LMFDB - Embedded newform 2070.2.e.b.1241.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2070,2,Mod(1241,2070)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2070, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("2070.1241");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2070 = 2 \cdot 3^{2} \cdot 5 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2070.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:	$16.5290332184$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} + 32x^{14} + 392x^{12} + 2324x^{10} + 6930x^{8} + 9856x^{6} + 5740x^{4} + 1108x^{2} + 1 $ x^16 + 32x^14 + 392x^12 + 2324x^10 + 6930x^8 + 9856x^6 + 5740x^4 + 1108*x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{23}]$
Coefficient ring index:	$ 2^{11} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1241.7
Root			$-2.59147i$ of defining polynomial
Character	$\chi$	$=$	2070.1241
Dual form			2070.2.e.b.1241.10

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q-1.00000i q^{2} -1.00000 q^{4} +1.00000 q^{5} +3.66489i q^{7} +1.00000i q^{8} -1.00000i q^{10} -4.11811 q^{11} +1.37333 q^{13} +3.66489 q^{14} +1.00000 q^{16} -5.07910 q^{17} -4.19067i q^{19} -1.00000 q^{20} +4.11811i q^{22} +(-3.46711 - 3.31348i) q^{23} +1.00000 q^{25} -1.37333i q^{26} -3.66489i q^{28} +8.77528i q^{29} -2.18748 q^{31} -1.00000i q^{32} +5.07910i q^{34} +3.66489i q^{35} -3.27275i q^{37} -4.19067 q^{38} +1.00000i q^{40} -7.34071i q^{41} -3.98306i q^{43} +4.11811 q^{44} +(-3.31348 + 3.46711i) q^{46} -13.3579i q^{47} -6.43143 q^{49} -1.00000i q^{50} -1.37333 q^{52} -9.16726 q^{53} -4.11811 q^{55} -3.66489 q^{56} +8.77528 q^{58} -7.31535i q^{59} +3.23953i q^{61} +2.18748i q^{62} -1.00000 q^{64} +1.37333 q^{65} -6.81459i q^{67} +5.07910 q^{68} +3.66489 q^{70} -10.4476i q^{71} -8.69614 q^{73} -3.27275 q^{74} +4.19067i q^{76} -15.0924i q^{77} +0.619189i q^{79} +1.00000 q^{80} -7.34071 q^{82} -7.82170 q^{83} -5.07910 q^{85} -3.98306 q^{86} -4.11811i q^{88} -0.592028 q^{89} +5.03309i q^{91} +(3.46711 + 3.31348i) q^{92} -13.3579 q^{94} -4.19067i q^{95} +11.4733i q^{97} +6.43143i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 16 q^{4} + 16 q^{5} - 24 q^{11} + 16 q^{16} - 16 q^{20} - 4 q^{23} + 16 q^{25} - 8 q^{31} - 8 q^{38} + 24 q^{44} - 4 q^{46} - 16 q^{49} + 8 q^{53} - 24 q^{55} + 16 q^{58} - 16 q^{64} + 32 q^{73}+ \cdots + 4 q^{92}+O(q^{100}) $ 16 * q - 16 * q^4 + 16 * q^5 - 24 * q^11 + 16 * q^16 - 16 * q^20 - 4 * q^23 + 16 * q^25 - 8 * q^31 - 8 * q^38 + 24 * q^44 - 4 * q^46 - 16 * q^49 + 8 * q^53 - 24 * q^55 + 16 * q^58 - 16 * q^64 + 32 * q^73 + 16 * q^80 + 32 * q^82 - 56 * q^83 - 8 * q^86 + 40 * q^89 + 4 * q^92

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2070\mathbb{Z}\right)^\times$.

$n$	$461$	$1657$	$1891$
$\chi(n)$	$-1$	$1$	$-1$

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 $

$2$		−	1.00000i		−	0.707107i
$2$		−	1.00000i		−	0.707107i
$3$		0			0
$3$		0			0
$4$	−1.00000			−0.500000
$5$	1.00000			0.447214
$5$	1.00000			0.447214
$6$		0			0
$7$			3.66489i			1.38520i	0.721322	+	0.692599i	$0.243535\pi$
$7$			3.66489i			1.38520i	−0.721322	+	0.692599i	$0.756465\pi$
$8$			1.00000i			0.353553i
$9$		0			0
$10$		−	1.00000i		−	0.316228i
$11$	−4.11811			−1.24166			−0.620829	−	0.783946i	$-0.713204\pi$
$11$	−4.11811			−1.24166			−0.620829	+	0.783946i	$0.713204\pi$
$12$		0			0
$13$	1.37333			0.380892			0.190446	−	0.981698i	$-0.439007\pi$
$13$	1.37333			0.380892			0.190446	+	0.981698i	$0.439007\pi$
$14$	3.66489			0.979483
$15$		0			0
$16$	1.00000			0.250000
$17$	−5.07910			−1.23186			−0.615932	−	0.787799i	$-0.711220\pi$
$17$	−5.07910			−1.23186			−0.615932	+	0.787799i	$0.711220\pi$
$18$		0			0
$19$		−	4.19067i		−	0.961405i	−0.876884	−	0.480702i	$-0.840382\pi$
$19$		−	4.19067i		−	0.961405i	0.876884	−	0.480702i	$-0.159618\pi$
$20$	−1.00000			−0.223607
$21$		0			0
$22$			4.11811i			0.877985i
$23$	−3.46711	−	3.31348i	−0.722943	−	0.690908i
$23$	−3.46711	−	3.31348i	−0.722943	−	0.690908i
$24$		0			0
$25$	1.00000			0.200000
$26$		−	1.37333i		−	0.269332i
$27$		0			0
$28$		−	3.66489i		−	0.692599i
$29$			8.77528i			1.62953i	0.579793	+	0.814764i	$0.303133\pi$
$29$			8.77528i			1.62953i	−0.579793	+	0.814764i	$0.696867\pi$
$30$		0			0
$31$	−2.18748			−0.392882			−0.196441	−	0.980516i	$-0.562938\pi$
$31$	−2.18748			−0.392882			−0.196441	+	0.980516i	$0.562938\pi$
$32$		−	1.00000i		−	0.176777i
$33$		0			0
$34$			5.07910i			0.871059i
$35$			3.66489i			0.619480i
$36$		0			0
$37$		−	3.27275i		−	0.538037i	−0.963135	−	0.269018i	$-0.913301\pi$
$37$		−	3.27275i		−	0.538037i	0.963135	−	0.269018i	$-0.0866992\pi$
$38$	−4.19067			−0.679816
$39$		0			0
$40$			1.00000i			0.158114i
$41$		−	7.34071i		−	1.14643i	−0.819406	−	0.573213i	$-0.805697\pi$
$41$		−	7.34071i		−	1.14643i	0.819406	−	0.573213i	$-0.194303\pi$
$42$		0			0
$43$		−	3.98306i		−	0.607412i	−0.952766	−	0.303706i	$-0.901776\pi$
$43$		−	3.98306i		−	0.607412i	0.952766	−	0.303706i	$-0.0982240\pi$
$44$	4.11811			0.620829
$45$		0			0
$46$	−3.31348	+	3.46711i	−0.488545	+	0.511198i
$47$		−	13.3579i		−	1.94845i	−0.225570	−	0.974227i	$-0.572424\pi$
$47$		−	13.3579i		−	1.94845i	0.225570	−	0.974227i	$-0.427576\pi$
$48$		0			0
$49$	−6.43143			−0.918776
$50$		−	1.00000i		−	0.141421i
$51$		0			0
$52$	−1.37333			−0.190446
$53$	−9.16726			−1.25922			−0.629610	−	0.776911i	$-0.716785\pi$
$53$	−9.16726			−1.25922			−0.629610	+	0.776911i	$0.716785\pi$
$54$		0			0
$55$	−4.11811			−0.555286
$56$	−3.66489			−0.489742
$57$		0			0
$58$	8.77528			1.15225
$59$		−	7.31535i		−	0.952377i	−0.879343	−	0.476189i	$-0.842018\pi$
$59$		−	7.31535i		−	0.952377i	0.879343	−	0.476189i	$-0.157982\pi$
$60$		0			0
$61$			3.23953i			0.414780i	0.978258	+	0.207390i	$0.0664969\pi$
