LMFDB - Embedded newform 2178.4.a.bt.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2178,4,Mod(1,2178)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2178, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("2178.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 2178 = 2 \cdot 3^{2} \cdot 11^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	2178.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:	$128.506159993$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.978025.2
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{3} - 99x^{2} + 100x + 2420 $ x^4 - 2x^3 - 99x^2 + 100*x + 2420
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 11 $
Twist minimal:	no (minimal twist has level 22)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-5.92695$ of defining polynomial
Character	$\chi$	$=$	2178.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.00000 q^{2} +4.00000 q^{4} +8.06215 q^{5} +26.0792 q^{7} -8.00000 q^{8} +O(q^{10})$	$q-2.00000 q^{2} +4.00000 q^{4} +8.06215 q^{5} +26.0792 q^{7} -8.00000 q^{8} -16.1243 q^{10} +3.26382 q^{13} -52.1583 q^{14} +16.0000 q^{16} -20.8717 q^{17} +125.970 q^{19} +32.2486 q^{20} -97.8394 q^{23} -60.0018 q^{25} -6.52764 q^{26} +104.317 q^{28} -263.834 q^{29} +199.364 q^{31} -32.0000 q^{32} +41.7433 q^{34} +210.254 q^{35} +365.643 q^{37} -251.939 q^{38} -64.4972 q^{40} +273.732 q^{41} -388.059 q^{43} +195.679 q^{46} +51.8541 q^{47} +337.122 q^{49} +120.004 q^{50} +13.0553 q^{52} +412.524 q^{53} -208.633 q^{56} +527.669 q^{58} -26.2834 q^{59} -164.149 q^{61} -398.727 q^{62} +64.0000 q^{64} +26.3134 q^{65} +276.961 q^{67} -83.4866 q^{68} -420.508 q^{70} +516.930 q^{71} +241.565 q^{73} -731.287 q^{74} +503.879 q^{76} +273.120 q^{79} +128.994 q^{80} -547.465 q^{82} +72.5940 q^{83} -168.270 q^{85} +776.118 q^{86} +1194.73 q^{89} +85.1177 q^{91} -391.358 q^{92} -103.708 q^{94} +1015.59 q^{95} +1463.63 q^{97} -674.244 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 8 q^{2} + 16 q^{4} - 25 q^{5} + 3 q^{7} - 32 q^{8}+O(q^{10}) $ 4 * q - 8 * q^2 + 16 * q^4 - 25 * q^5 + 3 * q^7 - 32 * q^8	$ 4 q - 8 q^{2} + 16 q^{4} - 25 q^{5} + 3 q^{7} - 32 q^{8} + 50 q^{10} - 41 q^{13} - 6 q^{14} + 64 q^{16} - 52 q^{17} + 16 q^{19} - 100 q^{20} - 314 q^{23} - 21 q^{25} + 82 q^{26} + 12 q^{28} - 561 q^{29} + 199 q^{31} - 128 q^{32} + 104 q^{34} + 714 q^{35} + 357 q^{37} - 32 q^{38} + 200 q^{40} - 32 q^{41} - 721 q^{43} + 628 q^{46} - 403 q^{47} + 823 q^{49} + 42 q^{50} - 164 q^{52} + 133 q^{53} - 24 q^{56} + 1122 q^{58} - 1016 q^{59} - 919 q^{61} - 398 q^{62} + 256 q^{64} - 69 q^{65} + 289 q^{67} - 208 q^{68} - 1428 q^{70} + 1205 q^{71} - 1234 q^{73} - 714 q^{74} + 64 q^{76} - 603 q^{79} - 400 q^{80} + 64 q^{82} - 1514 q^{83} + 717 q^{85} + 1442 q^{86} + 1101 q^{89} - 2306 q^{91} - 1256 q^{92} + 806 q^{94} + 1766 q^{95} + 2116 q^{97} - 1646 q^{98}+O(q^{100}) $ 4 * q - 8 * q^2 + 16 * q^4 - 25 * q^5 + 3 * q^7 - 32 * q^8 + 50 * q^10 - 41 * q^13 - 6 * q^14 + 64 * q^16 - 52 * q^17 + 16 * q^19 - 100 * q^20 - 314 * q^23 - 21 * q^25 + 82 * q^26 + 12 * q^28 - 561 * q^29 + 199 * q^31 - 128 * q^32 + 104 * q^34 + 714 * q^35 + 357 * q^37 - 32 * q^38 + 200 * q^40 - 32 * q^41 - 721 * q^43 + 628 * q^46 - 403 * q^47 + 823 * q^49 + 42 * q^50 - 164 * q^52 + 133 * q^53 - 24 * q^56 + 1122 * q^58 - 1016 * q^59 - 919 * q^61 - 398 * q^62 + 256 * q^64 - 69 * q^65 + 289 * q^67 - 208 * q^68 - 1428 * q^70 + 1205 * q^71 - 1234 * q^73 - 714 * q^74 + 64 * q^76 - 603 * q^79 - 400 * q^80 + 64 * q^82 - 1514 * q^83 + 717 * q^85 + 1442 * q^86 + 1101 * q^89 - 2306 * q^91 - 1256 * q^92 + 806 * q^94 + 1766 * q^95 + 2116 * q^97 - 1646 * 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 $

$2$	−2.00000	−0.707107
$2$	−2.00000	−0.707107
$3$	0	0
$3$	0	0
$4$	4.00000	0.500000
$5$	8.06215	0.721100	0.360550	−	0.932740i	$-0.382589\pi$
$5$	8.06215	0.721100	0.360550	+	0.932740i	$0.382589\pi$
$6$	0	0
$7$	26.0792	1.40814	0.704071	−	0.710130i	$-0.251364\pi$
$7$	26.0792	1.40814	0.704071	+	0.710130i	$0.251364\pi$
$8$	−8.00000	−0.353553
$9$	0	0
$10$	−16.1243	−0.509895
$11$	0	0
$11$	0	0
$12$	0	0
$13$	3.26382	0.0696324	0.0348162	−	0.999394i	$-0.488915\pi$
$13$	3.26382	0.0696324	0.0348162	+	0.999394i	$0.488915\pi$
$14$	−52.1583	−0.995707
$15$	0	0
$16$	16.0000	0.250000
$17$	−20.8717	−0.297772	−0.148886	−	0.988854i	$-0.547569\pi$
$17$	−20.8717	−0.297772	−0.148886	+	0.988854i	$0.547569\pi$
$18$	0	0
$19$	125.970	1.52102	0.760511	−	0.649325i	$-0.224948\pi$
$19$	125.970	1.52102	0.760511	+	0.649325i	$0.224948\pi$
$20$	32.2486	0.360550
$21$	0	0
$22$	0	0
$23$	−97.8394	−0.886997	−0.443498	−	0.896275i	$-0.646263\pi$
$23$	−97.8394	−0.886997	−0.443498	+	0.896275i	$0.646263\pi$
$24$	0	0
$25$	−60.0018	−0.480014
$26$	−6.52764	−0.0492375
$27$	0	0
$28$	104.317	0.704071
$29$	−263.834	−1.68941	−0.844703	−	0.535235i	$-0.820223\pi$
$29$	−263.834	−1.68941	−0.844703	+	0.535235i	$0.820223\pi$
$30$	0	0
$31$	199.364	1.15506	0.577528	−	0.816371i	$-0.304017\pi$
$31$	199.364	1.15506	0.577528	+	0.816371i	$0.304017\pi$
$32$	−32.0000	−0.176777
$33$	0	0
$34$	41.7433	0.210557
$35$	210.254	1.01541
$36$	0	0
$37$	365.643	1.62463	0.812316	−	0.583217i	$-0.198206\pi$
$37$	365.643	1.62463	0.812316	+	0.583217i	$0.198206\pi$
$38$	−251.939	−1.07553
$39$	0	0
$40$	−64.4972	−0.254947
