LMFDB - Embedded newform 1620.2.d.e.649.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1620,2,Mod(649,1620)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1620, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1620.649");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1620 = 2^{2} \cdot 3^{4} \cdot 5 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1620.d (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:	$12.9357651274$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.28356903014400.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 3x^{6} + 20x^{4} - 75x^{2} + 625 $ x^8 - 3x^6 + 20x^4 - 75*x^2 + 625
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			649.6
Root			$-1.20635 - 1.88274i$ of defining polynomial
Character	$\chi$	$=$	1620.649
Dual form			1620.2.d.e.649.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.20635 + 1.88274i) q^{5} -1.28148i q^{7} +O(q^{10})$	$q+(1.20635 + 1.88274i) q^{5} -1.28148i q^{7} +4.14474 q^{11} -6.52202i q^{13} -5.98507i q^{17} -7.17891 q^{19} -7.53098i q^{23} +(-2.08945 + 4.54249i) q^{25} -5.19615 q^{29} +5.17891 q^{31} +(2.41269 - 1.54591i) q^{35} +5.24054i q^{37} +0.680643 q^{41} +1.28148i q^{43} -5.31139i q^{47} +5.35782 q^{49} +2.21958i q^{53} +(5.00000 + 7.80350i) q^{55} +7.60885 q^{59} -2.17891 q^{61} +(12.2793 - 7.86782i) q^{65} -15.6070i q^{67} +5.50603 q^{71} +7.80350i q^{73} -5.31139i q^{77} -6.00000 q^{79} -9.75056i q^{83} +(11.2684 - 7.22007i) q^{85} +10.0215 q^{89} -8.35782 q^{91} +(-8.66025 - 13.5161i) q^{95} +14.3255i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q+O(q^{10}) $ 8 * q	$ 8 q - 12 q^{19} + 6 q^{25} - 4 q^{31} - 48 q^{49} + 40 q^{55} + 28 q^{61} - 48 q^{79} + 22 q^{85} + 24 q^{91}+O(q^{100}) $ 8 * q - 12 * q^19 + 6 * q^25 - 4 * q^31 - 48 * q^49 + 40 * q^55 + 28 * q^61 - 48 * q^79 + 22 * q^85 + 24 * q^91

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1620\mathbb{Z}\right)^\times$.

$n$	$811$	$1297$	$1541$
$\chi(n)$	$1$	$-1$	$1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$	1.20635	+	1.88274i	0.539495	+	0.841989i
$5$	1.20635	+	1.88274i	0.539495	+	0.841989i
$6$		0			0
$7$		−	1.28148i		−	0.484353i	−0.970232	−	0.242176i	$-0.922139\pi$
$7$		−	1.28148i		−	0.484353i	0.970232	−	0.242176i	$-0.0778613\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	4.14474			1.24969			0.624844	−	0.780750i	$-0.285163\pi$
$11$	4.14474			1.24969			0.624844	+	0.780750i	$0.285163\pi$
$12$		0			0
$13$		−	6.52202i		−	1.80888i	−0.426598	−	0.904441i	$-0.640288\pi$
$13$		−	6.52202i		−	1.80888i	0.426598	−	0.904441i	$-0.359712\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$		−	5.98507i		−	1.45159i	−0.687909	−	0.725797i	$-0.741471\pi$
$17$		−	5.98507i		−	1.45159i	0.687909	−	0.725797i	$-0.258529\pi$
$18$		0			0
$19$	−7.17891			−1.64695			−0.823477	−	0.567349i	$-0.807969\pi$
$19$	−7.17891			−1.64695			−0.823477	+	0.567349i	$0.807969\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$		−	7.53098i		−	1.57032i	−0.619295	−	0.785159i	$-0.712581\pi$
$23$		−	7.53098i		−	1.57032i	0.619295	−	0.785159i	$-0.287419\pi$
$24$		0			0
$25$	−2.08945	+	4.54249i	−0.417891	+	0.908497i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−5.19615			−0.964901			−0.482451	−	0.875923i	$-0.660253\pi$
$29$	−5.19615			−0.964901			−0.482451	+	0.875923i	$0.660253\pi$
$30$		0			0
$31$	5.17891			0.930159			0.465080	−	0.885269i	$-0.346026\pi$
$31$	5.17891			0.930159			0.465080	+	0.885269i	$0.346026\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	2.41269	−	1.54591i	0.407820	−	0.261306i
$36$		0			0
$37$			5.24054i			0.861540i	0.902462	+	0.430770i	$0.141758\pi$
$37$			5.24054i			0.861540i	−0.902462	+	0.430770i	$0.858242\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	0.680643			0.106299			0.0531493	−	0.998587i	$-0.483074\pi$
$41$	0.680643			0.106299			0.0531493	+	0.998587i	$0.483074\pi$
$42$		0			0
$43$			1.28148i			0.195423i	0.995215	+	0.0977117i	$0.0311523\pi$
$43$			1.28148i			0.195423i	−0.995215	+	0.0977117i	$0.968848\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		−	5.31139i		−	0.774747i	−0.921923	−	0.387373i	$-0.873382\pi$
$47$		−	5.31139i		−	0.774747i	0.921923	−	0.387373i	$-0.126618\pi$
$48$		0			0
$49$	5.35782			0.765402
$50$		0			0
$51$		0			0
$52$		0			0
$53$			2.21958i			0.304883i	0.988312	+	0.152442i	$0.0487136\pi$
$53$			2.21958i			0.304883i	−0.988312	+	0.152442i	$0.951286\pi$
$54$		0			0
$55$	5.00000	+	7.80350i	0.674200	+	1.05222i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	7.60885			0.990587			0.495294	−	0.868726i	$-0.335060\pi$
$59$	7.60885			0.990587			0.495294	+	0.868726i	$0.335060\pi$
$60$		0			0
$61$	−2.17891			−0.278981			−0.139490	−	0.990223i	$-0.544546\pi$
$61$	−2.17891			−0.278981			−0.139490	+	0.990223i	$0.544546\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	12.2793	−	7.86782i	1.52306	−	0.975883i
$66$		0			0
$67$		−	15.6070i		−	1.90670i	−0.301871	−	0.953349i	$-0.597611\pi$
$67$		−	15.6070i		−	1.90670i	0.301871	−	0.953349i	$-0.402389\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	5.50603			0.653446			0.326723	−	0.945120i	$-0.394056\pi$
$71$	5.50603			0.653446			0.326723	+	0.945120i	$0.394056\pi$
$72$		0			0
