LMFDB - Embedded newform 207.6.a.c.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [207,6,Mod(1,207)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(207, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0]))

N = Newforms(chi, 6, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("207.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 207 = 3^{2} \cdot 23 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	207.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$33.1994507013$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.5333.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 11x + 8 $ x^3 - x^2 - 11*x + 8
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			1.2
Root			$3.49331$ 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)

$f(q)$	$=$	$q+1.29009 q^{2} -30.3357 q^{4} -59.1123 q^{5} -213.331 q^{7} -80.4187 q^{8} +O(q^{10})$	\(q+1.29009 q^{2} -30.3357 q^{4} -59.1123 q^{5} -213.331 q^{7} -80.4187 q^{8} -76.2602 q^{10} -126.528 q^{11} -884.825 q^{13} -275.217 q^{14} +866.994 q^{16} +1179.98 q^{17} -1866.19 q^{19} +1793.21 q^{20} -163.233 q^{22} -529.000 q^{23} +369.263 q^{25} -1141.50 q^{26} +6471.55 q^{28} +6786.62 q^{29} -5146.34 q^{31} +3691.90 q^{32} +1522.28 q^{34} +12610.5 q^{35} +5137.07 q^{37} -2407.55 q^{38} +4753.73 q^{40} -12482.7 q^{41} +4198.66 q^{43} +3838.32 q^{44} -682.458 q^{46} -23006.9 q^{47} +28703.3 q^{49} +476.382 q^{50} +26841.8 q^{52} -25175.4 q^{53} +7479.38 q^{55} +17155.8 q^{56} +8755.36 q^{58} +37118.1 q^{59} +26410.4 q^{61} -6639.24 q^{62} -22980.9 q^{64} +52304.0 q^{65} -54398.5 q^{67} -35795.4 q^{68} +16268.7 q^{70} -35684.7 q^{71} +33937.4 q^{73} +6627.29 q^{74} +56612.0 q^{76} +26992.5 q^{77} -76625.8 q^{79} -51250.0 q^{80} -16103.8 q^{82} +96627.2 q^{83} -69751.1 q^{85} +5416.65 q^{86} +10175.2 q^{88} -30080.8 q^{89} +188761. q^{91} +16047.6 q^{92} -29680.9 q^{94} +110315. q^{95} -14637.2 q^{97} +37029.9 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 8 q^{2} + 22 q^{4} + 56 q^{5} - 114 q^{7} + 510 q^{8}+O(q^{10}) $ 3 * q + 8 * q^2 + 22 * q^4 + 56 * q^5 - 114 * q^7 + 510 * q^8	$ 3 q + 8 q^{2} + 22 q^{4} + 56 q^{5} - 114 q^{7} + 510 q^{8} + 282 q^{10} + 376 q^{11} - 858 q^{13} - 588 q^{14} + 2738 q^{16} + 2548 q^{17} - 2846 q^{19} + 4618 q^{20} - 5050 q^{22} - 1587 q^{23} + 753 q^{25} + 7788 q^{26} + 4736 q^{28} + 16370 q^{29} - 14756 q^{31} + 3878 q^{32} + 16520 q^{34} + 18520 q^{35} + 15874 q^{37} - 12438 q^{38} + 38270 q^{40} - 12606 q^{41} + 3154 q^{43} - 27114 q^{44} - 4232 q^{46} - 29928 q^{47} + 4471 q^{49} - 1452 q^{50} + 86856 q^{52} + 44084 q^{53} + 38360 q^{55} + 35704 q^{56} + 73316 q^{58} + 29300 q^{59} + 54010 q^{61} - 99908 q^{62} - 1582 q^{64} + 51216 q^{65} + 43390 q^{67} + 69840 q^{68} - 2476 q^{70} - 23424 q^{71} - 91402 q^{73} + 2294 q^{74} - 14274 q^{76} + 97208 q^{77} - 49398 q^{79} + 52626 q^{80} + 40152 q^{82} + 103936 q^{83} + 5888 q^{85} - 133634 q^{86} + 48898 q^{88} - 96112 q^{89} + 129228 q^{91} - 11638 q^{92} - 133688 q^{94} + 55928 q^{95} - 135318 q^{97} - 108440 q^{98}+O(q^{100}) $ 3 * q + 8 * q^2 + 22 * q^4 + 56 * q^5 - 114 * q^7 + 510 * q^8 + 282 * q^10 + 376 * q^11 - 858 * q^13 - 588 * q^14 + 2738 * q^16 + 2548 * q^17 - 2846 * q^19 + 4618 * q^20 - 5050 * q^22 - 1587 * q^23 + 753 * q^25 + 7788 * q^26 + 4736 * q^28 + 16370 * q^29 - 14756 * q^31 + 3878 * q^32 + 16520 * q^34 + 18520 * q^35 + 15874 * q^37 - 12438 * q^38 + 38270 * q^40 - 12606 * q^41 + 3154 * q^43 - 27114 * q^44 - 4232 * q^46 - 29928 * q^47 + 4471 * q^49 - 1452 * q^50 + 86856 * q^52 + 44084 * q^53 + 38360 * q^55 + 35704 * q^56 + 73316 * q^58 + 29300 * q^59 + 54010 * q^61 - 99908 * q^62 - 1582 * q^64 + 51216 * q^65 + 43390 * q^67 + 69840 * q^68 - 2476 * q^70 - 23424 * q^71 - 91402 * q^73 + 2294 * q^74 - 14274 * q^76 + 97208 * q^77 - 49398 * q^79 + 52626 * q^80 + 40152 * q^82 + 103936 * q^83 + 5888 * q^85 - 133634 * q^86 + 48898 * q^88 - 96112 * q^89 + 129228 * q^91 - 11638 * q^92 - 133688 * q^94 + 55928 * q^95 - 135318 * q^97 - 108440 * 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 $

$2$	1.29009	0.228058	0.114029	−	0.993477i	$-0.463624\pi$
$2$	1.29009	0.228058	0.114029	+	0.993477i	$0.463624\pi$
$3$	0	0
$3$	0	0
$4$	−30.3357	−0.947990
$5$	−59.1123	−1.05743	−0.528716	−	0.848799i	$-0.677326\pi$
$5$	−59.1123	−1.05743	−0.528716	+	0.848799i	$0.677326\pi$
$6$	0	0
$7$	−213.331	−1.64554	−0.822772	−	0.568371i	$-0.807574\pi$
$7$	−213.331	−1.64554	−0.822772	+	0.568371i	$0.807574\pi$
$8$	−80.4187	−0.444255
$9$	0	0
$10$	−76.2602	−0.241156
$11$	−126.528	−0.315287	−0.157643	−	0.987496i	$-0.550390\pi$
$11$	−126.528	−0.315287	−0.157643	+	0.987496i	$0.550390\pi$
$12$	0	0
$13$	−884.825	−1.45211	−0.726054	−	0.687638i	$-0.758648\pi$
$13$	−884.825	−1.45211	−0.726054	+	0.687638i	$0.758648\pi$
$14$	−275.217	−0.375280
$15$	0	0
$16$	866.994	0.846674
$17$	1179.98	0.990264	0.495132	−	0.868818i	$-0.335120\pi$
$17$	1179.98	0.990264	0.495132	+	0.868818i	$0.335120\pi$
$18$	0	0
$19$	−1866.19	−1.18596	−0.592981	−	0.805216i	$-0.702049\pi$
$19$	−1866.19	−1.18596	−0.592981	+	0.805216i	$0.702049\pi$
$20$	1793.21	1.00244
$21$	0	0
$22$	−163.233	−0.0719037
$23$	−529.000	−0.208514
$23$	−529.000	−0.208514
$24$	0	0
$25$	369.263	0.118164
$26$	−1141.50	−0.331165
$27$	0	0
$28$	6471.55	1.55996
$29$	6786.62	1.49851	0.749253	−	0.662284i	$-0.230413\pi$
$29$	6786.62	1.49851	0.749253	+	0.662284i	$0.230413\pi$
$30$	0	0
$31$	−5146.34	−0.961820	−0.480910	−	0.876770i	$-0.659694\pi$
$31$	−5146.34	−0.961820	−0.480910	+	0.876770i	$0.659694\pi$
