LMFDB - Embedded newform 672.2.h.c.575.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [672,2,Mod(575,672)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(672, 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("672.575");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 672 = 2^{5} \cdot 3 \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	672.h (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:	$5.36594701583$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{8})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 1 $ x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			575.3
Root			$-0.707107 + 0.707107i$ of defining polynomial
Character	$\chi$	$=$	672.575
Dual form			672.2.h.c.575.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.41421 - 1.00000i) q^{3} +1.41421i q^{5} +1.00000i q^{7} +(1.00000 - 2.82843i) q^{9} +O(q^{10})$	$q+(1.41421 - 1.00000i) q^{3} +1.41421i q^{5} +1.00000i q^{7} +(1.00000 - 2.82843i) q^{9} -1.41421 q^{11} +6.00000 q^{13} +(1.41421 + 2.00000i) q^{15} +1.41421i q^{17} -4.00000i q^{19} +(1.00000 + 1.41421i) q^{21} +7.07107 q^{23} +3.00000 q^{25} +(-1.41421 - 5.00000i) q^{27} +8.48528i q^{29} -2.00000i q^{31} +(-2.00000 + 1.41421i) q^{33} -1.41421 q^{35} -4.00000 q^{37} +(8.48528 - 6.00000i) q^{39} +4.24264i q^{41} +(4.00000 + 1.41421i) q^{45} -2.82843 q^{47} -1.00000 q^{49} +(1.41421 + 2.00000i) q^{51} -5.65685i q^{53} -2.00000i q^{55} +(-4.00000 - 5.65685i) q^{57} -11.3137 q^{59} -10.0000 q^{61} +(2.82843 + 1.00000i) q^{63} +8.48528i q^{65} -16.0000i q^{67} +(10.0000 - 7.07107i) q^{69} -4.24264 q^{71} +14.0000 q^{73} +(4.24264 - 3.00000i) q^{75} -1.41421i q^{77} +4.00000i q^{79} +(-7.00000 - 5.65685i) q^{81} -8.48528 q^{83} -2.00000 q^{85} +(8.48528 + 12.0000i) q^{87} +7.07107i q^{89} +6.00000i q^{91} +(-2.00000 - 2.82843i) q^{93} +5.65685 q^{95} +6.00000 q^{97} +(-1.41421 + 4.00000i) q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{9}+O(q^{10}) $ 4 * q + 4 * q^9	$ 4 q + 4 q^{9} + 24 q^{13} + 4 q^{21} + 12 q^{25} - 8 q^{33} - 16 q^{37} + 16 q^{45} - 4 q^{49} - 16 q^{57} - 40 q^{61} + 40 q^{69} + 56 q^{73} - 28 q^{81} - 8 q^{85} - 8 q^{93} + 24 q^{97}+O(q^{100}) $ 4 * q + 4 * q^9 + 24 * q^13 + 4 * q^21 + 12 * q^25 - 8 * q^33 - 16 * q^37 + 16 * q^45 - 4 * q^49 - 16 * q^57 - 40 * q^61 + 40 * q^69 + 56 * q^73 - 28 * q^81 - 8 * q^85 - 8 * q^93 + 24 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/672\mathbb{Z}\right)^\times$.

$n$	$127$	$421$	$449$	$577$
$\chi(n)$	$-1$	$1$	$-1$	$1$

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
$3$	1.41421	−	1.00000i	0.816497	−	0.577350i
$3$	1.41421	−	1.00000i	0.816497	−	0.577350i
$4$		0			0
$5$			1.41421i			0.632456i	0.948683	+	0.316228i	$0.102416\pi$
$5$			1.41421i			0.632456i	−0.948683	+	0.316228i	$0.897584\pi$
$6$		0			0
$7$			1.00000i			0.377964i
$7$			1.00000i			0.377964i
$8$		0			0
$9$	1.00000	−	2.82843i	0.333333	−	0.942809i
$10$		0			0
$11$	−1.41421			−0.426401			−0.213201	−	0.977008i	$-0.568389\pi$
$11$	−1.41421			−0.426401			−0.213201	+	0.977008i	$0.568389\pi$
$12$		0			0
$13$	6.00000			1.66410			0.832050	−	0.554700i	$-0.187167\pi$
$13$	6.00000			1.66410			0.832050	+	0.554700i	$0.187167\pi$
$14$		0			0
$15$	1.41421	+	2.00000i	0.365148	+	0.516398i
$16$		0			0
$17$			1.41421i			0.342997i	0.985184	+	0.171499i	$0.0548609\pi$
$17$			1.41421i			0.342997i	−0.985184	+	0.171499i	$0.945139\pi$
$18$		0			0
$19$		−	4.00000i		−	0.917663i	−0.888523	−	0.458831i	$-0.848268\pi$
$19$		−	4.00000i		−	0.917663i	0.888523	−	0.458831i	$-0.151732\pi$
$20$		0			0
$21$	1.00000	+	1.41421i	0.218218	+	0.308607i
$22$		0			0
$23$	7.07107			1.47442			0.737210	−	0.675664i	$-0.236143\pi$
$23$	7.07107			1.47442			0.737210	+	0.675664i	$0.236143\pi$
$24$		0			0
$25$	3.00000			0.600000
$26$		0			0
$27$	−1.41421	−	5.00000i	−0.272166	−	0.962250i
$28$		0			0
$29$			8.48528i			1.57568i	0.615882	+	0.787839i	$0.288800\pi$
$29$			8.48528i			1.57568i	−0.615882	+	0.787839i	$0.711200\pi$
$30$		0			0
$31$		−	2.00000i		−	0.359211i	−0.983739	−	0.179605i	$-0.942518\pi$
$31$		−	2.00000i		−	0.359211i	0.983739	−	0.179605i	$-0.0574821\pi$
$32$		0			0
$33$	−2.00000	+	1.41421i	−0.348155	+	0.246183i
$34$		0			0
$35$	−1.41421			−0.239046
$36$		0			0
$37$	−4.00000			−0.657596			−0.328798	−	0.944400i	$-0.606644\pi$
$37$	−4.00000			−0.657596			−0.328798	+	0.944400i	$0.606644\pi$
$38$		0			0
$39$	8.48528	−	6.00000i	1.35873	−	0.960769i
$40$		0			0
$41$			4.24264i			0.662589i	0.943527	+	0.331295i	$0.107485\pi$
$41$			4.24264i			0.662589i	−0.943527	+	0.331295i	$0.892515\pi$
$42$		0			0
$43$		0			0		1.00000			$0$
$43$		0			0		−1.00000			$\pi$
$44$		0			0
$45$	4.00000	+	1.41421i	0.596285	+	0.210819i
$46$		0			0
$47$	−2.82843			−0.412568			−0.206284	−	0.978492i	$-0.566137\pi$
$47$	−2.82843			−0.412568			−0.206284	+	0.978492i	$0.566137\pi$
$48$		0			0
$49$	−1.00000			−0.142857
$50$		0			0
$51$	1.41421	+	2.00000i	0.198030	+	0.280056i
$52$		0			0
$53$		−	5.65685i		−	0.777029i	−0.921443	−	0.388514i	$-0.872988\pi$
$53$		−	5.65685i		−	0.777029i	0.921443	−	0.388514i	$-0.127012\pi$
$54$		0			0
$55$		−	2.00000i		−	0.269680i
$56$		0			0
$57$	−4.00000	−	5.65685i	−0.529813	−	0.749269i
$58$		0			0
$59$	−11.3137			−1.47292			−0.736460	−	0.676481i	$-0.763504\pi$
$59$	−11.3137			−1.47292			−0.736460	+	0.676481i	$0.763504\pi$
$60$		0			0
$61$	−10.0000			−1.28037			−0.640184	−	0.768221i	$-0.721142\pi$
$61$	−10.0000			−1.28037			−0.640184	+	0.768221i	$0.721142\pi$
$62$		0			0
$63$	2.82843	+	1.00000i	0.356348	+	0.125988i
$64$		0			0
$65$			8.48528i			1.05247i
$66$		0			0
$67$		−	16.0000i		−	1.95471i	−0.211604	−	0.977356i	$-0.567869\pi$
