LMFDB - Embedded newform 2178.4.a.cf.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:	$1$
Dimension:	$6$
Coefficient field:	$\mathbb{Q}[x]/(x^{6} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - x^{5} - 331x^{4} + 48x^{3} + 23386x^{2} - 36820x - 100804 $ x^6 - x^5 - 331x^4 + 48x^3 + 23386x^2 - 36820x - 100804
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 11 $
Twist minimal:	no (minimal twist has level 198)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$3.35806$ 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)

$f(q)$	$=$	$q+2.00000 q^{2} +4.00000 q^{4} +4.05148 q^{5} -10.5792 q^{7} +8.00000 q^{8} +O(q^{10})$	$q+2.00000 q^{2} +4.00000 q^{4} +4.05148 q^{5} -10.5792 q^{7} +8.00000 q^{8} +8.10296 q^{10} +14.3043 q^{13} -21.1585 q^{14} +16.0000 q^{16} +85.9094 q^{17} -152.251 q^{19} +16.2059 q^{20} -42.0807 q^{23} -108.585 q^{25} +28.6087 q^{26} -42.3169 q^{28} -12.7177 q^{29} +71.2937 q^{31} +32.0000 q^{32} +171.819 q^{34} -42.8616 q^{35} -162.596 q^{37} -304.502 q^{38} +32.4119 q^{40} -112.291 q^{41} +55.3779 q^{43} -84.1613 q^{46} +345.768 q^{47} -231.080 q^{49} -217.171 q^{50} +57.2173 q^{52} -258.590 q^{53} -84.6339 q^{56} -25.4353 q^{58} -476.982 q^{59} -17.8848 q^{61} +142.587 q^{62} +64.0000 q^{64} +57.9537 q^{65} -41.5848 q^{67} +343.638 q^{68} -85.7232 q^{70} -168.628 q^{71} +570.437 q^{73} -325.191 q^{74} -609.005 q^{76} -1267.87 q^{79} +64.8237 q^{80} -224.583 q^{82} +1024.92 q^{83} +348.061 q^{85} +110.756 q^{86} -412.090 q^{89} -151.329 q^{91} -168.323 q^{92} +691.535 q^{94} -616.843 q^{95} -1045.21 q^{97} -462.160 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 12 q^{2} + 24 q^{4} - 17 q^{5} - 7 q^{7} + 48 q^{8}+O(q^{10}) $ 6 * q + 12 * q^2 + 24 * q^4 - 17 * q^5 - 7 * q^7 + 48 * q^8	$ 6 q + 12 q^{2} + 24 q^{4} - 17 q^{5} - 7 q^{7} + 48 q^{8} - 34 q^{10} - 68 q^{13} - 14 q^{14} + 96 q^{16} + 42 q^{17} - 98 q^{19} - 68 q^{20} - 210 q^{23} + 47 q^{25} - 136 q^{26} - 28 q^{28} + 13 q^{29} - 125 q^{31} + 192 q^{32} + 84 q^{34} + 534 q^{35} + 282 q^{37} - 196 q^{38} - 136 q^{40} - 170 q^{41} - 868 q^{43} - 420 q^{46} - 782 q^{47} - 439 q^{49} + 94 q^{50} - 272 q^{52} - 645 q^{53} - 56 q^{56} + 26 q^{58} - 507 q^{59} - 1772 q^{61} - 250 q^{62} + 384 q^{64} - 1856 q^{65} + 686 q^{67} + 168 q^{68} + 1068 q^{70} - 2782 q^{71} - 335 q^{73} + 564 q^{74} - 392 q^{76} - 127 q^{79} - 272 q^{80} - 340 q^{82} - 9 q^{83} - 370 q^{85} - 1736 q^{86} - 2526 q^{89} + 296 q^{91} - 840 q^{92} - 1564 q^{94} - 1194 q^{95} + 89 q^{97} - 878 q^{98}+O(q^{100}) $ 6 * q + 12 * q^2 + 24 * q^4 - 17 * q^5 - 7 * q^7 + 48 * q^8 - 34 * q^10 - 68 * q^13 - 14 * q^14 + 96 * q^16 + 42 * q^17 - 98 * q^19 - 68 * q^20 - 210 * q^23 + 47 * q^25 - 136 * q^26 - 28 * q^28 + 13 * q^29 - 125 * q^31 + 192 * q^32 + 84 * q^34 + 534 * q^35 + 282 * q^37 - 196 * q^38 - 136 * q^40 - 170 * q^41 - 868 * q^43 - 420 * q^46 - 782 * q^47 - 439 * q^49 + 94 * q^50 - 272 * q^52 - 645 * q^53 - 56 * q^56 + 26 * q^58 - 507 * q^59 - 1772 * q^61 - 250 * q^62 + 384 * q^64 - 1856 * q^65 + 686 * q^67 + 168 * q^68 + 1068 * q^70 - 2782 * q^71 - 335 * q^73 + 564 * q^74 - 392 * q^76 - 127 * q^79 - 272 * q^80 - 340 * q^82 - 9 * q^83 - 370 * q^85 - 1736 * q^86 - 2526 * q^89 + 296 * q^91 - 840 * q^92 - 1564 * q^94 - 1194 * q^95 + 89 * q^97 - 878 * 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
$5$	4.05148	0.362376	0.181188	−	0.983449i	$-0.442006\pi$
$5$	4.05148	0.362376	0.181188	+	0.983449i	$0.442006\pi$
$6$	0	0
$7$	−10.5792	−0.571225	−0.285612	−	0.958345i	$-0.592197\pi$
$7$	−10.5792	−0.571225	−0.285612	+	0.958345i	$0.592197\pi$
$8$	8.00000	0.353553
$9$	0	0
$10$	8.10296	0.256238
$11$	0	0
$11$	0	0
$12$	0	0
$13$	14.3043	0.305177	0.152589	−	0.988290i	$-0.451239\pi$
$13$	14.3043	0.305177	0.152589	+	0.988290i	$0.451239\pi$
$14$	−21.1585	−0.403917
$15$	0	0
$16$	16.0000	0.250000
$17$	85.9094	1.22565	0.612826	−	0.790218i	$-0.290033\pi$
$17$	85.9094	1.22565	0.612826	+	0.790218i	$0.290033\pi$
$18$	0	0
$19$	−152.251	−1.83836	−0.919179	−	0.393839i	$-0.871147\pi$
$19$	−152.251	−1.83836	−0.919179	+	0.393839i	$0.871147\pi$
$20$	16.2059	0.181188
$21$	0	0
$22$	0	0
$23$	−42.0807	−0.381497	−0.190748	−	0.981639i	$-0.561091\pi$
$23$	−42.0807	−0.381497	−0.190748	+	0.981639i	$0.561091\pi$
$24$	0	0
$25$	−108.585	−0.868684
$26$	28.6087	0.215793
$27$	0	0
$28$	−42.3169	−0.285612
$29$	−12.7177	−0.0814349	−0.0407174	−	0.999171i	$-0.512964\pi$
$29$	−12.7177	−0.0814349	−0.0407174	+	0.999171i	$0.512964\pi$
$30$	0	0
$31$	71.2937	0.413056	0.206528	−	0.978441i	$-0.433784\pi$
$31$	71.2937	0.413056	0.206528	+	0.978441i	$0.433784\pi$
$32$	32.0000	0.176777
$33$	0	0
$34$	171.819	0.866667
$35$	−42.8616	−0.206998
$36$	0	0
$37$	−162.596	−0.722447	−0.361224	−	0.932479i	$-0.617641\pi$
$37$	−162.596	−0.722447	−0.361224	+	0.932479i	$0.617641\pi$
$38$	−304.502	−1.29992
$39$	0	0
$40$	32.4119	0.128119
$41$	−112.291	−0.427731	−0.213866	−	0.976863i	$-0.568605\pi$
$41$	−112.291	−0.427731	−0.213866	+	0.976863i	$0.568605\pi$
$42$	0	0
$43$	55.3779	0.196397	0.0981983	−	0.995167i	$-0.468692\pi$
$43$	55.3779	0.196397	0.0981983	+	0.995167i	$0.468692\pi$
$44$	0	0
$45$	0	0
$46$	−84.1613	−0.269759
$47$	345.768	1.07309	0.536547	−	0.843871i	$-0.319729\pi$
