LMFDB - Embedded newform 5040.2.k.b.1889.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5040,2,Mod(1889,5040)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5040, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("5040.1889");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5040 = 2^{4} \cdot 3^{2} \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5040.k (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:	$40.2446026187$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{-2}, \sqrt{5})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 6x^{2} + 4 $ x^4 + 6*x^2 + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 630)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1889.4
Root			$-2.28825i$ of defining polynomial
Character	$\chi$	$=$	5040.1889
Dual form			5040.2.k.b.1889.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.23607 q^{5} +(2.23607 + 1.41421i) q^{7} +O(q^{10})$	$q+2.23607 q^{5} +(2.23607 + 1.41421i) q^{7} -5.65685i q^{11} +4.47214 q^{13} +3.16228i q^{17} +3.16228i q^{19} -4.00000 q^{23} +5.00000 q^{25} +2.82843i q^{29} +6.32456i q^{31} +(5.00000 + 3.16228i) q^{35} +9.89949i q^{37} +4.47214 q^{41} -1.41421i q^{43} +9.48683i q^{47} +(3.00000 + 6.32456i) q^{49} +4.00000 q^{53} -12.6491i q^{55} -4.47214 q^{59} -9.48683i q^{61} +10.0000 q^{65} -7.07107i q^{67} -1.41421i q^{71} +13.4164 q^{73} +(8.00000 - 12.6491i) q^{77} -6.00000 q^{79} -12.6491i q^{83} +7.07107i q^{85} -4.47214 q^{89} +(10.0000 + 6.32456i) q^{91} +7.07107i q^{95} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q+O(q^{10}) $ 4 * q	$ 4 q - 16 q^{23} + 20 q^{25} + 20 q^{35} + 12 q^{49} + 16 q^{53} + 40 q^{65} + 32 q^{77} - 24 q^{79} + 40 q^{91}+O(q^{100}) $ 4 * q - 16 * q^23 + 20 * q^25 + 20 * q^35 + 12 * q^49 + 16 * q^53 + 40 * q^65 + 32 * q^77 - 24 * q^79 + 40 * q^91

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/5040\mathbb{Z}\right)^\times$.

$n$	$2017$	$2801$	$3151$	$3601$	$3781$
$\chi(n)$	$-1$	$-1$	$1$	$-1$	$1$

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			0
$3$		0			0
$4$		0			0
$5$	2.23607			1.00000
$5$	2.23607			1.00000
$6$		0			0
$7$	2.23607	+	1.41421i	0.845154	+	0.534522i
$7$	2.23607	+	1.41421i	0.845154	+	0.534522i
$8$		0			0
$9$		0			0
$10$		0			0
$11$		−	5.65685i		−	1.70561i	−0.522233	−	0.852803i	$-0.674901\pi$
$11$		−	5.65685i		−	1.70561i	0.522233	−	0.852803i	$-0.325099\pi$
$12$		0			0
$13$	4.47214			1.24035			0.620174	−	0.784465i	$-0.287062\pi$
$13$	4.47214			1.24035			0.620174	+	0.784465i	$0.287062\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$			3.16228i			0.766965i	0.923548	+	0.383482i	$0.125275\pi$
$17$			3.16228i			0.766965i	−0.923548	+	0.383482i	$0.874725\pi$
$18$		0			0
$19$			3.16228i			0.725476i	0.931891	+	0.362738i	$0.118158\pi$
$19$			3.16228i			0.725476i	−0.931891	+	0.362738i	$0.881842\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−4.00000			−0.834058			−0.417029	−	0.908893i	$-0.636929\pi$
$23$	−4.00000			−0.834058			−0.417029	+	0.908893i	$0.636929\pi$
$24$		0			0
$25$	5.00000			1.00000
$26$		0			0
$27$		0			0
$28$		0			0
$29$			2.82843i			0.525226i	0.964901	+	0.262613i	$0.0845842\pi$
$29$			2.82843i			0.525226i	−0.964901	+	0.262613i	$0.915416\pi$
$30$		0			0
$31$			6.32456i			1.13592i	0.823055	+	0.567962i	$0.192268\pi$
$31$			6.32456i			1.13592i	−0.823055	+	0.567962i	$0.807732\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	5.00000	+	3.16228i	0.845154	+	0.534522i
$36$		0			0
$37$			9.89949i			1.62747i	0.581238	+	0.813733i	$0.302568\pi$
$37$			9.89949i			1.62747i	−0.581238	+	0.813733i	$0.697432\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	4.47214			0.698430			0.349215	−	0.937043i	$-0.386448\pi$
$41$	4.47214			0.698430			0.349215	+	0.937043i	$0.386448\pi$
$42$		0			0
$43$		−	1.41421i		−	0.215666i	−0.994169	−	0.107833i	$-0.965609\pi$
$43$		−	1.41421i		−	0.215666i	0.994169	−	0.107833i	$-0.0343911\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$			9.48683i			1.38380i	0.721995	+	0.691898i	$0.243225\pi$
$47$			9.48683i			1.38380i	−0.721995	+	0.691898i	$0.756775\pi$
$48$		0			0
$49$	3.00000	+	6.32456i	0.428571	+	0.903508i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	4.00000			0.549442			0.274721	−	0.961524i	$-0.411414\pi$
$53$	4.00000			0.549442			0.274721	+	0.961524i	$0.411414\pi$
$54$		0			0
$55$		−	12.6491i		−	1.70561i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−4.47214			−0.582223			−0.291111	−	0.956689i	$-0.594025\pi$
$59$	−4.47214			−0.582223			−0.291111	+	0.956689i	$0.594025\pi$
$60$		0			0
$61$		−	9.48683i		−	1.21466i	−0.794448	−	0.607332i	$-0.792240\pi$
$61$		−	9.48683i		−	1.21466i	0.794448	−	0.607332i	$-0.207760\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	10.0000			1.24035
$66$		0			0
$67$		−	7.07107i		−	0.863868i	−0.901905	−	0.431934i	$-0.857831\pi$
$67$		−	7.07107i		−	0.863868i	0.901905	−	0.431934i	$-0.142169\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		−	1.41421i		−	0.167836i	−0.996473	−	0.0839181i	$-0.973257\pi$
$71$		−	1.41421i		−	0.167836i	0.996473	−	0.0839181i	$-0.0267434\pi$
$72$		0			0
$73$	13.4164			1.57027			0.785136	−	0.619324i	$-0.212593\pi$
