LMFDB - Embedded newform 1344.4.p.a.223.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1344,4,Mod(223,1344)]

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, 1, 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("1344.223");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1344 = 2^{6} \cdot 3 \cdot 7 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1344.p (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:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 2 x^{15} + 2 x^{14} + 58 x^{13} + 178264 x^{12} - 331354 x^{11} + 307862 x^{10} + \cdots + 22\!\cdots\!01 $ x^16 - 2x^15 + 2x^14 + 58x^13 + 178264x^12 - 331354x^11 + 307862x^10 - 610x^9 + 8375926786x^8 - 15937543350x^7 + 15140222838x^6 - 113574152394x^5 + 45261122709708x^4 - 121918187031822x^3 + 150109311002562x^2 + 2604526546410882*x + 22595395600887201
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{17}\cdot 3^{4} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			223.3
Root			$5.50749 + 5.50749i$ of defining polynomial
Character	$\chi$	$=$	1344.223
Dual form			1344.4.p.a.223.11

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.00000i q^{3} -4.35176 q^{5} +(-7.09314 + 17.1081i) q^{7} -9.00000 q^{9} +O(q^{10})$	$q-3.00000i q^{3} -4.35176 q^{5} +(-7.09314 + 17.1081i) q^{7} -9.00000 q^{9} -43.1344 q^{11} +26.5241 q^{13} +13.0553i q^{15} -76.6921i q^{17} -58.7403i q^{19} +(51.3243 + 21.2794i) q^{21} +0.747396i q^{23} -106.062 q^{25} +27.0000i q^{27} -64.9778i q^{29} -165.456 q^{31} +129.403i q^{33} +(30.8676 - 74.4503i) q^{35} -389.771i q^{37} -79.5723i q^{39} +290.660i q^{41} +32.4066 q^{43} +39.1658 q^{45} +368.457 q^{47} +(-242.375 - 242.700i) q^{49} -230.076 q^{51} +588.016i q^{53} +187.710 q^{55} -176.221 q^{57} +471.583i q^{59} +448.047 q^{61} +(63.8382 - 153.973i) q^{63} -115.426 q^{65} +707.632 q^{67} +2.24219 q^{69} +997.655i q^{71} +1225.05i q^{73} +318.187i q^{75} +(305.958 - 737.947i) q^{77} -160.665i q^{79} +81.0000 q^{81} +1000.97i q^{83} +333.745i q^{85} -194.933 q^{87} -1138.80i q^{89} +(-188.139 + 453.777i) q^{91} +496.368i q^{93} +255.624i q^{95} -729.761i q^{97} +388.209 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 144 q^{9}+O(q^{10}) $ 16 * q - 144 * q^9	$ 16 q - 144 q^{9} - 56 q^{13} - 36 q^{21} + 80 q^{25} + 392 q^{49} + 336 q^{57} - 184 q^{61} - 1536 q^{65} - 864 q^{69} + 240 q^{77} + 1296 q^{81}+O(q^{100}) $ 16 * q - 144 * q^9 - 56 * q^13 - 36 * q^21 + 80 * q^25 + 392 * q^49 + 336 * q^57 - 184 * q^61 - 1536 * q^65 - 864 * q^69 + 240 * q^77 + 1296 * q^81

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
$5$	−4.35176			−0.389233			−0.194617	−	0.980879i	$-0.562346\pi$
$5$	−4.35176			−0.389233			−0.194617	+	0.980879i	$0.562346\pi$
$6$		0			0
$7$	−7.09314	+	17.1081i	−0.382993	+	0.923751i
$7$	−7.09314	+	17.1081i	−0.382993	+	0.923751i
$8$		0			0
$9$	−9.00000			−0.333333
$10$		0			0
$11$	−43.1344			−1.18232			−0.591159	−	0.806555i	$-0.701329\pi$
$11$	−43.1344			−1.18232			−0.591159	+	0.806555i	$0.701329\pi$
$12$		0			0
$13$	26.5241			0.565882			0.282941	−	0.959137i	$-0.408690\pi$
$13$	26.5241			0.565882			0.282941	+	0.959137i	$0.408690\pi$
$14$		0			0
$15$			13.0553i			0.224724i
$16$		0			0
$17$		−	76.6921i		−	1.09415i	−0.837083	−	0.547075i	$-0.815741\pi$
$17$		−	76.6921i		−	1.09415i	0.837083	−	0.547075i	$-0.184259\pi$
$18$		0			0
$19$		−	58.7403i		−	0.709261i	−0.935007	−	0.354630i	$-0.884607\pi$
$19$		−	58.7403i		−	0.709261i	0.935007	−	0.354630i	$-0.115393\pi$
$20$		0			0
$21$	51.3243	+	21.2794i	0.533328	+	0.221121i
$22$		0			0
$23$			0.747396i			0.00677578i	0.999994	+	0.00338789i	$0.00107840\pi$
$23$			0.747396i			0.00677578i	−0.999994	+	0.00338789i	$0.998922\pi$
$24$		0			0
$25$	−106.062			−0.848498
$26$		0			0
$27$			27.0000i			0.192450i
$28$		0			0
$29$		−	64.9778i		−	0.416071i	−0.978121	−	0.208036i	$-0.933293\pi$
$29$		−	64.9778i		−	0.416071i	0.978121	−	0.208036i	$-0.0667070\pi$
$30$		0			0
$31$	−165.456			−0.958605			−0.479302	−	0.877650i	$-0.659110\pi$
$31$	−165.456			−0.958605			−0.479302	+	0.877650i	$0.659110\pi$
$32$		0			0
$33$			129.403i			0.682612i
$34$		0			0
$35$	30.8676	−	74.4503i	0.149074	−	0.359554i
$36$		0			0
$37$		−	389.771i		−	1.73184i	−0.500184	−	0.865919i	$-0.666734\pi$
$37$		−	389.771i		−	1.73184i	0.500184	−	0.865919i	$-0.333266\pi$
$38$		0			0
$39$		−	79.5723i		−	0.326712i
$40$		0			0
$41$			290.660i			1.10716i	0.832796	+	0.553579i	$0.186738\pi$
$41$			290.660i			1.10716i	−0.832796	+	0.553579i	$0.813262\pi$
$42$		0			0
$43$	32.4066			0.114929			0.0574647	−	0.998348i	$-0.481698\pi$
$43$	32.4066			0.114929			0.0574647	+	0.998348i	$0.481698\pi$
$44$		0			0
$45$	39.1658			0.129744
$46$		0			0
$47$	368.457			1.14351			0.571756	−	0.820424i	$-0.306263\pi$
$47$	368.457			1.14351			0.571756	+	0.820424i	$0.306263\pi$
$48$		0			0
$49$	−242.375	−	242.700i	−0.706632	−	0.707581i
$50$		0			0
$51$	−230.076			−0.631708
$52$		0			0
$53$			588.016i			1.52397i	0.647596	+	0.761984i	$0.275774\pi$
$53$			588.016i			1.52397i	−0.647596	+	0.761984i	$0.724226\pi$
$54$		0			0
$55$	187.710			0.460197
$56$		0			0
$57$	−176.221			−0.409492
$58$		0			0
$59$			471.583i			1.04059i	0.853986	+	0.520296i	$0.174178\pi$
$59$			471.583i			1.04059i	−0.853986	+	0.520296i	$0.825822\pi$
$60$		0			0
$61$	448.047			0.940434			0.470217	−	0.882551i	$-0.344176\pi$
$61$	448.047			0.940434			0.470217	+	0.882551i	$0.344176\pi$
$62$		0			0
$63$	63.8382	−	153.973i	0.127664	−	0.307917i
$64$		0			0
$65$	−115.426			−0.220260
