LMFDB - Embedded newform 2240.2.g.j.449.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2240,2,Mod(449,2240)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2240, 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("2240.449");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2240 = 2^{6} \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2240.g (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$17.8864900528$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{6})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 9 $ x^4 + 9
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 70)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			449.3
Root			$-1.22474 + 1.22474i$ of defining polynomial
Character	$\chi$	$=$	2240.449
Dual form			2240.2.g.j.449.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.44949i q^{3} +(-2.22474 - 0.224745i) q^{5} +1.00000i q^{7} -3.00000 q^{9} +4.89898 q^{11} -4.44949i q^{13} +(0.550510 - 5.44949i) q^{15} -2.00000i q^{17} +1.55051 q^{19} -2.44949 q^{21} +2.89898i q^{23} +(4.89898 + 1.00000i) q^{25} +6.89898 q^{29} +8.89898 q^{31} +12.0000i q^{33} +(0.224745 - 2.22474i) q^{35} +2.00000i q^{37} +10.8990 q^{39} -1.10102 q^{41} +0.898979i q^{43} +(6.67423 + 0.674235i) q^{45} -8.89898i q^{47} -1.00000 q^{49} +4.89898 q^{51} +10.8990i q^{53} +(-10.8990 - 1.10102i) q^{55} +3.79796i q^{57} -1.55051 q^{59} -3.55051 q^{61} -3.00000i q^{63} +(-1.00000 + 9.89898i) q^{65} -8.00000i q^{67} -7.10102 q^{69} -1.10102 q^{71} +2.89898i q^{73} +(-2.44949 + 12.0000i) q^{75} +4.89898i q^{77} -6.89898 q^{79} -9.00000 q^{81} +2.44949i q^{83} +(-0.449490 + 4.44949i) q^{85} +16.8990i q^{87} +10.0000 q^{89} +4.44949 q^{91} +21.7980i q^{93} +(-3.44949 - 0.348469i) q^{95} -15.7980i q^{97} -14.6969 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{5} - 12 q^{9} + 12 q^{15} + 16 q^{19} + 8 q^{29} + 16 q^{31} - 4 q^{35} + 24 q^{39} - 24 q^{41} + 12 q^{45} - 4 q^{49} - 24 q^{55} - 16 q^{59} - 24 q^{61} - 4 q^{65} - 48 q^{69} - 24 q^{71} - 8 q^{79}+ \cdots - 4 q^{95}+O(q^{100}) $ 4 * q - 4 * q^5 - 12 * q^9 + 12 * q^15 + 16 * q^19 + 8 * q^29 + 16 * q^31 - 4 * q^35 + 24 * q^39 - 24 * q^41 + 12 * q^45 - 4 * q^49 - 24 * q^55 - 16 * q^59 - 24 * q^61 - 4 * q^65 - 48 * q^69 - 24 * q^71 - 8 * q^79 - 36 * q^81 + 8 * q^85 + 40 * q^89 + 8 * q^91 - 4 * q^95

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2240\mathbb{Z}\right)^\times$.

$n$	$897$	$1471$	$1541$	$1921$
$\chi(n)$	$-1$	$1$	$1$	$1$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$			2.44949i			1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$3$			2.44949i			1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$4$		0			0
$5$	−2.22474	−	0.224745i	−0.994936	−	0.100509i
$5$	−2.22474	−	0.224745i	−0.994936	−	0.100509i
$6$		0			0
$7$			1.00000i			0.377964i
$7$			1.00000i			0.377964i
$8$		0			0
$9$	−3.00000			−1.00000
$10$		0			0
$11$	4.89898			1.47710			0.738549	−	0.674200i	$-0.235511\pi$
$11$	4.89898			1.47710			0.738549	+	0.674200i	$0.235511\pi$
$12$		0			0
$13$		−	4.44949i		−	1.23407i	−0.786937	−	0.617033i	$-0.788334\pi$
$13$		−	4.44949i		−	1.23407i	0.786937	−	0.617033i	$-0.211666\pi$
$14$		0			0
$15$	0.550510	−	5.44949i	0.142141	−	1.40705i
$16$		0			0
$17$		−	2.00000i		−	0.485071i	−0.970143	−	0.242536i	$-0.922021\pi$
$17$		−	2.00000i		−	0.485071i	0.970143	−	0.242536i	$-0.0779791\pi$
$18$		0			0
$19$	1.55051			0.355711			0.177856	−	0.984057i	$-0.443084\pi$
$19$	1.55051			0.355711			0.177856	+	0.984057i	$0.443084\pi$
$20$		0			0
$21$	−2.44949			−0.534522
$22$		0			0
$23$			2.89898i			0.604479i	0.953232	+	0.302240i	$0.0977342\pi$
$23$			2.89898i			0.604479i	−0.953232	+	0.302240i	$0.902266\pi$
$24$		0			0
$25$	4.89898	+	1.00000i	0.979796	+	0.200000i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	6.89898			1.28111			0.640554	−	0.767913i	$-0.278705\pi$
$29$	6.89898			1.28111			0.640554	+	0.767913i	$0.278705\pi$
$30$		0			0
$31$	8.89898			1.59830			0.799152	−	0.601129i	$-0.205282\pi$
$31$	8.89898			1.59830			0.799152	+	0.601129i	$0.205282\pi$
$32$		0			0
$33$			12.0000i			2.08893i
$34$		0			0
$35$	0.224745	−	2.22474i	0.0379888	−	0.376051i
$36$		0			0
$37$			2.00000i			0.328798i	0.986394	+	0.164399i	$0.0525685\pi$
$37$			2.00000i			0.328798i	−0.986394	+	0.164399i	$0.947432\pi$
$38$		0			0
$39$	10.8990			1.74523
$40$		0			0
$41$	−1.10102			−0.171951			−0.0859753	−	0.996297i	$-0.527401\pi$
$41$	−1.10102			−0.171951			−0.0859753	+	0.996297i	$0.527401\pi$
$42$		0			0
$43$			0.898979i			0.137093i	0.997648	+	0.0685465i	$0.0218362\pi$
$43$			0.898979i			0.137093i	−0.997648	+	0.0685465i	$0.978164\pi$
$44$		0			0
$45$	6.67423	+	0.674235i	0.994936	+	0.100509i
$46$		0			0
$47$		−	8.89898i		−	1.29805i	−0.760767	−	0.649025i	$-0.775177\pi$
$47$		−	8.89898i		−	1.29805i	0.760767	−	0.649025i	$-0.224823\pi$
$48$		0			0
$49$	−1.00000			−0.142857
$50$		0			0
$51$	4.89898			0.685994
$52$		0			0
$53$			10.8990i			1.49709i	0.663084	+	0.748545i	$0.269247\pi$
$53$			10.8990i			1.49709i	−0.663084	+	0.748545i	$0.730753\pi$
$54$		0			0
$55$	−10.8990	−	1.10102i	−1.46962	−	0.148462i
$56$		0			0
$57$			3.79796i			0.503052i
$58$		0			0
$59$	−1.55051			−0.201859			−0.100930	−	0.994894i	$-0.532182\pi$
$59$	−1.55051			−0.201859			−0.100930	+	0.994894i	$0.532182\pi$
$60$		0			0
$61$	−3.55051			−0.454596			−0.227298	−	0.973825i	$-0.572989\pi$
$61$	−3.55051			−0.454596			−0.227298	+	0.973825i	$0.572989\pi$
$62$		0			0
$63$		−	3.00000i		−	0.377964i
$64$		0			0
$65$	−1.00000	+	9.89898i	−0.124035	+	1.22782i
$66$		0			0
$67$		−	8.00000i		−	0.977356i	−0.872464	−	0.488678i	$-0.837479\pi$
$67$		−	8.00000i		−	0.977356i	0.872464	−	0.488678i	$-0.162521\pi$
$68$		0			0
$69$	−7.10102			−0.854862
$70$		0			0
