LMFDB - Embedded newform 1856.4.a.bk.1.12

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:	$0$
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} - 245x^{10} + 21294x^{8} - 755514x^{6} + 8955005x^{4} - 27099393x^{2} + 23710340 $ x^12 - 245x^10 + 21294x^8 - 755514x^6 + 8955005x^4 - 27099393*x^2 + 23710340
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{19} $
Twist minimal:	no (minimal twist has level 928)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.12
Root			$9.17689$ 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)

$f(q)$	$=$	$q+9.17689 q^{3} +14.0091 q^{5} +13.6574 q^{7} +57.2153 q^{9} +O(q^{10})$	$q+9.17689 q^{3} +14.0091 q^{5} +13.6574 q^{7} +57.2153 q^{9} +4.50774 q^{11} -7.29514 q^{13} +128.560 q^{15} +28.0079 q^{17} +100.099 q^{19} +125.332 q^{21} -62.4954 q^{23} +71.2554 q^{25} +277.282 q^{27} +29.0000 q^{29} -156.926 q^{31} +41.3670 q^{33} +191.328 q^{35} +233.828 q^{37} -66.9466 q^{39} +7.39085 q^{41} +31.0965 q^{43} +801.535 q^{45} -226.708 q^{47} -156.475 q^{49} +257.026 q^{51} -254.020 q^{53} +63.1495 q^{55} +918.593 q^{57} -33.2971 q^{59} -644.725 q^{61} +781.412 q^{63} -102.198 q^{65} +939.189 q^{67} -573.513 q^{69} -842.541 q^{71} +664.251 q^{73} +653.903 q^{75} +61.5640 q^{77} +0.292394 q^{79} +999.774 q^{81} +283.786 q^{83} +392.367 q^{85} +266.130 q^{87} +1414.38 q^{89} -99.6326 q^{91} -1440.09 q^{93} +1402.29 q^{95} -642.913 q^{97} +257.911 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 10 q^{5} + 166 q^{9}+O(q^{10}) $ 12 * q - 10 * q^5 + 166 * q^9	$ 12 q - 10 q^{5} + 166 q^{9} - 30 q^{13} + 336 q^{17} - 56 q^{21} + 294 q^{25} + 348 q^{29} + 214 q^{33} - 196 q^{37} + 940 q^{41} - 1672 q^{45} + 972 q^{49} + 406 q^{53} + 1224 q^{57} - 400 q^{61} + 2998 q^{65} - 1004 q^{69} + 2252 q^{73} - 3032 q^{77} + 3932 q^{81} - 5564 q^{85} + 5212 q^{89} - 3722 q^{93} + 6164 q^{97}+O(q^{100}) $ 12 * q - 10 * q^5 + 166 * q^9 - 30 * q^13 + 336 * q^17 - 56 * q^21 + 294 * q^25 + 348 * q^29 + 214 * q^33 - 196 * q^37 + 940 * q^41 - 1672 * q^45 + 972 * q^49 + 406 * q^53 + 1224 * q^57 - 400 * q^61 + 2998 * q^65 - 1004 * q^69 + 2252 * q^73 - 3032 * q^77 + 3932 * q^81 - 5564 * q^85 + 5212 * q^89 - 3722 * q^93 + 6164 * q^97

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
$3$	9.17689	1.76609	0.883046	−	0.469286i	$-0.155489\pi$
$3$	9.17689	1.76609	0.883046	+	0.469286i	$0.155489\pi$
$4$	0	0
$5$	14.0091	1.25301	0.626507	−	0.779416i	$-0.284484\pi$
$5$	14.0091	1.25301	0.626507	+	0.779416i	$0.284484\pi$
$6$	0	0
$7$	13.6574	0.737431	0.368715	−	0.929542i	$-0.379798\pi$
$7$	13.6574	0.737431	0.368715	+	0.929542i	$0.379798\pi$
$8$	0	0
$9$	57.2153	2.11908
$10$	0	0
$11$	4.50774	0.123558	0.0617788	−	0.998090i	$-0.480323\pi$
$11$	4.50774	0.123558	0.0617788	+	0.998090i	$0.480323\pi$
$12$	0	0
$13$	−7.29514	−0.155639	−0.0778195	−	0.996967i	$-0.524796\pi$
$13$	−7.29514	−0.155639	−0.0778195	+	0.996967i	$0.524796\pi$
$14$	0	0
$15$	128.560	2.21294
$16$	0	0
$17$	28.0079	0.399584	0.199792	−	0.979838i	$-0.435973\pi$
$17$	28.0079	0.399584	0.199792	+	0.979838i	$0.435973\pi$
$18$	0	0
$19$	100.099	1.20864	0.604320	−	0.796741i	$-0.293445\pi$
$19$	100.099	1.20864	0.604320	+	0.796741i	$0.293445\pi$
$20$	0	0
$21$	125.332	1.30237
$22$	0	0
$23$	−62.4954	−0.566574	−0.283287	−	0.959035i	$-0.591425\pi$
$23$	−62.4954	−0.566574	−0.283287	+	0.959035i	$0.591425\pi$
$24$	0	0
$25$	71.2554	0.570043
$26$	0	0
$27$	277.282	1.97641
$28$	0	0
$29$	29.0000	0.185695
$29$	29.0000	0.185695
$30$	0	0
$31$	−156.926	−0.909183	−0.454591	−	0.890700i	$-0.650215\pi$
$31$	−156.926	−0.909183	−0.454591	+	0.890700i	$0.650215\pi$
$32$	0	0
$33$	41.3670	0.218214
$34$	0	0
$35$	191.328	0.924011
$36$	0	0
$37$	233.828	1.03895	0.519475	−	0.854486i	$-0.326128\pi$
$37$	233.828	1.03895	0.519475	+	0.854486i	$0.326128\pi$
$38$	0	0
$39$	−66.9466	−0.274873
$40$	0	0
$41$	7.39085	0.0281526	0.0140763	−	0.999901i	$-0.495519\pi$
$41$	7.39085	0.0281526	0.0140763	+	0.999901i	$0.495519\pi$
$42$	0	0
$43$	31.0965	0.110283	0.0551415	−	0.998479i	$-0.482439\pi$
$43$	31.0965	0.110283	0.0551415	+	0.998479i	$0.482439\pi$
$44$	0	0
$45$	801.535	2.65524
$46$	0	0
$47$	−226.708	−0.703590	−0.351795	−	0.936077i	$-0.614429\pi$
$47$	−226.708	−0.703590	−0.351795	+	0.936077i	$0.614429\pi$
$48$	0	0
$49$	−156.475	−0.456196
$50$	0	0
$51$	257.026	0.705702
$52$	0	0
$53$	−254.020	−0.658345	−0.329172	−	0.944270i	$-0.606770\pi$
$53$	−254.020	−0.658345	−0.329172	+	0.944270i	$0.606770\pi$
$54$	0	0
$55$	63.1495	0.154819
$56$	0	0
$57$	918.593	2.13457
$58$	0	0
$59$	−33.2971	−0.0734731	−0.0367365	−	0.999325i	$-0.511696\pi$
$59$	−33.2971	−0.0734731	−0.0367365	+	0.999325i	$0.511696\pi$
$60$	0	0
$61$	−644.725	−1.35325	−0.676627	−	0.736326i	$-0.736559\pi$
