LMFDB - Embedded newform 1344.4.p.b.223.12

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			223.12
Root			$-12.1800 + 12.1800i$ of defining polynomial
Character	$\chi$	$=$	1344.223
Dual form			1344.4.p.b.223.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.00000i q^{3} -10.4276 q^{5} +(9.40604 - 15.9539i) q^{7} -9.00000 q^{9} +O(q^{10})$	$q+3.00000i q^{3} -10.4276 q^{5} +(9.40604 - 15.9539i) q^{7} -9.00000 q^{9} +51.5119 q^{11} -46.4237 q^{13} -31.2827i q^{15} +99.1647i q^{17} +12.5160i q^{19} +(47.8617 + 28.2181i) q^{21} -129.614i q^{23} -16.2658 q^{25} -27.0000i q^{27} +207.130i q^{29} -139.565 q^{31} +154.536i q^{33} +(-98.0821 + 166.360i) q^{35} -152.824i q^{37} -139.271i q^{39} -317.226i q^{41} +366.573 q^{43} +93.8481 q^{45} +103.059 q^{47} +(-166.053 - 300.126i) q^{49} -297.494 q^{51} -24.2027i q^{53} -537.144 q^{55} -37.5479 q^{57} +339.691i q^{59} -25.3894 q^{61} +(-84.6543 + 143.585i) q^{63} +484.087 q^{65} +660.173 q^{67} +388.843 q^{69} -0.771730i q^{71} +342.435i q^{73} -48.7975i q^{75} +(484.523 - 821.816i) q^{77} -951.005i q^{79} +81.0000 q^{81} -244.673i q^{83} -1034.05i q^{85} -621.391 q^{87} -1172.22i q^{89} +(-436.663 + 740.639i) q^{91} -418.696i q^{93} -130.511i q^{95} +1407.75i q^{97} -463.607 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	−10.4276			−0.932670			−0.466335	−	0.884608i	$-0.654426\pi$
$5$	−10.4276			−0.932670			−0.466335	+	0.884608i	$0.654426\pi$
$6$		0			0
$7$	9.40604	−	15.9539i	0.507878	−	0.861429i
$7$	9.40604	−	15.9539i	0.507878	−	0.861429i
$8$		0			0
$9$	−9.00000			−0.333333
$10$		0			0
$11$	51.5119			1.41195			0.705974	−	0.708237i	$-0.250509\pi$
$11$	51.5119			1.41195			0.705974	+	0.708237i	$0.250509\pi$
$12$		0			0
$13$	−46.4237			−0.990433			−0.495216	−	0.868770i	$-0.664911\pi$
$13$	−46.4237			−0.990433			−0.495216	+	0.868770i	$0.664911\pi$
$14$		0			0
$15$		−	31.2827i		−	0.538477i
$16$		0			0
$17$			99.1647i			1.41476i	0.706832	+	0.707382i	$0.250124\pi$
$17$			99.1647i			1.41476i	−0.706832	+	0.707382i	$0.749876\pi$
$18$		0			0
$19$			12.5160i			0.151124i	0.997141	+	0.0755621i	$0.0240751\pi$
$19$			12.5160i			0.151124i	−0.997141	+	0.0755621i	$0.975925\pi$
$20$		0			0
$21$	47.8617	+	28.2181i	0.497346	+	0.293224i
$22$		0			0
$23$		−	129.614i		−	1.17506i	−0.809202	−	0.587531i	$-0.800100\pi$
$23$		−	129.614i		−	1.17506i	0.809202	−	0.587531i	$-0.199900\pi$
$24$		0			0
$25$	−16.2658			−0.130127
$26$		0			0
$27$		−	27.0000i		−	0.192450i
$28$		0			0
$29$			207.130i			1.32632i	0.748480	+	0.663158i	$0.230784\pi$
$29$			207.130i			1.32632i	−0.748480	+	0.663158i	$0.769216\pi$
$30$		0			0
$31$	−139.565			−0.808603			−0.404301	−	0.914626i	$-0.632485\pi$
$31$	−139.565			−0.808603			−0.404301	+	0.914626i	$0.632485\pi$
$32$		0			0
$33$			154.536i			0.815189i
$34$		0			0
$35$	−98.0821	+	166.360i	−0.473683	+	0.803429i
$36$		0			0
$37$		−	152.824i		−	0.679032i	−0.940600	−	0.339516i	$-0.889737\pi$
$37$		−	152.824i		−	0.679032i	0.940600	−	0.339516i	$-0.110263\pi$
$38$		0			0
$39$		−	139.271i		−	0.571827i
$40$		0			0
$41$		−	317.226i		−	1.20835i	−0.796851	−	0.604176i	$-0.793502\pi$
$41$		−	317.226i		−	1.20835i	0.796851	−	0.604176i	$-0.206498\pi$
$42$		0			0
$43$	366.573			1.30004			0.650022	−	0.759916i	$-0.274760\pi$
$43$	366.573			1.30004			0.650022	+	0.759916i	$0.274760\pi$
$44$		0			0
$45$	93.8481			0.310890
$46$		0			0
$47$	103.059			0.319845			0.159922	−	0.987130i	$-0.448876\pi$
$47$	103.059			0.319845			0.159922	+	0.987130i	$0.448876\pi$
$48$		0			0
$49$	−166.053	−	300.126i	−0.484119	−	0.875002i
$50$		0			0
$51$	−297.494			−0.816814
$52$		0			0
$53$		−	24.2027i		−	0.0627263i	−0.999508	−	0.0313631i	$-0.990015\pi$
$53$		−	24.2027i		−	0.0627263i	0.999508	−	0.0313631i	$-0.00998484\pi$
$54$		0			0
$55$	−537.144			−1.31688
$56$		0			0
$57$	−37.5479			−0.0872515
$58$		0			0
$59$			339.691i			0.749559i	0.927114	+	0.374780i	$0.122282\pi$
$59$			339.691i			0.749559i	−0.927114	+	0.374780i	$0.877718\pi$
$60$		0			0
$61$	−25.3894			−0.0532915			−0.0266458	−	0.999645i	$-0.508483\pi$
$61$	−25.3894			−0.0532915			−0.0266458	+	0.999645i	$0.508483\pi$
$62$		0			0
$63$	−84.6543	+	143.585i	−0.169293	+	0.287143i
$64$		0			0
$65$	484.087			0.923747
$66$		0			0
$67$	660.173			1.20378			0.601888	−	0.798581i	$-0.294416\pi$
$67$	660.173			1.20378			0.601888	+	0.798581i	$0.294416\pi$
$68$		0			0
$69$	388.843			0.678422
$70$		0			0
$71$		−	0.771730i		−	0.00128996i	−1.00000		0.000644982i	$-0.999795\pi$
$71$		−	0.771730i		−	0.00128996i	1.00000		0.000644982i	$-0.000205304\pi$
$72$		0			0
$73$			342.435i			0.549028i	0.961583	+	0.274514i	$0.0885169\pi$
$73$			342.435i			0.549028i	−0.961583	+	0.274514i	$0.911483\pi$
$74$		0			0
$75$		−	48.7975i		−	0.0751286i
$76$		0			0
$77$	484.523	−	821.816i	0.717098	−	1.21629i
$78$		0			0
$79$		−	951.005i		−	1.35438i	−0.735806	−	0.677192i	$-0.763197\pi$
$79$		−	951.005i		−	1.35438i	0.735806	−	0.677192i	$-0.236803\pi$
$80$		0			0
$81$	81.0000			0.111111
$82$		0			0
$83$		−	244.673i		−	0.323571i	−0.986826	−	0.161785i	$-0.948275\pi$
$83$		−	244.673i		−	0.323571i	0.986826	−	0.161785i	$-0.0517253\pi$
$84$		0			0
$85$		−	1034.05i		−	1.31951i
$86$		0			0
$87$	−621.391			−0.765749
$88$		0			0
