LMFDB - Embedded newform 1344.4.a.bh.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1344,4,Mod(1,1344)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1344, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("1344.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1344 = 2^{6} \cdot 3 \cdot 7 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1344.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:	$79.2985670477$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{37}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 9 $ x^2 - x - 9
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 672)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-2.54138$ of defining polynomial
Character	$\chi$	$=$	1344.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.00000 q^{3} -10.1655 q^{5} +7.00000 q^{7} +9.00000 q^{9} +O(q^{10})$	$q-3.00000 q^{3} -10.1655 q^{5} +7.00000 q^{7} +9.00000 q^{9} +36.1655 q^{11} +74.6621 q^{13} +30.4966 q^{15} -91.1587 q^{17} +104.331 q^{19} -21.0000 q^{21} +36.8276 q^{23} -21.6621 q^{25} -27.0000 q^{27} +262.993 q^{29} -310.979 q^{31} -108.497 q^{33} -71.1587 q^{35} -285.311 q^{37} -223.986 q^{39} +62.4966 q^{41} -386.648 q^{43} -91.4897 q^{45} +430.290 q^{47} +49.0000 q^{49} +273.476 q^{51} -111.959 q^{53} -367.642 q^{55} -312.993 q^{57} +479.642 q^{59} +602.993 q^{61} +63.0000 q^{63} -758.979 q^{65} -1020.97 q^{67} -110.483 q^{69} +284.442 q^{71} -566.331 q^{73} +64.9863 q^{75} +253.159 q^{77} -94.2900 q^{79} +81.0000 q^{81} -626.621 q^{83} +926.676 q^{85} -788.979 q^{87} +1532.40 q^{89} +522.635 q^{91} +932.938 q^{93} -1060.58 q^{95} +718.304 q^{97} +325.490 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 6 q^{3} + 4 q^{5} + 14 q^{7} + 18 q^{9}+O(q^{10}) $ 2 * q - 6 * q^3 + 4 * q^5 + 14 * q^7 + 18 * q^9	$ 2 q - 6 q^{3} + 4 q^{5} + 14 q^{7} + 18 q^{9} + 48 q^{11} + 52 q^{13} - 12 q^{15} - 12 q^{17} + 160 q^{19} - 42 q^{21} - 48 q^{23} + 54 q^{25} - 54 q^{27} + 380 q^{29} - 184 q^{31} - 144 q^{33} + 28 q^{35} - 84 q^{37} - 156 q^{39} + 52 q^{41} - 384 q^{43} + 36 q^{45} - 64 q^{47} + 98 q^{49} + 36 q^{51} + 652 q^{53} - 200 q^{55} - 480 q^{57} + 424 q^{59} + 1060 q^{61} + 126 q^{63} - 1080 q^{65} - 1312 q^{67} + 144 q^{69} - 672 q^{71} - 1084 q^{73} - 162 q^{75} + 336 q^{77} + 736 q^{79} + 162 q^{81} - 280 q^{83} + 2048 q^{85} - 1140 q^{87} + 948 q^{89} + 364 q^{91} + 552 q^{93} - 272 q^{95} + 804 q^{97} + 432 q^{99}+O(q^{100}) $ 2 * q - 6 * q^3 + 4 * q^5 + 14 * q^7 + 18 * q^9 + 48 * q^11 + 52 * q^13 - 12 * q^15 - 12 * q^17 + 160 * q^19 - 42 * q^21 - 48 * q^23 + 54 * q^25 - 54 * q^27 + 380 * q^29 - 184 * q^31 - 144 * q^33 + 28 * q^35 - 84 * q^37 - 156 * q^39 + 52 * q^41 - 384 * q^43 + 36 * q^45 - 64 * q^47 + 98 * q^49 + 36 * q^51 + 652 * q^53 - 200 * q^55 - 480 * q^57 + 424 * q^59 + 1060 * q^61 + 126 * q^63 - 1080 * q^65 - 1312 * q^67 + 144 * q^69 - 672 * q^71 - 1084 * q^73 - 162 * q^75 + 336 * q^77 + 736 * q^79 + 162 * q^81 - 280 * q^83 + 2048 * q^85 - 1140 * q^87 + 948 * q^89 + 364 * q^91 + 552 * q^93 - 272 * q^95 + 804 * q^97 + 432 * 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
$3$	−3.00000	−0.577350
$3$	−3.00000	−0.577350
$4$	0	0
$5$	−10.1655	−0.909232	−0.454616	−	0.890687i	$-0.650224\pi$
$5$	−10.1655	−0.909232	−0.454616	+	0.890687i	$0.650224\pi$
$6$	0	0
$7$	7.00000	0.377964
$7$	7.00000	0.377964
$8$	0	0
$9$	9.00000	0.333333
$10$	0	0
$11$	36.1655	0.991301	0.495651	−	0.868522i	$-0.334930\pi$
$11$	36.1655	0.991301	0.495651	+	0.868522i	$0.334930\pi$
$12$	0	0
$13$	74.6621	1.59289	0.796444	−	0.604712i	$-0.206712\pi$
$13$	74.6621	1.59289	0.796444	+	0.604712i	$0.206712\pi$
$14$	0	0
$15$	30.4966	0.524945
$16$	0	0
$17$	−91.1587	−1.30054	−0.650271	−	0.759702i	$-0.725345\pi$
$17$	−91.1587	−1.30054	−0.650271	+	0.759702i	$0.725345\pi$
$18$	0	0
$19$	104.331	1.25975	0.629873	−	0.776698i	$-0.283107\pi$
$19$	104.331	1.25975	0.629873	+	0.776698i	$0.283107\pi$
$20$	0	0
$21$	−21.0000	−0.218218
$22$	0	0
$23$	36.8276	0.333874	0.166937	−	0.985968i	$-0.446612\pi$
$23$	36.8276	0.333874	0.166937	+	0.985968i	$0.446612\pi$
$24$	0	0
$25$	−21.6621	−0.173297
$26$	0	0
$27$	−27.0000	−0.192450
$28$	0	0
$29$	262.993	1.68402	0.842010	−	0.539461i	$-0.181372\pi$
$29$	262.993	1.68402	0.842010	+	0.539461i	$0.181372\pi$
$30$	0	0
$31$	−310.979	−1.80173	−0.900864	−	0.434102i	$-0.857066\pi$
$31$	−310.979	−1.80173	−0.900864	+	0.434102i	$0.857066\pi$
$32$	0	0
$33$	−108.497	−0.572328
$34$	0	0
$35$	−71.1587	−0.343657
$36$	0	0
$37$	−285.311	−1.26770	−0.633848	−	0.773458i	$-0.718526\pi$
$37$	−285.311	−1.26770	−0.633848	+	0.773458i	$0.718526\pi$
$38$	0	0
$39$	−223.986	−0.919654
$40$	0	0
$41$	62.4966	0.238057	0.119028	−	0.992891i	$-0.462022\pi$
$41$	62.4966	0.238057	0.119028	+	0.992891i	$0.462022\pi$
$42$	0	0
$43$	−386.648	−1.37124	−0.685620	−	0.727960i	$-0.740469\pi$
$43$	−386.648	−1.37124	−0.685620	+	0.727960i	$0.740469\pi$
$44$	0	0