$47$	345.768	1.07309	0.536547	+	0.843871i	$0.319729\pi$
$48$	0	0
$49$	−231.080	−0.673702
$50$	−217.171	−0.614252
$51$	0	0
$52$	57.2173	0.152589
$53$	−258.590	−0.670189	−0.335095	−	0.942184i	$-0.608768\pi$
$53$	−258.590	−0.670189	−0.335095	+	0.942184i	$0.608768\pi$
$54$	0	0
$55$	0	0
$56$	−84.6339	−0.201958
$57$	0	0
$58$	−25.4353	−0.0575831
$59$	−476.982	−1.05250	−0.526252	−	0.850328i	$-0.676403\pi$
$59$	−476.982	−1.05250	−0.526252	+	0.850328i	$0.676403\pi$
$60$	0	0
$61$	−17.8848	−0.0375396	−0.0187698	−	0.999824i	$-0.505975\pi$
$61$	−17.8848	−0.0375396	−0.0187698	+	0.999824i	$0.505975\pi$
$62$	142.587	0.292074
$63$	0	0
$64$	64.0000	0.125000
$65$	57.9537	0.110589
$66$	0	0
$67$	−41.5848	−0.0758267	−0.0379133	−	0.999281i	$-0.512071\pi$
$67$	−41.5848	−0.0758267	−0.0379133	+	0.999281i	$0.512071\pi$
$68$	343.638	0.612826
$69$	0	0
$70$	−85.7232	−0.146370
$71$	−168.628	−0.281865	−0.140932	−	0.990019i	$-0.545010\pi$
$71$	−168.628	−0.281865	−0.140932	+	0.990019i	$0.545010\pi$
$72$	0	0
$73$	570.437	0.914583	0.457291	−	0.889317i	$-0.348820\pi$
$73$	570.437	0.914583	0.457291	+	0.889317i	$0.348820\pi$
$74$	−325.191	−0.510847
$75$	0	0
$76$	−609.005	−0.919179
$77$	0	0
$78$	0	0
$79$	−1267.87	−1.80565	−0.902827	−	0.430004i	$-0.858512\pi$
$79$	−1267.87	−1.80565	−0.902827	+	0.430004i	$0.858512\pi$
$80$	64.8237	0.0905939
$81$	0	0
$82$	−224.583	−0.302452
$83$	1024.92	1.35542	0.677711	−	0.735328i	$-0.262972\pi$
$83$	1024.92	1.35542	0.677711	+	0.735328i	$0.262972\pi$
$84$	0	0
$85$	348.061	0.444147
$86$	110.756	0.138873
$87$	0	0
$88$	0	0
$89$	−412.090	−0.490803	−0.245402	−	0.969421i	$-0.578920\pi$
$89$	−412.090	−0.490803	−0.245402	+	0.969421i	$0.578920\pi$
$90$	0	0
$91$	−151.329	−0.174325
$92$	−168.323	−0.190748
$93$	0	0
$94$	691.535	0.758792
$95$	−616.843	−0.666176
$96$	0	0
$97$	−1045.21	−1.09407	−0.547034	−	0.837111i	$-0.684243\pi$
$97$	−1045.21	−1.09407	−0.547034	+	0.837111i	$0.684243\pi$
$98$	−462.160	−0.476379
$99$	0	0
$100$	−434.342	−0.434342
$101$	1176.65	1.15922	0.579612	−	0.814893i	$-0.303204\pi$
$101$	1176.65	1.15922	0.579612	+	0.814893i	$0.303204\pi$
$102$	0	0
$103$	1282.79	1.22716	0.613579	−	0.789633i	$-0.289729\pi$
$103$	1282.79	1.22716	0.613579	+	0.789633i	$0.289729\pi$
$104$	114.435	0.107897
$105$	0	0
$106$	−517.180	−0.473895
$107$	1048.83	0.947608	0.473804	−	0.880630i	$-0.342880\pi$
$107$	1048.83	0.947608	0.473804	+	0.880630i	$0.342880\pi$
$108$	0	0
$109$	−1395.86	−1.22659	−0.613297	−	0.789853i	$-0.710157\pi$
$109$	−1395.86	−1.22659	−0.613297	+	0.789853i	$0.710157\pi$
$110$	0	0
$111$	0	0
$112$	−169.268	−0.142806
$113$	−1415.90	−1.17873	−0.589365	−	0.807867i	$-0.700622\pi$
$113$	−1415.90	−1.17873	−0.589365	+	0.807867i	$0.700622\pi$
$114$	0	0
$115$	−170.489	−0.138245
$116$	−50.8707	−0.0407174
$117$	0	0
$118$	−953.964	−0.744233
$119$	−908.856	−0.700123
$120$	0	0
$121$	0	0
$122$	−35.7696	−0.0265445
$123$	0	0
$124$	285.175	0.206528
$125$	−946.368	−0.677165
$126$	0	0
$127$	−246.142	−0.171981	−0.0859903	−	0.996296i	$-0.527405\pi$
$127$	−246.142	−0.171981	−0.0859903	+	0.996296i	$0.527405\pi$
$128$	128.000	0.0883883
$129$	0	0
$130$	115.907	0.0781981
$131$	−2469.49	−1.64703	−0.823514	−	0.567296i	$-0.807989\pi$
$131$	−2469.49	−1.64703	−0.823514	+	0.567296i	$0.807989\pi$
$132$	0	0
$133$	1610.70	1.05012
$134$	−83.1695	−0.0536175
$135$	0	0
$136$	687.275	0.433334
$137$	−2094.44	−1.30613	−0.653065	−	0.757302i	$-0.726517\pi$
$137$	−2094.44	−1.30613	−0.653065	+	0.757302i	$0.726517\pi$
$138$	0	0
$139$	277.516	0.169342	0.0846712	−	0.996409i	$-0.473016\pi$
$139$	277.516	0.169342	0.0846712	+	0.996409i	$0.473016\pi$
$140$	−171.446	−0.103499
$141$	0	0
$142$	−337.255	−0.199309
$143$	0	0
$144$	0	0
$145$	−51.5254	−0.0295100
$146$	1140.87	0.646708
$147$	0	0
$148$	−650.382	−0.361224
$149$	−3464.68	−1.90495	−0.952476	−	0.304614i	$-0.901473\pi$
$149$	−3464.68	−1.90495	−0.952476	+	0.304614i	$0.901473\pi$
$150$	0	0
$151$	−1367.57	−0.737027	−0.368514	−	0.929622i	$-0.620133\pi$
$151$	−1367.57	−0.737027	−0.368514	+	0.929622i	$0.620133\pi$
$152$	−1218.01	−0.649958
$153$	0	0
$154$	0	0
$155$	288.845	0.149681
$156$	0	0
$157$	−1700.51	−0.864432	−0.432216	−	0.901770i	$-0.642268\pi$
$157$	−1700.51	−0.864432	−0.432216	+	0.901770i	$0.642268\pi$
$158$	−2535.74	−1.27679
$159$	0	0
$160$	129.647	0.0640596
$161$	445.181	0.217920
$162$	0	0
$163$	2641.16	1.26915	0.634576	−	0.772860i	$-0.281175\pi$
$163$	2641.16	1.26915	0.634576	+	0.772860i	$0.281175\pi$
$164$	−449.166	−0.213866
$165$	0	0
$166$	2049.85	0.958428
$167$	1869.70	0.866356	0.433178	−	0.901308i	$-0.357392\pi$
$167$	1869.70	0.866356	0.433178	+	0.901308i	$0.357392\pi$
$168$	0	0
$169$	−1992.39	−0.906867
$170$	696.121	0.314059
$171$	0	0
$172$	221.512	0.0981983
$173$	−615.240	−0.270381	−0.135190	−	0.990820i	$-0.543165\pi$
$173$	−615.240	−0.270381	−0.135190	+	0.990820i	$0.543165\pi$
$174$	0	0
$175$	1148.75	0.496214
$176$	0	0
$177$	0	0
$178$	−824.181	−0.347050
$179$	1952.26	0.815191	0.407595	−	0.913163i	$-0.366367\pi$
$179$	1952.26	0.815191	0.407595	+	0.913163i	$0.366367\pi$
$180$	0	0
$181$	−4169.89	−1.71241	−0.856203	−	0.516639i	$-0.827183\pi$
$181$	−4169.89	−1.71241	−0.856203	+	0.516639i	$0.827183\pi$
$182$	−302.658	−0.123266
$183$	0	0