$73$			7.80350i			0.913330i	0.889639	+	0.456665i	$0.150956\pi$
$73$			7.80350i			0.913330i	−0.889639	+	0.456665i	$0.849044\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$		−	5.31139i		−	0.605290i
$78$		0			0
$79$	−6.00000			−0.675053			−0.337526	−	0.941316i	$-0.609590\pi$
$79$	−6.00000			−0.675053			−0.337526	+	0.941316i	$0.609590\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$		−	9.75056i		−	1.07026i	−0.844769	−	0.535132i	$-0.820262\pi$
$83$		−	9.75056i		−	1.07026i	0.844769	−	0.535132i	$-0.179738\pi$
$84$		0			0
$85$	11.2684	−	7.22007i	1.22223	−	0.783127i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	10.0215			1.06228			0.531141	−	0.847284i	$-0.321764\pi$
$89$	10.0215			1.06228			0.531141	+	0.847284i	$0.321764\pi$
$90$		0			0
$91$	−8.35782			−0.876137
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−8.66025	−	13.5161i	−0.888523	−	1.38672i
$96$		0			0
$97$			14.3255i			1.45454i	0.686354	+	0.727268i	$0.259210\pi$
$97$			14.3255i			1.45454i	−0.686354	+	0.727268i	$0.740790\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−0.680643			−0.0677265			−0.0338633	−	0.999426i	$-0.510781\pi$
$101$	−0.680643			−0.0677265			−0.0338633	+	0.999426i	$0.510781\pi$
$102$		0			0
$103$			2.56295i			0.252535i	0.991996	+	0.126268i	$0.0402998\pi$
$103$			2.56295i			0.252535i	−0.991996	+	0.126268i	$0.959700\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		0			0		1.00000			$0$
$107$		0			0		−1.00000			$\pi$
$108$		0			0
$109$	−7.00000			−0.670478			−0.335239	−	0.942133i	$-0.608817\pi$
$109$	−7.00000			−0.670478			−0.335239	+	0.942133i	$0.608817\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$			9.07688i			0.853881i	0.904280	+	0.426941i	$0.140409\pi$
$113$			9.07688i			0.853881i	−0.904280	+	0.426941i	$0.859591\pi$
$114$		0			0
$115$	14.1789	−	9.08497i	1.32219	−	0.847178i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−7.66973			−0.703083
$120$		0			0
$121$	6.17891			0.561719
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−11.0729	+	1.54591i	−0.990395	+	0.138270i
$126$		0			0
$127$		−	1.28148i		−	0.113713i	−0.998382	−	0.0568563i	$-0.981892\pi$
$127$		−	1.28148i		−	0.113713i	0.998382	−	0.0568563i	$-0.0181077\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	5.50603			0.481064			0.240532	−	0.970641i	$-0.422678\pi$
$131$	5.50603			0.481064			0.240532	+	0.970641i	$0.422678\pi$
$132$		0			0
$133$			9.19961i			0.797707i
$134$		0			0
$135$		0			0
$136$		0			0
$137$			11.2965i			0.965122i	0.875862	+	0.482561i	$0.160293\pi$
$137$			11.2965i			0.965122i	−0.875862	+	0.482561i	$0.839707\pi$
$138$		0			0
$139$	−17.1789			−1.45710			−0.728548	−	0.684995i	$-0.759804\pi$
$139$	−17.1789			−1.45710			−0.728548	+	0.684995i	$0.759804\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		−	27.0321i		−	2.26054i
$144$		0			0
$145$	−6.26836	−	9.78303i	−0.520559	−	0.812436i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	19.7332			1.61661			0.808303	−	0.588766i	$-0.200386\pi$
$149$	19.7332			1.61661			0.808303	+	0.588766i	$0.200386\pi$
$150$		0			0
$151$	5.17891			0.421454			0.210727	−	0.977545i	$-0.432417\pi$
$151$	5.17891			0.421454			0.210727	+	0.977545i	$0.432417\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	6.24756	+	9.75056i	0.501816	+	0.783184i
$156$		0			0
$157$		−	5.24054i		−	0.418241i	−0.977890	−	0.209120i	$-0.932940\pi$
$157$		−	5.24054i		−	0.418241i	0.977890	−	0.209120i	$-0.0670600\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−9.65078			−0.760588
$162$		0			0
$163$			11.7626i			0.921315i	0.887578	+	0.460657i	$0.152386\pi$
$163$			11.7626i			0.921315i	−0.887578	+	0.460657i	$0.847614\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		−	2.21958i		−	0.171757i	−0.996306	−	0.0858783i	$-0.972630\pi$
$167$		−	2.21958i		−	0.171757i	0.996306	−	0.0858783i	$-0.0273696\pi$
$168$		0			0
$169$	−29.5367			−2.27206
$170$		0			0
$171$		0			0
$172$		0			0
$173$			0.673677i			0.0512187i	0.999672	+	0.0256094i	$0.00815261\pi$
$173$			0.673677i			0.0512187i	−0.999672	+	0.0256094i	$0.991847\pi$
$174$		0			0
$175$	5.82109	+	2.67759i	0.440033	+	0.202407i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	14.5370			1.08655			0.543275	−	0.839555i	$-0.317184\pi$
$179$	14.5370			1.08655			0.543275	+	0.839555i	$0.317184\pi$
$180$		0			0
$181$	15.1789			1.12824			0.564120	−	0.825693i	$-0.309216\pi$
$181$	15.1789			1.12824			0.564120	+	0.825693i	$0.309216\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−9.86660	+	6.32191i	−0.725407	+	0.464796i
$186$		0			0
$187$		−	24.8066i		−	1.81404i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	26.2906			1.90232			0.951162	−	0.308692i	$-0.0998913\pi$
$191$	26.2906			1.90232			0.951162	+	0.308692i	$0.0998913\pi$
$192$		0			0
$193$		−	12.9294i		−	0.930679i	−0.885132	−	0.465339i	$-0.845932\pi$
$193$		−	12.9294i		−	0.930679i	0.885132	−	0.465339i	$-0.154068\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$			8.20466i			0.584558i	0.956333	+	0.292279i	$0.0944135\pi$
$197$			8.20466i			0.584558i	−0.956333	+	0.292279i	$0.905586\pi$
$198$		0			0
$199$	2.35782			0.167141			0.0835706	−	0.996502i	$-0.473368\pi$
