LMFDB - Embedded newform 207.6.a.a.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:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{29}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 7 $ x^2 - x - 7
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 69)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$3.19258$ 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)

$f(q)$	$=$	$q-10.7703 q^{2} +84.0000 q^{4} -52.3852 q^{5} -118.237 q^{7} -560.057 q^{8} +O(q^{10})$	\(q-10.7703 q^{2} +84.0000 q^{4} -52.3852 q^{5} -118.237 q^{7} -560.057 q^{8} +564.205 q^{10} -741.598 q^{11} -542.502 q^{13} +1273.45 q^{14} +3344.00 q^{16} +834.993 q^{17} -1309.79 q^{19} -4400.35 q^{20} +7987.25 q^{22} -529.000 q^{23} -380.795 q^{25} +5842.93 q^{26} -9931.89 q^{28} -5536.45 q^{29} -7267.14 q^{31} -18094.2 q^{32} -8993.15 q^{34} +6193.85 q^{35} -10446.7 q^{37} +14106.8 q^{38} +29338.7 q^{40} +4630.45 q^{41} -9201.23 q^{43} -62294.2 q^{44} +5697.50 q^{46} -15272.3 q^{47} -2827.06 q^{49} +4101.28 q^{50} -45570.2 q^{52} +36202.8 q^{53} +38848.7 q^{55} +66219.4 q^{56} +59629.4 q^{58} -6436.34 q^{59} -2637.86 q^{61} +78269.5 q^{62} +87872.0 q^{64} +28419.1 q^{65} +21068.5 q^{67} +70139.4 q^{68} -66709.9 q^{70} -74668.2 q^{71} +80742.0 q^{73} +112514. q^{74} -110022. q^{76} +87684.2 q^{77} +39107.7 q^{79} -175176. q^{80} -49871.4 q^{82} -101465. q^{83} -43741.2 q^{85} +99100.3 q^{86} +415337. q^{88} -22407.6 q^{89} +64143.7 q^{91} -44436.0 q^{92} +164488. q^{94} +68613.4 q^{95} -29724.8 q^{97} +30448.3 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 168 q^{4} - 94 q^{5} - 118 q^{7}+O(q^{10}) $ 2 * q + 168 * q^4 - 94 * q^5 - 118 * q^7	$ 2 q + 168 q^{4} - 94 q^{5} - 118 q^{7} + 116 q^{10} - 320 q^{11} - 288 q^{13} + 1276 q^{14} + 6688 q^{16} + 1810 q^{17} + 730 q^{19} - 7896 q^{20} + 12528 q^{22} - 1058 q^{23} - 1774 q^{25} + 8584 q^{26} - 9912 q^{28} - 8208 q^{29} + 1772 q^{31} + 1508 q^{34} + 6184 q^{35} - 23112 q^{37} + 36076 q^{38} + 6032 q^{40} - 5516 q^{41} + 10322 q^{43} - 26880 q^{44} - 42952 q^{47} - 19634 q^{49} - 10904 q^{50} - 24192 q^{52} + 25350 q^{53} + 21304 q^{55} + 66352 q^{56} + 30856 q^{58} - 18344 q^{59} + 37224 q^{61} + 175624 q^{62} + 175744 q^{64} + 17828 q^{65} - 7482 q^{67} + 152040 q^{68} - 66816 q^{70} - 126848 q^{71} + 137660 q^{73} - 23896 q^{74} + 61320 q^{76} + 87784 q^{77} + 62286 q^{79} - 314336 q^{80} - 159152 q^{82} - 83120 q^{83} - 84316 q^{85} + 309372 q^{86} + 651456 q^{88} - 69770 q^{89} + 64204 q^{91} - 88872 q^{92} - 133632 q^{94} - 16272 q^{95} - 170104 q^{97} - 150568 q^{98}+O(q^{100}) $ 2 * q + 168 * q^4 - 94 * q^5 - 118 * q^7 + 116 * q^10 - 320 * q^11 - 288 * q^13 + 1276 * q^14 + 6688 * q^16 + 1810 * q^17 + 730 * q^19 - 7896 * q^20 + 12528 * q^22 - 1058 * q^23 - 1774 * q^25 + 8584 * q^26 - 9912 * q^28 - 8208 * q^29 + 1772 * q^31 + 1508 * q^34 + 6184 * q^35 - 23112 * q^37 + 36076 * q^38 + 6032 * q^40 - 5516 * q^41 + 10322 * q^43 - 26880 * q^44 - 42952 * q^47 - 19634 * q^49 - 10904 * q^50 - 24192 * q^52 + 25350 * q^53 + 21304 * q^55 + 66352 * q^56 + 30856 * q^58 - 18344 * q^59 + 37224 * q^61 + 175624 * q^62 + 175744 * q^64 + 17828 * q^65 - 7482 * q^67 + 152040 * q^68 - 66816 * q^70 - 126848 * q^71 + 137660 * q^73 - 23896 * q^74 + 61320 * q^76 + 87784 * q^77 + 62286 * q^79 - 314336 * q^80 - 159152 * q^82 - 83120 * q^83 - 84316 * q^85 + 309372 * q^86 + 651456 * q^88 - 69770 * q^89 + 64204 * q^91 - 88872 * q^92 - 133632 * q^94 - 16272 * q^95 - 170104 * q^97 - 150568 * 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 $

$2$	−10.7703	−1.90394	−0.951972	−	0.306186i	$-0.900947\pi$
$2$	−10.7703	−1.90394	−0.951972	+	0.306186i	$0.900947\pi$
$3$	0	0
$3$	0	0
$4$	84.0000	2.62500
$5$	−52.3852	−0.937094	−0.468547	−	0.883438i	$-0.655222\pi$
$5$	−52.3852	−0.937094	−0.468547	+	0.883438i	$0.655222\pi$
$6$	0	0
$7$	−118.237	−0.912027	−0.456013	−	0.889973i	$-0.650723\pi$
$7$	−118.237	−0.912027	−0.456013	+	0.889973i	$0.650723\pi$
$8$	−560.057	−3.09391
$9$	0	0
$10$	564.205	1.78417
$11$	−741.598	−1.84794	−0.923968	−	0.382471i	$-0.875073\pi$
$11$	−741.598	−1.84794	−0.923968	+	0.382471i	$0.875073\pi$
$12$	0	0
$13$	−542.502	−0.890314	−0.445157	−	0.895453i	$-0.646852\pi$
$13$	−542.502	−0.890314	−0.445157	+	0.895453i	$0.646852\pi$
$14$	1273.45	1.73645
$15$	0	0
$16$	3344.00	3.26562
$17$	834.993	0.700746	0.350373	−	0.936610i	$-0.386055\pi$
$17$	834.993	0.700746	0.350373	+	0.936610i	$0.386055\pi$
$18$	0	0
$19$	−1309.79	−0.832370	−0.416185	−	0.909280i	$-0.636633\pi$
$19$	−1309.79	−0.832370	−0.416185	+	0.909280i	$0.636633\pi$
$20$	−4400.35	−2.45987
$21$	0	0
$22$	7987.25	3.51836
$23$	−529.000	−0.208514
$23$	−529.000	−0.208514
$24$	0	0
$25$	−380.795	−0.121854
$26$	5842.93	1.69511
$27$	0	0
$28$	−9931.89	−2.39407
$29$	−5536.45	−1.22247	−0.611233	−	0.791451i	$-0.709326\pi$
$29$	−5536.45	−1.22247	−0.611233	+	0.791451i	$0.709326\pi$
$30$	0	0
$31$	−7267.14	−1.35819	−0.679093	−	0.734052i	$-0.737627\pi$
$31$	−7267.14	−1.35819	−0.679093	+	0.734052i	$0.737627\pi$
$32$	−18094.2	−3.12366
$33$	0	0
$34$	−8993.15	−1.33418
$35$	6193.85	0.854655
$36$	0	0
$37$	−10446.7	−1.25451	−0.627253	−	0.778815i	$-0.715821\pi$
$37$	−10446.7	−1.25451	−0.627253	+	0.778815i	$0.715821\pi$
$38$	14106.8	1.58479
$39$	0	0
$40$	29338.7	2.89928
$41$	4630.45	0.430193	0.215096	−	0.976593i	$-0.430993\pi$
$41$	4630.45	0.430193	0.215096	+	0.976593i	$0.430993\pi$
$42$	0	0
