LMFDB - Embedded newform 198.6.a.m.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [198,6,Mod(1,198)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(198, 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("198.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 198 = 2 \cdot 3^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	198.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:	$31.7559963230$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{694}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 694 $ x^2 - 694
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$26.3439$ of defining polynomial
Character	$\chi$	$=$	198.1

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+4.00000 q^{2} +16.0000 q^{4} +32.6878 q^{5} +196.063 q^{7} +64.0000 q^{8} +O(q^{10})$	\(q+4.00000 q^{2} +16.0000 q^{4} +32.6878 q^{5} +196.063 q^{7} +64.0000 q^{8} +130.751 q^{10} -121.000 q^{11} +109.186 q^{13} +784.253 q^{14} +256.000 q^{16} +772.751 q^{17} +68.2448 q^{19} +523.004 q^{20} -484.000 q^{22} -1601.70 q^{23} -2056.51 q^{25} +436.743 q^{26} +3137.01 q^{28} +1885.87 q^{29} +5702.89 q^{31} +1024.00 q^{32} +3091.00 q^{34} +6408.87 q^{35} +1234.36 q^{37} +272.979 q^{38} +2092.02 q^{40} +4308.40 q^{41} +14316.4 q^{43} -1936.00 q^{44} -6406.80 q^{46} +990.999 q^{47} +21633.8 q^{49} -8226.04 q^{50} +1746.97 q^{52} +9787.27 q^{53} -3955.22 q^{55} +12548.0 q^{56} +7543.49 q^{58} -47657.8 q^{59} +16949.8 q^{61} +22811.6 q^{62} +4096.00 q^{64} +3569.04 q^{65} +56417.4 q^{67} +12364.0 q^{68} +25635.5 q^{70} -48526.1 q^{71} +20814.1 q^{73} +4937.45 q^{74} +1091.92 q^{76} -23723.7 q^{77} +54232.5 q^{79} +8368.07 q^{80} +17233.6 q^{82} -104154. q^{83} +25259.5 q^{85} +57265.5 q^{86} -7744.00 q^{88} -107890. q^{89} +21407.3 q^{91} -25627.2 q^{92} +3964.00 q^{94} +2230.77 q^{95} -77323.0 q^{97} +86535.2 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 8 q^{2} + 32 q^{4} - 40 q^{5} + 76 q^{7} + 128 q^{8}+O(q^{10}) $ 2 * q + 8 * q^2 + 32 * q^4 - 40 * q^5 + 76 * q^7 + 128 * q^8	$ 2 q + 8 q^{2} + 32 q^{4} - 40 q^{5} + 76 q^{7} + 128 q^{8} - 160 q^{10} - 242 q^{11} + 956 q^{13} + 304 q^{14} + 512 q^{16} + 1124 q^{17} + 2244 q^{19} - 640 q^{20} - 968 q^{22} + 1960 q^{23} + 102 q^{25} + 3824 q^{26} + 1216 q^{28} + 4404 q^{29} + 3608 q^{31} + 2048 q^{32} + 4496 q^{34} + 15136 q^{35} + 7316 q^{37} + 8976 q^{38} - 2560 q^{40} - 3396 q^{41} + 30108 q^{43} - 3872 q^{44} + 7840 q^{46} - 22992 q^{47} + 19242 q^{49} + 408 q^{50} + 15296 q^{52} - 13408 q^{53} + 4840 q^{55} + 4864 q^{56} + 17616 q^{58} - 6168 q^{59} + 58452 q^{61} + 14432 q^{62} + 8192 q^{64} - 57984 q^{65} + 39072 q^{67} + 17984 q^{68} + 60544 q^{70} - 36672 q^{71} + 107804 q^{73} + 29264 q^{74} + 35904 q^{76} - 9196 q^{77} + 65788 q^{79} - 10240 q^{80} - 13584 q^{82} - 131384 q^{83} - 272 q^{85} + 120432 q^{86} - 15488 q^{88} - 97760 q^{89} - 80264 q^{91} + 31360 q^{92} - 91968 q^{94} - 155920 q^{95} - 36204 q^{97} + 76968 q^{98}+O(q^{100}) $ 2 * q + 8 * q^2 + 32 * q^4 - 40 * q^5 + 76 * q^7 + 128 * q^8 - 160 * q^10 - 242 * q^11 + 956 * q^13 + 304 * q^14 + 512 * q^16 + 1124 * q^17 + 2244 * q^19 - 640 * q^20 - 968 * q^22 + 1960 * q^23 + 102 * q^25 + 3824 * q^26 + 1216 * q^28 + 4404 * q^29 + 3608 * q^31 + 2048 * q^32 + 4496 * q^34 + 15136 * q^35 + 7316 * q^37 + 8976 * q^38 - 2560 * q^40 - 3396 * q^41 + 30108 * q^43 - 3872 * q^44 + 7840 * q^46 - 22992 * q^47 + 19242 * q^49 + 408 * q^50 + 15296 * q^52 - 13408 * q^53 + 4840 * q^55 + 4864 * q^56 + 17616 * q^58 - 6168 * q^59 + 58452 * q^61 + 14432 * q^62 + 8192 * q^64 - 57984 * q^65 + 39072 * q^67 + 17984 * q^68 + 60544 * q^70 - 36672 * q^71 + 107804 * q^73 + 29264 * q^74 + 35904 * q^76 - 9196 * q^77 + 65788 * q^79 - 10240 * q^80 - 13584 * q^82 - 131384 * q^83 - 272 * q^85 + 120432 * q^86 - 15488 * q^88 - 97760 * q^89 - 80264 * q^91 + 31360 * q^92 - 91968 * q^94 - 155920 * q^95 - 36204 * q^97 + 76968 * 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 $

$2$	4.00000	0.707107
$2$	4.00000	0.707107
$3$	0	0
$3$	0	0
$4$	16.0000	0.500000
$5$	32.6878	0.584736	0.292368	−	0.956306i	$-0.405557\pi$
$5$	32.6878	0.584736	0.292368	+	0.956306i	$0.405557\pi$
$6$	0	0
$7$	196.063	1.51235	0.756173	−	0.654372i	$-0.227067\pi$
$7$	196.063	1.51235	0.756173	+	0.654372i	$0.227067\pi$
$8$	64.0000	0.353553
$9$	0	0
$10$	130.751	0.413471
$11$	−121.000	−0.301511
$11$	−121.000	−0.301511
$12$	0	0
$13$	109.186	0.179187	0.0895937	−	0.995978i	$-0.471443\pi$
$13$	109.186	0.179187	0.0895937	+	0.995978i	$0.471443\pi$
$14$	784.253	1.06939
$15$	0	0
$16$	256.000	0.250000
$17$	772.751	0.648511	0.324255	−	0.945970i	$-0.394886\pi$
$17$	772.751	0.648511	0.324255	+	0.945970i	$0.394886\pi$
$18$	0	0
$19$	68.2448	0.0433696	0.0216848	−	0.999765i	$-0.493097\pi$
$19$	68.2448	0.0433696	0.0216848	+	0.999765i	$0.493097\pi$
$20$	523.004	0.292368
$21$	0	0
$22$	−484.000	−0.213201
$23$	−1601.70	−0.631338	−0.315669	−	0.948869i	$-0.602229\pi$
$23$	−1601.70	−0.631338	−0.315669	+	0.948869i	$0.602229\pi$
$24$	0	0
$25$	−2056.51	−0.658083
$26$	436.743	0.126705
$27$	0	0
$28$	3137.01	0.756173
$29$	1885.87	0.416407	0.208203	−	0.978086i	$-0.433238\pi$
$29$	1885.87	0.416407	0.208203	+	0.978086i	$0.433238\pi$
$30$	0	0
$31$	5702.89	1.06584	0.532919	−	0.846166i	$-0.321095\pi$
$31$	5702.89	1.06584	0.532919	+	0.846166i	$0.321095\pi$
$32$	1024.00	0.176777
$33$	0	0
$34$	3091.00	0.458566
$35$	6408.87	0.884324
$36$	0	0
$37$	1234.36	0.148231	0.0741154	−	0.997250i	$-0.476387\pi$
