LMFDB - Embedded newform 1856.4.a.bd.1.5

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:	$6$
Coefficient field:	$\mathbb{Q}[x]/(x^{6} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 40x^{4} - 88x^{3} - 8x^{2} + 48x - 9 $ x^6 - 40x^4 - 88x^3 - 8x^2 + 48x - 9
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{7} $
Twist minimal:	no (minimal twist has level 232)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$0.215317$ 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)

$f(q)$	$=$	$q+6.36242 q^{3} -8.72860 q^{5} -8.76166 q^{7} +13.4804 q^{9} +O(q^{10})$	$q+6.36242 q^{3} -8.72860 q^{5} -8.76166 q^{7} +13.4804 q^{9} -39.5540 q^{11} +27.6290 q^{13} -55.5351 q^{15} +134.762 q^{17} +161.056 q^{19} -55.7454 q^{21} -163.135 q^{23} -48.8115 q^{25} -86.0171 q^{27} +29.0000 q^{29} -200.241 q^{31} -251.659 q^{33} +76.4771 q^{35} +208.370 q^{37} +175.788 q^{39} -83.2931 q^{41} +334.050 q^{43} -117.665 q^{45} -277.727 q^{47} -266.233 q^{49} +857.414 q^{51} +185.468 q^{53} +345.251 q^{55} +1024.71 q^{57} -190.892 q^{59} -622.436 q^{61} -118.111 q^{63} -241.163 q^{65} -627.283 q^{67} -1037.93 q^{69} +872.996 q^{71} -359.712 q^{73} -310.560 q^{75} +346.559 q^{77} -165.772 q^{79} -911.250 q^{81} +565.467 q^{83} -1176.29 q^{85} +184.510 q^{87} +15.8167 q^{89} -242.076 q^{91} -1274.02 q^{93} -1405.79 q^{95} -1436.13 q^{97} -533.206 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 5 q^{3} + 5 q^{5} - 38 q^{7} + 47 q^{9}+O(q^{10}) $ 6 * q + 5 * q^3 + 5 * q^5 - 38 * q^7 + 47 * q^9	$ 6 q + 5 q^{3} + 5 q^{5} - 38 q^{7} + 47 q^{9} + 19 q^{11} - 13 q^{13} - 191 q^{15} - 218 q^{17} + 290 q^{19} + 266 q^{21} - 196 q^{23} - 13 q^{25} + 437 q^{27} + 174 q^{29} - 675 q^{31} + 291 q^{33} + 466 q^{35} + 238 q^{37} - 1297 q^{39} - 464 q^{41} + 579 q^{43} + 148 q^{45} - 975 q^{47} + 914 q^{49} + 576 q^{51} - 515 q^{53} - 1605 q^{55} - 340 q^{57} + 108 q^{59} - 1158 q^{61} - 1136 q^{63} + 1239 q^{65} + 80 q^{67} - 2568 q^{69} + 438 q^{71} + 262 q^{73} - 1766 q^{75} - 194 q^{77} - 237 q^{79} + 2554 q^{81} + 1288 q^{83} - 3112 q^{85} + 145 q^{87} - 252 q^{89} - 2450 q^{91} - 2131 q^{93} + 180 q^{95} + 380 q^{97} - 2264 q^{99}+O(q^{100}) $ 6 * q + 5 * q^3 + 5 * q^5 - 38 * q^7 + 47 * q^9 + 19 * q^11 - 13 * q^13 - 191 * q^15 - 218 * q^17 + 290 * q^19 + 266 * q^21 - 196 * q^23 - 13 * q^25 + 437 * q^27 + 174 * q^29 - 675 * q^31 + 291 * q^33 + 466 * q^35 + 238 * q^37 - 1297 * q^39 - 464 * q^41 + 579 * q^43 + 148 * q^45 - 975 * q^47 + 914 * q^49 + 576 * q^51 - 515 * q^53 - 1605 * q^55 - 340 * q^57 + 108 * q^59 - 1158 * q^61 - 1136 * q^63 + 1239 * q^65 + 80 * q^67 - 2568 * q^69 + 438 * q^71 + 262 * q^73 - 1766 * q^75 - 194 * q^77 - 237 * q^79 + 2554 * q^81 + 1288 * q^83 - 3112 * q^85 + 145 * q^87 - 252 * q^89 - 2450 * q^91 - 2131 * q^93 + 180 * q^95 + 380 * q^97 - 2264 * 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
$3$	6.36242	1.22445	0.612225	−	0.790684i	$-0.290275\pi$
$3$	6.36242	1.22445	0.612225	+	0.790684i	$0.290275\pi$
$4$	0	0
$5$	−8.72860	−0.780710	−0.390355	−	0.920664i	$-0.627648\pi$
$5$	−8.72860	−0.780710	−0.390355	+	0.920664i	$0.627648\pi$
$6$	0	0
$7$	−8.76166	−0.473085	−0.236543	−	0.971621i	$-0.576014\pi$
$7$	−8.76166	−0.473085	−0.236543	+	0.971621i	$0.576014\pi$
$8$	0	0
$9$	13.4804	0.499276
$10$	0	0
$11$	−39.5540	−1.08418	−0.542090	−	0.840320i	$-0.682367\pi$
$11$	−39.5540	−1.08418	−0.542090	+	0.840320i	$0.682367\pi$
$12$	0	0
$13$	27.6290	0.589455	0.294727	−	0.955581i	$-0.404771\pi$
$13$	27.6290	0.589455	0.294727	+	0.955581i	$0.404771\pi$
$14$	0	0
$15$	−55.5351	−0.955940
$16$	0	0
$17$	134.762	1.92263	0.961313	−	0.275459i	$-0.0888299\pi$
$17$	134.762	1.92263	0.961313	+	0.275459i	$0.0888299\pi$
$18$	0	0
$19$	161.056	1.94467	0.972336	−	0.233585i	$-0.0750456\pi$
$19$	161.056	1.94467	0.972336	+	0.233585i	$0.0750456\pi$
$20$	0	0
$21$	−55.7454	−0.579269
$22$	0	0
$23$	−163.135	−1.47895	−0.739477	−	0.673181i	$-0.764927\pi$
$23$	−163.135	−1.47895	−0.739477	+	0.673181i	$0.764927\pi$
$24$	0	0
$25$	−48.8115	−0.390492
$26$	0	0
$27$	−86.0171	−0.613111
$28$	0	0
$29$	29.0000	0.185695
$29$	29.0000	0.185695
$30$	0	0
$31$	−200.241	−1.16014	−0.580069	−	0.814568i	$-0.696974\pi$
$31$	−200.241	−1.16014	−0.580069	+	0.814568i	$0.696974\pi$
$32$	0	0
$33$	−251.659	−1.32752
$34$	0	0
$35$	76.4771	0.369342
$36$	0	0
$37$	208.370	0.925833	0.462917	−	0.886402i	$-0.346803\pi$
$37$	208.370	0.925833	0.462917	+	0.886402i	$0.346803\pi$
$38$	0	0
$39$	175.788	0.721758
$40$	0	0
$41$	−83.2931	−0.317273	−0.158637	−	0.987337i	$-0.550710\pi$
$41$	−83.2931	−0.317273	−0.158637	+	0.987337i	$0.550710\pi$
$42$	0	0
$43$	334.050	1.18470	0.592351	−	0.805680i	$-0.298200\pi$
$43$	334.050	1.18470	0.592351	+	0.805680i	$0.298200\pi$
$44$	0	0
$45$	−117.665	−0.389790
$46$	0	0
$47$	−277.727	−0.861928	−0.430964	−	0.902369i	$-0.641826\pi$
$47$	−277.727	−0.861928	−0.430964	+	0.902369i	$0.641826\pi$