$184$	−336.645	−0.134879
$185$	−658.753	−0.261797
$186$	0	0
$187$	0	0
$188$	1383.07	0.536547
$189$	0	0
$190$	−1233.69	−0.471058
$191$	−2411.85	−0.913692	−0.456846	−	0.889546i	$-0.651021\pi$
$191$	−2411.85	−0.913692	−0.456846	+	0.889546i	$0.651021\pi$
$192$	0	0
$193$	−100.352	−0.0374276	−0.0187138	−	0.999825i	$-0.505957\pi$
$193$	−100.352	−0.0374276	−0.0187138	+	0.999825i	$0.505957\pi$
$194$	−2090.41	−0.773622
$195$	0	0
$196$	−924.319	−0.336851
$197$	−2367.77	−0.856327	−0.428164	−	0.903701i	$-0.640839\pi$
$197$	−2367.77	−0.856327	−0.428164	+	0.903701i	$0.640839\pi$
$198$	0	0
$199$	2765.67	0.985193	0.492597	−	0.870258i	$-0.336048\pi$
$199$	2765.67	0.985193	0.492597	+	0.870258i	$0.336048\pi$
$200$	−868.684	−0.307126
$201$	0	0
$202$	2353.31	0.819695
$203$	134.543	0.0465176
$204$	0	0
$205$	−454.947	−0.154999
$206$	2565.58	0.867731
$207$	0	0
$208$	228.869	0.0762944
$209$	0	0
$210$	0	0
$211$	−669.093	−0.218305	−0.109152	−	0.994025i	$-0.534814\pi$
$211$	−669.093	−0.218305	−0.109152	+	0.994025i	$0.534814\pi$
$212$	−1034.36	−0.335095
$213$	0	0
$214$	2097.66	0.670060
$215$	224.363	0.0711694
$216$	0	0
$217$	−754.233	−0.235948
$218$	−2791.71	−0.867333
$219$	0	0
$220$	0	0
$221$	1228.88	0.374042
$222$	0	0
$223$	2684.03	0.805992	0.402996	−	0.915202i	$-0.367969\pi$
$223$	2684.03	0.805992	0.402996	+	0.915202i	$0.367969\pi$
$224$	−338.535	−0.100979
$225$	0	0
$226$	−2831.80	−0.833488
$227$	535.913	0.156695	0.0783476	−	0.996926i	$-0.475036\pi$
$227$	535.913	0.156695	0.0783476	+	0.996926i	$0.475036\pi$
$228$	0	0
$229$	12.7392	0.00367613	0.00183806	−	0.999998i	$-0.499415\pi$
$229$	12.7392	0.00367613	0.00183806	+	0.999998i	$0.499415\pi$
$230$	−340.978	−0.0977541
$231$	0	0
$232$	−101.741	−0.0287916
$233$	6485.26	1.82345	0.911724	−	0.410804i	$-0.134752\pi$
$233$	6485.26	1.82345	0.911724	+	0.410804i	$0.134752\pi$
$234$	0	0
$235$	1400.87	0.388863
$236$	−1907.93	−0.526252
$237$	0	0
$238$	−1817.71	−0.495062
$239$	6717.97	1.81820	0.909099	−	0.416580i	$-0.136771\pi$
$239$	6717.97	1.81820	0.909099	+	0.416580i	$0.136771\pi$
$240$	0	0
$241$	−2252.84	−0.602149	−0.301075	−	0.953601i	$-0.597345\pi$
$241$	−2252.84	−0.602149	−0.301075	+	0.953601i	$0.597345\pi$
$242$	0	0
$243$	0	0
$244$	−71.5393	−0.0187698
$245$	−936.216	−0.244133
$246$	0	0
$247$	−2177.85	−0.561026
$248$	570.350	0.146037
$249$	0	0
$250$	−1892.74	−0.478828
$251$	−6008.91	−1.51107	−0.755535	−	0.655108i	$-0.772623\pi$
$251$	−6008.91	−1.51107	−0.755535	+	0.655108i	$0.772623\pi$
$252$	0	0
$253$	0	0
$254$	−492.283	−0.121609
$255$	0	0
$256$	256.000	0.0625000
$257$	−2969.20	−0.720675	−0.360338	−	0.932822i	$-0.617338\pi$
$257$	−2969.20	−0.720675	−0.360338	+	0.932822i	$0.617338\pi$
$258$	0	0
$259$	1720.14	0.412680
$260$	231.815	0.0552944
$261$	0	0
$262$	−4938.99	−1.16462
$263$	−4083.41	−0.957391	−0.478695	−	0.877981i	$-0.658890\pi$
$263$	−4083.41	−0.957391	−0.478695	+	0.877981i	$0.658890\pi$
$264$	0	0
$265$	−1047.67	−0.242860
$266$	3221.40	0.742544
$267$	0	0
$268$	−166.339	−0.0379133
$269$	−4833.74	−1.09561	−0.547804	−	0.836607i	$-0.684536\pi$
$269$	−4833.74	−1.09561	−0.547804	+	0.836607i	$0.684536\pi$
$270$	0	0
$271$	2549.77	0.571540	0.285770	−	0.958298i	$-0.407751\pi$
$271$	2549.77	0.571540	0.285770	+	0.958298i	$0.407751\pi$
$272$	1374.55	0.306413
$273$	0	0
$274$	−4188.87	−0.923573
$275$	0	0
$276$	0	0
$277$	−6014.45	−1.30460	−0.652298	−	0.757963i	$-0.726195\pi$
$277$	−6014.45	−1.30460	−0.652298	+	0.757963i	$0.726195\pi$
$278$	555.032	0.119743
$279$	0	0
$280$	−342.893	−0.0731848
$281$	−2629.67	−0.558266	−0.279133	−	0.960252i	$-0.590047\pi$
$281$	−2629.67	−0.558266	−0.279133	+	0.960252i	$0.590047\pi$
$282$	0	0
$283$	5244.73	1.10165	0.550825	−	0.834621i	$-0.314313\pi$
$283$	5244.73	1.10165	0.550825	+	0.834621i	$0.314313\pi$
$284$	−674.510	−0.140932
$285$	0	0
$286$	0	0
$287$	1187.96	0.244331
$288$	0	0
$289$	2467.43	0.502224
$290$	−103.051	−0.0208667
$291$	0	0
$292$	2281.75	0.457291
$293$	2800.58	0.558402	0.279201	−	0.960233i	$-0.409930\pi$
$293$	2800.58	0.558402	0.279201	+	0.960233i	$0.409930\pi$
$294$	0	0
$295$	−1932.48	−0.381402
$296$	−1300.76	−0.255424
$297$	0	0
$298$	−6929.36	−1.34700
$299$	−601.936	−0.116424
$300$	0	0
$301$	−585.856	−0.112187
$302$	−2735.14	−0.521157
$303$	0	0
$304$	−2436.02	−0.459590
$305$	−72.4600	−0.0136034
$306$	0	0
$307$	−1869.08	−0.347472	−0.173736	−	0.984792i	$-0.555584\pi$
$307$	−1869.08	−0.347472	−0.173736	+	0.984792i	$0.555584\pi$
$308$	0	0
$309$	0	0
$310$	577.690	0.105841
$311$	7026.00	1.28105	0.640527	−	0.767935i	$-0.278716\pi$
$311$	7026.00	1.28105	0.640527	+	0.767935i	$0.278716\pi$
$312$	0	0
$313$	6478.25	1.16988	0.584940	−	0.811077i	$-0.301118\pi$
$313$	6478.25	1.16988	0.584940	+	0.811077i	$0.301118\pi$
$314$	−3401.03	−0.611245
$315$	0	0
$316$	−5071.49	−0.902827
$317$	−7872.40	−1.39482	−0.697410	−	0.716672i	$-0.745664\pi$
$317$	−7872.40	−1.39482	−0.697410	+	0.716672i	$0.745664\pi$
$318$	0	0
$319$	0	0
$320$	259.295	0.0452970
$321$	0	0
$322$	890.362	0.154093
$323$	−13079.8	−2.25319
$324$	0	0
$325$	−1553.24	−0.265103
$326$	5282.33	0.897426
$327$	0	0
$328$	−898.331	−0.151226
$329$	−3657.96	−0.612978
$330$	0	0
$331$	3754.41	0.623447	0.311724	−	0.950173i	$-0.399094\pi$
$331$	3754.41	0.623447	0.311724	+	0.950173i	$0.399094\pi$
