LMFDB - Embedded newform 1344.4.c.c.673.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1344,4,Mod(673,1344)]

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, 1, 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("1344.673");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1344 = 2^{6} \cdot 3 \cdot 7 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1344.c (of order $2$, degree $1$, 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:	$6$
Coefficient field:	6.0.14024243776.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} - 17x^{4} - 164x^{3} + 299x^{2} + 2466x + 13042 $ x^6 - 2x^5 - 17x^4 - 164x^3 + 299x^2 + 2466*x + 13042
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			673.5
Root			$-1.73614 - 4.18492i$ of defining polynomial
Character	$\chi$	$=$	1344.673
Dual form			1344.4.c.c.673.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.00000i q^{3} -2.89754i q^{5} +7.00000 q^{7} -9.00000 q^{9} +O(q^{10})$	$q+3.00000i q^{3} -2.89754i q^{5} +7.00000 q^{7} -9.00000 q^{9} +23.3214i q^{11} -14.3298i q^{13} +8.69263 q^{15} +42.8175 q^{17} -76.6120i q^{19} +21.0000i q^{21} -79.0225 q^{23} +116.604 q^{25} -27.0000i q^{27} -197.648i q^{29} -266.873 q^{31} -69.9643 q^{33} -20.2828i q^{35} +92.7624i q^{37} +42.9895 q^{39} -90.8175 q^{41} +425.641i q^{43} +26.0779i q^{45} +550.816 q^{47} +49.0000 q^{49} +128.453i q^{51} -255.831i q^{53} +67.5749 q^{55} +229.836 q^{57} -736.228i q^{59} -549.256i q^{61} -63.0000 q^{63} -41.5213 q^{65} -438.114i q^{67} -237.067i q^{69} -463.675 q^{71} -518.392 q^{73} +349.813i q^{75} +163.250i q^{77} +722.003 q^{79} +81.0000 q^{81} +1230.42i q^{83} -124.066i q^{85} +592.944 q^{87} +36.2772 q^{89} -100.309i q^{91} -800.620i q^{93} -221.987 q^{95} +536.018 q^{97} -209.893i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 42 q^{7} - 54 q^{9}+O(q^{10}) $ 6 * q + 42 * q^7 - 54 * q^9	$ 6 q + 42 q^{7} - 54 q^{9} - 24 q^{15} - 72 q^{17} - 196 q^{23} + 246 q^{25} + 104 q^{31} + 408 q^{33} - 348 q^{39} - 216 q^{41} - 408 q^{47} + 294 q^{49} + 720 q^{55} + 336 q^{57} - 378 q^{63} - 2472 q^{65} - 1164 q^{71} - 276 q^{73} + 1352 q^{79} + 486 q^{81} + 720 q^{87} + 2520 q^{89} + 4648 q^{95} - 2340 q^{97}+O(q^{100}) $ 6 * q + 42 * q^7 - 54 * q^9 - 24 * q^15 - 72 * q^17 - 196 * q^23 + 246 * q^25 + 104 * q^31 + 408 * q^33 - 348 * q^39 - 216 * q^41 - 408 * q^47 + 294 * q^49 + 720 * q^55 + 336 * q^57 - 378 * q^63 - 2472 * q^65 - 1164 * q^71 - 276 * q^73 + 1352 * q^79 + 486 * q^81 + 720 * q^87 + 2520 * q^89 + 4648 * q^95 - 2340 * q^97

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
$3$	3.00000i	0.577350i
$3$	3.00000i	0.577350i
$4$	0	0
$5$	− 2.89754i	− 0.259164i	−0.991569	−	0.129582i	$-0.958636\pi$
$5$	− 2.89754i	− 0.259164i	0.991569	−	0.129582i	$-0.0413636\pi$
$6$	0	0
$7$	7.00000	0.377964
$7$	7.00000	0.377964
$8$	0	0
$9$	−9.00000	−0.333333
$10$	0	0
$11$	23.3214i	0.639244i	0.947545	+	0.319622i	$0.103556\pi$
$11$	23.3214i	0.639244i	−0.947545	+	0.319622i	$0.896444\pi$
$12$	0	0
$13$	− 14.3298i	− 0.305722i	−0.988248	−	0.152861i	$-0.951151\pi$
$13$	− 14.3298i	− 0.305722i	0.988248	−	0.152861i	$-0.0488486\pi$
$14$	0	0
$15$	8.69263	0.149628
$16$	0	0
$17$	42.8175	0.610869	0.305435	−	0.952213i	$-0.401198\pi$
$17$	42.8175	0.610869	0.305435	+	0.952213i	$0.401198\pi$
$18$	0	0
$19$	− 76.6120i	− 0.925053i	−0.886605	−	0.462526i	$-0.846943\pi$
$19$	− 76.6120i	− 0.925053i	0.886605	−	0.462526i	$-0.153057\pi$
$20$	0	0
$21$	21.0000i	0.218218i
$22$	0	0
$23$	−79.0225	−0.716405	−0.358203	−	0.933644i	$-0.616610\pi$
$23$	−79.0225	−0.716405	−0.358203	+	0.933644i	$0.616610\pi$
$24$	0	0
$25$	116.604	0.932834
$26$	0	0
$27$	− 27.0000i	− 0.192450i
$28$	0	0
$29$	− 197.648i	− 1.26560i	−0.774317	−	0.632798i	$-0.781906\pi$
$29$	− 197.648i	− 1.26560i	0.774317	−	0.632798i	$-0.218094\pi$
$30$	0	0
$31$	−266.873	−1.54619	−0.773094	−	0.634291i	$-0.781292\pi$
$31$	−266.873	−1.54619	−0.773094	+	0.634291i	$0.781292\pi$
$32$	0	0
$33$	−69.9643	−0.369067
$34$	0	0
$35$	− 20.2828i	− 0.0979548i
$36$	0	0
$37$	92.7624i	0.412163i	0.978535	+	0.206082i	$0.0660712\pi$
$37$	92.7624i	0.412163i	−0.978535	+	0.206082i	$0.933929\pi$
$38$	0	0
$39$	42.9895	0.176508
$40$	0	0
$41$	−90.8175	−0.345935	−0.172967	−	0.984928i	$-0.555335\pi$
$41$	−90.8175	−0.345935	−0.172967	+	0.984928i	$0.555335\pi$
$42$	0	0
$43$	425.641i	1.50953i	0.655997	+	0.754763i	$0.272248\pi$
$43$	425.641i	1.50953i	−0.655997	+	0.754763i	$0.727752\pi$
$44$	0	0
$45$	26.0779i	0.0863880i
$46$	0	0
$47$	550.816	1.70946	0.854732	−	0.519070i	$-0.173722\pi$
$47$	550.816	1.70946	0.854732	+	0.519070i	$0.173722\pi$
$48$	0	0
$49$	49.0000	0.142857
$50$	0	0
$51$	128.453i	0.352686i
$52$	0	0
$53$	− 255.831i	− 0.663040i	−0.943448	−	0.331520i	$-0.892438\pi$
$53$	− 255.831i	− 0.663040i	0.943448	−	0.331520i	$-0.107562\pi$
$54$	0	0
$55$	67.5749	0.165669
$56$	0	0
$57$	229.836	0.534080
$58$	0	0