$61$	−644.725	−1.35325	−0.676627	+	0.736326i	$0.736559\pi$
$62$	0	0
$63$	781.412	1.56268
$64$	0	0
$65$	−102.198	−0.195018
$66$	0	0
$67$	939.189	1.71254	0.856270	−	0.516528i	$-0.172776\pi$
$67$	939.189	1.71254	0.856270	+	0.516528i	$0.172776\pi$
$68$	0	0
$69$	−573.513	−1.00062
$70$	0	0
$71$	−842.541	−1.40833	−0.704164	−	0.710038i	$-0.748678\pi$
$71$	−842.541	−1.40833	−0.704164	+	0.710038i	$0.748678\pi$
$72$	0	0
$73$	664.251	1.06500	0.532498	−	0.846431i	$-0.321253\pi$
$73$	664.251	1.06500	0.532498	+	0.846431i	$0.321253\pi$
$74$	0	0
$75$	653.903	1.00675
$76$	0	0
$77$	61.5640	0.0911152
$78$	0	0
$79$	0.292394	0.000416416 0	0.000208208	−	1.00000i	$-0.499934\pi$
$79$	0.292394	0.000416416 0	0.000208208		1.00000i	$0.499934\pi$
$80$	0	0
$81$	999.774	1.37143
$82$	0	0
$83$	283.786	0.375296	0.187648	−	0.982236i	$-0.439914\pi$
$83$	283.786	0.375296	0.187648	+	0.982236i	$0.439914\pi$
$84$	0	0
$85$	392.367	0.500684
$86$	0	0
$87$	266.130	0.327955
$88$	0	0
$89$	1414.38	1.68454	0.842270	−	0.539056i	$-0.181219\pi$
$89$	1414.38	1.68454	0.842270	+	0.539056i	$0.181219\pi$
$90$	0	0
$91$	−99.6326	−0.114773
$92$	0	0
$93$	−1440.09	−1.60570
$94$	0	0
$95$	1402.29	1.51444
$96$	0	0
$97$	−642.913	−0.672968	−0.336484	−	0.941689i	$-0.609238\pi$
$97$	−642.913	−0.672968	−0.336484	+	0.941689i	$0.609238\pi$
$98$	0	0
$99$	257.911	0.261829
$100$	0	0
$101$	−1710.19	−1.68486	−0.842429	−	0.538808i	$-0.818875\pi$
$101$	−1710.19	−1.68486	−0.842429	+	0.538808i	$0.818875\pi$
$102$	0	0
$103$	110.293	0.105509	0.0527547	−	0.998608i	$-0.483200\pi$
$103$	110.293	0.105509	0.0527547	+	0.998608i	$0.483200\pi$
$104$	0	0
$105$	1755.80	1.63189
$106$	0	0
$107$	1062.68	0.960124	0.480062	−	0.877235i	$-0.340614\pi$
$107$	1062.68	0.960124	0.480062	+	0.877235i	$0.340614\pi$
$108$	0	0
$109$	−544.198	−0.478208	−0.239104	−	0.970994i	$-0.576854\pi$
$109$	−544.198	−0.478208	−0.239104	+	0.970994i	$0.576854\pi$
$110$	0	0
$111$	2145.82	1.83488
$112$	0	0
$113$	−815.771	−0.679126	−0.339563	−	0.940583i	$-0.610279\pi$
$113$	−815.771	−0.679126	−0.339563	+	0.940583i	$0.610279\pi$
$114$	0	0
$115$	−875.506	−0.709925
$116$	0	0
$117$	−417.393	−0.329812
$118$	0	0
$119$	382.516	0.294665
$120$	0	0
$121$	−1310.68	−0.984733
$122$	0	0
$123$	67.8250	0.0497201
$124$	0	0
$125$	−752.914	−0.538742
$126$	0	0
$127$	1448.39	1.01200	0.505998	−	0.862535i	$-0.331124\pi$
$127$	1448.39	1.01200	0.505998	+	0.862535i	$0.331124\pi$
$128$	0	0
$129$	285.369	0.194770
$130$	0	0
$131$	−2671.44	−1.78172	−0.890858	−	0.454281i	$-0.849896\pi$
$131$	−2671.44	−1.78172	−0.890858	+	0.454281i	$0.849896\pi$
$132$	0	0
$133$	1367.09	0.891289
$134$	0	0
$135$	3884.48	2.47646
$136$	0	0
$137$	856.732	0.534274	0.267137	−	0.963659i	$-0.413922\pi$
$137$	856.732	0.534274	0.267137	+	0.963659i	$0.413922\pi$
$138$	0	0
$139$	−545.813	−0.333059	−0.166530	−	0.986036i	$-0.553256\pi$
$139$	−545.813	−0.333059	−0.166530	+	0.986036i	$0.553256\pi$
$140$	0	0
$141$	−2080.47	−1.24261
$142$	0	0
$143$	−32.8846	−0.0192304
$144$	0	0
$145$	406.264	0.232679
$146$	0	0
$147$	−1435.96	−0.805685
$148$	0	0
$149$	3398.46	1.86854	0.934271	−	0.356564i	$-0.116052\pi$
$149$	3398.46	1.86854	0.934271	+	0.356564i	$0.116052\pi$
$150$	0	0
$151$	−2535.99	−1.36673	−0.683365	−	0.730077i	$-0.739484\pi$
$151$	−2535.99	−1.36673	−0.683365	+	0.730077i	$0.739484\pi$
$152$	0	0
$153$	1602.48	0.846752
$154$	0	0
$155$	−2198.39	−1.13922
$156$	0	0
$157$	1492.77	0.758826	0.379413	−	0.925227i	$-0.376126\pi$
$157$	1492.77	0.758826	0.379413	+	0.925227i	$0.376126\pi$
$158$	0	0
$159$	−2331.11	−1.16270
$160$	0	0
$161$	−853.525	−0.417809
$162$	0	0
$163$	−82.3321	−0.0395629	−0.0197814	−	0.999804i	$-0.506297\pi$
$163$	−82.3321	−0.0395629	−0.0197814	+	0.999804i	$0.506297\pi$
$164$	0	0
$165$	579.515	0.273426
$166$	0	0
$167$	3428.68	1.58874	0.794368	−	0.607437i	$-0.207802\pi$
$167$	3428.68	1.58874	0.794368	+	0.607437i	$0.207802\pi$
$168$	0	0
$169$	−2143.78	−0.975777
$170$	0	0
$171$	5727.16	2.56121
$172$	0	0
$173$	1766.93	0.776515	0.388257	−	0.921551i	$-0.373077\pi$
$173$	1766.93	0.776515	0.388257	+	0.921551i	$0.373077\pi$
$174$	0	0
$175$	973.164	0.420367
$176$	0	0
$177$	−305.564	−0.129760
$178$	0	0
$179$	−3627.06	−1.51452	−0.757261	−	0.653113i	$-0.773463\pi$
$179$	−3627.06	−1.51452	−0.757261	+	0.653113i	$0.773463\pi$
$180$	0	0
$181$	−848.353	−0.348385	−0.174192	−	0.984712i	$-0.555731\pi$
$181$	−848.353	−0.348385	−0.174192	+	0.984712i	$0.555731\pi$
$182$	0	0
$183$	−5916.57	−2.38997
$184$	0	0
$185$	3275.73	1.30182
$186$	0	0
$187$	126.253	0.0493716
$188$	0	0
$189$	3786.95	1.45746
$190$	0	0
$191$	−3990.26	−1.51165	−0.755825	−	0.654774i	$-0.772764\pi$
$191$	−3990.26	−1.51165	−0.755825	+	0.654774i	$0.772764\pi$
$192$	0	0
