LMFDB - Embedded newform 1344.4.b.j.895.2

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:	$24$
Twist minimal:	no (minimal twist has level 672)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			895.2
Character	$\chi$	$=$	1344.895
Dual form			1344.4.b.j.895.23

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.00000 q^{3} -15.7575i q^{5} +(15.0622 - 10.7764i) q^{7} +9.00000 q^{9} +O(q^{10})$	$q+3.00000 q^{3} -15.7575i q^{5} +(15.0622 - 10.7764i) q^{7} +9.00000 q^{9} +63.5030i q^{11} -34.9026i q^{13} -47.2725i q^{15} -39.0949i q^{17} +139.703 q^{19} +(45.1865 - 32.3293i) q^{21} +123.427i q^{23} -123.299 q^{25} +27.0000 q^{27} +131.511 q^{29} +225.225 q^{31} +190.509i q^{33} +(-169.810 - 237.342i) q^{35} +148.493 q^{37} -104.708i q^{39} -174.658i q^{41} +451.335i q^{43} -141.818i q^{45} -146.540 q^{47} +(110.737 - 324.633i) q^{49} -117.285i q^{51} +278.721 q^{53} +1000.65 q^{55} +419.109 q^{57} -26.4098 q^{59} +69.5922i q^{61} +(135.559 - 96.9879i) q^{63} -549.978 q^{65} +288.829i q^{67} +370.281i q^{69} +362.999i q^{71} -719.404i q^{73} -369.898 q^{75} +(684.336 + 956.492i) q^{77} -972.073i q^{79} +81.0000 q^{81} +494.011 q^{83} -616.038 q^{85} +394.533 q^{87} +88.7502i q^{89} +(-376.126 - 525.708i) q^{91} +675.676 q^{93} -2201.37i q^{95} -372.210i q^{97} +571.527i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q + 72 q^{3} - 20 q^{7} + 216 q^{9}+O(q^{10}) $ 24 * q + 72 * q^3 - 20 * q^7 + 216 * q^9	$ 24 q + 72 q^{3} - 20 q^{7} + 216 q^{9} - 56 q^{19} - 60 q^{21} - 432 q^{25} + 648 q^{27} + 464 q^{31} + 568 q^{35} - 504 q^{37} + 560 q^{47} - 128 q^{49} + 784 q^{53} + 424 q^{55} - 168 q^{57} + 800 q^{59} - 180 q^{63} + 560 q^{65} - 1296 q^{75} + 1568 q^{77} + 1944 q^{81} + 1936 q^{83} - 3000 q^{85} - 496 q^{91} + 1392 q^{93}+O(q^{100}) $ 24 * q + 72 * q^3 - 20 * q^7 + 216 * q^9 - 56 * q^19 - 60 * q^21 - 432 * q^25 + 648 * q^27 + 464 * q^31 + 568 * q^35 - 504 * q^37 + 560 * q^47 - 128 * q^49 + 784 * q^53 + 424 * q^55 - 168 * q^57 + 800 * q^59 - 180 * q^63 + 560 * q^65 - 1296 * q^75 + 1568 * q^77 + 1944 * q^81 + 1936 * q^83 - 3000 * q^85 - 496 * q^91 + 1392 * 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
$5$		−	15.7575i		−	1.40939i	−0.709508	−	0.704697i	$-0.751083\pi$
$5$		−	15.7575i		−	1.40939i	0.709508	−	0.704697i	$-0.248917\pi$
$6$		0			0
$7$	15.0622	−	10.7764i	0.813280	−	0.581873i
$7$	15.0622	−	10.7764i	0.813280	−	0.581873i
$8$		0			0
$9$	9.00000			0.333333
$10$		0			0
$11$			63.5030i			1.74062i	0.492500	+	0.870312i	$0.336083\pi$
$11$			63.5030i			1.74062i	−0.492500	+	0.870312i	$0.663917\pi$
$12$		0			0
$13$		−	34.9026i		−	0.744634i	−0.928106	−	0.372317i	$-0.878563\pi$
$13$		−	34.9026i		−	0.744634i	0.928106	−	0.372317i	$-0.121437\pi$
$14$		0			0
$15$		−	47.2725i		−	0.813714i
$16$		0			0
$17$		−	39.0949i		−	0.557759i	−0.960326	−	0.278879i	$-0.910037\pi$
$17$		−	39.0949i		−	0.557759i	0.960326	−	0.278879i	$-0.0899630\pi$
$18$		0			0
$19$	139.703			1.68685			0.843424	−	0.537249i	$-0.180536\pi$
$19$	139.703			1.68685			0.843424	+	0.537249i	$0.180536\pi$
$20$		0			0
$21$	45.1865	−	32.3293i	0.469547	−	0.335945i
$22$		0			0
$23$			123.427i			1.11897i	0.828840	+	0.559485i	$0.189001\pi$
$23$			123.427i			1.11897i	−0.828840	+	0.559485i	$0.810999\pi$
$24$		0			0
$25$	−123.299			−0.986394
$26$		0			0
$27$	27.0000			0.192450
$28$		0			0
$29$	131.511			0.842103			0.421051	−	0.907037i	$-0.361661\pi$
$29$	131.511			0.842103			0.421051	+	0.907037i	$0.361661\pi$
$30$		0			0
$31$	225.225			1.30489			0.652446	−	0.757835i	$-0.273743\pi$
$31$	225.225			1.30489			0.652446	+	0.757835i	$0.273743\pi$
$32$		0			0
$33$			190.509i			1.00495i
$34$		0			0
$35$	−169.810	−	237.342i	−0.820089	−	1.14623i
$36$		0			0
$37$	148.493			0.659787			0.329894	−	0.944018i	$-0.392987\pi$
$37$	148.493			0.659787			0.329894	+	0.944018i	$0.392987\pi$
$38$		0			0
$39$		−	104.708i		−	0.429915i
$40$		0			0
$41$		−	174.658i		−	0.665291i	−0.943052	−	0.332646i	$-0.892059\pi$
$41$		−	174.658i		−	0.665291i	0.943052	−	0.332646i	$-0.107941\pi$
$42$		0			0
$43$			451.335i			1.60065i	0.599567	+	0.800325i	$0.295340\pi$
$43$			451.335i			1.60065i	−0.599567	+	0.800325i	$0.704660\pi$
$44$		0			0
$45$		−	141.818i		−	0.469798i
$46$		0			0
$47$	−146.540			−0.454789			−0.227395	−	0.973803i	$-0.573021\pi$
$47$	−146.540			−0.454789			−0.227395	+	0.973803i	$0.573021\pi$
$48$		0			0
$49$	110.737	−	324.633i	0.322848	−	0.946451i
$50$		0			0
$51$		−	117.285i		−	0.322022i
$52$		0			0
$53$	278.721			0.722365			0.361182	−	0.932495i	$-0.382373\pi$
$53$	278.721			0.722365			0.361182	+	0.932495i	$0.382373\pi$
$54$		0			0
$55$	1000.65			2.45323
$56$		0			0
$57$	419.109			0.973902
$58$		0			0
$59$	−26.4098			−0.0582756			−0.0291378	−	0.999575i	$-0.509276\pi$
$59$	−26.4098			−0.0582756			−0.0291378	+	0.999575i	$0.509276\pi$
$60$		0			0
$61$			69.5922i			0.146072i	0.997329	+	0.0730358i	$0.0232687\pi$
$61$			69.5922i			0.146072i	−0.997329	+	0.0730358i	$0.976731\pi$
$62$		0			0
$63$	135.559	−	96.9879i	0.271093	−	0.193958i
$64$		0			0
$65$	−549.978			−1.04948
$66$		0			0
$67$			288.829i			0.526659i	0.964706	+	0.263329i	$0.0848206\pi$
$67$			288.829i			0.526659i	−0.964706	+	0.263329i	$0.915179\pi$
$68$		0			0
$69$			370.281i			0.646038i
$70$		0			0
$71$			362.999i			0.606761i	0.952870	+	0.303380i	$0.0981153\pi$
$71$			362.999i			0.606761i	−0.952870	+	0.303380i	$0.901885\pi$
$72$		0			0
$73$		−	719.404i		−	1.15342i	−0.816948	−	0.576712i	$-0.804336\pi$
