LMFDB - Embedded newform 928.4.a.h.1.12

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [928,4,Mod(1,928)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(928, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("928.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 928 = 2^{5} \cdot 29 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	928.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:	$54.7537724853$
Analytic rank:	$1$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 2 x^{11} - 217 x^{10} + 520 x^{9} + 16022 x^{8} - 37368 x^{7} - 509640 x^{6} + 989168 x^{5} + \cdots + 14751072 $ x^12 - 2x^11 - 217x^10 + 520x^9 + 16022x^8 - 37368x^7 - 509640x^6 + 989168x^5 + 7106592x^4 - 7979328x^3 - 38912400x^2 + 1083456*x + 14751072
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{17} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.12
Root			$-10.7228$ of defining polynomial
Character	$\chi$	$=$	928.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+9.72277 q^{3} -6.27199 q^{5} -31.0130 q^{7} +67.5323 q^{9} +O(q^{10})$	$q+9.72277 q^{3} -6.27199 q^{5} -31.0130 q^{7} +67.5323 q^{9} -8.24716 q^{11} +32.0907 q^{13} -60.9811 q^{15} +51.1369 q^{17} -158.789 q^{19} -301.533 q^{21} -129.494 q^{23} -85.6622 q^{25} +394.086 q^{27} +29.0000 q^{29} +14.6426 q^{31} -80.1853 q^{33} +194.513 q^{35} -257.398 q^{37} +312.011 q^{39} -26.0208 q^{41} -3.40346 q^{43} -423.562 q^{45} -482.228 q^{47} +618.808 q^{49} +497.192 q^{51} +169.104 q^{53} +51.7261 q^{55} -1543.87 q^{57} +569.745 q^{59} -500.616 q^{61} -2094.38 q^{63} -201.273 q^{65} -335.555 q^{67} -1259.04 q^{69} -958.137 q^{71} -946.043 q^{73} -832.874 q^{75} +255.769 q^{77} +179.326 q^{79} +2008.24 q^{81} +285.090 q^{83} -320.730 q^{85} +281.960 q^{87} +832.078 q^{89} -995.231 q^{91} +142.367 q^{93} +995.921 q^{95} -908.370 q^{97} -556.950 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 14 q^{3} - 10 q^{5} - 44 q^{7} + 130 q^{9}+O(q^{10}) $ 12 * q - 14 * q^3 - 10 * q^5 - 44 * q^7 + 130 * q^9	$ 12 q - 14 q^{3} - 10 q^{5} - 44 q^{7} + 130 q^{9} - 46 q^{11} - 34 q^{13} + 50 q^{15} + 36 q^{17} - 148 q^{19} - 92 q^{21} - 328 q^{23} + 486 q^{25} - 326 q^{27} + 348 q^{29} - 374 q^{31} + 710 q^{33} - 204 q^{35} - 340 q^{37} + 122 q^{39} + 32 q^{41} - 462 q^{43} - 1132 q^{45} - 434 q^{47} + 1508 q^{49} - 440 q^{51} + 610 q^{53} - 46 q^{55} - 932 q^{57} - 1240 q^{59} - 1228 q^{61} - 4240 q^{63} + 730 q^{65} - 1672 q^{67} - 528 q^{69} - 3220 q^{71} + 564 q^{73} - 6032 q^{75} + 644 q^{77} - 1862 q^{79} + 3040 q^{81} - 3736 q^{83} - 808 q^{85} - 406 q^{87} + 584 q^{89} - 4844 q^{91} - 3226 q^{93} - 2844 q^{95} + 904 q^{97} - 6832 q^{99}+O(q^{100}) $ 12 * q - 14 * q^3 - 10 * q^5 - 44 * q^7 + 130 * q^9 - 46 * q^11 - 34 * q^13 + 50 * q^15 + 36 * q^17 - 148 * q^19 - 92 * q^21 - 328 * q^23 + 486 * q^25 - 326 * q^27 + 348 * q^29 - 374 * q^31 + 710 * q^33 - 204 * q^35 - 340 * q^37 + 122 * q^39 + 32 * q^41 - 462 * q^43 - 1132 * q^45 - 434 * q^47 + 1508 * q^49 - 440 * q^51 + 610 * q^53 - 46 * q^55 - 932 * q^57 - 1240 * q^59 - 1228 * q^61 - 4240 * q^63 + 730 * q^65 - 1672 * q^67 - 528 * q^69 - 3220 * q^71 + 564 * q^73 - 6032 * q^75 + 644 * q^77 - 1862 * q^79 + 3040 * q^81 - 3736 * q^83 - 808 * q^85 - 406 * q^87 + 584 * q^89 - 4844 * q^91 - 3226 * q^93 - 2844 * q^95 + 904 * q^97 - 6832 * q^99

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 $

$2$	0	0
$2$	0	0
$3$	9.72277	1.87115	0.935574	−	0.353130i	$-0.114883\pi$
$3$	9.72277	1.87115	0.935574	+	0.353130i	$0.114883\pi$
$4$	0	0
$5$	−6.27199	−0.560984	−0.280492	−	0.959856i	$-0.590498\pi$
$5$	−6.27199	−0.560984	−0.280492	+	0.959856i	$0.590498\pi$
$6$	0	0
$7$	−31.0130	−1.67455	−0.837273	−	0.546785i	$-0.815851\pi$
$7$	−31.0130	−1.67455	−0.837273	+	0.546785i	$0.815851\pi$
$8$	0	0
$9$	67.5323	2.50120
$10$	0	0
$11$	−8.24716	−0.226056	−0.113028	−	0.993592i	$-0.536055\pi$
$11$	−8.24716	−0.226056	−0.113028	+	0.993592i	$0.536055\pi$
$12$	0	0
$13$	32.0907	0.684644	0.342322	−	0.939583i	$-0.388787\pi$
$13$	32.0907	0.684644	0.342322	+	0.939583i	$0.388787\pi$
$14$	0	0
$15$	−60.9811	−1.04968
$16$	0	0
$17$	51.1369	0.729560	0.364780	−	0.931094i	$-0.381144\pi$
$17$	51.1369	0.729560	0.364780	+	0.931094i	$0.381144\pi$
$18$	0	0
$19$	−158.789	−1.91730	−0.958648	−	0.284593i	$-0.908141\pi$
$19$	−158.789	−1.91730	−0.958648	+	0.284593i	$0.908141\pi$
$20$	0	0
$21$	−301.533	−3.13332
$22$	0	0
$23$	−129.494	−1.17398	−0.586988	−	0.809596i	$-0.699686\pi$
$23$	−129.494	−1.17398	−0.586988	+	0.809596i	$0.699686\pi$
$24$	0	0
$25$	−85.6622	−0.685297
$26$	0	0
$27$	394.086	2.80896
$28$	0	0
$29$	29.0000	0.185695
$29$	29.0000	0.185695
$30$	0	0
$31$	14.6426	0.0848353	0.0424177	−	0.999100i	$-0.486494\pi$
$31$	14.6426	0.0848353	0.0424177	+	0.999100i	$0.486494\pi$
$32$	0	0
$33$	−80.1853	−0.422984
$34$	0	0
$35$	194.513	0.939393
$36$	0	0
$37$	−257.398	−1.14367	−0.571837	−	0.820367i	$-0.693769\pi$