$43$	−9201.23	−0.758883	−0.379442	−	0.925216i	$-0.623884\pi$
$43$	−9201.23	−0.758883	−0.379442	+	0.925216i	$0.623884\pi$
$44$	−62294.2	−4.85083
$45$	0	0
$46$	5697.50	0.397000
$47$	−15272.3	−1.00846	−0.504231	−	0.863569i	$-0.668224\pi$
$47$	−15272.3	−1.00846	−0.504231	+	0.863569i	$0.668224\pi$
$48$	0	0
$49$	−2827.06	−0.168207
$50$	4101.28	0.232004
$51$	0	0
$52$	−45570.2	−2.33707
$53$	36202.8	1.77032	0.885161	−	0.465285i	$-0.154048\pi$
$53$	36202.8	1.77032	0.885161	+	0.465285i	$0.154048\pi$
$54$	0	0
$55$	38848.7	1.73169
$56$	66219.4	2.82173
$57$	0	0
$58$	59629.4	2.32751
$59$	−6436.34	−0.240718	−0.120359	−	0.992730i	$-0.538405\pi$
$59$	−6436.34	−0.240718	−0.120359	+	0.992730i	$0.538405\pi$
$60$	0	0
$61$	−2637.86	−0.0907668	−0.0453834	−	0.998970i	$-0.514451\pi$
$61$	−2637.86	−0.0907668	−0.0453834	+	0.998970i	$0.514451\pi$
$62$	78269.5	2.58591
$63$	0	0
$64$	87872.0	2.68164
$65$	28419.1	0.834308
$66$	0	0
$67$	21068.5	0.573384	0.286692	−	0.958023i	$-0.407444\pi$
$67$	21068.5	0.573384	0.286692	+	0.958023i	$0.407444\pi$
$68$	70139.4	1.83946
$69$	0	0
$70$	−66709.9	−1.62721
$71$	−74668.2	−1.75788	−0.878941	−	0.476930i	$-0.841750\pi$
$71$	−74668.2	−1.75788	−0.878941	+	0.476930i	$0.841750\pi$
$72$	0	0
$73$	80742.0	1.77334	0.886671	−	0.462402i	$-0.153012\pi$
$73$	80742.0	1.77334	0.886671	+	0.462402i	$0.153012\pi$
$74$	112514.	2.38851
$75$	0	0
$76$	−110022.	−2.18497
$77$	87684.2	1.68537
$78$	0	0
$79$	39107.7	0.705008	0.352504	−	0.935810i	$-0.385330\pi$
$79$	39107.7	0.705008	0.352504	+	0.935810i	$0.385330\pi$
$80$	−175176.	−3.06020
$81$	0	0
$82$	−49871.4	−0.819063
$83$	−101465.	−1.61666	−0.808331	−	0.588728i	$-0.799629\pi$
$83$	−101465.	−1.61666	−0.808331	+	0.588728i	$0.799629\pi$
$84$	0	0
$85$	−43741.2	−0.656665
$86$	99100.3	1.44487
$87$	0	0
$88$	415337.	5.71734
$89$	−22407.6	−0.299861	−0.149930	−	0.988697i	$-0.547905\pi$
$89$	−22407.6	−0.299861	−0.149930	+	0.988697i	$0.547905\pi$
$90$	0	0
$91$	64143.7	0.811990
$92$	−44436.0	−0.547350
$93$	0	0
$94$	164488.	1.92005
$95$	68613.4	0.780009
$96$	0	0
$97$	−29724.8	−0.320767	−0.160384	−	0.987055i	$-0.551273\pi$
$97$	−29724.8	−0.320767	−0.160384	+	0.987055i	$0.551273\pi$
$98$	30448.3	0.320257
$99$	0	0
$100$	−31986.7	−0.319867
$101$	−96859.7	−0.944800	−0.472400	−	0.881384i	$-0.656612\pi$
$101$	−96859.7	−0.944800	−0.472400	+	0.881384i	$0.656612\pi$
$102$	0	0
$103$	2081.11	0.0193287	0.00966434	−	0.999953i	$-0.496924\pi$
$103$	2081.11	0.0193287	0.00966434	+	0.999953i	$0.496924\pi$
$104$	303832.	2.75455
$105$	0	0
$106$	−389916.	−3.37059
$107$	76907.3	0.649393	0.324697	−	0.945818i	$-0.394738\pi$
$107$	76907.3	0.649393	0.324697	+	0.945818i	$0.394738\pi$
$108$	0	0
$109$	−27250.0	−0.219685	−0.109842	−	0.993949i	$-0.535035\pi$
$109$	−27250.0	−0.219685	−0.109842	+	0.993949i	$0.535035\pi$
$110$	−418414.	−3.29704
$111$	0	0
$112$	−395384.	−2.97834
$113$	120738.	0.889505	0.444753	−	0.895653i	$-0.353292\pi$
$113$	120738.	0.889505	0.444753	+	0.895653i	$0.353292\pi$
$114$	0	0
$115$	27711.8	0.195398
$116$	−465062.	−3.20897
$117$	0	0
$118$	69321.5	0.458314
$119$	−98726.9	−0.639099
$120$	0	0
$121$	388916.	2.41486
$122$	28410.6	0.172815
$123$	0	0
$124$	−610440.	−3.56524
$125$	183652.	1.05128
$126$	0	0
$127$	−140823.	−0.774755	−0.387377	−	0.921921i	$-0.626619\pi$
$127$	−140823.	−0.774755	−0.387377	+	0.921921i	$0.626619\pi$
$128$	−367397.	−1.98203
$129$	0	0
$130$	−306083.	−1.58848
$131$	137026.	0.697630	0.348815	−	0.937192i	$-0.386584\pi$
$131$	137026.	0.697630	0.348815	+	0.937192i	$0.386584\pi$
$132$	0	0
$133$	154865.	0.759144
$134$	−226914.	−1.09169
$135$	0	0
$136$	−467644.	−2.16804
$137$	−73627.5	−0.335149	−0.167575	−	0.985859i	$-0.553594\pi$
$137$	−73627.5	−0.335149	−0.167575	+	0.985859i	$0.553594\pi$
$138$	0	0
$139$	214252.	0.940562	0.470281	−	0.882517i	$-0.344152\pi$
$139$	214252.	0.940562	0.470281	+	0.882517i	$0.344152\pi$
$140$	520284.	2.24347
$141$	0	0
$142$	804201.	3.34691
$143$	402318.	1.64524
$144$	0	0
$145$	290028.	1.14557
$146$	−869618.	−3.37634
$147$	0	0
$148$	−877519.	−3.29308
$149$	−132470.	−0.488823	−0.244412	−	0.969672i	$-0.578595\pi$
$149$	−132470.	−0.488823	−0.244412	+	0.969672i	$0.578595\pi$
$150$	0	0
$151$	171660.	0.612669	0.306334	−	0.951924i	$-0.400897\pi$
$151$	171660.	0.612669	0.306334	+	0.951924i	$0.400897\pi$
$152$	733555.	2.57528
$153$	0	0
$154$	−944387.	−3.20884
$155$	380690.	1.27275
$156$	0	0
$157$	−359852.	−1.16513	−0.582565	−	0.812784i	$-0.697951\pi$
$157$	−359852.	−1.16513	−0.582565	+	0.812784i	$0.697951\pi$
$158$	−421202.	−1.34230
$159$	0	0
$160$	947865.	2.92716
$161$	62547.3	0.190171
$162$	0	0
$163$	454656.	1.34034	0.670168	−	0.742210i	$-0.266222\pi$
$163$	454656.	1.34034	0.670168	+	0.742210i	$0.266222\pi$
$164$	388957.	1.12926
$165$	0	0
$166$	1.09281e6	3.07803
$167$	359837.	0.998424	0.499212	−	0.866480i	$-0.333623\pi$
$167$	359837.	0.998424	0.499212	+	0.866480i	$0.333623\pi$
$168$	0	0
$169$	−76984.4	−0.207341
$170$	471108.	1.25025
$171$	0	0
$172$	−772904.	−1.99207
$173$	308025.	0.782476	0.391238	−	0.920289i	$-0.372047\pi$
$173$	308025.	0.782476	0.391238	+	0.920289i	$0.372047\pi$
$174$	0	0
$175$	45023.9	0.111134
$176$	−2.47990e6	−6.03466
$177$	0	0
$178$	241337.	0.570918
$179$	−628893.	−1.46705	−0.733524	−	0.679663i	$-0.762126\pi$
$179$	−628893.	−1.46705	−0.733524	+	0.679663i	$0.762126\pi$
$180$	0	0
$181$	−667015.	−1.51335	−0.756675	−	0.653791i	$-0.773177\pi$
$181$	−667015.	−1.51335	−0.756675	+	0.653791i	$0.773177\pi$
$182$	−690849.	−1.54598
$183$	0	0
