LMFDB - Embedded newform 1856.4.a.bk.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:	$0$
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			1.5
Root			$-1.50033$ 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)

$f(q)$	$=$	$q-1.50033 q^{3} -0.787262 q^{5} -27.3506 q^{7} -24.7490 q^{9} +O(q^{10})$	$q-1.50033 q^{3} -0.787262 q^{5} -27.3506 q^{7} -24.7490 q^{9} -62.2529 q^{11} +27.7949 q^{13} +1.18115 q^{15} +102.604 q^{17} -55.9390 q^{19} +41.0348 q^{21} -175.715 q^{23} -124.380 q^{25} +77.6404 q^{27} +29.0000 q^{29} -199.792 q^{31} +93.3997 q^{33} +21.5321 q^{35} -403.647 q^{37} -41.7014 q^{39} -118.286 q^{41} -354.315 q^{43} +19.4840 q^{45} +148.783 q^{47} +405.055 q^{49} -153.939 q^{51} -376.090 q^{53} +49.0093 q^{55} +83.9268 q^{57} -329.095 q^{59} -789.113 q^{61} +676.900 q^{63} -21.8818 q^{65} -147.681 q^{67} +263.630 q^{69} -254.191 q^{71} -402.746 q^{73} +186.611 q^{75} +1702.65 q^{77} +429.232 q^{79} +551.738 q^{81} -78.8317 q^{83} -80.7759 q^{85} -43.5095 q^{87} +1034.55 q^{89} -760.207 q^{91} +299.753 q^{93} +44.0386 q^{95} +1790.86 q^{97} +1540.70 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−1.50033	−0.288738	−0.144369	−	0.989524i	$-0.546115\pi$
$3$	−1.50033	−0.288738	−0.144369	+	0.989524i	$0.546115\pi$
$4$	0	0
$5$	−0.787262	−0.0704148	−0.0352074	−	0.999380i	$-0.511209\pi$
$5$	−0.787262	−0.0704148	−0.0352074	+	0.999380i	$0.511209\pi$
$6$	0	0
$7$	−27.3506	−1.47679	−0.738396	−	0.674367i	$-0.764417\pi$
$7$	−27.3506	−1.47679	−0.738396	+	0.674367i	$0.764417\pi$
$8$	0	0
$9$	−24.7490	−0.916630
$10$	0	0
$11$	−62.2529	−1.70636	−0.853180	−	0.521616i	$-0.825329\pi$
$11$	−62.2529	−1.70636	−0.853180	+	0.521616i	$0.825329\pi$
$12$	0	0
$13$	27.7949	0.592993	0.296497	−	0.955034i	$-0.404182\pi$
$13$	27.7949	0.592993	0.296497	+	0.955034i	$0.404182\pi$
$14$	0	0
$15$	1.18115	0.0203314
$16$	0	0
$17$	102.604	1.46383	0.731913	−	0.681398i	$-0.238628\pi$
$17$	102.604	1.46383	0.731913	+	0.681398i	$0.238628\pi$
$18$	0	0
$19$	−55.9390	−0.675436	−0.337718	−	0.941247i	$-0.609655\pi$
$19$	−55.9390	−0.675436	−0.337718	+	0.941247i	$0.609655\pi$
$20$	0	0
$21$	41.0348	0.426406
$22$	0	0
$23$	−175.715	−1.59300	−0.796502	−	0.604636i	$-0.793319\pi$
$23$	−175.715	−1.59300	−0.796502	+	0.604636i	$0.793319\pi$
$24$	0	0
$25$	−124.380	−0.995042
$26$	0	0
$27$	77.6404	0.553404
$28$	0	0
$29$	29.0000	0.185695
$29$	29.0000	0.185695
$30$	0	0
$31$	−199.792	−1.15754	−0.578769	−	0.815492i	$-0.696467\pi$
$31$	−199.792	−1.15754	−0.578769	+	0.815492i	$0.696467\pi$
$32$	0	0
$33$	93.3997	0.492691
$34$	0	0
$35$	21.5321	0.103988
$36$	0	0
$37$	−403.647	−1.79349	−0.896746	−	0.442546i	$-0.854075\pi$
$37$	−403.647	−1.79349	−0.896746	+	0.442546i	$0.854075\pi$
$38$	0	0
$39$	−41.7014	−0.171220
$40$	0	0
$41$	−118.286	−0.450565	−0.225282	−	0.974294i	$-0.572330\pi$
$41$	−118.286	−0.450565	−0.225282	+	0.974294i	$0.572330\pi$
$42$	0	0
$43$	−354.315	−1.25657	−0.628286	−	0.777982i	$-0.716243\pi$
$43$	−354.315	−1.25657	−0.628286	+	0.777982i	$0.716243\pi$
$44$	0	0
$45$	19.4840	0.0645444
$46$	0	0
$47$	148.783	0.461750	0.230875	−	0.972983i	$-0.425841\pi$
$47$	148.783	0.461750	0.230875	+	0.972983i	$0.425841\pi$
$48$	0	0
$49$	405.055	1.18092
$50$	0	0
$51$	−153.939	−0.422662
$52$	0	0
$53$	−376.090	−0.974716	−0.487358	−	0.873202i	$-0.662039\pi$
$53$	−376.090	−0.974716	−0.487358	+	0.873202i	$0.662039\pi$
$54$	0	0
$55$	49.0093	0.120153
$56$	0	0
$57$	83.9268	0.195024
$58$	0	0
$59$	−329.095	−0.726178	−0.363089	−	0.931754i	$-0.618278\pi$
$59$	−329.095	−0.726178	−0.363089	+	0.931754i	$0.618278\pi$
$60$	0	0
$61$	−789.113	−1.65632	−0.828160	−	0.560492i	$-0.810612\pi$
$61$	−789.113	−1.65632	−0.828160	+	0.560492i	$0.810612\pi$
$62$	0	0
$63$	676.900	1.35367
$64$	0	0
$65$	−21.8818	−0.0417555
$66$	0	0
$67$	−147.681	−0.269286	−0.134643	−	0.990894i	$-0.542989\pi$
$67$	−147.681	−0.269286	−0.134643	+	0.990894i	$0.542989\pi$
$68$	0	0
$69$	263.630	0.459961
$70$	0	0
$71$	−254.191	−0.424887	−0.212443	−	0.977173i	$-0.568142\pi$
$71$	−254.191	−0.424887	−0.212443	+	0.977173i	$0.568142\pi$
$72$	0	0
$73$	−402.746	−0.645725	−0.322862	−	0.946446i	$-0.604645\pi$
$73$	−402.746	−0.645725	−0.322862	+	0.946446i	$0.604645\pi$
$74$	0	0
$75$	186.611	0.287306
$76$	0	0
$77$	1702.65	2.51994
$78$	0	0
$79$	429.232	0.611295	0.305648	−	0.952145i	$-0.401127\pi$
$79$	429.232	0.611295	0.305648	+	0.952145i	$0.401127\pi$
$80$	0	0
$81$	551.738	0.756842
$82$	0	0
$83$	−78.8317	−0.104252	−0.0521259	−	0.998641i	$-0.516600\pi$
