LMFDB - Embedded newform 420.2.k.a.169.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [420,2,Mod(169,420)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(420, 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("420.169");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 420 = 2^{2} \cdot 3 \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	420.k (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:	$3.35371688489$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			169.1
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	420.169
Dual form			420.2.k.a.169.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{3} +(-2.00000 + 1.00000i) q^{5} -1.00000i q^{7} -1.00000 q^{9} +O(q^{10})$	$q-1.00000i q^{3} +(-2.00000 + 1.00000i) q^{5} -1.00000i q^{7} -1.00000 q^{9} -4.00000 q^{11} -6.00000i q^{13} +(1.00000 + 2.00000i) q^{15} -2.00000i q^{17} -6.00000 q^{19} -1.00000 q^{21} +2.00000i q^{23} +(3.00000 - 4.00000i) q^{25} +1.00000i q^{27} -6.00000 q^{29} -2.00000 q^{31} +4.00000i q^{33} +(1.00000 + 2.00000i) q^{35} +4.00000i q^{37} -6.00000 q^{39} +8.00000 q^{41} -4.00000i q^{43} +(2.00000 - 1.00000i) q^{45} -4.00000i q^{47} -1.00000 q^{49} -2.00000 q^{51} +6.00000i q^{53} +(8.00000 - 4.00000i) q^{55} +6.00000i q^{57} -4.00000 q^{59} +14.0000 q^{61} +1.00000i q^{63} +(6.00000 + 12.0000i) q^{65} -4.00000i q^{67} +2.00000 q^{69} -10.0000i q^{73} +(-4.00000 - 3.00000i) q^{75} +4.00000i q^{77} +1.00000 q^{81} -16.0000i q^{83} +(2.00000 + 4.00000i) q^{85} +6.00000i q^{87} -8.00000 q^{89} -6.00000 q^{91} +2.00000i q^{93} +(12.0000 - 6.00000i) q^{95} -10.0000i q^{97} +4.00000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 4 q^{5} - 2 q^{9}+O(q^{10}) $ 2 * q - 4 * q^5 - 2 * q^9	$ 2 q - 4 q^{5} - 2 q^{9} - 8 q^{11} + 2 q^{15} - 12 q^{19} - 2 q^{21} + 6 q^{25} - 12 q^{29} - 4 q^{31} + 2 q^{35} - 12 q^{39} + 16 q^{41} + 4 q^{45} - 2 q^{49} - 4 q^{51} + 16 q^{55} - 8 q^{59} + 28 q^{61} + 12 q^{65} + 4 q^{69} - 8 q^{75} + 2 q^{81} + 4 q^{85} - 16 q^{89} - 12 q^{91} + 24 q^{95} + 8 q^{99}+O(q^{100}) $ 2 * q - 4 * q^5 - 2 * q^9 - 8 * q^11 + 2 * q^15 - 12 * q^19 - 2 * q^21 + 6 * q^25 - 12 * q^29 - 4 * q^31 + 2 * q^35 - 12 * q^39 + 16 * q^41 + 4 * q^45 - 2 * q^49 - 4 * q^51 + 16 * q^55 - 8 * q^59 + 28 * q^61 + 12 * q^65 + 4 * q^69 - 8 * q^75 + 2 * q^81 + 4 * q^85 - 16 * q^89 - 12 * q^91 + 24 * q^95 + 8 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/420\mathbb{Z}\right)^\times$.

$n$	$211$	$241$	$281$	$337$
$\chi(n)$	$1$	$1$	$1$	$-1$

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
$3$		−	1.00000i		−	0.577350i
$3$		−	1.00000i		−	0.577350i
$4$		0			0
$5$	−2.00000	+	1.00000i	−0.894427	+	0.447214i
$5$	−2.00000	+	1.00000i	−0.894427	+	0.447214i
$6$		0			0
$7$		−	1.00000i		−	0.377964i
$7$		−	1.00000i		−	0.377964i
$8$		0			0
$9$	−1.00000			−0.333333
$10$		0			0
$11$	−4.00000			−1.20605			−0.603023	−	0.797724i	$-0.706037\pi$
$11$	−4.00000			−1.20605			−0.603023	+	0.797724i	$0.706037\pi$
$12$		0			0
$13$		−	6.00000i		−	1.66410i	−0.554700	−	0.832050i	$-0.687167\pi$
$13$		−	6.00000i		−	1.66410i	0.554700	−	0.832050i	$-0.312833\pi$
$14$		0			0
$15$	1.00000	+	2.00000i	0.258199	+	0.516398i
$16$		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
$19$	−6.00000			−1.37649			−0.688247	−	0.725476i	$-0.741620\pi$
$19$	−6.00000			−1.37649			−0.688247	+	0.725476i	$0.741620\pi$
$20$		0			0
$21$	−1.00000			−0.218218
$22$		0			0
$23$			2.00000i			0.417029i	0.978019	+	0.208514i	$0.0668628\pi$
$23$			2.00000i			0.417029i	−0.978019	+	0.208514i	$0.933137\pi$
$24$		0			0
$25$	3.00000	−	4.00000i	0.600000	−	0.800000i
$26$		0			0
$27$			1.00000i			0.192450i
$28$		0			0
$29$	−6.00000			−1.11417			−0.557086	−	0.830455i	$-0.688081\pi$
$29$	−6.00000			−1.11417			−0.557086	+	0.830455i	$0.688081\pi$
$30$		0			0
$31$	−2.00000			−0.359211			−0.179605	−	0.983739i	$-0.557482\pi$
$31$	−2.00000			−0.359211			−0.179605	+	0.983739i	$0.557482\pi$
$32$		0			0
$33$			4.00000i			0.696311i
$34$		0			0
$35$	1.00000	+	2.00000i	0.169031	+	0.338062i
$36$		0			0
$37$			4.00000i			0.657596i	0.944400	+	0.328798i	$0.106644\pi$
$37$			4.00000i			0.657596i	−0.944400	+	0.328798i	$0.893356\pi$
$38$		0			0
$39$	−6.00000			−0.960769
$40$		0			0
$41$	8.00000			1.24939			0.624695	−	0.780869i	$-0.285223\pi$
$41$	8.00000			1.24939			0.624695	+	0.780869i	$0.285223\pi$
$42$		0			0
$43$		−	4.00000i		−	0.609994i	−0.952353	−	0.304997i	$-0.901344\pi$
$43$		−	4.00000i		−	0.609994i	0.952353	−	0.304997i	$-0.0986555\pi$
$44$		0			0
$45$	2.00000	−	1.00000i	0.298142	−	0.149071i
$46$		0			0
$47$		−	4.00000i		−	0.583460i	−0.956501	−	0.291730i	$-0.905769\pi$
$47$		−	4.00000i		−	0.583460i	0.956501	−	0.291730i	$-0.0942309\pi$
$48$		0			0
$49$	−1.00000			−0.142857
$50$		0			0
$51$	−2.00000			−0.280056
$52$		0			0
$53$			6.00000i			0.824163i	0.911147	+	0.412082i	$0.135198\pi$
$53$			6.00000i			0.824163i	−0.911147	+	0.412082i	$0.864802\pi$
$54$		0			0
$55$	8.00000	−	4.00000i	1.07872	−	0.539360i
$56$		0			0
$57$			6.00000i			0.794719i
$58$		0			0
$59$	−4.00000			−0.520756			−0.260378	−	0.965507i	$-0.583847\pi$
$59$	−4.00000			−0.520756			−0.260378	+	0.965507i	$0.583847\pi$
$60$		0			0
$61$	14.0000			1.79252			0.896258	−	0.443533i	$-0.146275\pi$
$61$	14.0000			1.79252			0.896258	+	0.443533i	$0.146275\pi$
$62$		0			0
$63$			1.00000i			0.125988i
$64$		0			0
$65$	6.00000	+	12.0000i	0.744208	+	1.48842i
$66$		0			0
$67$		−	4.00000i		−	0.488678i	−0.969690	−	0.244339i	$-0.921429\pi$
$67$		−	4.00000i		−	0.488678i	0.969690	−	0.244339i	$-0.0785709\pi$
$68$		0			0
$69$	2.00000			0.240772
$70$		0			0