$71$	−1.10102			−0.130667			−0.0653335	−	0.997863i	$-0.520811\pi$
$71$	−1.10102			−0.130667			−0.0653335	+	0.997863i	$0.520811\pi$
$72$		0			0
$73$			2.89898i			0.339300i	0.985504	+	0.169650i	$0.0542637\pi$
$73$			2.89898i			0.339300i	−0.985504	+	0.169650i	$0.945736\pi$
$74$		0			0
$75$	−2.44949	+	12.0000i	−0.282843	+	1.38564i
$76$		0			0
$77$			4.89898i			0.558291i
$78$		0			0
$79$	−6.89898			−0.776196			−0.388098	−	0.921618i	$-0.626868\pi$
$79$	−6.89898			−0.776196			−0.388098	+	0.921618i	$0.626868\pi$
$80$		0			0
$81$	−9.00000			−1.00000
$82$		0			0
$83$			2.44949i			0.268866i	0.990923	+	0.134433i	$0.0429214\pi$
$83$			2.44949i			0.268866i	−0.990923	+	0.134433i	$0.957079\pi$
$84$		0			0
$85$	−0.449490	+	4.44949i	−0.0487540	+	0.482615i
$86$		0			0
$87$			16.8990i			1.81176i
$88$		0			0
$89$	10.0000			1.06000			0.529999	−	0.847998i	$-0.322192\pi$
$89$	10.0000			1.06000			0.529999	+	0.847998i	$0.322192\pi$
$90$		0			0
$91$	4.44949			0.466433
$92$		0			0
$93$			21.7980i			2.26034i
$94$		0			0
$95$	−3.44949	−	0.348469i	−0.353910	−	0.0357522i
$96$		0			0
$97$		−	15.7980i		−	1.60404i	−0.597297	−	0.802020i	$-0.703759\pi$
$97$		−	15.7980i		−	1.60404i	0.597297	−	0.802020i	$-0.296241\pi$
$98$		0			0
$99$	−14.6969			−1.47710
$100$		0			0
$101$	−3.55051			−0.353289			−0.176644	−	0.984275i	$-0.556524\pi$
$101$	−3.55051			−0.353289			−0.176644	+	0.984275i	$0.556524\pi$
$102$		0			0
$103$			12.8990i			1.27097i	0.772111	+	0.635487i	$0.219201\pi$
$103$			12.8990i			1.27097i	−0.772111	+	0.635487i	$0.780799\pi$
$104$		0			0
$105$	5.44949	+	0.550510i	0.531816	+	0.0537243i
$106$		0			0
$107$		−	8.00000i		−	0.773389i	−0.922208	−	0.386695i	$-0.873617\pi$
$107$		−	8.00000i		−	0.773389i	0.922208	−	0.386695i	$-0.126383\pi$
$108$		0			0
$109$	6.89898			0.660802			0.330401	−	0.943841i	$-0.392816\pi$
$109$	6.89898			0.660802			0.330401	+	0.943841i	$0.392816\pi$
$110$		0			0
$111$	−4.89898			−0.464991
$112$		0			0
$113$			19.7980i			1.86244i	0.364464	+	0.931218i	$0.381252\pi$
$113$			19.7980i			1.86244i	−0.364464	+	0.931218i	$0.618748\pi$
$114$		0			0
$115$	0.651531	−	6.44949i	0.0607556	−	0.601418i
$116$		0			0
$117$			13.3485i			1.23407i
$118$		0			0
$119$	2.00000			0.183340
$120$		0			0
$121$	13.0000			1.18182
$122$		0			0
$123$		−	2.69694i		−	0.243175i
$124$		0			0
$125$	−10.6742	−	3.32577i	−0.954733	−	0.297465i
$126$		0			0
$127$			14.8990i			1.32207i	0.750355	+	0.661035i	$0.229883\pi$
$127$			14.8990i			1.32207i	−0.750355	+	0.661035i	$0.770117\pi$
$128$		0			0
$129$	−2.20204			−0.193879
$130$		0			0
$131$	6.44949			0.563495			0.281747	−	0.959489i	$-0.409086\pi$
$131$	6.44949			0.563495			0.281747	+	0.959489i	$0.409086\pi$
$132$		0			0
$133$			1.55051i			0.134446i
$134$		0			0
$135$		0			0
$136$		0			0
$137$			1.79796i			0.153610i	0.997046	+	0.0768050i	$0.0244719\pi$
$137$			1.79796i			0.153610i	−0.997046	+	0.0768050i	$0.975528\pi$
$138$		0			0
$139$	1.55051			0.131513			0.0657563	−	0.997836i	$-0.479054\pi$
$139$	1.55051			0.131513			0.0657563	+	0.997836i	$0.479054\pi$
$140$		0			0
$141$	21.7980			1.83572
$142$		0			0
$143$		−	21.7980i		−	1.82284i
$144$		0			0
$145$	−15.3485	−	1.55051i	−1.27462	−	0.128763i
$146$		0			0
$147$		−	2.44949i		−	0.202031i
$148$		0			0
$149$	−3.79796			−0.311141			−0.155570	−	0.987825i	$-0.549722\pi$
$149$	−3.79796			−0.311141			−0.155570	+	0.987825i	$0.549722\pi$
$150$		0			0
$151$	19.5959			1.59469			0.797347	−	0.603522i	$-0.206236\pi$
$151$	19.5959			1.59469			0.797347	+	0.603522i	$0.206236\pi$
$152$		0			0
$153$			6.00000i			0.485071i
$154$		0			0
$155$	−19.7980	−	2.00000i	−1.59021	−	0.160644i
$156$		0			0
$157$			3.55051i			0.283362i	0.989912	+	0.141681i	$0.0452507\pi$
$157$			3.55051i			0.283362i	−0.989912	+	0.141681i	$0.954749\pi$
$158$		0			0
$159$	−26.6969			−2.11720
$160$		0			0
$161$	−2.89898			−0.228472
$162$		0			0
$163$			7.10102i			0.556195i	0.960553	+	0.278097i	$0.0897038\pi$
$163$			7.10102i			0.556195i	−0.960553	+	0.278097i	$0.910296\pi$
$164$		0			0
$165$	2.69694	−	26.6969i	0.209956	−	2.07835i
$166$		0			0
$167$			4.89898i			0.379094i	0.981872	+	0.189547i	$0.0607020\pi$
$167$			4.89898i			0.379094i	−0.981872	+	0.189547i	$0.939298\pi$
$168$		0			0
$169$	−6.79796			−0.522920
$170$		0			0
$171$	−4.65153			−0.355711
$172$		0			0
$173$			6.24745i			0.474985i	0.971389	+	0.237492i	$0.0763255\pi$
$173$			6.24745i			0.474985i	−0.971389	+	0.237492i	$0.923675\pi$
$174$		0			0
$175$	−1.00000	+	4.89898i	−0.0755929	+	0.370328i
$176$		0			0
$177$		−	3.79796i		−	0.285472i
$178$		0			0
$179$	13.7980			1.03131			0.515654	−	0.856797i	$-0.327549\pi$
$179$	13.7980			1.03131			0.515654	+	0.856797i	$0.327549\pi$
$180$		0			0
$181$	10.2474			0.761687			0.380843	−	0.924640i	$-0.375634\pi$
$181$	10.2474			0.761687			0.380843	+	0.924640i	$0.375634\pi$
$182$		0			0
$183$		−	8.69694i		−	0.642896i
$184$		0			0
$185$	0.449490	−	4.44949i	0.0330471	−	0.327133i
$186$		0			0
$187$		−	9.79796i		−	0.716498i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	12.6969			0.918718			0.459359	−	0.888251i	$-0.348079\pi$
$191$	12.6969			0.918718			0.459359	+	0.888251i	$0.348079\pi$
$192$		0			0
$193$		−	21.5959i		−	1.55451i	−0.629187	−	0.777254i	$-0.716612\pi$
$193$		−	21.5959i		−	1.55451i	0.629187	−	0.777254i	$-0.283388\pi$
$194$		0			0
$195$	−24.2474	−	2.44949i	−1.73640	−	0.175412i
$196$		0			0
$197$			18.8990i			1.34650i	0.739417	+	0.673248i	$0.235101\pi$
$197$			18.8990i			1.34650i	−0.739417	+	0.673248i	$0.764899\pi$
$198$		0			0
$199$	−16.8990			−1.19794			−0.598968	−	0.800773i	$-0.704423\pi$
$199$	−16.8990			−1.19794			−0.598968	+	0.800773i	$0.704423\pi$
$200$		0			0
$201$	19.5959			1.38219
$202$		0			0
$203$			6.89898i			0.484213i
$204$		0			0
$205$	2.44949	+	0.247449i	0.171080	+	0.0172826i
