LMFDB - Embedded newform 1344.2.h.f.575.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1344,2,Mod(575,1344)]

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, 1, 0]))

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("1344.575");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1344 = 2^{6} \cdot 3 \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1344.h (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:	$10.7318940317$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.40960000.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 7x^{4} + 1 $ x^8 + 7*x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	no (minimal twist has level 672)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			575.7
Root			$-1.14412 - 1.14412i$ of defining polynomial
Character	$\chi$	$=$	1344.575
Dual form			1344.2.h.f.575.6

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.707107 + 1.58114i) q^{3} -1.41421i q^{5} +1.00000i q^{7} +(-2.00000 + 2.23607i) q^{9} +O(q^{10})$	\(q+(0.707107 + 1.58114i) q^{3} -1.41421i q^{5} +1.00000i q^{7} +(-2.00000 + 2.23607i) q^{9} -4.47214 q^{11} +7.16228 q^{13} +(2.23607 - 1.00000i) q^{15} +7.30056i q^{17} -0.837722i q^{19} +(-1.58114 + 0.707107i) q^{21} -5.65685 q^{23} +3.00000 q^{25} +(-4.94975 - 1.58114i) q^{27} -1.64371i q^{29} +6.32456i q^{31} +(-3.16228 - 7.07107i) q^{33} +1.41421 q^{35} -4.32456 q^{37} +(5.06450 + 11.3246i) q^{39} +10.1290i q^{41} +8.32456i q^{43} +(3.16228 + 2.82843i) q^{45} +8.94427 q^{47} -1.00000 q^{49} +(-11.5432 + 5.16228i) q^{51} -1.18472i q^{53} +6.32456i q^{55} +(1.32456 - 0.592359i) q^{57} -1.41421 q^{59} -3.16228 q^{61} +(-2.23607 - 2.00000i) q^{63} -10.1290i q^{65} -2.00000i q^{67} +(-4.00000 - 8.94427i) q^{69} -10.1290 q^{71} +4.32456 q^{73} +(2.12132 + 4.74342i) q^{75} -4.47214i q^{77} +4.00000i q^{79} +(-1.00000 - 8.94427i) q^{81} +7.53006 q^{83} +10.3246 q^{85} +(2.59893 - 1.16228i) q^{87} +1.18472i q^{89} +7.16228i q^{91} +(-10.0000 + 4.47214i) q^{93} -1.18472 q^{95} -10.6491 q^{97} +(8.94427 - 10.0000i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 16 q^{9}+O(q^{10}) $ 8 * q - 16 * q^9	$ 8 q - 16 q^{9} + 32 q^{13} + 24 q^{25} + 16 q^{37} - 8 q^{49} - 40 q^{57} - 32 q^{69} - 16 q^{73} - 8 q^{81} + 32 q^{85} - 80 q^{93} + 16 q^{97}+O(q^{100}) $ 8 * q - 16 * q^9 + 32 * q^13 + 24 * q^25 + 16 * q^37 - 8 * q^49 - 40 * q^57 - 32 * q^69 - 16 * q^73 - 8 * q^81 + 32 * q^85 - 80 * q^93 + 16 * q^97

Display 100 coefficients 10 coefficients

Character values

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

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

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		0			0
$2$		0			0
$3$	0.707107	+	1.58114i	0.408248	+	0.912871i
$3$	0.707107	+	1.58114i	0.408248	+	0.912871i
$4$		0			0
$5$		−	1.41421i		−	0.632456i	−0.948683	−	0.316228i	$-0.897584\pi$
$5$		−	1.41421i		−	0.632456i	0.948683	−	0.316228i	$-0.102416\pi$
$6$		0			0
$7$			1.00000i			0.377964i
$7$			1.00000i			0.377964i
$8$		0			0
$9$	−2.00000	+	2.23607i	−0.666667	+	0.745356i
$10$		0			0
$11$	−4.47214			−1.34840			−0.674200	−	0.738549i	$-0.735511\pi$
$11$	−4.47214			−1.34840			−0.674200	+	0.738549i	$0.735511\pi$
$12$		0			0
$13$	7.16228			1.98646			0.993229	−	0.116171i	$-0.0370621\pi$
$13$	7.16228			1.98646			0.993229	+	0.116171i	$0.0370621\pi$
$14$		0			0
$15$	2.23607	−	1.00000i	0.577350	−	0.258199i
$16$		0			0
$17$			7.30056i			1.77065i	0.464976	+	0.885323i	$0.346063\pi$
$17$			7.30056i			1.77065i	−0.464976	+	0.885323i	$0.653937\pi$
$18$		0			0
$19$		−	0.837722i		−	0.192187i	−0.995372	−	0.0960933i	$-0.969365\pi$
$19$		−	0.837722i		−	0.192187i	0.995372	−	0.0960933i	$-0.0306347\pi$
$20$		0			0
$21$	−1.58114	+	0.707107i	−0.345033	+	0.154303i
$22$		0			0
$23$	−5.65685			−1.17954			−0.589768	−	0.807573i	$-0.700781\pi$
$23$	−5.65685			−1.17954			−0.589768	+	0.807573i	$0.700781\pi$
$24$		0			0
$25$	3.00000			0.600000
$26$		0			0
$27$	−4.94975	−	1.58114i	−0.952579	−	0.304290i
$28$		0			0
$29$		−	1.64371i		−	0.305229i	−0.988286	−	0.152615i	$-0.951231\pi$
$29$		−	1.64371i		−	0.305229i	0.988286	−	0.152615i	$-0.0487693\pi$
$30$		0			0
$31$			6.32456i			1.13592i	0.823055	+	0.567962i	$0.192268\pi$
$31$			6.32456i			1.13592i	−0.823055	+	0.567962i	$0.807732\pi$
$32$		0			0
$33$	−3.16228	−	7.07107i	−0.550482	−	1.23091i
$34$		0			0
$35$	1.41421			0.239046
$36$		0			0
$37$	−4.32456			−0.710953			−0.355476	−	0.934685i	$-0.615681\pi$
$37$	−4.32456			−0.710953			−0.355476	+	0.934685i	$0.615681\pi$
$38$		0			0
$39$	5.06450	+	11.3246i	0.810968	+	1.81338i
$40$		0			0
$41$			10.1290i			1.58188i	0.611892	+	0.790941i	$0.290409\pi$
$41$			10.1290i			1.58188i	−0.611892	+	0.790941i	$0.709591\pi$
$42$		0			0
$43$			8.32456i			1.26948i	0.772725	+	0.634741i	$0.218893\pi$
$43$			8.32456i			1.26948i	−0.772725	+	0.634741i	$0.781107\pi$
$44$		0			0
$45$	3.16228	+	2.82843i	0.471405	+	0.421637i
$46$		0			0
$47$	8.94427			1.30466			0.652328	−	0.757937i	$-0.273792\pi$
$47$	8.94427			1.30466			0.652328	+	0.757937i	$0.273792\pi$
$48$		0			0
$49$	−1.00000			−0.142857
$50$		0			0
$51$	−11.5432	+	5.16228i	−1.61637	+	0.722863i
$52$		0			0
$53$		−	1.18472i		−	0.162734i	−0.996684	−	0.0813668i	$-0.974071\pi$
$53$		−	1.18472i		−	0.162734i	0.996684	−	0.0813668i	$-0.0259285\pi$
$54$		0			0
$55$			6.32456i			0.852803i
$56$		0			0
$57$	1.32456	−	0.592359i	0.175442	−	0.0784599i
$58$		0			0
$59$	−1.41421			−0.184115			−0.0920575	−	0.995754i	$-0.529344\pi$
$59$	−1.41421			−0.184115			−0.0920575	+	0.995754i	$0.529344\pi$
$60$		0			0
$61$	−3.16228			−0.404888			−0.202444	−	0.979294i	$-0.564888\pi$
$61$	−3.16228			−0.404888			−0.202444	+	0.979294i	$0.564888\pi$
$62$		0			0
$63$	−2.23607	−	2.00000i	−0.281718	−	0.251976i
$64$		0			0