$73$	13.4164			1.57027			0.785136	+	0.619324i	$0.212593\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	8.00000	−	12.6491i	0.911685	−	1.44150i
$78$		0			0
$79$	−6.00000			−0.675053			−0.337526	−	0.941316i	$-0.609590\pi$
$79$	−6.00000			−0.675053			−0.337526	+	0.941316i	$0.609590\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$		−	12.6491i		−	1.38842i	−0.719772	−	0.694210i	$-0.755754\pi$
$83$		−	12.6491i		−	1.38842i	0.719772	−	0.694210i	$-0.244246\pi$
$84$		0			0
$85$			7.07107i			0.766965i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−4.47214			−0.474045			−0.237023	−	0.971504i	$-0.576172\pi$
$89$	−4.47214			−0.474045			−0.237023	+	0.971504i	$0.576172\pi$
$90$		0			0
$91$	10.0000	+	6.32456i	1.04828	+	0.662994i
$92$		0			0
$93$		0			0
$94$		0			0
$95$			7.07107i			0.725476i
$96$		0			0
$97$		0			0			−	1.00000i	$-0.5\pi$
$97$		0			0				1.00000i	$0.5\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−17.8885			−1.77998			−0.889988	−	0.455983i	$-0.849288\pi$
$101$	−17.8885			−1.77998			−0.889988	+	0.455983i	$0.849288\pi$
$102$		0			0
$103$	8.94427			0.881305			0.440653	−	0.897678i	$-0.354747\pi$
$103$	8.94427			0.881305			0.440653	+	0.897678i	$0.354747\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	18.0000			1.74013			0.870063	−	0.492941i	$-0.164078\pi$
$107$	18.0000			1.74013			0.870063	+	0.492941i	$0.164078\pi$
$108$		0			0
$109$	−10.0000			−0.957826			−0.478913	−	0.877862i	$-0.658969\pi$
$109$	−10.0000			−0.957826			−0.478913	+	0.877862i	$0.658969\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	6.00000			0.564433			0.282216	−	0.959351i	$-0.408930\pi$
$113$	6.00000			0.564433			0.282216	+	0.959351i	$0.408930\pi$
$114$		0			0
$115$	−8.94427			−0.834058
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−4.47214	+	7.07107i	−0.409960	+	0.648204i
$120$		0			0
$121$	−21.0000			−1.90909
$122$		0			0
$123$		0			0
$124$		0			0
$125$	11.1803			1.00000
$126$		0			0
$127$			14.1421i			1.25491i	0.778652	+	0.627456i	$0.215904\pi$
$127$			14.1421i			1.25491i	−0.778652	+	0.627456i	$0.784096\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	17.8885			1.56293			0.781465	−	0.623949i	$-0.214473\pi$
$131$	17.8885			1.56293			0.781465	+	0.623949i	$0.214473\pi$
$132$		0			0
$133$	−4.47214	+	7.07107i	−0.387783	+	0.613139i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−2.00000			−0.170872			−0.0854358	−	0.996344i	$-0.527228\pi$
$137$	−2.00000			−0.170872			−0.0854358	+	0.996344i	$0.527228\pi$
$138$		0			0
$139$		−	15.8114i		−	1.34110i	−0.741862	−	0.670552i	$-0.766057\pi$
$139$		−	15.8114i		−	1.34110i	0.741862	−	0.670552i	$-0.233943\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		−	25.2982i		−	2.11554i
$144$		0			0
$145$			6.32456i			0.525226i
$146$		0			0
$147$		0			0
$148$		0			0
$149$		−	16.9706i		−	1.39028i	−0.718873	−	0.695141i	$-0.755342\pi$
$149$		−	16.9706i		−	1.39028i	0.718873	−	0.695141i	$-0.244658\pi$
$150$		0			0
$151$	−10.0000			−0.813788			−0.406894	−	0.913475i	$-0.633388\pi$
$151$	−10.0000			−0.813788			−0.406894	+	0.913475i	$0.633388\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$			14.1421i			1.13592i
$156$		0			0
$157$	4.47214			0.356915			0.178458	−	0.983948i	$-0.442889\pi$
$157$	4.47214			0.356915			0.178458	+	0.983948i	$0.442889\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−8.94427	−	5.65685i	−0.704907	−	0.445823i
$162$		0			0
$163$			21.2132i			1.66155i	0.556611	+	0.830773i	$0.312101\pi$
$163$			21.2132i			1.66155i	−0.556611	+	0.830773i	$0.687899\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$			9.48683i			0.734113i	0.930199	+	0.367057i	$0.119634\pi$
$167$			9.48683i			0.734113i	−0.930199	+	0.367057i	$0.880366\pi$
$168$		0			0
$169$	7.00000			0.538462
$170$		0			0
$171$		0			0
$172$		0			0
$173$		−	25.2982i		−	1.92339i	−0.274125	−	0.961694i	$-0.588388\pi$
$173$		−	25.2982i		−	1.92339i	0.274125	−	0.961694i	$-0.411612\pi$
$174$		0			0
$175$	11.1803	+	7.07107i	0.845154	+	0.534522i
$176$		0			0
$177$		0			0
$178$		0			0
$179$		−	2.82843i		−	0.211407i	−0.994398	−	0.105703i	$-0.966291\pi$
$179$		−	2.82843i		−	0.211407i	0.994398	−	0.105703i	$-0.0337094\pi$
$180$		0			0
$181$			3.16228i			0.235050i	0.993070	+	0.117525i	$0.0374961\pi$
$181$			3.16228i			0.235050i	−0.993070	+	0.117525i	$0.962504\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$			22.1359i			1.62747i
$186$		0			0
$187$	17.8885			1.30814
$188$		0			0
$189$		0			0
$190$		0			0
$191$			1.41421i			0.102329i	0.998690	+	0.0511645i	$0.0162933\pi$
$191$			1.41421i			0.102329i	−0.998690	+	0.0511645i	$0.983707\pi$
$192$		0			0
$193$		−	8.48528i		−	0.610784i	−0.952227	−	0.305392i	$-0.901213\pi$
$193$		−	8.48528i		−	0.610784i	0.952227	−	0.305392i	$-0.0987875\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−12.0000			−0.854965			−0.427482	−	0.904024i	$-0.640599\pi$
$197$	−12.0000			−0.854965			−0.427482	+	0.904024i	$0.640599\pi$
$198$		0			0
$199$		0			0		1.00000			$0$
$199$		0			0		−1.00000			$\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−4.00000	+	6.32456i	−0.280745	+	0.443897i
$204$		0			0
$205$	10.0000			0.698430
$206$		0			0
$207$		0			0