$73$		−	719.404i		−	1.15342i	0.816948	−	0.576712i	$-0.195664\pi$
$74$		0			0
$75$	−369.898			−0.569495
$76$		0			0
$77$	684.336	+	956.492i	1.01282	+	1.41561i
$78$		0			0
$79$		−	972.073i		−	1.38439i	−0.721711	−	0.692195i	$-0.756644\pi$
$79$		−	972.073i		−	1.38439i	0.721711	−	0.692195i	$-0.243356\pi$
$80$		0			0
$81$	81.0000			0.111111
$82$		0			0
$83$	494.011			0.653310			0.326655	−	0.945144i	$-0.394079\pi$
$83$	494.011			0.653310			0.326655	+	0.945144i	$0.394079\pi$
$84$		0			0
$85$	−616.038			−0.786102
$86$		0			0
$87$	394.533			0.486188
$88$		0			0
$89$			88.7502i			0.105702i	0.998602	+	0.0528512i	$0.0168309\pi$
$89$			88.7502i			0.105702i	−0.998602	+	0.0528512i	$0.983169\pi$
$90$		0			0
$91$	−376.126	−	525.708i	−0.433282	−	0.605596i
$92$		0			0
$93$	675.676			0.753380
$94$		0			0
$95$		−	2201.37i		−	2.37743i
$96$		0			0
$97$		−	372.210i		−	0.389610i	−0.980842	−	0.194805i	$-0.937593\pi$
$97$		−	372.210i		−	0.389610i	0.980842	−	0.194805i	$-0.0624074\pi$
$98$		0			0
$99$			571.527i			0.580208i
$100$		0			0
$101$		−	353.995i		−	0.348751i	−0.984679	−	0.174375i	$-0.944209\pi$
$101$		−	353.995i		−	0.348751i	0.984679	−	0.174375i	$-0.0557906\pi$
$102$		0			0
$103$	−1469.01			−1.40530			−0.702650	−	0.711536i	$-0.748000\pi$
$103$	−1469.01			−1.40530			−0.702650	+	0.711536i	$0.748000\pi$
$104$		0			0
$105$	−509.430	−	712.026i	−0.473478	−	0.661777i
$106$		0			0
$107$			353.811i			0.319665i	0.987144	+	0.159833i	$0.0510954\pi$
$107$			353.811i			0.319665i	−0.987144	+	0.159833i	$0.948905\pi$
$108$		0			0
$109$	−1714.06			−1.50621			−0.753107	−	0.657898i	$-0.771446\pi$
$109$	−1714.06			−1.50621			−0.753107	+	0.657898i	$0.771446\pi$
$110$		0			0
$111$	445.480			0.380928
$112$		0			0
$113$	−2255.85			−1.87799			−0.938993	−	0.343936i	$-0.888240\pi$
$113$	−2255.85			−1.87799			−0.938993	+	0.343936i	$0.888240\pi$
$114$		0			0
$115$	1944.90			1.57707
$116$		0			0
$117$		−	314.123i		−	0.248211i
$118$		0			0
$119$	−421.303	−	588.853i	−0.324545	−	0.453614i
$120$		0			0
$121$	−2701.63			−2.02978
$122$		0			0
$123$		−	523.973i		−	0.384106i
$124$		0			0
$125$		−	26.8002i		−	0.0191767i
$126$		0			0
$127$		−	2394.07i		−	1.67275i	−0.548157	−	0.836376i	$-0.684670\pi$
$127$		−	2394.07i		−	1.67275i	0.548157	−	0.836376i	$-0.315330\pi$
$128$		0			0
$129$			1354.00i			0.924135i
$130$		0			0
$131$	−2873.08			−1.91620			−0.958100	−	0.286433i	$-0.907530\pi$
$131$	−2873.08			−1.91620			−0.958100	+	0.286433i	$0.907530\pi$
$132$		0			0
$133$	2104.23	−	1505.50i	1.37188	−	0.981531i
$134$		0			0
$135$		−	425.453i		−	0.271238i
$136$		0			0
$137$	3087.51			1.92543			0.962715	−	0.270519i	$-0.0871954\pi$
$137$	3087.51			1.92543			0.962715	+	0.270519i	$0.0871954\pi$
$138$		0			0
$139$	−358.649			−0.218850			−0.109425	−	0.993995i	$-0.534901\pi$
$139$	−358.649			−0.218850			−0.109425	+	0.993995i	$0.534901\pi$
$140$		0			0
$141$	−439.621			−0.262573
$142$		0			0
$143$	2216.42			1.29613
$144$		0			0
$145$		−	2072.29i		−	1.18686i
$146$		0			0
$147$	332.210	−	973.898i	0.186396	−	0.546434i
$148$		0			0
$149$	1933.03			1.06282			0.531409	−	0.847115i	$-0.321663\pi$
$149$	1933.03			1.06282			0.531409	+	0.847115i	$0.321663\pi$
$150$		0			0
$151$		−	2409.37i		−	1.29849i	−0.760580	−	0.649245i	$-0.775085\pi$
$151$		−	2409.37i		−	1.29849i	0.760580	−	0.649245i	$-0.224915\pi$
$152$		0			0
$153$		−	351.854i		−	0.185920i
$154$		0			0
$155$		−	3548.99i		−	1.83911i
$156$		0			0
$157$			2404.36i			1.22222i	0.791544	+	0.611112i	$0.209277\pi$
$157$			2404.36i			1.22222i	−0.791544	+	0.611112i	$0.790723\pi$
$158$		0			0
$159$	836.164			0.417058
$160$		0			0
$161$	1330.10	+	1859.08i	0.651099	+	0.910036i
$162$		0			0
$163$		−	2011.44i		−	0.966551i	−0.875468	−	0.483276i	$-0.839447\pi$
$163$		−	2011.44i		−	0.966551i	0.875468	−	0.483276i	$-0.160553\pi$
$164$		0			0
$165$	3001.95			1.41637
$166$		0			0
$167$	2013.56			0.933016			0.466508	−	0.884517i	$-0.345512\pi$
$167$	2013.56			0.933016			0.466508	+	0.884517i	$0.345512\pi$
$168$		0			0
$169$	978.808			0.445520
$170$		0			0
$171$	1257.33			0.562282
$172$		0			0
$173$			2439.55i			1.07211i	0.844182	+	0.536057i	$0.180087\pi$
$173$			2439.55i			1.07211i	−0.844182	+	0.536057i	$0.819913\pi$
$174$		0			0
$175$	−1857.15	+	1328.73i	−0.802214	+	0.573956i
$176$		0			0
$177$	−79.2293			−0.0336454
$178$		0			0
$179$			3292.25i			1.37472i	0.726319	+	0.687358i	$0.241229\pi$
$179$			3292.25i			1.37472i	−0.726319	+	0.687358i	$0.758771\pi$
$180$		0			0
$181$			471.688i			0.193703i	0.995299	+	0.0968517i	$0.0308773\pi$
$181$			471.688i			0.193703i	−0.995299	+	0.0968517i	$0.969123\pi$
$182$		0			0
$183$			208.777i			0.0843345i
$184$		0			0
$185$		−	2339.88i		−	0.929901i
$186$		0			0
$187$	2482.64			0.970849
$188$		0			0
$189$	406.678	−	290.964i	0.156516	−	0.111982i
$190$		0			0
$191$		−	427.664i		−	0.162014i	−0.996714	−	0.0810070i	$-0.974186\pi$
$191$		−	427.664i		−	0.162014i	0.996714	−	0.0810070i	$-0.0258136\pi$
$192$		0			0
$193$	156.240			0.0582714			0.0291357	−	0.999575i	$-0.490725\pi$
$193$	156.240			0.0582714			0.0291357	+	0.999575i	$0.490725\pi$
$194$		0			0
$195$	−1649.93			−0.605919
$196$		0			0
$197$	−2047.84			−0.740623			−0.370311	−	0.928908i	$-0.620749\pi$
$197$	−2047.84			−0.740623			−0.370311	+	0.928908i	$0.620749\pi$
$198$		0			0
$199$	2954.37			1.05241			0.526205	−	0.850358i	$-0.323614\pi$
$199$	2954.37			1.05241			0.526205	+	0.850358i	$0.323614\pi$
$200$		0			0
$201$			866.488i			0.304066i
$202$		0			0
$203$	1980.84	−	1417.22i	0.684865	−	0.489997i