$61$			3.23953i			0.414780i	−0.978258	+	0.207390i	$0.933503\pi$
$62$			2.18748i			0.277810i
$63$		0			0
$64$	−1.00000			−0.125000
$65$	1.37333			0.170340
$66$		0			0
$67$		−	6.81459i		−	0.832534i	−0.909242	−	0.416267i	$-0.863338\pi$
$67$		−	6.81459i		−	0.832534i	0.909242	−	0.416267i	$-0.136662\pi$
$68$	5.07910			0.615932
$69$		0			0
$70$	3.66489			0.438038
$71$		−	10.4476i		−	1.23990i	−0.784641	−	0.619950i	$-0.787153\pi$
$71$		−	10.4476i		−	1.23990i	0.784641	−	0.619950i	$-0.212847\pi$
$72$		0			0
$73$	−8.69614			−1.01781			−0.508903	−	0.860824i	$-0.669949\pi$
$73$	−8.69614			−1.01781			−0.508903	+	0.860824i	$0.669949\pi$
$74$	−3.27275			−0.380449
$75$		0			0
$76$			4.19067i			0.480702i
$77$		−	15.0924i		−	1.71994i
$78$		0			0
$79$			0.619189i			0.0696642i	0.999393	+	0.0348321i	$0.0110896\pi$
$79$			0.619189i			0.0696642i	−0.999393	+	0.0348321i	$0.988910\pi$
$80$	1.00000			0.111803
$81$		0			0
$82$	−7.34071			−0.810646
$83$	−7.82170			−0.858543			−0.429272	−	0.903175i	$-0.641230\pi$
$83$	−7.82170			−0.858543			−0.429272	+	0.903175i	$0.641230\pi$
$84$		0			0
$85$	−5.07910			−0.550906
$86$	−3.98306			−0.429505
$87$		0			0
$88$		−	4.11811i		−	0.438992i
$89$	−0.592028			−0.0627548			−0.0313774	−	0.999508i	$-0.509989\pi$
$89$	−0.592028			−0.0627548			−0.0313774	+	0.999508i	$0.509989\pi$
$90$		0			0
$91$			5.03309i			0.527612i
$92$	3.46711	+	3.31348i	0.361472	+	0.345454i
$93$		0			0
$94$	−13.3579			−1.37777
$95$		−	4.19067i		−	0.429953i
$96$		0			0
$97$			11.4733i			1.16493i	0.812854	+	0.582467i	$0.197913\pi$
$97$			11.4733i			1.16493i	−0.812854	+	0.582467i	$0.802087\pi$
$98$			6.43143i			0.649672i
$99$		0			0
$100$	−1.00000			−0.100000
$101$			9.81005i			0.976136i	0.872805	+	0.488068i	$0.162298\pi$
$101$			9.81005i			0.976136i	−0.872805	+	0.488068i	$0.837702\pi$
$102$		0			0
$103$			12.9748i			1.27845i	0.769020	+	0.639225i	$0.220745\pi$
$103$			12.9748i			1.27845i	−0.769020	+	0.639225i	$0.779255\pi$
$104$			1.37333i			0.134666i
$105$		0			0
$106$			9.16726i			0.890403i
$107$	17.2233			1.66504			0.832520	−	0.553994i	$-0.186897\pi$
$107$	17.2233			1.66504			0.832520	+	0.553994i	$0.186897\pi$
$108$		0			0
$109$			2.13239i			0.204246i	0.994772	+	0.102123i	$0.0325635\pi$
$109$			2.13239i			0.204246i	−0.994772	+	0.102123i	$0.967436\pi$
$110$			4.11811i			0.392647i
$111$		0			0
$112$			3.66489i			0.346300i
$113$	−10.6560			−1.00243			−0.501214	−	0.865323i	$-0.667113\pi$
$113$	−10.6560			−1.00243			−0.501214	+	0.865323i	$0.667113\pi$
$114$		0			0
$115$	−3.46711	−	3.31348i	−0.323310	−	0.308983i
$116$		−	8.77528i		−	0.814764i
$117$		0			0
$118$	−7.31535			−0.673432
$119$		−	18.6144i		−	1.70638i
$120$		0			0
$121$	5.95885			0.541714
$122$	3.23953			0.293294
$123$		0			0
$124$	2.18748			0.196441
$125$	1.00000			0.0894427
$126$		0			0
$127$	11.9442			1.05988			0.529938	−	0.848037i	$-0.322215\pi$
$127$	11.9442			1.05988			0.529938	+	0.848037i	$0.322215\pi$
$128$			1.00000i			0.0883883i
$129$		0			0
$130$		−	1.37333i		−	0.120449i
$131$			3.60510i			0.314979i	0.987521	+	0.157490i	$0.0503401\pi$
$131$			3.60510i			0.314979i	−0.987521	+	0.157490i	$0.949660\pi$
$132$		0			0
$133$	15.3583			1.33174
$134$	−6.81459			−0.588691
$135$		0			0
$136$		−	5.07910i		−	0.435530i
$137$	−15.5428			−1.32791			−0.663957	−	0.747771i	$-0.731124\pi$
$137$	−15.5428			−1.32791			−0.663957	+	0.747771i	$0.731124\pi$
$138$		0			0
$139$	−6.30232			−0.534556			−0.267278	−	0.963619i	$-0.586124\pi$
$139$	−6.30232			−0.534556			−0.267278	+	0.963619i	$0.586124\pi$
$140$		−	3.66489i		−	0.309740i
$141$		0			0
$142$	−10.4476			−0.876742
$143$	−5.65551			−0.472938
$144$		0			0
$145$			8.77528i			0.728747i
$146$			8.69614i			0.719698i
$147$		0			0
$148$			3.27275i			0.269018i
$149$	−6.79080			−0.556324			−0.278162	−	0.960534i	$-0.589725\pi$
$149$	−6.79080			−0.556324			−0.278162	+	0.960534i	$0.589725\pi$
$150$		0			0
$151$	19.1608			1.55929			0.779643	−	0.626224i	$-0.215401\pi$
$151$	19.1608			1.55929			0.779643	+	0.626224i	$0.215401\pi$
$152$	4.19067			0.339908
$153$		0			0
$154$	−15.0924			−1.21618
$155$	−2.18748			−0.175702
$156$		0			0
$157$			14.1815i			1.13181i	0.824471	+	0.565904i	$0.191473\pi$
$157$			14.1815i			1.13181i	−0.824471	+	0.565904i	$0.808527\pi$
$158$	0.619189			0.0492600
$159$		0			0
$160$		−	1.00000i		−	0.0790569i
$161$	12.1435	−	12.7066i	0.957044	−	1.00142i
$162$		0			0
$163$	−11.7476			−0.920140			−0.460070	−	0.887883i	$-0.652176\pi$
$163$	−11.7476			−0.920140			−0.460070	+	0.887883i	$0.652176\pi$
$164$			7.34071i			0.573213i
$165$		0			0
$166$			7.82170i			0.607082i
$167$			19.2009i			1.48581i	0.669396	+	0.742906i	$0.266553\pi$
$167$			19.2009i			1.48581i	−0.669396	+	0.742906i	$0.733447\pi$
$168$		0			0
$169$	−11.1140			−0.854921
$170$			5.07910i			0.389550i
$171$		0			0
$172$			3.98306i			0.303706i
$173$			3.67537i			0.279433i	0.990192	+	0.139717i	$0.0446191\pi$
$173$			3.67537i			0.279433i	−0.990192	+	0.139717i	$0.955381\pi$
$174$		0			0
$175$			3.66489i			0.277040i
$176$	−4.11811			−0.310414
$177$		0			0
$178$			0.592028i			0.0443744i
$179$		−	16.7414i		−	1.25131i	−0.780101	−	0.625654i	$-0.784832\pi$
$179$		−	16.7414i		−	1.25131i	0.780101	−	0.625654i	$-0.215168\pi$
$180$		0			0
$181$		−	1.04339i		−	0.0775547i	−0.999248	−	0.0387774i	$-0.987654\pi$
$181$		−	1.04339i		−	0.0775547i	0.999248	−	0.0387774i	$-0.0123463\pi$
$182$	5.03309			0.373078
$183$		0			0
$184$	3.31348	−	3.46711i	0.244273	−	0.255599i
$185$		−	3.27275i		−	0.240617i
$186$		0			0
$187$	20.9163			1.52955
$188$			13.3579i			0.974227i
$189$		0			0
$190$	−4.19067			−0.304023
$191$	8.60808			0.622859			0.311429	−	0.950269i	$-0.399192\pi$
$191$	8.60808			0.622859			0.311429	+	0.950269i	$0.399192\pi$
$192$		0			0
$193$	−24.8742			−1.79048			−0.895241	−	0.445581i	$-0.852997\pi$
