LMFDB - Embedded newform 207.6.a.h.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:	$10$
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			1.2
Root			$-7.82929$ 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)

$f(q)$	$=$	$q-8.82929 q^{2} +45.9564 q^{4} -87.3727 q^{5} -52.6709 q^{7} -123.225 q^{8} +O(q^{10})$	\(q-8.82929 q^{2} +45.9564 q^{4} -87.3727 q^{5} -52.6709 q^{7} -123.225 q^{8} +771.439 q^{10} -273.076 q^{11} +908.964 q^{13} +465.046 q^{14} -382.616 q^{16} +501.866 q^{17} -136.500 q^{19} -4015.33 q^{20} +2411.07 q^{22} -529.000 q^{23} +4508.99 q^{25} -8025.51 q^{26} -2420.56 q^{28} +1702.41 q^{29} +8489.64 q^{31} +7321.42 q^{32} -4431.12 q^{34} +4602.00 q^{35} -1225.28 q^{37} +1205.20 q^{38} +10766.5 q^{40} +6331.15 q^{41} +5249.14 q^{43} -12549.6 q^{44} +4670.69 q^{46} +18623.6 q^{47} -14032.8 q^{49} -39811.2 q^{50} +41772.7 q^{52} -38489.0 q^{53} +23859.4 q^{55} +6490.36 q^{56} -15031.1 q^{58} +15953.5 q^{59} -13225.0 q^{61} -74957.5 q^{62} -52399.3 q^{64} -79418.7 q^{65} -29127.2 q^{67} +23063.9 q^{68} -40632.4 q^{70} -61870.1 q^{71} +30941.0 q^{73} +10818.4 q^{74} -6273.06 q^{76} +14383.1 q^{77} -88782.4 q^{79} +33430.2 q^{80} -55899.6 q^{82} +16555.4 q^{83} -43849.4 q^{85} -46346.2 q^{86} +33649.7 q^{88} -60386.4 q^{89} -47875.9 q^{91} -24310.9 q^{92} -164433. q^{94} +11926.4 q^{95} -156203. q^{97} +123899. q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−8.82929	−1.56081	−0.780406	−	0.625273i	$-0.784988\pi$
$2$	−8.82929	−1.56081	−0.780406	+	0.625273i	$0.784988\pi$
$3$	0	0
$3$	0	0
$4$	45.9564	1.43614
$5$	−87.3727	−1.56297	−0.781485	−	0.623924i	$-0.785538\pi$
$5$	−87.3727	−1.56297	−0.781485	+	0.623924i	$0.785538\pi$
$6$	0	0
$7$	−52.6709	−0.406280	−0.203140	−	0.979150i	$-0.565115\pi$
$7$	−52.6709	−0.406280	−0.203140	+	0.979150i	$0.565115\pi$
$8$	−123.225	−0.680728
$9$	0	0
$10$	771.439	2.43950
$11$	−273.076	−0.680459	−0.340229	−	0.940343i	$-0.610505\pi$
$11$	−273.076	−0.680459	−0.340229	+	0.940343i	$0.610505\pi$
$12$	0	0
$13$	908.964	1.49172	0.745862	−	0.666101i	$-0.232038\pi$
$13$	908.964	1.49172	0.745862	+	0.666101i	$0.232038\pi$
$14$	465.046	0.634127
$15$	0	0
$16$	−382.616	−0.373648
$17$	501.866	0.421178	0.210589	−	0.977575i	$-0.432462\pi$
$17$	501.866	0.421178	0.210589	+	0.977575i	$0.432462\pi$
$18$	0	0
$19$	−136.500	−0.0867460	−0.0433730	−	0.999059i	$-0.513810\pi$
$19$	−136.500	−0.0867460	−0.0433730	+	0.999059i	$0.513810\pi$
$20$	−4015.33	−2.24464
$21$	0	0
$22$	2411.07	1.06207
$23$	−529.000	−0.208514
$23$	−529.000	−0.208514
$24$	0	0
$25$	4508.99	1.44288
$26$	−8025.51	−2.32830
$27$	0	0
$28$	−2420.56	−0.583473
$29$	1702.41	0.375898	0.187949	−	0.982179i	$-0.439816\pi$
$29$	1702.41	0.375898	0.187949	+	0.982179i	$0.439816\pi$
$30$	0	0
$31$	8489.64	1.58666	0.793332	−	0.608789i	$-0.208344\pi$
$31$	8489.64	1.58666	0.793332	+	0.608789i	$0.208344\pi$
$32$	7321.42	1.26392
$33$	0	0
$34$	−4431.12	−0.657380
$35$	4602.00	0.635004
$36$	0	0
$37$	−1225.28	−0.147140	−0.0735702	−	0.997290i	$-0.523439\pi$
$37$	−1225.28	−0.147140	−0.0735702	+	0.997290i	$0.523439\pi$
$38$	1205.20	0.135394
$39$	0	0
$40$	10766.5	1.06396
$41$	6331.15	0.588198	0.294099	−	0.955775i	$-0.404981\pi$
$41$	6331.15	0.588198	0.294099	+	0.955775i	$0.404981\pi$
$42$	0	0
$43$	5249.14	0.432929	0.216465	−	0.976290i	$-0.430547\pi$
$43$	5249.14	0.432929	0.216465	+	0.976290i	$0.430547\pi$
$44$	−12549.6	−0.977231
$45$	0	0
$46$	4670.69	0.325452
$47$	18623.6	1.22976	0.614878	−	0.788622i	$-0.289205\pi$
$47$	18623.6	1.22976	0.614878	+	0.788622i	$0.289205\pi$
$48$	0	0
$49$	−14032.8	−0.834937
$50$	−39811.2	−2.25206
$51$	0	0
$52$	41772.7	2.14232
$53$	−38489.0	−1.88212	−0.941059	−	0.338242i	$-0.890168\pi$
$53$	−38489.0	−1.88212	−0.941059	+	0.338242i	$0.890168\pi$
$54$	0	0
$55$	23859.4	1.06354
$56$	6490.36	0.276566
$57$	0	0
$58$	−15031.1	−0.586707
$59$	15953.5	0.596659	0.298330	−	0.954463i	$-0.403571\pi$
$59$	15953.5	0.596659	0.298330	+	0.954463i	$0.403571\pi$
$60$	0	0
$61$	−13225.0	−0.455063	−0.227531	−	0.973771i	$-0.573065\pi$
$61$	−13225.0	−0.455063	−0.227531	+	0.973771i	$0.573065\pi$
$62$	−74957.5	−2.47649
$63$	0	0
$64$	−52399.3	−1.59910
$65$	−79418.7	−2.33152
$66$	0	0
$67$	−29127.2	−0.792704	−0.396352	−	0.918099i	$-0.629724\pi$
$67$	−29127.2	−0.792704	−0.396352	+	0.918099i	$0.629724\pi$
$68$	23063.9	0.604869
$69$	0	0
$70$	−40632.4	−0.991122
$71$	−61870.1	−1.45658	−0.728291	−	0.685268i	$-0.759685\pi$
$71$	−61870.1	−1.45658	−0.728291	+	0.685268i	$0.759685\pi$
$72$	0	0
$73$	30941.0	0.679559	0.339779	−	0.940505i	$-0.389648\pi$
$73$	30941.0	0.679559	0.339779	+	0.940505i	$0.389648\pi$
$74$	10818.4	0.229659
$75$	0	0
$76$	−6273.06	−0.124579
$77$	14383.1	0.276457
$78$	0	0
$79$	−88782.4	−1.60051	−0.800256	−	0.599658i	$-0.795303\pi$
$79$	−88782.4	−1.60051	−0.800256	+	0.599658i	$0.795303\pi$
$80$	33430.2	0.584001
$81$	0	0
$82$	−55899.6	−0.918066
$83$	16555.4	0.263782	0.131891	−	0.991264i	$-0.457895\pi$
$83$	16555.4	0.263782	0.131891	+	0.991264i	$0.457895\pi$
