LMFDB - Embedded newform 207.6.a.g.1.5

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:	$6$
Coefficient field:	$\mathbb{Q}[x]/(x^{6} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} - 149x^{4} + 215x^{3} + 6182x^{2} - 4625x - 79150 $ x^6 - 2x^5 - 149x^4 + 215x^3 + 6182x^2 - 4625*x - 79150
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2\cdot 3^{2}\cdot 5 $
Twist minimal:	no (minimal twist has level 23)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$6.03655$ 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)

$f(q)$	$=$	$q+5.03655 q^{2} -6.63314 q^{4} -108.846 q^{5} -33.8609 q^{7} -194.578 q^{8} +O(q^{10})$	\(q+5.03655 q^{2} -6.63314 q^{4} -108.846 q^{5} -33.8609 q^{7} -194.578 q^{8} -548.208 q^{10} -88.4826 q^{11} +857.760 q^{13} -170.542 q^{14} -767.741 q^{16} +302.638 q^{17} +1481.71 q^{19} +721.991 q^{20} -445.647 q^{22} -529.000 q^{23} +8722.45 q^{25} +4320.15 q^{26} +224.604 q^{28} -2736.52 q^{29} -6971.35 q^{31} +2359.72 q^{32} +1524.25 q^{34} +3685.62 q^{35} +12506.1 q^{37} +7462.72 q^{38} +21179.0 q^{40} +15533.7 q^{41} -6897.06 q^{43} +586.918 q^{44} -2664.34 q^{46} -22897.6 q^{47} -15660.4 q^{49} +43931.1 q^{50} -5689.64 q^{52} +2332.22 q^{53} +9630.97 q^{55} +6588.58 q^{56} -13782.6 q^{58} -3636.94 q^{59} +2332.32 q^{61} -35111.6 q^{62} +36452.6 q^{64} -93363.7 q^{65} +19089.5 q^{67} -2007.44 q^{68} +18562.8 q^{70} +16361.1 q^{71} +9104.09 q^{73} +62987.8 q^{74} -9828.41 q^{76} +2996.10 q^{77} +58599.3 q^{79} +83565.5 q^{80} +78236.2 q^{82} +14544.6 q^{83} -32940.9 q^{85} -34737.4 q^{86} +17216.8 q^{88} +95942.7 q^{89} -29044.5 q^{91} +3508.93 q^{92} -115325. q^{94} -161278. q^{95} +53438.7 q^{97} -78874.6 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 4 q^{2} + 112 q^{4} - 42 q^{5} + 300 q^{7} - 393 q^{8}+O(q^{10}) $ 6 * q - 4 * q^2 + 112 * q^4 - 42 * q^5 + 300 * q^7 - 393 * q^8	$ 6 q - 4 q^{2} + 112 q^{4} - 42 q^{5} + 300 q^{7} - 393 q^{8} - 10 q^{10} + 58 q^{11} + 792 q^{13} + 2984 q^{14} - 1904 q^{16} + 400 q^{17} + 2738 q^{19} + 7124 q^{20} - 1972 q^{22} - 3174 q^{23} + 9966 q^{25} - 4511 q^{26} + 9570 q^{28} - 11244 q^{29} + 13748 q^{31} - 6600 q^{32} - 16226 q^{34} + 4296 q^{35} + 25426 q^{37} + 8028 q^{38} + 10230 q^{40} + 14268 q^{41} - 18082 q^{43} + 51146 q^{44} + 2116 q^{46} + 23084 q^{47} + 37422 q^{49} + 67436 q^{50} + 36807 q^{52} - 17522 q^{53} + 47576 q^{55} - 44946 q^{56} + 141001 q^{58} + 36392 q^{59} + 27062 q^{61} - 48971 q^{62} + 89451 q^{64} - 7108 q^{65} + 37138 q^{67} - 17260 q^{68} + 248380 q^{70} + 158556 q^{71} + 112228 q^{73} + 66878 q^{74} + 157816 q^{76} + 89760 q^{77} + 36844 q^{79} + 158530 q^{80} + 150039 q^{82} + 76350 q^{83} - 102132 q^{85} + 100578 q^{86} - 219028 q^{88} - 16100 q^{89} - 250592 q^{91} - 59248 q^{92} + 12887 q^{94} + 190096 q^{95} + 259432 q^{97} + 325816 q^{98}+O(q^{100}) $ 6 * q - 4 * q^2 + 112 * q^4 - 42 * q^5 + 300 * q^7 - 393 * q^8 - 10 * q^10 + 58 * q^11 + 792 * q^13 + 2984 * q^14 - 1904 * q^16 + 400 * q^17 + 2738 * q^19 + 7124 * q^20 - 1972 * q^22 - 3174 * q^23 + 9966 * q^25 - 4511 * q^26 + 9570 * q^28 - 11244 * q^29 + 13748 * q^31 - 6600 * q^32 - 16226 * q^34 + 4296 * q^35 + 25426 * q^37 + 8028 * q^38 + 10230 * q^40 + 14268 * q^41 - 18082 * q^43 + 51146 * q^44 + 2116 * q^46 + 23084 * q^47 + 37422 * q^49 + 67436 * q^50 + 36807 * q^52 - 17522 * q^53 + 47576 * q^55 - 44946 * q^56 + 141001 * q^58 + 36392 * q^59 + 27062 * q^61 - 48971 * q^62 + 89451 * q^64 - 7108 * q^65 + 37138 * q^67 - 17260 * q^68 + 248380 * q^70 + 158556 * q^71 + 112228 * q^73 + 66878 * q^74 + 157816 * q^76 + 89760 * q^77 + 36844 * q^79 + 158530 * q^80 + 150039 * q^82 + 76350 * q^83 - 102132 * q^85 + 100578 * q^86 - 219028 * q^88 - 16100 * q^89 - 250592 * q^91 - 59248 * q^92 + 12887 * q^94 + 190096 * q^95 + 259432 * q^97 + 325816 * 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 $

$2$	5.03655	0.890345	0.445173	−	0.895445i	$-0.353142\pi$
$2$	5.03655	0.890345	0.445173	+	0.895445i	$0.353142\pi$
$3$	0	0
$3$	0	0
$4$	−6.63314	−0.207286
$5$	−108.846	−1.94710	−0.973548	−	0.228483i	$-0.926624\pi$
$5$	−108.846	−1.94710	−0.973548	+	0.228483i	$0.926624\pi$
$6$	0	0
$7$	−33.8609	−0.261188	−0.130594	−	0.991436i	$-0.541688\pi$
$7$	−33.8609	−0.261188	−0.130594	+	0.991436i	$0.541688\pi$
$8$	−194.578	−1.07490
$9$	0	0
$10$	−548.208	−1.73359
$11$	−88.4826	−0.220484	−0.110242	−	0.993905i	$-0.535163\pi$
$11$	−88.4826	−0.220484	−0.110242	+	0.993905i	$0.535163\pi$
$12$	0	0
$13$	857.760	1.40769	0.703845	−	0.710353i	$-0.251465\pi$
$13$	857.760	1.40769	0.703845	+	0.710353i	$0.251465\pi$
$14$	−170.542	−0.232547
$15$	0	0
$16$	−767.741	−0.749747
$17$	302.638	0.253981	0.126990	−	0.991904i	$-0.459468\pi$
$17$	302.638	0.253981	0.126990	+	0.991904i	$0.459468\pi$
$18$	0	0
$19$	1481.71	0.941629	0.470815	−	0.882232i	$-0.343960\pi$
$19$	1481.71	0.941629	0.470815	+	0.882232i	$0.343960\pi$
$20$	721.991	0.403605
$21$	0	0
$22$	−445.647	−0.196306
$23$	−529.000	−0.208514
$23$	−529.000	−0.208514
$24$	0	0
$25$	8722.45	2.79118
$26$	4320.15	1.25333
$27$	0	0
$28$	224.604	0.0541405
$29$	−2736.52	−0.604232	−0.302116	−	0.953271i	$-0.597693\pi$
$29$	−2736.52	−0.604232	−0.302116	+	0.953271i	$0.597693\pi$
$30$	0	0
$31$	−6971.35	−1.30291	−0.651453	−	0.758689i	$-0.725840\pi$
$31$	−6971.35	−1.30291	−0.651453	+	0.758689i	$0.725840\pi$
$32$	2359.72	0.407367
$33$	0	0
$34$	1524.25	0.226131
$35$	3685.62	0.508558
$36$	0	0
$37$	12506.1	1.50182	0.750911	−	0.660403i	$-0.229615\pi$
$37$	12506.1	1.50182	0.750911	+	0.660403i	$0.229615\pi$
$38$	7462.72	0.838375