$37$	1234.36	0.148231	0.0741154	+	0.997250i	$0.476387\pi$
$38$	272.979	0.0306670
$39$	0	0
$40$	2092.02	0.206736
$41$	4308.40	0.400273	0.200137	−	0.979768i	$-0.435861\pi$
$41$	4308.40	0.400273	0.200137	+	0.979768i	$0.435861\pi$
$42$	0	0
$43$	14316.4	1.18076	0.590380	−	0.807125i	$-0.298978\pi$
$43$	14316.4	1.18076	0.590380	+	0.807125i	$0.298978\pi$
$44$	−1936.00	−0.150756
$45$	0	0
$46$	−6406.80	−0.446423
$47$	990.999	0.0654378	0.0327189	−	0.999465i	$-0.489583\pi$
$47$	990.999	0.0654378	0.0327189	+	0.999465i	$0.489583\pi$
$48$	0	0
$49$	21633.8	1.28719
$50$	−8226.04	−0.465335
$51$	0	0
$52$	1746.97	0.0895937
$53$	9787.27	0.478599	0.239300	−	0.970946i	$-0.423082\pi$
$53$	9787.27	0.478599	0.239300	+	0.970946i	$0.423082\pi$
$54$	0	0
$55$	−3955.22	−0.176305
$56$	12548.0	0.534695
$57$	0	0
$58$	7543.49	0.294444
$59$	−47657.8	−1.78240	−0.891198	−	0.453614i	$-0.850135\pi$
$59$	−47657.8	−1.78240	−0.891198	+	0.453614i	$0.850135\pi$
$60$	0	0
$61$	16949.8	0.583229	0.291614	−	0.956536i	$-0.405808\pi$
$61$	16949.8	0.583229	0.291614	+	0.956536i	$0.405808\pi$
$62$	22811.6	0.753661
$63$	0	0
$64$	4096.00	0.125000
$65$	3569.04	0.104777
$66$	0	0
$67$	56417.4	1.53542	0.767708	−	0.640799i	$-0.221397\pi$
$67$	56417.4	1.53542	0.767708	+	0.640799i	$0.221397\pi$
$68$	12364.0	0.324255
$69$	0	0
$70$	25635.5	0.625311
$71$	−48526.1	−1.14243	−0.571215	−	0.820801i	$-0.693528\pi$
$71$	−48526.1	−1.14243	−0.571215	+	0.820801i	$0.693528\pi$
$72$	0	0
$73$	20814.1	0.457141	0.228571	−	0.973527i	$-0.426595\pi$
$73$	20814.1	0.457141	0.228571	+	0.973527i	$0.426595\pi$
$74$	4937.45	0.104815
$75$	0	0
$76$	1091.92	0.0216848
$77$	−23723.7	−0.455989
$78$	0	0
$79$	54232.5	0.977670	0.488835	−	0.872376i	$-0.337422\pi$
$79$	54232.5	0.977670	0.488835	+	0.872376i	$0.337422\pi$
$80$	8368.07	0.146184
$81$	0	0
$82$	17233.6	0.283036
$83$	−104154.	−1.65951	−0.829757	−	0.558125i	$-0.811521\pi$
$83$	−104154.	−1.65951	−0.829757	+	0.558125i	$0.811521\pi$
$84$	0	0
$85$	25259.5	0.379208
$86$	57265.5	0.834924
$87$	0	0
$88$	−7744.00	−0.106600
$89$	−107890.	−1.44380	−0.721901	−	0.691997i	$-0.756731\pi$
$89$	−107890.	−1.44380	−0.721901	+	0.691997i	$0.756731\pi$
$90$	0	0
$91$	21407.3	0.270993
$92$	−25627.2	−0.315669
$93$	0	0
$94$	3964.00	0.0462715
$95$	2230.77	0.0253598
$96$	0	0
$97$	−77323.0	−0.834410	−0.417205	−	0.908812i	$-0.636990\pi$
$97$	−77323.0	−0.834410	−0.417205	+	0.908812i	$0.636990\pi$
$98$	86535.2	0.910181
$99$	0	0
$100$	−32904.2	−0.329042
$101$	140950.	1.37487	0.687434	−	0.726247i	$-0.258737\pi$
$101$	140950.	1.37487	0.687434	+	0.726247i	$0.258737\pi$
$102$	0	0
$103$	−108912.	−1.01154	−0.505768	−	0.862670i	$-0.668791\pi$
$103$	−108912.	−1.01154	−0.505768	+	0.862670i	$0.668791\pi$
$104$	6987.88	0.0633523
$105$	0	0
$106$	39149.1	0.338421
$107$	−5919.82	−0.0499861	−0.0249930	−	0.999688i	$-0.507956\pi$
$107$	−5919.82	−0.0499861	−0.0249930	+	0.999688i	$0.507956\pi$
$108$	0	0
$109$	−163854.	−1.32096	−0.660481	−	0.750843i	$-0.729647\pi$
$109$	−163854.	−1.32096	−0.660481	+	0.750843i	$0.729647\pi$
$110$	−15820.9	−0.124666
$111$	0	0
$112$	50192.2	0.378086
$113$	49644.2	0.365740	0.182870	−	0.983137i	$-0.441461\pi$
$113$	49644.2	0.365740	0.182870	+	0.983137i	$0.441461\pi$
$114$	0	0
$115$	−52356.0	−0.369166
$116$	30174.0	0.208203
$117$	0	0
$118$	−190631.	−1.26034
$119$	151508.	0.980773
$120$	0	0
$121$	14641.0	0.0909091
$122$	67799.0	0.412405
$123$	0	0
$124$	91246.3	0.532919
$125$	−169372.	−0.969542
$126$	0	0
$127$	104191.	0.573221	0.286611	−	0.958047i	$-0.407471\pi$
$127$	104191.	0.573221	0.286611	+	0.958047i	$0.407471\pi$
$128$	16384.0	0.0883883
$129$	0	0
$130$	14276.1	0.0740888
$131$	−30121.2	−0.153354	−0.0766768	−	0.997056i	$-0.524431\pi$
$131$	−30121.2	−0.153354	−0.0766768	+	0.997056i	$0.524431\pi$
$132$	0	0
$133$	13380.3	0.0655899
$134$	225670.	1.08570
$135$	0	0
$136$	49456.1	0.229283
$137$	−127899.	−0.582193	−0.291096	−	0.956694i	$-0.594020\pi$
$137$	−127899.	−0.582193	−0.291096	+	0.956694i	$0.594020\pi$
$138$	0	0
$139$	−180195.	−0.791053	−0.395527	−	0.918455i	$-0.629438\pi$
$139$	−180195.	−0.791053	−0.395527	+	0.918455i	$0.629438\pi$
$140$	102542.	0.442162
$141$	0	0
$142$	−194104.	−0.807820
$143$	−13211.5	−0.0540270
$144$	0	0
$145$	61645.0	0.243488
$146$	83256.3	0.323248
$147$	0	0
$148$	19749.8	0.0741154
$149$	−326010.	−1.20300	−0.601500	−	0.798873i	$-0.705430\pi$
$149$	−326010.	−1.20300	−0.601500	+	0.798873i	$0.705430\pi$
$150$	0	0
$151$	136433.	0.486942	0.243471	−	0.969908i	$-0.421714\pi$
$151$	136433.	0.486942	0.243471	+	0.969908i	$0.421714\pi$
$152$	4367.67	0.0153335
$153$	0	0
$154$	−94894.6	−0.322433
$155$	186415.	0.623234
$156$	0	0
$157$	−457291.	−1.48062	−0.740310	−	0.672266i	$-0.765321\pi$
$157$	−457291.	−1.48062	−0.740310	+	0.672266i	$0.765321\pi$
$158$	216930.	0.691317
$159$	0	0
$160$	33472.3	0.103368
$161$	−314035.	−0.954801
$162$	0	0
$163$	−352472.	−1.03910	−0.519549	−	0.854441i	$-0.673900\pi$
$163$	−352472.	−1.03910	−0.519549	+	0.854441i	$0.673900\pi$
$164$	68934.5	0.200137
$165$	0	0
$166$	−416616.	−1.17345
$167$	−187387.	−0.519934	−0.259967	−	0.965618i	$-0.583712\pi$
$167$	−187387.	−0.519934	−0.259967	+	0.965618i	$0.583712\pi$
$168$	0	0
$169$	−359371.	−0.967892
$170$	101038.	0.268141
$171$	0	0
$172$	229062.	0.590380
$173$	−127602.	−0.324148	−0.162074	−	0.986779i	$-0.551818\pi$
$173$	−127602.	−0.324148	−0.162074	+	0.986779i	$0.551818\pi$
$174$	0	0
$175$	−403206.	−0.995250
$176$	−30976.0	−0.0753778
$177$	0	0
$178$	−431561.	−1.02092
$179$	−260976.	−0.608790	−0.304395	−	0.952546i	$-0.598454\pi$