$83$	−78.8317	−0.104252	−0.0521259	+	0.998641i	$0.516600\pi$
$84$	0	0
$85$	−80.7759	−0.103075
$86$	0	0
$87$	−43.5095	−0.0536173
$88$	0	0
$89$	1034.55	1.23216	0.616081	−	0.787683i	$-0.288720\pi$
$89$	1034.55	1.23216	0.616081	+	0.787683i	$0.288720\pi$
$90$	0	0
$91$	−760.207	−0.875729
$92$	0	0
$93$	299.753	0.334225
$94$	0	0
$95$	44.0386	0.0475607
$96$	0	0
$97$	1790.86	1.87458	0.937289	−	0.348552i	$-0.113327\pi$
$97$	1790.86	1.87458	0.937289	+	0.348552i	$0.113327\pi$
$98$	0	0
$99$	1540.70	1.56410
$100$	0	0
$101$	−724.751	−0.714014	−0.357007	−	0.934102i	$-0.616203\pi$
$101$	−724.751	−0.714014	−0.357007	+	0.934102i	$0.616203\pi$
$102$	0	0
$103$	−178.280	−0.170548	−0.0852739	−	0.996358i	$-0.527177\pi$
$103$	−178.280	−0.170548	−0.0852739	+	0.996358i	$0.527177\pi$
$104$	0	0
$105$	−32.3051	−0.0300253
$106$	0	0
$107$	1815.82	1.64058	0.820290	−	0.571948i	$-0.193812\pi$
$107$	1815.82	1.64058	0.820290	+	0.571948i	$0.193812\pi$
$108$	0	0
$109$	−1002.06	−0.880548	−0.440274	−	0.897863i	$-0.645119\pi$
$109$	−1002.06	−0.880548	−0.440274	+	0.897863i	$0.645119\pi$
$110$	0	0
$111$	605.603	0.517849
$112$	0	0
$113$	85.9758	0.0715745	0.0357873	−	0.999359i	$-0.488606\pi$
$113$	85.9758	0.0715745	0.0357873	+	0.999359i	$0.488606\pi$
$114$	0	0
$115$	138.334	0.112171
$116$	0	0
$117$	−687.896	−0.543556
$118$	0	0
$119$	−2806.27	−2.16177
$120$	0	0
$121$	2544.43	1.91167
$122$	0	0
$123$	177.468	0.130095
$124$	0	0
$125$	196.327	0.140480
$126$	0	0
$127$	1766.02	1.23393	0.616965	−	0.786991i	$-0.288362\pi$
$127$	1766.02	1.23393	0.616965	+	0.786991i	$0.288362\pi$
$128$	0	0
$129$	531.589	0.362820
$130$	0	0
$131$	1040.00	0.693628	0.346814	−	0.937934i	$-0.387263\pi$
$131$	1040.00	0.693628	0.346814	+	0.937934i	$0.387263\pi$
$132$	0	0
$133$	1529.97	0.997480
$134$	0	0
$135$	−61.1233	−0.0389678
$136$	0	0
$137$	1620.70	1.01070	0.505351	−	0.862914i	$-0.331363\pi$
$137$	1620.70	1.01070	0.505351	+	0.862914i	$0.331363\pi$
$138$	0	0
$139$	606.981	0.370385	0.185192	−	0.982702i	$-0.440709\pi$
$139$	606.981	0.370385	0.185192	+	0.982702i	$0.440709\pi$
$140$	0	0
$141$	−223.223	−0.133325
$142$	0	0
$143$	−1730.31	−1.01186
$144$	0	0
$145$	−22.8306	−0.0130757
$146$	0	0
$147$	−607.714	−0.340976
$148$	0	0
$149$	−1203.11	−0.661494	−0.330747	−	0.943719i	$-0.607301\pi$
$149$	−1203.11	−0.661494	−0.330747	+	0.943719i	$0.607301\pi$
$150$	0	0
$151$	−1644.41	−0.886227	−0.443114	−	0.896465i	$-0.646126\pi$
$151$	−1644.41	−0.886227	−0.443114	+	0.896465i	$0.646126\pi$
$152$	0	0
$153$	−2539.34	−1.34179
$154$	0	0
$155$	157.288	0.0815078
$156$	0	0
$157$	−1262.40	−0.641725	−0.320862	−	0.947126i	$-0.603973\pi$
$157$	−1262.40	−0.641725	−0.320862	+	0.947126i	$0.603973\pi$
$158$	0	0
$159$	564.258	0.281437
$160$	0	0
$161$	4805.91	2.35254
$162$	0	0
$163$	−2456.18	−1.18026	−0.590131	−	0.807307i	$-0.700924\pi$
$163$	−2456.18	−1.18026	−0.590131	+	0.807307i	$0.700924\pi$
$164$	0	0
$165$	−73.5300	−0.0346928
$166$	0	0
$167$	775.298	0.359248	0.179624	−	0.983735i	$-0.442512\pi$
$167$	775.298	0.359248	0.179624	+	0.983735i	$0.442512\pi$
$168$	0	0
$169$	−1424.44	−0.648359
$170$	0	0
$171$	1384.44	0.619125
$172$	0	0
$173$	−823.000	−0.361685	−0.180842	−	0.983512i	$-0.557882\pi$
$173$	−823.000	−0.361685	−0.180842	+	0.983512i	$0.557882\pi$
$174$	0	0
$175$	3401.87	1.46947
$176$	0	0
$177$	493.750	0.209675
$178$	0	0
$179$	−3570.45	−1.49088	−0.745441	−	0.666571i	$-0.767761\pi$
$179$	−3570.45	−1.49088	−0.745441	+	0.666571i	$0.767761\pi$
$180$	0	0
$181$	−671.175	−0.275624	−0.137812	−	0.990458i	$-0.544007\pi$
$181$	−671.175	−0.275624	−0.137812	+	0.990458i	$0.544007\pi$
$182$	0	0
$183$	1183.93	0.478242
$184$	0	0
$185$	317.776	0.126288
$186$	0	0
$187$	−6387.38	−2.49782
$188$	0	0
$189$	−2123.51	−0.817263
$190$	0	0
$191$	−5081.79	−1.92516	−0.962579	−	0.271002i	$-0.912645\pi$
$191$	−5081.79	−1.92516	−0.962579	+	0.271002i	$0.912645\pi$
$192$	0	0
$193$	−2000.69	−0.746179	−0.373089	−	0.927795i	$-0.621702\pi$
$193$	−2000.69	−0.746179	−0.373089	+	0.927795i	$0.621702\pi$
$194$	0	0
$195$	32.8299	0.0120564
$196$	0	0
$197$	1144.50	0.413919	0.206960	−	0.978349i	$-0.433643\pi$
$197$	1144.50	0.413919	0.206960	+	0.978349i	$0.433643\pi$
$198$	0	0
$199$	1651.91	0.588445	0.294222	−	0.955737i	$-0.404939\pi$
$199$	1651.91	0.588445	0.294222	+	0.955737i	$0.404939\pi$
$200$	0	0
$201$	221.570	0.0777530
$202$	0	0
$203$	−793.167	−0.274234
$204$	0	0
$205$	93.1220	0.0317264
$206$	0	0
$207$	4348.77	1.46020
$208$	0	0
$209$	3482.37	1.15254
$210$	0	0
$211$	2761.98	0.901149	0.450574	−	0.892739i	$-0.351219\pi$