$65$		−	10.1290i		−	1.25635i
$66$		0			0
$67$		−	2.00000i		−	0.244339i	−0.992509	−	0.122169i	$-0.961015\pi$
$67$		−	2.00000i		−	0.244339i	0.992509	−	0.122169i	$-0.0389851\pi$
$68$		0			0
$69$	−4.00000	−	8.94427i	−0.481543	−	1.07676i
$70$		0			0
$71$	−10.1290			−1.20209			−0.601045	−	0.799215i	$-0.705249\pi$
$71$	−10.1290			−1.20209			−0.601045	+	0.799215i	$0.705249\pi$
$72$		0			0
$73$	4.32456			0.506151			0.253075	−	0.967447i	$-0.418558\pi$
$73$	4.32456			0.506151			0.253075	+	0.967447i	$0.418558\pi$
$74$		0			0
$75$	2.12132	+	4.74342i	0.244949	+	0.547723i
$76$		0			0
$77$		−	4.47214i		−	0.509647i
$78$		0			0
$79$			4.00000i			0.450035i	0.974355	+	0.225018i	$0.0722440\pi$
$79$			4.00000i			0.450035i	−0.974355	+	0.225018i	$0.927756\pi$
$80$		0			0
$81$	−1.00000	−	8.94427i	−0.111111	−	0.993808i
$82$		0			0
$83$	7.53006			0.826531			0.413266	−	0.910611i	$-0.364388\pi$
$83$	7.53006			0.826531			0.413266	+	0.910611i	$0.364388\pi$
$84$		0			0
$85$	10.3246			1.11986
$86$		0			0
$87$	2.59893	−	1.16228i	0.278635	−	0.124609i
$88$		0			0
$89$			1.18472i			0.125580i	0.998027	+	0.0627899i	$0.0199998\pi$
$89$			1.18472i			0.125580i	−0.998027	+	0.0627899i	$0.980000\pi$
$90$		0			0
$91$			7.16228i			0.750811i
$92$		0			0
$93$	−10.0000	+	4.47214i	−1.03695	+	0.463739i
$94$		0			0
$95$	−1.18472			−0.121550
$96$		0			0
$97$	−10.6491			−1.08125			−0.540627	−	0.841263i	$-0.681813\pi$
$97$	−10.6491			−1.08125			−0.540627	+	0.841263i	$0.681813\pi$
$98$		0			0
$99$	8.94427	−	10.0000i	0.898933	−	1.00504i
$100$		0			0
$101$		−	1.41421i		−	0.140720i	−0.997522	−	0.0703598i	$-0.977585\pi$
$101$		−	1.41421i		−	0.140720i	0.997522	−	0.0703598i	$-0.0224147\pi$
$102$		0			0
$103$		−	8.00000i		−	0.788263i	−0.919054	−	0.394132i	$-0.871045\pi$
$103$		−	8.00000i		−	0.788263i	0.919054	−	0.394132i	$-0.128955\pi$
$104$		0			0
$105$	1.00000	+	2.23607i	0.0975900	+	0.218218i
$106$		0			0
$107$	9.67000			0.934834			0.467417	−	0.884037i	$-0.345185\pi$
$107$	9.67000			0.934834			0.467417	+	0.884037i	$0.345185\pi$
$108$		0			0
$109$	4.32456			0.414217			0.207109	−	0.978318i	$-0.433595\pi$
$109$	4.32456			0.414217			0.207109	+	0.978318i	$0.433595\pi$
$110$		0			0
$111$	−3.05792	−	6.83772i	−0.290245	−	0.649008i
$112$		0			0
$113$			6.84157i			0.643601i	0.946807	+	0.321801i	$0.104288\pi$
$113$			6.84157i			0.643601i	−0.946807	+	0.321801i	$0.895712\pi$
$114$		0			0
$115$			8.00000i			0.746004i
$116$		0			0
$117$	−14.3246	+	16.0153i	−1.32431	+	1.48062i
$118$		0			0
$119$	−7.30056			−0.669242
$120$		0			0
$121$	9.00000			0.818182
$122$		0			0
$123$	−16.0153	+	7.16228i	−1.44405	+	0.645801i
$124$		0			0
$125$		−	11.3137i		−	1.01193i
$126$		0			0
$127$		−	6.00000i		−	0.532414i	−0.963916	−	0.266207i	$-0.914230\pi$
$127$		−	6.00000i		−	0.532414i	0.963916	−	0.266207i	$-0.0857705\pi$
$128$		0			0
$129$	−13.1623	+	5.88635i	−1.15887	+	0.518264i
$130$		0			0
$131$	4.70163			0.410783			0.205392	−	0.978680i	$-0.434153\pi$
$131$	4.70163			0.410783			0.205392	+	0.978680i	$0.434153\pi$
$132$		0			0
$133$	0.837722			0.0726397
$134$		0			0
$135$	−2.23607	+	7.00000i	−0.192450	+	0.602464i
$136$		0			0
$137$		−	17.8885i		−	1.52832i	−0.645026	−	0.764161i	$-0.723153\pi$
$137$		−	17.8885i		−	1.52832i	0.645026	−	0.764161i	$-0.276847\pi$
$138$		0			0
$139$			7.16228i			0.607496i	0.952752	+	0.303748i	$0.0982382\pi$
$139$			7.16228i			0.607496i	−0.952752	+	0.303748i	$0.901762\pi$
$140$		0			0
$141$	6.32456	+	14.1421i	0.532624	+	1.19098i
$142$		0			0
$143$	−32.0307			−2.67854
$144$		0			0
$145$	−2.32456			−0.193044
$146$		0			0
$147$	−0.707107	−	1.58114i	−0.0583212	−	0.130410i
$148$		0			0
$149$			4.47214i			0.366372i	0.983078	+	0.183186i	$0.0586410\pi$
$149$			4.47214i			0.366372i	−0.983078	+	0.183186i	$0.941359\pi$
$150$		0			0
$151$			10.0000i			0.813788i	0.913475	+	0.406894i	$0.133388\pi$
$151$			10.0000i			0.813788i	−0.913475	+	0.406894i	$0.866612\pi$
$152$		0			0
$153$	−16.3246	−	14.6011i	−1.31976	−	1.18043i
$154$		0			0
$155$	8.94427			0.718421
$156$		0			0
$157$	13.4868			1.07637			0.538183	−	0.842828i	$-0.319111\pi$
$157$	13.4868			1.07637			0.538183	+	0.842828i	$0.319111\pi$
$158$		0			0
$159$	1.87320	−	0.837722i	0.148555	−	0.0664357i
$160$		0			0
$161$		−	5.65685i		−	0.445823i
$162$		0			0
$163$			4.32456i			0.338725i	0.985554	+	0.169363i	$0.0541709\pi$
$163$			4.32456i			0.338725i	−0.985554	+	0.169363i	$0.945829\pi$
$164$		0			0
$165$	−10.0000	+	4.47214i	−0.778499	+	0.348155i
$166$		0			0
$167$	−0.458991			−0.0355178			−0.0177589	−	0.999842i	$-0.505653\pi$
$167$	−0.458991			−0.0355178			−0.0177589	+	0.999842i	$0.505653\pi$
$168$		0			0
$169$	38.2982			2.94602
$170$		0			0
$171$	1.87320	+	1.67544i	0.143247	+	0.128124i
$172$		0			0
$173$		−	3.78365i		−	0.287666i	−0.989602	−	0.143833i	$-0.954057\pi$
$173$		−	3.78365i		−	0.287666i	0.989602	−	0.143833i	$-0.0459427\pi$
$174$		0			0
$175$			3.00000i			0.226779i
$176$		0			0
$177$	−1.00000	−	2.23607i	−0.0751646	−	0.168073i
$178$		0			0
$179$	−19.5323			−1.45991			−0.729955	−	0.683496i	$-0.760459\pi$
$179$	−19.5323			−1.45991			−0.729955	+	0.683496i	$0.760459\pi$
$180$		0			0
$181$	−18.1359			−1.34803			−0.674017	−	0.738716i	$-0.735433\pi$
$181$	−18.1359			−1.34803			−0.674017	+	0.738716i	$0.735433\pi$
$182$		0			0
$183$	−2.23607	−	5.00000i	−0.165295	−	0.369611i
$184$		0			0
$185$			6.11584i			0.449646i
$186$		0			0
$187$		−	32.6491i		−	2.38754i
$188$		0			0
$189$	1.58114	−	4.94975i	0.115011	−	0.360041i
$190$		0			0
$191$	12.4984			0.904354			0.452177	−	0.891928i	$-0.350647\pi$
$191$	12.4984			0.904354			0.452177	+	0.891928i	$0.350647\pi$
$192$		0			0
$193$	16.0000			1.15171			0.575853	−	0.817554i	$-0.304670\pi$
$193$	16.0000			1.15171			0.575853	+	0.817554i	$0.304670\pi$
$194$		0			0
$195$	16.0153	−	7.16228i	1.14688	−	0.512901i
