LMFDB - Embedded newform 3078.2.b.a.3077.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3078,2,Mod(3077,3078)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3078, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("3078.3077");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3078 = 2 \cdot 3^{4} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3078.b (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:	$24.5779537422$
Analytic rank:	$0$
Dimension:	$20$
Coefficient field:	$\mathbb{Q}[x]/(x^{20} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} - x^{19} + x^{18} + 3 x^{17} - 15 x^{16} + 33 x^{15} - 42 x^{14} + 72 x^{12} - 243 x^{11} + \cdots + 59049 $ x^20 - x^19 + x^18 + 3x^17 - 15x^16 + 33x^15 - 42x^14 + 72x^12 - 243x^11 + 675x^10 - 729x^9 + 648x^8 - 3402x^6 + 8019x^5 - 10935x^4 + 6561x^3 + 6561x^2 - 19683*x + 59049
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{9}\cdot 3^{10} $
Twist minimal:	no (minimal twist has level 342)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			3077.2
Root			$1.01548 + 1.40314i$ of defining polynomial
Character	$\chi$	$=$	3078.3077
Dual form			3078.2.b.a.3077.19

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} +1.00000 q^{4} -3.20235i q^{5} -1.52069 q^{7} -1.00000 q^{8} +O(q^{10})$	\(q-1.00000 q^{2} +1.00000 q^{4} -3.20235i q^{5} -1.52069 q^{7} -1.00000 q^{8} +3.20235i q^{10} +4.26277i q^{11} -2.53743i q^{13} +1.52069 q^{14} +1.00000 q^{16} -4.67109i q^{17} +(-2.68816 + 3.43130i) q^{19} -3.20235i q^{20} -4.26277i q^{22} -6.39559i q^{23} -5.25502 q^{25} +2.53743i q^{26} -1.52069 q^{28} +6.16275 q^{29} +5.54784i q^{31} -1.00000 q^{32} +4.67109i q^{34} +4.86977i q^{35} +0.247975i q^{37} +(2.68816 - 3.43130i) q^{38} +3.20235i q^{40} +0.215591 q^{41} -2.80421 q^{43} +4.26277i q^{44} +6.39559i q^{46} -8.60891i q^{47} -4.68751 q^{49} +5.25502 q^{50} -2.53743i q^{52} -13.5109 q^{53} +13.6508 q^{55} +1.52069 q^{56} -6.16275 q^{58} -0.580945 q^{59} -7.73842 q^{61} -5.54784i q^{62} +1.00000 q^{64} -8.12574 q^{65} -6.47560i q^{67} -4.67109i q^{68} -4.86977i q^{70} +9.99801 q^{71} -13.3133 q^{73} -0.247975i q^{74} +(-2.68816 + 3.43130i) q^{76} -6.48233i q^{77} +3.88042i q^{79} -3.20235i q^{80} -0.215591 q^{82} -6.14953i q^{83} -14.9584 q^{85} +2.80421 q^{86} -4.26277i q^{88} -12.4040 q^{89} +3.85865i q^{91} -6.39559i q^{92} +8.60891i q^{94} +(10.9882 + 8.60842i) q^{95} +2.59452i q^{97} +4.68751 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q - 20 q^{2} + 20 q^{4} - 4 q^{7} - 20 q^{8}+O(q^{10}) $ 20 * q - 20 * q^2 + 20 * q^4 - 4 * q^7 - 20 * q^8	$ 20 q - 20 q^{2} + 20 q^{4} - 4 q^{7} - 20 q^{8} + 4 q^{14} + 20 q^{16} + q^{19} - 20 q^{25} - 4 q^{28} - 20 q^{32} - q^{38} - 6 q^{41} + 10 q^{43} + 24 q^{49} + 20 q^{50} + 4 q^{56} + 6 q^{59} - 28 q^{61} + 20 q^{64} + 24 q^{65} - 24 q^{71} + 34 q^{73} + q^{76} + 6 q^{82} - 10 q^{86} - 36 q^{89} - 24 q^{98}+O(q^{100}) $ 20 * q - 20 * q^2 + 20 * q^4 - 4 * q^7 - 20 * q^8 + 4 * q^14 + 20 * q^16 + q^19 - 20 * q^25 - 4 * q^28 - 20 * q^32 - q^38 - 6 * q^41 + 10 * q^43 + 24 * q^49 + 20 * q^50 + 4 * q^56 + 6 * q^59 - 28 * q^61 + 20 * q^64 + 24 * q^65 - 24 * q^71 + 34 * q^73 + q^76 + 6 * q^82 - 10 * q^86 - 36 * q^89 - 24 * q^98

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3078\mathbb{Z}\right)^\times$.

$n$	$325$	$1217$
$\chi(n)$	$-1$	$-1$

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 $

$2$	−1.00000			−0.707107
$2$	−1.00000			−0.707107
$3$		0			0
$3$		0			0
$4$	1.00000			0.500000
$5$		−	3.20235i		−	1.43213i	−0.698032	−	0.716066i	$-0.745941\pi$
$5$		−	3.20235i		−	1.43213i	0.698032	−	0.716066i	$-0.254059\pi$
$6$		0			0
$7$	−1.52069			−0.574766			−0.287383	−	0.957816i	$-0.592785\pi$
$7$	−1.52069			−0.574766			−0.287383	+	0.957816i	$0.592785\pi$
$8$	−1.00000			−0.353553
$9$		0			0
$10$			3.20235i			1.01267i
$11$			4.26277i			1.28527i	0.766172	+	0.642636i	$0.222159\pi$
$11$			4.26277i			1.28527i	−0.766172	+	0.642636i	$0.777841\pi$
$12$		0			0
$13$		−	2.53743i		−	0.703758i	−0.936046	−	0.351879i	$-0.885543\pi$
$13$		−	2.53743i		−	0.703758i	0.936046	−	0.351879i	$-0.114457\pi$
$14$	1.52069			0.406421
$15$		0			0
$16$	1.00000			0.250000
$17$		−	4.67109i		−	1.13291i	−0.824094	−	0.566453i	$-0.808315\pi$
$17$		−	4.67109i		−	1.13291i	0.824094	−	0.566453i	$-0.191685\pi$
$18$		0			0
$19$	−2.68816	+	3.43130i	−0.616707	+	0.787193i
$19$	−2.68816	+	3.43130i	−0.616707	+	0.787193i
$20$		−	3.20235i		−	0.716066i
$21$		0			0
$22$		−	4.26277i		−	0.908825i
$23$		−	6.39559i		−	1.33357i	−0.745249	−	0.666787i	$-0.767669\pi$
$23$		−	6.39559i		−	1.33357i	0.745249	−	0.666787i	$-0.232331\pi$
$24$		0			0
$25$	−5.25502			−1.05100
$26$			2.53743i			0.497632i
$27$		0			0
$28$	−1.52069			−0.287383
$29$	6.16275			1.14439			0.572197	−	0.820116i	$-0.306091\pi$
$29$	6.16275			1.14439			0.572197	+	0.820116i	$0.306091\pi$
$30$		0			0
$31$			5.54784i			0.996422i	0.867056	+	0.498211i	$0.166009\pi$
$31$			5.54784i			0.996422i	−0.867056	+	0.498211i	$0.833991\pi$
$32$	−1.00000			−0.176777
$33$		0			0
$34$			4.67109i			0.801085i
$35$			4.86977i			0.823141i
$36$		0			0
$37$			0.247975i			0.0407669i	0.999792	+	0.0203834i	$0.00648870\pi$
$37$			0.247975i			0.0407669i	−0.999792	+	0.0203834i	$0.993511\pi$
$38$	2.68816	−	3.43130i	0.436077	−	0.556630i
$39$		0			0
$40$			3.20235i			0.506335i
$41$	0.215591			0.0336697			0.0168348	−	0.999858i	$-0.494641\pi$
$41$	0.215591			0.0336697			0.0168348	+	0.999858i	$0.494641\pi$
$42$		0			0
$43$	−2.80421			−0.427638			−0.213819	−	0.976873i	$-0.568590\pi$
$43$	−2.80421			−0.427638			−0.213819	+	0.976873i	$0.568590\pi$
$44$			4.26277i			0.642636i
$45$		0			0
$46$			6.39559i			0.942979i
$47$		−	8.60891i		−	1.25574i	−0.778319	−	0.627869i	$-0.783927\pi$
$47$		−	8.60891i		−	1.25574i	0.778319	−	0.627869i	$-0.216073\pi$
$48$		0			0
$49$	−4.68751			−0.669644
$50$	5.25502			0.743171
$51$		0			0
$52$		−	2.53743i		−	0.351879i
$53$	−13.5109			−1.85586			−0.927932	−	0.372749i	$-0.878415\pi$