$48$	0	0
$49$	−266.233	−0.776190
$50$	0	0
$51$	857.414	2.35416
$52$	0	0
$53$	185.468	0.480678	0.240339	−	0.970689i	$-0.422741\pi$
$53$	185.468	0.480678	0.240339	+	0.970689i	$0.422741\pi$
$54$	0	0
$55$	345.251	0.846430
$56$	0	0
$57$	1024.71	2.38115
$58$	0	0
$59$	−190.892	−0.421220	−0.210610	−	0.977570i	$-0.567545\pi$
$59$	−190.892	−0.421220	−0.210610	+	0.977570i	$0.567545\pi$
$60$	0	0
$61$	−622.436	−1.30647	−0.653235	−	0.757155i	$-0.726589\pi$
$61$	−622.436	−1.30647	−0.653235	+	0.757155i	$0.726589\pi$
$62$	0	0
$63$	−118.111	−0.236200
$64$	0	0
$65$	−241.163	−0.460193
$66$	0	0
$67$	−627.283	−1.14380	−0.571901	−	0.820322i	$-0.693794\pi$
$67$	−627.283	−1.14380	−0.571901	+	0.820322i	$0.693794\pi$
$68$	0	0
$69$	−1037.93	−1.81090
$70$	0	0
$71$	872.996	1.45923	0.729616	−	0.683857i	$-0.239699\pi$
$71$	872.996	1.45923	0.729616	+	0.683857i	$0.239699\pi$
$72$	0	0
$73$	−359.712	−0.576728	−0.288364	−	0.957521i	$-0.593111\pi$
$73$	−359.712	−0.576728	−0.288364	+	0.957521i	$0.593111\pi$
$74$	0	0
$75$	−310.560	−0.478138
$76$	0	0
$77$	346.559	0.512910
$78$	0	0
$79$	−165.772	−0.236086	−0.118043	−	0.993008i	$-0.537662\pi$
$79$	−165.772	−0.236086	−0.118043	+	0.993008i	$0.537662\pi$
$80$	0	0
$81$	−911.250	−1.25000
$82$	0	0
$83$	565.467	0.747808	0.373904	−	0.927467i	$-0.378019\pi$
$83$	565.467	0.747808	0.373904	+	0.927467i	$0.378019\pi$
$84$	0	0
$85$	−1176.29	−1.50101
$86$	0	0
$87$	184.510	0.227375
$88$	0	0
$89$	15.8167	0.0188378	0.00941890	−	0.999956i	$-0.497002\pi$
$89$	15.8167	0.0188378	0.00941890	+	0.999956i	$0.497002\pi$
$90$	0	0
$91$	−242.076	−0.278862
$92$	0	0
$93$	−1274.02	−1.42053
$94$	0	0
$95$	−1405.79	−1.51823
$96$	0	0
$97$	−1436.13	−1.50327	−0.751634	−	0.659580i	$-0.770734\pi$
$97$	−1436.13	−1.50327	−0.751634	+	0.659580i	$0.770734\pi$
$98$	0	0
$99$	−533.206	−0.541305
$100$	0	0
$101$	−618.989	−0.609819	−0.304910	−	0.952381i	$-0.598626\pi$
$101$	−618.989	−0.609819	−0.304910	+	0.952381i	$0.598626\pi$
$102$	0	0
$103$	−1112.51	−1.06426	−0.532128	−	0.846664i	$-0.678608\pi$
$103$	−1112.51	−1.06426	−0.532128	+	0.846664i	$0.678608\pi$
$104$	0	0
$105$	486.580	0.452241
$106$	0	0
$107$	−985.275	−0.890188	−0.445094	−	0.895484i	$-0.646830\pi$
$107$	−985.275	−0.890188	−0.445094	+	0.895484i	$0.646830\pi$
$108$	0	0
$109$	−510.618	−0.448700	−0.224350	−	0.974509i	$-0.572026\pi$
$109$	−510.618	−0.448700	−0.224350	+	0.974509i	$0.572026\pi$
$110$	0	0
$111$	1325.74	1.13364
$112$	0	0
$113$	1119.33	0.931836	0.465918	−	0.884828i	$-0.345724\pi$
$113$	1119.33	0.931836	0.465918	+	0.884828i	$0.345724\pi$
$114$	0	0
$115$	1423.94	1.15463
$116$	0	0
$117$	372.452	0.294301
$118$	0	0
$119$	−1180.74	−0.909566
$120$	0	0
$121$	233.519	0.175446
$122$	0	0
$123$	−529.946	−0.388485
$124$	0	0
$125$	1517.13	1.08557
$126$	0	0
$127$	−2342.91	−1.63700	−0.818502	−	0.574503i	$-0.805195\pi$
$127$	−2342.91	−1.63700	−0.818502	+	0.574503i	$0.805195\pi$
$128$	0	0
$129$	2125.37	1.45061
$130$	0	0
$131$	−1140.68	−0.760778	−0.380389	−	0.924827i	$-0.624210\pi$
$131$	−1140.68	−0.760778	−0.380389	+	0.924827i	$0.624210\pi$
$132$	0	0
$133$	−1411.12	−0.919996
$134$	0	0
$135$	750.809	0.478662
$136$	0	0
$137$	−1859.11	−1.15937	−0.579687	−	0.814839i	$-0.696825\pi$
$137$	−1859.11	−1.15937	−0.579687	+	0.814839i	$0.696825\pi$
$138$	0	0
$139$	−2724.85	−1.66272	−0.831362	−	0.555731i	$-0.812438\pi$
$139$	−2724.85	−1.66272	−0.831362	+	0.555731i	$0.812438\pi$
$140$	0	0
$141$	−1767.02	−1.05539
$142$	0	0
$143$	−1092.84	−0.639075
$144$	0	0
$145$	−253.129	−0.144974
$146$	0	0
$147$	−1693.89	−0.950406
$148$	0	0
$149$	765.729	0.421013	0.210506	−	0.977592i	$-0.432489\pi$
$149$	765.729	0.421013	0.210506	+	0.977592i	$0.432489\pi$
$150$	0	0
$151$	3074.79	1.65710	0.828551	−	0.559913i	$-0.189165\pi$
$151$	3074.79	1.65710	0.828551	+	0.559913i	$0.189165\pi$
$152$	0	0
$153$	1816.65	0.959920
$154$	0	0
$155$	1747.82	0.905731
$156$	0	0
$157$	1934.86	0.983560	0.491780	−	0.870720i	$-0.336346\pi$
$157$	1934.86	0.983560	0.491780	+	0.870720i	$0.336346\pi$
$158$	0	0
$159$	1180.02	0.588566
$160$	0	0
$161$	1429.33	0.699672
$162$	0	0
$163$	684.358	0.328853	0.164427	−	0.986389i	$-0.447423\pi$
$163$	684.358	0.328853	0.164427	+	0.986389i	$0.447423\pi$
$164$	0	0
$165$	2196.63	1.03641
$166$	0	0
$167$	−2994.75	−1.38767	−0.693834	−	0.720135i	$-0.744080\pi$
$167$	−2994.75	−1.38767	−0.693834	+	0.720135i	$0.744080\pi$
$168$	0	0
$169$	−1433.64	−0.652543
$170$	0	0
$171$	2171.11	0.970928
$172$	0	0
$173$	2563.06	1.12639	0.563197	−	0.826323i	$-0.309571\pi$
$173$	2563.06	1.12639	0.563197	+	0.826323i	$0.309571\pi$
$174$	0	0
$175$	427.670	0.184736
$176$	0	0
$177$	−1214.53	−0.515762
$178$	0	0
$179$	−1480.22	−0.618085	−0.309042	−	0.951048i	$-0.600008\pi$
$179$	−1480.22	−0.618085	−0.309042	+	0.951048i	$0.600008\pi$
$180$	0	0
$181$	−2998.88	−1.23152	−0.615759	−	0.787935i	$-0.711150\pi$
$181$	−2998.88	−1.23152	−0.615759	+	0.787935i	$0.711150\pi$