$32$	3691.90	0.637345
$33$	0	0
$34$	1522.28	0.225838
$35$	12610.5	1.74005
$36$	0	0
$37$	5137.07	0.616895	0.308448	−	0.951241i	$-0.400191\pi$
$37$	5137.07	0.616895	0.308448	+	0.951241i	$0.400191\pi$
$38$	−2407.55	−0.270468
$39$	0	0
$40$	4753.73	0.469769
$41$	−12482.7	−1.15971	−0.579855	−	0.814720i	$-0.696891\pi$
$41$	−12482.7	−1.15971	−0.579855	+	0.814720i	$0.696891\pi$
$42$	0	0
$43$	4198.66	0.346290	0.173145	−	0.984896i	$-0.444607\pi$
$43$	4198.66	0.346290	0.173145	+	0.984896i	$0.444607\pi$
$44$	3838.32	0.298889
$45$	0	0
$46$	−682.458	−0.0475534
$47$	−23006.9	−1.51919	−0.759596	−	0.650395i	$-0.774603\pi$
$47$	−23006.9	−1.51919	−0.759596	+	0.650395i	$0.774603\pi$
$48$	0	0
$49$	28703.3	1.70782
$50$	476.382	0.0269483
$51$	0	0
$52$	26841.8	1.37658
$53$	−25175.4	−1.23108	−0.615541	−	0.788105i	$-0.711062\pi$
$53$	−25175.4	−1.23108	−0.615541	+	0.788105i	$0.711062\pi$
$54$	0	0
$55$	7479.38	0.333395
$56$	17155.8	0.731041
$57$	0	0
$58$	8755.36	0.341746
$59$	37118.1	1.38821	0.694107	−	0.719872i	$-0.255800\pi$
$59$	37118.1	1.38821	0.694107	+	0.719872i	$0.255800\pi$
$60$	0	0
$61$	26410.4	0.908761	0.454381	−	0.890808i	$-0.349861\pi$
$61$	26410.4	0.908761	0.454381	+	0.890808i	$0.349861\pi$
$62$	−6639.24	−0.219351
$63$	0	0
$64$	−22980.9	−0.701322
$65$	52304.0	1.53551
$66$	0	0
$67$	−54398.5	−1.48047	−0.740236	−	0.672347i	$-0.765286\pi$
$67$	−54398.5	−1.48047	−0.740236	+	0.672347i	$0.765286\pi$
$68$	−35795.4	−0.938760
$69$	0	0
$70$	16268.7	0.396833
$71$	−35684.7	−0.840110	−0.420055	−	0.907499i	$-0.637989\pi$
$71$	−35684.7	−0.840110	−0.420055	+	0.907499i	$0.637989\pi$
$72$	0	0
$73$	33937.4	0.745369	0.372684	−	0.927958i	$-0.378437\pi$
$73$	33937.4	0.745369	0.372684	+	0.927958i	$0.378437\pi$
$74$	6627.29	0.140688
$75$	0	0
$76$	56612.0	1.12428
$77$	26992.5	0.518819
$78$	0	0
$79$	−76625.8	−1.38136	−0.690681	−	0.723160i	$-0.742689\pi$
$79$	−76625.8	−1.38136	−0.690681	+	0.723160i	$0.742689\pi$
$80$	−51250.0	−0.895300
$81$	0	0
$82$	−16103.8	−0.264481
$83$	96627.2	1.53959	0.769794	−	0.638293i	$-0.220359\pi$
$83$	96627.2	1.53959	0.769794	+	0.638293i	$0.220359\pi$
$84$	0	0
$85$	−69751.1	−1.04714
$86$	5416.65	0.0789741
$87$	0	0
$88$	10175.2	0.140068
$89$	−30080.8	−0.402545	−0.201273	−	0.979535i	$-0.564508\pi$
$89$	−30080.8	−0.402545	−0.201273	+	0.979535i	$0.564508\pi$
$90$	0	0
$91$	188761.	2.38951
$92$	16047.6	0.197669
$93$	0	0
$94$	−29680.9	−0.346464
$95$	110315.	1.25408
$96$	0	0
$97$	−14637.2	−0.157953	−0.0789766	−	0.996876i	$-0.525165\pi$
$97$	−14637.2	−0.157953	−0.0789766	+	0.996876i	$0.525165\pi$
$98$	37029.9	0.389482
$99$	0	0
$100$	−11201.8	−0.112018
$101$	37693.7	0.367676	0.183838	−	0.982957i	$-0.441148\pi$
$101$	37693.7	0.367676	0.183838	+	0.982957i	$0.441148\pi$
$102$	0	0
$103$	125252.	1.16330	0.581649	−	0.813440i	$-0.302408\pi$
$103$	125252.	1.16330	0.581649	+	0.813440i	$0.302408\pi$
$104$	71156.5	0.645106
$105$	0	0
$106$	−32478.6	−0.280758
$107$	−28002.7	−0.236451	−0.118225	−	0.992987i	$-0.537721\pi$
$107$	−28002.7	−0.236451	−0.118225	+	0.992987i	$0.537721\pi$
$108$	0	0
$109$	−64151.4	−0.517178	−0.258589	−	0.965987i	$-0.583257\pi$
$109$	−64151.4	−0.517178	−0.258589	+	0.965987i	$0.583257\pi$
$110$	9649.08	0.0760333
$111$	0	0
$112$	−184957.	−1.39324
$113$	25226.7	0.185851	0.0929253	−	0.995673i	$-0.470378\pi$
$113$	25226.7	0.185851	0.0929253	+	0.995673i	$0.470378\pi$
$114$	0	0
$115$	31270.4	0.220490
$116$	−205877.	−1.42057
$117$	0	0
$118$	47885.8	0.316593
$119$	−251726.	−1.62952
$120$	0	0
$121$	−145042.	−0.900594
$122$	34071.8	0.207250
$123$	0	0
$124$	156118.	0.911796
$125$	162898.	0.932482
$126$	0	0
$127$	35783.9	0.196870	0.0984348	−	0.995144i	$-0.468616\pi$
$127$	35783.9	0.196870	0.0984348	+	0.995144i	$0.468616\pi$
$128$	−147788.	−0.797288
$129$	0	0
$130$	67477.0	0.350185
$131$	−78119.6	−0.397724	−0.198862	−	0.980028i	$-0.563725\pi$
$131$	−78119.6	−0.397724	−0.198862	+	0.980028i	$0.563725\pi$
$132$	0	0
$133$	398116.	1.95155
$134$	−70179.1	−0.337634
$135$	0	0
$136$	−94892.2	−0.439930
$137$	92816.7	0.422498	0.211249	−	0.977432i	$-0.432247\pi$
$137$	92816.7	0.422498	0.211249	+	0.977432i	$0.432247\pi$
$138$	0	0
$139$	−417118.	−1.83114	−0.915570	−	0.402159i	$-0.868260\pi$
$139$	−417118.	−1.83114	−0.915570	+	0.402159i	$0.868260\pi$
$140$	−382548.	−1.64955
$141$	0	0
$142$	−46036.5	−0.191594
$143$	111955.	0.457831
$144$	0	0
$145$	−401173.	−1.58457
$146$	43782.3	0.169987
$147$	0	0
$148$	−155836.	−0.584810
$149$	−445936.	−1.64553	−0.822766	−	0.568380i	$-0.807570\pi$
$149$	−445936.	−1.64553	−0.822766	+	0.568380i	$0.807570\pi$
$150$	0	0
$151$	−260458.	−0.929597	−0.464799	−	0.885416i	$-0.653873\pi$
$151$	−260458.	−0.929597	−0.464799	+	0.885416i	$0.653873\pi$
$152$	150076.	0.526869
$153$	0	0
$154$	34822.7	0.118321
$155$	304212.	1.01706
$156$	0	0
$157$	46067.4	0.149157	0.0745787	−	0.997215i	$-0.476239\pi$
$157$	46067.4	0.149157	0.0745787	+	0.997215i	$0.476239\pi$
$158$	−98854.3	−0.315031
$159$	0	0
$160$	−218237.	−0.673950
$161$	112852.	0.343120
$162$	0	0
$163$	188377.	0.555341	0.277671	−	0.960676i	$-0.410438\pi$
$163$	188377.	0.555341	0.277671	+	0.960676i	$0.410438\pi$
$164$	378671.	1.09939
$165$	0	0
$166$	124658.	0.351115
$167$	−203214.	−0.563849	−0.281924	−	0.959437i	$-0.590973\pi$
$167$	−203214.	−0.563849	−0.281924	+	0.959437i	$0.590973\pi$
$168$	0	0
$169$	411622.	1.10862
$170$	−89985.3	−0.238808
$171$	0	0
$172$	−127369.	−0.328279
$173$	−158051.	−0.401497	−0.200749	−	0.979643i	$-0.564337\pi$
$173$	−158051.	−0.401497	−0.200749	+	0.979643i	$0.564337\pi$
$174$	0	0
$175$	−78775.3	−0.194444
