LMFDB - Embedded newform 1120.2.g.b.449.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1120,2,Mod(449,1120)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1120, 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("1120.449");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1120 = 2^{5} \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1120.g (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:	$8.94324502638$
Analytic rank:	$0$
Dimension:	$10$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} + 13x^{8} + 56x^{6} + 97x^{4} + 61x^{2} + 4 $ x^10 + 13x^8 + 56x^6 + 97x^4 + 61x^2 + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			449.6
Root			$1.23118i$ of defining polynomial
Character	$\chi$	$=$	1120.449
Dual form			1120.2.g.b.449.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.746976i q^{3} +(-0.782984 + 2.09450i) q^{5} +1.00000i q^{7} +2.44203 q^{9} +O(q^{10})$	$q+0.746976i q^{3} +(-0.782984 + 2.09450i) q^{5} +1.00000i q^{7} +2.44203 q^{9} +5.90438 q^{11} -3.20933i q^{13} +(-1.56454 - 0.584870i) q^{15} -2.14941i q^{17} +3.56597 q^{19} -0.746976 q^{21} +3.75497i q^{23} +(-3.77387 - 3.27992i) q^{25} +4.06506i q^{27} +6.61177 q^{29} -5.79278 q^{31} +4.41043i q^{33} +(-2.09450 - 0.782984i) q^{35} +0.623035i q^{37} +2.39729 q^{39} -5.43076 q^{41} +12.6768i q^{43} +(-1.91207 + 5.11483i) q^{45} +4.31294i q^{47} -1.00000 q^{49} +1.60556 q^{51} -2.11699i q^{53} +(-4.62304 + 12.3667i) q^{55} +2.66369i q^{57} +7.01926 q^{59} +1.86479 q^{61} +2.44203i q^{63} +(6.72195 + 2.51285i) q^{65} +6.88405i q^{67} -2.80487 q^{69} -1.81816 q^{71} -6.11699i q^{73} +(2.45002 - 2.81899i) q^{75} +5.90438i q^{77} +4.35664 q^{79} +4.28958 q^{81} -13.1675i q^{83} +(4.50195 + 1.68295i) q^{85} +4.93883i q^{87} -12.1708 q^{89} +3.20933 q^{91} -4.32706i q^{93} +(-2.79209 + 7.46892i) q^{95} -9.65935i q^{97} +14.4187 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q - 2 q^{5} - 14 q^{9}+O(q^{10}) $ 10 * q - 2 * q^5 - 14 * q^9	$ 10 q - 2 q^{5} - 14 q^{9} - 8 q^{11} - 4 q^{15} + 24 q^{19} - 4 q^{21} + 6 q^{25} + 24 q^{29} - 24 q^{31} + 64 q^{39} - 4 q^{41} + 10 q^{45} - 10 q^{49} - 24 q^{51} - 16 q^{55} + 32 q^{59} - 20 q^{61} - 8 q^{65} - 8 q^{69} - 8 q^{71} - 64 q^{75} + 64 q^{79} + 2 q^{81} - 12 q^{85} - 4 q^{89} - 60 q^{95} + 80 q^{99}+O(q^{100}) $ 10 * q - 2 * q^5 - 14 * q^9 - 8 * q^11 - 4 * q^15 + 24 * q^19 - 4 * q^21 + 6 * q^25 + 24 * q^29 - 24 * q^31 + 64 * q^39 - 4 * q^41 + 10 * q^45 - 10 * q^49 - 24 * q^51 - 16 * q^55 + 32 * q^59 - 20 * q^61 - 8 * q^65 - 8 * q^69 - 8 * q^71 - 64 * q^75 + 64 * q^79 + 2 * q^81 - 12 * q^85 - 4 * q^89 - 60 * q^95 + 80 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1120\mathbb{Z}\right)^\times$.

$n$	$351$	$421$	$801$	$897$
$\chi(n)$	$1$	$1$	$1$	$-1$

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
$3$			0.746976i			0.431267i	0.976474	+	0.215633i	$0.0691816\pi$
$3$			0.746976i			0.431267i	−0.976474	+	0.215633i	$0.930818\pi$
$4$		0			0
$5$	−0.782984	+	2.09450i	−0.350161	+	0.936690i
$5$	−0.782984	+	2.09450i	−0.350161	+	0.936690i
$6$		0			0
$7$			1.00000i			0.377964i
$7$			1.00000i			0.377964i
$8$		0			0
$9$	2.44203			0.814009
$10$		0			0
$11$	5.90438			1.78024			0.890119	−	0.455728i	$-0.150621\pi$
$11$	5.90438			1.78024			0.890119	+	0.455728i	$0.150621\pi$
$12$		0			0
$13$		−	3.20933i		−	0.890108i	−0.895504	−	0.445054i	$-0.853184\pi$
$13$		−	3.20933i		−	0.890108i	0.895504	−	0.445054i	$-0.146816\pi$
$14$		0			0
$15$	−1.56454	−	0.584870i	−0.403963	−	0.151013i
$16$		0			0
$17$		−	2.14941i		−	0.521309i	−0.965432	−	0.260654i	$-0.916062\pi$
$17$		−	2.14941i		−	0.521309i	0.965432	−	0.260654i	$-0.0839383\pi$
$18$		0			0
$19$	3.56597			0.818089			0.409045	−	0.912514i	$-0.365862\pi$
$19$	3.56597			0.818089			0.409045	+	0.912514i	$0.365862\pi$
$20$		0			0
$21$	−0.746976			−0.163003
$22$		0			0
$23$			3.75497i			0.782965i	0.920185	+	0.391483i	$0.128038\pi$
$23$			3.75497i			0.782965i	−0.920185	+	0.391483i	$0.871962\pi$
$24$		0			0
$25$	−3.77387	−	3.27992i	−0.754775	−	0.655984i
$26$		0			0
$27$			4.06506i			0.782322i
$28$		0			0
$29$	6.61177			1.22777			0.613887	−	0.789394i	$-0.289605\pi$
$29$	6.61177			1.22777			0.613887	+	0.789394i	$0.289605\pi$
$30$		0			0
$31$	−5.79278			−1.04041			−0.520207	−	0.854040i	$-0.674145\pi$
$31$	−5.79278			−1.04041			−0.520207	+	0.854040i	$0.674145\pi$
$32$		0			0
$33$			4.41043i			0.767757i
$34$		0			0
$35$	−2.09450	−	0.782984i	−0.354035	−	0.132348i
$36$		0			0
$37$			0.623035i			0.102426i	0.998688	+	0.0512132i	$0.0163088\pi$
$37$			0.623035i			0.102426i	−0.998688	+	0.0512132i	$0.983691\pi$
$38$		0			0
$39$	2.39729			0.383874
$40$		0			0
$41$	−5.43076			−0.848142			−0.424071	−	0.905629i	$-0.639399\pi$
$41$	−5.43076			−0.848142			−0.424071	+	0.905629i	$0.639399\pi$
$42$		0			0
$43$			12.6768i			1.93320i	0.256293	+	0.966599i	$0.417499\pi$
$43$			12.6768i			1.93320i	−0.256293	+	0.966599i	$0.582501\pi$
$44$		0			0
$45$	−1.91207	+	5.11483i	−0.285034	+	0.762474i
$46$		0			0
$47$			4.31294i			0.629107i	0.949240	+	0.314554i	$0.101855\pi$
$47$			4.31294i			0.629107i	−0.949240	+	0.314554i	$0.898145\pi$
$48$		0			0
$49$	−1.00000			−0.142857
$50$		0			0
$51$	1.60556			0.224823
$52$		0			0
$53$		−	2.11699i		−	0.290791i	−0.989374	−	0.145395i	$-0.953555\pi$
$53$		−	2.11699i		−	0.290791i	0.989374	−	0.145395i	$-0.0464454\pi$
$54$		0			0
$55$	−4.62304	+	12.3667i	−0.623370	+	1.66753i
$56$		0			0
$57$			2.66369i			0.352815i
$58$		0			0
$59$	7.01926			0.913830			0.456915	−	0.889510i	$-0.348954\pi$
$59$	7.01926			0.913830			0.456915	+	0.889510i	$0.348954\pi$
$60$		0			0
$61$	1.86479			0.238762			0.119381	−	0.992849i	$-0.461909\pi$