$182$	0	0
$183$	−3960.20	−1.59971
$184$	0	0
$185$	−1818.78	−0.722807
$186$	0	0
$187$	−5330.38	−2.08447
$188$	0	0
$189$	753.653	0.290054
$190$	0	0
$191$	−423.297	−0.160360	−0.0801799	−	0.996780i	$-0.525549\pi$
$191$	−423.297	−0.160360	−0.0801799	+	0.996780i	$0.525549\pi$
$192$	0	0
$193$	306.989	0.114495	0.0572476	−	0.998360i	$-0.481768\pi$
$193$	306.989	0.114495	0.0572476	+	0.998360i	$0.481768\pi$
$194$	0	0
$195$	−1534.38	−0.563483
$196$	0	0
$197$	−2904.54	−1.05046	−0.525229	−	0.850961i	$-0.676020\pi$
$197$	−2904.54	−1.05046	−0.525229	+	0.850961i	$0.676020\pi$
$198$	0	0
$199$	266.678	0.0949963	0.0474982	−	0.998871i	$-0.484875\pi$
$199$	266.678	0.0949963	0.0474982	+	0.998871i	$0.484875\pi$
$200$	0	0
$201$	−3991.04	−1.40053
$202$	0	0
$203$	−254.088	−0.0878497
$204$	0	0
$205$	727.033	0.247698
$206$	0	0
$207$	−2199.13	−0.738406
$208$	0	0
$209$	−6370.41	−2.10838
$210$	0	0
$211$	−919.255	−0.299925	−0.149962	−	0.988692i	$-0.547915\pi$
$211$	−919.255	−0.299925	−0.149962	+	0.988692i	$0.547915\pi$
$212$	0	0
$213$	5554.37	1.78676
$214$	0	0
$215$	−2915.79	−0.924909
$216$	0	0
$217$	1754.44	0.548844
$218$	0	0
$219$	−2288.64	−0.706174
$220$	0	0
$221$	3723.35	1.13330
$222$	0	0
$223$	−2349.53	−0.705542	−0.352771	−	0.935710i	$-0.614761\pi$
$223$	−2349.53	−0.705542	−0.352771	+	0.935710i	$0.614761\pi$
$224$	0	0
$225$	−658.001	−0.194963
$226$	0	0
$227$	−1026.17	−0.300042	−0.150021	−	0.988683i	$-0.547934\pi$
$227$	−1026.17	−0.300042	−0.150021	+	0.988683i	$0.547934\pi$
$228$	0	0
$229$	471.496	0.136058	0.0680292	−	0.997683i	$-0.478329\pi$
$229$	471.496	0.136058	0.0680292	+	0.997683i	$0.478329\pi$
$230$	0	0
$231$	2204.95	0.628032
$232$	0	0
$233$	1412.43	0.397131	0.198566	−	0.980088i	$-0.436372\pi$
$233$	1412.43	0.397131	0.198566	+	0.980088i	$0.436372\pi$
$234$	0	0
$235$	2424.17	0.672916
$236$	0	0
$237$	−1054.71	−0.289075
$238$	0	0
$239$	−64.0933	−0.0173467	−0.00867333	−	0.999962i	$-0.502761\pi$
$239$	−64.0933	−0.0173467	−0.00867333	+	0.999962i	$0.502761\pi$
$240$	0	0
$241$	−3629.05	−0.969991	−0.484995	−	0.874517i	$-0.661179\pi$
$241$	−3629.05	−0.969991	−0.484995	+	0.874517i	$0.661179\pi$
$242$	0	0
$243$	−3475.29	−0.917450
$244$	0	0
$245$	2323.84	0.605979
$246$	0	0
$247$	4449.82	1.14630
$248$	0	0
$249$	3597.74	0.915653
$250$	0	0
$251$	−1919.48	−0.482696	−0.241348	−	0.970439i	$-0.577589\pi$
$251$	−1919.48	−0.482696	−0.241348	+	0.970439i	$0.577589\pi$
$252$	0	0
$253$	6452.63	1.60345
$254$	0	0
$255$	−7484.03	−1.83791
$256$	0	0
$257$	4978.66	1.20841	0.604203	−	0.796831i	$-0.293492\pi$
$257$	4978.66	1.20841	0.604203	+	0.796831i	$0.293492\pi$
$258$	0	0
$259$	−1825.67	−0.437998
$260$	0	0
$261$	390.933	0.0927132
$262$	0	0
$263$	6142.46	1.44015	0.720077	−	0.693894i	$-0.244107\pi$
$263$	6142.46	1.44015	0.720077	+	0.693894i	$0.244107\pi$
$264$	0	0
$265$	−1618.87	−0.375270
$266$	0	0
$267$	100.632	0.0230659
$268$	0	0
$269$	8276.23	1.87588	0.937938	−	0.346803i	$-0.112733\pi$
$269$	8276.23	1.87588	0.937938	+	0.346803i	$0.112733\pi$
$270$	0	0
$271$	−3102.84	−0.695514	−0.347757	−	0.937585i	$-0.613057\pi$
$271$	−3102.84	−0.695514	−0.347757	+	0.937585i	$0.613057\pi$
$272$	0	0
$273$	−1540.19	−0.341453
$274$	0	0
$275$	1930.69	0.423364
$276$	0	0
$277$	−934.063	−0.202608	−0.101304	−	0.994856i	$-0.532301\pi$
$277$	−934.063	−0.202608	−0.101304	+	0.994856i	$0.532301\pi$
$278$	0	0
$279$	−2699.33	−0.579229
$280$	0	0
$281$	−728.824	−0.154726	−0.0773630	−	0.997003i	$-0.524650\pi$
$281$	−728.824	−0.154726	−0.0773630	+	0.997003i	$0.524650\pi$
$282$	0	0
$283$	−4414.80	−0.927323	−0.463662	−	0.886012i	$-0.653465\pi$
$283$	−4414.80	−0.927323	−0.463662	+	0.886012i	$0.653465\pi$
$284$	0	0
$285$	−8944.26	−1.85899
$286$	0	0
$287$	729.786	0.150097
$288$	0	0
$289$	13247.9	2.69649
$290$	0	0
$291$	−9137.28	−1.84068
$292$	0	0
$293$	8255.06	1.64596	0.822979	−	0.568072i	$-0.192311\pi$
$293$	8255.06	1.64596	0.822979	+	0.568072i	$0.192311\pi$
$294$	0	0
$295$	1666.22	0.328850
$296$	0	0
$297$	3402.32	0.664723
$298$	0	0
$299$	−4507.26	−0.871777
$300$	0	0
$301$	−2926.84	−0.560465
$302$	0	0
$303$	−3938.27	−0.746693
$304$	0	0
$305$	5432.99	1.01997
$306$	0	0
$307$	−966.478	−0.179674	−0.0898368	−	0.995956i	$-0.528635\pi$
$307$	−966.478	−0.179674	−0.0898368	+	0.995956i	$0.528635\pi$
$308$	0	0
$309$	−7078.23	−1.30313
$310$	0	0
$311$	6837.36	1.24666	0.623330	−	0.781959i	$-0.285779\pi$
$311$	6837.36	1.24666	0.623330	+	0.781959i	$0.285779\pi$
$312$	0	0
$313$	9633.46	1.73967	0.869833	−	0.493346i	$-0.164226\pi$
$313$	9633.46	1.73967	0.869833	+	0.493346i	$0.164226\pi$
$314$	0	0
$315$	1030.95	0.184404
$316$	0	0
$317$	−7832.86	−1.38781	−0.693907	−	0.720065i	$-0.744112\pi$
$317$	−7832.86	−1.38781	−0.693907	+	0.720065i	$0.744112\pi$
$318$	0	0
$319$	−1147.07	−0.201327
$320$	0	0
$321$	−6268.74	−1.08999
$322$	0	0
$323$	21704.3	3.73888
$324$	0	0
$325$	−1348.61	−0.230177
$326$	0	0
$327$	−3248.77	−0.549411