$193$	3743.94	1.39635	0.698173	−	0.715929i	$-0.253997\pi$
$193$	3743.94	1.39635	0.698173	+	0.715929i	$0.253997\pi$
$194$	0	0
$195$	−937.864	−0.344420
$196$	0	0
$197$	−366.361	−0.132498	−0.0662490	−	0.997803i	$-0.521103\pi$
$197$	−366.361	−0.132498	−0.0662490	+	0.997803i	$0.521103\pi$
$198$	0	0
$199$	2961.91	1.05510	0.527549	−	0.849525i	$-0.323111\pi$
$199$	2961.91	1.05510	0.527549	+	0.849525i	$0.323111\pi$
$200$	0	0
$201$	8618.84	3.02451
$202$	0	0
$203$	396.065	0.136937
$204$	0	0
$205$	103.539	0.0352756
$206$	0	0
$207$	−3575.69	−1.20062
$208$	0	0
$209$	451.218	0.149337
$210$	0	0
$211$	1425.98	0.465252	0.232626	−	0.972566i	$-0.425268\pi$
$211$	1425.98	0.465252	0.232626	+	0.972566i	$0.425268\pi$
$212$	0	0
$213$	−7731.91	−2.48724
$214$	0	0
$215$	435.634	0.138186
$216$	0	0
$217$	−2143.20	−0.670459
$218$	0	0
$219$	6095.75	1.88088
$220$	0	0
$221$	−204.322	−0.0621908
$222$	0	0
$223$	843.229	0.253214	0.126607	−	0.991953i	$-0.459591\pi$
$223$	843.229	0.253214	0.126607	+	0.991953i	$0.459591\pi$
$224$	0	0
$225$	4076.90	1.20797
$226$	0	0
$227$	−5480.14	−1.60233	−0.801167	−	0.598441i	$-0.795787\pi$
$227$	−5480.14	−1.60233	−0.801167	+	0.598441i	$0.795787\pi$
$228$	0	0
$229$	1748.22	0.504479	0.252240	−	0.967665i	$-0.418833\pi$
$229$	1748.22	0.504479	0.252240	+	0.967665i	$0.418833\pi$
$230$	0	0
$231$	564.966	0.160918
$232$	0	0
$233$	3627.19	1.01985	0.509925	−	0.860219i	$-0.329673\pi$
$233$	3627.19	1.01985	0.509925	+	0.860219i	$0.329673\pi$
$234$	0	0
$235$	−3175.98	−0.881608
$236$	0	0
$237$	2.68327	0.000735430 0
$238$	0	0
$239$	97.1814	0.0263019	0.0131509	−	0.999914i	$-0.495814\pi$
$239$	97.1814	0.0263019	0.0131509	+	0.999914i	$0.495814\pi$
$240$	0	0
$241$	−5736.63	−1.53331	−0.766657	−	0.642057i	$-0.778081\pi$
$241$	−5736.63	−1.53331	−0.766657	+	0.642057i	$0.778081\pi$
$242$	0	0
$243$	1688.20	0.445671
$244$	0	0
$245$	−2192.08	−0.571620
$246$	0	0
$247$	−730.232	−0.188112
$248$	0	0
$249$	2604.27	0.662807
$250$	0	0
$251$	−6981.23	−1.75558	−0.877791	−	0.479044i	$-0.840983\pi$
$251$	−6981.23	−1.75558	−0.877791	+	0.479044i	$0.840983\pi$
$252$	0	0
$253$	−281.713	−0.0700045
$254$	0	0
$255$	3600.70	0.884254
$256$	0	0
$257$	5232.97	1.27013	0.635065	−	0.772458i	$-0.280973\pi$
$257$	5232.97	1.27013	0.635065	+	0.772458i	$0.280973\pi$
$258$	0	0
$259$	3193.49	0.766153
$260$	0	0
$261$	1659.24	0.393504
$262$	0	0
$263$	−2634.50	−0.617682	−0.308841	−	0.951114i	$-0.599941\pi$
$263$	−2634.50	−0.617682	−0.308841	+	0.951114i	$0.599941\pi$
$264$	0	0
$265$	−3558.59	−0.824915
$266$	0	0
$267$	12979.6	2.97505
$268$	0	0
$269$	−163.237	−0.0369990	−0.0184995	−	0.999829i	$-0.505889\pi$
$269$	−163.237	−0.0369990	−0.0184995	+	0.999829i	$0.505889\pi$
$270$	0	0
$271$	1044.55	0.234141	0.117070	−	0.993124i	$-0.462650\pi$
$271$	1044.55	0.234141	0.117070	+	0.993124i	$0.462650\pi$
$272$	0	0
$273$	−914.317	−0.202700
$274$	0	0
$275$	321.201	0.0704332
$276$	0	0
$277$	2943.00	0.638366	0.319183	−	0.947693i	$-0.396591\pi$
$277$	2943.00	0.638366	0.319183	+	0.947693i	$0.396591\pi$
$278$	0	0
$279$	−8978.54	−1.92663
$280$	0	0
$281$	4706.57	0.999184	0.499592	−	0.866261i	$-0.333483\pi$
$281$	4706.57	0.999184	0.499592	+	0.866261i	$0.333483\pi$
$282$	0	0
$283$	212.937	0.0447271	0.0223636	−	0.999750i	$-0.492881\pi$
$283$	212.937	0.0447271	0.0223636	+	0.999750i	$0.492881\pi$
$284$	0	0
$285$	12868.7	2.67465
$286$	0	0
$287$	100.940	0.0207606
$288$	0	0
$289$	−4128.56	−0.840333
$290$	0	0
$291$	−5899.94	−1.18852
$292$	0	0
$293$	−1474.73	−0.294044	−0.147022	−	0.989133i	$-0.546969\pi$
$293$	−1474.73	−0.294044	−0.147022	+	0.989133i	$0.546969\pi$
$294$	0	0
$295$	−466.463	−0.0920628
$296$	0	0
$297$	1249.92	0.244200
$298$	0	0
$299$	455.913	0.0881809
$300$	0	0
$301$	424.697	0.0813260
$302$	0	0
$303$	−15694.2	−2.97561
$304$	0	0
$305$	−9032.02	−1.69565
$306$	0	0
$307$	−6933.89	−1.28905	−0.644525	−	0.764583i	$-0.722945\pi$
$307$	−6933.89	−1.28905	−0.644525	+	0.764583i	$0.722945\pi$
$308$	0	0
$309$	1012.14	0.186339
$310$	0	0
$311$	−9011.82	−1.64313	−0.821565	−	0.570114i	$-0.806899\pi$
$311$	−9011.82	−1.64313	−0.821565	+	0.570114i	$0.806899\pi$
$312$	0	0
$313$	−4415.44	−0.797366	−0.398683	−	0.917089i	$-0.630533\pi$
$313$	−4415.44	−0.797366	−0.398683	+	0.917089i	$0.630533\pi$
$314$	0	0
$315$	10946.9	1.95806
$316$	0	0
$317$	−5219.14	−0.924720	−0.462360	−	0.886692i	$-0.652997\pi$
$317$	−5219.14	−0.924720	−0.462360	+	0.886692i	$0.652997\pi$
$318$	0	0
$319$	130.724	0.0229441
$320$	0	0
$321$	9752.10	1.69567
$322$	0	0
$323$	2803.55	0.482953
$324$	0	0
$325$	−519.818	−0.0887210
$326$	0	0
$327$	−4994.04	−0.844560
$328$	0	0
$329$	−3096.24	−0.518849
$330$	0	0
$331$	−4797.61	−0.796678	−0.398339	−	0.917238i	$-0.630413\pi$
$331$	−4797.61	−0.796678	−0.398339	+	0.917238i	$0.630413\pi$