$61$	1.86479			0.238762			0.119381	+	0.992849i	$0.461909\pi$
$62$		0			0
$63$			2.44203i			0.307667i
$64$		0			0
$65$	6.72195	+	2.51285i	0.833755	+	0.311681i
$66$		0			0
$67$			6.88405i			0.841021i	0.907288	+	0.420511i	$0.138149\pi$
$67$			6.88405i			0.841021i	−0.907288	+	0.420511i	$0.861851\pi$
$68$		0			0
$69$	−2.80487			−0.337667
$70$		0			0
$71$	−1.81816			−0.215776			−0.107888	−	0.994163i	$-0.534409\pi$
$71$	−1.81816			−0.215776			−0.107888	+	0.994163i	$0.534409\pi$
$72$		0			0
$73$		−	6.11699i		−	0.715939i	−0.933733	−	0.357970i	$-0.883469\pi$
$73$		−	6.11699i		−	0.715939i	0.933733	−	0.357970i	$-0.116531\pi$
$74$		0			0
$75$	2.45002	−	2.81899i	0.282904	−	0.325509i
$76$		0			0
$77$			5.90438i			0.672867i
$78$		0			0
$79$	4.35664			0.490160			0.245080	−	0.969503i	$-0.421186\pi$
$79$	4.35664			0.490160			0.245080	+	0.969503i	$0.421186\pi$
$80$		0			0
$81$	4.28958			0.476620
$82$		0			0
$83$		−	13.1675i		−	1.44532i	−0.691203	−	0.722661i	$-0.742919\pi$
$83$		−	13.1675i		−	1.44532i	0.691203	−	0.722661i	$-0.257081\pi$
$84$		0			0
$85$	4.50195	+	1.68295i	0.488305	+	0.182542i
$86$		0			0
$87$			4.93883i			0.529498i
$88$		0			0
$89$	−12.1708			−1.29010			−0.645050	−	0.764140i	$-0.723164\pi$
$89$	−12.1708			−1.29010			−0.645050	+	0.764140i	$0.723164\pi$
$90$		0			0
$91$	3.20933			0.336429
$92$		0			0
$93$		−	4.32706i		−	0.448696i
$94$		0			0
$95$	−2.79209	+	7.46892i	−0.286463	+	0.766295i
$96$		0			0
$97$		−	9.65935i		−	0.980759i	−0.871509	−	0.490379i	$-0.836858\pi$
$97$		−	9.65935i		−	0.980759i	0.871509	−	0.490379i	$-0.163142\pi$
$98$		0			0
$99$	14.4187			1.44913
$100$		0			0
$101$	−13.2307			−1.31650			−0.658252	−	0.752798i	$-0.728704\pi$
$101$	−13.2307			−1.31650			−0.658252	+	0.752798i	$0.728704\pi$
$102$		0			0
$103$			3.19700i			0.315010i	0.987518	+	0.157505i	$0.0503450\pi$
$103$			3.19700i			0.315010i	−0.987518	+	0.157505i	$0.949655\pi$
$104$		0			0
$105$	0.584870	−	1.56454i	0.0570775	−	0.152684i
$106$		0			0
$107$		−	12.5220i		−	1.21055i	−0.796016	−	0.605276i	$-0.793063\pi$
$107$		−	12.5220i		−	1.21055i	0.796016	−	0.605276i	$-0.206937\pi$
$108$		0			0
$109$	−14.1217			−1.35261			−0.676307	−	0.736620i	$-0.736421\pi$
$109$	−14.1217			−1.35261			−0.676307	+	0.736620i	$0.736421\pi$
$110$		0			0
$111$	−0.465392			−0.0441731
$112$		0			0
$113$			1.63513i			0.153820i	0.997038	+	0.0769102i	$0.0245055\pi$
$113$			1.63513i			0.153820i	−0.997038	+	0.0769102i	$0.975495\pi$
$114$		0			0
$115$	−7.86479	−	2.94008i	−0.733396	−	0.274164i
$116$		0			0
$117$		−	7.83727i		−	0.724556i
$118$		0			0
$119$	2.14941			0.197036
$120$		0			0
$121$	23.8617			2.16925
$122$		0			0
$123$		−	4.05665i		−	0.365775i
$124$		0			0
$125$	9.82468	−	5.33626i	0.878746	−	0.477289i
$126$		0			0
$127$		−	1.79563i		−	0.159336i	−0.996821	−	0.0796680i	$-0.974614\pi$
$127$		−	1.79563i		−	0.159336i	0.996821	−	0.0796680i	$-0.0253860\pi$
$128$		0			0
$129$	−9.46928			−0.833724
$130$		0			0
$131$	−12.3868			−1.08224			−0.541121	−	0.840945i	$-0.682000\pi$
$131$	−12.3868			−1.08224			−0.541121	+	0.840945i	$0.682000\pi$
$132$		0			0
$133$			3.56597i			0.309209i
$134$		0			0
$135$	−8.51428	−	3.18288i	−0.732793	−	0.273939i
$136$		0			0
$137$			14.3940i			1.22976i	0.788620	+	0.614881i	$0.210796\pi$
$137$			14.3940i			1.22976i	−0.788620	+	0.614881i	$0.789204\pi$
$138$		0			0
$139$	22.7488			1.92953			0.964766	−	0.263110i	$-0.0847481\pi$
$139$	22.7488			1.92953			0.964766	+	0.263110i	$0.0847481\pi$
$140$		0			0
$141$	−3.22166			−0.271313
$142$		0			0
$143$		−	18.9491i		−	1.58460i
$144$		0			0
$145$	−5.17691	+	13.8484i	−0.429919	+	1.15004i
$146$		0			0
$147$		−	0.746976i		−	0.0616095i
$148$		0			0
$149$	18.7560			1.53655			0.768276	−	0.640119i	$-0.221115\pi$
$149$	18.7560			1.53655			0.768276	+	0.640119i	$0.221115\pi$
$150$		0			0
$151$	−22.2824			−1.81332			−0.906658	−	0.421867i	$-0.861375\pi$
$151$	−22.2824			−1.81332			−0.906658	+	0.421867i	$0.861375\pi$
$152$		0			0
$153$		−	5.24892i		−	0.424350i
$154$		0			0
$155$	4.53565	−	12.1330i	0.364312	−	0.974544i
$156$		0			0
$157$			16.7110i			1.33369i	0.745198	+	0.666843i	$0.232355\pi$
$157$			16.7110i			1.33369i	−0.745198	+	0.666843i	$0.767645\pi$
$158$		0			0
$159$	1.58134			0.125408
$160$		0			0
$161$	−3.75497			−0.295933
$162$		0			0
$163$			16.3555i			1.28106i	0.767933	+	0.640530i	$0.221285\pi$
$163$			16.3555i			1.28106i	−0.767933	+	0.640530i	$0.778715\pi$
$164$		0			0
$165$	−9.23765	−	3.45330i	−0.719150	−	0.268839i
$166$		0			0
$167$			3.66452i			0.283569i	0.989898	+	0.141785i	$0.0452840\pi$
$167$			3.66452i			0.283569i	−0.989898	+	0.141785i	$0.954716\pi$
$168$		0			0
$169$	2.70019			0.207707
$170$		0			0
$171$	8.70819			0.665932
$172$		0			0
$173$		−	8.37622i		−	0.636832i	−0.947951	−	0.318416i	$-0.896849\pi$
$173$		−	8.37622i		−	0.636832i	0.947951	−	0.318416i	$-0.103151\pi$
$174$		0			0
$175$	3.27992	−	3.77387i	0.247939	−	0.285278i
$176$		0			0
$177$			5.24322i			0.394105i
$178$		0			0
$179$	1.73613			0.129764			0.0648822	−	0.997893i	$-0.479333\pi$
$179$	1.73613			0.129764			0.0648822	+	0.997893i	$0.479333\pi$
$180$		0			0
$181$	13.4621			1.00063			0.500316	−	0.865843i	$-0.333217\pi$
$181$	13.4621			1.00063			0.500316	+	0.865843i	$0.333217\pi$
$182$		0			0
$183$			1.39295i			0.102970i
$184$		0			0
$185$	−1.30495	−	0.487827i	−0.0959417	−	0.0358657i
$186$		0			0
$187$		−	12.6909i		−	0.928054i
$188$		0			0
$189$	−4.06506			−0.295690
$190$		0			0
$191$	−6.91933			−0.500665			−0.250333	−	0.968160i	$-0.580540\pi$
$191$	−6.91933			−0.500665			−0.250333	+	0.968160i	$0.580540\pi$
$192$		0			0
$193$			8.32421i			0.599190i	0.954066	+	0.299595i	$0.0968515\pi$
$193$			8.32421i			0.599190i	−0.954066	+	0.299595i	$0.903148\pi$
$194$		0			0
