LMFDB - Embedded newform 4004.2.e.a.3849.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4004,2,Mod(3849,4004)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4004, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1, 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("4004.3849");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4004 = 2^{2} \cdot 7 \cdot 11 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4004.e (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:	$31.9721009693$
Analytic rank:	$0$
Dimension:	$48$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			3849.5
Character	$\chi$	$=$	4004.3849
Dual form			4004.2.e.a.3849.44

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.95250i q^{3} +1.47372i q^{5} +(0.700125 + 2.55144i) q^{7} -5.71727 q^{9} +O(q^{10})$	$q-2.95250i q^{3} +1.47372i q^{5} +(0.700125 + 2.55144i) q^{7} -5.71727 q^{9} +(-2.46682 + 2.21693i) q^{11} +1.00000 q^{13} +4.35115 q^{15} -0.856767 q^{17} +2.02387 q^{19} +(7.53312 - 2.06712i) q^{21} -5.44295 q^{23} +2.82816 q^{25} +8.02273i q^{27} -7.20111i q^{29} -8.75886i q^{31} +(6.54550 + 7.28330i) q^{33} +(-3.76009 + 1.03178i) q^{35} -0.648463 q^{37} -2.95250i q^{39} +0.228272 q^{41} -3.56019i q^{43} -8.42562i q^{45} +0.0660373i q^{47} +(-6.01965 + 3.57265i) q^{49} +2.52960i q^{51} -8.15975 q^{53} +(-3.26713 - 3.63539i) q^{55} -5.97549i q^{57} +8.61123i q^{59} -2.74889 q^{61} +(-4.00280 - 14.5872i) q^{63} +1.47372i q^{65} +2.46150 q^{67} +16.0703i q^{69} -2.94553 q^{71} -10.9592 q^{73} -8.35016i q^{75} +(-7.38344 - 4.74181i) q^{77} +11.5879i q^{79} +6.53533 q^{81} -3.70251 q^{83} -1.26263i q^{85} -21.2613 q^{87} +1.53564i q^{89} +(0.700125 + 2.55144i) q^{91} -25.8605 q^{93} +2.98261i q^{95} -1.74605i q^{97} +(14.1035 - 12.6748i) q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 48 q - 4 q^{7} - 48 q^{9}+O(q^{10}) $ 48 * q - 4 * q^7 - 48 * q^9	$ 48 q - 4 q^{7} - 48 q^{9} + 2 q^{11} + 48 q^{13} + 8 q^{15} + 4 q^{17} + 10 q^{21} + 4 q^{23} - 44 q^{25} + 10 q^{33} - 14 q^{35} - 16 q^{37} + 10 q^{49} - 8 q^{53} - 12 q^{55} + 16 q^{61} + 16 q^{63} + 4 q^{67} - 16 q^{73} + 2 q^{77} + 64 q^{81} - 4 q^{83} - 12 q^{87} - 4 q^{91} + 36 q^{93} - 40 q^{99}+O(q^{100}) $ 48 * q - 4 * q^7 - 48 * q^9 + 2 * q^11 + 48 * q^13 + 8 * q^15 + 4 * q^17 + 10 * q^21 + 4 * q^23 - 44 * q^25 + 10 * q^33 - 14 * q^35 - 16 * q^37 + 10 * q^49 - 8 * q^53 - 12 * q^55 + 16 * q^61 + 16 * q^63 + 4 * q^67 - 16 * q^73 + 2 * q^77 + 64 * q^81 - 4 * q^83 - 12 * q^87 - 4 * q^91 + 36 * q^93 - 40 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/4004\mathbb{Z}\right)^\times$.

$n$	$365$	$925$	$2003$	$3433$
$\chi(n)$	$-1$	$1$	$1$	$-1$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$		−	2.95250i		−	1.70463i	−0.523031	−	0.852314i	$-0.675199\pi$
$3$		−	2.95250i		−	1.70463i	0.523031	−	0.852314i	$-0.324801\pi$
$4$		0			0
$5$			1.47372i			0.659065i	0.944144	+	0.329533i	$0.106891\pi$
$5$			1.47372i			0.659065i	−0.944144	+	0.329533i	$0.893109\pi$
$6$		0			0
$7$	0.700125	+	2.55144i	0.264622	+	0.964352i
$7$	0.700125	+	2.55144i	0.264622	+	0.964352i
$8$		0			0
$9$	−5.71727			−1.90576
$10$		0			0
$11$	−2.46682	+	2.21693i	−0.743775	+	0.668430i
$11$	−2.46682	+	2.21693i	−0.743775	+	0.668430i
$12$		0			0
$13$	1.00000			0.277350
$13$	1.00000			0.277350
$14$		0			0
$15$	4.35115			1.12346
$16$		0			0
$17$	−0.856767			−0.207796			−0.103898	−	0.994588i	$-0.533132\pi$
$17$	−0.856767			−0.207796			−0.103898	+	0.994588i	$0.533132\pi$
$18$		0			0
$19$	2.02387			0.464308			0.232154	−	0.972679i	$-0.425423\pi$
$19$	2.02387			0.464308			0.232154	+	0.972679i	$0.425423\pi$
$20$		0			0
$21$	7.53312	−	2.06712i	1.64386	−	0.451082i
$22$		0			0
$23$	−5.44295			−1.13493			−0.567467	−	0.823396i	$-0.692076\pi$
$23$	−5.44295			−1.13493			−0.567467	+	0.823396i	$0.692076\pi$
$24$		0			0
$25$	2.82816			0.565633
$26$		0			0
$27$			8.02273i			1.54398i
$28$		0			0
$29$		−	7.20111i		−	1.33721i	−0.743617	−	0.668606i	$-0.766891\pi$
$29$		−	7.20111i		−	1.33721i	0.743617	−	0.668606i	$-0.233109\pi$
$30$		0			0
$31$		−	8.75886i		−	1.57314i	−0.617503	−	0.786569i	$-0.711856\pi$
$31$		−	8.75886i		−	1.57314i	0.617503	−	0.786569i	$-0.288144\pi$
$32$		0			0
$33$	6.54550	+	7.28330i	1.13942	+	1.26786i
$34$		0			0
$35$	−3.76009	+	1.03178i	−0.635571	+	0.174403i
$36$		0			0
$37$	−0.648463			−0.106607			−0.0533033	−	0.998578i	$-0.516975\pi$
$37$	−0.648463			−0.106607			−0.0533033	+	0.998578i	$0.516975\pi$
$38$		0			0
$39$		−	2.95250i		−	0.472779i
$40$		0			0
$41$	0.228272			0.0356501			0.0178251	−	0.999841i	$-0.494326\pi$
$41$	0.228272			0.0356501			0.0178251	+	0.999841i	$0.494326\pi$
$42$		0			0
$43$		−	3.56019i		−	0.542924i	−0.962449	−	0.271462i	$-0.912493\pi$
$43$		−	3.56019i		−	0.542924i	0.962449	−	0.271462i	$-0.0875071\pi$
$44$		0			0
$45$		−	8.42562i		−	1.25602i
$46$		0			0
$47$			0.0660373i			0.00963252i	0.999988	+	0.00481626i	$0.00153307\pi$
$47$			0.0660373i			0.00963252i	−0.999988	+	0.00481626i	$0.998467\pi$
$48$		0			0
$49$	−6.01965	+	3.57265i	−0.859950	+	0.510378i
$50$		0			0
$51$			2.52960i			0.354215i
$52$		0			0
$53$	−8.15975			−1.12083			−0.560414	−	0.828213i	$-0.689358\pi$
$53$	−8.15975			−1.12083			−0.560414	+	0.828213i	$0.689358\pi$
$54$		0			0
$55$	−3.26713	−	3.63539i	−0.440539	−	0.490196i
$56$		0			0
$57$		−	5.97549i		−	0.791472i
$58$		0			0
$59$			8.61123i			1.12109i	0.828125	+	0.560543i	$0.189408\pi$
$59$			8.61123i			1.12109i	−0.828125	+	0.560543i	$0.810592\pi$
$60$		0			0
$61$	−2.74889			−0.351960			−0.175980	−	0.984394i	$-0.556309\pi$
$61$	−2.74889			−0.351960			−0.175980	+	0.984394i	$0.556309\pi$
$62$		0			0