$89$		−	1172.22i		−	1.39613i	−0.716037	−	0.698063i	$-0.754046\pi$
$89$		−	1172.22i		−	1.39613i	0.716037	−	0.698063i	$-0.245954\pi$
$90$		0			0
$91$	−436.663	+	740.639i	−0.503019	+	0.853187i
$92$		0			0
$93$		−	418.696i		−	0.466847i
$94$		0			0
$95$		−	130.511i		−	0.140949i
$96$		0			0
$97$			1407.75i			1.47356i	0.676134	+	0.736779i	$0.263654\pi$
$97$			1407.75i			1.47356i	−0.676134	+	0.736779i	$0.736346\pi$
$98$		0			0
$99$	−463.607			−0.470650
$100$		0			0
$101$	1166.30			1.14902			0.574509	−	0.818498i	$-0.305193\pi$
$101$	1166.30			1.14902			0.574509	+	0.818498i	$0.305193\pi$
$102$		0			0
$103$	1305.15			1.24855			0.624275	−	0.781205i	$-0.285394\pi$
$103$	1305.15			1.24855			0.624275	+	0.781205i	$0.285394\pi$
$104$		0			0
$105$	−499.081	−	294.246i	−0.463860	−	0.273481i
$106$		0			0
$107$	17.4629			0.0157776			0.00788882	−	0.999969i	$-0.497489\pi$
$107$	17.4629			0.0157776			0.00788882	+	0.999969i	$0.497489\pi$
$108$		0			0
$109$			1274.60i			1.12004i	0.828480	+	0.560019i	$0.189206\pi$
$109$			1274.60i			1.12004i	−0.828480	+	0.560019i	$0.810794\pi$
$110$		0			0
$111$	458.473			0.392039
$112$		0			0
$113$	1104.04			0.919113			0.459557	−	0.888148i	$-0.348008\pi$
$113$	1104.04			0.919113			0.459557	+	0.888148i	$0.348008\pi$
$114$		0			0
$115$			1351.56i			1.09595i
$116$		0			0
$117$	417.814			0.330144
$118$		0			0
$119$	1582.06	+	932.747i	1.21872	+	0.718527i
$120$		0			0
$121$	1322.48			0.993599
$122$		0			0
$123$	951.679			0.697642
$124$		0			0
$125$	1473.06			1.05404
$126$		0			0
$127$			64.0894i			0.0447797i	0.999749	+	0.0223898i	$0.00712750\pi$
$127$			64.0894i			0.0447797i	−0.999749	+	0.0223898i	$0.992872\pi$
$128$		0			0
$129$			1099.72i			0.750580i
$130$		0			0
$131$			334.724i			0.223244i	0.993751	+	0.111622i	$0.0356046\pi$
$131$			334.724i			0.223244i	−0.993751	+	0.111622i	$0.964395\pi$
$132$		0			0
$133$	199.678	+	117.726i	0.130183	+	0.0767526i
$134$		0			0
$135$			281.544i			0.179492i
$136$		0			0
$137$	1403.88			0.875486			0.437743	−	0.899100i	$-0.355778\pi$
$137$	1403.88			0.875486			0.437743	+	0.899100i	$0.355778\pi$
$138$		0			0
$139$			2483.10i			1.51521i	0.652714	+	0.757604i	$0.273630\pi$
$139$			2483.10i			1.51521i	−0.652714	+	0.757604i	$0.726370\pi$
$140$		0			0
$141$			309.177i			0.184662i
$142$		0			0
$143$	−2391.38			−1.39844
$144$		0			0
$145$		−	2159.87i		−	1.23701i
$146$		0			0
$147$	900.377	−	498.159i	0.505183	−	0.279507i
$148$		0			0
$149$		−	2626.69i		−	1.44421i	−0.691785	−	0.722103i	$-0.743175\pi$
$149$		−	2626.69i		−	1.44421i	0.691785	−	0.722103i	$-0.256825\pi$
$150$		0			0
$151$			3396.96i			1.83073i	0.402622	+	0.915366i	$0.368099\pi$
$151$			3396.96i			1.83073i	−0.402622	+	0.915366i	$0.631901\pi$
$152$		0			0
$153$		−	892.482i		−	0.471588i
$154$		0			0
$155$	1455.33			0.754160
$156$		0			0
$157$	−984.767			−0.500592			−0.250296	−	0.968169i	$-0.580528\pi$
$157$	−984.767			−0.500592			−0.250296	+	0.968169i	$0.580528\pi$
$158$		0			0
$159$	72.6080			0.0362150
$160$		0			0
$161$	−2067.85	−	1219.16i	−1.01223	−	0.596788i
$162$		0			0
$163$	3313.69			1.59232			0.796159	−	0.605087i	$-0.206862\pi$
$163$	3313.69			1.59232			0.796159	+	0.605087i	$0.206862\pi$
$164$		0			0
$165$		−	1611.43i		−	0.760302i
$166$		0			0
$167$	1502.47			0.696196			0.348098	−	0.937458i	$-0.386828\pi$
$167$	1502.47			0.696196			0.348098	+	0.937458i	$0.386828\pi$
$168$		0			0
$169$	−41.8371			−0.0190428
$170$		0			0
$171$		−	112.644i		−	0.0503747i
$172$		0			0
$173$	1526.50			0.670854			0.335427	−	0.942066i	$-0.391119\pi$
$173$	1526.50			0.670854			0.335427	+	0.942066i	$0.391119\pi$
$174$		0			0
$175$	−152.997	+	259.503i	−0.0660885	+	0.112095i
$176$		0			0
$177$	−1019.07			−0.432758
$178$		0			0
$179$	75.7519			0.0316311			0.0158155	−	0.999875i	$-0.494966\pi$
$179$	75.7519			0.0316311			0.0158155	+	0.999875i	$0.494966\pi$
$180$		0			0
$181$	4246.13			1.74372			0.871858	−	0.489759i	$-0.162915\pi$
$181$	4246.13			1.74372			0.871858	+	0.489759i	$0.162915\pi$
$182$		0			0
$183$		−	76.1683i		−	0.0307679i
$184$		0			0
$185$			1593.59i			0.633312i
$186$		0			0
$187$			5108.17i			1.99757i
$188$		0			0
$189$	−430.755	−	253.963i	−0.165782	−	0.0977412i
$190$		0			0
$191$		−	2284.87i		−	0.865588i	−0.901493	−	0.432794i	$-0.857528\pi$
$191$		−	2284.87i		−	0.865588i	0.901493	−	0.432794i	$-0.142472\pi$
$192$		0			0
$193$	−4674.77			−1.74351			−0.871755	−	0.489941i	$-0.837018\pi$
$193$	−4674.77			−1.74351			−0.871755	+	0.489941i	$0.837018\pi$
$194$		0			0
$195$			1452.26i			0.533326i
$196$		0			0
$197$			377.065i			0.136369i	0.997673	+	0.0681846i	$0.0217207\pi$
$197$			377.065i			0.136369i	−0.997673	+	0.0681846i	$0.978279\pi$
$198$		0			0
$199$	443.329			0.157923			0.0789616	−	0.996878i	$-0.474840\pi$
$199$	443.329			0.157923			0.0789616	+	0.996878i	$0.474840\pi$
$200$		0			0
$201$			1980.52i			0.695000i
$202$		0			0
$203$	3304.53	+	1948.28i	1.14253	+	0.673607i
$204$		0			0
$205$			3307.90i			1.12699i
$206$		0			0
$207$			1166.53i			0.391687i
$208$		0			0
$209$			644.721i			0.213379i
$210$		0			0
$211$	679.767			0.221787			0.110894	−	0.993832i	$-0.464629\pi$
$211$	679.767			0.221787			0.110894	+	0.993832i	$0.464629\pi$
$212$		0			0
$213$	2.31519			0.000744761