$184$	296270.	0.645124
$185$	547250.	1.17559
$186$	0	0
$187$	−619229.	−1.29493
$188$	−1.28287e6	−2.64721
$189$	0	0
$190$	−738989.	−1.48509
$191$	−573327.	−1.13715	−0.568577	−	0.822630i	$-0.692506\pi$
$191$	−573327.	−1.13715	−0.568577	+	0.822630i	$0.692506\pi$
$192$	0	0
$193$	21198.3	0.0409646	0.0204823	−	0.999790i	$-0.493480\pi$
$193$	21198.3	0.0409646	0.0204823	+	0.999790i	$0.493480\pi$
$194$	320146.	0.610722
$195$	0	0
$196$	−237473.	−0.441544
$197$	342207.	0.628237	0.314118	−	0.949384i	$-0.398291\pi$
$197$	342207.	0.628237	0.314118	+	0.949384i	$0.398291\pi$
$198$	0	0
$199$	−32335.5	−0.0578825	−0.0289412	−	0.999581i	$-0.509214\pi$
$199$	−32335.5	−0.0578825	−0.0289412	+	0.999581i	$0.509214\pi$
$200$	213267.	0.377006
$201$	0	0
$202$	1.04321e6	1.79885
$203$	654613.	1.11492
$204$	0	0
$205$	−242567.	−0.403131
$206$	−22414.3	−0.0368007
$207$	0	0
$208$	−1.81413e6	−2.90743
$209$	971335.	1.53817
$210$	0	0
$211$	−1.03399e6	−1.59886	−0.799430	−	0.600759i	$-0.794865\pi$
$211$	−1.03399e6	−1.59886	−0.799430	+	0.600759i	$0.794865\pi$
$212$	3.04103e6	4.64710
$213$	0	0
$214$	−828317.	−1.23641
$215$	482008.	0.711145
$216$	0	0
$217$	859243.	1.23870
$218$	293491.	0.418267
$219$	0	0
$220$	3.26329e6	4.54568
$221$	−452985.	−0.623884
$222$	0	0
$223$	990187.	1.33338	0.666692	−	0.745333i	$-0.267710\pi$
$223$	990187.	1.33338	0.666692	+	0.745333i	$0.267710\pi$
$224$	2.13940e6	2.84886
$225$	0	0
$226$	−1.30039e6	−1.69357
$227$	26685.3	0.0343723	0.0171861	−	0.999852i	$-0.494529\pi$
$227$	26685.3	0.0343723	0.0171861	+	0.999852i	$0.494529\pi$
$228$	0	0
$229$	150586.	0.189756	0.0948778	−	0.995489i	$-0.469754\pi$
$229$	150586.	0.189756	0.0948778	+	0.995489i	$0.469754\pi$
$230$	−298465.	−0.372026
$231$	0	0
$232$	3.10073e6	3.78220
$233$	−1.29421e6	−1.56176	−0.780881	−	0.624680i	$-0.785230\pi$
$233$	−1.29421e6	−1.56176	−0.780881	+	0.624680i	$0.785230\pi$
$234$	0	0
$235$	800041.	0.945024
$236$	−540652.	−0.631885
$237$	0	0
$238$	1.06332e6	1.21681
$239$	136898.	0.155025	0.0775123	−	0.996991i	$-0.475302\pi$
$239$	136898.	0.155025	0.0775123	+	0.996991i	$0.475302\pi$
$240$	0	0
$241$	−284796.	−0.315858	−0.157929	−	0.987450i	$-0.550482\pi$
$241$	−284796.	−0.315858	−0.157929	+	0.987450i	$0.550482\pi$
$242$	−4.18876e6	−4.59776
$243$	0	0
$244$	−221580.	−0.238263
$245$	148096.	0.157626
$246$	0	0
$247$	710562.	0.741071
$248$	4.07001e6	4.20210
$249$	0	0
$250$	−1.97799e6	−2.00158
$251$	−1.26148e6	−1.26385	−0.631927	−	0.775028i	$-0.717736\pi$
$251$	−1.26148e6	−1.26385	−0.631927	+	0.775028i	$0.717736\pi$
$252$	0	0
$253$	392305.	0.385321
$254$	1.51671e6	1.47509
$255$	0	0
$256$	1.14509e6	1.09204
$257$	−552720.	−0.522002	−0.261001	−	0.965338i	$-0.584053\pi$
$257$	−552720.	−0.522002	−0.261001	+	0.965338i	$0.584053\pi$
$258$	0	0
$259$	1.23518e6	1.14414
$260$	2.38720e6	2.19006
$261$	0	0
$262$	−1.47582e6	−1.32825
$263$	651514.	0.580811	0.290405	−	0.956904i	$-0.406210\pi$
$263$	651514.	0.580811	0.290405	+	0.956904i	$0.406210\pi$
$264$	0	0
$265$	−1.89649e6	−1.65896
$266$	−1.66795e6	−1.44537
$267$	0	0
$268$	1.76975e6	1.50513
$269$	−1.62659e6	−1.37056	−0.685280	−	0.728280i	$-0.740320\pi$
$269$	−1.62659e6	−1.37056	−0.685280	+	0.728280i	$0.740320\pi$
$270$	0	0
$271$	526548.	0.435527	0.217763	−	0.976002i	$-0.430124\pi$
$271$	526548.	0.435527	0.217763	+	0.976002i	$0.430124\pi$
$272$	2.79222e6	2.28837
$273$	0	0
$274$	792992.	0.638106
$275$	282396.	0.225179
$276$	0	0
$277$	102974.	0.0806358	0.0403179	−	0.999187i	$-0.487163\pi$
$277$	102974.	0.0806358	0.0403179	+	0.999187i	$0.487163\pi$
$278$	−2.30756e6	−1.79078
$279$	0	0
$280$	−3.46891e6	−2.64422
$281$	1.79209e6	1.35392	0.676960	−	0.736020i	$-0.263297\pi$
$281$	1.79209e6	1.35392	0.676960	+	0.736020i	$0.263297\pi$
$282$	0	0
$283$	−1.06565e6	−0.790946	−0.395473	−	0.918478i	$-0.629419\pi$
$283$	−1.06565e6	−0.790946	−0.395473	+	0.918478i	$0.629419\pi$
$284$	−6.27213e6	−4.61444
$285$	0	0
$286$	−4.33310e6	−3.13245
$287$	−547489.	−0.392347
$288$	0	0
$289$	−722644.	−0.508955
$290$	−3.12370e6	−2.18109
$291$	0	0
$292$	6.78233e6	4.65502
$293$	−1.85858e6	−1.26477	−0.632387	−	0.774653i	$-0.717925\pi$
$293$	−1.85858e6	−1.26477	−0.632387	+	0.774653i	$0.717925\pi$
$294$	0	0
$295$	337169.	0.225576
$296$	5.85072e6	3.88133
$297$	0	0
$298$	1.42675e6	0.930692
$299$	286984.	0.185643
$300$	0	0
$301$	1.08792e6	0.692122
$302$	−1.84883e6	−1.16649
$303$	0	0
$304$	−4.37993e6	−2.71821
$305$	138185.	0.0850571
$306$	0	0
$307$	−2.13805e6	−1.29471	−0.647355	−	0.762189i	$-0.724125\pi$
$307$	−2.13805e6	−1.29471	−0.647355	+	0.762189i	$0.724125\pi$
$308$	7.36547e6	4.42409
$309$	0	0
$310$	−4.10016e6	−2.42324
$311$	−1.35485e6	−0.794311	−0.397155	−	0.917751i	$-0.630003\pi$
$311$	−1.35485e6	−0.794311	−0.397155	+	0.917751i	$0.630003\pi$
$312$	0	0
$313$	−523789.	−0.302201	−0.151100	−	0.988518i	$-0.548282\pi$
$313$	−523789.	−0.302201	−0.151100	+	0.988518i	$0.548282\pi$
$314$	3.87572e6	2.21834
$315$	0	0
$316$	3.28504e6	1.85065
$317$	−2.59918e6	−1.45274	−0.726371	−	0.687302i	$-0.758795\pi$
$317$	−2.59918e6	−1.45274	−0.726371	+	0.687302i	$0.758795\pi$
$318$	0	0
$319$	4.10582e6	2.25904
$320$	−4.60319e6	−2.51295
$321$	0	0
$322$	−673655.	−0.362074
$323$	−1.09366e6	−0.583280
$324$	0	0
$325$	206582.	0.108489
$326$	−4.89679e6	−2.55192
$327$	0	0
$328$	−2.59331e6	−1.33098
$329$	1.80575e6	0.919744
$330$	0	0
$331$	−1.02887e6	−0.516168	−0.258084	−	0.966122i	$-0.583091\pi$
$331$	−1.02887e6	−0.516168	−0.258084	+	0.966122i	$0.583091\pi$