$179$	−260976.	−0.608790	−0.304395	+	0.952546i	$0.598454\pi$
$180$	0	0
$181$	494760.	1.12253	0.561265	−	0.827636i	$-0.310315\pi$
$181$	494760.	1.12253	0.561265	+	0.827636i	$0.310315\pi$
$182$	85629.2	0.191621
$183$	0	0
$184$	−102509.	−0.223212
$185$	40348.6	0.0866760
$186$	0	0
$187$	−93502.9	−0.195533
$188$	15856.0	0.0327189
$189$	0	0
$190$	8923.08	0.0179321
$191$	767302.	1.52189	0.760945	−	0.648817i	$-0.224736\pi$
$191$	767302.	1.52189	0.760945	+	0.648817i	$0.224736\pi$
$192$	0	0
$193$	−475601.	−0.919072	−0.459536	−	0.888159i	$-0.651984\pi$
$193$	−475601.	−0.919072	−0.459536	+	0.888159i	$0.651984\pi$
$194$	−309292.	−0.590017
$195$	0	0
$196$	346141.	0.643595
$197$	−722246.	−1.32593	−0.662964	−	0.748652i	$-0.730702\pi$
$197$	−722246.	−1.32593	−0.662964	+	0.748652i	$0.730702\pi$
$198$	0	0
$199$	646416.	1.15712	0.578562	−	0.815639i	$-0.303614\pi$
$199$	646416.	1.15712	0.578562	+	0.815639i	$0.303614\pi$
$200$	−131617.	−0.232668
$201$	0	0
$202$	563799.	0.972179
$203$	369751.	0.629751
$204$	0	0
$205$	140832.	0.234054
$206$	−435647.	−0.715264
$207$	0	0
$208$	27951.5	0.0447968
$209$	−8257.62	−0.0130764
$210$	0	0
$211$	308596.	0.477182	0.238591	−	0.971120i	$-0.423314\pi$
$211$	308596.	0.477182	0.238591	+	0.971120i	$0.423314\pi$
$212$	156596.	0.239300
$213$	0	0
$214$	−23679.3	−0.0353455
$215$	467970.	0.690434
$216$	0	0
$217$	1.11813e6	1.61192
$218$	−655415.	−0.934061
$219$	0	0
$220$	−63283.5	−0.0881523
$221$	84373.4	0.116205
$222$	0	0
$223$	−734582.	−0.989187	−0.494593	−	0.869124i	$-0.664683\pi$
$223$	−734582.	−0.989187	−0.494593	+	0.869124i	$0.664683\pi$
$224$	200769.	0.267348
$225$	0	0
$226$	198577.	0.258617
$227$	−1.06217e6	−1.36814	−0.684068	−	0.729419i	$-0.739791\pi$
$227$	−1.06217e6	−1.36814	−0.684068	+	0.729419i	$0.739791\pi$
$228$	0	0
$229$	402412.	0.507087	0.253544	−	0.967324i	$-0.418404\pi$
$229$	402412.	0.507087	0.253544	+	0.967324i	$0.418404\pi$
$230$	−209424.	−0.261040
$231$	0	0
$232$	120696.	0.147222
$233$	−937107.	−1.13084	−0.565418	−	0.824805i	$-0.691285\pi$
$233$	−937107.	−1.13084	−0.565418	+	0.824805i	$0.691285\pi$
$234$	0	0
$235$	32393.5	0.0382639
$236$	−762526.	−0.891198
$237$	0	0
$238$	606032.	0.693511
$239$	1.56502e6	1.77225	0.886126	−	0.463445i	$-0.153387\pi$
$239$	1.56502e6	1.77225	0.886126	+	0.463445i	$0.153387\pi$
$240$	0	0
$241$	53871.1	0.0597466	0.0298733	−	0.999554i	$-0.490490\pi$
$241$	53871.1	0.0597466	0.0298733	+	0.999554i	$0.490490\pi$
$242$	58564.0	0.0642824
$243$	0	0
$244$	271196.	0.291614
$245$	707161.	0.752667
$246$	0	0
$247$	7451.36	0.00777129
$248$	364985.	0.376831
$249$	0	0
$250$	−677488.	−0.685570
$251$	−542998.	−0.544019	−0.272009	−	0.962295i	$-0.587688\pi$
$251$	−542998.	−0.544019	−0.272009	+	0.962295i	$0.587688\pi$
$252$	0	0
$253$	193806.	0.190355
$254$	416765.	0.405329
$255$	0	0
$256$	65536.0	0.0625000
$257$	1.55305e6	1.46674	0.733368	−	0.679832i	$-0.237947\pi$
$257$	1.55305e6	1.46674	0.733368	+	0.679832i	$0.237947\pi$
$258$	0	0
$259$	242013.	0.224176
$260$	57104.6	0.0523887
$261$	0	0
$262$	−120485.	−0.108437
$263$	226686.	0.202085	0.101043	−	0.994882i	$-0.467782\pi$
$263$	226686.	0.202085	0.101043	+	0.994882i	$0.467782\pi$
$264$	0	0
$265$	319924.	0.279854
$266$	53521.2	0.0463790
$267$	0	0
$268$	902679.	0.767708
$269$	504596.	0.425171	0.212585	−	0.977143i	$-0.431812\pi$
$269$	504596.	0.425171	0.212585	+	0.977143i	$0.431812\pi$
$270$	0	0
$271$	2.10177e6	1.73845	0.869226	−	0.494415i	$-0.164618\pi$
$271$	2.10177e6	1.73845	0.869226	+	0.494415i	$0.164618\pi$
$272$	197824.	0.162128
$273$	0	0
$274$	−511597.	−0.411673
$275$	248838.	0.198420
$276$	0	0
$277$	1.34571e6	1.05378	0.526891	−	0.849933i	$-0.323358\pi$
$277$	1.34571e6	1.05378	0.526891	+	0.849933i	$0.323358\pi$
$278$	−720780.	−0.559359
$279$	0	0
$280$	410168.	0.312656
$281$	−266310.	−0.201197	−0.100599	−	0.994927i	$-0.532076\pi$
$281$	−266310.	−0.201197	−0.100599	+	0.994927i	$0.532076\pi$
$282$	0	0
$283$	1.73919e6	1.29087	0.645434	−	0.763816i	$-0.276677\pi$
$283$	1.73919e6	1.29087	0.645434	+	0.763816i	$0.276677\pi$
$284$	−776417.	−0.571215
$285$	0	0
$286$	−52845.9	−0.0382029
$287$	844720.	0.605352
$288$	0	0
$289$	−822713.	−0.579434
$290$	246580.	0.172172
$291$	0	0
$292$	333025.	0.228571
$293$	718674.	0.489061	0.244530	−	0.969642i	$-0.421366\pi$
$293$	718674.	0.489061	0.244530	+	0.969642i	$0.421366\pi$
$294$	0	0
$295$	−1.55783e6	−1.04223
$296$	78999.2	0.0524075
$297$	0	0
$298$	−1.30404e6	−0.850649
$299$	−174883.	−0.113128
$300$	0	0
$301$	2.80691e6	1.78572
$302$	545732.	0.344320
$303$	0	0
$304$	17470.7	0.0108424
$305$	554049.	0.341035
$306$	0	0
$307$	−943214.	−0.571169	−0.285584	−	0.958354i	$-0.592188\pi$
$307$	−943214.	−0.571169	−0.285584	+	0.958354i	$0.592188\pi$
$308$	−379579.	−0.227995
$309$	0	0
$310$	745659.	0.440693
$311$	2.85596e6	1.67437	0.837186	−	0.546918i	$-0.184199\pi$
$311$	2.85596e6	1.67437	0.837186	+	0.546918i	$0.184199\pi$
$312$	0	0
$313$	−2.09030e6	−1.20600	−0.603002	−	0.797740i	$-0.706029\pi$
$313$	−2.09030e6	−1.20600	−0.603002	+	0.797740i	$0.706029\pi$
$314$	−1.82916e6	−1.04696
$315$	0	0
$316$	867721.	0.488835
$317$	−205839.	−0.115048	−0.0575241	−	0.998344i	$-0.518321\pi$
$317$	−205839.	−0.115048	−0.0575241	+	0.998344i	$0.518321\pi$
$318$	0	0
$319$	−228191.	−0.125551
$320$	133889.	0.0730921
$321$	0	0
$322$	−1.25614e6	−0.675146
$323$	52736.2	0.0281257
$324$	0	0
$325$	−224541.	−0.117920
$326$	−1.40989e6	−0.734753
$327$	0	0
$328$	275738.	0.141518
$329$	194299.	0.0989646
$330$	0	0
$331$	−2.01994e6	−1.01337	−0.506685	−	0.862131i	$-0.669129\pi$
