LMFDB - Embedded newform 3584.2.b.i.1793.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3584,2,Mod(1793,3584)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3584, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("3584.1793");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3584 = 2^{9} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3584.b (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:	$28.6183840844$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 4 x^{11} + 4 x^{10} + 8 x^{9} - 24 x^{8} + 24 x^{7} + 176 x^{6} + 160 x^{5} + 145 x^{4} + \cdots + 8 $ x^12 - 4x^11 + 4x^10 + 8x^9 - 24x^8 + 24x^7 + 176x^6 + 160x^5 + 145x^4 + 108x^3 + 68x^2 + 32*x + 8
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{14} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1793.7
Root			$0.271901 - 0.656426i$ of defining polynomial
Character	$\chi$	$=$	3584.1793
Dual form			3584.2.b.i.1793.6

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.101362i q^{3} -2.19640i q^{5} -1.00000 q^{7} +2.98973 q^{9} +O(q^{10})$	$q+0.101362i q^{3} -2.19640i q^{5} -1.00000 q^{7} +2.98973 q^{9} +0.674013i q^{11} -5.30258i q^{13} +0.222631 q^{15} +3.87523 q^{17} -0.995183i q^{19} -0.101362i q^{21} +3.36590 q^{23} +0.175829 q^{25} +0.607128i q^{27} +0.134404i q^{29} -3.11512 q^{31} -0.0683190 q^{33} +2.19640i q^{35} -2.07795i q^{37} +0.537478 q^{39} +3.76710 q^{41} +3.61268i q^{43} -6.56663i q^{45} -6.68913 q^{47} +1.00000 q^{49} +0.392799i q^{51} -2.93656i q^{53} +1.48040 q^{55} +0.100873 q^{57} +6.92085i q^{59} +4.30265i q^{61} -2.98973 q^{63} -11.6466 q^{65} -13.9311i q^{67} +0.341173i q^{69} +10.6460 q^{71} -8.62368 q^{73} +0.0178223i q^{75} -0.674013i q^{77} +7.12273 q^{79} +8.90764 q^{81} -15.1833i q^{83} -8.51155i q^{85} -0.0136234 q^{87} -0.856726 q^{89} +5.30258i q^{91} -0.315754i q^{93} -2.18582 q^{95} -11.0125 q^{97} +2.01511i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 12 q^{7} - 20 q^{9}+O(q^{10}) $ 12 * q - 12 * q^7 - 20 * q^9	$ 12 q - 12 q^{7} - 20 q^{9} - 16 q^{15} - 4 q^{25} - 16 q^{31} + 8 q^{33} + 8 q^{41} + 16 q^{47} + 12 q^{49} - 32 q^{55} + 8 q^{57} + 20 q^{63} - 16 q^{65} + 16 q^{71} - 32 q^{73} - 48 q^{79} + 20 q^{81} + 16 q^{87} - 32 q^{89} - 32 q^{95} + 32 q^{97}+O(q^{100}) $ 12 * q - 12 * q^7 - 20 * q^9 - 16 * q^15 - 4 * q^25 - 16 * q^31 + 8 * q^33 + 8 * q^41 + 16 * q^47 + 12 * q^49 - 32 * q^55 + 8 * q^57 + 20 * q^63 - 16 * q^65 + 16 * q^71 - 32 * q^73 - 48 * q^79 + 20 * q^81 + 16 * q^87 - 32 * q^89 - 32 * q^95 + 32 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3584\mathbb{Z}\right)^\times$.

$n$	$1023$	$1025$	$1541$
$\chi(n)$	$1$	$1$	$-1$

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	0.101362i	0.0585211i	0.999572	+	0.0292606i	$0.00931526\pi$
$3$	0.101362i	0.0585211i	−0.999572	+	0.0292606i	$0.990685\pi$
$4$	0	0
$5$	− 2.19640i	− 0.982260i	−0.871086	−	0.491130i	$-0.836584\pi$
$5$	− 2.19640i	− 0.982260i	0.871086	−	0.491130i	$-0.163416\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	2.98973	0.996575
$10$	0	0
$11$	0.674013i	0.203223i	0.994824	+	0.101611i	$0.0323998\pi$
$11$	0.674013i	0.203223i	−0.994824	+	0.101611i	$0.967600\pi$
$12$	0	0
$13$	− 5.30258i	− 1.47067i	−0.677704	−	0.735335i	$-0.737025\pi$
$13$	− 5.30258i	− 1.47067i	0.677704	−	0.735335i	$-0.262975\pi$
$14$	0	0
$15$	0.222631	0.0574830
$16$	0	0
$17$	3.87523	0.939881	0.469941	−	0.882698i	$-0.344275\pi$
$17$	3.87523	0.939881	0.469941	+	0.882698i	$0.344275\pi$
$18$	0	0
$19$	− 0.995183i	− 0.228311i	−0.993463	−	0.114155i	$-0.963584\pi$
$19$	− 0.995183i	− 0.228311i	0.993463	−	0.114155i	$-0.0364162\pi$
$20$	0	0
$21$	− 0.101362i	− 0.0221189i
$22$	0	0
$23$	3.36590	0.701840	0.350920	−	0.936406i	$-0.385869\pi$
$23$	3.36590	0.701840	0.350920	+	0.936406i	$0.385869\pi$
$24$	0	0
$25$	0.175829	0.0351657
$26$	0	0
$27$	0.607128i	0.116842i
$28$	0	0
$29$	0.134404i	0.0249582i	0.999922	+	0.0124791i	$0.00397233\pi$
$29$	0.134404i	0.0249582i	−0.999922	+	0.0124791i	$0.996028\pi$
$30$	0	0
$31$	−3.11512	−0.559492	−0.279746	−	0.960074i	$-0.590250\pi$
$31$	−3.11512	−0.559492	−0.279746	+	0.960074i	$0.590250\pi$
$32$	0	0
$33$	−0.0683190	−0.0118928
$34$	0	0
$35$	2.19640i	0.371259i
$36$	0	0
$37$	− 2.07795i	− 0.341613i	−0.985305	−	0.170807i	$-0.945363\pi$
$37$	− 2.07795i	− 0.341613i	0.985305	−	0.170807i	$-0.0546373\pi$
$38$	0	0
$39$	0.537478	0.0860653
$40$	0	0
$41$	3.76710	0.588321	0.294161	−	0.955756i	$-0.404960\pi$
$41$	3.76710	0.588321	0.294161	+	0.955756i	$0.404960\pi$
$42$	0	0
$43$	3.61268i	0.550929i	0.961311	+	0.275464i	$0.0888316\pi$
$43$	3.61268i	0.550929i	−0.961311	+	0.275464i	$0.911168\pi$
$44$	0	0
$45$	− 6.56663i	− 0.978896i
$46$	0	0
$47$	−6.68913	−0.975709	−0.487855	−	0.872925i	$-0.662220\pi$
$47$	−6.68913	−0.975709	−0.487855	+	0.872925i	$0.662220\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	0.392799i	0.0550029i
$52$	0	0
$53$	− 2.93656i	− 0.403368i	−0.979451	−	0.201684i	$-0.935359\pi$