$211$	2761.98	0.901149	0.450574	+	0.892739i	$0.351219\pi$
$212$	0	0
$213$	381.370	0.122681
$214$	0	0
$215$	278.939	0.0884813
$216$	0	0
$217$	5464.42	1.70944
$218$	0	0
$219$	604.251	0.186445
$220$	0	0
$221$	2851.86	0.868039
$222$	0	0
$223$	701.622	0.210691	0.105345	−	0.994436i	$-0.466405\pi$
$223$	701.622	0.210691	0.105345	+	0.994436i	$0.466405\pi$
$224$	0	0
$225$	3078.29	0.912085
$226$	0	0
$227$	5281.91	1.54437	0.772187	−	0.635395i	$-0.219163\pi$
$227$	5281.91	1.54437	0.772187	+	0.635395i	$0.219163\pi$
$228$	0	0
$229$	5162.80	1.48982	0.744908	−	0.667168i	$-0.232493\pi$
$229$	5162.80	1.48982	0.744908	+	0.667168i	$0.232493\pi$
$230$	0	0
$231$	−2554.54	−0.727603
$232$	0	0
$233$	5656.98	1.59056	0.795281	−	0.606241i	$-0.207323\pi$
$233$	5656.98	1.59056	0.795281	+	0.606241i	$0.207323\pi$
$234$	0	0
$235$	−117.131	−0.0325140
$236$	0	0
$237$	−643.988	−0.176504
$238$	0	0
$239$	−4688.50	−1.26893	−0.634464	−	0.772953i	$-0.718779\pi$
$239$	−4688.50	−1.26893	−0.634464	+	0.772953i	$0.718779\pi$
$240$	0	0
$241$	−2254.32	−0.602545	−0.301273	−	0.953538i	$-0.597411\pi$
$241$	−2254.32	−0.602545	−0.301273	+	0.953538i	$0.597411\pi$
$242$	0	0
$243$	−2924.08	−0.771933
$244$	0	0
$245$	−318.884	−0.0831541
$246$	0	0
$247$	−1554.82	−0.400529
$248$	0	0
$249$	118.273	0.0301015
$250$	0	0
$251$	−906.394	−0.227933	−0.113966	−	0.993485i	$-0.536356\pi$
$251$	−906.394	−0.227933	−0.113966	+	0.993485i	$0.536356\pi$
$252$	0	0
$253$	10938.8	2.71824
$254$	0	0
$255$	121.190	0.0297617
$256$	0	0
$257$	−5953.47	−1.44501	−0.722504	−	0.691367i	$-0.757009\pi$
$257$	−5953.47	−1.44501	−0.722504	+	0.691367i	$0.757009\pi$
$258$	0	0
$259$	11040.0	2.64862
$260$	0	0
$261$	−717.722	−0.170214
$262$	0	0
$263$	2339.06	0.548414	0.274207	−	0.961671i	$-0.411585\pi$
$263$	2339.06	0.548414	0.274207	+	0.961671i	$0.411585\pi$
$264$	0	0
$265$	296.081	0.0686344
$266$	0	0
$267$	−1552.17	−0.355772
$268$	0	0
$269$	574.684	0.130257	0.0651285	−	0.997877i	$-0.479254\pi$
$269$	574.684	0.130257	0.0651285	+	0.997877i	$0.479254\pi$
$270$	0	0
$271$	−4985.61	−1.11754	−0.558772	−	0.829321i	$-0.688727\pi$
$271$	−4985.61	−1.11754	−0.558772	+	0.829321i	$0.688727\pi$
$272$	0	0
$273$	1140.56	0.252856
$274$	0	0
$275$	7743.03	1.69790
$276$	0	0
$277$	−2383.28	−0.516959	−0.258480	−	0.966017i	$-0.583221\pi$
$277$	−2383.28	−0.516959	−0.258480	+	0.966017i	$0.583221\pi$
$278$	0	0
$279$	4944.65	1.06103
$280$	0	0
$281$	−1020.93	−0.216739	−0.108370	−	0.994111i	$-0.534563\pi$
$281$	−1020.93	−0.216739	−0.108370	+	0.994111i	$0.534563\pi$
$282$	0	0
$283$	−2873.16	−0.603504	−0.301752	−	0.953387i	$-0.597571\pi$
$283$	−2873.16	−0.603504	−0.301752	+	0.953387i	$0.597571\pi$
$284$	0	0
$285$	−66.0723	−0.0137326
$286$	0	0
$287$	3235.19	0.665391
$288$	0	0
$289$	5614.51	1.14279
$290$	0	0
$291$	−2686.87	−0.541262
$292$	0	0
$293$	−2250.38	−0.448698	−0.224349	−	0.974509i	$-0.572026\pi$
$293$	−2250.38	−0.448698	−0.224349	+	0.974509i	$0.572026\pi$
$294$	0	0
$295$	259.084	0.0511337
$296$	0	0
$297$	−4833.35	−0.944307
$298$	0	0
$299$	−4883.98	−0.944641
$300$	0	0
$301$	9690.74	1.85570
$302$	0	0
$303$	1087.36	0.206163
$304$	0	0
$305$	621.238	0.116629
$306$	0	0
$307$	−4353.67	−0.809372	−0.404686	−	0.914456i	$-0.632619\pi$
$307$	−4353.67	−0.809372	−0.404686	+	0.914456i	$0.632619\pi$
$308$	0	0
$309$	267.478	0.0492436
$310$	0	0
$311$	−995.466	−0.181504	−0.0907519	−	0.995874i	$-0.528927\pi$
$311$	−995.466	−0.181504	−0.0907519	+	0.995874i	$0.528927\pi$
$312$	0	0
$313$	10687.1	1.92993	0.964966	−	0.262375i	$-0.0845059\pi$
$313$	10687.1	1.92993	0.964966	+	0.262375i	$0.0845059\pi$
$314$	0	0
$315$	−532.898	−0.0953187
$316$	0	0
$317$	−8547.86	−1.51450	−0.757249	−	0.653127i	$-0.773457\pi$
$317$	−8547.86	−1.51450	−0.757249	+	0.653127i	$0.773457\pi$
$318$	0	0
$319$	−1805.34	−0.316863
$320$	0	0
$321$	−2724.32	−0.473698
$322$	0	0
$323$	−5739.55	−0.988721
$324$	0	0
$325$	−3457.13	−0.590053
$326$	0	0
$327$	1503.41	0.254248
$328$	0	0
$329$	−4069.30	−0.681908
$330$	0	0
$331$	4474.69	0.743055	0.371528	−	0.928422i	$-0.378834\pi$
$331$	4474.69	0.743055	0.371528	+	0.928422i	$0.378834\pi$
$332$	0	0
$333$	9989.87	1.64397
$334$	0	0
$335$	116.264	0.0189617
$336$	0	0
$337$	7158.57	1.15713	0.578564	−	0.815637i	$-0.303613\pi$
$337$	7158.57	1.15713	0.578564	+	0.815637i	$0.303613\pi$
$338$	0	0
$339$	−128.992	−0.0206663
$340$	0	0
$341$	12437.6	1.97518
$342$	0	0
$343$	−1697.23	−0.267178
$344$	0	0
$345$	−207.546	−0.0323881
$346$	0	0
$347$	−1529.77	−0.236663	−0.118332	−	0.992974i	$-0.537755\pi$