$196$		0			0
$197$		−	10.1290i		−	0.721661i	−0.932631	−	0.360830i	$-0.882493\pi$
$197$		−	10.1290i		−	0.721661i	0.932631	−	0.360830i	$-0.117507\pi$
$198$		0			0
$199$		−	24.6491i		−	1.74733i	−0.486529	−	0.873665i	$-0.661737\pi$
$199$		−	24.6491i		−	1.74733i	0.486529	−	0.873665i	$-0.338263\pi$
$200$		0			0
$201$	3.16228	−	1.41421i	0.223050	−	0.0997509i
$202$		0			0
$203$	1.64371			0.115366
$204$		0			0
$205$	14.3246			1.00047
$206$		0			0
$207$	11.3137	−	12.6491i	0.786357	−	0.879174i
$208$		0			0
$209$			3.74641i			0.259144i
$210$		0			0
$211$			1.35089i			0.0929991i	0.998918	+	0.0464995i	$0.0148066\pi$
$211$			1.35089i			0.0929991i	−0.998918	+	0.0464995i	$0.985193\pi$
$212$		0			0
$213$	−7.16228	−	16.0153i	−0.490751	−	1.09735i
$214$		0			0
$215$	11.7727			0.802891
$216$		0			0
$217$	−6.32456			−0.429339
$218$		0			0
$219$	3.05792	+	6.83772i	0.206635	+	0.462050i
$220$		0			0
$221$			52.2887i			3.51732i
$222$		0			0
$223$		−	2.32456i		−	0.155664i	−0.996966	−	0.0778319i	$-0.975200\pi$
$223$		−	2.32456i		−	0.155664i	0.996966	−	0.0778319i	$-0.0247997\pi$
$224$		0			0
$225$	−6.00000	+	6.70820i	−0.400000	+	0.447214i
$226$		0			0
$227$	13.1869			0.875246			0.437623	−	0.899159i	$-0.355820\pi$
$227$	13.1869			0.875246			0.437623	+	0.899159i	$0.355820\pi$
$228$		0			0
$229$	−3.81139			−0.251864			−0.125932	−	0.992039i	$-0.540192\pi$
$229$	−3.81139			−0.251864			−0.125932	+	0.992039i	$0.540192\pi$
$230$		0			0
$231$	7.07107	−	3.16228i	0.465242	−	0.208063i
$232$		0			0
$233$			5.65685i			0.370593i	0.982683	+	0.185296i	$0.0593245\pi$
$233$			5.65685i			0.370593i	−0.982683	+	0.185296i	$0.940675\pi$
$234$		0			0
$235$		−	12.6491i		−	0.825137i
$236$		0			0
$237$	−6.32456	+	2.82843i	−0.410824	+	0.183726i
$238$		0			0
$239$	25.9148			1.67629			0.838146	−	0.545446i	$-0.183640\pi$
$239$	25.9148			1.67629			0.838146	+	0.545446i	$0.183640\pi$
$240$		0			0
$241$	14.0000			0.901819			0.450910	−	0.892570i	$-0.351100\pi$
$241$	14.0000			0.901819			0.450910	+	0.892570i	$0.351100\pi$
$242$		0			0
$243$	13.4350	−	7.90569i	0.861858	−	0.507151i
$244$		0			0
$245$			1.41421i			0.0903508i
$246$		0			0
$247$		−	6.00000i		−	0.381771i
$248$		0			0
$249$	5.32456	+	11.9061i	0.337430	+	0.754516i
$250$		0			0
$251$	18.8438			1.18941			0.594704	−	0.803945i	$-0.297269\pi$
$251$	18.8438			1.18941			0.594704	+	0.803945i	$0.297269\pi$
$252$		0			0
$253$	25.2982			1.59049
$254$		0			0
$255$	7.30056	+	16.3246i	0.457179	+	1.02228i
$256$		0			0
$257$		−	16.2448i		−	1.01332i	−0.862144	−	0.506662i	$-0.830879\pi$
$257$		−	16.2448i		−	1.01332i	0.862144	−	0.506662i	$-0.169121\pi$
$258$		0			0
$259$		−	4.32456i		−	0.268715i
$260$		0			0
$261$	3.67544	+	3.28742i	0.227504	+	0.203486i
$262$		0			0
$263$	2.10270			0.129658			0.0648290	−	0.997896i	$-0.479350\pi$
$263$	2.10270			0.129658			0.0648290	+	0.997896i	$0.479350\pi$
$264$		0			0
$265$	−1.67544			−0.102922
$266$		0			0
$267$	−1.87320	+	0.837722i	−0.114638	+	0.0512678i
$268$		0			0
$269$			6.61208i			0.403145i	0.979474	+	0.201573i	$0.0646052\pi$
$269$			6.61208i			0.403145i	−0.979474	+	0.201573i	$0.935395\pi$
$270$		0			0
$271$			1.67544i			0.101776i	0.998704	+	0.0508880i	$0.0162052\pi$
$271$			1.67544i			0.101776i	−0.998704	+	0.0508880i	$0.983795\pi$
$272$		0			0
$273$	−11.3246	+	5.06450i	−0.685393	+	0.306517i
$274$		0			0
$275$	−13.4164			−0.809040
$276$		0			0
$277$	−2.00000			−0.120168			−0.0600842	−	0.998193i	$-0.519137\pi$
$277$	−2.00000			−0.120168			−0.0600842	+	0.998193i	$0.519137\pi$
$278$		0			0
$279$	−14.1421	−	12.6491i	−0.846668	−	0.757282i
$280$		0			0
$281$			23.5454i			1.40460i	0.711881	+	0.702300i	$0.247844\pi$
$281$			23.5454i			1.40460i	−0.711881	+	0.702300i	$0.752156\pi$
$282$		0			0
$283$		−	6.51317i		−	0.387168i	−0.981084	−	0.193584i	$-0.937989\pi$
$283$		−	6.51317i		−	0.387168i	0.981084	−	0.193584i	$-0.0620111\pi$
$284$		0			0
$285$	−0.837722	−	1.87320i	−0.0496224	−	0.110959i
$286$		0			0
$287$	−10.1290			−0.597895
$288$		0			0
$289$	−36.2982			−2.13519
$290$		0			0
$291$	−7.53006	−	16.8377i	−0.441420	−	0.987045i
$292$		0			0
$293$		−	22.1312i		−	1.29292i	−0.762949	−	0.646459i	$-0.776249\pi$
$293$		−	22.1312i		−	1.29292i	0.762949	−	0.646459i	$-0.223751\pi$
$294$		0			0
$295$			2.00000i			0.116445i
$296$		0			0
$297$	22.1359	+	7.07107i	1.28446	+	0.410305i
$298$		0			0
$299$	−40.5160			−2.34310
$300$		0			0
$301$	−8.32456			−0.479819
$302$		0			0
$303$	2.23607	−	1.00000i	0.128459	−	0.0574485i
$304$		0			0
$305$			4.47214i			0.256074i
$306$		0			0
$307$			2.51317i			0.143434i	0.997425	+	0.0717170i	$0.0228478\pi$
$307$			2.51317i			0.143434i	−0.997425	+	0.0717170i	$0.977152\pi$
$308$		0			0
$309$	12.6491	−	5.65685i	0.719583	−	0.321807i
$310$		0			0
$311$	17.4296			0.988339			0.494170	−	0.869365i	$-0.335472\pi$
$311$	17.4296			0.988339			0.494170	+	0.869365i	$0.335472\pi$
$312$		0			0
$313$	−4.97367			−0.281128			−0.140564	−	0.990072i	$-0.544892\pi$
$313$	−4.97367			−0.281128			−0.140564	+	0.990072i	$0.544892\pi$
$314$		0			0
$315$	−2.82843	+	3.16228i	−0.159364	+	0.178174i
$316$		0			0
$317$		−	7.75955i		−	0.435820i	−0.975969	−	0.217910i	$-0.930076\pi$
$317$		−	7.75955i		−	0.435820i	0.975969	−	0.217910i	$-0.0699239\pi$
$318$		0			0
$319$			7.35089i			0.411571i
$320$		0			0
$321$	6.83772	+	15.2896i	0.381644	+	0.853383i
$322$		0			0
$323$	6.11584			0.340295
$324$		0			0
$325$	21.4868			1.19188
$326$		0			0
$327$	3.05792	+	6.83772i	0.169103	+	0.378127i
$328$		0			0
$329$			8.94427i			0.493114i
$330$		0			0
$331$		−	28.9737i		−	1.59254i	−0.604944	−	0.796268i	$-0.706804\pi$
$331$		−	28.9737i		−	1.59254i	0.604944	−	0.796268i	$-0.293196\pi$
$332$		0			0
$333$	8.64911	−	9.67000i	0.473968	−	0.529913i
$334$		0			0
$335$	−2.82843			−0.154533
$336$		0			0