$53$	− 2.93656i	− 0.403368i	0.979451	−	0.201684i	$-0.0646413\pi$
$54$	0	0
$55$	1.48040	0.199617
$56$	0	0
$57$	0.100873	0.0133610
$58$	0	0
$59$	6.92085i	0.901017i	0.892772	+	0.450509i	$0.148757\pi$
$59$	6.92085i	0.901017i	−0.892772	+	0.450509i	$0.851243\pi$
$60$	0	0
$61$	4.30265i	0.550898i	0.961316	+	0.275449i	$0.0888265\pi$
$61$	4.30265i	0.550898i	−0.961316	+	0.275449i	$0.911173\pi$
$62$	0	0
$63$	−2.98973	−0.376670
$64$	0	0
$65$	−11.6466	−1.44458
$66$	0	0
$67$	− 13.9311i	− 1.70196i	−0.525198	−	0.850980i	$-0.676009\pi$
$67$	− 13.9311i	− 1.70196i	0.525198	−	0.850980i	$-0.323991\pi$
$68$	0	0
$69$	0.341173i	0.0410724i
$70$	0	0
$71$	10.6460	1.26344	0.631721	−	0.775196i	$-0.282349\pi$
$71$	10.6460	1.26344	0.631721	+	0.775196i	$0.282349\pi$
$72$	0	0
$73$	−8.62368	−1.00932	−0.504662	−	0.863317i	$-0.668383\pi$
$73$	−8.62368	−1.00932	−0.504662	+	0.863317i	$0.668383\pi$
$74$	0	0
$75$	0.0178223i	0.00205794i
$76$	0	0
$77$	− 0.674013i	− 0.0768109i
$78$	0	0
$79$	7.12273	0.801369	0.400685	−	0.916216i	$-0.368772\pi$
$79$	7.12273	0.801369	0.400685	+	0.916216i	$0.368772\pi$
$80$	0	0
$81$	8.90764	0.989738
$82$	0	0
$83$	− 15.1833i	− 1.66658i	−0.552834	−	0.833291i	$-0.686454\pi$
$83$	− 15.1833i	− 1.66658i	0.552834	−	0.833291i	$-0.313546\pi$
$84$	0	0
$85$	− 8.51155i	− 0.923207i
$86$	0	0
$87$	−0.0136234	−0.00146058
$88$	0	0
$89$	−0.856726	−0.0908128	−0.0454064	−	0.998969i	$-0.514458\pi$
$89$	−0.856726	−0.0908128	−0.0454064	+	0.998969i	$0.514458\pi$
$90$	0	0
$91$	5.30258i	0.555861i
$92$	0	0
$93$	− 0.315754i	− 0.0327421i
$94$	0	0
$95$	−2.18582	−0.224260
$96$	0	0
$97$	−11.0125	−1.11815	−0.559074	−	0.829118i	$-0.688843\pi$
$97$	−11.0125	−1.11815	−0.559074	+	0.829118i	$0.688843\pi$
$98$	0	0
$99$	2.01511i	0.202527i
$100$	0	0
$101$	− 0.354276i	− 0.0352518i	−0.999845	−	0.0176259i	$-0.994389\pi$
$101$	− 0.354276i	− 0.0352518i	0.999845	−	0.0176259i	$-0.00561079\pi$
$102$	0	0
$103$	−16.5750	−1.63318	−0.816590	−	0.577218i	$-0.804138\pi$
$103$	−16.5750	−1.63318	−0.816590	+	0.577218i	$0.804138\pi$
$104$	0	0
$105$	−0.222631	−0.0217265
$106$	0	0
$107$	17.7782i	1.71869i	0.511400	+	0.859343i	$0.329127\pi$
$107$	17.7782i	1.71869i	−0.511400	+	0.859343i	$0.670873\pi$
$108$	0	0
$109$	− 10.1967i	− 0.976667i	−0.872657	−	0.488334i	$-0.837605\pi$
$109$	− 10.1967i	− 0.976667i	0.872657	−	0.488334i	$-0.162395\pi$
$110$	0	0
$111$	0.210624	0.0199916
$112$	0	0
$113$	9.69262	0.911805	0.455902	−	0.890030i	$-0.349317\pi$
$113$	9.69262	0.911805	0.455902	+	0.890030i	$0.349317\pi$
$114$	0	0
$115$	− 7.39287i	− 0.689389i
$116$	0	0
$117$	− 15.8533i	− 1.46563i
$118$	0	0
$119$	−3.87523	−0.355242
$120$	0	0
$121$	10.5457	0.958701
$122$	0	0
$123$	0.381839i	0.0344292i
$124$	0	0
$125$	− 11.3682i	− 1.01680i
$126$	0	0
$127$	20.2674	1.79844	0.899220	−	0.437497i	$-0.144135\pi$
$127$	20.2674	1.79844	0.899220	+	0.437497i	$0.144135\pi$
$128$	0	0
$129$	−0.366187	−0.0322410
$130$	0	0
$131$	− 17.8643i	− 1.56081i	−0.625272	−	0.780406i	$-0.715012\pi$
$131$	− 17.8643i	− 1.56081i	0.625272	−	0.780406i	$-0.284988\pi$
$132$	0	0
$133$	0.995183i	0.0862933i
$134$	0	0
$135$	1.33350	0.114769
$136$	0	0
$137$	−11.4894	−0.981606	−0.490803	−	0.871271i	$-0.663296\pi$
$137$	−11.4894	−0.981606	−0.490803	+	0.871271i	$0.663296\pi$
$138$	0	0
$139$	0.209585i	0.0177768i	0.999960	+	0.00888838i	$0.00282929\pi$
$139$	0.209585i	0.0177768i	−0.999960	+	0.00888838i	$0.997171\pi$
$140$	0	0
$141$	− 0.678020i	− 0.0570996i
$142$	0	0
$143$	3.57401	0.298873
$144$	0	0
$145$	0.295205	0.0245155
$146$	0	0
$147$	0.101362i	0.00836016i
$148$	0	0
$149$	− 10.5895i	− 0.867526i	−0.901027	−	0.433763i	$-0.857186\pi$
$149$	− 10.5895i	− 0.867526i	0.901027	−	0.433763i	$-0.142814\pi$
$150$	0	0
$151$	−11.4820	−0.934391	−0.467196	−	0.884154i	$-0.654736\pi$
$151$	−11.4820	−0.934391	−0.467196	+	0.884154i	$0.654736\pi$
$152$	0	0
$153$	11.5859	0.936662
$154$	0	0
$155$	6.84205i	0.549567i
$156$	0	0
$157$	− 9.98207i	− 0.796656i	−0.917243	−	0.398328i	$-0.869591\pi$
$157$	− 9.98207i	− 0.796656i	0.917243	−	0.398328i	$-0.130409\pi$
$158$	0	0
$159$	0.297654	0.0236055
$160$	0	0
$161$	−3.36590	−0.265270
$162$	0	0
$163$	3.76859i	0.295178i	0.989049	+	0.147589i	$0.0471513\pi$
$163$	3.76859i	0.295178i	−0.989049	+	0.147589i	$0.952849\pi$
$164$	0	0
$165$	0.150056i	0.0116818i
$166$	0	0
$167$	−8.45888	−0.654568	−0.327284	−	0.944926i	$-0.606133\pi$
$167$	−8.45888	−0.654568	−0.327284	+	0.944926i	$0.606133\pi$
$168$	0	0
$169$	−15.1173	−1.16287
$170$	0	0
$171$	− 2.97533i	− 0.227529i
$172$	0	0
$173$	1.41548i	0.107617i	0.998551	+	0.0538085i	$0.0171361\pi$
$173$	1.41548i	0.107617i	−0.998551	+	0.0538085i	$0.982864\pi$
$174$	0	0
$175$	−0.175829	−0.0132914
$176$	0	0
$177$	−0.701508	−0.0527286
$178$	0	0
$179$	17.2449i	1.28894i	0.764629	+	0.644470i	$0.222922\pi$
$179$	17.2449i	1.28894i	−0.764629	+	0.644470i	$0.777078\pi$
$180$	0	0
$181$	4.77705i	0.355075i	0.984114	+	0.177538i	$0.0568132\pi$
$181$	4.77705i	0.355075i	−0.984114	+	0.177538i	$0.943187\pi$
$182$	0	0
$183$	−0.436123	−0.0322392
