LMFDB - Embedded newform 920.2.e.c.369.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [920,2,Mod(369,920)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(920, 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("920.369");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 920 = 2^{3} \cdot 5 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	920.e (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:	$7.34623698596$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 2 x^{15} + 2 x^{14} + 6 x^{13} + 100 x^{12} - 196 x^{11} + 210 x^{10} + 702 x^{9} + 1572 x^{8} + \cdots + 1024 $ x^16 - 2x^15 + 2x^14 + 6x^13 + 100x^12 - 196x^11 + 210x^10 + 702x^9 + 1572x^8 - 1594x^7 + 2464x^6 + 9568x^5 + 15457x^4 + 4336x^3 + 128x^2 - 512*x + 1024
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			369.8
Root			$-0.485591 + 0.485591i$ of defining polynomial
Character	$\chi$	$=$	920.369
Dual form			920.2.e.c.369.9

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.296848i q^{3} +(-2.18271 + 0.485591i) q^{5} -3.46037i q^{7} +2.91188 q^{9} +O(q^{10})$	$q-0.296848i q^{3} +(-2.18271 + 0.485591i) q^{5} -3.46037i q^{7} +2.91188 q^{9} -3.11377 q^{11} +4.60394i q^{13} +(0.144147 + 0.647931i) q^{15} -5.49355i q^{17} -4.48919 q^{19} -1.02720 q^{21} +1.00000i q^{23} +(4.52840 - 2.11980i) q^{25} -1.75493i q^{27} -9.19670 q^{29} -5.89980 q^{31} +0.924314i q^{33} +(1.68033 + 7.55297i) q^{35} +6.95324i q^{37} +1.36667 q^{39} -9.03617 q^{41} -5.55051i q^{43} +(-6.35578 + 1.41398i) q^{45} -5.48499i q^{47} -4.97418 q^{49} -1.63075 q^{51} -2.74411i q^{53} +(6.79643 - 1.51202i) q^{55} +1.33261i q^{57} -9.33659 q^{59} -1.40206 q^{61} -10.0762i q^{63} +(-2.23563 - 10.0490i) q^{65} -3.49779i q^{67} +0.296848 q^{69} +4.28869 q^{71} +4.92048i q^{73} +(-0.629259 - 1.34425i) q^{75} +10.7748i q^{77} -2.12284 q^{79} +8.21470 q^{81} -16.0159i q^{83} +(2.66762 + 11.9908i) q^{85} +2.73002i q^{87} -11.9821 q^{89} +15.9314 q^{91} +1.75134i q^{93} +(9.79858 - 2.17991i) q^{95} +4.37324i q^{97} -9.06692 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 2 q^{5} - 22 q^{9}+O(q^{10}) $ 16 * q - 2 * q^5 - 22 * q^9	$ 16 q - 2 q^{5} - 22 q^{9} + 14 q^{11} + 6 q^{15} - 22 q^{19} + 12 q^{25} - 44 q^{29} + 18 q^{31} + 20 q^{35} + 14 q^{41} + 14 q^{45} - 78 q^{49} - 38 q^{51} + 30 q^{55} - 64 q^{59} + 34 q^{61} + 6 q^{65} + 6 q^{69} + 30 q^{71} + 56 q^{75} + 4 q^{79} + 48 q^{81} + 52 q^{85} - 92 q^{89} - 70 q^{91} + 38 q^{95} - 122 q^{99}+O(q^{100}) $ 16 * q - 2 * q^5 - 22 * q^9 + 14 * q^11 + 6 * q^15 - 22 * q^19 + 12 * q^25 - 44 * q^29 + 18 * q^31 + 20 * q^35 + 14 * q^41 + 14 * q^45 - 78 * q^49 - 38 * q^51 + 30 * q^55 - 64 * q^59 + 34 * q^61 + 6 * q^65 + 6 * q^69 + 30 * q^71 + 56 * q^75 + 4 * q^79 + 48 * q^81 + 52 * q^85 - 92 * q^89 - 70 * q^91 + 38 * q^95 - 122 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/920\mathbb{Z}\right)^\times$.

$n$	$231$	$281$	$461$	$737$
$\chi(n)$	$1$	$1$	$1$	$-1$

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
$3$		−	0.296848i		−	0.171385i	−0.996322	−	0.0856925i	$-0.972690\pi$
$3$		−	0.296848i		−	0.171385i	0.996322	−	0.0856925i	$-0.0273103\pi$
$4$		0			0
$5$	−2.18271	+	0.485591i	−0.976135	+	0.217163i
$5$	−2.18271	+	0.485591i	−0.976135	+	0.217163i
$6$		0			0
$7$		−	3.46037i		−	1.30790i	−0.756539	−	0.653949i	$-0.773111\pi$
$7$		−	3.46037i		−	1.30790i	0.756539	−	0.653949i	$-0.226889\pi$
$8$		0			0
$9$	2.91188			0.970627
$10$		0			0
$11$	−3.11377			−0.938836			−0.469418	−	0.882976i	$-0.655536\pi$
$11$	−3.11377			−0.938836			−0.469418	+	0.882976i	$0.655536\pi$
$12$		0			0
$13$			4.60394i			1.27690i	0.769662	+	0.638452i	$0.220425\pi$
$13$			4.60394i			1.27690i	−0.769662	+	0.638452i	$0.779575\pi$
$14$		0			0
$15$	0.144147	+	0.647931i	0.0372185	+	0.167295i
$16$		0			0
$17$		−	5.49355i		−	1.33238i	−0.745782	−	0.666190i	$-0.767924\pi$
$17$		−	5.49355i		−	1.33238i	0.745782	−	0.666190i	$-0.232076\pi$
$18$		0			0
$19$	−4.48919			−1.02989			−0.514946	−	0.857223i	$-0.672188\pi$
$19$	−4.48919			−1.02989			−0.514946	+	0.857223i	$0.672188\pi$
$20$		0			0
$21$	−1.02720			−0.224154
$22$		0			0
$23$			1.00000i			0.208514i
$23$			1.00000i			0.208514i
$24$		0			0
$25$	4.52840	−	2.11980i	0.905681	−	0.423961i
$26$		0			0
$27$		−	1.75493i		−	0.337736i
$28$		0			0
$29$	−9.19670			−1.70778			−0.853892	−	0.520450i	$-0.825764\pi$
$29$	−9.19670			−1.70778			−0.853892	+	0.520450i	$0.825764\pi$
$30$		0			0
$31$	−5.89980			−1.05964			−0.529818	−	0.848111i	$-0.677740\pi$
$31$	−5.89980			−1.05964			−0.529818	+	0.848111i	$0.677740\pi$
$32$		0			0
$33$			0.924314i			0.160902i
$34$		0			0
$35$	1.68033	+	7.55297i	0.284027	+	1.27669i
$36$		0			0
$37$			6.95324i			1.14311i	0.820565	+	0.571553i	$0.193659\pi$
$37$			6.95324i			1.14311i	−0.820565	+	0.571553i	$0.806341\pi$
$38$		0			0
$39$	1.36667			0.218842
$40$		0			0
$41$	−9.03617			−1.41121			−0.705607	−	0.708604i	$-0.749325\pi$
$41$	−9.03617			−1.41121			−0.705607	+	0.708604i	$0.749325\pi$
$42$		0			0
$43$		−	5.55051i		−	0.846444i	−0.906026	−	0.423222i	$-0.860899\pi$
$43$		−	5.55051i		−	0.846444i	0.906026	−	0.423222i	$-0.139101\pi$
$44$		0			0
$45$	−6.35578	+	1.41398i	−0.947464	+	0.210784i
$46$		0			0
$47$		−	5.48499i		−	0.800068i	−0.916500	−	0.400034i	$-0.868998\pi$
$47$		−	5.48499i		−	0.800068i	0.916500	−	0.400034i	$-0.131002\pi$
$48$		0			0
$49$	−4.97418			−0.710598
$50$		0			0
$51$	−1.63075			−0.228350
$52$		0			0
$53$		−	2.74411i		−	0.376932i	−0.982080	−	0.188466i	$-0.939648\pi$
$53$		−	2.74411i		−	0.376932i	0.982080	−	0.188466i	$-0.0603516\pi$
$54$		0			0
$55$	6.79643	−	1.51202i	0.916431	−	0.203880i
$56$		0			0
$57$			1.33261i			0.176508i
$58$		0			0
$59$	−9.33659			−1.21552			−0.607760	−	0.794120i	$-0.707932\pi$
