−0.823273	−0.411637	−	0.911348i	$-0.635043\pi$
$73$	−37484.5	−0.823273	−0.411637	+	0.911348i	$0.635043\pi$
$74$	−33861.6	−0.718834
$75$	0	0
$76$	−138416.	−2.74886
$77$	−80721.9	−1.55154
$78$	0	0
$79$	37368.8	0.673660	0.336830	−	0.941565i	$-0.390645\pi$
$79$	37368.8	0.673660	0.336830	+	0.941565i	$0.390645\pi$
$80$	6124.22	0.106986
$81$	0	0
$82$	150206.	2.46691
$83$	−76901.5	−1.22529	−0.612646	−	0.790357i	$-0.709895\pi$
$83$	−76901.5	−1.22529	−0.612646	+	0.790357i	$0.709895\pi$
$84$	0	0
$85$	16065.3	0.241181
$86$	−60309.1	−0.879300
$87$	0	0
$88$	106306.	1.46336
$89$	−62991.4	−0.842959	−0.421480	−	0.906838i	$-0.638489\pi$
$89$	−62991.4	−0.842959	−0.421480	+	0.906838i	$0.638489\pi$
$90$	0	0
$91$	−4467.60	−0.0565550
$92$	28561.4	0.351812
$93$	0	0
$94$	−192068.	−2.24200
$95$	−96118.9	−1.09270
$96$	0	0
$97$	61484.7	0.663495	0.331748	−	0.943368i	$-0.392362\pi$
$97$	61484.7	0.663495	0.331748	+	0.943368i	$0.392362\pi$
$98$	−66503.7	−0.699489
$99$	0	0
$100$	−92827.0	−0.928270
$101$	89228.8	0.870365	0.435183	−	0.900342i	$-0.356684\pi$
$101$	89228.8	0.870365	0.435183	+	0.900342i	$0.356684\pi$
$102$	0	0
$103$	76234.9	0.708045	0.354023	−	0.935237i	$-0.384814\pi$
$103$	76234.9	0.708045	0.354023	+	0.935237i	$0.384814\pi$
$104$	5883.59	0.0533407
$105$	0	0
$106$	318937.	2.75702
$107$	98757.7	0.833895	0.416947	−	0.908931i	$-0.363100\pi$
$107$	98757.7	0.833895	0.416947	+	0.908931i	$0.363100\pi$
$108$	0	0
$109$	242295.	1.95334	0.976672	−	0.214738i	$-0.0688898\pi$
$109$	242295.	1.95334	0.976672	+	0.214738i	$0.0688898\pi$
$110$	181240.	1.42814
$111$	0	0
$112$	25293.9	0.190533
$113$	−218460.	−1.60945	−0.804723	−	0.593650i	$-0.797686\pi$
$113$	−218460.	−1.60945	−0.804723	+	0.593650i	$0.797686\pi$
$114$	0	0
$115$	19833.7	0.139849
$116$	129639.	0.894523
$117$	0	0
$118$	−169693.	−1.12191
$119$	66352.1	0.429524
$120$	0	0
$121$	110692.	0.687307
$122$	245830.	1.49532
$123$	0	0
$124$	−115013.	−0.671726
$125$	−181626.	−1.03969
$126$	0	0
$127$	−199105.	−1.09540	−0.547700	−	0.836675i	$-0.684496\pi$
$127$	−199105.	−1.09540	−0.547700	+	0.836675i	$0.684496\pi$
$128$	319023.	1.72106
$129$	0	0
$130$	10030.8	0.0520570
$131$	−236710.	−1.20514	−0.602572	−	0.798064i	$-0.705858\pi$
$131$	−236710.	−1.20514	−0.602572	+	0.798064i	$0.705858\pi$
$132$	0	0
$133$	−396985.	−1.94601
$134$	244011.	1.17395
$135$	0	0
$136$	−87382.1	−0.405112
$137$	308407.	1.40386	0.701929	−	0.712247i	$-0.252322\pi$
$137$	308407.	1.40386	0.701929	+	0.712247i	$0.252322\pi$
$138$	0	0
$139$	−99198.8	−0.435481	−0.217741	−	0.976007i	$-0.569869\pi$
$139$	−99198.8	−0.435481	−0.217741	+	0.976007i	$0.569869\pi$
$140$	313461.	1.35165
$141$	0	0
$142$	365911.	1.52284
$143$	15039.8	0.0615037
$144$	0	0
$145$	90024.2	0.355582
$146$	347599.	1.34957
$147$	0	0
$148$	197154.	0.739862
$149$	10925.0	0.0403140	0.0201570	−	0.999797i	$-0.493583\pi$
$149$	10925.0	0.0403140	0.0201570	+	0.999797i	$0.493583\pi$
$150$	0	0
$151$	−488685.	−1.74416	−0.872081	−	0.489362i	$-0.837230\pi$
$151$	−488685.	−1.74416	−0.872081	+	0.489362i	$0.837230\pi$
$152$	522807.	1.83541
$153$	0	0
$154$	748546.	2.54341
$155$	−79867.4	−0.267018
$156$	0	0
$157$	−589168.	−1.90761	−0.953806	−	0.300423i	$-0.902872\pi$
$157$	−589168.	−1.90761	−0.953806	+	0.300423i	$0.902872\pi$
$158$	−346526.	−1.10432
$159$	0	0
$160$	187877.	0.580195
$161$	81915.9	0.249060
$162$	0	0
$163$	598674.	1.76490	0.882452	−	0.470402i	$-0.155891\pi$
$163$	598674.	1.76490	0.882452	+	0.470402i	$0.155891\pi$
$164$	−874552.	−2.53908
$165$	0	0
$166$	713119.	2.00859
$167$	−222576.	−0.617570	−0.308785	−	0.951132i	$-0.599922\pi$
$167$	−222576.	−0.617570	−0.308785	+	0.951132i	$0.599922\pi$
$168$	0	0
$169$	−370461.	−0.997758
$170$	−148976.	−0.395362
$171$	0	0
$172$	351140.	0.905021
$173$	−438731.	−1.11451	−0.557253	−	0.830343i	$-0.688145\pi$
$173$	−438731.	−1.11451	−0.557253	+	0.830343i	$0.688145\pi$
$174$	0	0
$175$	−266233.	−0.657154
$176$	−85149.5	−0.207205
$177$	0	0
$178$	584129.	1.38184
$179$	−159561.	−0.372216	−0.186108	−	0.982529i	$-0.559587\pi$
$179$	−159561.	−0.372216	−0.186108	+	0.982529i	$0.559587\pi$
$180$	0	0
$181$	448958.	1.01861	0.509306	−	0.860585i	$-0.329902\pi$
$181$	448958.	1.01861	0.509306	+	0.860585i	$0.329902\pi$
$182$	41428.8	0.0927094
$183$	0	0
$184$	−107879.	−0.234904
$185$	136908.	0.294103
$186$	0	0
$187$	−223368.	−0.467108
$188$	1.11829e6	2.30759
$189$	0	0
$190$	891326.	1.79123
$191$	−170763.	−0.338696	−0.169348	−	0.985556i	$-0.554166\pi$
$191$	−170763.	−0.338696	−0.169348	+	0.985556i	$0.554166\pi$
$192$	0	0
$193$	211013.	0.407771	0.203885	−	0.978995i	$-0.434643\pi$
$193$	211013.	0.407771	0.203885	+	0.978995i	$0.434643\pi$
$194$	−570157.	−1.08765
