LMFDB - Embedded newform 2178.4.a.by.1.3

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			1.3
Root			$8.19378$ 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)

$f(q)$	$=$	$q+2.00000 q^{2} +4.00000 q^{4} -5.40810 q^{5} -22.1498 q^{7} +8.00000 q^{8} +O(q^{10})$	$q+2.00000 q^{2} +4.00000 q^{4} -5.40810 q^{5} -22.1498 q^{7} +8.00000 q^{8} -10.8162 q^{10} +76.8269 q^{13} -44.2996 q^{14} +16.0000 q^{16} -59.2883 q^{17} +95.2626 q^{19} -21.6324 q^{20} -142.484 q^{23} -95.7524 q^{25} +153.654 q^{26} -88.5993 q^{28} +20.4183 q^{29} +213.304 q^{31} +32.0000 q^{32} -118.577 q^{34} +119.788 q^{35} -145.578 q^{37} +190.525 q^{38} -43.2648 q^{40} -82.8326 q^{41} -151.373 q^{43} -284.968 q^{46} -90.3643 q^{47} +147.615 q^{49} -191.505 q^{50} +307.308 q^{52} +234.849 q^{53} -177.199 q^{56} +40.8366 q^{58} -302.497 q^{59} -149.279 q^{61} +426.608 q^{62} +64.0000 q^{64} -415.488 q^{65} +826.236 q^{67} -237.153 q^{68} +239.577 q^{70} +898.965 q^{71} -137.993 q^{73} -291.155 q^{74} +381.050 q^{76} -304.459 q^{79} -86.5296 q^{80} -165.665 q^{82} +764.294 q^{83} +320.637 q^{85} -302.746 q^{86} +313.100 q^{89} -1701.70 q^{91} -569.936 q^{92} -180.729 q^{94} -515.189 q^{95} -582.505 q^{97} +295.229 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	−5.40810	−0.483715	−0.241858	−	0.970312i	$-0.577757\pi$
$5$	−5.40810	−0.483715	−0.241858	+	0.970312i	$0.577757\pi$
$6$	0	0
$7$	−22.1498	−1.19598	−0.597989	−	0.801504i	$-0.704033\pi$
$7$	−22.1498	−1.19598	−0.597989	+	0.801504i	$0.704033\pi$
$8$	8.00000	0.353553
$9$	0	0
$10$	−10.8162	−0.342038
$11$	0	0
$11$	0	0
$12$	0	0
$13$	76.8269	1.63907	0.819537	−	0.573027i	$-0.194231\pi$
$13$	76.8269	1.63907	0.819537	+	0.573027i	$0.194231\pi$
$14$	−44.2996	−0.845684
$15$	0	0
$16$	16.0000	0.250000
$17$	−59.2883	−0.845855	−0.422927	−	0.906164i	$-0.638997\pi$
$17$	−59.2883	−0.845855	−0.422927	+	0.906164i	$0.638997\pi$
$18$	0	0
$19$	95.2626	1.15025	0.575124	−	0.818066i	$-0.304954\pi$
$19$	95.2626	1.15025	0.575124	+	0.818066i	$0.304954\pi$
$20$	−21.6324	−0.241858
$21$	0	0
$22$	0	0
$23$	−142.484	−1.29174	−0.645868	−	0.763449i	$-0.723505\pi$
$23$	−142.484	−1.29174	−0.645868	+	0.763449i	$0.723505\pi$
$24$	0	0
$25$	−95.7524	−0.766020
$26$	153.654	1.15900
$27$	0	0
$28$	−88.5993	−0.597989
$29$	20.4183	0.130744	0.0653721	−	0.997861i	$-0.479177\pi$
$29$	20.4183	0.130744	0.0653721	+	0.997861i	$0.479177\pi$
$30$	0	0
$31$	213.304	1.23582	0.617912	−	0.786247i	$-0.287979\pi$
$31$	213.304	1.23582	0.617912	+	0.786247i	$0.287979\pi$
$32$	32.0000	0.176777
$33$	0	0
$34$	−118.577	−0.598110
$35$	119.788	0.578513
$36$	0	0
$37$	−145.578	−0.646832	−0.323416	−	0.946257i	$-0.604831\pi$
$37$	−145.578	−0.646832	−0.323416	+	0.946257i	$0.604831\pi$
$38$	190.525	0.813349
$39$	0	0
$40$	−43.2648	−0.171019
$41$	−82.8326	−0.315519	−0.157759	−	0.987478i	$-0.550427\pi$
$41$	−82.8326	−0.315519	−0.157759	+	0.987478i	$0.550427\pi$
$42$	0	0
$43$	−151.373	−0.536841	−0.268420	−	0.963302i	$-0.586502\pi$
$43$	−151.373	−0.536841	−0.268420	+	0.963302i	$0.586502\pi$
$44$	0	0
$45$	0	0
$46$	−284.968	−0.913396
$47$	−90.3643	−0.280447	−0.140223	−	0.990120i	$-0.544782\pi$
$47$	−90.3643	−0.280447	−0.140223	+	0.990120i	$0.544782\pi$
$48$	0	0
$49$	147.615	0.430363
$50$	−191.505	−0.541658
$51$	0	0
$52$	307.308	0.819537
$53$	234.849	0.608660	0.304330	−	0.952567i	$-0.401567\pi$
$53$	234.849	0.608660	0.304330	+	0.952567i	$0.401567\pi$
$54$	0	0
$55$	0	0
$56$	−177.199	−0.422842
$57$	0	0
$58$	40.8366	0.0924502
$59$	−302.497	−0.667488	−0.333744	−	0.942664i	$-0.608312\pi$
$59$	−302.497	−0.667488	−0.333744	+	0.942664i	$0.608312\pi$
$60$	0	0
$61$	−149.279	−0.313332	−0.156666	−	0.987652i	$-0.550075\pi$
$61$	−149.279	−0.313332	−0.156666	+	0.987652i	$0.550075\pi$
$62$	426.608	0.873860
$63$	0	0
$64$	64.0000	0.125000
$65$	−415.488	−0.792845
$66$	0	0
$67$	826.236	1.50658	0.753290	−	0.657689i	$-0.228466\pi$
$67$	826.236	1.50658	0.753290	+	0.657689i	$0.228466\pi$
$68$	−237.153	−0.422927
$69$	0	0
$70$	239.577	0.409070
$71$	898.965	1.50264	0.751321	−	0.659937i	$-0.229417\pi$
$71$	898.965	1.50264	0.751321	+	0.659937i	$0.229417\pi$
$72$	0	0
$73$	−137.993	−0.221245	−0.110623	−	0.993862i	$-0.535284\pi$
$73$	−137.993	−0.221245	−0.110623	+	0.993862i	$0.535284\pi$
$74$	−291.155	−0.457380
$75$	0	0
$76$	381.050	0.575124
$77$	0	0
$78$	0	0
$79$	−304.459	−0.433598	−0.216799	−	0.976216i	$-0.569562\pi$
$79$	−304.459	−0.433598	−0.216799	+	0.976216i	$0.569562\pi$
$80$	−86.5296	−0.120929
$81$	0	0
$82$	−165.665	−0.223106
$83$	764.294	1.01075	0.505374	−	0.862900i	$-0.331354\pi$
$83$	764.294	1.01075	0.505374	+	0.862900i	$0.331354\pi$
$84$	0	0
$85$	320.637	0.409153
$86$	−302.746	−0.379604
$87$	0	0
$88$	0	0
$89$	313.100	0.372905	0.186452	−	0.982464i	$-0.440301\pi$
$89$	313.100	0.372905	0.186452	+	0.982464i	$0.440301\pi$
$90$	0	0
