LMFDB - Embedded newform 1856.4.a.bg.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1856,4,Mod(1,1856)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1856, 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("1856.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1856 = 2^{6} \cdot 29 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1856.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:	$109.507544971$
Analytic rank:	$1$
Dimension:	$9$
Coefficient field:	$\mathbb{Q}[x]/(x^{9} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{9} - 4x^{8} - 138x^{7} + 394x^{6} + 5872x^{5} - 10822x^{4} - 85158x^{3} + 30654x^{2} + 439999x + 396802 $ x^9 - 4x^8 - 138x^7 + 394x^6 + 5872x^5 - 10822x^4 - 85158x^3 + 30654x^2 + 439999x + 396802
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{12} $
Twist minimal:	no (minimal twist has level 928)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-7.63776$ of defining polynomial
Character	$\chi$	$=$	1856.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-7.63776 q^{3} +7.65015 q^{5} -5.64550 q^{7} +31.3353 q^{9} +O(q^{10})$	$q-7.63776 q^{3} +7.65015 q^{5} -5.64550 q^{7} +31.3353 q^{9} +64.9460 q^{11} -54.0253 q^{13} -58.4300 q^{15} -20.0898 q^{17} -52.3646 q^{19} +43.1190 q^{21} +132.044 q^{23} -66.4752 q^{25} -33.1122 q^{27} +29.0000 q^{29} -135.023 q^{31} -496.042 q^{33} -43.1889 q^{35} +137.761 q^{37} +412.632 q^{39} -229.496 q^{41} +46.2145 q^{43} +239.720 q^{45} -327.225 q^{47} -311.128 q^{49} +153.441 q^{51} -32.9249 q^{53} +496.846 q^{55} +399.948 q^{57} +378.832 q^{59} +18.5312 q^{61} -176.904 q^{63} -413.301 q^{65} +998.295 q^{67} -1008.52 q^{69} -570.499 q^{71} -224.392 q^{73} +507.722 q^{75} -366.653 q^{77} +430.785 q^{79} -593.151 q^{81} +1390.04 q^{83} -153.690 q^{85} -221.495 q^{87} -131.569 q^{89} +305.000 q^{91} +1031.27 q^{93} -400.597 q^{95} +837.766 q^{97} +2035.10 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 9 q + 4 q^{3} - 10 q^{5} - 12 q^{7} + 49 q^{9}+O(q^{10}) $ 9 * q + 4 * q^3 - 10 * q^5 - 12 * q^7 + 49 * q^9	$ 9 q + 4 q^{3} - 10 q^{5} - 12 q^{7} + 49 q^{9} + 64 q^{11} - 70 q^{13} - 170 q^{15} - 66 q^{17} + 42 q^{19} - 76 q^{21} - 40 q^{23} + 111 q^{25} + 322 q^{27} + 261 q^{29} + 64 q^{31} - 52 q^{33} + 496 q^{35} + 54 q^{37} - 590 q^{39} - 378 q^{41} - 32 q^{43} - 1046 q^{45} - 1164 q^{47} - 351 q^{49} + 376 q^{51} - 278 q^{53} - 614 q^{55} + 28 q^{57} + 640 q^{59} - 1054 q^{61} - 1660 q^{63} - 708 q^{65} + 1184 q^{67} - 188 q^{69} - 1988 q^{71} - 750 q^{73} + 3126 q^{75} - 1260 q^{77} - 2916 q^{79} + 293 q^{81} + 2832 q^{83} - 56 q^{85} + 116 q^{87} - 370 q^{89} + 3016 q^{91} + 1696 q^{93} - 4412 q^{95} - 2234 q^{97} + 4118 q^{99}+O(q^{100}) $ 9 * q + 4 * q^3 - 10 * q^5 - 12 * q^7 + 49 * q^9 + 64 * q^11 - 70 * q^13 - 170 * q^15 - 66 * q^17 + 42 * q^19 - 76 * q^21 - 40 * q^23 + 111 * q^25 + 322 * q^27 + 261 * q^29 + 64 * q^31 - 52 * q^33 + 496 * q^35 + 54 * q^37 - 590 * q^39 - 378 * q^41 - 32 * q^43 - 1046 * q^45 - 1164 * q^47 - 351 * q^49 + 376 * q^51 - 278 * q^53 - 614 * q^55 + 28 * q^57 + 640 * q^59 - 1054 * q^61 - 1660 * q^63 - 708 * q^65 + 1184 * q^67 - 188 * q^69 - 1988 * q^71 - 750 * q^73 + 3126 * q^75 - 1260 * q^77 - 2916 * q^79 + 293 * q^81 + 2832 * q^83 - 56 * q^85 + 116 * q^87 - 370 * q^89 + 3016 * q^91 + 1696 * q^93 - 4412 * q^95 - 2234 * q^97 + 4118 * 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
$3$	−7.63776	−1.46989	−0.734943	−	0.678128i	$-0.762791\pi$
$3$	−7.63776	−1.46989	−0.734943	+	0.678128i	$0.762791\pi$
$4$	0	0
$5$	7.65015	0.684250	0.342125	−	0.939654i	$-0.388853\pi$
$5$	7.65015	0.684250	0.342125	+	0.939654i	$0.388853\pi$
$6$	0	0
$7$	−5.64550	−0.304829	−0.152414	−	0.988317i	$-0.548705\pi$
$7$	−5.64550	−0.304829	−0.152414	+	0.988317i	$0.548705\pi$
$8$	0	0
$9$	31.3353	1.16057
$10$	0	0
$11$	64.9460	1.78018	0.890089	−	0.455787i	$-0.150642\pi$
$11$	64.9460	1.78018	0.890089	+	0.455787i	$0.150642\pi$
$12$	0	0
$13$	−54.0253	−1.15261	−0.576304	−	0.817235i	$-0.695506\pi$
$13$	−54.0253	−1.15261	−0.576304	+	0.817235i	$0.695506\pi$
$14$	0	0
$15$	−58.4300	−1.00577
$16$	0	0
$17$	−20.0898	−0.286618	−0.143309	−	0.989678i	$-0.545774\pi$
$17$	−20.0898	−0.286618	−0.143309	+	0.989678i	$0.545774\pi$
$18$	0	0
$19$	−52.3646	−0.632277	−0.316139	−	0.948713i	$-0.602386\pi$
$19$	−52.3646	−0.632277	−0.316139	+	0.948713i	$0.602386\pi$
$20$	0	0
$21$	43.1190	0.448064
$22$	0	0
$23$	132.044	1.19709	0.598547	−	0.801088i	$-0.295745\pi$
$23$	132.044	1.19709	0.598547	+	0.801088i	$0.295745\pi$
$24$	0	0
$25$	−66.4752	−0.531802
$26$	0	0
$27$	−33.1122	−0.236016
$28$	0	0
$29$	29.0000	0.185695
$29$	29.0000	0.185695
$30$	0	0
$31$	−135.023	−0.782284	−0.391142	−	0.920330i	$-0.627920\pi$
$31$	−135.023	−0.782284	−0.391142	+	0.920330i	$0.627920\pi$
$32$	0	0
$33$	−496.042	−2.61666
$34$	0	0
$35$	−43.1889	−0.208579
$36$	0	0
$37$	137.761	0.612101	0.306050	−	0.952015i	$-0.400992\pi$