$41$	273.732	1.04268	0.521339	−	0.853350i	$-0.325433\pi$
$41$	273.732	1.04268	0.521339	+	0.853350i	$0.325433\pi$
$42$	0	0
$43$	−388.059	−1.37624	−0.688121	−	0.725596i	$-0.741564\pi$
$43$	−388.059	−1.37624	−0.688121	+	0.725596i	$0.741564\pi$
$44$	0	0
$45$	0	0
$46$	195.679	0.627201
$47$	51.8541	0.160930	0.0804649	−	0.996757i	$-0.474359\pi$
$47$	51.8541	0.160930	0.0804649	+	0.996757i	$0.474359\pi$
$48$	0	0
$49$	337.122	0.982863
$50$	120.004	0.339421
$51$	0	0
$52$	13.0553	0.0348162
$53$	412.524	1.06914	0.534572	−	0.845123i	$-0.320473\pi$
$53$	412.524	1.06914	0.534572	+	0.845123i	$0.320473\pi$
$54$	0	0
$55$	0	0
$56$	−208.633	−0.497853
$57$	0	0
$58$	527.669	1.19459
$59$	−26.2834	−0.0579967	−0.0289984	−	0.999579i	$-0.509232\pi$
$59$	−26.2834	−0.0579967	−0.0289984	+	0.999579i	$0.509232\pi$
$60$	0	0
$61$	−164.149	−0.344543	−0.172272	−	0.985050i	$-0.555111\pi$
$61$	−164.149	−0.344543	−0.172272	+	0.985050i	$0.555111\pi$
$62$	−398.727	−0.816748
$63$	0	0
$64$	64.0000	0.125000
$65$	26.3134	0.0502119
$66$	0	0
$67$	276.961	0.505017	0.252508	−	0.967595i	$-0.418744\pi$
$67$	276.961	0.505017	0.252508	+	0.967595i	$0.418744\pi$
$68$	−83.4866	−0.148886
$69$	0	0
$70$	−420.508	−0.718004
$71$	516.930	0.864060	0.432030	−	0.901859i	$-0.357797\pi$
$71$	516.930	0.864060	0.432030	+	0.901859i	$0.357797\pi$
$72$	0	0
$73$	241.565	0.387302	0.193651	−	0.981071i	$-0.437967\pi$
$73$	241.565	0.387302	0.193651	+	0.981071i	$0.437967\pi$
$74$	−731.287	−1.14879
$75$	0	0
$76$	503.879	0.760511
$77$	0	0
$78$	0	0
$79$	273.120	0.388967	0.194483	−	0.980906i	$-0.437697\pi$
$79$	273.120	0.388967	0.194483	+	0.980906i	$0.437697\pi$
$80$	128.994	0.180275
$81$	0	0
$82$	−547.465	−0.737285
$83$	72.5940	0.0960027	0.0480014	−	0.998847i	$-0.484715\pi$
$83$	72.5940	0.0960027	0.0480014	+	0.998847i	$0.484715\pi$
$84$	0	0
$85$	−168.270	−0.214723
$86$	776.118	0.973151
$87$	0	0
$88$	0	0
$89$	1194.73	1.42294	0.711470	−	0.702717i	$-0.248030\pi$
$89$	1194.73	1.42294	0.711470	+	0.702717i	$0.248030\pi$
$90$	0	0
$91$	85.1177	0.0980522
$92$	−391.358	−0.443498
$93$	0	0
$94$	−103.708	−0.113795
$95$	1015.59	1.09681
$96$	0	0
$97$	1463.63	1.53205	0.766026	−	0.642810i	$-0.222232\pi$
$97$	1463.63	1.53205	0.766026	+	0.642810i	$0.222232\pi$
$98$	−674.244	−0.694989
$99$	0	0
$100$	−240.007	−0.240007
$101$	−900.093	−0.886759	−0.443379	−	0.896334i	$-0.646221\pi$
$101$	−900.093	−0.886759	−0.443379	+	0.896334i	$0.646221\pi$
$102$	0	0
$103$	−417.125	−0.399035	−0.199517	−	0.979894i	$-0.563937\pi$
$103$	−417.125	−0.399035	−0.199517	+	0.979894i	$0.563937\pi$
$104$	−26.1106	−0.0246188
$105$	0	0
$106$	−825.049	−0.755998
$107$	1080.39	0.976120	0.488060	−	0.872810i	$-0.337705\pi$
$107$	1080.39	0.976120	0.488060	+	0.872810i	$0.337705\pi$
$108$	0	0
$109$	1472.08	1.29358	0.646789	−	0.762669i	$-0.276112\pi$
$109$	1472.08	1.29358	0.646789	+	0.762669i	$0.276112\pi$
$110$	0	0
$111$	0	0
$112$	417.266	0.352035
$113$	126.481	0.105295	0.0526473	−	0.998613i	$-0.483234\pi$
$113$	126.481	0.105295	0.0526473	+	0.998613i	$0.483234\pi$
$114$	0	0
$115$	−788.796	−0.639614
$116$	−1055.34	−0.844703
$117$	0	0
$118$	52.5668	0.0410099
$119$	−544.315	−0.419305
$120$	0	0
$121$	0	0
$122$	328.298	0.243629
$123$	0	0
$124$	797.454	0.577528
$125$	−1491.51	−1.06724
$126$	0	0
$127$	−1063.54	−0.743105	−0.371552	−	0.928412i	$-0.621174\pi$
$127$	−1063.54	−0.743105	−0.371552	+	0.928412i	$0.621174\pi$
$128$	−128.000	−0.0883883
$129$	0	0
$130$	−52.6268	−0.0355052
$131$	−1525.04	−1.01713	−0.508563	−	0.861025i	$-0.669823\pi$
$131$	−1525.04	−1.01713	−0.508563	+	0.861025i	$0.669823\pi$
$132$	0	0
$133$	3285.18	2.14182
$134$	−553.921	−0.357101
$135$	0	0
$136$	166.973	0.105278
$137$	2078.83	1.29640	0.648199	−	0.761471i	$-0.275522\pi$
$137$	2078.83	1.29640	0.648199	+	0.761471i	$0.275522\pi$
$138$	0	0
$139$	−1507.45	−0.919855	−0.459927	−	0.887957i	$-0.652125\pi$
$139$	−1507.45	−0.919855	−0.459927	+	0.887957i	$0.652125\pi$
$140$	841.016	0.507706
$141$	0	0
$142$	−1033.86	−0.610983
$143$	0	0
$144$	0	0
$145$	−2127.07	−1.21823
$146$	−483.130	−0.273864
$147$	0	0
$148$	1462.57	0.812316
$149$	−429.848	−0.236339	−0.118169	−	0.992993i	$-0.537703\pi$
$149$	−429.848	−0.236339	−0.118169	+	0.992993i	$0.537703\pi$
$150$	0	0
$151$	−891.682	−0.480556	−0.240278	−	0.970704i	$-0.577239\pi$
$151$	−891.682	−0.480556	−0.240278	+	0.970704i	$0.577239\pi$
$152$	−1007.76	−0.537763
$153$	0	0
$154$	0	0
$155$	1607.30	0.832912
$156$	0	0
$157$	541.031	0.275025	0.137513	−	0.990500i	$-0.456089\pi$
$157$	541.031	0.275025	0.137513	+	0.990500i	$0.456089\pi$
$158$	−546.240	−0.275041
$159$	0	0
$160$	−257.989	−0.127474
$161$	−2551.57	−1.24902
$162$	0	0
$163$	−2927.59	−1.40679	−0.703395	−	0.710799i	$-0.748334\pi$
$163$	−2927.59	−1.40679	−0.703395	+	0.710799i	$0.748334\pi$
$164$	1094.93	0.521339
$165$	0	0
$166$	−145.188	−0.0678842
$167$	1740.23	0.806366	0.403183	−	0.915119i	$-0.367904\pi$
$167$	1740.23	0.806366	0.403183	+	0.915119i	$0.367904\pi$
$168$	0	0
$169$	−2186.35	−0.995151
$170$	336.541	0.151832
$171$	0	0
$172$	−1552.24	−0.688121
$173$	−290.801	−0.127799	−0.0638995	−	0.997956i	$-0.520354\pi$
$173$	−290.801	−0.127799	−0.0638995	+	0.997956i	$0.520354\pi$
$174$	0	0
$175$	−1564.80	−0.675928
$176$	0	0
$177$	0	0
$178$	−2389.47	−1.00617
$179$	−2600.32	−1.08580	−0.542898	−	0.839799i	$-0.682673\pi$
$179$	−2600.32	−1.08580	−0.542898	+	0.839799i	$0.682673\pi$