$84$	0	0
$85$	−43849.4	−0.658289
$86$	−46346.2	−0.675722
$87$	0	0
$88$	33649.7	0.463207
$89$	−60386.4	−0.808098	−0.404049	−	0.914737i	$-0.632398\pi$
$89$	−60386.4	−0.808098	−0.404049	+	0.914737i	$0.632398\pi$
$90$	0	0
$91$	−47875.9	−0.606057
$92$	−24310.9	−0.299455
$93$	0	0
$94$	−164433.	−1.91942
$95$	11926.4	0.135581
$96$	0	0
$97$	−156203.	−1.68562	−0.842812	−	0.538207i	$-0.819102\pi$
$97$	−156203.	−1.68562	−0.842812	+	0.538207i	$0.819102\pi$
$98$	123899.	1.30318
$99$	0	0
$100$	207217.	2.07217
$101$	65965.8	0.643450	0.321725	−	0.946833i	$-0.395737\pi$
$101$	65965.8	0.643450	0.321725	+	0.946833i	$0.395737\pi$
$102$	0	0
$103$	−61735.2	−0.573376	−0.286688	−	0.958024i	$-0.592554\pi$
$103$	−61735.2	−0.573376	−0.286688	+	0.958024i	$0.592554\pi$
$104$	−112007.	−1.01546
$105$	0	0
$106$	339831.	2.93763
$107$	106350.	0.898001	0.449000	−	0.893532i	$-0.351780\pi$
$107$	106350.	0.898001	0.449000	+	0.893532i	$0.351780\pi$
$108$	0	0
$109$	133852.	1.07909	0.539547	−	0.841955i	$-0.318595\pi$
$109$	133852.	1.07909	0.539547	+	0.841955i	$0.318595\pi$
$110$	−210661.	−1.65998
$111$	0	0
$112$	20152.7	0.151806
$113$	181151.	1.33458	0.667289	−	0.744799i	$-0.267455\pi$
$113$	181151.	1.33458	0.667289	+	0.744799i	$0.267455\pi$
$114$	0	0
$115$	46220.2	0.325902
$116$	78236.8	0.539841
$117$	0	0
$118$	−140858.	−0.931273
$119$	−26433.7	−0.171116
$120$	0	0
$121$	−86480.6	−0.536976
$122$	116767.	0.710267
$123$	0	0
$124$	390153.	2.27867
$125$	−120923.	−0.692203
$126$	0	0
$127$	−164435.	−0.904657	−0.452329	−	0.891851i	$-0.649407\pi$
$127$	−164435.	−0.904657	−0.452329	+	0.891851i	$0.649407\pi$
$128$	228363.	1.23197
$129$	0	0
$130$	701210.	3.63907
$131$	299017.	1.52236	0.761180	−	0.648541i	$-0.224621\pi$
$131$	299017.	1.52236	0.761180	+	0.648541i	$0.224621\pi$
$132$	0	0
$133$	7189.59	0.0352432
$134$	257172.	1.23726
$135$	0	0
$136$	−61842.4	−0.286707
$137$	−256175.	−1.16610	−0.583049	−	0.812437i	$-0.698140\pi$
$137$	−256175.	−1.16610	−0.583049	+	0.812437i	$0.698140\pi$
$138$	0	0
$139$	−168988.	−0.741853	−0.370926	−	0.928662i	$-0.620960\pi$
$139$	−168988.	−0.741853	−0.370926	+	0.928662i	$0.620960\pi$
$140$	211491.	0.911952
$141$	0	0
$142$	546269.	2.27345
$143$	−248216.	−1.01506
$144$	0	0
$145$	−148745.	−0.587518
$146$	−273187.	−1.06066
$147$	0	0
$148$	−56309.6	−0.211314
$149$	−141284.	−0.521348	−0.260674	−	0.965427i	$-0.583945\pi$
$149$	−141284.	−0.521348	−0.260674	+	0.965427i	$0.583945\pi$
$150$	0	0
$151$	−80667.0	−0.287908	−0.143954	−	0.989584i	$-0.545982\pi$
$151$	−80667.0	−0.287908	−0.143954	+	0.989584i	$0.545982\pi$
$152$	16820.2	0.0590504
$153$	0	0
$154$	−126993.	−0.431497
$155$	−741763.	−2.47991
$156$	0	0
$157$	−39359.6	−0.127439	−0.0637194	−	0.997968i	$-0.520296\pi$
$157$	−39359.6	−0.127439	−0.0637194	+	0.997968i	$0.520296\pi$
$158$	783886.	2.49810
$159$	0	0
$160$	−639692.	−1.97547
$161$	27862.9	0.0847152
$162$	0	0
$163$	−669985.	−1.97513	−0.987566	−	0.157205i	$-0.949752\pi$
$163$	−669985.	−1.97513	−0.987566	+	0.157205i	$0.949752\pi$
$164$	290957.	0.844732
$165$	0	0
$166$	−146173.	−0.411715
$167$	−41710.2	−0.115732	−0.0578658	−	0.998324i	$-0.518430\pi$
$167$	−41710.2	−0.115732	−0.0578658	+	0.998324i	$0.518430\pi$
$168$	0	0
$169$	454923.	1.22524
$170$	387159.	1.02747
$171$	0	0
$172$	241231.	0.621746
$173$	222570.	0.565394	0.282697	−	0.959209i	$-0.408771\pi$
$173$	222570.	0.565394	0.282697	+	0.959209i	$0.408771\pi$
$174$	0	0
$175$	−237492.	−0.586212
$176$	104483.	0.254252
$177$	0	0
$178$	533169.	1.26129
$179$	115807.	0.270148	0.135074	−	0.990836i	$-0.456873\pi$
$179$	115807.	0.270148	0.135074	+	0.990836i	$0.456873\pi$
$180$	0	0
$181$	646392.	1.46656	0.733279	−	0.679928i	$-0.237989\pi$
$181$	646392.	1.46656	0.733279	+	0.679928i	$0.237989\pi$
$182$	422710.	0.945942
$183$	0	0
$184$	65186.0	0.141942
$185$	107056.	0.229976
$186$	0	0
$187$	−137048.	−0.286594
$188$	855874.	1.76610
$189$	0	0
$190$	−105302.	−0.211617
$191$	445860.	0.884332	0.442166	−	0.896933i	$-0.354210\pi$
$191$	445860.	0.884332	0.442166	+	0.896933i	$0.354210\pi$
$192$	0	0
$193$	446982.	0.863768	0.431884	−	0.901929i	$-0.357849\pi$
$193$	446982.	0.863768	0.431884	+	0.901929i	$0.357849\pi$
$194$	1.37916e6	2.63094
$195$	0	0
$196$	−644896.	−1.19908
$197$	−883473.	−1.62191	−0.810957	−	0.585106i	$-0.801053\pi$
$197$	−883473.	−1.62191	−0.810957	+	0.585106i	$0.801053\pi$
$198$	0	0
$199$	−431473.	−0.772361	−0.386181	−	0.922423i	$-0.626206\pi$
$199$	−431473.	−0.772361	−0.386181	+	0.922423i	$0.626206\pi$
$200$	−555620.	−0.982206
$201$	0	0
$202$	−582431.	−1.00431
$203$	−89667.6	−0.152720
$204$	0	0
$205$	−553170.	−0.919335
$206$	545078.	0.894933
$207$	0	0
$208$	−347784.	−0.557380
$209$	37274.9	0.0590271
$210$	0	0
$211$	1.07446e6	1.66144	0.830720	−	0.556690i	$-0.187929\pi$
$211$	1.07446e6	1.66144	0.830720	+	0.556690i	$0.187929\pi$
$212$	−1.76881e6	−2.70298
$213$	0	0
$214$	−938992.	−1.40161
$215$	−458632.	−0.676656
$216$	0	0
$217$	−447157.	−0.644630
$218$	−1.18182e6	−1.68427