$67$		−	16.0000i		−	1.95471i	0.211604	−	0.977356i	$-0.432131\pi$
$68$		0			0
$69$	10.0000	−	7.07107i	1.20386	−	0.851257i
$70$		0			0
$71$	−4.24264			−0.503509			−0.251754	−	0.967791i	$-0.581008\pi$
$71$	−4.24264			−0.503509			−0.251754	+	0.967791i	$0.581008\pi$
$72$		0			0
$73$	14.0000			1.63858			0.819288	−	0.573382i	$-0.194369\pi$
$73$	14.0000			1.63858			0.819288	+	0.573382i	$0.194369\pi$
$74$		0			0
$75$	4.24264	−	3.00000i	0.489898	−	0.346410i
$76$		0			0
$77$		−	1.41421i		−	0.161165i
$78$		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
$81$	−7.00000	−	5.65685i	−0.777778	−	0.628539i
$82$		0			0
$83$	−8.48528			−0.931381			−0.465690	−	0.884948i	$-0.654194\pi$
$83$	−8.48528			−0.931381			−0.465690	+	0.884948i	$0.654194\pi$
$84$		0			0
$85$	−2.00000			−0.216930
$86$		0			0
$87$	8.48528	+	12.0000i	0.909718	+	1.28654i
$88$		0			0
$89$			7.07107i			0.749532i	0.927119	+	0.374766i	$0.122277\pi$
$89$			7.07107i			0.749532i	−0.927119	+	0.374766i	$0.877723\pi$
$90$		0			0
$91$			6.00000i			0.628971i
$92$		0			0
$93$	−2.00000	−	2.82843i	−0.207390	−	0.293294i
$94$		0			0
$95$	5.65685			0.580381
$96$		0			0
$97$	6.00000			0.609208			0.304604	−	0.952479i	$-0.401476\pi$
$97$	6.00000			0.609208			0.304604	+	0.952479i	$0.401476\pi$
$98$		0			0
$99$	−1.41421	+	4.00000i	−0.142134	+	0.402015i
$100$		0			0
$101$			1.41421i			0.140720i	0.997522	+	0.0703598i	$0.0224147\pi$
$101$			1.41421i			0.140720i	−0.997522	+	0.0703598i	$0.977585\pi$
$102$		0			0
$103$			10.0000i			0.985329i	0.870219	+	0.492665i	$0.163977\pi$
$103$			10.0000i			0.985329i	−0.870219	+	0.492665i	$0.836023\pi$
$104$		0			0
$105$	−2.00000	+	1.41421i	−0.195180	+	0.138013i
$106$		0			0
$107$	−15.5563			−1.50389			−0.751945	−	0.659226i	$-0.770884\pi$
$107$	−15.5563			−1.50389			−0.751945	+	0.659226i	$0.770884\pi$
$108$		0			0
$109$	−14.0000			−1.34096			−0.670478	−	0.741929i	$-0.733911\pi$
$109$	−14.0000			−1.34096			−0.670478	+	0.741929i	$0.733911\pi$
$110$		0			0
$111$	−5.65685	+	4.00000i	−0.536925	+	0.379663i
$112$		0			0
$113$		0			0		1.00000			$0$
$113$		0			0		−1.00000			$\pi$
$114$		0			0
$115$			10.0000i			0.932505i
$116$		0			0
$117$	6.00000	−	16.9706i	0.554700	−	1.56893i
$118$		0			0
$119$	−1.41421			−0.129641
$120$		0			0
$121$	−9.00000			−0.818182
$122$		0			0
$123$	4.24264	+	6.00000i	0.382546	+	0.541002i
$124$		0			0
$125$			11.3137i			1.01193i
$126$		0			0
$127$			12.0000i			1.06483i	0.846484	+	0.532414i	$0.178715\pi$
$127$			12.0000i			1.06483i	−0.846484	+	0.532414i	$0.821285\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	19.7990			1.72985			0.864923	−	0.501905i	$-0.167367\pi$
$131$	19.7990			1.72985			0.864923	+	0.501905i	$0.167367\pi$
$132$		0			0
$133$	4.00000			0.346844
$134$		0			0
$135$	7.07107	−	2.00000i	0.608581	−	0.172133i
$136$		0			0
$137$		−	19.7990i		−	1.69154i	−0.533546	−	0.845771i	$-0.679141\pi$
$137$		−	19.7990i		−	1.69154i	0.533546	−	0.845771i	$-0.320859\pi$
$138$		0			0
$139$			6.00000i			0.508913i	0.967084	+	0.254457i	$0.0818966\pi$
$139$			6.00000i			0.508913i	−0.967084	+	0.254457i	$0.918103\pi$
$140$		0			0
$141$	−4.00000	+	2.82843i	−0.336861	+	0.238197i
$142$		0			0
$143$	−8.48528			−0.709575
$144$		0			0
$145$	−12.0000			−0.996546
$146$		0			0
$147$	−1.41421	+	1.00000i	−0.116642	+	0.0824786i
$148$		0			0
$149$		−	11.3137i		−	0.926855i	−0.886135	−	0.463428i	$-0.846619\pi$
$149$		−	11.3137i		−	0.926855i	0.886135	−	0.463428i	$-0.153381\pi$
$150$		0			0
$151$		−	8.00000i		−	0.651031i	−0.945537	−	0.325515i	$-0.894462\pi$
$151$		−	8.00000i		−	0.651031i	0.945537	−	0.325515i	$-0.105538\pi$
$152$		0			0
$153$	4.00000	+	1.41421i	0.323381	+	0.114332i
$154$		0			0
$155$	2.82843			0.227185
$156$		0			0
$157$	−10.0000			−0.798087			−0.399043	−	0.916932i	$-0.630658\pi$
$157$	−10.0000			−0.798087			−0.399043	+	0.916932i	$0.630658\pi$
$158$		0			0
$159$	−5.65685	−	8.00000i	−0.448618	−	0.634441i
$160$		0			0
$161$			7.07107i			0.557278i
$162$		0			0
$163$			4.00000i			0.313304i	0.987654	+	0.156652i	$0.0500701\pi$
$163$			4.00000i			0.313304i	−0.987654	+	0.156652i	$0.949930\pi$
$164$		0			0
$165$	−2.00000	−	2.82843i	−0.155700	−	0.220193i
$166$		0			0
$167$	−14.1421			−1.09435			−0.547176	−	0.837018i	$-0.684297\pi$
$167$	−14.1421			−1.09435			−0.547176	+	0.837018i	$0.684297\pi$
$168$		0			0
$169$	23.0000			1.76923
$170$		0			0
$171$	−11.3137	−	4.00000i	−0.865181	−	0.305888i
$172$		0			0
$173$		−	9.89949i		−	0.752645i	−0.926489	−	0.376322i	$-0.877189\pi$
$173$		−	9.89949i		−	0.752645i	0.926489	−	0.376322i	$-0.122811\pi$
$174$		0			0
$175$			3.00000i			0.226779i
$176$		0			0
$177$	−16.0000	+	11.3137i	−1.20263	+	0.850390i
$178$		0			0
$179$	−9.89949			−0.739923			−0.369961	−	0.929047i	$-0.620629\pi$
$179$	−9.89949			−0.739923			−0.369961	+	0.929047i	$0.620629\pi$
$180$		0			0
$181$	−2.00000			−0.148659			−0.0743294	−	0.997234i	$-0.523682\pi$
$181$	−2.00000			−0.148659			−0.0743294	+	0.997234i	$0.523682\pi$
$182$		0			0
$183$	−14.1421	+	10.0000i	−1.04542	+	0.739221i
$184$		0			0
$185$		−	5.65685i		−	0.415900i
$186$		0			0
$187$		−	2.00000i		−	0.146254i
$188$		0			0
$189$	5.00000	−	1.41421i	0.363696	−	0.102869i
$190$		0			0
$191$	−7.07107			−0.511645			−0.255822	−	0.966724i	$-0.582346\pi$
$191$	−7.07107			−0.511645			−0.255822	+	0.966724i	$0.582346\pi$
$192$		0			0
$193$	−20.0000			−1.43963			−0.719816	−	0.694165i	$-0.755774\pi$
$193$	−20.0000			−1.43963			−0.719816	+	0.694165i	$0.755774\pi$
$194$		0			0
$195$	8.48528	+	12.0000i	0.607644	+	0.859338i
$196$		0			0
$197$			16.9706i			1.20910i	0.796566	+	0.604551i	$0.206648\pi$
$197$			16.9706i			1.20910i	−0.796566	+	0.604551i	$0.793352\pi$