$208$		0			0
$209$	17.8885			1.23738
$210$		0			0
$211$	20.0000			1.37686			0.688428	−	0.725304i	$-0.258301\pi$
$211$	20.0000			1.37686			0.688428	+	0.725304i	$0.258301\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		−	3.16228i		−	0.215666i
$216$		0			0
$217$	−8.94427	+	14.1421i	−0.607177	+	0.960031i
$218$		0			0
$219$		0			0
$220$		0			0
$221$			14.1421i			0.951303i
$222$		0			0
$223$	−8.94427			−0.598953			−0.299476	−	0.954104i	$-0.596812\pi$
$223$	−8.94427			−0.598953			−0.299476	+	0.954104i	$0.596812\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$		−	18.9737i		−	1.25933i	−0.776868	−	0.629663i	$-0.783193\pi$
$227$		−	18.9737i		−	1.25933i	0.776868	−	0.629663i	$-0.216807\pi$
$228$		0			0
$229$			15.8114i			1.04485i	0.852686	+	0.522423i	$0.174972\pi$
$229$			15.8114i			1.04485i	−0.852686	+	0.522423i	$0.825028\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−14.0000			−0.917170			−0.458585	−	0.888650i	$-0.651644\pi$
$233$	−14.0000			−0.917170			−0.458585	+	0.888650i	$0.651644\pi$
$234$		0			0
$235$			21.2132i			1.38380i
$236$		0			0
$237$		0			0
$238$		0			0
$239$		−	4.24264i		−	0.274434i	−0.990541	−	0.137217i	$-0.956184\pi$
$239$		−	4.24264i		−	0.274434i	0.990541	−	0.137217i	$-0.0438157\pi$
$240$		0			0
$241$		−	25.2982i		−	1.62960i	−0.579741	−	0.814801i	$-0.696846\pi$
$241$		−	25.2982i		−	1.62960i	0.579741	−	0.814801i	$-0.303154\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	6.70820	+	14.1421i	0.428571	+	0.903508i
$246$		0			0
$247$			14.1421i			0.899843i
$248$		0			0
$249$		0			0
$250$		0			0
$251$		0			0			−	1.00000i	$-0.5\pi$
$251$		0			0				1.00000i	$0.5\pi$
$252$		0			0
$253$			22.6274i			1.42257i
$254$		0			0
$255$		0			0
$256$		0			0
$257$		−	15.8114i		−	0.986287i	−0.869948	−	0.493144i	$-0.835848\pi$
$257$		−	15.8114i		−	0.986287i	0.869948	−	0.493144i	$-0.164152\pi$
$258$		0			0
$259$	−14.0000	+	22.1359i	−0.869918	+	1.37546i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	16.0000			0.986602			0.493301	−	0.869859i	$-0.335790\pi$
$263$	16.0000			0.986602			0.493301	+	0.869859i	$0.335790\pi$
$264$		0			0
$265$	8.94427			0.549442
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−17.8885			−1.09068			−0.545342	−	0.838214i	$-0.683600\pi$
$269$	−17.8885			−1.09068			−0.545342	+	0.838214i	$0.683600\pi$
$270$		0			0
$271$			6.32456i			0.384189i	0.981376	+	0.192095i	$0.0615281\pi$
$271$			6.32456i			0.384189i	−0.981376	+	0.192095i	$0.938472\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		−	28.2843i		−	1.70561i
$276$		0			0
$277$			21.2132i			1.27458i	0.770625	+	0.637289i	$0.219944\pi$
$277$			21.2132i			1.27458i	−0.770625	+	0.637289i	$0.780056\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$			12.7279i			0.759284i	0.925133	+	0.379642i	$0.123953\pi$
$281$			12.7279i			0.759284i	−0.925133	+	0.379642i	$0.876047\pi$
$282$		0			0
$283$		0			0			−	1.00000i	$-0.5\pi$
$283$		0			0				1.00000i	$0.5\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	10.0000	+	6.32456i	0.590281	+	0.373327i
$288$		0			0
$289$	7.00000			0.411765
$290$		0			0
$291$		0			0
$292$		0			0
$293$		−	6.32456i		−	0.369484i	−0.982787	−	0.184742i	$-0.940855\pi$
$293$		−	6.32456i		−	0.369484i	0.982787	−	0.184742i	$-0.0591450\pi$
$294$		0			0
$295$	−10.0000			−0.582223
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−17.8885			−1.03452
$300$		0			0
$301$	2.00000	−	3.16228i	0.115278	−	0.182271i
$302$		0			0
$303$		0			0
$304$		0			0
$305$		−	21.2132i		−	1.21466i
$306$		0			0
$307$	26.8328			1.53143			0.765715	−	0.643180i	$-0.222385\pi$
$307$	26.8328			1.53143			0.765715	+	0.643180i	$0.222385\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−8.94427			−0.507183			−0.253592	−	0.967311i	$-0.581612\pi$
$311$	−8.94427			−0.507183			−0.253592	+	0.967311i	$0.581612\pi$
$312$		0			0
$313$	−13.4164			−0.758340			−0.379170	−	0.925327i	$-0.623790\pi$
$313$	−13.4164			−0.758340			−0.379170	+	0.925327i	$0.623790\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	18.0000			1.01098			0.505490	−	0.862832i	$-0.331312\pi$
$317$	18.0000			1.01098			0.505490	+	0.862832i	$0.331312\pi$
$318$		0			0
$319$	16.0000			0.895828
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−10.0000			−0.556415
$324$		0			0
$325$	22.3607			1.24035
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−13.4164	+	21.2132i	−0.739671	+	1.16952i
$330$		0			0
$331$	−20.0000			−1.09930			−0.549650	−	0.835395i	$-0.685239\pi$
$331$	−20.0000			−1.09930			−0.549650	+	0.835395i	$0.685239\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		−	15.8114i		−	0.863868i
$336$		0			0
$337$		−	25.4558i		−	1.38667i	−0.720616	−	0.693334i	$-0.756141\pi$
$337$		−	25.4558i		−	1.38667i	0.720616	−	0.693334i	$-0.243859\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	35.7771			1.93744
$342$		0			0
$343$	−2.23607	+	18.3848i	−0.120736	+	0.992685i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−12.0000			−0.644194			−0.322097	−	0.946707i	$-0.604388\pi$
$347$	−12.0000			−0.644194			−0.322097	+	0.946707i	$0.604388\pi$
$348$		0			0
$349$			15.8114i			0.846364i	0.906045	+	0.423182i	$0.139087\pi$