$204$		0			0
$205$	−2752.17			−0.937658
$206$		0			0
$207$			1110.84i			0.372990i
$208$		0			0
$209$			8871.57i			2.93617i
$210$		0			0
$211$		−	2145.80i		−	0.700110i	−0.936729	−	0.350055i	$-0.886163\pi$
$211$		−	2145.80i		−	0.700110i	0.936729	−	0.350055i	$-0.113837\pi$
$212$		0			0
$213$			1089.00i			0.350314i
$214$		0			0
$215$	7111.91			2.25595
$216$		0			0
$217$	3392.38	−	2427.13i	1.06124	−	0.759281i
$218$		0			0
$219$		−	2158.21i		−	0.665929i
$220$		0			0
$221$	−1364.51			−0.415326
$222$		0			0
$223$	−2824.19			−0.848080			−0.424040	−	0.905643i	$-0.639388\pi$
$223$	−2824.19			−0.848080			−0.424040	+	0.905643i	$0.639388\pi$
$224$		0			0
$225$	−1109.69			−0.328798
$226$		0			0
$227$	1950.36			0.570265			0.285132	−	0.958488i	$-0.407962\pi$
$227$	1950.36			0.570265			0.285132	+	0.958488i	$0.407962\pi$
$228$		0			0
$229$			302.636i			0.0873307i	0.999046	+	0.0436653i	$0.0139035\pi$
$229$			302.636i			0.0873307i	−0.999046	+	0.0436653i	$0.986096\pi$
$230$		0			0
$231$	2053.01	+	2869.48i	0.584753	+	0.817306i
$232$		0			0
$233$	−4882.80			−1.37289			−0.686444	−	0.727182i	$-0.740829\pi$
$233$	−4882.80			−1.37289			−0.686444	+	0.727182i	$0.740829\pi$
$234$		0			0
$235$			2309.11i			0.640978i
$236$		0			0
$237$		−	2916.22i		−	0.799278i
$238$		0			0
$239$		−	247.430i		−	0.0669662i	−0.999439	−	0.0334831i	$-0.989340\pi$
$239$		−	247.430i		−	0.0669662i	0.999439	−	0.0334831i	$-0.0106600\pi$
$240$		0			0
$241$		−	1249.21i		−	0.333894i	−0.985966	−	0.166947i	$-0.946609\pi$
$241$		−	1249.21i		−	0.333894i	0.985966	−	0.166947i	$-0.0533909\pi$
$242$		0			0
$243$	243.000			0.0641500
$244$		0			0
$245$	−5115.40	−	1744.94i	−1.33392	−	0.455020i
$246$		0			0
$247$		−	4876.00i		−	1.25608i
$248$		0			0
$249$	1482.03			0.377189
$250$		0			0
$251$	4760.98			1.19725			0.598626	−	0.801028i	$-0.295713\pi$
$251$	4760.98			1.19725			0.598626	+	0.801028i	$0.295713\pi$
$252$		0			0
$253$	−7837.99			−1.94771
$254$		0			0
$255$	−1848.11			−0.453856
$256$		0			0
$257$			3555.00i			0.862860i	0.902146	+	0.431430i	$0.141991\pi$
$257$			3555.00i			0.862860i	−0.902146	+	0.431430i	$0.858009\pi$
$258$		0			0
$259$	2236.63	−	1600.23i	0.536592	−	0.383912i
$260$		0			0
$261$	1183.60			0.280701
$262$		0			0
$263$			2726.20i			0.639182i	0.947556	+	0.319591i	$0.103545\pi$
$263$			2726.20i			0.639182i	−0.947556	+	0.319591i	$0.896455\pi$
$264$		0			0
$265$		−	4391.96i		−	1.01810i
$266$		0			0
$267$			266.251i			0.0610273i
$268$		0			0
$269$		−	2360.01i		−	0.534916i	−0.963570	−	0.267458i	$-0.913816\pi$
$269$		−	2360.01i		−	0.534916i	0.963570	−	0.267458i	$-0.0861837\pi$
$270$		0			0
$271$	135.448			0.0303613			0.0151806	−	0.999885i	$-0.495168\pi$
$271$	135.448			0.0303613			0.0151806	+	0.999885i	$0.495168\pi$
$272$		0			0
$273$	−1128.38	−	1577.12i	−0.250156	−	0.349641i
$274$		0			0
$275$		−	7829.87i		−	1.71694i
$276$		0			0
$277$	4298.01			0.932282			0.466141	−	0.884710i	$-0.345644\pi$
$277$	4298.01			0.932282			0.466141	+	0.884710i	$0.345644\pi$
$278$		0			0
$279$	2027.03			0.434964
$280$		0			0
$281$	−6617.98			−1.40497			−0.702483	−	0.711701i	$-0.747925\pi$
$281$	−6617.98			−1.40497			−0.702483	+	0.711701i	$0.747925\pi$
$282$		0			0
$283$	8380.43			1.76030			0.880150	−	0.474696i	$-0.157442\pi$
$283$	8380.43			1.76030			0.880150	+	0.474696i	$0.157442\pi$
$284$		0			0
$285$		−	6604.12i		−	1.37261i
$286$		0			0
$287$	−1882.19	−	2630.72i	−0.387115	−	0.541068i
$288$		0			0
$289$	3384.59			0.688905
$290$		0			0
$291$		−	1116.63i		−	0.224942i
$292$		0			0
$293$		−	8588.24i		−	1.71239i	−0.516652	−	0.856195i	$-0.672822\pi$
$293$		−	8588.24i		−	1.71239i	0.516652	−	0.856195i	$-0.327178\pi$
$294$		0			0
$295$			416.152i			0.0821333i
$296$		0			0
$297$			1714.58i			0.334983i
$298$		0			0
$299$	4307.92			0.833223
$300$		0			0
$301$	4863.78	+	6798.07i	0.931375	+	1.30178i
$302$		0			0
$303$		−	1061.99i		−	0.201351i
$304$		0			0
$305$	1096.60			0.205873
$306$		0			0
$307$	235.006			0.0436890			0.0218445	−	0.999761i	$-0.493046\pi$
$307$	235.006			0.0436890			0.0218445	+	0.999761i	$0.493046\pi$
$308$		0			0
$309$	−4407.03			−0.811350
$310$		0			0
$311$	−9544.48			−1.74025			−0.870125	−	0.492831i	$-0.835962\pi$
$311$	−9544.48			−1.74025			−0.870125	+	0.492831i	$0.835962\pi$
$312$		0			0
$313$			9235.69i			1.66783i	0.551890	+	0.833917i	$0.313907\pi$
$313$			9235.69i			1.66783i	−0.551890	+	0.833917i	$0.686093\pi$
$314$		0			0
$315$	−1528.29	−	2136.08i	−0.273363	−	0.382077i
$316$		0			0
$317$	226.546			0.0401390			0.0200695	−	0.999799i	$-0.493611\pi$
$317$	226.546			0.0401390			0.0200695	+	0.999799i	$0.493611\pi$
$318$		0			0
$319$			8351.34i			1.46579i
$320$		0			0
$321$			1061.43i			0.184559i
$322$		0			0
$323$		−	5461.68i		−	0.940854i
$324$		0			0
$325$			4303.46i			0.734502i
$326$		0			0
$327$	−5142.19			−0.869613
$328$		0			0
$329$	−2207.21	+	1579.18i	−0.369871	+	0.264630i
$330$		0			0
$331$			8498.15i			1.41118i	0.708621	+	0.705590i	$0.249318\pi$
$331$			8498.15i			1.41118i	−0.708621	+	0.705590i	$0.750682\pi$
$332$		0			0
$333$	1336.44			0.219929
$334$		0			0
$335$	4551.23			0.742270
$336$		0			0
$337$	8729.62			1.41108			0.705538	−	0.708672i	$-0.250705\pi$
$337$	8729.62			1.41108			0.705538	+	0.708672i	$0.250705\pi$
$338$		0			0
$339$	−6767.55			−1.08426
$340$		0			0
$341$			14302.5i			2.27133i
$342$		0			0
$343$	−1830.45	−	6083.01i	−0.288149	−	0.957586i
$344$		0			0
$345$	5834.71			0.910522
$346$		0			0
$347$		−	9307.97i		−	1.43999i	−0.693977	−	0.719997i	$-0.744143\pi$
