LMFDB - Embedded newform 1408.2.g.d.703.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1408,2,Mod(703,1408)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1408, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 1, 1]))

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

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

chi := DirichletCharacter("1408.703");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1408 = 2^{7} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1408.g (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:	$11.2429366046$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{-2}, \sqrt{-7})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{3} + 9x^{2} - 8x + 2 $ x^4 - 2x^3 + 9x^2 - 8*x + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			703.1
Root			$0.500000 + 2.73709i$ of defining polynomial
Character	$\chi$	$=$	1408.703
Dual form			1408.2.g.d.703.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} -2.64575i q^{5} -3.74166 q^{7} -2.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} -2.64575i q^{5} -3.74166 q^{7} -2.00000 q^{9} +(-3.00000 + 1.41421i) q^{11} +3.74166 q^{13} -2.64575i q^{15} +4.24264i q^{17} +4.24264i q^{19} -3.74166 q^{21} +2.64575i q^{23} -2.00000 q^{25} -5.00000 q^{27} -7.93725i q^{31} +(-3.00000 + 1.41421i) q^{33} +9.89949i q^{35} +7.93725i q^{37} +3.74166 q^{39} +1.41421i q^{41} +8.48528i q^{43} +5.29150i q^{45} +10.5830i q^{47} +7.00000 q^{49} +4.24264i q^{51} -5.29150i q^{53} +(3.74166 + 7.93725i) q^{55} +4.24264i q^{57} -9.00000 q^{59} -14.9666 q^{61} +7.48331 q^{63} -9.89949i q^{65} +3.00000 q^{67} +2.64575i q^{69} -2.64575i q^{71} -12.7279i q^{73} -2.00000 q^{75} +(11.2250 - 5.29150i) q^{77} -7.48331 q^{79} +1.00000 q^{81} +8.48528i q^{83} +11.2250 q^{85} +3.00000 q^{89} -14.0000 q^{91} -7.93725i q^{93} +11.2250 q^{95} -3.00000 q^{97} +(6.00000 - 2.82843i) q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{3} - 8 q^{9}+O(q^{10}) $ 4 * q + 4 * q^3 - 8 * q^9	$ 4 q + 4 q^{3} - 8 q^{9} - 12 q^{11} - 8 q^{25} - 20 q^{27} - 12 q^{33} + 28 q^{49} - 36 q^{59} + 12 q^{67} - 8 q^{75} + 4 q^{81} + 12 q^{89} - 56 q^{91} - 12 q^{97} + 24 q^{99}+O(q^{100}) $ 4 * q + 4 * q^3 - 8 * q^9 - 12 * q^11 - 8 * q^25 - 20 * q^27 - 12 * q^33 + 28 * q^49 - 36 * q^59 + 12 * q^67 - 8 * q^75 + 4 * q^81 + 12 * q^89 - 56 * q^91 - 12 * q^97 + 24 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1408\mathbb{Z}\right)^\times$.

$n$	$133$	$639$	$1025$
$\chi(n)$	$-1$	$-1$	$-1$

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
$3$	1.00000			0.577350			0.288675	−	0.957427i	$-0.406785\pi$
$3$	1.00000			0.577350			0.288675	+	0.957427i	$0.406785\pi$
$4$		0			0
$5$		−	2.64575i		−	1.18322i	−0.806226	−	0.591608i	$-0.798493\pi$
$5$		−	2.64575i		−	1.18322i	0.806226	−	0.591608i	$-0.201507\pi$
$6$		0			0
$7$	−3.74166			−1.41421			−0.707107	−	0.707107i	$-0.750000\pi$
$7$	−3.74166			−1.41421			−0.707107	+	0.707107i	$0.750000\pi$
$8$		0			0
$9$	−2.00000			−0.666667
$10$		0			0
$11$	−3.00000	+	1.41421i	−0.904534	+	0.426401i
$11$	−3.00000	+	1.41421i	−0.904534	+	0.426401i
$12$		0			0
$13$	3.74166			1.03775			0.518875	−	0.854850i	$-0.326351\pi$
$13$	3.74166			1.03775			0.518875	+	0.854850i	$0.326351\pi$
$14$		0			0
$15$		−	2.64575i		−	0.683130i
$16$		0			0
$17$			4.24264i			1.02899i	0.857493	+	0.514496i	$0.172021\pi$
$17$			4.24264i			1.02899i	−0.857493	+	0.514496i	$0.827979\pi$
$18$		0			0
$19$			4.24264i			0.973329i	0.873589	+	0.486664i	$0.161786\pi$
$19$			4.24264i			0.973329i	−0.873589	+	0.486664i	$0.838214\pi$
$20$		0			0
$21$	−3.74166			−0.816497
$22$		0			0
$23$			2.64575i			0.551677i	0.961204	+	0.275839i	$0.0889555\pi$
$23$			2.64575i			0.551677i	−0.961204	+	0.275839i	$0.911044\pi$
$24$		0			0
$25$	−2.00000			−0.400000
$26$		0			0
$27$	−5.00000			−0.962250
$28$		0			0
$29$		0			0			−	1.00000i	$-0.5\pi$
$29$		0			0				1.00000i	$0.5\pi$
$30$		0			0
$31$		−	7.93725i		−	1.42557i	−0.701381	−	0.712786i	$-0.747433\pi$
$31$		−	7.93725i		−	1.42557i	0.701381	−	0.712786i	$-0.252567\pi$
$32$		0			0
$33$	−3.00000	+	1.41421i	−0.522233	+	0.246183i
$34$		0			0
$35$			9.89949i			1.67332i
$36$		0			0
$37$			7.93725i			1.30488i	0.757842	+	0.652438i	$0.226254\pi$
$37$			7.93725i			1.30488i	−0.757842	+	0.652438i	$0.773746\pi$
$38$		0			0
$39$	3.74166			0.599145
$40$		0			0
$41$			1.41421i			0.220863i	0.993884	+	0.110432i	$0.0352233\pi$
$41$			1.41421i			0.220863i	−0.993884	+	0.110432i	$0.964777\pi$
$42$		0			0
$43$			8.48528i			1.29399i	0.762493	+	0.646997i	$0.223975\pi$
$43$			8.48528i			1.29399i	−0.762493	+	0.646997i	$0.776025\pi$
$44$		0			0
$45$			5.29150i			0.788811i
$46$		0			0
$47$			10.5830i			1.54369i	0.635811	+	0.771845i	$0.280666\pi$
$47$			10.5830i			1.54369i	−0.635811	+	0.771845i	$0.719334\pi$
$48$		0			0
$49$	7.00000			1.00000
$50$		0			0
$51$			4.24264i			0.594089i
$52$		0			0
$53$		−	5.29150i		−	0.726844i	−0.931625	−	0.363422i	$-0.881608\pi$
$53$		−	5.29150i		−	0.726844i	0.931625	−	0.363422i	$-0.118392\pi$
$54$		0			0
$55$	3.74166	+	7.93725i	0.504525	+	1.07026i
$56$		0			0
$57$			4.24264i			0.561951i
$58$		0			0
$59$	−9.00000			−1.17170			−0.585850	−	0.810419i	$-0.699239\pi$
$59$	−9.00000			−1.17170			−0.585850	+	0.810419i	$0.699239\pi$
$60$		0			0
$61$	−14.9666			−1.91628			−0.958140	−	0.286299i	$-0.907575\pi$
$61$	−14.9666			−1.91628			−0.958140	+	0.286299i	$0.907575\pi$
$62$		0			0