$328$	0	0
$329$	2433.35	0.407766
$330$	0	0
$331$	3002.68	0.498617	0.249308	−	0.968424i	$-0.419797\pi$
$331$	3002.68	0.498617	0.249308	+	0.968424i	$0.419797\pi$
$332$	0	0
$333$	2808.92	0.462246
$334$	0	0
$335$	5475.30	0.892978
$336$	0	0
$337$	−6707.78	−1.08426	−0.542131	−	0.840294i	$-0.682382\pi$
$337$	−6707.78	−1.08426	−0.542131	+	0.840294i	$0.682382\pi$
$338$	0	0
$339$	7121.64	1.14099
$340$	0	0
$341$	7920.31	1.25780
$342$	0	0
$343$	5337.90	0.840290
$344$	0	0
$345$	9059.70	1.41379
$346$	0	0
$347$	−9319.21	−1.44173	−0.720866	−	0.693074i	$-0.756256\pi$
$347$	−9319.21	−1.44173	−0.720866	+	0.693074i	$0.756256\pi$
$348$	0	0
$349$	−5673.56	−0.870197	−0.435098	−	0.900383i	$-0.643286\pi$
$349$	−5673.56	−0.870197	−0.435098	+	0.900383i	$0.643286\pi$
$350$	0	0
$351$	−2376.57	−0.361401
$352$	0	0
$353$	−6621.49	−0.998375	−0.499188	−	0.866494i	$-0.666368\pi$
$353$	−6621.49	−0.998375	−0.499188	+	0.866494i	$0.666368\pi$
$354$	0	0
$355$	−7620.03	−1.13924
$356$	0	0
$357$	−7512.38	−1.11372
$358$	0	0
$359$	−2432.83	−0.357660	−0.178830	−	0.983880i	$-0.557231\pi$
$359$	−2432.83	−0.357660	−0.178830	+	0.983880i	$0.557231\pi$
$360$	0	0
$361$	19080.0	2.78175
$362$	0	0
$363$	1485.74	0.214825
$364$	0	0
$365$	3139.78	0.450257
$366$	0	0
$367$	−6367.01	−0.905600	−0.452800	−	0.891612i	$-0.649575\pi$
$367$	−6367.01	−0.905600	−0.452800	+	0.891612i	$0.649575\pi$
$368$	0	0
$369$	−1122.83	−0.158407
$370$	0	0
$371$	−1625.00	−0.227402
$372$	0	0
$373$	6083.32	0.844456	0.422228	−	0.906490i	$-0.361248\pi$
$373$	6083.32	0.844456	0.422228	+	0.906490i	$0.361248\pi$
$374$	0	0
$375$	9652.63	1.32923
$376$	0	0
$377$	801.242	0.109459
$378$	0	0
$379$	3131.84	0.424464	0.212232	−	0.977219i	$-0.431927\pi$
$379$	3131.84	0.424464	0.212232	+	0.977219i	$0.431927\pi$
$380$	0	0
$381$	−14906.6	−2.00443
$382$	0	0
$383$	−2584.17	−0.344765	−0.172383	−	0.985030i	$-0.555147\pi$
$383$	−2584.17	−0.344765	−0.172383	+	0.985030i	$0.555147\pi$
$384$	0	0
$385$	−3024.97	−0.400434
$386$	0	0
$387$	4503.15	0.591493
$388$	0	0
$389$	−2718.60	−0.354341	−0.177170	−	0.984180i	$-0.556694\pi$
$389$	−2718.60	−0.354341	−0.177170	+	0.984180i	$0.556694\pi$
$390$	0	0
$391$	−21984.4	−2.84348
$392$	0	0
$393$	−7257.51	−0.931534
$394$	0	0
$395$	1446.96	0.184315
$396$	0	0
$397$	−8139.31	−1.02897	−0.514484	−	0.857500i	$-0.672017\pi$
$397$	−8139.31	−1.02897	−0.514484	+	0.857500i	$0.672017\pi$
$398$	0	0
$399$	−8978.14	−1.12649
$400$	0	0
$401$	−8000.82	−0.996364	−0.498182	−	0.867072i	$-0.665999\pi$
$401$	−8000.82	−0.996364	−0.498182	+	0.867072i	$0.665999\pi$
$402$	0	0
$403$	−5532.45	−0.683849
$404$	0	0
$405$	7953.93	0.975887
$406$	0	0
$407$	−8241.87	−1.00377
$408$	0	0
$409$	−3305.16	−0.399583	−0.199792	−	0.979838i	$-0.564027\pi$
$409$	−3305.16	−0.399583	−0.199792	+	0.979838i	$0.564027\pi$
$410$	0	0
$411$	−11828.4	−1.41960
$412$	0	0
$413$	1672.53	0.199273
$414$	0	0
$415$	−4935.74	−0.583821
$416$	0	0
$417$	−17336.6	−2.03592
$418$	0	0
$419$	−1456.05	−0.169768	−0.0848841	−	0.996391i	$-0.527052\pi$
$419$	−1456.05	−0.169768	−0.0848841	+	0.996391i	$0.527052\pi$
$420$	0	0
$421$	9020.41	1.04425	0.522123	−	0.852870i	$-0.325140\pi$
$421$	9020.41	1.04425	0.522123	+	0.852870i	$0.325140\pi$
$422$	0	0
$423$	−3743.88	−0.430340
$424$	0	0
$425$	−6577.95	−0.750770
$426$	0	0
$427$	5453.57	0.618072
$428$	0	0
$429$	−6953.10	−0.782515
$430$	0	0
$431$	10391.1	1.16130	0.580650	−	0.814153i	$-0.302798\pi$
$431$	10391.1	1.16130	0.580650	+	0.814153i	$0.302798\pi$
$432$	0	0
$433$	12704.4	1.41001	0.705003	−	0.709204i	$-0.250946\pi$
$433$	12704.4	1.41001	0.705003	+	0.709204i	$0.250946\pi$
$434$	0	0
$435$	−1610.52	−0.177514
$436$	0	0
$437$	−26273.8	−2.87608
$438$	0	0
$439$	−6317.12	−0.686788	−0.343394	−	0.939192i	$-0.611577\pi$
$439$	−6317.12	−0.686788	−0.343394	+	0.939192i	$0.611577\pi$
$440$	0	0
$441$	−3588.94	−0.387533
$442$	0	0
$443$	−12564.4	−1.34752	−0.673760	−	0.738951i	$-0.735322\pi$
$443$	−12564.4	−1.34752	−0.673760	+	0.738951i	$0.735322\pi$
$444$	0	0
$445$	−138.057	−0.0147069
$446$	0	0
$447$	4871.89	0.515509
$448$	0	0
$449$	502.664	0.0528334	0.0264167	−	0.999651i	$-0.491590\pi$
$449$	502.664	0.0528334	0.0264167	+	0.999651i	$0.491590\pi$
$450$	0	0
$451$	3294.58	0.343981
$452$	0	0
$453$	19563.1	2.02904
$454$	0	0
$455$	2112.99	0.217711
$456$	0	0
$457$	12772.7	1.30739	0.653697	−	0.756756i	$-0.273217\pi$
$457$	12772.7	1.30739	0.653697	+	0.756756i	$0.273217\pi$
$458$	0	0
$459$	−11591.9	−1.17878
$460$	0	0
$461$	5821.63	0.588157	0.294078	−	0.955781i	$-0.404987\pi$
$461$	5821.63	0.588157	0.294078	+	0.955781i	$0.404987\pi$
$462$	0	0
$463$	−8530.04	−0.856208	−0.428104	−	0.903729i	$-0.640818\pi$
$463$	−8530.04	−0.856208	−0.428104	+	0.903729i	$0.640818\pi$
$464$	0	0
$465$	11120.4	1.10902
$466$	0	0
$467$	19333.8	1.91576	0.957880	−	0.287169i	$-0.0927140\pi$
$467$	19333.8	1.91576	0.957880	+	0.287169i	$0.0927140\pi$
$468$	0	0
$469$	5496.04	0.541116
