LMFDB - Embedded newform 207.6.a.h.1.1

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			1.1
Root			$-10.1836$ 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)

$f(q)$	$=$	$q-11.1836 q^{2} +93.0738 q^{4} -30.2673 q^{5} +207.819 q^{7} -683.027 q^{8} +O(q^{10})$	\(q-11.1836 q^{2} +93.0738 q^{4} -30.2673 q^{5} +207.819 q^{7} -683.027 q^{8} +338.499 q^{10} +461.365 q^{11} -1009.51 q^{13} -2324.17 q^{14} +4660.37 q^{16} -1035.90 q^{17} -1405.37 q^{19} -2817.09 q^{20} -5159.74 q^{22} -529.000 q^{23} -2208.89 q^{25} +11290.0 q^{26} +19342.5 q^{28} +8519.52 q^{29} +3418.23 q^{31} -30263.0 q^{32} +11585.2 q^{34} -6290.11 q^{35} +696.523 q^{37} +15717.2 q^{38} +20673.4 q^{40} -4749.13 q^{41} +3562.06 q^{43} +42941.0 q^{44} +5916.15 q^{46} -6214.60 q^{47} +26381.6 q^{49} +24703.4 q^{50} -93958.8 q^{52} -6373.84 q^{53} -13964.3 q^{55} -141946. q^{56} -95279.2 q^{58} -15326.1 q^{59} -16413.0 q^{61} -38228.3 q^{62} +189319. q^{64} +30555.1 q^{65} -14535.9 q^{67} -96415.5 q^{68} +70346.3 q^{70} -34956.0 q^{71} -10547.1 q^{73} -7789.66 q^{74} -130803. q^{76} +95880.2 q^{77} -97648.3 q^{79} -141057. q^{80} +53112.5 q^{82} +44297.5 q^{83} +31354.0 q^{85} -39836.8 q^{86} -315125. q^{88} +5925.70 q^{89} -209795. q^{91} -49236.0 q^{92} +69501.8 q^{94} +42536.8 q^{95} +81185.3 q^{97} -295042. q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−11.1836	−1.97701	−0.988503	−	0.151199i	$-0.951687\pi$
$2$	−11.1836	−1.97701	−0.988503	+	0.151199i	$0.951687\pi$
$3$	0	0
$3$	0	0
$4$	93.0738	2.90856
$5$	−30.2673	−0.541438	−0.270719	−	0.962658i	$-0.587261\pi$
$5$	−30.2673	−0.541438	−0.270719	+	0.962658i	$0.587261\pi$
$6$	0	0
$7$	207.819	1.60302	0.801511	−	0.597980i	$-0.204030\pi$
$7$	207.819	1.60302	0.801511	+	0.597980i	$0.204030\pi$
$8$	−683.027	−3.77323
$9$	0	0
$10$	338.499	1.07043
$11$	461.365	1.14964	0.574821	−	0.818279i	$-0.305072\pi$
$11$	461.365	1.14964	0.574821	+	0.818279i	$0.305072\pi$
$12$	0	0
$13$	−1009.51	−1.65673	−0.828365	−	0.560189i	$-0.810729\pi$
$13$	−1009.51	−1.65673	−0.828365	+	0.560189i	$0.810729\pi$
$14$	−2324.17	−3.16918
$15$	0	0
$16$	4660.37	4.55114
$17$	−1035.90	−0.869355	−0.434678	−	0.900586i	$-0.643138\pi$
$17$	−1035.90	−0.869355	−0.434678	+	0.900586i	$0.643138\pi$
$18$	0	0
$19$	−1405.37	−0.893114	−0.446557	−	0.894755i	$-0.647350\pi$
$19$	−1405.37	−0.893114	−0.446557	+	0.894755i	$0.647350\pi$
$20$	−2817.09	−1.57480
$21$	0	0
$22$	−5159.74	−2.27285
$23$	−529.000	−0.208514
$23$	−529.000	−0.208514
$24$	0	0
$25$	−2208.89	−0.706845
$26$	11290.0	3.27537
$27$	0	0
$28$	19342.5	4.66248
$29$	8519.52	1.88114	0.940568	−	0.339606i	$-0.110294\pi$
$29$	8519.52	1.88114	0.940568	+	0.339606i	$0.110294\pi$
$30$	0	0
$31$	3418.23	0.638848	0.319424	−	0.947612i	$-0.396511\pi$
$31$	3418.23	0.638848	0.319424	+	0.947612i	$0.396511\pi$
$32$	−30263.0	−5.22441
$33$	0	0
$34$	11585.2	1.71872
$35$	−6290.11	−0.867937
$36$	0	0
$37$	696.523	0.0836433	0.0418217	−	0.999125i	$-0.486684\pi$
$37$	696.523	0.0836433	0.0418217	+	0.999125i	$0.486684\pi$
$38$	15717.2	1.76569
$39$	0	0
$40$	20673.4	2.04297
$41$	−4749.13	−0.441219	−0.220609	−	0.975362i	$-0.570805\pi$
$41$	−4749.13	−0.441219	−0.220609	+	0.975362i	$0.570805\pi$
$42$	0	0
$43$	3562.06	0.293785	0.146893	−	0.989152i	$-0.453073\pi$
$43$	3562.06	0.293785	0.146893	+	0.989152i	$0.453073\pi$
$44$	42941.0	3.34380
$45$	0	0
$46$	5916.15	0.412234
$47$	−6214.60	−0.410363	−0.205182	−	0.978724i	$-0.565778\pi$
$47$	−6214.60	−0.410363	−0.205182	+	0.978724i	$0.565778\pi$
$48$	0	0
$49$	26381.6	1.56968
$50$	24703.4	1.39744
$51$	0	0
$52$	−93958.8	−4.81869
$53$	−6373.84	−0.311682	−0.155841	−	0.987782i	$-0.549809\pi$
$53$	−6373.84	−0.311682	−0.155841	+	0.987782i	$0.549809\pi$
$54$	0	0
$55$	−13964.3	−0.622460
$56$	−141946.	−6.04857
$57$	0	0
$58$	−95279.2	−3.71902
$59$	−15326.1	−0.573195	−0.286598	−	0.958051i	$-0.592524\pi$
$59$	−15326.1	−0.573195	−0.286598	+	0.958051i	$0.592524\pi$
$60$	0	0
$61$	−16413.0	−0.564760	−0.282380	−	0.959303i	$-0.591124\pi$
$61$	−16413.0	−0.564760	−0.282380	+	0.959303i	$0.591124\pi$
$62$	−38228.3	−1.26301
$63$	0	0
$64$	189319.	5.77755
$65$	30555.1	0.897017
$66$	0	0
$67$	−14535.9	−0.395599	−0.197799	−	0.980243i	$-0.563379\pi$
$67$	−14535.9	−0.395599	−0.197799	+	0.980243i	$0.563379\pi$
$68$	−96415.5	−2.52857
$69$	0	0
$70$	70346.3	1.71592
$71$	−34956.0	−0.822955	−0.411478	−	0.911420i	$-0.634987\pi$
$71$	−34956.0	−0.822955	−0.411478	+	0.911420i	$0.634987\pi$
$72$	0	0
$73$	−10547.1	−0.231648	−0.115824	−	0.993270i	$-0.536951\pi$
$73$	−10547.1	−0.231648	−0.115824	+	0.993270i	$0.536951\pi$
$74$	−7789.66	−0.165363
$75$	0	0
$76$	−130803.	−2.59767
$77$	95880.2	1.84290
$78$	0	0
$79$	−97648.3	−1.76034	−0.880171	−	0.474658i	$-0.842572\pi$
$79$	−97648.3	−1.76034	−0.880171	+	0.474658i	$0.842572\pi$
$80$	−141057.	−2.46416
$81$	0	0
$82$	53112.5	0.872293
$83$	44297.5	0.705804	0.352902	−	0.935660i	$-0.385195\pi$
$83$	44297.5	0.705804	0.352902	+	0.935660i	$0.385195\pi$
$84$	0	0
$85$	31354.0	0.470702
$86$	−39836.8	−0.580816
$87$	0	0
$88$	−315125.	−4.33786
$89$	5925.70	0.0792985	0.0396492	−	0.999214i	$-0.487376\pi$
$89$	5925.70	0.0792985	0.0396492	+	0.999214i	$0.487376\pi$
$90$	0	0
$91$	−209795.	−2.65577