$332$	4099.70	0.677711
$333$	0	0
$334$	3739.39	0.612606
$335$	−168.480	−0.0274777
$336$	0	0
$337$	6498.15	1.05038	0.525189	−	0.850986i	$-0.323995\pi$
$337$	6498.15	1.05038	0.525189	+	0.850986i	$0.323995\pi$
$338$	−3984.77	−0.641252
$339$	0	0
$340$	1392.24	0.222073
$341$	0	0
$342$	0	0
$343$	6073.32	0.956060
$344$	443.024	0.0694367
$345$	0	0
$346$	−1230.48	−0.191188
$347$	−2822.59	−0.436671	−0.218335	−	0.975874i	$-0.570063\pi$
$347$	−2822.59	−0.436671	−0.218335	+	0.975874i	$0.570063\pi$
$348$	0	0
$349$	9128.39	1.40009	0.700045	−	0.714099i	$-0.253163\pi$
$349$	9128.39	1.40009	0.700045	+	0.714099i	$0.253163\pi$
$350$	2297.50	0.350876
$351$	0	0
$352$	0	0
$353$	8605.39	1.29750	0.648751	−	0.761000i	$-0.275291\pi$
$353$	8605.39	1.29750	0.648751	+	0.761000i	$0.275291\pi$
$354$	0	0
$355$	−683.192	−0.102141
$356$	−1648.36	−0.245402
$357$	0	0
$358$	3904.53	0.576427
$359$	−8031.13	−1.18069	−0.590344	−	0.807152i	$-0.701008\pi$
$359$	−8031.13	−1.18069	−0.590344	+	0.807152i	$0.701008\pi$
$360$	0	0
$361$	16321.4	2.37956
$362$	−8339.78	−1.21085
$363$	0	0
$364$	−605.315	−0.0871625
$365$	2311.11	0.331423
$366$	0	0
$367$	−9516.53	−1.35357	−0.676783	−	0.736182i	$-0.736627\pi$
$367$	−9516.53	−1.35357	−0.676783	+	0.736182i	$0.736627\pi$
$368$	−673.291	−0.0953742
$369$	0	0
$370$	−1317.51	−0.185119
$371$	2735.68	0.382829
$372$	0	0
$373$	12133.5	1.68431	0.842155	−	0.539235i	$-0.181286\pi$
$373$	12133.5	1.68431	0.842155	+	0.539235i	$0.181286\pi$
$374$	0	0
$375$	0	0
$376$	2766.14	0.379396
$377$	−181.918	−0.0248521
$378$	0	0
$379$	−1665.63	−0.225746	−0.112873	−	0.993609i	$-0.536005\pi$
$379$	−1665.63	−0.225746	−0.112873	+	0.993609i	$0.536005\pi$
$380$	−2467.37	−0.333088
$381$	0	0
$382$	−4823.70	−0.646078
$383$	−14614.0	−1.94972	−0.974858	−	0.222827i	$-0.928471\pi$
$383$	−14614.0	−1.94972	−0.974858	+	0.222827i	$0.928471\pi$
$384$	0	0
$385$	0	0
$386$	−200.705	−0.0264653
$387$	0	0
$388$	−4180.82	−0.547034
$389$	−9852.06	−1.28411	−0.642055	−	0.766658i	$-0.721918\pi$
$389$	−9852.06	−1.28411	−0.642055	+	0.766658i	$0.721918\pi$
$390$	0	0
$391$	−3615.13	−0.467583
$392$	−1848.64	−0.238190
$393$	0	0
$394$	−4735.54	−0.605515
$395$	−5136.76	−0.654325
$396$	0	0
$397$	−2933.26	−0.370822	−0.185411	−	0.982661i	$-0.559362\pi$
$397$	−2933.26	−0.370822	−0.185411	+	0.982661i	$0.559362\pi$
$398$	5531.35	0.696637
$399$	0	0
$400$	−1737.37	−0.217171
$401$	−8453.84	−1.05278	−0.526390	−	0.850243i	$-0.676455\pi$
$401$	−8453.84	−1.05278	−0.526390	+	0.850243i	$0.676455\pi$
$402$	0	0
$403$	1019.81	0.126055
$404$	4706.62	0.579612
$405$	0	0
$406$	269.086	0.0328929
$407$	0	0
$408$	0	0
$409$	1256.08	0.151856	0.0759279	−	0.997113i	$-0.475808\pi$
$409$	1256.08	0.151856	0.0759279	+	0.997113i	$0.475808\pi$
$410$	−909.893	−0.109601
$411$	0	0
$412$	5131.17	0.613579
$413$	5046.10	0.601217
$414$	0	0
$415$	4152.46	0.491172
$416$	457.738	0.0539483
$417$	0	0
$418$	0	0
$419$	−11397.1	−1.32885	−0.664423	−	0.747356i	$-0.731323\pi$
$419$	−11397.1	−1.32885	−0.664423	+	0.747356i	$0.731323\pi$
$420$	0	0
$421$	−14318.8	−1.65761	−0.828807	−	0.559535i	$-0.810980\pi$
$421$	−14318.8	−1.65761	−0.828807	+	0.559535i	$0.810980\pi$
$422$	−1338.19	−0.154365
$423$	0	0
$424$	−2068.72	−0.236948
$425$	−9328.52	−1.06470
$426$	0	0
$427$	189.208	0.0214436
$428$	4195.31	0.473804
$429$	0	0
$430$	448.726	0.0503243
$431$	13479.9	1.50650	0.753251	−	0.657734i	$-0.228485\pi$
$431$	13479.9	1.50650	0.753251	+	0.657734i	$0.228485\pi$
$432$	0	0
$433$	−8730.69	−0.968985	−0.484492	−	0.874796i	$-0.660996\pi$
$433$	−8730.69	−0.968985	−0.484492	+	0.874796i	$0.660996\pi$
$434$	−1508.47	−0.166840
$435$	0	0
$436$	−5583.42	−0.613297
$437$	6406.83	0.701328
$438$	0	0
$439$	8698.33	0.945669	0.472834	−	0.881151i	$-0.343231\pi$
$439$	8698.33	0.945669	0.472834	+	0.881151i	$0.343231\pi$
$440$	0	0
$441$	0	0
$442$	2457.75	0.264487
$443$	−10751.3	−1.15306	−0.576532	−	0.817074i	$-0.695595\pi$
$443$	−10751.3	−1.15306	−0.576532	+	0.817074i	$0.695595\pi$
$444$	0	0
$445$	−1669.58	−0.177855
$446$	5368.07	0.569923
$447$	0	0
$448$	−677.071	−0.0714031
$449$	8166.88	0.858394	0.429197	−	0.903211i	$-0.358797\pi$
$449$	8166.88	0.858394	0.429197	+	0.903211i	$0.358797\pi$
$450$	0	0
$451$	0	0
$452$	−5663.60	−0.589365
$453$	0	0
$454$	1071.83	0.110800
$455$	−613.106	−0.0631711
$456$	0	0
$457$	5359.76	0.548619	0.274310	−	0.961641i	$-0.411551\pi$
$457$	5359.76	0.548619	0.274310	+	0.961641i	$0.411551\pi$
$458$	25.4785	0.00259941
$459$	0	0
$460$	−681.956	−0.0691226
$461$	−16998.8	−1.71738	−0.858690	−	0.512495i	$-0.828721\pi$
$461$	−16998.8	−1.71738	−0.858690	+	0.512495i	$0.828721\pi$
$462$	0	0
$463$	−6298.12	−0.632178	−0.316089	−	0.948730i	$-0.602370\pi$
$463$	−6298.12	−0.632178	−0.316089	+	0.948730i	$0.602370\pi$
$464$	−203.483	−0.0203587
$465$	0	0
$466$	12970.5	1.28937
$467$	−1987.61	−0.196950	−0.0984752	−	0.995140i	$-0.531397\pi$
$467$	−1987.61	−0.196950	−0.0984752	+	0.995140i	$0.531397\pi$
$468$	0	0
$469$	439.935	0.0433141
$470$	2801.74	0.274968
$471$	0	0
$472$	−3815.86	−0.372117
$473$	0	0
$474$	0	0
$475$	16532.3	1.59695
$476$	−3635.42	−0.350062
$477$	0	0
$478$	13435.9	1.28566
$479$	−10489.4	−1.00057	−0.500283	−	0.865862i	$-0.666771\pi$