$180$	0	0
$181$	1855.83	0.762114	0.381057	−	0.924552i	$-0.375560\pi$
$181$	1855.83	0.762114	0.381057	+	0.924552i	$0.375560\pi$
$182$	−170.235	−0.0693334
$183$	0	0
$184$	782.715	0.313601
$185$	2947.87	1.17152
$186$	0	0
$187$	0	0
$188$	207.417	0.0804649
$189$	0	0
$190$	−2031.17	−0.775562
$191$	−1532.16	−0.580434	−0.290217	−	0.956961i	$-0.593727\pi$
$191$	−1532.16	−0.580434	−0.290217	+	0.956961i	$0.593727\pi$
$192$	0	0
$193$	−1051.58	−0.392197	−0.196099	−	0.980584i	$-0.562827\pi$
$193$	−1051.58	−0.392197	−0.196099	+	0.980584i	$0.562827\pi$
$194$	−2927.26	−1.08332
$195$	0	0
$196$	1348.49	0.491432
$197$	−1577.77	−0.570616	−0.285308	−	0.958436i	$-0.592096\pi$
$197$	−1577.77	−0.570616	−0.285308	+	0.958436i	$0.592096\pi$
$198$	0	0
$199$	3760.53	1.33958	0.669791	−	0.742550i	$-0.266384\pi$
$199$	3760.53	1.33958	0.669791	+	0.742550i	$0.266384\pi$
$200$	480.014	0.169711
$201$	0	0
$202$	1800.19	0.627033
$203$	−6880.57	−2.37892
$204$	0	0
$205$	2206.87	0.751876
$206$	834.251	0.282160
$207$	0	0
$208$	52.2211	0.0174081
$209$	0	0
$210$	0	0
$211$	−1420.58	−0.463492	−0.231746	−	0.972776i	$-0.574444\pi$
$211$	−1420.58	−0.463492	−0.231746	+	0.972776i	$0.574444\pi$
$212$	1650.10	0.534572
$213$	0	0
$214$	−2160.77	−0.690221
$215$	−3128.59	−0.992409
$216$	0	0
$217$	5199.23	1.62648
$218$	−2944.16	−0.914697
$219$	0	0
$220$	0	0
$221$	−68.1213	−0.0207346
$222$	0	0
$223$	5256.47	1.57847	0.789235	−	0.614091i	$-0.210477\pi$
$223$	5256.47	1.57847	0.789235	+	0.614091i	$0.210477\pi$
$224$	−834.533	−0.248927
$225$	0	0
$226$	−252.961	−0.0744545
$227$	2786.28	0.814677	0.407338	−	0.913277i	$-0.366457\pi$
$227$	2786.28	0.814677	0.407338	+	0.913277i	$0.366457\pi$
$228$	0	0
$229$	4482.59	1.29353	0.646763	−	0.762691i	$-0.276122\pi$
$229$	4482.59	1.29353	0.646763	+	0.762691i	$0.276122\pi$
$230$	1577.59	0.452275
$231$	0	0
$232$	2110.67	0.597296
$233$	315.613	0.0887404	0.0443702	−	0.999015i	$-0.485872\pi$
$233$	315.613	0.0887404	0.0443702	+	0.999015i	$0.485872\pi$
$234$	0	0
$235$	418.056	0.116047
$236$	−105.134	−0.0289984
$237$	0	0
$238$	1088.63	0.296493
$239$	806.382	0.218245	0.109122	−	0.994028i	$-0.465196\pi$
$239$	806.382	0.218245	0.109122	+	0.994028i	$0.465196\pi$
$240$	0	0
$241$	1009.91	0.269935	0.134967	−	0.990850i	$-0.456907\pi$
$241$	1009.91	0.269935	0.134967	+	0.990850i	$0.456907\pi$
$242$	0	0
$243$	0	0
$244$	−656.596	−0.172272
$245$	2717.93	0.708743
$246$	0	0
$247$	411.142	0.105912
$248$	−1594.91	−0.408374
$249$	0	0
$250$	2983.02	0.754652
$251$	3187.25	0.801503	0.400751	−	0.916187i	$-0.368749\pi$
$251$	3187.25	0.801503	0.400751	+	0.916187i	$0.368749\pi$
$252$	0	0
$253$	0	0
$254$	2127.09	0.525455
$255$	0	0
$256$	256.000	0.0625000
$257$	−2543.39	−0.617325	−0.308662	−	0.951172i	$-0.599881\pi$
$257$	−2543.39	−0.617325	−0.308662	+	0.951172i	$0.599881\pi$
$258$	0	0
$259$	9535.67	2.28771
$260$	105.254	0.0251060
$261$	0	0
$262$	3050.09	0.719217
$263$	2992.29	0.701568	0.350784	−	0.936456i	$-0.385915\pi$
$263$	2992.29	0.701568	0.350784	+	0.936456i	$0.385915\pi$
$264$	0	0
$265$	3325.83	0.770960
$266$	−6570.36	−1.51449
$267$	0	0
$268$	1107.84	0.252508
$269$	821.328	0.186161	0.0930804	−	0.995659i	$-0.470329\pi$
$269$	821.328	0.186161	0.0930804	+	0.995659i	$0.470329\pi$
$270$	0	0
$271$	−6439.74	−1.44349	−0.721746	−	0.692158i	$-0.756660\pi$
$271$	−6439.74	−1.44349	−0.721746	+	0.692158i	$0.756660\pi$
$272$	−333.947	−0.0744430
$273$	0	0
$274$	−4157.66	−0.916692
$275$	0	0
$276$	0	0
$277$	−514.466	−0.111593	−0.0557965	−	0.998442i	$-0.517770\pi$
$277$	−514.466	−0.111593	−0.0557965	+	0.998442i	$0.517770\pi$
$278$	3014.89	0.650436
$279$	0	0
$280$	−1682.03	−0.359002
$281$	7758.68	1.64713	0.823566	−	0.567220i	$-0.191981\pi$
$281$	7758.68	1.64713	0.823566	+	0.567220i	$0.191981\pi$
$282$	0	0
$283$	5847.37	1.22823	0.614116	−	0.789215i	$-0.289513\pi$
$283$	5847.37	1.22823	0.614116	+	0.789215i	$0.289513\pi$
$284$	2067.72	0.432030
$285$	0	0
$286$	0	0
$287$	7138.71	1.46824
$288$	0	0
$289$	−4477.37	−0.911332
$290$	4254.14	0.861420
$291$	0	0
$292$	966.259	0.193651
$293$	−8074.49	−1.60996	−0.804978	−	0.593305i	$-0.797823\pi$
$293$	−8074.49	−1.60996	−0.804978	+	0.593305i	$0.797823\pi$
$294$	0	0
$295$	−211.901	−0.0418215
$296$	−2925.15	−0.574394
$297$	0	0
$298$	859.695	0.167117
$299$	−319.330	−0.0617637
$300$	0	0
$301$	−10120.2	−1.93794
$302$	1783.36	0.339805
$303$	0	0
$304$	2015.51	0.380256
$305$	−1323.39	−0.248450
$306$	0	0
$307$	4210.64	0.782781	0.391391	−	0.920225i	$-0.371994\pi$
$307$	4210.64	0.782781	0.391391	+	0.920225i	$0.371994\pi$
$308$	0	0
$309$	0	0
$310$	−3214.60	−0.588958
$311$	1318.85	0.240468	0.120234	−	0.992746i	$-0.461636\pi$
$311$	1318.85	0.240468	0.120234	+	0.992746i	$0.461636\pi$
$312$	0	0
$313$	−4206.15	−0.759571	−0.379785	−	0.925075i	$-0.624002\pi$
$313$	−4206.15	−0.759571	−0.379785	+	0.925075i	$0.624002\pi$
$314$	−1082.06	−0.194472
$315$	0	0
$316$	1092.48	0.194483
$317$	2463.10	0.436409	0.218204	−	0.975903i	$-0.429980\pi$
$317$	2463.10	0.436409	0.218204	+	0.975903i	$0.429980\pi$
$318$	0	0
$319$	0	0
$320$	515.977	0.0901376
$321$	0	0
$322$	5103.14	0.883189
$323$	−2629.20	−0.452918
$324$	0	0
$325$	−195.835	−0.0334245
$326$	5855.19	0.994751
$327$	0	0
$328$	−2189.86	−0.368642
$329$	1352.31	0.226612
$330$	0	0
$331$	−3332.42	−0.553373	−0.276687	−	0.960960i	$-0.589236\pi$