$184$	0	0
$185$	−4.56401	−0.335553
$186$	0	0
$187$	2.61195i	0.191005i
$188$	0	0
$189$	− 0.607128i	− 0.0441621i
$190$	0	0
$191$	−8.90148	−0.644089	−0.322044	−	0.946725i	$-0.604370\pi$
$191$	−8.90148	−0.644089	−0.322044	+	0.946725i	$0.604370\pi$
$192$	0	0
$193$	−11.9910	−0.863129	−0.431565	−	0.902082i	$-0.642038\pi$
$193$	−11.9910	−0.863129	−0.431565	+	0.902082i	$0.642038\pi$
$194$	0	0
$195$	− 1.18052i	− 0.0845385i
$196$	0	0
$197$	− 18.5079i	− 1.31863i	−0.751865	−	0.659317i	$-0.770846\pi$
$197$	− 18.5079i	− 1.31863i	0.751865	−	0.659317i	$-0.229154\pi$
$198$	0	0
$199$	2.13252	0.151170	0.0755852	−	0.997139i	$-0.475918\pi$
$199$	2.13252	0.151170	0.0755852	+	0.997139i	$0.475918\pi$
$200$	0	0
$201$	1.41208	0.0996006
$202$	0	0
$203$	− 0.134404i	− 0.00943332i
$204$	0	0
$205$	− 8.27405i	− 0.577884i
$206$	0	0
$207$	10.0631	0.699436
$208$	0	0
$209$	0.670766	0.0463979
$210$	0	0
$211$	− 22.7237i	− 1.56436i	−0.623051	−	0.782181i	$-0.714107\pi$
$211$	− 22.7237i	− 1.56436i	0.623051	−	0.782181i	$-0.285893\pi$
$212$	0	0
$213$	1.07909i	0.0739381i
$214$	0	0
$215$	7.93489	0.541155
$216$	0	0
$217$	3.11512	0.211468
$218$	0	0
$219$	− 0.874109i	− 0.0590668i
$220$	0	0
$221$	− 20.5487i	− 1.38226i
$222$	0	0
$223$	−1.10025	−0.0736781	−0.0368391	−	0.999321i	$-0.511729\pi$
$223$	−1.10025	−0.0736781	−0.0368391	+	0.999321i	$0.511729\pi$
$224$	0	0
$225$	0.525679	0.0350453
$226$	0	0
$227$	2.63939i	0.175183i	0.996156	+	0.0875913i	$0.0279169\pi$
$227$	2.63939i	0.175183i	−0.996156	+	0.0875913i	$0.972083\pi$
$228$	0	0
$229$	− 19.2282i	− 1.27064i	−0.772250	−	0.635319i	$-0.780869\pi$
$229$	− 19.2282i	− 1.27064i	0.772250	−	0.635319i	$-0.219131\pi$
$230$	0	0
$231$	0.0683190	0.00449506
$232$	0	0
$233$	−8.69975	−0.569939	−0.284970	−	0.958537i	$-0.591984\pi$
$233$	−8.69975	−0.569939	−0.284970	+	0.958537i	$0.591984\pi$
$234$	0	0
$235$	14.6920i	0.958400i
$236$	0	0
$237$	0.721971i	0.0468970i
$238$	0	0
$239$	−1.72852	−0.111809	−0.0559044	−	0.998436i	$-0.517804\pi$
$239$	−1.72852	−0.111809	−0.0559044	+	0.998436i	$0.517804\pi$
$240$	0	0
$241$	17.6482	1.13682	0.568409	−	0.822746i	$-0.307559\pi$
$241$	17.6482	1.13682	0.568409	+	0.822746i	$0.307559\pi$
$242$	0	0
$243$	2.72428i	0.174762i
$244$	0	0
$245$	− 2.19640i	− 0.140323i
$246$	0	0
$247$	−5.27704	−0.335770
$248$	0	0
$249$	1.53900	0.0975303
$250$	0	0
$251$	− 14.6099i	− 0.922169i	−0.887356	−	0.461084i	$-0.847461\pi$
$251$	− 14.6099i	− 0.922169i	0.887356	−	0.461084i	$-0.152539\pi$
$252$	0	0
$253$	2.26866i	0.142630i
$254$	0	0
$255$	0.862744	0.0540271
$256$	0	0
$257$	−24.1929	−1.50911	−0.754556	−	0.656235i	$-0.772148\pi$
$257$	−24.1929	−1.50911	−0.754556	+	0.656235i	$0.772148\pi$
$258$	0	0
$259$	2.07795i	0.129118i
$260$	0	0
$261$	0.401832i	0.0248727i
$262$	0	0
$263$	23.8158	1.46854	0.734271	−	0.678856i	$-0.237524\pi$
$263$	23.8158	1.46854	0.734271	+	0.678856i	$0.237524\pi$
$264$	0	0
$265$	−6.44986	−0.396212
$266$	0	0
$267$	− 0.0868391i	− 0.00531446i
$268$	0	0
$269$	− 0.304478i	− 0.0185643i	−0.999957	−	0.00928216i	$-0.997045\pi$
$269$	− 0.304478i	− 0.0185643i	0.999957	−	0.00928216i	$-0.00295465\pi$
$270$	0	0
$271$	−17.2018	−1.04493	−0.522467	−	0.852659i	$-0.674988\pi$
$271$	−17.2018	−1.04493	−0.522467	+	0.852659i	$0.674988\pi$
$272$	0	0
$273$	−0.537478	−0.0325296
$274$	0	0
$275$	0.118511i	0.00714646i
$276$	0	0
$277$	21.1489i	1.27072i	0.772218	+	0.635358i	$0.219147\pi$
$277$	21.1489i	1.27072i	−0.772218	+	0.635358i	$0.780853\pi$
$278$	0	0
$279$	−9.31336	−0.557576
$280$	0	0
$281$	−11.0974	−0.662018	−0.331009	−	0.943628i	$-0.607389\pi$
$281$	−11.0974	−0.662018	−0.331009	+	0.943628i	$0.607389\pi$
$282$	0	0
$283$	23.8050i	1.41506i	0.706683	+	0.707530i	$0.250191\pi$
$283$	23.8050i	1.41506i	−0.706683	+	0.707530i	$0.749809\pi$
$284$	0	0
$285$	− 0.221558i	− 0.0131240i
$286$	0	0
$287$	−3.76710	−0.222365
$288$	0	0
$289$	−1.98260	−0.116624
$290$	0	0
$291$	− 1.11624i	− 0.0654353i
$292$	0	0
$293$	− 13.2713i	− 0.775316i	−0.921803	−	0.387658i	$-0.873284\pi$
$293$	− 13.2713i	− 0.775316i	0.921803	−	0.387658i	$-0.126716\pi$
$294$	0	0
$295$	15.2009	0.885033
$296$	0	0
$297$	−0.409212	−0.0237449
$298$	0	0
$299$	− 17.8480i	− 1.03217i
$300$	0	0
$301$	− 3.61268i	− 0.208231i
$302$	0	0
$303$	0.0359100	0.00206298
$304$	0	0
$305$	9.45034	0.541125
$306$	0	0
$307$	− 14.4325i	− 0.823705i	−0.911251	−	0.411852i	$-0.864882\pi$
$307$	− 14.4325i	− 0.823705i	0.911251	−	0.411852i	$-0.135118\pi$
$308$	0	0
$309$	− 1.68007i	− 0.0955756i
$310$	0	0
$311$	22.4304	1.27191	0.635956	−	0.771725i	$-0.280606\pi$
$311$	22.4304	1.27191	0.635956	+	0.771725i	$0.280606\pi$
$312$	0	0
$313$	24.6010	1.39053	0.695266	−	0.718753i	$-0.255287\pi$
$313$	24.6010	1.39053	0.695266	+	0.718753i	$0.255287\pi$
$314$	0	0
$315$	6.56663i	0.369988i
$316$	0	0
$317$	− 30.9518i	− 1.73842i	−0.494439	−	0.869212i	$-0.664626\pi$
$317$	− 30.9518i	− 1.73842i	0.494439	−	0.869212i	$-0.335374\pi$
$318$	0	0
$319$	−0.0905901	−0.00507207
$320$	0	0
$321$	−1.80203	−0.100579
$322$	0	0
$323$	− 3.85656i	− 0.214585i