$206$		0			0
$207$		−	8.69694i		−	0.604479i
$208$		0			0
$209$	7.59592			0.525421
$210$		0			0
$211$	−12.0000			−0.826114			−0.413057	−	0.910705i	$-0.635539\pi$
$211$	−12.0000			−0.826114			−0.413057	+	0.910705i	$0.635539\pi$
$212$		0			0
$213$		−	2.69694i		−	0.184791i
$214$		0			0
$215$	0.202041	−	2.00000i	0.0137791	−	0.136399i
$216$		0			0
$217$			8.89898i			0.604102i
$218$		0			0
$219$	−7.10102			−0.479842
$220$		0			0
$221$	−8.89898			−0.598610
$222$		0			0
$223$		−	4.00000i		−	0.267860i	−0.990991	−	0.133930i	$-0.957240\pi$
$223$		−	4.00000i		−	0.267860i	0.990991	−	0.133930i	$-0.0427597\pi$
$224$		0			0
$225$	−14.6969	−	3.00000i	−0.979796	−	0.200000i
$226$		0			0
$227$			7.34847i			0.487735i	0.969809	+	0.243868i	$0.0784162\pi$
$227$			7.34847i			0.487735i	−0.969809	+	0.243868i	$0.921584\pi$
$228$		0			0
$229$	−19.1464			−1.26523			−0.632616	−	0.774466i	$-0.718019\pi$
$229$	−19.1464			−1.26523			−0.632616	+	0.774466i	$0.718019\pi$
$230$		0			0
$231$	−12.0000			−0.789542
$232$		0			0
$233$			29.7980i			1.95213i	0.217481	+	0.976065i	$0.430216\pi$
$233$			29.7980i			1.95213i	−0.217481	+	0.976065i	$0.569784\pi$
$234$		0			0
$235$	−2.00000	+	19.7980i	−0.130466	+	1.29148i
$236$		0			0
$237$		−	16.8990i		−	1.09771i
$238$		0			0
$239$	6.20204			0.401177			0.200588	−	0.979676i	$-0.435715\pi$
$239$	6.20204			0.401177			0.200588	+	0.979676i	$0.435715\pi$
$240$		0			0
$241$	−8.69694			−0.560219			−0.280110	−	0.959968i	$-0.590371\pi$
$241$	−8.69694			−0.560219			−0.280110	+	0.959968i	$0.590371\pi$
$242$		0			0
$243$		−	22.0454i		−	1.41421i
$244$		0			0
$245$	2.22474	+	0.224745i	0.142134	+	0.0143584i
$246$		0			0
$247$		−	6.89898i		−	0.438972i
$248$		0			0
$249$	−6.00000			−0.380235
$250$		0			0
$251$	6.44949			0.407088			0.203544	−	0.979066i	$-0.434754\pi$
$251$	6.44949			0.407088			0.203544	+	0.979066i	$0.434754\pi$
$252$		0			0
$253$			14.2020i			0.892875i
$254$		0			0
$255$	−10.8990	−	1.10102i	−0.682521	−	0.0689486i
$256$		0			0
$257$			8.69694i			0.542500i	0.962509	+	0.271250i	$0.0874370\pi$
$257$			8.69694i			0.542500i	−0.962509	+	0.271250i	$0.912563\pi$
$258$		0			0
$259$	−2.00000			−0.124274
$260$		0			0
$261$	−20.6969			−1.28111
$262$		0			0
$263$			9.79796i			0.604168i	0.953281	+	0.302084i	$0.0976823\pi$
$263$			9.79796i			0.604168i	−0.953281	+	0.302084i	$0.902318\pi$
$264$		0			0
$265$	2.44949	−	24.2474i	0.150471	−	1.48951i
$266$		0			0
$267$			24.4949i			1.49906i
$268$		0			0
$269$	19.1464			1.16738			0.583689	−	0.811977i	$-0.301609\pi$
$269$	19.1464			1.16738			0.583689	+	0.811977i	$0.301609\pi$
$270$		0			0
$271$	12.0000			0.728948			0.364474	−	0.931214i	$-0.381249\pi$
$271$	12.0000			0.728948			0.364474	+	0.931214i	$0.381249\pi$
$272$		0			0
$273$			10.8990i			0.659636i
$274$		0			0
$275$	24.0000	+	4.89898i	1.44725	+	0.295420i
$276$		0			0
$277$		−	14.8990i		−	0.895193i	−0.894236	−	0.447596i	$-0.852280\pi$
$277$		−	14.8990i		−	0.895193i	0.894236	−	0.447596i	$-0.147720\pi$
$278$		0			0
$279$	−26.6969			−1.59830
$280$		0			0
$281$	−18.0000			−1.07379			−0.536895	−	0.843649i	$-0.680403\pi$
$281$	−18.0000			−1.07379			−0.536895	+	0.843649i	$0.680403\pi$
$282$		0			0
$283$		−	3.75255i		−	0.223066i	−0.993761	−	0.111533i	$-0.964424\pi$
$283$		−	3.75255i		−	0.223066i	0.993761	−	0.111533i	$-0.0355761\pi$
$284$		0			0
$285$	0.853572	−	8.44949i	0.0505612	−	0.500505i
$286$		0			0
$287$		−	1.10102i		−	0.0649912i
$288$		0			0
$289$	13.0000			0.764706
$290$		0			0
$291$	38.6969			2.26845
$292$		0			0
$293$		−	18.2474i		−	1.06603i	−0.846107	−	0.533014i	$-0.821059\pi$
$293$		−	18.2474i		−	1.06603i	0.846107	−	0.533014i	$-0.178941\pi$
$294$		0			0
$295$	3.44949	+	0.348469i	0.200837	+	0.0202887i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	12.8990			0.745967
$300$		0			0
$301$	−0.898979			−0.0518163
$302$		0			0
$303$		−	8.69694i		−	0.499626i
$304$		0			0
$305$	7.89898	+	0.797959i	0.452294	+	0.0456910i
$306$		0			0
$307$		−	20.2474i		−	1.15558i	−0.816184	−	0.577791i	$-0.803915\pi$
$307$		−	20.2474i		−	1.15558i	0.816184	−	0.577791i	$-0.196085\pi$
$308$		0			0
$309$	−31.5959			−1.79743
$310$		0			0
$311$	12.0000			0.680458			0.340229	−	0.940343i	$-0.389495\pi$
$311$	12.0000			0.680458			0.340229	+	0.940343i	$0.389495\pi$
$312$		0			0
$313$		−	21.5959i		−	1.22067i	−0.792142	−	0.610337i	$-0.791034\pi$
$313$		−	21.5959i		−	1.22067i	0.792142	−	0.610337i	$-0.208966\pi$
$314$		0			0
$315$	−0.674235	+	6.67423i	−0.0379888	+	0.376051i
$316$		0			0
$317$		−	22.4949i		−	1.26344i	−0.775197	−	0.631720i	$-0.782349\pi$
$317$		−	22.4949i		−	1.26344i	0.775197	−	0.631720i	$-0.217651\pi$
$318$		0			0
$319$	33.7980			1.89232
$320$		0			0
$321$	19.5959			1.09374
$322$		0			0
$323$		−	3.10102i		−	0.172545i
$324$		0			0
$325$	4.44949	−	21.7980i	0.246813	−	1.20913i
$326$		0			0
$327$			16.8990i			0.934516i
$328$		0			0
$329$	8.89898			0.490617
$330$		0			0
$331$	18.6969			1.02768			0.513838	−	0.857887i	$-0.328223\pi$
$331$	18.6969			1.02768			0.513838	+	0.857887i	$0.328223\pi$
$332$		0			0
$333$		−	6.00000i		−	0.328798i
$334$		0			0
$335$	−1.79796	+	17.7980i	−0.0982330	+	0.972406i
$336$		0			0
$337$		−	9.59592i		−	0.522723i	−0.965241	−	0.261361i	$-0.915829\pi$
$337$		−	9.59592i		−	0.522723i	0.965241	−	0.261361i	$-0.0841715\pi$
$338$		0			0
$339$	−48.4949			−2.63388
$340$		0			0
$341$	43.5959			2.36085
$342$		0			0
$343$		−	1.00000i		−	0.0539949i
$344$		0			0
$345$	15.7980	+	1.59592i	0.850534	+	0.0859213i
$346$		0			0
$347$			28.8990i			1.55138i	0.631115	+	0.775689i	$0.282598\pi$
$347$			28.8990i			1.55138i	−0.631115	+	0.775689i	$0.717402\pi$
$348$		0			0
$349$	−8.44949			−0.452291			−0.226145	−	0.974094i	$-0.572612\pi$
