LMFDB - Embedded newform 1520.2.d.e.609.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1520,2,Mod(609,1520)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1520, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("1520.609");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1520 = 2^{4} \cdot 5 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1520.d (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:	$12.1372611072$
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:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 190)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			609.2
Root			$-0.707107 - 0.707107i$ of defining polynomial
Character	$\chi$	$=$	1520.609
Dual form			1520.2.d.e.609.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.414214i q^{3} +(0.707107 + 2.12132i) q^{5} -4.41421i q^{7} +2.82843 q^{9} +O(q^{10})$	$q-0.414214i q^{3} +(0.707107 + 2.12132i) q^{5} -4.41421i q^{7} +2.82843 q^{9} +1.41421 q^{11} -5.82843i q^{13} +(0.878680 - 0.292893i) q^{15} +1.00000i q^{17} -1.00000 q^{19} -1.82843 q^{21} +0.757359i q^{23} +(-4.00000 + 3.00000i) q^{25} -2.41421i q^{27} +0.171573 q^{29} -6.24264 q^{31} -0.585786i q^{33} +(9.36396 - 3.12132i) q^{35} -8.48528i q^{37} -2.41421 q^{39} -4.24264 q^{41} +1.75736i q^{43} +(2.00000 + 6.00000i) q^{45} -12.4853 q^{49} +0.414214 q^{51} -5.48528i q^{53} +(1.00000 + 3.00000i) q^{55} +0.414214i q^{57} +6.89949 q^{59} +14.2426 q^{61} -12.4853i q^{63} +(12.3640 - 4.12132i) q^{65} -4.75736i q^{67} +0.313708 q^{69} +13.4142 q^{71} -11.4853i q^{73} +(1.24264 + 1.65685i) q^{75} -6.24264i q^{77} -6.48528 q^{79} +7.48528 q^{81} +14.4853i q^{83} +(-2.12132 + 0.707107i) q^{85} -0.0710678i q^{87} -7.07107 q^{89} -25.7279 q^{91} +2.58579i q^{93} +(-0.707107 - 2.12132i) q^{95} -0.343146i q^{97} +4.00000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q+O(q^{10}) $ 4 * q	$ 4 q + 12 q^{15} - 4 q^{19} + 4 q^{21} - 16 q^{25} + 12 q^{29} - 8 q^{31} + 12 q^{35} - 4 q^{39} + 8 q^{45} - 16 q^{49} - 4 q^{51} + 4 q^{55} - 12 q^{59} + 40 q^{61} + 24 q^{65} - 44 q^{69} + 48 q^{71} - 12 q^{75} + 8 q^{79} - 4 q^{81} - 52 q^{91} + 16 q^{99}+O(q^{100}) $ 4 * q + 12 * q^15 - 4 * q^19 + 4 * q^21 - 16 * q^25 + 12 * q^29 - 8 * q^31 + 12 * q^35 - 4 * q^39 + 8 * q^45 - 16 * q^49 - 4 * q^51 + 4 * q^55 - 12 * q^59 + 40 * q^61 + 24 * q^65 - 44 * q^69 + 48 * q^71 - 12 * q^75 + 8 * q^79 - 4 * q^81 - 52 * q^91 + 16 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1520\mathbb{Z}\right)^\times$.

$n$	$191$	$401$	$1141$	$1217$
$\chi(n)$	$1$	$1$	$1$	$-1$

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
$3$		−	0.414214i		−	0.239146i	−0.992825	−	0.119573i	$-0.961847\pi$
$3$		−	0.414214i		−	0.239146i	0.992825	−	0.119573i	$-0.0381526\pi$
$4$		0			0
$5$	0.707107	+	2.12132i	0.316228	+	0.948683i
$5$	0.707107	+	2.12132i	0.316228	+	0.948683i
$6$		0			0
$7$		−	4.41421i		−	1.66842i	−0.551450	−	0.834208i	$-0.685925\pi$
$7$		−	4.41421i		−	1.66842i	0.551450	−	0.834208i	$-0.314075\pi$
$8$		0			0
$9$	2.82843			0.942809
$10$		0			0
$11$	1.41421			0.426401			0.213201	−	0.977008i	$-0.431611\pi$
$11$	1.41421			0.426401			0.213201	+	0.977008i	$0.431611\pi$
$12$		0			0
$13$		−	5.82843i		−	1.61651i	−0.588829	−	0.808257i	$-0.700411\pi$
$13$		−	5.82843i		−	1.61651i	0.588829	−	0.808257i	$-0.299589\pi$
$14$		0			0
$15$	0.878680	−	0.292893i	0.226874	−	0.0756247i
$16$		0			0
$17$			1.00000i			0.242536i	0.992620	+	0.121268i	$0.0386960\pi$
$17$			1.00000i			0.242536i	−0.992620	+	0.121268i	$0.961304\pi$
$18$		0			0
$19$	−1.00000			−0.229416
$19$	−1.00000			−0.229416
$20$		0			0
$21$	−1.82843			−0.398996
$22$		0			0
$23$			0.757359i			0.157920i	0.996878	+	0.0789602i	$0.0251600\pi$
$23$			0.757359i			0.157920i	−0.996878	+	0.0789602i	$0.974840\pi$
$24$		0			0
$25$	−4.00000	+	3.00000i	−0.800000	+	0.600000i
$26$		0			0
$27$		−	2.41421i		−	0.464616i
$28$		0			0
$29$	0.171573			0.0318603			0.0159301	−	0.999873i	$-0.494929\pi$
$29$	0.171573			0.0318603			0.0159301	+	0.999873i	$0.494929\pi$
$30$		0			0
$31$	−6.24264			−1.12121			−0.560606	−	0.828083i	$-0.689432\pi$
$31$	−6.24264			−1.12121			−0.560606	+	0.828083i	$0.689432\pi$
$32$		0			0
$33$		−	0.585786i		−	0.101972i
$34$		0			0
$35$	9.36396	−	3.12132i	1.58280	−	0.527599i
$36$		0			0
$37$		−	8.48528i		−	1.39497i	−0.716599	−	0.697486i	$-0.754302\pi$
$37$		−	8.48528i		−	1.39497i	0.716599	−	0.697486i	$-0.245698\pi$
$38$		0			0
$39$	−2.41421			−0.386584
$40$		0			0
$41$	−4.24264			−0.662589			−0.331295	−	0.943527i	$-0.607485\pi$