$331$	−2.01994e6	−1.01337	−0.506685	+	0.862131i	$0.669129\pi$
$332$	−1.66647e6	−0.829757
$333$	0	0
$334$	−749548.	−0.367649
$335$	1.84416e6	0.897814
$336$	0	0
$337$	374069.	0.179423	0.0897113	−	0.995968i	$-0.471406\pi$
$337$	374069.	0.179423	0.0897113	+	0.995968i	$0.471406\pi$
$338$	−1.43749e6	−0.684403
$339$	0	0
$340$	404152.	0.189604
$341$	−690050.	−0.321362
$342$	0	0
$343$	946360.	0.434331
$344$	916248.	0.417462
$345$	0	0
$346$	−510409.	−0.229207
$347$	−2.44466e6	−1.08992	−0.544959	−	0.838463i	$-0.683455\pi$
$347$	−2.44466e6	−1.08992	−0.544959	+	0.838463i	$0.683455\pi$
$348$	0	0
$349$	−1.01556e6	−0.446314	−0.223157	−	0.974783i	$-0.571636\pi$
$349$	−1.01556e6	−0.446314	−0.223157	+	0.974783i	$0.571636\pi$
$350$	−1.61282e6	−0.703748
$351$	0	0
$352$	−123904.	−0.0533002
$353$	−3.28020e6	−1.40108	−0.700541	−	0.713612i	$-0.747058\pi$
$353$	−3.28020e6	−1.40108	−0.700541	+	0.713612i	$0.747058\pi$
$354$	0	0
$355$	−1.58621e6	−0.668020
$356$	−1.72624e6	−0.721901
$357$	0	0
$358$	−1.04390e6	−0.430480
$359$	1.69816e6	0.695411	0.347705	−	0.937604i	$-0.386961\pi$
$359$	1.69816e6	0.695411	0.347705	+	0.937604i	$0.386961\pi$
$360$	0	0
$361$	−2.47144e6	−0.998119
$362$	1.97904e6	0.793748
$363$	0	0
$364$	342517.	0.135497
$365$	680366.	0.267307
$366$	0	0
$367$	207265.	0.0803268	0.0401634	−	0.999193i	$-0.487212\pi$
$367$	207265.	0.0803268	0.0401634	+	0.999193i	$0.487212\pi$
$368$	−410035.	−0.157834
$369$	0	0
$370$	161394.	0.0612892
$371$	1.91892e6	0.723807
$372$	0	0
$373$	160124.	0.0595915	0.0297957	−	0.999556i	$-0.490514\pi$
$373$	160124.	0.0595915	0.0297957	+	0.999556i	$0.490514\pi$
$374$	−374012.	−0.138263
$375$	0	0
$376$	63423.9	0.0231357
$377$	205910.	0.0746148
$378$	0	0
$379$	1.79445e6	0.641702	0.320851	−	0.947130i	$-0.396031\pi$
$379$	1.79445e6	0.641702	0.320851	+	0.947130i	$0.396031\pi$
$380$	35692.3	0.0126799
$381$	0	0
$382$	3.06921e6	1.07614
$383$	−1.12808e6	−0.392954	−0.196477	−	0.980508i	$-0.562950\pi$
$383$	−1.12808e6	−0.392954	−0.196477	+	0.980508i	$0.562950\pi$
$384$	0	0
$385$	−775473.	−0.266634
$386$	−1.90240e6	−0.649882
$387$	0	0
$388$	−1.23717e6	−0.417205
$389$	−4.20240e6	−1.40807	−0.704033	−	0.710167i	$-0.748619\pi$
$389$	−4.20240e6	−1.40807	−0.704033	+	0.710167i	$0.748619\pi$
$390$	0	0
$391$	−1.23772e6	−0.409429
$392$	1.38456e6	0.455091
$393$	0	0
$394$	−2.88898e6	−0.937572
$395$	1.77274e6	0.571679
$396$	0	0
$397$	−1.64115e6	−0.522603	−0.261301	−	0.965257i	$-0.584152\pi$
$397$	−1.64115e6	−0.522603	−0.261301	+	0.965257i	$0.584152\pi$
$398$	2.58567e6	0.818210
$399$	0	0
$400$	−526467.	−0.164521
$401$	1.75479e6	0.544960	0.272480	−	0.962161i	$-0.412156\pi$
$401$	1.75479e6	0.544960	0.272480	+	0.962161i	$0.412156\pi$
$402$	0	0
$403$	622674.	0.190985
$404$	2.25520e6	0.687434
$405$	0	0
$406$	1.47900e6	0.445301
$407$	−149358.	−0.0446933
$408$	0	0
$409$	−2.92788e6	−0.865456	−0.432728	−	0.901525i	$-0.642449\pi$
$409$	−2.92788e6	−0.865456	−0.432728	+	0.901525i	$0.642449\pi$
$410$	563328.	0.165502
$411$	0	0
$412$	−1.74259e6	−0.505768
$413$	−9.34395e6	−2.69560
$414$	0	0
$415$	−3.40456e6	−0.970378
$416$	111806.	0.0316761
$417$	0	0
$418$	−33030.5	−0.00924643
$419$	213626.	0.0594454	0.0297227	−	0.999558i	$-0.490538\pi$
$419$	213626.	0.0594454	0.0297227	+	0.999558i	$0.490538\pi$
$420$	0	0
$421$	−7.02462e6	−1.93160	−0.965801	−	0.259284i	$-0.916513\pi$
$421$	−7.02462e6	−1.93160	−0.965801	+	0.259284i	$0.916513\pi$
$422$	1.23438e6	0.337419
$423$	0	0
$424$	626385.	0.169210
$425$	−1.58917e6	−0.426774
$426$	0	0
$427$	3.32322e6	0.882043
$428$	−94717.1	−0.0249930
$429$	0	0
$430$	1.87188e6	0.488210
$431$	5.27875e6	1.36879	0.684396	−	0.729110i	$-0.260066\pi$
$431$	5.27875e6	1.36879	0.684396	+	0.729110i	$0.260066\pi$
$432$	0	0
$433$	3.97409e6	1.01863	0.509317	−	0.860579i	$-0.329898\pi$
$433$	3.97409e6	1.01863	0.509317	+	0.860579i	$0.329898\pi$
$434$	4.47251e6	1.13980
$435$	0	0
$436$	−2.62166e6	−0.660481
$437$	−109308.	−0.0273809
$438$	0	0
$439$	−7.09464e6	−1.75699	−0.878495	−	0.477752i	$-0.841452\pi$
$439$	−7.09464e6	−1.75699	−0.878495	+	0.477752i	$0.841452\pi$
$440$	−253134.	−0.0623331
$441$	0	0
$442$	337493.	0.0821693
$443$	7.66467e6	1.85560	0.927799	−	0.373080i	$-0.121698\pi$
$443$	7.66467e6	1.85560	0.927799	+	0.373080i	$0.121698\pi$
$444$	0	0
$445$	−3.52669e6	−0.844243
$446$	−2.93833e6	−0.699461
$447$	0	0
$448$	803075.	0.189043
$449$	−7.14952e6	−1.67364	−0.836819	−	0.547480i	$-0.815587\pi$
$449$	−7.14952e6	−1.67364	−0.836819	+	0.547480i	$0.815587\pi$
$450$	0	0
$451$	−521317.	−0.120687
$452$	794308.	0.182870
$453$	0	0
$454$	−4.24868e6	−0.967418
$455$	699757.	0.158460
$456$	0	0
$457$	−3.71292e6	−0.831621	−0.415810	−	0.909451i	$-0.636502\pi$
$457$	−3.71292e6	−0.831621	−0.415810	+	0.909451i	$0.636502\pi$
$458$	1.60965e6	0.358565
$459$	0	0
$460$	−837696.	−0.184583
$461$	2.12899e6	0.466574	0.233287	−	0.972408i	$-0.425052\pi$
$461$	2.12899e6	0.466574	0.233287	+	0.972408i	$0.425052\pi$
$462$	0	0
$463$	3.78502e6	0.820570	0.410285	−	0.911957i	$-0.365429\pi$
$463$	3.78502e6	0.820570	0.410285	+	0.911957i	$0.365429\pi$
$464$	482784.	0.104102
$465$	0	0
$466$	−3.74843e6	−0.799621
$467$	6.77264e6	1.43703	0.718515	−	0.695511i	$-0.244822\pi$
$467$	6.77264e6	1.43703	0.718515	+	0.695511i	$0.244822\pi$
$468$	0	0
$469$	1.10614e7	2.32208
$470$	129574.	0.0270566
$471$	0	0
$472$	−3.05010e6	−0.630172
$473$	−1.73228e6	−0.356013
$474$	0	0
$475$	−140346.	−0.0285408
$476$	2.42413e6	0.490386
$477$	0	0
$478$	6.26008e6	1.25317
$479$	−6.22246e6	−1.23915	−0.619574	−	0.784938i	$-0.712695\pi$