$195$	0	0
$196$	387207.	0.719951
$197$	429130.	0.787813	0.393906	−	0.919151i	$-0.371123\pi$
$197$	429130.	0.787813	0.393906	+	0.919151i	$0.371123\pi$
$198$	0	0
$199$	−646856.	−1.15791	−0.578955	−	0.815359i	$-0.696539\pi$
$199$	−646856.	−1.15791	−0.578955	+	0.815359i	$0.696539\pi$
$200$	350614.	0.619804
$201$	0	0
$202$	−827432.	−1.42677
$203$	371813.	0.633263
$204$	0	0
$205$	−607308.	−1.00931
$206$	−706938.	−1.16068
$207$	0	0
$208$	−4712.66	−0.00755279
$209$	1.33641e6	2.11629
$210$	0	0
$211$	282510.	0.436845	0.218422	−	0.975854i	$-0.429909\pi$
$211$	282510.	0.436845	0.218422	+	0.975854i	$0.429909\pi$
$212$	−1.85696e6	−2.83767
$213$	0	0
$214$	−915795.	−1.36699
$215$	243839.	0.359755
$216$	0	0
$217$	−329864.	−0.475538
$218$	−2.24684e6	−3.20207
$219$	0	0
$220$	−1.05524e6	−1.46992
$221$	−12362.5	−0.0170265
$222$	0	0
$223$	−637440.	−0.858375	−0.429187	−	0.903215i	$-0.641200\pi$
$223$	−637440.	−0.858375	−0.429187	+	0.903215i	$0.641200\pi$
$224$	775959.	1.03328
$225$	0	0
$226$	2.02582e6	2.63833
$227$	−687712.	−0.885813	−0.442906	−	0.896568i	$-0.646053\pi$
$227$	−687712.	−0.885813	−0.442906	+	0.896568i	$0.646053\pi$
$228$	0	0
$229$	−606450.	−0.764199	−0.382100	−	0.924121i	$-0.624799\pi$
$229$	−606450.	−0.764199	−0.382100	+	0.924121i	$0.624799\pi$
$230$	−183921.	−0.229251
$231$	0	0
$232$	−489657.	−0.597271
$233$	−1.44933e6	−1.74895	−0.874475	−	0.485070i	$-0.838794\pi$
$233$	−1.44933e6	−1.74895	−0.874475	+	0.485070i	$0.838794\pi$
$234$	0	0
$235$	776561.	0.917289
$236$	988007.	1.15473
$237$	0	0
$238$	−615294.	−0.704109
$239$	−1.10668e6	−1.25322	−0.626610	−	0.779333i	$-0.715558\pi$
$239$	−1.10668e6	−1.25322	−0.626610	+	0.779333i	$0.715558\pi$
$240$	0	0
$241$	−1.02594e6	−1.13784	−0.568919	−	0.822394i	$-0.692638\pi$
$241$	−1.02594e6	−1.13784	−0.568919	+	0.822394i	$0.692638\pi$
$242$	−1.02646e6	−1.12669
$243$	0	0
$244$	−1.43130e6	−1.53906
$245$	268885.	0.286188
$246$	0	0
$247$	73964.6	0.0771403
$248$	434412.	0.448510
$249$	0	0
$250$	1.68424e6	1.70434
$251$	1.07943e6	1.08146	0.540730	−	0.841196i	$-0.318148\pi$
$251$	1.07943e6	1.08146	0.540730	+	0.841196i	$0.318148\pi$
$252$	0	0
$253$	−275762.	−0.270853
$254$	1.84633e6	1.79566
$255$	0	0
$256$	−1.30411e6	−1.24369
$257$	−1.24866e6	−1.17926	−0.589631	−	0.807673i	$-0.700727\pi$
$257$	−1.24866e6	−1.17926	−0.589631	+	0.807673i	$0.700727\pi$
$258$	0	0
$259$	565448.	0.523774
$260$	−58402.9	−0.0535798
$261$	0	0
$262$	2.19505e6	1.97557
$263$	−53460.0	−0.0476584	−0.0238292	−	0.999716i	$-0.507586\pi$
$263$	−53460.0	−0.0476584	−0.0238292	+	0.999716i	$0.507586\pi$
$264$	0	0
$265$	−1.28951e6	−1.12800
$266$	3.68130e6	3.19005
$267$	0	0
$268$	−1.42071e6	−1.20829
$269$	1.19910e6	1.01035	0.505177	−	0.863016i	$-0.331427\pi$
$269$	1.19910e6	1.01035	0.505177	+	0.863016i	$0.331427\pi$
$270$	0	0
$271$	595615.	0.492654	0.246327	−	0.969187i	$-0.420776\pi$
$271$	595615.	0.492654	0.246327	+	0.969187i	$0.420776\pi$
$272$	69991.6	0.0573619
$273$	0	0
$274$	−2.85991e6	−2.30131
$275$	896249.	0.714656
$276$	0	0
$277$	2.14463e6	1.67940	0.839698	−	0.543053i	$-0.182732\pi$
$277$	2.14463e6	1.67940	0.839698	+	0.543053i	$0.182732\pi$
$278$	919886.	0.713875
$279$	0	0
$280$	−1.18397e6	−0.902494
$281$	1.27455e6	0.962921	0.481461	−	0.876468i	$-0.340106\pi$
$281$	1.27455e6	0.962921	0.481461	+	0.876468i	$0.340106\pi$
$282$	0	0
$283$	1.33116e6	0.988014	0.494007	−	0.869458i	$-0.335532\pi$
$283$	1.33116e6	0.988014	0.494007	+	0.869458i	$0.335532\pi$
$284$	−2.13046e6	−1.56739
$285$	0	0
$286$	−139466.	−0.100822
$287$	−2.50827e6	−1.79750
$288$	0	0
$289$	−1.23625e6	−0.870687
$290$	−834808.	−0.582897
$291$	0	0
$292$	−2.02384e6	−1.38905
$293$	543720.	0.370004	0.185002	−	0.982738i	$-0.440771\pi$
$293$	543720.	0.370004	0.185002	+	0.982738i	$0.440771\pi$
$294$	0	0
$295$	686093.	0.459016
$296$	−744664.	−0.494005
$297$	0	0
$298$	−101309.	−0.0660858
$299$	−15262.2	−0.00987279
$300$	0	0
$301$	1.00709e6	0.640695
$302$	4.53165e6	2.85917
$303$	0	0
$304$	−418760.	−0.259885
$305$	−993926.	−0.611793
$306$	0	0
$307$	2.16140e6	1.30885	0.654424	−	0.756127i	$-0.272911\pi$
$307$	2.16140e6	1.30885	0.654424	+	0.756127i	$0.272911\pi$
$308$	−4.35828e6	−2.61781
$309$	0	0
$310$	740623.	0.437716
$311$	2.27827e6	1.33569	0.667843	−	0.744302i	$-0.267218\pi$
$311$	2.27827e6	1.33569	0.667843	+	0.744302i	$0.267218\pi$
$312$	0	0
$313$	−2.34170e6	−1.35105	−0.675524	−	0.737338i	$-0.736083\pi$
$313$	−2.34170e6	−1.35105	−0.675524	+	0.737338i	$0.736083\pi$
$314$	5.46345e6	3.12711
$315$	0	0
$316$	2.01759e6	1.13662
$317$	−1.70927e6	−0.955348	−0.477674	−	0.878537i	$-0.658520\pi$
$317$	−1.70927e6	−0.955348	−0.477674	+	0.878537i	$0.658520\pi$
$318$	0	0