$53$	−13.5109			−1.85586			−0.927932	+	0.372749i	$0.878415\pi$
$54$		0			0
$55$	13.6508			1.84068
$56$	1.52069			0.203210
$57$		0			0
$58$	−6.16275			−0.809209
$59$	−0.580945			−0.0756326			−0.0378163	−	0.999285i	$-0.512040\pi$
$59$	−0.580945			−0.0756326			−0.0378163	+	0.999285i	$0.512040\pi$
$60$		0			0
$61$	−7.73842			−0.990803			−0.495401	−	0.868664i	$-0.664979\pi$
$61$	−7.73842			−0.990803			−0.495401	+	0.868664i	$0.664979\pi$
$62$		−	5.54784i		−	0.704577i
$63$		0			0
$64$	1.00000			0.125000
$65$	−8.12574			−1.00787
$66$		0			0
$67$		−	6.47560i		−	0.791120i	−0.918440	−	0.395560i	$-0.870551\pi$
$67$		−	6.47560i		−	0.791120i	0.918440	−	0.395560i	$-0.129449\pi$
$68$		−	4.67109i		−	0.566453i
$69$		0			0
$70$		−	4.86977i		−	0.582048i
$71$	9.99801			1.18654			0.593272	−	0.805002i	$-0.297836\pi$
$71$	9.99801			1.18654			0.593272	+	0.805002i	$0.297836\pi$
$72$		0			0
$73$	−13.3133			−1.55820			−0.779101	−	0.626899i	$-0.784324\pi$
$73$	−13.3133			−1.55820			−0.779101	+	0.626899i	$0.784324\pi$
$74$		−	0.247975i		−	0.0288265i
$75$		0			0
$76$	−2.68816	+	3.43130i	−0.308353	+	0.393597i
$77$		−	6.48233i		−	0.738730i
$78$		0			0
$79$			3.88042i			0.436581i	0.975884	+	0.218291i	$0.0700481\pi$
$79$			3.88042i			0.436581i	−0.975884	+	0.218291i	$0.929952\pi$
$80$		−	3.20235i		−	0.358033i
$81$		0			0
$82$	−0.215591			−0.0238080
$83$		−	6.14953i		−	0.674999i	−0.941326	−	0.337499i	$-0.890419\pi$
$83$		−	6.14953i		−	0.674999i	0.941326	−	0.337499i	$-0.109581\pi$
$84$		0			0
$85$	−14.9584			−1.62247
$86$	2.80421			0.302385
$87$		0			0
$88$		−	4.26277i		−	0.454412i
$89$	−12.4040			−1.31482			−0.657410	−	0.753533i	$-0.728348\pi$
$89$	−12.4040			−1.31482			−0.657410	+	0.753533i	$0.728348\pi$
$90$		0			0
$91$			3.85865i			0.404496i
$92$		−	6.39559i		−	0.666787i
$93$		0			0
$94$			8.60891i			0.887941i
$95$	10.9882	+	8.60842i	1.12736	+	0.883205i
$96$		0			0
$97$			2.59452i			0.263433i	0.991287	+	0.131717i	$0.0420489\pi$
$97$			2.59452i			0.263433i	−0.991287	+	0.131717i	$0.957951\pi$
$98$	4.68751			0.473510
$99$		0			0
$100$	−5.25502			−0.525502
$101$			11.5990i			1.15414i	0.816694	+	0.577070i	$0.195804\pi$
$101$			11.5990i			1.15414i	−0.816694	+	0.577070i	$0.804196\pi$
$102$		0			0
$103$			10.1433i			0.999454i	0.866183	+	0.499727i	$0.166566\pi$
$103$			10.1433i			0.999454i	−0.866183	+	0.499727i	$0.833434\pi$
$104$			2.53743i			0.248816i
$105$		0			0
$106$	13.5109			1.31229
$107$	9.34489			0.903404			0.451702	−	0.892169i	$-0.350817\pi$
$107$	9.34489			0.903404			0.451702	+	0.892169i	$0.350817\pi$
$108$		0			0
$109$		−	4.86226i		−	0.465720i	−0.972510	−	0.232860i	$-0.925192\pi$
$109$		−	4.86226i		−	0.465720i	0.972510	−	0.232860i	$-0.0748084\pi$
$110$	−13.6508			−1.30156
$111$		0			0
$112$	−1.52069			−0.143691
$113$	5.19041			0.488273			0.244137	−	0.969741i	$-0.421495\pi$
$113$	5.19041			0.488273			0.244137	+	0.969741i	$0.421495\pi$
$114$		0			0
$115$	−20.4809			−1.90985
$116$	6.16275			0.572197
$117$		0			0
$118$	0.580945			0.0534803
$119$			7.10327i			0.651155i
$120$		0			0
$121$	−7.17117			−0.651924
$122$	7.73842			0.700603
$123$		0			0
$124$			5.54784i			0.498211i
$125$			0.816649i			0.0730433i
$126$		0			0
$127$			19.0260i			1.68829i	0.536117	+	0.844144i	$0.319891\pi$
$127$			19.0260i			1.68829i	−0.536117	+	0.844144i	$0.680109\pi$
$128$	−1.00000			−0.0883883
$129$		0			0
$130$	8.12574			0.712675
$131$			7.21609i			0.630473i	0.949013	+	0.315236i	$0.102084\pi$
$131$			7.21609i			0.630473i	−0.949013	+	0.315236i	$0.897916\pi$
$132$		0			0
$133$	4.08785	−	5.21793i	0.354462	−	0.452452i
$134$			6.47560i			0.559406i
$135$		0			0
$136$			4.67109i			0.400543i
$137$			19.5982i			1.67439i	0.546905	+	0.837195i	$0.315806\pi$
$137$			19.5982i			1.67439i	−0.546905	+	0.837195i	$0.684194\pi$
$138$		0			0
$139$	10.6317			0.901766			0.450883	−	0.892583i	$-0.351109\pi$
$139$	10.6317			0.901766			0.450883	+	0.892583i	$0.351109\pi$
$140$			4.86977i			0.411570i
$141$		0			0
$142$	−9.99801			−0.839014
$143$	10.8165			0.904520
$144$		0			0
$145$		−	19.7352i		−	1.63892i
$146$	13.3133			1.10181
$147$		0			0
$148$			0.247975i			0.0203834i
$149$			19.1410i			1.56809i	0.620702	+	0.784047i	$0.286848\pi$
$149$			19.1410i			1.56809i	−0.620702	+	0.784047i	$0.713152\pi$
$150$		0			0
$151$			15.6476i			1.27339i	0.771117	+	0.636693i	$0.219698\pi$
$151$			15.6476i			1.27339i	−0.771117	+	0.636693i	$0.780302\pi$
$152$	2.68816	−	3.43130i	0.218039	−	0.278315i
$153$		0			0
$154$			6.48233i			0.522361i
$155$	17.7661			1.42701
$156$		0			0
$157$	−11.3730			−0.907663			−0.453831	−	0.891088i	$-0.649943\pi$
$157$	−11.3730			−0.907663			−0.453831	+	0.891088i	$0.649943\pi$
$158$		−	3.88042i		−	0.308710i
$159$		0			0
$160$			3.20235i			0.253168i
$161$			9.72570i			0.766492i
$162$		0			0
$163$	−21.1943			−1.66007			−0.830034	−	0.557713i	$-0.811679\pi$
$163$	−21.1943			−1.66007			−0.830034	+	0.557713i	$0.811679\pi$
$164$	0.215591			0.0168348
$165$		0			0
$166$			6.14953i			0.477296i
$167$	−24.1337			−1.86752			−0.933762	−	0.357894i	$-0.883495\pi$
$167$	−24.1337			−1.86752			−0.933762	+	0.357894i	$0.883495\pi$
$168$		0			0
$169$	6.56142			0.504725
$170$	14.9584			1.14726
$171$		0			0
$172$	−2.80421			−0.213819
$173$	22.1422			1.68344			0.841720	−	0.539914i	$-0.181543\pi$
$173$	22.1422			1.68344			0.841720	+	0.539914i	$0.181543\pi$
$174$		0			0
$175$	7.99124			0.604081
$176$			4.26277i			0.321318i
$177$		0			0
$178$	12.4040			0.929718
$179$	−16.7184			−1.24959			−0.624796	−	0.780788i	$-0.714818\pi$
$179$	−16.7184			−1.24959			−0.624796	+	0.780788i	$0.714818\pi$
$180$		0			0
$181$		−	1.92544i		−	0.143116i	−0.997436	−	0.0715582i	$-0.977203\pi$
$181$		−	1.92544i		−	0.143116i	0.997436	−	0.0715582i	$-0.0227972\pi$
$182$		−	3.85865i		−	0.286022i
$183$		0			0
$184$			6.39559i			0.471489i
$185$	0.794103			0.0583836
$186$		0			0
$187$	19.9118			1.45609
$188$		−	8.60891i		−	0.627869i
$189$		0			0
$190$	−10.9882	−	8.60842i	−0.797167	−	0.624521i