$39$	0	0
$40$	21179.0	2.09294
$41$	15533.7	1.44316	0.721581	−	0.692330i	$-0.243416\pi$
$41$	15533.7	1.44316	0.721581	+	0.692330i	$0.243416\pi$
$42$	0	0
$43$	−6897.06	−0.568844	−0.284422	−	0.958699i	$-0.591802\pi$
$43$	−6897.06	−0.568844	−0.284422	+	0.958699i	$0.591802\pi$
$44$	586.918	0.0457031
$45$	0	0
$46$	−2664.34	−0.185650
$47$	−22897.6	−1.51198	−0.755990	−	0.654584i	$-0.772844\pi$
$47$	−22897.6	−1.51198	−0.755990	+	0.654584i	$0.772844\pi$
$48$	0	0
$49$	−15660.4	−0.931781
$50$	43931.1	2.48512
$51$	0	0
$52$	−5689.64	−0.291794
$53$	2332.22	0.114046	0.0570229	−	0.998373i	$-0.481839\pi$
$53$	2332.22	0.114046	0.0570229	+	0.998373i	$0.481839\pi$
$54$	0	0
$55$	9630.97	0.429303
$56$	6588.58	0.280751
$57$	0	0
$58$	−13782.6	−0.537975
$59$	−3636.94	−0.136021	−0.0680105	−	0.997685i	$-0.521665\pi$
$59$	−3636.94	−0.136021	−0.0680105	+	0.997685i	$0.521665\pi$
$60$	0	0
$61$	2332.32	0.0802533	0.0401266	−	0.999195i	$-0.487224\pi$
$61$	2332.32	0.0802533	0.0401266	+	0.999195i	$0.487224\pi$
$62$	−35111.6	−1.16004
$63$	0	0
$64$	36452.6	1.11244
$65$	−93363.7	−2.74091
$66$	0	0
$67$	19089.5	0.519526	0.259763	−	0.965672i	$-0.416356\pi$
$67$	19089.5	0.519526	0.259763	+	0.965672i	$0.416356\pi$
$68$	−2007.44	−0.0526466
$69$	0	0
$70$	18562.8	0.452792
$71$	16361.1	0.385182	0.192591	−	0.981279i	$-0.438311\pi$
$71$	16361.1	0.385182	0.192591	+	0.981279i	$0.438311\pi$
$72$	0	0
$73$	9104.09	0.199954	0.0999768	−	0.994990i	$-0.468123\pi$
$73$	9104.09	0.199954	0.0999768	+	0.994990i	$0.468123\pi$
$74$	62987.8	1.33714
$75$	0	0
$76$	−9828.41	−0.195186
$77$	2996.10	0.0575876
$78$	0	0
$79$	58599.3	1.05639	0.528195	−	0.849123i	$-0.322869\pi$
$79$	58599.3	1.05639	0.528195	+	0.849123i	$0.322869\pi$
$80$	83565.5	1.45983
$81$	0	0
$82$	78236.2	1.28491
$83$	14544.6	0.231743	0.115871	−	0.993264i	$-0.463034\pi$
$83$	14544.6	0.231743	0.115871	+	0.993264i	$0.463034\pi$
$84$	0	0
$85$	−32940.9	−0.494525
$86$	−34737.4	−0.506467
$87$	0	0
$88$	17216.8	0.236998
$89$	95942.7	1.28392	0.641958	−	0.766739i	$-0.278122\pi$
$89$	95942.7	1.28392	0.641958	+	0.766739i	$0.278122\pi$
$90$	0	0
$91$	−29044.5	−0.367672
$92$	3508.93	0.0432221
$93$	0	0
$94$	−115325.	−1.34618
$95$	−161278.	−1.83344
$96$	0	0
$97$	53438.7	0.576669	0.288334	−	0.957530i	$-0.406899\pi$
$97$	53438.7	0.576669	0.288334	+	0.957530i	$0.406899\pi$
$98$	−78874.6	−0.829606
$99$	0	0
$100$	−57857.2	−0.578572
$101$	43884.2	0.428060	0.214030	−	0.976827i	$-0.431341\pi$
$101$	43884.2	0.428060	0.214030	+	0.976827i	$0.431341\pi$
$102$	0	0
$103$	−7603.66	−0.0706203	−0.0353102	−	0.999376i	$-0.511242\pi$
$103$	−7603.66	−0.0706203	−0.0353102	+	0.999376i	$0.511242\pi$
$104$	−166901.	−1.51313
$105$	0	0
$106$	11746.3	0.101540
$107$	117308.	0.990528	0.495264	−	0.868743i	$-0.335071\pi$
$107$	117308.	0.990528	0.495264	+	0.868743i	$0.335071\pi$
$108$	0	0
$109$	203426.	1.63998	0.819992	−	0.572375i	$-0.193978\pi$
$109$	203426.	1.63998	0.819992	+	0.572375i	$0.193978\pi$
$110$	48506.9	0.382227
$111$	0	0
$112$	25996.4	0.195825
$113$	49446.6	0.364285	0.182142	−	0.983272i	$-0.441697\pi$
$113$	49446.6	0.364285	0.182142	+	0.983272i	$0.441697\pi$
$114$	0	0
$115$	57579.5	0.405998
$116$	18151.7	0.125249
$117$	0	0
$118$	−18317.6	−0.121106
$119$	−10247.6	−0.0663367
$120$	0	0
$121$	−153222.	−0.951387
$122$	11746.8	0.0714531
$123$	0	0
$124$	46242.0	0.270074
$125$	−609259.	−3.48760
$126$	0	0
$127$	−22532.8	−0.123967	−0.0619834	−	0.998077i	$-0.519743\pi$
$127$	−22532.8	−0.123967	−0.0619834	+	0.998077i	$0.519743\pi$
$128$	108084.	0.583092
$129$	0	0
$130$	−470231.	−2.44035
$131$	349601.	1.77990	0.889949	−	0.456061i	$-0.150740\pi$
$131$	349601.	1.77990	0.889949	+	0.456061i	$0.150740\pi$
$132$	0	0
$133$	−50172.1	−0.245942
$134$	96145.2	0.462557
$135$	0	0
$136$	−58886.6	−0.273004
$137$	272974.	1.24257	0.621283	−	0.783586i	$-0.286612\pi$
$137$	272974.	1.24257	0.621283	+	0.783586i	$0.286612\pi$
$138$	0	0
$139$	−179372.	−0.787438	−0.393719	−	0.919231i	$-0.628812\pi$
$139$	−179372.	−0.787438	−0.393719	+	0.919231i	$0.628812\pi$
$140$	−24447.3	−0.105417
$141$	0	0
$142$	82403.4	0.342945
$143$	−75896.8	−0.310373
$144$	0	0
$145$	297859.	1.17650
$146$	45853.2	0.178028
$147$	0	0
$148$	−82955.0	−0.311306
$149$	−105237.	−0.388333	−0.194167	−	0.980969i	$-0.562200\pi$
$149$	−105237.	−0.388333	−0.194167	+	0.980969i	$0.562200\pi$
$150$	0	0
$151$	124122.	0.443004	0.221502	−	0.975160i	$-0.428904\pi$
$151$	124122.	0.443004	0.221502	+	0.975160i	$0.428904\pi$
$152$	−288308.	−1.01216
$153$	0	0
$154$	15090.0	0.0512729
$155$	758804.	2.53688
$156$	0	0
$157$	−333172.	−1.07875	−0.539373	−	0.842067i	$-0.681339\pi$
$157$	−333172.	−1.07875	−0.539373	+	0.842067i	$0.681339\pi$
$158$	295138.	0.940552
$159$	0	0
$160$	−256847.	−0.793184
$161$	17912.4	0.0544615
$162$	0	0
$163$	−314046.	−0.925816	−0.462908	−	0.886406i	$-0.653194\pi$
$163$	−314046.	−0.925816	−0.462908	+	0.886406i	$0.653194\pi$
$164$	−103037.	−0.299147
$165$	0	0
$166$	73254.6	0.206331
$167$	−62785.5	−0.174208	−0.0871040	−	0.996199i	$-0.527761\pi$
$167$	−62785.5	−0.174208	−0.0871040	+	0.996199i	$0.527761\pi$
$168$	0	0
$169$	364459.	0.981593
$170$	−165909.	−0.440298
$171$	0	0
$172$	45749.2	0.117913
$173$	−164529.	−0.417953	−0.208977	−	0.977921i	$-0.567013\pi$
$173$	−164529.	−0.417953	−0.208977	+	0.977921i	$0.567013\pi$
$174$	0	0
$175$	−295350.	−0.729023
$176$	67931.7	0.165307
$177$	0	0
$178$	483220.	1.14313
$179$	76866.0	0.179309	0.0896545	−	0.995973i	$-0.471424\pi$
$179$	76866.0	0.179309	0.0896545	+	0.995973i	$0.471424\pi$
$180$	0	0
$181$	45321.0	0.102826	0.0514130	−	0.998677i	$-0.483628\pi$