$199$	2.35782			0.167141			0.0835706	+	0.996502i	$0.473368\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$			6.65875i			0.467353i
$204$		0			0
$205$	0.821092	+	1.28148i	0.0573475	+	0.0895022i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−29.7547			−2.05818
$210$		0			0
$211$	4.82109			0.331898			0.165949	−	0.986134i	$-0.446931\pi$
$211$	4.82109			0.331898			0.165949	+	0.986134i	$0.446931\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−2.41269	+	1.54591i	−0.164544	+	0.105430i
$216$		0			0
$217$		−	6.63665i		−	0.450525i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−39.0348			−2.62576
$222$		0			0
$223$		−	7.91813i		−	0.530237i	−0.964216	−	0.265119i	$-0.914589\pi$
$223$		−	7.91813i		−	0.530237i	0.964216	−	0.265119i	$-0.0854111\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$		−	21.7207i		−	1.44165i	−0.693115	−	0.720827i	$-0.743762\pi$
$227$		−	21.7207i		−	1.44165i	0.693115	−	0.720827i	$-0.256238\pi$
$228$		0			0
$229$	−18.1789			−1.20130			−0.600648	−	0.799514i	$-0.705091\pi$
$229$	−18.1789			−1.20130			−0.600648	+	0.799514i	$0.705091\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$			21.0470i			1.37884i	0.724363	+	0.689418i	$0.242134\pi$
$233$			21.0470i			1.37884i	−0.724363	+	0.689418i	$0.757866\pi$
$234$		0			0
$235$	10.0000	−	6.40739i	0.652328	−	0.417972i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	22.1459			1.43250			0.716249	−	0.697844i	$-0.245857\pi$
$239$	22.1459			1.43250			0.716249	+	0.697844i	$0.245857\pi$
$240$		0			0
$241$	19.0000			1.22390			0.611949	−	0.790897i	$-0.290386\pi$
$241$	19.0000			1.22390			0.611949	+	0.790897i	$0.290386\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	6.46339	+	10.0874i	0.412931	+	0.644460i
$246$		0			0
$247$			46.8210i			2.97915i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−17.3205			−1.09326			−0.546630	−	0.837374i	$-0.684090\pi$
$251$	−17.3205			−1.09326			−0.546630	+	0.837374i	$0.684090\pi$
$252$		0			0
$253$		−	31.2140i		−	1.96241i
$254$		0			0
$255$		0			0
$256$		0			0
$257$			6.85730i			0.427747i	0.976861	+	0.213873i	$0.0686080\pi$
$257$			6.85730i			0.427747i	−0.976861	+	0.213873i	$0.931392\pi$
$258$		0			0
$259$	6.71563			0.417289
$260$		0			0
$261$		0			0
$262$		0			0
$263$			15.0620i			0.928760i	0.885636	+	0.464380i	$0.153723\pi$
$263$			15.0620i			0.928760i	−0.885636	+	0.464380i	$0.846277\pi$
$264$		0			0
$265$	−4.17891	+	2.67759i	−0.256708	+	0.164483i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−9.28001			−0.565812			−0.282906	−	0.959148i	$-0.591298\pi$
$269$	−9.28001			−0.565812			−0.282906	+	0.959148i	$0.591298\pi$
$270$		0			0
$271$	0.357817			0.0217358			0.0108679	−	0.999941i	$-0.496541\pi$
$271$	0.357817			0.0217358			0.0108679	+	0.999941i	$0.496541\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−8.66025	+	18.8274i	−0.522233	+	1.13534i
$276$		0			0
$277$			11.7626i			0.706744i	0.935483	+	0.353372i	$0.114965\pi$
$277$			11.7626i			0.706744i	−0.935483	+	0.353372i	$0.885035\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−7.23808			−0.431788			−0.215894	−	0.976417i	$-0.569267\pi$
$281$	−7.23808			−0.431788			−0.215894	+	0.976417i	$0.569267\pi$
$282$		0			0
$283$			28.6510i			1.70313i	0.524252	+	0.851563i	$0.324345\pi$
$283$			28.6510i			1.70313i	−0.524252	+	0.851563i	$0.675655\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		−	0.872228i		−	0.0514860i
$288$		0			0
$289$	−18.8211			−1.10712
$290$		0			0
$291$		0			0
$292$		0			0
$293$		−	17.9552i		−	1.04895i	−0.851424	−	0.524477i	$-0.824261\pi$
$293$		−	17.9552i		−	1.04895i	0.851424	−	0.524477i	$-0.175739\pi$
$294$		0			0
$295$	9.17891	+	14.3255i	0.534417	+	0.834064i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−49.1172			−2.84052
$300$		0			0
$301$	1.64218			0.0946539
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−2.62852	−	4.10233i	−0.150509	−	0.234899i
$306$		0			0
$307$			16.8885i			0.963876i	0.876205	+	0.481938i	$0.160067\pi$
$307$			16.8885i			0.963876i	−0.876205	+	0.481938i	$0.839933\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	8.97013			0.508650			0.254325	−	0.967119i	$-0.418147\pi$
$311$	8.97013			0.508650			0.254325	+	0.967119i	$0.418147\pi$
$312$		0			0
$313$		−	9.08497i		−	0.513513i	−0.966476	−	0.256757i	$-0.917346\pi$
$313$		−	9.08497i		−	0.513513i	0.966476	−	0.256757i	$-0.0826538\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$			12.1687i			0.683462i	0.939798	+	0.341731i	$0.111013\pi$
$317$			12.1687i			0.683462i	−0.939798	+	0.341731i	$0.888987\pi$
$318$		0			0
$319$	−21.5367			−1.20583
$320$		0			0
$321$		0			0
$322$		0			0
$323$			42.9663i			2.39071i
$324$		0			0
$325$	29.6262	+	13.6275i	1.64336	+	0.755915i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−6.80643			−0.375251
$330$		0			0
$331$	1.53673			0.0844660			0.0422330	−	0.999108i	$-0.486553\pi$
$331$	1.53673			0.0844660			0.0422330	+	0.999108i	$0.486553\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	29.3840	−	18.8274i	1.60542	−	1.02865i
$336$		0			0
$337$			2.56295i			0.139613i	0.997561	+	0.0698065i	$0.0222382\pi$
$337$			2.56295i			0.139613i	−0.997561	+	0.0698065i	$0.977762\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	21.4653			1.16241
$342$		0			0
$343$		−	15.8363i		−	0.855078i