$214$		0			0
$215$	−3822.46			−1.21251
$216$		0			0
$217$	−1312.76	+	2226.61i	−0.410672	+	0.696554i
$218$		0			0
$219$	−1027.31			−0.316981
$220$		0			0
$221$		−	4603.60i		−	1.40123i
$222$		0			0
$223$	−3129.02			−0.939618			−0.469809	−	0.882768i	$-0.655677\pi$
$223$	−3129.02			−0.939618			−0.469809	+	0.882768i	$0.655677\pi$
$224$		0			0
$225$	146.392			0.0433755
$226$		0			0
$227$			5124.51i			1.49835i	0.662372	+	0.749175i	$0.269550\pi$
$227$			5124.51i			1.49835i	−0.662372	+	0.749175i	$0.730450\pi$
$228$		0			0
$229$	4051.18			1.16904			0.584519	−	0.811380i	$-0.301283\pi$
$229$	4051.18			1.16904			0.584519	+	0.811380i	$0.301283\pi$
$230$		0			0
$231$	2465.45	+	1453.57i	0.702227	+	0.414017i
$232$		0			0
$233$	−2032.72			−0.571537			−0.285769	−	0.958299i	$-0.592249\pi$
$233$	−2032.72			−0.571537			−0.285769	+	0.958299i	$0.592249\pi$
$234$		0			0
$235$	−1074.65			−0.298309
$236$		0			0
$237$	2853.01			0.781954
$238$		0			0
$239$		−	6609.52i		−	1.78885i	−0.447220	−	0.894424i	$-0.647586\pi$
$239$		−	6609.52i		−	1.78885i	0.447220	−	0.894424i	$-0.352414\pi$
$240$		0			0
$241$			876.509i			0.234278i	0.993116	+	0.117139i	$0.0373722\pi$
$241$			876.509i			0.234278i	−0.993116	+	0.117139i	$0.962628\pi$
$242$		0			0
$243$			243.000i			0.0641500i
$244$		0			0
$245$	1731.53	+	3129.58i	0.451524	+	0.816088i
$246$		0			0
$247$		−	581.038i		−	0.149678i
$248$		0			0
$249$	734.020			0.186814
$250$		0			0
$251$		−	668.178i		−	0.168028i	−0.996465	−	0.0840140i	$-0.973226\pi$
$251$		−	668.178i		−	0.168028i	0.996465	−	0.0840140i	$-0.0267740\pi$
$252$		0			0
$253$		−	6676.68i		−	1.65913i
$254$		0			0
$255$	3102.14			0.761818
$256$		0			0
$257$			4278.82i			1.03854i	0.854609	+	0.519272i	$0.173797\pi$
$257$			4278.82i			1.03854i	−0.854609	+	0.519272i	$0.826203\pi$
$258$		0			0
$259$	−2438.14	−	1437.47i	−0.584937	−	0.344865i
$260$		0			0
$261$		−	1864.17i		−	0.442105i
$262$		0			0
$263$			1411.24i			0.330878i	0.986220	+	0.165439i	$0.0529042\pi$
$263$			1411.24i			0.330878i	−0.986220	+	0.165439i	$0.947096\pi$
$264$		0			0
$265$			252.375i			0.0585029i
$266$		0			0
$267$	3516.66			0.806053
$268$		0			0
$269$	6033.09			1.36745			0.683724	−	0.729740i	$-0.260359\pi$
$269$	6033.09			1.36745			0.683724	+	0.729740i	$0.260359\pi$
$270$		0			0
$271$	8666.20			1.94256			0.971281	−	0.237936i	$-0.0764707\pi$
$271$	8666.20			1.94256			0.971281	+	0.237936i	$0.0764707\pi$
$272$		0			0
$273$	−2221.92	−	1309.99i	−0.492588	−	0.290418i
$274$		0			0
$275$	−837.884			−0.183732
$276$		0			0
$277$		−	6127.37i		−	1.32909i	−0.747248	−	0.664545i	$-0.768625\pi$
$277$		−	6127.37i		−	1.32909i	0.747248	−	0.664545i	$-0.231375\pi$
$278$		0			0
$279$	1256.09			0.269534
$280$		0			0
$281$	8672.23			1.84107			0.920537	−	0.390654i	$-0.127751\pi$
$281$	8672.23			1.84107			0.920537	+	0.390654i	$0.127751\pi$
$282$		0			0
$283$		−	2306.61i		−	0.484501i	−0.970214	−	0.242250i	$-0.922114\pi$
$283$		−	2306.61i		−	0.484501i	0.970214	−	0.242250i	$-0.0778855\pi$
$284$		0			0
$285$	391.533			0.0813769
$286$		0			0
$287$	−5060.99	−	2983.84i	−1.04091	−	0.613695i
$288$		0			0
$289$	−4920.64			−1.00155
$290$		0			0
$291$	−4223.24			−0.850759
$292$		0			0
$293$	−3052.63			−0.608656			−0.304328	−	0.952567i	$-0.598432\pi$
$293$	−3052.63			−0.608656			−0.304328	+	0.952567i	$0.598432\pi$
$294$		0			0
$295$		−	3542.15i		−	0.699091i
$296$		0			0
$297$		−	1390.82i		−	0.271730i
$298$		0			0
$299$			6017.18i			1.16382i
$300$		0			0
$301$	3448.00	−	5848.26i	0.660264	−	1.11989i
$302$		0			0
$303$			3498.89i			0.663386i
$304$		0			0
$305$	264.750			0.0497034
$306$		0			0
$307$		−	4218.34i		−	0.784212i	−0.919920	−	0.392106i	$-0.871747\pi$
$307$		−	4218.34i		−	0.784212i	0.919920	−	0.392106i	$-0.128253\pi$
$308$		0			0
$309$			3915.46i			0.720851i
$310$		0			0
$311$	814.934			0.148587			0.0742936	−	0.997236i	$-0.476330\pi$
$311$	814.934			0.148587			0.0742936	+	0.997236i	$0.476330\pi$
$312$		0			0
$313$		−	2003.63i		−	0.361826i	−0.983499	−	0.180913i	$-0.942095\pi$
$313$		−	2003.63i		−	0.361826i	0.983499	−	0.180913i	$-0.0579053\pi$
$314$		0			0
$315$	882.739	−	1497.24i	0.157894	−	0.267810i
$316$		0			0
$317$			2158.78i			0.382489i	0.981542	+	0.191245i	$0.0612524\pi$
$317$			2158.78i			0.382489i	−0.981542	+	0.191245i	$0.938748\pi$
$318$		0			0
$319$			10669.7i			1.87269i
$320$		0			0
$321$			52.3888i			0.00910922i
$322$		0			0
$323$	−1241.14			−0.213805
$324$		0			0
$325$	755.120			0.128882
$326$		0			0
$327$	−3823.79			−0.646654
$328$		0			0
$329$	969.376	−	1644.19i	0.162442	−	0.275523i
$330$		0			0
$331$	10865.4			1.80428			0.902140	−	0.431443i	$-0.141995\pi$
$331$	10865.4			1.80428			0.902140	+	0.431443i	$0.141995\pi$
$332$		0			0
$333$			1375.42i			0.226344i
$334$		0			0
$335$	−6884.00			−1.12273
$336$		0			0
$337$	8496.49			1.37339			0.686696	−	0.726944i	$-0.259060\pi$
$337$	8496.49			1.37339			0.686696	+	0.726944i	$0.259060\pi$
$338$		0			0
$339$			3312.13i			0.530650i
$340$		0			0
$341$	−7189.29			−1.14171
$342$		0			0
$343$	−6350.07	−	173.802i	−0.999626	−	0.0273599i
$344$		0			0
$345$	−4054.68			−0.632744
$346$		0			0
$347$	4377.26			0.677186			0.338593	−	0.940933i	$-0.390049\pi$
$347$	4377.26			0.677186			0.338593	+	0.940933i	$0.390049\pi$