$37$	−257.398	−1.14367	−0.571837	+	0.820367i	$0.693769\pi$
$38$	0	0
$39$	312.011	1.28107
$40$	0	0
$41$	−26.0208	−0.0991163	−0.0495582	−	0.998771i	$-0.515781\pi$
$41$	−26.0208	−0.0991163	−0.0495582	+	0.998771i	$0.515781\pi$
$42$	0	0
$43$	−3.40346	−0.0120703	−0.00603516	−	0.999982i	$-0.501921\pi$
$43$	−3.40346	−0.0120703	−0.00603516	+	0.999982i	$0.501921\pi$
$44$	0	0
$45$	−423.562	−1.40313
$46$	0	0
$47$	−482.228	−1.49660	−0.748300	−	0.663361i	$-0.769129\pi$
$47$	−482.228	−1.49660	−0.748300	+	0.663361i	$0.769129\pi$
$48$	0	0
$49$	618.808	1.80410
$50$	0	0
$51$	497.192	1.36511
$52$	0	0
$53$	169.104	0.438269	0.219135	−	0.975695i	$-0.429677\pi$
$53$	169.104	0.438269	0.219135	+	0.975695i	$0.429677\pi$
$54$	0	0
$55$	51.7261	0.126814
$56$	0	0
$57$	−1543.87	−3.58755
$58$	0	0
$59$	569.745	1.25719	0.628597	−	0.777731i	$-0.283630\pi$
$59$	569.745	1.25719	0.628597	+	0.777731i	$0.283630\pi$
$60$	0	0
$61$	−500.616	−1.05078	−0.525388	−	0.850863i	$-0.676080\pi$
$61$	−500.616	−1.05078	−0.525388	+	0.850863i	$0.676080\pi$
$62$	0	0
$63$	−2094.38	−4.18837
$64$	0	0
$65$	−201.273	−0.384074
$66$	0	0
$67$	−335.555	−0.611859	−0.305930	−	0.952054i	$-0.598967\pi$
$67$	−335.555	−0.611859	−0.305930	+	0.952054i	$0.598967\pi$
$68$	0	0
$69$	−1259.04	−2.19668
$70$	0	0
$71$	−958.137	−1.60155	−0.800774	−	0.598966i	$-0.795578\pi$
$71$	−958.137	−1.60155	−0.800774	+	0.598966i	$0.795578\pi$
$72$	0	0
$73$	−946.043	−1.51679	−0.758397	−	0.651793i	$-0.774017\pi$
$73$	−946.043	−1.51679	−0.758397	+	0.651793i	$0.774017\pi$
$74$	0	0
$75$	−832.874	−1.28229
$76$	0	0
$77$	255.769	0.378541
$78$	0	0
$79$	179.326	0.255389	0.127694	−	0.991814i	$-0.459242\pi$
$79$	179.326	0.255389	0.127694	+	0.991814i	$0.459242\pi$
$80$	0	0
$81$	2008.24	2.75479
$82$	0	0
$83$	285.090	0.377020	0.188510	−	0.982071i	$-0.439634\pi$
$83$	285.090	0.377020	0.188510	+	0.982071i	$0.439634\pi$
$84$	0	0
$85$	−320.730	−0.409271
$86$	0	0
$87$	281.960	0.347464
$88$	0	0
$89$	832.078	0.991013	0.495506	−	0.868604i	$-0.334983\pi$
$89$	832.078	0.991013	0.495506	+	0.868604i	$0.334983\pi$
$90$	0	0
$91$	−995.231	−1.14647
$92$	0	0
$93$	142.367	0.158740
$94$	0	0
$95$	995.921	1.07557
$96$	0	0
$97$	−908.370	−0.950835	−0.475417	−	0.879760i	$-0.657703\pi$
$97$	−908.370	−0.950835	−0.475417	+	0.879760i	$0.657703\pi$
$98$	0	0
$99$	−556.950	−0.565410
$100$	0	0
$101$	996.317	0.981557	0.490778	−	0.871284i	$-0.336712\pi$
$101$	996.317	0.981557	0.490778	+	0.871284i	$0.336712\pi$
$102$	0	0
$103$	−1295.46	−1.23928	−0.619639	−	0.784887i	$-0.712721\pi$
$103$	−1295.46	−1.23928	−0.619639	+	0.784887i	$0.712721\pi$
$104$	0	0
$105$	1891.21	1.75774
$106$	0	0
$107$	−1276.23	−1.15307	−0.576533	−	0.817074i	$-0.695595\pi$
$107$	−1276.23	−1.15307	−0.576533	+	0.817074i	$0.695595\pi$
$108$	0	0
$109$	1481.99	1.30228	0.651141	−	0.758957i	$-0.274291\pi$
$109$	1481.99	1.30228	0.651141	+	0.758957i	$0.274291\pi$
$110$	0	0
$111$	−2502.62	−2.13998
$112$	0	0
$113$	565.572	0.470837	0.235418	−	0.971894i	$-0.424354\pi$
$113$	565.572	0.470837	0.235418	+	0.971894i	$0.424354\pi$
$114$	0	0
$115$	812.187	0.658581
$116$	0	0
$117$	2167.16	1.71243
$118$	0	0
$119$	−1585.91	−1.22168
$120$	0	0
$121$	−1262.98	−0.948899
$122$	0	0
$123$	−252.995	−0.185461
$124$	0	0
$125$	1321.27	0.945424
$126$	0	0
$127$	2132.64	1.49009	0.745045	−	0.667014i	$-0.232428\pi$
$127$	2132.64	1.49009	0.745045	+	0.667014i	$0.232428\pi$
$128$	0	0
$129$	−33.0911	−0.0225853
$130$	0	0
$131$	−2518.91	−1.67999	−0.839993	−	0.542598i	$-0.817441\pi$
$131$	−2518.91	−1.67999	−0.839993	+	0.542598i	$0.817441\pi$
$132$	0	0
$133$	4924.52	3.21060
$134$	0	0
$135$	−2471.71	−1.57578
$136$	0	0
$137$	−57.5792	−0.0359075	−0.0179537	−	0.999839i	$-0.505715\pi$
$137$	−57.5792	−0.0359075	−0.0179537	+	0.999839i	$0.505715\pi$
$138$	0	0
$139$	1250.64	0.763148	0.381574	−	0.924338i	$-0.375382\pi$
$139$	1250.64	0.763148	0.381574	+	0.924338i	$0.375382\pi$
$140$	0	0
$141$	−4688.59	−2.80036
$142$	0	0
$143$	−264.658	−0.154768
$144$	0	0
$145$	−181.888	−0.104172
$146$	0	0
$147$	6016.52	3.37575
$148$	0	0
$149$	−1811.93	−0.996235	−0.498118	−	0.867109i	$-0.665975\pi$
$149$	−1811.93	−0.996235	−0.498118	+	0.867109i	$0.665975\pi$
$150$	0	0
$151$	−680.525	−0.366757	−0.183379	−	0.983042i	$-0.558703\pi$
$151$	−680.525	−0.366757	−0.183379	+	0.983042i	$0.558703\pi$
$152$	0	0
$153$	3453.39	1.82477
$154$	0	0
$155$	−91.8385	−0.0475912
$156$	0	0
$157$	−1422.01	−0.722858	−0.361429	−	0.932400i	$-0.617711\pi$
$157$	−1422.01	−0.722858	−0.361429	+	0.932400i	$0.617711\pi$
$158$	0	0
$159$	1644.16	0.820066
$160$	0	0
$161$	4016.01	1.96588
$162$	0	0
$163$	3154.01	1.51559	0.757794	−	0.652493i	$-0.226277\pi$
$163$	3154.01	1.51559	0.757794	+	0.652493i	$0.226277\pi$
$164$	0	0
$165$	502.921	0.237287
$166$	0	0
$167$	1539.64	0.713417	0.356708	−	0.934216i	$-0.383899\pi$
$167$	1539.64	0.713417	0.356708	+	0.934216i	$0.383899\pi$
$168$	0	0
$169$	−1167.18	−0.531263
$170$	0	0
$171$	−10723.4	−4.79554
$172$	0	0
$173$	2816.06	1.23758	0.618789	−	0.785557i	$-0.287624\pi$