$191$		−	0.736584i		−	0.0532973i	−0.999645	−	0.0266487i	$-0.991516\pi$
$191$		−	0.736584i		−	0.0532973i	0.999645	−	0.0266487i	$-0.00848354\pi$
$192$		0			0
$193$		−	9.11819i		−	0.656341i	−0.944618	−	0.328171i	$-0.893568\pi$
$193$		−	9.11819i		−	0.656341i	0.944618	−	0.328171i	$-0.106432\pi$
$194$		−	2.59452i		−	0.186275i
$195$		0			0
$196$	−4.68751			−0.334822
$197$		−	8.47801i		−	0.604033i	−0.953303	−	0.302017i	$-0.902340\pi$
$197$		−	8.47801i		−	0.604033i	0.953303	−	0.302017i	$-0.0976598\pi$
$198$		0			0
$199$	16.4335			1.16494			0.582470	−	0.812852i	$-0.302086\pi$
$199$	16.4335			1.16494			0.582470	+	0.812852i	$0.302086\pi$
$200$	5.25502			0.371586
$201$		0			0
$202$		−	11.5990i		−	0.816101i
$203$	−9.37161			−0.657758
$204$		0			0
$205$		−	0.690397i		−	0.0482194i
$206$		−	10.1433i		−	0.706720i
$207$		0			0
$208$		−	2.53743i		−	0.175939i
$209$	−14.6268	−	11.4590i	−1.01176	−	0.792636i
$210$		0			0
$211$		−	0.927889i		−	0.0638785i	−0.999490	−	0.0319393i	$-0.989832\pi$
$211$		−	0.927889i		−	0.0638785i	0.999490	−	0.0319393i	$-0.0101683\pi$
$212$	−13.5109			−0.927932
$213$		0			0
$214$	−9.34489			−0.638803
$215$			8.98004i			0.612434i
$216$		0			0
$217$		−	8.43653i		−	0.572709i
$218$			4.86226i			0.329314i
$219$		0			0
$220$	13.6508			0.920340
$221$	−11.8526			−0.797291
$222$		0			0
$223$			19.2520i			1.28921i	0.764516	+	0.644605i	$0.222978\pi$
$223$			19.2520i			1.28921i	−0.764516	+	0.644605i	$0.777022\pi$
$224$	1.52069			0.101605
$225$		0			0
$226$	−5.19041			−0.345261
$227$	−21.1219			−1.40191			−0.700955	−	0.713205i	$-0.747243\pi$
$227$	−21.1219			−1.40191			−0.700955	+	0.713205i	$0.747243\pi$
$228$		0			0
$229$	1.02591			0.0677939			0.0338970	−	0.999425i	$-0.489208\pi$
$229$	1.02591			0.0677939			0.0338970	+	0.999425i	$0.489208\pi$
$230$	20.4809			1.35047
$231$		0			0
$232$	−6.16275			−0.404604
$233$		−	5.08018i		−	0.332814i	−0.986057	−	0.166407i	$-0.946784\pi$
$233$		−	5.08018i		−	0.332814i	0.986057	−	0.166407i	$-0.0532165\pi$
$234$		0			0
$235$	−27.5687			−1.79838
$236$	−0.580945			−0.0378163
$237$		0			0
$238$		−	7.10327i		−	0.460436i
$239$		−	11.1467i		−	0.721019i	−0.932756	−	0.360510i	$-0.882603\pi$
$239$		−	11.1467i		−	0.721019i	0.932756	−	0.360510i	$-0.117397\pi$
$240$		0			0
$241$			11.0135i			0.709442i	0.934972	+	0.354721i	$0.115424\pi$
$241$			11.0135i			0.709442i	−0.934972	+	0.354721i	$0.884576\pi$
$242$	7.17117			0.460980
$243$		0			0
$244$	−7.73842			−0.495401
$245$			15.0110i			0.959019i
$246$		0			0
$247$	8.70669	+	6.82104i	0.553993	+	0.434012i
$248$		−	5.54784i		−	0.352288i
$249$		0			0
$250$		−	0.816649i		−	0.0516494i
$251$		−	19.9197i		−	1.25732i	−0.777679	−	0.628661i	$-0.783603\pi$
$251$		−	19.9197i		−	1.25732i	0.777679	−	0.628661i	$-0.216397\pi$
$252$		0			0
$253$	27.2629			1.71400
$254$		−	19.0260i		−	1.19380i
$255$		0			0
$256$	1.00000			0.0625000
$257$	−17.8096			−1.11093			−0.555466	−	0.831539i	$-0.687460\pi$
$257$	−17.8096			−1.11093			−0.555466	+	0.831539i	$0.687460\pi$
$258$		0			0
$259$		−	0.377093i		−	0.0234314i
$260$	−8.12574			−0.503937
$261$		0			0
$262$		−	7.21609i		−	0.445812i
$263$		−	6.07594i		−	0.374659i	−0.982297	−	0.187329i	$-0.940017\pi$
$263$		−	6.07594i		−	0.374659i	0.982297	−	0.187329i	$-0.0599832\pi$
$264$		0			0
$265$			43.2666i			2.65784i
$266$	−4.08785	+	5.21793i	−0.250642	+	0.319932i
$267$		0			0
$268$		−	6.47560i		−	0.395560i
$269$	1.92957			0.117648			0.0588240	−	0.998268i	$-0.481265\pi$
$269$	1.92957			0.117648			0.0588240	+	0.998268i	$0.481265\pi$
$270$		0			0
$271$	−11.8324			−0.718767			−0.359384	−	0.933190i	$-0.617013\pi$
$271$	−11.8324			−0.718767			−0.359384	+	0.933190i	$0.617013\pi$
$272$		−	4.67109i		−	0.283226i
$273$		0			0
$274$		−	19.5982i		−	1.18397i
$275$		−	22.4009i		−	1.35082i
$276$		0			0
$277$	−3.01423			−0.181108			−0.0905538	−	0.995892i	$-0.528864\pi$
$277$	−3.01423			−0.181108			−0.0905538	+	0.995892i	$0.528864\pi$
$278$	−10.6317			−0.637645
$279$		0			0
$280$		−	4.86977i		−	0.291024i
$281$	−16.4152			−0.979251			−0.489625	−	0.871933i	$-0.662866\pi$
$281$	−16.4152			−0.979251			−0.489625	+	0.871933i	$0.662866\pi$
$282$		0			0
$283$	15.4198			0.916611			0.458305	−	0.888795i	$-0.348457\pi$
$283$	15.4198			0.916611			0.458305	+	0.888795i	$0.348457\pi$
$284$	9.99801			0.593272
$285$		0			0
$286$	−10.8165			−0.639592
$287$	−0.327846			−0.0193522
$288$		0			0
$289$	−4.81908			−0.283475
$290$			19.7352i			1.15889i
$291$		0			0
$292$	−13.3133			−0.779101
$293$	−24.6567			−1.44046			−0.720231	−	0.693734i	$-0.755964\pi$
$293$	−24.6567			−1.44046			−0.720231	+	0.693734i	$0.755964\pi$
$294$		0			0
$295$			1.86039i			0.108316i
$296$		−	0.247975i		−	0.0144133i
$297$		0			0
$298$		−	19.1410i		−	1.10881i
$299$	−16.2284			−0.938512
$300$		0			0
$301$	4.26432			0.245792
$302$		−	15.6476i		−	0.900420i
$303$		0			0
$304$	−2.68816	+	3.43130i	−0.154177	+	0.196798i
$305$			24.7811i			1.41896i
$306$		0			0
$307$			18.4337i			1.05207i	0.850464	+	0.526033i	$0.176321\pi$
$307$			18.4337i			1.05207i	−0.850464	+	0.526033i	$0.823679\pi$
$308$		−	6.48233i		−	0.369365i
$309$		0			0
$310$	−17.7661			−1.00905
$311$			18.0078i			1.02113i	0.859839	+	0.510566i	$0.170564\pi$
$311$			18.0078i			1.02113i	−0.859839	+	0.510566i	$0.829436\pi$
$312$		0			0
$313$	−11.3994			−0.644335			−0.322167	−	0.946683i	$-0.604411\pi$
$313$	−11.3994			−0.644335			−0.322167	+	0.946683i	$0.604411\pi$
$314$	11.3730			0.641814
$315$		0			0
$316$			3.88042i			0.218291i
$317$	−30.3425			−1.70420			−0.852102	−	0.523375i	$-0.824673\pi$
$317$	−30.3425			−1.70420			−0.852102	+	0.523375i	$0.824673\pi$
$318$		0			0
$319$			26.2703i			1.47086i
$320$		−	3.20235i		−	0.179017i
$321$		0			0
$322$		−	9.72570i		−	0.541992i
$323$	16.0279	+	12.5566i	0.891816	+	0.698670i
$324$		0			0
$325$			13.3343i			0.739652i
$326$	21.1943			1.17385
$327$		0			0
$328$	−0.215591			−0.0119040
$329$			13.0915i			0.721756i
$330$		0			0
$331$		−	27.3689i		−	1.50433i	−0.658975	−	0.752165i	$-0.729010\pi$