$63$	7.48331			0.942809
$64$		0			0
$65$		−	9.89949i		−	1.22788i
$66$		0			0
$67$	3.00000			0.366508			0.183254	−	0.983066i	$-0.441337\pi$
$67$	3.00000			0.366508			0.183254	+	0.983066i	$0.441337\pi$
$68$		0			0
$69$			2.64575i			0.318511i
$70$		0			0
$71$		−	2.64575i		−	0.313993i	−0.987599	−	0.156996i	$-0.949819\pi$
$71$		−	2.64575i		−	0.313993i	0.987599	−	0.156996i	$-0.0501811\pi$
$72$		0			0
$73$		−	12.7279i		−	1.48969i	−0.667237	−	0.744845i	$-0.732523\pi$
$73$		−	12.7279i		−	1.48969i	0.667237	−	0.744845i	$-0.267477\pi$
$74$		0			0
$75$	−2.00000			−0.230940
$76$		0			0
$77$	11.2250	−	5.29150i	1.27920	−	0.603023i
$78$		0			0
$79$	−7.48331			−0.841939			−0.420969	−	0.907075i	$-0.638310\pi$
$79$	−7.48331			−0.841939			−0.420969	+	0.907075i	$0.638310\pi$
$80$		0			0
$81$	1.00000			0.111111
$82$		0			0
$83$			8.48528i			0.931381i	0.884948	+	0.465690i	$0.154194\pi$
$83$			8.48528i			0.931381i	−0.884948	+	0.465690i	$0.845806\pi$
$84$		0			0
$85$	11.2250			1.21752
$86$		0			0
$87$		0			0
$88$		0			0
$89$	3.00000			0.317999			0.159000	−	0.987279i	$-0.449173\pi$
$89$	3.00000			0.317999			0.159000	+	0.987279i	$0.449173\pi$
$90$		0			0
$91$	−14.0000			−1.46760
$92$		0			0
$93$		−	7.93725i		−	0.823055i
$94$		0			0
$95$	11.2250			1.15166
$96$		0			0
$97$	−3.00000			−0.304604			−0.152302	−	0.988334i	$-0.548669\pi$
$97$	−3.00000			−0.304604			−0.152302	+	0.988334i	$0.548669\pi$
$98$		0			0
$99$	6.00000	−	2.82843i	0.603023	−	0.284268i
$100$		0			0
$101$	11.2250			1.11693			0.558463	−	0.829529i	$-0.311391\pi$
$101$	11.2250			1.11693			0.558463	+	0.829529i	$0.311391\pi$
$102$		0			0
$103$		−	15.8745i		−	1.56416i	−0.623177	−	0.782081i	$-0.714158\pi$
$103$		−	15.8745i		−	1.56416i	0.623177	−	0.782081i	$-0.285842\pi$
$104$		0			0
$105$			9.89949i			0.966092i
$106$		0			0
$107$			7.07107i			0.683586i	0.939775	+	0.341793i	$0.111034\pi$
$107$			7.07107i			0.683586i	−0.939775	+	0.341793i	$0.888966\pi$
$108$		0			0
$109$	−7.48331			−0.716772			−0.358386	−	0.933574i	$-0.616673\pi$
$109$	−7.48331			−0.716772			−0.358386	+	0.933574i	$0.616673\pi$
$110$		0			0
$111$			7.93725i			0.753371i
$112$		0			0
$113$	3.00000			0.282216			0.141108	−	0.989994i	$-0.454933\pi$
$113$	3.00000			0.282216			0.141108	+	0.989994i	$0.454933\pi$
$114$		0			0
$115$	7.00000			0.652753
$116$		0			0
$117$	−7.48331			−0.691833
$118$		0			0
$119$		−	15.8745i		−	1.45521i
$120$		0			0
$121$	7.00000	−	8.48528i	0.636364	−	0.771389i
$122$		0			0
$123$			1.41421i			0.127515i
$124$		0			0
$125$		−	7.93725i		−	0.709930i
$126$		0			0
$127$	−18.7083			−1.66009			−0.830046	−	0.557695i	$-0.811686\pi$
$127$	−18.7083			−1.66009			−0.830046	+	0.557695i	$0.811686\pi$
$128$		0			0
$129$			8.48528i			0.747087i
$130$		0			0
$131$			21.2132i			1.85341i	0.375794	+	0.926703i	$0.377370\pi$
$131$			21.2132i			1.85341i	−0.375794	+	0.926703i	$0.622630\pi$
$132$		0			0
$133$		−	15.8745i		−	1.37649i
$134$		0			0
$135$			13.2288i			1.13855i
$136$		0			0
$137$	−15.0000			−1.28154			−0.640768	−	0.767734i	$-0.721384\pi$
$137$	−15.0000			−1.28154			−0.640768	+	0.767734i	$0.721384\pi$
$138$		0			0
$139$		−	12.7279i		−	1.07957i	−0.841803	−	0.539784i	$-0.818506\pi$
$139$		−	12.7279i		−	1.07957i	0.841803	−	0.539784i	$-0.181494\pi$
$140$		0			0
$141$			10.5830i			0.891250i
$142$		0			0
$143$	−11.2250	+	5.29150i	−0.938679	+	0.442498i
$144$		0			0
$145$		0			0
$146$		0			0
$147$	7.00000			0.577350
$148$		0			0
$149$	−11.2250			−0.919586			−0.459793	−	0.888026i	$-0.652076\pi$
$149$	−11.2250			−0.919586			−0.459793	+	0.888026i	$0.652076\pi$
$150$		0			0
$151$	−14.9666			−1.21797			−0.608984	−	0.793183i	$-0.708422\pi$
$151$	−14.9666			−1.21797			−0.608984	+	0.793183i	$0.708422\pi$
$152$		0			0
$153$		−	8.48528i		−	0.685994i
$154$		0			0
$155$	−21.0000			−1.68676
$156$		0			0
$157$			7.93725i			0.633462i	0.948515	+	0.316731i	$0.102585\pi$
$157$			7.93725i			0.633462i	−0.948515	+	0.316731i	$0.897415\pi$
$158$		0			0
$159$		−	5.29150i		−	0.419643i
$160$		0			0
$161$		−	9.89949i		−	0.780189i
$162$		0			0
$163$	−2.00000			−0.156652			−0.0783260	−	0.996928i	$-0.524958\pi$
$163$	−2.00000			−0.156652			−0.0783260	+	0.996928i	$0.524958\pi$
$164$		0			0
$165$	3.74166	+	7.93725i	0.291288	+	0.617914i
$166$		0			0
$167$	11.2250			0.868614			0.434307	−	0.900765i	$-0.356993\pi$
$167$	11.2250			0.868614			0.434307	+	0.900765i	$0.356993\pi$
$168$		0			0
$169$	1.00000			0.0769231
$170$		0			0
$171$		−	8.48528i		−	0.648886i
$172$		0			0
$173$		0			0			−	1.00000i	$-0.5\pi$
$173$		0			0				1.00000i	$0.5\pi$
$174$		0			0
$175$	7.48331			0.565685
$176$		0			0
$177$	−9.00000			−0.676481
$178$		0			0
$179$	21.0000			1.56961			0.784807	−	0.619740i	$-0.212762\pi$
$179$	21.0000			1.56961			0.784807	+	0.619740i	$0.212762\pi$
$180$		0			0
$181$			23.8118i			1.76991i	0.465672	+	0.884957i	$0.345812\pi$
$181$			23.8118i			1.76991i	−0.465672	+	0.884957i	$0.654188\pi$
$182$		0			0
$183$	−14.9666			−1.10637
$184$		0			0
$185$	21.0000			1.54395
$186$		0			0
$187$	−6.00000	−	12.7279i	−0.438763	−	0.930758i
$188$		0			0
$189$	18.7083			1.36083
$190$		0			0
$191$			2.64575i			0.191440i	0.995408	+	0.0957199i	$0.0305153\pi$
$191$			2.64575i			0.191440i	−0.995408	+	0.0957199i	$0.969485\pi$
$192$		0			0
$193$			16.9706i			1.22157i	0.791797	+	0.610784i	$0.209146\pi$
$193$			16.9706i			1.22157i	−0.791797	+	0.610784i	$0.790854\pi$
$194$		0			0
$195$		−	9.89949i		−	0.708918i