$324$	0	0
$325$	− 0.932345i	− 0.0517172i
$326$	0	0
$327$	1.03355	0.0571557
$328$	0	0
$329$	6.68913	0.368783
$330$	0	0
$331$	33.4297i	1.83746i	0.394884	+	0.918731i	$0.370785\pi$
$331$	33.4297i	1.83746i	−0.394884	+	0.918731i	$0.629215\pi$
$332$	0	0
$333$	− 6.21251i	− 0.340443i
$334$	0	0
$335$	−30.5984	−1.67177
$336$	0	0
$337$	27.1717	1.48014	0.740069	−	0.672530i	$-0.234793\pi$
$337$	27.1717	1.48014	0.740069	+	0.672530i	$0.234793\pi$
$338$	0	0
$339$	0.982459i	0.0533599i
$340$	0	0
$341$	− 2.09963i	− 0.113701i
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	0.749353	0.0403438
$346$	0	0
$347$	25.0784i	1.34628i	0.739516	+	0.673139i	$0.235054\pi$
$347$	25.0784i	1.34628i	−0.739516	+	0.673139i	$0.764946\pi$
$348$	0	0
$349$	23.7113i	1.26924i	0.772826	+	0.634618i	$0.218843\pi$
$349$	23.7113i	1.26924i	−0.772826	+	0.634618i	$0.781157\pi$
$350$	0	0
$351$	3.21934	0.171836
$352$	0	0
$353$	−20.6382	−1.09846	−0.549230	−	0.835672i	$-0.685079\pi$
$353$	−20.6382	−1.09846	−0.549230	+	0.835672i	$0.685079\pi$
$354$	0	0
$355$	− 23.3828i	− 1.24103i
$356$	0	0
$357$	− 0.392799i	− 0.0207891i
$358$	0	0
$359$	−27.2976	−1.44071	−0.720355	−	0.693605i	$-0.756021\pi$
$359$	−27.2976	−1.44071	−0.720355	+	0.693605i	$0.756021\pi$
$360$	0	0
$361$	18.0096	0.947874
$362$	0	0
$363$	1.06893i	0.0561042i
$364$	0	0
$365$	18.9410i	0.991419i
$366$	0	0
$367$	28.7125	1.49878	0.749390	−	0.662128i	$-0.230347\pi$
$367$	28.7125	1.49878	0.749390	+	0.662128i	$0.230347\pi$
$368$	0	0
$369$	11.2626	0.586307
$370$	0	0
$371$	2.93656i	0.152459i
$372$	0	0
$373$	29.1078i	1.50715i	0.657364	+	0.753573i	$0.271671\pi$
$373$	29.1078i	1.50715i	−0.657364	+	0.753573i	$0.728329\pi$
$374$	0	0
$375$	1.15230	0.0595044
$376$	0	0
$377$	0.712688	0.0367053
$378$	0	0
$379$	− 19.5975i	− 1.00666i	−0.864096	−	0.503328i	$-0.832109\pi$
$379$	− 19.5975i	− 1.00666i	0.864096	−	0.503328i	$-0.167891\pi$
$380$	0	0
$381$	2.05433i	0.105247i
$382$	0	0
$383$	19.5193	0.997389	0.498694	−	0.866778i	$-0.333813\pi$
$383$	19.5193	0.997389	0.498694	+	0.866778i	$0.333813\pi$
$384$	0	0
$385$	−1.48040	−0.0754482
$386$	0	0
$387$	10.8009i	0.549042i
$388$	0	0
$389$	10.7701i	0.546065i	0.962005	+	0.273033i	$0.0880267\pi$
$389$	10.7701i	0.546065i	−0.962005	+	0.273033i	$0.911973\pi$
$390$	0	0
$391$	13.0437	0.659646
$392$	0	0
$393$	1.81076	0.0913405
$394$	0	0
$395$	− 15.6444i	− 0.787153i
$396$	0	0
$397$	29.9845i	1.50488i	0.658661	+	0.752440i	$0.271123\pi$
$397$	29.9845i	1.50488i	−0.658661	+	0.752440i	$0.728877\pi$
$398$	0	0
$399$	−0.100873	−0.00504998
$400$	0	0
$401$	34.4739	1.72155	0.860773	−	0.508989i	$-0.169981\pi$
$401$	34.4739	1.72155	0.860773	+	0.508989i	$0.169981\pi$
$402$	0	0
$403$	16.5182i	0.822829i
$404$	0	0
$405$	− 19.5647i	− 0.972179i
$406$	0	0
$407$	1.40057	0.0694235
$408$	0	0
$409$	14.7405	0.728869	0.364435	−	0.931229i	$-0.381262\pi$
$409$	14.7405	0.728869	0.364435	+	0.931229i	$0.381262\pi$
$410$	0	0
$411$	− 1.16458i	− 0.0574447i
$412$	0	0
$413$	− 6.92085i	− 0.340553i
$414$	0	0
$415$	−33.3486	−1.63702
$416$	0	0
$417$	−0.0212438	−0.00104032
$418$	0	0
$419$	5.42033i	0.264800i	0.991196	+	0.132400i	$0.0422684\pi$
$419$	5.42033i	0.264800i	−0.991196	+	0.132400i	$0.957732\pi$
$420$	0	0
$421$	9.72201i	0.473822i	0.971531	+	0.236911i	$0.0761350\pi$
$421$	9.72201i	0.473822i	−0.971531	+	0.236911i	$0.923865\pi$
$422$	0	0
$423$	−19.9987	−0.972368
$424$	0	0
$425$	0.681376	0.0330516
$426$	0	0
$427$	− 4.30265i	− 0.208220i
$428$	0	0
$429$	0.362267i	0.0174904i
$430$	0	0
$431$	12.1520	0.585341	0.292670	−	0.956213i	$-0.405456\pi$
$431$	12.1520	0.585341	0.292670	+	0.956213i	$0.405456\pi$
$432$	0	0
$433$	30.0836	1.44573	0.722864	−	0.690991i	$-0.242825\pi$
$433$	30.0836	1.44573	0.722864	+	0.690991i	$0.242825\pi$
$434$	0	0
$435$	0.0299225i	0.00143467i
$436$	0	0
$437$	− 3.34969i	− 0.160238i
$438$	0	0
$439$	0.195719	0.00934117	0.00467058	−	0.999989i	$-0.498513\pi$
$439$	0.195719	0.00934117	0.00467058	+	0.999989i	$0.498513\pi$
$440$	0	0
$441$	2.98973	0.142368
$442$	0	0
$443$	− 28.1256i	− 1.33629i	−0.744032	−	0.668144i	$-0.767089\pi$
$443$	− 28.1256i	− 1.33629i	0.744032	−	0.668144i	$-0.232911\pi$
$444$	0	0
$445$	1.88171i	0.0892017i
$446$	0	0
$447$	1.07337	0.0507686
$448$	0	0
$449$	−12.7288	−0.600709	−0.300355	−	0.953828i	$-0.597105\pi$
$449$	−12.7288	−0.600709	−0.300355	+	0.953828i	$0.597105\pi$
$450$	0	0
$451$	2.53907i	0.119560i
$452$	0	0
$453$	− 1.16383i	− 0.0546816i
$454$	0	0
$455$	11.6466	0.546000
$456$	0	0
$457$	−3.64289	−0.170407	−0.0852035	−	0.996364i	$-0.527154\pi$
$457$	−3.64289	−0.170407	−0.0852035	+	0.996364i	$0.527154\pi$
$458$	0	0
$459$	2.35276i	0.109817i
$460$	0	0
$461$	11.4883i	0.535064i	0.963549	+	0.267532i	$0.0862081\pi$
$461$	11.4883i	0.535064i	−0.963549	+	0.267532i	$0.913792\pi$
$462$	0	0
$463$	−7.98371	−0.371035	−0.185517	−	0.982641i	$-0.559396\pi$
$463$	−7.98371	−0.371035	−0.185517	+	0.982641i	$0.559396\pi$
$464$	0	0
$465$	−0.693521	−0.0321613
$466$	0	0
$467$	− 27.5689i	− 1.27573i	−0.770146	−	0.637867i	$-0.779817\pi$