$479$	−10489.4	−1.00057	−0.500283	+	0.865862i	$0.666771\pi$
$480$	0	0
$481$	−2325.82	−0.220475
$482$	−4505.67	−0.425784
$483$	0	0
$484$	0	0
$485$	−4234.63	−0.396463
$486$	0	0
$487$	14379.3	1.33796	0.668980	−	0.743281i	$-0.266731\pi$
$487$	14379.3	1.33796	0.668980	+	0.743281i	$0.266731\pi$
$488$	−143.079	−0.0132723
$489$	0	0
$490$	−1872.43	−0.172628
$491$	12154.6	1.11717	0.558585	−	0.829447i	$-0.311344\pi$
$491$	12154.6	1.11717	0.558585	+	0.829447i	$0.311344\pi$
$492$	0	0
$493$	−1092.57	−0.0998108
$494$	−4355.70	−0.396705
$495$	0	0
$496$	1140.70	0.103264
$497$	1783.95	0.161008
$498$	0	0
$499$	−7864.79	−0.705564	−0.352782	−	0.935706i	$-0.614764\pi$
$499$	−7864.79	−0.705564	−0.352782	+	0.935706i	$0.614764\pi$
$500$	−3785.47	−0.338583
$501$	0	0
$502$	−12017.8	−1.06849
$503$	−8269.20	−0.733013	−0.366507	−	0.930415i	$-0.619446\pi$
$503$	−8269.20	−0.733013	−0.366507	+	0.930415i	$0.619446\pi$
$504$	0	0
$505$	4767.20	0.420074
$506$	0	0
$507$	0	0
$508$	−984.567	−0.0859903
$509$	15996.1	1.39296	0.696480	−	0.717577i	$-0.254749\pi$
$509$	15996.1	1.39296	0.696480	+	0.717577i	$0.254749\pi$
$510$	0	0
$511$	−6034.78	−0.522432
$512$	512.000	0.0441942
$513$	0	0
$514$	−5938.40	−0.509594
$515$	5197.21	0.444692
$516$	0	0
$517$	0	0
$518$	3440.27	0.291809
$519$	0	0
$520$	463.630	0.0390991
$521$	−10545.2	−0.886745	−0.443372	−	0.896337i	$-0.646218\pi$
$521$	−10545.2	−0.886745	−0.443372	+	0.896337i	$0.646218\pi$
$522$	0	0
$523$	−3525.37	−0.294749	−0.147375	−	0.989081i	$-0.547082\pi$
$523$	−3525.37	−0.294749	−0.147375	+	0.989081i	$0.547082\pi$
$524$	−9877.97	−0.823514
$525$	0	0
$526$	−8166.82	−0.676978
$527$	6124.80	0.506263
$528$	0	0
$529$	−10396.2	−0.854460
$530$	−2095.34	−0.171728
$531$	0	0
$532$	6442.80	0.525058
$533$	−1606.25	−0.130534
$534$	0	0
$535$	4249.31	0.343390
$536$	−332.678	−0.0268088
$537$	0	0
$538$	−9667.48	−0.774712
$539$	0	0
$540$	0	0
$541$	−2457.31	−0.195283	−0.0976413	−	0.995222i	$-0.531130\pi$
$541$	−2457.31	−0.195283	−0.0976413	+	0.995222i	$0.531130\pi$
$542$	5099.54	0.404140
$543$	0	0
$544$	2749.10	0.216667
$545$	−5655.28	−0.444488
$546$	0	0
$547$	19587.0	1.53104	0.765522	−	0.643410i	$-0.222481\pi$
$547$	19587.0	1.53104	0.765522	+	0.643410i	$0.222481\pi$
$548$	−8377.75	−0.653065
$549$	0	0
$550$	0	0
$551$	1936.28	0.149706
$552$	0	0
$553$	13413.1	1.03143
$554$	−12028.9	−0.922489
$555$	0	0
$556$	1110.06	0.0846712
$557$	5128.40	0.390121	0.195061	−	0.980791i	$-0.437510\pi$
$557$	5128.40	0.390121	0.195061	+	0.980791i	$0.437510\pi$
$558$	0	0
$559$	792.144	0.0599358
$560$	−685.785	−0.0517495
$561$	0	0
$562$	−5259.34	−0.394754
$563$	10130.4	0.758340	0.379170	−	0.925327i	$-0.376210\pi$
$563$	10130.4	0.758340	0.379170	+	0.925327i	$0.376210\pi$
$564$	0	0
$565$	−5736.49	−0.427143
$566$	10489.5	0.778984
$567$	0	0
$568$	−1349.02	−0.0996543
$569$	−12404.6	−0.913933	−0.456966	−	0.889484i	$-0.651064\pi$
$569$	−12404.6	−0.913933	−0.456966	+	0.889484i	$0.651064\pi$
$570$	0	0
$571$	15435.5	1.13127	0.565636	−	0.824655i	$-0.308631\pi$
$571$	15435.5	1.13127	0.565636	+	0.824655i	$0.308631\pi$
$572$	0	0
$573$	0	0
$574$	2375.91	0.172768
$575$	4569.35	0.331400
$576$	0	0
$577$	17813.7	1.28526	0.642629	−	0.766178i	$-0.277844\pi$
$577$	17813.7	1.28526	0.642629	+	0.766178i	$0.277844\pi$
$578$	4934.86	0.355126
$579$	0	0
$580$	−206.102	−0.0147550
$581$	−10842.9	−0.774251
$582$	0	0
$583$	0	0
$584$	4563.49	0.323354
$585$	0	0
$586$	5601.17	0.394850
$587$	−3302.38	−0.232204	−0.116102	−	0.993237i	$-0.537040\pi$
$587$	−3302.38	−0.232204	−0.116102	+	0.993237i	$0.537040\pi$
$588$	0	0
$589$	−10854.5	−0.759344
$590$	−3864.97	−0.269692
$591$	0	0
$592$	−2601.53	−0.180612
$593$	−358.816	−0.0248479	−0.0124239	−	0.999923i	$-0.503955\pi$
$593$	−358.816	−0.0248479	−0.0124239	+	0.999923i	$0.503955\pi$
$594$	0	0
$595$	−3682.21	−0.253708
$596$	−13858.7	−0.952476
$597$	0	0
$598$	−1203.87	−0.0823244
$599$	−10906.2	−0.743934	−0.371967	−	0.928246i	$-0.621316\pi$
$599$	−10906.2	−0.743934	−0.371967	+	0.928246i	$0.621316\pi$
$600$	0	0
$601$	25770.7	1.74910	0.874550	−	0.484936i	$-0.161157\pi$
$601$	25770.7	1.74910	0.874550	+	0.484936i	$0.161157\pi$
$602$	−1171.71	−0.0793280
$603$	0	0
$604$	−5470.27	−0.368514
$605$	0	0
$606$	0	0
$607$	−13681.1	−0.914827	−0.457413	−	0.889254i	$-0.651224\pi$
$607$	−13681.1	−0.914827	−0.457413	+	0.889254i	$0.651224\pi$
$608$	−4872.04	−0.324979
$609$	0	0
$610$	−144.920	−0.00961908
$611$	4945.97	0.327484
$612$	0	0
$613$	25648.4	1.68994	0.844968	−	0.534817i	$-0.179619\pi$
$613$	25648.4	1.68994	0.844968	+	0.534817i	$0.179619\pi$
$614$	−3738.16	−0.245700
$615$	0	0
$616$	0	0
$617$	3399.28	0.221799	0.110899	−	0.993832i	$-0.464627\pi$
$617$	3399.28	0.221799	0.110899	+	0.993832i	$0.464627\pi$
$618$	0	0
$619$	7953.51	0.516443	0.258222	−	0.966086i	$-0.416864\pi$
$619$	7953.51	0.516443	0.258222	+	0.966086i	$0.416864\pi$
$620$	1155.38	0.0748407
$621$	0	0
$622$	14052.0	0.905842
$623$	4359.60	0.280359
$624$	0	0
$625$	9738.99	0.623296
$626$	12956.5	0.827230
$627$	0	0
$628$	−6802.05	−0.432216
$629$	−13968.5	−0.885469
$630$	0	0
$631$	−4065.79	−0.256508	−0.128254	−	0.991741i	$-0.540937\pi$
$631$	−4065.79	−0.256508	−0.128254	+	0.991741i	$0.540937\pi$
$632$	−10143.0	−0.638395
$633$	0	0
$634$	−15744.8	−0.986287
$635$	−997.239	−0.0623216