$45$	−91.4897	−0.303077
$46$	0	0
$47$	430.290	1.33541	0.667705	−	0.744426i	$-0.267277\pi$
$47$	430.290	1.33541	0.667705	+	0.744426i	$0.267277\pi$
$48$	0	0
$49$	49.0000	0.142857
$50$	0	0
$51$	273.476	0.750869
$52$	0	0
$53$	−111.959	−0.290165	−0.145082	−	0.989420i	$-0.546345\pi$
$53$	−111.959	−0.290165	−0.145082	+	0.989420i	$0.546345\pi$
$54$	0	0
$55$	−367.642	−0.901323
$56$	0	0
$57$	−312.993	−0.727315
$58$	0	0
$59$	479.642	1.05837	0.529187	−	0.848506i	$-0.322497\pi$
$59$	479.642	1.05837	0.529187	+	0.848506i	$0.322497\pi$
$60$	0	0
$61$	602.993	1.26566	0.632831	−	0.774290i	$-0.281893\pi$
$61$	602.993	1.26566	0.632831	+	0.774290i	$0.281893\pi$
$62$	0	0
$63$	63.0000	0.125988
$64$	0	0
$65$	−758.979	−1.44830
$66$	0	0
$67$	−1020.97	−1.86165	−0.930827	−	0.365460i	$-0.880912\pi$
$67$	−1020.97	−1.86165	−0.930827	+	0.365460i	$0.880912\pi$
$68$	0	0
$69$	−110.483	−0.192762
$70$	0	0
$71$	284.442	0.475451	0.237726	−	0.971332i	$-0.423598\pi$
$71$	284.442	0.475451	0.237726	+	0.971332i	$0.423598\pi$
$72$	0	0
$73$	−566.331	−0.908000	−0.454000	−	0.891002i	$-0.650003\pi$
$73$	−566.331	−0.908000	−0.454000	+	0.891002i	$0.650003\pi$
$74$	0	0
$75$	64.9863	0.100053
$76$	0	0
$77$	253.159	0.374677
$78$	0	0
$79$	−94.2900	−0.134284	−0.0671421	−	0.997743i	$-0.521388\pi$
$79$	−94.2900	−0.134284	−0.0671421	+	0.997743i	$0.521388\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	0	0
$83$	−626.621	−0.828682	−0.414341	−	0.910122i	$-0.635988\pi$
$83$	−626.621	−0.828682	−0.414341	+	0.910122i	$0.635988\pi$
$84$	0	0
$85$	926.676	1.18250
$86$	0	0
$87$	−788.979	−0.972270
$88$	0	0
$89$	1532.40	1.82510	0.912551	−	0.408962i	$-0.134109\pi$
$89$	1532.40	1.82510	0.912551	+	0.408962i	$0.134109\pi$
$90$	0	0
$91$	522.635	0.602055
$92$	0	0
$93$	932.938	1.04023
$94$	0	0
$95$	−1060.58	−1.14540
$96$	0	0
$97$	718.304	0.751883	0.375942	−	0.926643i	$-0.377319\pi$
$97$	718.304	0.751883	0.375942	+	0.926643i	$0.377319\pi$
$98$	0	0
$99$	325.490	0.330434
$100$	0	0
$101$	1420.12	1.39909	0.699543	−	0.714591i	$-0.253387\pi$
$101$	1420.12	1.39909	0.699543	+	0.714591i	$0.253387\pi$
$102$	0	0
$103$	1106.24	1.05826	0.529129	−	0.848541i	$-0.322519\pi$
$103$	1106.24	1.05826	0.529129	+	0.848541i	$0.322519\pi$
$104$	0	0
$105$	213.476	0.198411
$106$	0	0
$107$	−157.213	−0.142041	−0.0710206	−	0.997475i	$-0.522626\pi$
$107$	−157.213	−0.142041	−0.0710206	+	0.997475i	$0.522626\pi$
$108$	0	0
$109$	2153.89	1.89271	0.946355	−	0.323130i	$-0.104735\pi$
$109$	2153.89	1.89271	0.946355	+	0.323130i	$0.104735\pi$
$110$	0	0
$111$	855.932	0.731905
$112$	0	0
$113$	−1253.23	−1.04331	−0.521654	−	0.853157i	$-0.674685\pi$
$113$	−1253.23	−1.04331	−0.521654	+	0.853157i	$0.674685\pi$
$114$	0	0
$115$	−374.372	−0.303569
$116$	0	0
$117$	671.959	0.530963
$118$	0	0
$119$	−638.111	−0.491559
$120$	0	0
$121$	−23.0548	−0.0173214
$122$	0	0
$123$	−187.490	−0.137442
$124$	0	0
$125$	1490.90	1.06680
$126$	0	0
$127$	1381.02	0.964927	0.482464	−	0.875916i	$-0.339742\pi$
$127$	1381.02	0.964927	0.482464	+	0.875916i	$0.339742\pi$
$128$	0	0
$129$	1159.95	0.791686
$130$	0	0
$131$	−413.269	−0.275630	−0.137815	−	0.990458i	$-0.544008\pi$
$131$	−413.269	−0.275630	−0.137815	+	0.990458i	$0.544008\pi$
$132$	0	0
$133$	730.317	0.476139
$134$	0	0
$135$	274.469	0.174982
$136$	0	0
$137$	147.655	0.0920806	0.0460403	−	0.998940i	$-0.485340\pi$
$137$	147.655	0.0920806	0.0460403	+	0.998940i	$0.485340\pi$
$138$	0	0
$139$	−1949.82	−1.18980	−0.594898	−	0.803801i	$-0.702808\pi$
$139$	−1949.82	−1.18980	−0.594898	+	0.803801i	$0.702808\pi$
$140$	0	0
$141$	−1290.87	−0.770999
$142$	0	0
$143$	2700.19	1.57903
$144$	0	0
$145$	−2673.46	−1.53117
$146$	0	0
$147$	−147.000	−0.0824786
$148$	0	0
$149$	1004.07	0.552057	0.276029	−	0.961149i	$-0.410982\pi$
$149$	1004.07	0.552057	0.276029	+	0.961149i	$0.410982\pi$
$150$	0	0
$151$	−1285.79	−0.692957	−0.346478	−	0.938058i	$-0.612623\pi$
$151$	−1285.79	−0.692957	−0.346478	+	0.938058i	$0.612623\pi$
$152$	0	0
$153$	−820.428	−0.433514
$154$	0	0
$155$	3161.27	1.63819
$156$	0	0
$157$	408.345	0.207576	0.103788	−	0.994599i	$-0.466904\pi$
$157$	408.345	0.207576	0.103788	+	0.994599i	$0.466904\pi$
$158$	0	0
$159$	335.877	0.167527
$160$	0	0
$161$	257.793	0.126192
$162$	0	0
$163$	1816.83	0.873037	0.436518	−	0.899695i	$-0.356211\pi$
$163$	1816.83	0.873037	0.436518	+	0.899695i	$0.356211\pi$
$164$	0	0
$165$	1102.92	0.520379
$166$	0	0
$167$	−3522.15	−1.63205	−0.816025	−	0.578016i	$-0.803827\pi$
$167$	−3522.15	−1.63205	−0.816025	+	0.578016i	$0.803827\pi$
$168$	0	0
$169$	3377.43	1.53729
$170$	0	0
$171$	938.979	0.419916
$172$	0	0
$173$	2395.57	1.05279	0.526393	−	0.850241i	$-0.323544\pi$
$173$	2395.57	1.05279	0.526393	+	0.850241i	$0.323544\pi$
$174$	0	0
$175$	−151.635	−0.0655000
$176$	0	0
$177$	−1438.92	−0.611052
$178$	0	0
$179$	1505.08	0.628462	0.314231	−	0.949347i	$-0.398253\pi$
$179$	1505.08	0.628462	0.314231	+	0.949347i	$0.398253\pi$
$180$	0	0