$219$	0	0
$220$	1.09649e6	1.52738
$221$	456178.	0.628281
$222$	0	0
$223$	−178733.	−0.240681	−0.120341	−	0.992733i	$-0.538399\pi$
$223$	−178733.	−0.240681	−0.120341	+	0.992733i	$0.538399\pi$
$224$	−385626.	−0.513506
$225$	0	0
$226$	−1.59943e6	−2.08303
$227$	590716.	0.760876	0.380438	−	0.924806i	$-0.375773\pi$
$227$	590716.	0.760876	0.380438	+	0.924806i	$0.375773\pi$
$228$	0	0
$229$	264066.	0.332754	0.166377	−	0.986062i	$-0.446793\pi$
$229$	264066.	0.332754	0.166377	+	0.986062i	$0.446793\pi$
$230$	−408091.	−0.508672
$231$	0	0
$232$	−209780.	−0.255884
$233$	−1.36878e6	−1.65174	−0.825872	−	0.563858i	$-0.809317\pi$
$233$	−1.36878e6	−1.65174	−0.825872	+	0.563858i	$0.809317\pi$
$234$	0	0
$235$	−1.62720e6	−1.92207
$236$	733166.	0.856884
$237$	0	0
$238$	233391.	0.267080
$239$	−109767.	−0.124302	−0.0621509	−	0.998067i	$-0.519796\pi$
$239$	−109767.	−0.124302	−0.0621509	+	0.998067i	$0.519796\pi$
$240$	0	0
$241$	851594.	0.944474	0.472237	−	0.881472i	$-0.343447\pi$
$241$	851594.	0.944474	0.472237	+	0.881472i	$0.343447\pi$
$242$	763562.	0.838119
$243$	0	0
$244$	−607773.	−0.653532
$245$	1.22608e6	1.30498
$246$	0	0
$247$	−124074.	−0.129401
$248$	−1.04613e6	−1.08009
$249$	0	0
$250$	1.06766e6	1.08040
$251$	−906290.	−0.907994	−0.453997	−	0.891003i	$-0.650002\pi$
$251$	−906290.	−0.907994	−0.453997	+	0.891003i	$0.650002\pi$
$252$	0	0
$253$	144457.	0.141885
$254$	1.45184e6	1.41200
$255$	0	0
$256$	−339505.	−0.323777
$257$	−462699.	−0.436984	−0.218492	−	0.975839i	$-0.570114\pi$
$257$	−462699.	−0.436984	−0.218492	+	0.975839i	$0.570114\pi$
$258$	0	0
$259$	64536.7	0.0597802
$260$	−3.64979e6	−3.34838
$261$	0	0
$262$	−2.64010e6	−2.37612
$263$	−1.34981e6	−1.20333	−0.601664	−	0.798749i	$-0.705495\pi$
$263$	−1.34981e6	−1.20333	−0.601664	+	0.798749i	$0.705495\pi$
$264$	0	0
$265$	3.36289e6	2.94170
$266$	−63479.0	−0.0550080
$267$	0	0
$268$	−1.33858e6	−1.13843
$269$	−1.90332e6	−1.60373	−0.801866	−	0.597504i	$-0.796159\pi$
$269$	−1.90332e6	−1.60373	−0.801866	+	0.597504i	$0.796159\pi$
$270$	0	0
$271$	1.60349e6	1.32630	0.663151	−	0.748485i	$-0.269219\pi$
$271$	1.60349e6	1.32630	0.663151	+	0.748485i	$0.269219\pi$
$272$	−192022.	−0.157372
$273$	0	0
$274$	2.26184e6	1.82006
$275$	−1.23130e6	−0.981818
$276$	0	0
$277$	2.08579e6	1.63332	0.816659	−	0.577121i	$-0.195824\pi$
$277$	2.08579e6	1.63332	0.816659	+	0.577121i	$0.195824\pi$
$278$	1.49204e6	1.15789
$279$	0	0
$280$	−567080.	−0.432264
$281$	2.06095e6	1.55705	0.778523	−	0.627616i	$-0.215969\pi$
$281$	2.06095e6	1.55705	0.778523	+	0.627616i	$0.215969\pi$
$282$	0	0
$283$	577290.	0.428477	0.214239	−	0.976781i	$-0.431273\pi$
$283$	577290.	0.428477	0.214239	+	0.976781i	$0.431273\pi$
$284$	−2.84332e6	−2.09185
$285$	0	0
$286$	2.19157e6	1.58431
$287$	−333467.	−0.238973
$288$	0	0
$289$	−1.16799e6	−0.822609
$290$	1.31331e6	0.917005
$291$	0	0
$292$	1.42194e6	0.975939
$293$	−1.57599e6	−1.07247	−0.536235	−	0.844069i	$-0.680154\pi$
$293$	−1.57599e6	−1.07247	−0.536235	+	0.844069i	$0.680154\pi$
$294$	0	0
$295$	−1.39390e6	−0.932561
$296$	150985.	0.100163
$297$	0	0
$298$	1.24744e6	0.813726
$299$	−480842.	−0.311046
$300$	0	0
$301$	−276477.	−0.175890
$302$	712233.	0.449370
$303$	0	0
$304$	52227.2	0.0324125
$305$	1.15550e6	0.711249
$306$	0	0
$307$	1.81058e6	1.09641	0.548204	−	0.836345i	$-0.315312\pi$
$307$	1.81058e6	1.09641	0.548204	+	0.836345i	$0.315312\pi$
$308$	660997.	0.397029
$309$	0	0
$310$	6.54924e6	3.87067
$311$	1.22366e6	0.717396	0.358698	−	0.933454i	$-0.383221\pi$
$311$	1.22366e6	0.717396	0.358698	+	0.933454i	$0.383221\pi$
$312$	0	0
$313$	−2.43795e6	−1.40658	−0.703288	−	0.710905i	$-0.748286\pi$
$313$	−2.43795e6	−1.40658	−0.703288	+	0.710905i	$0.748286\pi$
$314$	347517.	0.198908
$315$	0	0
$316$	−4.08012e6	−2.29856
$317$	−2.57574e6	−1.43964	−0.719819	−	0.694162i	$-0.755775\pi$
$317$	−2.57574e6	−1.43964	−0.719819	+	0.694162i	$0.755775\pi$
$318$	0	0
$319$	−464888.	−0.255783
$320$	4.57826e6	2.49934
$321$	0	0
$322$	−246010.	−0.132225
$323$	−68504.9	−0.0365355
$324$	0	0
$325$	4.09851e6	2.15237
$326$	5.91549e6	3.08281
$327$	0	0
$328$	−780156.	−0.400402
$329$	−980922.	−0.499625
$330$	0	0
$331$	−2.00077e6	−1.00375	−0.501877	−	0.864939i	$-0.667357\pi$
$331$	−2.00077e6	−1.00375	−0.501877	+	0.864939i	$0.667357\pi$
$332$	760828.	0.378827
$333$	0	0
$334$	368272.	0.180635
$335$	2.54492e6	1.23897
$336$	0	0
$337$	−3.32602e6	−1.59533	−0.797665	−	0.603100i	$-0.793932\pi$
$337$	−3.32602e6	−1.59533	−0.797665	+	0.603100i	$0.793932\pi$
$338$	−4.01665e6	−1.91237
$339$	0	0
$340$	−2.01516e6	−0.945392
$341$	−2.31832e6	−1.07966
$342$	0	0
$343$	1.62436e6	0.745498
$344$	−646825.	−0.294707
$345$	0	0
$346$	−1.96513e6	−0.882474
$347$	−2.52993e6	−1.12794	−0.563969	−	0.825796i	$-0.690726\pi$
$347$	−2.52993e6	−1.12794	−0.563969	+	0.825796i	$0.690726\pi$
$348$	0	0
$349$	2.06149e6	0.905977	0.452989	−	0.891516i	$-0.350358\pi$
$349$	2.06149e6	0.905977	0.452989	+	0.891516i	$0.350358\pi$
$350$	2.09689e6	0.914967
$351$	0	0
$352$	−1.99930e6	−0.860047