$66$		0			0
$67$	707.632			1.29031			0.645157	−	0.764050i	$-0.276792\pi$
$67$	707.632			1.29031			0.645157	+	0.764050i	$0.276792\pi$
$68$		0			0
$69$	2.24219			0.00391200
$70$		0			0
$71$			997.655i			1.66760i	0.552064	+	0.833802i	$0.313840\pi$
$71$			997.655i			1.66760i	−0.552064	+	0.833802i	$0.686160\pi$
$72$		0			0
$73$			1225.05i			1.96413i	0.188553	+	0.982063i	$0.439620\pi$
$73$			1225.05i			1.96413i	−0.188553	+	0.982063i	$0.560380\pi$
$74$		0			0
$75$			318.187i			0.489880i
$76$		0			0
$77$	305.958	−	737.947i	0.452820	−	1.09217i
$78$		0			0
$79$		−	160.665i		−	0.228813i	−0.993434	−	0.114407i	$-0.963503\pi$
$79$		−	160.665i		−	0.228813i	0.993434	−	0.114407i	$-0.0364966\pi$
$80$		0			0
$81$	81.0000			0.111111
$82$		0			0
$83$			1000.97i			1.32375i	0.749616	+	0.661873i	$0.230238\pi$
$83$			1000.97i			1.32375i	−0.749616	+	0.661873i	$0.769762\pi$
$84$		0			0
$85$			333.745i			0.425879i
$86$		0			0
$87$	−194.933			−0.240219
$88$		0			0
$89$		−	1138.80i		−	1.35632i	−0.734915	−	0.678160i	$-0.762778\pi$
$89$		−	1138.80i		−	1.35632i	0.734915	−	0.678160i	$-0.237222\pi$
$90$		0			0
$91$	−188.139	+	453.777i	−0.216729	+	0.522734i
$92$		0			0
$93$			496.368i			0.553451i
$94$		0			0
$95$			255.624i			0.276068i
$96$		0			0
$97$		−	729.761i		−	0.763876i	−0.924188	−	0.381938i	$-0.875257\pi$
$97$		−	729.761i		−	0.763876i	0.924188	−	0.381938i	$-0.124743\pi$
$98$		0			0
$99$	388.209			0.394106
$100$		0			0
$101$	313.536			0.308891			0.154446	−	0.988001i	$-0.450641\pi$
$101$	313.536			0.308891			0.154446	+	0.988001i	$0.450641\pi$
$102$		0			0
$103$	−141.702			−0.135557			−0.0677784	−	0.997700i	$-0.521591\pi$
$103$	−141.702			−0.135557			−0.0677784	+	0.997700i	$0.521591\pi$
$104$		0			0
$105$	−223.351	−	92.6028i	−0.207589	−	0.0860677i
$106$		0			0
$107$	960.650			0.867940			0.433970	−	0.900927i	$-0.357112\pi$
$107$	960.650			0.867940			0.433970	+	0.900927i	$0.357112\pi$
$108$		0			0
$109$		−	1044.39i		−	0.917750i	−0.888501	−	0.458875i	$-0.848253\pi$
$109$		−	1044.39i		−	0.917750i	0.888501	−	0.458875i	$-0.151747\pi$
$110$		0			0
$111$	−1169.31			−0.999877
$112$		0			0
$113$	45.9468			0.0382506			0.0191253	−	0.999817i	$-0.493912\pi$
$113$	45.9468			0.0382506			0.0191253	+	0.999817i	$0.493912\pi$
$114$		0			0
$115$		−	3.25249i		−	0.00263736i
$116$		0			0
$117$	−238.717			−0.188627
$118$		0			0
$119$	1312.06	+	543.987i	1.01072	+	0.419052i
$120$		0			0
$121$	529.573			0.397876
$122$		0			0
$123$	871.981			0.639219
$124$		0			0
$125$	1005.53			0.719496
$126$		0			0
$127$			340.925i			0.238206i	0.992882	+	0.119103i	$0.0380019\pi$
$127$			340.925i			0.238206i	−0.992882	+	0.119103i	$0.961998\pi$
$128$		0			0
$129$		−	97.2199i		−	0.0663545i
$130$		0			0
$131$			732.678i			0.488659i	0.969692	+	0.244330i	$0.0785679\pi$
$131$			732.678i			0.488659i	−0.969692	+	0.244330i	$0.921432\pi$
$132$		0			0
$133$	1004.94	+	416.653i	0.655180	+	0.271642i
$134$		0			0
$135$		−	117.497i		−	0.0749079i
$136$		0			0
$137$	292.821			0.182609			0.0913044	−	0.995823i	$-0.470896\pi$
$137$	292.821			0.182609			0.0913044	+	0.995823i	$0.470896\pi$
$138$		0			0
$139$			1618.66i			0.987717i	0.869542	+	0.493858i	$0.164414\pi$
$139$			1618.66i			0.987717i	−0.869542	+	0.493858i	$0.835586\pi$
$140$		0			0
$141$		−	1105.37i		−	0.660206i
$142$		0			0
$143$	−1144.10			−0.669052
$144$		0			0
$145$			282.767i			0.161949i
$146$		0			0
$147$	−728.101	+	727.124i	−0.408522	+	0.407974i
$148$		0			0
$149$		−	1405.12i		−	0.772561i	−0.922381	−	0.386280i	$-0.873760\pi$
$149$		−	1405.12i		−	0.772561i	0.922381	−	0.386280i	$-0.126240\pi$
$150$		0			0
$151$		−	463.962i		−	0.250044i	−0.992154	−	0.125022i	$-0.960100\pi$
$151$		−	463.962i		−	0.250044i	0.992154	−	0.125022i	$-0.0399002\pi$
$152$		0			0
$153$			690.229i			0.364717i
$154$		0			0
$155$	720.024			0.373121
$156$		0			0
$157$	3761.11			1.91191			0.955953	−	0.293521i	$-0.0948272\pi$
$157$	3761.11			1.91191			0.955953	+	0.293521i	$0.0948272\pi$
$158$		0			0
$159$	1764.05			0.879863
$160$		0			0
$161$	−12.7865	−	5.30139i	−0.00625913	−	0.00259508i
$162$		0			0
$163$	−1828.77			−0.878775			−0.439388	−	0.898298i	$-0.644805\pi$
$163$	−1828.77			−0.878775			−0.439388	+	0.898298i	$0.644805\pi$
$164$		0			0
$165$		−	563.131i		−	0.265695i
$166$		0			0
$167$	376.850			0.174620			0.0873100	−	0.996181i	$-0.472173\pi$
$167$	376.850			0.174620			0.0873100	+	0.996181i	$0.472173\pi$
$168$		0			0
$169$	−1493.47			−0.679778
$170$		0			0
$171$			528.663i			0.236420i
$172$		0			0
$173$	3388.91			1.48933			0.744665	−	0.667438i	$-0.232609\pi$
$173$	3388.91			1.48933			0.744665	+	0.667438i	$0.232609\pi$
$174$		0			0
$175$	752.314	−	1814.52i	0.324969	−	0.783801i
$176$		0			0
$177$	1414.75			0.600786
$178$		0			0
$179$	−326.185			−0.136202			−0.0681011	−	0.997678i	$-0.521694\pi$
$179$	−326.185			−0.136202			−0.0681011	+	0.997678i	$0.521694\pi$
$180$		0			0
$181$	1651.06			0.678024			0.339012	−	0.940782i	$-0.389907\pi$
$181$	1651.06			0.678024			0.339012	+	0.940782i	$0.389907\pi$
$182$		0			0
$183$		−	1344.14i		−	0.542960i
$184$		0			0
$185$			1696.19i			0.674089i
$186$		0			0
$187$			3308.06i			1.29363i
$188$		0			0
$189$	−461.919	−	191.515i	−0.177776	−	0.0737071i
$190$		0			0
$191$		−	1762.41i		−	0.667664i	−0.942633	−	0.333832i	$-0.891658\pi$
$191$		−	1762.41i		−	0.667664i	0.942633	−	0.333832i	$-0.108342\pi$
$192$		0			0
$193$	4353.22			1.62358			0.811792	−	0.583946i	$-0.198492\pi$
$193$	4353.22			1.62358			0.811792	+	0.583946i	$0.198492\pi$
$194$		0			0
$195$			346.279i			0.127167i