$193$	−24.8742			−1.79048			−0.895241	+	0.445581i	$0.852997\pi$
$194$	11.4733			0.823733
$195$		0			0
$196$	6.43143			0.459388
$197$			23.0649i			1.64330i	0.569990	+	0.821651i	$0.306947\pi$
$197$			23.0649i			1.64330i	−0.569990	+	0.821651i	$0.693053\pi$
$198$		0			0
$199$			10.1323i			0.718259i	0.933288	+	0.359130i	$0.116926\pi$
$199$			10.1323i			0.718259i	−0.933288	+	0.359130i	$0.883074\pi$
$200$			1.00000i			0.0707107i
$201$		0			0
$202$	9.81005			0.690233
$203$	−32.1604			−2.25722
$204$		0			0
$205$		−	7.34071i		−	0.512698i
$206$	12.9748			0.904000
$207$		0			0
$208$	1.37333			0.0952231
$209$			17.2576i			1.19374i
$210$		0			0
$211$	−17.7381			−1.22114			−0.610571	−	0.791961i	$-0.709060\pi$
$211$	−17.7381			−1.22114			−0.610571	+	0.791961i	$0.709060\pi$
$212$	9.16726			0.629610
$213$		0			0
$214$		−	17.2233i		−	1.17736i
$215$		−	3.98306i		−	0.271643i
$216$		0			0
$217$		−	8.01686i		−	0.544220i
$218$	2.13239			0.144424
$219$		0			0
$220$	4.11811			0.277643
$221$	−6.97527			−0.469207
$222$		0			0
$223$	−5.53297			−0.370515			−0.185258	−	0.982690i	$-0.559312\pi$
$223$	−5.53297			−0.370515			−0.185258	+	0.982690i	$0.559312\pi$
$224$	3.66489			0.244871
$225$		0			0
$226$			10.6560i			0.708824i
$227$	−14.0021			−0.929351			−0.464676	−	0.885481i	$-0.653829\pi$
$227$	−14.0021			−0.929351			−0.464676	+	0.885481i	$0.653829\pi$
$228$		0			0
$229$		−	6.28684i		−	0.415446i	−0.978188	−	0.207723i	$-0.933395\pi$
$229$		−	6.28684i		−	0.415446i	0.978188	−	0.207723i	$-0.0666053\pi$
$230$	−3.31348	+	3.46711i	−0.218484	+	0.228615i
$231$		0			0
$232$	−8.77528			−0.576125
$233$			17.7054i			1.15992i	0.814646	+	0.579959i	$0.196931\pi$
$233$			17.7054i			1.15992i	−0.814646	+	0.579959i	$0.803069\pi$
$234$		0			0
$235$		−	13.3579i		−	0.871375i
$236$			7.31535i			0.476189i
$237$		0			0
$238$	−18.6144			−1.20659
$239$		−	7.37705i		−	0.477182i	−0.971120	−	0.238591i	$-0.923314\pi$
$239$		−	7.37705i		−	0.477182i	0.971120	−	0.238591i	$-0.0766855\pi$
$240$		0			0
$241$		−	18.9364i		−	1.21980i	−0.792478	−	0.609901i	$-0.791209\pi$
$241$		−	18.9364i		−	1.21980i	0.792478	−	0.609901i	$-0.208791\pi$
$242$		−	5.95885i		−	0.383050i
$243$		0			0
$244$		−	3.23953i		−	0.207390i
$245$	−6.43143			−0.410889
$246$		0			0
$247$		−	5.75516i		−	0.366192i
$248$		−	2.18748i		−	0.138905i
$249$		0			0
$250$		−	1.00000i		−	0.0632456i
$251$	18.9716			1.19748			0.598739	−	0.800944i	$-0.295669\pi$
$251$	18.9716			1.19748			0.598739	+	0.800944i	$0.295669\pi$
$252$		0			0
$253$	14.2780	+	13.6453i	0.897648	+	0.857871i
$254$		−	11.9442i		−	0.749445i
$255$		0			0
$256$	1.00000			0.0625000
$257$		−	20.1868i		−	1.25922i	−0.776913	−	0.629608i	$-0.783216\pi$
$257$		−	20.1868i		−	1.25922i	0.776913	−	0.629608i	$-0.216784\pi$
$258$		0			0
$259$	11.9943			0.745288
$260$	−1.37333			−0.0851701
$261$		0			0
$262$	3.60510			0.222724
$263$	10.9944			0.677942			0.338971	−	0.940797i	$-0.389921\pi$
$263$	10.9944			0.677942			0.338971	+	0.940797i	$0.389921\pi$
$264$		0			0
$265$	−9.16726			−0.563140
$266$		−	15.3583i		−	0.941680i
$267$		0			0
$268$			6.81459i			0.416267i
$269$			14.9281i			0.910182i	0.890445	+	0.455091i	$0.150393\pi$
$269$			14.9281i			0.910182i	−0.890445	+	0.455091i	$0.849607\pi$
$270$		0			0
$271$	13.5649			0.824011			0.412006	−	0.911181i	$-0.364828\pi$
$271$	13.5649			0.824011			0.412006	+	0.911181i	$0.364828\pi$
$272$	−5.07910			−0.307966
$273$		0			0
$274$			15.5428i			0.938977i
$275$	−4.11811			−0.248332
$276$		0			0
$277$	−2.59763			−0.156077			−0.0780383	−	0.996950i	$-0.524866\pi$
$277$	−2.59763			−0.156077			−0.0780383	+	0.996950i	$0.524866\pi$
$278$			6.30232i			0.377988i
$279$		0			0
$280$	−3.66489			−0.219019
$281$	26.8084			1.59925			0.799627	−	0.600496i	$-0.205030\pi$
$281$	26.8084			1.59925			0.799627	+	0.600496i	$0.205030\pi$
$282$		0			0
$283$		−	18.9945i		−	1.12911i	−0.825397	−	0.564553i	$-0.809049\pi$
$283$		−	18.9945i		−	1.12911i	0.825397	−	0.564553i	$-0.190951\pi$
$284$			10.4476i			0.619950i
$285$		0			0
$286$			5.65551i			0.334418i
$287$	26.9029			1.58803
$288$		0			0
$289$	8.79731			0.517489
$290$	8.77528			0.515302
$291$		0			0
$292$	8.69614			0.508903
$293$	−11.3576			−0.663520			−0.331760	−	0.943364i	$-0.607642\pi$
$293$	−11.3576			−0.663520			−0.331760	+	0.943364i	$0.607642\pi$
$294$		0			0
$295$		−	7.31535i		−	0.425916i
$296$	3.27275			0.190225
$297$		0			0
$298$			6.79080i			0.393380i
$299$	−4.76148	−	4.55049i	−0.275363	−	0.263161i
$300$		0			0
$301$	14.5975			0.841386
$302$		−	19.1608i		−	1.10258i
$303$		0			0
$304$		−	4.19067i		−	0.240351i
$305$			3.23953i			0.185495i
$306$		0			0
$307$	13.9099			0.793878			0.396939	−	0.917845i	$-0.370073\pi$
$307$	13.9099			0.793878			0.396939	+	0.917845i	$0.370073\pi$
$308$			15.0924i			0.859971i
$309$		0			0
$310$			2.18748i			0.124240i
$311$			0.0885379i			0.00502053i	0.999997	+	0.00251026i	$0.000799042\pi$
$311$			0.0885379i			0.00502053i	−0.999997	+	0.00251026i	$0.999201\pi$
$312$		0			0
$313$		−	9.11501i		−	0.515211i	−0.966250	−	0.257605i	$-0.917067\pi$
$313$		−	9.11501i		−	0.515211i	0.966250	−	0.257605i	$-0.0829334\pi$
$314$	14.1815			0.800309
$315$		0			0
$316$		−	0.619189i		−	0.0348321i
$317$		−	29.1559i		−	1.63756i	−0.574110	−	0.818778i	$-0.694652\pi$
$317$		−	29.1559i		−	1.63756i	0.574110	−	0.818778i	$-0.305348\pi$
$318$		0			0
$319$		−	36.1376i		−	2.02332i
$320$	−1.00000			−0.0559017
$321$		0			0
$322$	−12.7066	−	12.1435i	−0.708111	−	0.676733i
$323$			21.2848i			1.18432i
$324$		0			0
$325$	1.37333			0.0761785
$326$			11.7476i			0.650637i
$327$		0			0
$328$	7.34071			0.405323
$329$	48.9554			2.69900
$330$		0			0
$331$	−18.5024			−1.01699			−0.508493	−	0.861066i	$-0.669797\pi$
$331$	−18.5024			−1.01699			−0.508493	+	0.861066i	$0.669797\pi$