$71$		0			0			−	1.00000i	$-0.5\pi$
$71$		0			0				1.00000i	$0.5\pi$
$72$		0			0
$73$		−	10.0000i		−	1.17041i	−0.810885	−	0.585206i	$-0.801014\pi$
$73$		−	10.0000i		−	1.17041i	0.810885	−	0.585206i	$-0.198986\pi$
$74$		0			0
$75$	−4.00000	−	3.00000i	−0.461880	−	0.346410i
$76$		0			0
$77$			4.00000i			0.455842i
$78$		0			0
$79$		0			0			−	1.00000i	$-0.5\pi$
$79$		0			0				1.00000i	$0.5\pi$
$80$		0			0
$81$	1.00000			0.111111
$82$		0			0
$83$		−	16.0000i		−	1.75623i	−0.478451	−	0.878114i	$-0.658802\pi$
$83$		−	16.0000i		−	1.75623i	0.478451	−	0.878114i	$-0.341198\pi$
$84$		0			0
$85$	2.00000	+	4.00000i	0.216930	+	0.433861i
$86$		0			0
$87$			6.00000i			0.643268i
$88$		0			0
$89$	−8.00000			−0.847998			−0.423999	−	0.905663i	$-0.639374\pi$
$89$	−8.00000			−0.847998			−0.423999	+	0.905663i	$0.639374\pi$
$90$		0			0
$91$	−6.00000			−0.628971
$92$		0			0
$93$			2.00000i			0.207390i
$94$		0			0
$95$	12.0000	−	6.00000i	1.23117	−	0.615587i
$96$		0			0
$97$		−	10.0000i		−	1.01535i	−0.861550	−	0.507673i	$-0.830506\pi$
$97$		−	10.0000i		−	1.01535i	0.861550	−	0.507673i	$-0.169494\pi$
$98$		0			0
$99$	4.00000			0.402015
$100$		0			0
$101$	16.0000			1.59206			0.796030	−	0.605257i	$-0.206930\pi$
$101$	16.0000			1.59206			0.796030	+	0.605257i	$0.206930\pi$
$102$		0			0
$103$			16.0000i			1.57653i	0.615338	+	0.788263i	$0.289020\pi$
$103$			16.0000i			1.57653i	−0.615338	+	0.788263i	$0.710980\pi$
$104$		0			0
$105$	2.00000	−	1.00000i	0.195180	−	0.0975900i
$106$		0			0
$107$			6.00000i			0.580042i	0.957020	+	0.290021i	$0.0936623\pi$
$107$			6.00000i			0.580042i	−0.957020	+	0.290021i	$0.906338\pi$
$108$		0			0
$109$	14.0000			1.34096			0.670478	−	0.741929i	$-0.266089\pi$
$109$	14.0000			1.34096			0.670478	+	0.741929i	$0.266089\pi$
$110$		0			0
$111$	4.00000			0.379663
$112$		0			0
$113$			18.0000i			1.69330i	0.532152	+	0.846649i	$0.321383\pi$
$113$			18.0000i			1.69330i	−0.532152	+	0.846649i	$0.678617\pi$
$114$		0			0
$115$	−2.00000	−	4.00000i	−0.186501	−	0.373002i
$116$		0			0
$117$			6.00000i			0.554700i
$118$		0			0
$119$	−2.00000			−0.183340
$120$		0			0
$121$	5.00000			0.454545
$122$		0			0
$123$		−	8.00000i		−	0.721336i
$124$		0			0
$125$	−2.00000	+	11.0000i	−0.178885	+	0.983870i
$126$		0			0
$127$			16.0000i			1.41977i	0.704317	+	0.709885i	$0.251253\pi$
$127$			16.0000i			1.41977i	−0.704317	+	0.709885i	$0.748747\pi$
$128$		0			0
$129$	−4.00000			−0.352180
$130$		0			0
$131$	−12.0000			−1.04844			−0.524222	−	0.851581i	$-0.675644\pi$
$131$	−12.0000			−1.04844			−0.524222	+	0.851581i	$0.675644\pi$
$132$		0			0
$133$			6.00000i			0.520266i
$134$		0			0
$135$	−1.00000	−	2.00000i	−0.0860663	−	0.172133i
$136$		0			0
$137$		−	18.0000i		−	1.53784i	−0.639343	−	0.768922i	$-0.720793\pi$
$137$		−	18.0000i		−	1.53784i	0.639343	−	0.768922i	$-0.279207\pi$
$138$		0			0
$139$	−10.0000			−0.848189			−0.424094	−	0.905618i	$-0.639408\pi$
$139$	−10.0000			−0.848189			−0.424094	+	0.905618i	$0.639408\pi$
$140$		0			0
$141$	−4.00000			−0.336861
$142$		0			0
$143$			24.0000i			2.00698i
$144$		0			0
$145$	12.0000	−	6.00000i	0.996546	−	0.498273i
$146$		0			0
$147$			1.00000i			0.0824786i
$148$		0			0
$149$	−22.0000			−1.80231			−0.901155	−	0.433497i	$-0.857280\pi$
$149$	−22.0000			−1.80231			−0.901155	+	0.433497i	$0.857280\pi$
$150$		0			0
$151$	16.0000			1.30206			0.651031	−	0.759051i	$-0.274337\pi$
$151$	16.0000			1.30206			0.651031	+	0.759051i	$0.274337\pi$
$152$		0			0
$153$			2.00000i			0.161690i
$154$		0			0
$155$	4.00000	−	2.00000i	0.321288	−	0.160644i
$156$		0			0
$157$		−	10.0000i		−	0.798087i	−0.916932	−	0.399043i	$-0.869342\pi$
$157$		−	10.0000i		−	0.798087i	0.916932	−	0.399043i	$-0.130658\pi$
$158$		0			0
$159$	6.00000			0.475831
$160$		0			0
$161$	2.00000			0.157622
$162$		0			0
$163$		−	8.00000i		−	0.626608i	−0.949653	−	0.313304i	$-0.898564\pi$
$163$		−	8.00000i		−	0.626608i	0.949653	−	0.313304i	$-0.101436\pi$
$164$		0			0
$165$	−4.00000	−	8.00000i	−0.311400	−	0.622799i
$166$		0			0
$167$		−	24.0000i		−	1.85718i	−0.371113	−	0.928588i	$-0.621024\pi$
$167$		−	24.0000i		−	1.85718i	0.371113	−	0.928588i	$-0.378976\pi$
$168$		0			0
$169$	−23.0000			−1.76923
$170$		0			0
$171$	6.00000			0.458831
$172$		0			0
$173$		−	2.00000i		−	0.152057i	−0.997106	−	0.0760286i	$-0.975776\pi$
$173$		−	2.00000i		−	0.152057i	0.997106	−	0.0760286i	$-0.0242240\pi$
$174$		0			0
$175$	−4.00000	−	3.00000i	−0.302372	−	0.226779i
$176$		0			0
$177$			4.00000i			0.300658i
$178$		0			0
$179$	−4.00000			−0.298974			−0.149487	−	0.988764i	$-0.547762\pi$
$179$	−4.00000			−0.298974			−0.149487	+	0.988764i	$0.547762\pi$
$180$		0			0
$181$	−10.0000			−0.743294			−0.371647	−	0.928374i	$-0.621207\pi$
$181$	−10.0000			−0.743294			−0.371647	+	0.928374i	$0.621207\pi$
$182$		0			0
$183$		−	14.0000i		−	1.03491i
$184$		0			0
$185$	−4.00000	−	8.00000i	−0.294086	−	0.588172i
$186$		0			0
$187$			8.00000i			0.585018i
$188$		0			0
$189$	1.00000			0.0727393
$190$		0			0
$191$		0			0			−	1.00000i	$-0.5\pi$
$191$		0			0				1.00000i	$0.5\pi$
$192$		0			0
$193$			8.00000i			0.575853i	0.957653	+	0.287926i	$0.0929658\pi$
$193$			8.00000i			0.575853i	−0.957653	+	0.287926i	$0.907034\pi$
$194$		0			0
$195$	12.0000	−	6.00000i	0.859338	−	0.429669i
$196$		0			0
$197$		−	10.0000i		−	0.712470i	−0.934396	−	0.356235i	$-0.884060\pi$
$197$		−	10.0000i		−	0.712470i	0.934396	−	0.356235i	$-0.115940\pi$
$198$		0			0
$199$	−18.0000			−1.27599			−0.637993	−	0.770042i	$-0.720235\pi$
$199$	−18.0000			−1.27599			−0.637993	+	0.770042i	$0.720235\pi$
$200$		0			0
$201$	−4.00000			−0.282138
$202$		0			0
$203$			6.00000i			0.421117i
$204$		0			0