$337$	−7.35089			−0.400428			−0.200214	−	0.979752i	$-0.564164\pi$
$337$	−7.35089			−0.400428			−0.200214	+	0.979752i	$0.564164\pi$
$338$		0			0
$339$	−10.8175	+	4.83772i	−0.587525	+	0.262749i
$340$		0			0
$341$		−	28.2843i		−	1.53168i
$342$		0			0
$343$		−	1.00000i		−	0.0539949i
$344$		0			0
$345$	−12.6491	+	5.65685i	−0.681005	+	0.304555i
$346$		0			0
$347$	21.4427			1.15110			0.575552	−	0.817765i	$-0.304787\pi$
$347$	21.4427			1.15110			0.575552	+	0.817765i	$0.304787\pi$
$348$		0			0
$349$	16.8377			0.901303			0.450651	−	0.892700i	$-0.351192\pi$
$349$	16.8377			0.901303			0.450651	+	0.892700i	$0.351192\pi$
$350$		0			0
$351$	−35.4515	−	11.3246i	−1.89226	−	0.604460i
$352$		0			0
$353$		−	12.9574i		−	0.689654i	−0.938666	−	0.344827i	$-0.887938\pi$
$353$		−	12.9574i		−	0.689654i	0.938666	−	0.344827i	$-0.112062\pi$
$354$		0			0
$355$			14.3246i			0.760268i
$356$		0			0
$357$	−5.16228	−	11.5432i	−0.273217	−	0.610931i
$358$		0			0
$359$	12.2317			0.645564			0.322782	−	0.946473i	$-0.395382\pi$
$359$	12.2317			0.645564			0.322782	+	0.946473i	$0.395382\pi$
$360$		0			0
$361$	18.2982			0.963064
$362$		0			0
$363$	6.36396	+	14.2302i	0.334021	+	0.746894i
$364$		0			0
$365$		−	6.11584i		−	0.320118i
$366$		0			0
$367$		−	18.3246i		−	0.956534i	−0.878214	−	0.478267i	$-0.841265\pi$
$367$		−	18.3246i		−	0.956534i	0.878214	−	0.478267i	$-0.158735\pi$
$368$		0			0
$369$	−22.6491	−	20.2580i	−1.17907	−	1.05459i
$370$		0			0
$371$	1.18472			0.0615075
$372$		0			0
$373$	−30.6491			−1.58695			−0.793475	−	0.608602i	$-0.791730\pi$
$373$	−30.6491			−1.58695			−0.793475	+	0.608602i	$0.791730\pi$
$374$		0			0
$375$	17.8885	−	8.00000i	0.923760	−	0.413118i
$376$		0			0
$377$		−	11.7727i		−	0.606325i
$378$		0			0
$379$		−	24.3246i		−	1.24947i	−0.780837	−	0.624734i	$-0.785207\pi$
$379$		−	24.3246i		−	1.24947i	0.780837	−	0.624734i	$-0.214793\pi$
$380$		0			0
$381$	9.48683	−	4.24264i	0.486025	−	0.217357i
$382$		0			0
$383$	22.6274			1.15621			0.578103	−	0.815963i	$-0.303793\pi$
$383$	22.6274			1.15621			0.578103	+	0.815963i	$0.303793\pi$
$384$		0			0
$385$	−6.32456			−0.322329
$386$		0			0
$387$	−18.6143	−	16.6491i	−0.946217	−	0.846322i
$388$		0			0
$389$			6.38258i			0.323610i	0.986823	+	0.161805i	$0.0517315\pi$
$389$			6.38258i			0.323610i	−0.986823	+	0.161805i	$0.948268\pi$
$390$		0			0
$391$		−	41.2982i		−	2.08854i
$392$		0			0
$393$	3.32456	+	7.43393i	0.167702	+	0.374992i
$394$		0			0
$395$	5.65685			0.284627
$396$		0			0
$397$	−23.1623			−1.16248			−0.581241	−	0.813732i	$-0.697433\pi$
$397$	−23.1623			−1.16248			−0.581241	+	0.813732i	$0.697433\pi$
$398$		0			0
$399$	0.592359	+	1.32456i	0.0296550	+	0.0663107i
$400$		0			0
$401$		−	21.4427i		−	1.07080i	−0.844599	−	0.535399i	$-0.820161\pi$
$401$		−	21.4427i		−	1.07080i	0.844599	−	0.535399i	$-0.179839\pi$
$402$		0			0
$403$			45.2982i			2.25647i
$404$		0			0
$405$	−12.6491	+	1.41421i	−0.628539	+	0.0702728i
$406$		0			0
$407$	19.3400			0.958648
$408$		0			0
$409$	−11.6754			−0.577314			−0.288657	−	0.957433i	$-0.593209\pi$
$409$	−11.6754			−0.577314			−0.288657	+	0.957433i	$0.593209\pi$
$410$		0			0
$411$	28.2843	−	12.6491i	1.39516	−	0.623935i
$412$		0			0
$413$		−	1.41421i		−	0.0695889i
$414$		0			0
$415$		−	10.6491i		−	0.522744i
$416$		0			0
$417$	−11.3246	+	5.06450i	−0.554566	+	0.248009i
$418$		0			0
$419$	−27.7880			−1.35753			−0.678767	−	0.734353i	$-0.737485\pi$
$419$	−27.7880			−1.35753			−0.678767	+	0.734353i	$0.737485\pi$
$420$		0			0
$421$	22.6491			1.10385			0.551925	−	0.833894i	$-0.313893\pi$
$421$	22.6491			1.10385			0.551925	+	0.833894i	$0.313893\pi$
$422$		0			0
$423$	−17.8885	+	20.0000i	−0.869771	+	0.972433i
$424$		0			0
$425$			21.9017i			1.06239i
$426$		0			0
$427$		−	3.16228i		−	0.153033i
$428$		0			0
$429$	−22.6491	−	50.6450i	−1.09351	−	2.44516i
$430$		0			0
$431$	14.6011			0.703311			0.351656	−	0.936129i	$-0.385619\pi$
$431$	14.6011			0.703311			0.351656	+	0.936129i	$0.385619\pi$
$432$		0			0
$433$	6.00000			0.288342			0.144171	−	0.989553i	$-0.453949\pi$
$433$	6.00000			0.288342			0.144171	+	0.989553i	$0.453949\pi$
$434$		0			0
$435$	−1.64371	−	3.67544i	−0.0788098	−	0.176224i
$436$		0			0
$437$			4.73887i			0.226691i
$438$		0			0
$439$			0.649111i			0.0309804i	0.999880	+	0.0154902i	$0.00493087\pi$
$439$			0.649111i			0.0309804i	−0.999880	+	0.0154902i	$0.995069\pi$
$440$		0			0
$441$	2.00000	−	2.23607i	0.0952381	−	0.106479i
$442$		0			0
$443$	−4.01315			−0.190670			−0.0953351	−	0.995445i	$-0.530392\pi$
$443$	−4.01315			−0.190670			−0.0953351	+	0.995445i	$0.530392\pi$
$444$		0			0
$445$	1.67544			0.0794237
$446$		0			0
$447$	−7.07107	+	3.16228i	−0.334450	+	0.149571i
$448$		0			0
$449$			16.9706i			0.800890i	0.916321	+	0.400445i	$0.131145\pi$
$449$			16.9706i			0.800890i	−0.916321	+	0.400445i	$0.868855\pi$
$450$		0			0
$451$		−	45.2982i		−	2.13301i
$452$		0			0
$453$	−15.8114	+	7.07107i	−0.742884	+	0.332228i
$454$		0			0
$455$	10.1290			0.474854
$456$		0			0
$457$	24.6491			1.15304			0.576518	−	0.817084i	$-0.304411\pi$
$457$	24.6491			1.15304			0.576518	+	0.817084i	$0.304411\pi$
$458$		0			0
$459$	11.5432	−	36.1359i	0.538791	−	1.68668i
$460$		0			0
$461$			32.9859i			1.53631i	0.640266	+	0.768153i	$0.278824\pi$
$461$			32.9859i			1.53631i	−0.640266	+	0.768153i	$0.721176\pi$
$462$		0			0
$463$			40.6491i			1.88912i	0.328333	+	0.944562i	$0.393513\pi$
$463$			40.6491i			1.88912i	−0.328333	+	0.944562i	$0.606487\pi$
$464$		0			0
$465$	6.32456	+	14.1421i	0.293294	+	0.655826i
$466$		0			0
$467$	−27.3290			−1.26464			−0.632319	−	0.774708i	$-0.717897\pi$
$467$	−27.3290			−1.26464			−0.632319	+	0.774708i	$0.717897\pi$
$468$		0			0
$469$	2.00000			0.0923514
$470$		0			0
$471$	9.53663	+	21.3246i	0.439425	+	0.982584i
$472$		0			0
$473$		−	37.2285i		−	1.71177i