$332$	0	0
$333$	13378.5	2.20162
$334$	0	0
$335$	13157.2	2.14584
$336$	0	0
$337$	4182.15	0.676013	0.338006	−	0.941144i	$-0.390247\pi$
$337$	4182.15	0.676013	0.338006	+	0.941144i	$0.390247\pi$
$338$	0	0
$339$	−7486.24	−1.19940
$340$	0	0
$341$	−707.380	−0.112337
$342$	0	0
$343$	−6821.54	−1.07384
$344$	0	0
$345$	−8034.42	−1.25379
$346$	0	0
$347$	−1391.64	−0.215294	−0.107647	−	0.994189i	$-0.534332\pi$
$347$	−1391.64	−0.215294	−0.107647	+	0.994189i	$0.534332\pi$
$348$	0	0
$349$	11159.2	1.71157	0.855787	−	0.517328i	$-0.173073\pi$
$349$	11159.2	1.71157	0.855787	+	0.517328i	$0.173073\pi$
$350$	0	0
$351$	−2022.81	−0.307606
$352$	0	0
$353$	12702.3	1.91523	0.957613	−	0.288059i	$-0.0930098\pi$
$353$	12702.3	1.91523	0.957613	+	0.288059i	$0.0930098\pi$
$354$	0	0
$355$	−11803.3	−1.76465
$356$	0	0
$357$	3510.30	0.520406
$358$	0	0
$359$	−727.922	−0.107015	−0.0535073	−	0.998567i	$-0.517040\pi$
$359$	−727.922	−0.107015	−0.0535073	+	0.998567i	$0.517040\pi$
$360$	0	0
$361$	3160.71	0.460813
$362$	0	0
$363$	−12028.0	−1.73913
$364$	0	0
$365$	9305.57	1.33445
$366$	0	0
$367$	6792.19	0.966075	0.483038	−	0.875600i	$-0.339533\pi$
$367$	6792.19	0.966075	0.483038	+	0.875600i	$0.339533\pi$
$368$	0	0
$369$	422.869	0.0596577
$370$	0	0
$371$	−3469.25	−0.485484
$372$	0	0
$373$	−13659.3	−1.89611	−0.948057	−	0.318100i	$-0.896955\pi$
$373$	−13659.3	−1.89611	−0.948057	+	0.318100i	$0.896955\pi$
$374$	0	0
$375$	−6909.41	−0.951468
$376$	0	0
$377$	−211.559	−0.0289014
$378$	0	0
$379$	199.856	0.0270869	0.0135434	−	0.999908i	$-0.495689\pi$
$379$	199.856	0.0270869	0.0135434	+	0.999908i	$0.495689\pi$
$380$	0	0
$381$	13291.7	1.78728
$382$	0	0
$383$	6689.13	0.892424	0.446212	−	0.894927i	$-0.352773\pi$
$383$	6689.13	0.892424	0.446212	+	0.894927i	$0.352773\pi$
$384$	0	0
$385$	862.458	0.114169
$386$	0	0
$387$	1779.19	0.233699
$388$	0	0
$389$	11082.6	1.44450	0.722248	−	0.691634i	$-0.243109\pi$
$389$	11082.6	1.44450	0.722248	+	0.691634i	$0.243109\pi$
$390$	0	0
$391$	−1750.37	−0.226394
$392$	0	0
$393$	−24515.5	−3.14668
$394$	0	0
$395$	4.09618	0.000521775 0
$396$	0	0
$397$	2941.58	0.371873	0.185937	−	0.982562i	$-0.440468\pi$
$397$	2941.58	0.371873	0.185937	+	0.982562i	$0.440468\pi$
$398$	0	0
$399$	12545.6	1.57410
$400$	0	0
$401$	15343.4	1.91075	0.955375	−	0.295395i	$-0.0954513\pi$
$401$	15343.4	1.91075	0.955375	+	0.295395i	$0.0954513\pi$
$402$	0	0
$403$	1144.79	0.141504
$404$	0	0
$405$	14006.0	1.71842
$406$	0	0
$407$	1054.04	0.128370
$408$	0	0
$409$	5573.49	0.673818	0.336909	−	0.941537i	$-0.390619\pi$
$409$	5573.49	0.673818	0.336909	+	0.941537i	$0.390619\pi$
$410$	0	0
$411$	7862.14	0.943578
$412$	0	0
$413$	−454.752	−0.0541813
$414$	0	0
$415$	3975.59	0.470251
$416$	0	0
$417$	−5008.86	−0.588213
$418$	0	0
$419$	3169.81	0.369583	0.184792	−	0.982778i	$-0.440839\pi$
$419$	3169.81	0.369583	0.184792	+	0.982778i	$0.440839\pi$
$420$	0	0
$421$	−29.4016	−0.00340367	−0.00170183	−	0.999999i	$-0.500542\pi$
$421$	−29.4016	−0.00340367	−0.00170183	+	0.999999i	$0.500542\pi$
$422$	0	0
$423$	−12971.2	−1.49097
$424$	0	0
$425$	1995.72	0.227780
$426$	0	0
$427$	−8805.27	−0.997931
$428$	0	0
$429$	−301.778	−0.0339627
$430$	0	0
$431$	12774.5	1.42767	0.713833	−	0.700316i	$-0.246958\pi$
$431$	12774.5	1.42767	0.713833	+	0.700316i	$0.246958\pi$
$432$	0	0
$433$	1549.79	0.172005	0.0860023	−	0.996295i	$-0.472591\pi$
$433$	1549.79	0.172005	0.0860023	+	0.996295i	$0.472591\pi$
$434$	0	0
$435$	3728.24	0.410932
$436$	0	0
$437$	−6255.70	−0.684784
$438$	0	0
$439$	−9161.15	−0.995986	−0.497993	−	0.867181i	$-0.665930\pi$
$439$	−9161.15	−0.995986	−0.497993	+	0.867181i	$0.665930\pi$
$440$	0	0
$441$	−8952.77	−0.966718
$442$	0	0
$443$	12192.8	1.30767	0.653833	−	0.756639i	$-0.273160\pi$
$443$	12192.8	1.30767	0.653833	+	0.756639i	$0.273160\pi$
$444$	0	0
$445$	19814.2	2.11075
$446$	0	0
$447$	31187.3	3.30002
$448$	0	0
$449$	6856.53	0.720667	0.360334	−	0.932824i	$-0.382663\pi$
$449$	6856.53	0.720667	0.360334	+	0.932824i	$0.382663\pi$
$450$	0	0
$451$	33.3160	0.00347847
$452$	0	0
$453$	−23272.5	−2.41377
$454$	0	0
$455$	−1395.77	−0.143812
$456$	0	0
$457$	14578.3	1.49222	0.746111	−	0.665821i	$-0.231919\pi$
$457$	14578.3	1.49222	0.746111	+	0.665821i	$0.231919\pi$
$458$	0	0
$459$	7766.10	0.789740
$460$	0	0
$461$	6740.91	0.681031	0.340516	−	0.940239i	$-0.389398\pi$
$461$	6740.91	0.681031	0.340516	+	0.940239i	$0.389398\pi$
$462$	0	0
$463$	12058.0	1.21033	0.605164	−	0.796101i	$-0.293107\pi$
$463$	12058.0	1.21033	0.605164	+	0.796101i	$0.293107\pi$
$464$	0	0
$465$	−20174.4	−2.01197
$466$	0	0
$467$	−1459.40	−0.144610	−0.0723052	−	0.997383i	$-0.523036\pi$
$467$	−1459.40	−0.144610	−0.0723052	+	0.997383i	$0.523036\pi$
$468$	0	0
$469$	12826.9	1.26288
$470$	0	0
$471$	13698.9	1.34016