$198$		0			0
$199$		−	8.00000i		−	0.567105i	−0.958957	−	0.283552i	$-0.908487\pi$
$199$		−	8.00000i		−	0.567105i	0.958957	−	0.283552i	$-0.0915130\pi$
$200$		0			0
$201$	−16.0000	−	22.6274i	−1.12855	−	1.59601i
$202$		0			0
$203$	−8.48528			−0.595550
$204$		0			0
$205$	−6.00000			−0.419058
$206$		0			0
$207$	7.07107	−	20.0000i	0.491473	−	1.39010i
$208$		0			0
$209$			5.65685i			0.391293i
$210$		0			0
$211$		0			0		1.00000			$0$
$211$		0			0		−1.00000			$\pi$
$212$		0			0
$213$	−6.00000	+	4.24264i	−0.411113	+	0.290701i
$214$		0			0
$215$		0			0
$216$		0			0
$217$	2.00000			0.135769
$218$		0			0
$219$	19.7990	−	14.0000i	1.33789	−	0.946032i
$220$		0			0
$221$			8.48528i			0.570782i
$222$		0			0
$223$			24.0000i			1.60716i	0.595198	+	0.803579i	$0.297074\pi$
$223$			24.0000i			1.60716i	−0.595198	+	0.803579i	$0.702926\pi$
$224$		0			0
$225$	3.00000	−	8.48528i	0.200000	−	0.565685i
$226$		0			0
$227$	11.3137			0.750917			0.375459	−	0.926839i	$-0.377485\pi$
$227$	11.3137			0.750917			0.375459	+	0.926839i	$0.377485\pi$
$228$		0			0
$229$	10.0000			0.660819			0.330409	−	0.943838i	$-0.392813\pi$
$229$	10.0000			0.660819			0.330409	+	0.943838i	$0.392813\pi$
$230$		0			0
$231$	−1.41421	−	2.00000i	−0.0930484	−	0.131590i
$232$		0			0
$233$		−	19.7990i		−	1.29707i	−0.761183	−	0.648537i	$-0.775381\pi$
$233$		−	19.7990i		−	1.29707i	0.761183	−	0.648537i	$-0.224619\pi$
$234$		0			0
$235$		−	4.00000i		−	0.260931i
$236$		0			0
$237$	4.00000	+	5.65685i	0.259828	+	0.367452i
$238$		0			0
$239$	1.41421			0.0914779			0.0457389	−	0.998953i	$-0.485436\pi$
$239$	1.41421			0.0914779			0.0457389	+	0.998953i	$0.485436\pi$
$240$		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
$243$	−15.5563	−	1.00000i	−0.997940	−	0.0641500i
$244$		0			0
$245$		−	1.41421i		−	0.0903508i
$246$		0			0
$247$		−	24.0000i		−	1.52708i
$248$		0			0
$249$	−12.0000	+	8.48528i	−0.760469	+	0.537733i
$250$		0			0
$251$	5.65685			0.357057			0.178529	−	0.983935i	$-0.442866\pi$
$251$	5.65685			0.357057			0.178529	+	0.983935i	$0.442866\pi$
$252$		0			0
$253$	−10.0000			−0.628695
$254$		0			0
$255$	−2.82843	+	2.00000i	−0.177123	+	0.125245i
$256$		0			0
$257$			26.8701i			1.67611i	0.545587	+	0.838054i	$0.316307\pi$
$257$			26.8701i			1.67611i	−0.545587	+	0.838054i	$0.683693\pi$
$258$		0			0
$259$		−	4.00000i		−	0.248548i
$260$		0			0
$261$	24.0000	+	8.48528i	1.48556	+	0.525226i
$262$		0			0
$263$	9.89949			0.610429			0.305215	−	0.952284i	$-0.401272\pi$
$263$	9.89949			0.610429			0.305215	+	0.952284i	$0.401272\pi$
$264$		0			0
$265$	8.00000			0.491436
$266$		0			0
$267$	7.07107	+	10.0000i	0.432742	+	0.611990i
$268$		0			0
$269$			32.5269i			1.98320i	0.129339	+	0.991600i	$0.458714\pi$
$269$			32.5269i			1.98320i	−0.129339	+	0.991600i	$0.541286\pi$
$270$		0			0
$271$		−	26.0000i		−	1.57939i	−0.613501	−	0.789694i	$-0.710239\pi$
$271$		−	26.0000i		−	1.57939i	0.613501	−	0.789694i	$-0.289761\pi$
$272$		0			0
$273$	6.00000	+	8.48528i	0.363137	+	0.513553i
$274$		0			0
$275$	−4.24264			−0.255841
$276$		0			0
$277$	−16.0000			−0.961347			−0.480673	−	0.876900i	$-0.659608\pi$
$277$	−16.0000			−0.961347			−0.480673	+	0.876900i	$0.659608\pi$
$278$		0			0
$279$	−5.65685	−	2.00000i	−0.338667	−	0.119737i
$280$		0			0
$281$		−	25.4558i		−	1.51857i	−0.650759	−	0.759284i	$-0.725549\pi$
$281$		−	25.4558i		−	1.51857i	0.650759	−	0.759284i	$-0.274451\pi$
$282$		0			0
$283$			28.0000i			1.66443i	0.554455	+	0.832214i	$0.312927\pi$
$283$			28.0000i			1.66443i	−0.554455	+	0.832214i	$0.687073\pi$
$284$		0			0
$285$	8.00000	−	5.65685i	0.473879	−	0.335083i
$286$		0			0
$287$	−4.24264			−0.250435
$288$		0			0
$289$	15.0000			0.882353
$290$		0			0
$291$	8.48528	−	6.00000i	0.497416	−	0.351726i
$292$		0			0
$293$		−	26.8701i		−	1.56977i	−0.619644	−	0.784883i	$-0.712723\pi$
$293$		−	26.8701i		−	1.56977i	0.619644	−	0.784883i	$-0.287277\pi$
$294$		0			0
$295$		−	16.0000i		−	0.931556i
$296$		0			0
$297$	2.00000	+	7.07107i	0.116052	+	0.410305i
$298$		0			0
$299$	42.4264			2.45358
$300$		0			0
$301$		0			0
$302$		0			0
$303$	1.41421	+	2.00000i	0.0812444	+	0.114897i
$304$		0			0
$305$		−	14.1421i		−	0.809776i
$306$		0			0
$307$			12.0000i			0.684876i	0.939540	+	0.342438i	$0.111253\pi$
$307$			12.0000i			0.684876i	−0.939540	+	0.342438i	$0.888747\pi$
$308$		0			0
$309$	10.0000	+	14.1421i	0.568880	+	0.804518i
$310$		0			0
$311$	−19.7990			−1.12270			−0.561349	−	0.827579i	$-0.689717\pi$
$311$	−19.7990			−1.12270			−0.561349	+	0.827579i	$0.689717\pi$
$312$		0			0
$313$	2.00000			0.113047			0.0565233	−	0.998401i	$-0.481998\pi$
$313$	2.00000			0.113047			0.0565233	+	0.998401i	$0.481998\pi$
$314$		0			0
$315$	−1.41421	+	4.00000i	−0.0796819	+	0.225374i
$316$		0			0
$317$		−	22.6274i		−	1.27088i	−0.772149	−	0.635441i	$-0.780818\pi$
$317$		−	22.6274i		−	1.27088i	0.772149	−	0.635441i	$-0.219182\pi$
$318$		0			0
$319$		−	12.0000i		−	0.671871i
$320$		0			0
$321$	−22.0000	+	15.5563i	−1.22792	+	0.868271i
$322$		0			0
$323$	5.65685			0.314756
$324$		0			0
$325$	18.0000			0.998460
$326$		0			0
$327$	−19.7990	+	14.0000i	−1.09489	+	0.774202i
$328$		0			0
$329$		−	2.82843i		−	0.155936i
$330$		0			0
$331$			4.00000i			0.219860i	0.993939	+	0.109930i	$0.0350627\pi$
$331$			4.00000i			0.219860i	−0.993939	+	0.109930i	$0.964937\pi$
$332$		0			0
$333$	−4.00000	+	11.3137i	−0.219199	+	0.619987i
$334$		0			0
$335$	22.6274			1.23627
$336$		0			0
$337$	30.0000			1.63420			0.817102	−	0.576493i	$-0.195579\pi$
$337$	30.0000			1.63420			0.817102	+	0.576493i	$0.195579\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$			2.82843i			0.153168i
$342$		0			0
$343$		−	1.00000i		−	0.0539949i