$59$	−9.33659			−1.21552			−0.607760	+	0.794120i	$0.707932\pi$
$60$		0			0
$61$	−1.40206			−0.179515			−0.0897576	−	0.995964i	$-0.528609\pi$
$61$	−1.40206			−0.179515			−0.0897576	+	0.995964i	$0.528609\pi$
$62$		0			0
$63$		−	10.0762i		−	1.26948i
$64$		0			0
$65$	−2.23563	−	10.0490i	−0.277296	−	1.24643i
$66$		0			0
$67$		−	3.49779i		−	0.427323i	−0.976908	−	0.213662i	$-0.931461\pi$
$67$		−	3.49779i		−	0.427323i	0.976908	−	0.213662i	$-0.0685390\pi$
$68$		0			0
$69$	0.296848			0.0357362
$70$		0			0
$71$	4.28869			0.508974			0.254487	−	0.967076i	$-0.418093\pi$
$71$	4.28869			0.508974			0.254487	+	0.967076i	$0.418093\pi$
$72$		0			0
$73$			4.92048i			0.575899i	0.957646	+	0.287949i	$0.0929735\pi$
$73$			4.92048i			0.575899i	−0.957646	+	0.287949i	$0.907026\pi$
$74$		0			0
$75$	−0.629259	−	1.34425i	−0.0726605	−	0.155220i
$76$		0			0
$77$			10.7748i			1.22790i
$78$		0			0
$79$	−2.12284			−0.238838			−0.119419	−	0.992844i	$-0.538103\pi$
$79$	−2.12284			−0.238838			−0.119419	+	0.992844i	$0.538103\pi$
$80$		0			0
$81$	8.21470			0.912744
$82$		0			0
$83$		−	16.0159i		−	1.75797i	−0.476845	−	0.878987i	$-0.658220\pi$
$83$		−	16.0159i		−	1.75797i	0.476845	−	0.878987i	$-0.341780\pi$
$84$		0			0
$85$	2.66762	+	11.9908i	0.289344	+	1.30058i
$86$		0			0
$87$			2.73002i			0.292689i
$88$		0			0
$89$	−11.9821			−1.27010			−0.635048	−	0.772473i	$-0.719020\pi$
$89$	−11.9821			−1.27010			−0.635048	+	0.772473i	$0.719020\pi$
$90$		0			0
$91$	15.9314			1.67006
$92$		0			0
$93$			1.75134i			0.181606i
$94$		0			0
$95$	9.79858	−	2.17991i	1.00531	−	0.223654i
$96$		0			0
$97$			4.37324i			0.444035i	0.975043	+	0.222018i	$0.0712643\pi$
$97$			4.37324i			0.444035i	−0.975043	+	0.222018i	$0.928736\pi$
$98$		0			0
$99$	−9.06692			−0.911259
$100$		0			0
$101$	10.4175			1.03658			0.518288	−	0.855206i	$-0.326570\pi$
$101$	10.4175			1.03658			0.518288	+	0.855206i	$0.326570\pi$
$102$		0			0
$103$			7.21376i			0.710793i	0.934716	+	0.355396i	$0.115654\pi$
$103$			7.21376i			0.710793i	−0.934716	+	0.355396i	$0.884346\pi$
$104$		0			0
$105$	2.24208	−	0.498801i	0.218805	−	0.0486780i
$106$		0			0
$107$		−	7.07230i		−	0.683705i	−0.939754	−	0.341853i	$-0.888946\pi$
$107$		−	7.07230i		−	0.683705i	0.939754	−	0.341853i	$-0.111054\pi$
$108$		0			0
$109$	9.92058			0.950219			0.475110	−	0.879927i	$-0.342408\pi$
$109$	9.92058			0.950219			0.475110	+	0.879927i	$0.342408\pi$
$110$		0			0
$111$	2.06405			0.195911
$112$		0			0
$113$			12.9797i			1.22103i	0.792007	+	0.610513i	$0.209036\pi$
$113$			12.9797i			1.22103i	−0.792007	+	0.610513i	$0.790964\pi$
$114$		0			0
$115$	−0.485591	−	2.18271i	−0.0452816	−	0.203538i
$116$		0			0
$117$			13.4061i			1.23940i
$118$		0			0
$119$	−19.0097			−1.74262
$120$		0			0
$121$	−1.30446			−0.118587
$122$		0			0
$123$			2.68237i			0.241861i
$124$		0			0
$125$	−8.85481	+	6.82586i	−0.791998	+	0.610523i
$126$		0			0
$127$			14.9664i			1.32805i	0.747709	+	0.664027i	$0.231154\pi$
$127$			14.9664i			1.32805i	−0.747709	+	0.664027i	$0.768846\pi$
$128$		0			0
$129$	−1.64765			−0.145068
$130$		0			0
$131$	18.5945			1.62461			0.812305	−	0.583233i	$-0.198212\pi$
$131$	18.5945			1.62461			0.812305	+	0.583233i	$0.198212\pi$
$132$		0			0
$133$			15.5343i			1.34699i
$134$		0			0
$135$	0.852177	+	3.83049i	0.0733437	+	0.329676i
$136$		0			0
$137$		−	19.9298i		−	1.70272i	−0.524583	−	0.851359i	$-0.675779\pi$
$137$		−	19.9298i		−	1.70272i	0.524583	−	0.851359i	$-0.324221\pi$
$138$		0			0
$139$	0.346731			0.0294094			0.0147047	−	0.999892i	$-0.495319\pi$
$139$	0.346731			0.0294094			0.0147047	+	0.999892i	$0.495319\pi$
$140$		0			0
$141$	−1.62821			−0.137120
$142$		0			0
$143$		−	14.3356i		−	1.19880i
$144$		0			0
$145$	20.0737	−	4.46584i	1.66703	−	0.370867i
$146$		0			0
$147$			1.47657i			0.121786i
$148$		0			0
$149$	1.87953			0.153977			0.0769884	−	0.997032i	$-0.475470\pi$
$149$	1.87953			0.153977			0.0769884	+	0.997032i	$0.475470\pi$
$150$		0			0
$151$	9.84743			0.801373			0.400686	−	0.916215i	$-0.368772\pi$
$151$	9.84743			0.801373			0.400686	+	0.916215i	$0.368772\pi$
$152$		0			0
$153$		−	15.9966i		−	1.29324i
$154$		0			0
$155$	12.8775	−	2.86489i	1.03435	−	0.230114i
$156$		0			0
$157$		−	16.5630i		−	1.32187i	−0.750442	−	0.660936i	$-0.770159\pi$
$157$		−	16.5630i		−	1.32187i	0.750442	−	0.660936i	$-0.229841\pi$
$158$		0			0
$159$	−0.814582			−0.0646006
$160$		0			0
$161$	3.46037			0.272716
$162$		0			0
$163$		−	9.48888i		−	0.743226i	−0.928388	−	0.371613i	$-0.878805\pi$
$163$		−	9.48888i		−	0.743226i	0.928388	−	0.371613i	$-0.121195\pi$
$164$		0			0
$165$	−0.448839	−	2.01750i	−0.0349420	−	0.157062i
$166$		0			0
$167$			9.28434i			0.718444i	0.933252	+	0.359222i	$0.116958\pi$
$167$			9.28434i			0.718444i	−0.933252	+	0.359222i	$0.883042\pi$
$168$		0			0
$169$	−8.19629			−0.630484
$170$		0			0
$171$	−13.0720			−0.999640
$172$		0			0
$173$			9.34023i			0.710124i	0.934843	+	0.355062i	$0.115540\pi$
$173$			9.34023i			0.710124i	−0.934843	+	0.355062i	$0.884460\pi$
$174$		0			0
$175$	−7.33531	−	15.6700i	−0.554498	−	1.18454i
$176$		0			0
$177$			2.77154i			0.208322i
$178$		0			0
$179$	0.255383			0.0190882			0.00954411	−	0.999954i	$-0.496962\pi$
$179$	0.255383			0.0190882			0.00954411	+	0.999954i	$0.496962\pi$
$180$		0			0
$181$	−4.02663			−0.299297			−0.149649	−	0.988739i	$-0.547814\pi$
$181$	−4.02663			−0.299297			−0.149649	+	0.988739i	$0.547814\pi$
$182$		0			0
$183$			0.416198i			0.0307662i
$184$		0			0
$185$	−3.37643	−	15.1769i	−0.248240	−	1.11583i
$186$		0			0
$187$			17.1056i			1.25089i
$188$		0			0
$189$	−6.07270			−0.441724
$190$		0			0
$191$	−24.7376			−1.78995			−0.894974	−	0.446118i	$-0.852806\pi$
$191$	−24.7376			−1.78995			−0.894974	+	0.446118i	$0.852806\pi$