$332$	−8.52302e6	−4.24374
$333$	0	0
$334$	−3.87557e6	−1.90094
$335$	−1.10367e6	−0.537315
$336$	0	0
$337$	1.13501e6	0.544406	0.272203	−	0.962240i	$-0.412248\pi$
$337$	1.13501e6	0.544406	0.272203	+	0.962240i	$0.412248\pi$
$338$	829147.	0.394766
$339$	0	0
$340$	−3.67426e6	−1.72375
$341$	5.38929e6	2.50984
$342$	0	0
$343$	2.32147e6	1.06544
$344$	5.15322e6	2.34791
$345$	0	0
$346$	−3.31753e6	−1.48979
$347$	−1.08736e6	−0.484787	−0.242393	−	0.970178i	$-0.577932\pi$
$347$	−1.08736e6	−0.484787	−0.242393	+	0.970178i	$0.577932\pi$
$348$	0	0
$349$	147242.	0.0647097	0.0323548	−	0.999476i	$-0.489699\pi$
$349$	147242.	0.0647097	0.0323548	+	0.999476i	$0.489699\pi$
$350$	−484923.	−0.211593
$351$	0	0
$352$	1.34186e7	5.77232
$353$	−1.58645e6	−0.677623	−0.338812	−	0.940854i	$-0.610025\pi$
$353$	−1.58645e6	−0.677623	−0.338812	+	0.940854i	$0.610025\pi$
$354$	0	0
$355$	3.91151e6	1.64730
$356$	−1.88224e6	−0.787135
$357$	0	0
$358$	6.77339e6	2.79318
$359$	1.85640e6	0.760211	0.380106	−	0.924943i	$-0.375888\pi$
$359$	1.85640e6	0.760211	0.380106	+	0.924943i	$0.375888\pi$
$360$	0	0
$361$	−760559.	−0.307160
$362$	7.18397e6	2.88133
$363$	0	0
$364$	5.38807e6	2.13147
$365$	−4.22968e6	−1.66179
$366$	0	0
$367$	−2.01535e6	−0.781061	−0.390531	−	0.920590i	$-0.627708\pi$
$367$	−2.01535e6	−0.781061	−0.390531	+	0.920590i	$0.627708\pi$
$368$	−1.76898e6	−0.680930
$369$	0	0
$370$	−5.89406e6	−2.23826
$371$	−4.28050e6	−1.61458
$372$	0	0
$373$	−1.33526e6	−0.496928	−0.248464	−	0.968641i	$-0.579926\pi$
$373$	−1.33526e6	−0.496928	−0.248464	+	0.968641i	$0.579926\pi$
$374$	6.66930e6	2.46548
$375$	0	0
$376$	8.55336e6	3.12009
$377$	3.00354e6	1.08838
$378$	0	0
$379$	−2.84308e6	−1.01669	−0.508347	−	0.861152i	$-0.669743\pi$
$379$	−2.84308e6	−1.01669	−0.508347	+	0.861152i	$0.669743\pi$
$380$	5.76352e6	2.04752
$381$	0	0
$382$	6.17492e6	2.16508
$383$	−3.29481e6	−1.14771	−0.573857	−	0.818956i	$-0.694553\pi$
$383$	−3.29481e6	−1.14771	−0.573857	+	0.818956i	$0.694553\pi$
$384$	0	0
$385$	−4.59335e6	−1.57935
$386$	−228313.	−0.0779942
$387$	0	0
$388$	−2.49688e6	−0.842014
$389$	4.92017e6	1.64857	0.824283	−	0.566177i	$-0.191578\pi$
$389$	4.92017e6	1.64857	0.824283	+	0.566177i	$0.191578\pi$
$390$	0	0
$391$	−441711.	−0.146116
$392$	1.58331e6	0.520417
$393$	0	0
$394$	−3.68568e6	−1.19613
$395$	−2.04866e6	−0.660659
$396$	0	0
$397$	−4.13215e6	−1.31583	−0.657915	−	0.753092i	$-0.728561\pi$
$397$	−4.13215e6	−1.31583	−0.657915	+	0.753092i	$0.728561\pi$
$398$	348264.	0.110205
$399$	0	0
$400$	−1.27338e6	−0.397930
$401$	3.07921e6	0.956266	0.478133	−	0.878287i	$-0.341314\pi$
$401$	3.07921e6	0.956266	0.478133	+	0.878287i	$0.341314\pi$
$402$	0	0
$403$	3.94244e6	1.20921
$404$	−8.13622e6	−2.48010
$405$	0	0
$406$	−7.05039e6	−2.12275
$407$	7.74722e6	2.31825
$408$	0	0
$409$	−483481.	−0.142913	−0.0714564	−	0.997444i	$-0.522765\pi$
$409$	−483481.	−0.142913	−0.0714564	+	0.997444i	$0.522765\pi$
$410$	2.61252e6	0.767539
$411$	0	0
$412$	174813.	0.0507378
$413$	761012.	0.219541
$414$	0	0
$415$	5.31524e6	1.51496
$416$	9.81612e6	2.78103
$417$	0	0
$418$	−1.04616e7	−2.92858
$419$	−1.51752e6	−0.422279	−0.211140	−	0.977456i	$-0.567718\pi$
$419$	−1.51752e6	−0.422279	−0.211140	+	0.977456i	$0.567718\pi$
$420$	0	0
$421$	3.76256e6	1.03461	0.517306	−	0.855800i	$-0.326935\pi$
$421$	3.76256e6	1.03461	0.517306	+	0.855800i	$0.326935\pi$
$422$	1.11364e7	3.04414
$423$	0	0
$424$	−2.02756e7	−5.47721
$425$	−317961.	−0.0853888
$426$	0	0
$427$	311892.	0.0827818
$428$	6.46021e6	1.70466
$429$	0	0
$430$	−5.19139e6	−1.35398
$431$	−2.80235e6	−0.726658	−0.363329	−	0.931661i	$-0.618360\pi$
$431$	−2.80235e6	−0.726658	−0.363329	+	0.931661i	$0.618360\pi$
$432$	0	0
$433$	3.06756e6	0.786272	0.393136	−	0.919480i	$-0.371390\pi$
$433$	3.06756e6	0.786272	0.393136	+	0.919480i	$0.371390\pi$
$434$	−9.25433e6	−2.35842
$435$	0	0
$436$	−2.28900e6	−0.576672
$437$	692877.	0.173561
$438$	0	0
$439$	−4.01436e6	−0.994158	−0.497079	−	0.867705i	$-0.665594\pi$
$439$	−4.01436e6	−0.994158	−0.497079	+	0.867705i	$0.665594\pi$
$440$	−2.17575e7	−5.35769
$441$	0	0
$442$	4.87880e6	1.18784
$443$	−4.74625e6	−1.14906	−0.574528	−	0.818485i	$-0.694814\pi$
$443$	−4.74625e6	−1.14906	−0.574528	+	0.818485i	$0.694814\pi$
$444$	0	0
$445$	1.17382e6	0.280998
$446$	−1.06646e7	−2.53869
$447$	0	0
$448$	−1.03897e7	−2.44573
$449$	−1.57070e6	−0.367687	−0.183843	−	0.982956i	$-0.558854\pi$
$449$	−1.57070e6	−0.367687	−0.183843	+	0.982956i	$0.558854\pi$
$450$	0	0
$451$	−3.43393e6	−0.794968
$452$	1.01420e7	2.33495
$453$	0	0
$454$	−287410.	−0.0654429
$455$	−3.36018e6	−0.760911
$456$	0	0
$457$	4.94573e6	1.10775	0.553873	−	0.832601i	$-0.313149\pi$
$457$	4.94573e6	1.10775	0.553873	+	0.832601i	$0.313149\pi$
$458$	−1.62186e6	−0.361284
$459$	0	0
$460$	2.32779e6	0.512919
$461$	−4.86491e6	−1.06616	−0.533080	−	0.846065i	$-0.678965\pi$
$461$	−4.86491e6	−1.06616	−0.533080	+	0.846065i	$0.678965\pi$
$462$	0	0
$463$	−2.04379e6	−0.443082	−0.221541	−	0.975151i	$-0.571109\pi$
$463$	−2.04379e6	−0.443082	−0.221541	+	0.975151i	$0.571109\pi$
$464$	−1.85139e7	−3.99211
$465$	0	0
$466$	1.39391e7	2.97351
$467$	6.33570e6	1.34432	0.672159	−	0.740407i	$-0.265367\pi$
$467$	6.33570e6	1.34432	0.672159	+	0.740407i	$0.265367\pi$
$468$	0	0
$469$	−2.49107e6	−0.522942
$470$	−8.61671e6	−1.79927
$471$	0	0
$472$	3.60472e6	0.744760
$473$	6.82362e6	1.40237
$474$	0	0
$475$	498759.	0.101428
$476$	−8.29306e6	−1.67763
$477$	0	0
$478$	−1.47443e6	−0.295158
$479$	−2.39044e6	−0.476036	−0.238018	−	0.971261i	$-0.576498\pi$