$205$	−16.0000	+	8.00000i	−1.11749	+	0.558744i
$206$		0			0
$207$		−	2.00000i		−	0.139010i
$208$		0			0
$209$	24.0000			1.66011
$210$		0			0
$211$	−24.0000			−1.65223			−0.826114	−	0.563503i	$-0.809453\pi$
$211$	−24.0000			−1.65223			−0.826114	+	0.563503i	$0.809453\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	4.00000	+	8.00000i	0.272798	+	0.545595i
$216$		0			0
$217$			2.00000i			0.135769i
$218$		0			0
$219$	−10.0000			−0.675737
$220$		0			0
$221$	−12.0000			−0.807207
$222$		0			0
$223$		−	16.0000i		−	1.07144i	−0.844396	−	0.535720i	$-0.820040\pi$
$223$		−	16.0000i		−	1.07144i	0.844396	−	0.535720i	$-0.179960\pi$
$224$		0			0
$225$	−3.00000	+	4.00000i	−0.200000	+	0.266667i
$226$		0			0
$227$		−	20.0000i		−	1.32745i	−0.747978	−	0.663723i	$-0.768975\pi$
$227$		−	20.0000i		−	1.32745i	0.747978	−	0.663723i	$-0.231025\pi$
$228$		0			0
$229$	−2.00000			−0.132164			−0.0660819	−	0.997814i	$-0.521050\pi$
$229$	−2.00000			−0.132164			−0.0660819	+	0.997814i	$0.521050\pi$
$230$		0			0
$231$	4.00000			0.263181
$232$		0			0
$233$		−	26.0000i		−	1.70332i	−0.524097	−	0.851658i	$-0.675597\pi$
$233$		−	26.0000i		−	1.70332i	0.524097	−	0.851658i	$-0.324403\pi$
$234$		0			0
$235$	4.00000	+	8.00000i	0.260931	+	0.521862i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	16.0000			1.03495			0.517477	−	0.855697i	$-0.326871\pi$
$239$	16.0000			1.03495			0.517477	+	0.855697i	$0.326871\pi$
$240$		0			0
$241$	10.0000			0.644157			0.322078	−	0.946713i	$-0.395619\pi$
$241$	10.0000			0.644157			0.322078	+	0.946713i	$0.395619\pi$
$242$		0			0
$243$		−	1.00000i		−	0.0641500i
$244$		0			0
$245$	2.00000	−	1.00000i	0.127775	−	0.0638877i
$246$		0			0
$247$			36.0000i			2.29063i
$248$		0			0
$249$	−16.0000			−1.01396
$250$		0			0
$251$	−12.0000			−0.757433			−0.378717	−	0.925513i	$-0.623635\pi$
$251$	−12.0000			−0.757433			−0.378717	+	0.925513i	$0.623635\pi$
$252$		0			0
$253$		−	8.00000i		−	0.502956i
$254$		0			0
$255$	4.00000	−	2.00000i	0.250490	−	0.125245i
$256$		0			0
$257$		−	6.00000i		−	0.374270i	−0.982334	−	0.187135i	$-0.940080\pi$
$257$		−	6.00000i		−	0.374270i	0.982334	−	0.187135i	$-0.0599201\pi$
$258$		0			0
$259$	4.00000			0.248548
$260$		0			0
$261$	6.00000			0.371391
$262$		0			0
$263$		−	10.0000i		−	0.616626i	−0.951285	−	0.308313i	$-0.900236\pi$
$263$		−	10.0000i		−	0.616626i	0.951285	−	0.308313i	$-0.0997645\pi$
$264$		0			0
$265$	−6.00000	−	12.0000i	−0.368577	−	0.737154i
$266$		0			0
$267$			8.00000i			0.489592i
$268$		0			0
$269$	12.0000			0.731653			0.365826	−	0.930683i	$-0.380786\pi$
$269$	12.0000			0.731653			0.365826	+	0.930683i	$0.380786\pi$
$270$		0			0
$271$	−14.0000			−0.850439			−0.425220	−	0.905090i	$-0.639803\pi$
$271$	−14.0000			−0.850439			−0.425220	+	0.905090i	$0.639803\pi$
$272$		0			0
$273$			6.00000i			0.363137i
$274$		0			0
$275$	−12.0000	+	16.0000i	−0.723627	+	0.964836i
$276$		0			0
$277$			28.0000i			1.68236i	0.540758	+	0.841178i	$0.318138\pi$
$277$			28.0000i			1.68236i	−0.540758	+	0.841178i	$0.681862\pi$
$278$		0			0
$279$	2.00000			0.119737
$280$		0			0
$281$	−6.00000			−0.357930			−0.178965	−	0.983855i	$-0.557275\pi$
$281$	−6.00000			−0.357930			−0.178965	+	0.983855i	$0.557275\pi$
$282$		0			0
$283$			28.0000i			1.66443i	0.554455	+	0.832214i	$0.312927\pi$
$283$			28.0000i			1.66443i	−0.554455	+	0.832214i	$0.687073\pi$
$284$		0			0
$285$	−6.00000	−	12.0000i	−0.355409	−	0.710819i
$286$		0			0
$287$		−	8.00000i		−	0.472225i
$288$		0			0
$289$	13.0000			0.764706
$290$		0			0
$291$	−10.0000			−0.586210
$292$		0			0
$293$		−	6.00000i		−	0.350524i	−0.984522	−	0.175262i	$-0.943923\pi$
$293$		−	6.00000i		−	0.350524i	0.984522	−	0.175262i	$-0.0560772\pi$
$294$		0			0
$295$	8.00000	−	4.00000i	0.465778	−	0.232889i
$296$		0			0
$297$		−	4.00000i		−	0.232104i
$298$		0			0
$299$	12.0000			0.693978
$300$		0			0
$301$	−4.00000			−0.230556
$302$		0			0
$303$		−	16.0000i		−	0.919176i
$304$		0			0
$305$	−28.0000	+	14.0000i	−1.60328	+	0.801638i
$306$		0			0
$307$			12.0000i			0.684876i	0.939540	+	0.342438i	$0.111253\pi$
$307$			12.0000i			0.684876i	−0.939540	+	0.342438i	$0.888747\pi$
$308$		0			0
$309$	16.0000			0.910208
$310$		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
$313$			14.0000i			0.791327i	0.918396	+	0.395663i	$0.129485\pi$
$313$			14.0000i			0.791327i	−0.918396	+	0.395663i	$0.870515\pi$
$314$		0			0
$315$	−1.00000	−	2.00000i	−0.0563436	−	0.112687i
$316$		0			0
$317$		−	14.0000i		−	0.786318i	−0.919470	−	0.393159i	$-0.871382\pi$
$317$		−	14.0000i		−	0.786318i	0.919470	−	0.393159i	$-0.128618\pi$
$318$		0			0
$319$	24.0000			1.34374
$320$		0			0
$321$	6.00000			0.334887
$322$		0			0
$323$			12.0000i			0.667698i
$324$		0			0
$325$	−24.0000	−	18.0000i	−1.33128	−	0.998460i
$326$		0			0
$327$		−	14.0000i		−	0.774202i
$328$		0			0
$329$	−4.00000			−0.220527
$330$		0			0
$331$	−8.00000			−0.439720			−0.219860	−	0.975531i	$-0.570560\pi$
$331$	−8.00000			−0.439720			−0.219860	+	0.975531i	$0.570560\pi$
$332$		0			0
$333$		−	4.00000i		−	0.219199i
$334$		0			0
$335$	4.00000	+	8.00000i	0.218543	+	0.437087i
$336$		0			0
$337$		−	8.00000i		−	0.435788i	−0.975972	−	0.217894i	$-0.930081\pi$
$337$		−	8.00000i		−	0.435788i	0.975972	−	0.217894i	$-0.0699187\pi$
$338$		0			0
$339$	18.0000			0.977626
$340$		0			0
$341$	8.00000			0.433224
$342$		0			0
$343$			1.00000i			0.0539949i
$344$		0			0
$345$	−4.00000	+	2.00000i	−0.215353	+	0.107676i
$346$		0			0
$347$		−	18.0000i		−	0.966291i	−0.875540	−	0.483145i	$-0.839494\pi$
$347$		−	18.0000i		−	0.966291i	0.875540	−	0.483145i	$-0.160506\pi$
$348$		0			0
$349$	−22.0000			−1.17763			−0.588817	−	0.808267i	$-0.700406\pi$