$181$	−3034.42	−1.24611	−0.623056	−	0.782177i	$-0.714109\pi$
$181$	−3034.42	−1.24611	−0.623056	+	0.782177i	$0.714109\pi$
$182$	0	0
$183$	−1808.98	−0.730730
$184$	0	0
$185$	2900.33	1.15263
$186$	0	0
$187$	−3296.80	−1.28923
$188$	0	0
$189$	−189.000	−0.0727393
$190$	0	0
$191$	−641.462	−0.243008	−0.121504	−	0.992591i	$-0.538772\pi$
$191$	−641.462	−0.243008	−0.121504	+	0.992591i	$0.538772\pi$
$192$	0	0
$193$	2892.62	1.07884	0.539418	−	0.842038i	$-0.318644\pi$
$193$	2892.62	1.07884	0.539418	+	0.842038i	$0.318644\pi$
$194$	0	0
$195$	2276.94	0.836179
$196$	0	0
$197$	3337.86	1.20717	0.603586	−	0.797298i	$-0.293738\pi$
$197$	3337.86	1.20717	0.603586	+	0.797298i	$0.293738\pi$
$198$	0	0
$199$	1401.82	0.499359	0.249680	−	0.968328i	$-0.419675\pi$
$199$	1401.82	0.499359	0.249680	+	0.968328i	$0.419675\pi$
$200$	0	0
$201$	3062.90	1.07483
$202$	0	0
$203$	1840.95	0.636500
$204$	0	0
$205$	−635.311	−0.216449
$206$	0	0
$207$	331.449	0.111291
$208$	0	0
$209$	3773.19	1.24879
$210$	0	0
$211$	−4967.62	−1.62078	−0.810391	−	0.585890i	$-0.800745\pi$
$211$	−4967.62	−1.62078	−0.810391	+	0.585890i	$0.800745\pi$
$212$	0	0
$213$	−853.325	−0.274502
$214$	0	0
$215$	3930.48	1.24678
$216$	0	0
$217$	−2176.86	−0.680989
$218$	0	0
$219$	1698.99	0.524234
$220$	0	0
$221$	−6806.10	−2.07162
$222$	0	0
$223$	242.758	0.0728981	0.0364491	−	0.999336i	$-0.488395\pi$
$223$	242.758	0.0728981	0.0364491	+	0.999336i	$0.488395\pi$
$224$	0	0
$225$	−194.959	−0.0577656
$226$	0	0
$227$	5873.30	1.71729	0.858644	−	0.512572i	$-0.171307\pi$
$227$	5873.30	1.71729	0.858644	+	0.512572i	$0.171307\pi$
$228$	0	0
$229$	3979.27	1.14829	0.574143	−	0.818755i	$-0.305335\pi$
$229$	3979.27	1.14829	0.574143	+	0.818755i	$0.305335\pi$
$230$	0	0
$231$	−759.476	−0.216320
$232$	0	0
$233$	601.724	0.169186	0.0845928	−	0.996416i	$-0.473041\pi$
$233$	601.724	0.169186	0.0845928	+	0.996416i	$0.473041\pi$
$234$	0	0
$235$	−4374.12	−1.21420
$236$	0	0
$237$	282.870	0.0775290
$238$	0	0
$239$	−5140.22	−1.39119	−0.695593	−	0.718436i	$-0.744858\pi$
$239$	−5140.22	−1.39119	−0.695593	+	0.718436i	$0.744858\pi$
$240$	0	0
$241$	2036.43	0.544306	0.272153	−	0.962254i	$-0.412264\pi$
$241$	2036.43	0.544306	0.272153	+	0.962254i	$0.412264\pi$
$242$	0	0
$243$	−243.000	−0.0641500
$244$	0	0
$245$	−498.111	−0.129890
$246$	0	0
$247$	7789.58	2.00663
$248$	0	0
$249$	1879.86	0.478440
$250$	0	0
$251$	6583.70	1.65561	0.827807	−	0.561013i	$-0.189588\pi$
$251$	6583.70	1.65561	0.827807	+	0.561013i	$0.189588\pi$
$252$	0	0
$253$	1331.89	0.330969
$254$	0	0
$255$	−2780.03	−0.682714
$256$	0	0
$257$	1277.78	0.310139	0.155069	−	0.987904i	$-0.450440\pi$
$257$	1277.78	0.310139	0.155069	+	0.987904i	$0.450440\pi$
$258$	0	0
$259$	−1997.17	−0.479144
$260$	0	0
$261$	2366.94	0.561340
$262$	0	0
$263$	4266.18	1.00024	0.500121	−	0.865955i	$-0.333289\pi$
$263$	4266.18	1.00024	0.500121	+	0.865955i	$0.333289\pi$
$264$	0	0
$265$	1138.12	0.263827
$266$	0	0
$267$	−4597.20	−1.05372
$268$	0	0
$269$	317.560	0.0719777	0.0359889	−	0.999352i	$-0.488542\pi$
$269$	317.560	0.0719777	0.0359889	+	0.999352i	$0.488542\pi$
$270$	0	0
$271$	896.687	0.200996	0.100498	−	0.994937i	$-0.467956\pi$
$271$	896.687	0.200996	0.100498	+	0.994937i	$0.467956\pi$
$272$	0	0
$273$	−1567.90	−0.347597
$274$	0	0
$275$	−783.421	−0.171789
$276$	0	0
$277$	−1919.24	−0.416304	−0.208152	−	0.978097i	$-0.566745\pi$
$277$	−1919.24	−0.416304	−0.208152	+	0.978097i	$0.566745\pi$
$278$	0	0
$279$	−2798.82	−0.600576
$280$	0	0
$281$	−744.541	−0.158063	−0.0790313	−	0.996872i	$-0.525183\pi$
$281$	−744.541	−0.158063	−0.0790313	+	0.996872i	$0.525183\pi$
$282$	0	0
$283$	2447.23	0.514038	0.257019	−	0.966406i	$-0.417260\pi$
$283$	2447.23	0.514038	0.257019	+	0.966406i	$0.417260\pi$
$284$	0	0
$285$	3181.74	0.661298
$286$	0	0
$287$	437.476	0.0899770
$288$	0	0
$289$	3396.90	0.691411
$290$	0	0
$291$	−2154.91	−0.434100
$292$	0	0
$293$	−1825.72	−0.364027	−0.182014	−	0.983296i	$-0.558261\pi$
$293$	−1825.72	−0.364027	−0.182014	+	0.983296i	$0.558261\pi$
$294$	0	0
$295$	−4875.81	−0.962307
$296$	0	0
$297$	−976.469	−0.190776
$298$	0	0
$299$	2749.63	0.531823
$300$	0	0
$301$	−2706.54	−0.518280
$302$	0	0
$303$	−4260.37	−0.807763
$304$	0	0
$305$	−6129.74	−1.15078
$306$	0	0
$307$	−905.267	−0.168294	−0.0841471	−	0.996453i	$-0.526817\pi$
$307$	−905.267	−0.168294	−0.0841471	+	0.996453i	$0.526817\pi$
$308$	0	0
$309$	−3318.71	−0.610986
$310$	0	0
$311$	1943.29	0.354320	0.177160	−	0.984182i	$-0.443309\pi$
$311$	1943.29	0.354320	0.177160	+	0.984182i	$0.443309\pi$
$312$	0	0
$313$	8197.97	1.48044	0.740219	−	0.672366i	$-0.234722\pi$
$313$	8197.97	1.48044	0.740219	+	0.672366i	$0.234722\pi$
$314$	0	0
$315$	−640.428	−0.114552
$316$	0	0
$317$	−3657.78	−0.648080	−0.324040	−	0.946043i	$-0.605041\pi$
$317$	−3657.78	−0.648080	−0.324040	+	0.946043i	$0.605041\pi$
$318$	0	0
$319$	9511.29	1.66937
$320$	0	0
$321$	471.640	0.0820075
$322$	0	0
$323$	−9510.68	−1.63835
$324$	0	0
$325$	−1617.34	−0.276042
