LMFDB - Embedded newform 1920.2.h.f.1151.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1920,2,Mod(1151,1920)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1920, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 0, 1, 0]))

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

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

chi := DirichletCharacter("1920.1151");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1920 = 2^{7} \cdot 3 \cdot 5 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1920.h (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$15.3312771881$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{5})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 3x^{2} + 1 $ x^4 + 3*x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1151.4
Root			$0.618034i$ of defining polynomial
Character	$\chi$	$=$	1920.1151
Dual form			1920.2.h.f.1151.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+(0.618034 + 1.61803i) q^{3} +1.00000i q^{5} +2.00000i q^{7} +(-2.23607 + 2.00000i) q^{9} -2.47214 q^{11} -1.23607 q^{13} +(-1.61803 + 0.618034i) q^{15} +0.763932i q^{17} +5.23607i q^{19} +(-3.23607 + 1.23607i) q^{21} +0.472136 q^{23} -1.00000 q^{25} +(-4.61803 - 2.38197i) q^{27} -8.47214i q^{29} -4.76393i q^{31} +(-1.52786 - 4.00000i) q^{33} -2.00000 q^{35} -7.70820 q^{37} +(-0.763932 - 2.00000i) q^{39} +1.52786i q^{41} +9.70820i q^{43} +(-2.00000 - 2.23607i) q^{45} +4.47214 q^{47} +3.00000 q^{49} +(-1.23607 + 0.472136i) q^{51} -4.47214i q^{53} -2.47214i q^{55} +(-8.47214 + 3.23607i) q^{57} -6.47214 q^{59} -12.4721 q^{61} +(-4.00000 - 4.47214i) q^{63} -1.23607i q^{65} +11.2361i q^{67} +(0.291796 + 0.763932i) q^{69} +4.00000 q^{71} -0.472136 q^{73} +(-0.618034 - 1.61803i) q^{75} -4.94427i q^{77} -8.18034i q^{79} +(1.00000 - 8.94427i) q^{81} +11.7082 q^{83} -0.763932 q^{85} +(13.7082 - 5.23607i) q^{87} -1.52786i q^{89} -2.47214i q^{91} +(7.70820 - 2.94427i) q^{93} -5.23607 q^{95} -12.4721 q^{97} +(5.52786 - 4.94427i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 2 q^{3} + 8 q^{11} + 4 q^{13} - 2 q^{15} - 4 q^{21} - 16 q^{23} - 4 q^{25} - 14 q^{27} - 24 q^{33} - 8 q^{35} - 4 q^{37} - 12 q^{39} - 8 q^{45} + 12 q^{49} + 4 q^{51} - 16 q^{57} - 8 q^{59} - 32 q^{61}+ \cdots + 40 q^{99}+O(q^{100}) $ 4 * q - 2 * q^3 + 8 * q^11 + 4 * q^13 - 2 * q^15 - 4 * q^21 - 16 * q^23 - 4 * q^25 - 14 * q^27 - 24 * q^33 - 8 * q^35 - 4 * q^37 - 12 * q^39 - 8 * q^45 + 12 * q^49 + 4 * q^51 - 16 * q^57 - 8 * q^59 - 32 * q^61 - 16 * q^63 + 28 * q^69 + 16 * q^71 + 16 * q^73 + 2 * q^75 + 4 * q^81 + 20 * q^83 - 12 * q^85 + 28 * q^87 + 4 * q^93 - 12 * q^95 - 32 * q^97 + 40 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1920\mathbb{Z}\right)^\times$.

$n$	$511$	$641$	$901$	$1537$
$\chi(n)$	$-1$	$-1$	$1$	$1$

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
$3$	0.618034	+	1.61803i	0.356822	+	0.934172i
$3$	0.618034	+	1.61803i	0.356822	+	0.934172i
$4$		0			0
$5$			1.00000i			0.447214i
$5$			1.00000i			0.447214i
$6$		0			0
$7$			2.00000i			0.755929i	0.925820	+	0.377964i	$0.123376\pi$
$7$			2.00000i			0.755929i	−0.925820	+	0.377964i	$0.876624\pi$
$8$		0			0
$9$	−2.23607	+	2.00000i	−0.745356	+	0.666667i
$10$		0			0
$11$	−2.47214			−0.745377			−0.372689	−	0.927957i	$-0.621564\pi$
$11$	−2.47214			−0.745377			−0.372689	+	0.927957i	$0.621564\pi$
$12$		0			0
$13$	−1.23607			−0.342824			−0.171412	−	0.985199i	$-0.554833\pi$
$13$	−1.23607			−0.342824			−0.171412	+	0.985199i	$0.554833\pi$
$14$		0			0
$15$	−1.61803	+	0.618034i	−0.417775	+	0.159576i
$16$		0			0
$17$			0.763932i			0.185281i	0.995700	+	0.0926404i	$0.0295307\pi$
$17$			0.763932i			0.185281i	−0.995700	+	0.0926404i	$0.970469\pi$
$18$		0			0
$19$			5.23607i			1.20124i	0.799536	+	0.600618i	$0.205079\pi$
$19$			5.23607i			1.20124i	−0.799536	+	0.600618i	$0.794921\pi$
$20$		0			0
$21$	−3.23607	+	1.23607i	−0.706168	+	0.269732i
$22$		0			0
$23$	0.472136			0.0984472			0.0492236	−	0.998788i	$-0.484325\pi$
$23$	0.472136			0.0984472			0.0492236	+	0.998788i	$0.484325\pi$
$24$		0			0
$25$	−1.00000			−0.200000
$26$		0			0
$27$	−4.61803	−	2.38197i	−0.888741	−	0.458410i
$28$		0			0
$29$		−	8.47214i		−	1.57324i	−0.617440	−	0.786618i	$-0.711830\pi$
$29$		−	8.47214i		−	1.57324i	0.617440	−	0.786618i	$-0.288170\pi$
$30$		0			0
$31$		−	4.76393i		−	0.855627i	−0.903867	−	0.427814i	$-0.859284\pi$
$31$		−	4.76393i		−	0.855627i	0.903867	−	0.427814i	$-0.140716\pi$
$32$		0			0
$33$	−1.52786	−	4.00000i	−0.265967	−	0.696311i
$34$		0			0
$35$	−2.00000			−0.338062
$36$		0			0
$37$	−7.70820			−1.26722			−0.633610	−	0.773652i	$-0.718428\pi$
$37$	−7.70820			−1.26722			−0.633610	+	0.773652i	$0.718428\pi$
$38$		0			0
$39$	−0.763932	−	2.00000i	−0.122327	−	0.320256i
$40$		0			0
$41$			1.52786i			0.238612i	0.992858	+	0.119306i	$0.0380670\pi$
$41$			1.52786i			0.238612i	−0.992858	+	0.119306i	$0.961933\pi$
$42$		0			0
$43$			9.70820i			1.48049i	0.672339	+	0.740244i	$0.265290\pi$
$43$			9.70820i			1.48049i	−0.672339	+	0.740244i	$0.734710\pi$
$44$		0			0
$45$	−2.00000	−	2.23607i	−0.298142	−	0.333333i
$46$		0			0
$47$	4.47214			0.652328			0.326164	−	0.945313i	$-0.394244\pi$
$47$	4.47214			0.652328			0.326164	+	0.945313i	$0.394244\pi$
$48$		0			0
$49$	3.00000			0.428571
$50$		0			0
$51$	−1.23607	+	0.472136i	−0.173084	+	0.0661123i
$52$		0			0
$53$		−	4.47214i		−	0.614295i	−0.951662	−	0.307148i	$-0.900625\pi$
$53$		−	4.47214i		−	0.614295i	0.951662	−	0.307148i	$-0.0993745\pi$
$54$		0			0
$55$		−	2.47214i		−	0.333343i
$56$		0			0
$57$	−8.47214	+	3.23607i	−1.12216	+	0.428628i
$58$		0			0
$59$	−6.47214			−0.842600			−0.421300	−	0.906921i	$-0.638426\pi$
$59$	−6.47214			−0.842600			−0.421300	+	0.906921i	$0.638426\pi$
$60$		0			0
$61$	−12.4721			−1.59689			−0.798447	−	0.602066i	$-0.794345\pi$
$61$	−12.4721			−1.59689			−0.798447	+	0.602066i	$0.794345\pi$