$349$			15.8114i			0.846364i	−0.906045	+	0.423182i	$0.860913\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$			3.16228i			0.168311i	0.996453	+	0.0841555i	$0.0268193\pi$
$353$			3.16228i			0.168311i	−0.996453	+	0.0841555i	$0.973181\pi$
$354$		0			0
$355$		−	3.16228i		−	0.167836i
$356$		0			0
$357$		0			0
$358$		0			0
$359$		−	18.3848i		−	0.970311i	−0.874428	−	0.485156i	$-0.838763\pi$
$359$		−	18.3848i		−	0.970311i	0.874428	−	0.485156i	$-0.161237\pi$
$360$		0			0
$361$	9.00000			0.473684
$362$		0			0
$363$		0			0
$364$		0			0
$365$	30.0000			1.57027
$366$		0			0
$367$	−13.4164			−0.700331			−0.350165	−	0.936688i	$-0.613875\pi$
$367$	−13.4164			−0.700331			−0.350165	+	0.936688i	$0.613875\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	8.94427	+	5.65685i	0.464363	+	0.293689i
$372$		0			0
$373$			21.2132i			1.09838i	0.835698	+	0.549189i	$0.185063\pi$
$373$			21.2132i			1.09838i	−0.835698	+	0.549189i	$0.814937\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$			12.6491i			0.651462i
$378$		0			0
$379$		0			0			−	1.00000i	$-0.5\pi$
$379$		0			0				1.00000i	$0.5\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		−	9.48683i		−	0.484755i	−0.970182	−	0.242377i	$-0.922073\pi$
$383$		−	9.48683i		−	0.484755i	0.970182	−	0.242377i	$-0.0779272\pi$
$384$		0			0
$385$	17.8885	−	28.2843i	0.911685	−	1.44150i
$386$		0			0
$387$		0			0
$388$		0			0
$389$		−	16.9706i		−	0.860442i	−0.902724	−	0.430221i	$-0.858436\pi$
$389$		−	16.9706i		−	0.860442i	0.902724	−	0.430221i	$-0.141564\pi$
$390$		0			0
$391$		−	12.6491i		−	0.639693i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−13.4164			−0.675053
$396$		0			0
$397$	−4.47214			−0.224450			−0.112225	−	0.993683i	$-0.535798\pi$
$397$	−4.47214			−0.224450			−0.112225	+	0.993683i	$0.535798\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$		−	1.41421i		−	0.0706225i	−0.999376	−	0.0353112i	$-0.988758\pi$
$401$		−	1.41421i		−	0.0706225i	0.999376	−	0.0353112i	$-0.0112422\pi$
$402$		0			0
$403$			28.2843i			1.40894i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	56.0000			2.77582
$408$		0			0
$409$		−	37.9473i		−	1.87637i	−0.346128	−	0.938187i	$-0.612504\pi$
$409$		−	37.9473i		−	1.87637i	0.346128	−	0.938187i	$-0.387496\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−10.0000	−	6.32456i	−0.492068	−	0.311211i
$414$		0			0
$415$		−	28.2843i		−	1.38842i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−13.4164			−0.655434			−0.327717	−	0.944776i	$-0.606279\pi$
$419$	−13.4164			−0.655434			−0.327717	+	0.944776i	$0.606279\pi$
$420$		0			0
$421$	38.0000			1.85201			0.926003	−	0.377515i	$-0.123221\pi$
$421$	38.0000			1.85201			0.926003	+	0.377515i	$0.123221\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$			15.8114i			0.766965i
$426$		0			0
$427$	13.4164	−	21.2132i	0.649265	−	1.02658i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		−	15.5563i		−	0.749323i	−0.927162	−	0.374661i	$-0.877759\pi$
$431$		−	15.5563i		−	0.749323i	0.927162	−	0.374661i	$-0.122241\pi$
$432$		0			0
$433$	−26.8328			−1.28950			−0.644751	−	0.764392i	$-0.723039\pi$
$433$	−26.8328			−1.28950			−0.644751	+	0.764392i	$0.723039\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		−	12.6491i		−	0.605089i
$438$		0			0
$439$			6.32456i			0.301855i	0.988545	+	0.150927i	$0.0482259\pi$
$439$			6.32456i			0.301855i	−0.988545	+	0.150927i	$0.951774\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	4.00000			0.190046			0.0950229	−	0.995475i	$-0.469708\pi$
$443$	4.00000			0.190046			0.0950229	+	0.995475i	$0.469708\pi$
$444$		0			0
$445$	−10.0000			−0.474045
$446$		0			0
$447$		0			0
$448$		0			0
$449$			9.89949i			0.467186i	0.972334	+	0.233593i	$0.0750483\pi$
$449$			9.89949i			0.467186i	−0.972334	+	0.233593i	$0.924952\pi$
$450$		0			0
$451$		−	25.2982i		−	1.19125i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	22.3607	+	14.1421i	1.04828	+	0.662994i
$456$		0			0
$457$		−	16.9706i		−	0.793849i	−0.917851	−	0.396925i	$-0.870077\pi$
$457$		−	16.9706i		−	0.793849i	0.917851	−	0.396925i	$-0.129923\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−8.94427			−0.416576			−0.208288	−	0.978068i	$-0.566789\pi$
$461$	−8.94427			−0.416576			−0.208288	+	0.978068i	$0.566789\pi$
$462$		0			0
$463$		0			0		1.00000			$0$
$463$		0			0		−1.00000			$\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$			12.6491i			0.585331i	0.956215	+	0.292666i	$0.0945422\pi$
$467$			12.6491i			0.585331i	−0.956215	+	0.292666i	$0.905458\pi$
$468$		0			0
$469$	10.0000	−	15.8114i	0.461757	−	0.730102i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−8.00000			−0.367840
$474$		0			0
$475$			15.8114i			0.725476i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	17.8885			0.817348			0.408674	−	0.912680i	$-0.365991\pi$
$479$	17.8885			0.817348			0.408674	+	0.912680i	$0.365991\pi$
$480$		0			0
$481$			44.2719i			2.01862i
$482$		0			0
$483$		0			0
$484$		0			0
$485$		0			0
$486$		0			0
$487$		−	25.4558i		−	1.15351i	−0.816916	−	0.576757i	$-0.804318\pi$
$487$		−	25.4558i		−	1.15351i	0.816916	−	0.576757i	$-0.195682\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$			19.7990i			0.893516i	0.894655	+	0.446758i	$0.147421\pi$