$41$	−4.24264			−0.662589			−0.331295	+	0.943527i	$0.607485\pi$
$42$		0			0
$43$			1.75736i			0.267995i	0.990982	+	0.133997i	$0.0427814\pi$
$43$			1.75736i			0.267995i	−0.990982	+	0.133997i	$0.957219\pi$
$44$		0			0
$45$	2.00000	+	6.00000i	0.298142	+	0.894427i
$46$		0			0
$47$		0			0		1.00000			$0$
$47$		0			0		−1.00000			$\pi$
$48$		0			0
$49$	−12.4853			−1.78361
$50$		0			0
$51$	0.414214			0.0580015
$52$		0			0
$53$		−	5.48528i		−	0.753461i	−0.926323	−	0.376731i	$-0.877048\pi$
$53$		−	5.48528i		−	0.753461i	0.926323	−	0.376731i	$-0.122952\pi$
$54$		0			0
$55$	1.00000	+	3.00000i	0.134840	+	0.404520i
$56$		0			0
$57$			0.414214i			0.0548639i
$58$		0			0
$59$	6.89949			0.898238			0.449119	−	0.893472i	$-0.351738\pi$
$59$	6.89949			0.898238			0.449119	+	0.893472i	$0.351738\pi$
$60$		0			0
$61$	14.2426			1.82358			0.911792	−	0.410653i	$-0.134699\pi$
$61$	14.2426			1.82358			0.911792	+	0.410653i	$0.134699\pi$
$62$		0			0
$63$		−	12.4853i		−	1.57300i
$64$		0			0
$65$	12.3640	−	4.12132i	1.53356	−	0.511187i
$66$		0			0
$67$		−	4.75736i		−	0.581204i	−0.956844	−	0.290602i	$-0.906144\pi$
$67$		−	4.75736i		−	0.581204i	0.956844	−	0.290602i	$-0.0938555\pi$
$68$		0			0
$69$	0.313708			0.0377661
$70$		0			0
$71$	13.4142			1.59197			0.795987	−	0.605314i	$-0.206952\pi$
$71$	13.4142			1.59197			0.795987	+	0.605314i	$0.206952\pi$
$72$		0			0
$73$		−	11.4853i		−	1.34425i	−0.740437	−	0.672125i	$-0.765382\pi$
$73$		−	11.4853i		−	1.34425i	0.740437	−	0.672125i	$-0.234618\pi$
$74$		0			0
$75$	1.24264	+	1.65685i	0.143488	+	0.191317i
$76$		0			0
$77$		−	6.24264i		−	0.711415i
$78$		0			0
$79$	−6.48528			−0.729651			−0.364826	−	0.931076i	$-0.618871\pi$
$79$	−6.48528			−0.729651			−0.364826	+	0.931076i	$0.618871\pi$
$80$		0			0
$81$	7.48528			0.831698
$82$		0			0
$83$			14.4853i			1.58997i	0.606632	+	0.794983i	$0.292520\pi$
$83$			14.4853i			1.58997i	−0.606632	+	0.794983i	$0.707480\pi$
$84$		0			0
$85$	−2.12132	+	0.707107i	−0.230089	+	0.0766965i
$86$		0			0
$87$		−	0.0710678i		−	0.00761927i
$88$		0			0
$89$	−7.07107			−0.749532			−0.374766	−	0.927119i	$-0.622277\pi$
$89$	−7.07107			−0.749532			−0.374766	+	0.927119i	$0.622277\pi$
$90$		0			0
$91$	−25.7279			−2.69702
$92$		0			0
$93$			2.58579i			0.268134i
$94$		0			0
$95$	−0.707107	−	2.12132i	−0.0725476	−	0.217643i
$96$		0			0
$97$		−	0.343146i		−	0.0348412i	−0.999848	−	0.0174206i	$-0.994455\pi$
$97$		−	0.343146i		−	0.0348412i	0.999848	−	0.0174206i	$-0.00554543\pi$
$98$		0			0
$99$	4.00000			0.402015
$100$		0			0
$101$	13.0711			1.30062			0.650310	−	0.759669i	$-0.274639\pi$
$101$	13.0711			1.30062			0.650310	+	0.759669i	$0.274639\pi$
$102$		0			0
$103$			4.24264i			0.418040i	0.977911	+	0.209020i	$0.0670273\pi$
$103$			4.24264i			0.418040i	−0.977911	+	0.209020i	$0.932973\pi$
$104$		0			0
$105$	−1.29289	−	3.87868i	−0.126173	−	0.378520i
$106$		0			0
$107$		−	19.7279i		−	1.90717i	−0.301124	−	0.953585i	$-0.597362\pi$
$107$		−	19.7279i		−	1.90717i	0.301124	−	0.953585i	$-0.402638\pi$
$108$		0			0
$109$	17.9706			1.72127			0.860634	−	0.509224i	$-0.170068\pi$
$109$	17.9706			1.72127			0.860634	+	0.509224i	$0.170068\pi$
$110$		0			0
$111$	−3.51472			−0.333602
$112$		0			0
$113$			10.2426i			0.963547i	0.876296	+	0.481773i	$0.160007\pi$
$113$			10.2426i			0.963547i	−0.876296	+	0.481773i	$0.839993\pi$
$114$		0			0
$115$	−1.60660	+	0.535534i	−0.149816	+	0.0499388i
$116$		0			0
$117$		−	16.4853i		−	1.52406i
$118$		0			0
$119$	4.41421			0.404650
$120$		0			0
$121$	−9.00000			−0.818182
$122$		0			0
$123$			1.75736i			0.158456i
$124$		0			0
$125$	−9.19239	−	6.36396i	−0.822192	−	0.569210i
$126$		0			0
$127$			2.48528i			0.220533i	0.993902	+	0.110267i	$0.0351704\pi$
$127$			2.48528i			0.220533i	−0.993902	+	0.110267i	$0.964830\pi$
$128$		0			0
$129$	0.727922			0.0640900
$130$		0			0
$131$	16.9706			1.48272			0.741362	−	0.671105i	$-0.234180\pi$
$131$	16.9706			1.48272			0.741362	+	0.671105i	$0.234180\pi$
$132$		0			0
$133$			4.41421i			0.382761i
$134$		0			0
$135$	5.12132	−	1.70711i	0.440773	−	0.146924i
$136$		0			0
$137$			13.0000i			1.11066i	0.831628	+	0.555332i	$0.187409\pi$
$137$			13.0000i			1.11066i	−0.831628	+	0.555332i	$0.812591\pi$
$138$		0			0
$139$	−12.0000			−1.01783			−0.508913	−	0.860818i	$-0.669953\pi$
$139$	−12.0000			−1.01783			−0.508913	+	0.860818i	$0.669953\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		−	8.24264i		−	0.689284i
$144$		0			0
$145$	0.121320	+	0.363961i	0.0100751	+	0.0302253i
$146$		0			0
$147$			5.17157i			0.426544i
$148$		0			0