$467$	− 27.5689i	− 1.27573i	0.770146	−	0.637867i	$-0.220183\pi$
$468$	0	0
$469$	13.9311i	0.643280i
$470$	0	0
$471$	1.01180	0.0466212
$472$	0	0
$473$	−2.43499	−0.111961
$474$	0	0
$475$	− 0.174982i	− 0.00802871i
$476$	0	0
$477$	− 8.77951i	− 0.401986i
$478$	0	0
$479$	−12.9612	−0.592210	−0.296105	−	0.955155i	$-0.595688\pi$
$479$	−12.9612	−0.592210	−0.296105	+	0.955155i	$0.595688\pi$
$480$	0	0
$481$	−11.0185	−0.502401
$482$	0	0
$483$	− 0.341173i	− 0.0155239i
$484$	0	0
$485$	24.1878i	1.09831i
$486$	0	0
$487$	−8.86309	−0.401625	−0.200812	−	0.979630i	$-0.564358\pi$
$487$	−8.86309	−0.401625	−0.200812	+	0.979630i	$0.564358\pi$
$488$	0	0
$489$	−0.381990	−0.0172742
$490$	0	0
$491$	− 17.5466i	− 0.791865i	−0.918280	−	0.395933i	$-0.870421\pi$
$491$	− 17.5466i	− 0.791865i	0.918280	−	0.395933i	$-0.129579\pi$
$492$	0	0
$493$	0.520847i	0.0234578i
$494$	0	0
$495$	4.42599	0.198934
$496$	0	0
$497$	−10.6460	−0.477536
$498$	0	0
$499$	0.153293i	0.00686231i	0.999994	+	0.00343116i	$0.00109217\pi$
$499$	0.153293i	0.00686231i	−0.999994	+	0.00343116i	$0.998908\pi$
$500$	0	0
$501$	− 0.857406i	− 0.0383061i
$502$	0	0
$503$	−15.5136	−0.691719	−0.345860	−	0.938286i	$-0.612413\pi$
$503$	−15.5136	−0.691719	−0.345860	+	0.938286i	$0.612413\pi$
$504$	0	0
$505$	−0.778133	−0.0346264
$506$	0	0
$507$	− 1.53232i	− 0.0680526i
$508$	0	0
$509$	27.0386i	1.19846i	0.800575	+	0.599232i	$0.204527\pi$
$509$	27.0386i	1.19846i	−0.800575	+	0.599232i	$0.795473\pi$
$510$	0	0
$511$	8.62368	0.381489
$512$	0	0
$513$	0.604204	0.0266762
$514$	0	0
$515$	36.4053i	1.60421i
$516$	0	0
$517$	− 4.50856i	− 0.198286i
$518$	0	0
$519$	−0.143475	−0.00629787
$520$	0	0
$521$	15.7310	0.689190	0.344595	−	0.938752i	$-0.388016\pi$
$521$	15.7310	0.689190	0.344595	+	0.938752i	$0.388016\pi$
$522$	0	0
$523$	− 43.4999i	− 1.90212i	−0.309009	−	0.951059i	$-0.599997\pi$
$523$	− 43.4999i	− 1.90212i	0.309009	−	0.951059i	$-0.400003\pi$
$524$	0	0
$525$	− 0.0178223i	0 0.000777827i
$526$	0	0
$527$	−12.0718	−0.525856
$528$	0	0
$529$	−11.6707	−0.507421
$530$	0	0
$531$	20.6914i	0.897932i
$532$	0	0
$533$	− 19.9753i	− 0.865227i
$534$	0	0
$535$	39.0481	1.68820
$536$	0	0
$537$	−1.74797	−0.0754303
$538$	0	0
$539$	0.674013i	0.0290318i
$540$	0	0
$541$	30.1374i	1.29571i	0.761765	+	0.647853i	$0.224333\pi$
$541$	30.1374i	1.29571i	−0.761765	+	0.647853i	$0.775667\pi$
$542$	0	0
$543$	−0.484209	−0.0207794
$544$	0	0
$545$	−22.3960	−0.959341
$546$	0	0
$547$	− 22.1918i	− 0.948852i	−0.880295	−	0.474426i	$-0.842656\pi$
$547$	− 22.1918i	− 0.948852i	0.880295	−	0.474426i	$-0.157344\pi$
$548$	0	0
$549$	12.8637i	0.549011i
$550$	0	0
$551$	0.133757	0.00569823
$552$	0	0
$553$	−7.12273	−0.302889
$554$	0	0
$555$	− 0.462616i	− 0.0196369i
$556$	0	0
$557$	− 4.64011i	− 0.196608i	−0.995156	−	0.0983038i	$-0.968658\pi$
$557$	− 4.64011i	− 0.196608i	0.995156	−	0.0983038i	$-0.0313417\pi$
$558$	0	0
$559$	19.1565	0.810235
$560$	0	0
$561$	−0.264752	−0.0111778
$562$	0	0
$563$	42.9470i	1.81000i	0.425411	+	0.905000i	$0.360130\pi$
$563$	42.9470i	1.81000i	−0.425411	+	0.905000i	$0.639870\pi$
$564$	0	0
$565$	− 21.2889i	− 0.895629i
$566$	0	0
$567$	−8.90764	−0.374086
$568$	0	0
$569$	−28.2672	−1.18502	−0.592512	−	0.805561i	$-0.701864\pi$
$569$	−28.2672	−1.18502	−0.592512	+	0.805561i	$0.701864\pi$
$570$	0	0
$571$	22.6294i	0.947012i	0.880791	+	0.473506i	$0.157012\pi$
$571$	22.6294i	0.947012i	−0.880791	+	0.473506i	$0.842988\pi$
$572$	0	0
$573$	− 0.902268i	− 0.0376928i
$574$	0	0
$575$	0.591822	0.0246807
$576$	0	0
$577$	12.2701	0.510809	0.255405	−	0.966834i	$-0.417791\pi$
$577$	12.2701	0.510809	0.255405	+	0.966834i	$0.417791\pi$
$578$	0	0
$579$	− 1.21542i	− 0.0505113i
$580$	0	0
$581$	15.1833i	0.629909i
$582$	0	0
$583$	1.97928	0.0819734
$584$	0	0
$585$	−34.8201	−1.43963
$586$	0	0
$587$	21.0007i	0.866793i	0.901203	+	0.433396i	$0.142685\pi$
$587$	21.0007i	0.866793i	−0.901203	+	0.433396i	$0.857315\pi$
$588$	0	0
$589$	3.10012i	0.127738i
$590$	0	0
$591$	1.87599	0.0771679
$592$	0	0
$593$	4.30581	0.176819	0.0884093	−	0.996084i	$-0.471822\pi$
$593$	4.30581	0.176819	0.0884093	+	0.996084i	$0.471822\pi$
$594$	0	0
$595$	8.51155i	0.348940i
$596$	0	0
$597$	0.216156i	0.00884666i
$598$	0	0
$599$	−2.42319	−0.0990087	−0.0495044	−	0.998774i	$-0.515764\pi$
$599$	−2.42319	−0.0990087	−0.0495044	+	0.998774i	$0.515764\pi$
$600$	0	0
$601$	−6.18973	−0.252485	−0.126242	−	0.991999i	$-0.540292\pi$
$601$	−6.18973	−0.252485	−0.126242	+	0.991999i	$0.540292\pi$
$602$	0	0
$603$	− 41.6503i	− 1.69613i
$604$	0	0
$605$	− 23.1626i	− 0.941693i
$606$	0	0
$607$	−8.80892	−0.357543	−0.178772	−	0.983891i	$-0.557212\pi$
$607$	−8.80892	−0.357543	−0.178772	+	0.983891i	$0.557212\pi$
$608$	0	0
$609$	0.0136234	0.000552049 0
$610$	0	0
$611$	35.4696i	1.43495i
$612$	0	0
$613$	19.7718i	0.798577i	0.916825	+	0.399289i	$0.130743\pi$
$613$	19.7718i	0.798577i	−0.916825	+	0.399289i	$0.869257\pi$
$614$	0	0
$615$	0.838670	0.0338185
$616$	0	0
$617$	−12.9420	−0.521026	−0.260513	−	0.965470i	$-0.583892\pi$