$181$	45321.0	0.102826	0.0514130	+	0.998677i	$0.483628\pi$
$182$	−146284.	−0.327355
$183$	0	0
$184$	102932.	0.224132
$185$	−1.36124e6	−2.92419
$186$	0	0
$187$	−26778.2	−0.0559986
$188$	151883.	0.313412
$189$	0	0
$190$	−812287.	−1.63240
$191$	−604355.	−1.19870	−0.599348	−	0.800489i	$-0.704573\pi$
$191$	−604355.	−1.19870	−0.599348	+	0.800489i	$0.704573\pi$
$192$	0	0
$193$	−18922.6	−0.0365669	−0.0182834	−	0.999833i	$-0.505820\pi$
$193$	−18922.6	−0.0365669	−0.0182834	+	0.999833i	$0.505820\pi$
$194$	269147.	0.513434
$195$	0	0
$196$	103878.	0.193145
$197$	−24405.8	−0.0448051	−0.0224026	−	0.999749i	$-0.507132\pi$
$197$	−24405.8	−0.0448051	−0.0224026	+	0.999749i	$0.507132\pi$
$198$	0	0
$199$	439784.	0.787238	0.393619	−	0.919274i	$-0.371223\pi$
$199$	439784.	0.787238	0.393619	+	0.919274i	$0.371223\pi$
$200$	−1.69719e6	−3.00024
$201$	0	0
$202$	221025.	0.381121
$203$	92661.1	0.157818
$204$	0	0
$205$	−1.69078e6	−2.80997
$206$	−38296.2	−0.0628765
$207$	0	0
$208$	−658537.	−1.05541
$209$	−131106.	−0.207614
$210$	0	0
$211$	−69368.1	−0.107264	−0.0536320	−	0.998561i	$-0.517080\pi$
$211$	−69368.1	−0.107264	−0.0536320	+	0.998561i	$0.517080\pi$
$212$	−15469.9	−0.0236401
$213$	0	0
$214$	590826.	0.881911
$215$	750718.	1.10759
$216$	0	0
$217$	236056.	0.340303
$218$	1.02456e6	1.46015
$219$	0	0
$220$	−63883.6	−0.0889883
$221$	259591.	0.357527
$222$	0	0
$223$	586002.	0.789108	0.394554	−	0.918873i	$-0.370899\pi$
$223$	586002.	0.789108	0.394554	+	0.918873i	$0.370899\pi$
$224$	−79902.4	−0.106399
$225$	0	0
$226$	249041.	0.324339
$227$	−987055.	−1.27138	−0.635692	−	0.771943i	$-0.719285\pi$
$227$	−987055.	−1.27138	−0.635692	+	0.771943i	$0.719285\pi$
$228$	0	0
$229$	−452130.	−0.569737	−0.284869	−	0.958567i	$-0.591950\pi$
$229$	−452130.	−0.569737	−0.284869	+	0.958567i	$0.591950\pi$
$230$	290002.	0.361478
$231$	0	0
$232$	532467.	0.649490
$233$	465677.	0.561946	0.280973	−	0.959716i	$-0.409343\pi$
$233$	465677.	0.561946	0.280973	+	0.959716i	$0.409343\pi$
$234$	0	0
$235$	2.49231e6	2.94397
$236$	24124.3	0.0281952
$237$	0	0
$238$	−51612.5	−0.0590626
$239$	−605482.	−0.685656	−0.342828	−	0.939398i	$-0.611385\pi$
$239$	−605482.	−0.685656	−0.342828	+	0.939398i	$0.611385\pi$
$240$	0	0
$241$	464027.	0.514637	0.257319	−	0.966327i	$-0.417161\pi$
$241$	464027.	0.514637	0.257319	+	0.966327i	$0.417161\pi$
$242$	−771710.	−0.847063
$243$	0	0
$244$	−15470.6	−0.0166354
$245$	1.70458e6	1.81427
$246$	0	0
$247$	1.27095e6	1.32552
$248$	1.35647e6	1.40049
$249$	0	0
$250$	−3.06857e6	−3.10517
$251$	1.13870e6	1.14084	0.570418	−	0.821354i	$-0.306781\pi$
$251$	1.13870e6	1.14084	0.570418	+	0.821354i	$0.306781\pi$
$252$	0	0
$253$	46807.3	0.0459740
$254$	−113488.	−0.110373
$255$	0	0
$256$	−622111.	−0.593292
$257$	−157200.	−0.148463	−0.0742316	−	0.997241i	$-0.523650\pi$
$257$	−157200.	−0.148463	−0.0742316	+	0.997241i	$0.523650\pi$
$258$	0	0
$259$	−423469.	−0.392258
$260$	619295.	0.568151
$261$	0	0
$262$	1.76079e6	1.58472
$263$	2.19871e6	1.96010	0.980051	−	0.198744i	$-0.0636864\pi$
$263$	2.19871e6	1.96010	0.980051	+	0.198744i	$0.0636864\pi$
$264$	0	0
$265$	−253853.	−0.222058
$266$	−252694.	−0.218973
$267$	0	0
$268$	−126623.	−0.107690
$269$	622228.	0.524287	0.262143	−	0.965029i	$-0.415571\pi$
$269$	622228.	0.524287	0.262143	+	0.965029i	$0.415571\pi$
$270$	0	0
$271$	1.04460e6	0.864030	0.432015	−	0.901866i	$-0.357803\pi$
$271$	1.04460e6	0.864030	0.432015	+	0.901866i	$0.357803\pi$
$272$	−232347.	−0.190421
$273$	0	0
$274$	1.37485e6	1.10631
$275$	−771785.	−0.615410
$276$	0	0
$277$	−1.23380e6	−0.966149	−0.483075	−	0.875579i	$-0.660480\pi$
$277$	−1.23380e6	−0.966149	−0.483075	+	0.875579i	$0.660480\pi$
$278$	−903414.	−0.701092
$279$	0	0
$280$	−717140.	−0.546649
$281$	799854.	0.604289	0.302145	−	0.953262i	$-0.402297\pi$
$281$	799854.	0.604289	0.302145	+	0.953262i	$0.402297\pi$
$282$	0	0
$283$	2.57580e6	1.91181	0.955907	−	0.293668i	$-0.0948761\pi$
$283$	2.57580e6	1.91181	0.955907	+	0.293668i	$0.0948761\pi$
$284$	−108525.	−0.0798428
$285$	0	0
$286$	−382258.	−0.276339
$287$	−525984.	−0.376936
$288$	0	0
$289$	−1.32827e6	−0.935494
$290$	1.50018e6	1.04749
$291$	0	0
$292$	−60388.7	−0.0414475
$293$	44188.8	0.0300706	0.0150353	−	0.999887i	$-0.495214\pi$
$293$	44188.8	0.0300706	0.0150353	+	0.999887i	$0.495214\pi$
$294$	0	0
$295$	395866.	0.264846
$296$	−2.43342e6	−1.61431
$297$	0	0
$298$	−530033.	−0.345750
$299$	−453755.	−0.293524
$300$	0	0
$301$	233541.	0.148575
$302$	625149.	0.394427
$303$	0	0
$304$	−1.13757e6	−0.705983
$305$	−253863.	−0.156261
$306$	0	0
$307$	869175.	0.526334	0.263167	−	0.964750i	$-0.415233\pi$
$307$	869175.	0.526334	0.263167	+	0.964750i	$0.415233\pi$
$308$	−19873.6	−0.0119371
$309$	0	0
$310$	3.82175e6	2.25870
$311$	1.91412e6	1.12220	0.561098	−	0.827749i	$-0.310379\pi$
$311$	1.91412e6	1.12220	0.561098	+	0.827749i	$0.310379\pi$
$312$	0	0
$313$	1.40630e6	0.811364	0.405682	−	0.914014i	$-0.367034\pi$
$313$	1.40630e6	0.811364	0.405682	+	0.914014i	$0.367034\pi$
$314$	−1.67804e6	−0.960456
$315$	0	0
$316$	−388697.	−0.218975
$317$	2.84799e6	1.59181	0.795905	−	0.605422i	$-0.206996\pi$
$317$	2.84799e6	1.59181	0.795905	+	0.605422i	$0.206996\pi$
$318$	0	0
$319$	242135.	0.133223
$320$	−3.96772e6	−2.16604
$321$	0	0
$322$	90216.8	0.0484895
$323$	448422.	0.239156
$324$	0	0
$325$	7.48176e6	3.92912
$326$	−1.58171e6	−0.824296
$327$	0	0
$328$	−3.02251e6	−1.55126
$329$	775334.	0.394911
$330$	0	0
$331$	−2.81506e6	−1.41227	−0.706136	−	0.708076i	$-0.749563\pi$
$331$	−2.81506e6	−1.41227	−0.706136	+	0.708076i	$0.749563\pi$