$173$	2816.06	1.23758	0.618789	+	0.785557i	$0.287624\pi$
$174$	0	0
$175$	2656.64	1.14756
$176$	0	0
$177$	5539.50	2.35240
$178$	0	0
$179$	3564.28	1.48830	0.744152	−	0.668010i	$-0.232854\pi$
$179$	3564.28	1.48830	0.744152	+	0.668010i	$0.232854\pi$
$180$	0	0
$181$	−261.021	−0.107191	−0.0535953	−	0.998563i	$-0.517068\pi$
$181$	−261.021	−0.107191	−0.0535953	+	0.998563i	$0.517068\pi$
$182$	0	0
$183$	−4867.38	−1.96616
$184$	0	0
$185$	1614.40	0.641583
$186$	0	0
$187$	−421.734	−0.164921
$188$	0	0
$189$	−12221.8	−4.70373
$190$	0	0
$191$	2242.54	0.849551	0.424776	−	0.905299i	$-0.360353\pi$
$191$	2242.54	0.849551	0.424776	+	0.905299i	$0.360353\pi$
$192$	0	0
$193$	120.858	0.0450752	0.0225376	−	0.999746i	$-0.492825\pi$
$193$	120.858	0.0450752	0.0225376	+	0.999746i	$0.492825\pi$
$194$	0	0
$195$	−1956.93	−0.718660
$196$	0	0
$197$	−864.591	−0.312688	−0.156344	−	0.987703i	$-0.549971\pi$
$197$	−864.591	−0.312688	−0.156344	+	0.987703i	$0.549971\pi$
$198$	0	0
$199$	4885.39	1.74028	0.870140	−	0.492804i	$-0.164028\pi$
$199$	4885.39	1.74028	0.870140	+	0.492804i	$0.164028\pi$
$200$	0	0
$201$	−3262.52	−1.14488
$202$	0	0
$203$	−899.378	−0.310955
$204$	0	0
$205$	163.202	0.0556027
$206$	0	0
$207$	−8745.06	−2.93634
$208$	0	0
$209$	1309.56	0.433416
$210$	0	0
$211$	−5261.82	−1.71677	−0.858386	−	0.513005i	$-0.828532\pi$
$211$	−5261.82	−1.71677	−0.858386	+	0.513005i	$0.828532\pi$
$212$	0	0
$213$	−9315.75	−2.99673
$214$	0	0
$215$	21.3465	0.00677125
$216$	0	0
$217$	−454.112	−0.142061
$218$	0	0
$219$	−9198.16	−2.83815
$220$	0	0
$221$	1641.02	0.499489
$222$	0	0
$223$	−2726.49	−0.818740	−0.409370	−	0.912368i	$-0.634252\pi$
$223$	−2726.49	−0.818740	−0.409370	+	0.912368i	$0.634252\pi$
$224$	0	0
$225$	−5784.96	−1.71406
$226$	0	0
$227$	−6079.66	−1.77763	−0.888813	−	0.458270i	$-0.848469\pi$
$227$	−6079.66	−1.77763	−0.888813	+	0.458270i	$0.848469\pi$
$228$	0	0
$229$	538.146	0.155291	0.0776456	−	0.996981i	$-0.475260\pi$
$229$	538.146	0.155291	0.0776456	+	0.996981i	$0.475260\pi$
$230$	0	0
$231$	2486.79	0.708306
$232$	0	0
$233$	2834.78	0.797051	0.398525	−	0.917157i	$-0.369522\pi$
$233$	2834.78	0.797051	0.398525	+	0.917157i	$0.369522\pi$
$234$	0	0
$235$	3024.53	0.839568
$236$	0	0
$237$	1743.54	0.477870
$238$	0	0
$239$	354.937	0.0960627	0.0480313	−	0.998846i	$-0.484705\pi$
$239$	354.937	0.0960627	0.0480313	+	0.998846i	$0.484705\pi$
$240$	0	0
$241$	4887.24	1.30628	0.653142	−	0.757235i	$-0.273450\pi$
$241$	4887.24	1.30628	0.653142	+	0.757235i	$0.273450\pi$
$242$	0	0
$243$	8885.33	2.34565
$244$	0	0
$245$	−3881.15	−1.01207
$246$	0	0
$247$	−5095.65	−1.31267
$248$	0	0
$249$	2771.86	0.705461
$250$	0	0
$251$	250.836	0.0630783	0.0315391	−	0.999503i	$-0.489959\pi$
$251$	250.836	0.0630783	0.0315391	+	0.999503i	$0.489959\pi$
$252$	0	0
$253$	1067.96	0.265384
$254$	0	0
$255$	−3118.38	−0.765807
$256$	0	0
$257$	721.700	0.175169	0.0875845	−	0.996157i	$-0.472085\pi$
$257$	721.700	0.175169	0.0875845	+	0.996157i	$0.472085\pi$
$258$	0	0
$259$	7982.68	1.91513
$260$	0	0
$261$	1958.44	0.464461
$262$	0	0
$263$	3765.00	0.882737	0.441368	−	0.897326i	$-0.354493\pi$
$263$	3765.00	0.882737	0.441368	+	0.897326i	$0.354493\pi$
$264$	0	0
$265$	−1060.62	−0.245862
$266$	0	0
$267$	8090.11	1.85433
$268$	0	0
$269$	552.384	0.125202	0.0626012	−	0.998039i	$-0.480060\pi$
$269$	552.384	0.125202	0.0626012	+	0.998039i	$0.480060\pi$
$270$	0	0
$271$	5370.32	1.20378	0.601888	−	0.798580i	$-0.294415\pi$
$271$	5370.32	1.20378	0.601888	+	0.798580i	$0.294415\pi$
$272$	0	0
$273$	−9676.40	−2.14521
$274$	0	0
$275$	706.470	0.154915
$276$	0	0
$277$	−3413.04	−0.740324	−0.370162	−	0.928967i	$-0.620698\pi$
$277$	−3413.04	−0.740324	−0.370162	+	0.928967i	$0.620698\pi$
$278$	0	0
$279$	988.851	0.212190
$280$	0	0
$281$	−6727.56	−1.42823	−0.714115	−	0.700028i	$-0.753171\pi$
$281$	−6727.56	−1.42823	−0.714115	+	0.700028i	$0.753171\pi$
$282$	0	0
$283$	−8718.25	−1.83126	−0.915630	−	0.402023i	$-0.868307\pi$
$283$	−8718.25	−1.83126	−0.915630	+	0.402023i	$0.868307\pi$
$284$	0	0
$285$	9683.11	2.01256
$286$	0	0
$287$	806.984	0.165975
$288$	0	0
$289$	−2298.02	−0.467742
$290$	0	0
$291$	−8831.87	−1.77915
$292$	0	0
$293$	1279.00	0.255018	0.127509	−	0.991837i	$-0.459302\pi$
$293$	1279.00	0.255018	0.127509	+	0.991837i	$0.459302\pi$
$294$	0	0
$295$	−3573.43	−0.705265
$296$	0	0
$297$	−3250.09	−0.634982
$298$	0	0
$299$	−4155.57	−0.803755
$300$	0	0
$301$	105.552	0.0202123
$302$	0	0
$303$	9686.96	1.83664
$304$	0	0
$305$	3139.86	0.589468
$306$	0	0
$307$	8483.52	1.57713	0.788567	−	0.614949i	$-0.210823\pi$
$307$	8483.52	1.57713	0.788567	+	0.614949i	$0.210823\pi$
$308$	0	0
$309$	−12595.5	−2.31887
$310$	0	0
$311$	7130.60	1.30013	0.650063	−	0.759880i	$-0.274743\pi$
$311$	7130.60	1.30013	0.650063	+	0.759880i	$0.274743\pi$
$312$	0	0
$313$	3430.86	0.619565	0.309782	−	0.950808i	$-0.399744\pi$
$313$	3430.86	0.619565	0.309782	+	0.950808i	$0.399744\pi$
$314$	0	0
$315$	13135.9	2.34961
$316$	0	0