$332$	7.82170			0.429272
$333$		0			0
$334$	19.2009			1.05063
$335$		−	6.81459i		−	0.372321i
$336$		0			0
$337$			6.18235i			0.336774i	0.985721	+	0.168387i	$0.0538559\pi$
$337$			6.18235i			0.336774i	−0.985721	+	0.168387i	$0.946144\pi$
$338$			11.1140i			0.604520i
$339$		0			0
$340$	5.07910			0.275453
$341$	9.00827			0.487825
$342$		0			0
$343$			2.08375i			0.112512i
$344$	3.98306			0.214752
$345$		0			0
$346$	3.67537			0.197589
$347$			9.17696i			0.492645i	0.969188	+	0.246323i	$0.0792223\pi$
$347$			9.17696i			0.492645i	−0.969188	+	0.246323i	$0.920778\pi$
$348$		0			0
$349$	−13.7288			−0.734885			−0.367442	−	0.930046i	$-0.619766\pi$
$349$	−13.7288			−0.734885			−0.367442	+	0.930046i	$0.619766\pi$
$350$	3.66489			0.195897
$351$		0			0
$352$			4.11811i			0.219496i
$353$			23.6643i			1.25952i	0.776788	+	0.629762i	$0.216848\pi$
$353$			23.6643i			1.25952i	−0.776788	+	0.629762i	$0.783152\pi$
$354$		0			0
$355$		−	10.4476i		−	0.554500i
$356$	0.592028			0.0313774
$357$		0			0
$358$	−16.7414			−0.884809
$359$	−25.0529			−1.32224			−0.661122	−	0.750279i	$-0.729919\pi$
$359$	−25.0529			−1.32224			−0.661122	+	0.750279i	$0.729919\pi$
$360$		0			0
$361$	1.43831			0.0757004
$362$	−1.04339			−0.0548395
$363$		0			0
$364$		−	5.03309i		−	0.263806i
$365$	−8.69614			−0.455177
$366$		0			0
$367$			30.4227i			1.58805i	0.607883	+	0.794026i	$0.292019\pi$
$367$			30.4227i			1.58805i	−0.607883	+	0.794026i	$0.707981\pi$
$368$	−3.46711	−	3.31348i	−0.180736	−	0.172727i
$369$		0			0
$370$	−3.27275			−0.170142
$371$		−	33.5970i		−	1.74427i
$372$		0			0
$373$			10.9564i			0.567301i	0.958928	+	0.283651i	$0.0915456\pi$
$373$			10.9564i			0.567301i	−0.958928	+	0.283651i	$0.908454\pi$
$374$		−	20.9163i		−	1.08156i
$375$		0			0
$376$	13.3579			0.688883
$377$			12.0513i			0.620675i
$378$		0			0
$379$		−	29.2567i		−	1.50282i	−0.659837	−	0.751409i	$-0.729375\pi$
$379$		−	29.2567i		−	1.50282i	0.659837	−	0.751409i	$-0.270625\pi$
$380$			4.19067i			0.214977i
$381$		0			0
$382$		−	8.60808i		−	0.440428i
$383$	−14.1993			−0.725552			−0.362776	−	0.931876i	$-0.618171\pi$
$383$	−14.1993			−0.725552			−0.362776	+	0.931876i	$0.618171\pi$
$384$		0			0
$385$		−	15.0924i		−	0.769182i
$386$			24.8742i			1.26606i
$387$		0			0
$388$		−	11.4733i		−	0.582467i
$389$	16.1749			0.820101			0.410050	−	0.912063i	$-0.365511\pi$
$389$	16.1749			0.820101			0.410050	+	0.912063i	$0.365511\pi$
$390$		0			0
$391$	17.6098	+	16.8295i	0.890567	+	0.851104i
$392$		−	6.43143i		−	0.324836i
$393$		0			0
$394$	23.0649			1.16199
$395$			0.619189i			0.0311548i
$396$		0			0
$397$	−5.37770			−0.269899			−0.134950	−	0.990852i	$-0.543087\pi$
$397$	−5.37770			−0.269899			−0.134950	+	0.990852i	$0.543087\pi$
$398$	10.1323			0.507886
$399$		0			0
$400$	1.00000			0.0500000
$401$	18.3723			0.917470			0.458735	−	0.888573i	$-0.348303\pi$
$401$	18.3723			0.917470			0.458735	+	0.888573i	$0.348303\pi$
$402$		0			0
$403$	−3.00412			−0.149646
$404$		−	9.81005i		−	0.488068i
$405$		0			0
$406$			32.1604i			1.59610i
$407$			13.4776i			0.668058i
$408$		0			0
$409$	30.0999			1.48835			0.744173	−	0.667987i	$-0.232844\pi$
$409$	30.0999			1.48835			0.744173	+	0.667987i	$0.232844\pi$
$410$	−7.34071			−0.362532
$411$		0			0
$412$		−	12.9748i		−	0.639225i
$413$	26.8100			1.31923
$414$		0			0
$415$	−7.82170			−0.383952
$416$		−	1.37333i		−	0.0673329i
$417$		0			0
$418$	17.2576			0.844099
$419$	−36.0650			−1.76189			−0.880945	−	0.473219i	$-0.843092\pi$
$419$	−36.0650			−1.76189			−0.880945	+	0.473219i	$0.843092\pi$
$420$		0			0
$421$			5.92958i			0.288990i	0.989506	+	0.144495i	$0.0461558\pi$
$421$			5.92958i			0.288990i	−0.989506	+	0.144495i	$0.953844\pi$
$422$			17.7381i			0.863478i
$423$		0			0
$424$		−	9.16726i		−	0.445202i
$425$	−5.07910			−0.246373
$426$		0			0
$427$	−11.8725			−0.574553
$428$	−17.2233			−0.832520
$429$		0			0
$430$	−3.98306			−0.192080
$431$	−31.2612			−1.50580			−0.752900	−	0.658135i	$-0.771346\pi$
$431$	−31.2612			−1.50580			−0.752900	+	0.658135i	$0.771346\pi$
$432$		0			0
$433$		−	19.8707i		−	0.954924i	−0.878652	−	0.477462i	$-0.841557\pi$
$433$		−	19.8707i		−	0.954924i	0.878652	−	0.477462i	$-0.158443\pi$
$434$	−8.01686			−0.384822
$435$		0			0
$436$		−	2.13239i		−	0.102123i
$437$	−13.8857	+	14.5295i	−0.664242	+	0.695041i
$438$		0			0
$439$	15.0653			0.719029			0.359515	−	0.933139i	$-0.382942\pi$
$439$	15.0653			0.719029			0.359515	+	0.933139i	$0.382942\pi$
$440$		−	4.11811i		−	0.196323i
$441$		0			0
$442$			6.97527i			0.331780i
$443$			38.3156i			1.82043i	0.414137	+	0.910214i	$0.364083\pi$
$443$			38.3156i			1.82043i	−0.414137	+	0.910214i	$0.635917\pi$
$444$		0			0
$445$	−0.592028			−0.0280648
$446$			5.53297i			0.261994i
$447$		0			0
$448$		−	3.66489i		−	0.173150i
$449$		−	13.0185i		−	0.614380i	−0.951648	−	0.307190i	$-0.900611\pi$
$449$		−	13.0185i		−	0.614380i	0.951648	−	0.307190i	$-0.0993886\pi$
$450$		0			0
$451$			30.2299i			1.42347i
$452$	10.6560			0.501214
$453$		0			0
$454$			14.0021i			0.657150i
$455$			5.03309i			0.235955i
$456$		0			0
$457$		−	15.4017i		−	0.720461i	−0.932863	−	0.360230i	$-0.882698\pi$
$457$		−	15.4017i		−	0.720461i	0.932863	−	0.360230i	$-0.117302\pi$
$458$	−6.28684			−0.293765
$459$		0			0
$460$	3.46711	+	3.31348i	0.161655	+	0.154492i
$461$		−	19.9043i		−	0.927035i	−0.886088	−	0.463517i	$-0.846587\pi$
$461$		−	19.9043i		−	0.927035i	0.886088	−	0.463517i	$-0.153413\pi$
$462$		0			0
$463$	25.1321			1.16799			0.583994	−	0.811758i	$-0.301489\pi$
$463$	25.1321			1.16799			0.583994	+	0.811758i	$0.301489\pi$
$464$			8.77528i			0.407382i
$465$		0			0
$466$	17.7054			0.820186
$467$	−26.1235			−1.20885			−0.604425	−	0.796662i	$-0.706597\pi$
$467$	−26.1235			−1.20885			−0.604425	+	0.796662i	$0.706597\pi$
$468$		0			0