$332$	−96476.4	−0.0480370
$333$	0	0
$334$	−316222.	−0.155105
$335$	−2.07781e6	−1.01157
$336$	0	0
$337$	−2.61013e6	−1.25195	−0.625977	−	0.779842i	$-0.715299\pi$
$337$	−2.61013e6	−1.25195	−0.625977	+	0.779842i	$0.715299\pi$
$338$	1.83561e6	0.873956
$339$	0	0
$340$	218502.	0.102508
$341$	616843.	0.287269
$342$	0	0
$343$	1.09938e6	0.504558
$344$	1.34202e6	0.611451
$345$	0	0
$346$	−828659.	−0.372122
$347$	2.20275e6	0.982066	0.491033	−	0.871141i	$-0.336619\pi$
$347$	2.20275e6	0.982066	0.491033	+	0.871141i	$0.336619\pi$
$348$	0	0
$349$	204307.	0.0897885	0.0448942	−	0.998992i	$-0.485705\pi$
$349$	204307.	0.0897885	0.0448942	+	0.998992i	$0.485705\pi$
$350$	−1.48754e6	−0.649082
$351$	0	0
$352$	−208795.	−0.0898178
$353$	2.12969e6	0.909661	0.454831	−	0.890578i	$-0.349700\pi$
$353$	2.12969e6	0.909661	0.454831	+	0.890578i	$0.349700\pi$
$354$	0	0
$355$	−1.78084e6	−0.749986
$356$	−636402.	−0.266138
$357$	0	0
$358$	387140.	0.159647
$359$	−1.07164e6	−0.438848	−0.219424	−	0.975630i	$-0.570418\pi$
$359$	−1.07164e6	−0.438848	−0.219424	+	0.975630i	$0.570418\pi$
$360$	0	0
$361$	−280628.	−0.113335
$362$	228262.	0.0915507
$363$	0	0
$364$	192656.	0.0762131
$365$	−990943.	−0.389329
$366$	0	0
$367$	−3.82254e6	−1.48145	−0.740726	−	0.671808i	$-0.765518\pi$
$367$	−3.82254e6	−1.48145	−0.740726	+	0.671808i	$0.765518\pi$
$368$	406135.	0.156333
$369$	0	0
$370$	−6.85596e6	−2.60354
$371$	−78971.0	−0.0297874
$372$	0	0
$373$	1.96127e6	0.729903	0.364951	−	0.931027i	$-0.381086\pi$
$373$	1.96127e6	0.729903	0.364951	+	0.931027i	$0.381086\pi$
$374$	−134870.	−0.0498581
$375$	0	0
$376$	4.45537e6	1.62523
$377$	−2.34728e6	−0.850572
$378$	0	0
$379$	2.24978e6	0.804531	0.402266	−	0.915523i	$-0.368223\pi$
$379$	2.24978e6	0.804531	0.402266	+	0.915523i	$0.368223\pi$
$380$	1.06978e6	0.380046
$381$	0	0
$382$	−3.04387e6	−1.06725
$383$	−2.93484e6	−1.02232	−0.511160	−	0.859486i	$-0.670784\pi$
$383$	−2.93484e6	−1.02232	−0.511160	+	0.859486i	$0.670784\pi$
$384$	0	0
$385$	−326113.	−0.112129
$386$	−95304.7	−0.0325571
$387$	0	0
$388$	−354466.	−0.119535
$389$	452068.	0.151471	0.0757356	−	0.997128i	$-0.475870\pi$
$389$	452068.	0.151471	0.0757356	+	0.997128i	$0.475870\pi$
$390$	0	0
$391$	−160095.	−0.0529587
$392$	3.04717e6	1.00157
$393$	0	0
$394$	−122921.	−0.0398920
$395$	−6.37829e6	−2.05689
$396$	0	0
$397$	−406766.	−0.129530	−0.0647648	−	0.997901i	$-0.520630\pi$
$397$	−406766.	−0.129530	−0.0647648	+	0.997901i	$0.520630\pi$
$398$	2.21499e6	0.700914
$399$	0	0
$400$	−6.69658e6	−2.09268
$401$	−959888.	−0.298098	−0.149049	−	0.988830i	$-0.547621\pi$
$401$	−959888.	−0.298098	−0.149049	+	0.988830i	$0.547621\pi$
$402$	0	0
$403$	−5.97975e6	−1.83409
$404$	−291090.	−0.0887308
$405$	0	0
$406$	466692.	0.140513
$407$	−1.10657e6	−0.331127
$408$	0	0
$409$	3.85815e6	1.14044	0.570218	−	0.821493i	$-0.306859\pi$
$409$	3.85815e6	1.14044	0.570218	+	0.821493i	$0.306859\pi$
$410$	−8.51570e6	−2.50185
$411$	0	0
$412$	50436.2	0.0146386
$413$	123150.	0.0355270
$414$	0	0
$415$	−1.58312e6	−0.451226
$416$	2.02408e6	0.573447
$417$	0	0
$418$	−660321.	−0.184848
$419$	3.30735e6	0.920334	0.460167	−	0.887832i	$-0.347790\pi$
$419$	3.30735e6	0.920334	0.460167	+	0.887832i	$0.347790\pi$
$420$	0	0
$421$	1.93171e6	0.531174	0.265587	−	0.964087i	$-0.414434\pi$
$421$	1.93171e6	0.531174	0.265587	+	0.964087i	$0.414434\pi$
$422$	−349376.	−0.0955019
$423$	0	0
$424$	−453798.	−0.122588
$425$	2.63974e6	0.708907
$426$	0	0
$427$	−78974.3	−0.0209612
$428$	−778118.	−0.205322
$429$	0	0
$430$	3.78103e6	0.986141
$431$	−4.69563e6	−1.21759	−0.608795	−	0.793328i	$-0.708347\pi$
$431$	−4.69563e6	−1.21759	−0.608795	+	0.793328i	$0.708347\pi$
$432$	0	0
$433$	1.81189e6	0.464421	0.232211	−	0.972666i	$-0.425404\pi$
$433$	1.81189e6	0.464421	0.232211	+	0.972666i	$0.425404\pi$
$434$	1.18891e6	0.302987
$435$	0	0
$436$	−1.34935e6	−0.339945
$437$	−783826.	−0.196343
$438$	0	0
$439$	4.92151e6	1.21881	0.609406	−	0.792858i	$-0.291408\pi$
$439$	4.92151e6	1.21881	0.609406	+	0.792858i	$0.291408\pi$
$440$	−1.87397e6	−0.461458
$441$	0	0
$442$	1.30744e6	0.318322
$443$	−6.89554e6	−1.66939	−0.834697	−	0.550710i	$-0.814357\pi$
$443$	−6.89554e6	−1.66939	−0.834697	+	0.550710i	$0.814357\pi$
$444$	0	0
$445$	−1.04430e7	−2.49991
$446$	2.95143e6	0.702579
$447$	0	0
$448$	−1.23432e6	−0.290557
$449$	2.30681e6	0.540003	0.270001	−	0.962860i	$-0.412976\pi$
$449$	2.30681e6	0.540003	0.270001	+	0.962860i	$0.412976\pi$
$450$	0	0
$451$	−1.37446e6	−0.318193
$452$	−327987.	−0.0755110
$453$	0	0
$454$	−4.97135e6	−1.13197
$455$	3.16138e6	0.715892
$456$	0	0
$457$	−8.75227e6	−1.96034	−0.980168	−	0.198170i	$-0.936500\pi$
$457$	−8.75227e6	−1.96034	−0.980168	+	0.198170i	$0.936500\pi$
$458$	−2.27718e6	−0.507263
$459$	0	0
$460$	−381933.	−0.0841575
$461$	−544322.	−0.119290	−0.0596449	−	0.998220i	$-0.518997\pi$
$461$	−544322.	−0.119290	−0.0596449	+	0.998220i	$0.518997\pi$
$462$	0	0
$463$	−7.40626e6	−1.60563	−0.802817	−	0.596226i	$-0.796666\pi$
$463$	−7.40626e6	−1.60563	−0.802817	+	0.596226i	$0.796666\pi$
$464$	2.10094e6	0.453021
$465$	0	0
$466$	2.34540e6	0.500326
$467$	−3.51121e6	−0.745014	−0.372507	−	0.928029i	$-0.621502\pi$
$467$	−3.51121e6	−0.745014	−0.372507	+	0.928029i	$0.621502\pi$
$468$	0	0
$469$	−646387.	−0.135694
$470$	1.25527e7	2.62115
$471$	0	0
$472$	707667.	0.146209
$473$	610270.	0.125421
$474$	0	0
$475$	1.29242e7	2.62826
$476$	67973.7	0.0137507
$477$	0	0
$478$	−3.04954e6	−0.610470
$479$	4.63723e6	0.923464	0.461732	−	0.887019i	$-0.347228\pi$
$479$	4.63723e6	0.923464	0.461732	+	0.887019i	$0.347228\pi$
$480$	0	0
$481$	1.07273e7	2.11410