$196$		0			0
$197$	−22.4499			−1.59949			−0.799746	−	0.600338i	$-0.795033\pi$
$197$	−22.4499			−1.59949			−0.799746	+	0.600338i	$0.795033\pi$
$198$		0			0
$199$		0			0		1.00000			$0$
$199$		0			0		−1.00000			$\pi$
$200$		0			0
$201$	3.00000			0.211604
$202$		0			0
$203$		0			0
$204$		0			0
$205$	3.74166			0.261329
$206$		0			0
$207$		−	5.29150i		−	0.367785i
$208$		0			0
$209$	−6.00000	−	12.7279i	−0.415029	−	0.880409i
$210$		0			0
$211$		0			0		1.00000			$0$
$211$		0			0		−1.00000			$\pi$
$212$		0			0
$213$		−	2.64575i		−	0.181284i
$214$		0			0
$215$	22.4499			1.53107
$216$		0			0
$217$			29.6985i			2.01606i
$218$		0			0
$219$		−	12.7279i		−	0.860073i
$220$		0			0
$221$			15.8745i			1.06783i
$222$		0			0
$223$		−	7.93725i		−	0.531518i	−0.964040	−	0.265759i	$-0.914377\pi$
$223$		−	7.93725i		−	0.531518i	0.964040	−	0.265759i	$-0.0856225\pi$
$224$		0			0
$225$	4.00000			0.266667
$226$		0			0
$227$		−	15.5563i		−	1.03251i	−0.856435	−	0.516256i	$-0.827325\pi$
$227$		−	15.5563i		−	1.03251i	0.856435	−	0.516256i	$-0.172675\pi$
$228$		0			0
$229$		−	7.93725i		−	0.524509i	−0.964999	−	0.262254i	$-0.915534\pi$
$229$		−	7.93725i		−	0.524509i	0.964999	−	0.262254i	$-0.0844659\pi$
$230$		0			0
$231$	11.2250	−	5.29150i	0.738549	−	0.348155i
$232$		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
$235$	28.0000			1.82652
$236$		0			0
$237$	−7.48331			−0.486094
$238$		0			0
$239$		0			0			−	1.00000i	$-0.5\pi$
$239$		0			0				1.00000i	$0.5\pi$
$240$		0			0
$241$		−	16.9706i		−	1.09317i	−0.837404	−	0.546585i	$-0.815928\pi$
$241$		−	16.9706i		−	1.09317i	0.837404	−	0.546585i	$-0.184072\pi$
$242$		0			0
$243$	16.0000			1.02640
$244$		0			0
$245$		−	18.5203i		−	1.18322i
$246$		0			0
$247$			15.8745i			1.01007i
$248$		0			0
$249$			8.48528i			0.537733i
$250$		0			0
$251$	−3.00000			−0.189358			−0.0946792	−	0.995508i	$-0.530183\pi$
$251$	−3.00000			−0.189358			−0.0946792	+	0.995508i	$0.530183\pi$
$252$		0			0
$253$	−3.74166	−	7.93725i	−0.235236	−	0.499011i
$254$		0			0
$255$	11.2250			0.702935
$256$		0			0
$257$	−12.0000			−0.748539			−0.374270	−	0.927320i	$-0.622107\pi$
$257$	−12.0000			−0.748539			−0.374270	+	0.927320i	$0.622107\pi$
$258$		0			0
$259$		−	29.6985i		−	1.84537i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−11.2250			−0.692161			−0.346081	−	0.938205i	$-0.612488\pi$
$263$	−11.2250			−0.692161			−0.346081	+	0.938205i	$0.612488\pi$
$264$		0			0
$265$	−14.0000			−0.860013
$266$		0			0
$267$	3.00000			0.183597
$268$		0			0
$269$			5.29150i			0.322629i	0.986903	+	0.161314i	$0.0515733\pi$
$269$			5.29150i			0.322629i	−0.986903	+	0.161314i	$0.948427\pi$
$270$		0			0
$271$	18.7083			1.13645			0.568224	−	0.822874i	$-0.307631\pi$
$271$	18.7083			1.13645			0.568224	+	0.822874i	$0.307631\pi$
$272$		0			0
$273$	−14.0000			−0.847319
$274$		0			0
$275$	6.00000	−	2.82843i	0.361814	−	0.170561i
$276$		0			0
$277$	7.48331			0.449629			0.224814	−	0.974402i	$-0.427822\pi$
$277$	7.48331			0.449629			0.224814	+	0.974402i	$0.427822\pi$
$278$		0			0
$279$			15.8745i			0.950382i
$280$		0			0
$281$			8.48528i			0.506189i	0.967442	+	0.253095i	$0.0814484\pi$
$281$			8.48528i			0.506189i	−0.967442	+	0.253095i	$0.918552\pi$
$282$		0			0
$283$		−	16.9706i		−	1.00880i	−0.863472	−	0.504398i	$-0.831715\pi$
$283$		−	16.9706i		−	1.00880i	0.863472	−	0.504398i	$-0.168285\pi$
$284$		0			0
$285$	11.2250			0.664910
$286$		0			0
$287$		−	5.29150i		−	0.312348i
$288$		0			0
$289$	−1.00000			−0.0588235
$290$		0			0
$291$	−3.00000			−0.175863
$292$		0			0
$293$	11.2250			0.655770			0.327885	−	0.944718i	$-0.393664\pi$
$293$	11.2250			0.655770			0.327885	+	0.944718i	$0.393664\pi$
$294$		0			0
$295$			23.8118i			1.38637i
$296$		0			0
$297$	15.0000	−	7.07107i	0.870388	−	0.410305i
$298$		0			0
$299$			9.89949i			0.572503i
$300$		0			0
$301$		−	31.7490i		−	1.82998i
$302$		0			0
$303$	11.2250			0.644858
$304$		0			0
$305$			39.5980i			2.26737i
$306$		0			0
$307$		−	12.7279i		−	0.726421i	−0.931707	−	0.363210i	$-0.881681\pi$
$307$		−	12.7279i		−	0.726421i	0.931707	−	0.363210i	$-0.118319\pi$
$308$		0			0
$309$		−	15.8745i		−	0.903069i
$310$		0			0
$311$		−	10.5830i		−	0.600107i	−0.953922	−	0.300054i	$-0.902995\pi$
$311$		−	10.5830i		−	0.600107i	0.953922	−	0.300054i	$-0.0970046\pi$
$312$		0			0
$313$	−17.0000			−0.960897			−0.480448	−	0.877023i	$-0.659526\pi$
$313$	−17.0000			−0.960897			−0.480448	+	0.877023i	$0.659526\pi$
$314$		0			0
$315$		−	19.7990i		−	1.11555i
$316$		0			0
$317$			13.2288i			0.743001i	0.928433	+	0.371500i	$0.121157\pi$
$317$			13.2288i			0.743001i	−0.928433	+	0.371500i	$0.878843\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$			7.07107i			0.394669i
$322$		0			0
$323$	−18.0000			−1.00155
$324$		0			0
$325$	−7.48331			−0.415100
$326$		0			0
$327$	−7.48331			−0.413828
$328$		0			0
$329$		−	39.5980i		−	2.18311i
$330$		0			0
$331$	19.0000			1.04433			0.522167	−	0.852843i	$-0.325124\pi$
$331$	19.0000			1.04433			0.522167	+	0.852843i	$0.325124\pi$
$332$		0			0
$333$		−	15.8745i		−	0.869918i
$334$		0			0
$335$		−	7.93725i		−	0.433659i
$336$		0			0
$337$		−	4.24264i		−	0.231111i	−0.993301	−	0.115556i	$-0.963135\pi$
$337$		−	4.24264i		−	0.231111i	0.993301	−	0.115556i	$-0.0368649\pi$
$338$		0			0
$339$	3.00000			0.162938
$340$		0			0
$341$	11.2250	+	23.8118i	0.607866	+	1.28948i
$342$		0			0
$343$		0			0
$344$		0			0
$345$	7.00000			0.376867
$346$		0			0