$469$	24.9747			1.15323
$470$	−13.3579			−0.616155
$471$		0			0
$472$	7.31535			0.336716
$473$			16.4027i			0.754197i
$474$		0			0
$475$		−	4.19067i		−	0.192281i
$476$			18.6144i			0.853188i
$477$		0			0
$478$	−7.37705			−0.337419
$479$	−23.1310			−1.05688			−0.528441	−	0.848970i	$-0.677223\pi$
$479$	−23.1310			−1.05688			−0.528441	+	0.848970i	$0.677223\pi$
$480$		0			0
$481$		−	4.49455i		−	0.204934i
$482$	−18.9364			−0.862530
$483$		0			0
$484$	−5.95885			−0.270857
$485$			11.4733i			0.520975i
$486$		0			0
$487$	39.2504			1.77860			0.889302	−	0.457321i	$-0.151191\pi$
$487$	39.2504			1.77860			0.889302	+	0.457321i	$0.151191\pi$
$488$	−3.23953			−0.146647
$489$		0			0
$490$			6.43143i			0.290542i
$491$		−	16.7695i		−	0.756797i	−0.925643	−	0.378398i	$-0.876475\pi$
$491$		−	16.7695i		−	0.756797i	0.925643	−	0.378398i	$-0.123525\pi$
$492$		0			0
$493$		−	44.5705i		−	2.00736i
$494$	−5.75516			−0.258937
$495$		0			0
$496$	−2.18748			−0.0982205
$497$	38.2893			1.71751
$498$		0			0
$499$	−9.29413			−0.416063			−0.208031	−	0.978122i	$-0.566706\pi$
$499$	−9.29413			−0.416063			−0.208031	+	0.978122i	$0.566706\pi$
$500$	−1.00000			−0.0447214
$501$		0			0
$502$		−	18.9716i		−	0.846745i
$503$	1.26001			0.0561809			0.0280904	−	0.999605i	$-0.491057\pi$
$503$	1.26001			0.0561809			0.0280904	+	0.999605i	$0.491057\pi$
$504$		0			0
$505$			9.81005i			0.436541i
$506$	13.6453	−	14.2780i	0.606606	−	0.634733i
$507$		0			0
$508$	−11.9442			−0.529938
$509$		−	35.6885i		−	1.58187i	−0.611902	−	0.790934i	$-0.709595\pi$
$509$		−	35.6885i		−	1.58187i	0.611902	−	0.790934i	$-0.290405\pi$
$510$		0			0
$511$		−	31.8704i		−	1.40986i
$512$		−	1.00000i		−	0.0441942i
$513$		0			0
$514$	−20.1868			−0.890400
$515$			12.9748i			0.571740i
$516$		0			0
$517$			55.0095i			2.41931i
$518$		−	11.9943i		−	0.526998i
$519$		0			0
$520$			1.37333i			0.0602244i
$521$	−11.7144			−0.513216			−0.256608	−	0.966516i	$-0.582605\pi$
$521$	−11.7144			−0.513216			−0.256608	+	0.966516i	$0.582605\pi$
$522$		0			0
$523$		−	26.4487i		−	1.15652i	−0.815853	−	0.578260i	$-0.803732\pi$
$523$		−	26.4487i		−	1.15652i	0.815853	−	0.578260i	$-0.196268\pi$
$524$		−	3.60510i		−	0.157490i
$525$		0			0
$526$		−	10.9944i		−	0.479377i
$527$	11.1104			0.483977
$528$		0			0
$529$	1.04175	+	22.9764i	0.0452934	+	0.998974i
$530$			9.16726i			0.398200i
$531$		0			0
$532$	−15.3583			−0.665868
$533$		−	10.0812i		−	0.436665i
$534$		0			0
$535$	17.2233			0.744629
$536$	6.81459			0.294345
$537$		0			0
$538$	14.9281			0.643596
$539$	26.4854			1.14080
$540$		0			0
$541$	−38.8791			−1.67154			−0.835772	−	0.549077i	$-0.814979\pi$
$541$	−38.8791			−1.67154			−0.835772	+	0.549077i	$0.814979\pi$
$542$		−	13.5649i		−	0.582664i
$543$		0			0
$544$			5.07910i			0.217765i
$545$			2.13239i			0.0913416i
$546$		0			0
$547$	−6.97599			−0.298272			−0.149136	−	0.988817i	$-0.547649\pi$
$547$	−6.97599			−0.298272			−0.149136	+	0.988817i	$0.547649\pi$
$548$	15.5428			0.663957
$549$		0			0
$550$			4.11811i			0.175597i
$551$	36.7743			1.56664
$552$		0			0
$553$	−2.26926			−0.0964988
$554$			2.59763i			0.110363i
$555$		0			0
$556$	6.30232			0.267278
$557$	38.0409			1.61185			0.805923	−	0.592020i	$-0.201670\pi$
$557$	38.0409			1.61185			0.805923	+	0.592020i	$0.201670\pi$
$558$		0			0
$559$		−	5.47005i		−	0.231358i
$560$			3.66489i			0.154870i
$561$		0			0
$562$		−	26.8084i		−	1.13084i
$563$	−19.8362			−0.835998			−0.417999	−	0.908447i	$-0.637268\pi$
$563$	−19.8362			−0.835998			−0.417999	+	0.908447i	$0.637268\pi$
$564$		0			0
$565$	−10.6560			−0.448300
$566$	−18.9945			−0.798398
$567$		0			0
$568$	10.4476			0.438371
$569$	39.1504			1.64127			0.820634	−	0.571454i	$-0.193620\pi$
$569$	39.1504			1.64127			0.820634	+	0.571454i	$0.193620\pi$
$570$		0			0
$571$		−	22.3013i		−	0.933280i	−0.884447	−	0.466640i	$-0.845464\pi$
$571$		−	22.3013i		−	0.933280i	0.884447	−	0.466640i	$-0.154536\pi$
$572$	5.65551			0.236469
$573$		0			0
$574$		−	26.9029i		−	1.12291i
$575$	−3.46711	−	3.31348i	−0.144589	−	0.138182i
$576$		0			0
$577$	9.84297			0.409768			0.204884	−	0.978786i	$-0.434318\pi$
$577$	9.84297			0.409768			0.204884	+	0.978786i	$0.434318\pi$
$578$		−	8.79731i		−	0.365920i
$579$		0			0
$580$		−	8.77528i		−	0.364374i
$581$		−	28.6657i		−	1.18925i
$582$		0			0
$583$	37.7518			1.56352
$584$		−	8.69614i		−	0.359849i
$585$		0			0
$586$			11.3576i			0.469179i
$587$		−	26.7112i		−	1.10249i	−0.834344	−	0.551245i	$-0.814153\pi$
$587$		−	26.7112i		−	1.10249i	0.834344	−	0.551245i	$-0.185847\pi$
$588$		0			0
$589$			9.16698i			0.377719i
$590$	−7.31535			−0.301168
$591$		0			0
$592$		−	3.27275i		−	0.134509i
$593$			9.09562i			0.373512i	0.982406	+	0.186756i	$0.0597974\pi$
$593$			9.09562i			0.373512i	−0.982406	+	0.186756i	$0.940203\pi$
$594$		0			0
$595$		−	18.6144i		−	0.763115i
$596$	6.79080			0.278162
$597$		0			0
$598$	−4.55049	+	4.76148i	−0.186083	+	0.194711i
$599$			3.72692i			0.152278i	0.997097	+	0.0761388i	$0.0242592\pi$
$599$			3.72692i			0.152278i	−0.997097	+	0.0761388i	$0.975741\pi$
$600$		0			0
$601$	32.7678			1.33663			0.668314	−	0.743880i	$-0.267016\pi$
$601$	32.7678			1.33663			0.668314	+	0.743880i	$0.267016\pi$
$602$		−	14.5975i		−	0.594950i
$603$		0			0
$604$	−19.1608			−0.779643
$605$	5.95885			0.242262
$606$		0			0
$607$	−9.38225			−0.380814			−0.190407	−	0.981705i	$-0.560981\pi$
$607$	−9.38225			−0.380814			−0.190407	+	0.981705i	$0.560981\pi$
$608$	−4.19067			−0.169954
$609$		0			0
$610$	3.23953			0.131165
$611$		−	18.3448i		−	0.742151i
$612$		0			0
$613$		−	9.98992i		−	0.403489i	−0.979438	−	0.201744i	$-0.935339\pi$
$613$		−	9.98992i		−	0.403489i	0.979438	−	0.201744i	$-0.0646610\pi$
$614$		−	13.9099i		−	0.561356i
$615$		0			0
$616$	15.0924			0.608092
$617$	−37.8743			−1.52476			−0.762380	−	0.647129i	$-0.775969\pi$