$331$		−	27.3689i		−	1.50433i	0.658975	−	0.752165i	$-0.270990\pi$
$332$		−	6.14953i		−	0.337499i
$333$		0			0
$334$	24.1337			1.32054
$335$	−20.7371			−1.13299
$336$		0			0
$337$		−	10.6004i		−	0.577442i	−0.957413	−	0.288721i	$-0.906770\pi$
$337$		−	10.6004i		−	0.577442i	0.957413	−	0.288721i	$-0.0932301\pi$
$338$	−6.56142			−0.356894
$339$		0			0
$340$	−14.9584			−0.811235
$341$	−23.6491			−1.28067
$342$		0			0
$343$	17.7730			0.959654
$344$	2.80421			0.151193
$345$		0			0
$346$	−22.1422			−1.19037
$347$		−	5.99303i		−	0.321723i	−0.986977	−	0.160861i	$-0.948573\pi$
$347$		−	5.99303i		−	0.321723i	0.986977	−	0.160861i	$-0.0514272\pi$
$348$		0			0
$349$	1.34086			0.0717745			0.0358872	−	0.999356i	$-0.488574\pi$
$349$	1.34086			0.0717745			0.0358872	+	0.999356i	$0.488574\pi$
$350$	−7.99124			−0.427150
$351$		0			0
$352$		−	4.26277i		−	0.227206i
$353$		−	11.4733i		−	0.610663i	−0.952246	−	0.305332i	$-0.901233\pi$
$353$		−	11.4733i		−	0.610663i	0.952246	−	0.305332i	$-0.0987673\pi$
$354$		0			0
$355$		−	32.0171i		−	1.69929i
$356$	−12.4040			−0.657410
$357$		0			0
$358$	16.7184			0.883594
$359$		−	2.97199i		−	0.156856i	−0.996920	−	0.0784278i	$-0.975010\pi$
$359$		−	2.97199i		−	0.156856i	0.996920	−	0.0784278i	$-0.0249900\pi$
$360$		0			0
$361$	−4.54757	−	18.4478i	−0.239346	−	0.970934i
$362$			1.92544i			0.101199i
$363$		0			0
$364$			3.85865i			0.202248i
$365$			42.6337i			2.23155i
$366$		0			0
$367$	13.7275			0.716571			0.358286	−	0.933612i	$-0.383361\pi$
$367$	13.7275			0.716571			0.358286	+	0.933612i	$0.383361\pi$
$368$		−	6.39559i		−	0.333393i
$369$		0			0
$370$	−0.794103			−0.0412834
$371$	20.5458			1.06669
$372$		0			0
$373$		−	15.0917i		−	0.781421i	−0.920514	−	0.390711i	$-0.872229\pi$
$373$		−	15.0917i		−	0.781421i	0.920514	−	0.390711i	$-0.127771\pi$
$374$	−19.9118			−1.02961
$375$		0			0
$376$			8.60891i			0.443971i
$377$		−	15.6376i		−	0.805376i
$378$		0			0
$379$			1.29330i			0.0664322i	0.999448	+	0.0332161i	$0.0105750\pi$
$379$			1.29330i			0.0664322i	−0.999448	+	0.0332161i	$0.989425\pi$
$380$	10.9882	+	8.60842i	0.563682	+	0.441603i
$381$		0			0
$382$			0.736584i			0.0376869i
$383$	−2.05850			−0.105185			−0.0525923	−	0.998616i	$-0.516748\pi$
$383$	−2.05850			−0.105185			−0.0525923	+	0.998616i	$0.516748\pi$
$384$		0			0
$385$	−20.7587			−1.05796
$386$			9.11819i			0.464103i
$387$		0			0
$388$			2.59452i			0.131717i
$389$		−	22.5996i		−	1.14585i	−0.819609	−	0.572923i	$-0.805809\pi$
$389$		−	22.5996i		−	1.14585i	0.819609	−	0.572923i	$-0.194191\pi$
$390$		0			0
$391$	−29.8744			−1.51081
$392$	4.68751			0.236755
$393$		0			0
$394$			8.47801i			0.427116i
$395$	12.4264			0.625242
$396$		0			0
$397$	−21.6329			−1.08572			−0.542862	−	0.839822i	$-0.682659\pi$
$397$	−21.6329			−1.08572			−0.542862	+	0.839822i	$0.682659\pi$
$398$	−16.4335			−0.823737
$399$		0			0
$400$	−5.25502			−0.262751
$401$	34.5347			1.72458			0.862291	−	0.506412i	$-0.169029\pi$
$401$	34.5347			1.72458			0.862291	+	0.506412i	$0.169029\pi$
$402$		0			0
$403$	14.0773			0.701240
$404$			11.5990i			0.577070i
$405$		0			0
$406$	9.37161			0.465105
$407$	−1.05706			−0.0523965
$408$		0			0
$409$		−	18.1554i		−	0.897725i	−0.893601	−	0.448862i	$-0.851829\pi$
$409$		−	18.1554i		−	0.897725i	0.893601	−	0.448862i	$-0.148171\pi$
$410$			0.690397i			0.0340963i
$411$		0			0
$412$			10.1433i			0.499727i
$413$	0.883436			0.0434711
$414$		0			0
$415$	−19.6929			−0.966688
$416$			2.53743i			0.124408i
$417$		0			0
$418$	14.6268	+	11.4590i	0.715420	+	0.560478i
$419$		−	15.0840i		−	0.736900i	−0.929647	−	0.368450i	$-0.879888\pi$
$419$		−	15.0840i		−	0.736900i	0.929647	−	0.368450i	$-0.120112\pi$
$420$		0			0
$421$			25.8311i			1.25893i	0.777028	+	0.629466i	$0.216726\pi$
$421$			25.8311i			1.25893i	−0.777028	+	0.629466i	$0.783274\pi$
$422$			0.927889i			0.0451689i
$423$		0			0
$424$	13.5109			0.656147
$425$			24.5467i			1.19069i
$426$		0			0
$427$	11.7677			0.569479
$428$	9.34489			0.451702
$429$		0			0
$430$		−	8.98004i		−	0.433056i
$431$	−18.7560			−0.903444			−0.451722	−	0.892159i	$-0.649190\pi$
$431$	−18.7560			−0.903444			−0.451722	+	0.892159i	$0.649190\pi$
$432$		0			0
$433$			21.8593i			1.05049i	0.850951	+	0.525245i	$0.176026\pi$
$433$			21.8593i			1.05049i	−0.850951	+	0.525245i	$0.823974\pi$
$434$			8.43653i			0.404967i
$435$		0			0
$436$		−	4.86226i		−	0.232860i
$437$	21.9452	+	17.1924i	1.04978	+	0.822423i
$438$		0			0
$439$		−	22.6494i		−	1.08100i	−0.841345	−	0.540499i	$-0.818236\pi$
$439$		−	22.6494i		−	1.08100i	0.841345	−	0.540499i	$-0.181764\pi$
$440$	−13.6508			−0.650779
$441$		0			0
$442$	11.8526			0.563770
$443$			8.13524i			0.386517i	0.981148	+	0.193258i	$0.0619056\pi$
$443$			8.13524i			0.386517i	−0.981148	+	0.193258i	$0.938094\pi$
$444$		0			0
$445$			39.7218i			1.88300i
$446$		−	19.2520i		−	0.911609i
$447$		0			0
$448$	−1.52069			−0.0718457
$449$	−34.4488			−1.62574			−0.812870	−	0.582445i	$-0.802096\pi$
$449$	−34.4488			−1.62574			−0.812870	+	0.582445i	$0.802096\pi$
$450$		0			0
$451$			0.919014i			0.0432747i
$452$	5.19041			0.244137
$453$		0			0
$454$	21.1219			0.991301
$455$	12.3567			0.579292
$456$		0			0
$457$	28.7479			1.34477			0.672385	−	0.740202i	$-0.265270\pi$
$457$	28.7479			1.34477			0.672385	+	0.740202i	$0.265270\pi$
$458$	−1.02591			−0.0479375
$459$		0			0
$460$	−20.4809			−0.954927
$461$			7.62779i			0.355262i	0.984097	+	0.177631i	$0.0568433\pi$
$461$			7.62779i			0.355262i	−0.984097	+	0.177631i	$0.943157\pi$
$462$		0			0
$463$	3.84491			0.178688			0.0893441	−	0.996001i	$-0.471523\pi$
$463$	3.84491			0.178688			0.0893441	+	0.996001i	$0.471523\pi$
$464$	6.16275			0.286098
$465$		0			0
$466$			5.08018i			0.235335i
$467$			17.3946i			0.804926i	0.915436	+	0.402463i	$0.131846\pi$
$467$			17.3946i			0.804926i	−0.915436	+	0.402463i	$0.868154\pi$
$468$		0			0
$469$			9.84736i			0.454709i
$470$	27.5687			1.27165
$471$		0			0
$472$	0.580945			0.0267402
$473$		−	11.9537i		−	0.549631i