$344$		0			0
$345$	10.0000	+	14.1421i	0.538382	+	0.761387i
$346$		0			0
$347$	35.3553			1.89797			0.948987	−	0.315315i	$-0.102110\pi$
$347$	35.3553			1.89797			0.948987	+	0.315315i	$0.102110\pi$
$348$		0			0
$349$	6.00000			0.321173			0.160586	−	0.987022i	$-0.448662\pi$
$349$	6.00000			0.321173			0.160586	+	0.987022i	$0.448662\pi$
$350$		0			0
$351$	−8.48528	−	30.0000i	−0.452911	−	1.60128i
$352$		0			0
$353$		−	7.07107i		−	0.376355i	−0.982135	−	0.188177i	$-0.939742\pi$
$353$		−	7.07107i		−	0.376355i	0.982135	−	0.188177i	$-0.0602580\pi$
$354$		0			0
$355$		−	6.00000i		−	0.318447i
$356$		0			0
$357$	−2.00000	+	1.41421i	−0.105851	+	0.0748481i
$358$		0			0
$359$	1.41421			0.0746393			0.0373197	−	0.999303i	$-0.488118\pi$
$359$	1.41421			0.0746393			0.0373197	+	0.999303i	$0.488118\pi$
$360$		0			0
$361$	3.00000			0.157895
$362$		0			0
$363$	−12.7279	+	9.00000i	−0.668043	+	0.472377i
$364$		0			0
$365$			19.7990i			1.03633i
$366$		0			0
$367$			8.00000i			0.417597i	0.977959	+	0.208798i	$0.0669552\pi$
$367$			8.00000i			0.417597i	−0.977959	+	0.208798i	$0.933045\pi$
$368$		0			0
$369$	12.0000	+	4.24264i	0.624695	+	0.220863i
$370$		0			0
$371$	5.65685			0.293689
$372$		0			0
$373$	14.0000			0.724893			0.362446	−	0.932005i	$-0.381942\pi$
$373$	14.0000			0.724893			0.362446	+	0.932005i	$0.381942\pi$
$374$		0			0
$375$	11.3137	+	16.0000i	0.584237	+	0.826236i
$376$		0			0
$377$			50.9117i			2.62209i
$378$		0			0
$379$		−	20.0000i		−	1.02733i	−0.857991	−	0.513665i	$-0.828287\pi$
$379$		−	20.0000i		−	1.02733i	0.857991	−	0.513665i	$-0.171713\pi$
$380$		0			0
$381$	12.0000	+	16.9706i	0.614779	+	0.869428i
$382$		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
$385$	2.00000			0.101929
$386$		0			0
$387$		0			0
$388$		0			0
$389$		−	36.7696i		−	1.86429i	−0.362085	−	0.932145i	$-0.617935\pi$
$389$		−	36.7696i		−	1.86429i	0.362085	−	0.932145i	$-0.382065\pi$
$390$		0			0
$391$			10.0000i			0.505722i
$392$		0			0
$393$	28.0000	−	19.7990i	1.41241	−	0.998727i
$394$		0			0
$395$	−5.65685			−0.284627
$396$		0			0
$397$	10.0000			0.501886			0.250943	−	0.968002i	$-0.419259\pi$
$397$	10.0000			0.501886			0.250943	+	0.968002i	$0.419259\pi$
$398$		0			0
$399$	5.65685	−	4.00000i	0.283197	−	0.200250i
$400$		0			0
$401$		−	2.82843i		−	0.141245i	−0.997503	−	0.0706225i	$-0.977501\pi$
$401$		−	2.82843i		−	0.141245i	0.997503	−	0.0706225i	$-0.0224986\pi$
$402$		0			0
$403$		−	12.0000i		−	0.597763i
$404$		0			0
$405$	8.00000	−	9.89949i	0.397523	−	0.491910i
$406$		0			0
$407$	5.65685			0.280400
$408$		0			0
$409$	−2.00000			−0.0988936			−0.0494468	−	0.998777i	$-0.515746\pi$
$409$	−2.00000			−0.0988936			−0.0494468	+	0.998777i	$0.515746\pi$
$410$		0			0
$411$	−19.7990	−	28.0000i	−0.976612	−	1.38114i
$412$		0			0
$413$		−	11.3137i		−	0.556711i
$414$		0			0
$415$		−	12.0000i		−	0.589057i
$416$		0			0
$417$	6.00000	+	8.48528i	0.293821	+	0.415526i
$418$		0			0
$419$	16.9706			0.829066			0.414533	−	0.910034i	$-0.363945\pi$
$419$	16.9706			0.829066			0.414533	+	0.910034i	$0.363945\pi$
$420$		0			0
$421$	−24.0000			−1.16969			−0.584844	−	0.811146i	$-0.698844\pi$
$421$	−24.0000			−1.16969			−0.584844	+	0.811146i	$0.698844\pi$
$422$		0			0
$423$	−2.82843	+	8.00000i	−0.137523	+	0.388973i
$424$		0			0
$425$			4.24264i			0.205798i
$426$		0			0
$427$		−	10.0000i		−	0.483934i
$428$		0			0
$429$	−12.0000	+	8.48528i	−0.579365	+	0.409673i
$430$		0			0
$431$	−35.3553			−1.70301			−0.851503	−	0.524349i	$-0.824309\pi$
$431$	−35.3553			−1.70301			−0.851503	+	0.524349i	$0.824309\pi$
$432$		0			0
$433$	−30.0000			−1.44171			−0.720854	−	0.693087i	$-0.756250\pi$
$433$	−30.0000			−1.44171			−0.720854	+	0.693087i	$0.756250\pi$
$434$		0			0
$435$	−16.9706	+	12.0000i	−0.813676	+	0.575356i
$436$		0			0
$437$		−	28.2843i		−	1.35302i
$438$		0			0
$439$		−	16.0000i		−	0.763638i	−0.924237	−	0.381819i	$-0.875298\pi$
$439$		−	16.0000i		−	0.763638i	0.924237	−	0.381819i	$-0.124702\pi$
$440$		0			0
$441$	−1.00000	+	2.82843i	−0.0476190	+	0.134687i
$442$		0			0
$443$	9.89949			0.470339			0.235170	−	0.971954i	$-0.424435\pi$
$443$	9.89949			0.470339			0.235170	+	0.971954i	$0.424435\pi$
$444$		0			0
$445$	−10.0000			−0.474045
$446$		0			0
$447$	−11.3137	−	16.0000i	−0.535120	−	0.756774i
$448$		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
$451$		−	6.00000i		−	0.282529i
$452$		0			0
$453$	−8.00000	−	11.3137i	−0.375873	−	0.531564i
$454$		0			0
$455$	−8.48528			−0.397796
$456$		0			0
$457$	−10.0000			−0.467780			−0.233890	−	0.972263i	$-0.575146\pi$
$457$	−10.0000			−0.467780			−0.233890	+	0.972263i	$0.575146\pi$
$458$		0			0
$459$	7.07107	−	2.00000i	0.330049	−	0.0933520i
$460$		0			0
$461$			4.24264i			0.197599i	0.995107	+	0.0987997i	$0.0315003\pi$
$461$			4.24264i			0.197599i	−0.995107	+	0.0987997i	$0.968500\pi$
$462$		0			0
$463$			24.0000i			1.11537i	0.830051	+	0.557687i	$0.188311\pi$
$463$			24.0000i			1.11537i	−0.830051	+	0.557687i	$0.811689\pi$
$464$		0			0
$465$	4.00000	−	2.82843i	0.185496	−	0.131165i
$466$		0			0
$467$	−22.6274			−1.04707			−0.523536	−	0.852004i	$-0.675387\pi$
$467$	−22.6274			−1.04707			−0.523536	+	0.852004i	$0.675387\pi$
$468$		0			0
$469$	16.0000			0.738811
$470$		0			0
$471$	−14.1421	+	10.0000i	−0.651635	+	0.460776i
$472$		0			0
$473$		0			0
$474$		0			0
$475$		−	12.0000i		−	0.550598i
$476$		0			0
$477$	−16.0000	−	5.65685i	−0.732590	−	0.259010i
$478$		0			0
$479$	31.1127			1.42158			0.710788	−	0.703407i	$-0.248339\pi$
$479$	31.1127			1.42158			0.710788	+	0.703407i	$0.248339\pi$
$480$		0			0
$481$	−24.0000			−1.09431
$482$		0			0
$483$	7.07107	+	10.0000i	0.321745	+	0.455016i
$484$		0			0