$482$	2.33710e6	0.458205
$483$	0	0
$484$	1.01634e6	0.197209
$485$	−5.81658e6	−1.12283
$486$	0	0
$487$	2.05026e6	0.391731	0.195865	−	0.980631i	$-0.437248\pi$
$487$	2.05026e6	0.391731	0.195865	+	0.980631i	$0.437248\pi$
$488$	−453817.	−0.0862643
$489$	0	0
$490$	8.58519e6	1.61532
$491$	−3.10489e6	−0.581222	−0.290611	−	0.956841i	$-0.593859\pi$
$491$	−3.10489e6	−0.581222	−0.290611	+	0.956841i	$0.593859\pi$
$492$	0	0
$493$	−828175.	−0.153463
$494$	6.40122e6	1.18017
$495$	0	0
$496$	5.35219e6	0.976849
$497$	−554001.	−0.100605
$498$	0	0
$499$	−6.39520e6	−1.14975	−0.574874	−	0.818242i	$-0.694949\pi$
$499$	−6.39520e6	−1.14975	−0.574874	+	0.818242i	$0.694949\pi$
$500$	4.04131e6	0.722931
$501$	0	0
$502$	5.73510e6	1.01574
$503$	7.97883e6	1.40611	0.703054	−	0.711136i	$-0.251819\pi$
$503$	7.97883e6	1.40611	0.703054	+	0.711136i	$0.251819\pi$
$504$	0	0
$505$	−4.77662e6	−0.833474
$506$	235747.	0.0409327
$507$	0	0
$508$	149463.	0.0256966
$509$	−142812.	−0.0244325	−0.0122163	−	0.999925i	$-0.503889\pi$
$509$	−142812.	−0.0244325	−0.0122163	+	0.999925i	$0.503889\pi$
$510$	0	0
$511$	−308272.	−0.0522255
$512$	−6.59199e6	−1.11133
$513$	0	0
$514$	−791744.	−0.132183
$515$	827628.	0.137505
$516$	0	0
$517$	2.02604e6	0.333366
$518$	−2.13282e6	−0.349245
$519$	0	0
$520$	1.81665e7	2.94621
$521$	1.06044e7	1.71156	0.855780	−	0.517340i	$-0.173078\pi$
$521$	1.06044e7	1.71156	0.855780	+	0.517340i	$0.173078\pi$
$522$	0	0
$523$	−3.76110e6	−0.601257	−0.300629	−	0.953741i	$-0.597196\pi$
$523$	−3.76110e6	−0.601257	−0.300629	+	0.953741i	$0.597196\pi$
$524$	−2.31896e6	−0.368947
$525$	0	0
$526$	1.10739e7	1.74517
$527$	−2.10980e6	−0.330913
$528$	0	0
$529$	279841.	0.0434783
$530$	−1.27854e6	−0.197708
$531$	0	0
$532$	332799.	0.0509803
$533$	1.33242e7	2.03153
$534$	0	0
$535$	−1.27685e7	−1.92865
$536$	−3.71439e6	−0.558439
$537$	0	0
$538$	3.13389e6	0.466796
$539$	1.38568e6	0.205442
$540$	0	0
$541$	9.68021e6	1.42197	0.710987	−	0.703206i	$-0.248249\pi$
$541$	9.68021e6	1.42197	0.710987	+	0.703206i	$0.248249\pi$
$542$	5.26121e6	0.769285
$543$	0	0
$544$	714142.	0.103464
$545$	−2.21421e7	−3.19321
$546$	0	0
$547$	−978123.	−0.139774	−0.0698868	−	0.997555i	$-0.522264\pi$
$547$	−978123.	−0.139774	−0.0698868	+	0.997555i	$0.522264\pi$
$548$	−1.81067e6	−0.257566
$549$	0	0
$550$	−3.88713e6	−0.547927
$551$	−4.05474e6	−0.568963
$552$	0	0
$553$	−1.98422e6	−0.275916
$554$	−6.21408e6	−0.860206
$555$	0	0
$556$	1.18980e6	0.163225
$557$	−8.24343e6	−1.12582	−0.562911	−	0.826517i	$-0.690319\pi$
$557$	−8.24343e6	−1.12582	−0.562911	+	0.826517i	$0.690319\pi$
$558$	0	0
$559$	−5.91602e6	−0.800756
$560$	−2.82960e6	−0.381290
$561$	0	0
$562$	4.02851e6	0.538026
$563$	7.30485e6	0.971270	0.485635	−	0.874162i	$-0.338588\pi$
$563$	7.30485e6	0.971270	0.485635	+	0.874162i	$0.338588\pi$
$564$	0	0
$565$	−5.38207e6	−0.709297
$566$	1.29731e7	1.70217
$567$	0	0
$568$	−3.18350e6	−0.414033
$569$	3.20986e6	0.415629	0.207814	−	0.978168i	$-0.433365\pi$
$569$	3.20986e6	0.415629	0.207814	+	0.978168i	$0.433365\pi$
$570$	0	0
$571$	−1.02719e7	−1.31844	−0.659219	−	0.751951i	$-0.729113\pi$
$571$	−1.02719e7	−1.31844	−0.659219	+	0.751951i	$0.729113\pi$
$572$	503434.	0.0643358
$573$	0	0
$574$	−2.64915e6	−0.335603
$575$	−4.61417e6	−0.582002
$576$	0	0
$577$	1.39706e6	0.174693	0.0873466	−	0.996178i	$-0.472161\pi$
$577$	1.39706e6	0.174693	0.0873466	+	0.996178i	$0.472161\pi$
$578$	−6.68989e6	−0.832912
$579$	0	0
$580$	−1.97574e6	−0.243871
$581$	−492493.	−0.0605285
$582$	0	0
$583$	−206361.	−0.0251452
$584$	−1.77145e6	−0.214930
$585$	0	0
$586$	222559.	0.0267732
$587$	−142524.	−0.0170723	−0.00853615	−	0.999964i	$-0.502717\pi$
$587$	−142524.	−0.0170723	−0.00853615	+	0.999964i	$0.502717\pi$
$588$	0	0
$589$	−1.03295e7	−1.22685
$590$	1.99380e6	0.235804
$591$	0	0
$592$	−9.60146e6	−1.12599
$593$	3.64903e6	0.426128	0.213064	−	0.977038i	$-0.431656\pi$
$593$	3.64903e6	0.426128	0.213064	+	0.977038i	$0.431656\pi$
$594$	0	0
$595$	1.11541e6	0.129164
$596$	698055.	0.0804959
$597$	0	0
$598$	−2.28536e6	−0.261337
$599$	−2.97083e6	−0.338307	−0.169153	−	0.985590i	$-0.554103\pi$
$599$	−2.97083e6	−0.338307	−0.169153	+	0.985590i	$0.554103\pi$
$600$	0	0
$601$	1.34022e7	1.51353	0.756763	−	0.653689i	$-0.226780\pi$
$601$	1.34022e7	1.51353	0.756763	+	0.653689i	$0.226780\pi$
$602$	1.17624e6	0.132283
$603$	0	0
$604$	−823322.	−0.0918285
$605$	1.66776e7	1.85244
$606$	0	0
$607$	1.23572e7	1.36128	0.680640	−	0.732618i	$-0.261702\pi$
$607$	1.23572e7	1.36128	0.680640	+	0.732618i	$0.261702\pi$
$608$	3.49643e6	0.383589
$609$	0	0
$610$	−1.27860e6	−0.139126
$611$	−1.96407e7	−2.12840
$612$	0	0
$613$	6.32585e6	0.679936	0.339968	−	0.940437i	$-0.389584\pi$
$613$	6.32585e6	0.679936	0.339968	+	0.940437i	$0.389584\pi$
$614$	4.37764e6	0.468619
$615$	0	0
$616$	−582974.	−0.0619010
$617$	1.27877e7	1.35232	0.676158	−	0.736756i	$-0.263644\pi$
$617$	1.27877e7	1.35232	0.676158	+	0.736756i	$0.263644\pi$
$618$	0	0
$619$	−1.40070e6	−0.146932	−0.0734662	−	0.997298i	$-0.523406\pi$
$619$	−1.40070e6	−0.146932	−0.0734662	+	0.997298i	$0.523406\pi$
$620$	−5.03325e6	−0.525859
$621$	0	0
$622$	9.64058e6	0.999142
$623$	−3.24870e6	−0.335344
$624$	0	0
$625$	3.90578e7	3.99952
$626$	7.08288e6	0.722394
$627$	0	0
$628$	2.20998e6	0.223609
$629$	3.78483e6	0.381434
$630$	0	0
$631$	2.14447e6	0.214411	0.107206	−	0.994237i	$-0.465810\pi$
$631$	2.14447e6	0.214411	0.107206	+	0.994237i	$0.465810\pi$
$632$	−1.14021e7	−1.13551
$633$	0	0
$634$	1.43441e7	1.41726
$635$	2.45260e6	0.241375
$636$	0	0
$637$	−1.34329e7	−1.31166
$638$	1.21952e6	0.118615
$639$	0	0
$640$	−1.17645e7	−1.13534