$347$	−1529.77	−0.236663	−0.118332	+	0.992974i	$0.537755\pi$
$348$	0	0
$349$	−1291.38	−0.198068	−0.0990340	−	0.995084i	$-0.531575\pi$
$349$	−1291.38	−0.198068	−0.0990340	+	0.995084i	$0.531575\pi$
$350$	0	0
$351$	2158.01	0.328165
$352$	0	0
$353$	−1268.27	−0.191228	−0.0956139	−	0.995418i	$-0.530481\pi$
$353$	−1268.27	−0.191228	−0.0956139	+	0.995418i	$0.530481\pi$
$354$	0	0
$355$	200.115	0.0299183
$356$	0	0
$357$	4210.32	0.624185
$358$	0	0
$359$	11996.6	1.76366	0.881832	−	0.471564i	$-0.156310\pi$
$359$	11996.6	1.76366	0.881832	+	0.471564i	$0.156310\pi$
$360$	0	0
$361$	−3729.83	−0.543786
$362$	0	0
$363$	−3817.47	−0.551971
$364$	0	0
$365$	317.067	0.0454686
$366$	0	0
$367$	−11662.6	−1.65881	−0.829406	−	0.558646i	$-0.811321\pi$
$367$	−11662.6	−1.65881	−0.829406	+	0.558646i	$0.811321\pi$
$368$	0	0
$369$	2927.46	0.413001
$370$	0	0
$371$	10286.3	1.43945
$372$	0	0
$373$	9970.91	1.38411	0.692057	−	0.721843i	$-0.256705\pi$
$373$	9970.91	1.38411	0.692057	+	0.721843i	$0.256705\pi$
$374$	0	0
$375$	−294.555	−0.0405621
$376$	0	0
$377$	806.052	0.110116
$378$	0	0
$379$	−7518.92	−1.01905	−0.509526	−	0.860455i	$-0.670179\pi$
$379$	−7518.92	−1.01905	−0.509526	+	0.860455i	$0.670179\pi$
$380$	0	0
$381$	−2649.61	−0.356282
$382$	0	0
$383$	6929.25	0.924460	0.462230	−	0.886760i	$-0.347049\pi$
$383$	6929.25	0.924460	0.462230	+	0.886760i	$0.347049\pi$
$384$	0	0
$385$	−1340.43	−0.177441
$386$	0	0
$387$	8768.96	1.15181
$388$	0	0
$389$	296.982	0.0387084	0.0193542	−	0.999813i	$-0.493839\pi$
$389$	296.982	0.0387084	0.0193542	+	0.999813i	$0.493839\pi$
$390$	0	0
$391$	−18029.0	−2.33188
$392$	0	0
$393$	−1560.34	−0.200277
$394$	0	0
$395$	−337.918	−0.0430442
$396$	0	0
$397$	14349.1	1.81401	0.907004	−	0.421123i	$-0.138364\pi$
$397$	14349.1	1.81401	0.907004	+	0.421123i	$0.138364\pi$
$398$	0	0
$399$	−2295.45	−0.288010
$400$	0	0
$401$	−250.164	−0.0311536	−0.0155768	−	0.999879i	$-0.504958\pi$
$401$	−250.164	−0.0311536	−0.0155768	+	0.999879i	$0.504958\pi$
$402$	0	0
$403$	−5553.19	−0.686412
$404$	0	0
$405$	−434.362	−0.0532929
$406$	0	0
$407$	25128.2	3.06034
$408$	0	0
$409$	−13128.8	−1.58723	−0.793616	−	0.608419i	$-0.791804\pi$
$409$	−13128.8	−1.58723	−0.793616	+	0.608419i	$0.791804\pi$
$410$	0	0
$411$	−2431.59	−0.291828
$412$	0	0
$413$	9000.94	1.07241
$414$	0	0
$415$	62.0612	0.00734088
$416$	0	0
$417$	−910.670	−0.106944
$418$	0	0
$419$	−11430.9	−1.33278	−0.666389	−	0.745604i	$-0.732161\pi$
$419$	−11430.9	−1.33278	−0.666389	+	0.745604i	$0.732161\pi$
$420$	0	0
$421$	−13092.7	−1.51567	−0.757835	−	0.652446i	$-0.773743\pi$
$421$	−13092.7	−1.51567	−0.757835	+	0.652446i	$0.773743\pi$
$422$	0	0
$423$	−3682.23	−0.423254
$424$	0	0
$425$	−12761.9	−1.45657
$426$	0	0
$427$	21582.7	2.44604
$428$	0	0
$429$	2596.04	0.292163
$430$	0	0
$431$	−2761.82	−0.308660	−0.154330	−	0.988019i	$-0.549322\pi$
$431$	−2761.82	−0.308660	−0.154330	+	0.988019i	$0.549322\pi$
$432$	0	0
$433$	−12593.9	−1.39774	−0.698872	−	0.715247i	$-0.746314\pi$
$433$	−12593.9	−1.39774	−0.698872	+	0.715247i	$0.746314\pi$
$434$	0	0
$435$	34.2533	0.00377545
$436$	0	0
$437$	9829.32	1.07597
$438$	0	0
$439$	15492.1	1.68428	0.842138	−	0.539261i	$-0.181297\pi$
$439$	15492.1	1.68428	0.842138	+	0.539261i	$0.181297\pi$
$440$	0	0
$441$	−10024.7	−1.08246
$442$	0	0
$443$	8097.69	0.868472	0.434236	−	0.900799i	$-0.357018\pi$
$443$	8097.69	0.868472	0.434236	+	0.900799i	$0.357018\pi$
$444$	0	0
$445$	−814.463	−0.0867624
$446$	0	0
$447$	1805.06	0.190998
$448$	0	0
$449$	7625.66	0.801508	0.400754	−	0.916186i	$-0.368748\pi$
$449$	7625.66	0.801508	0.400754	+	0.916186i	$0.368748\pi$
$450$	0	0
$451$	7363.65	0.768826
$452$	0	0
$453$	2467.15	0.255887
$454$	0	0
$455$	598.481	0.0616643
$456$	0	0
$457$	3035.58	0.310719	0.155360	−	0.987858i	$-0.450346\pi$
$457$	3035.58	0.310719	0.155360	+	0.987858i	$0.450346\pi$
$458$	0	0
$459$	7966.19	0.810087
$460$	0	0
$461$	−18036.1	−1.82218	−0.911088	−	0.412213i	$-0.864756\pi$
$461$	−18036.1	−1.82218	−0.911088	+	0.412213i	$0.864756\pi$
$462$	0	0
$463$	−2284.08	−0.229266	−0.114633	−	0.993408i	$-0.536569\pi$
$463$	−2284.08	−0.229266	−0.114633	+	0.993408i	$0.536569\pi$
$464$	0	0
$465$	−235.984	−0.0235344
$466$	0	0
$467$	−8813.46	−0.873316	−0.436658	−	0.899628i	$-0.643838\pi$
$467$	−8813.46	−0.873316	−0.436658	+	0.899628i	$0.643838\pi$
$468$	0	0
$469$	4039.17	0.397679
$470$	0	0
$471$	1894.02	0.185290
$472$	0	0
$473$	22057.2	2.14417
$474$	0	0
$475$	6957.71	0.672087
$476$	0	0
$477$	9307.86	0.893454
$478$	0	0