$59$	− 736.228i	− 1.62455i	−0.583271	−	0.812277i	$-0.698228\pi$
$59$	− 736.228i	− 1.62455i	0.583271	−	0.812277i	$-0.301772\pi$
$60$	0	0
$61$	− 549.256i	− 1.15287i	−0.817143	−	0.576435i	$-0.804443\pi$
$61$	− 549.256i	− 1.15287i	0.817143	−	0.576435i	$-0.195557\pi$
$62$	0	0
$63$	−63.0000	−0.125988
$64$	0	0
$65$	−41.5213	−0.0792321
$66$	0	0
$67$	− 438.114i	− 0.798867i	−0.916762	−	0.399434i	$-0.869207\pi$
$67$	− 438.114i	− 0.798867i	0.916762	−	0.399434i	$-0.130793\pi$
$68$	0	0
$69$	− 237.067i	− 0.413617i
$70$	0	0
$71$	−463.675	−0.775044	−0.387522	−	0.921860i	$-0.626669\pi$
$71$	−463.675	−0.775044	−0.387522	+	0.921860i	$0.626669\pi$
$72$	0	0
$73$	−518.392	−0.831140	−0.415570	−	0.909561i	$-0.636418\pi$
$73$	−518.392	−0.831140	−0.415570	+	0.909561i	$0.636418\pi$
$74$	0	0
$75$	349.813i	0.538572i
$76$	0	0
$77$	163.250i	0.241611i
$78$	0	0
$79$	722.003	1.02825	0.514125	−	0.857716i	$-0.328117\pi$
$79$	722.003	1.02825	0.514125	+	0.857716i	$0.328117\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	0	0
$83$	1230.42i	1.62718i	0.581438	+	0.813591i	$0.302490\pi$
$83$	1230.42i	1.62718i	−0.581438	+	0.813591i	$0.697510\pi$
$84$	0	0
$85$	− 124.066i	− 0.158315i
$86$	0	0
$87$	592.944	0.730693
$88$	0	0
$89$	36.2772	0.0432064	0.0216032	−	0.999767i	$-0.493123\pi$
$89$	36.2772	0.0432064	0.0216032	+	0.999767i	$0.493123\pi$
$90$	0	0
$91$	− 100.309i	− 0.115552i
$92$	0	0
$93$	− 800.620i	− 0.892692i
$94$	0	0
$95$	−221.987	−0.239741
$96$	0	0
$97$	536.018	0.561076	0.280538	−	0.959843i	$-0.409487\pi$
$97$	536.018	0.561076	0.280538	+	0.959843i	$0.409487\pi$
$98$	0	0
$99$	− 209.893i	− 0.213081i
$100$	0	0
$101$	− 1502.19i	− 1.47993i	−0.672645	−	0.739965i	$-0.734842\pi$
$101$	− 1502.19i	− 1.47993i	0.672645	−	0.739965i	$-0.265158\pi$
$102$	0	0
$103$	−788.165	−0.753983	−0.376991	−	0.926217i	$-0.623041\pi$
$103$	−788.165	−0.753983	−0.376991	+	0.926217i	$0.623041\pi$
$104$	0	0
$105$	60.8484	0.0565543
$106$	0	0
$107$	− 85.1395i	− 0.0769228i	−0.999260	−	0.0384614i	$-0.987754\pi$
$107$	− 85.1395i	− 0.0769228i	0.999260	−	0.0384614i	$-0.0122457\pi$
$108$	0	0
$109$	− 852.759i	− 0.749353i	−0.927156	−	0.374677i	$-0.877754\pi$
$109$	− 852.759i	− 0.749353i	0.927156	−	0.374677i	$-0.122246\pi$
$110$	0	0
$111$	−278.287	−0.237963
$112$	0	0
$113$	−1046.24	−0.870989	−0.435494	−	0.900191i	$-0.643426\pi$
$113$	−1046.24	−0.870989	−0.435494	+	0.900191i	$0.643426\pi$
$114$	0	0
$115$	228.971i	0.185667i
$116$	0	0
$117$	128.969i	0.101907i
$118$	0	0
$119$	299.723	0.230887
$120$	0	0
$121$	787.110	0.591368
$122$	0	0
$123$	− 272.453i	− 0.199725i
$124$	0	0
$125$	− 700.059i	− 0.500921i
$126$	0	0
$127$	2345.56	1.63886	0.819430	−	0.573180i	$-0.194290\pi$
$127$	2345.56	1.63886	0.819430	+	0.573180i	$0.194290\pi$
$128$	0	0
$129$	−1276.92	−0.871526
$130$	0	0
$131$	− 897.289i	− 0.598446i	−0.954183	−	0.299223i	$-0.903273\pi$
$131$	− 897.289i	− 0.598446i	0.954183	−	0.299223i	$-0.0967275\pi$
$132$	0	0
$133$	− 536.284i	− 0.349637i
$134$	0	0
$135$	−78.2337	−0.0498762
$136$	0	0
$137$	228.878	0.142733	0.0713664	−	0.997450i	$-0.477264\pi$
$137$	228.878	0.142733	0.0713664	+	0.997450i	$0.477264\pi$
$138$	0	0
$139$	− 1273.53i	− 0.777115i	−0.921424	−	0.388558i	$-0.872973\pi$
$139$	− 1273.53i	− 0.777115i	0.921424	−	0.388558i	$-0.127027\pi$
$140$	0	0
$141$	1652.45i	0.986959i
$142$	0	0
$143$	334.192	0.195431
$144$	0	0
$145$	−572.693	−0.327997
$146$	0	0
$147$	147.000i	0.0824786i
$148$	0	0
$149$	− 1542.40i	− 0.848042i	−0.905652	−	0.424021i	$-0.860618\pi$
$149$	− 1542.40i	− 0.848042i	0.905652	−	0.424021i	$-0.139382\pi$
$150$	0	0
$151$	2981.33	1.60674	0.803368	−	0.595483i	$-0.203039\pi$
$151$	2981.33	1.60674	0.803368	+	0.595483i	$0.203039\pi$
$152$	0	0
$153$	−385.358	−0.203623
$154$	0	0
$155$	773.277i	0.400717i
$156$	0	0
$157$	667.153i	0.339138i	0.985518	+	0.169569i	$0.0542375\pi$
$157$	667.153i	0.339138i	−0.985518	+	0.169569i	$0.945763\pi$
$158$	0	0
$159$	767.494	0.382807
$160$	0	0
$161$	−553.157	−0.270776
$162$	0	0
$163$	− 212.700i	− 0.102208i	−0.998693	−	0.0511042i	$-0.983726\pi$
$163$	− 212.700i	− 0.102208i	0.998693	−	0.0511042i	$-0.0162741\pi$
$164$	0	0
$165$	202.725i	0.0956490i
$166$	0	0
$167$	1880.82	0.871508	0.435754	−	0.900066i	$-0.356482\pi$
$167$	1880.82	0.871508	0.435754	+	0.900066i	$0.356482\pi$
$168$	0	0
$169$	1991.66	0.906534
$170$	0	0
$171$	689.508i	0.308351i
$172$	0	0
$173$	− 323.494i	− 0.142166i	−0.997470	−	0.0710832i	$-0.977354\pi$
$173$	− 323.494i	− 0.142166i	0.997470	−	0.0710832i	$-0.0226456\pi$
$174$	0	0
$175$	816.230	0.352578
$176$	0	0
$177$	2208.68	0.937937
$178$	0	0
$179$	− 919.312i	− 0.383869i	−0.981408	−	0.191935i	$-0.938524\pi$
$179$	− 919.312i	− 0.383869i	0.981408	−	0.191935i	$-0.0614762\pi$
$180$	0	0
$181$	− 2660.43i	− 1.09253i	−0.837611	−	0.546267i	$-0.816049\pi$
$181$	− 2660.43i	− 1.09253i	0.837611	−	0.546267i	$-0.183951\pi$
$182$	0	0
$183$	1647.77	0.665610
$184$	0	0
$185$	268.783	0.106818
$186$	0	0
$187$	998.567i	0.390494i