$479$	−6.22246e6	−1.23915	−0.619574	+	0.784938i	$0.712695\pi$
$480$	0	0
$481$	134775.	0.0265611
$482$	215484.	0.0422472
$483$	0	0
$484$	234256.	0.0454545
$485$	−2.52752e6	−0.487910
$486$	0	0
$487$	7.92206e6	1.51362	0.756808	−	0.653637i	$-0.226758\pi$
$487$	7.92206e6	1.51362	0.756808	+	0.653637i	$0.226758\pi$
$488$	1.08478e6	0.206202
$489$	0	0
$490$	2.82864e6	0.532216
$491$	1.31504e6	0.246170	0.123085	−	0.992396i	$-0.460721\pi$
$491$	1.31504e6	0.246170	0.123085	+	0.992396i	$0.460721\pi$
$492$	0	0
$493$	1.45731e6	0.270044
$494$	29805.4	0.00549513
$495$	0	0
$496$	1.45994e6	0.266459
$497$	−9.51418e6	−1.72775
$498$	0	0
$499$	1.10305e6	0.198309	0.0991544	−	0.995072i	$-0.468386\pi$
$499$	1.10305e6	0.198309	0.0991544	+	0.995072i	$0.468386\pi$
$500$	−2.70995e6	−0.484771
$501$	0	0
$502$	−2.17199e6	−0.384679
$503$	5.45915e6	0.962066	0.481033	−	0.876703i	$-0.340262\pi$
$503$	5.45915e6	0.962066	0.481033	+	0.876703i	$0.340262\pi$
$504$	0	0
$505$	4.60733e6	0.803936
$506$	775223.	0.134602
$507$	0	0
$508$	1.66706e6	0.286611
$509$	−309185.	−0.0528962	−0.0264481	−	0.999650i	$-0.508420\pi$
$509$	−309185.	−0.0528962	−0.0264481	+	0.999650i	$0.508420\pi$
$510$	0	0
$511$	4.08088e6	0.691355
$512$	262144.	0.0441942
$513$	0	0
$514$	6.21219e6	1.03714
$515$	−3.56008e6	−0.591482
$516$	0	0
$517$	−119911.	−0.0197302
$518$	968053.	0.158517
$519$	0	0
$520$	228418.	0.0370444
$521$	9.13859e6	1.47498	0.737488	−	0.675360i	$-0.236012\pi$
$521$	9.13859e6	1.47498	0.737488	+	0.675360i	$0.236012\pi$
$522$	0	0
$523$	6.00202e6	0.959496	0.479748	−	0.877406i	$-0.340728\pi$
$523$	6.00202e6	0.959496	0.479748	+	0.877406i	$0.340728\pi$
$524$	−481939.	−0.0766768
$525$	0	0
$526$	906743.	0.142896
$527$	4.40692e6	0.691207
$528$	0	0
$529$	−3.87090e6	−0.601413
$530$	1.27970e6	0.197887
$531$	0	0
$532$	214085.	0.0327949
$533$	470416.	0.0717239
$534$	0	0
$535$	−193506.	−0.0292287
$536$	3.61072e6	0.542852
$537$	0	0
$538$	2.01838e6	0.300641
$539$	−2.61769e6	−0.388103
$540$	0	0
$541$	3.75533e6	0.551638	0.275819	−	0.961210i	$-0.411051\pi$
$541$	3.75533e6	0.551638	0.275819	+	0.961210i	$0.411051\pi$
$542$	8.40709e6	1.22927
$543$	0	0
$544$	791297.	0.114642
$545$	−5.35601e6	−0.772414
$546$	0	0
$547$	−5.34623e6	−0.763975	−0.381988	−	0.924167i	$-0.624760\pi$
$547$	−5.34623e6	−0.763975	−0.381988	+	0.924167i	$0.624760\pi$
$548$	−2.04639e6	−0.291096
$549$	0	0
$550$	995351.	0.140304
$551$	128701.	0.0180594
$552$	0	0
$553$	1.06330e7	1.47858
$554$	5.38282e6	0.745136
$555$	0	0
$556$	−2.88312e6	−0.395527
$557$	−1.10602e7	−1.51051	−0.755256	−	0.655430i	$-0.772488\pi$
$557$	−1.10602e7	−1.51051	−0.755256	+	0.655430i	$0.772488\pi$
$558$	0	0
$559$	1.56314e6	0.211577
$560$	1.64067e6	0.221081
$561$	0	0
$562$	−1.06524e6	−0.142268
$563$	−7.45472e6	−0.991198	−0.495599	−	0.868551i	$-0.665051\pi$
$563$	−7.45472e6	−0.991198	−0.495599	+	0.868551i	$0.665051\pi$
$564$	0	0
$565$	1.62276e6	0.213862
$566$	6.95677e6	0.912781
$567$	0	0
$568$	−3.10567e6	−0.403910
$569$	618768.	0.0801211	0.0400606	−	0.999197i	$-0.487245\pi$
$569$	618768.	0.0801211	0.0400606	+	0.999197i	$0.487245\pi$
$570$	0	0
$571$	−1.26963e7	−1.62962	−0.814811	−	0.579726i	$-0.803160\pi$
$571$	−1.26963e7	−1.62962	−0.814811	+	0.579726i	$0.803160\pi$
$572$	−211383.	−0.0270135
$573$	0	0
$574$	3.37888e6	0.428049
$575$	3.29391e6	0.415473
$576$	0	0
$577$	−811954.	−0.101529	−0.0507647	−	0.998711i	$-0.516166\pi$
$577$	−811954.	−0.101529	−0.0507647	+	0.998711i	$0.516166\pi$
$578$	−3.29085e6	−0.409721
$579$	0	0
$580$	986320.	0.121744
$581$	−2.04208e7	−2.50976
$582$	0	0
$583$	−1.18426e6	−0.144303
$584$	1.33210e6	0.161624
$585$	0	0
$586$	2.87469e6	0.345818
$587$	5.09023e6	0.609737	0.304868	−	0.952395i	$-0.401388\pi$
$587$	5.09023e6	0.609737	0.304868	+	0.952395i	$0.401388\pi$
$588$	0	0
$589$	389193.	0.0462250
$590$	−6.23131e6	−0.736969
$591$	0	0
$592$	315997.	0.0370577
$593$	−1.35083e7	−1.57748	−0.788740	−	0.614727i	$-0.789266\pi$
$593$	−1.35083e7	−1.57748	−0.788740	+	0.614727i	$0.789266\pi$
$594$	0	0
$595$	4.95246e6	0.573494
$596$	−5.21616e6	−0.601500
$597$	0	0
$598$	−699531.	−0.0799934
$599$	2.28475e6	0.260178	0.130089	−	0.991502i	$-0.458474\pi$
$599$	2.28475e6	0.260178	0.130089	+	0.991502i	$0.458474\pi$
$600$	0	0
$601$	−3.04689e6	−0.344088	−0.172044	−	0.985089i	$-0.555037\pi$
$601$	−3.04689e6	−0.344088	−0.172044	+	0.985089i	$0.555037\pi$
$602$	1.12277e7	1.26269
$603$	0	0
$604$	2.18293e6	0.243471
$605$	478581.	0.0531579
$606$	0	0
$607$	3.63159e6	0.400060	0.200030	−	0.979790i	$-0.435896\pi$
$607$	3.63159e6	0.400060	0.200030	+	0.979790i	$0.435896\pi$
$608$	69882.7	0.00766674
$609$	0	0
$610$	2.21620e6	0.241148
$611$	108203.	0.0117256
$612$	0	0
$613$	7.10222e6	0.763383	0.381692	−	0.924290i	$-0.375342\pi$
$613$	7.10222e6	0.763383	0.381692	+	0.924290i	$0.375342\pi$
$614$	−3.77286e6	−0.403877
$615$	0	0
$616$	−1.51831e6	−0.161217
$617$	6.37671e6	0.674348	0.337174	−	0.941442i	$-0.390529\pi$
$617$	6.37671e6	0.674348	0.337174	+	0.941442i	$0.390529\pi$
$618$	0	0
$619$	3.59853e6	0.377484	0.188742	−	0.982027i	$-0.439559\pi$
$619$	3.59853e6	0.377484	0.188742	+	0.982027i	$0.439559\pi$
$620$	2.98264e6	0.311617
$621$	0	0
$622$	1.14239e7	1.18396
$623$	−2.11533e7	−2.18353
$624$	0	0
$625$	890205.	0.0911570
$626$	−8.36121e6	−0.852773
$627$	0	0
$628$	−7.31666e6	−0.740310
$629$	953855.	0.0961293
$630$	0	0
$631$	−8.78387e6	−0.878239	−0.439119	−	0.898429i	$-0.644709\pi$
$631$	−8.78387e6	−0.878239	−0.439119	+	0.898429i	$0.644709\pi$
$632$	3.47088e6	0.345659
$633$	0	0