$479$	−18983.6	−1.81082	−0.905411	−	0.424537i	$-0.860437\pi$
$479$	−18983.6	−1.81082	−0.905411	+	0.424537i	$0.860437\pi$
$480$	0	0
$481$	−11219.3	−1.06353
$482$	0	0
$483$	−7210.43	−0.679267
$484$	0	0
$485$	−1409.87	−0.131998
$486$	0	0
$487$	−17208.6	−1.60122	−0.800610	−	0.599186i	$-0.795491\pi$
$487$	−17208.6	−1.60122	−0.800610	+	0.599186i	$0.795491\pi$
$488$	0	0
$489$	3685.07	0.340787
$490$	0	0
$491$	20652.2	1.89821	0.949106	−	0.314957i	$-0.101990\pi$
$491$	20652.2	1.89821	0.949106	+	0.314957i	$0.101990\pi$
$492$	0	0
$493$	2975.51	0.271826
$494$	0	0
$495$	−1212.93	−0.110136
$496$	0	0
$497$	6952.28	0.627469
$498$	0	0
$499$	−19866.7	−1.78228	−0.891139	−	0.453730i	$-0.850093\pi$
$499$	−19866.7	−1.78228	−0.891139	+	0.453730i	$0.850093\pi$
$500$	0	0
$501$	−1163.20	−0.103729
$502$	0	0
$503$	5377.97	0.476723	0.238362	−	0.971176i	$-0.423390\pi$
$503$	5377.97	0.476723	0.238362	+	0.971176i	$0.423390\pi$
$504$	0	0
$505$	570.568	0.0502771
$506$	0	0
$507$	2137.13	0.187206
$508$	0	0
$509$	701.222	0.0610631	0.0305315	−	0.999534i	$-0.490280\pi$
$509$	701.222	0.0610631	0.0305315	+	0.999534i	$0.490280\pi$
$510$	0	0
$511$	11015.4	0.953602
$512$	0	0
$513$	−4343.13	−0.373789
$514$	0	0
$515$	140.353	0.0120091
$516$	0	0
$517$	−9262.18	−0.787911
$518$	0	0
$519$	1234.77	0.104432
$520$	0	0
$521$	−21334.1	−1.79398	−0.896990	−	0.442051i	$-0.854251\pi$
$521$	−21334.1	−1.79398	−0.896990	+	0.442051i	$0.854251\pi$
$522$	0	0
$523$	−12288.4	−1.02741	−0.513703	−	0.857968i	$-0.671727\pi$
$523$	−12288.4	−1.02741	−0.513703	+	0.857968i	$0.671727\pi$
$524$	0	0
$525$	−5103.92	−0.424292
$526$	0	0
$527$	−20499.4	−1.69443
$528$	0	0
$529$	18708.7	1.53766
$530$	0	0
$531$	8144.78	0.665637
$532$	0	0
$533$	−3287.74	−0.267182
$534$	0	0
$535$	−1429.53	−0.115521
$536$	0	0
$537$	5356.84	0.430474
$538$	0	0
$539$	−25215.8	−2.01507
$540$	0	0
$541$	−1517.00	−0.120556	−0.0602780	−	0.998182i	$-0.519199\pi$
$541$	−1517.00	−0.120556	−0.0602780	+	0.998182i	$0.519199\pi$
$542$	0	0
$543$	1006.98	0.0795832
$544$	0	0
$545$	788.882	0.0620036
$546$	0	0
$547$	−15366.1	−1.20111	−0.600554	−	0.799584i	$-0.705053\pi$
$547$	−15366.1	−1.20111	−0.600554	+	0.799584i	$0.705053\pi$
$548$	0	0
$549$	19529.8	1.51823
$550$	0	0
$551$	−1622.23	−0.125425
$552$	0	0
$553$	−11739.7	−0.902757
$554$	0	0
$555$	−476.768	−0.0364642
$556$	0	0
$557$	7717.53	0.587078	0.293539	−	0.955947i	$-0.405167\pi$
$557$	7717.53	0.587078	0.293539	+	0.955947i	$0.405167\pi$
$558$	0	0
$559$	−9848.16	−0.745139
$560$	0	0
$561$	9583.15	0.721214
$562$	0	0
$563$	−7026.17	−0.525964	−0.262982	−	0.964801i	$-0.584706\pi$
$563$	−7026.17	−0.525964	−0.262982	+	0.964801i	$0.584706\pi$
$564$	0	0
$565$	−67.6854	−0.00503991
$566$	0	0
$567$	−15090.3	−1.11770
$568$	0	0
$569$	−10743.0	−0.791510	−0.395755	−	0.918356i	$-0.629517\pi$
$569$	−10743.0	−0.791510	−0.395755	+	0.918356i	$0.629517\pi$
$570$	0	0
$571$	9274.65	0.679741	0.339871	−	0.940472i	$-0.389617\pi$
$571$	9274.65	0.679741	0.339871	+	0.940472i	$0.389617\pi$
$572$	0	0
$573$	7624.34	0.555866
$574$	0	0
$575$	21855.5	1.58511
$576$	0	0
$577$	−14538.6	−1.04896	−0.524479	−	0.851423i	$-0.675740\pi$
$577$	−14538.6	−1.04896	−0.524479	+	0.851423i	$0.675740\pi$
$578$	0	0
$579$	3001.68	0.215450
$580$	0	0
$581$	2156.09	0.153958
$582$	0	0
$583$	23412.7	1.66322
$584$	0	0
$585$	541.554	0.0382744
$586$	0	0
$587$	15075.6	1.06003	0.530014	−	0.847989i	$-0.322187\pi$
$587$	15075.6	1.06003	0.530014	+	0.847989i	$0.322187\pi$
$588$	0	0
$589$	11176.2	0.781843
$590$	0	0
$591$	−1717.12	−0.119514
$592$	0	0
$593$	10204.6	0.706668	0.353334	−	0.935497i	$-0.385048\pi$
$593$	10204.6	0.706668	0.353334	+	0.935497i	$0.385048\pi$
$594$	0	0
$595$	2209.27	0.152220
$596$	0	0
$597$	−2478.40	−0.169906
$598$	0	0
$599$	−12544.9	−0.855713	−0.427856	−	0.903847i	$-0.640731\pi$
$599$	−12544.9	−0.855713	−0.427856	+	0.903847i	$0.640731\pi$
$600$	0	0
$601$	−18702.6	−1.26937	−0.634687	−	0.772770i	$-0.718871\pi$
$601$	−18702.6	−1.26937	−0.634687	+	0.772770i	$0.718871\pi$
$602$	0	0
$603$	3654.97	0.246835
$604$	0	0
$605$	−2003.13	−0.134610
$606$	0	0
$607$	5121.68	0.342475	0.171238	−	0.985230i	$-0.445223\pi$
$607$	5121.68	0.342475	0.171238	+	0.985230i	$0.445223\pi$
$608$	0	0
$609$	1190.01	0.0791817
$610$	0	0
$611$	4135.41	0.273814
$612$	0	0
$613$	−3382.24	−0.222850	−0.111425	−	0.993773i	$-0.535542\pi$
$613$	−3382.24	−0.222850	−0.111425	+	0.993773i	$0.535542\pi$
$614$	0	0
$615$	−139.713	−0.00916063
$616$	0	0
$617$	−571.111	−0.0372643	−0.0186321	−	0.999826i	$-0.505931\pi$