$474$		0			0
$475$		−	2.51317i		−	0.115312i
$476$		0			0
$477$	2.64911	+	2.36944i	0.121294	+	0.108489i
$478$		0			0
$479$	5.65685			0.258468			0.129234	−	0.991614i	$-0.458748\pi$
$479$	5.65685			0.258468			0.129234	+	0.991614i	$0.458748\pi$
$480$		0			0
$481$	−30.9737			−1.41228
$482$		0			0
$483$	8.94427	−	4.00000i	0.406978	−	0.182006i
$484$		0			0
$485$			15.0601i			0.683845i
$486$		0			0
$487$			23.2982i			1.05574i	0.849324	+	0.527872i	$0.177010\pi$
$487$			23.2982i			1.05574i	−0.849324	+	0.527872i	$0.822990\pi$
$488$		0			0
$489$	−6.83772	+	3.05792i	−0.309212	+	0.138284i
$490$		0			0
$491$	−29.9280			−1.35063			−0.675315	−	0.737529i	$-0.735992\pi$
$491$	−29.9280			−1.35063			−0.675315	+	0.737529i	$0.735992\pi$
$492$		0			0
$493$	12.0000			0.540453
$494$		0			0
$495$	−14.1421	−	12.6491i	−0.635642	−	0.568535i
$496$		0			0
$497$		−	10.1290i		−	0.454347i
$498$		0			0
$499$			2.00000i			0.0895323i	0.998997	+	0.0447661i	$0.0142543\pi$
$499$			2.00000i			0.0895323i	−0.998997	+	0.0447661i	$0.985746\pi$
$500$		0			0
$501$	−0.324555	−	0.725728i	−0.0145001	−	0.0324231i
$502$		0			0
$503$	−11.7727			−0.524919			−0.262459	−	0.964943i	$-0.584534\pi$
$503$	−11.7727			−0.524919			−0.262459	+	0.964943i	$0.584534\pi$
$504$		0			0
$505$	−2.00000			−0.0889988
$506$		0			0
$507$	27.0809	+	60.5548i	1.20271	+	2.68933i
$508$		0			0
$509$		−	4.70163i		−	0.208396i	−0.994557	−	0.104198i	$-0.966772\pi$
$509$		−	4.70163i		−	0.208396i	0.994557	−	0.104198i	$-0.0332276\pi$
$510$		0			0
$511$			4.32456i			0.191307i
$512$		0			0
$513$	−1.32456	+	4.14651i	−0.0584805	+	0.183073i
$514$		0			0
$515$	−11.3137			−0.498542
$516$		0			0
$517$	−40.0000			−1.75920
$518$		0			0
$519$	5.98248	−	2.67544i	0.262602	−	0.117439i
$520$		0			0
$521$		−	2.10270i		−	0.0921209i	−0.998939	−	0.0460605i	$-0.985333\pi$
$521$		−	2.10270i		−	0.0921209i	0.998939	−	0.0460605i	$-0.0146667\pi$
$522$		0			0
$523$			39.8114i			1.74083i	0.492318	+	0.870415i	$0.336150\pi$
$523$			39.8114i			1.74083i	−0.492318	+	0.870415i	$0.663850\pi$
$524$		0			0
$525$	−4.74342	+	2.12132i	−0.207020	+	0.0925820i
$526$		0			0
$527$	−46.1728			−2.01132
$528$		0			0
$529$	9.00000			0.391304
$530$		0			0
$531$	2.82843	−	3.16228i	0.122743	−	0.137231i
$532$		0			0
$533$			72.5466i			3.14234i
$534$		0			0
$535$		−	13.6754i		−	0.591241i
$536$		0			0
$537$	−13.8114	−	30.8832i	−0.596005	−	1.33271i
$538$		0			0
$539$	4.47214			0.192629
$540$		0			0
$541$	2.00000			0.0859867			0.0429934	−	0.999075i	$-0.486311\pi$
$541$	2.00000			0.0859867			0.0429934	+	0.999075i	$0.486311\pi$
$542$		0			0
$543$	−12.8240	−	28.6754i	−0.550333	−	1.23058i
$544$		0			0
$545$		−	6.11584i		−	0.261974i
$546$		0			0
$547$			36.3246i			1.55313i	0.630040	+	0.776563i	$0.283039\pi$
$547$			36.3246i			1.55313i	−0.630040	+	0.776563i	$0.716961\pi$
$548$		0			0
$549$	6.32456	−	7.07107i	0.269925	−	0.301786i
$550$		0			0
$551$	−1.37697			−0.0586610
$552$		0			0
$553$	−4.00000			−0.170097
$554$		0			0
$555$	−9.67000	+	4.32456i	−0.410469	+	0.183567i
$556$		0			0
$557$		−	32.7564i		−	1.38793i	−0.720007	−	0.693967i	$-0.755861\pi$
$557$		−	32.7564i		−	1.38793i	0.720007	−	0.693967i	$-0.244139\pi$
$558$		0			0
$559$			59.6228i			2.52177i
$560$		0			0
$561$	51.6228	−	23.0864i	2.17952	−	0.974709i
$562$		0			0
$563$	22.1312			0.932718			0.466359	−	0.884595i	$-0.345565\pi$
$563$	22.1312			0.932718			0.466359	+	0.884595i	$0.345565\pi$
$564$		0			0
$565$	9.67544			0.407049
$566$		0			0
$567$	8.94427	−	1.00000i	0.375624	−	0.0419961i
$568$		0			0
$569$		−	29.4690i		−	1.23540i	−0.786412	−	0.617702i	$-0.788064\pi$
$569$		−	29.4690i		−	1.23540i	0.786412	−	0.617702i	$-0.211936\pi$
$570$		0			0
$571$		−	12.3246i		−	0.515767i	−0.966176	−	0.257883i	$-0.916975\pi$
$571$		−	12.3246i		−	0.515767i	0.966176	−	0.257883i	$-0.0830250\pi$
$572$		0			0
$573$	8.83772	+	19.7617i	0.369201	+	0.825559i
$574$		0			0
$575$	−16.9706			−0.707721
$576$		0			0
$577$	35.2982			1.46948			0.734742	−	0.678347i	$-0.237303\pi$
$577$	35.2982			1.46948			0.734742	+	0.678347i	$0.237303\pi$
$578$		0			0
$579$	11.3137	+	25.2982i	0.470182	+	1.05136i
$580$		0			0
$581$			7.53006i			0.312399i
$582$		0			0
$583$			5.29822i			0.219430i
$584$		0			0
$585$	22.6491	+	20.2580i	0.936425	+	0.837564i
$586$		0			0
$587$	−9.44050			−0.389651			−0.194826	−	0.980838i	$-0.562414\pi$
$587$	−9.44050			−0.389651			−0.194826	+	0.980838i	$0.562414\pi$
$588$		0			0
$589$	5.29822			0.218309
$590$		0			0
$591$	16.0153	−	7.16228i	0.658783	−	0.294617i
$592$		0			0
$593$			24.2711i			0.996696i	0.866977	+	0.498348i	$0.166060\pi$
$593$			24.2711i			0.996696i	−0.866977	+	0.498348i	$0.833940\pi$
$594$		0			0
$595$			10.3246i			0.423266i
$596$		0			0
$597$	38.9737	−	17.4296i	1.59509	−	0.713344i
$598$		0			0
$599$	−24.7301			−1.01045			−0.505223	−	0.862989i	$-0.668590\pi$
$599$	−24.7301			−1.01045			−0.505223	+	0.862989i	$0.668590\pi$
$600$		0			0
$601$	−29.6228			−1.20834			−0.604169	−	0.796856i	$-0.706495\pi$
$601$	−29.6228			−1.20834			−0.604169	+	0.796856i	$0.706495\pi$
$602$		0			0
$603$	4.47214	+	4.00000i	0.182119	+	0.162893i
$604$		0			0
$605$		−	12.7279i		−	0.517464i
$606$		0			0
$607$			17.6754i			0.717424i	0.933448	+	0.358712i	$0.116784\pi$
$607$			17.6754i			0.717424i	−0.933448	+	0.358712i	$0.883216\pi$
$608$		0			0
$609$	1.16228	+	2.59893i	0.0470979	+	0.105314i
$610$		0			0
$611$	64.0614			2.59165
$612$		0			0
$613$	−19.6754			−0.794684			−0.397342	−	0.917671i	$-0.630067\pi$
$613$	−19.6754			−0.794684			−0.397342	+	0.917671i	$0.630067\pi$
$614$		0			0
$615$	10.1290	+	22.6491i	0.408440	+	0.913300i
$616$		0			0
$617$			27.0996i			1.09099i	0.838115	+	0.545493i	$0.183658\pi$
$617$			27.0996i			1.09099i	−0.838115	+	0.545493i	$0.816342\pi$
$618$		0			0