$353$	366431.	0.156515	0.0782573	−	0.996933i	$-0.475064\pi$
$353$	366431.	0.156515	0.0782573	+	0.996933i	$0.475064\pi$
$354$	0	0
$355$	5.40576e6	2.27659
$356$	−2.77514e6	−1.16054
$357$	0	0
$358$	−1.02249e6	−0.421651
$359$	−528866.	−0.216576	−0.108288	−	0.994120i	$-0.534537\pi$
$359$	−528866.	−0.216576	−0.108288	+	0.994120i	$0.534537\pi$
$360$	0	0
$361$	−2.45747e6	−0.992475
$362$	−5.70718e6	−2.28902
$363$	0	0
$364$	−2.20020e6	−0.870381
$365$	−2.70340e6	−1.06213
$366$	0	0
$367$	−4.99920e6	−1.93747	−0.968737	−	0.248092i	$-0.920197\pi$
$367$	−4.99920e6	−1.93747	−0.968737	+	0.248092i	$0.920197\pi$
$368$	202404.	0.0779110
$369$	0	0
$370$	−945231.	−0.358950
$371$	2.02725e6	0.764667
$372$	0	0
$373$	−1.55818e6	−0.579891	−0.289946	−	0.957043i	$-0.593637\pi$
$373$	−1.55818e6	−0.579891	−0.289946	+	0.957043i	$0.593637\pi$
$374$	1.21003e6	0.447320
$375$	0	0
$376$	−2.29489e6	−0.837129
$377$	1.54743e6	0.560736
$378$	0	0
$379$	−1.29305e6	−0.462400	−0.231200	−	0.972906i	$-0.574265\pi$
$379$	−1.29305e6	−0.462400	−0.231200	+	0.972906i	$0.574265\pi$
$380$	548094.	0.194714
$381$	0	0
$382$	−3.93663e6	−1.38028
$383$	1.17169e6	0.408145	0.204072	−	0.978956i	$-0.434582\pi$
$383$	1.17169e6	0.408145	0.204072	+	0.978956i	$0.434582\pi$
$384$	0	0
$385$	−1.25669e6	−0.432094
$386$	−3.94654e6	−1.34818
$387$	0	0
$388$	−7.17854e6	−2.42079
$389$	−2.57762e6	−0.863665	−0.431833	−	0.901954i	$-0.642133\pi$
$389$	−2.57762e6	−0.863665	−0.431833	+	0.901954i	$0.642133\pi$
$390$	0	0
$391$	−265487.	−0.0878217
$392$	1.72919e6	0.568364
$393$	0	0
$394$	7.80044e6	2.53150
$395$	7.75716e6	2.50155
$396$	0	0
$397$	−1.54677e6	−0.492547	−0.246274	−	0.969200i	$-0.579206\pi$
$397$	−1.54677e6	−0.492547	−0.246274	+	0.969200i	$0.579206\pi$
$398$	3.80960e6	1.20551
$399$	0	0
$400$	−1.72521e6	−0.539128
$401$	4.58300e6	1.42328	0.711638	−	0.702546i	$-0.247954\pi$
$401$	4.58300e6	1.42328	0.711638	+	0.702546i	$0.247954\pi$
$402$	0	0
$403$	7.71678e6	2.36686
$404$	3.03155e6	0.924083
$405$	0	0
$406$	791702.	0.238367
$407$	334595.	0.100123
$408$	0	0
$409$	3.57554e6	1.05690	0.528449	−	0.848965i	$-0.322774\pi$
$409$	3.57554e6	1.05690	0.528449	+	0.848965i	$0.322774\pi$
$410$	4.88410e6	1.43491
$411$	0	0
$412$	−2.83712e6	−0.823446
$413$	−840285.	−0.242411
$414$	0	0
$415$	−1.44649e6	−0.412284
$416$	6.65491e6	1.88542
$417$	0	0
$418$	−329111.	−0.0921302
$419$	−1.90507e6	−0.530123	−0.265062	−	0.964231i	$-0.585392\pi$
$419$	−1.90507e6	−0.530123	−0.265062	+	0.964231i	$0.585392\pi$
$420$	0	0
$421$	−3.79685e6	−1.04404	−0.522021	−	0.852932i	$-0.674822\pi$
$421$	−3.79685e6	−1.04404	−0.522021	+	0.852932i	$0.674822\pi$
$422$	−9.48673e6	−2.59320
$423$	0	0
$424$	4.74280e6	1.28121
$425$	2.26291e6	0.607708
$426$	0	0
$427$	696572.	0.184883
$428$	4.88745e6	1.28965
$429$	0	0
$430$	4.04939e6	1.05613
$431$	4.30617e6	1.11660	0.558301	−	0.829639i	$-0.311454\pi$
$431$	4.30617e6	1.11660	0.558301	+	0.829639i	$0.311454\pi$
$432$	0	0
$433$	−6.07379e6	−1.55682	−0.778412	−	0.627753i	$-0.783975\pi$
$433$	−6.07379e6	−1.55682	−0.778412	+	0.627753i	$0.783975\pi$
$434$	3.94808e6	1.00615
$435$	0	0
$436$	6.15137e6	1.54973
$437$	72208.6	0.0180878
$438$	0	0
$439$	352319.	0.0872520	0.0436260	−	0.999048i	$-0.486109\pi$
$439$	352319.	0.0872520	0.0436260	+	0.999048i	$0.486109\pi$
$440$	−2.94007e6	−0.723979
$441$	0	0
$442$	−4.02773e6	−0.980629
$443$	−7.10541e6	−1.72020	−0.860101	−	0.510123i	$-0.829600\pi$
$443$	−7.10541e6	−1.72020	−0.860101	+	0.510123i	$0.829600\pi$
$444$	0	0
$445$	5.27612e6	1.26303
$446$	1.57808e6	0.375658
$447$	0	0
$448$	2.75991e6	0.649682
$449$	−3.92088e6	−0.917842	−0.458921	−	0.888477i	$-0.651764\pi$
$449$	−3.92088e6	−0.917842	−0.458921	+	0.888477i	$0.651764\pi$
$450$	0	0
$451$	−1.72889e6	−0.400244
$452$	8.32503e6	1.91664
$453$	0	0
$454$	−5.21560e6	−1.18759
$455$	4.18305e6	0.947250
$456$	0	0
$457$	−650772.	−0.145760	−0.0728800	−	0.997341i	$-0.523219\pi$
$457$	−650772.	−0.145760	−0.0728800	+	0.997341i	$0.523219\pi$
$458$	−2.33151e6	−0.519366
$459$	0	0
$460$	2.12411e6	0.468040
$461$	−3.92665e6	−0.860539	−0.430269	−	0.902701i	$-0.641581\pi$
$461$	−3.92665e6	−0.860539	−0.430269	+	0.902701i	$0.641581\pi$
$462$	0	0
$463$	6.38938e6	1.38518	0.692590	−	0.721332i	$-0.256470\pi$
$463$	6.38938e6	1.38518	0.692590	+	0.721332i	$0.256470\pi$
$464$	−651371.	−0.140454
$465$	0	0
$466$	1.20853e7	2.57806
$467$	−3.06502e6	−0.650341	−0.325171	−	0.945655i	$-0.605422\pi$
$467$	−3.06502e6	−0.650341	−0.325171	+	0.945655i	$0.605422\pi$
$468$	0	0
$469$	1.53415e6	0.322060
$470$	1.43670e7	3.00000
$471$	0	0
$472$	−1.96587e6	−0.406162
$473$	−1.43341e6	−0.294590
$474$	0	0
$475$	−615478.	−0.125164
$476$	−1.21480e6	−0.245746
$477$	0	0
$478$	969166.	0.194012
$479$	−2.44104e6	−0.486112	−0.243056	−	0.970012i	$-0.578150\pi$
$479$	−2.44104e6	−0.486112	−0.243056	+	0.970012i	$0.578150\pi$
$480$	0	0
$481$	−1.11374e6	−0.219493
$482$	−7.51897e6	−1.47415
$483$	0	0
$484$	−3.97433e6	−0.771171
$485$	1.36479e7	2.63458
$486$	0	0
$487$	8.62602e6	1.64812	0.824059	−	0.566504i	$-0.191704\pi$