$92$	−49236.0	−0.606476
$93$	0	0
$94$	69501.8	0.811291
$95$	42536.8	0.483566
$96$	0	0
$97$	81185.3	0.876089	0.438044	−	0.898953i	$-0.355671\pi$
$97$	81185.3	0.876089	0.438044	+	0.898953i	$0.355671\pi$
$98$	−295042.	−3.10327
$99$	0	0
$100$	−205590.	−2.05590
$101$	704.822	0.00687505	0.00343753	−	0.999994i	$-0.498906\pi$
$101$	704.822	0.00687505	0.00343753	+	0.999994i	$0.498906\pi$
$102$	0	0
$103$	−32424.9	−0.301152	−0.150576	−	0.988598i	$-0.548113\pi$
$103$	−32424.9	−0.301152	−0.150576	+	0.988598i	$0.548113\pi$
$104$	689522.	6.25122
$105$	0	0
$106$	71282.7	0.616197
$107$	−94280.5	−0.796091	−0.398045	−	0.917366i	$-0.630311\pi$
$107$	−94280.5	−0.796091	−0.398045	+	0.917366i	$0.630311\pi$
$108$	0	0
$109$	−137626.	−1.10952	−0.554759	−	0.832011i	$-0.687190\pi$
$109$	−137626.	−1.10952	−0.554759	+	0.832011i	$0.687190\pi$
$110$	156171.	1.23061
$111$	0	0
$112$	968511.	7.29558
$113$	96721.8	0.712571	0.356286	−	0.934377i	$-0.384043\pi$
$113$	96721.8	0.712571	0.356286	+	0.934377i	$0.384043\pi$
$114$	0	0
$115$	16011.4	0.112898
$116$	792944.	5.47139
$117$	0	0
$118$	171402.	1.13321
$119$	−215280.	−1.39360
$120$	0	0
$121$	51806.4	0.321677
$122$	183557.	1.11653
$123$	0	0
$124$	318148.	1.85812
$125$	161443.	0.924151
$126$	0	0
$127$	−151779.	−0.835030	−0.417515	−	0.908670i	$-0.637099\pi$
$127$	−151779.	−0.835030	−0.417515	+	0.908670i	$0.637099\pi$
$128$	−1.14886e6	−6.19785
$129$	0	0
$130$	−341717.	−1.77341
$131$	−223649.	−1.13865	−0.569323	−	0.822114i	$-0.692795\pi$
$131$	−223649.	−1.13865	−0.569323	+	0.822114i	$0.692795\pi$
$132$	0	0
$133$	−292062.	−1.43168
$134$	162564.	0.782101
$135$	0	0
$136$	707551.	3.28028
$137$	−107815.	−0.490771	−0.245385	−	0.969426i	$-0.578915\pi$
$137$	−107815.	−0.490771	−0.245385	+	0.969426i	$0.578915\pi$
$138$	0	0
$139$	208183.	0.913919	0.456960	−	0.889487i	$-0.348938\pi$
$139$	208183.	0.913919	0.456960	+	0.889487i	$0.348938\pi$
$140$	−585445.	−2.52444
$141$	0	0
$142$	390936.	1.62699
$143$	−465752.	−1.90465
$144$	0	0
$145$	−257863.	−1.01852
$146$	117955.	0.457969
$147$	0	0
$148$	64828.0	0.243281
$149$	53529.1	0.197526	0.0987630	−	0.995111i	$-0.468511\pi$
$149$	53529.1	0.197526	0.0987630	+	0.995111i	$0.468511\pi$
$150$	0	0
$151$	79746.7	0.284623	0.142312	−	0.989822i	$-0.454546\pi$
$151$	79746.7	0.284623	0.142312	+	0.989822i	$0.454546\pi$
$152$	959907.	3.36992
$153$	0	0
$154$	−1.07229e6	−3.64343
$155$	−103461.	−0.345897
$156$	0	0
$157$	−246025.	−0.796581	−0.398291	−	0.917259i	$-0.630396\pi$
$157$	−246025.	−0.796581	−0.398291	+	0.917259i	$0.630396\pi$
$158$	1.09206e6	3.48021
$159$	0	0
$160$	915980.	2.82869
$161$	−109936.	−0.334253
$162$	0	0
$163$	−284186.	−0.837786	−0.418893	−	0.908036i	$-0.637582\pi$
$163$	−284186.	−0.837786	−0.418893	+	0.908036i	$0.637582\pi$
$164$	−442019.	−1.28331
$165$	0	0
$166$	−495407.	−1.39538
$167$	−313664.	−0.870309	−0.435154	−	0.900356i	$-0.643306\pi$
$167$	−313664.	−0.870309	−0.435154	+	0.900356i	$0.643306\pi$
$168$	0	0
$169$	647815.	1.74475
$170$	−350652.	−0.930581
$171$	0	0
$172$	331535.	0.854491
$173$	−541439.	−1.37542	−0.687708	−	0.725987i	$-0.741383\pi$
$173$	−541439.	−1.37542	−0.687708	+	0.725987i	$0.741383\pi$
$174$	0	0
$175$	−459048.	−1.13309
$176$	2.15013e6	5.23218
$177$	0	0
$178$	−66270.9	−0.156774
$179$	128471.	0.299691	0.149845	−	0.988709i	$-0.452122\pi$
$179$	128471.	0.299691	0.149845	+	0.988709i	$0.452122\pi$
$180$	0	0
$181$	−580733.	−1.31759	−0.658795	−	0.752323i	$-0.728933\pi$
$181$	−580733.	−1.31759	−0.658795	+	0.752323i	$0.728933\pi$
$182$	2.34627e6	5.25048
$183$	0	0
$184$	361321.	0.786772
$185$	−21081.9	−0.0452877
$186$	0	0
$187$	−477930.	−0.999447
$188$	−578416.	−1.19356
$189$	0	0
$190$	−475716.	−0.956014
$191$	−422899.	−0.838789	−0.419395	−	0.907804i	$-0.637758\pi$
$191$	−422899.	−0.838789	−0.419395	+	0.907804i	$0.637758\pi$
$192$	0	0
$193$	−482285.	−0.931989	−0.465994	−	0.884788i	$-0.654303\pi$
$193$	−482285.	−0.931989	−0.465994	+	0.884788i	$0.654303\pi$
$194$	−907947.	−1.73203
$195$	0	0
$196$	2.45543e6	4.56550
$197$	−524730.	−0.963320	−0.481660	−	0.876358i	$-0.659966\pi$
$197$	−524730.	−0.963320	−0.481660	+	0.876358i	$0.659966\pi$
$198$	0	0
$199$	797258.	1.42714	0.713569	−	0.700585i	$-0.247078\pi$
$199$	797258.	1.42714	0.713569	+	0.700585i	$0.247078\pi$
$200$	1.50873e6	2.66709
$201$	0	0
$202$	−7882.48	−0.0135920
$203$	1.77052e6	3.01550
$204$	0	0
$205$	143743.	0.238893
$206$	362628.	0.595379
$207$	0	0
$208$	−4.70468e6	−7.54001
$209$	−648389.	−1.02676
$210$	0	0
$211$	−223232.	−0.345183	−0.172592	−	0.984993i	$-0.555214\pi$
$211$	−223232.	−0.345183	−0.172592	+	0.984993i	$0.555214\pi$
$212$	−593237.	−0.906543
$213$	0	0
$214$	1.05440e6	1.57388
$215$	−107814.	−0.159067
$216$	0	0
$217$	710373.	1.02409
$218$	1.53916e6	2.19353
$219$	0	0
$220$	−1.29971e6	−1.81046
$221$	1.04575e6	1.44029
$222$	0	0
$223$	−51440.1	−0.0692691	−0.0346346	−	0.999400i	$-0.511027\pi$
$223$	−51440.1	−0.0692691	−0.0346346	+	0.999400i	$0.511027\pi$
$224$	−6.28922e6	−8.37484
$225$	0	0
$226$	−1.08170e6	−1.40876
$227$	−1.20143e6	−1.54751	−0.773755	−	0.633485i	$-0.781624\pi$
$227$	−1.20143e6	−1.54751	−0.773755	+	0.633485i	$0.781624\pi$
$228$	0	0
$229$	807299.	1.01729	0.508646	−	0.860976i	$-0.330146\pi$