$319$	−1.25167e6	−0.688675
$320$	−1.93819e6	−1.05809
$321$	0	0
$322$	−759618.	−0.408278
$323$	−1.09851e6	−0.585865
$324$	0	0
$325$	49603.5	0.0260498
$326$	−5.55159e6	−2.89317
$327$	0	0
$328$	3.30325e6	1.69534
$329$	3.20731e6	1.63362
$330$	0	0
$331$	3.09295e6	1.55168	0.775840	−	0.630930i	$-0.217326\pi$
$331$	3.09295e6	1.55168	0.775840	+	0.630930i	$0.217326\pi$
$332$	−4.15202e6	−2.06735
$333$	0	0
$334$	2.06398e6	1.01237
$335$	−986575.	−0.480306
$336$	0	0
$337$	−691524.	−0.331690	−0.165845	−	0.986152i	$-0.553035\pi$
$337$	−691524.	−0.331690	−0.165845	+	0.986152i	$0.553035\pi$
$338$	3.43534e6	1.63560
$339$	0	0
$340$	867390.	0.406928
$341$	1.11045e6	0.517148
$342$	0	0
$343$	−1.49204e6	−0.684770
$344$	−1.32628e6	−0.604281
$345$	0	0
$346$	4.06842e6	1.82699
$347$	4.15931e6	1.85437	0.927187	−	0.374599i	$-0.122220\pi$
$347$	4.15931e6	1.85437	0.927187	+	0.374599i	$0.122220\pi$
$348$	0	0
$349$	−3.44531e6	−1.51414	−0.757068	−	0.653337i	$-0.773369\pi$
$349$	−3.44531e6	−1.51414	−0.757068	+	0.653337i	$0.773369\pi$
$350$	2.46882e6	1.07726
$351$	0	0
$352$	−2.61219e6	−1.12370
$353$	2.27651e6	0.972373	0.486186	−	0.873855i	$-0.338388\pi$
$353$	2.27651e6	0.972373	0.486186	+	0.873855i	$0.338388\pi$
$354$	0	0
$355$	−1.47943e6	−0.623053
$356$	−3.40099e6	−1.42227
$357$	0	0
$358$	1.47964e6	0.610165
$359$	602616.	0.246777	0.123388	−	0.992358i	$-0.460624\pi$
$359$	602616.	0.246777	0.123388	+	0.992358i	$0.460624\pi$
$360$	0	0
$361$	4.09629e6	1.65433
$362$	−4.16326e6	−1.66979
$363$	0	0
$364$	−241212.	−0.0954214
$365$	−1.40540e6	−0.552162
$366$	0	0
$367$	2.69133e6	1.04304	0.521521	−	0.853238i	$-0.325365\pi$
$367$	2.69133e6	1.04304	0.521521	+	0.853238i	$0.325365\pi$
$368$	86409.0	0.0332613
$369$	0	0
$370$	−1.26957e6	−0.482116
$371$	−5.32586e6	−2.00889
$372$	0	0
$373$	−2.93803e6	−1.09341	−0.546707	−	0.837324i	$-0.684119\pi$
$373$	−2.93803e6	−1.09341	−0.546707	+	0.837324i	$0.684119\pi$
$374$	2.07133e6	0.765720
$375$	0	0
$376$	−4.22384e6	−1.54077
$377$	−69274.6	−0.0251027
$378$	0	0
$379$	2.78962e6	0.997578	0.498789	−	0.866723i	$-0.333778\pi$
$379$	2.78962e6	0.997578	0.498789	+	0.866723i	$0.333778\pi$
$380$	−5.18959e6	−1.84363
$381$	0	0
$382$	1.58351e6	0.555216
$383$	−3.28779e6	−1.14527	−0.572634	−	0.819811i	$-0.694078\pi$
$383$	−3.28779e6	−1.14527	−0.572634	+	0.819811i	$0.694078\pi$
$384$	0	0
$385$	−3.02649e6	−1.04061
$386$	−1.95676e6	−0.668450
$387$	0	0
$388$	3.31964e6	1.11947
$389$	2.17987e6	0.730392	0.365196	−	0.930931i	$-0.381002\pi$
$389$	2.17987e6	0.730392	0.365196	+	0.930931i	$0.381002\pi$
$390$	0	0
$391$	226672.	0.0749819
$392$	−1.46251e6	−0.480710
$393$	0	0
$394$	−3.97939e6	−1.29144
$395$	1.40106e6	0.451818
$396$	0	0
$397$	−3.30454e6	−1.05229	−0.526144	−	0.850396i	$-0.676363\pi$
$397$	−3.30454e6	−1.05229	−0.526144	+	0.850396i	$0.676363\pi$
$398$	5.99839e6	1.89814
$399$	0	0
$400$	−280836.	−0.0877613
$401$	−3.56153e6	−1.10605	−0.553026	−	0.833164i	$-0.686527\pi$
$401$	−3.56153e6	−1.10605	−0.553026	+	0.833164i	$0.686527\pi$
$402$	0	0
$403$	61458.9	0.0188505
$404$	4.81758e6	1.46851
$405$	0	0
$406$	−3.44788e6	−1.03809
$407$	−1.90353e6	−0.569605
$408$	0	0
$409$	−2.96716e6	−0.877066	−0.438533	−	0.898715i	$-0.644502\pi$
$409$	−2.96716e6	−0.877066	−0.438533	+	0.898715i	$0.644502\pi$
$410$	5.63166e6	1.65454
$411$	0	0
$412$	4.11603e6	1.19464
$413$	2.83366e6	0.817472
$414$	0	0
$415$	−2.88325e6	−0.821792
$416$	−144574.	−0.0409596
$417$	0	0
$418$	−1.23928e7	−3.46918
$419$	1.51390e6	0.421272	0.210636	−	0.977565i	$-0.432447\pi$
$419$	1.51390e6	0.421272	0.210636	+	0.977565i	$0.432447\pi$
$420$	0	0
$421$	746530.	0.205278	0.102639	−	0.994719i	$-0.467271\pi$
$421$	746530.	0.205278	0.102639	+	0.994719i	$0.467271\pi$
$422$	−2.61975e6	−0.716109
$423$	0	0
$424$	7.01386e6	1.89471
$425$	−736703.	−0.197843
$426$	0	0
$427$	−4.10506e6	−1.08956
$428$	5.33206e6	1.40697
$429$	0	0
$430$	−2.26116e6	−0.589738
$431$	−2.15954e6	−0.559976	−0.279988	−	0.960004i	$-0.590330\pi$
$431$	−2.15954e6	−0.559976	−0.279988	+	0.960004i	$0.590330\pi$
$432$	0	0
$433$	5.74239e6	1.47188	0.735941	−	0.677045i	$-0.236740\pi$
$433$	5.74239e6	1.47188	0.735941	+	0.677045i	$0.236740\pi$
$434$	3.05888e6	0.779538
$435$	0	0
$436$	1.30818e7	3.29574
$437$	−1.35618e6	−0.339714
$438$	0	0
$439$	3.04352e6	0.753728	0.376864	−	0.926269i	$-0.377002\pi$
$439$	3.04352e6	0.753728	0.376864	+	0.926269i	$0.377002\pi$
$440$	3.98571e6	0.981464
$441$	0	0
$442$	114639.	0.0279111
$443$	2.95192e6	0.714652	0.357326	−	0.933980i	$-0.383689\pi$
$443$	2.95192e6	0.714652	0.357326	+	0.933980i	$0.383689\pi$
$444$	0	0
$445$	−2.36172e6	−0.565365
$446$	5.91108e6	1.40711
$447$	0	0
$448$	−8.00499e6	−1.88437