$487$	8.62602e6	1.64812	0.824059	+	0.566504i	$0.191704\pi$
$488$	1.62965e6	0.309774
$489$	0	0
$490$	−1.08254e7	−2.03683
$491$	−7.57600e6	−1.41820	−0.709098	−	0.705110i	$-0.750898\pi$
$491$	−7.57600e6	−1.41820	−0.709098	+	0.705110i	$0.750898\pi$
$492$	0	0
$493$	854384.	0.158320
$494$	1.09548e6	0.201971
$495$	0	0
$496$	−3.24827e6	−0.592854
$497$	3.25875e6	0.591780
$498$	0	0
$499$	−4.34014e6	−0.780284	−0.390142	−	0.920755i	$-0.627574\pi$
$499$	−4.34014e6	−0.780284	−0.390142	+	0.920755i	$0.627574\pi$
$500$	−5.55718e6	−0.994099
$501$	0	0
$502$	8.00190e6	1.41721
$503$	−6.41560e6	−1.13062	−0.565311	−	0.824878i	$-0.691244\pi$
$503$	−6.41560e6	−1.13062	−0.565311	+	0.824878i	$0.691244\pi$
$504$	0	0
$505$	−5.76361e6	−1.00569
$506$	−1.27545e6	−0.221457
$507$	0	0
$508$	−7.55682e6	−1.29921
$509$	−4.84397e6	−0.828718	−0.414359	−	0.910113i	$-0.635994\pi$
$509$	−4.84397e6	−0.828718	−0.414359	+	0.910113i	$0.635994\pi$
$510$	0	0
$511$	−1.62969e6	−0.276091
$512$	−4.31002e6	−0.726615
$513$	0	0
$514$	4.08531e6	0.682051
$515$	5.39397e6	0.896170
$516$	0	0
$517$	−5.08566e6	−0.836799
$518$	−569814.	−0.0933057
$519$	0	0
$520$	9.78635e6	1.58713
$521$	2.22762e6	0.359540	0.179770	−	0.983709i	$-0.442465\pi$
$521$	2.22762e6	0.359540	0.179770	+	0.983709i	$0.442465\pi$
$522$	0	0
$523$	1.08981e7	1.74219	0.871094	−	0.491116i	$-0.163411\pi$
$523$	1.08981e7	1.74219	0.871094	+	0.491116i	$0.163411\pi$
$524$	1.37417e7	2.18632
$525$	0	0
$526$	1.19179e7	1.87817
$527$	4.26066e6	0.668268
$528$	0	0
$529$	279841.	0.0434783
$530$	−2.96919e7	−4.59144
$531$	0	0
$532$	330407.	0.0506140
$533$	5.75479e6	0.877428
$534$	0	0
$535$	−9.29206e6	−1.40355
$536$	3.58919e6	0.539616
$537$	0	0
$538$	1.68050e7	2.50313
$539$	3.83201e6	0.568140
$540$	0	0
$541$	1.70357e6	0.250246	0.125123	−	0.992141i	$-0.460067\pi$
$541$	1.70357e6	0.250246	0.125123	+	0.992141i	$0.460067\pi$
$542$	−1.41577e7	−2.07011
$543$	0	0
$544$	3.67437e6	0.532336
$545$	−1.16950e7	−1.68659
$546$	0	0
$547$	−1.29001e7	−1.84342	−0.921712	−	0.387876i	$-0.873209\pi$
$547$	−1.29001e7	−1.84342	−0.921712	+	0.387876i	$0.873209\pi$
$548$	−1.17729e7	−1.67468
$549$	0	0
$550$	1.08715e7	1.53243
$551$	−232380.	−0.0326077
$552$	0	0
$553$	4.67625e6	0.650256
$554$	−1.84160e7	−2.54930
$555$	0	0
$556$	−7.76606e6	−1.06540
$557$	−1.60076e6	−0.218619	−0.109310	−	0.994008i	$-0.534864\pi$
$557$	−1.60076e6	−0.218619	−0.109310	+	0.994008i	$0.534864\pi$
$558$	0	0
$559$	4.77128e6	0.645811
$560$	−1.76080e6	−0.237268
$561$	0	0
$562$	−1.81967e7	−2.43026
$563$	1.28586e7	1.70971	0.854853	−	0.518870i	$-0.173647\pi$
$563$	1.28586e7	1.70971	0.854853	+	0.518870i	$0.173647\pi$
$564$	0	0
$565$	−1.58276e7	−2.08591
$566$	−5.09706e6	−0.668773
$567$	0	0
$568$	7.62393e6	0.991535
$569$	3.23968e6	0.419490	0.209745	−	0.977756i	$-0.432737\pi$
$569$	3.23968e6	0.419490	0.209745	+	0.977756i	$0.432737\pi$
$570$	0	0
$571$	−4.67802e6	−0.600443	−0.300222	−	0.953869i	$-0.597061\pi$
$571$	−4.67802e6	−0.600443	−0.300222	+	0.953869i	$0.597061\pi$
$572$	−1.14071e7	−1.45776
$573$	0	0
$574$	2.94428e6	0.372992
$575$	−2.38526e6	−0.300861
$576$	0	0
$577$	−5.46173e6	−0.682953	−0.341476	−	0.939890i	$-0.610927\pi$
$577$	−5.46173e6	−0.682953	−0.341476	+	0.939890i	$0.610927\pi$
$578$	1.03125e7	1.28394
$579$	0	0
$580$	−6.83576e6	−0.843756
$581$	−871989.	−0.107169
$582$	0	0
$583$	1.05104e7	1.28070
$584$	−3.81270e6	−0.462594
$585$	0	0
$586$	1.39149e7	1.67392
$587$	−1.16753e6	−0.139853	−0.0699265	−	0.997552i	$-0.522276\pi$
$587$	−1.16753e6	−0.139853	−0.0699265	+	0.997552i	$0.522276\pi$
$588$	0	0
$589$	−1.15884e6	−0.137637
$590$	1.23072e7	1.45555
$591$	0	0
$592$	468813.	0.0549788
$593$	−3.68142e6	−0.429911	−0.214955	−	0.976624i	$-0.568961\pi$
$593$	−3.68142e6	−0.429911	−0.214955	+	0.976624i	$0.568961\pi$
$594$	0	0
$595$	2.30959e6	0.267449
$596$	−6.49290e6	−0.748727
$597$	0	0
$598$	4.24549e6	0.485484
$599$	8.99880e6	1.02475	0.512374	−	0.858762i	$-0.328766\pi$
$599$	8.99880e6	1.02475	0.512374	+	0.858762i	$0.328766\pi$
$600$	0	0
$601$	−1.72600e7	−1.94919	−0.974596	−	0.223972i	$-0.928098\pi$
$601$	−1.72600e7	−1.94919	−0.974596	+	0.223972i	$0.928098\pi$
$602$	2.44109e6	0.274532
$603$	0	0
$604$	−3.70716e6	−0.413475
$605$	7.55604e6	0.839278
$606$	0	0
$607$	−1.19516e6	−0.131660	−0.0658300	−	0.997831i	$-0.520970\pi$
$607$	−1.19516e6	−0.131660	−0.0658300	+	0.997831i	$0.520970\pi$
$608$	−999376.	−0.109640
$609$	0	0
$610$	−1.02023e7	−1.11013
$611$	1.69282e7	1.83446
$612$	0	0
$613$	1.08207e6	0.116306	0.0581531	−	0.998308i	$-0.481479\pi$
$613$	1.08207e6	0.116306	0.0581531	+	0.998308i	$0.481479\pi$
$614$	−1.59861e7	−1.71129
$615$	0	0
$616$	−1.77236e6	−0.188192
$617$	7.50358e6	0.793516	0.396758	−	0.917923i	$-0.370135\pi$
$617$	7.50358e6	0.793516	0.396758	+	0.917923i	$0.370135\pi$
$618$	0	0
$619$	8.05552e6	0.845019	0.422510	−	0.906358i	$-0.361149\pi$
$619$	8.05552e6	0.845019	0.422510	+	0.906358i	$0.361149\pi$
$620$	−3.40887e7	−3.56149
$621$	0	0
$622$	−1.08040e7	−1.11972