$195$	−1.87704	+	5.02113i	−0.134418	+	0.359571i
$196$		0			0
$197$		−	17.6044i		−	1.25426i	−0.778914	−	0.627130i	$-0.784229\pi$
$197$		−	17.6044i		−	1.25426i	0.778914	−	0.627130i	$-0.215771\pi$
$198$		0			0
$199$	4.56924			0.323905			0.161952	−	0.986799i	$-0.448221\pi$
$199$	4.56924			0.323905			0.161952	+	0.986799i	$0.448221\pi$
$200$		0			0
$201$	−5.14222			−0.362704
$202$		0			0
$203$			6.61177i			0.464055i
$204$		0			0
$205$	4.25220	−	11.3747i	0.296986	−	0.794446i
$206$		0			0
$207$			9.16974i			0.637341i
$208$		0			0
$209$	21.0548			1.45639
$210$		0			0
$211$	−28.2582			−1.94537			−0.972687	−	0.232120i	$-0.925434\pi$
$211$	−28.2582			−1.94537			−0.972687	+	0.232120i	$0.925434\pi$
$212$		0			0
$213$		−	1.35812i		−	0.0930571i
$214$		0			0
$215$	−26.5516	−	9.92575i	−1.81081	−	0.676931i
$216$		0			0
$217$		−	5.79278i		−	0.393239i
$218$		0			0
$219$	4.56924			0.308761
$220$		0			0
$221$	−6.89817			−0.464021
$222$		0			0
$223$		−	25.2130i		−	1.68839i	−0.536039	−	0.844193i	$-0.680080\pi$
$223$		−	25.2130i		−	1.68839i	0.536039	−	0.844193i	$-0.319920\pi$
$224$		0			0
$225$	−9.21590	−	8.00966i	−0.614393	−	0.533977i
$226$		0			0
$227$		−	17.5030i		−	1.16171i	−0.814006	−	0.580857i	$-0.802718\pi$
$227$		−	17.5030i		−	1.16171i	0.814006	−	0.580857i	$-0.197282\pi$
$228$		0			0
$229$	0.226808			0.0149879			0.00749394	−	0.999972i	$-0.497615\pi$
$229$	0.226808			0.0149879			0.00749394	+	0.999972i	$0.497615\pi$
$230$		0			0
$231$	−4.41043			−0.290185
$232$		0			0
$233$		−	27.3836i		−	1.79396i	−0.442075	−	0.896978i	$-0.645757\pi$
$233$		−	27.3836i		−	1.79396i	0.442075	−	0.896978i	$-0.354243\pi$
$234$		0			0
$235$	−9.03347	−	3.37696i	−0.589278	−	0.220289i
$236$		0			0
$237$			3.25430i			0.211390i
$238$		0			0
$239$	10.2504			0.663044			0.331522	−	0.943448i	$-0.392438\pi$
$239$	10.2504			0.663044			0.331522	+	0.943448i	$0.392438\pi$
$240$		0			0
$241$	−20.9407			−1.34891			−0.674455	−	0.738316i	$-0.735621\pi$
$241$	−20.9407			−1.34891			−0.674455	+	0.738316i	$0.735621\pi$
$242$		0			0
$243$			15.3994i			0.987872i
$244$		0			0
$245$	0.782984	−	2.09450i	0.0500230	−	0.133813i
$246$		0			0
$247$		−	11.4444i		−	0.728188i
$248$		0			0
$249$	9.83581			0.623319
$250$		0			0
$251$	−4.25670			−0.268681			−0.134340	−	0.990935i	$-0.542892\pi$
$251$	−4.25670			−0.268681			−0.134340	+	0.990935i	$0.542892\pi$
$252$		0			0
$253$			22.1708i			1.39387i
$254$		0			0
$255$	−1.25713	+	3.36284i	−0.0787243	+	0.210590i
$256$		0			0
$257$			0.636942i			0.0397314i	0.999803	+	0.0198657i	$0.00632386\pi$
$257$			0.636942i			0.0397314i	−0.999803	+	0.0198657i	$0.993676\pi$
$258$		0			0
$259$	−0.623035			−0.0387135
$260$		0			0
$261$	16.1461			0.999419
$262$		0			0
$263$		−	24.7560i		−	1.52652i	−0.646091	−	0.763261i	$-0.723597\pi$
$263$		−	24.7560i		−	1.52652i	0.646091	−	0.763261i	$-0.276403\pi$
$264$		0			0
$265$	4.43403	+	1.65757i	0.272380	+	0.101823i
$266$		0			0
$267$		−	9.09128i		−	0.556377i
$268$		0			0
$269$	0.594054			0.0362201			0.0181100	−	0.999836i	$-0.494235\pi$
$269$	0.594054			0.0362201			0.0181100	+	0.999836i	$0.494235\pi$
$270$		0			0
$271$	28.4774			1.72988			0.864939	−	0.501877i	$-0.167357\pi$
$271$	28.4774			1.72988			0.864939	+	0.501877i	$0.167357\pi$
$272$		0			0
$273$			2.39729i			0.145091i
$274$		0			0
$275$	−22.2824	−	19.3659i	−1.34368	−	1.16781i
$276$		0			0
$277$		−	13.1049i		−	0.787397i	−0.919240	−	0.393698i	$-0.871195\pi$
$277$		−	13.1049i		−	0.787397i	0.919240	−	0.393698i	$-0.128805\pi$
$278$		0			0
$279$	−14.1461			−0.846906
$280$		0			0
$281$	1.24420			0.0742228			0.0371114	−	0.999311i	$-0.488184\pi$
$281$	1.24420			0.0742228			0.0371114	+	0.999311i	$0.488184\pi$
$282$		0			0
$283$		−	0.476558i		−	0.0283284i	−0.999900	−	0.0141642i	$-0.995491\pi$
$283$		−	0.476558i		−	0.0283284i	0.999900	−	0.0141642i	$-0.00450876\pi$
$284$		0			0
$285$	−5.57911	−	2.08563i	−0.330478	−	0.123542i
$286$		0			0
$287$		−	5.43076i		−	0.320568i
$288$		0			0
$289$	12.3800			0.728237
$290$		0			0
$291$	7.21530			0.422969
$292$		0			0
$293$			32.4190i			1.89394i	0.321328	+	0.946968i	$0.395871\pi$
$293$			32.4190i			1.89394i	−0.321328	+	0.946968i	$0.604129\pi$
$294$		0			0
$295$	−5.49597	+	14.7019i	−0.319988	+	0.855975i
$296$		0			0
$297$			24.0017i			1.39272i
$298$		0			0
$299$	12.0509			0.696924
$300$		0			0
$301$	−12.6768			−0.730680
$302$		0			0
$303$		−	9.88301i		−	0.567764i
$304$		0			0
$305$	−1.46010	+	3.90581i	−0.0836052	+	0.223646i
$306$		0			0
$307$		−	0.718737i		−	0.0410205i	−0.999790	−	0.0205102i	$-0.993471\pi$
$307$		−	0.718737i		−	0.0410205i	0.999790	−	0.0205102i	$-0.00652907\pi$
$308$		0			0
$309$	−2.38808			−0.135853
$310$		0			0
$311$	−19.1237			−1.08441			−0.542204	−	0.840247i	$-0.682410\pi$
$311$	−19.1237			−1.08441			−0.542204	+	0.840247i	$0.682410\pi$
$312$		0			0
$313$			24.4560i			1.38234i	0.722694	+	0.691168i	$0.242904\pi$
$313$			24.4560i			1.38234i	−0.722694	+	0.691168i	$0.757096\pi$
$314$		0			0
$315$	−5.11483	−	1.91207i	−0.288188	−	0.107733i
$316$		0			0
$317$		−	4.08203i		−	0.229270i	−0.993408	−	0.114635i	$-0.963430\pi$
$317$		−	4.08203i		−	0.229270i	0.993408	−	0.114635i	$-0.0365698\pi$
$318$		0			0
$319$	39.0384			2.18573
$320$		0			0
$321$	9.35366			0.522070
$322$		0			0
$323$		−	7.66473i		−	0.426477i
$324$		0			0
$325$	−10.5264	+	12.1116i	−0.583897	+	0.671831i
$326$		0			0
$327$		−	10.5486i		−	0.583337i
$328$		0			0
$329$	−4.31294			−0.237780
$330$		0			0
$331$	5.70539			0.313597			0.156798	−	0.987631i	$-0.449883\pi$
$331$	5.70539			0.313597			0.156798	+	0.987631i	$0.449883\pi$
$332$		0			0
$333$			1.52147i			0.0833760i
$334$		0			0
$335$	−14.4187	−	5.39010i	−0.787776	−	0.294493i
$336$		0			0
$337$		−	19.5071i		−	1.06262i	−0.847178	−	0.531309i	$-0.821700\pi$