$449$	−3.29085e6	−0.770356	−0.385178	−	0.922842i	$-0.625860\pi$
$449$	−3.29085e6	−0.770356	−0.385178	+	0.922842i	$0.625860\pi$
$450$	0	0
$451$	8.44384e6	1.95478
$452$	−1.17950e7	−2.71551
$453$	0	0
$454$	6.37726e6	1.45209
$455$	−167503.	−0.0379309
$456$	0	0
$457$	5.82436e6	1.30454	0.652271	−	0.757986i	$-0.273816\pi$
$457$	5.82436e6	1.30454	0.652271	+	0.757986i	$0.273816\pi$
$458$	5.62371e6	1.25273
$459$	0	0
$460$	1.07085e6	0.235957
$461$	1.73714e6	0.380700	0.190350	−	0.981716i	$-0.439038\pi$
$461$	1.73714e6	0.380700	0.190350	+	0.981716i	$0.439038\pi$
$462$	0	0
$463$	3.08948e6	0.669780	0.334890	−	0.942257i	$-0.391301\pi$
$463$	3.08948e6	0.669780	0.334890	+	0.942257i	$0.391301\pi$
$464$	392207.	0.0845708
$465$	0	0
$466$	1.34399e7	2.86702
$467$	1.75231e6	0.371808	0.185904	−	0.982568i	$-0.440479\pi$
$467$	1.75231e6	0.371808	0.185904	+	0.982568i	$0.440479\pi$
$468$	0	0
$469$	−4.07469e6	−0.855387
$470$	−7.20117e6	−1.50369
$471$	0	0
$472$	−3.73178e6	−0.771011
$473$	−3.39027e6	−0.696757
$474$	0	0
$475$	4.40769e6	0.896349
$476$	3.58244e6	0.724706
$477$	0	0
$478$	1.02624e7	2.05438
$479$	5.38341e6	1.07206	0.536030	−	0.844199i	$-0.319924\pi$
$479$	5.38341e6	1.07206	0.536030	+	0.844199i	$0.319924\pi$
$480$	0	0
$481$	−105352.	−0.0207625
$482$	9.51372e6	1.86523
$483$	0	0
$484$	5.97639e6	1.15965
$485$	2.30523e6	0.445000
$486$	0	0
$487$	−6.11771e6	−1.16887	−0.584436	−	0.811440i	$-0.698684\pi$
$487$	−6.11771e6	−1.16887	−0.584436	+	0.811440i	$0.698684\pi$
$488$	5.40613e6	1.02763
$489$	0	0
$490$	−2.49341e6	−0.469141
$491$	−5.22411e6	−0.977932	−0.488966	−	0.872303i	$-0.662626\pi$
$491$	−5.22411e6	−0.977932	−0.488966	+	0.872303i	$0.662626\pi$
$492$	0	0
$493$	1.02886e6	0.190650
$494$	−685885.	−0.126454
$495$	0	0
$496$	−347957.	−0.0635070
$497$	−6.11027e6	−1.10961
$498$	0	0
$499$	8.89404e6	1.59900	0.799498	−	0.600668i	$-0.205099\pi$
$499$	8.89404e6	1.59900	0.799498	+	0.600668i	$0.205099\pi$
$500$	−9.80623e6	−1.75419
$501$	0	0
$502$	−1.00097e7	−1.77281
$503$	3.62219e6	0.638339	0.319170	−	0.947698i	$-0.396596\pi$
$503$	3.62219e6	0.638339	0.319170	+	0.947698i	$0.396596\pi$
$504$	0	0
$505$	3.34543e6	0.583746
$506$	2.55718e6	0.444003
$507$	0	0
$508$	−1.07499e7	−1.84819
$509$	−454923.	−0.0778294	−0.0389147	−	0.999243i	$-0.512390\pi$
$509$	−454923.	−0.0778294	−0.0389147	+	0.999243i	$0.512390\pi$
$510$	0	0
$511$	−5.80448e6	−0.983357
$512$	1.88446e6	0.317696
$513$	0	0
$514$	1.15790e7	1.93314
$515$	2.85826e6	0.474879
$516$	0	0
$517$	−1.07971e7	−1.77656
$518$	−5.24349e6	−0.858610
$519$	0	0
$520$	220592.	0.0357751
$521$	−4.46566e6	−0.720761	−0.360380	−	0.932805i	$-0.617353\pi$
$521$	−4.46566e6	−0.720761	−0.360380	+	0.932805i	$0.617353\pi$
$522$	0	0
$523$	−2.91793e6	−0.466467	−0.233234	−	0.972421i	$-0.574931\pi$
$523$	−2.91793e6	−0.466467	−0.233234	+	0.972421i	$0.574931\pi$
$524$	−1.27803e7	−2.03336
$525$	0	0
$526$	495743.	0.0781254
$527$	−912777.	−0.143165
$528$	0	0
$529$	279841.	0.0434783
$530$	1.19578e7	1.84911
$531$	0	0
$532$	−2.14337e7	−3.28336
$533$	467330.	0.0712534
$534$	0	0
$535$	3.70270e6	0.559286
$536$	5.36614e6	0.806771
$537$	0	0
$538$	−1.11194e7	−1.65625
$539$	−3.73850e6	−0.554275
$540$	0	0
$541$	−2.73195e6	−0.401310	−0.200655	−	0.979662i	$-0.564307\pi$
$541$	−2.73195e6	−0.401310	−0.200655	+	0.979662i	$0.564307\pi$
$542$	−5.52323e6	−0.807597
$543$	0	0
$544$	2.14718e6	0.311080
$545$	9.08432e6	1.31009
$546$	0	0
$547$	−5.55388e6	−0.793648	−0.396824	−	0.917895i	$-0.629888\pi$
$547$	−5.55388e6	−0.793648	−0.396824	+	0.917895i	$0.629888\pi$
$548$	1.66513e7	2.36863
$549$	0	0
$550$	−8.31105e6	−1.17152
$551$	−6.15564e6	−0.863763
$552$	0	0
$553$	5.78657e6	0.804652
$554$	−1.98875e7	−2.75300
$555$	0	0
$556$	−5.35588e6	−0.734757
$557$	1.69772e6	0.231862	0.115931	−	0.993257i	$-0.463015\pi$
$557$	1.69772e6	0.231862	0.115931	+	0.993257i	$0.463015\pi$
$558$	0	0
$559$	−187637.	−0.0253974
$560$	948338.	0.127789
$561$	0	0
$562$	−1.18191e7	−1.57849
$563$	−4.59153e6	−0.610501	−0.305250	−	0.952272i	$-0.598740\pi$
$563$	−4.59153e6	−0.610501	−0.305250	+	0.952272i	$0.598740\pi$
$564$	0	0
$565$	−8.19068e6	−1.07944
$566$	−1.23440e7	−1.61963
$567$	0	0
$568$	8.04689e6	1.04654
$569$	1.10517e7	1.43103	0.715517	−	0.698595i	$-0.246191\pi$
$569$	1.10517e7	1.43103	0.715517	+	0.698595i	$0.246191\pi$
$570$	0	0
$571$	−1.35952e7	−1.74500	−0.872500	−	0.488613i	$-0.837503\pi$
$571$	−1.35952e7	−1.74500	−0.872500	+	0.488613i	$0.837503\pi$
$572$	812018.	0.103771
$573$	0	0
$574$	2.32595e7	2.94660
$575$	−909506.	−0.114719
$576$	0	0
$577$	4.14819e6	0.518703	0.259351	−	0.965783i	$-0.416491\pi$
$577$	4.14819e6	0.518703	0.259351	+	0.965783i	$0.416491\pi$
$578$	1.14640e7	1.42730
$579$	0	0