$347$		−	9307.97i		−	1.43999i	0.693977	−	0.719997i	$-0.255857\pi$
$348$		0			0
$349$		−	1055.71i		−	0.161922i	−0.996717	−	0.0809610i	$-0.974201\pi$
$349$		−	1055.71i		−	0.161922i	0.996717	−	0.0809610i	$-0.0257989\pi$
$350$		0			0
$351$		−	942.370i		−	0.143305i
$352$		0			0
$353$			77.4512i			0.0116779i	0.999983	+	0.00583897i	$0.00185861\pi$
$353$			77.4512i			0.0116779i	−0.999983	+	0.00583897i	$0.998141\pi$
$354$		0			0
$355$	5719.96			0.855166
$356$		0			0
$357$	−1263.91	−	1766.56i	−0.187376	−	0.261894i
$358$		0			0
$359$		−	2486.10i		−	0.365491i	−0.983160	−	0.182746i	$-0.941502\pi$
$359$		−	2486.10i		−	0.365491i	0.983160	−	0.182746i	$-0.0584985\pi$
$360$		0			0
$361$	12658.0			1.84545
$362$		0			0
$363$	−8104.89			−1.17189
$364$		0			0
$365$	−11336.0			−1.62563
$366$		0			0
$367$	−11699.5			−1.66406			−0.832032	−	0.554728i	$-0.812822\pi$
$367$	−11699.5			−1.66406			−0.832032	+	0.554728i	$0.812822\pi$
$368$		0			0
$369$		−	1571.92i		−	0.221764i
$370$		0			0
$371$	4198.15	−	3003.63i	0.587485	−	0.420325i
$372$		0			0
$373$	−6932.47			−0.962331			−0.481166	−	0.876630i	$-0.659786\pi$
$373$	−6932.47			−0.962331			−0.481166	+	0.876630i	$0.659786\pi$
$374$		0			0
$375$		−	80.4007i		−	0.0110717i
$376$		0			0
$377$		−	4590.08i		−	0.627058i
$378$		0			0
$379$			3574.75i			0.484492i	0.970215	+	0.242246i	$0.0778842\pi$
$379$			3574.75i			0.484492i	−0.970215	+	0.242246i	$0.922116\pi$
$380$		0			0
$381$		−	7182.21i		−	0.965764i
$382$		0			0
$383$	5336.96			0.712026			0.356013	−	0.934481i	$-0.384136\pi$
$383$	5336.96			0.712026			0.356013	+	0.934481i	$0.384136\pi$
$384$		0			0
$385$	15071.9	−	10783.4i	1.99516	−	1.42747i
$386$		0			0
$387$			4062.01i			0.533550i
$388$		0			0
$389$	10949.8			1.42719			0.713593	−	0.700560i	$-0.247067\pi$
$389$	10949.8			1.42719			0.713593	+	0.700560i	$0.247067\pi$
$390$		0			0
$391$	4825.36			0.624115
$392$		0			0
$393$	−8619.24			−1.10632
$394$		0			0
$395$	−15317.5			−1.95115
$396$		0			0
$397$		−	14793.2i		−	1.87015i	−0.354449	−	0.935075i	$-0.615332\pi$
$397$		−	14793.2i		−	1.87015i	0.354449	−	0.935075i	$-0.384668\pi$
$398$		0			0
$399$	6312.69	−	4516.51i	0.792054	−	0.566687i
$400$		0			0
$401$	9526.62			1.18638			0.593188	−	0.805064i	$-0.297869\pi$
$401$	9526.62			1.18638			0.593188	+	0.805064i	$0.297869\pi$
$402$		0			0
$403$		−	7860.95i		−	0.971667i
$404$		0			0
$405$		−	1276.36i		−	0.156599i
$406$		0			0
$407$			9429.76i			1.14844i
$408$		0			0
$409$		−	4573.84i		−	0.552963i	−0.961019	−	0.276481i	$-0.910832\pi$
$409$		−	4573.84i		−	0.552963i	0.961019	−	0.276481i	$-0.0891684\pi$
$410$		0			0
$411$	9262.53			1.11165
$412$		0			0
$413$	−397.788	+	284.603i	−0.0473943	+	0.0339090i
$414$		0			0
$415$		−	7784.38i		−	0.920771i
$416$		0			0
$417$	−1075.95			−0.126353
$418$		0			0
$419$	−14014.3			−1.63400			−0.816999	−	0.576639i	$-0.804364\pi$
$419$	−14014.3			−1.63400			−0.816999	+	0.576639i	$0.804364\pi$
$420$		0			0
$421$	11324.2			1.31095			0.655474	−	0.755218i	$-0.272469\pi$
$421$	11324.2			1.31095			0.655474	+	0.755218i	$0.272469\pi$
$422$		0			0
$423$	−1318.86			−0.151596
$424$		0			0
$425$			4820.37i			0.550170i
$426$		0			0
$427$	749.956	+	1048.21i	0.0849951	+	0.118797i
$428$		0			0
$429$	6649.26			0.748320
$430$		0			0
$431$			11531.3i			1.28873i	0.764718	+	0.644365i	$0.222878\pi$
$431$			11531.3i			1.28873i	−0.764718	+	0.644365i	$0.777122\pi$
$432$		0			0
$433$			5632.81i			0.625163i	0.949891	+	0.312582i	$0.101194\pi$
$433$			5632.81i			0.625163i	−0.949891	+	0.312582i	$0.898806\pi$
$434$		0			0
$435$		−	6216.86i		−	0.685231i
$436$		0			0
$437$			17243.1i			1.88753i
$438$		0			0
$439$	−8607.97			−0.935846			−0.467923	−	0.883769i	$-0.654997\pi$
$439$	−8607.97			−0.935846			−0.467923	+	0.883769i	$0.654997\pi$
$440$		0			0
$441$	996.631	−	2921.69i	0.107616	−	0.315484i
$442$		0			0
$443$			6396.61i			0.686032i	0.939330	+	0.343016i	$0.111449\pi$
$443$			6396.61i			0.686032i	−0.939330	+	0.343016i	$0.888551\pi$
$444$		0			0
$445$	1398.48			0.148976
$446$		0			0
$447$	5799.09			0.613619
$448$		0			0
$449$	−11275.2			−1.18510			−0.592549	−	0.805534i	$-0.701879\pi$
$449$	−11275.2			−1.18510			−0.592549	+	0.805534i	$0.701879\pi$
$450$		0			0
$451$	11091.3			1.15802
$452$		0			0
$453$		−	7228.12i		−	0.749683i
$454$		0			0
$455$	−8283.85	+	5926.81i	−0.853523	+	0.610666i
$456$		0			0
$457$	−11187.2			−1.14511			−0.572554	−	0.819867i	$-0.694047\pi$
$457$	−11187.2			−1.14511			−0.572554	+	0.819867i	$0.694047\pi$
$458$		0			0
$459$		−	1055.56i		−	0.107341i
$460$		0			0
$461$		−	13198.3i		−	1.33341i	−0.745320	−	0.666707i	$-0.767703\pi$
$461$		−	13198.3i		−	1.33341i	0.745320	−	0.666707i	$-0.232297\pi$
$462$		0			0
$463$			13072.9i			1.31220i	0.754675	+	0.656098i	$0.227794\pi$
$463$			13072.9i			1.31220i	−0.754675	+	0.656098i	$0.772206\pi$
$464$		0			0
$465$		−	10647.0i		−	1.06181i
$466$		0			0
$467$	−9255.58			−0.917125			−0.458563	−	0.888662i	$-0.651635\pi$
$467$	−9255.58			−0.917125			−0.458563	+	0.888662i	$0.651635\pi$
$468$		0			0
$469$	3112.55	+	4350.39i	0.306448	+	0.428321i
$470$		0			0
$471$			7213.09i			0.705651i
$472$		0			0
$473$	−28661.1			−2.78613
$474$		0			0
$475$	−17225.3			−1.66390
$476$		0			0
$477$	2508.49			0.240788
$478$		0			0
$479$	−12030.0			−1.14752			−0.573762	−	0.819022i	$-0.694517\pi$
$479$	−12030.0			−1.14752			−0.573762	+	0.819022i	$0.694517\pi$
$480$		0			0
$481$		−	5182.80i		−	0.491300i
$482$		0			0
$483$	3990.31	+	5577.23i	0.375912	+	0.525409i
$484$		0			0