$63$	−4.00280	−	14.5872i	−0.504305	−	1.83782i
$64$		0			0
$65$			1.47372i			0.182792i
$66$		0			0
$67$	2.46150			0.300721			0.150360	−	0.988631i	$-0.451957\pi$
$67$	2.46150			0.300721			0.150360	+	0.988631i	$0.451957\pi$
$68$		0			0
$69$			16.0703i			1.93464i
$70$		0			0
$71$	−2.94553			−0.349571			−0.174785	−	0.984607i	$-0.555923\pi$
$71$	−2.94553			−0.349571			−0.174785	+	0.984607i	$0.555923\pi$
$72$		0			0
$73$	−10.9592			−1.28268			−0.641341	−	0.767256i	$-0.721622\pi$
$73$	−10.9592			−1.28268			−0.641341	+	0.767256i	$0.721622\pi$
$74$		0			0
$75$		−	8.35016i		−	0.964193i
$76$		0			0
$77$	−7.38344	−	4.74181i	−0.841421	−	0.540379i
$78$		0			0
$79$			11.5879i			1.30374i	0.758329	+	0.651872i	$0.226016\pi$
$79$			11.5879i			1.30374i	−0.758329	+	0.651872i	$0.773984\pi$
$80$		0			0
$81$	6.53533			0.726148
$82$		0			0
$83$	−3.70251			−0.406403			−0.203201	−	0.979137i	$-0.565135\pi$
$83$	−3.70251			−0.406403			−0.203201	+	0.979137i	$0.565135\pi$
$84$		0			0
$85$		−	1.26263i		−	0.136951i
$86$		0			0
$87$	−21.2613			−2.27945
$88$		0			0
$89$			1.53564i			0.162777i	0.996682	+	0.0813887i	$0.0259355\pi$
$89$			1.53564i			0.162777i	−0.996682	+	0.0813887i	$0.974064\pi$
$90$		0			0
$91$	0.700125	+	2.55144i	0.0733930	+	0.267463i
$92$		0			0
$93$	−25.8605			−2.68161
$94$		0			0
$95$			2.98261i			0.306009i
$96$		0			0
$97$		−	1.74605i		−	0.177285i	−0.996064	−	0.0886424i	$-0.971747\pi$
$97$		−	1.74605i		−	0.177285i	0.996064	−	0.0886424i	$-0.0282528\pi$
$98$		0			0
$99$	14.1035	−	12.6748i	1.41745	−	1.27386i
$100$		0			0
$101$	−17.0955			−1.70107			−0.850534	−	0.525920i	$-0.823721\pi$
$101$	−17.0955			−1.70107			−0.850534	+	0.525920i	$0.823721\pi$
$102$		0			0
$103$			13.1693i			1.29761i	0.760955	+	0.648805i	$0.224731\pi$
$103$			13.1693i			1.29761i	−0.760955	+	0.648805i	$0.775269\pi$
$104$		0			0
$105$	3.04635	+	11.1017i	0.297293	+	1.08341i
$106$		0			0
$107$			3.06264i			0.296077i	0.988982	+	0.148038i	$0.0472959\pi$
$107$			3.06264i			0.296077i	−0.988982	+	0.148038i	$0.952704\pi$
$108$		0			0
$109$		−	18.8598i		−	1.80644i	−0.429180	−	0.903219i	$-0.641197\pi$
$109$		−	18.8598i		−	1.80644i	0.429180	−	0.903219i	$-0.358803\pi$
$110$		0			0
$111$			1.91459i			0.181725i
$112$		0			0
$113$	−14.6842			−1.38138			−0.690688	−	0.723153i	$-0.742692\pi$
$113$	−14.6842			−1.38138			−0.690688	+	0.723153i	$0.742692\pi$
$114$		0			0
$115$		−	8.02135i		−	0.747995i
$116$		0			0
$117$	−5.71727			−0.528561
$118$		0			0
$119$	−0.599843	−	2.18598i	−0.0549876	−	0.200389i
$120$		0			0
$121$	1.17042	−	10.9376i	0.106402	−	0.994323i
$122$		0			0
$123$		−	0.673974i		−	0.0607702i
$124$		0			0
$125$			11.5365i			1.03185i
$126$		0			0
$127$			3.85222i			0.341829i	0.985286	+	0.170914i	$0.0546722\pi$
$127$			3.85222i			0.341829i	−0.985286	+	0.170914i	$0.945328\pi$
$128$		0			0
$129$	−10.5115			−0.925483
$130$		0			0
$131$	1.96074			0.171310			0.0856552	−	0.996325i	$-0.472702\pi$
$131$	1.96074			0.171310			0.0856552	+	0.996325i	$0.472702\pi$
$132$		0			0
$133$	1.41696	+	5.16378i	0.122866	+	0.447756i
$134$		0			0
$135$	−11.8232			−1.01758
$136$		0			0
$137$	−18.2373			−1.55812			−0.779059	−	0.626951i	$-0.784303\pi$
$137$	−18.2373			−1.55812			−0.779059	+	0.626951i	$0.784303\pi$
$138$		0			0
$139$	−0.828888			−0.0703054			−0.0351527	−	0.999382i	$-0.511192\pi$
$139$	−0.828888			−0.0703054			−0.0351527	+	0.999382i	$0.511192\pi$
$140$		0			0
$141$	0.194975			0.0164199
$142$		0			0
$143$	−2.46682	+	2.21693i	−0.206286	+	0.185389i
$144$		0			0
$145$	10.6124			0.881311
$146$		0			0
$147$	10.5482	+	17.7730i	0.870005	+	1.46589i
$148$		0			0
$149$		−	7.10513i		−	0.582075i	−0.956712	−	0.291037i	$-0.906000\pi$
$149$		−	7.10513i		−	0.582075i	0.956712	−	0.291037i	$-0.0940004\pi$
$150$		0			0
$151$		−	15.2537i		−	1.24133i	−0.784077	−	0.620664i	$-0.786863\pi$
$151$		−	15.2537i		−	1.24133i	0.784077	−	0.620664i	$-0.213137\pi$
$152$		0			0
$153$	4.89836			0.396009
$154$		0			0
$155$	12.9081			1.03680
$156$		0			0
$157$		−	15.5331i		−	1.23967i	−0.784730	−	0.619837i	$-0.787199\pi$
$157$		−	15.5331i		−	1.23967i	0.784730	−	0.619837i	$-0.212801\pi$
$158$		0			0
$159$			24.0917i			1.91059i
$160$		0			0
$161$	−3.81074	−	13.8873i	−0.300329	−	1.09448i
$162$		0			0
$163$	−6.21570			−0.486851			−0.243426	−	0.969920i	$-0.578271\pi$
$163$	−6.21570			−0.486851			−0.243426	+	0.969920i	$0.578271\pi$
$164$		0			0
$165$	−10.7335	+	9.64620i	−0.835602	+	0.750955i
$166$		0			0
$167$	−6.90519			−0.534340			−0.267170	−	0.963649i	$-0.586088\pi$
$167$	−6.90519			−0.534340			−0.267170	+	0.963649i	$0.586088\pi$
$168$		0			0
$169$	1.00000			0.0769231
$170$		0			0
$171$	−11.5710			−0.884858
$172$		0			0
$173$	2.76814			0.210458			0.105229	−	0.994448i	$-0.466442\pi$
$173$	2.76814			0.210458			0.105229	+	0.994448i	$0.466442\pi$
$174$		0			0
$175$	1.98007	+	7.21588i	0.149679	+	0.545469i
$176$		0			0
$177$	25.4247			1.91104
$178$		0			0
$179$	−4.49063			−0.335645			−0.167823	−	0.985817i	$-0.553674\pi$
$179$	−4.49063			−0.335645			−0.167823	+	0.985817i	$0.553674\pi$
$180$		0			0
$181$			4.94157i			0.367304i	0.982991	+	0.183652i	$0.0587920\pi$
$181$			4.94157i			0.367304i	−0.982991	+	0.183652i	$0.941208\pi$
$182$		0			0
$183$			8.11611i			0.599960i
$184$		0			0
$185$		−	0.955649i		−	0.0702607i
$186$		0			0
$187$	2.11349	−	1.89939i	0.154554	−	0.138897i
$188$		0			0
$189$	−20.4695	+	5.61691i	−1.48894	+	0.408570i
$190$		0			0
$191$	−1.44020			−0.104209			−0.0521045	−	0.998642i	$-0.516593\pi$
$191$	−1.44020			−0.104209			−0.0521045	+	0.998642i	$0.516593\pi$
$192$		0			0
$193$			2.87450i			0.206911i	0.994634	+	0.103455i	$0.0329899\pi$
$193$			2.87450i			0.206911i	−0.994634	+	0.103455i	$0.967010\pi$