$192$		0			0
$193$		−	7.34062i		−	0.528390i	−0.964469	−	0.264195i	$-0.914894\pi$
$193$		−	7.34062i		−	0.528390i	0.964469	−	0.264195i	$-0.0851062\pi$
$194$		0			0
$195$	−2.98304	+	0.663642i	−0.213620	+	0.0475244i
$196$		0			0
$197$		−	16.2033i		−	1.15444i	−0.816589	−	0.577220i	$-0.804138\pi$
$197$		−	16.2033i		−	1.15444i	0.816589	−	0.577220i	$-0.195862\pi$
$198$		0			0
$199$	13.8498			0.981786			0.490893	−	0.871220i	$-0.336671\pi$
$199$	13.8498			0.981786			0.490893	+	0.871220i	$0.336671\pi$
$200$		0			0
$201$	−1.03831			−0.0732368
$202$		0			0
$203$			31.8240i			2.23361i
$204$		0			0
$205$	19.7233	−	4.38789i	1.37754	−	0.306463i
$206$		0			0
$207$			2.91188i			0.202390i
$208$		0			0
$209$	13.9783			0.966899
$210$		0			0
$211$	19.8811			1.36867			0.684335	−	0.729168i	$-0.260093\pi$
$211$	19.8811			1.36867			0.684335	+	0.729168i	$0.260093\pi$
$212$		0			0
$213$		−	1.27309i		−	0.0872305i
$214$		0			0
$215$	2.69528	+	12.1151i	0.183816	+	0.826244i
$216$		0			0
$217$			20.4155i			1.38590i
$218$		0			0
$219$	1.46063			0.0987004
$220$		0			0
$221$	25.2920			1.70132
$222$		0			0
$223$		−	14.3441i		−	0.960550i	−0.877118	−	0.480275i	$-0.840537\pi$
$223$		−	14.3441i		−	0.960550i	0.877118	−	0.480275i	$-0.159463\pi$
$224$		0			0
$225$	13.1862	−	6.17262i	0.879078	−	0.411508i
$226$		0			0
$227$			0.219192i			0.0145483i	0.999974	+	0.00727413i	$0.00231545\pi$
$227$			0.219192i			0.0145483i	−0.999974	+	0.00727413i	$0.997685\pi$
$228$		0			0
$229$	−0.919300			−0.0607491			−0.0303745	−	0.999539i	$-0.509670\pi$
$229$	−0.919300			−0.0607491			−0.0303745	+	0.999539i	$0.509670\pi$
$230$		0			0
$231$	3.19847			0.210444
$232$		0			0
$233$			22.5267i			1.47578i	0.674923	+	0.737888i	$0.264177\pi$
$233$			22.5267i			1.47578i	−0.674923	+	0.737888i	$0.735823\pi$
$234$		0			0
$235$	2.66346	+	11.9721i	0.173745	+	0.780975i
$236$		0			0
$237$			0.630160i			0.0409333i
$238$		0			0
$239$	11.4614			0.741376			0.370688	−	0.928758i	$-0.379122\pi$
$239$	11.4614			0.741376			0.370688	+	0.928758i	$0.379122\pi$
$240$		0			0
$241$	15.3035			0.985784			0.492892	−	0.870090i	$-0.335940\pi$
$241$	15.3035			0.985784			0.492892	+	0.870090i	$0.335940\pi$
$242$		0			0
$243$		−	7.70330i		−	0.494167i
$244$		0			0
$245$	10.8572	−	2.41542i	0.693640	−	0.154315i
$246$		0			0
$247$		−	20.6680i		−	1.31507i
$248$		0			0
$249$	−4.75428			−0.301290
$250$		0			0
$251$	21.1450			1.33466			0.667329	−	0.744763i	$-0.267437\pi$
$251$	21.1450			1.33466			0.667329	+	0.744763i	$0.267437\pi$
$252$		0			0
$253$		−	3.11377i		−	0.195761i
$254$		0			0
$255$	3.55944	−	0.791876i	0.222901	−	0.0495892i
$256$		0			0
$257$		−	2.81665i		−	0.175698i	−0.996134	−	0.0878489i	$-0.972001\pi$
$257$		−	2.81665i		−	0.175698i	0.996134	−	0.0878489i	$-0.0279993\pi$
$258$		0			0
$259$	24.0608			1.49507
$260$		0			0
$261$	−26.7797			−1.65762
$262$		0			0
$263$			16.8627i			1.03980i	0.854228	+	0.519899i	$0.174030\pi$
$263$			16.8627i			1.03980i	−0.854228	+	0.519899i	$0.825970\pi$
$264$		0			0
$265$	1.33252	+	5.98958i	0.0818558	+	0.367937i
$266$		0			0
$267$			3.55684i			0.217675i
$268$		0			0
$269$	−27.8171			−1.69604			−0.848019	−	0.529966i	$-0.822205\pi$
$269$	−27.8171			−1.69604			−0.848019	+	0.529966i	$0.822205\pi$
$270$		0			0
$271$	−4.73892			−0.287869			−0.143934	−	0.989587i	$-0.545975\pi$
$271$	−4.73892			−0.287869			−0.143934	+	0.989587i	$0.545975\pi$
$272$		0			0
$273$		−	4.72919i		−	0.286223i
$274$		0			0
$275$	−14.1004	+	6.60057i	−0.850285	+	0.398030i
$276$		0			0
$277$			0.983771i			0.0591091i	0.999563	+	0.0295545i	$0.00940888\pi$
$277$			0.983771i			0.0591091i	−0.999563	+	0.0295545i	$0.990591\pi$
$278$		0			0
$279$	−17.1795			−1.02851
$280$		0			0
$281$	24.1462			1.44044			0.720219	−	0.693746i	$-0.244041\pi$
$281$	24.1462			1.44044			0.720219	+	0.693746i	$0.244041\pi$
$282$		0			0
$283$		−	22.6523i		−	1.34654i	−0.739397	−	0.673269i	$-0.764890\pi$
$283$		−	22.6523i		−	1.34654i	0.739397	−	0.673269i	$-0.235110\pi$
$284$		0			0
$285$	−0.647101	−	2.90868i	−0.0383310	−	0.172296i
$286$		0			0
$287$			31.2685i			1.84572i
$288$		0			0
$289$	−13.1791			−0.775238
$290$		0			0
$291$	1.29819			0.0761010
$292$		0			0
$293$			12.9668i			0.757529i	0.925493	+	0.378764i	$0.123651\pi$
$293$			12.9668i			0.757529i	−0.925493	+	0.378764i	$0.876349\pi$
$294$		0			0
$295$	20.3790	−	4.53377i	1.18651	−	0.263966i
$296$		0			0
$297$			5.46443i			0.317079i
$298$		0			0
$299$	−4.60394			−0.266253
$300$		0			0
$301$	−19.2068			−1.10706
$302$		0			0
$303$		−	3.09240i		−	0.177654i
$304$		0			0
$305$	3.06028	−	0.680827i	0.175231	−	0.0389841i
$306$		0			0
$307$		−	27.4380i		−	1.56597i	−0.622041	−	0.782984i	$-0.713696\pi$
$307$		−	27.4380i		−	1.56597i	0.622041	−	0.782984i	$-0.286304\pi$
$308$		0			0
$309$	2.14139			0.121819
$310$		0			0
$311$	−30.8655			−1.75022			−0.875112	−	0.483920i	$-0.839212\pi$
$311$	−30.8655			−1.75022			−0.875112	+	0.483920i	$0.839212\pi$
$312$		0			0
$313$		−	3.64274i		−	0.205900i	−0.994687	−	0.102950i	$-0.967172\pi$
$313$		−	3.64274i		−	0.205900i	0.994687	−	0.102950i	$-0.0328281\pi$
$314$		0			0
$315$	4.89291	+	21.9934i	0.275684	+	1.23919i
$316$		0			0
$317$		−	24.0344i		−	1.34990i	−0.737862	−	0.674952i	$-0.764164\pi$
$317$		−	24.0344i		−	1.34990i	0.737862	−	0.674952i	$-0.235836\pi$
$318$		0			0
$319$	28.6364			1.60333
$320$		0			0
$321$	−2.09940			−0.117177
$322$		0			0
$323$			24.6616i			1.37221i
$324$		0			0
$325$	9.75946	+	20.8485i	0.541357	+	1.15647i
$326$		0			0
$327$		−	2.94490i		−	0.162853i
$328$		0			0
$329$	−18.9801			−1.04641
$330$		0			0
$331$	−21.7999			−1.19823			−0.599115	−	0.800663i	$-0.704481\pi$
$331$	−21.7999			−1.19823			−0.599115	+	0.800663i	$0.704481\pi$
$332$		0			0