$349$	−8.44949			−0.452291			−0.226145	+	0.974094i	$0.572612\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$			22.8990i			1.21879i	0.792867	+	0.609395i	$0.208588\pi$
$353$			22.8990i			1.21879i	−0.792867	+	0.609395i	$0.791412\pi$
$354$		0			0
$355$	2.44949	+	0.247449i	0.130005	+	0.0131332i
$356$		0			0
$357$			4.89898i			0.259281i
$358$		0			0
$359$	27.5959			1.45646			0.728228	−	0.685334i	$-0.240344\pi$
$359$	27.5959			1.45646			0.728228	+	0.685334i	$0.240344\pi$
$360$		0			0
$361$	−16.5959			−0.873469
$362$		0			0
$363$			31.8434i			1.67134i
$364$		0			0
$365$	0.651531	−	6.44949i	0.0341027	−	0.337582i
$366$		0			0
$367$		−	32.0000i		−	1.67039i	−0.549957	−	0.835193i	$-0.685356\pi$
$367$		−	32.0000i		−	1.67039i	0.549957	−	0.835193i	$-0.314644\pi$
$368$		0			0
$369$	3.30306			0.171951
$370$		0			0
$371$	−10.8990			−0.565847
$372$		0			0
$373$			4.69694i			0.243198i	0.992579	+	0.121599i	$0.0388022\pi$
$373$			4.69694i			0.243198i	−0.992579	+	0.121599i	$0.961198\pi$
$374$		0			0
$375$	8.14643	−	26.1464i	0.420680	−	1.35020i
$376$		0			0
$377$		−	30.6969i		−	1.58097i
$378$		0			0
$379$	30.6969			1.57680			0.788398	−	0.615166i	$-0.210911\pi$
$379$	30.6969			1.57680			0.788398	+	0.615166i	$0.210911\pi$
$380$		0			0
$381$	−36.4949			−1.86969
$382$		0			0
$383$		−	7.10102i		−	0.362845i	−0.983405	−	0.181423i	$-0.941930\pi$
$383$		−	7.10102i		−	0.362845i	0.983405	−	0.181423i	$-0.0580702\pi$
$384$		0			0
$385$	1.10102	−	10.8990i	0.0561132	−	0.555463i
$386$		0			0
$387$		−	2.69694i		−	0.137093i
$388$		0			0
$389$	13.1010			0.664248			0.332124	−	0.943236i	$-0.392235\pi$
$389$	13.1010			0.664248			0.332124	+	0.943236i	$0.392235\pi$
$390$		0			0
$391$	5.79796			0.293215
$392$		0			0
$393$			15.7980i			0.796902i
$394$		0			0
$395$	15.3485	+	1.55051i	0.772265	+	0.0780146i
$396$		0			0
$397$		−	2.65153i		−	0.133077i	−0.997784	−	0.0665383i	$-0.978805\pi$
$397$		−	2.65153i		−	0.133077i	0.997784	−	0.0665383i	$-0.0211954\pi$
$398$		0			0
$399$	−3.79796			−0.190136
$400$		0			0
$401$	−29.3939			−1.46786			−0.733930	−	0.679225i	$-0.762316\pi$
$401$	−29.3939			−1.46786			−0.733930	+	0.679225i	$0.762316\pi$
$402$		0			0
$403$		−	39.5959i		−	1.97241i
$404$		0			0
$405$	20.0227	+	2.02270i	0.994936	+	0.100509i
$406$		0			0
$407$			9.79796i			0.485667i
$408$		0			0
$409$	−34.4949			−1.70566			−0.852831	−	0.522186i	$-0.825117\pi$
$409$	−34.4949			−1.70566			−0.852831	+	0.522186i	$0.825117\pi$
$410$		0			0
$411$	−4.40408			−0.217237
$412$		0			0
$413$		−	1.55051i		−	0.0762956i
$414$		0			0
$415$	0.550510	−	5.44949i	0.0270235	−	0.267505i
$416$		0			0
$417$			3.79796i			0.185987i
$418$		0			0
$419$	−1.55051			−0.0757474			−0.0378737	−	0.999283i	$-0.512058\pi$
$419$	−1.55051			−0.0757474			−0.0378737	+	0.999283i	$0.512058\pi$
$420$		0			0
$421$	4.20204			0.204795			0.102397	−	0.994744i	$-0.467349\pi$
$421$	4.20204			0.204795			0.102397	+	0.994744i	$0.467349\pi$
$422$		0			0
$423$			26.6969i			1.29805i
$424$		0			0
$425$	2.00000	−	9.79796i	0.0970143	−	0.475271i
$426$		0			0
$427$		−	3.55051i		−	0.171821i
$428$		0			0
$429$	53.3939			2.57788
$430$		0			0
$431$	−1.79796			−0.0866046			−0.0433023	−	0.999062i	$-0.513788\pi$
$431$	−1.79796			−0.0866046			−0.0433023	+	0.999062i	$0.513788\pi$
$432$		0			0
$433$		−	0.202041i		−	0.00970947i	−0.999988	−	0.00485474i	$-0.998455\pi$
$433$		−	0.202041i		−	0.00970947i	0.999988	−	0.00485474i	$-0.00154532\pi$
$434$		0			0
$435$	3.79796	−	37.5959i	0.182098	−	1.80259i
$436$		0			0
$437$			4.49490i			0.215020i
$438$		0			0
$439$	21.3939			1.02107			0.510537	−	0.859856i	$-0.329447\pi$
$439$	21.3939			1.02107			0.510537	+	0.859856i	$0.329447\pi$
$440$		0			0
$441$	3.00000			0.142857
$442$		0			0
$443$		−	9.79796i		−	0.465515i	−0.972535	−	0.232758i	$-0.925225\pi$
$443$		−	9.79796i		−	0.465515i	0.972535	−	0.232758i	$-0.0747749\pi$
$444$		0			0
$445$	−22.2474	−	2.24745i	−1.05463	−	0.106539i
$446$		0			0
$447$		−	9.30306i		−	0.440020i
$448$		0			0
$449$	10.0000			0.471929			0.235965	−	0.971762i	$-0.424175\pi$
$449$	10.0000			0.471929			0.235965	+	0.971762i	$0.424175\pi$
$450$		0			0
$451$	−5.39388			−0.253988
$452$		0			0
$453$			48.0000i			2.25524i
$454$		0			0
$455$	−9.89898	−	1.00000i	−0.464071	−	0.0468807i
$456$		0			0
$457$		−	29.5959i		−	1.38444i	−0.721687	−	0.692219i	$-0.756633\pi$
$457$		−	29.5959i		−	1.38444i	0.721687	−	0.692219i	$-0.243367\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−17.3485			−0.807999			−0.403999	−	0.914759i	$-0.632380\pi$
$461$	−17.3485			−0.807999			−0.403999	+	0.914759i	$0.632380\pi$
$462$		0			0
$463$			3.59592i			0.167116i	0.996503	+	0.0835582i	$0.0266285\pi$
$463$			3.59592i			0.167116i	−0.996503	+	0.0835582i	$0.973372\pi$
$464$		0			0
$465$	4.89898	−	48.4949i	0.227185	−	2.24890i
$466$		0			0
$467$			10.4495i			0.483545i	0.970333	+	0.241772i	$0.0777287\pi$
$467$			10.4495i			0.483545i	−0.970333	+	0.241772i	$0.922271\pi$
$468$		0			0
$469$	8.00000			0.369406
$470$		0			0
$471$	−8.69694			−0.400734
$472$		0			0
$473$			4.40408i			0.202500i
$474$		0			0
$475$	7.59592	+	1.55051i	0.348525	+	0.0711423i
$476$		0			0
$477$		−	32.6969i		−	1.49709i
$478$		0			0
$479$	−9.30306			−0.425068			−0.212534	−	0.977154i	$-0.568172\pi$
$479$	−9.30306			−0.425068			−0.212534	+	0.977154i	$0.568172\pi$
$480$		0			0
$481$	8.89898			0.405759
$482$		0			0
$483$		−	7.10102i		−	0.323108i
$484$		0			0
$485$	−3.55051	+	35.1464i	−0.161220	+	1.59592i
$486$		0			0
$487$			7.30306i			0.330933i	0.986215	+	0.165467i	$0.0529130\pi$
$487$			7.30306i			0.330933i	−0.986215	+	0.165467i	$0.947087\pi$
$488$		0			0
$489$	−17.3939			−0.786578
$490$		0			0
$491$	−19.5959			−0.884351			−0.442176	−	0.896928i	$-0.645793\pi$
$491$	−19.5959			−0.884351			−0.442176	+	0.896928i	$0.645793\pi$
$492$		0			0
$493$		−	13.7980i		−	0.621429i