$485$			8.48528i			0.385297i
$486$		0			0
$487$			8.00000i			0.362515i	0.983436	+	0.181257i	$0.0580167\pi$
$487$			8.00000i			0.362515i	−0.983436	+	0.181257i	$0.941983\pi$
$488$		0			0
$489$	4.00000	+	5.65685i	0.180886	+	0.255812i
$490$		0			0
$491$	−1.41421			−0.0638226			−0.0319113	−	0.999491i	$-0.510159\pi$
$491$	−1.41421			−0.0638226			−0.0319113	+	0.999491i	$0.510159\pi$
$492$		0			0
$493$	−12.0000			−0.540453
$494$		0			0
$495$	−5.65685	−	2.00000i	−0.254257	−	0.0898933i
$496$		0			0
$497$		−	4.24264i		−	0.190308i
$498$		0			0
$499$		−	20.0000i		−	0.895323i	−0.894203	−	0.447661i	$-0.852257\pi$
$499$		−	20.0000i		−	0.895323i	0.894203	−	0.447661i	$-0.147743\pi$
$500$		0			0
$501$	−20.0000	+	14.1421i	−0.893534	+	0.631824i
$502$		0			0
$503$		0			0			−	1.00000i	$-0.5\pi$
$503$		0			0				1.00000i	$0.5\pi$
$504$		0			0
$505$	−2.00000			−0.0889988
$506$		0			0
$507$	32.5269	−	23.0000i	1.44457	−	1.02147i
$508$		0			0
$509$		−	7.07107i		−	0.313420i	−0.987645	−	0.156710i	$-0.949911\pi$
$509$		−	7.07107i		−	0.313420i	0.987645	−	0.156710i	$-0.0500887\pi$
$510$		0			0
$511$			14.0000i			0.619324i
$512$		0			0
$513$	−20.0000	+	5.65685i	−0.883022	+	0.249756i
$514$		0			0
$515$	−14.1421			−0.623177
$516$		0			0
$517$	4.00000			0.175920
$518$		0			0
$519$	−9.89949	−	14.0000i	−0.434540	−	0.614532i
$520$		0			0
$521$		−	35.3553i		−	1.54895i	−0.632607	−	0.774473i	$-0.718015\pi$
$521$		−	35.3553i		−	1.54895i	0.632607	−	0.774473i	$-0.281985\pi$
$522$		0			0
$523$			26.0000i			1.13690i	0.822718	+	0.568450i	$0.192457\pi$
$523$			26.0000i			1.13690i	−0.822718	+	0.568450i	$0.807543\pi$
$524$		0			0
$525$	3.00000	+	4.24264i	0.130931	+	0.185164i
$526$		0			0
$527$	2.82843			0.123208
$528$		0			0
$529$	27.0000			1.17391
$530$		0			0
$531$	−11.3137	+	32.0000i	−0.490973	+	1.38868i
$532$		0			0
$533$			25.4558i			1.10262i
$534$		0			0
$535$		−	22.0000i		−	0.951143i
$536$		0			0
$537$	−14.0000	+	9.89949i	−0.604145	+	0.427195i
$538$		0			0
$539$	1.41421			0.0609145
$540$		0			0
$541$	16.0000			0.687894			0.343947	−	0.938989i	$-0.388236\pi$
$541$	16.0000			0.687894			0.343947	+	0.938989i	$0.388236\pi$
$542$		0			0
$543$	−2.82843	+	2.00000i	−0.121379	+	0.0858282i
$544$		0			0
$545$		−	19.7990i		−	0.848096i
$546$		0			0
$547$			8.00000i			0.342055i	0.985266	+	0.171028i	$0.0547087\pi$
$547$			8.00000i			0.342055i	−0.985266	+	0.171028i	$0.945291\pi$
$548$		0			0
$549$	−10.0000	+	28.2843i	−0.426790	+	1.20714i
$550$		0			0
$551$	33.9411			1.44594
$552$		0			0
$553$	−4.00000			−0.170097
$554$		0			0
$555$	−5.65685	−	8.00000i	−0.240120	−	0.339581i
$556$		0			0
$557$		−	11.3137i		−	0.479377i	−0.970850	−	0.239689i	$-0.922955\pi$
$557$		−	11.3137i		−	0.479377i	0.970850	−	0.239689i	$-0.0770453\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$	−2.00000	−	2.82843i	−0.0844401	−	0.119416i
$562$		0			0
$563$	−11.3137			−0.476816			−0.238408	−	0.971165i	$-0.576626\pi$
$563$	−11.3137			−0.476816			−0.238408	+	0.971165i	$0.576626\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	5.65685	−	7.00000i	0.237566	−	0.293972i
$568$		0			0
$569$			2.82843i			0.118574i	0.998241	+	0.0592869i	$0.0188827\pi$
$569$			2.82843i			0.118574i	−0.998241	+	0.0592869i	$0.981117\pi$
$570$		0			0
$571$			40.0000i			1.67395i	0.547243	+	0.836974i	$0.315677\pi$
$571$			40.0000i			1.67395i	−0.547243	+	0.836974i	$0.684323\pi$
$572$		0			0
$573$	−10.0000	+	7.07107i	−0.417756	+	0.295398i
$574$		0			0
$575$	21.2132			0.884652
$576$		0			0
$577$	−34.0000			−1.41544			−0.707719	−	0.706494i	$-0.750276\pi$
$577$	−34.0000			−1.41544			−0.707719	+	0.706494i	$0.750276\pi$
$578$		0			0
$579$	−28.2843	+	20.0000i	−1.17545	+	0.831172i
$580$		0			0
$581$		−	8.48528i		−	0.352029i
$582$		0			0
$583$			8.00000i			0.331326i
$584$		0			0
$585$	24.0000	+	8.48528i	0.992278	+	0.350823i
$586$		0			0
$587$	−16.9706			−0.700450			−0.350225	−	0.936666i	$-0.613895\pi$
$587$	−16.9706			−0.700450			−0.350225	+	0.936666i	$0.613895\pi$
$588$		0			0
$589$	−8.00000			−0.329634
$590$		0			0
$591$	16.9706	+	24.0000i	0.698076	+	0.987228i
$592$		0			0
$593$			18.3848i			0.754972i	0.926015	+	0.377486i	$0.123211\pi$
$593$			18.3848i			0.754972i	−0.926015	+	0.377486i	$0.876789\pi$
$594$		0			0
$595$		−	2.00000i		−	0.0819920i
$596$		0			0
$597$	−8.00000	−	11.3137i	−0.327418	−	0.463039i
$598$		0			0
$599$	18.3848			0.751182			0.375591	−	0.926786i	$-0.377440\pi$
$599$	18.3848			0.751182			0.375591	+	0.926786i	$0.377440\pi$
$600$		0			0
$601$	−6.00000			−0.244745			−0.122373	−	0.992484i	$-0.539050\pi$
$601$	−6.00000			−0.244745			−0.122373	+	0.992484i	$0.539050\pi$
$602$		0			0
$603$	−45.2548	−	16.0000i	−1.84292	−	0.651570i
$604$		0			0
$605$		−	12.7279i		−	0.517464i
$606$		0			0
$607$			8.00000i			0.324710i	0.986732	+	0.162355i	$0.0519090\pi$
$607$			8.00000i			0.324710i	−0.986732	+	0.162355i	$0.948091\pi$
$608$		0			0
$609$	−12.0000	+	8.48528i	−0.486265	+	0.343841i
$610$		0			0
$611$	−16.9706			−0.686555
$612$		0			0
$613$	10.0000			0.403896			0.201948	−	0.979396i	$-0.435273\pi$
$613$	10.0000			0.403896			0.201948	+	0.979396i	$0.435273\pi$
$614$		0			0
$615$	−8.48528	+	6.00000i	−0.342160	+	0.241943i
$616$		0			0
$617$		−	42.4264i		−	1.70802i	−0.520254	−	0.854011i	$-0.674163\pi$
$617$		−	42.4264i		−	1.70802i	0.520254	−	0.854011i	$-0.325837\pi$
$618$		0			0
$619$			2.00000i			0.0803868i	0.999192	+	0.0401934i	$0.0127974\pi$
$619$			2.00000i			0.0803868i	−0.999192	+	0.0401934i	$0.987203\pi$
$620$		0			0
$621$	−10.0000	−	35.3553i	−0.401286	−	1.41876i
$622$		0			0
$623$	−7.07107			−0.283296
$624$		0			0
$625$	−1.00000			−0.0400000
$626$		0			0
$627$	5.65685	+	8.00000i	0.225913	+	0.319489i