$62$		0			0
$63$	−4.00000	−	4.47214i	−0.503953	−	0.563436i
$64$		0			0
$65$		−	1.23607i		−	0.153315i
$66$		0			0
$67$			11.2361i			1.37270i	0.727269	+	0.686352i	$0.240789\pi$
$67$			11.2361i			1.37270i	−0.727269	+	0.686352i	$0.759211\pi$
$68$		0			0
$69$	0.291796	+	0.763932i	0.0351281	+	0.0919666i
$70$		0			0
$71$	4.00000			0.474713			0.237356	−	0.971423i	$-0.423719\pi$
$71$	4.00000			0.474713			0.237356	+	0.971423i	$0.423719\pi$
$72$		0			0
$73$	−0.472136			−0.0552593			−0.0276297	−	0.999618i	$-0.508796\pi$
$73$	−0.472136			−0.0552593			−0.0276297	+	0.999618i	$0.508796\pi$
$74$		0			0
$75$	−0.618034	−	1.61803i	−0.0713644	−	0.186834i
$76$		0			0
$77$		−	4.94427i		−	0.563452i
$78$		0			0
$79$		−	8.18034i		−	0.920360i	−0.887826	−	0.460180i	$-0.847785\pi$
$79$		−	8.18034i		−	0.920360i	0.887826	−	0.460180i	$-0.152215\pi$
$80$		0			0
$81$	1.00000	−	8.94427i	0.111111	−	0.993808i
$82$		0			0
$83$	11.7082			1.28514			0.642571	−	0.766226i	$-0.277868\pi$
$83$	11.7082			1.28514			0.642571	+	0.766226i	$0.277868\pi$
$84$		0			0
$85$	−0.763932			−0.0828601
$86$		0			0
$87$	13.7082	−	5.23607i	1.46967	−	0.561365i
$88$		0			0
$89$		−	1.52786i		−	0.161953i	−0.996716	−	0.0809766i	$-0.974196\pi$
$89$		−	1.52786i		−	0.161953i	0.996716	−	0.0809766i	$-0.0258039\pi$
$90$		0			0
$91$		−	2.47214i		−	0.259150i
$92$		0			0
$93$	7.70820	−	2.94427i	0.799304	−	0.305307i
$94$		0			0
$95$	−5.23607			−0.537209
$96$		0			0
$97$	−12.4721			−1.26635			−0.633177	−	0.774007i	$-0.718249\pi$
$97$	−12.4721			−1.26635			−0.633177	+	0.774007i	$0.718249\pi$
$98$		0			0
$99$	5.52786	−	4.94427i	0.555571	−	0.496918i
$100$		0			0
$101$			17.4164i			1.73300i	0.499179	+	0.866499i	$0.333635\pi$
$101$			17.4164i			1.73300i	−0.499179	+	0.866499i	$0.666365\pi$
$102$		0			0
$103$		−	0.472136i		−	0.0465209i	−0.999729	−	0.0232605i	$-0.992595\pi$
$103$		−	0.472136i		−	0.0465209i	0.999729	−	0.0232605i	$-0.00740471\pi$
$104$		0			0
$105$	−1.23607	−	3.23607i	−0.120628	−	0.315808i
$106$		0			0
$107$	13.2361			1.27958			0.639789	−	0.768550i	$-0.279022\pi$
$107$	13.2361			1.27958			0.639789	+	0.768550i	$0.279022\pi$
$108$		0			0
$109$	−1.05573			−0.101120			−0.0505602	−	0.998721i	$-0.516101\pi$
$109$	−1.05573			−0.101120			−0.0505602	+	0.998721i	$0.516101\pi$
$110$		0			0
$111$	−4.76393	−	12.4721i	−0.452172	−	1.18380i
$112$		0			0
$113$		−	21.1246i		−	1.98724i	−0.112796	−	0.993618i	$-0.535981\pi$
$113$		−	21.1246i		−	1.98724i	0.112796	−	0.993618i	$-0.464019\pi$
$114$		0			0
$115$			0.472136i			0.0440269i
$116$		0			0
$117$	2.76393	−	2.47214i	0.255526	−	0.228549i
$118$		0			0
$119$	−1.52786			−0.140059
$120$		0			0
$121$	−4.88854			−0.444413
$122$		0			0
$123$	−2.47214	+	0.944272i	−0.222905	+	0.0851421i
$124$		0			0
$125$		−	1.00000i		−	0.0894427i
$126$		0			0
$127$			10.0000i			0.887357i	0.896186	+	0.443678i	$0.146327\pi$
$127$			10.0000i			0.887357i	−0.896186	+	0.443678i	$0.853673\pi$
$128$		0			0
$129$	−15.7082	+	6.00000i	−1.38303	+	0.528271i
$130$		0			0
$131$	−21.8885			−1.91241			−0.956205	−	0.292696i	$-0.905448\pi$
$131$	−21.8885			−1.91241			−0.956205	+	0.292696i	$0.905448\pi$
$132$		0			0
$133$	−10.4721			−0.908049
$134$		0			0
$135$	2.38197	−	4.61803i	0.205007	−	0.397457i
$136$		0			0
$137$			18.6525i			1.59359i	0.604251	+	0.796794i	$0.293473\pi$
$137$			18.6525i			1.59359i	−0.604251	+	0.796794i	$0.706527\pi$
$138$		0			0
$139$			14.1803i			1.20276i	0.798963	+	0.601380i	$0.205382\pi$
$139$			14.1803i			1.20276i	−0.798963	+	0.601380i	$0.794618\pi$
$140$		0			0
$141$	2.76393	+	7.23607i	0.232765	+	0.609387i
$142$		0			0
$143$	3.05573			0.255533
$144$		0			0
$145$	8.47214			0.703573
$146$		0			0
$147$	1.85410	+	4.85410i	0.152924	+	0.400360i
$148$		0			0
$149$			10.0000i			0.819232i	0.912258	+	0.409616i	$0.134337\pi$
$149$			10.0000i			0.819232i	−0.912258	+	0.409616i	$0.865663\pi$
$150$		0			0
$151$			3.23607i			0.263347i	0.991293	+	0.131674i	$0.0420351\pi$
$151$			3.23607i			0.263347i	−0.991293	+	0.131674i	$0.957965\pi$
$152$		0			0
$153$	−1.52786	−	1.70820i	−0.123520	−	0.138100i
$154$		0			0
$155$	4.76393			0.382648
$156$		0			0
$157$	11.7082			0.934416			0.467208	−	0.884147i	$-0.345260\pi$
$157$	11.7082			0.934416			0.467208	+	0.884147i	$0.345260\pi$
$158$		0			0
$159$	7.23607	−	2.76393i	0.573858	−	0.219194i
$160$		0			0
$161$			0.944272i			0.0744191i
$162$		0			0
$163$		−	3.23607i		−	0.253468i	−0.991937	−	0.126734i	$-0.959550\pi$
$163$		−	3.23607i		−	0.253468i	0.991937	−	0.126734i	$-0.0404495\pi$
$164$		0			0
$165$	4.00000	−	1.52786i	0.311400	−	0.118944i
$166$		0			0
$167$	2.94427			0.227835			0.113917	−	0.993490i	$-0.463660\pi$
$167$	2.94427			0.227835			0.113917	+	0.993490i	$0.463660\pi$
$168$		0			0
$169$	−11.4721			−0.882472
$170$		0			0
$171$	−10.4721	−	11.7082i	−0.800824	−	0.895349i
$172$		0			0
$173$		−	20.4721i		−	1.55647i	−0.627975	−	0.778234i	$-0.716116\pi$
$173$		−	20.4721i		−	1.55647i	0.627975	−	0.778234i	$-0.283884\pi$
$174$		0			0
$175$		−	2.00000i		−	0.151186i
$176$		0			0
$177$	−4.00000	−	10.4721i	−0.300658	−	0.787134i
$178$		0			0
$179$	−1.52786			−0.114198			−0.0570990	−	0.998369i	$-0.518185\pi$
$179$	−1.52786			−0.114198			−0.0570990	+	0.998369i	$0.518185\pi$
$180$		0			0
$181$	−13.4164			−0.997234			−0.498617	−	0.866822i	$-0.666159\pi$
$181$	−13.4164			−0.997234			−0.498617	+	0.866822i	$0.666159\pi$
$182$		0			0
$183$	−7.70820	−	20.1803i	−0.569807	−	1.49177i
$184$		0			0
$185$		−	7.70820i		−	0.566718i
$186$		0			0
$187$		−	1.88854i		−	0.138104i
$188$		0			0
$189$	4.76393	−	9.23607i	0.346525	−	0.671825i
$190$		0			0
$191$	14.4721			1.04717			0.523584	−	0.851974i	$-0.324595\pi$
$191$	14.4721			1.04717			0.523584	+	0.851974i	$0.324595\pi$
$192$		0			0
$193$	2.94427			0.211933			0.105967	−	0.994370i	$-0.466206\pi$
$193$	2.94427			0.211933			0.105967	+	0.994370i	$0.466206\pi$