$617$	−12.9420	−0.521026	−0.260513	+	0.965470i	$0.583892\pi$
$618$	0	0
$619$	17.1723i	0.690213i	0.938563	+	0.345107i	$0.112157\pi$
$619$	17.1723i	0.690213i	−0.938563	+	0.345107i	$0.887843\pi$
$620$	0	0
$621$	2.04354i	0.0820042i
$622$	0	0
$623$	0.856726	0.0343240
$624$	0	0
$625$	−24.0899	−0.963598
$626$	0	0
$627$	0.0679899i	0.00271526i
$628$	0	0
$629$	− 8.05254i	− 0.321076i
$630$	0	0
$631$	−13.7312	−0.546629	−0.273314	−	0.961925i	$-0.588120\pi$
$631$	−13.7312	−0.546629	−0.273314	+	0.961925i	$0.588120\pi$
$632$	0	0
$633$	2.30331	0.0915482
$634$	0	0
$635$	− 44.5153i	− 1.76654i
$636$	0	0
$637$	− 5.30258i	− 0.210096i
$638$	0	0
$639$	31.8285	1.25912
$640$	0	0
$641$	−11.7414	−0.463759	−0.231879	−	0.972745i	$-0.574487\pi$
$641$	−11.7414	−0.463759	−0.231879	+	0.972745i	$0.574487\pi$
$642$	0	0
$643$	2.59560i	0.102361i	0.998689	+	0.0511803i	$0.0162983\pi$
$643$	2.59560i	0.102361i	−0.998689	+	0.0511803i	$0.983702\pi$
$644$	0	0
$645$	0.804293i	0.0316690i
$646$	0	0
$647$	−29.0272	−1.14118	−0.570588	−	0.821236i	$-0.693285\pi$
$647$	−29.0272	−1.14118	−0.570588	+	0.821236i	$0.693285\pi$
$648$	0	0
$649$	−4.66474	−0.183107
$650$	0	0
$651$	0.315754i	0.0123754i
$652$	0	0
$653$	− 39.2320i	− 1.53527i	−0.640890	−	0.767633i	$-0.721435\pi$
$653$	− 39.2320i	− 1.53527i	0.640890	−	0.767633i	$-0.278565\pi$
$654$	0	0
$655$	−39.2372	−1.53312
$656$	0	0
$657$	−25.7824	−1.00587
$658$	0	0
$659$	18.2689i	0.711655i	0.934552	+	0.355828i	$0.115801\pi$
$659$	18.2689i	0.711655i	−0.934552	+	0.355828i	$0.884199\pi$
$660$	0	0
$661$	39.5936i	1.54001i	0.638036	+	0.770007i	$0.279747\pi$
$661$	39.5936i	1.54001i	−0.638036	+	0.770007i	$0.720253\pi$
$662$	0	0
$663$	2.08285	0.0808911
$664$	0	0
$665$	2.18582	0.0847625
$666$	0	0
$667$	0.452392i	0.0175167i
$668$	0	0
$669$	− 0.111523i	− 0.00431173i
$670$	0	0
$671$	−2.90004	−0.111955
$672$	0	0
$673$	−8.38015	−0.323031	−0.161516	−	0.986870i	$-0.551638\pi$
$673$	−8.38015	−0.323031	−0.161516	+	0.986870i	$0.551638\pi$
$674$	0	0
$675$	0.106750i	0.00410883i
$676$	0	0
$677$	− 20.5355i	− 0.789243i	−0.918844	−	0.394621i	$-0.870876\pi$
$677$	− 20.5355i	− 0.789243i	0.918844	−	0.394621i	$-0.129124\pi$
$678$	0	0
$679$	11.0125	0.422620
$680$	0	0
$681$	−0.267533	−0.0102519
$682$	0	0
$683$	− 24.2522i	− 0.927983i	−0.885840	−	0.463991i	$-0.846417\pi$
$683$	− 24.2522i	− 0.927983i	0.885840	−	0.463991i	$-0.153583\pi$
$684$	0	0
$685$	25.2353i	0.964192i
$686$	0	0
$687$	1.94900	0.0743592
$688$	0	0
$689$	−15.5713	−0.593221
$690$	0	0
$691$	− 30.8579i	− 1.17389i	−0.809627	−	0.586945i	$-0.800331\pi$
$691$	− 30.8579i	− 1.17389i	0.809627	−	0.586945i	$-0.199669\pi$
$692$	0	0
$693$	− 2.01511i	− 0.0765478i
$694$	0	0
$695$	0.460332	0.0174614
$696$	0	0
$697$	14.5984	0.552952
$698$	0	0
$699$	− 0.881820i	− 0.0333535i
$700$	0	0
$701$	49.9348i	1.88601i	0.332775	+	0.943006i	$0.392015\pi$
$701$	49.9348i	1.88601i	−0.332775	+	0.943006i	$0.607985\pi$
$702$	0	0
$703$	−2.06794	−0.0779940
$704$	0	0
$705$	−1.48920	−0.0560867
$706$	0	0
$707$	0.354276i	0.0133239i
$708$	0	0
$709$	30.1773i	1.13333i	0.823948	+	0.566665i	$0.191767\pi$
$709$	30.1773i	1.13333i	−0.823948	+	0.566665i	$0.808233\pi$
$710$	0	0
$711$	21.2950	0.798625
$712$	0	0
$713$	−10.4852	−0.392674
$714$	0	0
$715$	− 7.84994i	− 0.293571i
$716$	0	0
$717$	− 0.175206i	− 0.00654318i
$718$	0	0
$719$	25.5563	0.953089	0.476544	−	0.879150i	$-0.341889\pi$
$719$	25.5563	0.953089	0.476544	+	0.879150i	$0.341889\pi$
$720$	0	0
$721$	16.5750	0.617284
$722$	0	0
$723$	1.78885i	0.0665279i
$724$	0	0
$725$	0.0236321i	0 0.000877674i
$726$	0	0
$727$	−11.4845	−0.425937	−0.212969	−	0.977059i	$-0.568313\pi$
$727$	−11.4845	−0.425937	−0.212969	+	0.977059i	$0.568313\pi$
$728$	0	0
$729$	26.4468	0.979510
$730$	0	0
$731$	14.0000i	0.517807i
$732$	0	0
$733$	52.0835i	1.92375i	0.273495	+	0.961873i	$0.411820\pi$
$733$	52.0835i	1.92375i	−0.273495	+	0.961873i	$0.588180\pi$
$734$	0	0
$735$	0.222631	0.00821185
$736$	0	0
$737$	9.38977	0.345877
$738$	0	0
$739$	25.2269i	0.927986i	0.885839	+	0.463993i	$0.153584\pi$
$739$	25.2269i	0.927986i	−0.885839	+	0.463993i	$0.846416\pi$
$740$	0	0
$741$	− 0.534889i	− 0.0196496i
$742$	0	0
$743$	−23.8950	−0.876623	−0.438311	−	0.898823i	$-0.644423\pi$
$743$	−23.8950	−0.876623	−0.438311	+	0.898823i	$0.644423\pi$
$744$	0	0
$745$	−23.2588	−0.852136
$746$	0	0
$747$	− 45.3939i	− 1.66087i
$748$	0	0
$749$	− 17.7782i	− 0.649602i
$750$	0	0
$751$	29.7013	1.08381	0.541907	−	0.840438i	$-0.317702\pi$
$751$	29.7013	1.08381	0.541907	+	0.840438i	$0.317702\pi$
$752$	0	0
$753$	1.48088	0.0539664
$754$	0	0
$755$	25.2190i	0.917815i
$756$	0	0
$757$	47.1188i	1.71256i	0.516511	+	0.856280i	$0.327230\pi$
$757$	47.1188i	1.71256i	−0.516511	+	0.856280i	$0.672770\pi$
$758$	0	0
$759$	−0.229955	−0.00834685
$760$	0	0
$761$	−11.5153	−0.417430	−0.208715	−	0.977977i	$-0.566928\pi$
$761$	−11.5153	−0.417430	−0.208715	+	0.977977i	$0.566928\pi$
$762$	0	0
$763$	10.1967i	0.369145i
$764$	0	0
$765$	− 25.4472i	− 0.920046i
$766$	0	0
$767$	36.6983	1.32510