$188$	0	0
$189$	− 189.000i	− 0.0727393i
$190$	0	0
$191$	2127.87	0.806111	0.403055	−	0.915176i	$-0.367948\pi$
$191$	2127.87	0.806111	0.403055	+	0.915176i	$0.367948\pi$
$192$	0	0
$193$	−1014.67	−0.378434	−0.189217	−	0.981935i	$-0.560595\pi$
$193$	−1014.67	−0.378434	−0.189217	+	0.981935i	$0.560595\pi$
$194$	0	0
$195$	− 124.564i	− 0.0457447i
$196$	0	0
$197$	− 3373.89i	− 1.22020i	−0.792324	−	0.610101i	$-0.791129\pi$
$197$	− 3373.89i	− 1.22020i	0.792324	−	0.610101i	$-0.208871\pi$
$198$	0	0
$199$	2143.58	0.763591	0.381796	−	0.924247i	$-0.375306\pi$
$199$	2143.58	0.763591	0.381796	+	0.924247i	$0.375306\pi$
$200$	0	0
$201$	1314.34	0.461226
$202$	0	0
$203$	− 1383.54i	− 0.478351i
$204$	0	0
$205$	263.148i	0.0896538i
$206$	0	0
$207$	711.202	0.238802
$208$	0	0
$209$	1786.70	0.591334
$210$	0	0
$211$	1797.10i	0.586337i	0.956061	+	0.293169i	$0.0947097\pi$
$211$	1797.10i	0.586337i	−0.956061	+	0.293169i	$0.905290\pi$
$212$	0	0
$213$	− 1391.03i	− 0.447472i
$214$	0	0
$215$	1233.31	0.391215
$216$	0	0
$217$	−1868.11	−0.584404
$218$	0	0
$219$	− 1555.18i	− 0.479859i
$220$	0	0
$221$	− 613.568i	− 0.186756i
$222$	0	0
$223$	2899.76	0.870772	0.435386	−	0.900244i	$-0.356612\pi$
$223$	2899.76	0.870772	0.435386	+	0.900244i	$0.356612\pi$
$224$	0	0
$225$	−1049.44	−0.310945
$226$	0	0
$227$	2165.99i	0.633311i	0.948541	+	0.316655i	$0.102560\pi$
$227$	2165.99i	0.633311i	−0.948541	+	0.316655i	$0.897440\pi$
$228$	0	0
$229$	− 2365.76i	− 0.682680i	−0.939940	−	0.341340i	$-0.889119\pi$
$229$	− 2365.76i	− 0.682680i	0.939940	−	0.341340i	$-0.110881\pi$
$230$	0	0
$231$	−489.750	−0.139494
$232$	0	0
$233$	−2623.79	−0.737727	−0.368863	−	0.929484i	$-0.620253\pi$
$233$	−2623.79	−0.737727	−0.368863	+	0.929484i	$0.620253\pi$
$234$	0	0
$235$	− 1596.01i	− 0.443032i
$236$	0	0
$237$	2166.01i	0.593660i
$238$	0	0
$239$	2048.13	0.554319	0.277160	−	0.960824i	$-0.410607\pi$
$239$	2048.13	0.554319	0.277160	+	0.960824i	$0.410607\pi$
$240$	0	0
$241$	620.856	0.165945	0.0829727	−	0.996552i	$-0.473559\pi$
$241$	620.856	0.165945	0.0829727	+	0.996552i	$0.473559\pi$
$242$	0	0
$243$	243.000i	0.0641500i
$244$	0	0
$245$	− 141.980i	− 0.0370234i
$246$	0	0
$247$	−1097.84	−0.282809
$248$	0	0
$249$	−3691.26	−0.939454
$250$	0	0
$251$	2047.50i	0.514889i	0.966293	+	0.257444i	$0.0828804\pi$
$251$	2047.50i	0.514889i	−0.966293	+	0.257444i	$0.917120\pi$
$252$	0	0
$253$	− 1842.92i	− 0.457958i
$254$	0	0
$255$	372.197	0.0914035
$256$	0	0
$257$	7026.27	1.70540	0.852698	−	0.522404i	$-0.174965\pi$
$257$	7026.27	1.70540	0.852698	+	0.522404i	$0.174965\pi$
$258$	0	0
$259$	649.337i	0.155783i
$260$	0	0
$261$	1778.83i	0.421866i
$262$	0	0
$263$	−2670.59	−0.626143	−0.313072	−	0.949730i	$-0.601358\pi$
$263$	−2670.59	−0.626143	−0.313072	+	0.949730i	$0.601358\pi$
$264$	0	0
$265$	−741.282	−0.171836
$266$	0	0
$267$	108.831i	0.0249452i
$268$	0	0
$269$	− 6182.51i	− 1.40132i	−0.713497	−	0.700658i	$-0.752890\pi$
$269$	− 6182.51i	− 1.40132i	0.713497	−	0.700658i	$-0.247110\pi$
$270$	0	0
$271$	5620.24	1.25980	0.629899	−	0.776677i	$-0.283096\pi$
$271$	5620.24	1.25980	0.629899	+	0.776677i	$0.283096\pi$
$272$	0	0
$273$	300.927	0.0667139
$274$	0	0
$275$	2719.38i	0.596308i
$276$	0	0
$277$	2170.28i	0.470756i	0.971904	+	0.235378i	$0.0756329\pi$
$277$	2170.28i	0.470756i	−0.971904	+	0.235378i	$0.924367\pi$
$278$	0	0
$279$	2401.86	0.515396
$280$	0	0
$281$	−1338.93	−0.284248	−0.142124	−	0.989849i	$-0.545393\pi$
$281$	−1338.93	−0.284248	−0.142124	+	0.989849i	$0.545393\pi$
$282$	0	0
$283$	490.929i	0.103119i	0.998670	+	0.0515595i	$0.0164192\pi$
$283$	490.929i	0.103119i	−0.998670	+	0.0515595i	$0.983581\pi$
$284$	0	0
$285$	− 665.960i	− 0.138414i
$286$	0	0
$287$	−635.723	−0.130751
$288$	0	0
$289$	−3079.66	−0.626838
$290$	0	0
$291$	1608.05i	0.323937i
$292$	0	0
$293$	− 2137.88i	− 0.426266i	−0.977023	−	0.213133i	$-0.931633\pi$
$293$	− 2137.88i	− 0.426266i	0.977023	−	0.213133i	$-0.0683668\pi$
$294$	0	0
$295$	−2133.25	−0.421026
$296$	0	0
$297$	629.679	0.123022
$298$	0	0
$299$	1132.38i	0.219021i
$300$	0	0
$301$	2979.49i	0.570547i
$302$	0	0
$303$	4506.56	0.854438
$304$	0	0
$305$	−1591.49	−0.298782
$306$	0	0
$307$	− 4250.11i	− 0.790119i	−0.918656	−	0.395059i	$-0.870724\pi$
$307$	− 4250.11i	− 0.790119i	0.918656	−	0.395059i	$-0.129276\pi$
$308$	0	0
$309$	− 2364.50i	− 0.435312i
$310$	0	0
$311$	−4040.33	−0.736676	−0.368338	−	0.929692i	$-0.620073\pi$
$311$	−4040.33	−0.736676	−0.368338	+	0.929692i	$0.620073\pi$
$312$	0	0
$313$	4977.80	0.898919	0.449459	−	0.893301i	$-0.351617\pi$
$313$	4977.80	0.898919	0.449459	+	0.893301i	$0.351617\pi$
$314$	0	0
$315$	182.545i	0.0326516i
$316$	0	0
$317$	− 5261.97i	− 0.932309i	−0.884703	−	0.466154i	$-0.845639\pi$
$317$	− 5261.97i	− 0.932309i	0.884703	−	0.466154i	$-0.154361\pi$
$318$	0	0
$319$	4609.44	0.809025
$320$	0	0
$321$	255.418	0.0444114
$322$	0	0
$323$	− 3280.34i	− 0.565087i
$324$	0	0
$325$	− 1670.92i	− 0.285187i
$326$	0	0
$327$	2558.28	0.432639