$470$	0	0
$471$	12310.4	1.20432
$472$	0	0
$473$	−13213.0	−1.28443
$474$	0	0
$475$	−7861.39	−0.759380
$476$	0	0
$477$	2500.19	0.239991
$478$	0	0
$479$	15650.9	1.49292	0.746458	−	0.665433i	$-0.231753\pi$
$479$	15650.9	1.49292	0.746458	+	0.665433i	$0.231753\pi$
$480$	0	0
$481$	5757.06	0.545737
$482$	0	0
$483$	9094.02	0.856713
$484$	0	0
$485$	12535.4	1.17362
$486$	0	0
$487$	−20860.9	−1.94106	−0.970532	−	0.240971i	$-0.922534\pi$
$487$	−20860.9	−1.94106	−0.970532	+	0.240971i	$0.922534\pi$
$488$	0	0
$489$	4354.18	0.402664
$490$	0	0
$491$	3936.88	0.361851	0.180926	−	0.983497i	$-0.442091\pi$
$491$	3936.88	0.361851	0.180926	+	0.983497i	$0.442091\pi$
$492$	0	0
$493$	3908.10	0.357023
$494$	0	0
$495$	4654.14	0.422602
$496$	0	0
$497$	−7648.89	−0.690342
$498$	0	0
$499$	−7186.66	−0.644728	−0.322364	−	0.946616i	$-0.604477\pi$
$499$	−7186.66	−0.644728	−0.322364	+	0.946616i	$0.604477\pi$
$500$	0	0
$501$	−19053.9	−1.69913
$502$	0	0
$503$	−6398.26	−0.567166	−0.283583	−	0.958948i	$-0.591523\pi$
$503$	−6398.26	−0.567166	−0.283583	+	0.958948i	$0.591523\pi$
$504$	0	0
$505$	5402.91	0.476092
$506$	0	0
$507$	−9121.41	−0.799006
$508$	0	0
$509$	−5903.72	−0.514101	−0.257051	−	0.966398i	$-0.582751\pi$
$509$	−5903.72	−0.514101	−0.257051	+	0.966398i	$0.582751\pi$
$510$	0	0
$511$	3151.68	0.272841
$512$	0	0
$513$	−13853.6	−1.19230
$514$	0	0
$515$	9710.62	0.830876
$516$	0	0
$517$	10985.2	0.934485
$518$	0	0
$519$	16307.3	1.37921
$520$	0	0
$521$	−22463.3	−1.88894	−0.944469	−	0.328601i	$-0.893423\pi$
$521$	−22463.3	−1.88894	−0.944469	+	0.328601i	$0.893423\pi$
$522$	0	0
$523$	−204.414	−0.0170906	−0.00854531	−	0.999963i	$-0.502720\pi$
$523$	−204.414	−0.0170906	−0.00854531	+	0.999963i	$0.502720\pi$
$524$	0	0
$525$	2721.02	0.226200
$526$	0	0
$527$	−26984.9	−2.23051
$528$	0	0
$529$	14446.0	1.18731
$530$	0	0
$531$	−2573.30	−0.210305
$532$	0	0
$533$	−2301.31	−0.187018
$534$	0	0
$535$	8600.07	0.694979
$536$	0	0
$537$	−9417.82	−0.756813
$538$	0	0
$539$	10530.6	0.841530
$540$	0	0
$541$	−2499.56	−0.198640	−0.0993202	−	0.995056i	$-0.531667\pi$
$541$	−2499.56	−0.198640	−0.0993202	+	0.995056i	$0.531667\pi$
$542$	0	0
$543$	−19080.1	−1.50793
$544$	0	0
$545$	4456.98	0.350305
$546$	0	0
$547$	15423.7	1.20562	0.602808	−	0.797886i	$-0.294049\pi$
$547$	15423.7	1.20562	0.602808	+	0.797886i	$0.294049\pi$
$548$	0	0
$549$	−8390.71	−0.652289
$550$	0	0
$551$	4670.62	0.361117
$552$	0	0
$553$	1452.44	0.111689
$554$	0	0
$555$	−11571.8	−0.885041
$556$	0	0
$557$	5007.02	0.380887	0.190444	−	0.981698i	$-0.439007\pi$
$557$	5007.02	0.380887	0.190444	+	0.981698i	$0.439007\pi$
$558$	0	0
$559$	9229.48	0.698328
$560$	0	0
$561$	−33914.2	−2.55233
$562$	0	0
$563$	14132.0	1.05789	0.528946	−	0.848655i	$-0.322587\pi$
$563$	14132.0	1.05789	0.528946	+	0.848655i	$0.322587\pi$
$564$	0	0
$565$	−9770.17	−0.727494
$566$	0	0
$567$	7984.06	0.591356
$568$	0	0
$569$	−2334.74	−0.172017	−0.0860083	−	0.996294i	$-0.527411\pi$
$569$	−2334.74	−0.172017	−0.0860083	+	0.996294i	$0.527411\pi$
$570$	0	0
$571$	14655.5	1.07410	0.537052	−	0.843549i	$-0.319538\pi$
$571$	14655.5	1.07410	0.537052	+	0.843549i	$0.319538\pi$
$572$	0	0
$573$	−2693.20	−0.196352
$574$	0	0
$575$	7962.86	0.577520
$576$	0	0
$577$	−3608.50	−0.260353	−0.130177	−	0.991491i	$-0.541554\pi$
$577$	−3608.50	−0.260353	−0.130177	+	0.991491i	$0.541554\pi$
$578$	0	0
$579$	1953.20	0.140194
$580$	0	0
$581$	−4954.43	−0.353777
$582$	0	0
$583$	−7335.98	−0.521141
$584$	0	0
$585$	−3250.98	−0.229763
$586$	0	0
$587$	−5916.60	−0.416021	−0.208010	−	0.978127i	$-0.566699\pi$
$587$	−5916.60	−0.416021	−0.208010	+	0.978127i	$0.566699\pi$
$588$	0	0
$589$	−32249.9	−2.25609
$590$	0	0
$591$	−18479.9	−1.28623
$592$	0	0
$593$	2017.60	0.139718	0.0698589	−	0.997557i	$-0.477745\pi$
$593$	2017.60	0.139718	0.0698589	+	0.997557i	$0.477745\pi$
$594$	0	0
$595$	10306.2	0.710107
$596$	0	0
$597$	1696.72	0.116318
$598$	0	0
$599$	−11250.3	−0.767403	−0.383702	−	0.923457i	$-0.625351\pi$
$599$	−11250.3	−0.767403	−0.383702	+	0.923457i	$0.625351\pi$
$600$	0	0
$601$	−2924.39	−0.198483	−0.0992416	−	0.995063i	$-0.531642\pi$
$601$	−2924.39	−0.198483	−0.0992416	+	0.995063i	$0.531642\pi$
$602$	0	0
$603$	−8456.05	−0.571073
$604$	0	0
$605$	−2038.29	−0.136972
$606$	0	0
$607$	242.410	0.0162094	0.00810472	−	0.999967i	$-0.497420\pi$
$607$	242.410	0.0162094	0.00810472	+	0.999967i	$0.497420\pi$
$608$	0	0
$609$	−1616.62	−0.107568
$610$	0	0
$611$	−7673.32	−0.508068
$612$	0	0
$613$	−13987.6	−0.921622	−0.460811	−	0.887498i	$-0.652441\pi$
$613$	−13987.6	−0.921622	−0.460811	+	0.887498i	$0.652441\pi$
$614$	0	0
$615$	4625.69	0.303294
$616$	0	0
$617$	23904.2	1.55972	0.779861	−	0.625953i	$-0.215290\pi$
$617$	23904.2	1.55972	0.779861	+	0.625953i	$0.215290\pi$
$618$	0	0
$619$	360.437	0.0234042	0.0117021	−	0.999932i	$-0.496275\pi$
$619$	360.437	0.0234042	0.0117021	+	0.999932i	$0.496275\pi$
$620$	0	0
$621$	14032.4	0.906764
$622$	0	0