$326$	0	0
$327$	−6461.67	−1.09276
$328$	0	0
$329$	3012.03	0.504737
$330$	0	0
$331$	−123.087	−0.0204395	−0.0102197	−	0.999948i	$-0.503253\pi$
$331$	−123.087	−0.0204395	−0.0102197	+	0.999948i	$0.503253\pi$
$332$	0	0
$333$	−2567.79	−0.422565
$334$	0	0
$335$	10378.7	1.69268
$336$	0	0
$337$	6104.63	0.986766	0.493383	−	0.869812i	$-0.335760\pi$
$337$	6104.63	0.986766	0.493383	+	0.869812i	$0.335760\pi$
$338$	0	0
$339$	3759.68	0.602354
$340$	0	0
$341$	−11246.7	−1.78606
$342$	0	0
$343$	343.000	0.0539949
$344$	0	0
$345$	1123.12	0.175265
$346$	0	0
$347$	−3875.62	−0.599579	−0.299790	−	0.954005i	$-0.596917\pi$
$347$	−3875.62	−0.599579	−0.299790	+	0.954005i	$0.596917\pi$
$348$	0	0
$349$	2621.15	0.402025	0.201013	−	0.979589i	$-0.435577\pi$
$349$	2621.15	0.402025	0.201013	+	0.979589i	$0.435577\pi$
$350$	0	0
$351$	−2015.88	−0.306551
$352$	0	0
$353$	−168.784	−0.0254490	−0.0127245	−	0.999919i	$-0.504050\pi$
$353$	−168.784	−0.0254490	−0.0127245	+	0.999919i	$0.504050\pi$
$354$	0	0
$355$	−2891.50	−0.432295
$356$	0	0
$357$	1914.33	0.283802
$358$	0	0
$359$	−2644.83	−0.388826	−0.194413	−	0.980920i	$-0.562280\pi$
$359$	−2644.83	−0.388826	−0.194413	+	0.980920i	$0.562280\pi$
$360$	0	0
$361$	4025.97	0.586961
$362$	0	0
$363$	69.1644	0.0100005
$364$	0	0
$365$	5757.05	0.825583
$366$	0	0
$367$	−4909.52	−0.698296	−0.349148	−	0.937068i	$-0.613529\pi$
$367$	−4909.52	−0.698296	−0.349148	+	0.937068i	$0.613529\pi$
$368$	0	0
$369$	562.469	0.0793522
$370$	0	0
$371$	−783.712	−0.109672
$372$	0	0
$373$	5574.80	0.773867	0.386933	−	0.922108i	$-0.373534\pi$
$373$	5574.80	0.773867	0.386933	+	0.922108i	$0.373534\pi$
$374$	0	0
$375$	−4472.69	−0.615917
$376$	0	0
$377$	19635.6	2.68246
$378$	0	0
$379$	8099.04	1.09768	0.548839	−	0.835928i	$-0.315070\pi$
$379$	8099.04	1.09768	0.548839	+	0.835928i	$0.315070\pi$
$380$	0	0
$381$	−4143.06	−0.557101
$382$	0	0
$383$	−4779.21	−0.637614	−0.318807	−	0.947820i	$-0.603282\pi$
$383$	−4779.21	−0.637614	−0.318807	+	0.947820i	$0.603282\pi$
$384$	0	0
$385$	−2573.49	−0.340668
$386$	0	0
$387$	−3479.84	−0.457080
$388$	0	0
$389$	11688.8	1.52351	0.761757	−	0.647863i	$-0.224337\pi$
$389$	11688.8	1.52351	0.761757	+	0.647863i	$0.224337\pi$
$390$	0	0
$391$	−3357.16	−0.434217
$392$	0	0
$393$	1239.81	0.159135
$394$	0	0
$395$	958.507	0.122095
$396$	0	0
$397$	−2563.24	−0.324044	−0.162022	−	0.986787i	$-0.551802\pi$
$397$	−2563.24	−0.324044	−0.162022	+	0.986787i	$0.551802\pi$
$398$	0	0
$399$	−2190.95	−0.274899
$400$	0	0
$401$	−4139.96	−0.515560	−0.257780	−	0.966204i	$-0.582991\pi$
$401$	−4139.96	−0.515560	−0.257780	+	0.966204i	$0.582991\pi$
$402$	0	0
$403$	−23218.4	−2.86995
$404$	0	0
$405$	−823.408	−0.101026
$406$	0	0
$407$	−10318.4	−1.25667
$408$	0	0
$409$	−2015.71	−0.243693	−0.121847	−	0.992549i	$-0.538882\pi$
$409$	−2015.71	−0.243693	−0.121847	+	0.992549i	$0.538882\pi$
$410$	0	0
$411$	−442.966	−0.0531627
$412$	0	0
$413$	3357.49	0.400027
$414$	0	0
$415$	6369.93	0.753464
$416$	0	0
$417$	5849.47	0.686929
$418$	0	0
$419$	14830.8	1.72919	0.864594	−	0.502471i	$-0.167575\pi$
$419$	14830.8	1.72919	0.864594	+	0.502471i	$0.167575\pi$
$420$	0	0
$421$	9103.21	1.05383	0.526916	−	0.849917i	$-0.323348\pi$
$421$	9103.21	1.05383	0.526916	+	0.849917i	$0.323348\pi$
$422$	0	0
$423$	3872.61	0.445136
$424$	0	0
$425$	1974.69	0.225380
$426$	0	0
$427$	4220.95	0.478375
$428$	0	0
$429$	−8100.58	−0.911655
$430$	0	0
$431$	14680.9	1.64072	0.820362	−	0.571845i	$-0.193772\pi$
$431$	14680.9	1.64072	0.820362	+	0.571845i	$0.193772\pi$
$432$	0	0
$433$	−5929.64	−0.658107	−0.329053	−	0.944311i	$-0.606730\pi$
$433$	−5929.64	−0.658107	−0.329053	+	0.944311i	$0.606730\pi$
$434$	0	0
$435$	8020.39	0.884019
$436$	0	0
$437$	3842.26	0.420596
$438$	0	0
$439$	16070.6	1.74717	0.873587	−	0.486669i	$-0.161788\pi$
$439$	16070.6	1.74717	0.873587	+	0.486669i	$0.161788\pi$
$440$	0	0
$441$	441.000	0.0476190
$442$	0	0
$443$	−610.567	−0.0654829	−0.0327415	−	0.999464i	$-0.510424\pi$
$443$	−610.567	−0.0654829	−0.0327415	+	0.999464i	$0.510424\pi$
$444$	0	0
$445$	−15577.7	−1.65944
$446$	0	0
$447$	−3012.21	−0.318730
$448$	0	0
$449$	1092.21	0.114798	0.0573991	−	0.998351i	$-0.481719\pi$
$449$	1092.21	0.114798	0.0573991	+	0.998351i	$0.481719\pi$
$450$	0	0
$451$	2260.22	0.235986
$452$	0	0
$453$	3857.38	0.400079
$454$	0	0
$455$	−5312.86	−0.547408
$456$	0	0
$457$	672.511	0.0688375	0.0344188	−	0.999407i	$-0.489042\pi$
$457$	672.511	0.0688375	0.0344188	+	0.999407i	$0.489042\pi$
$458$	0	0
$459$	2461.28	0.250290
$460$	0	0
$461$	−10504.7	−1.06128	−0.530642	−	0.847596i	$-0.678049\pi$
$461$	−10504.7	−1.06128	−0.530642	+	0.847596i	$0.678049\pi$
$462$	0	0
$463$	−6034.32	−0.605699	−0.302849	−	0.953038i	$-0.597938\pi$
$463$	−6034.32	−0.605699	−0.302849	+	0.953038i	$0.597938\pi$
$464$	0	0
$465$	−9483.81	−0.945809
$466$	0	0
$467$	16545.7	1.63949	0.819747	−	0.572726i	$-0.194114\pi$
$467$	16545.7	1.63949	0.819747	+	0.572726i	$0.194114\pi$
$468$	0	0
$469$	−7146.76	−0.703639