$91$	−1701.70	−1.96030
$92$	−569.936	−0.645868
$93$	0	0
$94$	−180.729	−0.198306
$95$	−515.189	−0.556393
$96$	0	0
$97$	−582.505	−0.609737	−0.304868	−	0.952394i	$-0.598612\pi$
$97$	−582.505	−0.609737	−0.304868	+	0.952394i	$0.598612\pi$
$98$	295.229	0.304313
$99$	0	0
$100$	−383.010	−0.383010
$101$	141.554	0.139457	0.0697286	−	0.997566i	$-0.477787\pi$
$101$	141.554	0.139457	0.0697286	+	0.997566i	$0.477787\pi$
$102$	0	0
$103$	841.380	0.804890	0.402445	−	0.915444i	$-0.368161\pi$
$103$	841.380	0.804890	0.402445	+	0.915444i	$0.368161\pi$
$104$	614.615	0.579500
$105$	0	0
$106$	469.698	0.430387
$107$	64.4319	0.0582137	0.0291069	−	0.999576i	$-0.490734\pi$
$107$	64.4319	0.0582137	0.0291069	+	0.999576i	$0.490734\pi$
$108$	0	0
$109$	1559.04	1.36999	0.684995	−	0.728547i	$-0.259804\pi$
$109$	1559.04	1.36999	0.684995	+	0.728547i	$0.259804\pi$
$110$	0	0
$111$	0	0
$112$	−354.397	−0.298994
$113$	2239.14	1.86408	0.932039	−	0.362359i	$-0.118029\pi$
$113$	2239.14	1.86408	0.932039	+	0.362359i	$0.118029\pi$
$114$	0	0
$115$	770.567	0.624833
$116$	81.6732	0.0653721
$117$	0	0
$118$	−604.995	−0.471986
$119$	1313.23	1.01162
$120$	0	0
$121$	0	0
$122$	−298.559	−0.221559
$123$	0	0
$124$	853.217	0.617912
$125$	1193.85	0.854251
$126$	0	0
$127$	1093.46	0.764004	0.382002	−	0.924162i	$-0.375235\pi$
$127$	1093.46	0.764004	0.382002	+	0.924162i	$0.375235\pi$
$128$	128.000	0.0883883
$129$	0	0
$130$	−830.976	−0.560626
$131$	2466.16	1.64481	0.822403	−	0.568906i	$-0.192633\pi$
$131$	2466.16	1.64481	0.822403	+	0.568906i	$0.192633\pi$
$132$	0	0
$133$	−2110.05	−1.37567
$134$	1652.47	1.06531
$135$	0	0
$136$	−474.307	−0.299055
$137$	973.286	0.606959	0.303480	−	0.952838i	$-0.401852\pi$
$137$	973.286	0.606959	0.303480	+	0.952838i	$0.401852\pi$
$138$	0	0
$139$	−1263.04	−0.770717	−0.385358	−	0.922767i	$-0.625922\pi$
$139$	−1263.04	−0.770717	−0.385358	+	0.922767i	$0.625922\pi$
$140$	479.154	0.289256
$141$	0	0
$142$	1797.93	1.06253
$143$	0	0
$144$	0	0
$145$	−110.424	−0.0632430
$146$	−275.987	−0.156444
$147$	0	0
$148$	−582.310	−0.323416
$149$	489.340	0.269049	0.134525	−	0.990910i	$-0.457049\pi$
$149$	489.340	0.269049	0.134525	+	0.990910i	$0.457049\pi$
$150$	0	0
$151$	1707.77	0.920376	0.460188	−	0.887822i	$-0.347782\pi$
$151$	1707.77	0.920376	0.460188	+	0.887822i	$0.347782\pi$
$152$	762.100	0.406674
$153$	0	0
$154$	0	0
$155$	−1153.57	−0.597787
$156$	0	0
$157$	2561.73	1.30222	0.651108	−	0.758985i	$-0.274304\pi$
$157$	2561.73	1.30222	0.651108	+	0.758985i	$0.274304\pi$
$158$	−608.917	−0.306600
$159$	0	0
$160$	−173.059	−0.0855096
$161$	3155.99	1.54489
$162$	0	0
$163$	1816.09	0.872681	0.436340	−	0.899782i	$-0.356274\pi$
$163$	1816.09	0.872681	0.436340	+	0.899782i	$0.356274\pi$
$164$	−331.330	−0.157759
$165$	0	0
$166$	1528.59	0.714707
$167$	−378.925	−0.175581	−0.0877907	−	0.996139i	$-0.527981\pi$
$167$	−378.925	−0.175581	−0.0877907	+	0.996139i	$0.527981\pi$
$168$	0	0
$169$	3705.38	1.68656
$170$	641.275	0.289315
$171$	0	0
$172$	−605.491	−0.268420
$173$	−50.7220	−0.0222909	−0.0111454	−	0.999938i	$-0.503548\pi$
$173$	−50.7220	−0.0222909	−0.0111454	+	0.999938i	$0.503548\pi$
$174$	0	0
$175$	2120.90	0.916142
$176$	0	0
$177$	0	0
$178$	626.199	0.263683
$179$	−20.8678	−0.00871359	−0.00435679	−	0.999991i	$-0.501387\pi$
$179$	−20.8678	−0.00871359	−0.00435679	+	0.999991i	$0.501387\pi$
$180$	0	0
$181$	−176.153	−0.0723391	−0.0361695	−	0.999346i	$-0.511516\pi$
$181$	−176.153	−0.0723391	−0.0361695	+	0.999346i	$0.511516\pi$
$182$	−3403.41	−1.38614
$183$	0	0
$184$	−1139.87	−0.456698
$185$	787.298	0.312883
$186$	0	0
$187$	0	0
$188$	−361.457	−0.140223
$189$	0	0
$190$	−1030.38	−0.393429
$191$	1159.22	0.439153	0.219576	−	0.975595i	$-0.429532\pi$
$191$	1159.22	0.439153	0.219576	+	0.975595i	$0.429532\pi$
$192$	0	0
$193$	−1284.94	−0.479232	−0.239616	−	0.970868i	$-0.577022\pi$
$193$	−1284.94	−0.479232	−0.239616	+	0.970868i	$0.577022\pi$
$194$	−1165.01	−0.431149
$195$	0	0
$196$	590.458	0.215181
$197$	2685.06	0.971078	0.485539	−	0.874215i	$-0.338623\pi$
$197$	2685.06	0.971078	0.485539	+	0.874215i	$0.338623\pi$
$198$	0	0
$199$	−1333.54	−0.475036	−0.237518	−	0.971383i	$-0.576334\pi$
$199$	−1333.54	−0.475036	−0.237518	+	0.971383i	$0.576334\pi$
$200$	−766.020	−0.270829
$201$	0	0
$202$	283.109	0.0986112
$203$	−452.262	−0.156367
$204$	0	0
$205$	447.967	0.152621
$206$	1682.76	0.569143
$207$	0	0
$208$	1229.23	0.409768
$209$	0	0
$210$	0	0
$211$	−1628.17	−0.531221	−0.265611	−	0.964080i	$-0.585574\pi$
$211$	−1628.17	−0.531221	−0.265611	+	0.964080i	$0.585574\pi$
$212$	939.395	0.304330
$213$	0	0
$214$	128.864	0.0411633
$215$	818.640	0.259678
$216$	0	0
$217$	−4724.65	−1.47802
$218$	3118.08	0.968730
$219$	0	0
$220$	0	0
$221$	−4554.94	−1.38642
$222$	0	0
$223$	−2628.19	−0.789224	−0.394612	−	0.918848i	$-0.629121\pi$
$223$	−2628.19	−0.789224	−0.394612	+	0.918848i	$0.629121\pi$