$229$	807299.	1.01729	0.508646	+	0.860976i	$0.330146\pi$
$230$	−179066.	−0.223199
$231$	0	0
$232$	−5.81906e6	−7.09795
$233$	920840.	1.11121	0.555603	−	0.831448i	$-0.312488\pi$
$233$	920840.	1.11121	0.555603	+	0.831448i	$0.312488\pi$
$234$	0	0
$235$	188099.	0.222186
$236$	−1.42646e6	−1.66717
$237$	0	0
$238$	2.40762e6	2.75515
$239$	−425777.	−0.482156	−0.241078	−	0.970506i	$-0.577501\pi$
$239$	−425777.	−0.482156	−0.241078	+	0.970506i	$0.577501\pi$
$240$	0	0
$241$	1.08916e6	1.20795	0.603973	−	0.797005i	$-0.293584\pi$
$241$	1.08916e6	1.20795	0.603973	+	0.797005i	$0.293584\pi$
$242$	−579384.	−0.635958
$243$	0	0
$244$	−1.52762e6	−1.64263
$245$	−798500.	−0.849884
$246$	0	0
$247$	1.41873e6	1.47965
$248$	−2.33475e6	−2.41052
$249$	0	0
$250$	−1.80551e6	−1.82705
$251$	944926.	0.946702	0.473351	−	0.880874i	$-0.343044\pi$
$251$	944926.	0.946702	0.473351	+	0.880874i	$0.343044\pi$
$252$	0	0
$253$	−244062.	−0.239717
$254$	1.69744e6	1.65086
$255$	0	0
$256$	6.79020e6	6.47564
$257$	−1.56185e6	−1.47505	−0.737524	−	0.675321i	$-0.764005\pi$
$257$	−1.56185e6	−1.47505	−0.737524	+	0.675321i	$0.764005\pi$
$258$	0	0
$259$	144750.	0.134082
$260$	2.84388e6	2.60902
$261$	0	0
$262$	2.50121e6	2.25111
$263$	1.47638e6	1.31616	0.658079	−	0.752949i	$-0.271369\pi$
$263$	1.47638e6	1.31616	0.658079	+	0.752949i	$0.271369\pi$
$264$	0	0
$265$	192919.	0.168756
$266$	3.26632e6	2.83044
$267$	0	0
$268$	−1.35291e6	−1.15062
$269$	−707436.	−0.596082	−0.298041	−	0.954553i	$-0.596333\pi$
$269$	−707436.	−0.596082	−0.298041	+	0.954553i	$0.596333\pi$
$270$	0	0
$271$	−559321.	−0.462634	−0.231317	−	0.972878i	$-0.574303\pi$
$271$	−559321.	−0.462634	−0.231317	+	0.972878i	$0.574303\pi$
$272$	−4.82769e6	−3.95656
$273$	0	0
$274$	1.20577e6	0.970257
$275$	−1.01910e6	−0.812619
$276$	0	0
$277$	179587.	0.140629	0.0703146	−	0.997525i	$-0.477600\pi$
$277$	179587.	0.140629	0.0703146	+	0.997525i	$0.477600\pi$
$278$	−2.32824e6	−1.80682
$279$	0	0
$280$	4.29632e6	3.27492
$281$	−2.29041e6	−1.73040	−0.865201	−	0.501425i	$-0.832809\pi$
$281$	−2.29041e6	−1.73040	−0.865201	+	0.501425i	$0.832809\pi$
$282$	0	0
$283$	−2.32523e6	−1.72584	−0.862920	−	0.505340i	$-0.831367\pi$
$283$	−2.32523e6	−1.72584	−0.862920	+	0.505340i	$0.831367\pi$
$284$	−3.25349e6	−2.39361
$285$	0	0
$286$	5.20880e6	3.76550
$287$	−986957.	−0.707283
$288$	0	0
$289$	−346760.	−0.244222
$290$	2.88385e6	2.01362
$291$	0	0
$292$	−981663.	−0.673760
$293$	2.10635e6	1.43338	0.716690	−	0.697392i	$-0.245656\pi$
$293$	2.10635e6	1.43338	0.716690	+	0.697392i	$0.245656\pi$
$294$	0	0
$295$	463881.	0.310350
$296$	−475744.	−0.315605
$297$	0	0
$298$	−598650.	−0.390510
$299$	534030.	0.345452
$300$	0	0
$301$	740263.	0.470944
$302$	−891859.	−0.562702
$303$	0	0
$304$	−6.54955e6	−4.06469
$305$	496778.	0.305782
$306$	0	0
$307$	2.40376e6	1.45561	0.727805	−	0.685784i	$-0.240541\pi$
$307$	2.40376e6	1.45561	0.727805	+	0.685784i	$0.240541\pi$
$308$	8.92393e6	5.36018
$309$	0	0
$310$	1.15707e6	0.683840
$311$	−1.17216e6	−0.687207	−0.343603	−	0.939115i	$-0.611648\pi$
$311$	−1.17216e6	−0.687207	−0.343603	+	0.939115i	$0.611648\pi$
$312$	0	0
$313$	−2.90462e6	−1.67583	−0.837913	−	0.545804i	$-0.816224\pi$
$313$	−2.90462e6	−1.67583	−0.837913	+	0.545804i	$0.816224\pi$
$314$	2.75146e6	1.57485
$315$	0	0
$316$	−9.08849e6	−5.12005
$317$	422969.	0.236407	0.118204	−	0.992989i	$-0.462286\pi$
$317$	422969.	0.236407	0.118204	+	0.992989i	$0.462286\pi$
$318$	0	0
$319$	3.93061e6	2.16263
$320$	−5.73017e6	−3.12819
$321$	0	0
$322$	1.22949e6	0.660821
$323$	1.45583e6	0.776434
$324$	0	0
$325$	2.22989e6	1.17105
$326$	3.17823e6	1.65631
$327$	0	0
$328$	3.24378e6	1.66482
$329$	−1.29151e6	−0.657821
$330$	0	0
$331$	−1.53755e6	−0.771364	−0.385682	−	0.922632i	$-0.626034\pi$
$331$	−1.53755e6	−0.771364	−0.385682	+	0.922632i	$0.626034\pi$
$332$	4.12294e6	2.05287
$333$	0	0
$334$	3.50790e6	1.72061
$335$	439963.	0.214192
$336$	0	0
$337$	2.34240e6	1.12354	0.561768	−	0.827295i	$-0.310121\pi$
$337$	2.34240e6	1.12354	0.561768	+	0.827295i	$0.310121\pi$
$338$	−7.24493e6	−3.44939
$339$	0	0
$340$	2.91824e6	1.36906
$341$	1.57705e6	0.734446
$342$	0	0
$343$	1.98978e6	0.913207
$344$	−2.43298e6	−1.10852
$345$	0	0
$346$	6.05526e6	2.71921
$347$	−809438.	−0.360878	−0.180439	−	0.983586i	$-0.557752\pi$
$347$	−809438.	−0.360878	−0.180439	+	0.983586i	$0.557752\pi$
$348$	0	0
$349$	2.79641e6	1.22896	0.614480	−	0.788932i	$-0.289366\pi$
$349$	2.79641e6	1.22896	0.614480	+	0.788932i	$0.289366\pi$
$350$	5.13383e6	2.24012
$351$	0	0
$352$	−1.39623e7	−6.00620
$353$	−4.39812e6	−1.87858	−0.939291	−	0.343122i	$-0.888515\pi$
$353$	−4.39812e6	−1.87858	−0.939291	+	0.343122i	$0.888515\pi$
$354$	0	0
$355$	1.05802e6	0.445579
$356$	551528.	0.230644
$357$	0	0
$358$	−1.43678e6	−0.592491
$359$	−3.01194e6	−1.23342	−0.616709	−	0.787191i	$-0.711534\pi$
$359$	−3.01194e6	−1.23342	−0.616709	+	0.787191i	$0.711534\pi$
$360$	0	0
$361$	−501031.	−0.202347
$362$	6.49471e6	2.60488
$363$	0	0
$364$	−1.95264e7	−7.72447
$365$	319234.	0.125423
$366$	0	0
$367$	−4.11877e6	−1.59625	−0.798127	−	0.602489i	$-0.794176\pi$
$367$	−4.11877e6	−1.59625	−0.798127	+	0.602489i	$0.794176\pi$
$368$	−2.46533e6	−0.948978
$369$	0	0
$370$	235772.	0.0895340
$371$	−1.32460e6	−0.499632
$372$	0	0