$194$		0			0
$195$	2.00000	−	0.763932i	0.143223	−	0.0547063i
$196$		0			0
$197$			10.0000i			0.712470i	0.934396	+	0.356235i	$0.115940\pi$
$197$			10.0000i			0.712470i	−0.934396	+	0.356235i	$0.884060\pi$
$198$		0			0
$199$			17.7082i			1.25530i	0.778495	+	0.627651i	$0.215983\pi$
$199$			17.7082i			1.25530i	−0.778495	+	0.627651i	$0.784017\pi$
$200$		0			0
$201$	−18.1803	+	6.94427i	−1.28234	+	0.489811i
$202$		0			0
$203$	16.9443			1.18925
$204$		0			0
$205$	−1.52786			−0.106711
$206$		0			0
$207$	−1.05573	+	0.944272i	−0.0733782	+	0.0656314i
$208$		0			0
$209$		−	12.9443i		−	0.895374i
$210$		0			0
$211$			0.291796i			0.0200881i	0.999950	+	0.0100440i	$0.00319717\pi$
$211$			0.291796i			0.0200881i	−0.999950	+	0.0100440i	$0.996803\pi$
$212$		0			0
$213$	2.47214	+	6.47214i	0.169388	+	0.443463i
$214$		0			0
$215$	−9.70820			−0.662094
$216$		0			0
$217$	9.52786			0.646794
$218$		0			0
$219$	−0.291796	−	0.763932i	−0.0197178	−	0.0516217i
$220$		0			0
$221$		−	0.944272i		−	0.0635186i
$222$		0			0
$223$		−	2.94427i		−	0.197163i	−0.995129	−	0.0985815i	$-0.968569\pi$
$223$		−	2.94427i		−	0.197163i	0.995129	−	0.0985815i	$-0.0314305\pi$
$224$		0			0
$225$	2.23607	−	2.00000i	0.149071	−	0.133333i
$226$		0			0
$227$	−19.7082			−1.30808			−0.654040	−	0.756460i	$-0.726927\pi$
$227$	−19.7082			−1.30808			−0.654040	+	0.756460i	$0.726927\pi$
$228$		0			0
$229$	−14.9443			−0.987545			−0.493773	−	0.869591i	$-0.664382\pi$
$229$	−14.9443			−0.987545			−0.493773	+	0.869591i	$0.664382\pi$
$230$		0			0
$231$	8.00000	−	3.05573i	0.526361	−	0.201052i
$232$		0			0
$233$			9.70820i			0.636006i	0.948090	+	0.318003i	$0.103012\pi$
$233$			9.70820i			0.636006i	−0.948090	+	0.318003i	$0.896988\pi$
$234$		0			0
$235$			4.47214i			0.291730i
$236$		0			0
$237$	13.2361	−	5.05573i	0.859775	−	0.328405i
$238$		0			0
$239$	23.4164			1.51468			0.757341	−	0.653020i	$-0.226498\pi$
$239$	23.4164			1.51468			0.757341	+	0.653020i	$0.226498\pi$
$240$		0			0
$241$	22.9443			1.47797			0.738985	−	0.673722i	$-0.235305\pi$
$241$	22.9443			1.47797			0.738985	+	0.673722i	$0.235305\pi$
$242$		0			0
$243$	15.0902	−	3.90983i	0.968035	−	0.250816i
$244$		0			0
$245$			3.00000i			0.191663i
$246$		0			0
$247$		−	6.47214i		−	0.411812i
$248$		0			0
$249$	7.23607	+	18.9443i	0.458567	+	1.20054i
$250$		0			0
$251$	−10.4721			−0.660995			−0.330498	−	0.943807i	$-0.607217\pi$
$251$	−10.4721			−0.660995			−0.330498	+	0.943807i	$0.607217\pi$
$252$		0			0
$253$	−1.16718			−0.0733802
$254$		0			0
$255$	−0.472136	−	1.23607i	−0.0295663	−	0.0774056i
$256$		0			0
$257$			3.23607i			0.201860i	0.994894	+	0.100930i	$0.0321819\pi$
$257$			3.23607i			0.201860i	−0.994894	+	0.100930i	$0.967818\pi$
$258$		0			0
$259$		−	15.4164i		−	0.957929i
$260$		0			0
$261$	16.9443	+	18.9443i	1.04882	+	1.17262i
$262$		0			0
$263$	−23.8885			−1.47303			−0.736515	−	0.676421i	$-0.763530\pi$
$263$	−23.8885			−1.47303			−0.736515	+	0.676421i	$0.763530\pi$
$264$		0			0
$265$	4.47214			0.274721
$266$		0			0
$267$	2.47214	−	0.944272i	0.151292	−	0.0577885i
$268$		0			0
$269$			14.0000i			0.853595i	0.904347	+	0.426798i	$0.140358\pi$
$269$			14.0000i			0.853595i	−0.904347	+	0.426798i	$0.859642\pi$
$270$		0			0
$271$			28.1803i			1.71183i	0.517113	+	0.855917i	$0.327007\pi$
$271$			28.1803i			1.71183i	−0.517113	+	0.855917i	$0.672993\pi$
$272$		0			0
$273$	4.00000	−	1.52786i	0.242091	−	0.0924705i
$274$		0			0
$275$	2.47214			0.149075
$276$		0			0
$277$	−12.6525			−0.760214			−0.380107	−	0.924943i	$-0.624113\pi$
$277$	−12.6525			−0.760214			−0.380107	+	0.924943i	$0.624113\pi$
$278$		0			0
$279$	9.52786	+	10.6525i	0.570418	+	0.637747i
$280$		0			0
$281$			5.52786i			0.329765i	0.986313	+	0.164882i	$0.0527244\pi$
$281$			5.52786i			0.329765i	−0.986313	+	0.164882i	$0.947276\pi$
$282$		0			0
$283$			11.2361i			0.667915i	0.942588	+	0.333957i	$0.108384\pi$
$283$			11.2361i			0.667915i	−0.942588	+	0.333957i	$0.891616\pi$
$284$		0			0
$285$	−3.23607	−	8.47214i	−0.191688	−	0.501846i
$286$		0			0
$287$	−3.05573			−0.180374
$288$		0			0
$289$	16.4164			0.965671
$290$		0			0
$291$	−7.70820	−	20.1803i	−0.451863	−	1.18299i
$292$		0			0
$293$		−	29.4164i		−	1.71852i	−0.511535	−	0.859262i	$-0.670923\pi$
$293$		−	29.4164i		−	1.71852i	0.511535	−	0.859262i	$-0.329077\pi$
$294$		0			0
$295$		−	6.47214i		−	0.376822i
$296$		0			0
$297$	11.4164	+	5.88854i	0.662447	+	0.341688i
$298$		0			0
$299$	−0.583592			−0.0337500
$300$		0			0
$301$	−19.4164			−1.11914
$302$		0			0
$303$	−28.1803	+	10.7639i	−1.61892	+	0.618372i
$304$		0			0
$305$		−	12.4721i		−	0.714152i
$306$		0			0
$307$			17.1246i			0.977353i	0.872465	+	0.488677i	$0.162520\pi$
$307$			17.1246i			0.977353i	−0.872465	+	0.488677i	$0.837480\pi$
$308$		0			0
$309$	0.763932	−	0.291796i	0.0434586	−	0.0165997i
$310$		0			0
$311$	−20.0000			−1.13410			−0.567048	−	0.823685i	$-0.691915\pi$
$311$	−20.0000			−1.13410			−0.567048	+	0.823685i	$0.691915\pi$
$312$		0			0
$313$	23.8885			1.35026			0.675130	−	0.737699i	$-0.264087\pi$
$313$	23.8885			1.35026			0.675130	+	0.737699i	$0.264087\pi$
$314$		0			0
$315$	4.47214	−	4.00000i	0.251976	−	0.225374i
$316$		0			0
$317$			7.88854i			0.443065i	0.975153	+	0.221532i	$0.0711059\pi$
$317$			7.88854i			0.443065i	−0.975153	+	0.221532i	$0.928894\pi$
$318$		0			0
$319$			20.9443i			1.17265i
$320$		0			0
$321$	8.18034	+	21.4164i	0.456582	+	1.19535i
$322$		0			0
$323$	−4.00000			−0.222566
$324$		0			0
$325$	1.23607			0.0685647
$326$		0			0
$327$	−0.652476	−	1.70820i	−0.0360820	−	0.0944639i
$328$		0			0
$329$			8.94427i			0.493114i
$330$		0			0
$331$		−	2.76393i		−	0.151919i	−0.997111	−	0.0759597i	$-0.975798\pi$
$331$		−	2.76393i		−	0.151919i	0.997111	−	0.0759597i	$-0.0242020\pi$
$332$		0			0
$333$	17.2361	−	15.4164i	0.944531	−	0.844814i
$334$		0			0
$335$	−11.2361			−0.613892