$641$	1.24431e7	1.19614	0.598071	−	0.801443i	$-0.295934\pi$
$641$	1.24431e7	1.19614	0.598071	+	0.801443i	$0.295934\pi$
$642$	0	0
$643$	8.25180e6	0.787084	0.393542	−	0.919307i	$-0.371250\pi$
$643$	8.25180e6	0.787084	0.393542	+	0.919307i	$0.371250\pi$
$644$	−118816.	−0.0112891
$645$	0	0
$646$	2.25850e6	0.212931
$647$	−1.82649e7	−1.71536	−0.857681	−	0.514183i	$-0.828095\pi$
$647$	−1.82649e7	−1.71536	−0.857681	+	0.514183i	$0.828095\pi$
$648$	0	0
$649$	321806.	0.0299904
$650$	3.76823e7	3.49827
$651$	0	0
$652$	2.08312e6	0.191909
$653$	6.86993e6	0.630477	0.315239	−	0.949012i	$-0.397915\pi$
$653$	6.86993e6	0.630477	0.315239	+	0.949012i	$0.397915\pi$
$654$	0	0
$655$	−3.80527e7	−3.46563
$656$	−1.19258e7	−1.08201
$657$	0	0
$658$	3.90501e6	0.351607
$659$	4.39836e6	0.394528	0.197264	−	0.980350i	$-0.436794\pi$
$659$	4.39836e6	0.394528	0.197264	+	0.980350i	$0.436794\pi$
$660$	0	0
$661$	−8.73583e6	−0.777679	−0.388840	−	0.921305i	$-0.627124\pi$
$661$	−8.73583e6	−0.777679	−0.388840	+	0.921305i	$0.627124\pi$
$662$	−1.41782e7	−1.25741
$663$	0	0
$664$	−2.83006e6	−0.249101
$665$	5.46103e6	0.478873
$666$	0	0
$667$	1.44762e6	0.125991
$668$	416465.	0.0361108
$669$	0	0
$670$	−1.04650e7	−0.900643
$671$	−206369.	−0.0176945
$672$	0	0
$673$	8.38413e6	0.713543	0.356772	−	0.934192i	$-0.383877\pi$
$673$	8.38413e6	0.713543	0.356772	+	0.934192i	$0.383877\pi$
$674$	−1.31461e7	−1.11467
$675$	0	0
$676$	−2.41751e6	−0.203470
$677$	−2.12039e7	−1.77805	−0.889027	−	0.457855i	$-0.848618\pi$
$677$	−2.12039e7	−1.77805	−0.889027	+	0.457855i	$0.848618\pi$
$678$	0	0
$679$	−1.80948e6	−0.150619
$680$	6.40957e6	0.531565
$681$	0	0
$682$	3.10676e6	0.255769
$683$	−1.15827e7	−0.950075	−0.475038	−	0.879966i	$-0.657566\pi$
$683$	−1.15827e7	−0.950075	−0.475038	+	0.879966i	$0.657566\pi$
$684$	0	0
$685$	−2.97121e7	−2.41939
$686$	5.53706e6	0.449231
$687$	0	0
$688$	5.29516e6	0.426489
$689$	2.00048e6	0.160541
$690$	0	0
$691$	−1.68594e7	−1.34322	−0.671611	−	0.740904i	$-0.734397\pi$
$691$	−1.68594e7	−1.34322	−0.671611	+	0.740904i	$0.734397\pi$
$692$	1.09135e6	0.0866357
$693$	0	0
$694$	1.10942e7	0.874377
$695$	1.95239e7	1.53322
$696$	0	0
$697$	4.70108e6	0.366535
$698$	1.02901e6	0.0799427
$699$	0	0
$700$	1.95910e6	0.151116
$701$	−2.16111e7	−1.66105	−0.830523	−	0.556985i	$-0.811958\pi$
$701$	−2.16111e7	−1.66105	−0.830523	+	0.556985i	$0.811958\pi$
$702$	0	0
$703$	1.85305e7	1.41416
$704$	−3.22542e6	−0.245276
$705$	0	0
$706$	1.07263e7	0.809912
$707$	−1.48596e6	−0.111804
$708$	0	0
$709$	1.07627e7	0.804088	0.402044	−	0.915620i	$-0.368300\pi$
$709$	1.07627e7	0.804088	0.402044	+	0.915620i	$0.368300\pi$
$710$	−8.96928e6	−0.667747
$711$	0	0
$712$	−1.86683e7	−1.38008
$713$	3.68785e6	0.271675
$714$	0	0
$715$	8.26106e6	0.604325
$716$	−509864.	−0.0371682
$717$	0	0
$718$	−5.39738e6	−0.390726
$719$	7.69330e6	0.554997	0.277498	−	0.960726i	$-0.410495\pi$
$719$	7.69330e6	0.554997	0.277498	+	0.960726i	$0.410495\pi$
$720$	0	0
$721$	257467.	0.0184452
$722$	−1.41340e6	−0.100907
$723$	0	0
$724$	−300621.	−0.0213144
$725$	−2.38692e7	−1.68652
$726$	0	0
$727$	1.42775e7	1.00188	0.500941	−	0.865481i	$-0.332987\pi$
$727$	1.42775e7	1.00188	0.500941	+	0.865481i	$0.332987\pi$
$728$	5.65142e6	0.395211
$729$	0	0
$730$	−4.99094e6	−0.346637
$731$	−2.08731e6	−0.144475
$732$	0	0
$733$	996838.	0.0685275	0.0342637	−	0.999413i	$-0.489091\pi$
$733$	996838.	0.0685275	0.0342637	+	0.999413i	$0.489091\pi$
$734$	−1.92524e7	−1.31900
$735$	0	0
$736$	−1.24829e6	−0.0849420
$737$	−1.68909e6	−0.114547
$738$	0	0
$739$	2.33398e6	0.157212	0.0786059	−	0.996906i	$-0.474953\pi$
$739$	2.33398e6	0.157212	0.0786059	+	0.996906i	$0.474953\pi$
$740$	9.02931e6	0.606143
$741$	0	0
$742$	−397741.	−0.0265211
$743$	−1.93861e7	−1.28830	−0.644151	−	0.764899i	$-0.722789\pi$
$743$	−1.93861e7	−1.28830	−0.644151	+	0.764899i	$0.722789\pi$
$744$	0	0
$745$	1.14547e7	0.756122
$746$	9.87803e6	0.649865
$747$	0	0
$748$	177624.	0.0116077
$749$	−3.97214e6	−0.258714
$750$	0	0
$751$	−1.54380e7	−0.998831	−0.499415	−	0.866363i	$-0.666452\pi$
$751$	−1.54380e7	−0.998831	−0.499415	+	0.866363i	$0.666452\pi$
$752$	1.75794e7	1.13360
$753$	0	0
$754$	−1.18222e7	−0.757303
$755$	−1.35102e7	−0.862572
$756$	0	0
$757$	−2.29164e7	−1.45347	−0.726737	−	0.686916i	$-0.758964\pi$
$757$	−2.29164e7	−1.45347	−0.726737	+	0.686916i	$0.758964\pi$
$758$	1.13312e7	0.716310
$759$	0	0
$760$	3.13812e7	1.97077
$761$	1.17153e7	0.733316	0.366658	−	0.930356i	$-0.380502\pi$
$761$	1.17153e7	0.733316	0.366658	+	0.930356i	$0.380502\pi$
$762$	0	0
$763$	−6.88817e6	−0.428344
$764$	4.00878e6	0.248473
$765$	0	0
$766$	−1.47815e7	−0.910218
$767$	−3.11962e6	−0.191475
$768$	0	0
$769$	2.14122e7	1.30571	0.652853	−	0.757485i	$-0.273572\pi$
$769$	2.14122e7	1.30571	0.652853	+	0.757485i	$0.273572\pi$
$770$	−1.64249e6	−0.0998332
$771$	0	0
$772$	125516.	0.00757979
$773$	−5.38793e6	−0.324320	−0.162160	−	0.986765i	$-0.551846\pi$
$773$	−5.38793e6	−0.324320	−0.162160	+	0.986765i	$0.551846\pi$
$774$	0	0
$775$	−6.08072e7	−3.63665
$776$	−1.03980e7	−0.619862
$777$	0	0
$778$	2.27686e6	0.134862
$779$	2.30165e7	1.35892
$780$	0	0
$781$	−1.44767e6	−0.0849263
$782$	−806329.	−0.0471515
$783$	0	0
$784$	1.20232e7	0.698600
$785$	3.62644e7	2.10042
$786$	0	0
$787$	−6.16548e6	−0.354838	−0.177419	−	0.984135i	$-0.556775\pi$
$787$	−6.16548e6	−0.354838	−0.177419	+	0.984135i	$0.556775\pi$
$788$	161887.	0.00928746
$789$	0	0
$790$	−3.21246e7	−1.83134
$791$	−1.67431e6	−0.0951467
$792$	0	0
$793$	2.00057e6	0.112972
$794$	−2.04870e6	−0.115326
$795$	0	0
$796$	−2.91715e6	−0.163183