$491$			19.7990i			0.893516i	−0.894655	+	0.446758i	$0.852579\pi$
$492$		0			0
$493$	−8.94427			−0.402830
$494$		0			0
$495$		0			0
$496$		0			0
$497$	2.00000	−	3.16228i	0.0897123	−	0.141848i
$498$		0			0
$499$	20.0000			0.895323			0.447661	−	0.894203i	$-0.352257\pi$
$499$	20.0000			0.895323			0.447661	+	0.894203i	$0.352257\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		−	9.48683i		−	0.422997i	−0.977378	−	0.211498i	$-0.932166\pi$
$503$		−	9.48683i		−	0.422997i	0.977378	−	0.211498i	$-0.0678343\pi$
$504$		0			0
$505$	−40.0000			−1.77998
$506$		0			0
$507$		0			0
$508$		0			0
$509$	13.4164			0.594672			0.297336	−	0.954773i	$-0.403902\pi$
$509$	13.4164			0.594672			0.297336	+	0.954773i	$0.403902\pi$
$510$		0			0
$511$	30.0000	+	18.9737i	1.32712	+	0.839346i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	20.0000			0.881305
$516$		0			0
$517$	53.6656			2.36021
$518$		0			0
$519$		0			0
$520$		0			0
$521$	13.4164			0.587784			0.293892	−	0.955839i	$-0.405049\pi$
$521$	13.4164			0.587784			0.293892	+	0.955839i	$0.405049\pi$
$522$		0			0
$523$	−17.8885			−0.782211			−0.391106	−	0.920346i	$-0.627907\pi$
$523$	−17.8885			−0.782211			−0.391106	+	0.920346i	$0.627907\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−20.0000			−0.871214
$528$		0			0
$529$	−7.00000			−0.304348
$530$		0			0
$531$		0			0
$532$		0			0
$533$	20.0000			0.866296
$534$		0			0
$535$	40.2492			1.74013
$536$		0			0
$537$		0			0
$538$		0			0
$539$	35.7771	−	16.9706i	1.54103	−	0.730974i
$540$		0			0
$541$	−22.0000			−0.945854			−0.472927	−	0.881102i	$-0.656803\pi$
$541$	−22.0000			−0.945854			−0.472927	+	0.881102i	$0.656803\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−22.3607			−0.957826
$546$		0			0
$547$			21.2132i			0.907011i	0.891253	+	0.453506i	$0.149827\pi$
$547$			21.2132i			0.907011i	−0.891253	+	0.453506i	$0.850173\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−8.94427			−0.381039
$552$		0			0
$553$	−13.4164	−	8.48528i	−0.570524	−	0.360831i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	38.0000			1.61011			0.805056	−	0.593199i	$-0.202135\pi$
$557$	38.0000			1.61011			0.805056	+	0.593199i	$0.202135\pi$
$558$		0			0
$559$		−	6.32456i		−	0.267500i
$560$		0			0
$561$		0			0
$562$		0			0
$563$			18.9737i			0.799645i	0.916593	+	0.399822i	$0.130928\pi$
$563$			18.9737i			0.799645i	−0.916593	+	0.399822i	$0.869072\pi$
$564$		0			0
$565$	13.4164			0.564433
$566$		0			0
$567$		0			0
$568$		0			0
$569$		−	24.0416i		−	1.00788i	−0.863739	−	0.503939i	$-0.831884\pi$
$569$		−	24.0416i		−	1.00788i	0.863739	−	0.503939i	$-0.168116\pi$
$570$		0			0
$571$	−28.0000			−1.17176			−0.585882	−	0.810397i	$-0.699252\pi$
$571$	−28.0000			−1.17176			−0.585882	+	0.810397i	$0.699252\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−20.0000			−0.834058
$576$		0			0
$577$	−22.3607			−0.930887			−0.465444	−	0.885078i	$-0.654105\pi$
$577$	−22.3607			−0.930887			−0.465444	+	0.885078i	$0.654105\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	17.8885	−	28.2843i	0.742142	−	1.17343i
$582$		0			0
$583$		−	22.6274i		−	0.937132i
$584$		0			0
$585$		0			0
$586$		0			0
$587$			31.6228i			1.30521i	0.757698	+	0.652606i	$0.226324\pi$
$587$			31.6228i			1.30521i	−0.757698	+	0.652606i	$0.773676\pi$
$588$		0			0
$589$	−20.0000			−0.824086
$590$		0			0
$591$		0			0
$592$		0			0
$593$			9.48683i			0.389578i	0.980845	+	0.194789i	$0.0624021\pi$
$593$			9.48683i			0.389578i	−0.980845	+	0.194789i	$0.937598\pi$
$594$		0			0
$595$	−10.0000	+	15.8114i	−0.409960	+	0.648204i
$596$		0			0
$597$		0			0
$598$		0			0
$599$			32.5269i			1.32901i	0.747282	+	0.664507i	$0.231358\pi$
$599$			32.5269i			1.32901i	−0.747282	+	0.664507i	$0.768642\pi$
$600$		0			0
$601$			25.2982i			1.03194i	0.856608	+	0.515968i	$0.172568\pi$
$601$			25.2982i			1.03194i	−0.856608	+	0.515968i	$0.827432\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−46.9574			−1.90909
$606$		0			0
$607$	−8.94427			−0.363037			−0.181518	−	0.983388i	$-0.558101\pi$
$607$	−8.94427			−0.363037			−0.181518	+	0.983388i	$0.558101\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$			42.4264i			1.71639i
$612$		0			0
$613$			35.3553i			1.42799i	0.700151	+	0.713994i	$0.253116\pi$
$613$			35.3553i			1.42799i	−0.700151	+	0.713994i	$0.746884\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−38.0000			−1.52982			−0.764911	−	0.644136i	$-0.777217\pi$
$617$	−38.0000			−1.52982			−0.764911	+	0.644136i	$0.777217\pi$
$618$		0			0
$619$		−	28.4605i		−	1.14392i	−0.820280	−	0.571962i	$-0.806182\pi$
$619$		−	28.4605i		−	1.14392i	0.820280	−	0.571962i	$-0.193818\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−10.0000	−	6.32456i	−0.400642	−	0.253388i
$624$		0			0
$625$	25.0000			1.00000
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−31.3050			−1.24821
$630$		0			0
$631$	8.00000			0.318475			0.159237	−	0.987240i	$-0.449096\pi$
$631$	8.00000			0.318475			0.159237	+	0.987240i	$0.449096\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$			31.6228i			1.25491i
$636$		0			0
$637$	13.4164	+	28.2843i	0.531577	+	1.12066i
$638$		0			0
$639$		0			0
$640$		0			0
$641$			1.41421i			0.0558581i	0.999610	+	0.0279290i	$0.00889125\pi$