$194$		0			0
$195$	4.35115			0.311592
$196$		0			0
$197$			1.53318i			0.109234i	0.998507	+	0.0546172i	$0.0173939\pi$
$197$			1.53318i			0.109234i	−0.998507	+	0.0546172i	$0.982606\pi$
$198$		0			0
$199$		−	5.10475i		−	0.361866i	−0.983495	−	0.180933i	$-0.942088\pi$
$199$		−	5.10475i		−	0.361866i	0.983495	−	0.180933i	$-0.0579118\pi$
$200$		0			0
$201$		−	7.26759i		−	0.512617i
$202$		0			0
$203$	18.3732	−	5.04167i	1.28954	−	0.353856i
$204$		0			0
$205$			0.336408i			0.0234958i
$206$		0			0
$207$	31.1188			2.16290
$208$		0			0
$209$	−4.99253	+	4.48679i	−0.345341	+	0.310358i
$210$		0			0
$211$		−	2.28121i		−	0.157045i	−0.996912	−	0.0785223i	$-0.974980\pi$
$211$		−	2.28121i		−	0.157045i	0.996912	−	0.0785223i	$-0.0250202\pi$
$212$		0			0
$213$			8.69670i			0.595888i
$214$		0			0
$215$	5.24671			0.357822
$216$		0			0
$217$	22.3477	−	6.13229i	1.51706	−	0.416287i
$218$		0			0
$219$			32.3572i			2.18650i
$220$		0			0
$221$	−0.856767			−0.0576324
$222$		0			0
$223$		−	15.1693i		−	1.01581i	−0.861413	−	0.507905i	$-0.830420\pi$
$223$		−	15.1693i		−	1.01581i	0.861413	−	0.507905i	$-0.169580\pi$
$224$		0			0
$225$	−16.1694			−1.07796
$226$		0			0
$227$	20.6744			1.37221			0.686103	−	0.727505i	$-0.259320\pi$
$227$	20.6744			1.37221			0.686103	+	0.727505i	$0.259320\pi$
$228$		0			0
$229$		−	7.93913i		−	0.524633i	−0.964982	−	0.262316i	$-0.915514\pi$
$229$		−	7.93913i		−	0.524633i	0.964982	−	0.262316i	$-0.0844864\pi$
$230$		0			0
$231$	−14.0002	+	21.7996i	−0.921146	+	1.43431i
$232$		0			0
$233$		−	20.1780i		−	1.32190i	−0.750429	−	0.660951i	$-0.770153\pi$
$233$		−	20.1780i		−	1.32190i	0.750429	−	0.660951i	$-0.229847\pi$
$234$		0			0
$235$	−0.0973201			−0.00634846
$236$		0			0
$237$	34.2134			2.22240
$238$		0			0
$239$			17.7041i			1.14518i	0.819841	+	0.572591i	$0.194062\pi$
$239$			17.7041i			1.14518i	−0.819841	+	0.572591i	$0.805938\pi$
$240$		0			0
$241$	14.3287			0.922995			0.461498	−	0.887141i	$-0.347312\pi$
$241$	14.3287			0.922995			0.461498	+	0.887141i	$0.347312\pi$
$242$		0			0
$243$			4.77262i			0.306163i
$244$		0			0
$245$	−5.26506	−	8.87125i	−0.336373	−	0.566763i
$246$		0			0
$247$	2.02387			0.128776
$248$		0			0
$249$			10.9317i			0.692765i
$250$		0			0
$251$			5.99369i			0.378318i	0.981946	+	0.189159i	$0.0605762\pi$
$251$			5.99369i			0.378318i	−0.981946	+	0.189159i	$0.939424\pi$
$252$		0			0
$253$	13.4268	−	12.0666i	0.844135	−	0.758623i
$254$		0			0
$255$	−3.72792			−0.233451
$256$		0			0
$257$		−	26.4394i		−	1.64924i	−0.565684	−	0.824622i	$-0.691388\pi$
$257$		−	26.4394i		−	1.64924i	0.565684	−	0.824622i	$-0.308612\pi$
$258$		0			0
$259$	−0.454005	−	1.65451i	−0.0282105	−	0.102806i
$260$		0			0
$261$			41.1707i			2.54840i
$262$		0			0
$263$		−	7.29077i		−	0.449568i	−0.974409	−	0.224784i	$-0.927832\pi$
$263$		−	7.29077i		−	0.449568i	0.974409	−	0.224784i	$-0.0721676\pi$
$264$		0			0
$265$		−	12.0251i		−	0.738699i
$266$		0			0
$267$	4.53398			0.277475
$268$		0			0
$269$			17.2572i			1.05219i	0.850425	+	0.526096i	$0.176345\pi$
$269$			17.2572i			1.05219i	−0.850425	+	0.526096i	$0.823655\pi$
$270$		0			0
$271$	−23.3871			−1.42067			−0.710334	−	0.703865i	$-0.751456\pi$
$271$	−23.3871			−1.42067			−0.710334	+	0.703865i	$0.751456\pi$
$272$		0			0
$273$	7.53312	−	2.06712i	0.455925	−	0.125108i
$274$		0			0
$275$	−6.97658	+	6.26985i	−0.420703	+	0.378086i
$276$		0			0
$277$			11.2295i			0.674713i	0.941377	+	0.337357i	$0.109533\pi$
$277$			11.2295i			0.674713i	−0.941377	+	0.337357i	$0.890467\pi$
$278$		0			0
$279$			50.0767i			2.99802i
$280$		0			0
$281$		−	11.7032i		−	0.698154i	−0.937094	−	0.349077i	$-0.886495\pi$
$281$		−	11.7032i		−	0.698154i	0.937094	−	0.349077i	$-0.113505\pi$
$282$		0			0
$283$	−19.2534			−1.14450			−0.572249	−	0.820080i	$-0.693929\pi$
$283$	−19.2534			−1.14450			−0.572249	+	0.820080i	$0.693929\pi$
$284$		0			0
$285$	8.80616			0.521632
$286$		0			0
$287$	0.159819	+	0.582422i	0.00943381	+	0.0343793i
$288$		0			0
$289$	−16.2660			−0.956821
$290$		0			0
$291$	−5.15522			−0.302205
$292$		0			0
$293$	8.21332			0.479827			0.239914	−	0.970794i	$-0.422881\pi$
$293$	8.21332			0.479827			0.239914	+	0.970794i	$0.422881\pi$
$294$		0			0
$295$	−12.6905			−0.738869
$296$		0			0
$297$	−17.7859	−	19.7907i	−1.03204	−	1.14837i
$298$		0			0
$299$	−5.44295			−0.314774
$300$		0			0
$301$	9.08360	−	2.49258i	0.523570	−	0.143670i
$302$		0			0
$303$			50.4746i			2.89969i
$304$		0			0
$305$		−	4.05109i		−	0.231965i
$306$		0			0
$307$	1.59390			0.0909689			0.0454844	−	0.998965i	$-0.485517\pi$
$307$	1.59390			0.0909689			0.0454844	+	0.998965i	$0.485517\pi$
$308$		0			0
$309$	38.8824			2.21194
$310$		0			0
$311$			2.89268i			0.164029i	0.996631	+	0.0820144i	$0.0261353\pi$
$311$			2.89268i			0.164029i	−0.996631	+	0.0820144i	$0.973865\pi$
$312$		0			0
$313$			7.14296i			0.403744i	0.979412	+	0.201872i	$0.0647025\pi$
$313$			7.14296i			0.403744i	−0.979412	+	0.201872i	$0.935298\pi$
$314$		0			0
$315$	21.4974	−	5.89899i	1.21124	−	0.332370i
$316$		0			0
$317$	15.0222			0.843733			0.421866	−	0.906658i	$-0.361375\pi$
$317$	15.0222			0.843733			0.421866	+	0.906658i	$0.361375\pi$
$318$		0			0
$319$	15.9644	+	17.7639i	0.893833	+	0.994585i
$320$		0			0
$321$	9.04246			0.504701
$322$		0			0
$323$	−1.73399			−0.0964815
$324$		0			0
$325$	2.82816			0.156878
$326$		0			0
$327$	−55.6835			−3.07930
$328$		0			0
$329$	−0.168490	+	0.0462343i	−0.00928915	+	0.00254898i
$330$		0			0
$331$	−2.34816			−0.129067			−0.0645333	−	0.997916i	$-0.520556\pi$
$331$	−2.34816			−0.129067			−0.0645333	+	0.997916i	$0.520556\pi$
$332$		0			0
$333$	3.70743			0.203166
$334$		0			0
$335$			3.62756i			0.198195i
$336$		0			0