$470$	0	0
$471$	−1225.03	−0.119844
$472$	0	0
$473$	−13983.3	−1.35931
$474$	0	0
$475$	−2260.03	−0.218310
$476$	0	0
$477$	−1007.63	−0.0967216
$478$	0	0
$479$	−11914.9	−1.13655	−0.568274	−	0.822839i	$-0.692389\pi$
$479$	−11914.9	−1.13655	−0.568274	+	0.822839i	$0.692389\pi$
$480$	0	0
$481$	−21301.9	−2.01930
$482$	0	0
$483$	−773.380	−0.0728572
$484$	0	0
$485$	−7301.93	−0.683637
$486$	0	0
$487$	943.059	0.0877497	0.0438748	−	0.999037i	$-0.486030\pi$
$487$	943.059	0.0877497	0.0438748	+	0.999037i	$0.486030\pi$
$488$	0	0
$489$	−5450.49	−0.504048
$490$	0	0
$491$	7491.83	0.688598	0.344299	−	0.938860i	$-0.388117\pi$
$491$	7491.83	0.688598	0.344299	+	0.938860i	$0.388117\pi$
$492$	0	0
$493$	−23974.1	−2.19014
$494$	0	0
$495$	−3308.77	−0.300441
$496$	0	0
$497$	1991.09	0.179704
$498$	0	0
$499$	−19551.1	−1.75396	−0.876980	−	0.480527i	$-0.840446\pi$
$499$	−19551.1	−1.75396	−0.876980	+	0.480527i	$0.840446\pi$
$500$	0	0
$501$	10566.5	0.942265
$502$	0	0
$503$	−16168.0	−1.43319	−0.716596	−	0.697488i	$-0.754301\pi$
$503$	−16168.0	−1.43319	−0.716596	+	0.697488i	$0.754301\pi$
$504$	0	0
$505$	−14436.3	−1.27209
$506$	0	0
$507$	−10132.3	−0.887556
$508$	0	0
$509$	6152.87	0.535798	0.267899	−	0.963447i	$-0.413671\pi$
$509$	6152.87	0.535798	0.267899	+	0.963447i	$0.413671\pi$
$510$	0	0
$511$	−3964.32	−0.343192
$512$	0	0
$513$	−2816.94	−0.242438
$514$	0	0
$515$	−11245.5	−0.962202
$516$	0	0
$517$	15561.7	1.32379
$518$	0	0
$519$	−7186.72	−0.607826
$520$	0	0
$521$	−21054.8	−1.77049	−0.885245	−	0.465125i	$-0.846009\pi$
$521$	−21054.8	−1.77049	−0.885245	+	0.465125i	$0.846009\pi$
$522$	0	0
$523$	7707.01	0.644367	0.322184	−	0.946677i	$-0.395583\pi$
$523$	7707.01	0.644367	0.322184	+	0.946677i	$0.395583\pi$
$524$	0	0
$525$	454.904	0.0378165
$526$	0	0
$527$	28348.5	2.34322
$528$	0	0
$529$	−10810.7	−0.888528
$530$	0	0
$531$	4316.77	0.352791
$532$	0	0
$533$	4666.13	0.379198
$534$	0	0
$535$	1598.16	0.129148
$536$	0	0
$537$	−4515.23	−0.362843
$538$	0	0
$539$	1772.11	0.141614
$540$	0	0
$541$	20382.6	1.61981	0.809905	−	0.586561i	$-0.199519\pi$
$541$	20382.6	1.61981	0.809905	+	0.586561i	$0.199519\pi$
$542$	0	0
$543$	9103.25	0.719443
$544$	0	0
$545$	−21895.4	−1.72091
$546$	0	0
$547$	−2378.87	−0.185947	−0.0929736	−	0.995669i	$-0.529637\pi$
$547$	−2378.87	−0.185947	−0.0929736	+	0.995669i	$0.529637\pi$
$548$	0	0
$549$	5426.94	0.421887
$550$	0	0
$551$	27438.4	2.12144
$552$	0	0
$553$	−660.030	−0.0507546
$554$	0	0
$555$	−8700.99	−0.665471
$556$	0	0
$557$	−9909.32	−0.753808	−0.376904	−	0.926252i	$-0.623011\pi$
$557$	−9909.32	−0.753808	−0.376904	+	0.926252i	$0.623011\pi$
$558$	0	0
$559$	−28868.0	−2.18423
$560$	0	0
$561$	9890.40	0.744337
$562$	0	0
$563$	11964.2	0.895612	0.447806	−	0.894131i	$-0.352206\pi$
$563$	11964.2	0.895612	0.447806	+	0.894131i	$0.352206\pi$
$564$	0	0
$565$	12739.7	0.948609
$566$	0	0
$567$	567.000	0.0419961
$568$	0	0
$569$	1008.95	0.0743365	0.0371683	−	0.999309i	$-0.488166\pi$
$569$	1008.95	0.0743365	0.0371683	+	0.999309i	$0.488166\pi$
$570$	0	0
$571$	4128.83	0.302603	0.151301	−	0.988488i	$-0.451654\pi$
$571$	4128.83	0.302603	0.151301	+	0.988488i	$0.451654\pi$
$572$	0	0
$573$	1924.39	0.140301
$574$	0	0
$575$	−797.764	−0.0578592
$576$	0	0
$577$	1693.04	0.122153	0.0610763	−	0.998133i	$-0.480547\pi$
$577$	1693.04	0.122153	0.0610763	+	0.998133i	$0.480547\pi$
$578$	0	0
$579$	−8677.86	−0.622866
$580$	0	0
$581$	−4386.35	−0.313212
$582$	0	0
$583$	−4049.05	−0.287641
$584$	0	0
$585$	−6830.82	−0.482768
$586$	0	0
$587$	10416.6	0.732433	0.366217	−	0.930530i	$-0.380653\pi$
$587$	10416.6	0.732433	0.366217	+	0.930530i	$0.380653\pi$
$588$	0	0
$589$	−32444.8	−2.26972
$590$	0	0
$591$	−10013.6	−0.696961
$592$	0	0
$593$	12079.7	0.836513	0.418257	−	0.908329i	$-0.362641\pi$
$593$	12079.7	0.836513	0.418257	+	0.908329i	$0.362641\pi$
$594$	0	0
$595$	6486.73	0.446941
$596$	0	0
$597$	−4205.47	−0.288305
$598$	0	0
$599$	6531.39	0.445518	0.222759	−	0.974874i	$-0.428494\pi$
$599$	6531.39	0.445518	0.222759	+	0.974874i	$0.428494\pi$
$600$	0	0
$601$	−14001.5	−0.950303	−0.475151	−	0.879904i	$-0.657607\pi$
$601$	−14001.5	−0.950303	−0.475151	+	0.879904i	$0.657607\pi$
$602$	0	0
$603$	−9188.69	−0.620551
$604$	0	0
$605$	234.364	0.0157492
$606$	0	0
$607$	−1272.28	−0.0850745	−0.0425372	−	0.999095i	$-0.513544\pi$
$607$	−1272.28	−0.0850745	−0.0425372	+	0.999095i	$0.513544\pi$
$608$	0	0
$609$	−5522.86	−0.367483
$610$	0	0
$611$	32126.4	2.12716
$612$	0	0
$613$	−6823.22	−0.449571	−0.224786	−	0.974408i	$-0.572168\pi$
$613$	−6823.22	−0.449571	−0.224786	+	0.974408i	$0.572168\pi$
$614$	0	0
$615$	1905.93	0.124967
$616$	0	0
$617$	−3787.85	−0.247153	−0.123576	−	0.992335i	$-0.539436\pi$
$617$	−3787.85	−0.247153	−0.123576	+	0.992335i	$0.539436\pi$
$618$	0	0
$619$	−30311.6	−1.96822	−0.984108	−	0.177570i	$-0.943176\pi$
$619$	−30311.6	−1.96822	−0.984108	+	0.177570i	$0.943176\pi$
$620$	0	0
$621$	−994.346	−0.0642540
$622$	0	0