$479$	−2.39044e6	−0.476036	−0.238018	+	0.971261i	$0.576498\pi$
$480$	0	0
$481$	5.66733e6	1.11690
$482$	3.06735e6	0.601375
$483$	0	0
$484$	3.26690e7	6.33902
$485$	1.55714e6	0.300589
$486$	0	0
$487$	−8.69235e6	−1.66079	−0.830395	−	0.557175i	$-0.811885\pi$
$487$	−8.69235e6	−1.66079	−0.830395	+	0.557175i	$0.811885\pi$
$488$	1.47735e6	0.280824
$489$	0	0
$490$	−1.59504e6	−0.300111
$491$	−3.02753e6	−0.566742	−0.283371	−	0.959010i	$-0.591453\pi$
$491$	−3.02753e6	−0.566742	−0.283371	+	0.959010i	$0.591453\pi$
$492$	0	0
$493$	−4.62290e6	−0.856638
$494$	−7.65299e6	−1.41096
$495$	0	0
$496$	−2.43013e7	−4.43533
$497$	8.82853e6	1.60324
$498$	0	0
$499$	3.24808e6	0.583950	0.291975	−	0.956426i	$-0.405688\pi$
$499$	3.24808e6	0.583950	0.291975	+	0.956426i	$0.405688\pi$
$500$	1.54267e7	2.75962
$501$	0	0
$502$	1.35866e7	2.40631
$503$	1.14587e6	0.201937	0.100969	−	0.994890i	$-0.467806\pi$
$503$	1.14587e6	0.201937	0.100969	+	0.994890i	$0.467806\pi$
$504$	0	0
$505$	5.07401e6	0.885367
$506$	−4.22526e6	−0.733630
$507$	0	0
$508$	−1.18291e7	−2.03373
$509$	7.47756e6	1.27928	0.639640	−	0.768675i	$-0.279083\pi$
$509$	7.47756e6	1.27928	0.639640	+	0.768675i	$0.279083\pi$
$510$	0	0
$511$	−9.54667e6	−1.61733
$512$	−576256.	−0.0971494
$513$	0	0
$514$	5.95298e6	0.993863
$515$	−109019.	−0.0181128
$516$	0	0
$517$	1.13259e7	1.86357
$518$	−1.33033e7	−2.17838
$519$	0	0
$520$	−1.59163e7	−2.58127
$521$	568216.	0.0917106	0.0458553	−	0.998948i	$-0.485399\pi$
$521$	568216.	0.0917106	0.0458553	+	0.998948i	$0.485399\pi$
$522$	0	0
$523$	6.11824e6	0.978076	0.489038	−	0.872262i	$-0.337348\pi$
$523$	6.11824e6	0.978076	0.489038	+	0.872262i	$0.337348\pi$
$524$	1.15102e7	1.83128
$525$	0	0
$526$	−7.01703e6	−1.10583
$527$	−6.06801e6	−0.951743
$528$	0	0
$529$	279841.	0.0434783
$530$	2.04258e7	3.15856
$531$	0	0
$532$	1.30087e7	1.99275
$533$	−2.51203e6	−0.383007
$534$	0	0
$535$	−4.02880e6	−0.608543
$536$	−1.17995e7	−1.77400
$537$	0	0
$538$	1.75189e7	2.60947
$539$	2.09654e6	0.310836
$540$	0	0
$541$	−4.31434e6	−0.633755	−0.316878	−	0.948466i	$-0.602634\pi$
$541$	−4.31434e6	−0.633755	−0.316878	+	0.948466i	$0.602634\pi$
$542$	−5.67110e6	−0.829219
$543$	0	0
$544$	−1.51085e7	−2.18889
$545$	1.42749e6	0.205865
$546$	0	0
$547$	7.34052e6	1.04896	0.524480	−	0.851423i	$-0.324260\pi$
$547$	7.34052e6	1.04896	0.524480	+	0.851423i	$0.324260\pi$
$548$	−6.18471e6	−0.879767
$549$	0	0
$550$	−3.04150e6	−0.428728
$551$	7.25157e6	1.01754
$552$	0	0
$553$	−4.62396e6	−0.642986
$554$	−1.10906e6	−0.153526
$555$	0	0
$556$	1.79972e7	2.46898
$557$	−8.10617e6	−1.10708	−0.553538	−	0.832824i	$-0.686723\pi$
$557$	−8.10617e6	−1.10708	−0.553538	+	0.832824i	$0.686723\pi$
$558$	0	0
$559$	4.99169e6	0.675644
$560$	2.07123e7	2.79098
$561$	0	0
$562$	−1.93013e7	−2.57779
$563$	−1.15177e7	−1.53142	−0.765709	−	0.643187i	$-0.777612\pi$
$563$	−1.15177e7	−1.53142	−0.765709	+	0.643187i	$0.777612\pi$
$564$	0	0
$565$	−6.32489e6	−0.833550
$566$	1.14774e7	1.50592
$567$	0	0
$568$	4.18185e7	5.43873
$569$	−248635.	−0.0321945	−0.0160972	−	0.999870i	$-0.505124\pi$
$569$	−248635.	−0.0321945	−0.0160972	+	0.999870i	$0.505124\pi$
$570$	0	0
$571$	2.55456e6	0.327888	0.163944	−	0.986470i	$-0.447578\pi$
$571$	2.55456e6	0.327888	0.163944	+	0.986470i	$0.447578\pi$
$572$	3.37947e7	4.31876
$573$	0	0
$574$	5.89664e6	0.747007
$575$	201440.	0.0254084
$576$	0	0
$577$	3.96849e6	0.496233	0.248116	−	0.968730i	$-0.420188\pi$
$577$	3.96849e6	0.496233	0.248116	+	0.968730i	$0.420188\pi$
$578$	7.78311e6	0.969022
$579$	0	0
$580$	2.43624e7	3.00711
$581$	1.19968e7	1.47444
$582$	0	0
$583$	−2.68479e7	−3.27144
$584$	−4.52201e7	−5.48655
$585$	0	0
$586$	2.00175e7	2.40806
$587$	−7.38560e6	−0.884689	−0.442344	−	0.896845i	$-0.645853\pi$
$587$	−7.38560e6	−0.884689	−0.442344	+	0.896845i	$0.645853\pi$
$588$	0	0
$589$	9.51840e6	1.13051
$590$	−3.63142e6	−0.429483
$591$	0	0
$592$	−3.49336e7	−4.09675
$593$	1.34366e7	1.56910	0.784552	−	0.620063i	$-0.212893\pi$
$593$	1.34366e7	1.56910	0.784552	+	0.620063i	$0.212893\pi$
$594$	0	0
$595$	5.17182e6	0.598896
$596$	−1.11275e7	−1.28316
$597$	0	0
$598$	−3.09091e6	−0.353454
$599$	−1.65576e6	−0.188552	−0.0942760	−	0.995546i	$-0.530054\pi$
$599$	−1.65576e6	−0.188552	−0.0942760	+	0.995546i	$0.530054\pi$
$600$	0	0
$601$	−8.29015e6	−0.936216	−0.468108	−	0.883671i	$-0.655064\pi$
$601$	−8.29015e6	−0.936216	−0.468108	+	0.883671i	$0.655064\pi$
$602$	−1.17173e7	−1.31776
$603$	0	0
$604$	1.44194e7	1.60826
$605$	−2.03734e7	−2.26296
$606$	0	0
$607$	−1.24477e7	−1.37125	−0.685624	−	0.727956i	$-0.740470\pi$
$607$	−1.24477e7	−1.37125	−0.685624	+	0.727956i	$0.740470\pi$
$608$	2.36995e7	2.60004
$609$	0	0
$610$	−1.48830e6	−0.161944
$611$	8.28525e6	0.897848
$612$	0	0
$613$	−9.98753e6	−1.07351	−0.536756	−	0.843737i	$-0.680351\pi$
$613$	−9.98753e6	−1.07351	−0.536756	+	0.843737i	$0.680351\pi$
$614$	2.30275e7	2.46505
$615$	0	0
$616$	−4.91081e7	−5.21437
$617$	1.14994e7	1.21608	0.608042	−	0.793905i	$-0.291955\pi$
$617$	1.14994e7	1.21608	0.608042	+	0.793905i	$0.291955\pi$
$618$	0	0
$619$	−5.70831e6	−0.598799	−0.299400	−	0.954128i	$-0.596786\pi$
$619$	−5.70831e6	−0.598799	−0.299400	+	0.954128i	$0.596786\pi$
$620$	3.19780e7	3.34096
$621$	0	0
$622$	1.45922e7	1.51232
$623$	2.64940e6	0.273481
$624$	0	0
$625$	−8.43064e6	−0.863297
$626$	5.64138e6	0.575373
$627$	0	0
$628$	−3.02275e7	−3.05847
$629$	−8.72288e6	−0.879090
$630$	0	0
$631$	6.75550e6	0.675436	0.337718	−	0.941247i	$-0.390345\pi$
