LMFDB - Embedded newform 1344.4.b.g.895.11

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1344,4,Mod(895,1344)]

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([1, 0, 0, 1]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1344.895");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1344 = 2^{6} \cdot 3 \cdot 7 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1344.b (of order $2$, degree $1$, not minimal)

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:	no
Analytic conductor:	$79.2985670477$
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} - x^{11} - 2x^{10} - 6x^{9} + 56x^{7} - 448x^{6} + 448x^{5} - 3072x^{3} - 8192x^{2} - 32768x + 262144 $ x^12 - x^11 - 2x^10 - 6x^9 + 56x^7 - 448x^6 + 448x^5 - 3072x^3 - 8192x^2 - 32768x + 262144
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{30} $
Twist minimal:	no (minimal twist has level 84)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			895.11
Root			$2.82801 - 0.0488466i$ of defining polynomial
Character	$\chi$	$=$	1344.895
Dual form			1344.4.b.g.895.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.00000 q^{3} +16.6517i q^{5} +(-15.0420 - 10.8045i) q^{7} +9.00000 q^{9} +O(q^{10})$	$q-3.00000 q^{3} +16.6517i q^{5} +(-15.0420 - 10.8045i) q^{7} +9.00000 q^{9} -64.0450i q^{11} -28.6879i q^{13} -49.9552i q^{15} +82.9041i q^{17} -17.1236 q^{19} +(45.1261 + 32.4134i) q^{21} -95.0501i q^{23} -152.281 q^{25} -27.0000 q^{27} +197.365 q^{29} +153.515 q^{31} +192.135i q^{33} +(179.913 - 250.476i) q^{35} -10.7262 q^{37} +86.0638i q^{39} +41.1342i q^{41} +412.497i q^{43} +149.866i q^{45} -477.704 q^{47} +(109.526 + 325.043i) q^{49} -248.712i q^{51} +35.2304 q^{53} +1066.46 q^{55} +51.3707 q^{57} +494.608 q^{59} +294.084i q^{61} +(-135.378 - 97.2403i) q^{63} +477.704 q^{65} +207.870i q^{67} +285.150i q^{69} +534.040i q^{71} -582.270i q^{73} +456.842 q^{75} +(-691.973 + 963.368i) q^{77} -311.277i q^{79} +81.0000 q^{81} -1319.63 q^{83} -1380.50 q^{85} -592.095 q^{87} -616.091i q^{89} +(-309.958 + 431.525i) q^{91} -460.545 q^{93} -285.137i q^{95} +104.076i q^{97} -576.405i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 36 q^{3} - 10 q^{7} + 108 q^{9}+O(q^{10}) $ 12 * q - 36 * q^3 - 10 * q^7 + 108 * q^9	$ 12 q - 36 q^{3} - 10 q^{7} + 108 q^{9} - 84 q^{19} + 30 q^{21} - 216 q^{25} - 324 q^{27} - 200 q^{29} + 384 q^{31} + 84 q^{35} + 244 q^{37} - 280 q^{47} - 424 q^{49} + 16 q^{53} - 212 q^{55} + 252 q^{57} + 1168 q^{59} - 90 q^{63} + 280 q^{65} + 648 q^{75} - 968 q^{77} + 972 q^{81} - 968 q^{83} + 852 q^{85} + 600 q^{87} + 1648 q^{91} - 1152 q^{93}+O(q^{100}) $ 12 * q - 36 * q^3 - 10 * q^7 + 108 * q^9 - 84 * q^19 + 30 * q^21 - 216 * q^25 - 324 * q^27 - 200 * q^29 + 384 * q^31 + 84 * q^35 + 244 * q^37 - 280 * q^47 - 424 * q^49 + 16 * q^53 - 212 * q^55 + 252 * q^57 + 1168 * q^59 - 90 * q^63 + 280 * q^65 + 648 * q^75 - 968 * q^77 + 972 * q^81 - 968 * q^83 + 852 * q^85 + 600 * q^87 + 1648 * q^91 - 1152 * q^93

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1344\mathbb{Z}\right)^\times$.

$n$	$127$	$449$	$577$	$1093$
$\chi(n)$	$-1$	$1$	$-1$	$1$

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
$5$			16.6517i			1.48938i	0.667412	+	0.744689i	$0.267402\pi$
$5$			16.6517i			1.48938i	−0.667412	+	0.744689i	$0.732598\pi$
$6$		0			0
$7$	−15.0420	−	10.8045i	−0.812194	−	0.583387i
$7$	−15.0420	−	10.8045i	−0.812194	−	0.583387i
$8$		0			0
$9$	9.00000			0.333333
$10$		0			0
$11$		−	64.0450i		−	1.75548i	−0.479136	−	0.877741i	$-0.659050\pi$
$11$		−	64.0450i		−	1.75548i	0.479136	−	0.877741i	$-0.340950\pi$
$12$		0			0
$13$		−	28.6879i		−	0.612046i	−0.952024	−	0.306023i	$-0.901002\pi$
$13$		−	28.6879i		−	0.612046i	0.952024	−	0.306023i	$-0.0989985\pi$
$14$		0			0
$15$		−	49.9552i		−	0.859892i
$16$		0			0
$17$			82.9041i			1.18278i	0.806387	+	0.591388i	$0.201420\pi$
$17$			82.9041i			1.18278i	−0.806387	+	0.591388i	$0.798580\pi$
$18$		0			0
$19$	−17.1236			−0.206759			−0.103379	−	0.994642i	$-0.532966\pi$
$19$	−17.1236			−0.206759			−0.103379	+	0.994642i	$0.532966\pi$
$20$		0			0
$21$	45.1261	+	32.4134i	0.468921	+	0.336819i
$22$		0			0
$23$		−	95.0501i		−	0.861710i	−0.902421	−	0.430855i	$-0.858212\pi$
$23$		−	95.0501i		−	0.861710i	0.902421	−	0.430855i	$-0.141788\pi$
$24$		0			0
$25$	−152.281			−1.21824
$26$		0			0
$27$	−27.0000			−0.192450
$28$		0			0
$29$	197.365			1.26378			0.631892	−	0.775056i	$-0.282278\pi$
$29$	197.365			1.26378			0.631892	+	0.775056i	$0.282278\pi$
$30$		0			0
$31$	153.515			0.889422			0.444711	−	0.895674i	$-0.353306\pi$
$31$	153.515			0.889422			0.444711	+	0.895674i	$0.353306\pi$
$32$		0			0
$33$			192.135i			1.01353i
$34$		0			0
$35$	179.913	−	250.476i	0.868883	−	1.20966i
$36$		0			0
$37$	−10.7262			−0.0476589			−0.0238294	−	0.999716i	$-0.507586\pi$
$37$	−10.7262			−0.0476589			−0.0238294	+	0.999716i	$0.507586\pi$
$38$		0			0
$39$			86.0638i			0.353365i
$40$		0			0
$41$			41.1342i			0.156685i	0.996927	+	0.0783425i	$0.0249628\pi$
$41$			41.1342i			0.156685i	−0.996927	+	0.0783425i	$0.975037\pi$
$42$		0			0
$43$			412.497i			1.46291i	0.681890	+	0.731455i	$0.261158\pi$
$43$			412.497i			1.46291i	−0.681890	+	0.731455i	$0.738842\pi$
$44$		0			0
$45$			149.866i			0.496459i
$46$		0			0
$47$	−477.704			−1.48256			−0.741280	−	0.671196i	$-0.765781\pi$
$47$	−477.704			−1.48256			−0.741280	+	0.671196i	$0.765781\pi$
$48$		0			0
$49$	109.526	+	325.043i	0.319319	+	0.947647i
$50$		0			0
$51$		−	248.712i		−	0.682877i
$52$		0			0
$53$	35.2304			0.0913069			0.0456534	−	0.998957i	$-0.485463\pi$
$53$	35.2304			0.0913069			0.0456534	+	0.998957i	$0.485463\pi$
$54$		0			0
$55$	1066.46			2.61457