$176$	−109699.	−0.266945
$177$	0	0
$178$	−38807.0	−0.0918037
$179$	−166444.	−0.388271	−0.194136	−	0.980975i	$-0.562190\pi$
$179$	−166444.	−0.388271	−0.194136	+	0.980975i	$0.562190\pi$
$180$	0	0
$181$	185620.	0.421141	0.210571	−	0.977579i	$-0.432468\pi$
$181$	185620.	0.421141	0.210571	+	0.977579i	$0.432468\pi$
$182$	243519.	0.544947
$183$	0	0
$184$	42541.5	0.0926335
$185$	−303664.	−0.652325
$186$	0	0
$187$	−149300.	−0.312217
$188$	697928.	1.44018
$189$	0	0
$190$	142316.	0.286002
$191$	769733.	1.52671	0.763356	−	0.645978i	$-0.223550\pi$
$191$	769733.	1.52671	0.763356	+	0.645978i	$0.223550\pi$
$192$	0	0
$193$	−384145.	−0.742338	−0.371169	−	0.928565i	$-0.621043\pi$
$193$	−384145.	−0.742338	−0.371169	+	0.928565i	$0.621043\pi$
$194$	−18883.3	−0.0360225
$195$	0	0
$196$	−870734.	−1.61899
$197$	−266048.	−0.488421	−0.244211	−	0.969722i	$-0.578529\pi$
$197$	−266048.	−0.488421	−0.244211	+	0.969722i	$0.578529\pi$
$198$	0	0
$199$	−706070.	−1.26391	−0.631953	−	0.775007i	$-0.717746\pi$
$199$	−706070.	−1.26391	−0.631953	+	0.775007i	$0.717746\pi$
$200$	−29695.6	−0.0524949
$201$	0	0
$202$	48628.3	0.0838515
$203$	−1.44780e6	−2.46586
$204$	0	0
$205$	737882.	1.22631
$206$	161586.	0.265300
$207$	0	0
$208$	−767138.	−1.22946
$209$	236125.	0.373918
$210$	0	0
$211$	−290636.	−0.449410	−0.224705	−	0.974427i	$-0.572142\pi$
$211$	−290636.	−0.449410	−0.224705	+	0.974427i	$0.572142\pi$
$212$	763712.	1.16705
$213$	0	0
$214$	−36126.1	−0.0539245
$215$	−248192.	−0.366178
$216$	0	0
$217$	1.09788e6	1.58272
$218$	−82761.1	−0.117947
$219$	0	0
$220$	−226892.	−0.316055
$221$	−1.04407e6	−1.43797
$222$	0	0
$223$	−574915.	−0.774179	−0.387089	−	0.922042i	$-0.626519\pi$
$223$	−574915.	−0.774179	−0.387089	+	0.922042i	$0.626519\pi$
$224$	−787598.	−1.04878
$225$	0	0
$226$	32544.7	0.0423847
$227$	1.37363e6	1.76931	0.884655	−	0.466246i	$-0.154394\pi$
$227$	1.37363e6	1.76931	0.884655	+	0.466246i	$0.154394\pi$
$228$	0	0
$229$	870725.	1.09722	0.548608	−	0.836080i	$-0.315158\pi$
$229$	870725.	1.09722	0.548608	+	0.836080i	$0.315158\pi$
$230$	40341.7	0.0502845
$231$	0	0
$232$	−545771.	−0.665718
$233$	1.61732e6	1.95166	0.975832	−	0.218522i	$-0.0701235\pi$
$233$	1.61732e6	1.95166	0.975832	+	0.218522i	$0.0701235\pi$
$234$	0	0
$235$	1.35999e6	1.60644
$236$	−1.12600e6	−1.31601
$237$	0	0
$238$	−324750.	−0.371626
$239$	1.65281e6	1.87167	0.935833	−	0.352444i	$-0.114649\pi$
$239$	1.65281e6	1.87167	0.935833	+	0.352444i	$0.114649\pi$
$240$	0	0
$241$	928466.	1.02973	0.514865	−	0.857271i	$-0.327842\pi$
$241$	928466.	1.02973	0.514865	+	0.857271i	$0.327842\pi$
$242$	−187117.	−0.205388
$243$	0	0
$244$	−801176.	−0.861496
$245$	−1.69672e6	−1.80590
$246$	0	0
$247$	1.65125e6	1.72215
$248$	413862.	0.427293
$249$	0	0
$250$	210153.	0.212660
$251$	909693.	0.911403	0.455702	−	0.890133i	$-0.349388\pi$
$251$	909693.	0.911403	0.455702	+	0.890133i	$0.349388\pi$
$252$	0	0
$253$	66933.5	0.0657419
$254$	46164.5	0.0448977
$255$	0	0
$256$	544729.	0.519494
$257$	−52666.7	−0.0497397	−0.0248699	−	0.999691i	$-0.507917\pi$
$257$	−52666.7	−0.0497397	−0.0248699	+	0.999691i	$0.507917\pi$
$258$	0	0
$259$	−1.09590e6	−1.01513
$260$	−1.58668e6	−1.45564
$261$	0	0
$262$	−100781.	−0.0907042
$263$	−464764.	−0.414327	−0.207164	−	0.978306i	$-0.566423\pi$
$263$	−464764.	−0.414327	−0.207164	+	0.978306i	$0.566423\pi$
$264$	0	0
$265$	1.48818e6	1.30179
$266$	513606.	0.445068
$267$	0	0
$268$	1.65022e6	1.40347
$269$	−968420.	−0.815987	−0.407993	−	0.912985i	$-0.633771\pi$
$269$	−968420.	−0.815987	−0.407993	+	0.912985i	$0.633771\pi$
$270$	0	0
$271$	2.29262e6	1.89630	0.948152	−	0.317817i	$-0.102950\pi$
$271$	2.29262e6	1.89630	0.948152	+	0.317817i	$0.102950\pi$
$272$	1.02303e6	0.838431
$273$	0	0
$274$	119742.	0.0963541
$275$	−46722.2	−0.0372556
$276$	0	0
$277$	290954.	0.227837	0.113919	−	0.993490i	$-0.463660\pi$
$277$	290954.	0.227837	0.113919	+	0.993490i	$0.463660\pi$
$278$	−538120.	−0.417606
$279$	0	0
$280$	−1.01412e6	−0.773027
$281$	391978.	0.296139	0.148070	−	0.988977i	$-0.452694\pi$
$281$	391978.	0.296139	0.148070	+	0.988977i	$0.452694\pi$
$282$	0	0
$283$	732768.	0.543877	0.271938	−	0.962315i	$-0.412335\pi$
$283$	732768.	0.543877	0.271938	+	0.962315i	$0.412335\pi$
$284$	1.08252e6	0.796415
$285$	0	0
$286$	144433.	0.104412
$287$	2.66295e6	1.90835
$288$	0	0
$289$	−27511.7	−0.0193764
$290$	−517549.	−0.361374
$291$	0	0
$292$	−1.02951e6	−0.706602
$293$	−1.62587e6	−1.10641	−0.553207	−	0.833044i	$-0.686596\pi$
$293$	−1.62587e6	−1.10641	−0.553207	+	0.833044i	$0.686596\pi$
$294$	0	0
$295$	−2.19414e6	−1.46794
$296$	−413117.	−0.274059
$297$	0	0
$298$	−575298.	−0.375277
$299$	468072.	0.302785
$300$	0	0
$301$	−895706.	−0.569835
$302$	−336014.	−0.212002
$303$	0	0
$304$	−1.61797e6	−1.00412
$305$	−1.56118e6	−0.960954
$306$	0	0
$307$	1.24581e6	0.754406	0.377203	−	0.926131i	$-0.376886\pi$
$307$	1.24581e6	0.754406	0.377203	+	0.926131i	$0.376886\pi$
$308$	−818834.	−0.491835
$309$	0	0
$310$	392461.	0.231949
$311$	3.26852e6	1.91624	0.958119	−	0.286369i	$-0.0924483\pi$
$311$	3.26852e6	1.91624	0.958119	+	0.286369i	$0.0924483\pi$
$312$	0	0
$313$	1.70161e6	0.981744	0.490872	−	0.871232i	$-0.336678\pi$
$313$	1.70161e6	0.981744	0.490872	+	0.871232i	$0.336678\pi$
$314$	59431.2	0.0340166
$315$	0	0
$316$	2.32450e6	1.30952
$317$	908317.	0.507679	0.253840	−	0.967246i	$-0.418306\pi$
$317$	908317.	0.507679	0.253840	+	0.967246i	$0.418306\pi$
$318$	0	0
$319$	−858699.	−0.472459
$320$	1.35845e6	0.741601
$321$	0	0
$322$	145590.	0.0782512
$323$	−2.20206e6	−1.17442
$324$	0	0
$325$	−326733.	−0.171587
$326$	243024.	0.126650