$623$	−138.580	−0.00891189
$624$	0	0
$625$	−7141.00	−0.457024
$626$	0	0
$627$	−40531.2	−2.58160
$628$	0	0
$629$	28080.4	1.78003
$630$	0	0
$631$	6056.06	0.382073	0.191036	−	0.981583i	$-0.438815\pi$
$631$	6056.06	0.382073	0.191036	+	0.981583i	$0.438815\pi$
$632$	0	0
$633$	−5848.69	−0.367243
$634$	0	0
$635$	20450.3	1.27803
$636$	0	0
$637$	−7355.77	−0.457529
$638$	0	0
$639$	11768.4	0.728560
$640$	0	0
$641$	23049.6	1.42029	0.710143	−	0.704057i	$-0.248630\pi$
$641$	23049.6	1.42029	0.710143	+	0.704057i	$0.248630\pi$
$642$	0	0
$643$	2776.47	0.170285	0.0851426	−	0.996369i	$-0.472865\pi$
$643$	2776.47	0.170285	0.0851426	+	0.996369i	$0.472865\pi$
$644$	0	0
$645$	−18551.5	−1.13250
$646$	0	0
$647$	16585.2	1.00777	0.503887	−	0.863769i	$-0.331903\pi$
$647$	16585.2	1.00777	0.503887	+	0.863769i	$0.331903\pi$
$648$	0	0
$649$	7550.53	0.456678
$650$	0	0
$651$	11162.5	0.672032
$652$	0	0
$653$	−18704.2	−1.12091	−0.560453	−	0.828187i	$-0.689373\pi$
$653$	−18704.2	−1.12091	−0.560453	+	0.828187i	$0.689373\pi$
$654$	0	0
$655$	9956.57	0.593947
$656$	0	0
$657$	−4849.08	−0.287946
$658$	0	0
$659$	−23883.7	−1.41180	−0.705901	−	0.708310i	$-0.749458\pi$
$659$	−23883.7	−1.41180	−0.705901	+	0.708310i	$0.749458\pi$
$660$	0	0
$661$	19069.9	1.12214	0.561068	−	0.827770i	$-0.310391\pi$
$661$	19069.9	1.12214	0.561068	+	0.827770i	$0.310391\pi$
$662$	0	0
$663$	23689.5	1.38767
$664$	0	0
$665$	12317.1	0.718250
$666$	0	0
$667$	−4730.91	−0.274635
$668$	0	0
$669$	−14948.7	−0.863901
$670$	0	0
$671$	24619.8	1.41645
$672$	0	0
$673$	−31295.3	−1.79249	−0.896245	−	0.443560i	$-0.853715\pi$
$673$	−31295.3	−1.79249	−0.896245	+	0.443560i	$0.853715\pi$
$674$	0	0
$675$	4198.63	0.239415
$676$	0	0
$677$	12955.4	0.735474	0.367737	−	0.929930i	$-0.380133\pi$
$677$	12955.4	0.735474	0.367737	+	0.929930i	$0.380133\pi$
$678$	0	0
$679$	12582.9	0.711174
$680$	0	0
$681$	−6528.95	−0.367386
$682$	0	0
$683$	−29725.3	−1.66531	−0.832655	−	0.553791i	$-0.813180\pi$
$683$	−29725.3	−1.66531	−0.832655	+	0.553791i	$0.813180\pi$
$684$	0	0
$685$	16227.4	0.905135
$686$	0	0
$687$	2999.86	0.166597
$688$	0	0
$689$	5124.29	0.283338
$690$	0	0
$691$	−26123.3	−1.43817	−0.719085	−	0.694922i	$-0.755439\pi$
$691$	−26123.3	−1.43817	−0.719085	+	0.694922i	$0.755439\pi$
$692$	0	0
$693$	4671.77	0.256083
$694$	0	0
$695$	23784.1	1.29810
$696$	0	0
$697$	−11224.8	−0.609998
$698$	0	0
$699$	8986.50	0.486267
$700$	0	0
$701$	9382.25	0.505510	0.252755	−	0.967530i	$-0.418663\pi$
$701$	9382.25	0.505510	0.252755	+	0.967530i	$0.418663\pi$
$702$	0	0
$703$	33559.3	1.80044
$704$	0	0
$705$	15423.6	0.823951
$706$	0	0
$707$	5423.38	0.288497
$708$	0	0
$709$	22719.7	1.20347	0.601733	−	0.798697i	$-0.294477\pi$
$709$	22719.7	1.20347	0.601733	+	0.798697i	$0.294477\pi$
$710$	0	0
$711$	−2234.68	−0.117872
$712$	0	0
$713$	32666.2	1.71579
$714$	0	0
$715$	9538.95	0.498932
$716$	0	0
$717$	−407.789	−0.0212401
$718$	0	0
$719$	3371.69	0.174886	0.0874430	−	0.996170i	$-0.472130\pi$
$719$	3371.69	0.174886	0.0874430	+	0.996170i	$0.472130\pi$
$720$	0	0
$721$	9747.40	0.503484
$722$	0	0
$723$	−23089.6	−1.18770
$724$	0	0
$725$	−1415.53	−0.0725126
$726$	0	0
$727$	−27147.6	−1.38493	−0.692467	−	0.721449i	$-0.743476\pi$
$727$	−27147.6	−1.38493	−0.692467	+	0.721449i	$0.743476\pi$
$728$	0	0
$729$	2492.44	0.126629
$730$	0	0
$731$	45017.3	2.27774
$732$	0	0
$733$	12826.1	0.646305	0.323152	−	0.946347i	$-0.395257\pi$
$733$	12826.1	0.646305	0.323152	+	0.946347i	$0.395257\pi$
$734$	0	0
$735$	14785.3	0.741991
$736$	0	0
$737$	24811.5	1.24009
$738$	0	0
$739$	30957.7	1.54100	0.770500	−	0.637440i	$-0.220007\pi$
$739$	30957.7	1.54100	0.770500	+	0.637440i	$0.220007\pi$
$740$	0	0
$741$	28311.7	1.40358
$742$	0	0
$743$	12335.4	0.609076	0.304538	−	0.952500i	$-0.401498\pi$
$743$	12335.4	0.609076	0.304538	+	0.952500i	$0.401498\pi$
$744$	0	0
$745$	−6683.74	−0.328689
$746$	0	0
$747$	7622.75	0.373362
$748$	0	0
$749$	8632.65	0.421135
$750$	0	0
$751$	−19311.8	−0.938346	−0.469173	−	0.883106i	$-0.655448\pi$
$751$	−19311.8	−0.938346	−0.469173	+	0.883106i	$0.655448\pi$
$752$	0	0
$753$	−12212.6	−0.591036
$754$	0	0
$755$	−26838.6	−1.29372
$756$	0	0
$757$	−5018.13	−0.240934	−0.120467	−	0.992717i	$-0.538439\pi$
$757$	−5018.13	−0.240934	−0.120467	+	0.992717i	$0.538439\pi$
$758$	0	0
$759$	41054.4	1.96335
$760$	0	0
$761$	−34735.3	−1.65460	−0.827301	−	0.561759i	$-0.810125\pi$
$761$	−34735.3	−1.65460	−0.827301	+	0.561759i	$0.810125\pi$
$762$	0	0
$763$	4473.86	0.212274
$764$	0	0
$765$	−15856.9	−0.749419
$766$	0	0
$767$	−5274.15	−0.248290
$768$	0	0
$769$	2219.79	0.104093	0.0520466	−	0.998645i	$-0.483426\pi$
$769$	2219.79	0.104093	0.0520466	+	0.998645i	$0.483426\pi$
$770$	0	0
$771$	31676.3	1.47963
$772$	0	0
$773$	20986.4	0.976491	0.488246	−	0.872706i	$-0.337637\pi$
$773$	20986.4	0.976491	0.488246	+	0.872706i	$0.337637\pi$
$774$	0	0
$775$	9774.04	0.453025
$776$	0	0
$777$	−11615.7	−0.536306
$778$	0	0