$37$	137.761	0.612101	0.306050	+	0.952015i	$0.400992\pi$
$38$	0	0
$39$	412.632	1.69420
$40$	0	0
$41$	−229.496	−0.874176	−0.437088	−	0.899419i	$-0.643990\pi$
$41$	−229.496	−0.874176	−0.437088	+	0.899419i	$0.643990\pi$
$42$	0	0
$43$	46.2145	0.163899	0.0819493	−	0.996637i	$-0.473885\pi$
$43$	46.2145	0.163899	0.0819493	+	0.996637i	$0.473885\pi$
$44$	0	0
$45$	239.720	0.794118
$46$	0	0
$47$	−327.225	−1.01555	−0.507773	−	0.861491i	$-0.669531\pi$
$47$	−327.225	−1.01555	−0.507773	+	0.861491i	$0.669531\pi$
$48$	0	0
$49$	−311.128	−0.907080
$50$	0	0
$51$	153.441	0.421295
$52$	0	0
$53$	−32.9249	−0.0853318	−0.0426659	−	0.999089i	$-0.513585\pi$
$53$	−32.9249	−0.0853318	−0.0426659	+	0.999089i	$0.513585\pi$
$54$	0	0
$55$	496.846	1.21809
$56$	0	0
$57$	399.948	0.929376
$58$	0	0
$59$	378.832	0.835929	0.417964	−	0.908463i	$-0.362744\pi$
$59$	378.832	0.835929	0.417964	+	0.908463i	$0.362744\pi$
$60$	0	0
$61$	18.5312	0.0388964	0.0194482	−	0.999811i	$-0.493809\pi$
$61$	18.5312	0.0388964	0.0194482	+	0.999811i	$0.493809\pi$
$62$	0	0
$63$	−176.904	−0.353774
$64$	0	0
$65$	−413.301	−0.788673
$66$	0	0
$67$	998.295	1.82032	0.910158	−	0.414261i	$-0.135960\pi$
$67$	998.295	1.82032	0.910158	+	0.414261i	$0.135960\pi$
$68$	0	0
$69$	−1008.52	−1.75959
$70$	0	0
$71$	−570.499	−0.953602	−0.476801	−	0.879011i	$-0.658204\pi$
$71$	−570.499	−0.953602	−0.476801	+	0.879011i	$0.658204\pi$
$72$	0	0
$73$	−224.392	−0.359768	−0.179884	−	0.983688i	$-0.557572\pi$
$73$	−224.392	−0.359768	−0.179884	+	0.983688i	$0.557572\pi$
$74$	0	0
$75$	507.722	0.781689
$76$	0	0
$77$	−366.653	−0.542649
$78$	0	0
$79$	430.785	0.613507	0.306754	−	0.951789i	$-0.400757\pi$
$79$	430.785	0.613507	0.306754	+	0.951789i	$0.400757\pi$
$80$	0	0
$81$	−593.151	−0.813650
$82$	0	0
$83$	1390.04	1.83828	0.919138	−	0.393935i	$-0.128886\pi$
$83$	1390.04	1.83828	0.919138	+	0.393935i	$0.128886\pi$
$84$	0	0
$85$	−153.690	−0.196118
$86$	0	0
$87$	−221.495	−0.272951
$88$	0	0
$89$	−131.569	−0.156700	−0.0783501	−	0.996926i	$-0.524965\pi$
$89$	−131.569	−0.156700	−0.0783501	+	0.996926i	$0.524965\pi$
$90$	0	0
$91$	305.000	0.351348
$92$	0	0
$93$	1031.27	1.14987
$94$	0	0
$95$	−400.597	−0.432636
$96$	0	0
$97$	837.766	0.876930	0.438465	−	0.898748i	$-0.355522\pi$
$97$	837.766	0.876930	0.438465	+	0.898748i	$0.355522\pi$
$98$	0	0
$99$	2035.10	2.06602
$100$	0	0
$101$	1595.03	1.57140	0.785702	−	0.618605i	$-0.212302\pi$
$101$	1595.03	1.57140	0.785702	+	0.618605i	$0.212302\pi$
$102$	0	0
$103$	1565.54	1.49765	0.748824	−	0.662769i	$-0.230619\pi$
$103$	1565.54	1.49765	0.748824	+	0.662769i	$0.230619\pi$
$104$	0	0
$105$	329.867	0.306587
$106$	0	0
$107$	−1243.19	−1.12321	−0.561604	−	0.827406i	$-0.689816\pi$
$107$	−1243.19	−1.12321	−0.561604	+	0.827406i	$0.689816\pi$
$108$	0	0
$109$	−1158.97	−1.01843	−0.509217	−	0.860638i	$-0.670065\pi$
$109$	−1158.97	−1.01843	−0.509217	+	0.860638i	$0.670065\pi$
$110$	0	0
$111$	−1052.18	−0.899719
$112$	0	0
$113$	−341.404	−0.284217	−0.142109	−	0.989851i	$-0.545388\pi$
$113$	−341.404	−0.284217	−0.142109	+	0.989851i	$0.545388\pi$
$114$	0	0
$115$	1010.16	0.819111
$116$	0	0
$117$	−1692.90	−1.33768
$118$	0	0
$119$	113.417	0.0873692
$120$	0	0
$121$	2886.98	2.16903
$122$	0	0
$123$	1752.83	1.28494
$124$	0	0
$125$	−1464.81	−1.04814
$126$	0	0
$127$	258.090	0.180329	0.0901644	−	0.995927i	$-0.471261\pi$
$127$	258.090	0.180329	0.0901644	+	0.995927i	$0.471261\pi$
$128$	0	0
$129$	−352.975	−0.240912
$130$	0	0
$131$	−728.188	−0.485665	−0.242832	−	0.970068i	$-0.578077\pi$
$131$	−728.188	−0.485665	−0.242832	+	0.970068i	$0.578077\pi$
$132$	0	0
$133$	295.625	0.192736
$134$	0	0
$135$	−253.313	−0.161494
$136$	0	0
$137$	82.1655	0.0512400	0.0256200	−	0.999672i	$-0.491844\pi$
$137$	82.1655	0.0512400	0.0256200	+	0.999672i	$0.491844\pi$
$138$	0	0
$139$	−593.000	−0.361853	−0.180927	−	0.983497i	$-0.557910\pi$
$139$	−593.000	−0.361853	−0.180927	+	0.983497i	$0.557910\pi$
$140$	0	0
$141$	2499.27	1.49274
$142$	0	0
$143$	−3508.73	−2.05185
$144$	0	0
$145$	221.854	0.127062
$146$	0	0
$147$	2376.32	1.33330
$148$	0	0
$149$	−728.700	−0.400654	−0.200327	−	0.979729i	$-0.564200\pi$
$149$	−728.700	−0.400654	−0.200327	+	0.979729i	$0.564200\pi$
$150$	0	0
$151$	1203.47	0.648590	0.324295	−	0.945956i	$-0.394873\pi$
$151$	1203.47	0.648590	0.324295	+	0.945956i	$0.394873\pi$
$152$	0	0
$153$	−629.521	−0.332639
$154$	0	0
$155$	−1032.94	−0.535278
$156$	0	0
$157$	−2704.74	−1.37492	−0.687458	−	0.726224i	$-0.741273\pi$
$157$	−2704.74	−1.37492	−0.687458	+	0.726224i	$0.741273\pi$
$158$	0	0
$159$	251.473	0.125428
$160$	0	0
$161$	−745.457	−0.364908
$162$	0	0
$163$	−1310.50	−0.629732	−0.314866	−	0.949136i	$-0.601960\pi$
$163$	−1310.50	−0.629732	−0.314866	+	0.949136i	$0.601960\pi$
$164$	0	0
$165$	−3794.79	−1.79045
$166$	0	0
$167$	−2891.17	−1.33967	−0.669837	−	0.742508i	$-0.733636\pi$
$167$	−2891.17	−1.33967	−0.669837	+	0.742508i	$0.733636\pi$
$168$	0	0
$169$	721.730	0.328507
$170$	0	0
$171$	−1640.86	−0.733800
$172$	0	0
$173$	−2915.04	−1.28108	−0.640539	−	0.767925i	$-0.721289\pi$