$347$		−	12.7279i		−	0.683271i	−0.939833	−	0.341635i	$-0.889019\pi$
$347$		−	12.7279i		−	0.683271i	0.939833	−	0.341635i	$-0.110981\pi$
$348$		0			0
$349$	3.74166			0.200286			0.100143	−	0.994973i	$-0.468070\pi$
$349$	3.74166			0.200286			0.100143	+	0.994973i	$0.468070\pi$
$350$		0			0
$351$	−18.7083			−0.998574
$352$		0			0
$353$	33.0000			1.75641			0.878206	−	0.478282i	$-0.158740\pi$
$353$	33.0000			1.75641			0.878206	+	0.478282i	$0.158740\pi$
$354$		0			0
$355$	−7.00000			−0.371521
$356$		0			0
$357$		−	15.8745i		−	0.840168i
$358$		0			0
$359$	−11.2250			−0.592431			−0.296216	−	0.955121i	$-0.595725\pi$
$359$	−11.2250			−0.592431			−0.296216	+	0.955121i	$0.595725\pi$
$360$		0			0
$361$	1.00000			0.0526316
$362$		0			0
$363$	7.00000	−	8.48528i	0.367405	−	0.445362i
$364$		0			0
$365$	−33.6749			−1.76263
$366$		0			0
$367$		−	7.93725i		−	0.414321i	−0.978307	−	0.207161i	$-0.933578\pi$
$367$		−	7.93725i		−	0.414321i	0.978307	−	0.207161i	$-0.0664223\pi$
$368$		0			0
$369$		−	2.82843i		−	0.147242i
$370$		0			0
$371$			19.7990i			1.02791i
$372$		0			0
$373$	26.1916			1.35615			0.678075	−	0.734993i	$-0.262815\pi$
$373$	26.1916			1.35615			0.678075	+	0.734993i	$0.262815\pi$
$374$		0			0
$375$		−	7.93725i		−	0.409878i
$376$		0			0
$377$		0			0
$378$		0			0
$379$	11.0000			0.565032			0.282516	−	0.959263i	$-0.408831\pi$
$379$	11.0000			0.565032			0.282516	+	0.959263i	$0.408831\pi$
$380$		0			0
$381$	−18.7083			−0.958455
$382$		0			0
$383$			29.1033i			1.48711i	0.668676	+	0.743554i	$0.266861\pi$
$383$			29.1033i			1.48711i	−0.668676	+	0.743554i	$0.733139\pi$
$384$		0			0
$385$	−14.0000	−	29.6985i	−0.713506	−	1.51357i
$386$		0			0
$387$		−	16.9706i		−	0.862662i
$388$		0			0
$389$			34.3948i			1.74388i	0.489609	+	0.871942i	$0.337139\pi$
$389$			34.3948i			1.74388i	−0.489609	+	0.871942i	$0.662861\pi$
$390$		0			0
$391$	−11.2250			−0.567671
$392$		0			0
$393$			21.2132i			1.07006i
$394$		0			0
$395$			19.7990i			0.996195i
$396$		0			0
$397$			15.8745i			0.796719i	0.917229	+	0.398359i	$0.130420\pi$
$397$			15.8745i			0.796719i	−0.917229	+	0.398359i	$0.869580\pi$
$398$		0			0
$399$		−	15.8745i		−	0.794719i
$400$		0			0
$401$	36.0000			1.79775			0.898877	−	0.438201i	$-0.144384\pi$
$401$	36.0000			1.79775			0.898877	+	0.438201i	$0.144384\pi$
$402$		0			0
$403$		−	29.6985i		−	1.47939i
$404$		0			0
$405$		−	2.64575i		−	0.131468i
$406$		0			0
$407$	−11.2250	−	23.8118i	−0.556401	−	1.18031i
$408$		0			0
$409$		−	25.4558i		−	1.25871i	−0.777118	−	0.629355i	$-0.783319\pi$
$409$		−	25.4558i		−	1.25871i	0.777118	−	0.629355i	$-0.216681\pi$
$410$		0			0
$411$	−15.0000			−0.739895
$412$		0			0
$413$	33.6749			1.65703
$414$		0			0
$415$	22.4499			1.10202
$416$		0			0
$417$		−	12.7279i		−	0.623289i
$418$		0			0
$419$	6.00000			0.293119			0.146560	−	0.989202i	$-0.453180\pi$
$419$	6.00000			0.293119			0.146560	+	0.989202i	$0.453180\pi$
$420$		0			0
$421$			15.8745i			0.773676i	0.922148	+	0.386838i	$0.126433\pi$
$421$			15.8745i			0.773676i	−0.922148	+	0.386838i	$0.873567\pi$
$422$		0			0
$423$		−	21.1660i		−	1.02913i
$424$		0			0
$425$		−	8.48528i		−	0.411597i
$426$		0			0
$427$	56.0000			2.71003
$428$		0			0
$429$	−11.2250	+	5.29150i	−0.541947	+	0.255476i
$430$		0			0
$431$	−33.6749			−1.62206			−0.811032	−	0.585002i	$-0.801094\pi$
$431$	−33.6749			−1.62206			−0.811032	+	0.585002i	$0.801094\pi$
$432$		0			0
$433$	25.0000			1.20142			0.600712	−	0.799466i	$-0.294884\pi$
$433$	25.0000			1.20142			0.600712	+	0.799466i	$0.294884\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−11.2250			−0.536963
$438$		0			0
$439$	14.9666			0.714318			0.357159	−	0.934044i	$-0.383745\pi$
$439$	14.9666			0.714318			0.357159	+	0.934044i	$0.383745\pi$
$440$		0			0
$441$	−14.0000			−0.666667
$442$		0			0
$443$	−15.0000			−0.712672			−0.356336	−	0.934358i	$-0.615974\pi$
$443$	−15.0000			−0.712672			−0.356336	+	0.934358i	$0.615974\pi$
$444$		0			0
$445$		−	7.93725i		−	0.376262i
$446$		0			0
$447$	−11.2250			−0.530923
$448$		0			0
$449$	−21.0000			−0.991051			−0.495526	−	0.868593i	$-0.665025\pi$
$449$	−21.0000			−0.991051			−0.495526	+	0.868593i	$0.665025\pi$
$450$		0			0
$451$	−2.00000	−	4.24264i	−0.0941763	−	0.199778i
$452$		0			0
$453$	−14.9666			−0.703194
$454$		0			0
$455$			37.0405i			1.73649i
$456$		0			0
$457$			33.9411i			1.58770i	0.608114	+	0.793849i	$0.291926\pi$
$457$			33.9411i			1.58770i	−0.608114	+	0.793849i	$0.708074\pi$
$458$		0			0
$459$		−	21.2132i		−	0.990148i
$460$		0			0
$461$	−22.4499			−1.04560			−0.522799	−	0.852456i	$-0.675112\pi$
$461$	−22.4499			−1.04560			−0.522799	+	0.852456i	$0.675112\pi$
$462$		0			0
$463$		−	23.8118i		−	1.10663i	−0.832973	−	0.553313i	$-0.813363\pi$
$463$		−	23.8118i		−	1.10663i	0.832973	−	0.553313i	$-0.186637\pi$
$464$		0			0
$465$	−21.0000			−0.973852
$466$		0			0
$467$	−3.00000			−0.138823			−0.0694117	−	0.997588i	$-0.522112\pi$
$467$	−3.00000			−0.138823			−0.0694117	+	0.997588i	$0.522112\pi$
$468$		0			0
$469$	−11.2250			−0.518321
$470$		0			0
$471$			7.93725i			0.365729i
$472$		0			0
$473$	−12.0000	−	25.4558i	−0.551761	−	1.17046i
$474$		0			0
$475$		−	8.48528i		−	0.389331i
$476$		0			0
$477$			10.5830i			0.484563i
$478$		0			0
$479$	−33.6749			−1.53865			−0.769323	−	0.638860i	$-0.779406\pi$
$479$	−33.6749			−1.53865			−0.769323	+	0.638860i	$0.779406\pi$
$480$		0			0
$481$			29.6985i			1.35413i
$482$		0			0
$483$		−	9.89949i		−	0.450443i
$484$		0			0