$336$		0			0
$337$	7.52786			0.410069			0.205034	−	0.978755i	$-0.434269\pi$
$337$	7.52786			0.410069			0.205034	+	0.978755i	$0.434269\pi$
$338$		0			0
$339$	34.1803	−	13.0557i	1.85642	−	0.709090i
$340$		0			0
$341$			11.7771i			0.637765i
$342$		0			0
$343$			20.0000i			1.07990i
$344$		0			0
$345$	−0.763932	+	0.291796i	−0.0411287	+	0.0157098i
$346$		0			0
$347$	10.7639			0.577838			0.288919	−	0.957354i	$-0.406704\pi$
$347$	10.7639			0.577838			0.288919	+	0.957354i	$0.406704\pi$
$348$		0			0
$349$	17.0557			0.912972			0.456486	−	0.889731i	$-0.349108\pi$
$349$	17.0557			0.912972			0.456486	+	0.889731i	$0.349108\pi$
$350$		0			0
$351$	5.70820	+	2.94427i	0.304681	+	0.157154i
$352$		0			0
$353$			4.76393i			0.253559i	0.991931	+	0.126779i	$0.0404640\pi$
$353$			4.76393i			0.253559i	−0.991931	+	0.126779i	$0.959536\pi$
$354$		0			0
$355$			4.00000i			0.212298i
$356$		0			0
$357$	−0.944272	−	2.47214i	−0.0499762	−	0.130839i
$358$		0			0
$359$	−31.4164			−1.65809			−0.829047	−	0.559178i	$-0.811117\pi$
$359$	−31.4164			−1.65809			−0.829047	+	0.559178i	$0.811117\pi$
$360$		0			0
$361$	−8.41641			−0.442969
$362$		0			0
$363$	−3.02129	−	7.90983i	−0.158576	−	0.415158i
$364$		0			0
$365$		−	0.472136i		−	0.0247127i
$366$		0			0
$367$			27.3050i			1.42531i	0.701516	+	0.712653i	$0.252507\pi$
$367$			27.3050i			1.42531i	−0.701516	+	0.712653i	$0.747493\pi$
$368$		0			0
$369$	−3.05573	−	3.41641i	−0.159075	−	0.177851i
$370$		0			0
$371$	8.94427			0.464363
$372$		0			0
$373$	37.5967			1.94669			0.973343	−	0.229355i	$-0.0736616\pi$
$373$	37.5967			1.94669			0.973343	+	0.229355i	$0.0736616\pi$
$374$		0			0
$375$	1.61803	−	0.618034i	0.0835549	−	0.0319151i
$376$		0			0
$377$			10.4721i			0.539342i
$378$		0			0
$379$			8.29180i			0.425921i	0.977061	+	0.212960i	$0.0683106\pi$
$379$			8.29180i			0.425921i	−0.977061	+	0.212960i	$0.931689\pi$
$380$		0			0
$381$	−16.1803	+	6.18034i	−0.828944	+	0.316628i
$382$		0			0
$383$	−2.94427			−0.150445			−0.0752226	−	0.997167i	$-0.523967\pi$
$383$	−2.94427			−0.150445			−0.0752226	+	0.997167i	$0.523967\pi$
$384$		0			0
$385$	4.94427			0.251983
$386$		0			0
$387$	−19.4164	−	21.7082i	−0.986991	−	1.10349i
$388$		0			0
$389$			2.94427i			0.149281i	0.997211	+	0.0746403i	$0.0237808\pi$
$389$			2.94427i			0.149281i	−0.997211	+	0.0746403i	$0.976219\pi$
$390$		0			0
$391$			0.360680i			0.0182404i
$392$		0			0
$393$	−13.5279	−	35.4164i	−0.682390	−	1.78652i
$394$		0			0
$395$	8.18034			0.411598
$396$		0			0
$397$	2.18034			0.109428			0.0547141	−	0.998502i	$-0.482575\pi$
$397$	2.18034			0.109428			0.0547141	+	0.998502i	$0.482575\pi$
$398$		0			0
$399$	−6.47214	−	16.9443i	−0.324012	−	0.848275i
$400$		0			0
$401$			34.8328i			1.73947i	0.493521	+	0.869734i	$0.335710\pi$
$401$			34.8328i			1.73947i	−0.493521	+	0.869734i	$0.664290\pi$
$402$		0			0
$403$			5.88854i			0.293329i
$404$		0			0
$405$	8.94427	+	1.00000i	0.444444	+	0.0496904i
$406$		0			0
$407$	19.0557			0.944557
$408$		0			0
$409$	29.4164			1.45455			0.727274	−	0.686347i	$-0.240787\pi$
$409$	29.4164			1.45455			0.727274	+	0.686347i	$0.240787\pi$
$410$		0			0
$411$	−30.1803	+	11.5279i	−1.48869	+	0.568628i
$412$		0			0
$413$		−	12.9443i		−	0.636946i
$414$		0			0
$415$			11.7082i			0.574733i
$416$		0			0
$417$	−22.9443	+	8.76393i	−1.12359	+	0.429172i
$418$		0			0
$419$	−36.9443			−1.80485			−0.902423	−	0.430851i	$-0.858213\pi$
$419$	−36.9443			−1.80485			−0.902423	+	0.430851i	$0.858213\pi$
$420$		0			0
$421$	39.3050			1.91561			0.957803	−	0.287425i	$-0.0927993\pi$
$421$	39.3050			1.91561			0.957803	+	0.287425i	$0.0927993\pi$
$422$		0			0
$423$	−10.0000	+	8.94427i	−0.486217	+	0.434885i
$424$		0			0
$425$		−	0.763932i		−	0.0370561i
$426$		0			0
$427$		−	24.9443i		−	1.20714i
$428$		0			0
$429$	1.88854	+	4.94427i	0.0911798	+	0.238712i
$430$		0			0
$431$	28.0000			1.34871			0.674356	−	0.738406i	$-0.264421\pi$
$431$	28.0000			1.34871			0.674356	+	0.738406i	$0.264421\pi$
$432$		0			0
$433$	18.3607			0.882358			0.441179	−	0.897419i	$-0.354560\pi$
$433$	18.3607			0.882358			0.441179	+	0.897419i	$0.354560\pi$
$434$		0			0
$435$	5.23607	+	13.7082i	0.251050	+	0.657258i
$436$		0			0
$437$			2.47214i			0.118258i
$438$		0			0
$439$			6.65248i			0.317505i	0.987318	+	0.158753i	$0.0507472\pi$
$439$			6.65248i			0.317505i	−0.987318	+	0.158753i	$0.949253\pi$
$440$		0			0
$441$	−6.70820	+	6.00000i	−0.319438	+	0.285714i
$442$		0			0
$443$	32.6525			1.55137			0.775683	−	0.631123i	$-0.217406\pi$
$443$	32.6525			1.55137			0.775683	+	0.631123i	$0.217406\pi$
$444$		0			0
$445$	1.52786			0.0724277
$446$		0			0
$447$	−16.1803	+	6.18034i	−0.765304	+	0.292320i
$448$		0			0
$449$		−	19.0557i		−	0.899295i	−0.893206	−	0.449648i	$-0.851550\pi$
$449$		−	19.0557i		−	0.899295i	0.893206	−	0.449648i	$-0.148450\pi$
$450$		0			0
$451$		−	3.77709i		−	0.177856i
$452$		0			0
$453$	−5.23607	+	2.00000i	−0.246012	+	0.0939682i
$454$		0			0
$455$	2.47214			0.115896
$456$		0			0
$457$	29.4164			1.37604			0.688021	−	0.725691i	$-0.258480\pi$
$457$	29.4164			1.37604			0.688021	+	0.725691i	$0.258480\pi$
$458$		0			0
$459$	1.81966	−	3.52786i	0.0849345	−	0.164667i
$460$		0			0
$461$		−	8.47214i		−	0.394587i	−0.980344	−	0.197293i	$-0.936785\pi$
$461$		−	8.47214i		−	0.394587i	0.980344	−	0.197293i	$-0.0632151\pi$
$462$		0			0
$463$		−	18.3607i		−	0.853293i	−0.904418	−	0.426647i	$-0.859695\pi$
$463$		−	18.3607i		−	0.853293i	0.904418	−	0.426647i	$-0.140305\pi$
$464$		0			0
$465$	2.94427	+	7.70820i	0.136537	+	0.357459i
$466$		0			0
$467$	−3.34752			−0.154905			−0.0774525	−	0.996996i	$-0.524679\pi$
$467$	−3.34752			−0.154905			−0.0774525	+	0.996996i	$0.524679\pi$
$468$		0			0
$469$	−22.4721			−1.03767
$470$		0			0
$471$	7.23607	+	18.9443i	0.333420	+	0.872906i
$472$		0			0
$473$		−	24.0000i		−	1.10352i
$474$		0			0
$475$		−	5.23607i		−	0.240247i