$344$		0			0
$345$		0			0
$346$		0			0
$347$		−	2.21958i		−	0.119153i	−0.998224	−	0.0595767i	$-0.981025\pi$
$347$		−	2.21958i		−	0.119153i	0.998224	−	0.0595767i	$-0.0189751\pi$
$348$		0			0
$349$	−8.82109			−0.472182			−0.236091	−	0.971731i	$-0.575866\pi$
$349$	−8.82109			−0.472182			−0.236091	+	0.971731i	$0.575866\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$			27.9043i			1.48520i	0.669737	+	0.742599i	$0.266407\pi$
$353$			27.9043i			1.48520i	−0.669737	+	0.742599i	$0.733593\pi$
$354$		0			0
$355$	6.64218	+	10.3664i	0.352530	+	0.550194i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	19.3624			1.02191			0.510955	−	0.859607i	$-0.329292\pi$
$359$	19.3624			1.02191			0.510955	+	0.859607i	$0.329292\pi$
$360$		0			0
$361$	32.5367			1.71246
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−14.6920	+	9.41372i	−0.769014	+	0.492737i
$366$		0			0
$367$			2.56295i			0.133785i	0.997760	+	0.0668926i	$0.0213085\pi$
$367$			2.56295i			0.133785i	−0.997760	+	0.0668926i	$0.978692\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	2.84434			0.147671
$372$		0			0
$373$			16.8885i			0.874452i	0.899352	+	0.437226i	$0.144039\pi$
$373$			16.8885i			0.874452i	−0.899352	+	0.437226i	$0.855961\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$			33.8894i			1.74539i
$378$		0			0
$379$	−22.3578			−1.14844			−0.574222	−	0.818700i	$-0.694695\pi$
$379$	−22.3578			−1.14844			−0.574222	+	0.818700i	$0.694695\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$			11.9701i			0.611646i	0.952088	+	0.305823i	$0.0989316\pi$
$383$			11.9701i			0.611646i	−0.952088	+	0.305823i	$0.901068\pi$
$384$		0			0
$385$	10.0000	−	6.40739i	0.509647	−	0.326551i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−12.4951			−0.633528			−0.316764	−	0.948504i	$-0.602596\pi$
$389$	−12.4951			−0.633528			−0.316764	+	0.948504i	$0.602596\pi$
$390$		0			0
$391$	−45.0735			−2.27946
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−7.23808	−	11.2965i	−0.364187	−	0.568387i
$396$		0			0
$397$		−	24.6920i		−	1.23925i	−0.784896	−	0.619627i	$-0.787284\pi$
$397$		−	24.6920i		−	1.23925i	0.784896	−	0.619627i	$-0.212716\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	12.0635			0.602421			0.301210	−	0.953558i	$-0.402609\pi$
$401$	12.0635			0.602421			0.301210	+	0.953558i	$0.402609\pi$
$402$		0			0
$403$		−	33.7769i		−	1.68255i
$404$		0			0
$405$		0			0
$406$		0			0
$407$			21.7207i			1.07666i
$408$		0			0
$409$	−18.1789			−0.898889			−0.449445	−	0.893308i	$-0.648378\pi$
$409$	−18.1789			−0.898889			−0.449445	+	0.893308i	$0.648378\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		−	9.75056i		−	0.479794i
$414$		0			0
$415$	18.3578	−	11.7626i	0.901150	−	0.577401i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−9.65078			−0.471471			−0.235736	−	0.971817i	$-0.575750\pi$
$419$	−9.65078			−0.471471			−0.235736	+	0.971817i	$0.575750\pi$
$420$		0			0
$421$	−15.7156			−0.765933			−0.382967	−	0.923762i	$-0.625098\pi$
$421$	−15.7156			−0.765933			−0.382967	+	0.923762i	$0.625098\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	27.1871	+	12.5055i	1.31877	+	0.606608i
$426$		0			0
$427$			2.79222i			0.135125i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−31.1160			−1.49881			−0.749403	−	0.662114i	$-0.769659\pi$
$431$	−31.1160			−1.49881			−0.749403	+	0.662114i	$0.769659\pi$
$432$		0			0
$433$		−	16.7738i		−	0.806099i	−0.915178	−	0.403050i	$-0.867950\pi$
$433$		−	16.7738i		−	0.806099i	0.915178	−	0.403050i	$-0.132050\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$			54.0642i			2.58624i
$438$		0			0
$439$	15.5367			0.741527			0.370764	−	0.928727i	$-0.379096\pi$
$439$	15.5367			0.741527			0.370764	+	0.928727i	$0.379096\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$		−	20.3734i		−	0.967967i	−0.875077	−	0.483984i	$-0.839189\pi$
$443$		−	20.3734i		−	0.967967i	0.875077	−	0.483984i	$-0.160811\pi$
$444$		0			0
$445$	12.0895	+	18.8680i	0.573095	+	0.894429i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−33.8386			−1.59694			−0.798471	−	0.602033i	$-0.794358\pi$
$449$	−33.8386			−1.59694			−0.798471	+	0.602033i	$0.794358\pi$
$450$		0			0
$451$	2.82109			0.132840
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−10.0824	−	15.7356i	−0.472671	−	0.737698i
$456$		0			0
$457$		−	0.114633i		−	0.00536233i	−0.999996	−	0.00268116i	$-0.999147\pi$
$457$		−	0.114633i		−	0.00536233i	0.999996	−	0.00268116i	$-0.000853442\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−8.97013			−0.417781			−0.208890	−	0.977939i	$-0.566985\pi$
$461$	−8.97013			−0.417781			−0.208890	+	0.977939i	$0.566985\pi$
$462$		0			0
$463$		−	7.68886i		−	0.357332i	−0.983910	−	0.178666i	$-0.942822\pi$
$463$		−	7.68886i		−	0.357332i	0.983910	−	0.178666i	$-0.0571781\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		−	29.2517i		−	1.35361i	−0.736164	−	0.676803i	$-0.763365\pi$
$467$		−	29.2517i		−	1.35361i	0.736164	−	0.676803i	$-0.236635\pi$
$468$		0			0
$469$	−20.0000			−0.923514
$470$		0			0
$471$		0			0
$472$		0			0
$473$			5.31139i			0.244218i
$474$		0			0
$475$	15.0000	−	32.6101i	0.688247	−	1.49625i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−31.8576			−1.45561			−0.727804	−	0.685785i	$-0.759459\pi$