$333$			20.2470i			1.10953i
$334$		0			0
$335$	1.69850	+	7.63465i	0.0927988	+	0.417125i
$336$		0			0
$337$		−	2.82526i		−	0.153902i	−0.997035	−	0.0769510i	$-0.975482\pi$
$337$		−	2.82526i		−	0.153902i	0.997035	−	0.0769510i	$-0.0245185\pi$
$338$		0			0
$339$	3.85298			0.209265
$340$		0			0
$341$	18.3706			0.994824
$342$		0			0
$343$		−	7.01008i		−	0.378509i
$344$		0			0
$345$	−0.647931	+	0.144147i	−0.0348834	+	0.00776059i
$346$		0			0
$347$			12.5727i			0.674939i	0.941337	+	0.337469i	$0.109571\pi$
$347$			12.5727i			0.674939i	−0.941337	+	0.337469i	$0.890429\pi$
$348$		0			0
$349$	−7.15382			−0.382935			−0.191468	−	0.981499i	$-0.561325\pi$
$349$	−7.15382			−0.382935			−0.191468	+	0.981499i	$0.561325\pi$
$350$		0			0
$351$	8.07959			0.431256
$352$		0			0
$353$		−	20.7013i		−	1.10182i	−0.834565	−	0.550909i	$-0.814281\pi$
$353$		−	20.7013i		−	1.10182i	0.834565	−	0.550909i	$-0.185719\pi$
$354$		0			0
$355$	−9.36094	+	2.08255i	−0.496827	+	0.110530i
$356$		0			0
$357$			5.64299i			0.298659i
$358$		0			0
$359$	−28.6077			−1.50985			−0.754927	−	0.655808i	$-0.772328\pi$
$359$	−28.6077			−1.50985			−0.754927	+	0.655808i	$0.772328\pi$
$360$		0			0
$361$	1.15284			0.0606756
$362$		0			0
$363$			0.387226i			0.0203241i
$364$		0			0
$365$	−2.38934	−	10.7400i	−0.125064	−	0.562155i
$366$		0			0
$367$			20.4541i			1.06769i	0.845581	+	0.533847i	$0.179254\pi$
$367$			20.4541i			1.06769i	−0.845581	+	0.533847i	$0.820746\pi$
$368$		0			0
$369$	−26.3123			−1.36976
$370$		0			0
$371$	−9.49564			−0.492989
$372$		0			0
$373$		−	3.75830i		−	0.194598i	−0.995255	−	0.0972988i	$-0.968980\pi$
$373$		−	3.75830i		−	0.194598i	0.995255	−	0.0972988i	$-0.0310203\pi$
$374$		0			0
$375$	2.02624	+	2.62853i	0.104635	+	0.135737i
$376$		0			0
$377$		−	42.3411i		−	2.18068i
$378$		0			0
$379$	−28.9884			−1.48903			−0.744516	−	0.667605i	$-0.767320\pi$
$379$	−28.9884			−1.48903			−0.744516	+	0.667605i	$0.767320\pi$
$380$		0			0
$381$	4.44274			0.227608
$382$		0			0
$383$		−	29.8354i		−	1.52452i	−0.647273	−	0.762258i	$-0.724091\pi$
$383$		−	29.8354i		−	1.52452i	0.647273	−	0.762258i	$-0.275909\pi$
$384$		0			0
$385$	−5.23214	−	23.5182i	−0.266655	−	1.19860i
$386$		0			0
$387$		−	16.1624i		−	0.821582i
$388$		0			0
$389$	4.02138			0.203892			0.101946	−	0.994790i	$-0.467493\pi$
$389$	4.02138			0.203892			0.101946	+	0.994790i	$0.467493\pi$
$390$		0			0
$391$	5.49355			0.277821
$392$		0			0
$393$		−	5.51974i		−	0.278434i
$394$		0			0
$395$	4.63354	−	1.03083i	0.233139	−	0.0518668i
$396$		0			0
$397$		−	5.86060i		−	0.294135i	−0.989126	−	0.147068i	$-0.953017\pi$
$397$		−	5.86060i		−	0.294135i	0.989126	−	0.147068i	$-0.0469835\pi$
$398$		0			0
$399$	4.61131			0.230854
$400$		0			0
$401$	22.3492			1.11607			0.558034	−	0.829818i	$-0.311556\pi$
$401$	22.3492			1.11607			0.558034	+	0.829818i	$0.311556\pi$
$402$		0			0
$403$		−	27.1624i		−	1.35305i
$404$		0			0
$405$	−17.9303	+	3.98898i	−0.890962	+	0.198214i
$406$		0			0
$407$		−	21.6508i		−	1.07319i
$408$		0			0
$409$	36.4740			1.80352			0.901762	−	0.432234i	$-0.142275\pi$
$409$	36.4740			1.80352			0.901762	+	0.432234i	$0.142275\pi$
$410$		0			0
$411$	−5.91611			−0.291820
$412$		0			0
$413$			32.3081i			1.58978i
$414$		0			0
$415$	7.77718	+	34.9580i	0.381767	+	1.71602i
$416$		0			0
$417$		−	0.102926i		−	0.00504032i
$418$		0			0
$419$	−29.0555			−1.41945			−0.709727	−	0.704477i	$-0.751182\pi$
$419$	−29.0555			−1.41945			−0.709727	+	0.704477i	$0.751182\pi$
$420$		0			0
$421$	28.3055			1.37952			0.689762	−	0.724036i	$-0.257715\pi$
$421$	28.3055			1.37952			0.689762	+	0.724036i	$0.257715\pi$
$422$		0			0
$423$		−	15.9716i		−	0.776568i
$424$		0			0
$425$	−11.6452	−	24.8770i	−0.564877	−	1.20671i
$426$		0			0
$427$			4.85165i			0.234788i
$428$		0			0
$429$	−4.25549			−0.205457
$430$		0			0
$431$	−26.4640			−1.27472			−0.637362	−	0.770564i	$-0.719974\pi$
$431$	−26.4640			−1.27472			−0.637362	+	0.770564i	$0.719974\pi$
$432$		0			0
$433$		−	7.95631i		−	0.382356i	−0.981555	−	0.191178i	$-0.938769\pi$
$433$		−	7.95631i		−	0.382356i	0.981555	−	0.191178i	$-0.0612307\pi$
$434$		0			0
$435$	−1.32567	−	5.95882i	−0.0635611	−	0.285704i
$436$		0			0
$437$		−	4.48919i		−	0.214747i
$438$		0			0
$439$	−20.0799			−0.958361			−0.479181	−	0.877716i	$-0.659066\pi$
$439$	−20.0799			−0.958361			−0.479181	+	0.877716i	$0.659066\pi$
$440$		0			0
$441$	−14.4842			−0.689725
$442$		0			0
$443$			12.8355i			0.609834i	0.952379	+	0.304917i	$0.0986287\pi$
$443$			12.8355i			0.609834i	−0.952379	+	0.304917i	$0.901371\pi$
$444$		0			0
$445$	26.1533	−	5.81838i	1.23979	−	0.275818i
$446$		0			0
$447$		−	0.557933i		−	0.0263893i
$448$		0			0
$449$	−4.18739			−0.197615			−0.0988075	−	0.995107i	$-0.531503\pi$
$449$	−4.18739			−0.197615			−0.0988075	+	0.995107i	$0.531503\pi$
$450$		0			0
$451$	28.1365			1.32490
$452$		0			0
$453$		−	2.92319i		−	0.137343i
$454$		0			0
$455$	−34.7735	+	7.73613i	−1.63021	+	0.362675i
$456$		0			0
$457$		−	42.1962i		−	1.97386i	−0.161158	−	0.986929i	$-0.551523\pi$
$457$		−	42.1962i		−	1.97386i	0.161158	−	0.986929i	$-0.448477\pi$
$458$		0			0
$459$	−9.64078			−0.449993
$460$		0			0
$461$	−10.4930			−0.488709			−0.244355	−	0.969686i	$-0.578576\pi$
$461$	−10.4930			−0.488709			−0.244355	+	0.969686i	$0.578576\pi$
$462$		0			0
$463$			39.5643i			1.83871i	0.393432	+	0.919354i	$0.371288\pi$
$463$			39.5643i			1.83871i	−0.393432	+	0.919354i	$0.628712\pi$
$464$		0			0
$465$	−0.850436	−	3.82266i	−0.0394380	−	0.177272i
$466$		0			0
$467$			27.2819i			1.26246i	0.775598	+	0.631228i	$0.217449\pi$
$467$			27.2819i			1.26246i	−0.775598	+	0.631228i	$0.782551\pi$
$468$		0			0
$469$	−12.1037			−0.558895
$470$		0			0
$471$	−4.91669			−0.226549
$472$		0			0
$473$			17.2830i			0.794672i