$485$			7.93725i			0.360412i
$486$		0			0
$487$		−	39.6863i		−	1.79836i	−0.437582	−	0.899178i	$-0.644165\pi$
$487$		−	39.6863i		−	1.79836i	0.437582	−	0.899178i	$-0.355835\pi$
$488$		0			0
$489$	−2.00000			−0.0904431
$490$		0			0
$491$			39.5980i			1.78703i	0.449032	+	0.893516i	$0.351769\pi$
$491$			39.5980i			1.78703i	−0.449032	+	0.893516i	$0.648231\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$	−7.48331	−	15.8745i	−0.336350	−	0.713506i
$496$		0			0
$497$			9.89949i			0.444053i
$498$		0			0
$499$	−22.0000			−0.984855			−0.492428	−	0.870353i	$-0.663890\pi$
$499$	−22.0000			−0.984855			−0.492428	+	0.870353i	$0.663890\pi$
$500$		0			0
$501$	11.2250			0.501495
$502$		0			0
$503$	22.4499			1.00099			0.500497	−	0.865738i	$-0.333151\pi$
$503$	22.4499			1.00099			0.500497	+	0.865738i	$0.333151\pi$
$504$		0			0
$505$		−	29.6985i		−	1.32157i
$506$		0			0
$507$	1.00000			0.0444116
$508$		0			0
$509$			2.64575i			0.117271i	0.998279	+	0.0586354i	$0.0186749\pi$
$509$			2.64575i			0.117271i	−0.998279	+	0.0586354i	$0.981325\pi$
$510$		0			0
$511$			47.6235i			2.10674i
$512$		0			0
$513$		−	21.2132i		−	0.936586i
$514$		0			0
$515$	−42.0000			−1.85074
$516$		0			0
$517$	−14.9666	−	31.7490i	−0.658232	−	1.39632i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−27.0000			−1.18289			−0.591446	−	0.806345i	$-0.701443\pi$
$521$	−27.0000			−1.18289			−0.591446	+	0.806345i	$0.701443\pi$
$522$		0			0
$523$			21.2132i			0.927589i	0.885943	+	0.463794i	$0.153512\pi$
$523$			21.2132i			0.927589i	−0.885943	+	0.463794i	$0.846488\pi$
$524$		0			0
$525$	7.48331			0.326599
$526$		0			0
$527$	33.6749			1.46690
$528$		0			0
$529$	16.0000			0.695652
$530$		0			0
$531$	18.0000			0.781133
$532$		0			0
$533$			5.29150i			0.229200i
$534$		0			0
$535$	18.7083			0.808830
$536$		0			0
$537$	21.0000			0.906217
$538$		0			0
$539$	−21.0000	+	9.89949i	−0.904534	+	0.426401i
$540$		0			0
$541$	3.74166			0.160866			0.0804332	−	0.996760i	$-0.474370\pi$
$541$	3.74166			0.160866			0.0804332	+	0.996760i	$0.474370\pi$
$542$		0			0
$543$			23.8118i			1.02186i
$544$		0			0
$545$			19.7990i			0.848096i
$546$		0			0
$547$		−	8.48528i		−	0.362804i	−0.983409	−	0.181402i	$-0.941936\pi$
$547$		−	8.48528i		−	0.362804i	0.983409	−	0.181402i	$-0.0580636\pi$
$548$		0			0
$549$	29.9333			1.27752
$550$		0			0
$551$		0			0
$552$		0			0
$553$	28.0000			1.19068
$554$		0			0
$555$	21.0000			0.891400
$556$		0			0
$557$	33.6749			1.42685			0.713426	−	0.700731i	$-0.247142\pi$
$557$	33.6749			1.42685			0.713426	+	0.700731i	$0.247142\pi$
$558$		0			0
$559$			31.7490i			1.34284i
$560$		0			0
$561$	−6.00000	−	12.7279i	−0.253320	−	0.537373i
$562$		0			0
$563$			14.1421i			0.596020i	0.954563	+	0.298010i	$0.0963229\pi$
$563$			14.1421i			0.596020i	−0.954563	+	0.298010i	$0.903677\pi$
$564$		0			0
$565$		−	7.93725i		−	0.333923i
$566$		0			0
$567$	−3.74166			−0.157135
$568$		0			0
$569$			42.4264i			1.77861i	0.457317	+	0.889304i	$0.348810\pi$
$569$			42.4264i			1.77861i	−0.457317	+	0.889304i	$0.651190\pi$
$570$		0			0
$571$			16.9706i			0.710196i	0.934829	+	0.355098i	$0.115552\pi$
$571$			16.9706i			0.710196i	−0.934829	+	0.355098i	$0.884448\pi$
$572$		0			0
$573$			2.64575i			0.110528i
$574$		0			0
$575$		−	5.29150i		−	0.220671i
$576$		0			0
$577$	27.0000			1.12402			0.562012	−	0.827129i	$-0.310027\pi$
$577$	27.0000			1.12402			0.562012	+	0.827129i	$0.310027\pi$
$578$		0			0
$579$			16.9706i			0.705273i
$580$		0			0
$581$		−	31.7490i		−	1.31717i
$582$		0			0
$583$	7.48331	+	15.8745i	0.309927	+	0.657455i
$584$		0			0
$585$			19.7990i			0.818587i
$586$		0			0
$587$	24.0000			0.990586			0.495293	−	0.868726i	$-0.335061\pi$
$587$	24.0000			0.990586			0.495293	+	0.868726i	$0.335061\pi$
$588$		0			0
$589$	33.6749			1.38755
$590$		0			0
$591$	−22.4499			−0.923467
$592$		0			0
$593$		−	25.4558i		−	1.04535i	−0.852533	−	0.522673i	$-0.824935\pi$
$593$		−	25.4558i		−	1.04535i	0.852533	−	0.522673i	$-0.175065\pi$
$594$		0			0
$595$	−42.0000			−1.72183
$596$		0			0
$597$		0			0
$598$		0			0
$599$		−	10.5830i		−	0.432410i	−0.976348	−	0.216205i	$-0.930632\pi$
$599$		−	10.5830i		−	0.432410i	0.976348	−	0.216205i	$-0.0693679\pi$
$600$		0			0
$601$		−	12.7279i		−	0.519183i	−0.965719	−	0.259591i	$-0.916412\pi$
$601$		−	12.7279i		−	0.519183i	0.965719	−	0.259591i	$-0.0835879\pi$
$602$		0			0
$603$	−6.00000			−0.244339
$604$		0			0
$605$	−22.4499	−	18.5203i	−0.912720	−	0.752956i
$606$		0			0
$607$	29.9333			1.21495			0.607477	−	0.794337i	$-0.292182\pi$
$607$	29.9333			1.21495			0.607477	+	0.794337i	$0.292182\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$			39.5980i			1.60196i
$612$		0			0
$613$	−37.4166			−1.51124			−0.755621	−	0.655010i	$-0.772665\pi$
$613$	−37.4166			−1.51124			−0.755621	+	0.655010i	$0.772665\pi$
$614$		0			0
$615$	3.74166			0.150878
$616$		0			0
$617$	−12.0000			−0.483102			−0.241551	−	0.970388i	$-0.577656\pi$
$617$	−12.0000			−0.483102			−0.241551	+	0.970388i	$0.577656\pi$
$618$		0			0
$619$	27.0000			1.08522			0.542611	−	0.839984i	$-0.317436\pi$
$619$	27.0000			1.08522			0.542611	+	0.839984i	$0.317436\pi$
$620$		0			0
$621$		−	13.2288i		−	0.530852i
$622$		0			0
$623$	−11.2250			−0.449719
$624$		0			0
$625$	−31.0000			−1.24000
$626$		0			0
$627$	−6.00000	−	12.7279i	−0.239617	−	0.508304i
$628$		0			0
$629$	−33.6749			−1.34271
$630$		0			0
$631$		−	23.8118i		−	0.947931i	−0.880543	−	0.473966i	$-0.842822\pi$