$617$	−37.8743			−1.52476			−0.762380	+	0.647129i	$0.775969\pi$
$618$		0			0
$619$			22.4368i			0.901811i	0.892572	+	0.450905i	$0.148899\pi$
$619$			22.4368i			0.901811i	−0.892572	+	0.450905i	$0.851101\pi$
$620$	2.18748			0.0878511
$621$		0			0
$622$	0.0885379			0.00355005
$623$		−	2.16972i		−	0.0869279i
$624$		0			0
$625$	1.00000			0.0400000
$626$	−9.11501			−0.364309
$627$		0			0
$628$		−	14.1815i		−	0.565904i
$629$			16.6226i			0.662788i
$630$		0			0
$631$		−	3.71783i		−	0.148004i	−0.997258	−	0.0740022i	$-0.976423\pi$
$631$		−	3.71783i		−	0.148004i	0.997258	−	0.0740022i	$-0.0235772\pi$
$632$	−0.619189			−0.0246300
$633$		0			0
$634$	−29.1559			−1.15793
$635$	11.9442			0.473991
$636$		0			0
$637$	−8.83245			−0.349955
$638$	−36.1376			−1.43070
$639$		0			0
$640$			1.00000i			0.0395285i
$641$	−17.9718			−0.709845			−0.354922	−	0.934896i	$-0.615493\pi$
$641$	−17.9718			−0.709845			−0.354922	+	0.934896i	$0.615493\pi$
$642$		0			0
$643$			11.9598i			0.471650i	0.971796	+	0.235825i	$0.0757792\pi$
$643$			11.9598i			0.471650i	−0.971796	+	0.235825i	$0.924221\pi$
$644$	−12.1435	+	12.7066i	−0.478522	+	0.500710i
$645$		0			0
$646$	21.2848			0.837441
$647$			33.6779i			1.32401i	0.749498	+	0.662007i	$0.230295\pi$
$647$			33.6779i			1.32401i	−0.749498	+	0.662007i	$0.769705\pi$
$648$		0			0
$649$			30.1254i			1.18253i
$650$		−	1.37333i		−	0.0538663i
$651$		0			0
$652$	11.7476			0.460070
$653$		−	8.47244i		−	0.331552i	−0.986163	−	0.165776i	$-0.946987\pi$
$653$		−	8.47244i		−	0.331552i	0.986163	−	0.165776i	$-0.0530129\pi$
$654$		0			0
$655$			3.60510i			0.140863i
$656$		−	7.34071i		−	0.286607i
$657$		0			0
$658$		−	48.9554i		−	1.90848i
$659$	23.1047			0.900032			0.450016	−	0.893021i	$-0.351418\pi$
$659$	23.1047			0.900032			0.450016	+	0.893021i	$0.351418\pi$
$660$		0			0
$661$		−	6.19923i		−	0.241122i	−0.992706	−	0.120561i	$-0.961531\pi$
$661$		−	6.19923i		−	0.241122i	0.992706	−	0.120561i	$-0.0384693\pi$
$662$			18.5024i			0.719118i
$663$		0			0
$664$		−	7.82170i		−	0.303541i
$665$	15.3583			0.595571
$666$		0			0
$667$	29.0767	−	30.4249i	1.12585	−	1.17806i
$668$		−	19.2009i		−	0.742906i
$669$		0			0
$670$	−6.81459			−0.263270
$671$		−	13.3408i		−	0.515015i
$672$		0			0
$673$	−28.9418			−1.11563			−0.557813	−	0.829967i	$-0.688359\pi$
$673$	−28.9418			−1.11563			−0.557813	+	0.829967i	$0.688359\pi$
$674$	6.18235			0.238135
$675$		0			0
$676$	11.1140			0.427461
$677$	−17.8256			−0.685095			−0.342547	−	0.939501i	$-0.611290\pi$
$677$	−17.8256			−0.685095			−0.342547	+	0.939501i	$0.611290\pi$
$678$		0			0
$679$	−42.0483			−1.61367
$680$		−	5.07910i		−	0.194775i
$681$		0			0
$682$		−	9.00827i		−	0.344944i
$683$			11.3520i			0.434370i	0.976130	+	0.217185i	$0.0696875\pi$
$683$			11.3520i			0.434370i	−0.976130	+	0.217185i	$0.930312\pi$
$684$		0			0
$685$	−15.5428			−0.593861
$686$	2.08375			0.0795580
$687$		0			0
$688$		−	3.98306i		−	0.151853i
$689$	−12.5896			−0.479627
$690$		0			0
$691$	40.3088			1.53342			0.766709	−	0.641995i	$-0.221893\pi$
$691$	40.3088			1.53342			0.766709	+	0.641995i	$0.221893\pi$
$692$		−	3.67537i		−	0.139717i
$693$		0			0
$694$	9.17696			0.348353
$695$	−6.30232			−0.239061
$696$		0			0
$697$			37.2842i			1.41224i
$698$			13.7288i			0.519642i
$699$		0			0
$700$		−	3.66489i		−	0.138520i
$701$	44.7271			1.68932			0.844660	−	0.535303i	$-0.179803\pi$
$701$	44.7271			1.68932			0.844660	+	0.535303i	$0.179803\pi$
$702$		0			0
$703$	−13.7150			−0.517271
$704$	4.11811			0.155207
$705$		0			0
$706$	23.6643			0.890618
$707$	−35.9528			−1.35214
$708$		0			0
$709$			1.75818i			0.0660297i	0.999455	+	0.0330148i	$0.0105109\pi$
$709$			1.75818i			0.0660297i	−0.999455	+	0.0330148i	$0.989489\pi$
$710$	−10.4476			−0.392091
$711$		0			0
$712$		−	0.592028i		−	0.0221872i
$713$	7.58422	+	7.24815i	0.284031	+	0.271445i
$714$		0			0
$715$	−5.65551			−0.211504
$716$			16.7414i			0.625654i
$717$		0			0
$718$			25.0529i			0.934967i
$719$			13.4033i			0.499860i	0.968264	+	0.249930i	$0.0804076\pi$
$719$			13.4033i			0.499860i	−0.968264	+	0.249930i	$0.919592\pi$
$720$		0			0
$721$	−47.5514			−1.77091
$722$		−	1.43831i		−	0.0535283i
$723$		0			0
$724$			1.04339i			0.0387774i
$725$			8.77528i			0.325906i
$726$		0			0
$727$		−	48.0291i		−	1.78130i	−0.454687	−	0.890651i	$-0.650249\pi$
$727$		−	48.0291i		−	1.78130i	0.454687	−	0.890651i	$-0.349751\pi$
$728$	−5.03309			−0.186539
$729$		0			0
$730$			8.69614i			0.321859i
$731$			20.2304i			0.748248i
$732$		0			0
$733$		−	13.9287i		−	0.514466i	−0.966349	−	0.257233i	$-0.917189\pi$
$733$		−	13.9287i		−	0.514466i	0.966349	−	0.257233i	$-0.0828109\pi$
$734$	30.4227			1.12292
$735$		0			0
$736$	−3.31348	+	3.46711i	−0.122136	+	0.127799i
$737$			28.0632i			1.03372i
$738$		0			0
$739$	−6.85157			−0.252039			−0.126020	−	0.992028i	$-0.540220\pi$
$739$	−6.85157			−0.252039			−0.126020	+	0.992028i	$0.540220\pi$
$740$			3.27275i			0.120309i
$741$		0			0
$742$	−33.5970			−1.23339
$743$	−21.2451			−0.779409			−0.389704	−	0.920940i	$-0.627423\pi$
$743$	−21.2451			−0.779409			−0.389704	+	0.920940i	$0.627423\pi$
$744$		0			0
$745$	−6.79080			−0.248796
$746$	10.9564			0.401142
$747$		0			0
$748$	−20.9163			−0.764777
$749$			63.1216i			2.30641i
$750$		0			0
$751$		−	8.14617i		−	0.297258i	−0.988893	−	0.148629i	$-0.952514\pi$
$751$		−	8.14617i		−	0.297258i	0.988893	−	0.148629i	$-0.0474860\pi$
$752$		−	13.3579i		−	0.487113i
$753$		0			0
$754$	12.0513			0.438883
$755$	19.1608			0.697334
$756$		0			0
$757$		−	22.2646i		−	0.809221i	−0.914489	−	0.404610i	$-0.867407\pi$
$757$		−	22.2646i		−	0.809221i	0.914489	−	0.404610i	$-0.132593\pi$
$758$	−29.2567			−1.06265
$759$		0			0
$760$	4.19067			0.152011
$761$			2.46950i			0.0895193i	0.998998	+	0.0447597i	$0.0142522\pi$