$373$	1.89375e6	0.704774	0.352387	−	0.935854i	$-0.385370\pi$
$373$	1.89375e6	0.704774	0.352387	+	0.935854i	$0.385370\pi$
$374$	5.34499e6	1.97591
$375$	0	0
$376$	4.24474e6	1.54839
$377$	−8.60053e6	−3.11653
$378$	0	0
$379$	1.98839e6	0.711057	0.355529	−	0.934665i	$-0.384301\pi$
$379$	1.98839e6	0.711057	0.355529	+	0.934665i	$0.384301\pi$
$380$	3.95906e6	1.40648
$381$	0	0
$382$	4.72955e6	1.65829
$383$	−2.61731e6	−0.911715	−0.455857	−	0.890053i	$-0.650667\pi$
$383$	−2.61731e6	−0.911715	−0.455857	+	0.890053i	$0.650667\pi$
$384$	0	0
$385$	−2.90204e6	−0.997817
$386$	5.39370e6	1.84255
$387$	0	0
$388$	7.55623e6	2.54815
$389$	3.47567e6	1.16457	0.582284	−	0.812986i	$-0.302159\pi$
$389$	3.47567e6	1.16457	0.582284	+	0.812986i	$0.302159\pi$
$390$	0	0
$391$	547993.	0.181273
$392$	−1.80193e7	−5.92275
$393$	0	0
$394$	5.86839e6	1.90449
$395$	2.95555e6	0.953116
$396$	0	0
$397$	−1.78310e6	−0.567806	−0.283903	−	0.958853i	$-0.591629\pi$
$397$	−1.78310e6	−0.567806	−0.283903	+	0.958853i	$0.591629\pi$
$398$	−8.91624e6	−2.82146
$399$	0	0
$400$	−1.02942e7	−3.21695
$401$	4.15508e6	1.29038	0.645190	−	0.764022i	$-0.276778\pi$
$401$	4.15508e6	1.29038	0.645190	+	0.764022i	$0.276778\pi$
$402$	0	0
$403$	−3.45074e6	−1.05840
$404$	65600.5	0.0199965
$405$	0	0
$406$	−1.98008e7	−5.96167
$407$	321351.	0.0961599
$408$	0	0
$409$	−1.26125e6	−0.372814	−0.186407	−	0.982473i	$-0.559684\pi$
$409$	−1.26125e6	−0.372814	−0.186407	+	0.982473i	$0.559684\pi$
$410$	−1.60757e6	−0.472293
$411$	0	0
$412$	−3.01791e6	−0.875916
$413$	−3.18506e6	−0.918844
$414$	0	0
$415$	−1.34077e6	−0.382149
$416$	3.05508e7	8.65543
$417$	0	0
$418$	7.25135e6	2.02992
$419$	3.81513e6	1.06163	0.530816	−	0.847487i	$-0.321885\pi$
$419$	3.81513e6	1.06163	0.530816	+	0.847487i	$0.321885\pi$
$420$	0	0
$421$	581066.	0.159779	0.0798896	−	0.996804i	$-0.474543\pi$
$421$	581066.	0.159779	0.0798896	+	0.996804i	$0.474543\pi$
$422$	2.49654e6	0.682430
$423$	0	0
$424$	4.35350e6	1.17605
$425$	2.28820e6	0.614499
$426$	0	0
$427$	−3.41093e6	−0.905322
$428$	−8.77505e6	−2.31547
$429$	0	0
$430$	1.20575e6	0.314476
$431$	992560.	0.257373	0.128687	−	0.991685i	$-0.458924\pi$
$431$	992560.	0.257373	0.128687	+	0.991685i	$0.458924\pi$
$432$	0	0
$433$	4.49265e6	1.15155	0.575775	−	0.817608i	$-0.304700\pi$
$433$	4.49265e6	1.15155	0.575775	+	0.817608i	$0.304700\pi$
$434$	−7.94455e6	−2.02463
$435$	0	0
$436$	−1.28094e7	−3.22710
$437$	743441.	0.186227
$438$	0	0
$439$	1.79028e6	0.443364	0.221682	−	0.975119i	$-0.428845\pi$
$439$	1.79028e6	0.443364	0.221682	+	0.975119i	$0.428845\pi$
$440$	9.53798e6	2.34868
$441$	0	0
$442$	−1.16953e7	−2.84746
$443$	5.97606e6	1.44679	0.723396	−	0.690434i	$-0.242580\pi$
$443$	5.97606e6	1.44679	0.723396	+	0.690434i	$0.242580\pi$
$444$	0	0
$445$	−179355.	−0.0429352
$446$	575288.	0.136946
$447$	0	0
$448$	3.93440e7	9.26154
$449$	−2.27980e6	−0.533679	−0.266840	−	0.963741i	$-0.585979\pi$
$449$	−2.27980e6	−0.533679	−0.266840	+	0.963741i	$0.585979\pi$
$450$	0	0
$451$	−2.19108e6	−0.507244
$452$	9.00227e6	2.07255
$453$	0	0
$454$	1.34364e7	3.05944
$455$	6.34992e6	1.43794
$456$	0	0
$457$	846039.	0.189496	0.0947479	−	0.995501i	$-0.469795\pi$
$457$	846039.	0.189496	0.0947479	+	0.995501i	$0.469795\pi$
$458$	−9.02854e6	−2.01119
$459$	0	0
$460$	1.49024e6	0.328369
$461$	−5.62754e6	−1.23329	−0.616647	−	0.787240i	$-0.711509\pi$
$461$	−5.62754e6	−1.23329	−0.616647	+	0.787240i	$0.711509\pi$
$462$	0	0
$463$	7.02874e6	1.52379	0.761895	−	0.647700i	$-0.224269\pi$
$463$	7.02874e6	1.52379	0.761895	+	0.647700i	$0.224269\pi$
$464$	3.97041e7	8.56131
$465$	0	0
$466$	−1.02983e7	−2.19686
$467$	7.42129e6	1.57466	0.787331	−	0.616530i	$-0.211462\pi$
$467$	7.42129e6	1.57466	0.787331	+	0.616530i	$0.211462\pi$
$468$	0	0
$469$	−3.02083e6	−0.634153
$470$	−2.10363e6	−0.439264
$471$	0	0
$472$	1.04682e7	2.16280
$473$	1.64341e6	0.337748
$474$	0	0
$475$	3.10431e6	0.631293
$476$	−2.00369e7	−4.05335
$477$	0	0
$478$	4.76174e6	0.953226
$479$	2.07782e6	0.413779	0.206890	−	0.978364i	$-0.433666\pi$
$479$	2.07782e6	0.413779	0.206890	+	0.978364i	$0.433666\pi$
$480$	0	0
$481$	−703146.	−0.138574
$482$	−1.21807e7	−2.38812
$483$	0	0
$484$	4.82182e6	0.935616
$485$	−2.45726e6	−0.474348
$486$	0	0
$487$	−2.43700e6	−0.465622	−0.232811	−	0.972522i	$-0.574792\pi$
$487$	−2.43700e6	−0.465622	−0.232811	+	0.972522i	$0.574792\pi$
$488$	1.12105e7	2.13097
$489$	0	0
$490$	8.93013e6	1.68023
$491$	−5.74298e6	−1.07506	−0.537531	−	0.843244i	$-0.680643\pi$
$491$	−5.74298e6	−1.07506	−0.537531	+	0.843244i	$0.680643\pi$
$492$	0	0
$493$	−8.82541e6	−1.63538
$494$	−1.58666e7	−2.92528
$495$	0	0
$496$	1.59302e7	2.90749
$497$	−7.26451e6	−1.31921
$498$	0	0
$499$	−1.93853e6	−0.348514	−0.174257	−	0.984700i	$-0.555752\pi$
$499$	−1.93853e6	−0.348514	−0.174257	+	0.984700i	$0.555752\pi$
$500$	1.50261e7	2.68794
$501$	0	0
$502$	−1.05677e7	−1.87164
$503$	4.26532e6	0.751677	0.375839	−	0.926685i	$-0.377355\pi$
$503$	4.26532e6	0.751677	0.375839	+	0.926685i	$0.377355\pi$
$504$	0	0
$505$	−21333.1	−0.00372242
$506$	2.72950e6	0.473922
$507$	0	0
$508$	−1.41266e7	−2.42873
$509$	−7.65299e6	−1.30929	−0.654646	−	0.755935i	$-0.727182\pi$
$509$	−7.65299e6	−1.30929	−0.654646	+	0.755935i	$0.727182\pi$
$510$	0	0
$511$	−2.19189e6	−0.371336