$634$	−823356.	−0.0813513
$635$	3.40578e6	0.335183
$636$	0	0
$637$	2.36210e6	0.230648
$638$	−912763.	−0.0887782
$639$	0	0
$640$	535556.	0.0516839
$641$	1.95178e7	1.87623	0.938114	−	0.346326i	$-0.112571\pi$
$641$	1.95178e7	1.87623	0.938114	+	0.346326i	$0.112571\pi$
$642$	0	0
$643$	1.14311e7	1.09034	0.545169	−	0.838326i	$-0.316465\pi$
$643$	1.14311e7	1.09034	0.545169	+	0.838326i	$0.316465\pi$
$644$	−5.02455e6	−0.477400
$645$	0	0
$646$	210945.	0.0198879
$647$	−1.03213e7	−0.969336	−0.484668	−	0.874698i	$-0.661060\pi$
$647$	−1.03213e7	−0.969336	−0.484668	+	0.874698i	$0.661060\pi$
$648$	0	0
$649$	5.76660e6	0.537413
$650$	−898166.	−0.0833822
$651$	0	0
$652$	−5.63956e6	−0.519549
$653$	1.79523e7	1.64754	0.823772	−	0.566921i	$-0.191866\pi$
$653$	1.79523e7	1.64754	0.823772	+	0.566921i	$0.191866\pi$
$654$	0	0
$655$	−984594.	−0.0896714
$656$	1.10295e6	0.100068
$657$	0	0
$658$	777194.	0.0699785
$659$	1.08453e7	0.972813	0.486407	−	0.873733i	$-0.338307\pi$
$659$	1.08453e7	0.972813	0.486407	+	0.873733i	$0.338307\pi$
$660$	0	0
$661$	6.52939e6	0.581258	0.290629	−	0.956836i	$-0.406135\pi$
$661$	6.52939e6	0.581258	0.290629	+	0.956836i	$0.406135\pi$
$662$	−8.07975e6	−0.716561
$663$	0	0
$664$	−6.66586e6	−0.586727
$665$	437372.	0.0383528
$666$	0	0
$667$	−3.02060e6	−0.262893
$668$	−2.99819e6	−0.259967
$669$	0	0
$670$	7.37664e6	0.634850
$671$	−2.05092e6	−0.175850
$672$	0	0
$673$	−1.21636e7	−1.03520	−0.517599	−	0.855623i	$-0.673174\pi$
$673$	−1.21636e7	−1.03520	−0.517599	+	0.855623i	$0.673174\pi$
$674$	1.49628e6	0.126871
$675$	0	0
$676$	−5.74994e6	−0.483946
$677$	−1.04586e7	−0.877002	−0.438501	−	0.898731i	$-0.644490\pi$
$677$	−1.04586e7	−0.877002	−0.438501	+	0.898731i	$0.644490\pi$
$678$	0	0
$679$	−1.51602e7	−1.26192
$680$	1.61661e6	0.134070
$681$	0	0
$682$	−2.76020e6	−0.227237
$683$	1.32344e7	1.08556	0.542778	−	0.839876i	$-0.317372\pi$
$683$	1.32344e7	1.08556	0.542778	+	0.839876i	$0.317372\pi$
$684$	0	0
$685$	−4.18074e6	−0.340429
$686$	3.78544e6	0.307119
$687$	0	0
$688$	3.66499e6	0.295190
$689$	1.06863e6	0.0857589
$690$	0	0
$691$	−7.09181e6	−0.565018	−0.282509	−	0.959265i	$-0.591167\pi$
$691$	−7.09181e6	−0.565018	−0.282509	+	0.959265i	$0.591167\pi$
$692$	−2.04164e6	−0.162074
$693$	0	0
$694$	−9.77862e6	−0.770689
$695$	−5.89017e6	−0.462558
$696$	0	0
$697$	3.32932e6	0.259582
$698$	−4.06222e6	−0.315591
$699$	0	0
$700$	−6.45130e6	−0.497625
$701$	2.24791e7	1.72776	0.863879	−	0.503699i	$-0.168028\pi$
$701$	2.24791e7	1.72776	0.863879	+	0.503699i	$0.168028\pi$
$702$	0	0
$703$	84238.9	0.00642872
$704$	−495616.	−0.0376889
$705$	0	0
$706$	−1.31208e7	−0.990715
$707$	2.76351e7	2.07928
$708$	0	0
$709$	−2.28777e7	−1.70921	−0.854606	−	0.519277i	$-0.826201\pi$
$709$	−2.28777e7	−1.70921	−0.854606	+	0.519277i	$0.826201\pi$
$710$	−6.34484e6	−0.472362
$711$	0	0
$712$	−6.90498e6	−0.510461
$713$	−9.13433e6	−0.672903
$714$	0	0
$715$	−431853.	−0.0315916
$716$	−4.17561e6	−0.304395
$717$	0	0
$718$	6.79262e6	0.491730
$719$	2.59546e7	1.87237	0.936186	−	0.351506i	$-0.114330\pi$
$719$	2.59546e7	1.87237	0.936186	+	0.351506i	$0.114330\pi$
$720$	0	0
$721$	−2.13536e7	−1.52979
$722$	−9.88577e6	−0.705777
$723$	0	0
$724$	7.91615e6	0.561265
$725$	−3.87832e6	−0.274030
$726$	0	0
$727$	2.30772e7	1.61937	0.809686	−	0.586864i	$-0.199638\pi$
$727$	2.30772e7	1.61937	0.809686	+	0.586864i	$0.199638\pi$
$728$	1.37007e6	0.0958106
$729$	0	0
$730$	2.72146e6	0.189015
$731$	1.10630e7	0.765736
$732$	0	0
$733$	−1.16927e7	−0.803816	−0.401908	−	0.915680i	$-0.631653\pi$
$733$	−1.16927e7	−0.803816	−0.401908	+	0.915680i	$0.631653\pi$
$734$	829059.	0.0567996
$735$	0	0
$736$	−1.64014e6	−0.111606
$737$	−6.82651e6	−0.462946
$738$	0	0
$739$	−2.08322e6	−0.140321	−0.0701606	−	0.997536i	$-0.522351\pi$
$739$	−2.08322e6	−0.140321	−0.0701606	+	0.997536i	$0.522351\pi$
$740$	645577.	0.0433380
$741$	0	0
$742$	7.67570e6	0.511809
$743$	−2.28741e7	−1.52010	−0.760049	−	0.649866i	$-0.774825\pi$
$743$	−2.28741e7	−1.52010	−0.760049	+	0.649866i	$0.774825\pi$
$744$	0	0
$745$	−1.06565e7	−0.703438
$746$	640496.	0.0421375
$747$	0	0
$748$	−1.49605e6	−0.0977667
$749$	−1.16066e6	−0.0755962
$750$	0	0
$751$	3.81147e6	0.246599	0.123300	−	0.992369i	$-0.460652\pi$
$751$	3.81147e6	0.246599	0.123300	+	0.992369i	$0.460652\pi$
$752$	253696.	0.0163594
$753$	0	0
$754$	823642.	0.0527606
$755$	4.45969e6	0.284733
$756$	0	0
$757$	3.70093e6	0.234732	0.117366	−	0.993089i	$-0.462555\pi$
$757$	3.70093e6	0.234732	0.117366	+	0.993089i	$0.462555\pi$
$758$	7.17780e6	0.453752
$759$	0	0
$760$	142769.	0.00896604
$761$	2.27349e7	1.42309	0.711544	−	0.702642i	$-0.247996\pi$
$761$	2.27349e7	1.42309	0.711544	+	0.702642i	$0.247996\pi$
$762$	0	0
$763$	−3.21257e7	−1.99775
$764$	1.22768e7	0.760945
$765$	0	0
$766$	−4.51230e6	−0.277860
$767$	−5.20355e6	−0.319383
$768$	0	0
$769$	3.11247e7	1.89797	0.948985	−	0.315322i	$-0.102113\pi$
$769$	3.11247e7	1.89797	0.948985	+	0.315322i	$0.102113\pi$
$770$	−3.10189e6	−0.188538
$771$	0	0
$772$	−7.60961e6	−0.459536
$773$	−1.57774e7	−0.949702	−0.474851	−	0.880066i	$-0.657498\pi$
$773$	−1.57774e7	−0.949702	−0.474851	+	0.880066i	$0.657498\pi$
$774$	0	0
$775$	−1.17281e7	−0.701410
$776$	−4.94867e6	−0.295009
$777$	0	0
$778$	−1.68096e7	−0.995653
$779$	294026.	0.0173597
$780$	0	0
$781$	5.87166e6	0.344455
$782$	−4.95086e6	−0.289510
$783$	0	0
$784$	5.53826e6	0.321798
$785$	−1.49478e7	−0.865773
$786$	0	0
$787$	−9.81692e6	−0.564987	−0.282493	−	0.959269i	$-0.591162\pi$
$787$	−9.81692e6	−0.564987	−0.282493	+	0.959269i	$0.591162\pi$
$788$	−1.15559e7	−0.662964