$327$	0	0
$328$	1.00384e6	0.515206
$329$	4.90809e6	2.49990
$330$	0	0
$331$	−2.14345e6	−1.07533	−0.537666	−	0.843158i	$-0.680694\pi$
$331$	−2.14345e6	−1.07533	−0.537666	+	0.843158i	$0.680694\pi$
$332$	−2.93125e6	−1.45951
$333$	0	0
$334$	−262165.	−0.128590
$335$	3.21562e6	1.56550
$336$	0	0
$337$	817498.	0.392114	0.196057	−	0.980593i	$-0.437186\pi$
$337$	817498.	0.392114	0.196057	+	0.980593i	$0.437186\pi$
$338$	531030.	0.252829
$339$	0	0
$340$	2.11595e6	0.992676
$341$	651157.	0.303249
$342$	0	0
$343$	−2.53785e6	−1.16475
$344$	−337651.	−0.153841
$345$	0	0
$346$	−203900.	−0.0915647
$347$	−3.20207e6	−1.42760	−0.713801	−	0.700349i	$-0.753028\pi$
$347$	−3.20207e6	−1.42760	−0.713801	+	0.700349i	$0.753028\pi$
$348$	0	0
$349$	−3.23916e6	−1.42354	−0.711768	−	0.702415i	$-0.752105\pi$
$349$	−3.23916e6	−1.42354	−0.711768	+	0.702415i	$0.752105\pi$
$350$	−101627.	−0.0443446
$351$	0	0
$352$	−467130.	−0.200947
$353$	1.72676e6	0.737555	0.368777	−	0.929518i	$-0.379776\pi$
$353$	1.72676e6	0.737555	0.368777	+	0.929518i	$0.379776\pi$
$354$	0	0
$355$	2.10940e6	0.888360
$356$	912522.	0.381609
$357$	0	0
$358$	−214728.	−0.0885484
$359$	−3.12338e6	−1.27905	−0.639527	−	0.768769i	$-0.720870\pi$
$359$	−3.12338e6	−1.27905	−0.639527	+	0.768769i	$0.720870\pi$
$360$	0	0
$361$	1.00655e6	0.406507
$362$	239466.	0.0960446
$363$	0	0
$364$	−5.72619e6	−2.26523
$365$	−2.00612e6	−0.788177
$366$	0	0
$367$	1.17031e6	0.453560	0.226780	−	0.973946i	$-0.427180\pi$
$367$	1.17031e6	0.453560	0.226780	+	0.973946i	$0.427180\pi$
$368$	−458640.	−0.176544
$369$	0	0
$370$	−391754.	−0.148768
$371$	5.37070e6	2.02580
$372$	0	0
$373$	−692891.	−0.257865	−0.128933	−	0.991653i	$-0.541155\pi$
$373$	−692891.	−0.257865	−0.128933	+	0.991653i	$0.541155\pi$
$374$	−192611.	−0.0712037
$375$	0	0
$376$	1.85018e6	0.674908
$377$	−6.00497e6	−2.17599
$378$	0	0
$379$	2.12163e6	0.758702	0.379351	−	0.925253i	$-0.376147\pi$
$379$	2.12163e6	0.758702	0.379351	+	0.925253i	$0.376147\pi$
$380$	−3.34646e6	−1.18885
$381$	0	0
$382$	993026.	0.348179
$383$	287079.	0.100001	0.0500005	−	0.998749i	$-0.484078\pi$
$383$	287079.	0.100001	0.0500005	+	0.998749i	$0.484078\pi$
$384$	0	0
$385$	−1.59559e6	−0.548616
$386$	−495582.	−0.169296
$387$	0	0
$388$	444029.	0.149738
$389$	−679811.	−0.227779	−0.113890	−	0.993493i	$-0.536331\pi$
$389$	−679811.	−0.227779	−0.113890	+	0.993493i	$0.536331\pi$
$390$	0	0
$391$	−624208.	−0.206484
$392$	−2.30828e6	−0.758706
$393$	0	0
$394$	−343226.	−0.111388
$395$	4.52953e6	1.46070
$396$	0	0
$397$	−5.39917e6	−1.71930	−0.859648	−	0.510886i	$-0.829317\pi$
$397$	−5.39917e6	−1.71930	−0.859648	+	0.510886i	$0.829317\pi$
$398$	−910894.	−0.288244
$399$	0	0
$400$	320148.	0.100046
$401$	2.17808e6	0.676416	0.338208	−	0.941071i	$-0.390179\pi$
$401$	2.17808e6	0.676416	0.338208	+	0.941071i	$0.390179\pi$
$402$	0	0
$403$	4.55361e6	1.39667
$404$	−1.14346e6	−0.348553
$405$	0	0
$406$	−1.86779e6	−0.562359
$407$	−649985.	−0.194499
$408$	0	0
$409$	−356567.	−0.105398	−0.0526991	−	0.998610i	$-0.516782\pi$
$409$	−356567.	−0.105398	−0.0526991	+	0.998610i	$0.516782\pi$
$410$	951934.	0.279671
$411$	0	0
$412$	−3.79960e6	−1.10279
$413$	−7.91847e6	−2.28437
$414$	0	0
$415$	−5.71186e6	−1.62801
$416$	−3.26668e6	−0.925495
$417$	0	0
$418$	304623.	0.0852751
$419$	−3.82634e6	−1.06475	−0.532376	−	0.846508i	$-0.678701\pi$
$419$	−3.82634e6	−1.06475	−0.532376	+	0.846508i	$0.678701\pi$
$420$	0	0
$421$	−3.00814e6	−0.827167	−0.413584	−	0.910466i	$-0.635723\pi$
$421$	−3.00814e6	−0.827167	−0.413584	+	0.910466i	$0.635723\pi$
$422$	−374947.	−0.102492
$423$	0	0
$424$	2.02457e6	0.546914
$425$	435721.	0.117014
$426$	0	0
$427$	−5.63416e6	−1.49541
$428$	849481.	0.224153
$429$	0	0
$430$	−320191.	−0.0835098
$431$	−577809.	−0.149827	−0.0749136	−	0.997190i	$-0.523868\pi$
$431$	−577809.	−0.149827	−0.0749136	+	0.997190i	$0.523868\pi$
$432$	0	0
$433$	7.21909e6	1.85039	0.925194	−	0.379494i	$-0.123902\pi$
$433$	7.21909e6	1.85039	0.925194	+	0.379494i	$0.123902\pi$
$434$	1.41636e6	0.360952
$435$	0	0
$436$	1.94607e6	0.490279
$437$	987212.	0.247290
$438$	0	0
$439$	−760935.	−0.188446	−0.0942229	−	0.995551i	$-0.530037\pi$
$439$	−760935.	−0.188446	−0.0942229	+	0.995551i	$0.530037\pi$
$440$	−601482.	−0.148112
$441$	0	0
$442$	−1.34695e6	−0.327941
$443$	6.96892e6	1.68716	0.843580	−	0.537003i	$-0.180444\pi$
$443$	6.96892e6	1.68716	0.843580	+	0.537003i	$0.180444\pi$
$444$	0	0
$445$	1.77815e6	0.425665
$446$	−741692.	−0.176558
$447$	0	0
$448$	4.90255e6	1.15406
$449$	−5.48897e6	−1.28492	−0.642458	−	0.766321i	$-0.722085\pi$
$449$	−5.48897e6	−1.28492	−0.642458	+	0.766321i	$0.722085\pi$
$450$	0	0
$451$	1.57942e6	0.365641
$452$	−765268.	−0.176184
$453$	0	0
$454$	1.77210e6	0.403506
$455$	−1.11581e7	−2.52675
$456$	0	0
$457$	−2.49968e6	−0.559879	−0.279940	−	0.960018i	$-0.590314\pi$
$457$	−2.49968e6	−0.559879	−0.279940	+	0.960018i	$0.590314\pi$
$458$	1.12331e6	0.250229
$459$	0	0
$460$	−948608.	−0.209022
$461$	8.10339e6	1.77588	0.887942	−	0.459956i	$-0.152135\pi$
$461$	8.10339e6	1.77588	0.887942	+	0.459956i	$0.152135\pi$
$462$	0	0
$463$	−4.71421e6	−1.02201	−0.511007	−	0.859577i	$-0.670727\pi$
$463$	−4.71421e6	−1.02201	−0.511007	+	0.859577i	$0.670727\pi$
$464$	5.88396e6	1.26875
$465$	0	0
$466$	2.08649e6	0.445093
$467$	1.03346e6	0.219281	0.109641	−	0.993971i	$-0.465030\pi$
$467$	1.03346e6	0.219281	0.109641	+	0.993971i	$0.465030\pi$
$468$	0	0
$469$	1.16049e7	2.43618
$470$	1.75451e6	0.366362
$471$	0	0
$472$	−2.98499e6	−0.616720
$473$	−531249.	−0.109181
$474$	0	0
$475$	−689113.	−0.140138
$476$	7.63628e6	1.54477
$477$	0	0