$479$	−31.8576			−1.45561			−0.727804	+	0.685785i	$0.759459\pi$
$480$		0			0
$481$	34.1789			1.55842
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−26.9713	+	17.2815i	−1.22470	+	0.784714i
$486$		0			0
$487$		−	33.7769i		−	1.53058i	−0.643686	−	0.765290i	$-0.722596\pi$
$487$		−	33.7769i		−	1.53058i	0.643686	−	0.765290i	$-0.277404\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−23.4463			−1.05812			−0.529058	−	0.848586i	$-0.677455\pi$
$491$	−23.4463			−1.05812			−0.529058	+	0.848586i	$0.677455\pi$
$492$		0			0
$493$			31.0993i			1.40064i
$494$		0			0
$495$		0			0
$496$		0			0
$497$		−	7.05585i		−	0.316498i
$498$		0			0
$499$	7.17891			0.321372			0.160686	−	0.987006i	$-0.448629\pi$
$499$	7.17891			0.321372			0.160686	+	0.987006i	$0.448629\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		−	40.7467i		−	1.81681i	−0.418096	−	0.908403i	$-0.637302\pi$
$503$		−	40.7467i		−	1.81681i	0.418096	−	0.908403i	$-0.362698\pi$
$504$		0			0
$505$	−0.821092	−	1.28148i	−0.0365381	−	0.0570250i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	31.7967			1.40936			0.704681	−	0.709524i	$-0.251090\pi$
$509$	31.7967			1.40936			0.704681	+	0.709524i	$0.251090\pi$
$510$		0			0
$511$	10.0000			0.442374
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−4.82539	+	3.09181i	−0.212632	+	0.136242i
$516$		0			0
$517$		−	22.0144i		−	0.968191i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	22.1459			0.970229			0.485115	−	0.874451i	$-0.338778\pi$
$521$	22.1459			0.970229			0.485115	+	0.874451i	$0.338778\pi$
$522$		0			0
$523$			41.6951i			1.82320i	0.411081	+	0.911599i	$0.365151\pi$
$523$			41.6951i			1.82320i	−0.411081	+	0.911599i	$0.634849\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		−	30.9961i		−	1.35021i
$528$		0			0
$529$	−33.7156			−1.46590
$530$		0			0
$531$		0			0
$532$		0			0
$533$		−	4.43917i		−	0.192282i
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$	22.2068			0.956514
$540$		0			0
$541$	−19.3578			−0.832258			−0.416129	−	0.909306i	$-0.636613\pi$
$541$	−19.3578			−0.832258			−0.416129	+	0.909306i	$0.636613\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−8.44443	−	13.1792i	−0.361720	−	0.564535i
$546$		0			0
$547$		−	5.12591i		−	0.219168i	−0.993978	−	0.109584i	$-0.965048\pi$
$547$		−	5.12591i		−	0.219168i	0.993978	−	0.109584i	$-0.0349519\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	37.3027			1.58915
$552$		0			0
$553$			7.68886i			0.326964i
$554$		0			0
$555$		0			0
$556$		0			0
$557$			2.41813i			0.102460i	0.998687	+	0.0512298i	$0.0163141\pi$
$557$			2.41813i			0.102460i	−0.998687	+	0.0512298i	$0.983686\pi$
$558$		0			0
$559$	8.35782			0.353498
$560$		0			0
$561$		0			0
$562$		0			0
$563$			5.31139i			0.223849i	0.993717	+	0.111924i	$0.0357014\pi$
$563$			5.31139i			0.223849i	−0.993717	+	0.111924i	$0.964299\pi$
$564$		0			0
$565$	−17.0895	+	10.9499i	−0.718959	+	0.460665i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−21.7751			−0.912861			−0.456430	−	0.889759i	$-0.650872\pi$
$569$	−21.7751			−0.912861			−0.456430	+	0.889759i	$0.650872\pi$
$570$		0			0
$571$	−6.82109			−0.285454			−0.142727	−	0.989762i	$-0.545587\pi$
$571$	−6.82109			−0.285454			−0.142727	+	0.989762i	$0.545587\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	34.2094	+	15.7356i	1.42663	+	0.656221i
$576$		0			0
$577$			5.24054i			0.218167i	0.994033	+	0.109083i	$0.0347915\pi$
$577$			5.24054i			0.218167i	−0.994033	+	0.109083i	$0.965208\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−12.4951			−0.518385
$582$		0			0
$583$			9.19961i			0.381009i
$584$		0			0
$585$		0			0
$586$		0			0
$587$			47.4055i			1.95663i	0.207117	+	0.978316i	$0.433592\pi$
$587$			47.4055i			1.95663i	−0.207117	+	0.978316i	$0.566408\pi$
$588$		0			0
$589$	−37.1789			−1.53193
$590$		0			0
$591$		0			0
$592$		0			0
$593$		−	11.2965i		−	0.463890i	−0.972729	−	0.231945i	$-0.925491\pi$
$593$		−	11.2965i		−	0.463890i	0.972729	−	0.231945i	$-0.0745090\pi$
$594$		0			0
$595$	−9.25236	−	14.4401i	−0.379310	−	0.591988i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	36.6829			1.49882			0.749412	−	0.662104i	$-0.230336\pi$
$599$	36.6829			1.49882			0.749412	+	0.662104i	$0.230336\pi$
$600$		0			0
$601$	19.3578			0.789622			0.394811	−	0.918762i	$-0.370810\pi$
$601$	19.3578			0.789622			0.394811	+	0.918762i	$0.370810\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	7.45391	+	11.6333i	0.303044	+	0.472961i
$606$		0			0
$607$			1.28148i			0.0520135i	0.999662	+	0.0260068i	$0.00827915\pi$
$607$			1.28148i			0.0520135i	−0.999662	+	0.0260068i	$0.991721\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−34.6410			−1.40143
$612$		0			0
$613$			38.9028i			1.57127i	0.618690	+	0.785636i	$0.287664\pi$
$613$			38.9028i			1.57127i	−0.618690	+	0.785636i	$0.712336\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		−	30.7976i		−	1.23986i	−0.784655	−	0.619932i	$-0.787160\pi$
$617$		−	30.7976i		−	1.23986i	0.784655	−	0.619932i	$-0.212840\pi$
$618$		0			0
$619$	−34.3578			−1.38096			−0.690479	−	0.723353i	$-0.742600\pi$
$619$	−34.3578			−1.38096			−0.690479	+	0.723353i	$0.742600\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		−	12.8424i		−	0.514519i