$494$		0			0
$495$	32.6969	+	3.30306i	1.46962	+	0.148462i
$496$		0			0
$497$		−	1.10102i		−	0.0493875i
$498$		0			0
$499$	6.20204			0.277641			0.138821	−	0.990318i	$-0.455669\pi$
$499$	6.20204			0.277641			0.138821	+	0.990318i	$0.455669\pi$
$500$		0			0
$501$	−12.0000			−0.536120
$502$		0			0
$503$		−	4.00000i		−	0.178351i	−0.996016	−	0.0891756i	$-0.971577\pi$
$503$		−	4.00000i		−	0.178351i	0.996016	−	0.0891756i	$-0.0284232\pi$
$504$		0			0
$505$	7.89898	+	0.797959i	0.351500	+	0.0355087i
$506$		0			0
$507$		−	16.6515i		−	0.739520i
$508$		0			0
$509$	−31.5505			−1.39845			−0.699226	−	0.714901i	$-0.746472\pi$
$509$	−31.5505			−1.39845			−0.699226	+	0.714901i	$0.746472\pi$
$510$		0			0
$511$	−2.89898			−0.128243
$512$		0			0
$513$		0			0
$514$		0			0
$515$	2.89898	−	28.6969i	0.127744	−	1.26454i
$516$		0			0
$517$		−	43.5959i		−	1.91735i
$518$		0			0
$519$	−15.3031			−0.671730
$520$		0			0
$521$	32.6969			1.43248			0.716239	−	0.697855i	$-0.245862\pi$
$521$	32.6969			1.43248			0.716239	+	0.697855i	$0.245862\pi$
$522$		0			0
$523$			33.1464i			1.44939i	0.689069	+	0.724696i	$0.258020\pi$
$523$			33.1464i			1.44939i	−0.689069	+	0.724696i	$0.741980\pi$
$524$		0			0
$525$	−12.0000	−	2.44949i	−0.523723	−	0.106904i
$526$		0			0
$527$		−	17.7980i		−	0.775291i
$528$		0			0
$529$	14.5959			0.634605
$530$		0			0
$531$	4.65153			0.201859
$532$		0			0
$533$			4.89898i			0.212198i
$534$		0			0
$535$	−1.79796	+	17.7980i	−0.0777325	+	0.769473i
$536$		0			0
$537$			33.7980i			1.45849i
$538$		0			0
$539$	−4.89898			−0.211014
$540$		0			0
$541$	−9.59592			−0.412561			−0.206280	−	0.978493i	$-0.566136\pi$
$541$	−9.59592			−0.412561			−0.206280	+	0.978493i	$0.566136\pi$
$542$		0			0
$543$			25.1010i			1.07719i
$544$		0			0
$545$	−15.3485	−	1.55051i	−0.657456	−	0.0664166i
$546$		0			0
$547$		−	18.6969i		−	0.799423i	−0.916641	−	0.399712i	$-0.869110\pi$
$547$		−	18.6969i		−	0.799423i	0.916641	−	0.399712i	$-0.130890\pi$
$548$		0			0
$549$	10.6515			0.454596
$550$		0			0
$551$	10.6969			0.455705
$552$		0			0
$553$		−	6.89898i		−	0.293374i
$554$		0			0
$555$	10.8990	+	1.10102i	0.462636	+	0.0467357i
$556$		0			0
$557$			12.6969i			0.537987i	0.963142	+	0.268993i	$0.0866909\pi$
$557$			12.6969i			0.537987i	−0.963142	+	0.268993i	$0.913309\pi$
$558$		0			0
$559$	4.00000			0.169182
$560$		0			0
$561$	24.0000			1.01328
$562$		0			0
$563$			30.0454i			1.26626i	0.774044	+	0.633131i	$0.218231\pi$
$563$			30.0454i			1.26626i	−0.774044	+	0.633131i	$0.781769\pi$
$564$		0			0
$565$	4.44949	−	44.0454i	0.187191	−	1.85300i
$566$		0			0
$567$		−	9.00000i		−	0.377964i
$568$		0			0
$569$	−33.7980			−1.41688			−0.708442	−	0.705769i	$-0.750602\pi$
$569$	−33.7980			−1.41688			−0.708442	+	0.705769i	$0.750602\pi$
$570$		0			0
$571$	11.1010			0.464563			0.232282	−	0.972649i	$-0.425381\pi$
$571$	11.1010			0.464563			0.232282	+	0.972649i	$0.425381\pi$
$572$		0			0
$573$			31.1010i			1.29926i
$574$		0			0
$575$	−2.89898	+	14.2020i	−0.120896	+	0.592266i
$576$		0			0
$577$			2.49490i			0.103864i	0.998651	+	0.0519320i	$0.0165379\pi$
$577$			2.49490i			0.103864i	−0.998651	+	0.0519320i	$0.983462\pi$
$578$		0			0
$579$	52.8990			2.19841
$580$		0			0
$581$	−2.44949			−0.101622
$582$		0			0
$583$			53.3939i			2.21135i
$584$		0			0
$585$	3.00000	−	29.6969i	0.124035	−	1.22782i
$586$		0			0
$587$			1.14643i			0.0473182i	0.999720	+	0.0236591i	$0.00753162\pi$
$587$			1.14643i			0.0473182i	−0.999720	+	0.0236591i	$0.992468\pi$
$588$		0			0
$589$	13.7980			0.568535
$590$		0			0
$591$	−46.2929			−1.90423
$592$		0			0
$593$		−	10.8990i		−	0.447567i	−0.974639	−	0.223784i	$-0.928159\pi$
$593$		−	10.8990i		−	0.447567i	0.974639	−	0.223784i	$-0.0718409\pi$
$594$		0			0
$595$	−4.44949	−	0.449490i	−0.182411	−	0.0184273i
$596$		0			0
$597$		−	41.3939i		−	1.69414i
$598$		0			0
$599$	13.1010			0.535293			0.267647	−	0.963517i	$-0.413754\pi$
$599$	13.1010			0.535293			0.267647	+	0.963517i	$0.413754\pi$
$600$		0			0
$601$	−39.3939			−1.60691			−0.803455	−	0.595366i	$-0.797007\pi$
$601$	−39.3939			−1.60691			−0.803455	+	0.595366i	$0.797007\pi$
$602$		0			0
$603$			24.0000i			0.977356i
$604$		0			0
$605$	−28.9217	−	2.92168i	−1.17583	−	0.118783i
$606$		0			0
$607$		−	33.3939i		−	1.35542i	−0.735331	−	0.677708i	$-0.762973\pi$
$607$		−	33.3939i		−	1.35542i	0.735331	−	0.677708i	$-0.237027\pi$
$608$		0			0
$609$	−16.8990			−0.684781
$610$		0			0
$611$	−39.5959			−1.60188
$612$		0			0
$613$			27.7980i			1.12275i	0.827562	+	0.561374i	$0.189727\pi$
$613$			27.7980i			1.12275i	−0.827562	+	0.561374i	$0.810273\pi$
$614$		0			0
$615$	−0.606123	+	6.00000i	−0.0244412	+	0.241943i
$616$		0			0
$617$		−	29.5959i		−	1.19149i	−0.803175	−	0.595743i	$-0.796858\pi$
$617$		−	29.5959i		−	1.19149i	0.803175	−	0.595743i	$-0.203142\pi$
$618$		0			0
$619$	−41.5505			−1.67006			−0.835028	−	0.550207i	$-0.814549\pi$
$619$	−41.5505			−1.67006			−0.835028	+	0.550207i	$0.814549\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$			10.0000i			0.400642i
$624$		0			0
$625$	23.0000	+	9.79796i	0.920000	+	0.391918i
$626$		0			0
$627$			18.6061i			0.743057i
$628$		0			0
$629$	4.00000			0.159490
$630$		0			0
$631$	−42.4949			−1.69170			−0.845848	−	0.533425i	$-0.820905\pi$
$631$	−42.4949			−1.69170			−0.845848	+	0.533425i	$0.820905\pi$
$632$		0			0
$633$		−	29.3939i		−	1.16830i
$634$		0			0
$635$	3.34847	−	33.1464i	0.132880	−	1.31538i
$636$		0			0
$637$			4.44949i			0.176295i
$638$		0			0
$639$	3.30306			0.130667
$640$		0			0
$641$	25.7980			1.01896			0.509479	−	0.860483i	$-0.329838\pi$
$641$	25.7980			1.01896			0.509479	+	0.860483i	$0.329838\pi$
$642$		0			0
$643$		−	25.1464i		−	0.991678i	−0.868414	−	0.495839i	$-0.834861\pi$
$643$		−	25.1464i		−	0.991678i	0.868414	−	0.495839i	$-0.165139\pi$
