LMFDB - Embedded newform 1520.2.d.e.609.1

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			609.1
Root			$0.707107 - 0.707107i$ of defining polynomial
Character	$\chi$	$=$	1520.609
Dual form			1520.2.d.e.609.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.41421i q^{3} +(-0.707107 + 2.12132i) q^{5} +1.58579i q^{7} -2.82843 q^{9} +O(q^{10})$	$q-2.41421i q^{3} +(-0.707107 + 2.12132i) q^{5} +1.58579i q^{7} -2.82843 q^{9} -1.41421 q^{11} +0.171573i q^{13} +(5.12132 + 1.70711i) q^{15} -1.00000i q^{17} -1.00000 q^{19} +3.82843 q^{21} -9.24264i q^{23} +(-4.00000 - 3.00000i) q^{25} -0.414214i q^{27} +5.82843 q^{29} +2.24264 q^{31} +3.41421i q^{33} +(-3.36396 - 1.12132i) q^{35} -8.48528i q^{37} +0.414214 q^{39} +4.24264 q^{41} -10.2426i q^{43} +(2.00000 - 6.00000i) q^{45} +4.48528 q^{49} -2.41421 q^{51} -11.4853i q^{53} +(1.00000 - 3.00000i) q^{55} +2.41421i q^{57} -12.8995 q^{59} +5.75736 q^{61} -4.48528i q^{63} +(-0.363961 - 0.121320i) q^{65} +13.2426i q^{67} -22.3137 q^{69} +10.5858 q^{71} -5.48528i q^{73} +(-7.24264 + 9.65685i) q^{75} -2.24264i q^{77} +10.4853 q^{79} -9.48528 q^{81} +2.48528i q^{83} +(2.12132 + 0.707107i) q^{85} -14.0711i q^{87} +7.07107 q^{89} -0.272078 q^{91} -5.41421i q^{93} +(0.707107 - 2.12132i) q^{95} +11.6569i q^{97} +4.00000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$		−	2.41421i		−	1.39385i	−0.717146	−	0.696923i	$-0.754552\pi$
$3$		−	2.41421i		−	1.39385i	0.717146	−	0.696923i	$-0.245448\pi$
$4$		0			0
$5$	−0.707107	+	2.12132i	−0.316228	+	0.948683i
$5$	−0.707107	+	2.12132i	−0.316228	+	0.948683i
$6$		0			0
$7$			1.58579i			0.599371i	0.954038	+	0.299685i	$0.0968817\pi$
$7$			1.58579i			0.599371i	−0.954038	+	0.299685i	$0.903118\pi$
$8$		0			0
$9$	−2.82843			−0.942809
$10$		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
$13$			0.171573i			0.0475858i	0.999717	+	0.0237929i	$0.00757422\pi$
$13$			0.171573i			0.0475858i	−0.999717	+	0.0237929i	$0.992426\pi$
$14$		0			0
$15$	5.12132	+	1.70711i	1.32232	+	0.440773i
$16$		0			0
$17$		−	1.00000i		−	0.242536i	−0.992620	−	0.121268i	$-0.961304\pi$
$17$		−	1.00000i		−	0.242536i	0.992620	−	0.121268i	$-0.0386960\pi$
$18$		0			0
$19$	−1.00000			−0.229416
$19$	−1.00000			−0.229416
$20$		0			0
$21$	3.82843			0.835431
$22$		0			0
$23$		−	9.24264i		−	1.92722i	−0.267305	−	0.963612i	$-0.586133\pi$
$23$		−	9.24264i		−	1.92722i	0.267305	−	0.963612i	$-0.413867\pi$
$24$		0			0
$25$	−4.00000	−	3.00000i	−0.800000	−	0.600000i
$26$		0			0
$27$		−	0.414214i		−	0.0797154i
$28$		0			0
$29$	5.82843			1.08231			0.541156	−	0.840922i	$-0.317987\pi$
$29$	5.82843			1.08231			0.541156	+	0.840922i	$0.317987\pi$
$30$		0			0
$31$	2.24264			0.402790			0.201395	−	0.979510i	$-0.435452\pi$
$31$	2.24264			0.402790			0.201395	+	0.979510i	$0.435452\pi$
$32$		0			0
$33$			3.41421i			0.594338i
$34$		0			0
$35$	−3.36396	−	1.12132i	−0.568613	−	0.189538i
$36$		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
$39$	0.414214			0.0663273
$40$		0			0
$41$	4.24264			0.662589			0.331295	−	0.943527i	$-0.392515\pi$
$41$	4.24264			0.662589			0.331295	+	0.943527i	$0.392515\pi$
$42$		0			0
$43$		−	10.2426i		−	1.56199i	−0.624538	−	0.780994i	$-0.714713\pi$
$43$		−	10.2426i		−	1.56199i	0.624538	−	0.780994i	$-0.285287\pi$
$44$		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
$49$	4.48528			0.640754
$50$		0			0
$51$	−2.41421			−0.338058
$52$		0			0
$53$		−	11.4853i		−	1.57762i	−0.614634	−	0.788812i	$-0.710696\pi$
$53$		−	11.4853i		−	1.57762i	0.614634	−	0.788812i	$-0.289304\pi$
$54$		0			0
$55$	1.00000	−	3.00000i	0.134840	−	0.404520i
$56$		0			0
$57$			2.41421i			0.319770i
$58$		0			0
$59$	−12.8995			−1.67937			−0.839686	−	0.543073i	$-0.817261\pi$
$59$	−12.8995			−1.67937			−0.839686	+	0.543073i	$0.817261\pi$
$60$		0			0
$61$	5.75736			0.737154			0.368577	−	0.929597i	$-0.379845\pi$
$61$	5.75736			0.737154			0.368577	+	0.929597i	$0.379845\pi$
$62$		0			0
$63$		−	4.48528i		−	0.565092i
$64$		0			0
$65$	−0.363961	−	0.121320i	−0.0451438	−	0.0150479i
$66$		0			0
$67$			13.2426i			1.61785i	0.587915	+	0.808923i	$0.299949\pi$
$67$			13.2426i			1.61785i	−0.587915	+	0.808923i	$0.700051\pi$
$68$		0			0
$69$	−22.3137			−2.68625
$70$		0			0
$71$	10.5858			1.25630			0.628151	−	0.778092i	$-0.283812\pi$
$71$	10.5858			1.25630			0.628151	+	0.778092i	$0.283812\pi$
$72$		0			0
$73$		−	5.48528i		−	0.642004i	−0.947079	−	0.321002i	$-0.895980\pi$
$73$		−	5.48528i		−	0.642004i	0.947079	−	0.321002i	$-0.104020\pi$
$74$		0			0
$75$	−7.24264	+	9.65685i	−0.836308	+	1.11508i
$76$		0			0
$77$		−	2.24264i		−	0.255573i
$78$		0			0
$79$	10.4853			1.17969			0.589843	−	0.807518i	$-0.299190\pi$
$79$	10.4853			1.17969			0.589843	+	0.807518i	$0.299190\pi$
$80$		0			0
$81$	−9.48528			−1.05392
$82$		0			0
$83$			2.48528i			0.272795i	0.990654	+	0.136398i	$0.0435524\pi$
$83$			2.48528i			0.272795i	−0.990654	+	0.136398i	$0.956448\pi$
$84$		0			0
$85$	2.12132	+	0.707107i	0.230089	+	0.0766965i
$86$		0			0
$87$		−	14.0711i		−	1.50858i
$88$		0			0
$89$	7.07107			0.749532			0.374766	−	0.927119i	$-0.377723\pi$
$89$	7.07107			0.749532			0.374766	+	0.927119i	$0.377723\pi$
$90$		0			0
$91$	−0.272078			−0.0285215
$92$		0			0
$93$		−	5.41421i		−	0.561428i
$94$		0			0