$624$		0			0
$625$	−16.2684	−	18.9826i	−0.650735	−	0.759305i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	31.3650			1.25061
$630$		0			0
$631$	11.5367			0.459270			0.229635	−	0.973277i	$-0.426247\pi$
$631$	11.5367			0.459270			0.229635	+	0.973277i	$0.426247\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	2.41269	−	1.54591i	0.0957448	−	0.0613474i
$636$		0			0
$637$		−	34.9438i		−	1.38452i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	7.91872			0.312771			0.156385	−	0.987696i	$-0.450016\pi$
$641$	7.91872			0.312771			0.156385	+	0.987696i	$0.450016\pi$
$642$		0			0
$643$		−	9.19961i		−	0.362797i	−0.983410	−	0.181399i	$-0.941938\pi$
$643$		−	9.19961i		−	0.362797i	0.983410	−	0.181399i	$-0.0580624\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$			13.7146i			0.539177i	0.962976	+	0.269588i	$0.0868876\pi$
$647$			13.7146i			0.539177i	−0.962976	+	0.269588i	$0.913112\pi$
$648$		0			0
$649$	31.5367			1.23792
$650$		0			0
$651$		0			0
$652$		0			0
$653$			42.0941i			1.64727i	0.567122	+	0.823634i	$0.308057\pi$
$653$			42.0941i			1.64727i	−0.567122	+	0.823634i	$0.691943\pi$
$654$		0			0
$655$	6.64218	+	10.3664i	0.259532	+	0.405051i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	24.1269			0.939852			0.469926	−	0.882706i	$-0.344281\pi$
$659$	24.1269			0.939852			0.469926	+	0.882706i	$0.344281\pi$
$660$		0			0
$661$	44.4313			1.72818			0.864088	−	0.503341i	$-0.167896\pi$
$661$	44.4313			1.72818			0.864088	+	0.503341i	$0.167896\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−17.3205	+	11.0979i	−0.671660	+	0.430359i
$666$		0			0
$667$			39.1321i			1.51520i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−9.03102			−0.348639
$672$		0			0
$673$			32.6101i			1.25703i	0.777799	+	0.628513i	$0.216336\pi$
$673$			32.6101i			1.25703i	−0.777799	+	0.628513i	$0.783664\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$		−	50.9724i		−	1.95903i	−0.201376	−	0.979514i	$-0.564541\pi$
$677$		−	50.9724i		−	1.95903i	0.201376	−	0.979514i	$-0.435459\pi$
$678$		0			0
$679$	18.3578			0.704508
$680$		0			0
$681$		0			0
$682$		0			0
$683$			28.3795i			1.08591i	0.839762	+	0.542955i	$0.182695\pi$
$683$			28.3795i			1.08591i	−0.839762	+	0.542955i	$0.817305\pi$
$684$		0			0
$685$	−21.2684	+	13.6275i	−0.812622	+	0.520678i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	14.4762			0.551498
$690$		0			0
$691$	14.7156			0.559809			0.279905	−	0.960028i	$-0.409697\pi$
$691$	14.7156			0.559809			0.279905	+	0.960028i	$0.409697\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−20.7237	−	32.3435i	−0.786096	−	1.22686i
$696$		0			0
$697$		−	4.07370i		−	0.154302i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	14.2272			0.537353			0.268676	−	0.963230i	$-0.413414\pi$
$701$	14.2272			0.537353			0.268676	+	0.963230i	$0.413414\pi$
$702$		0			0
$703$		−	37.6214i		−	1.41892i
$704$		0			0
$705$		0			0
$706$		0			0
$707$			0.872228i			0.0328035i
$708$		0			0
$709$	18.5367			0.696161			0.348081	−	0.937465i	$-0.386834\pi$
$709$	18.5367			0.696161			0.348081	+	0.937465i	$0.386834\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		−	39.0022i		−	1.46065i
$714$		0			0
$715$	50.8945	−	32.6101i	1.90335	−	1.21955i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	31.7358			1.18355			0.591773	−	0.806105i	$-0.298428\pi$
$719$	31.7358			1.18355			0.591773	+	0.806105i	$0.298428\pi$
$720$		0			0
$721$	3.28437			0.122316
$722$		0			0
$723$		0			0
$724$		0			0
$725$	10.8571	−	23.6035i	0.403223	−	0.876610i
$726$		0			0
$727$			31.4432i			1.16617i	0.812413	+	0.583083i	$0.198154\pi$
$727$			31.4432i			1.16617i	−0.812413	+	0.583083i	$0.801846\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	7.66973			0.283675
$732$		0			0
$733$		−	5.12591i		−	0.189330i	−0.995509	−	0.0946649i	$-0.969822\pi$
$733$		−	5.12591i		−	0.189330i	0.995509	−	0.0946649i	$-0.0301780\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		−	64.6870i		−	2.38278i
$738$		0			0
$739$	−33.8945			−1.24683			−0.623415	−	0.781891i	$-0.714255\pi$
$739$	−33.8945			−1.24683			−0.623415	+	0.781891i	$0.714255\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		−	38.1300i		−	1.39885i	−0.714704	−	0.699427i	$-0.753438\pi$
$743$		−	38.1300i		−	1.39885i	0.714704	−	0.699427i	$-0.246562\pi$
$744$		0			0
$745$	23.8051	+	37.1526i	0.872151	+	1.36117i
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	16.3578			0.596905			0.298453	−	0.954424i	$-0.403530\pi$
$751$	16.3578			0.596905			0.298453	+	0.954424i	$0.403530\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	6.24756	+	9.75056i	0.227372	+	0.354859i
$756$		0			0
$757$			28.6510i			1.04134i	0.853758	+	0.520670i	$0.174318\pi$
$757$			28.6510i			1.04134i	−0.853758	+	0.520670i	$0.825682\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	1.73205			0.0627868			0.0313934	−	0.999507i	$-0.490006\pi$
$761$	1.73205			0.0627868			0.0313934	+	0.999507i	$0.490006\pi$
$762$		0			0
$763$			8.97034i			0.324748i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		−	49.6250i		−	1.79186i
$768$		0			0
$769$	19.7156			0.710964			0.355482	−	0.934683i	$-0.384317\pi$
$769$	19.7156			0.710964			0.355482	+	0.934683i	$0.384317\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$			4.63772i			0.166807i	0.996516	+	0.0834036i	$0.0265791\pi$