$474$		0			0
$475$	14.1263	−	18.0315i	0.648161	−	0.827342i
$476$			7.10327i			0.325578i
$477$		0			0
$478$			11.1467i			0.509838i
$479$		−	29.5315i		−	1.34933i	−0.738125	−	0.674664i	$-0.764288\pi$
$479$		−	29.5315i		−	1.34933i	0.738125	−	0.674664i	$-0.235712\pi$
$480$		0			0
$481$	0.629221			0.0286900
$482$		−	11.0135i		−	0.501651i
$483$		0			0
$484$	−7.17117			−0.325962
$485$	8.30854			0.377271
$486$		0			0
$487$		−	15.8426i		−	0.717895i	−0.933358	−	0.358948i	$-0.883136\pi$
$487$		−	15.8426i		−	0.717895i	0.933358	−	0.358948i	$-0.116864\pi$
$488$	7.73842			0.350302
$489$		0			0
$490$		−	15.0110i		−	0.678129i
$491$			4.37074i			0.197249i	0.995125	+	0.0986245i	$0.0314443\pi$
$491$			4.37074i			0.197249i	−0.995125	+	0.0986245i	$0.968556\pi$
$492$		0			0
$493$		−	28.7868i		−	1.29649i
$494$	−8.70669	−	6.82104i	−0.391732	−	0.306893i
$495$		0			0
$496$			5.54784i			0.249105i
$497$	−15.2038			−0.681985
$498$		0			0
$499$	−20.4879			−0.917165			−0.458582	−	0.888652i	$-0.651643\pi$
$499$	−20.4879			−0.917165			−0.458582	+	0.888652i	$0.651643\pi$
$500$			0.816649i			0.0365217i
$501$		0			0
$502$			19.9197i			0.889061i
$503$			37.5977i			1.67640i	0.545366	+	0.838198i	$0.316391\pi$
$503$			37.5977i			1.67640i	−0.545366	+	0.838198i	$0.683609\pi$
$504$		0			0
$505$	37.1439			1.65288
$506$	−27.2629			−1.21198
$507$		0			0
$508$			19.0260i			0.844144i
$509$	3.69850			0.163933			0.0819666	−	0.996635i	$-0.473880\pi$
$509$	3.69850			0.163933			0.0819666	+	0.996635i	$0.473880\pi$
$510$		0			0
$511$	20.2453			0.895601
$512$	−1.00000			−0.0441942
$513$		0			0
$514$	17.8096			0.785548
$515$	32.4825			1.43135
$516$		0			0
$517$	36.6978			1.61397
$518$			0.377093i			0.0165685i
$519$		0			0
$520$	8.12574			0.356337
$521$	13.3408			0.584472			0.292236	−	0.956346i	$-0.405601\pi$
$521$	13.3408			0.584472			0.292236	+	0.956346i	$0.405601\pi$
$522$		0			0
$523$			16.5133i			0.722078i	0.932551	+	0.361039i	$0.117578\pi$
$523$			16.5133i			0.722078i	−0.932551	+	0.361039i	$0.882422\pi$
$524$			7.21609i			0.315236i
$525$		0			0
$526$			6.07594i			0.264924i
$527$	25.9145			1.12885
$528$		0			0
$529$	−17.9036			−0.778417
$530$		−	43.2666i		−	1.87938i
$531$		0			0
$532$	4.08785	−	5.21793i	0.177231	−	0.226226i
$533$		−	0.547048i		−	0.0236953i
$534$		0			0
$535$		−	29.9256i		−	1.29379i
$536$			6.47560i			0.279703i
$537$		0			0
$538$	−1.92957			−0.0831897
$539$		−	19.9818i		−	0.860675i
$540$		0			0
$541$	33.5888			1.44410			0.722049	−	0.691842i	$-0.243201\pi$
$541$	33.5888			1.44410			0.722049	+	0.691842i	$0.243201\pi$
$542$	11.8324			0.508245
$543$		0			0
$544$			4.67109i			0.200271i
$545$	−15.5706			−0.666973
$546$		0			0
$547$			31.4956i			1.34666i	0.739344	+	0.673328i	$0.235136\pi$
$547$			31.4956i			1.34666i	−0.739344	+	0.673328i	$0.764864\pi$
$548$			19.5982i			0.837195i
$549$		0			0
$550$			22.4009i			0.955178i
$551$	−16.5665	+	21.1462i	−0.705755	+	0.900859i
$552$		0			0
$553$		−	5.90091i		−	0.250932i
$554$	3.01423			0.128062
$555$		0			0
$556$	10.6317			0.450883
$557$			28.7677i			1.21893i	0.792814	+	0.609464i	$0.208615\pi$
$557$			28.7677i			1.21893i	−0.792814	+	0.609464i	$0.791385\pi$
$558$		0			0
$559$			7.11549i			0.300953i
$560$			4.86977i			0.205785i
$561$		0			0
$562$	16.4152			0.692435
$563$	−28.6495			−1.20743			−0.603716	−	0.797200i	$-0.706314\pi$
$563$	−28.6495			−1.20743			−0.603716	+	0.797200i	$0.706314\pi$
$564$		0			0
$565$		−	16.6215i		−	0.699272i
$566$	−15.4198			−0.648142
$567$		0			0
$568$	−9.99801			−0.419507
$569$	26.7583			1.12177			0.560883	−	0.827895i	$-0.310462\pi$
$569$	26.7583			1.12177			0.560883	+	0.827895i	$0.310462\pi$
$570$		0			0
$571$	−8.16762			−0.341804			−0.170902	−	0.985288i	$-0.554668\pi$
$571$	−8.16762			−0.341804			−0.170902	+	0.985288i	$0.554668\pi$
$572$	10.8165			0.452260
$573$		0			0
$574$	0.327846			0.0136840
$575$			33.6089i			1.40159i
$576$		0			0
$577$	−19.9431			−0.830242			−0.415121	−	0.909766i	$-0.636261\pi$
$577$	−19.9431			−0.830242			−0.415121	+	0.909766i	$0.636261\pi$
$578$	4.81908			0.200447
$579$		0			0
$580$		−	19.7352i		−	0.819462i
$581$			9.35152i			0.387966i
$582$		0			0
$583$		−	57.5938i		−	2.38529i
$584$	13.3133			0.550907
$585$		0			0
$586$	24.6567			1.01856
$587$		−	24.5806i		−	1.01455i	−0.861785	−	0.507274i	$-0.830653\pi$
$587$		−	24.5806i		−	1.01455i	0.861785	−	0.507274i	$-0.169347\pi$
$588$		0			0
$589$	−19.0363	−	14.9135i	−0.784376	−	0.614500i
$590$		−	1.86039i		−	0.0765909i
$591$		0			0
$592$			0.247975i			0.0101917i
$593$		−	35.9084i		−	1.47458i	−0.675575	−	0.737292i	$-0.736104\pi$
$593$		−	35.9084i		−	1.47458i	0.675575	−	0.737292i	$-0.263896\pi$
$594$		0			0
$595$	22.7471			0.932541
$596$			19.1410i			0.784047i
$597$		0			0
$598$	16.2284			0.663629
$599$	5.41265			0.221155			0.110577	−	0.993868i	$-0.464730\pi$
$599$	5.41265			0.221155			0.110577	+	0.993868i	$0.464730\pi$
$600$		0			0
$601$		−	44.4279i		−	1.81225i	−0.423009	−	0.906125i	$-0.639026\pi$
$601$		−	44.4279i		−	1.81225i	0.423009	−	0.906125i	$-0.360974\pi$
$602$	−4.26432			−0.173801
$603$		0			0
$604$			15.6476i			0.636693i
$605$			22.9646i			0.933642i
$606$		0			0
$607$		−	38.7380i		−	1.57233i	−0.618019	−	0.786163i	$-0.712065\pi$
$607$		−	38.7380i		−	1.57233i	0.618019	−	0.786163i	$-0.287935\pi$
$608$	2.68816	−	3.43130i	0.109019	−	0.139157i
$609$		0			0
$610$		−	24.7811i		−	1.00336i
$611$	−21.8446			−0.883736
$612$		0			0
$613$	12.7971			0.516870			0.258435	−	0.966029i	$-0.416793\pi$
$613$	12.7971			0.516870			0.258435	+	0.966029i	$0.416793\pi$
$614$		−	18.4337i		−	0.743923i
$615$		0			0
$616$			6.48233i			0.261181i
$617$		−	23.1532i		−	0.932115i	−0.884755	−	0.466057i	$-0.845674\pi$
$617$		−	23.1532i		−	0.932115i	0.884755	−	0.466057i	$-0.154326\pi$
$618$		0			0
$619$	−36.6250			−1.47208			−0.736042	−	0.676936i	$-0.763307\pi$
$619$	−36.6250			−1.47208			−0.736042	+	0.676936i	$0.763307\pi$
$620$	17.7661			0.713504
$621$		0			0
$622$		−	18.0078i		−	0.722049i