$196$		0			0
$197$		−	1469.72i		−	0.531541i	−0.964036	−	0.265770i	$-0.914374\pi$
$197$		−	1469.72i		−	0.531541i	0.964036	−	0.265770i	$-0.0856263\pi$
$198$		0			0
$199$	−2812.82			−1.00199			−0.500994	−	0.865450i	$-0.667032\pi$
$199$	−2812.82			−1.00199			−0.500994	+	0.865450i	$0.667032\pi$
$200$		0			0
$201$		−	2122.90i		−	0.744963i
$202$		0			0
$203$	1111.65	+	460.896i	0.384346	+	0.159353i
$204$		0			0
$205$		−	1264.88i		−	0.430943i
$206$		0			0
$207$		−	6.72657i		−	0.00225859i
$208$		0			0
$209$			2533.73i			0.838572i
$210$		0			0
$211$	2434.21			0.794208			0.397104	−	0.917774i	$-0.370015\pi$
$211$	2434.21			0.794208			0.397104	+	0.917774i	$0.370015\pi$
$212$		0			0
$213$	2992.96			0.962791
$214$		0			0
$215$	−141.026			−0.0447343
$216$		0			0
$217$	1173.60	−	2830.64i	0.367139	−	0.885512i
$218$		0			0
$219$	3675.15			1.13399
$220$		0			0
$221$		−	2034.19i		−	0.619160i
$222$		0			0
$223$	849.910			0.255221			0.127610	−	0.991824i	$-0.459269\pi$
$223$	849.910			0.255221			0.127610	+	0.991824i	$0.459269\pi$
$224$		0			0
$225$	954.560			0.282833
$226$		0			0
$227$			2287.25i			0.668768i	0.942437	+	0.334384i	$0.108528\pi$
$227$			2287.25i			0.668768i	−0.942437	+	0.334384i	$0.891472\pi$
$228$		0			0
$229$	2263.21			0.653088			0.326544	−	0.945182i	$-0.394116\pi$
$229$	2263.21			0.653088			0.326544	+	0.945182i	$0.394116\pi$
$230$		0			0
$231$	−2213.84	−	917.874i	−0.630563	−	0.261436i
$232$		0			0
$233$	−6311.21			−1.77451			−0.887255	−	0.461279i	$-0.847391\pi$
$233$	−6311.21			−1.77451			−0.887255	+	0.461279i	$0.847391\pi$
$234$		0			0
$235$	−1603.44			−0.445092
$236$		0			0
$237$	−481.995			−0.132105
$238$		0			0
$239$			2083.06i			0.563774i	0.959448	+	0.281887i	$0.0909603\pi$
$239$			2083.06i			0.563774i	−0.959448	+	0.281887i	$0.909040\pi$
$240$		0			0
$241$		−	2487.56i		−	0.664887i	−0.943123	−	0.332443i	$-0.892127\pi$
$241$		−	2487.56i		−	0.664887i	0.943123	−	0.332443i	$-0.107873\pi$
$242$		0			0
$243$		−	243.000i		−	0.0641500i
$244$		0			0
$245$	1054.76	+	1056.17i	0.275045	+	0.275414i
$246$		0			0
$247$		−	1558.03i		−	0.401358i
$248$		0			0
$249$	3002.91			0.764265
$250$		0			0
$251$			7731.44i			1.94424i	0.234483	+	0.972120i	$0.424660\pi$
$251$			7731.44i			1.94424i	−0.234483	+	0.972120i	$0.575340\pi$
$252$		0			0
$253$		−	32.2385i		−	0.00801113i
$254$		0			0
$255$	1001.24			0.245882
$256$		0			0
$257$			515.804i			0.125194i	0.998039	+	0.0625972i	$0.0199383\pi$
$257$			515.804i			0.125194i	−0.998039	+	0.0625972i	$0.980062\pi$
$258$		0			0
$259$	6668.25	+	2764.70i	1.59979	+	0.663283i
$260$		0			0
$261$			584.800i			0.138690i
$262$		0			0
$263$			2690.82i			0.630886i	0.948945	+	0.315443i	$0.102153\pi$
$263$			2690.82i			0.630886i	−0.948945	+	0.315443i	$0.897847\pi$
$264$		0			0
$265$		−	2558.90i		−	0.593178i
$266$		0			0
$267$	−3416.40			−0.783071
$268$		0			0
$269$	−292.293			−0.0662507			−0.0331254	−	0.999451i	$-0.510546\pi$
$269$	−292.293			−0.0662507			−0.0331254	+	0.999451i	$0.510546\pi$
$270$		0			0
$271$	−2946.93			−0.660565			−0.330282	−	0.943882i	$-0.607144\pi$
$271$	−2946.93			−0.660565			−0.330282	+	0.943882i	$0.607144\pi$
$272$		0			0
$273$	1361.33	+	564.417i	0.301800	+	0.125128i
$274$		0			0
$275$	4574.93			1.00319
$276$		0			0
$277$		−	2058.88i		−	0.446593i	−0.974751	−	0.223296i	$-0.928318\pi$
$277$		−	2058.88i		−	0.446593i	0.974751	−	0.223296i	$-0.0716818\pi$
$278$		0			0
$279$	1489.10			0.319535
$280$		0			0
$281$	−5425.14			−1.15173			−0.575866	−	0.817544i	$-0.695335\pi$
$281$	−5425.14			−1.15173			−0.575866	+	0.817544i	$0.695335\pi$
$282$		0			0
$283$			3570.12i			0.749901i	0.927045	+	0.374950i	$0.122340\pi$
$283$			3570.12i			0.749901i	−0.927045	+	0.374950i	$0.877660\pi$
$284$		0			0
$285$	766.871			0.159388
$286$		0			0
$287$	−4972.65	−	2061.69i	−1.02274	−	0.424035i
$288$		0			0
$289$	−968.671			−0.197165
$290$		0			0
$291$	−2189.28			−0.441024
$292$		0			0
$293$	1942.50			0.387311			0.193655	−	0.981070i	$-0.437966\pi$
$293$	1942.50			0.387311			0.193655	+	0.981070i	$0.437966\pi$
$294$		0			0
$295$		−	2052.22i		−	0.405033i
$296$		0			0
$297$		−	1164.63i		−	0.227537i
$298$		0			0
$299$			19.8240i			0.00383429i
$300$		0			0
$301$	−229.865	+	554.416i	−0.0440172	+	0.106166i
$302$		0			0
$303$		−	940.609i		−	0.178338i
$304$		0			0
$305$	−1949.79			−0.366048
$306$		0			0
$307$		−	766.366i		−	0.142472i	−0.997460	−	0.0712358i	$-0.977306\pi$
$307$		−	766.366i		−	0.142472i	0.997460	−	0.0712358i	$-0.0226943\pi$
$308$		0			0
$309$			425.107i			0.0782638i
$310$		0			0
$311$	6733.22			1.22767			0.613836	−	0.789434i	$-0.289626\pi$
$311$	6733.22			1.22767			0.613836	+	0.789434i	$0.289626\pi$
$312$		0			0
$313$			7627.66i			1.37745i	0.725024	+	0.688723i	$0.241829\pi$
$313$			7627.66i			1.37745i	−0.725024	+	0.688723i	$0.758171\pi$
$314$		0			0
$315$	−277.809	+	670.053i	−0.0496912	+	0.119851i
$316$		0			0
$317$			8243.80i			1.46062i	0.683114	+	0.730312i	$0.260625\pi$
$317$			8243.80i			1.46062i	−0.683114	+	0.730312i	$0.739375\pi$
$318$		0			0
$319$			2802.77i			0.491929i
$320$		0			0
$321$		−	2881.95i		−	0.501105i
$322$		0			0
$323$	−4504.92			−0.776038
$324$		0			0
$325$	−2813.20			−0.480149
$326$		0			0
$327$	−3133.18			−0.529863
$328$		0			0
$329$	−2613.52	+	6303.61i	−0.437957	+	1.05632i
$330$		0			0
$331$	−1133.57			−0.188237			−0.0941187	−	0.995561i	$-0.530003\pi$
$331$	−1133.57			−0.188237			−0.0941187	+	0.995561i	$0.530003\pi$
$332$		0			0
$333$			3507.94i			0.577279i
$334$		0			0
$335$	−3079.44			−0.502233
$336$		0			0