$631$	6.75550e6	0.675436	0.337718	+	0.941247i	$0.390345\pi$
$632$	−2.19025e7	−2.18123
$633$	0	0
$634$	2.79941e7	2.76594
$635$	7.37703e6	0.726018
$636$	0	0
$637$	1.53368e6	0.149757
$638$	−4.42211e7	−4.30108
$639$	0	0
$640$	1.92462e7	1.85735
$641$	−355815.	−0.0342042	−0.0171021	−	0.999854i	$-0.505444\pi$
$641$	−355815.	−0.0342042	−0.0171021	+	0.999854i	$0.505444\pi$
$642$	0	0
$643$	1.12269e6	0.107086	0.0535432	−	0.998566i	$-0.482949\pi$
$643$	1.12269e6	0.107086	0.0535432	+	0.998566i	$0.482949\pi$
$644$	5.25397e6	0.499198
$645$	0	0
$646$	1.17791e7	1.11053
$647$	1.91791e7	1.80123	0.900613	−	0.434621i	$-0.143118\pi$
$647$	1.91791e7	1.80123	0.900613	+	0.434621i	$0.143118\pi$
$648$	0	0
$649$	4.77317e6	0.444831
$650$	−2.22495e6	−0.206556
$651$	0	0
$652$	3.81911e7	3.51838
$653$	9.62631e6	0.883440	0.441720	−	0.897153i	$-0.354368\pi$
$653$	9.62631e6	0.883440	0.441720	+	0.897153i	$0.354368\pi$
$654$	0	0
$655$	−7.17813e6	−0.653745
$656$	1.54842e7	1.40485
$657$	0	0
$658$	−1.94485e7	−1.75114
$659$	6.82930e6	0.612579	0.306290	−	0.951938i	$-0.400912\pi$
$659$	6.82930e6	0.612579	0.306290	+	0.951938i	$0.400912\pi$
$660$	0	0
$661$	9.20374e6	0.819333	0.409667	−	0.912235i	$-0.365645\pi$
$661$	9.20374e6	0.819333	0.409667	+	0.912235i	$0.365645\pi$
$662$	1.10813e7	0.982755
$663$	0	0
$664$	5.68260e7	5.00180
$665$	−8.11263e6	−0.711389
$666$	0	0
$667$	2.92878e6	0.254902
$668$	3.02263e7	2.62086
$669$	0	0
$670$	1.18869e7	1.02302
$671$	1.95623e6	0.167731
$672$	0	0
$673$	−9.25649e6	−0.787787	−0.393893	−	0.919156i	$-0.628872\pi$
$673$	−9.25649e6	−0.787787	−0.393893	+	0.919156i	$0.628872\pi$
$674$	−1.22244e7	−1.03652
$675$	0	0
$676$	−6.46669e6	−0.544271
$677$	4.89629e6	0.410578	0.205289	−	0.978701i	$-0.434187\pi$
$677$	4.89629e6	0.410578	0.205289	+	0.978701i	$0.434187\pi$
$678$	0	0
$679$	3.51457e6	0.292548
$680$	2.44976e7	2.03166
$681$	0	0
$682$	−5.80445e7	−4.77859
$683$	1.01443e6	0.0832086	0.0416043	−	0.999134i	$-0.486753\pi$
$683$	1.01443e6	0.0832086	0.0416043	+	0.999134i	$0.486753\pi$
$684$	0	0
$685$	3.85699e6	0.314067
$686$	−2.50030e7	−2.02853
$687$	0	0
$688$	−3.07689e7	−2.47823
$689$	−1.96401e7	−1.57614
$690$	0	0
$691$	−9.64799e6	−0.768673	−0.384336	−	0.923193i	$-0.625570\pi$
$691$	−9.64799e6	−0.768673	−0.384336	+	0.923193i	$0.625570\pi$
$692$	2.58741e7	2.05400
$693$	0	0
$694$	1.17113e7	0.923007
$695$	−1.12236e7	−0.881396
$696$	0	0
$697$	3.86639e6	0.301456
$698$	−1.58585e6	−0.123204
$699$	0	0
$700$	3.78201e6	0.291728
$701$	1.47825e7	1.13620	0.568098	−	0.822961i	$-0.307679\pi$
$701$	1.47825e7	1.13620	0.568098	+	0.822961i	$0.307679\pi$
$702$	0	0
$703$	1.36829e7	1.04421
$704$	−6.51657e7	−4.95550
$705$	0	0
$706$	1.70865e7	1.29016
$707$	1.14524e7	0.861683
$708$	0	0
$709$	−585189.	−0.0437201	−0.0218600	−	0.999761i	$-0.506959\pi$
$709$	−585189.	−0.0437201	−0.0218600	+	0.999761i	$0.506959\pi$
$710$	−4.21282e7	−3.13637
$711$	0	0
$712$	1.25495e7	0.927742
$713$	3.84432e6	0.283201
$714$	0	0
$715$	−2.10755e7	−1.54175
$716$	−5.28271e7	−3.85100
$717$	0	0
$718$	−1.99940e7	−1.44740
$719$	3.42448e6	0.247043	0.123521	−	0.992342i	$-0.460581\pi$
$719$	3.42448e6	0.247043	0.123521	+	0.992342i	$0.460581\pi$
$720$	0	0
$721$	−246064.	−0.0176283
$722$	8.19147e6	0.584816
$723$	0	0
$724$	−5.60293e7	−3.97254
$725$	2.10825e6	0.148963
$726$	0	0
$727$	−1.91984e7	−1.34719	−0.673595	−	0.739100i	$-0.735251\pi$
$727$	−1.91984e7	−1.34719	−0.673595	+	0.739100i	$0.735251\pi$
$728$	−3.59242e7	−2.51222
$729$	0	0
$730$	4.55551e7	3.16395
$731$	−7.68297e6	−0.531784
$732$	0	0
$733$	2.66362e7	1.83110	0.915552	−	0.402200i	$-0.131754\pi$
$733$	2.66362e7	1.83110	0.915552	+	0.402200i	$0.131754\pi$
$734$	2.17060e7	1.48710
$735$	0	0
$736$	9.57181e6	0.651327
$737$	−1.56243e7	−1.05958
$738$	0	0
$739$	−6.81746e6	−0.459210	−0.229605	−	0.973284i	$-0.573743\pi$
$739$	−6.81746e6	−0.459210	−0.229605	+	0.973284i	$0.573743\pi$
$740$	4.59690e7	3.08593
$741$	0	0
$742$	4.61024e7	3.07407
$743$	377932.	0.0251155	0.0125577	−	0.999921i	$-0.496003\pi$
$743$	377932.	0.0251155	0.0125577	+	0.999921i	$0.496003\pi$
$744$	0	0
$745$	6.93946e6	0.458073
$746$	1.43812e7	0.946122
$747$	0	0
$748$	−5.20152e7	−3.39920
$749$	−9.09327e6	−0.592264
$750$	0	0
$751$	−8.02610e6	−0.519283	−0.259642	−	0.965705i	$-0.583604\pi$
$751$	−8.02610e6	−0.519283	−0.259642	+	0.965705i	$0.583604\pi$
$752$	−5.10705e7	−3.29326
$753$	0	0
$754$	−3.23491e7	−2.07221
$755$	−8.99242e6	−0.574129
$756$	0	0
$757$	−4.55894e6	−0.289150	−0.144575	−	0.989494i	$-0.546182\pi$
$757$	−4.55894e6	−0.289150	−0.144575	+	0.989494i	$0.546182\pi$
$758$	3.06209e7	1.93573
$759$	0	0
$760$	−3.84274e7	−2.41328
$761$	−2.77860e7	−1.73926	−0.869629	−	0.493705i	$-0.835642\pi$
$761$	−2.77860e7	−1.73926	−0.869629	+	0.493705i	$0.835642\pi$
$762$	0	0
$763$	3.22195e6	0.200358
$764$	−4.81595e7	−2.98503
$765$	0	0
$766$	3.54862e7	2.18518
$767$	3.49173e6	0.214315
$768$	0	0
$769$	1.47056e7	0.896742	0.448371	−	0.893847i	$-0.352004\pi$
$769$	1.47056e7	0.896742	0.448371	+	0.893847i	$0.352004\pi$
$770$	4.94719e7	3.00699
$771$	0	0
$772$	1.78066e6	0.107532
$773$	−1.46941e7	−0.884495	−0.442248	−	0.896893i	$-0.645819\pi$
$773$	−1.46941e7	−0.884495	−0.442248	+	0.896893i	$0.645819\pi$
$774$	0	0
$775$	2.76729e6	0.165501
$776$	1.66476e7	0.992424
$777$	0	0
$778$	−5.29919e7	−3.13878
$779$	−6.06489e6	−0.358080
$780$	0	0
$781$	5.53738e7	3.24845
$782$	4.75738e6	0.278196
$783$	0	0
$784$	−9.45368e6	−0.549301
$785$	1.88509e7	1.09184
$786$	0	0
$787$	1.02770e7	0.591465	0.295732	−	0.955271i	$-0.404436\pi$