$474$		0			0
$475$	−20.3289	+	9.51621i	−0.932752	+	0.436634i
$476$		0			0
$477$		−	7.99052i		−	0.365861i
$478$		0			0
$479$	−2.88476			−0.131808			−0.0659041	−	0.997826i	$-0.520993\pi$
$479$	−2.88476			−0.131808			−0.0659041	+	0.997826i	$0.520993\pi$
$480$		0			0
$481$	−32.0123			−1.45964
$482$		0			0
$483$		−	1.02720i		−	0.0467394i
$484$		0			0
$485$	−2.12361	−	9.54550i	−0.0964280	−	0.433439i
$486$		0			0
$487$			1.55229i			0.0703409i	0.999381	+	0.0351704i	$0.0111974\pi$
$487$			1.55229i			0.0703409i	−0.999381	+	0.0351704i	$0.988803\pi$
$488$		0			0
$489$	−2.81675			−0.127378
$490$		0			0
$491$	0.415377			0.0187457			0.00937286	−	0.999956i	$-0.497016\pi$
$491$	0.415377			0.0187457			0.00937286	+	0.999956i	$0.497016\pi$
$492$		0			0
$493$			50.5225i			2.27542i
$494$		0			0
$495$	19.7904	−	4.40281i	0.889513	−	0.197892i
$496$		0			0
$497$		−	14.8405i		−	0.665686i
$498$		0			0
$499$	−13.8674			−0.620789			−0.310394	−	0.950608i	$-0.600461\pi$
$499$	−13.8674			−0.620789			−0.310394	+	0.950608i	$0.600461\pi$
$500$		0			0
$501$	2.75603			0.123130
$502$		0			0
$503$		−	34.5409i		−	1.54010i	−0.637982	−	0.770051i	$-0.720231\pi$
$503$		−	34.5409i		−	1.54010i	0.637982	−	0.770051i	$-0.279769\pi$
$504$		0			0
$505$	−22.7382	+	5.05863i	−1.01184	+	0.225106i
$506$		0			0
$507$			2.43305i			0.108056i
$508$		0			0
$509$	−15.5389			−0.688748			−0.344374	−	0.938833i	$-0.611909\pi$
$509$	−15.5389			−0.688748			−0.344374	+	0.938833i	$0.611909\pi$
$510$		0			0
$511$	17.0267			0.753217
$512$		0			0
$513$			7.87821i			0.347831i
$514$		0			0
$515$	−3.50294	−	15.7455i	−0.154358	−	0.693830i
$516$		0			0
$517$			17.0790i			0.751133i
$518$		0			0
$519$	2.77262			0.121705
$520$		0			0
$521$	−5.67959			−0.248827			−0.124414	−	0.992230i	$-0.539705\pi$
$521$	−5.67959			−0.248827			−0.124414	+	0.992230i	$0.539705\pi$
$522$		0			0
$523$			18.4095i			0.804992i	0.915422	+	0.402496i	$0.131857\pi$
$523$			18.4095i			0.804992i	−0.915422	+	0.402496i	$0.868143\pi$
$524$		0			0
$525$	−4.65159	+	2.17747i	−0.203012	+	0.0950326i
$526$		0			0
$527$			32.4108i			1.41184i
$528$		0			0
$529$	−1.00000			−0.0434783
$530$		0			0
$531$	−27.1871			−1.17982
$532$		0			0
$533$		−	41.6020i		−	1.80198i
$534$		0			0
$535$	3.43425	+	15.4367i	0.148475	+	0.667389i
$536$		0			0
$537$		−	0.0758098i		−	0.00327143i
$538$		0			0
$539$	15.4884			0.667134
$540$		0			0
$541$	−17.5169			−0.753110			−0.376555	−	0.926394i	$-0.622891\pi$
$541$	−17.5169			−0.753110			−0.376555	+	0.926394i	$0.622891\pi$
$542$		0			0
$543$			1.19530i			0.0512951i
$544$		0			0
$545$	−21.6537	+	4.81734i	−0.927542	+	0.206352i
$546$		0			0
$547$			38.1355i			1.63056i	0.579070	+	0.815278i	$0.303416\pi$
$547$			38.1355i			1.63056i	−0.579070	+	0.815278i	$0.696584\pi$
$548$		0			0
$549$	−4.08263			−0.174242
$550$		0			0
$551$	41.2857			1.75883
$552$		0			0
$553$			7.34582i			0.312376i
$554$		0			0
$555$	−4.50522	+	1.00229i	−0.191236	+	0.0425447i
$556$		0			0
$557$			9.09458i			0.385350i	0.981263	+	0.192675i	$0.0617162\pi$
$557$			9.09458i			0.385350i	−0.981263	+	0.192675i	$0.938284\pi$
$558$		0			0
$559$	25.5542			1.08083
$560$		0			0
$561$	5.07776			0.214383
$562$		0			0
$563$		−	37.7282i		−	1.59005i	−0.606574	−	0.795027i	$-0.707457\pi$
$563$		−	37.7282i		−	1.59005i	0.606574	−	0.795027i	$-0.292543\pi$
$564$		0			0
$565$	−6.30281	−	28.3308i	−0.265161	−	1.19189i
$566$		0			0
$567$		−	28.4259i		−	1.19378i
$568$		0			0
$569$	34.0456			1.42727			0.713633	−	0.700520i	$-0.247048\pi$
$569$	34.0456			1.42727			0.713633	+	0.700520i	$0.247048\pi$
$570$		0			0
$571$	−11.4449			−0.478953			−0.239477	−	0.970902i	$-0.576976\pi$
$571$	−11.4449			−0.478953			−0.239477	+	0.970902i	$0.576976\pi$
$572$		0			0
$573$			7.34329i			0.306770i
$574$		0			0
$575$	2.11980	+	4.52840i	0.0884020	+	0.188847i
$576$		0			0
$577$			40.8683i			1.70137i	0.525675	+	0.850686i	$0.323813\pi$
$577$			40.8683i			1.70137i	−0.525675	+	0.850686i	$0.676187\pi$
$578$		0			0
$579$	−2.17905			−0.0905581
$580$		0			0
$581$	−55.4210			−2.29925
$582$		0			0
$583$			8.54452i			0.353878i
$584$		0			0
$585$	−6.50990	−	29.2616i	−0.269151	−	1.20982i
$586$		0			0
$587$			28.3630i			1.17067i	0.810793	+	0.585334i	$0.199036\pi$
$587$			28.3630i			1.17067i	−0.810793	+	0.585334i	$0.800964\pi$
$588$		0			0
$589$	26.4853			1.09131
$590$		0			0
$591$	−4.80992			−0.197854
$592$		0			0
$593$		−	25.2847i		−	1.03832i	−0.854678	−	0.519159i	$-0.826245\pi$
$593$		−	25.2847i		−	1.03832i	0.854678	−	0.519159i	$-0.173755\pi$
$594$		0			0
$595$	41.4926	−	9.23095i	1.70103	−	0.378432i
$596$		0			0
$597$		−	4.11128i		−	0.168263i
$598$		0			0
$599$	25.9826			1.06162			0.530811	−	0.847490i	$-0.321888\pi$
$599$	25.9826			1.06162			0.530811	+	0.847490i	$0.321888\pi$
$600$		0			0
$601$	−10.8318			−0.441840			−0.220920	−	0.975292i	$-0.570906\pi$
$601$	−10.8318			−0.441840			−0.220920	+	0.975292i	$0.570906\pi$
$602$		0			0
$603$		−	10.1852i		−	0.414772i
$604$		0			0
$605$	2.84726	−	0.633435i	0.115757	−	0.0257528i
$606$		0			0
$607$		−	14.5105i		−	0.588962i	−0.955657	−	0.294481i	$-0.904853\pi$
$607$		−	14.5105i		−	0.588962i	0.955657	−	0.294481i	$-0.0951468\pi$
$608$		0			0
$609$	9.44688			0.382807
$610$		0			0
$611$	25.2526			1.02161
$612$		0			0
$613$			3.87278i			0.156420i	0.996937	+	0.0782101i	$0.0249205\pi$
$613$			3.87278i			0.156420i	−0.996937	+	0.0782101i	$0.975080\pi$
$614$		0			0
$615$	−1.30253	−	5.85481i	−0.0525232	−	0.236089i
$616$		0			0
$617$			26.3999i			1.06282i	0.847114	+	0.531410i	$0.178338\pi$
$617$			26.3999i			1.06282i	−0.847114	+	0.531410i	$0.821662\pi$
$618$		0			0
$619$	−46.0115			−1.84936			−0.924679	−	0.380748i	$-0.875666\pi$
$619$	−46.0115			−1.84936			−0.924679	+	0.380748i	$0.875666\pi$