$173$	−2915.04	−1.28108	−0.640539	+	0.767925i	$0.721289\pi$
$174$	0	0
$175$	375.286	0.162108
$176$	0	0
$177$	−2893.43	−1.22872
$178$	0	0
$179$	−2414.92	−1.00838	−0.504188	−	0.863594i	$-0.668208\pi$
$179$	−2414.92	−1.00838	−0.504188	+	0.863594i	$0.668208\pi$
$180$	0	0
$181$	−100.467	−0.0412576	−0.0206288	−	0.999787i	$-0.506567\pi$
$181$	−100.467	−0.0412576	−0.0206288	+	0.999787i	$0.506567\pi$
$182$	0	0
$183$	−141.537	−0.0571733
$184$	0	0
$185$	1053.89	0.418830
$186$	0	0
$187$	−1304.75	−0.510230
$188$	0	0
$189$	186.935	0.0719445
$190$	0	0
$191$	−4186.06	−1.58583	−0.792914	−	0.609334i	$-0.791437\pi$
$191$	−4186.06	−1.58583	−0.792914	+	0.609334i	$0.791437\pi$
$192$	0	0
$193$	2681.27	1.00001	0.500006	−	0.866022i	$-0.333331\pi$
$193$	2681.27	1.00001	0.500006	+	0.866022i	$0.333331\pi$
$194$	0	0
$195$	3156.69	1.15926
$196$	0	0
$197$	−2684.48	−0.970869	−0.485435	−	0.874273i	$-0.661339\pi$
$197$	−2684.48	−0.970869	−0.485435	+	0.874273i	$0.661339\pi$
$198$	0	0
$199$	−3671.70	−1.30794	−0.653970	−	0.756520i	$-0.726898\pi$
$199$	−3671.70	−1.30794	−0.653970	+	0.756520i	$0.726898\pi$
$200$	0	0
$201$	−7624.74	−2.67566
$202$	0	0
$203$	−163.720	−0.0566052
$204$	0	0
$205$	−1755.68	−0.598155
$206$	0	0
$207$	4137.65	1.38931
$208$	0	0
$209$	−3400.87	−1.12557
$210$	0	0
$211$	4641.03	1.51423	0.757113	−	0.653284i	$-0.226609\pi$
$211$	4641.03	1.51423	0.757113	+	0.653284i	$0.226609\pi$
$212$	0	0
$213$	4357.33	1.40169
$214$	0	0
$215$	353.547	0.112148
$216$	0	0
$217$	762.272	0.238462
$218$	0	0
$219$	1713.85	0.528819
$220$	0	0
$221$	1085.36	0.330358
$222$	0	0
$223$	−1275.71	−0.383086	−0.191543	−	0.981484i	$-0.561349\pi$
$223$	−1275.71	−0.383086	−0.191543	+	0.981484i	$0.561349\pi$
$224$	0	0
$225$	−2083.02	−0.617192
$226$	0	0
$227$	876.500	0.256279	0.128140	−	0.991756i	$-0.459099\pi$
$227$	876.500	0.256279	0.128140	+	0.991756i	$0.459099\pi$
$228$	0	0
$229$	1083.31	0.312606	0.156303	−	0.987709i	$-0.450042\pi$
$229$	1083.31	0.312606	0.156303	+	0.987709i	$0.450042\pi$
$230$	0	0
$231$	2800.41	0.797633
$232$	0	0
$233$	5786.21	1.62690	0.813448	−	0.581637i	$-0.197588\pi$
$233$	5786.21	1.62690	0.813448	+	0.581637i	$0.197588\pi$
$234$	0	0
$235$	−2503.32	−0.694888
$236$	0	0
$237$	−3290.23	−0.901786
$238$	0	0
$239$	1971.04	0.533456	0.266728	−	0.963772i	$-0.414057\pi$
$239$	1971.04	0.533456	0.266728	+	0.963772i	$0.414057\pi$
$240$	0	0
$241$	−4272.70	−1.14203	−0.571014	−	0.820941i	$-0.693450\pi$
$241$	−4272.70	−1.14203	−0.571014	+	0.820941i	$0.693450\pi$
$242$	0	0
$243$	5424.37	1.43199
$244$	0	0
$245$	−2380.18	−0.620669
$246$	0	0
$247$	2829.01	0.728768
$248$	0	0
$249$	−10616.8	−2.70206
$250$	0	0
$251$	2804.44	0.705238	0.352619	−	0.935767i	$-0.385291\pi$
$251$	2804.44	0.705238	0.352619	+	0.935767i	$0.385291\pi$
$252$	0	0
$253$	8575.75	2.13104
$254$	0	0
$255$	1173.85	0.288271
$256$	0	0
$257$	3591.88	0.871811	0.435906	−	0.899992i	$-0.356428\pi$
$257$	3591.88	0.871811	0.435906	+	0.899992i	$0.356428\pi$
$258$	0	0
$259$	−777.729	−0.186586
$260$	0	0
$261$	908.724	0.215512
$262$	0	0
$263$	−4143.91	−0.971576	−0.485788	−	0.874077i	$-0.661467\pi$
$263$	−4143.91	−0.971576	−0.485788	+	0.874077i	$0.661467\pi$
$264$	0	0
$265$	−251.880	−0.0583883
$266$	0	0
$267$	1004.89	0.230331
$268$	0	0
$269$	−4024.31	−0.912143	−0.456072	−	0.889943i	$-0.650744\pi$
$269$	−4024.31	−0.912143	−0.456072	+	0.889943i	$0.650744\pi$
$270$	0	0
$271$	−6233.96	−1.39737	−0.698683	−	0.715431i	$-0.746230\pi$
$271$	−6233.96	−1.39737	−0.698683	+	0.715431i	$0.746230\pi$
$272$	0	0
$273$	−2329.52	−0.516442
$274$	0	0
$275$	−4317.30	−0.946702
$276$	0	0
$277$	435.106	0.0943789	0.0471895	−	0.998886i	$-0.484974\pi$
$277$	435.106	0.0943789	0.0471895	+	0.998886i	$0.484974\pi$
$278$	0	0
$279$	−4230.98	−0.907893
$280$	0	0
$281$	−3805.46	−0.807882	−0.403941	−	0.914785i	$-0.632360\pi$
$281$	−3805.46	−0.807882	−0.403941	+	0.914785i	$0.632360\pi$
$282$	0	0
$283$	−239.164	−0.0502361	−0.0251180	−	0.999684i	$-0.507996\pi$
$283$	−239.164	−0.0502361	−0.0251180	+	0.999684i	$0.507996\pi$
$284$	0	0
$285$	3059.66	0.635926
$286$	0	0
$287$	1295.62	0.266474
$288$	0	0
$289$	−4509.40	−0.917850
$290$	0	0
$291$	−6398.65	−1.28899
$292$	0	0
$293$	−719.126	−0.143385	−0.0716925	−	0.997427i	$-0.522840\pi$
$293$	−719.126	−0.143385	−0.0716925	+	0.997427i	$0.522840\pi$
$294$	0	0
$295$	2898.12	0.571984
$296$	0	0
$297$	−2150.50	−0.420151
$298$	0	0
$299$	−7133.73	−1.37978
$300$	0	0
$301$	−260.904	−0.0499610
$302$	0	0
$303$	−12182.5	−2.30979
$304$	0	0
$305$	141.767	0.0266148
$306$	0	0
$307$	−9702.09	−1.80367	−0.901836	−	0.432078i	$-0.857780\pi$
$307$	−9702.09	−1.80367	−0.901836	+	0.432078i	$0.857780\pi$
$308$	0	0
$309$	−11957.2	−2.20137
$310$	0	0
$311$	−3369.02	−0.614275	−0.307137	−	0.951665i	$-0.599371\pi$
$311$	−3369.02	−0.614275	−0.307137	+	0.951665i	$0.599371\pi$
$312$	0	0
$313$	−5269.30	−0.951561	−0.475781	−	0.879564i	$-0.657834\pi$
$313$	−5269.30	−0.951561	−0.475781	+	0.879564i	$0.657834\pi$