$623$	10726.8	0.689824
$624$	0	0
$625$	−12448.0	−0.796671
$626$	0	0
$627$	−11319.6	−0.720988
$628$	0	0
$629$	26008.5	1.64869
$630$	0	0
$631$	−4357.18	−0.274892	−0.137446	−	0.990509i	$-0.543889\pi$
$631$	−4357.18	−0.274892	−0.137446	+	0.990509i	$0.543889\pi$
$632$	0	0
$633$	14902.8	0.935758
$634$	0	0
$635$	−14038.8	−0.877343
$636$	0	0
$637$	3658.44	0.227555
$638$	0	0
$639$	2559.98	0.158484
$640$	0	0
$641$	−7843.40	−0.483301	−0.241650	−	0.970363i	$-0.577689\pi$
$641$	−7843.40	−0.483301	−0.241650	+	0.970363i	$0.577689\pi$
$642$	0	0
$643$	13730.3	0.842101	0.421050	−	0.907037i	$-0.361662\pi$
$643$	13730.3	0.842101	0.421050	+	0.907037i	$0.361662\pi$
$644$	0	0
$645$	−11791.5	−0.719826
$646$	0	0
$647$	−18378.8	−1.11676	−0.558380	−	0.829585i	$-0.688577\pi$
$647$	−18378.8	−1.11676	−0.558380	+	0.829585i	$0.688577\pi$
$648$	0	0
$649$	17346.5	1.04917
$650$	0	0
$651$	6530.57	0.393169
$652$	0	0
$653$	−29631.0	−1.77573	−0.887864	−	0.460107i	$-0.847811\pi$
$653$	−29631.0	−1.77573	−0.887864	+	0.460107i	$0.847811\pi$
$654$	0	0
$655$	4201.10	0.250612
$656$	0	0
$657$	−5096.98	−0.302667
$658$	0	0
$659$	11311.1	0.668616	0.334308	−	0.942464i	$-0.391497\pi$
$659$	11311.1	0.668616	0.334308	+	0.942464i	$0.391497\pi$
$660$	0	0
$661$	12315.4	0.724682	0.362341	−	0.932046i	$-0.381978\pi$
$661$	12315.4	0.724682	0.362341	+	0.932046i	$0.381978\pi$
$662$	0	0
$663$	20418.3	1.19605
$664$	0	0
$665$	−7424.06	−0.432921
$666$	0	0
$667$	9685.41	0.562250
$668$	0	0
$669$	−728.274	−0.0420877
$670$	0	0
$671$	21807.6	1.25465
$672$	0	0
$673$	−14615.5	−0.837127	−0.418564	−	0.908187i	$-0.637466\pi$
$673$	−14615.5	−0.837127	−0.418564	+	0.908187i	$0.637466\pi$
$674$	0	0
$675$	584.877	0.0333510
$676$	0	0
$677$	12450.9	0.706835	0.353418	−	0.935466i	$-0.385019\pi$
$677$	12450.9	0.706835	0.353418	+	0.935466i	$0.385019\pi$
$678$	0	0
$679$	5028.13	0.284185
$680$	0	0
$681$	−17619.9	−0.991477
$682$	0	0
$683$	−22985.2	−1.28771	−0.643853	−	0.765149i	$-0.722665\pi$
$683$	−22985.2	−1.28771	−0.643853	+	0.765149i	$0.722665\pi$
$684$	0	0
$685$	−1500.99	−0.0837226
$686$	0	0
$687$	−11937.8	−0.662963
$688$	0	0
$689$	−8359.09	−0.462200
$690$	0	0
$691$	6824.13	0.375691	0.187845	−	0.982199i	$-0.439850\pi$
$691$	6824.13	0.375691	0.187845	+	0.982199i	$0.439850\pi$
$692$	0	0
$693$	2278.43	0.124892
$694$	0	0
$695$	19821.0	1.08180
$696$	0	0
$697$	−5697.11	−0.309603
$698$	0	0
$699$	−1805.17	−0.0976793
$700$	0	0
$701$	443.637	0.0239029	0.0119515	−	0.999929i	$-0.496196\pi$
$701$	443.637	0.0239029	0.0119515	+	0.999929i	$0.496196\pi$
$702$	0	0
$703$	−29766.7	−1.59698
$704$	0	0
$705$	13122.4	0.701017
$706$	0	0
$707$	9940.87	0.528805
$708$	0	0
$709$	−24146.1	−1.27902	−0.639510	−	0.768783i	$-0.720863\pi$
$709$	−24146.1	−1.27902	−0.639510	+	0.768783i	$0.720863\pi$
$710$	0	0
$711$	−848.610	−0.0447614
$712$	0	0
$713$	−11452.6	−0.601549
$714$	0	0
$715$	−27448.9	−1.43571
$716$	0	0
$717$	15420.7	0.803201
$718$	0	0
$719$	19205.6	0.996171	0.498085	−	0.867128i	$-0.334037\pi$
$719$	19205.6	0.996171	0.498085	+	0.867128i	$0.334037\pi$
$720$	0	0
$721$	7743.65	0.399984
$722$	0	0
$723$	−6109.28	−0.314255
$724$	0	0
$725$	−5696.98	−0.291835
$726$	0	0
$727$	6045.94	0.308434	0.154217	−	0.988037i	$-0.450714\pi$
$727$	6045.94	0.308434	0.154217	+	0.988037i	$0.450714\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	0	0
$731$	35246.4	1.78336
$732$	0	0
$733$	3004.60	0.151402	0.0757008	−	0.997131i	$-0.475881\pi$
$733$	3004.60	0.151402	0.0757008	+	0.997131i	$0.475881\pi$
$734$	0	0
$735$	1494.33	0.0749922
$736$	0	0
$737$	−36923.8	−1.84546
$738$	0	0
$739$	−31957.4	−1.59076	−0.795380	−	0.606111i	$-0.792729\pi$
$739$	−31957.4	−1.59076	−0.795380	+	0.606111i	$0.792729\pi$
$740$	0	0
$741$	−23368.7	−1.15853
$742$	0	0
$743$	16854.1	0.832190	0.416095	−	0.909321i	$-0.363398\pi$
$743$	16854.1	0.832190	0.416095	+	0.909321i	$0.363398\pi$
$744$	0	0
$745$	−10206.9	−0.501948
$746$	0	0
$747$	−5639.59	−0.276227
$748$	0	0
$749$	−1100.49	−0.0536865
$750$	0	0
$751$	−13020.8	−0.632671	−0.316335	−	0.948647i	$-0.602453\pi$
$751$	−13020.8	−0.632671	−0.316335	+	0.948647i	$0.602453\pi$
$752$	0	0
$753$	−19751.1	−0.955869
$754$	0	0
$755$	13070.8	0.630059
$756$	0	0
$757$	17832.8	0.856203	0.428102	−	0.903731i	$-0.359183\pi$
$757$	17832.8	0.856203	0.428102	+	0.903731i	$0.359183\pi$
$758$	0	0
$759$	−3995.67	−0.191085
$760$	0	0
$761$	11732.4	0.558869	0.279434	−	0.960165i	$-0.409853\pi$
$761$	11732.4	0.558869	0.279434	+	0.960165i	$0.409853\pi$
$762$	0	0
$763$	15077.2	0.715377
$764$	0	0
$765$	8340.08	0.394165
$766$	0	0
$767$	35811.0	1.68587
$768$	0	0
$769$	12233.1	0.573651	0.286826	−	0.957983i	$-0.407400\pi$
$769$	12233.1	0.573651	0.286826	+	0.957983i	$0.407400\pi$
$770$	0	0
$771$	−3833.34	−0.179059
$772$	0	0
$773$	7475.30	0.347824	0.173912	−	0.984761i	$-0.444359\pi$
$773$	7475.30	0.347824	0.173912	+	0.984761i	$0.444359\pi$
$774$	0	0
$775$	6736.47	0.312234
$776$	0	0
$777$	5991.52	0.276634