$337$		−	12.3255i		−	0.671411i	−0.941967	−	0.335705i	$-0.891025\pi$
$337$		−	12.3255i		−	0.671411i	0.941967	−	0.335705i	$-0.108975\pi$
$338$		0			0
$339$			43.3552i			2.35473i
$340$		0			0
$341$	19.4178	+	21.6066i	1.05153	+	1.17006i
$342$		0			0
$343$	−13.3299	−	12.8575i	−0.719746	−	0.694237i
$344$		0			0
$345$	−23.6831			−1.27505
$346$		0			0
$347$		−	12.3593i		−	0.663481i	−0.943371	−	0.331741i	$-0.892364\pi$
$347$		−	12.3593i		−	0.663481i	0.943371	−	0.331741i	$-0.107636\pi$
$348$		0			0
$349$	−19.8213			−1.06101			−0.530504	−	0.847682i	$-0.677997\pi$
$349$	−19.8213			−1.06101			−0.530504	+	0.847682i	$0.677997\pi$
$350$		0			0
$351$			8.02273i			0.428222i
$352$		0			0
$353$		−	16.4577i		−	0.875953i	−0.898986	−	0.437977i	$-0.855695\pi$
$353$		−	16.4577i		−	0.875953i	0.898986	−	0.437977i	$-0.144305\pi$
$354$		0			0
$355$		−	4.34088i		−	0.230390i
$356$		0			0
$357$	−6.45412	+	1.77104i	−0.341588	+	0.0937333i
$358$		0			0
$359$			7.99193i			0.421798i	0.977508	+	0.210899i	$0.0676391\pi$
$359$			7.99193i			0.421798i	−0.977508	+	0.210899i	$0.932361\pi$
$360$		0			0
$361$	−14.9039			−0.784418
$362$		0			0
$363$	−32.2931	−	3.45568i	−1.69495	−	0.181376i
$364$		0			0
$365$		−	16.1508i		−	0.845372i
$366$		0			0
$367$			35.4920i			1.85267i	0.376706	+	0.926333i	$0.377057\pi$
$367$			35.4920i			1.85267i	−0.376706	+	0.926333i	$0.622943\pi$
$368$		0			0
$369$	−1.30509			−0.0679404
$370$		0			0
$371$	−5.71284	−	20.8191i	−0.296596	−	1.08087i
$372$		0			0
$373$			37.0347i			1.91759i	0.284107	+	0.958793i	$0.408303\pi$
$373$			37.0347i			1.91759i	−0.284107	+	0.958793i	$0.591697\pi$
$374$		0			0
$375$	34.0615			1.75893
$376$		0			0
$377$		−	7.20111i		−	0.370876i
$378$		0			0
$379$	−0.783377			−0.0402394			−0.0201197	−	0.999798i	$-0.506405\pi$
$379$	−0.783377			−0.0402394			−0.0201197	+	0.999798i	$0.506405\pi$
$380$		0			0
$381$	11.3737			0.582691
$382$		0			0
$383$			17.5468i			0.896597i	0.893884	+	0.448299i	$0.147970\pi$
$383$			17.5468i			0.896597i	−0.893884	+	0.448299i	$0.852030\pi$
$384$		0			0
$385$	6.98808	−	10.8811i	0.356145	−	0.554552i
$386$		0			0
$387$			20.3546i			1.03468i
$388$		0			0
$389$	12.8069			0.649335			0.324668	−	0.945828i	$-0.394748\pi$
$389$	12.8069			0.649335			0.324668	+	0.945828i	$0.394748\pi$
$390$		0			0
$391$	4.66334			0.235835
$392$		0			0
$393$		−	5.78908i		−	0.292020i
$394$		0			0
$395$	−17.0773			−0.859253
$396$		0			0
$397$			1.26500i			0.0634885i	0.999496	+	0.0317443i	$0.0101062\pi$
$397$			1.26500i			0.0634885i	−0.999496	+	0.0317443i	$0.989894\pi$
$398$		0			0
$399$	15.2461	−	4.18358i	0.763258	−	0.209441i
$400$		0			0
$401$	−30.7858			−1.53737			−0.768686	−	0.639627i	$-0.779089\pi$
$401$	−30.7858			−1.53737			−0.768686	+	0.639627i	$0.779089\pi$
$402$		0			0
$403$		−	8.75886i		−	0.436310i
$404$		0			0
$405$			9.63122i			0.478579i
$406$		0			0
$407$	1.59964	−	1.43760i	0.0792913	−	0.0712591i
$408$		0			0
$409$	−0.819909			−0.0405419			−0.0202709	−	0.999795i	$-0.506453\pi$
$409$	−0.819909			−0.0405419			−0.0202709	+	0.999795i	$0.506453\pi$
$410$		0			0
$411$			53.8457i			2.65601i
$412$		0			0
$413$	−21.9710	+	6.02893i	−1.08112	+	0.296664i
$414$		0			0
$415$		−	5.45644i		−	0.267846i
$416$		0			0
$417$			2.44729i			0.119844i
$418$		0			0
$419$			10.0496i			0.490953i	0.969403	+	0.245477i	$0.0789445\pi$
$419$			10.0496i			0.490953i	−0.969403	+	0.245477i	$0.921056\pi$
$420$		0			0
$421$	16.0393			0.781706			0.390853	−	0.920453i	$-0.372180\pi$
$421$	16.0393			0.781706			0.390853	+	0.920453i	$0.372180\pi$
$422$		0			0
$423$		−	0.377553i		−	0.0183572i
$424$		0			0
$425$	−2.42308			−0.117536
$426$		0			0
$427$	−1.92457	−	7.01363i	−0.0931364	−	0.339413i
$428$		0			0
$429$	6.54550	+	7.28330i	0.316019	+	0.351641i
$430$		0			0
$431$		−	1.13586i		−	0.0547124i	−0.999626	−	0.0273562i	$-0.991291\pi$
$431$		−	1.13586i		−	0.0547124i	0.999626	−	0.0273562i	$-0.00870883\pi$
$432$		0			0
$433$			12.2166i			0.587090i	0.955945	+	0.293545i	$0.0948351\pi$
$433$			12.2166i			0.587090i	−0.955945	+	0.293545i	$0.905165\pi$
$434$		0			0
$435$		−	31.3331i		−	1.50231i
$436$		0			0
$437$	−11.0158			−0.526959
$438$		0			0
$439$	33.5089			1.59929			0.799646	−	0.600472i	$-0.205021\pi$
$439$	33.5089			1.59929			0.799646	+	0.600472i	$0.205021\pi$
$440$		0			0
$441$	34.4159	−	20.4258i	1.63885	−	0.972656i
$442$		0			0
$443$	−27.2804			−1.29613			−0.648067	−	0.761584i	$-0.724422\pi$
$443$	−27.2804			−1.29613			−0.648067	+	0.761584i	$0.724422\pi$
$444$		0			0
$445$	−2.26310			−0.107281
$446$		0			0
$447$	−20.9779			−0.992221
$448$		0			0
$449$	11.1353			0.525508			0.262754	−	0.964863i	$-0.415369\pi$
$449$	11.1353			0.525508			0.262754	+	0.964863i	$0.415369\pi$
$450$		0			0
$451$	−0.563107	+	0.506064i	−0.0265157	+	0.0238296i
$452$		0			0
$453$	−45.0366			−2.11600
$454$		0			0
$455$	−3.76009	+	1.03178i	−0.176276	+	0.0483708i
$456$		0			0
$457$		−	7.69353i		−	0.359888i	−0.983677	−	0.179944i	$-0.942408\pi$
$457$		−	7.69353i		−	0.359888i	0.983677	−	0.179944i	$-0.0575917\pi$
$458$		0			0
$459$		−	6.87361i		−	0.320833i
$460$		0			0
$461$	−6.60419			−0.307588			−0.153794	−	0.988103i	$-0.549149\pi$
$461$	−6.60419			−0.307588			−0.153794	+	0.988103i	$0.549149\pi$
$462$		0			0
$463$	37.0831			1.72340			0.861698	−	0.507421i	$-0.169401\pi$
$463$	37.0831			1.72340			0.861698	+	0.507421i	$0.169401\pi$
$464$		0			0
$465$		−	38.1111i		−	1.76736i
$466$		0			0
$467$		−	20.7065i		−	0.958183i	−0.877765	−	0.479092i	$-0.840966\pi$
$467$		−	20.7065i		−	0.958183i	0.877765	−	0.479092i	$-0.159034\pi$
$468$		0			0
$469$	1.72336	+	6.28037i	0.0795774	+	0.290001i
$470$		0			0
$471$	−45.8615			−2.11318
$472$		0			0