$314$	0	0
$315$	−1353.34	−0.242070
$316$	0	0
$317$	7664.88	1.35805	0.679026	−	0.734114i	$-0.262402\pi$
$317$	7664.88	1.35805	0.679026	+	0.734114i	$0.262402\pi$
$318$	0	0
$319$	1883.43	0.330571
$320$	0	0
$321$	9495.15	1.65099
$322$	0	0
$323$	1052.00	0.181222
$324$	0	0
$325$	3591.34	0.612960
$326$	0	0
$327$	8851.94	1.49698
$328$	0	0
$329$	1847.35	0.309568
$330$	0	0
$331$	−6587.05	−1.09383	−0.546914	−	0.837189i	$-0.684198\pi$
$331$	−6587.05	−1.09383	−0.546914	+	0.837189i	$0.684198\pi$
$332$	0	0
$333$	4316.78	0.710385
$334$	0	0
$335$	7637.11	1.24555
$336$	0	0
$337$	11419.9	1.84595	0.922973	−	0.384865i	$-0.125752\pi$
$337$	11419.9	1.84595	0.922973	+	0.384865i	$0.125752\pi$
$338$	0	0
$339$	2607.56	0.417767
$340$	0	0
$341$	−8769.19	−1.39260
$342$	0	0
$343$	3692.88	0.581332
$344$	0	0
$345$	−7715.35	−1.20400
$346$	0	0
$347$	−2175.03	−0.336488	−0.168244	−	0.985745i	$-0.553810\pi$
$347$	−2175.03	−0.336488	−0.168244	+	0.985745i	$0.553810\pi$
$348$	0	0
$349$	−5659.99	−0.868115	−0.434058	−	0.900885i	$-0.642919\pi$
$349$	−5659.99	−0.868115	−0.434058	+	0.900885i	$0.642919\pi$
$350$	0	0
$351$	1788.89	0.272034
$352$	0	0
$353$	−2506.30	−0.377895	−0.188948	−	0.981987i	$-0.560508\pi$
$353$	−2506.30	−0.377895	−0.188948	+	0.981987i	$0.560508\pi$
$354$	0	0
$355$	−4364.40	−0.652502
$356$	0	0
$357$	−866.253	−0.128423
$358$	0	0
$359$	−4848.73	−0.712831	−0.356415	−	0.934328i	$-0.616001\pi$
$359$	−4848.73	−0.712831	−0.356415	+	0.934328i	$0.616001\pi$
$360$	0	0
$361$	−4116.95	−0.600226
$362$	0	0
$363$	−22050.1	−3.18823
$364$	0	0
$365$	−1716.63	−0.246171
$366$	0	0
$367$	−9034.96	−1.28507	−0.642535	−	0.766256i	$-0.722117\pi$
$367$	−9034.96	−1.28507	−0.642535	+	0.766256i	$0.722117\pi$
$368$	0	0
$369$	−7191.32	−1.01454
$370$	0	0
$371$	185.878	0.0260116
$372$	0	0
$373$	−4060.90	−0.563715	−0.281857	−	0.959456i	$-0.590950\pi$
$373$	−4060.90	−0.563715	−0.281857	+	0.959456i	$0.590950\pi$
$374$	0	0
$375$	11187.9	1.54064
$376$	0	0
$377$	−1566.73	−0.214034
$378$	0	0
$379$	6094.76	0.826034	0.413017	−	0.910723i	$-0.364475\pi$
$379$	6094.76	0.826034	0.413017	+	0.910723i	$0.364475\pi$
$380$	0	0
$381$	−1971.23	−0.265063
$382$	0	0
$383$	−4602.40	−0.614025	−0.307013	−	0.951705i	$-0.599329\pi$
$383$	−4602.40	−0.614025	−0.307013	+	0.951705i	$0.599329\pi$
$384$	0	0
$385$	−2804.95	−0.371308
$386$	0	0
$387$	1448.15	0.190215
$388$	0	0
$389$	−9393.43	−1.22433	−0.612167	−	0.790729i	$-0.709702\pi$
$389$	−9393.43	−1.22433	−0.612167	+	0.790729i	$0.709702\pi$
$390$	0	0
$391$	−2652.75	−0.343108
$392$	0	0
$393$	5561.72	0.713872
$394$	0	0
$395$	3295.57	0.419792
$396$	0	0
$397$	−13398.1	−1.69378	−0.846889	−	0.531769i	$-0.821527\pi$
$397$	−13398.1	−1.69378	−0.846889	+	0.531769i	$0.821527\pi$
$398$	0	0
$399$	−2257.91	−0.283300
$400$	0	0
$401$	−12917.9	−1.60870	−0.804348	−	0.594159i	$-0.797485\pi$
$401$	−12917.9	−1.60870	−0.804348	+	0.594159i	$0.797485\pi$
$402$	0	0
$403$	7294.64	0.901667
$404$	0	0
$405$	−4537.69	−0.556740
$406$	0	0
$407$	8947.01	1.08965
$408$	0	0
$409$	−5221.70	−0.631288	−0.315644	−	0.948878i	$-0.602220\pi$
$409$	−5221.70	−0.631288	−0.315644	+	0.948878i	$0.602220\pi$
$410$	0	0
$411$	−627.560	−0.0753169
$412$	0	0
$413$	−2138.70	−0.254815
$414$	0	0
$415$	10634.0	1.25784
$416$	0	0
$417$	4529.19	0.531883
$418$	0	0
$419$	−6246.54	−0.728314	−0.364157	−	0.931338i	$-0.618643\pi$
$419$	−6246.54	−0.728314	−0.364157	+	0.931338i	$0.618643\pi$
$420$	0	0
$421$	−583.420	−0.0675396	−0.0337698	−	0.999430i	$-0.510751\pi$
$421$	−583.420	−0.0675396	−0.0337698	+	0.999430i	$0.510751\pi$
$422$	0	0
$423$	−10253.7	−1.17861
$424$	0	0
$425$	1335.48	0.152424
$426$	0	0
$427$	−104.618	−0.0118567
$428$	0	0
$429$	26798.8	3.01599
$430$	0	0
$431$	−4845.37	−0.541516	−0.270758	−	0.962647i	$-0.587274\pi$
$431$	−4845.37	−0.541516	−0.270758	+	0.962647i	$0.587274\pi$
$432$	0	0
$433$	2159.85	0.239714	0.119857	−	0.992791i	$-0.461756\pi$
$433$	2159.85	0.239714	0.119857	+	0.992791i	$0.461756\pi$
$434$	0	0
$435$	−1694.47	−0.186767
$436$	0	0
$437$	−6914.45	−0.756895
$438$	0	0
$439$	17016.0	1.84995	0.924976	−	0.380026i	$-0.124085\pi$
$439$	17016.0	1.84995	0.924976	+	0.380026i	$0.124085\pi$
$440$	0	0
$441$	−9749.31	−1.05273
$442$	0	0
$443$	−3676.92	−0.394347	−0.197173	−	0.980369i	$-0.563176\pi$
$443$	−3676.92	−0.394347	−0.197173	+	0.980369i	$0.563176\pi$
$444$	0	0
$445$	−1006.52	−0.107222
$446$	0	0
$447$	5565.63	0.588916
$448$	0	0
$449$	11074.1	1.16396	0.581979	−	0.813204i	$-0.302279\pi$
$449$	11074.1	1.16396	0.581979	+	0.813204i	$0.302279\pi$
$450$	0	0
$451$	−14904.8	−1.55619
$452$	0	0
$453$	−9191.82	−0.953354
$454$	0	0
$455$	2333.29	0.240410
$456$	0	0
$457$	−1146.80	−0.117386	−0.0586928	−	0.998276i	$-0.518693\pi$
$457$	−1146.80	−0.117386	−0.0586928	+	0.998276i	$0.518693\pi$
$458$	0	0
$459$	665.217	0.0676464
$460$	0	0
$461$	−15788.0	−1.59505	−0.797526	−	0.603284i	$-0.793858\pi$
$461$	−15788.0	−1.59505	−0.797526	+	0.603284i	$0.793858\pi$
$462$	0	0