$331$	−3332.42	−0.553373	−0.276687	+	0.960960i	$0.589236\pi$
$332$	290.376	0.0480014
$333$	0	0
$334$	−3480.46	−0.570187
$335$	2232.90	0.364168
$336$	0	0
$337$	−8323.48	−1.34543	−0.672714	−	0.739903i	$-0.734871\pi$
$337$	−8323.48	−1.34543	−0.672714	+	0.739903i	$0.734871\pi$
$338$	4372.69	0.703678
$339$	0	0
$340$	−673.082	−0.107362
$341$	0	0
$342$	0	0
$343$	−153.291	−0.0241311
$344$	3104.47	0.486575
$345$	0	0
$346$	581.603	0.0903675
$347$	3622.50	0.560420	0.280210	−	0.959939i	$-0.409596\pi$
$347$	3622.50	0.560420	0.280210	+	0.959939i	$0.409596\pi$
$348$	0	0
$349$	−2086.66	−0.320047	−0.160023	−	0.987113i	$-0.551157\pi$
$349$	−2086.66	−0.320047	−0.160023	+	0.987113i	$0.551157\pi$
$350$	3129.59	0.477953
$351$	0	0
$352$	0	0
$353$	−7582.15	−1.14322	−0.571611	−	0.820525i	$-0.693681\pi$
$353$	−7582.15	−1.14322	−0.571611	+	0.820525i	$0.693681\pi$
$354$	0	0
$355$	4167.56	0.623074
$356$	4778.94	0.711470
$357$	0	0
$358$	5200.65	0.767773
$359$	4532.99	0.666413	0.333206	−	0.942854i	$-0.391869\pi$
$359$	4532.99	0.666413	0.333206	+	0.942854i	$0.391869\pi$
$360$	0	0
$361$	9009.36	1.31351
$362$	−3711.66	−0.538896
$363$	0	0
$364$	340.471	0.0490261
$365$	1947.53	0.279283
$366$	0	0
$367$	−2893.05	−0.411488	−0.205744	−	0.978606i	$-0.565961\pi$
$367$	−2893.05	−0.411488	−0.205744	+	0.978606i	$0.565961\pi$
$368$	−1565.43	−0.221749
$369$	0	0
$370$	−5895.74	−0.828392
$371$	10758.3	1.50550
$372$	0	0
$373$	3389.46	0.470508	0.235254	−	0.971934i	$-0.424408\pi$
$373$	3389.46	0.470508	0.235254	+	0.971934i	$0.424408\pi$
$374$	0	0
$375$	0	0
$376$	−414.833	−0.0568973
$377$	−861.108	−0.117637
$378$	0	0
$379$	8063.43	1.09285	0.546426	−	0.837508i	$-0.315988\pi$
$379$	8063.43	1.09285	0.546426	+	0.837508i	$0.315988\pi$
$380$	4062.34	0.548405
$381$	0	0
$382$	3064.31	0.410429
$383$	−5254.51	−0.701026	−0.350513	−	0.936558i	$-0.613993\pi$
$383$	−5254.51	−0.701026	−0.350513	+	0.936558i	$0.613993\pi$
$384$	0	0
$385$	0	0
$386$	2103.15	0.277325
$387$	0	0
$388$	5854.51	0.766026
$389$	11334.5	1.47733	0.738666	−	0.674071i	$-0.235456\pi$
$389$	11334.5	1.47733	0.738666	+	0.674071i	$0.235456\pi$
$390$	0	0
$391$	2042.07	0.264123
$392$	−2696.98	−0.347495
$393$	0	0
$394$	3155.54	0.403486
$395$	2201.93	0.280484
$396$	0	0
$397$	9896.10	1.25106	0.625530	−	0.780200i	$-0.284883\pi$
$397$	9896.10	1.25106	0.625530	+	0.780200i	$0.284883\pi$
$398$	−7521.05	−0.947227
$399$	0	0
$400$	−960.028	−0.120004
$401$	14853.7	1.84978	0.924888	−	0.380239i	$-0.124158\pi$
$401$	14853.7	1.84978	0.924888	+	0.380239i	$0.124158\pi$
$402$	0	0
$403$	650.687	0.0804293
$404$	−3600.37	−0.443379
$405$	0	0
$406$	13761.1	1.68215
$407$	0	0
$408$	0	0
$409$	−8342.08	−1.00853	−0.504265	−	0.863549i	$-0.668237\pi$
$409$	−8342.08	−1.00853	−0.504265	+	0.863549i	$0.668237\pi$
$410$	−4413.74	−0.531656
$411$	0	0
$412$	−1668.50	−0.199517
$413$	−685.449	−0.0816676
$414$	0	0
$415$	585.263	0.0692276
$416$	−104.442	−0.0123094
$417$	0	0
$418$	0	0
$419$	13082.4	1.52534	0.762670	−	0.646788i	$-0.223888\pi$
$419$	13082.4	1.52534	0.762670	+	0.646788i	$0.223888\pi$
$420$	0	0
$421$	−5555.51	−0.643133	−0.321567	−	0.946887i	$-0.604209\pi$
$421$	−5555.51	−0.643133	−0.321567	+	0.946887i	$0.604209\pi$
$422$	2841.16	0.327738
$423$	0	0
$424$	−3300.20	−0.377999
$425$	1252.34	0.142935
$426$	0	0
$427$	−4280.87	−0.485165
$428$	4321.55	0.488060
$429$	0	0
$430$	6257.18	0.701739
$431$	−359.818	−0.0402130	−0.0201065	−	0.999798i	$-0.506401\pi$
$431$	−359.818	−0.0402130	−0.0201065	+	0.999798i	$0.506401\pi$
$432$	0	0
$433$	14968.3	1.66127	0.830634	−	0.556819i	$-0.187978\pi$
$433$	14968.3	1.66127	0.830634	+	0.556819i	$0.187978\pi$
$434$	−10398.5	−1.15010
$435$	0	0
$436$	5888.33	0.646789
$437$	−12324.8	−1.34914
$438$	0	0
$439$	−15893.9	−1.72796	−0.863979	−	0.503527i	$-0.832035\pi$
$439$	−15893.9	−1.72796	−0.863979	+	0.503527i	$0.832035\pi$
$440$	0	0
$441$	0	0
$442$	136.243	0.0146615
$443$	2628.46	0.281900	0.140950	−	0.990017i	$-0.454984\pi$
$443$	2628.46	0.281900	0.140950	+	0.990017i	$0.454984\pi$
$444$	0	0
$445$	9632.13	1.02608
$446$	−10512.9	−1.11615
$447$	0	0
$448$	1669.07	0.176018
$449$	1297.47	0.136373	0.0681865	−	0.997673i	$-0.478279\pi$
$449$	1297.47	0.136373	0.0681865	+	0.997673i	$0.478279\pi$
$450$	0	0
$451$	0	0
$452$	505.922	0.0526473
$453$	0	0
$454$	−5572.55	−0.576063
$455$	686.231	0.0707055
$456$	0	0
$457$	−2251.96	−0.230508	−0.115254	−	0.993336i	$-0.536768\pi$
$457$	−2251.96	−0.230508	−0.115254	+	0.993336i	$0.536768\pi$
$458$	−8965.17	−0.914662
$459$	0	0
$460$	−3155.18	−0.319807
$461$	16772.0	1.69447	0.847233	−	0.531222i	$-0.178267\pi$
$461$	16772.0	1.69447	0.847233	+	0.531222i	$0.178267\pi$
$462$	0	0
$463$	7726.06	0.775509	0.387754	−	0.921763i	$-0.373251\pi$
$463$	7726.06	0.775509	0.387754	+	0.921763i	$0.373251\pi$
$464$	−4221.35	−0.422352
$465$	0	0
$466$	−631.227	−0.0627489
$467$	7555.52	0.748668	0.374334	−	0.927294i	$-0.377871\pi$
$467$	7555.52	0.748668	0.374334	+	0.927294i	$0.377871\pi$
$468$	0	0
$469$	7222.90	0.711135
$470$	−836.111	−0.0820573
$471$	0	0
$472$	210.267	0.0205049
$473$	0	0
$474$	0	0
$475$	−7558.40	−0.730112
$476$	−2177.26	−0.209652
$477$	0	0
$478$	−1612.76	−0.154322
$479$	13342.6	1.27273	0.636367	−	0.771387i	$-0.280437\pi$
$479$	13342.6	1.27273	0.636367	+	0.771387i	$0.280437\pi$
$480$	0	0
$481$	1193.39	0.113127
$482$	−2019.83	−0.190873
$483$	0	0
$484$	0	0
$485$	11800.0	1.10476