$348$		0			0
$349$	68.6024			0.0105221			0.00526103	−	0.999986i	$-0.498325\pi$
$349$	68.6024			0.0105221			0.00526103	+	0.999986i	$0.498325\pi$
$350$		0			0
$351$			1253.44i			0.190609i
$352$		0			0
$353$		−	11573.3i		−	1.74501i	−0.488610	−	0.872503i	$-0.662496\pi$
$353$		−	11573.3i		−	1.74501i	0.488610	−	0.872503i	$-0.337504\pi$
$354$		0			0
$355$			8.04727i			0.00120311i
$356$		0			0
$357$	−2798.24	+	4746.19i	−0.414842	+	0.703627i
$358$		0			0
$359$		−	7032.26i		−	1.03384i	−0.856034	−	0.516920i	$-0.827079\pi$
$359$		−	7032.26i		−	1.03384i	0.856034	−	0.516920i	$-0.172921\pi$
$360$		0			0
$361$	6702.35			0.977162
$362$		0			0
$363$			3967.44i			0.573655i
$364$		0			0
$365$		−	3570.77i		−	0.512062i
$366$		0			0
$367$	6607.00			0.939734			0.469867	−	0.882737i	$-0.344302\pi$
$367$	6607.00			0.939734			0.469867	+	0.882737i	$0.344302\pi$
$368$		0			0
$369$			2855.04i			0.402784i
$370$		0			0
$371$	−386.127	−	227.651i	−0.0540342	−	0.0318573i
$372$		0			0
$373$			2866.56i			0.397921i	0.980007	+	0.198961i	$0.0637566\pi$
$373$			2866.56i			0.397921i	−0.980007	+	0.198961i	$0.936243\pi$
$374$		0			0
$375$			4419.18i			0.608548i
$376$		0			0
$377$		−	9615.77i		−	1.31363i
$378$		0			0
$379$	−11899.6			−1.61278			−0.806389	−	0.591386i	$-0.798581\pi$
$379$	−11899.6			−1.61278			−0.806389	+	0.591386i	$0.798581\pi$
$380$		0			0
$381$	−192.268			−0.0258536
$382$		0			0
$383$	−5013.77			−0.668908			−0.334454	−	0.942412i	$-0.608552\pi$
$383$	−5013.77			−0.668908			−0.334454	+	0.942412i	$0.608552\pi$
$384$		0			0
$385$	−5052.40	+	8569.54i	−0.668816	+	1.13440i
$386$		0			0
$387$	−3299.16			−0.433348
$388$		0			0
$389$			8609.83i			1.12220i	0.827748	+	0.561100i	$0.189622\pi$
$389$			8609.83i			1.12220i	−0.827748	+	0.561100i	$0.810378\pi$
$390$		0			0
$391$	12853.2			1.66243
$392$		0			0
$393$	−1004.17			−0.128890
$394$		0			0
$395$			9916.67i			1.26319i
$396$		0			0
$397$	−589.928			−0.0745784			−0.0372892	−	0.999305i	$-0.511872\pi$
$397$	−589.928			−0.0745784			−0.0372892	+	0.999305i	$0.511872\pi$
$398$		0			0
$399$	−353.177	+	599.035i	−0.0443132	+	0.0751610i
$400$		0			0
$401$	10725.7			1.33570			0.667850	−	0.744296i	$-0.267215\pi$
$401$	10725.7			1.33570			0.667850	+	0.744296i	$0.267215\pi$
$402$		0			0
$403$	6479.15			0.800867
$404$		0			0
$405$	−844.633			−0.103630
$406$		0			0
$407$		−	7872.28i		−	0.958758i
$408$		0			0
$409$		−	8629.15i		−	1.04324i	−0.853179	−	0.521619i	$-0.825328\pi$
$409$		−	8629.15i		−	1.04324i	0.853179	−	0.521619i	$-0.174672\pi$
$410$		0			0
$411$			4211.64i			0.505462i
$412$		0			0
$413$	5419.39	+	3195.15i	0.645692	+	0.380685i
$414$		0			0
$415$			2551.35i			0.301785i
$416$		0			0
$417$	−7449.31			−0.874806
$418$		0			0
$419$		−	14290.8i		−	1.66623i	−0.553100	−	0.833115i	$-0.686555\pi$
$419$		−	14290.8i		−	1.66623i	0.553100	−	0.833115i	$-0.313445\pi$
$420$		0			0
$421$			6067.23i			0.702372i	0.936306	+	0.351186i	$0.114222\pi$
$421$			6067.23i			0.702372i	−0.936306	+	0.351186i	$0.885778\pi$
$422$		0			0
$423$	−927.530			−0.106615
$424$		0			0
$425$		−	1613.00i		−	0.184098i
$426$		0			0
$427$	−238.814	+	405.060i	−0.0270656	+	0.0459069i
$428$		0			0
$429$		−	7174.13i		−	0.807390i
$430$		0			0
$431$		−	142.686i		−	0.0159465i	−0.999968	−	0.00797326i	$-0.997462\pi$
$431$		−	142.686i		−	0.0159465i	0.999968	−	0.00797326i	$-0.00253799\pi$
$432$		0			0
$433$			3447.06i			0.382575i	0.981534	+	0.191287i	$0.0612663\pi$
$433$			3447.06i			0.382575i	−0.981534	+	0.191287i	$0.938734\pi$
$434$		0			0
$435$	6479.60			0.714191
$436$		0			0
$437$	1622.25			0.177580
$438$		0			0
$439$	−17481.2			−1.90053			−0.950265	−	0.311442i	$-0.899188\pi$
$439$	−17481.2			−1.90053			−0.950265	+	0.311442i	$0.899188\pi$
$440$		0			0
$441$	1494.48	+	2701.13i	0.161373	+	0.291667i
$442$		0			0
$443$	−8678.02			−0.930712			−0.465356	−	0.885124i	$-0.654074\pi$
$443$	−8678.02			−0.930712			−0.465356	+	0.885124i	$0.654074\pi$
$444$		0			0
$445$			12223.4i			1.30212i
$446$		0			0
$447$	7880.07			0.833813
$448$		0			0
$449$	17412.4			1.83016			0.915082	−	0.403267i	$-0.132126\pi$
$449$	17412.4			1.83016			0.915082	+	0.403267i	$0.132126\pi$
$450$		0			0
$451$		−	16340.9i		−	1.70613i
$452$		0			0
$453$	−10190.9			−1.05697
$454$		0			0
$455$	4553.34	−	7723.06i	0.469151	−	0.795742i
$456$		0			0
$457$	−9994.06			−1.02298			−0.511491	−	0.859289i	$-0.670907\pi$
$457$	−9994.06			−1.02298			−0.511491	+	0.859289i	$0.670907\pi$
$458$		0			0
$459$	2677.45			0.272271
$460$		0			0
$461$	−5785.14			−0.584471			−0.292235	−	0.956346i	$-0.594399\pi$
$461$	−5785.14			−0.584471			−0.292235	+	0.956346i	$0.594399\pi$
$462$		0			0
$463$			2300.81i			0.230945i	0.993311	+	0.115473i	$0.0368383\pi$
$463$			2300.81i			0.230945i	−0.993311	+	0.115473i	$0.963162\pi$
$464$		0			0
$465$			4365.98i			0.435414i
$466$		0			0
$467$		−	7724.99i		−	0.765461i	−0.923860	−	0.382730i	$-0.874984\pi$
$467$		−	7724.99i		−	0.765461i	0.923860	−	0.382730i	$-0.125016\pi$
$468$		0			0
$469$	6209.61	−	10532.3i	0.611371	−	1.03697i
$470$		0			0
$471$		−	2954.30i		−	0.289017i
$472$		0			0
$473$	18882.9			1.83559
$474$		0			0
$475$		−	203.582i		−	0.0196653i
$476$		0			0
$477$			217.824i			0.0209088i
$478$		0			0