$512$	−3.91757e7	−6.60453
$513$	0	0
$514$	1.74671e7	2.91618
$515$	981414.	0.163055
$516$	0	0
$517$	−2.86720e6	−0.471771
$518$	−1.61884e6	−0.265081
$519$	0	0
$520$	−2.08700e7	−3.38465
$521$	1.60053e6	0.258326	0.129163	−	0.991623i	$-0.458771\pi$
$521$	1.60053e6	0.258326	0.129163	+	0.991623i	$0.458771\pi$
$522$	0	0
$523$	−5.91321e6	−0.945299	−0.472650	−	0.881250i	$-0.656702\pi$
$523$	−5.91321e6	−0.945299	−0.472650	+	0.881250i	$0.656702\pi$
$524$	−2.08159e7	−3.31181
$525$	0	0
$526$	−1.65113e7	−2.60205
$527$	−3.54096e6	−0.555386
$528$	0	0
$529$	279841.	0.0434783
$530$	−2.15754e6	−0.333632
$531$	0	0
$532$	−2.71833e7	−4.16413
$533$	4.79429e6	0.730981
$534$	0	0
$535$	2.85362e6	0.431034
$536$	9.92841e6	1.49268
$537$	0	0
$538$	7.91171e6	1.17846
$539$	1.21715e7	1.80457
$540$	0	0
$541$	1.27500e7	1.87291	0.936455	−	0.350788i	$-0.114086\pi$
$541$	1.27500e7	1.87291	0.936455	+	0.350788i	$0.114086\pi$
$542$	6.25524e6	0.914631
$543$	0	0
$544$	3.13496e7	4.54187
$545$	4.16557e6	0.600736
$546$	0	0
$547$	5.69349e6	0.813599	0.406800	−	0.913517i	$-0.366645\pi$
$547$	5.69349e6	0.813599	0.406800	+	0.913517i	$0.366645\pi$
$548$	−1.00348e7	−1.42743
$549$	0	0
$550$	1.13973e7	1.60655
$551$	−1.19731e7	−1.68007
$552$	0	0
$553$	−2.02931e7	−2.82186
$554$	−2.00844e6	−0.278025
$555$	0	0
$556$	1.93764e7	2.65819
$557$	1.44995e6	0.198023	0.0990114	−	0.995086i	$-0.468432\pi$
$557$	1.44995e6	0.198023	0.0990114	+	0.995086i	$0.468432\pi$
$558$	0	0
$559$	−3.59593e6	−0.486723
$560$	−2.93142e7	−3.95010
$561$	0	0
$562$	2.56151e7	3.42102
$563$	−9.95975e6	−1.32427	−0.662137	−	0.749383i	$-0.730350\pi$
$563$	−9.95975e6	−1.32427	−0.662137	+	0.749383i	$0.730350\pi$
$564$	0	0
$565$	−2.92751e6	−0.385813
$566$	2.60046e7	3.41200
$567$	0	0
$568$	2.38759e7	3.10520
$569$	−6.34532e6	−0.821624	−0.410812	−	0.911720i	$-0.634755\pi$
$569$	−6.34532e6	−0.821624	−0.410812	+	0.911720i	$0.634755\pi$
$570$	0	0
$571$	−600766.	−0.0771108	−0.0385554	−	0.999256i	$-0.512276\pi$
$571$	−600766.	−0.0771108	−0.0385554	+	0.999256i	$0.512276\pi$
$572$	−4.33493e7	−5.53977
$573$	0	0
$574$	1.10378e7	1.39830
$575$	1.16850e6	0.147387
$576$	0	0
$577$	−8.08792e6	−1.01134	−0.505670	−	0.862727i	$-0.668755\pi$
$577$	−8.08792e6	−1.01134	−0.505670	+	0.862727i	$0.668755\pi$
$578$	3.87803e6	0.482828
$579$	0	0
$580$	−2.40003e7	−2.96242
$581$	9.20585e6	1.13142
$582$	0	0
$583$	−2.94066e6	−0.358322
$584$	7.20399e6	0.874059
$585$	0	0
$586$	−2.35567e7	−2.83380
$587$	1.39739e7	1.67387	0.836936	−	0.547301i	$-0.184345\pi$
$587$	1.39739e7	1.67387	0.836936	+	0.547301i	$0.184345\pi$
$588$	0	0
$589$	−4.80389e6	−0.570564
$590$	−5.18787e6	−0.613563
$591$	0	0
$592$	3.24605e6	0.380672
$593$	1.19611e7	1.39680	0.698399	−	0.715709i	$-0.253896\pi$
$593$	1.19611e7	1.39680	0.698399	+	0.715709i	$0.253896\pi$
$594$	0	0
$595$	6.51595e6	0.754546
$596$	4.98216e6	0.574515
$597$	0	0
$598$	−5.97240e6	−0.682961
$599$	5.90056e6	0.671933	0.335967	−	0.941874i	$-0.390937\pi$
$599$	5.90056e6	0.671933	0.335967	+	0.941874i	$0.390937\pi$
$600$	0	0
$601$	−2.03654e6	−0.229988	−0.114994	−	0.993366i	$-0.536685\pi$
$601$	−2.03654e6	−0.229988	−0.114994	+	0.993366i	$0.536685\pi$
$602$	−8.27883e6	−0.931060
$603$	0	0
$604$	7.42233e6	0.827843
$605$	−1.56804e6	−0.174168
$606$	0	0
$607$	−1.20672e7	−1.32933	−0.664666	−	0.747141i	$-0.731426\pi$
$607$	−1.20672e7	−1.32933	−0.664666	+	0.747141i	$0.731426\pi$
$608$	4.25308e7	4.66599
$609$	0	0
$610$	−5.55578e6	−0.604534
$611$	6.27369e6	0.679861
$612$	0	0
$613$	4.90686e6	0.527415	0.263708	−	0.964603i	$-0.415055\pi$
$613$	4.90686e6	0.527415	0.263708	+	0.964603i	$0.415055\pi$
$614$	−2.68828e7	−2.87775
$615$	0	0
$616$	−6.54888e7	−6.95369
$617$	1.10702e7	1.17070	0.585348	−	0.810782i	$-0.300958\pi$
$617$	1.10702e7	1.17070	0.585348	+	0.810782i	$0.300958\pi$
$618$	0	0
$619$	−1.26986e7	−1.33208	−0.666038	−	0.745918i	$-0.732011\pi$
$619$	−1.26986e7	−1.33208	−0.666038	+	0.745918i	$0.732011\pi$
$620$	−9.62948e6	−1.00606
$621$	0	0
$622$	1.31091e7	1.35861
$623$	1.23147e6	0.127117
$624$	0	0
$625$	2.01635e6	0.206474
$626$	3.24842e7	3.31312
$627$	0	0
$628$	−2.28985e7	−2.31690
$629$	−721531.	−0.0727157
$630$	0	0
$631$	−7.58417e6	−0.758289	−0.379145	−	0.925337i	$-0.623782\pi$
$631$	−7.58417e6	−0.758289	−0.379145	+	0.925337i	$0.623782\pi$
$632$	6.66964e7	6.64217
$633$	0	0
$634$	−4.73033e6	−0.467378
$635$	4.59394e6	0.452117
$636$	0	0
$637$	−2.66324e7	−2.60053
$638$	−4.39585e7	−4.27554
$639$	0	0
$640$	3.47728e7	3.35575
$641$	6.75938e6	0.649773	0.324886	−	0.945753i	$-0.394674\pi$
$641$	6.75938e6	0.649773	0.324886	+	0.945753i	$0.394674\pi$
$642$	0	0
$643$	8.10870e6	0.773435	0.386717	−	0.922198i	$-0.373609\pi$
$643$	8.10870e6	0.773435	0.386717	+	0.922198i	$0.373609\pi$
$644$	−1.02322e7	−0.972194
$645$	0	0
$646$	−1.62815e7	−1.53501
$647$	−1.57471e7	−1.47890	−0.739450	−	0.673211i	$-0.764914\pi$
$647$	−1.57471e7	−1.47890	−0.739450	+	0.673211i	$0.764914\pi$
$648$	0	0
$649$	−7.07094e6	−0.658969
$650$	−2.49383e7	−2.31518
$651$	0	0
$652$	−2.64502e7	−2.43675
$653$	2.57497e6	0.236314	0.118157	−	0.992995i	$-0.462301\pi$
$653$	2.57497e6	0.236314	0.118157	+	0.992995i	$0.462301\pi$
$654$	0	0
$655$	6.76925e6	0.616506
$656$	−2.21327e7	−2.00805