$463$	2709.43	0.271961	0.135981	−	0.990711i	$-0.456582\pi$
$463$	2709.43	0.271961	0.135981	+	0.990711i	$0.456582\pi$
$464$	0	0
$465$	7889.37	0.786798
$466$	0	0
$467$	16216.0	1.60682	0.803411	−	0.595425i	$-0.203016\pi$
$467$	16216.0	1.60682	0.803411	+	0.595425i	$0.203016\pi$
$468$	0	0
$469$	−5635.88	−0.554884
$470$	0	0
$471$	20658.2	2.02097
$472$	0	0
$473$	3001.44	0.291769
$474$	0	0
$475$	3480.95	0.336246
$476$	0	0
$477$	−1031.71	−0.0990333
$478$	0	0
$479$	−12106.9	−1.15487	−0.577433	−	0.816438i	$-0.695945\pi$
$479$	−12106.9	−1.15487	−0.577433	+	0.816438i	$0.695945\pi$
$480$	0	0
$481$	−7442.56	−0.705513
$482$	0	0
$483$	5693.62	0.536374
$484$	0	0
$485$	6409.03	0.600039
$486$	0	0
$487$	14106.0	1.31253	0.656267	−	0.754529i	$-0.272135\pi$
$487$	14106.0	1.31253	0.656267	+	0.754529i	$0.272135\pi$
$488$	0	0
$489$	10009.3	0.925635
$490$	0	0
$491$	16587.4	1.52460	0.762299	−	0.647225i	$-0.224070\pi$
$491$	16587.4	1.52460	0.762299	+	0.647225i	$0.224070\pi$
$492$	0	0
$493$	−582.605	−0.0532235
$494$	0	0
$495$	15568.8	1.41367
$496$	0	0
$497$	3220.75	0.290685
$498$	0	0
$499$	5294.94	0.475018	0.237509	−	0.971385i	$-0.423669\pi$
$499$	5294.94	0.475018	0.237509	+	0.971385i	$0.423669\pi$
$500$	0	0
$501$	22082.1	1.96917
$502$	0	0
$503$	5856.03	0.519101	0.259550	−	0.965730i	$-0.416426\pi$
$503$	5856.03	0.519101	0.259550	+	0.965730i	$0.416426\pi$
$504$	0	0
$505$	12202.2	1.07523
$506$	0	0
$507$	−5512.40	−0.482868
$508$	0	0
$509$	−3680.23	−0.320478	−0.160239	−	0.987078i	$-0.551227\pi$
$509$	−3680.23	−0.320478	−0.160239	+	0.987078i	$0.551227\pi$
$510$	0	0
$511$	1266.81	0.109668
$512$	0	0
$513$	1733.91	0.149228
$514$	0	0
$515$	11976.6	1.02477
$516$	0	0
$517$	−21252.0	−1.80785
$518$	0	0
$519$	22264.4	1.88304
$520$	0	0
$521$	15679.1	1.31845	0.659227	−	0.751944i	$-0.270884\pi$
$521$	15679.1	1.31845	0.659227	+	0.751944i	$0.270884\pi$
$522$	0	0
$523$	−17311.4	−1.44737	−0.723686	−	0.690130i	$-0.757553\pi$
$523$	−17311.4	−1.44737	−0.723686	+	0.690130i	$0.757553\pi$
$524$	0	0
$525$	−2866.35	−0.238281
$526$	0	0
$527$	2712.58	0.224216
$528$	0	0
$529$	5268.72	0.433034
$530$	0	0
$531$	11870.8	0.970152
$532$	0	0
$533$	12398.6	1.00758
$534$	0	0
$535$	−9510.55	−0.768555
$536$	0	0
$537$	18444.6	1.48220
$538$	0	0
$539$	−20206.5	−1.61476
$540$	0	0
$541$	−7300.99	−0.580211	−0.290105	−	0.956995i	$-0.593690\pi$
$541$	−7300.99	−0.580211	−0.290105	+	0.956995i	$0.593690\pi$
$542$	0	0
$543$	767.340	0.0606441
$544$	0	0
$545$	−8866.30	−0.696863
$546$	0	0
$547$	12610.1	0.985684	0.492842	−	0.870119i	$-0.335958\pi$
$547$	12610.1	0.985684	0.492842	+	0.870119i	$0.335958\pi$
$548$	0	0
$549$	580.682	0.0451419
$550$	0	0
$551$	−1518.57	−0.117411
$552$	0	0
$553$	−2432.00	−0.187015
$554$	0	0
$555$	−8049.36	−0.615633
$556$	0	0
$557$	2845.32	0.216446	0.108223	−	0.994127i	$-0.465484\pi$
$557$	2845.32	0.216446	0.108223	+	0.994127i	$0.465484\pi$
$558$	0	0
$559$	−2496.75	−0.188911
$560$	0	0
$561$	9965.39	0.749981
$562$	0	0
$563$	3877.50	0.290261	0.145131	−	0.989412i	$-0.453640\pi$
$563$	3877.50	0.290261	0.145131	+	0.989412i	$0.453640\pi$
$564$	0	0
$565$	−2611.79	−0.194476
$566$	0	0
$567$	3348.64	0.248024
$568$	0	0
$569$	−22052.8	−1.62478	−0.812391	−	0.583113i	$-0.801835\pi$
$569$	−22052.8	−1.62478	−0.812391	+	0.583113i	$0.801835\pi$
$570$	0	0
$571$	3017.56	0.221158	0.110579	−	0.993867i	$-0.464730\pi$
$571$	3017.56	0.221158	0.110579	+	0.993867i	$0.464730\pi$
$572$	0	0
$573$	31972.1	2.33099
$574$	0	0
$575$	−8777.68	−0.636617
$576$	0	0
$577$	9766.04	0.704620	0.352310	−	0.935883i	$-0.385396\pi$
$577$	9766.04	0.704620	0.352310	+	0.935883i	$0.385396\pi$
$578$	0	0
$579$	−20478.9	−1.46990
$580$	0	0
$581$	−7847.49	−0.560359
$582$	0	0
$583$	−2138.34	−0.151906
$584$	0	0
$585$	−12950.9	−0.915308
$586$	0	0
$587$	20684.2	1.45439	0.727195	−	0.686431i	$-0.240824\pi$
$587$	20684.2	1.45439	0.727195	+	0.686431i	$0.240824\pi$
$588$	0	0
$589$	7070.41	0.494620
$590$	0	0
$591$	20503.4	1.42707
$592$	0	0
$593$	20556.7	1.42354	0.711772	−	0.702410i	$-0.247893\pi$
$593$	20556.7	1.42354	0.711772	+	0.702410i	$0.247893\pi$
$594$	0	0
$595$	867.658	0.0597824
$596$	0	0
$597$	28043.6	1.92252
$598$	0	0
$599$	−11055.7	−0.754131	−0.377066	−	0.926186i	$-0.623067\pi$
$599$	−11055.7	−0.754131	−0.377066	+	0.926186i	$0.623067\pi$
$600$	0	0
$601$	−8326.30	−0.565119	−0.282560	−	0.959250i	$-0.591184\pi$
$601$	−8326.30	−0.565119	−0.282560	+	0.959250i	$0.591184\pi$
$602$	0	0
$603$	31281.9	2.11260
$604$	0	0
$605$	22085.8	1.48416
$606$	0	0
$607$	−22922.1	−1.53275	−0.766374	−	0.642395i	$-0.777941\pi$
$607$	−22922.1	−1.53275	−0.766374	+	0.642395i	$0.777941\pi$
$608$	0	0
$609$	1250.45	0.0832033
$610$	0	0
$611$	17678.4	1.17053
$612$	0	0
$613$	8197.48	0.540119	0.270060	−	0.962844i	$-0.412957\pi$
$613$	8197.48	0.540119	0.270060	+	0.962844i	$0.412957\pi$
$614$	0	0
$615$	13409.4	0.879220
$616$	0	0
$617$	−27147.0	−1.77131	−0.885656	−	0.464343i	$-0.846291\pi$
$617$	−27147.0	−1.77131	−0.885656	+	0.464343i	$0.846291\pi$
$618$	0	0