$478$	2.13228e6	0.426849
$479$	−1.13020e6	−0.225069	−0.112534	−	0.993648i	$-0.535897\pi$
$479$	−1.13020e6	−0.225069	−0.112534	+	0.993648i	$0.535897\pi$
$480$	0	0
$481$	−4.54541e6	−0.895798
$482$	1.19781e6	0.234838
$483$	0	0
$484$	4.39993e6	0.853754
$485$	865238.	0.167025
$486$	0	0
$487$	−793366.	−0.151583	−0.0757916	−	0.997124i	$-0.524148\pi$
$487$	−793366.	−0.151583	−0.0757916	+	0.997124i	$0.524148\pi$
$488$	−2.12389e6	−0.403721
$489$	0	0
$490$	−2.18892e6	−0.411851
$491$	3.66618e6	0.686294	0.343147	−	0.939282i	$-0.388507\pi$
$491$	3.66618e6	0.686294	0.343147	+	0.939282i	$0.388507\pi$
$492$	0	0
$493$	8.00805e6	1.48392
$494$	2.13026e6	0.392749
$495$	0	0
$496$	−4.46184e6	−0.814348
$497$	7.61267e6	1.38244
$498$	0	0
$499$	−5.08658e6	−0.914480	−0.457240	−	0.889343i	$-0.651162\pi$
$499$	−5.08658e6	−0.914480	−0.457240	+	0.889343i	$0.651162\pi$
$500$	−4.94162e6	−0.883983
$501$	0	0
$502$	1.17359e6	0.207853
$503$	−6.49233e6	−1.14414	−0.572072	−	0.820203i	$-0.693860\pi$
$503$	−6.49233e6	−1.14414	−0.572072	+	0.820203i	$0.693860\pi$
$504$	0	0
$505$	−2.22816e6	−0.388793
$506$	86350.3	0.0149930
$507$	0	0
$508$	−1.08553e6	−0.186630
$509$	5.58035e6	0.954700	0.477350	−	0.878713i	$-0.341597\pi$
$509$	5.58035e6	0.954700	0.477350	+	0.878713i	$0.341597\pi$
$510$	0	0
$511$	−7.23991e6	−1.22654
$512$	5.43197e6	0.915762
$513$	0	0
$514$	−67944.9	−0.0113435
$515$	−7.40393e6	−1.23011
$516$	0	0
$517$	2.91102e6	0.478981
$518$	−1.41381e6	−0.231508
$519$	0	0
$520$	−4.20622e6	−0.682156
$521$	4.02579e6	0.649766	0.324883	−	0.945754i	$-0.394675\pi$
$521$	4.02579e6	0.649766	0.324883	+	0.945754i	$0.394675\pi$
$522$	0	0
$523$	−9.41059e6	−1.50440	−0.752198	−	0.658937i	$-0.771007\pi$
$523$	−9.41059e6	−1.50440	−0.752198	+	0.658937i	$0.771007\pi$
$524$	2.36981e6	0.377038
$525$	0	0
$526$	−599588.	−0.0944906
$527$	−6.07256e6	−0.952457
$528$	0	0
$529$	279841.	0.0434783
$530$	1.91988e6	0.296883
$531$	0	0
$532$	−1.20771e7	−1.85005
$533$	1.10450e7	1.68402
$534$	0	0
$535$	1.65531e6	0.250031
$536$	4.37466e6	0.657707
$537$	0	0
$538$	−1.24935e6	−0.186092
$539$	−3.63178e6	−0.538453
$540$	0	0
$541$	−2.68147e6	−0.393894	−0.196947	−	0.980414i	$-0.563103\pi$
$541$	−2.68147e6	−0.393894	−0.196947	+	0.980414i	$0.563103\pi$
$542$	2.95768e6	0.432467
$543$	0	0
$544$	4.35636e6	0.631141
$545$	3.79214e6	0.546881
$546$	0	0
$547$	8.74676e6	1.24991	0.624955	−	0.780660i	$-0.285117\pi$
$547$	8.74676e6	1.24991	0.624955	+	0.780660i	$0.285117\pi$
$548$	−2.81566e6	−0.400524
$549$	0	0
$550$	−60275.8	−0.00849643
$551$	−1.26651e7	−1.77717
$552$	0	0
$553$	1.63467e7	2.27309
$554$	375357.	0.0519601
$555$	0	0
$556$	1.26535e7	1.73590
$557$	−2.07905e6	−0.283940	−0.141970	−	0.989871i	$-0.545344\pi$
$557$	−2.07905e6	−0.283940	−0.141970	+	0.989871i	$0.545344\pi$
$558$	0	0
$559$	−3.71508e6	−0.502850
$560$	1.09332e7	1.47326
$561$	0	0
$562$	505687.	0.0675369
$563$	−8.52157e6	−1.13305	−0.566524	−	0.824045i	$-0.691712\pi$
$563$	−8.52157e6	−1.13305	−0.566524	+	0.824045i	$0.691712\pi$
$564$	0	0
$565$	−1.49121e6	−0.196524
$566$	945338.	0.124035
$567$	0	0
$568$	2.86972e6	0.373223
$569$	−6.58526e6	−0.852692	−0.426346	−	0.904560i	$-0.640199\pi$
$569$	−6.58526e6	−0.852692	−0.426346	+	0.904560i	$0.640199\pi$
$570$	0	0
$571$	−6.31614e6	−0.810702	−0.405351	−	0.914161i	$-0.632851\pi$
$571$	−6.31614e6	−0.810702	−0.405351	+	0.914161i	$0.632851\pi$
$572$	−3.39624e6	−0.434019
$573$	0	0
$574$	3.43545e6	0.435216
$575$	−195340.	−0.0246389
$576$	0	0
$577$	4.48677e6	0.561040	0.280520	−	0.959848i	$-0.409493\pi$
$577$	4.48677e6	0.561040	0.280520	+	0.959848i	$0.409493\pi$
$578$	−35492.6	−0.00441894
$579$	0	0
$580$	1.21698e7	1.50216
$581$	−2.06136e7	−2.53346
$582$	0	0
$583$	3.18540e6	0.388144
$584$	−2.72920e6	−0.331134
$585$	0	0
$586$	−2.09752e6	−0.252326
$587$	7.42328e6	0.889203	0.444601	−	0.895729i	$-0.353345\pi$
$587$	7.42328e6	0.889203	0.444601	+	0.895729i	$0.353345\pi$
$588$	0	0
$589$	9.60402e6	1.14068
$590$	−2.83064e6	−0.334776
$591$	0	0
$592$	4.45381e6	0.522309
$593$	−7.02182e6	−0.819998	−0.409999	−	0.912086i	$-0.634471\pi$
$593$	−7.02182e6	−0.819998	−0.409999	+	0.912086i	$0.634471\pi$
$594$	0	0
$595$	1.48801e7	1.72311
$596$	1.35278e7	1.55995
$597$	0	0
$598$	603856.	0.0690527
$599$	−5.02165e6	−0.571846	−0.285923	−	0.958253i	$-0.592300\pi$
$599$	−5.02165e6	−0.571846	−0.285923	+	0.958253i	$0.592300\pi$
$600$	0	0
$601$	4.66921e6	0.527300	0.263650	−	0.964618i	$-0.415074\pi$
$601$	4.66921e6	0.527300	0.263650	+	0.964618i	$0.415074\pi$
$602$	−1.15554e6	−0.129955
$603$	0	0
$604$	7.90116e6	0.881248
$605$	8.57374e6	0.952318
$606$	0	0
$607$	−1.48942e7	−1.64076	−0.820380	−	0.571819i	$-0.806238\pi$
$607$	−1.48942e7	−1.64076	−0.820380	+	0.571819i	$0.806238\pi$
$608$	−6.88977e6	−0.755868
$609$	0	0
$610$	−2.01406e6	−0.219153
$611$	2.03570e7	2.20603
$612$	0	0
$613$	1.09408e7	1.17598	0.587989	−	0.808869i	$-0.299920\pi$
$613$	1.09408e7	1.17598	0.587989	+	0.808869i	$0.299920\pi$
$614$	1.60721e6	0.172048
$615$	0	0
$616$	−2.17070e6	−0.230488
$617$	6.42199e6	0.679136	0.339568	−	0.940582i	$-0.389719\pi$
$617$	6.42199e6	0.679136	0.339568	+	0.940582i	$0.389719\pi$
$618$	0	0
$619$	−3.05345e6	−0.320305	−0.160152	−	0.987092i	$-0.551199\pi$
$619$	−3.05345e6	−0.320305	−0.160152	+	0.987092i	$0.551199\pi$
$620$	−9.22847e6	−0.964163
$621$	0	0
$622$	4.21668e6	0.437014
$623$	6.41719e6	0.662406
$624$	0	0
$625$	−1.07832e7	−1.10420
$626$	2.19523e6	0.223895
$627$	0	0
$628$	−1.39749e6	−0.141400
$629$	6.06162e6	0.610889
$630$	0	0
$631$	1.05012e7	1.04994	0.524971	−	0.851120i	$-0.324076\pi$