$95$	0.707107	−	2.12132i	0.0725476	−	0.217643i
$96$		0			0
$97$			11.6569i			1.18357i	0.806094	+	0.591787i	$0.201577\pi$
$97$			11.6569i			1.18357i	−0.806094	+	0.591787i	$0.798423\pi$
$98$		0			0
$99$	4.00000			0.402015
$100$		0			0
$101$	−1.07107			−0.106575			−0.0532876	−	0.998579i	$-0.516970\pi$
$101$	−1.07107			−0.106575			−0.0532876	+	0.998579i	$0.516970\pi$
$102$		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
$105$	−2.70711	+	8.12132i	−0.264187	+	0.792560i
$106$		0			0
$107$		−	5.72792i		−	0.553739i	−0.960908	−	0.276870i	$-0.910703\pi$
$107$		−	5.72792i		−	0.553739i	0.960908	−	0.276870i	$-0.0892970\pi$
$108$		0			0
$109$	−15.9706			−1.52970			−0.764851	−	0.644207i	$-0.777188\pi$
$109$	−15.9706			−1.52970			−0.764851	+	0.644207i	$0.777188\pi$
$110$		0			0
$111$	−20.4853			−1.94438
$112$		0			0
$113$		−	1.75736i		−	0.165318i	−0.996578	−	0.0826592i	$-0.973659\pi$
$113$		−	1.75736i		−	0.165318i	0.996578	−	0.0826592i	$-0.0263413\pi$
$114$		0			0
$115$	19.6066	+	6.53553i	1.82833	+	0.609442i
$116$		0			0
$117$		−	0.485281i		−	0.0448643i
$118$		0			0
$119$	1.58579			0.145369
$120$		0			0
$121$	−9.00000			−0.818182
$122$		0			0
$123$		−	10.2426i		−	0.923548i
$124$		0			0
$125$	9.19239	−	6.36396i	0.822192	−	0.569210i
$126$		0			0
$127$			14.4853i			1.28536i	0.766134	+	0.642680i	$0.222178\pi$
$127$			14.4853i			1.28536i	−0.766134	+	0.642680i	$0.777822\pi$
$128$		0			0
$129$	−24.7279			−2.17717
$130$		0			0
$131$	−16.9706			−1.48272			−0.741362	−	0.671105i	$-0.765820\pi$
$131$	−16.9706			−1.48272			−0.741362	+	0.671105i	$0.765820\pi$
$132$		0			0
$133$		−	1.58579i		−	0.137505i
$134$		0			0
$135$	0.878680	+	0.292893i	0.0756247	+	0.0252082i
$136$		0			0
$137$		−	13.0000i		−	1.11066i	−0.831628	−	0.555332i	$-0.812591\pi$
$137$		−	13.0000i		−	1.11066i	0.831628	−	0.555332i	$-0.187409\pi$
$138$		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
$143$		−	0.242641i		−	0.0202906i
$144$		0			0
$145$	−4.12132	+	12.3640i	−0.342257	+	1.02677i
$146$		0			0
$147$		−	10.8284i		−	0.893114i
$148$		0			0
$149$	−6.34315			−0.519651			−0.259825	−	0.965656i	$-0.583665\pi$
$149$	−6.34315			−0.519651			−0.259825	+	0.965656i	$0.583665\pi$
$150$		0			0
$151$	−6.48528			−0.527765			−0.263882	−	0.964555i	$-0.585003\pi$
$151$	−6.48528			−0.527765			−0.263882	+	0.964555i	$0.585003\pi$
$152$		0			0
$153$			2.82843i			0.228665i
$154$		0			0
$155$	−1.58579	+	4.75736i	−0.127373	+	0.382120i
$156$		0			0
$157$		−	0.343146i		−	0.0273860i	−0.999906	−	0.0136930i	$-0.995641\pi$
$157$		−	0.343146i		−	0.0273860i	0.999906	−	0.0136930i	$-0.00435876\pi$
$158$		0			0
$159$	−27.7279			−2.19897
$160$		0			0
$161$	14.6569			1.15512
$162$		0			0
$163$			1.75736i			0.137647i	0.997629	+	0.0688235i	$0.0219245\pi$
$163$			1.75736i			0.137647i	−0.997629	+	0.0688235i	$0.978075\pi$
$164$		0			0
$165$	−7.24264	−	2.41421i	−0.563839	−	0.187946i
$166$		0			0
$167$		−	9.75736i		−	0.755047i	−0.926000	−	0.377524i	$-0.876776\pi$
$167$		−	9.75736i		−	0.755047i	0.926000	−	0.377524i	$-0.123224\pi$
$168$		0			0
$169$	12.9706			0.997736
$170$		0			0
$171$	2.82843			0.216295
$172$		0			0
$173$		−	16.4853i		−	1.25335i	−0.779280	−	0.626676i	$-0.784415\pi$
$173$		−	16.4853i		−	1.25335i	0.779280	−	0.626676i	$-0.215585\pi$
$174$		0			0
$175$	4.75736	−	6.34315i	0.359623	−	0.479497i
$176$		0			0
$177$			31.1421i			2.34079i
$178$		0			0
$179$	−0.343146			−0.0256479			−0.0128240	−	0.999918i	$-0.504082\pi$
$179$	−0.343146			−0.0256479			−0.0128240	+	0.999918i	$0.504082\pi$
$180$		0			0
$181$	8.48528			0.630706			0.315353	−	0.948974i	$-0.397877\pi$
$181$	8.48528			0.630706			0.315353	+	0.948974i	$0.397877\pi$
$182$		0			0
$183$		−	13.8995i		−	1.02748i
$184$		0			0
$185$	18.0000	+	6.00000i	1.32339	+	0.441129i
$186$		0			0
$187$			1.41421i			0.103418i
$188$		0			0
$189$	0.656854			0.0477791
$190$		0			0
$191$	18.5563			1.34269			0.671345	−	0.741145i	$-0.265717\pi$
$191$	18.5563			1.34269			0.671345	+	0.741145i	$0.265717\pi$
$192$		0			0
$193$		−	11.6569i		−	0.839079i	−0.907737	−	0.419539i	$-0.862192\pi$
$193$		−	11.6569i		−	0.839079i	0.907737	−	0.419539i	$-0.137808\pi$
$194$		0			0
$195$	−0.292893	+	0.878680i	−0.0209745	+	0.0629236i
$196$		0			0
$197$			20.2426i			1.44223i	0.692816	+	0.721114i	$0.256370\pi$
$197$			20.2426i			1.44223i	−0.692816	+	0.721114i	$0.743630\pi$
$198$		0			0
$199$	0.757359			0.0536878			0.0268439	−	0.999640i	$-0.491454\pi$
$199$	0.757359			0.0536878			0.0268439	+	0.999640i	$0.491454\pi$
$200$		0			0
$201$	31.9706			2.25503
$202$		0			0
$203$			9.24264i			0.648706i
$204$		0			0
$205$	−3.00000	+	9.00000i	−0.209529	+	0.628587i
$206$		0			0
$207$			26.1421i			1.81700i
$208$		0			0
$209$	1.41421			0.0978232
$210$		0			0
$211$	−5.72792			−0.394326			−0.197163	−	0.980371i	$-0.563173\pi$
$211$	−5.72792			−0.394326			−0.197163	+	0.980371i	$0.563173\pi$
$212$		0			0
$213$		−	25.5563i		−	1.75109i
$214$		0			0
$215$	21.7279	+	7.24264i	1.48183	+	0.493944i
$216$		0			0
$217$			3.55635i			0.241421i
$218$		0			0
$219$	−13.2426			−0.894855
$220$		0			0
$221$	0.171573			0.0115412
$222$		0			0
$223$			20.8284i			1.39477i	0.716694	+	0.697387i	$0.245654\pi$
$223$			20.8284i			1.39477i	−0.716694	+	0.697387i	$0.754346\pi$
$224$		0			0