$337$	7101.55			1.14791			0.573956	−	0.818886i	$-0.305408\pi$
$337$	7101.55			1.14791			0.573956	+	0.818886i	$0.305408\pi$
$338$		0			0
$339$		−	137.841i		−	0.0220840i
$340$		0			0
$341$	7136.83			1.13338
$342$		0			0
$343$	5871.34	−	2425.07i	0.924264	−	0.381753i
$344$		0			0
$345$	−9.75746			−0.00152268
$346$		0			0
$347$	6829.95			1.05663			0.528315	−	0.849048i	$-0.322824\pi$
$347$	6829.95			1.05663			0.528315	+	0.849048i	$0.322824\pi$
$348$		0			0
$349$	−5310.28			−0.814478			−0.407239	−	0.913322i	$-0.633508\pi$
$349$	−5310.28			−0.814478			−0.407239	+	0.913322i	$0.633508\pi$
$350$		0			0
$351$			716.151i			0.108904i
$352$		0			0
$353$		−	147.157i		−	0.0221880i	−0.999938	−	0.0110940i	$-0.996469\pi$
$353$		−	147.157i		−	0.0221880i	0.999938	−	0.0110940i	$-0.00353140\pi$
$354$		0			0
$355$		−	4341.55i		−	0.649086i
$356$		0			0
$357$	1631.96	−	3936.17i	0.241940	−	0.583541i
$358$		0			0
$359$			9618.54i			1.41406i	0.707184	+	0.707030i	$0.249965\pi$
$359$			9618.54i			1.41406i	−0.707184	+	0.707030i	$0.750035\pi$
$360$		0			0
$361$	3408.58			0.496949
$362$		0			0
$363$		−	1588.72i		−	0.229714i
$364$		0			0
$365$		−	5331.12i		−	0.764503i
$366$		0			0
$367$	7096.51			1.00936			0.504680	−	0.863307i	$-0.331611\pi$
$367$	7096.51			1.00936			0.504680	+	0.863307i	$0.331611\pi$
$368$		0			0
$369$		−	2615.94i		−	0.369053i
$370$		0			0
$371$	−10059.8	−	4170.88i	−1.40777	−	0.583669i
$372$		0			0
$373$		−	12908.5i		−	1.79190i	−0.444160	−	0.895948i	$-0.646498\pi$
$373$		−	12908.5i		−	1.79190i	0.444160	−	0.895948i	$-0.353502\pi$
$374$		0			0
$375$		−	3016.58i		−	0.415401i
$376$		0			0
$377$		−	1723.48i		−	0.235447i
$378$		0			0
$379$	9161.34			1.24165			0.620826	−	0.783948i	$-0.286797\pi$
$379$	9161.34			1.24165			0.620826	+	0.783948i	$0.286797\pi$
$380$		0			0
$381$	1022.78			0.137529
$382$		0			0
$383$	9816.05			1.30960			0.654800	−	0.755802i	$-0.272753\pi$
$383$	9816.05			1.30960			0.654800	+	0.755802i	$0.272753\pi$
$384$		0			0
$385$	−1331.45	+	3211.37i	−0.176253	+	0.425108i
$386$		0			0
$387$	−291.660			−0.0383098
$388$		0			0
$389$		−	4106.75i		−	0.535271i	−0.963520	−	0.267636i	$-0.913758\pi$
$389$		−	4106.75i		−	0.535271i	0.963520	−	0.267636i	$-0.0862423\pi$
$390$		0			0
$391$	57.3194			0.00741372
$392$		0			0
$393$	2198.03			0.282127
$394$		0			0
$395$			699.175i			0.0890616i
$396$		0			0
$397$	6582.72			0.832184			0.416092	−	0.909323i	$-0.363399\pi$
$397$	6582.72			0.832184			0.416092	+	0.909323i	$0.363399\pi$
$398$		0			0
$399$	1249.96	−	3014.81i	0.156833	−	0.378269i
$400$		0			0
$401$	−9585.89			−1.19376			−0.596878	−	0.802332i	$-0.703592\pi$
$401$	−9585.89			−1.19376			−0.596878	+	0.802332i	$0.703592\pi$
$402$		0			0
$403$	−4388.57			−0.542457
$404$		0			0
$405$	−352.492			−0.0432481
$406$		0			0
$407$			16812.5i			2.04758i
$408$		0			0
$409$		−	7390.53i		−	0.893492i	−0.894661	−	0.446746i	$-0.852583\pi$
$409$		−	7390.53i		−	0.893492i	0.894661	−	0.446746i	$-0.147417\pi$
$410$		0			0
$411$		−	878.463i		−	0.105429i
$412$		0			0
$413$	−8067.89	−	3345.00i	−0.961247	−	0.398540i
$414$		0			0
$415$		−	4355.99i		−	0.515246i
$416$		0			0
$417$	4855.97			0.570259
$418$		0			0
$419$		−	5141.45i		−	0.599466i	−0.954023	−	0.299733i	$-0.903102\pi$
$419$		−	5141.45i		−	0.599466i	0.954023	−	0.299733i	$-0.0968976\pi$
$420$		0			0
$421$			10171.4i			1.17749i	0.808319	+	0.588744i	$0.200377\pi$
$421$			10171.4i			1.17749i	−0.808319	+	0.588744i	$0.799623\pi$
$422$		0			0
$423$	−3316.12			−0.381170
$424$		0			0
$425$			8134.13i			0.928384i
$426$		0			0
$427$	−3178.06	+	7665.23i	−0.360180	+	0.868727i
$428$		0			0
$429$			3432.30i			0.386277i
$430$		0			0
$431$			16075.1i			1.79655i	0.439438	+	0.898273i	$0.355178\pi$
$431$			16075.1i			1.79655i	−0.439438	+	0.898273i	$0.644822\pi$
$432$		0			0
$433$		−	120.502i		−	0.0133740i	−0.999978	−	0.00668702i	$-0.997871\pi$
$433$		−	120.502i		−	0.0133740i	0.999978	−	0.00668702i	$-0.00212856\pi$
$434$		0			0
$435$	848.302			0.0935011
$436$		0			0
$437$	43.9023			0.00480579
$438$		0			0
$439$	−15873.0			−1.72569			−0.862844	−	0.505470i	$-0.831319\pi$
$439$	−15873.0			−1.72569			−0.862844	+	0.505470i	$0.831319\pi$
$440$		0			0
$441$	2181.37	+	2184.30i	0.235544	+	0.235860i
$442$		0			0
$443$	−9195.34			−0.986194			−0.493097	−	0.869974i	$-0.664135\pi$
$443$	−9195.34			−0.986194			−0.493097	+	0.869974i	$0.664135\pi$
$444$		0			0
$445$			4955.77i			0.527924i
$446$		0			0
$447$	−4215.35			−0.446038
$448$		0			0
$449$	−5130.99			−0.539302			−0.269651	−	0.962958i	$-0.586908\pi$
$449$	−5130.99			−0.539302			−0.269651	+	0.962958i	$0.586908\pi$
$450$		0			0
$451$		−	12537.4i		−	1.30901i
$452$		0			0
$453$	−1391.89			−0.144363
$454$		0			0
$455$	818.736	−	1974.73i	0.0843581	−	0.203465i
$456$		0			0
$457$	−4512.80			−0.461925			−0.230963	−	0.972963i	$-0.574188\pi$
$457$	−4512.80			−0.461925			−0.230963	+	0.972963i	$0.574188\pi$
$458$		0			0
$459$	2070.69			0.210569
$460$		0			0
$461$	1593.06			0.160946			0.0804732	−	0.996757i	$-0.474357\pi$
$461$	1593.06			0.160946			0.0804732	+	0.996757i	$0.474357\pi$
$462$		0			0
$463$			2589.59i			0.259931i	0.991518	+	0.129966i	$0.0414867\pi$
$463$			2589.59i			0.259931i	−0.991518	+	0.129966i	$0.958513\pi$
$464$		0			0
$465$		−	2160.07i		−	0.215421i
$466$		0			0
$467$		−	18926.4i		−	1.87539i	−0.347459	−	0.937695i	$-0.612956\pi$
$467$		−	18926.4i		−	1.87539i	0.347459	−	0.937695i	$-0.387044\pi$
$468$		0			0
$469$	−5019.33	+	12106.2i	−0.494182	+	1.19193i
$470$		0			0
$471$		−	11283.3i		−	1.10384i