$224$	−708.794	−0.211421
$225$	0	0
$226$	4478.28	1.31810
$227$	−1702.56	−0.497809	−0.248904	−	0.968528i	$-0.580070\pi$
$227$	−1702.56	−0.497809	−0.248904	+	0.968528i	$0.580070\pi$
$228$	0	0
$229$	5395.95	1.55709	0.778546	−	0.627587i	$-0.215957\pi$
$229$	5395.95	1.55709	0.778546	+	0.627587i	$0.215957\pi$
$230$	1541.13	0.441823
$231$	0	0
$232$	163.346	0.0462251
$233$	6054.18	1.70224	0.851122	−	0.524969i	$-0.175923\pi$
$233$	6054.18	1.70224	0.851122	+	0.524969i	$0.175923\pi$
$234$	0	0
$235$	488.699	0.135656
$236$	−1209.99	−0.333744
$237$	0	0
$238$	2626.45	0.715326
$239$	−3136.60	−0.848911	−0.424456	−	0.905449i	$-0.639534\pi$
$239$	−3136.60	−0.848911	−0.424456	+	0.905449i	$0.639534\pi$
$240$	0	0
$241$	−6499.26	−1.73715	−0.868577	−	0.495555i	$-0.834965\pi$
$241$	−6499.26	−1.73715	−0.868577	+	0.495555i	$0.834965\pi$
$242$	0	0
$243$	0	0
$244$	−597.118	−0.156666
$245$	−798.314	−0.208173
$246$	0	0
$247$	7318.73	1.88534
$248$	1706.43	0.436930
$249$	0	0
$250$	2387.70	0.604046
$251$	−5565.79	−1.39964	−0.699820	−	0.714320i	$-0.746736\pi$
$251$	−5565.79	−1.39964	−0.699820	+	0.714320i	$0.746736\pi$
$252$	0	0
$253$	0	0
$254$	2186.91	0.540232
$255$	0	0
$256$	256.000	0.0625000
$257$	6036.31	1.46512	0.732558	−	0.680705i	$-0.238327\pi$
$257$	6036.31	1.46512	0.732558	+	0.680705i	$0.238327\pi$
$258$	0	0
$259$	3224.52	0.773597
$260$	−1661.95	−0.396422
$261$	0	0
$262$	4932.32	1.16305
$263$	708.442	0.166100	0.0830502	−	0.996545i	$-0.473534\pi$
$263$	708.442	0.166100	0.0830502	+	0.996545i	$0.473534\pi$
$264$	0	0
$265$	−1270.09	−0.294418
$266$	−4220.10	−0.972747
$267$	0	0
$268$	3304.95	0.753290
$269$	4809.71	1.09016	0.545080	−	0.838384i	$-0.316499\pi$
$269$	4809.71	1.09016	0.545080	+	0.838384i	$0.316499\pi$
$270$	0	0
$271$	−5240.88	−1.17476	−0.587382	−	0.809310i	$-0.699841\pi$
$271$	−5240.88	−1.17476	−0.587382	+	0.809310i	$0.699841\pi$
$272$	−948.613	−0.211464
$273$	0	0
$274$	1946.57	0.429185
$275$	0	0
$276$	0	0
$277$	−5339.75	−1.15825	−0.579123	−	0.815240i	$-0.696605\pi$
$277$	−5339.75	−1.15825	−0.579123	+	0.815240i	$0.696605\pi$
$278$	−2526.08	−0.544979
$279$	0	0
$280$	958.308	0.204535
$281$	3279.97	0.696322	0.348161	−	0.937435i	$-0.386806\pi$
$281$	3279.97	0.696322	0.348161	+	0.937435i	$0.386806\pi$
$282$	0	0
$283$	−4895.49	−1.02829	−0.514146	−	0.857703i	$-0.671891\pi$
$283$	−4895.49	−1.02829	−0.514146	+	0.857703i	$0.671891\pi$
$284$	3595.86	0.751321
$285$	0	0
$286$	0	0
$287$	1834.73	0.377354
$288$	0	0
$289$	−1397.89	−0.284529
$290$	−220.848	−0.0447195
$291$	0	0
$292$	−551.974	−0.110623
$293$	−6705.80	−1.33705	−0.668527	−	0.743688i	$-0.733075\pi$
$293$	−6705.80	−1.33705	−0.668527	+	0.743688i	$0.733075\pi$
$294$	0	0
$295$	1635.94	0.322874
$296$	−1164.62	−0.228690
$297$	0	0
$298$	978.680	0.190246
$299$	−10946.6	−2.11725
$300$	0	0
$301$	3352.88	0.642049
$302$	3415.55	0.650804
$303$	0	0
$304$	1524.20	0.287562
$305$	807.318	0.151564
$306$	0	0
$307$	9507.29	1.76746	0.883730	−	0.467998i	$-0.155025\pi$
$307$	9507.29	1.76746	0.883730	+	0.467998i	$0.155025\pi$
$308$	0	0
$309$	0	0
$310$	−2307.14	−0.422699
$311$	−5768.07	−1.05170	−0.525848	−	0.850579i	$-0.676252\pi$
$311$	−5768.07	−1.05170	−0.525848	+	0.850579i	$0.676252\pi$
$312$	0	0
$313$	5977.75	1.07950	0.539748	−	0.841826i	$-0.318519\pi$
$313$	5977.75	1.07950	0.539748	+	0.841826i	$0.318519\pi$
$314$	5123.45	0.920806
$315$	0	0
$316$	−1217.83	−0.216799
$317$	−5231.93	−0.926987	−0.463493	−	0.886100i	$-0.653404\pi$
$317$	−5231.93	−0.926987	−0.463493	+	0.886100i	$0.653404\pi$
$318$	0	0
$319$	0	0
$320$	−346.118	−0.0604644
$321$	0	0
$322$	6311.99	1.09240
$323$	−5647.96	−0.972944
$324$	0	0
$325$	−7356.37	−1.25556
$326$	3632.17	0.617078
$327$	0	0
$328$	−662.661	−0.111553
$329$	2001.55	0.335408
$330$	0	0
$331$	−3963.72	−0.658205	−0.329102	−	0.944294i	$-0.606746\pi$
$331$	−3963.72	−0.658205	−0.329102	+	0.944294i	$0.606746\pi$
$332$	3057.17	0.505374
$333$	0	0
$334$	−757.850	−0.124155
$335$	−4468.37	−0.728756
$336$	0	0
$337$	5106.60	0.825443	0.412721	−	0.910857i	$-0.364578\pi$
$337$	5106.60	0.825443	0.412721	+	0.910857i	$0.364578\pi$
$338$	7410.75	1.19258
$339$	0	0
$340$	1282.55	0.204576
$341$	0	0
$342$	0	0
$343$	4327.75	0.681273
$344$	−1210.98	−0.189802
$345$	0	0
$346$	−101.444	−0.0157620
$347$	−7545.75	−1.16737	−0.583685	−	0.811980i	$-0.698390\pi$
$347$	−7545.75	−1.16737	−0.583685	+	0.811980i	$0.698390\pi$
$348$	0	0
$349$	−3256.71	−0.499507	−0.249753	−	0.968309i	$-0.580350\pi$
$349$	−3256.71	−0.499507	−0.249753	+	0.968309i	$0.580350\pi$
$350$	4241.80	0.647811
$351$	0	0
$352$	0	0
$353$	6506.83	0.981087	0.490543	−	0.871417i	$-0.336798\pi$
$353$	6506.83	0.981087	0.490543	+	0.871417i	$0.336798\pi$
$354$	0	0
$355$	−4861.69	−0.726851
$356$	1252.40	0.186452
$357$	0	0
$358$	−41.7356	−0.00616144
$359$	−4066.21	−0.597789	−0.298895	−	0.954286i	$-0.596618\pi$