$486$	0	0
$487$	18820.1	1.75117	0.875584	−	0.483066i	$-0.160477\pi$
$487$	18820.1	1.75117	0.875584	+	0.483066i	$0.160477\pi$
$488$	1313.19	0.121814
$489$	0	0
$490$	−5435.86	−0.501157
$491$	3139.16	0.288530	0.144265	−	0.989539i	$-0.453918\pi$
$491$	3139.16	0.288530	0.144265	+	0.989539i	$0.453918\pi$
$492$	0	0
$493$	5506.66	0.503058
$494$	−822.285	−0.0748914
$495$	0	0
$496$	3189.82	0.288764
$497$	13481.1	1.21672
$498$	0	0
$499$	−5575.30	−0.500170	−0.250085	−	0.968224i	$-0.580459\pi$
$499$	−5575.30	−0.500170	−0.250085	+	0.968224i	$0.580459\pi$
$500$	−5966.05	−0.533619
$501$	0	0
$502$	−6374.49	−0.566748
$503$	2359.08	0.209117	0.104559	−	0.994519i	$-0.466657\pi$
$503$	2359.08	0.209117	0.104559	+	0.994519i	$0.466657\pi$
$504$	0	0
$505$	−7256.68	−0.639442
$506$	0	0
$507$	0	0
$508$	−4254.18	−0.371552
$509$	7623.13	0.663829	0.331915	−	0.943309i	$-0.392305\pi$
$509$	7623.13	0.663829	0.331915	+	0.943309i	$0.392305\pi$
$510$	0	0
$511$	6299.80	0.545376
$512$	−512.000	−0.0441942
$513$	0	0
$514$	5086.78	0.436514
$515$	−3362.93	−0.287744
$516$	0	0
$517$	0	0
$518$	−19071.3	−1.61766
$519$	0	0
$520$	−210.507	−0.0177526
$521$	−20944.8	−1.76125	−0.880623	−	0.473818i	$-0.842875\pi$
$521$	−20944.8	−1.76125	−0.880623	+	0.473818i	$0.842875\pi$
$522$	0	0
$523$	5979.83	0.499962	0.249981	−	0.968251i	$-0.419576\pi$
$523$	5979.83	0.499962	0.249981	+	0.968251i	$0.419576\pi$
$524$	−6100.17	−0.508563
$525$	0	0
$526$	−5984.58	−0.496084
$527$	−4161.05	−0.343943
$528$	0	0
$529$	−2594.45	−0.213237
$530$	−6651.67	−0.545151
$531$	0	0
$532$	13140.7	1.07091
$533$	893.413	0.0726042
$534$	0	0
$535$	8710.23	0.703881
$536$	−2215.68	−0.178550
$537$	0	0
$538$	−1642.66	−0.131636
$539$	0	0
$540$	0	0
$541$	8452.29	0.671705	0.335852	−	0.941915i	$-0.390976\pi$
$541$	8452.29	0.671705	0.335852	+	0.941915i	$0.390976\pi$
$542$	12879.5	1.02070
$543$	0	0
$544$	667.893	0.0526391
$545$	11868.1	0.932799
$546$	0	0
$547$	−1216.63	−0.0950995	−0.0475498	−	0.998869i	$-0.515141\pi$
$547$	−1216.63	−0.0950995	−0.0475498	+	0.998869i	$0.515141\pi$
$548$	8315.33	0.648199
$549$	0	0
$550$	0	0
$551$	−33235.1	−2.56963
$552$	0	0
$553$	7122.73	0.547720
$554$	1028.93	0.0789082
$555$	0	0
$556$	−6029.78	−0.459927
$557$	6620.02	0.503589	0.251795	−	0.967781i	$-0.418979\pi$
$557$	6620.02	0.503589	0.251795	+	0.967781i	$0.418979\pi$
$558$	0	0
$559$	−1266.55	−0.0958310
$560$	3364.06	0.253853
$561$	0	0
$562$	−15517.4	−1.16470
$563$	971.347	0.0727129	0.0363565	−	0.999339i	$-0.488425\pi$
$563$	971.347	0.0727129	0.0363565	+	0.999339i	$0.488425\pi$
$564$	0	0
$565$	1019.70	0.0759280
$566$	−11694.7	−0.868492
$567$	0	0
$568$	−4135.44	−0.305491
$569$	4202.74	0.309645	0.154823	−	0.987942i	$-0.450519\pi$
$569$	4202.74	0.309645	0.154823	+	0.987942i	$0.450519\pi$
$570$	0	0
$571$	11418.5	0.836862	0.418431	−	0.908248i	$-0.362580\pi$
$571$	11418.5	0.836862	0.418431	+	0.908248i	$0.362580\pi$
$572$	0	0
$573$	0	0
$574$	−14277.4	−1.03820
$575$	5870.54	0.425771
$576$	0	0
$577$	−1453.97	−0.104904	−0.0524518	−	0.998623i	$-0.516704\pi$
$577$	−1453.97	−0.104904	−0.0524518	+	0.998623i	$0.516704\pi$
$578$	8954.75	0.644409
$579$	0	0
$580$	−8508.28	−0.609116
$581$	1893.19	0.135185
$582$	0	0
$583$	0	0
$584$	−1932.52	−0.136932
$585$	0	0
$586$	16149.0	1.13841
$587$	−9754.51	−0.685880	−0.342940	−	0.939357i	$-0.611423\pi$
$587$	−9754.51	−0.685880	−0.342940	+	0.939357i	$0.611423\pi$
$588$	0	0
$589$	25113.8	1.75687
$590$	423.801	0.0295722
$591$	0	0
$592$	5850.29	0.406158
$593$	−22963.3	−1.59020	−0.795102	−	0.606476i	$-0.792583\pi$
$593$	−22963.3	−1.59020	−0.795102	+	0.606476i	$0.792583\pi$
$594$	0	0
$595$	−4388.35	−0.302361
$596$	−1719.39	−0.118169
$597$	0	0
$598$	638.660	0.0436735
$599$	−9988.67	−0.681345	−0.340673	−	0.940182i	$-0.610655\pi$
$599$	−9988.67	−0.681345	−0.340673	+	0.940182i	$0.610655\pi$
$600$	0	0
$601$	473.946	0.0321675	0.0160837	−	0.999871i	$-0.494880\pi$
$601$	473.946	0.0321675	0.0160837	+	0.999871i	$0.494880\pi$
$602$	20240.5	1.37033
$603$	0	0
$604$	−3566.73	−0.240278
$605$	0	0
$606$	0	0
$607$	3994.04	0.267073	0.133536	−	0.991044i	$-0.457367\pi$
$607$	3994.04	0.267073	0.133536	+	0.991044i	$0.457367\pi$
$608$	−4031.03	−0.268881
$609$	0	0
$610$	2646.79	0.175681
$611$	169.243	0.0112059
$612$	0	0
$613$	−10552.2	−0.695270	−0.347635	−	0.937630i	$-0.613015\pi$
$613$	−10552.2	−0.695270	−0.347635	+	0.937630i	$0.613015\pi$
$614$	−8421.28	−0.553510
$615$	0	0
$616$	0	0
$617$	14598.0	0.952500	0.476250	−	0.879310i	$-0.341996\pi$
$617$	14598.0	0.952500	0.476250	+	0.879310i	$0.341996\pi$
$618$	0	0
$619$	−13995.3	−0.908752	−0.454376	−	0.890810i	$-0.650138\pi$
$619$	−13995.3	−0.908752	−0.454376	+	0.890810i	$0.650138\pi$
$620$	6429.19	0.416456
$621$	0	0
$622$	−2637.71	−0.170036
$623$	31157.7	2.00370
$624$	0	0
$625$	−4524.57	−0.289572
$626$	8412.30	0.537097
$627$	0	0
$628$	2164.12	0.137513
$629$	−7631.59	−0.483770
$630$	0	0
$631$	−8163.63	−0.515038	−0.257519	−	0.966273i	$-0.582905\pi$
$631$	−8163.63	−0.515038	−0.257519	+	0.966273i	$0.582905\pi$
$632$	−2184.96	−0.137521
$633$	0	0
$634$	−4926.20	−0.308587
$635$	−8574.45	−0.535853
$636$	0	0
$637$	1100.31	0.0684391
$638$	0	0
$639$	0	0
$640$	−1031.95	−0.0637369
$641$	−17660.4	−1.08821	−0.544105	−	0.839017i	$-0.683131\pi$
$641$	−17660.4	−1.08821	−0.544105	+	0.839017i	$0.683131\pi$
$642$	0	0
$643$	−18030.1	−1.10581	−0.552907	−	0.833243i	$-0.686482\pi$