$472$		0			0
$473$	−1397.84			−0.135883
$474$		0			0
$475$			6230.13i			0.601806i
$476$		0			0
$477$		−	5292.15i		−	0.507989i
$478$		0			0
$479$	6302.34			0.601172			0.300586	−	0.953755i	$-0.402818\pi$
$479$	6302.34			0.601172			0.300586	+	0.953755i	$0.402818\pi$
$480$		0			0
$481$		−	10338.3i		−	0.980015i
$482$		0			0
$483$	−15.9042	+	38.3596i	−0.00149827	+	0.00361371i
$484$		0			0
$485$			3175.74i			0.297326i
$486$		0			0
$487$		−	4607.23i		−	0.428693i	−0.976758	−	0.214347i	$-0.931238\pi$
$487$		−	4607.23i		−	0.428693i	0.976758	−	0.214347i	$-0.0687622\pi$
$488$		0			0
$489$			5486.31i			0.507361i
$490$		0			0
$491$	3378.29			0.310510			0.155255	−	0.987874i	$-0.450380\pi$
$491$	3378.29			0.310510			0.155255	+	0.987874i	$0.450380\pi$
$492$		0			0
$493$	−4983.28			−0.455245
$494$		0			0
$495$	−1689.39			−0.153399
$496$		0			0
$497$	−17068.0	−	7076.50i	−1.54045	−	0.638681i
$498$		0			0
$499$	−15145.2			−1.35870			−0.679350	−	0.733815i	$-0.737738\pi$
$499$	−15145.2			−1.35870			−0.679350	+	0.733815i	$0.737738\pi$
$500$		0			0
$501$		−	1130.55i		−	0.100817i
$502$		0			0
$503$	19755.1			1.75117			0.875583	−	0.483068i	$-0.160477\pi$
$503$	19755.1			1.75117			0.875583	+	0.483068i	$0.160477\pi$
$504$		0			0
$505$	−1364.43			−0.120231
$506$		0			0
$507$			4480.42i			0.392470i
$508$		0			0
$509$	−13777.1			−1.19973			−0.599863	−	0.800103i	$-0.704778\pi$
$509$	−13777.1			−1.19973			−0.599863	+	0.800103i	$0.704778\pi$
$510$		0			0
$511$	−20958.3	−	8689.44i	−1.81436	−	0.752247i
$512$		0			0
$513$	1585.99			0.136497
$514$		0			0
$515$	616.655			0.0527632
$516$		0			0
$517$	−15893.2			−1.35199
$518$		0			0
$519$		−	10166.7i		−	0.859866i
$520$		0			0
$521$		−	3629.45i		−	0.305200i	−0.988288	−	0.152600i	$-0.951235\pi$
$521$		−	3629.45i		−	0.305200i	0.988288	−	0.152600i	$-0.0487646\pi$
$522$		0			0
$523$		−	10629.9i		−	0.888740i	−0.895843	−	0.444370i	$-0.853427\pi$
$523$		−	10629.9i		−	0.888740i	0.895843	−	0.444370i	$-0.146573\pi$
$524$		0			0
$525$	−5443.57	−	2256.94i	−0.452527	−	0.187621i
$526$		0			0
$527$			12689.1i			1.04886i
$528$		0			0
$529$	12166.4			0.999954
$530$		0			0
$531$		−	4244.25i		−	0.346864i
$532$		0			0
$533$			7709.50i			0.626521i
$534$		0			0
$535$	−4180.52			−0.337831
$536$		0			0
$537$			978.554i			0.0786363i
$538$		0			0
$539$	10454.7	+	10468.7i	0.835464	+	0.836586i
$540$		0			0
$541$		−	2515.95i		−	0.199943i	−0.994990	−	0.0999714i	$-0.968125\pi$
$541$		−	2515.95i		−	0.199943i	0.994990	−	0.0999714i	$-0.0318751\pi$
$542$		0			0
$543$		−	4953.18i		−	0.391457i
$544$		0			0
$545$			4544.95i			0.357219i
$546$		0			0
$547$	−11030.6			−0.862222			−0.431111	−	0.902299i	$-0.641878\pi$
$547$	−11030.6			−0.862222			−0.431111	+	0.902299i	$0.641878\pi$
$548$		0			0
$549$	−4032.42			−0.313478
$550$		0			0
$551$	−3816.81			−0.295103
$552$		0			0
$553$	2748.68	+	1139.62i	0.211366	+	0.0876339i
$554$		0			0
$555$	5088.57			0.389185
$556$		0			0
$557$			18189.0i			1.38365i	0.722066	+	0.691824i	$0.243193\pi$
$557$			18189.0i			1.38365i	−0.722066	+	0.691824i	$0.756807\pi$
$558$		0			0
$559$	859.557			0.0650364
$560$		0			0
$561$	9924.19			0.746880
$562$		0			0
$563$		−	23074.7i		−	1.72732i	−0.504074	−	0.863660i	$-0.668166\pi$
$563$		−	23074.7i		−	1.72732i	0.504074	−	0.863660i	$-0.331834\pi$
$564$		0			0
$565$	−199.950			−0.0148884
$566$		0			0
$567$	−574.544	+	1385.76i	−0.0425548	+	0.102639i
$568$		0			0
$569$	5699.50			0.419922			0.209961	−	0.977710i	$-0.432666\pi$
$569$	5699.50			0.419922			0.209961	+	0.977710i	$0.432666\pi$
$570$		0			0
$571$	7695.91			0.564035			0.282018	−	0.959409i	$-0.408996\pi$
$571$	7695.91			0.564035			0.282018	+	0.959409i	$0.408996\pi$
$572$		0			0
$573$	−5287.24			−0.385476
$574$		0			0
$575$		−	79.2705i		−	0.00574923i
$576$		0			0
$577$			12406.6i			0.895136i	0.894250	+	0.447568i	$0.147710\pi$
$577$			12406.6i			0.895136i	−0.894250	+	0.447568i	$0.852290\pi$
$578$		0			0
$579$		−	13059.7i		−	0.937377i
$580$		0			0
$581$	−17124.7	−	7100.03i	−1.22281	−	0.506986i
$582$		0			0
$583$		−	25363.7i		−	1.80181i
$584$		0			0
$585$	1038.84			0.0734199
$586$		0			0
$587$			4773.97i			0.335678i	0.985814	+	0.167839i	$0.0536788\pi$
$587$			4773.97i			0.335678i	−0.985814	+	0.167839i	$0.946321\pi$
$588$		0			0
$589$			9718.93i			0.679901i
$590$		0			0
$591$	−4409.17			−0.306885
$592$		0			0
$593$			14978.5i			1.03726i	0.854999	+	0.518630i	$0.173558\pi$
$593$			14978.5i			1.03726i	−0.854999	+	0.518630i	$0.826442\pi$
$594$		0			0
$595$	−5709.75	−	2367.30i	−0.393407	−	0.163109i
$596$		0			0
$597$			8438.47i			0.578499i
$598$		0			0
$599$			13952.5i			0.951724i	0.879520	+	0.475862i	$0.157864\pi$
$599$			13952.5i			0.951724i	−0.879520	+	0.475862i	$0.842136\pi$
$600$		0			0
$601$			3345.06i			0.227035i	0.993536	+	0.113517i	$0.0362117\pi$
$601$			3345.06i			0.227035i	−0.993536	+	0.113517i	$0.963788\pi$
$602$		0			0
$603$	−6368.69			−0.430105
$604$		0			0
$605$	−2304.57			−0.154866
$606$		0			0
$607$	25460.4			1.70248			0.851239	−	0.524778i	$-0.175852\pi$
$607$	25460.4			1.70248			0.851239	+	0.524778i	$0.175852\pi$
$608$		0			0
$609$	1382.69	−	3334.94i	0.0920022	−	0.221902i
$610$		0			0
$611$	9773.00			0.647092
$612$		0			0
$613$			8938.60i			0.588950i	0.955659	+	0.294475i	$0.0951448\pi$
$613$			8938.60i			0.588950i	−0.955659	+	0.294475i	$0.904855\pi$
$614$		0			0
$615$	−3794.65			−0.248805
$616$		0			0
$617$	−5770.64			−0.376527			−0.188264	−	0.982119i	$-0.560286\pi$
$617$	−5770.64			−0.376527			−0.188264	+	0.982119i	$0.560286\pi$