$225$	11.3137	+	8.48528i	0.754247	+	0.565685i
$226$		0			0
$227$		−	25.2426i		−	1.67541i	−0.546121	−	0.837706i	$-0.683896\pi$
$227$		−	25.2426i		−	1.67541i	0.546121	−	0.837706i	$-0.316104\pi$
$228$		0			0
$229$	18.9706			1.25361			0.626805	−	0.779176i	$-0.284362\pi$
$229$	18.9706			1.25361			0.626805	+	0.779176i	$0.284362\pi$
$230$		0			0
$231$	−5.41421			−0.356229
$232$		0			0
$233$			8.97056i			0.587681i	0.955854	+	0.293841i	$0.0949335\pi$
$233$			8.97056i			0.587681i	−0.955854	+	0.293841i	$0.905067\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		−	25.3137i		−	1.64430i
$238$		0			0
$239$	12.8995			0.834399			0.417199	−	0.908815i	$-0.363012\pi$
$239$	12.8995			0.834399			0.417199	+	0.908815i	$0.363012\pi$
$240$		0			0
$241$	−24.9706			−1.60850			−0.804248	−	0.594294i	$-0.797431\pi$
$241$	−24.9706			−1.60850			−0.804248	+	0.594294i	$0.797431\pi$
$242$		0			0
$243$			21.6569i			1.38929i
$244$		0			0
$245$	−3.17157	+	9.51472i	−0.202624	+	0.607873i
$246$		0			0
$247$		−	0.171573i		−	0.0109169i
$248$		0			0
$249$	6.00000			0.380235
$250$		0			0
$251$	27.5563			1.73934			0.869671	−	0.493632i	$-0.164331\pi$
$251$	27.5563			1.73934			0.869671	+	0.493632i	$0.164331\pi$
$252$		0			0
$253$			13.0711i			0.821771i
$254$		0			0
$255$	1.70711	−	5.12132i	0.106903	−	0.320710i
$256$		0			0
$257$			4.72792i			0.294920i	0.989068	+	0.147460i	$0.0471097\pi$
$257$			4.72792i			0.294920i	−0.989068	+	0.147460i	$0.952890\pi$
$258$		0			0
$259$	13.4558			0.836105
$260$		0			0
$261$	−16.4853			−1.02041
$262$		0			0
$263$		−	6.97056i		−	0.429823i	−0.976633	−	0.214912i	$-0.931054\pi$
$263$		−	6.97056i		−	0.429823i	0.976633	−	0.214912i	$-0.0689464\pi$
$264$		0			0
$265$	24.3640	+	8.12132i	1.49667	+	0.498889i
$266$		0			0
$267$		−	17.0711i		−	1.04473i
$268$		0			0
$269$	−28.6274			−1.74544			−0.872722	−	0.488217i	$-0.837647\pi$
$269$	−28.6274			−1.74544			−0.872722	+	0.488217i	$0.837647\pi$
$270$		0			0
$271$	18.7574			1.13943			0.569714	−	0.821843i	$-0.307054\pi$
$271$	18.7574			1.13943			0.569714	+	0.821843i	$0.307054\pi$
$272$		0			0
$273$			0.656854i			0.0397546i
$274$		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
$279$	−6.34315			−0.379754
$280$		0			0
$281$	−4.24264			−0.253095			−0.126547	−	0.991961i	$-0.540390\pi$
$281$	−4.24264			−0.253095			−0.126547	+	0.991961i	$0.540390\pi$
$282$		0			0
$283$		−	3.85786i		−	0.229326i	−0.993404	−	0.114663i	$-0.963421\pi$
$283$		−	3.85786i		−	0.229326i	0.993404	−	0.114663i	$-0.0365789\pi$
$284$		0			0
$285$	−5.12132	−	1.70711i	−0.303361	−	0.101120i
$286$		0			0
$287$			6.72792i			0.397137i
$288$		0			0
$289$	16.0000			0.941176
$290$		0			0
$291$	28.1421			1.64972
$292$		0			0
$293$			11.4853i			0.670977i	0.942044	+	0.335489i	$0.108901\pi$
$293$			11.4853i			0.670977i	−0.942044	+	0.335489i	$0.891099\pi$
$294$		0			0
$295$	9.12132	−	27.3640i	0.531064	−	1.59319i
$296$		0			0
$297$			0.585786i			0.0339908i
$298$		0			0
$299$	1.58579			0.0917084
$300$		0			0
$301$	16.2426			0.936210
$302$		0			0
$303$			2.58579i			0.148550i
$304$		0			0
$305$	−4.07107	+	12.2132i	−0.233109	+	0.699326i
$306$		0			0
$307$		−	6.34315i		−	0.362022i	−0.983481	−	0.181011i	$-0.942063\pi$
$307$		−	6.34315i		−	0.362022i	0.983481	−	0.181011i	$-0.0579370\pi$
$308$		0			0
$309$	10.2426			0.582683
$310$		0			0
$311$	13.2426			0.750921			0.375461	−	0.926838i	$-0.377485\pi$
$311$	13.2426			0.750921			0.375461	+	0.926838i	$0.377485\pi$
$312$		0			0
$313$			25.9706i			1.46794i	0.679180	+	0.733971i	$0.262335\pi$
$313$			25.9706i			1.46794i	−0.679180	+	0.733971i	$0.737665\pi$
$314$		0			0
$315$	9.51472	+	3.17157i	0.536094	+	0.178698i
$316$		0			0
$317$		−	7.48528i		−	0.420415i	−0.977657	−	0.210208i	$-0.932586\pi$
$317$		−	7.48528i		−	0.420415i	0.977657	−	0.210208i	$-0.0674140\pi$
$318$		0			0
$319$	−8.24264			−0.461499
$320$		0			0
$321$	−13.8284			−0.771828
$322$		0			0
$323$			1.00000i			0.0556415i
$324$		0			0
$325$	0.514719	−	0.686292i	0.0285515	−	0.0380686i
$326$		0			0
$327$			38.5563i			2.13217i
$328$		0			0
$329$		0			0
$330$		0			0
$331$	10.7574			0.591278			0.295639	−	0.955300i	$-0.404467\pi$
$331$	10.7574			0.591278			0.295639	+	0.955300i	$0.404467\pi$
$332$		0			0
$333$			24.0000i			1.31519i
$334$		0			0
$335$	−28.0919	−	9.36396i	−1.53482	−	0.511608i
$336$		0			0
$337$		−	33.8995i		−	1.84662i	−0.384052	−	0.923312i	$-0.625472\pi$
$337$		−	33.8995i		−	1.84662i	0.384052	−	0.923312i	$-0.374528\pi$
$338$		0			0
$339$	−4.24264			−0.230429
$340$		0			0
$341$	−3.17157			−0.171750
$342$		0			0
$343$			18.2132i			0.983421i
$344$		0			0
$345$	15.7782	−	47.3345i	0.849468	−	2.54841i
$346$		0			0
$347$		−	22.4853i		−	1.20707i	−0.797335	−	0.603537i	$-0.793758\pi$
$347$		−	22.4853i		−	1.20707i	0.797335	−	0.603537i	$-0.206242\pi$
$348$		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
$351$	0.0710678			0.00379332
$352$		0			0
$353$		−	19.4853i		−	1.03710i	−0.855048	−	0.518548i	$-0.826473\pi$
$353$		−	19.4853i		−	1.03710i	0.855048	−	0.518548i	$-0.173527\pi$
$354$		0			0
$355$	−7.48528	+	22.4558i	−0.397277	+	1.19183i
$356$		0			0
$357$		−	3.82843i		−	0.202622i
$358$		0			0
$359$	−31.2426			−1.64892			−0.824462	−	0.565918i	$-0.808522\pi$
$359$	−31.2426			−1.64892			−0.824462	+	0.565918i	$0.808522\pi$