$149$	−17.6569			−1.44651			−0.723253	−	0.690583i	$-0.757354\pi$
$149$	−17.6569			−1.44651			−0.723253	+	0.690583i	$0.757354\pi$
$150$		0			0
$151$	10.4853			0.853280			0.426640	−	0.904422i	$-0.359697\pi$
$151$	10.4853			0.853280			0.426640	+	0.904422i	$0.359697\pi$
$152$		0			0
$153$			2.82843i			0.228665i
$154$		0			0
$155$	−4.41421	−	13.2426i	−0.354558	−	1.06367i
$156$		0			0
$157$			11.6569i			0.930318i	0.885227	+	0.465159i	$0.154003\pi$
$157$			11.6569i			0.930318i	−0.885227	+	0.465159i	$0.845997\pi$
$158$		0			0
$159$	−2.27208			−0.180188
$160$		0			0
$161$	3.34315			0.263477
$162$		0			0
$163$		−	10.2426i		−	0.802266i	−0.916020	−	0.401133i	$-0.868617\pi$
$163$		−	10.2426i		−	0.802266i	0.916020	−	0.401133i	$-0.131383\pi$
$164$		0			0
$165$	1.24264	−	0.414214i	0.0967394	−	0.0322465i
$166$		0			0
$167$			18.2426i			1.41166i	0.708382	+	0.705829i	$0.249425\pi$
$167$			18.2426i			1.41166i	−0.708382	+	0.705829i	$0.750575\pi$
$168$		0			0
$169$	−20.9706			−1.61312
$170$		0			0
$171$	−2.82843			−0.216295
$172$		0			0
$173$		−	0.485281i		−	0.0368953i	−0.999830	−	0.0184476i	$-0.994128\pi$
$173$		−	0.485281i		−	0.0368953i	0.999830	−	0.0184476i	$-0.00587240\pi$
$174$		0			0
$175$	13.2426	+	17.6569i	1.00105	+	1.33473i
$176$		0			0
$177$		−	2.85786i		−	0.214810i
$178$		0			0
$179$	−11.6569			−0.871274			−0.435637	−	0.900122i	$-0.643477\pi$
$179$	−11.6569			−0.871274			−0.435637	+	0.900122i	$0.643477\pi$
$180$		0			0
$181$	−8.48528			−0.630706			−0.315353	−	0.948974i	$-0.602123\pi$
$181$	−8.48528			−0.630706			−0.315353	+	0.948974i	$0.602123\pi$
$182$		0			0
$183$		−	5.89949i		−	0.436103i
$184$		0			0
$185$	18.0000	−	6.00000i	1.32339	−	0.441129i
$186$		0			0
$187$			1.41421i			0.103418i
$188$		0			0
$189$	−10.6569			−0.775172
$190$		0			0
$191$	−12.5563			−0.908546			−0.454273	−	0.890863i	$-0.650101\pi$
$191$	−12.5563			−0.908546			−0.454273	+	0.890863i	$0.650101\pi$
$192$		0			0
$193$			0.343146i			0.0247002i	0.999924	+	0.0123501i	$0.00393125\pi$
$193$			0.343146i			0.0247002i	−0.999924	+	0.0123501i	$0.996069\pi$
$194$		0			0
$195$	−1.70711	−	5.12132i	−0.122248	−	0.366745i
$196$		0			0
$197$		−	11.7574i		−	0.837677i	−0.908061	−	0.418839i	$-0.862437\pi$
$197$		−	11.7574i		−	0.837677i	0.908061	−	0.418839i	$-0.137563\pi$
$198$		0			0
$199$	9.24264			0.655193			0.327597	−	0.944818i	$-0.393761\pi$
$199$	9.24264			0.655193			0.327597	+	0.944818i	$0.393761\pi$
$200$		0			0
$201$	−1.97056			−0.138993
$202$		0			0
$203$		−	0.757359i		−	0.0531562i
$204$		0			0
$205$	−3.00000	−	9.00000i	−0.209529	−	0.628587i
$206$		0			0
$207$			2.14214i			0.148889i
$208$		0			0
$209$	−1.41421			−0.0978232
$210$		0			0
$211$	19.7279			1.35813			0.679063	−	0.734080i	$-0.262386\pi$
$211$	19.7279			1.35813			0.679063	+	0.734080i	$0.262386\pi$
$212$		0			0
$213$		−	5.55635i		−	0.380715i
$214$		0			0
$215$	−3.72792	+	1.24264i	−0.254242	+	0.0847474i
$216$		0			0
$217$			27.5563i			1.87065i
$218$		0			0
$219$	−4.75736			−0.321473
$220$		0			0
$221$	5.82843			0.392062
$222$		0			0
$223$		−	15.1716i		−	1.01596i	−0.861368	−	0.507982i	$-0.830392\pi$
$223$		−	15.1716i		−	1.01596i	0.861368	−	0.507982i	$-0.169608\pi$
$224$		0			0
$225$	−11.3137	+	8.48528i	−0.754247	+	0.565685i
$226$		0			0
$227$			16.7574i			1.11223i	0.831107	+	0.556113i	$0.187708\pi$
$227$			16.7574i			1.11223i	−0.831107	+	0.556113i	$0.812292\pi$
$228$		0			0
$229$	−14.9706			−0.989283			−0.494641	−	0.869097i	$-0.664701\pi$
$229$	−14.9706			−0.989283			−0.494641	+	0.869097i	$0.664701\pi$
$230$		0			0
$231$	−2.58579			−0.170132
$232$		0			0
$233$			24.9706i			1.63588i	0.575306	+	0.817938i	$0.304883\pi$
$233$			24.9706i			1.63588i	−0.575306	+	0.817938i	$0.695117\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$			2.68629i			0.174493i
$238$		0			0
$239$	−6.89949			−0.446291			−0.223146	−	0.974785i	$-0.571633\pi$
$239$	−6.89949			−0.446291			−0.223146	+	0.974785i	$0.571633\pi$
$240$		0			0
$241$	8.97056			0.577845			0.288922	−	0.957353i	$-0.406703\pi$
$241$	8.97056			0.577845			0.288922	+	0.957353i	$0.406703\pi$
$242$		0			0
$243$		−	10.3431i		−	0.663513i
$244$		0			0
$245$	−8.82843	−	26.4853i	−0.564028	−	1.69208i
$246$		0			0
$247$			5.82843i			0.370854i
$248$		0			0
$249$	6.00000			0.380235
$250$		0			0
$251$	−3.55635			−0.224475			−0.112237	−	0.993681i	$-0.535802\pi$
$251$	−3.55635			−0.224475			−0.112237	+	0.993681i	$0.535802\pi$
$252$		0			0
$253$			1.07107i			0.0673375i
$254$		0			0
$255$	0.292893	+	0.878680i	0.0183417	+	0.0550251i
$256$		0			0
$257$			20.7279i			1.29297i	0.762926	+	0.646486i	$0.223762\pi$