$620$		0			0
$621$	1.75493			0.0704228
$622$		0			0
$623$			41.4624i			1.66116i
$624$		0			0
$625$	16.0129	−	19.1987i	0.640514	−	0.767946i
$626$		0			0
$627$		−	4.14942i		−	0.165712i
$628$		0			0
$629$	38.1979			1.52305
$630$		0			0
$631$	1.15900			0.0461389			0.0230695	−	0.999734i	$-0.492656\pi$
$631$	1.15900			0.0461389			0.0230695	+	0.999734i	$0.492656\pi$
$632$		0			0
$633$		−	5.90165i		−	0.234569i
$634$		0			0
$635$	−7.26755	−	32.6672i	−0.288404	−	1.29636i
$636$		0			0
$637$		−	22.9009i		−	0.907365i
$638$		0			0
$639$	12.4882			0.494024
$640$		0			0
$641$	−5.15529			−0.203622			−0.101811	−	0.994804i	$-0.532464\pi$
$641$	−5.15529			−0.203622			−0.101811	+	0.994804i	$0.532464\pi$
$642$		0			0
$643$		−	8.72811i		−	0.344203i	−0.985079	−	0.172102i	$-0.944944\pi$
$643$		−	8.72811i		−	0.344203i	0.985079	−	0.172102i	$-0.0550557\pi$
$644$		0			0
$645$	3.59634	−	0.800086i	0.141606	−	0.0315034i
$646$		0			0
$647$			10.4205i			0.409672i	0.978796	+	0.204836i	$0.0656662\pi$
$647$			10.4205i			0.409672i	−0.978796	+	0.204836i	$0.934334\pi$
$648$		0			0
$649$	29.0720			1.14117
$650$		0			0
$651$	6.06030			0.237522
$652$		0			0
$653$		−	36.2315i		−	1.41785i	−0.705285	−	0.708924i	$-0.749181\pi$
$653$		−	36.2315i		−	1.41785i	0.705285	−	0.708924i	$-0.250819\pi$
$654$		0			0
$655$	−40.5863	+	9.02933i	−1.58584	+	0.352805i
$656$		0			0
$657$			14.3279i			0.558983i
$658$		0			0
$659$	−13.4717			−0.524783			−0.262392	−	0.964961i	$-0.584511\pi$
$659$	−13.4717			−0.524783			−0.262392	+	0.964961i	$0.584511\pi$
$660$		0			0
$661$	−18.9448			−0.736867			−0.368434	−	0.929654i	$-0.620106\pi$
$661$	−18.9448			−0.736867			−0.368434	+	0.929654i	$0.620106\pi$
$662$		0			0
$663$		−	7.50786i		−	0.291581i
$664$		0			0
$665$	−7.54331	−	33.9067i	−0.292517	−	1.31485i
$666$		0			0
$667$		−	9.19670i		−	0.356098i
$668$		0			0
$669$	−4.25800			−0.164624
$670$		0			0
$671$	4.36568			0.168535
$672$		0			0
$673$		−	27.3807i		−	1.05545i	−0.849415	−	0.527725i	$-0.823045\pi$
$673$		−	27.3807i		−	1.05545i	0.849415	−	0.527725i	$-0.176955\pi$
$674$		0			0
$675$	−3.72010	−	7.94702i	−0.143187	−	0.305881i
$676$		0			0
$677$			23.9986i			0.922342i	0.887311	+	0.461171i	$0.152571\pi$
$677$			23.9986i			0.922342i	−0.887311	+	0.461171i	$0.847429\pi$
$678$		0			0
$679$	15.1330			0.580753
$680$		0			0
$681$	0.0650665			0.00249335
$682$		0			0
$683$			28.6152i			1.09493i	0.836829	+	0.547464i	$0.184407\pi$
$683$			28.6152i			1.09493i	−0.836829	+	0.547464i	$0.815593\pi$
$684$		0			0
$685$	9.67774	+	43.5009i	0.369767	+	1.66208i
$686$		0			0
$687$			0.272892i			0.0104115i
$688$		0			0
$689$	12.6337			0.481307
$690$		0			0
$691$	−24.0645			−0.915457			−0.457728	−	0.889092i	$-0.651337\pi$
$691$	−24.0645			−0.915457			−0.457728	+	0.889092i	$0.651337\pi$
$692$		0			0
$693$			31.3749i			1.19183i
$694$		0			0
$695$	−0.756812	+	0.168370i	−0.0287075	+	0.00638662i
$696$		0			0
$697$			49.6406i			1.88027i
$698$		0			0
$699$	6.68701			0.252926
$700$		0			0
$701$	41.5376			1.56885			0.784427	−	0.620221i	$-0.212957\pi$
$701$	41.5376			1.56885			0.784427	+	0.620221i	$0.212957\pi$
$702$		0			0
$703$		−	31.2144i		−	1.17727i
$704$		0			0
$705$	3.55389	−	0.790643i	0.133847	−	0.0297773i
$706$		0			0
$707$		−	36.0483i		−	1.35574i
$708$		0			0
$709$	0.239337			0.00898849			0.00449424	−	0.999990i	$-0.498569\pi$
$709$	0.239337			0.00898849			0.00449424	+	0.999990i	$0.498569\pi$
$710$		0			0
$711$	−6.18146			−0.231823
$712$		0			0
$713$		−	5.89980i		−	0.220949i
$714$		0			0
$715$	6.96124	+	31.2904i	0.260336	+	1.17019i
$716$		0			0
$717$		−	3.40229i		−	0.127061i
$718$		0			0
$719$	−18.2322			−0.679948			−0.339974	−	0.940435i	$-0.610418\pi$
$719$	−18.2322			−0.679948			−0.339974	+	0.940435i	$0.610418\pi$
$720$		0			0
$721$	24.9623			0.929645
$722$		0			0
$723$		−	4.54280i		−	0.168949i
$724$		0			0
$725$	−41.6464	+	19.4952i	−1.54671	+	0.724034i
$726$		0			0
$727$		−	1.16149i		−	0.0430772i	−0.999768	−	0.0215386i	$-0.993144\pi$
$727$		−	1.16149i		−	0.0430772i	0.999768	−	0.0215386i	$-0.00685648\pi$
$728$		0			0
$729$	22.3574			0.828052
$730$		0			0
$731$	−30.4920			−1.12779
$732$		0			0
$733$			16.9474i			0.625967i	0.949759	+	0.312984i	$0.101329\pi$
$733$			16.9474i			0.625967i	−0.949759	+	0.312984i	$0.898671\pi$
$734$		0			0
$735$	−0.717011	−	3.22293i	−0.0264474	−	0.118879i
$736$		0			0
$737$			10.8913i			0.401186i
$738$		0			0
$739$	30.7566			1.13140			0.565700	−	0.824611i	$-0.308606\pi$
$739$	30.7566			1.13140			0.565700	+	0.824611i	$0.308606\pi$
$740$		0			0
$741$	−6.13524			−0.225384
$742$		0			0
$743$			6.66705i			0.244591i	0.992494	+	0.122295i	$0.0390255\pi$
$743$			6.66705i			0.244591i	−0.992494	+	0.122295i	$0.960975\pi$
$744$		0			0
$745$	−4.10245	+	0.912682i	−0.150302	+	0.0334381i
$746$		0			0
$747$		−	46.6364i		−	1.70634i
$748$		0			0
$749$	−24.4728			−0.894217
$750$		0			0
$751$	−8.32271			−0.303700			−0.151850	−	0.988404i	$-0.548523\pi$
$751$	−8.32271			−0.303700			−0.151850	+	0.988404i	$0.548523\pi$
$752$		0			0
$753$		−	6.27683i		−	0.228740i
$754$		0			0
$755$	−21.4940	+	4.78183i	−0.782248	+	0.174028i
$756$		0			0
$757$		−	34.5436i		−	1.25551i	−0.778412	−	0.627754i	$-0.783974\pi$
$757$		−	34.5436i		−	1.25551i	0.778412	−	0.627754i	$-0.216026\pi$
$758$		0			0
$759$	−0.924314			−0.0335505
$760$		0			0
$761$	−42.1781			−1.52896			−0.764478	−	0.644650i	$-0.777003\pi$
$761$	−42.1781			−1.52896			−0.764478	+	0.644650i	$0.777003\pi$
$762$		0			0
$763$		−	34.3289i		−	1.24279i
$764$		0			0
$765$	7.76779	+	34.9158i	0.280845	+	1.26238i
$766$		0			0
$767$		−	42.9851i		−	1.55210i
$768$		0			0
$769$	−13.0704			−0.471332			−0.235666	−	0.971834i	$-0.575727\pi$