$657$	0	0
$658$	1.44438e7	1.30052
$659$	8.95517e6	0.803268	0.401634	−	0.915800i	$-0.368442\pi$
$659$	8.95517e6	0.803268	0.401634	+	0.915800i	$0.368442\pi$
$660$	0	0
$661$	2.12702e7	1.89351	0.946755	−	0.321956i	$-0.104340\pi$
$661$	2.12702e7	1.89351	0.946755	+	0.321956i	$0.104340\pi$
$662$	1.71954e7	1.52499
$663$	0	0
$664$	−3.02564e7	−2.66316
$665$	8.83994e6	0.775167
$666$	0	0
$667$	−4.50683e6	−0.392244
$668$	−2.91939e7	−2.53134
$669$	0	0
$670$	−4.92038e6	−0.423460
$671$	−7.57238e6	−0.649271
$672$	0	0
$673$	1.18368e7	1.00739	0.503694	−	0.863882i	$-0.331974\pi$
$673$	1.18368e7	1.00739	0.503694	+	0.863882i	$0.331974\pi$
$674$	−2.61966e7	−2.22124
$675$	0	0
$676$	6.02946e7	5.07472
$677$	−1.25467e7	−1.05210	−0.526051	−	0.850453i	$-0.676328\pi$
$677$	−1.25467e7	−1.05210	−0.526051	+	0.850453i	$0.676328\pi$
$678$	0	0
$679$	1.68718e7	1.40439
$680$	−2.14157e7	−1.77607
$681$	0	0
$682$	−1.76372e7	−1.45201
$683$	1.53179e7	1.25646	0.628230	−	0.778028i	$-0.283780\pi$
$683$	1.53179e7	1.25646	0.628230	+	0.778028i	$0.283780\pi$
$684$	0	0
$685$	3.26328e6	0.265722
$686$	−2.22530e7	−1.80542
$687$	0	0
$688$	1.66005e7	1.33706
$689$	6.43444e6	0.516372
$690$	0	0
$691$	1.27756e7	1.01785	0.508927	−	0.860810i	$-0.330042\pi$
$691$	1.27756e7	1.01785	0.508927	+	0.860810i	$0.330042\pi$
$692$	−5.03938e7	−4.00048
$693$	0	0
$694$	9.05247e6	0.713458
$695$	−6.30113e6	−0.494831
$696$	0	0
$697$	4.91964e6	0.383576
$698$	−3.12741e7	−2.42966
$699$	0	0
$700$	−4.27254e7	−3.29565
$701$	2.07822e7	1.59733	0.798667	−	0.601774i	$-0.205539\pi$
$701$	2.07822e7	1.59733	0.798667	+	0.601774i	$0.205539\pi$
$702$	0	0
$703$	−978873.	−0.0747030
$704$	8.73450e7	6.64212
$705$	0	0
$706$	4.91870e7	3.71397
$707$	146475.	0.0110209
$708$	0	0
$709$	2.11536e7	1.58041	0.790205	−	0.612843i	$-0.209974\pi$
$709$	2.11536e7	1.58041	0.790205	+	0.612843i	$0.209974\pi$
$710$	−1.18326e7	−0.880913
$711$	0	0
$712$	−4.04742e6	−0.299211
$713$	−1.80825e6	−0.133209
$714$	0	0
$715$	1.40971e7	1.03125
$716$	1.19573e7	0.871667
$717$	0	0
$718$	3.36845e7	2.43848
$719$	−6.09514e6	−0.439705	−0.219853	−	0.975533i	$-0.570558\pi$
$719$	−6.09514e6	−0.439705	−0.219853	+	0.975533i	$0.570558\pi$
$720$	0	0
$721$	−6.73849e6	−0.482753
$722$	5.60335e6	0.400041
$723$	0	0
$724$	−5.40510e7	−3.83228
$725$	−1.88187e7	−1.32967
$726$	0	0
$727$	6.96668e6	0.488866	0.244433	−	0.969666i	$-0.421398\pi$
$727$	6.96668e6	0.488866	0.244433	+	0.969666i	$0.421398\pi$
$728$	1.43296e8	10.0208
$729$	0	0
$730$	−3.57020e6	−0.247962
$731$	−3.68995e6	−0.255404
$732$	0	0
$733$	1.65360e7	1.13676	0.568381	−	0.822765i	$-0.307570\pi$
$733$	1.65360e7	1.13676	0.568381	+	0.822765i	$0.307570\pi$
$734$	4.60628e7	3.15581
$735$	0	0
$736$	1.60091e7	1.08936
$737$	−6.70635e6	−0.454797
$738$	0	0
$739$	120477.	0.00811510	0.00405755	−	0.999992i	$-0.498708\pi$
$739$	120477.	0.00811510	0.00405755	+	0.999992i	$0.498708\pi$
$740$	−1.96217e6	−0.131722
$741$	0	0
$742$	1.48139e7	0.987777
$743$	−30872.8	−0.00205165	−0.00102583	−	0.999999i	$-0.500327\pi$
$743$	−30872.8	−0.00205165	−0.00102583	+	0.999999i	$0.500327\pi$
$744$	0	0
$745$	−1.62018e6	−0.106948
$746$	−2.11790e7	−1.39334
$747$	0	0
$748$	−4.44827e7	−2.90695
$749$	−1.95933e7	−1.27615
$750$	0	0
$751$	4.54723e6	0.294203	0.147101	−	0.989121i	$-0.453006\pi$
$751$	4.54723e6	0.294203	0.147101	+	0.989121i	$0.453006\pi$
$752$	−2.89623e7	−1.86762
$753$	0	0
$754$	9.61852e7	6.16141
$755$	−2.41372e6	−0.154106
$756$	0	0
$757$	−1.56951e7	−0.995464	−0.497732	−	0.867331i	$-0.665834\pi$
$757$	−1.56951e7	−0.995464	−0.497732	+	0.867331i	$0.665834\pi$
$758$	−2.22375e7	−1.40577
$759$	0	0
$760$	−2.90538e7	−1.82461
$761$	−1.33501e6	−0.0835647	−0.0417823	−	0.999127i	$-0.513304\pi$
$761$	−1.33501e6	−0.0835647	−0.0417823	+	0.999127i	$0.513304\pi$
$762$	0	0
$763$	−2.86013e7	−1.77858
$764$	−3.93608e7	−2.43967
$765$	0	0
$766$	2.92711e7	1.80247
$767$	1.54719e7	0.949629
$768$	0	0
$769$	−1.36782e7	−0.834089	−0.417044	−	0.908886i	$-0.636934\pi$
$769$	−1.36782e7	−0.834089	−0.417044	+	0.908886i	$0.636934\pi$
$770$	3.24553e7	1.97269
$771$	0	0
$772$	−4.48881e7	−2.71074
$773$	5.44551e6	0.327786	0.163893	−	0.986478i	$-0.447595\pi$
$773$	5.44551e6	0.327786	0.163893	+	0.986478i	$0.447595\pi$
$774$	0	0
$775$	−7.55050e6	−0.451566
$776$	−5.54518e7	−3.30568
$777$	0	0
$778$	−3.88706e7	−2.30236
$779$	6.67429e6	0.394059
$780$	0	0
$781$	−1.61275e7	−0.946104
$782$	−6.12856e6	−0.358378
$783$	0	0
$784$	1.22948e8	7.14383
$785$	7.44652e6	0.431300
$786$	0	0
$787$	2.17109e7	1.24951	0.624757	−	0.780820i	$-0.285198\pi$
$787$	2.17109e7	1.24951	0.624757	+	0.780820i	$0.285198\pi$
$788$	−4.88386e7	−2.80187
$789$	0	0
$790$	−3.30538e7	−1.88432
$791$	2.01006e7	1.14227
$792$	0	0
$793$	1.65691e7	0.935654
$794$	1.99416e7	1.12256
$795$	0	0
$796$	7.42038e7	4.15091
$797$	2.17776e6	0.121441	0.0607204	−	0.998155i	$-0.480660\pi$
$797$	2.17776e6	0.121441	0.0607204	+	0.998155i	$0.480660\pi$
$798$	0	0
$799$	6.43773e6	0.356751
$800$	6.68476e7	3.69285
$801$	0	0
$802$	−4.64689e7	−2.55109
$803$	−4.86608e6	−0.266312
$804$	0	0
$805$	3.32747e6	0.180977
$806$	3.85918e7	2.09246
$807$	0	0
$808$	−481413.	−0.0259411
$809$	8.61138e6	0.462596	0.231298	−	0.972883i	$-0.425703\pi$