$636$	0	0
$637$	−3305.44	−0.205599
$638$	0	0
$639$	0	0
$640$	518.590	0.0320298
$641$	−14820.5	−0.913218	−0.456609	−	0.889667i	$-0.650936\pi$
$641$	−14820.5	−0.913218	−0.456609	+	0.889667i	$0.650936\pi$
$642$	0	0
$643$	−4672.16	−0.286550	−0.143275	−	0.989683i	$-0.545763\pi$
$643$	−4672.16	−0.286550	−0.143275	+	0.989683i	$0.545763\pi$
$644$	1780.72	0.108960
$645$	0	0
$646$	−26159.6	−1.59325
$647$	16818.1	1.02193	0.510964	−	0.859602i	$-0.329289\pi$
$647$	16818.1	1.02193	0.510964	+	0.859602i	$0.329289\pi$
$648$	0	0
$649$	0	0
$650$	−3106.48	−0.187456
$651$	0	0
$652$	10564.7	0.634576
$653$	32163.6	1.92750	0.963750	−	0.266807i	$-0.0859686\pi$
$653$	32163.6	1.92750	0.963750	+	0.266807i	$0.0859686\pi$
$654$	0	0
$655$	−10005.1	−0.596843
$656$	−1796.66	−0.106933
$657$	0	0
$658$	−7315.91	−0.433441
$659$	17816.7	1.05317	0.526585	−	0.850122i	$-0.323472\pi$
$659$	17816.7	1.05317	0.526585	+	0.850122i	$0.323472\pi$
$660$	0	0
$661$	19855.6	1.16837	0.584186	−	0.811620i	$-0.301414\pi$
$661$	19855.6	1.16837	0.584186	+	0.811620i	$0.301414\pi$
$662$	7508.82	0.440844
$663$	0	0
$664$	8199.39	0.479214
$665$	6525.73	0.380537
$666$	0	0
$667$	535.168	0.0310671
$668$	7478.79	0.433178
$669$	0	0
$670$	−336.960	−0.0194297
$671$	0	0
$672$	0	0
$673$	25414.2	1.45564	0.727819	−	0.685769i	$-0.240534\pi$
$673$	25414.2	1.45564	0.727819	+	0.685769i	$0.240534\pi$
$674$	12996.3	0.742729
$675$	0	0
$676$	−7969.54	−0.453433
$677$	21739.8	1.23416	0.617080	−	0.786900i	$-0.288315\pi$
$677$	21739.8	1.23416	0.617080	+	0.786900i	$0.288315\pi$
$678$	0	0
$679$	11057.5	0.624958
$680$	2784.48	0.157030
$681$	0	0
$682$	0	0
$683$	6498.39	0.364062	0.182031	−	0.983293i	$-0.441733\pi$
$683$	6498.39	0.364062	0.182031	+	0.983293i	$0.441733\pi$
$684$	0	0
$685$	−8485.57	−0.473310
$686$	12146.6	0.676037
$687$	0	0
$688$	886.047	0.0490992
$689$	−3698.95	−0.204527
$690$	0	0
$691$	20318.6	1.11860	0.559302	−	0.828964i	$-0.311069\pi$
$691$	20318.6	1.11860	0.559302	+	0.828964i	$0.311069\pi$
$692$	−2460.96	−0.135190
$693$	0	0
$694$	−5645.19	−0.308773
$695$	1124.35	0.0613655
$696$	0	0
$697$	−9646.89	−0.524250
$698$	18256.8	0.990013
$699$	0	0
$700$	4595.00	0.248107
$701$	31563.2	1.70061	0.850304	−	0.526291i	$-0.176418\pi$
$701$	31563.2	1.70061	0.850304	+	0.526291i	$0.176418\pi$
$702$	0	0
$703$	24755.4	1.32812
$704$	0	0
$705$	0	0
$706$	17210.8	0.917473
$707$	−12448.1	−0.662177
$708$	0	0
$709$	−11022.2	−0.583845	−0.291923	−	0.956442i	$-0.594295\pi$
$709$	−11022.2	−0.583845	−0.291923	+	0.956442i	$0.594295\pi$
$710$	−1366.38	−0.0722246
$711$	0	0
$712$	−3296.72	−0.173525
$713$	−3000.09	−0.157579
$714$	0	0
$715$	0	0
$716$	7809.06	0.407595
$717$	0	0
$718$	−16062.3	−0.834872
$719$	29587.2	1.53465	0.767327	−	0.641257i	$-0.221587\pi$
$719$	29587.2	1.53465	0.767327	+	0.641257i	$0.221587\pi$
$720$	0	0
$721$	−13571.0	−0.700983
$722$	32642.8	1.68260
$723$	0	0
$724$	−16679.6	−0.856203
$725$	1380.95	0.0707412
$726$	0	0
$727$	22025.2	1.12362	0.561809	−	0.827267i	$-0.310106\pi$
$727$	22025.2	1.12362	0.561809	+	0.827267i	$0.310106\pi$
$728$	−1210.63	−0.0616332
$729$	0	0
$730$	4622.23	0.234351
$731$	4757.49	0.240714
$732$	0	0
$733$	−19945.2	−1.00504	−0.502519	−	0.864566i	$-0.667593\pi$
$733$	−19945.2	−1.00504	−0.502519	+	0.864566i	$0.667593\pi$
$734$	−19033.1	−0.957116
$735$	0	0
$736$	−1346.58	−0.0674397
$737$	0	0
$738$	0	0
$739$	−24432.3	−1.21618	−0.608091	−	0.793867i	$-0.708064\pi$
$739$	−24432.3	−1.21618	−0.608091	+	0.793867i	$0.708064\pi$
$740$	−2635.01	−0.130899
$741$	0	0
$742$	5471.36	0.270701
$743$	−8083.34	−0.399124	−0.199562	−	0.979885i	$-0.563952\pi$
$743$	−8083.34	−0.399124	−0.199562	+	0.979885i	$0.563952\pi$
$744$	0	0
$745$	−14037.1	−0.690308
$746$	24267.0	1.19099
$747$	0	0
$748$	0	0
$749$	−11095.8	−0.541297
$750$	0	0
$751$	−23769.3	−1.15493	−0.577467	−	0.816414i	$-0.695959\pi$
$751$	−23769.3	−1.15493	−0.577467	+	0.816414i	$0.695959\pi$
$752$	5532.28	0.268273
$753$	0	0
$754$	−363.835	−0.0175731
$755$	−5540.68	−0.267081
$756$	0	0
$757$	−1579.34	−0.0758282	−0.0379141	−	0.999281i	$-0.512071\pi$
$757$	−1579.34	−0.0758282	−0.0379141	+	0.999281i	$0.512071\pi$
$758$	−3331.26	−0.159626
$759$	0	0
$760$	−4934.74	−0.235529
$761$	−19422.8	−0.925200	−0.462600	−	0.886567i	$-0.653083\pi$
$761$	−19422.8	−0.925200	−0.462600	+	0.886567i	$0.653083\pi$
$762$	0	0
$763$	14767.1	0.700661
$764$	−9647.39	−0.456846
$765$	0	0
$766$	−29228.0	−1.37866
$767$	−6822.91	−0.321201
$768$	0	0
$769$	−17063.4	−0.800159	−0.400080	−	0.916480i	$-0.631018\pi$
$769$	−17063.4	−0.800159	−0.400080	+	0.916480i	$0.631018\pi$
$770$	0	0
$771$	0	0
$772$	−401.410	−0.0187138
$773$	12960.4	0.603044	0.301522	−	0.953459i	$-0.402505\pi$
$773$	12960.4	0.603044	0.301522	+	0.953459i	$0.402505\pi$
$774$	0	0
$775$	−7741.46	−0.358815
$776$	−8361.64	−0.386811
$777$	0	0
$778$	−19704.1	−0.908003
$779$	17096.5	0.786323
$780$	0	0
$781$	0	0
$782$	−7230.25	−0.330631
$783$	0	0
$784$	−3697.28	−0.168426
$785$	−6889.60	−0.313249
$786$	0	0
$787$	38382.6	1.73849	0.869246	−	0.494380i	$-0.164605\pi$
$787$	38382.6	1.73849	0.869246	+	0.494380i	$0.164605\pi$
$788$	−9471.07	−0.428164
$789$	0	0
$790$	−10273.5	−0.462678
$791$	14979.1	0.673320
$792$	0	0
$793$	−255.830	−0.0114562
$794$	−5866.52	−0.262211