$328$	0	0
$329$	3855.71	0.646116
$330$	0	0
$331$	− 3277.26i	− 0.544213i	−0.962267	−	0.272106i	$-0.912280\pi$
$331$	− 3277.26i	− 0.544213i	0.962267	−	0.272106i	$-0.0877202\pi$
$332$	0	0
$333$	− 834.861i	− 0.137388i
$334$	0	0
$335$	−1269.45	−0.207038
$336$	0	0
$337$	−5734.49	−0.926936	−0.463468	−	0.886114i	$-0.653395\pi$
$337$	−5734.49	−0.926936	−0.463468	+	0.886114i	$0.653395\pi$
$338$	0	0
$339$	− 3138.71i	− 0.502866i
$340$	0	0
$341$	− 6223.87i	− 0.988391i
$342$	0	0
$343$	343.000	0.0539949
$344$	0	0
$345$	−686.913	−0.107195
$346$	0	0
$347$	− 2507.23i	− 0.387882i	−0.981013	−	0.193941i	$-0.937873\pi$
$347$	− 2507.23i	− 0.387882i	0.981013	−	0.193941i	$-0.0621270\pi$
$348$	0	0
$349$	7914.51i	1.21391i	0.794737	+	0.606954i	$0.207609\pi$
$349$	7914.51i	1.21391i	−0.794737	+	0.606954i	$0.792391\pi$
$350$	0	0
$351$	−386.906	−0.0588362
$352$	0	0
$353$	−4249.64	−0.640752	−0.320376	−	0.947290i	$-0.603809\pi$
$353$	−4249.64	−0.640752	−0.320376	+	0.947290i	$0.603809\pi$
$354$	0	0
$355$	1343.52i	0.200864i
$356$	0	0
$357$	899.169i	0.133303i
$358$	0	0
$359$	−9073.37	−1.33391	−0.666956	−	0.745098i	$-0.732403\pi$
$359$	−9073.37	−1.33391	−0.666956	+	0.745098i	$0.732403\pi$
$360$	0	0
$361$	989.597	0.144277
$362$	0	0
$363$	2361.33i	0.341426i
$364$	0	0
$365$	1502.06i	0.215402i
$366$	0	0
$367$	1275.39	0.181402	0.0907012	−	0.995878i	$-0.471089\pi$
$367$	1275.39	0.181402	0.0907012	+	0.995878i	$0.471089\pi$
$368$	0	0
$369$	817.358	0.115312
$370$	0	0
$371$	− 1790.82i	− 0.250606i
$372$	0	0
$373$	− 3863.95i	− 0.536375i	−0.963367	−	0.268187i	$-0.913575\pi$
$373$	− 3863.95i	− 0.536375i	0.963367	−	0.268187i	$-0.0864246\pi$
$374$	0	0
$375$	2100.18	0.289207
$376$	0	0
$377$	−2832.26	−0.386920
$378$	0	0
$379$	14365.9i	1.94703i	0.228624	+	0.973515i	$0.426578\pi$
$379$	14365.9i	1.94703i	−0.228624	+	0.973515i	$0.573422\pi$
$380$	0	0
$381$	7036.69i	0.946196i
$382$	0	0
$383$	−13592.4	−1.81342	−0.906710	−	0.421754i	$-0.861415\pi$
$383$	−13592.4	−1.81342	−0.906710	+	0.421754i	$0.861415\pi$
$384$	0	0
$385$	473.024	0.0626170
$386$	0	0
$387$	− 3830.77i	− 0.503176i
$388$	0	0
$389$	− 8501.30i	− 1.10805i	−0.832499	−	0.554027i	$-0.813090\pi$
$389$	− 8501.30i	− 1.10805i	0.832499	−	0.554027i	$-0.186910\pi$
$390$	0	0
$391$	−3383.55	−0.437630
$392$	0	0
$393$	2691.87	0.345513
$394$	0	0
$395$	− 2092.03i	− 0.266485i
$396$	0	0
$397$	5410.16i	0.683950i	0.939709	+	0.341975i	$0.111096\pi$
$397$	5410.16i	0.683950i	−0.939709	+	0.341975i	$0.888904\pi$
$398$	0	0
$399$	1608.85	0.201863
$400$	0	0
$401$	−3259.80	−0.405952	−0.202976	−	0.979184i	$-0.565061\pi$
$401$	−3259.80	−0.405952	−0.202976	+	0.979184i	$0.565061\pi$
$402$	0	0
$403$	3824.25i	0.472703i
$404$	0	0
$405$	− 234.701i	− 0.0287960i
$406$	0	0
$407$	−2163.35	−0.263473
$408$	0	0
$409$	−3186.08	−0.385188	−0.192594	−	0.981279i	$-0.561690\pi$
$409$	−3186.08	−0.385188	−0.192594	+	0.981279i	$0.561690\pi$
$410$	0	0
$411$	686.635i	0.0824069i
$412$	0	0
$413$	− 5153.60i	− 0.614024i
$414$	0	0
$415$	3565.19	0.421707
$416$	0	0
$417$	3820.58	0.448668
$418$	0	0
$419$	− 2190.87i	− 0.255444i	−0.991810	−	0.127722i	$-0.959233\pi$
$419$	− 2190.87i	− 0.255444i	0.991810	−	0.127722i	$-0.0407665\pi$
$420$	0	0
$421$	6704.97i	0.776200i	0.921617	+	0.388100i	$0.126869\pi$
$421$	6704.97i	0.776200i	−0.921617	+	0.388100i	$0.873131\pi$
$422$	0	0
$423$	−4957.34	−0.569821
$424$	0	0
$425$	4992.71	0.569840
$426$	0	0
$427$	− 3844.79i	− 0.435744i
$428$	0	0
$429$	1002.58i	0.112832i
$430$	0	0
$431$	3518.19	0.393191	0.196596	−	0.980485i	$-0.437011\pi$
$431$	3518.19	0.393191	0.196596	+	0.980485i	$0.437011\pi$
$432$	0	0
$433$	15484.0	1.71851	0.859254	−	0.511550i	$-0.170928\pi$
$433$	15484.0	1.71851	0.859254	+	0.511550i	$0.170928\pi$
$434$	0	0
$435$	− 1718.08i	− 0.189369i
$436$	0	0
$437$	6054.07i	0.662713i
$438$	0	0
$439$	−3761.89	−0.408987	−0.204493	−	0.978868i	$-0.565555\pi$
$439$	−3761.89	−0.408987	−0.204493	+	0.978868i	$0.565555\pi$
$440$	0	0
$441$	−441.000	−0.0476190
$442$	0	0
$443$	− 14228.4i	− 1.52598i	−0.646409	−	0.762991i	$-0.723730\pi$
$443$	− 14228.4i	− 1.52598i	0.646409	−	0.762991i	$-0.276270\pi$
$444$	0	0
$445$	− 105.115i	− 0.0111976i
$446$	0	0
$447$	4627.20	0.489617
$448$	0	0
$449$	6496.42	0.682818	0.341409	−	0.939915i	$-0.389096\pi$
$449$	6496.42	0.682818	0.341409	+	0.939915i	$0.389096\pi$
$450$	0	0
$451$	− 2118.00i	− 0.221136i
$452$	0	0
$453$	8943.99i	0.927649i
$454$	0	0
$455$	−290.649	−0.0299469
$456$	0	0
$457$	−10638.2	−1.08892	−0.544459	−	0.838788i	$-0.683265\pi$
$457$	−10638.2	−1.08892	−0.544459	+	0.838788i	$0.683265\pi$
$458$	0	0
$459$	− 1156.07i	− 0.117562i
$460$	0	0
$461$	− 14207.6i	− 1.43539i	−0.696357	−	0.717696i	$-0.745197\pi$
$461$	− 14207.6i	− 1.43539i	0.696357	−	0.717696i	$-0.254803\pi$
$462$	0	0
$463$	1452.15	0.145760	0.0728800	−	0.997341i	$-0.476781\pi$
$463$	1452.15	0.145760	0.0728800	+	0.997341i	$0.476781\pi$
$464$	0	0
$465$	−2319.83	−0.231354
$466$	0	0