$617$	−571.111	−0.0372643	−0.0186321	+	0.999826i	$0.505931\pi$
$618$	0	0
$619$	−10978.5	−0.712867	−0.356433	−	0.934321i	$-0.616007\pi$
$619$	−10978.5	−0.712867	−0.356433	+	0.934321i	$0.616007\pi$
$620$	0	0
$621$	−13642.6	−0.881575
$622$	0	0
$623$	−28295.6	−1.81965
$624$	0	0
$625$	15393.0	0.985150
$626$	0	0
$627$	−5224.69	−0.332782
$628$	0	0
$629$	−41415.7	−2.62536
$630$	0	0
$631$	−2688.48	−0.169615	−0.0848073	−	0.996397i	$-0.527027\pi$
$631$	−2688.48	−0.169615	−0.0848073	+	0.996397i	$0.527027\pi$
$632$	0	0
$633$	−4143.87	−0.260196
$634$	0	0
$635$	−1390.32	−0.0868869
$636$	0	0
$637$	11258.4	0.700276
$638$	0	0
$639$	6290.98	0.389464
$640$	0	0
$641$	−12783.8	−0.787724	−0.393862	−	0.919170i	$-0.628861\pi$
$641$	−12783.8	−0.787724	−0.393862	+	0.919170i	$0.628861\pi$
$642$	0	0
$643$	−21418.6	−1.31363	−0.656816	−	0.754051i	$-0.728097\pi$
$643$	−21418.6	−1.31363	−0.656816	+	0.754051i	$0.728097\pi$
$644$	0	0
$645$	−418.500	−0.0255479
$646$	0	0
$647$	−1439.68	−0.0874799	−0.0437400	−	0.999043i	$-0.513927\pi$
$647$	−1439.68	−0.0874799	−0.0437400	+	0.999043i	$0.513927\pi$
$648$	0	0
$649$	20487.1	1.23912
$650$	0	0
$651$	−8198.42	−0.493581
$652$	0	0
$653$	11505.7	0.689515	0.344757	−	0.938692i	$-0.387961\pi$
$653$	11505.7	0.689515	0.344757	+	0.938692i	$0.387961\pi$
$654$	0	0
$655$	−818.753	−0.0488417
$656$	0	0
$657$	9967.58	0.591891
$658$	0	0
$659$	3275.96	0.193647	0.0968234	−	0.995302i	$-0.469132\pi$
$659$	3275.96	0.193647	0.0968234	+	0.995302i	$0.469132\pi$
$660$	0	0
$661$	5455.49	0.321019	0.160510	−	0.987034i	$-0.448686\pi$
$661$	5455.49	0.321019	0.160510	+	0.987034i	$0.448686\pi$
$662$	0	0
$663$	−4278.72	−0.250636
$664$	0	0
$665$	−1204.48	−0.0702373
$666$	0	0
$667$	−5095.73	−0.295813
$668$	0	0
$669$	−1052.66	−0.0608345
$670$	0	0
$671$	49124.6	2.82628
$672$	0	0
$673$	535.453	0.0306689	0.0153345	−	0.999882i	$-0.495119\pi$
$673$	535.453	0.0306689	0.0153345	+	0.999882i	$0.495119\pi$
$674$	0	0
$675$	−9656.93	−0.550660
$676$	0	0
$677$	−16698.6	−0.947976	−0.473988	−	0.880531i	$-0.657186\pi$
$677$	−16698.6	−0.947976	−0.473988	+	0.880531i	$0.657186\pi$
$678$	0	0
$679$	−48981.0	−2.76836
$680$	0	0
$681$	−7924.59	−0.445919
$682$	0	0
$683$	1082.40	0.0606396	0.0303198	−	0.999540i	$-0.490347\pi$
$683$	1082.40	0.0606396	0.0303198	+	0.999540i	$0.490347\pi$
$684$	0	0
$685$	−1275.92	−0.0711683
$686$	0	0
$687$	−7745.89	−0.430166
$688$	0	0
$689$	−10453.4	−0.578000
$690$	0	0
$691$	−9761.34	−0.537393	−0.268697	−	0.963225i	$-0.586593\pi$
$691$	−9761.34	−0.537393	−0.268697	+	0.963225i	$0.586593\pi$
$692$	0	0
$693$	−42139.0	−2.30985
$694$	0	0
$695$	−477.853	−0.0260806
$696$	0	0
$697$	−12136.6	−0.659549
$698$	0	0
$699$	−8487.31	−0.459256
$700$	0	0
$701$	3032.05	0.163365	0.0816827	−	0.996658i	$-0.473971\pi$
$701$	3032.05	0.163365	0.0816827	+	0.996658i	$0.473971\pi$
$702$	0	0
$703$	22579.6	1.21139
$704$	0	0
$705$	175.735	0.00938803
$706$	0	0
$707$	19822.4	1.05445
$708$	0	0
$709$	4640.54	0.245810	0.122905	−	0.992418i	$-0.460779\pi$
$709$	4640.54	0.245810	0.122905	+	0.992418i	$0.460779\pi$
$710$	0	0
$711$	−10623.1	−0.560332
$712$	0	0
$713$	35106.4	1.84396
$714$	0	0
$715$	1362.21	0.0712500
$716$	0	0
$717$	7034.27	0.366387
$718$	0	0
$719$	20575.7	1.06724	0.533619	−	0.845725i	$-0.320832\pi$
$719$	20575.7	1.06724	0.533619	+	0.845725i	$0.320832\pi$
$720$	0	0
$721$	4876.06	0.251864
$722$	0	0
$723$	3382.21	0.173978
$724$	0	0
$725$	−3607.03	−0.184775
$726$	0	0
$727$	13890.0	0.708600	0.354300	−	0.935132i	$-0.384719\pi$
$727$	13890.0	0.708600	0.354300	+	0.935132i	$0.384719\pi$
$728$	0	0
$729$	−10509.8	−0.533955
$730$	0	0
$731$	−36354.1	−1.83940
$732$	0	0
$733$	−6253.36	−0.315107	−0.157553	−	0.987510i	$-0.550361\pi$
$733$	−6253.36	−0.315107	−0.157553	+	0.987510i	$0.550361\pi$
$734$	0	0
$735$	478.430	0.0240097
$736$	0	0
$737$	9193.59	0.459498
$738$	0	0
$739$	4938.33	0.245818	0.122909	−	0.992418i	$-0.460778\pi$
$739$	4938.33	0.245818	0.122909	+	0.992418i	$0.460778\pi$
$740$	0	0
$741$	2332.74	0.115648
$742$	0	0
$743$	17443.8	0.861307	0.430653	−	0.902517i	$-0.358283\pi$
$743$	17443.8	0.861307	0.430653	+	0.902517i	$0.358283\pi$
$744$	0	0
$745$	947.162	0.0465790
$746$	0	0
$747$	1951.01	0.0955605
$748$	0	0
$749$	−49663.8	−2.42280
$750$	0	0
$751$	13989.1	0.679718	0.339859	−	0.940476i	$-0.389621\pi$
$751$	13989.1	0.679718	0.339859	+	0.940476i	$0.389621\pi$
$752$	0	0
$753$	1359.89	0.0658128
$754$	0	0
$755$	1294.58	0.0624035
$756$	0	0
$757$	−35278.7	−1.69382	−0.846912	−	0.531733i	$-0.821541\pi$
$757$	−35278.7	−1.69382	−0.846912	+	0.531733i	$0.821541\pi$