$623$	18.8626			0.755713
$624$		0			0
$625$	−23.6599			−0.946395
$626$	11.3994			0.455613
$627$		0			0
$628$	−11.3730			−0.453831
$629$	1.15831			0.0461850
$630$		0			0
$631$	−34.7571			−1.38366			−0.691830	−	0.722060i	$-0.743195\pi$
$631$	−34.7571			−1.38366			−0.691830	+	0.722060i	$0.743195\pi$
$632$		−	3.88042i		−	0.154355i
$633$		0			0
$634$	30.3425			1.20505
$635$	60.9279			2.41785
$636$		0			0
$637$			11.8943i			0.471267i
$638$		−	26.2703i		−	1.04005i
$639$		0			0
$640$			3.20235i			0.126584i
$641$	10.6594			0.421020			0.210510	−	0.977592i	$-0.432488\pi$
$641$	10.6594			0.421020			0.210510	+	0.977592i	$0.432488\pi$
$642$		0			0
$643$	24.6969			0.973951			0.486976	−	0.873416i	$-0.338100\pi$
$643$	24.6969			0.973951			0.486976	+	0.873416i	$0.338100\pi$
$644$			9.72570i			0.383246i
$645$		0			0
$646$	−16.0279	−	12.5566i	−0.630609	−	0.494035i
$647$			6.34810i			0.249569i	0.992184	+	0.124785i	$0.0398240\pi$
$647$			6.34810i			0.249569i	−0.992184	+	0.124785i	$0.960176\pi$
$648$		0			0
$649$		−	2.47643i		−	0.0972085i
$650$		−	13.3343i		−	0.523013i
$651$		0			0
$652$	−21.1943			−0.830034
$653$		−	22.6159i		−	0.885028i	−0.896762	−	0.442514i	$-0.854087\pi$
$653$		−	22.6159i		−	0.885028i	0.896762	−	0.442514i	$-0.145913\pi$
$654$		0			0
$655$	23.1084			0.902920
$656$	0.215591			0.00841741
$657$		0			0
$658$		−	13.0915i		−	0.510358i
$659$	−26.7234			−1.04099			−0.520497	−	0.853863i	$-0.674253\pi$
$659$	−26.7234			−1.04099			−0.520497	+	0.853863i	$0.674253\pi$
$660$		0			0
$661$		−	5.05743i		−	0.196711i	−0.995151	−	0.0983556i	$-0.968642\pi$
$661$		−	5.05743i		−	0.196711i	0.995151	−	0.0983556i	$-0.0313583\pi$
$662$			27.3689i			1.06372i
$663$		0			0
$664$			6.14953i			0.238648i
$665$	−16.7096	−	13.0907i	−0.647971	−	0.507636i
$666$		0			0
$667$		−	39.4144i		−	1.52613i
$668$	−24.1337			−0.933762
$669$		0			0
$670$	20.7371			0.801144
$671$		−	32.9870i		−	1.27345i
$672$		0			0
$673$		−	17.6462i		−	0.680210i	−0.940387	−	0.340105i	$-0.889537\pi$
$673$		−	17.6462i		−	0.680210i	0.940387	−	0.340105i	$-0.110463\pi$
$674$			10.6004i			0.408313i
$675$		0			0
$676$	6.56142			0.252362
$677$	−9.75616			−0.374960			−0.187480	−	0.982268i	$-0.560032\pi$
$677$	−9.75616			−0.374960			−0.187480	+	0.982268i	$0.560032\pi$
$678$		0			0
$679$		−	3.94545i		−	0.151412i
$680$	14.9584			0.573630
$681$		0			0
$682$	23.6491			0.905573
$683$	8.11524			0.310521			0.155261	−	0.987874i	$-0.450378\pi$
$683$	8.11524			0.310521			0.155261	+	0.987874i	$0.450378\pi$
$684$		0			0
$685$	62.7603			2.39795
$686$	−17.7730			−0.678578
$687$		0			0
$688$	−2.80421			−0.106909
$689$			34.2830i			1.30608i
$690$		0			0
$691$	−21.5955			−0.821531			−0.410766	−	0.911741i	$-0.634739\pi$
$691$	−21.5955			−0.821531			−0.410766	+	0.911741i	$0.634739\pi$
$692$	22.1422			0.841720
$693$		0			0
$694$			5.99303i			0.227492i
$695$		−	34.0463i		−	1.29145i
$696$		0			0
$697$		−	1.00704i		−	0.0381445i
$698$	−1.34086			−0.0507522
$699$		0			0
$700$	7.99124			0.302040
$701$			32.3415i			1.22152i	0.791815	+	0.610761i	$0.209137\pi$
$701$			32.3415i			1.22152i	−0.791815	+	0.610761i	$0.790863\pi$
$702$		0			0
$703$	−0.850876	−	0.666598i	−0.0320914	−	0.0251412i
$704$			4.26277i			0.160659i
$705$		0			0
$706$			11.4733i			0.431804i
$707$		−	17.6384i		−	0.663361i
$708$		0			0
$709$	−9.70529			−0.364490			−0.182245	−	0.983253i	$-0.558336\pi$
$709$	−9.70529			−0.364490			−0.182245	+	0.983253i	$0.558336\pi$
$710$			32.0171i			1.20158i
$711$		0			0
$712$	12.4040			0.464859
$713$	35.4817			1.32880
$714$		0			0
$715$		−	34.6381i		−	1.29539i
$716$	−16.7184			−0.624796
$717$		0			0
$718$			2.97199i			0.110914i
$719$		−	33.3349i		−	1.24318i	−0.783342	−	0.621591i	$-0.786487\pi$
$719$		−	33.3349i		−	1.24318i	0.783342	−	0.621591i	$-0.213513\pi$
$720$		0			0
$721$		−	15.4249i		−	0.574452i
$722$	4.54757	+	18.4478i	0.169243	+	0.686554i
$723$		0			0
$724$		−	1.92544i		−	0.0715582i
$725$	−32.3853			−1.20276
$726$		0			0
$727$	37.8889			1.40522			0.702611	−	0.711574i	$-0.252017\pi$
$727$	37.8889			1.40522			0.702611	+	0.711574i	$0.252017\pi$
$728$		−	3.85865i		−	0.143011i
$729$		0			0
$730$		−	42.6337i		−	1.57794i
$731$			13.0987i			0.484473i
$732$		0			0
$733$	0.722670			0.0266924			0.0133462	−	0.999911i	$-0.495752\pi$
$733$	0.722670			0.0266924			0.0133462	+	0.999911i	$0.495752\pi$
$734$	−13.7275			−0.506692
$735$		0			0
$736$			6.39559i			0.235745i
$737$	27.6039			1.01680
$738$		0			0
$739$	−18.6408			−0.685713			−0.342856	−	0.939388i	$-0.611394\pi$
$739$	−18.6408			−0.685713			−0.342856	+	0.939388i	$0.611394\pi$
$740$	0.794103			0.0291918
$741$		0			0
$742$	−20.5458			−0.754262
$743$	8.40218			0.308246			0.154123	−	0.988052i	$-0.450745\pi$
$743$	8.40218			0.308246			0.154123	+	0.988052i	$0.450745\pi$
$744$		0			0
$745$	61.2962			2.24572
$746$			15.0917i			0.552548i
$747$		0			0
$748$	19.9118			0.728046
$749$	−14.2106			−0.519246
$750$		0			0
$751$			10.3814i			0.378822i	0.981898	+	0.189411i	$0.0606578\pi$
$751$			10.3814i			0.378822i	−0.981898	+	0.189411i	$0.939342\pi$
$752$		−	8.60891i		−	0.313935i
$753$		0			0
$754$			15.6376i			0.569487i
$755$	50.1091			1.82366
$756$		0			0
$757$	−6.21886			−0.226028			−0.113014	−	0.993593i	$-0.536051\pi$
$757$	−6.21886			−0.226028			−0.113014	+	0.993593i	$0.536051\pi$
$758$		−	1.29330i		−	0.0469747i
$759$		0			0
$760$	−10.9882	−	8.60842i	−0.398584	−	0.312260i
$761$		−	40.7565i		−	1.47742i	−0.674022	−	0.738711i	$-0.735435\pi$
$761$		−	40.7565i		−	1.47742i	0.674022	−	0.738711i	$-0.264565\pi$
$762$		0			0
$763$			7.39398i			0.267680i
$764$		−	0.736584i		−	0.0266487i
$765$		0			0
$766$	2.05850			0.0743767
$767$			1.47411i			0.0532271i
$768$		0			0
$769$	−0.612317			−0.0220807			−0.0110404	−	0.999939i	$-0.503514\pi$
$769$	−0.612317			−0.0220807			−0.0110404	+	0.999939i	$0.503514\pi$
$770$	20.7587			0.748091
$771$		0			0
$772$		−	9.11819i		−	0.328171i
$773$	−23.8826			−0.858998			−0.429499	−	0.903067i	$-0.641310\pi$