$795$	0	0
$796$	11062.7	0.492597
$797$	37871.1	1.68314	0.841571	−	0.540146i	$-0.181631\pi$
$797$	37871.1	1.68314	0.841571	+	0.540146i	$0.181631\pi$
$798$	0	0
$799$	29704.7	1.31524
$800$	−3474.74	−0.153563
$801$	0	0
$802$	−16907.7	−0.744428
$803$	0	0
$804$	0	0
$805$	1803.64	0.0789691
$806$	2039.62	0.0891345
$807$	0	0
$808$	9413.24	0.409847
$809$	3528.60	0.153349	0.0766743	−	0.997056i	$-0.475570\pi$
$809$	3528.60	0.153349	0.0766743	+	0.997056i	$0.475570\pi$
$810$	0	0
$811$	−19000.1	−0.822669	−0.411334	−	0.911485i	$-0.634937\pi$
$811$	−19000.1	−0.822669	−0.411334	+	0.911485i	$0.634937\pi$
$812$	538.173	0.0232588
$813$	0	0
$814$	0	0
$815$	10700.6	0.459910
$816$	0	0
$817$	−8431.36	−0.361048
$818$	2512.16	0.107378
$819$	0	0
$820$	−1819.79	−0.0774996
$821$	5731.14	0.243627	0.121814	−	0.992553i	$-0.461129\pi$
$821$	5731.14	0.243627	0.121814	+	0.992553i	$0.461129\pi$
$822$	0	0
$823$	26892.1	1.13900	0.569501	−	0.821990i	$-0.307136\pi$
$823$	26892.1	1.13900	0.569501	+	0.821990i	$0.307136\pi$
$824$	10262.3	0.433866
$825$	0	0
$826$	10092.2	0.425125
$827$	−33406.7	−1.40467	−0.702337	−	0.711844i	$-0.747860\pi$
$827$	−33406.7	−1.40467	−0.702337	+	0.711844i	$0.747860\pi$
$828$	0	0
$829$	−29242.0	−1.22511	−0.612556	−	0.790427i	$-0.709859\pi$
$829$	−29242.0	−1.22511	−0.612556	+	0.790427i	$0.709859\pi$
$830$	8304.92	0.347311
$831$	0	0
$832$	915.477	0.0381472
$833$	−19851.9	−0.825725
$834$	0	0
$835$	7575.04	0.313946
$836$	0	0
$837$	0	0
$838$	−22794.3	−0.939637
$839$	−4368.65	−0.179765	−0.0898824	−	0.995952i	$-0.528649\pi$
$839$	−4368.65	−0.179765	−0.0898824	+	0.995952i	$0.528649\pi$
$840$	0	0
$841$	−24227.3	−0.993368
$842$	−28637.6	−1.17211
$843$	0	0
$844$	−2676.37	−0.109152
$845$	−8072.12	−0.328626
$846$	0	0
$847$	0	0
$848$	−4137.44	−0.167547
$849$	0	0
$850$	−18657.0	−0.752860
$851$	6842.13	0.275611
$852$	0	0
$853$	−3321.66	−0.133331	−0.0666656	−	0.997775i	$-0.521236\pi$
$853$	−3321.66	−0.133331	−0.0666656	+	0.997775i	$0.521236\pi$
$854$	378.415	0.0151629
$855$	0	0
$856$	8390.63	0.335030
$857$	44811.9	1.78617	0.893083	−	0.449892i	$-0.148537\pi$
$857$	44811.9	1.78617	0.893083	+	0.449892i	$0.148537\pi$
$858$	0	0
$859$	−1369.55	−0.0543985	−0.0271993	−	0.999630i	$-0.508659\pi$
$859$	−1369.55	−0.0543985	−0.0271993	+	0.999630i	$0.508659\pi$
$860$	897.451	0.0355847
$861$	0	0
$862$	26959.7	1.06526
$863$	25851.4	1.01969	0.509844	−	0.860267i	$-0.329703\pi$
$863$	25851.4	1.01969	0.509844	+	0.860267i	$0.329703\pi$
$864$	0	0
$865$	−2492.63	−0.0979793
$866$	−17461.4	−0.685176
$867$	0	0
$868$	−3016.93	−0.117974
$869$	0	0
$870$	0	0
$871$	−594.842	−0.0231406
$872$	−11166.8	−0.433666
$873$	0	0
$874$	12813.7	0.495914
$875$	10011.8	0.386814
$876$	0	0
$877$	33283.1	1.28152	0.640759	−	0.767742i	$-0.278620\pi$
$877$	33283.1	1.28152	0.640759	+	0.767742i	$0.278620\pi$
$878$	17396.7	0.668689
$879$	0	0
$880$	0	0
$881$	14423.9	0.551592	0.275796	−	0.961216i	$-0.411059\pi$
$881$	14423.9	0.551592	0.275796	+	0.961216i	$0.411059\pi$
$882$	0	0
$883$	23600.8	0.899467	0.449734	−	0.893163i	$-0.351519\pi$
$883$	23600.8	0.899467	0.449734	+	0.893163i	$0.351519\pi$
$884$	4915.51	0.187021
$885$	0	0
$886$	−21502.5	−0.815340
$887$	−29741.3	−1.12583	−0.562917	−	0.826514i	$-0.690321\pi$
$887$	−29741.3	−1.12583	−0.562917	+	0.826514i	$0.690321\pi$
$888$	0	0
$889$	2603.99	0.0982396
$890$	−3339.15	−0.125763
$891$	0	0
$892$	10736.1	0.402996
$893$	−52643.5	−1.97273
$894$	0	0
$895$	7909.57	0.295405
$896$	−1354.14	−0.0504896
$897$	0	0
$898$	16333.8	0.606976
$899$	−906.689	−0.0336371
$900$	0	0
$901$	−22215.3	−0.821419
$902$	0	0
$903$	0	0
$904$	−11327.2	−0.416744
$905$	−16894.2	−0.620534
$906$	0	0
$907$	−29527.3	−1.08097	−0.540484	−	0.841354i	$-0.681759\pi$
$907$	−29527.3	−1.08097	−0.540484	+	0.841354i	$0.681759\pi$
$908$	2143.65	0.0783476
$909$	0	0
$910$	−1226.21	−0.0446687
$911$	43163.4	1.56978	0.784888	−	0.619638i	$-0.212720\pi$
$911$	43163.4	1.56978	0.784888	+	0.619638i	$0.212720\pi$
$912$	0	0
$913$	0	0
$914$	10719.5	0.387932
$915$	0	0
$916$	50.9570	0.00183806
$917$	26125.3	0.940823
$918$	0	0
$919$	22365.0	0.802780	0.401390	−	0.915907i	$-0.368527\pi$
$919$	22365.0	0.802780	0.401390	+	0.915907i	$0.368527\pi$
$920$	−1363.91	−0.0488770
$921$	0	0
$922$	−33997.6	−1.21437
$923$	−2412.10	−0.0860188
$924$	0	0
$925$	17655.5	0.627578
$926$	−12596.2	−0.447017
$927$	0	0
$928$	−406.965	−0.0143958
$929$	26590.1	0.939066	0.469533	−	0.882915i	$-0.344422\pi$
$929$	26590.1	0.939066	0.469533	+	0.882915i	$0.344422\pi$
$930$	0	0
$931$	35182.2	1.23851
$932$	25941.0	0.911724
$933$	0	0
$934$	−3975.23	−0.139265
$935$	0	0
$936$	0	0
$937$	−39891.6	−1.39082	−0.695412	−	0.718611i	$-0.744778\pi$
$937$	−39891.6	−1.39082	−0.695412	+	0.718611i	$0.744778\pi$
$938$	879.870	0.0306277
$939$	0	0
$940$	5603.48	0.194431
$941$	−6601.96	−0.228712	−0.114356	−	0.993440i	$-0.536480\pi$
$941$	−6601.96	−0.228712	−0.114356	+	0.993440i	$0.536480\pi$
$942$	0	0
$943$	4725.30	0.163178
$944$	−7631.71	−0.263126
$945$	0	0
$946$	0	0
$947$	−9073.67	−0.311357	−0.155678	−	0.987808i	$-0.549756\pi$
$947$	−9073.67	−0.311357	−0.155678	+	0.987808i	$0.549756\pi$
$948$	0	0
$949$	8159.71	0.279110
$950$	33064.5	1.12922
$951$	0	0
$952$	−7270.85	−0.247531
$953$	10799.7	0.367088	0.183544	−	0.983011i	$-0.441243\pi$