$485$	−5865.10			−0.549114
$486$		0			0
$487$		−	9122.55i		−	0.848834i	−0.905467	−	0.424417i	$-0.860479\pi$
$487$		−	9122.55i		−	0.848834i	0.905467	−	0.424417i	$-0.139521\pi$
$488$		0			0
$489$		−	6034.31i		−	0.558039i
$490$		0			0
$491$			19318.0i			1.77558i	0.460252	+	0.887788i	$0.347759\pi$
$491$			19318.0i			1.77558i	−0.460252	+	0.887788i	$0.652241\pi$
$492$		0			0
$493$		−	5141.41i		−	0.469690i
$494$		0			0
$495$	9005.84			0.817743
$496$		0			0
$497$	3911.83	+	5467.54i	0.353058	+	0.493466i
$498$		0			0
$499$		−	5219.96i		−	0.468292i	−0.972201	−	0.234146i	$-0.924771\pi$
$499$		−	5219.96i		−	0.468292i	0.972201	−	0.234146i	$-0.0752294\pi$
$500$		0			0
$501$	6040.67			0.538677
$502$		0			0
$503$	544.887			0.0483008			0.0241504	−	0.999708i	$-0.492312\pi$
$503$	544.887			0.0483008			0.0241504	+	0.999708i	$0.492312\pi$
$504$		0			0
$505$	−5578.08			−0.491528
$506$		0			0
$507$	2936.42			0.257221
$508$		0			0
$509$			1777.65i			0.154800i	0.997000	+	0.0773999i	$0.0246618\pi$
$509$			1777.65i			0.154800i	−0.997000	+	0.0773999i	$0.975338\pi$
$510$		0			0
$511$	−7752.62	−	10835.8i	−0.671146	−	0.938056i
$512$		0			0
$513$	3771.98			0.324634
$514$		0			0
$515$			23147.9i			1.98062i
$516$		0			0
$517$		−	9305.75i		−	0.791618i
$518$		0			0
$519$			7318.66i			0.618985i
$520$		0			0
$521$			15239.0i			1.28144i	0.767773	+	0.640722i	$0.221365\pi$
$521$			15239.0i			1.28144i	−0.767773	+	0.640722i	$0.778635\pi$
$522$		0			0
$523$	−17032.9			−1.42408			−0.712042	−	0.702137i	$-0.752229\pi$
$523$	−17032.9			−1.42408			−0.712042	+	0.702137i	$0.752229\pi$
$524$		0			0
$525$	−5571.45	+	3986.18i	−0.463158	+	0.331374i
$526$		0			0
$527$		−	8805.15i		−	0.727815i
$528$		0			0
$529$	−3067.23			−0.252094
$530$		0			0
$531$	−237.688			−0.0194252
$532$		0			0
$533$	−6096.01			−0.495398
$534$		0			0
$535$	5575.18			0.450534
$536$		0			0
$537$			9876.74i			0.793692i
$538$		0			0
$539$	20615.1	+	7032.11i	1.64742	+	0.561957i
$540$		0			0
$541$	−6828.08			−0.542628			−0.271314	−	0.962491i	$-0.587458\pi$
$541$	−6828.08			−0.542628			−0.271314	+	0.962491i	$0.587458\pi$
$542$		0			0
$543$			1415.06i			0.111835i
$544$		0			0
$545$			27009.4i			2.12285i
$546$		0			0
$547$			18137.0i			1.41770i	0.705359	+	0.708850i	$0.250786\pi$
$547$			18137.0i			1.41770i	−0.705359	+	0.708850i	$0.749214\pi$
$548$		0			0
$549$			626.330i			0.0486905i
$550$		0			0
$551$	18372.5			1.42050
$552$		0			0
$553$	−10475.5	−	14641.5i	−0.805539	−	1.12590i
$554$		0			0
$555$		−	7019.65i		−	0.536878i
$556$		0			0
$557$	−3823.57			−0.290862			−0.145431	−	0.989368i	$-0.546457\pi$
$557$	−3823.57			−0.290862			−0.145431	+	0.989368i	$0.546457\pi$
$558$		0			0
$559$	15752.8			1.19190
$560$		0			0
$561$	7447.92			0.560520
$562$		0			0
$563$	−2485.52			−0.186061			−0.0930303	−	0.995663i	$-0.529655\pi$
$563$	−2485.52			−0.186061			−0.0930303	+	0.995663i	$0.529655\pi$
$564$		0			0
$565$			35546.6i			2.64682i
$566$		0			0
$567$	1220.03	−	872.892i	0.0903644	−	0.0646526i
$568$		0			0
$569$	6007.73			0.442631			0.221316	−	0.975202i	$-0.428965\pi$
$569$	6007.73			0.442631			0.221316	+	0.975202i	$0.428965\pi$
$570$		0			0
$571$			10646.6i			0.780289i	0.920754	+	0.390145i	$0.127575\pi$
$571$			10646.6i			0.780289i	−0.920754	+	0.390145i	$0.872425\pi$
$572$		0			0
$573$		−	1282.99i		−	0.0935388i
$574$		0			0
$575$		−	15218.5i		−	1.10375i
$576$		0			0
$577$		−	6678.64i		−	0.481864i	−0.970542	−	0.240932i	$-0.922547\pi$
$577$		−	6678.64i		−	0.481864i	0.970542	−	0.240932i	$-0.0774531\pi$
$578$		0			0
$579$	468.719			0.0336430
$580$		0			0
$581$	7440.86	−	5323.68i	0.531324	−	0.380143i
$582$		0			0
$583$			17699.7i			1.25737i
$584$		0			0
$585$	−4949.80			−0.349828
$586$		0			0
$587$	5989.29			0.421132			0.210566	−	0.977580i	$-0.432469\pi$
$587$	5989.29			0.421132			0.210566	+	0.977580i	$0.432469\pi$
$588$		0			0
$589$	31464.7			2.20115
$590$		0			0
$591$	−6143.53			−0.427599
$592$		0			0
$593$			418.327i			0.0289690i	0.999895	+	0.0144845i	$0.00461073\pi$
$593$			418.327i			0.0289690i	−0.999895	+	0.0144845i	$0.995389\pi$
$594$		0			0
$595$	−9278.85	+	6638.69i	−0.639321	+	0.457412i
$596$		0			0
$597$	8863.10			0.607609
$598$		0			0
$599$			6666.21i			0.454715i	0.973811	+	0.227357i	$0.0730086\pi$
$599$			6666.21i			0.454715i	−0.973811	+	0.227357i	$0.926991\pi$
$600$		0			0
$601$		−	10811.3i		−	0.733778i	−0.930265	−	0.366889i	$-0.880423\pi$
$601$		−	10811.3i		−	0.733778i	0.930265	−	0.366889i	$-0.119577\pi$
$602$		0			0
$603$			2599.46i			0.175553i
$604$		0			0
$605$			42571.0i			2.86075i
$606$		0			0
$607$	−15095.4			−1.00939			−0.504697	−	0.863297i	$-0.668396\pi$
$607$	−15095.4			−1.00939			−0.504697	+	0.863297i	$0.668396\pi$
$608$		0			0
$609$	5942.52	−	4251.66i	0.395407	−	0.282900i
$610$		0			0
$611$			5114.64i			0.338651i
$612$		0			0
$613$	−8809.28			−0.580430			−0.290215	−	0.956961i	$-0.593727\pi$
$613$	−8809.28			−0.580430			−0.290215	+	0.956961i	$0.593727\pi$
$614$		0			0
$615$	−8256.51			−0.541357
$616$		0			0
$617$	638.273			0.0416465			0.0208233	−	0.999783i	$-0.493371\pi$
$617$	638.273			0.0416465			0.0208233	+	0.999783i	$0.493371\pi$
$618$		0			0
$619$	12363.8			0.802815			0.401408	−	0.915900i	$-0.368521\pi$
$619$	12363.8			0.802815			0.401408	+	0.915900i	$0.368521\pi$
$620$		0			0
$621$			3332.53i			0.215346i
$622$		0			0
$623$	956.412	+	1336.77i	0.0615053	+	0.0859656i
$624$		0			0
$625$	−15834.7			−1.01342
$626$		0			0
$627$			26614.7i			1.69520i
$628$		0			0
$629$		−	5805.32i		−	0.368002i