$473$	7.89270	+	8.78236i	0.362907	+	0.403813i
$474$		0			0
$475$	5.72384			0.262628
$476$		0			0
$477$	46.6514			2.13602
$478$		0			0
$479$	17.5562			0.802165			0.401083	−	0.916042i	$-0.368634\pi$
$479$	17.5562			0.802165			0.401083	+	0.916042i	$0.368634\pi$
$480$		0			0
$481$	−0.648463			−0.0295674
$482$		0			0
$483$	−41.0024	+	11.2512i	−1.86567	+	0.511948i
$484$		0			0
$485$	2.57318			0.116842
$486$		0			0
$487$	−16.8169			−0.762048			−0.381024	−	0.924565i	$-0.624428\pi$
$487$	−16.8169			−0.762048			−0.381024	+	0.924565i	$0.624428\pi$
$488$		0			0
$489$			18.3519i			0.829900i
$490$		0			0
$491$		−	20.7261i		−	0.935357i	−0.883899	−	0.467678i	$-0.845091\pi$
$491$		−	20.7261i		−	0.935357i	0.883899	−	0.467678i	$-0.154909\pi$
$492$		0			0
$493$			6.16967i			0.277868i
$494$		0			0
$495$	18.6790	+	20.7845i	0.839560	+	0.934194i
$496$		0			0
$497$	−2.06224	−	7.51534i	−0.0925042	−	0.337109i
$498$		0			0
$499$	−26.4557			−1.18432			−0.592160	−	0.805821i	$-0.701725\pi$
$499$	−26.4557			−1.18432			−0.592160	+	0.805821i	$0.701725\pi$
$500$		0			0
$501$			20.3876i			0.910850i
$502$		0			0
$503$	29.3512			1.30871			0.654353	−	0.756189i	$-0.272941\pi$
$503$	29.3512			1.30871			0.654353	+	0.756189i	$0.272941\pi$
$504$		0			0
$505$		−	25.1939i		−	1.12112i
$506$		0			0
$507$		−	2.95250i		−	0.131125i
$508$		0			0
$509$		−	8.00613i		−	0.354866i	−0.984133	−	0.177433i	$-0.943221\pi$
$509$		−	8.00613i		−	0.354866i	0.984133	−	0.177433i	$-0.0567793\pi$
$510$		0			0
$511$	−7.67284	−	27.9618i	−0.339426	−	1.23696i
$512$		0			0
$513$			16.2370i			0.716880i
$514$		0			0
$515$	−19.4078			−0.855210
$516$		0			0
$517$	−0.146400	−	0.162902i	−0.00643867	−	0.00716443i
$518$		0			0
$519$		−	8.17294i		−	0.358752i
$520$		0			0
$521$		−	9.65689i		−	0.423076i	−0.977370	−	0.211538i	$-0.932153\pi$
$521$		−	9.65689i		−	0.423076i	0.977370	−	0.211538i	$-0.0678472\pi$
$522$		0			0
$523$	30.1008			1.31622			0.658109	−	0.752923i	$-0.271357\pi$
$523$	30.1008			1.31622			0.658109	+	0.752923i	$0.271357\pi$
$524$		0			0
$525$	21.3049	−	5.84615i	0.929822	−	0.255147i
$526$		0			0
$527$			7.50430i			0.326892i
$528$		0			0
$529$	6.62568			0.288073
$530$		0			0
$531$		−	49.2327i		−	2.13652i
$532$		0			0
$533$	0.228272			0.00988756
$534$		0			0
$535$	−4.51346			−0.195134
$536$		0			0
$537$			13.2586i			0.572150i
$538$		0			0
$539$	6.92909	−	22.1582i	0.298457	−	0.954423i
$540$		0			0
$541$		−	11.9929i		−	0.515617i	−0.966196	−	0.257808i	$-0.917000\pi$
$541$		−	11.9929i		−	0.515617i	0.966196	−	0.257808i	$-0.0830003\pi$
$542$		0			0
$543$	14.5900			0.626117
$544$		0			0
$545$	27.7939			1.19056
$546$		0			0
$547$			30.8291i			1.31816i	0.752074	+	0.659078i	$0.229053\pi$
$547$			30.8291i			1.31816i	−0.752074	+	0.659078i	$0.770947\pi$
$548$		0			0
$549$	15.7162			0.670749
$550$		0			0
$551$		−	14.5741i		−	0.620879i
$552$		0			0
$553$	−29.5659	+	8.11300i	−1.25727	+	0.345000i
$554$		0			0
$555$	−2.82156			−0.119768
$556$		0			0
$557$			29.8366i			1.26422i	0.774881	+	0.632108i	$0.217810\pi$
$557$			29.8366i			1.26422i	−0.774881	+	0.632108i	$0.782190\pi$
$558$		0			0
$559$		−	3.56019i		−	0.150580i
$560$		0			0
$561$	−5.60796	−	6.24009i	−0.236768	−	0.263457i
$562$		0			0
$563$	39.3859			1.65992			0.829958	−	0.557825i	$-0.188364\pi$
$563$	39.3859			1.65992			0.829958	+	0.557825i	$0.188364\pi$
$564$		0			0
$565$		−	21.6404i		−	0.910417i
$566$		0			0
$567$	4.57555	+	16.6745i	0.192155	+	0.700263i
$568$		0			0
$569$		−	1.50539i		−	0.0631092i	−0.999502	−	0.0315546i	$-0.989954\pi$
$569$		−	1.50539i		−	0.0631092i	0.999502	−	0.0315546i	$-0.0100458\pi$
$570$		0			0
$571$			0.801576i			0.0335449i	0.999859	+	0.0167725i	$0.00533909\pi$
$571$			0.801576i			0.0335449i	−0.999859	+	0.0167725i	$0.994661\pi$
$572$		0			0
$573$			4.25218i			0.177638i
$574$		0			0
$575$	−15.3935			−0.641955
$576$		0			0
$577$		−	47.2677i		−	1.96778i	−0.178775	−	0.983890i	$-0.557213\pi$
$577$		−	47.2677i		−	1.96778i	0.178775	−	0.983890i	$-0.442787\pi$
$578$		0			0
$579$	8.48696			0.352706
$580$		0			0
$581$	−2.59222	−	9.44670i	−0.107543	−	0.391915i
$582$		0			0
$583$	20.1286	−	18.0896i	0.833643	−	0.749195i
$584$		0			0
$585$		−	8.42562i		−	0.348357i
$586$		0			0
$587$		−	13.5882i		−	0.560845i	−0.959877	−	0.280422i	$-0.909525\pi$
$587$		−	13.5882i		−	0.560845i	0.959877	−	0.280422i	$-0.0904745\pi$
$588$		0			0
$589$		−	17.7268i		−	0.730421i
$590$		0			0
$591$	4.52671			0.186204
$592$		0			0
$593$	12.8670			0.528384			0.264192	−	0.964470i	$-0.414895\pi$
$593$	12.8670			0.528384			0.264192	+	0.964470i	$0.414895\pi$
$594$		0			0
$595$	3.22152	−	0.883998i	0.132069	−	0.0362404i
$596$		0			0
$597$	−15.0718			−0.616848
$598$		0			0
$599$	−46.0667			−1.88223			−0.941117	−	0.338081i	$-0.890222\pi$
$599$	−46.0667			−1.88223			−0.941117	+	0.338081i	$0.890222\pi$
$600$		0			0
$601$	31.4166			1.28151			0.640754	−	0.767746i	$-0.278622\pi$
$601$	31.4166			1.28151			0.640754	+	0.767746i	$0.278622\pi$
$602$		0			0
$603$	−14.0731			−0.573100
$604$		0			0
$605$	16.1188	+	1.72487i	0.655324	+	0.0701261i
$606$		0			0
$607$	32.2878			1.31052			0.655260	−	0.755404i	$-0.272559\pi$
$607$	32.2878			1.31052			0.655260	+	0.755404i	$0.272559\pi$
$608$		0			0
$609$	−14.8856	−	54.2468i	−0.603193	−	2.19819i
$610$		0			0
$611$			0.0660373i			0.00267158i
$612$		0			0
$613$			12.7672i			0.515662i	0.966190	+	0.257831i	$0.0830078\pi$
$613$			12.7672i			0.515662i	−0.966190	+	0.257831i	$0.916992\pi$
$614$		0			0
$615$	0.993245			0.0400515
$616$		0			0
$617$	−10.9361			−0.440270			−0.220135	−	0.975469i	$-0.570650\pi$
$617$	−10.9361			−0.440270			−0.220135	+	0.975469i	$0.570650\pi$
$618$		0			0