$56$		0			0
$57$	51.3707			0.119372
$58$		0			0
$59$	494.608			1.09140			0.545699	−	0.837981i	$-0.316264\pi$
$59$	494.608			1.09140			0.545699	+	0.837981i	$0.316264\pi$
$60$		0			0
$61$			294.084i			0.617273i	0.951180	+	0.308636i	$0.0998727\pi$
$61$			294.084i			0.617273i	−0.951180	+	0.308636i	$0.900127\pi$
$62$		0			0
$63$	−135.378	−	97.2403i	−0.270731	−	0.194462i
$64$		0			0
$65$	477.704			0.911567
$66$		0			0
$67$			207.870i			0.379036i	0.981877	+	0.189518i	$0.0606926\pi$
$67$			207.870i			0.379036i	−0.981877	+	0.189518i	$0.939307\pi$
$68$		0			0
$69$			285.150i			0.497508i
$70$		0			0
$71$			534.040i			0.892660i	0.894869	+	0.446330i	$0.147269\pi$
$71$			534.040i			0.892660i	−0.894869	+	0.446330i	$0.852731\pi$
$72$		0			0
$73$		−	582.270i		−	0.933555i	−0.884375	−	0.466778i	$-0.845415\pi$
$73$		−	582.270i		−	0.933555i	0.884375	−	0.466778i	$-0.154585\pi$
$74$		0			0
$75$	456.842			0.703354
$76$		0			0
$77$	−691.973	+	963.368i	−1.02413	+	1.42579i
$78$		0			0
$79$		−	311.277i		−	0.443309i	−0.975125	−	0.221654i	$-0.928854\pi$
$79$		−	311.277i		−	0.443309i	0.975125	−	0.221654i	$-0.0711457\pi$
$80$		0			0
$81$	81.0000			0.111111
$82$		0			0
$83$	−1319.63			−1.74515			−0.872577	−	0.488477i	$-0.837553\pi$
$83$	−1319.63			−1.74515			−0.872577	+	0.488477i	$0.837553\pi$
$84$		0			0
$85$	−1380.50			−1.76160
$86$		0			0
$87$	−592.095			−0.729646
$88$		0			0
$89$		−	616.091i		−	0.733770i	−0.930266	−	0.366885i	$-0.880424\pi$
$89$		−	616.091i		−	0.733770i	0.930266	−	0.366885i	$-0.119576\pi$
$90$		0			0
$91$	−309.958	+	431.525i	−0.357060	+	0.497100i
$92$		0			0
$93$	−460.545			−0.513508
$94$		0			0
$95$		−	285.137i		−	0.307941i
$96$		0			0
$97$			104.076i			0.108941i	0.998515	+	0.0544706i	$0.0173471\pi$
$97$			104.076i			0.108941i	−0.998515	+	0.0544706i	$0.982653\pi$
$98$		0			0
$99$		−	576.405i		−	0.585161i
$100$		0			0
$101$		−	1754.34i		−	1.72835i	−0.503196	−	0.864173i	$-0.667842\pi$
$101$		−	1754.34i		−	1.72835i	0.503196	−	0.864173i	$-0.332158\pi$
$102$		0			0
$103$	−310.370			−0.296910			−0.148455	−	0.988919i	$-0.547430\pi$
$103$	−310.370			−0.296910			−0.148455	+	0.988919i	$0.547430\pi$
$104$		0			0
$105$	−539.740	+	751.429i	−0.501650	+	0.698400i
$106$		0			0
$107$		−	482.444i		−	0.435884i	−0.975962	−	0.217942i	$-0.930066\pi$
$107$		−	482.444i		−	0.435884i	0.975962	−	0.217942i	$-0.0699344\pi$
$108$		0			0
$109$	72.5584			0.0637600			0.0318800	−	0.999492i	$-0.489851\pi$
$109$	72.5584			0.0637600			0.0318800	+	0.999492i	$0.489851\pi$
$110$		0			0
$111$	32.1786			0.0275159
$112$		0			0
$113$	−1545.99			−1.28703			−0.643514	−	0.765434i	$-0.722524\pi$
$113$	−1545.99			−1.28703			−0.643514	+	0.765434i	$0.722524\pi$
$114$		0			0
$115$	1582.75			1.28341
$116$		0			0
$117$		−	258.191i		−	0.204015i
$118$		0			0
$119$	895.736	−	1247.05i	0.690017	−	0.960645i
$120$		0			0
$121$	−2770.76			−2.08172
$122$		0			0
$123$		−	123.403i		−	0.0904621i
$124$		0			0
$125$		−	454.269i		−	0.325048i
$126$		0			0
$127$			1982.75i			1.38536i	0.721246	+	0.692679i	$0.243570\pi$
$127$			1982.75i			1.38536i	−0.721246	+	0.692679i	$0.756430\pi$
$128$		0			0
$129$		−	1237.49i		−	0.844612i
$130$		0			0
$131$	403.984			0.269437			0.134719	−	0.990884i	$-0.456987\pi$
$131$	403.984			0.269437			0.134719	+	0.990884i	$0.456987\pi$
$132$		0			0
$133$	257.573	+	185.011i	0.167928	+	0.120620i
$134$		0			0
$135$		−	449.597i		−	0.286631i
$136$		0			0
$137$	−1366.78			−0.852350			−0.426175	−	0.904641i	$-0.640139\pi$
$137$	−1366.78			−0.852350			−0.426175	+	0.904641i	$0.640139\pi$
$138$		0			0
$139$	−1770.10			−1.08013			−0.540065	−	0.841623i	$-0.681600\pi$
$139$	−1770.10			−1.08013			−0.540065	+	0.841623i	$0.681600\pi$
$140$		0			0
$141$	1433.11			0.855956
$142$		0			0
$143$	−1837.32			−1.07444
$144$		0			0
$145$			3286.47i			1.88225i
$146$		0			0
$147$	−328.579	−	975.129i	−0.184359	−	0.547124i
$148$		0			0
$149$	−1991.97			−1.09522			−0.547612	−	0.836732i	$-0.684463\pi$
$149$	−1991.97			−1.09522			−0.547612	+	0.836732i	$0.684463\pi$
$150$		0			0
$151$		−	1117.45i		−	0.602229i	−0.953588	−	0.301115i	$-0.902641\pi$
$151$		−	1117.45i		−	0.602229i	0.953588	−	0.301115i	$-0.0973586\pi$
$152$		0			0
$153$			746.137i			0.394259i
$154$		0			0
$155$			2556.29i			1.32468i
$156$		0			0
$157$			758.016i			0.385327i	0.981265	+	0.192663i	$0.0617125\pi$
$157$			758.016i			0.385327i	−0.981265	+	0.192663i	$0.938287\pi$
$158$		0			0
$159$	−105.691			−0.0527160
$160$		0			0
$161$	−1026.97	+	1429.75i	−0.502710	+	0.699876i
$162$		0			0
$163$			147.846i			0.0710443i	0.999369	+	0.0355222i	$0.0113094\pi$
$163$			147.846i			0.0710443i	−0.999369	+	0.0355222i	$0.988691\pi$
$164$		0			0
$165$	−3199.38			−1.50953
$166$		0			0
$167$	1741.92			0.807148			0.403574	−	0.914947i	$-0.367768\pi$
$167$	1741.92			0.807148			0.403574	+	0.914947i	$0.367768\pi$
$168$		0			0
$169$	1374.00			0.625400
$170$		0			0
$171$	−154.112			−0.0689195
$172$		0			0
$173$			2686.73i			1.18074i	0.807131	+	0.590372i	$0.201019\pi$
$173$			2686.73i			1.18074i	−0.807131	+	0.590372i	$0.798981\pi$
$174$		0			0
$175$	2290.61	+	1645.31i	0.989451	+	0.710708i
$176$		0			0
$177$	−1483.82			−0.630119
$178$		0			0
$179$		−	1032.53i		−	0.431146i	−0.976488	−	0.215573i	$-0.930838\pi$
$179$		−	1032.53i		−	0.431146i	0.976488	−	0.215573i	$-0.0691619\pi$
$180$		0			0
$181$			1608.53i			0.660558i	0.943883	+	0.330279i	$0.107143\pi$
$181$			1608.53i			0.660558i	−0.943883	+	0.330279i	$0.892857\pi$
$182$		0			0
$183$		−	882.253i		−	0.356383i
$184$		0			0
$185$		−	178.610i		−	0.0709821i
$186$		0			0
$187$	5309.60			2.07634
$188$		0			0
$189$	406.135	+	291.721i	0.156307	+	0.112273i
$190$		0			0