$623$	3.18060e6	0.328314
$624$	0	0
$625$	−3.52523e6	−0.360983
$626$	2.15253e7	2.19540
$627$	0	0
$628$	−1.80882e6	−0.183019
$629$	−614928.	−0.0619723
$630$	0	0
$631$	−8.40142e6	−0.840000	−0.420000	−	0.907524i	$-0.637970\pi$
$631$	−8.40142e6	−0.840000	−0.420000	+	0.907524i	$0.637970\pi$
$632$	1.09402e7	1.08951
$633$	0	0
$634$	2.27419e7	2.24701
$635$	1.43671e7	1.41395
$636$	0	0
$637$	−1.27553e7	−1.24549
$638$	4.10463e6	0.399230
$639$	0	0
$640$	−1.99527e7	−1.92553
$641$	−1.85131e7	−1.77965	−0.889823	−	0.456305i	$-0.849173\pi$
$641$	−1.85131e7	−1.77965	−0.889823	+	0.456305i	$0.849173\pi$
$642$	0	0
$643$	−1.80326e7	−1.72001	−0.860007	−	0.510283i	$-0.829541\pi$
$643$	−1.80326e7	−1.72001	−0.860007	+	0.510283i	$0.829541\pi$
$644$	1.28048e6	0.121663
$645$	0	0
$646$	604849.	0.0570251
$647$	9.56924e6	0.898704	0.449352	−	0.893355i	$-0.351655\pi$
$647$	9.56924e6	0.898704	0.449352	+	0.893355i	$0.351655\pi$
$648$	0	0
$649$	−4.35652e6	−0.406002
$650$	−3.61869e7	−3.35945
$651$	0	0
$652$	−3.07901e7	−2.83656
$653$	−4.37914e6	−0.401889	−0.200944	−	0.979603i	$-0.564401\pi$
$653$	−4.37914e6	−0.401889	−0.200944	+	0.979603i	$0.564401\pi$
$654$	0	0
$655$	−2.61259e7	−2.37940
$656$	−2.42240e6	−0.219779
$657$	0	0
$658$	8.66084e6	0.779822
$659$	1.62939e7	1.46154	0.730772	−	0.682622i	$-0.239160\pi$
$659$	1.62939e7	1.46154	0.730772	+	0.682622i	$0.239160\pi$
$660$	0	0
$661$	1.99949e7	1.77998	0.889991	−	0.455979i	$-0.150711\pi$
$661$	1.99949e7	1.77998	0.889991	+	0.455979i	$0.150711\pi$
$662$	1.76654e7	1.56667
$663$	0	0
$664$	−2.04004e6	−0.179564
$665$	−628174.	−0.0550840
$666$	0	0
$667$	−900577.	−0.0783802
$668$	−1.91685e6	−0.166206
$669$	0	0
$670$	−2.24698e7	−1.93381
$671$	3.61143e6	0.309651
$672$	0	0
$673$	−4.71166e6	−0.400992	−0.200496	−	0.979694i	$-0.564255\pi$
$673$	−4.71166e6	−0.400992	−0.200496	+	0.979694i	$0.564255\pi$
$674$	2.93664e7	2.49001
$675$	0	0
$676$	2.09066e7	1.75961
$677$	−1.66019e7	−1.39215	−0.696075	−	0.717969i	$-0.745072\pi$
$677$	−1.66019e7	−1.39215	−0.696075	+	0.717969i	$0.745072\pi$
$678$	0	0
$679$	8.22736e6	0.684836
$680$	5.40334e6	0.448115
$681$	0	0
$682$	2.04691e7	1.68515
$683$	−1.41708e7	−1.16237	−0.581183	−	0.813773i	$-0.697410\pi$
$683$	−1.41708e7	−1.16237	−0.581183	+	0.813773i	$0.697410\pi$
$684$	0	0
$685$	2.23827e7	1.82258
$686$	−1.43419e7	−1.16358
$687$	0	0
$688$	−2.00840e6	−0.161763
$689$	−3.49851e7	−2.80760
$690$	0	0
$691$	−4.95611e6	−0.394863	−0.197431	−	0.980317i	$-0.563260\pi$
$691$	−4.95611e6	−0.394863	−0.197431	+	0.980317i	$0.563260\pi$
$692$	1.02285e7	0.811983
$693$	0	0
$694$	2.23375e7	1.76050
$695$	1.47649e7	1.15949
$696$	0	0
$697$	3.17739e6	0.247736
$698$	−1.82015e7	−1.41406
$699$	0	0
$700$	−1.09143e7	−0.841880
$701$	8.69065e6	0.667971	0.333985	−	0.942578i	$-0.391606\pi$
$701$	8.69065e6	0.667971	0.333985	+	0.942578i	$0.391606\pi$
$702$	0	0
$703$	167252.	0.0127639
$704$	1.43090e7	1.08812
$705$	0	0
$706$	−3.23532e6	−0.244290
$707$	−3.47447e6	−0.261421
$708$	0	0
$709$	1.29901e7	0.970503	0.485252	−	0.874375i	$-0.338728\pi$
$709$	1.29901e7	0.970503	0.485252	+	0.874375i	$0.338728\pi$
$710$	−4.77290e7	−3.55334
$711$	0	0
$712$	7.44111e6	0.550095
$713$	−4.49102e6	−0.330842
$714$	0	0
$715$	2.16873e7	1.58650
$716$	5.32206e6	0.387970
$717$	0	0
$718$	4.66951e6	0.338034
$719$	−1.91925e7	−1.38455	−0.692275	−	0.721634i	$-0.743392\pi$
$719$	−1.91925e7	−1.38455	−0.692275	+	0.721634i	$0.743392\pi$
$720$	0	0
$721$	3.25164e6	0.232951
$722$	2.16977e7	1.54907
$723$	0	0
$724$	2.97058e7	2.10618
$725$	7.67617e6	0.542375
$726$	0	0
$727$	−5.17672e6	−0.363261	−0.181631	−	0.983367i	$-0.558138\pi$
$727$	−5.17672e6	−0.363261	−0.181631	+	0.983367i	$0.558138\pi$
$728$	5.89950e6	0.412560
$729$	0	0
$730$	2.38691e7	1.65779
$731$	2.63437e6	0.182340
$732$	0	0
$733$	1.44290e7	0.991921	0.495961	−	0.868345i	$-0.334816\pi$
$733$	1.44290e7	0.991921	0.495961	+	0.868345i	$0.334816\pi$
$734$	4.41394e7	3.02403
$735$	0	0
$736$	−3.87303e6	−0.263546
$737$	7.95393e6	0.539402
$738$	0	0
$739$	−4.68188e6	−0.315362	−0.157681	−	0.987490i	$-0.550402\pi$
$739$	−4.68188e6	−0.315362	−0.157681	+	0.987490i	$0.550402\pi$
$740$	4.91992e6	0.330277
$741$	0	0
$742$	−1.78992e7	−1.19350
$743$	1.47103e7	0.977571	0.488785	−	0.872404i	$-0.337440\pi$
$743$	1.47103e7	0.977571	0.488785	+	0.872404i	$0.337440\pi$
$744$	0	0
$745$	1.23444e7	0.814851
$746$	1.37577e7	0.905102
$747$	0	0
$748$	−6.29821e6	−0.411588
$749$	−5.60153e6	−0.364840
$750$	0	0
$751$	6.37383e6	0.412383	0.206191	−	0.978512i	$-0.433893\pi$
$751$	6.37383e6	0.412383	0.206191	+	0.978512i	$0.433893\pi$
$752$	−7.12569e6	−0.459496
$753$	0	0
$754$	−1.36627e7	−0.875204
$755$	7.04810e6	0.449992
$756$	0	0
$757$	1.74667e7	1.10782	0.553912	−	0.832575i	$-0.313134\pi$
$757$	1.74667e7	1.10782	0.553912	+	0.832575i	$0.313134\pi$
$758$	1.14167e7	0.721719
$759$	0	0
$760$	−1.46963e6	−0.0922940
$761$	−2.41165e7	−1.50957	−0.754783	−	0.655975i	$-0.772258\pi$
$761$	−2.41165e7	−1.50957	−0.754783	+	0.655975i	$0.772258\pi$
$762$	0	0