$472$	0	0
$473$	140.175	0.0136263
$474$	0	0
$475$	7132.56	0.688978
$476$	0	0
$477$	−14533.8	−1.39509
$478$	0	0
$479$	−14648.8	−1.39733	−0.698666	−	0.715448i	$-0.746223\pi$
$479$	−14648.8	−1.39733	−0.698666	+	0.715448i	$0.746223\pi$
$480$	0	0
$481$	−1705.81	−0.161701
$482$	0	0
$483$	−7832.70	−0.737889
$484$	0	0
$485$	−9006.64	−0.843238
$486$	0	0
$487$	4557.53	0.424069	0.212034	−	0.977262i	$-0.431991\pi$
$487$	4557.53	0.424069	0.212034	+	0.977262i	$0.431991\pi$
$488$	0	0
$489$	−755.552	−0.0698717
$490$	0	0
$491$	−7563.44	−0.695179	−0.347590	−	0.937647i	$-0.613000\pi$
$491$	−7563.44	−0.695179	−0.347590	+	0.937647i	$0.613000\pi$
$492$	0	0
$493$	812.230	0.0742008
$494$	0	0
$495$	3613.11	0.328075
$496$	0	0
$497$	−11506.9	−1.03854
$498$	0	0
$499$	2933.62	0.263180	0.131590	−	0.991304i	$-0.457992\pi$
$499$	2933.62	0.263180	0.131590	+	0.991304i	$0.457992\pi$
$500$	0	0
$501$	31464.6	2.80586
$502$	0	0
$503$	−4013.31	−0.355755	−0.177877	−	0.984053i	$-0.556923\pi$
$503$	−4013.31	−0.355755	−0.177877	+	0.984053i	$0.556923\pi$
$504$	0	0
$505$	−23958.3	−2.11115
$506$	0	0
$507$	−19673.2	−1.72331
$508$	0	0
$509$	21058.1	1.83376	0.916880	−	0.399163i	$-0.130699\pi$
$509$	21058.1	1.83376	0.916880	+	0.399163i	$0.130699\pi$
$510$	0	0
$511$	9071.94	0.785360
$512$	0	0
$513$	27755.5	2.38876
$514$	0	0
$515$	1545.10	0.132205
$516$	0	0
$517$	−1021.94	−0.0869340
$518$	0	0
$519$	16214.9	1.37140
$520$	0	0
$521$	6743.14	0.567029	0.283515	−	0.958968i	$-0.408500\pi$
$521$	6743.14	0.567029	0.283515	+	0.958968i	$0.408500\pi$
$522$	0	0
$523$	−1918.97	−0.160441	−0.0802206	−	0.996777i	$-0.525562\pi$
$523$	−1918.97	−0.160441	−0.0802206	+	0.996777i	$0.525562\pi$
$524$	0	0
$525$	8930.62	0.742408
$526$	0	0
$527$	−4395.16	−0.363295
$528$	0	0
$529$	−8261.32	−0.678994
$530$	0	0
$531$	−1905.10	−0.155696
$532$	0	0
$533$	−53.9172	−0.00438164
$534$	0	0
$535$	14887.2	1.20305
$536$	0	0
$537$	−33285.1	−2.67479
$538$	0	0
$539$	−705.350	−0.0563665
$540$	0	0
$541$	−13747.5	−1.09252	−0.546259	−	0.837617i	$-0.683948\pi$
$541$	−13747.5	−1.09252	−0.546259	+	0.837617i	$0.683948\pi$
$542$	0	0
$543$	−7785.24	−0.615280
$544$	0	0
$545$	−7623.74	−0.599202
$546$	0	0
$547$	9785.34	0.764883	0.382441	−	0.923980i	$-0.375083\pi$
$547$	9785.34	0.764883	0.382441	+	0.923980i	$0.375083\pi$
$548$	0	0
$549$	−36888.1	−2.86766
$550$	0	0
$551$	2902.86	0.224439
$552$	0	0
$553$	3.99334	0.000307078 0
$554$	0	0
$555$	30061.0	2.29913
$556$	0	0
$557$	−21524.5	−1.63738	−0.818692	−	0.574232i	$-0.805301\pi$
$557$	−21524.5	−1.63738	−0.818692	+	0.574232i	$0.805301\pi$
$558$	0	0
$559$	−226.853	−0.0171643
$560$	0	0
$561$	1158.61	0.0871949
$562$	0	0
$563$	−7302.37	−0.546640	−0.273320	−	0.961923i	$-0.588122\pi$
$563$	−7302.37	−0.546640	−0.273320	+	0.961923i	$0.588122\pi$
$564$	0	0
$565$	−11428.2	−0.850955
$566$	0	0
$567$	13654.3	1.01134
$568$	0	0
$569$	−13132.2	−0.967541	−0.483771	−	0.875195i	$-0.660733\pi$
$569$	−13132.2	−0.967541	−0.483771	+	0.875195i	$0.660733\pi$
$570$	0	0
$571$	−11235.4	−0.823447	−0.411724	−	0.911309i	$-0.635073\pi$
$571$	−11235.4	−0.823447	−0.411724	+	0.911309i	$0.635073\pi$
$572$	0	0
$573$	−36618.2	−2.66971
$574$	0	0
$575$	−4453.14	−0.322972
$576$	0	0
$577$	21896.6	1.57984	0.789919	−	0.613211i	$-0.210123\pi$
$577$	21896.6	1.57984	0.789919	+	0.613211i	$0.210123\pi$
$578$	0	0
$579$	34357.7	2.46608
$580$	0	0
$581$	3875.78	0.276755
$582$	0	0
$583$	−1145.05	−0.0813436
$584$	0	0
$585$	−5847.31	−0.413259
$586$	0	0
$587$	1582.58	0.111278	0.0556388	−	0.998451i	$-0.482280\pi$
$587$	1582.58	0.111278	0.0556388	+	0.998451i	$0.482280\pi$
$588$	0	0
$589$	−15708.0	−1.09888
$590$	0	0
$591$	−3362.05	−0.234004
$592$	0	0
$593$	−2738.39	−0.189633	−0.0948165	−	0.995495i	$-0.530226\pi$
$593$	−2738.39	−0.189633	−0.0948165	+	0.995495i	$0.530226\pi$
$594$	0	0
$595$	5358.71	0.369220
$596$	0	0
$597$	27181.2	1.86340
$598$	0	0
$599$	−1868.85	−0.127478	−0.0637390	−	0.997967i	$-0.520303\pi$
$599$	−1868.85	−0.127478	−0.0637390	+	0.997967i	$0.520303\pi$
$600$	0	0
$601$	−13016.9	−0.883481	−0.441741	−	0.897143i	$-0.645639\pi$
$601$	−13016.9	−0.883481	−0.441741	+	0.897143i	$0.645639\pi$
$602$	0	0
$603$	53736.0	3.62902
$604$	0	0
$605$	−18361.5	−1.23388
$606$	0	0
$607$	8352.64	0.558523	0.279261	−	0.960215i	$-0.409910\pi$
$607$	8352.64	0.558523	0.279261	+	0.960215i	$0.409910\pi$
$608$	0	0
$609$	3634.64	0.241844
$610$	0	0
$611$	1653.86	0.109506
$612$	0	0
$613$	−8937.94	−0.588907	−0.294453	−	0.955666i	$-0.595138\pi$
$613$	−8937.94	−0.588907	−0.294453	+	0.955666i	$0.595138\pi$
$614$	0	0
$615$	950.168	0.0622999
$616$	0	0
$617$	4243.94	0.276912	0.138456	−	0.990369i	$-0.455786\pi$
$617$	4243.94	0.276912	0.138456	+	0.990369i	$0.455786\pi$
$618$	0	0