$619$		−	1.64615i		−	0.0661642i	−0.999453	−	0.0330821i	$-0.989468\pi$
$619$		−	1.64615i		−	0.0661642i	0.999453	−	0.0330821i	$-0.0105323\pi$
$620$		0			0
$621$		−	43.6673i		−	1.75231i
$622$		0			0
$623$	−3.91809	+	1.07514i	−0.156975	+	0.0430745i
$624$		0			0
$625$	−2.86067			−0.114427
$626$		0			0
$627$	13.2472	+	14.7405i	0.529044	+	0.588677i
$628$		0			0
$629$	0.555581			0.0221525
$630$		0			0
$631$	7.20401			0.286787			0.143394	−	0.989666i	$-0.454199\pi$
$631$	7.20401			0.286787			0.143394	+	0.989666i	$0.454199\pi$
$632$		0			0
$633$	−6.73526			−0.267703
$634$		0			0
$635$	−5.67707			−0.225288
$636$		0			0
$637$	−6.01965	+	3.57265i	−0.238507	+	0.141553i
$638$		0			0
$639$	16.8404			0.666196
$640$		0			0
$641$	−43.6216			−1.72295			−0.861474	−	0.507802i	$-0.830458\pi$
$641$	−43.6216			−1.72295			−0.861474	+	0.507802i	$0.830458\pi$
$642$		0			0
$643$		−	34.6802i		−	1.36765i	−0.729645	−	0.683827i	$-0.760315\pi$
$643$		−	34.6802i		−	1.36765i	0.729645	−	0.683827i	$-0.239685\pi$
$644$		0			0
$645$		−	15.4909i		−	0.609954i
$646$		0			0
$647$			28.8059i			1.13248i	0.824241	+	0.566239i	$0.191602\pi$
$647$			28.8059i			1.13248i	−0.824241	+	0.566239i	$0.808398\pi$
$648$		0			0
$649$	−19.0905	−	21.2424i	−0.749368	−	0.833836i
$650$		0			0
$651$	−18.1056	−	65.9815i	−0.709615	−	2.58602i
$652$		0			0
$653$	35.3733			1.38426			0.692132	−	0.721771i	$-0.256672\pi$
$653$	35.3733			1.38426			0.692132	+	0.721771i	$0.256672\pi$
$654$		0			0
$655$			2.88957i			0.112905i
$656$		0			0
$657$	62.6569			2.44448
$658$		0			0
$659$		−	9.45540i		−	0.368330i	−0.982895	−	0.184165i	$-0.941042\pi$
$659$		−	9.45540i		−	0.368330i	0.982895	−	0.184165i	$-0.0589581\pi$
$660$		0			0
$661$		−	26.5769i		−	1.03372i	−0.856070	−	0.516860i	$-0.827101\pi$
$661$		−	26.5769i		−	1.03372i	0.856070	−	0.516860i	$-0.172899\pi$
$662$		0			0
$663$			2.52960i			0.0982417i
$664$		0			0
$665$	−7.60994	+	2.08820i	−0.295101	+	0.0809769i
$666$		0			0
$667$			39.1953i			1.51765i
$668$		0			0
$669$	−44.7873			−1.73158
$670$		0			0
$671$	6.78103	−	6.09411i	0.261779	−	0.235261i
$672$		0			0
$673$		−	48.0712i		−	1.85301i	−0.376284	−	0.926504i	$-0.622798\pi$
$673$		−	48.0712i		−	1.85301i	0.376284	−	0.926504i	$-0.377202\pi$
$674$		0			0
$675$			22.6896i			0.873323i
$676$		0			0
$677$	24.0979			0.926157			0.463078	−	0.886317i	$-0.346745\pi$
$677$	24.0979			0.926157			0.463078	+	0.886317i	$0.346745\pi$
$678$		0			0
$679$	4.45494	−	1.22245i	0.170965	−	0.0469135i
$680$		0			0
$681$		−	61.0411i		−	2.33910i
$682$		0			0
$683$	15.7386			0.602221			0.301110	−	0.953589i	$-0.402643\pi$
$683$	15.7386			0.602221			0.301110	+	0.953589i	$0.402643\pi$
$684$		0			0
$685$		−	26.8766i		−	1.02690i
$686$		0			0
$687$	−23.4403			−0.894303
$688$		0			0
$689$	−8.15975			−0.310862
$690$		0			0
$691$		−	1.92518i		−	0.0732373i	−0.999329	−	0.0366187i	$-0.988341\pi$
$691$		−	1.92518i		−	0.0732373i	0.999329	−	0.0366187i	$-0.0116587\pi$
$692$		0			0
$693$	42.2131	+	27.1102i	1.60354	+	1.02983i
$694$		0			0
$695$		−	1.22154i		−	0.0463358i
$696$		0			0
$697$	−0.195576			−0.00740796
$698$		0			0
$699$	−59.5755			−2.25335
$700$		0			0
$701$			16.5215i			0.624010i	0.950080	+	0.312005i	$0.101001\pi$
$701$			16.5215i			0.624010i	−0.950080	+	0.312005i	$0.898999\pi$
$702$		0			0
$703$	−1.31241			−0.0494983
$704$		0			0
$705$			0.287338i			0.0108218i
$706$		0			0
$707$	−11.9690	−	43.6181i	−0.450140	−	1.64043i
$708$		0			0
$709$	−38.1169			−1.43151			−0.715754	−	0.698352i	$-0.753917\pi$
$709$	−38.1169			−1.43151			−0.715754	+	0.698352i	$0.753917\pi$
$710$		0			0
$711$		−	66.2513i		−	2.48462i
$712$		0			0
$713$			47.6740i			1.78541i
$714$		0			0
$715$	−3.26713	−	3.63539i	−0.122184	−	0.135956i
$716$		0			0
$717$	52.2713			1.95211
$718$		0			0
$719$			15.9417i			0.594526i	0.954796	+	0.297263i	$0.0960739\pi$
$719$			15.9417i			0.594526i	−0.954796	+	0.297263i	$0.903926\pi$
$720$		0			0
$721$	−33.6006	+	9.22016i	−1.25135	+	0.343377i
$722$		0			0
$723$		−	42.3056i		−	1.57336i
$724$		0			0
$725$		−	20.3659i		−	0.756371i
$726$		0			0
$727$			7.22826i			0.268081i	0.990976	+	0.134041i	$0.0427953\pi$
$727$			7.22826i			0.268081i	−0.990976	+	0.134041i	$0.957205\pi$
$728$		0			0
$729$	33.6972			1.24804
$730$		0			0
$731$			3.05025i			0.112818i
$732$		0			0
$733$	−41.5642			−1.53521			−0.767604	−	0.640924i	$-0.778551\pi$
$733$	−41.5642			−1.53521			−0.767604	+	0.640924i	$0.778551\pi$
$734$		0			0
$735$	−26.1924	+	15.5451i	−0.966121	+	0.573390i
$736$		0			0
$737$	−6.07209	+	5.45699i	−0.223668	+	0.201011i
$738$		0			0
$739$			43.4940i			1.59995i	0.600033	+	0.799975i	$0.295154\pi$
$739$			43.4940i			1.59995i	−0.600033	+	0.799975i	$0.704846\pi$
$740$		0			0
$741$		−	5.97549i		−	0.219515i
$742$		0			0
$743$		−	29.5863i		−	1.08541i	−0.839922	−	0.542707i	$-0.817399\pi$
$743$		−	29.5863i		−	1.08541i	0.839922	−	0.542707i	$-0.182601\pi$
$744$		0			0
$745$	10.4709			0.383625
$746$		0			0
$747$	21.1682			0.774504
$748$		0			0
$749$	−7.81414	+	2.14423i	−0.285522	+	0.0783486i
$750$		0			0
$751$	15.0235			0.548216			0.274108	−	0.961699i	$-0.411617\pi$
$751$	15.0235			0.548216			0.274108	+	0.961699i	$0.411617\pi$
$752$		0			0
$753$	17.6964			0.644892
$754$		0			0
$755$	22.4796			0.818116
$756$		0			0
$757$	12.7543			0.463563			0.231782	−	0.972768i	$-0.425544\pi$
$757$	12.7543			0.463563			0.231782	+	0.972768i	$0.425544\pi$
$758$		0			0
$759$	−35.6268	−	39.6426i	−1.29317	−	1.43894i
$760$		0			0
$761$	−10.9948			−0.398562			−0.199281	−	0.979942i	$-0.563861\pi$
$761$	−10.9948			−0.398562			−0.199281	+	0.979942i	$0.563861\pi$
$762$		0			0
$763$	48.1195	−	13.2042i	1.74204	−	0.478024i
$764$		0			0
$765$			7.21879i			0.260996i