$773$	−23.8826			−0.858998			−0.429499	+	0.903067i	$0.641310\pi$
$774$		0			0
$775$		−	29.1540i		−	1.04724i
$776$		−	2.59452i		−	0.0931377i
$777$		0			0
$778$			22.5996i			0.810235i
$779$	−0.579543	+	0.739756i	−0.0207643	+	0.0265045i
$780$		0			0
$781$			42.6191i			1.52503i
$782$	29.8744			1.06831
$783$		0			0
$784$	−4.68751			−0.167411
$785$			36.4202i			1.29989i
$786$		0			0
$787$		−	23.8364i		−	0.849675i	−0.905270	−	0.424838i	$-0.860331\pi$
$787$		−	23.8364i		−	0.849675i	0.905270	−	0.424838i	$-0.139669\pi$
$788$		−	8.47801i		−	0.302017i
$789$		0			0
$790$	−12.4264			−0.442113
$791$	−7.89300			−0.280643
$792$		0			0
$793$			19.6357i			0.697285i
$794$	21.6329			0.767723
$795$		0			0
$796$	16.4335			0.582470
$797$	−42.5223			−1.50622			−0.753109	−	0.657896i	$-0.771447\pi$
$797$	−42.5223			−1.50622			−0.753109	+	0.657896i	$0.771447\pi$
$798$		0			0
$799$	−40.2130			−1.42263
$800$	5.25502			0.185793
$801$		0			0
$802$	−34.5347			−1.21946
$803$		−	56.7514i		−	2.00271i
$804$		0			0
$805$	31.1450			1.09772
$806$	−14.0773			−0.495851
$807$		0			0
$808$		−	11.5990i		−	0.408050i
$809$			7.46870i			0.262585i	0.991344	+	0.131293i	$0.0419128\pi$
$809$			7.46870i			0.262585i	−0.991344	+	0.131293i	$0.958087\pi$
$810$		0			0
$811$		−	12.1478i		−	0.426568i	−0.976990	−	0.213284i	$-0.931584\pi$
$811$		−	12.1478i		−	0.426568i	0.976990	−	0.213284i	$-0.0684160\pi$
$812$	−9.37161			−0.328879
$813$		0			0
$814$	1.05706			0.0370499
$815$			67.8716i			2.37744i
$816$		0			0
$817$	7.53816	−	9.62206i	0.263727	−	0.336633i
$818$			18.1554i			0.634787i
$819$		0			0
$820$		−	0.690397i		−	0.0241097i
$821$		−	32.5477i		−	1.13592i	−0.823055	−	0.567962i	$-0.807732\pi$
$821$		−	32.5477i		−	1.13592i	0.823055	−	0.567962i	$-0.192268\pi$
$822$		0			0
$823$	12.1203			0.422487			0.211243	−	0.977433i	$-0.432249\pi$
$823$	12.1203			0.422487			0.211243	+	0.977433i	$0.432249\pi$
$824$		−	10.1433i		−	0.353360i
$825$		0			0
$826$	−0.883436			−0.0307387
$827$	9.56163			0.332490			0.166245	−	0.986084i	$-0.446836\pi$
$827$	9.56163			0.332490			0.166245	+	0.986084i	$0.446836\pi$
$828$		0			0
$829$		−	26.0509i		−	0.904786i	−0.891819	−	0.452393i	$-0.850570\pi$
$829$		−	26.0509i		−	0.904786i	0.891819	−	0.452393i	$-0.149430\pi$
$830$	19.6929			0.683552
$831$		0			0
$832$		−	2.53743i		−	0.0879697i
$833$			21.8958i			0.758644i
$834$		0			0
$835$			77.2846i			2.67454i
$836$	−14.6268	−	11.4590i	−0.505879	−	0.396318i
$837$		0			0
$838$			15.0840i			0.521067i
$839$	33.1961			1.14606			0.573029	−	0.819535i	$-0.305768\pi$
$839$	33.1961			1.14606			0.573029	+	0.819535i	$0.305768\pi$
$840$		0			0
$841$	8.97947			0.309637
$842$		−	25.8311i		−	0.890200i
$843$		0			0
$844$		−	0.927889i		−	0.0319393i
$845$		−	21.0119i		−	0.722833i
$846$		0			0
$847$	10.9051			0.374704
$848$	−13.5109			−0.463966
$849$		0			0
$850$		−	24.5467i		−	0.841943i
$851$	1.58595			0.0543656
$852$		0			0
$853$	41.3775			1.41674			0.708369	−	0.705842i	$-0.249431\pi$
$853$	41.3775			1.41674			0.708369	+	0.705842i	$0.249431\pi$
$854$	−11.7677			−0.402683
$855$		0			0
$856$	−9.34489			−0.319402
$857$	−17.6293			−0.602206			−0.301103	−	0.953592i	$-0.597355\pi$
$857$	−17.6293			−0.602206			−0.301103	+	0.953592i	$0.597355\pi$
$858$		0			0
$859$	45.4974			1.55235			0.776176	−	0.630516i	$-0.217157\pi$
$859$	45.4974			1.55235			0.776176	+	0.630516i	$0.217157\pi$
$860$			8.98004i			0.306217i
$861$		0			0
$862$	18.7560			0.638832
$863$	19.9241			0.678223			0.339111	−	0.940746i	$-0.389874\pi$
$863$	19.9241			0.678223			0.339111	+	0.940746i	$0.389874\pi$
$864$		0			0
$865$		−	70.9070i		−	2.41091i
$866$		−	21.8593i		−	0.742809i
$867$		0			0
$868$		−	8.43653i		−	0.286355i
$869$	−16.5413			−0.561126
$870$		0			0
$871$	−16.4314			−0.556757
$872$			4.86226i			0.164657i
$873$		0			0
$874$	−21.9452	−	17.1924i	−0.742306	−	0.581541i
$875$		−	1.24187i		−	0.0419828i
$876$		0			0
$877$			27.7886i			0.938355i	0.883104	+	0.469178i	$0.155450\pi$
$877$			27.7886i			0.938355i	−0.883104	+	0.469178i	$0.844550\pi$
$878$			22.6494i			0.764381i
$879$		0			0
$880$	13.6508			0.460170
$881$		−	52.4731i		−	1.76786i	−0.467615	−	0.883932i	$-0.654887\pi$
$881$		−	52.4731i		−	1.76786i	0.467615	−	0.883932i	$-0.345113\pi$
$882$		0			0
$883$	38.6910			1.30206			0.651028	−	0.759054i	$-0.274338\pi$
$883$	38.6910			1.30206			0.651028	+	0.759054i	$0.274338\pi$
$884$	−11.8526			−0.398646
$885$		0			0
$886$		−	8.13524i		−	0.273309i
$887$	14.4952			0.486701			0.243350	−	0.969938i	$-0.421754\pi$
$887$	14.4952			0.486701			0.243350	+	0.969938i	$0.421754\pi$
$888$		0			0
$889$		−	28.9326i		−	0.970370i
$890$		−	39.7218i		−	1.33148i
$891$		0			0
$892$			19.2520i			0.644605i
$893$	29.5397	+	23.1421i	0.988509	+	0.774422i
$894$		0			0
$895$			53.5381i			1.78958i
$896$	1.52069			0.0508026
$897$		0			0
$898$	34.4488			1.14957
$899$			34.1899i			1.14030i
$900$		0			0
$901$			63.1106i			2.10252i
$902$		−	0.919014i		−	0.0305998i
$903$		0			0
$904$	−5.19041			−0.172631
$905$	−6.16591			−0.204962
$906$		0			0
$907$			44.5497i			1.47925i	0.673019	+	0.739625i	$0.264997\pi$
$907$			44.5497i			1.47925i	−0.673019	+	0.739625i	$0.735003\pi$
$908$	−21.1219			−0.700955
$909$		0			0
$910$	−12.3567			−0.409621
$911$	−18.9344			−0.627324			−0.313662	−	0.949535i	$-0.601556\pi$
$911$	−18.9344			−0.627324			−0.313662	+	0.949535i	$0.601556\pi$
$912$		0			0
$913$	26.2140			0.867557
$914$	−28.7479			−0.950896
$915$		0			0
$916$	1.02591			0.0338970
$917$		−	10.9734i		−	0.362374i
$918$		0			0
$919$	11.7145			0.386424			0.193212	−	0.981157i	$-0.438109\pi$
$919$	11.7145			0.386424			0.193212	+	0.981157i	$0.438109\pi$
$920$	20.4809			0.675235
$921$		0			0
$922$		−	7.62779i		−	0.251208i
$923$		−	25.3693i		−	0.835040i
$924$		0			0
$925$		−	1.30311i		−	0.0428461i
$926$	−3.84491			−0.126352
$927$		0			0
$928$	−6.16275			−0.202302
$929$		−	18.5101i		−	0.607298i	−0.952784	−	0.303649i	$-0.901795\pi$
$929$		−	18.5101i		−	0.607298i	0.952784	−	0.303649i	$-0.0982051\pi$