$644$		0			0
$645$	4.89898	+	0.494897i	0.192897	+	0.0194866i
$646$		0			0
$647$			46.2929i			1.81996i	0.414652	+	0.909980i	$0.363903\pi$
$647$			46.2929i			1.81996i	−0.414652	+	0.909980i	$0.636097\pi$
$648$		0			0
$649$	−7.59592			−0.298166
$650$		0			0
$651$	−21.7980			−0.854329
$652$		0			0
$653$			20.2020i			0.790567i	0.918559	+	0.395283i	$0.129354\pi$
$653$			20.2020i			0.790567i	−0.918559	+	0.395283i	$0.870646\pi$
$654$		0			0
$655$	−14.3485	−	1.44949i	−0.560641	−	0.0566363i
$656$		0			0
$657$		−	8.69694i		−	0.339300i
$658$		0			0
$659$	16.8990			0.658291			0.329145	−	0.944279i	$-0.393239\pi$
$659$	16.8990			0.658291			0.329145	+	0.944279i	$0.393239\pi$
$660$		0			0
$661$	40.9444			1.59255			0.796276	−	0.604933i	$-0.206800\pi$
$661$	40.9444			1.59255			0.796276	+	0.604933i	$0.206800\pi$
$662$		0			0
$663$		−	21.7980i		−	0.846563i
$664$		0			0
$665$	0.348469	−	3.44949i	0.0135131	−	0.133765i
$666$		0			0
$667$			20.0000i			0.774403i
$668$		0			0
$669$	9.79796			0.378811
$670$		0			0
$671$	−17.3939			−0.671483
$672$		0			0
$673$		−	17.7980i		−	0.686061i	−0.939324	−	0.343030i	$-0.888547\pi$
$673$		−	17.7980i		−	0.686061i	0.939324	−	0.343030i	$-0.111453\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$		−	36.4495i		−	1.40087i	−0.713717	−	0.700434i	$-0.752990\pi$
$677$		−	36.4495i		−	1.40087i	0.713717	−	0.700434i	$-0.247010\pi$
$678$		0			0
$679$	15.7980			0.606270
$680$		0			0
$681$	−18.0000			−0.689761
$682$		0			0
$683$		−	3.59592i		−	0.137594i	−0.997631	−	0.0687970i	$-0.978084\pi$
$683$		−	3.59592i		−	0.137594i	0.997631	−	0.0687970i	$-0.0219161\pi$
$684$		0			0
$685$	0.404082	−	4.00000i	0.0154392	−	0.152832i
$686$		0			0
$687$		−	46.8990i		−	1.78931i
$688$		0			0
$689$	48.4949			1.84751
$690$		0			0
$691$	−21.1464			−0.804448			−0.402224	−	0.915541i	$-0.631763\pi$
$691$	−21.1464			−0.804448			−0.402224	+	0.915541i	$0.631763\pi$
$692$		0			0
$693$		−	14.6969i		−	0.558291i
$694$		0			0
$695$	−3.44949	−	0.348469i	−0.130847	−	0.0132182i
$696$		0			0
$697$			2.20204i			0.0834083i
$698$		0			0
$699$	−72.9898			−2.76073
$700$		0			0
$701$	−11.3031			−0.426911			−0.213455	−	0.976953i	$-0.568472\pi$
$701$	−11.3031			−0.426911			−0.213455	+	0.976953i	$0.568472\pi$
$702$		0			0
$703$			3.10102i			0.116957i
$704$		0			0
$705$	−48.4949	−	4.89898i	−1.82642	−	0.184506i
$706$		0			0
$707$		−	3.55051i		−	0.133531i
$708$		0			0
$709$	−28.2929			−1.06256			−0.531280	−	0.847196i	$-0.678289\pi$
$709$	−28.2929			−1.06256			−0.531280	+	0.847196i	$0.678289\pi$
$710$		0			0
$711$	20.6969			0.776196
$712$		0			0
$713$			25.7980i			0.966141i
$714$		0			0
$715$	−4.89898	+	48.4949i	−0.183211	+	1.81361i
$716$		0			0
$717$			15.1918i			0.567350i
$718$		0			0
$719$	4.49490			0.167631			0.0838157	−	0.996481i	$-0.473289\pi$
$719$	4.49490			0.167631			0.0838157	+	0.996481i	$0.473289\pi$
$720$		0			0
$721$	−12.8990			−0.480383
$722$		0			0
$723$		−	21.3031i		−	0.792269i
$724$		0			0
$725$	33.7980	+	6.89898i	1.25522	+	0.256222i
$726$		0			0
$727$		−	22.6969i		−	0.841783i	−0.907111	−	0.420891i	$-0.861717\pi$
$727$		−	22.6969i		−	0.841783i	0.907111	−	0.420891i	$-0.138283\pi$
$728$		0			0
$729$	27.0000			1.00000
$730$		0			0
$731$	1.79796			0.0664999
$732$		0			0
$733$		−	39.6413i		−	1.46419i	−0.681205	−	0.732093i	$-0.738544\pi$
$733$		−	39.6413i		−	1.46419i	0.681205	−	0.732093i	$-0.261456\pi$
$734$		0			0
$735$	−0.550510	+	5.44949i	−0.0203059	+	0.201007i
$736$		0			0
$737$		−	39.1918i		−	1.44365i
$738$		0			0
$739$	4.49490			0.165347			0.0826737	−	0.996577i	$-0.473654\pi$
$739$	4.49490			0.165347			0.0826737	+	0.996577i	$0.473654\pi$
$740$		0			0
$741$	16.8990			0.620800
$742$		0			0
$743$		−	44.6969i		−	1.63977i	−0.572527	−	0.819886i	$-0.694037\pi$
$743$		−	44.6969i		−	1.63977i	0.572527	−	0.819886i	$-0.305963\pi$
$744$		0			0
$745$	8.44949	+	0.853572i	0.309565	+	0.0312725i
$746$		0			0
$747$		−	7.34847i		−	0.268866i
$748$		0			0
$749$	8.00000			0.292314
$750$		0			0
$751$	−41.7980			−1.52523			−0.762615	−	0.646853i	$-0.776085\pi$
$751$	−41.7980			−1.52523			−0.762615	+	0.646853i	$0.776085\pi$
$752$		0			0
$753$			15.7980i			0.575710i
$754$		0			0
$755$	−43.5959	−	4.40408i	−1.58662	−	0.160281i
$756$		0			0
$757$		−	51.7980i		−	1.88263i	−0.337531	−	0.941314i	$-0.609592\pi$
$757$		−	51.7980i		−	1.88263i	0.337531	−	0.941314i	$-0.390408\pi$
$758$		0			0
$759$	−34.7878			−1.26272
$760$		0			0
$761$	−21.1010			−0.764911			−0.382456	−	0.923974i	$-0.624922\pi$
$761$	−21.1010			−0.764911			−0.382456	+	0.923974i	$0.624922\pi$
$762$		0			0
$763$			6.89898i			0.249760i
$764$		0			0
$765$	1.34847	−	13.3485i	0.0487540	−	0.482615i
$766$		0			0
$767$			6.89898i			0.249108i
$768$		0			0
$769$	40.6969			1.46757			0.733785	−	0.679382i	$-0.237752\pi$
$769$	40.6969			1.46757			0.733785	+	0.679382i	$0.237752\pi$
$770$		0			0
$771$	−21.3031			−0.767211
$772$		0			0
$773$		−	1.34847i		−	0.0485011i	−0.999706	−	0.0242505i	$-0.992280\pi$
$773$		−	1.34847i		−	0.0485011i	0.999706	−	0.0242505i	$-0.00771994\pi$
$774$		0			0
$775$	43.5959	+	8.89898i	1.56601	+	0.319661i
$776$		0			0
$777$		−	4.89898i		−	0.175750i
$778$		0			0
$779$	−1.70714			−0.0611648
$780$		0			0
$781$	−5.39388			−0.193008
$782$		0			0
$783$		0			0
$784$		0			0
$785$	0.797959	−	7.89898i	0.0284804	−	0.281927i
$786$		0			0
$787$			50.4495i			1.79833i	0.437610	+	0.899165i	$0.355825\pi$
$787$			50.4495i			1.79833i	−0.437610	+	0.899165i	$0.644175\pi$
$788$		0			0
$789$	−24.0000			−0.854423
$790$		0			0
$791$	−19.7980			−0.703934
$792$		0			0
$793$			15.7980i			0.561002i
$794$		0			0
$795$	59.3939	+	6.00000i	2.10648	+	0.212798i
$796$		0			0
$797$		−	0.944387i		−	0.0334519i	−0.999860	−	0.0167260i	$-0.994676\pi$
$797$		−	0.944387i		−	0.0334519i	0.999860	−	0.0167260i	$-0.00532429\pi$