$337$		−	19.5071i		−	1.06262i	0.847178	−	0.531309i	$-0.178300\pi$
$338$		0			0
$339$	−1.22140			−0.0663376
$340$		0			0
$341$	−34.2028			−1.85218
$342$		0			0
$343$		−	1.00000i		−	0.0539949i
$344$		0			0
$345$	2.19617	−	5.87481i	0.118238	−	0.316289i
$346$		0			0
$347$		−	30.5488i		−	1.63994i	−0.572403	−	0.819972i	$-0.693989\pi$
$347$		−	30.5488i		−	1.63994i	0.572403	−	0.819972i	$-0.306011\pi$
$348$		0			0
$349$	21.1043			1.12969			0.564844	−	0.825198i	$-0.308936\pi$
$349$	21.1043			1.12969			0.564844	+	0.825198i	$0.308936\pi$
$350$		0			0
$351$	13.0461			0.696351
$352$		0			0
$353$		−	25.4989i		−	1.35717i	−0.734523	−	0.678584i	$-0.762594\pi$
$353$		−	25.4989i		−	1.35717i	0.734523	−	0.678584i	$-0.237406\pi$
$354$		0			0
$355$	1.42359	−	3.80815i	0.0755564	−	0.202115i
$356$		0			0
$357$			1.60556i			0.0849752i
$358$		0			0
$359$	22.3658			1.18042			0.590210	−	0.807250i	$-0.299045\pi$
$359$	22.3658			1.18042			0.590210	+	0.807250i	$0.299045\pi$
$360$		0			0
$361$	−6.28388			−0.330730
$362$		0			0
$363$			17.8241i			0.935524i
$364$		0			0
$365$	12.8120	+	4.78950i	0.670613	+	0.250694i
$366$		0			0
$367$			18.0126i			0.940252i	0.882599	+	0.470126i	$0.155791\pi$
$367$			18.0126i			0.940252i	−0.882599	+	0.470126i	$0.844209\pi$
$368$		0			0
$369$	−13.2621			−0.690395
$370$		0			0
$371$	2.11699			0.109908
$372$		0			0
$373$		−	22.8397i		−	1.18259i	−0.806454	−	0.591297i	$-0.798616\pi$
$373$		−	22.8397i		−	1.18259i	0.806454	−	0.591297i	$-0.201384\pi$
$374$		0			0
$375$	3.98606	+	7.33880i	0.205839	+	0.378974i
$376$		0			0
$377$		−	21.2193i		−	1.09285i
$378$		0			0
$379$	−5.98775			−0.307570			−0.153785	−	0.988104i	$-0.549146\pi$
$379$	−5.98775			−0.307570			−0.153785	+	0.988104i	$0.549146\pi$
$380$		0			0
$381$	1.34129			0.0687163
$382$		0			0
$383$		−	25.6298i		−	1.30962i	−0.755793	−	0.654810i	$-0.772749\pi$
$383$		−	25.6298i		−	1.30962i	0.755793	−	0.654810i	$-0.227251\pi$
$384$		0			0
$385$	−12.3667	−	4.62304i	−0.630267	−	0.235612i
$386$		0			0
$387$			30.9572i			1.57364i
$388$		0			0
$389$	−24.3043			−1.23228			−0.616138	−	0.787639i	$-0.711303\pi$
$389$	−24.3043			−1.23228			−0.616138	+	0.787639i	$0.711303\pi$
$390$		0			0
$391$	8.07098			0.408167
$392$		0			0
$393$		−	9.25266i		−	0.466735i
$394$		0			0
$395$	−3.41118	+	9.12498i	−0.171635	+	0.459128i
$396$		0			0
$397$			3.65219i			0.183298i	0.995791	+	0.0916490i	$0.0292138\pi$
$397$			3.65219i			0.183298i	−0.995791	+	0.0916490i	$0.970786\pi$
$398$		0			0
$399$	−2.66369			−0.133351
$400$		0			0
$401$	16.3543			0.816696			0.408348	−	0.912826i	$-0.366105\pi$
$401$	16.3543			0.816696			0.408348	+	0.912826i	$0.366105\pi$
$402$		0			0
$403$			18.5909i			0.926080i
$404$		0			0
$405$	−3.35867	+	8.98453i	−0.166894	+	0.446445i
$406$		0			0
$407$			3.67864i			0.182343i
$408$		0			0
$409$	−35.8675			−1.77353			−0.886767	−	0.462217i	$-0.847054\pi$
$409$	−35.8675			−1.77353			−0.886767	+	0.462217i	$0.847054\pi$
$410$		0			0
$411$	−10.7520			−0.530355
$412$		0			0
$413$			7.01926i			0.345395i
$414$		0			0
$415$	27.5794	+	10.3099i	1.35382	+	0.506095i
$416$		0			0
$417$			16.9928i			0.832143i
$418$		0			0
$419$	−22.5741			−1.10282			−0.551408	−	0.834236i	$-0.685909\pi$
$419$	−22.5741			−1.10282			−0.551408	+	0.834236i	$0.685909\pi$
$420$		0			0
$421$	19.5931			0.954910			0.477455	−	0.878656i	$-0.341559\pi$
$421$	19.5931			0.954910			0.477455	+	0.878656i	$0.341559\pi$
$422$		0			0
$423$			10.5323i			0.512099i
$424$		0			0
$425$	−7.04990	+	8.11161i	−0.341970	+	0.393471i
$426$		0			0
$427$			1.86479i			0.0902436i
$428$		0			0
$429$	14.1545			0.683387
$430$		0			0
$431$	−4.32304			−0.208234			−0.104117	−	0.994565i	$-0.533202\pi$
$431$	−4.32304			−0.208234			−0.104117	+	0.994565i	$0.533202\pi$
$432$		0			0
$433$			3.71479i			0.178521i	0.996008	+	0.0892606i	$0.0284504\pi$
$433$			3.71479i			0.178521i	−0.996008	+	0.0892606i	$0.971550\pi$
$434$		0			0
$435$	−10.3444	−	3.86702i	−0.495975	−	0.185410i
$436$		0			0
$437$			13.3901i			0.640535i
$438$		0			0
$439$	11.0528			0.527519			0.263760	−	0.964588i	$-0.415037\pi$
$439$	11.0528			0.527519			0.263760	+	0.964588i	$0.415037\pi$
$440$		0			0
$441$	−2.44203			−0.116287
$442$		0			0
$443$		−	0.538500i		−	0.0255849i	−0.999918	−	0.0127925i	$-0.995928\pi$
$443$		−	0.538500i		−	0.0255849i	0.999918	−	0.0127925i	$-0.00407208\pi$
$444$		0			0
$445$	9.52952	−	25.4917i	0.451743	−	1.20842i
$446$		0			0
$447$			14.0103i			0.662664i
$448$		0			0
$449$	9.38003			0.442671			0.221335	−	0.975198i	$-0.428958\pi$
$449$	9.38003			0.442671			0.221335	+	0.975198i	$0.428958\pi$
$450$		0			0
$451$	−32.0653			−1.50989
$452$		0			0
$453$		−	16.6444i		−	0.782022i
$454$		0			0
$455$	−2.51285	+	6.72195i	−0.117804	+	0.315130i
$456$		0			0
$457$		−	23.7242i		−	1.10977i	−0.831926	−	0.554886i	$-0.812762\pi$
$457$		−	23.7242i		−	1.10977i	0.831926	−	0.554886i	$-0.187238\pi$
$458$		0			0
$459$	8.73749			0.407831
$460$		0			0
$461$	36.8198			1.71487			0.857435	−	0.514592i	$-0.172057\pi$
$461$	36.8198			1.71487			0.857435	+	0.514592i	$0.172057\pi$
$462$		0			0
$463$		−	14.3287i		−	0.665912i	−0.942942	−	0.332956i	$-0.891954\pi$
$463$		−	14.3287i		−	0.665912i	0.942942	−	0.332956i	$-0.108046\pi$
$464$		0			0
$465$	9.06304	+	3.38802i	0.420288	+	0.157116i
$466$		0			0
$467$			18.3887i			0.850927i	0.904975	+	0.425464i	$0.139889\pi$
$467$			18.3887i			0.850927i	−0.904975	+	0.425464i	$0.860111\pi$
$468$		0			0
$469$	−6.88405			−0.317876
$470$		0			0
$471$	−12.4827			−0.575174
$472$		0			0
$473$			74.8488i			3.44155i
$474$		0			0
$475$	−13.4575	−	11.6961i	−0.617473	−	0.536654i
$476$		0			0
$477$		−	5.16974i		−	0.236706i
$478$		0			0
$479$	26.1400			1.19437			0.597184	−	0.802104i	$-0.296286\pi$