$769$	−13.0704			−0.471332			−0.235666	+	0.971834i	$0.575727\pi$
$770$		0			0
$771$	−0.836115			−0.0301120
$772$		0			0
$773$		−	7.14112i		−	0.256848i	−0.991719	−	0.128424i	$-0.959008\pi$
$773$		−	7.14112i		−	0.256848i	0.991719	−	0.128424i	$-0.0409919\pi$
$774$		0			0
$775$	−26.7167	+	12.5064i	−0.959692	+	0.449244i
$776$		0			0
$777$		−	7.14239i		−	0.256232i
$778$		0			0
$779$	40.5651			1.45340
$780$		0			0
$781$	−13.3540			−0.477843
$782$		0			0
$783$			16.1395i			0.576780i
$784$		0			0
$785$	8.04285	+	36.1522i	0.287062	+	1.29033i
$786$		0			0
$787$			30.5980i			1.09070i	0.838208	+	0.545351i	$0.183604\pi$
$787$			30.5980i			1.09070i	−0.838208	+	0.545351i	$0.816396\pi$
$788$		0			0
$789$	5.00565			0.178206
$790$		0			0
$791$	44.9145			1.59698
$792$		0			0
$793$		−	6.45500i		−	0.229224i
$794$		0			0
$795$	1.77799	−	0.395554i	0.0630589	−	0.0140289i
$796$		0			0
$797$			32.8338i			1.16303i	0.813535	+	0.581516i	$0.197540\pi$
$797$			32.8338i			1.16303i	−0.813535	+	0.581516i	$0.802460\pi$
$798$		0			0
$799$	−30.1321			−1.06600
$800$		0			0
$801$	−34.8903			−1.23279
$802$		0			0
$803$		−	15.3212i		−	0.540674i
$804$		0			0
$805$	−7.55297	+	1.68033i	−0.266207	+	0.0592237i
$806$		0			0
$807$			8.25743i			0.290675i
$808$		0			0
$809$	43.7461			1.53803			0.769015	−	0.639230i	$-0.220747\pi$
$809$	43.7461			1.53803			0.769015	+	0.639230i	$0.220747\pi$
$810$		0			0
$811$	38.0475			1.33603			0.668015	−	0.744148i	$-0.267144\pi$
$811$	38.0475			1.33603			0.668015	+	0.744148i	$0.267144\pi$
$812$		0			0
$813$			1.40674i			0.0493364i
$814$		0			0
$815$	4.60772	+	20.7114i	0.161401	+	0.725490i
$816$		0			0
$817$			24.9173i			0.871746i
$818$		0			0
$819$	46.3902			1.62101
$820$		0			0
$821$	9.79962			0.342009			0.171004	−	0.985270i	$-0.445299\pi$
$821$	9.79962			0.342009			0.171004	+	0.985270i	$0.445299\pi$
$822$		0			0
$823$		−	33.2526i		−	1.15911i	−0.814932	−	0.579557i	$-0.803226\pi$
$823$		−	33.2526i		−	1.15911i	0.814932	−	0.579557i	$-0.196774\pi$
$824$		0			0
$825$	1.95936	+	4.18566i	0.0682163	+	0.145726i
$826$		0			0
$827$			11.9711i			0.416275i	0.978100	+	0.208138i	$0.0667402\pi$
$827$			11.9711i			0.416275i	−0.978100	+	0.208138i	$0.933260\pi$
$828$		0			0
$829$	7.99628			0.277722			0.138861	−	0.990312i	$-0.455656\pi$
$829$	7.99628			0.277722			0.138861	+	0.990312i	$0.455656\pi$
$830$		0			0
$831$	0.292030			0.0101304
$832$		0			0
$833$			27.3259i			0.946787i
$834$		0			0
$835$	−4.50839	−	20.2650i	−0.156019	−	0.701298i
$836$		0			0
$837$			10.3537i			0.357877i
$838$		0			0
$839$	−8.94284			−0.308741			−0.154371	−	0.988013i	$-0.549335\pi$
$839$	−8.94284			−0.308741			−0.154371	+	0.988013i	$0.549335\pi$
$840$		0			0
$841$	55.5793			1.91653
$842$		0			0
$843$		−	7.16773i		−	0.246870i
$844$		0			0
$845$	17.8901	−	3.98005i	0.615438	−	0.136918i
$846$		0			0
$847$			4.51393i			0.155100i
$848$		0			0
$849$	−6.72427			−0.230777
$850$		0			0
$851$	−6.95324			−0.238354
$852$		0			0
$853$			44.8144i			1.53442i	0.641398	+	0.767208i	$0.278355\pi$
$853$			44.8144i			1.53442i	−0.641398	+	0.767208i	$0.721645\pi$
$854$		0			0
$855$	28.5323	−	6.34764i	0.975784	−	0.217085i
$856$		0			0
$857$			10.2025i			0.348510i	0.984701	+	0.174255i	$0.0557517\pi$
$857$			10.2025i			0.348510i	−0.984701	+	0.174255i	$0.944248\pi$
$858$		0			0
$859$	30.0587			1.02559			0.512796	−	0.858511i	$-0.328610\pi$
$859$	30.0587			1.02559			0.512796	+	0.858511i	$0.328610\pi$
$860$		0			0
$861$	9.28199			0.316329
$862$		0			0
$863$			19.9013i			0.677449i	0.940886	+	0.338724i	$0.109995\pi$
$863$			19.9013i			0.677449i	−0.940886	+	0.338724i	$0.890005\pi$
$864$		0			0
$865$	−4.53553	−	20.3870i	−0.154213	−	0.693177i
$866$		0			0
$867$			3.91217i			0.132864i
$868$		0			0
$869$	6.61003			0.224230
$870$		0			0
$871$	16.1036			0.545651
$872$		0			0
$873$			12.7344i			0.430993i
$874$		0			0
$875$	23.6200	+	30.6409i	0.798503	+	1.03585i
$876$		0			0
$877$		−	49.0835i		−	1.65743i	−0.559668	−	0.828717i	$-0.689071\pi$
$877$		−	49.0835i		−	1.65743i	0.559668	−	0.828717i	$-0.310929\pi$
$878$		0			0
$879$	3.84917			0.129829
$880$		0			0
$881$	25.0658			0.844489			0.422245	−	0.906482i	$-0.361242\pi$
$881$	25.0658			0.844489			0.422245	+	0.906482i	$0.361242\pi$
$882$		0			0
$883$		−	35.7237i		−	1.20220i	−0.799174	−	0.601100i	$-0.794729\pi$
$883$		−	35.7237i		−	1.20220i	0.799174	−	0.601100i	$-0.205271\pi$
$884$		0			0
$885$	−1.34584	−	6.04946i	−0.0452398	−	0.203351i
$886$		0			0
$887$		−	27.4000i		−	0.920001i	−0.887919	−	0.460001i	$-0.847849\pi$
$887$		−	27.4000i		−	0.920001i	0.887919	−	0.460001i	$-0.152151\pi$
$888$		0			0
$889$	51.7893			1.73696
$890$		0			0
$891$	−25.5786			−0.856917
$892$		0			0
$893$			24.6232i			0.823983i
$894$		0			0
$895$	−0.557426	+	0.124012i	−0.0186327	+	0.00414525i
$896$		0			0
$897$			1.36667i			0.0456318i
$898$		0			0
$899$	54.2587			1.80963
$900$		0			0
$901$	−15.0749			−0.502218
$902$		0			0
$903$			5.70150i			0.189734i
$904$		0			0
$905$	8.78895	−	1.95530i	0.292155	−	0.0649963i
$906$		0			0
$907$			53.9804i			1.79239i	0.443661	+	0.896195i	$0.353679\pi$
$907$			53.9804i			1.79239i	−0.443661	+	0.896195i	$0.646321\pi$
$908$		0			0
$909$	30.3344			1.00613
$910$		0			0
$911$	8.29792			0.274922			0.137461	−	0.990507i	$-0.456106\pi$
$911$	8.29792			0.274922			0.137461	+	0.990507i	$0.456106\pi$
$912$		0			0
$913$			49.8698i			1.65045i
$914$		0			0
$915$	−0.202102	−	0.908437i	−0.00668128	−	0.0300320i
$916$		0			0
$917$		−	64.3440i		−	2.12482i
$918$		0			0
$919$	42.3852			1.39816			0.699080	−	0.715043i	$-0.253593\pi$
$919$	42.3852			1.39816			0.699080	+	0.715043i	$0.253593\pi$
$920$		0			0
$921$	−8.14490			−0.268384
$922$		0			0
$923$			19.7449i			0.649911i
$924$		0			0
$925$	14.7395	+	31.4871i	0.484632	+	1.03529i