$628$		0			0
$629$		−	5.65685i		−	0.225554i
$630$		0			0
$631$			48.0000i			1.91085i	0.295234	+	0.955425i	$0.404602\pi$
$631$			48.0000i			1.91085i	−0.295234	+	0.955425i	$0.595398\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−16.9706			−0.673456
$636$		0			0
$637$	−6.00000			−0.237729
$638$		0			0
$639$	−4.24264	+	12.0000i	−0.167836	+	0.474713i
$640$		0			0
$641$		−	31.1127i		−	1.22888i	−0.788964	−	0.614439i	$-0.789382\pi$
$641$		−	31.1127i		−	1.22888i	0.788964	−	0.614439i	$-0.210618\pi$
$642$		0			0
$643$		−	36.0000i		−	1.41970i	−0.704352	−	0.709851i	$-0.748762\pi$
$643$		−	36.0000i		−	1.41970i	0.704352	−	0.709851i	$-0.251238\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−36.7696			−1.44556			−0.722780	−	0.691078i	$-0.757136\pi$
$647$	−36.7696			−1.44556			−0.722780	+	0.691078i	$0.757136\pi$
$648$		0			0
$649$	16.0000			0.628055
$650$		0			0
$651$	2.82843	−	2.00000i	0.110855	−	0.0783862i
$652$		0			0
$653$		−	31.1127i		−	1.21753i	−0.793349	−	0.608767i	$-0.791664\pi$
$653$		−	31.1127i		−	1.21753i	0.793349	−	0.608767i	$-0.208336\pi$
$654$		0			0
$655$			28.0000i			1.09405i
$656$		0			0
$657$	14.0000	−	39.5980i	0.546192	−	1.54486i
$658$		0			0
$659$	12.7279			0.495809			0.247905	−	0.968784i	$-0.420258\pi$
$659$	12.7279			0.495809			0.247905	+	0.968784i	$0.420258\pi$
$660$		0			0
$661$	−10.0000			−0.388955			−0.194477	−	0.980907i	$-0.562301\pi$
$661$	−10.0000			−0.388955			−0.194477	+	0.980907i	$0.562301\pi$
$662$		0			0
$663$	8.48528	+	12.0000i	0.329541	+	0.466041i
$664$		0			0
$665$			5.65685i			0.219363i
$666$		0			0
$667$			60.0000i			2.32321i
$668$		0			0
$669$	24.0000	+	33.9411i	0.927894	+	1.31224i
$670$		0			0
$671$	14.1421			0.545951
$672$		0			0
$673$	−24.0000			−0.925132			−0.462566	−	0.886585i	$-0.653071\pi$
$673$	−24.0000			−0.925132			−0.462566	+	0.886585i	$0.653071\pi$
$674$		0			0
$675$	−4.24264	−	15.0000i	−0.163299	−	0.577350i
$676$		0			0
$677$			26.8701i			1.03270i	0.856378	+	0.516350i	$0.172710\pi$
$677$			26.8701i			1.03270i	−0.856378	+	0.516350i	$0.827290\pi$
$678$		0			0
$679$			6.00000i			0.230259i
$680$		0			0
$681$	16.0000	−	11.3137i	0.613121	−	0.433542i
$682$		0			0
$683$	−35.3553			−1.35283			−0.676417	−	0.736519i	$-0.736468\pi$
$683$	−35.3553			−1.35283			−0.676417	+	0.736519i	$0.736468\pi$
$684$		0			0
$685$	28.0000			1.06983
$686$		0			0
$687$	14.1421	−	10.0000i	0.539556	−	0.381524i
$688$		0			0
$689$		−	33.9411i		−	1.29305i
$690$		0			0
$691$		−	2.00000i		−	0.0760836i	−0.999276	−	0.0380418i	$-0.987888\pi$
$691$		−	2.00000i		−	0.0760836i	0.999276	−	0.0380418i	$-0.0121120\pi$
$692$		0			0
$693$	−4.00000	−	1.41421i	−0.151947	−	0.0537215i
$694$		0			0
$695$	−8.48528			−0.321865
$696$		0			0
$697$	−6.00000			−0.227266
$698$		0			0
$699$	−19.7990	−	28.0000i	−0.748867	−	1.05906i
$700$		0			0
$701$			19.7990i			0.747798i	0.927470	+	0.373899i	$0.121979\pi$
$701$			19.7990i			0.747798i	−0.927470	+	0.373899i	$0.878021\pi$
$702$		0			0
$703$			16.0000i			0.603451i
$704$		0			0
$705$	−4.00000	−	5.65685i	−0.150649	−	0.213049i
$706$		0			0
$707$	−1.41421			−0.0531870
$708$		0			0
$709$	−10.0000			−0.375558			−0.187779	−	0.982211i	$-0.560129\pi$
$709$	−10.0000			−0.375558			−0.187779	+	0.982211i	$0.560129\pi$
$710$		0			0
$711$	11.3137	+	4.00000i	0.424297	+	0.150012i
$712$		0			0
$713$		−	14.1421i		−	0.529627i
$714$		0			0
$715$		−	12.0000i		−	0.448775i
$716$		0			0
$717$	2.00000	−	1.41421i	0.0746914	−	0.0528148i
$718$		0			0
$719$	11.3137			0.421930			0.210965	−	0.977494i	$-0.432339\pi$
$719$	11.3137			0.421930			0.210965	+	0.977494i	$0.432339\pi$
$720$		0			0
$721$	−10.0000			−0.372419
$722$		0			0
$723$	19.7990	−	14.0000i	0.736332	−	0.520666i
$724$		0			0
$725$			25.4558i			0.945406i
$726$		0			0
$727$		−	18.0000i		−	0.667583i	−0.942647	−	0.333792i	$-0.891672\pi$
$727$		−	18.0000i		−	0.667583i	0.942647	−	0.333792i	$-0.108328\pi$
$728$		0			0
$729$	−23.0000	+	14.1421i	−0.851852	+	0.523783i
$730$		0			0
$731$		0			0
$732$		0			0
$733$	26.0000			0.960332			0.480166	−	0.877178i	$-0.340576\pi$
$733$	26.0000			0.960332			0.480166	+	0.877178i	$0.340576\pi$
$734$		0			0
$735$	−1.41421	−	2.00000i	−0.0521641	−	0.0737711i
$736$		0			0
$737$			22.6274i			0.833492i
$738$		0			0
$739$		−	44.0000i		−	1.61857i	−0.587419	−	0.809283i	$-0.699856\pi$
$739$		−	44.0000i		−	1.61857i	0.587419	−	0.809283i	$-0.300144\pi$
$740$		0			0
$741$	−24.0000	−	33.9411i	−0.881662	−	1.24686i
$742$		0			0
$743$	38.1838			1.40083			0.700413	−	0.713738i	$-0.252999\pi$
$743$	38.1838			1.40083			0.700413	+	0.713738i	$0.252999\pi$
$744$		0			0
$745$	16.0000			0.586195
$746$		0			0
$747$	−8.48528	+	24.0000i	−0.310460	+	0.878114i
$748$		0			0
$749$		−	15.5563i		−	0.568417i
$750$		0			0
$751$			20.0000i			0.729810i	0.931045	+	0.364905i	$0.118899\pi$
$751$			20.0000i			0.729810i	−0.931045	+	0.364905i	$0.881101\pi$
$752$		0			0
$753$	8.00000	−	5.65685i	0.291536	−	0.206147i
$754$		0			0
$755$	11.3137			0.411748
$756$		0			0
$757$	6.00000			0.218074			0.109037	−	0.994038i	$-0.465223\pi$
$757$	6.00000			0.218074			0.109037	+	0.994038i	$0.465223\pi$
$758$		0			0
$759$	−14.1421	+	10.0000i	−0.513327	+	0.362977i
$760$		0			0
$761$			26.8701i			0.974039i	0.873391	+	0.487019i	$0.161916\pi$
$761$			26.8701i			0.974039i	−0.873391	+	0.487019i	$0.838084\pi$
$762$		0			0
$763$		−	14.0000i		−	0.506834i
$764$		0			0
$765$	−2.00000	+	5.65685i	−0.0723102	+	0.204524i
$766$		0			0
$767$	−67.8823			−2.45109
$768$		0			0
$769$	6.00000			0.216366			0.108183	−	0.994131i	$-0.465497\pi$
$769$	6.00000			0.216366			0.108183	+	0.994131i	$0.465497\pi$
$770$		0			0