$761$			2.46950i			0.0895193i	−0.998998	+	0.0447597i	$0.985748\pi$
$762$		0			0
$763$	−7.81498			−0.282921
$764$	−8.60808			−0.311429
$765$		0			0
$766$			14.1993i			0.513043i
$767$		−	10.0464i		−	0.362753i
$768$		0			0
$769$			9.27287i			0.334388i	0.985924	+	0.167194i	$0.0534707\pi$
$769$			9.27287i			0.334388i	−0.985924	+	0.167194i	$0.946529\pi$
$770$	−15.0924			−0.543894
$771$		0			0
$772$	24.8742			0.895241
$773$	−37.6990			−1.35594			−0.677969	−	0.735090i	$-0.737140\pi$
$773$	−37.6990			−1.35594			−0.677969	+	0.735090i	$0.737140\pi$
$774$		0			0
$775$	−2.18748			−0.0785764
$776$	−11.4733			−0.411867
$777$		0			0
$778$		−	16.1749i		−	0.579899i
$779$	−30.7625			−1.10218
$780$		0			0
$781$			43.0243i			1.53953i
$782$	16.8295	−	17.6098i	0.601821	−	0.629726i
$783$		0			0
$784$	−6.43143			−0.229694
$785$			14.1815i			0.506160i
$786$		0			0
$787$		−	16.9689i		−	0.604875i	−0.953169	−	0.302437i	$-0.902200\pi$
$787$		−	16.9689i		−	0.604875i	0.953169	−	0.302437i	$-0.0978003\pi$
$788$		−	23.0649i		−	0.821651i
$789$		0			0
$790$	0.619189			0.0220298
$791$		−	39.0529i		−	1.38856i
$792$		0			0
$793$			4.44894i			0.157986i
$794$			5.37770i			0.190848i
$795$		0			0
$796$		−	10.1323i		−	0.359130i
$797$	45.4466			1.60980			0.804900	−	0.593410i	$-0.202219\pi$
$797$	45.4466			1.60980			0.804900	+	0.593410i	$0.202219\pi$
$798$		0			0
$799$			67.8463i			2.40023i
$800$		−	1.00000i		−	0.0353553i
$801$		0			0
$802$		−	18.3723i		−	0.648749i
$803$	35.8117			1.26377
$804$		0			0
$805$	12.1435	−	12.7066i	0.428003	−	0.447849i
$806$			3.00412i			0.105816i
$807$		0			0
$808$	−9.81005			−0.345116
$809$			40.4973i			1.42381i	0.702276	+	0.711905i	$0.252167\pi$
$809$			40.4973i			1.42381i	−0.702276	+	0.711905i	$0.747833\pi$
$810$		0			0
$811$	−21.1875			−0.743994			−0.371997	−	0.928234i	$-0.621327\pi$
$811$	−21.1875			−0.743994			−0.371997	+	0.928234i	$0.621327\pi$
$812$	32.1604			1.12861
$813$		0			0
$814$	13.4776			0.472388
$815$	−11.7476			−0.411499
$816$		0			0
$817$	−16.6917			−0.583969
$818$		−	30.0999i		−	1.05242i
$819$		0			0
$820$			7.34071i			0.256349i
$821$			16.7811i			0.585666i	0.956164	+	0.292833i	$0.0945979\pi$
$821$			16.7811i			0.585666i	−0.956164	+	0.292833i	$0.905402\pi$
$822$		0			0
$823$	−43.7002			−1.52329			−0.761647	−	0.647992i	$-0.775609\pi$
$823$	−43.7002			−1.52329			−0.761647	+	0.647992i	$0.775609\pi$
$824$	−12.9748			−0.452000
$825$		0			0
$826$		−	26.8100i		−	0.932837i
$827$	−43.6631			−1.51831			−0.759157	−	0.650907i	$-0.774389\pi$
$827$	−43.6631			−1.51831			−0.759157	+	0.650907i	$0.774389\pi$
$828$		0			0
$829$	21.6081			0.750479			0.375240	−	0.926928i	$-0.377560\pi$
$829$	21.6081			0.750479			0.375240	+	0.926928i	$0.377560\pi$
$830$			7.82170i			0.271495i
$831$		0			0
$832$	−1.37333			−0.0476115
$833$	32.6659			1.13181
$834$		0			0
$835$			19.2009i			0.664475i
$836$		−	17.2576i		−	0.596868i
$837$		0			0
$838$			36.0650i			1.24584i
$839$	34.3173			1.18477			0.592383	−	0.805657i	$-0.298187\pi$
$839$	34.3173			1.18477			0.592383	+	0.805657i	$0.298187\pi$
$840$		0			0
$841$	−48.0055			−1.65536
$842$	5.92958			0.204347
$843$		0			0
$844$	17.7381			0.610571
$845$	−11.1140			−0.382332
$846$		0			0
$847$			21.8386i			0.750381i
$848$	−9.16726			−0.314805
$849$		0			0
$850$			5.07910i			0.174212i
$851$	−10.8442	+	11.3470i	−0.371734	+	0.388970i
$852$		0			0
$853$	−3.87711			−0.132750			−0.0663749	−	0.997795i	$-0.521143\pi$
$853$	−3.87711			−0.132750			−0.0663749	+	0.997795i	$0.521143\pi$
$854$			11.8725i			0.406270i
$855$		0			0
$856$			17.2233i			0.588681i
$857$			1.20652i			0.0412138i	0.999788	+	0.0206069i	$0.00655985\pi$
$857$			1.20652i			0.0412138i	−0.999788	+	0.0206069i	$0.993440\pi$
$858$		0			0
$859$	39.9727			1.36385			0.681925	−	0.731422i	$-0.261143\pi$
$859$	39.9727			1.36385			0.681925	+	0.731422i	$0.261143\pi$
$860$			3.98306i			0.135821i
$861$		0			0
$862$			31.2612i			1.06476i
$863$		−	5.03369i		−	0.171349i	−0.996323	−	0.0856744i	$-0.972696\pi$
$863$		−	5.03369i		−	0.171349i	0.996323	−	0.0856744i	$-0.0273045\pi$
$864$		0			0
$865$			3.67537i			0.124966i
$866$	−19.8707			−0.675233
$867$		0			0
$868$			8.01686i			0.272110i
$869$		−	2.54989i		−	0.0864991i
$870$		0			0
$871$		−	9.35865i		−	0.317106i
$872$	−2.13239			−0.0722119
$873$		0			0
$874$	14.5295	+	13.8857i	0.491468	+	0.469690i
$875$			3.66489i			0.123896i
$876$		0			0
$877$	12.6274			0.426396			0.213198	−	0.977009i	$-0.431612\pi$
$877$	12.6274			0.426396			0.213198	+	0.977009i	$0.431612\pi$
$878$		−	15.0653i		−	0.508430i
$879$		0			0
$880$	−4.11811			−0.138822
$881$	37.2185			1.25392			0.626962	−	0.779050i	$-0.284298\pi$
$881$	37.2185			1.25392			0.626962	+	0.779050i	$0.284298\pi$
$882$		0			0
$883$	−12.1568			−0.409108			−0.204554	−	0.978855i	$-0.565574\pi$
$883$	−12.1568			−0.409108			−0.204554	+	0.978855i	$0.565574\pi$
$884$	6.97527			0.234604
$885$		0			0
$886$	38.3156			1.28724
$887$		−	57.7006i		−	1.93740i	−0.248240	−	0.968699i	$-0.579852\pi$
$887$		−	57.7006i		−	1.93740i	0.248240	−	0.968699i	$-0.420148\pi$
$888$		0			0
$889$			43.7742i			1.46814i
$890$			0.592028i			0.0198448i
$891$		0			0
$892$	5.53297			0.185258
$893$	−55.9786			−1.87325
$894$		0			0
$895$		−	16.7414i		−	0.559602i
$896$	−3.66489			−0.122435
$897$		0			0
$898$	−13.0185			−0.434432
$899$		−	19.1957i		−	0.640212i
$900$		0			0
$901$	46.5615			1.55119
$902$	30.2299			1.00654
$903$		0			0
$904$		−	10.6560i		−	0.354412i
$905$		−	1.04339i		−	0.0346835i
$906$		0			0
$907$			15.2751i			0.507202i	0.967309	+	0.253601i	$0.0816150\pi$
$907$			15.2751i			0.507202i	−0.967309	+	0.253601i	$0.918385\pi$
$908$	14.0021			0.464676
$909$		0			0
$910$	5.03309			0.166845
$911$	12.2177			0.404789			0.202394	−	0.979304i	$-0.435128\pi$