$257$			20.7279i			1.29297i	−0.762926	+	0.646486i	$0.776238\pi$
$258$		0			0
$259$	−37.4558			−2.32739
$260$		0			0
$261$	0.485281			0.0300382
$262$		0			0
$263$		−	26.9706i		−	1.66308i	−0.555468	−	0.831538i	$-0.687461\pi$
$263$		−	26.9706i		−	1.66308i	0.555468	−	0.831538i	$-0.312539\pi$
$264$		0			0
$265$	11.6360	−	3.87868i	0.714796	−	0.238265i
$266$		0			0
$267$			2.92893i			0.179248i
$268$		0			0
$269$	16.6274			1.01379			0.506896	−	0.862007i	$-0.330793\pi$
$269$	16.6274			1.01379			0.506896	+	0.862007i	$0.330793\pi$
$270$		0			0
$271$	27.2426			1.65487			0.827436	−	0.561560i	$-0.189798\pi$
$271$	27.2426			1.65487			0.827436	+	0.561560i	$0.189798\pi$
$272$		0			0
$273$			10.6569i			0.644982i
$274$		0			0
$275$	−5.65685	+	4.24264i	−0.341121	+	0.255841i
$276$		0			0
$277$		0			0		1.00000			$0$
$277$		0			0		−1.00000			$\pi$
$278$		0			0
$279$	−17.6569			−1.05709
$280$		0			0
$281$	4.24264			0.253095			0.126547	−	0.991961i	$-0.459610\pi$
$281$	4.24264			0.253095			0.126547	+	0.991961i	$0.459610\pi$
$282$		0			0
$283$			32.1421i			1.91065i	0.295555	+	0.955326i	$0.404496\pi$
$283$			32.1421i			1.91065i	−0.295555	+	0.955326i	$0.595504\pi$
$284$		0			0
$285$	−0.878680	+	0.292893i	−0.0520485	+	0.0173495i
$286$		0			0
$287$			18.7279i			1.10547i
$288$		0			0
$289$	16.0000			0.941176
$290$		0			0
$291$	−0.142136			−0.00833214
$292$		0			0
$293$			5.48528i			0.320454i	0.987080	+	0.160227i	$0.0512226\pi$
$293$			5.48528i			0.320454i	−0.987080	+	0.160227i	$0.948777\pi$
$294$		0			0
$295$	4.87868	+	14.6360i	0.284048	+	0.852143i
$296$		0			0
$297$		−	3.41421i		−	0.198113i
$298$		0			0
$299$	4.41421			0.255281
$300$		0			0
$301$	7.75736			0.447127
$302$		0			0
$303$		−	5.41421i		−	0.311038i
$304$		0			0
$305$	10.0711	+	30.2132i	0.576668	+	1.73000i
$306$		0			0
$307$			17.6569i			1.00773i	0.863782	+	0.503865i	$0.168089\pi$
$307$			17.6569i			1.00773i	−0.863782	+	0.503865i	$0.831911\pi$
$308$		0			0
$309$	1.75736			0.0999727
$310$		0			0
$311$	4.75736			0.269765			0.134883	−	0.990862i	$-0.456934\pi$
$311$	4.75736			0.269765			0.134883	+	0.990862i	$0.456934\pi$
$312$		0			0
$313$			7.97056i			0.450523i	0.974298	+	0.225261i	$0.0723236\pi$
$313$			7.97056i			0.450523i	−0.974298	+	0.225261i	$0.927676\pi$
$314$		0			0
$315$	26.4853	−	8.82843i	1.49228	−	0.497426i
$316$		0			0
$317$		−	9.48528i		−	0.532746i	−0.963870	−	0.266373i	$-0.914175\pi$
$317$		−	9.48528i		−	0.532746i	0.963870	−	0.266373i	$-0.0858254\pi$
$318$		0			0
$319$	0.242641			0.0135853
$320$		0			0
$321$	−8.17157			−0.456093
$322$		0			0
$323$		−	1.00000i		−	0.0556415i
$324$		0			0
$325$	17.4853	+	23.3137i	0.969909	+	1.29321i
$326$		0			0
$327$		−	7.44365i		−	0.411635i
$328$		0			0
$329$		0			0
$330$		0			0
$331$	19.2426			1.05767			0.528836	−	0.848724i	$-0.322629\pi$
$331$	19.2426			1.05767			0.528836	+	0.848724i	$0.322629\pi$
$332$		0			0
$333$		−	24.0000i		−	1.31519i
$334$		0			0
$335$	10.0919	−	3.36396i	0.551378	−	0.183793i
$336$		0			0
$337$			14.1005i			0.768103i	0.923312	+	0.384052i	$0.125472\pi$
$337$			14.1005i			0.768103i	−0.923312	+	0.384052i	$0.874528\pi$
$338$		0			0
$339$	4.24264			0.230429
$340$		0			0
$341$	−8.82843			−0.478086
$342$		0			0
$343$			24.2132i			1.30739i
$344$		0			0
$345$	0.221825	+	0.665476i	0.0119427	+	0.0358280i
$346$		0			0
$347$			5.51472i			0.296046i	0.988984	+	0.148023i	$0.0472909\pi$
$347$			5.51472i			0.296046i	−0.988984	+	0.148023i	$0.952709\pi$
$348$		0			0
$349$	−6.00000			−0.321173			−0.160586	−	0.987022i	$-0.551338\pi$
$349$	−6.00000			−0.321173			−0.160586	+	0.987022i	$0.551338\pi$
$350$		0			0
$351$	−14.0711			−0.751058
$352$		0			0
$353$			2.51472i			0.133845i	0.997758	+	0.0669225i	$0.0213180\pi$
$353$			2.51472i			0.133845i	−0.997758	+	0.0669225i	$0.978682\pi$
$354$		0			0
$355$	9.48528	+	28.4558i	0.503426	+	1.51028i
$356$		0			0
$357$		−	1.82843i		−	0.0967706i
$358$		0			0
$359$	−22.7574			−1.20109			−0.600544	−	0.799592i	$-0.705049\pi$
$359$	−22.7574			−1.20109			−0.600544	+	0.799592i	$0.705049\pi$
$360$		0			0
$361$	1.00000			0.0526316
$362$		0			0
$363$			3.72792i			0.195665i
$364$		0			0
$365$	24.3640	−	8.12132i	1.27527	−	0.425089i
$366$		0			0
$367$			25.4558i			1.32878i	0.747384	+	0.664392i	$0.231309\pi$
$367$			25.4558i			1.32878i	−0.747384	+	0.664392i	$0.768691\pi$
$368$		0			0
$369$	−12.0000			−0.624695
$370$		0			0
$371$	−24.2132			−1.25709
$372$		0			0
$373$		−	9.00000i		−	0.466002i	−0.972476	−	0.233001i	$-0.925145\pi$
$373$		−	9.00000i		−	0.466002i	0.972476	−	0.233001i	$-0.0748546\pi$