$467$	8782.32i	0.870230i	0.900375	+	0.435115i	$0.143292\pi$
$467$	8782.32i	0.870230i	−0.900375	+	0.435115i	$0.856708\pi$
$468$	0	0
$469$	− 3066.80i	− 0.301944i
$470$	0	0
$471$	−2001.46	−0.195801
$472$	0	0
$473$	−9926.56	−0.964955
$474$	0	0
$475$	− 8933.29i	− 0.862921i
$476$	0	0
$477$	2302.48i	0.221013i
$478$	0	0
$479$	−937.413	−0.0894185	−0.0447093	−	0.999000i	$-0.514236\pi$
$479$	−937.413	−0.0894185	−0.0447093	+	0.999000i	$0.514236\pi$
$480$	0	0
$481$	1329.27	0.126007
$482$	0	0
$483$	− 1659.47i	− 0.156332i
$484$	0	0
$485$	− 1553.13i	− 0.145411i
$486$	0	0
$487$	−12472.1	−1.16050	−0.580249	−	0.814439i	$-0.697045\pi$
$487$	−12472.1	−1.16050	−0.580249	+	0.814439i	$0.697045\pi$
$488$	0	0
$489$	638.100	0.0590100
$490$	0	0
$491$	19959.7i	1.83456i	0.398244	+	0.917280i	$0.369620\pi$
$491$	19959.7i	1.83456i	−0.398244	+	0.917280i	$0.630380\pi$
$492$	0	0
$493$	− 8462.80i	− 0.773114i
$494$	0	0
$495$	−608.174	−0.0552230
$496$	0	0
$497$	−3245.73	−0.292939
$498$	0	0
$499$	− 9473.91i	− 0.849921i	−0.905212	−	0.424960i	$-0.860288\pi$
$499$	− 9473.91i	− 0.849921i	0.905212	−	0.424960i	$-0.139712\pi$
$500$	0	0
$501$	5642.45i	0.503166i
$502$	0	0
$503$	−22033.1	−1.95309	−0.976547	−	0.215307i	$-0.930925\pi$
$503$	−22033.1	−1.95309	−0.976547	+	0.215307i	$0.930925\pi$
$504$	0	0
$505$	−4352.65	−0.383545
$506$	0	0
$507$	5974.97i	0.523388i
$508$	0	0
$509$	− 3706.41i	− 0.322758i	−0.986892	−	0.161379i	$-0.948406\pi$
$509$	− 3706.41i	− 0.322758i	0.986892	−	0.161379i	$-0.0515942\pi$
$510$	0	0
$511$	−3628.75	−0.314141
$512$	0	0
$513$	−2068.52	−0.178027
$514$	0	0
$515$	2283.74i	0.195405i
$516$	0	0
$517$	12845.8i	1.09276i
$518$	0	0
$519$	970.482	0.0820798
$520$	0	0
$521$	−774.967	−0.0651668	−0.0325834	−	0.999469i	$-0.510373\pi$
$521$	−774.967	−0.0651668	−0.0325834	+	0.999469i	$0.510373\pi$
$522$	0	0
$523$	8016.92i	0.670278i	0.942169	+	0.335139i	$0.108783\pi$
$523$	8016.92i	0.670278i	−0.942169	+	0.335139i	$0.891217\pi$
$524$	0	0
$525$	2448.69i	0.203561i
$526$	0	0
$527$	−11426.9	−0.944519
$528$	0	0
$529$	−5922.45	−0.486763
$530$	0	0
$531$	6626.05i	0.541518i
$532$	0	0
$533$	1301.40i	0.105760i
$534$	0	0
$535$	−246.695	−0.0199356
$536$	0	0
$537$	2757.94	0.221627
$538$	0	0
$539$	1142.75i	0.0913205i
$540$	0	0
$541$	18999.3i	1.50988i	0.655793	+	0.754941i	$0.272334\pi$
$541$	18999.3i	1.50988i	−0.655793	+	0.754941i	$0.727666\pi$
$542$	0	0
$543$	7981.30	0.630774
$544$	0	0
$545$	−2470.91	−0.194205
$546$	0	0
$547$	2672.36i	0.208888i	0.994531	+	0.104444i	$0.0333064\pi$
$547$	2672.36i	0.208888i	−0.994531	+	0.104444i	$0.966694\pi$
$548$	0	0
$549$	4943.31i	0.384290i
$550$	0	0
$551$	−15142.2	−1.17074
$552$	0	0
$553$	5054.02	0.388642
$554$	0	0
$555$	806.349i	0.0616714i
$556$	0	0
$557$	20190.2i	1.53588i	0.640521	+	0.767941i	$0.278719\pi$
$557$	20190.2i	1.53588i	−0.640521	+	0.767941i	$0.721281\pi$
$558$	0	0
$559$	6099.37	0.461495
$560$	0	0
$561$	−2995.70	−0.225452
$562$	0	0
$563$	2232.11i	0.167091i	0.996504	+	0.0835454i	$0.0266244\pi$
$563$	2232.11i	0.167091i	−0.996504	+	0.0835454i	$0.973376\pi$
$564$	0	0
$565$	3031.52i	0.225729i
$566$	0	0
$567$	567.000	0.0419961
$568$	0	0
$569$	24584.7	1.81132	0.905661	−	0.424002i	$-0.139375\pi$
$569$	24584.7	1.81132	0.905661	+	0.424002i	$0.139375\pi$
$570$	0	0
$571$	− 2629.53i	− 0.192719i	−0.995347	−	0.0963595i	$-0.969280\pi$
$571$	− 2629.53i	− 0.192719i	0.995347	−	0.0963595i	$-0.0307198\pi$
$572$	0	0
$573$	6383.60i	0.465408i
$574$	0	0
$575$	−9214.35	−0.668287
$576$	0	0
$577$	−8537.54	−0.615983	−0.307992	−	0.951389i	$-0.599657\pi$
$577$	−8537.54	−0.615983	−0.307992	+	0.951389i	$0.599657\pi$
$578$	0	0
$579$	− 3044.02i	− 0.218489i
$580$	0	0
$581$	8612.93i	0.615017i
$582$	0	0
$583$	5966.36	0.423844
$584$	0	0
$585$	373.692	0.0264107
$586$	0	0
$587$	− 19239.4i	− 1.35281i	−0.736532	−	0.676403i	$-0.763538\pi$
$587$	− 19239.4i	− 1.35281i	0.736532	−	0.676403i	$-0.236462\pi$
$588$	0	0
$589$	20445.7i	1.43031i
$590$	0	0
$591$	10121.7	0.704484
$592$	0	0
$593$	6000.24	0.415515	0.207757	−	0.978180i	$-0.433384\pi$
$593$	6000.24	0.415515	0.207757	+	0.978180i	$0.433384\pi$
$594$	0	0
$595$	− 868.460i	− 0.0598376i
$596$	0	0
$597$	6430.75i	0.440860i
$598$	0	0
$599$	−28332.4	−1.93261	−0.966303	−	0.257406i	$-0.917132\pi$
$599$	−28332.4	−1.93261	−0.966303	+	0.257406i	$0.917132\pi$
$600$	0	0
$601$	15058.7	1.02206	0.511028	−	0.859564i	$-0.329265\pi$
$601$	15058.7	1.02206	0.511028	+	0.859564i	$0.329265\pi$
$602$	0	0
$603$	3943.02i	0.266289i
$604$	0	0
$605$	− 2280.69i	− 0.153261i
$606$	0	0
$607$	24364.0	1.62917	0.814584	−	0.580045i	$-0.196965\pi$
$607$	24364.0	1.62917	0.814584	+	0.580045i	$0.196965\pi$
$608$	0	0
$609$	4150.61	0.276176
$610$	0	0
$611$	− 7893.10i	− 0.522620i
$612$	0	0
$613$	12077.9i	0.795793i	0.917430	+	0.397897i	$0.130260\pi$
$613$	12077.9i	0.795793i	−0.917430	+	0.397897i	$0.869740\pi$
$614$	0	0
$615$	−789.443	−0.0517617
$616$	0	0
$617$	−28110.7	−1.83419	−0.917096	−	0.398667i	$-0.869473\pi$