$580$	4.86053e6	0.599948
$581$	−1.19082e7	−1.46355
$582$	0	0
$583$	1.79290e7	2.18467
$584$	7.64418e6	0.927468
$585$	0	0
$586$	−5.04200e6	−0.606539
$587$	55407.2	0.00663699	0.00331849	−	0.999994i	$-0.498944\pi$
$587$	55407.2	0.00663699	0.00331849	+	0.999994i	$0.498944\pi$
$588$	0	0
$589$	5.46114e6	0.648628
$590$	−6.36225e6	−0.752455
$591$	0	0
$592$	596464.	0.0699487
$593$	1.44004e7	1.68166	0.840829	−	0.541300i	$-0.182068\pi$
$593$	1.44004e7	1.68166	0.840829	+	0.541300i	$0.182068\pi$
$594$	0	0
$595$	2.48773e6	0.288078
$596$	589856.	0.0680190
$597$	0	0
$598$	141529.	0.0161842
$599$	−1.33225e6	−0.151712	−0.0758559	−	0.997119i	$-0.524169\pi$
$599$	−1.33225e6	−0.151712	−0.0758559	+	0.997119i	$0.524169\pi$
$600$	0	0
$601$	−4.11211e6	−0.464386	−0.232193	−	0.972670i	$-0.574590\pi$
$601$	−4.11211e6	−0.464386	−0.232193	+	0.972670i	$0.574590\pi$
$602$	−9.33889e6	−1.05028
$603$	0	0
$604$	−2.63848e7	−2.94280
$605$	4.15013e6	0.460971
$606$	0	0
$607$	−5.38575e6	−0.593300	−0.296650	−	0.954986i	$-0.595869\pi$
$607$	−5.38575e6	−0.593300	−0.296650	+	0.954986i	$0.595869\pi$
$608$	−1.28466e7	−1.40938
$609$	0	0
$610$	9.21683e6	1.00290
$611$	−597572.	−0.0647571
$612$	0	0
$613$	−2.17889e6	−0.234199	−0.117100	−	0.993120i	$-0.537360\pi$
$613$	−2.17889e6	−0.234199	−0.117100	+	0.993120i	$0.537360\pi$
$614$	−2.00430e7	−2.14557
$615$	0	0
$616$	1.64616e7	1.74791
$617$	−1.40873e7	−1.48976	−0.744880	−	0.667199i	$-0.767493\pi$
$617$	−1.40873e7	−1.48976	−0.744880	+	0.667199i	$0.767493\pi$
$618$	0	0
$619$	−5.63132e6	−0.590722	−0.295361	−	0.955386i	$-0.595440\pi$
$619$	−5.63132e6	−0.590722	−0.295361	+	0.955386i	$0.595440\pi$
$620$	−4.31215e6	−0.450521
$621$	0	0
$622$	−2.11268e7	−2.18956
$623$	−9.75425e6	−1.00687
$624$	0	0
$625$	−1.43687e6	−0.147135
$626$	2.17150e7	2.21474
$627$	0	0
$628$	−3.18100e7	−3.21858
$629$	1.56467e6	0.157687
$630$	0	0
$631$	−1.47000e7	−1.46976	−0.734878	−	0.678200i	$-0.762760\pi$
$631$	−1.47000e7	−1.46976	−0.734878	+	0.678200i	$0.762760\pi$
$632$	−7.62059e6	−0.758919
$633$	0	0
$634$	1.58503e7	1.56608
$635$	−7.46499e6	−0.734675
$636$	0	0
$637$	−206910.	−0.0202038
$638$	1.16070e7	1.12893
$639$	0	0
$640$	1.19610e7	1.15430
$641$	1.20465e7	1.15802	0.579012	−	0.815319i	$-0.303439\pi$
$641$	1.20465e7	1.15802	0.579012	+	0.815319i	$0.303439\pi$
$642$	0	0
$643$	4.72652e6	0.450831	0.225416	−	0.974263i	$-0.427626\pi$
$643$	4.72652e6	0.450831	0.225416	+	0.974263i	$0.427626\pi$
$644$	4.42275e6	0.420221
$645$	0	0
$646$	1.01867e7	0.960396
$647$	1.24621e7	1.17039	0.585196	−	0.810892i	$-0.301018\pi$
$647$	1.24621e7	1.17039	0.585196	+	0.810892i	$0.301018\pi$
$648$	0	0
$649$	−9.53926e6	−0.889002
$650$	−459981.	−0.0427028
$651$	0	0
$652$	3.23232e7	2.97780
$653$	7.75118e6	0.711353	0.355676	−	0.934609i	$-0.384251\pi$
$653$	7.75118e6	0.711353	0.355676	+	0.934609i	$0.384251\pi$
$654$	0	0
$655$	−8.87493e6	−0.808280
$656$	−2.64585e6	−0.240052
$657$	0	0
$658$	−2.97418e7	−2.67795
$659$	−1.52713e7	−1.36981	−0.684907	−	0.728630i	$-0.740157\pi$
$659$	−1.52713e7	−1.36981	−0.684907	+	0.728630i	$0.740157\pi$
$660$	0	0
$661$	2.21977e7	1.97608	0.988041	−	0.154190i	$-0.0492767\pi$
$661$	2.21977e7	1.97608	0.988041	+	0.154190i	$0.0492767\pi$
$662$	−2.86814e7	−2.54363
$663$	0	0
$664$	1.56825e7	1.38037
$665$	−1.48841e7	−1.30517
$666$	0	0
$667$	1.27019e6	0.110549
$668$	−1.20172e7	−1.04198
$669$	0	0
$670$	9.14866e6	0.787355
$671$	1.38193e7	1.18489
$672$	0	0
$673$	6.38649e6	0.543531	0.271766	−	0.962363i	$-0.412392\pi$
$673$	6.38649e6	0.543531	0.271766	+	0.962363i	$0.412392\pi$
$674$	6.41261e6	0.543732
$675$	0	0
$676$	−2.00017e7	−1.68345
$677$	−6.89077e6	−0.577825	−0.288913	−	0.957356i	$-0.593294\pi$
$677$	−6.89077e6	−0.577825	−0.288913	+	0.957356i	$0.593294\pi$
$678$	0	0
$679$	9.52093e6	0.792510
$680$	−3.27620e6	−0.271705
$681$	0	0
$682$	−1.02974e7	−0.847749
$683$	2.36161e7	1.93712	0.968558	−	0.248788i	$-0.0800321\pi$
$683$	2.36161e7	1.93712	0.968558	+	0.248788i	$0.0800321\pi$
$684$	0	0
$685$	1.15630e7	0.941555
$686$	1.38359e7	1.12253
$687$	0	0
$688$	1.06233e6	0.0855634
$689$	992293.	0.0796328
$690$	0	0
$691$	1.22060e7	0.972472	0.486236	−	0.873828i	$-0.338370\pi$
$691$	1.22060e7	0.972472	0.486236	+	0.873828i	$0.338370\pi$
$692$	−2.36877e7	−1.88043
$693$	0	0
$694$	−3.85699e7	−3.03983
$695$	−3.71924e6	−0.292073
$696$	0	0
$697$	−6.94071e6	−0.541155
$698$	3.19489e7	2.48209
$699$	0	0
$700$	−1.43743e7	−1.10877
$701$	−6.09955e6	−0.468816	−0.234408	−	0.972138i	$-0.575315\pi$
$701$	−6.09955e6	−0.468816	−0.234408	+	0.972138i	$0.575315\pi$
$702$	0	0
$703$	−9.36143e6	−0.714420
$704$	2.69481e7	2.04925
$705$	0	0
$706$	−2.11104e7	−1.59399
$707$	1.38171e7	1.03961
$708$	0	0