$374$		0			0
$375$	−2.63604	+	3.80761i	−0.136124	+	0.196624i
$376$		0			0
$377$		−	1.00000i		−	0.0515026i
$378$		0			0
$379$	11.2426			0.577496			0.288748	−	0.957405i	$-0.406761\pi$
$379$	11.2426			0.577496			0.288748	+	0.957405i	$0.406761\pi$
$380$		0			0
$381$	1.02944			0.0527397
$382$		0			0
$383$			12.2426i			0.625570i	0.949824	+	0.312785i	$0.101262\pi$
$383$			12.2426i			0.625570i	−0.949824	+	0.312785i	$0.898738\pi$
$384$		0			0
$385$	13.2426	−	4.41421i	0.674907	−	0.224969i
$386$		0			0
$387$			4.97056i			0.252668i
$388$		0			0
$389$	−22.9289			−1.16254			−0.581272	−	0.813710i	$-0.697445\pi$
$389$	−22.9289			−1.16254			−0.581272	+	0.813710i	$0.697445\pi$
$390$		0			0
$391$	−0.757359			−0.0383013
$392$		0			0
$393$		−	7.02944i		−	0.354588i
$394$		0			0
$395$	−4.58579	−	13.7574i	−0.230736	−	0.692208i
$396$		0			0
$397$			24.0000i			1.20453i	0.798298	+	0.602263i	$0.205734\pi$
$397$			24.0000i			1.20453i	−0.798298	+	0.602263i	$0.794266\pi$
$398$		0			0
$399$	1.82843			0.0915358
$400$		0			0
$401$	−25.4142			−1.26913			−0.634563	−	0.772871i	$-0.718820\pi$
$401$	−25.4142			−1.26913			−0.634563	+	0.772871i	$0.718820\pi$
$402$		0			0
$403$			36.3848i			1.81245i
$404$		0			0
$405$	5.29289	+	15.8787i	0.263006	+	0.789018i
$406$		0			0
$407$		−	12.0000i		−	0.594818i
$408$		0			0
$409$	−25.2132			−1.24671			−0.623356	−	0.781938i	$-0.714231\pi$
$409$	−25.2132			−1.24671			−0.623356	+	0.781938i	$0.714231\pi$
$410$		0			0
$411$	5.38478			0.265611
$412$		0			0
$413$		−	30.4558i		−	1.49863i
$414$		0			0
$415$	−30.7279	+	10.2426i	−1.50837	+	0.502791i
$416$		0			0
$417$			4.97056i			0.243410i
$418$		0			0
$419$	19.4142			0.948446			0.474223	−	0.880405i	$-0.342729\pi$
$419$	19.4142			0.948446			0.474223	+	0.880405i	$0.342729\pi$
$420$		0			0
$421$	19.4853			0.949655			0.474827	−	0.880079i	$-0.342511\pi$
$421$	19.4853			0.949655			0.474827	+	0.880079i	$0.342511\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−3.00000	−	4.00000i	−0.145521	−	0.194029i
$426$		0			0
$427$		−	62.8701i		−	3.04250i
$428$		0			0
$429$	−3.41421			−0.164840
$430$		0			0
$431$	6.38478			0.307544			0.153772	−	0.988106i	$-0.450858\pi$
$431$	6.38478			0.307544			0.153772	+	0.988106i	$0.450858\pi$
$432$		0			0
$433$			9.55635i			0.459249i	0.973279	+	0.229624i	$0.0737498\pi$
$433$			9.55635i			0.459249i	−0.973279	+	0.229624i	$0.926250\pi$
$434$		0			0
$435$	0.150758	−	0.0502525i	0.00722827	−	0.00240942i
$436$		0			0
$437$		−	0.757359i		−	0.0362294i
$438$		0			0
$439$	−5.75736			−0.274784			−0.137392	−	0.990517i	$-0.543872\pi$
$439$	−5.75736			−0.274784			−0.137392	+	0.990517i	$0.543872\pi$
$440$		0			0
$441$	−35.3137			−1.68161
$442$		0			0
$443$			4.24264i			0.201574i	0.994908	+	0.100787i	$0.0321361\pi$
$443$			4.24264i			0.201574i	−0.994908	+	0.100787i	$0.967864\pi$
$444$		0			0
$445$	−5.00000	−	15.0000i	−0.237023	−	0.711068i
$446$		0			0
$447$			7.31371i			0.345927i
$448$		0			0
$449$	−17.3137			−0.817084			−0.408542	−	0.912739i	$-0.633963\pi$
$449$	−17.3137			−0.817084			−0.408542	+	0.912739i	$0.633963\pi$
$450$		0			0
$451$	−6.00000			−0.282529
$452$		0			0
$453$		−	4.34315i		−	0.204059i
$454$		0			0
$455$	−18.1924	−	54.5772i	−0.852872	−	2.55862i
$456$		0			0
$457$		−	3.00000i		−	0.140334i	−0.997535	−	0.0701670i	$-0.977647\pi$
$457$		−	3.00000i		−	0.140334i	0.997535	−	0.0701670i	$-0.0223532\pi$
$458$		0			0
$459$	2.41421			0.112686
$460$		0			0
$461$	−3.55635			−0.165636			−0.0828178	−	0.996565i	$-0.526392\pi$
$461$	−3.55635			−0.165636			−0.0828178	+	0.996565i	$0.526392\pi$
$462$		0			0
$463$			14.1421i			0.657241i	0.944462	+	0.328620i	$0.106584\pi$
$463$			14.1421i			0.657241i	−0.944462	+	0.328620i	$0.893416\pi$
$464$		0			0
$465$	−5.48528	+	1.82843i	−0.254374	+	0.0847913i
$466$		0			0
$467$		−	0.727922i		−	0.0336842i	−0.999858	−	0.0168421i	$-0.994639\pi$
$467$		−	0.727922i		−	0.0336842i	0.999858	−	0.0168421i	$-0.00536126\pi$
$468$		0			0
$469$	−21.0000			−0.969690
$470$		0			0
$471$	4.82843			0.222482
$472$		0			0
$473$			2.48528i			0.114273i
$474$		0			0
$475$	4.00000	−	3.00000i	0.183533	−	0.137649i
$476$		0			0
$477$		−	15.5147i		−	0.710370i
$478$		0			0
$479$	−31.1127			−1.42158			−0.710788	−	0.703407i	$-0.751661\pi$
$479$	−31.1127			−1.42158			−0.710788	+	0.703407i	$0.751661\pi$
$480$		0			0
$481$	−49.4558			−2.25499
$482$		0			0
$483$		−	1.38478i		−	0.0630095i
$484$		0			0
$485$	0.727922	−	0.242641i	0.0330532	−	0.0110177i
$486$		0			0
$487$		−	37.7990i		−	1.71284i	−0.516283	−	0.856418i	$-0.672685\pi$