$768$	0	0
$769$	27.5195	0.992378	0.496189	−	0.868215i	$-0.334732\pi$
$769$	27.5195	0.992378	0.496189	+	0.868215i	$0.334732\pi$
$770$	0	0
$771$	− 2.45223i	− 0.0883150i
$772$	0	0
$773$	13.6192i	0.489850i	0.969542	+	0.244925i	$0.0787634\pi$
$773$	13.6192i	0.489850i	−0.969542	+	0.244925i	$0.921237\pi$
$774$	0	0
$775$	−0.547727	−0.0196749
$776$	0	0
$777$	−0.210624	−0.00755611
$778$	0	0
$779$	− 3.74895i	− 0.134320i
$780$	0	0
$781$	7.17551i	0.256760i
$782$	0	0
$783$	−0.0816005	−0.00291616
$784$	0	0
$785$	−21.9246	−0.782523
$786$	0	0
$787$	34.6258i	1.23428i	0.786855	+	0.617138i	$0.211708\pi$
$787$	34.6258i	1.23428i	−0.786855	+	0.617138i	$0.788292\pi$
$788$	0	0
$789$	2.41400i	0.0859408i
$790$	0	0
$791$	−9.69262	−0.344630
$792$	0	0
$793$	22.8151	0.810189
$794$	0	0
$795$	− 0.653768i	− 0.0231868i
$796$	0	0
$797$	35.0863i	1.24282i	0.783486	+	0.621410i	$0.213440\pi$
$797$	35.0863i	1.24282i	−0.783486	+	0.621410i	$0.786560\pi$
$798$	0	0
$799$	−25.9219	−0.917051
$800$	0	0
$801$	−2.56138	−0.0905017
$802$	0	0
$803$	− 5.81247i	− 0.205118i
$804$	0	0
$805$	7.39287i	0.260565i
$806$	0	0
$807$	0.0308623	0.00108641
$808$	0	0
$809$	28.3561	0.996947	0.498473	−	0.866905i	$-0.333894\pi$
$809$	28.3561	0.996947	0.498473	+	0.866905i	$0.333894\pi$
$810$	0	0
$811$	2.29432i	0.0805646i	0.999188	+	0.0402823i	$0.0128257\pi$
$811$	2.29432i	0.0805646i	−0.999188	+	0.0402823i	$0.987174\pi$
$812$	0	0
$813$	− 1.74360i	− 0.0611508i
$814$	0	0
$815$	8.27732	0.289942
$816$	0	0
$817$	3.59528	0.125783
$818$	0	0
$819$	15.8533i	0.553958i
$820$	0	0
$821$	41.7766i	1.45801i	0.684506	+	0.729007i	$0.260018\pi$
$821$	41.7766i	1.45801i	−0.684506	+	0.729007i	$0.739982\pi$
$822$	0	0
$823$	−45.2110	−1.57596	−0.787979	−	0.615702i	$-0.788872\pi$
$823$	−45.2110	−1.57596	−0.787979	+	0.615702i	$0.788872\pi$
$824$	0	0
$825$	−0.0120124	−0.000418219 0
$826$	0	0
$827$	5.06276i	0.176049i	0.996118	+	0.0880246i	$0.0280554\pi$
$827$	5.06276i	0.176049i	−0.996118	+	0.0880246i	$0.971945\pi$
$828$	0	0
$829$	51.1701i	1.77721i	0.458671	+	0.888606i	$0.348326\pi$
$829$	51.1701i	1.77721i	−0.458671	+	0.888606i	$0.651674\pi$
$830$	0	0
$831$	−2.14369	−0.0743637
$832$	0	0
$833$	3.87523	0.134269
$834$	0	0
$835$	18.5791i	0.642956i
$836$	0	0
$837$	− 1.89128i	− 0.0653721i
$838$	0	0
$839$	40.5224	1.39899	0.699495	−	0.714638i	$-0.253408\pi$
$839$	40.5224	1.39899	0.699495	+	0.714638i	$0.253408\pi$
$840$	0	0
$841$	28.9819	0.999377
$842$	0	0
$843$	− 1.12485i	− 0.0387420i
$844$	0	0
$845$	33.2037i	1.14224i
$846$	0	0
$847$	−10.5457	−0.362355
$848$	0	0
$849$	−2.41291	−0.0828109
$850$	0	0
$851$	− 6.99419i	− 0.239758i
$852$	0	0
$853$	8.02166i	0.274656i	0.990526	+	0.137328i	$0.0438515\pi$
$853$	8.02166i	0.274656i	−0.990526	+	0.137328i	$0.956148\pi$
$854$	0	0
$855$	−6.53500	−0.223492
$856$	0	0
$857$	0.785744	0.0268405	0.0134203	−	0.999910i	$-0.495728\pi$
$857$	0.785744	0.0268405	0.0134203	+	0.999910i	$0.495728\pi$
$858$	0	0
$859$	− 17.1625i	− 0.585577i	−0.956177	−	0.292789i	$-0.905417\pi$
$859$	− 17.1625i	− 0.585577i	0.956177	−	0.292789i	$-0.0945833\pi$
$860$	0	0
$861$	− 0.381839i	− 0.0130130i
$862$	0	0
$863$	8.01096	0.272696	0.136348	−	0.990661i	$-0.456463\pi$
$863$	8.01096	0.272696	0.136348	+	0.990661i	$0.456463\pi$
$864$	0	0
$865$	3.10896	0.105708
$866$	0	0
$867$	− 0.200959i	− 0.00682494i
$868$	0	0
$869$	4.80081i	0.162856i
$870$	0	0
$871$	−73.8710	−2.50302
$872$	0	0
$873$	−32.9243	−1.11432
$874$	0	0
$875$	11.3682i	0.384315i
$876$	0	0
$877$	7.20234i	0.243206i	0.992579	+	0.121603i	$0.0388034\pi$
$877$	7.20234i	0.243206i	−0.992579	+	0.121603i	$0.961197\pi$
$878$	0	0
$879$	1.34520	0.0453724
$880$	0	0
$881$	28.3868	0.956377	0.478188	−	0.878257i	$-0.341294\pi$
$881$	28.3868	0.956377	0.478188	+	0.878257i	$0.341294\pi$
$882$	0	0
$883$	− 27.6390i	− 0.930126i	−0.885278	−	0.465063i	$-0.846032\pi$
$883$	− 27.6390i	− 0.930126i	0.885278	−	0.465063i	$-0.153968\pi$
$884$	0	0
$885$	1.54079i	0.0517931i
$886$	0	0
$887$	6.99105	0.234737	0.117368	−	0.993088i	$-0.462554\pi$
$887$	6.99105	0.234737	0.117368	+	0.993088i	$0.462554\pi$
$888$	0	0
$889$	−20.2674	−0.679746
$890$	0	0
$891$	6.00386i	0.201137i
$892$	0	0
$893$	6.65691i	0.222765i
$894$	0	0
$895$	37.8766	1.26607
$896$	0	0
$897$	1.80910	0.0604040
$898$	0	0
$899$	− 0.418685i	− 0.0139639i
$900$	0	0
$901$	− 11.3798i	− 0.379118i
$902$	0	0
$903$	0.366187	0.0121859
$904$	0	0
$905$	10.4923	0.348776
$906$	0	0
$907$	8.75000i	0.290539i	0.989392	+	0.145270i	$0.0464049\pi$
$907$	8.75000i	0.290539i	−0.989392	+	0.145270i	$0.953595\pi$
$908$	0	0
$909$	− 1.05919i	− 0.0351311i
$910$	0	0
$911$	−23.5428	−0.780009	−0.390005	−	0.920813i	$-0.627527\pi$
$911$	−23.5428	−0.780009	−0.390005	+	0.920813i	$0.627527\pi$
$912$	0	0
$913$	10.2337	0.338687
$914$	0	0
$915$	0.957901i	0.0316672i
$916$	0	0
$917$	17.8643i	0.589932i
$918$	0	0
$919$	53.5349	1.76595	0.882976	−	0.469417i	$-0.155536\pi$
$919$	53.5349	1.76595	0.882976	+	0.469417i	$0.155536\pi$
$920$	0	0
$921$	1.46290	0.0482041
$922$	0	0
$923$	− 56.4510i	− 1.85811i