$476$		0			0
$477$	8.94427	+	10.0000i	0.409530	+	0.457869i
$478$		0			0
$479$	−12.3607			−0.564774			−0.282387	−	0.959301i	$-0.591126\pi$
$479$	−12.3607			−0.564774			−0.282387	+	0.959301i	$0.591126\pi$
$480$		0			0
$481$	9.52786			0.434433
$482$		0			0
$483$	−1.52786	+	0.583592i	−0.0695202	+	0.0265544i
$484$		0			0
$485$		−	12.4721i		−	0.566331i
$486$		0			0
$487$			7.52786i			0.341120i	0.985347	+	0.170560i	$0.0545577\pi$
$487$			7.52786i			0.341120i	−0.985347	+	0.170560i	$0.945442\pi$
$488$		0			0
$489$	5.23607	−	2.00000i	0.236783	−	0.0904431i
$490$		0			0
$491$	20.0000			0.902587			0.451294	−	0.892375i	$-0.350963\pi$
$491$	20.0000			0.902587			0.451294	+	0.892375i	$0.350963\pi$
$492$		0			0
$493$	6.47214			0.291490
$494$		0			0
$495$	4.94427	+	5.52786i	0.222228	+	0.248459i
$496$		0			0
$497$			8.00000i			0.358849i
$498$		0			0
$499$			4.29180i			0.192127i	0.995375	+	0.0960636i	$0.0306252\pi$
$499$			4.29180i			0.192127i	−0.995375	+	0.0960636i	$0.969375\pi$
$500$		0			0
$501$	1.81966	+	4.76393i	0.0812964	+	0.212837i
$502$		0			0
$503$	−23.8885			−1.06514			−0.532569	−	0.846387i	$-0.678773\pi$
$503$	−23.8885			−1.06514			−0.532569	+	0.846387i	$0.678773\pi$
$504$		0			0
$505$	−17.4164			−0.775020
$506$		0			0
$507$	−7.09017	−	18.5623i	−0.314886	−	0.824381i
$508$		0			0
$509$			31.3050i			1.38757i	0.720183	+	0.693784i	$0.244058\pi$
$509$			31.3050i			1.38757i	−0.720183	+	0.693784i	$0.755942\pi$
$510$		0			0
$511$		−	0.944272i		−	0.0417721i
$512$		0			0
$513$	12.4721	−	24.1803i	0.550658	−	1.06759i
$514$		0			0
$515$	0.472136			0.0208048
$516$		0			0
$517$	−11.0557			−0.486230
$518$		0			0
$519$	33.1246	−	12.6525i	1.45401	−	0.555382i
$520$		0			0
$521$			20.0000i			0.876216i	0.898922	+	0.438108i	$0.144351\pi$
$521$			20.0000i			0.876216i	−0.898922	+	0.438108i	$0.855649\pi$
$522$		0			0
$523$			18.6525i			0.815616i	0.913068	+	0.407808i	$0.133707\pi$
$523$			18.6525i			0.815616i	−0.913068	+	0.407808i	$0.866293\pi$
$524$		0			0
$525$	3.23607	−	1.23607i	0.141234	−	0.0539464i
$526$		0			0
$527$	3.63932			0.158531
$528$		0			0
$529$	−22.7771			−0.990308
$530$		0			0
$531$	14.4721	−	12.9443i	0.628037	−	0.561734i
$532$		0			0
$533$		−	1.88854i		−	0.0818019i
$534$		0			0
$535$			13.2361i			0.572245i
$536$		0			0
$537$	−0.944272	−	2.47214i	−0.0407483	−	0.106681i
$538$		0			0
$539$	−7.41641			−0.319447
$540$		0			0
$541$	−18.3607			−0.789387			−0.394694	−	0.918813i	$-0.629149\pi$
$541$	−18.3607			−0.789387			−0.394694	+	0.918813i	$0.629149\pi$
$542$		0			0
$543$	−8.29180	−	21.7082i	−0.355835	−	0.931588i
$544$		0			0
$545$		−	1.05573i		−	0.0452224i
$546$		0			0
$547$		−	26.0689i		−	1.11462i	−0.830303	−	0.557312i	$-0.811833\pi$
$547$		−	26.0689i		−	1.11462i	0.830303	−	0.557312i	$-0.188167\pi$
$548$		0			0
$549$	27.8885	−	24.9443i	1.19025	−	1.06460i
$550$		0			0
$551$	44.3607			1.88983
$552$		0			0
$553$	16.3607			0.695727
$554$		0			0
$555$	12.4721	−	4.76393i	0.529413	−	0.202218i
$556$		0			0
$557$			44.2492i			1.87490i	0.348120	+	0.937450i	$0.386820\pi$
$557$			44.2492i			1.87490i	−0.348120	+	0.937450i	$0.613180\pi$
$558$		0			0
$559$		−	12.0000i		−	0.507546i
$560$		0			0
$561$	3.05573	−	1.16718i	0.129013	−	0.0492786i
$562$		0			0
$563$	44.0689			1.85728			0.928641	−	0.370980i	$-0.120978\pi$
$563$	44.0689			1.85728			0.928641	+	0.370980i	$0.120978\pi$
$564$		0			0
$565$	21.1246			0.888719
$566$		0			0
$567$	17.8885	+	2.00000i	0.751248	+	0.0839921i
$568$		0			0
$569$		−	19.0557i		−	0.798858i	−0.916764	−	0.399429i	$-0.869208\pi$
$569$		−	19.0557i		−	0.798858i	0.916764	−	0.399429i	$-0.130792\pi$
$570$		0			0
$571$		−	28.6525i		−	1.19907i	−0.800349	−	0.599534i	$-0.795352\pi$
$571$		−	28.6525i		−	1.19907i	0.800349	−	0.599534i	$-0.204648\pi$
$572$		0			0
$573$	8.94427	+	23.4164i	0.373652	+	0.978234i
$574$		0			0
$575$	−0.472136			−0.0196894
$576$		0			0
$577$	−10.3607			−0.431321			−0.215660	−	0.976468i	$-0.569190\pi$
$577$	−10.3607			−0.431321			−0.215660	+	0.976468i	$0.569190\pi$
$578$		0			0
$579$	1.81966	+	4.76393i	0.0756225	+	0.197982i
$580$		0			0
$581$			23.4164i			0.971476i
$582$		0			0
$583$			11.0557i			0.457881i
$584$		0			0
$585$	2.47214	+	2.76393i	0.102210	+	0.114275i
$586$		0			0
$587$	−12.6525			−0.522224			−0.261112	−	0.965309i	$-0.584089\pi$
$587$	−12.6525			−0.522224			−0.261112	+	0.965309i	$0.584089\pi$
$588$		0			0
$589$	24.9443			1.02781
$590$		0			0
$591$	−16.1803	+	6.18034i	−0.665570	+	0.254225i
$592$		0			0
$593$			4.76393i			0.195631i	0.995205	+	0.0978156i	$0.0311855\pi$
$593$			4.76393i			0.195631i	−0.995205	+	0.0978156i	$0.968814\pi$
$594$		0			0
$595$		−	1.52786i		−	0.0626363i
$596$		0			0
$597$	−28.6525	+	10.9443i	−1.17267	+	0.447919i
$598$		0			0
$599$	−16.5836			−0.677587			−0.338794	−	0.940861i	$-0.610019\pi$
$599$	−16.5836			−0.677587			−0.338794	+	0.940861i	$0.610019\pi$
$600$		0			0
$601$	−35.5279			−1.44921			−0.724606	−	0.689163i	$-0.757978\pi$
$601$	−35.5279			−1.44921			−0.724606	+	0.689163i	$0.757978\pi$
$602$		0			0
$603$	−22.4721	−	25.1246i	−0.915136	−	1.02315i
$604$		0			0
$605$		−	4.88854i		−	0.198748i
$606$		0			0
$607$		−	3.52786i		−	0.143192i	−0.997434	−	0.0715958i	$-0.977191\pi$
$607$		−	3.52786i		−	0.143192i	0.997434	−	0.0715958i	$-0.0228092\pi$
$608$		0			0
$609$	10.4721	+	27.4164i	0.424352	+	1.11097i
$610$		0			0
$611$	−5.52786			−0.223633
$612$		0			0
$613$	−10.7639			−0.434751			−0.217376	−	0.976088i	$-0.569750\pi$
$613$	−10.7639			−0.434751			−0.217376	+	0.976088i	$0.569750\pi$
$614$		0			0
$615$	−0.944272	−	2.47214i	−0.0380767	−	0.0996861i
$616$		0			0
$617$			3.23607i			0.130279i	0.997876	+	0.0651396i	$0.0207493\pi$
$617$			3.23607i			0.130279i	−0.997876	+	0.0651396i	$0.979251\pi$
$618$		0			0
$619$		−	32.6525i		−	1.31241i	−0.754581	−	0.656207i	$-0.772160\pi$
$619$		−	32.6525i		−	1.31241i	0.754581	−	0.656207i	$-0.227840\pi$