$619$	−25500.0	−1.65579	−0.827894	−	0.560885i	$-0.810461\pi$
$619$	−25500.0	−1.65579	−0.827894	+	0.560885i	$0.810461\pi$
$620$	0	0
$621$	−17328.9	−1.11978
$622$	0	0
$623$	19316.8	1.24223
$624$	0	0
$625$	−19454.6	−1.24509
$626$	0	0
$627$	4140.78	0.263743
$628$	0	0
$629$	6549.05	0.415147
$630$	0	0
$631$	−13923.2	−0.878405	−0.439203	−	0.898388i	$-0.644739\pi$
$631$	−13923.2	−0.878405	−0.439203	+	0.898388i	$0.644739\pi$
$632$	0	0
$633$	13086.0	0.821679
$634$	0	0
$635$	20290.6	1.26804
$636$	0	0
$637$	1141.51	0.0710019
$638$	0	0
$639$	−48206.2	−2.98436
$640$	0	0
$641$	15525.7	0.956672	0.478336	−	0.878177i	$-0.341240\pi$
$641$	15525.7	0.956672	0.478336	+	0.878177i	$0.341240\pi$
$642$	0	0
$643$	30205.1	1.85252	0.926261	−	0.376883i	$-0.123004\pi$
$643$	30205.1	1.85252	0.926261	+	0.376883i	$0.123004\pi$
$644$	0	0
$645$	3997.77	0.244049
$646$	0	0
$647$	−3920.67	−0.238234	−0.119117	−	0.992880i	$-0.538006\pi$
$647$	−3920.67	−0.238234	−0.119117	+	0.992880i	$0.538006\pi$
$648$	0	0
$649$	−150.095	−0.00907816
$650$	0	0
$651$	−19667.9	−1.18409
$652$	0	0
$653$	−21252.6	−1.27363	−0.636815	−	0.771017i	$-0.719748\pi$
$653$	−21252.6	−1.27363	−0.636815	+	0.771017i	$0.719748\pi$
$654$	0	0
$655$	−37424.5	−2.23252
$656$	0	0
$657$	38005.3	2.25681
$658$	0	0
$659$	−6413.43	−0.379107	−0.189554	−	0.981870i	$-0.560704\pi$
$659$	−6413.43	−0.379107	−0.189554	+	0.981870i	$0.560704\pi$
$660$	0	0
$661$	−516.403	−0.0303869	−0.0151934	−	0.999885i	$-0.504836\pi$
$661$	−516.403	−0.0303869	−0.0151934	+	0.999885i	$0.504836\pi$
$662$	0	0
$663$	−1875.04	−0.109835
$664$	0	0
$665$	19151.7	1.11680
$666$	0	0
$667$	−1812.37	−0.105210
$668$	0	0
$669$	7738.21	0.447200
$670$	0	0
$671$	−2906.25	−0.167205
$672$	0	0
$673$	−16043.0	−0.918886	−0.459443	−	0.888207i	$-0.651951\pi$
$673$	−16043.0	−0.918886	−0.459443	+	0.888207i	$0.651951\pi$
$674$	0	0
$675$	19757.9	1.12664
$676$	0	0
$677$	−10152.6	−0.576360	−0.288180	−	0.957576i	$-0.593050\pi$
$677$	−10152.6	−0.576360	−0.288180	+	0.957576i	$0.593050\pi$
$678$	0	0
$679$	−8780.52	−0.496267
$680$	0	0
$681$	−50290.6	−2.82987
$682$	0	0
$683$	−17659.3	−0.989332	−0.494666	−	0.869083i	$-0.664710\pi$
$683$	−17659.3	−0.989332	−0.494666	+	0.869083i	$0.664710\pi$
$684$	0	0
$685$	12002.1	0.669453
$686$	0	0
$687$	16043.2	0.890957
$688$	0	0
$689$	1853.11	0.102464
$690$	0	0
$691$	21407.0	1.17853	0.589264	−	0.807941i	$-0.299418\pi$
$691$	21407.0	1.17853	0.589264	+	0.807941i	$0.299418\pi$
$692$	0	0
$693$	3522.40	0.193081
$694$	0	0
$695$	−7646.35	−0.417328
$696$	0	0
$697$	207.002	0.0112493
$698$	0	0
$699$	33286.3	1.80115
$700$	0	0
$701$	4871.91	0.262496	0.131248	−	0.991350i	$-0.458102\pi$
$701$	4871.91	0.262496	0.131248	+	0.991350i	$0.458102\pi$
$702$	0	0
$703$	23405.9	1.25572
$704$	0	0
$705$	−29145.6	−1.55700
$706$	0	0
$707$	−23356.8	−1.24247
$708$	0	0
$709$	−16561.0	−0.877240	−0.438620	−	0.898673i	$-0.644533\pi$
$709$	−16561.0	−0.877240	−0.438620	+	0.898673i	$0.644533\pi$
$710$	0	0
$711$	16.7294	0.000882421 0
$712$	0	0
$713$	9807.13	0.515119
$714$	0	0
$715$	−460.684	−0.0240959
$716$	0	0
$717$	891.823	0.0464515
$718$	0	0
$719$	−20089.0	−1.04199	−0.520996	−	0.853559i	$-0.674439\pi$
$719$	−20089.0	−1.04199	−0.520996	+	0.853559i	$0.674439\pi$
$720$	0	0
$721$	1506.31	0.0778059
$722$	0	0
$723$	−52644.4	−2.70797
$724$	0	0
$725$	2066.41	0.105854
$726$	0	0
$727$	−1212.39	−0.0618501	−0.0309250	−	0.999522i	$-0.509845\pi$
$727$	−1212.39	−0.0618501	−0.0309250	+	0.999522i	$0.509845\pi$
$728$	0	0
$729$	−11501.5	−0.584336
$730$	0	0
$731$	870.948	0.0440673
$732$	0	0
$733$	2250.65	0.113410	0.0567050	−	0.998391i	$-0.481941\pi$
$733$	2250.65	0.113410	0.0567050	+	0.998391i	$0.481941\pi$
$734$	0	0
$735$	−20116.5	−1.00953
$736$	0	0
$737$	4233.62	0.211598
$738$	0	0
$739$	−33601.0	−1.67257	−0.836287	−	0.548292i	$-0.815278\pi$
$739$	−33601.0	−1.67257	−0.836287	+	0.548292i	$0.815278\pi$
$740$	0	0
$741$	−6701.26	−0.332223
$742$	0	0
$743$	−1205.51	−0.0595236	−0.0297618	−	0.999557i	$-0.509475\pi$
$743$	−1205.51	−0.0595236	−0.0297618	+	0.999557i	$0.509475\pi$
$744$	0	0
$745$	47609.4	2.34131
$746$	0	0
$747$	16236.9	0.795283
$748$	0	0
$749$	14513.5	0.708024
$750$	0	0
$751$	24033.9	1.16779	0.583895	−	0.811829i	$-0.301528\pi$
$751$	24033.9	1.16779	0.583895	+	0.811829i	$0.301528\pi$
$752$	0	0
$753$	−64065.9	−3.10052
$754$	0	0
$755$	−35527.0	−1.71253
$756$	0	0
$757$	33709.6	1.61849	0.809244	−	0.587472i	$-0.199877\pi$
$757$	33709.6	1.61849	0.809244	+	0.587472i	$0.199877\pi$
$758$	0	0
$759$	−2585.25	−0.123634
$760$	0	0
$761$	−5472.43	−0.260677	−0.130339	−	0.991470i	$-0.541606\pi$
$761$	−5472.43	−0.260677	−0.130339	+	0.991470i	$0.541606\pi$
$762$	0	0
$763$	−7432.33	−0.352646
$764$	0	0
$765$	22449.4	1.06099
$766$	0	0
$767$	242.907	0.0114353