$641$			1.41421i			0.0558581i	−0.999610	+	0.0279290i	$0.991109\pi$
$642$		0			0
$643$	−44.7214			−1.76364			−0.881819	−	0.471588i	$-0.843681\pi$
$643$	−44.7214			−1.76364			−0.881819	+	0.471588i	$0.843681\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$			34.7851i			1.36754i	0.729697	+	0.683771i	$0.239661\pi$
$647$			34.7851i			1.36754i	−0.729697	+	0.683771i	$0.760339\pi$
$648$		0			0
$649$			25.2982i			0.993042i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	14.0000			0.547862			0.273931	−	0.961749i	$-0.411676\pi$
$653$	14.0000			0.547862			0.273931	+	0.961749i	$0.411676\pi$
$654$		0			0
$655$	40.0000			1.56293
$656$		0			0
$657$		0			0
$658$		0			0
$659$			16.9706i			0.661079i	0.943792	+	0.330540i	$0.107231\pi$
$659$			16.9706i			0.661079i	−0.943792	+	0.330540i	$0.892769\pi$
$660$		0			0
$661$		−	3.16228i		−	0.122998i	−0.998107	−	0.0614992i	$-0.980412\pi$
$661$		−	3.16228i		−	0.122998i	0.998107	−	0.0614992i	$-0.0195882\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−10.0000	+	15.8114i	−0.387783	+	0.613139i
$666$		0			0
$667$		−	11.3137i		−	0.438069i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−53.6656			−2.07174
$672$		0			0
$673$			19.7990i			0.763195i	0.924329	+	0.381597i	$0.124626\pi$
$673$			19.7990i			0.763195i	−0.924329	+	0.381597i	$0.875374\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$			37.9473i			1.45843i	0.684282	+	0.729217i	$0.260116\pi$
$677$			37.9473i			1.45843i	−0.684282	+	0.729217i	$0.739884\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−14.0000			−0.535695			−0.267848	−	0.963461i	$-0.586312\pi$
$683$	−14.0000			−0.535695			−0.267848	+	0.963461i	$0.586312\pi$
$684$		0			0
$685$	−4.47214			−0.170872
$686$		0			0
$687$		0			0
$688$		0			0
$689$	17.8885			0.681499
$690$		0			0
$691$		−	28.4605i		−	1.08269i	−0.840801	−	0.541344i	$-0.817916\pi$
$691$		−	28.4605i		−	1.08269i	0.840801	−	0.541344i	$-0.182084\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		−	35.3553i		−	1.34110i
$696$		0			0
$697$			14.1421i			0.535672i
$698$		0			0
$699$		0			0
$700$		0			0
$701$		−	19.7990i		−	0.747798i	−0.927470	−	0.373899i	$-0.878021\pi$
$701$		−	19.7990i		−	0.747798i	0.927470	−	0.373899i	$-0.121979\pi$
$702$		0			0
$703$	−31.3050			−1.18069
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−40.0000	−	25.2982i	−1.50435	−	0.951438i
$708$		0			0
$709$	−30.0000			−1.12667			−0.563337	−	0.826227i	$-0.690483\pi$
$709$	−30.0000			−1.12667			−0.563337	+	0.826227i	$0.690483\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		−	25.2982i		−	0.947426i
$714$		0			0
$715$		−	56.5685i		−	2.11554i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	44.7214			1.66783			0.833913	−	0.551896i	$-0.186096\pi$
$719$	44.7214			1.66783			0.833913	+	0.551896i	$0.186096\pi$
$720$		0			0
$721$	20.0000	+	12.6491i	0.744839	+	0.471077i
$722$		0			0
$723$		0			0
$724$		0			0
$725$			14.1421i			0.525226i
$726$		0			0
$727$	26.8328			0.995174			0.497587	−	0.867414i	$-0.334220\pi$
$727$	26.8328			0.995174			0.497587	+	0.867414i	$0.334220\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	4.47214			0.165408
$732$		0			0
$733$	40.2492			1.48664			0.743319	−	0.668937i	$-0.233250\pi$
$733$	40.2492			1.48664			0.743319	+	0.668937i	$0.233250\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−40.0000			−1.47342
$738$		0			0
$739$	−16.0000			−0.588570			−0.294285	−	0.955718i	$-0.595081\pi$
$739$	−16.0000			−0.588570			−0.294285	+	0.955718i	$0.595081\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−36.0000			−1.32071			−0.660356	−	0.750953i	$-0.729595\pi$
$743$	−36.0000			−1.32071			−0.660356	+	0.750953i	$0.729595\pi$
$744$		0			0
$745$		−	37.9473i		−	1.39028i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	40.2492	+	25.4558i	1.47067	+	0.930136i
$750$		0			0
$751$	−40.0000			−1.45962			−0.729810	−	0.683650i	$-0.760392\pi$
$751$	−40.0000			−1.45962			−0.729810	+	0.683650i	$0.760392\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−22.3607			−0.813788
$756$		0			0
$757$		−	32.5269i		−	1.18221i	−0.806594	−	0.591105i	$-0.798692\pi$
$757$		−	32.5269i		−	1.18221i	0.806594	−	0.591105i	$-0.201308\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−31.3050			−1.13480			−0.567402	−	0.823441i	$-0.692051\pi$
$761$	−31.3050			−1.13480			−0.567402	+	0.823441i	$0.692051\pi$
$762$		0			0
$763$	−22.3607	−	14.1421i	−0.809511	−	0.511980i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−20.0000			−0.722158
$768$		0			0
$769$		−	31.6228i		−	1.14035i	−0.821524	−	0.570173i	$-0.806876\pi$
$769$		−	31.6228i		−	1.14035i	0.821524	−	0.570173i	$-0.193124\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$		−	31.6228i		−	1.13739i	−0.822548	−	0.568696i	$-0.807448\pi$
$773$		−	31.6228i		−	1.13739i	0.822548	−	0.568696i	$-0.192552\pi$
$774$		0			0
$775$			31.6228i			1.13592i
$776$		0			0
$777$		0			0
$778$		0			0
$779$			14.1421i			0.506695i
$780$		0			0
$781$	−8.00000			−0.286263
$782$		0			0
$783$		0			0
$784$		0			0
$785$	10.0000			0.356915
$786$		0			0
$787$	26.8328			0.956487			0.478243	−	0.878227i	$-0.341274\pi$