$618$		0			0
$619$		−	6297.57i		−	0.408918i	−0.978875	−	0.204459i	$-0.934456\pi$
$619$		−	6297.57i		−	0.408918i	0.978875	−	0.204459i	$-0.0655436\pi$
$620$		0			0
$621$	−20.1797			−0.00130400
$622$		0			0
$623$	19482.7	+	8077.65i	1.25290	+	0.519461i
$624$		0			0
$625$	8881.97			0.568446
$626$		0			0
$627$	7601.18			0.484150
$628$		0			0
$629$	−29892.4			−1.89489
$630$		0			0
$631$		−	9060.26i		−	0.571606i	−0.958288	−	0.285803i	$-0.907740\pi$
$631$		−	9060.26i		−	0.571606i	0.958288	−	0.285803i	$-0.0922603\pi$
$632$		0			0
$633$		−	7302.63i		−	0.458536i
$634$		0			0
$635$		−	1483.62i		−	0.0927178i
$636$		0			0
$637$	−6428.77	−	6437.41i	−0.399870	−	0.400407i
$638$		0			0
$639$		−	8978.89i		−	0.555868i
$640$		0			0
$641$	−9635.66			−0.593737			−0.296869	−	0.954918i	$-0.595942\pi$
$641$	−9635.66			−0.593737			−0.296869	+	0.954918i	$0.595942\pi$
$642$		0			0
$643$		−	22309.6i		−	1.36828i	−0.729351	−	0.684140i	$-0.760178\pi$
$643$		−	22309.6i		−	1.36828i	0.729351	−	0.684140i	$-0.239822\pi$
$644$		0			0
$645$			423.077i			0.0258274i
$646$		0			0
$647$	−8248.87			−0.501231			−0.250616	−	0.968087i	$-0.580633\pi$
$647$	−8248.87			−0.501231			−0.250616	+	0.968087i	$0.580633\pi$
$648$		0			0
$649$		−	20341.4i		−	1.23031i
$650$		0			0
$651$	−8491.91	−	3520.80i	−0.511251	−	0.211968i
$652$		0			0
$653$		−	13530.5i		−	0.810859i	−0.914126	−	0.405429i	$-0.867122\pi$
$653$		−	13530.5i		−	0.810859i	0.914126	−	0.405429i	$-0.132878\pi$
$654$		0			0
$655$		−	3188.44i		−	0.190202i
$656$		0			0
$657$		−	11025.4i		−	0.654709i
$658$		0			0
$659$	−15900.8			−0.939917			−0.469959	−	0.882689i	$-0.655731\pi$
$659$	−15900.8			−0.939917			−0.469959	+	0.882689i	$0.655731\pi$
$660$		0			0
$661$	−15518.3			−0.913152			−0.456576	−	0.889684i	$-0.650924\pi$
$661$	−15518.3			−0.913152			−0.456576	+	0.889684i	$0.650924\pi$
$662$		0			0
$663$	−6102.56			−0.357472
$664$		0			0
$665$	−4373.24	−	1813.17i	−0.255018	−	0.105732i
$666$		0			0
$667$	48.5641			0.00281921
$668$		0			0
$669$		−	2549.73i		−	0.147352i
$670$		0			0
$671$	−19326.2			−1.11189
$672$		0			0
$673$	8192.08			0.469215			0.234607	−	0.972090i	$-0.424620\pi$
$673$	8192.08			0.469215			0.234607	+	0.972090i	$0.424620\pi$
$674$		0			0
$675$		−	2863.68i		−	0.163293i
$676$		0			0
$677$	17289.6			0.981524			0.490762	−	0.871294i	$-0.336718\pi$
$677$	17289.6			0.981524			0.490762	+	0.871294i	$0.336718\pi$
$678$		0			0
$679$	12484.8	+	5176.29i	0.705631	+	0.292560i
$680$		0			0
$681$	6861.76			0.386113
$682$		0			0
$683$	−31297.3			−1.75338			−0.876689	−	0.481058i	$-0.840253\pi$
$683$	−31297.3			−1.75338			−0.876689	+	0.481058i	$0.840253\pi$
$684$		0			0
$685$	−1274.29			−0.0710774
$686$		0			0
$687$		−	6789.63i		−	0.377060i
$688$		0			0
$689$			15596.6i			0.862385i
$690$		0			0
$691$		−	32710.9i		−	1.80084i	−0.435021	−	0.900420i	$-0.643259\pi$
$691$		−	32710.9i		−	1.80084i	0.435021	−	0.900420i	$-0.356741\pi$
$692$		0			0
$693$	−2753.62	+	6641.53i	−0.150940	+	0.364056i
$694$		0			0
$695$		−	7044.00i		−	0.384452i
$696$		0			0
$697$	22291.3			1.21140
$698$		0			0
$699$			18933.6i			1.02451i
$700$		0			0
$701$			4111.40i			0.221520i	0.993847	+	0.110760i	$0.0353285\pi$
$701$			4111.40i			0.221520i	−0.993847	+	0.110760i	$0.964672\pi$
$702$		0			0
$703$	−22895.3			−1.22832
$704$		0			0
$705$			4810.31i			0.256974i
$706$		0			0
$707$	−2223.96	+	5364.01i	−0.118303	+	0.285339i
$708$		0			0
$709$			23791.2i			1.26022i	0.776506	+	0.630110i	$0.216990\pi$
$709$			23791.2i			1.26022i	−0.776506	+	0.630110i	$0.783010\pi$
$710$		0			0
$711$			1445.99i			0.0762710i
$712$		0			0
$713$		−	123.661i		−	0.00649529i
$714$		0			0
$715$	4978.85			0.260417
$716$		0			0
$717$	6249.18			0.325495
$718$		0			0
$719$	706.487			0.0366447			0.0183223	−	0.999832i	$-0.494167\pi$
$719$	706.487			0.0366447			0.0183223	+	0.999832i	$0.494167\pi$
$720$		0			0
$721$	1005.11	−	2424.26i	0.0519174	−	0.125221i
$722$		0			0
$723$	−7462.68			−0.383873
$724$		0			0
$725$			6891.68i			0.353035i
$726$		0			0
$727$	−24189.9			−1.23405			−0.617024	−	0.786944i	$-0.711662\pi$
$727$	−24189.9			−1.23405			−0.617024	+	0.786944i	$0.711662\pi$
$728$		0			0
$729$	−729.000			−0.0370370
$730$		0			0
$731$		−	2485.33i		−	0.125750i
$732$		0			0
$733$	23762.4			1.19739			0.598693	−	0.800979i	$-0.295687\pi$
$733$	23762.4			1.19739			0.598693	+	0.800979i	$0.295687\pi$
$734$		0			0
$735$	3168.52	−	3164.27i	0.159010	−	0.158797i
$736$		0			0
$737$	−30523.3			−1.52556
$738$		0			0
$739$	−22183.4			−1.10424			−0.552119	−	0.833766i	$-0.686180\pi$
$739$	−22183.4			−1.10424			−0.552119	+	0.833766i	$0.686180\pi$
$740$		0			0
$741$	−4674.10			−0.231724
$742$		0			0
$743$			22419.7i			1.10700i	0.832851	+	0.553498i	$0.186707\pi$
$743$			22419.7i			1.10700i	−0.832851	+	0.553498i	$0.813293\pi$
$744$		0			0
$745$			6114.72i			0.300706i
$746$		0			0
$747$		−	9008.74i		−	0.441249i
$748$		0			0
$749$	−6814.02	+	16434.9i	−0.332415	+	0.801760i
$750$		0			0
$751$		−	4193.10i		−	0.203739i	−0.994798	−	0.101870i	$-0.967518\pi$
$751$		−	4193.10i		−	0.203739i	0.994798	−	0.101870i	$-0.0324825\pi$
$752$		0			0
$753$	23194.3			1.12251
$754$		0			0
$755$			2019.05i			0.0973256i
$756$		0			0
$757$		−	37722.4i		−	1.81115i	−0.424181	−	0.905577i	$-0.639438\pi$
$757$		−	37722.4i		−	1.81115i	0.424181	−	0.905577i	$-0.360562\pi$
$758$		0			0
$759$	−96.7154			−0.00462523
$760$		0			0
$761$		−	26381.1i		−	1.25666i	−0.777948	−	0.628328i	$-0.783739\pi$
$761$		−	26381.1i		−	1.25666i	0.777948	−	0.628328i	$-0.216261\pi$