$487$		−	37.7990i		−	1.71284i	0.516283	−	0.856418i	$-0.327315\pi$
$488$		0			0
$489$	−4.24264			−0.191859
$490$		0			0
$491$	33.5563			1.51438			0.757188	−	0.653197i	$-0.226572\pi$
$491$	33.5563			1.51438			0.757188	+	0.653197i	$0.226572\pi$
$492$		0			0
$493$			0.171573i			0.00772725i
$494$		0			0
$495$	2.82843	+	8.48528i	0.127128	+	0.381385i
$496$		0			0
$497$		−	59.2132i		−	2.65608i
$498$		0			0
$499$	−25.7574			−1.15306			−0.576529	−	0.817077i	$-0.695593\pi$
$499$	−25.7574			−1.15306			−0.576529	+	0.817077i	$0.695593\pi$
$500$		0			0
$501$	7.55635			0.337593
$502$		0			0
$503$			14.2721i			0.636361i	0.948030	+	0.318180i	$0.103072\pi$
$503$			14.2721i			0.636361i	−0.948030	+	0.318180i	$0.896928\pi$
$504$		0			0
$505$	9.24264	+	27.7279i	0.411292	+	1.23388i
$506$		0			0
$507$			8.68629i			0.385772i
$508$		0			0
$509$	−28.9706			−1.28410			−0.642049	−	0.766664i	$-0.721915\pi$
$509$	−28.9706			−1.28410			−0.642049	+	0.766664i	$0.721915\pi$
$510$		0			0
$511$	−50.6985			−2.24277
$512$		0			0
$513$			2.41421i			0.106590i
$514$		0			0
$515$	−9.00000	+	3.00000i	−0.396587	+	0.132196i
$516$		0			0
$517$		0			0
$518$		0			0
$519$	−0.201010			−0.00882337
$520$		0			0
$521$	23.3137			1.02139			0.510696	−	0.859761i	$-0.329388\pi$
$521$	23.3137			1.02139			0.510696	+	0.859761i	$0.329388\pi$
$522$		0			0
$523$		−	2.27208i		−	0.0993510i	−0.998765	−	0.0496755i	$-0.984181\pi$
$523$		−	2.27208i		−	0.0993510i	0.998765	−	0.0496755i	$-0.0158187\pi$
$524$		0			0
$525$	7.31371	−	5.48528i	0.319196	−	0.239397i
$526$		0			0
$527$		−	6.24264i		−	0.271934i
$528$		0			0
$529$	22.4264			0.975061
$530$		0			0
$531$	19.5147			0.846867
$532$		0			0
$533$			24.7279i			1.07109i
$534$		0			0
$535$	41.8492	−	13.9497i	1.80930	−	0.603100i
$536$		0			0
$537$			4.82843i			0.208362i
$538$		0			0
$539$	−17.6569			−0.760535
$540$		0			0
$541$	9.75736			0.419502			0.209751	−	0.977755i	$-0.432735\pi$
$541$	9.75736			0.419502			0.209751	+	0.977755i	$0.432735\pi$
$542$		0			0
$543$			3.51472i			0.150831i
$544$		0			0
$545$	12.7071	+	38.1213i	0.544313	+	1.63294i
$546$		0			0
$547$			17.3137i			0.740281i	0.928976	+	0.370140i	$0.120690\pi$
$547$			17.3137i			0.740281i	−0.928976	+	0.370140i	$0.879310\pi$
$548$		0			0
$549$	40.2843			1.71929
$550$		0			0
$551$	−0.171573			−0.00730925
$552$		0			0
$553$			28.6274i			1.21736i
$554$		0			0
$555$	−2.48528	−	7.45584i	−0.105494	−	0.316483i
$556$		0			0
$557$			16.0000i			0.677942i	0.940797	+	0.338971i	$0.110079\pi$
$557$			16.0000i			0.677942i	−0.940797	+	0.338971i	$0.889921\pi$
$558$		0			0
$559$	10.2426			0.433218
$560$		0			0
$561$	0.585786			0.0247319
$562$		0			0
$563$			4.97056i			0.209484i	0.994499	+	0.104742i	$0.0334017\pi$
$563$			4.97056i			0.209484i	−0.994499	+	0.104742i	$0.966598\pi$
$564$		0			0
$565$	−21.7279	+	7.24264i	−0.914101	+	0.304700i
$566$		0			0
$567$		−	33.0416i		−	1.38762i
$568$		0			0
$569$	28.2843			1.18574			0.592869	−	0.805299i	$-0.297995\pi$
$569$	28.2843			1.18574			0.592869	+	0.805299i	$0.297995\pi$
$570$		0			0
$571$	2.24264			0.0938516			0.0469258	−	0.998898i	$-0.485058\pi$
$571$	2.24264			0.0938516			0.0469258	+	0.998898i	$0.485058\pi$
$572$		0			0
$573$			5.20101i			0.217275i
$574$		0			0
$575$	−2.27208	−	3.02944i	−0.0947522	−	0.126336i
$576$		0			0
$577$		−	20.3137i		−	0.845671i	−0.906207	−	0.422835i	$-0.861035\pi$
$577$		−	20.3137i		−	0.845671i	0.906207	−	0.422835i	$-0.138965\pi$
$578$		0			0
$579$	0.142136			0.00590695
$580$		0			0
$581$	63.9411			2.65272
$582$		0			0
$583$		−	7.75736i		−	0.321277i
$584$		0			0
$585$	34.9706	−	11.6569i	1.44585	−	0.481952i
$586$		0			0
$587$		−	30.2426i		−	1.24825i	−0.781326	−	0.624124i	$-0.785456\pi$
$587$		−	30.2426i		−	1.24825i	0.781326	−	0.624124i	$-0.214544\pi$
$588$		0			0
$589$	6.24264			0.257224
$590$		0			0
$591$	−4.87006			−0.200327
$592$		0			0
$593$		−	22.0000i		−	0.903432i	−0.892162	−	0.451716i	$-0.850812\pi$
$593$		−	22.0000i		−	0.903432i	0.892162	−	0.451716i	$-0.149188\pi$
$594$		0			0
$595$	3.12132	+	9.36396i	0.127962	+	0.383885i
$596$		0			0
$597$		−	3.82843i		−	0.156687i
$598$		0			0
$599$	−15.2132			−0.621595			−0.310797	−	0.950476i	$-0.600596\pi$
$599$	−15.2132			−0.621595			−0.310797	+	0.950476i	$0.600596\pi$
$600$		0			0
$601$	−20.2426			−0.825715			−0.412857	−	0.910796i	$-0.635469\pi$
$601$	−20.2426			−0.825715			−0.412857	+	0.910796i	$0.635469\pi$
$602$		0			0
$603$		−	13.4558i		−	0.547964i
$604$		0			0
$605$	−6.36396	−	19.0919i	−0.258732	−	0.776195i
$606$		0			0
$607$		−	3.17157i		−	0.128730i	−0.997926	−	0.0643651i	$-0.979498\pi$