$797$	−8.44524e6	−0.470941	−0.235470	−	0.971882i	$-0.575663\pi$
$797$	−8.44524e6	−0.470941	−0.235470	+	0.971882i	$0.575663\pi$
$798$	0	0
$799$	−6.92969e6	−0.384014
$800$	2.05826e7	1.13704
$801$	0	0
$802$	−4.83453e6	−0.265410
$803$	−805553.	−0.0440865
$804$	0	0
$805$	−1.94969e6	−0.106042
$806$	−3.01173e7	−1.63297
$807$	0	0
$808$	−8.53890e6	−0.460122
$809$	1.72869e7	0.928636	0.464318	−	0.885669i	$-0.346300\pi$
$809$	1.72869e7	0.928636	0.464318	+	0.885669i	$0.346300\pi$
$810$	0	0
$811$	1.70374e7	0.909599	0.454800	−	0.890594i	$-0.349711\pi$
$811$	1.70374e7	0.909599	0.454800	+	0.890594i	$0.349711\pi$
$812$	−614634.	−0.0327135
$813$	0	0
$814$	−5.57332e6	−0.294817
$815$	3.41827e7	1.80265
$816$	0	0
$817$	−1.02195e7	−0.535640
$818$	1.94318e7	1.01538
$819$	0	0
$820$	1.12152e7	0.582468
$821$	−2.35799e7	−1.22091	−0.610455	−	0.792051i	$-0.709014\pi$
$821$	−2.35799e7	−1.22091	−0.610455	+	0.792051i	$0.709014\pi$
$822$	0	0
$823$	3.52171e7	1.81240	0.906200	−	0.422848i	$-0.138970\pi$
$823$	3.52171e7	1.81240	0.906200	+	0.422848i	$0.138970\pi$
$824$	1.47950e6	0.0759099
$825$	0	0
$826$	620251.	0.0316313
$827$	−5.11451e6	−0.260040	−0.130020	−	0.991511i	$-0.541504\pi$
$827$	−5.11451e6	−0.260040	−0.130020	+	0.991511i	$0.541504\pi$
$828$	0	0
$829$	−2.61366e7	−1.32088	−0.660439	−	0.750879i	$-0.729630\pi$
$829$	−2.61366e7	−1.32088	−0.660439	+	0.750879i	$0.729630\pi$
$830$	−7.97347e6	−0.401747
$831$	0	0
$832$	3.12676e7	1.56598
$833$	−4.73944e6	−0.236655
$834$	0	0
$835$	6.83395e6	0.339200
$836$	869643.	0.0430354
$837$	0	0
$838$	1.66577e7	0.819415
$839$	1.29851e7	0.636854	0.318427	−	0.947947i	$-0.396845\pi$
$839$	1.29851e7	0.636854	0.318427	+	0.947947i	$0.396845\pi$
$840$	0	0
$841$	−1.30226e7	−0.634903
$842$	9.72916e6	0.472928
$843$	0	0
$844$	460129.	0.0222343
$845$	−3.96698e7	−1.91126
$846$	0	0
$847$	5.18823e6	0.248491
$848$	−1.79054e6	−0.0855055
$849$	0	0
$850$	1.32952e7	0.631172
$851$	−6.61574e6	−0.313152
$852$	0	0
$853$	1.49057e7	0.701425	0.350712	−	0.936483i	$-0.385939\pi$
$853$	1.49057e7	0.701425	0.350712	+	0.936483i	$0.385939\pi$
$854$	−397758.	−0.0186627
$855$	0	0
$856$	−2.28255e7	−1.06472
$857$	−8.45793e6	−0.393380	−0.196690	−	0.980466i	$-0.563019\pi$
$857$	−8.45793e6	−0.393380	−0.196690	+	0.980466i	$0.563019\pi$
$858$	0	0
$859$	−2.84116e7	−1.31375	−0.656875	−	0.753999i	$-0.728122\pi$
$859$	−2.84116e7	−1.31375	−0.656875	+	0.753999i	$0.728122\pi$
$860$	−4.97962e6	−0.229588
$861$	0	0
$862$	−2.36498e7	−1.08407
$863$	2.78133e7	1.27123	0.635617	−	0.772005i	$-0.280746\pi$
$863$	2.78133e7	1.27123	0.635617	+	0.772005i	$0.280746\pi$
$864$	0	0
$865$	1.79083e7	0.813795
$866$	9.12567e6	0.413495
$867$	0	0
$868$	−1.56579e6	−0.0705400
$869$	−5.18501e6	−0.232917
$870$	0	0
$871$	1.63742e7	0.731332
$872$	−3.95821e7	−1.76282
$873$	0	0
$874$	−3.94778e6	−0.174813
$875$	2.06301e7	0.910920
$876$	0	0
$877$	1.05659e7	0.463882	0.231941	−	0.972730i	$-0.425492\pi$
$877$	1.05659e7	0.463882	0.231941	+	0.972730i	$0.425492\pi$
$878$	2.47874e7	1.08516
$879$	0	0
$880$	−7.39409e6	−0.321868
$881$	2.66383e7	1.15629	0.578144	−	0.815934i	$-0.303777\pi$
$881$	2.66383e7	1.15629	0.578144	+	0.815934i	$0.303777\pi$
$882$	0	0
$883$	−1.40812e7	−0.607767	−0.303883	−	0.952709i	$-0.598283\pi$
$883$	−1.40812e7	−0.607767	−0.303883	+	0.952709i	$0.598283\pi$
$884$	−1.72190e6	−0.0741102
$885$	0	0
$886$	−3.47297e7	−1.48634
$887$	−2.77934e7	−1.18613	−0.593066	−	0.805154i	$-0.702083\pi$
$887$	−2.77934e7	−1.18613	−0.593066	+	0.805154i	$0.702083\pi$
$888$	0	0
$889$	762980.	0.0323786
$890$	−5.25966e7	−2.22578
$891$	0	0
$892$	−3.88703e6	−0.163571
$893$	−3.39277e7	−1.42372
$894$	0	0
$895$	−8.36656e6	−0.349132
$896$	−3.65982e6	−0.152297
$897$	0	0
$898$	1.16184e7	0.480789
$899$	1.90773e7	0.787258
$900$	0	0
$901$	705818.	0.0289655
$902$	−6.92254e6	−0.283302
$903$	0	0
$904$	−9.62122e6	−0.391570
$905$	−4.93301e6	−0.200212
$906$	0	0
$907$	3.73068e7	1.50581	0.752905	−	0.658129i	$-0.228652\pi$
$907$	3.73068e7	1.50581	0.752905	+	0.658129i	$0.228652\pi$
$908$	6.54728e6	0.263540
$909$	0	0
$910$	1.59224e7	0.637391
$911$	3.28509e7	1.31145	0.655724	−	0.755001i	$-0.272364\pi$
$911$	3.28509e7	1.31145	0.655724	+	0.755001i	$0.272364\pi$
$912$	0	0
$913$	−1.28694e6	−0.0510955
$914$	−4.40813e7	−1.74537
$915$	0	0
$916$	2.99904e6	0.118098
$917$	−1.18378e7	−0.464888
$918$	0	0
$919$	−2.67326e7	−1.04412	−0.522062	−	0.852908i	$-0.674837\pi$
$919$	−2.67326e7	−1.04412	−0.522062	+	0.852908i	$0.674837\pi$
$920$	−1.12037e7	−0.436407
$921$	0	0
$922$	−2.74151e6	−0.106209
$923$	1.40339e7	0.542217
$924$	0	0
$925$	1.09084e8	4.19186
$926$	−3.73020e7	−1.42957
$927$	0	0
$928$	−6.45744e6	−0.246145
$929$	−4.41532e7	−1.67851	−0.839253	−	0.543740i	$-0.817008\pi$
$929$	−4.41532e7	−1.67851	−0.839253	+	0.543740i	$0.817008\pi$
$930$	0	0
$931$	−2.32043e7	−0.877392
$932$	−3.08890e6	−0.116483
$933$	0	0
$934$	−1.76844e7	−0.663319
$935$	2.91470e6	0.109035
$936$	0	0
$937$	−2.97371e7	−1.10649	−0.553247	−	0.833017i	$-0.686611\pi$
$937$	−2.97371e7	−1.10649	−0.553247	+	0.833017i	$0.686611\pi$
$938$	−3.25556e6	−0.120814
$939$	0	0
$940$	−1.65319e7	−0.610243
$941$	2.47003e7	0.909345	0.454672	−	0.890659i	$-0.349756\pi$
$941$	2.47003e7	0.909345	0.454672	+	0.890659i	$0.349756\pi$
$942$	0	0
$943$	−8.21732e6	−0.300920
$944$	2.79222e6	0.101981
$945$	0	0
$946$	3.07366e6	0.111668
$947$	−3.57425e7	−1.29512	−0.647559	−	0.762015i	$-0.724210\pi$
$947$	−3.57425e7	−1.29512	−0.647559	+	0.762015i	$0.724210\pi$
$948$	0	0
$949$	7.80912e6	0.281473
$950$	6.50932e7	2.34006
$951$	0	0
$952$	1.99395e6	0.0713054