$787$	1.02770e7	0.591465	0.295732	+	0.955271i	$0.404436\pi$
$788$	2.87454e7	1.64912
$789$	0	0
$790$	2.20648e7	1.25786
$791$	−1.42757e7	−0.811253
$792$	0	0
$793$	1.43105e6	0.0808110
$794$	4.45046e7	2.50527
$795$	0	0
$796$	−2.71618e6	−0.151942
$797$	−2.13182e7	−1.18879	−0.594395	−	0.804173i	$-0.702608\pi$
$797$	−2.13182e7	−1.18879	−0.594395	+	0.804173i	$0.702608\pi$
$798$	0	0
$799$	−1.27523e7	−0.706675
$800$	6.89015e6	0.380631
$801$	0	0
$802$	−3.31641e7	−1.82068
$803$	−5.98781e7	−3.27702
$804$	0	0
$805$	−3.27655e6	−0.178208
$806$	−4.24614e7	−2.30227
$807$	0	0
$808$	5.42470e7	2.92312
$809$	−2.64965e7	−1.42337	−0.711685	−	0.702499i	$-0.752068\pi$
$809$	−2.64965e7	−1.42337	−0.711685	+	0.702499i	$0.752068\pi$
$810$	0	0
$811$	1.59727e7	0.852758	0.426379	−	0.904545i	$-0.359789\pi$
$811$	1.59727e7	0.852758	0.426379	+	0.904545i	$0.359789\pi$
$812$	5.49875e7	2.92667
$813$	0	0
$814$	−8.34401e7	−4.41381
$815$	−2.38172e7	−1.25602
$816$	0	0
$817$	1.20517e7	0.631672
$818$	5.20725e6	0.272098
$819$	0	0
$820$	−2.03756e7	−1.05822
$821$	−2.19797e7	−1.13806	−0.569029	−	0.822317i	$-0.692681\pi$
$821$	−2.19797e7	−1.13806	−0.569029	+	0.822317i	$0.692681\pi$
$822$	0	0
$823$	−3.47354e7	−1.78761	−0.893806	−	0.448454i	$-0.851975\pi$
$823$	−3.47354e7	−1.78761	−0.893806	+	0.448454i	$0.851975\pi$
$824$	−1.16554e6	−0.0598012
$825$	0	0
$826$	−8.19635e6	−0.417994
$827$	1.25679e7	0.638997	0.319499	−	0.947587i	$-0.396485\pi$
$827$	1.25679e7	0.638997	0.319499	+	0.947587i	$0.396485\pi$
$828$	0	0
$829$	−3.35386e7	−1.69495	−0.847477	−	0.530831i	$-0.821880\pi$
$829$	−3.35386e7	−1.69495	−0.847477	+	0.530831i	$0.821880\pi$
$830$	−5.72469e7	−2.88441
$831$	0	0
$832$	−4.76708e7	−2.38750
$833$	−2.36057e6	−0.117870
$834$	0	0
$835$	−1.88501e7	−0.935617
$836$	8.15921e7	4.03769
$837$	0	0
$838$	1.63442e7	0.803996
$839$	−1.15158e6	−0.0564791	−0.0282395	−	0.999601i	$-0.508990\pi$
$839$	−1.15158e6	−0.0564791	−0.0282395	+	0.999601i	$0.508990\pi$
$840$	0	0
$841$	1.01412e7	0.494422
$842$	−4.05240e7	−1.96984
$843$	0	0
$844$	−8.68552e7	−4.19701
$845$	4.03284e6	0.194298
$846$	0	0
$847$	−4.59842e7	−2.20242
$848$	1.21062e8	5.78121
$849$	0	0
$850$	3.42454e6	0.162576
$851$	5.52628e6	0.261583
$852$	0	0
$853$	−1.19102e7	−0.560465	−0.280232	−	0.959932i	$-0.590411\pi$
$853$	−1.19102e7	−0.560465	−0.280232	+	0.959932i	$0.590411\pi$
$854$	−3.35918e6	−0.157612
$855$	0	0
$856$	−4.30725e7	−2.00916
$857$	1.59627e7	0.742430	0.371215	−	0.928547i	$-0.378941\pi$
$857$	1.59627e7	0.742430	0.371215	+	0.928547i	$0.378941\pi$
$858$	0	0
$859$	1.22935e7	0.568452	0.284226	−	0.958757i	$-0.408263\pi$
$859$	1.22935e7	0.568452	0.284226	+	0.958757i	$0.408263\pi$
$860$	4.04887e7	1.86676
$861$	0	0
$862$	3.01823e7	1.38351
$863$	2.14243e7	0.979219	0.489609	−	0.871942i	$-0.337139\pi$
$863$	2.14243e7	0.979219	0.489609	+	0.871942i	$0.337139\pi$
$864$	0	0
$865$	−1.61360e7	−0.733254
$866$	−3.30386e7	−1.49702
$867$	0	0
$868$	7.21764e7	3.25159
$869$	−2.90022e7	−1.30281
$870$	0	0
$871$	−1.14297e7	−0.510492
$872$	1.52615e7	0.679684
$873$	0	0
$874$	−7.46251e6	−0.330451
$875$	−2.17144e7	−0.958799
$876$	0	0
$877$	−3.70233e6	−0.162546	−0.0812730	−	0.996692i	$-0.525899\pi$
$877$	−3.70233e6	−0.162546	−0.0812730	+	0.996692i	$0.525899\pi$
$878$	4.32360e7	1.89282
$879$	0	0
$880$	1.29910e8	5.65505
$881$	−5.90856e6	−0.256473	−0.128237	−	0.991744i	$-0.540932\pi$
$881$	−5.90856e6	−0.256473	−0.128237	+	0.991744i	$0.540932\pi$
$882$	0	0
$883$	1.83087e7	0.790233	0.395117	−	0.918631i	$-0.370704\pi$
$883$	1.83087e7	0.790233	0.395117	+	0.918631i	$0.370704\pi$
$884$	−3.80508e7	−1.63769
$885$	0	0
$886$	5.11186e7	2.18774
$887$	−2.01751e7	−0.861005	−0.430503	−	0.902589i	$-0.641664\pi$
$887$	−2.01751e7	−0.861005	−0.430503	+	0.902589i	$0.641664\pi$
$888$	0	0
$889$	1.66505e7	0.706597
$890$	−1.26425e7	−0.535004
$891$	0	0
$892$	8.31757e7	3.50013
$893$	2.00034e7	0.839413
$894$	0	0
$895$	3.29447e7	1.37476
$896$	4.34399e7	1.80767
$897$	0	0
$898$	1.69170e7	0.700055
$899$	4.02342e7	1.66034
$900$	0	0
$901$	3.02291e7	1.24055
$902$	3.69845e7	1.51357
$903$	0	0
$904$	−6.76203e7	−2.75205
$905$	3.49417e7	1.41815
$906$	0	0
$907$	−3.53066e7	−1.42508	−0.712538	−	0.701634i	$-0.752454\pi$
$907$	−3.53066e7	−1.42508	−0.712538	+	0.701634i	$0.752454\pi$
$908$	2.24157e6	0.0902272
$909$	0	0
$910$	3.61902e7	1.44873
$911$	1.91557e7	0.764720	0.382360	−	0.924014i	$-0.375111\pi$
$911$	1.91557e7	0.764720	0.382360	+	0.924014i	$0.375111\pi$
$912$	0	0
$913$	7.52459e7	2.98749
$914$	−5.32672e7	−2.10909
$915$	0	0
$916$	1.26492e7	0.498108
$917$	−1.62015e7	−0.636257
$918$	0	0
$919$	−3.36053e7	−1.31256	−0.656279	−	0.754518i	$-0.727871\pi$
$919$	−3.36053e7	−1.31256	−0.656279	+	0.754518i	$0.727871\pi$
$920$	−1.55202e7	−0.604542
$921$	0	0
$922$	5.23966e7	2.02991
$923$	4.05077e7	1.56507
$924$	0	0
$925$	3.97803e6	0.152867
$926$	2.20123e7	0.843602
$927$	0	0
$928$	1.00177e8	3.81856
$929$	2.51698e7	0.956842	0.478421	−	0.878131i	$-0.341209\pi$
$929$	2.51698e7	0.956842	0.478421	+	0.878131i	$0.341209\pi$
$930$	0	0
$931$	3.70284e6	0.140011
$932$	−1.08714e8	−4.09963
$933$	0	0
$934$	−6.82375e7	−2.55951
$935$	3.24384e7	1.21347
$936$	0	0
$937$	−2.32062e7	−0.863484	−0.431742	−	0.901997i	$-0.642101\pi$
$937$	−2.32062e7	−0.863484	−0.431742	+	0.901997i	$0.642101\pi$
$938$	2.68296e7	0.995651
$939$	0	0
$940$	6.72035e7	2.48069
$941$	1.44758e7	0.532930	0.266465	−	0.963845i	$-0.414144\pi$
$941$	1.44758e7	0.532930	0.266465	+	0.963845i	$0.414144\pi$