$631$		−	23.8118i		−	0.947931i	0.880543	−	0.473966i	$-0.157178\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$			49.4975i			1.96425i
$636$		0			0
$637$	26.1916			1.03775
$638$		0			0
$639$			5.29150i			0.209329i
$640$		0			0
$641$	−21.0000			−0.829450			−0.414725	−	0.909947i	$-0.636122\pi$
$641$	−21.0000			−0.829450			−0.414725	+	0.909947i	$0.636122\pi$
$642$		0			0
$643$	−45.0000			−1.77463			−0.887313	−	0.461167i	$-0.847431\pi$
$643$	−45.0000			−1.77463			−0.887313	+	0.461167i	$0.847431\pi$
$644$		0			0
$645$	22.4499			0.883966
$646$		0			0
$647$		−	2.64575i		−	0.104015i	−0.998647	−	0.0520076i	$-0.983438\pi$
$647$		−	2.64575i		−	0.104015i	0.998647	−	0.0520076i	$-0.0165620\pi$
$648$		0			0
$649$	27.0000	−	12.7279i	1.05984	−	0.499615i
$650$		0			0
$651$			29.6985i			1.16398i
$652$		0			0
$653$			13.2288i			0.517681i	0.965920	+	0.258841i	$0.0833404\pi$
$653$			13.2288i			0.517681i	−0.965920	+	0.258841i	$0.916660\pi$
$654$		0			0
$655$	56.1249			2.19298
$656$		0			0
$657$			25.4558i			0.993127i
$658$		0			0
$659$			28.2843i			1.10180i	0.834572	+	0.550899i	$0.185715\pi$
$659$			28.2843i			1.10180i	−0.834572	+	0.550899i	$0.814285\pi$
$660$		0			0
$661$		−	23.8118i		−	0.926170i	−0.886314	−	0.463085i	$-0.846742\pi$
$661$		−	23.8118i		−	0.926170i	0.886314	−	0.463085i	$-0.153258\pi$
$662$		0			0
$663$			15.8745i			0.616515i
$664$		0			0
$665$	−42.0000			−1.62869
$666$		0			0
$667$		0			0
$668$		0			0
$669$		−	7.93725i		−	0.306872i
$670$		0			0
$671$	44.8999	−	21.1660i	1.73334	−	0.817105i
$672$		0			0
$673$			8.48528i			0.327084i	0.986536	+	0.163542i	$0.0522919\pi$
$673$			8.48528i			0.327084i	−0.986536	+	0.163542i	$0.947708\pi$
$674$		0			0
$675$	10.0000			0.384900
$676$		0			0
$677$	33.6749			1.29423			0.647116	−	0.762391i	$-0.275975\pi$
$677$	33.6749			1.29423			0.647116	+	0.762391i	$0.275975\pi$
$678$		0			0
$679$	11.2250			0.430775
$680$		0			0
$681$		−	15.5563i		−	0.596121i
$682$		0			0
$683$	18.0000			0.688751			0.344375	−	0.938832i	$-0.388091\pi$
$683$	18.0000			0.688751			0.344375	+	0.938832i	$0.388091\pi$
$684$		0			0
$685$			39.6863i			1.51633i
$686$		0			0
$687$		−	7.93725i		−	0.302825i
$688$		0			0
$689$		−	19.7990i		−	0.754281i
$690$		0			0
$691$	−33.0000			−1.25538			−0.627690	−	0.778464i	$-0.715999\pi$
$691$	−33.0000			−1.25538			−0.627690	+	0.778464i	$0.715999\pi$
$692$		0			0
$693$	−22.4499	+	10.5830i	−0.852803	+	0.402015i
$694$		0			0
$695$	−33.6749			−1.27736
$696$		0			0
$697$	−6.00000			−0.227266
$698$		0			0
$699$			5.65685i			0.213962i
$700$		0			0
$701$	−11.2250			−0.423961			−0.211981	−	0.977274i	$-0.567991\pi$
$701$	−11.2250			−0.423961			−0.211981	+	0.977274i	$0.567991\pi$
$702$		0			0
$703$	−33.6749			−1.27007
$704$		0			0
$705$	28.0000			1.05454
$706$		0			0
$707$	−42.0000			−1.57957
$708$		0			0
$709$		−	7.93725i		−	0.298090i	−0.988830	−	0.149045i	$-0.952380\pi$
$709$		−	7.93725i		−	0.298090i	0.988830	−	0.149045i	$-0.0476199\pi$
$710$		0			0
$711$	14.9666			0.561292
$712$		0			0
$713$	21.0000			0.786456
$714$		0			0
$715$	14.0000	+	29.6985i	0.523570	+	1.11066i
$716$		0			0
$717$		0			0
$718$		0			0
$719$			13.2288i			0.493349i	0.969098	+	0.246675i	$0.0793379\pi$
$719$			13.2288i			0.493349i	−0.969098	+	0.246675i	$0.920662\pi$
$720$		0			0
$721$			59.3970i			2.21206i
$722$		0			0
$723$		−	16.9706i		−	0.631142i
$724$		0			0
$725$		0			0
$726$		0			0
$727$			7.93725i			0.294376i	0.989109	+	0.147188i	$0.0470223\pi$
$727$			7.93725i			0.294376i	−0.989109	+	0.147188i	$0.952978\pi$
$728$		0			0
$729$	13.0000			0.481481
$730$		0			0
$731$	−36.0000			−1.33151
$732$		0			0
$733$	−29.9333			−1.10561			−0.552805	−	0.833311i	$-0.686443\pi$
$733$	−29.9333			−1.10561			−0.552805	+	0.833311i	$0.686443\pi$
$734$		0			0
$735$		−	18.5203i		−	0.683130i
$736$		0			0
$737$	−9.00000	+	4.24264i	−0.331519	+	0.156280i
$738$		0			0
$739$		−	16.9706i		−	0.624272i	−0.950037	−	0.312136i	$-0.898955\pi$
$739$		−	16.9706i		−	0.624272i	0.950037	−	0.312136i	$-0.101045\pi$
$740$		0			0
$741$			15.8745i			0.583165i
$742$		0			0
$743$	−22.4499			−0.823609			−0.411804	−	0.911272i	$-0.635101\pi$
$743$	−22.4499			−0.823609			−0.411804	+	0.911272i	$0.635101\pi$
$744$		0			0
$745$			29.6985i			1.08807i
$746$		0			0
$747$		−	16.9706i		−	0.620920i
$748$		0			0
$749$		−	26.4575i		−	0.966736i
$750$		0			0
$751$		−	23.8118i		−	0.868904i	−0.900695	−	0.434452i	$-0.856942\pi$
$751$		−	23.8118i		−	0.868904i	0.900695	−	0.434452i	$-0.143058\pi$
$752$		0			0
$753$	−3.00000			−0.109326
$754$		0			0
$755$			39.5980i			1.44112i
$756$		0			0
$757$			47.6235i			1.73091i	0.500990	+	0.865453i	$0.332969\pi$
$757$			47.6235i			1.73091i	−0.500990	+	0.865453i	$0.667031\pi$
$758$		0			0
$759$	−3.74166	−	7.93725i	−0.135814	−	0.288104i
$760$		0			0
$761$			38.1838i			1.38416i	0.721821	+	0.692080i	$0.243306\pi$
$761$			38.1838i			1.38416i	−0.721821	+	0.692080i	$0.756694\pi$
$762$		0			0
$763$	28.0000			1.01367
$764$		0			0
$765$	−22.4499			−0.811679
$766$		0			0
$767$	−33.6749			−1.21593
$768$		0			0
$769$		−	38.1838i		−	1.37694i	−0.725264	−	0.688471i	$-0.758282\pi$
$769$		−	38.1838i		−	1.37694i	0.725264	−	0.688471i	$-0.241718\pi$
$770$		0			0
$771$	−12.0000			−0.432169
$772$		0			0
$773$			21.1660i			0.761288i	0.924722	+	0.380644i	$0.124298\pi$
$773$			21.1660i			0.761288i	−0.924722	+	0.380644i	$0.875702\pi$
$774$		0			0
$775$			15.8745i			0.570229i
$776$		0			0
$777$		−	29.6985i		−	1.06543i