$617$	−28110.7	−1.83419	−0.917096	+	0.398667i	$0.869473\pi$
$618$	0	0
$619$	2753.22i	0.178774i	0.995997	+	0.0893872i	$0.0284908\pi$
$619$	2753.22i	0.178774i	−0.995997	+	0.0893872i	$0.971509\pi$
$620$	0	0
$621$	2133.61i	0.137872i
$622$	0	0
$623$	253.940	0.0163305
$624$	0	0
$625$	12547.1	0.803013
$626$	0	0
$627$	5360.11i	0.341407i
$628$	0	0
$629$	3971.86i	0.251778i
$630$	0	0
$631$	−7249.48	−0.457365	−0.228682	−	0.973501i	$-0.573442\pi$
$631$	−7249.48	−0.457365	−0.228682	+	0.973501i	$0.573442\pi$
$632$	0	0
$633$	−5391.29	−0.338522
$634$	0	0
$635$	− 6796.37i	− 0.424734i
$636$	0	0
$637$	− 702.162i	− 0.0436745i
$638$	0	0
$639$	4173.08	0.258348
$640$	0	0
$641$	−16851.3	−1.03835	−0.519177	−	0.854667i	$-0.673761\pi$
$641$	−16851.3	−1.03835	−0.519177	+	0.854667i	$0.673761\pi$
$642$	0	0
$643$	19280.2i	1.18248i	0.806495	+	0.591241i	$0.201362\pi$
$643$	19280.2i	1.18248i	−0.806495	+	0.591241i	$0.798638\pi$
$644$	0	0
$645$	3699.94i	0.225868i
$646$	0	0
$647$	−29423.4	−1.78787	−0.893937	−	0.448192i	$-0.852068\pi$
$647$	−29423.4	−1.78787	−0.893937	+	0.448192i	$0.852068\pi$
$648$	0	0
$649$	17169.9	1.03849
$650$	0	0
$651$	− 5604.34i	− 0.337406i
$652$	0	0
$653$	− 5046.63i	− 0.302435i	−0.988501	−	0.151218i	$-0.951681\pi$
$653$	− 5046.63i	− 0.302435i	0.988501	−	0.151218i	$-0.0483194\pi$
$654$	0	0
$655$	−2599.93	−0.155096
$656$	0	0
$657$	4665.53	0.277047
$658$	0	0
$659$	9708.45i	0.573881i	0.957948	+	0.286941i	$0.0926382\pi$
$659$	9708.45i	0.573881i	−0.957948	+	0.286941i	$0.907362\pi$
$660$	0	0
$661$	20983.4i	1.23473i	0.786676	+	0.617366i	$0.211800\pi$
$661$	20983.4i	1.23473i	−0.786676	+	0.617366i	$0.788200\pi$
$662$	0	0
$663$	1840.71	0.107824
$664$	0	0
$665$	−1553.91	−0.0906134
$666$	0	0
$667$	15618.6i	0.906680i
$668$	0	0
$669$	8699.27i	0.502740i
$670$	0	0
$671$	12809.4	0.736964
$672$	0	0
$673$	−15351.0	−0.879256	−0.439628	−	0.898180i	$-0.644890\pi$
$673$	−15351.0	−0.879256	−0.439628	+	0.898180i	$0.644890\pi$
$674$	0	0
$675$	− 3148.31i	− 0.179524i
$676$	0	0
$677$	6601.02i	0.374738i	0.982290	+	0.187369i	$0.0599960\pi$
$677$	6601.02i	0.374738i	−0.982290	+	0.187369i	$0.940004\pi$
$678$	0	0
$679$	3752.12	0.212067
$680$	0	0
$681$	−6497.96	−0.365642
$682$	0	0
$683$	6066.43i	0.339862i	0.985456	+	0.169931i	$0.0543544\pi$
$683$	6066.43i	0.339862i	−0.985456	+	0.169931i	$0.945646\pi$
$684$	0	0
$685$	− 663.185i	− 0.0369912i
$686$	0	0
$687$	7097.27	0.394145
$688$	0	0
$689$	−3666.02	−0.202706
$690$	0	0
$691$	− 10254.6i	− 0.564547i	−0.959334	−	0.282273i	$-0.908911\pi$
$691$	− 10254.6i	− 0.564547i	0.959334	−	0.282273i	$-0.0910885\pi$
$692$	0	0
$693$	− 1469.25i	− 0.0805371i
$694$	0	0
$695$	−3690.09	−0.201400
$696$	0	0
$697$	−3888.58	−0.211321
$698$	0	0
$699$	− 7871.38i	− 0.425927i
$700$	0	0
$701$	− 33015.6i	− 1.77886i	−0.457070	−	0.889430i	$-0.651101\pi$
$701$	− 33015.6i	− 1.77886i	0.457070	−	0.889430i	$-0.348899\pi$
$702$	0	0
$703$	7106.71	0.381273
$704$	0	0
$705$	4788.04	0.255784
$706$	0	0
$707$	− 10515.3i	− 0.559361i
$708$	0	0
$709$	23288.8i	1.23361i	0.787116	+	0.616804i	$0.211573\pi$
$709$	23288.8i	1.23361i	−0.787116	+	0.616804i	$0.788427\pi$
$710$	0	0
$711$	−6498.03	−0.342750
$712$	0	0
$713$	21089.0	1.10770
$714$	0	0
$715$	− 968.337i	− 0.0506486i
$716$	0	0
$717$	6144.38i	0.320036i
$718$	0	0
$719$	14536.4	0.753985	0.376992	−	0.926216i	$-0.376958\pi$
$719$	14536.4	0.753985	0.376992	+	0.926216i	$0.376958\pi$
$720$	0	0
$721$	−5517.16	−0.284979
$722$	0	0
$723$	1862.57i	0.0958087i
$724$	0	0
$725$	− 23046.6i	− 1.18059i
$726$	0	0
$727$	−11078.8	−0.565186	−0.282593	−	0.959240i	$-0.591195\pi$
$727$	−11078.8	−0.565186	−0.282593	+	0.959240i	$0.591195\pi$
$728$	0	0
$729$	−729.000	−0.0370370
$730$	0	0
$731$	18224.9i	0.922124i
$732$	0	0
$733$	24144.6i	1.21664i	0.793691	+	0.608322i	$0.208157\pi$
$733$	24144.6i	1.21664i	−0.793691	+	0.608322i	$0.791843\pi$
$734$	0	0
$735$	425.939	0.0213755
$736$	0	0
$737$	10217.4	0.510671
$738$	0	0
$739$	− 28167.5i	− 1.40211i	−0.713109	−	0.701054i	$-0.752713\pi$
$739$	− 28167.5i	− 1.40211i	0.713109	−	0.701054i	$-0.247287\pi$
$740$	0	0
$741$	− 3293.51i	− 0.163280i
$742$	0	0
$743$	−29204.2	−1.44199	−0.720994	−	0.692941i	$-0.756315\pi$
$743$	−29204.2	−1.44199	−0.720994	+	0.692941i	$0.756315\pi$
$744$	0	0
$745$	−4469.17	−0.219782
$746$	0	0
$747$	− 11073.8i	− 0.542394i
$748$	0	0
$749$	− 595.976i	− 0.0290741i
$750$	0	0
$751$	−29966.4	−1.45604	−0.728021	−	0.685554i	$-0.759560\pi$
$751$	−29966.4	−1.45604	−0.728021	+	0.685554i	$0.759560\pi$
$752$	0	0
$753$	−6142.50	−0.297271
$754$	0	0
$755$	− 8638.53i	− 0.416408i
$756$	0	0
$757$	− 9682.16i	− 0.464866i	−0.972612	−	0.232433i	$-0.925331\pi$
$757$	− 9682.16i	− 0.464866i	0.972612	−	0.232433i	$-0.0746687\pi$
$758$	0	0
$759$	5528.75	0.264402
$760$	0	0
$761$	24893.6	1.18580	0.592899	−	0.805277i	$-0.297983\pi$
$761$	24893.6	1.18580	0.592899	+	0.805277i	$0.297983\pi$
$762$	0	0
$763$	− 5969.31i	− 0.283229i
$764$	0	0