$360$		0			0
$361$	1.00000			0.0526316
$362$		0			0
$363$			21.7279i			1.14042i
$364$		0			0
$365$	11.6360	+	3.87868i	0.609058	+	0.203019i
$366$		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
$371$	18.2132			0.945582
$372$		0			0
$373$			9.00000i			0.466002i	0.972476	+	0.233001i	$0.0748546\pi$
$373$			9.00000i			0.466002i	−0.972476	+	0.233001i	$0.925145\pi$
$374$		0			0
$375$	−15.3640	−	22.1924i	−0.793392	−	1.14601i
$376$		0			0
$377$			1.00000i			0.0515026i
$378$		0			0
$379$	2.75736			0.141636			0.0708180	−	0.997489i	$-0.477439\pi$
$379$	2.75736			0.141636			0.0708180	+	0.997489i	$0.477439\pi$
$380$		0			0
$381$	34.9706			1.79160
$382$		0			0
$383$		−	3.75736i		−	0.191992i	−0.995382	−	0.0959960i	$-0.969396\pi$
$383$		−	3.75736i		−	0.191992i	0.995382	−	0.0959960i	$-0.0306036\pi$
$384$		0			0
$385$	4.75736	+	1.58579i	0.242457	+	0.0808192i
$386$		0			0
$387$			28.9706i			1.47266i
$388$		0			0
$389$	−37.0711			−1.87958			−0.939789	−	0.341756i	$-0.888979\pi$
$389$	−37.0711			−1.87958			−0.939789	+	0.341756i	$0.888979\pi$
$390$		0			0
$391$	−9.24264			−0.467420
$392$		0			0
$393$			40.9706i			2.06669i
$394$		0			0
$395$	−7.41421	+	22.2426i	−0.373050	+	1.11915i
$396$		0			0
$397$		−	24.0000i		−	1.20453i	−0.798298	−	0.602263i	$-0.794266\pi$
$397$		−	24.0000i		−	1.20453i	0.798298	−	0.602263i	$-0.205734\pi$
$398$		0			0
$399$	−3.82843			−0.191661
$400$		0			0
$401$	−22.5858			−1.12788			−0.563940	−	0.825816i	$-0.690715\pi$
$401$	−22.5858			−1.12788			−0.563940	+	0.825816i	$0.690715\pi$
$402$		0			0
$403$			0.384776i			0.0191671i
$404$		0			0
$405$	6.70711	−	20.1213i	0.333279	−	0.999836i
$406$		0			0
$407$			12.0000i			0.594818i
$408$		0			0
$409$	17.2132			0.851138			0.425569	−	0.904926i	$-0.360074\pi$
$409$	17.2132			0.851138			0.425569	+	0.904926i	$0.360074\pi$
$410$		0			0
$411$	−31.3848			−1.54810
$412$		0			0
$413$		−	20.4558i		−	1.00657i
$414$		0			0
$415$	−5.27208	−	1.75736i	−0.258796	−	0.0862654i
$416$		0			0
$417$			28.9706i			1.41869i
$418$		0			0
$419$	16.5858			0.810269			0.405134	−	0.914257i	$-0.367225\pi$
$419$	16.5858			0.810269			0.405134	+	0.914257i	$0.367225\pi$
$420$		0			0
$421$	2.51472			0.122560			0.0612799	−	0.998121i	$-0.480482\pi$
$421$	2.51472			0.122560			0.0612799	+	0.998121i	$0.480482\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−3.00000	+	4.00000i	−0.145521	+	0.194029i
$426$		0			0
$427$			9.12994i			0.441829i
$428$		0			0
$429$	−0.585786			−0.0282820
$430$		0			0
$431$	−30.3848			−1.46358			−0.731792	−	0.681528i	$-0.761316\pi$
$431$	−30.3848			−1.46358			−0.731792	+	0.681528i	$0.761316\pi$
$432$		0			0
$433$			21.5563i			1.03593i	0.855401	+	0.517966i	$0.173311\pi$
$433$			21.5563i			1.03593i	−0.855401	+	0.517966i	$0.826689\pi$
$434$		0			0
$435$	29.8492	+	9.94975i	1.43116	+	0.477054i
$436$		0			0
$437$			9.24264i			0.442135i
$438$		0			0
$439$	−14.2426			−0.679764			−0.339882	−	0.940468i	$-0.610387\pi$
$439$	−14.2426			−0.679764			−0.339882	+	0.940468i	$0.610387\pi$
$440$		0			0
$441$	−12.6863			−0.604109
$442$		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
$447$			15.3137i			0.724314i
$448$		0			0
$449$	5.31371			0.250769			0.125385	−	0.992108i	$-0.459983\pi$
$449$	5.31371			0.250769			0.125385	+	0.992108i	$0.459983\pi$
$450$		0			0
$451$	−6.00000			−0.282529
$452$		0			0
$453$			15.6569i			0.735623i
$454$		0			0
$455$	0.192388	−	0.577164i	0.00901930	−	0.0270579i
$456$		0			0
$457$			3.00000i			0.140334i	0.997535	+	0.0701670i	$0.0223532\pi$
$457$			3.00000i			0.140334i	−0.997535	+	0.0701670i	$0.977647\pi$
$458$		0			0
$459$	−0.414214			−0.0193338
$460$		0			0
$461$	27.5563			1.28343			0.641714	−	0.766944i	$-0.278224\pi$
$461$	27.5563			1.28343			0.641714	+	0.766944i	$0.278224\pi$
$462$		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
$465$	11.4853	+	3.82843i	0.532617	+	0.177539i
$466$		0			0
$467$		−	24.7279i		−	1.14427i	−0.820159	−	0.572136i	$-0.806115\pi$
$467$		−	24.7279i		−	1.14427i	0.820159	−	0.572136i	$-0.193885\pi$
$468$		0			0
$469$	−21.0000			−0.969690
$470$		0			0
$471$	−0.828427			−0.0381719
$472$		0			0
$473$			14.4853i			0.666034i
$474$		0			0
$475$	4.00000	+	3.00000i	0.183533	+	0.137649i
$476$		0			0
$477$			32.4853i			1.48740i
$478$		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
$481$	1.45584			0.0663808
$482$		0			0
$483$		−	35.3848i		−	1.61006i
$484$		0			0
$485$	−24.7279	−	8.24264i	−1.12284	−	0.374279i
$486$		0			0
$487$		−	1.79899i		−	0.0815200i	−0.999169	−	0.0407600i	$-0.987022\pi$
$487$		−	1.79899i		−	0.0815200i	0.999169	−	0.0407600i	$-0.0129779\pi$
$488$		0			0
$489$	4.24264			0.191859
$490$		0			0
$491$	2.44365			0.110280			0.0551402	−	0.998479i	$-0.482439\pi$
$491$	2.44365			0.110280			0.0551402	+	0.998479i	$0.482439\pi$
$492$		0			0
$493$		−	5.82843i		−	0.262499i
$494$		0			0
$495$	−2.82843	+	8.48528i	−0.127128	+	0.381385i
$496$		0			0
$497$			16.7868i			0.752991i
$498$		0			0
$499$	−34.2426			−1.53291			−0.766456	−	0.642297i	$-0.777981\pi$
$499$	−34.2426			−1.53291			−0.766456	+	0.642297i	$0.777981\pi$
$500$		0			0
$501$	−23.5563			−1.05242
$502$		0			0
$503$		−	39.7279i		−	1.77138i	−0.464277	−	0.885690i	$-0.653686\pi$