$787$	26.8328			0.956487			0.478243	+	0.878227i	$0.341274\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	13.4164	+	8.48528i	0.477033	+	0.301702i
$792$		0			0
$793$		−	42.4264i		−	1.50661i
$794$		0			0
$795$		0			0
$796$		0			0
$797$			25.2982i			0.896109i	0.894006	+	0.448054i	$0.147883\pi$
$797$			25.2982i			0.896109i	−0.894006	+	0.448054i	$0.852117\pi$
$798$		0			0
$799$	−30.0000			−1.06132
$800$		0			0
$801$		0			0
$802$		0			0
$803$		−	75.8947i		−	2.67826i
$804$		0			0
$805$	−20.0000	−	12.6491i	−0.704907	−	0.445823i
$806$		0			0
$807$		0			0
$808$		0			0
$809$			18.3848i			0.646374i	0.946335	+	0.323187i	$0.104754\pi$
$809$			18.3848i			0.646374i	−0.946335	+	0.323187i	$0.895246\pi$
$810$		0			0
$811$		−	22.1359i		−	0.777298i	−0.921386	−	0.388649i	$-0.872942\pi$
$811$		−	22.1359i		−	0.777298i	0.921386	−	0.388649i	$-0.127058\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$			47.4342i			1.66155i
$816$		0			0
$817$	4.47214			0.156460
$818$		0			0
$819$		0			0
$820$		0			0
$821$			19.7990i			0.690990i	0.938421	+	0.345495i	$0.112289\pi$
$821$			19.7990i			0.690990i	−0.938421	+	0.345495i	$0.887711\pi$
$822$		0			0
$823$			28.2843i			0.985928i	0.870050	+	0.492964i	$0.164087\pi$
$823$			28.2843i			0.985928i	−0.870050	+	0.492964i	$0.835913\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−12.0000			−0.417281			−0.208640	−	0.977992i	$-0.566904\pi$
$827$	−12.0000			−0.417281			−0.208640	+	0.977992i	$0.566904\pi$
$828$		0			0
$829$		−	41.1096i		−	1.42780i	−0.700250	−	0.713898i	$-0.746928\pi$
$829$		−	41.1096i		−	1.42780i	0.700250	−	0.713898i	$-0.253072\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−20.0000	+	9.48683i	−0.692959	+	0.328699i
$834$		0			0
$835$			21.2132i			0.734113i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−17.8885			−0.617581			−0.308791	−	0.951130i	$-0.599924\pi$
$839$	−17.8885			−0.617581			−0.308791	+	0.951130i	$0.599924\pi$
$840$		0			0
$841$	21.0000			0.724138
$842$		0			0
$843$		0			0
$844$		0			0
$845$	15.6525			0.538462
$846$		0			0
$847$	−46.9574	−	29.6985i	−1.61348	−	1.02045i
$848$		0			0
$849$		0			0
$850$		0			0
$851$		−	39.5980i		−	1.35740i
$852$		0			0
$853$	−49.1935			−1.68435			−0.842177	−	0.539202i	$-0.818726\pi$
$853$	−49.1935			−1.68435			−0.842177	+	0.539202i	$0.818726\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		−	41.1096i		−	1.40428i	−0.712040	−	0.702139i	$-0.752229\pi$
$857$		−	41.1096i		−	1.40428i	0.712040	−	0.702139i	$-0.247771\pi$
$858$		0			0
$859$			34.7851i			1.18685i	0.804889	+	0.593425i	$0.202225\pi$
$859$			34.7851i			1.18685i	−0.804889	+	0.593425i	$0.797775\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−36.0000			−1.22545			−0.612727	−	0.790295i	$-0.709928\pi$
$863$	−36.0000			−1.22545			−0.612727	+	0.790295i	$0.709928\pi$
$864$		0			0
$865$		−	56.5685i		−	1.92339i
$866$		0			0
$867$		0			0
$868$		0			0
$869$			33.9411i			1.15137i
$870$		0			0
$871$		−	31.6228i		−	1.07150i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	25.0000	+	15.8114i	0.845154	+	0.534522i
$876$		0			0
$877$		−	21.2132i		−	0.716319i	−0.933660	−	0.358159i	$-0.883404\pi$
$877$		−	21.2132i		−	0.716319i	0.933660	−	0.358159i	$-0.116596\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	4.47214			0.150670			0.0753350	−	0.997158i	$-0.475997\pi$
$881$	4.47214			0.150670			0.0753350	+	0.997158i	$0.475997\pi$
$882$		0			0
$883$		−	12.7279i		−	0.428329i	−0.976798	−	0.214164i	$-0.931297\pi$
$883$		−	12.7279i		−	0.428329i	0.976798	−	0.214164i	$-0.0687028\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		−	9.48683i		−	0.318537i	−0.987235	−	0.159268i	$-0.949086\pi$
$887$		−	9.48683i		−	0.318537i	0.987235	−	0.159268i	$-0.0509135\pi$
$888$		0			0
$889$	−20.0000	+	31.6228i	−0.670778	+	1.06059i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−30.0000			−1.00391
$894$		0			0
$895$		−	6.32456i		−	0.211407i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−17.8885			−0.596616
$900$		0			0
$901$			12.6491i			0.421403i
$902$		0			0
$903$		0			0
$904$		0			0
$905$			7.07107i			0.235050i
$906$		0			0
$907$		−	46.6690i		−	1.54962i	−0.632194	−	0.774810i	$-0.717845\pi$
$907$		−	46.6690i		−	1.54962i	0.632194	−	0.774810i	$-0.282155\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		−	43.8406i		−	1.45250i	−0.687428	−	0.726252i	$-0.741260\pi$
$911$		−	43.8406i		−	1.45250i	0.687428	−	0.726252i	$-0.258740\pi$
$912$		0			0
$913$	−71.5542			−2.36810
$914$		0			0
$915$		0			0
$916$		0			0
$917$	40.0000	+	25.2982i	1.32092	+	0.835421i
$918$		0			0
$919$	40.0000			1.31948			0.659739	−	0.751495i	$-0.270667\pi$
$919$	40.0000			1.31948			0.659739	+	0.751495i	$0.270667\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		−	6.32456i		−	0.208175i
$924$		0			0
$925$			49.4975i			1.62747i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	22.3607			0.733630			0.366815	−	0.930294i	$-0.380448\pi$
$929$	22.3607			0.733630			0.366815	+	0.930294i	$0.380448\pi$
$930$		0			0
$931$	−20.0000	+	9.48683i	−0.655474	+	0.310918i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	40.0000			1.30814
$936$		0			0
$937$	−53.6656			−1.75318			−0.876590	−	0.481238i	$-0.840187\pi$