$359$	−4066.21	−0.597789	−0.298895	+	0.954286i	$0.596618\pi$
$360$	0	0
$361$	2215.95	0.323073
$362$	−352.307	−0.0511514
$363$	0	0
$364$	−6806.81	−0.980148
$365$	746.282	0.107020
$366$	0	0
$367$	−1392.27	−0.198028	−0.0990138	−	0.995086i	$-0.531569\pi$
$367$	−1392.27	−0.198028	−0.0990138	+	0.995086i	$0.531569\pi$
$368$	−2279.74	−0.322934
$369$	0	0
$370$	1574.60	0.221241
$371$	−5201.86	−0.727944
$372$	0	0
$373$	9440.30	1.31046	0.655228	−	0.755431i	$-0.272572\pi$
$373$	9440.30	1.31046	0.655228	+	0.755431i	$0.272572\pi$
$374$	0	0
$375$	0	0
$376$	−722.915	−0.0991529
$377$	1568.68	0.214299
$378$	0	0
$379$	1290.35	0.174884	0.0874418	−	0.996170i	$-0.472131\pi$
$379$	1290.35	0.174884	0.0874418	+	0.996170i	$0.472131\pi$
$380$	−2060.76	−0.278196
$381$	0	0
$382$	2318.44	0.310528
$383$	5652.89	0.754176	0.377088	−	0.926177i	$-0.376925\pi$
$383$	5652.89	0.754176	0.377088	+	0.926177i	$0.376925\pi$
$384$	0	0
$385$	0	0
$386$	−2569.88	−0.338868
$387$	0	0
$388$	−2330.02	−0.304868
$389$	12838.0	1.67330	0.836648	−	0.547741i	$-0.184512\pi$
$389$	12838.0	1.67330	0.836648	+	0.547741i	$0.184512\pi$
$390$	0	0
$391$	8447.63	1.09262
$392$	1180.92	0.152156
$393$	0	0
$394$	5370.12	0.686656
$395$	1646.54	0.209738
$396$	0	0
$397$	7691.26	0.972326	0.486163	−	0.873868i	$-0.338396\pi$
$397$	7691.26	0.972326	0.486163	+	0.873868i	$0.338396\pi$
$398$	−2667.08	−0.335901
$399$	0	0
$400$	−1532.04	−0.191505
$401$	−7786.17	−0.969633	−0.484817	−	0.874616i	$-0.661114\pi$
$401$	−7786.17	−0.969633	−0.484817	+	0.874616i	$0.661114\pi$
$402$	0	0
$403$	16387.5	2.02561
$404$	566.217	0.0697286
$405$	0	0
$406$	−904.523	−0.110568
$407$	0	0
$408$	0	0
$409$	−8859.88	−1.07113	−0.535566	−	0.844493i	$-0.679902\pi$
$409$	−8859.88	−1.07113	−0.535566	+	0.844493i	$0.679902\pi$
$410$	895.934	0.107920
$411$	0	0
$412$	3365.52	0.402445
$413$	6700.26	0.798301
$414$	0	0
$415$	−4133.38	−0.488914
$416$	2458.46	0.289750
$417$	0	0
$418$	0	0
$419$	−9469.08	−1.10405	−0.552023	−	0.833829i	$-0.686144\pi$
$419$	−9469.08	−1.10405	−0.552023	+	0.833829i	$0.686144\pi$
$420$	0	0
$421$	5177.94	0.599424	0.299712	−	0.954030i	$-0.403109\pi$
$421$	5177.94	0.599424	0.299712	+	0.954030i	$0.403109\pi$
$422$	−3256.34	−0.375630
$423$	0	0
$424$	1878.79	0.215194
$425$	5677.00	0.647941
$426$	0	0
$427$	3306.51	0.374739
$428$	257.728	0.0291069
$429$	0	0
$430$	1637.28	0.183620
$431$	−10463.6	−1.16940	−0.584700	−	0.811249i	$-0.698788\pi$
$431$	−10463.6	−1.16940	−0.584700	+	0.811249i	$0.698788\pi$
$432$	0	0
$433$	15838.2	1.75782	0.878911	−	0.476986i	$-0.158271\pi$
$433$	15838.2	1.75782	0.878911	+	0.476986i	$0.158271\pi$
$434$	−9449.30	−1.04512
$435$	0	0
$436$	6236.16	0.684995
$437$	−13573.4	−1.48582
$438$	0	0
$439$	−4824.70	−0.524534	−0.262267	−	0.964995i	$-0.584470\pi$
$439$	−4824.70	−0.524534	−0.262267	+	0.964995i	$0.584470\pi$
$440$	0	0
$441$	0	0
$442$	−9109.88	−0.980346
$443$	−3583.38	−0.384315	−0.192158	−	0.981364i	$-0.561549\pi$
$443$	−3583.38	−0.384315	−0.192158	+	0.981364i	$0.561549\pi$
$444$	0	0
$445$	−1693.27	−0.180380
$446$	−5256.39	−0.558066
$447$	0	0
$448$	−1417.59	−0.149497
$449$	7795.98	0.819410	0.409705	−	0.912218i	$-0.365632\pi$
$449$	7795.98	0.819410	0.409705	+	0.912218i	$0.365632\pi$
$450$	0	0
$451$	0	0
$452$	8956.57	0.932039
$453$	0	0
$454$	−3405.11	−0.352004
$455$	9202.98	0.948225
$456$	0	0
$457$	3348.02	0.342700	0.171350	−	0.985210i	$-0.445187\pi$
$457$	3348.02	0.342700	0.171350	+	0.985210i	$0.445187\pi$
$458$	10791.9	1.10103
$459$	0	0
$460$	3082.27	0.312416
$461$	−7171.96	−0.724580	−0.362290	−	0.932065i	$-0.618005\pi$
$461$	−7171.96	−0.724580	−0.362290	+	0.932065i	$0.618005\pi$
$462$	0	0
$463$	4034.74	0.404990	0.202495	−	0.979283i	$-0.435095\pi$
$463$	4034.74	0.404990	0.202495	+	0.979283i	$0.435095\pi$
$464$	326.693	0.0326861
$465$	0	0
$466$	12108.4	1.20367
$467$	11821.8	1.17141	0.585706	−	0.810523i	$-0.300817\pi$
$467$	11821.8	1.17141	0.585706	+	0.810523i	$0.300817\pi$
$468$	0	0
$469$	−18301.0	−1.80184
$470$	977.399	0.0959235
$471$	0	0
$472$	−2419.98	−0.235993
$473$	0	0
$474$	0	0
$475$	−9121.62	−0.881113
$476$	5252.90	0.505812
$477$	0	0
$478$	−6273.20	−0.600271
$479$	8659.94	0.826060	0.413030	−	0.910717i	$-0.364470\pi$
$479$	8659.94	0.826060	0.413030	+	0.910717i	$0.364470\pi$
$480$	0	0
$481$	−11184.3	−1.06021
$482$	−12998.5	−1.22835
$483$	0	0
$484$	0	0
$485$	3150.25	0.294939
$486$	0	0
$487$	7427.66	0.691128	0.345564	−	0.938395i	$-0.387688\pi$
$487$	7427.66	0.691128	0.345564	+	0.938395i	$0.387688\pi$
$488$	−1194.24	−0.110780
$489$	0	0
$490$	−1596.63	−0.147201
$491$	5543.89	0.509556	0.254778	−	0.967000i	$-0.417998\pi$
$491$	5543.89	0.509556	0.254778	+	0.967000i	$0.417998\pi$
$492$	0	0
$493$	−1210.57	−0.110591
$494$	14637.5	1.33314
$495$	0	0
$496$	3412.87	0.308956
$497$	−19911.9	−1.79713
$498$	0	0
$499$	20412.0	1.83120	0.915599	−	0.402093i	$-0.131717\pi$