$503$		−	39.7279i		−	1.77138i	0.464277	−	0.885690i	$-0.346314\pi$
$504$		0			0
$505$	0.757359	−	2.27208i	0.0337020	−	0.101106i
$506$		0			0
$507$		−	31.3137i		−	1.39069i
$508$		0			0
$509$	4.97056			0.220316			0.110158	−	0.993914i	$-0.464864\pi$
$509$	4.97056			0.220316			0.110158	+	0.993914i	$0.464864\pi$
$510$		0			0
$511$	8.69848			0.384798
$512$		0			0
$513$			0.414214i			0.0182880i
$514$		0			0
$515$	−9.00000	−	3.00000i	−0.396587	−	0.132196i
$516$		0			0
$517$		0			0
$518$		0			0
$519$	−39.7990			−1.74698
$520$		0			0
$521$	0.686292			0.0300670			0.0150335	−	0.999887i	$-0.495215\pi$
$521$	0.686292			0.0300670			0.0150335	+	0.999887i	$0.495215\pi$
$522$		0			0
$523$			27.7279i			1.21246i	0.795290	+	0.606229i	$0.207318\pi$
$523$			27.7279i			1.21246i	−0.795290	+	0.606229i	$0.792682\pi$
$524$		0			0
$525$	−15.3137	−	11.4853i	−0.668345	−	0.501259i
$526$		0			0
$527$		−	2.24264i		−	0.0976910i
$528$		0			0
$529$	−62.4264			−2.71419
$530$		0			0
$531$	36.4853			1.58333
$532$		0			0
$533$			0.727922i			0.0315298i
$534$		0			0
$535$	12.1508	+	4.05025i	0.525323	+	0.175108i
$536$		0			0
$537$			0.828427i			0.0357493i
$538$		0			0
$539$	−6.34315			−0.273219
$540$		0			0
$541$	18.2426			0.784312			0.392156	−	0.919899i	$-0.371729\pi$
$541$	18.2426			0.784312			0.392156	+	0.919899i	$0.371729\pi$
$542$		0			0
$543$		−	20.4853i		−	0.879108i
$544$		0			0
$545$	11.2929	−	33.8787i	0.483734	−	1.45120i
$546$		0			0
$547$			5.31371i			0.227198i	0.993527	+	0.113599i	$0.0362379\pi$
$547$			5.31371i			0.227198i	−0.993527	+	0.113599i	$0.963762\pi$
$548$		0			0
$549$	−16.2843			−0.694996
$550$		0			0
$551$	−5.82843			−0.248299
$552$		0			0
$553$			16.6274i			0.707070i
$554$		0			0
$555$	14.4853	−	43.4558i	0.614866	−	1.84460i
$556$		0			0
$557$		−	16.0000i		−	0.677942i	−0.940797	−	0.338971i	$-0.889921\pi$
$557$		−	16.0000i		−	0.677942i	0.940797	−	0.338971i	$-0.110079\pi$
$558$		0			0
$559$	1.75736			0.0743284
$560$		0			0
$561$	3.41421			0.144148
$562$		0			0
$563$			28.9706i			1.22096i	0.792030	+	0.610482i	$0.209024\pi$
$563$			28.9706i			1.22096i	−0.792030	+	0.610482i	$0.790976\pi$
$564$		0			0
$565$	3.72792	+	1.24264i	0.156835	+	0.0522783i
$566$		0			0
$567$		−	15.0416i		−	0.631689i
$568$		0			0
$569$	−28.2843			−1.18574			−0.592869	−	0.805299i	$-0.702005\pi$
$569$	−28.2843			−1.18574			−0.592869	+	0.805299i	$0.702005\pi$
$570$		0			0
$571$	−6.24264			−0.261246			−0.130623	−	0.991432i	$-0.541698\pi$
$571$	−6.24264			−0.261246			−0.130623	+	0.991432i	$0.541698\pi$
$572$		0			0
$573$		−	44.7990i		−	1.87150i
$574$		0			0
$575$	−27.7279	+	36.9706i	−1.15633	+	1.54178i
$576$		0			0
$577$		−	2.31371i		−	0.0963209i	−0.998840	−	0.0481605i	$-0.984664\pi$
$577$		−	2.31371i		−	0.0963209i	0.998840	−	0.0481605i	$-0.0153359\pi$
$578$		0			0
$579$	−28.1421			−1.16955
$580$		0			0
$581$	−3.94113			−0.163505
$582$		0			0
$583$			16.2426i			0.672701i
$584$		0			0
$585$	1.02944	+	0.343146i	0.0425620	+	0.0141873i
$586$		0			0
$587$			21.7574i			0.898022i	0.893526	+	0.449011i	$0.148224\pi$
$587$			21.7574i			0.898022i	−0.893526	+	0.449011i	$0.851776\pi$
$588$		0			0
$589$	−2.24264			−0.0924064
$590$		0			0
$591$	48.8701			2.01025
$592$		0			0
$593$			22.0000i			0.903432i	0.892162	+	0.451716i	$0.149188\pi$
$593$			22.0000i			0.903432i	−0.892162	+	0.451716i	$0.850812\pi$
$594$		0			0
$595$	−1.12132	+	3.36396i	−0.0459697	+	0.137909i
$596$		0			0
$597$		−	1.82843i		−	0.0748325i
$598$		0			0
$599$	27.2132			1.11190			0.555951	−	0.831215i	$-0.312354\pi$
$599$	27.2132			1.11190			0.555951	+	0.831215i	$0.312354\pi$
$600$		0			0
$601$	−11.7574			−0.479593			−0.239796	−	0.970823i	$-0.577081\pi$
$601$	−11.7574			−0.479593			−0.239796	+	0.970823i	$0.577081\pi$
$602$		0			0
$603$		−	37.4558i		−	1.52532i
$604$		0			0
$605$	6.36396	−	19.0919i	0.258732	−	0.776195i
$606$		0			0
$607$			8.82843i			0.358335i	0.983819	+	0.179167i	$0.0573404\pi$
$607$			8.82843i			0.358335i	−0.983819	+	0.179167i	$0.942660\pi$
$608$		0			0
$609$	22.3137			0.904197
$610$		0			0
$611$		0			0
$612$		0			0
$613$			47.6985i			1.92652i	0.268564	+	0.963262i	$0.413451\pi$
$613$			47.6985i			1.92652i	−0.268564	+	0.963262i	$0.586549\pi$
$614$		0			0
$615$	21.7279	+	7.24264i	0.876154	+	0.292051i
$616$		0			0
$617$		−	12.4853i		−	0.502639i	−0.967904	−	0.251319i	$-0.919136\pi$
$617$		−	12.4853i		−	0.502639i	0.967904	−	0.251319i	$-0.0808644\pi$
$618$		0			0
$619$	16.2426			0.652847			0.326423	−	0.945224i	$-0.394156\pi$
$619$	16.2426			0.652847			0.326423	+	0.945224i	$0.394156\pi$
$620$		0			0
$621$	−3.82843			−0.153629
$622$		0			0
$623$			11.2132i			0.449248i
$624$		0			0
$625$	7.00000	+	24.0000i	0.280000	+	0.960000i
$626$		0			0
$627$		−	3.41421i		−	0.136351i
$628$		0			0
$629$	−8.48528			−0.338330
$630$		0			0
$631$	5.02944			0.200219			0.100109	−	0.994976i	$-0.468081\pi$
$631$	5.02944			0.200219			0.100109	+	0.994976i	$0.468081\pi$
$632$		0			0
$633$			13.8284i			0.549631i
$634$		0			0
$635$	−30.7279	−	10.2426i	−1.21940	−	0.406467i
$636$		0			0
$637$			0.769553i			0.0304908i
$638$		0			0
$639$	−29.9411			−1.18445
$640$		0			0
$641$	30.0416			1.18657			0.593287	−	0.804991i	$-0.297830\pi$
$641$	30.0416			1.18657			0.593287	+	0.804991i	$0.297830\pi$
$642$		0			0