$930$		0			0
$931$	12.6008	−	16.0842i	0.412974	−	0.527139i
$932$		−	5.08018i		−	0.166407i
$933$		0			0
$934$		−	17.3946i		−	0.569169i
$935$		−	63.7643i		−	2.08532i
$936$		0			0
$937$	−28.6238			−0.935100			−0.467550	−	0.883967i	$-0.654863\pi$
$937$	−28.6238			−0.935100			−0.467550	+	0.883967i	$0.654863\pi$
$938$		−	9.84736i		−	0.321528i
$939$		0			0
$940$	−27.5687			−0.899192
$941$	−9.52179			−0.310401			−0.155201	−	0.987883i	$-0.549602\pi$
$941$	−9.52179			−0.310401			−0.155201	+	0.987883i	$0.549602\pi$
$942$		0			0
$943$		−	1.37883i		−	0.0449009i
$944$	−0.580945			−0.0189082
$945$		0			0
$946$			11.9537i			0.388648i
$947$			41.8960i			1.36144i	0.732545	+	0.680719i	$0.238332\pi$
$947$			41.8960i			1.36144i	−0.732545	+	0.680719i	$0.761668\pi$
$948$		0			0
$949$			33.7816i			1.09660i
$950$	−14.1263	+	18.0315i	−0.458319	+	0.585019i
$951$		0			0
$952$		−	7.10327i		−	0.230218i
$953$	29.1035			0.942755			0.471378	−	0.881932i	$-0.343757\pi$
$953$	29.1035			0.942755			0.471378	+	0.881932i	$0.343757\pi$
$954$		0			0
$955$	−2.35880			−0.0763288
$956$		−	11.1467i		−	0.360510i
$957$		0			0
$958$			29.5315i			0.954119i
$959$		−	29.8028i		−	0.962382i
$960$		0			0
$961$	0.221458			0.00714381
$962$	−0.629221			−0.0202869
$963$		0			0
$964$			11.0135i			0.354721i
$965$	−29.1996			−0.939968
$966$		0			0
$967$	3.33579			0.107272			0.0536359	−	0.998561i	$-0.482919\pi$
$967$	3.33579			0.107272			0.0536359	+	0.998561i	$0.482919\pi$
$968$	7.17117			0.230490
$969$		0			0
$970$	−8.30854			−0.266771
$971$	−7.01147			−0.225009			−0.112504	−	0.993651i	$-0.535887\pi$
$971$	−7.01147			−0.225009			−0.112504	+	0.993651i	$0.535887\pi$
$972$		0			0
$973$	−16.1674			−0.518304
$974$			15.8426i			0.507629i
$975$		0			0
$976$	−7.73842			−0.247701
$977$	35.1927			1.12591			0.562957	−	0.826487i	$-0.309664\pi$
$977$	35.1927			1.12591			0.562957	+	0.826487i	$0.309664\pi$
$978$		0			0
$979$		−	52.8753i		−	1.68990i
$980$			15.0110i			0.479510i
$981$		0			0
$982$		−	4.37074i		−	0.139476i
$983$	−2.49754			−0.0796592			−0.0398296	−	0.999206i	$-0.512682\pi$
$983$	−2.49754			−0.0796592			−0.0398296	+	0.999206i	$0.512682\pi$
$984$		0			0
$985$	−27.1495			−0.865056
$986$			28.7868i			0.916757i
$987$		0			0
$988$	8.70669	+	6.82104i	0.276997	+	0.217006i
$989$			17.9346i			0.570286i
$990$		0			0
$991$			1.66534i			0.0529012i	0.999650	+	0.0264506i	$0.00842047\pi$
$991$			1.66534i			0.0529012i	−0.999650	+	0.0264506i	$0.991580\pi$
$992$		−	5.54784i		−	0.176144i
$993$		0			0
$994$	15.2038			0.482237
$995$		−	52.6257i		−	1.66835i
$996$		0			0
$997$	−28.6124			−0.906163			−0.453081	−	0.891469i	$-0.649675\pi$
$997$	−28.6124			−0.906163			−0.453081	+	0.891469i	$0.649675\pi$
$998$	20.4879			0.648533
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3078.2.b.a.3077.2		20
3.2	odd	2		3078.2.b.c.3077.19		20
9.2	odd	6		342.2.p.a.113.6	✓	20
9.4	even	3		342.2.p.b.227.5	yes	20
9.5	odd	6		1026.2.p.a.683.10		20
9.7	even	3		1026.2.p.b.341.10		20
19.18	odd	2		3078.2.b.c.3077.2		20
57.56	even	2	inner	3078.2.b.a.3077.19		20
171.56	even	6		342.2.p.b.113.5	yes	20
171.94	odd	6		342.2.p.a.227.6	yes	20
171.113	even	6		1026.2.p.b.683.10		20
171.151	odd	6		1026.2.p.a.341.10		20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
342.2.p.a.113.6	✓	20	9.2	odd	6
342.2.p.a.227.6	yes	20	171.94	odd	6
342.2.p.b.113.5	yes	20	171.56	even	6
342.2.p.b.227.5	yes	20	9.4	even	3
1026.2.p.a.341.10		20	171.151	odd	6
1026.2.p.a.683.10		20	9.5	odd	6
1026.2.p.b.341.10		20	9.7	even	3
1026.2.p.b.683.10		20	171.113	even	6
3078.2.b.a.3077.2		20	1.1	even	1	trivial
3078.2.b.a.3077.19		20	57.56	even	2	inner
3078.2.b.c.3077.2		20	19.18	odd	2
3078.2.b.c.3077.19		20	3.2	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:	\( N \)	\(=\)	\( 3078 = 2 \cdot 3^{4} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3078.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000 q^{4} -3.20235i q^{5} -1.52069 q^{7} -1.00000 q^{8} +O(q^{10})\)	\(q-1.00000 q^{2} +1.00000 q^{4} -3.20235i q^{5} -1.52069 q^{7} -1.00000 q^{8} +3.20235i q^{10} +4.26277i q^{11} -2.53743i q^{13} +1.52069 q^{14} +1.00000 q^{16} -4.67109i q^{17} +(-2.68816 + 3.43130i) q^{19} -3.20235i q^{20} -4.26277i q^{22} -6.39559i q^{23} -5.25502 q^{25} +2.53743i q^{26} -1.52069 q^{28} +6.16275 q^{29} +5.54784i q^{31} -1.00000 q^{32} +4.67109i q^{34} +4.86977i q^{35} +0.247975i q^{37} +(2.68816 - 3.43130i) q^{38} +3.20235i q^{40} +0.215591 q^{41} -2.80421 q^{43} +4.26277i q^{44} +6.39559i q^{46} -8.60891i q^{47} -4.68751 q^{49} +5.25502 q^{50} -2.53743i q^{52} -13.5109 q^{53} +13.6508 q^{55} +1.52069 q^{56} -6.16275 q^{58} -0.580945 q^{59} -7.73842 q^{61} -5.54784i q^{62} +1.00000 q^{64} -8.12574 q^{65} -6.47560i q^{67} -4.67109i q^{68} -4.86977i q^{70} +9.99801 q^{71} -13.3133 q^{73} -0.247975i q^{74} +(-2.68816 + 3.43130i) q^{76} -6.48233i q^{77} +3.88042i q^{79} -3.20235i q^{80} -0.215591 q^{82} -6.14953i q^{83} -14.9584 q^{85} +2.80421 q^{86} -4.26277i q^{88} -12.4040 q^{89} +3.85865i q^{91} -6.39559i q^{92} +8.60891i q^{94} +(10.9882 + 8.60842i) q^{95} +2.59452i q^{97} +4.68751 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q - 20 q^{2} + 20 q^{4} - 4 q^{7} - 20 q^{8}+O(q^{10}) \) 20 * q - 20 * q^2 + 20 * q^4 - 4 * q^7 - 20 * q^8	\( 20 q - 20 q^{2} + 20 q^{4} - 4 q^{7} - 20 q^{8} + 4 q^{14} + 20 q^{16} + q^{19} - 20 q^{25} - 4 q^{28} - 20 q^{32} - q^{38} - 6 q^{41} + 10 q^{43} + 24 q^{49} + 20 q^{50} + 4 q^{56} + 6 q^{59} - 28 q^{61} + 20 q^{64} + 24 q^{65} - 24 q^{71} + 34 q^{73} + q^{76} + 6 q^{82} - 10 q^{86} - 36 q^{89} - 24 q^{98}+O(q^{100}) \) 20 * q - 20 * q^2 + 20 * q^4 - 4 * q^7 - 20 * q^8 + 4 * q^14 + 20 * q^16 + q^19 - 20 * q^25 - 4 * q^28 - 20 * q^32 - q^38 - 6 * q^41 + 10 * q^43 + 24 * q^49 + 20 * q^50 + 4 * q^56 + 6 * q^59 - 28 * q^61 + 20 * q^64 + 24 * q^65 - 24 * q^71 + 34 * q^73 + q^76 + 6 * q^82 - 10 * q^86 - 36 * q^89 - 24 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more