$620$		0			0
$621$	−2.18034	−	1.12461i	−0.0874940	−	0.0451291i
$622$		0			0
$623$	3.05573			0.122425
$624$		0			0
$625$	1.00000			0.0400000
$626$		0			0
$627$	20.9443	−	8.00000i	0.836434	−	0.319489i
$628$		0			0
$629$		−	5.88854i		−	0.234792i
$630$		0			0
$631$			2.29180i			0.0912350i	0.998959	+	0.0456175i	$0.0145255\pi$
$631$			2.29180i			0.0912350i	−0.998959	+	0.0456175i	$0.985474\pi$
$632$		0			0
$633$	−0.472136	+	0.180340i	−0.0187657	+	0.00716787i
$634$		0			0
$635$	−10.0000			−0.396838
$636$		0			0
$637$	−3.70820			−0.146924
$638$		0			0
$639$	−8.94427	+	8.00000i	−0.353830	+	0.316475i
$640$		0			0
$641$			37.3050i			1.47346i	0.676189	+	0.736729i	$0.263630\pi$
$641$			37.3050i			1.47346i	−0.676189	+	0.736729i	$0.736370\pi$
$642$		0			0
$643$		−	6.29180i		−	0.248124i	−0.992274	−	0.124062i	$-0.960408\pi$
$643$		−	6.29180i		−	0.248124i	0.992274	−	0.124062i	$-0.0395922\pi$
$644$		0			0
$645$	−6.00000	−	15.7082i	−0.236250	−	0.618510i
$646$		0			0
$647$	−0.111456			−0.00438179			−0.00219090	−	0.999998i	$-0.500697\pi$
$647$	−0.111456			−0.00438179			−0.00219090	+	0.999998i	$0.500697\pi$
$648$		0			0
$649$	16.0000			0.628055
$650$		0			0
$651$	5.88854	+	15.4164i	0.230790	+	0.604217i
$652$		0			0
$653$		−	27.5279i		−	1.07725i	−0.842546	−	0.538624i	$-0.818944\pi$
$653$		−	27.5279i		−	1.07725i	0.842546	−	0.538624i	$-0.181056\pi$
$654$		0			0
$655$		−	21.8885i		−	0.855256i
$656$		0			0
$657$	1.05573	−	0.944272i	0.0411879	−	0.0368396i
$658$		0			0
$659$	14.4721			0.563754			0.281877	−	0.959450i	$-0.409043\pi$
$659$	14.4721			0.563754			0.281877	+	0.959450i	$0.409043\pi$
$660$		0			0
$661$	40.4721			1.57418			0.787092	−	0.616836i	$-0.211586\pi$
$661$	40.4721			1.57418			0.787092	+	0.616836i	$0.211586\pi$
$662$		0			0
$663$	1.52786	−	0.583592i	0.0593373	−	0.0226648i
$664$		0			0
$665$		−	10.4721i		−	0.406092i
$666$		0			0
$667$		−	4.00000i		−	0.154881i
$668$		0			0
$669$	4.76393	−	1.81966i	0.184184	−	0.0703521i
$670$		0			0
$671$	30.8328			1.19029
$672$		0			0
$673$	−17.4164			−0.671353			−0.335677	−	0.941977i	$-0.608965\pi$
$673$	−17.4164			−0.671353			−0.335677	+	0.941977i	$0.608965\pi$
$674$		0			0
$675$	4.61803	+	2.38197i	0.177748	+	0.0916819i
$676$		0			0
$677$		−	14.3607i		−	0.551926i	−0.961168	−	0.275963i	$-0.911003\pi$
$677$		−	14.3607i		−	0.551926i	0.961168	−	0.275963i	$-0.0889967\pi$
$678$		0			0
$679$		−	24.9443i		−	0.957273i
$680$		0			0
$681$	−12.1803	−	31.8885i	−0.466752	−	1.22197i
$682$		0			0
$683$	37.2361			1.42480			0.712399	−	0.701774i	$-0.247609\pi$
$683$	37.2361			1.42480			0.712399	+	0.701774i	$0.247609\pi$
$684$		0			0
$685$	−18.6525			−0.712674
$686$		0			0
$687$	−9.23607	−	24.1803i	−0.352378	−	0.922538i
$688$		0			0
$689$			5.52786i			0.210595i
$690$		0			0
$691$			24.0689i			0.915623i	0.889049	+	0.457812i	$0.151367\pi$
$691$			24.0689i			0.915623i	−0.889049	+	0.457812i	$0.848633\pi$
$692$		0			0
$693$	9.88854	+	11.0557i	0.375635	+	0.419972i
$694$		0			0
$695$	−14.1803			−0.537891
$696$		0			0
$697$	−1.16718			−0.0442103
$698$		0			0
$699$	−15.7082	+	6.00000i	−0.594139	+	0.226941i
$700$		0			0
$701$			0.111456i			0.00420964i	0.999998	+	0.00210482i	$0.000669986\pi$
$701$			0.111456i			0.00420964i	−0.999998	+	0.00210482i	$0.999330\pi$
$702$		0			0
$703$		−	40.3607i		−	1.52223i
$704$		0			0
$705$	−7.23607	+	2.76393i	−0.272526	+	0.104096i
$706$		0			0
$707$	−34.8328			−1.31002
$708$		0			0
$709$	−10.0000			−0.375558			−0.187779	−	0.982211i	$-0.560129\pi$
$709$	−10.0000			−0.375558			−0.187779	+	0.982211i	$0.560129\pi$
$710$		0			0
$711$	16.3607	+	18.2918i	0.613573	+	0.685996i
$712$		0			0
$713$		−	2.24922i		−	0.0842341i
$714$		0			0
$715$			3.05573i			0.114278i
$716$		0			0
$717$	14.4721	+	37.8885i	0.540472	+	1.41497i
$718$		0			0
$719$	−19.0557			−0.710659			−0.355329	−	0.934741i	$-0.615631\pi$
$719$	−19.0557			−0.710659			−0.355329	+	0.934741i	$0.615631\pi$
$720$		0			0
$721$	0.944272			0.0351665
$722$		0			0
$723$	14.1803	+	37.1246i	0.527373	+	1.38068i
$724$		0			0
$725$			8.47214i			0.314647i
$726$		0			0
$727$		−	22.5836i		−	0.837579i	−0.908083	−	0.418790i	$-0.862455\pi$
$727$		−	22.5836i		−	0.837579i	0.908083	−	0.418790i	$-0.137545\pi$
$728$		0			0
$729$	15.6525	+	22.0000i	0.579721	+	0.814815i
$730$		0			0
$731$	−7.41641			−0.274306
$732$		0			0
$733$	13.2361			0.488885			0.244443	−	0.969664i	$-0.421395\pi$
$733$	13.2361			0.488885			0.244443	+	0.969664i	$0.421395\pi$
$734$		0			0
$735$	−4.85410	+	1.85410i	−0.179046	+	0.0683896i
$736$		0			0
$737$		−	27.7771i		−	1.02318i
$738$		0			0
$739$		−	0.652476i		−	0.0240017i	−0.999928	−	0.0120009i	$-0.996180\pi$
$739$		−	0.652476i		−	0.0240017i	0.999928	−	0.0120009i	$-0.00382008\pi$
$740$		0			0
$741$	10.4721	−	4.00000i	0.384704	−	0.146944i
$742$		0			0
$743$	6.58359			0.241529			0.120764	−	0.992681i	$-0.461465\pi$
$743$	6.58359			0.241529			0.120764	+	0.992681i	$0.461465\pi$
$744$		0			0
$745$	−10.0000			−0.366372
$746$		0			0
$747$	−26.1803	+	23.4164i	−0.957889	+	0.856762i
$748$		0			0
$749$			26.4721i			0.967271i
$750$		0			0
$751$			42.0689i			1.53512i	0.640980	+	0.767558i	$0.278528\pi$
$751$			42.0689i			1.53512i	−0.640980	+	0.767558i	$0.721472\pi$
$752$		0			0
$753$	−6.47214	−	16.9443i	−0.235858	−	0.617484i
$754$		0			0
$755$	−3.23607			−0.117773
$756$		0			0
$757$	−41.5967			−1.51186			−0.755930	−	0.654653i	$-0.772815\pi$
$757$	−41.5967			−1.51186			−0.755930	+	0.654653i	$0.772815\pi$
$758$		0			0
$759$	−0.721360	−	1.88854i	−0.0261837	−	0.0685498i
$760$		0			0
$761$		−	20.0000i		−	0.724999i	−0.931984	−	0.362500i	$-0.881923\pi$
$761$		−	20.0000i		−	0.724999i	0.931984	−	0.362500i	$-0.118077\pi$
$762$		0			0
$763$		−	2.11146i		−	0.0764398i
$764$		0			0
$765$	1.70820	−	1.52786i	0.0617602	−	0.0552400i
$766$		0			0
$767$	8.00000			0.288863
$768$		0			0