$758$	0	0
$759$	−16411.7	−0.784859
$760$	0	0
$761$	−15090.1	−0.718814	−0.359407	−	0.933181i	$-0.617021\pi$
$761$	−15090.1	−0.718814	−0.359407	+	0.933181i	$0.617021\pi$
$762$	0	0
$763$	27406.9	1.30039
$764$	0	0
$765$	1999.12	0.0944817
$766$	0	0
$767$	−9147.16	−0.430619
$768$	0	0
$769$	−4238.61	−0.198762	−0.0993812	−	0.995049i	$-0.531686\pi$
$769$	−4238.61	−0.198762	−0.0993812	+	0.995049i	$0.531686\pi$
$770$	0	0
$771$	8932.14	0.417229
$772$	0	0
$773$	−36539.4	−1.70017	−0.850084	−	0.526647i	$-0.823449\pi$
$773$	−36539.4	−1.70017	−0.850084	+	0.526647i	$0.823449\pi$
$774$	0	0
$775$	24850.1	1.15180
$776$	0	0
$777$	−16563.6	−0.764756
$778$	0	0
$779$	6616.80	0.304328
$780$	0	0
$781$	15824.1	0.725010
$782$	0	0
$783$	2251.57	0.102765
$784$	0	0
$785$	993.842	0.0451869
$786$	0	0
$787$	25556.5	1.15755	0.578774	−	0.815488i	$-0.303531\pi$
$787$	25556.5	1.15755	0.578774	+	0.815488i	$0.303531\pi$
$788$	0	0
$789$	−3509.36	−0.158348
$790$	0	0
$791$	−2351.49	−0.105701
$792$	0	0
$793$	−21933.3	−0.982187
$794$	0	0
$795$	−444.218	−0.0198174
$796$	0	0
$797$	−16208.1	−0.720352	−0.360176	−	0.932884i	$-0.617283\pi$
$797$	−16208.1	−0.720352	−0.360176	+	0.932884i	$0.617283\pi$
$798$	0	0
$799$	15265.7	0.675921
$800$	0	0
$801$	−25604.2	−1.12944
$802$	0	0
$803$	25072.2	1.10184
$804$	0	0
$805$	−3783.51	−0.165653
$806$	0	0
$807$	−862.214	−0.0376101
$808$	0	0
$809$	24649.6	1.07124	0.535621	−	0.844459i	$-0.320078\pi$
$809$	24649.6	1.07124	0.535621	+	0.844459i	$0.320078\pi$
$810$	0	0
$811$	43969.2	1.90378	0.951892	−	0.306435i	$-0.0991361\pi$
$811$	43969.2	1.90378	0.951892	+	0.306435i	$0.0991361\pi$
$812$	0	0
$813$	7480.05	0.322677
$814$	0	0
$815$	1933.66	0.0831080
$816$	0	0
$817$	19820.1	0.848735
$818$	0	0
$819$	18814.4	0.802719
$820$	0	0
$821$	−14759.0	−0.627398	−0.313699	−	0.949523i	$-0.601568\pi$
$821$	−14759.0	−0.627398	−0.313699	+	0.949523i	$0.601568\pi$
$822$	0	0
$823$	36849.9	1.56076	0.780381	−	0.625304i	$-0.215025\pi$
$823$	36849.9	1.56076	0.780381	+	0.625304i	$0.215025\pi$
$824$	0	0
$825$	−11617.1	−0.490248
$826$	0	0
$827$	−7780.68	−0.327159	−0.163580	−	0.986530i	$-0.552304\pi$
$827$	−7780.68	−0.327159	−0.163580	+	0.986530i	$0.552304\pi$
$828$	0	0
$829$	−40014.8	−1.67644	−0.838221	−	0.545330i	$-0.816404\pi$
$829$	−40014.8	−1.67644	−0.838221	+	0.545330i	$0.816404\pi$
$830$	0	0
$831$	3575.70	0.149266
$832$	0	0
$833$	41560.1	1.72866
$834$	0	0
$835$	−610.363	−0.0252964
$836$	0	0
$837$	−15511.9	−0.640586
$838$	0	0
$839$	2466.60	0.101498	0.0507488	−	0.998711i	$-0.483839\pi$
$839$	2466.60	0.101498	0.0507488	+	0.998711i	$0.483839\pi$
$840$	0	0
$841$	841.000	0.0344828
$842$	0	0
$843$	1531.73	0.0625809
$844$	0	0
$845$	1121.41	0.0456541
$846$	0	0
$847$	−69591.6	−2.82314
$848$	0	0
$849$	4310.68	0.174254
$850$	0	0
$851$	70926.8	2.85704
$852$	0	0
$853$	7269.34	0.291791	0.145895	−	0.989300i	$-0.453394\pi$
$853$	7269.34	0.291791	0.145895	+	0.989300i	$0.453394\pi$
$854$	0	0
$855$	−1089.91	−0.0435956
$856$	0	0
$857$	−13378.5	−0.533256	−0.266628	−	0.963800i	$-0.585910\pi$
$857$	−13378.5	−0.533256	−0.266628	+	0.963800i	$0.585910\pi$
$858$	0	0
$859$	35532.4	1.41135	0.705675	−	0.708536i	$-0.250644\pi$
$859$	35532.4	1.41135	0.705675	+	0.708536i	$0.250644\pi$
$860$	0	0
$861$	−4853.84	−0.192124
$862$	0	0
$863$	−9556.27	−0.376940	−0.188470	−	0.982079i	$-0.560353\pi$
$863$	−9556.27	−0.376940	−0.188470	+	0.982079i	$0.560353\pi$
$864$	0	0
$865$	647.916	0.0254680
$866$	0	0
$867$	−8423.60	−0.329966
$868$	0	0
$869$	−26720.9	−1.04309
$870$	0	0
$871$	−4104.78	−0.159685
$872$	0	0
$873$	−44322.0	−1.71830
$874$	0	0
$875$	−5369.67	−0.207461
$876$	0	0
$877$	18138.4	0.698394	0.349197	−	0.937049i	$-0.386454\pi$
$877$	18138.4	0.698394	0.349197	+	0.937049i	$0.386454\pi$
$878$	0	0
$879$	3376.30	0.129556
$880$	0	0
$881$	38547.5	1.47412	0.737058	−	0.675829i	$-0.236214\pi$
$881$	38547.5	1.47412	0.737058	+	0.675829i	$0.236214\pi$
$882$	0	0
$883$	−12228.3	−0.466040	−0.233020	−	0.972472i	$-0.574861\pi$
$883$	−12228.3	−0.466040	−0.233020	+	0.972472i	$0.574861\pi$
$884$	0	0
$885$	−388.710	−0.0147642
$886$	0	0
$887$	35387.4	1.33956	0.669782	−	0.742558i	$-0.266387\pi$
$887$	35387.4	1.33956	0.669782	+	0.742558i	$0.266387\pi$
$888$	0	0
$889$	−48301.7	−1.82226
$890$	0	0
$891$	−34347.3	−1.29144
$892$	0	0
$893$	−8322.77	−0.311882
$894$	0	0
$895$	2810.88	0.104980
$896$	0	0
$897$	7327.56	0.272754
$898$	0	0
$899$	−5793.96	−0.214949
$900$	0	0
$901$	−38588.2	−1.42681
$902$	0	0
$903$	−14539.3	−0.535810
$904$	0	0