$763$	−7.05012e6	−0.438415
$764$	2.04901e7	1.27002
$765$	0	0
$766$	−1.03451e7	−0.637037
$767$	1.45012e7	0.890050
$768$	0	0
$769$	−8.31857e6	−0.507263	−0.253631	−	0.967301i	$-0.581625\pi$
$769$	−8.31857e6	−0.507263	−0.253631	+	0.967301i	$0.581625\pi$
$770$	1.10957e7	0.674417
$771$	0	0
$772$	2.05417e7	1.24049
$773$	1.37431e7	0.827249	0.413625	−	0.910447i	$-0.364263\pi$
$773$	1.37431e7	0.827249	0.413625	+	0.910447i	$0.364263\pi$
$774$	0	0
$775$	3.82797e7	2.28936
$776$	1.92481e7	1.14745
$777$	0	0
$778$	2.27586e7	1.34802
$779$	−864204.	−0.0510238
$780$	0	0
$781$	1.68952e7	0.991143
$782$	2.34406e6	0.137073
$783$	0	0
$784$	5.36916e6	0.311973
$785$	3.43895e6	0.199183
$786$	0	0
$787$	2.59508e7	1.49353	0.746765	−	0.665088i	$-0.231606\pi$
$787$	2.59508e7	1.49353	0.746765	+	0.665088i	$0.231606\pi$
$788$	−4.06012e7	−2.32929
$789$	0	0
$790$	−6.84902e7	−3.90446
$791$	−9.54137e6	−0.542212
$792$	0	0
$793$	−1.20211e7	−0.678828
$794$	1.36568e7	0.768774
$795$	0	0
$796$	−1.98289e7	−1.10922
$797$	2.75089e7	1.53401	0.767003	−	0.641644i	$-0.221747\pi$
$797$	2.75089e7	1.53401	0.767003	+	0.641644i	$0.221747\pi$
$798$	0	0
$799$	9.34656e6	0.517946
$800$	3.30122e7	1.82368
$801$	0	0
$802$	−4.04647e7	−2.22147
$803$	−8.44924e6	−0.462412
$804$	0	0
$805$	−2.43446e6	−0.132407
$806$	−6.81337e7	−3.69423
$807$	0	0
$808$	−8.12862e6	−0.438015
$809$	1.95854e7	1.05211	0.526056	−	0.850450i	$-0.323670\pi$
$809$	1.95854e7	1.05211	0.526056	+	0.850450i	$0.323670\pi$
$810$	0	0
$811$	1.55546e7	0.830437	0.415219	−	0.909722i	$-0.363705\pi$
$811$	1.55546e7	0.830437	0.415219	+	0.909722i	$0.363705\pi$
$812$	−4.12080e6	−0.219327
$813$	0	0
$814$	−2.95424e6	−0.156273
$815$	5.85384e7	3.08707
$816$	0	0
$817$	−716509.	−0.0375549
$818$	−3.15695e7	−1.64962
$819$	0	0
$820$	−2.54217e7	−1.32029
$821$	1.60179e7	0.829370	0.414685	−	0.909965i	$-0.363892\pi$
$821$	1.60179e7	0.829370	0.414685	+	0.909965i	$0.363892\pi$
$822$	0	0
$823$	−2.04719e7	−1.05356	−0.526779	−	0.850003i	$-0.676600\pi$
$823$	−2.04719e7	−1.05356	−0.526779	+	0.850003i	$0.676600\pi$
$824$	7.60731e6	0.390313
$825$	0	0
$826$	7.41912e6	0.378358
$827$	−1.77905e7	−0.904534	−0.452267	−	0.891883i	$-0.649385\pi$
$827$	−1.77905e7	−0.904534	−0.452267	+	0.891883i	$0.649385\pi$
$828$	0	0
$829$	3.66614e7	1.85278	0.926388	−	0.376570i	$-0.122897\pi$
$829$	3.66614e7	1.85278	0.926388	+	0.376570i	$0.122897\pi$
$830$	1.27715e7	0.643498
$831$	0	0
$832$	−4.76290e7	−2.38541
$833$	−7.04258e6	−0.351657
$834$	0	0
$835$	3.64434e6	0.180885
$836$	1.71302e6	0.0847709
$837$	0	0
$838$	1.68205e7	0.827423
$839$	3.57649e7	1.75409	0.877047	−	0.480405i	$-0.159510\pi$
$839$	3.57649e7	1.75409	0.877047	+	0.480405i	$0.159510\pi$
$840$	0	0
$841$	−1.76129e7	−0.858701
$842$	3.35235e7	1.62956
$843$	0	0
$844$	4.93784e7	2.38606
$845$	−3.97478e7	−1.91501
$846$	0	0
$847$	4.55501e6	0.218163
$848$	1.47265e7	0.703250
$849$	0	0
$850$	−1.99799e7	−0.948518
$851$	648175.	0.0306809
$852$	0	0
$853$	3.91315e7	1.84143	0.920713	−	0.390240i	$-0.127608\pi$
$853$	3.91315e7	1.84143	0.920713	+	0.390240i	$0.127608\pi$
$854$	−6.15024e6	−0.288567
$855$	0	0
$856$	−1.31049e7	−0.611294
$857$	−1.26056e7	−0.586290	−0.293145	−	0.956068i	$-0.594702\pi$
$857$	−1.26056e7	−0.586290	−0.293145	+	0.956068i	$0.594702\pi$
$858$	0	0
$859$	−9.93410e6	−0.459352	−0.229676	−	0.973267i	$-0.573767\pi$
$859$	−9.93410e6	−0.459352	−0.229676	+	0.973267i	$0.573767\pi$
$860$	−2.10770e7	−0.971770
$861$	0	0
$862$	−3.80204e7	−1.74281
$863$	−3.39364e6	−0.155110	−0.0775548	−	0.996988i	$-0.524711\pi$
$863$	−3.39364e6	−0.155110	−0.0775548	+	0.996988i	$0.524711\pi$
$864$	0	0
$865$	−1.94465e7	−0.883694
$866$	5.36272e7	2.42991
$867$	0	0
$868$	−2.05497e7	−0.925776
$869$	2.42443e7	1.08908
$870$	0	0
$871$	−2.64756e7	−1.18250
$872$	−1.64939e7	−0.734570
$873$	0	0
$874$	−637551.	−0.0282317
$875$	6.36912e6	0.281228
$876$	0	0
$877$	696304.	0.0305703	0.0152852	−	0.999883i	$-0.495134\pi$
$877$	696304.	0.0305703	0.0152852	+	0.999883i	$0.495134\pi$
$878$	−3.11073e6	−0.136184
$879$	0	0
$880$	−9.12898e6	−0.397389
$881$	−3.45736e6	−0.150074	−0.0750370	−	0.997181i	$-0.523907\pi$
$881$	−3.45736e6	−0.150074	−0.0750370	+	0.997181i	$0.523907\pi$
$882$	0	0
$883$	−1.85957e7	−0.802620	−0.401310	−	0.915942i	$-0.631445\pi$
$883$	−1.85957e7	−0.802620	−0.401310	+	0.915942i	$0.631445\pi$
$884$	2.09643e7	0.902297
$885$	0	0
$886$	6.27357e7	2.68491
$887$	−3.65335e7	−1.55913	−0.779565	−	0.626321i	$-0.784560\pi$
$887$	−3.65335e7	−1.55913	−0.779565	+	0.626321i	$0.784560\pi$
$888$	0	0
$889$	8.66092e6	0.367544
$890$	−4.65844e7	−1.97136
$891$	0	0
$892$	−8.21391e6	−0.345651
$893$	−2.54213e6	−0.106677
$894$	0	0
$895$	−1.01184e7	−0.422233
$896$	−1.20281e7	−0.500525
$897$	0	0
$898$	3.46186e7	1.43258
$899$	1.44529e7	0.596424
$900$	0	0
$901$	−1.93163e7	−0.792707
$902$	1.52648e7	0.624706
$903$	0	0
$904$	−2.23223e7	−0.908484
$905$	−5.64770e7	−2.29219
$906$	0	0
$907$	−3.73827e7	−1.50887	−0.754437	−	0.656372i	$-0.772090\pi$
$907$	−3.73827e7	−1.50887	−0.754437	+	0.656372i	$0.772090\pi$