$499$	20412.0	1.83120	0.915599	+	0.402093i	$0.131717\pi$
$500$	4775.41	0.427125
$501$	0	0
$502$	−11131.6	−0.989694
$503$	182.742	0.0161989	0.00809946	−	0.999967i	$-0.497422\pi$
$503$	182.742	0.0161989	0.00809946	+	0.999967i	$0.497422\pi$
$504$	0	0
$505$	−765.540	−0.0674576
$506$	0	0
$507$	0	0
$508$	4373.82	0.382002
$509$	−4755.29	−0.414095	−0.207048	−	0.978331i	$-0.566386\pi$
$509$	−4755.29	−0.414095	−0.207048	+	0.978331i	$0.566386\pi$
$510$	0	0
$511$	3056.53	0.264604
$512$	512.000	0.0441942
$513$	0	0
$514$	12072.6	1.03599
$515$	−4550.27	−0.389337
$516$	0	0
$517$	0	0
$518$	6449.03	0.547016
$519$	0	0
$520$	−3323.90	−0.280313
$521$	11731.1	0.986470	0.493235	−	0.869896i	$-0.335814\pi$
$521$	11731.1	0.986470	0.493235	+	0.869896i	$0.335814\pi$
$522$	0	0
$523$	−343.839	−0.0287477	−0.0143738	−	0.999897i	$-0.504575\pi$
$523$	−343.839	−0.0287477	−0.0143738	+	0.999897i	$0.504575\pi$
$524$	9864.65	0.822403
$525$	0	0
$526$	1416.88	0.117451
$527$	−12646.4	−1.04533
$528$	0	0
$529$	8134.66	0.668584
$530$	−2540.17	−0.208185
$531$	0	0
$532$	−8440.19	−0.687836
$533$	−6363.77	−0.517159
$534$	0	0
$535$	−348.454	−0.0281589
$536$	6609.89	0.532656
$537$	0	0
$538$	9619.41	0.770859
$539$	0	0
$540$	0	0
$541$	−17243.8	−1.37037	−0.685183	−	0.728371i	$-0.740278\pi$
$541$	−17243.8	−1.37037	−0.685183	+	0.728371i	$0.740278\pi$
$542$	−10481.8	−0.830683
$543$	0	0
$544$	−1897.23	−0.149527
$545$	−8431.45	−0.662685
$546$	0	0
$547$	−7643.21	−0.597441	−0.298720	−	0.954341i	$-0.596560\pi$
$547$	−7643.21	−0.597441	−0.298720	+	0.954341i	$0.596560\pi$
$548$	3893.14	0.303480
$549$	0	0
$550$	0	0
$551$	1945.10	0.150388
$552$	0	0
$553$	6743.70	0.518574
$554$	−10679.5	−0.819004
$555$	0	0
$556$	−5052.16	−0.385358
$557$	−26106.2	−1.98591	−0.992957	−	0.118471i	$-0.962201\pi$
$557$	−26106.2	−1.98591	−0.992957	+	0.118471i	$0.962201\pi$
$558$	0	0
$559$	−11629.5	−0.879921
$560$	1916.62	0.144628
$561$	0	0
$562$	6559.94	0.492374
$563$	18073.9	1.35297	0.676487	−	0.736455i	$-0.263502\pi$
$563$	18073.9	1.35297	0.676487	+	0.736455i	$0.263502\pi$
$564$	0	0
$565$	−12109.5	−0.901682
$566$	−9790.98	−0.727112
$567$	0	0
$568$	7191.72	0.531264
$569$	−7805.16	−0.575060	−0.287530	−	0.957772i	$-0.592834\pi$
$569$	−7805.16	−0.575060	−0.287530	+	0.957772i	$0.592834\pi$
$570$	0	0
$571$	−13039.6	−0.955672	−0.477836	−	0.878449i	$-0.658579\pi$
$571$	−13039.6	−0.955672	−0.477836	+	0.878449i	$0.658579\pi$
$572$	0	0
$573$	0	0
$574$	3669.45	0.266829
$575$	13643.2	0.989496
$576$	0	0
$577$	−9569.43	−0.690434	−0.345217	−	0.938523i	$-0.612195\pi$
$577$	−9569.43	−0.690434	−0.345217	+	0.938523i	$0.612195\pi$
$578$	−2795.79	−0.201193
$579$	0	0
$580$	−441.697	−0.0316215
$581$	−16929.0	−1.20883
$582$	0	0
$583$	0	0
$584$	−1103.95	−0.0782220
$585$	0	0
$586$	−13411.6	−0.945440
$587$	−9727.00	−0.683946	−0.341973	−	0.939710i	$-0.611095\pi$
$587$	−9727.00	−0.683946	−0.341973	+	0.939710i	$0.611095\pi$
$588$	0	0
$589$	20319.9	1.42151
$590$	3271.87	0.228307
$591$	0	0
$592$	−2329.24	−0.161708
$593$	−6926.77	−0.479677	−0.239838	−	0.970813i	$-0.577094\pi$
$593$	−6926.77	−0.479677	−0.239838	+	0.970813i	$0.577094\pi$
$594$	0	0
$595$	−7102.06	−0.489338
$596$	1957.36	0.134525
$597$	0	0
$598$	−21893.2	−1.49712
$599$	21984.7	1.49961	0.749807	−	0.661656i	$-0.230146\pi$
$599$	21984.7	1.49961	0.749807	+	0.661656i	$0.230146\pi$
$600$	0	0
$601$	−7363.27	−0.499757	−0.249878	−	0.968277i	$-0.580391\pi$
$601$	−7363.27	−0.499757	−0.249878	+	0.968277i	$0.580391\pi$
$602$	6705.76	0.453998
$603$	0	0
$604$	6831.10	0.460188
$605$	0	0
$606$	0	0
$607$	10168.4	0.679938	0.339969	−	0.940437i	$-0.389583\pi$
$607$	10168.4	0.679938	0.339969	+	0.940437i	$0.389583\pi$
$608$	3048.40	0.203337
$609$	0	0
$610$	1614.64	0.107172
$611$	−6942.41	−0.459673
$612$	0	0
$613$	25084.5	1.65278	0.826389	−	0.563099i	$-0.190391\pi$
$613$	25084.5	1.65278	0.826389	+	0.563099i	$0.190391\pi$
$614$	19014.6	1.24978
$615$	0	0
$616$	0	0
$617$	26335.5	1.71836	0.859178	−	0.511677i	$-0.170976\pi$
$617$	26335.5	1.71836	0.859178	+	0.511677i	$0.170976\pi$
$618$	0	0
$619$	−24643.2	−1.60015	−0.800075	−	0.599901i	$-0.795207\pi$
$619$	−24643.2	−1.60015	−0.800075	+	0.599901i	$0.795207\pi$
$620$	−4614.28	−0.298894
$621$	0	0
$622$	−11536.1	−0.743661
$623$	−6935.10	−0.445986
$624$	0	0
$625$	5512.59	0.352806
$626$	11955.5	0.763319
$627$	0	0
$628$	10246.9	0.651108
$629$	8631.05	0.547126
$630$	0	0
$631$	7455.32	0.470351	0.235176	−	0.971953i	$-0.424433\pi$
$631$	7455.32	0.470351	0.235176	+	0.971953i	$0.424433\pi$
$632$	−2435.67	−0.153300
$633$	0	0
$634$	−10463.9	−0.655478
$635$	−5913.52	−0.369560
$636$	0	0
$637$	11340.8	0.705397
$638$	0	0
$639$	0	0
$640$	−692.237	−0.0427548
$641$	2778.67	0.171218	0.0856090	−	0.996329i	$-0.472716\pi$
$641$	2778.67	0.171218	0.0856090	+	0.996329i	$0.472716\pi$
$642$	0	0