$630$		0			0
$631$		−	7405.51i		−	0.467209i	−0.972332	−	0.233604i	$-0.924948\pi$
$631$		−	7405.51i		−	0.467209i	0.972332	−	0.233604i	$-0.0750521\pi$
$632$		0			0
$633$		−	6437.41i		−	0.404209i
$634$		0			0
$635$	−37724.6			−2.35757
$636$		0			0
$637$	−11330.5	−	3865.00i	−0.704759	−	0.240403i
$638$		0			0
$639$			3266.99i			0.202254i
$640$		0			0
$641$	10714.3			0.660199			0.330100	−	0.943946i	$-0.392918\pi$
$641$	10714.3			0.660199			0.330100	+	0.943946i	$0.392918\pi$
$642$		0			0
$643$	2855.49			0.175132			0.0875659	−	0.996159i	$-0.472091\pi$
$643$	2855.49			0.175132			0.0875659	+	0.996159i	$0.472091\pi$
$644$		0			0
$645$	21335.7			1.30247
$646$		0			0
$647$	7315.44			0.444513			0.222256	−	0.974988i	$-0.428658\pi$
$647$	7315.44			0.444513			0.222256	+	0.974988i	$0.428658\pi$
$648$		0			0
$649$		−	1677.10i		−	0.101436i
$650$		0			0
$651$	10177.1	−	7281.38i	0.612708	−	0.438371i
$652$		0			0
$653$	−20654.7			−1.23780			−0.618898	−	0.785472i	$-0.712420\pi$
$653$	−20654.7			−1.23780			−0.618898	+	0.785472i	$0.712420\pi$
$654$		0			0
$655$			45272.6i			2.70068i
$656$		0			0
$657$		−	6474.64i		−	0.384474i
$658$		0			0
$659$			15304.8i			0.904688i	0.891843	+	0.452344i	$0.149412\pi$
$659$			15304.8i			0.904688i	−0.891843	+	0.452344i	$0.850588\pi$
$660$		0			0
$661$			13407.2i			0.788928i	0.918912	+	0.394464i	$0.129070\pi$
$661$			13407.2i			0.788928i	−0.918912	+	0.394464i	$0.870930\pi$
$662$		0			0
$663$	−4093.54			−0.239789
$664$		0			0
$665$	−23723.0	−	33157.4i	−1.38336	−	1.93352i
$666$		0			0
$667$			16232.0i			0.942288i
$668$		0			0
$669$	−8472.58			−0.489639
$670$		0			0
$671$	−4419.31			−0.254256
$672$		0			0
$673$	14785.5			0.846863			0.423431	−	0.905928i	$-0.360825\pi$
$673$	14785.5			0.846863			0.423431	+	0.905928i	$0.360825\pi$
$674$		0			0
$675$	−3329.08			−0.189832
$676$		0			0
$677$			34340.9i			1.94953i	0.223240	+	0.974763i	$0.428337\pi$
$677$			34340.9i			1.94953i	−0.223240	+	0.974763i	$0.571663\pi$
$678$		0			0
$679$	−4011.10	−	5606.28i	−0.226704	−	0.316862i
$680$		0			0
$681$	5851.09			0.329243
$682$		0			0
$683$		−	3806.83i		−	0.213271i	−0.994298	−	0.106636i	$-0.965992\pi$
$683$		−	3806.83i		−	0.213271i	0.994298	−	0.106636i	$-0.0340079\pi$
$684$		0			0
$685$		−	48651.5i		−	2.71369i
$686$		0			0
$687$			907.907i			0.0504204i
$688$		0			0
$689$		−	9728.11i		−	0.537897i
$690$		0			0
$691$	−32647.0			−1.79732			−0.898660	−	0.438645i	$-0.855459\pi$
$691$	−32647.0			−1.79732			−0.898660	+	0.438645i	$0.855459\pi$
$692$		0			0
$693$	6159.03	+	8608.43i	0.337608	+	0.471872i
$694$		0			0
$695$			5651.41i			0.308446i
$696$		0			0
$697$	−6828.22			−0.371072
$698$		0			0
$699$	−14648.4			−0.792638
$700$		0			0
$701$	−10926.7			−0.588722			−0.294361	−	0.955694i	$-0.595107\pi$
$701$	−10926.7			−0.588722			−0.294361	+	0.955694i	$0.595107\pi$
$702$		0			0
$703$	20745.0			1.11296
$704$		0			0
$705$			6927.33i			0.370069i
$706$		0			0
$707$	−3814.81	−	5331.93i	−0.202929	−	0.283632i
$708$		0			0
$709$	22837.0			1.20968			0.604840	−	0.796347i	$-0.293237\pi$
$709$	22837.0			1.20968			0.604840	+	0.796347i	$0.293237\pi$
$710$		0			0
$711$		−	8748.66i		−	0.461463i
$712$		0			0
$713$			27798.9i			1.46014i
$714$		0			0
$715$		−	34925.3i		−	1.82676i
$716$		0			0
$717$		−	742.290i		−	0.0386630i
$718$		0			0
$719$	−11476.6			−0.595277			−0.297638	−	0.954679i	$-0.596199\pi$
$719$	−11476.6			−0.595277			−0.297638	+	0.954679i	$0.596199\pi$
$720$		0			0
$721$	−22126.4	+	15830.7i	−1.14290	+	0.817706i
$722$		0			0
$723$		−	3747.62i		−	0.192774i
$724$		0			0
$725$	−16215.2			−0.830645
$726$		0			0
$727$	28268.3			1.44211			0.721054	−	0.692879i	$-0.243658\pi$
$727$	28268.3			1.44211			0.721054	+	0.692879i	$0.243658\pi$
$728$		0			0
$729$	729.000			0.0370370
$730$		0			0
$731$	17644.9			0.892776
$732$		0			0
$733$		−	19519.1i		−	0.983569i	−0.870717	−	0.491784i	$-0.836345\pi$
$733$		−	19519.1i		−	0.983569i	0.870717	−	0.491784i	$-0.163655\pi$
$734$		0			0
$735$	−15346.2	−	5234.81i	−0.770141	−	0.262706i
$736$		0			0
$737$	−18341.5			−0.916715
$738$		0			0
$739$			21461.4i			1.06830i	0.845391	+	0.534149i	$0.179368\pi$
$739$			21461.4i			1.06830i	−0.845391	+	0.534149i	$0.820632\pi$
$740$		0			0
$741$		−	14628.0i		−	0.725200i
$742$		0			0
$743$		−	21132.8i		−	1.04346i	−0.853112	−	0.521729i	$-0.825287\pi$
$743$		−	21132.8i		−	1.04346i	0.853112	−	0.521729i	$-0.174713\pi$
$744$		0			0
$745$		−	30459.7i		−	1.49793i
$746$		0			0
$747$	4446.10			0.217770
$748$		0			0
$749$	3812.82	+	5329.15i	0.186004	+	0.259977i
$750$		0			0
$751$		−	14378.9i		−	0.698659i	−0.937000	−	0.349329i	$-0.886409\pi$
$751$		−	14378.9i		−	0.698659i	0.937000	−	0.349329i	$-0.113591\pi$
$752$		0			0
$753$	14282.9			0.691234
$754$		0			0
$755$	−37965.7			−1.83008
$756$		0			0
$757$	−9210.36			−0.442215			−0.221107	−	0.975249i	$-0.570967\pi$
$757$	−9210.36			−0.442215			−0.221107	+	0.975249i	$0.570967\pi$
$758$		0			0
$759$	−23514.0			−1.12451
$760$		0			0
$761$			25087.6i			1.19504i	0.801854	+	0.597519i	$0.203847\pi$
$761$			25087.6i			1.19504i	−0.801854	+	0.597519i	$0.796153\pi$
$762$		0			0
$763$	−25817.5	+	18471.5i	−1.22497	+	0.876425i
$764$		0			0
$765$	−5544.34			−0.262034
$766$		0			0
$767$			921.770i			0.0433940i
$768$		0			0
$769$			330.201i			0.0154842i	0.999970	+	0.00774211i	$0.00246441\pi$
$769$			330.201i			0.0154842i	−0.999970	+	0.00774211i	$0.997536\pi$
$770$		0			0
$771$			10665.0i			0.498172i
$772$		0			0
$773$			8703.46i			0.404970i	0.979285	+	0.202485i	$0.0649017\pi$