$191$			1970.43i			0.746469i	0.927737	+	0.373235i	$0.121751\pi$
$191$			1970.43i			0.746469i	−0.927737	+	0.373235i	$0.878249\pi$
$192$		0			0
$193$	966.622			0.360513			0.180256	−	0.983620i	$-0.442307\pi$
$193$	966.622			0.360513			0.180256	+	0.983620i	$0.442307\pi$
$194$		0			0
$195$	−1433.11			−0.526294
$196$		0			0
$197$	−1068.53			−0.386443			−0.193222	−	0.981155i	$-0.561894\pi$
$197$	−1068.53			−0.386443			−0.193222	+	0.981155i	$0.561894\pi$
$198$		0			0
$199$	−252.660			−0.0900029			−0.0450015	−	0.998987i	$-0.514329\pi$
$199$	−252.660			−0.0900029			−0.0450015	+	0.998987i	$0.514329\pi$
$200$		0			0
$201$		−	623.611i		−	0.218837i
$202$		0			0
$203$	−2968.77	−	2132.43i	−1.02644	−	0.737276i
$204$		0			0
$205$	−684.956			−0.233363
$206$		0			0
$207$		−	855.451i		−	0.287237i
$208$		0			0
$209$			1096.68i			0.362961i
$210$		0			0
$211$			4606.20i			1.50286i	0.659811	+	0.751432i	$0.270636\pi$
$211$			4606.20i			1.50286i	−0.659811	+	0.751432i	$0.729364\pi$
$212$		0			0
$213$		−	1602.12i		−	0.515377i
$214$		0			0
$215$	−6868.79			−2.17883
$216$		0			0
$217$	−2309.18	−	1658.65i	−0.722383	−	0.518877i
$218$		0			0
$219$			1746.81i			0.538988i
$220$		0			0
$221$	2378.35			0.723914
$222$		0			0
$223$	−1927.65			−0.578857			−0.289428	−	0.957200i	$-0.593465\pi$
$223$	−1927.65			−0.578857			−0.289428	+	0.957200i	$0.593465\pi$
$224$		0			0
$225$	−1370.52			−0.406081
$226$		0			0
$227$	−4063.72			−1.18819			−0.594093	−	0.804396i	$-0.702489\pi$
$227$	−4063.72			−1.18819			−0.594093	+	0.804396i	$0.702489\pi$
$228$		0			0
$229$			4096.49i			1.18211i	0.806631	+	0.591056i	$0.201289\pi$
$229$			4096.49i			1.18211i	−0.806631	+	0.591056i	$0.798711\pi$
$230$		0			0
$231$	2075.92	−	2890.10i	0.591279	−	0.823181i
$232$		0			0
$233$	1950.08			0.548301			0.274150	−	0.961687i	$-0.411603\pi$
$233$	1950.08			0.548301			0.274150	+	0.961687i	$0.411603\pi$
$234$		0			0
$235$		−	7954.60i		−	2.20809i
$236$		0			0
$237$			933.831i			0.255944i
$238$		0			0
$239$			2861.53i			0.774465i	0.921982	+	0.387232i	$0.126569\pi$
$239$			2861.53i			0.774465i	−0.921982	+	0.387232i	$0.873431\pi$
$240$		0			0
$241$			4320.12i			1.15470i	0.816496	+	0.577351i	$0.195914\pi$
$241$			4320.12i			1.15470i	−0.816496	+	0.577351i	$0.804086\pi$
$242$		0			0
$243$	−243.000			−0.0641500
$244$		0			0
$245$	−5412.53	+	1823.81i	−1.41140	+	0.475586i
$246$		0			0
$247$			491.239i			0.126546i
$248$		0			0
$249$	3958.88			1.00756
$250$		0			0
$251$	1366.74			0.343698			0.171849	−	0.985123i	$-0.445026\pi$
$251$	1366.74			0.343698			0.171849	+	0.985123i	$0.445026\pi$
$252$		0			0
$253$	−6087.49			−1.51272
$254$		0			0
$255$	4141.50			1.01706
$256$		0			0
$257$			5993.21i			1.45465i	0.686291	+	0.727327i	$0.259238\pi$
$257$			5993.21i			1.45465i	−0.686291	+	0.727327i	$0.740762\pi$
$258$		0			0
$259$	161.344	+	115.891i	0.0387083	+	0.0278036i
$260$		0			0
$261$	1776.28			0.421262
$262$		0			0
$263$		−	1636.95i		−	0.383798i	−0.981415	−	0.191899i	$-0.938535\pi$
$263$		−	1636.95i		−	0.383798i	0.981415	−	0.191899i	$-0.0614646\pi$
$264$		0			0
$265$			586.647i			0.135990i
$266$		0			0
$267$			1848.27i			0.423642i
$268$		0			0
$269$			7442.41i			1.68688i	0.537220	+	0.843442i	$0.319474\pi$
$269$			7442.41i			1.68688i	−0.537220	+	0.843442i	$0.680526\pi$
$270$		0			0
$271$	−5619.33			−1.25959			−0.629797	−	0.776760i	$-0.716862\pi$
$271$	−5619.33			−1.25959			−0.629797	+	0.776760i	$0.716862\pi$
$272$		0			0
$273$	929.874	−	1294.58i	0.206149	−	0.287001i
$274$		0			0
$275$			9752.81i			2.13861i
$276$		0			0
$277$	698.240			0.151456			0.0757278	−	0.997129i	$-0.475872\pi$
$277$	698.240			0.151456			0.0757278	+	0.997129i	$0.475872\pi$
$278$		0			0
$279$	1381.63			0.296474
$280$		0			0
$281$	−5026.99			−1.06721			−0.533604	−	0.845735i	$-0.679163\pi$
$281$	−5026.99			−1.06721			−0.533604	+	0.845735i	$0.679163\pi$
$282$		0			0
$283$	−3596.17			−0.755371			−0.377686	−	0.925934i	$-0.623280\pi$
$283$	−3596.17			−0.755371			−0.377686	+	0.925934i	$0.623280\pi$
$284$		0			0
$285$			855.411i			0.177790i
$286$		0			0
$287$	444.434	−	618.743i	0.0914080	−	0.127259i
$288$		0			0
$289$	−1960.10			−0.398961
$290$		0			0
$291$		−	312.227i		−	0.0628972i
$292$		0			0
$293$			4095.02i			0.816497i	0.912871	+	0.408248i	$0.133860\pi$
$293$			4095.02i			0.816497i	−0.912871	+	0.408248i	$0.866140\pi$
$294$		0			0
$295$			8236.09i			1.62550i
$296$		0			0
$297$			1729.22i			0.337843i
$298$		0			0
$299$	−2726.79			−0.527406
$300$		0			0
$301$	4456.81	−	6204.79i	0.853443	−	1.18817i
$302$		0			0
$303$			5263.01i			0.997861i
$304$		0			0
$305$	−4897.02			−0.919352
$306$		0			0
$307$	6214.27			1.15527			0.577633	−	0.816296i	$-0.303976\pi$
$307$	6214.27			1.15527			0.577633	+	0.816296i	$0.303976\pi$
$308$		0			0
$309$	931.110			0.171421
$310$		0			0
$311$	4044.93			0.737514			0.368757	−	0.929526i	$-0.379784\pi$
$311$	4044.93			0.737514			0.368757	+	0.929526i	$0.379784\pi$
$312$		0			0
$313$		−	6931.84i		−	1.25179i	−0.779907	−	0.625896i	$-0.784734\pi$
$313$		−	6931.84i		−	1.25179i	0.779907	−	0.625896i	$-0.215266\pi$
$314$		0			0
$315$	1619.22	−	2254.29i	0.289628	−	0.403221i
$316$		0			0
$317$	−3966.39			−0.702759			−0.351379	−	0.936233i	$-0.614287\pi$
$317$	−3966.39			−0.702759			−0.351379	+	0.936233i	$0.614287\pi$
$318$		0			0
$319$		−	12640.2i		−	2.21855i
$320$		0			0
$321$			1447.33i			0.251658i
$322$		0			0
$323$		−	1419.61i		−	0.244549i
$324$		0			0
$325$			4368.61i			0.745622i
$326$		0			0
$327$	−217.675			−0.0368118
$328$		0			0
$329$	7185.65	+	5161.34i	1.20413	+	0.864906i
$330$		0			0
$331$		−	9949.84i		−	1.65224i	−0.563491	−	0.826122i	$-0.690542\pi$
$331$		−	9949.84i		−	1.65224i	0.563491	−	0.826122i	$-0.309458\pi$