$768$	0	0
$769$	−3939.61	−0.184741	−0.0923706	−	0.995725i	$-0.529444\pi$
$769$	−3939.61	−0.184741	−0.0923706	+	0.995725i	$0.529444\pi$
$770$	0	0
$771$	48022.4	2.24317
$772$	0	0
$773$	299.990	0.0139585	0.00697924	−	0.999976i	$-0.497778\pi$
$773$	299.990	0.0139585	0.00697924	+	0.999976i	$0.497778\pi$
$774$	0	0
$775$	−11181.8	−0.518274
$776$	0	0
$777$	29306.3	1.35310
$778$	0	0
$779$	739.813	0.0340264
$780$	0	0
$781$	−3797.96	−0.174010
$782$	0	0
$783$	8041.18	0.367009
$784$	0	0
$785$	20912.3	0.950819
$786$	0	0
$787$	−39130.1	−1.77235	−0.886174	−	0.463352i	$-0.846646\pi$
$787$	−39130.1	−1.77235	−0.886174	+	0.463352i	$0.846646\pi$
$788$	0	0
$789$	−24176.5	−1.09088
$790$	0	0
$791$	−11141.3	−0.500809
$792$	0	0
$793$	4703.35	0.210619
$794$	0	0
$795$	−32656.8	−1.45688
$796$	0	0
$797$	15591.2	0.692933	0.346466	−	0.938062i	$-0.387381\pi$
$797$	15591.2	0.692933	0.346466	+	0.938062i	$0.387381\pi$
$798$	0	0
$799$	−6349.62	−0.281143
$800$	0	0
$801$	80924.1	3.56968
$802$	0	0
$803$	2994.27	0.131588
$804$	0	0
$805$	−11957.1	−0.523520
$806$	0	0
$807$	−1498.01	−0.0653437
$808$	0	0
$809$	10085.3	0.438296	0.219148	−	0.975692i	$-0.429672\pi$
$809$	10085.3	0.438296	0.219148	+	0.975692i	$0.429672\pi$
$810$	0	0
$811$	4240.62	0.183611	0.0918053	−	0.995777i	$-0.470736\pi$
$811$	4240.62	0.183611	0.0918053	+	0.995777i	$0.470736\pi$
$812$	0	0
$813$	9585.75	0.413514
$814$	0	0
$815$	−1153.40	−0.0495728
$816$	0	0
$817$	3112.71	0.133292
$818$	0	0
$819$	−5700.51	−0.243213
$820$	0	0
$821$	−2096.00	−0.0890999	−0.0445500	−	0.999007i	$-0.514185\pi$
$821$	−2096.00	−0.0890999	−0.0445500	+	0.999007i	$0.514185\pi$
$822$	0	0
$823$	35843.2	1.51812	0.759062	−	0.651018i	$-0.225658\pi$
$823$	35843.2	1.51812	0.759062	+	0.651018i	$0.225658\pi$
$824$	0	0
$825$	2947.62	0.124392
$826$	0	0
$827$	−34088.3	−1.43333	−0.716667	−	0.697416i	$-0.754333\pi$
$827$	−34088.3	−1.43333	−0.716667	+	0.697416i	$0.754333\pi$
$828$	0	0
$829$	5277.98	0.221124	0.110562	−	0.993869i	$-0.464735\pi$
$829$	5277.98	0.221124	0.110562	+	0.993869i	$0.464735\pi$
$830$	0	0
$831$	27007.5	1.12741
$832$	0	0
$833$	−4382.55	−0.182289
$834$	0	0
$835$	48032.7	1.99071
$836$	0	0
$837$	−43512.7	−1.79691
$838$	0	0
$839$	−13449.4	−0.553426	−0.276713	−	0.960953i	$-0.589245\pi$
$839$	−13449.4	−0.553426	−0.276713	+	0.960953i	$0.589245\pi$
$840$	0	0
$841$	841.000	0.0344828
$842$	0	0
$843$	43191.7	1.76465
$844$	0	0
$845$	−30032.5	−1.22266
$846$	0	0
$847$	−17900.5	−0.726173
$848$	0	0
$849$	1954.10	0.0789923
$850$	0	0
$851$	−14613.2	−0.588641
$852$	0	0
$853$	−484.372	−0.0194426	−0.00972132	−	0.999953i	$-0.503094\pi$
$853$	−484.372	−0.0194426	−0.00972132	+	0.999953i	$0.503094\pi$
$854$	0	0
$855$	80232.5	3.20923
$856$	0	0
$857$	3781.84	0.150741	0.0753705	−	0.997156i	$-0.475986\pi$
$857$	3781.84	0.150741	0.0753705	+	0.997156i	$0.475986\pi$
$858$	0	0
$859$	3541.87	0.140683	0.0703417	−	0.997523i	$-0.477591\pi$
$859$	3541.87	0.140683	0.0703417	+	0.997523i	$0.477591\pi$
$860$	0	0
$861$	926.313	0.0366651
$862$	0	0
$863$	−43405.8	−1.71211	−0.856056	−	0.516883i	$-0.827092\pi$
$863$	−43405.8	−1.71211	−0.856056	+	0.516883i	$0.827092\pi$
$864$	0	0
$865$	24753.1	0.972984
$866$	0	0
$867$	−37887.3	−1.48411
$868$	0	0
$869$	1.31804	5.14514e−5 0
$870$	0	0
$871$	−6851.51	−0.266538
$872$	0	0
$873$	−36784.4	−1.42608
$874$	0	0
$875$	−10282.9	−0.397284
$876$	0	0
$877$	−14767.0	−0.568583	−0.284292	−	0.958738i	$-0.591758\pi$
$877$	−14767.0	−0.568583	−0.284292	+	0.958738i	$0.591758\pi$
$878$	0	0
$879$	−13533.5	−0.519308
$880$	0	0
$881$	18112.9	0.692668	0.346334	−	0.938111i	$-0.387426\pi$
$881$	18112.9	0.692668	0.346334	+	0.938111i	$0.387426\pi$
$882$	0	0
$883$	−28230.9	−1.07593	−0.537964	−	0.842968i	$-0.680807\pi$
$883$	−28230.9	−1.07593	−0.537964	+	0.842968i	$0.680807\pi$
$884$	0	0
$885$	−4280.68	−0.162591
$886$	0	0
$887$	2372.78	0.0898200	0.0449100	−	0.998991i	$-0.485700\pi$
$887$	2372.78	0.0898200	0.0449100	+	0.998991i	$0.485700\pi$
$888$	0	0
$889$	19781.2	0.746276
$890$	0	0
$891$	4506.72	0.169451
$892$	0	0
$893$	−22693.1	−0.850388
$894$	0	0
$895$	−50811.9	−1.89772
$896$	0	0
$897$	4183.86	0.155736
$898$	0	0
$899$	−4550.84	−0.168831
$900$	0	0
$901$	−7114.57	−0.263064
$902$	0	0
$903$	3897.40	0.143629
$904$	0	0
$905$	−11884.7	−0.436531
$906$	0	0
$907$	−38489.1	−1.40905	−0.704525	−	0.709679i	$-0.748840\pi$
$907$	−38489.1	−1.40905	−0.704525	+	0.709679i	$0.748840\pi$
$908$	0	0
$909$	−97849.1	−3.57035
$910$	0	0
$911$	29547.5	1.07459	0.537295	−	0.843394i	$-0.319446\pi$
$911$	29547.5	1.07459	0.537295	+	0.843394i	$0.319446\pi$
$912$	0	0
$913$	1279.23	0.0463707
$914$	0	0
$915$	−82885.9	−2.99467
$916$	0	0
$917$	−36484.9	−1.31389
$918$	0	0
$919$	−25068.6	−0.899822	−0.449911	−	0.893073i	$-0.648544\pi$