$809$	8.61138e6	0.462596	0.231298	+	0.972883i	$0.425703\pi$
$810$	0	0
$811$	−2.23446e7	−1.19294	−0.596472	−	0.802634i	$-0.703431\pi$
$811$	−2.23446e7	−1.19294	−0.596472	+	0.802634i	$0.703431\pi$
$812$	1.64789e8	8.77075
$813$	0	0
$814$	−3.59388e6	−0.190109
$815$	8.60154e6	0.453609
$816$	0	0
$817$	−5.00602e6	−0.262384
$818$	1.41053e7	0.737055
$819$	0	0
$820$	1.33787e7	0.694833
$821$	−1.71095e7	−0.885889	−0.442945	−	0.896549i	$-0.646066\pi$
$821$	−1.71095e7	−0.885889	−0.442945	+	0.896549i	$0.646066\pi$
$822$	0	0
$823$	−2.07720e7	−1.06900	−0.534501	−	0.845168i	$-0.679501\pi$
$823$	−2.07720e7	−1.06900	−0.534501	+	0.845168i	$0.679501\pi$
$824$	2.21471e7	1.13631
$825$	0	0
$826$	3.56205e7	1.81656
$827$	8.57119e6	0.435790	0.217895	−	0.975972i	$-0.430081\pi$
$827$	8.57119e6	0.435790	0.217895	+	0.975972i	$0.430081\pi$
$828$	0	0
$829$	2.50024e7	1.26356	0.631779	−	0.775149i	$-0.282325\pi$
$829$	2.50024e7	1.26356	0.631779	+	0.775149i	$0.282325\pi$
$830$	1.49946e7	0.755511
$831$	0	0
$832$	−1.91119e8	−9.57184
$833$	−2.73288e7	−1.36461
$834$	0	0
$835$	9.49376e6	0.471218
$836$	−6.03480e7	−2.98639
$837$	0	0
$838$	−4.26670e7	−2.09885
$839$	−1.24284e7	−0.609550	−0.304775	−	0.952424i	$-0.598581\pi$
$839$	−1.24284e7	−0.609550	−0.304775	+	0.952424i	$0.598581\pi$
$840$	0	0
$841$	5.20711e7	2.53867
$842$	−6.49843e6	−0.315884
$843$	0	0
$844$	−2.07770e7	−1.00399
$845$	−1.96076e7	−0.944677
$846$	0	0
$847$	1.07663e7	0.515655
$848$	−2.97044e7	−1.41851
$849$	0	0
$850$	−2.55904e7	−1.21487
$851$	−368461.	−0.0174408
$852$	0	0
$853$	2.79770e7	1.31652	0.658261	−	0.752790i	$-0.271292\pi$
$853$	2.79770e7	1.31652	0.658261	+	0.752790i	$0.271292\pi$
$854$	3.81466e7	1.78983
$855$	0	0
$856$	6.43962e7	3.00383
$857$	−3.81515e7	−1.77443	−0.887217	−	0.461352i	$-0.847365\pi$
$857$	−3.81515e7	−1.77443	−0.887217	+	0.461352i	$0.847365\pi$
$858$	0	0
$859$	2.68510e7	1.24159	0.620794	−	0.783974i	$-0.286810\pi$
$859$	2.68510e7	1.24159	0.620794	+	0.783974i	$0.286810\pi$
$860$	−1.00347e7	−0.462654
$861$	0	0
$862$	−1.11004e7	−0.508829
$863$	3.13336e7	1.43213	0.716067	−	0.698032i	$-0.245941\pi$
$863$	3.13336e7	1.43213	0.716067	+	0.698032i	$0.245941\pi$
$864$	0	0
$865$	1.63879e7	0.744703
$866$	−5.02442e7	−2.27662
$867$	0	0
$868$	6.61171e7	2.97861
$869$	−4.50515e7	−2.02376
$870$	0	0
$871$	1.46741e7	0.655400
$872$	9.40024e7	4.18647
$873$	0	0
$874$	−8.31438e6	−0.368172
$875$	3.35508e7	1.48143
$876$	0	0
$877$	2.30631e7	1.01256	0.506278	−	0.862371i	$-0.331021\pi$
$877$	2.30631e7	1.01256	0.506278	+	0.862371i	$0.331021\pi$
$878$	−2.00219e7	−0.876533
$879$	0	0
$880$	−6.50786e7	−2.83290
$881$	−1.92775e7	−0.836779	−0.418390	−	0.908268i	$-0.637405\pi$
$881$	−1.92775e7	−0.836779	−0.418390	+	0.908268i	$0.637405\pi$
$882$	0	0
$883$	−8.88633e6	−0.383549	−0.191774	−	0.981439i	$-0.561424\pi$
$883$	−8.88633e6	−0.383549	−0.191774	+	0.981439i	$0.561424\pi$
$884$	9.73323e7	4.18915
$885$	0	0
$886$	−6.68341e7	−2.86032
$887$	−2.71081e7	−1.15689	−0.578443	−	0.815723i	$-0.696339\pi$
$887$	−2.71081e7	−1.15689	−0.578443	+	0.815723i	$0.696339\pi$
$888$	0	0
$889$	−3.15425e7	−1.33857
$890$	2.00584e6	0.0848832
$891$	0	0
$892$	−4.78773e6	−0.201473
$893$	8.73382e6	0.366501
$894$	0	0
$895$	−3.88848e6	−0.162264
$896$	−2.38754e8	−9.93528
$897$	0	0
$898$	2.54964e7	1.05509
$899$	2.91217e7	1.20176
$900$	0	0
$901$	6.60268e6	0.270962
$902$	2.45042e7	1.00282
$903$	0	0
$904$	−6.60636e7	−2.68869
$905$	1.75772e7	0.713393
$906$	0	0
$907$	1.04286e7	0.420930	0.210465	−	0.977601i	$-0.432502\pi$
$907$	1.04286e7	0.420930	0.210465	+	0.977601i	$0.432502\pi$
$908$	−1.11822e8	−4.50102
$909$	0	0
$910$	−7.10153e7	−2.84281
$911$	4.42057e7	1.76475	0.882373	−	0.470550i	$-0.155944\pi$
$911$	4.42057e7	1.76475	0.882373	+	0.470550i	$0.155944\pi$
$912$	0	0
$913$	2.04373e7	0.811422
$914$	−9.46179e6	−0.374635
$915$	0	0
$916$	7.51384e7	2.95885
$917$	−4.64784e7	−1.82527
$918$	0	0
$919$	−1.95075e7	−0.761924	−0.380962	−	0.924591i	$-0.624407\pi$
$919$	−1.95075e7	−0.761924	−0.380962	+	0.924591i	$0.624407\pi$
$920$	−1.09362e7	−0.425989
$921$	0	0
$922$	6.29364e7	2.43823
$923$	3.52884e7	1.36341
$924$	0	0
$925$	−1.53854e6	−0.0591228
$926$	−7.86069e7	−3.01254
$927$	0	0
$928$	−2.57826e8	−9.82782
$929$	1.23158e7	0.468192	0.234096	−	0.972213i	$-0.424787\pi$
$929$	1.23158e7	0.468192	0.234096	+	0.972213i	$0.424787\pi$
$930$	0	0
$931$	−3.70759e7	−1.40190
$932$	8.57061e7	3.23200
$933$	0	0
$934$	−8.29971e7	−3.11312
$935$	1.44656e7	0.541139
$936$	0	0
$937$	−3.39707e7	−1.26403	−0.632013	−	0.774958i	$-0.717771\pi$
$937$	−3.39707e7	−1.26403	−0.632013	+	0.774958i	$0.717771\pi$
$938$	3.37839e7	1.25373
$939$	0	0
$940$	1.75071e7	0.646241
$941$	−4.85861e7	−1.78870	−0.894351	−	0.447366i	$-0.852362\pi$
$941$	−4.85861e7	−1.78870	−0.894351	+	0.447366i	$0.852362\pi$
$942$	0	0
$943$	2.51229e6	0.0920005
$944$	−7.14254e7	−2.60869
$945$	0	0
$946$	−1.83793e7	−0.667730
$947$	5.13197e7	1.85956	0.929779	−	0.368119i	$-0.119998\pi$
$947$	5.13197e7	1.85956	0.929779	+	0.368119i	$0.119998\pi$
$948$	0	0
$949$	1.06474e7	0.383778
$950$	−3.47175e7	−1.24807
$951$	0	0
$952$	1.47042e8	5.25835
$953$	−5.02504e7	−1.79228	−0.896142	−	0.443767i	$-0.853642\pi$
$953$	−5.02504e7	−1.79228	−0.896142	+	0.443767i	$0.853642\pi$
$954$	0	0