$771$	26.8701	+	38.0000i	0.967701	+	1.36854i
$772$		0			0
$773$			26.8701i			0.966449i	0.875497	+	0.483224i	$0.160534\pi$
$773$			26.8701i			0.966449i	−0.875497	+	0.483224i	$0.839466\pi$
$774$		0			0
$775$		−	6.00000i		−	0.215526i
$776$		0			0
$777$	−4.00000	−	5.65685i	−0.143499	−	0.202939i
$778$		0			0
$779$	16.9706			0.608034
$780$		0			0
$781$	6.00000			0.214697
$782$		0			0
$783$	42.4264	−	12.0000i	1.51620	−	0.428845i
$784$		0			0
$785$		−	14.1421i		−	0.504754i
$786$		0			0
$787$			26.0000i			0.926800i	0.886149	+	0.463400i	$0.153371\pi$
$787$			26.0000i			0.926800i	−0.886149	+	0.463400i	$0.846629\pi$
$788$		0			0
$789$	14.0000	−	9.89949i	0.498413	−	0.352431i
$790$		0			0
$791$		0			0
$792$		0			0
$793$	−60.0000			−2.13066
$794$		0			0
$795$	11.3137	−	8.00000i	0.401256	−	0.283731i
$796$		0			0
$797$			4.24264i			0.150282i	0.997173	+	0.0751410i	$0.0239407\pi$
$797$			4.24264i			0.150282i	−0.997173	+	0.0751410i	$0.976059\pi$
$798$		0			0
$799$		−	4.00000i		−	0.141510i
$800$		0			0
$801$	20.0000	+	7.07107i	0.706665	+	0.249844i
$802$		0			0
$803$	−19.7990			−0.698691
$804$		0			0
$805$	−10.0000			−0.352454
$806$		0			0
$807$	32.5269	+	46.0000i	1.14500	+	1.61928i
$808$		0			0
$809$			28.2843i			0.994422i	0.867630	+	0.497211i	$0.165643\pi$
$809$			28.2843i			0.994422i	−0.867630	+	0.497211i	$0.834357\pi$
$810$		0			0
$811$		−	30.0000i		−	1.05344i	−0.850038	−	0.526721i	$-0.823421\pi$
$811$		−	30.0000i		−	1.05344i	0.850038	−	0.526721i	$-0.176579\pi$
$812$		0			0
$813$	−26.0000	−	36.7696i	−0.911860	−	1.28956i
$814$		0			0
$815$	−5.65685			−0.198151
$816$		0			0
$817$		0			0
$818$		0			0
$819$	16.9706	+	6.00000i	0.592999	+	0.209657i
$820$		0			0
$821$			5.65685i			0.197426i	0.995116	+	0.0987128i	$0.0314725\pi$
$821$			5.65685i			0.197426i	−0.995116	+	0.0987128i	$0.968527\pi$
$822$		0			0
$823$			44.0000i			1.53374i	0.641800	+	0.766872i	$0.278188\pi$
$823$			44.0000i			1.53374i	−0.641800	+	0.766872i	$0.721812\pi$
$824$		0			0
$825$	−6.00000	+	4.24264i	−0.208893	+	0.147710i
$826$		0			0
$827$	−12.7279			−0.442593			−0.221297	−	0.975207i	$-0.571029\pi$
$827$	−12.7279			−0.442593			−0.221297	+	0.975207i	$0.571029\pi$
$828$		0			0
$829$	−14.0000			−0.486240			−0.243120	−	0.969996i	$-0.578171\pi$
$829$	−14.0000			−0.486240			−0.243120	+	0.969996i	$0.578171\pi$
$830$		0			0
$831$	−22.6274	+	16.0000i	−0.784936	+	0.555034i
$832$		0			0
$833$		−	1.41421i		−	0.0489996i
$834$		0			0
$835$		−	20.0000i		−	0.692129i
$836$		0			0
$837$	−10.0000	+	2.82843i	−0.345651	+	0.0977647i
$838$		0			0
$839$	25.4558			0.878833			0.439417	−	0.898283i	$-0.355185\pi$
$839$	25.4558			0.878833			0.439417	+	0.898283i	$0.355185\pi$
$840$		0			0
$841$	−43.0000			−1.48276
$842$		0			0
$843$	−25.4558	−	36.0000i	−0.876746	−	1.23991i
$844$		0			0
$845$			32.5269i			1.11896i
$846$		0			0
$847$		−	9.00000i		−	0.309244i
$848$		0			0
$849$	28.0000	+	39.5980i	0.960958	+	1.35900i
$850$		0			0
$851$	−28.2843			−0.969572
$852$		0			0
$853$	−6.00000			−0.205436			−0.102718	−	0.994711i	$-0.532754\pi$
$853$	−6.00000			−0.205436			−0.102718	+	0.994711i	$0.532754\pi$
$854$		0			0
$855$	5.65685	−	16.0000i	0.193460	−	0.547188i
$856$		0			0
$857$			49.4975i			1.69080i	0.534133	+	0.845401i	$0.320638\pi$
$857$			49.4975i			1.69080i	−0.534133	+	0.845401i	$0.679362\pi$
$858$		0			0
$859$			20.0000i			0.682391i	0.939992	+	0.341196i	$0.110832\pi$
$859$			20.0000i			0.682391i	−0.939992	+	0.341196i	$0.889168\pi$
$860$		0			0
$861$	−6.00000	+	4.24264i	−0.204479	+	0.144589i
$862$		0			0
$863$	−43.8406			−1.49235			−0.746176	−	0.665749i	$-0.768112\pi$
$863$	−43.8406			−1.49235			−0.746176	+	0.665749i	$0.768112\pi$
$864$		0			0
$865$	14.0000			0.476014
$866$		0			0
$867$	21.2132	−	15.0000i	0.720438	−	0.509427i
$868$		0			0
$869$		−	5.65685i		−	0.191896i
$870$		0			0
$871$		−	96.0000i		−	3.25284i
$872$		0			0
$873$	6.00000	−	16.9706i	0.203069	−	0.574367i
$874$		0			0
$875$	−11.3137			−0.382473
$876$		0			0
$877$	34.0000			1.14810			0.574049	−	0.818821i	$-0.305372\pi$
$877$	34.0000			1.14810			0.574049	+	0.818821i	$0.305372\pi$
$878$		0			0
$879$	−26.8701	−	38.0000i	−0.906305	−	1.28171i
$880$		0			0
$881$		−	7.07107i		−	0.238230i	−0.992880	−	0.119115i	$-0.961994\pi$
$881$		−	7.07107i		−	0.238230i	0.992880	−	0.119115i	$-0.0380058\pi$
$882$		0			0
$883$		−	36.0000i		−	1.21150i	−0.795656	−	0.605748i	$-0.792874\pi$
$883$		−	36.0000i		−	1.21150i	0.795656	−	0.605748i	$-0.207126\pi$
$884$		0			0
$885$	−16.0000	−	22.6274i	−0.537834	−	0.760612i
$886$		0			0
$887$	48.0833			1.61448			0.807239	−	0.590225i	$-0.200961\pi$
$887$	48.0833			1.61448			0.807239	+	0.590225i	$0.200961\pi$
$888$		0			0
$889$	−12.0000			−0.402467
$890$		0			0
$891$	9.89949	+	8.00000i	0.331646	+	0.268010i
$892$		0			0
$893$			11.3137i			0.378599i
$894$		0			0
$895$		−	14.0000i		−	0.467968i
$896$		0			0
$897$	60.0000	−	42.4264i	2.00334	−	1.41658i
$898$		0			0
$899$	16.9706			0.566000
$900$		0			0
$901$	8.00000			0.266519
$902$		0			0
$903$		0			0
$904$		0			0
$905$		−	2.82843i		−	0.0940201i
$906$		0			0
$907$		−	4.00000i		−	0.132818i	−0.997792	−	0.0664089i	$-0.978846\pi$
$907$		−	4.00000i		−	0.132818i	0.997792	−	0.0664089i	$-0.0211542\pi$
$908$		0			0
$909$	4.00000	+	1.41421i	0.132672	+	0.0469065i
$910$		0			0
$911$	12.7279			0.421695			0.210847	−	0.977519i	$-0.432378\pi$
$911$	12.7279			0.421695			0.210847	+	0.977519i	$0.432378\pi$
$912$		0			0
$913$	12.0000			0.397142
$914$		0			0
$915$	−14.1421	−	20.0000i	−0.467525	−	0.661180i
$916$		0			0
$917$			19.7990i			0.653820i
$918$		0			0
$919$		−	4.00000i		−	0.131948i	−0.997821	−	0.0659739i	$-0.978985\pi$