$942$	0	0
$943$	−2.44951e6	−0.0897014
$944$	−2.15231e7	−0.786095
$945$	0	0
$946$	−7.34926e7	−2.67003
$947$	3.57094e7	1.29392	0.646960	−	0.762524i	$-0.276040\pi$
$947$	3.57094e7	1.29392	0.646960	+	0.762524i	$0.276040\pi$
$948$	0	0
$949$	−4.38027e7	−1.57883
$950$	−5.37180e6	−0.193113
$951$	0	0
$952$	5.52927e7	1.97731
$953$	6.09472e6	0.217381	0.108691	−	0.994076i	$-0.465334\pi$
$953$	6.09472e6	0.217381	0.108691	+	0.994076i	$0.465334\pi$
$954$	0	0
$955$	3.00338e7	1.06562
$956$	1.14994e7	0.406940
$957$	0	0
$958$	2.57459e7	0.906345
$959$	8.70548e6	0.305665
$960$	0	0
$961$	2.41822e7	0.844669
$962$	−6.10391e7	−2.12652
$963$	0	0
$964$	−2.39229e7	−0.829127
$965$	−1.11048e6	−0.0383877
$966$	0	0
$967$	1.48405e7	0.510366	0.255183	−	0.966893i	$-0.417864\pi$
$967$	1.48405e7	0.510366	0.255183	+	0.966893i	$0.417864\pi$
$968$	−2.17815e8	−7.47137
$969$	0	0
$970$	−1.67709e7	−0.572304
$971$	−1.73779e7	−0.591493	−0.295746	−	0.955267i	$-0.595568\pi$
$971$	−1.73779e7	−0.591493	−0.295746	+	0.955267i	$0.595568\pi$
$972$	0	0
$973$	−2.53325e7	−0.857818
$974$	9.36194e7	3.16205
$975$	0	0
$976$	−8.82100e6	−0.296410
$977$	1.30124e7	0.436136	0.218068	−	0.975934i	$-0.430024\pi$
$977$	1.30124e7	0.436136	0.218068	+	0.975934i	$0.430024\pi$
$978$	0	0
$979$	1.66174e7	0.554123
$980$	1.24400e7	0.413768
$981$	0	0
$982$	3.26075e7	1.07904
$983$	−1.94435e7	−0.641787	−0.320893	−	0.947115i	$-0.603983\pi$
$983$	−1.94435e7	−0.641787	−0.320893	+	0.947115i	$0.603983\pi$
$984$	0	0
$985$	−1.79266e7	−0.588717
$986$	4.97902e7	1.63099
$987$	0	0
$988$	5.96872e7	1.94531
$989$	4.86745e6	0.158238
$990$	0	0
$991$	−3.56706e6	−0.115379	−0.0576894	−	0.998335i	$-0.518373\pi$
$991$	−3.56706e6	−0.115379	−0.0576894	+	0.998335i	$0.518373\pi$
$992$	1.31493e8	4.24251
$993$	0	0
$994$	−9.50862e7	−3.05247
$995$	1.69390e6	0.0542414
$996$	0	0
$997$	3.36336e7	1.07161	0.535804	−	0.844342i	$-0.320009\pi$
$997$	3.36336e7	1.07161	0.535804	+	0.844342i	$0.320009\pi$
$998$	−3.49829e7	−1.11181
$999$	0	0

Twists

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

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

\(f(q)\)	\(=\)	\(q-10.7703 q^{2} +84.0000 q^{4} -52.3852 q^{5} -118.237 q^{7} -560.057 q^{8} +O(q^{10})\)	\(q-10.7703 q^{2} +84.0000 q^{4} -52.3852 q^{5} -118.237 q^{7} -560.057 q^{8} +564.205 q^{10} -741.598 q^{11} -542.502 q^{13} +1273.45 q^{14} +3344.00 q^{16} +834.993 q^{17} -1309.79 q^{19} -4400.35 q^{20} +7987.25 q^{22} -529.000 q^{23} -380.795 q^{25} +5842.93 q^{26} -9931.89 q^{28} -5536.45 q^{29} -7267.14 q^{31} -18094.2 q^{32} -8993.15 q^{34} +6193.85 q^{35} -10446.7 q^{37} +14106.8 q^{38} +29338.7 q^{40} +4630.45 q^{41} -9201.23 q^{43} -62294.2 q^{44} +5697.50 q^{46} -15272.3 q^{47} -2827.06 q^{49} +4101.28 q^{50} -45570.2 q^{52} +36202.8 q^{53} +38848.7 q^{55} +66219.4 q^{56} +59629.4 q^{58} -6436.34 q^{59} -2637.86 q^{61} +78269.5 q^{62} +87872.0 q^{64} +28419.1 q^{65} +21068.5 q^{67} +70139.4 q^{68} -66709.9 q^{70} -74668.2 q^{71} +80742.0 q^{73} +112514. q^{74} -110022. q^{76} +87684.2 q^{77} +39107.7 q^{79} -175176. q^{80} -49871.4 q^{82} -101465. q^{83} -43741.2 q^{85} +99100.3 q^{86} +415337. q^{88} -22407.6 q^{89} +64143.7 q^{91} -44436.0 q^{92} +164488. q^{94} +68613.4 q^{95} -29724.8 q^{97} +30448.3 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 168 q^{4} - 94 q^{5} - 118 q^{7}+O(q^{10}) \) 2 * q + 168 * q^4 - 94 * q^5 - 118 * q^7	\( 2 q + 168 q^{4} - 94 q^{5} - 118 q^{7} + 116 q^{10} - 320 q^{11} - 288 q^{13} + 1276 q^{14} + 6688 q^{16} + 1810 q^{17} + 730 q^{19} - 7896 q^{20} + 12528 q^{22} - 1058 q^{23} - 1774 q^{25} + 8584 q^{26} - 9912 q^{28} - 8208 q^{29} + 1772 q^{31} + 1508 q^{34} + 6184 q^{35} - 23112 q^{37} + 36076 q^{38} + 6032 q^{40} - 5516 q^{41} + 10322 q^{43} - 26880 q^{44} - 42952 q^{47} - 19634 q^{49} - 10904 q^{50} - 24192 q^{52} + 25350 q^{53} + 21304 q^{55} + 66352 q^{56} + 30856 q^{58} - 18344 q^{59} + 37224 q^{61} + 175624 q^{62} + 175744 q^{64} + 17828 q^{65} - 7482 q^{67} + 152040 q^{68} - 66816 q^{70} - 126848 q^{71} + 137660 q^{73} - 23896 q^{74} + 61320 q^{76} + 87784 q^{77} + 62286 q^{79} - 314336 q^{80} - 159152 q^{82} - 83120 q^{83} - 84316 q^{85} + 309372 q^{86} + 651456 q^{88} - 69770 q^{89} + 64204 q^{91} - 88872 q^{92} - 133632 q^{94} - 16272 q^{95} - 170104 q^{97} - 150568 q^{98}+O(q^{100}) \) 2 * q + 168 * q^4 - 94 * q^5 - 118 * q^7 + 116 * q^10 - 320 * q^11 - 288 * q^13 + 1276 * q^14 + 6688 * q^16 + 1810 * q^17 + 730 * q^19 - 7896 * q^20 + 12528 * q^22 - 1058 * q^23 - 1774 * q^25 + 8584 * q^26 - 9912 * q^28 - 8208 * q^29 + 1772 * q^31 + 1508 * q^34 + 6184 * q^35 - 23112 * q^37 + 36076 * q^38 + 6032 * q^40 - 5516 * q^41 + 10322 * q^43 - 26880 * q^44 - 42952 * q^47 - 19634 * q^49 - 10904 * q^50 - 24192 * q^52 + 25350 * q^53 + 21304 * q^55 + 66352 * q^56 + 30856 * q^58 - 18344 * q^59 + 37224 * q^61 + 175624 * q^62 + 175744 * q^64 + 17828 * q^65 - 7482 * q^67 + 152040 * q^68 - 66816 * q^70 - 126848 * q^71 + 137660 * q^73 - 23896 * q^74 + 61320 * q^76 + 87784 * q^77 + 62286 * q^79 - 314336 * q^80 - 159152 * q^82 - 83120 * q^83 - 84316 * q^85 + 309372 * q^86 + 651456 * q^88 - 69770 * q^89 + 64204 * q^91 - 88872 * q^92 - 133632 * q^94 - 16272 * q^95 - 170104 * q^97 - 150568 * q^98

Introduction

L-functions

Modular forms

Varieties

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Groups

Database

Properties

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