$926$		0			0
$927$			21.0056i			0.689915i
$928$		0			0
$929$	−22.9844			−0.754092			−0.377046	−	0.926194i	$-0.623060\pi$
$929$	−22.9844			−0.754092			−0.377046	+	0.926194i	$0.623060\pi$
$930$		0			0
$931$	22.3301			0.731838
$932$		0			0
$933$			9.16236i			0.299962i
$934$		0			0
$935$	−8.30634	−	37.3365i	−0.271646	−	1.22103i
$936$		0			0
$937$		−	37.8592i		−	1.23681i	−0.785861	−	0.618403i	$-0.787780\pi$
$937$		−	37.8592i		−	1.23681i	0.785861	−	0.618403i	$-0.212220\pi$
$938$		0			0
$939$	−1.08134			−0.0352881
$940$		0			0
$941$	−7.13773			−0.232683			−0.116342	−	0.993209i	$-0.537117\pi$
$941$	−7.13773			−0.232683			−0.116342	+	0.993209i	$0.537117\pi$
$942$		0			0
$943$		−	9.03617i		−	0.294258i
$944$		0			0
$945$	13.2549	−	2.94885i	0.431183	−	0.0959261i
$946$		0			0
$947$		−	23.1645i		−	0.752745i	−0.926468	−	0.376372i	$-0.877171\pi$
$947$		−	23.1645i		−	0.752745i	0.926468	−	0.376372i	$-0.122829\pi$
$948$		0			0
$949$	−22.6536			−0.735368
$950$		0			0
$951$	−7.13454			−0.231353
$952$		0			0
$953$		−	37.0608i		−	1.20052i	−0.799806	−	0.600258i	$-0.795065\pi$
$953$		−	37.0608i		−	1.20052i	0.799806	−	0.600258i	$-0.204935\pi$
$954$		0			0
$955$	53.9948	−	12.0123i	1.74723	−	0.388710i
$956$		0			0
$957$		−	8.50064i		−	0.274787i
$958$		0			0
$959$	−68.9646			−2.22698
$960$		0			0
$961$	3.80769			0.122829
$962$		0			0
$963$		−	20.5937i		−	0.663623i
$964$		0			0
$965$	3.56454	+	16.0224i	0.114747	+	0.515780i
$966$		0			0
$967$		−	47.2555i		−	1.51964i	−0.650136	−	0.759818i	$-0.725288\pi$
$967$		−	47.2555i		−	1.51964i	0.650136	−	0.759818i	$-0.274712\pi$
$968$		0			0
$969$	7.32073			0.235176
$970$		0			0
$971$	−53.8987			−1.72969			−0.864845	−	0.502038i	$-0.832584\pi$
$971$	−53.8987			−1.72969			−0.864845	+	0.502038i	$0.832584\pi$
$972$		0			0
$973$		−	1.19982i		−	0.0384644i
$974$		0			0
$975$	6.18883	−	2.89707i	0.198201	−	0.0927805i
$976$		0			0
$977$			61.0840i			1.95425i	0.212662	+	0.977126i	$0.431787\pi$
$977$			61.0840i			1.95425i	−0.212662	+	0.977126i	$0.568213\pi$
$978$		0			0
$979$	37.3093			1.19241
$980$		0			0
$981$	28.8876			0.922308
$982$		0			0
$983$		−	53.4981i		−	1.70632i	−0.521645	−	0.853162i	$-0.674682\pi$
$983$		−	53.4981i		−	1.70632i	0.521645	−	0.853162i	$-0.325318\pi$
$984$		0			0
$985$	7.86819	+	35.3671i	0.250701	+	1.12689i
$986$		0			0
$987$			5.63420i			0.179339i
$988$		0			0
$989$	5.55051			0.176496
$990$		0			0
$991$	18.2274			0.579013			0.289506	−	0.957176i	$-0.406509\pi$
$991$	18.2274			0.579013			0.289506	+	0.957176i	$0.406509\pi$
$992$		0			0
$993$			6.47125i			0.205359i
$994$		0			0
$995$	−30.2300	+	6.72533i	−0.958356	+	0.213207i
$996$		0			0
$997$		−	55.7823i		−	1.76664i	−0.468767	−	0.883322i	$-0.655302\pi$
$997$		−	55.7823i		−	1.76664i	0.468767	−	0.883322i	$-0.344698\pi$
$998$		0			0
$999$	12.2024			0.386068

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	920.2.e.c.369.8	✓	16
4.3	odd	2		1840.2.e.h.369.9		16
5.2	odd	4		4600.2.a.bj.1.5		8
5.3	odd	4		4600.2.a.bk.1.4		8
5.4	even	2	inner	920.2.e.c.369.9	yes	16
20.3	even	4		9200.2.a.dd.1.5		8
20.7	even	4		9200.2.a.de.1.4		8
20.19	odd	2		1840.2.e.h.369.8		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
920.2.e.c.369.8	✓	16	1.1	even	1	trivial
920.2.e.c.369.9	yes	16	5.4	even	2	inner
1840.2.e.h.369.8		16	20.19	odd	2
1840.2.e.h.369.9		16	4.3	odd	2
4600.2.a.bj.1.5		8	5.2	odd	4
4600.2.a.bk.1.4		8	5.3	odd	4
9200.2.a.dd.1.5		8	20.3	even	4
9200.2.a.de.1.4		8	20.7	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:	\( N \)	\(=\)	\( 920 = 2^{3} \cdot 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	920.e (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-0.296848i q^{3} +(-2.18271 + 0.485591i) q^{5} -3.46037i q^{7} +2.91188 q^{9} +O(q^{10})\)	\(q-0.296848i q^{3} +(-2.18271 + 0.485591i) q^{5} -3.46037i q^{7} +2.91188 q^{9} -3.11377 q^{11} +4.60394i q^{13} +(0.144147 + 0.647931i) q^{15} -5.49355i q^{17} -4.48919 q^{19} -1.02720 q^{21} +1.00000i q^{23} +(4.52840 - 2.11980i) q^{25} -1.75493i q^{27} -9.19670 q^{29} -5.89980 q^{31} +0.924314i q^{33} +(1.68033 + 7.55297i) q^{35} +6.95324i q^{37} +1.36667 q^{39} -9.03617 q^{41} -5.55051i q^{43} +(-6.35578 + 1.41398i) q^{45} -5.48499i q^{47} -4.97418 q^{49} -1.63075 q^{51} -2.74411i q^{53} +(6.79643 - 1.51202i) q^{55} +1.33261i q^{57} -9.33659 q^{59} -1.40206 q^{61} -10.0762i q^{63} +(-2.23563 - 10.0490i) q^{65} -3.49779i q^{67} +0.296848 q^{69} +4.28869 q^{71} +4.92048i q^{73} +(-0.629259 - 1.34425i) q^{75} +10.7748i q^{77} -2.12284 q^{79} +8.21470 q^{81} -16.0159i q^{83} +(2.66762 + 11.9908i) q^{85} +2.73002i q^{87} -11.9821 q^{89} +15.9314 q^{91} +1.75134i q^{93} +(9.79858 - 2.17991i) q^{95} +4.37324i q^{97} -9.06692 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 2 q^{5} - 22 q^{9}+O(q^{10}) \) 16 * q - 2 * q^5 - 22 * q^9	\( 16 q - 2 q^{5} - 22 q^{9} + 14 q^{11} + 6 q^{15} - 22 q^{19} + 12 q^{25} - 44 q^{29} + 18 q^{31} + 20 q^{35} + 14 q^{41} + 14 q^{45} - 78 q^{49} - 38 q^{51} + 30 q^{55} - 64 q^{59} + 34 q^{61} + 6 q^{65} + 6 q^{69} + 30 q^{71} + 56 q^{75} + 4 q^{79} + 48 q^{81} + 52 q^{85} - 92 q^{89} - 70 q^{91} + 38 q^{95} - 122 q^{99}+O(q^{100}) \) 16 * q - 2 * q^5 - 22 * q^9 + 14 * q^11 + 6 * q^15 - 22 * q^19 + 12 * q^25 - 44 * q^29 + 18 * q^31 + 20 * q^35 + 14 * q^41 + 14 * q^45 - 78 * q^49 - 38 * q^51 + 30 * q^55 - 64 * q^59 + 34 * q^61 + 6 * q^65 + 6 * q^69 + 30 * q^71 + 56 * q^75 + 4 * q^79 + 48 * q^81 + 52 * q^85 - 92 * q^89 - 70 * q^91 + 38 * q^95 - 122 * q^99

Introduction

L-functions

Modular forms

Varieties

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Representations

Groups

Database

Properties

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