$332$		0			0
$333$	−96.5359			−0.0158863
$334$		0			0
$335$	−3461.41			−0.564528
$336$		0			0
$337$	4947.26			0.799686			0.399843	−	0.916584i	$-0.369065\pi$
$337$	4947.26			0.799686			0.399843	+	0.916584i	$0.369065\pi$
$338$		0			0
$339$	4637.96			0.743066
$340$		0			0
$341$		−	9831.86i		−	1.56136i
$342$		0			0
$343$	1864.42	−	6072.69i	0.293496	−	0.955960i
$344$		0			0
$345$	−4748.25			−0.740978
$346$		0			0
$347$			5246.44i			0.811653i	0.913950	+	0.405826i	$0.133016\pi$
$347$			5246.44i			0.811653i	−0.913950	+	0.405826i	$0.866984\pi$
$348$		0			0
$349$		−	8703.25i		−	1.33488i	−0.744662	−	0.667442i	$-0.767389\pi$
$349$		−	8703.25i		−	1.33488i	0.744662	−	0.667442i	$-0.232611\pi$
$350$		0			0
$351$			774.574i			0.117788i
$352$		0			0
$353$			1402.85i			0.211519i	0.994392	+	0.105760i	$0.0337274\pi$
$353$			1402.85i			0.211519i	−0.994392	+	0.105760i	$0.966273\pi$
$354$		0			0
$355$	−8892.69			−1.32951
$356$		0			0
$357$	−2687.21	+	3741.14i	−0.398381	+	0.554628i
$358$		0			0
$359$			9556.07i			1.40487i	0.711746	+	0.702437i	$0.247905\pi$
$359$			9556.07i			1.40487i	−0.711746	+	0.702437i	$0.752095\pi$
$360$		0			0
$361$	−6565.78			−0.957251
$362$		0			0
$363$	8312.29			1.20188
$364$		0			0
$365$	9695.81			1.39042
$366$		0			0
$367$	−1036.73			−0.147458			−0.0737289	−	0.997278i	$-0.523490\pi$
$367$	−1036.73			−0.147458			−0.0737289	+	0.997278i	$0.523490\pi$
$368$		0			0
$369$			370.208i			0.0522283i
$370$		0			0
$371$	−529.937	−	380.646i	−0.0741589	−	0.0532672i
$372$		0			0
$373$	−10769.4			−1.49496			−0.747481	−	0.664283i	$-0.768737\pi$
$373$	−10769.4			−1.49496			−0.747481	+	0.664283i	$0.768737\pi$
$374$		0			0
$375$			1362.81i			0.187667i
$376$		0			0
$377$		−	5661.99i		−	0.773494i
$378$		0			0
$379$			3268.34i			0.442963i	0.975165	+	0.221482i	$0.0710893\pi$
$379$			3268.34i			0.442963i	−0.975165	+	0.221482i	$0.928911\pi$
$380$		0			0
$381$		−	5948.25i		−	0.799837i
$382$		0			0
$383$	5143.39			0.686201			0.343101	−	0.939299i	$-0.388523\pi$
$383$	5143.39			0.686201			0.343101	+	0.939299i	$0.388523\pi$
$384$		0			0
$385$	−16041.8	−	11522.6i	−2.12354	−	1.52531i
$386$		0			0
$387$			3712.47i			0.487637i
$388$		0			0
$389$	5140.27			0.669979			0.334990	−	0.942222i	$-0.391267\pi$
$389$	5140.27			0.669979			0.334990	+	0.942222i	$0.391267\pi$
$390$		0			0
$391$	7880.05			1.01921
$392$		0			0
$393$	−1211.95			−0.155560
$394$		0			0
$395$	5183.30			0.660254
$396$		0			0
$397$		−	6774.24i		−	0.856396i	−0.903685	−	0.428198i	$-0.859149\pi$
$397$		−	6774.24i		−	0.856396i	0.903685	−	0.428198i	$-0.140851\pi$
$398$		0			0
$399$	−772.720	−	555.033i	−0.0969533	−	0.0696401i
$400$		0			0
$401$	−8827.27			−1.09928			−0.549642	−	0.835401i	$-0.685236\pi$
$401$	−8827.27			−1.09928			−0.549642	+	0.835401i	$0.685236\pi$
$402$		0			0
$403$		−	4404.02i		−	0.544367i
$404$		0			0
$405$			1348.79i			0.165486i
$406$		0			0
$407$			686.961i			0.0836643i
$408$		0			0
$409$			11986.2i			1.44910i	0.689223	+	0.724549i	$0.257952\pi$
$409$			11986.2i			1.44910i	−0.689223	+	0.724549i	$0.742048\pi$
$410$		0			0
$411$	4100.34			0.492105
$412$		0			0
$413$	−7439.92	−	5343.98i	−0.886427	−	0.636708i
$414$		0			0
$415$		−	21974.1i		−	2.59919i
$416$		0			0
$417$	5310.31			0.623614
$418$		0			0
$419$	−4098.42			−0.477854			−0.238927	−	0.971038i	$-0.576796\pi$
$419$	−4098.42			−0.477854			−0.238927	+	0.971038i	$0.576796\pi$
$420$		0			0
$421$	−318.996			−0.0369285			−0.0184643	−	0.999830i	$-0.505878\pi$
$421$	−318.996			−0.0369285			−0.0184643	+	0.999830i	$0.505878\pi$
$422$		0			0
$423$	−4299.34			−0.494186
$424$		0			0
$425$		−	12624.7i		−	1.44091i
$426$		0			0
$427$	3177.43	−	4423.63i	0.360109	−	0.501346i
$428$		0			0
$429$	5511.96			0.620326
$430$		0			0
$431$			9939.59i			1.11084i	0.831569	+	0.555422i	$0.187443\pi$
$431$			9939.59i			1.11084i	−0.831569	+	0.555422i	$0.812557\pi$
$432$		0			0
$433$		−	1254.43i		−	0.139225i	−0.997574	−	0.0696123i	$-0.977824\pi$
$433$		−	1254.43i		−	0.139225i	0.997574	−	0.0696123i	$-0.0221762\pi$
$434$		0			0
$435$		−	9859.41i		−	1.08672i
$436$		0			0
$437$			1627.60i			0.178166i
$438$		0			0
$439$	12933.4			1.40610			0.703049	−	0.711141i	$-0.251821\pi$
$439$	12933.4			1.40610			0.703049	+	0.711141i	$0.251821\pi$
$440$		0			0
$441$	985.738	+	2925.39i	0.106440	+	0.315882i
$442$		0			0
$443$		−	596.175i		−	0.0639394i	−0.999489	−	0.0319697i	$-0.989822\pi$
$443$		−	596.175i		−	0.0639394i	0.999489	−	0.0319697i	$-0.0101780\pi$
$444$		0			0
$445$	10259.0			1.09286
$446$		0			0
$447$	5975.91			0.632328
$448$		0			0
$449$	−3428.69			−0.360378			−0.180189	−	0.983632i	$-0.557671\pi$
$449$	−3428.69			−0.360378			−0.180189	+	0.983632i	$0.557671\pi$
$450$		0			0
$451$	2634.44			0.275058
$452$		0			0
$453$			3352.34i			0.347697i
$454$		0			0
$455$	−7185.65	−	5161.34i	−0.740370	−	0.531797i
$456$		0			0
$457$	19011.5			1.94600			0.973000	−	0.230807i	$-0.0741367\pi$
$457$	19011.5			1.94600			0.973000	+	0.230807i	$0.0741367\pi$
$458$		0			0
$459$		−	2238.41i		−	0.227626i
$460$		0			0
$461$		−	13187.0i		−	1.33227i	−0.745830	−	0.666137i	$-0.767947\pi$
$461$		−	13187.0i		−	1.33227i	0.745830	−	0.666137i	$-0.232053\pi$
$462$		0			0
$463$		−	7425.75i		−	0.745364i	−0.927959	−	0.372682i	$-0.878438\pi$
$463$		−	7425.75i		−	0.745364i	0.927959	−	0.372682i	$-0.121562\pi$
$464$		0			0
$465$		−	7668.87i		−	0.764807i
$466$		0			0
$467$	−11474.1			−1.13695			−0.568476	−	0.822700i	$-0.692467\pi$
$467$	−11474.1			−1.13695			−0.568476	+	0.822700i	$0.692467\pi$
$468$		0			0
$469$	2245.93	−	3126.80i	0.221125	−	0.307851i
$470$		0			0
$471$		−	2274.05i		−	0.222468i
$472$		0			0
$473$	26418.4			2.56811