$349$	−22.0000			−1.17763			−0.588817	+	0.808267i	$0.700406\pi$
$350$		0			0
$351$	6.00000			0.320256
$352$		0			0
$353$		−	2.00000i		−	0.106449i	−0.998583	−	0.0532246i	$-0.983050\pi$
$353$		−	2.00000i		−	0.106449i	0.998583	−	0.0532246i	$-0.0169499\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$			2.00000i			0.105851i
$358$		0			0
$359$		0			0			−	1.00000i	$-0.5\pi$
$359$		0			0				1.00000i	$0.5\pi$
$360$		0			0
$361$	17.0000			0.894737
$362$		0			0
$363$		−	5.00000i		−	0.262432i
$364$		0			0
$365$	10.0000	+	20.0000i	0.523424	+	1.04685i
$366$		0			0
$367$			16.0000i			0.835193i	0.908633	+	0.417597i	$0.137127\pi$
$367$			16.0000i			0.835193i	−0.908633	+	0.417597i	$0.862873\pi$
$368$		0			0
$369$	−8.00000			−0.416463
$370$		0			0
$371$	6.00000			0.311504
$372$		0			0
$373$		−	24.0000i		−	1.24267i	−0.783544	−	0.621336i	$-0.786590\pi$
$373$		−	24.0000i		−	1.24267i	0.783544	−	0.621336i	$-0.213410\pi$
$374$		0			0
$375$	11.0000	+	2.00000i	0.568038	+	0.103280i
$376$		0			0
$377$			36.0000i			1.85409i
$378$		0			0
$379$	28.0000			1.43826			0.719132	−	0.694874i	$-0.244540\pi$
$379$	28.0000			1.43826			0.719132	+	0.694874i	$0.244540\pi$
$380$		0			0
$381$	16.0000			0.819705
$382$		0			0
$383$			28.0000i			1.43073i	0.698749	+	0.715367i	$0.253740\pi$
$383$			28.0000i			1.43073i	−0.698749	+	0.715367i	$0.746260\pi$
$384$		0			0
$385$	−4.00000	−	8.00000i	−0.203859	−	0.407718i
$386$		0			0
$387$			4.00000i			0.203331i
$388$		0			0
$389$	−2.00000			−0.101404			−0.0507020	−	0.998714i	$-0.516146\pi$
$389$	−2.00000			−0.101404			−0.0507020	+	0.998714i	$0.516146\pi$
$390$		0			0
$391$	4.00000			0.202289
$392$		0			0
$393$			12.0000i			0.605320i
$394$		0			0
$395$		0			0
$396$		0			0
$397$			10.0000i			0.501886i	0.968002	+	0.250943i	$0.0807406\pi$
$397$			10.0000i			0.501886i	−0.968002	+	0.250943i	$0.919259\pi$
$398$		0			0
$399$	6.00000			0.300376
$400$		0			0
$401$	22.0000			1.09863			0.549314	−	0.835616i	$-0.314889\pi$
$401$	22.0000			1.09863			0.549314	+	0.835616i	$0.314889\pi$
$402$		0			0
$403$			12.0000i			0.597763i
$404$		0			0
$405$	−2.00000	+	1.00000i	−0.0993808	+	0.0496904i
$406$		0			0
$407$		−	16.0000i		−	0.793091i
$408$		0			0
$409$	10.0000			0.494468			0.247234	−	0.968956i	$-0.420478\pi$
$409$	10.0000			0.494468			0.247234	+	0.968956i	$0.420478\pi$
$410$		0			0
$411$	−18.0000			−0.887875
$412$		0			0
$413$			4.00000i			0.196827i
$414$		0			0
$415$	16.0000	+	32.0000i	0.785409	+	1.57082i
$416$		0			0
$417$			10.0000i			0.489702i
$418$		0			0
$419$	40.0000			1.95413			0.977064	−	0.212946i	$-0.0683059\pi$
$419$	40.0000			1.95413			0.977064	+	0.212946i	$0.0683059\pi$
$420$		0			0
$421$	−2.00000			−0.0974740			−0.0487370	−	0.998812i	$-0.515520\pi$
$421$	−2.00000			−0.0974740			−0.0487370	+	0.998812i	$0.515520\pi$
$422$		0			0
$423$			4.00000i			0.194487i
$424$		0			0
$425$	−8.00000	−	6.00000i	−0.388057	−	0.291043i
$426$		0			0
$427$		−	14.0000i		−	0.677507i
$428$		0			0
$429$	24.0000			1.15873
$430$		0			0
$431$	−16.0000			−0.770693			−0.385346	−	0.922772i	$-0.625918\pi$
$431$	−16.0000			−0.770693			−0.385346	+	0.922772i	$0.625918\pi$
$432$		0			0
$433$		−	10.0000i		−	0.480569i	−0.970702	−	0.240285i	$-0.922759\pi$
$433$		−	10.0000i		−	0.480569i	0.970702	−	0.240285i	$-0.0772408\pi$
$434$		0			0
$435$	−6.00000	−	12.0000i	−0.287678	−	0.575356i
$436$		0			0
$437$		−	12.0000i		−	0.574038i
$438$		0			0
$439$	30.0000			1.43182			0.715911	−	0.698192i	$-0.246012\pi$
$439$	30.0000			1.43182			0.715911	+	0.698192i	$0.246012\pi$
$440$		0			0
$441$	1.00000			0.0476190
$442$		0			0
$443$		−	30.0000i		−	1.42534i	−0.701498	−	0.712672i	$-0.747485\pi$
$443$		−	30.0000i		−	1.42534i	0.701498	−	0.712672i	$-0.252515\pi$
$444$		0			0
$445$	16.0000	−	8.00000i	0.758473	−	0.379236i
$446$		0			0
$447$			22.0000i			1.04056i
$448$		0			0
$449$	−18.0000			−0.849473			−0.424736	−	0.905317i	$-0.639633\pi$
$449$	−18.0000			−0.849473			−0.424736	+	0.905317i	$0.639633\pi$
$450$		0			0
$451$	−32.0000			−1.50682
$452$		0			0
$453$		−	16.0000i		−	0.751746i
$454$		0			0
$455$	12.0000	−	6.00000i	0.562569	−	0.281284i
$456$		0			0
$457$		−	20.0000i		−	0.935561i	−0.883845	−	0.467780i	$-0.845054\pi$
$457$		−	20.0000i		−	0.935561i	0.883845	−	0.467780i	$-0.154946\pi$
$458$		0			0
$459$	2.00000			0.0933520
$460$		0			0
$461$		0			0			−	1.00000i	$-0.5\pi$
$461$		0			0				1.00000i	$0.5\pi$
$462$		0			0
$463$		−	24.0000i		−	1.11537i	−0.830051	−	0.557687i	$-0.811689\pi$
$463$		−	24.0000i		−	1.11537i	0.830051	−	0.557687i	$-0.188311\pi$
$464$		0			0
$465$	−2.00000	−	4.00000i	−0.0927478	−	0.185496i
$466$		0			0
$467$		−	12.0000i		−	0.555294i	−0.960683	−	0.277647i	$-0.910445\pi$
$467$		−	12.0000i		−	0.555294i	0.960683	−	0.277647i	$-0.0895545\pi$
$468$		0			0
$469$	−4.00000			−0.184703
$470$		0			0
$471$	−10.0000			−0.460776
$472$		0			0
$473$			16.0000i			0.735681i
$474$		0			0
$475$	−18.0000	+	24.0000i	−0.825897	+	1.10120i
$476$		0			0
$477$		−	6.00000i		−	0.274721i
$478$		0			0
$479$	4.00000			0.182765			0.0913823	−	0.995816i	$-0.470871\pi$
$479$	4.00000			0.182765			0.0913823	+	0.995816i	$0.470871\pi$
$480$		0			0
$481$	24.0000			1.09431
$482$		0			0
$483$		−	2.00000i		−	0.0910032i
$484$		0			0
$485$	10.0000	+	20.0000i	0.454077	+	0.908153i
$486$		0			0
$487$			12.0000i			0.543772i	0.962329	+	0.271886i	$0.0876473\pi$
$487$			12.0000i			0.543772i	−0.962329	+	0.271886i	$0.912353\pi$
$488$		0			0
$489$	−8.00000			−0.361773
$490$		0			0
$491$	12.0000			0.541552			0.270776	−	0.962642i	$-0.412720\pi$
$491$	12.0000			0.541552			0.270776	+	0.962642i	$0.412720\pi$