$619$	472.406	0.0306747	0.0153373	−	0.999882i	$-0.495118\pi$
$619$	472.406	0.0306747	0.0153373	+	0.999882i	$0.495118\pi$
$620$	0	0
$621$	−4372.27	−0.282534
$622$	0	0
$623$	742.775	0.0477667
$624$	0	0
$625$	−2896.64	−0.185385
$626$	0	0
$627$	25975.0	1.65445
$628$	0	0
$629$	−2767.59	−0.175439
$630$	0	0
$631$	−2414.00	−0.152297	−0.0761487	−	0.997096i	$-0.524262\pi$
$631$	−2414.00	−0.152297	−0.0761487	+	0.997096i	$0.524262\pi$
$632$	0	0
$633$	−35447.1	−2.22574
$634$	0	0
$635$	1974.42	0.123390
$636$	0	0
$637$	16808.8	1.04551
$638$	0	0
$639$	−17876.8	−1.10672
$640$	0	0
$641$	−10379.6	−0.639575	−0.319788	−	0.947489i	$-0.603612\pi$
$641$	−10379.6	−0.639575	−0.319788	+	0.947489i	$0.603612\pi$
$642$	0	0
$643$	−19509.5	−1.19655	−0.598275	−	0.801291i	$-0.704147\pi$
$643$	−19509.5	−1.19655	−0.598275	+	0.801291i	$0.704147\pi$
$644$	0	0
$645$	−2700.31	−0.164844
$646$	0	0
$647$	−15125.8	−0.919096	−0.459548	−	0.888153i	$-0.651989\pi$
$647$	−15125.8	−0.919096	−0.459548	+	0.888153i	$0.651989\pi$
$648$	0	0
$649$	24603.7	1.48810
$650$	0	0
$651$	−5822.04	−0.350513
$652$	0	0
$653$	22790.8	1.36581	0.682905	−	0.730507i	$-0.260716\pi$
$653$	22790.8	1.36581	0.682905	+	0.730507i	$0.260716\pi$
$654$	0	0
$655$	−5570.75	−0.332316
$656$	0	0
$657$	−7031.39	−0.417535
$658$	0	0
$659$	−13542.4	−0.800514	−0.400257	−	0.916403i	$-0.631079\pi$
$659$	−13542.4	−0.800514	−0.400257	+	0.916403i	$0.631079\pi$
$660$	0	0
$661$	23999.0	1.41218	0.706092	−	0.708120i	$-0.250456\pi$
$661$	23999.0	1.41218	0.706092	+	0.708120i	$0.250456\pi$
$662$	0	0
$663$	−8289.70	−0.485589
$664$	0	0
$665$	2261.57	0.131880
$666$	0	0
$667$	3829.29	0.222295
$668$	0	0
$669$	9743.59	0.563092
$670$	0	0
$671$	1203.53	0.0692425
$672$	0	0
$673$	34631.5	1.98357	0.991787	−	0.127901i	$-0.0408240\pi$
$673$	34631.5	1.98357	0.991787	+	0.127901i	$0.0408240\pi$
$674$	0	0
$675$	2201.14	0.125514
$676$	0	0
$677$	−5200.08	−0.295207	−0.147604	−	0.989047i	$-0.547156\pi$
$677$	−5200.08	−0.295207	−0.147604	+	0.989047i	$0.547156\pi$
$678$	0	0
$679$	−4729.61	−0.267313
$680$	0	0
$681$	−6694.50	−0.376701
$682$	0	0
$683$	−11690.2	−0.654926	−0.327463	−	0.944864i	$-0.606194\pi$
$683$	−11690.2	−0.654926	−0.327463	+	0.944864i	$0.606194\pi$
$684$	0	0
$685$	628.578	0.0350609
$686$	0	0
$687$	−8274.03	−0.459496
$688$	0	0
$689$	1778.78	0.0983542
$690$	0	0
$691$	13188.4	0.726064	0.363032	−	0.931777i	$-0.381742\pi$
$691$	13188.4	0.726064	0.363032	+	0.931777i	$0.381742\pi$
$692$	0	0
$693$	−11489.2	−0.629781
$694$	0	0
$695$	−4536.54	−0.247598
$696$	0	0
$697$	4610.53	0.250554
$698$	0	0
$699$	−44193.6	−2.39135
$700$	0	0
$701$	19761.3	1.06473	0.532364	−	0.846516i	$-0.321304\pi$
$701$	19761.3	1.06473	0.532364	+	0.846516i	$0.321304\pi$
$702$	0	0
$703$	−7213.79	−0.387017
$704$	0	0
$705$	19119.8	1.02141
$706$	0	0
$707$	−9004.77	−0.479009
$708$	0	0
$709$	16955.0	0.898106	0.449053	−	0.893505i	$-0.351761\pi$
$709$	16955.0	0.898106	0.449053	+	0.893505i	$0.351761\pi$
$710$	0	0
$711$	13498.8	0.712017
$712$	0	0
$713$	−17829.0	−0.936467
$714$	0	0
$715$	−26842.3	−1.40398
$716$	0	0
$717$	−15054.3	−0.784120
$718$	0	0
$719$	8924.11	0.462884	0.231442	−	0.972849i	$-0.425656\pi$
$719$	8924.11	0.462884	0.231442	+	0.972849i	$0.425656\pi$
$720$	0	0
$721$	−8838.29	−0.456526
$722$	0	0
$723$	32633.8	1.67865
$724$	0	0
$725$	−1927.78	−0.0987531
$726$	0	0
$727$	14644.9	0.747112	0.373556	−	0.927608i	$-0.378138\pi$
$727$	14644.9	0.747112	0.373556	+	0.927608i	$0.378138\pi$
$728$	0	0
$729$	−25415.0	−1.29121
$730$	0	0
$731$	−928.440	−0.0469762
$732$	0	0
$733$	−31105.6	−1.56741	−0.783704	−	0.621134i	$-0.786672\pi$
$733$	−31105.6	−1.56741	−0.783704	+	0.621134i	$0.786672\pi$
$734$	0	0
$735$	18179.2	0.912313
$736$	0	0
$737$	64835.3	3.24049
$738$	0	0
$739$	−29071.3	−1.44710	−0.723549	−	0.690273i	$-0.757490\pi$
$739$	−29071.3	−1.44710	−0.723549	+	0.690273i	$0.757490\pi$
$740$	0	0
$741$	−21607.3	−1.07121
$742$	0	0
$743$	8095.53	0.399726	0.199863	−	0.979824i	$-0.435950\pi$
$743$	8095.53	0.399726	0.199863	+	0.979824i	$0.435950\pi$
$744$	0	0
$745$	−5574.66	−0.274147
$746$	0	0
$747$	43557.4	2.13344
$748$	0	0
$749$	7018.41	0.342386
$750$	0	0
$751$	−6109.17	−0.296840	−0.148420	−	0.988924i	$-0.547419\pi$
$751$	−6109.17	−0.296840	−0.148420	+	0.988924i	$0.547419\pi$
$752$	0	0
$753$	−21419.6	−1.03662
$754$	0	0
$755$	9206.73	0.443798
$756$	0	0
$757$	−17269.3	−0.829146	−0.414573	−	0.910016i	$-0.636069\pi$
$757$	−17269.3	−0.829146	−0.414573	+	0.910016i	$0.636069\pi$
$758$	0	0
$759$	−65499.5	−3.13239
$760$	0	0
$761$	39975.0	1.90420	0.952098	−	0.305792i	$-0.0989212\pi$
$761$	39975.0	1.90420	0.952098	+	0.305792i	$0.0989212\pi$
$762$	0	0
$763$	6542.97	0.310448
$764$	0	0
$765$	−4815.93	−0.227608
$766$	0	0
$767$	−20466.5	−0.963499
$768$	0	0
$769$	−29413.2	−1.37928	−0.689640	−	0.724152i	$-0.742231\pi$
$769$	−29413.2	−1.37928	−0.689640	+	0.724152i	$0.742231\pi$
$770$	0	0
$771$	−27433.9	−1.28146
$772$	0	0
$773$	−29797.9	−1.38649	−0.693245	−	0.720702i	$-0.743820\pi$