$765$	1116.59i	0.0527718i
$766$	0	0
$767$	−10550.0	−0.496662
$768$	0	0
$769$	−15786.4	−0.740277	−0.370139	−	0.928976i	$-0.620690\pi$
$769$	−15786.4	−0.740277	−0.370139	+	0.928976i	$0.620690\pi$
$770$	0	0
$771$	21078.8i	0.984611i
$772$	0	0
$773$	− 1.91752i	0 8.92219e-5i	−1.00000		4.46109e-5i	$-0.999986\pi$
$773$	− 1.91752i	0 8.92219e-5i	1.00000		4.46109e-5i	$-1.42001e-5\pi$
$774$	0	0
$775$	−31118.6	−1.44234
$776$	0	0
$777$	−1948.01	−0.0899414
$778$	0	0
$779$	6957.72i	0.320008i
$780$	0	0
$781$	− 10813.6i	− 0.495442i
$782$	0	0
$783$	−5336.49	−0.243564
$784$	0	0
$785$	1933.10	0.0878923
$786$	0	0
$787$	− 1907.02i	− 0.0863762i	−0.999067	−	0.0431881i	$-0.986249\pi$
$787$	− 1907.02i	− 0.0863762i	0.999067	−	0.0431881i	$-0.0137515\pi$
$788$	0	0
$789$	− 8011.77i	− 0.361504i
$790$	0	0
$791$	−7323.66	−0.329203
$792$	0	0
$793$	−7870.75	−0.352457
$794$	0	0
$795$	− 2223.85i	− 0.0992097i
$796$	0	0
$797$	4127.45i	0.183440i	0.995785	+	0.0917200i	$0.0292365\pi$
$797$	4127.45i	0.183440i	−0.995785	+	0.0917200i	$0.970764\pi$
$798$	0	0
$799$	23584.6	1.04426
$800$	0	0
$801$	−326.494	−0.0144021
$802$	0	0
$803$	− 12089.7i	− 0.531301i
$804$	0	0
$805$	1602.80i	0.0701754i
$806$	0	0
$807$	18547.5	0.809050
$808$	0	0
$809$	−13459.5	−0.584935	−0.292468	−	0.956275i	$-0.594476\pi$
$809$	−13459.5	−0.584935	−0.292468	+	0.956275i	$0.594476\pi$
$810$	0	0
$811$	− 24277.3i	− 1.05116i	−0.850744	−	0.525580i	$-0.823848\pi$
$811$	− 24277.3i	− 1.05116i	0.850744	−	0.525580i	$-0.176152\pi$
$812$	0	0
$813$	16860.7i	0.727344i
$814$	0	0
$815$	−616.308	−0.0264887
$816$	0	0
$817$	32609.2	1.39639
$818$	0	0
$819$	902.780i	0.0385173i
$820$	0	0
$821$	7544.40i	0.320708i	0.987060	+	0.160354i	$0.0512636\pi$
$821$	7544.40i	0.320708i	−0.987060	+	0.160354i	$0.948736\pi$
$822$	0	0
$823$	34480.4	1.46040	0.730200	−	0.683233i	$-0.239427\pi$
$823$	34480.4	1.46040	0.730200	+	0.683233i	$0.239427\pi$
$824$	0	0
$825$	−8158.14	−0.344279
$826$	0	0
$827$	20857.4i	0.877006i	0.898730	+	0.438503i	$0.144491\pi$
$827$	20857.4i	0.877006i	−0.898730	+	0.438503i	$0.855509\pi$
$828$	0	0
$829$	24673.3i	1.03370i	0.856076	+	0.516850i	$0.172896\pi$
$829$	24673.3i	1.03370i	−0.856076	+	0.516850i	$0.827104\pi$
$830$	0	0
$831$	−6510.84	−0.271791
$832$	0	0
$833$	2098.06	0.0872671
$834$	0	0
$835$	− 5449.75i	− 0.225864i
$836$	0	0
$837$	7205.58i	0.297564i
$838$	0	0
$839$	−19668.7	−0.809343	−0.404672	−	0.914462i	$-0.632614\pi$
$839$	−19668.7	−0.809343	−0.404672	+	0.914462i	$0.632614\pi$
$840$	0	0
$841$	−14675.7	−0.601735
$842$	0	0
$843$	− 4016.78i	− 0.164111i
$844$	0	0
$845$	− 5770.91i	− 0.234941i
$846$	0	0
$847$	5509.77	0.223516
$848$	0	0
$849$	−1472.79	−0.0595358
$850$	0	0
$851$	− 7330.31i	− 0.295276i
$852$	0	0
$853$	− 1798.73i	− 0.0722007i	−0.999348	−	0.0361003i	$-0.988506\pi$
$853$	− 1798.73i	− 0.0722007i	0.999348	−	0.0361003i	$-0.0114936\pi$
$854$	0	0
$855$	1997.88	0.0799135
$856$	0	0
$857$	−18898.6	−0.753285	−0.376642	−	0.926359i	$-0.622921\pi$
$857$	−18898.6	−0.753285	−0.376642	+	0.926359i	$0.622921\pi$
$858$	0	0
$859$	− 38907.3i	− 1.54540i	−0.634771	−	0.772701i	$-0.718905\pi$
$859$	− 38907.3i	− 1.54540i	0.634771	−	0.772701i	$-0.281095\pi$
$860$	0	0
$861$	− 1907.17i	− 0.0754891i
$862$	0	0
$863$	39499.2	1.55802	0.779009	−	0.627012i	$-0.215722\pi$
$863$	39499.2	1.55802	0.779009	+	0.627012i	$0.215722\pi$
$864$	0	0
$865$	−937.337	−0.0368444
$866$	0	0
$867$	− 9238.97i	− 0.361905i
$868$	0	0
$869$	16838.1i	0.657302i
$870$	0	0
$871$	−6278.10	−0.244231
$872$	0	0
$873$	−4824.16	−0.187025
$874$	0	0
$875$	− 4900.41i	− 0.189330i
$876$	0	0
$877$	− 26654.6i	− 1.02630i	−0.858300	−	0.513149i	$-0.828479\pi$
$877$	− 26654.6i	− 1.02630i	0.858300	−	0.513149i	$-0.171521\pi$
$878$	0	0
$879$	6413.63	0.246105
$880$	0	0
$881$	14147.6	0.541028	0.270514	−	0.962716i	$-0.412806\pi$
$881$	14147.6	0.541028	0.270514	+	0.962716i	$0.412806\pi$
$882$	0	0
$883$	48974.2i	1.86649i	0.359238	+	0.933246i	$0.383037\pi$
$883$	48974.2i	1.86649i	−0.359238	+	0.933246i	$0.616963\pi$
$884$	0	0
$885$	− 6399.76i	− 0.243080i
$886$	0	0
$887$	−14319.3	−0.542046	−0.271023	−	0.962573i	$-0.587362\pi$
$887$	−14319.3	−0.542046	−0.271023	+	0.962573i	$0.587362\pi$
$888$	0	0
$889$	16418.9	0.619431
$890$	0	0
$891$	1889.04i	0.0710271i
$892$	0	0
$893$	− 42199.1i	− 1.58134i
$894$	0	0
$895$	−2663.75	−0.0994852
$896$	0	0
$897$	−3397.14	−0.126452
$898$	0	0
$899$	52746.9i	1.95685i
$900$	0	0
$901$	− 10954.1i	− 0.405031i
$902$	0	0
$903$	−8938.46	−0.329406
$904$	0	0
$905$	−7708.72	−0.283145
$906$	0	0
$907$	45757.6i	1.67515i	0.546325	+	0.837573i	$0.316026\pi$
$907$	45757.6i	1.67515i	−0.546325	+	0.837573i	$0.683974\pi$
$908$	0	0
$909$	13519.7i	0.493310i
$910$	0	0
$911$	−16805.6	−0.611191	−0.305595	−	0.952161i	$-0.598855\pi$
$911$	−16805.6	−0.611191	−0.305595	+	0.952161i	$0.598855\pi$
$912$	0	0
$913$	−28695.1	−1.04017
$914$	0	0
$915$	− 4774.48i	− 0.172502i
$916$	0	0
$917$	− 6281.02i	− 0.226191i
$918$	0	0