$643$		−	2.48528i		−	0.0980099i	−0.998799	−	0.0490050i	$-0.984395\pi$
$643$		−	2.48528i		−	0.0980099i	0.998799	−	0.0490050i	$-0.0156050\pi$
$644$		0			0
$645$	17.4853	−	52.4558i	0.688482	−	2.06545i
$646$		0			0
$647$		−	27.2426i		−	1.07102i	−0.844529	−	0.535509i	$-0.820120\pi$
$647$		−	27.2426i		−	1.07102i	0.844529	−	0.535509i	$-0.179880\pi$
$648$		0			0
$649$	18.2426			0.716086
$650$		0			0
$651$	8.58579			0.336504
$652$		0			0
$653$			4.97056i			0.194513i	0.995259	+	0.0972566i	$0.0310067\pi$
$653$			4.97056i			0.194513i	−0.995259	+	0.0972566i	$0.968993\pi$
$654$		0			0
$655$	12.0000	−	36.0000i	0.468879	−	1.40664i
$656$		0			0
$657$			15.5147i			0.605287i
$658$		0			0
$659$	−0.899495			−0.0350393			−0.0175197	−	0.999847i	$-0.505577\pi$
$659$	−0.899495			−0.0350393			−0.0175197	+	0.999847i	$0.505577\pi$
$660$		0			0
$661$	32.4558			1.26239			0.631193	−	0.775626i	$-0.282566\pi$
$661$	32.4558			1.26239			0.631193	+	0.775626i	$0.282566\pi$
$662$		0			0
$663$		−	0.414214i		−	0.0160867i
$664$		0			0
$665$	3.36396	+	1.12132i	0.130449	+	0.0434829i
$666$		0			0
$667$		−	53.8701i		−	2.08586i
$668$		0			0
$669$	50.2843			1.94410
$670$		0			0
$671$	−8.14214			−0.314324
$672$		0			0
$673$		−	12.0000i		−	0.462566i	−0.972887	−	0.231283i	$-0.925708\pi$
$673$		−	12.0000i		−	0.462566i	0.972887	−	0.231283i	$-0.0742923\pi$
$674$		0			0
$675$	−1.24264	+	1.65685i	−0.0478293	+	0.0637723i
$676$		0			0
$677$		−	26.9411i		−	1.03543i	−0.855553	−	0.517716i	$-0.826782\pi$
$677$		−	26.9411i		−	1.03543i	0.855553	−	0.517716i	$-0.173218\pi$
$678$		0			0
$679$	−18.4853			−0.709400
$680$		0			0
$681$	−60.9411			−2.33527
$682$		0			0
$683$		−	12.0000i		−	0.459167i	−0.973289	−	0.229584i	$-0.926264\pi$
$683$		−	12.0000i		−	0.459167i	0.973289	−	0.229584i	$-0.0737364\pi$
$684$		0			0
$685$	27.5772	+	9.19239i	1.05367	+	0.351223i
$686$		0			0
$687$		−	45.7990i		−	1.74734i
$688$		0			0
$689$	1.97056			0.0750725
$690$		0			0
$691$	−5.51472			−0.209790			−0.104895	−	0.994483i	$-0.533451\pi$
$691$	−5.51472			−0.209790			−0.104895	+	0.994483i	$0.533451\pi$
$692$		0			0
$693$			6.34315i			0.240956i
$694$		0			0
$695$	8.48528	−	25.4558i	0.321865	−	0.965595i
$696$		0			0
$697$		−	4.24264i		−	0.160701i
$698$		0			0
$699$	21.6569			0.819137
$700$		0			0
$701$	−16.9706			−0.640969			−0.320485	−	0.947254i	$-0.603846\pi$
$701$	−16.9706			−0.640969			−0.320485	+	0.947254i	$0.603846\pi$
$702$		0			0
$703$			8.48528i			0.320028i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		−	1.69848i		−	0.0638781i
$708$		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
$711$	−29.6569			−1.11222
$712$		0			0
$713$		−	20.7279i		−	0.776267i
$714$		0			0
$715$	0.514719	+	0.171573i	0.0192494	+	0.00641646i
$716$		0			0
$717$		−	31.1421i		−	1.16302i
$718$		0			0
$719$	−24.8995			−0.928594			−0.464297	−	0.885679i	$-0.653693\pi$
$719$	−24.8995			−0.928594			−0.464297	+	0.885679i	$0.653693\pi$
$720$		0			0
$721$	−6.72792			−0.250561
$722$		0			0
$723$			60.2843i			2.24200i
$724$		0			0
$725$	−23.3137	−	17.4853i	−0.865849	−	0.649387i
$726$		0			0
$727$		−	15.7279i		−	0.583316i	−0.956523	−	0.291658i	$-0.905793\pi$
$727$		−	15.7279i		−	0.583316i	0.956523	−	0.291658i	$-0.0942070\pi$
$728$		0			0
$729$	23.8284			0.882534
$730$		0			0
$731$	−10.2426			−0.378838
$732$		0			0
$733$			12.0000i			0.443230i	0.975134	+	0.221615i	$0.0711328\pi$
$733$			12.0000i			0.443230i	−0.975134	+	0.221615i	$0.928867\pi$
$734$		0			0
$735$	22.9706	+	7.65685i	0.847282	+	0.282427i
$736$		0			0
$737$		−	18.7279i		−	0.689852i
$738$		0			0
$739$	26.7279			0.983203			0.491601	−	0.870820i	$-0.336412\pi$
$739$	26.7279			0.983203			0.491601	+	0.870820i	$0.336412\pi$
$740$		0			0
$741$	−0.414214			−0.0152165
$742$		0			0
$743$			18.7279i			0.687061i	0.939142	+	0.343530i	$0.111623\pi$
$743$			18.7279i			0.687061i	−0.939142	+	0.343530i	$0.888377\pi$
$744$		0			0
$745$	4.48528	−	13.4558i	0.164328	−	0.492984i
$746$		0			0
$747$		−	7.02944i		−	0.257194i
$748$		0			0
$749$	9.08326			0.331895
$750$		0			0
$751$	1.27208			0.0464188			0.0232094	−	0.999731i	$-0.492612\pi$
$751$	1.27208			0.0464188			0.0232094	+	0.999731i	$0.492612\pi$
$752$		0			0
$753$		−	66.5269i		−	2.42438i
$754$		0			0
$755$	4.58579	−	13.7574i	0.166894	−	0.500682i
$756$		0			0
$757$		−	23.6569i		−	0.859823i	−0.902871	−	0.429911i	$-0.858545\pi$
$757$		−	23.6569i		−	0.859823i	0.902871	−	0.429911i	$-0.141455\pi$
$758$		0			0
$759$	31.5563			1.14542
$760$		0			0
$761$	10.0294			0.363567			0.181783	−	0.983339i	$-0.441813\pi$
$761$	10.0294			0.363567			0.181783	+	0.983339i	$0.441813\pi$
$762$		0			0
$763$		−	25.3259i		−	0.916859i
$764$		0			0
$765$	−6.00000	−	2.00000i	−0.216930	−	0.0723102i
$766$		0			0
$767$		−	2.21320i		−	0.0799141i
$768$		0			0
$769$	−14.4558			−0.521291			−0.260646	−	0.965435i	$-0.583935\pi$
$769$	−14.4558			−0.521291			−0.260646	+	0.965435i	$0.583935\pi$
$770$		0			0
$771$	11.4142			0.411073
$772$		0			0
$773$		−	19.9706i		−	0.718291i	−0.933282	−	0.359146i	$-0.883068\pi$
$773$		−	19.9706i		−	0.718291i	0.933282	−	0.359146i	$-0.116932\pi$
$774$		0			0
$775$	−8.97056	−	6.72792i	−0.322232	−	0.241674i
$776$		0			0
$777$		−	32.4853i		−	1.16540i
$778$		0			0
$779$	−4.24264			−0.152008