$709$	2.32369e7	1.73605	0.868026	−	0.496518i	$-0.165388\pi$
$709$	2.32369e7	1.73605	0.868026	+	0.496518i	$0.165388\pi$
$710$	1.37190e7	1.02136
$711$	0	0
$712$	1.28458e7	0.949645
$713$	−1.12688e6	−0.0830145
$714$	0	0
$715$	563883.	0.0412500
$716$	−8.61493e6	−0.628014
$717$	0	0
$718$	−5.58815e6	−0.404536
$719$	1.64452e7	1.18636	0.593179	−	0.805071i	$-0.297873\pi$
$719$	1.64452e7	1.18636	0.593179	+	0.805071i	$0.297873\pi$
$720$	0	0
$721$	1.18050e7	0.845723
$722$	−3.79855e7	−2.71191
$723$	0	0
$724$	2.42399e7	1.71863
$725$	−4.12821e6	−0.291687
$726$	0	0
$727$	−3.66575e6	−0.257233	−0.128617	−	0.991694i	$-0.541054\pi$
$727$	−3.66575e6	−0.257233	−0.128617	+	0.991694i	$0.541054\pi$
$728$	911076.	0.0637127
$729$	0	0
$730$	1.30324e7	0.905146
$731$	2.78675e6	0.192888
$732$	0	0
$733$	−2.15654e7	−1.48251	−0.741256	−	0.671223i	$-0.765769\pi$
$733$	−2.15654e7	−1.48251	−0.741256	+	0.671223i	$0.765769\pi$
$734$	−2.49571e7	−1.70984
$735$	0	0
$736$	2.65083e6	0.180380
$737$	1.37171e7	0.930235
$738$	0	0
$739$	2.21913e7	1.49476	0.747380	−	0.664397i	$-0.231311\pi$
$739$	2.21913e7	1.49476	0.747380	+	0.664397i	$0.231311\pi$
$740$	7.39184e6	0.496219
$741$	0	0
$742$	4.93875e7	3.29312
$743$	6.61440e6	0.439560	0.219780	−	0.975549i	$-0.429466\pi$
$743$	6.61440e6	0.439560	0.219780	+	0.975549i	$0.429466\pi$
$744$	0	0
$745$	409609.	0.0270382
$746$	2.72448e7	1.79241
$747$	0	0
$748$	−1.20600e7	−0.788119
$749$	1.52927e7	0.996044
$750$	0	0
$751$	−8.17370e6	−0.528834	−0.264417	−	0.964409i	$-0.585179\pi$
$751$	−8.17370e6	−0.528834	−0.264417	+	0.964409i	$0.585179\pi$
$752$	3.38323e6	0.218166
$753$	0	0
$754$	642394.	0.0411503
$755$	−1.83222e7	−1.16979
$756$	0	0
$757$	−6.07796e6	−0.385495	−0.192747	−	0.981248i	$-0.561740\pi$
$757$	−6.07796e6	−0.385495	−0.192747	+	0.981248i	$0.561740\pi$
$758$	−2.58686e7	−1.63531
$759$	0	0
$760$	1.96015e7	1.23099
$761$	−2.93696e6	−0.183838	−0.0919192	−	0.995766i	$-0.529300\pi$
$761$	−2.93696e6	−0.183838	−0.0919192	+	0.995766i	$0.529300\pi$
$762$	0	0
$763$	3.75195e7	2.33317
$764$	−9.21971e6	−0.571457
$765$	0	0
$766$	3.04882e7	1.87741
$767$	−527957.	−0.0324048
$768$	0	0
$769$	6.47116e6	0.394608	0.197304	−	0.980342i	$-0.436781\pi$
$769$	6.47116e6	0.394608	0.197304	+	0.980342i	$0.436781\pi$
$770$	2.80651e7	1.70584
$771$	0	0
$772$	1.13929e7	0.688003
$773$	−2.52231e7	−1.51827	−0.759136	−	0.650932i	$-0.774378\pi$
$773$	−2.52231e7	−1.51827	−0.759136	+	0.650932i	$0.774378\pi$
$774$	0	0
$775$	3.66245e6	0.219037
$776$	−1.25385e7	−0.747468
$777$	0	0
$778$	−2.02143e7	−1.19732
$779$	4.15263e7	2.45177
$780$	0	0
$781$	2.05697e7	1.20670
$782$	−2.10197e6	−0.122916
$783$	0	0
$784$	1.17145e6	0.0680663
$785$	−2.20895e7	−1.27942
$786$	0	0
$787$	−1.74674e7	−1.00529	−0.502644	−	0.864493i	$-0.667639\pi$
$787$	−1.74674e7	−1.00529	−0.502644	+	0.864493i	$0.667639\pi$
$788$	2.31693e7	1.32922
$789$	0	0
$790$	−1.29922e7	−0.740655
$791$	−3.38287e7	−1.92240
$792$	0	0
$793$	764838.	0.0431903
$794$	3.06435e7	1.72499
$795$	0	0
$796$	−3.49246e7	−1.95366
$797$	−2.12790e6	−0.118660	−0.0593302	−	0.998238i	$-0.518896\pi$
$797$	−2.12790e6	−0.118660	−0.0593302	+	0.998238i	$0.518896\pi$
$798$	0	0
$799$	8.87505e6	0.491817
$800$	−8.61542e6	−0.475939
$801$	0	0
$802$	3.30266e7	1.81312
$803$	1.95402e7	1.06940
$804$	0	0
$805$	3.07125e6	0.167042
$806$	−569917.	−0.0309011
$807$	0	0
$808$	−1.81964e7	−0.980519
$809$	5.17917e6	0.278220	0.139110	−	0.990277i	$-0.455576\pi$
$809$	5.17917e6	0.278220	0.139110	+	0.990277i	$0.455576\pi$
$810$	0	0
$811$	2.72737e7	1.45610	0.728050	−	0.685524i	$-0.240427\pi$
$811$	2.72737e7	1.45610	0.728050	+	0.685524i	$0.240427\pi$
$812$	2.00747e7	1.06846
$813$	0	0
$814$	1.76517e7	0.933740
$815$	2.24459e7	1.18371
$816$	0	0
$817$	−1.66731e7	−0.873900
$818$	2.75149e7	1.43775
$819$	0	0
$820$	−3.27894e7	−1.70294
$821$	−2.28417e7	−1.18269	−0.591346	−	0.806418i	$-0.701403\pi$
$821$	−2.28417e7	−1.18269	−0.591346	+	0.806418i	$0.701403\pi$
$822$	0	0
$823$	1.38673e7	0.713664	0.356832	−	0.934169i	$-0.383857\pi$
$823$	1.38673e7	0.713664	0.356832	+	0.934169i	$0.383857\pi$
$824$	−1.55465e7	−0.797656
$825$	0	0
$826$	−2.62770e7	−1.34006
$827$	7.98777e6	0.406127	0.203064	−	0.979166i	$-0.434910\pi$
$827$	7.98777e6	0.406127	0.203064	+	0.979166i	$0.434910\pi$
$828$	0	0
$829$	−8.88714e6	−0.449134	−0.224567	−	0.974459i	$-0.572097\pi$
$829$	−8.88714e6	−0.449134	−0.224567	+	0.974459i	$0.572097\pi$
$830$	2.67368e7	1.34715
$831$	0	0
$832$	1.49146e6	0.0746969
$833$	3.07299e6	0.153444
$834$	0	0
$835$	−8.34498e6	−0.414199
$836$	7.21547e7	3.57066
$837$	0	0
$838$	−1.40386e7	−0.690581
$839$	2.25661e7	1.10676	0.553379	−	0.832930i	$-0.313338\pi$