$479$	5805.40			0.553770			0.276885	−	0.960903i	$-0.410698\pi$
$479$	5805.40			0.553770			0.276885	+	0.960903i	$0.410698\pi$
$480$		0			0
$481$			7094.68i			0.672535i
$482$		0			0
$483$	3657.47	−	6203.55i	0.344556	−	0.584413i
$484$		0			0
$485$		−	14679.4i		−	1.37434i
$486$		0			0
$487$		−	248.163i		−	0.0230910i	−0.999933	−	0.0115455i	$-0.996325\pi$
$487$		−	248.163i		−	0.0230910i	0.999933	−	0.0115455i	$-0.00367513\pi$
$488$		0			0
$489$			9941.06i			0.919325i
$490$		0			0
$491$	16760.0			1.54046			0.770232	−	0.637764i	$-0.220140\pi$
$491$	16760.0			1.54046			0.770232	+	0.637764i	$0.220140\pi$
$492$		0			0
$493$	−20540.0			−1.87642
$494$		0			0
$495$	4834.30			0.438961
$496$		0			0
$497$	−12.3121	−	7.25892i	−0.00111121	−	0.000655145i
$498$		0			0
$499$	−16544.8			−1.48426			−0.742131	−	0.670255i	$-0.766185\pi$
$499$	−16544.8			−1.48426			−0.742131	+	0.670255i	$0.766185\pi$
$500$		0			0
$501$			4507.41i			0.401949i
$502$		0			0
$503$	−1889.10			−0.167457			−0.0837286	−	0.996489i	$-0.526683\pi$
$503$	−1889.10			−0.167457			−0.0837286	+	0.996489i	$0.526683\pi$
$504$		0			0
$505$	−12161.6			−1.07166
$506$		0			0
$507$		−	125.511i		−	0.0109944i
$508$		0			0
$509$	−18116.8			−1.57763			−0.788816	−	0.614629i	$-0.789306\pi$
$509$	−18116.8			−1.57763			−0.788816	+	0.614629i	$0.789306\pi$
$510$		0			0
$511$	5463.18	+	3220.96i	0.472948	+	0.278839i
$512$		0			0
$513$	337.931			0.0290838
$514$		0			0
$515$	−13609.6			−1.16449
$516$		0			0
$517$	5308.77			0.451604
$518$		0			0
$519$			4579.51i			0.387318i
$520$		0			0
$521$		−	298.274i		−	0.0250818i	−0.999921	−	0.0125409i	$-0.996008\pi$
$521$		−	298.274i		−	0.0250818i	0.999921	−	0.0125409i	$-0.00399200\pi$
$522$		0			0
$523$			14187.2i			1.18616i	0.805143	+	0.593081i	$0.202089\pi$
$523$			14187.2i			1.18616i	−0.805143	+	0.593081i	$0.797911\pi$
$524$		0			0
$525$	−778.509	−	458.991i	−0.0647180	−	0.0381562i
$526$		0			0
$527$		−	13840.0i		−	1.14398i
$528$		0			0
$529$	−4632.84			−0.380771
$530$		0			0
$531$		−	3057.22i		−	0.249853i
$532$		0			0
$533$			14726.8i			1.19679i
$534$		0			0
$535$	−182.096			−0.0147153
$536$		0			0
$537$			227.256i			0.0182622i
$538$		0			0
$539$	−8553.71	−	15460.1i	−0.683552	−	1.23546i
$540$		0			0
$541$			18394.5i			1.46181i	0.682478	+	0.730906i	$0.260902\pi$
$541$			18394.5i			1.46181i	−0.682478	+	0.730906i	$0.739098\pi$
$542$		0			0
$543$			12738.4i			1.00673i
$544$		0			0
$545$		−	13290.9i		−	1.04463i
$546$		0			0
$547$	−19598.5			−1.53194			−0.765970	−	0.642877i	$-0.777741\pi$
$547$	−19598.5			−1.53194			−0.765970	+	0.642877i	$0.777741\pi$
$548$		0			0
$549$	228.505			0.0177638
$550$		0			0
$551$	−2592.44			−0.200438
$552$		0			0
$553$	−15172.2	−	8945.18i	−1.16671	−	0.687862i
$554$		0			0
$555$	−4780.76			−0.365643
$556$		0			0
$557$		−	1640.60i		−	0.124801i	−0.998051	−	0.0624007i	$-0.980124\pi$
$557$		−	1640.60i		−	0.124801i	0.998051	−	0.0624007i	$-0.0198757\pi$
$558$		0			0
$559$	−17017.7			−1.28761
$560$		0			0
$561$	−15324.5			−1.15330
$562$		0			0
$563$		−	20377.4i		−	1.52541i	−0.646747	−	0.762705i	$-0.723871\pi$
$563$		−	20377.4i		−	1.52541i	0.646747	−	0.762705i	$-0.276129\pi$
$564$		0			0
$565$	−11512.5			−0.857229
$566$		0			0
$567$	761.889	−	1292.26i	0.0564309	−	0.0957143i
$568$		0			0
$569$	3016.11			0.222218			0.111109	−	0.993808i	$-0.464560\pi$
$569$	3016.11			0.222218			0.111109	+	0.993808i	$0.464560\pi$
$570$		0			0
$571$	3655.22			0.267892			0.133946	−	0.990989i	$-0.457235\pi$
$571$	3655.22			0.267892			0.133946	+	0.990989i	$0.457235\pi$
$572$		0			0
$573$	6854.61			0.499747
$574$		0			0
$575$			2108.28i			0.152907i
$576$		0			0
$577$			13481.2i			0.972665i	0.873774	+	0.486333i	$0.161666\pi$
$577$			13481.2i			0.972665i	−0.873774	+	0.486333i	$0.838334\pi$
$578$		0			0
$579$		−	14024.3i		−	1.00662i
$580$		0			0
$581$	−3903.49	−	2301.41i	−0.278733	−	0.164335i
$582$		0			0
$583$		−	1246.73i		−	0.0885663i
$584$		0			0
$585$	−4356.78			−0.307916
$586$		0			0
$587$			7198.62i			0.506165i	0.967445	+	0.253083i	$0.0814444\pi$
$587$			7198.62i			0.506165i	−0.967445	+	0.253083i	$0.918556\pi$
$588$		0			0
$589$		−	1746.80i		−	0.122199i
$590$		0			0
$591$	−1131.19			−0.0787328
$592$		0			0
$593$		−	19738.0i		−	1.36685i	−0.730019	−	0.683426i	$-0.760489\pi$
$593$		−	19738.0i		−	1.36685i	0.730019	−	0.683426i	$-0.239511\pi$
$594$		0			0
$595$	−16497.1	−	9726.28i	−1.13666	−	0.670149i
$596$		0			0
$597$			1329.99i			0.0911770i
$598$		0			0
$599$		−	7832.94i		−	0.534299i	−0.963655	−	0.267150i	$-0.913918\pi$
$599$		−	7832.94i		−	0.534299i	0.963655	−	0.267150i	$-0.0860818\pi$
$600$		0			0
$601$			12726.9i			0.863793i	0.901923	+	0.431897i	$0.142155\pi$
$601$			12726.9i			0.863793i	−0.901923	+	0.431897i	$0.857845\pi$
$602$		0			0
$603$	−5941.56			−0.401258
$604$		0			0
$605$	−13790.3			−0.926700
$606$		0			0
$607$	14748.7			0.986212			0.493106	−	0.869969i	$-0.335861\pi$
$607$	14748.7			0.986212			0.493106	+	0.869969i	$0.335861\pi$
$608$		0			0
$609$	−5844.83	+	9913.60i	−0.388907	+	0.659638i
$610$		0			0
$611$	−4784.38			−0.316784
$612$		0			0
$613$			2054.02i			0.135337i	0.997708	+	0.0676683i	$0.0215559\pi$
$613$			2054.02i			0.135337i	−0.997708	+	0.0676683i	$0.978444\pi$
$614$		0			0