$778$		0			0
$779$	−6.00000			−0.214972
$780$		0			0
$781$	3.74166	+	7.93725i	0.133887	+	0.284017i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	21.0000			0.749522
$786$		0			0
$787$		−	16.9706i		−	0.604935i	−0.953160	−	0.302468i	$-0.902190\pi$
$787$		−	16.9706i		−	0.604935i	0.953160	−	0.302468i	$-0.0978104\pi$
$788$		0			0
$789$	−11.2250			−0.399620
$790$		0			0
$791$	−11.2250			−0.399114
$792$		0			0
$793$	−56.0000			−1.98862
$794$		0			0
$795$	−14.0000			−0.496529
$796$		0			0
$797$		−	13.2288i		−	0.468587i	−0.972166	−	0.234293i	$-0.924722\pi$
$797$		−	13.2288i		−	0.468587i	0.972166	−	0.234293i	$-0.0752776\pi$
$798$		0			0
$799$	−44.8999			−1.58844
$800$		0			0
$801$	−6.00000			−0.212000
$802$		0			0
$803$	18.0000	+	38.1838i	0.635206	+	1.34748i
$804$		0			0
$805$	−26.1916			−0.923133
$806$		0			0
$807$			5.29150i			0.186270i
$808$		0			0
$809$			32.5269i			1.14359i	0.820398	+	0.571793i	$0.193752\pi$
$809$			32.5269i			1.14359i	−0.820398	+	0.571793i	$0.806248\pi$
$810$		0			0
$811$			50.9117i			1.78775i	0.448315	+	0.893876i	$0.352024\pi$
$811$			50.9117i			1.78775i	−0.448315	+	0.893876i	$0.647976\pi$
$812$		0			0
$813$	18.7083			0.656128
$814$		0			0
$815$			5.29150i			0.185353i
$816$		0			0
$817$	−36.0000			−1.25948
$818$		0			0
$819$	28.0000			0.978399
$820$		0			0
$821$	22.4499			0.783508			0.391754	−	0.920070i	$-0.371868\pi$
$821$	22.4499			0.783508			0.391754	+	0.920070i	$0.371868\pi$
$822$		0			0
$823$			23.8118i			0.830026i	0.909816	+	0.415013i	$0.136223\pi$
$823$			23.8118i			0.830026i	−0.909816	+	0.415013i	$0.863777\pi$
$824$		0			0
$825$	6.00000	−	2.82843i	0.208893	−	0.0984732i
$826$		0			0
$827$		−	4.24264i		−	0.147531i	−0.997276	−	0.0737655i	$-0.976498\pi$
$827$		−	4.24264i		−	0.147531i	0.997276	−	0.0737655i	$-0.0235016\pi$
$828$		0			0
$829$			23.8118i			0.827017i	0.910500	+	0.413508i	$0.135697\pi$
$829$			23.8118i			0.827017i	−0.910500	+	0.413508i	$0.864303\pi$
$830$		0			0
$831$	7.48331			0.259593
$832$		0			0
$833$			29.6985i			1.02899i
$834$		0			0
$835$		−	29.6985i		−	1.02776i
$836$		0			0
$837$			39.6863i			1.37176i
$838$		0			0
$839$			34.3948i			1.18744i	0.804672	+	0.593720i	$0.202341\pi$
$839$			34.3948i			1.18744i	−0.804672	+	0.593720i	$0.797659\pi$
$840$		0			0
$841$	−29.0000			−1.00000
$842$		0			0
$843$			8.48528i			0.292249i
$844$		0			0
$845$		−	2.64575i		−	0.0910166i
$846$		0			0
$847$	−26.1916	+	31.7490i	−0.899954	+	1.09091i
$848$		0			0
$849$		−	16.9706i		−	0.582428i
$850$		0			0
$851$	−21.0000			−0.719871
$852$		0			0
$853$	29.9333			1.02490			0.512448	−	0.858718i	$-0.328739\pi$
$853$	29.9333			1.02490			0.512448	+	0.858718i	$0.328739\pi$
$854$		0			0
$855$	−22.4499			−0.767772
$856$		0			0
$857$		−	22.6274i		−	0.772938i	−0.922302	−	0.386469i	$-0.873695\pi$
$857$		−	22.6274i		−	0.772938i	0.922302	−	0.386469i	$-0.126305\pi$
$858$		0			0
$859$	−31.0000			−1.05771			−0.528853	−	0.848713i	$-0.677378\pi$
$859$	−31.0000			−1.05771			−0.528853	+	0.848713i	$0.677378\pi$
$860$		0			0
$861$		−	5.29150i		−	0.180334i
$862$		0			0
$863$		−	52.9150i		−	1.80125i	−0.434599	−	0.900624i	$-0.643110\pi$
$863$		−	52.9150i		−	1.80125i	0.434599	−	0.900624i	$-0.356890\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	−1.00000			−0.0339618
$868$		0			0
$869$	22.4499	−	10.5830i	0.761562	−	0.359004i
$870$		0			0
$871$	11.2250			0.380344
$872$		0			0
$873$	6.00000			0.203069
$874$		0			0
$875$			29.6985i			1.00399i
$876$		0			0
$877$	−3.74166			−0.126347			−0.0631734	−	0.998003i	$-0.520122\pi$
$877$	−3.74166			−0.126347			−0.0631734	+	0.998003i	$0.520122\pi$
$878$		0			0
$879$	11.2250			0.378609
$880$		0			0
$881$	−3.00000			−0.101073			−0.0505363	−	0.998722i	$-0.516093\pi$
$881$	−3.00000			−0.101073			−0.0505363	+	0.998722i	$0.516093\pi$
$882$		0			0
$883$		0			0			−	1.00000i	$-0.5\pi$
$883$		0			0				1.00000i	$0.5\pi$
$884$		0			0
$885$			23.8118i			0.800424i
$886$		0			0
$887$	22.4499			0.753795			0.376898	−	0.926255i	$-0.376991\pi$
$887$	22.4499			0.753795			0.376898	+	0.926255i	$0.376991\pi$
$888$		0			0
$889$	70.0000			2.34772
$890$		0			0
$891$	−3.00000	+	1.41421i	−0.100504	+	0.0473779i
$892$		0			0
$893$	−44.8999			−1.50252
$894$		0			0
$895$		−	55.5608i		−	1.85719i
$896$		0			0
$897$			9.89949i			0.330535i
$898$		0			0
$899$		0			0
$900$		0			0
$901$	22.4499			0.747916
$902$		0			0
$903$		−	31.7490i		−	1.05654i
$904$		0			0
$905$	63.0000			2.09419
$906$		0			0
$907$	42.0000			1.39459			0.697294	−	0.716786i	$-0.254387\pi$
$907$	42.0000			1.39459			0.697294	+	0.716786i	$0.254387\pi$
$908$		0			0
$909$	−22.4499			−0.744618
$910$		0			0
$911$		−	52.9150i		−	1.75315i	−0.481263	−	0.876577i	$-0.659822\pi$
$911$		−	52.9150i		−	1.75315i	0.481263	−	0.876577i	$-0.340178\pi$
$912$		0			0
$913$	−12.0000	−	25.4558i	−0.397142	−	0.842465i
$914$		0			0
$915$			39.5980i			1.30907i
$916$		0			0
$917$		−	79.3725i		−	2.62111i
$918$		0			0
$919$	−14.9666			−0.493704			−0.246852	−	0.969053i	$-0.579396\pi$
$919$	−14.9666			−0.493704			−0.246852	+	0.969053i	$0.579396\pi$
$920$		0			0
$921$		−	12.7279i		−	0.419399i
$922$		0			0
$923$		−	9.89949i		−	0.325846i
$924$		0			0
$925$		−	15.8745i		−	0.521951i
$926$		0			0
$927$			31.7490i			1.04277i
$928$		0			0
$929$	12.0000			0.393707			0.196854	−	0.980433i	$-0.436928\pi$
$929$	12.0000			0.393707			0.196854	+	0.980433i	$0.436928\pi$
$930$		0			0
$931$			29.6985i			0.973329i