$798$		0			0
$799$	−17.7980			−0.629647
$800$		0			0
$801$	−30.0000			−1.06000
$802$		0			0
$803$			14.2020i			0.501179i
$804$		0			0
$805$	6.44949	+	0.651531i	0.227315	+	0.0229634i
$806$		0			0
$807$			46.8990i			1.65092i
$808$		0			0
$809$	−47.5959			−1.67338			−0.836692	−	0.547674i	$-0.815513\pi$
$809$	−47.5959			−1.67338			−0.836692	+	0.547674i	$0.815513\pi$
$810$		0			0
$811$	−14.9444			−0.524768			−0.262384	−	0.964963i	$-0.584509\pi$
$811$	−14.9444			−0.524768			−0.262384	+	0.964963i	$0.584509\pi$
$812$		0			0
$813$			29.3939i			1.03089i
$814$		0			0
$815$	1.59592	−	15.7980i	0.0559026	−	0.553378i
$816$		0			0
$817$			1.39388i			0.0487656i
$818$		0			0
$819$	−13.3485			−0.466433
$820$		0			0
$821$	−8.20204			−0.286253			−0.143127	−	0.989704i	$-0.545716\pi$
$821$	−8.20204			−0.286253			−0.143127	+	0.989704i	$0.545716\pi$
$822$		0			0
$823$		−	39.1918i		−	1.36614i	−0.730352	−	0.683071i	$-0.760644\pi$
$823$		−	39.1918i		−	1.36614i	0.730352	−	0.683071i	$-0.239356\pi$
$824$		0			0
$825$	−12.0000	+	58.7878i	−0.417786	+	2.04673i
$826$		0			0
$827$		−	15.5959i		−	0.542323i	−0.962534	−	0.271162i	$-0.912592\pi$
$827$		−	15.5959i		−	0.542323i	0.962534	−	0.271162i	$-0.0874078\pi$
$828$		0			0
$829$	−43.6413			−1.51573			−0.757863	−	0.652414i	$-0.773756\pi$
$829$	−43.6413			−1.51573			−0.757863	+	0.652414i	$0.773756\pi$
$830$		0			0
$831$	36.4949			1.26599
$832$		0			0
$833$			2.00000i			0.0692959i
$834$		0			0
$835$	1.10102	−	10.8990i	0.0381024	−	0.377175i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−36.8990			−1.27389			−0.636947	−	0.770907i	$-0.719803\pi$
$839$	−36.8990			−1.27389			−0.636947	+	0.770907i	$0.719803\pi$
$840$		0			0
$841$	18.5959			0.641239
$842$		0			0
$843$		−	44.0908i		−	1.51857i
$844$		0			0
$845$	15.1237	+	1.52781i	0.520272	+	0.0525581i
$846$		0			0
$847$			13.0000i			0.446685i
$848$		0			0
$849$	9.19184			0.315463
$850$		0			0
$851$	−5.79796			−0.198751
$852$		0			0
$853$			33.8434i			1.15877i	0.815052	+	0.579387i	$0.196708\pi$
$853$			33.8434i			1.15877i	−0.815052	+	0.579387i	$0.803292\pi$
$854$		0			0
$855$	10.3485	+	1.04541i	0.353910	+	0.0357522i
$856$		0			0
$857$			53.1918i			1.81700i	0.417886	+	0.908499i	$0.362771\pi$
$857$			53.1918i			1.81700i	−0.417886	+	0.908499i	$0.637229\pi$
$858$		0			0
$859$	53.6413			1.83022			0.915109	−	0.403206i	$-0.132104\pi$
$859$	53.6413			1.83022			0.915109	+	0.403206i	$0.132104\pi$
$860$		0			0
$861$	2.69694			0.0919114
$862$		0			0
$863$		−	45.3939i		−	1.54523i	−0.634878	−	0.772613i	$-0.718949\pi$
$863$		−	45.3939i		−	1.54523i	0.634878	−	0.772613i	$-0.281051\pi$
$864$		0			0
$865$	1.40408	−	13.8990i	0.0477402	−	0.472579i
$866$		0			0
$867$			31.8434i			1.08146i
$868$		0			0
$869$	−33.7980			−1.14652
$870$		0			0
$871$	−35.5959			−1.20612
$872$		0			0
$873$			47.3939i			1.60404i
$874$		0			0
$875$	3.32577	−	10.6742i	0.112431	−	0.360855i
$876$		0			0
$877$		−	39.3939i		−	1.33024i	−0.746738	−	0.665118i	$-0.768381\pi$
$877$		−	39.3939i		−	1.33024i	0.746738	−	0.665118i	$-0.231619\pi$
$878$		0			0
$879$	44.6969			1.50759
$880$		0			0
$881$	8.20204			0.276334			0.138167	−	0.990409i	$-0.455879\pi$
$881$	8.20204			0.276334			0.138167	+	0.990409i	$0.455879\pi$
$882$		0			0
$883$		−	22.2020i		−	0.747158i	−0.927598	−	0.373579i	$-0.878130\pi$
$883$		−	22.2020i		−	0.747158i	0.927598	−	0.373579i	$-0.121870\pi$
$884$		0			0
$885$	−0.853572	+	8.44949i	−0.0286925	+	0.284026i
$886$		0			0
$887$		−	2.69694i		−	0.0905543i	−0.998974	−	0.0452772i	$-0.985583\pi$
$887$		−	2.69694i		−	0.0905543i	0.998974	−	0.0452772i	$-0.0144171\pi$
$888$		0			0
$889$	−14.8990			−0.499696
$890$		0			0
$891$	−44.0908			−1.47710
$892$		0			0
$893$		−	13.7980i		−	0.461731i
$894$		0			0
$895$	−30.6969	−	3.10102i	−1.02609	−	0.103656i
$896$		0			0
$897$			31.5959i			1.05496i
$898$		0			0
$899$	61.3939			2.04760
$900$		0			0
$901$	21.7980			0.726195
$902$		0			0
$903$		−	2.20204i		−	0.0732793i
$904$		0			0
$905$	−22.7980	−	2.30306i	−0.757830	−	0.0765564i
$906$		0			0
$907$		−	41.7980i		−	1.38788i	−0.720034	−	0.693939i	$-0.755874\pi$
$907$		−	41.7980i		−	1.38788i	0.720034	−	0.693939i	$-0.244126\pi$
$908$		0			0
$909$	10.6515			0.353289
$910$		0			0
$911$	−35.5959			−1.17935			−0.589673	−	0.807642i	$-0.700743\pi$
$911$	−35.5959			−1.17935			−0.589673	+	0.807642i	$0.700743\pi$
$912$		0			0
$913$			12.0000i			0.397142i
$914$		0			0
$915$	−1.95459	+	19.3485i	−0.0646168	+	0.639641i
$916$		0			0
$917$			6.44949i			0.212981i
$918$		0			0
$919$	26.8990			0.887315			0.443658	−	0.896196i	$-0.353681\pi$
$919$	26.8990			0.887315			0.443658	+	0.896196i	$0.353681\pi$
$920$		0			0
$921$	49.5959			1.63424
$922$		0			0
$923$			4.89898i			0.161252i
$924$		0			0
$925$	−2.00000	+	9.79796i	−0.0657596	+	0.322155i
$926$		0			0
$927$		−	38.6969i		−	1.27097i
$928$		0			0
$929$	28.2929			0.928259			0.464129	−	0.885767i	$-0.346367\pi$
$929$	28.2929			0.928259			0.464129	+	0.885767i	$0.346367\pi$
$930$		0			0
$931$	−1.55051			−0.0508159
$932$		0			0
$933$			29.3939i			0.962312i
$934$		0			0
$935$	−2.20204	+	21.7980i	−0.0720144	+	0.712869i
$936$		0			0
$937$			41.1010i			1.34271i	0.741135	+	0.671356i	$0.234288\pi$
$937$			41.1010i			1.34271i	−0.741135	+	0.671356i	$0.765712\pi$
$938$		0			0
$939$	52.8990			1.72629
$940$		0			0
$941$	19.5505			0.637328			0.318664	−	0.947868i	$-0.396766\pi$
$941$	19.5505			0.637328			0.318664	+	0.947868i	$0.396766\pi$
$942$		0			0
$943$		−	3.19184i		−	0.103940i
$944$		0			0
$945$		0			0
$946$		0			0
$947$			44.0908i			1.43276i	0.697711	+	0.716379i	$0.254202\pi$
$947$			44.0908i			1.43276i	−0.697711	+	0.716379i	$0.745798\pi$
$948$		0			0
$949$	12.8990			0.418719
$950$		0			0
$951$	55.1010			1.78677
$952$		0			0