$317$	7352.21	1.30265	0.651327	−	0.758797i	$-0.274213\pi$
$317$	7352.21	1.30265	0.651327	+	0.758797i	$0.274213\pi$
$318$	0	0
$319$	−239.168	−0.0419775
$320$	0	0
$321$	−12408.5	−2.15756
$322$	0	0
$323$	−8119.96	−1.39878
$324$	0	0
$325$	−2748.96	−0.469185
$326$	0	0
$327$	14409.0	2.43676
$328$	0	0
$329$	14955.3	2.50612
$330$	0	0
$331$	−5975.85	−0.992333	−0.496166	−	0.868227i	$-0.665259\pi$
$331$	−5975.85	−0.992333	−0.496166	+	0.868227i	$0.665259\pi$
$332$	0	0
$333$	−17382.7	−2.86055
$334$	0	0
$335$	2104.60	0.343243
$336$	0	0
$337$	6455.18	1.04343	0.521715	−	0.853120i	$-0.325292\pi$
$337$	6455.18	1.04343	0.521715	+	0.853120i	$0.325292\pi$
$338$	0	0
$339$	5498.93	0.881005
$340$	0	0
$341$	−120.760	−0.0191775
$342$	0	0
$343$	−8553.62	−1.34651
$344$	0	0
$345$	7896.71	1.23230
$346$	0	0
$347$	−12624.0	−1.95300	−0.976502	−	0.215509i	$-0.930859\pi$
$347$	−12624.0	−1.95300	−0.976502	+	0.215509i	$0.930859\pi$
$348$	0	0
$349$	−35.6888	−0.00547386	−0.00273693	−	0.999996i	$-0.500871\pi$
$349$	−35.6888	−0.00547386	−0.00273693	+	0.999996i	$0.500871\pi$
$350$	0	0
$351$	12646.5	1.92314
$352$	0	0
$353$	−639.322	−0.0963957	−0.0481978	−	0.998838i	$-0.515348\pi$
$353$	−639.322	−0.0963957	−0.0481978	+	0.998838i	$0.515348\pi$
$354$	0	0
$355$	6009.42	0.898443
$356$	0	0
$357$	−15419.4	−2.28595
$358$	0	0
$359$	−7152.10	−1.05146	−0.525729	−	0.850652i	$-0.676207\pi$
$359$	−7152.10	−1.05146	−0.525729	+	0.850652i	$0.676207\pi$
$360$	0	0
$361$	18354.9	2.67603
$362$	0	0
$363$	−12279.7	−1.77553
$364$	0	0
$365$	5933.57	0.850897
$366$	0	0
$367$	6894.06	0.980564	0.490282	−	0.871564i	$-0.336894\pi$
$367$	6894.06	0.980564	0.490282	+	0.871564i	$0.336894\pi$
$368$	0	0
$369$	−1757.25	−0.247909
$370$	0	0
$371$	−5244.43	−0.733902
$372$	0	0
$373$	−8479.89	−1.17714	−0.588569	−	0.808447i	$-0.700308\pi$
$373$	−8479.89	−1.17714	−0.588569	+	0.808447i	$0.700308\pi$
$374$	0	0
$375$	12846.4	1.76903
$376$	0	0
$377$	930.631	0.127135
$378$	0	0
$379$	4935.89	0.668970	0.334485	−	0.942401i	$-0.391438\pi$
$379$	4935.89	0.668970	0.334485	+	0.942401i	$0.391438\pi$
$380$	0	0
$381$	20735.2	2.78818
$382$	0	0
$383$	−1790.95	−0.238938	−0.119469	−	0.992838i	$-0.538119\pi$
$383$	−1790.95	−0.238938	−0.119469	+	0.992838i	$0.538119\pi$
$384$	0	0
$385$	−1604.18	−0.212355
$386$	0	0
$387$	−229.844	−0.0301902
$388$	0	0
$389$	−6332.31	−0.825350	−0.412675	−	0.910878i	$-0.635405\pi$
$389$	−6332.31	−0.825350	−0.412675	+	0.910878i	$0.635405\pi$
$390$	0	0
$391$	−6621.94	−0.856486
$392$	0	0
$393$	−24490.8	−3.14350
$394$	0	0
$395$	−1124.73	−0.143269
$396$	0	0
$397$	14324.5	1.81089	0.905446	−	0.424462i	$-0.139537\pi$
$397$	14324.5	1.81089	0.905446	+	0.424462i	$0.139537\pi$
$398$	0	0
$399$	47880.0	6.00751
$400$	0	0
$401$	6430.30	0.800783	0.400392	−	0.916344i	$-0.368874\pi$
$401$	6430.30	0.800783	0.400392	+	0.916344i	$0.368874\pi$
$402$	0	0
$403$	469.893	0.0580820
$404$	0	0
$405$	−12595.7	−1.54539
$406$	0	0
$407$	2122.80	0.258534
$408$	0	0
$409$	44.4605	0.00537513	0.00268757	−	0.999996i	$-0.499145\pi$
$409$	44.4605	0.00537513	0.00268757	+	0.999996i	$0.499145\pi$
$410$	0	0
$411$	−559.830	−0.0671882
$412$	0	0
$413$	−17669.5	−2.10523
$414$	0	0
$415$	−1788.08	−0.211502
$416$	0	0
$417$	12159.7	1.42796
$418$	0	0
$419$	23.4862	0.00273837	0.00136918	−	0.999999i	$-0.499564\pi$
$419$	23.4862	0.00273837	0.00136918	+	0.999999i	$0.499564\pi$
$420$	0	0
$421$	3365.39	0.389594	0.194797	−	0.980844i	$-0.437595\pi$
$421$	3365.39	0.389594	0.194797	+	0.980844i	$0.437595\pi$
$422$	0	0
$423$	−32566.0	−3.74329
$424$	0	0
$425$	−4380.50	−0.499965
$426$	0	0
$427$	15525.6	1.75957
$428$	0	0
$429$	−2573.21	−0.289593
$430$	0	0
$431$	9557.80	1.06817	0.534087	−	0.845429i	$-0.320655\pi$
$431$	9557.80	1.06817	0.534087	+	0.845429i	$0.320655\pi$
$432$	0	0
$433$	11660.3	1.29413	0.647065	−	0.762435i	$-0.275996\pi$
$433$	11660.3	1.29413	0.647065	+	0.762435i	$0.275996\pi$
$434$	0	0
$435$	−1768.45	−0.194921
$436$	0	0
$437$	20562.3	2.25086
$438$	0	0
$439$	−16716.4	−1.81738	−0.908689	−	0.417474i	$-0.862916\pi$
$439$	−16716.4	−1.81738	−0.908689	+	0.417474i	$0.862916\pi$
$440$	0	0
$441$	41789.5	4.51242
$442$	0	0
$443$	367.759	0.0394419	0.0197209	−	0.999806i	$-0.493722\pi$
$443$	367.759	0.0394419	0.0197209	+	0.999806i	$0.493722\pi$
$444$	0	0
$445$	−5218.79	−0.555942
$446$	0	0
$447$	−17617.0	−1.86410
$448$	0	0
$449$	−4823.82	−0.507015	−0.253508	−	0.967333i	$-0.581584\pi$
$449$	−4823.82	−0.507015	−0.253508	+	0.967333i	$0.581584\pi$
$450$	0	0
$451$	214.598	0.0224058
$452$	0	0
$453$	−6616.59	−0.686257
$454$	0	0
$455$	6242.08	0.643150
$456$	0	0
$457$	−2785.87	−0.285159	−0.142579	−	0.989783i	$-0.545540\pi$
$457$	−2785.87	−0.285159	−0.142579	+	0.989783i	$0.545540\pi$
$458$	0	0
$459$	20152.4	2.04931
$460$	0	0
$461$	−9052.88	−0.914608	−0.457304	−	0.889310i	$-0.651185\pi$
$461$	−9052.88	−0.914608	−0.457304	+	0.889310i	$0.651185\pi$
$462$	0	0
$463$	−3825.06	−0.383943	−0.191972	−	0.981400i	$-0.561488\pi$
$463$	−3825.06	−0.383943	−0.191972	+	0.981400i	$0.561488\pi$