$762$		0			0
$763$	17867.6	+	7408.03i	0.847773	+	0.351492i
$764$		0			0
$765$		−	3003.71i		−	0.141960i
$766$		0			0
$767$			12508.3i			0.588851i
$768$		0			0
$769$		−	11955.6i		−	0.560637i	−0.959907	−	0.280318i	$-0.909560\pi$
$769$		−	11955.6i		−	0.560637i	0.959907	−	0.280318i	$-0.0904400\pi$
$770$		0			0
$771$	1547.41			0.0722810
$772$		0			0
$773$	11231.8			0.522614			0.261307	−	0.965256i	$-0.415847\pi$
$773$	11231.8			0.522614			0.261307	+	0.965256i	$0.415847\pi$
$774$		0			0
$775$	17548.6			0.813374
$776$		0			0
$777$	8294.10	−	20004.8i	0.382946	−	0.923638i
$778$		0			0
$779$	17073.5			0.785264
$780$		0			0
$781$		−	43033.2i		−	1.97164i
$782$		0			0
$783$	1754.40			0.0800730
$784$		0			0
$785$	−16367.4			−0.744177
$786$		0			0
$787$		−	28526.7i		−	1.29208i	−0.763303	−	0.646041i	$-0.776424\pi$
$787$		−	28526.7i		−	1.29208i	0.763303	−	0.646041i	$-0.223576\pi$
$788$		0			0
$789$	8072.45			0.364242
$790$		0			0
$791$	−325.907	+	786.064i	−0.0146497	+	0.0353340i
$792$		0			0
$793$	11884.0			0.532174
$794$		0			0
$795$	−7676.71			−0.342472
$796$		0			0
$797$	−20195.5			−0.897568			−0.448784	−	0.893640i	$-0.648143\pi$
$797$	−20195.5			−0.897568			−0.448784	+	0.893640i	$0.648143\pi$
$798$		0			0
$799$		−	28257.7i		−	1.25117i
$800$		0			0
$801$			10249.2i			0.452106i
$802$		0			0
$803$		−	52841.7i		−	2.32222i
$804$		0			0
$805$	55.6439	+	23.0703i	0.00243626	+	0.00101009i
$806$		0			0
$807$			876.880i			0.0382499i
$808$		0			0
$809$	−20974.4			−0.911521			−0.455760	−	0.890102i	$-0.650633\pi$
$809$	−20974.4			−0.911521			−0.455760	+	0.890102i	$0.650633\pi$
$810$		0			0
$811$			33677.8i			1.45818i	0.684416	+	0.729092i	$0.260057\pi$
$811$			33677.8i			1.45818i	−0.684416	+	0.729092i	$0.739943\pi$
$812$		0			0
$813$			8840.78i			0.381377i
$814$		0			0
$815$	7958.37			0.342048
$816$		0			0
$817$		−	1903.58i		−	0.0815149i
$818$		0			0
$819$	1693.25	−	4083.99i	0.0722430	−	0.174245i
$820$		0			0
$821$			12454.1i			0.529418i	0.964328	+	0.264709i	$0.0852760\pi$
$821$			12454.1i			0.529418i	−0.964328	+	0.264709i	$0.914724\pi$
$822$		0			0
$823$			27909.6i			1.18210i	0.806635	+	0.591050i	$0.201286\pi$
$823$			27909.6i			1.18210i	−0.806635	+	0.591050i	$0.798714\pi$
$824$		0			0
$825$		−	13724.8i		−	0.579194i
$826$		0			0
$827$	7350.79			0.309083			0.154542	−	0.987986i	$-0.450610\pi$
$827$	7350.79			0.309083			0.154542	+	0.987986i	$0.450610\pi$
$828$		0			0
$829$	−13101.8			−0.548905			−0.274453	−	0.961601i	$-0.588497\pi$
$829$	−13101.8			−0.548905			−0.274453	+	0.961601i	$0.588497\pi$
$830$		0			0
$831$	−6176.64			−0.257840
$832$		0			0
$833$	−18613.2	+	18588.2i	−0.774200	+	0.773162i
$834$		0			0
$835$	−1639.96			−0.0679679
$836$		0			0
$837$		−	4467.31i		−	0.184484i
$838$		0			0
$839$	21819.6			0.897850			0.448925	−	0.893569i	$-0.351807\pi$
$839$	21819.6			0.897850			0.448925	+	0.893569i	$0.351807\pi$
$840$		0			0
$841$	20166.9			0.826885
$842$		0			0
$843$			16275.4i			0.664953i
$844$		0			0
$845$	6499.23			0.264592
$846$		0			0
$847$	−3756.33	+	9059.99i	−0.152384	+	0.367538i
$848$		0			0
$849$	10710.4			0.432955
$850$		0			0
$851$	291.314			0.0117346
$852$		0			0
$853$	−34494.6			−1.38461			−0.692306	−	0.721604i	$-0.743405\pi$
$853$	−34494.6			−1.38461			−0.692306	+	0.721604i	$0.743405\pi$
$854$		0			0
$855$		−	2300.61i		−	0.0920226i
$856$		0			0
$857$			41632.1i			1.65942i	0.558191	+	0.829712i	$0.311495\pi$
$857$			41632.1i			1.65942i	−0.558191	+	0.829712i	$0.688505\pi$
$858$		0			0
$859$		−	5117.73i		−	0.203277i	−0.994821	−	0.101638i	$-0.967592\pi$
$859$		−	5117.73i		−	0.203277i	0.994821	−	0.101638i	$-0.0324084\pi$
$860$		0			0
$861$	−6185.08	+	14917.9i	−0.244816	+	0.590479i
$862$		0			0
$863$			16578.1i			0.653913i	0.945039	+	0.326956i	$0.106023\pi$
$863$			16578.1i			0.653913i	−0.945039	+	0.326956i	$0.893977\pi$
$864$		0			0
$865$	−14747.7			−0.579697
$866$		0			0
$867$			2906.01i			0.113833i
$868$		0			0
$869$			6930.18i			0.270530i
$870$		0			0
$871$	18769.3			0.730165
$872$		0			0
$873$			6567.85i			0.254625i
$874$		0			0
$875$	−7132.34	+	17202.7i	−0.275562	+	0.664636i
$876$		0			0
$877$			11830.6i			0.455521i	0.973717	+	0.227760i	$0.0731403\pi$
$877$			11830.6i			0.455521i	−0.973717	+	0.227760i	$0.926860\pi$
$878$		0			0
$879$		−	5827.50i		−	0.223614i
$880$		0			0
$881$			37236.4i			1.42398i	0.702190	+	0.711989i	$0.252206\pi$
$881$			37236.4i			1.42398i	−0.702190	+	0.711989i	$0.747794\pi$
$882$		0			0
$883$	7687.64			0.292989			0.146495	−	0.989211i	$-0.453201\pi$
$883$	7687.64			0.292989			0.146495	+	0.989211i	$0.453201\pi$
$884$		0			0
$885$	−6156.65			−0.233846
$886$		0			0
$887$	−830.437			−0.0314356			−0.0157178	−	0.999876i	$-0.505003\pi$
$887$	−830.437			−0.0314356			−0.0157178	+	0.999876i	$0.505003\pi$
$888$		0			0
$889$	−5832.58	−	2418.23i	−0.220043	−	0.0912315i
$890$		0			0
$891$	−3493.88			−0.131369
$892$		0			0
$893$		−	21643.3i		−	0.811047i
$894$		0			0
$895$	1419.48			0.0530144
$896$		0			0
$897$	59.4720			0.00221373
$898$		0			0
$899$			10750.9i			0.398848i
$900$		0			0
$901$	45096.2			1.66745
$902$		0			0
$903$	1663.25	+	689.594i	0.0612951	+	0.0254133i
$904$		0			0
$905$	−7185.01			−0.263909
$906$		0			0
$907$	30689.8			1.12353			0.561763	−	0.827298i	$-0.310123\pi$
$907$	30689.8			1.12353			0.561763	+	0.827298i	$0.310123\pi$
$908$		0			0
$909$	−2821.83			−0.102964
$910$		0			0
$911$			8924.46i			0.324567i	0.986744	+	0.162284i	$0.0518859\pi$
$911$			8924.46i			0.324567i	−0.986744	+	0.162284i	$0.948114\pi$
$912$		0			0