$789$	0	0
$790$	7.09096e6	0.404238
$791$	9.73341e6	0.553126
$792$	0	0
$793$	1.85067e6	0.104507
$794$	−6.56460e6	−0.369536
$795$	0	0
$796$	1.03427e7	0.578562
$797$	1.97609e7	1.10195	0.550974	−	0.834523i	$-0.314256\pi$
$797$	1.97609e7	1.10195	0.550974	+	0.834523i	$0.314256\pi$
$798$	0	0
$799$	765796.	0.0424371
$800$	−2.10587e6	−0.116334
$801$	0	0
$802$	7.01917e6	0.385345
$803$	−2.51850e6	−0.137833
$804$	0	0
$805$	−1.02651e7	−0.558307
$806$	2.49070e6	0.135047
$807$	0	0
$808$	9.02079e6	0.486089
$809$	−2.26858e7	−1.21866	−0.609330	−	0.792917i	$-0.708562\pi$
$809$	−2.26858e7	−1.21866	−0.609330	+	0.792917i	$0.708562\pi$
$810$	0	0
$811$	2.91154e6	0.155443	0.0777215	−	0.996975i	$-0.475236\pi$
$811$	2.91154e6	0.155443	0.0777215	+	0.996975i	$0.475236\pi$
$812$	5.91601e6	0.314875
$813$	0	0
$814$	−597432.	−0.0316029
$815$	−1.15215e7	−0.607598
$816$	0	0
$817$	977018.	0.0512091
$818$	−1.17115e7	−0.611970
$819$	0	0
$820$	2.25331e6	0.117027
$821$	−2.48798e7	−1.28822	−0.644109	−	0.764934i	$-0.722772\pi$
$821$	−2.48798e7	−1.28822	−0.644109	+	0.764934i	$0.722772\pi$
$822$	0	0
$823$	−2.04870e7	−1.05434	−0.527169	−	0.849761i	$-0.676746\pi$
$823$	−2.04870e7	−1.05434	−0.527169	+	0.849761i	$0.676746\pi$
$824$	−6.97035e6	−0.357632
$825$	0	0
$826$	−3.73758e7	−1.90608
$827$	2.26717e7	1.15271	0.576356	−	0.817199i	$-0.304474\pi$
$827$	2.26717e7	1.15271	0.576356	+	0.817199i	$0.304474\pi$
$828$	0	0
$829$	9.92625e6	0.501648	0.250824	−	0.968033i	$-0.419298\pi$
$829$	9.92625e6	0.501648	0.250824	+	0.968033i	$0.419298\pi$
$830$	−1.36183e7	−0.686161
$831$	0	0
$832$	447225.	0.0223984
$833$	1.67175e7	0.834757
$834$	0	0
$835$	−6.12526e6	−0.304024
$836$	−132122.	−0.00653822
$837$	0	0
$838$	854502.	0.0420342
$839$	3.60157e7	1.76639	0.883196	−	0.469004i	$-0.155387\pi$
$839$	3.60157e7	1.76639	0.883196	+	0.469004i	$0.155387\pi$
$840$	0	0
$841$	−1.69546e7	−0.826606
$842$	−2.80985e7	−1.36585
$843$	0	0
$844$	4.93753e6	0.238591
$845$	−1.17470e7	−0.565962
$846$	0	0
$847$	2.87056e6	0.137486
$848$	2.50554e6	0.119650
$849$	0	0
$850$	−6.35668e6	−0.301775
$851$	−1.97708e6	−0.0935837
$852$	0	0
$853$	−2.68083e7	−1.26153	−0.630764	−	0.775975i	$-0.717258\pi$
$853$	−2.68083e7	−1.26153	−0.630764	+	0.775975i	$0.717258\pi$
$854$	1.32929e7	0.623699
$855$	0	0
$856$	−378868.	−0.0176727
$857$	3.21518e7	1.49539	0.747694	−	0.664044i	$-0.231161\pi$
$857$	3.21518e7	1.49539	0.747694	+	0.664044i	$0.231161\pi$
$858$	0	0
$859$	3.18182e7	1.47127	0.735635	−	0.677378i	$-0.236884\pi$
$859$	3.18182e7	1.47127	0.735635	+	0.677378i	$0.236884\pi$
$860$	7.48752e6	0.345217
$861$	0	0
$862$	2.11150e7	0.967882
$863$	−6.29142e6	−0.287556	−0.143778	−	0.989610i	$-0.545925\pi$
$863$	−6.29142e6	−0.287556	−0.143778	+	0.989610i	$0.545925\pi$
$864$	0	0
$865$	−4.17103e6	−0.189541
$866$	1.58964e7	0.720283
$867$	0	0
$868$	1.78901e7	0.805958
$869$	−6.56214e6	−0.294779
$870$	0	0
$871$	6.15998e6	0.275127
$872$	−1.04866e7	−0.467030
$873$	0	0
$874$	−437231.	−0.0193612
$875$	−3.32076e7	−1.46628
$876$	0	0
$877$	2.30304e7	1.01112	0.505560	−	0.862791i	$-0.331286\pi$
$877$	2.30304e7	1.01112	0.505560	+	0.862791i	$0.331286\pi$
$878$	−2.83786e7	−1.24238
$879$	0	0
$880$	−1.01254e6	−0.0440762
$881$	−4.54687e6	−0.197366	−0.0986831	−	0.995119i	$-0.531463\pi$
$881$	−4.54687e6	−0.197366	−0.0986831	+	0.995119i	$0.531463\pi$
$882$	0	0
$883$	4.42384e7	1.90940	0.954702	−	0.297564i	$-0.0961743\pi$
$883$	4.42384e7	1.90940	0.954702	+	0.297564i	$0.0961743\pi$
$884$	1.34997e6	0.0581025
$885$	0	0
$886$	3.06587e7	1.31211
$887$	9.27511e6	0.395831	0.197916	−	0.980219i	$-0.436583\pi$
$887$	9.27511e6	0.395831	0.197916	+	0.980219i	$0.436583\pi$
$888$	0	0
$889$	2.04281e7	0.866909
$890$	−1.41068e7	−0.596970
$891$	0	0
$892$	−1.17533e7	−0.494593
$893$	67630.5	0.00283801
$894$	0	0
$895$	−8.53071e6	−0.355982
$896$	3.21230e6	0.133674
$897$	0	0
$898$	−2.85981e7	−1.18344
$899$	1.07549e7	0.443822
$900$	0	0
$901$	7.56312e6	0.310377
$902$	−2.08527e6	−0.0853386
$903$	0	0
$904$	3.17723e6	0.129309
$905$	1.61726e7	0.656384
$906$	0	0
$907$	9.92137e6	0.400455	0.200227	−	0.979749i	$-0.435832\pi$
$907$	9.92137e6	0.400455	0.200227	+	0.979749i	$0.435832\pi$
$908$	−1.69947e7	−0.684068
$909$	0	0
$910$	2.79903e6	0.112048
$911$	7.60600e6	0.303641	0.151821	−	0.988408i	$-0.451486\pi$
$911$	7.60600e6	0.303641	0.151821	+	0.988408i	$0.451486\pi$
$912$	0	0
$913$	1.26026e7	0.500362
$914$	−1.48517e7	−0.588045
$915$	0	0
$916$	6.43860e6	0.253544
$917$	−5.90566e6	−0.231924
$918$	0	0
$919$	3.70300e7	1.44632	0.723160	−	0.690680i	$-0.242689\pi$
$919$	3.70300e7	1.44632	0.723160	+	0.690680i	$0.242689\pi$
$920$	−3.35078e6	−0.130520
$921$	0	0
$922$	8.51595e6	0.329918
$923$	−5.29835e6	−0.204709
$924$	0	0
$925$	−2.53848e6	−0.0975483
$926$	1.51401e7	0.580231
$927$	0	0
$928$	1.93113e6	0.0736110
$929$	9.24519e6	0.351460	0.175730	−	0.984438i	$-0.443771\pi$
$929$	9.24519e6	0.351460	0.175730	+	0.984438i	$0.443771\pi$
$930$	0	0
$931$	1.47640e6	0.0558250
$932$	−1.49937e7	−0.565418
$933$	0	0
$934$	2.70906e7	1.01613
$935$	−3.05640e6	−0.114335
$936$	0	0
$937$	3.92084e7	1.45892	0.729459	−	0.684025i	$-0.239772\pi$
$937$	3.92084e7	1.45892	0.729459	+	0.684025i	$0.239772\pi$
$938$	4.42455e7	1.64196
$939$	0	0
$940$	518297.	0.0191319
$941$	−1.20478e7	−0.443539	−0.221770	−	0.975099i	$-0.571183\pi$
$941$	−1.20478e7	−0.443539	−0.221770	+	0.975099i	$0.571183\pi$
$942$	0	0
$943$	−6.90077e6	−0.252708
$944$	−1.22004e7	−0.445599
$945$	0	0
$946$	−6.92912e6	−0.251739
$947$	−1.44556e7	−0.523796	−0.261898	−	0.965096i	$-0.584348\pi$