$919$	−25068.6	−0.899822	−0.449911	+	0.893073i	$0.648544\pi$
$920$	0	0
$921$	−63631.6	−2.27658
$922$	0	0
$923$	6146.45	0.219191
$924$	0	0
$925$	16661.5	0.592246
$926$	0	0
$927$	6310.43	0.223583
$928$	0	0
$929$	30066.2	1.06183	0.530915	−	0.847425i	$-0.321848\pi$
$929$	30066.2	1.06183	0.530915	+	0.847425i	$0.321848\pi$
$930$	0	0
$931$	−15662.9	−0.551377
$932$	0	0
$933$	−82700.5	−2.90192
$934$	0	0
$935$	1768.69	0.0618634
$936$	0	0
$937$	−14590.0	−0.508682	−0.254341	−	0.967115i	$-0.581858\pi$
$937$	−14590.0	−0.508682	−0.254341	+	0.967115i	$0.581858\pi$
$938$	0	0
$939$	−40520.0	−1.40822
$940$	0	0
$941$	−51836.7	−1.79578	−0.897889	−	0.440221i	$-0.854900\pi$
$941$	−51836.7	−1.79578	−0.897889	+	0.440221i	$0.854900\pi$
$942$	0	0
$943$	−461.894	−0.0159505
$944$	0	0
$945$	53051.9	1.82622
$946$	0	0
$947$	−29471.8	−1.01130	−0.505652	−	0.862738i	$-0.668748\pi$
$947$	−29471.8	−1.01130	−0.505652	+	0.862738i	$0.668748\pi$
$948$	0	0
$949$	−4845.80	−0.165755
$950$	0	0
$951$	−47895.5	−1.63314
$952$	0	0
$953$	−43125.7	−1.46588	−0.732938	−	0.680295i	$-0.761851\pi$
$953$	−43125.7	−1.46588	−0.732938	+	0.680295i	$0.761851\pi$
$954$	0	0
$955$	−55900.0	−1.89412
$956$	0	0
$957$	1199.64	0.0405214
$958$	0	0
$959$	11700.7	0.393990
$960$	0	0
$961$	−5165.36	−0.173387
$962$	0	0
$963$	60801.6	2.03458
$964$	0	0
$965$	52449.3	1.74964
$966$	0	0
$967$	−12445.3	−0.413872	−0.206936	−	0.978354i	$-0.566349\pi$
$967$	−12445.3	−0.413872	−0.206936	+	0.978354i	$0.566349\pi$
$968$	0	0
$969$	25727.9	0.852940
$970$	0	0
$971$	−43174.2	−1.42691	−0.713453	−	0.700704i	$-0.752870\pi$
$971$	−43174.2	−1.42691	−0.713453	+	0.700704i	$0.752870\pi$
$972$	0	0
$973$	−7454.38	−0.245608
$974$	0	0
$975$	−4770.31	−0.156689
$976$	0	0
$977$	−8505.17	−0.278510	−0.139255	−	0.990257i	$-0.544471\pi$
$977$	−8505.17	−0.278510	−0.139255	+	0.990257i	$0.544471\pi$
$978$	0	0
$979$	6375.66	0.208138
$980$	0	0
$981$	−31136.4	−1.01336
$982$	0	0
$983$	−7035.65	−0.228283	−0.114142	−	0.993464i	$-0.536412\pi$
$983$	−7035.65	−0.228283	−0.114142	+	0.993464i	$0.536412\pi$
$984$	0	0
$985$	−5132.39	−0.166022
$986$	0	0
$987$	−28413.9	−0.916335
$988$	0	0
$989$	−1943.39	−0.0624834
$990$	0	0
$991$	−37661.1	−1.20721	−0.603605	−	0.797283i	$-0.706270\pi$
$991$	−37661.1	−1.20721	−0.603605	+	0.797283i	$0.706270\pi$
$992$	0	0
$993$	−44027.1	−1.40701
$994$	0	0
$995$	41493.8	1.32205
$996$	0	0
$997$	−17472.0	−0.555009	−0.277504	−	0.960724i	$-0.589507\pi$
$997$	−17472.0	−0.555009	−0.277504	+	0.960724i	$0.589507\pi$
$998$	0	0
$999$	64836.4	2.05339

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1856.4.a.bk.1.12		12
4.3	odd	2	inner	1856.4.a.bk.1.1		12
8.3	odd	2		928.4.a.i.1.12	yes	12
8.5	even	2		928.4.a.i.1.1	✓	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
928.4.a.i.1.1	✓	12	8.5	even	2
928.4.a.i.1.12	yes	12	8.3	odd	2
1856.4.a.bk.1.1		12	4.3	odd	2	inner
1856.4.a.bk.1.12		12	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)

\(f(q)\)	\(=\)	\(q+9.17689 q^{3} +14.0091 q^{5} +13.6574 q^{7} +57.2153 q^{9} +O(q^{10})\)	\(q+9.17689 q^{3} +14.0091 q^{5} +13.6574 q^{7} +57.2153 q^{9} +4.50774 q^{11} -7.29514 q^{13} +128.560 q^{15} +28.0079 q^{17} +100.099 q^{19} +125.332 q^{21} -62.4954 q^{23} +71.2554 q^{25} +277.282 q^{27} +29.0000 q^{29} -156.926 q^{31} +41.3670 q^{33} +191.328 q^{35} +233.828 q^{37} -66.9466 q^{39} +7.39085 q^{41} +31.0965 q^{43} +801.535 q^{45} -226.708 q^{47} -156.475 q^{49} +257.026 q^{51} -254.020 q^{53} +63.1495 q^{55} +918.593 q^{57} -33.2971 q^{59} -644.725 q^{61} +781.412 q^{63} -102.198 q^{65} +939.189 q^{67} -573.513 q^{69} -842.541 q^{71} +664.251 q^{73} +653.903 q^{75} +61.5640 q^{77} +0.292394 q^{79} +999.774 q^{81} +283.786 q^{83} +392.367 q^{85} +266.130 q^{87} +1414.38 q^{89} -99.6326 q^{91} -1440.09 q^{93} +1402.29 q^{95} -642.913 q^{97} +257.911 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 10 q^{5} + 166 q^{9}+O(q^{10}) \) 12 * q - 10 * q^5 + 166 * q^9	\( 12 q - 10 q^{5} + 166 q^{9} - 30 q^{13} + 336 q^{17} - 56 q^{21} + 294 q^{25} + 348 q^{29} + 214 q^{33} - 196 q^{37} + 940 q^{41} - 1672 q^{45} + 972 q^{49} + 406 q^{53} + 1224 q^{57} - 400 q^{61} + 2998 q^{65} - 1004 q^{69} + 2252 q^{73} - 3032 q^{77} + 3932 q^{81} - 5564 q^{85} + 5212 q^{89} - 3722 q^{93} + 6164 q^{97}+O(q^{100}) \) 12 * q - 10 * q^5 + 166 * q^9 - 30 * q^13 + 336 * q^17 - 56 * q^21 + 294 * q^25 + 348 * q^29 + 214 * q^33 - 196 * q^37 + 940 * q^41 - 1672 * q^45 + 972 * q^49 + 406 * q^53 + 1224 * q^57 - 400 * q^61 + 2998 * q^65 - 1004 * q^69 + 2252 * q^73 - 3032 * q^77 + 3932 * q^81 - 5564 * q^85 + 5212 * q^89 - 3722 * q^93 + 6164 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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