$643$	7557.71	0.463526	0.231763	−	0.972772i	$-0.425551\pi$
$643$	7557.71	0.463526	0.231763	+	0.972772i	$0.425551\pi$
$644$	12624.0	0.772444
$645$	0	0
$646$	−11295.9	−0.687975
$647$	27430.7	1.66679	0.833394	−	0.552679i	$-0.186395\pi$
$647$	27430.7	1.66679	0.833394	+	0.552679i	$0.186395\pi$
$648$	0	0
$649$	0	0
$650$	−14712.7	−0.887817
$651$	0	0
$652$	7264.35	0.436340
$653$	−284.728	−0.0170632	−0.00853160	−	0.999964i	$-0.502716\pi$
$653$	−284.728	−0.0170632	−0.00853160	+	0.999964i	$0.502716\pi$
$654$	0	0
$655$	−13337.2	−0.795617
$656$	−1325.32	−0.0788797
$657$	0	0
$658$	4003.11	0.237169
$659$	−20747.0	−1.22639	−0.613193	−	0.789933i	$-0.710115\pi$
$659$	−20747.0	−1.22639	−0.613193	+	0.789933i	$0.710115\pi$
$660$	0	0
$661$	−18908.8	−1.11266	−0.556328	−	0.830963i	$-0.687790\pi$
$661$	−18908.8	−1.11266	−0.556328	+	0.830963i	$0.687790\pi$
$662$	−7927.44	−0.465421
$663$	0	0
$664$	6114.35	0.357354
$665$	11411.4	0.665434
$666$	0	0
$667$	−2909.28	−0.168887
$668$	−1515.70	−0.0877907
$669$	0	0
$670$	−8936.74	−0.515308
$671$	0	0
$672$	0	0
$673$	29865.9	1.71062	0.855309	−	0.518118i	$-0.173367\pi$
$673$	29865.9	1.71062	0.855309	+	0.518118i	$0.173367\pi$
$674$	10213.2	0.583676
$675$	0	0
$676$	14821.5	0.843281
$677$	17280.7	0.981024	0.490512	−	0.871435i	$-0.336810\pi$
$677$	17280.7	0.981024	0.490512	+	0.871435i	$0.336810\pi$
$678$	0	0
$679$	12902.4	0.729232
$680$	2565.10	0.144657
$681$	0	0
$682$	0	0
$683$	−25104.0	−1.40641	−0.703204	−	0.710988i	$-0.748248\pi$
$683$	−25104.0	−1.40641	−0.703204	+	0.710988i	$0.748248\pi$
$684$	0	0
$685$	−5263.63	−0.293595
$686$	8655.51	0.481733
$687$	0	0
$688$	−2421.97	−0.134210
$689$	18042.7	0.997638
$690$	0	0
$691$	−26163.8	−1.44040	−0.720201	−	0.693765i	$-0.755950\pi$
$691$	−26163.8	−1.44040	−0.720201	+	0.693765i	$0.755950\pi$
$692$	−202.888	−0.0111454
$693$	0	0
$694$	−15091.5	−0.825455
$695$	6830.64	0.372807
$696$	0	0
$697$	4911.00	0.266883
$698$	−6513.42	−0.353205
$699$	0	0
$700$	8483.60	0.458071
$701$	1519.45	0.0818669	0.0409334	−	0.999162i	$-0.486967\pi$
$701$	1519.45	0.0818669	0.0409334	+	0.999162i	$0.486967\pi$
$702$	0	0
$703$	−13868.1	−0.744018
$704$	0	0
$705$	0	0
$706$	13013.7	0.693733
$707$	−3135.40	−0.166788
$708$	0	0
$709$	−21733.9	−1.15125	−0.575623	−	0.817715i	$-0.695240\pi$
$709$	−21733.9	−1.15125	−0.575623	+	0.817715i	$0.695240\pi$
$710$	−9723.39	−0.513961
$711$	0	0
$712$	2504.80	0.131842
$713$	−30392.4	−1.59636
$714$	0	0
$715$	0	0
$716$	−83.4712	−0.00435679
$717$	0	0
$718$	−8132.42	−0.422701
$719$	14071.8	0.729890	0.364945	−	0.931029i	$-0.381088\pi$
$719$	14071.8	0.729890	0.364945	+	0.931029i	$0.381088\pi$
$720$	0	0
$721$	−18636.4	−0.962630
$722$	4431.91	0.228447
$723$	0	0
$724$	−704.613	−0.0361695
$725$	−1955.10	−0.100153
$726$	0	0
$727$	10409.2	0.531027	0.265513	−	0.964107i	$-0.414459\pi$
$727$	10409.2	0.531027	0.265513	+	0.964107i	$0.414459\pi$
$728$	−13613.6	−0.693069
$729$	0	0
$730$	1492.56	0.0756744
$731$	8974.65	0.454089
$732$	0	0
$733$	−19394.9	−0.977307	−0.488653	−	0.872478i	$-0.662512\pi$
$733$	−19394.9	−0.977307	−0.488653	+	0.872478i	$0.662512\pi$
$734$	−2784.55	−0.140027
$735$	0	0
$736$	−4559.48	−0.228349
$737$	0	0
$738$	0	0
$739$	−6672.24	−0.332128	−0.166064	−	0.986115i	$-0.553106\pi$
$739$	−6672.24	−0.332128	−0.166064	+	0.986115i	$0.553106\pi$
$740$	3149.19	0.156441
$741$	0	0
$742$	−10403.7	−0.514734
$743$	−6565.18	−0.324163	−0.162082	−	0.986777i	$-0.551821\pi$
$743$	−6565.18	−0.324163	−0.162082	+	0.986777i	$0.551821\pi$
$744$	0	0
$745$	−2646.40	−0.130143
$746$	18880.6	0.926633
$747$	0	0
$748$	0	0
$749$	−1427.16	−0.0696223
$750$	0	0
$751$	18938.7	0.920216	0.460108	−	0.887863i	$-0.347811\pi$
$751$	18938.7	0.920216	0.460108	+	0.887863i	$0.347811\pi$
$752$	−1445.83	−0.0701117
$753$	0	0
$754$	3137.35	0.151533
$755$	−9235.82	−0.445200
$756$	0	0
$757$	−10958.4	−0.526141	−0.263070	−	0.964777i	$-0.584735\pi$
$757$	−10958.4	−0.526141	−0.263070	+	0.964777i	$0.584735\pi$
$758$	2580.70	0.123661
$759$	0	0
$760$	−4121.52	−0.196715
$761$	35442.2	1.68828	0.844138	−	0.536126i	$-0.180113\pi$
$761$	35442.2	1.68828	0.844138	+	0.536126i	$0.180113\pi$
$762$	0	0
$763$	−34532.5	−1.63848
$764$	4636.88	0.219576
$765$	0	0
$766$	11305.8	0.533283
$767$	−23240.0	−1.09406
$768$	0	0
$769$	4444.21	0.208404	0.104202	−	0.994556i	$-0.466771\pi$
$769$	4444.21	0.208404	0.104202	+	0.994556i	$0.466771\pi$
$770$	0	0
$771$	0	0
$772$	−5139.75	−0.239616
$773$	−10160.2	−0.472750	−0.236375	−	0.971662i	$-0.575959\pi$
$773$	−10160.2	−0.472750	−0.236375	+	0.971662i	$0.575959\pi$
$774$	0	0
$775$	−20424.4	−0.946666
$776$	−4660.04	−0.215575
$777$	0	0
$778$	25676.0	1.18320
$779$	−7890.84	−0.362925
$780$	0	0
$781$	0	0
$782$	16895.3	0.772600
$783$	0	0
$784$	2361.83	0.107591
$785$	−13854.1	−0.629902
$786$	0	0
$787$	−18270.1	−0.827520	−0.413760	−	0.910386i	$-0.635785\pi$