$953$	10799.7	0.367088	0.183544	+	0.983011i	$0.441243\pi$
$954$	0	0
$955$	−9771.56	−0.331100
$956$	26871.9	0.909099
$957$	0	0
$958$	−20978.7	−0.707507
$959$	22157.5	0.746094
$960$	0	0
$961$	−24708.2	−0.829385
$962$	−4651.64	−0.155899
$963$	0	0
$964$	−9011.35	−0.301075
$965$	−406.576	−0.0135629
$966$	0	0
$967$	−23167.7	−0.770448	−0.385224	−	0.922823i	$-0.625876\pi$
$967$	−23167.7	−0.770448	−0.385224	+	0.922823i	$0.625876\pi$
$968$	0	0
$969$	0	0
$970$	−8469.26	−0.280342
$971$	−8045.07	−0.265889	−0.132945	−	0.991123i	$-0.542443\pi$
$971$	−8045.07	−0.265889	−0.132945	+	0.991123i	$0.542443\pi$
$972$	0	0
$973$	−2935.90	−0.0967325
$974$	28758.5	0.946080
$975$	0	0
$976$	−286.157	−0.00938490
$977$	35161.1	1.15138	0.575692	−	0.817666i	$-0.304733\pi$
$977$	35161.1	1.15138	0.575692	+	0.817666i	$0.304733\pi$
$978$	0	0
$979$	0	0
$980$	−3744.86	−0.122067
$981$	0	0
$982$	24309.3	0.789959
$983$	32507.0	1.05474	0.527372	−	0.849635i	$-0.323178\pi$
$983$	32507.0	1.05474	0.527372	+	0.849635i	$0.323178\pi$
$984$	0	0
$985$	−9592.97	−0.310312
$986$	−2185.13	−0.0705769
$987$	0	0
$988$	−8711.40	−0.280513
$989$	−2330.34	−0.0749247
$990$	0	0
$991$	−12751.9	−0.408758	−0.204379	−	0.978892i	$-0.565517\pi$
$991$	−12751.9	−0.408758	−0.204379	+	0.978892i	$0.565517\pi$
$992$	2281.40	0.0730186
$993$	0	0
$994$	3567.90	0.113850
$995$	11205.1	0.357010
$996$	0	0
$997$	21597.1	0.686044	0.343022	−	0.939327i	$-0.388550\pi$
$997$	21597.1	0.686044	0.343022	+	0.939327i	$0.388550\pi$
$998$	−15729.6	−0.498909
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2178.4.a.cf.1.4		6
3.2	odd	2		2178.4.a.ce.1.3		6
11.3	even	5		198.4.f.g.163.2	✓	12
11.4	even	5		198.4.f.g.181.2	yes	12
11.10	odd	2		2178.4.a.cd.1.4		6
33.14	odd	10		198.4.f.h.163.2	yes	12
33.26	odd	10		198.4.f.h.181.2	yes	12
33.32	even	2		2178.4.a.cg.1.3		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
198.4.f.g.163.2	✓	12	11.3	even	5
198.4.f.g.181.2	yes	12	11.4	even	5
198.4.f.h.163.2	yes	12	33.14	odd	10
198.4.f.h.181.2	yes	12	33.26	odd	10
2178.4.a.cd.1.4		6	11.10	odd	2
2178.4.a.ce.1.3		6	3.2	odd	2
2178.4.a.cf.1.4		6	1.1	even	1	trivial
2178.4.a.cg.1.3		6	33.32	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:	\( N \)	\(=\)	\( 2178 = 2 \cdot 3^{2} \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2178.a (trivial)

\(f(q)\)	\(=\)	\(q+2.00000 q^{2} +4.00000 q^{4} +4.05148 q^{5} -10.5792 q^{7} +8.00000 q^{8} +O(q^{10})\)	\(q+2.00000 q^{2} +4.00000 q^{4} +4.05148 q^{5} -10.5792 q^{7} +8.00000 q^{8} +8.10296 q^{10} +14.3043 q^{13} -21.1585 q^{14} +16.0000 q^{16} +85.9094 q^{17} -152.251 q^{19} +16.2059 q^{20} -42.0807 q^{23} -108.585 q^{25} +28.6087 q^{26} -42.3169 q^{28} -12.7177 q^{29} +71.2937 q^{31} +32.0000 q^{32} +171.819 q^{34} -42.8616 q^{35} -162.596 q^{37} -304.502 q^{38} +32.4119 q^{40} -112.291 q^{41} +55.3779 q^{43} -84.1613 q^{46} +345.768 q^{47} -231.080 q^{49} -217.171 q^{50} +57.2173 q^{52} -258.590 q^{53} -84.6339 q^{56} -25.4353 q^{58} -476.982 q^{59} -17.8848 q^{61} +142.587 q^{62} +64.0000 q^{64} +57.9537 q^{65} -41.5848 q^{67} +343.638 q^{68} -85.7232 q^{70} -168.628 q^{71} +570.437 q^{73} -325.191 q^{74} -609.005 q^{76} -1267.87 q^{79} +64.8237 q^{80} -224.583 q^{82} +1024.92 q^{83} +348.061 q^{85} +110.756 q^{86} -412.090 q^{89} -151.329 q^{91} -168.323 q^{92} +691.535 q^{94} -616.843 q^{95} -1045.21 q^{97} -462.160 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 12 q^{2} + 24 q^{4} - 17 q^{5} - 7 q^{7} + 48 q^{8}+O(q^{10}) \) 6 * q + 12 * q^2 + 24 * q^4 - 17 * q^5 - 7 * q^7 + 48 * q^8	\( 6 q + 12 q^{2} + 24 q^{4} - 17 q^{5} - 7 q^{7} + 48 q^{8} - 34 q^{10} - 68 q^{13} - 14 q^{14} + 96 q^{16} + 42 q^{17} - 98 q^{19} - 68 q^{20} - 210 q^{23} + 47 q^{25} - 136 q^{26} - 28 q^{28} + 13 q^{29} - 125 q^{31} + 192 q^{32} + 84 q^{34} + 534 q^{35} + 282 q^{37} - 196 q^{38} - 136 q^{40} - 170 q^{41} - 868 q^{43} - 420 q^{46} - 782 q^{47} - 439 q^{49} + 94 q^{50} - 272 q^{52} - 645 q^{53} - 56 q^{56} + 26 q^{58} - 507 q^{59} - 1772 q^{61} - 250 q^{62} + 384 q^{64} - 1856 q^{65} + 686 q^{67} + 168 q^{68} + 1068 q^{70} - 2782 q^{71} - 335 q^{73} + 564 q^{74} - 392 q^{76} - 127 q^{79} - 272 q^{80} - 340 q^{82} - 9 q^{83} - 370 q^{85} - 1736 q^{86} - 2526 q^{89} + 296 q^{91} - 840 q^{92} - 1564 q^{94} - 1194 q^{95} + 89 q^{97} - 878 q^{98}+O(q^{100}) \) 6 * q + 12 * q^2 + 24 * q^4 - 17 * q^5 - 7 * q^7 + 48 * q^8 - 34 * q^10 - 68 * q^13 - 14 * q^14 + 96 * q^16 + 42 * q^17 - 98 * q^19 - 68 * q^20 - 210 * q^23 + 47 * q^25 - 136 * q^26 - 28 * q^28 + 13 * q^29 - 125 * q^31 + 192 * q^32 + 84 * q^34 + 534 * q^35 + 282 * q^37 - 196 * q^38 - 136 * q^40 - 170 * q^41 - 868 * q^43 - 420 * q^46 - 782 * q^47 - 439 * q^49 + 94 * q^50 - 272 * q^52 - 645 * q^53 - 56 * q^56 + 26 * q^58 - 507 * q^59 - 1772 * q^61 - 250 * q^62 + 384 * q^64 - 1856 * q^65 + 686 * q^67 + 168 * q^68 + 1068 * q^70 - 2782 * q^71 - 335 * q^73 + 564 * q^74 - 392 * q^76 - 127 * q^79 - 272 * q^80 - 340 * q^82 - 9 * q^83 - 370 * q^85 - 1736 * q^86 - 2526 * q^89 + 296 * q^91 - 840 * q^92 - 1564 * q^94 - 1194 * q^95 + 89 * q^97 - 878 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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