$905$	528.390	0.0194080
$906$	0	0
$907$	−25552.8	−0.935465	−0.467733	−	0.883870i	$-0.654929\pi$
$907$	−25552.8	−0.935465	−0.467733	+	0.883870i	$0.654929\pi$
$908$	0	0
$909$	17936.9	0.654487
$910$	0	0
$911$	−11127.5	−0.404688	−0.202344	−	0.979314i	$-0.564856\pi$
$911$	−11127.5	−0.404688	−0.202344	+	0.979314i	$0.564856\pi$
$912$	0	0
$913$	4907.51	0.177891
$914$	0	0
$915$	−932.060	−0.0336754
$916$	0	0
$917$	−28444.6	−1.02435
$918$	0	0
$919$	−34243.6	−1.22915	−0.614577	−	0.788857i	$-0.710673\pi$
$919$	−34243.6	−1.22915	−0.614577	+	0.788857i	$0.710673\pi$
$920$	0	0
$921$	6531.93	0.233697
$922$	0	0
$923$	−7065.22	−0.251955
$924$	0	0
$925$	50205.7	1.78460
$926$	0	0
$927$	4412.25	0.156329
$928$	0	0
$929$	5315.50	0.187724	0.0938621	−	0.995585i	$-0.470079\pi$
$929$	5315.50	0.187724	0.0938621	+	0.995585i	$0.470079\pi$
$930$	0	0
$931$	−22658.4	−0.797635
$932$	0	0
$933$	1493.52	0.0524071
$934$	0	0
$935$	5028.54	0.175883
$936$	0	0
$937$	−31475.4	−1.09739	−0.548697	−	0.836021i	$-0.684876\pi$
$937$	−31475.4	−1.09739	−0.548697	+	0.836021i	$0.684876\pi$
$938$	0	0
$939$	−16034.1	−0.557245
$940$	0	0
$941$	16014.6	0.554794	0.277397	−	0.960755i	$-0.410528\pi$
$941$	16014.6	0.554794	0.277397	+	0.960755i	$0.410528\pi$
$942$	0	0
$943$	20784.6	0.717752
$944$	0	0
$945$	1671.76	0.0575474
$946$	0	0
$947$	28388.8	0.974140	0.487070	−	0.873363i	$-0.338066\pi$
$947$	28388.8	0.974140	0.487070	+	0.873363i	$0.338066\pi$
$948$	0	0
$949$	−11194.3	−0.382911
$950$	0	0
$951$	12824.6	0.437293
$952$	0	0
$953$	32633.4	1.10924	0.554618	−	0.832105i	$-0.312865\pi$
$953$	32633.4	1.10924	0.554618	+	0.832105i	$0.312865\pi$
$954$	0	0
$955$	4000.69	0.135560
$956$	0	0
$957$	2708.59	0.0914905
$958$	0	0
$959$	−44327.2	−1.49260
$960$	0	0
$961$	10125.7	0.339893
$962$	0	0
$963$	−44939.8	−1.50380
$964$	0	0
$965$	1575.06	0.0525421
$966$	0	0
$967$	36126.7	1.20140	0.600702	−	0.799473i	$-0.294888\pi$
$967$	36126.7	1.20140	0.600702	+	0.799473i	$0.294888\pi$
$968$	0	0
$969$	8611.20	0.285481
$970$	0	0
$971$	−45941.5	−1.51836	−0.759182	−	0.650878i	$-0.774401\pi$
$971$	−45941.5	−1.51836	−0.759182	+	0.650878i	$0.774401\pi$
$972$	0	0
$973$	−16601.3	−0.546982
$974$	0	0
$975$	5186.83	0.170371
$976$	0	0
$977$	−37251.1	−1.21982	−0.609911	−	0.792470i	$-0.708795\pi$
$977$	−37251.1	−1.21982	−0.609911	+	0.792470i	$0.708795\pi$
$978$	0	0
$979$	−64403.9	−2.10251
$980$	0	0
$981$	24800.0	0.807137
$982$	0	0
$983$	−52551.7	−1.70513	−0.852563	−	0.522624i	$-0.824953\pi$
$983$	−52551.7	−1.70513	−0.852563	+	0.522624i	$0.824953\pi$
$984$	0	0
$985$	−901.020	−0.0291461
$986$	0	0
$987$	6105.28	0.196893
$988$	0	0
$989$	62258.5	2.00172
$990$	0	0
$991$	10063.1	0.322569	0.161284	−	0.986908i	$-0.448436\pi$
$991$	10063.1	0.322569	0.161284	+	0.986908i	$0.448436\pi$
$992$	0	0
$993$	−6713.50	−0.214548
$994$	0	0
$995$	−1300.48	−0.0414352
$996$	0	0
$997$	−2111.64	−0.0670776	−0.0335388	−	0.999437i	$-0.510678\pi$
$997$	−2111.64	−0.0670776	−0.0335388	+	0.999437i	$0.510678\pi$
$998$	0	0
$999$	−31339.3	−0.992525

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1856.4.a.bk.1.5		12
4.3	odd	2	inner	1856.4.a.bk.1.8		12
8.3	odd	2		928.4.a.i.1.5	✓	12
8.5	even	2		928.4.a.i.1.8	yes	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
928.4.a.i.1.5	✓	12	8.3	odd	2
928.4.a.i.1.8	yes	12	8.5	even	2
1856.4.a.bk.1.5		12	1.1	even	1	trivial
1856.4.a.bk.1.8		12	4.3	odd	2	inner

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)

\(f(q)\)	\(=\)	\(q-1.50033 q^{3} -0.787262 q^{5} -27.3506 q^{7} -24.7490 q^{9} +O(q^{10})\)	\(q-1.50033 q^{3} -0.787262 q^{5} -27.3506 q^{7} -24.7490 q^{9} -62.2529 q^{11} +27.7949 q^{13} +1.18115 q^{15} +102.604 q^{17} -55.9390 q^{19} +41.0348 q^{21} -175.715 q^{23} -124.380 q^{25} +77.6404 q^{27} +29.0000 q^{29} -199.792 q^{31} +93.3997 q^{33} +21.5321 q^{35} -403.647 q^{37} -41.7014 q^{39} -118.286 q^{41} -354.315 q^{43} +19.4840 q^{45} +148.783 q^{47} +405.055 q^{49} -153.939 q^{51} -376.090 q^{53} +49.0093 q^{55} +83.9268 q^{57} -329.095 q^{59} -789.113 q^{61} +676.900 q^{63} -21.8818 q^{65} -147.681 q^{67} +263.630 q^{69} -254.191 q^{71} -402.746 q^{73} +186.611 q^{75} +1702.65 q^{77} +429.232 q^{79} +551.738 q^{81} -78.8317 q^{83} -80.7759 q^{85} -43.5095 q^{87} +1034.55 q^{89} -760.207 q^{91} +299.753 q^{93} +44.0386 q^{95} +1790.86 q^{97} +1540.70 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 10 q^{5} + 166 q^{9}+O(q^{10}) \) 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

Related objects

Downloads

Learn more