$932$		0			0
$933$		−	10.5830i		−	0.346472i
$934$		0			0
$935$	−33.6749	+	15.8745i	−1.10129	+	0.519152i
$936$		0			0
$937$		−	21.2132i		−	0.693005i	−0.938049	−	0.346503i	$-0.887369\pi$
$937$		−	21.2132i		−	0.693005i	0.938049	−	0.346503i	$-0.112631\pi$
$938$		0			0
$939$	−17.0000			−0.554774
$940$		0			0
$941$	22.4499			0.731847			0.365924	−	0.930645i	$-0.380753\pi$
$941$	22.4499			0.731847			0.365924	+	0.930645i	$0.380753\pi$
$942$		0			0
$943$	−3.74166			−0.121845
$944$		0			0
$945$		−	49.4975i		−	1.61015i
$946$		0			0
$947$	9.00000			0.292461			0.146230	−	0.989251i	$-0.453286\pi$
$947$	9.00000			0.292461			0.146230	+	0.989251i	$0.453286\pi$
$948$		0			0
$949$		−	47.6235i		−	1.54592i
$950$		0			0
$951$			13.2288i			0.428972i
$952$		0			0
$953$		−	43.8406i		−	1.42014i	−0.704133	−	0.710068i	$-0.748664\pi$
$953$		−	43.8406i		−	1.42014i	0.704133	−	0.710068i	$-0.251336\pi$
$954$		0			0
$955$	7.00000			0.226515
$956$		0			0
$957$		0			0
$958$		0			0
$959$	56.1249			1.81237
$960$		0			0
$961$	−32.0000			−1.03226
$962$		0			0
$963$		−	14.1421i		−	0.455724i
$964$		0			0
$965$	44.8999			1.44538
$966$		0			0
$967$	−7.48331			−0.240647			−0.120324	−	0.992735i	$-0.538393\pi$
$967$	−7.48331			−0.240647			−0.120324	+	0.992735i	$0.538393\pi$
$968$		0			0
$969$	−18.0000			−0.578243
$970$		0			0
$971$	−45.0000			−1.44412			−0.722059	−	0.691831i	$-0.756804\pi$
$971$	−45.0000			−1.44412			−0.722059	+	0.691831i	$0.756804\pi$
$972$		0			0
$973$			47.6235i			1.52674i
$974$		0			0
$975$	−7.48331			−0.239658
$976$		0			0
$977$	−27.0000			−0.863807			−0.431903	−	0.901920i	$-0.642158\pi$
$977$	−27.0000			−0.863807			−0.431903	+	0.901920i	$0.642158\pi$
$978$		0			0
$979$	−9.00000	+	4.24264i	−0.287641	+	0.135595i
$980$		0			0
$981$	14.9666			0.477848
$982$		0			0
$983$		−	18.5203i		−	0.590705i	−0.955388	−	0.295352i	$-0.904563\pi$
$983$		−	18.5203i		−	0.590705i	0.955388	−	0.295352i	$-0.0954370\pi$
$984$		0			0
$985$			59.3970i			1.89254i
$986$		0			0
$987$		−	39.5980i		−	1.26042i
$988$		0			0
$989$	−22.4499			−0.713867
$990$		0			0
$991$			31.7490i			1.00854i	0.863546	+	0.504270i	$0.168238\pi$
$991$			31.7490i			1.00854i	−0.863546	+	0.504270i	$0.831762\pi$
$992$		0			0
$993$	19.0000			0.602947
$994$		0			0
$995$		0			0
$996$		0			0
$997$	−48.6415			−1.54049			−0.770247	−	0.637746i	$-0.779867\pi$
$997$	−48.6415			−1.54049			−0.770247	+	0.637746i	$0.779867\pi$
$998$		0			0
$999$		−	39.6863i		−	1.25562i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1408.2.g.d.703.1	yes	4
4.3	odd	2		1408.2.g.a.703.2	yes	4
8.3	odd	2	inner	1408.2.g.d.703.4	yes	4
8.5	even	2		1408.2.g.a.703.3	yes	4
11.10	odd	2	inner	1408.2.g.d.703.2	yes	4
16.3	odd	4		2816.2.e.n.2815.1		8
16.5	even	4		2816.2.e.n.2815.4		8
16.11	odd	4		2816.2.e.n.2815.7		8
16.13	even	4		2816.2.e.n.2815.6		8
44.43	even	2		1408.2.g.a.703.1	✓	4
88.21	odd	2		1408.2.g.a.703.4	yes	4
88.43	even	2	inner	1408.2.g.d.703.3	yes	4
176.21	odd	4		2816.2.e.n.2815.3		8
176.43	even	4		2816.2.e.n.2815.8		8
176.109	odd	4		2816.2.e.n.2815.5		8
176.131	even	4		2816.2.e.n.2815.2		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1408.2.g.a.703.1	✓	4	44.43	even	2
1408.2.g.a.703.2	yes	4	4.3	odd	2
1408.2.g.a.703.3	yes	4	8.5	even	2
1408.2.g.a.703.4	yes	4	88.21	odd	2
1408.2.g.d.703.1	yes	4	1.1	even	1	trivial
1408.2.g.d.703.2	yes	4	11.10	odd	2	inner
1408.2.g.d.703.3	yes	4	88.43	even	2	inner
1408.2.g.d.703.4	yes	4	8.3	odd	2	inner
2816.2.e.n.2815.1		8	16.3	odd	4
2816.2.e.n.2815.2		8	176.131	even	4
2816.2.e.n.2815.3		8	176.21	odd	4
2816.2.e.n.2815.4		8	16.5	even	4
2816.2.e.n.2815.5		8	176.109	odd	4
2816.2.e.n.2815.6		8	16.13	even	4
2816.2.e.n.2815.7		8	16.11	odd	4
2816.2.e.n.2815.8		8	176.43	even	4

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:	\( N \)	\(=\)	\( 1408 = 2^{7} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1408.g (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} -2.64575i q^{5} -3.74166 q^{7} -2.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} -2.64575i q^{5} -3.74166 q^{7} -2.00000 q^{9} +(-3.00000 + 1.41421i) q^{11} +3.74166 q^{13} -2.64575i q^{15} +4.24264i q^{17} +4.24264i q^{19} -3.74166 q^{21} +2.64575i q^{23} -2.00000 q^{25} -5.00000 q^{27} -7.93725i q^{31} +(-3.00000 + 1.41421i) q^{33} +9.89949i q^{35} +7.93725i q^{37} +3.74166 q^{39} +1.41421i q^{41} +8.48528i q^{43} +5.29150i q^{45} +10.5830i q^{47} +7.00000 q^{49} +4.24264i q^{51} -5.29150i q^{53} +(3.74166 + 7.93725i) q^{55} +4.24264i q^{57} -9.00000 q^{59} -14.9666 q^{61} +7.48331 q^{63} -9.89949i q^{65} +3.00000 q^{67} +2.64575i q^{69} -2.64575i q^{71} -12.7279i q^{73} -2.00000 q^{75} +(11.2250 - 5.29150i) q^{77} -7.48331 q^{79} +1.00000 q^{81} +8.48528i q^{83} +11.2250 q^{85} +3.00000 q^{89} -14.0000 q^{91} -7.93725i q^{93} +11.2250 q^{95} -3.00000 q^{97} +(6.00000 - 2.82843i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{3} - 8 q^{9}+O(q^{10}) \) 4 * q + 4 * q^3 - 8 * q^9	\( 4 q + 4 q^{3} - 8 q^{9} - 12 q^{11} - 8 q^{25} - 20 q^{27} - 12 q^{33} + 28 q^{49} - 36 q^{59} + 12 q^{67} - 8 q^{75} + 4 q^{81} + 12 q^{89} - 56 q^{91} - 12 q^{97} + 24 q^{99}+O(q^{100}) \) 4 * q + 4 * q^3 - 8 * q^9 - 12 * q^11 - 8 * q^25 - 20 * q^27 - 12 * q^33 + 28 * q^49 - 36 * q^59 + 12 * q^67 - 8 * q^75 + 4 * q^81 + 12 * q^89 - 56 * q^91 - 12 * q^97 + 24 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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