$919$		−	4.00000i		−	0.131948i	0.997821	−	0.0659739i	$-0.0210154\pi$
$920$		0			0
$921$	12.0000	+	16.9706i	0.395413	+	0.559199i
$922$		0			0
$923$	−25.4558			−0.837889
$924$		0			0
$925$	−12.0000			−0.394558
$926$		0			0
$927$	28.2843	+	10.0000i	0.928977	+	0.328443i
$928$		0			0
$929$			15.5563i			0.510387i	0.966890	+	0.255194i	$0.0821392\pi$
$929$			15.5563i			0.510387i	−0.966890	+	0.255194i	$0.917861\pi$
$930$		0			0
$931$			4.00000i			0.131095i
$932$		0			0
$933$	−28.0000	+	19.7990i	−0.916679	+	0.648190i
$934$		0			0
$935$	2.82843			0.0924995
$936$		0			0
$937$	2.00000			0.0653372			0.0326686	−	0.999466i	$-0.489599\pi$
$937$	2.00000			0.0653372			0.0326686	+	0.999466i	$0.489599\pi$
$938$		0			0
$939$	2.82843	−	2.00000i	0.0923022	−	0.0652675i
$940$		0			0
$941$		−	26.8701i		−	0.875939i	−0.898990	−	0.437969i	$-0.855698\pi$
$941$		−	26.8701i		−	0.875939i	0.898990	−	0.437969i	$-0.144302\pi$
$942$		0			0
$943$			30.0000i			0.976934i
$944$		0			0
$945$	2.00000	+	7.07107i	0.0650600	+	0.230022i
$946$		0			0
$947$	−1.41421			−0.0459558			−0.0229779	−	0.999736i	$-0.507315\pi$
$947$	−1.41421			−0.0459558			−0.0229779	+	0.999736i	$0.507315\pi$
$948$		0			0
$949$	84.0000			2.72676
$950$		0			0
$951$	−22.6274	−	32.0000i	−0.733744	−	1.03767i
$952$		0			0
$953$			33.9411i			1.09946i	0.835342	+	0.549730i	$0.185270\pi$
$953$			33.9411i			1.09946i	−0.835342	+	0.549730i	$0.814730\pi$
$954$		0			0
$955$		−	10.0000i		−	0.323592i
$956$		0			0
$957$	−12.0000	−	16.9706i	−0.387905	−	0.548580i
$958$		0			0
$959$	19.7990			0.639343
$960$		0			0
$961$	27.0000			0.870968
$962$		0			0
$963$	−15.5563	+	44.0000i	−0.501296	+	1.41788i
$964$		0			0
$965$		−	28.2843i		−	0.910503i
$966$		0			0
$967$		−	36.0000i		−	1.15768i	−0.815440	−	0.578841i	$-0.803505\pi$
$967$		−	36.0000i		−	1.15768i	0.815440	−	0.578841i	$-0.196495\pi$
$968$		0			0
$969$	8.00000	−	5.65685i	0.256997	−	0.181724i
$970$		0			0
$971$	19.7990			0.635380			0.317690	−	0.948195i	$-0.397093\pi$
$971$	19.7990			0.635380			0.317690	+	0.948195i	$0.397093\pi$
$972$		0			0
$973$	−6.00000			−0.192351
$974$		0			0
$975$	25.4558	−	18.0000i	0.815239	−	0.576461i
$976$		0			0
$977$			14.1421i			0.452447i	0.974075	+	0.226224i	$0.0726380\pi$
$977$			14.1421i			0.452447i	−0.974075	+	0.226224i	$0.927362\pi$
$978$		0			0
$979$		−	10.0000i		−	0.319601i
$980$		0			0
$981$	−14.0000	+	39.5980i	−0.446986	+	1.26427i
$982$		0			0
$983$	28.2843			0.902128			0.451064	−	0.892492i	$-0.351045\pi$
$983$	28.2843			0.902128			0.451064	+	0.892492i	$0.351045\pi$
$984$		0			0
$985$	−24.0000			−0.764704
$986$		0			0
$987$	−2.82843	−	4.00000i	−0.0900298	−	0.127321i
$988$		0			0
$989$		0			0
$990$		0			0
$991$			20.0000i			0.635321i	0.948205	+	0.317660i	$0.102897\pi$
$991$			20.0000i			0.635321i	−0.948205	+	0.317660i	$0.897103\pi$
$992$		0			0
$993$	4.00000	+	5.65685i	0.126936	+	0.179515i
$994$		0			0
$995$	11.3137			0.358669
$996$		0			0
$997$	−22.0000			−0.696747			−0.348373	−	0.937356i	$-0.613266\pi$
$997$	−22.0000			−0.696747			−0.348373	+	0.937356i	$0.613266\pi$
$998$		0			0
$999$	5.65685	+	20.0000i	0.178975	+	0.632772i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	672.2.h.c.575.3	yes	4
3.2	odd	2	inner	672.2.h.c.575.1	✓	4
4.3	odd	2	inner	672.2.h.c.575.2	yes	4
8.3	odd	2		1344.2.h.b.575.3		4
8.5	even	2		1344.2.h.b.575.2		4
12.11	even	2	inner	672.2.h.c.575.4	yes	4
24.5	odd	2		1344.2.h.b.575.4		4
24.11	even	2		1344.2.h.b.575.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
672.2.h.c.575.1	✓	4	3.2	odd	2	inner
672.2.h.c.575.2	yes	4	4.3	odd	2	inner
672.2.h.c.575.3	yes	4	1.1	even	1	trivial
672.2.h.c.575.4	yes	4	12.11	even	2	inner
1344.2.h.b.575.1		4	24.11	even	2
1344.2.h.b.575.2		4	8.5	even	2
1344.2.h.b.575.3		4	8.3	odd	2
1344.2.h.b.575.4		4	24.5	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 672 = 2^{5} \cdot 3 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	672.h (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(1.41421 - 1.00000i) q^{3} +1.41421i q^{5} +1.00000i q^{7} +(1.00000 - 2.82843i) q^{9} +O(q^{10})\)	\(q+(1.41421 - 1.00000i) q^{3} +1.41421i q^{5} +1.00000i q^{7} +(1.00000 - 2.82843i) q^{9} -1.41421 q^{11} +6.00000 q^{13} +(1.41421 + 2.00000i) q^{15} +1.41421i q^{17} -4.00000i q^{19} +(1.00000 + 1.41421i) q^{21} +7.07107 q^{23} +3.00000 q^{25} +(-1.41421 - 5.00000i) q^{27} +8.48528i q^{29} -2.00000i q^{31} +(-2.00000 + 1.41421i) q^{33} -1.41421 q^{35} -4.00000 q^{37} +(8.48528 - 6.00000i) q^{39} +4.24264i q^{41} +(4.00000 + 1.41421i) q^{45} -2.82843 q^{47} -1.00000 q^{49} +(1.41421 + 2.00000i) q^{51} -5.65685i q^{53} -2.00000i q^{55} +(-4.00000 - 5.65685i) q^{57} -11.3137 q^{59} -10.0000 q^{61} +(2.82843 + 1.00000i) q^{63} +8.48528i q^{65} -16.0000i q^{67} +(10.0000 - 7.07107i) q^{69} -4.24264 q^{71} +14.0000 q^{73} +(4.24264 - 3.00000i) q^{75} -1.41421i q^{77} +4.00000i q^{79} +(-7.00000 - 5.65685i) q^{81} -8.48528 q^{83} -2.00000 q^{85} +(8.48528 + 12.0000i) q^{87} +7.07107i q^{89} +6.00000i q^{91} +(-2.00000 - 2.82843i) q^{93} +5.65685 q^{95} +6.00000 q^{97} +(-1.41421 + 4.00000i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{9}+O(q^{10}) \) 4 * q + 4 * q^9	\( 4 q + 4 q^{9} + 24 q^{13} + 4 q^{21} + 12 q^{25} - 8 q^{33} - 16 q^{37} + 16 q^{45} - 4 q^{49} - 16 q^{57} - 40 q^{61} + 40 q^{69} + 56 q^{73} - 28 q^{81} - 8 q^{85} - 8 q^{93} + 24 q^{97}+O(q^{100}) \) 4 * q + 4 * q^9 + 24 * q^13 + 4 * q^21 + 12 * q^25 - 8 * q^33 - 16 * q^37 + 16 * q^45 - 4 * q^49 - 16 * q^57 - 40 * q^61 + 40 * q^69 + 56 * q^73 - 28 * q^81 - 8 * q^85 - 8 * q^93 + 24 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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