$607$		−	3.17157i		−	0.128730i	0.997926	−	0.0643651i	$-0.0205022\pi$
$608$		0			0
$609$	−0.313708			−0.0127121
$610$		0			0
$611$		0			0
$612$		0			0
$613$			11.6985i			0.472497i	0.971693	+	0.236249i	$0.0759180\pi$
$613$			11.6985i			0.472497i	−0.971693	+	0.236249i	$0.924082\pi$
$614$		0			0
$615$	−3.72792	+	1.24264i	−0.150324	+	0.0501081i
$616$		0			0
$617$		−	4.48528i		−	0.180571i	−0.995916	−	0.0902853i	$-0.971222\pi$
$617$		−	4.48528i		−	0.180571i	0.995916	−	0.0902853i	$-0.0287779\pi$
$618$		0			0
$619$	7.75736			0.311795			0.155897	−	0.987773i	$-0.450173\pi$
$619$	7.75736			0.311795			0.155897	+	0.987773i	$0.450173\pi$
$620$		0			0
$621$	1.82843			0.0733723
$622$		0			0
$623$			31.2132i			1.25053i
$624$		0			0
$625$	7.00000	−	24.0000i	0.280000	−	0.960000i
$626$		0			0
$627$			0.585786i			0.0233941i
$628$		0			0
$629$	8.48528			0.338330
$630$		0			0
$631$	38.9706			1.55139			0.775697	−	0.631106i	$-0.217399\pi$
$631$	38.9706			1.55139			0.775697	+	0.631106i	$0.217399\pi$
$632$		0			0
$633$		−	8.17157i		−	0.324791i
$634$		0			0
$635$	−5.27208	+	1.75736i	−0.209216	+	0.0697387i
$636$		0			0
$637$			72.7696i			2.88323i
$638$		0			0
$639$	37.9411			1.50093
$640$		0			0
$641$	−18.0416			−0.712602			−0.356301	−	0.934371i	$-0.615962\pi$
$641$	−18.0416			−0.712602			−0.356301	+	0.934371i	$0.615962\pi$
$642$		0			0
$643$		−	14.4853i		−	0.571244i	−0.958342	−	0.285622i	$-0.907800\pi$
$643$		−	14.4853i		−	0.571244i	0.958342	−	0.285622i	$-0.0922001\pi$
$644$		0			0
$645$	0.514719	+	1.54416i	0.0202670	+	0.0608011i
$646$		0			0
$647$			18.7574i			0.737428i	0.929543	+	0.368714i	$0.120202\pi$
$647$			18.7574i			0.737428i	−0.929543	+	0.368714i	$0.879798\pi$
$648$		0			0
$649$	9.75736			0.383010
$650$		0			0
$651$	11.4142			0.447358
$652$		0			0
$653$			28.9706i			1.13371i	0.823819	+	0.566853i	$0.191839\pi$
$653$			28.9706i			1.13371i	−0.823819	+	0.566853i	$0.808161\pi$
$654$		0			0
$655$	12.0000	+	36.0000i	0.468879	+	1.40664i
$656$		0			0
$657$		−	32.4853i		−	1.26737i
$658$		0			0
$659$	18.8995			0.736220			0.368110	−	0.929782i	$-0.380005\pi$
$659$	18.8995			0.736220			0.368110	+	0.929782i	$0.380005\pi$
$660$		0			0
$661$	−18.4558			−0.717849			−0.358925	−	0.933367i	$-0.616856\pi$
$661$	−18.4558			−0.717849			−0.358925	+	0.933367i	$0.616856\pi$
$662$		0			0
$663$		−	2.41421i		−	0.0937603i
$664$		0			0
$665$	−9.36396	+	3.12132i	−0.363119	+	0.121040i
$666$		0			0
$667$			0.129942i			0.00503139i
$668$		0			0
$669$	−6.28427			−0.242964
$670$		0			0
$671$	20.1421			0.777579
$672$		0			0
$673$			12.0000i			0.462566i	0.972887	+	0.231283i	$0.0742923\pi$
$673$			12.0000i			0.462566i	−0.972887	+	0.231283i	$0.925708\pi$
$674$		0			0
$675$	7.24264	+	9.65685i	0.278769	+	0.371692i
$676$		0			0
$677$		−	40.9411i		−	1.57350i	−0.617275	−	0.786748i	$-0.711763\pi$
$677$		−	40.9411i		−	1.57350i	0.617275	−	0.786748i	$-0.288237\pi$
$678$		0			0
$679$	−1.51472			−0.0581296
$680$		0			0
$681$	6.94113			0.265985
$682$		0			0
$683$			12.0000i			0.459167i	0.973289	+	0.229584i	$0.0737364\pi$
$683$			12.0000i			0.459167i	−0.973289	+	0.229584i	$0.926264\pi$
$684$		0			0
$685$	−27.5772	+	9.19239i	−1.05367	+	0.351223i
$686$		0			0
$687$			6.20101i			0.236583i
$688$		0			0
$689$	−31.9706			−1.21798
$690$		0			0
$691$	−22.4853			−0.855380			−0.427690	−	0.903925i	$-0.640673\pi$
$691$	−22.4853			−0.855380			−0.427690	+	0.903925i	$0.640673\pi$
$692$		0			0
$693$		−	17.6569i		−	0.670728i
$694$		0			0
$695$	−8.48528	−	25.4558i	−0.321865	−	0.965595i
$696$		0			0
$697$		−	4.24264i		−	0.160701i
$698$		0			0
$699$	10.3431			0.391214
$700$		0			0
$701$	16.9706			0.640969			0.320485	−	0.947254i	$-0.396154\pi$
$701$	16.9706			0.640969			0.320485	+	0.947254i	$0.396154\pi$
$702$		0			0
$703$			8.48528i			0.320028i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		−	57.6985i		−	2.16997i
$708$		0			0
$709$	34.0000			1.27690			0.638448	−	0.769665i	$-0.279577\pi$
$709$	34.0000			1.27690			0.638448	+	0.769665i	$0.279577\pi$
$710$		0			0
$711$	−18.3431			−0.687922
$712$		0			0
$713$		−	4.72792i		−	0.177062i
$714$		0			0
$715$	17.4853	−	5.82843i	0.653912	−	0.217971i
$716$		0			0
$717$			2.85786i			0.106729i
$718$		0			0
$719$	−5.10051			−0.190217			−0.0951084	−	0.995467i	$-0.530320\pi$
$719$	−5.10051			−0.190217			−0.0951084	+	0.995467i	$0.530320\pi$
$720$		0			0
$721$	18.7279			0.697464
$722$		0			0
$723$		−	3.71573i		−	0.138189i
$724$		0			0
$725$	−0.686292	+	0.514719i	−0.0254882	+	0.0191162i
$726$		0			0