$643$	−18030.1	−1.10581	−0.552907	+	0.833243i	$0.686482\pi$
$644$	−10206.3	−0.624509
$645$	0	0
$646$	5258.39	0.320261
$647$	−9871.78	−0.599845	−0.299922	−	0.953964i	$-0.596961\pi$
$647$	−9871.78	−0.599845	−0.299922	+	0.953964i	$0.596961\pi$
$648$	0	0
$649$	0	0
$650$	391.670	0.0236347
$651$	0	0
$652$	−11710.4	−0.703395
$653$	−9962.52	−0.597034	−0.298517	−	0.954404i	$-0.596492\pi$
$653$	−9962.52	−0.597034	−0.298517	+	0.954404i	$0.596492\pi$
$654$	0	0
$655$	−12295.1	−0.733451
$656$	4379.72	0.260670
$657$	0	0
$658$	−2704.62	−0.160239
$659$	−15778.5	−0.932692	−0.466346	−	0.884602i	$-0.654430\pi$
$659$	−15778.5	−0.932692	−0.466346	+	0.884602i	$0.654430\pi$
$660$	0	0
$661$	9698.70	0.570704	0.285352	−	0.958423i	$-0.407889\pi$
$661$	9698.70	0.570704	0.285352	+	0.958423i	$0.407889\pi$
$662$	6664.85	0.391294
$663$	0	0
$664$	−580.752	−0.0339421
$665$	26485.6	1.54446
$666$	0	0
$667$	25813.4	1.49850
$668$	6960.93	0.403183
$669$	0	0
$670$	−4465.79	−0.257506
$671$	0	0
$672$	0	0
$673$	27028.8	1.54812	0.774058	−	0.633115i	$-0.218224\pi$
$673$	27028.8	1.54812	0.774058	+	0.633115i	$0.218224\pi$
$674$	16647.0	0.951361
$675$	0	0
$676$	−8745.39	−0.497576
$677$	3475.64	0.197311	0.0986555	−	0.995122i	$-0.468546\pi$
$677$	3475.64	0.197311	0.0986555	+	0.995122i	$0.468546\pi$
$678$	0	0
$679$	38170.2	2.15735
$680$	1346.16	0.0759162
$681$	0	0
$682$	0	0
$683$	2691.57	0.150790	0.0753952	−	0.997154i	$-0.475978\pi$
$683$	2691.57	0.150790	0.0753952	+	0.997154i	$0.475978\pi$
$684$	0	0
$685$	16759.8	0.934833
$686$	306.583	0.0170632
$687$	0	0
$688$	−6208.94	−0.344061
$689$	1346.41	0.0744470
$690$	0	0
$691$	7178.64	0.395207	0.197604	−	0.980282i	$-0.436684\pi$
$691$	7178.64	0.395207	0.197604	+	0.980282i	$0.436684\pi$
$692$	−1163.21	−0.0638995
$693$	0	0
$694$	−7244.99	−0.396277
$695$	−12153.2	−0.663308
$696$	0	0
$697$	−5713.25	−0.310480
$698$	4173.32	0.226307
$699$	0	0
$700$	−6259.18	−0.337964
$701$	4109.27	0.221405	0.110703	−	0.993854i	$-0.464690\pi$
$701$	4109.27	0.221405	0.110703	+	0.993854i	$0.464690\pi$
$702$	0	0
$703$	46060.0	2.47110
$704$	0	0
$705$	0	0
$706$	15164.3	0.808380
$707$	−23473.7	−1.24868
$708$	0	0
$709$	−16036.5	−0.849456	−0.424728	−	0.905321i	$-0.639630\pi$
$709$	−16036.5	−0.849456	−0.424728	+	0.905321i	$0.639630\pi$
$710$	−8335.13	−0.440580
$711$	0	0
$712$	−9557.88	−0.503085
$713$	−19505.6	−1.02453
$714$	0	0
$715$	0	0
$716$	−10401.3	−0.542898
$717$	0	0
$718$	−9065.98	−0.471225
$719$	36813.7	1.90949	0.954743	−	0.297432i	$-0.0961303\pi$
$719$	36813.7	1.90949	0.954743	+	0.297432i	$0.0961303\pi$
$720$	0	0
$721$	−10878.3	−0.561898
$722$	−18018.7	−0.928791
$723$	0	0
$724$	7423.31	0.381057
$725$	15830.5	0.810939
$726$	0	0
$727$	−21685.1	−1.10627	−0.553133	−	0.833093i	$-0.686568\pi$
$727$	−21685.1	−1.10627	−0.553133	+	0.833093i	$0.686568\pi$
$728$	−680.941	−0.0346667
$729$	0	0
$730$	−3895.06	−0.197483
$731$	8099.44	0.409806
$732$	0	0
$733$	−24131.3	−1.21598	−0.607988	−	0.793946i	$-0.708023\pi$
$733$	−24131.3	−1.21598	−0.607988	+	0.793946i	$0.708023\pi$
$734$	5786.11	0.290966
$735$	0	0
$736$	3130.86	0.156800
$737$	0	0
$738$	0	0
$739$	36063.3	1.79514	0.897570	−	0.440872i	$-0.145331\pi$
$739$	36063.3	1.79514	0.897570	+	0.440872i	$0.145331\pi$
$740$	11791.5	0.585762
$741$	0	0
$742$	−21516.6	−1.06455
$743$	14060.8	0.694268	0.347134	−	0.937816i	$-0.387155\pi$
$743$	14060.8	0.694268	0.347134	+	0.937816i	$0.387155\pi$
$744$	0	0
$745$	−3465.49	−0.170424
$746$	−6778.92	−0.332699
$747$	0	0
$748$	0	0
$749$	28175.6	1.37452
$750$	0	0
$751$	−28942.9	−1.40632	−0.703158	−	0.711034i	$-0.748227\pi$
$751$	−28942.9	−1.40632	−0.703158	+	0.711034i	$0.748227\pi$
$752$	829.666	0.0402325
$753$	0	0
$754$	1722.22	0.0831822
$755$	−7188.87	−0.346529
$756$	0	0
$757$	−21622.7	−1.03817	−0.519083	−	0.854724i	$-0.673726\pi$
$757$	−21622.7	−1.03817	−0.519083	+	0.854724i	$0.673726\pi$
$758$	−16126.9	−0.772763
$759$	0	0
$760$	−8124.69	−0.387781
$761$	21593.7	1.02861	0.514305	−	0.857607i	$-0.328050\pi$
$761$	21593.7	1.02861	0.514305	+	0.857607i	$0.328050\pi$
$762$	0	0
$763$	38390.7	1.82154
$764$	−6128.62	−0.290217
$765$	0	0
$766$	10509.0	0.495700
$767$	−85.7843	−0.00403845
$768$	0	0
$769$	−30161.8	−1.41439	−0.707194	−	0.707020i	$-0.750039\pi$
$769$	−30161.8	−1.41439	−0.707194	+	0.707020i	$0.750039\pi$
$770$	0	0
$771$	0	0
$772$	−4206.30	−0.196099
$773$	−31010.7	−1.44292	−0.721461	−	0.692455i	$-0.756529\pi$
$773$	−31010.7	−1.44292	−0.721461	+	0.692455i	$0.756529\pi$
$774$	0	0
$775$	−11962.2	−0.554443
$776$	−11709.0	−0.541662
$777$	0	0
$778$	−22669.0	−1.04463
$779$	34482.0	1.58594
$780$	0	0
$781$	0	0
$782$	−4084.14	−0.186763
$783$	0	0
$784$	5393.95	0.245716
$785$	4361.87	0.198321
$786$	0	0
$787$	19518.7	0.884072	0.442036	−	0.896997i	$-0.354256\pi$
$787$	19518.7	0.884072	0.442036	+	0.896997i	$0.354256\pi$
$788$	−6311.07	−0.285308
$789$	0	0
$790$	−4403.86	−0.198332
$791$	3298.51	0.148270
$792$	0	0
$793$	−535.753	−0.0239913
$794$	−19792.2	−0.884633
$795$	0	0
$796$	15042.1	0.669791
$797$	−21887.8	−0.972781	−0.486391	−	0.873741i	$-0.661687\pi$
$797$	−21887.8	−0.972781	−0.486391	+	0.873741i	$0.661687\pi$
$798$	0	0
$799$	−1082.28	−0.0479204
$800$	1920.06	0.0848553
$801$	0	0
$802$	−29707.5	−1.30799
$803$	0	0
$804$	0	0
$805$	−20571.1	−0.900667
$806$	−1301.37	−0.0568721
$807$	0	0
$808$	7200.75	0.313517