$492$		0			0
$493$			12.0000i			0.540453i
$494$		0			0
$495$	−8.00000	+	4.00000i	−0.359573	+	0.179787i
$496$		0			0
$497$		0			0
$498$		0			0
$499$	36.0000			1.61158			0.805791	−	0.592200i	$-0.201741\pi$
$499$	36.0000			1.61158			0.805791	+	0.592200i	$0.201741\pi$
$500$		0			0
$501$	−24.0000			−1.07224
$502$		0			0
$503$			36.0000i			1.60516i	0.596544	+	0.802580i	$0.296540\pi$
$503$			36.0000i			1.60516i	−0.596544	+	0.802580i	$0.703460\pi$
$504$		0			0
$505$	−32.0000	+	16.0000i	−1.42398	+	0.711991i
$506$		0			0
$507$			23.0000i			1.02147i
$508$		0			0
$509$	−12.0000			−0.531891			−0.265945	−	0.963988i	$-0.585684\pi$
$509$	−12.0000			−0.531891			−0.265945	+	0.963988i	$0.585684\pi$
$510$		0			0
$511$	−10.0000			−0.442374
$512$		0			0
$513$		−	6.00000i		−	0.264906i
$514$		0			0
$515$	−16.0000	−	32.0000i	−0.705044	−	1.41009i
$516$		0			0
$517$			16.0000i			0.703679i
$518$		0			0
$519$	−2.00000			−0.0877903
$520$		0			0
$521$	−44.0000			−1.92767			−0.963837	−	0.266491i	$-0.914136\pi$
$521$	−44.0000			−1.92767			−0.963837	+	0.266491i	$0.914136\pi$
$522$		0			0
$523$		−	20.0000i		−	0.874539i	−0.899331	−	0.437269i	$-0.855946\pi$
$523$		−	20.0000i		−	0.874539i	0.899331	−	0.437269i	$-0.144054\pi$
$524$		0			0
$525$	−3.00000	+	4.00000i	−0.130931	+	0.174574i
$526$		0			0
$527$			4.00000i			0.174243i
$528$		0			0
$529$	19.0000			0.826087
$530$		0			0
$531$	4.00000			0.173585
$532$		0			0
$533$		−	48.0000i		−	2.07911i
$534$		0			0
$535$	−6.00000	−	12.0000i	−0.259403	−	0.518805i
$536$		0			0
$537$			4.00000i			0.172613i
$538$		0			0
$539$	4.00000			0.172292
$540$		0			0
$541$	10.0000			0.429934			0.214967	−	0.976621i	$-0.431036\pi$
$541$	10.0000			0.429934			0.214967	+	0.976621i	$0.431036\pi$
$542$		0			0
$543$			10.0000i			0.429141i
$544$		0			0
$545$	−28.0000	+	14.0000i	−1.19939	+	0.599694i
$546$		0			0
$547$		−	20.0000i		−	0.855138i	−0.903983	−	0.427569i	$-0.859370\pi$
$547$		−	20.0000i		−	0.855138i	0.903983	−	0.427569i	$-0.140630\pi$
$548$		0			0
$549$	−14.0000			−0.597505
$550$		0			0
$551$	36.0000			1.53365
$552$		0			0
$553$		0			0
$554$		0			0
$555$	−8.00000	+	4.00000i	−0.339581	+	0.169791i
$556$		0			0
$557$		−	6.00000i		−	0.254228i	−0.991888	−	0.127114i	$-0.959429\pi$
$557$		−	6.00000i		−	0.254228i	0.991888	−	0.127114i	$-0.0405714\pi$
$558$		0			0
$559$	−24.0000			−1.01509
$560$		0			0
$561$	8.00000			0.337760
$562$		0			0
$563$			40.0000i			1.68580i	0.538071	+	0.842900i	$0.319153\pi$
$563$			40.0000i			1.68580i	−0.538071	+	0.842900i	$0.680847\pi$
$564$		0			0
$565$	−18.0000	−	36.0000i	−0.757266	−	1.51453i
$566$		0			0
$567$		−	1.00000i		−	0.0419961i
$568$		0			0
$569$	10.0000			0.419222			0.209611	−	0.977785i	$-0.432780\pi$
$569$	10.0000			0.419222			0.209611	+	0.977785i	$0.432780\pi$
$570$		0			0
$571$	−8.00000			−0.334790			−0.167395	−	0.985890i	$-0.553535\pi$
$571$	−8.00000			−0.334790			−0.167395	+	0.985890i	$0.553535\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	8.00000	+	6.00000i	0.333623	+	0.250217i
$576$		0			0
$577$		−	38.0000i		−	1.58196i	−0.611842	−	0.790980i	$-0.709571\pi$
$577$		−	38.0000i		−	1.58196i	0.611842	−	0.790980i	$-0.290429\pi$
$578$		0			0
$579$	8.00000			0.332469
$580$		0			0
$581$	−16.0000			−0.663792
$582$		0			0
$583$		−	24.0000i		−	0.993978i
$584$		0			0
$585$	−6.00000	−	12.0000i	−0.248069	−	0.496139i
$586$		0			0
$587$		0			0		1.00000			$0$
$587$		0			0		−1.00000			$\pi$
$588$		0			0
$589$	12.0000			0.494451
$590$		0			0
$591$	−10.0000			−0.411345
$592$		0			0
$593$			18.0000i			0.739171i	0.929197	+	0.369586i	$0.120500\pi$
$593$			18.0000i			0.739171i	−0.929197	+	0.369586i	$0.879500\pi$
$594$		0			0
$595$	4.00000	−	2.00000i	0.163984	−	0.0819920i
$596$		0			0
$597$			18.0000i			0.736691i
$598$		0			0
$599$		0			0			−	1.00000i	$-0.5\pi$
$599$		0			0				1.00000i	$0.5\pi$
$600$		0			0
$601$	−34.0000			−1.38689			−0.693444	−	0.720510i	$-0.743908\pi$
$601$	−34.0000			−1.38689			−0.693444	+	0.720510i	$0.743908\pi$
$602$		0			0
$603$			4.00000i			0.162893i
$604$		0			0
$605$	−10.0000	+	5.00000i	−0.406558	+	0.203279i
$606$		0			0
$607$		−	40.0000i		−	1.62355i	−0.583970	−	0.811775i	$-0.698502\pi$
$607$		−	40.0000i		−	1.62355i	0.583970	−	0.811775i	$-0.301498\pi$
$608$		0			0
$609$	6.00000			0.243132
$610$		0			0
$611$	−24.0000			−0.970936
$612$		0			0
$613$			44.0000i			1.77714i	0.458738	+	0.888572i	$0.348302\pi$
$613$			44.0000i			1.77714i	−0.458738	+	0.888572i	$0.651698\pi$
$614$		0			0
$615$	8.00000	+	16.0000i	0.322591	+	0.645182i
$616$		0			0
$617$			34.0000i			1.36879i	0.729112	+	0.684394i	$0.239933\pi$
$617$			34.0000i			1.36879i	−0.729112	+	0.684394i	$0.760067\pi$
$618$		0			0
$619$	−26.0000			−1.04503			−0.522514	−	0.852631i	$-0.675006\pi$
$619$	−26.0000			−1.04503			−0.522514	+	0.852631i	$0.675006\pi$
$620$		0			0
$621$	−2.00000			−0.0802572
$622$		0			0
$623$			8.00000i			0.320513i
$624$		0			0
$625$	−7.00000	−	24.0000i	−0.280000	−	0.960000i
$626$		0			0
$627$		−	24.0000i		−	0.958468i
$628$		0			0
$629$	8.00000			0.318981
$630$		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
$633$			24.0000i			0.953914i
$634$		0			0
$635$	−16.0000	−	32.0000i	−0.634941	−	1.26988i
$636$		0			0
$637$			6.00000i			0.237729i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	2.00000			0.0789953			0.0394976	−	0.999220i	$-0.487424\pi$
$641$	2.00000			0.0789953			0.0394976	+	0.999220i	$0.487424\pi$
$642$		0			0
$643$		−	12.0000i		−	0.473234i	−0.971603	−	0.236617i	$-0.923961\pi$
$643$		−	12.0000i		−	0.473234i	0.971603	−	0.236617i	$-0.0760386\pi$
$644$		0			0
$645$	8.00000	−	4.00000i	0.315000	−	0.157500i
$646$		0			0