$779$	−13414.9	−0.616993
$780$	0	0
$781$	−34530.5	−1.58207
$782$	0	0
$783$	−2494.50	−0.113852
$784$	0	0
$785$	−16888.6	−0.767875
$786$	0	0
$787$	21717.5	0.983665	0.491833	−	0.870690i	$-0.336327\pi$
$787$	21717.5	0.983665	0.491833	+	0.870690i	$0.336327\pi$
$788$	0	0
$789$	39080.9	1.76339
$790$	0	0
$791$	−9807.17	−0.440838
$792$	0	0
$793$	−17197.3	−0.770106
$794$	0	0
$795$	−10300.0	−0.459499
$796$	0	0
$797$	27242.7	1.21077	0.605387	−	0.795931i	$-0.293018\pi$
$797$	27242.7	1.21077	0.605387	+	0.795931i	$0.293018\pi$
$798$	0	0
$799$	−37427.1	−1.65716
$800$	0	0
$801$	213.216	0.00940526
$802$	0	0
$803$	14228.1	0.625277
$804$	0	0
$805$	−12476.1	−0.546241
$806$	0	0
$807$	52656.9	2.29691
$808$	0	0
$809$	27019.9	1.17425	0.587125	−	0.809496i	$-0.300260\pi$
$809$	27019.9	1.17425	0.587125	+	0.809496i	$0.300260\pi$
$810$	0	0
$811$	34289.1	1.48465	0.742327	−	0.670038i	$-0.233722\pi$
$811$	34289.1	1.48465	0.742327	+	0.670038i	$0.233722\pi$
$812$	0	0
$813$	−19741.6	−0.851622
$814$	0	0
$815$	−5973.49	−0.256739
$816$	0	0
$817$	53800.8	2.30386
$818$	0	0
$819$	−3263.30	−0.139229
$820$	0	0
$821$	−43635.4	−1.85492	−0.927458	−	0.373928i	$-0.878011\pi$
$821$	−43635.4	−1.85492	−0.927458	+	0.373928i	$0.878011\pi$
$822$	0	0
$823$	−6114.42	−0.258974	−0.129487	−	0.991581i	$-0.541333\pi$
$823$	−6114.42	−0.258974	−0.129487	+	0.991581i	$0.541333\pi$
$824$	0	0
$825$	12283.9	0.518387
$826$	0	0
$827$	14994.4	0.630477	0.315239	−	0.949012i	$-0.397915\pi$
$827$	14994.4	0.630477	0.315239	+	0.949012i	$0.397915\pi$
$828$	0	0
$829$	−6543.43	−0.274141	−0.137070	−	0.990561i	$-0.543769\pi$
$829$	−6543.43	−0.274141	−0.137070	+	0.990561i	$0.543769\pi$
$830$	0	0
$831$	−5942.91	−0.248083
$832$	0	0
$833$	−35878.2	−1.49232
$834$	0	0
$835$	26140.0	1.08337
$836$	0	0
$837$	17224.1	0.711293
$838$	0	0
$839$	9312.42	0.383195	0.191597	−	0.981474i	$-0.438633\pi$
$839$	9312.42	0.383195	0.191597	+	0.981474i	$0.438633\pi$
$840$	0	0
$841$	841.000	0.0344828
$842$	0	0
$843$	−4637.09	−0.189454
$844$	0	0
$845$	12513.6	0.509447
$846$	0	0
$847$	−2046.01	−0.0830009
$848$	0	0
$849$	−28088.8	−1.13546
$850$	0	0
$851$	−33992.4	−1.36927
$852$	0	0
$853$	−24580.0	−0.986638	−0.493319	−	0.869849i	$-0.664216\pi$
$853$	−24580.0	−0.986638	−0.493319	+	0.869849i	$0.664216\pi$
$854$	0	0
$855$	−18950.7	−0.758013
$856$	0	0
$857$	−11348.7	−0.452352	−0.226176	−	0.974086i	$-0.572622\pi$
$857$	−11348.7	−0.452352	−0.226176	+	0.974086i	$0.572622\pi$
$858$	0	0
$859$	−6307.05	−0.250516	−0.125258	−	0.992124i	$-0.539976\pi$
$859$	−6307.05	−0.250516	−0.125258	+	0.992124i	$0.539976\pi$
$860$	0	0
$861$	4643.21	0.183787
$862$	0	0
$863$	−4313.70	−0.170151	−0.0850753	−	0.996375i	$-0.527113\pi$
$863$	−4313.70	−0.170151	−0.0850753	+	0.996375i	$0.527113\pi$
$864$	0	0
$865$	−22372.0	−0.879387
$866$	0	0
$867$	84288.5	3.30171
$868$	0	0
$869$	6556.94	0.255959
$870$	0	0
$871$	−17331.2	−0.674220
$872$	0	0
$873$	−19359.7	−0.750545
$874$	0	0
$875$	−13292.6	−0.513568
$876$	0	0
$877$	−21034.0	−0.809883	−0.404941	−	0.914343i	$-0.632708\pi$
$877$	−21034.0	−0.809883	−0.404941	+	0.914343i	$0.632708\pi$
$878$	0	0
$879$	52522.2	2.01539
$880$	0	0
$881$	10244.5	0.391767	0.195884	−	0.980627i	$-0.437243\pi$
$881$	10244.5	0.391767	0.195884	+	0.980627i	$0.437243\pi$
$882$	0	0
$883$	2962.63	0.112911	0.0564555	−	0.998405i	$-0.482020\pi$
$883$	2962.63	0.112911	0.0564555	+	0.998405i	$0.482020\pi$
$884$	0	0
$885$	10601.2	0.402661
$886$	0	0
$887$	−16888.9	−0.639318	−0.319659	−	0.947533i	$-0.603568\pi$
$887$	−16888.9	−0.639318	−0.319659	+	0.947533i	$0.603568\pi$
$888$	0	0
$889$	20527.8	0.774443
$890$	0	0
$891$	36043.6	1.35522
$892$	0	0
$893$	−44729.6	−1.67617
$894$	0	0
$895$	12920.3	0.482545
$896$	0	0
$897$	−28677.1	−1.06745
$898$	0	0
$899$	−5806.98	−0.215432
$900$	0	0
$901$	24994.0	0.924164
$902$	0	0
$903$	−18621.8	−0.686261
$904$	0	0
$905$	26176.0	0.961458
$906$	0	0
$907$	−24646.4	−0.902284	−0.451142	−	0.892452i	$-0.648983\pi$
$907$	−24646.4	−0.902284	−0.451142	+	0.892452i	$0.648983\pi$
$908$	0	0
$909$	−8344.25	−0.304468
$910$	0	0
$911$	27585.7	1.00324	0.501622	−	0.865087i	$-0.332737\pi$
$911$	27585.7	1.00324	0.501622	+	0.865087i	$0.332737\pi$
$912$	0	0
$913$	−22366.5	−0.810758
$914$	0	0
$915$	34567.0	1.24891
$916$	0	0
$917$	9994.28	0.359913
$918$	0	0
$919$	−35694.5	−1.28123	−0.640616	−	0.767861i	$-0.721321\pi$
$919$	−35694.5	−1.28123	−0.640616	+	0.767861i	$0.721321\pi$
$920$	0	0
$921$	−6149.14	−0.220001
$922$	0	0
$923$	24120.0	0.860152
$924$	0	0
$925$	−10170.9	−0.361531
$926$	0	0
$927$	−14997.1	−0.531358
$928$	0	0
$929$	24741.4	0.873776	0.436888	−	0.899516i	$-0.356081\pi$
$929$	24741.4	0.873776	0.436888	+	0.899516i	$0.356081\pi$
$930$	0	0
$931$	−42878.5	−1.50944
$932$	0	0
$933$	43502.2	1.52647
$934$	0	0
$935$	46526.8	1.62737
$936$	0	0
$937$	20600.1	0.718222	0.359111	−	0.933295i	$-0.383080\pi$
$937$	20600.1	0.718222	0.359111	+	0.933295i	$0.383080\pi$