$615$	−9923.69			−0.650670
$616$		0			0
$617$	−9381.71			−0.612145			−0.306073	−	0.952008i	$-0.599015\pi$
$617$	−9381.71			−0.612145			−0.306073	+	0.952008i	$0.599015\pi$
$618$		0			0
$619$		−	10324.6i		−	0.670404i	−0.942146	−	0.335202i	$-0.891195\pi$
$619$		−	10324.6i		−	0.670404i	0.942146	−	0.335202i	$-0.108805\pi$
$620$		0			0
$621$	−3499.58			−0.226141
$622$		0			0
$623$	−18701.5	−	11025.9i	−1.20266	−	0.709062i
$624$		0			0
$625$	−13327.2			−0.852940
$626$		0			0
$627$	−1934.16			−0.123195
$628$		0			0
$629$	15154.8			0.960669
$630$		0			0
$631$			19075.7i			1.20347i	0.798695	+	0.601736i	$0.205524\pi$
$631$			19075.7i			1.20347i	−0.798695	+	0.601736i	$0.794476\pi$
$632$		0			0
$633$			2039.30i			0.128049i
$634$		0			0
$635$		−	668.297i		−	0.0417646i
$636$		0			0
$637$	7708.80	+	13933.0i	0.479488	+	0.866631i
$638$		0			0
$639$			6.94557i			0.000429988i
$640$		0			0
$641$	7606.96			0.468731			0.234366	−	0.972149i	$-0.424699\pi$
$641$	7606.96			0.468731			0.234366	+	0.972149i	$0.424699\pi$
$642$		0			0
$643$		−	26087.8i		−	1.60000i	−0.599998	−	0.800002i	$-0.704832\pi$
$643$		−	26087.8i		−	1.60000i	0.599998	−	0.800002i	$-0.295168\pi$
$644$		0			0
$645$		−	11467.4i		−	0.700044i
$646$		0			0
$647$	−24770.7			−1.50515			−0.752577	−	0.658504i	$-0.771190\pi$
$647$	−24770.7			−1.50515			−0.752577	+	0.658504i	$0.771190\pi$
$648$		0			0
$649$			17498.1i			1.05834i
$650$		0			0
$651$	−6679.83	−	3938.27i	−0.402156	−	0.237101i
$652$		0			0
$653$			10693.7i			0.640854i	0.947273	+	0.320427i	$0.103826\pi$
$653$			10693.7i			0.640854i	−0.947273	+	0.320427i	$0.896174\pi$
$654$		0			0
$655$		−	3490.36i		−	0.208213i
$656$		0			0
$657$		−	3081.92i		−	0.183009i
$658$		0			0
$659$	−12741.7			−0.753178			−0.376589	−	0.926380i	$-0.622903\pi$
$659$	−12741.7			−0.753178			−0.376589	+	0.926380i	$0.622903\pi$
$660$		0			0
$661$	−27121.9			−1.59594			−0.797972	−	0.602694i	$-0.794094\pi$
$661$	−27121.9			−1.59594			−0.797972	+	0.602694i	$0.794094\pi$
$662$		0			0
$663$	13810.8			0.808999
$664$		0			0
$665$	−2082.16	−	1227.59i	−0.121417	−	0.0715849i
$666$		0			0
$667$	26847.0			1.55850
$668$		0			0
$669$		−	9387.06i		−	0.542488i
$670$		0			0
$671$	−1307.86			−0.0752449
$672$		0			0
$673$	−17710.1			−1.01437			−0.507187	−	0.861836i	$-0.669315\pi$
$673$	−17710.1			−1.01437			−0.507187	+	0.861836i	$0.669315\pi$
$674$		0			0
$675$			439.177i			0.0250429i
$676$		0			0
$677$	−29650.4			−1.68324			−0.841622	−	0.540067i	$-0.818399\pi$
$677$	−29650.4			−1.68324			−0.841622	+	0.540067i	$0.818399\pi$
$678$		0			0
$679$	22459.0	+	13241.3i	1.26937	+	0.748388i
$680$		0			0
$681$	−15373.5			−0.865073
$682$		0			0
$683$	−1961.33			−0.109880			−0.0549401	−	0.998490i	$-0.517497\pi$
$683$	−1961.33			−0.109880			−0.0549401	+	0.998490i	$0.517497\pi$
$684$		0			0
$685$	−14639.1			−0.816540
$686$		0			0
$687$			12153.5i			0.674944i
$688$		0			0
$689$			1123.58i			0.0621262i
$690$		0			0
$691$			20655.9i			1.13718i	0.822622	+	0.568588i	$0.192510\pi$
$691$			20655.9i			1.13718i	−0.822622	+	0.568588i	$0.807490\pi$
$692$		0			0
$693$	−4360.71	+	7396.34i	−0.239033	+	0.405431i
$694$		0			0
$695$		−	25892.7i		−	1.41319i
$696$		0			0
$697$	31457.6			1.70953
$698$		0			0
$699$		−	6098.17i		−	0.329977i
$700$		0			0
$701$			19012.5i			1.02438i	0.858872	+	0.512190i	$0.171166\pi$
$701$			19012.5i			1.02438i	−0.858872	+	0.512190i	$0.828834\pi$
$702$		0			0
$703$	1912.74			0.102618
$704$		0			0
$705$		−	3223.96i		−	0.172229i
$706$		0			0
$707$	10970.2	−	18607.0i	0.583562	−	0.989798i
$708$		0			0
$709$			8682.66i			0.459921i	0.973200	+	0.229961i	$0.0738597\pi$
$709$			8682.66i			0.459921i	−0.973200	+	0.229961i	$0.926140\pi$
$710$		0			0
$711$			8559.04i			0.451462i
$712$		0			0
$713$			18089.7i			0.950159i
$714$		0			0
$715$	24936.2			1.30428
$716$		0			0
$717$	19828.6			1.03279
$718$		0			0
$719$	23035.6			1.19483			0.597416	−	0.801931i	$-0.296194\pi$
$719$	23035.6			1.19483			0.597416	+	0.801931i	$0.296194\pi$
$720$		0			0
$721$	12276.3	−	20822.3i	0.634111	−	1.07554i
$722$		0			0
$723$	−2629.53			−0.135260
$724$		0			0
$725$		−	3369.15i		−	0.172589i
$726$		0			0
$727$	29814.9			1.52101			0.760504	−	0.649333i	$-0.224952\pi$
$727$	29814.9			1.52101			0.760504	+	0.649333i	$0.224952\pi$
$728$		0			0
$729$	−729.000			−0.0370370
$730$		0			0
$731$			36351.1i			1.83925i
$732$		0			0
$733$	−34361.3			−1.73147			−0.865733	−	0.500507i	$-0.833147\pi$
$733$	−34361.3			−1.73147			−0.865733	+	0.500507i	$0.833147\pi$
$734$		0			0
$735$	−9388.74	+	5194.59i	−0.471169	+	0.260687i
$736$		0			0
$737$	34006.8			1.69967
$738$		0			0
$739$	−12514.2			−0.622928			−0.311464	−	0.950258i	$-0.600819\pi$
$739$	−12514.2			−0.622928			−0.311464	+	0.950258i	$0.600819\pi$
$740$		0			0
$741$	1743.11			0.0864168
$742$		0			0
$743$			21810.2i			1.07690i	0.842656	+	0.538452i	$0.180991\pi$
$743$			21810.2i			1.07690i	−0.842656	+	0.538452i	$0.819009\pi$
$744$		0			0
$745$			27390.0i			1.34697i
$746$		0			0
$747$			2202.06i			0.107857i
$748$		0			0
$749$	164.257	−	278.602i	0.00801312	−	0.0135913i
$750$		0			0
$751$			18834.3i			0.915145i	0.889172	+	0.457572i	$0.151281\pi$
$751$			18834.3i			0.915145i	−0.889172	+	0.457572i	$0.848719\pi$