$955$	1.28000e7	0.454153
$956$	−3.96287e7	−1.40238
$957$	0	0
$958$	−2.32376e7	−0.818044
$959$	−2.24060e7	−0.786716
$960$	0	0
$961$	−1.69448e7	−0.591873
$962$	7.86373e6	0.273962
$963$	0	0
$964$	1.01372e8	3.51338
$965$	1.45975e7	0.504614
$966$	0	0
$967$	3.16466e6	0.108833	0.0544165	−	0.998518i	$-0.482670\pi$
$967$	3.16466e6	0.108833	0.0544165	+	0.998518i	$0.482670\pi$
$968$	−3.53852e7	−1.21376
$969$	0	0
$970$	2.74811e7	0.937789
$971$	−3.41793e7	−1.16336	−0.581681	−	0.813417i	$-0.697605\pi$
$971$	−3.41793e7	−1.16336	−0.581681	+	0.813417i	$0.697605\pi$
$972$	0	0
$973$	4.32643e7	1.46503
$974$	2.72546e7	0.920538
$975$	0	0
$976$	−7.64907e7	−2.57030
$977$	4.49152e7	1.50542	0.752709	−	0.658353i	$-0.228747\pi$
$977$	4.49152e7	1.50542	0.752709	+	0.658353i	$0.228747\pi$
$978$	0	0
$979$	2.73391e6	0.0911649
$980$	−7.43194e7	−2.47193
$981$	0	0
$982$	6.42274e7	2.12541
$983$	1.80359e7	0.595324	0.297662	−	0.954671i	$-0.403793\pi$
$983$	1.80359e7	0.595324	0.297662	+	0.954671i	$0.403793\pi$
$984$	0	0
$985$	1.58822e7	0.521578
$986$	9.87002e7	3.23315
$987$	0	0
$988$	1.32047e8	4.30364
$989$	−1.88433e6	−0.0612585
$990$	0	0
$991$	−3.55656e7	−1.15039	−0.575197	−	0.818015i	$-0.695075\pi$
$991$	−3.55656e7	−1.15039	−0.575197	+	0.818015i	$0.695075\pi$
$992$	−1.03446e8	−3.33760
$993$	0	0
$994$	8.12437e7	2.60810
$995$	−2.41308e7	−0.772707
$996$	0	0
$997$	−1.79570e7	−0.572133	−0.286066	−	0.958210i	$-0.592348\pi$
$997$	−1.79570e7	−0.572133	−0.286066	+	0.958210i	$0.592348\pi$
$998$	2.16798e7	0.689015
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	207.6.a.h.1.1	✓	10
3.2	odd	2		207.6.a.i.1.10	yes	10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
207.6.a.h.1.1	✓	10	1.1	even	1	trivial
207.6.a.i.1.10	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)

\(f(q)\)	\(=\)	\(q-11.1836 q^{2} +93.0738 q^{4} -30.2673 q^{5} +207.819 q^{7} -683.027 q^{8} +O(q^{10})\)	\(q-11.1836 q^{2} +93.0738 q^{4} -30.2673 q^{5} +207.819 q^{7} -683.027 q^{8} +338.499 q^{10} +461.365 q^{11} -1009.51 q^{13} -2324.17 q^{14} +4660.37 q^{16} -1035.90 q^{17} -1405.37 q^{19} -2817.09 q^{20} -5159.74 q^{22} -529.000 q^{23} -2208.89 q^{25} +11290.0 q^{26} +19342.5 q^{28} +8519.52 q^{29} +3418.23 q^{31} -30263.0 q^{32} +11585.2 q^{34} -6290.11 q^{35} +696.523 q^{37} +15717.2 q^{38} +20673.4 q^{40} -4749.13 q^{41} +3562.06 q^{43} +42941.0 q^{44} +5916.15 q^{46} -6214.60 q^{47} +26381.6 q^{49} +24703.4 q^{50} -93958.8 q^{52} -6373.84 q^{53} -13964.3 q^{55} -141946. q^{56} -95279.2 q^{58} -15326.1 q^{59} -16413.0 q^{61} -38228.3 q^{62} +189319. q^{64} +30555.1 q^{65} -14535.9 q^{67} -96415.5 q^{68} +70346.3 q^{70} -34956.0 q^{71} -10547.1 q^{73} -7789.66 q^{74} -130803. q^{76} +95880.2 q^{77} -97648.3 q^{79} -141057. q^{80} +53112.5 q^{82} +44297.5 q^{83} +31354.0 q^{85} -39836.8 q^{86} -315125. q^{88} +5925.70 q^{89} -209795. q^{91} -49236.0 q^{92} +69501.8 q^{94} +42536.8 q^{95} +81185.3 q^{97} -295042. q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q - 8 q^{2} + 192 q^{4} - 100 q^{5} + 20 q^{7} - 384 q^{8}+O(q^{10}) \) 10 * q - 8 * q^2 + 192 * q^4 - 100 * q^5 + 20 * q^7 - 384 * q^8	\( 10 q - 8 q^{2} + 192 q^{4} - 100 q^{5} + 20 q^{7} - 384 q^{8} - 250 q^{10} - 460 q^{11} + 464 q^{13} - 3676 q^{14} + 4612 q^{16} - 4756 q^{17} - 1780 q^{19} - 10314 q^{20} - 4214 q^{22} - 5290 q^{23} + 1330 q^{25} + 5152 q^{26} + 7072 q^{28} - 4048 q^{29} + 2816 q^{31} - 27436 q^{32} + 420 q^{34} - 9452 q^{35} + 2872 q^{37} - 31038 q^{38} + 2618 q^{40} - 34056 q^{41} + 7316 q^{43} - 33562 q^{44} + 4232 q^{46} - 49300 q^{47} + 45118 q^{49} - 44764 q^{50} - 25120 q^{52} - 86676 q^{53} - 2120 q^{55} - 290684 q^{56} - 87408 q^{58} - 67100 q^{59} - 40432 q^{61} - 230992 q^{62} + 136776 q^{64} - 184000 q^{65} - 50108 q^{67} - 270592 q^{68} + 117456 q^{70} - 238584 q^{71} - 13804 q^{73} - 150074 q^{74} - 197622 q^{76} - 116248 q^{77} - 9228 q^{79} - 313010 q^{80} - 68604 q^{82} - 155300 q^{83} + 80444 q^{85} + 80914 q^{86} - 237738 q^{88} - 213732 q^{89} - 264352 q^{91} - 101568 q^{92} + 140280 q^{94} + 123612 q^{95} + 42516 q^{97} + 151388 q^{98}+O(q^{100}) \) 10 * q - 8 * q^2 + 192 * q^4 - 100 * q^5 + 20 * q^7 - 384 * q^8 - 250 * q^10 - 460 * q^11 + 464 * q^13 - 3676 * q^14 + 4612 * q^16 - 4756 * q^17 - 1780 * q^19 - 10314 * q^20 - 4214 * q^22 - 5290 * q^23 + 1330 * q^25 + 5152 * q^26 + 7072 * q^28 - 4048 * q^29 + 2816 * q^31 - 27436 * q^32 + 420 * q^34 - 9452 * q^35 + 2872 * q^37 - 31038 * q^38 + 2618 * q^40 - 34056 * q^41 + 7316 * q^43 - 33562 * q^44 + 4232 * q^46 - 49300 * q^47 + 45118 * q^49 - 44764 * q^50 - 25120 * q^52 - 86676 * q^53 - 2120 * q^55 - 290684 * q^56 - 87408 * q^58 - 67100 * q^59 - 40432 * q^61 - 230992 * q^62 + 136776 * q^64 - 184000 * q^65 - 50108 * q^67 - 270592 * q^68 + 117456 * q^70 - 238584 * q^71 - 13804 * q^73 - 150074 * q^74 - 197622 * q^76 - 116248 * q^77 - 9228 * q^79 - 313010 * q^80 - 68604 * q^82 - 155300 * q^83 + 80444 * q^85 + 80914 * q^86 - 237738 * q^88 - 213732 * q^89 - 264352 * q^91 - 101568 * q^92 + 140280 * q^94 + 123612 * q^95 + 42516 * q^97 + 151388 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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