$938$	0	0
$939$	61292.2	2.13013
$940$	0	0
$941$	31347.7	1.08598	0.542989	−	0.839740i	$-0.317292\pi$
$941$	31347.7	1.08598	0.542989	+	0.839740i	$0.317292\pi$
$942$	0	0
$943$	13588.0	0.469233
$944$	0	0
$945$	−6578.34	−0.226448
$946$	0	0
$947$	26650.1	0.914479	0.457240	−	0.889343i	$-0.348838\pi$
$947$	26650.1	0.914479	0.457240	+	0.889343i	$0.348838\pi$
$948$	0	0
$949$	−9938.50	−0.339955
$950$	0	0
$951$	−49836.0	−1.69931
$952$	0	0
$953$	25259.5	0.858591	0.429295	−	0.903164i	$-0.358762\pi$
$953$	25259.5	0.858591	0.429295	+	0.903164i	$0.358762\pi$
$954$	0	0
$955$	3694.79	0.125194
$956$	0	0
$957$	−7298.12	−0.246515
$958$	0	0
$959$	16288.9	0.548483
$960$	0	0
$961$	10305.3	0.345919
$962$	0	0
$963$	−13281.9	−0.444449
$964$	0	0
$965$	−2679.59	−0.0893875
$966$	0	0
$967$	−52945.9	−1.76073	−0.880365	−	0.474296i	$-0.842703\pi$
$967$	−52945.9	−1.76073	−0.880365	+	0.474296i	$0.842703\pi$
$968$	0	0
$969$	138092.	4.57807
$970$	0	0
$971$	−38466.0	−1.27130	−0.635650	−	0.771978i	$-0.719268\pi$
$971$	−38466.0	−1.27130	−0.635650	+	0.771978i	$0.719268\pi$
$972$	0	0
$973$	23874.2	0.786610
$974$	0	0
$975$	−8580.46	−0.281841
$976$	0	0
$977$	12800.3	0.419159	0.209580	−	0.977792i	$-0.432790\pi$
$977$	12800.3	0.419159	0.209580	+	0.977792i	$0.432790\pi$
$978$	0	0
$979$	−625.613	−0.0204236
$980$	0	0
$981$	−6883.36	−0.224025
$982$	0	0
$983$	−46863.4	−1.52056	−0.760280	−	0.649596i	$-0.774938\pi$
$983$	−46863.4	−1.52056	−0.760280	+	0.649596i	$0.774938\pi$
$984$	0	0
$985$	25352.6	0.820103
$986$	0	0
$987$	15482.0	0.499288
$988$	0	0
$989$	−54495.2	−1.75212
$990$	0	0
$991$	−5189.58	−0.166350	−0.0831749	−	0.996535i	$-0.526506\pi$
$991$	−5189.58	−0.166350	−0.0831749	+	0.996535i	$0.526506\pi$
$992$	0	0
$993$	19104.3	0.610531
$994$	0	0
$995$	−2327.72	−0.0741646
$996$	0	0
$997$	49632.0	1.57659	0.788295	−	0.615298i	$-0.210964\pi$
$997$	49632.0	1.57659	0.788295	+	0.615298i	$0.210964\pi$
$998$	0	0
$999$	−17923.4	−0.567639

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1856.4.a.bd.1.5		6
4.3	odd	2		1856.4.a.bc.1.2		6
8.3	odd	2		464.4.a.n.1.5		6
8.5	even	2		232.4.a.e.1.2	✓	6
24.5	odd	2		2088.4.a.l.1.2		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
232.4.a.e.1.2	✓	6	8.5	even	2
464.4.a.n.1.5		6	8.3	odd	2
1856.4.a.bc.1.2		6	4.3	odd	2
1856.4.a.bd.1.5		6	1.1	even	1	trivial
2088.4.a.l.1.2		6	24.5	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 1856 = 2^{6} \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1856.a (trivial)

\(f(q)\)	\(=\)	\(q+6.36242 q^{3} -8.72860 q^{5} -8.76166 q^{7} +13.4804 q^{9} +O(q^{10})\)	\(q+6.36242 q^{3} -8.72860 q^{5} -8.76166 q^{7} +13.4804 q^{9} -39.5540 q^{11} +27.6290 q^{13} -55.5351 q^{15} +134.762 q^{17} +161.056 q^{19} -55.7454 q^{21} -163.135 q^{23} -48.8115 q^{25} -86.0171 q^{27} +29.0000 q^{29} -200.241 q^{31} -251.659 q^{33} +76.4771 q^{35} +208.370 q^{37} +175.788 q^{39} -83.2931 q^{41} +334.050 q^{43} -117.665 q^{45} -277.727 q^{47} -266.233 q^{49} +857.414 q^{51} +185.468 q^{53} +345.251 q^{55} +1024.71 q^{57} -190.892 q^{59} -622.436 q^{61} -118.111 q^{63} -241.163 q^{65} -627.283 q^{67} -1037.93 q^{69} +872.996 q^{71} -359.712 q^{73} -310.560 q^{75} +346.559 q^{77} -165.772 q^{79} -911.250 q^{81} +565.467 q^{83} -1176.29 q^{85} +184.510 q^{87} +15.8167 q^{89} -242.076 q^{91} -1274.02 q^{93} -1405.79 q^{95} -1436.13 q^{97} -533.206 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 5 q^{3} + 5 q^{5} - 38 q^{7} + 47 q^{9}+O(q^{10}) \) 6 * q + 5 * q^3 + 5 * q^5 - 38 * q^7 + 47 * q^9	\( 6 q + 5 q^{3} + 5 q^{5} - 38 q^{7} + 47 q^{9} + 19 q^{11} - 13 q^{13} - 191 q^{15} - 218 q^{17} + 290 q^{19} + 266 q^{21} - 196 q^{23} - 13 q^{25} + 437 q^{27} + 174 q^{29} - 675 q^{31} + 291 q^{33} + 466 q^{35} + 238 q^{37} - 1297 q^{39} - 464 q^{41} + 579 q^{43} + 148 q^{45} - 975 q^{47} + 914 q^{49} + 576 q^{51} - 515 q^{53} - 1605 q^{55} - 340 q^{57} + 108 q^{59} - 1158 q^{61} - 1136 q^{63} + 1239 q^{65} + 80 q^{67} - 2568 q^{69} + 438 q^{71} + 262 q^{73} - 1766 q^{75} - 194 q^{77} - 237 q^{79} + 2554 q^{81} + 1288 q^{83} - 3112 q^{85} + 145 q^{87} - 252 q^{89} - 2450 q^{91} - 2131 q^{93} + 180 q^{95} + 380 q^{97} - 2264 q^{99}+O(q^{100}) \) 6 * q + 5 * q^3 + 5 * q^5 - 38 * q^7 + 47 * q^9 + 19 * q^11 - 13 * q^13 - 191 * q^15 - 218 * q^17 + 290 * q^19 + 266 * q^21 - 196 * q^23 - 13 * q^25 + 437 * q^27 + 174 * q^29 - 675 * q^31 + 291 * q^33 + 466 * q^35 + 238 * q^37 - 1297 * q^39 - 464 * q^41 + 579 * q^43 + 148 * q^45 - 975 * q^47 + 914 * q^49 + 576 * q^51 - 515 * q^53 - 1605 * q^55 - 340 * q^57 + 108 * q^59 - 1158 * q^61 - 1136 * q^63 + 1239 * q^65 + 80 * q^67 - 2568 * q^69 + 438 * q^71 + 262 * q^73 - 1766 * q^75 - 194 * q^77 - 237 * q^79 + 2554 * q^81 + 1288 * q^83 - 3112 * q^85 + 145 * q^87 - 252 * q^89 - 2450 * q^91 - 2131 * q^93 + 180 * q^95 + 380 * q^97 - 2264 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more