$752$		0			0
$753$	2004.53			0.0970110
$754$		0			0
$755$		−	35422.0i		−	1.70747i
$756$		0			0
$757$		−	8178.85i		−	0.392689i	−0.980535	−	0.196344i	$-0.937093\pi$
$757$		−	8178.85i		−	0.392689i	0.980535	−	0.196344i	$-0.0629070\pi$
$758$		0			0
$759$	20030.0			0.957898
$760$		0			0
$761$		−	12711.4i		−	0.605502i	−0.953070	−	0.302751i	$-0.902095\pi$
$761$		−	12711.4i		−	0.605502i	0.953070	−	0.302751i	$-0.0979050\pi$
$762$		0			0
$763$	20334.8	+	11988.9i	0.964833	+	0.568843i
$764$		0			0
$765$			9306.42i			0.439836i
$766$		0			0
$767$		−	15769.7i		−	0.742388i
$768$		0			0
$769$		−	28075.1i		−	1.31653i	−0.752786	−	0.658266i	$-0.771290\pi$
$769$		−	28075.1i		−	1.31653i	0.752786	−	0.658266i	$-0.228710\pi$
$770$		0			0
$771$	−12836.5			−0.599603
$772$		0			0
$773$	1473.33			0.0685535			0.0342768	−	0.999412i	$-0.489087\pi$
$773$	1473.33			0.0685535			0.0342768	+	0.999412i	$0.489087\pi$
$774$		0			0
$775$	2270.15			0.105221
$776$		0			0
$777$	4312.41	−	7314.43i	0.199108	−	0.337714i
$778$		0			0
$779$	3970.39			0.182611
$780$		0			0
$781$		−	39.7533i		−	0.00182136i
$782$		0			0
$783$	5592.52			0.255250
$784$		0			0
$785$	10268.7			0.466887
$786$		0			0
$787$		−	11221.1i		−	0.508244i	−0.967172	−	0.254122i	$-0.918213\pi$
$787$		−	11221.1i		−	0.508244i	0.967172	−	0.254122i	$-0.0817865\pi$
$788$		0			0
$789$	−4233.73			−0.191033
$790$		0			0
$791$	10384.7	−	17613.8i	0.466798	−	0.791751i
$792$		0			0
$793$	1178.67			0.0527817
$794$		0			0
$795$	−757.125			−0.0337767
$796$		0			0
$797$	20859.6			0.927082			0.463541	−	0.886075i	$-0.346579\pi$
$797$	20859.6			0.927082			0.463541	+	0.886075i	$0.346579\pi$
$798$		0			0
$799$			10219.8i			0.452504i
$800$		0			0
$801$			10550.0i			0.465375i
$802$		0			0
$803$			17639.5i			0.775199i
$804$		0			0
$805$	21562.7	+	12712.8i	0.944079	+	0.556607i
$806$		0			0
$807$			18099.3i			0.789497i
$808$		0			0
$809$	12125.8			0.526973			0.263486	−	0.964663i	$-0.415128\pi$
$809$	12125.8			0.526973			0.263486	+	0.964663i	$0.415128\pi$
$810$		0			0
$811$		−	20324.0i		−	0.879991i	−0.898000	−	0.439995i	$-0.854980\pi$
$811$		−	20324.0i		−	0.879991i	0.898000	−	0.439995i	$-0.145020\pi$
$812$		0			0
$813$			25998.6i			1.12154i
$814$		0			0
$815$	−34553.7			−1.48511
$816$		0			0
$817$			4588.01i			0.196468i
$818$		0			0
$819$	3929.97	−	6665.75i	0.167673	−	0.284396i
$820$		0			0
$821$			31423.3i			1.33578i	0.744258	+	0.667892i	$0.232803\pi$
$821$			31423.3i			1.33578i	−0.744258	+	0.667892i	$0.767197\pi$
$822$		0			0
$823$			14350.9i			0.607827i	0.952700	+	0.303913i	$0.0982933\pi$
$823$			14350.9i			0.607827i	−0.952700	+	0.303913i	$0.901707\pi$
$824$		0			0
$825$		−	2513.65i		−	0.106078i
$826$		0			0
$827$	−34780.0			−1.46242			−0.731209	−	0.682153i	$-0.761044\pi$
$827$	−34780.0			−1.46242			−0.731209	+	0.682153i	$0.761044\pi$
$828$		0			0
$829$	−26803.0			−1.12293			−0.561464	−	0.827501i	$-0.689762\pi$
$829$	−26803.0			−1.12293			−0.561464	+	0.827501i	$0.689762\pi$
$830$		0			0
$831$	18382.1			0.767350
$832$		0			0
$833$	29761.9	−	16466.6i	1.23792	−	0.684914i
$834$		0			0
$835$	−15667.1			−0.649321
$836$		0			0
$837$			3768.27i			0.155616i
$838$		0			0
$839$	−29291.6			−1.20531			−0.602657	−	0.798000i	$-0.705891\pi$
$839$	−29291.6			−1.20531			−0.602657	+	0.798000i	$0.705891\pi$
$840$		0			0
$841$	−18514.0			−0.759113
$842$		0			0
$843$			26016.7i			1.06295i
$844$		0			0
$845$	436.259			0.0177607
$846$		0			0
$847$	12439.3	−	21098.7i	0.504627	−	0.855915i
$848$		0			0
$849$	6919.83			0.279727
$850$		0			0
$851$	−19808.2			−0.797904
$852$		0			0
$853$	−813.999			−0.0326738			−0.0163369	−	0.999867i	$-0.505200\pi$
$853$	−813.999			−0.0326738			−0.0163369	+	0.999867i	$0.505200\pi$
$854$		0			0
$855$			1174.60i			0.0469830i
$856$		0			0
$857$		−	7451.12i		−	0.296996i	−0.988913	−	0.148498i	$-0.952556\pi$
$857$		−	7451.12i		−	0.296996i	0.988913	−	0.148498i	$-0.0474438\pi$
$858$		0			0
$859$			45141.7i			1.79303i	0.443013	+	0.896515i	$0.353910\pi$
$859$			45141.7i			1.79303i	−0.443013	+	0.896515i	$0.646090\pi$
$860$		0			0
$861$	8951.52	−	15183.0i	0.354317	−	0.600969i
$862$		0			0
$863$			9660.92i			0.381068i	0.981681	+	0.190534i	$0.0610219\pi$
$863$			9660.92i			0.381068i	−0.981681	+	0.190534i	$0.938978\pi$
$864$		0			0
$865$	−15917.7			−0.625686
$866$		0			0
$867$		−	14761.9i		−	0.578248i
$868$		0			0
$869$		−	48988.1i		−	1.91232i
$870$		0			0
$871$	−30647.7			−1.19226
$872$		0			0
$873$		−	12669.7i		−	0.491186i
$874$		0			0
$875$	13855.6	−	23501.0i	0.535322	−	0.907976i
$876$		0			0
$877$		−	46120.0i		−	1.77578i	−0.460052	−	0.887892i	$-0.652169\pi$
$877$		−	46120.0i		−	1.77578i	0.460052	−	0.887892i	$-0.347831\pi$
$878$		0			0
$879$		−	9157.88i		−	0.351408i
$880$		0			0
$881$			31118.8i			1.19003i	0.803713	+	0.595017i	$0.202855\pi$
$881$			31118.8i			1.19003i	−0.803713	+	0.595017i	$0.797145\pi$
$882$		0			0
$883$	26337.2			1.00376			0.501878	−	0.864938i	$-0.332643\pi$
$883$	26337.2			1.00376			0.501878	+	0.864938i	$0.332643\pi$
$884$		0			0
$885$	10626.5			0.403621
$886$		0			0
$887$	−39333.5			−1.48894			−0.744470	−	0.667656i	$-0.767298\pi$
$887$	−39333.5			−1.48894			−0.744470	+	0.667656i	$0.767298\pi$
$888$		0			0
$889$	1022.48	+	602.827i	0.0385745	+	0.0227426i