$839$	2.25661e7	1.10676	0.553379	+	0.832930i	$0.313338\pi$
$840$	0	0
$841$	−1.47458e7	−0.718917
$842$	−6.92268e6	−0.336507
$843$	0	0
$844$	1.52531e7	0.737057
$845$	−1.38896e7	−0.669187
$846$	0	0
$847$	1.71406e7	0.820953
$848$	−5.61799e6	−0.268282
$849$	0	0
$850$	6.83156e6	0.324319
$851$	1.93169e6	0.0914350
$852$	0	0
$853$	−3.35094e7	−1.57686	−0.788431	−	0.615124i	$-0.789106\pi$
$853$	−3.35094e7	−1.57686	−0.788431	+	0.615124i	$0.789106\pi$
$854$	3.80668e7	1.78608
$855$	0	0
$856$	−2.01396e7	−0.939433
$857$	1.84965e7	0.860276	0.430138	−	0.902763i	$-0.358465\pi$
$857$	1.84965e7	0.860276	0.430138	+	0.902763i	$0.358465\pi$
$858$	0	0
$859$	3.55324e7	1.64302	0.821508	−	0.570198i	$-0.193133\pi$
$859$	3.55324e7	1.64302	0.821508	+	0.570198i	$0.193133\pi$
$860$	1.31652e7	0.606989
$861$	0	0
$862$	2.00258e7	0.917955
$863$	1.37196e7	0.627067	0.313534	−	0.949577i	$-0.398487\pi$
$863$	1.37196e7	0.627067	0.313534	+	0.949577i	$0.398487\pi$
$864$	0	0
$865$	−1.64492e7	−0.747489
$866$	−5.32501e7	−2.41282
$867$	0	0
$868$	−1.78098e7	−0.802342
$869$	−1.94799e7	−0.875060
$870$	0	0
$871$	759180.	0.0339078
$872$	−4.94111e7	−2.20056
$873$	0	0
$874$	1.25761e7	0.556886
$875$	−2.81248e7	−1.24185
$876$	0	0
$877$	−1.74346e7	−0.765443	−0.382721	−	0.923864i	$-0.625013\pi$
$877$	−1.74346e7	−0.765443	−0.382721	+	0.923864i	$0.625013\pi$
$878$	−2.82230e7	−1.23557
$879$	0	0
$880$	−3.19249e6	−0.138971
$881$	−3.33087e7	−1.44583	−0.722916	−	0.690936i	$-0.757199\pi$
$881$	−3.33087e7	−1.44583	−0.722916	+	0.690936i	$0.757199\pi$
$882$	0	0
$883$	3.36375e6	0.145185	0.0725925	−	0.997362i	$-0.476873\pi$
$883$	3.36375e6	0.145185	0.0725925	+	0.997362i	$0.476873\pi$
$884$	−667466.	−0.0287276
$885$	0	0
$886$	−2.73736e7	−1.17151
$887$	1.07106e7	0.457095	0.228547	−	0.973533i	$-0.426602\pi$
$887$	1.07106e7	0.457095	0.228547	+	0.973533i	$0.426602\pi$
$888$	0	0
$889$	−3.08315e7	−1.30840
$890$	2.19006e7	0.926790
$891$	0	0
$892$	−3.44162e7	−1.44828
$893$	−5.30994e7	−2.22824
$894$	0	0
$895$	−5.98239e6	−0.249642
$896$	4.94008e7	2.05572
$897$	0	0
$898$	3.05165e7	1.26283
$899$	−5.11486e6	−0.211074
$900$	0	0
$901$	−1.47374e7	−0.604795
$902$	−7.83010e7	−3.20443
$903$	0	0
$904$	4.45505e7	1.81314
$905$	1.68327e7	0.683174
$906$	0	0
$907$	1.90456e7	0.768735	0.384367	−	0.923180i	$-0.374420\pi$
$907$	1.90456e7	0.768735	0.384367	+	0.923180i	$0.374420\pi$
$908$	−3.71305e7	−1.49457
$909$	0	0
$910$	1.55328e6	0.0621793
$911$	−1.05019e6	−0.0419249	−0.0209624	−	0.999780i	$-0.506673\pi$
$911$	−1.05019e6	−0.0419249	−0.0209624	+	0.999780i	$0.506673\pi$
$912$	0	0
$913$	4.00879e7	1.59161
$914$	−5.40102e7	−2.13851
$915$	0	0
$916$	−3.27431e7	−1.28938
$917$	−3.66547e7	−1.43948
$918$	0	0
$919$	−1.19851e7	−0.468114	−0.234057	−	0.972223i	$-0.575200\pi$
$919$	−1.19851e7	−0.468114	−0.234057	+	0.972223i	$0.575200\pi$
$920$	−4.04467e6	−0.157548
$921$	0	0
$922$	−1.61088e7	−0.624073
$923$	1.13844e6	0.0439852
$924$	0	0
$925$	−6.27813e6	−0.241255
$926$	−2.86492e7	−1.09796
$927$	0	0
$928$	1.20320e7	0.458636
$929$	−1.29362e7	−0.491777	−0.245888	−	0.969298i	$-0.579080\pi$
$929$	−1.29362e7	−0.491777	−0.245888	+	0.969298i	$0.579080\pi$
$930$	0	0
$931$	−1.83857e7	−0.695194
$932$	−7.82513e7	−2.95088
$933$	0	0
$934$	−1.62494e7	−0.609496
$935$	−8.37469e6	−0.313285
$936$	0	0
$937$	−1.25390e7	−0.466567	−0.233283	−	0.972409i	$-0.574947\pi$
$937$	−1.25390e7	−0.466567	−0.233283	+	0.972409i	$0.574947\pi$
$938$	3.77852e7	1.40222
$939$	0	0
$940$	4.19276e7	1.54768
$941$	2.45328e7	0.903178	0.451589	−	0.892226i	$-0.350857\pi$
$941$	2.45328e7	0.903178	0.451589	+	0.892226i	$0.350857\pi$
$942$	0	0
$943$	−8.56874e6	−0.313789
$944$	2.98909e6	0.109171
$945$	0	0
$946$	3.14385e7	1.14218
$947$	2.08008e7	0.753713	0.376856	−	0.926272i	$-0.377005\pi$
$947$	2.08008e7	0.753713	0.376856	+	0.926272i	$0.377005\pi$
$948$	0	0
$949$	1.08147e6	0.0389806
$950$	−4.08732e7	−1.46937
$951$	0	0
$952$	−1.35311e7	−0.483885
$953$	9.76608e6	0.348328	0.174164	−	0.984717i	$-0.444278\pi$
$953$	9.76608e6	0.348328	0.174164	+	0.984717i	$0.444278\pi$
$954$	0	0
$955$	−6.40236e6	−0.227160
$956$	−5.97512e7	−2.11447
$957$	0	0
$958$	−4.99212e7	−1.75740
$959$	4.77570e7	1.67684
$960$	0	0
$961$	−2.40914e7	−0.841498
$962$	976946.	0.0340356
$963$	0	0
$964$	−5.53920e7	−1.91979
$965$	7.91147e6	0.273488
$966$	0	0
$967$	−2.35516e7	−0.809942	−0.404971	−	0.914329i	$-0.632718\pi$
$967$	−2.35516e7	−0.809942	−0.404971	+	0.914329i	$0.632718\pi$
$968$	−2.25733e7	−0.774294
$969$	0	0
$970$	−2.13768e7	−0.729479
$971$	−2.32199e7	−0.790336	−0.395168	−	0.918609i	$-0.629314\pi$
$971$	−2.32199e7	−0.790336	−0.395168	+	0.918609i	$0.629314\pi$
$972$	0	0
$973$	−1.53610e7	−0.520160
$974$	5.67305e7	1.91610
$975$	0	0