$474$		0			0
$475$	2607.58			0.251882
$476$		0			0
$477$	317.073			0.0304356
$478$		0			0
$479$	−17924.0			−1.70975			−0.854873	−	0.518837i	$-0.826365\pi$
$479$	−17924.0			−1.70975			−0.854873	+	0.518837i	$0.826365\pi$
$480$		0			0
$481$			307.713i			0.0291694i
$482$		0			0
$483$	3080.90	−	4289.25i	0.290240	−	0.404073i
$484$		0			0
$485$	−1733.04			−0.162254
$486$		0			0
$487$			8161.93i			0.759450i	0.925099	+	0.379725i	$0.123981\pi$
$487$			8161.93i			0.759450i	−0.925099	+	0.379725i	$0.876019\pi$
$488$		0			0
$489$		−	443.539i		−	0.0410174i
$490$		0			0
$491$			17622.4i			1.61973i	0.586616	+	0.809865i	$0.300460\pi$
$491$			17622.4i			1.61973i	−0.586616	+	0.809865i	$0.699540\pi$
$492$		0			0
$493$			16362.4i			1.49478i
$494$		0			0
$495$	9598.15			0.871525
$496$		0			0
$497$	5770.02	−	8033.05i	0.520766	−	0.725013i
$498$		0			0
$499$		−	6772.04i		−	0.607531i	−0.952747	−	0.303766i	$-0.901756\pi$
$499$		−	6772.04i		−	0.607531i	0.952747	−	0.303766i	$-0.0982440\pi$
$500$		0			0
$501$	−5225.75			−0.466007
$502$		0			0
$503$	−15340.8			−1.35987			−0.679935	−	0.733273i	$-0.737992\pi$
$503$	−15340.8			−1.35987			−0.679935	+	0.733273i	$0.737992\pi$
$504$		0			0
$505$	29212.7			2.57416
$506$		0			0
$507$	−4122.01			−0.361075
$508$		0			0
$509$			7729.91i			0.673128i	0.941661	+	0.336564i	$0.109265\pi$
$509$			7729.91i			0.673128i	−0.941661	+	0.336564i	$0.890735\pi$
$510$		0			0
$511$	−6291.12	+	8758.53i	−0.544624	+	0.758228i
$512$		0			0
$513$	462.336			0.0397907
$514$		0			0
$515$		−	5168.20i		−	0.442210i
$516$		0			0
$517$			30594.6i			2.60261i
$518$		0			0
$519$		−	8060.20i		−	0.681703i
$520$		0			0
$521$		−	9454.94i		−	0.795064i	−0.917588	−	0.397532i	$-0.869867\pi$
$521$		−	9454.94i		−	0.795064i	0.917588	−	0.397532i	$-0.130133\pi$
$522$		0			0
$523$	−20468.5			−1.71133			−0.855665	−	0.517531i	$-0.826851\pi$
$523$	−20468.5			−1.71133			−0.855665	+	0.517531i	$0.826851\pi$
$524$		0			0
$525$	−6871.83	−	4935.94i	−0.571260	−	0.410327i
$526$		0			0
$527$			12727.0i			1.05199i
$528$		0			0
$529$	3132.47			0.257456
$530$		0			0
$531$	4451.47			0.363799
$532$		0			0
$533$	1180.05			0.0958984
$534$		0			0
$535$	8033.53			0.649196
$536$		0			0
$537$			3097.60i			0.248922i
$538$		0			0
$539$	20817.4	−	7014.62i	1.66358	−	0.560559i
$540$		0			0
$541$	14593.7			1.15976			0.579882	−	0.814700i	$-0.303099\pi$
$541$	14593.7			1.15976			0.579882	+	0.814700i	$0.303099\pi$
$542$		0			0
$543$		−	4825.58i		−	0.381373i
$544$		0			0
$545$			1208.22i			0.0949627i
$546$		0			0
$547$			8758.57i			0.684624i	0.939586	+	0.342312i	$0.111210\pi$
$547$			8758.57i			0.684624i	−0.939586	+	0.342312i	$0.888790\pi$
$548$		0			0
$549$			2646.76i			0.205758i
$550$		0			0
$551$	−3379.59			−0.261298
$552$		0			0
$553$	−3363.19	+	4682.24i	−0.258621	+	0.360053i
$554$		0			0
$555$			535.831i			0.0409815i
$556$		0			0
$557$	3314.26			0.252118			0.126059	−	0.992023i	$-0.459767\pi$
$557$	3314.26			0.252118			0.126059	+	0.992023i	$0.459767\pi$
$558$		0			0
$559$	11833.7			0.895369
$560$		0			0
$561$	−15928.8			−1.19878
$562$		0			0
$563$	−21115.0			−1.58062			−0.790310	−	0.612707i	$-0.790081\pi$
$563$	−21115.0			−1.58062			−0.790310	+	0.612707i	$0.790081\pi$
$564$		0			0
$565$		−	25743.4i		−	1.91687i
$566$		0			0
$567$	−1218.41	−	875.163i	−0.0902438	−	0.0648208i
$568$		0			0
$569$	−8726.10			−0.642912			−0.321456	−	0.946924i	$-0.604172\pi$
$569$	−8726.10			−0.642912			−0.321456	+	0.946924i	$0.604172\pi$
$570$		0			0
$571$			19948.3i			1.46202i	0.682369	+	0.731008i	$0.260950\pi$
$571$			19948.3i			1.46202i	−0.682369	+	0.731008i	$0.739050\pi$
$572$		0			0
$573$		−	5911.30i		−	0.430974i
$574$		0			0
$575$			14474.3i			1.04977i
$576$		0			0
$577$			4516.90i			0.325894i	0.986635	+	0.162947i	$0.0521000\pi$
$577$			4516.90i			0.325894i	−0.986635	+	0.162947i	$0.947900\pi$
$578$		0			0
$579$	−2899.87			−0.208142
$580$		0			0
$581$	19849.9	+	14257.9i	1.41740	+	1.01810i
$582$		0			0
$583$		−	2256.33i		−	0.160288i
$584$		0			0
$585$	4299.34			0.303856
$586$		0			0
$587$	−1629.66			−0.114588			−0.0572942	−	0.998357i	$-0.518247\pi$
$587$	−1629.66			−0.114588			−0.0572942	+	0.998357i	$0.518247\pi$
$588$		0			0
$589$	−2628.72			−0.183896
$590$		0			0
$591$	3205.58			0.223113
$592$		0			0
$593$			6205.32i			0.429717i	0.976645	+	0.214858i	$0.0689290\pi$
$593$			6205.32i			0.429717i	−0.976645	+	0.214858i	$0.931071\pi$
$594$		0			0
$595$	20765.5	+	14915.6i	1.43076	+	1.02770i
$596$		0			0
$597$	757.980			0.0519632
$598$		0			0
$599$		−	752.600i		−	0.0513362i	−0.999671	−	0.0256681i	$-0.991829\pi$
$599$		−	752.600i		−	0.0513362i	0.999671	−	0.0256681i	$-0.00817131\pi$
$600$		0			0
$601$			21945.7i			1.48949i	0.667348	+	0.744746i	$0.267430\pi$
$601$			21945.7i			1.48949i	−0.667348	+	0.744746i	$0.732570\pi$
$602$		0			0
$603$			1870.83i			0.126345i
$604$		0			0
$605$		−	46138.1i		−	3.10046i
$606$		0			0
$607$	−1770.10			−0.118363			−0.0591815	−	0.998247i	$-0.518849\pi$
$607$	−1770.10			−0.118363			−0.0591815	+	0.998247i	$0.518849\pi$
$608$		0			0
$609$	8906.32	+	6397.28i	0.592615	+	0.425666i
$610$		0			0
$611$			13704.3i			0.907395i
$612$		0			0
$613$	22582.9			1.48795			0.743977	−	0.668205i	$-0.232937\pi$
$613$	22582.9			1.48795			0.743977	+	0.668205i	$0.232937\pi$
$614$		0			0
$615$	2054.87			0.134732
$616$		0			0
$617$	12035.2			0.785279			0.392639	−	0.919692i	$-0.371562\pi$
$617$	12035.2			0.785279			0.392639	+	0.919692i	$0.371562\pi$
$618$		0			0
$619$	875.313			0.0568365			0.0284183	−	0.999596i	$-0.490953\pi$
$619$	875.313			0.0568365			0.0284183	+	0.999596i	$0.490953\pi$
$620$		0			0
$621$			2566.35i			0.165836i
$622$		0			0