$911$	12.2177			0.404789			0.202394	+	0.979304i	$0.435128\pi$
$912$		0			0
$913$	32.2107			1.06602
$914$	−15.4017			−0.509443
$915$		0			0
$916$			6.28684i			0.207723i
$917$	−13.2123			−0.436309
$918$		0			0
$919$			14.3237i			0.472494i	0.971693	+	0.236247i	$0.0759175\pi$
$919$			14.3237i			0.472494i	−0.971693	+	0.236247i	$0.924083\pi$
$920$	3.31348	−	3.46711i	0.109242	−	0.114307i
$921$		0			0
$922$	−19.9043			−0.655513
$923$		−	14.3480i		−	0.472269i
$924$		0			0
$925$		−	3.27275i		−	0.107607i
$926$		−	25.1321i		−	0.825893i
$927$		0			0
$928$	8.77528			0.288063
$929$			51.0889i			1.67617i	0.545538	+	0.838086i	$0.316326\pi$
$929$			51.0889i			1.67617i	−0.545538	+	0.838086i	$0.683674\pi$
$930$		0			0
$931$			26.9520i			0.883315i
$932$		−	17.7054i		−	0.579959i
$933$		0			0
$934$			26.1235i			0.854786i
$935$	20.9163			0.684037
$936$		0			0
$937$			43.0286i			1.40568i	0.711347	+	0.702841i	$0.248086\pi$
$937$			43.0286i			1.40568i	−0.711347	+	0.702841i	$0.751914\pi$
$938$		−	24.9747i		−	0.815453i
$939$		0			0
$940$			13.3579i			0.435688i
$941$	51.8215			1.68933			0.844667	−	0.535292i	$-0.179798\pi$
$941$	51.8215			1.68933			0.844667	+	0.535292i	$0.179798\pi$
$942$		0			0
$943$	−24.3233	+	25.4511i	−0.792075	+	0.828801i
$944$		−	7.31535i		−	0.238094i
$945$		0			0
$946$	16.4027			0.533298
$947$			22.3758i			0.727117i	0.931571	+	0.363559i	$0.118438\pi$
$947$			22.3758i			0.727117i	−0.931571	+	0.363559i	$0.881562\pi$
$948$		0			0
$949$	−11.9426			−0.387675
$950$	−4.19067			−0.135963
$951$		0			0
$952$	18.6144			0.603295
$953$	6.43104			0.208322			0.104161	−	0.994560i	$-0.466784\pi$
$953$	6.43104			0.208322			0.104161	+	0.994560i	$0.466784\pi$
$954$		0			0
$955$	8.60808			0.278551
$956$			7.37705i			0.238591i
$957$		0			0
$958$			23.1310i			0.747328i
$959$		−	56.9628i		−	1.83942i
$960$		0			0
$961$	−26.2150			−0.845644
$962$	−4.49455			−0.144910
$963$		0			0
$964$			18.9364i			0.609901i
$965$	−24.8742			−0.800728
$966$		0			0
$967$	2.71460			0.0872956			0.0436478	−	0.999047i	$-0.486102\pi$
$967$	2.71460			0.0872956			0.0436478	+	0.999047i	$0.486102\pi$
$968$			5.95885i			0.191525i
$969$		0			0
$970$	11.4733			0.368385
$971$	39.4161			1.26492			0.632461	−	0.774592i	$-0.282045\pi$
$971$	39.4161			1.26492			0.632461	+	0.774592i	$0.282045\pi$
$972$		0			0
$973$		−	23.0973i		−	0.740466i
$974$		−	39.2504i		−	1.25766i
$975$		0			0
$976$			3.23953i			0.103695i
$977$	21.5755			0.690261			0.345130	−	0.938555i	$-0.387835\pi$
$977$	21.5755			0.690261			0.345130	+	0.938555i	$0.387835\pi$
$978$		0			0
$979$	2.43804			0.0779200
$980$	6.43143			0.205444
$981$		0			0
$982$	−16.7695			−0.535136
$983$	−41.5756			−1.32605			−0.663027	−	0.748595i	$-0.730729\pi$
$983$	−41.5756			−1.32605			−0.663027	+	0.748595i	$0.730729\pi$
$984$		0			0
$985$			23.0649i			0.734907i
$986$	−44.5705			−1.41942
$987$		0			0
$988$			5.75516i			0.183096i
$989$	−13.1978	+	13.8097i	−0.419665	+	0.439124i
$990$		0			0
$991$	16.6208			0.527977			0.263988	−	0.964526i	$-0.414962\pi$
$991$	16.6208			0.527977			0.263988	+	0.964526i	$0.414962\pi$
$992$			2.18748i			0.0694524i
$993$		0			0
$994$		−	38.2893i		−	1.21446i
$995$			10.1323i			0.321215i
$996$		0			0
$997$	−23.1337			−0.732650			−0.366325	−	0.930487i	$-0.619384\pi$
$997$	−23.1337			−0.732650			−0.366325	+	0.930487i	$0.619384\pi$
$998$			9.29413i			0.294201i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2070.2.e.b.1241.7	yes	16
3.2	odd	2		2070.2.e.a.1241.15	yes	16
23.22	odd	2		2070.2.e.a.1241.2	✓	16
69.68	even	2	inner	2070.2.e.b.1241.10	yes	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2070.2.e.a.1241.2	✓	16	23.22	odd	2
2070.2.e.a.1241.15	yes	16	3.2	odd	2
2070.2.e.b.1241.7	yes	16	1.1	even	1	trivial
2070.2.e.b.1241.10	yes	16	69.68	even	2	inner

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:	\( N \)	\(=\)	\( 2070 = 2 \cdot 3^{2} \cdot 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2070.e (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.00000i q^{2} -1.00000 q^{4} +1.00000 q^{5} +3.66489i q^{7} +1.00000i q^{8} -1.00000i q^{10} -4.11811 q^{11} +1.37333 q^{13} +3.66489 q^{14} +1.00000 q^{16} -5.07910 q^{17} -4.19067i q^{19} -1.00000 q^{20} +4.11811i q^{22} +(-3.46711 - 3.31348i) q^{23} +1.00000 q^{25} -1.37333i q^{26} -3.66489i q^{28} +8.77528i q^{29} -2.18748 q^{31} -1.00000i q^{32} +5.07910i q^{34} +3.66489i q^{35} -3.27275i q^{37} -4.19067 q^{38} +1.00000i q^{40} -7.34071i q^{41} -3.98306i q^{43} +4.11811 q^{44} +(-3.31348 + 3.46711i) q^{46} -13.3579i q^{47} -6.43143 q^{49} -1.00000i q^{50} -1.37333 q^{52} -9.16726 q^{53} -4.11811 q^{55} -3.66489 q^{56} +8.77528 q^{58} -7.31535i q^{59} +3.23953i q^{61} +2.18748i q^{62} -1.00000 q^{64} +1.37333 q^{65} -6.81459i q^{67} +5.07910 q^{68} +3.66489 q^{70} -10.4476i q^{71} -8.69614 q^{73} -3.27275 q^{74} +4.19067i q^{76} -15.0924i q^{77} +0.619189i q^{79} +1.00000 q^{80} -7.34071 q^{82} -7.82170 q^{83} -5.07910 q^{85} -3.98306 q^{86} -4.11811i q^{88} -0.592028 q^{89} +5.03309i q^{91} +(3.46711 + 3.31348i) q^{92} -13.3579 q^{94} -4.19067i q^{95} +11.4733i q^{97} +6.43143i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 16 q^{4} + 16 q^{5} - 24 q^{11} + 16 q^{16} - 16 q^{20} - 4 q^{23} + 16 q^{25} - 8 q^{31} - 8 q^{38} + 24 q^{44} - 4 q^{46} - 16 q^{49} + 8 q^{53} - 24 q^{55} + 16 q^{58} - 16 q^{64} + 32 q^{73}+ \cdots + 4 q^{92}+O(q^{100}) \) 16 * q - 16 * q^4 + 16 * q^5 - 24 * q^11 + 16 * q^16 - 16 * q^20 - 4 * q^23 + 16 * q^25 - 8 * q^31 - 8 * q^38 + 24 * q^44 - 4 * q^46 - 16 * q^49 + 8 * q^53 - 24 * q^55 + 16 * q^58 - 16 * q^64 + 32 * q^73 + 16 * q^80 + 32 * q^82 - 56 * q^83 - 8 * q^86 + 40 * q^89 + 4 * q^92

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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