$773$			4.63772i			0.166807i	−0.996516	+	0.0834036i	$0.973421\pi$
$774$		0			0
$775$	−10.8211	+	23.5251i	−0.388705	+	0.845047i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−4.88627			−0.175069
$780$		0			0
$781$	22.8211			0.816603
$782$		0			0
$783$		0			0
$784$		0			0
$785$	9.86660	−	6.32191i	0.352154	−	0.225639i
$786$		0			0
$787$		−	45.7688i		−	1.63148i	−0.578419	−	0.815740i	$-0.696330\pi$
$787$		−	45.7688i		−	1.63148i	0.578419	−	0.815740i	$-0.303670\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	11.6318			0.413580
$792$		0			0
$793$			14.2109i			0.504643i
$794$		0			0
$795$		0			0
$796$		0			0
$797$		−	26.3584i		−	0.933663i	−0.884346	−	0.466832i	$-0.845395\pi$
$797$		−	26.3584i		−	0.933663i	0.884346	−	0.466832i	$-0.154605\pi$
$798$		0			0
$799$	−31.7891			−1.12462
$800$		0			0
$801$		0			0
$802$		0			0
$803$			32.3435i			1.14138i
$804$		0			0
$805$	−11.6422	−	18.1699i	−0.410333	−	0.640406i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−5.19615			−0.182687			−0.0913435	−	0.995819i	$-0.529116\pi$
$809$	−5.19615			−0.182687			−0.0913435	+	0.995819i	$0.529116\pi$
$810$		0			0
$811$	−49.8945			−1.75203			−0.876017	−	0.482280i	$-0.839809\pi$
$811$	−49.8945			−1.75203			−0.876017	+	0.482280i	$0.839809\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−22.1459	+	14.1897i	−0.775737	+	0.497044i
$816$		0			0
$817$		−	9.19961i		−	0.321853i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	0.248992			0.00868988			0.00434494	−	0.999991i	$-0.498617\pi$
$821$	0.248992			0.00868988			0.00434494	+	0.999991i	$0.498617\pi$
$822$		0			0
$823$		−	26.0881i		−	0.909373i	−0.890652	−	0.454687i	$-0.849751\pi$
$823$		−	26.0881i		−	0.909373i	0.890652	−	0.454687i	$-0.150249\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$			11.9701i			0.416243i	0.978103	+	0.208121i	$0.0667349\pi$
$827$			11.9701i			0.416243i	−0.978103	+	0.208121i	$0.933265\pi$
$828$		0			0
$829$	−7.17891			−0.249334			−0.124667	−	0.992199i	$-0.539786\pi$
$829$	−7.17891			−0.249334			−0.124667	+	0.992199i	$0.539786\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		−	32.0669i		−	1.11105i
$834$		0			0
$835$	4.17891	−	2.67759i	0.144617	−	0.0926617i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	22.0850			0.762459			0.381230	−	0.924480i	$-0.375501\pi$
$839$	22.0850			0.762459			0.381230	+	0.924480i	$0.375501\pi$
$840$		0			0
$841$	−2.00000			−0.0689655
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−35.6315	−	55.6101i	−1.22576	−	1.91305i
$846$		0			0
$847$		−	7.91813i		−	0.272070i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	39.4664			1.35289
$852$		0			0
$853$		−	40.4136i		−	1.38373i	−0.722025	−	0.691867i	$-0.756788\pi$
$853$		−	40.4136i		−	1.38373i	0.722025	−	0.691867i	$-0.243212\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$			2.02103i			0.0690371i	0.999404	+	0.0345186i	$0.0109898\pi$
$857$			2.02103i			0.0690371i	−0.999404	+	0.0345186i	$0.989010\pi$
$858$		0			0
$859$	−4.46327			−0.152285			−0.0761425	−	0.997097i	$-0.524260\pi$
$859$	−4.46327			−0.152285			−0.0761425	+	0.997097i	$0.524260\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		−	2.21958i		−	0.0755555i	−0.999286	−	0.0377777i	$-0.987972\pi$
$863$		−	2.21958i		−	0.0755555i	0.999286	−	0.0377777i	$-0.0120279\pi$
$864$		0			0
$865$	−1.26836	+	0.812689i	−0.0431256	+	0.0276322i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−24.8685			−0.843605
$870$		0			0
$871$	−101.789			−3.44899
$872$		0			0
$873$		0			0
$874$		0			0
$875$	1.98104	+	14.1897i	0.0669715	+	0.479700i
$876$		0			0
$877$			9.31424i			0.314520i	0.987557	+	0.157260i	$0.0502660\pi$
$877$			9.31424i			0.314520i	−0.987557	+	0.157260i	$0.949734\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	43.6111			1.46930			0.734648	−	0.678448i	$-0.237347\pi$
$881$	43.6111			1.46930			0.734648	+	0.678448i	$0.237347\pi$
$882$		0			0
$883$			1.51074i			0.0508406i	0.999677	+	0.0254203i	$0.00809240\pi$
$883$			1.51074i			0.0508406i	−0.999677	+	0.0254203i	$0.991908\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$			6.65875i			0.223579i	0.993732	+	0.111789i	$0.0356582\pi$
$887$			6.65875i			0.223579i	−0.993732	+	0.111789i	$0.964342\pi$
$888$		0			0
$889$	−1.64218			−0.0550771
$890$		0			0
$891$		0			0
$892$		0			0
$893$			38.1300i			1.27597i
$894$		0			0
$895$	17.5367	+	27.3696i	0.586188	+	0.914863i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−26.9104			−0.897512
$900$		0			0
$901$	13.2844			0.442566
$902$		0			0
$903$		0			0
$904$		0			0
$905$	18.3110	+	28.5780i	0.608679	+	0.949965i
$906$		0			0
$907$			15.3777i			0.510609i	0.966861	+	0.255304i	$0.0821757\pi$
$907$			15.3777i			0.510609i	−0.966861	+	0.255304i	$0.917824\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−23.4463			−0.776810			−0.388405	−	0.921489i	$-0.626974\pi$
$911$	−23.4463			−0.776810			−0.388405	+	0.921489i	$0.626974\pi$
$912$		0			0
$913$		−	40.4136i		−	1.33749i
$914$		0			0
$915$		0			0
$916$		0			0
$917$		−	7.05585i		−	0.233005i
$918$		0			0
$919$	31.1789			1.02850			0.514249	−	0.857641i	$-0.328071\pi$