$780$		0			0
$781$	−14.9706			−0.535689
$782$		0			0
$783$		−	2.41421i		−	0.0862770i
$784$		0			0
$785$	0.727922	+	0.242641i	0.0259807	+	0.00866022i
$786$		0			0
$787$			35.1838i			1.25417i	0.778953	+	0.627083i	$0.215751\pi$
$787$			35.1838i			1.25417i	−0.778953	+	0.627083i	$0.784249\pi$
$788$		0			0
$789$	−16.8284			−0.599108
$790$		0			0
$791$	2.78680			0.0990871
$792$		0			0
$793$			0.987807i			0.0350780i
$794$		0			0
$795$	19.6066	−	58.8198i	0.695375	−	2.08612i
$796$		0			0
$797$			30.5147i			1.08089i	0.841380	+	0.540443i	$0.181743\pi$
$797$			30.5147i			1.08089i	−0.841380	+	0.540443i	$0.818257\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$	−20.0000			−0.706665
$802$		0			0
$803$			7.75736i			0.273751i
$804$		0			0
$805$	−10.3640	+	31.0919i	−0.365282	+	1.09584i
$806$		0			0
$807$			69.1127i			2.43288i
$808$		0			0
$809$	−10.7990			−0.379672			−0.189836	−	0.981816i	$-0.560796\pi$
$809$	−10.7990			−0.379672			−0.189836	+	0.981816i	$0.560796\pi$
$810$		0			0
$811$	6.69848			0.235216			0.117608	−	0.993060i	$-0.462477\pi$
$811$	6.69848			0.235216			0.117608	+	0.993060i	$0.462477\pi$
$812$		0			0
$813$		−	45.2843i		−	1.58819i
$814$		0			0
$815$	−3.72792	−	1.24264i	−0.130583	−	0.0435278i
$816$		0			0
$817$			10.2426i			0.358345i
$818$		0			0
$819$	0.769553			0.0268903
$820$		0			0
$821$	17.6569			0.616228			0.308114	−	0.951349i	$-0.400302\pi$
$821$	17.6569			0.616228			0.308114	+	0.951349i	$0.400302\pi$
$822$		0			0
$823$		−	9.72792i		−	0.339094i	−0.985522	−	0.169547i	$-0.945770\pi$
$823$		−	9.72792i		−	0.339094i	0.985522	−	0.169547i	$-0.0542305\pi$
$824$		0			0
$825$	10.2426	−	13.6569i	0.356603	−	0.475471i
$826$		0			0
$827$			38.6985i			1.34568i	0.739789	+	0.672839i	$0.234925\pi$
$827$			38.6985i			1.34568i	−0.739789	+	0.672839i	$0.765075\pi$
$828$		0			0
$829$	40.4558			1.40509			0.702545	−	0.711640i	$-0.252047\pi$
$829$	40.4558			1.40509			0.702545	+	0.711640i	$0.252047\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		−	4.48528i		−	0.155406i
$834$		0			0
$835$	20.6985	+	6.89949i	0.716301	+	0.238767i
$836$		0			0
$837$		−	0.928932i		−	0.0321086i
$838$		0			0
$839$	−22.6274			−0.781185			−0.390593	−	0.920564i	$-0.627730\pi$
$839$	−22.6274			−0.781185			−0.390593	+	0.920564i	$0.627730\pi$
$840$		0			0
$841$	4.97056			0.171399
$842$		0			0
$843$			10.2426i			0.352775i
$844$		0			0
$845$	−9.17157	+	27.5147i	−0.315512	+	0.946535i
$846$		0			0
$847$		−	14.2721i		−	0.490394i
$848$		0			0
$849$	−9.31371			−0.319646
$850$		0			0
$851$	−78.4264			−2.68842
$852$		0			0
$853$			39.9411i			1.36756i	0.729689	+	0.683779i	$0.239665\pi$
$853$			39.9411i			1.36756i	−0.729689	+	0.683779i	$0.760335\pi$
$854$		0			0
$855$	−2.00000	+	6.00000i	−0.0683986	+	0.205196i
$856$		0			0
$857$		0			0		1.00000			$0$
$857$		0			0		−1.00000			$\pi$
$858$		0			0
$859$	−20.7279			−0.707228			−0.353614	−	0.935392i	$-0.615047\pi$
$859$	−20.7279			−0.707228			−0.353614	+	0.935392i	$0.615047\pi$
$860$		0			0
$861$	16.2426			0.553548
$862$		0			0
$863$			24.7279i			0.841748i	0.907119	+	0.420874i	$0.138277\pi$
$863$			24.7279i			0.841748i	−0.907119	+	0.420874i	$0.861723\pi$
$864$		0			0
$865$	34.9706	+	11.6569i	1.18903	+	0.396345i
$866$		0			0
$867$		−	38.6274i		−	1.31186i
$868$		0			0
$869$	−14.8284			−0.503020
$870$		0			0
$871$	−2.27208			−0.0769864
$872$		0			0
$873$		−	32.9706i		−	1.11588i
$874$		0			0
$875$	10.0919	+	14.5772i	0.341168	+	0.492798i
$876$		0			0
$877$			47.1421i			1.59188i	0.605378	+	0.795938i	$0.293022\pi$
$877$			47.1421i			1.59188i	−0.605378	+	0.795938i	$0.706978\pi$
$878$		0			0
$879$	27.7279			0.935240
$880$		0			0
$881$	39.5980			1.33409			0.667045	−	0.745018i	$-0.267559\pi$
$881$	39.5980			1.33409			0.667045	+	0.745018i	$0.267559\pi$
$882$		0			0
$883$			2.48528i			0.0836364i	0.999125	+	0.0418182i	$0.0133150\pi$
$883$			2.48528i			0.0836364i	−0.999125	+	0.0418182i	$0.986685\pi$
$884$		0			0
$885$	−66.0624	−	22.0208i	−2.22066	−	0.740222i
$886$		0			0
$887$		−	2.78680i		−	0.0935715i	−0.998905	−	0.0467857i	$-0.985102\pi$
$887$		−	2.78680i		−	0.0935715i	0.998905	−	0.0467857i	$-0.0148978\pi$
$888$		0			0
$889$	−22.9706			−0.770408
$890$		0			0
$891$	13.4142			0.449393
$892$		0			0
$893$		0			0
$894$		0			0
$895$	0.242641	−	0.727922i	0.00811058	−	0.0243318i
$896$		0			0
$897$		−	3.82843i		−	0.127827i
$898$		0			0
$899$	13.0711			0.435945
$900$		0			0
$901$	−11.4853			−0.382630
$902$		0			0
$903$		−	39.2132i		−	1.30493i
$904$		0			0
$905$	−6.00000	+	18.0000i	−0.199447	+	0.598340i
$906$		0			0
$907$		−	20.6985i		−	0.687282i	−0.939101	−	0.343641i	$-0.888340\pi$
$907$		−	20.6985i		−	0.687282i	0.939101	−	0.343641i	$-0.111660\pi$
$908$		0			0
$909$	3.02944			0.100480
$910$		0			0
$911$	16.2843			0.539522			0.269761	−	0.962927i	$-0.413055\pi$
$911$	16.2843			0.539522			0.269761	+	0.962927i	$0.413055\pi$
$912$		0			0
$913$		−	3.51472i		−	0.116320i
$914$		0			0
$915$	29.4853	+	9.82843i	0.974753	+	0.324918i
$916$		0			0
$917$		−	26.9117i		−	0.888702i
$918$		0			0
$919$	36.2132			1.19456			0.597282	−	0.802032i	$-0.296247\pi$
$919$	36.2132			1.19456			0.597282	+	0.802032i	$0.296247\pi$
$920$		0			0
$921$	−15.3137			−0.504604
$922$		0			0
$923$			1.81623i			0.0597821i
$924$		0			0