$773$			8703.46i			0.404970i	−0.979285	+	0.202485i	$0.935098\pi$
$774$		0			0
$775$	−27770.1			−1.28714
$776$		0			0
$777$	6709.88	−	4800.68i	0.309801	−	0.221652i
$778$		0			0
$779$		−	24400.2i		−	1.12224i
$780$		0			0
$781$	−23051.5			−1.05614
$782$		0			0
$783$	3550.80			0.162063
$784$		0			0
$785$	37886.8			1.72260
$786$		0			0
$787$	41633.7			1.88574			0.942872	−	0.333156i	$-0.108114\pi$
$787$	41633.7			1.88574			0.942872	+	0.333156i	$0.108114\pi$
$788$		0			0
$789$			8178.61i			0.369032i
$790$		0			0
$791$	−33977.9	+	24310.0i	−1.52733	+	1.09275i
$792$		0			0
$793$	2428.95			0.108770
$794$		0			0
$795$		−	13175.9i		−	0.587799i
$796$		0			0
$797$			16494.2i			0.733068i	0.930405	+	0.366534i	$0.119456\pi$
$797$			16494.2i			0.733068i	−0.930405	+	0.366534i	$0.880544\pi$
$798$		0			0
$799$			5728.97i			0.253663i
$800$		0			0
$801$			798.752i			0.0352341i
$802$		0			0
$803$	45684.3			2.00768
$804$		0			0
$805$	29294.4	−	20959.1i	1.28260	−	0.917655i
$806$		0			0
$807$		−	7080.03i		−	0.308834i
$808$		0			0
$809$	−14458.7			−0.628359			−0.314179	−	0.949364i	$-0.601729\pi$
$809$	−14458.7			−0.628359			−0.314179	+	0.949364i	$0.601729\pi$
$810$		0			0
$811$	−26005.5			−1.12599			−0.562994	−	0.826461i	$-0.690351\pi$
$811$	−26005.5			−1.12599			−0.562994	+	0.826461i	$0.690351\pi$
$812$		0			0
$813$	406.345			0.0175291
$814$		0			0
$815$	−31695.2			−1.36225
$816$		0			0
$817$			63052.9i			2.70005i
$818$		0			0
$819$	−3385.13	−	4731.37i	−0.144427	−	0.201865i
$820$		0			0
$821$	29654.2			1.26058			0.630291	−	0.776359i	$-0.282935\pi$
$821$	29654.2			1.26058			0.630291	+	0.776359i	$0.282935\pi$
$822$		0			0
$823$		−	21383.1i		−	0.905672i	−0.891594	−	0.452836i	$-0.850412\pi$
$823$		−	21383.1i		−	0.905672i	0.891594	−	0.452836i	$-0.149588\pi$
$824$		0			0
$825$		−	23489.6i		−	0.991277i
$826$		0			0
$827$		−	4239.01i		−	0.178240i	−0.996021	−	0.0891201i	$-0.971594\pi$
$827$		−	4239.01i		−	0.178240i	0.996021	−	0.0891201i	$-0.0284055\pi$
$828$		0			0
$829$		−	18040.1i		−	0.755799i	−0.925847	−	0.377900i	$-0.876646\pi$
$829$		−	18040.1i		−	0.755799i	0.925847	−	0.377900i	$-0.123354\pi$
$830$		0			0
$831$	12894.0			0.538253
$832$		0			0
$833$	−12691.5	−	4329.24i	−0.527891	−	0.180071i
$834$		0			0
$835$		−	31728.6i		−	1.31499i
$836$		0			0
$837$	6081.08			0.251127
$838$		0			0
$839$	−8064.28			−0.331836			−0.165918	−	0.986140i	$-0.553059\pi$
$839$	−8064.28			−0.331836			−0.165918	+	0.986140i	$0.553059\pi$
$840$		0			0
$841$	−7093.85			−0.290863
$842$		0			0
$843$	−19853.9			−0.811157
$844$		0			0
$845$		−	15423.6i		−	0.627914i
$846$		0			0
$847$	−40692.4	+	29114.0i	−1.65077	+	1.18107i
$848$		0			0
$849$	25141.3			1.01631
$850$		0			0
$851$			18328.1i			0.738282i
$852$		0			0
$853$		−	41230.1i		−	1.65497i	−0.561486	−	0.827486i	$-0.689770\pi$
$853$		−	41230.1i		−	1.65497i	0.561486	−	0.827486i	$-0.310230\pi$
$854$		0			0
$855$		−	19812.4i		−	0.792478i
$856$		0			0
$857$			1919.57i			0.0765124i	0.999268	+	0.0382562i	$0.0121803\pi$
$857$			1919.57i			0.0765124i	−0.999268	+	0.0382562i	$0.987820\pi$
$858$		0			0
$859$	−8841.06			−0.351168			−0.175584	−	0.984464i	$-0.556181\pi$
$859$	−8841.06			−0.351168			−0.175584	+	0.984464i	$0.556181\pi$
$860$		0			0
$861$	−5646.56	−	7892.16i	−0.223501	−	0.312386i
$862$		0			0
$863$			30598.2i			1.20692i	0.797391	+	0.603462i	$0.206213\pi$
$863$			30598.2i			1.20692i	−0.797391	+	0.603462i	$0.793787\pi$
$864$		0			0
$865$	38441.3			1.51103
$866$		0			0
$867$	10153.8			0.397740
$868$		0			0
$869$	61729.6			2.40970
$870$		0			0
$871$	10080.9			0.392168
$872$		0			0
$873$		−	3349.89i		−	0.129870i
$874$		0			0
$875$	−288.811	−	403.669i	−0.0111584	−	0.0155960i
$876$		0			0
$877$	−13839.9			−0.532885			−0.266442	−	0.963851i	$-0.585848\pi$
$877$	−13839.9			−0.532885			−0.266442	+	0.963851i	$0.585848\pi$
$878$		0			0
$879$		−	25764.7i		−	0.988649i
$880$		0			0
$881$		−	7609.12i		−	0.290985i	−0.989359	−	0.145492i	$-0.953523\pi$
$881$		−	7609.12i		−	0.290985i	0.989359	−	0.145492i	$-0.0464766\pi$
$882$		0			0
$883$			18613.2i			0.709382i	0.934984	+	0.354691i	$0.115414\pi$
$883$			18613.2i			0.709382i	−0.934984	+	0.354691i	$0.884586\pi$
$884$		0			0
$885$			1248.46i			0.0474197i
$886$		0			0
$887$	16661.0			0.630691			0.315346	−	0.948977i	$-0.397880\pi$
$887$	16661.0			0.630691			0.315346	+	0.948977i	$0.397880\pi$
$888$		0			0
$889$	−25799.6	−	36059.9i	−0.973329	−	1.36041i
$890$		0			0
$891$			5143.74i			0.193403i
$892$		0			0
$893$	−20472.1			−0.767160
$894$		0			0
$895$	51877.6			1.93752
$896$		0			0
$897$	12923.8			0.481062
$898$		0			0
$899$	29619.6			1.09885
$900$		0			0
$901$		−	10896.6i		−	0.402905i
$902$		0			0
$903$	14591.3	+	20394.2i	0.537729	+	0.751580i
$904$		0			0
$905$	7432.63			0.273005
$906$		0			0
$907$		−	45091.3i		−	1.65075i	−0.564584	−	0.825376i	$-0.690963\pi$
$907$		−	45091.3i		−	1.65075i	0.564584	−	0.825376i	$-0.309037\pi$
$908$		0			0
$909$		−	3185.96i		−	0.116250i
$910$		0			0
$911$			4614.00i			0.167803i	0.996474	+	0.0839015i	$0.0267381\pi$
$911$			4614.00i			0.167803i	−0.996474	+	0.0839015i	$0.973262\pi$
$912$		0			0
$913$			31371.2i			1.13717i
$914$		0			0
$915$	3289.80			0.118861
$916$		0			0
$917$	−43274.8	+	30961.6i	−1.55841	+	1.11499i
$918$		0			0
$919$			38342.0i			1.37626i	0.725586	+	0.688131i	$0.241569\pi$
$919$			38342.0i			1.37626i	−0.725586	+	0.688131i	$0.758431\pi$
$920$		0			0
$921$	705.019			0.0252238
$922$		0			0
$923$	12669.6			0.451815
$924$		0			0