$619$		−	8.18861i		−	0.329128i	−0.986366	−	0.164564i	$-0.947378\pi$
$619$		−	8.18861i		−	0.329128i	0.986366	−	0.164564i	$-0.0526217\pi$
$620$		0			0
$621$	28.0000	+	8.94427i	1.12360	+	0.358921i
$622$		0			0
$623$	−1.18472			−0.0474647
$624$		0			0
$625$	−1.00000			−0.0400000
$626$		0			0
$627$	−5.92359	+	2.64911i	−0.236565	+	0.105795i
$628$		0			0
$629$		−	31.5717i		−	1.25885i
$630$		0			0
$631$		−	37.9473i		−	1.51066i	−0.655345	−	0.755330i	$-0.727477\pi$
$631$		−	37.9473i		−	1.51066i	0.655345	−	0.755330i	$-0.272523\pi$
$632$		0			0
$633$	−2.13594	+	0.955223i	−0.0848961	+	0.0379667i
$634$		0			0
$635$	−8.48528			−0.336728
$636$		0			0
$637$	−7.16228			−0.283780
$638$		0			0
$639$	20.2580	−	22.6491i	0.801393	−	0.895985i
$640$		0			0
$641$		−	12.4984i		−	0.493658i	−0.969059	−	0.246829i	$-0.920611\pi$
$641$		−	12.4984i		−	0.493658i	0.969059	−	0.246829i	$-0.0793886\pi$
$642$		0			0
$643$			4.83772i			0.190781i	0.995440	+	0.0953906i	$0.0304100\pi$
$643$			4.83772i			0.190781i	−0.995440	+	0.0953906i	$0.969590\pi$
$644$		0			0
$645$	8.32456	+	18.6143i	0.327779	+	0.732936i
$646$		0			0
$647$	−13.2242			−0.519895			−0.259948	−	0.965623i	$-0.583705\pi$
$647$	−13.2242			−0.519895			−0.259948	+	0.965623i	$0.583705\pi$
$648$		0			0
$649$	6.32456			0.248261
$650$		0			0
$651$	−4.47214	−	10.0000i	−0.175277	−	0.391931i
$652$		0			0
$653$			47.8165i			1.87121i	0.353055	+	0.935603i	$0.385143\pi$
$653$			47.8165i			1.87121i	−0.353055	+	0.935603i	$0.614857\pi$
$654$		0			0
$655$		−	6.64911i		−	0.259802i
$656$		0			0
$657$	−8.64911	+	9.67000i	−0.337434	+	0.377263i
$658$		0			0
$659$	−16.7038			−0.650689			−0.325344	−	0.945596i	$-0.605480\pi$
$659$	−16.7038			−0.650689			−0.325344	+	0.945596i	$0.605480\pi$
$660$		0			0
$661$	−22.5132			−0.875661			−0.437830	−	0.899058i	$-0.644253\pi$
$661$	−22.5132			−0.875661			−0.437830	+	0.899058i	$0.644253\pi$
$662$		0			0
$663$	−82.6756	+	36.9737i	−3.21086	+	1.43594i
$664$		0			0
$665$		−	1.18472i		−	0.0459414i
$666$		0			0
$667$			9.29822i			0.360029i
$668$		0			0
$669$	3.67544	−	1.64371i	0.142101	−	0.0635495i
$670$		0			0
$671$	14.1421			0.545951
$672$		0			0
$673$	27.2982			1.05227			0.526135	−	0.850401i	$-0.323641\pi$
$673$	27.2982			1.05227			0.526135	+	0.850401i	$0.323641\pi$
$674$		0			0
$675$	−14.8492	−	4.74342i	−0.571548	−	0.182574i
$676$		0			0
$677$			8.44804i			0.324685i	0.986735	+	0.162342i	$0.0519049\pi$
$677$			8.44804i			0.324685i	−0.986735	+	0.162342i	$0.948095\pi$
$678$		0			0
$679$		−	10.6491i		−	0.408675i
$680$		0			0
$681$	9.32456	+	20.8503i	0.357318	+	0.798987i
$682$		0			0
$683$	51.1039			1.95544			0.977719	−	0.209918i	$-0.0673197\pi$
$683$	51.1039			1.95544			0.977719	+	0.209918i	$0.0673197\pi$
$684$		0			0
$685$	−25.2982			−0.966595
$686$		0			0
$687$	−2.69506	−	6.02633i	−0.102823	−	0.229919i
$688$		0			0
$689$		−	8.48528i		−	0.323263i
$690$		0			0
$691$			15.1623i			0.576800i	0.957510	+	0.288400i	$0.0931233\pi$
$691$			15.1623i			0.576800i	−0.957510	+	0.288400i	$0.906877\pi$
$692$		0			0
$693$	10.0000	+	8.94427i	0.379869	+	0.339765i
$694$		0			0
$695$	10.1290			0.384214
$696$		0			0
$697$	−73.9473			−2.80095
$698$		0			0
$699$	−8.94427	+	4.00000i	−0.338303	+	0.151294i
$700$		0			0
$701$		−	26.6406i		−	1.00620i	−0.864228	−	0.503100i	$-0.832193\pi$
$701$		−	26.6406i		−	1.00620i	0.864228	−	0.503100i	$-0.167807\pi$
$702$		0			0
$703$			3.62278i			0.136636i
$704$		0			0
$705$	20.0000	−	8.94427i	0.753244	−	0.336861i
$706$		0			0
$707$	1.41421			0.0531870
$708$		0			0
$709$	3.02633			0.113656			0.0568282	−	0.998384i	$-0.481901\pi$
$709$	3.02633			0.113656			0.0568282	+	0.998384i	$0.481901\pi$
$710$		0			0
$711$	−8.94427	−	8.00000i	−0.335436	−	0.300023i
$712$		0			0
$713$		−	35.7771i		−	1.33986i
$714$		0			0
$715$			45.2982i			1.69406i
$716$		0			0
$717$	18.3246	+	40.9750i	0.684343	+	1.53024i
$718$		0			0
$719$	−12.2317			−0.456165			−0.228083	−	0.973642i	$-0.573246\pi$
$719$	−12.2317			−0.456165			−0.228083	+	0.973642i	$0.573246\pi$
$720$		0			0
$721$	8.00000			0.297936
$722$		0			0
$723$	9.89949	+	22.1359i	0.368166	+	0.823245i
$724$		0			0
$725$		−	4.93113i		−	0.183137i
$726$		0			0
$727$		−	16.6491i		−	0.617481i	−0.951146	−	0.308741i	$-0.900092\pi$
$727$		−	16.6491i		−	0.617481i	0.951146	−	0.308741i	$-0.0999075\pi$
$728$		0			0
$729$	22.0000	+	15.6525i	0.814815	+	0.579721i
$730$		0			0
$731$	−60.7739			−2.24781
$732$		0			0
$733$	−46.1359			−1.70407			−0.852035	−	0.523485i	$-0.824632\pi$
$733$	−46.1359			−1.70407			−0.852035	+	0.523485i	$0.824632\pi$
$734$		0			0
$735$	−2.23607	+	1.00000i	−0.0824786	+	0.0368856i
$736$		0			0
$737$			8.94427i			0.329466i
$738$		0			0
$739$			32.9737i			1.21296i	0.795100	+	0.606478i	$0.207418\pi$
$739$			32.9737i			1.21296i	−0.795100	+	0.606478i	$0.792582\pi$
$740$		0			0
$741$	9.48683	−	4.24264i	0.348508	−	0.155857i
$742$		0			0
$743$	−23.5454			−0.863797			−0.431898	−	0.901922i	$-0.642156\pi$
$743$	−23.5454			−0.863797			−0.431898	+	0.901922i	$0.642156\pi$
$744$		0			0
$745$	6.32456			0.231714
$746$		0			0
$747$	−15.0601	+	16.8377i	−0.551021	+	0.616060i
$748$		0			0
$749$			9.67000i			0.353334i
$750$		0			0
$751$		−	17.3509i		−	0.633143i	−0.948569	−	0.316571i	$-0.897468\pi$
$751$		−	17.3509i		−	0.633143i	0.948569	−	0.316571i	$-0.102532\pi$
$752$		0			0
$753$	13.3246	+	29.7946i	0.485574	+	1.08578i
$754$		0			0
$755$	14.1421			0.514685
$756$		0			0
$757$	20.3246			0.738709			0.369354	−	0.929289i	$-0.379579\pi$
$757$	20.3246			0.738709			0.369354	+	0.929289i	$0.379579\pi$
$758$		0			0
$759$	17.8885	+	40.0000i	0.649313	+	1.45191i
$760$		0			0
$761$		−	4.47214i		−	0.162115i	−0.996709	−	0.0810574i	$-0.974170\pi$
$761$		−	4.47214i		−	0.162115i	0.996709	−	0.0810574i	$-0.0258297\pi$
$762$		0			0
$763$			4.32456i			0.156559i
$764$		0			0