$464$	0	0
$465$	−892.924	−0.0890503
$466$	0	0
$467$	−3943.34	−0.390741	−0.195371	−	0.980729i	$-0.562591\pi$
$467$	−3943.34	−0.390741	−0.195371	+	0.980729i	$0.562591\pi$
$468$	0	0
$469$	10406.6	1.02459
$470$	0	0
$471$	−13825.9	−1.35258
$472$	0	0
$473$	28.0689	0.00272856
$474$	0	0
$475$	13602.2	1.31392
$476$	0	0
$477$	11420.0	1.09620
$478$	0	0
$479$	−7621.40	−0.726995	−0.363497	−	0.931595i	$-0.618417\pi$
$479$	−7621.40	−0.726995	−0.363497	+	0.931595i	$0.618417\pi$
$480$	0	0
$481$	−8260.09	−0.783009
$482$	0	0
$483$	39046.8	3.67845
$484$	0	0
$485$	5697.28	0.533403
$486$	0	0
$487$	−2939.21	−0.273488	−0.136744	−	0.990606i	$-0.543664\pi$
$487$	−2939.21	−0.273488	−0.136744	+	0.990606i	$0.543664\pi$
$488$	0	0
$489$	30665.7	2.83589
$490$	0	0
$491$	21365.5	1.96377	0.981886	−	0.189473i	$-0.0606780\pi$
$491$	21365.5	1.96377	0.981886	+	0.189473i	$0.0606780\pi$
$492$	0	0
$493$	1482.97	0.135476
$494$	0	0
$495$	3493.18	0.317186
$496$	0	0
$497$	29714.7	2.68187
$498$	0	0
$499$	−5450.33	−0.488958	−0.244479	−	0.969655i	$-0.578617\pi$
$499$	−5450.33	−0.488958	−0.244479	+	0.969655i	$0.578617\pi$
$500$	0	0
$501$	14969.5	1.33491
$502$	0	0
$503$	−19531.8	−1.73137	−0.865686	−	0.500587i	$-0.833118\pi$
$503$	−19531.8	−1.73137	−0.865686	+	0.500587i	$0.833118\pi$
$504$	0	0
$505$	−6248.89	−0.550637
$506$	0	0
$507$	−11348.3	−0.994072
$508$	0	0
$509$	7726.61	0.672841	0.336420	−	0.941712i	$-0.390784\pi$
$509$	7726.61	0.672841	0.336420	+	0.941712i	$0.390784\pi$
$510$	0	0
$511$	29339.7	2.53994
$512$	0	0
$513$	−62576.5	−5.38561
$514$	0	0
$515$	8125.12	0.695215
$516$	0	0
$517$	3977.01	0.338315
$518$	0	0
$519$	27379.9	2.31569
$520$	0	0
$521$	−7915.18	−0.665586	−0.332793	−	0.943000i	$-0.607991\pi$
$521$	−7915.18	−0.665586	−0.332793	+	0.943000i	$0.607991\pi$
$522$	0	0
$523$	−8078.88	−0.675459	−0.337729	−	0.941243i	$-0.609659\pi$
$523$	−8078.88	−0.675459	−0.337729	+	0.941243i	$0.609659\pi$
$524$	0	0
$525$	25829.9	2.14726
$526$	0	0
$527$	748.779	0.0618925
$528$	0	0
$529$	4601.80	0.378220
$530$	0	0
$531$	38476.2	3.14449
$532$	0	0
$533$	−835.028	−0.0678594
$534$	0	0
$535$	8004.51	0.646851
$536$	0	0
$537$	34654.6	2.78484
$538$	0	0
$539$	−5103.41	−0.407828
$540$	0	0
$541$	−12220.5	−0.971163	−0.485582	−	0.874191i	$-0.661392\pi$
$541$	−12220.5	−0.971163	−0.485582	+	0.874191i	$0.661392\pi$
$542$	0	0
$543$	−2537.84	−0.200570
$544$	0	0
$545$	−9295.01	−0.730559
$546$	0	0
$547$	−6172.25	−0.482462	−0.241231	−	0.970468i	$-0.577551\pi$
$547$	−6172.25	−0.482462	−0.241231	+	0.970468i	$0.577551\pi$
$548$	0	0
$549$	−33807.8	−2.62820
$550$	0	0
$551$	−4604.87	−0.356033
$552$	0	0
$553$	−5561.43	−0.427660
$554$	0	0
$555$	15696.4	1.20050
$556$	0	0
$557$	−3103.59	−0.236092	−0.118046	−	0.993008i	$-0.537663\pi$
$557$	−3103.59	−0.236092	−0.118046	+	0.993008i	$0.537663\pi$
$558$	0	0
$559$	−109.220	−0.00826386
$560$	0	0
$561$	−4100.43	−0.308592
$562$	0	0
$563$	−21390.9	−1.60128	−0.800639	−	0.599147i	$-0.795506\pi$
$563$	−21390.9	−1.60128	−0.800639	+	0.599147i	$0.795506\pi$
$564$	0	0
$565$	−3547.26	−0.264132
$566$	0	0
$567$	−62281.6	−4.61302
$568$	0	0
$569$	−23212.7	−1.71024	−0.855120	−	0.518431i	$-0.826516\pi$
$569$	−23212.7	−1.71024	−0.855120	+	0.518431i	$0.826516\pi$
$570$	0	0
$571$	11439.8	0.838424	0.419212	−	0.907888i	$-0.362306\pi$
$571$	11439.8	0.838424	0.419212	+	0.907888i	$0.362306\pi$
$572$	0	0
$573$	21803.7	1.58964
$574$	0	0
$575$	11092.8	0.804523
$576$	0	0
$577$	−8568.11	−0.618189	−0.309095	−	0.951031i	$-0.600026\pi$
$577$	−8568.11	−0.618189	−0.309095	+	0.951031i	$0.600026\pi$
$578$	0	0
$579$	1175.07	0.0843425
$580$	0	0
$581$	−8841.50	−0.631338
$582$	0	0
$583$	−1394.63	−0.0990732
$584$	0	0
$585$	−13592.4	−0.960645
$586$	0	0
$587$	2184.34	0.153590	0.0767950	−	0.997047i	$-0.475531\pi$
$587$	2184.34	0.153590	0.0767950	+	0.997047i	$0.475531\pi$
$588$	0	0
$589$	−2325.09	−0.162655
$590$	0	0
$591$	−8406.22	−0.585086
$592$	0	0
$593$	−8518.37	−0.589895	−0.294947	−	0.955513i	$-0.595302\pi$
$593$	−8518.37	−0.589895	−0.294947	+	0.955513i	$0.595302\pi$
$594$	0	0
$595$	9946.81	0.685343
$596$	0	0
$597$	47499.5	3.25632
$598$	0	0
$599$	17697.5	1.20718	0.603591	−	0.797294i	$-0.293736\pi$
$599$	17697.5	1.20718	0.603591	+	0.797294i	$0.293736\pi$
$600$	0	0
$601$	25751.4	1.74779	0.873894	−	0.486116i	$-0.161587\pi$
$601$	25751.4	1.74779	0.873894	+	0.486116i	$0.161587\pi$
$602$	0	0
$603$	−22660.8	−1.53038
$604$	0	0
$605$	7921.42	0.532317
$606$	0	0
$607$	−12644.9	−0.845538	−0.422769	−	0.906238i	$-0.638942\pi$
$607$	−12644.9	−0.845538	−0.422769	+	0.906238i	$0.638942\pi$
$608$	0	0
$609$	−8744.44	−0.581844
$610$	0	0
$611$	−15475.1	−1.02464
$612$	0	0
$613$	18792.6	1.23821	0.619107	−	0.785307i	$-0.287495\pi$
$613$	18792.6	1.23821	0.619107	+	0.785307i	$0.287495\pi$
$614$	0	0
$615$	1586.78	0.104041
$616$	0	0
$617$	−8630.30	−0.563116	−0.281558	−	0.959544i	$-0.590851\pi$
$617$	−8630.30	−0.563116	−0.281558	+	0.959544i	$0.590851\pi$
$618$	0	0