$809$	−37165.5	−1.61517	−0.807584	−	0.589752i	$-0.799225\pi$
$809$	−37165.5	−1.61517	−0.807584	+	0.589752i	$0.799225\pi$
$810$	0	0
$811$	−28558.8	−1.23654	−0.618272	−	0.785965i	$-0.712167\pi$
$811$	−28558.8	−1.23654	−0.618272	+	0.785965i	$0.712167\pi$
$812$	−27522.3	−1.18946
$813$	0	0
$814$	0	0
$815$	−23602.7	−1.01444
$816$	0	0
$817$	−48883.7	−2.09330
$818$	16684.2	0.713139
$819$	0	0
$820$	8827.48	0.375938
$821$	6293.15	0.267518	0.133759	−	0.991014i	$-0.457295\pi$
$821$	6293.15	0.267518	0.133759	+	0.991014i	$0.457295\pi$
$822$	0	0
$823$	14939.0	0.632733	0.316367	−	0.948637i	$-0.397537\pi$
$823$	14939.0	0.632733	0.316367	+	0.948637i	$0.397537\pi$
$824$	3337.00	0.141080
$825$	0	0
$826$	1370.90	0.0577477
$827$	34659.0	1.45733	0.728665	−	0.684870i	$-0.240141\pi$
$827$	34659.0	1.45733	0.728665	+	0.684870i	$0.240141\pi$
$828$	0	0
$829$	−29323.7	−1.22853	−0.614266	−	0.789099i	$-0.710548\pi$
$829$	−29323.7	−1.22853	−0.614266	+	0.789099i	$0.710548\pi$
$830$	−1170.53	−0.0489513
$831$	0	0
$832$	208.884	0.00870405
$833$	−7036.30	−0.292669
$834$	0	0
$835$	14030.0	0.581471
$836$	0	0
$837$	0	0
$838$	−26164.8	−1.07858
$839$	−5888.98	−0.242324	−0.121162	−	0.992633i	$-0.538662\pi$
$839$	−5888.98	−0.242324	−0.121162	+	0.992633i	$0.538662\pi$
$840$	0	0
$841$	45219.5	1.85410
$842$	11111.0	0.454764
$843$	0	0
$844$	−5682.32	−0.231746
$845$	−17626.7	−0.717604
$846$	0	0
$847$	0	0
$848$	6600.39	0.267286
$849$	0	0
$850$	−2504.67	−0.101070
$851$	−35774.3	−1.44104
$852$	0	0
$853$	−38138.0	−1.53085	−0.765427	−	0.643522i	$-0.777472\pi$
$853$	−38138.0	−1.53085	−0.765427	+	0.643522i	$0.777472\pi$
$854$	8561.73	0.343064
$855$	0	0
$856$	−8643.09	−0.345111
$857$	12704.1	0.506377	0.253188	−	0.967417i	$-0.418521\pi$
$857$	12704.1	0.506377	0.253188	+	0.967417i	$0.418521\pi$
$858$	0	0
$859$	−11123.7	−0.441833	−0.220916	−	0.975293i	$-0.570905\pi$
$859$	−11123.7	−0.441833	−0.220916	+	0.975293i	$0.570905\pi$
$860$	−12514.4	−0.496205
$861$	0	0
$862$	719.636	0.0284349
$863$	13680.5	0.539617	0.269808	−	0.962914i	$-0.413040\pi$
$863$	13680.5	0.539617	0.269808	+	0.962914i	$0.413040\pi$
$864$	0	0
$865$	−2344.48	−0.0921559
$866$	−29936.5	−1.17469
$867$	0	0
$868$	20796.9	0.813242
$869$	0	0
$870$	0	0
$871$	903.950	0.0351655
$872$	−11776.7	−0.457349
$873$	0	0
$874$	24649.6	0.953988
$875$	−38897.4	−1.50282
$876$	0	0
$877$	−45407.1	−1.74833	−0.874166	−	0.485627i	$-0.838591\pi$
$877$	−45407.1	−1.74833	−0.874166	+	0.485627i	$0.838591\pi$
$878$	31787.8	1.22185
$879$	0	0
$880$	0	0
$881$	−13620.9	−0.520884	−0.260442	−	0.965490i	$-0.583868\pi$
$881$	−13620.9	−0.520884	−0.260442	+	0.965490i	$0.583868\pi$
$882$	0	0
$883$	−7945.73	−0.302826	−0.151413	−	0.988471i	$-0.548382\pi$
$883$	−7945.73	−0.302826	−0.151413	+	0.988471i	$0.548382\pi$
$884$	−272.485	−0.0103673
$885$	0	0
$886$	−5256.91	−0.199333
$887$	−41541.1	−1.57251	−0.786253	−	0.617904i	$-0.787982\pi$
$887$	−41541.1	−1.57251	−0.786253	+	0.617904i	$0.787982\pi$
$888$	0	0
$889$	−27736.3	−1.04640
$890$	−19264.3	−0.725550
$891$	0	0
$892$	21025.9	0.789235
$893$	6532.05	0.244778
$894$	0	0
$895$	−20964.2	−0.782968
$896$	−3338.13	−0.124463
$897$	0	0
$898$	−2594.94	−0.0964303
$899$	−52598.9	−1.95136
$900$	0	0
$901$	−8610.07	−0.318361
$902$	0	0
$903$	0	0
$904$	−1011.84	−0.0372273
$905$	14962.0	0.549561
$906$	0	0
$907$	43231.1	1.58265	0.791326	−	0.611394i	$-0.209391\pi$
$907$	43231.1	1.58265	0.791326	+	0.611394i	$0.209391\pi$
$908$	11145.1	0.407338
$909$	0	0
$910$	−1372.46	−0.0499963
$911$	33500.8	1.21837	0.609183	−	0.793030i	$-0.291497\pi$
$911$	33500.8	1.21837	0.609183	+	0.793030i	$0.291497\pi$
$912$	0	0
$913$	0	0
$914$	4503.92	0.162994
$915$	0	0
$916$	17930.3	0.646763
$917$	−39771.8	−1.43226
$918$	0	0
$919$	−18900.5	−0.678421	−0.339211	−	0.940710i	$-0.610160\pi$
$919$	−18900.5	−0.678421	−0.339211	+	0.940710i	$0.610160\pi$
$920$	6310.37	0.226138
$921$	0	0
$922$	−33544.0	−1.19817
$923$	1687.17	0.0601665
$924$	0	0
$925$	−21939.3	−0.779847
$926$	−15452.1	−0.548367
$927$	0	0
$928$	8442.70	0.298648
$929$	−29149.2	−1.02944	−0.514722	−	0.857357i	$-0.672105\pi$
$929$	−29149.2	−1.02944	−0.514722	+	0.857357i	$0.672105\pi$
$930$	0	0
$931$	42467.2	1.49496
$932$	1262.45	0.0443702
$933$	0	0
$934$	−15111.0	−0.529388
$935$	0	0
$936$	0	0
$937$	32193.0	1.12241	0.561205	−	0.827677i	$-0.310338\pi$
$937$	32193.0	1.12241	0.561205	+	0.827677i	$0.310338\pi$
$938$	−14445.8	−0.502849
$939$	0	0
$940$	1672.22	0.0580233
$941$	37395.9	1.29550	0.647752	−	0.761851i	$-0.275709\pi$
$941$	37395.9	1.29550	0.647752	+	0.761851i	$0.275709\pi$
$942$	0	0
$943$	−26781.8	−0.924852
$944$	−420.534	−0.0144992
$945$	0	0
$946$	0	0
$947$	−3065.34	−0.105185	−0.0525925	−	0.998616i	$-0.516748\pi$
$947$	−3065.34	−0.105185	−0.0525925	+	0.998616i	$0.516748\pi$
$948$	0	0
$949$	788.424	0.0269687
$950$	15116.8	0.516267
$951$	0	0
$952$	4354.52	0.148247
$953$	16385.4	0.556953	0.278476	−	0.960443i	$-0.410171\pi$
$953$	16385.4	0.556953	0.278476	+	0.960443i	$0.410171\pi$
$954$	0	0
$955$	−12352.5	−0.418551
$956$	3225.53	0.109122
$957$	0	0
$958$	−26685.2	−0.899958
$959$	54214.2	1.82551
$960$	0	0
$961$	9954.82	0.334155
$962$	−2386.79	−0.0799929
$963$	0	0
$964$	4039.65	0.134967
$965$	−8477.95	−0.282813
$966$	0	0
$967$	−16183.5	−0.538187	−0.269094	−	0.963114i	$-0.586724\pi$
$967$	−16183.5	−0.538187	−0.269094	+	0.963114i	$0.586724\pi$