$778$	0	0
$779$	6520.33	0.299891
$780$	0	0
$781$	10287.0	0.471315
$782$	0	0
$783$	−7100.82	−0.324090
$784$	0	0
$785$	−4151.04	−0.188735
$786$	0	0
$787$	30616.2	1.38672	0.693360	−	0.720591i	$-0.256129\pi$
$787$	30616.2	1.38672	0.693360	+	0.720591i	$0.256129\pi$
$788$	0	0
$789$	−12798.5	−0.577491
$790$	0	0
$791$	−8772.60	−0.394333
$792$	0	0
$793$	45020.7	2.01606
$794$	0	0
$795$	−3414.36	−0.152321
$796$	0	0
$797$	−32993.8	−1.46637	−0.733187	−	0.680027i	$-0.761968\pi$
$797$	−32993.8	−1.46637	−0.733187	+	0.680027i	$0.761968\pi$
$798$	0	0
$799$	−39224.7	−1.73676
$800$	0	0
$801$	13791.6	0.608368
$802$	0	0
$803$	−20481.7	−0.900102
$804$	0	0
$805$	−2620.61	−0.114738
$806$	0	0
$807$	−952.681	−0.0415564
$808$	0	0
$809$	40725.3	1.76987	0.884936	−	0.465713i	$-0.154202\pi$
$809$	40725.3	1.76987	0.884936	+	0.465713i	$0.154202\pi$
$810$	0	0
$811$	22387.3	0.969327	0.484663	−	0.874701i	$-0.338942\pi$
$811$	22387.3	0.969327	0.484663	+	0.874701i	$0.338942\pi$
$812$	0	0
$813$	−2690.06	−0.116045
$814$	0	0
$815$	−18469.0	−0.793793
$816$	0	0
$817$	−40339.4	−1.72742
$818$	0	0
$819$	4703.71	0.200685
$820$	0	0
$821$	12101.2	0.514416	0.257208	−	0.966356i	$-0.417197\pi$
$821$	12101.2	0.514416	0.257208	+	0.966356i	$0.417197\pi$
$822$	0	0
$823$	17732.5	0.751051	0.375526	−	0.926812i	$-0.377462\pi$
$823$	17732.5	0.751051	0.375526	+	0.926812i	$0.377462\pi$
$824$	0	0
$825$	2350.26	0.0991826
$826$	0	0
$827$	−3150.77	−0.132482	−0.0662412	−	0.997804i	$-0.521101\pi$
$827$	−3150.77	−0.132482	−0.0662412	+	0.997804i	$0.521101\pi$
$828$	0	0
$829$	−17769.5	−0.744465	−0.372232	−	0.928140i	$-0.621408\pi$
$829$	−17769.5	−0.744465	−0.372232	+	0.928140i	$0.621408\pi$
$830$	0	0
$831$	5757.73	0.240353
$832$	0	0
$833$	−4466.78	−0.185792
$834$	0	0
$835$	35804.5	1.48391
$836$	0	0
$837$	8396.45	0.346743
$838$	0	0
$839$	19970.0	0.821741	0.410871	−	0.911694i	$-0.365225\pi$
$839$	19970.0	0.821741	0.410871	+	0.911694i	$0.365225\pi$
$840$	0	0
$841$	44776.4	1.83593
$842$	0	0
$843$	2233.62	0.0912575
$844$	0	0
$845$	−34333.3	−1.39775
$846$	0	0
$847$	−161.384	−0.00654688
$848$	0	0
$849$	−7341.69	−0.296780
$850$	0	0
$851$	−10507.3	−0.423250
$852$	0	0
$853$	37094.0	1.48895	0.744476	−	0.667650i	$-0.232700\pi$
$853$	37094.0	1.48895	0.744476	+	0.667650i	$0.232700\pi$
$854$	0	0
$855$	−9545.22	−0.381801
$856$	0	0
$857$	28630.4	1.14119	0.570593	−	0.821233i	$-0.306714\pi$
$857$	28630.4	1.14119	0.570593	+	0.821233i	$0.306714\pi$
$858$	0	0
$859$	−33096.6	−1.31460	−0.657301	−	0.753628i	$-0.728302\pi$
$859$	−33096.6	−1.31460	−0.657301	+	0.753628i	$0.728302\pi$
$860$	0	0
$861$	−1312.43	−0.0519482
$862$	0	0
$863$	−19554.7	−0.771323	−0.385661	−	0.922640i	$-0.626027\pi$
$863$	−19554.7	−0.771323	−0.385661	+	0.922640i	$0.626027\pi$
$864$	0	0
$865$	−24352.2	−0.957227
$866$	0	0
$867$	−10190.7	−0.399187
$868$	0	0
$869$	−3410.05	−0.133116
$870$	0	0
$871$	−76227.4	−2.96541
$872$	0	0
$873$	6464.73	0.250628
$874$	0	0
$875$	10436.3	0.403212
$876$	0	0
$877$	−8043.01	−0.309684	−0.154842	−	0.987939i	$-0.549487\pi$
$877$	−8043.01	−0.309684	−0.154842	+	0.987939i	$0.549487\pi$
$878$	0	0
$879$	5477.17	0.210171
$880$	0	0
$881$	−27751.2	−1.06125	−0.530625	−	0.847606i	$-0.678043\pi$
$881$	−27751.2	−1.06125	−0.530625	+	0.847606i	$0.678043\pi$
$882$	0	0
$883$	20355.2	0.775771	0.387885	−	0.921708i	$-0.373206\pi$
$883$	20355.2	0.775771	0.387885	+	0.921708i	$0.373206\pi$
$884$	0	0
$885$	14627.4	0.555588
$886$	0	0
$887$	−11001.2	−0.416443	−0.208221	−	0.978082i	$-0.566767\pi$
$887$	−11001.2	−0.416443	−0.208221	+	0.978082i	$0.566767\pi$
$888$	0	0
$889$	9667.14	0.364708
$890$	0	0
$891$	2929.41	0.110145
$892$	0	0
$893$	44892.6	1.68228
$894$	0	0
$895$	−15299.9	−0.571418
$896$	0	0
$897$	−8248.88	−0.307048
$898$	0	0
$899$	−81785.5	−3.03415
$900$	0	0
$901$	10206.0	0.377372
$902$	0	0
$903$	8119.62	0.299229
$904$	0	0
$905$	30846.4	1.13301
$906$	0	0
$907$	11666.9	0.427116	0.213558	−	0.976930i	$-0.431495\pi$
$907$	11666.9	0.427116	0.213558	+	0.976930i	$0.431495\pi$
$908$	0	0
$909$	12781.1	0.466362
$910$	0	0
$911$	3478.55	0.126509	0.0632543	−	0.997997i	$-0.479852\pi$
$911$	3478.55	0.126509	0.0632543	+	0.997997i	$0.479852\pi$
$912$	0	0
$913$	−22662.1	−0.821473
$914$	0	0
$915$	18389.2	0.664403
$916$	0	0
$917$	−2892.89	−0.104178
$918$	0	0
$919$	−14907.0	−0.535079	−0.267540	−	0.963547i	$-0.586211\pi$
$919$	−14907.0	−0.535079	−0.267540	+	0.963547i	$0.586211\pi$
$920$	0	0
$921$	2715.80	0.0971647
$922$	0	0
$923$	21237.0	0.757340
$924$	0	0
$925$	6180.42	0.219688
$926$	0	0
$927$	9956.12	0.352753
$928$	0	0
$929$	31359.5	1.10751	0.553753	−	0.832681i	$-0.313195\pi$
$929$	31359.5	1.10751	0.553753	+	0.832681i	$0.313195\pi$
$930$	0	0
$931$	5112.22	0.179964
$932$	0	0
$933$	−5829.86	−0.204567
$934$	0	0
$935$	33513.7	1.17221
$936$	0	0
$937$	28359.5	0.988757	0.494378	−	0.869247i	$-0.335396\pi$
$937$	28359.5	0.988757	0.494378	+	0.869247i	$0.335396\pi$
$938$	0	0
$939$	−24593.9	−0.854731