$924$	0	0
$925$	− 0.365363i	− 0.0120131i
$926$	0	0
$927$	−49.5546	−1.62759
$928$	0	0
$929$	43.7405	1.43508	0.717540	−	0.696518i	$-0.245268\pi$
$929$	43.7405	1.43508	0.717540	+	0.696518i	$0.245268\pi$
$930$	0	0
$931$	− 0.995183i	− 0.0326158i
$932$	0	0
$933$	2.27358i	0.0744337i
$934$	0	0
$935$	5.73689	0.187617
$936$	0	0
$937$	8.90798	0.291011	0.145506	−	0.989357i	$-0.453519\pi$
$937$	8.90798	0.291011	0.145506	+	0.989357i	$0.453519\pi$
$938$	0	0
$939$	2.49360i	0.0813755i
$940$	0	0
$941$	− 24.1765i	− 0.788133i	−0.919082	−	0.394066i	$-0.871068\pi$
$941$	− 24.1765i	− 0.788133i	0.919082	−	0.394066i	$-0.128932\pi$
$942$	0	0
$943$	12.6797	0.412907
$944$	0	0
$945$	−1.33350	−0.0433786
$946$	0	0
$947$	8.55120i	0.277877i	0.990301	+	0.138938i	$0.0443690\pi$
$947$	8.55120i	0.277877i	−0.990301	+	0.138938i	$0.955631\pi$
$948$	0	0
$949$	45.7277i	1.48438i
$950$	0	0
$951$	3.13732	0.101735
$952$	0	0
$953$	23.7559	0.769529	0.384765	−	0.923015i	$-0.374283\pi$
$953$	23.7559	0.769529	0.384765	+	0.923015i	$0.374283\pi$
$954$	0	0
$955$	19.5512i	0.632662i
$956$	0	0
$957$	− 0.00918236i	0 0.000296823i
$958$	0	0
$959$	11.4894	0.371012
$960$	0	0
$961$	−21.2960	−0.686968
$962$	0	0
$963$	53.1520i	1.71280i
$964$	0	0
$965$	26.3370i	0.847817i
$966$	0	0
$967$	56.9403	1.83108	0.915538	−	0.402231i	$-0.131765\pi$
$967$	56.9403	1.83108	0.915538	+	0.402231i	$0.131765\pi$
$968$	0	0
$969$	0.390907	0.0125578
$970$	0	0
$971$	− 25.9137i	− 0.831610i	−0.909454	−	0.415805i	$-0.863500\pi$
$971$	− 25.9137i	− 0.831610i	0.909454	−	0.415805i	$-0.136500\pi$
$972$	0	0
$973$	− 0.209585i	− 0.00671898i
$974$	0	0
$975$	0.0945039	0.00302655
$976$	0	0
$977$	13.2222	0.423014	0.211507	−	0.977376i	$-0.432163\pi$
$977$	13.2222	0.423014	0.211507	+	0.977376i	$0.432163\pi$
$978$	0	0
$979$	− 0.577444i	− 0.0184552i
$980$	0	0
$981$	− 30.4854i	− 0.973322i
$982$	0	0
$983$	39.0935	1.24689	0.623445	−	0.781867i	$-0.285732\pi$
$983$	39.0935	1.24689	0.623445	+	0.781867i	$0.285732\pi$
$984$	0	0
$985$	−40.6508	−1.29524
$986$	0	0
$987$	0.678020i	0.0215816i
$988$	0	0
$989$	12.1599i	0.386664i
$990$	0	0
$991$	−21.0291	−0.668012	−0.334006	−	0.942571i	$-0.608401\pi$
$991$	−21.0291	−0.668012	−0.334006	+	0.942571i	$0.608401\pi$
$992$	0	0
$993$	−3.38849	−0.107530
$994$	0	0
$995$	− 4.68387i	− 0.148489i
$996$	0	0
$997$	26.9937i	0.854899i	0.904039	+	0.427449i	$0.140588\pi$
$997$	26.9937i	0.854899i	−0.904039	+	0.427449i	$0.859412\pi$
$998$	0	0
$999$	1.26158	0.0399147

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3584.2.b.i.1793.7		12
4.3	odd	2		3584.2.b.k.1793.6		12
8.3	odd	2		3584.2.b.k.1793.7		12
8.5	even	2	inner	3584.2.b.i.1793.6		12
16.3	odd	4		3584.2.a.e.1.4	✓	6
16.5	even	4		3584.2.a.f.1.4	yes	6
16.11	odd	4		3584.2.a.k.1.3	yes	6
16.13	even	4		3584.2.a.l.1.3	yes	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3584.2.a.e.1.4	✓	6	16.3	odd	4
3584.2.a.f.1.4	yes	6	16.5	even	4
3584.2.a.k.1.3	yes	6	16.11	odd	4
3584.2.a.l.1.3	yes	6	16.13	even	4
3584.2.b.i.1793.6		12	8.5	even	2	inner
3584.2.b.i.1793.7		12	1.1	even	1	trivial
3584.2.b.k.1793.6		12	4.3	odd	2
3584.2.b.k.1793.7		12	8.3	odd	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:	\( N \)	\(=\)	\( 3584 = 2^{9} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3584.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+0.101362i q^{3} -2.19640i q^{5} -1.00000 q^{7} +2.98973 q^{9} +O(q^{10})\)	\(q+0.101362i q^{3} -2.19640i q^{5} -1.00000 q^{7} +2.98973 q^{9} +0.674013i q^{11} -5.30258i q^{13} +0.222631 q^{15} +3.87523 q^{17} -0.995183i q^{19} -0.101362i q^{21} +3.36590 q^{23} +0.175829 q^{25} +0.607128i q^{27} +0.134404i q^{29} -3.11512 q^{31} -0.0683190 q^{33} +2.19640i q^{35} -2.07795i q^{37} +0.537478 q^{39} +3.76710 q^{41} +3.61268i q^{43} -6.56663i q^{45} -6.68913 q^{47} +1.00000 q^{49} +0.392799i q^{51} -2.93656i q^{53} +1.48040 q^{55} +0.100873 q^{57} +6.92085i q^{59} +4.30265i q^{61} -2.98973 q^{63} -11.6466 q^{65} -13.9311i q^{67} +0.341173i q^{69} +10.6460 q^{71} -8.62368 q^{73} +0.0178223i q^{75} -0.674013i q^{77} +7.12273 q^{79} +8.90764 q^{81} -15.1833i q^{83} -8.51155i q^{85} -0.0136234 q^{87} -0.856726 q^{89} +5.30258i q^{91} -0.315754i q^{93} -2.18582 q^{95} -11.0125 q^{97} +2.01511i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 12 q^{7} - 20 q^{9}+O(q^{10}) \) 12 * q - 12 * q^7 - 20 * q^9	\( 12 q - 12 q^{7} - 20 q^{9} - 16 q^{15} - 4 q^{25} - 16 q^{31} + 8 q^{33} + 8 q^{41} + 16 q^{47} + 12 q^{49} - 32 q^{55} + 8 q^{57} + 20 q^{63} - 16 q^{65} + 16 q^{71} - 32 q^{73} - 48 q^{79} + 20 q^{81} + 16 q^{87} - 32 q^{89} - 32 q^{95} + 32 q^{97}+O(q^{100}) \) 12 * q - 12 * q^7 - 20 * q^9 - 16 * q^15 - 4 * q^25 - 16 * q^31 + 8 * q^33 + 8 * q^41 + 16 * q^47 + 12 * q^49 - 32 * q^55 + 8 * q^57 + 20 * q^63 - 16 * q^65 + 16 * q^71 - 32 * q^73 - 48 * q^79 + 20 * q^81 + 16 * q^87 - 32 * q^89 - 32 * q^95 + 32 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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