$773$	−29797.9	−1.38649	−0.693245	+	0.720702i	$0.743820\pi$
$774$	0	0
$775$	8975.67	0.416020
$776$	0	0
$777$	5940.11	0.274260
$778$	0	0
$779$	12017.5	0.552721
$780$	0	0
$781$	−37051.6	−1.69758
$782$	0	0
$783$	−960.253	−0.0438271
$784$	0	0
$785$	−20691.7	−0.940786
$786$	0	0
$787$	29580.7	1.33982	0.669910	−	0.742443i	$-0.266333\pi$
$787$	29580.7	1.33982	0.669910	+	0.742443i	$0.266333\pi$
$788$	0	0
$789$	31650.2	1.42811
$790$	0	0
$791$	1927.40	0.0866375
$792$	0	0
$793$	−1001.15	−0.0448323
$794$	0	0
$795$	1923.80	0.0858242
$796$	0	0
$797$	−35669.3	−1.58528	−0.792642	−	0.609688i	$-0.791295\pi$
$797$	−35669.3	−1.58528	−0.792642	+	0.609688i	$0.791295\pi$
$798$	0	0
$799$	6573.90	0.291073
$800$	0	0
$801$	−4122.77	−0.181861
$802$	0	0
$803$	−14573.4	−0.640451
$804$	0	0
$805$	−5702.86	−0.249689
$806$	0	0
$807$	30736.7	1.34075
$808$	0	0
$809$	−27538.1	−1.19677	−0.598386	−	0.801208i	$-0.704191\pi$
$809$	−27538.1	−1.19677	−0.598386	+	0.801208i	$0.704191\pi$
$810$	0	0
$811$	−428.113	−0.0185365	−0.00926824	−	0.999957i	$-0.502950\pi$
$811$	−428.113	−0.0185365	−0.00926824	+	0.999957i	$0.502950\pi$
$812$	0	0
$813$	47613.5	2.05397
$814$	0	0
$815$	−10025.5	−0.430894
$816$	0	0
$817$	−2420.00	−0.103629
$818$	0	0
$819$	9557.27	0.407763
$820$	0	0
$821$	22381.3	0.951415	0.475707	−	0.879604i	$-0.342192\pi$
$821$	22381.3	0.951415	0.475707	+	0.879604i	$0.342192\pi$
$822$	0	0
$823$	4884.52	0.206882	0.103441	−	0.994636i	$-0.467015\pi$
$823$	4884.52	0.206882	0.103441	+	0.994636i	$0.467015\pi$
$824$	0	0
$825$	32974.5	1.39154
$826$	0	0
$827$	20727.9	0.871558	0.435779	−	0.900054i	$-0.356473\pi$
$827$	20727.9	0.871558	0.435779	+	0.900054i	$0.356473\pi$
$828$	0	0
$829$	7720.98	0.323475	0.161738	−	0.986834i	$-0.448290\pi$
$829$	7720.98	0.323475	0.161738	+	0.986834i	$0.448290\pi$
$830$	0	0
$831$	−3323.23	−0.138726
$832$	0	0
$833$	6250.51	0.259985
$834$	0	0
$835$	−22117.9	−0.916671
$836$	0	0
$837$	4470.90	0.184632
$838$	0	0
$839$	−39526.8	−1.62648	−0.813239	−	0.581929i	$-0.802298\pi$
$839$	−39526.8	−1.62648	−0.813239	+	0.581929i	$0.802298\pi$
$840$	0	0
$841$	841.000	0.0344828
$842$	0	0
$843$	29065.2	1.18749
$844$	0	0
$845$	5521.34	0.224781
$846$	0	0
$847$	−16298.5	−0.661183
$848$	0	0
$849$	1826.67	0.0738414
$850$	0	0
$851$	18190.5	0.732742
$852$	0	0
$853$	6398.26	0.256826	0.128413	−	0.991721i	$-0.459012\pi$
$853$	6398.26	0.256826	0.128413	+	0.991721i	$0.459012\pi$
$854$	0	0
$855$	−12552.8	−0.502103
$856$	0	0
$857$	18780.2	0.748563	0.374282	−	0.927315i	$-0.377889\pi$
$857$	18780.2	0.748563	0.374282	+	0.927315i	$0.377889\pi$
$858$	0	0
$859$	18787.8	0.746253	0.373127	−	0.927780i	$-0.378286\pi$
$859$	18787.8	0.746253	0.373127	+	0.927780i	$0.378286\pi$
$860$	0	0
$861$	−9895.62	−0.391686
$862$	0	0
$863$	28372.1	1.11912	0.559559	−	0.828791i	$-0.310971\pi$
$863$	28372.1	1.11912	0.559559	+	0.828791i	$0.310971\pi$
$864$	0	0
$865$	−22300.5	−0.876578
$866$	0	0
$867$	34441.7	1.34914
$868$	0	0
$869$	27977.7	1.09215
$870$	0	0
$871$	−53933.2	−2.09811
$872$	0	0
$873$	26251.7	1.01774
$874$	0	0
$875$	8269.61	0.319502
$876$	0	0
$877$	−3003.79	−0.115656	−0.0578282	−	0.998327i	$-0.518418\pi$
$877$	−3003.79	−0.115656	−0.0578282	+	0.998327i	$0.518418\pi$
$878$	0	0
$879$	5492.51	0.210760
$880$	0	0
$881$	28184.2	1.07781	0.538904	−	0.842367i	$-0.318838\pi$
$881$	28184.2	1.07781	0.538904	+	0.842367i	$0.318838\pi$
$882$	0	0
$883$	−12092.6	−0.460871	−0.230435	−	0.973088i	$-0.574015\pi$
$883$	−12092.6	−0.460871	−0.230435	+	0.973088i	$0.574015\pi$
$884$	0	0
$885$	−22135.2	−0.840752
$886$	0	0
$887$	−41525.1	−1.57190	−0.785951	−	0.618289i	$-0.787826\pi$
$887$	−41525.1	−1.57190	−0.785951	+	0.618289i	$0.787826\pi$
$888$	0	0
$889$	−1457.05	−0.0549694
$890$	0	0
$891$	−38522.8	−1.44844
$892$	0	0
$893$	17135.0	0.642107
$894$	0	0
$895$	−18474.5	−0.689982
$896$	0	0
$897$	54485.7	2.02812
$898$	0	0
$899$	−3915.66	−0.145266
$900$	0	0
$901$	661.456	0.0244576
$902$	0	0
$903$	1992.72	0.0734370
$904$	0	0
$905$	−768.585	−0.0282305
$906$	0	0
$907$	−17464.3	−0.639353	−0.319676	−	0.947527i	$-0.603574\pi$
$907$	−17464.3	−0.639353	−0.319676	+	0.947527i	$0.603574\pi$
$908$	0	0
$909$	49980.9	1.82372
$910$	0	0
$911$	−8847.73	−0.321776	−0.160888	−	0.986973i	$-0.551436\pi$
$911$	−8847.73	−0.321776	−0.160888	+	0.986973i	$0.551436\pi$
$912$	0	0
$913$	90277.7	3.27246
$914$	0	0
$915$	−1082.78	−0.0391208
$916$	0	0
$917$	4110.99	0.148044
$918$	0	0
$919$	−45221.9	−1.62321	−0.811606	−	0.584205i	$-0.801406\pi$
$919$	−45221.9	−1.62321	−0.811606	+	0.584205i	$0.801406\pi$
$920$	0	0
$921$	74102.2	2.65119
$922$	0	0
$923$	30821.4	1.09913
$924$	0	0
$925$	−9157.68	−0.325516
$926$	0	0
$927$	49056.9	1.73812
$928$	0	0
$929$	−5013.65	−0.177064	−0.0885320	−	0.996073i	$-0.528218\pi$
$929$	−5013.65	−0.177064	−0.0885320	+	0.996073i	$0.528218\pi$
$930$	0	0
$931$	16292.1	0.573526
$932$	0	0
$933$	25731.7	0.902915
$934$	0	0
$935$	−9981.56	−0.349125
$936$	0	0
$937$	−39065.1	−1.36201	−0.681004	−	0.732280i	$-0.738456\pi$