$623$	−6656.54	+	9267.27i	−0.428072	+	0.595964i
$624$		0			0
$625$	−11470.7			−0.734125
$626$		0			0
$627$		−	3290.03i		−	0.209556i
$628$		0			0
$629$		−	889.248i		−	0.0563698i
$630$		0			0
$631$			18162.8i			1.14588i	0.819597	+	0.572940i	$0.194197\pi$
$631$			18162.8i			1.14588i	−0.819597	+	0.572940i	$0.805803\pi$
$632$		0			0
$633$		−	13818.6i		−	0.867679i
$634$		0			0
$635$	−33016.2			−2.06332
$636$		0			0
$637$	9324.81	−	3142.09i	0.580004	−	0.195438i
$638$		0			0
$639$			4806.36i			0.297553i
$640$		0			0
$641$	2528.87			0.155826			0.0779128	−	0.996960i	$-0.475174\pi$
$641$	2528.87			0.155826			0.0779128	+	0.996960i	$0.475174\pi$
$642$		0			0
$643$	22947.8			1.40742			0.703712	−	0.710485i	$-0.251524\pi$
$643$	22947.8			1.40742			0.703712	+	0.710485i	$0.251524\pi$
$644$		0			0
$645$	20606.4			1.25795
$646$		0			0
$647$	−8556.53			−0.519926			−0.259963	−	0.965619i	$-0.583710\pi$
$647$	−8556.53			−0.519926			−0.259963	+	0.965619i	$0.583710\pi$
$648$		0			0
$649$		−	31677.2i		−	1.91593i
$650$		0			0
$651$	6927.53	+	4975.95i	0.417068	+	0.299574i
$652$		0			0
$653$	18532.9			1.11064			0.555319	−	0.831637i	$-0.312596\pi$
$653$	18532.9			1.11064			0.555319	+	0.831637i	$0.312596\pi$
$654$		0			0
$655$			6727.04i			0.401293i
$656$		0			0
$657$		−	5240.43i		−	0.311185i
$658$		0			0
$659$		−	20893.8i		−	1.23506i	−0.786546	−	0.617532i	$-0.788133\pi$
$659$		−	20893.8i		−	1.23506i	0.786546	−	0.617532i	$-0.211867\pi$
$660$		0			0
$661$		−	5459.18i		−	0.321237i	−0.987017	−	0.160618i	$-0.948651\pi$
$661$		−	5459.18i		−	0.321237i	0.987017	−	0.160618i	$-0.0513488\pi$
$662$		0			0
$663$	−7135.04			−0.417952
$664$		0			0
$665$	−3080.76	+	4289.04i	−0.179649	+	0.250108i
$666$		0			0
$667$		−	18759.6i		−	1.08902i
$668$		0			0
$669$	5782.95			0.334203
$670$		0			0
$671$	18834.6			1.08361
$672$		0			0
$673$	1716.03			0.0982882			0.0491441	−	0.998792i	$-0.484351\pi$
$673$	1716.03			0.0982882			0.0491441	+	0.998792i	$0.484351\pi$
$674$		0			0
$675$	4111.57			0.234451
$676$		0			0
$677$			17235.8i			0.978474i	0.872151	+	0.489237i	$0.162725\pi$
$677$			17235.8i			0.978474i	−0.872151	+	0.489237i	$0.837275\pi$
$678$		0			0
$679$	1124.48	−	1565.51i	0.0635548	−	0.0884813i
$680$		0			0
$681$	12191.1			0.686000
$682$		0			0
$683$			19454.4i			1.08990i	0.838468	+	0.544951i	$0.183452\pi$
$683$			19454.4i			1.08990i	−0.838468	+	0.544951i	$0.816548\pi$
$684$		0			0
$685$		−	22759.3i		−	1.26947i
$686$		0			0
$687$		−	12289.5i		−	0.682492i
$688$		0			0
$689$		−	1010.69i		−	0.0558840i
$690$		0			0
$691$	17230.7			0.948604			0.474302	−	0.880362i	$-0.342700\pi$
$691$	17230.7			0.948604			0.474302	+	0.880362i	$0.342700\pi$
$692$		0			0
$693$	−6227.76	+	8670.31i	−0.341375	+	0.475264i
$694$		0			0
$695$		−	29475.3i		−	1.60872i
$696$		0			0
$697$	−3410.20			−0.185323
$698$		0			0
$699$	−5850.25			−0.316562
$700$		0			0
$701$	22169.2			1.19447			0.597233	−	0.802068i	$-0.296267\pi$
$701$	22169.2			1.19447			0.597233	+	0.802068i	$0.296267\pi$
$702$		0			0
$703$	183.671			0.00985388
$704$		0			0
$705$			23863.8i			1.27484i
$706$		0			0
$707$	−18954.7	+	26388.8i	−1.00829	+	1.40375i
$708$		0			0
$709$	−15665.5			−0.829800			−0.414900	−	0.909867i	$-0.636183\pi$
$709$	−15665.5			−0.829800			−0.414900	+	0.909867i	$0.636183\pi$
$710$		0			0
$711$		−	2801.49i		−	0.147770i
$712$		0			0
$713$		−	14591.6i		−	0.766424i
$714$		0			0
$715$		−	30594.6i		−	1.60024i
$716$		0			0
$717$		−	8584.59i		−	0.447137i
$718$		0			0
$719$	−35225.3			−1.82709			−0.913547	−	0.406733i	$-0.866668\pi$
$719$	−35225.3			−1.82709			−0.913547	+	0.406733i	$0.866668\pi$
$720$		0			0
$721$	4668.60	+	3353.39i	0.241148	+	0.173213i
$722$		0			0
$723$		−	12960.3i		−	0.666667i
$724$		0			0
$725$	−30054.8			−1.53960
$726$		0			0
$727$	−12778.0			−0.651872			−0.325936	−	0.945392i	$-0.605679\pi$
$727$	−12778.0			−0.651872			−0.325936	+	0.945392i	$0.605679\pi$
$728$		0			0
$729$	729.000			0.0370370
$730$		0			0
$731$	−34197.7			−1.73030
$732$		0			0
$733$		−	9725.54i		−	0.490069i	−0.969514	−	0.245035i	$-0.921201\pi$
$733$		−	9725.54i		−	0.490069i	0.969514	−	0.245035i	$-0.0787994\pi$
$734$		0			0
$735$	16237.6	−	5471.42i	0.814875	−	0.274580i
$736$		0			0
$737$	13313.1			0.665391
$738$		0			0
$739$		−	18977.4i		−	0.944648i	−0.881425	−	0.472324i	$-0.843415\pi$
$739$		−	18977.4i		−	0.944648i	0.881425	−	0.472324i	$-0.156585\pi$
$740$		0			0
$741$		−	1473.72i		−	0.0730612i
$742$		0			0
$743$		−	18688.2i		−	0.922752i	−0.887205	−	0.461376i	$-0.847356\pi$
$743$		−	18688.2i		−	0.922752i	0.887205	−	0.461376i	$-0.152644\pi$
$744$		0			0
$745$		−	33169.7i		−	1.63120i
$746$		0			0
$747$	−11876.6			−0.581718
$748$		0			0
$749$	−5212.56	+	7256.95i	−0.254289	+	0.354023i
$750$		0			0
$751$			19229.5i			0.934345i	0.884166	+	0.467173i	$0.154727\pi$
$751$			19229.5i			0.934345i	−0.884166	+	0.467173i	$0.845273\pi$
$752$		0			0
$753$	−4100.23			−0.198434
$754$		0			0
$755$	18607.4			0.896946
$756$		0			0
$757$	−35890.6			−1.72321			−0.861603	−	0.507584i	$-0.830539\pi$
$757$	−35890.6			−1.72321			−0.861603	+	0.507584i	$0.830539\pi$
$758$		0			0
$759$	18262.5			0.873367
$760$		0			0
$761$			16455.3i			0.783843i	0.919999	+	0.391922i	$0.128190\pi$
$761$			16455.3i			0.783843i	−0.919999	+	0.391922i	$0.871810\pi$
$762$		0			0
$763$	−1091.43	−	783.956i	−0.0517855	−	0.0371968i
$764$		0			0
$765$	−12424.5			−0.587200
$766$		0			0
$767$		−	14189.3i		−	0.667986i
$768$		0			0
$769$		−	32125.7i		−	1.50648i	−0.657746	−	0.753240i	$-0.728490\pi$
$769$		−	32125.7i		−	1.50648i	0.657746	−	0.753240i	$-0.271510\pi$
$770$		0			0
$771$		−	17979.6i		−	0.839845i
$772$		0			0
$773$			302.967i			0.0140970i	0.999975	+	0.00704850i	$0.00224363\pi$