$479$	26.1400			1.19437			0.597184	+	0.802104i	$0.296286\pi$
$480$		0			0
$481$	1.99953			0.0911706
$482$		0			0
$483$		−	2.80487i		−	0.127626i
$484$		0			0
$485$	20.2315	+	7.56312i	0.918666	+	0.343423i
$486$		0			0
$487$			7.27567i			0.329692i	0.986319	+	0.164846i	$0.0527127\pi$
$487$			7.27567i			0.329692i	−0.986319	+	0.164846i	$0.947287\pi$
$488$		0			0
$489$	−12.2171			−0.552478
$490$		0			0
$491$	−7.73616			−0.349128			−0.174564	−	0.984646i	$-0.555852\pi$
$491$	−7.73616			−0.349128			−0.174564	+	0.984646i	$0.555852\pi$
$492$		0			0
$493$		−	14.2114i		−	0.640050i
$494$		0			0
$495$	−11.2896	+	30.1999i	−0.507429	+	1.35738i
$496$		0			0
$497$		−	1.81816i		−	0.0815558i
$498$		0			0
$499$	−8.66694			−0.387986			−0.193993	−	0.981003i	$-0.562144\pi$
$499$	−8.66694			−0.387986			−0.193993	+	0.981003i	$0.562144\pi$
$500$		0			0
$501$	−2.73731			−0.122294
$502$		0			0
$503$			8.28220i			0.369285i	0.982806	+	0.184643i	$0.0591127\pi$
$503$			8.28220i			0.369285i	−0.982806	+	0.184643i	$0.940887\pi$
$504$		0			0
$505$	10.3594	−	27.7117i	0.460988	−	1.23316i
$506$		0			0
$507$			2.01698i			0.0895772i
$508$		0			0
$509$	−34.9114			−1.54742			−0.773711	−	0.633539i	$-0.781602\pi$
$509$	−34.9114			−1.54742			−0.773711	+	0.633539i	$0.781602\pi$
$510$		0			0
$511$	6.11699			0.270600
$512$		0			0
$513$			14.4959i			0.640009i
$514$		0			0
$515$	−6.69612	−	2.50320i	−0.295066	−	0.110304i
$516$		0			0
$517$			25.4653i			1.11996i
$518$		0			0
$519$	6.25683			0.274645
$520$		0			0
$521$	12.9125			0.565705			0.282853	−	0.959163i	$-0.408719\pi$
$521$	12.9125			0.565705			0.282853	+	0.959163i	$0.408719\pi$
$522$		0			0
$523$			24.4885i			1.07081i	0.844596	+	0.535405i	$0.179841\pi$
$523$			24.4885i			1.07081i	−0.844596	+	0.535405i	$0.820159\pi$
$524$		0			0
$525$	2.81899	+	2.45002i	0.123031	+	0.106928i
$526$		0			0
$527$			12.4511i			0.542377i
$528$		0			0
$529$	8.90020			0.386965
$530$		0			0
$531$	17.1412			0.743866
$532$		0			0
$533$			17.4291i			0.754938i
$534$		0			0
$535$	26.2274	+	9.80455i	1.13391	+	0.423888i
$536$		0			0
$537$			1.29685i			0.0559631i
$538$		0			0
$539$	−5.90438			−0.254320
$540$		0			0
$541$	−10.0318			−0.431299			−0.215650	−	0.976471i	$-0.569187\pi$
$541$	−10.0318			−0.431299			−0.215650	+	0.976471i	$0.569187\pi$
$542$		0			0
$543$			10.0559i			0.431539i
$544$		0			0
$545$	11.0571	−	29.5779i	0.473633	−	1.26698i
$546$		0			0
$547$			21.3310i			0.912046i	0.889968	+	0.456023i	$0.150726\pi$
$547$			21.3310i			0.912046i	−0.889968	+	0.456023i	$0.849274\pi$
$548$		0			0
$549$	4.55387			0.194354
$550$		0			0
$551$	23.5773			1.00443
$552$		0			0
$553$			4.35664i			0.185263i
$554$		0			0
$555$	0.364395	−	0.974765i	0.0154677	−	0.0413765i
$556$		0			0
$557$		−	32.4179i		−	1.37359i	−0.726851	−	0.686795i	$-0.759017\pi$
$557$		−	32.4179i		−	1.37359i	0.726851	−	0.686795i	$-0.240983\pi$
$558$		0			0
$559$	40.6841			1.72076
$560$		0			0
$561$	9.47983			0.400239
$562$		0			0
$563$			44.8971i			1.89219i	0.323894	+	0.946093i	$0.395008\pi$
$563$			44.8971i			1.89219i	−0.323894	+	0.946093i	$0.604992\pi$
$564$		0			0
$565$	−3.42479	−	1.28028i	−0.144082	−	0.0538619i
$566$		0			0
$567$			4.28958i			0.180145i
$568$		0			0
$569$	16.1454			0.676850			0.338425	−	0.940993i	$-0.390106\pi$
$569$	16.1454			0.676850			0.338425	+	0.940993i	$0.390106\pi$
$570$		0			0
$571$	−14.1954			−0.594061			−0.297031	−	0.954868i	$-0.595996\pi$
$571$	−14.1954			−0.594061			−0.297031	+	0.954868i	$0.595996\pi$
$572$		0			0
$573$		−	5.16857i		−	0.215920i
$574$		0			0
$575$	12.3160	−	14.1708i	0.513613	−	0.590962i
$576$		0			0
$577$		−	14.0481i		−	0.584831i	−0.956291	−	0.292415i	$-0.905541\pi$
$577$		−	14.0481i		−	0.584831i	0.956291	−	0.292415i	$-0.0944590\pi$
$578$		0			0
$579$	−6.21798			−0.258411
$580$		0			0
$581$	13.1675			0.546280
$582$		0			0
$583$		−	12.4995i		−	0.517676i
$584$		0			0
$585$	16.4152	+	6.13646i	0.678684	+	0.253711i
$586$		0			0
$587$		−	20.4742i		−	0.845061i	−0.906349	−	0.422531i	$-0.861142\pi$
$587$		−	20.4742i		−	0.845061i	0.906349	−	0.422531i	$-0.138858\pi$
$588$		0			0
$589$	−20.6568			−0.851151
$590$		0			0
$591$	13.1501			0.540921
$592$		0			0
$593$		−	16.1372i		−	0.662674i	−0.943513	−	0.331337i	$-0.892500\pi$
$593$		−	16.1372i		−	0.662674i	0.943513	−	0.331337i	$-0.107500\pi$
$594$		0			0
$595$	−1.68295	+	4.50195i	−0.0689944	+	0.184562i
$596$		0			0
$597$			3.41311i			0.139689i
$598$		0			0
$599$	−39.5597			−1.61637			−0.808183	−	0.588931i	$-0.799549\pi$
$599$	−39.5597			−1.61637			−0.808183	+	0.588931i	$0.799549\pi$
$600$		0			0
$601$	−29.4061			−1.19950			−0.599750	−	0.800188i	$-0.704733\pi$
$601$	−29.4061			−1.19950			−0.599750	+	0.800188i	$0.704733\pi$
$602$		0			0
$603$			16.8110i			0.684599i
$604$		0			0
$605$	−18.6833	+	49.9784i	−0.759586	+	2.03191i
$606$		0			0
$607$			22.1165i			0.897680i	0.893612	+	0.448840i	$0.148163\pi$
$607$			22.1165i			0.897680i	−0.893612	+	0.448840i	$0.851837\pi$
$608$		0			0
$609$	−4.93883			−0.200131
$610$		0			0
$611$	13.8417			0.559974
$612$		0			0
$613$		−	27.1411i		−	1.09622i	−0.836407	−	0.548109i	$-0.815348\pi$
$613$		−	27.1411i		−	1.09622i	0.836407	−	0.548109i	$-0.184652\pi$
$614$		0			0
$615$	8.49665	+	3.17629i	0.342618	+	0.128080i
$616$		0			0
$617$		−	1.66337i		−	0.0669648i	−0.999439	−	0.0334824i	$-0.989340\pi$
$617$		−	1.66337i		−	0.0669648i	0.999439	−	0.0334824i	$-0.0106598\pi$
$618$		0			0
$619$	19.8283			0.796969			0.398484	−	0.917175i	$-0.369536\pi$
$619$	19.8283			0.796969			0.398484	+	0.917175i	$0.369536\pi$
$620$		0			0
$621$	−15.2642			−0.612531
$622$		0			0
$623$		−	12.1708i		−	0.487612i
$624$		0			0
$625$	3.48423	+	24.7560i	0.139369	+	0.990240i
$626$		0			0