$647$			24.0000i			0.943537i	0.881722	+	0.471769i	$0.156384\pi$
$647$			24.0000i			0.943537i	−0.881722	+	0.471769i	$0.843616\pi$
$648$		0			0
$649$	16.0000			0.628055
$650$		0			0
$651$	2.00000			0.0783862
$652$		0			0
$653$		−	34.0000i		−	1.33052i	−0.746611	−	0.665261i	$-0.768320\pi$
$653$		−	34.0000i		−	1.33052i	0.746611	−	0.665261i	$-0.231680\pi$
$654$		0			0
$655$	24.0000	−	12.0000i	0.937758	−	0.468879i
$656$		0			0
$657$			10.0000i			0.390137i
$658$		0			0
$659$	−4.00000			−0.155818			−0.0779089	−	0.996960i	$-0.524824\pi$
$659$	−4.00000			−0.155818			−0.0779089	+	0.996960i	$0.524824\pi$
$660$		0			0
$661$	14.0000			0.544537			0.272268	−	0.962221i	$-0.412226\pi$
$661$	14.0000			0.544537			0.272268	+	0.962221i	$0.412226\pi$
$662$		0			0
$663$			12.0000i			0.466041i
$664$		0			0
$665$	−6.00000	−	12.0000i	−0.232670	−	0.465340i
$666$		0			0
$667$		−	12.0000i		−	0.464642i
$668$		0			0
$669$	−16.0000			−0.618596
$670$		0			0
$671$	−56.0000			−2.16186
$672$		0			0
$673$			12.0000i			0.462566i	0.972887	+	0.231283i	$0.0742923\pi$
$673$			12.0000i			0.462566i	−0.972887	+	0.231283i	$0.925708\pi$
$674$		0			0
$675$	4.00000	+	3.00000i	0.153960	+	0.115470i
$676$		0			0
$677$		−	38.0000i		−	1.46046i	−0.683202	−	0.730229i	$-0.739413\pi$
$677$		−	38.0000i		−	1.46046i	0.683202	−	0.730229i	$-0.260587\pi$
$678$		0			0
$679$	−10.0000			−0.383765
$680$		0			0
$681$	−20.0000			−0.766402
$682$		0			0
$683$			26.0000i			0.994862i	0.867503	+	0.497431i	$0.165723\pi$
$683$			26.0000i			0.994862i	−0.867503	+	0.497431i	$0.834277\pi$
$684$		0			0
$685$	18.0000	+	36.0000i	0.687745	+	1.37549i
$686$		0			0
$687$			2.00000i			0.0763048i
$688$		0			0
$689$	36.0000			1.37149
$690$		0			0
$691$	−2.00000			−0.0760836			−0.0380418	−	0.999276i	$-0.512112\pi$
$691$	−2.00000			−0.0760836			−0.0380418	+	0.999276i	$0.512112\pi$
$692$		0			0
$693$		−	4.00000i		−	0.151947i
$694$		0			0
$695$	20.0000	−	10.0000i	0.758643	−	0.379322i
$696$		0			0
$697$		−	16.0000i		−	0.606043i
$698$		0			0
$699$	−26.0000			−0.983410
$700$		0			0
$701$	38.0000			1.43524			0.717620	−	0.696435i	$-0.245231\pi$
$701$	38.0000			1.43524			0.717620	+	0.696435i	$0.245231\pi$
$702$		0			0
$703$		−	24.0000i		−	0.905177i
$704$		0			0
$705$	8.00000	−	4.00000i	0.301297	−	0.150649i
$706$		0			0
$707$		−	16.0000i		−	0.601742i
$708$		0			0
$709$	6.00000			0.225335			0.112667	−	0.993633i	$-0.464061\pi$
$709$	6.00000			0.225335			0.112667	+	0.993633i	$0.464061\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		−	4.00000i		−	0.149801i
$714$		0			0
$715$	−24.0000	−	48.0000i	−0.897549	−	1.79510i
$716$		0			0
$717$		−	16.0000i		−	0.597531i
$718$		0			0
$719$	−44.0000			−1.64092			−0.820462	−	0.571702i	$-0.806283\pi$
$719$	−44.0000			−1.64092			−0.820462	+	0.571702i	$0.806283\pi$
$720$		0			0
$721$	16.0000			0.595871
$722$		0			0
$723$		−	10.0000i		−	0.371904i
$724$		0			0
$725$	−18.0000	+	24.0000i	−0.668503	+	0.891338i
$726$		0			0
$727$		0			0		1.00000			$0$
$727$		0			0		−1.00000			$\pi$
$728$		0			0
$729$	−1.00000			−0.0370370
$730$		0			0
$731$	−8.00000			−0.295891
$732$		0			0
$733$			34.0000i			1.25582i	0.778287	+	0.627909i	$0.216089\pi$
$733$			34.0000i			1.25582i	−0.778287	+	0.627909i	$0.783911\pi$
$734$		0			0
$735$	−1.00000	−	2.00000i	−0.0368856	−	0.0737711i
$736$		0			0
$737$			16.0000i			0.589368i
$738$		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
$741$	36.0000			1.32249
$742$		0			0
$743$		−	54.0000i		−	1.98107i	−0.137268	−	0.990534i	$-0.543832\pi$
$743$		−	54.0000i		−	1.98107i	0.137268	−	0.990534i	$-0.456168\pi$
$744$		0			0
$745$	44.0000	−	22.0000i	1.61204	−	0.806018i
$746$		0			0
$747$			16.0000i			0.585409i
$748$		0			0
$749$	6.00000			0.219235
$750$		0			0
$751$	20.0000			0.729810			0.364905	−	0.931045i	$-0.381101\pi$
$751$	20.0000			0.729810			0.364905	+	0.931045i	$0.381101\pi$
$752$		0			0
$753$			12.0000i			0.437304i
$754$		0			0
$755$	−32.0000	+	16.0000i	−1.16460	+	0.582300i
$756$		0			0
$757$		−	8.00000i		−	0.290765i	−0.989376	−	0.145382i	$-0.953559\pi$
$757$		−	8.00000i		−	0.290765i	0.989376	−	0.145382i	$-0.0464413\pi$
$758$		0			0
$759$	−8.00000			−0.290382
$760$		0			0
$761$	−24.0000			−0.869999			−0.435000	−	0.900431i	$-0.643252\pi$
$761$	−24.0000			−0.869999			−0.435000	+	0.900431i	$0.643252\pi$
$762$		0			0
$763$		−	14.0000i		−	0.506834i
$764$		0			0
$765$	−2.00000	−	4.00000i	−0.0723102	−	0.144620i
$766$		0			0
$767$			24.0000i			0.866590i
$768$		0			0
$769$	−30.0000			−1.08183			−0.540914	−	0.841078i	$-0.681921\pi$
$769$	−30.0000			−1.08183			−0.540914	+	0.841078i	$0.681921\pi$
$770$		0			0
$771$	−6.00000			−0.216085
$772$		0			0
$773$		−	6.00000i		−	0.215805i	−0.994161	−	0.107903i	$-0.965587\pi$
$773$		−	6.00000i		−	0.215805i	0.994161	−	0.107903i	$-0.0344134\pi$
$774$		0			0
$775$	−6.00000	+	8.00000i	−0.215526	+	0.287368i
$776$		0			0
$777$		−	4.00000i		−	0.143499i
$778$		0			0
$779$	−48.0000			−1.71978
$780$		0			0
$781$		0			0
$782$		0			0
$783$		−	6.00000i		−	0.214423i
$784$		0			0
$785$	10.0000	+	20.0000i	0.356915	+	0.713831i
$786$		0			0
$787$			4.00000i			0.142585i	0.997455	+	0.0712923i	$0.0227123\pi$
$787$			4.00000i			0.142585i	−0.997455	+	0.0712923i	$0.977288\pi$
$788$		0			0
$789$	−10.0000			−0.356009
$790$		0			0
$791$	18.0000			0.640006
$792$		0			0
$793$		−	84.0000i		−	2.98293i
$794$		0			0
$795$	−12.0000	+	6.00000i	−0.425596	+	0.212798i
$796$		0			0
$797$			34.0000i			1.20434i	0.798367	+	0.602171i	$0.205697\pi$
$797$			34.0000i			1.20434i	−0.798367	+	0.602171i	$0.794303\pi$
$798$		0			0
$799$	−8.00000			−0.283020
$800$		0			0
$801$	8.00000			0.282666