$773$			302.967i			0.0140970i	−0.999975	+	0.00704850i	$0.997756\pi$
$774$		0			0
$775$	−23377.3			−1.08353
$776$		0			0
$777$	−484.033	−	347.674i	−0.0223482	−	0.0160524i
$778$		0			0
$779$		−	704.364i		−	0.0323959i
$780$		0			0
$781$	34202.6			1.56705
$782$		0			0
$783$	−5328.85			−0.243215
$784$		0			0
$785$	−12622.3			−0.573897
$786$		0			0
$787$	23519.9			1.06531			0.532653	−	0.846334i	$-0.321195\pi$
$787$	23519.9			1.06531			0.532653	+	0.846334i	$0.321195\pi$
$788$		0			0
$789$			4910.86i			0.221586i
$790$		0			0
$791$	23254.8	+	16703.6i	1.04532	+	0.750835i
$792$		0			0
$793$	8436.67			0.377800
$794$		0			0
$795$		−	1759.94i		−	0.0785141i
$796$		0			0
$797$		−	7210.03i		−	0.320442i	−0.987081	−	0.160221i	$-0.948779\pi$
$797$		−	7210.03i		−	0.320442i	0.987081	−	0.160221i	$-0.0512207\pi$
$798$		0			0
$799$		−	39603.6i		−	1.75354i
$800$		0			0
$801$		−	5544.82i		−	0.244590i
$802$		0			0
$803$	−37291.5			−1.63884
$804$		0			0
$805$	−23807.8	−	17100.8i	−1.04238	−	0.748725i
$806$		0			0
$807$		−	22327.2i		−	0.973923i
$808$		0			0
$809$	40393.4			1.75544			0.877722	−	0.479170i	$-0.159062\pi$
$809$	40393.4			1.75544			0.877722	+	0.479170i	$0.159062\pi$
$810$		0			0
$811$	29435.6			1.27450			0.637252	−	0.770655i	$-0.280071\pi$
$811$	29435.6			1.27450			0.637252	+	0.770655i	$0.280071\pi$
$812$		0			0
$813$	16858.0			0.727227
$814$		0			0
$815$	−2461.90			−0.105812
$816$		0			0
$817$		−	7063.41i		−	0.302469i
$818$		0			0
$819$	−2789.62	+	3883.73i	−0.119020	+	0.165700i
$820$		0			0
$821$	−12341.4			−0.524624			−0.262312	−	0.964983i	$-0.584485\pi$
$821$	−12341.4			−0.524624			−0.262312	+	0.964983i	$0.584485\pi$
$822$		0			0
$823$			22102.8i			0.936153i	0.883688	+	0.468077i	$0.155053\pi$
$823$			22102.8i			0.936153i	−0.883688	+	0.468077i	$0.844947\pi$
$824$		0			0
$825$		−	29258.4i		−	1.23472i
$826$		0			0
$827$		−	3867.63i		−	0.162625i	−0.996689	−	0.0813124i	$-0.974089\pi$
$827$		−	3867.63i		−	0.162625i	0.996689	−	0.0813124i	$-0.0259111\pi$
$828$		0			0
$829$			11642.7i			0.487778i	0.969803	+	0.243889i	$0.0784233\pi$
$829$			11642.7i			0.487778i	−0.969803	+	0.243889i	$0.921577\pi$
$830$		0			0
$831$	−2094.72			−0.0874429
$832$		0			0
$833$	−26947.4	+	9080.19i	−1.12086	+	0.377683i
$834$		0			0
$835$			29006.0i			1.20215i
$836$		0			0
$837$	−4144.90			−0.171169
$838$		0			0
$839$	−1522.64			−0.0626546			−0.0313273	−	0.999509i	$-0.509973\pi$
$839$	−1522.64			−0.0626546			−0.0313273	+	0.999509i	$0.509973\pi$
$840$		0			0
$841$	14563.9			0.597152
$842$		0			0
$843$	15081.0			0.616152
$844$		0			0
$845$			22879.5i			0.931456i
$846$		0			0
$847$	41678.0	+	29936.7i	1.69076	+	1.21445i
$848$		0			0
$849$	10788.5			0.436114
$850$		0			0
$851$			1019.53i			0.0410681i
$852$		0			0
$853$		−	16334.5i		−	0.655664i	−0.944736	−	0.327832i	$-0.893682\pi$
$853$		−	16334.5i		−	0.655664i	0.944736	−	0.327832i	$-0.106318\pi$
$854$		0			0
$855$		−	2566.23i		−	0.102647i
$856$		0			0
$857$		−	40489.7i		−	1.61389i	−0.590627	−	0.806945i	$-0.701120\pi$
$857$		−	40489.7i		−	1.61389i	0.590627	−	0.806945i	$-0.298880\pi$
$858$		0			0
$859$	−3963.14			−0.157416			−0.0787080	−	0.996898i	$-0.525079\pi$
$859$	−3963.14			−0.157416			−0.0787080	+	0.996898i	$0.525079\pi$
$860$		0			0
$861$	−1333.30	+	1856.23i	−0.0527744	+	0.0734728i
$862$		0			0
$863$			38486.3i			1.51806i	0.651054	+	0.759031i	$0.274327\pi$
$863$			38486.3i			1.51806i	−0.651054	+	0.759031i	$0.725673\pi$
$864$		0			0
$865$	−44738.8			−1.75857
$866$		0			0
$867$	5880.29			0.230340
$868$		0			0
$869$	−19935.7			−0.778221
$870$		0			0
$871$	5963.37			0.231988
$872$		0			0
$873$			936.681i			0.0363137i
$874$		0			0
$875$	−4908.14	+	6833.13i	−0.189629	+	0.264002i
$876$		0			0
$877$	7715.19			0.297062			0.148531	−	0.988908i	$-0.452546\pi$
$877$	7715.19			0.297062			0.148531	+	0.988908i	$0.452546\pi$
$878$		0			0
$879$		−	12285.1i		−	0.471405i
$880$		0			0
$881$			10746.0i			0.410946i	0.978663	+	0.205473i	$0.0658732\pi$
$881$			10746.0i			0.410946i	−0.978663	+	0.205473i	$0.934127\pi$
$882$		0			0
$883$		−	3332.31i		−	0.127000i	−0.997982	−	0.0635000i	$-0.979774\pi$
$883$		−	3332.31i		−	0.127000i	0.997982	−	0.0635000i	$-0.0202263\pi$
$884$		0			0
$885$		−	24708.3i		−	0.938485i
$886$		0			0
$887$	−29194.6			−1.10514			−0.552569	−	0.833467i	$-0.686353\pi$
$887$	−29194.6			−1.10514			−0.552569	+	0.833467i	$0.686353\pi$
$888$		0			0
$889$	21422.6	−	29824.6i	0.808200	−	1.12518i
$890$		0			0
$891$		−	5187.65i		−	0.195054i
$892$		0			0
$893$	8179.99			0.306532
$894$		0			0
$895$	17193.5			0.642139
$896$		0			0
$897$	8180.37			0.304498
$898$		0			0
$899$	30298.5			1.12404
$900$		0			0
$901$			2920.74i			0.107996i
$902$		0			0
$903$	−13370.4	+	18614.4i	−0.492736	+	0.685989i
$904$		0			0
$905$	−26784.8			−0.983820
$906$		0			0
$907$			42775.0i			1.56595i	0.622051	+	0.782976i	$0.286299\pi$
$907$			42775.0i			1.56595i	−0.622051	+	0.782976i	$0.713701\pi$
$908$		0			0
$909$		−	15789.0i		−	0.576115i
$910$		0			0
$911$		−	43757.7i		−	1.59139i	−0.605697	−	0.795696i	$-0.707105\pi$
$911$		−	43757.7i		−	1.59139i	0.605697	−	0.795696i	$-0.292895\pi$
$912$		0			0
$913$			84515.5i			3.06359i
$914$		0			0
$915$	14691.1			0.530788
$916$		0			0
$917$	−6076.75	−	4364.84i	−0.218835	−	0.157186i
$918$		0			0
$919$		−	2327.16i		−	0.0835322i	−0.999127	−	0.0417661i	$-0.986702\pi$
$919$		−	2327.16i		−	0.0835322i	0.999127	−	0.0417661i	$-0.0132984\pi$
$920$		0			0
$921$	−18642.8			−0.666994
$922$		0			0
$923$	15320.5			0.546349
$924$		0			0
$925$	1633.39			0.0580602
$926$		0			0
$927$	−2793.33			−0.0989698
$928$		0			0
$929$			44503.1i			1.57169i	0.618423	+	0.785845i	$0.287772\pi$