$627$			15.7275i			0.628094i
$628$		0			0
$629$	1.33916			0.0533958
$630$		0			0
$631$	−20.1113			−0.800619			−0.400309	−	0.916380i	$-0.631097\pi$
$631$	−20.1113			−0.800619			−0.400309	+	0.916380i	$0.631097\pi$
$632$		0			0
$633$		−	21.1082i		−	0.838975i
$634$		0			0
$635$	3.76094	+	1.40595i	0.149248	+	0.0557933i
$636$		0			0
$637$			3.20933i			0.127158i
$638$		0			0
$639$	−4.44000			−0.175644
$640$		0			0
$641$	−38.5135			−1.52119			−0.760597	−	0.649225i	$-0.775093\pi$
$641$	−38.5135			−1.52119			−0.760597	+	0.649225i	$0.775093\pi$
$642$		0			0
$643$			39.3844i			1.55317i	0.630013	+	0.776584i	$0.283049\pi$
$643$			39.3844i			1.55317i	−0.630013	+	0.776584i	$0.716951\pi$
$644$		0			0
$645$	7.41430	−	19.8334i	0.291938	−	0.780941i
$646$		0			0
$647$		−	22.1010i		−	0.868882i	−0.900700	−	0.434441i	$-0.856946\pi$
$647$		−	22.1010i		−	0.868882i	0.900700	−	0.434441i	$-0.143054\pi$
$648$		0			0
$649$	41.4444			1.62684
$650$		0			0
$651$	4.32706			0.169591
$652$		0			0
$653$		−	26.5824i		−	1.04025i	−0.854090	−	0.520124i	$-0.825886\pi$
$653$		−	26.5824i		−	1.04025i	0.854090	−	0.520124i	$-0.174114\pi$
$654$		0			0
$655$	9.69869	−	25.9442i	0.378959	−	1.01373i
$656$		0			0
$657$		−	14.9378i		−	0.582781i
$658$		0			0
$659$	12.6953			0.494541			0.247270	−	0.968947i	$-0.420466\pi$
$659$	12.6953			0.494541			0.247270	+	0.968947i	$0.420466\pi$
$660$		0			0
$661$	2.67371			0.103995			0.0519976	−	0.998647i	$-0.483441\pi$
$661$	2.67371			0.103995			0.0519976	+	0.998647i	$0.483441\pi$
$662$		0			0
$663$		−	5.15277i		−	0.200117i
$664$		0			0
$665$	−7.46892	−	2.79209i	−0.289632	−	0.108273i
$666$		0			0
$667$			24.8270i			0.961305i
$668$		0			0
$669$	18.8335			0.728145
$670$		0			0
$671$	11.0104			0.425053
$672$		0			0
$673$			25.2235i			0.972296i	0.873877	+	0.486148i	$0.161598\pi$
$673$			25.2235i			0.972296i	−0.873877	+	0.486148i	$0.838402\pi$
$674$		0			0
$675$	13.3331	−	15.3410i	0.513191	−	0.590476i
$676$		0			0
$677$			8.88750i			0.341574i	0.985308	+	0.170787i	$0.0546310\pi$
$677$			8.88750i			0.341574i	−0.985308	+	0.170787i	$0.945369\pi$
$678$		0			0
$679$	9.65935			0.370692
$680$		0			0
$681$	13.0743			0.501009
$682$		0			0
$683$			36.7217i			1.40512i	0.711626	+	0.702558i	$0.247959\pi$
$683$			36.7217i			1.40512i	−0.711626	+	0.702558i	$0.752041\pi$
$684$		0			0
$685$	−30.1482	−	11.2703i	−1.15191	−	0.430615i
$686$		0			0
$687$			0.169420i			0.00646377i
$688$		0			0
$689$	−6.79411			−0.258835
$690$		0			0
$691$	−22.9721			−0.873898			−0.436949	−	0.899486i	$-0.643941\pi$
$691$	−22.9721			−0.873898			−0.436949	+	0.899486i	$0.643941\pi$
$692$		0			0
$693$			14.4187i			0.547720i
$694$		0			0
$695$	−17.8120	+	47.6475i	−0.675647	+	1.80737i
$696$		0			0
$697$			11.6729i			0.442144i
$698$		0			0
$699$	20.4549			0.773674
$700$		0			0
$701$	−2.86785			−0.108317			−0.0541586	−	0.998532i	$-0.517248\pi$
$701$	−2.86785			−0.108317			−0.0541586	+	0.998532i	$0.517248\pi$
$702$		0			0
$703$			2.22172i			0.0837939i
$704$		0			0
$705$	2.52251	−	6.74778i	0.0950033	−	0.254136i
$706$		0			0
$707$		−	13.2307i		−	0.497592i
$708$		0			0
$709$	34.9590			1.31291			0.656457	−	0.754363i	$-0.272054\pi$
$709$	34.9590			1.31291			0.656457	+	0.754363i	$0.272054\pi$
$710$		0			0
$711$	10.6390			0.398995
$712$		0			0
$713$		−	21.7517i		−	0.814608i
$714$		0			0
$715$	39.6890	+	14.8369i	1.48428	+	0.554867i
$716$		0			0
$717$			7.65681i			0.285949i
$718$		0			0
$719$	−35.5609			−1.32620			−0.663099	−	0.748532i	$-0.730759\pi$
$719$	−35.5609			−1.32620			−0.663099	+	0.748532i	$0.730759\pi$
$720$		0			0
$721$	−3.19700			−0.119062
$722$		0			0
$723$		−	15.6422i		−	0.581740i
$724$		0			0
$725$	−24.9520	−	21.6861i	−0.926693	−	0.805401i
$726$		0			0
$727$		−	29.2096i		−	1.08333i	−0.840596	−	0.541663i	$-0.817795\pi$
$727$		−	29.2096i		−	1.08333i	0.840596	−	0.541663i	$-0.182205\pi$
$728$		0			0
$729$	1.36576			0.0505836
$730$		0			0
$731$	27.2477			1.00779
$732$		0			0
$733$			12.5651i			0.464103i	0.972703	+	0.232052i	$0.0745438\pi$
$733$			12.5651i			0.464103i	−0.972703	+	0.232052i	$0.925456\pi$
$734$		0			0
$735$	1.56454	+	0.584870i	0.0577090	+	0.0215733i
$736$		0			0
$737$			40.6461i			1.49722i
$738$		0			0
$739$	−24.5041			−0.901398			−0.450699	−	0.892676i	$-0.648825\pi$
$739$	−24.5041			−0.901398			−0.450699	+	0.892676i	$0.648825\pi$
$740$		0			0
$741$	8.54867			0.314043
$742$		0			0
$743$		−	22.9661i		−	0.842544i	−0.906934	−	0.421272i	$-0.861584\pi$
$743$		−	22.9661i		−	0.842544i	0.906934	−	0.421272i	$-0.138416\pi$
$744$		0			0
$745$	−14.6857	+	39.2845i	−0.538041	+	1.43927i
$746$		0			0
$747$		−	32.1554i		−	1.17650i
$748$		0			0
$749$	12.5220			0.457545
$750$		0			0
$751$	8.06696			0.294368			0.147184	−	0.989109i	$-0.452979\pi$
$751$	8.06696			0.294368			0.147184	+	0.989109i	$0.452979\pi$
$752$		0			0
$753$		−	3.17965i		−	0.115873i
$754$		0			0
$755$	17.4467	−	46.6705i	0.634952	−	1.69851i
$756$		0			0
$757$			43.5310i			1.58216i	0.611711	+	0.791081i	$0.290481\pi$
$757$			43.5310i			1.58216i	−0.611711	+	0.791081i	$0.709519\pi$
$758$		0			0
$759$	−16.5610			−0.601128
$760$		0			0
$761$	14.5242			0.526501			0.263250	−	0.964728i	$-0.415205\pi$
$761$	14.5242			0.526501			0.263250	+	0.964728i	$0.415205\pi$
$762$		0			0
$763$		−	14.1217i		−	0.511240i
$764$		0			0
$765$	10.9939	+	4.10982i	0.397484	+	0.148591i
$766$		0			0
$767$		−	22.5271i		−	0.813408i
$768$		0			0
$769$	12.4577			0.449235			0.224617	−	0.974447i	$-0.427887\pi$
$769$	12.4577			0.449235			0.224617	+	0.974447i	$0.427887\pi$
$770$		0			0
$771$	−0.475780			−0.0171348
$772$		0			0
$773$		−	18.8821i		−	0.679142i	−0.940580	−	0.339571i	$-0.889718\pi$
$773$		−	18.8821i		−	0.679142i	0.940580	−	0.339571i	$-0.110282\pi$