$968$	0	0
$969$	0	0
$970$	−23600.0	−0.781185
$971$	−4694.04	−0.155138	−0.0775690	−	0.996987i	$-0.524716\pi$
$971$	−4694.04	−0.155138	−0.0775690	+	0.996987i	$0.524716\pi$
$972$	0	0
$973$	−39312.9	−1.29529
$974$	−37640.1	−1.23826
$975$	0	0
$976$	−2626.38	−0.0861358
$977$	7663.58	0.250952	0.125476	−	0.992097i	$-0.459954\pi$
$977$	7663.58	0.250952	0.125476	+	0.992097i	$0.459954\pi$
$978$	0	0
$979$	0	0
$980$	10871.7	0.354372
$981$	0	0
$982$	−6278.33	−0.204022
$983$	−37397.3	−1.21342	−0.606708	−	0.794925i	$-0.707510\pi$
$983$	−37397.3	−1.21342	−0.606708	+	0.794925i	$0.707510\pi$
$984$	0	0
$985$	−12720.2	−0.411471
$986$	−11013.3	−0.355716
$987$	0	0
$988$	1644.57	0.0529562
$989$	37967.5	1.22072
$990$	0	0
$991$	15536.1	0.498002	0.249001	−	0.968503i	$-0.419898\pi$
$991$	15536.1	0.498002	0.249001	+	0.968503i	$0.419898\pi$
$992$	−6379.63	−0.204187
$993$	0	0
$994$	−26962.2	−0.860350
$995$	30317.9	0.965973
$996$	0	0
$997$	41167.8	1.30772	0.653860	−	0.756615i	$-0.273148\pi$
$997$	41167.8	1.30772	0.653860	+	0.756615i	$0.273148\pi$
$998$	11150.6	0.353674
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2178.4.a.bt.1.4		4
3.2	odd	2		242.4.a.o.1.2		4
11.2	odd	10		198.4.f.d.37.1		8
11.6	odd	10		198.4.f.d.91.1		8
11.10	odd	2		2178.4.a.by.1.4		4
12.11	even	2		1936.4.a.bm.1.3		4
33.2	even	10		22.4.c.b.15.1	yes	8
33.5	odd	10		242.4.c.q.3.1		8
33.8	even	10		242.4.c.r.9.2		8
33.14	odd	10		242.4.c.n.9.2		8
33.17	even	10		22.4.c.b.3.1	✓	8
33.20	odd	10		242.4.c.q.81.1		8
33.26	odd	10		242.4.c.n.27.2		8
33.29	even	10		242.4.c.r.27.2		8
33.32	even	2		242.4.a.n.1.2		4
132.35	odd	10		176.4.m.b.81.2		8
132.83	odd	10		176.4.m.b.113.2		8
132.131	odd	2		1936.4.a.bn.1.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
22.4.c.b.3.1	✓	8	33.17	even	10
22.4.c.b.15.1	yes	8	33.2	even	10
176.4.m.b.81.2		8	132.35	odd	10
176.4.m.b.113.2		8	132.83	odd	10
198.4.f.d.37.1		8	11.2	odd	10
198.4.f.d.91.1		8	11.6	odd	10
242.4.a.n.1.2		4	33.32	even	2
242.4.a.o.1.2		4	3.2	odd	2
242.4.c.n.9.2		8	33.14	odd	10
242.4.c.n.27.2		8	33.26	odd	10
242.4.c.q.3.1		8	33.5	odd	10
242.4.c.q.81.1		8	33.20	odd	10
242.4.c.r.9.2		8	33.8	even	10
242.4.c.r.27.2		8	33.29	even	10
1936.4.a.bm.1.3		4	12.11	even	2
1936.4.a.bn.1.3		4	132.131	odd	2
2178.4.a.bt.1.4		4	1.1	even	1	trivial
2178.4.a.by.1.4		4	11.10	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:	\( N \)	\(=\)	\( 2178 = 2 \cdot 3^{2} \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2178.a (trivial)

\(f(q)\)	\(=\)	\(q-2.00000 q^{2} +4.00000 q^{4} +8.06215 q^{5} +26.0792 q^{7} -8.00000 q^{8} +O(q^{10})\)	\(q-2.00000 q^{2} +4.00000 q^{4} +8.06215 q^{5} +26.0792 q^{7} -8.00000 q^{8} -16.1243 q^{10} +3.26382 q^{13} -52.1583 q^{14} +16.0000 q^{16} -20.8717 q^{17} +125.970 q^{19} +32.2486 q^{20} -97.8394 q^{23} -60.0018 q^{25} -6.52764 q^{26} +104.317 q^{28} -263.834 q^{29} +199.364 q^{31} -32.0000 q^{32} +41.7433 q^{34} +210.254 q^{35} +365.643 q^{37} -251.939 q^{38} -64.4972 q^{40} +273.732 q^{41} -388.059 q^{43} +195.679 q^{46} +51.8541 q^{47} +337.122 q^{49} +120.004 q^{50} +13.0553 q^{52} +412.524 q^{53} -208.633 q^{56} +527.669 q^{58} -26.2834 q^{59} -164.149 q^{61} -398.727 q^{62} +64.0000 q^{64} +26.3134 q^{65} +276.961 q^{67} -83.4866 q^{68} -420.508 q^{70} +516.930 q^{71} +241.565 q^{73} -731.287 q^{74} +503.879 q^{76} +273.120 q^{79} +128.994 q^{80} -547.465 q^{82} +72.5940 q^{83} -168.270 q^{85} +776.118 q^{86} +1194.73 q^{89} +85.1177 q^{91} -391.358 q^{92} -103.708 q^{94} +1015.59 q^{95} +1463.63 q^{97} -674.244 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 8 q^{2} + 16 q^{4} - 25 q^{5} + 3 q^{7} - 32 q^{8}+O(q^{10}) \) 4 * q - 8 * q^2 + 16 * q^4 - 25 * q^5 + 3 * q^7 - 32 * q^8	\( 4 q - 8 q^{2} + 16 q^{4} - 25 q^{5} + 3 q^{7} - 32 q^{8} + 50 q^{10} - 41 q^{13} - 6 q^{14} + 64 q^{16} - 52 q^{17} + 16 q^{19} - 100 q^{20} - 314 q^{23} - 21 q^{25} + 82 q^{26} + 12 q^{28} - 561 q^{29} + 199 q^{31} - 128 q^{32} + 104 q^{34} + 714 q^{35} + 357 q^{37} - 32 q^{38} + 200 q^{40} - 32 q^{41} - 721 q^{43} + 628 q^{46} - 403 q^{47} + 823 q^{49} + 42 q^{50} - 164 q^{52} + 133 q^{53} - 24 q^{56} + 1122 q^{58} - 1016 q^{59} - 919 q^{61} - 398 q^{62} + 256 q^{64} - 69 q^{65} + 289 q^{67} - 208 q^{68} - 1428 q^{70} + 1205 q^{71} - 1234 q^{73} - 714 q^{74} + 64 q^{76} - 603 q^{79} - 400 q^{80} + 64 q^{82} - 1514 q^{83} + 717 q^{85} + 1442 q^{86} + 1101 q^{89} - 2306 q^{91} - 1256 q^{92} + 806 q^{94} + 1766 q^{95} + 2116 q^{97} - 1646 q^{98}+O(q^{100}) \) 4 * q - 8 * q^2 + 16 * q^4 - 25 * q^5 + 3 * q^7 - 32 * q^8 + 50 * q^10 - 41 * q^13 - 6 * q^14 + 64 * q^16 - 52 * q^17 + 16 * q^19 - 100 * q^20 - 314 * q^23 - 21 * q^25 + 82 * q^26 + 12 * q^28 - 561 * q^29 + 199 * q^31 - 128 * q^32 + 104 * q^34 + 714 * q^35 + 357 * q^37 - 32 * q^38 + 200 * q^40 - 32 * q^41 - 721 * q^43 + 628 * q^46 - 403 * q^47 + 823 * q^49 + 42 * q^50 - 164 * q^52 + 133 * q^53 - 24 * q^56 + 1122 * q^58 - 1016 * q^59 - 919 * q^61 - 398 * q^62 + 256 * q^64 - 69 * q^65 + 289 * q^67 - 208 * q^68 - 1428 * q^70 + 1205 * q^71 - 1234 * q^73 - 714 * q^74 + 64 * q^76 - 603 * q^79 - 400 * q^80 + 64 * q^82 - 1514 * q^83 + 717 * q^85 + 1442 * q^86 + 1101 * q^89 - 2306 * q^91 - 1256 * q^92 + 806 * q^94 + 1766 * q^95 + 2116 * q^97 - 1646 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more