$619$	−26553.4	−1.72419	−0.862094	−	0.506748i	$-0.830847\pi$
$619$	−26553.4	−1.72419	−0.862094	+	0.506748i	$0.830847\pi$
$620$	0	0
$621$	−51032.0	−3.29765
$622$	0	0
$623$	−25805.3	−1.65950
$624$	0	0
$625$	2420.78	0.154930
$626$	0	0
$627$	12732.5	0.810985
$628$	0	0
$629$	−13162.5	−0.834379
$630$	0	0
$631$	−18310.6	−1.15521	−0.577603	−	0.816318i	$-0.696012\pi$
$631$	−18310.6	−1.15521	−0.577603	+	0.816318i	$0.696012\pi$
$632$	0	0
$633$	−51159.5	−3.21233
$634$	0	0
$635$	−13375.9	−0.835916
$636$	0	0
$637$	19858.0	1.23517
$638$	0	0
$639$	−64705.2	−4.00579
$640$	0	0
$641$	−7753.98	−0.477790	−0.238895	−	0.971045i	$-0.576785\pi$
$641$	−7753.98	−0.477790	−0.238895	+	0.971045i	$0.576785\pi$
$642$	0	0
$643$	24720.7	1.51616	0.758078	−	0.652163i	$-0.226138\pi$
$643$	24720.7	1.51616	0.758078	+	0.652163i	$0.226138\pi$
$644$	0	0
$645$	207.547	0.0126700
$646$	0	0
$647$	−10318.0	−0.626957	−0.313478	−	0.949595i	$-0.601494\pi$
$647$	−10318.0	−0.626957	−0.313478	+	0.949595i	$0.601494\pi$
$648$	0	0
$649$	−4698.78	−0.284196
$650$	0	0
$651$	−4415.23	−0.265817
$652$	0	0
$653$	−15277.3	−0.915542	−0.457771	−	0.889070i	$-0.651352\pi$
$653$	−15277.3	−0.915542	−0.457771	+	0.889070i	$0.651352\pi$
$654$	0	0
$655$	15798.6	0.942444
$656$	0	0
$657$	−63888.5	−3.79380
$658$	0	0
$659$	−9600.60	−0.567506	−0.283753	−	0.958897i	$-0.591580\pi$
$659$	−9600.60	−0.567506	−0.283753	+	0.958897i	$0.591580\pi$
$660$	0	0
$661$	6812.43	0.400866	0.200433	−	0.979707i	$-0.435765\pi$
$661$	6812.43	0.400866	0.200433	+	0.979707i	$0.435765\pi$
$662$	0	0
$663$	15955.3	0.934617
$664$	0	0
$665$	−30886.5	−1.80109
$666$	0	0
$667$	−3755.34	−0.218002
$668$	0	0
$669$	−26509.0	−1.53198
$670$	0	0
$671$	4128.66	0.237534
$672$	0	0
$673$	9459.01	0.541780	0.270890	−	0.962610i	$-0.412682\pi$
$673$	9459.01	0.541780	0.270890	+	0.962610i	$0.412682\pi$
$674$	0	0
$675$	−33758.3	−1.92497
$676$	0	0
$677$	−25231.8	−1.43241	−0.716203	−	0.697892i	$-0.754121\pi$
$677$	−25231.8	−1.43241	−0.716203	+	0.697892i	$0.754121\pi$
$678$	0	0
$679$	28171.3	1.59222
$680$	0	0
$681$	−59111.1	−3.32620
$682$	0	0
$683$	16443.6	0.921226	0.460613	−	0.887601i	$-0.347630\pi$
$683$	16443.6	0.921226	0.460613	+	0.887601i	$0.347630\pi$
$684$	0	0
$685$	361.136	0.0201435
$686$	0	0
$687$	5232.27	0.290573
$688$	0	0
$689$	5426.68	0.300058
$690$	0	0
$691$	−8693.86	−0.478625	−0.239313	−	0.970943i	$-0.576922\pi$
$691$	−8693.86	−0.478625	−0.239313	+	0.970943i	$0.576922\pi$
$692$	0	0
$693$	17272.7	0.946805
$694$	0	0
$695$	−7843.98	−0.428114
$696$	0	0
$697$	−1330.62	−0.0723113
$698$	0	0
$699$	27562.0	1.49140
$700$	0	0
$701$	−2006.88	−0.108130	−0.0540648	−	0.998537i	$-0.517218\pi$
$701$	−2006.88	−0.108130	−0.0540648	+	0.998537i	$0.517218\pi$
$702$	0	0
$703$	40871.9	2.19276
$704$	0	0
$705$	29406.8	1.57096
$706$	0	0
$707$	−30898.8	−1.64366
$708$	0	0
$709$	−10597.0	−0.561322	−0.280661	−	0.959807i	$-0.590554\pi$
$709$	−10597.0	−0.561322	−0.280661	+	0.959807i	$0.590554\pi$
$710$	0	0
$711$	12110.3	0.638778
$712$	0	0
$713$	−1896.14	−0.0995947
$714$	0	0
$715$	1659.93	0.0868221
$716$	0	0
$717$	3450.97	0.179748
$718$	0	0
$719$	2248.05	0.116604	0.0583020	−	0.998299i	$-0.481431\pi$
$719$	2248.05	0.116604	0.0583020	+	0.998299i	$0.481431\pi$
$720$	0	0
$721$	40176.2	2.07523
$722$	0	0
$723$	47517.5	2.44425
$724$	0	0
$725$	−2484.20	−0.127257
$726$	0	0
$727$	26960.8	1.37541	0.687704	−	0.725991i	$-0.258619\pi$
$727$	26960.8	1.37541	0.687704	+	0.725991i	$0.258619\pi$
$728$	0	0
$729$	32167.5	1.63428
$730$	0	0
$731$	−174.043	−0.00880601
$732$	0	0
$733$	10345.6	0.521317	0.260658	−	0.965431i	$-0.416060\pi$
$733$	10345.6	0.521317	0.260658	+	0.965431i	$0.416060\pi$
$734$	0	0
$735$	−37735.6	−1.89374
$736$	0	0
$737$	2767.38	0.138314
$738$	0	0
$739$	−11942.7	−0.594479	−0.297239	−	0.954803i	$-0.596066\pi$
$739$	−11942.7	−0.594479	−0.297239	+	0.954803i	$0.596066\pi$
$740$	0	0
$741$	−49543.8	−2.45619
$742$	0	0
$743$	−1334.16	−0.0658754	−0.0329377	−	0.999457i	$-0.510486\pi$
$743$	−1334.16	−0.0658754	−0.0329377	+	0.999457i	$0.510486\pi$
$744$	0	0
$745$	11364.4	0.558872
$746$	0	0
$747$	19252.8	0.943002
$748$	0	0
$749$	39579.8	1.93086
$750$	0	0
$751$	17347.0	0.842876	0.421438	−	0.906857i	$-0.361526\pi$
$751$	17347.0	0.842876	0.421438	+	0.906857i	$0.361526\pi$
$752$	0	0
$753$	2438.82	0.118029
$754$	0	0
$755$	4268.25	0.205745
$756$	0	0
$757$	5359.59	0.257328	0.128664	−	0.991688i	$-0.458931\pi$
$757$	5359.59	0.257328	0.128664	+	0.991688i	$0.458931\pi$
$758$	0	0
$759$	10383.5	0.496573
$760$	0	0
$761$	10355.3	0.493272	0.246636	−	0.969108i	$-0.420675\pi$
$761$	10355.3	0.493272	0.246636	+	0.969108i	$0.420675\pi$
$762$	0	0
$763$	−45960.9	−2.18073
$764$	0	0
$765$	−21659.6	−1.02367
$766$	0	0
$767$	18283.5	0.860730
$768$	0	0
$769$	−9339.42	−0.437956	−0.218978	−	0.975730i	$-0.570272\pi$
$769$	−9339.42	−0.437956	−0.218978	+	0.975730i	$0.570272\pi$
$770$	0	0
$771$	7016.93	0.327767
$772$	0	0
$773$	21743.9	1.01174	0.505869	−	0.862610i	$-0.331172\pi$