$631$	1.05012e7	1.04994	0.524971	+	0.851120i	$0.324076\pi$
$632$	6.16215e6	0.613677
$633$	0	0
$634$	1.17181e6	0.115780
$635$	−2.11527e6	−0.208176
$636$	0	0
$637$	−2.53974e7	−2.47994
$638$	−1.10780e6	−0.107748
$639$	0	0
$640$	8.73610e6	0.843078
$641$	1.57928e7	1.51815	0.759073	−	0.651006i	$-0.225653\pi$
$641$	1.57928e7	1.51815	0.759073	+	0.651006i	$0.225653\pi$
$642$	0	0
$643$	551159.	0.0525714	0.0262857	−	0.999654i	$-0.491632\pi$
$643$	551159.	0.0525714	0.0262857	+	0.999654i	$0.491632\pi$
$644$	−3.42345e6	−0.325274
$645$	0	0
$646$	−2.84085e6	−0.267835
$647$	−1.12498e7	−1.05653	−0.528267	−	0.849078i	$-0.677158\pi$
$647$	−1.12498e7	−1.05653	−0.528267	+	0.849078i	$0.677158\pi$
$648$	0	0
$649$	−4.69649e6	−0.437685
$650$	−421515.	−0.0391318
$651$	0	0
$652$	−5.71456e6	−0.526458
$653$	2.56046e6	0.234982	0.117491	−	0.993074i	$-0.462515\pi$
$653$	2.56046e6	0.234982	0.117491	+	0.993074i	$0.462515\pi$
$654$	0	0
$655$	4.61783e6	0.420566
$656$	−1.08224e7	−0.981895
$657$	0	0
$658$	6.33188e6	0.570122
$659$	734143.	0.0658518	0.0329259	−	0.999458i	$-0.489517\pi$
$659$	734143.	0.0658518	0.0329259	+	0.999458i	$0.489517\pi$
$660$	0	0
$661$	1.92354e7	1.71237	0.856185	−	0.516670i	$-0.172828\pi$
$661$	1.92354e7	1.71237	0.856185	+	0.516670i	$0.172828\pi$
$662$	−2.76524e6	−0.245238
$663$	0	0
$664$	−7.77064e6	−0.683969
$665$	−2.35336e7	−2.06364
$666$	0	0
$667$	−3.59012e6	−0.312460
$668$	6.16464e6	0.534523
$669$	0	0
$670$	4.14845e6	0.357025
$671$	−3.34166e6	−0.286520
$672$	0	0
$673$	1.20397e7	1.02465	0.512326	−	0.858791i	$-0.328784\pi$
$673$	1.20397e7	1.02465	0.512326	+	0.858791i	$0.328784\pi$
$674$	1.05465e6	0.0894247
$675$	0	0
$676$	−1.24868e7	−1.05096
$677$	−99259.2	−0.00832337	−0.00416168	−	0.999991i	$-0.501325\pi$
$677$	−99259.2	−0.00832337	−0.00416168	+	0.999991i	$0.501325\pi$
$678$	0	0
$679$	3.12257e6	0.259919
$680$	5.60929e6	0.465196
$681$	0	0
$682$	840052.	0.0691585
$683$	−9.22059e6	−0.756323	−0.378161	−	0.925740i	$-0.623444\pi$
$683$	−9.22059e6	−0.756323	−0.378161	+	0.925740i	$0.623444\pi$
$684$	0	0
$685$	−5.48661e6	−0.446763
$686$	−3.27406e6	−0.265630
$687$	0	0
$688$	3.64021e6	0.293194
$689$	2.22758e7	1.78766
$690$	0	0
$691$	−1.77887e7	−1.41726	−0.708629	−	0.705581i	$-0.750686\pi$
$691$	−1.77887e7	−1.41726	−0.708629	+	0.705581i	$0.750686\pi$
$692$	4.79459e6	0.380615
$693$	0	0
$694$	−4.13096e6	−0.325576
$695$	2.46568e7	1.93631
$696$	0	0
$697$	−1.47293e7	−1.14842
$698$	−4.17881e6	−0.324649
$699$	0	0
$700$	2.38970e6	0.184331
$701$	−1.60390e7	−1.23277	−0.616384	−	0.787446i	$-0.711403\pi$
$701$	−1.60390e7	−1.23277	−0.616384	+	0.787446i	$0.711403\pi$
$702$	0	0
$703$	−9.58673e6	−0.731614
$704$	2.90774e6	0.221118
$705$	0	0
$706$	2.22767e6	0.168205
$707$	−8.04125e6	−0.605028
$708$	0	0
$709$	−1.64673e7	−1.23029	−0.615145	−	0.788414i	$-0.710903\pi$
$709$	−1.64673e7	−1.23029	−0.615145	+	0.788414i	$0.710903\pi$
$710$	2.72132e6	0.202598
$711$	0	0
$712$	2.41906e6	0.178833
$713$	2.72241e6	0.200553
$714$	0	0
$715$	−6.61794e6	−0.484125
$716$	5.04919e6	0.368077
$717$	0	0
$718$	−4.02945e6	−0.291698
$719$	−5.48538e6	−0.395717	−0.197859	−	0.980231i	$-0.563399\pi$
$719$	−5.48538e6	−0.395717	−0.197859	+	0.980231i	$0.563399\pi$
$720$	0	0
$721$	−2.67202e7	−1.91426
$722$	1.29854e6	0.0927071
$723$	0	0
$724$	−5.63090e6	−0.399237
$725$	2.50604e6	0.177070
$726$	0	0
$727$	−2.75484e7	−1.93313	−0.966563	−	0.256431i	$-0.917453\pi$
$727$	−2.75484e7	−1.93313	−0.966563	+	0.256431i	$0.917453\pi$
$728$	−1.51799e7	−1.06155
$729$	0	0
$730$	−2.58807e6	−0.179750
$731$	4.95432e6	0.342918
$732$	0	0
$733$	1.51233e7	1.03965	0.519825	−	0.854273i	$-0.325997\pi$
$733$	1.51233e7	1.03965	0.519825	+	0.854273i	$0.325997\pi$
$734$	1.50980e6	0.103438
$735$	0	0
$736$	−1.95301e6	−0.132896
$737$	6.88295e6	0.466773
$738$	0	0
$739$	−1.56004e7	−1.05081	−0.525405	−	0.850852i	$-0.676086\pi$
$739$	−1.56004e7	−1.05081	−0.525405	+	0.850852i	$0.676086\pi$
$740$	9.21185e6	0.618397
$741$	0	0
$742$	6.92870e6	0.462000
$743$	−1.77279e6	−0.117811	−0.0589053	−	0.998264i	$-0.518761\pi$
$743$	−1.77279e6	−0.117811	−0.0589053	+	0.998264i	$0.518761\pi$
$744$	0	0
$745$	2.63603e7	1.74004
$746$	−893892.	−0.0588082
$747$	0	0
$748$	4.52913e6	0.295979
$749$	5.97386e6	0.389090
$750$	0	0
$751$	5.07089e6	0.328083	0.164042	−	0.986453i	$-0.447547\pi$
$751$	5.07089e6	0.328083	0.164042	+	0.986453i	$0.447547\pi$
$752$	−1.99468e7	−1.28626
$753$	0	0
$754$	−7.74696e6	−0.496253
$755$	1.53962e7	0.982987
$756$	0	0
$757$	−733561.	−0.0465261	−0.0232631	−	0.999729i	$-0.507406\pi$
$757$	−733561.	−0.0465261	−0.0232631	+	0.999729i	$0.507406\pi$
$758$	2.73709e6	0.173028
$759$	0	0
$760$	−8.87135e6	−0.557129
$761$	−2.90077e6	−0.181573	−0.0907866	−	0.995870i	$-0.528938\pi$
$761$	−2.90077e6	−0.181573	−0.0907866	+	0.995870i	$0.528938\pi$
$762$	0	0
$763$	1.36855e7	0.851039
$764$	−2.33504e7	−1.44731
$765$	0	0
$766$	370358.	0.0228060
$767$	−3.28431e7	−2.01584
$768$	0	0
$769$	−6.02882e6	−0.367635	−0.183817	−	0.982960i	$-0.558846\pi$
$769$	−6.02882e6	−0.367635	−0.183817	+	0.982960i	$0.558846\pi$
$770$	−2.05845e6	−0.125116
$771$	0	0
$772$	1.16533e7	0.703729
$773$	−1.48827e7	−0.895844	−0.447922	−	0.894073i	$-0.647836\pi$
$773$	−1.48827e7	−0.895844	−0.447922	+	0.894073i	$0.647836\pi$
$774$	0	0
$775$	−1.90035e6	−0.113653
$776$	1.17710e6	0.0701715
$777$	0	0
$778$	−877019.	−0.0519469
$779$	2.32951e7	1.37537
$780$	0	0
$781$	4.51512e6	0.264876
$782$	−805285.	−0.0470904
$783$	0	0
$784$	2.48856e7	1.44596
$785$	−2.72315e6	−0.157724
$786$	0	0
$787$	2.19846e7	1.26526	0.632631	−	0.774453i	$-0.281975\pi$