$953$			2.20204i			0.0713311i	0.999364	+	0.0356656i	$0.0113551\pi$
$953$			2.20204i			0.0713311i	−0.999364	+	0.0356656i	$0.988645\pi$
$954$		0			0
$955$	−28.2474	−	2.85357i	−0.914066	−	0.0923394i
$956$		0			0
$957$			82.7878i			2.67615i
$958$		0			0
$959$	−1.79796			−0.0580591
$960$		0			0
$961$	48.1918			1.55458
$962$		0			0
$963$			24.0000i			0.773389i
$964$		0			0
$965$	−4.85357	+	48.0454i	−0.156242	+	1.54664i
$966$		0			0
$967$			36.2929i			1.16710i	0.812077	+	0.583550i	$0.198337\pi$
$967$			36.2929i			1.16710i	−0.812077	+	0.583550i	$0.801663\pi$
$968$		0			0
$969$	7.59592			0.244016
$970$		0			0
$971$	9.55051			0.306490			0.153245	−	0.988188i	$-0.451028\pi$
$971$	9.55051			0.306490			0.153245	+	0.988188i	$0.451028\pi$
$972$		0			0
$973$			1.55051i			0.0497071i
$974$		0			0
$975$	53.3939	+	10.8990i	1.70997	+	0.349047i
$976$		0			0
$977$			29.3939i			0.940393i	0.882562	+	0.470197i	$0.155817\pi$
$977$			29.3939i			0.940393i	−0.882562	+	0.470197i	$0.844183\pi$
$978$		0			0
$979$	48.9898			1.56572
$980$		0			0
$981$	−20.6969			−0.660802
$982$		0			0
$983$		−	13.3031i		−	0.424302i	−0.977237	−	0.212151i	$-0.931953\pi$
$983$		−	13.3031i		−	0.424302i	0.977237	−	0.212151i	$-0.0680468\pi$
$984$		0			0
$985$	4.24745	−	42.0454i	0.135335	−	1.33968i
$986$		0			0
$987$			21.7980i			0.693837i
$988$		0			0
$989$	−2.60612			−0.0828699
$990$		0			0
$991$	31.3031			0.994375			0.497187	−	0.867643i	$-0.334366\pi$
$991$	31.3031			0.994375			0.497187	+	0.867643i	$0.334366\pi$
$992$		0			0
$993$			45.7980i			1.45335i
$994$		0			0
$995$	37.5959	+	3.79796i	1.19187	+	0.120403i
$996$		0			0
$997$			57.3485i			1.81624i	0.418705	+	0.908122i	$0.362484\pi$
$997$			57.3485i			1.81624i	−0.418705	+	0.908122i	$0.637516\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2240.2.g.j.449.3		4
4.3	odd	2		2240.2.g.i.449.1		4
5.4	even	2	inner	2240.2.g.j.449.1		4
8.3	odd	2		560.2.g.e.449.4		4
8.5	even	2		70.2.c.a.29.3	yes	4
20.19	odd	2		2240.2.g.i.449.3		4
24.5	odd	2		630.2.g.g.379.1		4
24.11	even	2		5040.2.t.t.1009.1		4
40.3	even	4		2800.2.a.bl.1.1		2
40.13	odd	4		350.2.a.h.1.2		2
40.19	odd	2		560.2.g.e.449.2		4
40.27	even	4		2800.2.a.bm.1.2		2
40.29	even	2		70.2.c.a.29.2	✓	4
40.37	odd	4		350.2.a.g.1.1		2
56.5	odd	6		490.2.i.f.459.3		8
56.13	odd	2		490.2.c.e.99.4		4
56.37	even	6		490.2.i.c.459.4		8
56.45	odd	6		490.2.i.f.79.2		8
56.53	even	6		490.2.i.c.79.1		8
120.29	odd	2		630.2.g.g.379.3		4
120.53	even	4		3150.2.a.bs.1.2		2
120.59	even	2		5040.2.t.t.1009.2		4
120.77	even	4		3150.2.a.bt.1.2		2
280.13	even	4		2450.2.a.bq.1.1		2
280.69	odd	2		490.2.c.e.99.1		4
280.109	even	6		490.2.i.c.79.4		8
280.149	even	6		490.2.i.c.459.1		8
280.229	odd	6		490.2.i.f.459.2		8
280.237	even	4		2450.2.a.bl.1.2		2
280.269	odd	6		490.2.i.f.79.3		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
70.2.c.a.29.2	✓	4	40.29	even	2
70.2.c.a.29.3	yes	4	8.5	even	2
350.2.a.g.1.1		2	40.37	odd	4
350.2.a.h.1.2		2	40.13	odd	4
490.2.c.e.99.1		4	280.69	odd	2
490.2.c.e.99.4		4	56.13	odd	2
490.2.i.c.79.1		8	56.53	even	6
490.2.i.c.79.4		8	280.109	even	6
490.2.i.c.459.1		8	280.149	even	6
490.2.i.c.459.4		8	56.37	even	6
490.2.i.f.79.2		8	56.45	odd	6
490.2.i.f.79.3		8	280.269	odd	6
490.2.i.f.459.2		8	280.229	odd	6
490.2.i.f.459.3		8	56.5	odd	6
560.2.g.e.449.2		4	40.19	odd	2
560.2.g.e.449.4		4	8.3	odd	2
630.2.g.g.379.1		4	24.5	odd	2
630.2.g.g.379.3		4	120.29	odd	2
2240.2.g.i.449.1		4	4.3	odd	2
2240.2.g.i.449.3		4	20.19	odd	2
2240.2.g.j.449.1		4	5.4	even	2	inner
2240.2.g.j.449.3		4	1.1	even	1	trivial
2450.2.a.bl.1.2		2	280.237	even	4
2450.2.a.bq.1.1		2	280.13	even	4
2800.2.a.bl.1.1		2	40.3	even	4
2800.2.a.bm.1.2		2	40.27	even	4
3150.2.a.bs.1.2		2	120.53	even	4
3150.2.a.bt.1.2		2	120.77	even	4
5040.2.t.t.1009.1		4	24.11	even	2
5040.2.t.t.1009.2		4	120.59	even	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:	\( N \)	\(=\)	\( 2240 = 2^{6} \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2240.g (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+2.44949i q^{3} +(-2.22474 - 0.224745i) q^{5} +1.00000i q^{7} -3.00000 q^{9} +4.89898 q^{11} -4.44949i q^{13} +(0.550510 - 5.44949i) q^{15} -2.00000i q^{17} +1.55051 q^{19} -2.44949 q^{21} +2.89898i q^{23} +(4.89898 + 1.00000i) q^{25} +6.89898 q^{29} +8.89898 q^{31} +12.0000i q^{33} +(0.224745 - 2.22474i) q^{35} +2.00000i q^{37} +10.8990 q^{39} -1.10102 q^{41} +0.898979i q^{43} +(6.67423 + 0.674235i) q^{45} -8.89898i q^{47} -1.00000 q^{49} +4.89898 q^{51} +10.8990i q^{53} +(-10.8990 - 1.10102i) q^{55} +3.79796i q^{57} -1.55051 q^{59} -3.55051 q^{61} -3.00000i q^{63} +(-1.00000 + 9.89898i) q^{65} -8.00000i q^{67} -7.10102 q^{69} -1.10102 q^{71} +2.89898i q^{73} +(-2.44949 + 12.0000i) q^{75} +4.89898i q^{77} -6.89898 q^{79} -9.00000 q^{81} +2.44949i q^{83} +(-0.449490 + 4.44949i) q^{85} +16.8990i q^{87} +10.0000 q^{89} +4.44949 q^{91} +21.7980i q^{93} +(-3.44949 - 0.348469i) q^{95} -15.7980i q^{97} -14.6969 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{5} - 12 q^{9} + 12 q^{15} + 16 q^{19} + 8 q^{29} + 16 q^{31} - 4 q^{35} + 24 q^{39} - 24 q^{41} + 12 q^{45} - 4 q^{49} - 24 q^{55} - 16 q^{59} - 24 q^{61} - 4 q^{65} - 48 q^{69} - 24 q^{71} - 8 q^{79}+ \cdots - 4 q^{95}+O(q^{100}) \) 4 * q - 4 * q^5 - 12 * q^9 + 12 * q^15 + 16 * q^19 + 8 * q^29 + 16 * q^31 - 4 * q^35 + 24 * q^39 - 24 * q^41 + 12 * q^45 - 4 * q^49 - 24 * q^55 - 16 * q^59 - 24 * q^61 - 4 * q^65 - 48 * q^69 - 24 * q^71 - 8 * q^79 - 36 * q^81 + 8 * q^85 + 40 * q^89 + 8 * q^91 - 4 * q^95

Introduction

L-functions

Modular forms

Varieties

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Representations

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Database

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