$773$	21743.9	1.01174	0.505869	+	0.862610i	$0.331172\pi$
$774$	0	0
$775$	−1254.32	−0.0581374
$776$	0	0
$777$	77613.8	3.58350
$778$	0	0
$779$	4131.81	0.190035
$780$	0	0
$781$	7901.91	0.362039
$782$	0	0
$783$	11428.5	0.521611
$784$	0	0
$785$	8918.83	0.405512
$786$	0	0
$787$	−26378.6	−1.19478	−0.597391	−	0.801950i	$-0.703796\pi$
$787$	−26378.6	−1.19478	−0.597391	+	0.801950i	$0.703796\pi$
$788$	0	0
$789$	36606.2	1.65173
$790$	0	0
$791$	−17540.1	−0.788438
$792$	0	0
$793$	−16065.1	−0.719407
$794$	0	0
$795$	−10312.2	−0.460044
$796$	0	0
$797$	−36134.8	−1.60597	−0.802986	−	0.595999i	$-0.796756\pi$
$797$	−36134.8	−1.60597	−0.802986	+	0.595999i	$0.796756\pi$
$798$	0	0
$799$	−24659.6	−1.09186
$800$	0	0
$801$	56192.2	2.47872
$802$	0	0
$803$	7802.17	0.342880
$804$	0	0
$805$	−25188.4	−1.10282
$806$	0	0
$807$	5370.70	0.234272
$808$	0	0
$809$	368.546	0.0160166	0.00800828	−	0.999968i	$-0.497451\pi$
$809$	368.546	0.0160166	0.00800828	+	0.999968i	$0.497451\pi$
$810$	0	0
$811$	−24252.1	−1.05007	−0.525034	−	0.851081i	$-0.675948\pi$
$811$	−24252.1	−1.05007	−0.525034	+	0.851081i	$0.675948\pi$
$812$	0	0
$813$	52214.4	2.25244
$814$	0	0
$815$	−19781.9	−0.850221
$816$	0	0
$817$	540.432	0.0231424
$818$	0	0
$819$	−67210.2	−2.86754
$820$	0	0
$821$	8693.57	0.369559	0.184779	−	0.982780i	$-0.440843\pi$
$821$	8693.57	0.369559	0.184779	+	0.982780i	$0.440843\pi$
$822$	0	0
$823$	−4674.45	−0.197984	−0.0989922	−	0.995088i	$-0.531562\pi$
$823$	−4674.45	−0.197984	−0.0989922	+	0.995088i	$0.531562\pi$
$824$	0	0
$825$	6868.84	0.289870
$826$	0	0
$827$	−18877.2	−0.793740	−0.396870	−	0.917875i	$-0.629904\pi$
$827$	−18877.2	−0.793740	−0.396870	+	0.917875i	$0.629904\pi$
$828$	0	0
$829$	30674.6	1.28513	0.642565	−	0.766231i	$-0.277870\pi$
$829$	30674.6	1.28513	0.642565	+	0.766231i	$0.277870\pi$
$830$	0	0
$831$	−33184.2	−1.38526
$832$	0	0
$833$	31643.9	1.31620
$834$	0	0
$835$	−9656.58	−0.400215
$836$	0	0
$837$	5770.46	0.238299
$838$	0	0
$839$	6238.05	0.256688	0.128344	−	0.991730i	$-0.459034\pi$
$839$	6238.05	0.256688	0.128344	+	0.991730i	$0.459034\pi$
$840$	0	0
$841$	841.000	0.0344828
$842$	0	0
$843$	−65410.5	−2.67243
$844$	0	0
$845$	7320.57	0.298030
$846$	0	0
$847$	39169.0	1.58897
$848$	0	0
$849$	−84765.6	−3.42656
$850$	0	0
$851$	33331.6	1.34265
$852$	0	0
$853$	−16138.1	−0.647780	−0.323890	−	0.946095i	$-0.604991\pi$
$853$	−16138.1	−0.647780	−0.323890	+	0.946095i	$0.604991\pi$
$854$	0	0
$855$	67256.9	2.69022
$856$	0	0
$857$	47596.5	1.89716	0.948579	−	0.316541i	$-0.102522\pi$
$857$	47596.5	1.89716	0.948579	+	0.316541i	$0.102522\pi$
$858$	0	0
$859$	−572.644	−0.0227455	−0.0113727	−	0.999935i	$-0.503620\pi$
$859$	−572.644	−0.0227455	−0.0113727	+	0.999935i	$0.503620\pi$
$860$	0	0
$861$	7846.13	0.310564
$862$	0	0
$863$	−23203.3	−0.915237	−0.457619	−	0.889149i	$-0.651297\pi$
$863$	−23203.3	−0.915237	−0.457619	+	0.889149i	$0.651297\pi$
$864$	0	0
$865$	−17662.3	−0.694261
$866$	0	0
$867$	−22343.1	−0.875215
$868$	0	0
$869$	−1478.93	−0.0577321
$870$	0	0
$871$	−10768.2	−0.418906
$872$	0	0
$873$	−61344.3	−2.37822
$874$	0	0
$875$	−40976.6	−1.58316
$876$	0	0
$877$	−41334.1	−1.59151	−0.795755	−	0.605619i	$-0.792926\pi$
$877$	−41334.1	−1.59151	−0.795755	+	0.605619i	$0.792926\pi$
$878$	0	0
$879$	12435.5	0.477176
$880$	0	0
$881$	−30870.9	−1.18055	−0.590276	−	0.807202i	$-0.700981\pi$
$881$	−30870.9	−1.18055	−0.590276	+	0.807202i	$0.700981\pi$
$882$	0	0
$883$	11615.6	0.442691	0.221346	−	0.975195i	$-0.428955\pi$
$883$	11615.6	0.442691	0.221346	+	0.975195i	$0.428955\pi$
$884$	0	0
$885$	−34743.7	−1.31966
$886$	0	0
$887$	9853.82	0.373009	0.186505	−	0.982454i	$-0.440284\pi$
$887$	9853.82	0.373009	0.186505	+	0.982454i	$0.440284\pi$
$888$	0	0
$889$	−66139.7	−2.49522
$890$	0	0
$891$	−16562.3	−0.622735
$892$	0	0
$893$	76572.4	2.86942
$894$	0	0
$895$	−22355.1	−0.834914
$896$	0	0
$897$	−40403.7	−1.50395
$898$	0	0
$899$	424.637	0.0157535
$900$	0	0
$901$	8647.47	0.319744
$902$	0	0
$903$	1026.26	0.0378202
$904$	0	0
$905$	1637.12	0.0601322
$906$	0	0
$907$	10244.0	0.375024	0.187512	−	0.982262i	$-0.439958\pi$
$907$	10244.0	0.375024	0.187512	+	0.982262i	$0.439958\pi$
$908$	0	0
$909$	67283.6	2.45507
$910$	0	0
$911$	−20813.9	−0.756966	−0.378483	−	0.925608i	$-0.623554\pi$
$911$	−20813.9	−0.756966	−0.378483	+	0.925608i	$0.623554\pi$
$912$	0	0
$913$	−2351.18	−0.0852276
$914$	0	0
$915$	30528.1	1.10298
$916$	0	0
$917$	78119.0	2.81321
$918$	0	0
$919$	9794.71	0.351575	0.175788	−	0.984428i	$-0.443753\pi$
$919$	9794.71	0.351575	0.175788	+	0.984428i	$0.443753\pi$
$920$	0	0
$921$	82483.3	2.95105
$922$	0	0
$923$	−30747.3	−1.09649
$924$	0	0
$925$	22049.3	0.783757
$926$	0	0
$927$	−87485.5	−3.09968
$928$	0	0
$929$	−29395.4	−1.03814	−0.519070	−	0.854732i	$-0.673722\pi$
$929$	−29395.4	−1.03814	−0.519070	+	0.854732i	$0.673722\pi$
$930$	0	0
$931$	−98259.7	−3.45900
$932$	0	0
$933$	69329.2	2.43273
$934$	0	0
$935$	2645.11	0.0925181
$936$	0	0
$937$	50174.1	1.74932	0.874661	−	0.484735i	$-0.161084\pi$