$787$	2.19846e7	1.26526	0.632631	+	0.774453i	$0.281975\pi$
$788$	8.07075e6	0.463018
$789$	0	0
$790$	5.84350e6	0.333124
$791$	−5.38164e6	−0.305825
$792$	0	0
$793$	−2.33685e7	−1.31962
$794$	−6.96542e6	−0.392099
$795$	0	0
$796$	2.14191e7	1.19817
$797$	2.92578e6	0.163153	0.0815765	−	0.996667i	$-0.474005\pi$
$797$	2.92578e6	0.163153	0.0815765	+	0.996667i	$0.474005\pi$
$798$	0	0
$799$	−2.71476e7	−1.50440
$800$	1.36328e6	0.0753113
$801$	0	0
$802$	2.80993e6	0.154262
$803$	−4.29404e6	−0.235005
$804$	0	0
$805$	−6.67096e6	−0.362826
$806$	5.87457e6	0.318521
$807$	0	0
$808$	−3.03128e6	−0.163342
$809$	2.36363e6	0.126972	0.0634860	−	0.997983i	$-0.479778\pi$
$809$	2.36363e6	0.126972	0.0634860	+	0.997983i	$0.479778\pi$
$810$	0	0
$811$	−1.40226e7	−0.748648	−0.374324	−	0.927298i	$-0.622125\pi$
$811$	−1.40226e7	−0.748648	−0.374324	+	0.927298i	$0.622125\pi$
$812$	4.39200e7	2.33761
$813$	0	0
$814$	−838540.	−0.0443570
$815$	−1.11354e7	−0.587236
$816$	0	0
$817$	−7.83548e6	−0.410686
$818$	−460004.	−0.0240369
$819$	0	0
$820$	−2.23841e7	−1.16253
$821$	−3.57170e7	−1.84934	−0.924670	−	0.380769i	$-0.875659\pi$
$821$	−3.57170e7	−1.84934	−0.924670	+	0.380769i	$0.875659\pi$
$822$	0	0
$823$	−1.15100e7	−0.592344	−0.296172	−	0.955135i	$-0.595710\pi$
$823$	−1.15100e7	−0.592344	−0.296172	+	0.955135i	$0.595710\pi$
$824$	−1.00726e7	−0.516801
$825$	0	0
$826$	−1.02155e7	−0.520968
$827$	1.24864e7	0.634854	0.317427	−	0.948283i	$-0.397181\pi$
$827$	1.24864e7	0.634854	0.317427	+	0.948283i	$0.397181\pi$
$828$	0	0
$829$	−1.13398e7	−0.573087	−0.286544	−	0.958067i	$-0.592506\pi$
$829$	−1.13398e7	−0.573087	−0.286544	+	0.958067i	$0.592506\pi$
$830$	−7.36882e6	−0.371281
$831$	0	0
$832$	2.03341e7	1.01840
$833$	3.38692e7	1.69119
$834$	0	0
$835$	1.20125e7	0.596232
$836$	−7.16302e6	−0.354471
$837$	0	0
$838$	−4.93633e6	−0.242825
$839$	−2.91430e7	−1.42932	−0.714659	−	0.699473i	$-0.753418\pi$
$839$	−2.91430e7	−1.42932	−0.714659	+	0.699473i	$0.753418\pi$
$840$	0	0
$841$	2.55471e7	1.24552
$842$	−3.88078e6	−0.188642
$843$	0	0
$844$	8.81663e6	0.426036
$845$	−2.43319e7	−1.17229
$846$	0	0
$847$	3.09419e7	1.48197
$848$	−2.18269e7	−1.04232
$849$	0	0
$850$	562120.	0.0266859
$851$	−2.71751e6	−0.128632
$852$	0	0
$853$	−9.77420e6	−0.459948	−0.229974	−	0.973197i	$-0.573864\pi$
$853$	−9.77420e6	−0.459948	−0.229974	+	0.973197i	$0.573864\pi$
$854$	−7.26858e6	−0.341040
$855$	0	0
$856$	2.25194e6	0.105044
$857$	2.33262e7	1.08491	0.542453	−	0.840086i	$-0.317496\pi$
$857$	2.33262e7	1.08491	0.542453	+	0.840086i	$0.317496\pi$
$858$	0	0
$859$	−1.14340e7	−0.528708	−0.264354	−	0.964426i	$-0.585159\pi$
$859$	−1.14340e7	−0.528708	−0.264354	+	0.964426i	$0.585159\pi$
$860$	7.52908e6	0.347133
$861$	0	0
$862$	−745426.	−0.0341693
$863$	7.70458e6	0.352145	0.176073	−	0.984377i	$-0.443661\pi$
$863$	7.70458e6	0.352145	0.176073	+	0.984377i	$0.443661\pi$
$864$	0	0
$865$	9.34277e6	0.424556
$866$	9.31329e6	0.421996
$867$	0	0
$868$	−3.33048e7	−1.50040
$869$	9.69534e6	0.435525
$870$	0	0
$871$	4.81332e7	2.14981
$872$	5.15897e6	0.229759
$873$	0	0
$874$	1.27359e6	0.0563965
$875$	−3.47513e7	−1.53444
$876$	0	0
$877$	1.42476e7	0.625520	0.312760	−	0.949832i	$-0.398746\pi$
$877$	1.42476e7	0.625520	0.312760	+	0.949832i	$0.398746\pi$
$878$	−981676.	−0.0429766
$879$	0	0
$880$	6.48457e6	0.282276
$881$	1.76114e7	0.764459	0.382230	−	0.924067i	$-0.375156\pi$
$881$	1.76114e7	0.764459	0.382230	+	0.924067i	$0.375156\pi$
$882$	0	0
$883$	−9.40461e6	−0.405919	−0.202959	−	0.979187i	$-0.565056\pi$
$883$	−9.40461e6	−0.405919	−0.202959	+	0.979187i	$0.565056\pi$
$884$	3.16726e7	1.36318
$885$	0	0
$886$	8.99054e6	0.384770
$887$	−1.38034e7	−0.589086	−0.294543	−	0.955638i	$-0.595167\pi$
$887$	−1.38034e7	−0.589086	−0.294543	+	0.955638i	$0.595167\pi$
$888$	0	0
$889$	−7.63383e6	−0.323958
$890$	2.29397e6	0.0970762
$891$	0	0
$892$	1.74404e7	0.733913
$893$	4.29351e7	1.80170
$894$	0	0
$895$	9.83888e6	0.410571
$896$	3.15279e7	1.31197
$897$	0	0
$898$	−7.08127e6	−0.293036
$899$	−3.49262e7	−1.44129
$900$	0	0
$901$	−2.97064e7	−1.21910
$902$	2.03759e6	0.0833874
$903$	0	0
$904$	−2.02870e6	−0.0825650
$905$	−1.09724e7	−0.445328
$906$	0	0
$907$	802157.	0.0323773	0.0161887	−	0.999869i	$-0.494847\pi$
$907$	802157.	0.0323773	0.0161887	+	0.999869i	$0.494847\pi$
$908$	−4.16699e7	−1.67729
$909$	0	0
$910$	−1.43950e7	−0.576245
$911$	−4.80457e6	−0.191804	−0.0959022	−	0.995391i	$-0.530574\pi$
$911$	−4.80457e6	−0.191804	−0.0959022	+	0.995391i	$0.530574\pi$
$912$	0	0
$913$	−1.22261e7	−0.485412
$914$	−3.22482e6	−0.127685
$915$	0	0
$916$	−2.64140e7	−1.04015
$917$	1.66654e7	0.654473
$918$	0	0
$919$	2.84366e7	1.11068	0.555339	−	0.831624i	$-0.312588\pi$
$919$	2.84366e7	1.11068	0.555339	+	0.831624i	$0.312588\pi$
$920$	−2.51472e6	−0.0979537
$921$	0	0
$922$	1.04541e7	0.405005
$923$	3.15747e7	1.21993
$924$	0	0
$925$	1.89693e6	0.0728948
$926$	−6.08176e6	−0.233078
$927$	0	0
$928$	2.50555e7	0.955066
$929$	−5.06040e6	−0.192374	−0.0961868	−	0.995363i	$-0.530665\pi$
$929$	−5.06040e6	−0.192374	−0.0961868	+	0.995363i	$0.530665\pi$
$930$	0	0
$931$	−5.35657e7	−2.02541
$932$	−4.90624e7	−1.85016
$933$	0	0
$934$	1.33326e6	0.0500089
$935$	8.82549e6	0.330149
$936$	0	0
$937$	1.27197e7	0.473291	0.236646	−	0.971596i	$-0.423952\pi$
$937$	1.27197e7	0.473291	0.236646	+	0.971596i	$0.423952\pi$
$938$	1.49714e7	0.555591
$939$	0	0
$940$	−4.12561e7	−1.52289
$941$	1.66122e7	0.611581	0.305790	−	0.952099i	$-0.401079\pi$
$941$	1.66122e7	0.611581	0.305790	+	0.952099i	$0.401079\pi$
$942$	0	0
$943$	6.60335e6	0.241816
$944$	3.21812e7	1.17536
$945$	0	0
$946$	−685360.	−0.0248995