$929$			44503.1i			1.57169i	−0.618423	+	0.785845i	$0.712228\pi$
$930$		0			0
$931$	−1875.48	−	5565.89i	−0.0660219	−	0.195934i
$932$		0			0
$933$	−12134.8			−0.425804
$934$		0			0
$935$			88414.0i			3.09246i
$936$		0			0
$937$			19995.0i			0.697128i	0.937285	+	0.348564i	$0.113331\pi$
$937$			19995.0i			0.697128i	−0.937285	+	0.348564i	$0.886669\pi$
$938$		0			0
$939$			20795.5i			0.722722i
$940$		0			0
$941$			5492.38i			0.190272i	0.995464	+	0.0951362i	$0.0303287\pi$
$941$			5492.38i			0.190272i	−0.995464	+	0.0951362i	$0.969671\pi$
$942$		0			0
$943$	3909.81			0.135017
$944$		0			0
$945$	−4857.66	+	6762.86i	−0.167217	+	0.232800i
$946$		0			0
$947$		−	25553.9i		−	0.876863i	−0.898765	−	0.438432i	$-0.855534\pi$
$947$		−	25553.9i		−	0.876863i	0.898765	−	0.438432i	$-0.144466\pi$
$948$		0			0
$949$	−16704.1			−0.571379
$950$		0			0
$951$	11899.2			0.405738
$952$		0			0
$953$	−39066.4			−1.32790			−0.663948	−	0.747779i	$-0.731120\pi$
$953$	−39066.4			−1.32790			−0.663948	+	0.747779i	$0.731120\pi$
$954$		0			0
$955$	−32811.2			−1.11177
$956$		0			0
$957$			37920.7i			1.28088i
$958$		0			0
$959$	20559.2	+	14767.4i	0.692274	+	0.497250i
$960$		0			0
$961$	−6224.19			−0.208928
$962$		0			0
$963$		−	4342.00i		−	0.145295i
$964$		0			0
$965$			16095.9i			0.536939i
$966$		0			0
$967$		−	17718.4i		−	0.589229i	−0.955616	−	0.294614i	$-0.904809\pi$
$967$		−	17718.4i		−	0.589229i	0.955616	−	0.294614i	$-0.0951912\pi$
$968$		0			0
$969$			4258.84i			0.141191i
$970$		0			0
$971$	48730.8			1.61055			0.805276	−	0.592900i	$-0.202017\pi$
$971$	48730.8			1.61055			0.805276	+	0.592900i	$0.202017\pi$
$972$		0			0
$973$	26626.0	+	19125.0i	0.877276	+	0.630134i
$974$		0			0
$975$		−	13105.8i		−	0.430485i
$976$		0			0
$977$	18189.4			0.595629			0.297815	−	0.954624i	$-0.403742\pi$
$977$	18189.4			0.595629			0.297815	+	0.954624i	$0.403742\pi$
$978$		0			0
$979$	−39457.6			−1.28812
$980$		0			0
$981$	653.026			0.0212533
$982$		0			0
$983$	−8921.53			−0.289474			−0.144737	−	0.989470i	$-0.546234\pi$
$983$	−8921.53			−0.289474			−0.144737	+	0.989470i	$0.546234\pi$
$984$		0			0
$985$		−	17792.8i		−	0.575560i
$986$		0			0
$987$	−21556.9	−	15484.0i	−0.695203	−	0.499354i
$988$		0			0
$989$	39207.9			1.26060
$990$		0			0
$991$		−	38882.1i		−	1.24635i	−0.782083	−	0.623174i	$-0.785843\pi$
$991$		−	38882.1i		−	1.24635i	0.782083	−	0.623174i	$-0.214157\pi$
$992$		0			0
$993$			29849.5i			0.953924i
$994$		0			0
$995$		−	4207.23i		−	0.134048i
$996$		0			0
$997$			37847.7i			1.20225i	0.799153	+	0.601127i	$0.205281\pi$
$997$			37847.7i			1.20225i	−0.799153	+	0.601127i	$0.794719\pi$
$998$		0			0
$999$	289.608			0.00917196

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1344.4.b.g.895.11		12
4.3	odd	2		1344.4.b.h.895.11		12
7.6	odd	2		1344.4.b.h.895.2		12
8.3	odd	2		84.4.b.a.55.12	yes	12
8.5	even	2		84.4.b.b.55.11	yes	12
24.5	odd	2		252.4.b.e.55.2		12
24.11	even	2		252.4.b.f.55.1		12
28.27	even	2	inner	1344.4.b.g.895.2		12
56.13	odd	2		84.4.b.a.55.11	✓	12
56.27	even	2		84.4.b.b.55.12	yes	12
168.83	odd	2		252.4.b.e.55.1		12
168.125	even	2		252.4.b.f.55.2		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
84.4.b.a.55.11	✓	12	56.13	odd	2
84.4.b.a.55.12	yes	12	8.3	odd	2
84.4.b.b.55.11	yes	12	8.5	even	2
84.4.b.b.55.12	yes	12	56.27	even	2
252.4.b.e.55.1		12	168.83	odd	2
252.4.b.e.55.2		12	24.5	odd	2
252.4.b.f.55.1		12	24.11	even	2
252.4.b.f.55.2		12	168.125	even	2
1344.4.b.g.895.2		12	28.27	even	2	inner
1344.4.b.g.895.11		12	1.1	even	1	trivial
1344.4.b.h.895.2		12	7.6	odd	2
1344.4.b.h.895.11		12	4.3	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.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-3.00000 q^{3} +16.6517i q^{5} +(-15.0420 - 10.8045i) q^{7} +9.00000 q^{9} +O(q^{10})\)	\(q-3.00000 q^{3} +16.6517i q^{5} +(-15.0420 - 10.8045i) q^{7} +9.00000 q^{9} -64.0450i q^{11} -28.6879i q^{13} -49.9552i q^{15} +82.9041i q^{17} -17.1236 q^{19} +(45.1261 + 32.4134i) q^{21} -95.0501i q^{23} -152.281 q^{25} -27.0000 q^{27} +197.365 q^{29} +153.515 q^{31} +192.135i q^{33} +(179.913 - 250.476i) q^{35} -10.7262 q^{37} +86.0638i q^{39} +41.1342i q^{41} +412.497i q^{43} +149.866i q^{45} -477.704 q^{47} +(109.526 + 325.043i) q^{49} -248.712i q^{51} +35.2304 q^{53} +1066.46 q^{55} +51.3707 q^{57} +494.608 q^{59} +294.084i q^{61} +(-135.378 - 97.2403i) q^{63} +477.704 q^{65} +207.870i q^{67} +285.150i q^{69} +534.040i q^{71} -582.270i q^{73} +456.842 q^{75} +(-691.973 + 963.368i) q^{77} -311.277i q^{79} +81.0000 q^{81} -1319.63 q^{83} -1380.50 q^{85} -592.095 q^{87} -616.091i q^{89} +(-309.958 + 431.525i) q^{91} -460.545 q^{93} -285.137i q^{95} +104.076i q^{97} -576.405i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 36 q^{3} - 10 q^{7} + 108 q^{9}+O(q^{10}) \) 12 * q - 36 * q^3 - 10 * q^7 + 108 * q^9	\( 12 q - 36 q^{3} - 10 q^{7} + 108 q^{9} - 84 q^{19} + 30 q^{21} - 216 q^{25} - 324 q^{27} - 200 q^{29} + 384 q^{31} + 84 q^{35} + 244 q^{37} - 280 q^{47} - 424 q^{49} + 16 q^{53} - 212 q^{55} + 252 q^{57} + 1168 q^{59} - 90 q^{63} + 280 q^{65} + 648 q^{75} - 968 q^{77} + 972 q^{81} - 968 q^{83} + 852 q^{85} + 600 q^{87} + 1648 q^{91} - 1152 q^{93}+O(q^{100}) \) 12 * q - 36 * q^3 - 10 * q^7 + 108 * q^9 - 84 * q^19 + 30 * q^21 - 216 * q^25 - 324 * q^27 - 200 * q^29 + 384 * q^31 + 84 * q^35 + 244 * q^37 - 280 * q^47 - 424 * q^49 + 16 * q^53 - 212 * q^55 + 252 * q^57 + 1168 * q^59 - 90 * q^63 + 280 * q^65 + 648 * q^75 - 968 * q^77 + 972 * q^81 - 968 * q^83 + 852 * q^85 + 600 * q^87 + 1648 * q^91 - 1152 * q^93

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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