$919$	−20260.0	−0.727221	−0.363610	−	0.931551i	$-0.618456\pi$
$919$	−20260.0	−0.727221	−0.363610	+	0.931551i	$0.618456\pi$
$920$	0	0
$921$	12750.3	0.456175
$922$	0	0
$923$	6644.39i	0.236948i
$924$	0	0
$925$	10816.5i	0.384480i
$926$	0	0
$927$	7093.49	0.251328
$928$	0	0
$929$	−16992.8	−0.600123	−0.300062	−	0.953920i	$-0.597007\pi$
$929$	−16992.8	−0.600123	−0.300062	+	0.953920i	$0.597007\pi$
$930$	0	0
$931$	− 3753.99i	− 0.132150i
$932$	0	0
$933$	− 12121.0i	− 0.425320i
$934$	0	0
$935$	2893.39	0.101202
$936$	0	0
$937$	32347.4	1.12779	0.563897	−	0.825845i	$-0.309302\pi$
$937$	32347.4	1.12779	0.563897	+	0.825845i	$0.309302\pi$
$938$	0	0
$939$	14933.4i	0.518991i
$940$	0	0
$941$	− 42834.0i	− 1.48390i	−0.670455	−	0.741950i	$-0.733901\pi$
$941$	− 42834.0i	− 1.48390i	0.670455	−	0.741950i	$-0.266099\pi$
$942$	0	0
$943$	7176.63	0.247829
$944$	0	0
$945$	−547.636	−0.0188514
$946$	0	0
$947$	− 11224.4i	− 0.385159i	−0.981281	−	0.192579i	$-0.938315\pi$
$947$	− 11224.4i	− 0.385159i	0.981281	−	0.192579i	$-0.0616853\pi$
$948$	0	0
$949$	7428.48i	0.254098i
$950$	0	0
$951$	15785.9	0.538269
$952$	0	0
$953$	−44248.5	−1.50404	−0.752019	−	0.659141i	$-0.770920\pi$
$953$	−44248.5	−1.50404	−0.752019	+	0.659141i	$0.770920\pi$
$954$	0	0
$955$	− 6165.59i	− 0.208915i
$956$	0	0
$957$	13828.3i	0.467091i
$958$	0	0
$959$	1602.15	0.0539480
$960$	0	0
$961$	41430.3	1.39070
$962$	0	0
$963$	766.255i	0.0256409i
$964$	0	0
$965$	2940.06i	0.0980766i
$966$	0	0
$967$	−6647.14	−0.221052	−0.110526	−	0.993873i	$-0.535254\pi$
$967$	−6647.14	−0.221052	−0.110526	+	0.993873i	$0.535254\pi$
$968$	0	0
$969$	9841.02	0.326253
$970$	0	0
$971$	− 28118.2i	− 0.929304i	−0.885493	−	0.464652i	$-0.846179\pi$
$971$	− 28118.2i	− 0.929304i	0.885493	−	0.464652i	$-0.153821\pi$
$972$	0	0
$973$	− 8914.68i	− 0.293722i
$974$	0	0
$975$	5012.76	0.164653
$976$	0	0
$977$	40876.6	1.33854	0.669272	−	0.743018i	$-0.266606\pi$
$977$	40876.6	1.33854	0.669272	+	0.743018i	$0.266606\pi$
$978$	0	0
$979$	846.035i	0.0276194i
$980$	0	0
$981$	7674.83i	0.249784i
$982$	0	0
$983$	21738.8	0.705350	0.352675	−	0.935746i	$-0.385272\pi$
$983$	21738.8	0.705350	0.352675	+	0.935746i	$0.385272\pi$
$984$	0	0
$985$	−9776.00	−0.316233
$986$	0	0
$987$	11567.1i	0.373035i
$988$	0	0
$989$	− 33635.2i	− 1.08143i
$990$	0	0
$991$	13680.9	0.438536	0.219268	−	0.975665i	$-0.429633\pi$
$991$	13680.9	0.438536	0.219268	+	0.975665i	$0.429633\pi$
$992$	0	0
$993$	9831.77	0.314201
$994$	0	0
$995$	− 6211.13i	− 0.197895i
$996$	0	0
$997$	− 16598.4i	− 0.527257i	−0.964624	−	0.263629i	$-0.915081\pi$
$997$	− 16598.4i	− 0.527257i	0.964624	−	0.263629i	$-0.0849193\pi$
$998$	0	0
$999$	2504.58	0.0793209

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1344.4.c.c.673.5	yes	6
4.3	odd	2		1344.4.c.b.673.2	✓	6
8.3	odd	2		1344.4.c.b.673.5	yes	6
8.5	even	2	inner	1344.4.c.c.673.2	yes	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1344.4.c.b.673.2	✓	6	4.3	odd	2
1344.4.c.b.673.5	yes	6	8.3	odd	2
1344.4.c.c.673.2	yes	6	8.5	even	2	inner
1344.4.c.c.673.5	yes	6	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1344 = 2^{6} \cdot 3 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1344.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+3.00000i q^{3} -2.89754i q^{5} +7.00000 q^{7} -9.00000 q^{9} +O(q^{10})\)	\(q+3.00000i q^{3} -2.89754i q^{5} +7.00000 q^{7} -9.00000 q^{9} +23.3214i q^{11} -14.3298i q^{13} +8.69263 q^{15} +42.8175 q^{17} -76.6120i q^{19} +21.0000i q^{21} -79.0225 q^{23} +116.604 q^{25} -27.0000i q^{27} -197.648i q^{29} -266.873 q^{31} -69.9643 q^{33} -20.2828i q^{35} +92.7624i q^{37} +42.9895 q^{39} -90.8175 q^{41} +425.641i q^{43} +26.0779i q^{45} +550.816 q^{47} +49.0000 q^{49} +128.453i q^{51} -255.831i q^{53} +67.5749 q^{55} +229.836 q^{57} -736.228i q^{59} -549.256i q^{61} -63.0000 q^{63} -41.5213 q^{65} -438.114i q^{67} -237.067i q^{69} -463.675 q^{71} -518.392 q^{73} +349.813i q^{75} +163.250i q^{77} +722.003 q^{79} +81.0000 q^{81} +1230.42i q^{83} -124.066i q^{85} +592.944 q^{87} +36.2772 q^{89} -100.309i q^{91} -800.620i q^{93} -221.987 q^{95} +536.018 q^{97} -209.893i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 42 q^{7} - 54 q^{9}+O(q^{10}) \) 6 * q + 42 * q^7 - 54 * q^9	\( 6 q + 42 q^{7} - 54 q^{9} - 24 q^{15} - 72 q^{17} - 196 q^{23} + 246 q^{25} + 104 q^{31} + 408 q^{33} - 348 q^{39} - 216 q^{41} - 408 q^{47} + 294 q^{49} + 720 q^{55} + 336 q^{57} - 378 q^{63} - 2472 q^{65} - 1164 q^{71} - 276 q^{73} + 1352 q^{79} + 486 q^{81} + 720 q^{87} + 2520 q^{89} + 4648 q^{95} - 2340 q^{97}+O(q^{100}) \) 6 * q + 42 * q^7 - 54 * q^9 - 24 * q^15 - 72 * q^17 - 196 * q^23 + 246 * q^25 + 104 * q^31 + 408 * q^33 - 348 * q^39 - 216 * q^41 - 408 * q^47 + 294 * q^49 + 720 * q^55 + 336 * q^57 - 378 * q^63 - 2472 * q^65 - 1164 * q^71 - 276 * q^73 + 1352 * q^79 + 486 * q^81 + 720 * q^87 + 2520 * q^89 + 4648 * q^95 - 2340 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more