$947$	−4.77751e7	−1.73112	−0.865558	−	0.500808i	$-0.833036\pi$
$947$	−4.77751e7	−1.73112	−0.865558	+	0.500808i	$0.833036\pi$
$948$	0	0
$949$	−3.00286e7	−1.08236
$950$	−889018.	−0.0319596
$951$	0	0
$952$	2.02435e7	0.723924
$953$	3.77139e7	1.34514	0.672572	−	0.740032i	$-0.265190\pi$
$953$	3.77139e7	1.34514	0.672572	+	0.740032i	$0.265190\pi$
$954$	0	0
$955$	−4.55007e7	−1.61439
$956$	−5.01391e7	−1.77432
$957$	0	0
$958$	−1.45806e6	−0.0513287
$959$	−1.98007e7	−0.695239
$960$	0	0
$961$	−2.14436e6	−0.0749014
$962$	−5.86399e6	−0.204294
$963$	0	0
$964$	−2.81656e7	−0.976173
$965$	2.27077e7	0.784973
$966$	0	0
$967$	−1.36407e7	−0.469105	−0.234553	−	0.972103i	$-0.575362\pi$
$967$	−1.36407e7	−0.469105	−0.234553	+	0.972103i	$0.575362\pi$
$968$	1.16641e7	0.400093
$969$	0	0
$970$	1.11624e6	0.0380914
$971$	3.96602e7	1.34992	0.674958	−	0.737856i	$-0.264162\pi$
$971$	3.96602e7	1.34992	0.674958	+	0.737856i	$0.264162\pi$
$972$	0	0
$973$	8.89843e7	3.01322
$974$	−1.02351e6	−0.0345698
$975$	0	0
$976$	2.28976e7	0.769424
$977$	9.70600e6	0.325315	0.162657	−	0.986683i	$-0.447993\pi$
$977$	9.70600e6	0.325315	0.162657	+	0.986683i	$0.447993\pi$
$978$	0	0
$979$	3.80608e6	0.126917
$980$	5.14711e7	1.71198
$981$	0	0
$982$	4.72971e6	0.156515
$983$	2.88398e7	0.951937	0.475969	−	0.879462i	$-0.342098\pi$
$983$	2.88398e7	0.951937	0.475969	+	0.879462i	$0.342098\pi$
$984$	0	0
$985$	1.57267e7	0.516473
$986$	1.03311e7	0.338419
$987$	0	0
$988$	−5.00917e7	−1.63258
$989$	−2.22109e6	−0.0722064
$990$	0	0
$991$	3.94595e7	1.27634	0.638172	−	0.769894i	$-0.279691\pi$
$991$	3.94595e7	1.27634	0.638172	+	0.769894i	$0.279691\pi$
$992$	−1.89998e7	−0.613012
$993$	0	0
$994$	9.82104e6	0.315276
$995$	4.17374e7	1.33650
$996$	0	0
$997$	3.61342e7	1.15128	0.575639	−	0.817704i	$-0.304753\pi$
$997$	3.61342e7	1.15128	0.575639	+	0.817704i	$0.304753\pi$
$998$	−6.56215e6	−0.208555
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	207.6.a.c.1.2		3
3.2	odd	2		69.6.a.b.1.2	✓	3
12.11	even	2		1104.6.a.i.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
69.6.a.b.1.2	✓	3	3.2	odd	2
207.6.a.c.1.2		3	1.1	even	1	trivial
1104.6.a.i.1.3		3	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)

\(f(q)\)	\(=\)	\(q+1.29009 q^{2} -30.3357 q^{4} -59.1123 q^{5} -213.331 q^{7} -80.4187 q^{8} +O(q^{10})\)	\(q+1.29009 q^{2} -30.3357 q^{4} -59.1123 q^{5} -213.331 q^{7} -80.4187 q^{8} -76.2602 q^{10} -126.528 q^{11} -884.825 q^{13} -275.217 q^{14} +866.994 q^{16} +1179.98 q^{17} -1866.19 q^{19} +1793.21 q^{20} -163.233 q^{22} -529.000 q^{23} +369.263 q^{25} -1141.50 q^{26} +6471.55 q^{28} +6786.62 q^{29} -5146.34 q^{31} +3691.90 q^{32} +1522.28 q^{34} +12610.5 q^{35} +5137.07 q^{37} -2407.55 q^{38} +4753.73 q^{40} -12482.7 q^{41} +4198.66 q^{43} +3838.32 q^{44} -682.458 q^{46} -23006.9 q^{47} +28703.3 q^{49} +476.382 q^{50} +26841.8 q^{52} -25175.4 q^{53} +7479.38 q^{55} +17155.8 q^{56} +8755.36 q^{58} +37118.1 q^{59} +26410.4 q^{61} -6639.24 q^{62} -22980.9 q^{64} +52304.0 q^{65} -54398.5 q^{67} -35795.4 q^{68} +16268.7 q^{70} -35684.7 q^{71} +33937.4 q^{73} +6627.29 q^{74} +56612.0 q^{76} +26992.5 q^{77} -76625.8 q^{79} -51250.0 q^{80} -16103.8 q^{82} +96627.2 q^{83} -69751.1 q^{85} +5416.65 q^{86} +10175.2 q^{88} -30080.8 q^{89} +188761. q^{91} +16047.6 q^{92} -29680.9 q^{94} +110315. q^{95} -14637.2 q^{97} +37029.9 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 8 q^{2} + 22 q^{4} + 56 q^{5} - 114 q^{7} + 510 q^{8}+O(q^{10}) \) 3 * q + 8 * q^2 + 22 * q^4 + 56 * q^5 - 114 * q^7 + 510 * q^8	\( 3 q + 8 q^{2} + 22 q^{4} + 56 q^{5} - 114 q^{7} + 510 q^{8} + 282 q^{10} + 376 q^{11} - 858 q^{13} - 588 q^{14} + 2738 q^{16} + 2548 q^{17} - 2846 q^{19} + 4618 q^{20} - 5050 q^{22} - 1587 q^{23} + 753 q^{25} + 7788 q^{26} + 4736 q^{28} + 16370 q^{29} - 14756 q^{31} + 3878 q^{32} + 16520 q^{34} + 18520 q^{35} + 15874 q^{37} - 12438 q^{38} + 38270 q^{40} - 12606 q^{41} + 3154 q^{43} - 27114 q^{44} - 4232 q^{46} - 29928 q^{47} + 4471 q^{49} - 1452 q^{50} + 86856 q^{52} + 44084 q^{53} + 38360 q^{55} + 35704 q^{56} + 73316 q^{58} + 29300 q^{59} + 54010 q^{61} - 99908 q^{62} - 1582 q^{64} + 51216 q^{65} + 43390 q^{67} + 69840 q^{68} - 2476 q^{70} - 23424 q^{71} - 91402 q^{73} + 2294 q^{74} - 14274 q^{76} + 97208 q^{77} - 49398 q^{79} + 52626 q^{80} + 40152 q^{82} + 103936 q^{83} + 5888 q^{85} - 133634 q^{86} + 48898 q^{88} - 96112 q^{89} + 129228 q^{91} - 11638 q^{92} - 133688 q^{94} + 55928 q^{95} - 135318 q^{97} - 108440 q^{98}+O(q^{100}) \) 3 * q + 8 * q^2 + 22 * q^4 + 56 * q^5 - 114 * q^7 + 510 * q^8 + 282 * q^10 + 376 * q^11 - 858 * q^13 - 588 * q^14 + 2738 * q^16 + 2548 * q^17 - 2846 * q^19 + 4618 * q^20 - 5050 * q^22 - 1587 * q^23 + 753 * q^25 + 7788 * q^26 + 4736 * q^28 + 16370 * q^29 - 14756 * q^31 + 3878 * q^32 + 16520 * q^34 + 18520 * q^35 + 15874 * q^37 - 12438 * q^38 + 38270 * q^40 - 12606 * q^41 + 3154 * q^43 - 27114 * q^44 - 4232 * q^46 - 29928 * q^47 + 4471 * q^49 - 1452 * q^50 + 86856 * q^52 + 44084 * q^53 + 38360 * q^55 + 35704 * q^56 + 73316 * q^58 + 29300 * q^59 + 54010 * q^61 - 99908 * q^62 - 1582 * q^64 + 51216 * q^65 + 43390 * q^67 + 69840 * q^68 - 2476 * q^70 - 23424 * q^71 - 91402 * q^73 + 2294 * q^74 - 14274 * q^76 + 97208 * q^77 - 49398 * q^79 + 52626 * q^80 + 40152 * q^82 + 103936 * q^83 + 5888 * q^85 - 133634 * q^86 + 48898 * q^88 - 96112 * q^89 + 129228 * q^91 - 11638 * q^92 - 133688 * q^94 + 55928 * q^95 - 135318 * q^97 - 108440 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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