$765$	−20.6491	+	23.0864i	−0.746570	+	0.834691i
$766$		0			0
$767$	−10.1290			−0.365737
$768$		0			0
$769$	−27.2982			−0.984399			−0.492200	−	0.870482i	$-0.663807\pi$
$769$	−27.2982			−0.984399			−0.492200	+	0.870482i	$0.663807\pi$
$770$		0			0
$771$	25.6853	−	11.4868i	0.925035	−	0.413688i
$772$		0			0
$773$			45.6766i			1.64287i	0.570300	+	0.821436i	$0.306827\pi$
$773$			45.6766i			1.64287i	−0.570300	+	0.821436i	$0.693173\pi$
$774$		0			0
$775$			18.9737i			0.681554i
$776$		0			0
$777$	6.83772	−	3.05792i	0.245302	−	0.109702i
$778$		0			0
$779$	8.48528			0.304017
$780$		0			0
$781$	45.2982			1.62090
$782$		0			0
$783$	−2.59893	+	8.13594i	−0.0928782	+	0.290755i
$784$		0			0
$785$		−	19.0733i		−	0.680754i
$786$		0			0
$787$		−	39.1623i		−	1.39598i	−0.716105	−	0.697992i	$-0.754077\pi$
$787$		−	39.1623i		−	1.39598i	0.716105	−	0.697992i	$-0.245923\pi$
$788$		0			0
$789$	1.48683	+	3.32466i	0.0529327	+	0.118361i
$790$		0			0
$791$	−6.84157			−0.243258
$792$		0			0
$793$	−22.6491			−0.804294
$794$		0			0
$795$	−1.18472	−	2.64911i	−0.0420176	−	0.0939543i
$796$		0			0
$797$		−	17.9258i		−	0.634964i	−0.948264	−	0.317482i	$-0.897163\pi$
$797$		−	17.9258i		−	0.634964i	0.948264	−	0.317482i	$-0.102837\pi$
$798$		0			0
$799$			65.2982i			2.31009i
$800$		0			0
$801$	−2.64911	−	2.36944i	−0.0936017	−	0.0837199i
$802$		0			0
$803$	−19.3400			−0.682494
$804$		0			0
$805$	−8.00000			−0.281963
$806$		0			0
$807$	−10.4546	+	4.67544i	−0.368020	+	0.164583i
$808$		0			0
$809$			44.9881i			1.58170i	0.612012	+	0.790849i	$0.290360\pi$
$809$			44.9881i			1.58170i	−0.612012	+	0.790849i	$0.709640\pi$
$810$		0			0
$811$			33.4868i			1.17588i	0.808904	+	0.587941i	$0.200061\pi$
$811$			33.4868i			1.17588i	−0.808904	+	0.587941i	$0.799939\pi$
$812$		0			0
$813$	−2.64911	+	1.18472i	−0.0929084	+	0.0415499i
$814$		0			0
$815$	6.11584			0.214229
$816$		0			0
$817$	6.97367			0.243978
$818$		0			0
$819$	−16.0153	−	14.3246i	−0.559621	−	0.500540i
$820$		0			0
$821$		−	22.3607i		−	0.780393i	−0.920732	−	0.390197i	$-0.872407\pi$
$821$		−	22.3607i		−	0.780393i	0.920732	−	0.390197i	$-0.127593\pi$
$822$		0			0
$823$			21.9473i			0.765036i	0.923948	+	0.382518i	$0.124943\pi$
$823$			21.9473i			0.765036i	−0.923948	+	0.382518i	$0.875057\pi$
$824$		0			0
$825$	−9.48683	−	21.2132i	−0.330289	−	0.738549i
$826$		0			0
$827$	−4.93113			−0.171472			−0.0857360	−	0.996318i	$-0.527324\pi$
$827$	−4.93113			−0.171472			−0.0857360	+	0.996318i	$0.527324\pi$
$828$		0			0
$829$	−35.1623			−1.22124			−0.610618	−	0.791925i	$-0.709079\pi$
$829$	−35.1623			−1.22124			−0.610618	+	0.791925i	$0.709079\pi$
$830$		0			0
$831$	−1.41421	−	3.16228i	−0.0490585	−	0.109698i
$832$		0			0
$833$		−	7.30056i		−	0.252950i
$834$		0			0
$835$			0.649111i			0.0224634i
$836$		0			0
$837$	10.0000	−	31.3050i	0.345651	−	1.08206i
$838$		0			0
$839$	49.0012			1.69171			0.845855	−	0.533412i	$-0.179091\pi$
$839$	49.0012			1.69171			0.845855	+	0.533412i	$0.179091\pi$
$840$		0			0
$841$	26.2982			0.906835
$842$		0			0
$843$	−37.2285	+	16.6491i	−1.28222	+	0.573426i
$844$		0			0
$845$		−	54.1619i		−	1.86322i
$846$		0			0
$847$			9.00000i			0.309244i
$848$		0			0
$849$	10.2982	−	4.60550i	0.353434	−	0.158061i
$850$		0			0
$851$	24.4634			0.838594
$852$		0			0
$853$	26.1359			0.894878			0.447439	−	0.894315i	$-0.352336\pi$
$853$	26.1359			0.894878			0.447439	+	0.894315i	$0.352336\pi$
$854$		0			0
$855$	2.36944	−	2.64911i	0.0810330	−	0.0905977i
$856$		0			0
$857$		−	19.0733i		−	0.651530i	−0.945451	−	0.325765i	$-0.894378\pi$
$857$		−	19.0733i		−	0.651530i	0.945451	−	0.325765i	$-0.105622\pi$
$858$		0			0
$859$		−	48.4605i		−	1.65345i	−0.562606	−	0.826725i	$-0.690201\pi$
$859$		−	48.4605i		−	1.65345i	0.562606	−	0.826725i	$-0.309799\pi$
$860$		0			0
$861$	−7.16228	−	16.0153i	−0.244090	−	0.545801i
$862$		0			0
$863$	11.0470			0.376043			0.188022	−	0.982165i	$-0.439792\pi$
$863$	11.0470			0.376043			0.188022	+	0.982165i	$0.439792\pi$
$864$		0			0
$865$	−5.35089			−0.181936
$866$		0			0
$867$	−25.6667	−	57.3925i	−0.871687	−	1.94915i
$868$		0			0
$869$		−	17.8885i		−	0.606827i
$870$		0			0
$871$		−	14.3246i		−	0.485369i
$872$		0			0
$873$	21.2982	−	23.8121i	0.720836	−	0.805919i
$874$		0			0
$875$	11.3137			0.382473
$876$		0			0
$877$	−40.9737			−1.38358			−0.691791	−	0.722098i	$-0.743178\pi$
$877$	−40.9737			−1.38358			−0.691791	+	0.722098i	$0.743178\pi$
$878$		0			0
$879$	34.9925	−	15.6491i	1.18027	−	0.527831i
$880$		0			0
$881$			24.2711i			0.817715i	0.912598	+	0.408858i	$0.134073\pi$
$881$			24.2711i			0.817715i	−0.912598	+	0.408858i	$0.865927\pi$
$882$		0			0
$883$		−	18.0000i		−	0.605748i	−0.953031	−	0.302874i	$-0.902054\pi$
$883$		−	18.0000i		−	0.605748i	0.953031	−	0.302874i	$-0.0979462\pi$
$884$		0			0
$885$	−3.16228	+	1.41421i	−0.106299	+	0.0475383i
$886$		0			0
$887$	−16.5116			−0.554404			−0.277202	−	0.960812i	$-0.589407\pi$
$887$	−16.5116			−0.554404			−0.277202	+	0.960812i	$0.589407\pi$
$888$		0			0
$889$	6.00000			0.201234
$890$		0			0
$891$	4.47214	+	40.0000i	0.149822	+	1.34005i
$892$		0			0
$893$		−	7.49282i		−	0.250738i
$894$		0			0
$895$			27.6228i			0.923328i
$896$		0			0
$897$	−28.6491	−	64.0614i	−0.956566	−	2.13895i
$898$		0			0
$899$	10.3957			0.346717
$900$		0			0
$901$	8.64911			0.288144
$902$		0			0
$903$	−5.88635	−	13.1623i	−0.195885	−	0.438013i
$904$		0			0
$905$			25.6481i			0.852572i
$906$		0			0
$907$		−	27.9473i		−	0.927976i	−0.885842	−	0.463988i	$-0.846418\pi$
$907$		−	27.9473i		−	0.927976i	0.885842	−	0.463988i	$-0.153582\pi$
$908$		0			0
$909$	3.16228	+	2.82843i	0.104886	+	0.0938130i
$910$		0			0
$911$	−13.6831			−0.453343			−0.226671	−	0.973971i	$-0.572784\pi$
$911$	−13.6831			−0.453343			−0.226671	+	0.973971i	$0.572784\pi$
$912$		0			0
$913$	−33.6754			−1.11449