$937$	−53.6656			−1.75318			−0.876590	+	0.481238i	$0.840187\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−26.8328			−0.874725			−0.437362	−	0.899285i	$-0.644087\pi$
$941$	−26.8328			−0.874725			−0.437362	+	0.899285i	$0.644087\pi$
$942$		0			0
$943$	−17.8885			−0.582531
$944$		0			0
$945$		0			0
$946$		0			0
$947$	42.0000			1.36482			0.682408	−	0.730971i	$-0.260933\pi$
$947$	42.0000			1.36482			0.682408	+	0.730971i	$0.260933\pi$
$948$		0			0
$949$	60.0000			1.94768
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−54.0000			−1.74923			−0.874616	−	0.484817i	$-0.838886\pi$
$953$	−54.0000			−1.74923			−0.874616	+	0.484817i	$0.838886\pi$
$954$		0			0
$955$			3.16228i			0.102329i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−4.47214	−	2.82843i	−0.144413	−	0.0913347i
$960$		0			0
$961$	−9.00000			−0.290323
$962$		0			0
$963$		0			0
$964$		0			0
$965$		−	18.9737i		−	0.610784i
$966$		0			0
$967$			42.4264i			1.36434i	0.731193	+	0.682171i	$0.238964\pi$
$967$			42.4264i			1.36434i	−0.731193	+	0.682171i	$0.761036\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−35.7771			−1.14814			−0.574071	−	0.818806i	$-0.694637\pi$
$971$	−35.7771			−1.14814			−0.574071	+	0.818806i	$0.694637\pi$
$972$		0			0
$973$	22.3607	−	35.3553i	0.716850	−	1.13344i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−18.0000			−0.575871			−0.287936	−	0.957650i	$-0.592969\pi$
$977$	−18.0000			−0.575871			−0.287936	+	0.957650i	$0.592969\pi$
$978$		0			0
$979$			25.2982i			0.808535i
$980$		0			0
$981$		0			0
$982$		0			0
$983$		−	9.48683i		−	0.302583i	−0.988489	−	0.151291i	$-0.951657\pi$
$983$		−	9.48683i		−	0.302583i	0.988489	−	0.151291i	$-0.0483432\pi$
$984$		0			0
$985$	−26.8328			−0.854965
$986$		0			0
$987$		0			0
$988$		0			0
$989$			5.65685i			0.179878i
$990$		0			0
$991$	−40.0000			−1.27064			−0.635321	−	0.772248i	$-0.719132\pi$
$991$	−40.0000			−1.27064			−0.635321	+	0.772248i	$0.719132\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$	13.4164			0.424902			0.212451	−	0.977172i	$-0.431855\pi$
$997$	13.4164			0.424902			0.212451	+	0.977172i	$0.431855\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5040.2.k.b.1889.4		4
3.2	odd	2		5040.2.k.c.1889.2		4
4.3	odd	2		630.2.d.b.629.3	yes	4
5.4	even	2		5040.2.k.c.1889.3		4
7.6	odd	2	inner	5040.2.k.b.1889.2		4
12.11	even	2		630.2.d.c.629.1	yes	4
15.14	odd	2	inner	5040.2.k.b.1889.1		4
20.3	even	4		3150.2.b.b.251.6		8
20.7	even	4		3150.2.b.b.251.3		8
20.19	odd	2		630.2.d.c.629.4	yes	4
21.20	even	2		5040.2.k.c.1889.4		4
28.27	even	2		630.2.d.b.629.1	✓	4
35.34	odd	2		5040.2.k.c.1889.1		4
60.23	odd	4		3150.2.b.b.251.2		8
60.47	odd	4		3150.2.b.b.251.7		8
60.59	even	2		630.2.d.b.629.2	yes	4
84.83	odd	2		630.2.d.c.629.3	yes	4
105.104	even	2	inner	5040.2.k.b.1889.3		4
140.27	odd	4		3150.2.b.b.251.4		8
140.83	odd	4		3150.2.b.b.251.5		8
140.139	even	2		630.2.d.c.629.2	yes	4
420.83	even	4		3150.2.b.b.251.1		8
420.167	even	4		3150.2.b.b.251.8		8
420.419	odd	2		630.2.d.b.629.4	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
630.2.d.b.629.1	✓	4	28.27	even	2
630.2.d.b.629.2	yes	4	60.59	even	2
630.2.d.b.629.3	yes	4	4.3	odd	2
630.2.d.b.629.4	yes	4	420.419	odd	2
630.2.d.c.629.1	yes	4	12.11	even	2
630.2.d.c.629.2	yes	4	140.139	even	2
630.2.d.c.629.3	yes	4	84.83	odd	2
630.2.d.c.629.4	yes	4	20.19	odd	2
3150.2.b.b.251.1		8	420.83	even	4
3150.2.b.b.251.2		8	60.23	odd	4
3150.2.b.b.251.3		8	20.7	even	4
3150.2.b.b.251.4		8	140.27	odd	4
3150.2.b.b.251.5		8	140.83	odd	4
3150.2.b.b.251.6		8	20.3	even	4
3150.2.b.b.251.7		8	60.47	odd	4
3150.2.b.b.251.8		8	420.167	even	4
5040.2.k.b.1889.1		4	15.14	odd	2	inner
5040.2.k.b.1889.2		4	7.6	odd	2	inner
5040.2.k.b.1889.3		4	105.104	even	2	inner
5040.2.k.b.1889.4		4	1.1	even	1	trivial
5040.2.k.c.1889.1		4	35.34	odd	2
5040.2.k.c.1889.2		4	3.2	odd	2
5040.2.k.c.1889.3		4	5.4	even	2
5040.2.k.c.1889.4		4	21.20	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:	\( N \)	\(=\)	\( 5040 = 2^{4} \cdot 3^{2} \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5040.k (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+2.23607 q^{5} +(2.23607 + 1.41421i) q^{7} +O(q^{10})\)	\(q+2.23607 q^{5} +(2.23607 + 1.41421i) q^{7} -5.65685i q^{11} +4.47214 q^{13} +3.16228i q^{17} +3.16228i q^{19} -4.00000 q^{23} +5.00000 q^{25} +2.82843i q^{29} +6.32456i q^{31} +(5.00000 + 3.16228i) q^{35} +9.89949i q^{37} +4.47214 q^{41} -1.41421i q^{43} +9.48683i q^{47} +(3.00000 + 6.32456i) q^{49} +4.00000 q^{53} -12.6491i q^{55} -4.47214 q^{59} -9.48683i q^{61} +10.0000 q^{65} -7.07107i q^{67} -1.41421i q^{71} +13.4164 q^{73} +(8.00000 - 12.6491i) q^{77} -6.00000 q^{79} -12.6491i q^{83} +7.07107i q^{85} -4.47214 q^{89} +(10.0000 + 6.32456i) q^{91} +7.07107i q^{95} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q+O(q^{10}) \) 4 * q	\( 4 q - 16 q^{23} + 20 q^{25} + 20 q^{35} + 12 q^{49} + 16 q^{53} + 40 q^{65} + 32 q^{77} - 24 q^{79} + 40 q^{91}+O(q^{100}) \) 4 * q - 16 * q^23 + 20 * q^25 + 20 * q^35 + 12 * q^49 + 16 * q^53 + 40 * q^65 + 32 * q^77 - 24 * q^79 + 40 * q^91

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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