$953$	2.18718e7	0.780102	0.390051	−	0.920793i	$-0.372457\pi$
$953$	2.18718e7	0.780102	0.390051	+	0.920793i	$0.372457\pi$
$954$	0	0
$955$	6.57817e7	2.33398
$956$	4.01625e6	0.142127
$957$	0	0
$958$	2.33557e7	0.822202
$959$	−9.24313e6	−0.324543
$960$	0	0
$961$	1.99706e7	0.697562
$962$	5.40284e7	1.88228
$963$	0	0
$964$	−3.07796e6	−0.106677
$965$	2.05965e6	0.0711992
$966$	0	0
$967$	−3.56066e7	−1.22452	−0.612258	−	0.790658i	$-0.709739\pi$
$967$	−3.56066e7	−1.22452	−0.612258	+	0.790658i	$0.709739\pi$
$968$	2.98136e7	1.02265
$969$	0	0
$970$	−2.92955e7	−0.999705
$971$	2.43167e7	0.827668	0.413834	−	0.910352i	$-0.364189\pi$
$971$	2.43167e7	0.827668	0.413834	+	0.910352i	$0.364189\pi$
$972$	0	0
$973$	6.07368e6	0.205669
$974$	1.03263e7	0.348775
$975$	0	0
$976$	−1.79061e6	−0.0601696
$977$	−2.27487e7	−0.762466	−0.381233	−	0.924479i	$-0.624500\pi$
$977$	−2.27487e7	−0.762466	−0.381233	+	0.924479i	$0.624500\pi$
$978$	0	0
$979$	−8.48926e6	−0.283083
$980$	−1.13067e7	−0.376072
$981$	0	0
$982$	−1.56379e7	−0.517488
$983$	2.60467e7	0.859745	0.429872	−	0.902890i	$-0.358559\pi$
$983$	2.60467e7	0.859745	0.429872	+	0.902890i	$0.358559\pi$
$984$	0	0
$985$	2.65647e6	0.0872398
$986$	−4.17115e6	−0.136635
$987$	0	0
$988$	−8.43041e6	−0.274762
$989$	3.64855e6	0.118612
$990$	0	0
$991$	−2.30370e7	−0.745146	−0.372573	−	0.928003i	$-0.621524\pi$
$991$	−2.30370e7	−0.745146	−0.372573	+	0.928003i	$0.621524\pi$
$992$	−1.64505e7	−0.530761
$993$	0	0
$994$	−2.79025e6	−0.0895731
$995$	−4.78687e7	−1.53283
$996$	0	0
$997$	−3.11789e7	−0.993398	−0.496699	−	0.867923i	$-0.665455\pi$
$997$	−3.11789e7	−0.993398	−0.496699	+	0.867923i	$0.665455\pi$
$998$	−3.22098e7	−1.02367
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	207.6.a.g.1.5		6
3.2	odd	2		23.6.a.b.1.2	✓	6
12.11	even	2		368.6.a.h.1.6		6
15.14	odd	2		575.6.a.c.1.5		6
69.68	even	2		529.6.a.c.1.2		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
23.6.a.b.1.2	✓	6	3.2	odd	2
207.6.a.g.1.5		6	1.1	even	1	trivial
368.6.a.h.1.6		6	12.11	even	2
529.6.a.c.1.2		6	69.68	even	2
575.6.a.c.1.5		6	15.14	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)

\(f(q)\)	\(=\)	\(q+5.03655 q^{2} -6.63314 q^{4} -108.846 q^{5} -33.8609 q^{7} -194.578 q^{8} +O(q^{10})\)	\(q+5.03655 q^{2} -6.63314 q^{4} -108.846 q^{5} -33.8609 q^{7} -194.578 q^{8} -548.208 q^{10} -88.4826 q^{11} +857.760 q^{13} -170.542 q^{14} -767.741 q^{16} +302.638 q^{17} +1481.71 q^{19} +721.991 q^{20} -445.647 q^{22} -529.000 q^{23} +8722.45 q^{25} +4320.15 q^{26} +224.604 q^{28} -2736.52 q^{29} -6971.35 q^{31} +2359.72 q^{32} +1524.25 q^{34} +3685.62 q^{35} +12506.1 q^{37} +7462.72 q^{38} +21179.0 q^{40} +15533.7 q^{41} -6897.06 q^{43} +586.918 q^{44} -2664.34 q^{46} -22897.6 q^{47} -15660.4 q^{49} +43931.1 q^{50} -5689.64 q^{52} +2332.22 q^{53} +9630.97 q^{55} +6588.58 q^{56} -13782.6 q^{58} -3636.94 q^{59} +2332.32 q^{61} -35111.6 q^{62} +36452.6 q^{64} -93363.7 q^{65} +19089.5 q^{67} -2007.44 q^{68} +18562.8 q^{70} +16361.1 q^{71} +9104.09 q^{73} +62987.8 q^{74} -9828.41 q^{76} +2996.10 q^{77} +58599.3 q^{79} +83565.5 q^{80} +78236.2 q^{82} +14544.6 q^{83} -32940.9 q^{85} -34737.4 q^{86} +17216.8 q^{88} +95942.7 q^{89} -29044.5 q^{91} +3508.93 q^{92} -115325. q^{94} -161278. q^{95} +53438.7 q^{97} -78874.6 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 4 q^{2} + 112 q^{4} - 42 q^{5} + 300 q^{7} - 393 q^{8}+O(q^{10}) \) 6 * q - 4 * q^2 + 112 * q^4 - 42 * q^5 + 300 * q^7 - 393 * q^8	\( 6 q - 4 q^{2} + 112 q^{4} - 42 q^{5} + 300 q^{7} - 393 q^{8} - 10 q^{10} + 58 q^{11} + 792 q^{13} + 2984 q^{14} - 1904 q^{16} + 400 q^{17} + 2738 q^{19} + 7124 q^{20} - 1972 q^{22} - 3174 q^{23} + 9966 q^{25} - 4511 q^{26} + 9570 q^{28} - 11244 q^{29} + 13748 q^{31} - 6600 q^{32} - 16226 q^{34} + 4296 q^{35} + 25426 q^{37} + 8028 q^{38} + 10230 q^{40} + 14268 q^{41} - 18082 q^{43} + 51146 q^{44} + 2116 q^{46} + 23084 q^{47} + 37422 q^{49} + 67436 q^{50} + 36807 q^{52} - 17522 q^{53} + 47576 q^{55} - 44946 q^{56} + 141001 q^{58} + 36392 q^{59} + 27062 q^{61} - 48971 q^{62} + 89451 q^{64} - 7108 q^{65} + 37138 q^{67} - 17260 q^{68} + 248380 q^{70} + 158556 q^{71} + 112228 q^{73} + 66878 q^{74} + 157816 q^{76} + 89760 q^{77} + 36844 q^{79} + 158530 q^{80} + 150039 q^{82} + 76350 q^{83} - 102132 q^{85} + 100578 q^{86} - 219028 q^{88} - 16100 q^{89} - 250592 q^{91} - 59248 q^{92} + 12887 q^{94} + 190096 q^{95} + 259432 q^{97} + 325816 q^{98}+O(q^{100}) \) 6 * q - 4 * q^2 + 112 * q^4 - 42 * q^5 + 300 * q^7 - 393 * q^8 - 10 * q^10 + 58 * q^11 + 792 * q^13 + 2984 * q^14 - 1904 * q^16 + 400 * q^17 + 2738 * q^19 + 7124 * q^20 - 1972 * q^22 - 3174 * q^23 + 9966 * q^25 - 4511 * q^26 + 9570 * q^28 - 11244 * q^29 + 13748 * q^31 - 6600 * q^32 - 16226 * q^34 + 4296 * q^35 + 25426 * q^37 + 8028 * q^38 + 10230 * q^40 + 14268 * q^41 - 18082 * q^43 + 51146 * q^44 + 2116 * q^46 + 23084 * q^47 + 37422 * q^49 + 67436 * q^50 + 36807 * q^52 - 17522 * q^53 + 47576 * q^55 - 44946 * q^56 + 141001 * q^58 + 36392 * q^59 + 27062 * q^61 - 48971 * q^62 + 89451 * q^64 - 7108 * q^65 + 37138 * q^67 - 17260 * q^68 + 248380 * q^70 + 158556 * q^71 + 112228 * q^73 + 66878 * q^74 + 157816 * q^76 + 89760 * q^77 + 36844 * q^79 + 158530 * q^80 + 150039 * q^82 + 76350 * q^83 - 102132 * q^85 + 100578 * q^86 - 219028 * q^88 - 16100 * q^89 - 250592 * q^91 - 59248 * q^92 + 12887 * q^94 + 190096 * q^95 + 259432 * q^97 + 325816 * q^98

Introduction

L-functions

Modular forms

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Database

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