$766$		0			0
$767$			8.61123i			0.310933i
$768$		0			0
$769$	50.6109			1.82507			0.912537	−	0.408994i	$-0.134120\pi$
$769$	50.6109			1.82507			0.912537	+	0.408994i	$0.134120\pi$
$770$		0			0
$771$	−78.0623			−2.81135
$772$		0			0
$773$			18.8767i			0.678949i	0.940615	+	0.339474i	$0.110249\pi$
$773$			18.8767i			0.678949i	−0.940615	+	0.339474i	$0.889751\pi$
$774$		0			0
$775$		−	24.7715i		−	0.889818i
$776$		0			0
$777$	−4.88495	+	1.34045i	−0.175246	+	0.0480884i
$778$		0			0
$779$	0.461993			0.0165526
$780$		0			0
$781$	7.26611	−	6.53005i	0.260002	−	0.233664i
$782$		0			0
$783$	57.7726			2.06462
$784$		0			0
$785$	22.8913			0.817027
$786$		0			0
$787$	43.7110			1.55813			0.779065	−	0.626944i	$-0.215694\pi$
$787$	43.7110			1.55813			0.779065	+	0.626944i	$0.215694\pi$
$788$		0			0
$789$	−21.5260			−0.766346
$790$		0			0
$791$	−10.2808	−	37.4659i	−0.365543	−	1.33213i
$792$		0			0
$793$	−2.74889			−0.0976161
$794$		0			0
$795$	−35.5043			−1.25921
$796$		0			0
$797$			32.8904i			1.16504i	0.812817	+	0.582519i	$0.197933\pi$
$797$			32.8904i			1.16504i	−0.812817	+	0.582519i	$0.802067\pi$
$798$		0			0
$799$		−	0.0565785i		−	0.00200160i
$800$		0			0
$801$		−	8.77966i		−	0.310214i
$802$		0			0
$803$	27.0345	−	24.2959i	0.954027	−	0.857383i
$804$		0			0
$805$	20.4660	−	5.61595i	0.721331	−	0.197936i
$806$		0			0
$807$	50.9520			1.79359
$808$		0			0
$809$		−	30.3773i		−	1.06801i	−0.845481	−	0.534005i	$-0.820686\pi$
$809$		−	30.3773i		−	1.06801i	0.845481	−	0.534005i	$-0.179314\pi$
$810$		0			0
$811$	52.2430			1.83450			0.917251	−	0.398310i	$-0.130403\pi$
$811$	52.2430			1.83450			0.917251	+	0.398310i	$0.130403\pi$
$812$		0			0
$813$			69.0506i			2.42171i
$814$		0			0
$815$		−	9.16017i		−	0.320867i
$816$		0			0
$817$		−	7.20537i		−	0.252084i
$818$		0			0
$819$	−4.00280	−	14.5872i	−0.139869	−	0.509719i
$820$		0			0
$821$		−	26.9825i		−	0.941694i	−0.882215	−	0.470847i	$-0.843948\pi$
$821$		−	26.9825i		−	0.941694i	0.882215	−	0.470847i	$-0.156052\pi$
$822$		0			0
$823$	−37.6961			−1.31400			−0.657002	−	0.753889i	$-0.728176\pi$
$823$	−37.6961			−1.31400			−0.657002	+	0.753889i	$0.728176\pi$
$824$		0			0
$825$	18.5117	+	20.5984i	0.644496	+	0.717143i
$826$		0			0
$827$			39.1052i			1.35982i	0.733296	+	0.679910i	$0.237981\pi$
$827$			39.1052i			1.35982i	−0.733296	+	0.679910i	$0.762019\pi$
$828$		0			0
$829$		−	16.0282i		−	0.556683i	−0.960482	−	0.278342i	$-0.910215\pi$
$829$		−	16.0282i		−	0.556683i	0.960482	−	0.278342i	$-0.0897848\pi$
$830$		0			0
$831$	33.1550			1.15014
$832$		0			0
$833$	5.15744	−	3.06092i	0.178695	−	0.106055i
$834$		0			0
$835$		−	10.1763i		−	0.352165i
$836$		0			0
$837$	70.2700			2.42889
$838$		0			0
$839$			46.8246i			1.61656i	0.588795	+	0.808282i	$0.299602\pi$
$839$			46.8246i			1.61656i	−0.588795	+	0.808282i	$0.700398\pi$
$840$		0			0
$841$	−22.8560			−0.788137
$842$		0			0
$843$	−34.5537			−1.19009
$844$		0			0
$845$			1.47372i			0.0506973i
$846$		0			0
$847$	28.7259	−	4.67139i	0.987034	−	0.160511i
$848$		0			0
$849$			56.8458i			1.95094i
$850$		0			0
$851$	3.52955			0.120991
$852$		0			0
$853$	−50.0385			−1.71329			−0.856643	−	0.515910i	$-0.827454\pi$
$853$	−50.0385			−1.71329			−0.856643	+	0.515910i	$0.827454\pi$
$854$		0			0
$855$		−	17.0524i		−	0.583179i
$856$		0			0
$857$	−44.7480			−1.52856			−0.764281	−	0.644883i	$-0.776906\pi$
$857$	−44.7480			−1.52856			−0.764281	+	0.644883i	$0.776906\pi$
$858$		0			0
$859$			41.0785i			1.40158i	0.713367	+	0.700791i	$0.247169\pi$
$859$			41.0785i			1.40158i	−0.713367	+	0.700791i	$0.752831\pi$
$860$		0			0
$861$	1.71960	−	0.471866i	0.0586038	−	0.0160811i
$862$		0			0
$863$	−54.5445			−1.85672			−0.928358	−	0.371687i	$-0.878780\pi$
$863$	−54.5445			−1.85672			−0.928358	+	0.371687i	$0.878780\pi$
$864$		0			0
$865$			4.07945i			0.138705i
$866$		0			0
$867$			48.0252i			1.63102i
$868$		0			0
$869$	−25.6897	−	28.5854i	−0.871462	−	0.969693i
$870$		0			0
$871$	2.46150			0.0834049
$872$		0			0
$873$			9.98265i			0.337862i
$874$		0			0
$875$	−29.4346	+	8.07698i	−0.995071	+	0.273052i
$876$		0			0
$877$			36.1329i			1.22012i	0.792354	+	0.610061i	$0.208855\pi$
$877$			36.1329i			1.22012i	−0.792354	+	0.610061i	$0.791145\pi$
$878$		0			0
$879$		−	24.2498i		−	0.817926i
$880$		0			0
$881$			5.53394i			0.186443i	0.995645	+	0.0932217i	$0.0297165\pi$
$881$			5.53394i			0.186443i	−0.995645	+	0.0932217i	$0.970283\pi$
$882$		0			0
$883$	−43.5089			−1.46419			−0.732095	−	0.681202i	$-0.761457\pi$
$883$	−43.5089			−1.46419			−0.732095	+	0.681202i	$0.761457\pi$
$884$		0			0
$885$			37.4687i			1.25950i
$886$		0			0
$887$	−20.3975			−0.684881			−0.342441	−	0.939539i	$-0.611254\pi$
$887$	−20.3975			−0.684881			−0.342441	+	0.939539i	$0.611254\pi$
$888$		0			0
$889$	−9.82868	+	2.69703i	−0.329643	+	0.0904555i
$890$		0			0
$891$	−16.1215	+	14.4884i	−0.540091	+	0.485379i
$892$		0			0
$893$			0.133651i			0.00447246i
$894$		0			0
$895$		−	6.61791i		−	0.221212i
$896$		0			0
$897$			16.0703i			0.536572i
$898$		0			0
$899$	−63.0735			−2.10362
$900$		0			0
$901$	6.99100			0.232904
$902$		0			0
$903$	−7.35934	−	26.8193i	−0.244903	−	0.892492i
$904$		0			0
$905$	−7.28247			−0.242077
$906$		0			0
$907$	−22.4657			−0.745962			−0.372981	−	0.927839i	$-0.621664\pi$
$907$	−22.4657			−0.745962			−0.372981	+	0.927839i	$0.621664\pi$
$908$		0			0
$909$	97.7396			3.24182
$910$		0			0
$911$	−18.7061			−0.619760			−0.309880	−	0.950776i	$-0.600289\pi$
$911$	−18.7061			−0.619760			−0.309880	+	0.950776i	$0.600289\pi$
$912$		0			0
$913$	9.13342	−	8.20820i	0.302272	−	0.271652i
$914$		0			0
$915$	−11.9608			−0.395413
$916$		0			0
$917$	1.37276	+	5.00270i	0.0453325	+	0.165204i