$976$	−4.33022e6	−0.145508
$977$	3.29920e7	1.10579	0.552895	−	0.833251i	$-0.313523\pi$
$977$	3.29920e7	1.10579	0.552895	+	0.833251i	$0.313523\pi$
$978$	0	0
$979$	3.28368e7	1.09497
$980$	1.45175e7	0.482864
$981$	0	0
$982$	4.84440e7	1.60310
$983$	2.82144e7	0.931294	0.465647	−	0.884970i	$-0.345822\pi$
$983$	2.82144e7	0.931294	0.465647	+	0.884970i	$0.345822\pi$
$984$	0	0
$985$	1.60893e7	0.528379
$986$	−9.54074e6	−0.312529
$987$	0	0
$988$	3.99345e6	0.130153
$989$	3.44042e6	0.111846
$990$	0	0
$991$	−3.77541e7	−1.22118	−0.610590	−	0.791947i	$-0.709068\pi$
$991$	−3.77541e7	−1.22118	−0.610590	+	0.791947i	$0.709068\pi$
$992$	−1.06745e7	−0.344405
$993$	0	0
$994$	5.66615e7	1.81896
$995$	−2.42524e7	−0.776600
$996$	0	0
$997$	3.33776e7	1.06345	0.531726	−	0.846917i	$-0.321544\pi$
$997$	3.33776e7	1.06345	0.531726	+	0.846917i	$0.321544\pi$
$998$	−8.24758e7	−2.62120
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	207.6.a.f.1.2		5
3.2	odd	2		69.6.a.e.1.4	✓	5
12.11	even	2		1104.6.a.r.1.2		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
69.6.a.e.1.4	✓	5	3.2	odd	2
207.6.a.f.1.2		5	1.1	even	1	trivial
1104.6.a.r.1.2		5	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)

\(f(q)\)	\(=\)	\(q-9.27315 q^{2} +53.9914 q^{4} +37.4928 q^{5} +154.850 q^{7} -203.929 q^{8} +O(q^{10})\)	\(q-9.27315 q^{2} +53.9914 q^{4} +37.4928 q^{5} +154.850 q^{7} -203.929 q^{8} -347.676 q^{10} -521.289 q^{11} -28.8511 q^{13} -1435.95 q^{14} +163.344 q^{16} +428.492 q^{17} -2563.67 q^{19} +2024.29 q^{20} +4834.00 q^{22} +529.000 q^{23} -1719.29 q^{25} +267.541 q^{26} +8360.58 q^{28} +2401.11 q^{29} -2130.21 q^{31} +5011.02 q^{32} -3973.47 q^{34} +5805.77 q^{35} +3651.58 q^{37} +23773.3 q^{38} -7645.87 q^{40} -16198.0 q^{41} +6503.63 q^{43} -28145.1 q^{44} -4905.50 q^{46} +20712.3 q^{47} +7171.64 q^{49} +15943.3 q^{50} -1557.71 q^{52} -34393.6 q^{53} -19544.6 q^{55} -31578.5 q^{56} -22265.9 q^{58} +18299.4 q^{59} -26509.8 q^{61} +19753.7 q^{62} -51695.0 q^{64} -1081.71 q^{65} -26313.7 q^{67} +23134.9 q^{68} -53837.8 q^{70} -39459.2 q^{71} -37484.5 q^{73} -33861.6 q^{74} -138416. q^{76} -80721.9 q^{77} +37368.8 q^{79} +6124.22 q^{80} +150206. q^{82} -76901.5 q^{83} +16065.3 q^{85} -60309.1 q^{86} +106306. q^{88} -62991.4 q^{89} -4467.60 q^{91} +28561.4 q^{92} -192068. q^{94} -96118.9 q^{95} +61484.7 q^{97} -66503.7 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - 8 q^{2} + 118 q^{4} - 94 q^{5} + 272 q^{7} - 258 q^{8}+O(q^{10}) \) 5 * q - 8 * q^2 + 118 * q^4 - 94 * q^5 + 272 * q^7 - 258 * q^8	\( 5 q - 8 q^{2} + 118 q^{4} - 94 q^{5} + 272 q^{7} - 258 q^{8} - 172 q^{10} - 1100 q^{11} - 978 q^{13} + 344 q^{14} + 1218 q^{16} - 2522 q^{17} + 2060 q^{19} - 7720 q^{20} - 2572 q^{22} + 2645 q^{23} + 12035 q^{25} - 9280 q^{26} + 8072 q^{28} - 1526 q^{29} - 7392 q^{31} + 5086 q^{32} - 15608 q^{34} - 6056 q^{35} - 8210 q^{37} + 14276 q^{38} - 37472 q^{40} - 21250 q^{41} - 4548 q^{43} + 4260 q^{44} - 4232 q^{46} - 536 q^{47} - 27979 q^{49} + 81872 q^{50} - 76380 q^{52} + 11482 q^{53} - 77064 q^{55} + 28624 q^{56} - 79680 q^{58} - 74676 q^{59} - 44618 q^{61} - 64880 q^{62} - 137382 q^{64} + 24388 q^{65} - 1412 q^{67} - 80196 q^{68} - 222304 q^{70} - 37912 q^{71} + 46546 q^{73} - 111604 q^{74} - 79548 q^{76} - 157008 q^{77} + 50544 q^{79} - 69424 q^{80} - 233720 q^{82} - 89588 q^{83} + 147892 q^{85} - 77428 q^{86} + 54484 q^{88} - 280410 q^{89} - 27416 q^{91} + 62422 q^{92} + 113632 q^{94} - 203120 q^{95} + 90074 q^{97} - 32976 q^{98}+O(q^{100}) \) 5 * q - 8 * q^2 + 118 * q^4 - 94 * q^5 + 272 * q^7 - 258 * q^8 - 172 * q^10 - 1100 * q^11 - 978 * q^13 + 344 * q^14 + 1218 * q^16 - 2522 * q^17 + 2060 * q^19 - 7720 * q^20 - 2572 * q^22 + 2645 * q^23 + 12035 * q^25 - 9280 * q^26 + 8072 * q^28 - 1526 * q^29 - 7392 * q^31 + 5086 * q^32 - 15608 * q^34 - 6056 * q^35 - 8210 * q^37 + 14276 * q^38 - 37472 * q^40 - 21250 * q^41 - 4548 * q^43 + 4260 * q^44 - 4232 * q^46 - 536 * q^47 - 27979 * q^49 + 81872 * q^50 - 76380 * q^52 + 11482 * q^53 - 77064 * q^55 + 28624 * q^56 - 79680 * q^58 - 74676 * q^59 - 44618 * q^61 - 64880 * q^62 - 137382 * q^64 + 24388 * q^65 - 1412 * q^67 - 80196 * q^68 - 222304 * q^70 - 37912 * q^71 + 46546 * q^73 - 111604 * q^74 - 79548 * q^76 - 157008 * q^77 + 50544 * q^79 - 69424 * q^80 - 233720 * q^82 - 89588 * q^83 + 147892 * q^85 - 77428 * q^86 + 54484 * q^88 - 280410 * q^89 - 27416 * q^91 + 62422 * q^92 + 113632 * q^94 - 203120 * q^95 + 90074 * q^97 - 32976 * q^98

Introduction

L-functions

Modular forms

Varieties

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Database

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