$940$	0	0
$941$	4772.62	0.165338	0.0826690	−	0.996577i	$-0.473656\pi$
$941$	4772.62	0.165338	0.0826690	+	0.996577i	$0.473656\pi$
$942$	0	0
$943$	2301.60	0.0794808
$944$	0	0
$945$	1921.28	0.0661369
$946$	0	0
$947$	−50905.8	−1.74679	−0.873397	−	0.487008i	$-0.838088\pi$
$947$	−50905.8	−1.74679	−0.873397	+	0.487008i	$0.838088\pi$
$948$	0	0
$949$	−42283.5	−1.44634
$950$	0	0
$951$	10973.3	0.374169
$952$	0	0
$953$	16818.3	0.571666	0.285833	−	0.958279i	$-0.407730\pi$
$953$	16818.3	0.571666	0.285833	+	0.958279i	$0.407730\pi$
$954$	0	0
$955$	6520.80	0.220951
$956$	0	0
$957$	−28533.9	−0.963813
$958$	0	0
$959$	1033.59	0.0348032
$960$	0	0
$961$	66917.2	2.24622
$962$	0	0
$963$	−1414.92	−0.0473470
$964$	0	0
$965$	−29405.0	−0.980913
$966$	0	0
$967$	−45160.6	−1.50183	−0.750913	−	0.660401i	$-0.770386\pi$
$967$	−45160.6	−1.50183	−0.750913	+	0.660401i	$0.770386\pi$
$968$	0	0
$969$	28532.0	0.945904
$970$	0	0
$971$	−29892.4	−0.987943	−0.493972	−	0.869478i	$-0.664455\pi$
$971$	−29892.4	−0.987943	−0.493972	+	0.869478i	$0.664455\pi$
$972$	0	0
$973$	−13648.8	−0.449701
$974$	0	0
$975$	4852.01	0.159373
$976$	0	0
$977$	36169.1	1.18439	0.592197	−	0.805793i	$-0.298261\pi$
$977$	36169.1	1.18439	0.592197	+	0.805793i	$0.298261\pi$
$978$	0	0
$979$	55420.1	1.80923
$980$	0	0
$981$	19385.0	0.630903
$982$	0	0
$983$	13732.3	0.445567	0.222784	−	0.974868i	$-0.428486\pi$
$983$	13732.3	0.445567	0.222784	+	0.974868i	$0.428486\pi$
$984$	0	0
$985$	−33931.1	−1.09760
$986$	0	0
$987$	−9036.09	−0.291410
$988$	0	0
$989$	−14239.3	−0.457821
$990$	0	0
$991$	−31284.6	−1.00281	−0.501407	−	0.865211i	$-0.667184\pi$
$991$	−31284.6	−1.00281	−0.501407	+	0.865211i	$0.667184\pi$
$992$	0	0
$993$	369.260	0.0118007
$994$	0	0
$995$	−14250.3	−0.454034
$996$	0	0
$997$	−20015.1	−0.635790	−0.317895	−	0.948126i	$-0.602976\pi$
$997$	−20015.1	−0.635790	−0.317895	+	0.948126i	$0.602976\pi$
$998$	0	0
$999$	7703.38	0.243968

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1344.4.a.bh.1.1		2
4.3	odd	2		1344.4.a.bp.1.1		2
8.3	odd	2		672.4.a.f.1.2	✓	2
8.5	even	2		672.4.a.k.1.2	yes	2
24.5	odd	2		2016.4.a.n.1.1		2
24.11	even	2		2016.4.a.m.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
672.4.a.f.1.2	✓	2	8.3	odd	2
672.4.a.k.1.2	yes	2	8.5	even	2
1344.4.a.bh.1.1		2	1.1	even	1	trivial
1344.4.a.bp.1.1		2	4.3	odd	2
2016.4.a.m.1.1		2	24.11	even	2
2016.4.a.n.1.1		2	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:	\( N \)	\(=\)	\( 1344 = 2^{6} \cdot 3 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1344.a (trivial)

\(f(q)\)	\(=\)	\(q-3.00000 q^{3} -10.1655 q^{5} +7.00000 q^{7} +9.00000 q^{9} +O(q^{10})\)	\(q-3.00000 q^{3} -10.1655 q^{5} +7.00000 q^{7} +9.00000 q^{9} +36.1655 q^{11} +74.6621 q^{13} +30.4966 q^{15} -91.1587 q^{17} +104.331 q^{19} -21.0000 q^{21} +36.8276 q^{23} -21.6621 q^{25} -27.0000 q^{27} +262.993 q^{29} -310.979 q^{31} -108.497 q^{33} -71.1587 q^{35} -285.311 q^{37} -223.986 q^{39} +62.4966 q^{41} -386.648 q^{43} -91.4897 q^{45} +430.290 q^{47} +49.0000 q^{49} +273.476 q^{51} -111.959 q^{53} -367.642 q^{55} -312.993 q^{57} +479.642 q^{59} +602.993 q^{61} +63.0000 q^{63} -758.979 q^{65} -1020.97 q^{67} -110.483 q^{69} +284.442 q^{71} -566.331 q^{73} +64.9863 q^{75} +253.159 q^{77} -94.2900 q^{79} +81.0000 q^{81} -626.621 q^{83} +926.676 q^{85} -788.979 q^{87} +1532.40 q^{89} +522.635 q^{91} +932.938 q^{93} -1060.58 q^{95} +718.304 q^{97} +325.490 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 6 q^{3} + 4 q^{5} + 14 q^{7} + 18 q^{9}+O(q^{10}) \) 2 * q - 6 * q^3 + 4 * q^5 + 14 * q^7 + 18 * q^9	\( 2 q - 6 q^{3} + 4 q^{5} + 14 q^{7} + 18 q^{9} + 48 q^{11} + 52 q^{13} - 12 q^{15} - 12 q^{17} + 160 q^{19} - 42 q^{21} - 48 q^{23} + 54 q^{25} - 54 q^{27} + 380 q^{29} - 184 q^{31} - 144 q^{33} + 28 q^{35} - 84 q^{37} - 156 q^{39} + 52 q^{41} - 384 q^{43} + 36 q^{45} - 64 q^{47} + 98 q^{49} + 36 q^{51} + 652 q^{53} - 200 q^{55} - 480 q^{57} + 424 q^{59} + 1060 q^{61} + 126 q^{63} - 1080 q^{65} - 1312 q^{67} + 144 q^{69} - 672 q^{71} - 1084 q^{73} - 162 q^{75} + 336 q^{77} + 736 q^{79} + 162 q^{81} - 280 q^{83} + 2048 q^{85} - 1140 q^{87} + 948 q^{89} + 364 q^{91} + 552 q^{93} - 272 q^{95} + 804 q^{97} + 432 q^{99}+O(q^{100}) \) 2 * q - 6 * q^3 + 4 * q^5 + 14 * q^7 + 18 * q^9 + 48 * q^11 + 52 * q^13 - 12 * q^15 - 12 * q^17 + 160 * q^19 - 42 * q^21 - 48 * q^23 + 54 * q^25 - 54 * q^27 + 380 * q^29 - 184 * q^31 - 144 * q^33 + 28 * q^35 - 84 * q^37 - 156 * q^39 + 52 * q^41 - 384 * q^43 + 36 * q^45 - 64 * q^47 + 98 * q^49 + 36 * q^51 + 652 * q^53 - 200 * q^55 - 480 * q^57 + 424 * q^59 + 1060 * q^61 + 126 * q^63 - 1080 * q^65 - 1312 * q^67 + 144 * q^69 - 672 * q^71 - 1084 * q^73 - 162 * q^75 + 336 * q^77 + 736 * q^79 + 162 * q^81 - 280 * q^83 + 2048 * q^85 - 1140 * q^87 + 948 * q^89 + 364 * q^91 + 552 * q^93 - 272 * q^95 + 804 * q^97 + 432 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more