$908$	2.71472e7	1.09272
$909$	0	0
$910$	−3.69334e7	−1.47848
$911$	−9.67529e6	−0.386249	−0.193125	−	0.981174i	$-0.561862\pi$
$911$	−9.67529e6	−0.386249	−0.193125	+	0.981174i	$0.561862\pi$
$912$	0	0
$913$	−4.52089e6	−0.179493
$914$	5.74585e6	0.227504
$915$	0	0
$916$	1.21355e7	0.477880
$917$	−1.57495e7	−0.618504
$918$	0	0
$919$	1.43849e7	0.561847	0.280924	−	0.959730i	$-0.409359\pi$
$919$	1.43849e7	0.561847	0.280924	+	0.959730i	$0.409359\pi$
$920$	−5.69547e6	−0.221850
$921$	0	0
$922$	3.46696e7	1.34314
$923$	−5.62377e7	−2.17282
$924$	0	0
$925$	−5.52479e6	−0.212306
$926$	−5.64137e7	−2.16201
$927$	0	0
$928$	1.24641e7	0.475106
$929$	−1.16122e7	−0.441442	−0.220721	−	0.975337i	$-0.570841\pi$
$929$	−1.16122e7	−0.441442	−0.220721	+	0.975337i	$0.570841\pi$
$930$	0	0
$931$	1.91548e6	0.0724274
$932$	−6.29040e7	−2.37213
$933$	0	0
$934$	2.70620e7	1.01506
$935$	1.19742e7	0.447938
$936$	0	0
$937$	−3.54313e7	−1.31837	−0.659186	−	0.751980i	$-0.729099\pi$
$937$	−3.54313e7	−1.31837	−0.659186	+	0.751980i	$0.729099\pi$
$938$	−1.35455e7	−0.502675
$939$	0	0
$940$	−7.47800e7	−2.76036
$941$	−2.83015e7	−1.04192	−0.520961	−	0.853580i	$-0.674426\pi$
$941$	−2.83015e7	−1.04192	−0.520961	+	0.853580i	$0.674426\pi$
$942$	0	0
$943$	−3.34918e6	−0.122648
$944$	−6.10407e6	−0.222941
$945$	0	0
$946$	1.26560e7	0.459801
$947$	3.31990e7	1.20296	0.601479	−	0.798889i	$-0.294578\pi$
$947$	3.31990e7	1.20296	0.601479	+	0.798889i	$0.294578\pi$
$948$	0	0
$949$	2.81243e7	1.01371
$950$	5.43424e6	0.195357
$951$	0	0
$952$	3.25729e6	0.116483
$953$	−3.48861e6	−0.124429	−0.0622143	−	0.998063i	$-0.519816\pi$
$953$	−3.48861e6	−0.124429	−0.0622143	+	0.998063i	$0.519816\pi$
$954$	0	0
$955$	−3.89560e7	−1.38218
$956$	−5.04450e6	−0.178514
$957$	0	0
$958$	2.15527e7	0.758729
$959$	1.34930e7	0.473762
$960$	0	0
$961$	4.34448e7	1.51750
$962$	9.83352e6	0.342587
$963$	0	0
$964$	3.91362e7	1.35639
$965$	−3.90541e7	−1.35004
$966$	0	0
$967$	−5.79306e6	−0.199224	−0.0996120	−	0.995026i	$-0.531760\pi$
$967$	−5.79306e6	−0.199224	−0.0996120	+	0.995026i	$0.531760\pi$
$968$	1.06566e7	0.365535
$969$	0	0
$970$	−1.20501e8	−4.11209
$971$	−2.58348e7	−0.879342	−0.439671	−	0.898159i	$-0.644905\pi$
$971$	−2.58348e7	−0.879342	−0.439671	+	0.898159i	$0.644905\pi$
$972$	0	0
$973$	8.90072e6	0.301400
$974$	−7.61617e7	−2.57240
$975$	0	0
$976$	5.06009e6	0.170033
$977$	3.02885e6	0.101517	0.0507587	−	0.998711i	$-0.483836\pi$
$977$	3.02885e6	0.101517	0.0507587	+	0.998711i	$0.483836\pi$
$978$	0	0
$979$	1.64901e7	0.549877
$980$	5.63463e7	1.87413
$981$	0	0
$982$	6.68907e7	2.21354
$983$	316215.	0.0104376	0.00521878	−	0.999986i	$-0.498339\pi$
$983$	316215.	0.0104376	0.00521878	+	0.999986i	$0.498339\pi$
$984$	0	0
$985$	7.71915e7	2.53500
$986$	−7.54360e6	−0.247108
$987$	0	0
$988$	−5.70198e6	−0.185838
$989$	−2.77680e6	−0.0902720
$990$	0	0
$991$	2.54167e7	0.822121	0.411060	−	0.911608i	$-0.365159\pi$
$991$	2.54167e7	0.822121	0.411060	+	0.911608i	$0.365159\pi$
$992$	6.21562e7	2.00542
$993$	0	0
$994$	−2.87725e7	−0.923657
$995$	3.76989e7	1.20718
$996$	0	0
$997$	−5.93187e7	−1.88996	−0.944982	−	0.327123i	$-0.893921\pi$
$997$	−5.93187e7	−1.88996	−0.944982	+	0.327123i	$0.893921\pi$
$998$	3.83204e7	1.21788
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	207.6.a.h.1.2	✓	10
3.2	odd	2		207.6.a.i.1.9	yes	10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
207.6.a.h.1.2	✓	10	1.1	even	1	trivial
207.6.a.i.1.9	yes	10	3.2	odd	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)

\(f(q)\)	\(=\)	\(q-8.82929 q^{2} +45.9564 q^{4} -87.3727 q^{5} -52.6709 q^{7} -123.225 q^{8} +O(q^{10})\)	\(q-8.82929 q^{2} +45.9564 q^{4} -87.3727 q^{5} -52.6709 q^{7} -123.225 q^{8} +771.439 q^{10} -273.076 q^{11} +908.964 q^{13} +465.046 q^{14} -382.616 q^{16} +501.866 q^{17} -136.500 q^{19} -4015.33 q^{20} +2411.07 q^{22} -529.000 q^{23} +4508.99 q^{25} -8025.51 q^{26} -2420.56 q^{28} +1702.41 q^{29} +8489.64 q^{31} +7321.42 q^{32} -4431.12 q^{34} +4602.00 q^{35} -1225.28 q^{37} +1205.20 q^{38} +10766.5 q^{40} +6331.15 q^{41} +5249.14 q^{43} -12549.6 q^{44} +4670.69 q^{46} +18623.6 q^{47} -14032.8 q^{49} -39811.2 q^{50} +41772.7 q^{52} -38489.0 q^{53} +23859.4 q^{55} +6490.36 q^{56} -15031.1 q^{58} +15953.5 q^{59} -13225.0 q^{61} -74957.5 q^{62} -52399.3 q^{64} -79418.7 q^{65} -29127.2 q^{67} +23063.9 q^{68} -40632.4 q^{70} -61870.1 q^{71} +30941.0 q^{73} +10818.4 q^{74} -6273.06 q^{76} +14383.1 q^{77} -88782.4 q^{79} +33430.2 q^{80} -55899.6 q^{82} +16555.4 q^{83} -43849.4 q^{85} -46346.2 q^{86} +33649.7 q^{88} -60386.4 q^{89} -47875.9 q^{91} -24310.9 q^{92} -164433. q^{94} +11926.4 q^{95} -156203. q^{97} +123899. q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q - 8 q^{2} + 192 q^{4} - 100 q^{5} + 20 q^{7} - 384 q^{8}+O(q^{10}) \) 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

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Groups

Database

Properties

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