$802$		0			0
$803$			40.0000i			1.41157i
$804$		0			0
$805$	−4.00000	+	2.00000i	−0.140981	+	0.0704907i
$806$		0			0
$807$		−	12.0000i		−	0.422420i
$808$		0			0
$809$	18.0000			0.632846			0.316423	−	0.948618i	$-0.397518\pi$
$809$	18.0000			0.632846			0.316423	+	0.948618i	$0.397518\pi$
$810$		0			0
$811$	34.0000			1.19390			0.596951	−	0.802278i	$-0.296379\pi$
$811$	34.0000			1.19390			0.596951	+	0.802278i	$0.296379\pi$
$812$		0			0
$813$			14.0000i			0.491001i
$814$		0			0
$815$	8.00000	+	16.0000i	0.280228	+	0.560456i
$816$		0			0
$817$			24.0000i			0.839654i
$818$		0			0
$819$	6.00000			0.209657
$820$		0			0
$821$	10.0000			0.349002			0.174501	−	0.984657i	$-0.444169\pi$
$821$	10.0000			0.349002			0.174501	+	0.984657i	$0.444169\pi$
$822$		0			0
$823$			24.0000i			0.836587i	0.908312	+	0.418294i	$0.137372\pi$
$823$			24.0000i			0.836587i	−0.908312	+	0.418294i	$0.862628\pi$
$824$		0			0
$825$	16.0000	+	12.0000i	0.557048	+	0.417786i
$826$		0			0
$827$			22.0000i			0.765015i	0.923952	+	0.382507i	$0.124939\pi$
$827$			22.0000i			0.765015i	−0.923952	+	0.382507i	$0.875061\pi$
$828$		0			0
$829$	−6.00000			−0.208389			−0.104194	−	0.994557i	$-0.533226\pi$
$829$	−6.00000			−0.208389			−0.104194	+	0.994557i	$0.533226\pi$
$830$		0			0
$831$	28.0000			0.971309
$832$		0			0
$833$			2.00000i			0.0692959i
$834$		0			0
$835$	24.0000	+	48.0000i	0.830554	+	1.66111i
$836$		0			0
$837$		−	2.00000i		−	0.0691301i
$838$		0			0
$839$	32.0000			1.10476			0.552381	−	0.833592i	$-0.313719\pi$
$839$	32.0000			1.10476			0.552381	+	0.833592i	$0.313719\pi$
$840$		0			0
$841$	7.00000			0.241379
$842$		0			0
$843$			6.00000i			0.206651i
$844$		0			0
$845$	46.0000	−	23.0000i	1.58245	−	0.791224i
$846$		0			0
$847$		−	5.00000i		−	0.171802i
$848$		0			0
$849$	28.0000			0.960958
$850$		0			0
$851$	−8.00000			−0.274236
$852$		0			0
$853$		−	2.00000i		−	0.0684787i	−0.999414	−	0.0342393i	$-0.989099\pi$
$853$		−	2.00000i		−	0.0684787i	0.999414	−	0.0342393i	$-0.0109009\pi$
$854$		0			0
$855$	−12.0000	+	6.00000i	−0.410391	+	0.205196i
$856$		0			0
$857$		−	6.00000i		−	0.204956i	−0.994735	−	0.102478i	$-0.967323\pi$
$857$		−	6.00000i		−	0.204956i	0.994735	−	0.102478i	$-0.0326771\pi$
$858$		0			0
$859$	46.0000			1.56950			0.784750	−	0.619813i	$-0.212791\pi$
$859$	46.0000			1.56950			0.784750	+	0.619813i	$0.212791\pi$
$860$		0			0
$861$	−8.00000			−0.272639
$862$		0			0
$863$			18.0000i			0.612727i	0.951915	+	0.306364i	$0.0991123\pi$
$863$			18.0000i			0.612727i	−0.951915	+	0.306364i	$0.900888\pi$
$864$		0			0
$865$	2.00000	+	4.00000i	0.0680020	+	0.136004i
$866$		0			0
$867$		−	13.0000i		−	0.441503i
$868$		0			0
$869$		0			0
$870$		0			0
$871$	−24.0000			−0.813209
$872$		0			0
$873$			10.0000i			0.338449i
$874$		0			0
$875$	11.0000	+	2.00000i	0.371868	+	0.0676123i
$876$		0			0
$877$			44.0000i			1.48577i	0.669417	+	0.742887i	$0.266544\pi$
$877$			44.0000i			1.48577i	−0.669417	+	0.742887i	$0.733456\pi$
$878$		0			0
$879$	−6.00000			−0.202375
$880$		0			0
$881$	36.0000			1.21287			0.606435	−	0.795133i	$-0.292599\pi$
$881$	36.0000			1.21287			0.606435	+	0.795133i	$0.292599\pi$
$882$		0			0
$883$		−	24.0000i		−	0.807664i	−0.914833	−	0.403832i	$-0.867678\pi$
$883$		−	24.0000i		−	0.807664i	0.914833	−	0.403832i	$-0.132322\pi$
$884$		0			0
$885$	−4.00000	−	8.00000i	−0.134459	−	0.268917i
$886$		0			0
$887$		−	12.0000i		−	0.402921i	−0.979497	−	0.201460i	$-0.935431\pi$
$887$		−	12.0000i		−	0.402921i	0.979497	−	0.201460i	$-0.0645687\pi$
$888$		0			0
$889$	16.0000			0.536623
$890$		0			0
$891$	−4.00000			−0.134005
$892$		0			0
$893$			24.0000i			0.803129i
$894$		0			0
$895$	8.00000	−	4.00000i	0.267411	−	0.133705i
$896$		0			0
$897$		−	12.0000i		−	0.400668i
$898$		0			0
$899$	12.0000			0.400222
$900$		0			0
$901$	12.0000			0.399778
$902$		0			0
$903$			4.00000i			0.133112i
$904$		0			0
$905$	20.0000	−	10.0000i	0.664822	−	0.332411i
$906$		0			0
$907$		−	24.0000i		−	0.796907i	−0.917189	−	0.398453i	$-0.869547\pi$
$907$		−	24.0000i		−	0.796907i	0.917189	−	0.398453i	$-0.130453\pi$
$908$		0			0
$909$	−16.0000			−0.530687
$910$		0			0
$911$	−24.0000			−0.795155			−0.397578	−	0.917568i	$-0.630149\pi$
$911$	−24.0000			−0.795155			−0.397578	+	0.917568i	$0.630149\pi$
$912$		0			0
$913$			64.0000i			2.11809i
$914$		0			0
$915$	14.0000	+	28.0000i	0.462826	+	0.925651i
$916$		0			0
$917$			12.0000i			0.396275i
$918$		0			0
$919$	−36.0000			−1.18753			−0.593765	−	0.804638i	$-0.702359\pi$
$919$	−36.0000			−1.18753			−0.593765	+	0.804638i	$0.702359\pi$
$920$		0			0
$921$	12.0000			0.395413
$922$		0			0
$923$		0			0
$924$		0			0
$925$	16.0000	+	12.0000i	0.526077	+	0.394558i
$926$		0			0
$927$		−	16.0000i		−	0.525509i
$928$		0			0
$929$	−24.0000			−0.787414			−0.393707	−	0.919236i	$-0.628808\pi$
$929$	−24.0000			−0.787414			−0.393707	+	0.919236i	$0.628808\pi$
$930$		0			0
$931$	6.00000			0.196642
$932$		0			0
$933$		−	12.0000i		−	0.392862i
$934$		0			0
$935$	−8.00000	−	16.0000i	−0.261628	−	0.523256i
$936$		0			0
$937$			14.0000i			0.457360i	0.973502	+	0.228680i	$0.0734410\pi$
$937$			14.0000i			0.457360i	−0.973502	+	0.228680i	$0.926559\pi$
$938$		0			0
$939$	14.0000			0.456873
$940$		0			0
$941$	44.0000			1.43436			0.717180	−	0.696888i	$-0.245433\pi$
$941$	44.0000			1.43436			0.717180	+	0.696888i	$0.245433\pi$
$942$		0			0
$943$			16.0000i			0.521032i
$944$		0			0
$945$	−2.00000	+	1.00000i	−0.0650600	+	0.0325300i
$946$		0			0
$947$		−	6.00000i		−	0.194974i	−0.995237	−	0.0974869i	$-0.968920\pi$
$947$		−	6.00000i		−	0.194974i	0.995237	−	0.0974869i	$-0.0310804\pi$
$948$		0			0
$949$	−60.0000			−1.94768