$769$	−30.3607			−1.09483			−0.547417	−	0.836860i	$-0.684389\pi$
$769$	−30.3607			−1.09483			−0.547417	+	0.836860i	$0.684389\pi$
$770$		0			0
$771$	−5.23607	+	2.00000i	−0.188572	+	0.0720282i
$772$		0			0
$773$		−	46.0000i		−	1.65451i	−0.561830	−	0.827253i	$-0.689903\pi$
$773$		−	46.0000i		−	1.65451i	0.561830	−	0.827253i	$-0.310097\pi$
$774$		0			0
$775$			4.76393i			0.171125i
$776$		0			0
$777$	24.9443	−	9.52786i	0.894871	−	0.341810i
$778$		0			0
$779$	−8.00000			−0.286630
$780$		0			0
$781$	−9.88854			−0.353840
$782$		0			0
$783$	−20.1803	+	39.1246i	−0.721187	+	1.39820i
$784$		0			0
$785$			11.7082i			0.417884i
$786$		0			0
$787$			5.70820i			0.203475i	0.994811	+	0.101738i	$0.0324403\pi$
$787$			5.70820i			0.203475i	−0.994811	+	0.101738i	$0.967560\pi$
$788$		0			0
$789$	−14.7639	−	38.6525i	−0.525610	−	1.37606i
$790$		0			0
$791$	42.2492			1.50221
$792$		0			0
$793$	15.4164			0.547453
$794$		0			0
$795$	2.76393	+	7.23607i	0.0980266	+	0.256637i
$796$		0			0
$797$		−	13.0557i		−	0.462458i	−0.972899	−	0.231229i	$-0.925725\pi$
$797$		−	13.0557i		−	0.462458i	0.972899	−	0.231229i	$-0.0742746\pi$
$798$		0			0
$799$			3.41641i			0.120864i
$800$		0			0
$801$	3.05573	+	3.41641i	0.107969	+	0.120713i
$802$		0			0
$803$	1.16718			0.0411890
$804$		0			0
$805$	−0.944272			−0.0332812
$806$		0			0
$807$	−22.6525	+	8.65248i	−0.797405	+	0.304582i
$808$		0			0
$809$		−	17.3050i		−	0.608410i	−0.952607	−	0.304205i	$-0.901609\pi$
$809$		−	17.3050i		−	0.608410i	0.952607	−	0.304205i	$-0.0983907\pi$
$810$		0			0
$811$			41.2361i			1.44799i	0.689803	+	0.723997i	$0.257697\pi$
$811$			41.2361i			1.44799i	−0.689803	+	0.723997i	$0.742303\pi$
$812$		0			0
$813$	−45.5967	+	17.4164i	−1.59915	+	0.610820i
$814$		0			0
$815$	3.23607			0.113355
$816$		0			0
$817$	−50.8328			−1.77842
$818$		0			0
$819$	4.94427	+	5.52786i	0.172767	+	0.193159i
$820$		0			0
$821$		−	47.8885i		−	1.67132i	−0.549246	−	0.835661i	$-0.685085\pi$
$821$		−	47.8885i		−	1.67132i	0.549246	−	0.835661i	$-0.314915\pi$
$822$		0			0
$823$		−	18.3607i		−	0.640013i	−0.947415	−	0.320007i	$-0.896315\pi$
$823$		−	18.3607i		−	0.640013i	0.947415	−	0.320007i	$-0.103685\pi$
$824$		0			0
$825$	1.52786	+	4.00000i	0.0531934	+	0.139262i
$826$		0			0
$827$	−27.1246			−0.943215			−0.471608	−	0.881809i	$-0.656326\pi$
$827$	−27.1246			−0.943215			−0.471608	+	0.881809i	$0.656326\pi$
$828$		0			0
$829$	−30.0000			−1.04194			−0.520972	−	0.853574i	$-0.674430\pi$
$829$	−30.0000			−1.04194			−0.520972	+	0.853574i	$0.674430\pi$
$830$		0			0
$831$	−7.81966	−	20.4721i	−0.271261	−	0.710171i
$832$		0			0
$833$			2.29180i			0.0794060i
$834$		0			0
$835$			2.94427i			0.101891i
$836$		0			0
$837$	−11.3475	+	22.0000i	−0.392228	+	0.760431i
$838$		0			0
$839$	−21.5279			−0.743224			−0.371612	−	0.928388i	$-0.621195\pi$
$839$	−21.5279			−0.743224			−0.371612	+	0.928388i	$0.621195\pi$
$840$		0			0
$841$	−42.7771			−1.47507
$842$		0			0
$843$	−8.94427	+	3.41641i	−0.308057	+	0.117667i
$844$		0			0
$845$		−	11.4721i		−	0.394653i
$846$		0			0
$847$		−	9.77709i		−	0.335945i
$848$		0			0
$849$	−18.1803	+	6.94427i	−0.623948	+	0.238327i
$850$		0			0
$851$	−3.63932			−0.124754
$852$		0			0
$853$	−0.875388			−0.0299727			−0.0149864	−	0.999888i	$-0.504770\pi$
$853$	−0.875388			−0.0299727			−0.0149864	+	0.999888i	$0.504770\pi$
$854$		0			0
$855$	11.7082	−	10.4721i	0.400412	−	0.358139i
$856$		0			0
$857$		−	55.2361i		−	1.88683i	−0.331617	−	0.943414i	$-0.607594\pi$
$857$		−	55.2361i		−	1.88683i	0.331617	−	0.943414i	$-0.392406\pi$
$858$		0			0
$859$		−	13.8197i		−	0.471521i	−0.971811	−	0.235760i	$-0.924242\pi$
$859$		−	13.8197i		−	0.471521i	0.971811	−	0.235760i	$-0.0757580\pi$
$860$		0			0
$861$	−1.88854	−	4.94427i	−0.0643614	−	0.168500i
$862$		0			0
$863$	34.3607			1.16965			0.584826	−	0.811159i	$-0.301163\pi$
$863$	34.3607			1.16965			0.584826	+	0.811159i	$0.301163\pi$
$864$		0			0
$865$	20.4721			0.696074
$866$		0			0
$867$	10.1459	+	26.5623i	0.344573	+	0.902103i
$868$		0			0
$869$			20.2229i			0.686015i
$870$		0			0
$871$		−	13.8885i		−	0.470595i
$872$		0			0
$873$	27.8885	−	24.9443i	0.943884	−	0.844236i
$874$		0			0
$875$	2.00000			0.0676123
$876$		0			0
$877$	19.7082			0.665499			0.332749	−	0.943015i	$-0.392024\pi$
$877$	19.7082			0.665499			0.332749	+	0.943015i	$0.392024\pi$
$878$		0			0
$879$	47.5967	−	18.1803i	1.60540	−	0.613208i
$880$		0			0
$881$			44.3607i			1.49455i	0.664515	+	0.747275i	$0.268638\pi$
$881$			44.3607i			1.49455i	−0.664515	+	0.747275i	$0.731362\pi$
$882$		0			0
$883$			8.54102i			0.287428i	0.989619	+	0.143714i	$0.0459046\pi$
$883$			8.54102i			0.287428i	−0.989619	+	0.143714i	$0.954095\pi$
$884$		0			0
$885$	10.4721	−	4.00000i	0.352017	−	0.134459i
$886$		0			0
$887$	−48.8328			−1.63965			−0.819823	−	0.572617i	$-0.805928\pi$
$887$	−48.8328			−1.63965			−0.819823	+	0.572617i	$0.805928\pi$
$888$		0			0
$889$	−20.0000			−0.670778
$890$		0			0
$891$	−2.47214	+	22.1115i	−0.0828197	+	0.740762i
$892$		0			0
$893$			23.4164i			0.783600i
$894$		0			0
$895$		−	1.52786i		−	0.0510709i
$896$		0			0
$897$	−0.360680	−	0.944272i	−0.0120427	−	0.0315283i
$898$		0			0
$899$	−40.3607			−1.34610
$900$		0			0
$901$	3.41641			0.113817
$902$		0			0
$903$	−12.0000	−	31.4164i	−0.399335	−	1.04547i
$904$		0			0
$905$		−	13.4164i		−	0.445976i
$906$		0			0
$907$		−	0.403252i		−	0.0133898i	−0.999978	−	0.00669489i	$-0.997869\pi$
$907$		−	0.403252i		−	0.0133898i	0.999978	−	0.00669489i	$-0.00213106\pi$
$908$		0			0
$909$	−34.8328	−	38.9443i	−1.15533	−	1.29170i
$910$		0			0
$911$	−7.05573			−0.233767			−0.116883	−	0.993146i	$-0.537290\pi$
$911$	−7.05573			−0.233767			−0.116883	+	0.993146i	$0.537290\pi$
$912$		0			0
$913$	−28.9443			−0.957916
$914$		0			0
$915$	20.1803	−	7.70820i	0.667141	−	0.254825i
$916$		0			0
$917$		−	43.7771i		−	1.44565i
$918$		0			0