$890$		0			0
$891$	4172.47			0.156883
$892$		0			0
$893$			1289.88i			0.0483362i
$894$		0			0
$895$	−789.908			−0.0295014
$896$		0			0
$897$	−18051.5			−0.671932
$898$		0			0
$899$		−	28908.2i		−	1.07246i
$900$		0			0
$901$	2400.05			0.0887428
$902$		0			0
$903$	17544.8	+	10344.0i	0.646571	+	0.381203i
$904$		0			0
$905$	−44276.8			−1.62631
$906$		0			0
$907$	−23943.1			−0.876534			−0.438267	−	0.898845i	$-0.644408\pi$
$907$	−23943.1			−0.876534			−0.438267	+	0.898845i	$0.644408\pi$
$908$		0			0
$909$	−10496.7			−0.383006
$910$		0			0
$911$		−	19935.2i		−	0.725009i	−0.931982	−	0.362505i	$-0.881922\pi$
$911$		−	19935.2i		−	0.725009i	0.931982	−	0.362505i	$-0.118078\pi$
$912$		0			0
$913$		−	12603.6i		−	0.456866i
$914$		0			0
$915$			794.250i			0.0286963i
$916$		0			0
$917$	5340.15	+	3148.43i	0.192309	+	0.113381i
$918$		0			0
$919$		−	13368.0i		−	0.479836i	−0.970793	−	0.239918i	$-0.922879\pi$
$919$		−	13368.0i		−	0.479836i	0.970793	−	0.239918i	$-0.0771206\pi$
$920$		0			0
$921$	12655.0			0.452765
$922$		0			0
$923$			35.8266i			0.00127762i
$924$		0			0
$925$			2485.81i			0.0883601i
$926$		0			0
$927$	−11746.4			−0.416183
$928$		0			0
$929$		−	32288.3i		−	1.14031i	−0.821538	−	0.570153i	$-0.806884\pi$
$929$		−	32288.3i		−	1.14031i	0.821538	−	0.570153i	$-0.193116\pi$
$930$		0			0
$931$	3756.36	−	2078.31i	0.132234	−	0.0731621i
$932$		0			0
$933$			2444.80i			0.0857869i
$934$		0			0
$935$		−	53265.7i		−	1.86308i
$936$		0			0
$937$		−	25670.9i		−	0.895019i	−0.894279	−	0.447509i	$-0.852311\pi$
$937$		−	25670.9i		−	0.895019i	0.894279	−	0.447509i	$-0.147689\pi$
$938$		0			0
$939$	6010.88			0.208900
$940$		0			0
$941$	15069.6			0.522055			0.261027	−	0.965331i	$-0.415939\pi$
$941$	15069.6			0.522055			0.261027	+	0.965331i	$0.415939\pi$
$942$		0			0
$943$	−41117.0			−1.41989
$944$		0			0
$945$	4491.73	+	2648.22i	0.154620	+	0.0911603i
$946$		0			0
$947$	20296.9			0.696475			0.348237	−	0.937406i	$-0.386780\pi$
$947$	20296.9			0.696475			0.348237	+	0.937406i	$0.386780\pi$
$948$		0			0
$949$		−	15897.1i		−	0.543775i
$950$		0			0
$951$	−6476.33			−0.220830
$952$		0			0
$953$	20571.8			0.699250			0.349625	−	0.936890i	$-0.386309\pi$
$953$	20571.8			0.699250			0.349625	+	0.936890i	$0.386309\pi$
$954$		0			0
$955$			23825.6i			0.807308i
$956$		0			0
$957$	−32009.1			−1.08120
$958$		0			0
$959$	13204.9	−	22397.3i	0.444640	−	0.754169i
$960$		0			0
$961$	−10312.5			−0.346161
$962$		0			0
$963$	−157.167			−0.00525921
$964$		0			0
$965$	48746.5			1.62612
$966$		0			0
$967$			18483.0i			0.614658i	0.951603	+	0.307329i	$0.0994351\pi$
$967$			18483.0i			0.614658i	−0.951603	+	0.307329i	$0.900565\pi$
$968$		0			0
$969$		−	3723.42i		−	0.123440i
$970$		0			0
$971$			42988.0i			1.42075i	0.703822	+	0.710376i	$0.251475\pi$
$971$			42988.0i			1.42075i	−0.703822	+	0.710376i	$0.748525\pi$
$972$		0			0
$973$	39615.1	+	23356.2i	1.30524	+	0.769541i
$974$		0			0
$975$			2265.36i			0.0744098i
$976$		0			0
$977$	20691.5			0.677565			0.338782	−	0.940865i	$-0.389985\pi$
$977$	20691.5			0.677565			0.338782	+	0.940865i	$0.389985\pi$
$978$		0			0
$979$		−	60383.4i		−	1.97126i
$980$		0			0
$981$		−	11471.4i		−	0.373346i
$982$		0			0
$983$	41099.7			1.33355			0.666773	−	0.745260i	$-0.267675\pi$
$983$	41099.7			1.33355			0.666773	+	0.745260i	$0.267675\pi$
$984$		0			0
$985$		−	3931.87i		−	0.127188i
$986$		0			0
$987$	4932.57	+	2908.13i	0.159073	+	0.0937860i
$988$		0			0
$989$		−	47513.1i		−	1.52763i
$990$		0			0
$991$			3899.50i			0.124997i	0.998045	+	0.0624983i	$0.0199068\pi$
$991$			3899.50i			0.124997i	−0.998045	+	0.0624983i	$0.980093\pi$
$992$		0			0
$993$			32596.2i			1.04170i
$994$		0			0
$995$	−4622.84			−0.147290
$996$		0			0
$997$	9066.39			0.287999			0.144000	−	0.989578i	$-0.454004\pi$
$997$	9066.39			0.287999			0.144000	+	0.989578i	$0.454004\pi$
$998$		0			0
$999$	−4126.26			−0.130680

Twists

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

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

\(f(q)\)	\(=\)	\(q+3.00000i q^{3} -10.4276 q^{5} +(9.40604 - 15.9539i) q^{7} -9.00000 q^{9} +O(q^{10})\)	\(q+3.00000i q^{3} -10.4276 q^{5} +(9.40604 - 15.9539i) q^{7} -9.00000 q^{9} +51.5119 q^{11} -46.4237 q^{13} -31.2827i q^{15} +99.1647i q^{17} +12.5160i q^{19} +(47.8617 + 28.2181i) q^{21} -129.614i q^{23} -16.2658 q^{25} -27.0000i q^{27} +207.130i q^{29} -139.565 q^{31} +154.536i q^{33} +(-98.0821 + 166.360i) q^{35} -152.824i q^{37} -139.271i q^{39} -317.226i q^{41} +366.573 q^{43} +93.8481 q^{45} +103.059 q^{47} +(-166.053 - 300.126i) q^{49} -297.494 q^{51} -24.2027i q^{53} -537.144 q^{55} -37.5479 q^{57} +339.691i q^{59} -25.3894 q^{61} +(-84.6543 + 143.585i) q^{63} +484.087 q^{65} +660.173 q^{67} +388.843 q^{69} -0.771730i q^{71} +342.435i q^{73} -48.7975i q^{75} +(484.523 - 821.816i) q^{77} -951.005i q^{79} +81.0000 q^{81} -244.673i q^{83} -1034.05i q^{85} -621.391 q^{87} -1172.22i q^{89} +(-436.663 + 740.639i) q^{91} -418.696i q^{93} -130.511i q^{95} +1407.75i q^{97} -463.607 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 144 q^{9}+O(q^{10}) \) 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

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