$913$		−	43176.3i		−	1.56509i
$914$		0			0
$915$			5849.37i			0.211338i
$916$		0			0
$917$	−12534.7	−	5196.98i	−0.451399	−	0.187153i
$918$		0			0
$919$			18566.2i			0.666422i	0.942852	+	0.333211i	$0.108132\pi$
$919$			18566.2i			0.666422i	−0.942852	+	0.333211i	$0.891868\pi$
$920$		0			0
$921$	−2299.10			−0.0822560
$922$		0			0
$923$			26461.9i			0.943666i
$924$		0			0
$925$			41340.0i			1.46946i
$926$		0			0
$927$	1275.32			0.0451856
$928$		0			0
$929$			4047.86i			0.142956i	0.997442	+	0.0714779i	$0.0227716\pi$
$929$			4047.86i			0.142956i	−0.997442	+	0.0714779i	$0.977228\pi$
$930$		0			0
$931$	−14256.3	+	14237.2i	−0.501859	+	0.501186i
$932$		0			0
$933$		−	20199.7i		−	0.708797i
$934$		0			0
$935$		−	14395.9i		−	0.503525i
$936$		0			0
$937$			28211.1i			0.983581i	0.870713	+	0.491791i	$0.163657\pi$
$937$			28211.1i			0.983581i	−0.870713	+	0.491791i	$0.836343\pi$
$938$		0			0
$939$	22883.0			0.795269
$940$		0			0
$941$	43313.0			1.50049			0.750247	−	0.661158i	$-0.229935\pi$
$941$	43313.0			1.50049			0.750247	+	0.661158i	$0.229935\pi$
$942$		0			0
$943$	−217.238			−0.00750186
$944$		0			0
$945$	2010.16	+	833.426i	0.0691963	+	0.0286892i
$946$		0			0
$947$	−36121.7			−1.23949			−0.619745	−	0.784804i	$-0.712764\pi$
$947$	−36121.7			−1.23949			−0.619745	+	0.784804i	$0.712764\pi$
$948$		0			0
$949$			32493.3i			1.11146i
$950$		0			0
$951$	24731.4			0.843292
$952$		0			0
$953$	36180.3			1.22979			0.614897	−	0.788607i	$-0.289198\pi$
$953$	36180.3			1.22979			0.614897	+	0.788607i	$0.289198\pi$
$954$		0			0
$955$			7669.60i			0.259877i
$956$		0			0
$957$	8408.32			0.284015
$958$		0			0
$959$	−2077.02	+	5009.62i	−0.0699379	+	0.168685i
$960$		0			0
$961$	−2415.37			−0.0810770
$962$		0			0
$963$	−8645.85			−0.289313
$964$		0			0
$965$	−18944.2			−0.631953
$966$		0			0
$967$		−	3564.98i		−	0.118554i	−0.998242	−	0.0592772i	$-0.981120\pi$
$967$		−	3564.98i		−	0.118554i	0.998242	−	0.0592772i	$-0.0188796\pi$
$968$		0			0
$969$			13514.7i			0.448046i
$970$		0			0
$971$			11726.3i			0.387555i	0.981046	+	0.193777i	$0.0620739\pi$
$971$			11726.3i			0.387555i	−0.981046	+	0.193777i	$0.937926\pi$
$972$		0			0
$973$	−27692.2	−	11481.4i	−0.912405	−	0.378289i
$974$		0			0
$975$			8439.61i			0.277214i
$976$		0			0
$977$	−43817.1			−1.43483			−0.717416	−	0.696645i	$-0.754676\pi$
$977$	−43817.1			−1.43483			−0.717416	+	0.696645i	$0.754676\pi$
$978$		0			0
$979$			49121.3i			1.60360i
$980$		0			0
$981$			9399.54i			0.305917i
$982$		0			0
$983$	−36423.2			−1.18181			−0.590906	−	0.806741i	$-0.701230\pi$
$983$	−36423.2			−1.18181			−0.590906	+	0.806741i	$0.701230\pi$
$984$		0			0
$985$			6395.89i			0.206893i
$986$		0			0
$987$	18910.8	+	7840.55i	0.609866	+	0.252855i
$988$		0			0
$989$			24.2206i			0.000778736i
$990$		0			0
$991$		−	17147.5i		−	0.549656i	−0.961493	−	0.274828i	$-0.911379\pi$
$991$		−	17147.5i		−	0.549656i	0.961493	−	0.274828i	$-0.0886209\pi$
$992$		0			0
$993$			3400.71i			0.108679i
$994$		0			0
$995$	12240.7			0.390007
$996$		0			0
$997$	37323.2			1.18560			0.592798	−	0.805351i	$-0.298023\pi$
$997$	37323.2			1.18560			0.592798	+	0.805351i	$0.298023\pi$
$998$		0			0
$999$	10523.8			0.333292

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1344.4.p.a.223.3	✓	16
4.3	odd	2	inner	1344.4.p.a.223.12	yes	16
7.6	odd	2		1344.4.p.b.223.14	yes	16
8.3	odd	2		1344.4.p.b.223.6	yes	16
8.5	even	2		1344.4.p.b.223.13	yes	16
28.27	even	2		1344.4.p.b.223.5	yes	16
56.13	odd	2	inner	1344.4.p.a.223.4	yes	16
56.27	even	2	inner	1344.4.p.a.223.11	yes	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1344.4.p.a.223.3	✓	16	1.1	even	1	trivial
1344.4.p.a.223.4	yes	16	56.13	odd	2	inner
1344.4.p.a.223.11	yes	16	56.27	even	2	inner
1344.4.p.a.223.12	yes	16	4.3	odd	2	inner
1344.4.p.b.223.5	yes	16	28.27	even	2
1344.4.p.b.223.6	yes	16	8.3	odd	2
1344.4.p.b.223.13	yes	16	8.5	even	2
1344.4.p.b.223.14	yes	16	7.6	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.p (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-3.00000i q^{3} -4.35176 q^{5} +(-7.09314 + 17.1081i) q^{7} -9.00000 q^{9} +O(q^{10})\)	\(q-3.00000i q^{3} -4.35176 q^{5} +(-7.09314 + 17.1081i) q^{7} -9.00000 q^{9} -43.1344 q^{11} +26.5241 q^{13} +13.0553i q^{15} -76.6921i q^{17} -58.7403i q^{19} +(51.3243 + 21.2794i) q^{21} +0.747396i q^{23} -106.062 q^{25} +27.0000i q^{27} -64.9778i q^{29} -165.456 q^{31} +129.403i q^{33} +(30.8676 - 74.4503i) q^{35} -389.771i q^{37} -79.5723i q^{39} +290.660i q^{41} +32.4066 q^{43} +39.1658 q^{45} +368.457 q^{47} +(-242.375 - 242.700i) q^{49} -230.076 q^{51} +588.016i q^{53} +187.710 q^{55} -176.221 q^{57} +471.583i q^{59} +448.047 q^{61} +(63.8382 - 153.973i) q^{63} -115.426 q^{65} +707.632 q^{67} +2.24219 q^{69} +997.655i q^{71} +1225.05i q^{73} +318.187i q^{75} +(305.958 - 737.947i) q^{77} -160.665i q^{79} +81.0000 q^{81} +1000.97i q^{83} +333.745i q^{85} -194.933 q^{87} -1138.80i q^{89} +(-188.139 + 453.777i) q^{91} +496.368i q^{93} +255.624i q^{95} -729.761i q^{97} +388.209 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 144 q^{9}+O(q^{10}) \) 16 * q - 144 * q^9	\( 16 q - 144 q^{9} - 56 q^{13} - 36 q^{21} + 80 q^{25} + 392 q^{49} + 336 q^{57} - 184 q^{61} - 1536 q^{65} - 864 q^{69} + 240 q^{77} + 1296 q^{81}+O(q^{100}) \) 16 * q - 144 * q^9 - 56 * q^13 - 36 * q^21 + 80 * q^25 + 392 * q^49 + 336 * q^57 - 184 * q^61 - 1536 * q^65 - 864 * q^69 + 240 * q^77 + 1296 * q^81

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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