$774$		0			0
$775$	21.8612	+	18.9998i	0.785277	+	0.682495i
$776$		0			0
$777$		−	0.465392i		−	0.0166959i
$778$		0			0
$779$	−19.3659			−0.693856
$780$		0			0
$781$	−10.7351			−0.384133
$782$		0			0
$783$			26.8772i			0.960514i
$784$		0			0
$785$	−35.0013	−	13.0845i	−1.24925	−	0.467005i
$786$		0			0
$787$			36.1080i			1.28711i	0.765399	+	0.643555i	$0.222541\pi$
$787$			36.1080i			1.28711i	−0.765399	+	0.643555i	$0.777459\pi$
$788$		0			0
$789$	18.4921			0.658338
$790$		0			0
$791$	−1.63513			−0.0581386
$792$		0			0
$793$		−	5.98473i		−	0.212524i
$794$		0			0
$795$	−1.23816	+	3.31211i	−0.0439131	+	0.117469i
$796$		0			0
$797$		−	26.4489i		−	0.936866i	−0.883499	−	0.468433i	$-0.844819\pi$
$797$		−	26.4489i		−	0.936866i	0.883499	−	0.468433i	$-0.155181\pi$
$798$		0			0
$799$	9.27029			0.327959
$800$		0			0
$801$	−29.7214			−1.05015
$802$		0			0
$803$		−	36.1170i		−	1.27454i
$804$		0			0
$805$	2.94008	−	7.86479i	0.103624	−	0.277197i
$806$		0			0
$807$			0.443744i			0.0156205i
$808$		0			0
$809$	−25.7779			−0.906302			−0.453151	−	0.891434i	$-0.649700\pi$
$809$	−25.7779			−0.906302			−0.453151	+	0.891434i	$0.649700\pi$
$810$		0			0
$811$	−28.3586			−0.995805			−0.497902	−	0.867233i	$-0.665896\pi$
$811$	−28.3586			−0.995805			−0.497902	+	0.867233i	$0.665896\pi$
$812$		0			0
$813$			21.2719i			0.746039i
$814$		0			0
$815$	−34.2566	−	12.8061i	−1.19995	−	0.448577i
$816$		0			0
$817$			45.2052i			1.58153i
$818$		0			0
$819$	7.83727			0.273856
$820$		0			0
$821$	−5.00452			−0.174659			−0.0873296	−	0.996179i	$-0.527833\pi$
$821$	−5.00452			−0.174659			−0.0873296	+	0.996179i	$0.527833\pi$
$822$		0			0
$823$			8.69074i			0.302940i	0.988462	+	0.151470i	$0.0484007\pi$
$823$			8.69074i			0.302940i	−0.988462	+	0.151470i	$0.951599\pi$
$824$		0			0
$825$	14.4659	−	16.6444i	0.503637	−	0.579484i
$826$		0			0
$827$		−	7.27223i		−	0.252880i	−0.991974	−	0.126440i	$-0.959645\pi$
$827$		−	7.27223i		−	0.252880i	0.991974	−	0.126440i	$-0.0403551\pi$
$828$		0			0
$829$	−9.68058			−0.336220			−0.168110	−	0.985768i	$-0.553766\pi$
$829$	−9.68058			−0.336220			−0.168110	+	0.985768i	$0.553766\pi$
$830$		0			0
$831$	9.78904			0.339578
$832$		0			0
$833$			2.14941i			0.0744727i
$834$		0			0
$835$	−7.67534	−	2.86926i	−0.265616	−	0.0992948i
$836$		0			0
$837$		−	23.5480i		−	0.813938i
$838$		0			0
$839$	−37.4160			−1.29174			−0.645872	−	0.763446i	$-0.723506\pi$
$839$	−37.4160			−1.29174			−0.645872	+	0.763446i	$0.723506\pi$
$840$		0			0
$841$	14.7155			0.507430
$842$		0			0
$843$			0.929388i			0.0320098i
$844$		0			0
$845$	−2.11421	+	5.65556i	−0.0727310	+	0.194557i
$846$		0			0
$847$			23.8617i			0.819899i
$848$		0			0
$849$	0.355977			0.0122171
$850$		0			0
$851$	−2.33948			−0.0801963
$852$		0			0
$853$			28.3650i			0.971200i	0.874181	+	0.485600i	$0.161399\pi$
$853$			28.3650i			0.971200i	−0.874181	+	0.485600i	$0.838601\pi$
$854$		0			0
$855$	−6.81837	+	18.2393i	−0.233183	+	0.623771i
$856$		0			0
$857$		−	13.3622i		−	0.456444i	−0.973609	−	0.228222i	$-0.926709\pi$
$857$		−	13.3622i		−	0.456444i	0.973609	−	0.228222i	$-0.0732913\pi$
$858$		0			0
$859$	−11.6961			−0.399066			−0.199533	−	0.979891i	$-0.563942\pi$
$859$	−11.6961			−0.399066			−0.199533	+	0.979891i	$0.563942\pi$
$860$		0			0
$861$	4.05665			0.138250
$862$		0			0
$863$		−	18.0895i		−	0.615773i	−0.951423	−	0.307886i	$-0.900378\pi$
$863$		−	18.0895i		−	0.615773i	0.951423	−	0.307886i	$-0.0996217\pi$
$864$		0			0
$865$	17.5440	+	6.55844i	0.596514	+	0.222994i
$866$		0			0
$867$			9.24758i			0.314064i
$868$		0			0
$869$	25.7232			0.872601
$870$		0			0
$871$	22.0932			0.748600
$872$		0			0
$873$		−	23.5884i		−	0.798346i
$874$		0			0
$875$	5.33626	+	9.82468i	0.180398	+	0.332135i
$876$		0			0
$877$		−	45.9438i		−	1.55141i	−0.631094	−	0.775706i	$-0.717394\pi$
$877$		−	45.9438i		−	1.55141i	0.631094	−	0.775706i	$-0.282606\pi$
$878$		0			0
$879$	−24.2162			−0.816791
$880$		0			0
$881$	−16.1280			−0.543368			−0.271684	−	0.962387i	$-0.587580\pi$
$881$	−16.1280			−0.543368			−0.271684	+	0.962387i	$0.587580\pi$
$882$		0			0
$883$		−	15.1120i		−	0.508558i	−0.967131	−	0.254279i	$-0.918162\pi$
$883$		−	15.1120i		−	0.508558i	0.967131	−	0.254279i	$-0.0818381\pi$
$884$		0			0
$885$	−10.9819	−	4.10536i	−0.369154	−	0.138000i
$886$		0			0
$887$			41.8450i			1.40502i	0.711676	+	0.702508i	$0.247937\pi$
$887$			41.8450i			1.40502i	−0.711676	+	0.702508i	$0.752063\pi$
$888$		0			0
$889$	1.79563			0.0602234
$890$		0			0
$891$	25.3273			0.848497
$892$		0			0
$893$			15.3798i			0.514666i
$894$		0			0
$895$	−1.35936	+	3.63633i	−0.0454384	+	0.121549i
$896$		0			0
$897$			9.00176i			0.300560i
$898$		0			0
$899$	−38.3005			−1.27739
$900$		0			0
$901$	−4.55028			−0.151592
$902$		0			0
$903$		−	9.46928i		−	0.315118i
$904$		0			0
$905$	−10.5406	+	28.1964i	−0.350382	+	0.937281i
$906$		0			0
$907$		−	10.1402i		−	0.336701i	−0.985727	−	0.168351i	$-0.946156\pi$
$907$		−	10.1402i		−	0.336701i	0.985727	−	0.168351i	$-0.0538441\pi$
$908$		0			0
$909$	−32.3097			−1.07165
$910$		0			0
$911$	−34.7638			−1.15178			−0.575888	−	0.817529i	$-0.695344\pi$
$911$	−34.7638			−1.15178			−0.575888	+	0.817529i	$0.695344\pi$
$912$		0			0
$913$		−	77.7460i		−	2.57302i
$914$		0			0
$915$	−2.91754	−	1.09066i	−0.0964510	−	0.0360561i
$916$		0			0
$917$		−	12.3868i		−	0.409049i
$918$		0			0
$919$	−19.8017			−0.653199			−0.326599	−	0.945163i	$-0.605903\pi$
$919$	−19.8017			−0.653199			−0.326599	+	0.945163i	$0.605903\pi$
$920$		0			0
$921$	0.536879			0.0176908
$922$		0			0
$923$			5.83509i			0.192064i
$924$		0			0
$925$	2.04351	−	2.35126i	0.0671901	−	0.0773088i
$926$		0			0
$927$			7.80715i			0.256421i
$928$		0			0
$929$	−16.2758			−0.533992			−0.266996	−	0.963698i	$-0.586031\pi$