$937$	50174.1	1.74932	0.874661	+	0.484735i	$0.161084\pi$
$938$	0	0
$939$	33357.5	1.15930
$940$	0	0
$941$	−37984.8	−1.31591	−0.657955	−	0.753058i	$-0.728578\pi$
$941$	−37984.8	−1.31591	−0.657955	+	0.753058i	$0.728578\pi$
$942$	0	0
$943$	3369.55	0.116360
$944$	0	0
$945$	76655.0	2.63872
$946$	0	0
$947$	−15217.3	−0.522171	−0.261085	−	0.965316i	$-0.584080\pi$
$947$	−15217.3	−0.522171	−0.261085	+	0.965316i	$0.584080\pi$
$948$	0	0
$949$	−30359.2	−1.03846
$950$	0	0
$951$	71483.9	2.43746
$952$	0	0
$953$	17012.1	0.578253	0.289126	−	0.957291i	$-0.406635\pi$
$953$	17012.1	0.578253	0.289126	+	0.957291i	$0.406635\pi$
$954$	0	0
$955$	−14065.2	−0.476584
$956$	0	0
$957$	−2325.37	−0.0785461
$958$	0	0
$959$	1785.71	0.0601287
$960$	0	0
$961$	−29576.6	−0.992803
$962$	0	0
$963$	−86186.9	−2.88404
$964$	0	0
$965$	−758.018	−0.0252865
$966$	0	0
$967$	37192.8	1.23686	0.618428	−	0.785841i	$-0.287770\pi$
$967$	37192.8	1.23686	0.618428	+	0.785841i	$0.287770\pi$
$968$	0	0
$969$	−78948.5	−2.61733
$970$	0	0
$971$	−55741.1	−1.84224	−0.921121	−	0.389276i	$-0.872725\pi$
$971$	−55741.1	−1.84224	−0.921121	+	0.389276i	$0.872725\pi$
$972$	0	0
$973$	−38786.0	−1.27793
$974$	0	0
$975$	−26727.5	−0.877914
$976$	0	0
$977$	7356.33	0.240890	0.120445	−	0.992720i	$-0.461568\pi$
$977$	7356.33	0.240890	0.120445	+	0.992720i	$0.461568\pi$
$978$	0	0
$979$	−6862.29	−0.224024
$980$	0	0
$981$	100082.	3.25726
$982$	0	0
$983$	−1951.93	−0.0633337	−0.0316668	−	0.999498i	$-0.510082\pi$
$983$	−1951.93	−0.0633337	−0.0316668	+	0.999498i	$0.510082\pi$
$984$	0	0
$985$	5422.70	0.175413
$986$	0	0
$987$	145407.	4.68933
$988$	0	0
$989$	440.729	0.0141703
$990$	0	0
$991$	31606.2	1.01312	0.506561	−	0.862204i	$-0.330917\pi$
$991$	31606.2	1.01312	0.506561	+	0.862204i	$0.330917\pi$
$992$	0	0
$993$	−58101.8	−1.85680
$994$	0	0
$995$	−30641.1	−0.976269
$996$	0	0
$997$	−6443.52	−0.204682	−0.102341	−	0.994749i	$-0.532633\pi$
$997$	−6443.52	−0.204682	−0.102341	+	0.994749i	$0.532633\pi$
$998$	0	0
$999$	−101437.	−3.21254

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	928.4.a.h.1.12	✓	12
4.3	odd	2		928.4.a.j.1.1	yes	12
8.3	odd	2		1856.4.a.bj.1.12		12
8.5	even	2		1856.4.a.bl.1.1		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
928.4.a.h.1.12	✓	12	1.1	even	1	trivial
928.4.a.j.1.1	yes	12	4.3	odd	2
1856.4.a.bj.1.12		12	8.3	odd	2
1856.4.a.bl.1.1		12	8.5	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:	\( N \)	\(=\)	\( 928 = 2^{5} \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	928.a (trivial)

\(f(q)\)	\(=\)	\(q+9.72277 q^{3} -6.27199 q^{5} -31.0130 q^{7} +67.5323 q^{9} +O(q^{10})\)	\(q+9.72277 q^{3} -6.27199 q^{5} -31.0130 q^{7} +67.5323 q^{9} -8.24716 q^{11} +32.0907 q^{13} -60.9811 q^{15} +51.1369 q^{17} -158.789 q^{19} -301.533 q^{21} -129.494 q^{23} -85.6622 q^{25} +394.086 q^{27} +29.0000 q^{29} +14.6426 q^{31} -80.1853 q^{33} +194.513 q^{35} -257.398 q^{37} +312.011 q^{39} -26.0208 q^{41} -3.40346 q^{43} -423.562 q^{45} -482.228 q^{47} +618.808 q^{49} +497.192 q^{51} +169.104 q^{53} +51.7261 q^{55} -1543.87 q^{57} +569.745 q^{59} -500.616 q^{61} -2094.38 q^{63} -201.273 q^{65} -335.555 q^{67} -1259.04 q^{69} -958.137 q^{71} -946.043 q^{73} -832.874 q^{75} +255.769 q^{77} +179.326 q^{79} +2008.24 q^{81} +285.090 q^{83} -320.730 q^{85} +281.960 q^{87} +832.078 q^{89} -995.231 q^{91} +142.367 q^{93} +995.921 q^{95} -908.370 q^{97} -556.950 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 14 q^{3} - 10 q^{5} - 44 q^{7} + 130 q^{9}+O(q^{10}) \) 12 * q - 14 * q^3 - 10 * q^5 - 44 * q^7 + 130 * q^9	\( 12 q - 14 q^{3} - 10 q^{5} - 44 q^{7} + 130 q^{9} - 46 q^{11} - 34 q^{13} + 50 q^{15} + 36 q^{17} - 148 q^{19} - 92 q^{21} - 328 q^{23} + 486 q^{25} - 326 q^{27} + 348 q^{29} - 374 q^{31} + 710 q^{33} - 204 q^{35} - 340 q^{37} + 122 q^{39} + 32 q^{41} - 462 q^{43} - 1132 q^{45} - 434 q^{47} + 1508 q^{49} - 440 q^{51} + 610 q^{53} - 46 q^{55} - 932 q^{57} - 1240 q^{59} - 1228 q^{61} - 4240 q^{63} + 730 q^{65} - 1672 q^{67} - 528 q^{69} - 3220 q^{71} + 564 q^{73} - 6032 q^{75} + 644 q^{77} - 1862 q^{79} + 3040 q^{81} - 3736 q^{83} - 808 q^{85} - 406 q^{87} + 584 q^{89} - 4844 q^{91} - 3226 q^{93} - 2844 q^{95} + 904 q^{97} - 6832 q^{99}+O(q^{100}) \) 12 * q - 14 * q^3 - 10 * q^5 - 44 * q^7 + 130 * q^9 - 46 * q^11 - 34 * q^13 + 50 * q^15 + 36 * q^17 - 148 * q^19 - 92 * q^21 - 328 * q^23 + 486 * q^25 - 326 * q^27 + 348 * q^29 - 374 * q^31 + 710 * q^33 - 204 * q^35 - 340 * q^37 + 122 * q^39 + 32 * q^41 - 462 * q^43 - 1132 * q^45 - 434 * q^47 + 1508 * q^49 - 440 * q^51 + 610 * q^53 - 46 * q^55 - 932 * q^57 - 1240 * q^59 - 1228 * q^61 - 4240 * q^63 + 730 * q^65 - 1672 * q^67 - 528 * q^69 - 3220 * q^71 + 564 * q^73 - 6032 * q^75 + 644 * q^77 - 1862 * q^79 + 3040 * q^81 - 3736 * q^83 - 808 * q^85 - 406 * q^87 + 584 * q^89 - 4844 * q^91 - 3226 * q^93 - 2844 * q^95 + 904 * q^97 - 6832 * q^99

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