$950$		0			0
$951$	−14.0000			−0.453981
$952$		0			0
$953$		−	14.0000i		−	0.453504i	−0.973952	−	0.226752i	$-0.927189\pi$
$953$		−	14.0000i		−	0.453504i	0.973952	−	0.226752i	$-0.0728108\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		−	24.0000i		−	0.775810i
$958$		0			0
$959$	−18.0000			−0.581250
$960$		0			0
$961$	−27.0000			−0.870968
$962$		0			0
$963$		−	6.00000i		−	0.193347i
$964$		0			0
$965$	−8.00000	−	16.0000i	−0.257529	−	0.515058i
$966$		0			0
$967$			20.0000i			0.643157i	0.946883	+	0.321578i	$0.104213\pi$
$967$			20.0000i			0.643157i	−0.946883	+	0.321578i	$0.895787\pi$
$968$		0			0
$969$	12.0000			0.385496
$970$		0			0
$971$	−24.0000			−0.770197			−0.385098	−	0.922876i	$-0.625832\pi$
$971$	−24.0000			−0.770197			−0.385098	+	0.922876i	$0.625832\pi$
$972$		0			0
$973$			10.0000i			0.320585i
$974$		0			0
$975$	−18.0000	+	24.0000i	−0.576461	+	0.768615i
$976$		0			0
$977$			18.0000i			0.575871i	0.957650	+	0.287936i	$0.0929689\pi$
$977$			18.0000i			0.575871i	−0.957650	+	0.287936i	$0.907031\pi$
$978$		0			0
$979$	32.0000			1.02272
$980$		0			0
$981$	−14.0000			−0.446986
$982$		0			0
$983$		−	8.00000i		−	0.255160i	−0.991828	−	0.127580i	$-0.959279\pi$
$983$		−	8.00000i		−	0.255160i	0.991828	−	0.127580i	$-0.0407210\pi$
$984$		0			0
$985$	10.0000	+	20.0000i	0.318626	+	0.637253i
$986$		0			0
$987$			4.00000i			0.127321i
$988$		0			0
$989$	8.00000			0.254385
$990$		0			0
$991$	52.0000			1.65183			0.825917	−	0.563791i	$-0.190658\pi$
$991$	52.0000			1.65183			0.825917	+	0.563791i	$0.190658\pi$
$992$		0			0
$993$			8.00000i			0.253872i
$994$		0			0
$995$	36.0000	−	18.0000i	1.14128	−	0.570638i
$996$		0			0
$997$			18.0000i			0.570066i	0.958518	+	0.285033i	$0.0920045\pi$
$997$			18.0000i			0.570066i	−0.958518	+	0.285033i	$0.907995\pi$
$998$		0			0
$999$	−4.00000			−0.126554

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	420.2.k.a.169.1	✓	2
3.2	odd	2		1260.2.k.d.1009.1		2
4.3	odd	2		1680.2.t.a.1009.2		2
5.2	odd	4		2100.2.a.e.1.1		1
5.3	odd	4		2100.2.a.j.1.1		1
5.4	even	2	inner	420.2.k.a.169.2	yes	2
7.2	even	3		2940.2.bb.h.949.2		4
7.3	odd	6		2940.2.bb.c.1549.2		4
7.4	even	3		2940.2.bb.h.1549.1		4
7.5	odd	6		2940.2.bb.c.949.1		4
7.6	odd	2		2940.2.k.d.589.2		2
12.11	even	2		5040.2.t.o.1009.1		2
15.2	even	4		6300.2.a.bc.1.1		1
15.8	even	4		6300.2.a.n.1.1		1
15.14	odd	2		1260.2.k.d.1009.2		2
20.3	even	4		8400.2.a.bh.1.1		1
20.7	even	4		8400.2.a.cd.1.1		1
20.19	odd	2		1680.2.t.a.1009.1		2
35.4	even	6		2940.2.bb.h.1549.2		4
35.9	even	6		2940.2.bb.h.949.1		4
35.19	odd	6		2940.2.bb.c.949.2		4
35.24	odd	6		2940.2.bb.c.1549.1		4
35.34	odd	2		2940.2.k.d.589.1		2
60.59	even	2		5040.2.t.o.1009.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
420.2.k.a.169.1	✓	2	1.1	even	1	trivial
420.2.k.a.169.2	yes	2	5.4	even	2	inner
1260.2.k.d.1009.1		2	3.2	odd	2
1260.2.k.d.1009.2		2	15.14	odd	2
1680.2.t.a.1009.1		2	20.19	odd	2
1680.2.t.a.1009.2		2	4.3	odd	2
2100.2.a.e.1.1		1	5.2	odd	4
2100.2.a.j.1.1		1	5.3	odd	4
2940.2.k.d.589.1		2	35.34	odd	2
2940.2.k.d.589.2		2	7.6	odd	2
2940.2.bb.c.949.1		4	7.5	odd	6
2940.2.bb.c.949.2		4	35.19	odd	6
2940.2.bb.c.1549.1		4	35.24	odd	6
2940.2.bb.c.1549.2		4	7.3	odd	6
2940.2.bb.h.949.1		4	35.9	even	6
2940.2.bb.h.949.2		4	7.2	even	3
2940.2.bb.h.1549.1		4	7.4	even	3
2940.2.bb.h.1549.2		4	35.4	even	6
5040.2.t.o.1009.1		2	12.11	even	2
5040.2.t.o.1009.2		2	60.59	even	2
6300.2.a.n.1.1		1	15.8	even	4
6300.2.a.bc.1.1		1	15.2	even	4
8400.2.a.bh.1.1		1	20.3	even	4
8400.2.a.cd.1.1		1	20.7	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:	\( N \)	\(=\)	\( 420 = 2^{2} \cdot 3 \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	420.k (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.00000i q^{3} +(-2.00000 + 1.00000i) q^{5} -1.00000i q^{7} -1.00000 q^{9} +O(q^{10})\)	\(q-1.00000i q^{3} +(-2.00000 + 1.00000i) q^{5} -1.00000i q^{7} -1.00000 q^{9} -4.00000 q^{11} -6.00000i q^{13} +(1.00000 + 2.00000i) q^{15} -2.00000i q^{17} -6.00000 q^{19} -1.00000 q^{21} +2.00000i q^{23} +(3.00000 - 4.00000i) q^{25} +1.00000i q^{27} -6.00000 q^{29} -2.00000 q^{31} +4.00000i q^{33} +(1.00000 + 2.00000i) q^{35} +4.00000i q^{37} -6.00000 q^{39} +8.00000 q^{41} -4.00000i q^{43} +(2.00000 - 1.00000i) q^{45} -4.00000i q^{47} -1.00000 q^{49} -2.00000 q^{51} +6.00000i q^{53} +(8.00000 - 4.00000i) q^{55} +6.00000i q^{57} -4.00000 q^{59} +14.0000 q^{61} +1.00000i q^{63} +(6.00000 + 12.0000i) q^{65} -4.00000i q^{67} +2.00000 q^{69} -10.0000i q^{73} +(-4.00000 - 3.00000i) q^{75} +4.00000i q^{77} +1.00000 q^{81} -16.0000i q^{83} +(2.00000 + 4.00000i) q^{85} +6.00000i q^{87} -8.00000 q^{89} -6.00000 q^{91} +2.00000i q^{93} +(12.0000 - 6.00000i) q^{95} -10.0000i q^{97} +4.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 4 q^{5} - 2 q^{9}+O(q^{10}) \) 2 * q - 4 * q^5 - 2 * q^9	\( 2 q - 4 q^{5} - 2 q^{9} - 8 q^{11} + 2 q^{15} - 12 q^{19} - 2 q^{21} + 6 q^{25} - 12 q^{29} - 4 q^{31} + 2 q^{35} - 12 q^{39} + 16 q^{41} + 4 q^{45} - 2 q^{49} - 4 q^{51} + 16 q^{55} - 8 q^{59} + 28 q^{61} + 12 q^{65} + 4 q^{69} - 8 q^{75} + 2 q^{81} + 4 q^{85} - 16 q^{89} - 12 q^{91} + 24 q^{95} + 8 q^{99}+O(q^{100}) \) 2 * q - 4 * q^5 - 2 * q^9 - 8 * q^11 + 2 * q^15 - 12 * q^19 - 2 * q^21 + 6 * q^25 - 12 * q^29 - 4 * q^31 + 2 * q^35 - 12 * q^39 + 16 * q^41 + 4 * q^45 - 2 * q^49 - 4 * q^51 + 16 * q^55 - 8 * q^59 + 28 * q^61 + 12 * q^65 + 4 * q^69 - 8 * q^75 + 2 * q^81 + 4 * q^85 - 16 * q^89 - 12 * q^91 + 24 * q^95 + 8 * q^99

Introduction

L-functions

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Database

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