$947$	−1.44556e7	−0.523796	−0.261898	+	0.965096i	$0.584348\pi$
$948$	0	0
$949$	2.27260e6	0.0819139
$950$	−561385.	−0.0201814
$951$	0	0
$952$	9.69652e6	0.346756
$953$	4.64014e7	1.65500	0.827502	−	0.561463i	$-0.189761\pi$
$953$	4.64014e7	1.65500	0.827502	+	0.561463i	$0.189761\pi$
$954$	0	0
$955$	2.50814e7	0.889904
$956$	2.50403e7	0.886126
$957$	0	0
$958$	−2.48898e7	−0.876210
$959$	−2.50764e7	−0.880477
$960$	0	0
$961$	3.89385e6	0.136010
$962$	539099.	0.0187815
$963$	0	0
$964$	861937.	0.0298733
$965$	−1.55463e7	−0.537415
$966$	0	0
$967$	3.19191e6	0.109770	0.0548850	−	0.998493i	$-0.482521\pi$
$967$	3.19191e6	0.109770	0.0548850	+	0.998493i	$0.482521\pi$
$968$	937024.	0.0321412
$969$	0	0
$970$	−1.01101e7	−0.345004
$971$	−1.10936e7	−0.377594	−0.188797	−	0.982016i	$-0.560459\pi$
$971$	−1.10936e7	−0.377594	−0.188797	+	0.982016i	$0.560459\pi$
$972$	0	0
$973$	−3.53296e7	−1.19635
$974$	3.16883e7	1.07029
$975$	0	0
$976$	4.33914e6	0.145807
$977$	1.83858e7	0.616236	0.308118	−	0.951348i	$-0.400301\pi$
$977$	1.83858e7	0.616236	0.308118	+	0.951348i	$0.400301\pi$
$978$	0	0
$979$	1.30547e7	0.435322
$980$	1.13146e7	0.376334
$981$	0	0
$982$	5.26017e6	0.174069
$983$	−2.74603e7	−0.906405	−0.453202	−	0.891408i	$-0.649718\pi$
$983$	−2.74603e7	−0.906405	−0.453202	+	0.891408i	$0.649718\pi$
$984$	0	0
$985$	−2.36086e7	−0.775318
$986$	5.82924e6	0.190950
$987$	0	0
$988$	119222.	0.00388564
$989$	−2.29305e7	−0.745458
$990$	0	0
$991$	1.05596e7	0.341556	0.170778	−	0.985309i	$-0.445372\pi$
$991$	1.05596e7	0.341556	0.170778	+	0.985309i	$0.445372\pi$
$992$	5.83976e6	0.188415
$993$	0	0
$994$	−3.80567e7	−1.22170
$995$	2.11299e7	0.676612
$996$	0	0
$997$	2.73338e7	0.870886	0.435443	−	0.900216i	$-0.356592\pi$
$997$	2.73338e7	0.870886	0.435443	+	0.900216i	$0.356592\pi$
$998$	4.41218e6	0.140225
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	198.6.a.m.1.2	yes	2
3.2	odd	2		198.6.a.l.1.1	✓	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
198.6.a.l.1.1	✓	2	3.2	odd	2
198.6.a.m.1.2	yes	2	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 198 = 2 \cdot 3^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	198.a (trivial)

\(f(q)\)	\(=\)	\(q+4.00000 q^{2} +16.0000 q^{4} +32.6878 q^{5} +196.063 q^{7} +64.0000 q^{8} +O(q^{10})\)	\(q+4.00000 q^{2} +16.0000 q^{4} +32.6878 q^{5} +196.063 q^{7} +64.0000 q^{8} +130.751 q^{10} -121.000 q^{11} +109.186 q^{13} +784.253 q^{14} +256.000 q^{16} +772.751 q^{17} +68.2448 q^{19} +523.004 q^{20} -484.000 q^{22} -1601.70 q^{23} -2056.51 q^{25} +436.743 q^{26} +3137.01 q^{28} +1885.87 q^{29} +5702.89 q^{31} +1024.00 q^{32} +3091.00 q^{34} +6408.87 q^{35} +1234.36 q^{37} +272.979 q^{38} +2092.02 q^{40} +4308.40 q^{41} +14316.4 q^{43} -1936.00 q^{44} -6406.80 q^{46} +990.999 q^{47} +21633.8 q^{49} -8226.04 q^{50} +1746.97 q^{52} +9787.27 q^{53} -3955.22 q^{55} +12548.0 q^{56} +7543.49 q^{58} -47657.8 q^{59} +16949.8 q^{61} +22811.6 q^{62} +4096.00 q^{64} +3569.04 q^{65} +56417.4 q^{67} +12364.0 q^{68} +25635.5 q^{70} -48526.1 q^{71} +20814.1 q^{73} +4937.45 q^{74} +1091.92 q^{76} -23723.7 q^{77} +54232.5 q^{79} +8368.07 q^{80} +17233.6 q^{82} -104154. q^{83} +25259.5 q^{85} +57265.5 q^{86} -7744.00 q^{88} -107890. q^{89} +21407.3 q^{91} -25627.2 q^{92} +3964.00 q^{94} +2230.77 q^{95} -77323.0 q^{97} +86535.2 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 8 q^{2} + 32 q^{4} - 40 q^{5} + 76 q^{7} + 128 q^{8}+O(q^{10}) \) 2 * q + 8 * q^2 + 32 * q^4 - 40 * q^5 + 76 * q^7 + 128 * q^8	\( 2 q + 8 q^{2} + 32 q^{4} - 40 q^{5} + 76 q^{7} + 128 q^{8} - 160 q^{10} - 242 q^{11} + 956 q^{13} + 304 q^{14} + 512 q^{16} + 1124 q^{17} + 2244 q^{19} - 640 q^{20} - 968 q^{22} + 1960 q^{23} + 102 q^{25} + 3824 q^{26} + 1216 q^{28} + 4404 q^{29} + 3608 q^{31} + 2048 q^{32} + 4496 q^{34} + 15136 q^{35} + 7316 q^{37} + 8976 q^{38} - 2560 q^{40} - 3396 q^{41} + 30108 q^{43} - 3872 q^{44} + 7840 q^{46} - 22992 q^{47} + 19242 q^{49} + 408 q^{50} + 15296 q^{52} - 13408 q^{53} + 4840 q^{55} + 4864 q^{56} + 17616 q^{58} - 6168 q^{59} + 58452 q^{61} + 14432 q^{62} + 8192 q^{64} - 57984 q^{65} + 39072 q^{67} + 17984 q^{68} + 60544 q^{70} - 36672 q^{71} + 107804 q^{73} + 29264 q^{74} + 35904 q^{76} - 9196 q^{77} + 65788 q^{79} - 10240 q^{80} - 13584 q^{82} - 131384 q^{83} - 272 q^{85} + 120432 q^{86} - 15488 q^{88} - 97760 q^{89} - 80264 q^{91} + 31360 q^{92} - 91968 q^{94} - 155920 q^{95} - 36204 q^{97} + 76968 q^{98}+O(q^{100}) \) 2 * q + 8 * q^2 + 32 * q^4 - 40 * q^5 + 76 * q^7 + 128 * q^8 - 160 * q^10 - 242 * q^11 + 956 * q^13 + 304 * q^14 + 512 * q^16 + 1124 * q^17 + 2244 * q^19 - 640 * q^20 - 968 * q^22 + 1960 * q^23 + 102 * q^25 + 3824 * q^26 + 1216 * q^28 + 4404 * q^29 + 3608 * q^31 + 2048 * q^32 + 4496 * q^34 + 15136 * q^35 + 7316 * q^37 + 8976 * q^38 - 2560 * q^40 - 3396 * q^41 + 30108 * q^43 - 3872 * q^44 + 7840 * q^46 - 22992 * q^47 + 19242 * q^49 + 408 * q^50 + 15296 * q^52 - 13408 * q^53 + 4840 * q^55 + 4864 * q^56 + 17616 * q^58 - 6168 * q^59 + 58452 * q^61 + 14432 * q^62 + 8192 * q^64 - 57984 * q^65 + 39072 * q^67 + 17984 * q^68 + 60544 * q^70 - 36672 * q^71 + 107804 * q^73 + 29264 * q^74 + 35904 * q^76 - 9196 * q^77 + 65788 * q^79 - 10240 * q^80 - 13584 * q^82 - 131384 * q^83 - 272 * q^85 + 120432 * q^86 - 15488 * q^88 - 97760 * q^89 - 80264 * q^91 + 31360 * q^92 - 91968 * q^94 - 155920 * q^95 - 36204 * q^97 + 76968 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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