$727$		−	9.72792i		−	0.360789i	−0.983594	−	0.180394i	$-0.942263\pi$
$727$		−	9.72792i		−	0.360789i	0.983594	−	0.180394i	$-0.0577374\pi$
$728$		0			0
$729$	18.1716			0.673021
$730$		0			0
$731$	−1.75736			−0.0649983
$732$		0			0
$733$		−	12.0000i		−	0.443230i	−0.975134	−	0.221615i	$-0.928867\pi$
$733$		−	12.0000i		−	0.443230i	0.975134	−	0.221615i	$-0.0711328\pi$
$734$		0			0
$735$	−10.9706	+	3.65685i	−0.404655	+	0.134885i
$736$		0			0
$737$		−	6.72792i		−	0.247826i
$738$		0			0
$739$	1.27208			0.0467941			0.0233971	−	0.999726i	$-0.492552\pi$
$739$	1.27208			0.0467941			0.0233971	+	0.999726i	$0.492552\pi$
$740$		0			0
$741$	2.41421			0.0886884
$742$		0			0
$743$			6.72792i			0.246824i	0.992356	+	0.123412i	$0.0393836\pi$
$743$			6.72792i			0.246824i	−0.992356	+	0.123412i	$0.960616\pi$
$744$		0			0
$745$	−12.4853	−	37.4558i	−0.457425	−	1.37228i
$746$		0			0
$747$			40.9706i			1.49903i
$748$		0			0
$749$	−87.0833			−3.18195
$750$		0			0
$751$	26.7279			0.975316			0.487658	−	0.873035i	$-0.337851\pi$
$751$	26.7279			0.975316			0.487658	+	0.873035i	$0.337851\pi$
$752$		0			0
$753$			1.47309i			0.0536823i
$754$		0			0
$755$	7.41421	+	22.2426i	0.269831	+	0.809493i
$756$		0			0
$757$			12.3431i			0.448619i	0.974518	+	0.224310i	$0.0720127\pi$
$757$			12.3431i			0.448619i	−0.974518	+	0.224310i	$0.927987\pi$
$758$		0			0
$759$	0.443651			0.0161035
$760$		0			0
$761$	43.9706			1.59393			0.796966	−	0.604024i	$-0.206437\pi$
$761$	43.9706			1.59393			0.796966	+	0.604024i	$0.206437\pi$
$762$		0			0
$763$		−	79.3259i		−	2.87179i
$764$		0			0
$765$	−6.00000	+	2.00000i	−0.216930	+	0.0723102i
$766$		0			0
$767$		−	40.2132i		−	1.45201i
$768$		0			0
$769$	36.4558			1.31463			0.657316	−	0.753615i	$-0.271692\pi$
$769$	36.4558			1.31463			0.657316	+	0.753615i	$0.271692\pi$
$770$		0			0
$771$	8.58579			0.309210
$772$		0			0
$773$		−	13.9706i		−	0.502486i	−0.967924	−	0.251243i	$-0.919161\pi$
$773$		−	13.9706i		−	0.502486i	0.967924	−	0.251243i	$-0.0808394\pi$
$774$		0			0
$775$	24.9706	−	18.7279i	0.896969	−	0.672727i
$776$		0			0
$777$			15.5147i			0.556587i
$778$		0			0
$779$	4.24264			0.152008
$780$		0			0
$781$	18.9706			0.678820
$782$		0			0
$783$		−	0.414214i		−	0.0148028i
$784$		0			0
$785$	−24.7279	+	8.24264i	−0.882577	+	0.294192i
$786$		0			0
$787$			41.1838i			1.46804i	0.679126	+	0.734021i	$0.262359\pi$
$787$			41.1838i			1.46804i	−0.679126	+	0.734021i	$0.737641\pi$
$788$		0			0
$789$	−11.1716			−0.397719
$790$		0			0
$791$	45.2132			1.60760
$792$		0			0
$793$		−	83.0122i		−	2.94785i
$794$		0			0
$795$	−1.60660	−	4.81981i	−0.0569803	−	0.170941i
$796$		0			0
$797$		−	47.4853i		−

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:	\( N \)	\(=\)	\( 1520 = 2^{4} \cdot 5 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1520.d (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-0.414214i q^{3} +(0.707107 + 2.12132i) q^{5} -4.41421i q^{7} +2.82843 q^{9} +O(q^{10})\)	\(q-0.414214i q^{3} +(0.707107 + 2.12132i) q^{5} -4.41421i q^{7} +2.82843 q^{9} +1.41421 q^{11} -5.82843i q^{13} +(0.878680 - 0.292893i) q^{15} +1.00000i q^{17} -1.00000 q^{19} -1.82843 q^{21} +0.757359i q^{23} +(-4.00000 + 3.00000i) q^{25} -2.41421i q^{27} +0.171573 q^{29} -6.24264 q^{31} -0.585786i q^{33} +(9.36396 - 3.12132i) q^{35} -8.48528i q^{37} -2.41421 q^{39} -4.24264 q^{41} +1.75736i q^{43} +(2.00000 + 6.00000i) q^{45} -12.4853 q^{49} +0.414214 q^{51} -5.48528i q^{53} +(1.00000 + 3.00000i) q^{55} +0.414214i q^{57} +6.89949 q^{59} +14.2426 q^{61} -12.4853i q^{63} +(12.3640 - 4.12132i) q^{65} -4.75736i q^{67} +0.313708 q^{69} +13.4142 q^{71} -11.4853i q^{73} +(1.24264 + 1.65685i) q^{75} -6.24264i q^{77} -6.48528 q^{79} +7.48528 q^{81} +14.4853i q^{83} +(-2.12132 + 0.707107i) q^{85} -0.0710678i q^{87} -7.07107 q^{89} -25.7279 q^{91} +2.58579i q^{93} +(-0.707107 - 2.12132i) q^{95} -0.343146i q^{97} +4.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q+O(q^{10}) \) 4 * q	\( 4 q + 12 q^{15} - 4 q^{19} + 4 q^{21} - 16 q^{25} + 12 q^{29} - 8 q^{31} + 12 q^{35} - 4 q^{39} + 8 q^{45} - 16 q^{49} - 4 q^{51} + 4 q^{55} - 12 q^{59} + 40 q^{61} + 24 q^{65} - 44 q^{69} + 48 q^{71} - 12 q^{75} + 8 q^{79} - 4 q^{81} - 52 q^{91} + 16 q^{99}+O(q^{100}) \) 4 * q + 12 * q^15 - 4 * q^19 + 4 * q^21 - 16 * q^25 + 12 * q^29 - 8 * q^31 + 12 * q^35 - 4 * q^39 + 8 * q^45 - 16 * q^49 - 4 * q^51 + 4 * q^55 - 12 * q^59 + 40 * q^61 + 24 * q^65 - 44 * q^69 + 48 * q^71 - 12 * q^75 + 8 * q^79 - 4 * q^81 - 52 * q^91 + 16 * q^99

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