$919$			21.7082i			0.716088i	0.933705	+	0.358044i	$0.116556\pi$
$919$			21.7082i			0.716088i	−0.933705	+	0.358044i	$0.883444\pi$
$920$		0			0
$921$	−27.7082	+	10.5836i	−0.913016	+	0.348741i
$922$		0			0
$923$	−4.94427			−0.162743
$924$		0			0
$925$	7.70820			0.253444
$926$		0			0
$927$	0.944272	+	1.05573i	0.0310140	+	0.0346747i
$928$		0			0
$929$		−	1.88854i		−	0.0619611i	−0.999520	−	0.0309806i	$-0.990137\pi$
$929$		−	1.88854i		−	0.0619611i	0.999520	−	0.0309806i	$-0.00986300\pi$
$930$		0			0
$931$			15.7082i			0.514816i
$932$		0			0
$933$	−12.3607	−	32.3607i	−0.404670	−	1.05944i
$934$		0			0
$935$	1.88854			0.0617620
$936$		0			0
$937$	35.8885			1.17243			0.586214	−	0.810156i	$-0.300618\pi$
$937$	35.8885			1.17243			0.586214	+	0.810156i	$0.300618\pi$
$938$		0			0
$939$	14.7639	+	38.6525i	0.481803	+	1.26138i
$940$		0			0
$941$			24.4721i			0.797769i	0.917001	+	0.398884i	$0.130603\pi$
$941$			24.4721i			0.797769i	−0.917001	+	0.398884i	$0.869397\pi$
$942$		0			0
$943$			0.721360i			0.0234907i
$944$		0			0
$945$	9.23607	+	4.76393i	0.300449	+	0.154971i
$946$		0			0
$947$	29.2361			0.950045			0.475022	−	0.879974i	$-0.342440\pi$
$947$	29.2361			0.950045			0.475022	+	0.879974i	$0.342440\pi$
$948$		0			0
$949$	0.583592			0.0189442
$950$		0			0
$951$	−12.7639	+	4.87539i	−0.413899	+	0.158095i
$952$		0			0
$953$		−	15.5967i		−	0.505228i	−0.967567	−	0.252614i	$-0.918710\pi$
$953$		−	15.5967i		−	0.505228i	0.967567	−	0.252614i	$-0.0812903\pi$
$954$		0			0
$955$			14.4721i			0.468307i
$956$		0			0
$957$	−33.8885	+	12.9443i	−1.09546	+	0.418429i
$958$		0			0
$959$	−37.3050			−1.20464
$960$		0			0
$961$	8.30495			0.267902
$962$		0			0
$963$	−29.5967	+	26.4721i	−0.953742	+	0.853053i
$964$		0			0
$965$			2.94427i			0.0947795i
$966$		0			0
$967$		−	32.4721i		−	1.04423i	−0.852874	−	0.522117i	$-0.825143\pi$
$967$		−	32.4721i		−	1.04423i	0.852874	−	0.522117i	$-0.174857\pi$
$968$		0			0
$969$	−2.47214	−	6.47214i	−0.0794164	−	0.207915i
$970$		0			0
$971$	18.4721			0.592799			0.296400	−	0.955064i	$-0.404214\pi$
$971$	18.4721			0.592799			0.296400	+	0.955064i	$0.404214\pi$
$972$		0			0
$973$	−28.3607			−0.909202
$974$		0			0
$975$	0.763932	+	2.00000i	0.0244654	+	0.0640513i
$976$		0			0
$977$			21.7082i			0.694507i	0.937771	+	0.347253i	$0.112886\pi$
$977$			21.7082i			0.694507i	−0.937771	+	0.347253i	$0.887114\pi$
$978$		0			0
$979$			3.77709i			0.120716i
$980$		0			0
$981$	2.36068	−	2.11146i	0.0753707	−	0.0674136i
$982$		0			0
$983$	44.2492			1.41133			0.705666	−	0.708545i	$-0.250648\pi$
$983$	44.2492			1.41133			0.705666	+	0.708545i	$0.250648\pi$
$984$		0			0
$985$	−10.0000			−0.318626
$986$		0			0
$987$	−14.4721	+	5.52786i	−0.460653	+	0.175954i
$988$		0			0
$989$			4.58359i			0.145750i
$990$		0			0
$991$		−	0.763932i		−	0.0242671i	−0.999926	−	0.0121336i	$-0.996138\pi$
$991$		−	0.763932i		−	0.0242671i	0.999926	−	0.0121336i	$-0.00386232\pi$
$992$		0			0
$993$	4.47214	−	1.70820i	0.141919	−	0.0542082i
$994$		0			0
$995$	−17.7082			−0.561388
$996$		0			0
$997$	33.8197			1.07108			0.535540	−	0.844510i	$-0.320108\pi$
$997$	33.8197			1.07108			0.535540	+	0.844510i	$0.320108\pi$
$998$		0			0
$999$	35.5967	+	18.3607i	1.12623	+	0.580906i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1920.2.h.f.1151.4	yes	4
3.2	odd	2		1920.2.h.l.1151.2	yes	4
4.3	odd	2		1920.2.h.l.1151.1	yes	4
8.3	odd	2		1920.2.h.e.1151.4	yes	4
8.5	even	2		1920.2.h.k.1151.1	yes	4
12.11	even	2	inner	1920.2.h.f.1151.3	yes	4
24.5	odd	2		1920.2.h.e.1151.3	✓	4
24.11	even	2		1920.2.h.k.1151.2	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1920.2.h.e.1151.3	✓	4	24.5	odd	2
1920.2.h.e.1151.4	yes	4	8.3	odd	2
1920.2.h.f.1151.3	yes	4	12.11	even	2	inner
1920.2.h.f.1151.4	yes	4	1.1	even	1	trivial
1920.2.h.k.1151.1	yes	4	8.5	even	2
1920.2.h.k.1151.2	yes	4	24.11	even	2
1920.2.h.l.1151.1	yes	4	4.3	odd	2
1920.2.h.l.1151.2	yes	4	3.2	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1920 = 2^{7} \cdot 3 \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1920.h (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(0.618034 + 1.61803i) q^{3} +1.00000i q^{5} +2.00000i q^{7} +(-2.23607 + 2.00000i) q^{9} -2.47214 q^{11} -1.23607 q^{13} +(-1.61803 + 0.618034i) q^{15} +0.763932i q^{17} +5.23607i q^{19} +(-3.23607 + 1.23607i) q^{21} +0.472136 q^{23} -1.00000 q^{25} +(-4.61803 - 2.38197i) q^{27} -8.47214i q^{29} -4.76393i q^{31} +(-1.52786 - 4.00000i) q^{33} -2.00000 q^{35} -7.70820 q^{37} +(-0.763932 - 2.00000i) q^{39} +1.52786i q^{41} +9.70820i q^{43} +(-2.00000 - 2.23607i) q^{45} +4.47214 q^{47} +3.00000 q^{49} +(-1.23607 + 0.472136i) q^{51} -4.47214i q^{53} -2.47214i q^{55} +(-8.47214 + 3.23607i) q^{57} -6.47214 q^{59} -12.4721 q^{61} +(-4.00000 - 4.47214i) q^{63} -1.23607i q^{65} +11.2361i q^{67} +(0.291796 + 0.763932i) q^{69} +4.00000 q^{71} -0.472136 q^{73} +(-0.618034 - 1.61803i) q^{75} -4.94427i q^{77} -8.18034i q^{79} +(1.00000 - 8.94427i) q^{81} +11.7082 q^{83} -0.763932 q^{85} +(13.7082 - 5.23607i) q^{87} -1.52786i q^{89} -2.47214i q^{91} +(7.70820 - 2.94427i) q^{93} -5.23607 q^{95} -12.4721 q^{97} +(5.52786 - 4.94427i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 2 q^{3} + 8 q^{11} + 4 q^{13} - 2 q^{15} - 4 q^{21} - 16 q^{23} - 4 q^{25} - 14 q^{27} - 24 q^{33} - 8 q^{35} - 4 q^{37} - 12 q^{39} - 8 q^{45} + 12 q^{49} + 4 q^{51} - 16 q^{57} - 8 q^{59} - 32 q^{61}+ \cdots + 40 q^{99}+O(q^{100}) \) 4 * q - 2 * q^3 + 8 * q^11 + 4 * q^13 - 2 * q^15 - 4 * q^21 - 16 * q^23 - 4 * q^25 - 14 * q^27 - 24 * q^33 - 8 * q^35 - 4 * q^37 - 12 * q^39 - 8 * q^45 + 12 * q^49 + 4 * q^51 - 16 * q^57 - 8 * q^59 - 32 * q^61 - 16 * q^63 + 28 * q^69 + 16 * q^71 + 16 * q^73 + 2 * q^75 + 4 * q^81 + 20 * q^83 - 12 * q^85 + 28 * q^87 + 4 * q^93 - 12 * q^95 - 32 * q^97 + 40 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more