$918$		0			0
$919$		−	39.0850i		−	1.28930i	−0.764480	−	0.644648i	$-0.777004\pi$
$919$		−	39.0850i		−	1.28930i	0.764480	−	0.644648i	$-0.222996\pi$
$920$		0			0
$921$		−	4.70600i		−	0.155068i
$922$		0			0
$923$	−2.94553			−0.0969535
$924$		0			0
$925$	−1.83396			−0.0603002
$926$		0			0
$927$		−	75.2924i		−	2.47293i
$928$		0			0
$929$		−	30.5265i		−	1.00154i	−0.865580	−	0.500771i	$-0.833050\pi$
$929$		−	30.5265i		−	1.00154i	0.865580	−	0.500771i	$-0.166950\pi$
$930$		0			0
$931$	−12.1830	+	7.23058i	−0.399282	+	0.236973i
$932$		0			0
$933$	8.54064			0.279608
$934$		0			0
$935$	2.79916	+	3.11468i	0.0915425	+	0.101861i
$936$		0			0
$937$	40.2748			1.31572			0.657860	−	0.753140i	$-0.271462\pi$
$937$	40.2748			1.31572			0.657860	+	0.753140i	$0.271462\pi$
$938$		0			0
$939$	21.0896			0.688233
$940$		0			0
$941$	32.0883			1.04605			0.523024	−	0.852318i	$-0.324804\pi$
$941$	32.0883			1.04605			0.523024	+	0.852318i	$0.324804\pi$
$942$		0			0
$943$	−1.24247			−0.0404605
$944$		0			0
$945$	−8.27773	−	30.1662i	−0.269275	−	0.981306i
$946$		0			0
$947$	−31.1602			−1.01257			−0.506286	−	0.862366i	$-0.668982\pi$
$947$	−31.1602			−1.01257			−0.506286	+	0.862366i	$0.668982\pi$
$948$		0			0
$949$	−10.9592			−0.355752
$950$		0			0
$951$		−	44.3532i		−	1.43825i
$952$		0			0
$953$			33.3755i			1.08114i	0.841300	+	0.540569i	$0.181791\pi$
$953$			33.3755i			1.08114i	−0.841300	+	0.540569i	$0.818209\pi$
$954$		0			0
$955$		−	2.12244i		−	0.0686806i
$956$		0			0
$957$	52.4478	−	47.1348i	1.69540	−	1.52365i
$958$		0			0
$959$	−12.7684	−	46.5313i	−0.412313	−	1.50257i
$960$		0			0
$961$	−45.7176			−1.47476
$962$		0			0
$963$		−	17.5100i		−	0.564250i
$964$		0			0
$965$	−4.23619			−0.136368
$966$		0			0
$967$			59.6906i			1.91952i	0.280821	+	0.959760i	$0.409393\pi$
$967$			59.6906i			1.91952i	−0.280821	+	0.959760i	$0.590607\pi$
$968$		0			0
$969$			5.11960i			0.164465i
$970$		0			0
$971$			10.9185i			0.350391i	0.984534	+	0.175196i	$0.0560558\pi$
$971$			10.9185i			0.350391i	−0.984534	+	0.175196i	$0.943944\pi$
$972$		0			0
$973$	−0.580325	−	2.11485i	−0.0186044	−	0.0677991i
$974$		0			0
$975$		−	8.35016i		−	0.267419i
$976$		0			0
$977$	26.3796			0.843959			0.421979	−	0.906605i	$-0.361335\pi$
$977$	26.3796			0.843959			0.421979	+	0.906605i	$0.361335\pi$
$978$		0			0
$979$	−3.40441	−	3.78815i	−0.108805	−	0.121070i
$980$		0			0
$981$			107.826i			3.44263i
$982$		0			0
$983$		−	37.9014i		−	1.20887i	−0.796655	−	0.604434i	$-0.793399\pi$
$983$		−	37.9014i		−	1.20887i	0.796655	−	0.604434i	$-0.206601\pi$
$984$		0			0
$985$	−2.25947			−0.0719927
$986$		0			0
$987$	0.136507	+	0.497467i	0.00434506	+	0.0158345i
$988$		0			0
$989$			19.3779i			0.616182i
$990$		0			0
$991$	−45.0161			−1.42998			−0.714991	−	0.699134i	$-0.753569\pi$
$991$	−45.0161			−1.42998			−0.714991	+	0.699134i	$0.753569\pi$
$992$		0			0
$993$			6.93295i			0.220011i
$994$		0			0
$995$	7.52295			0.238494
$996$		0			0
$997$	17.4078			0.551311			0.275656	−	0.961256i	$-0.411105\pi$
$997$	17.4078			0.551311			0.275656	+	0.961256i	$0.411105\pi$
$998$		0			0
$999$		−	5.20244i		−	0.164598i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4004.2.e.a.3849.5	✓	48
7.6	odd	2		4004.2.e.b.3849.44	yes	48
11.10	odd	2		4004.2.e.b.3849.5	yes	48
77.76	even	2	inner	4004.2.e.a.3849.44	yes	48

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4004.2.e.a.3849.5	✓	48	1.1	even	1	trivial
4004.2.e.a.3849.44	yes	48	77.76	even	2	inner
4004.2.e.b.3849.5	yes	48	11.10	odd	2
4004.2.e.b.3849.44	yes	48	7.6	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:	\( N \)	\(=\)	\( 4004 = 2^{2} \cdot 7 \cdot 11 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4004.e (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-2.95250i q^{3} +1.47372i q^{5} +(0.700125 + 2.55144i) q^{7} -5.71727 q^{9} +O(q^{10})\)	\(q-2.95250i q^{3} +1.47372i q^{5} +(0.700125 + 2.55144i) q^{7} -5.71727 q^{9} +(-2.46682 + 2.21693i) q^{11} +1.00000 q^{13} +4.35115 q^{15} -0.856767 q^{17} +2.02387 q^{19} +(7.53312 - 2.06712i) q^{21} -5.44295 q^{23} +2.82816 q^{25} +8.02273i q^{27} -7.20111i q^{29} -8.75886i q^{31} +(6.54550 + 7.28330i) q^{33} +(-3.76009 + 1.03178i) q^{35} -0.648463 q^{37} -2.95250i q^{39} +0.228272 q^{41} -3.56019i q^{43} -8.42562i q^{45} +0.0660373i q^{47} +(-6.01965 + 3.57265i) q^{49} +2.52960i q^{51} -8.15975 q^{53} +(-3.26713 - 3.63539i) q^{55} -5.97549i q^{57} +8.61123i q^{59} -2.74889 q^{61} +(-4.00280 - 14.5872i) q^{63} +1.47372i q^{65} +2.46150 q^{67} +16.0703i q^{69} -2.94553 q^{71} -10.9592 q^{73} -8.35016i q^{75} +(-7.38344 - 4.74181i) q^{77} +11.5879i q^{79} +6.53533 q^{81} -3.70251 q^{83} -1.26263i q^{85} -21.2613 q^{87} +1.53564i q^{89} +(0.700125 + 2.55144i) q^{91} -25.8605 q^{93} +2.98261i q^{95} -1.74605i q^{97} +(14.1035 - 12.6748i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q - 4 q^{7} - 48 q^{9}+O(q^{10}) \) 48 * q - 4 * q^7 - 48 * q^9	\( 48 q - 4 q^{7} - 48 q^{9} + 2 q^{11} + 48 q^{13} + 8 q^{15} + 4 q^{17} + 10 q^{21} + 4 q^{23} - 44 q^{25} + 10 q^{33} - 14 q^{35} - 16 q^{37} + 10 q^{49} - 8 q^{53} - 12 q^{55} + 16 q^{61} + 16 q^{63} + 4 q^{67} - 16 q^{73} + 2 q^{77} + 64 q^{81} - 4 q^{83} - 12 q^{87} - 4 q^{91} + 36 q^{93} - 40 q^{99}+O(q^{100}) \) 48 * q - 4 * q^7 - 48 * q^9 + 2 * q^11 + 48 * q^13 + 8 * q^15 + 4 * q^17 + 10 * q^21 + 4 * q^23 - 44 * q^25 + 10 * q^33 - 14 * q^35 - 16 * q^37 + 10 * q^49 - 8 * q^53 - 12 * q^55 + 16 * q^61 + 16 * q^63 + 4 * q^67 - 16 * q^73 + 2 * q^77 + 64 * q^81 - 4 * q^83 - 12 * q^87 - 4 * q^91 + 36 * q^93 - 40 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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