$929$	−16.2758			−0.533992			−0.266996	+	0.963698i	$0.586031\pi$
$930$		0			0
$931$	−3.56597			−0.116870
$932$		0			0
$933$		−	14.2850i		−	0.467669i
$934$		0			0
$935$	26.5812	+	9.93681i	0.869299	+	0.324968i
$936$		0			0
$937$		−	2.52172i		−	0.0823809i	−0.999151	−	0.0411904i	$-0.986885\pi$
$937$		−	2.52172i		−	0.0823809i	0.999151	−	0.0411904i	$-0.0131150\pi$
$938$		0			0
$939$	−18.2681			−0.596155
$940$		0			0
$941$	43.7285			1.42551			0.712755	−	0.701413i	$-0.247447\pi$
$941$	43.7285			1.42551			0.712755	+	0.701413i	$0.247447\pi$
$942$		0			0
$943$		−	20.3923i		−	0.664066i
$944$		0			0
$945$	3.18288	−	8.51428i	0.103539	−	0.276970i
$946$		0			0
$947$		−	40.9743i		−	1.33149i	−0.746181	−	0.665743i	$-0.768115\pi$
$947$		−	40.9743i		−	1.33149i	0.746181	−	0.665743i	$-0.231885\pi$
$948$		0			0
$949$	−19.6314			−0.637263
$950$		0			0
$951$	3.04918			0.0988764
$952$		0			0
$953$			5.09948i			0.165188i	0.996583	+	0.0825942i	$0.0263205\pi$
$953$			5.09948i			0.165188i	−0.996583	+	0.0825942i	$0.973679\pi$
$954$		0			0
$955$	5.41772	−	14.4925i	0.175313	−	0.468968i
$956$		0			0
$957$			29.1607i			0.942633i
$958$		0			0
$959$	−14.3940			−0.464806
$960$		0			0
$961$	2.55625			0.0824595
$962$		0			0
$963$		−	30.5792i		−	0.985400i
$964$		0			0
$965$	−17.4351	−	6.51772i	−0.561255	−	0.209813i
$966$		0			0
$967$		−	21.7426i		−	0.699193i	−0.936900	−	0.349597i	$-0.886319\pi$
$967$		−	21.7426i		−	0.699193i	0.936900	−	0.349597i	$-0.113681\pi$
$968$		0			0
$969$	5.72537			0.183925
$970$		0			0
$971$	57.9688			1.86031			0.930153	−	0.367172i	$-0.119674\pi$
$971$	57.9688			1.86031			0.930153	+	0.367172i	$0.119674\pi$
$972$		0			0
$973$			22.7488i			0.729295i
$974$		0			0
$975$	−9.04708	−	7.86293i	−0.289738	−	0.251815i
$976$		0			0
$977$		−	4.09815i		−	0.131111i	−0.997849	−	0.0655557i	$-0.979118\pi$
$977$		−	4.09815i		−	0.131111i	0.997849	−	0.0655557i	$-0.0208820\pi$
$978$		0			0
$979$	−71.8609			−2.29669
$980$		0			0
$981$	−34.4856			−1.10104
$982$		0			0
$983$		−	21.8461i		−	0.696781i	−0.937349	−	0.348391i	$-0.886728\pi$
$983$		−	21.8461i		−	0.696781i	0.937349	−	0.348391i	$-0.113272\pi$
$984$		0			0
$985$	36.8724	+	13.7840i	1.17485	+	0.439193i
$986$		0			0
$987$		−	3.22166i		−	0.102547i
$988$		0			0
$989$	−47.6011			−1.51363
$990$		0			0
$991$	16.4720			0.523250			0.261625	−	0.965170i	$-0.415742\pi$
$991$	16.4720			0.523250			0.261625	+	0.965170i	$0.415742\pi$
$992$		0			0
$993$			4.26179i			0.135244i
$994$		0			0
$995$	−3.57764	+	9.57028i	−0.113419	+	0.303398i
$996$		0			0
$997$			23.1301i			0.732539i	0.930509	+	0.366270i	$0.119365\pi$
$997$			23.1301i			0.732539i	−0.930509	+	0.366270i	$0.880635\pi$
$998$		0			0
$999$	−2.53268			−0.0801304

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1120.2.g.b.449.6	yes	10
4.3	odd	2		1120.2.g.c.449.5	yes	10
5.2	odd	4		5600.2.a.bw.1.3		5
5.3	odd	4		5600.2.a.bu.1.3		5
5.4	even	2	inner	1120.2.g.b.449.5	✓	10
8.3	odd	2		2240.2.g.n.449.6		10
8.5	even	2		2240.2.g.o.449.5		10
20.3	even	4		5600.2.a.bx.1.3		5
20.7	even	4		5600.2.a.bv.1.3		5
20.19	odd	2		1120.2.g.c.449.6	yes	10
40.19	odd	2		2240.2.g.n.449.5		10
40.29	even	2		2240.2.g.o.449.6		10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1120.2.g.b.449.5	✓	10	5.4	even	2	inner
1120.2.g.b.449.6	yes	10	1.1	even	1	trivial
1120.2.g.c.449.5	yes	10	4.3	odd	2
1120.2.g.c.449.6	yes	10	20.19	odd	2
2240.2.g.n.449.5		10	40.19	odd	2
2240.2.g.n.449.6		10	8.3	odd	2
2240.2.g.o.449.5		10	8.5	even	2
2240.2.g.o.449.6		10	40.29	even	2
5600.2.a.bu.1.3		5	5.3	odd	4
5600.2.a.bv.1.3		5	20.7	even	4
5600.2.a.bw.1.3		5	5.2	odd	4
5600.2.a.bx.1.3		5	20.3	even	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1120 = 2^{5} \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1120.g (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+0.746976i q^{3} +(-0.782984 + 2.09450i) q^{5} +1.00000i q^{7} +2.44203 q^{9} +O(q^{10})\)	\(q+0.746976i q^{3} +(-0.782984 + 2.09450i) q^{5} +1.00000i q^{7} +2.44203 q^{9} +5.90438 q^{11} -3.20933i q^{13} +(-1.56454 - 0.584870i) q^{15} -2.14941i q^{17} +3.56597 q^{19} -0.746976 q^{21} +3.75497i q^{23} +(-3.77387 - 3.27992i) q^{25} +4.06506i q^{27} +6.61177 q^{29} -5.79278 q^{31} +4.41043i q^{33} +(-2.09450 - 0.782984i) q^{35} +0.623035i q^{37} +2.39729 q^{39} -5.43076 q^{41} +12.6768i q^{43} +(-1.91207 + 5.11483i) q^{45} +4.31294i q^{47} -1.00000 q^{49} +1.60556 q^{51} -2.11699i q^{53} +(-4.62304 + 12.3667i) q^{55} +2.66369i q^{57} +7.01926 q^{59} +1.86479 q^{61} +2.44203i q^{63} +(6.72195 + 2.51285i) q^{65} +6.88405i q^{67} -2.80487 q^{69} -1.81816 q^{71} -6.11699i q^{73} +(2.45002 - 2.81899i) q^{75} +5.90438i q^{77} +4.35664 q^{79} +4.28958 q^{81} -13.1675i q^{83} +(4.50195 + 1.68295i) q^{85} +4.93883i q^{87} -12.1708 q^{89} +3.20933 q^{91} -4.32706i q^{93} +(-2.79209 + 7.46892i) q^{95} -9.65935i q^{97} +14.4187 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q - 2 q^{5} - 14 q^{9}+O(q^{10}) \) 10 * q - 2 * q^5 - 14 * q^9	\( 10 q - 2 q^{5} - 14 q^{9} - 8 q^{11} - 4 q^{15} + 24 q^{19} - 4 q^{21} + 6 q^{25} + 24 q^{29} - 24 q^{31} + 64 q^{39} - 4 q^{41} + 10 q^{45} - 10 q^{49} - 24 q^{51} - 16 q^{55} + 32 q^{59} - 20 q^{61} - 8 q^{65} - 8 q^{69} - 8 q^{71} - 64 q^{75} + 64 q^{79} + 2 q^{81} - 12 q^{85} - 4 q^{89} - 60 q^{95} + 80 q^{99}+O(q^{100}) \) 10 * q - 2 * q^5 - 14 * q^9 - 8 * q^11 - 4 * q^15 + 24 * q^19 - 4 * q^21 + 6 * q^25 + 24 * q^29 - 24 * q^31 + 64 * q^39 - 4 * q^41 + 10 * q^45 - 10 * q^49 - 24 * q^51 - 16 * q^55 + 32 * q^59 - 20 * q^61 - 8 * q^65 - 8 * q^69 - 8 * q^71 - 64 * q^75 + 64 * q^79 + 2 * q^81 - 12 * q^85 - 4 * q^89 - 60 * q^95 + 80 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more