$925$	−18309.1			−0.650810
$926$		0			0
$927$	−13221.1			−0.468433
$928$		0			0
$929$			30005.0i			1.05967i	0.848101	+	0.529834i	$0.177746\pi$
$929$			30005.0i			1.05967i	−0.848101	+	0.529834i	$0.822254\pi$
$930$		0			0
$931$	15470.3	−	45352.2i	0.544595	−	1.59652i
$932$		0			0
$933$	−28633.4			−1.00473
$934$		0			0
$935$		−	39120.3i		−	1.36831i
$936$		0			0
$937$			12167.2i			0.424212i	0.977247	+	0.212106i	$0.0680322\pi$
$937$			12167.2i			0.424212i	−0.977247	+	0.212106i	$0.931968\pi$
$938$		0			0
$939$			27707.1i			0.962924i
$940$		0			0
$941$			166.924i			0.00578275i	0.999996	+	0.00289138i	$0.000920355\pi$
$941$			166.924i			0.00578275i	−0.999996	+	0.00289138i	$0.999080\pi$
$942$		0			0
$943$	21557.5			0.744441
$944$		0			0
$945$	−4584.87	−	6408.23i	−0.157826	−	0.220592i
$946$		0			0
$947$		−	29354.6i		−	1.00728i	−0.863913	−	0.503642i	$-0.831993\pi$
$947$		−	29354.6i		−	1.00728i	0.863913	−	0.503642i	$-0.168007\pi$
$948$		0			0
$949$	−25109.1			−0.858878
$950$		0			0
$951$	679.637			0.0231743
$952$		0			0
$953$	3636.06			0.123592			0.0617961	−	0.998089i	$-0.480317\pi$
$953$	3636.06			0.123592			0.0617961	+	0.998089i	$0.480317\pi$
$954$		0			0
$955$	−6738.92			−0.228342
$956$		0			0
$957$			25054.0i			0.846272i
$958$		0			0
$959$	46504.5	−	33272.4i	1.56591	−	1.12036i
$960$		0			0
$961$	20935.4			0.702743
$962$		0			0
$963$			3184.30i			0.106555i
$964$		0			0
$965$		−	2461.95i		−	0.0821274i
$966$		0			0
$967$			40090.0i			1.33320i	0.745414	+	0.666602i	$0.232252\pi$
$967$			40090.0i			1.33320i	−0.745414	+	0.666602i	$0.767748\pi$
$968$		0			0
$969$		−	16385.0i		−	0.543202i
$970$		0			0
$971$	54004.4			1.78484			0.892421	−	0.451203i	$-0.149005\pi$
$971$	54004.4			1.78484			0.892421	+	0.451203i	$0.149005\pi$
$972$		0			0
$973$	−5402.02	+	3864.96i	−0.177986	+	0.127343i
$974$		0			0
$975$			12910.4i			0.424065i
$976$		0			0
$977$	−10178.4			−0.333303			−0.166652	−	0.986016i	$-0.553296\pi$
$977$	−10178.4			−0.333303			−0.166652	+	0.986016i	$0.553296\pi$
$978$		0			0
$979$	−5635.91			−0.183988
$980$		0			0
$981$	−15426.6			−0.502071
$982$		0			0
$983$	−11555.3			−0.374932			−0.187466	−	0.982271i	$-0.560027\pi$
$983$	−11555.3			−0.374932			−0.187466	+	0.982271i	$0.560027\pi$
$984$		0			0
$985$			32268.9i			1.04383i
$986$		0			0
$987$	−6621.63	+	4737.55i	−0.213545	+	0.152784i
$988$		0			0
$989$	−55706.9			−1.79108
$990$		0			0
$991$			16946.8i			0.543222i	0.962407	+	0.271611i	$0.0875564\pi$
$991$			16946.8i			0.543222i	−0.962407	+	0.271611i	$0.912444\pi$
$992$		0			0
$993$			25494.4i			0.814745i
$994$		0			0
$995$		−	46553.5i		−	1.48326i
$996$		0			0
$997$		−	18666.0i		−	0.592937i	−0.955043	−	0.296468i	$-0.904191\pi$
$997$		−	18666.0i		−	0.592937i	0.955043	−	0.296468i	$-0.0958090\pi$
$998$		0			0
$999$	4009.32			0.126976

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1344.4.b.j.895.2		24
4.3	odd	2		1344.4.b.i.895.2		24
7.6	odd	2		1344.4.b.i.895.23		24
8.3	odd	2		672.4.b.b.223.23	yes	24
8.5	even	2		672.4.b.a.223.23	yes	24
28.27	even	2	inner	1344.4.b.j.895.23		24
56.13	odd	2		672.4.b.b.223.2	yes	24
56.27	even	2		672.4.b.a.223.2	✓	24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
672.4.b.a.223.2	✓	24	56.27	even	2
672.4.b.a.223.23	yes	24	8.5	even	2
672.4.b.b.223.2	yes	24	56.13	odd	2
672.4.b.b.223.23	yes	24	8.3	odd	2
1344.4.b.i.895.2		24	4.3	odd	2
1344.4.b.i.895.23		24	7.6	odd	2
1344.4.b.j.895.2		24	1.1	even	1	trivial
1344.4.b.j.895.23		24	28.27	even	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q+3.00000 q^{3} -15.7575i q^{5} +(15.0622 - 10.7764i) q^{7} +9.00000 q^{9} +O(q^{10})\)	\(q+3.00000 q^{3} -15.7575i q^{5} +(15.0622 - 10.7764i) q^{7} +9.00000 q^{9} +63.5030i q^{11} -34.9026i q^{13} -47.2725i q^{15} -39.0949i q^{17} +139.703 q^{19} +(45.1865 - 32.3293i) q^{21} +123.427i q^{23} -123.299 q^{25} +27.0000 q^{27} +131.511 q^{29} +225.225 q^{31} +190.509i q^{33} +(-169.810 - 237.342i) q^{35} +148.493 q^{37} -104.708i q^{39} -174.658i q^{41} +451.335i q^{43} -141.818i q^{45} -146.540 q^{47} +(110.737 - 324.633i) q^{49} -117.285i q^{51} +278.721 q^{53} +1000.65 q^{55} +419.109 q^{57} -26.4098 q^{59} +69.5922i q^{61} +(135.559 - 96.9879i) q^{63} -549.978 q^{65} +288.829i q^{67} +370.281i q^{69} +362.999i q^{71} -719.404i q^{73} -369.898 q^{75} +(684.336 + 956.492i) q^{77} -972.073i q^{79} +81.0000 q^{81} +494.011 q^{83} -616.038 q^{85} +394.533 q^{87} +88.7502i q^{89} +(-376.126 - 525.708i) q^{91} +675.676 q^{93} -2201.37i q^{95} -372.210i q^{97} +571.527i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q + 72 q^{3} - 20 q^{7} + 216 q^{9}+O(q^{10}) \) 24 * q + 72 * q^3 - 20 * q^7 + 216 * q^9	\( 24 q + 72 q^{3} - 20 q^{7} + 216 q^{9} - 56 q^{19} - 60 q^{21} - 432 q^{25} + 648 q^{27} + 464 q^{31} + 568 q^{35} - 504 q^{37} + 560 q^{47} - 128 q^{49} + 784 q^{53} + 424 q^{55} - 168 q^{57} + 800 q^{59} - 180 q^{63} + 560 q^{65} - 1296 q^{75} + 1568 q^{77} + 1944 q^{81} + 1936 q^{83} - 3000 q^{85} - 496 q^{91} + 1392 q^{93}+O(q^{100}) \) 24 * q + 72 * q^3 - 20 * q^7 + 216 * q^9 - 56 * q^19 - 60 * q^21 - 432 * q^25 + 648 * q^27 + 464 * q^31 + 568 * q^35 - 504 * q^37 + 560 * q^47 - 128 * q^49 + 784 * q^53 + 424 * q^55 - 168 * q^57 + 800 * q^59 - 180 * q^63 + 560 * q^65 - 1296 * q^75 + 1568 * q^77 + 1944 * q^81 + 1936 * q^83 - 3000 * q^85 - 496 * q^91 + 1392 * q^93

Introduction

L-functions

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Database

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