$787$	−18270.1	−0.827520	−0.413760	+	0.910386i	$0.635785\pi$
$788$	10740.2	0.485539
$789$	0	0
$790$	3293.09	0.148307
$791$	−49596.6	−2.22939
$792$	0	0
$793$	−11468.7	−0.513575
$794$	15382.5	0.687538
$795$	0	0
$796$	−5334.16	−0.237518
$797$	−17984.0	−0.799279	−0.399639	−	0.916672i	$-0.630865\pi$
$797$	−17984.0	−0.799279	−0.399639	+	0.916672i	$0.630865\pi$
$798$	0	0
$799$	5357.55	0.237217
$800$	−3064.08	−0.135414
$801$	0	0
$802$	−15572.3	−0.685634
$803$	0	0
$804$	0	0
$805$	−17067.9	−0.747286
$806$	32775.0	1.43232
$807$	0	0
$808$	1132.43	0.0493056
$809$	−5144.24	−0.223562	−0.111781	−	0.993733i	$-0.535656\pi$
$809$	−5144.24	−0.223562	−0.111781	+	0.993733i	$0.535656\pi$
$810$	0	0
$811$	−8255.99	−0.357469	−0.178734	−	0.983897i	$-0.557200\pi$
$811$	−8255.99	−0.357469	−0.178734	+	0.983897i	$0.557200\pi$
$812$	−1809.05	−0.0781836
$813$	0	0
$814$	0	0
$815$	−9821.58	−0.422129
$816$	0	0
$817$	−14420.2	−0.617500
$818$	−17719.8	−0.757405
$819$	0	0
$820$	1791.87	0.0763106
$821$	−247.250	−0.0105105	−0.00525523	−	0.999986i	$-0.501673\pi$
$821$	−247.250	−0.0105105	−0.00525523	+	0.999986i	$0.501673\pi$
$822$	0	0
$823$	11794.8	0.499564	0.249782	−	0.968302i	$-0.419641\pi$
$823$	11794.8	0.499564	0.249782	+	0.968302i	$0.419641\pi$
$824$	6731.04	0.284571
$825$	0	0
$826$	13400.5	0.564484
$827$	2771.66	0.116542	0.0582709	−	0.998301i	$-0.481441\pi$
$827$	2771.66	0.116542	0.0582709	+	0.998301i	$0.481441\pi$
$828$	0	0
$829$	−27298.0	−1.14367	−0.571834	−	0.820370i	$-0.693768\pi$
$829$	−27298.0	−1.14367	−0.571834	+	0.820370i	$0.693768\pi$
$830$	−8266.75	−0.345715
$831$	0	0
$832$	4916.92	0.204884
$833$	−8751.82	−0.364025
$834$	0	0
$835$	2049.26	0.0849314
$836$	0	0
$837$	0	0
$838$	−18938.2	−0.780678
$839$	−8754.74	−0.360247	−0.180123	−	0.983644i	$-0.557650\pi$
$839$	−8754.74	−0.360247	−0.180123	+	0.983644i	$0.557650\pi$
$840$	0	0
$841$	−23972.1	−0.982906
$842$	10355.9	0.423857
$843$	0	0
$844$	−6512.67	−0.265611
$845$	−20039.1	−0.815816
$846$	0	0
$847$	0	0
$848$	3757.58	0.152165
$849$	0	0
$850$	11354.0	0.458164
$851$	20742.5	0.835537
$852$	0	0
$853$	41184.3	1.65313	0.826566	−	0.562839i	$-0.190291\pi$
$853$	41184.3	1.65313	0.826566	+	0.562839i	$0.190291\pi$
$854$	6613.03	0.264980
$855$	0	0
$856$	515.455	0.0205817
$857$	18020.0	0.718265	0.359132	−	0.933287i	$-0.383073\pi$
$857$	18020.0	0.718265	0.359132	+	0.933287i	$0.383073\pi$
$858$	0	0
$859$	6159.11	0.244640	0.122320	−	0.992491i	$-0.460967\pi$
$859$	6159.11	0.244640	0.122320	+	0.992491i	$0.460967\pi$
$860$	3274.56	0.129839
$861$	0	0
$862$	−20927.1	−0.826891
$863$	−28932.6	−1.14122	−0.570612	−	0.821220i	$-0.693294\pi$
$863$	−28932.6	−1.14122	−0.570612	+	0.821220i	$0.693294\pi$
$864$	0	0
$865$	274.309	0.0107824
$866$	31676.5	1.24297
$867$	0	0
$868$	−18898.6	−0.739009
$869$	0	0
$870$	0	0
$871$	63477.2	2.46940
$872$	12472.3	0.484365
$873$	0	0
$874$	−27146.8	−1.05063
$875$	−26443.6	−1.02166
$876$	0	0
$877$	−45152.6	−1.73854	−0.869268	−	0.494341i	$-0.835409\pi$
$877$	−45152.6	−1.73854	−0.869268	+	0.494341i	$0.835409\pi$
$878$	−9649.40	−0.370902
$879$	0	0
$880$	0	0
$881$	−22263.5	−0.851391	−0.425696	−	0.904866i	$-0.639971\pi$
$881$	−22263.5	−0.851391	−0.425696	+	0.904866i	$0.639971\pi$
$882$	0	0
$883$	38215.5	1.45646	0.728230	−	0.685332i	$-0.240343\pi$
$883$	38215.5	1.45646	0.728230	+	0.685332i	$0.240343\pi$
$884$	−18219.8	−0.693209
$885$	0	0
$886$	−7166.77	−0.271752
$887$	−48160.6	−1.82308	−0.911541	−	0.411208i	$-0.865107\pi$
$887$	−48160.6	−1.82308	−0.911541	+	0.411208i	$0.865107\pi$
$888$	0	0
$889$	−24219.8	−0.913731
$890$	−3386.55	−0.127548
$891$	0	0
$892$	−10512.8	−0.394612
$893$	−8608.34	−0.322583
$894$	0	0
$895$	112.855	0.00421490
$896$	−2835.18	−0.105711
$897$	0	0
$898$	15592.0	0.579411
$899$	4355.31	0.161577
$900$	0	0
$901$	−13923.8	−0.514838
$902$	0	0
$903$	0	0
$904$	17913.1	0.659051
$905$	952.655	0.0349915
$906$	0	0
$907$	8509.88	0.311539	0.155770	−	0.987793i	$-0.450214\pi$
$907$	8509.88	0.311539	0.155770	+	0.987793i	$0.450214\pi$
$908$	−6810.22	−0.248904
$909$	0	0
$910$	18406.0	0.670496
$911$	−11283.5	−0.410360	−0.205180	−	0.978724i	$-0.565778\pi$
$911$	−11283.5	−0.410360	−0.205180	+	0.978724i	$0.565778\pi$
$912$	0	0
$913$	0	0
$914$	6696.04	0.242325
$915$	0	0
$916$	21583.8	0.778546
$917$	−54625.0	−1.96715
$918$	0	0
$919$	26556.9	0.953245	0.476622	−	0.879108i	$-0.341861\pi$
$919$	26556.9	0.953245	0.476622	+	0.879108i	$0.341861\pi$
$920$	6164.54	0.220912
$921$	0	0
$922$	−14343.9	−0.512355
$923$	69064.7	2.46294
$924$	0	0
$925$	13939.4	0.495486
$926$	8069.49	0.286371
$927$	0	0
$928$	653.386	0.0231125
$929$	47913.1	1.69212	0.846059	−	0.533089i	$-0.178969\pi$
$929$	47913.1	1.69212	0.846059	+	0.533089i	$0.178969\pi$
$930$	0	0
$931$	14062.1	0.495025
$932$	24216.7	0.851122
$933$	0	0
$934$	23643.7	0.828314
$935$	0	0
$936$	0	0