$937$	−39065.1	−1.36201	−0.681004	+	0.732280i	$0.738456\pi$
$938$	0	0
$939$	40245.7	1.39869
$940$	0	0
$941$	−18526.0	−0.641796	−0.320898	−	0.947114i	$-0.603985\pi$
$941$	−18526.0	−0.641796	−0.320898	+	0.947114i	$0.603985\pi$
$942$	0	0
$943$	−30303.6	−1.04647
$944$	0	0
$945$	1430.08	0.0492280
$946$	0	0
$947$	−30773.5	−1.05597	−0.527986	−	0.849253i	$-0.677053\pi$
$947$	−30773.5	−1.05597	−0.527986	+	0.849253i	$0.677053\pi$
$948$	0	0
$949$	12122.8	0.414672
$950$	0	0
$951$	−58542.5	−1.99618
$952$	0	0
$953$	−32135.7	−1.09232	−0.546158	−	0.837682i	$-0.683910\pi$
$953$	−32135.7	−1.09232	−0.546158	+	0.837682i	$0.683910\pi$
$954$	0	0
$955$	−32024.0	−1.08510
$956$	0	0
$957$	−14385.2	−0.485902
$958$	0	0
$959$	−463.866	−0.0156194
$960$	0	0
$961$	−11559.9	−0.388032
$962$	0	0
$963$	−38955.6	−1.30356
$964$	0	0
$965$	20512.1	0.684258
$966$	0	0
$967$	−10300.8	−0.342555	−0.171278	−	0.985223i	$-0.554789\pi$
$967$	−10300.8	−0.342555	−0.171278	+	0.985223i	$0.554789\pi$
$968$	0	0
$969$	−8034.89	−0.266375
$970$	0	0
$971$	26187.1	0.865485	0.432742	−	0.901518i	$-0.357546\pi$
$971$	26187.1	0.865485	0.432742	+	0.901518i	$0.357546\pi$
$972$	0	0
$973$	3347.78	0.110303
$974$	0	0
$975$	−27429.8	−0.900981
$976$	0	0
$977$	21578.3	0.706604	0.353302	−	0.935509i	$-0.385059\pi$
$977$	21578.3	0.706604	0.353302	+	0.935509i	$0.385059\pi$
$978$	0	0
$979$	−8544.90	−0.278954
$980$	0	0
$981$	−36316.7	−1.18196
$982$	0	0
$983$	2362.28	0.0766481	0.0383240	−	0.999265i	$-0.487798\pi$
$983$	2362.28	0.0766481	0.0383240	+	0.999265i	$0.487798\pi$
$984$	0	0
$985$	−20536.7	−0.664317
$986$	0	0
$987$	−14109.6	−0.455029
$988$	0	0
$989$	6102.36	0.196202
$990$	0	0
$991$	15492.6	0.496609	0.248304	−	0.968682i	$-0.420127\pi$
$991$	15492.6	0.496609	0.248304	+	0.968682i	$0.420127\pi$
$992$	0	0
$993$	50310.3	1.60780
$994$	0	0
$995$	−28089.1	−0.894958
$996$	0	0
$997$	−30991.6	−0.984467	−0.492233	−	0.870463i	$-0.663819\pi$
$997$	−30991.6	−0.984467	−0.492233	+	0.870463i	$0.663819\pi$
$998$	0	0
$999$	−4561.56	−0.144466

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1856.4.a.bg.1.1		9
4.3	odd	2		1856.4.a.bf.1.9		9
8.3	odd	2		928.4.a.e.1.1	yes	9
8.5	even	2		928.4.a.d.1.9	✓	9

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
928.4.a.d.1.9	✓	9	8.5	even	2
928.4.a.e.1.1	yes	9	8.3	odd	2
1856.4.a.bf.1.9		9	4.3	odd	2
1856.4.a.bg.1.1		9	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:	\( N \)	\(=\)	\( 1856 = 2^{6} \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1856.a (trivial)

\(f(q)\)	\(=\)	\(q-7.63776 q^{3} +7.65015 q^{5} -5.64550 q^{7} +31.3353 q^{9} +O(q^{10})\)	\(q-7.63776 q^{3} +7.65015 q^{5} -5.64550 q^{7} +31.3353 q^{9} +64.9460 q^{11} -54.0253 q^{13} -58.4300 q^{15} -20.0898 q^{17} -52.3646 q^{19} +43.1190 q^{21} +132.044 q^{23} -66.4752 q^{25} -33.1122 q^{27} +29.0000 q^{29} -135.023 q^{31} -496.042 q^{33} -43.1889 q^{35} +137.761 q^{37} +412.632 q^{39} -229.496 q^{41} +46.2145 q^{43} +239.720 q^{45} -327.225 q^{47} -311.128 q^{49} +153.441 q^{51} -32.9249 q^{53} +496.846 q^{55} +399.948 q^{57} +378.832 q^{59} +18.5312 q^{61} -176.904 q^{63} -413.301 q^{65} +998.295 q^{67} -1008.52 q^{69} -570.499 q^{71} -224.392 q^{73} +507.722 q^{75} -366.653 q^{77} +430.785 q^{79} -593.151 q^{81} +1390.04 q^{83} -153.690 q^{85} -221.495 q^{87} -131.569 q^{89} +305.000 q^{91} +1031.27 q^{93} -400.597 q^{95} +837.766 q^{97} +2035.10 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 9 q + 4 q^{3} - 10 q^{5} - 12 q^{7} + 49 q^{9}+O(q^{10}) \) 9 * q + 4 * q^3 - 10 * q^5 - 12 * q^7 + 49 * q^9	\( 9 q + 4 q^{3} - 10 q^{5} - 12 q^{7} + 49 q^{9} + 64 q^{11} - 70 q^{13} - 170 q^{15} - 66 q^{17} + 42 q^{19} - 76 q^{21} - 40 q^{23} + 111 q^{25} + 322 q^{27} + 261 q^{29} + 64 q^{31} - 52 q^{33} + 496 q^{35} + 54 q^{37} - 590 q^{39} - 378 q^{41} - 32 q^{43} - 1046 q^{45} - 1164 q^{47} - 351 q^{49} + 376 q^{51} - 278 q^{53} - 614 q^{55} + 28 q^{57} + 640 q^{59} - 1054 q^{61} - 1660 q^{63} - 708 q^{65} + 1184 q^{67} - 188 q^{69} - 1988 q^{71} - 750 q^{73} + 3126 q^{75} - 1260 q^{77} - 2916 q^{79} + 293 q^{81} + 2832 q^{83} - 56 q^{85} + 116 q^{87} - 370 q^{89} + 3016 q^{91} + 1696 q^{93} - 4412 q^{95} - 2234 q^{97} + 4118 q^{99}+O(q^{100}) \) 9 * q + 4 * q^3 - 10 * q^5 - 12 * q^7 + 49 * q^9 + 64 * q^11 - 70 * q^13 - 170 * q^15 - 66 * q^17 + 42 * q^19 - 76 * q^21 - 40 * q^23 + 111 * q^25 + 322 * q^27 + 261 * q^29 + 64 * q^31 - 52 * q^33 + 496 * q^35 + 54 * q^37 - 590 * q^39 - 378 * q^41 - 32 * q^43 - 1046 * q^45 - 1164 * q^47 - 351 * q^49 + 376 * q^51 - 278 * q^53 - 614 * q^55 + 28 * q^57 + 640 * q^59 - 1054 * q^61 - 1660 * q^63 - 708 * q^65 + 1184 * q^67 - 188 * q^69 - 1988 * q^71 - 750 * q^73 + 3126 * q^75 - 1260 * q^77 - 2916 * q^79 + 293 * q^81 + 2832 * q^83 - 56 * q^85 + 116 * q^87 - 370 * q^89 + 3016 * q^91 + 1696 * q^93 - 4412 * q^95 - 2234 * q^97 + 4118 * q^99

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