$919$	31.1789			1.02850			0.514249	+	0.857641i	$0.328071\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		−	35.9104i		−	1.18201i
$924$		0			0
$925$	−23.8051	−	10.9499i	−0.782706	−	0.360030i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	26.6005			0.872735			0.436367	−	0.899769i	$-0.356265\pi$
$929$	26.6005			0.872735			0.436367	+	0.899769i	$0.356265\pi$
$930$		0			0
$931$	−38.4633			−1.26058
$932$		0			0
$933$		0			0
$934$		0			0
$935$	46.7045	−	29.9254i	1.52740	−	0.978664i
$936$		0			0
$937$			3.95906i			0.129337i	0.997907	+	0.0646685i	$0.0205990\pi$
$937$			3.95906i			0.129337i	−0.997907	+	0.0646685i	$0.979401\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	31.3650			1.02247			0.511235	−	0.859441i	$-0.329188\pi$
$941$	31.3650			1.02247			0.511235	+	0.859441i	$0.329188\pi$
$942$		0			0
$943$		−	5.12591i		−	0.166923i
$944$		0			0
$945$		0			0
$946$		0			0
$947$			27.0321i			0.878425i	0.898383	+	0.439213i	$0.144743\pi$
$947$			27.0321i			0.878425i	−0.898383	+	0.439213i	$0.855257\pi$
$948$		0			0
$949$	50.8945			1.65211
$950$		0			0
$951$		0			0
$952$		0			0
$953$		−	26.3584i		−	0.853833i	−0.904291	−	0.426917i	$-0.859600\pi$
$953$		−	26.3584i		−	0.853833i	0.904291	−	0.426917i	$-0.140400\pi$
$954$		0			0
$955$	31.7156	+	49.4986i	1.02629	+	1.60174i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	14.4762			0.467460
$960$		0			0
$961$	−4.17891			−0.134803
$962$		0			0
$963$		0			0
$964$		0			0
$965$	24.3428	−	15.5973i	0.783621	−	0.502096i
$966$		0			0
$967$		−	32.4955i		−	1.04498i	−0.852644	−	0.522492i	$-0.825003\pi$
$967$		−	32.4955i		−	1.04498i	0.852644	−	0.522492i	$-0.174997\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−37.3027			−1.19710			−0.598550	−	0.801085i	$-0.704256\pi$
$971$	−37.3027			−1.19710			−0.598550	+	0.801085i	$0.704256\pi$
$972$		0			0
$973$			22.0144i			0.705748i
$974$		0			0
$975$		0			0
$976$		0			0
$977$			18.1538i			0.580790i	0.956907	+	0.290395i	$0.0937868\pi$
$977$			18.1538i			0.580790i	−0.956907	+	0.290395i	$0.906213\pi$
$978$		0			0
$979$	41.5367			1.32752
$980$		0			0
$981$		0			0
$982$		0			0
$983$		−	48.7528i		−	1.55497i	−0.628900	−	0.777487i	$-0.716494\pi$
$983$		−	48.7528i		−	1.55497i	0.628900	−	0.777487i	$-0.283506\pi$
$984$		0			0
$985$	−15.4473	+	9.89766i	−0.492191	+	0.315366i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	9.65078			0.306877
$990$		0			0
$991$	1.53673			0.0488157			0.0244078	−	0.999702i	$-0.492230\pi$
$991$	1.53673			0.0488157			0.0244078	+	0.999702i	$0.492230\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	2.84434	+	4.43917i	0.0901718	+	0.140731i
$996$		0			0
$997$			44.1434i			1.39804i	0.715105	+	0.699018i	$0.246379\pi$
$997$			44.1434i			1.39804i	−0.715105	+	0.699018i	$0.753621\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1620.2.d.e.649.6	yes	8
3.2	odd	2	inner	1620.2.d.e.649.3	✓	8
5.2	odd	4		8100.2.a.be.1.6		8
5.3	odd	4		8100.2.a.be.1.4		8
5.4	even	2	inner	1620.2.d.e.649.5	yes	8
9.2	odd	6		1620.2.r.h.109.8		16
9.4	even	3		1620.2.r.h.1189.6		16
9.5	odd	6		1620.2.r.h.1189.3		16
9.7	even	3		1620.2.r.h.109.1		16
15.2	even	4		8100.2.a.be.1.5		8
15.8	even	4		8100.2.a.be.1.3		8
15.14	odd	2	inner	1620.2.d.e.649.4	yes	8
45.4	even	6		1620.2.r.h.1189.1		16
45.14	odd	6		1620.2.r.h.1189.8		16
45.29	odd	6		1620.2.r.h.109.3		16
45.34	even	6		1620.2.r.h.109.6		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1620.2.d.e.649.3	✓	8	3.2	odd	2	inner
1620.2.d.e.649.4	yes	8	15.14	odd	2	inner
1620.2.d.e.649.5	yes	8	5.4	even	2	inner
1620.2.d.e.649.6	yes	8	1.1	even	1	trivial
1620.2.r.h.109.1		16	9.7	even	3
1620.2.r.h.109.3		16	45.29	odd	6
1620.2.r.h.109.6		16	45.34	even	6
1620.2.r.h.109.8		16	9.2	odd	6
1620.2.r.h.1189.1		16	45.4	even	6
1620.2.r.h.1189.3		16	9.5	odd	6
1620.2.r.h.1189.6		16	9.4	even	3
1620.2.r.h.1189.8		16	45.14	odd	6
8100.2.a.be.1.3		8	15.8	even	4
8100.2.a.be.1.4		8	5.3	odd	4
8100.2.a.be.1.5		8	15.2	even	4
8100.2.a.be.1.6		8	5.2	odd	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1620 = 2^{2} \cdot 3^{4} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1620.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(1.20635 + 1.88274i) q^{5} -1.28148i q^{7} +O(q^{10})\)	\(q+(1.20635 + 1.88274i) q^{5} -1.28148i q^{7} +4.14474 q^{11} -6.52202i q^{13} -5.98507i q^{17} -7.17891 q^{19} -7.53098i q^{23} +(-2.08945 + 4.54249i) q^{25} -5.19615 q^{29} +5.17891 q^{31} +(2.41269 - 1.54591i) q^{35} +5.24054i q^{37} +0.680643 q^{41} +1.28148i q^{43} -5.31139i q^{47} +5.35782 q^{49} +2.21958i q^{53} +(5.00000 + 7.80350i) q^{55} +7.60885 q^{59} -2.17891 q^{61} +(12.2793 - 7.86782i) q^{65} -15.6070i q^{67} +5.50603 q^{71} +7.80350i q^{73} -5.31139i q^{77} -6.00000 q^{79} -9.75056i q^{83} +(11.2684 - 7.22007i) q^{85} +10.0215 q^{89} -8.35782 q^{91} +(-8.66025 - 13.5161i) q^{95} +14.3255i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q+O(q^{10}) \) 8 * q	\( 8 q - 12 q^{19} + 6 q^{25} - 4 q^{31} - 48 q^{49} + 40 q^{55} + 28 q^{61} - 48 q^{79} + 22 q^{85} + 24 q^{91}+O(q^{100}) \) 8 * q - 12 * q^19 + 6 * q^25 - 4 * q^31 - 48 * q^49 + 40 * q^55 + 28 * q^61 - 48 * q^79 + 22 * q^85 + 24 * q^91

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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