$914$		0			0
$915$	−7.07107	+	3.16228i	−0.233762	+	0.104542i
$916$		0			0
$917$			4.70163i			0.155262i
$918$		0			0
$919$		−	20.6491i		−	0.681151i	−0.940217	−	0.340576i	$-0.889378\pi$
$919$		−	20.6491i		−	0.681151i	0.940217	−	0.340576i	$-0.110622\pi$
$920$		0			0
$921$	−3.97367	+	1.77708i	−0.130937	+	0.0585567i
$922$		0			0
$923$	−72.5466			−2.38790
$924$		0			0
$925$	−12.9737			−0.426572
$926$		0			0
$927$	17.8885	+	16.0000i	0.587537	+	0.525509i
$928$		0			0
$929$			43.0777i			1.41333i	0.707547	+	0.706666i	$0.249802\pi$
$929$			43.0777i			1.41333i	−0.707547	+	0.706666i	$0.750198\pi$
$930$		0			0
$931$			0.837722i			0.0274552i
$932$		0			0
$933$	12.3246	+	27.5585i	0.403488	+	0.902226i
$934$		0			0
$935$	−46.1728			−1.51001
$936$		0			0
$937$	−7.67544			−0.250746			−0.125373	−	0.992110i	$-0.540013\pi$
$937$	−7.67544			−0.250746			−0.125373	+	0.992110i	$0.540013\pi$
$938$		0			0
$939$	−3.51691	−	7.86406i	−0.114770	−	0.256634i
$940$		0			0
$941$			23.0492i			0.751381i	0.926745	+	0.375691i	$0.122594\pi$
$941$			23.0492i			0.751381i	−0.926745	+	0.375691i	$0.877406\pi$
$942$		0			0
$943$		−	57.2982i		−	1.86589i
$944$		0			0
$945$	−7.00000	−	2.23607i	−0.227710	−	0.0727393i
$946$		0			0
$947$	−4.47214			−0.145325			−0.0726624	−	0.997357i	$-0.523150\pi$
$947$	−4.47214			−0.145325			−0.0726624	+	0.997357i	$0.523150\pi$
$948$		0			0
$949$	30.9737			1.00545
$950$		0			0
$951$	12.2689	−	5.48683i	0.397847	−	0.177923i
$952$		0			0
$953$		−	26.8328i		−	0.869200i	−0.900624	−	0.434600i	$-0.856890\pi$
$953$		−	26.8328i		−	0.869200i	0.900624	−	0.434600i	$-0.143110\pi$
$954$		0			0
$955$		−	17.6754i		−	0.571964i
$956$		0			0
$957$	−11.6228	+	5.19786i	−0.375711	+	0.168023i
$958$		0			0
$959$	17.8885			0.577651
$960$		0			0
$961$	−9.00000			−0.290323
$962$		0			0
$963$	−19.3400	+	21.6228i	−0.623223	+	0.696784i
$964$		0			0
$965$		−	22.6274i		−	0.728402i
$966$		0			0
$967$			34.6491i			1.11424i	0.830432	+	0.557120i	$0.188094\pi$
$967$			34.6491i			1.11424i	−0.830432	+	0.557120i	$0.811906\pi$
$968$		0			0
$969$	4.32456	+	9.67000i	0.138925	+	0.310645i
$970$		0			0
$971$	43.8406			1.40691			0.703456	−	0.710739i	$-0.251639\pi$
$971$	43.8406			1.40691			0.703456	+	0.710739i	$0.251639\pi$
$972$		0			0
$973$	−7.16228			−0.229612
$974$		0			0
$975$	15.1935	+	33.9737i	0.486581	+	1.08803i
$976$		0			0
$977$			49.4602i			1.58237i	0.611575	+	0.791187i	$0.290536\pi$
$977$			49.4602i			1.58237i	−0.611575	+	0.791187i	$0.709464\pi$
$978$		0			0
$979$		−	5.29822i		−	0.169332i
$980$		0			0
$981$	−8.64911	+	9.67000i	−0.276145	+	0.308739i
$982$		0			0
$983$	16.5116			0.526637			0.263319	−	0.964709i	$-0.415183\pi$
$983$	16.5116			0.526637			0.263319	+	0.964709i	$0.415183\pi$
$984$		0			0
$985$	−14.3246			−0.456418
$986$		0			0
$987$	−14.1421	+	6.32456i	−0.450149	+	0.201313i
$988$		0			0
$989$		−	47.0908i		−	1.49740i
$990$		0			0
$991$		−	46.5964i		−	1.48018i	−0.672505	−	0.740092i	$-0.734782\pi$
$991$		−	46.5964i		−	1.48018i	0.672505	−	0.740092i	$-0.265218\pi$
$992$		0			0
$993$	45.8114	−	20.4875i	1.45378	−	0.650150i
$994$		0			0
$995$	−34.8591			−1.10511
$996$		0			0
$997$	25.4868			0.807176			0.403588	−	0.914941i	$-0.367763\pi$
$997$	25.4868			0.807176			0.403588	+	0.914941i	$0.367763\pi$
$998$		0			0
$999$	21.4055	+	6.83772i	0.677239	+	0.216336i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1344.2.h.f.575.7		8
3.2	odd	2	inner	1344.2.h.f.575.4		8
4.3	odd	2	inner	1344.2.h.f.575.1		8
8.3	odd	2		672.2.h.e.575.8	yes	8
8.5	even	2		672.2.h.e.575.2	✓	8
12.11	even	2	inner	1344.2.h.f.575.6		8
24.5	odd	2		672.2.h.e.575.5	yes	8
24.11	even	2		672.2.h.e.575.3	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
672.2.h.e.575.2	✓	8	8.5	even	2
672.2.h.e.575.3	yes	8	24.11	even	2
672.2.h.e.575.5	yes	8	24.5	odd	2
672.2.h.e.575.8	yes	8	8.3	odd	2
1344.2.h.f.575.1		8	4.3	odd	2	inner
1344.2.h.f.575.4		8	3.2	odd	2	inner
1344.2.h.f.575.6		8	12.11	even	2	inner
1344.2.h.f.575.7		8	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:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1344.h (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+(0.707107 + 1.58114i) q^{3} -1.41421i q^{5} +1.00000i q^{7} +(-2.00000 + 2.23607i) q^{9} +O(q^{10})\)	\(q+(0.707107 + 1.58114i) q^{3} -1.41421i q^{5} +1.00000i q^{7} +(-2.00000 + 2.23607i) q^{9} -4.47214 q^{11} +7.16228 q^{13} +(2.23607 - 1.00000i) q^{15} +7.30056i q^{17} -0.837722i q^{19} +(-1.58114 + 0.707107i) q^{21} -5.65685 q^{23} +3.00000 q^{25} +(-4.94975 - 1.58114i) q^{27} -1.64371i q^{29} +6.32456i q^{31} +(-3.16228 - 7.07107i) q^{33} +1.41421 q^{35} -4.32456 q^{37} +(5.06450 + 11.3246i) q^{39} +10.1290i q^{41} +8.32456i q^{43} +(3.16228 + 2.82843i) q^{45} +8.94427 q^{47} -1.00000 q^{49} +(-11.5432 + 5.16228i) q^{51} -1.18472i q^{53} +6.32456i q^{55} +(1.32456 - 0.592359i) q^{57} -1.41421 q^{59} -3.16228 q^{61} +(-2.23607 - 2.00000i) q^{63} -10.1290i q^{65} -2.00000i q^{67} +(-4.00000 - 8.94427i) q^{69} -10.1290 q^{71} +4.32456 q^{73} +(2.12132 + 4.74342i) q^{75} -4.47214i q^{77} +4.00000i q^{79} +(-1.00000 - 8.94427i) q^{81} +7.53006 q^{83} +10.3246 q^{85} +(2.59893 - 1.16228i) q^{87} +1.18472i q^{89} +7.16228i q^{91} +(-10.0000 + 4.47214i) q^{93} -1.18472 q^{95} -10.6491 q^{97} +(8.94427 - 10.0000i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 16 q^{9}+O(q^{10}) \) 8 * q - 16 * q^9	\( 8 q - 16 q^{9} + 32 q^{13} + 24 q^{25} + 16 q^{37} - 8 q^{49} - 40 q^{57} - 32 q^{69} - 16 q^{73} - 8 q^{81} + 32 q^{85} - 80 q^{93} + 16 q^{97}+O(q^{100}) \) 8 * q - 16 * q^9 + 32 * q^13 + 24 * q^25 + 16 * q^37 - 8 * q^49 - 40 * q^57 - 32 * q^69 - 16 * q^73 - 8 * q^81 + 32 * q^85 - 80 * q^93 + 16 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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