$937$	5718.95	0.199391	0.0996957	−	0.995018i	$-0.468213\pi$
$937$	5718.95	0.199391	0.0996957	+	0.995018i	$0.468213\pi$
$938$	−36602.0	−1.27409
$939$	0	0
$940$	1954.80	0.0678281
$941$	11105.6	0.384732	0.192366	−	0.981323i	$-0.438384\pi$
$941$	11105.6	0.384732	0.192366	+	0.981323i	$0.438384\pi$
$942$	0	0
$943$	11802.3	0.407567
$944$	−4839.96	−0.166872
$945$	0	0
$946$	0	0
$947$	−39759.1	−1.36430	−0.682152	−	0.731210i	$-0.738956\pi$
$947$	−39759.1	−1.36430	−0.682152	+	0.731210i	$0.738956\pi$
$948$	0	0
$949$	−10601.6	−0.362637
$950$	−18243.2	−0.623041
$951$	0	0
$952$	10505.8	0.357663
$953$	15932.0	0.541539	0.270770	−	0.962644i	$-0.412722\pi$
$953$	15932.0	0.541539	0.270770	+	0.962644i	$0.412722\pi$
$954$	0	0
$955$	−6269.18	−0.212425
$956$	−12546.4	−0.424456
$957$	0	0
$958$	17319.9	0.584113
$959$	−21558.1	−0.725910
$960$	0	0
$961$	15707.7	0.527262
$962$	−22368.6	−0.749679
$963$	0	0
$964$	−25997.0	−0.868577
$965$	6949.07	0.231812
$966$	0	0
$967$	14952.9	0.497264	0.248632	−	0.968598i	$-0.420019\pi$
$967$	14952.9	0.497264	0.248632	+	0.968598i	$0.420019\pi$
$968$	0	0
$969$	0	0
$970$	6300.50	0.208553
$971$	−27937.3	−0.923327	−0.461663	−	0.887055i	$-0.652747\pi$
$971$	−27937.3	−0.923327	−0.461663	+	0.887055i	$0.652747\pi$
$972$	0	0
$973$	27976.1	0.921760
$974$	14855.3	0.488701
$975$	0	0
$976$	−2388.47	−0.0783331
$977$	−33359.2	−1.09238	−0.546191	−	0.837661i	$-0.683923\pi$
$977$	−33359.2	−1.09238	−0.546191	+	0.837661i	$0.683923\pi$
$978$	0	0
$979$	0	0
$980$	−3193.26	−0.104087
$981$	0	0
$982$	11087.8	0.360311
$983$	−12350.1	−0.400718	−0.200359	−	0.979723i	$-0.564211\pi$
$983$	−12350.1	−0.400718	−0.200359	+	0.979723i	$0.564211\pi$
$984$	0	0
$985$	−14521.1	−0.469725
$986$	−2421.13	−0.0781994
$987$	0	0
$988$	29274.9	0.942671
$989$	21568.2	0.693457
$990$	0	0
$991$	−30154.0	−0.966571	−0.483286	−	0.875463i	$-0.660557\pi$
$991$	−30154.0	−0.966571	−0.483286	+	0.875463i	$0.660557\pi$
$992$	6825.73	0.218465
$993$	0	0
$994$	−39823.8	−1.27076
$995$	7211.91	0.229782
$996$	0	0
$997$	2093.09	0.0664884	0.0332442	−	0.999447i	$-0.489416\pi$
$997$	2093.09	0.0664884	0.0332442	+	0.999447i	$0.489416\pi$
$998$	40824.0	1.29485
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2178.4.a.by.1.3		4
3.2	odd	2		242.4.a.n.1.3		4
11.3	even	5		198.4.f.d.163.2		8
11.4	even	5		198.4.f.d.181.2		8
11.10	odd	2		2178.4.a.bt.1.3		4
12.11	even	2		1936.4.a.bn.1.2		4
33.2	even	10		242.4.c.n.81.2		8
33.5	odd	10		242.4.c.r.3.2		8
33.8	even	10		242.4.c.q.9.1		8
33.14	odd	10		22.4.c.b.9.1	yes	8
33.17	even	10		242.4.c.n.3.2		8
33.20	odd	10		242.4.c.r.81.2		8
33.26	odd	10		22.4.c.b.5.1	✓	8
33.29	even	10		242.4.c.q.27.1		8
33.32	even	2		242.4.a.o.1.3		4
132.47	even	10		176.4.m.b.97.2		8
132.59	even	10		176.4.m.b.49.2		8
132.131	odd	2		1936.4.a.bm.1.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
22.4.c.b.5.1	✓	8	33.26	odd	10
22.4.c.b.9.1	yes	8	33.14	odd	10
176.4.m.b.49.2		8	132.59	even	10
176.4.m.b.97.2		8	132.47	even	10
198.4.f.d.163.2		8	11.3	even	5
198.4.f.d.181.2		8	11.4	even	5
242.4.a.n.1.3		4	3.2	odd	2
242.4.a.o.1.3		4	33.32	even	2
242.4.c.n.3.2		8	33.17	even	10
242.4.c.n.81.2		8	33.2	even	10
242.4.c.q.9.1		8	33.8	even	10
242.4.c.q.27.1		8	33.29	even	10
242.4.c.r.3.2		8	33.5	odd	10
242.4.c.r.81.2		8	33.20	odd	10
1936.4.a.bm.1.2		4	132.131	odd	2
1936.4.a.bn.1.2		4	12.11	even	2
2178.4.a.bt.1.3		4	11.10	odd	2
2178.4.a.by.1.3		4	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:	\( N \)	\(=\)	\( 2178 = 2 \cdot 3^{2} \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2178.a (trivial)

\(f(q)\)	\(=\)	\(q+2.00000 q^{2} +4.00000 q^{4} -5.40810 q^{5} -22.1498 q^{7} +8.00000 q^{8} +O(q^{10})\)	\(q+2.00000 q^{2} +4.00000 q^{4} -5.40810 q^{5} -22.1498 q^{7} +8.00000 q^{8} -10.8162 q^{10} +76.8269 q^{13} -44.2996 q^{14} +16.0000 q^{16} -59.2883 q^{17} +95.2626 q^{19} -21.6324 q^{20} -142.484 q^{23} -95.7524 q^{25} +153.654 q^{26} -88.5993 q^{28} +20.4183 q^{29} +213.304 q^{31} +32.0000 q^{32} -118.577 q^{34} +119.788 q^{35} -145.578 q^{37} +190.525 q^{38} -43.2648 q^{40} -82.8326 q^{41} -151.373 q^{43} -284.968 q^{46} -90.3643 q^{47} +147.615 q^{49} -191.505 q^{50} +307.308 q^{52} +234.849 q^{53} -177.199 q^{56} +40.8366 q^{58} -302.497 q^{59} -149.279 q^{61} +426.608 q^{62} +64.0000 q^{64} -415.488 q^{65} +826.236 q^{67} -237.153 q^{68} +239.577 q^{70} +898.965 q^{71} -137.993 q^{73} -291.155 q^{74} +381.050 q^{76} -304.459 q^{79} -86.5296 q^{80} -165.665 q^{82} +764.294 q^{83} +320.637 q^{85} -302.746 q^{86} +313.100 q^{89} -1701.70 q^{91} -569.936 q^{92} -180.729 q^{94} -515.189 q^{95} -582.505 q^{97} +295.229 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 8 q^{2} + 16 q^{4} - 25 q^{5} - 3 q^{7} + 32 q^{8}+O(q^{10}) \) 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

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