$925$	−25.4558	+	33.9411i	−0.836983	+	1.11598i
$926$		0			0
$927$		−	12.0000i		−	0.394132i
$928$		0			0
$929$	17.8284			0.584932			0.292466	−	0.956276i	$-0.405524\pi$
$929$	17.8284			0.584932			0.292466	+	0.956276i	$0.405524\pi$
$930$		0			0
$931$	−4.48528			−0.146999
$932$		0			0
$933$		−	31.9706i		−	1.04667i
$934$		0			0
$935$	−3.00000	−	1.00000i	−0.0981105	−	0.0327035i
$936$		0			0
$937$		−	27.0000i		−	0.882052i	−0.897494	−	0.441026i	$-0.854615\pi$
$937$		−	27.0000i		−	0.882052i	0.897494	−	0.441026i	$-0.145385\pi$
$938$		0			0
$939$	62.6985			2.04609
$940$		0			0
$941$	24.8579			0.810343			0.405172	−	0.914241i	$-0.367212\pi$
$941$	24.8579			0.810343			0.405172	+	0.914241i	$0.367212\pi$
$942$		0			0
$943$		−	39.2132i		−	1.27696i
$944$		0			0
$945$	−0.464466	+	1.39340i	−0.0151091	+	0.0453272i
$946$		0			0
$947$		−	12.0000i		−	0.389948i	−0.980808	−	0.194974i	$-0.937538\pi$
$947$		−	12.0000i		−	0.389948i	0.980808	−	0.194974i	$-0.0624622\pi$
$948$		0			0
$949$	0.941125			0.0305502
$950$		0			0
$951$	−18.0711			−0.585995
$952$		0			0
$953$		−	6.72792i		−	0.217939i	−0.994045	−	0.108969i	$-0.965245\pi$
$953$		−	6.72792i		−	0.217939i	0.994045	−	0.108969i	$-0.0347550\pi$
$954$		0			0
$955$	−13.1213	+	39.3640i	−0.424596	+	1.27379i
$956$		0			0
$957$			19.8995i			0.643259i
$958$		0			0
$959$	20.6152			0.665700
$960$		0			0
$961$	−25.9706			−0.837760
$962$		0			0
$963$			16.2010i			0.522070i
$964$		0			0
$965$	24.7279	+	8.24264i	0.796020	+	0.265340i
$966$		0			0
$967$		−	19.1127i		−	0.614623i	−0.951609	−	0.307311i	$-0.900571\pi$
$967$		−	19.1127i		−	0.614623i	0.951609	−	0.307311i	$-0.0994293\pi$
$968$		0			0
$969$	2.41421			0.0775557
$970$		0			0
$971$	−30.3431			−0.973758			−0.486879	−	0.873469i	$-0.661865\pi$
$971$	−30.3431			−0.973758			−0.486879	+	0.873469i	$0.661865\pi$
$972$		0			0
$973$		−	19.0294i		−	0.610056i
$974$		0			0
$975$	−1.65685	−	1.24264i	−0.0530618	−	0.0397964i
$976$		0			0
$977$			8.24264i			0.263705i	0.991269	+	0.131853i	$0.0420926\pi$
$977$			8.24264i			0.263705i	−0.991269	+	0.131853i	$0.957907\pi$
$978$		0			0
$979$	−10.0000			−0.319601
$980$		0			0
$981$	45.1716			1.44222
$982$		0			0
$983$			0.544156i			0.0173559i	0.999962	+	0.00867794i	$0.00276231\pi$
$983$			0.544156i			0.0173559i	−0.999962	+	0.00867794i	$0.997238\pi$
$984$		0			0
$985$	−42.9411	−	14.3137i	−1.36822	−	0.456073i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−94.6690			−3.01030
$990$		0			0
$991$	−22.2426			−0.706561			−0.353280	−	0.935517i	$-0.614934\pi$
$991$	−22.2426			−0.706561			−0.353280	+	0.935517i	$0.614934\pi$
$992$		0			0
$993$		−	25.9706i		−	0.824151i
$994$		0			0
$995$	−0.535534	+	1.60660i	−0.0169776	+	0.0509327i
$996$		0			0
$997$		−	23.2721i		−	0.737034i	−0.929621	−	0.368517i	$-0.879866\pi$
$997$		−	23.2721i		−	0.737034i	0.929621	−	0.368517i	$-0.120134\pi$
$998$		0			0
$999$	−3.51472			−0.111201

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1520.2.d.e.609.1		4
4.3	odd	2		190.2.b.a.39.4	yes	4
5.2	odd	4		7600.2.a.v.1.1		2
5.3	odd	4		7600.2.a.bg.1.2		2
5.4	even	2	inner	1520.2.d.e.609.4		4
12.11	even	2		1710.2.d.c.1369.2		4
20.3	even	4		950.2.a.g.1.1		2
20.7	even	4		950.2.a.f.1.2		2
20.19	odd	2		190.2.b.a.39.1	✓	4
60.23	odd	4		8550.2.a.bn.1.2		2
60.47	odd	4		8550.2.a.cb.1.1		2
60.59	even	2		1710.2.d.c.1369.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
190.2.b.a.39.1	✓	4	20.19	odd	2
190.2.b.a.39.4	yes	4	4.3	odd	2
950.2.a.f.1.2		2	20.7	even	4
950.2.a.g.1.1		2	20.3	even	4
1520.2.d.e.609.1		4	1.1	even	1	trivial
1520.2.d.e.609.4		4	5.4	even	2	inner
1710.2.d.c.1369.2		4	12.11	even	2
1710.2.d.c.1369.4		4	60.59	even	2
7600.2.a.v.1.1		2	5.2	odd	4
7600.2.a.bg.1.2		2	5.3	odd	4
8550.2.a.bn.1.2		2	60.23	odd	4
8550.2.a.cb.1.1		2	60.47	odd	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:	\( N \)	\(=\)	\( 1520 = 2^{4} \cdot 5 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1520.d (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-2.41421i q^{3} +(-0.707107 + 2.12132i) q^{5} +1.58579i q^{7} -2.82843 q^{9} +O(q^{10})\)	\(q-2.41421i q^{3} +(-0.707107 + 2.12132i) q^{5} +1.58579i q^{7} -2.82843 q^{9} -1.41421 q^{11} +0.171573i q^{13} +(5.12132 + 1.70711i) q^{15} -1.00000i q^{17} -1.00000 q^{19} +3.82843 q^{21} -9.24264i q^{23} +(-4.00000 - 3.00000i) q^{25} -0.414214i q^{27} +5.82843 q^{29} +2.24264 q^{31} +3.41421i q^{33} +(-3.36396 - 1.12132i) q^{35} -8.48528i q^{37} +0.414214 q^{39} +4.24264 q^{41} -10.2426i q^{43} +(2.00000 - 6.00000i) q^{45} +4.48528 q^{49} -2.41421 q^{51} -11.4853i q^{53} +(1.00000 - 3.00000i) q^{55} +2.41421i q^{57} -12.8995 q^{59} +5.75736 q^{61} -4.48528i q^{63} +(-0.363961 - 0.121320i) q^{65} +13.2426i q^{67} -22.3137 q^{69} +10.5858 q^{71} -5.48528i q^{73} +(-7.24264 + 9.65685i) q^{75} -2.24264i q^{77} +10.4853 q^{79} -9.48528 q^{81} +2.48528i q^{83} +(2.12132 + 0.707107i) q^{85} -14.0711i q^{87} +7.07107 q^{89} -0.272078 q^{91} -5.41421i q^{93} +(0.707107 - 2.12132i) q^{95} +11.6569i q^{97} +4.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q+O(q^{10}) \) 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

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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